Commun. Math. Phys. 292, 1–28 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0839-8
Communications in
Mathematical Physics
Right Limits and Reflectionless Measures for CMV Matrices Jonathan Breuer, Eric Ryckman, Maxim Zinchenko Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125-0001, USA. E-mail:
[email protected];
[email protected];
[email protected] Received: 24 November 2008 / Accepted: 10 February 2009 Published online: 29 May 2009 – © Springer-Verlag 2009
Abstract: We study CMV matrices by focusing on their right-limit sets. We prove a CMV version of a recent result of Remling dealing with the implications of the existence of absolutely continuous spectrum, and we study some of its consequences. We further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of Khrushchev. To demonstrate the analogy with the Jacobi case, we recover corresponding previous results of Simon using the same approach.
1. Introduction This paper considers some issues in the spectral theory of CMV matrices viewed through the lens of the notion of right limits. In particular, a central theme will be the fact that one may use the properties of right limits of a given CMV matrix to deduce relations between the asymptotics of its entries and its spectral measure. CMV matrices (see Definition 1.2 below) were named after Cantero, Moral and Velázquez [4] and may be described as the unitary analog of Jacobi matrices: they arise naturally in the theory of orthogonal polynomials on the unit circle (OPUC) in much the same way that Jacobi matrices arise in the theory of orthogonal polynomials on the real line (OPRL). Two related topics will be at the focus of our discussion. The first is the extension to the CMV setting of a collection of results, proven recently by Remling [33], describing various consequences of the existence of absolutely continuous spectrum of Jacobi matrices. The second topic is the simplification of various elements of Khrushchev’s theory of weak limits of spectral measures, through the understanding that the matrices at the center of attention have right limits in a very special class. As we shall see, these two subjects are intimately connected through the notion of reflectionless whole-line CMV matrices. This is a concept that has been extensively
2
J. Breuer, E. Ryckman, M. Zinchenko
investigated in recent years, in the context of both CMV and Jacobi matrices ([6,7,9– 11,14,17–22,24–29,32,33,38–42]) and was seen to have numerous applications in their spectral theory. There are various definitions of this notion, all of which turn out to be equivalent in the Jacobi matrix case. We shall show that this is not true in the CMV case. In particular, we construct an example of a whole-line CMV matrix that is not reflectionless in the spectral-theoretic sense, while all of its diagonal spectral measures are reflectionless in the measure-theoretic sense. We will show, however, that this may only happen for a very limited class of CMV matrices. Their existence in the CMV case, together with Remling’s Theorem (Theorem 1.4 below), provides for a simple proof of Khrushchev’s Theorem (Theorem 1.9 below). We should remark that ours is not the first paper to deal with right limits of CMV matrices. For other examples and related results, see for instance [15,23]. In order to describe our results, some notation is needed: given a probability measure, ∞ µ, on the boundary of the unit disc, ∂D, we let {n (z)}∞ n=0 and {ϕn (z)}n=0 denote the monic orthogonal and the orthonormal polynomials one gets by applying the Gram– Schmidt procedure to 1, z, z 2 , . . . (we assume throughout that the support of µ is an infinite set so the polynomial sequences are indeed infinite). The n satisfy the Szeg˝o recurrence equation: n+1 (z) = zn (z) − αn ∗n (z),
(1.1)
∗ n where {αn }∞ n=0 is a sequence of parameters satisfying |αn | < 1 and n (z) = z n (1/z). ∞ We call {αn }n=0 the Verblunsky coefficients associated with µ. As is well known [36], the sequence {αn }∞ n=0 may be used to construct a semi-infinite 5-diagonal matrix, C, (called the CMV matrix) such that the operator of multiplication by z on L 2 (∂D, dµ) is unitarily equivalent to the operator C on 2 (Z+ ) (Z+ = {0, 1, 2, . . .}). Explicitly, C is given by ⎛ ⎞ α¯ 0 α¯ 1 ρ0 ρ1 ρ0 0 0 ... ⎜ ρ0 −α¯ 1 α0 −ρ1 α0 0 0 ... ⎟ ⎜ ⎟ α¯ 2 ρ1 −α¯ 2 α1 α¯ 3 ρ2 ρ3 ρ2 ... ⎟ ⎜ 0 (1.2) C=⎜ ρ2 ρ1 −ρ2 α1 −α¯ 3 α2 −ρ3 α2 . . . ⎟ ⎜ 0 ⎟ ⎝ 0 ⎠ 0 0 α¯ 4 ρ3 −α¯ 4 α3 . . . ... ... ... ... ... ...
1/2 with ρn = 1 − |αn |2 . Now, for a probability measure µ on ∂D, let dµ(θ ) = w(θ )dθ + dµsing (θ ) be the decomposition into its absolutely continuous and singular parts (with respect to Lebesgue measure). If C is the corresponding CMV matrix, we define the essential support of the absolutely continuous spectrum of C to be the set ac (C) ≡ {θ | w(θ ) > 0}. Clearly, ac (C) is only defined up to sets of Lebesgue measure zero, so the symbol and the name should be understood as representing elements in an equivalence class of sets rather than a particular set. For the sake of simplicity we will ignore this point in our discussion. The first part of this paper deals with proving the analog of Remling’s Theorem (Theorem 1.4 in [33]) for CMV matrices and deriving some consequences. In a nutshell, Remling’s Theorem for CMV matrices says that for any given CMV matrix, C, all of its right limits are reflectionless on ac (C) (see Definitions 1.2 and 1.3 and Theorem 1.4 below). Here is a consequence that will also provide the link to Khrushchev’s theory (the Jacobi analog was stated and proved in [33]):
Right Limits for CMV Matrices
3
Theorem 1.1. Let {αn }∞ αn }∞ n=0 and { n=0 be two sequences of Verblunsky coefficients such that (with n = αn − αn ): (i) |αn | < 1, | αn | < 1 for all n. ∞ (ii) There exist sequences {m j }∞ j=1 , {n j } j=1 with n j − m j → ∞ so that
lim
sup
j→∞ m j ≤n
| n | = 0.
(iii) lim sup j→∞ | n j | > 0. Furthermore, let C and C denote the CMV matrices of {αn }∞ αn }∞ n=0 and { n=0 respectively. Then
ac (C) ∩ ac C = 0, (1.3) where | · | denotes Lebesgue measure. In particular, if C is associated with a sequence of Verblunsky coefficients satisfying ∀k ≥ 1 lim αn αn+k = 0, n→∞
lim sup |αn | > 0,
(1.4)
n→∞
then C has purely singular spectrum. In order to state Remling’s Theorem we need some more terminology. Definition 1.2. Given a sequence of Verblunsky coefficients {αn }∞ n=0 , a doubly-infinite sequence of parameters { αn }n∈Z with | αn | ≤ 1 is called a right limit of {αn }∞ n=0 if there is a sequence of integers n j → ∞ such that ∀n ∈ Z, αn = lim j→∞ αn+n j . Since a sequence of Verblunsky coefficients is always bounded, by compactness it always has at least one (and perhaps many) right limits. Given a doubly infinite sequence { αn }n∈Z , one may also define a corresponding unitary matrix on 2 (Z), extending the half-line matrices to the left and top (see (3.1) for the precise form). We call such a matrix the corresponding whole-line CMV matrix and denote it by E. For this reason, we shall often refer to a doubly infinite sequence of numbers { αn }n∈Z with | αn | ≤ 1 as a (doubly infinite) sequence of Verblunsky coefficients. If { αn }n∈Z is a right limit of {αn }∞ n=0 , we refer to the corresponding whole-line CMV matrix as a right limit of the half-line CMV matrix associated with {αn }∞ n=0 . Recall that any probability measure µ on ∂D may be naturally associated with a Schur function f (an analytic function on D satisfying supz∈D | f (z)| ≤ 1) and a Carathéodory function F (an analytic function on D satisfying F(0) = 1 and Re F(z) > 0 on D). This is given by 2π iθ 1 + z f (z) e +z = F(z) = dµ(θ ). (1.5) 1 − z f (z) eiθ − z 0 The correspondence is 1-1 and onto. By a classical result, limr ↑1 F(r eiθ ) and limr ↑1 f (r eiθ ) exist Lebesgue a.e. on ∂D. We denote them by F(eiθ ) and f (eiθ ) respectively and, when there is no danger of confusion, simply by F(z) or f (z) for z ∈ ∂D. Given a Schur function, f , let f 0 = f and define a sequence of Schur functions f n and parameters γn ∈ D by z f n+1 (z) =
f n (z) − γn , 1 − γ n f n (z)
γn = f n (0).
4
J. Breuer, E. Ryckman, M. Zinchenko
If for some n, |γn | = 1, we stop and the Schur function is a finite Blaschke product. Otherwise, we continue. It is known [36] that this process (known as the Schur algorithm) sets up a 1-1 correspondence between Schur functions f and parameter sequences γn ∈ D, so given any sequence γn ∈ D there is a unique Schur function f (z; γ0 , γ1 , . . . ) associated to it in the above way. The γ ’s are frequently termed the Schur parameters associated to f (or equivalently, to µ or F). Geronimus’s Theorem [8] says that γn = αn (the Verblunsky coefficients of µ appearing above). Finally, note that by definition f n (z; α0 , α1 , . . . ) = f (z; αn , αn+1 , . . . ).
(1.6)
For a doubly infinite sequence of Verblunsky coefficients, {αn }n∈Z (some of which may lie on ∂D), we define two sequences of Schur functions: f + (z, n) = f (z; αn , αn+1 , . . . ) and f − (z, n) = f (z; −αn−1 , −αn−2 , . . . ),
(1.7)
where as usual, if one of the α’s lies in ∂D then we stop the Schur algorithm at that point and the corresponding Schur function is a finite Blaschke product. Definition 1.3. Let {αn }n∈Z be a doubly-infinite sequence of Verblunsky coefficients and let E be the associated whole-line CMV matrix. Given a Borel set A ⊆ ∂D, we will say E is reflectionless on A if for all n ∈ Z, z f + (z, n) = f − (z, n) for Lebesgue almost every z ∈ A. By the Schur algorithm, one can easily see that “for all n ∈ Z” may be replaced with “for some n ∈ Z.” Remark. The analogous definition for whole-line Jacobi matrices involves a similar relationship between the left and right m-functions. The following is the CMV version of Remling’s Theorem: Theorem 1.4. (Remling’s Theorem for CMV matrices) Let C be a half-line CMV matrix, and let ac (C) be the essential support of the absolutely continuous part of the spectral measure. Then every right limit of C is reflectionless on ac (C). Remling’s proof in the Jacobi case relies on previous work by Breimesser and Pearson [1,2] concerning convergence of boundary values for Herglotz functions. We will prove Theorem 1.4 using the analogous theory for Schur functions. For a CMV matrix, C, recall that its essential spectrum, σess (C), is its spectrum with the isolated points removed. The following extension of a celebrated theorem of Rakhmanov is a simple corollary of Theorem 1.4: Theorem 1.5. Assume ac (C) = σess (C) = A, where σess (C) is the essential spectrum of C. Then for any right limit E of C, σ (E) = A and E is reflectionless on A. Remark. In the case that A is a finite union of intervals, the class of whole-line CMV matrices, T A , described in the above theorem is called the isospectral torus of A. Much the same as in the Jacobi case ([33, Sect. 6, 42, Sect. 2]) the torus structure is naturally introduced by considering the gaps ∂D\A = ∪j=1 (a j , b j ). The torus is obtained by taking two copies of each interval, “gluing” them at the edges and taking the Cartesian product. It then turns out that each point in this torus corresponds to a unique matrix in T A . More formally, one identifies CMV matrices E ∈ T A with -tuples (tˆ1 , . . . , tˆ ),
Right Limits for CMV Matrices
5
tˆj = (t j , ε j ), where t j ∈ [a j , b j ], ε j ∈ {±1}, and we identify (t j , 1) and (t j , −1) if t j = a j or b j . As [28, Thm. 1.4] has a detailed discussion of this identification in a slightly more general context (see also [12] for the case of a single arc spectrum), we simply sketch the argument here. The key is a circular analogue of Craig’s formula [6] for Carathéodory with Im F = 0 a.e. on A, Re F = 0 on ∂D\A, and having no zeros in functions, F, has a single singularity in each gap (a simple ∂D\A. By this formula, every such F pole in the interior of a gap or a square root singularity at an edge), and is uniquely determined up to a positive multiplicative constant by the locations of these singularities is not necessarily normalized to have F(0) = 1, and in general t j ∈ [a j , b j ] (note that F F(0) is not real). Given the ε j ’s, F can be uniquely decomposed into a sum of two ± having no common poles, F + (0) > 0, and F + = F − a.e. on Carathéodory functions F to either F + or F − according to the sign of A. This is done by assigning each pole of F the corresponding ε j ’s (the “gluing” comes from the fact that both will have a square and F ± by F + (0) = 1, it root singularity at t j if it is at an edge). After normalizing F ± . These, in turn, follows that every -tuple (tˆ1 , . . . , tˆ ) corresponds to a unique pair F + (z) = 1+z f+ (z) and F − (z) = 1+ f− (z) with z f + = f − define two Schur functions f ± by F 1−z f + (z) 1− f − (z) a.e. on A, which then determine a CMV matrix E ∈ T A . Conversely, it is easy to see by reversing the above steps that every element of T A corresponds to a unique -tuple. If A = ∂D, the isospectral torus is known to consist of a single point—the CMV matrix with Verblunsky coefficients all equal to zero [12]. Thus, one gets Rakhmanov’s Theorem [30,31] as a corollary. Proof of Theorem 1.5. Let E be a right limit of C and {δn }n∈Z be the standard orthonormal basis for 2 (Z). For ψ = n∈Z 2−|n| δn , let dµψ (θ ) = wψ (θ )dθ + dµψ,sing be the spectral measure of ψ and E. Let ac (E) = {θ | wψ (θ ) > 0} (defined, again, up to sets of Lebesgue measure zero). By Theorem 1.4, A ⊆ ac (E) (up to a set of Lebesgue measure zero), since the reflectionless condition implies positivity of the real part of the Carathéodory function associated with dµψ . Also, σ (E) ⊆ σess (C) = A by approximate-eigenvector arguments (see for instance [23]). Since obviously ac (E) ⊆ σ (E), we have equality throughout. The reflectionless condition now follows from Theorem 1.4. Remark. Using Theorem 1.4 and a bit of work, one can also derive parts of Kotani theory for ergodic CMV matrices (see for instance [37, Sect. 10.11]). Remling also obtains deterministic versions of these results for Jacobi matrices (see [33, Thm’s 1.1 and 1.2]). His proofs extend directly to the CMV case we are considering, so we will not pursue this here. Corresponding to the notion of reflectionless operators, there is also the notion of reflectionless measures: Definition 1.6. A probability measure µ on ∂D is said to be reflectionless on a Borel set A ⊆ ∂D if the corresponding Carathéodory function F has Im F(eiθ ) = 0 for Lebesgue a.e. eiθ ∈ A. Remark. The analogous definition for measures on the real line involves the vanishing of the real part of the Borel (a.k.a. Cauchy or Stieltjes) transform of µ (see for instance [43]).
6
J. Breuer, E. Ryckman, M. Zinchenko
Remark. There is also a natural dynamical notion for when an operator is reflectionless. For the relationship between this and spectral theory see [3]. Reflectionless Jacobi matrices and reflectionless measures on R are related in the following way: given a whole-line Jacobi matrix, H , let µn be the spectral measure of H and δn (δn ∈ 2 (Z) is defined by δn ( j) = δn, j with δn, j the Kronecker delta). Then H is reflectionless on A ⊆ R if and only if µn are reflectionless on A for all n ∈ Z (again, see [43]). A fact we would like to emphasize in this paper is that the analogous statement does not hold for CMV matrices. Example 1.7. Fix j0 ∈ Z and some 0 < |β| < 1, and let {αn }n∈Z be the sequence of Verblunsky coefficients defined by
β n = j0 αn = 0 otherwise. Let E be the CMV matrix for these α’s. From the Schur algorithm we see f (z; 0, 0, . . . ) = 0 and f (z; β, 0, 0, . . . ) = β,
(1.8)
so E is not reflectionless anywhere. On the other hand, let µn be the spectral measure of E and δn , and let f (z, n) be its corresponding Schur function. It is shown in [13] (see also [18] for the analogous formula in the half-line case) that f (z, n) = f + (z, n) f − (z, n), z ∈ D, n ∈ Z.
(1.9)
dθ . In particular, µn is reflectionless on Thus, for any n ∈ Z, (1.8) implies dµn (θ ) = 2π all of ∂D while E is not reflectionless on any subset of positive Lebesgue measure.
We will show, however, that this is the only example of such behavior: Theorem 1.8. Let E be the whole-line CMV matrix corresponding to the sequence {αn }n∈Z , satisfying αn = 0 for at least two different n ∈ Z. Then E is reflectionless on A ⊆ ∂D if and only if µn is reflectionless on A for all n. The connection between the above result and Khrushchev’s theory of weak limits comes from the fact that, together with Example 1.7, Theorem 1.1 provides for a particularly simple proof of the following theorem of Khrushchev. Theorem 1.9 (Khrushchev [18]). Let C be a CMV matrix with Verblunsky coefficients iθ 2 {αn }∞ n=0 and measure µ, and let dµn (θ ) = |ϕn (e )| dµ(θ ). Then dµn (θ ) →
dθ 2π
weakly if and only if ∀k ≥ 1 lim αn αn+k = 0. n→∞
Furthermore, these conditions imply that either αn → 0 or µ is purely singular.
Right Limits for CMV Matrices
7
This theorem is naturally a part of a larger discussion dealing with weak limits of µn . In particular, Khrushchev’s theory deals with the cases in which such weak limits exist. We will show that the analysis of these cases becomes simple when performed using right limits. The reason for this is that the µn above are actually the spectral measures of C and δn and, along a proper subsequence, these converge weakly to the corresponding spectral measures of the right limit. Thus, if µn converges weakly to ν as n → ∞, all of the diagonal measures of any right limit are ν. This leads naturally to Definition 1.10. We say that a whole-line CMV matrix, E, belongs to Khrushchev Class if µn = µm for all n, m ∈ Z, where µ j is the spectral measure of E and δ j . By the discussion above, Proposition 1.11. If C is a CMV matrix such that the sequence µn has a weak limit as n → ∞ then all right limits of C belong to Khrushchev Class. Thus, Khrushchev theory reduces to the analysis of Khrushchev Class. Since Simon analyzed the analogous Jacobi case [35], we feel the following is fitting: Definition 1.12. We say that a whole-line Jacobi matrix, H , belongs to Simon Class if µn = µm for all n, m ∈ Z, where µ j is the spectral measure of H and δ j . The final section of this paper will be devoted to the analysis of these two classes. In particular, we rederive all of the main results of [35] and even extend some of those of [19]. We conclude with an amusing (and easy) fact: Proposition 1.13. Any H in the Simon Class is either periodic, and so reflectionless on its spectrum, or decomposes into a direct sum of finite (in fact 2 × 2 matrices), and so has pure point spectrum of infinite multiplicity. Similarly, any E in the Khrushchev Class that does not belong to the class introduced in Example 1.7 is either reflectionless on its spectrum or has pure point spectrum. The rest of this paper is structured as follows. Section 2 contains the proof of Theorems 1.4 and 1.1 as well as an application to random perturbations of CMV matrices. Section 3 contains a proof of Theorems 1.8 and 1.9, and Sect. 4 contains our analysis of the operators in the Khrushchev and Simon Classes and their relevance to Khrushchev’s theory of weak limits and ratio asymptotics. 2. The Proof of Remling’s Theorem for OPUC Our proof will parallel that of Remling [33] quite closely, so we will content ourselves with presenting the parts that differ significantly, but only sketching those parts that are similar. We will first need some definitions. Let z ∈ D and let S ⊂ ∂D be a Borel set, and define iθ e + z dθ ωz (S) = Re iθ . e − z 2π S (Here, and numerous times below, we have made use of the standard identification of ∂D with [0, 2π ) in that the integration is actually over the set {θ ∈ [0, 2π ) : eiθ ∈ S}. We trust this will not cause any confusion.) If f : D → D is a Schur function, define ω f (eiθ ) (S) = lim ω f (r eiθ ) (S). r ↑1
8
J. Breuer, E. Ryckman, M. Zinchenko
As z → ω f (z) (S) is a non-negative harmonic function in D, Fatou’s Theorem implies that this limit exists for (Lebesgue) almost every θ . Given Schur functions f n (z) and f (z), we will say that f n converges to f in the sense of Pearson if for all Borel sets A, S ⊆ ∂D, dθ dθ = ω fn (eiθ ) (S) ω f (eiθ ) (S) . lim n→∞ A 2π 2π A (We note here that in [1,2,33] this mode of convergence was called convergence in value distribution. However, since this term had already been used in [26] for a completely different concept, we will use the above name instead.) The next lemma relates this type of convergence to a more standard one: Lemma 2.1. Let f , f n , n ∈ N, be Schur functions. Then f n converges to f in the sense of Pearson if and only if f n (z) converges to f (z) uniformly on compact subsets of D. Of course, in this case it is well-known that the associated spectral measures then converge weakly as well. Proof. We simply sketch the proof since the full details may be found in [33]. For the forward implication, we may use compactness to pick a subsequence where g(z) := limk→∞ f n k exists (uniformly on compact subsets of D) and defines an analytic function. By uniqueness of limits in the sense of Pearson, we then must have g = f . For the opposite direction, one may either use spectral averaging (as in [33]), or simply appeal to Lemma 2.4 below. The basic result behind Theorem 1.4 is the following analog of a result of Breimesser and Pearson [1]: Theorem 2.2. Let C be a half-line CMV matrix. For all Borel sets S ⊆ ∂D and A ⊆ ac (C) we have dθ ∗ dθ lim − = 0, ω f+ (eiθ ) (S) ωeiθ f− (eiθ ) (S ) n→∞ 2π 2π A A where S ∗ = {z : z ∈ S}. Assuming Theorem 2.2 for a moment, we can prove Theorem 1.4: Proof of Theorem 1.4. Let E be a right limit of C, so there is a sequence n j ↑ ∞ such that lim j→∞ αn+n j (C) = αn (E) for the corresponding sequences of Verblunsky coefficients. Thus, if f ± (z) are the Schur functions of E defined by (1.7) for n = 0, then f ± (z, n j ) → f ± (z) as j → ∞ uniformly on compact subsets of D. By Lemma 2.1 and Theorem 2.2 we now have dθ dθ = ω f+ (eiθ ) (S) ωeiθ f− (eiθ ) (S ∗ ) 2π 2π A A for all Borel sets A ⊆ ac (C), S ⊆ ∂D. Now Lebesgue’s differentiation theorem and the fact that ωz (S ∗ ) = ωz (S) shows f + (eiθ ) = e−iθ f − (eiθ ) almost everywhere on ac (C). Thus, E is reflectionless on ac (C).
Right Limits for CMV Matrices
9
We now turn to the proof of Theorem 2.2. We will need a few preparatory results. Lemma 2.3. For any Schur function f (z), Borel set S ⊆ ∂D, and z ∈ D we have 2π ω f (z) (S) = ω f (eiθ ) (S)dωz (eiθ ). 0
In particular, for any Borel set A ⊆ ∂D, 2π dθ dθ ω f (r eiθ ) (S) ω f (eiθ ) (S)ωr eiθ (A) . = 2π 2π A 0 Proof. For the first statement, just note that both sides are harmonic functions of z with the same boundary values. The second statement follows by writing dωr eiθ (eiφ ) = and applying Fubini’s theorem.
dφ 1 − r2 2 1 + r − 2r cos(φ − θ ) 2π
Lemma 2.4. Let A ⊆ ∂D be a Borel subset. Then
dθ dθ
− lim sup
ω f (r eiθ ) (S) ω f (eiθ ) (S)
= 0, r ↑1 f,S 2π 2π A A where the supremum is taken over all Schur functions f (z) and all Borel sets S ⊆ ∂D. Proof. This follows from Lemma 2.3 and analyzing (the f -independent quantity) dθ ωr eiθ (Ac ) . 2π A For more details, see Lemma A.1 in [33] whose proof is nearly identical.
We will need a notion of pseudohyperbolic distance on D. Given w1 , w2 ∈ D define |w1 − w2 | γ (w1 , w2 ) = . 1 − |w1 |2 1 − |w2 |2 This is an increasing function of the hyperbolic distance on D. As such, if F : D → D is analytic, then γ (F(w1 ), F(w2 )) ≤ γ (w1 , w2 ) and if F is an automorphism with respect to hyperbolic distance on D (written “F ∈ Aut(D)”) then we have equality above. Taking F(z) to be the analytic function whose real part is ωz (S), we see that for all z, ζ ∈ D and all Borel sets S ⊆ ∂D, |ωz (S) − ωζ (S)| |ωz (S) − ωζ (S)| ≤ ≤ γ (F(z), F(ζ )) ≤ γ (z, ζ ). 1 − |ωz (S)|2 1 − |ωζ (S)|2 (2.1)
10
J. Breuer, E. Ryckman, M. Zinchenko
Now let {αn }n∈Z be a sequence of Verblunsky coefficients (some of which may lie on ∂D). Recall the two sequences of Schur functions defined by (1.7): f + (z, n) = f (z; αn , αn+1 , . . . ),
f − (z, n) = f (z; −αn−1 , −αn−2 , . . . ).
Since the Schur algorithm terminates at any αk ∈ ∂D, we see that for a half-line sequence of α’s (recall α−1 = −1) we have f − (z, n = 0) = −α−1 = 1. Viewing matrix arithmetic projectively (that is, identifying an automorphism of D with its coefficient matrix, see for instance [33]), the Schur algorithm shows f ± (z, n + 1) = T± (z, αn ) f ± (z, n), where
1 T+ (z, α) = −zα
−α z
−α . 1
z and T− (z, α) = −zα
By elementary manipulations we see that for any z ∈ C, 1 0 1 0 T+ (z, α) = . T (z, α) 0 z − 0 z
(2.2)
We will let P± (z, n) = T± (z, αn−1 ) · · · T± (z, α0 ) so that f ± (z, n) = P± (z, n) f ± (z, n = 0). We have the following mapping properties of T± (z, α): Lemma 2.5. Let α ∈ D. (1) If z ∈ ∂D, then T± (z, α) ∈ Aut(D). (2) If z ∈ D, then T− (z, α) : D → D and γ (T− (z, α)w1 , T− (z, α)w2 ) ≤ |z|γ (w1 , w2 ) for all w1 , w2 ∈ D. Proof. Let
1 S(α) = −α
−α 1
z and M(z) = 0
0 1
so that T+ (z, α) = M(z −1 )S(α) and T− (z, α) = S(α)M(z). Because α ∈ D we have S(α) ∈ Aut(D). If z ∈ ∂D then M(z) ∈ Aut(D) as well, while a straightforward calculation shows that if z ∈ D then γ (M(z)w1 , M(z)w2 ) ≤ |z|γ (w1 , w2 ). This proves (1) and (2).
Right Limits for CMV Matrices
11
With these preliminaries in hand we are ready for the proof of Theorem 2.2. We emphasize again that we are following the proof of Theorem 3.1 from [33]. Proof of Theorem 2.2. Subdivide A = A0 ∪ A1 ∪ · · · ∪ A N in such a way that 1. |A0 | < ε. N Ak , limr ↑1 f + (r eiθ , 0) exists and lies in D. 2. On k=1 3. For each 1 ≤ k ≤ N , there is a point m k ∈ D such that γ f + (eiθ , 0), m k < ε for all eiθ ∈ Ak . The construction of such a decomposition is identical to that given in [33], so we do not review it here. To deal with A0 , we note that for any z ∈ D and any Borel set S ⊆ ∂D, we have |ωz (S)| ≤ 1. Thus
dθ ∗ dθ
− ω f+ (eiθ ,n) (S) ωeiθ f− (eiθ ,n) (S ) < 2ε.
2π 2π A0 A0 N Now we consider A1 , . . . , A N . Notice that if eiθ ∈ k=1 Ak , then for all n ∈ N we iθ also have that limr ↑1 f + (r e , n) exists and lies in D. As P+ (eiθ , n) ∈ Aut(D) we see γ f + (eiθ , n), P+ (eiθ , n)m k < ε for all eiθ ∈ Ak and all n ∈ N. Using (2.1) and integrating we find
dθ dθ
< ε|Ak |. ω (S) ω (S) − iθ iθ f + (e ,n) P+ (e ,n)m k
2π 2π
Ak
Ak
By (2.2) and the fact that ωz (S ∗ ) = ωz (S), we can rewrite this as
dθ ∗ dθ
− ω f+ (eiθ ,n) (S) ωeiθ P− (eiθ ,n)(e−iθ m k ) (S ) < ε|Ak |
2π 2π Ak Ak
(2.3)
(and notice that because T− (z, α) = S(α)M(z), we have that z P− (z, n)(z −1 m k ) is indeed a Schur function). By Lemma 2.5 there is an n 0 ∈ N so that for all n ≥ n 0 , γ z P− (z, n)(z −1 wk ), z f − (z, n) < ε. As before, using (2.1) and integrating shows
∗ dθ ∗ dθ
− ωz P− (z,n)(z −1 wk ) (S ) ωz f− (z,n) (S ) < ε|Ak |.
2π 2π Ak Ak
(2.4)
Now use Lemma 2.4 to find an r < 1 so that
dθ dθ
− ω f (eiθ ) (S) ω f (r eiθ ) (S) < ε|Ak |
2π 2π Ak
Ak
for all Schur functions f (z), all Borel sets S ⊆ ∂D, and k = 1, . . . , N . Applying this to (2.3) and (2.4) shows
dθ ∗ dθ
− ω f+ (eiθ ,n) (S) ωeiθ f− (eiθ ,n) (S ) < 4ε|Ak |.
2π 2π Ak Ak
12
J. Breuer, E. Ryckman, M. Zinchenko
Now summing in k shows
∗ dθ
ω iθ (S) dθ − ωeiθ f− (eiθ ,n) (S ) < 4ε|A| + 2ε f + (e ,n)
2π 2π A A for all n ≥ n 0 .
Next, we illustrate Theorem 1.4 by a simple example of constant coefficients CMV matrices: Example 2.6. Let C be the half-line CMV matrix associated with the constant Verblunsky coefficients αn = a, n ≥ 0, for some a ∈ (0, 1). It follows from the Schur algorithm that the corresponding Schur function f a satisfies the quadratic equation az f a (z)2 + (1 − z) f a (z) − a = 0, and hence is given by
(1 − z)2 + 4a 2 z , z ∈ D, 2az √ where the square root is defined so that eiθ = eiθ/2 for θ ∈ (−π, π ). Using the 1+z f a (z) Carathéodory function Fa (z) = 1−z f a (z) we compute f a (z) =
−(1 − z) +
ac (C) = {eiθ : Re Fa (eiθ ) > 0} = {eiθ : | f a (eiθ )| < 1} = {eiθ : 2 arcsin(a) < θ < 2π − 2 arcsin(a)}. The half-line CMV matrix C has exactly one right limit E which is the whole-line CMV matrix associated with the constant coefficients αn = a, n ∈ Z. It follows from (1.7) that the two Schur functions for E are given by f + (z, n) = f a (z) and f − (z, n) = f −a (z) = − f a (z), n ∈ Z. Since for all eiθ ∈ ac (C), ⎛ ⎞ 2 i sin(θ/2) a ⎝1 − 1 − ⎠ f a (eiθ ) = aeiθ/2 sin(θ/2) and the expression under the square root is positive, one easily verifies the reflectionless property of E on ac (C), eiθ f + (eiθ , n) = eiθ f a (eiθ ) = − f a (eiθ ) = f − (eiθ , n), eiθ ∈ ac (C), thus confirming the claim of Theorem 1.4. Note that adding a decaying perturbation to the Verblunsky coefficients of C does not change the uniqueness of the right limit, nor does it change the limiting operator. Moreover, if the decay is sufficiently fast (e.g. 1 ), ac (C) does not change either. The following is one of the reasons reflectionless operators are so useful: Lemma 2.7. Let {αn }n∈Z , {βn }n∈Z be two sequences of Verblunsky coefficients such that their corresponding whole-line CMV matrices are both reflectionless on some common set A with |A| > 0. If αn = βn for all n < 0, then αn = βn for all n.
Right Limits for CMV Matrices
13
Proof. By the Schur algorithm, {αn }n<0 determines f − (z, 0). This, by Definition 1.3, determines f + (z, 0) on A. But the values of a Schur function on a set of positive Lebesgue measure on ∂D determine the Schur function. Thus, f + (z, 0) is determined throughout D by {αn }n<0 . But, by the Schur algorithm again, this determines {αn }n≥0 . Proof of Theorem 1.1. Take a subsequence {n jk }∞ k=1 , of n j , such that both lim αn+n jk ≡ βn
k→∞
and n lim αn+n jk ≡ β
k→∞
n for all n < 0 exist for every n ∈ Z. By conditions (ii) and (iii) of the theorem, βn = β but β0 = β0 . By Theorem 1.4 the corresponding whole-line CMV matrices, E and E, are reflectionless on ac (C) and ac (C) respectively. Thus, by Lemma 2.7 these two sets cannot intersect each other. Viewing a CMV matrix satisfying (1.4) as a perturbation of the CMV matrix with all dθ Verblunsky coefficients equal to zero (this matrix has spectral measure 2π ), we see by the above analysis that such a matrix cannot have any absolutely continuous spectrum, since (1.4) is easily seen to imply conditions (ii) and (iii) of the theorem. We conclude this section with an application of Theorem 1.4 to random CMV matrices. Theorem 2.8. Let {βn (ω)}∞ n=1 be a sequence of random Verblunsky coefficients of the form: βn (ω) = αn + sn X n (ω), where X n (ω) is a sequence of independent, identically distributed random variables whose common distribution is not supported at a single point, and sn is a bounded sequence such that |αn + sn | < 1. Let C(ω) be the corresponding random CMV matrix. If sn → 0 as n → ∞ then ac (C(ω)) = ∅ almost surely. We first need a lemma: Lemma 2.9. Let {βn (ω)}∞ n=1 be a sequence of independent random Verblunsky coefficients and let C(ω) be the corresponding family of CMV matrices. Then there exists a set A ⊆ ∂D such that with probability one, ac (C(ω)) = A. Proof. This is the CMV version of a theorem of Jakši´c and Last [16, Cor. 1.1.3] for Jacobi matrices. The proof is the same: Since the absolutely continuous spectrum is stable under finite rank perturbations, it is easily seen to be a tail event. Thus, the result is implied by Kolmogorov’s 0-1 Law. For details see [16]. Proof of Theorem 2.8. Pick a sequence, n j , such that lim j→∞ sn+n j ≡ Sn exists for every n ∈ Z and S0 = 0. Such a sequence exists by the assumptions on sn . By restricting to a subsequence, we may assume that lim j→∞ αn+n j + sn+n j = αn + Sn also exists for any n ∈ Z.
14
J. Breuer, E. Ryckman, M. Zinchenko
Let X 1 = X 2 be two points in the support of the common distribution of X n (ω). By the Borel-Cantelli Lemma, with probability one, there exist two subsequences n jk (ω) and n jl (ω) , such that lim X n+n jk (ω) (ω) = X 1 , n ∈ Z
k→∞
and
lim X n+n jl (ω) (ω) =
l→∞
X1 X2
n < 0, n ≥ 0.
Let E1 and E2 be the two whole-line CMV matrices corresponding to the sequences n1 ≡ β αn + Sn X 1 , n ∈ Z and
αn + Sn X 1 n < 0, 2 βn = αn + Sn X 2 n ≥ 0, respectively. If it were not true that ac (C(ω)) = ∅ almost surely, then by Lemma 2.9, the essential support of the absolutely continuous spectrum would be some deterministic set A = ∅. By Theorem 1.4, since E1 and E2 are both right limits of CMV matrices with absolutely continuous spectrum on A, they are both reflectionless on A. This contradicts Lemma 2.7 so we see that ac (C(ω)) = ∅ almost surely. 3. Reflectionless Matrices and Reflectionless Measures In this section we verify that Example 1.7 is the only example where the two notions of reflectionless (cf. Definitions 1.3 and 1.6) are not equivalent. As an application of this fact we give a short proof of Theorem 1.9, a special case of Khrushchev’s results. We start by introducing the whole-line unitary 5-diagonal CMV matrix E associated to {αn }n∈Z by ⎛ ⎜ ⎜ ⎜ ⎜ E =⎜ ⎜ ⎜ ⎝
..
..
. 0
⎞ .. .. .. . . . . ⎟ ⎟ α 0 ρ−1 −α 0 α−1 α 1 ρ0 ρ1 ρ0 ⎟ ρ0 ρ−1 −ρ0 α−1 −α 1 α0 −ρ1 α0 0 ⎟ ⎟, 0 α 2 ρ1 −α 2 α1 α 3 ρ2 ρ3 ρ2 ⎟ ρ2 ρ1 −ρ2 α1 −α 3 α2 −ρ3 α2 0 ⎟ ⎠ .. .. .. .. .. . . . . .
0
(3.1)
0
where ρn = (1 − |αn |2 )1/2 . Here the diagonal elements are given by En,n = −α n αn−1 . It is known [4,36] that CMV matrices have the following LM factorization: E = LM,
(3.2)
Right Limits for CMV Matrices
where
⎛
15
⎞
..
⎜ . ⎜ θ2k−2 L=⎜ ⎜ ⎝
0
0
θ2k
..
.
⎟ ⎟ ⎟, ⎟ ⎠
such that, for all k ∈ Z L2k,2k L2k,2k+1 = θ2k , L2k+1,2k L2k+1,2k+1 αk θk = ρk
⎛
⎞
..
⎜ . ⎜ θ2k−1 M=⎜ ⎜ ⎝
0
M2k−1,2k−1 M2k,2k−1 ρk . −αk
θ2k+1
0
M2k−1,2k M2k,2k
..
.
⎟ ⎟ ⎟, ⎟ ⎠
= θ2k−1 ,
Next, we introduce the diagonal Schur function f (z, n) associated with the diagonal spectral measure µn (i.e., the spectral measure of E and δn ) by 2π iθ e +z 1 + z f (z, n) = dµn (θ ) = δn , (E + z I )(E − z I )−1 δn , n ∈ Z. 1 − z f (z, n) eiθ − z 0 Then it follows from Definition 1.6 that the measure µn is reflectionless on A ⊆ ∂D if and only if Im(z f (z, n)) = 0 for a.e. z ∈ A. Recall from (1.9) that the diagonal and half-line Schur functions are related by f (z, n) = f + (z, n) f − (z, n). Thus, if a whole-line CMV matrix, E, is reflectionless on A, it follows that all diagonal measures µn , n ∈ Z, are reflectionless on A as well. As indicated in Example 1.7, the converse is not true in general. Nevertheless, one can show that this example is the only exceptional case: whenever all α’s are identically zero, or there are at least two nonzero α’s, the converse holds. We start by showing that a reflectionless CMV matrix E with a finite number of nonzero α’s can have no more than one nonzero Verblunsky coefficient. Lemma 3.1. Suppose E is a whole-line CMV matrix of the form (3.1) such that αm = 0, αn = 0 for some m, n ∈ Z, m < n, and µn is reflectionless on a set A ⊆ ∂D of positive Lebesgue measure. Then there are infinitely many nonzero α’s. Proof. Assume that there are finitely many nonzero α’s. Then it follows from the Schur algorithm that the Schur functions f ± (z, n) are rational functions of z with finitely many zeros in D and poles in C \ D. By (1.9) the same holds for f (z, n). Let B be a neighborhood of the finite set of zeros of f (z, n) on ∂D such that A \ B has positive Lebesgue measure. Then log(z f (z, n)) is a well-defined analytic function on some open neighborhood of ∂D \ B. The reflectionless assumption implies that Im log(z f (z, n)) ∈ {0, π } Lebesgue a.e. on d A\B, and hence by the Cauchy–Riemann equations, the analytic function dz log(z f (z, n)) is zero on accumulation points of A \ B. Since A \ B is of positive Lebesgue measure, the set of its accumulation points is also of positive Lebesgue measure, and hence d dz log(z f (z, n)) is identically zero in the neighborhood of ∂D \ B. This implies that z f (z, n) is a nonzero constant in D \ B. This is a contradiction since f (z, n) is analytic in D.
16
J. Breuer, E. Ryckman, M. Zinchenko
Proof of Theorem 1.8. Assume E is reflectionless in the sense of Definition 1.3. Then it follows from the discussion at the beginning of this section that µn is reflectionless for all n. Now assume µn is reflectionless in the sense of Definition 1.6 for all n. Since two α’s are not zero, it follows from Lemma 3.1 that there are infinitely many nonzero α’s. Let αn −1 , αn 0 , αn 1 , αn 2 , denote four consecutive nonzero α’s, that is, four non-zero values with possibly some zero values between them. For n ∈ Z, introduce g+ (z, n) = z f + (z, n) and g− (z, n) = f − (z, n). Then g± (z, n j ), j = 0, 1, 2, are not identically zero functions and it follows from the Schur algorithm that for z ∈ ∂D and n ∈ Z, g± (z, n − 1) = z
g± (z, n) + αn−1 g± (z, n)/z − αn , g± (z, n + 1) = . (3.3) α n−1 g± (z, n) + 1 −α n g± (z, n)/z + 1
The reflectionless condition at n 1 implies g+ (z, n 1 )g− (z, n 1 ) = z f (z, n 1 ) ∈ R \ {0} for a.e. z ∈ A, and hence g± (z, n 1 ) = s± (z)eit (z) with s± (z) ∈ R and t (z) ∈ [0, π ) for a.e. z ∈ A. In order to check that E is reflectionless in the sense of Definition 1.3 it remains to check that s+ (z) = s− (z) for a.e. z ∈ A. By construction all α’s between αn 0 and αn 1 are zero, hence it follows from (3.3) that g± (z, n 0 ) = z n 1 −n 0 eit (z)
s± (z) + e−it (z) αn 0 for a.e. z ∈ A. α n 0 eit (z) s± (z) + 1
Since for every γ ∈ D \ R the function h γ : [−1, 1] → (− π2 , π2 ) defined by x+γ h(x) = arg γ γx + 1
(3.4)
is one-to-one, (3.3) and the reflectionless condition at n 0 (i.e., g+ (z, n 0 )g− (z, n 0 ) = z f (z, n 0 ) ∈ R \ {0} for a.e. z ∈ A) imply s+ (z) = s− (z) for a.e. z ∈ A such that e−it (z) αn 0 ∈ D \ R.
(3.5)
Similarly, since by construction all α’s between αn 1 and αn 2 are zero, it follows from (3.3) that g± (z, n 2 ) = z n 1 −n 2 eit (z)
s± (z) − e−it (z) z n 2 −n 1 αn 1 for a.e. z ∈ A, −α n 1 eit (z) z n 1 −n 2 s± (z) + 1
and hence, the reflectionless condition at n 2 (i.e., g+ (z, n 2 )g− (z, n 2 ) = z f (z, n 2 ) ∈ R \ {0} for a.e. z ∈ A), together with the injectivity of h γ , implies s+ (z) = s− (z) for a.e. z ∈ A such that e−it (z) z n 2 −n 1 αn 2 ∈ D \ R.
(3.6)
Since e−it (z) αn 0 ∈ R and e−it (z) z n 2 −n 1 αn 2 ∈ R may hold simultaneously only on a finite set, it follows from (3.5) and (3.6) that s+ (z) = s− (z) for a.e. z ∈ A, and hence g+ (z, n 1 ) = g− (z, n 1 ) for a.e. z ∈ A. That is, E is reflectionless on A according to Definition 1.3.
Right Limits for CMV Matrices
17
Remark. We note that the case of identically zero Verblunsky coefficients corresponds to f ± (z, n) ≡ 0 for all n ∈ Z, so that the associated CMV matrix is reflectionless on ∂D in the sense of Definition 1.3 and hence all its diagonal spectral measures are reflectionless on ∂D in the sense of Definition 1.6. The case of a single nonzero coefficient, discussed in Example 1.7, corresponds to one of f + (z, n) or f − (z, n) being nonzero and the other being identically zero for each n ∈ Z, so that the associated CMV matrix is not reflectionless on any subset of ∂D of positive Lebesgue measure, yet all the diagonal measures are reflectionless on ∂D. Proof of Theorem 1.9. Let E be a right limit of C. Then all the diagonal measures of dθ E are identical and equal to 2π . The corresponding diagonal Schur functions in this case are f (z, n) ≡ 0 for all n ∈ Z. Hence by (1.9) for each n ∈ Z either f − (z, n) ≡ 0 or f + (z, n) ≡ 0 or both. The latter case corresponds to E having identically zero Verblunsky coefficients and the other two cases correspond to E having exactly one nonzero Verblunsky coefficient. Since this holds for all right limits of C, we conclude that for all k ∈ N, lim αn αn+k = 0.
(3.7)
n→∞
Conversely, (3.7) implies that all right limits of C may have at most one nonzero Verblunsky coefficient. Hence all right limits of C have identical diagonal spectral meadθ sures equal to 2π . This implies that the diagonal measures dµn of C converge weakly dθ to 2π as n → ∞. The final statement follows since if αn → 0 then clearly (1.4) holds, which implies, by Theorem 1.1, that µ is purely singular. 4. The Simon and Khrushchev Classes In this section we extend the discussion of the previous section to include all cases where µn has a weak limit. We consider both the Jacobi and CMV cases, but begin with the Jacobi case since it is technically simpler. 4.1. The Simon Class. A half-line Jacobi matrix is a semi-infinite matrix of the form: ⎛ ⎞ 0 ... b1 a1 0 ⎜ a1 b2 a2 0 . . . ⎟ J {an , bn }∞ . (4.1) n=1 = ⎝ 0 a2 b3 a3 . . . ⎠ ... ... ... ... ... Its whole-line counterpart is defined by ⎞ ⎛ .. .. .. . . . ⎟ ⎜ a−1 b0 a0 ⎟ ⎜ ⎟ ⎜ a0 b1 a1 H ({an , bn }n∈Z ) = ⎜ ⎟, ⎟ ⎜ a1 b2 a2 ⎠ ⎝ .. .. .. . . .
0
0
(we assume bn ∈ R, an ≥ 0 in both cases).
(4.2)
18
J. Breuer, E. Ryckman, M. Zinchenko
These matrices may be viewed as operators on 2 (N) and 2 (Z), respectively, and both are clearly symmetric. As is well known, the theory of half-line Jacobi matrices with an > 0 is intimately related to that of orthogonal polynomials on the real line (OPRL) through the recursion formula for the polynomials (see e.g. [34]). In particular, in the self-adjoint case (to which we shall henceforth restrict our discussion), µ—the spectral measure for J and δ1 —is the unique solution to the corresponding moment problem. Furthermore, dµn (x) = | pn−1 (x)|2 dµ(x) is the spectral measure for J and δn , where { pn (x)} are the orthonormal polynomials corresponding to µ. The whole-line matrices enter naturally into this framework as right limits (see e.g. [33] for the definition). The problem at the center of our discussion is that of the identification of right limits of half-line Jacobi matrices with the property that µn has a weak limit as n → ∞. As is clear from the discussion in the Introduction, all these right limits belong to Simon Class (recall Definition 1.12). Theorem 4.1. Let H ({an , bn }n∈Z ) be a whole-line Jacobi matrix. The following are equivalent: (i) H belongs to Simon Class. (ii) For all m, n ∈ Z, xdµn (x) = xdµm (x) and x2 dµn (x) = x2 dµm (x). (iii) a2n = a, a2n+1 = c, bn = b for some numbers, a, c ≥ 0 and b ∈ R. Remark. In particular, this shows that if H has constant first and second moments, then H belongs to Simon Class. Note, however, that the values of these moments do not determine the element of the class itself (not even up to translation; see (4.3) and (4.4) below). Thus, it makes sense to define S(A, B) to be the set of all matrices in the Simon Class having a 2 + c2 = A and b = B, where a, b, c are as in (iii) above. Proof. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii): Noting that xdµn (x) = bn
(4.3)
2 x2 dµn (x) = an−1 + bn2 + an2 ,
(4.4)
and
the result follows immediately for the diagonal elements. With this in hand, comparing 2 2 the second moment of µn and µn+1 we see that an−1 + an2 = an2 + an+1 from which it follows that an−1 = an+1 , and we are done. (iii) ⇒ (i): Clearly, by symmetry, µn = µn+2 for all n, and if a = c also µn = µn+1 . Thus, we are left with showing µ0 = µ1 under the assumption a = c (so at least one is nonzero). We shall show that for any z ∈ C+ ,
dµ0 (x) = x−z
dµ1 (x) . x−z
Fix z ∈ C+ and let {u(n)} be a sequence satisfying an u(n+1)+bn u(n)+an−1 u(n−1) = zu(n) that is 2 at ∞. This sequence is unique up to a constant factor. By the symmetry
Right Limits for CMV Matrices
19
of H , note that v(n) ≡ u(1 − n) satisfies the same equation and is 2 at −∞. Now write dµ0 (x) u(0)v(0) = δ0 , (H − z)−1 δ0 = x−z v(1)u(0) − v(0)u(1) v(1)u(1) u(0)u(1) = = v(1)u(0) − v(0)u(1) v(1)u(0) − v(0)u(1) dµ1 (x) = δ1 , (H − z)−1 δ1 = , x−z from which, by standard results, it follows that µ0 = µ1 .
We immediately get the following: Corollary 4.2. Let J be a self-adjoint half-line Jacobi matrix and let µ be its spectral measure. For n ≥ 1, let dµn (x) = | pn−1 (x)|2 dµ(x) be the spectral measure of J and δn . If ˜ (4.5) xdµn (x) = B, lim n→∞
lim
n→∞
˜ x2 dµn (x) = A,
(4.6)
˜ then J is bounded and all right limits of J are in S( A˜ − B˜ 2 , B). Proof. That J is bounded follows from (4.3) and (4.4) applied to J , together with the fact that these moments converge. Since all right limits of J have constant first and second moments, it follows from Theorem 4.1 that they all belong to Simon Class. The rest follows from combining (4.3), (4.4), (4.5) and (4.6) together with the definition of S(A, B). Corollary 4.2 is closely related to Theorem 2 in [35]—both our assumptions and conclusions are weaker. We also note that our proof is not that much different from the corresponding parts in Simon’s proof. However, we believe that the “right limit point of view” makes various ideas especially transparent and clear. In particular, we would like to emphasize the following subtle point. While convergence of the first and second moments does not imply weak convergence, by Corollary 4.2 it does imply a certain weak form of weak convergence: it holds along any subsequence on which J has a right limit. Also, it is now clear that any additional condition forcing weak convergence of µn ˜ (up is equivalent to a condition that distinguishes a particular element of S( A˜ − B˜ 2 , B) to a shift). Computing powers of J shows that the third moment is not enough, but the fourth moment is. Thus we get Theorem 4.3 (Simon [35]). If (4.5) and (4.6) hold and limn→∞ x4 dµn (x) exists, then J is bounded, dµn converge weakly and J has a unique right limit (up to a shift) in ˜ S( A˜ − B˜ 2 , B). We next want to demonstrate how ratio asymptotics also fit naturally into this framework.
20
J. Breuer, E. Ryckman, M. Zinchenko
Definition 4.4. Let µ be a probability measure on the real line. We say µ is ratio asymptotic if lim
n→∞
Pn+1 (z) an+1 pn+1 (z) ≡ lim n→∞ Pn (z) pn (z)
exists for all z ∈ C \ R, where Pn are the monic orthogonal polynomials, pn are the orthonormal polynomials, and an are the off-diagonal Jacobi parameters corresponding to µ. As above, our strategy for dealing with ratio asymptotic measures consists of identifying the appropriate class of right limits by analyzing their invariants. We also want to emphasize the relationship with weak asymptotic convergence. Let H ({an , bn }n∈Z ) ∞ + be a whole-line Jacobi matrix. Let Jn = J {a j+n , b j+n } j=1 be the half-line Jacobi 2 matrix one gets when restrictingH to ( j > n) with Dirichlet boundary conditions, ∞ − and Jn = J {an− j , bn+1− j } j=1 , the half-line Jacobi matrix one gets when restricting
H to 2 ( j ≤ n) with Dirichlet boundary conditions. Jn+ and Jn− have spectral measures associated with them which we denote by µ+n and µ− n . Finally, for z ∈ C \ R, let dµ±n (x) m ± (n; z) = x−z be the corresponding Borel-Stieltjes transforms. We are interested in H for which these are constants in n. The reason for this is the fact that if H is a right (z) 1 limit of J , then − m − (0;z) is a limit of PPn+1 along an appropriate subsequence (see e.g. n (z) [33]—note that his m − is our −1/m − ). Thus, Theorem 4.5. Let H ({an , bn }n∈Z ) be a whole-line Jacobi matrix. Then the following are equivalent: (i) H belongs to Simon Class and its spectrum is a single interval. (ii) an = a, bn = b for some numbers, a ≥ 0 and b ∈ R and all n ∈ Z. (iii) m − (n; z) = m − (n + 1; z) for all z ∈ C \ R, n ∈ Z. (iv) m + (n; z) = m + (n + 1; z) for all z ∈ C \ R, n ∈ Z. (v) m − (n; z) = m − (n + 1; z) for some z ∈ C \ R and all n ∈ Z. (vi) m + (n; z) = m + (n + 1; z) for some z ∈ C \ R and all n ∈ Z. Proof. (i) ⇔ (ii) follows from the theory of periodic Jacobi matrices (see [43, Sect. 7.4]). (ii) ⇒ (iii) ⇒ (v) and (ii) ⇒ (iv) ⇒ (vi) are clear by periodicity. Thus we are left with showing (v) ⇒ (ii) and (vi) ⇒ (ii). Writing down the continued fraction expansion for m − (n; z): −
1 = z − bn+1 + an2 m − (n; z), n ∈ Z, m − (n + 1; z)
one sees that (v) implies m − (n; z) satisfies a quadratic equation. an and bn+1 are then determined from this equation by taking imaginary and real parts, and so we get (v) ⇒ (ii) (see the proof of Theorem 2.2 in [35] for details). The same can be done for m + (n; z) to get (vi) ⇒ (ii). By the above discussion and Theorem 4.5, it follows that µ is ratio asymptotic if and only if its Jacobi matrix has a unique right limit in Simon Class with constant off-diagonal elements. Moreover, (v) in Theorem 4.5 implies that it is enough to require ratio asymptotics at a single z ∈ C \ R. This is precisely the content of Theorem 1 in [35]. We shall show below that the same strategy can be applied in the OPUC case in order to get a strengthening of corresponding results by Khrushchev.
Right Limits for CMV Matrices
21
4.2. The Khrushchev Class. We now turn to the discussion of the analogous theory for half-line CMV matrices. Namely, we study CMV matrices with the property that dµn (θ ) = |ϕn (eiθ )|2 dµ(θ ) has a weak limit as n → ∞. Again, as is clear from the discussion in the Introduction, all these right limits belong to Khrushchev Class (recall Definition 1.10) and so the analysis is mainly the analysis of properties of that class. Since nontrivial CMV matrices can have many powers with zero diagonal, the computations are substantially more complicated. Here is the analog of Theorem 4.1: Theorem 4.6. Let E be a whole-line CMV matrix and k ∈ N ∪ {∞}. Then the following are equivalent: (i) E belongs to Khrushchev Class with [E ]n,n = 0 for = 1, . . . , k − 1 and all n ∈ Z, and in the case k < ∞, [E k ]n,n = c for some c ∈ D \ {0} and all n ∈ Z. (ii) For = 1, . . . , k − 1, 2π eiθ dµn (θ ) = 0, n ∈ Z, 0
and if k < ∞ then additionally, for some c ∈ D \ {0}, 2π eikθ dµn (θ ) = c, n ∈ Z. 0
(iii) There exist n 0 ∈ N, a, b ∈ (0, 1], and t ∈ [0, 2π ) such that in the case k < ∞, αn 0 +nk+ j
|αn 0 +2nk | = a, |αn 0 +(2n+1)k | = b, = 0, arg(α n 0 +(n+1)k αn 0 +nk ) = t, n ∈ Z, j = 1, . . . , k − 1,
and in the case k = ∞, α j = 0,
j ∈ Z \ {n 0 }.
Remark. In particular, this shows that the constancy of the first k moments, where the k th moment is the first nonzero one, implies that E belongs to Khrushchev Class. Note, however, that the value of the k th moment does not determine the element of the class itself (again, not even up to translation; see Theorem 4.8 below). Thus, it makes sense to define K(c, k), for k < ∞, to be the set of all matrices in the Khrushchev Class with [E ]n,n = 0 for all n ∈ Z, = 1, . . . , k − 1, and [E k ]n,n = c = 0 for all n ∈ Z. In the case k = ∞, let K(∞) be the set of all matrices with [E ]n,n = 0 for all n ∈ Z, ≥ 1. We note that every CMV matrix E from the Khrushchev Class belongs to one of K(c, k), c ∈ D \ {0}, k ∈ N, or to K(∞). Proof. (i) ⇒ (ii): Follows from 2π eiθ dµn (θ ) = [E ]n,n for all ∈ N, n ∈ Z.
(4.7)
0
(ii) ⇒ (iii): First, observe that (iii) is equivalent to the following López-type condition: there exists n 0 ∈ Z such that for all n ∈ Z, = 1, . . . , k, j = 0, . . . , − 1,
abeit j = 0, = k, (4.8) α n 0 +n+ j αn 0 +(n−1)+ j = 0 otherwise.
22
J. Breuer, E. Ryckman, M. Zinchenko
We will show that (4.8) holds with abeit = −c by verifying inductively with respect to that
2π iθ e dµn 0 +n j = 0, − α n 0 +n+ j αn 0 +(n−1)+ j = 0 (4.9) 0 otherwise for some n 0 ∈ Z and all n ∈ Z, = 1, . . . , k, j = 0, . . . , − 1. The case = 1 trivially follows from (3.1) since the first moment of µn is exactly the diagonal element En,n for all n ∈ Z. Now suppose (4.9) holds for = 1, . . . , p − 1 for some p ≤ k. In view of (4.7), we want to compute [E p ]n,n = [(LM) p ]n,n . To do this, it turns out to be useful to separate the diagonal and off-diagonal elements of L and M and identify the contributions to the product. Thus, let the diagonal matrices X −1 = diagL and X 1 = diagM be the diagonals of L and M respectively. Furthermore, define Y−1 and Y1 through L = X −1 + Y−1 , M = X 1 + Y1 . Expressed in this notation, our objective is to compute the diagonal elements of (4.10) E p = (X −1 + Y−1 ) X (−1)2 + Y(−1)2 · · · X (−1)2 p + Y(−1)2 p . First, it is a direct computation to verify that for any two s, r ∈ N, diagY(−1)s Y(−1)s+1 · · · Y(−1)s+r = 0.
(4.11)
Now, using (ii) and the induction hypothesis ((4.9) for ≤ p − 1) one verifies that [Y(−1) j−s Y(−1) j−s+1 · · · Y(−1) j−1 X (−1) j Y(−1) j+1 · · · Y(−1) j+s−1 Y(−1) j+s ]n,n
2 α n+s ρn2 · · · ρn+s−1 n + s + j is odd, = 2 2 −αn−s−1 ρn−s · · · ρn−1 n + s + j is even
α n+s n + s + j is odd, = n, j ∈ Z, s = 0, . . . , p − 1, (4.12) −αn−s−1 n + s + j is even, and [Y(−1) j−s Y(−1) j−s+1 · · · Y(−1) j−1 X (−1) j Y(−1) j+1 · · · Y(−1) j+s−1 Y(−1) j+s ]n,m = 0 (4.13) whenever n = m. This identity combined with the induction hypothesis, (ii), (4.7), (4.10), and (4.11) j = Y(−1) j , implies that (for notational simplicity we let Y X j = X (−1) j ) 2π ei pθ dµn (θ ) = [E p ]n,n 0
=
p 1 · · · Y −1 p+−1 2 p ]n,n +1 · · · Y p++1 · · · Y [Y X Y X p+ Y =1
p α n+ p− αn− n is odd, =− α n+−1 αn− p+−1 n is even =−
=1 p =1
α n+ p− αn− , n ∈ Z.
Right Limits for CMV Matrices
23
The idea behind the computation is that all summands containing no X , a single X , or two X ’s that are a distance greater than or less than p apart do not contribute to the diagonal. This follows from the induction hypothesis and (4.11)–(4.13). Now, observe that the sum in the above equality may have at most one nonzero term. Indeed, if there are no nonzero terms we are done (this may happen only if p < k), otherwise let n 0 ∈ Z be such that α n 0 αn 0 − p = 0. Then combining the induction hypothesis (4.9) with (ii) yields, α n+ αn = 0, n ∈ Z, = 1, . . . , p − 1 which together with α n 0 αn 0 − p = 0 implies αn 0 +np+ = 0, n ∈ Z, = 1, . . . , p − 1. Hence, when p < k,
2π
0= 0
ei pθ dµn 0 +np (θ ) = −α n 0 +np αn 0 +(n−1) p , n ∈ Z,
and carrying the induction up to k, c= 0
2π
eikθ dµn 0 +nk (θ ) = −α n 0 +nk αn 0 +(n−1)k , n ∈ Z.
Thus, (4.9) holds for = k, and hence (iii) follows from (4.8) with abeit = −c. (iii) ⇒ (i): First, note that if k = ∞ then there is at most one nonzero Verblunsky dθ coefficient and hence by Example 1.7, E is in the Khrushchev Class with dµn (θ ) = 2π for all n ∈ Z, and hence it follows from (4.7) that [E ]n,n = 0 for all ∈ N and n ∈ Z. Next suppose k < ∞. It follows from (iii) that there are t0 , t ∈ [0, 2π ) such that αn 0 +n = |αn 0 +n |ei(t0 +tn) for all n ∈ Z. Then, using the Schur algorithm, one finds the following relations between the functions f ± associated with α = {αn }n∈Z and |α| = {|αn |}n∈Z , respectively, f + (z, n 0 + n; α) = ei(t0 +tn) f + (e−it z, n 0 + n; |α|), f − (z, n 0 + n; α) = e−i(t0 +t (n−1)) f − (e−it z, n 0 + n; |α|), n ∈ Z, z ∈ D. Hence by (1.9) the diagonal Schur functions associated with α and |α| are related by f (z, n 0 + n; α) = eit f (e−it z, n 0 + n; |α|), n ∈ Z, z ∈ D.
(4.14)
Now, the conditions in (iii) imply, f + (z, n 0 + nk + j; |α|) = z k− j f + (z, n 0 + (n + 1)k; |α|), f − (z, n 0 + nk + j; |α|) = z j−1 f − (z, n 0 + nk + 1; |α|), f + (z, n 0 + nk; |α|) = f + (z, n 0 + (n mod 2)k; |α|), f − (z, n 0 + nk + 1; |α|) = − f + (z, n 0 + nk; |α|), n ∈ Z, j = 1, . . . , k, z ∈ D.
24
J. Breuer, E. Ryckman, M. Zinchenko
These identities together with (1.9) and (4.14) yield f (z, n 0 + nk + j; α) = e−it (k−2) z k−1 f + (e−it z, n 0 + (n + 1)k; |α|) f − (e−it z, n 0 + nk + 1; |α|)
= −e−it (k−2) z k−1 f + (e−it z, n 0 + (n + 1)k; |α|) f + (e−it z, n 0 + nk; |α|) = −e−it (k−2) z k−1 f + (e−it z, n 0 + k; |α|) f + (e−it z, n 0 ; |α|)
for all n ∈ Z, j = 1, . . . , k, z ∈ D. Hence f (·, n; α) = f (·, m; α) which is equivalent to µm = µn for all m, n ∈ Z. The presence of the factor z k−1 implies that the first k − 1 moments of µn , n ∈ Z, are zero. This follows from the relationship (1.5) between Schur functions and Carathéodory functions and the fact that Taylor coefficients of F are twice the complex conjugates of the moments of µ. Moreover, since z −k+1 f (z, n 0 + nk + j; α) is nonzero at the origin, the k th moment of µn , n ∈ Z, is nonzero, and hence one gets (i) from (4.7). Corollary 4.7. Let C be a half-line CMV matrix and let µ be its spectral measure. For n ≥ 0, let dµn (θ ) = |ϕn (eiθ )|2 dµ(θ ) be the spectral measure of C and δn . If for some c ∈ D \ {0} and k ∈ N ∪ {∞}, lim
n→∞ 0
2π
e
iθ
0 = 1, . . . , k − 1, dµn (θ ) = c = k, k < ∞,
(4.15)
then all right limits of C are in K(c, k) if k < ∞ or in K(∞) if k = ∞. The analogy with Corollary 4.2 should be clear. Corollary 4.7 is a variant of Theorem E in [19] with weaker assumptions and weaker conclusions. Our proof is new and based on a completely different approach. We also note that much the same as in the Jacobi case, convergence of the first k-moments does not imply weak convergence, but by Corollary 4.7 it does imply the same weak form of weak convergence: convergence holds along any subsequence on which C has a right limit. A notable difference between the OPUC and OPRL cases is the fact that multiplication of the Verblunsky coefficients by a constant phase does not change the spectral measures. Thus, even when µn converges weakly, it is not possible to deduce uniqueness of a right limit (even up to a shift). Note that the phase ambiguity is equivalent to a choice of t0 in the proof of Theorem 4.6 and there is no way to determine this t0 from information on µn alone. In the case k = ∞ even |αn 0 | cannot be determined from the information on the measure and so the indeterminacy is, in a sense, even more severe. That said, as in the Jacobi case, it is clear that when k < ∞ any condition forcing weak convergence of µn (in addition to those in Corollary 4.7) is equivalent to a condition that distinguishes an element of K(c, k) (up to a shift and multiplication by an arbitrary phase). In particular, a somewhat tedious computation (along the lines of the argument in (ii) ⇒ (iii) above) shows that the following result holds: 2π Theorem 4.8. Suppose that k < ∞, (4.15) holds, and limn→∞ 0 e2ikθ dµn (θ ) exists. Then dµn converge weakly and C has a unique right limit (up to a shift and multiplication by a constant phase) in K(c, k).
Right Limits for CMV Matrices
25
Remark. We note that on the level imply for k < ∞, ⎧ ⎪ ⎨a lim |αn 0 +2nk+ j | = b n→∞ ⎪ ⎩0
of Verblunsky coefficients, Theorems 4.6 and 4.8 j = 0, j = k, j ∈ {1, . . . , k − 1, k + 1, . . . , 2k − 1},
(4.16)
lim α n 0 +(n+1)k αn 0 +nk = −c, for some n 0 ∈ Z and ab = |c|,
n→∞
and similarly, Theorem 4.6 and Corollary 4.7 imply for k = ∞, lim |αn 0 +n αn | = 0 for any n 0 ∈ Z.
n→∞
(4.17)
This extends [19, Thm. E], where the stronger condition of weak convergence for the measures dµn is assumed. Next, we use right limits to study ratio asymptotics. It is convenient to introduce 1) as the subclass of K(c, 1) consisting of CMV matrices with Verblunsky coeffiK(c, cients of constant absolute value. Definition 4.9. Let µ be a probability measure on the unit circle. We say µ is ratio asymptotic if ∗n+1 (z) n→∞ ∗ n (z) lim
exists for all z ∈ D, where, as usual, n (z) is the degree n monic orthogonal polynomial associated to µ. In particular, we say ratio asymptotics holds at z ∈ D with limit G(z) if ∗n+1 (z) = G(z). n→∞ ∗ n (z) lim
(4.18)
Theorem 4.10. Let n be the monic orthogonal polynomials associated with a half-line CMV matrix C. If either all right limits of C are in K(∞) or C has a unique right limit 1), then µ is ratio asymptotic. (up to a multiplication by a constant phase) in K(c, Conversely, if ratio asymptotics holds at some point z 0 ∈ D\{0} with limit G(z 0 ) = 1, then all right limits of C are in K(∞). If ratio asymptotics holds at two points z 1 , z 2 ∈ D \ {0} and the limit is not 1 at either point, then C has a unique right limit (up to 1) for some c ∈ D \ {0}. multiplication by a constant phase) in K(c, Proof. First, observe that it follows from the Szeg˝o recursion (1.1) that for all n ∈ Z+ and z ∈ D, 1−
∗n+1 (z) n (z) = zαn ∗ = zαn f (z; −α n−1 , −α n−1 , . . . , −α 0 , 1). ∗ n (z) n (z)
(4.19)
We refer to [37, Prop. 9.2.3] for the details on the second equality. Abbreviating by f n (z) = f (z; −α n−1 , −α n−1 , . . . , −α 0 , 1), we see that ratio asymptotics (4.18) at z ∈ D \ {0} is equivalent to limn→∞ αn f n (z) = g(z) ≡ (1 − G(z))/z. Let E be a right limit of C and βn , f ± (·, n), n ∈ Z, be the Verblunsky coefficients and Schur functions associated with E. Then, βn f − (z, n) = lim j→∞ αn+n j f n+n j (z) for all n ∈ Z, z ∈ D, and some sequence {n j } j∈N .
26
J. Breuer, E. Ryckman, M. Zinchenko
By Theorem 4.6, if E ∈ K(∞) then at most √one βn is nonzero, and hence β0 f − (z, 0) = 1) then |βn | = |c| and β n+1 βn = −c for all n ∈ Z. Then 0 for all z ∈ D. If E ∈ K(c, the Schur algorithm implies that β0 f − (z, 0) is a function that depends only on the value of c. Since in both cases β0 f − (z, 0) is independent of the sequence n j , it follows that limn→∞ αn f n (z) = β0 f − (z, 0) for all z ∈ D. Thus, by (4.19), ratio asymptotics holds for all z ∈ D. Conversely, by (4.19), ratio asymptotics at z ∈ D \ {0} implies βn f − (z, n) = g(z) for all n ∈ Z. By the Schur algorithm we have f − (z, n + 1)[1 − zg(z)] = z f − (z, n) − β n for all n ∈ Z.
(4.20)
If (4.18) holds at z 0 = 0 and the limit is 1, then by (4.19) g(z 0 ) = 0. Thus, by (4.20) there is at most one nonzero βn since βn 0 = 0 implies inductively that f − (z 0 , n) = 0 and hence βn = 0 (since g(z 0 ) = 0) for all n > n 0 . By Theorem 4.6, E ∈ K(∞). Finally consider the case where ratio asymptotics holds at two different points z 1 , z 2 ∈ D \ {0} and the limit is not 1 at either point. Then by (4.19), g(z 1 ) = 0 and g(z 2 ) = 0 and hence βn = 0 for all n ∈ Z. We also see that z 1 g(z 1 ) = z 2 g(z 2 ) since otherwise it follows from (4.20) that z 1 = z 2 . Thus, multiplying (4.20) by βn+1 /(zg(z)), substituting z = z j , j = 1, 2, and subtracting the results then yields βn+1 β n =
1 z1
− g(z 1 ) − 1 z 1 g(z 1 )
−
1 z2
+ g(z 2 )
1 z 1 g(z 1 )
for all n ∈ Z.
(4.21)
Similarly, multiplying (4.20) by |βn |2 βn+1 and evaluating at z = z 1 one obtains, |βn |2 =
z 1 g(z 1 )βn+1 β n g(z 1 )(1 − z 1 g(z 1 )) + βn+1 β n
for all n ∈ Z.
(4.22)
By (4.21) the right-hand side of (4.22) is n-independent, and hence |βn | as well as βn+1 β n are n-independent constants uniquely determined by the ratio asymptotics (4.18) at z 1 1) with c = −β n+1 βn = 0. Since c is determined by the ratio and z 2 . Thus, E ∈ K(c, asymptotics at z 1 and z 2 , all right limits are the same up to multiplication by a constant phase. Remark. This theorem extends an earlier result of Khrushchev [19, Thm. A]. We conclude with the Proof of Theorem 1.13. Let H belong to the Simon Class and let a, b, c be as in (iii) of Theorem 4.1. If a, c > 0 then H is a periodic whole-line Jacobi matrix which is well known to be reflectionless on its spectrum ([43]). If a = 0 or c = 0 then H is a direct sum of identical 2 × 2 (or 1 × 1) self-adjoint matrices and so has pure point spectrum of infinite multiplicity supported on at most two points. For E in the Khrushchev Class with k < ∞ the same analysis goes through: as long as |a|, |b| < 1 we get a reflectionless operator. If one of them or both are unimodular then it is easy to see that E is a direct sum of 2 × 2 (or 1 × 1) matrices. If k = ∞ then E is either reflectionless or belongs to the class of matrices from Example 1.7. Acknowledgements. We would like to thank Barry Simon for helpful discussions, as well as the referees for their useful comments.
Right Limits for CMV Matrices
27
References 1. Breimesser, S.V., Pearson, D.B.: Asymptotic value distribution for solutions of the Schrödinger equation Math. Phys. Anal. Geom. 3, 385–403 (2000) 2. Breimesser, S.V., Pearson, D.B.: Geometrical aspects of spectral theory and value distribution for Herglotz functions. Math. Phys. Anal. Geom. 6, 29–57 (2003) 3. Breuer, J., Ryckman, E., Simon, B.: Equality of the spectral and dynamical definitions of reflection. (2009, preprint) 4. Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Lin. Alg. Appl. 362, 29–56 (2003) 5. De Concini, C., Johnson, R.A.: The algebraic-geometric AKNS potentials. Ergod. Th. Dyn. Syst. 7, 1–24 (1987) 6. Craig, W.: The trace formula for Schrödinger operators on the line. Commun. Math. Phys. 126, 379–407 (1989) 7. Deift, P., Simon, B.: Almost periodic Schrödinger operators III. The absolutely continuous spectrum in one dimension. Commun. Math. Phys. 90, 389–411 (1983) 8. Geronimus, Ya.L.: On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carathéodory and Schur functions. Mat. Sb. 15, 99–130 (1944) 9. Gesztesy, F., Krishna, M., Teschl, G.: On isospectral sets of Jacobi operators. Commun. Math. Phys. 181, 631–645 (1996) 10. Gesztesy, F., Makarov, K.A., Zinchenko, M.: Local AC spectrum for reflectionless Jacobi, CMV, and Schrödinger operators. Acta Appl. Math. 103, 315–339 (2008) 11. Gesztesy, F., Yuditskii, P.: Spectral properties of a class of reflectionless Schrödinger operators. J. Funct. Anal. 241, 486–527 (2006) 12. Gesztesy, F., Zinchenko, M.: A Borg-type theorem associated with orthogonal polynomials on the unit circle. J. Lond. Math. Soc. (2) 74, 757–777 (2006) 13. Gesztesy, F., Zinchenko, M.: Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. J. Approx. Theory 139, 172–213 (2006) 14. Gesztesy, F., Zinchenko, M.: Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators. J. Diff. Eqs. 246, 78–107 (2009) 15. Golinskii, L., Nevai, P.: Szego difference equations, transfer matrices and orthogonal polynomials on the unit circle. Commun. Math. Phys. 223, 223–259 (2001) 16. Jakši´c, V., Last, Y.: Spectral structure of Anderson type Hamiltonians. Invent. Math. 141, 561–577 (2000) 17. Johnson, R.A.: The recurrent Hill’s equation. J. Diff. Eqs. 46, 165–193 (1982) 18. Khrushchev, S.: Schur’s algorithm, orthogonal polynomials, and convergence of Wall’s continued fractions in L 2 (T). J. Approx. Theory 108, 161–248 (2001) 19. Khrushchev, S.: Classification theorems for general orthogonal polynomials on the unit circle. J. Approx. Theory 116, 268–342 (2002) 20. Kotani, S.: Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. In: Stochastic Analysis, K. Itˇo (ed.), Amsterdam: North-Holland, 1984, pp. 225–247 21. Kotani, S.: One-dimensional random Schrödinger operators and Herglotz functions. In: Probabilistic Methods in Mathematical Physics, K. Itˇo, N. Ikeda (eds.), New York: Academic Press, 1987, pp. 219– 250 22. Kotani, S., Krishna, M.: Almost periodicity of some random potentials. J. Funct. Anal. 78, 390–405 (1988) 23. Last, Y., Simon, B.: The essential spectrum of Schrödinger, Jacobi, and CMV operators. J. Anal. Math. 98, 183–220 (2006) 24. Melnikov, M., Poltoratski, A., Volberg, A.: Uniqueness theorems for Cauchy integrals. http://arxiv.org/ abs/0704.0621v1[math.cv], 2007 25. Nazarov, F., Volberg, A., Yuditskii, P.: Reflectionless measures with a point mass and singular continuous component. http://arxiv.org/abs/0711.0948v1[math-ph], 2007 26. Nevanlinna, R.: Analytic Functions. Translated from the second German edition by Phillip Emig, Die Grundlehren der mathematischen Wissenschaften, Band 162, New York-Berlin: Springer-Verlag, 1970 27. Peherstorfer, F., Yuditskii, P.: Asymptotic behavior of polynomials orthonormal on a homogeneous set. J. Anal. Math. 89, 113–154 (2003) 28. Peherstorfer, F., Yuditskii, P.: Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces. J. Approx. Theory 139, 91–106 (2006) 29. Poltoratski, A., Remling, C.: Reflectionless Herglotz functions and Jacobi matrices. To appear in Commun. Math. Phys., DOI:10.1007/s00220-008-0696-x, 2009 30. Rakhmanov, E.A.: On the asymptotics of the ratio of orthogonal polynomials. Math. USSR Sb. 32, 199–213 (1977)
28
J. Breuer, E. Ryckman, M. Zinchenko
31. Rakhmanov, E.A.: On the asymptotics of the ratio of orthogonal polynomials, II. Math. USSR Sb. 46, 105–117 (1983) 32. Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators. Math. Phys. Anal. Geom. 10, 359–373 (2007) 33. Remling, C.: The absolutely continuous spectrum of Jacobi matrices. http://arxiv.org/abs/0706. 1101v1[math.SP], 2007 34. Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998) 35. Simon, B.: Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line. J. Approx. Theory 126, 198–217 (2004) 36. Simon B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. AMS Colloquium Series, 54.1, Providence, RI: Amer. Math. Soc., 2005 37. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series, 54.2, American Mathematical Society, Providence, RI, 2005 38. Sims, R.: Reflectionless Sturm–Liouville equations. J. Comp. Appl. Math. 208, 207–225 (2007) 39. Sodin, M., Yuditskii, P.: Almost periodic Sturm–Liouville operators with Cantor homogeneous spectrum and pseudoextendible Weyl functions. Russ. Acad. Sci. Dokl. Math. 50, 512–515 (1995) 40. Sodin, M., Yuditskii, P.: Almost periodic Sturm–Liouville operators with Cantor homogeneous spectrum. Comment. Math. Helvetici 70, 639–658 (1995) 41. Sodin, M., Yuditskii, P.: Almost-periodic Sturm–Liouville operators with homogeneous spectrum. In: Algebraic and Geometric Methods in Mathematical Physics. A. Boutel de Monvel and A. Marchenko (eds.), Dordrecht: Kluwer, 1996, pp. 455–462 42. Sodin, M., Yuditskii, P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7, 387–435 (1997) 43. Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, 72, Providence, RI: Amer. Math. Soc., 2000 Communicated by B. Simon
Commun. Math. Phys. 292, 29–54 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0883-4
Communications in
Mathematical Physics
Eigenvalue Estimates for Schrödinger Operators with Complex Potentials Ari Laptev1 , Oleg Safronov2 1 Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate,
London SW7 2AZ, UK. E-mail:
[email protected]
2 University of North Carolina at Charlotte, Mathematics and Statistics, 9201 University City Blvd.,
Charlotte, NC 28223, USA. E-mail:
[email protected] Received: 27 November 2008 / Accepted: 8 June 2009 Published online: 31 July 2009 – © Springer-Verlag 2009
Abstract: We discuss properties of eigenvalues of non-self-adjoint Schrödinger operators with complex-valued potential V . Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L p -norm of V . 1. Introduction Throughout the paper, f ± denotes either the positive or the negative part of f , which is either a function or a self-adjoint operator. The symbols z and z denote the real and the imaginary part ofz. If a is a function on Rd , then a(i∇) is the operator whose integral kernel is (2π )−d eiξ(x−y) a(ξ )dξ . Let H be a non-self-adjoint Schrödinger operator in L 2 (Rd ), H = − + V (x), with a complex-valued potential V . We call λ an eigenvalue of H if there is a solution of the equation H ψ = λψ for some ψ ∈ L 2 , ψ = 0. We deal with operators that have countably many eigenvalues lying in the cut plane C\[0, ∞). We denote them by λ j , j = 1, 2, 3, . . . A given number λ ∈ C\[0, ∞) may occur several times in this list according to the dimension of the generalized eigenspace {ψ : (H − λ)k ψ = 0 for some k ∈ N}, which is called the algebraic multiplicity. In principle, a generalized eigenspace could have infinite dimension, but, as we shall see, this will not occur in the situations considered in this paper. The main result of [9] tells us, that for any t > 0, the eigenvalues λ j of H lying outside the sector {z : |z| < t z} satisfy the estimate |λ j |γ ≤ C |V (x)|γ +d/2 d x, γ ≥ 1, where the constant C may depend on t, γ and d.
30
A. Laptev, O. Safronov
In this paper we study inequalities for the eigenvalues that might be close to the positive half-line. In particular, our results provide some information about the rate of accumulation of eigenvalues to the set R+ = [0, ∞). Theorem 1. Let V ≥ 0 be a bounded function. Assume that V ∈ L p (Rd ), where p > d/2 if d ≥ 2 and p ≥ 1 if d = 1. Then the eigenvalues λ j of the operator H = − + V satisfy the estimate p λ j p ≤ C (V )+ (x) d x. (1.1) |λ j + 1|2 + 1 + Rd j
The constant C can be computed explicitly: −d C = (2π )
Rd
dξ . (ξ 2 + 1) p
(1.2)
Note that the right-hand side of (1.1) is independent of the real part of the potential ∞ (Rd ) (since we V and therefore the statement is true for an arbitrary V ≥ 0 from L loc only need to check that (3.2) holds). It is not the case when we try to obtain an estimate p of the sum j (λ j /(|λ j + 1|2 + 1))+ , where we allow p ≤ d/2 and where a certain regularity of V is required. Theorem 2. Let V ≥ 0 and V be two bounded real valued functions. Assume that V ∈ L p (Rd ), where p > d/4 if d ≥ 4 and p ≥ 1 if d ≤ 3. Then the eigenvalues λ j of the operator H = − + V satisfy the estimate p λ j p 2p ≤ C (1 + ||V || ) (V )+ (x) d x, (1.3) ∞ |λ j + 1|2 + 1 + Rd j
where C = (2π )−d
Rd
((ξ 2
dξ . + 1)2 + 1) p
(1.4)
Next Theorems 3-4 give sufficient conditions on V that guarantee convergence of the sum |λ j |γ < ∞ a<λ j
for 0 ≤ a < b < ∞. Let us introduce W = (|V |2 + 4V )+ and let b (W ) = W d/4−1/2+r d x + bd/2−1 Rd
Rd
W r d x, d ≥ 2.
Theorem 3. Assume that γ > 3/2 and r ∈ (γ − 21 , γ ) and let V ∈ L d/2−1+2r (Rd ) ∩ L r (Rd ), d ≥ 2. Then the eigenvalues λ j of the operator H lying inside the semi-infinite strip b = {z : 0 < z < b, z > 0} satisfy the inequality 2γ −1 1 |λ j |γ ≤ C|b (W )| 2r −1 (b + |b (W )| 2r −1 ). (1.5) λ j ∈b
The constant C in this inequality depends on d, γ and r .
Eigenvalues of Schrödinger Operators with Complex Potentials
31
Applying the same method we also prove: Theorem 4. Let λ j be the eigenvalues of the operator H = −d 2 /d x 2 + V lying inside the semi-infinite strip a,b = {z : a < z < b, z > 0} with a > 0. Then for any γ > 3/2 and r ∈ (γ − 21 , γ ) the condition V ∈ L 2r (R) ∩ L r (R) implies
2γ −1
1
|λ j |γ ≤ C| a (W )| 2r −1 (b + | a (W )| 2r −1 ),
λ j ∈a,b
where a (W ) = a −1/2
R
W r d x.
The constant in this inequality depends on γ and r . Note that in Theorems 3 and 4 the inequality γ ≥ 3/2 is required in any dimension while in Theorems 1 and 2 the values of p could be smaller in lower dimensions. One should mention that the paper [9] had been motivated by a question of E.B. Davies (see [1] and [7]). He obtained that if d = 1 and V ∈ L 1 (R), then all eigenvalues λ of H which do not belong to R+ satisfy 1 |λ| ≤ 4
2 |V (x)|d x
.
The question was raised if a similar estimate holds in dimension d ≥ 2. The following conjecture seems to be reasonable Conjecture. Let d ≥ 2, 0 < γ ≤ d/2 and let V ∈ L d/2+γ (Rd ) be a complex-valued potential. Then for any eigenvalue λ ∈ / R+ of the operator H = − + V , |λ|γ ≤ C |V (x)|d/2+γ d x, (1.6) Rd
for every complex valued potential and every eigenvalue λ ∈ / R+ of the operator H = − + V . We carefully avoid the case γ > d/2, since the operator H in this case might have arbitrary large positive eigenvalues due to the Wigner-Von Neumann example [18] (we are grateful to S. Molchanov for drawing our attention to this circumstance). So far, we are able to prove only the following result related to this conjecture: Theorem 5. Let V be a function from L p (Rd ), where p ≥ d/2, if d ≥ 3; p > 1, if d = 2, and p ≥ 1, if d = 1. Then every eigenvalue λ of the operator H = − + V with the property λ > 0 satisfies the estimate p−1 d/2−1 |λ| ≤ |λ| C |V | p d x. (1.7) Rd
The constant C in this inequality depends only on d and p. Moreover, C = 1/2 for p = d = 1.
32
A. Laptev, O. Safronov
The inequality (1.7) was established in [1] in the case d = p = 1. We prove it in higher dimensions and in dimension d = 1 for p > 1. We also show the elementary estimate (see Theorem 16) √ 2γ | λ| ≤ C |V |3/2+γ d x, γ > 0, d = 3, R3
however it is not quite the same as (1.6). While we are not able to prove Conjecture 1.1, we find some information about the location of eigenvalues of the operator − + i V with a positive V ≥ 0, see Thorem 13. In particular, in Theorem 15 we prove that if d = 3 and V d x is small and λ ∈ / R+ is an eigenvalue of − + i V , then |λ| must be large. It might seem that eigenvalues do not exist at all for small values of V d x, however their presence in such cases can be easily established using scaling. Proposition 1. Let d ≥ 3. Then there is a sequence of positive functions Vn ≥ 0 such that the “largest modulus” eigenvalue λn ∈ / R+ of the operator − + i Vn satisfies |λn | → ∞ as n → ∞, while limn→∞ Vn (x)d x = 0. Proof. If λ is an eigenvalue n 2 λ is an eigenvalue of −+n 2 i V (nx). 2of −+i V (x), then 2−d It remains to note that n V (nx)d x = Cn . The idea of the proof of existence of a non-real eigenvalue of − + i V (x) at least for one V ≥ 0 is to start with the onedimensional case, when V (x) = δ(x) + δ(x − ). In this case, there is an eigenvalue of H that behaves like 1 + iα + O( 2 ) as → 0. If V is spherically symmetric, then the multi-dimensional case can be reduced to the one-dimensional case by separation of variables.
Remark. Note that our results also imply that the eigenvalues of − + i V can not accumulate to zero in d = 3, if V ≥ 0 is integrable (Corollary 5). Complex potentials are used in physics (see, for example, [2,10 and 13]) to describe different phenomena. In quantum mechanics and nuclear physics, the imaginary part of the potential is used to describe dissipation. Unlike the selfadjoint case, where the L 2 -norm of the wave function is constant, the L 2 -norm of the wave function in systems with a dissipation might change in time. The real part of the potential describes usual scattering whereas the imaginary part describes the absorption. 2. Preliminaries In what follows, the inner products and the norms in various spaces are denoted by (·, ·) and || · || respectively. 1. Let a[·, ·] be a sesquilinear form in a Hilbert space H. We assume that its domain d[a] is dense in H and a is semibounded from below and closed on d[a]. The form a induces the selfadjoint operator A in H. Fix the value of γ ∈ R, such that aγ := a + γ ≥ 1, i.e. aγ [x, x] = a[x, x] + γ ||x||2 = ||x||2 ,
x ∈ d[a],
and denote by Hγ [a] the (complete) Hilbert space d[a] with the metric form aγ [x, x] = ||(A + γ I )1/2 x||2 ,
x ∈ d[a].
Eigenvalues of Schrödinger Operators with Complex Potentials
33
Let V : H → H be a selfadjoint linear operator, satisfying D(|V |1/2 ) ⊃ d[a] and G := |V |1/2 (A + γ I )−1/2 ∈ S∞ ,
(2.1)
where S∞ denotes the space of compact operators in H. Put V 1/2 x, |V | y . v[x, y] = |V |1/2
(2.2)
Then the form v is compact on d[a]. This means that the form v is continuous on Hγ [a] and the corresponding operator Q (determined by the relations aγ [Qx, y] = v[x, y] for x, y ∈ d[a]) is compact on Hγ [a]. Define the operator H by setting H + γ I = (A + γ I )(I + i Q),
(2.3)
on the domain D(H ) = (I +i Q)−1 D(A). It is clear that the operator H can be interpreted as the sum H = A + i V. Proposition 2. The operator H defined in (2.3) is densely defined and closed. Proof. Let us first prove that H is densely defined. Assume the opposite, that there is a non-zero vector h ∈ d[a] such that aγ [(I + i Q)−1 u, h] = 0 for all vectors u ∈ D(A). Then aγ [u, (I − i Q)−1 h] = 0 for all u ∈ d[a], which implies that (I − i Q)−1 h = 0. The latter relation contradicts the assumption that h = 0. In order to prove that H is closed, it is sufficient to observe that H + γ I is invertible and prove that the inverse is bounded. But this follows from the relation (H + γ I )−1 = (I + i Q)−1 (A + γ I )−1 , and the fact that (A + γ I )−1 maps continuously H to Hγ [a].
Remark. The condition that the sesquilinear form v is generated by a self-adjoint operator V is excessive. We can always define H by (2.3), as soon as we know that v[u, u] = aγ [Qu, u], where Q is compact in the space Hγ [a]. This remark allows one to consider the case when the elliptic operator A = 2 is perturbed by a differential operator of first order. Under the above assumptions, the difference between the resolvents of the operators A and H is compact. Hence, the spectrum σ (H ) of the operator H is discrete in C\σ (A). 2. Let H be as described above. In order to develop the perturbation theory suitable for non-selfadjoint operators, we consider a contour C which encloses a finite number of eigenvalues λ1 , λ2 , . . . , λm of the operator H and has no intersection with the spectrum of H . Then the projection onto the span of the corresponding root vectors is given by the formula i P= (H − z)−1 dz. (2.4) 2π C Consequently, if Hn is a family of closed operators in H having the property that σ (Hn )\R is discrete and satisfying the condition (Hn − z)−1 − (H − z)−1 → 0,
(2.5)
34
A. Laptev, O. Safronov
as n → ∞, uniformly for z ∈ C, then the sequence of projections Pn , defined by (2.4) with Hn instead of H , will converge to P. That means that the norm ||P − Pn || will be small for sufficiently large n. Now, in order to draw a conclusion about eigenvalues of Hn , we can apply the following statement. Lemma 1 (see, for example, [14]). If P and P0 are two projections such that rank P = rank P0 , then ||P − P0 || ≥ 1. Note that if Hn is a family of closed operators in H having the property that σ (Hn )\R is discrete and satisfying the condition (Hn − z 0 )−1 − (H − z 0 )−1 → 0,
(2.6)
as n → ∞, for some point z 0 , then (2.5) holds for all z outside of the spectrum of H , because of the formula (H − z)−1 = (I + (z − z 0 )(H − z 0 )−1 )−1 (H − z 0 )−1 and a similar formula for (Hn − z)−1 . The convergence in (2.5) is uniform on compact subsets of C\σ (H ). We conclude that every non-real eigenvalue λ j of the operator H is the limit of a sequence λ j (n) of the eigenvalues of the operators Hn . The algebraic multiplicities of the eigenvalues are preserved in the obvious manner, and we omit the discussion of that. Let us formulate a condition that guarantees (2.5). Let Vn be a sequence of self-adjoint operators in H such that (2.1) holds with Vn instead of V . Let the sesquilinear form vn be the same as (2.2) with V replaced by Vn . Suppose that the sequence of operators Q n determined by aγ [Q n x, y] = vn [x, y] for x, y ∈ d[a], converges to the operator Q (acting in the space Hγ [a]). If we define Hn by the formula Hn + γ I = (A + γ I )(I + i Q n ), then we will obtain a sequence of operators Hn having the property (2.6) with z 0 = −γ and H defined by (2.3). Proposition 3. Let Hn + γ I = (A + γ I )(I + i Q n ), where the sequence of compact selfadjoint operators Q n in Hγ [a] converges to Q (as operators in Hγ [a]). Let H +γ I = (A + γ I )(I + i Q). Then every non-real eigenvalue λ j of the operator H is the limit of a sequence λ j (n) of the eigenvalues of the operators Hn as n → ∞. Proposition 4. The operators Q and Q n are unitarily equivalent to the operators (A + γ )−1/2 V (A + γ )−1/2 and (A + γ )−1/2 Vn (A + γ )−1/2 acting in H. Convergence of Q n to Q (as operators in Hγ [a]) is equivalent to convergence of (A + γ )−1/2 Vn (A + γ )−1/2 to (A + γ )−1/2 V (A + γ )−1/2 . Proof. Since aγ [(A + γ )−1/2 u, (A + γ )−1/2 v] = (u, v),
∀u, v ∈ H,
we conclude that the operator U = (A + γ )−1/2 is a unitary mapping from H to Hγ [a]. It remains to prove that U (A + γ )−1/2 V (A + γ )−1/2 U ∗ = Q. In order to show this, we simply note that aγ [U (A + γ )−1/2 V (A + γ )−1/2 U ∗ u, v] = v[u, v] = aγ [Qu, v]. That proves one of the statements of the proposition. The other statements are obvious.
Eigenvalues of Schrödinger Operators with Complex Potentials
35
3. We already know that A is semibounded from below. Suppose also that the negative spectrum of A is discrete. Then the operator H has only a discrete set of eigenvalues in the left half-plane Cle f t = {z : z < 0}. Moreover, suppose that λ j ∈ Cle f t are eigenvalues of the operator H , and τ j are negative eigenvalues of A enumerated in the order of increasing real parts. Then n n λj ≤ |τ j | 1
1
for all n. Indeed, let P be the orthogonal projection onto the span of eigenvectors x j corresponding to λ j , 1 ≤ j ≤ n. Then tr H P =
n
λj.
1
Consequently,
n 1
λj =
n (Ax j , x j ) 1
≥ min tr ((A + γ )1/2 P)∗ A(A + γ )−1 (A + γ )1/2 P , P
(2.7)
where the minimum is taken over all orthogonal projections P of rank n with the property RanP ⊂ d[a]. Thus, n 1
λ j ≥
n
τj,
(2.8)
1
since the minimum in the right-hand side of (2.7) coincides with the sum in the right-hand side of (2.8). Corollary 1. Let γ > 0. Then n n (λ j + γ )− ≤ (τ j + γ )− . 1
(2.9)
1
Proof. We can find a positive integer m such that λm + γ < 0, but λm+1 + γ ≥ 0. There are two possibilities: either m ≤ n or m > n. If m ≤ n, then n m m n (λ j + γ )− = (λ j + γ )− ≤ (τ j + γ )− ≤ (τ j + γ )− . 1
1
1
1
If m > n then (2.9) is just obvious, since, in this case, n 1
(λ j + γ )− = −
n n n (λ j + γ ) ≤ − (τ j + γ ) ≤ (τ j + γ )− . 1
1
1
36
A. Laptev, O. Safronov
Actually, (2.8) holds for all n if and only if (2.9) holds for all n and γ > 0. Indeed, let us fix n and choose γ in (2.9) so small that all terms in both sides of (2.9) are different from zero. Then γ cancels out and we obtain (2.8). 4. Let T be a bounded operator in a Hilbert space, whose spectrum outside the unit circle {z : |z| > 1} is discrete. Suppose also that the essential spectrum of the operator (T ∗ T )1/2 is contained in [0, 1]. Let λ j be the eigenvalues of the operator T lying outside of the unit circle, and let s j > 1 be the eigenvalues of (T ∗ T )1/2 . If we enumerate the sequences |λ j | and s j in the decreasing order, then n
|λ j | ≤
1
n
sj
(2.10)
1
for all values of n. One should mention also that, if one of the sequences ends at j = j0 , we extend it by setting it equal to 1 for j > j0 . This inequality was discovered for compact operators by H. Weyl (see [19]). Weyl’s proof is carried over to the case of bounded operators. Indeed, let P be the orthogonal projection onto the span of eigenvectors corresponding to λ j , 1 ≤ j ≤ n. Then for any α > 0, det (I + P(αT ∗ T − I )P) = α n
n
|λ j |2 .
1
Consequently, αn
n
|λ j |2 ≤ det (I + P(αT ∗ T − I )+ P)
1
≤ det (I + (αT ∗ T − I )+ P(αT ∗ T − I )+ ). 1/2
1/2
Since (αT ∗ T − I )+ P(αT ∗ T − I )+ ) ≤ (αT ∗ T − I )+ ) we can remove the orthogonal projection P in the right-hand side and obtain 1/2
αn
1/2
n
|λ j |2 ≤ det (I + (αT ∗ T − I )+ ) =
αs 2j .
αs 2j >1
1
It remains to choose α = sn−2 . Note that if the number of s j > 1 is finite, we can take α = 1 to obtain that n
|λ j |2 ≤
1
s 2j
s 2j >1
for all n. Corollary 2. Let γ ≥ 1. Then n 1
for all n.
(|λ j |2 − 1)γ ≤
n (s 2j − 1)γ 1
Eigenvalues of Schrödinger Operators with Complex Potentials
37
Proof. Our arguments are quite standard and can be compared with the ones in the book by Birman and Solomyak [4], which contains a survey on different inequalities for compact operators. It is sufficient to consider the case γ > 1, because the proof in the case γ = 1 is obtained by passing to the limit as γ → 1. As a consequence of (2.10), we obtain that n
log |λ j | ≤
1
n
log s j .
(2.11)
1
Moreover, n n (log |λ j | − η)+ ≤ (log s j − η)+ j=1
(2.12)
j=1
for any −∞ < η < ∞. (To prove (2.12) one has to repeat the arguments of the proof of Corollary 1.) Note now that the function φ(t) = (e2t − 1)γ is representable in the form ∞ (λ − t)+ φ (t) dt and φ (t) ≥ 0 for t ≥ 0. φ(λ) = 0
Since φ(log |λ|) = (|λ|2 − 1)γ , the statement of Corollary 2 for γ > 1 follows from (2.12).
5. Let T be a compact operator in √ a Hilbert space and let n(s, T ) be the counting function of its s-numbers (eigenvalues of T ∗ T ) n(s, T ) = card{ j :
s j > s}, s > 0.
Then by the Ky Fan inequality (see [11]) for any pair of compact operators T1 and T2 and s1 , s2 > 0, n(s1 + s2 , T1 + T2 ) ≤ n(s1 , T1 ) + n(s2 , T2 ). The class of operators T for which p
[T ] p := sup s p n(s, T ) < ∞ s>0
is called the weak Neumann-Schatten class p . Let F be the Fourier transform F f (ξ ) = e−i xξ f (x) d x. Rd
Theorem 6 (M. Cwikel [5]). Let α and β be the operators of multiplication by the functions α(ξ ) and β(x). Suppose that β ∈ L q (Rd ), q > 2, and let q
[α]q = sup t q meas{ξ ∈ (Rd : |α(ξ )| > t} < ∞. t>0
Then the operator T = βF ∗ α (as well as the operator αFβ) is in p and q q [T ]q ≤ C[α]q |β(x)|q d x.
(2.13)
38
A. Laptev, O. Safronov
Proposition 5 (Birman-Schwinger principle [3,15]). Let A and V be two positive self-adjoint operators √ acting in the same Hilbert space. Suppose that V is bounded and the operator V (A + I )−1/2 is compact. Then for E > 0, the number N (E) of eigenvalues of the operator A − V lying to the left of −E satisfies the relation √ √ N (E) = n(1, V (A + E)−1 V ). In applications, A is a differential operator with constant coefficients and V is the operator √ of multiplication by a function. Then applying Theorem 6 to the operator T = V (A + I )−1/2 one obtains sharp inequalities for N (E). 6. In order to state the next result we need to introduce one more Neumann-Schatten class S p of compact operators. Namely, we say that T ∈ S p , p ≥ 1, if ||T || p := tr (T ∗ T ) p/2 = p
p
s j < ∞.
j
It is easy to see that S p is a Banach space. The next theorem gives us a sufficient condition guaranteeing that an operator of the form β(x)α(i∇) belongs to the class S p . Theorem 7. Let α and β be the operators of multiplication by α(ξ ) and β(x). Suppose that α, β ∈ L p (Rd ), where p ≥ 2. Then T = βF ∗ α ∈ S p and p (2.14) ||T || p ≤ (2π )−d |α(ξ )| p dξ |β(x)| p d x. This theorem can be found in [16]. See also [12] and [17]. 7. We now formulate a statement about eigenvalue estimates for a certain operator with constant coefficients perturbed by a potential V . It is one of the consequences of the inequality (2.13). Proposition 6. Let α(ξ ) = (|ξ |2 − µ)2 , V (x) ≥ 0 and p > 1/2. Suppose that V ∈ L p+d/4 (Rd ) ∩ L p+1/2 (Rd ) if d ≥ 2, or V ∈ L p+1/2 (R) if d = 1. Let N (E) be the number of eigenvalues of the operator α(i∇) − V (x) lying to the left of the point −E, where E > 0. Then C N (E) ≤ p V p+d/4 d x + µd/2−1 V p+1/2 d x , if d ≥ 2; (2.15) E Rd Rd C V p+1/2 d x, if d = 1. (2.16) N (E) ≤ p 1/2 E µ Rd Proof. It is an elementary application of the Cwikel estimate. Indeed, according to the Birman-Schwinger principle N (E) = n(1, X ), where X is the compact operator defined by the equality √ √ X = V (α(i∇) + E)−1 V .
Eigenvalues of Schrödinger Operators with Complex Potentials
39
Let χ be the characteristic function of the ball {|ξ | ∈ Rd : |ξ |2 ≤ µ}. Let us split X such that X = X 1 + X 2 , where √ √ X 2 = V (α(i∇) + E)−1 χ (i∇) V . According to the Ky Fan inequality, n(1, X ) ≤ n(1, 2X 1 ) + n(1, 2X 2 ).
(2.17)
Therefore it is sufficient to estimate each term in the right-hand side of (2.17) separately. We begin with the first term. Set q1 = p + d/4. Then according to (2.13), dξ n(1, 2X 1 ) ≤ C0 V q1 d x 2 2 q |ξ |2 >µ ((|ξ | − µ) + E) 1 ∞ ∞ d/2−1 s d/2−1 ds s ds q1 q1 ≤ C1 V d x ≤ C2 V d x 2 + E)q1 2 + E)q1 ((s − µ) (s µ 0 C = p V p+d/4 d x. E Rd In order to estimate the second term in (2.17) we set q2 = p + 1/2. Using (2.13) again we find dξ q2 n(1, 2X 2 ) ≤ C3 V d x 2 2 q |ξ |2 <µ ((|ξ | − µ) + E) 2 µ ∞ d/2−1 s d/2−1 ds µ ds q2 V ≤ C 4 V q2 d x ≤ C d x 5 2 + E)q2 2 + E)q2 ((s − µ) (s 0 −∞ Cµd/2−1 = V p+1/2 d x, Ep Rd which completes the proof of (2.15). √ √ In order to prove (2.16) we note that (|ξ |2 − µ)2 = (|ξ | − µ)2 (|ξ | + µ)2 ≥ √ 2 (|ξ | − µ) µ. Consequently, for any q > 1,
∞
∞
dξ ≤ 2 ((ξ − µ)2 + E)q
∞
dξ C = √ q−1/2 , √ 2 q ((|ξ | − µ) µ + E) µE
∞
where C=
∞
−∞
(s 2
ds . + 1)q
If now q = p + 1/2, then by using (2.13) we arrive at C V q dx C V p+1/2 d x N (E) ≤ √ = , √ µ E q−1/2 µ Ep which means that (2.16) is also proven.
40
A. Laptev, O. Safronov
3. Proof of Theorem 1 The main tool of the proof is the linear fractional mapping that takes the upper half-plane {z : z > 0} into the compliment of the unit disk {z : |z| > 1} given by the formula z →
z+i +1 . z−i +1
Insert the operator H = − + V instead of z into this formula, i.e. consider the operator U = (H + I + i)(H + I − i)−1 = I + 2i(H + I − i)−1 . Obviously z ∈ / R is an eigenvalue of the operator H if and only if (z + i + 1)/(z − i + 1) is an eigenvalue of U . Clearly U ∗ = I − 2i(H ∗ + I + i)−1 , and therefore U ∗ U = I + 2i(H + I − i)−1 − 2i(H ∗ + I + i)−1 + 4(H ∗ + I + i)−1 (H + I − i)−1 . Using the Hilbert identity, we obtain U ∗ U = I + 2i(H ∗ + I + i)−1 (H ∗ − H )(H + I − i)−1 , and since H ∗ − H = −i V , U ∗ U = I + 4(H ∗ + I + i)−1 V (H + I − i)−1 . In particular, this implies U ∗ U − I ≤ 4Y ∗ Y,
√ where Y = V + (H + I −i)−1 . By using Corollary 2 the eigenvalues λ j of the operator H satisfy the inequality p λ j + 1 + i 2 p 2p ≤ tr (U ∗ U − I )+ ≤ 4 p tr (Y ∗ Y ) p = 4 p ||Y ||2 p . λ + 1 − i − 1 j
j
+
It follows from this inequality that j
λ j |λ j + 1|2 + 1
p
2p
≤ ||Y ||2 p . +
Indeed, denote a = 2λ j /(|λ j + 1|2 + 1) and suppose that λ j > 0. Then λ j + 1 + i 2 − 1 = 1 + a − 1 ≥ 2a. λ + 1 − i 1−a j We come to the conclusion that one needs to estimate the norm of the operator
Y = V + (H + I − i)−1
(3.1)
Eigenvalues of Schrödinger Operators with Complex Potentials
41
in the class S2 p . Let us represent this operator in the form Y =
V + (− + I )−1/2 B,
where B = (− + I )1/2 (H + I − i)−1 .
We will show that the operator B is bounded and its norm does not exceed 1. In other words, we will show that ||(− + I )1/2 (H + I − i)−1 f ||2 ≤ || f ||2 ,
(3.2)
for all f ∈ L 2 . Denote u = (H + I − i)−1 f . It is obvious that
(|∇u| + (1 + V (x))|u| ) d x = 2
Rd
2
Rd
f u¯ d x.
Due to the condition V ≥ 0, we obtain from this relation that
1 (|∇u| + |u| ) d x ≤ (| f |2 + |u|2 ) d x. 2 Rd Rd 2
2
The latter inequality can be written in the form
(2|∇u| + |u| ) d x ≤ 2
Rd
2
Rd
| f |2 d x.
Replacing 2 by a smaller number we will make the inequality weaker. As a result we obtain the estimate ||(− + I )1/2 u||2 ≤ || f ||2 .
(3.3)
It remains to note that (3.3) is equivalent to (3.2). Let us summarize the results. Since Y =
V + (H + I − i)−1 =
V + (− + I )−1/2 B and ||B|| = 1,
we obtain
||Y ||2 p ≤ || V + (− + I )−1/2 ||2 p .
(3.4)
On the other side, according to Theorem 7,
p −1/2 2 p −d || V + (− + I ) ||2 p ≤ (2π ) C0 V + d x, where C0 = Rd (ξ 2 + 1)− p dξ . Combining (3.4) with (3.1), we complete the proof of Theorem 1.
42
A. Laptev, O. Safronov
4. Proof of Theorem 2 The main arguments in the proof of this result remain the same apart from the estimate of the norm ||Y ||2 p of the operator Y . Recall that j
λ j |λ j + 1|2 + 1
p
2p
≤ ||Y ||2 p ,
(4.1)
+
√ where Y = V + (H + I − i)−1 . In order to find a bound for the s-numbers of the operator Y we represent it in the form Y =
V + (− + I − i)−1 (I − V (H + I − i)−1 ).
In the previous section we have found that (H + I − i)−1 = (− + I )−1/2 B
and ||B|| ≤ 1.
Consequently, ||(H + I − i)−1 || ≤ 1, and this means that
||Y ||2 p ≤ || V + (− + I − i)−1 ||2 p (1 + ||V ||∞ ). By using Theorem 7 we obtain that for any p > d/4,
2p p || V + (− + I − i)−1 ||2 p ≤ (2π )−d C0 V + d x, where C0 =
((ξ 2
dξ . + 1)2 + 1) p
Consequently, ||Y ||2 p ≤ (2π )−d (1 + ||V ||∞ )2 p C0 2p
Consequently (4.1) and (4.2) imply (1.3).
p
V + d x.
(4.2)
Eigenvalues of Schrödinger Operators with Complex Potentials
43
5. Proof of Theorem 3 and Some Related Results Proof of Theorem 3. Assume that λ j ∈ b are enumerated in the order of decreasing imaginary parts. Note that the theorem would be proven, if instead of the infinite sum in the left-hand side of (1.5), we estimated a partial sum m
2γ −1
1
|λ j |γ ≤ C|b (W )| 2r −1 (b + |b (W )| 2r −1 ),
λ j ∈ b .
(5.1)
j=1
On the other hand, it is sufficient to prove the estimate (5.1) for the case when V ∈ C0∞ (Rd ). Indeed, if V ∈ / C0∞ (Rd ) , then we can always find a sequence Vn of C0∞ -functions that converges to V in L d/2−1+2r (Rd ) ∩ L r (Rd ). Obviously the corresponding sequence of quantities b (Wn ) (here Wn = (|Vn |2 + 4Vn )+ ) will converge to b (W ) in this case. Moreover, due to (trivially modified) Propositions 3 and 4, the non-real eigenvalues λ j of H will be the limits of the sequences of non-real eigenvalues λ j (n) of Hn = −+ Vn , which implies that mj=1 |λ j |γ = limn→∞ mj=1 |λ j (n)|γ . Note that convergence of (− + γ )−1/2 Vn (− + γ )−1/2 to (− + γ )−1/2 V (− + γ )−1/2 is guaranteed by Theorem 7. Corollary 1 plays an essential role in the proof, as well as a trick relating the eigenvalues of the operator H = −+V and the eigenvalues of the operator (−+2i −µ+V )2 , µ > 0, lying to the left of z = −4. Indeed, let λ j be eigenvalues of the operator −+V lying in the hyperbolic domain Dµ = {z : (z + 2)2 − (z − µ)2 ≥ 4, z > 0}, then (λ j − µ + 2i)2 are eigenvalues of the operator (− − µ + 2i + V )2 , and it is easy to see that (λ j − µ + 2i)2 = (λ j − µ)2 − (λ j + 2)2 ≤ −4,
∀λ j ∈ Dµ .
Consequently, due to Corollary 1, n n 2 sj , (λ j − µ + 2i) + 4 ≤ 1
(5.2)
1
where s j are eigenvalues of the operator T1 = (− − µ)2 + V1 (− − µ) + (− − µ)V1 + V12 − V22 − 4V2 , where V1 = V and V2 = V are the real and the imaginary parts of the potential. The inequality (5.2) takes care of all eigenvalues from the domain Dµ . It turns out that we do not need all of them, but only the eigenvalues λ j lying inside the domain µ = {z : (z + 1)2 − (z − µ)2 ≥ 1, z > 0}. Note that the boundaries of both domains Dµ and µ touch the real line at the point z = µ. Note also that µ ⊂ Dµ and therefore this might imply that bounds on eigenvalues lying in µ are better than those in Dµ . It turns out that the imaginary parts of eigenvalues in µ can be estimated in terms of real parts of eigenvalues of the operator (H − µ + 2i)2 + 4. A similar trick was used by Davies in [6] to obtain individual inequalities for the eigenvalues of the operator H .
44
A. Laptev, O. Safronov
Let us study the relation between the spectra of the operators H and (H − µ + 2i)2 in more detail. Assume that λ j ∈ µ and λ j > s. Then 2(λ j − s) ≤ (λ j + 1)2 − (λ j − µ)2 − 1 + 2(λ j − s) = (λ j + 2)2 − (λ j − µ)2 − 4 − 2s = −(λ j − µ + 2i)2 − 4 − 2s. Due to Corollary 1 it means that 2 (λ j − s)+ ≤ tr (H − µ + 2i)2 + 4 + 2s ≤ tr (T1 + 2s)− . −
λ j ∈µ
Now, we represent the operator T1 in the form T1 =
1 (− − µ)2 + 2
√ 1 √ (− − µ) + 2V1 2
2 − 4V2 − V12 − V22 .
Since the operator
√ 1 √ (− − µ) + 2V1 2
2 ≥0
is positive, we obtain that the spectrum of the operator T1 can be estimated by the spectrum of the operator 1 (− − µ)2 − |V |2 − 4V2 . 2
T2 = Thus,
2
(λ j − s)+ ≤ tr (T2 + 2s)− .
(5.3)
λ j ∈µ
Let τ j be negative eigenvalues of T2 . In order to estimate the right-hand side of (5.3) we apply Proposition 6 according to which the number N (E) of eigenvalues of T2 lying to the left of the point −E satisfies the inequality C N (E) ≤ p W d/4+ p d x + µd/2−1 W 1/2+ p d x (5.4) E Rd Rd with p > 1/2 and d ≥ 2. If now q > p > 1/2 then ∞ |τ j |q = q E q−1 N (E)d E 0
j
≤C
Rd
W
d/4+ p
dx + µ
d/2−1
Rd
W
1/2+ p
d x |λ1 |q− p .
From (5.4) it follows that the lowest eigenvalue τ1 satisfies the inequality |τ1 |r −1/2 ≤ C W d/4+r −1/2 d x + µd/2−1 W r d x = Cµ (W ). Rd
Rd
Eigenvalues of Schrödinger Operators with Complex Potentials
45
Hence for q > p > 1/2 and r > 1 we arrive at |τ j |q ≤ C W d/4+ p d x + µd/2−1 W 1/2+ p d x |µ (W )|2(q− p)/(2r −1) . Rd
j
Rd
Recall that
2
(λ j − s)+ ≤
λ j ∈µ
and therefore (λ j − s)+ ≤ C λ j ∈µ
(W − 2s)+
+µ
(τ j + 2s)− ,
j
d/4+ p
Rd
d/2−1
Rd
dx
1/2+ p (W −2s)+ dx
|µ (W )|
2(1− p) 2r −1
=: F(s, µ)
(5.5)
with 1/2 < p < 1 and r > 1. Let now b be the semi-infinite strip {z : 0 < z < b, z > 0}. Since the boundary of a domain µ touches the real line parabolically, it is obvious, that for small values of s < ε0 , the set b (s) of √ all points z ∈ b whose z > s > 0 can be covered by not more than m(b) = [Cb/ s] + 1 sets of the form µ . Since µ contains the sector z > |z − µ|, we obtain that the number of domains µ covering √ the√set b (s) can be also estimated by [b/s] + 1 for any s > 0. Finally, note that 1/ s ≥ ε0 /s for s ≥ ε0 . Therefore without loss of generality one can assume that √ m(b) = [Cb/ s] + 1, ∀s > 0. 2
Since λ j ≤ C|b (W )| 2r −1 for any λ j ∈ b , we obtain
(λ j − s)+ ≤
m(b)
λ j ∈b
1
(λ j − s)+ ≤ C
l=1 λ j ∈µl
Obviously
γ
|λ j | = γ (γ − 1)
λ j ∈b
which leads to
λ j ∈b
1
∞
(b + |b (W )| 2r −1 ) F(s, b). √ s
(λ j − s)+ s γ −2 ds,
0
|λ j |γ ≤ (b + |b (W )| 2r −1 )C
∞
s γ −5/2 F(s, b)ds.
0
λ j ∈b
The integral in the right-hand side converges only if γ > 3/2 and by using the notation introduced in (5.5) we finally obtain 2δ 1 |λ j |γ ≤ C|b (W )| 2r −1 (b + |b (W )| 2r −1 ) λ j ∈b
×
Rd
|W |d/4−1/2+γ −δ d x + +bd/2−1
Rd
|W |γ −δ d x ,
where 0 < δ < 1/2. It remains to set r = γ − δ to complete the proof.
46
A. Laptev, O. Safronov
We have proved inequalities for |λ j |γ with γ > 3/2. However, (5.5) allows us to obtain a bound on eigenvalues belonging to µ with γ = 1. Corollary 3. Let λ j be the eigenvalues of the operator − + V lying inside µ = {z : (z + 1)2 − (z − µ)2 ≥ 1, z > 0} and let d ≥ 2. Then 1− p d/4+ p d/2−1 1/2+ p |λ j | ≤ C W dx + µ W d x ||W ||∞ Rd
j
Rd
for any 1/2 < p < 1. Similarly we can show Corollary 4. Let d = 1 and let λ j be the eigenvalues of the operator −d 2 /d x 2 + V lying inside µ = {z : (z + 1)2 − (z − µ)2 ≥ 1, z > 0}. Then 1− p |λ j | ≤ C||W ||∞ µ−1/2 W 1/2+ p d x Rd
j
for any 1/2 < p < 1. Unfortunately, if d = 1 then in order to obtain similar results we have to avoid the point z = 0 and in this case we deal with the strip a < z < b, a > 0. However, this is no longer true if γ > 7/4. Indeed: Theorem 8. Let λ j be the eigenvalues of the operator H = −d 2 /d x 2 + V lying inside the semi-infinite strip b = {z : 0 < z < b, z > 0}. Then for any γ > 7/4, r ∈ (γ − 21 , γ ) and V ∈ L ∞ (Rd ) such that W = (|V |2 + 4V )+ ∈ L r −1/4 we have
γ
|λ j | ≤
γ −r C||W ||∞ (b
λ j ∈b
1/2 + W ∞ )
r −1/4
Rd
|W |
dx .
Proof. The inequality (5.5) could be easily modified and we can obtain that for 1/2 < p < 1, 1/2+ p 1− p (λ j − s) ≤ C µ−1/2 (W − 2s)+ d x W ∞ =: F(s, µ), (5.6) Rd
λ j ∈µ 1− p
where W ∞ appears when we estimate the lowest eigenvalue λ1 of the operator T2 = 21 (−d 2 /d x 2 − µ)2 − W . Now consider the part of the strip b = {z : 0 < z < b, z > 0} satisfying z > s > 0. We cover it by sets µ , µ ∈ R+ . While doing this, we avoid the value µ = √0 taking µ as large as possible. The optimal choice of such µ would be µ = µ0 = s 2 + 2s, so that µ0 satisfies the equation (s + 1)2 − µ20 = 1. Thus, without √ loss of generality, we can assume that µ ≥ s 2 + 2s. Arguing as in the proof of Theorem 3, we find that√set of all points z ∈ b whose z > s can be covered by not more than m(b) = [Cb/ s] + 1 sets of the form µ .
Eigenvalues of Schrödinger Operators with Complex Potentials
47
Since there is no λ j ∈ b satisfying λ j > W ∞ , we obtain
(λ j − s)+ ≤
λ j ∈b
m(b)
1/2
b + ||W ||∞ F(s, µ0 ). √ s
(λ j − s)+ ≤ C
l=1 λ j ∈µl
Therefore
|λ j |γ = γ (γ − 1)
∞
(λ j − s)+ s γ −2 ds
λ j ∈b 0
λ j ∈b
1/2
≤ (b + ||W ||∞ )C
∞
s γ −5/2 F(s,
√
s) ds.
0
The integral in the right-hand side converges only if γ > 7/4 and using (5.6) we arrive at 1− p 1/2 |λ j |γ ≤ C||W ||∞ (b + ||W ||∞ ) |W |γ −5/4+ p d x Rd
λ j ∈b
with 1/2 < p < 1. It remains to set r = γ + p − 1 to complete the proof.
6. Proof of Theorem 5 Theorem 5 has been already proved before for d = p = 1 (see [1,7]). Consider first the case when p > max{1, d/2}. By using Birman-Schwinger principle, we find that the value λ ∈ / R+ is an eigenvalue of the operator H if and only if 1 is an eigenvalue of the operator X = |V |1/2 (− − λ)−1 |V |−1/2 V, and thus X ≥ 1. Note now that X ≤ X p ≤ Q22 p , where Q = |V |1/2 | − − λ|−1/2 . Using Theorem 7 we obtain that 1≤
2p ||Q||2 p
≤ (2π )
−d
|V | d x p
Rd
Rd
dξ . ||ξ |2 − λ| p
Assuming that p > d/2 there is a constant C such that dξ dξ d/2− p = |λ| ≤ C |λ|d/2− p | sin φ|1− p , J= 2 − λ| p 2 − eiφ | p d d ||ξ | ||ξ | R R where φ = arg λ and consequently, J ≤ C |λ|1− p |λ|d/2−1 .
48
A. Laptev, O. Safronov
It remains to note that 1 ≤ (2π )−d J
Rd
|V | p d x.
In order to prove Theorem 5 for p = d/2 > 1 we use just Theorem 6 instead of Theorem 7. Indeed, let a(ξ ) =
1 ||ξ |2 − λ|
and p = d/2 > 1.
Then, using homogeneity, we obtain p
p
[a] p = [a0 ] p ,
where a0 =
There is a constant C > 0 such that p [a] p
=
p [a0 ] p
≤ C| sin φ|
1− p
1 . |ξ 2 − eiφ |
1− p λ = C . λ
It remains to note that, if λ is an eigenvalue of H , then p 1 ≤ C[a] p |V | p d x p = d/2. Rd
The proof is complete.
7. Individual Eigenvalue Estimates Let us now consider a Schrödinger operatorH = − + i V (x) whose potential is pure imaginary. Besides we assume that V ≥ 0 and lim|x|→∞ V (x) = 0. Our first statement concerns the case d = 3. / R+ be an eigenvalue of Theorem 9. Let V ∈ L 1 (R3 ), V ≥ 0 and let z = k 2 ∈ H = − + i V (x). Then k V (x) d x ≥ 1. (7.1) 4π R3 In particular, this shows that if R3 V d x is small, then the real part of the square root of the eigenvalue of H is large. That implies that non-real eigenvalues of − + it V escape any compact subset of C, as t → 0. It does not necessary imply that the eigenvalues tend to infinity as t → 0, because they might simply reach the positive real semi-axis for some t > 0 (see Theorem 15). Proof of Theorem 9. By using the Birman-Schwinger principle we find that z = k 2 ∈ R+ is an eigenvalue of the operator H = − + i V if and only if the operator √ √ X = −i V (− − z)−1 V (7.2) has an eigenvalue 1.
Eigenvalues of Schrödinger Operators with Complex Potentials
49
Suppose that z > 0. Then the real part of the operator X is positive and, consequently, the spectrum of this operator lies in the right half plane. Therefore if z is an eigenvalue of H , then ζ j ≥ 1, j
where ζ j are eigenvalues of X . On the other side, ζ j ≤ tr X = τ (x, x) d x, R3
j
where τ (x, y) is the integral kernel of the operator X . Since the kernel of the operator (− − z)−1 equals g(x, y) =
eik|x−y| , 4π |x − y|
we obtain that the kernel of the operator (− − z)−1 equals g0 (x, y) = (2i)−1 (g(x, y) − g(y, x)) whose diagonal values are g0 (x, x) =
k k + k¯ = . 8π 4π
Finally
k tr X = V (x) g0 (x, x) d x = V (x) d x 4π R3 R3 implies (7.1).
Corollary 5. Let d = 3 and let V ∈ L 1 (R3 ) be a positive function. Then non-real eigenvalues of − + i V do not accumulate to zero. Using the same approach we obtain the following two results in dimensions d = 1 and d = 2. Theorem 10. Let d = 1, z = k 2 ∈ R+ be an eigenvalue of the operator H = − + i V , V ≥ 0, V ∈ L 1 (R). Then k V (x) d x ≥ 1, 2|k|2 R which means that k lies inside the circle of radius 4−1 V (x) d x with the centre at 4−1 R V (x) d x. It is interesting to observe that if d = 2 then the eigenvalues do not appear at all if the integral of V is small.
50
A. Laptev, O. Safronov
Theorem 11. Let d = 2, z ∈ / R+ be an eigenvalue of H = −+i V , V ≥ 0, V ∈ L 1 (R2 ). Then 1 π + arctan( z/ z) V (x) d x ≥ 1. 2 2 In particular, the spectrum of H is real if π V (x) d x < 1. 2 Proof. In order to prove this statement we just notice that if X is the Birman-Schwinger operator (7.2) defined in the proof of Theorem 9, then 1 tr X = V (x) d x dξ. 2π(|ξ |2 − z) R2
The next result deals with some properties of complex eigenvalues of Schrödinger operators in higher dimensions d ≥ 4. Theorem 12. Let d ≥ 4 and let z ∈ / R+ be an eigenvalue of H = − + i V with V ≥ 0. Then −d+1 (d−2)/2 ωd−1 |z + 2V ∞ | (2π ) V (x) d x ≥ 2, (7.3) where ωd−1 is the area of the unit sphere Sd−1 . Proof. If as before X is the Birman-Schwinger operator introduced in (7.2) and z is an eigenvalue of the operator H , then 1/2 is an eigenvalue of the operator X − 1/2. Consequently, tr (X − 1/2)+ ≥ 1/2.
(7.4)
Indeed, for the eigenvalues λ j of the operator X we have (λ j − 1/2)+ ≤ tr ( X − 1/2)+ . Therefore the eigenvalue sum in the left-hand side is not less than 1/2. Obviously, V (X − 1/2)+ ≤ X − 2||V ||∞ + √ √ 1 = V (− − z)−1 − V. 2||V ||i n f t y + Concequently, using (7.4) we have −d 1/2 ≤ (2π ) V (x) d x Rd
Rd
z 1 − 2 2 2 (|ξ | − z) + (z) 2V ∞
dξ. +
Eigenvalues of Schrödinger Operators with Complex Potentials
51
The integration in the last integral is carried out over the domain where
|ξ |2 ≤ z + (2V − z)+ z ≤ z + 2V ∞ . Therefore −1 ωd−1
≤2 and we obtain (7.3).
Rd −1
z 1 − 2 2 2 (|ξ | − z) + ( z) 2V ∞
dξ +
π | z + 2V ∞ |(d−2)/2 ,
(7.5)
We now obtain some results involving L p norms of potentials with p > 1. Theorem 13. Let d ≥ 3 and let V ≥ 0. Suppose that z ∈ / R is an eigenvalue of H = − + i V . Then there are positive constants C1 and C2 depending only on d and γ ≥ 0 such that z d/2−1 γ (7.6) |z| ≤ C1 + C2 ( ) V d/2+γ d x. z Proof. Let as before √ √ X = −i V (− − z)−1 V . If z is an eigenvalue of H , then there is at least one eigenvalue of the operator X that is not less than 1. If by s j we denote the eigenvalues of the operator X , then this implies sup s −(d/2+γ ) card{ j : s j > s} ≥ 1. s>0
This supremum is related to the norm in the weak Neumann-Schatten class d/2+γ and, due to Theorem 6, it can be estimated by
V d/2+γ d x
Rd
z (ξ 2 − z)2 + z 2
d/2+γ dξ.
(7.7)
We conclude the proof by estimating the latter integral,
d/2+γ z dξ 2 2 2 Rd (ξ − z) + z d/2+γ d/2+γ ∞ ∞ z z d/2−1 s ds + C |z|d/2−1 ds ≤C 2 + z 2 2 + z 2 s s −∞ −∞ d/2−1 z ≤ C1 + C2 |z|−γ . z
52
A. Laptev, O. Safronov
Applying this result for the case γ = 0 we obtain:
Corollary 6. Let d ≥ 3 and let C1 be the constant in (7.6). If C1 V d/2 d x < 1, then the eigenvalues of − + i V belong to the conical sector {z : 0 ≤ arg z ≤ α}, where α satisfies the equation d/2−1 (C1 + C2 ( cot α) ) V d/2 d x = 1. If γ > 0 then in the proof of Theorem 13 one can apply Theorem 6 even if d = 2 and obtain Theorem 14. Let d = 2 and let V ≥ 0. Suppose that z ∈ / R is an eigenvalue of H = − + i V . Then there is a positive constant C depending only on γ > 0 such that |z|γ ≤ C V 1+γ d x, γ > 0.
8. Additional Remarks Concluding this paper, we mention two rather obvious facts, that are valid for an arbitrary complex potential V . For the sake of simplicity, we restrict our study to the case d = 3. As before, H = − + V is the Schrödinger operator and ω2 is the area of the unit sphere S2 . Theorem 15. Let d = 3. If V ∈ L ∞ ∩ L 1 and let ω2 V ∞ + 2V 1 < 8π . Then the spectrum of the operator H is real. The same statement is true if |V (y)| sup dy < 4π. x R3 |x − y| / R+ be an eigenvalue of the operator Theorem 16. Let d = 3 and let z = k 2 ∈ H = − + V , k > 0. Then there is a positive constant C depending only on γ > 0, such that 2γ (k) ≤ C |V |3/2+γ d x. R3
Proof of both theorems. Suppose that z = k 2 is an eigenvalue of the operator H . Then the norm of the operator X = |V |1/2 (− − z)−1 |V |1/2 is not smaller than 1. By Schur’s inequality if G is an integral operator with the kernel g(x, y) then G ≤ m 1 m 2 , where
m 1 = sup x
|g(x, y)|
dy and m 2 = sup ρ(x, y) y
(8.1) |g(x, y)|ρ(x, y) d x
and ρ is a positive weight. Since the kernel of the operator X equals |V (x)|1/2
eik|x−y| |V (y)|1/2 , 4π |x − y|
Eigenvalues of Schrödinger Operators with Complex Potentials
53
√ then applying (8.1) with the weight ρ = V (x)/V (y), we obtain that −k|x−y| e 1 sup |V (y)| dy. ||X || ≤ 4π x |x − y| The statement of Theorem 15 follows from the trivial estimate 1 1 |V (y)| 1 ≤ X ≤ sup dy ≤ (ω2 V ∞ + 2V 1 ). 4π x R3 |x − y| 8π We obtain the statement of Theorem 16 using the Hölder inequality |V (y)| −k|x−y| 1 e dy 1≤ 4π R3 |x − y| 1/q ||V || p e−qk|y| dy =C , ≤ C0 ||V || p q 2γ / p 3 |y| (k) R where p = 3/2 + γ and q = p/( p − 1).
Remark. By using similar arguments one can show that √ γ +1/2 z √ 2γ √ | z| ≤ C |V |1/2+γ d x, γ ≥ 1/2, z R for eigenvalues z ∈ / R+ of the one-dimensional Schrödinger operator H = −d 2 /d x 2 +V . The constant C in this inequality can be computed explicitly 1 2γ − 1 γ −1/2 C= . 2 2γ + 1 Acknowledgement. The authors would like to thank Grigori Rozenblioum, Rupert Frank, Stanislav Molchanov and Robert Seiringer for their remarks.
References 1. Abramov, A.A., Aslanyan, A., Davies, E.B.: Bounds on complex eigenvalues and resonances. J. Phys. A 34, 57–72 (2001) 2. Austern, N.: The use of complex potentials in nuclear physics. Ann. Phys. 45(1), 113–131 (1967) 3. Birman, M.: On the spectrum of singular boundary value problems. Matem. Sb. 55, 125–174 (1961) 4. Birman, M.Sh., Solomjak, M.Z.: Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). Dordrecht: D. Reidel Publishing Co., 1987 5. Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977) 6. Davies, E.B.: Linear operators and their spectra, Cambridge Studies in Advanced Mathematics 106, Cambridge: Cambridge University Press, 2007 7. Davies, E.B., Nath, J.: Schrödinger operators with slowly decaying potentials. J. Comput. Appl. Math. 148(1), 1–28 (2002) 8. Demuth, M., Hansmann, M., Katriel, G.: On the discrete spectrum of non-selfadjoint operators. Preprint, TV clausthal, 2008 9. Frank, R.L., Laptev, A., Lieb, E.H., Seiringer, R.: Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77(3), 309–316 (2006) 10. Ge, J.-Y., Zhang, J.: Use of negative complex potential as absorbing potential. J. Chem. Phys. 108(4), 1429–1433 (1998)
54
A. Laptev, O. Safronov
11. Fan, K.: Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Nat. Acad. Sci., U. S. A. 37, 760–766 (1951) 12. Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics (Essays in Honor of Valentine Bargmann), Princeton, NJ: Princeton Univ. Press, 1976, pp. 269–303 13. Nimtz, G., Spieker, H., Brodowsky, H.M.: Tunneling with dissipation. J. Phys. I France 4, 1379– 1382 (1994) 14. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New YorkLondon: Academic Press [Harcourt Brace Jovanovich, Publishers], 1978 15. Schwinger, J.: On the bound states of a given potential. Proc. Nat. Acad. Sci. USA 47, 122–129 (1967) 16. Seiler, E., Simon, B.: Bounds in the Yukawa2 quantum field theory: upper bound on the pressure, Hamiltonian bound and linear lower bound. Commun. Math. Phys. 45, 99–114 (1975) 17. Simon, B.: Trace ideals and their applications, London Mathematical Society Lecture Note Series 35, Cambidge: Cambidge University Press, 1979 18. Von Neumann, J., Wigner, E.: Uber merkwurdige diskrete Eigenwerte. Physik. Zeitschr. 30, 465 (1929) 19. Weyl, H.: Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. Nat. Acad Sci. USA 35, 408–411 (1949) Communicated by B. Simon
Commun. Math. Phys. 292, 55–97 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0887-0
Communications in
Mathematical Physics
Entanglement Transmission and Generation under Channel Uncertainty: Universal Quantum Channel Coding Igor Bjelakovi´c1,2 , Holger Boche1,2 , Janis Nötzel1 1 Heinrich-Hertz-Lehrstuhl für Mobilkommunikation (HFT 6), Technische Universität Berlin,
Einsteinufer 25, 10587 Berlin, Germany. E-mail:
[email protected];
[email protected];
[email protected] 2 Institut für Mathematik, Technische Universität Berlin, Strasse des 17, Juni 136, 10623 Berlin, Germany. Received: 28 November 2008 / Accepted: 20 May 2009 Published online: 13 August 2009 – © Springer-Verlag 2009
Abstract: We determine the optimal rates of universal quantum codes for entanglement transmission and generation under channel uncertainty. In the simplest scenario the sender and receiver are provided merely with the information that the channel they use belongs to a given set of channels, so that they are forced to use quantum codes that are reliable for the whole set of channels. This is precisely the quantum analog of the compound channel coding problem. We determine the entanglement transmission and entanglement-generating capacities of compound quantum channels and show that they are equal. Moreover, we investigate two variants of that basic scenario, namely the cases of informed decoder or informed encoder, and derive corresponding capacity results. Contents 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . One-Shot Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Part of the Coding Theorem for Finitely Many Channels . . . . . . . Finite Approximations in the Set of Quantum Channels . . . . . . . . . . . Direct Parts of the Coding Theorems for General Quantum Compound Channels Converse Parts of the Coding Theorems for General Quantum Compound Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Continuity of Compound Capacity . . . . . . . . . . . . . . . . . . . . . . 9. Entanglement-Generating Capacity of Compound Channels . . . . . . . . . 10. Conclusion and Further Remarks . . . . . . . . . . . . . . . . . . . . . . . A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55 60 62 74 79 80 87 91 92 94 94
1. Introduction The determination of capacities of quantum channels in various settings has been a field of intense work over the last decade. In contrast to classical information theory, to any
56
I. Bjelakovi´c, H. Boche, J. Nötzel
quantum channel we can associate in a natural way different notions of capacity depending on what is to be transmitted over the channel and which figure of merit is chosen as the criterion for the success of the particular quantum communication task. For example we may try to determine the maximum number of classical messages that can be reliably distinguished at the output of the channel leading to the notion of classical capacity of a quantum channel. We might as well wish to establish secure classical communication over a quantum channel, giving rise to the definition of a channel’s private capacity. On the other hand, in the realm of quantum communication, one may ask e.g. the question what the maximal amount of entanglement is that we can generate or transmit over a given quantum channel, leading to the notions of entanglement-generating and entanglement transmission capacities. Other examples of quantum capacities are the subspace transmission and average subspace transmission capacities. Such quantum communication tasks are needed, for example, to support computation in quantum circuits or to provide the best possible supply of pure entanglement in a noisy environment. Fortunately, these genuinely quantum mechanical capacities are shown to be equal for perfectly known single user channels [1,21]. First results indicating that coherent information was to play a role in the determination of the quantum capacity of memoryless channels were established by Schumacher and Nielsen [26] and, independently, by Lloyd [23] who was the first to conjecture that indeed the regularized coherent information would give the correct formula for the quantum capacity and gave strong heuristic evidence to his claim. In 1998 Barnum, Knill, and Nielsen and Barnum, Nielsen, and Schumacher [1] gave the first upper bound on the capacity of a memoryless channel in terms of the regularized coherent information. Later on, Shor [29] and Devetak [10] offered two independent approaches to the achievability part of the coding theorem. Despite the fact that the regularized coherent information was identified as the capacity of memoryless quantum channels many other approaches to the coding theorem have been offered subsequently, for example Devetak and Winter [11] and Hayden, Shor, and Winter [14]. Of particular interest for our paper are the developments by Klesse [20] and Hayden, Horodecki, Winter, and Yard [13] based on the decoupling idea which can be traced back to Schumacher and Westmoreland [28]. In fact, the main purpose of our work is to show that the decoupling idea can be utilized to prove the existence of reliable universal quantum codes for entanglement transmission and generation. On the other hand, the classical capacity of memoryless quantum channels has been determined in the pioneering work by Holevo [15] and Schumacher and Westmoreland [27]. Their results have been substantially sharpened by Winter [31] and Ogawa and Nagaoka [25] who gave independent proofs of the strong converse to the coding theorem. However, most of the work done so far on quantum channel capacities relies on the assumption that the channel is perfectly known to the sender and receiver. Such a requirement is hardly fulfilled in many situations. In this paper we consider compound quantum channels which are among the simplest non-trivial models with channel uncertainty. A rough description of this communication scenario is that the sender and receiver do not know the memoryless channel they have to use. The prior knowledge they have access to is merely that the actual channel belongs to a set I of channels which in turn is known to the sender and receiver. It is important to notice that we impose no restrictions on the set I, i.e. it can be finite, countably-infinite or uncountable. Our intention is to identify the best rates of quantum codes for entanglement transmission and generation that are reliable for the whole set of channels I simultaneously. This is, in some sense, a quantum channel counterpart of the universal quantum data compression result discovered by Jozsa and the Horodecki family [18].
Universal Quantum Channel Coding
57
While the classical capacity of compound quantum channels has been determined only recently in [3], in this paper we will focus on entanglement-generating and entanglement transmission capacities of compound quantum channels. Specifically we will determine both of them and show that they are equal. The investigation of their relation to other possible definitions of quantum capacity of compound quantum channels in spirit of [1,21] will be given elsewhere. 1.1. Related Work. The capacity of compound channels in the classical setting was determined by Wolfowitz [32,33] and Blackwell, Breiman, and Thomasian [5]. The full coding theorem for transmission of classical information via compound quantum channels was proven in [3]. Subsequently, Hayashi [12] obtained a closely related result with a completely different proof technique based on the Schur-Weyl duality from representation theory and the packing lemma from [7]. In our previous paper [4] we determined the entanglement transmission capacity of finite quantum compound channels (i.e. |I| < ∞). Moreover, we were able to prove the coding theorem for arbitrary I with informed decoder. It is important to remark here that we used a different notion of codes in [4], following [20], which is motivated by the theory of quantum error correction. In the cases of an informed decoder and uninformed users this change does not appear to be of importance. In the case of an informed encoder it is of crucial importance in the proof of the direct part of the coding result. In our former paper, the strategy of proof was as follows. First, we derived a modification of Klesse’s one-shot coding result [20] that was adapted to arithmetic averages of channels. Application of this theorem combined with a discretization technique based on τ -nets yielded the coding result for quantum compound channels with informed decoder and arbitrary I. With the help of the channel-estimation technique developed by Datta and Dorlas [9] we were able to show that in the case of√a finite compound channel it is asymptotically of no relevance if one spends the first l transmissions for channel estimation, thus turning an uninformed decoder into an informed decoder. Since for an informed decoder we had already proven the existence of good codes, we were able to obtain the full coding result in the case |I| < ∞. Unfortunately, the speed at which one can gain channel knowledge using the channel estimation technique we employed is highly dependent on the number of channels. Due to this fact, the combination of channel estimation and approximation of general compound channels through finite ones did not seem to work in the other two cases. In this paper, we use a more direct strategy. First, we derive one-shot coding results for finite compound channels with uninformed users and informed encoder. In order to evaluate the dependence of the derived bounds on the block length we have to project onto typical subspaces of suitable output states of the individual channels. Therefore, it turns out that we effectively end up in the scenario with informed decoder. Now, instead of employing a channel estimation strategy we study the impact of these projections onto the typical subspaces on the entanglement fidelity of the entire encoding-decoding procedure. It turns out that these projections can simply be removed without decreasing the entanglement fidelity too much and we have got a universal (i.e. uninformed) decoder for our coding problem. Then, again, using the discretization technique based on τ -nets we can convert these results for finite I to arbitrary compound quantum channels. Another difference to our previous paper [4] is that we determine the optimal rates in all the scenarios described above for entanglement generation over compound quantum channels and show that they coincide with the entanglement transmission capacities.
58
I. Bjelakovi´c, H. Boche, J. Nötzel
1.2. Outline. Section 2 contains the fundamental definitions of codes and capacities for entanglement transmission in all three different settings. Moreover, the reader can find there the statement of our main result. It is followed by a section on one-shot results containing the one-shot result of Klesse [20], as well as our modifications thereof. The modified coding results guarantee the existence of unitary encodings as well as recovery operations for finite arithmetic averaged channels in all three different cases and establish a relation between the rate of the code and its entanglement fidelity. We also give an estimate relating the entanglement fidelity of a coding-decoding procedure to that of a disturbed version, where disturbance means that the application of the channel is followed by a projection. With these one-shot results at hand, in Sect. 4 we are able to prove the existence of codes for entanglement transmission of sufficiently high rates and entanglement fidelity asymptotically approaching one exponentially fast in the case of finite compound channels. Section 5 states the basic properties of finite size nets in the set of quantum channels. They are used to approximate general sets of quantum channels and provide the link between finite and general compound channels. The construction is such that their size depends polynomially on the approximation parameter. We use the coding results for finite compound channels and the properties of finite nets in Sect. 6 to derive sharp lower bounds on the entanglement transmission capacity of general compound channels. This section also contains variants of the BSST Lemma [2], where BSST stands for Bennett, Shor, Smolin, and Thapliyal. The proofs rely heavily on the difference in the polynomial growth of nets versus exponentially fast convergence to the entanglement fidelity one for the codes in the finite setting. The next Sect. 7 contains the converse parts of the coding theorems for general compound channels. Since the converse must hold for arbitrary encoding schemes and since we explicitly allow the code space to be larger than the input space of the channels, we deviate from the usual structure and instead employ the converse part for the case of entanglement generation that was developed by Devetak [10]. We also use a recent continuity result due to Leung and Smith [22] that connects the difference in coherent information between nearby channels. In Sect. 8 we show, once again using the work of Leung and Smith [22], that the entanglement transmission capacities of compound quantum channels are continuous with respect to the Hausdorff metric. In the final Sect. 9 we apply the results obtained so far to determine the entanglement-generating capacities of compound quantum channels. It is not very surprising that it turns out that they coincide with their counterparts for entanglement transmission. 1.3. Notation and Conventions. All Hilbert spaces are assumed to have finite dimension and are over the field C. S(H) is the set of states, i.e. positive semi-definite operators with trace 1 acting on the Hilbert space H. Pure states are given by projections onto onedimensional subspaces. A vector of unit length spanning such a subspace will therefore be referred to as a state vector. To each subspace F of H we can associate unique projection qF whose range is the subspace F and we write πF for the maximally mixed state qF on F, i.e. πF := tr(q . The set of completely positive trace preserving (CPTP) maps F) between the operator spaces B(H) and B(K) is denoted by C(H, K). Thus H plays the role of the input Hilbert space to the channel (traditionally owned by Alice) and K is the channel’s output Hilbert space (usually in Bob’s possession). C ↓ (H, K) stands for the set of completely positive trace decreasing maps between B(H) and B(K). U(H)
Universal Quantum Channel Coding
59
will denote in what follows the group of unitary operators acting on H. For a Hilbert space G ⊂ H we will always identify U(G) with a subgroup of U(H) in the canonical way. For any projection q ∈ B(H) we set q ⊥ := 1H − q. Each projection q ∈ B(H) defines a completely positive trace decreasing map Q given by Q(a) := qaq for all a ∈ B(H). In a similar fashion any u ∈ U(H) defines a U ∈ C(H, H) by U(a) := uau ∗ for a ∈ B(H). We use the base two logarithm which is denoted by log. The von Neumann entropy of a state ρ ∈ S(H) is given by S(ρ) := −tr(ρ log ρ). The coherent information for N ∈ C(H, K) and ρ ∈ S(H) is defined by Ic (ρ, N ) := S(N (ρ)) − S((idH ⊗ N )(|ψ ψ|)), where ψ ∈ H ⊗ H is an arbitrary purification of the state ρ. Following the usual conventions we let Se (ρ, N ) := S((idH ⊗ N )(|ψ ψ|)) denote the entropy exchange. A useful equivalent definition of Ic (ρ, N ) is given in terms of N ∈ C(H, K) and the complementary channel N ∈ C(H, He ), where He denotes the Hilbert space of the environment: Due to Stinespring’s dilation theorem N can be represented as N (ρ) = trHe (vρv ∗ ) for ρ ∈ S(H), where v : H → K ⊗ He is a linear isometry. The complementary channel N ∈ C(H, He ) to N is given by N (ρ) := trH (vρv ∗ )
(ρ ∈ S(H)).
The coherent information can then be written as Ic (ρ, N ) = S(N (ρ)) − S(N (ρ)). As of closeness between two states ρ, σ ∈ S(H) we use the fidelity F(ρ, σ ) := √ √a measure || ρ σ ||21 . The fidelity is symmetric in the input and for a pure state ρ = |φ φ| we have F(|φ φ|, σ ) = φ, σ φ . A closely related quantity is the entanglement fidelity. For ρ ∈ S(H) and N ∈ C ↓ (H, K) it is given by Fe (ρ, N ) := ψ, (idH ⊗ N )(|ψ ψ|)ψ , with ψ ∈ H ⊗ H being an arbitrary purification of the state ρ. For the approximation of arbitrary compound channels by finite ones we use the diamond norm || · ||♦, which is given by ||N ||♦ := sup
max
n n∈N a∈B(C ⊗H),||a||1 =1
||(idn ⊗ N )(a)||1 ,
where idn : B(Cn ) → B(Cn ) is the identity channel, and N : B(H) → B(K) is any linear map, not necessarily completely positive. The merits of ||·||♦ are due to the following facts (cf. [19]). First, ||N ||♦ = 1 for all N ∈ C(H, K). Thus, C(H, K) ⊂ S♦, where S♦ denotes the unit sphere of the normed space (B(B(H), B(K)), || · ||♦). Moreover, ||N1 ⊗ N2 ||♦ = ||N1 ||♦||N2 ||♦ for arbitrary linear maps N1 , N2 : B(H) → B(K). We further use the diamond norm to define the function D♦(·, ·) on {(I, I ) : I, I ⊂ C(H, K)}, which is for I, I ⊂ C(H, K) given by D♦(I, I ) := max{ sup inf ||N − N ||♦, sup inf ||N − N ||♦}.
N ∈I N ∈I
N ∈I N ∈I
60
I. Bjelakovi´c, H. Boche, J. Nötzel
For I ⊂ C(H, K) let I¯ denote the closure of I in || · ||♦. Then D♦ defines a metric on ¯ I = I¯ } which is basically the Hausdorff distance {(I, I ) : I, I ⊂ C(H, K), I = I, induced by the diamond norm. Obviously, for arbitrary I, I ⊂ C(H, K), D♦(I, I ) ≤ implies that for every N ∈ I (N ∈ I ) there exists N ∈ I (N ∈ I) such that ||N − N ||♦ ≤ 2. If ¯ I = I¯ holds we even have ||N − N ||♦ ≤ . In this way D♦ gives a measure I = I, of distance between two compound channels. Finally, for any set I ⊂ C(H, K) and l ∈ N we set I⊗l := {N ⊗l : N ∈ I}. 2. Definitions and Main Result Let I ⊂ C(H, K). The memoryless compound channel associated with I is given by the family {N ⊗l : S(H⊗l ) → S(K⊗l )}ł∈N,N ∈I. In the rest of the paper we will simply write I for that family. Each compound channel can be used in three different scenarios: 1. the informed decoder 2. the informed encoder 3. the case of uninformed users. In the following three subsections we will give definitions of codes and capacity for these cases. 2.1. The Informed Decoder. An (l, kl )-code for I with informed decoder is a pair (P l , {RlN : N ∈ I}) where: 1. P l : B(Fl ) → B(H)⊗l is a CPTP map for some Hilbert space Fl with kl = dim Fl . 2. RlN : B(K)⊗l → B(Fl ) is a CPTP map for each N ∈ I, where the Hilbert space Fl satisfies Fl ⊂ Fl . In what follows the operations RlN are referred to as recovery (or decoding) operations. Since the decoder knows which channel is actually used during transmission, they are allowed to depend on the channel. Note at this point that we deviate from the standard assumption that Fl = Fl . We allow Fl Fl for convenience only since it allows more flexibility in code construction. It is readily seen from the definition of achievable rates and capacity below that the assumption Fl Fl cannot lead to a higher capacity of I in any of the three cases that we are dealing with. A non-negative number R is called an achievable rate for I with informed decoder if there is a sequence of (l, kl )-codes such that 1. lim inf l→∞ 1l log kl ≥ R, and 2. liml→∞ inf N ∈I Fe (πFl , RlN ◦ N ⊗l ◦ P l ) = 1 holds. The capacity Q I D (I) of the compound channel I with informed decoder is given by Q I D (I) := sup{R ∈ R+ : R is achievable for I with informed decoder}.
Universal Quantum Channel Coding
61
l : 2.2. The Informed Encoder. An (l, kl )-code for I with informed encoder is a pair ({PN N ∈ I}, Rl ) where: l : B(F ) → B(H)⊗l is a CPTP map for each N ∈ I for some Hilbert space 1. PN l l are the encoding operations which we allow to Fl with kl = dim Fl . The maps PN depend on N since the encoder knows which channel is in use. 2. Rl : B(K)⊗l → B(Fl ) is a CPTP map where the Hilbert space Fl satisfies Fl ⊂ Fl .
A non-negative number R is called an achievable rate for I with informed encoder if there is a sequence of (l, kl )-codes such that 1. lim inf l→∞ 1l log kl ≥ R, and l )=1 2. liml→∞ inf N ∈I Fe (πFl , Rl ◦ N ⊗l ◦ PN holds. The capacity Q I E (I) of the compound channel I with informed encoder is given by Q I E (I) := sup{R ∈ R+ : R is achievable for I with informed encoder}. 2.3. The Case of Uninformed Users. Codes and capacity for the compound channel I with uninformed users are defined in a similar fashion. The only change is that we do not allow the encoding operations to depend on N . I.e. an (l, kl )− code for I is a pair (P l , Rl ) of CPTP maps P l ∈ C(Fl , H⊗l ), where Fl is a Hilbert space with kl = dim Fl and Rl ∈ C(K⊗l , Fl ) with Fl ⊂ Fl . A non-negative number R is called an achievable rate for I if there is a sequence of (l, kl )-codes such that 1. lim inf l→∞ 1l log kl ≥ R, and 2. liml→∞ inf N ∈I Fe (πFl , Rl ◦ N ⊗l ◦ P l ) = 1. The capacity Q(I) of the compound channel I is given by Q(I) := sup{R ∈ R+ : R is achievable for I}. A first simple consequence of these definitions is the following relation among the capacities of I. Q(I) ≤ min{Q I D (I), Q I E (I)}. 2.4. Main Result. With these definitions at our disposal, we are ready now to state the main result of the paper. Theorem 1. Let I ⊂ C(H, K) be an arbitrary set of quantum channels, where H and K are finite dimensional Hilbert spaces. 1. Then Q(I) = Q I D (I) = lim
l→∞
1 max inf Ic (ρ, N ⊗l ), l ρ∈S (H⊗l ) N ∈I
and Q I E (I) = lim
l→∞
1 inf max Ic (ρ, N ⊗l ). l N ∈I ρ∈S (H⊗l )
62
I. Bjelakovi´c, H. Boche, J. Nötzel
2. Moreover, for the corresponding entanglement-generating capacities E(I), E I D (I), and E I E (I) (defined in Sect. 9) we have E(I) = E I D (I) = Q(I) and E I E (I) = Q I E (I). The rest of the paper contains a step-by-step proof of Theorem 1. 3. One-Shot Results In this section we will establish the basic building blocks for the achievability parts of the coding theorems for compound channels with and without channel knowledge. The results are formulated as one-shot statements in order to simplify the notation.
3.1. One-Shot Coding Result for a Single Channel. Before we turn our attention to quantum compound channels we will shortly describe a part of recent developments in coding theory for single (i.e. perfectly known) channels as given in [20] and [13]. Both approaches are based on a decoupling idea which is closely related to approximate error correction. In order to state this decoupling lemma we need some notational preparation. Let ρ ∈ S(H) be given and consider any purification ψ ∈ Ha ⊗ H, Ha = H, of ρ. According to Stinespring’s representation theorem any N ∈ C ↓ (H, K) is given by N ( · ) = trHe ((1H ⊗ pe )v( · )v ∗ ),
(1)
where He is a suitable finite-dimensional Hilbert space, pe is a projection onto a subspace of He , and v : H → K ⊗ He is an isometry. Let us define a pure state on Ha ⊗ K ⊗ He by the formula 1 ψ := √ (1Ha ⊗K ⊗ pe )(1Ha ⊗ v)ψ. tr(N (πF )) We set
:= trK (|ψ ψ |), ρ := trHa ⊗He (|ψ ψ |), ρae
and ρa := trK⊗He (|ψ ψ |), ρe := trHa ⊗K (|ψ ψ |). The announced decoupling lemma can now be stated as follows. Lemma 1 (cf. [13,20]). For ρ ∈ S(H) and N ∈ C ↓ (H, K) there exists a recovery operation R ∈ C(K, H) with
− wρa ⊗ ρe ||1 , Fe (ρ, R ◦ N ) ≥ w − ||wρae
where w = tr(N (ρ)).
Universal Quantum Channel Coding
63
−ρ ⊗ The striking implication of Lemma 1 is that if the so called quantum error ||ρae a
ρe ||1 for ρ ∈ S(H) and N ∈ C(H, K) is small then almost perfect error correction is possible via R. Lemma 1 was Klesse’s [20] starting point for his highly interesting proof of the following theorem which is a one-shot version of the achievability part of the coding theorem. In the statement of the result we will use the following notation.
Fc,e (ρ, N ) :=
max
R∈C (K,H)
Fe (ρ, R ◦ N ),
where ρ ∈ S(H) and N ∈ C ↓ (H, K). Theorem 2 (Klesse [20]). Let the Hilbert space H be given and consider subspaces E ⊂ G ⊂ H with dim E = k. Then for any N ∈ C ↓ (H, K) allowing a representation with n Kraus operators we have √ Fc,e (uπE u ∗ , N )du ≥ tr(N (πG )) − k · n||N (πG )||2 , U(G )
where U(G) denotes the group of unitaries acting on G and du indicates that the integration is with respect to the Haar measure on U(G). We will indicate briefly how Klesse [20] derived the direct part of the coding theorem for memoryless quantum channels from Theorem 2. Let us choose for each l ∈ N subspaces El ⊂ G ⊗l ⊂ H⊗l with dim El =: kl = 2l(Ic (πG ,N )−3) . To given N ∈ C(H, K) and πG Klesse constructed a reduced version Nl of N ⊗l in such a way that Nl has a Kraus representation with nl ≤ 2l(Se (πG ,N )+) Kraus operators. Let ql ∈ B(K⊗l ) be the entropy-typical projection of the state (N (πG ))⊗l and set Nl (·) := ql Nl (·)ql . Then we have the following properties (some of which are stated once more for completeness): 1. 2. 3. 4.
kl = 2l(Ic (πG ,N )−3) , tr(Nl (πG⊗l )) ≥ 1 − o(l 0 ),1 nl ≤ 2l(Se (πG ,N )+) , and ||Nl (πG⊗l )||22 ≤ 2−l(S(πG )−) .
An application of Theorem 2 to Nl shows heuristically the existence of a unitary u ∈ U(G ⊗l ) and a recovery operation Rl ∈ C(K⊗l , H⊗l ) with l
Fe (uπEl u ∗ , Rl ◦ Nl ) ≥ 1 − o(l 0 ) − 2− 2 . This in turn can be converted into Fe (uπEl u ∗ , Rl ◦ N ⊗l ) ≥ 1 − o(l 0 ), which is the achievability of Ic (πG , N ). The passage from πG to arbitrary states ρ is then accomplished via the Bennett, Shor, Smolin, and Thapliyal Lemma from [2] and the rest is by regularization. 1 Here, o(l 0 ) denotes simply a non-specified sequence tending to 0 as l → ∞, i.e. we (ab)use the BachmannLandau little-o notation.
64
I. Bjelakovi´c, H. Boche, J. Nötzel
3.2. One-Shot Coding Result for Uninformed Users. Our goal in this section is to establish a variant of Theorem 2 that works for finite sets of channels. Since the entanglement fidelity depends affinely on the channel it is easily seen that for each set I = {N1 , . . . , N N } any good coding scheme with uninformed users is also good for the channel N :=
N 1 Ni N i=1
and vice versa. Since it is easier to deal with a single channel and we do not loose anything if passing to averages, we will formulate our next theorem for arithmetic averages of completely positive trace decreasing maps instead of the set {N1 , . . . , N N }. Theorem 3 (One-Shot Result: Uninformed Users and Averaged Channel). Let the Hilbert space H be given and consider subspaces E ⊂ G ⊂ H with dim E = k. For any choice of N1 , . . . N N ∈ C ↓ (H, K) each allowing a representation with n j Kraus operators, j = 1, . . . , N , we set N :=
N 1 Nj, N j=1
and for any u ∈ U(G), Nu :=
N 1 N j ◦ U. N j=1
Then U(G )
Fc,e (πE , Nu )du ≥ tr(N (πG )) − 2
N
kn j ||N j (πG )||2 ,
j=1
where the integration is with respect to the normalized Haar measure on U(G). Remark 1. It is worth noting that the average in this theorem is no more over maximally mixed states like in Theorem 2, but rather over encoding operations. Proof. The proof is easily reduced to that of the corresponding theorem in our previous paper [4]. Most of the details can also be seen in the proof of Theorem 4 in the next subsection. 3.3. One-Shot Coding Result for Informed Encoder. Before stating the main result of this section we recall a useful lemma from [4] which will be needed in the proof of Theorem 4. Lemma 2. Let L and D be N × N matrices with non-negative entries which satisfy L jl ≤ L j j , L jl ≤ L ll ,
(2)
Universal Quantum Channel Coding
65
and D jl ≤ max{D j j , Dll }
(3)
for all j, l ∈ {1, . . . , N }. Then N N 1 L jl D jl ≤ 2 L jj Djj. N
j,l=1
j=1
Proof. The proof of this lemma is elementary. The details can be picked up in our previous paper [4]. We will focus now on the scenario where the sender or encoder knows which channel is in use. Consequently, the encoding operation can depend on the individual channel. The idea behind the next theorem is that we perform an independent, randomized selection of unitary encoders for each channel in the finite set I = {N1 , . . . , N N }. This explains why the averaging in (4) is with respect to products of Haar measures instead of averaging over one single Haar measure as in Theorem 3. Theorem 4 (One-Shot Result: Informed Encoder and Averaged Channel). Let the finitedimensional Hilbert spaces H and K be given. Consider subspaces E, G1 , . . . , G N ⊂ H with dim E = k such that for all i ∈ {1, . . . , N } the dimension relation k ≤ dim Gi holds. Let N1 , . . . N N ∈ C ↓ (H, K) each allowing a representation with N n j Kraus operators, j = 1, . . . , N . Let {vi }i=1 ⊂ U(H) be any fixed set of unitary operators such that vi E ⊂ Gi holds for every i ∈ {1, . . . , N }. For an arbitrary set N ⊂ U(H), define {u i }i=1 Nu 1 ,...,u N :=
N 1 Ni ◦ Ui ◦ Vi . N i=1
Then U(G1 )×...×U(G N )
Fc,e (πE , Nu 1 ,...,u N )du 1 . . . du N ≥
N 1 j=1
N
tr(N j (πG j ))
− 2 kn j ||N j (πG j )||2 , (4)
where the integration is with respect to the product of the normalized Haar measures on U(G1 ), . . . , U(G N ). Proof. Our first step in the proof is to show briefly that Fc,e (πE , Nu 1 ,...,u N ) depends measurably on (u 1 , . . . , u N ) ∈ U(G1 ) × · · · × U(G N ). For each recovery operation R ∈ C(K, H) we define a function f R : U(G1 ) × · · · × U(G N ) → [0, 1] by f R (u 1 , . . . , u N ) := Fe (πE , R ◦ Nu 1 ,...,u N ). Clearly, f R is continuous for each fixed R ∈ C(K, H). Thus, the function Fc,e (πE , Nu 1 ,...,u N ) =
max
R∈C (K,H)
f R (u 1 , . . . , u N )
is lower semicontinuous, and consequently measurable.
66
I. Bjelakovi´c, H. Boche, J. Nötzel
We turn now to the proof of inequality (4). From Lemma 1 we know that there is a recovery operation R such that
Fe (πE , R ◦ Nu 1 ,...,u N ) ≥ w − ||wρae − wρa ⊗ ρe ||1 ,
(5)
where we have used the notation introduced in the paragraph preceding Lemma 1, and w = w(u 1 , . . . , u N ) = tr(Nu 1 ,...,u N (πE )). n
j For each j ∈ {1, . . . , N } let {b j,i }i=1 be the set of Kraus operators of N j . Clearly, for nj every set u 1 , . . . , u N of unitary matrices, N j ◦ U j ◦ V j has Kraus operators {a j,i }i=1 given by a j,i = b j,i u j v j . Utilizing the very same calculation that was used in the proof of Theorem 3 in [4], which in turn is almost identical to the corresponding calculation in [20], we can reformulate inequality (5) as
Fe (πE , R ◦ Nu 1 ,...,u N ) ≥ w − ||D(u 1 , . . . , u N )||1 ,
(6)
with w = tr(Nu 1 ,...,u N (πE )) and D(u 1 , . . . , u N ) :=
n j ,nl N 1 D(i j)(rl) (u j , u l ) ⊗ |ei er | ⊗ | f j fl |, N i,r =1
j,l=1
where D(i j)(rl) (u j , u l ) :=
1 k
1 ∗ pa j,i al,r p − tr( pa ∗j,i al,r p) p , k
and p := kπE is the projection onto E. Let us define n j ,nl
D j,l (u j , u l ) :=
D(i j)(kl) (u j , u l ) ⊗ |ei ek | ⊗ | f j fl |.
(7)
i=1,k=1
The triangle inequality for the trace norm yields ||D(u 1 , . . . , u N )||1 ≤
N 1 ||D j,l (u j , u l )||1 N
j,l=1
≤
N 1 k min{n j , nl }||D j,l (u j , u l )||2 , N
j,l=1
N 1 = k min{n j , nl }||D j,l (u j , u l )||22 , N
(8)
j,l=1
√ where the second line follows from ||a||1 ≤ d||a||2 , d being the number of non-zero singular values of a. In the next step we will compute ||D j,l (u j , u l )||22 . We set pl := vl pvl∗ which defines N with supp( p ) ⊂ G for every l ∈ {1, . . . , N }. A glance at (7) new projections { pl }l=1 l l shows that n j ,nl ∗
(D j,l (u j , u l )) =
i=1,k=1
(D(i j)(kl) (u j , u l ))∗ ⊗ |ek ei | ⊗ | fl f j |,
(9)
Universal Quantum Channel Coding
67
and consequently we obtain ||D j,l (u j , u l )||22 = tr((D j,l (u j , u l ))∗ D j,l (u j , u l )) n j ,nl
=
tr((D(i j)(kl) (u j , u l ))∗ D(i j)(kl) (u j , u l ))
i=1,r =1 n j ,nl
=
1 k2
{tr( p(a ∗j,i al,r )∗ pa ∗j,i al,r )
i=1,r =1
1 − |tr( pa ∗j,i al,r )|2 } k n j ,nl 1 ∗ {tr( pl u l∗ bl,r b j,i u j p j u ∗j b∗j,i bl,r u l ) = 2 k i=1,r =1
1 − |tr( pv ∗j u ∗j b∗j,i bl,r u l vl )|2 }. k
(10)
It is apparent from the last two lines in (10) that ||D j,l (u j , u l )||22 depends measurably on (u 1 , . . . , u N ) ∈ U(G1 ) × · · · × U(G N ). Let U1 , . . . , U N be independent random variables taking values in U(Gi ) according to the normalized Haar measure on U(Gi ) (i ∈ {1, . . . , N }). Then using Jensen’s inequality and abbreviating L jl := k min{n j , nl } we can infer from (8) that E(||D(U1 , . . . , U N )||1 ) ≤
N 1 L jl E(||D j,l (U j , Ul )||22 ). N
(11)
j,l=1
Note that the expectations on the RHS of (11) are only with respect to pairs of random variables U1 , . . . , U N . Our next goal is to upper-bound E(||D j,l (U j , Ul )||22 ). Case j = l. Since the last term in (10) is non-negative and the random variables U j and Ul are independent we obtain the following chain of inequalities: E(||D j,l (U j , Ul )||22 ) =
1 k2 −
≤ = =
1 k2 1 k2 1 k2
n j ,nl
∗ Etr( pl Ul∗ bl,r b j,i U j p j U ∗j b∗j,i bl,r Ul )
i=1,r =1
1 E|tr( pv ∗j U ∗j b∗j,i bl,r Ul vl )|2 k
n j ,nl
∗ Etr( pl Ul∗ bl,r b j,i U j p j U ∗j b∗j,i bl,r Ul )
i=1,r =1 n j ,nl
∗ Etr(Ul pl Ul∗ bl,r b j,i U j p j U ∗j b∗j,i bl,r )
i=1,r =1 n j ,nl
i=1,r =1
∗ tr(E(Ul pl Ul∗ )bl,r b j,i E(U j p j U ∗j )b∗j,i bl,r )
68
I. Bjelakovi´c, H. Boche, J. Nötzel
=
1 k2
n j ,nl
∗ tr(k · πGl bl,r b j,i k · πG j b∗j,i bl,r )
i=1,r =1
= N j (πG j ), Nl (πGl ) H S ,
(12)
where · , · H S denotes the Hilbert-Schmidt inner product, and we used the fact that E(Ul pl Ul∗ ) = k · πGl and E(U j p j U ∗j ) = k · πG j . Case j = l. In this case we obtain
E(||D j, j (U j , U j )||22 ) =
1 k2
Etr( p j U ∗j b∗j,r b j,i U j p j U ∗j b∗j,i b j,r U j )
n j ,n j
i=1,r =1
1 − E|tr( pv ∗j U ∗j b∗j,i b j,r U j v j )|2 k 1 = 2 k
n j ,n j
Etr(U j p j U ∗j b∗j,r b j,i U j p j U ∗j b∗j,i b j,r )
i=1,r =1
1 ∗ ∗ 2 − E|tr(U j p j U j b j,i b j,r )| . k
(13)
Thus, the problem reduces to the evaluation of E{bU pU ∗ (x, y)},
(x, y ∈ B(H)),
where p is an orthogonal projection with tr( p) = k and 1 bU pU ∗ (x, y) := tr(U pU ∗ x ∗ U pU ∗ y) − tr(U pU ∗ x ∗ )tr(U pU ∗ y), k for a Haar distributed random variable U with values in U(G) where supp( p) ⊂ G ⊂ H. Here we can refer to [20] where the corresponding calculation is carried out via the theory of group invariants and explicit evaluations of appropriate integrals with respect to row-distributions of random unitary matrices. The result is
E{bU pU ∗ (x, y)} =
k2 − 1 1 − k2 ∗ tr( p tr( pG x ∗ )tr( pG y), x p y) + G G d2 − 1 d(d 2 − 1)
(14)
for all x, y ∈ B(H) where pG denotes the projection onto G with tr( pG ) = d. In Appendix A we will give an elementary derivation of (14) for the sake of completeness.
Universal Quantum Channel Coding
69
Inserting (14) with x = y = b∗j,i b j,r into (13) yields with d j := tr( pG j ): E(||D j, j (U j , U j )||22 ) =
≤
1− d 2j
−1
1−
⎡
n j ,n j
⎣
1 d 2j
tr( pG j b∗j,r b j,i pG j b∗j,i b j,r )
i=1,r =1
⎤ 1 − |tr(( pG j b∗j,i b j,r )|2 ⎦ dj
n j ,n j
1 k2
d 2j − 1 i=1,r =1
1 ≤ 2 dj =
1 k2
tr( pG j b∗j,r b j,i pG j b∗j,i b j,r )
n j ,n j
tr( pG j b∗j,r b j,i pG j b∗j,i b j,r )
i=1,r =1 n j ,n j
tr(b j,r pG j b∗j,r b j,i pG j b∗j,i )
i=1,r =1
= N j (πG j ), N j (πG j ) H S . Summarizing, we obtain E(||D j, j (U j , U j )||22 ) ≤ N j (πG j ), N j (πG j ) H S = ||N j (πG j )||22 .
(15)
Similarly E(tr(NU1 ,...,U N (πE ))) =
N 1 1 E(tr(N j (U j p j U ∗j ))) N k j=1
=
N 1 tr(N j (πG j )). N
(16)
j=1
Equations (6), (8), (12), (15), and (16) show that N 1 E(Fc,e (πE , NU1 ,...,U N )) ≥ tr(N j (πG j )) N j=1
−
N 1 L jl D jl , N
j,l=1
where for j, l ∈ {1, . . . , N } we introduced the abbreviation D jl := N j (πG j ), Nl (πGl ) H S , and, as before, L jl = k min{n j , nl }.
(17)
70
I. Bjelakovi´c, H. Boche, J. Nötzel
It is obvious that L jl ≤ L j j and L jl ≤ L ll hold. Moreover, the Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product shows that D jl = N j (πG j ), Nl (πGl ) H S ≤ ||N j (πG j )||2 ||Nl (πGl )||2 ≤ max{||N j (πG j )||22 , ||Nl (πGl )||22 } = max{D j j , Dll }. Therefore, an application of Lemma 2 allows us to conclude from (17) that E(Fc,e (πE , NU1 ,...,U N )) ≥
N 1 tr(N j (πG j )) N j=1
−2
N
kn j ||N j (πG j )||2 ,
j=1
and we are done.
3.4. Entanglement Fidelity. The purpose of this subsection is to develop a tool which will enable us to convert a special kind of recovery maps depending on the channel into such that are universal, at least for finite compound channels. Anticipating constructions in Sect. 4 below, the situation we will be faced with is as follows. For finite set I = {N1 , . . . , N N } of channels, block length l ∈ N, and small > 0 we will be able to find one single recovery map Rl and a unitary encoder W l such that for each i ∈ {1, . . . , N }, Fe (πFl , Rl ◦ Ql,i ◦ Ni⊗l ◦ W l ) ≥ 1 − , where Ql,i (·) := ql,i (·)ql,i with suitable projections ql,i acting on K⊗l . Thus we will effectively end up with the recovery maps Rli := Rl ◦ Ql,i . Consequently, it turns out that the decoder is informed. Lemma 3 below shows how to get rid of the maps Ql,i ensuring the existence of a universal recovery map for the whole set I while decreasing the entanglement fidelity only slightly. Lemma 3. Let ρ ∈ S(H) for some Hilbert space H. Let, for some other Hilbert space K, A ∈ C(H, K), D ∈ C(K, H), q ∈ B(K) be an orthogonal projection. 1. Denoting by Q⊥ the completely positive map induced by q ⊥ := 1K − q we have Fe (ρ, D ◦ A) ≥ Fe (ρ, D ◦ Q ◦ A)(1 − 2Fe (ρ, D ◦ Q⊥ ◦ A)).
(18)
2. If for some > 0 the relation Fe (ρ, D ◦ Q ◦ A) ≥ 1 − holds, then Fe (ρ, D ◦ Q⊥ ◦ A) ≤ , and (18) implies Fe (ρ, D ◦ A) ≥ (1 − )(1 − 2) ≥ 1 − 3.
(19)
Universal Quantum Channel Coding
71
3. If for some > 0 merely the relation tr{qA(ρ)} ≥ 1 − holds then we can conclude that Fe (ρ, D ◦ A) ≥ Fe (ρ, D ◦ Q ◦ A) − 2.
(20)
The following Lemma 4 contains two inequalities one of which will be needed in the proof of Lemma 3. Lemma 4. Let D ∈ C(K, H) and x1 ⊥ x2 , z 1 ⊥ z 2 be state vectors, x1 , x2 ∈ K, z 1 , z 2 ∈ H. Then | z 1 , D(|x1 x2 |)z 1 | ≤ | z 1 , D(|x1 x1 |)z 1 | · | z 1 , D(|x2 x2 |)z 1 | ≤ 1, (21) and | z 1 , D(|x1 x2 |)z 2 | ≤
| z 1 , D(|x1 x1 |)z 1 | · | z 2 , D(|x2 x2 |)z 2 | ≤ 1.
(22)
We will utilize only (21) in the proof of Lemma 3. But the inequality (22) might prove useful in other contexts so that we state it here for completeness. Proof of Lemma 4. Let dim H = h, dim K = κ. Extend {x1 , x2 } to an orthonormal basis {x1 , x2 , . . . , xκ } of K and {z 1 , z 2 } to an orthonormal basis {z 1 , z 2 , . . . , z h } on H. Since x1 ⊥ x2 and z 1 ⊥ z 2 , this can always be done. By the theorem of Choi [6], a linear map from B(H) to B(K) is completely positive if and only if its Choi matrix is positive. h ij Write D(|xi x j |) = k,l=1 Dkl |z k zl |. Then the Choi matrix of D is, with respect to the bases {x1 , . . . , xk } and {z 1 , . . . , z h }, written as CHOI(D) =
κ
|xi x j | ⊗
i, j=1
h
ij
Dkl |z k zl |.
k,l=1
If CHOI(D) is positive, then all principal minors of CHOI(D) are positive (cf. Corollary 7.1.5 in [17]) and thus ij ii | · |D j j | |Dkl | ≤ |Dkk ll for every suitable choice of i, j, k, l. Thus 12 | z 1 |D(|x1 x2 |)z 2 | = |D12 | 11 | · |D 22 | ≤ |D11 22 = | z 1 , D(|x1 x1 |)z 1 | · | z 2 , D(|x2 x2 |)z 2 |,
and similarly | z 1 , D(|x1 x2 |)z 1 | ≤
| z 1 , D(|x1 x1 |)z 1 | · | z 1 , D(|x2 x2 |)z 1 |.
The fact that D is trace preserving gives us the estimate z i , D(|x j x j |)z i ≤ 1 (i, j suitably chosen) and we are done.
72
I. Bjelakovi´c, H. Boche, J. Nötzel
Proof of Lemma 3. Let dim H = h, dim K = κ, |ψ ψ| ∈ Ha ⊗ H be a purification of ρ (w.l.o.g. Ha = H). Set D˜ := idHa ⊗ D, A˜ := idHa ⊗ A, q˜ := 1Ha ⊗ q and, as usual, q˜ ⊥ the orthocomplement of q˜ within Ha ⊗ K. Obviously, ˜ Fe (ρ, D ◦ A) = ψ, D˜ ◦ A(|ψ ψ|)ψ ˜ ˜ q˜ + q˜ ⊥ ]A(|ψ ψ|)[ q˜ + q˜ ⊥ ])ψ = ψ, D([ ˜ ˜ q˜ A(|ψ ψ|) ˜ ˜ q˜ ⊥ A(|ψ ψ|) q˜ ⊥ )ψ = ψ, D( q)ψ ˜ + ψ, D( ⊥ ⊥ ˜ ˜ q˜ A(|ψ ψ|) ˜ ˜ q˜ A(|ψ ψ|) q)ψ ˜ + ψ, D( q˜ )ψ + ψ, D( ˜ q˜ A(|ψ ψ|) ˜ ˜ q˜ A(|ψ ψ|) ˜ ≥ ψ, D( q)ψ ˜ + 2{ ψ, D( q˜ ⊥ )ψ } ˜ q˜ A(|ψ ψ|) ˜ ˜ q˜ A(|ψ ψ|) ˜ ≥ ψ, D( q)ψ ˜ − 2| ψ, D( q˜ ⊥ )ψ | ˜ q˜ A(|ψ ψ|) ˜ q˜ ⊥ )ψ |. = Fe (ρ, D ◦ Q ◦ A) − 2| ψ, D(
(23)
We establish a lower bound on the second term on the RHS of (23). Let ˜ A(|ψ ψ|) =
κ·h
λi |ai ai |,
i=1
where {a1 , . . . , aκ·h } are assumed to form an orthonormal basis. Now every ai can be written as ai = αi xi + βi yi where xi ∈ supp(q) ˜ and yi ∈ supp(q˜ ⊥ ), i ∈ {1, ..., κ · h}, ˜ then are state vectors and αi , βi ∈ C. Define σ := A(|ψ ψ|), σ =
κ·h
λ j (|α j |2 |x j x j | + α j β ∗j |x j y j | + β j α ∗j |y j x j | + |β j |2 |y j y j |).
(24)
j=1
˜ q˜ A(|ψ ψ|) ˜ Set X := | ψ, D( q˜ ⊥ )ψ |. Then ˜ qσ X = | ψ, D( ˜ q ⊥ )ψ | a
=|
κ·h
˜ q|a λi ψ, D( ˜ i ai |q˜ ⊥ )ψ |
i=1
=|
κ·h
˜ i yi |)ψ | λi αi βi∗ ψ, D(|x
i=1
≤
κ·h
˜ i yi |)ψ | |λi αi βi∗ | · | ψ, D(|x
i=1
κ·h b ˜ ˜ i yi |)ψ β ∗ | ≤ | λi | ψ, D(|xi xi |)ψ αi λi ψ, D(|y i i=1 c
≤
κ·h i=1
˜ i xi |)ψ λi |αi |2 ψ, D(|x
κ·h
˜ j y j |)ψ . λ j |β j |2 ψ, D(|y
(25)
j=1
˜ Here, a follows from using the convex decomposition of A(|ψ ψ|), b from utilizing inequality (21) from Lemma 4 and c is an application of the Cauchy-Schwarz inequality.
Universal Quantum Channel Coding
73
Now, employing the representation (24) it is easily seen that ˜ q˜ A(|ψ ψ|) ˜ Fe (ρ, D ◦ Q ◦ A) = ψ, D( q)ψ ˜ =
κ·h
˜ i , xi |)ψ λi |αi |2 ψ, D(|x
(26)
i=1
and similarly Fe (ρ, D ◦ Q ◦ A) =
κ·h
˜ j y j |)ψ . λ j |β j |2 ψ, D(|y
(27)
j=1
The inequalities (27), (26), (25), and (23) yield Fe (ρ, D ◦ A) ≥ Fe (ρ, D ◦ Q ◦ A) − 2Fe (ρ, D ◦ Q ◦ A)Fe (ρ, D ◦ Q⊥ ◦ A) = Fe (ρ, D ◦ Q ◦ A)(1 − 2Fe (ρ, D ◦ Q⊥ ◦ A))
(28)
which establishes (18). Let us turn now to the other assertions stated in the lemma. Let tr{qA(ρ)} ≥ 1 − . This implies tr(q ⊥ A(ρ)) ≤ . A direct calculation yields tr(q˜ ⊥ σ ) = tr Ha (tr K ((1Ha ⊗ q ⊥ )idHa ⊗ A(|ψ ψ|))) = tr K (q ⊥ A(tr Ha (|ψ ψ|))) = tr K (q ⊥ A(ρ)) ≤ . Using (24), we get the useful inequality ≥ tr(q˜ ⊥ σ ) =
κ·h
λi |βi |2 tr(q˜ ⊥ |yi yi |)
i=1
=
κ·h
λi |βi |2 .
(29)
i=1
Using Lemma 4 and (29) we get X ≤
κ·h i=1
λi |αi |2
κ·h
λ j |β j |2
j=1
≤ , thus by Eq. (23) we have Fe (ρ, D ◦ A) ≥ Fe (ρ, D ◦ Q ◦ A) − 2. In case that Fe (ρ, D ◦ Q ◦ A) ≥ 1 − , we note that the linear maps Q and Q⊥ are elements of C ↓ (K, K) whilst Q + Q⊥ ∈ C(K, K) and since Fe is affine in the operation Fe (ρ, D ◦ Q ◦ A) + Fe (ρ, D ◦ Q⊥ ◦ A) = Fe (ρ, D ◦ (Q + Q⊥ ) ◦ A) ≤ 1
74
I. Bjelakovi´c, H. Boche, J. Nötzel
has to hold. This in turn implies Fe (ρ, D ◦ Q⊥ ◦ A) ≤ . Using this, our assumption that Fe (ρ, D ◦ Q ◦ A) ≥ 1 − , and (28) we obtain that Fe (ρ, D ◦ A) ≥ Fe (ρ, D ◦ Q ◦ A)(1 − 2Fe (ρ, D ◦ Q⊥ ◦ A)) ≥ (1 − )(1 − 2) ≥ 1 − 3, which is the claim we made in (19).
4. Direct Part of the Coding Theorem for Finitely Many Channels 4.1. Typical Projections and Kraus Operators. In this subsection we recall briefly the well-known properties of frequency typical projections and reduced operations. A more detailed description can be found in [4] and references therein. Lemma 5. There is a real number c > 0 such that for every Hilbert space H there exist functions h : N → R+ , ϕ : (0, 1/2) → R+ with liml→∞ h(l) = 0 and limδ→0 ϕ(δ) = 0 such that for any ρ ∈ S(H), δ ∈ (0, 1/2), l ∈ N there is an orthogonal projection qδ,l ∈ B(H)⊗l called frequency-typical projection that satisfies 1. tr(ρ ⊗l qδ,l ) ≥ 1 − 2−l(cδ −h(l)) , 2. qδ,l ρ ⊗l qδ,l ≤ 2−l(S(ρ)−ϕ(δ)) qδ,l . 2
The inequality 2 implies ||qδ,l ρ ⊗l qδ,l ||22 ≤ 2−l(S(ρ)−ϕ(δ)) . Moreover, setting d := dim H, ϕ and h are given by h(l) =
d δ log(l + 1) ∀l ∈ N, ϕ(δ) = −δ log ∀δ ∈ (0, 1/2). l d
Lemma 6. Let H, K be finite dimensional Hilbert spaces. There are functions γ : (0, 1/2) → R+ , h : N → R+ satisfying limδ→0 γ (δ) = 0 and h (l) 0 such that for each N ∈ C(H, K), δ ∈ (0, 1/2), l ∈ N and maximally mixed state πG on some subspace G ⊂ H there is an operation Nδ,l ∈ C ↓ (H⊗l , K⊗l ) called the reduced operation with respect to N and πG that satisfies
2
1. tr(Nδ,l (πG⊗l )) ≥ 1 − 2−l(c δ −h (l)) , with a universal positive constant c > 0,
2. Nδ,l has a Kraus representation with at most n δ,l ≤ 2l(Se (πG ,N )+γ (δ)+h (l)) Kraus operators. 3. For every state ρ ∈ S(H⊗l ) and every two channels I ∈ C ↓ (H⊗l , H⊗l ) and L ∈ C ↓ (K⊗l , H⊗l ) the inequality Fe (ρ, L ◦ Nδ,l ◦ I) ≤ Fe (ρ, L ◦ N ⊗l ◦ I) is fulfilled. Setting d := dim H and κ := dim K, the function h : N → R+ is given by h (l) = d·κ δ l log(l + 1) ∀l ∈ N and γ by γ (δ) = −δ log d·κ , ∀δ ∈ (0, 1/2).
Universal Quantum Channel Coding
75
4.2. The Case of Uninformed Users. Let us consider a compound channel given by a finite set I := {N1 , . . . , N N } ⊂ C(H, K) and a subspace G ⊂ H. For every l ∈ N, we choose a subspace El ⊂ G ⊗l . As usual, πEl and πG denote the maximally mixed states on El , respectively G while kl := dim El gives the dimension of El . For j ∈ {1, . . . , N }, δ ∈ (0, 1/2), l ∈ N and states N j (πG ), let q j,δ,l ∈ B(K)⊗l be the frequency-typical projection of N j (πG ) and N j,δ,l be the reduced operation associated with N j and πG as defined in Subsect. 4.1. These quantities enable us to define a new set of channels that is more adapted to our problem than the original one. We set for an arbitrary unitary operation u l ∈ B(H⊗l ), l l Nˆ j,u l ,δ := Q j,δ,l ◦ N j,δ,l ◦ U
and, accordingly, N 1 ˆl Nˆ ul l ,δ := N j,ul ,δ . N j=1
We will show the existence of good codes for the reduced channels Q j,δ,l ◦ N j,δ,l in the limit of large l ∈ N. An application of Lemma 3 and Lemma 6 will then show that these codes are also good for the original compound channel. Let U l be a random variable taking values in U(G ⊗l ) which is distributed according to the Haar measure. Application of Theorem 3 yields EFc,e (πEl , Nˆ Ul l ,δ ) ≥ tr(Nˆ δl (πG⊗l )) − 2
N
l kl n j,δ,l ||Nˆ j,δ (πG⊗l )||2 ,
(30)
j=1
where n j,δ,l stands for the number of Kraus operators of the reduced operation N j,δ,l ( j ∈ {1, . . . , N }) and l Nˆ j,δ := Q j,δ,l ◦ N j,δ,l , N 1 ˆl N j,δ . Nˆ δl := N j=1
Notice that Q j,δ,l ◦ N j,δ,l trivially has a Kraus representation containing exactly n j,δ,l elements. We will use inequality (30) in the proof of the following theorem. Theorem 5 (Direct Part: Uninformed Users and |I| < ∞). Let I = {N1 , . . . , N N } ⊂ C(H, K) be a compound channel and πG the maximally mixed state associated to a subspace G ⊂ H. Then Q(I) ≥ min Ic (πG , Ni ). Ni ∈I
Proof We show that for every > 0 the number minNi ∈I Ic (πG , Ni )− is an achievable rate for I. 1) If minNi ∈I Ic (πG , Ni ) − ≤ 0, there is nothing to prove. 2) Let minNi ∈I Ic (πG , Ni ) − > 0.
76
I. Bjelakovi´c, H. Boche, J. Nötzel
Choose δ ∈ (0, 1/2) and l0 ∈ N satisfying γ (δ) + ϕ(δ) + h (l0 ) ≤ /2 with functions γ , ϕ, h from Lemma 5 and 6. Now choose for every l ∈ N a subspace El ⊂ G ⊗l such that dim El =: kl = 2l(minNi ∈I Ic (πG ,Ni )−) . By S(πG ) ≥ Ic (πG , N j ) (see [1]), this is always possible. Obviously, min Ic (πG , Ni ) − − o(l 0 ) ≤
Ni ∈I
1 log kl ≤ min Ic (πG , Ni ) − . l Ni ∈I
We will now give lower bounds on the terms in (30), thereby making use of Lemma 5 and Lemma 6: 2
2
tr(Nˆ l (π ⊗l )) ≥ 1 − 2−l(cδ −h(l)) − 2−l(c δ −h (l)) . (31) δ
G
A more detailed calculation can be found in [4] or [20]. Further, and additionally using the inequality ||A + B||22 ≥ ||A||22 + ||B||22 valid for non-negative operators A, B ∈ B(K⊗l ) (see [20]), we get the inequality l ||Nˆ j,δ (πG⊗l )||22 ≤ 2−l(S(N j (πG ))−ϕ(δ)) .
(32)
From (30), (31), (32) and our specific choice of kl it follows that 2
2
EFc,e (πEl , Nˆ Ul l ,δ ) ≥ 1 − 2−l(cδ −h(l)) − 2−l(c δ −h (l)) N 1
−2 2l( l log kl +γ (δ)+ϕ(δ)+h (l)−Ic (πG ,N j )
j=1
2
≥ 1 − 2−l(cδ −h(l)) − 2−l(c δ −h (l))
−2N 2−l(−γ (δ)−ϕ(δ)−h (l)) . 2
Since − γ (δ) − ϕ(δ) − h (l) ≥ ε/2 for every l ≥ l0 , this shows the existence of at least one sequence of (l, kl )−codes for I with uninformed users and lim inf l→∞
1 log kl = min Ic (πG , Ni ) − , l Ni ∈I
as well as (using that entanglement fidelity is affine in the channel), for every l ∈ N, min
j∈{1,...,N }
1 l Fe (πFl , Rl ◦ Nˆ j,δ ◦ W l ) ≥ 1 − N l , 3
where wl ∈ U(G ⊗l ) ∀l ∈ N and l = 3 · (2−l(cδ
2 −h(l))
2 −h (l))
+ 2−l(c δ
+ 2N 2−l(−γ (δ)−ϕ(δ)−h (l)) ).
(33)
(34)
Note that liml→∞ l = 0 exponentially fast, as can be seen from our choice of δ and l0 . For every j ∈ {1, . . . , N } and l ∈ N we thus have, by property 3 of Lemma 6, l construction of Nˆ j,w j ,δ , and Eq. (33), Fe (πFl , Rl ◦ Q j,δ,l ◦ N j⊗l ◦ W l ) ≥ Fe (πFl , Rl ◦ Q j,δ,l ◦ N j,δ,l ◦ W l ) l = Fe (πFl , Rl ◦ Nˆ j,w j ,δ )
1 ≥ 1 − N l . 3
Universal Quantum Channel Coding
77
By the first two parts of Lemma 3, this immediately implies min Fe (πFl , Rl ◦ N j⊗l ◦ W l ) ≥ 1 − N l ∀l ∈ N.
N j ∈I
(35)
Since > 0 was arbitrary, we have shown that minNi ∈I Ic (πG , Ni ) is an achievable rate. 4.3. The Informed Encoder. In this subsection we shall prove the following theorem: Theorem 6 (Direct Part: Informed Encoder and |I| < ∞). For every finite compound channel I = {N1 , . . . , N N } ⊂ C(H, K) and any set {πG1 , . . . , πG N } of maximally mixed states on subspaces {G1 , . . . , G N } with Gi ⊂ H for all i ∈ {1, . . . , N } we have Q I E (I) ≥ min Ic (πGi , Ni ). Ni ∈I
Proof Let a compound channel be given by a finite set I := {N1 , . . . , N N } ⊂ C(H, K) and let G1 , . . . , G N be arbitrary subspaces of H. We will prove that for every > 0 the value R() := min Ic (πGi , Ni ) − 1≤i≤N
is achievable. If R() ≤ 0, there is nothing to prove. Hence we assume R() > 0. For every l ∈ N and all i ∈ {1, . . . , N } we choose the following. First, a subspace El ⊂ H⊗l of dimension kl := dim El that satisfies kl ≤ dim Gi⊗l . Second, a set {v1l , . . . , vlN } of unitary operators with the property vil El ⊂ Gi⊗l . Again, the maximally mixed states associated to the above mentioned subspaces are denoted by πEl on El and πGi on Gi . For j ∈ {1, . . . , N }, δ ∈ (0, 1/2), l ∈ N and states N j (πG j ) let q j,δ,l ∈ B(K)⊗l be the frequency-typical projection of N j (πG j ) and N j,δ,l be the reduced operation associated with N j and πG j as considered in Sect. 4.1. Let, for the moment, l ∈ N be fixed. We define a new set of channels that is more adapted to our problem than the original one. We set, for an arbitrary set {u l1 , . . . , u lN } of unitary operators on H⊗l : l N˜ j,δ := Q j,δ,l ◦ N j,δ,l , l l l ˜l Nˆ j,u l ,δ := N j,δ ◦ U j ◦ V j , j
and, accordingly, Nˆ ul l ,...,ul 1
N ,δ
:=
N 1 ˆl N j,ul ,δ . N j j=1
We will first show the existence of good unitary encodings and recovery operation for l ,...,N ˜ l }. Like in the previous subsection, application of Lemma 3 will enable {N˜ 1,δ N ,δ us to show the existence of reliable encodings and recovery operation for the original compound channel I.
78
I. Bjelakovi´c, H. Boche, J. Nötzel
Let U1l , . . . , U Nl be independent random variables such that each Uil takes on values in U(Gi⊗l ) and is distributed according to the Haar measure on U(Gi⊗l ) (i ∈ {1, . . . , N }). By Theorem 4 we get the lower bound EFc,e (πEl , Nˆ Ul l ,...,U l ,δ ) ≥ 1
N
N 1 l l [ tr(N˜ j,δ (πG ⊗l )) − 2 kl n j,δ,l ||N˜ j,δ (πG ⊗l )||2 ], j j N
(36)
j=1
where n j,δ,l denotes the number of Kraus operators in the operations N˜ j,δ,l ( j ∈ {1, . . . , N }). By Lemmas 5, 6 for every j ∈ {1, . . . , N } the corresponding term in the above sum can be bounded from below through 1 1 2
2
l (πG ⊗l )) ≥ (1 − 2−l(cδ −h(l)) − 2−l(c δ −h (l)) ) tr(N˜ j,δ j N N and
l(− min1≤ j≤N Ic (πG j ,N j )+γ (δ)+ϕ(δ)+h (l)) l ˜ . −2 kl n j,δ,l ||N j,δ (πG ⊗l )||2 ≥ −2 kl · 2
j
Set kl :=
2l R() .
Obviously, for any j ∈ {1, . . . , N }, l(− min1≤ j≤N Ic (πG j ,N j ))
kl · 2
≤ 2−l .
This implies
2
EFc,e (πEl , Nˆ Ul l ,...,U l ,δ ) ≥ 1 − 2−l(cδ −h(l)) − 2−l(c δ −h (l)) 1 N
−2N 2l(−+γ (δ)+ϕ(δ)+h (l)) . 2
Now choosing both the approximation parameter δ and an integer l0 ∈ N such that − + γ (δ) + ϕ(δ) + h (l) < − 21 holds for every l ≥ l0 and setting 2
2
l := 2−l(cδ −h(l)) + 2−l(c δ −h (l)) + 2N 2l(−+γ (δ)+ϕ(δ)+h (l)) we see that EFc,e (πEl , Nˆ Ul l ,...,U l ,δ ) ≥ 1 − l , 1
N
where again l 0 and our choice of δ and l0 again shows that the speed of convergence is exponentially fast. Thus, there exist unitary operators w1l , . . . , wlN ⊂ U(H⊗l ) and a recovery operation Rl such that, passing to the individual channels, we have for every j ∈ {1, . . . , N }, Fe (πEl , Rl ◦ Q j,δ,l ◦ N j,δ,l ◦ W lj ) ≥ 1 − N l . By property 3 of Lemma 6 and Lemma 3, we immediately see that Fe (πEl , Rl ◦ N j⊗l ◦ W lj ) ≥ 1 − 3N l ∀ j ∈ {1, . . . , N } is valid as well. We finally get the desired result: For every set {πG1 , . . . , πG N } of maximally mixed states on subspaces G1 , . . . , G N ⊂ H and every > 0 there exists a sequence of (l, kl ) codes for I with informed encoder with the properties 1. lim inf l→∞ 1l log kl = minN j ∈I Ic (πG j , N j ) − , 2. minN j ∈I Fe (πEl , Rl ◦ N j⊗l ◦ W lj ) ≥ 1 − 3N l . Since > 0 was arbitrary and l 0, we are done.
Universal Quantum Channel Coding
79
5. Finite Approximations in the Set of Quantum Channels Our goal in this section is to discretize a given set of channels I ∈ C(H, K) in such a way that the results derived so far for finite sets can be employed to derive general versions of coding theorems for compound channels. The first concept we will need is that of a τ -net in the set C(H, K) and we will give an upper bound on the cardinality of the best τ -net in that set. Best τ -nets characterize the degree of compactness of C(H, K). N with the property that for each N ∈ C(H, K) A τ -net in C(H, K) is a finite set {Ni }i=1 there is at least one i ∈ {1, . . . , N } with ||N −Ni ||♦ < τ . Existence of τ -nets in C(H, K) is guaranteed by the compactness of C(H, K). The next lemma contains a crude upper bound on the cardinality of minimal τ -nets.
2
N in C(H, K) with N ≤ ( 3 )2(d·d ) , Lemma 7 For any τ ∈ (0, 1] there is a τ −net {Ni }i=1 τ
where d = dim H and d = dim K.
Proof The assertion of the lemma follows from the standard volume argument (cf. Lemma 2.6 in [24]). The details can be found in our previous paper [4]. N with Let I ⊆ C(H, K) be an arbitrary set. Starting from a τ/2−net N := {Ni }i=1 6 2(d·d )2
N ≤ (τ ) as in Lemma 7 we can build a τ/2−net Iτ that is adapted to the set I given by I τ := Ni ∈ N : ∃N ∈ I with ||N − Ni ||♦ < τ/2 , (37)
i.e. we select only those members of the τ/2-net that are contained in the τ/2neighborhood of I. Let T ∈ C(H, K) be the useless channel given by T (ρ) := dim1 K 1K , ρ ∈ S(H), and consider τ τ (38) Iτ := (1 − )N + T : N ∈ I τ , 2 2 where I τ is defined in (37). For I ⊆ C(H, K) we set Ic (ρ, I) := inf Ic (ρ, N ), N ∈I
for ρ ∈ S(H). We list a few more or less obvious results in the following lemma that will be needed in the following. Lemma 8 Let I ⊆ C(H, K). For each positive τ ≤ defined in (38).
1 e
let Iτ be the finite set of channels
2
1. |Iτ | ≤ ( τ6 )2(d·d ) with d = dim H and d = dim K. 2. For N ∈ I there is Ni ∈ Iτ with ||N ⊗l − Ni⊗l ||♦ < lτ.
(39)
Consequently, for N , Ni , and any CPTP maps P : B(F) → B(H)⊗l and R : B(K)⊗l → B(F ) the relation |Fe (ρ, R ◦ N ⊗l ◦ P) − Fe (ρ, R ◦ Ni⊗l ◦ P)| < lτ holds for all ρ ∈ S(H⊗l ) and l ∈ N.
(40)
80
I. Bjelakovi´c, H. Boche, J. Nötzel
3. For all ρ ∈ S(H) we have d |Ic (ρ, I) − Ic (ρ, Iτ )| ≤ τ + 3τ log . τ
(41)
Proof The proofs of the assertions claimed here are either identical to those given in [4] or can be obtained by trivial modifications thereof. 6. Direct Parts of the Coding Theorems for General Quantum Compound Channels 6.1. The Case of Informed Decoder and Uninformed Users. The main step towards the direct part of the coding theorem for quantum compound channels with uninformed users is the following theorem. Lemma 9 Let I ∈ C(H, K) be an arbitrary compound channel and let πG be the maximally mixed state associated with a subspace G ⊂ H. Then Q(I) ≥ inf Ic (πG , N ). N ∈I
Proof We consider two subspaces El , G ⊗l of H⊗l with El ⊂ G ⊗l ⊂ H⊗l . Let kl := dim El and we denote as before the associated maximally mixed states on El and G by πEl and πG . If inf N ∈I Ic (πG , N ) ≤ 0 there is nothing to prove. Therefore we will suppose in the following that inf Ic (πG , N ) > 0
N ∈I
holds. We will show that for each ε ∈ (0, inf N ∈I Ic (πG , N )) the number inf Ic (πG , N ) − ε
N ∈I
is an achievable rate. For each l ∈ N let us choose some τl > 0 with τl ≤ 1e , liml→∞ lτl = 0, and such that Nτl grows sub-exponentially with l. E.g. we may choose τl := min{1/e, 1/l 2 }. We consider, for each l ∈ N, the finite set of channels Iτl := {N1 , . . . , N Nτl } associated to I given in (38) with the properties listed in Lemma 8. We can conclude from the proof of Theorem 5 that for each l ∈ N there is a subspace Fl ⊂ G ⊗l of dimension ε
kl = 2l(mini∈{1,...,Nτ } Ic (πG ,Ni )− 2 ) ,
(42)
a recovery operation R, and a unitary encoder W l such that min
i∈{1,...,Nτl }
Fe (πFl , R ◦ Ni⊗l ◦ W l ) ≥ 1 − Nτl l ,
(43)
where l is defined in (34) (with the approximation parameter ε replaced by ε/2), and we have chosen l, l0 ∈ N with l ≥ l0 large enough and δ > 0 small enough to ensure that both ε min Ic (πG , Ni ) − > 0, i∈{1,...,Nτl } 2
Universal Quantum Channel Coding
81
and
ε − γ (δ) − ϕ(δ) − h (l0 ) > ε/4 > 0. 2 By our construction of Iτl we can find to each N ∈ I at least one Ni ∈ Iτl with |Fe (πFl , R ◦ Ni⊗l ◦ W l ) − Fe (πFl , R ◦ N ⊗l ◦ W l )| ≤ l · τl ,
(44)
according to Lemma 8. Moreover, by the last claim of Lemma 8 we obtain the following estimate on the dimension kl of the subspace Fl : l(inf N ∈I Ic (πG ,N )− 2ε −τl −2τl log
kl ≥ 2
d τl
.
(45)
The inequalities (43) and (44) show that inf Fe (πFl , R ◦ N ⊗l ◦ W l ) ≥ 1 − Nτl l − lτl ,
N ∈I
which in turn with (45) shows that inf N ∈I Ic (πG , N ) is an achievable rate.
In order to pass from the maximally mixed state πG to an arbitrary one we have to employ the compound generalization of Bennett, Shor, Smolin, and Thapliyal Lemma (BSST Lemma for short) from [2] and [16]. For the proof of this generalized BSST Lemma we refer to [4]. Lemma 10 (Compound BSST Lemma). Let I ⊂ C(H, K) be an arbitrary set of channels. For any ρ ∈ S(H) let qδ,l ∈ B(H⊗l ) be the frequency-typical projection of ρ and set qδ,l ∈ S(H⊗l ). πδ,l := tr(qδ,l ) Then there is a positive sequence (δl )l∈N satisfying liml→∞ δl = 0 with 1 inf Ic (πδl ,l , N ⊗l ) = inf Ic (ρ, N ). l→∞ l N ∈I N ∈I With these preparations it is easy now to finish the proof of the direct part of the coding theorem for the quantum compound channel with uninformed users. First notice that for each k ∈ N, lim
Q(I⊗k ) = k Q(I) S(H⊗m ) let q
(46)
B(H⊗ml ) be the frequency-typical projection
holds. For any fixed ρ ∈ δ,l ∈ q of ρ and set πδ,l = tr(qδ,lδ,l ) . Lemma 9 implies that for any δ ∈ (0, 1/2) we have Q(I⊗ml ) ≥ Ic (πδ,l , I⊗ml ),
(47)
for all m, l ∈ N. Utilizing (46), (47) and Lemma 10 we arrive at 1 1 lim Q(I⊗ml ) m l→∞ l 1 1 lim inf Ic (πδl ,l , (N ⊗m )⊗l ) ≥ m l→∞ l N ∈I 1 = Ic (ρ, I⊗m ). (48) m From (48) and since Q I D (I) ≥ Q(I) trivially holds we get without further ado the direct part of the coding theorem. Q(I) =
82
I. Bjelakovi´c, H. Boche, J. Nötzel
Theorem 7 (Direct Part: Informed Decoder and Uninformed Users). Let I ⊂ C(H, K) be an arbitrary set. Then 1 max inf Ic (ρ, N ⊗l ). l→∞ l ρ∈S (H⊗l ) N ∈I
Q I D (I) ≥ Q(I) ≥ lim
(49)
Remark 2. It is quite easy to see that the limit in (49) exists. Indeed it holds that max
inf Ic (ρ, N ⊗l+k ) ≥
ρ∈S (H⊗l+k ) N ∈I
max
inf Ic (ρ, N ⊗l )
ρ∈S (H⊗l ) N ∈I
+
max
inf Ic (ρ, N ⊗k ),
ρ∈S (H⊗k ) N ∈I
which implies the existence of the limit via standard arguments. 6.2. The Informed Encoder. The main result of this section will rely on an appropriate variant of the BSST Lemma. To this end we first recall Holevo’s version of that result. For δ > 0, l ∈ N, and ρ ∈ S(H), let qδ,l ∈ B(H⊗l ) denote the frequency typical projection of ρ ⊗l . Set πδ,l = πδ,l (ρ) :=
qδ,l . tr(qδ,l )
(50)
Moreover, let λmin (ρ) := min{λ ∈ σ (ρ) : λ > 0}, where σ (ρ) stands for the spectrum of the density operator ρ. 1 Lemma 11 (BSST Lemma [2,16]). For any δ ∈ (0, 2 dim H ), any N ∈ C(H, K), and every ρ ∈ S(H) with associated state πδ,l = πδ,l (ρ) ∈ S(H⊗l ) we have 1 S(N ⊗l (πδ,l )) − S(N (ρ)) ≤ θl (δ, λmin (ρ), λmin (N (ρ))), (51) l
where θl (δ, λmin (ρ), λmin (N (ρ))) =
dim H log(l + 1) − dim H · δ log δ l − dim H · δ · (log λmin (ρ) + log λmin (N (ρ))).
(52)
Before we present our extended version of the BSST Lemma we introduce some notation. For t ∈ (0, 1e ) and any set I ⊂ C(H, K) let us define I(t) := {N (t) = (1 − t)N + tTK : N ∈ I} = (1 − t)I + tTK ,
(53)
tr(x) where T ∈ C(H, K) is given by TK (x) := dim K 1K . On the other hand, to each N ∈ I ⊂ C(H, K) we can associate a complementary channel Nc ∈ C(H, He ) where we assume w.l.o.g. that He = Cdim H·dim K . Let I ⊂ C(H, He ) denote the set of channels complementary to I and set
I (t) := (I )
(t)
= {Nc(t) = (1 − t)Nc + tTHe : Nc ∈ I } = (1 − t)I + tTHe , (54)
Universal Quantum Channel Coding
83
where THe ∈ C(H, He ) is defined in a similar way as TK . Finally, for N ∈ I let ρN := arg max Ic (ρ, N ), ρ∈S (H)
and for t ∈ (0, 1e ), δ > 0, and l ∈ N define
(t) (t) πδ,l,N := πδ,l ρN ,
(55)
where we have used the notation from (50) and (t)
ρN := (1 − t)ρN +
t 1H . dim H
(56)
1 1 ), and δ ∈ (0, 2 dim Lemma 12 (Uniform BSST-Lemma). 1. Let l ∈ N, t ∈ (0, l·e H ). Then with the notation introduced in the preceding paragraph we have 1 inf Ic (π (t) , N ⊗l ) − inf max Ic (ρ, N ) ≤ ∆l (δ, t), l N ∈I δ,l,N N ∈I ρ∈S (H)
with
∆l (δ, t) = 2θl δ,
t t t t , + 2θl δ, , dim H dim K dim H dim He t lt −4t log − 2lt log , dim K · dim He dim K · He where θl δ, dimt H , dimt K and θl δ, dimt H , dimt He are from Lemma 11. 2. Consequently, choosing suitable positive sequences (δl )l∈N , (tl )l∈N with 1. liml→∞ δl = 0 = liml→∞ ltl , and 2. liml→∞ δl log tl = 0, we see that for νl := ∆l (δl , tl ), 1 inf Ic (π (tl ) , N ⊗l ) − inf max Ic (ρ, N ) ≤ νl l N ∈I δl ,l,N N ∈I ρ∈S (H)
(57)
holds with liml→∞ νl = 0. Proof. Our proof strategy is to reduce the claim to the BSST Lemma 11. Let t > 0 be 1 small enough to ensure that l · t ∈ (0, 1e ) and let δ ∈ (0, 2 dim H ) be given. From (53) and (54) we obtain that λmin (N (t) (ρ)) ≥
t , dim K
λmin (Nc(t) (ρ)) ≥
t dim He
∀ ρ ∈ S(H),
(58)
and (56) yields that (t) )≥ λmin (ρN
t dim H
for all N ∈ I. The bounds (58) and (59) along with Lemma 11 show that 1 S((N (t) )⊗l (π (t) )) − S(N (t) (ρ (t) )) ≤ θl δ, t , t , l N δ,l,N dim H dim K
(59)
(60)
84
and
I. Bjelakovi´c, H. Boche, J. Nötzel
1 t S((N (t) )⊗l (π (t) )) − S(N (t) (ρ (t) )) ≤ θl δ, t , . c c l N δ,l,N dim H dim He
(61)
On the other hand, by definition we have ||N (t) − N ||♦ ≤ t,
||(N (t) )⊗l − N ⊗l ||♦ ≤ l · t,
(62)
||Nc(t) − Nc ||♦ ≤ t,
||(Nc(t) )⊗l − Nc⊗l ||♦ ≤ l · t,
(63)
and similarly
for all N ∈ I. Since l · t ∈ (0, 1e ) we obtain from this by Fannes inequality t , dim K t (t) (t) , |S(Nc(t) (ρN )) − S(Nc (ρN ))| ≤ −t log dim He (t)
(t)
|S(N (t) (ρN )) − S(N (ρN ))| ≤ −t log
and
1 S((N (t) )⊗l (π (t) )) − 1 S(N ⊗l (π (t) )) ≤ −l · t log l · t , l N δ,l,N δ,l, l dim K
as well as 1 S((N (t) )⊗l (π (t) )) − 1 S(N ⊗l (π (t) )) ≤ −l · t log l · t , c c l N δ,l,N δ,l, l dim He
(64) (65)
(66)
(67)
for all N ∈ I. Since (t)
(t)
(t)
Ic (ρN , N ) = S(N (ρN )) − S(Nc (ρN )) and (t) (t) (t) ⊗l ⊗l ⊗l Ic (πδ,l, N , N ) = S(N (πδ,l,N )) − S(Nc (πδ,l,N )),
the inequalities (60),(61), (64), (65), (66), (67) and triangle inequality show that uniformly in N ∈ I we have 1 t t Ic (π (t) , N ⊗l )− Ic (ρ (t) , N ) ≤ θl δ, t , t δ, +θ , l l N δ,l,N dim H dim K dim H dim He t l ·t − t log −l · t log . dim K · dim He dim K · He (68) Now, by (56) we have (t)
||ρN − ρN ||1 ≤ t, which implies (t)
||N (ρN ) − N (ρN )||1 ≤ t,
(t)
||Nc (ρN ) − Nc (ρN )||1 ≤ t,
Universal Quantum Channel Coding
85
since the trace distance of two states can only decrease after applying a trace preserving completely positive map to both states. Thus Fannes inequality leads us to the conclusion that t (t) . Ic (ρN , N ) − Ic (ρN , N ) ≤ −t log dim K · dim He This and (68) shows that uniformly in N ∈ I, 1 t t Ic (π (t) , N ⊗l )− Ic (ρN , N ) ≤ θl δ, t , t δ, +θ , l l δ,l,N dim H dim K dim H dim He t l ·t −2t log − l · t log dim K · dim He dim K · He ∆l (δ, t) . (69) =: 2 Finally, it is clear from the uniform estimate in (69) that 1 inf Ic (π (t) , N ⊗l ) − inf max Ic (ρ, N ) = 1 inf Ic (π (t) , N ⊗l ) l N ∈I l N ∈I δ,l,N δ,l,N N ∈I ρ∈S (H) − inf Ic (ρN , N ) N ∈I ≤ ∆l (δ, t), which concludes the proof.
(70)
Lemma 12 and Theorem 6 easily imply the following result. Lemma 13. Let I ⊂ C(H, K) be an arbitrary set of quantum channels. Then Q I E (I) ≥ inf
max Ic (ρ, N ).
N ∈I ρ∈S (H)
Proof. Take any set {πGN }N ∈I of maximally mixed states on subspaces GN ⊂ H. In a first step we will show that Q I E (I) ≥ inf Ic (πGN , N ) N ∈I
(71)
holds. Notice that we can assume w.l.o.g. that inf N ∈I Ic (πGN , N ) > 0. Denote, for every τ > 0, by Iτ a τ -net for I as given in (38) of cardinality Nτ :=
2 |Iτ | ≤ ( τ6 )2(d·d ) , where d, d are the dimensions of H, K. Starting from this set Iτ it is easy to construct a finite set I◦τ with the following properties: 1. I◦τ ⊂ I,
2 2. |I◦τ | ≤ ( τ6 )2(d·d ) , and 3. to each N ∈ I there is at least one N ∈ I◦τ with ||N − N ||♦ ≤ 2τ . Let (τl )l∈N be defined by τl := l12 and consider the sets I◦τl , l ∈ N. Take any η ∈ (0, inf N ∈I Ic (πGN , N )) and set R(η) := inf Ic (πGN , N ) − η, N ∈I
86
I. Bjelakovi´c, H. Boche, J. Nötzel
and Rl (η) := min Ic (πGN , N ) − η. N ∈I◦τl
Then for every l ∈ N, Rl (η) ≥ R(η)
(72)
I◦τl
⊂ I. since Fix some δ ∈ (0, 1/2) such that γ (δ ) + ϕ(δ ) < η/4. For every l ∈ N, choose a subspace El ⊂ H⊗l of dimension kl (η) := dim El = 2l Rl (η) . The proof of Theorem 6 then shows the existence of a recovery operation Rl and for l such that for each l ∈ N, each N ∈ I◦τl a unitary encoder WN
l ∀ N ∈ I◦τl , Fe (πEl , Rl ◦ N ⊗l ◦ WN
) ≥ 1 − 3 · Nτl · εl 3η
2
2
where εl := 2−l(cδ −h(l)) + 2−l(c δ −h (l)) + 2Nτl 2l(− 4 +h (l)) ). From Lemma 8 along ◦ with the properties of Iτl and our specific choice of (τl )l∈N it follows that there exist l (for every l ∈ N and each N ∈ I), such that unitary encodings WN
2 ∀l ∈ N, N ∈ I. l l ) = 1 and (72) implies for each η ∈ (0, inf Clearly, liml→∞ Fe (πEl , Rl ◦N ⊗l ◦WN N ∈I Ic (πGN , N )) that l ) ≥ 1 − 3 · Nτl · εl − Fe (πEl , Rl ◦ N ⊗l ◦ WN
1 1 log kl (η) = lim inf log dim El ≥ R(η). l→∞ l l→∞ l Consequently inf N ∈I Ic (πGN , N ) is achievable. We proceed by repeated application of the inequality 1 Q I E (I) ≥ Q I E (I⊗l ) (∀l ∈ N). (73) l l } From (71) and (73) we get that for each l ∈ N and every set {πN N ∈I of maximally ⊗l mixed states on subspaces of H , 1 l Q I E (I) ≥ inf Ic (πN , N ⊗l ). l N ∈I lim inf
l , namely, for every N ∈ I and l ∈ N, set We now make a specific choice of the states πN (tl ) l := π (tl ) πN δl ,l,N with πδl ,l,N taken from the second part of Lemma 12. By an application of the second part of Lemma 12 it follows 1 l inf Ic (πN Q I E (I) ≥ lim , N ⊗l ) l→∞ l N ∈I ≥ lim ( inf Ic (ρN , N ) − νl ) l→∞ N ∈I
= inf Ic (ρN , N ) N ∈I
= inf
max Ic (ρ, N ).
N ∈I ρ∈S (H)
Universal Quantum Channel Coding
87
Employing inequality (73) one more time we obtain from Lemma 13 applied to I⊗l , 1 Q I E (I⊗l ) l 1 ≥ inf max Ic (ρ, N ⊗l ). l N ∈I ρ∈S (H⊗l )
Q I E (I) ≥
Consequently we obtain the desired achievability result. Theorem 8 (Direct Part: Informed Encoder). For any I ∈ C(H, K) we have Q I E (I) ≥ lim
l→∞
1 inf max Ic (ρ, N ⊗l ). l N ∈I ρ∈S (H⊗l )
(74)
Remark 3. Note that the limit in (74) exists. Indeed, set Cl (N ) :=
max
ρ∈S (H⊗l )
Ic (ρ, N ⊗l ).
Then it is clear that Cl+k (N ) ≥ Cl (N ) + Ck (N ) and consequently inf Cl+k (N ) ≥ inf (Cl (N ) + Ck (N ))
N ∈I
N ∈I
≥ inf Cl (N ) + inf Ck (N ), N ∈I
N ∈I
which implies the existence of the limit in (74). 7. Converse Parts of the Coding Theorems for General Quantum Compound Channels In this section we prove the converse parts of the coding theorems for general quantum compound channels in the three different settings concerned with entanglement transmission that are treated in this paper. The proofs deviate from the usual approach due to our more general definitions of codes. 7.1. Converse for Informed Decoder and Uninformed Users. We first prove the converse part in the case of a finite compound channel, then use a recent result [22] that gives a more convenient estimate for the difference in coherent information of two nearby channels in order to pass on to the general case. For the converse part in the case of a finite compound channel we need the following lemma that is due to Devetak [10]: Lemma 14 (cf. [10]). For two states σ, ρ ∈ S(H1 ⊗ H2 ) where dim H1 ⊗ H2 = b with fidelity f = F(σ, ρ), 2 |∆S(ρ) − ∆S(σ )| ≤ + 4 log(b) 1 − f , e where ∆S( · ) := S(tr H1 [ · ]) − S( · ). We shall now embark on the proof of the following theorem.
88
I. Bjelakovi´c, H. Boche, J. Nötzel
Theorem 9 (Converse Part: Informed Decoder, Uninformed Users, |I| < ∞). Let I = {N1 , . . . , N N } ⊂ C(H, K) be a finite compound channel. The capacities Q I D (I) and Q(I) of I with informed decoder and uninformed users are bounded from above by Q(I) ≤ Q I D (I) ≤ lim
max
min
l→∞ ρ∈S (H⊗l ) Ni ∈I
1 Ic (ρ, Ni⊗l ). l
Proof. The inequality Q(I) ≤ Q(I) I D is obvious from the definition of codes. We give a proof for the second inequality. Let for arbitrary l ∈ N an (l, kl ) code for a compound channel I = {N1 , . . . , N N } with informed decoder and the property min1≤i≤N Fe (πFl , Rli ◦ Ni⊗l ◦ P l ) ≥ 1 − l be given, where l ∈ [0, 1]. Let |ψl ψl | ∈ S(El ⊗Fl ) be a purification of πFl , where El is just a copy of Fl . We use the abbreviation N Dl := N1 i=1 Rli ◦ Ni⊗l . Obviously, the above code then satisfies ψl , idEl ⊗ Dl (idEl ⊗ P l (|ψl ψl |))ψl =
N 1 Fe (πFl , Rli ◦ Ni⊗l ◦ P l ) N i=1
≥ 1 − l .
(75)
Let σP l := idEl ⊗ P l (|ψ l ψ l |) and consider any convex decomposition σP l = (dim Fl )2 λi |ei ei | of σP l into pure states |ei ei | ∈ S(Fl ⊗ H⊗l ). By (75) there i=1 is at least one i ∈ {1, . . . , (dim Fl )2 } such that ψl , idEl ⊗ Dl (|ei ei |)ψl ≥ 1 − l
(76)
holds. Without loss of generality, i = 1. Turning back to the individual channels, we get ψl , idEl ⊗ Rli ◦ Ni⊗l (|e1 e1 |)ψl ≥ 1 − N l ∀i ∈ {1, . . . , N }.
(77)
We define the state ρ l := tr El (|e1 e1 |) ∈ S(H⊗l ) and note that |e1 e1 | is a purification of ρ l . Application of recovery operation and individual channels to ρ l now defines the states σkl := idEl ⊗ Rlk ◦ Nk⊗l (|e1 e1 |) (k ∈ {1, . . . , N }) which have independently of k the property F(ψ l , σkl ) = ψl , idEl ⊗ Rlk ◦ Nk⊗l (|ei ei |)ψl ≥ 1 − N l , and thus put us into position for an application of Lemma 14, which together with the data processing inequality for coherent information [26] establishes the following chain of inequalities for every k ∈ {1, . . . , N }: log dim Fl = S(πFl ) = ∆S(|ψ l ψ l |) 2 ≤ ∆S(σkl ) + + 4 log((dim Fl )2 ) N l e = S(tr El (idEl ⊗ Rlk ◦ N ⊗l (|e1 e1 |)) − S(idEl ⊗ Rlk ◦ Nk⊗l (|e1 e1 |)) 2 + + 4 log((dim Fl )2 ) N l e 2 = Ic (ρ l , Rlk ◦ Nk⊗l ) + + 4 log((dim Fl )2 ) N l e 2 ≤ Ic (ρ l , Nk⊗l ) + + 4 log((dim Fl )2 ) N l . (78) e
Universal Quantum Channel Coding
89
Thus, 2 + 4 log((dim Fl )2 ) N l k∈{1,...,N } e 2 ≤ max min Ic (ρ, Nk⊗l ) + + 8 log(dim Fl ) N l . e ρ∈S (H⊗l ) k∈{1,...,N }
log dim Fl ≤
min
Ic (ρ l , Nk⊗l ) +
(79)
Let a sequence of (l, kl ) codes for I with informed decoder be given such that lim inf l→∞ 1l log dim Fl = R ∈ R and liml→∞ l = 0. Then by (79) we get 1 log dim Fl l 1 max ≤ lim inf min Ic (ρ, Nk⊗l ) l→∞ l ρ∈S (H⊗l ) k∈{1,...,N } 12 + lim inf + lim inf 8 log(dim Fl ) N l l→∞ l e l→∞ 1 = lim min Ic (ρ, Nk⊗l ). max l→∞ l ρ∈S (H⊗l ) k∈{1,...,N }
R = lim inf l→∞
Let us now focus on the general case. We shall prove the following theorem: Theorem 10 (Converse Part: Informed Decoder, Uninformed Users). Let I ⊂ C(H, K) be a compound channel. The capacities Q I D (I) and Q(I) for I with informed decoder and with uninformed users are bounded from above by Q(I) ≤ Q I D (I) ≤ lim
max
inf
l→∞ ρ∈S (H⊗l ) N ∈I
1 Ic (ρ, N ⊗l ). l
For the proof of this theorem, we will make use of the following lemma: Lemma 15 (cf. [22]). Let N , Ni ∈ C(H, K) and dK = dim K. Let Hr be an additional Hilbert space , l ∈ N and φ ∈ S(Hr ⊗ H⊗l ). If ||N − Ni ||♦ ≤ , then |S(idHr ⊗ N ⊗l (φ)) − S(idHr ⊗ Ni⊗l (φ))| ≤ l(4 log(dK ) + 2h()). Here, h(·) denotes the binary entropy. This result immediately implies the following lemma: Lemma 16. Let H, K be finite dimensional Hilbert spaces. There is a function ν : [0, 1] → R+ with lim x→0 ν(x) = 0 such that for every I, I ⊆ C(H, K) with D♦(I, I ) ≤ τ ≤ 1/2 and every l ∈ N we have the estimates 1. 1 1 | Ic (ρ, I⊗l ) − Ic (ρ, I ⊗l )| ≤ ν(2τ ) ∀ρ ∈ S(H⊗l ). l l 2. |
1 1 inf max Ic (ρ, N ⊗l ) − inf max Ic (ρ, N ⊗l )| ≤ ν(2τ ). l N ∈I ρ∈S (H⊗l ) l N ∈I ρ∈S (H⊗l )
90
I. Bjelakovi´c, H. Boche, J. Nötzel
The function ν is given by ν(x) = x + 8x log(dK ) + 4h(x). Again, h(·) denotes the binary entropy. Proof of Theorem 10. Again, the first inequality is easily seen to be true from the very definition of codes in the two cases, so we concentrate on the second. Let I ⊂ C(H, K) be a compound channel and let for every l ∈ N an (l, kl ) code for I with informed decoder be given such that lim inf l→∞ 1l log kl = R, and liml→∞ inf N ∈I Fe (πFl , RlN ◦ N ⊗l ◦ P l ) = 1 hold. Take any 0 < τ ≤ 1/2. Then it is easily seen that starting with a τ2 -net in C(H, K) 2 we can find a set I τ = {N1 , . . . , N Nτ } ⊂ I with |Nτ | ≤ ( τ6 )2(dim H·dim K) such that for
each N ∈ I there is Ni ∈ Iτ with ||N − Ni ||♦ ≤ τ. Clearly, the above sequence of codes satisfies for each i ∈ {1, . . . , Nτ }: 1. lim inf l→∞ 1l log kl = R, and 2. liml→∞ minNi ∈Iτ Fe (πFl , Rl ◦ Ni⊗l ◦ P l ) = 1. From Theorem 9 it is immediately clear then, that R ≤ lim
max
min
l→∞ ρ∈S (H⊗l ) Ni ∈I τ
1 Ic (ρ, Ni⊗l ), l
and from the first estimate in Lemma 16 we get by noting that D♦(I, I τ ) ≤ τ holds, R ≤ lim
max
inf
l→∞ ρ∈S (H⊗l ) N ∈I
Taking the limit τ → 0 proves the theorem.
1 Ic (ρ, N ⊗l ) + ν(2τ ). l
7.2. The Informed Encoder. The case of an informed encoder can be treated in the same manner as the other two cases. We will just state the theorem and very briefly indicate the central ideas of the proof. Theorem 11 (Converse Part: Informed Encoder). Let I ⊂ C(H, K) be a compound channel. The capacity Q I E (I) for I with informed encoder is bounded from above by 1 inf max Ic (ρ, N ⊗l ). l→∞ l N ∈I ρ∈S (H⊗l )
Q I E (I) ≤ lim
Proof. The proof of this theorem is a trivial modification of the one for Theorem 10. Again, the first part of the proof is the converse in the finite case, while the second part uses the second estimate in Lemma 16. For the proof in the finite case note the following: due to the data processing inequality, the structure of the proof is entirely independent from the decoder. A change from an informed decoder to an uninformed decoder does not change our estimate. The only important change is that there will be a whole set {ei11 , . . . , eiNN } of vector states satisfying Eq. (76), one for each channel in I. This causes the state ρ l in Eq. (78) to depend on the channel.
Universal Quantum Channel Coding
91
8. Continuity of Compound Capacity This section is devoted to a question that has been answered only recently in [22] for single-channel capacities, namely that of continuity of capacities of quantum channels. The question is relevant not only from a mathematical point of view, but might also have a strong impact on applications. It seems a hard task in general to compute the regularized capacity formulas obtained so far for quantum channels. There are, however, cases where the regularized capacity formula can be reduced to a one-shot quantity (see for example [8] and references therein) that can be calculated using standard optimization techniques. Knowing that capacity is a continuous quantity one could raise the question how close an arbitrary (compound) channel is to a (compound) channel with one-shot capacity and thereby get an estimate on arbitrary capacities. We will now state the main result of this section. Theorem 12 (Continuity of Compound Capacity). The compound capacities Q( · ), Q I D ( · ) and Q I E ( · ) are continuous. To be more precise, let I, I ⊂ C(H, K) be two compound channels with D♦(I, I ) ≤ ≤ 1/2. Then |Q(I) − Q(I )| = |Q I D (I) − Q I D (I )| ≤ ν(2), |Q I E (I) − Q I E (I )| ≤ ν(2), where the function ν is taken from Lemma 16. ¯ = 0, implying that the three different Remark 4. Let I ⊂ C(H, K). Then D(I, I) ¯ We may thus define the equivalence relation capacities of I coincide with those for I. I ∼ I ⇔ I¯ = I¯ and even use D♦ as a metric on the set of equivalence classes without losing any information about our channels. Proof. Let D♦(I, I ) ≤ . By the first estimate in Lemma 16 and the capacity formula Q I D (I) = Q(I) = liml→∞ 1l maxρ∈S (H⊗l ) Ic (ρ, I⊗l ) we get |Q(I) − Q(I )| = |Q I D (I) − Q I D (I )| 1 max Ic (ρ, I⊗l ) − max Ic (ρ, I ⊗l ) = lim l→∞ l ρ∈S (H⊗l ) ρ∈S (H⊗l ) 1 1 ⊗l
⊗l = lim max Ic (ρ, I ) − max Ic (ρ, I ) l→∞ l ρ∈S (H⊗l ) l ρ∈S (H⊗l ) ≤ lim ν(2) l→∞
= ν(2). For the proof in the case of an informed encoder let us first note that Q I E (I) = liml→∞ inf N ∈I maxρ∈S (H⊗l ) Ic (ρ, N ⊗l ) holds. The second estimate in Lemma 16 justifies the following inequality: 1 inf max Ic (ρ, N ⊗l ) − inf |Q I E (I) − Q I E (I )| = lim max I (ρ, N ⊗l ) c
⊗l ⊗l l→∞ l N ∈I ρ∈S (H ) N ∈I ρ∈S (H ) 1 1 = lim inf max Ic (ρ, N ⊗l ) − max I (ρ, N ⊗l ) inf c
⊗l ⊗l l→∞ l N ∈I ρ∈S (H ) l N ∈I ρ∈S (H )
92
I. Bjelakovi´c, H. Boche, J. Nötzel
≤ lim ν(2) l→∞
= ν(2).
9. Entanglement-Generating Capacity of Compound Channels In this last section we will use the results obtained so far to achieve our main goal. Namely, we will determine the entanglement-generating capacity of quantum compound channels. We give the definitions of codes and capacity only for the most interesting case of uninformed users because there is no doubt that the reader will easily guess the definitions in the remaining cases. Nevertheless, we will state the coding result in all three cases. An entanglement-generating (l, kl )-code for the compound channel I ⊂ C(H, K) with uninformed users consists of a pair (Rl , ϕl ), where Rl ∈ C(K⊗l , Fl ) with kl = dim Fl and ϕl is a pure state on Fl ⊗ H⊗l . R ∈ R+ is called an achievable rate for I with uninformed users if there is a sequence of (l, kl ) entanglement-generating codes with 1. lim inf l→∞ 1l log kl ≥ R, and 2. liml→∞ inf N ∈I F(|ψl ψl |, (idFl ⊗ Rl ◦ N ⊗l )(|ϕl ϕl |)) = 1, where ψl denotes the standard maximally entangled state on Fl ⊗ Fl and F(·, ·) is the fidelity. The entanglement-generating capacity of I with uninformed users is then defined as the least upper bound of all achievable rates and is denoted by E(I). The entanglement-generating capacities E I D (I) and E I E (I) of I with informed decoder or informed encoder are obtained if we allow the decoder or preparator to choose Rl or ϕl in dependence of N ∈ I. Recall from the proof of Theorem 9 that to each subspace G ⊂ H and > 0 we always can find a subspace Fl ⊂ G ⊗l ⊂ H⊗l , a recovery operation Rl ∈ C(K⊗l , Fl ), and a unitary operation U l ∈ C(H⊗l , H⊗l ) with
kl = dim Fl ≥ 2l(inf N ∈I Ic (πG ,N )− 2 −o(l
0 ))
,
(80)
and inf Fe (πFl , Rl ◦ N ⊗l ◦ U l ) = 1 − o(l 0 ).
N ∈I
(81)
Notice that the maximally entangled state ψl in Fl ⊗ Fl purifies the maximally mixed state πFl on Fl and defining |ϕl ϕl | := U l (|ψl ψl |), the relation (81) can be rewritten as inf F(|ψl ψl |, idFl ⊗ Rl ◦ N ⊗l (|ϕl ϕl |)) = 1 − o(l 0 ).
N ∈I
(82)
This together with (80) shows that E(I) ≥ inf Ic (πG , N ). N ∈I
(83)
Thus, using the compound BSST Lemma 10 and arguing as in the proof of Theorem 7, we can conclude that 1 max inf Ic (ρ, N ⊗l ). E(I) ≥ Q(I) = lim (84) l→∞ l ρ∈S (H⊗l ) N ∈I
Universal Quantum Channel Coding
93
Since E(I) ≤ E I D (I) holds it suffices to show E I D (I) ≤ Q(I) = lim
l→∞
1 max inf Ic (ρ, N ⊗l ) l ρ∈S (H⊗l ) N ∈I
(85)
in order to establish the coding theorem for E I D (I) and E(I) simultaneously. The proof of (85) relies on Lemma 14 and the data processing inequality. Indeed, let R ∈ R+ be an achievable entanglement generation rate for I with informed decoder and let ((RlN )N ∈I, ϕl )l∈N be a corresponding sequence of (l, kl )-codes, i.e we have 1. lim inf l→∞ 1l log kl ≥ R, and 2. inf N ∈I F(|ψl ψl |, (idFl ⊗ RlN ◦ N ⊗l )(|ϕl ϕl |)) = 1 − l , where liml→∞ l = 0 and ψl denotes the standard maximally entangled state on Fl ⊗ Fl with Schmidt rank kl . Set ρ l := trFl (|ϕl ϕl |) and l := idFl ⊗ RlN ◦ N ⊗l (|ϕl ϕl |). σN
Then the data processing inequality and Lemma 14 imply for each N ∈ I, Ic (ρ l , N ⊗l ) ≥ Ic (ρ l , RlN ◦ N ⊗l ) l ) = ∆(σN
2 √ − 8 log(kl ) l e 2 √ = log kl − − 8 log(kl ) l . e ≥ ∆(|ψl ψl |) −
Consequently, 1 2 √ 1 max inf Ic (ρ, N ⊗l ) + , (1 − 8 l ) log kl ≤ l l ρ∈S (H⊗l ) N ∈I le
(86)
and we end up with R ≤ lim sup l→∞
1 1 log kl ≤ lim max inf Ic (ρ, N ⊗l ), l→∞ l ρ∈S (H⊗l ) N ∈I l
which implies (85). The expression for E I E (I) is obtained in a similar fashion. We summarize the results in the following theorem. Theorem 13 (Entanglement-Generating Capacities of I). For arbitrary compound channels I ⊂ C(H, K) we have E(I) = E I D (I) = Q(I) = lim
l→∞
1 max inf Ic (ρ, N ⊗l ), l ρ∈S (H⊗l ) N ∈I
and E I E (I) = Q I E (I) = lim
l→∞
1 inf max Ic (ρ, N ⊗l ). l N ∈I ρ∈S (H⊗l )
94
I. Bjelakovi´c, H. Boche, J. Nötzel
10. Conclusion and Further Remarks We have demonstrated that universal codes in the sense of compound quantum channels exist, and we determined the best achievable rates. The results are analogous to those well known related results from the classical information theory obtained by Wolfowitz [32,33], and Blackwell, Breiman and Thomasian [5]. In contrast to the classical results on compound channels there is, in general, no single-letter description of the quantum capacities for entanglement transmission and generation over compound quantum channels. Notice, however, that for compound channels with classical input and quantum output (cq-channels) a single-letter characterization of the capacity is always possible according to the results of [3]. Natural candidates of compound quantum channels that might admit a single-letter capacity formula are given by sets of quantum channels consisting entirely of degradable channels. While it is quite easy to see from the results in [8] that the degradable compound quantum channels with informed encoder have a single-letter capacity formula for entanglement transmission and generation, the corresponding statement in the uninformed case seems to be less obvious. This and related questions will be addressed in a future work. Another issue we left open in this paper is the relation of the capacities considered here to other quantum communication tasks, for example to the subspace transmission and average subspace transmission and even to the randomized versions thereof. Again, we hope to come back to this point at some later time. Acknowledgement. We would like to thank Mary Beth Ruskai and the referee for many helpful suggestions and advice that led to significant improvement of the overall structure and readability of the paper. I.B. is supported by the Deutsche Forschungsgemeinschaft (DFG) via project “Entropie und Kodierung großer Quanten-Informationssysteme” at the TU Berlin. H.B. and J.N. are grateful for the support by TU Berlin through the fund for basic research.
A. Appendix Let E and G be subspaces of H with E ⊂ G ⊂ H, where k := dim E, dG := dim G. p and pG will denote the orthogonal projections onto E and G. For a Haar distributed random variable U with values in U(G) and x, y ∈ B(H) we define a random sesquilinear form 1 bU pU ∗ (x, y) := tr(U pU ∗ x ∗ U pU ∗ y) − tr(U pU ∗ x ∗ )tr(U pU ∗ y). k In this appendix we will give an elementary derivation of the formula E{bU pU ∗ (x, y)} =
k2 − 1 1 − k2 ∗ x p y) + tr( p tr( pG x ∗ )tr( pG y) G G dG2 − 1 dG (dG2 − 1)
(87)
for all x, y ∈ B(H) and where the expectation is taken with respect to the random variable U . Let us set pU := U pU ∗ . Since tr( pU x ∗ pU y) and tr( pU x ∗ )tr( pU y) depend sesquilinearly on (x, y) ∈ B(H) × B(H) it suffices to consider operators of the form x = | f 1 g1 | and y = | f 2 g2 |
(88)
Universal Quantum Channel Coding
95
with suitable f 1 , f 2 , g1 , g2 ∈ H. With x, y as in (88) we obtain tr( pU x ∗ pU y) = f 1 , pU f 2 g2 , pU g1 = f 1 ⊗ g2 , (U ⊗ U )( p ⊗ p)(U ∗ ⊗ U ∗ ) f 2 ⊗ g1 ,
(89)
tr( pU x ∗ )tr( pU y) = tr(( pU ⊗ pU )(|g1 f 1 | ⊗ | f 2 g2 )) = f 1 ⊗ g2 , (U ⊗ U )( p ⊗ p)(U ∗ ⊗ U ∗ )g1 ⊗ f 2 .
(90)
and
Since the range of the random projection (U ⊗U )( p⊗ p)(U ∗ ⊗U ∗ ) is contained in G ⊗G we see from (89) and (90) that we may (and will) w.l.o.g. assume that f 1 , f 2 , g1 , g2 ∈ G. Moreover, (89) and (90) show, due to the linearity of expectation, that the whole task of computing the average in (87) is boiled down to the determination of A( p) := E((U ⊗ U )( p ⊗ p)(U ∗ ⊗ U ∗ )) = (u ⊗ u)( p ⊗ p)(u ∗ ⊗ u ∗ )du. U(G )
(91)
Obviously, A( p) is u ⊗ u-invariant, i.e. A( p)(u ⊗ u) = (u ⊗ u)A( p) for all u ∈ U(G). It is fairly standard (and proven by elementary means in [30]) that then A( p) = αΠs + βΠa ,
(92)
where Πs and Πa denote the projections onto the symmetric and antisymmetric subspaces of G ⊗ G. More specifically Πs :=
1 (id + F), 2
Πa =
1 (id − F), 2
with id( f ⊗ g) = f ⊗ g and F( f ⊗ g) = g ⊗ f , for all f, g ∈ G. Since Πs and Πa are obviously u ⊗ u-invariant, and Πs Πa = Πa Πs = 0 holds, the coefficients α and β in (92) are given by α=
1 2 tr(( p ⊗ p)Πs ) = tr(( p ⊗ p)Πs ), tr(Πs ) dG (dG + 1)
(93)
β=
2 1 tr(( p ⊗ p)Πa ) = tr(( p ⊗ p)Πa ), tr(Πa ) dG (dG − 1)
(94)
and
where dG = dim G and we have used the facts that tr(Πs ) = dim ran(Πs ) =
dG (dG + 1) 2
tr(Πa ) = dim ran(Πa ) =
dG (dG − 1) . 2
and
It is easily seen by an explicit computation with a suitable basis that tr(( p ⊗ p)Πs ) =
1 2 1 (k + k) and tr(( p ⊗ p)Πa ) = (k 2 − k). 2 2
(95)
96
I. Bjelakovi´c, H. Boche, J. Nötzel
For example choosing any orthonormal basis {e1 , . . . , edG } of G with e1 , . . . , ek ∈ ran( p) we obtain tr(( p ⊗ p)Πs ) =
dG
ei ⊗ e j , ( p ⊗ p)Πs ei ⊗ e j
i, j=1
=
k
ei ⊗ e j , ( p ⊗ p)Πs ei ⊗ e j
i, j=1
⎛
=
1⎝ 2
k
⎞ ei , ei e j , e j + ei , e j e j , ei ⎠
i, j=1
=
1 2 (k + k), 2
with a similar calculation for tr(( p ⊗ p)Πa ). Utilizing (93), (94), (95), and (92) we end up with A( p) =
k2 − k k2 + k Πs + Πa . dG (dG + 1) dG (dG − 1)
(96)
Now, (96), (91), (90), (89), and some simple algebra show that 1 k2 − 1 tr(x ∗ y) E{tr(U pU ∗ x ∗ U pU ∗ y) − tr(U pU ∗ x ∗ )tr(U pU ∗ y)} = 2 k dG − 1 +
1 − k2 tr(x ∗ )tr(y). dG (dG2 − 1)
References 1. Barnum, H., Knill, E., Nielsen, M.A.: On Quantum Fidelities and Channel Capacities, IEEE Trans. Inf. Th. 46, 1317–1329 (2000); Barnum, H., Nielsen, M.A., Schumacher, B.: Information transmission through a noisy quantum channel, Phys. Rev. A 57, No. 6, 4153 (1998) 2. Bennett, C.H., Shor, P.W., Smolin, J.A., Thapliyal, A.V.: Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Th. 48, 2637–2655 (2002) 3. Bjelakovi´c, I., Boche, H.: Classical Capacities of Averaged and Compound Quantum Channels. IEEE Trans. Inf. Th. 57(7), 3360–3374 (2009) 4. Bjelakovi´c, I., Boche, H., Nötzel, J.: Quantum capacity of a class of compound channels. Phys. Rev. A 78(4), 042331 (2008) 5. Blackwell, D., Breiman, L., Thomasian, A.J.: The capacity of a class of channels. Ann. Math. Stat. 30(4), 1229–1241 (1959) 6. Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Alg. Appl. 10, 285–290 (1975) 7. Csizsar, I., Körner, J.: Information Theory; Coding Theorems for Discrete Memoryless Systems. Budapest: Akadémiai Kiadó, New York: Academic Press Inc., 1981 8. Cubitt, T., Ruskai, M., Smith, G.: The structure of degradable quantum channels. J. Math. Phys. 49(10), 102104 (2008) 9. Datta, N., Dorlas, T.C.: The coding theorem for a class of quantum channels with long-term memory. J. Phys. A: Math. Theor. 40, 8147–8164 (2007) 10. Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Th. 51(1), 44–55 (2005) 11. Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A 461, 207–235 (2005)
Universal Quantum Channel Coding
97
12. Hayashi, M.: Universal coding for classical-quantum channel. http://arxiv.org/abs/0805.4092v2 [quawt-ph], 2008 13. Hayden, P., Horodecki, M., Winter, A., Yard, J.: A decoupling approach to the quantum capacity. Open. Syst. Inf. Dyn. 15, 7–19 (2008) 14. Hayden, P., Shor, P.W., Winter, A.: Random quantum codes from gaussian ensembles and an uncertainty relation. Open. Syst. Inf. Dyn. 15, 71–89 (2008) 15. Holevo, A.S.: The Capacity of the quantum channel with general signal states. IEEE Trans. Inf. Th. 44(1), 269–273 (1998) 16. Holevo, A.S.: On entanglement-assisted classical capacity. J. Math. Phys. 43(9), 4326–4333 (2002) 17. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge: Cambridge University Press, 1999 18. Jozsa, R., Horodecki, M., Horodecki, P., Horodecki, R.: Universal quantum information compression. Phys. Rev. Lett. 81(8), 1714–1717 (1998) 19. Kitaev, A.Yu., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Graduate Studies in Mathematics 47, Providence, RI: Amer. Math. Soc., 2002 20. Klesse, R.: Approximate Quantum Error Correction, Random Codes, and Quantum Channel Capacity. Phys. Rev. A 75, 062315 (2007) 21. Kretschmann, D., Werner, R.F.: Tema con variazioni: quantum channel capacity. New J. Phys. 6, 26–59 (2004) 22. Leung, D., Smith, G.: Continuity of quantum channel capacities, http://arxiv.org/abs/0810.4931v1 [quawt-ph], 2009 23. Lloyd, S.: Capacity of the noisy quantum channel. Phys. Rev. A 55(3), 1613–1622 (1997) 24. Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Mathematics 1200, Berlin: Springer-Verlag, corrected second printing, 2001 25. Ogawa, T., Nagaoka, H.: Strong converse to the quantum channel coding theorem. IEEE Trans. Inf. Th. 45(7), 2486–2489 (1999) 26. Schumacher, B., Nielsen, M.A.: Quantum data processing and error correction. Phys. Rev. A 54(4), 2629–2635 (1996) 27. Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997) 28. Schumacher, B., Westmoreland, M.D.: Approximate quantum error correction. Quant. Inf. Proc. 1, 5–12 (2002) 29. Shor, P.: Quntum error correction. Unpublished talk manuscript. Available at: http://www.msri.org/ publications/ln/msri/2002/quantumcrypto/shor/1/ 30. Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40(8), 4277–4281 (1989) 31. Winter, A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inf. Th. 45(7), 2481– 2485 (1999) 32. Wolfowitz, J.: Simultaneous channels. Arch. Rat. Mech. Anal. 4(4), 371–386 (1960) 33. Wolfowitz, J.: Coding Theorems of Information Theory. Erg. Math. Grenzgebiete. 31, 3rd Edition, Berlin: Springer-Verlag, 1978 Communicated by M.B. Ruskai
Commun. Math. Phys. 292, 99–129 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0837-x
Communications in
Mathematical Physics
Meixner Class of Non-Commutative Generalized Stochastic Processes with Freely Independent Values I. A Characterization Marek Bo˙zejko1 , Eugene Lytvynov2 1 Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4,
50-384 Wrocław, Poland. E-mail:
[email protected]
2 Department of Mathematics, Swansea University, Singleton Park,
Swansea SA2 8PP, U.K. E-mail:
[email protected] Received: 1 December 2008 / Accepted: 23 February 2009 Published online: 22 May 2009 – © Springer-Verlag 2009
Abstract: Let T be an underlying space with a non-atomic measure σ on it (e.g. T = Rd and σ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of T , with freely independent values. Such a process (field), ω = ω(t), t∈ T , is given a rigorous meaning through smearing out with test functions on T , with T σ (dt) f (t)ω(t) being a (bounded) linear operator in a full Fock space. We define a set CP of all continuous polynomials of ω, and then define a non-commutative L 2 -space L 2 (τ ) by taking the closure of CP in the norm P L 2 (τ ) := P, where is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between L 2 (τ ) 2 n and a (Fock-space-type) Hilbert space F = R ⊕ ∞ n=1 L (T , γn ), with explicitly given measures γn . We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set CP invariant. (Note that, in the general case, the projection of a continuous monomial of order n onto the n th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions λ and η ≥ 0 on T , such that, in the F space, ω has representation ω(t) = ∂t† + λ(t)∂t† ∂t + ∂t + η(t)∂t† ∂t2 , where ∂t† and ∂t are the usual creation and annihilation operators at point t.
1. Introduction In his classical work [30], Meixner searched for all probability measures µ on R with infinite support whose system of monic orthogonal polynomials ( p (n) )∞ n=0 has an (exponential) generating function of the exponential type: ∞ ∞ p (n) (t) n 1 z = exp(tψ(z) + φ(z)) = (tψ(z) + φ(z))k . n! k! n=0
k=0
(1.1)
100
M. Bo˙zejko, E. Lytvynov
Meixner discovered that this (essentially) holds if and only if there exist λ ∈ R and η ≥ 0 such that the polynomials ( p (n) )∞ n=0 satisfy the recursive relation t p (n) (t) = p (n+1) (t) + λnp (n) (t) + (n + ηn(n − 1)) p (n−1) (t).
(1.2)
(We refer to [35] for a modern presentation of this result.) From (1.2) one concludes that the measure µ can be either Gaussian, or Poisson, or gamma, or Pascal (negative binomial), or Meixner. We may now introduce in L 2 (R, µ) creation (raising) and annihilation (lowering) operators through ∂ † p (n) := p (n+1) and ∂ p (n) := np (n−1) , respectively. Then, by (1.2), the action of the operator of multiplication by t in L 2 (R, µ) has a representation t· = ∂ † + λ∂ † ∂ + ∂ + η∂ † ∂∂.
(1.3)
Since Meixner’s laws are infinitely divisible, they appear as distributions of increments of corresponding Lévy processes. These are exactly Brownian motion, Poisson, gamma, Pascal, and Meixner processes. Note the first two of these processes correspond to the case η = 0, while the latter three correspond to η > 0. We will refer to all of them as the Meixner class of Lévy processes. From numerous applications of these processes let us mention that, for η > 0, they naturally appear in the study of a realization of the renormalized square of white noise, see [1,28,36] and the references therein. In [26] (see also [14,23,24,33]), Meixner-type generalized stochastic processes with independent values were constructed and studied. More precisely, consider a standard triple of the form S ⊂ L 2 (R, dt) ⊂ S , where S is a nuclear space of smooth functions, and S is the dual of S with respect to the central space L 2 (R, dt), i.e., S is a space of generalized functions (distributions). Let λ, η be parameters as in (1.2), or even more generally, let λ(·) and η(·) be smooth functions on R, which give, at each t ∈ R, parameters λ(t), η(t). Then, there exists a probability measure µ on the space S which is a generalized stochastic process with independent values (in the sense of [21]), and the operator of multiplication by a monomial f, ω , f ∈ S, ω ∈ S , has a representation f, ω · = dt f (t)(∂t† + λ(t)∂t† ∂t + ∂t + η(t)∂t† ∂t ∂t ), R
that is, ω(t)· = ∂t† + λ(t)∂t† ∂t + ∂t + η(t)∂t† ∂t ∂t
(1.4)
(compare with (1.3)). In (1.4), the operators ∂t† and ∂t are defined by analogy with the one-dimensional case, although on infinite-dimensional orthogonal polynomials of ω ∈ S , so that ∂t† and ∂t are the usual creation and annihilation operators at point t. As a result, one has a unitary isomorphism between the L 2 -space L 2 (S , µ) and some Hilbert ∞ space F = n=0 F (n) , where F (0) = R, while for each n ∈ N, F (n) = L 2sym (Rn , θn ) — the space of all symmetric functions on Rn which are square integrable with respect to some measure θn (depending on λ and η). In the special case where η ≡ 0, the space F reduces to the usual symmetric Fock space over L 2 (R, dt), whereas in the general case the space F is wider than the Fock space, which is why, in [26], F was called an extended Fock space. As follows from [13,27], the Meixner class may be characterized between all generalized stochastic processes with independent values as exactly those processes whose orthogonal polynomials remain continuous polynomials. Recall that, in infinite dimensions, orthogonalization of polynomials means: first, decomposing the L 2 -space into the
Meixner Class of Non-Commutative Generalized Stochastic Processes
101
infinite orthogonal sum of its subspaces generated by polynomials, and second, taking the projection of each monomial of order n onto the n th space. This is why, although the initial monomials are continuous functions of ω ∈ S , their orthogonal projections do not need to retain this property. The result of [13,27] also means that it is only for the Meixner-type processes that the multiplication operator ω(t)· can be represented through the operators ∂t† , ∂t . In free probability, Meixner’s systems of polynomials (on R) were introduced by Anshelevich [2] and Saitoh, Yoshida [34]. (In fact, such polynomials had already occurred in many places in the literature even before [2,34], see [17, p. 62] and [5, p. 864] for bibliographical references.) The free Meixner polynomials (q (n) )∞ n=0 have a (usual) generating function of the resolvent type: ∞ n=0
q (n) (t)z n = (1 − (tψ(z) + φ(z)))−1 =
∞ (tψ(z) + φ(z))k
(1.5)
k=0
(compare with (1.1)). Recall the following notation from q-analysis: for each q ∈ [−1, 1], we define [0]q := 0 and [n]q := 1 + q + q 2 + · · · + q n−1 for n ∈ N. In particular, for q = 0, we have [0]0 = 0 and [n]0 = 1 for n ∈ N. Then, by [2], equality (1.5) (essentially) holds if and only if there exist λ ∈ R and η ≥ 0 such that the polynomials (q (n) )∞ n=0 satisfy the recursive relation tq (n) (t) = q (n+1) (t) + λ[n]0 q (n) (t) + ([n]0 + η[n]0 [n − 1]0 )q (n−1) (t),
(1.6)
or, equivalently, equality (1.3) holds in which ∂ † and ∂ are defined through ∂ † q (n) := q (n+1) and ∂q (n) := [n]0 q (n−1) . Each measure of orthogonality of a free Meixner system of polynomials (which has an infinite support) is freely infinitely divisible, and therefore there exist corresponding free Lévy processes. A characterization of these processes in terms of a regression problem was given in [17]. These processes also appeared in the study of a realization of the renormalized square of free white noise [36]. A deep study of free Meixner polynomials of d (d ∈ N) non-commutative variables has been carried out by Anshelevich in [3,5–7]. The aim of the present paper is to introduce and study the Meixner class of noncommutative generalized stochastic processes with freely independent values, or equivalently Meixner-type free polynomials of infinitely-many (non-commutative) variables. We “translate” the aforementioned results of the theory of classical generalized stochastic processes with independent values into the language of free probability. In particular, we derive representation (1.4) for these processes in which ∂t† and ∂t are the creation and annihilation operators, as in the full Fock space, at point t. The main result of the paper—Theorem 4.1—is the characterization of the Meixner class as exactly those noncommutative generalized stochastic processes with freely independent values whose orthogonal polynomials are continuous in ω. It should be stressed that, generally speaking, the orthogonal polynomials we consider resemble one-dimensional free Meixner polynomials only in the infinitesimal sense, i.e., at each point of the underlying space. The paper is organized as follows. We start, in Sect. 2, with a discussion of processes of Gauss–Poisson type. We fix an underlying space T and a non-atomic measure σ on it. (Although the most importanant case is when T is either R or [0, ∞) and σ is the Lebesgue measure, we prefer to deal with a general space to stress that its structure does not play any significant role.) We fix a function λ ∈ C(T ), and consider a process (noise)
102
M. Bo˙zejko, E. Lytvynov
of the form ω(t) = ∂t† + ∂t + λ(t)∂t† ∂t in the full Fock space over L 2 (T, σ ). A sense to this process is given through smearing out with a test function f on T . We introduce a free expectation τ and the corresponding (non-commutative) L 2 -space L 2 (τ ). In terms of the expansion through orthogonal polynomials of ω, the space L 2 (τ ) is unitarily isomorphic to the original Fock space. We prove that the procedure of orthogonalization in L 2 (τ ) is equivalent to the procedure of free Wick (normal) ordering of the operators ∂t† and ∂t . This, in particular, generalizes a corresponding result of [16, p. 137], which was proved in the Gaussian case, i.e., when λ ≡ 0 (compare also with [4, p. 186]). We note, however, that in [16], the authors did not use the Wick ordering in the infinitesimal sense, which is only possible when λ ≡ 0. We then derive theorems giving a Wick rule for the product ω(t1 ) · · · ω(tn ), as well as a Wick rule for a product of Wick products. The latter theorems present a free counterpart of results of [29], see also [4, Prop. 6] for a q-case. In Sect. 3, we study (quite) general non-commutative generalized stochastic processes with freely independent values. They are described by assigning to each t ∈ T , a compactly supported probability measure µ(t, ds) on R, so that µ(t, {0}) is the diffusion coefficient of the process, while outside zero ν(t, ds) := s12 µ(t, ds) is the Lévy measure of “jumps” at point t (compare with [8–10]). We prove that the set of continuous polynomials of ω is dense in the corresponding space L 2 (τ ), introduce orthogonal polynomials, decompose any element of L 2 (τ ) into an infinite sum of orthogonal polynomials, and thusderive a unitary isomorphism between L 2 (τ ) and an extended full (n) (n) = L 2 (T n , γ ) with some measure γ on T n . Fock space F = ∞ n n n=0 F , where F We also present an explicit form of the action of the operators of (left) multiplication by f, ω realized in the space F. These operators have a clear Jacobi-field structure (compare with [12,13,18,25]). To derive our results, we produce an expansion of L 2 (τ ) in multiple stochastic integrals, by analogy with the Nualart and Schoutens result [31] in the classical case. In fact, Anshelevich [4] extended the result of [31] to the case of general q-Lévy processes. Comparing our result in this section with that of [4], we note that, first, we do not assume the process to be stationary, i.e., we allow the Lévy measure to depend on t, and second, what is much more important, our main results in this section—Theorems 3.3 and 3.4—are new even in the stationary case (when q = 0). Finally, in Sect. 4, we derive the Meixner class of free processes as exactly those non-commutative generalized stochastic processes with freely independent values for which orthogonal polynomials are continuous in ω, and thus we derive a counterpart of formula (1.4) in the free case. In the second part of this paper, which is currently in preparation, we will discuss the generating function for the orthogonal polynomials of ω from the Meixner class and other related problems, and we will also mention some open problems. 2. Free Gauss-Poisson Process Let T be a locally compact, second countable Hausdorff topological space. Recall that such a space is known to be Polish. A subset of T is called bounded if it is relatively compact in T . We will additionally assume that T does not possess isolated points, i.e., for every t ∈ T , there exists a sequence {tn }∞ n=1 ⊂ T such that tn = t for all n ∈ N, and tn → t as n → ∞. We denote by B(T ) the Borel σ -algebra in T , and by B0 (T ) the collection of all relatively compact sets from B(T ). Let D := C0 (T ) denote the set of all real-valued continuous functions on T with compact support. Analogously, we define D(n) := C0 (T n ), n ∈ N, and D(0) := R.
Meixner Class of Non-Commutative Generalized Stochastic Processes
103
For a real, separable space H we denote by F(H) the full Fock space over Hilbert ⊗n , where H⊗0 := R. As usual, we will identify each H⊗n H, i.e., F(H) := ∞ H n=0 with the corresponding subspace of F(H). We denote by Ffin (D) the subset of F(H) consisting of all sequences f = ( f (0) , f (1) , . . . , f (n) , 0, 0, . . . ) such that f (i) ∈ D(i) , i = 0, 1, . . . , n, n ∈ N0 := N ∪ {0}. The element :=(1, 0, 0, . . . ) ∈ Ffin (D) is called the vacuum. Let σ be a Radon, non-atomic measure on (T, B(T )). We will assume that the measure σ satisfies σ (O) > 0 for each open, non-empty set O in T . Let H:=L 2 (T, σ ) be the real L 2 -space over T with respect to the measure σ , and thus we get the Fock space F(H) = F(L 2 (T, σ )). For each f ∈ D, we denote by a + ( f ), a − ( f ), and a 0 ( f ) the corresponding creation, annihilation, and neutral operators, respectively. These are bounded linear operators on F(H) given through a + ( f ) = f ⊗ g (n) , g (n) ∈ H⊗n , n ∈ N0 , a − ( f ) g1 ⊗ · · · ⊗ gn = ( f, g1 )H g2 ⊗ · · · ⊗ gn , a 0 ( f ) g1 ⊗ · · · ⊗ gn = ( f g1 ) ⊗ g2 ⊗ · · · ⊗ gn , g1 , . . . , gn ∈ H, n ∈ N, a − ( f ) = a 0 ( f ) = 0. The operator a + ( f ) is the adjoint of a − ( f ), whereas a 0 ( f ) is self-adjoint. Note that a + ( f ) and a − (g), f , g ∈ D, satisfy the free commutation relation a − (g)a + ( f ) = (g, f )H ,
(2.1)
where, as usual, a constant is understood as the constant times the identity operator 1. Throughout the paper, we will heavily use the following standard notations. For each t ∈ T , we define ∂t as the annihilation operator at point t. More precisely, we set ∂t := 0, and for each f (n) ∈ D(n) , n ∈ N, we set (∂t f (n) )(t1 , . . . , tn−1 ) := f (n) (t, t1 , . . . , tn−1 ). Clearly, ∂t f (n) ∈ D(n−1) . Extending by linearity, we see that ∂t maps Ffin (D) into itself. If we introduce the “delta-function” δt : δt , f = f (t) for f ∈ D, then the operator ∂t can be thought of as a − (δt ). Next, we heuristically define ∂t† as the creation operator at point t, i.e., ∂t† is the “adjoint” of ∂t , so that ∂t† = a + (δt ). A rigorous meaning to formulas involving ∂t† will be given through smearing with test functions. In particular, for each f ∈ D, we get: a+( f ) = σ (dt) f (t)∂t† , a − ( f ) = σ (dt) f (t)∂t , a 0 ( f ) = σ (dt) f (t)∂t† ∂t . T
T
T
(2.2) Note that the relation (2.1) can now be written down in the form ∂s ∂t† = δ(s, t), where
σ (dt)δ(s, t) f (2) (s, t) :=
σ (ds) T
T
σ (dt) f (2) (t, t), T
(2.3)
f (2) ∈ D(2) .
(2.4)
104
M. Bo˙zejko, E. Lytvynov
We now fix λ ∈ C(T ) — the space of all continuous functions on T , and define, for each f ∈ D a self-adjoint operator x( f ) := a + ( f ) + a − ( f ) + a 0 (λ f ), so that
x( f ) = T
σ (dt) f (t) ∂t† + ∂t + λ(t)∂t† ∂t .
(2.5)
As we will see below, if λ ≡ 0, then (x( f )) f ∈D is a free Gaussian process, and if λ ≡ 1, then (x( f )) f ∈D is a free (centered) Poisson process. In view of (2.5), we denote ω(t) := ∂t† + ∂t + λ(t)∂t† ∂t , so that (2.5) becomes
σ (dt) f (t)ω(t).
x( f ) = T
Thus, ω := (ω(t))t∈T can be interpreted as the corresponding free noise. Lemma 2.1. The vacuum vector is cyclic for the operator family (x( f )) f ∈D , i.e., c. l. s.{, x( f 1 ) · · · x( f n ) | f 1 , . . . , f n ∈ D, n ∈ N} = F(H). Here and below, c. l. s. stands for the closed linear span. Proof. The statement follows by induction from the fact that we have a Jacobi field, i.e., each operator a( f ) has a three-diagonal structure, with a + ( f ), f ∈ D, being the usual creation operators (compare with e.g. [12,25]). We can naturally extend the definition of x( f ) to the case where f ∈ B0 (T ) — the space of all real-valued bounded measurable functions on T with compact support. Let A denote the real algebra generated by (x( f )) f ∈B0 (T ) . We define a free expectation on A by τ (a) := (a, )F (H) , a ∈ A. Recall that a set partition π of a set X is a collection of disjoint subsets of X whose union equals X . Let N C(n) denote the collection of all non-crossing partitions of {1, . . . , n}, i.e., all set partitions π = {A1 , . . . , Ak }, k ≥ 1, of {1, . . . , n} such that there do not exist Ai , A j ∈ π , Ai = A j , for which the following inequalities hold: x1 < y1 < x2 < y2 for some x1 , x2 ∈ Ai and y1 , y2 ∈ A j . For each n ∈ N, we define a free cumulant C (n) as the n-linear mapping C (n) : B0 (T )n → R defined recurrently by the following formula, which connects the free cumulants with moments: C(A, f 1 , . . . , f n ), (2.6) τ (x( f 1 )x( f 2 ) · · · x( f n )) = π ∈N C(n) A∈π
where for each A = {a1 , . . . , ak } ⊂ {1, 2, . . . , n}, a1 < a2 < · · · < ak , C(A, f 1 , . . . , f n ) := C (k) ( f a1 , . . . , f ak ).
Meixner Class of Non-Commutative Generalized Stochastic Processes
As easily seen, C (1) ≡ 0 and f 1 (t) · · · f n (t)λn−2 (t) σ (dt), C (n) ( f 1 , . . . , f n ) =
105
f 1 , . . . , f n ∈ B0 (T ), n ≥ 2.
T
(2.7) By (2.6) and (2.7), the expectation τ on A is tracial, i.e., for any a, b ∈ A, τ (ab) = τ (ba). Proposition 2.1. Let f 1 , . . . , f n ∈ B0 (T ) be such that f i f j = 0 σ -a.e. for all 1 ≤ i < j ≤ n.
(2.8)
Then x( f i ), i = 1, . . . , n, are freely independent with respect to τ . Proof. By (2.7) and (2.8), for each k ≥ 2 and any indices i 1 , . . . , i k ∈ {1, . . . , n} such that il = i m for some l, m ∈ {1, . . . , k}, C (k) ( f i1 , . . . , f ik ) = 0. Using e.g. [37], we conclude from here the statement. Let B0 (T )C denote the complexification of B0 (T ). We extend C (n) by linearity to the n-linear mapping C (n) : B0 (T )nC → C. For each f ∈ B0 (T )C , we denote C (n) ( f ) := (n) C (n) ( f, . . . , f ), and define the free cumulant transform C( f ) := ∞ n=1 C ( f ), provided that the latter series converges absolutely. By (2.7) and the dominated convergence theorem, we get: Proposition 2.2. Let f ∈ B0 (T )C be such that there exists ε ∈ (0, 1) for which | f (t)| < (where
1−ε 0
1−ε for all t ∈ T λ(t)
(2.9)
:= +∞). Then C( f ) = T
f 2 (t) σ (dt). 1 − λ(t) f (t)
(2.10)
Remark 2.1. Note that, for f ∈ D, condition (2.9) is equivalent to | f (t)| < 1/|λ(t)| for all t ∈ T . For each ∈ B0 (T ), we define x() := x(χ ) =
σ (dt)ω(t),
where χ denotes the indicator function of . Then, by Proposition 2.1, for any mutually disjoint sets 1 , . . . , n ∈ B0 (T ), the operators x(1 ), . . . , x(n ) are freely independent, and so by analogy with the classical case (see e.g. [21]), we can interpret ω as a non-commutative generalized stochastic process with freely independent values. For each f (n) ∈ B0 (T n ), we define a monomial of ω by σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn )ω(t1 ) · · · ω(tn ) f (n) , ω⊗n := Tn = σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn )(∂t†1 + ∂t1 + λ(t1 )∂t†1 ∂t1 ) Tn
× · · · × (∂t†n + ∂tn + λ(tn )∂t†n ∂tn ).
(2.11)
106
M. Bo˙zejko, E. Lytvynov
In fact, the presence of ∂t†i in (2.11) just means the creation of a function in the ti -variable, the presence of λ(ti )∂t†i ∂ti means the identification of the ti -variable with the previous ti−1 -variable and additional multiplication by λ(ti ), whereas the presence of ∂ti means integration in the ti -variable. For example, for f (4) ∈ B0 (T 4 ) and g (2) ∈ D(2) ,
σ (dt1 ) · · · σ (dt4 ) f (4) (t1 , . . . , t4 )∂t†1 ∂t2 λ(t3 )∂t†3 ∂t3 ∂t†4 g (2) (s1 , s2 , s3 ) T4 = σ (dt)λ(t) f (4) (s1 , t, t, t)g (2) (s2 , s3 ). T
Using the Cauchy–Schwarz inequality, we easily conclude that (2.11) indeed identifies a bounded linear operator in F(H). In particular, if f (n) = f 1 ⊗ · · · ⊗ f n with f 1 , . . . , f n ∈ B0 (T ), then f 1 ⊗ · · · ⊗ f n , ω⊗n = f 1 , ω · · · f n , ω = x( f 1 ) · · · x( f n ). We will also interpret constants as monomials of order 0. Let P and CP denote the set of all non-commutative polynomials (finite sums of monomials) with kernels f (n) ∈ B0 (T n ) and f (n) ∈ D(n) , respectively. (CP stands for “continuous polynomials.”) Clearly, CP ⊂ P and A ⊂ P. Lemma 2.2. We have CP = Ffin (D). Proof. Clearly, CP ⊂ Ffin (D). On the other hand, for each f (n) ∈ D(n) , f (n) = f (n) , ω⊗n − g (n−1) , where g (n−1) ∈
n−1 i=0
D(i) . From here, by induction, we conclude that Ffin (D) ⊂ CP.
We now naturally extend the free expectation τ to the set CP, and define an inner product (P1 , P2 ) L 2 (τ ) := τ (P2 P1 ) = (P1 , P2 )F (H) ,
P1 , P2 ∈ CP.
LetP ∈ CP and P = 0. Then P = 0 as an element of L 2 (τ ). Indeed, let n (i) ⊗i (n) = 0 (we then call P a polynomial of order n). P = i=0 f , ω , where f ⊗n Then the H -th component of P is f (n) , which implies that (P, P) L 2 (τ ) > 0. Hence, we can define a real Hilbert space L 2 (τ ) as the closure of CP with respect to the norm generated by the inner product (·, ·) L 2 (τ ) . As we will see below, we can naturally embed P (and so also A) into L 2 (τ ). Furthermore, we will also show that every element of L 2 (τ ) may be understood as (generally speaking, unbounded) Hermitian operator in F(H). Let CP(n) denote the subset of CP consisting of all continuous polynomials of order ≤ n. Let MP(n) denote the closure of CP(n) in L 2 (τ ). (MP stands for “measurable polynomials.”) Let OP(n) := MP(n) MP(n−1) , n ∈ N, OP(0) := R, where the sign denotes the orthogonal difference in L 2 (τ ). (OP stands polynomials.”) for “orthogonal (n) Thus, we get the orthogonal decomposition L 2 (τ ) = ∞ n=0 OP . Proposition 2.3. Consider a linear operator I : CP → Ffin (D) given by I P = P for P ∈ CP. Then, I extends to a unitary operator I : L 2 (τ ) → F(H). Furthermore, I OP(n) = H⊗n .
Meixner Class of Non-Commutative Generalized Stochastic Processes
107
Proof. The first statement of the proposition directly follows Lemma 2.2. Next, n from it follows from the proof of Lemma 2.2 that I CP(n) = i=0 D(i) , so that I MP(n) = n ⊗i i=0 H . From here, the second statement follows. For f (n) ∈ D(n) , let P( f (n) ) denote the orthogonal projection of f (n) , ω⊗n onto OP(n) , i.e., by the results proved above, P( f (n) ) = I −1 f (n) . Theorem 2.1. For each f (n) ∈ D(n) , we have P( f (n) ) ∈ CP. Before proving Theorem 2.1, we have to introduce some notations. Let N C(n, ±1) denote the collection of all κ = {(A1 , m 1 ), . . . (Ak , m k )}, k ∈ N,
(2.12)
such that π(κ) := {A1 , . . . , Ak } is an element of N C(n), m 1 , . . . , m k ∈ {−1, +1}, and if for some i ∈ {1, . . . , k}, the set Ai has only one element, then m i = 1. For each j ∈ {1, . . . , k}, we will interpret m j as the mark of the element A j of the non-crossing partition π(κ). Finally, we denote by G n the subset of N C(n, ±1) consisting of all κ as in (2.12) such that there do not exist i, j ∈ {1, . . . , k}, i = j, for which min Ai < min A j ≤ max A j < max Ai with m j = +1, i.e., an element of a non-crossing partition with mark +1 cannot be “within” any other element of this partition. (Note that, in [3,4], elements of G n were called extended partitions, with classes labeled +1 called “classes open on the left”.) Let n ∈ N and let us fix an arbitrary κ ∈ G n as in (2.12). We then define W (κ)ω(t1 ) · · · ω(tn ) as follows. For each i ∈ {1, . . . , k}, let Ai = { j1 , j2 , . . . , jl }, j1 < j2 , · · · < jl . If m i = −1 (and so l ≥ 2), then replace the factors ω(t j1 ), ω(t j2 ), . . . , ω(t jl ) in the product ω(t1 )ω(t2 ) · · · ω(tn ) by the “function” λl−2 (t j1 )δ(t j1 , t j2 , . . . , t jl ). If m i = +1, then leave the factor ω(t j1 ) without changes, and if l ≥ 2 then additionally replace the factors ω(t j2 ), ω(t j3 ), . . . , ω(t jl ) in the product ω(t1 )ω(t2 ) · · · ω(tn ) by the “function’ λl−1 (t j1 )δ(t j1 , t j2 , . . . , t jl ). Here, analogously to (2.4), we have set, for k ≥ 2, (k) σ (t1 ) · · · σ (tk ) f (t1 , . . . , tk )δ(t1 , t2 , . . . , tk ) := σ (dt) f (k) (t, t . . . , t). Tk
T
For example, if n = 8, and κ = {({1, 2}, +1), ({3, 4, 8}, +1), ({5, 6, 7}, −1)} , then W (κ)ω(t1 ) · · · ω(t8 ) = λ(t1 )δ(t1 , t2 )λ2 (t3 )δ(t3 , t4 , t8 )λ(t5 )δ(t5 , t6 , t7 )ω(t1 )ω(t3 ). Next, we denote by Int(n) the collection of all interval partitions of {1, . . . , n}, all of whose elements are intervals of consecutive integers. Clearly, Int(n) ⊂ N C(n). We will denote by Int(n, ±1) the corresponding subset of N C(n, ±1). Note that Int(n, ±1) ⊂ G n .
108
M. Bo˙zejko, E. Lytvynov
Proof of Theorem 2.1. For any f ∈ D, denote by f, ω · the operator of left multiplication by f, ω acting on CP. Clearly, under I , f, ω · goes over into the operator f, ω acting on Ffin (D). Now, for any f 1 , . . . , f n ∈ D, n ≥ 2, f 1 , ω f 2 ⊗ · · · ⊗ f n = f 1 ⊗ · · · ⊗ f n + (λ f 1 f 2 ) ⊗ f 3 ⊗ · · · ⊗ f n +( f 1 , f 2 )H f 3 ⊗ · · · ⊗ f n . Therefore, applying I −1 to the above equality, we get P( f 1 ⊗ · · · ⊗ f n ) = f 1 , ω P( f 2 ⊗ · · · ⊗ f n ) − P((λ f 1 f 2 ) ⊗ f 3 ⊗ · · · ⊗ f n ) −( f 1 , f 2 )H P( f 3 ⊗ · · · ⊗ f n ). (2.13) (n) denote the subset of D(n) consisting of finite sums of functions of the form Let Dalg f 1 ⊗ · · · ⊗ f n with f 1 , . . . , f n ∈ D. Then, it follows by induction from (2.13) that, for (n) each f (n) ∈ Dalg ,
P( f
(n)
)=
c(κ)
κ∈Int(n,±1)
Tn
σ (dt1 ) · · · σ (tn ) f (n) (t1 , . . . , tn )W (κ)ω(t1 ) · · · ω(tn ), (2.14)
where c(κ) ∈ R (compare with [20, Sect. 4] and [3, Sect. 3]). (n) (n) Now, let us fix an arbitrary f (n) ∈ D(n) . Choose a sequence { f k }∞ k=1 ⊂ Dalg such ∞ (n) (n) (n) that the set k=1 supp f k is in B0 (T ), f k are uniformly bounded and f k → f (n) (n) point-wise as k → ∞. Hence f k , ω⊗n → f (n) , ω⊗n in L 2 (τ ), which implies that (n) P( f k ) → P( f (n) ) in L 2 (τ ). On the other hand, for each κ ∈ G n ,
(n)
σ (dt1 ) · · · σ (dtn ) f k (t1 , . . . , tn )W (κ)ω(t1 ) · · · ω(tn ) → σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn )W (κ)ω(t1 ) · · · ω(tn )
Tn
Tn
in L 2 (τ ) as n → ∞. This implies that (2.14) holds for each f (n) ∈ D(n) , and therefore P( f (n) ) ∈ CP. For each n ∈ N, we define (free) Wick product of ω(t1 ), . . . , ω(tn ), denoted by :ω(t1 ) · · · ω(tn ): as follows: first we formally evaluate the product ω(t1 ) · · · ω(tn ) = (∂t†1 + ∂t1 + λ(t1 )∂t†1 ∂t1 ) · · · (∂t†n + ∂tn + λ(tn )∂t†n ∂tn ), and then remove all the terms containing ∂ti ∂t†i+1 for some i ∈ {1, . . . , n − 1}. We clearly have the following recursive formula: :ω(t1 ) := ω(t1 ), :ω(t1 ) · · · ω(tn ) := ∂t†1 :ω(t2 ) · · · ω(tn ): +λ(t1 )∂t†1 ∂t1 ∂t2 · · · ∂tn + ∂t1 ∂t2 · · · ∂tn , n ≥ 2.
(2.15)
Meixner Class of Non-Commutative Generalized Stochastic Processes
109
Furthermore, as easily seen, :ω(t1 ) · · · ω(tn ) := ∂t†1 ∂t†2 · · · ∂t†n +
n (∂t†1 · · · ∂t†i−1 ∂ti · · · ∂tn + ∂t†1 · · · ∂t†i−1 λ(ti )∂t†i ∂ti ∂ti+1 · · · ∂tn ).
(2.16)
i=1
Theorem 2.2. For each f (n) ∈ D(n) , n ∈ N, (n) P( f ) = σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn ) :ω(t1 ) · · · ω(tn ):.
(2.17)
Tn
Proof. Analogously to the proof of Theorem 2.1, it suffices to prove formula (2.17) in the case f (n) = f 1 ⊗ · · · ⊗ f n with f 1 , . . . , f n ∈ D. Using (2.3) and (2.15), we have: ω(t1 ) :ω(t2 ) · · · ω(tn ) := ∂t†1 :ω(t2 ) · · · ω(tn ): + (λ(t1 )∂t†1 ∂t1 + ∂t1 )(∂t†2 :ω(t3 ) · · · ω(tn ): +λ(t2 )∂t†2 ∂t2 ∂t3 · · · ∂tn + ∂t2 ∂t3 · · · ∂tn ) = ∂t†1 :ω(t2 ) · · · ω(tn ): + λ(t1 )δ(t1 , t2 )∂t†2 :ω(t3 ) · · · ω(tn ): +λ(t1 )δ(t1 , t2 )λ(t2 )∂t†2 ∂t2 ∂t3 · · · ∂tn + λ(t1 )∂t†1 ∂t1 ∂t2 · · · ∂tn +δ(t1 , t2 ) :ω(t3 ) · · · ω(tn ): + λ(t1 )δ(t1 , t2 )∂t2 ∂t3 · · · ∂tn + ∂t1 ∂t2 · · · ∂tn =: ω(t1 ) · · · ω(tn ): + λ(t1 )δ(t1 , t2 ) :ω(t2 ) · · · ω(tn ): + δ(t1 , t2 ) :ω(t3 ) · · · ω(tn ):, (2.18) the calculations taking rigorous meaning after smearing out with the f (n) as above. By virtue of (2.13), we see that (2.18) implies the statement of the theorem. Taking Theorem 2.2 into account, for each f (n) ∈ D(n) we will write f (n) , :ω⊗n : for P( f (n) ). More generally, for each f (n) ∈ H⊗n , we will denote by f (n) , :ω⊗n : the element of L 2 (τ ) defined as I −1 f (n) . Thus, each element F ∈ L 2 (τ ) admits a unique representation F=
∞
f (n) , :ω⊗n : ,
n=0
where f =
( f (n) )
∈ F(H).
Remark 2.2. With each F ∈ L 2 (τ ) one can associate a Hermitian (i.e., densely defined and symmetric, possibly unbounded) operator in F(H) with domain Ffin (D). Indeed, let us fix arbitrary f (n) ∈ D(n) and g (m) ∈ D(m) . By virtue of (2.16), n∧m (2) (3) Lk ( f (n) ) + Lk ( f (n) ) g (m) , f (n) , :ω⊗n : g (m) = L(1) ( f (n) )g (m) + k=1
where L(1) ( f (n) ) = (2)
Lk ( f (n) ) = (n) L(3) ) k (f
=
Tn
Tn
σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn )∂t†1 · · · ∂t†n , σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn )∂t†1 · · · ∂t†n−k ∂tn−k+1 · · · ∂tn , σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn )
Tn ×∂t†1
· · · ∂t†n−k λ(tn−k+1 )∂t†n−k+1 ∂tn−k+1 ∂tn−k+2 · · · ∂tn .
(2.19)
110
M. Bo˙zejko, E. Lytvynov
Note that L(1) ( f (n) )g (m) ∈ H⊗(n+m) , (2)
(3)
Lk ( f (n) )g (m) ∈ H⊗(n+m−2k) , Lk ( f (n) )g (m) ∈ H⊗(n+m−2k+1) .
(2.20)
Using (2.19) and the Cauchy–Schwarz inequality, we conclude that the vectors in (2.20) are well-defined for each f (n) ∈ H⊗n (independently of the choice of a version of f (n) ), and the F(H)-norm of each such vector is bounded by C f (n) H⊗n , where the constant depends on g (m) and is independent of n. Therefore, for each ∞C >(n)0 only ⊗n F = n=0 f , :ω : ∈ L 2 (τ ), ∞
∞ m (2) (3) Fg (m) := Lk ( f (n) ) + Lk ( f (n) ) g (m) , L(1) ( f (n) )g (m) + n=0
k=1
n=k
which is a vector in F(H). Indeed, by (2.20), 2 ∞ L(1) ( f (n) )g (m) n=0
F (H )
=
∞
L(1) ( f (n) )g (m) 2F (H) ≤ C 2
n=0
∞
f (n) 2F (H) < ∞,
n=0
and analogously we deal with the other sums. Extending F by linearity to the whole Ffin (D), we thus get a Hermitian operator in F(H) with domain Ffin (D). The following theorem gives a rule of representation of a monomial through a sum of orthogonal polynomials. Theorem 2.3 (Wick rule for a product of free noises). For each n ∈ N, we have: ω(t1 ) · · · ω(tn ) = :W (κ)ω(t1 ) · · · ω(tn ): , (2.21) κ∈G n
the formula making sense after smearing out with a function f (n) ∈ D(n) . Proof. We prove (2.21) by induction. Formula (2.21) trivially holds for n = 1. Assume that it also holds for n − 1, n ≥ 2. Then ω(t1 ) · · · ω(tn ) = ω(t1 ) :W (κ)ω(t2 ) · · · ω(tn ): κ∈G n−1
=
3
:W (κ (i) )ω(t1 ) · · · ω(tn ):.
κ∈G n−1 i=1
Here, κ (i) , i = 1, 2, 3, are the elements of G n that are obtained by first taking the marked partition κ of {2, 3, . . . , n}, and then for i = 1, by adding {1} as a singleton element with mark +1, for i = 2, by adding 1 to the first (from the left-hand side) element of κ which has mark +1 (if there is no such an element, then this term is zero), and for i = 3, by adding 1 to the first element of κ that has mark +1 and changing the mark to −1 (again this term becomes zero if no element of κ has mark +1). From here the statement of the theorem follows.
Meixner Class of Non-Commutative Generalized Stochastic Processes
Remark 2.3. For each f (n) ∈ B0 (T n ), σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn ) :W (κ)ω(t1 ) · · · ω(tn ):
111
(2.22)
n κ∈G n T
is clearly an element of L 2 (τ ), and it follows from Theorem 2.3 and Remark 2.2 that (2.22) is associated with the operator f (n) , ω⊗n (first on Ffin (D), and then it is extended by continuity to the whole F(H)). Thus, we get the inclusion of P into L 2 (τ ). The following theorem generalizes Theorem 2.3. Theorem 2.4 (Wick rule for a product of normal products of free noises). For any k1 , . . . , kl ∈ N, l ∈ N, we have :ω(t1 ) · · · ω(tk1 ): :ω(tk1 +1 ) · · · ω(tk1 +k2 ): · · · :ω(tk1 +k2 +···+kl−1 +1 ) · · · ω(tn ): = :W (κ)ω(t1 )ω(t2 ) · · · ω(tn ): , (2.23) where n := k1 + k2 + · · · + kl and the summation in (2.23) is over all κ ∈ G n such that each element of the induced partition π(κ) of {1, . . . , n} contains maximum one element of each of the sets {1, . . . , k1 }, {k1 + 1, . . . , k1 + k2 }, …, {k1 + k2 + · · · + kl−1 + 1, . . . , n}. Formula (2.23) makes sense after smearing out with a function f (n) ∈ D(n) . Proof. Analogously to the proof of Theorem 2.3, it suffices to show that, for any k1 , k2 ∈ N, :ω(t1 ) · · · ω(tk1 ): :ω(tk1 +1 ) · · · ω(tk1 +k2 ): = :ω(t1 ) · · · ω(tk1 )ω(tk1 +1 ) · · · ω(tk1 +k2 ): +:ω(t1 ) · · · ω(tk1 −1 )δ(tk1 , tk1 +1 )ω(tk1 +2 ) · · · ω(tk1 +k2 ): +:ω(t1 ) · · · ω(tk1 −1 )λ(tk1 )δ(tk1 , tk1 +1 )ω(tk1 +1 ) · · · ω(tk1 +k2 ):.
(2.24)
To show (2.24), represent :ω(t1 ) · · · ω(tk1 ): in the form (2.16) and represent :ω(tk1 +1 ) · · · ω(tk1 +k2 ): in the form (2.15), then use the free commutation relation (2.3) whenever ∂tk1 ∂t†k +1 enters, and finally collect the terms in order to get the right-hand side of (2.24). 1 We leave these long, but quite simple calculations to the interested reader. 3. Non-Commutative Generalized Stochastic Processes with Freely Independent Values Let the space T and the measure σ be as in Sect. 2. For each t ∈ T , let µ(t, ·) be a probability measure on (R, B(R)) with compact support. We will assume that, for each A ∈ B(R), the mapping T t → µ(t, A) is measurable, and for each ∈ B0 (T ) there exists R = R() > 0 such that, for all t ∈ , the measure µ(t, ·) has support in [−R, R]. We denote T˜ := T × R, and define a measure σ˜ (dt, ds) := σ (dt)µ(t, ds) on (T˜ , B(T˜ )). Clearly, σ˜ ( × R) < ∞ for all ∈ B0 (T ).
(3.1)
We denote H := L 2 (T˜ , σ˜ ). Let λ ∈ C(T˜ ) be chosen as λ(t, s) := s. Let L 2 (τ ) be the Hilbert space as in Sect. 2 which corresponds to T˜ , σ˜ , and λ. By Proposition 2.3, we have a unitary operator I : L 2 (τ ) → F(H).
112
M. Bo˙zejko, E. Lytvynov
Remark 3.1. In view of (3.1), we will call a subset of T˜ bounded if it is a subset of a set × R, where ∈ B0 (T ). We then define B0 (T˜ ) and C0 (T˜ ) as the set of all bounded measurable functions on T˜ with bounded support, and the set of all bounded continuous functions on T˜ with bounded support, respectively. All the respective definitions and results of Sect. 2 evidently remain true for these spaces. For each f : T → R and g : R → R, we denote by f ⊗ g the function on T˜ given by ( f ⊗ g)(t, s) := f (t)g(s). If f ∈ D = C0 (T ) and g is continuous, then f ⊗ g is continuous, has bounded support, but is not necessarily bounded. Still we will identify this function with any f ⊗ g¯ ∈ C0 (T˜ ), where g¯ : R → R is continuous, bounded, and coincides with g on [−R, R]. Here R = R(supp f ) > 0, i.e., R is chosen so that, for each t from the support of f , µ(t, ·) has support in [−R, R]. We will analogously proceed in the case where f ∈ B0 (T ). Now, for each f ∈ B0 (T ) we define X ( f ) as the element of L 2 (τ ) given by X ( f ) := x( f ⊗ 1) = a + ( f ⊗ 1) + a − ( f ⊗ 1) + a 0 ( f ⊗ s).
(3.2)
(Here and below, if g(s) = s l , l ∈ N0 , we write the function f ⊗ g as f ⊗ s l .) Thus, † † X( f ) = σ˜ (dt, ds) f (t)(∂(t,s) + ∂(t,s) + s∂(t,s) ∂(t,s) ) T˜ † † = σ (dt) f (t) µ(t, ds)(∂(t,s) + ∂(t,s) + s∂(t,s) ∂(t,s) ) R T = σ (dt) f (t)ω(t), (3.3) T
where
ω(t) :=
R
† † µ(t, ds) (∂(t,s) + ∂(t,s) + s∂(t,s) ∂(t,s) ) =
R
µ(t, ds) (t, s)
(3.4)
with † † (t, s) = ∂(t,s) + ∂(t,s) + s∂(t,s) ∂(t,s) .
(3.5)
Also for ∈ B0 (T ), we set X () := X (χ ). By Proposition 2.1, for any f 1 , . . . , f n ∈ B0 (T ) such that f i f j = 0 σ -a.e. for all 1 ≤ i < j ≤ n, X ( f 1 ), . . . , X ( f n ) are freely independent with respect to the state τ . In particular, for any mutually disjoint sets 1 , . . . , n ∈ B0 (T ), the operators X (1 ), . . . , X (n ) are freely independent. Hence, we may interpret ω as a non-commutative generalized stochastic process with freely independent values (compare with [15]). Remark 3.2. Let us derive an equivalent representation of the free random field (X ( f )) f ∈B0 (T ) . For each t ∈ T , denote c(t) := µ(t, {0}), and let ν(t, ·) denote the measure on R \ {0} given by ν(t, ds) := s12 µ(t, ds). Then, we define a unitary operator U : H → L 2 (T, c(t)σ (dt)) ⊕ L 2 (T × (R \ {0}), σ (dt)ν(t, ds)) := G by H f → U f := ( f (t, 0), f (t, s)s) ∈ G.
Meixner Class of Non-Commutative Generalized Stochastic Processes
113
We naturally extend U to a unitary operator U : F(H) → F(G). As easily seen, for each f ∈ B0 (T ), U X ( f )U −1 = a + ( f, 0) + a − ( f, 0) + a + (0, f ⊗ s) + a 0 (0, f ⊗ s) + a − (0, f ⊗ s). (3.6) In (3.6), the operator B( f ) := a + ( f, 0) + a − ( f, 0) describes the Brownian part of the process, while the operator J ( f ) := a + (0, f ⊗ s) + a 0 (0, f ⊗ s) + a − (0, f ⊗ s)
(3.7)
describes the “jump” part of the process. Thus, ν(t, ·) is the Lévy measure of the process at point t, and it describes the value and intensity of “jumps” (compare with e.g. [32] in the bosonic (classical) case, and with [8–10] in the free case). Analogously to Sect. 2, we define the free cumulants C (n) : B0 (T )nC → C through τ (X ( f 1 )X ( f 2 ) · · · X ( f n )) = C(A, f 1 , . . . , f n ), f 1 , f 2 , . . . , f n ∈ B0 (T ), π ∈N C(n) A∈π
and then we define the free cumulant transform C( f ) :=
∞
C (n) ( f ),
f ∈ B0 (T )C
n=1
(we have used obvious notations). By (the proof of) Proposition 2.2 and using the notations introduced in Remark 3.2, we get: Proposition 3.1. Let f ∈ B0 (T )C be such that there exists ε ∈ (0, 1) for which | f (t)| <
1−ε for all t ∈ T, R
where R = R(supp f ) > 0, i.e., R is such that, for each t ∈ supp f , the measure µ(t, ·) has support in [−R, R]. Then f 2 (t) C( f ) = σ (dt) µ(t, ds) 1 − s f (t) T R f 2 (t)s 2 = σ (dt)c(t) f 2 (t) + σ (dt) ν(t, ds) 1 − s f (t) T T R\{0} ∞ = σ (dt)c(t) f 2 (t) + σ (dt) ν(t, ds) s n f n (t). T
T
R\{0}
n=2
Next, we have: Proposition 3.2. The vacuum vector in F(H) is cyclic for the operator family (X ( f )) f ∈D .
114
M. Bo˙zejko, E. Lytvynov
Proof. It can be easily shown by approximation that it suffices to prove that is cyclic for the operator family (X ( f )) f ∈B0 (T ) . We first state that the linear span of the set {χ ⊗ s n | ∈ B0 (T ), n ∈ N0 } is dense in L 2 (T˜ , σ˜ ). Indeed, let g ∈ L 2 (T˜ , σ˜ ) be orthogonal to all elements of this set, i.e., σ (dt) µ(t, ds)s n g(t, s) = 0 for all ∈ B0 (T ) and n ∈ N0 . (3.8) R
Since R µ(·, ds)s n g(·, s) ∈ L 1 (, σ ) for each ∈ B0 (T ), we conclude from (3.8) that, for σ -a.e. t ∈ T , µ(t, ds)s n g(t, s) = 0 for all n ∈ N0 . R
But, for each t ∈ T , µ(t, ·) is a probability measure on R with compact support, and hence the set of all polynomials on R is dense in L 2 (R, µ(t, ·)). Therefore, for σ -a.e. t ∈ T and for µ(t, ·)-a.e. s ∈ R, g(t, s) = 0. Hence g(t, s) = 0 for σ˜ -a.e. (t, s) ∈ T˜ . Since the measure σ is non-atomic, we can analogously prove the following lemma: Lemma 3.1. For each n ∈ N, H⊗n = c. l. s.({χ1 ⊗ s l1 ) ⊗ (χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χn ⊗ s ln ) | l1 , . . . , ln ∈ N0 , for each j = 1, . . . , n − 1: j ∩ j+1 = ∅}. Below, we denote by M the set of all multi-indices of the form (l1 , . . . , li ) ∈ Ni0 , i ∈ N. Lemma 3.2. For each n ∈ N, we define the following subsets of F(H): R(n) := c. l. s. {, X ( f 1 ) · · · X ( f i ) | f 1 , . . . , f i ∈ B0 (T ), i ∈ {1, . . . , n}} , S (n) := c. l. s.{, (χ1 ⊗ s l1 ) ⊗ · · · ⊗ (χi ⊗ s li ) | (l1 , . . . , li ) ∈ M, l1 + · · · + li + i ≤ n, for each j = 1, . . . , i − 1: j ∩ j+1 = ∅}.
(3.9)
Then R(n) = S (n) . Proof. First, we note by approximation that, for each n ∈ N, S (n) = c. l. s. , f (i) (t1 , . . . , ti )s1l1 · · · sili | f (i) ∈ B0 (T i ), (l1 , . . . , li ) ∈ M, l1 + · · · + li + i ≤ n}
(3.10)
(we are using obvious notations for elements of F(H)). From (3.2) and (3.10), the inclusion R(n) ⊂ S (n) follows by induction. Next, let us prove that S (n) ⊂ R(n) . For n = 1, this is trivially true. Assume now that this is true for n ∈ {1, . . . , N }, and let us show it for n = N + 1. Thus, we have to show
Meixner Class of Non-Commutative Generalized Stochastic Processes
115
that, for any 1 , . . . , i ∈ B0 (T ) such that j ∩ j+1 = ∅ for all j = 1, . . . , i − 1, and any (l1 , . . . , li ) ∈ M such that l1 + · · · + li + i = N + 1, (χ1 ⊗ s l1 ) ⊗ · · · ⊗ (χi ⊗ s li ) ∈ R(N +1) .
(3.11)
If l1 = 0, then (χ1 ⊗ 1) ⊗ (χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χi ⊗ s li ) = a + (χ1 ⊗ 1)((χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χi ⊗ s li )) = X (1 )((χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χi ⊗ s li )). Hence, in the case l1 = 0, (3.11) holds. Now, for l1 ≥ 1, we have: (χ1 ⊗ s l1 ) ⊗ · · · ⊗ (χi ⊗ s li ) = a 0 (χ1 ⊗ s)((χ1 ⊗ s l1 −1 ) ⊗ (χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χi ⊗ s li )) = X (1 )((χ1 ⊗ s l1 −1 ) ⊗ (χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χi ⊗ s li )) − (χ1 ⊗ 1) ⊗ (χ1 ⊗ s l1 −1 ) ⊗ (χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χi ⊗ s li )
σ˜ (dt, ds)χ1 (t)s l1 −1 (χ2 ⊗ s l2 ) ⊗ · · · ⊗ (χi ⊗ s li ). − T˜
By the results proved above and the induction’s assumption, we therefore conclude that (3.11) holds for l1 ≥ 1. From Lemmas 3.1 and 3.2 the proposition follows.
For each f (n) ∈ B0 (T n ), we define a monomial of ω by σ (dt1 ) · · · σ (dtn ) f (n) (t1 , . . . , tn )ω1 (t1 ) · · · ωn (tn ) f (n) , ω⊗n : = Tn = σ˜ (dt1 , ds1 ) · · · σ˜ (dtn , dsn ) f (n) (t1 , . . . , tn ) (t1 , s1 ) · · · (tn , sn ) T˜ n
(recall (3.3)–(3.5)). We clearly have, for f (n) = f 1 ⊗ · · · ⊗ f n with f 1 , . . . , f n ∈ B0 (T ): f 1 ⊗ · · · ⊗ f n , ω⊗n = f 1 , ω · · · f n , ω = X ( f 1 ) · · · X ( f n ).
(3.12)
With some abuse of notations, we will denote by P and CP the set of all polynomials in ω with kernels f (n) ∈ B0 (T n ) and f (n) ∈ D(n) , respectively. (Note that below we will not use polynomials in the variable, so keeping the same notations as in Sect. 2 for rather different objects should not lead to a contradiction, and will be justified below.) From Proposition 3.2, we now conclude: Proposition 3.3. The set CP is dense in L 2 (τ ). Let CP(n) denote the subset of CP consisting of all continuous polynomials in ω of order ≤ n. Let MP(n) denote the closure of CP(n) in L 2 (τ ). Let OP(n) := MP(n) MP(n−1) , n ∈ N, OP(0) := R. Thus, we get:
116
M. Bo˙zejko, E. Lytvynov
Theorem 3.1. We have the following orthogonal decomposition of L 2 (τ ): L 2 (τ ) =
∞
OP(n) .
n=0
Let us recall that, in the case of a classical Lévy process, Nualart and Schoutens [31] derived an orthogonal decomposition of any square-integrable functional of the process in multiple stochastic integrals with respect to orthogonalized power jump processes (see also [27] and [4] for extensions of this result). Our next aim is to derive a free counterpart of [27,31]. Fix any t ∈ T . Denote by ( p (n) (t, ·))n≥0 the system of monic polynomials on R which are orthogonal with respect to µ(t, ·). If the support of µ(t, ·) is an infinite set, then by Favard’s theorem, the following recursive formula holds: sp (0) (t, s) = p (1) (t, s) + b(0) (t), sp (n) (t, s) = p (n+1) (t, s) + b(n) (t) p (n) (t, s) + a (n) (t) p (n−1) (t, s), n ∈ N,
(3.13)
where p (0) (t, s) = 1, a (n) (t) > 0 for n ∈ N, and b(n) (t) ∈ R for n ∈ N0 . If, however, the support of µ(t, ·) is a finite set consisting of N points (N ∈ N), then we have a finite N −1 system of monic orthogonal polynomials ( p (n) (t, ·))n=0 satisfying (3.13) for n ≤ N −2, and, for n = N − 1, we have: sp (N −1) (t, s) = b(N −1) (t, s) p (N −1) (t, s) + a (N −1) (t, s) p (N −2) (t, s). For technical reasons, we set, in this case, p (n) (t, s) := 0, a (n) (t) := 0, n ≥ N , (b(n) (t), n ≥ N being arbitrary), so that recursive relation (3.13) now always holds. For each n ∈ N0 , we denote g (l) (t) := µ(t, ds)| p (l) (t, s)|2 , t ∈ T, (3.14) R
and then we define a measure on (T, B(T )) by σ (l) (dt) := g (l) (t)σ (dt).
(3.15)
Note that σ (0) = σ . For each (l1 , . . . , li ) ∈ M, we define H(l1 ,...,li ) := L 2 (T i , σ (l1 ) ⊗ · · · ⊗ σ (li ) ).
(3.16)
Then, clearly, the following mapping is an isometry: H(l1 ,...,li ) f (i) → K (l1 ,...,li ) f (i) = (K (l1 ,...,li ) f (i) )(t1 , s1 , . . . , ti , si ) := f (i) (t1 , . . . , ti ) p (l1 ) (t1 , s1 ) · · · p (li ) (ti , si ) ∈ H⊗i . We denote by H(l1 ,...,li ) the range of the isometry K (l1 ,...,li ) .
(3.17)
Meixner Class of Non-Commutative Generalized Stochastic Processes
Lemma 3.3. We have
F(H ) = R ⊕
H(l1 ,...,li ) .
117
(3.18)
(l1 ,...,li )∈M
Furthermore, for each (l1 , . . . , li ) ∈ M, we have: H(l1 ,...,li ) = c. l. s.{(χ1 × p (l1 ) ) ⊗ · · · ⊗ (χi × p (li ) ) | 1 , . . . , i ∈ B0 (T ), for all j = 1, . . . , i − 1: j ∩ j+1 = ∅}. (3.19) Here, (χ × p (l) )(t, s) := χ (t) p (l) (t, s). Proof. Fix any f (i) ∈ B0 (T i ). Let ∈ B0 (T ) be such that the support of f (i) is a subset of i . Choose R = R() > 0 such that, for each t ∈ , µ(t, ·) has support in [−R, R]. Recall the recursive formula (3.13). We have, for each t ∈ , |b(n) (t)| ≤ R and a (n) (t) ≤ R 2 , which easily follows from the theory of Jacobi matrices (see e.g. [11]). Therefore, by (3.13), each p (n) (t, s) is bounded as a function of (t, s) ∈ × [−R, R]. Therefore, for each f (i) ∈ B0 (T i ), f (i) (t1 , . . . , ti ) p (l1 ) (t1 , s1 ) · · · p (li ) (ti , si ) ∈ H(l1 ,...,li ) . From here equality (3.19) easily follows (recall that the measure σ is non-atomic, which allows us to choose only those sets 1 , . . . , i in (3.19) for which j ∩ j+1 = ∅ for j = 1, . . . , i − 1). Formula (3.18) can now be proven analogously to the proof of Lemma 3.1. Recall that by Proposition 2.3, we have a unitary operator I : L 2 (τ ) → F(H). For each (l1 , . . . , li ) ∈ M, denote H(l1 ,...,li ) . := I −1 H(l1 ,...,li ) . For any ∈ B0 (T ) and l ∈ N0 , denote X (l) () := σ˜ (dt, ds)χ (t) p (l) (t, s) (t, s). T˜
For arbitrary (l1 , . . . , li ) ∈ M and 1 , . . . , i ∈ B0 (T ) such that j ∩ j+1 = ∅ for j = 1, . . . , i − 1, we clearly have: X (l1 ) (1 ) · · · X (li ) (i ) = (χ1 × p (l1 ) ) ⊗ · · · ⊗ (χi × p (li ) ). Therefore, by (3.19), H(l1 ,...,li ) = c. l. s.{X (l1 ) (1 ) · · · X (li ) (i ) | 1 , . . . , i ∈ B0 (T ), for all j = 1, . . . , i − 1: j ∩ j+1 = ∅}. For each f (l1 ,...,li ) ∈ H(l1 ,...,li ) (recall (3.16)), we can easily define a non-commutative multiple stochastic integral f (l1 ,...,li ) (t1 , . . . , ti )X (l1 ) (dt1 ) · · · X (li ) (dti ) (3.20) Ti
as an element of H(l1 ,...,li ) . Indeed, for each f (l1 ,...,li ) of the form f (l1 ,...,li ) (t1 , . . . , ti ) = χ1 (t1 ) · · · χi (ti )
118
M. Bo˙zejko, E. Lytvynov
with 1 , . . . , i ∈ B0 (T ) such that j ∩ j+1 = ∅, j = 1, . . . , i − 1, we define (3.20) as X (l1 ) (1 ) · · · X (li ) (i ). We then extend this definition by linearity to the linear span of such functions, and finally we extend it by continuity to obtain a unitary operator H(l1 ,...,li ) f (l1 ,...,li ) → f (l1 ,...,li ) (t1 , . . . , ti )X (l1 ) (dt1 ) · · · X (li ) (dti ) ∈ H(l1 ,...,li ) . Ti
Taking (3.18) into account, we thus derive Theorem 3.2. Denote F := R ⊕
H(l1 ,...,li ) .
(l1 ,...,li )∈M
Then, the following unitary operator gives an orthogonal expansion of L 2 (τ ) in non-commutative multiple stochastic integrals: F F = (c, ( f (l1 ,...,li ) )(l1 ,...,li )∈M ) f (l1 ,...,li ) (t1 , . . . , ti )X (l1 ) (dt1 ) · · · X (li ) (dti ) ∈ L 2 (τ ). → J F := c1 + Ti
(l1 ,...,li )∈M
(3.21) In terms of this orthogonal expansion, we have: L 2 (τ ) = R ⊕ H(l1 ,...,li ) .
(3.22)
(l1 ,...,li )∈M
(Note that, in (3.22), R denotes the space of all operators c1, where c ∈ R.) Remark 3.3. For each l ∈ N0 and ∈ B0 (T ), define Y (l) () ∈ L 2 (τ ) by Y (l) () : = σ˜ (dt, ds)χ (t)s l (t, s) T˜ +
= a (χ ⊗ s l ) + a 0 (χ ⊗ s l+1 ) + a − (χ ⊗ s l ) (recall (3.5)). Clearly, Y (0) () = X (). Recall now the unitary operator U : F(H) → F(G) from Remark 3.2. Then, for each l ∈ N, we have: U Y (l) ()U −1 = a + (0, χ ⊗ s l+1 ) + a 0 (0, χ ⊗ s l+1 ) + a − (0, χ ⊗ s l+1 ) (compare with (3.6) and (3.7)). Hence, by analogy with the classical case (see [31]), Y (l) (·), l ∈ N0 , may be treated as “power jump processes” (recall that s describes the value of “jumps”). For any l1 , l2 ∈ N0 , l1 < l2 , and any 1 , 2 ∈ B0 (T ), τ (Y (l1 ) (1 )X (l2 ) (2 )) = (X (l2 ) (2 ), Y (l1 ) (1 ))F (H) = σ (dt) µ(t, ds) p (l2 ) (t, s)s l1 = 0. 1 ∩2
R
Therefore, X (l) (·), l ∈ N0 , may be thought of as the orthogonalized power jump processes Y (l) (·), l ∈ N0 . The following theorem describes a connection between Theorems 3.1 and 3.2.
Meixner Class of Non-Commutative Generalized Stochastic Processes
Theorem 3.3. For each n ∈ N,
OP(n) =
119
H(l1 ,...,li ) .
(l1 ,...,li )∈M, l1 +···+li +i=n
Proof. We have to show that, for each n ∈ N, MP(n) = R ⊕
H(l1 ,...,li ) ,
(l1 ,...,li )∈M, l1 +···+li +i≤n
or, equivalently, I MP(n) = Z (n) , where
Z (n) := R ⊕
H(l1 ,...,li ) .
(3.23)
(l1 ,...,li )∈M, l1 +···+li +i≤n
As easily seen, I MP(n) = R(n) (see (3.9)). Hence, by Lemma 3.2 and (3.10), I MP(n) = c. l. s.{, f (i) (t1 , . . . , ti )s1l1 · · · sili | f (i) ∈ B0 (T i ), (l1 , . . . , li ) ∈ M, l1 + · · · + li + i ≤ n}.
(3.24)
Furthermore, (3.19) implies that Z (n) = c. l. s.{, f (i) (t1 , . . . , ti ) p (l1 ) (t1 , s1 ) · · · p (li ) (ti , si ) | f (i) ∈ B0 (T i ), (l1 , . . . , li ) ∈ M, l1 + · · · + li + i ≤ n}. (3.25) It follows from the proof of Lemma 3.3 that each p (l) has a representation p (l) (t, s) =
l
α (l, j) (t)s j ,
j=0
where α (l, j) ’s are measurable functions on T which are bounded on each ∈ B0 (T ). By (3.24) and (3.25), we therefore get the inclusion Z (n) ⊂ I MP(n) . Next, for each t ∈ T , denote by P (i) (t, ·), i ∈ N0 , the system of normalized orthogonal polynomials in L 2 (R, µ(t, ·)). We then have an expansion s = l
l
β (l, j) (t)P ( j) (t, s),
(3.26)
j=0
where the functions β (l, j) are measurable. For each ∈ B0 (T ) and each t ∈ , we have: l (β (l, j) (t))2 = s l 2L 2 (R,µ(t,ds)) = s l 2L 2 ([−R,R],µ(t,ds)) ≤ R 2l , j=0
where R = R(). Thus, the functions β (l, i) (·) are locally bounded on T . Define Y (n) := c. l. s.{, f (i) (t1 , . . . , ti )P (l1 ) (t1 , s1 ) · · · P (li ) (ti , si ) | f (i) ∈ B0 (T i ), (l1 , . . . , li ) ∈ M, l1 + · · · + li + i ≤ n}.
120
M. Bo˙zejko, E. Lytvynov
Then, by (3.24) and (3.26), I MP(n) ⊂ Y (n) . Set u (l) (t) := p (l) (t, ·)−1 . (In the L 2 (R, µ(t,·))
case where p (l) (t, ·) = 0, set u (l) (t) := 0.) To show that I MP(n) ⊂ Z (n) , it only remains to show that, for each f (i) ∈ B0 (T i ) and each (l1 , . . . , li ) ∈ M, l1 + · · · + li + i ≤ n, the function f (i) (t1 , . . . , ti )u (l1 ) (t1 ) · · · u (li ) (ti ) p (l1 ) (t1 , s1 ) · · · p (li ) (ti , si ) belongs to Z (n) . But this easily follows through approximation of f (i) (t1 , . . . , ti )u (l1 ) (t1 ) · · · u (li ) (ti )
by functions from B0 (T i ).
Recall that we have constructed the following chain of unitary isomorphisms: J
I
F −→ L 2 (τ ) −→ F(H) (see, in particular, Theorem 3.2). Thus, K := I J : F → F(H) is a unitary operator. Note that the restriction of K to each space H(l1 ,...,li ) is K (l1 ,...,li ) , see (3.17). We will preserve the notation for the vector in F defined as K −1 . For each n ∈ N0 , we denote
so that F =
∞
F(n) := J −1 OP(n) ,
(n) n=0 F ,
and by Theorem 3.3, for each n ∈ N, F(n) = H(l1 ,...,li ) . (l1 ,...,li )∈M, l1 +···+li +i=n
For each f ∈ B0 (T ), we will preserve the notation X ( f ) for the image of this operator under K −1 , i.e., for the equivalent realization of X ( f ) in F. Corollary 3.1. For each f ∈ B0 (T ), we have X ( f ) = X + ( f ) + X 0 ( f ) + X − ( f ), where X + ( f ) : F(n) → F(n+1) , X 0 ( f ) : F(n) → F(n) , and X − ( f ) : F(n) → F(n−1) . Furthermore, X ± ( f ) = X 1± ( f ) + X 2± ( f ), and for each (l1 , . . . , li ) ∈ M, H(l1 ,...,li ) g → (X 1+ ( f )g)(t1 , . . . , ti+1 ) := f (t1 )g(t2 , . . . , ti+1 ) ∈ H(0,l1 ,...,li ) , (3.27) H(l1 ,...,li ) g → (X 2+ ( f )g)(t1 , . . . , ti ) := f (t1 )g(t1 , . . . , ti ) ∈ H(l1 +1,l2 ,...,li ) , (3.28) H(l1 ,...,li ) g → (X 1− ( f )g)(t1 , . . . , ti−1 ) σ (dt) f (t)g(t, t1 , . . . , ti−1 ) ∈ H(l2 ,...,li ) , := δl1 , 0 T
H(l1 ,...,li ) g → (X 2− ( f )g)(t1 , . . . , ti ) := (1 − δl1 , 0 )a (l1 ) (t1 ) f (t1 )g(t1 , . . . , ti ) ∈ H(l1 −1,...,li ) , H(l1 ,...,li ) g → (X 0 ( f )g)(t1 , . . . , ti ) := b(l1 ) (t1 ) f (t1 )g(t1 , . . . , ti ) ∈ H(l1 ,...,li ) , and X 0 ( f ) = X − ( f ) = 0, X + ( f ) = f ∈ H(0) . Here, δl1 , 0 is equal to 1 if l1 = 0, and equal to 0, otherwise.
Meixner Class of Non-Commutative Generalized Stochastic Processes
121
Proof. We fix any f ∈ B0 (T ) and g (i) ∈ H(l1 ,...,li ) . Then, by (3.13) and (3.17), we have: a + ( f ⊗ 1) + a − ( f ⊗ 1) + a 0 ( f ⊗ s) K g (i) = a + ( f ⊗ 1) + a − ( f ⊗ 1) + a 0 ( f ⊗ s) g (i) (t1 , . . . , ti ) p (l1 ) (t1 , s1 ) · · · p (li ) (ti , si ) = f (t1 )g (i) (t2 , . . . , ti+1 ) p (0) (t1 , s1 ) p (l1 ) (t2 , s2 ) · · · p (li ) (ti+1 , si+1 ) +δl1 ,0 σ (dt) f (t)g (i) (t, t1 , . . . , ti−1 ) p (l2 ) (t1 , s1 ) · · · p (li ) (ti−1 , si−1 ) T (i)
+ f (t1 )g (t1 , . . . , ti )( p (l1 +1) (t1 , s1 ) + b(l1 ) (t1 ) p (l1 ) (t1 , s1 ) + a (l1 ) (t1 ) p (l1 −1) (t1 , s1 )) × p (l2 ) (t2 , s2 ) · · · p (li ) (ti , si ). Applying the operator K −1 to the above element of F(H), we easily conclude the statement. For f (n) ∈ D(n) , let P( f (n) ) denote the orthogonal projection of f (n) , ω⊗n onto OP(n) . Remark 3.4. By Proposition 3.3, the set CP is dense in L 2 (τ ). From here it follows that the linear span of the set {P( f (n) ) | f (n) ∈ D(n) , n ∈ N0 } is also dense in L 2 (τ ). In fact, for each n ∈ N, the set {P( f (n) ) | f (n) ∈ D(n) } is dense in OP(n) . Indeed, by definition, the set CP(n) is dense in MP(n) . Therefore, the set of all projections of P ∈ CP(n) onto OP(n) is dense in OP(n) . But the projection of each P ∈ CP(n−1) onto OP(n) equals zero, from which the statement follows. Corollary 3.2. Let n ∈ N and let (l1 , . . . , li ) ∈ M, l1 + · · · + li + i = n. For each f (n) ∈ D(n) , the H(l1 ,...,li ) -coordinate of the vector J −1 P( f (n) ) in F(n) is given by f ( t1 , . . . , t1 , t2 , . . . , t2 , . . . , ti , . . . , ti ). (l1 + 1) times (l2 + 1) times
(li + 1) times
Proof. By approximation, it suffices to check the statement in the case where f (n) = f 1 ⊗ · · · ⊗ f n , f 1 , . . . , f n ∈ D. Then, by (3.12), J −1 f (n) , ω⊗n = X ( f 1 ) · · · X ( f n ). Hence, J −1 P( f (n) ) is equal to the projection of X ( f 1 ) · · · X ( f n ) onto F(n) . Therefore, by Corollary 3.1, J −1 P( f (n) ) = X + ( f 1 ) · · · X + ( f n ). The statement now follows from (3.27) and (3.28).
In view of Remark 3.4 and Corollary 3.2, we will now give an equivalent interpretation of the F(n) spaces. So, we fix any n ∈ N. For each (l1 , . . . , li ) ∈ M, l1 +· · ·+li +i = n, we define T (l1 ,...,li ) := (t1 , . . . , tn ) ∈ T n | t1 = t2 = · · · = tl1 +1 , tl1 +2 = tl1 +3 = · · · = tl1 +l2 +2 , . . . , tl1 +l2 +···+li−1 +i = tl1 +l2 +···+li−1 +i+1 = · · · = tn , tl1 +1 = tl1 +l2 +2 , tl1 +l2 +2 = tl1 +l2 +l3 +3 . . . , tl1 +···+li−1 +i−1 = tn . The T (l1 ,...,li ) sets with (l1 , . . . , li ) ∈ M, l1 + · · · + li + i = n, form a set partition of T n .
122
M. Bo˙zejko, E. Lytvynov
We define B(T (l1 ,...,li ) ) as the trace σ -algebra of B(T n ) on T (l1 ,...,li ) . Now, consider the measurable mapping T (l1 ,...,li ) (t1 , . . . , tn ) → (tl1 +1 , tl1 +l2 +2 , tl1 +l2 +l3 +3 , . . . , tn ) ∈ T i .
(3.29)
Since σ is a non-atomic measure, the image of T (l1 ,...,li ) under the mapping (3.29) is of full σ (l1 ) ⊗ · · · ⊗ σ (li ) measure. We denote by γ (l1 ,...,li ) the pre-image of the measure σ (l1 ) ⊗ · · · ⊗ σ (li ) under the mapping (3.29). We then extend γ (l1 ,...,li ) by zero to the whole space T n . Note that, for different (l1 , . . . , li ) and (l1 , . . . , l j ) from M for which
l1 +· · ·+li +i = l1 +· · ·+l j + j = n, the measures γ (l1 ,...,li ) and γ (l1 ,...,l j ) are concentrated on disjoint sets in T n . We then define a measure on (T n , B(T n )) as follows: γn := γ (l1 ,...,li ) . (3.30) (l1 ,...,li )∈M, l1 +···+li +i=n
Recall that, by Remark 3.4, the set {J −1 P( f n ) | f (n) ∈ D(n) } is dense in F(n) , while the set D(n) is clearly dense in L 2 (T n , γn ). Therefore, by Corollary 3.2 the mapping L 2 (T n , γn ) ⊃ D(n) f (n) → J −1 P( f (n) ) ∈ F(n) extends to a unitary operator. In terms of this unitary isomorphism, we will, in what follows, identify F(n) with L 2 (T n , γn ), so that the space F becomes F=R⊕
∞
L 2 (T n , γn ).
n=1
By analogy with [26,27], we call F a free extended Fock space. Since, for each n ∈ N, D(n) ⊂ L 2 (T n , γn ), we have an evident inclusion of Ffin (D) into F. Corollaries 3.1 and 3.2 can now be reformulated as the following theorem, which is the main result of this section. Theorem 3.4. The following mapping J
F ⊃ Ffin (D) ( f (0) , f (1) , f (2) , . . . ) −→
∞
P( f (n) ) ∈ L 2 (τ )
(3.31)
n=0
(the sum being, in fact, finite) extends to the unitary operator J : F → L 2 (τ ). In particular, for any f (n) , g (n) ∈ D(n) , n ∈ N, (P( f (n) ), P(g (n) )) L 2 (τ ) = ( f (n) , g (n) ) L 2 (T n ,γn ) = ( f (n) g (n) )(t1 , . . . , t1 , . . . , ti , . . . , ti ) Ti (l1 ,...,li )∈M, l1 +···+li +i=n
×g
(l1 )
(t1 ) · · · g
(li )
l1 + 1 times
(ti ) σ (dt1 ) · · · σ (dti ),
where the functions g (l) are given by (3.14).
li + 1 times
(3.32)
Meixner Class of Non-Commutative Generalized Stochastic Processes
123
For each f ∈ D, X ( f ) = X + ( f ) + X 0 ( f ) + X − ( f ), where X + ( f ) : F(n) → F(n+1) , : F(n) → F(n) , and X − ( f ) : F(n) → F(n−1) . Furthermore, for each n ∈ N and (n) each g ∈ D(n) , X 0( f )
(X + ( f )g (n) )(t1 , . . . , tn+1 ) = f (t1 )g(t2 , . . . , tn+1 ), (t1 , . . . , tn+1 ) ∈ T n+1 ; for each (l1 , . . . , li ) ∈ M, l1 + . . . , li + i = n, and each (t1 , . . . , tn ) ∈
(3.33)
T (l1 ,...,li ) ,
(X 0 ( f )g (n) )(t1 , . . . , tn ) = b(l1 ) (t1 ) f (t1 )g (n) (t1 , . . . , tn );
(3.34)
and for each (l1 , . . . , li ) ∈ M, l1 +. . . , li +i = n−1, and each (t1 , . . . , tn−1 ) ∈ (X − ( f )g (n) )(t1 , . . . , tn−1 ) = σ (dt) f (t)g (n) (t, t1 , . . . , tn−1 )
T (l1 ,...,li ) ,
T
+a (l1 +1) (t1 ) f (t1 )g (n) (t1 , t1 , t2 , . . . , tn−1 ) (3.35) (the second addend on the right-hand side of (3.35) being equal to zero for n = 1). Additionally, X + ( f ) = f , X 0 ( f ) = 0, X − ( f ) = 0. Remark 3.5. For the reader’s convenience, let us quickly summarize the constructed spaces and the established unitary isomorphisms. We first have the following commutative diagram: L 2 (τ ) = R ⊕
(l1 ,...,li )∈M
J
F=R⊕
- F (H ) = R ⊕
I
H(l1 ,...,li )
H(l1 ,...,li )
(l1 ,...,li )∈M
K H(l1 ,...,li )
(l1 ,...,li )∈M
Here, the spaces H(l1 ,...,li ) are defined by (3.16), the isomorphism I is established in Proposition 2.3, K is given through (3.17), J is given by (3.21), the spaces H(l1 ,...,li ) and H(l1 ,...,li ) are the images of H(l1 ,...,li ) under K and J , respectively. Furthermore, we have realized each space F(n) = H(l1 ,...,li ) , n ∈ N, (l1 ,...,li )∈M, l1 +···+li +i=n
as L 2 (T n , γn ), and derived the following commutative diagram: L 2 (τ ) =
∞
I
OP(n)
n=0
J F=R⊕
∞
- F(H) = R ⊕
K
∞
H(n)
n=1
L 2 (T n , γn )
n=1
where H(n) :=
H(l1 ,...,li ) , n ∈ N.
(l1 ,...,li )∈M, l1 +···+li +i=n
Formula (3.31) gives the action of J in terms of the latter diagram, while formulas (3.33)–(3.35) give the action of X ( f ) in F.
124
M. Bo˙zejko, E. Lytvynov
4. The Free Meixner Class As we saw in Theorem 2.1, the free Gauss–Poisson processes have the property that, for each f (n) ∈ D(n) , the orthogonal polynomial P( f (n) ) is a continuous polynomial. We will now search for all the free processes as in Sect. 3 for which this property remains true. So, as in Sect. 3, we fix a free process (X ( f )) f ∈D — a family of bounded linear operators in the free extended Fock space F. Theorem 4.1. The following statements are equivalent: i) For each f (n) ∈ D(n) , P( f (n) ) ∈ CP. ii) For each f ∈ D, X ( f ) maps Ffin (D) into itself. iii) There exist λ and η from C(T ), η(t) ≥ 0 for all t ∈ T , such that b(l) (t) = λ(t), t ∈ T, l ∈ N0 , a (l) (t) = η(t), t ∈ T, l ∈ N. In this case, for each f ∈ D and g (n) ∈ D(n) , n ∈ N, (X + ( f )g (n) )(t1 , . . . , tn+1 ) = f (t1 )g(t2 , . . . , tn+1 ), (t1 , . . . , tn+1 ) ∈ T n+1 , (X ( f )g
(n)
(n)
)(t1 , . . . , tn ) = λ(t1 ) f (t1 )g (t1 , t2 , . . . , tn ), (t1 , . . . , tn ) ∈ T , − (n) (X ( f )g )(t1 , . . . , tn−1 ) = σ (dt) f (t)g (n) (t, t1 , . . . , tn−1 ) 0
n
(4.1) (4.2)
T
+η(t1 ) f (t1 )g (n) (t1 , t1 , t2 , . . . , tn−1 ), (t1 , . . . , tn−1 ) ∈ T (n−1)
(4.3)
(the second addend on the right-hand side of (4.3) being equal to zero for n = 1). Proof. Assume that i) holds. Hence, for any n ∈ N, there exist linear operators Ui,n : D(n) → D(i) , i = 0, 1, . . . , n, such that P( f (n) ) =
n Ui,n f (n) , ω⊗i ,
f (n) ∈ D(n) .
(4.4)
i=0
Applying the orthogonal projection of L 2 (τ ) onto OP(n) to both right-and left-hand sides of (4.4), we get P( f (n) ) = P(Un,n f (n) ). Hence, Un,n is the identity operator, so that (4.4) becomes P( f (n) ) = f (n) , ω⊗n +
n−1 Ui,n f (n) , ω⊗i ,
f (n) ∈ D(n) .
(4.5)
i=0
From here it follows that, for any n ∈ N, there exist linear operators Vi,n : D(n) → D(i) , i = 0, 1, . . . , n − 1, such that f (n) , ω⊗n = P( f (n) ) +
n−1 i=0
P(Vi,n f (n) ),
f (n) ∈ D(n) .
(4.6)
Meixner Class of Non-Commutative Generalized Stochastic Processes
125
Indeed, for n = 1, (4.6) clearly holds. Assume that (4.6) holds for all n = 1, . . . , N , N ∈ N. Then, by (4.5) and by (4.6) for n ≤ N , we have, for each f (N +1) ∈ D(N +1) , f (N +1) , ω⊗(N +1) = P( f (N +1) ) −
N
Ui,N +1 f (N +1) , ω⊗i
i=0
= P( f (N +1) )−
N
⎛ ⎝P(Ui,N +1 f (N +1) )+
i=0
i−1
⎞ P(V j,i Ui,N +1 f (N +1) )⎠,
j=0
from which (4.6) holds for n = N + 1. Now, for each f ∈ D and g (n) ∈ D(n) , by (4.5) and (4.6), f, ω P(g (n) ) = f ⊗ g (n) , ω(n+1) +
n f ⊗ (Ui−1,n g (n) ), ω⊗i i=1
= P( f ⊗ g (n) ) +
+
n
n
P(V j,n+1 ( f ⊗ g (n) ))
j=0
P( f ⊗ (Ui−1,n g
i=1
(n)
)) +
i−1
P(Vk,i ( f ⊗ (Ui−1,n g
(n)
))
k=0
= P( f ⊗ g (n) ) +
n
P(Z j,n+1 ( f, g (n) )),
(4.7)
j=0
where Z j,n+1 ( f, g (n) ) ∈ D( j) . Thus, by (4.7), f, ω P(g (n) ) ∈ CP(n+1) , and so ii) holds. (Note that, in view of symmetricity, Z j,n+1 ( f, g (n) ) = 0 for j ≤ n − 2.) Let us now prove that ii) implies iii). For each t ∈ T , denote λ(t) := b(0) (t) and η(t) := a (1) (t). Fix any open set O ∈ B0 (T ). Let f, g ∈ D be such that f (t) = g(t) = 1 for all t ∈ O. Then, by (3.34), for each t ∈ O, (X 0 ( f )g)(t) = λ(t).
(4.8)
By ii), (4.8) implies that λ(t) continuously depends on t ∈ O. Hence, λ ∈ C(T ). Next, let a set O and functions f , g be as above, and assume additionally that f ≥ 0 and g ≥ 0 on T . Further, choose any h ∈ D such that h ≥ 0 on T , with h(t) = 0 for all t ∈ O, and f h ≡ 0. Choose ε < 0 such that σ (dt) f (t)(g(t) + εh(t)) = 0. T
Set g (2) (t1 , t2 ) := (g(t1 ) + εh(t1 ))g(t2 ), (t1 , t2 ) ∈ T 2 . Then, by (3.35), for each t ∈ O, (X − ( f )g (2) )(t) = η(t), which implies that η is continuous on O. Hence, η ∈ C(T ).
126
M. Bo˙zejko, E. Lytvynov
Next, fix any t ∈ T , and let f ∈ D and g (n) ∈ D(n) , n ≥ 2, be such that f (t) = 1 and = 1. By (3.34), for any (t1 , . . . , tn−1 ) ∈ T n−1 such that t = t1 , t1 = t2 , …, tn−2 = tn−1 , we have g (n) (t, t, . . . , t)
(X 0 ( f )g (n) )(t, t1 , t2 . . . , tn−1 ) = λ(t)g (n) (t, t1 , t2 , . . . , tn−1 ),
(4.9)
(X 0 ( f )g (n) )(t, t, . . . , t) = b(n−1) (t).
(4.10)
whereas
By ii), lim
(t1 ,t2 ,...,tn−1 )→(t,t,...,t)
(X 0 ( f )g (n) )(t, t1 , t2 , . . . , tn−1 ) = (X 0 ( f )g (n) )(t, t, . . . , t).
Hence, by (4.9) and (4.10), b(n−1) (t) = λ(t). Thus, for all t ∈ T and all n ∈ N0 , b(n) (t) = λ(t). Completely analogously, we then also deduce from (3.35) that, for all t ∈ T and all n ∈ N, a (n) (t) = η(t). Formulas (4.2) and (4.3) now follow from (3.34) and (3.35), respectively. Thus, iii) holds. Finally, we prove that iii) implies i). Analogously to (2.13), we now have, for any f 1 , . . . , f n ∈ D, n ≥ 2: P( f 1 ⊗ · · · ⊗ f n ) = f 1 , ω P( f 2 ⊗ · · · ⊗ f n ) − P((λ f 1 f 2 ) ⊗ f 3 ⊗ · · · ⊗ f n ) − σ (dt) f 1 (t) f 2 (t)P( f 3 ⊗ · · · ⊗ f n ) − P((η f 1 f 2 f 3 ) ⊗ f 4 ⊗ · · · ⊗ f n ) T
(compare with (2.13)). From here we conclude statement i) by an easy generalization of the proof of Theorem 2.1. Remark 4.1. As easily seen by approximation, formulas (4.1)–(4.3) remain true for any f ∈ B0 (T ) and g (n) ∈ F(n) = L 2 (T n , γn ). The set of all free processes as in Theorem 4.1, iii) will be called the Meixner class of free processes. We note that, if, for t ∈ T , η(t) = 0, then the measure µ(t, ·) is concentrated at one point, namely λ(t). Hence, g (0) (t) = 1 and g (l) (t) = 0 for all l ∈ N (see (3.14) and (3.15)). In particular, if η(t) = 0 for all t ∈ T , the measure γn becomes σ ⊗n (see (3.30)). Thus, F = F(H) and X ( f ) = x( f ), f ∈ D, where (x( f )) f ∈D is the free process as in Sect. 2, which corresponds to the function λ ∈ C(T ). If, however, η(t) > 0, then µ(t, ·) has an infinite support. Recall that µ(t, ·) is the measure of orthogonality of monic polynomials ( p (n) (t, ·))∞ n=0 satisfying sp (n) (t, s) = p (n+1) (t, s) + λ(t) p (n) (t, s) + η(t) p (n−1) (t, s), n ∈ N0 ,
(4.11)
where p (−1) (t, s) := 0. Hence, µ(t, ·) is Wigner’s semicircle law with mean λ(t) and variance η(t): −1 √ √ µ(t, ds) = χ[−2 η(t)+λ(t), 2 η(t)+λ(t)] (s) (4π η(t)) 4η(t) − (s − λ(t))2 ds (compare with [34] and [17]). By (3.14) and (4.11), we have: g (l) (t) = ηl (t), l ∈ N0 .
(4.12)
Meixner Class of Non-Commutative Generalized Stochastic Processes
127
Substituting (4.12) into (3.32), we get the explicit form of the inner product in the free extended Fock space F. Assume that, for some ∈ B0 (T ), the functions λ(·) and η(·) are constant on , i.e., λ(t) = λ, η(t) = η for all t ∈ , where λ ∈ R and η ≥ 0. Then, by (4.1)–(4.3) (see Remark 4.1), we have: ⊗(n+1)
⊗n X ()χ = χ
⊗(n−1)
⊗n + [n]0 λχ + ([n]0 σ () + [n]0 [n − 1]0 η)χ
, n ∈ N0 , (4.13)
⊗0 ⊗n ⊗n ⊗n where χ := . Denote P(χ ) := J χ . Then, by (4.13), P(χ ) = q (n) (X ()), where (q (n) )∞ n=0 is the system of monic polynomials on R satisfying the recursive relation (1.6). By Favard’s theorem, (q (n) )∞ n=0 is a system of polynomials which are orthogonal with respect to some probability measure ρλ,η,σ () . For an explicit form of this measure, we refer to e.g. [34].
Corollary 4.1. Let (X ( f )) f ∈B0 (T ) be as in Theorem 4.1 iii). Then, for each ∈ B0 (T ), there exists r = r () > 0 such that, for each f ∈ B0 (T )C satisfying | f (t)| < r χ (t) for all t ∈ T, we have
−1 2 2 σ (dt) 2 f (t) 1 − λ(t) f (t) + (1 − λ(t) f (t)) − 4 f (t)η(t) .
C( f ) =
2
T
Proof. The result directly follows from Proposition 3.1 and the following formula which holds for z ∈ C from a neighborhood of zero:
−1 1 2 2 = 2 1 − λ(t)z + (1 − λ(t)z) − 4z η(t) µ(t, ds) , 1 − sz R see [2,34]. Recall that Ffin (D) is a dense subset of F. Analogously to Sect. 2, we can therefore interpret smeared, Wick ordered products of operators ∂t† and ∂t as operators in F. Corollary 4.2. Let (X ( f )) f ∈B0 (T ) be as in Theorem 4.1 iii). Then, using the same notations as in Sect. 2, we may represent the action of each X ( f ) in F as follows: X( f ) = σ (dt) f (t)ω(t), T
where ω(t) = ∂t† + λ(t)∂t† ∂t + ∂t + η(t)∂t† ∂t ∂t . Proof. The statement directly follows from (4.1)–(4.3) if we note that, for each g (n) ∈ D(n) ,
† (n) (t1 , . . . , tn−1 ) = η(t1 ) f (t1 )g (n) (t1 , t1 , t2 , . . . , tn−1 ). σ (dt) f (t)η(t)∂t ∂t ∂t g T
Acknowledgements. We would like to thank the referee for a careful reading of the manuscript and making very useful comments and suggestions. The authors acknowledge the financial support of the SFB 701 “Spectral structures and topological methods in mathematics”, Bielefeld University. MB was partially supported by the KBN grant no. 1P03A 01330. EL was partially supported by the PTDC/MAT/67965/2006 grant, University of Madeira.
128
M. Bo˙zejko, E. Lytvynov
References 1. Accardi, L., Franz, U., Skeide, M.: Renormalized squares of white noise and other non-Gaussian noises as Lévy processes on real Lie algebras. Commun. Math. Phys. 228, 123–150 (2002) 2. Anshelevich, M.: Free martingale polynomials. J. Funct. Anal. 201, 228–261 (2003) 3. Anshelevich, M.: Appell polynomials and their relatives. Int. Math. Res. Not. 2004(65), 3469–3531 (2004) 4. Anshelevich, M.: q-Lévy processes. J. Reine Angew. Math. 576, 181–207 (2004) 5. Anshelevich, M.: Free Meixner states. Commun. Math. Phys. 276, 863–899 (2007) 6. Anshelevich, M.: Orthogonal polynomials with a resolvent-type generating function. Trans. Amer. Math. Soc. 360, 4125–4143 (2008) 7. Anshelevich, M.: Monic non-commutative orthogonal polynomials. Proc. Amer. Math. Soc. 136, 2395–2405 (2008) 8. Barndorff-Nielsen, O.E., Thorbjørnsen, S.: Lévy laws in free probability. Proc. Natl. Acad. Sci. USA 99, 16568–16575 (2002) (electronic) 9. Barndorff-Nielsen, O.E., Thorbjørnsen, S.: Lévy processes in free probability. Proc. Natl. Acad. Sci. USA 99, 16576–16580 (2002) (electronic) 10. Barndorff-Nielsen, O.E., Thorbjørnsen, S.: The Lévy-Itô decomposition in free probability. Probab. Theory Related Fields 131(2), 197–228 (2005) 11. Berezansky, Ju.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Providence, RI: Amer. Math. Soc., 1968 12. Berezansky, Yu.M.: Commutative Jacobi fields in Fock space. Integral Equations Operator Theory 30, 163–190 (1998) 13. Berezansky, Yu.M., Lytvynov, E., Mierzejewski, D.A.: The Jacobi field of a Lévy process. Ukrainian Math. J. 55, 853–858 (2003) 14. Berezansky, Yu.M., Mierzejewski, D.A.: The structure of the extended symmetric Fock space. Methods Funct. Anal. Topology 6(4), 1–13 (2000) 15. Biane, P.: Processes with free increments. Math. Z. 227, 143–174 (1998) 16. Bozejko, M., Kümmerer, B., Speicher, R.: q-Gaussian processes: non-commutative and classical aspects. Commun. Math. Phys. 185, 129–154 (1997) 17. Bozejko, M., Bryc, W.: On a class of free Lévy laws related to a regression problem. J. Funct. Anal. 236, 59–77 (2006) 18. Brüning, E.: When is a field a Jacobi field? A characterization of states on tensor algebras. Publ. Res. Inst. Math. Sci. 22, 209–246 (1986) 19. Donati-Martin, C.: Stochastic integration with respect to q Brownian motion. Probab. Theory Related Fields 125, 77–95 (2003) 20. Effros, E.G., Popa, M.: Feynman diagrams and Wick products associated with q-Fock space. Proc. Natl. Acad. Sci. USA 100, 8629–8633 (2003) (electronic) 21. Gel’fand, I.M., Vilenkin, N.Ya.: Generalized Functions, Vol. IV, New York-London: Academic Press, 1964 22. Hida, T., Kuo, H.-H., Potthoff, J., Streit, L.: White Noise: An Infinite Dimensional Calculus. DordrechtBoston-London: Kluwer Acad. Publ., 1993 23. Kondratiev, Yu.G., Lytvynov, E.W.: Operators of gamma white noise calculus. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3, 303–335 (2000) 24. Kondratiev, Yu.G., da Silva, J.L., Streit, L., Us, G.F.: Analysis on Poisson and gamma spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, 91–117 (1998) 25. Lytvynov, E.W.: Multiple Wiener integrals and non-Gaussian white noises: a Jacobi field approach. Meth. Func. Anal. and Topol 1, 61–85 (1995) 26. Lytvynov, E.: Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures. J. Funct. Anal. 200, 118–149 (2003) 27. Lytvynov, E.: Orthogonal decompositions for Lévy processes with an application to the gamma, Pascal, and Meixner processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, 73–102 (2003) 28. Lytvynov, E.: The square of white noise as a Jacobi field. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 619–629 (2004) 29. Lytvynov, E.W., Rebenko, A.L., Shchepan’uk, G.V.: Wick theorems in non-Gaussian white noise calculus. Rep. Math. Phys. 37, 217–232 (1996) 30. Meixner, J.: Orthogonale Polynomsysteme mit einem besonderen Gestalt der erzeugenden Funktion. J. London Math. Soc. 9, 6–13 (1934) 31. Nualart, D., Schoutens, W.: Chaotic and predictable representations for Lévy processes. Stochastic Process. Appl. 90, 109–122 (2000) 32. Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Basel: Birkhäuser Verlag, 1992 33. Rodionova, I.: Analysis connected with generating functions of exponential type in one and infinite dimensions. Methods Funct. Anal. Topology 11, 275–297 (2005)
Meixner Class of Non-Commutative Generalized Stochastic Processes
129
34. Saitoh, N., Yoshida, H.: The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Statist. 21, 159–170 (2001) 35. Schoutens, W.: Stochastic Processes and Orthogonal Polynomials. New York: Springer-Verlag, 2000 ´ 36. Sniady, P.: Quadratic bosonic and free white noises. Commun. Math. Phys. 211, 615–628 (2000) 37. Speicher, R.: Free probability theory and non-crossing partitions. Sém. Lothar. Combin. 39, Art. B39c, 38 pp. (1997) (electronic) Communicated by Y. Kawahigashi
Commun. Math. Phys. 292, 131–177 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0857-6
Communications in
Mathematical Physics
How Hot Can a Heat Bath Get? Martin Hairer1,2 1 Mathematics Institute, The University of Warwick, Coventry CV4 7AL,
United Kingdom. E-mail:
[email protected]
2 Courant Institute, New York University, New York, NY 10012, USA.
E-mail:
[email protected] Received: 2 December 2008 / Accepted: 31 March 2009 Published online: 25 June 2009 – © Springer-Verlag 2009
Abstract: We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme’ non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence/non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is ‘too stiff’, then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds. Contents 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heuristic Derivation of the Main Results . . . . . . . . . . . . . A Potpourri of Test Function Techniques . . . . . . . . . . . . . Existence and Non-existence of an Invariant Probability Measure Integrability Properties of the Invariant Measure . . . . . . . . . Convergence Speed Towards the Invariant Measure . . . . . . . . The Case of a Weak Pinning Potential . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
131 138 141 146 156 167 170
1. Introduction The aim of this work is to provide a detailed investigation of the dynamic and the long-time behaviour of the following model. Consider two point particles moving in a
132
M. Hairer
potential V1 and interacting through a harmonic force, that is the Hamiltonian system with Hamiltonian H ( p, q) =
p02 + p12 α + V1 (q0 ) + V1 (q1 ) + V2 (q0 − q1 ), V2 (q) = q 2 . 2 2
(1.1)
We assume that the first particle is in contact with a Langevin heat path at temperature T > 0. The second particle is also assumed to have a stochastic force acting on it, but no corresponding friction term, so that it is at ‘infinite temperature’. The corresponding equations of motion are dqi = pi dt, i = {0, 1}, dp0 = −V1 (q0 ) dt + α(q1 − q0 ) dt − γ p0 dt + 2γ T dw0 (t), dp1 = −V1 (q1 ) dt + α(q0 − q1 ) dt + 2γ T∞ dw1 (t),
(1.2)
where w0 and w1 are two independent Wiener processes. Although we use the symbol T∞ in the diffusion coefficient appearing in the second oscillator, this should not be interpreted as a physical temperature since the corresponding friction term is missing, so that detailed balance does not hold, even if T∞ = T . We also assume without further mention throughout this article that the parameters α, γ , T and T∞ appearing in the model (1.2) are all strictly positive. The equations of motion (1.2) determine a diffusion on R4 with generator L given by (1.3) L = X H − γ p0 ∂ p0 + γ T0 ∂ 2p0 + T∞ ∂ 2p1 , where X H is the Liouville operator associated to H , i.e. the first-order differential operator corresponding to the Hamiltonian vector field. It is easy to show that (1.2) has a unique global solution for every initial condition since the evolution of the total energy is controlled by LH = γ (T0 + T∞ ) − γ p02 ,
(1.4)
and so EH (t) ≤ H (0) exp (γ (T0 + T∞ )t). Schematically, the system under consideration can thus be depicted as follows, where we show the three terms contributing to the change of the total energy:
This model is very closely related to the toy model of heat conduction previously studied by various authors in [EPR99b,EPR99a,EH00,RT00,RT02,EH03,Car07,HM08a] consisting of a chain of N anharmonic oscillators coupled at its endpoints to two heat
How Hot Can a Heat Bath Get?
133
baths at possibly different temperatures. The main difference is that the present model does not have any friction term on the second particle. This is similar in spirit to the system considered in [MTVE02,DMP+ 07], where the authors study the stationary state of a ‘resonant duo’ with forcing on one degree of freedom and dissipation on another one. Because of this lack of dissipation, even the existence of a stationary state is not obvious at all in such a system. Indeed, if the coupling constant α is equal to zero, one can easily check that the invariant measure for (1.2) is (formally) given by exp(−( p02 /2+ V1 (q0 ))/T ) dp0 dp1 dq0 dq1 , which is obviously not integrable. One of the main questions of interest for such a system is therefore to understand the mechanism of energy dissipation. In this sense, this is a prime example of a ‘hypocoercive’ system where the dissipation mechanism does not act on all the degrees of freedom of the system directly, but is transmitted to them indirectly through the dynamic [Vil07,Vil08]. This is somewhat analogous to ‘hypoelliptic’ systems, where it is the smoothing mechanism that is transmitted to all degrees of freedom through the dynamic. The system under consideration happens to be hypoelliptic as well, but this is not going to cause any particular difficulty and will not be the main focus of the present work. Furthermore, since one of the heat baths is at ‘infinite’ temperature, even if a stationary state exists, one would not necessarily expect it to behave even roughly like exp(−β H ) for some effective inverse temperature β. It is therefore of independent interest to study the tail behaviour of the energy of (1.2) in its stationary state. In order to simplify our analysis, we are going to limit our investigation to one of the simplest possible cases, where V1 is a perturbation of a homogeneous potential. More precisely, we assume that V1 is an even function of class C 2 such that V1 (x) =
|x|2k + R1 (x), 2k
with a remainder term R1 such that (m)
|R1 (x)| < ∞. 2k−1−m x∈R\[−1,1] m≤2 |x| sup
sup
Here, k ∈ R is a parameter describing the ‘stiffness’ of the individual oscillators. (In the case k = 0, we assume that V1 (x) = C + R1 (x) for some constant C.)
In the case where both ends of the chain are at finite temperature (which would correspond to the situation depicted above), it was shown in [EPR99b,EH00,RT02,Car07] that, provided that the coupling potential V2 grows at least as fast at infinity as the pinning potential V1 and that the latter grows at least linearly (i.e. provided that 21 ≤ k ≤ 1 with our notations), the Markov semigroup associated to the model has a unique invariant
134
M. Hairer
measure µ and its transition probabilities converge to µ at exponential speed. One can actually show even more, namely that the Markov semigroup consists of compact operators in some suitably weighted space of functions. Intuitively, the condition that V2 grows at least as fast as V1 can be understood by the fact that in this case, at high energies, the interaction dominates so that no energy can get ‘trapped’ in the system. Therefore, the system is sufficiently stiff so that if the energy of any one of its oscillators is large, then the energy of all of the oscillators must be large after a very short time. As a consequence, the system behaves like a ‘molecule’ at some effective temperature that moves in the global potential V1 . While the arguments presented in [RT02,Car07] do not cover the case of one of the heat baths being at infinite temperature, it is nevertheless possible to show that in this case, the Markov semigroup Pt generated by solutions to (1.2) behave qualitatively like in the case of finite temperature. In particular, if V1 grows at least linearly at infinity, the system possesses a spectral gap in a space of functions weighted by a weight function ‘close to’ exp(β0 H ) for some β0 > 0. This discussion suggests that: 1. If V2 V1 1, our toy model can sustain arbitrarily large energy currents. 2. In this case, even though the heat bath to the right is at infinite temperature, the system stabilises at some finite ‘effective temperature’, as expressed by the fact that H has finite exponential moments under the invariant measure. This is in stark contrast with the behaviour encountered when V1 grows faster than V2 at infinity. In this case, the interaction between neighbouring particles is suppressed at high energies, which precisely favours the trapping of energy in the bulk of the chain. It was shown in [HM08a] that this can lead in many cases to a loss of compactness of the semigroup generated by the dynamic and the appearance of essential spectrum at 1. This is a manifestation of the fact that energy transport is very weak in such systems, due to the appearance of ‘breathers’, localised structures that only decay very slowly [MA94]. In this case, one expects that the long-time behaviour of (1.2) depends much more strongly on the fine details of the model. For example, regarding the finiteness of the ‘temperature’ of the second oscillator, one may introduce the following notions by increasing order of strength: 1. There exists an invariant probability measure µ for (1.2), that is a positive solution to L∗ µ = 0. 2. There exists an invariant probability measure µ and the average energy of the second oscillator is finite under µ . 3. There exists an invariant probability measure µ and the energy of the second oscillator has some finite exponential moment under µ . We will show that it is possible to find parameters such that the second oscillator does not have finite temperature according to any of these notions of finiteness. On the other hand, it is also possible to find parameters such that it does have finite temperature according to some notions and not to others. It turns out that, maybe rather surprisingly for such a simple model, there are five different critical values for the strength k of the pinning potential V1 that separate between qualitatively different behaviours regarding both the integrability properties of the invariant measure µ and the speed of convergence of transition probabilities towards it. These critical values are k = 0, k = 21 , k = 1, k = 43 , and k = 2. More precisely, there exists
How Hot Can a Heat Bath Get?
135
a constant Cˆ > 0 such that, setting 3 α 2 Cˆ − T∞ 2 , κ = − 1, 4 T∞ k the results in this article can be summarised as follows: ζ =
(1.5)
Theorem 1.1. The integrability properties of the invariant measure µ for (1.2) and the speed of convergence of transition probabilities of (1.2) toward µ can be described by the following table: Parameter range k>2 k = 2, T∞ > α 2 Cˆ k = 2, T∞ < α 2 Cˆ 4 ≤k<2 3
Integrability of µ — — H ζ ±ε exp(γ± H κ )
1 < k ≤ 43 k=1
exp(γ± H κ ) exp(γ± H )
1 2 ≤k<1 0 < k ≤ 21
exp(γ± H )
k≤0
exp(γ± H ) —
Convergence speed — — t −ζ±ε exp −γ± t κ/(1−κ)
Prefactor — — H ζ +ε+1 exp(δ H κ )
exp(−γ± t) exp(−γ± t)
exp(δ H 1−κ ) Hε
exp(−γ ± t) exp −γ± t k/(1−k) —
exp(δ H ) —
1
exp(δ H k −1 )
Here, the symbol ‘—’ means that no invariant probability measure exists for the corresponding range of parameters. Whenever there exists a (necessarily unique) invariant measure µ , we indicate integrability functions I± (H ), convergence speeds ψ± (t) and a prefactor K (H ). The constant ε can be made arbitrarily small, whereas the constants γ+ , γ− and δ are fixed and depend on the fine details of the model. For each line in this table, the following statements hold: • One has R4 I+ (H (x)) µ (d x) = +∞, but R4 I− (H (x)) µ (d x) < +∞. • There exists a constant C such that Pt (x, · ) − µ TV ≤ C K (H (x))ψ+ (t),
(1.6)
for every initial condition x ∈ and every time t ≥ 0. • For every initial condition x ∈ R , there exists a constant C x and a sequence of times tn increasing to infinity such that R4 4
Ptn (x, · ) − µ TV ≥ C x ψ− (tn ), for every n. Remark 1.2. This table is valid only in the case of a ‘chain’ consisting of two oscillators. However, combining the heuristics of Sect. 2 with the formal calculation from [HM08a, Sect. 2], we can conjecture that in the general case of a chain of n + 1 oscillators, one obtains a similar table with κ = 2n k + 1 − 2n. In particular, one would then expect to 2n have non-existence of an invariant probability measure as soon as k > 2n−1 . Remark 1.3. The case k = 2 and T∞ = α 2 Cˆ is not covered by these results. We expect that the system admits no invariant probability measure in this case. In the regime k ≈ 2, the heavy tails of the invariant measure, as well as the slow relaxation speed suggest the appearance of intermittent behaviour. This can be verified numerically (see Fig. 1) and dovetails nicely with the intermittent behaviour that was already observed in [MTVE02,DMP+ 07].
136
M. Hairer
100
Energy
50
0 0
0.2
0.4
0.6
0.8
1
Time (× 105) 1.2
0.2
0.4
0.6
0.8
1
Time ( × 105) 1.2
Energy 3,000 2,000 1,000
0 0
Energy (in millions) 1
0.5
0
Time ( × 105) 0
0.2
0.4
0.6
0.8
1
1.2
Fig. 1. Numerical simulation of the time evolution of the total energy in the case k = 1.1 (top), k = 1.8 (middle) and k = 2.5 (bottom). One clearly sees the appearance of intermittency at k ≈ 2, followed by the lack of a stationary solution for k > 2. The numerics was performed with a Störmer-Verlet scheme that was modified to take into account the damping and the stochastic forcing
Remark 1.4. For k ∈ (0, 21 ), even the gradient dynamic fails to exhibit a spectral gap. It is therefore not surprising (see for example [HN05]) that in this case we see again subexponential relaxation speeds. Remark 1.5. This table exhibits a symmetry κ ↔ k and H κ ↔ H around k = 1 (indicated by a grayed row in the table). The reason for this symmetry will be explained in Sect. 2 below. If we had chosen V1 (x) = K log x + R1 (x) in the case k = 0, this K symmetry would have extended to this case, via the correspondence ζ ↔ T +T − 21 . ∞ Remark 1.6. It follows from (1.6) that the time it takes for the transition probabilities 2 starting from x to satisfy Pt (x, · ) − µ TV ≤ 21 , say, is bounded by H (x)2− k for 1
k ∈ (1, 2) and by H (x) k −1 for k ∈ (0, 1). These bounds are expected to be sharp in view of the heuristics given in Sect. 2 below. Remark 1.7. Instead of considering only distances in total variation between probability measures, we could also have obtained bounds in weighted norms, similarly to [DFG06].
How Hot Can a Heat Bath Get?
137
Remark 1.8. The operator (1.3) appears to be very closely related to the kinetic Fokker-Planck operator LV,2 = p ∂q − ∇V (q) ∂ p − γ p ∂ p + ∂ 2p , for the potential V (q0 , q1 ) = V1 (q0 ) + V1 (q1 ) + α2 (q0 − q1 )2 . The fundamental difference however is that there is a lack of friction on the second degree of freedom. The effect of this is dramatic, since the results from [HN04] (see also [DV01]) show that one has exponential return to equilibrium for the kinetic Fokker-Planck operator in the case k ≥ 1, which is clearly not the case here. Finally, the techniques presented in this article also shed some light on the mechanisms at play in the Helffer-Nier conjecture [HN05, Conj. 1.2], namely that the long-time behaviour of the Fokker-Planck operator without inertia LV,1 = −∇V (q) ∂q + ∂q2 , is qualitatively the same as that of the kinetic Fokker-Planck operator. If V grows faster than quadratically at infinity (so that in particular LV,1 has a spectral gap), then the deterministic motion on the energy levels gets increasingly fast at high energies, so that the angular variables get washed out and the heuristics from Sect. 2.1 below suggests that the total energy of the system behaves like the square of an Ornstein-Uhlenbeck process, thus leading to a spectral gap for LV,2 as well. If on the other hand V grows slower than quadratically at infinity, then the motion of the momentum variable happens on a faster timescale at high energies than that of the position variable. The heuristics from Sect. 2.2 below then suggests that the dynamic corresponding to LV,2 is indeed very well approximated at high energies by that corresponding to LV,1 . These considerations suggest that any counterexample to the Helffer-Nier conjecture would come from a potential that has very irregular (oscillating) behaviour at infinity, so that none of these two arguments quite works. On the other hand, any proof of the conjecture would have to carefully glue together both arguments. The structure of the remainder of this article is the following. First, in Sect. 2, we derive in a heuristic way reduced equations for the energies of the two oscillators. While this section is very far from rigorous, it allows to understand the results presented above by linking the behaviour of (1.2) to that of the diffusion √ d X = −ηX σ dt + 2 dW (t), X ≥ 1, for suitable constants η and σ . The remainder of the article is devoted to the proof of Theorem 1.1, which is broken into five sections. In Sect. 3, we introduce the technical tools that are used to obtain the above statements. These tools are technically quite straightforward and are all based on the existence of test functions with certain properties. The whole art is to construct suitable test functions in a relatively systematic manner. This is done by refining the techniques developed in [HM08a] and based on ideas from homogenisation theory. In Sect. 4, we proceed to showing that k = 2 and T∞ = α 2 Cˆ is the borderline case for the existence of an invariant measure. In Sect. 5, we then show sharp integrability properties of the invariant measure for the regime k > 1 when it exists. This will imply in particular that even though the effective temperature of the first oscillator is always finite (for whatever measure of finiteness), the one of the second oscillator need not
138
M. Hairer
necessarily be. In particular, note that it follows from Theorem 1.1 that the borderline case for the integrability of the energy of the second oscillator in the invariant measure ˆ These two sections form the ‘meat’ of the paper. is given by k = 2 and T∞ = 73 α 2 C. In Sect. 6, we make use of the integrability results obtained previously in order to obtain bounds both from above and from below on the convergence of transition probabilities towards the invariant measure. The upper bounds are based on a recent criterion from [DFG06,BCG08], while the lower bounds are based on a simple criterion that exploits the knowledge that certain functions of the energy fail to be integrable in the invariant measure. Finally, in Sect. 7, we obtain the results for the case k ≤ 1. While these final results are based on the same techniques as the remainder of the article, the construction of the relevant test functions in this case in inspired by the arguments presented in [RT02,Car07]. 1.1. Notations. In the remainder of this article, we will use the symbol C to denote a generic strictly positive constant that, unless stated explicitly, depends only on the details of the model (1.2) and can change from line to line even within the same block of equations. 2. Heuristic Derivation of the Main Results In this section, we give a heuristic derivation of the results of Theorem 1.1. Since we are interested in the tail behaviour of the energy in the stationary state, an important ingredient of the analysis is to isolate the ‘worst-case’ degree of freedom of (1.2), that would be some degree of freedom X which dominates the behaviour of the energy at infinity. The aim of this section is to argue that it is always possible to find such a degree of freedom (but what X really describes depends on the details of the model, and in particular on the value of k) and that, for large values of X , it satisfies asymptotically an equation of the type √ d X = −ηX σ dt + 2 dW (t), (2.1) for some exponent σ and some constant η > 0. Before we proceed with this programme, let us consider the model (2.1) on the set {X ≥ 1} with reflected boundary conditions at X = 1. The invariant measure µ for (2.1) then has density proportional to exp(−ηX σ +1 /(σ + 1)) for σ > −1 and to X −η for σ = −1. In particular, (2.1) admits an invariant probability measure if and only if σ > −1 or σ = −1 and η > 1. For such a model, we have the following result, which is a slight refinement of the results obtained in [Ver00,VK04,Ver06]. Theorem 2.1. The long-time behaviour of (2.1) is described by the following table: Parameter range σ < −1 σ = −1, η ≤ 1
Integrability of µ — —
Convergence speed — —
σ = −1, η > 1 −1 < σ < 0
X η−1±ε exp γ± X σ +1 exp γ± X σ +1 exp γ± X σ +1 exp γ± X σ +1
t 2 ±ε exp −γ± t (1+σ )/(1−σ )
0≤σ <1 σ =1 σ >1
1−η
Prefactor — —
exp(−γ± t)
X η+1+ε exp δ X σ +1 exp δ X 1−σ
exp(−γ± t)
Xε
exp(−γ± t)
1
How Hot Can a Heat Bath Get?
139
The entries of this table have the same meaning as in Theorem 1.1, with the exception that the lower bounds on the convergence speed toward the invariant measure hold for all t > 0 rather than only for a subsequence. Proof. The case 0 ≤ σ ≤ 1 is very well-known (one can simply apply Theorem 3.4 below with either V (X ) = exp(δ X 1−σ ) for δ small enough in the case σ < 1 or with X ε in the case σ = 1). The case σ > 1 follows from the fact that in this case one can find a constant C > 0 such that EX (1) ≤ C, independently of the initial condition. The bounds for σ = −1 and η > 1 can be found in [Ver00,FR05,Ver06] (a slightly weaker upper bound can also be found in [RW01]). However, as shown in [BCG08], the upper bound can also be retrieved by using Theorem 3.5 below with a test function behaving like X η+1+ε for an arbitrarily small value of ε. The lower bound on the other hand can be obtained from Theorem 3.6 by using a test function behaving like X α , but with α 1. (These bounds could actually be slightly improved by choosing test functions of the form X η+1 (log X )β for the upper bound and exp((log X )β ) for the lower bound.) The upper bound for the case σ ∈ (−1, 0) can be found in [VK04] and more recently in [DFG06,BCG08]. This and the corresponding lower bound can be obtained similarly to above from Theorems 3.5 and 3.6 by considering test functions of the form exp(a X σ +1 ) for suitable values of a (small for the upper bound and large for the lower bound). Returning to the problem of interest, it was already noted in [EH00,RT02] that k = 1 is a boundary between two types of completely different behaviours for the dynamic (1.2). The remainder of this section is therefore divided into two subsections where we analyse the behaviour of these two regimes.
2.1. The case k > 1. When k > 1, the pinning potential V1 is stronger than the coupling potential V2 . Therefore, in this regime, one would expect the dynamic of the two oscillators to approximately decouple at very high energies [HM08a]. This suggests that one should be able to find functions H0 and H1 describing the energies of the two oscillators such that H0 is distributed approximately according to exp(−H0 /T ), while the distribution of H1 has heavier tails since that oscillator is not directly damped. In order to guess the behaviour of H1 at high energies, note first that since H0 is expected to have exponential tails, the regime of interest is that where H1 is very large, while H0 is of order one. In this regime, the second oscillator feels mainly its pinning potential, so that its motion is well approximated by the motion of a single free oscillator moving in the potential |q|2k /2k. A simple calculation shows that such a motion 1
is periodic with frequency proportional to H12
1 − 2k
and with amplitude proportional to
1 2k
H1 . In other words, one can find periodic functions P and Q such that in the regime of interest, one has (up to phases) 1
1
q1 (t) ≈ H12k Q(H12
1 − 2k
1
t),
1
p1 (t) ≈ H12 P(H12
1 − 2k
t).
(2.2)
˙ q = p, ˙ p = Q that average out to Let now p and q be the unique solutions to zero over one period. It is apparent from the equations of motion (1.2) that if we assume
140
M. Hairer
that (2.2) is a good model for the dynamic of the second oscillator, then the motion of the first oscillator can, at least to lowest order, be described by 1
p0 (t) = p˜ 0 (t) − α H1k
− 21
1
p (H12
1 − 2k
3
t),
q0 (t) = q˜0 (t) − α H12k
−1
1
q (H12
1 − 2k
t), (2.3)
where the functions p˜ 0 and q˜0 do not show any highly oscillatory behaviour anymore. Furthermore, they then satisfy, at least to lowest order, the decoupled Langevin equation d q˜0 ≈ p˜ 0 dt, d p˜ 0 ≈ −V1 (q˜0 ) dt − α q˜0 dt − γ p˜ 0 dt + 2γ T dw0 (t), (2.4) that indeed has exp(−H0 /T ) as invariant measure, provided that we set H0 =
p˜ 02 α + V1 (q˜0 ) + q˜02 . 2 2
Let us now return to the question of the behaviour of energy dissipation. The average rate of change of the total energy of our system is described by (1.4). Plugging our ansatz (2.3) into this equation and using the fact that p is highly oscillatory and averages out to 0, we obtain 2
LH ≈ γ (T + T∞ ) − γ p˜ 02 − γ α 2 H1k
−1
2p .
On the other hand, it follows from (2.4) that one has LH0 ≈ γ T − γ p˜ 02 ≈ C1 − C2 H0 ,
(2.5)
so that one expects to obtain for the energy of the second oscillator the expression 2
LH1 ≈ γ T∞ − γ α 2 2p H1k
−1
.
This suggests that, at least in the regime of interest, and since the p-dependence of H1 probably goes like of the type
p12 2 ,
the energy of the second oscillator follows a decoupled equation
2 −1 d H1 ≈ γ T∞ − γ α 2 2p H1k dt + 2γ T∞ K H1 dw1 (t),
(2.6)
where K is the average of p12 over one period of the free dynamic at energy 1, which will be shown in (5.10) below to be given by K = 2k/(1 + k). In order to analyse (2.6), it is convenient to introduce the variable X given by X 2 = 4H1 /(γ T∞ K ), so that its evolution is given by dX =
√ 2 2 2 3 γ α p γ T∞ K X 2 k − 2 √ 1 2 −1 − √ + 2 dw1 (t). K X 4 T∞ K
(2.7)
This shows that there is a transition at k = 2. For k > 2, we recover (2.1) with σ = −1 and η = 1 − K2 < 1, so that one does not expect to have an invariant measure, thus recovering the corresponding statement in Theorem 1.1.
How Hot Can a Heat Bath Get?
141
At k = 2, we still have σ = −1, but we obtain η =1−
3 α 2 2p 1 2 2α 2 2p + = − , K T∞ K 2 T∞ 2
so that one expects to have existence of an invariant probability measure if and only if T∞ < α 2 2p . Furthermore, we recover from Theorem 2.1 the integrability results and convergence rates of Theorem 1.1, noting that one has the formal correspondence ζ = (η − 1)/2. This correspondence comes from the fact that H ≈ X 2 in the regime of interest and that X η−1 is the borderline for non-integrability with respect to µ in Theorem 2.1. In the regime k ∈ (1, 2), the first term in the right-hand side of (2.7) is negligible, so that we have the case σ = k4 − 3. Applying Theorem 2.1 then immediately allows to derive the corresponding integrability and convergence results from Theorem 1.1, noting that one has the formal correspondence κ = (σ + 1)/2. 2.2. The case k < 1. This case is much more straightforward to analyse. When k < 1, the coupling potential V2 is stiffer than the pinning potential V1 . Therefore, one expects the two particles to behave like a single particle moving in the potential V1 . This suggests that the ‘worst case’ degree of freedom should be the centre of mass of the system, thus motivating the change of coordinates Q=
q1 − q0 q0 + q1 , q= . 2 2
Fixing Q and writing y = (q, p0 , p1 ) for the remaining coordinates, we see that there exist matrices A and B and a vector v such that y approximately satisfies the equation dy ≈ Ay dt + V1 (Q)v dt + B dw(t). Here, we made the approximation V1 (q0 ) ≈ V1 (q1 ) ≈ V1 (Q), which is expected to be justified in the regime of interest (Q large and y of order one). This shows that for Q fixed, the law of y is approximately Gaussian with covariance of order one and mean proportional to V1 (Q). Since d Q = ( p0 + p1 )/2 dt, we thus expect that over sufficiently long time intervals, the dynamic of Q is well approximated by d Q ≈ −C1 V1 (Q) dt + C2 dW (t) ≈ −C1 Q|Q|2k−2 dt + C2 dW (t), for some positive constants C1 , C2 and some Wiener process W . We are therefore reduced again to the case of Theorem 2.1 with X ∝ |Q| and σ = 2k − 1. Since in the regime considered here one has H ≈ X 2k , this immediately allows to recover the results of Theorem 1.1 for the case k < 1. 3. A Potpourri of Test Function Techniques In this section, we present the abstract results on which all the integrability and nonintegrability results in this article are based, as well as the techniques allowing to obtain upper and lower bounds on convergence rates toward the invariant measure. All of these results without exception are based on the existence of test functions with certain properties. In this sense, we follow to its bitter end the Lyapunov function-based approach advocated in [BCG08,CGWW07,CGGR08] and use it to derive not only upper bounds on convergence rates, but also lower bounds.
142
M. Hairer
While most of these results from this section are known in the literature (except for the one giving the lower bounds on the convergence of transition probabilities which appears to be new despite its relative triviality), the main interest of the present article is to provide tools for the construction of suitable test functions in problems where different timescales are present at the regimes relevant for the tail behaviour of the invariant measure. The general framework of this section is that of a Stratonovich diffusion on Rn with smooth coefficients: d x(t) = f 0 (x) dt +
m
f i (x) ◦ dwi (t), x(0) = x0 ∈ Rn .
(3.1)
i=1
Here, we assume that f j : Rn → Rn are C ∞ vector fields on Rn and the wi are independent standard Wiener processes. Denote by L the generator of (3.1), that is the differential operator given by 1 2 Xi , 2 m
L = X0 +
X j = f j (x)∇x .
i=1
We make the following two standing assumptions which can easily be verified in the context of the model presented in the introduction: Assumption 1. There exists a smooth function H : Rn → R+ with compact level sets and a constant C > 0 such that the bound LH ≤ C(1 + H ) holds. This assumption ensures that (3.1) has a unique global strong solution. We furthermore assume that: Assumption 2. Hörmander’s ‘bracket condition’ holds at every point in Rn . In other words, consider the families Ak (with k ≥ 0) of vector fields defined recursively by A0 = { f 1 , . . . , f m } and Ak+1 = Ak ∪ {[ f j , g], g ∈ Ak ,
j = 0, . . . , m}.
Define furthermore the subspaces A∞ (x) = span{g(x) : ∃k > 0 with g ∈ Ak }. We then assume that A∞ (x) = Rn for every x ∈ Rn . As a consequence of Hörmander’s celebrated ‘sums of squares’ theorem [Hör67, Hör85], this assumption ensures that transition probabilities for (3.1) have smooth densities pt (x, y) with respect to Lebesgue measure. In our case, Assumption 2 can be seen to hold because the coupling potential is harmonic. Assumption 3. The origin is reachable for the control problem associated to (3.1). That is, given any x0 ∈ Rn and any r > 0 there exists a time T > 0 and a smooth control u ∈ C ∞ ([0, T ], Rm ) such that the solution to the ordinary differential equation dz f i (z(t))u i (t), z(0) = x0 , = f 0 (z(t)) + dt m
i=1
satisfies z(T ) ≤ r .
How Hot Can a Heat Bath Get?
143
The fact that Assumption 3 also holds in our case is an immediate consequence of the results in [EPR99a,Hai05]. Assumptions 2 and 3 taken together imply that: 1. The operator L satisfies a strong maximum principle in the following sense. Let D ⊂ Rn be a compact domain with smooth boundary such that 0 ∈ D. Let furthermore u ∈ C 2 (D) be such that Lu(x) ≤ 0 for x in the interior of D and u(x) ≥ 0 for x ∈ ∂ D. Then, one has u(x) ≥ 0 for all x ∈ D, see [Bon69, Theorem 3.2]. 2. The Markov semigroup associated to (3.1) admits at most one invariant probability measure [DPZ96]. Furthermore, if such an invariant measure exists, then it has a smooth density with respect to Lebesgue measure. 3.1. Integrability properties of the invariant measure. Throughout the article, we are going to use the following criterion for the existence of an invariant measure with certain integrability properties: Theorem 3.1. Consider the diffusion (3.1) and let Assumptions 2 and 3 hold. If there exists a C 2 function V : Rn → [1, ∞) such that lim sup|x|→∞ LV (x) < 0, then there exists a unique invariant probability measure µ for (3.1). Furthermore, |LV | is integrable against µ and LV (x)µ (d x) = 0. Proof. The proof is a continuous-time version of the results in [MT93, Chap. 14]. See also for example [HM08a]. The condition given in Theorem 3.1 is actually an if and only if condition, but the other implication does not appear at first sight to be directly useful. However, it is possible to combine the strong maximum principle with a Lyapunov-type criterion to rule out in certain cases the existence of a function V as in Theorem 3.1. This is the content of the next theorem which provides a constructive criterion for the non-existence of an invariant probability measure with certain integrability properties: Theorem 3.2. Consider the diffusion (3.1) and let Assumptions 1, 2 and 3 hold. Let furthermore F : Rn → [1, ∞) be a continuous weight function. Assume that there exist two C 2 functions W1 and W2 such that: • The function W1 grows in some direction, that is lim sup|x|→∞ W1 (x) = ∞. • There exists R > 0 such that W2 (x) > 0 for |x| > R. • The function W2 is substantially larger than W1 in the sense that there exists a positive function H with lim|x|→∞ H (x) = +∞ and such that lim sup R→∞
sup H (x)=R W1 (x) inf H (x)=R W2 (x)
= 0.
• There exists R > 0 such that LW1 (x) ≥ 0 and LW2 (x) ≤ F(x) for |x| > R. Then the Markov process generated by solutions to (3.1) does not admit any invariant measure µ such that F(x) µ (d x) < ∞. Proof. The existence of an invariant measure that integrates F is equivalent to the existence of a positive C 2 function V such that LV ≤ − F outside of some compact set [MT93, Chap. 14]. The proof of the claim is then a straightforward extension of the proof given for the case F ≡ 1 by Wonham in [Won66]. Remark 3.3. If one is able to choose F ≡ 1 in Theorem 3.2, then its conclusion is that the system under consideration does not admit any invariant probability measure.
144
M. Hairer
3.2. Convergence speed toward the invariant measure: upper bounds. We still assume in this section that we are in the same setting as previously and that Assumptions 2–3 hold. The strongest kind of convergence result that one can hope to obtain is exponential convergence toward a unique invariant measure. In order to formulate a result of this type, given a positive function V , we define a weighted norm on measurable functions by ϕV = sup
x∈Rn
|ϕ(x)| . 1 + V (x)
We denote the corresponding Banach space by Bb (Rn ; V ). Furthermore, given a Markov semigroup Pt over Rn , we say that Pt has a spectral gap in Bb (Rn ; V ) if there exists a probability measure µ on Rn and constants C and γ > 0 such that the bound Pt ϕ − µ (ϕ)V ≤ Ce−γ t ϕ − µ (V )V , holds for every ϕ ∈ Bb (Rn ; V ). We will also say that a C 2 function V : Rn → R+ is a Lyapunov function for (3.1) if lim|x|→∞ V (x) = ∞ and there exists a strictly positive constant c such that LV ≤ −cV, holds outside of some compact set. With this notation, we have the following version of Harris’ theorem [MT93] (see also [HM08b] for an elementary proof): Theorem 3.4. Consider the diffusion (3.1) and let Assumptions 2 and 3 hold. If there exists a Lyapunov function V for (3.1), then Pt admitsa spectral gap in Bb (Rn ; V ). In particular, (3.1) admits a unique invariant measure µ , V dµ < ∞, and convergence of transition probabilities towards µ is exponential with prefactor V . However, there are situations where exponential convergence does simply not take place. In such situations, one cannot hope to be able to find a Lyapunov function as above, but it is still possible in general to find a ϕ-Lyapunov function V in the following sense. Given a function ϕ : R+ → R+ , we say that a C 2 function V : Rn → R+ is a ϕ-Lyapunov function if the bound LV ≤ −ϕ(V ), holds outside of some compact set and if lim|x|→∞ V (x) = ∞. If such a ϕ-Lyapunov function exists, upper bounds on convergence rates toward the invariant measure can be obtained by applying the following criterion from [DFG06,BCG08] (see also [FR05]): Theorem 3.5. Consider the diffusion (3.1) and let Assumptions 2 and 3 hold. Assume that there exists a ϕ-Lyapunov function V for (3.1), where ϕ is some increasing smooth concave function that is strictly sublinear. Then (3.1) admits a unique invariant measure µ and there exists a positive constant c such that for all x ∈ Rn , the bound Pt (x, ·) − µ TV ≤ cV (x)ψ(t), t holds, where ψ(t) = 1/(ϕ ◦ Hϕ−1 )(t) and Hϕ (t) = 1 (1/ϕ(s)) ds.
How Hot Can a Heat Bath Get?
145
3.3. Convergence speed toward the invariant measure: lower bounds. In order to obtain lower bounds on the rate of convergence towards the invariant measure µ , we are going to make use of the following mechanism. Suppose that we know of some function G that on the one hand it has very heavy (non-integrable) tails under the invariant measure of some Markov process but, on the other hand, its moments do not grow too fast. Then, this should give a lower bound on the speed of convergence towards the invariant measure since the moment bounds prevent the process from exploring its heavy tails too quickly. This is made precise by the following elementary result: Theorem 3.6. Let X t be a Markov process on a Polish space X with invariant measure µ and let G : X → [1, ∞) be such that: • There exists a function f : [1, ∞) → [0, 1] such that the function Id · f : y → y f (y) is increasing to infinity and such that µ (G ≥ y) ≥ f (y) for every y ≥ 1. • There exists a function g : X × R+ → [1, ∞) increasing in its second argument and such that E(G(X t ) | X 0 = x0 ) ≤ g(x0 , t). Then, one has the bound µtn − µ TV ≥
1 f (Id · f )−1 (2g(x0 , tn )) , 2
(3.2)
where µt is the law of X t with initial condition x0 ∈ X . Proof. It follows from the definition of the total variation distance and from Chebyshev’s inequality that, for every t ≥ 0 and every y ≥ 1, one has the lower bound µt − µ TV ≥ µ (G(x) ≥ y) − µt (G(x) ≥ y) ≥ f (y) −
g(x0 , t) . y
Choosing y to be the unique solution to the equation y f (y) = 2g(x0 , t), the result follows. The problem is that in our case, we do not in general have sufficiently good information on the tail behaviour of µ to be able to apply Theorem 3.6 as it stands. However, it follows immediately from the proof that the bound (3.2) still holds for a subsequence of times tn converging to ∞, provided that the bound µ (G ≥ yn ) ≥ f (yn ) holds for a sequence yn converging to infinity. This observation allows to obtain the following corollary that is of more use to us: Corollary 3.7. Let X t be a Markov process on a Polish space X with invariant measure µ and let W : X → [1, ∞) be such that W (x) µ (d x) = ∞. Assume that there exist F : [1, ∞) → R and h : [1, ∞) → R such that: ∞ • h is decreasing and 1 h(s) ds < ∞. • F · h is increasing and lims→∞ F(s)h(s) = ∞. • There exists a function g : X × R+ → R+ increasing in its second argument and such that E((F ◦ W )(X t ) | X 0 = x0 ) ≤ g(x0 , t). Then, for every x0 ∈ X , there exists a sequence of times tn increasing to infinity such that the bound µtn − µ TV ≥ h (F · h)−1 (g(x0 , tn )) holds, where µt is the law of X t with initial condition x0 ∈ X .
146
M. Hairer
Proof. Since W (x) µ (d x) = ∞, there exists a sequence wn increasing to infinity such that µ (W (x) ≥ wn ) ≥ 2h(wn ), for otherwise we would have the bound ∞ ∞ W (x) µ (d x) = 1 + µ (W (x) ≥ w) dw ≤ 1 + 2 h(w) dw < ∞, 1
1
thus leading to a contradiction. Applying Theorem 3.6 with G = F ◦W and f = 2h◦F −1 concludes the proof. 4. Existence and Non-existence of an Invariant Probability Measure 4.1. Non-existence of an invariant probability measure. The aim of this section is to show that (1.2) does not admit any invariant probability measure if k > 2 or k = 2 and ˆ Note first that one has an upper bound on the evolution of the total energy T∞ > α 2 C. of the system given by LH = γ T + γ T∞ − γ p02 , which suggests that H is a natural choice for the function W2 in Wonham’s criterion for the non-existence of an invariant probability measure. It therefore remains to find a function W1 that grows to infinity in some direction (not necessarily all), that is dominated by the energy in the sense that lim
E→∞
1 E
sup
H ( p,q)=E
W1 ( p, q) = 0,
(4.1)
and such that LW1 ≥ 0 outside of some compact region K. In order to construct W1 , we use some of the ideas introduced in [HM08a]. The technique used there was to make a change of variables such that, in the new variables, the motion of the ‘fast’ oscillator decouples from that of the ‘slow oscillator’. In the situation at hand, we wish to show that the energy of the second oscillator grows, so that the relevant regime is the one where that energy is very high. One is then tempted to set W1 = H −ζ (H − H0 ),
(4.2)
for some (typically small) exponent ζ ∈ (0, 1), where H0 is a multiple of the energy of the first oscillator, expressed in the ‘right’ set of variables. In order to compute LW1 , we make use of the following ‘chain rule’ for L: L( f ◦ g) = (∂i f ◦ g)Lgi + (∂i2j f ◦ g)(gi , g j ),
(4.3)
(summation over repeated indices is implied), where we defined the ‘carré du champ’ operator (gi , g j ) = γ T ∂ p0 gi ∂ p0 g j + γ T∞ ∂ p1 gi ∂ p1 g j . (Note that it differs by a factor two from the usual definition in order to keep expressions as compact as possible.) This allows us to obtain the identity LW1 = H −ζ (γ T + γ T∞ − γ p02 − LH0 ) −γ ζ H −ζ −1 (H − H0 )(T + T∞ − p02 )
−2γ ζ H −ζ −1 T p0 ( p0 − ∂ p0 H0 ) + T∞ p1 ( p1 − ∂ p1 H0 ) +γ ζ (ζ + 1)H −ζ −2 (H − H0 ) T p02 + T∞ p12 .
(4.4)
How Hot Can a Heat Bath Get?
147
Following our heuristic calculation in Sect. 2.1, we expect that at high energies, one has LH0 ≈ γ T −γ p˜ 02 , where p˜ 0 denotes the correct variable in which to express the motion of the oscillator. One would then like to first choose our compact set K sufficiently large so that the expression on the first line of (4.4) is larger than δ H −ζ (1+ p02 ) for some δ > 0. Then, by choosing ζ sufficiently close to zero, one would like to make the remaining terms sufficiently small so that LW1 > 0 outside of a compact set. This is made precise by the following lemma: Lemma 4.1. Let L be as in (1.3). Assume that there exist a C 2 function H0 : R4 → R and strictly positive constants c and C such that, outside of some compact subset of R4 , it satisfies the bounds LH0 ≤ γ (T + T∞ − p02 ) − c(1 + p02 ), |H0 | + |∂ p0 H0 |2 + |∂ p1 H0 |2 ≤ C H. If the function H0 furthermore satisfies lim sup E→∞
1 E
inf
H (x)=E
H0 (x) < 1,
(4.5)
then (1.2) admits no invariant probability measure. Proof. Setting W1 as in (4.2), we see from (4.4) and the assumptions on H0 that there exists a constant C > 0 independent of ζ ∈ (0, 1) such that the bound LW1 ≥ cH −ζ (1 + p02 ) − ζ C H −ζ (1 + p02 ) holds outside of some compact set. Choosing ζ < c/C, it follows that LW1 > 0 outside of some compact subset of R4 . Assumption (4.5) makes sure that W1 grows to +∞ in some direction and rules out the trivial choice H0 ∝ H . Since it follows furthermore from the assumptions that W1 ≤ C H 1−ζ , (4.1) holds so that the assumptions of Wonham’s criterion are satisfied. The remainder of this section is devoted to the construction of such a function H0 , thus giving rise to the following result: Theorem 4.2. There exists a constant Cˆ such that, if either k > 2, or k = 2 and ˆ the model (1.2) admits no invariant probability measure. T∞ > α 2 C, Remark 4.3. As will be seen from the construction, the constant Cˆ is really equal to the constant 2p from Sect. 2.1. Proof. As in [HM08a], we define the Hamiltonian Hf (P, Q) =
P 2 |Q|2k + 2 2k
of a ‘free’ oscillator on R2 and its generator L0 = P∂ Q − Q|Q|2k−1 ∂ P .
(4.6)
These definitions will be used for all of the remainder of this article, except for Sect. 7. The variables (P, Q) should be thought of as ‘dummy variables’ that will be replaced by for example ( p1 , q1 ) or ( p0 , q0 ) when needed.
148
M. Hairer
We also define as the unique centred1 solution to the Poisson equation L0 = Q − R(P, Q), where R : R2 → R is a smooth function averaging out to zero on level sets of Hf , and such that R = 0 outside of a compact set and R = Q inside an open set containing the origin. The reason for introducing the correction term R is so that the function is smooth everywhere including the origin, which would not be the case otherwise. It 1
follows from [HM08a, Prop. 3.7] that scales like Hfk
− 21
in the sense that, outside
1 1 k −2
a compact set, it can be written as = Hf 0 (ω), where ω is the angle variable conjugate to Hf . Inspired by the formal calculation from Sect. 2.1, we then define p˜ 0 = p0 − α( p1 , q1 ), so that the equations of motion for the first oscillator turn into dq0 = p˜ 0 dt + α dt,
(4.7) d p˜ 0 = −q0 |q0 |2k−2 dt − αq0 dt − γ p0 dt + 2γ T dw0 (t) 2 2 +αR dt − α (q0 − q1 )∂ P dt − α 2γ T∞ ∂ P dw1 (t) − αγ T∞ ∂ P dt +α R1 (q1 )∂ P dt − R1 (q0 ) dt. Here, we omitted the argument ( p1 , q1 ) from , its partial derivatives, and R in order to make the expressions shorter. Setting H˜ 0 =
p˜ 02 + Veff (q0 ) + θ p˜ 0 q0 , 2
Veff (q) = V1 (q) + α
q2 , 2
(4.8)
we obtain the following identity: L H˜ 0 = γ T − (γ − θ ) p02 − θ |q0 |2k − αθ |q0 |2 − γ θ p0 q0 +α 2 (γ − θ )2 + α(γ − θ ) p˜ 0 + αVeff (q0 ) +α p˜ 0 R − α 2 p˜ 0 (q0 − q1 )∂ P − αγ T∞ p˜ 0 ∂ P2 + α 2 γ T∞ (∂ P )2 +θq0 αR − α 2 (q0 − q1 )∂ P − αγ T∞ ∂ P2 +( p˜ 0 + θq0 )α R1 (q1 )∂ P .
(4.9)
All the terms on lines 3 to 5 (and also the terms on line 2 provided that k > 2) are of the form f ( p0 , q0 )g( p1 , q1 ) with g a function going to 0 at infinity and f a function such that f ( p0 , q0 )/Hf ( p0 , q0 ) goes to 0 at infinity. It follows that, for every ε > 0, there exists a compact set K ε ⊂ R4 such that, outside of K ε , one has the inequality θγ 2 + ε p02 − (θ − ε)|q0 |2k L H˜ 0 ≤ γ T − γ p02 + ε + θ + 4α +α 2 (γ − θ )2 + α(γ − θ ) p˜ 0 + αVeff (q0 ).
(4.10)
1 We say that a function on R2 is centred if it averages to 0 along orbits of the Hamiltonian system with Hamiltonian Hf .
How Hot Can a Heat Bath Get?
149
Here, we also used the fact that γ θ p0 q0 ≤ αθ |q0 |2 + γ4αθ p02 . If k > 2, then the function also converges to 0 at infinity, so that the bound θγ 2 L H˜ 0 ≤ γ T − γ p02 + ε + θ + + ε p02 − (θ − ε)|q0 |2k , 4α 2
holds outside of a sufficiently large compact set. It follows that the conditions of Lemma 4.1 are satisfied by H0 = (1 + δ) H˜ 0 for δ > 0 sufficiently small whenever T∞ > 0, provided that one also chooses both θ and ε sufficiently small. The case k = 2 is slightly more subtle and we assume that k = 2 for the remainder of this proof. In particular, this implies that scales like a constant outside of some compact set. This suggests that the term 2 should average out to a constant, whereas the terms p˜ 0 and Veff (q0 ) should average out to zero, modulo some lower-order corrections. It turns out that these corrections will have the unfortunate property that they grow faster than Hf in the ( p0 , q0 ) variables. On the other hand, we notice that both p˜ 0 and Veff (q0 ) do grow slower than Hf at infinity. As a consequence, it is sufficient to compensate these terms for ‘low’ values of ( p0 , q0 ). Before giving the precise expression for a function H0 that satisfies the assumptions of Lemma 4.1 for the case k = 2, we make some preliminary calculations. We denote by ψ : R → R+ a smooth decreasing ‘cutoff function’ such that ψ(x) = 1 for x ≤ 1 and ψ(x) = 0 for x ≥ 2. Given a positive constant E, we also set Hf Hf Hf ( p˜ 0 , q0 ) 1 1 , ψ E = ψ , ψ E = 2 ψ . ψ E ( p˜ 0 , q0 ) = ψ E E E E E Definition 4.4. We will say that a function f : R+ × R4 → R is negligible if, for every 4 ε > 0, there exists E ε > 0 and, for every E > E ε there
exists a compact set K E,ε R such that the bound | f (E; p, q)| ≤ ε 1 + Hf ( p˜ 0 , q0 ) holds for every ( p, q) ∈ K E,ε . With this definition at hand, we introduce the notations f g,
f ∼ g,
(4.11)
to mean that there exists a negligible function h such that f ≤ g + h or f = g + h respectively. With this notation, we can rewrite (4.10) as L H˜ 0 γ T − γθ p02 − θ |q0 |2k + α 2 (γ − θ )2 + f θ , where we introduced the constant γθ = γ − θ (1 + θ ) p˜ 0 + αVeff (q0 ).
γ2 4α )
(4.12)
and the function f θ = α(γ −
Lemma 4.5. Let a, b ≥ 0 and let f, g : R2 → R be functions that scale like Hfa and Hf−b respectively. Then, the following functions are negligible:
i) ii) ii’) iii) iv)
f ( p˜ 0 , q0 )g( p1 , q1 )ψ E ( p˜ 0 , q0 ), provided that b > 0. f ( p˜ 0 , q0 )g( p1 , q1 )ψ E ( p˜ 0 , q0 ), provided that b > 0 or a < 2.
2 f ( p˜ 0 , q0 )g( p1 , q1 ) ψ E ( p˜ 0 , q0 ) , provided that b > 0 or a < 3. f ( p˜ 0 , q0 )g( p1 , q1 )ψ E ( p˜ 0 , q0 ), provided that b > 0 or a < 3. f ( p˜ 0 , q0 )g( p1 , q1 )(1 − ψ E ( p˜ 0 , q0 )), provided that a < 1.
150
M. Hairer
Proof. We assume without loss of generality that the bounds f ( p, q) ≤ 1 ∨ Hfa ( p, q)
and g( p, q) ≤ 1 ∧ Hf−b ( p, q) hold for every ( p, q) ∈ R2 . In the case i), we take E ε = 1 and choose for K E,ε the set of points such that either Hf ( p˜ 0 , q0 ) ≥ 2E, in which case the expression vanishes, or Hf ( p1 , q1 ) ≥ (2E)a/b ε−1/b in which case the expression is smaller than ε. The case ii) with b > 0 follows exactly like the case i), so we consider the case a < 2 and b = 0. Since ψ E = 0 if Hf ( p˜ 0 , q0 ) ≥ 2E and is smaller than 1/E otherwise, we have the bounds | f ( p˜ 0 , q0 )g( p1 , q1 )ψ E ( p˜ 0 , q0 )| ≤ (1 + Hf ( p˜ 0 , q0 ))E (0∨a−1)−1 . Since the exponent of E appearing in this expression is negative provided that a < 2, this is shown to be negligible by choosing E ε sufficiently large and setting K E,ε = φ. Cases ii’) and iii) follow in a nearly identical manner. In the case iv), we use the fact that since a < 1, for fixed ε > 0, we can find a constant Cε such that | f ( p˜ 0 , q0 )| ≤ 2ε Hf ( p˜ 0 , q0 ) + Cε . We then set E ε = 2Cε /ε, so that Hf ( p˜ 0 , q0 ) ≥ E ε implies Hf ( p˜ 0 , q0 ) ≥ 2Cε ε . Since g is bounded by 1 by assumption and since 1 − ψ E vanishes for Hf ( p˜ 0 , q0 ) ≤ E, it follows that the expression iv) is uniformly bounded by ε Hf ( p˜ 0 , q0 ) for E ≥ E ε . Remark 4.6. In the case where both b > 0 and a < 1, the function f ( p˜ 0 , q0 )g( p1 , q1 ) is negligible, which can be seen from cases i) and iv) above. Corollary 4.7. In the setting of Lemma 4.5, the following functions are negligible: v) f ( p˜ 0 , q0 )g( p1 , q1 )∂ p0 ψ E ( p˜ 0 , q0 ) provided that b > 0 or a < 3/2. vi) f ( p˜ 0 , q0 )g( p1 , q1 )∂ p1 ψ E ( p˜ 0 , q0 ). vii) f ( p˜ 0 , q0 )g( p1 , q1 )Lψ E ( p˜ 0 , q0 ) provided that b > 21 − k1 or b = 21 − k1 and a < 1. Proof. We can write f ( p˜ 0 , q0 )g( p1 , q1 )∂ p0 ψ E ( p˜ 0 , q0 ) = p˜ 0 f ( p˜ 0 , q0 )g( p1 , q1 )ψ E , f ( p˜ 0 , q0 )g( p1 , q1 )∂ p1 ψ E ( p˜ 0 , q0 ) = p˜ 0 f ( p˜ 0 , q0 )g( p1 , q1 )∂ P ( p1 , q1 )ψ E , so that the first two cases can be reduced to case ii) of Lemma 4.5. For case vii), we use the fact that Lψ E = ψ E LHf + γ T0 + T∞ (∂ P )2 p˜ 02 ψ E , (4.13) and that LHf consists of terms that all scale like Hfc ( p˜ 0 , q0 )Hfd ( p1 , q1 ) with c ≤ 1 and d ≤ k1 − 21 (see (4.9)) to reduce ourselves to cases ii) and iii) of Lemma 4.5. Before we proceed with the proof of Theorem 4.2, we state two further preliminary results that will turn out to be useful also for the analysis of the case k ∈ (1, 2): Lemma 4.8. Let k ∈ (1, 2] and let f : R2 → R be a function that scales like Hfa for some a ∈ R. Then, the function g = L( f ( p˜ 0 , q0 )) consists of terms that are bounded by 1 1 multiples of Hfc ( p˜ 0 , q0 )Hfd ( p1 , q1 ) with either c ≤ a + 21 − 2k and d ≤ 0 or c ≤ a − 2k 1 1 and d ≤ k − 2 .
How Hot Can a Heat Bath Get?
151
Proof. It follows from (4.7) that g = −αq0 − γ ( p˜ 0 + α) + αR − α 2 (q0 − q1 )∂ P − αγ T∞ ∂ P2 ∂ P f
+ α R1 (q1 )∂ P − R1 (q0 ) ∂ P f +γ T + T∞ (∂ P )2 ∂ P2 f + ( p˜ 0 + α)∂ Q f − q0 |q0 |2k−2 ∂ P f, from which the claim follows by simple powercounting.
Lemma 4.9. Let k ∈ (1, 2] and let f : R2 → R be a function that scales like Hf−b for some b ∈ R. Then, the function g = L( f ( p1 , q1 ))−(L0 f )( p1 , q1 ) consists of terms that 1 are bounded by multiples of Hfc ( p˜ 0 , q0 )Hfd ( p1 , q1 ) with either c ≤ 2k and d ≤ −b − 21 1 1 or c ≤ 0 and d ≤ −b − 2 + 2k . Proof. It follows from (1.2) that g = α(q0 − q1 )∂ P f − R1 (q1 )∂ P f + γ T∞ ∂ P2 f, from which the claim follows.
(4.14)
We now return to the proof of Theorem 4.2. We define as the unique centred solution to the equation L0 = . One can see in a similar way as before that scales like −1 H 4 . Since scales like a constant, there exists some constant Cˆ such that 2 averages f
to Cˆ outside a compact set. While the constant Cˆ can not be expressed in simple terms, it is easy to compute it numerically: Cˆ ≈ 0.6354699.2 ˆ : R+ → R+ with compact support and such In particular, there exists a function R ˆ f (P, Q)) is centred. Denote by the centred solution to the equation that 2 − Cˆ + R(H ˆ f (P, Q)), L0 = 2 − Cˆ + R(H − 14
so that scales like Hf
(4.15)
, just like does. With these definitions at hand, we set
H0 = H˜ 0 − α 2 (γ − θ )( p1 , q1 ) + f θ ( p1 , q1 ) ψ E ( p˜ 0 , q0 ),
(4.16)
where we used the function f θ introduced in (4.12). Recalling that f θ consists of terms scaling like Hfa ( p˜ 0 , q0 ) with a ≤ 34 , we obtain from Lemmas 4.9 and 4.5 that f θ L( ( p1 , q1 ))ψ E = f θ − f θ (1 − ψ E ) + f θ (L − L0 )ψ E ∼ f θ . Similarly, we obtain that ˆ E ∼ 2 − C. ˆ L(( p1 , q1 ))ψ E = 2 − Cˆ − 2 − Cˆ (1 − ψ E ) + Rψ 2 All displayed digits are accurate.
152
M. Hairer
It therefore follows from (4.12), the facts that ∂ p0 p˜ 0 = 1 and ∂ p1 p˜ 0 = −α∂ P ( p1 , q1 ), and the multiplication rule for L, that one has the bound LH0 γ T − γθ p02 − θ |q0 |2k + α 2 (γ − θ )Cˆ −(α 2 (γ − θ ) + f θ )Lψ E − L f θ ψ E +C|∂ P ∂ p1 ψ E | + C| f θ ∂ P ∂ p1 ψ E | +C| ∂ P f θ 1 + α 2 (∂ P )2 ψ E | + C|∂ P ∂ P f θ ∂ P ψ E |. The terms on the second and third line are negligible by Lemma 4.8 and Corollary 4.7. The terms on the last line are negligible by Lemma 4.5, so that we finally obtain the bound ˆ ˆ − γθ p02 − θ |q0 |2k − α 2 θ C. LH0 γ (T + α 2 C)
(4.17)
Since the constant γθ can be made arbitrarily close to γ by choosing θ sufficiently small, ˆ it is possible to choose θ small enough we see as before that, provided that T∞ > α 2 C, and E large enough so that the choice H0 = (1+δ)H0 with δ > 0 sufficiently small again allows to satisfy the conditions of Lemma 4.1. This concludes the proof of Theorem 4.2.
4.2. Existence of an invariant measure. Theorem 4.2 has the following converse: ˆ the model (1.2) admits a Theorem 4.10. If either 1 < k < 2, or k = 2 and T∞ < α 2 C, unique invariant probability measure µ . The constant Cˆ is the same as in Theorem 4.2. Proof. Somewhat surprisingly given that the two statements are almost diametrically opposite, it is possible to prove this positive result in very similar way to the previous negative result by constructing the right kind of Lyapunov function. As before, the case k = 2 will be treated somewhat differently. The case k = 2. Similarly to what we did in (4.2), the idea is to look at the function V = H − cH0 for a suitable constant c, but this time we choose it in such a way that lim|( p,q)|→∞ V = ∞ and lim sup|( p,q)|→∞ LV < 0, so that we can apply Theorem 3.1. Note that, with the same notations as in the proof of Theorem 4.2, one has from (4.9), L H˜ 0 ∼ γ T − (γ − θ ) p02 − θ |q0 |2k − αθ |q0 |2 − γ θ p0 q0 + α 2 (γ − θ )2 + f θ , so that, provided this time that we choose θ < 0 in the definition of H˜ 0 (and therefore of H0 ), we have the bound LH0 ∼ γ T + α 2 (γ − θ )Cˆ − (γ − θ ) p02 − θ |q0 |2k − αθ |q0 |2 − γ θ p0 q0 γ T + α 2 (γ − θ )Cˆ − γθ p02 − θ |q0 |2k , where we set γθ = γ − θ (1 + γ4α ) as before. Here, the function H0 is as in (4.16) and depends on a large parameter E as above. If we choose c < 1, the function 2
V = H − cH0
(4.18)
How Hot Can a Heat Bath Get?
153
does then indeed grow to infinity in all directions and we have LV γ T (1 − c) + γ T∞ − cα 2 (γ − θ )Cˆ − c|θ ||q0 |2k − (γ − cγθ )| p0 |2 . If the assumption α 2 Cˆ > T∞ is satisfied, we can find a constant β > 0 such that γ T (1 − c) + γ T∞ − cα 2 (γ − θ )Cˆ ≤ −β for all θ sufficiently small and all c sufficiently close to 1. By fixing c and making θ sufficiently small, we can furthermore ensure that γ − cγθ > 0. This shows that, by first choosing c sufficiently close to 1, then making θ very small and finally choosing E very large, we have constructed a function V satisfying the assumptions of Theorem 3.1, thus concluding the proof in the case k = 2. The case k < 2. Even though one would expect this to be the easier case, it turns out to be tricky because of the fact that the approximate decoupling of the oscillators at high energies is not such a good description of the dynamic anymore. The idea is to consider again the variable p˜ 0 introduced previously but, because of the fact that the function is now no longer bounded, we are going to multiply certain correction terms by a ‘cutoff function’. Since we are following a similar line of proof to the non-existence result and since we expect from (2.5) and (2.6) to be able to find a function V close to H and such that it 2
−1
asymptotically satisfies a bound of the type LV ≈ −Hf ( p˜ 0 , q0 ) − Hfk ( p1 , q1 ), this suggests that we should introduce the following notion of a negligible function suited to this particular case: Definition 4.11. A function f : R4 → R is negligible exists a if, for every ε > 0, there 2
compact set K ε such that the bound | f ( p, q)| ≤ ε Hf ( p˜ 0 , q0 ) + Hfk
−1
( p1 , q1 ) holds
for every ( p, q) ∈ K ε . We also introduce the notations ∼ and similarly to before. For θ > 0, we then set Vˆ = H + θ p˜ 0 q0 , so that (4.7) yields LVˆ = γ (T + T∞ ) − γ p02 + αθ p˜ 0 + θ p˜ 02 − θ |q0 |2k − αθ |q0 |2 − γ θ p0 q0 +θ αq0 R − α(q0 − q1 )∂ P − γ T∞ ∂ P2
+θq0 α R1 (q1 )∂ P − R1 (q0 ) . It is straightforward to check that all of the terms on the second and third lines are negligible. Using the definition of p˜ 0 and completing the square for the term α|q0 |2 + γ p0 q0 , we thus obtain the bound LVˆ −γθ p˜ 02 − θ |q0 |2k + cθ p˜ 0 − αθ 2 . Here, we defined the constants γθ def αθ = αγ α − , 4 in order to shorten the expressions.
cθ = (αθ − 2αγ + 21 γ 2 θ ), def
(4.19)
154
M. Hairer
¯ : R2 → R+ As before, we see that there exists a positive constant C¯ and a function R 2 −1 ¯ is smooth, centred, and vanishes in with compact support such that 2 − C¯ Hfk + R a neighbourhood of the origin. Similarly to (4.15), we define as the unique centred solution to 2
−1 ¯ L0 (P, Q) = 2 (P, Q) − C¯ Hfk (P, Q) + R(P, Q), 5
and as the unique centred solution to L0 = . Note that scales like Hf2k 3 2k −1
that scales like Hf
− 23
and
.
At this stage, we would like to define V = Vˆ + αθ ( p1 , q1 ) − cθ p˜ 0 ( p1 , q1 ) in order to compensate for the last two terms in (4.19). The problem is that when applying the generator to p˜ 0 , we obtain an unwanted term of the type q0 |q0 |2k−2 , which grows too fast in the q0 direction. We note however that the term p˜ 0 only needs to be compensated when | p˜ 0 | , which is the regime in which the description (4.7) is expected to be relevant. We therefore consider the same cutoff function ψ as before and we set 1 + Hf ( p˜ 0 , q0 ) ˆ V = V + αθ ( p1 , q1 ) − cθ p˜ 0 ( p1 , q1 )ψ , (4.20) (1 + Hf ( p1 , q1 ))η for a positive exponent η to be determined later. In order to obtain bounds on LV , we make use of the fact that Lemma 4.9 still applies to the present situation. In particular, we can apply it to the function , thus obtaining the bound 2 k −1 LV −Cθ Hf ( p˜ 0 , q0 ) + Hf ( p1 , q1 ) + cθ ( p˜ 0 − L( p˜ 0 ( p1 , q1 )ψ)),
for some constant Cθ , where it is understood that the function ψ is composed with the ratio appearing in (4.20). Using the fact that L0 = by definition and applying the chain rule (4.3) for L, we thus obtain 2 k −1 LV −Cθ Hf ( p˜ 0 , q0 ) + Hf ( p1 , q1 ) −cθ ((L p˜ 0 ) ψ + p˜ 0 Lψ + p˜ 0 (L − L0 ) + p˜ 0 L (ψ − 1)))
−cθ T ∂ p1 p˜ 0 ∂ P ψ + ∂ p1 p˜ 0 ∂ p1 ψ + p˜ 0 ∂ P ∂ p1 ψ − cθ T∞ ∂ p0 ψ. (4.21) We claim that all the terms appearing on the second and the third line of this expression are negligible, thus concluding the proof. The most tricky part of showing this is to obtain bounds on Lψ. Define E 0 = 1 + Hf ( p˜ 0 , q0 ) and E 1 = 1 + Hf ( p1 , q1 ) as a shorthand. Our main tool in bounding LV is then the following result which shows that the terms containing Lψ are negligible: Proposition 4.12. Provided that η ∈ [2 − k, k], there exists a constant C such that η 1 Lψ E 0η ≤ C, ∂ p ψ E 0η ≤ C E − 2 , ∂ p ψ E 0η ≤ C E − 2 . 0 1 1 1 E1 E1 E1
How Hot Can a Heat Bath Get?
155
Proof. Define the function f : R2+ → R+ by f (x, y) = ψ((1 + x)/(1 + y)η ). It can then be checked by induction that, for every pair of positive integers m and n with m + n > 0 and for every real number β, there exists a constant C such that the bound |∂xm ∂ yn f | ≤ (1 + x)−m+β (1 + y)−n−ηβ
(4.22)
holds uniformly in x and y. It furthermore follows from (4.7) and (1.2) that 1 1 1 1− 2k k−2 , E1 |LE 0 | ≤ C E 0 + E 0 1 1 1 1 + |LE 1 | ≤ C E 12 2k + E 02k E 12 , 1/2
|∂ p0 E 0 | ≤ C E 0 , 1 2
1 k −1
|∂ p1 E 0 | ≤ C E 0 E 1
|∂ p0 E 1 | = 0, ,
1
|∂ p1 E 1 | ≤ C E 12 .
Combining these two bounds with (4.22) and the chain rule (4.3), the required bounds follow. Let us now return to the bound on LV . It is straightforward to check that 1 1 1 1− 2k k −2 , + E1 |L p˜ 0 | ≤ C E 0 for some constant C, so that 3 5 1− 1 −1 −3 | ( p1 , q1 )L p˜ 0 | ≤ C E 0 2k E 12k + E 12k 2 , which is negligible. Combining Proposition 4.12 with the scaling behaviours of and , one can check in a similar way that the term p˜ 0 Lψ, as well as all the terms appearing on the third line of (4.21) are also negligible. It therefore remains to bound p˜ 0 (L − L0 ) and p˜ 0 L (ψ − 1). It follows from (4.14) that 1 1 3 3 1 2 3 2k − 2 2k 2k 2k k −2 E0 + E1 ≤ C E0 + E1 , (4.23) |(L − L0 ) | ≤ C E 1 1
so that | p˜ 0 (L − L0 ) | is negligible as well. Since we know that L0 scales like E 1k it follows from (4.23) that 1 3 1 −3 1− 1 | p˜ 0 L | ≤ C E 02 E 12k 2 E 02k + E 1 2k .
− 21
,
This term has of course no chance of being negligible: we have to use the fact that it is η multiplied by 1 − ψ. The function 1 − ψ is non-vanishing only when E 0 ≥ E 1 , so that we obtain 1 1 1 3 3 1 1 1 1 + + ( − ) + ( − ) | p˜ 0 L (1 − ψ)| ≤ C E 02k 2 η 2k 2 + E 02 η k 2 .
156
M. Hairer
We see that both exponents are strictly smaller than 1, provided that η > 2k − 1. Combining all of these estimates with (4.21), we see that, provided that η ∈ ( 2k − 1, k), there exists a constant C such that 2 −1 . LV −C E 0 + E 1k In particular, using the scaling of , we deduce the existence of a constant c such that the bound 2
LV ≤ −cV k −1 ,
(4.24)
holds outside of a sufficiently large compact set (we can choose such a set so that V is positive outside), thus concluding the proof of Theorem 4.10 by applying Theorem 3.1.
5. Integrability Properties of the Invariant Measure The aim of this section is to explore the integrability properties of the invariant measure µ when it exists. First of all, we show the completely unsurprising fact that: Proposition 5.1. For all ranges of parameters for which there exists an invariant measure µ , one has exp (β H (x)) µ (d x) = ∞ for every β > 1/T . Proof. Choose β > β2 > 1/T . Setting W2 (x) = exp(β2 H (x)), we have LW2 = γβ2 W2 T + T∞ − p02 + β2 T p02 + T∞ p12 ≤ exp(β H ), outside of a sufficiently large compact set. Setting similarly W1 = exp(H (x)/T ), we see immediately from a similar calculation that LW1 ≥ 0, so that the result follows from Theorem 3.2. Remark 5.2. Actually, one can show similarly a slightly stronger result, namely that there exists some exponent α < 1 such that H α exp(H/T ) is not integrable against µ . 5.1. Energy of the first oscillator. What is maybe slightly more surprising is that the tail behaviour of the distribution of the energy of the first oscillator is not very strongly influenced by the presence of an infinite-temperature heat bath just next to it, provided that we look at the correct set of variables. Indeed, we have: Proposition 5.3. Let either 23 ≤ k < 2 or k = 2 and T∞ be such that there exists an invariant probability measure µ . Then exp β Hf ( p˜ 0 , q0 ) µ (d x) < ∞ for every β < 1/T . Remark 5.4. When k = 2, is bounded and the exponential integrability of Hf ( p˜ 0 , q0 ) is equivalent to that of Hf ( p0 , q0 ). This is however not the case when k < 2. Remark 5.5. The borderline case k = 23 is expected to be optimal if we restrict ourselves to the variables ( p˜ 0 , q0 ). This is because for k < 23 one would have to add additional correction terms taking into account the nonlinearity of the pinning potential.
How Hot Can a Heat Bath Get?
157
The main ingredient in the proof of Proposition 5.3 is the following proposition, which is also going to be very useful for the non-integrability results later in this section. Proposition 5.6. For every θ > 0, there exist functions H0 , pˆ 0 : R4 → R and a constant Cθ such that • For every ε > 0, there exists a constant Cε such that the bounds 0 ≤ H0 ≤ (1 + ε)H + Cε ,
(5.1)
hold. • Provided that k ≥ 23 , for every ε > 0 there exists a constant Cε such that the bound (1 − ε)Hf ( p˜ 0 , q0 ) − Cε ≤ H0 ≤ (1 + ε)Hf ( p˜ 0 , q0 ) + Cε ,
(5.2)
holds. • One has the bounds (∂ p0 H0 − pˆ 0 )2 ≤ Cθ + θ 4 H0 ,
(5.3a)
(∂ p1 H0 ) ≤ Cθ + θ H0 .
(5.3b)
2
4
• If furthermore k ≥ 23 , the bound LH0 ≤ Cθ − (γ − 2θ ) pˆ 02 − θ H0 holds. • If k ∈ (4/3, 3/2) then, for every δ > (2k − 1)( k3 − 2), one has the bound LH0 ≤ Cθ − (γ − 2θ ) pˆ 02 − θ H0 + θ 2 Hfδ ( p1 , q1 ). Remark 5.7. The presence of p˜ 0 rather than pˆ 0 in (5.2) is not a typographical mistake. Proof. We start by defining the differential operator K acting on functions F : R2 → R as
KF = γ T∞ ∂ P2 F ( p1 , q1 ) + α qˆ0 − q ( p1 , q1 ) − q1 − R1 (q1 ) (∂ P F)( p1 , q1 ), so that KF = L(F( p1 , q1 )) − (L0 F)( p1 , q1 ). Setting pˆ 0 = p0 + p ( p1 , q1 ),
η
qˆ0 = q0 + q ( p1 , q1 )ψ(E 0 /E 1 ),
(5.4)
for some yet to be defined functions p and q and for E i and ψ as in Proposition 4.12, we then obtain
d qˆ0 = pˆ 0 dt + L0 q − p dt +ψKq dt + (ψ − 1)L0 q dt + q Lψ dt + γ T∞ ∂ p1 ψ∂ P q dt
+ 2γ T∞ ψ∂ P q + q ∂ p1 ψ dw1 (t) + 2γ T q ∂ p0 ψ dw0 (t), d pˆ 0 = −Veff (qˆ0 ) dt − γ pˆ 0 dt + 2γ T dw0
+ L0 p − αq1 + γ p dt +K p dt + Veff (qˆ0 ) − Veff (q0 ) dt + 2γ T∞ ∂ P p dw1 (t), (5.5) 2
where we defined as before the effective potential Veff (q) = V1 (q) + α q2 . (1) Let E > 0 and set p as the unique centred solution to
L0 (1) p = α Q 1 − ψ Hf (P, Q)/E ,
(5.6)
158
M. Hairer (2)
where ψ is the same cutoff function already used previously. We then define p by (2) (1) (1) (2) L0 p = γ p and we set p = p + p . This ensures that one has the identity L0 p − αq1 + γ p = R p , 3 where the function R p consists of terms that scale like Hfa with a ≤ 2k − 1. We furthermore set q to be the unique centred solution to L0 q = p . Note that p consists of terms scaling like Hfa with a ≤ k1 − 21 and that q consists of terms scaling like Hfa 3 with a ≤ 2k − 1. The introduction of the parameter E in (5.6) ensures that we can make functions scaling like a negative power of Hf arbitrarily small in the supremum norm. 1
It follows indeed that one has for example |∂ P p | ≤ C E k −1 . With these definitions at hand, it follows from (5.5) that
d qˆ0 = pˆ 0 dt + 2γ T∞ ψ∂ P q + q ∂ p1 ψ dw1 (t) + 2γ T q ∂ p0 ψ dw0 (t) +ψKq dt + (ψ − 1) p dt + q Lψ dt + γ T∞ ∂ p1 ψ∂ P q dt, d pˆ 0 = −Veff (qˆ0 ) dt − γ pˆ 0 dt + 2γ T dw0 +R p dt + K p dt + Veff (qˆ0 ) − Veff (q0 ) dt + 2γ T∞ ∂ P p dw1 (t). (5.7) Let now H0 be defined by pˆ 02 + Veff (qˆ0 ) + θ pˆ 0 qˆ0 + C0 , 2 where C0 is a sufficiently large constant so that H0 ≥ 1. Note that, as a consequence of the definitions of pˆ 0 and qˆ0 , if k ≥ 3/2 then | pˆ 0 − p˜ 0 | and |qˆ0 − q0 | are bounded so that the two-sided bound (5.2) does indeed hold. Showing that the weaker one-sided bound (5.1) holds for every k ∈ [ 23 , 2] is straightforward to check. Before we turn to the proof of (5.3), let us recall the definitions of E 0 and E 1 from the proof of the case k < 2 of Theorem 4.10, and define similarly Eˆ 0 = 1 + Hf ( pˆ 0 , qˆ0 ). If k ≥ 23 , then Eˆ 0 and E 0 are equivalent in the sense that they are bounded by multiples of each other. If k < 23 , this is not the case, but it follows from the definitions of p and q that E 0 ≤ C Eˆ 0 + E 13−2k . Eˆ 0 ≤ C E 0 + E 13−2k , H0 =
It follows that, provided that we impose the condition η > 3 − 2k, where η is the exponent appearing in (5.4), then one has the implications E0 ≤ C E1
η
⇒
E0 ≥
η C E1
⇒
η Eˆ 0 ≤ C˜ E 1 , η Eˆ 0 ≥ C˜ E 1 ,
(5.8a) (5.8b)
for some constant C˜ depending on C. We will assume from now on that the condition η > 3 − 2k is indeed satisfied. Let us now show that (5.3a) holds. We have the identity ∂ p0 H0 − pˆ 0 = θ qˆ0 + Veff (qˆ0 ) + θ pˆ 0 q ∂ p0 ψ. Since the term θ qˆ0 satisfies the required bound, we only need to worry about the second term. It follows from Proposition 4.12 and from the scaling of q that this term is
How Hot Can a Heat Bath Get?
159 1
3
η
1− −1− 2 bounded by a multiple of Eˆ 0 2k E 12k . Since the bounds (5.8) hold on the support 3
η
−1− 2k 1/2 of ∂ p0 ψ, this in turn is bounded by a multiple of Eˆ 0 E 12k , so that the requested bound follows, provided again that the condition η > 3 − 2k holds. Turning to (5.3b), we have the identity ∂ p1 H0 = ( pˆ 0 + θ qˆ0 )∂ P p + Veff (qˆ0 ) + θ pˆ 0 ∂ p1 (q ψ).
Making use of the parameter E introduced in (5.6), it follows that the first term is 2k 1/2 1 bounded by Eˆ 0 E k −1 , which can be made sufficiently small by choosing E θ 1−k . In order to bound the second term, we expand the last factor into q ∂ p1 ψ + ψ∂ P q . The first term can be bounded just as we did for ∂ p0 H0 , noting that the bound on ∂ p1 ψ in Proposition 4.12 is better than the bound on ∂ p0 ψ. Using the fact that (5.8a) holds 1
η−3
(1− 1 )
k , which on the support of ψ, the second term yields a bound of the form Eˆ 02 E 1 2 yields the required bound provided that η < 3. It therefore remains to show the bound on LH0 . It follows from (5.7) that one has the identity
LH0 = γ T − (γ − θ ) pˆ 02 − θ |qˆ0 |2k − αθ |qˆ0 |2 − γ θ pˆ 0 qˆ0
2 +γ T∞ (∂ P p )2 + Veff (qˆ0 ) ∂ p1 (q ψ) + 2θ ∂ P p ∂ p1 (q ψ) +γ T Veff (qˆ0 )(q ∂ p0 ψ)2 + 2θ q ∂ p0 ψ +( pˆ 0 + θ qˆ0 ) R p + K p + Veff (qˆ0 ) − Veff (q0 ) + Veff (qˆ0 ) + θ pˆ 0
× ψKq + (ψ − 1) p + q Lψ + γ T∞ ∂ p1 ψ∂ P q . (5.9) We now use the following notion of a negligible function. A function f : R+ × R4 → R is negligible if, for every ε > 0 there exists a constant E ε and, for every E > E ε , there exists a constant Cε such that the bound | f (E; p, q)| ≤ Cε + ε Eˆ 0 + E 1δ holds, where δ is as in the statement of the proposition. (Set δ = 0 for k ≥ 3/2.) With this notation, the required bounds follow if we can show that all the terms appearing in (5.9) are negligible, except for those on the first line. The terms appearing in the second line are all smaller than the last term appearing in ∂ p1 H0 and so they are negligible. Similarly, the terms appearing in the third line are smaller than those appearing in ∂ p1 H0 − pˆ 0 . It is easy to see that the first termon the fourth line is negligible. Concerning the 1 3 −1 , so that this term is also seen to second term, we see that |K p | ≤ C Eˆ 02k + E 12k be negligible by power counting. Note now that the definitions of Veff and qˆ0 imply that one has the bound 3 −1 Veff (qˆ0 ) − Veff (q0 ) ≤ C 1 + |qˆ0 |2k−2 + |q0 |2k−2 Hf2k ( p1 , q1 ) 3 3 (2k−2)( 2k −1) −1 E 12k ≤ C 1 + |qˆ0 |2k−2 + E 1 3 3 1− 1 −1 (2k−1)( 2k −1) . ≤ C Eˆ 0 k E 12k + E 1
160
M. Hairer
η Furthermore, one has Veff (qˆ0 ) = Veff (q0 ), unless Eˆ 0 ≤ E 1 , so that we have the bound 3 1 η( 1 − 1 )+ 3 −1 (2k−1)( 2k −1) . pˆ 0 Veff (qˆ0 ) − Veff (q0 ) ≤ C Eˆ 0 E 1 2 k 2k + Eˆ 02 E 1
The second term is always negligible. Furthermore, if η > (3 − 2k)/(2 − k) the first term is also negligible. We now turn to the last line in (5.9). In order to bound the term involving Kq , note that the functions q ∂ P q , ∂ P2 q , and Q∂ P ϕq are bounded provided that k ≥ 4 3 , so that the terms involving these expressions are negligible. Concerning the term Veff (qˆ0 )qˆ0 ∂ P q , we use the fact that ∂ P q can be made arbitrarily small by choosing E large enough in (5.6) to conclude that it is also negligible. The term involving p is 1−
1
+1(1−1)
bounded by a multiple of Eˆ 0 2k η k 2 , so that it is negligible provided that η > 2 − k. The term involving q Lψ is bounded similarly, using the fact that Lψ is bounded by Proposition 4.12 and that q scales like a smaller power of Hf than p . Finally, the last term is negligible since ∂ p1 ψ∂ P q is bounded, thus concluding the proof of Proposition 5.6. Note that the choice η = 2 for example allows to satisfy all the conditions that we had to impose on η in the interval k ∈ [4/3, 2]. We are now able to give the Proof of Proposition 5.3. It follows from (5.2) that if we can show that exp(βH0 ) is integrable with respect to µ for every β < 1/T , then the same is also true for exp(β Hf ( p˜ 0 , q0 )), provided that we restrict ourselves to the range k ≥ 23 . Before we proceed, we also note that (5.3a) implies that for θ sufficiently small, one has the bound (∂ p0 H0 )2 ≤ (1 + θ ) pˆ 02 + C˜ θ + θ 2 H0 , for some constant C˜ θ . Setting W = exp(βH0 ), we thus have the bound LW = LH0 + γβ T (∂ p0 H0 )2 + T∞ (∂ p1 H0 )2 βW ≤ Cθ − (γ − 2θ ) pˆ 02 − θ H0 + γβ(1 + θ ) pˆ 02 + Cθ 2 H0 , for some constant C independent of θ . Since we assumed that β < 1/T , we can make θ sufficiently small so that −(γ − 2θ ) + γβ(1 + θ ) < 0 and Cθ 2 − θ < 0. The claim then follows from Theorem 3.1. 5.2. Integrability and non-integrability in the case k = 2. We next show that if k = 2 ˆ then the invariant measure is heavy-tailed in the sense that there exists and T∞ ≤ α 2 C, an exponent ζ such that H ζ (x) µ (d x) = ∞. Our precise result is given by: ˆ one has H ζ (x) µ (d x) = ∞, provided that Theorem 5.8. If k = 2 and T∞ ≤ α 2 C, ζ > ζ = def
Conversely, one has
3 α 2 Cˆ − T∞ . 4 T∞
H ζ (x) µ (d x) < ∞ for ζ < ζ .
How Hot Can a Heat Bath Get?
161
Proof. We first show the positive result, namely that H ζ is integrable with respect to µ for any ζ < ζ . Fixing such a ζ , our aim is to construct a smooth function W bounded from below such that, for some small value ε > 0, the bound LW ≤ −ε H ζ holds outside of some compact set. This then immediately implies the required integrability by Theorem 3.1. Consider the function V defined in (4.18). Note that this function depends on parameters E, θ and c and that, for any given value of ε > 0, it is possible to choose first θ sufficiently small and c sufficiently close to 1, and then E sufficiently large, so that the bound LV ≤ γ T∞ − α 2 γ Cˆ + ε, holds outside of some compact set. Let us now turn to the behaviour of ∂ p0 V and ∂ p1 V . It follows from the definitions, Lemma 4.5, and Corollary 4.7 that one has the identity
2 ∂ p0 V = (1 − c)2 p02 + R0 , where the function R0 can be bounded by an arbitrarily small multiple of V outside of some sufficiently large compact set. Furthermore, it follows from the definition of V and the construction of H0 that one has the bound V ≥ 1−c 2 H outside of some compact set, so that we have the bound
2 ∂ p0 V ≤ 4(1 − c)V + R0 . Ensuring first that 1 − c ≤ ε/8 and then choosing E sufficiently large, it follows that
2 we can ensure that ∂ p0 V ≤ εV outside of a sufficiently large compact set. It follows in a similar way that, by possibly choosing E even larger, the bound
2 ∂ p1 V ≤ p12 + εV holds outside of some compact set. Note now that since L0 (P Q) = 3P 2 − 4Hf ,
(5.10)
˜ : R2 → R be a centred compactly the function P 2 − 43 Hf is centred. Let furthermore R 4 2 ˜ supported function such that P − 3 Hf + R vanishes in a neighbourhood of the origin ˜ be the centred solution to and let ˜ = P2 − L0
4 ˜ Hf + R, 3
(5.11)
so that we have the identity ˜ p1 , q1 ) = p12 − L(
4 ˜ p1 , q1 ) + α(q0 − q1 ) − R1 (q1 ) ∂ P ˜ ( p1 , q1 ). Hf ( p1 , q1 ) + R( 3
Furthermore, it follows at once from the definition of V and the scaling behaviours of and that the bound Hf ( p1 , q1 ) ≤ (1 + ε)V,
162
M. Hairer
˜ is bounded and scales like holds outside of some compact set. Since furthermore R 3
Hf4 , it follows that the bound 4 ˜ p1 , q1 ) ≥ p12 − (1 + ε)V, L( 3 holds outside of some (possibly larger) compact set. Finally, it follows from the scaling ˜ that the bounds of ˜ ˜ ≤ εV, |LV | ≤ εV and |∂ p1 V ∂ p1 |
(5.12)
hold outside of some sufficiently large compact set. With all these definitions at hand, we consider the function ˜ p1 , q1 ). W = V ζ +1 − γ ζ (ζ + 1)T∞ V ζ (
(5.13)
Note that V is positive outside of a compact set, so that W is well-defined there. Since we do not care about compactly supported modifications of W , we can assume that (5.13) makes sense globally. We then have the identity
2
2 ˜ LW = (ζ + 1)V ζ LV + ζ γ (ζ + 1)V ζ −1 T ∂ p0 V + T∞ ∂ p1 V − T∞ L ˜ . ˜ + γ T∞ ∂ p1 V ∂ p1 − γ ζ 2 (ζ + 1)T∞ V ζ −1 LV Collecting all of the bounds obtained above, this in turn yields the bound 4 ζ 2 ˆ ζ LW ≤ (ζ + 1)V γ T∞ − α γ C + ε + ζ γ (ζ + 1)V T ε + T∞ ε + T∞ (1 + ε) 3 − γ εζ 2 (ζ + 1)T∞ V ζ 4 ≤ γ (ζ + 1) T∞ − α 2 Cˆ + ζ T∞ + K ε V ζ , 3 holding for some constant K > 0 independent of ε outside of some sufficiently large compact set. It follows that if ζ < ζ , it is possible to choose ε sufficiently small so that the prefactor in this expression is negative, thus yielding the desired result. We now prove the ‘negative result’, namely that H ζ is not integrable with respect to µ if ζ > ζ . In order to show this, we are going to apply Wonham’s criterion with W2 = H 1+ζ . It therefore suffices to find a function W1 growing to infinity in some direction, such that LW1 > 0 outside of some compact set, and such that sup
H ( p,q)=E
W1 ( p, q)E −1−ζ → 0
(5.14)
as E → ∞. We are going to construct W1 in a way very similar to the construction in the proof of the positive result above. Fix some arbitrarily small ε > 0 as before. Setting V as above, note first that it follows immediately from (4.17) that, by choosing first θ sufficiently small, then c sufficiently close to 1 and finally E large enough, we can ensure that the bound LV ≥ γ T∞ − α 2 γ Cˆ − ε(1 + p02 )
How Hot Can a Heat Bath Get?
163
holds outside of some sufficiently large compact set. Similarly as before, we can also ensure that the bound
2 ∂ p1 V ≤ p12 − εV holds. Fix now some ζ˜ ∈ (ζ , ζ ) and define W0 as in (5.13), but with ζ˜ replacing ζ . It follows that the bound 4 ˜ 2 ˆ 2 ˜ ˜ LW0 ≥ γ (ζ + 1) T∞ − α C + ζ T∞ − K ε(1 + p0 ) V ζ , 3 holds for some constant K > 0 outside of some compact set. The problem is that the right hand side of this expression is not everywhere positive because of the appearance of the term p02 . This can however be dealt with by setting ˜
W1 = W0 − K ε H 1+ζ , so that
(5.15)
4 ˜ LW1 ≥ γ (ζ˜ + 1) T∞ − α 2 Cˆ + ζ˜ T∞ − K˜ ε V ζ , 3
for some different constant K˜ . Since ζ˜ > ζ , we can ensure that this term is uniformly positive by choosing ε sufficiently small. By possibly making ε even smaller, we can furthermore guarantee that W1 grows in some direction, despite the presence of the ˜ term −K ε H 1+ζ in (5.15). Finally, the condition (5.14) is guaranteed to hold because we ˜ choose ζ < ζ . As a corollary of Theorem 5.8, we obtain: ˆ then even though the system admits Corollary 5.9. If k = 2 and α 2 Cˆ > T∞ > 37 α 2 C, , the average kinetic energy of the second oscillator is a unique invariant measure µ infinite, that is p12 µ (d x) = ∞. Proof. The proof is very similar to the proof of the “negative part” of Theorem 5.8. However, instead of choosing W2 = H 2 , we choose W2 = H 2 + K p1 q1 for some constant K . Since this additional term does not change the behaviour of W2 at infinity, the conclusions of Wonham’s criterion still apply, showing that (LW2 )+ is not integrable with respect to µ . A simple explicit calculation shows that, provided that K is large
enough, there exists a positive constant C such that LW2 ≤ C 1 + Hf ( p0 , q0 ) + p12 . On the other hand, we know that the expectation of Hf ( p0 , q0 ) is finite under µ by the remark following Proposition 5.3, so that the expectation of p12 under µ necessarily diverges. 5.3. Integrability and non-integrability in the case k < 2. In this case, we show that the exponential of a suitable fractional power of H is integrable with respect to the invariant measure. Our positive result is given by: Theorem 5.10. For every k ∈ (1, 2) there exists δ > 0 such that 2 exp δ H k −1 (x) µ (d x) < ∞, R4
where µ is the unique invariant measure for (1.2).
(5.16)
164
M. Hairer
Proof. Define W = exp(δV κ ) for a (small) constant δ > 0 and an exponent κ ∈ (0, 1] to be determined later (the optimal exponent will turn out to be κ = 2k − 1). Here, V is the function that was previously defined in (4.20). Since V and H are equivalent in the sense that there exist positive constants C1 and C2 such that C1−1 V − C2 ≤ H ≤ C1 V + C2 , showing the integrability of W implies (5.16) for a possibly different constant δ. Applying the chain rule (4.3), we obtain outside of a sufficiently large compact set the bound LW = δκ W V κ−1 LV + (δκ V 2κ−2 + (κ − 1)V κ−2 )(V, V ) ≤ δκ W V κ−1 LV + 2δκ V κ−1 (V, V ) . (5.17) Note now that it follows immediately from (4.20) and Proposition 4.12 that, outside of some compact set, one has the bounds 1 1 3 α 3 1 1 √ 2k −1− 2 2k −1 2 2 2 2 ≤ C E0 + E1 ≤ C V , |∂ p0 V | ≤ C E 0 + E 0 E 1 + E1 1 1 1 5 3 3 1 1 √ 2k 2k −2 2k − 2 2 2 2 2 ≤ C E0 + E1 ≤ C V , + E0 E1 |∂ p1 V | ≤ C E 1 + E 0 + E 1 so that (V, V ) ≤ C V . Combining this with (4.24), we obtain the existence of constants c and C (possibly depending on κ, but not depending on δ) such that 2 (5.18) LW ≤ δW V κ−1 C + CδV κ − cV k −1 , thus concluding the proof.
We have the following partial converse to Theorem 5.10: Theorem 5.11. Let k ∈ ( 43 , 2). Then, there exists > 0 such that 2 exp H k −1 (x) µ (d x) = ∞,
(5.19)
R4
where µ is the unique invariant measure for (1.2). Proof. We are again going to make use of Wonham’s criterion. Let K˜ be a (sufficiently large) constant, define κ = 2k − 1 ∈ (0, 21 ), set F(x) = exp(H κ (x)), and set W2 (x) = exp K˜ H κ (x) . We then have the bound LW2 = H κ−1 T + T∞ − p02 + H κ−2 (κ − 1 + κ K˜ H κ ) T p02 + T∞ p12 γ κ K˜ W2 ≤ C 1 + H 2κ−1 , for some constant C > 0. In particular, we have LV ≤ F outside of some compact set, provided that we choose > K˜ .
How Hot Can a Heat Bath Get?
165
˜ the centred solution to Similarly to 5.11, we denote by ˜ = Hf − L0
k+1 2 ˜ P + R, 2k
˜ ensuring that the right hand side vanishes in for some compactly supported function R a neighbourhood of the origin. Let now K be any constant smaller than K˜ , let M be a (large) positive constant to be determined later, and set def ˜ p1 , q 1 ) = exp(K H1 ), W1 = exp K H κ − 2H κ−1 H0 + M H 2κ−2 ( ˜ was defined above. Note that the where H0 is the function from Proposition 5.6 and properties of H0 imply that, outside of some compact set, one has the bounds 1 ≤ H0 ≤ (1 + ε)H. It is clear that W2 is much larger than W1 at infinity, so that it remains to show that LW1 > 0 outside of a compact set for K sufficiently large. We are actually going to show that there exists a constant C such that (LW1 )/W1 ≥ C H 2κ−1 outside of some compact set. Therefore, we call a function f negligible if, for every ε > 0, there exists a compact set such that | f | ≤ ε H 2κ−1 outside of this set. Note that since we consider the range of parameters such that κ < 21 , bounded functions are not negligible in general. Using the chain rule (4.3), we have the identity LW1 = LH1 + γ K T (∂ p0 H1 )2 + T∞ (∂ p1 H1 )2 K W1 ≥ LH1 + γ K T∞ (∂ p1 H1 )2 . We first turn to the estimate of LH1 . Using again (4.3), we have the identity ˜ LH1 = κ H κ−1 + 2(1 − κ)H κ−2 H0 LH − 2H κ−1 LH0 + M H 2κ−2 L ˜ ˜ + γ M(2κ − 2)(2κ − 3)H 2κ−4 T p02 + T∞ p12 + 2(κ − 1)M H 2κ−3 LH + γ (κ − 1)H κ−3 (κ H + 2(2 − κ)H0 ) T p02 + T∞ p12
+ 2γ (1 − κ)H κ−2 T p0 ∂ p0 H0 + T∞ p1 ∂ p1 H0 ˜ + γ M T∞ (2κ − 2)H 2κ−3 p1 ∂ P . ˜ scales like a power of We see immediately that since κ is strictly positive and since the energy strictly smaller than one, all terms except for the ones on the first line are negligible. Furthermore, it follows from (5.1) that κ κ−1 κ H κ−1 + 2(1 − κ)H κ−2 H0 ≤ 2 − H , 2 say. Combining this with Proposition 5.6 and the fact that the inequality κ > (2k − 1)( k3 − 2) holds in the range of parameters under consideration, we obtain the lower bound κ ˜ − 2 p02 + 2(γ − 2θ ) pˆ 02 + 2θ H0 + M H 2κ−2 L. LH1 H κ−1 γ 2
166
M. Hairer
Using the definition of pˆ 0 , and choosing θ < γ κ/8, we obtain the existence of a constant C such that ˜ LH1 H κ−1 2θ H0 − C Hfκ ( p1 , q1 ) + M H 2κ−2 L0 . ˜ in order to replace L ˜ by L0 . ˜ Here, we also made use of the scaling properties of Note that the constant C appearing in the expression above can be made independent of θ provided that we restrict ourselves to θ ≤ γ κ/16, say. At this point, we make the choice M = 2C and we set c = k+1 2k , so that we have the lower bound M κ Hf ( p1 , q1 ) + M H 2κ−2 (Hf ( p1 , q1 ) − cp12 ) LH1 H κ−1 2θ H0 − 2 M κ−1 κ Hf ( p1 , q1 ) + M H 2κ−2 (H0 + Hf ( p1 , q1 ) − cp12 ), − H 2 where we made use of the fact that, since κ < 1, for every constant C, there is a compact set such that H κ−1 H0 ≥ C H 2κ−2 H0 outside of that compact set. From the definitions of H and H0 , we see that there exists a constant C and a compact set outside of which CH0 + Hf ( p1 , q1 ) > 43 H , say, so that we finally obtain the lower bound M 2κ−1 H − cM H 2κ−2 p12 . 4
LH1
(5.20)
Let us now turn to the term (∂ p1 H1 )2 . We have the identity ˜ p1 ∂ p1 H1 = H κ−2 (κ H + 2(1 − κ)H0 ) p1 + 2M(κ − 1)H 2κ−3 ˜ − 2H κ−1 ∂ p1 H0 + M H 2κ−2 ∂ P . 2
Using the inequality (a + b)2 ≥ a2 − b2 , as well as the bound (5.3b), it follows that there exists a constant C such that the bound (∂ p1 H1 )2 ≥
κ 2 2κ−2 2 H p1 − 16θ 4 H 2κ−2 H0 − C H 2κ−2 , 2
holds. Combining this bound with (5.20), we obtain the lower bound LW1 M 2κ−1 H + K W1 4
γ K T∞ κ 2 − cM H 2κ−2 p12 − 16γ K T∞ θ 4 H 2κ−2 H0 . 2
We now choose K = 2cM/(γ T∞ κ 2 ) so that the second term vanishes. The prefactor of the last term is then given by 32Mcθ 4 /κ 2 . Choosing θ small enough so that θ 4 < κ 2 /(256c), say, we finally obtain the lower bound LW1 ≥
M K 2κ−1 W1 > 0, H 16
valid outside of some sufficiently large compact set, as required.
How Hot Can a Heat Bath Get?
167
6. Convergence Speed Towards the Invariant Measure In this section, we are concerned with the convergence rates towards the invariant measure in the case 1 < k ≤ 2, where it exists. Our main result will be that k = 43 is the threshold separating between exponential convergence and stretched exponential convergence. 6.1. Upper bounds. Our main tool for upper bounds will be the integrability bounds obtained in the previous section, together with the results recently obtained in [DFG06, BCG08]. The results obtained in Sect. 5 suggest that it is natural to work in spaces of functions weighted by exp(δV ε ), where V was defined in (4.18). For ε > 0 and δ > 0, we therefore define the space B(ε, δ) as the closure of the space of all smooth compactly supported functions under the norm
ϕ(ε,δ) = sup |ϕ(x)| exp −δ H ε (x) , x∈R4
where we used the letter x to denote the coordinates ( p0 , q0 , p1 , q1 ). Note that the dual norm on measures is a weighted total variation norm with weight exp(δ H ε (x)). We also say that a Markov semigroup Pt with invariant measure µ has a spectral gap in a Banach space B containing constants if there exist constants C and γˆ such that Pt ϕ − µ (ϕ)B ≤ Ce−γˆ t ϕB , ∀ϕ ∈ B. As a consequence of the bounds of Sect. 5, we obtain: Theorem 6.1. Let k ∈ (1, 2] and set κ = 2k − 1. Then, the semigroup Pt extends to a
C0 -semigroup on the space B(ε, δ), provided that ε ≤ max 21 , 1 − κ . Furthermore: a. If 1 < k < 43 then, for every ε ∈ [1 − κ, κ) and every δ > 0, the semigroup Pt has a spectral gap in B(ε, δ). Furthermore, there exists δ0 > 0 such that it has a spectral gap in B(κ, δ) for every δ ≤ δ0 . In particular, for every δ > 0 there exist constants C > 0 and γˆ > 0 such that the bound Pt (x, · ) − µ TV ≤ C exp(δ H 1−κ (x))e−γˆ t ,
(6.1)
holds uniformly over all initial conditions x and all times t ≥ 0. b. If k = 43 , then there exists δ0 > 0 such that the semigroup Pt has a spectral gap in B( 21 , δ) for every δ ≤ δ0 . In particular, there exists δ > 0 such that the convergence result (6.1) holds. c. For 43 < k < 2, there exist positive constants δ, C and γˆ such that the bound Pt (x, · ) − µ TV ≤ C exp(δ H κ (x))e−γˆ t
κ/(1−κ)
(6.2)
holds uniformly over all initial conditions x and all times t ≥ 0. d. For the case k = 2, set ζ as in Theorem 5.8. Then, for every T∞ < α 2 Cˆ and every ζ < ζ , there exists C > 0 such that the bound Pt (x, · ) − µ TV ≤ C H 1+ζ (x)t −ζ , holds uniformly over all initial conditions x and all times t ≥ 0.
(6.3)
168
M. Hairer
Proof. The set of bounded continuous functions is dense in B(ε, δ) and is mapped into itself by Pt . Therefore, in order to show that it extends to a C0 -semigroup on B(ε, δ), it remains to verify that: 1. There exists a constant C such that Pt ϕ(ε,δ) ≤ Cϕ(ε,δ) for every t ∈ [0, 1] and every bounded continuous function ϕ. 2. For every ϕ ∈ C0∞ , one has limt→0 Pt ϕ − ϕ(ε,δ) = 0. Using the a priori bounds on the solutions given by the bound LH ≤ γ (T + T∞ ), it is possible to check that the second statement holds for every (ε, δ). The first claim then follows from [MT93] and (5.18). It remains to show claims a to d. Claims a and b follow immediately from (5.18). To show that claim c also holds, we use the fact that, by using (5.18) in the case ε = 2k − 1, we can find δ > 0 such that the bound k
1
LW ≤ −δ 2 V 2κ−1 W = −δ 2−k W (log(W ))2− κ , holds outside of some compact subset of R4 . Since we are considering a regular Markov process, every compact set is petite. This shows that there exists a constant δ such that, in the terminology of [BCG08], W is a ϕ-Lyapunov function for our model with k
1
ϕ(t) = δ 2−k t (log t)2− κ . In particular, this yields the identity log(t) t k 1 1−κ ds = δ − 2−k s κ −2 ds = C(log t) κ , Hϕ (t) = ϕ(s) 1 0 for some constant C depending on δ and κ. It follows from the results in [BCG08] that the convergence rate to the invariant measure is given by ψ(t) =
1 (ϕ ◦
Hϕ−1 )(t)
= Ct
1−2κ 1−κ
e−γ t
κ/(1−κ)
,
for some positive constants C and γ , so that (6.2) follows. The case k = 2 can be treated in a very similar way. It follows from the first part of the proof of Theorem 5.8 that there exists β > 0 and a function W growing like H 1+ζ at infinity such that one has the bound LW ≤ −β H ζ outside of some sufficiently large ζ
compact set. Therefore, W is a ϕ-Lyapunov function for ϕ(t) = −βt 1+ζ . Following the same calculations as before, we obtain ψ(t) = Ct −ζ , so that the required bound follows at once. 6.2. Lower bounds. In order to be able to use Theorem 3.7, we need upper bounds on the moments of some observable that is not integrable with respect to the invariant measure. This is achieved by the following proposition: Proposition 6.2. For every α > 0 and every κ ∈ [0, 21 ] there exist constants Cα and Cκ such that the bounds
Pt H α (x) ≤ (H (x) + Cα t)α ,
Pt exp α H κ (x) ≤ exp α H κ (x) + Cκ (1 + t)κ/(1−κ) , hold for every t > 0 and every x ∈ R4 .
How Hot Can a Heat Bath Get?
169
Proof. Note first that LH ≤ γ (T + T∞ ) and that T (∂ p0 H )2 + T∞ (∂ p1 H )2 = T p02 + T∞ p12 ≤ 2(T + T∞ )H.
(6.4)
It follows that for α ≥ 1, there exists C > 0 such that one has the bound
d Pt H α (x) = Pt (LH α ) (x) dt = α Pt (H α−1 LH + γ (α − 1)H α−2 (T p02 + T∞ p12 )) (x)
1− 1 ≤ C Pt H α−1 (x) ≤ C Pt H α (x) α . 1
The last inequality followed from the concavity of x → x 1− α . Setting Cα = C/α, the bound on Pt H α now follows from a simple differential inequality. The corresponding bound for α ∈ (0, 1) follows by a simple application of Jensen’s inequality. The bounds on the exponential of the energy are obtained in a similar way. Set 1 f κ (x) = x(log x)2− κ and note that there exists a constant K κ such that, provided that 1 κ ∈ (0, 2 ], f κ is concave for x ≥ exp(α K κκ ). It then follows as before from (6.4) and the bound on LH that there exists a constant C such that
d Pt exp α(K κ + H )κ (x) ≤ C Pt (K κ + H )2κ−1 exp α(K κ + H )κ (x) dt
= C Pt f κ (exp α(K κ + H )κ ) (x)
(6.5) ≤ C f κ Pt exp α(K κ + H )κ (x) . The result then follows again from a simple differential inequality.
As a consequence, we have the following result in the case k = 2: Theorem 6.3. For every ζ > ζ and every x0 ∈ R4 , there exists a constant C and a −ζ sequence tn increasing to infinity such that µ − µtn ≥ Ctn . Proof. Let ζ˜ ∈ (ζ , ζ ), and let ε > 0, α > ζ (1+ε). It then follows from Theorem 5.8 and ˜ Proposition 6.2 that the assumptions of Theorem 3.7 are satisfied with W (x) = H ζ (x), ˜ h(s) = s −1−ε , F(s) = s α/ζ , and g(x0 , t) = (H (x0 ) + Ct)α . Applying Theorem 3.7 yields the lower bound ˜ α−ζ −εζ
ζ − (1+ε)α ˜ ˜
µ − µtn ≥ Ctn
,
for some C > 0 and some sequence tn increasing to infinity. Choosing ε sufficiently small and α sufficiently large, we can ensure that the exponent appearing in this expression is larger than −ζ , so that the claim follows. Furthermore, we have Theorem 6.4. Let k ∈ ( 43 , 2) and define κ = 2k − 1. Then, there exists a constant c such that, for every initial condition x0 ∈ R4 there exists a constant C and a sequence of κ/(1−κ) times tn increasing to infinity such that µ − µtn ≥ C exp(−ctn ).
170
M. Hairer
Proof. We apply Theorem 3.7 in a similar way to above, but it turns out that we don’t need to make such ‘sharp’ choices for h and F. Take h(s) = s −2 , F(s) = s 3 , and let W = exp(K H κ ) with the constant K large enough so that W is not integrable with respect to µ . It then follows
from Proposition 6.2 that we can choose g(x, t) = exp 3K H κ (x) + C(1 + t)κ/(1−κ) for a suitable constant C. The requested bound follows at once, noting that h ◦ (F · h) ◦ g = 1/g 2 . 7. The Case of a Weak Pinning Potential In this section, we are going to study the case k ≤ 1, that is when we have either V1 ≈ V2 or V1 " V2 at infinity. This case was studied extensively in the previous works [EH00,RT02,EH03,Car07], but the results and techniques obtained there do not seem to cover the situation at hand where one of the heat baths is at ‘infinite temperature’. Furthermore, these works do not cover the case k < 1/2, where one does not have a spectral gap and exponential convergence fails. One further interest of the present work is that, unlike in the above-mentioned works, we are able to work with the generator L instead of having to obtain bounds on the semigroup Pt . This makes the argument somewhat cleaner. We divide this part into two subsections. We first treat the case where one can find a spectral gap, which is relatively easy in the present setting. In the second part, we then treat the case where the spectral gap fails to hold, which follows more closely the heuristics set out in Sect. 2.2. There, we also show that, rather unsurprisingly, no invariant measure exists in the case where k ≤ 0. 7.1. The case k > 1/2. Our aim is to find a modified version Hˆ of the energy function H such that, for a sufficiently small constant β0 , one has exp(−β0 Hˆ )L exp(β0 Hˆ ) " 0 at infinity. This is achieved by the following result: Theorem 7.1. Let k ∈ ( 21 , 1) and let δ ∈ [ k1 −1, 1]. Then, there exist constants c, C > 0, β0 > 0 and a function Hˆ : R4 → R such that • The bounds cH ≤ Hˆ ≤ C H hold outside of some compact set. • For any t > 0, the operator Pt admits a spectral gap in the space of measurable functions weighted by exp(β0 Hˆ δ ). Remark 7.2. Combining this result with Proposition 5.1 shows the existence of constants c, C > 0 such that exp(cH ) dµ < ∞, but exp(C H ) dµ = ∞. Remark 7.3. The technique used in the proof of Theorem 7.1 is more robust than that used in the previous sections. In particular, it applies to chains of arbitrary length. It would also not be too difficult to modify it to suit the more general class of potentials considered in [RT02,Car07]. Proof. Define the variable y = (q, p0 , p1 ) with q = (q0 − q1 )/2 and let A and B be the matrices defined by ⎞ ⎛ ⎛ ⎞ 1 0 − 21 0 √0 2 def def A = ⎝−2α −γ B = 2γ ⎝ T √0 ⎠. (7.1) 0 ⎠, 2α 0 0 T∞ 0
How Hot Can a Heat Bath Get?
171
With this notation, we can write the equations of motion for y following from (1.2) as dy = Ay dt + F(y, Q) dt + B dw(t),
(7.2)
where we defined the centre of mass Q = (q0 + q1 )/2 and F : R → R is a vectorvalued function whose components are all bounded by C + |V1 (q0 )| + |V1 (q0 )| for some constant C. Since det A = −γ α < 0 and we know from a simple contradiction argument [RT02, Car07] that the energy of the system converges to zero under the deterministic equation y˙ = Ay, we conclude that all eigenvalues of A have strictly negative real part. As a consequence, there exists γ˜ > 0 such that the strictly positive definite symmetric quadratic form ∞ def y, Sy = eγ˜ t e At y2 dt (7.3) 4
3
0
is well-defined. A simple change of variable shows that one then has the bound e At , Se At y ≤ e−γ˜ t y, Sy.
(7.4)
For any given (small) value ε > 0, let now G ε : R → R be a smooth function such that: • There exists a constant Cε such that the bounds G ε (q)V1 (q) ≤ Cε − |V1 (q)|2 and |G ε (q)|2 ≤ Cε + |V1 (q)|2 hold for every q ∈ R. • One has |G ε (q)| ≤ ε for every q ∈ R. Since we assumed that k < 1, it is possible to construct a function G ε satisfying these conditions by choosing Rε sufficiently large, setting G ε (q) = −V1 (q) for |q| ≥ 2Rε , G ε (q) = q|Rε |2k−2 for |q| ≤ Rε , and interpolating smoothly in between. For large values of Rε , one can then guarantee that |G ε (q)| ≤ C Rε2k−2 , which does indeed go to 0 for large values of Rε . We now define, for a (large) constant ξ to be determined, Hˆ = H + y, Sy − ξ( p0 + p1 )(G ε (q0 ) + G ε (q1 )). Before we bound L Hˆ , we note that we have the bound
(G ε (q0 ) + G ε (q1 )) V1 (q0 ) + V1 (q1 ) q1
= 2 G ε (q0 )V1 (q0 ) + G ε (q1 )V1 (q1 ) + G ε (q) dq V1 (q0 ) − V1 (q1 ) q0 2 2 ≤ 2Cε − 2 |V1 (q0 )| + |V1 (q1 )| + Cε(q0 − q1 )2 , for some constant C independent of ε. It therefore follows from (7.4), (7.2), (1.2) and the properties of G ε that there exist constants Ci independent of ξ and ε such that we have the bound L Hˆ ≤ C1 − γ p02 − γ˜ y, Sy + 2y, S F(y, Q)
+ξ (G ε (q0 ) + G ε (q1 )) V1 (q0 ) + V1 (q1 )
+γ ξ p0 (G ε (q0 ) + G ε (q1 )) − ξ( p0 + p1 ) G ε (q0 ) p0 + G ε (q1 ) p1 ≤ C2 Cε + |V1 (q0 )|2 + |V1 (q1 )|2 γ˜ − C3 εξ y, Sy − 2ξ |V1 (q0 )|2 + |V1 (q1 )|2 . − 2
172
M. Hairer
It follows that, by first making ξ sufficiently large and then making ε sufficiently small, it is possible to obtain the bound γ˜ 1 + y, Sy + |V1 (q0 )|2 + |V1 (q1 )|2 , (7.5) L Hˆ ≤ C − 2 for some constant C. (The constant C depends of course on the choice of ξ and of ε, but assume those to be fixed from now on.) Furthermore, it follows immediately from the definition of Hˆ that 1 ( Hˆ , Hˆ ) ≤ C 1 + y, Sy + |V1 (q0 )|2 + |V1 (q1 )|2 ≤ C Hˆ 2− k , (7.6) where we used the scaling behaviour of V1 in order to obtain the second bound. Set now W = exp(β0 Hˆ δ ) for a constant β0 to be determined. It follows from (5.17) that the bound LW ≤ β0 δW Hˆ δ−1 L Hˆ + 2β0 δ Hˆ δ−1 ( Hˆ , Hˆ ) , holds outside of some sufficiently large compact set. Combining this with (7.6) and (7.5), we see that if δ ∈ [ k1 − 1, 1] and β0 is sufficiently small, then the bound LW ≤ −C W ( Hˆ )δ+1− k ≤ −C W, 1
holds outside of some compact set. The claim then follows immediately from Theorem 3.4. The case k = 1 can be shown similarly, but the result that we obtain is slightly stronger in the sense that one has a spectral gap in spaces weighted by H δ for any δ > 0: Theorem 7.4. Let k = 1 and let δ > 0. Then, for any t > 0, the operator Pt admits a spectral gap in the space of measurable functions weighted by H δ . Proof. The proof is similar to the above, but this time by setting y˜ = (q0 , q1 , p0 , p1 ), ⎞ ⎛ ⎛ ⎞ 0 0 0 0 1 0 ⎜ 0 0 ⎟ 0 0 1⎟ def ⎜ 0 def √ ⎟ , B˜ = 2γ ⎜ A˜ = ⎝ ⎝ T 0 ⎠, −α α −γ 0⎠ √ α −α 0 0 T∞ 0 and noting that d y˜ = A˜ y˜ dt + F( y˜ ) dt + B˜ dw(t), for some bounded function F. It then suffices to construct S˜ similarly to above and to set Hˆ = y˜ , S˜ y˜ , without requiring any correction term. This yields the existence of constants C1 and C2 such that one has the bounds L Hˆ ≤ −C1 Hˆ ,
( Hˆ , Hˆ ) ≤ C2 Hˆ ,
outside of some compact set. The existence of a spectral gap in spaces weighted by Hˆ δ follows at once. The claim then follows from the fact that Hˆ is bounded from above and from below by multiples of H .
How Hot Can a Heat Bath Get?
173
7.2. The case k ≤ 1/2. This case is slightly more subtle since the function V (q) is either bounded or even converges to zero at infinity, so that bounds of the type (7.5) are not very useful. We nevertheless have the following result: Theorem 7.5. Let k ∈ (0, 21 ]. Then, (1.2) admits a unique invariant probability measure µ and there exist constants c, C > 0, β0 > 0, and a function Hˆ : R4 → R such that • The bounds cH ≤ Hˆ ≤ C H hold outside of some compact set. • If k = 21 , then Pt admits a spectral gap in the space of measurable functions weighted by exp(β0 Hˆ ). • If k < 21 , then there exist positive constants C and γˆ such that the bound Pt (x, · ) − µ TV ≤ C exp(β0 H (x))e−γˆ t
k/(1−k)
,
(7.7)
holds uniformly over all initial conditions x and all times t ≥ 0. Proof. Define again y, A and B as in (7.1) but let us be slightly more careful about the remainder term. We define as before the center of mass Q = (q0 + q1 )/2 and the displacement q = (q0 − q1 )/2 and write V1 (q0 ) = V1 (Q) + R0 (q, Q), V1 (q1 ) = V1 (Q) + R1 (q, Q). With this notation, defining furthermore the vector 1 = (0, 1, 1), the equation of motion for y = (q, p0 , p1 ) is given by dy = Ay dt − V1 (Q)1 dt + R(Q, y) dt + B dw(t), R = (0, −R0 (Q, y), −R1 (Q, y)). This suggests the introduction, for fixed Q ∈ R, of the reduced generator L Q acting on functions from R3 to R by 1 L Q = Ay, ∂ y − V1 (Q)1, ∂ y + B ∗ ∂ y , B ∗ ∂ y . 2 Following the usual procedure in the theory of homogenisation, we wish to correct the ‘slow variable’ Q in order to obtain an effective equation that takes into account the behaviour of the ‘fast variable’ y. Since the equation of motion for Q is given by Q˙ = ( p0 + p1 )/2 = 1, y/2, this can be achieved by finding a function ψ(Q) such that 1, y/2 − ψ(Q) is centred with respect to the invariant measure for L Q and then solving the Poisson equation L Q ϕ Q = 1, y/2 − ψ(Q). Since all the coefficients of L Q are linear (remember that Q is a constant there), this can be solved explicitly, yielding 2 ψ(Q) = − V1 (Q), γ
ϕ Q (y) = −a, y, a = (1, 1/γ , 1/γ ).
We now introduce the corrected variable Qˆ = Q + a, y, so that the equations of motion for Qˆ are given by 2T 2T∞ 2 dw0 (t) + dw1 (t). d Qˆ = − V1 (Q) dt + a, R(Q, y) dt + γ γ γ
174
Defining γˆ =
M. Hairer 2 γ,
the ‘mean temperature’ Tˆ = (T + T∞ )/2, and ˆ − V1 (Q) , Rˆ = a, R(Q, y) + γˆ V1 ( Q)
(7.8)
we thus see that there exists a Wiener process W such that Qˆ satisfies the equation ˆ ˆ ˆ d Q = −γˆ V1 ( Q) dt + R dt + 2γˆ Tˆ dW (t). Setting again S as in (7.3), this suggests that in order to extract the tail behaviour of ˆ + y, Sy. This the invariant measure for (1.2), a good test function would be V1 ( Q) ˆ function however turns out not to be suitable in the regime where Q is large and y is small, because of the constant appearing when applying L to y, Sy. In order to avoid this, let us introduce a smooth increasing function χ : R+ → [0, 1] such that χ (t) = 1 ˆ = 1 + Qˆ 2 , so that for t ≥ 2 and χ (t) = 0 for χ ≤ 1. We also define the function Q ˆ ≤ C Q ˆ 2k−1 and similarly for V ( Q). ˆ |V1 ( Q)| 1 Note that, since we are considering the regime where V1 is a bounded function, there exists a constant C S such that −C S − 2γ˜ y, Sy ≤ Ly, Sy ≤ C S −
γ˜ y, Sy, 2
where γ˜ is as in (7.4). Furthermore, we note that since all terms contained in Rˆ are of ˆ − V ( Qˆ + b, y) for some vector b ∈ R3 , there exists a constant C such the form V1 ( Q) 1 that the bound ˆ ≤ C|y|, C | Q| ˆ ≤ (7.9) | R| 2k−2 ˆ ≥ C|y|, ˆ | Q| C|y| Q ˆ y). (In particular Rˆ is bounded.) We now set holds for every pair ( Q, ˆ , W = exp (β0 y, Sy) + exp β0 λV1 ( Q) for some positive constants β0 and λ to be determined. Since we are only interested in bounds that hold outside of a compact set, we use in the remainder of this proof the notation f g to signify that there exists a constant c > 0 such that the bound f ≤ cg holds outside of a sufficiently large compact set. With this notation, one can check in a straightforward way that there exist constants βi depending on β0 and λ such that the two-sided bound exp(β1 H ) W exp(β2 H ), holds. It follows then from the chain rule that there exist constants Ci > 0 such that one has the upper bound γ˜ LW ≤ β0 C S − ( − C1 β0 )y, Sy exp (β0 y, Sy) 2 ˆ R| ˆ + C4 V1 ( Q) ˆ exp β0 λV1 ( Q) ˆ . + β0 λ −(1 − C2 β0 λ)|V1 (Q)|2 + C3 |V1 ( Q)|| (7.10)
How Hot Can a Heat Bath Get?
175
Choosing β0 sufficiently small, we obtain the existence of a constant Cˆ such that the bound ˆ R| ˆ − Q ˆ 2k−1 exp β0 λV1 ( Q) ˆ 2k−1 C| ˆ , LW Cˆ − y, Sy exp (β0 y, Sy)+ Q holds. ˆ ≥ y, Sy ≥ Cˆ and We now consider three separate cases. In the regime λV1 ( Q) provided that λ is chosen sufficiently small, it follows from (7.9) that we have the bound ˆ 2k−1 C| ˆ R| ˆ − Q ˆ 2k−1 exp β0 λV1 ( Q) ˆ − Q ˆ 4k−2 W. LW Q ˆ ≥ y, Sy but y, Sy ≤ C, ˆ we similarly have In the regime where λV1 ( Q) ˆ − Q ˆ 4k−2 W. ˆ − Q ˆ 4k−2 exp β0 λV1 ( Q) LW Cˆ exp(β0 C) ˆ ≤ y, Sy, we have the bound Finally, in the regime where λV1 ( Q) LW −y, Sy exp (β0 y, Sy) −|y|2 W. Combining all of these bounds, we have 1
LW −(log W )2− k W, so that the upper bounds on the transition probabilities follow just as in the proof of Theorem 6.1 with κ replaced by k. Before we obtain lower bounds on the convergence speed, we show the following non-integrability result: ˆ dµ = ∞. Lemma 7.6. In the case k < 21 there exists β > 0 such that exp(βV1 ( Q)) Proof. We are going to construct functions W1 and W2 satisfying Wonham’s criterion. ˆ S and y be as in the proof of the previous result and set Let Q, ˆ + exp(εy, Sy), W2 = exp(2βV1 ( Q)) for constants β > 0 and ε > 0 to be determined. It follows from the boundedness of V1 , V1 and Rˆ that, whatever the choice of β, one has ˆ LW2 exp(βV1 ( Q)), provided that we choose ε sufficiently small. Setting ˆ − exp(εy, Sy), W1 = exp(βV1 ( Q)) we have similarly to (7.10) the bound ˆ 2 − C3 |V1 ( Q)|| ˆ R| ˆ + C4 V1 ( Q) ˆ exp(βV1 ( Q)) ˆ LW1 ≥ β (C2 β − 1)|V1 ( Q)| +ε((γ˜ /2 − C1 ε)y, Sy − C S ) exp(εy, Sy), so that an analysis similar to before shows that LW1 ≥ 0 outside of some compact set, provided that ε < γ˜ /(2C1 ) and β > 1/C2 , thus concluding the proof. Remark 7.7. The proof of Lemma 7.6 does not require k > 0. It therefore shows that there exists no invariant probabilitymeasure for (1.2) if k ≤ 0.
176
M. Hairer
We now use this result in order to obtain the following lower bound on the convergence of the transition probabilities towards the invariant measure: Theorem 7.8. Let k ∈ (0, 21 ). Then, there exists a constant c such that, for every initial condition x0 ∈ R4 there exists a constant C and a sequence of times tn increasing to k/(1−k) infinity such that µ − µtn ≥ C exp(−ctn ). Proof. We use the same notations as above. Let β be sufficiently large so that the funcˆ is not integrable with respect to the invariant measure. We also fix tion exp(βV1 ( Q)) some small ε > 0 and we set ˆ + exp(εy, Sy). W = exp(βV1 ( Q)) We then obtain in a very similar way to before the upper bound ˆ 2 + C3 |V1 ( Q)|| ˆ R| ˆ + C4 V1 ( Q) ˆ exp(βV1 ( Q)) ˆ LW ≤ β (C2 β − 1)|V1 ( Q)| + ε(C S − (γ˜ /2 − C1 ε)y, Sy) exp(εy, Sy). It follows again from a similar analysis that there exists a constant C > 0 such that the bound 1
LW ≤ C(log W )2− k W, holds outside of some compact set. As in the proof of Proposition 6.2, this implies the existence of a constant C > 0 such that one has the pointwise bound Pt W ≤ W exp C(1 + t)k/(1−k) . Combining this with Lemma 7.6, the rest of the proof is identical to that of Theorem 6.4.
Acknowledgements. The author would like to thank Jean-Pierre Eckmann, Xue-Mei Li, Jonathan Mattingly, and Eric Vanden-Eijnden for stimulating discussions on this and closely related problems, as well as Charles Manson for discovering several mistakes in an earlier version. This work was supported by an EPSRC Advanced Research Fellowship (grant number EP/D071593/1).
References [BCG08] [Bon69] [Car07] [CGGR08] [CGWW07] [DFG06]
Bakry, D., Cattiaux, P., Guillin, A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254(3), 727–759 (2008) Bony, J.-M.: Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19, no. fasc. 1, 277–304 xii (1969) Carmona, P.: Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths. Stoch. Process. Appl. 117(8), 1076–1092 (2007) Cattiaux, P., Gozlan, N., Guillin, A., Roberto, C.: Functional inequalities for heavy tails distributions and application to isoperimetry. http://arxiv.org/abs/0807.3112v1[math.PR], 2008 Cattiaux, P., Guillin, A., Wang, F.-Y., Wu, L.: Lyapunov conditions for logarithmic Sobolev and super Poincaré inequality, http://arxiv.org/abs/0712.0235[math.PR], 2007 Douc, R., Fort, G., Guillin, A.: Subgeometric rates of convergence of f -ergodic strong Markov processes, http://arxiv.org/abs/math/0605791v1[math.ST], 2006
How Hot Can a Heat Bath Get? [DMP+ 07] [DPZ96] [DV01] [EH00] [EH03] [EPR99a] [EPR99b] [FR05] [Hai05] [HM08a] [HM08b] [HN04] [HN05] [Hör67] [Hör85] [MA94] [MT93] [MTVE02] [RT00] [RT02] [RW01] [Ver00] [Ver06] [Vil07] [Vil08] [VK04] [Won66]
177
DeVille, R.E.L., Milewski, P.A., Pignol, R.J., Tabak, E.G., Vanden-Eijnden, E.: Nonequilibrium statistics of a reduced model for energy transfer in waves. Comm. Pure Appl. Math. 60(3), 439–461 (2007) Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems, Vol. 229 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1996 Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54(1), 1–42 (2001) Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000) Eckmann, J.-P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235(2), 233–253 (2003) Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Entropy production in nonlinear, thermally driven hamiltonian systems. J. Statist. Phys. 95(1-2), 305–331 (1999) Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201(3), 657–697 (1999) Fort, G., Roberts, G.O.: Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15(2), 1565–1589 (2005) Hairer, M.: A probabilistic argument for the controllability of conservative systems. http:// arxiv.org/abs/math-ph/0506064v2, 2005 Hairer, M., Mattingly, J.: Slow energy dissipation in anharmonic oscillator chains. http:// arxiv.org/abs/0712.3889v2[math-ph], 2009 Hairer, M., Mattingly J.: Yet another look at Harris’ ergodic theorem for Markov chains. http://arxiv.org/abs/0810.2777v1[math.PR], 2008 Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the fokker-planck equation with a high-degree potential. Arch. Rat. Mech. Anal. 171(2), 151–218 (2004) Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Vol. 1862 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2005 Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967) Hörmander, L.: The Analysis of Linear Partial Differential Operators. III, Vol. 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, 1985 MacKay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or hamiltonian networks of weakly coupled oscillators. Nonlinearity 7(6), 1623–1643 (1994) Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Communications and Control Engineering Series. London: Springer-Verlag London Ltd., 1993 Milewski, P.A., Tabak, E.G., Vanden-Eijnden, E.: Resonant wave interaction with random forcing and dissipation. Stud. Appl. Math. 108(1), 123–144 (2002) Rey-Bellet, L., Thomas, L.E.: Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Commun. Math. Phys. 215(1), 1–24 (2000) Rey-Bellet, L., Thomas, L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225(2), 305–329 (2002) Röckner, M., Wang, F.-Y.: Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups. J. Funct. Anal. 185(2), 564–603 (2001) Veretennikov, A.Y.: On polynomial mixing estimates for stochastic differential equations with a gradient drift. Teor. Veroyatnost. i Primenen. 45(1), 163–166 (2000) Veretennikov, A.Y.: On lower bounds for mixing coefficients of Markov diffusions. In: From Stochastic Calculus to Mathematical Finance. Berlin: Springer, 2006, pp. 623–633 Villani, C.: Hypocoercive diffusion operators. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10(2), 257–275 (2007) Villani, C.: Hypocoercivity, 2008 To appear in Memoirs Amer. Math. Soc. Veretennikov, A.Y., Klokov, S.A.: On the subexponential rate of mixing for Markov processes. Teor. Veroyatn. Primen. 49(1), 21–35 (2004) Wonham, W.M.: Liapunov criteria for weak stochastic stability. J. Diff. Eqs. 2, 195–207 (1966)
Communicated by A. Kupiainen
Commun. Math. Phys. 292, 179–199 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0836-y
Communications in
Mathematical Physics
Crystal Melting and Toric Calabi-Yau Manifolds Hirosi Ooguri1,2 , Masahito Yamazaki1,2,3 1 California Institute of Technology, 452-48, Pasadena, CA 91125, USA 2 Institute for the Physics and Mathematics of the Universe, University of Tokyo,
Kashiwa, Chiba 277-8586, Japan
3 Department of Physics, University of Tokyo, Hongo 7-3-1,
Tokyo 113-0033, Japan. E-mail:
[email protected] Received: 9 December 2008 / Accepted: 12 February 2009 Published online: 19 May 2009 – © Springer-Verlag 2009
Abstract: We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.
1. Introduction In type IIA superstring theory, supersymmetric bound states of D branes wrapping holomorphic cycles on a Calabi-Yau manifold give rise to BPS particles in four dimensions. In the past few years, remarkable connections have been found between the counting of such bound states and the topological string theory: (1) When the D brane charges are such that bound states become large black holes with smooth event horizons, the OSV conjecture [1] states that the generating function Z BH of a suitable index for black hole microstates is equal to the absolute value squared of the topological string partition function Z top , 2 Z BH = Z top , to all orders in the string coupling expansion.
(1.1)
180
H. Ooguri, M. Yamazaki
(2) When there is a single D6 brane with D0 and D2 branes bound on it, it has been proposed [2] that the bound states are counted by the Donaldson-Thomas invariants [3,4] of the moduli space of ideal sheaves on the D6 brane. For a non-compact toric Calabi-Yau manifold, the Donaldson-Thomas invariants are related to the topological string partition function [2,5,6] using the topological vertex construction [7]. Recently the connection between the topological string theory and the Donaldson-Thomas theory for toric Calabi-Yau manifolds was proven mathematically in [8]. Given the conjectural relation between the counting of D brane bound states and the Donaldson-Thomas theory, it is natural to expect the relation, Z BH = Z top .
(1.2)
The purpose of this paper is to understand the case (2) better. We start with the lefthand side of the relation, namely the counting of BPS states. Recently, the non-commutative version of the Donaldson-Thomas theory is formulated by Szendröi [9] for the conifold and by Mozgovoy and Reineke [10] for general toric Calabi-Yau manifolds.1 In this paper, we will establish a direct connection between the non-commutative Donaldson-Thomas theory and the counting of BPS bound states of D0 and D2 branes on a single D6 brane. Using this correspondence, we will find a statistical model of crystal melting which counts the BPS states. The crystal melting description has been found earlier in the topological string theory on the right-hand side of (1.2). It was shown in [2,5] that the topological string partition function on C3 , the simplest toric Calabi-Yau manifold, and the topological vertex can be expressed as sums of three-dimensional Young diagrams, which can be regarded as complements of molten crystals with the cubic lattice structure.2 Since the topological vertex can be used to compute the topological string partition function for a general non-compact toric Calabi-Yau manifold, it is natural to expect that a crystal melting description exists for any such manifold. To our knowledge, however, this idea has not been made explicit. The crystal melting model defined in this paper appears to be different from the one suggested by the topological vertex construction. The low energy effective theory of D0 and D2 branes bound on a single D6 brane is a one-dimensional supersymmetric gauge theory, which is a dimensional reduction of an N = 1 gauge theory in four dimensions. The field content of the gauge theory is encoded in a quiver diagram and the superpotential can be found by the brane tiling [25–28].3 From these gauge theory data, we define a crystalline structure in three dimensions. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram and carries a particular combination of D0 and D2 charges. The chemical bond is dictated by the arrows of the quiver diagram. There is a special crystal configuration, whose exterior shape lines up with the toric diagram of the Calabi-Yau manifold. Such a crystal corresponds to a single D6 brane with no D0 and D2 charges. We define a rule to remove atoms from the crystal, which basically says that the crystal melts from its peak. By using the non-commutative Donaldson-Thomas theory [10], we show that there is a one-to-one correspondence between molten crystal configurations and BPS bound states carrying non-zero D0 and D2 charges. The statistical model of crystal melting computes the index of D brane bound states. The number of BPS states depends on the choice of the stability condition, and the BPS countings for different stability conditions are related to each other by the wall 1 See [11–16] for further developments. 2 See [17–24] for further developments. 3 See [29,30] for reviews of the quiver gauge theory and the brane tiling method.
Crystal Melting and Toric Calabi-Yau Manifolds
181
crossing formulae. In this paper, we find that, under a certain stability condition, BPS bound states of D branes are counted by the non-commutative Donaldson-Thomas theory. We can use the wall crossing formulae recently derived in [13,14] to relate this result to the commutative Donaldson-Thomas theory. Since the topological string theory is equivalent to the commutative Donaldson-Thomas theory for a general toric Calabi-Yau manifold [8], the relation (1.2) is indeed true for some choice of the stability condition, as expected in [2]. In general, the topological string partition function and the partition function of the crystal melting model are not identical, but their relation involves the wall crossing, Z crystal melting ∼ Z top (modulo wall crossings).
(1.3)
This does not contradict the result in [2,5] since there is no wall crossing phenomenon for C3 . In general, however, a proper understanding of the relation between the topological string theory and the crystal melting requires that we take the wall crossing phenomena into account. In Sect. 2, we will summarize the computation of D brane bound states from the gauge theory perspective. In Sect. 3, we will discuss how this is related to the recent mathematical results on the non-commutative Donaldson-Thomas invariants. In Sect. 4, we will formulate the statistical model of crystal melting for a general toric Calabi-Yau manifold. The final section is devoted to summary of our result and discussion on the wall crossing phenomena. The Appendix explains the equivalence of a configuration of molten crystal with a perfect matching of the bipartite graph. 2. Quiver Quantum Mechanics In the classic paper by Douglas and Moore [31], it was shown that the low energy effective theories of D branes on some orbifolds are described by gauge theories associated to quiver diagrams. Subsequently, this result has been generalized to an arbitrary non-compact toric Calabi-Yau threefold. A toric Calabi-Yau threefold X is a fiber bundle of T 2 × R over R3 , where the fibers are special Lagrangian submanifolds. The toric diagram tells us where and how the fiber degenerates. For a given X and a set of D0 and D2 branes on X , the following procedure determines the field content and superpotential of the gauge theory on the branes. We will add a single D6 brane to the system later in this section.
2.1. Quiver diagram and field content. The low energy gauge theory is a one-dimensional theory given by dimensional reduction of an N = 1 supersymmetric gauge theory in four dimensions. The field content of the theory is encoded in a quiver diagram, which is determined from the toric data and the set of D branes, as described in the following. A quiver diagram Q = (Q 0 , Q 1 ) consists of a set Q 0 of nodes, with a rank Ni > 0 associated to each node i ∈ Q 0 , and a set Q 1 of arrows connecting the nodes. The corresponding gauge theory has a vector multiplet of gauge group U (Ni ) at each node i. There is also a chiral multiplet in the bifundamental representation associated to each arrow connecting a pair of nodes. In the following, we will explain how to identify the quiver diagram. The reader may want to consult Fig. 1, which describes the procedure for the Suspended Pinched Point singularity, which is a Calabi-Yau manifold defined by the toric diagram in Fig. 1-(a) or
182
H. Ooguri, M. Yamazaki
(a)
(b)
(c)
Fig. 1. (a) The toric diagram for the Suspended Pinched Point singularity. (b) The configuration of D2 and NS5 branes after the T-duality on T2 . The green exterior lines are periodically identified. The red lines representing NS5 branes separate the fundamental domain into several domains. The T-dual of D0 branes wrap the entire fundamental domain, and fractional D2 branes are suspended between the red lines. The white domains contain D2 branes only. In each shaded domain, there is an additional NS5 brane. There are two types of shades depending on the NS5 brane orientation. The white domains are connected by arrows through the vertices, and the directions of the arrows are determined by the orientation of the NS5 branes. (c) The quiver diagram obtained by replacing the white domains of (b) by the nodes
equivalently by the equation, x y = zw 2 ,
(2.1)
in C4 . To identify the quiver diagram, we take T-dual of the toric Calabi-Yau manifold along the T2 fibers [26,32]. The fibers degenerate at loci specified by the toric diagram , and the T-duality replaces the singular fibers by NS5 branes [33]. Some of these NS5 branes divide T2 into domains as shown in the red lines in Fig. 1-(b) [30,34–36]. The D0 branes become D2 branes wrapping the whole T2 . The original D2 are still D2 branes after the T-duality, but each of them is in a particular domain of T2 suspended between NS5 branes. In addition, there are some domains that contain NS5 branes stretched two-dimensionally in parallel with D2 branes.4 Let us denote the domains without NS5 branes by i ∈ Q 0 and the domains with NS5 branes by a ∈ I . In Fig. 1-(b), the Q 0 -type domains are shown in white, and the I -type domains are shown with shade. There are two types of shades, corresponding to two different orientations of NS5 branes. This distinction will become relevant when we discuss the superpotential. The Q 0 -type domains are identified with nodes of the quiver diagram since open strings ending on them can contain massless excitations. The rank Ni of the node i ∈ Q 0 is the number of D2 branes in the corresponding domain. On the other hand, I -type domains give rise to the superpotential constraints as we shall see below. Though two domains i, j ∈ Q 0 never share an edge, they can touch each other at a vertex. In that case, open strings going between i and j contain massless modes. We draw an arrow from i → j or i ← j depending on the orientation of the massless open string modes, which is determined by the orientation of NS5 branes. Note that the quiver gauge theory we consider in this paper are in general chiral. This completes the specification of the quiver diagram. 4 The NS5 branes are also filling the four-dimensional spacetime R1,3 while the D2 branes are localized along a timelike path in four dimensions.
Crystal Melting and Toric Calabi-Yau Manifolds
183
As another example, the quiver diagram for the conifold geometry has two nodes connected by two sets of arrows in both directions. The ranks of the gauge groups are n 0 and n 0 − n 2 , where n 0 and n 2 are the numbers of D0 and D2 branes. The gauge theory is a dimensional reduction of the Klebanov-Witten theory [37] when n 2 = 0 and the Klebanov-Strassler theory [38] when n 2 > 0. 2.2. Superpotential and brane tiling. Each domain a ∈ I containing an NS5 brane is surrounded by domains i 1 , i 2 , . . . , i n ∈ Q 0 without NS5 branes, as in Fig. 1-(b). By studying the geometry T-dual to X in more detail, one finds that the domain is contractible. Since open strings can end on the domains i 1 , i 2 , . . . , i n , the domain a can give rise to worldsheet instanton corrections to the superpotential. This fact, combined with the requirement that the moduli space of the gauge theory agrees with the geometric expectation for D branes on X , determines the superpotential. Depending on the NS5 brane orientation, the I -type domains are further classified into two types, I+ and I− , and thus the regions of torus are divided into three types Q 0 , I+ and I− . Such a brane configuration, or a classification of regions of T2 , is called the brane tiling.5 In Fig. 1-(b), the brane tiling is shown by the two different shades. The superpotential W is then given by ⎛ + ⎛ − ⎞ ⎞ na na W = Tr ⎝ Ai (a) ,i (a) ⎠ − Tr ⎝ Ai (a) , j (a) ⎠ , (2.2) a∈I+
q=1
q,+ q+1,+
a∈I−
q=1
q,−
(a)
q+1,−
(a)
where the domain a ∈ I± are surrounded by the arrows i 1,± → i 2,± → · · · → (a)
(a)
(a)
(a)
i n ± +1,± → i 1,± . For each arrow i q,± → i q+1,± (1 ≤ q ≤ n a± ), the corresponding a bifundamental field is denoted by Ai (a) ,i (a) . This formula is tested in many examq,± q+1,±
ples. In particular, it has been shown that the formula reproduces the toric Calabi-Yau manifold X as the moduli space of the quiver gauge theory [40]. In the literature of brane tiling, bipartite graphs are often used in place of brane configurations as in Fig. 1-(b). A bipartite graph is a graph consisting of vertices colored either black or white and edges connecting black and white vertices. Since bipartite graphs will also play roles in the following sections, it would be useful to explain how it is related to our story so far. For a given brane configuration, we can draw a bipartite graph on T2 as follows. In each domain in I+ (I− ), place a white (black) vertex. Draw a line connecting a white vertex in a domain i ∈ I+ and a black vertex in a neighboring domain j ∈ I− . The resulting graph is bipartite. See Fig. 2 for the comparison of the brane configuration and the bipartite graph in the case of the Suspended Pinched Point singularity. We can turn this into a form that is more commonly found in the literature, for example in [26], by choosing a different fundamental region as in Fig. 3.
2.3. D-term constraints and the moduli space. The F-term constraints are given by derivatives of the superpotential, which can be determined as in the above. The moduli space of solutions to the D-term constraints is then described by a set of gauge invariant observables divided out by the complexified gauge group G C [41]. The theorem by King 5 In the literature the word brane tiling refers to the bipartite graph explained below. Here the word brane tiling refers to a brane configuration as shown in Fig. 1-(b). Such a graph is called the fivebrane diagram in [39].
184
H. Ooguri, M. Yamazaki
Fig. 2. The correspondence between the brane configuration on T2 and the bipartite graph. The white (black) vertex of the bipartite graph corresponds to the region I+ (I− ) in light (dark) shade. The edge of the bipartite graph corresponds to an intersection of I− and in I+ . From this construction, it automatically follows that the graph so obtained is bipartite
Fig. 3. By choosing a different fundamental region of T2 , we find a bipartite graph which is more commonly found in the literature
[42] states that an orbit of G C contains a solution to the D-term conditions if and only if we start with a point that satisfies the θ -stability, a condition defined in the next section. Thus, we can think of the moduli space as a set of solutions to the F-term constraints obeying the θ -stability condition, modulo the action of G C . 2.4. Adding a single D6 brane. To make contact with the Donaldson-Thomas theory, we need to include one D6 brane. Since the D6 brane fills the entire Calabi-Yau manifold, which is non-compact, it behaves as a flavor brane. In the low energy limit, the open string between the D6 brane and another D brane gives rise to one chiral multiplet in the fundamental representation for the D brane on the other end. The D6 brane then enlarges the quiver diagram by one node and one arrow from the new node. To understand why we only get one arrow from the D6 brane, let us take T-duality along the T2 fiber again. The D6 brane is mapped into a D4 brane which is a point in some region in T2 . This means that we only have one new arrow from the new node corresponding to the D6 brane to the node corresponding to the D2 branes in the region. See [23,43] for related discussion in the literature. 3. Non-commutative Donaldson-Thomas Theory In the previous section, we discussed how to construct the moduli space of solutions to the F-term and D-term constraints in the quiver gauge theory corresponding to a toric
Crystal Melting and Toric Calabi-Yau Manifolds
185
Calabi-Yau manifold X with a set of D0/D2 branes and a single D6 brane. In this section, we will review and interpret the mathematical formulation of the non-commutative Donaldson-Thomas invariant in [9,10] for X . We find that it is identical to the Euler number of the gauge theory moduli space. 3.1. Path algebra and its module. For the purpose of this paper, modules are the same as representations. Consider a set of all open paths on the quiver diagram Q = (Q 0 , Q 1 ). By introducing a product as an operation to join a head of a path to a tail of another (the product is supposed to vanish if the head and the tail do not match on the same node) and by allowing formal sums of paths, the set of open oriented paths can be made into an algebra CQ called the path algebra. We would like to point out that there is a one-to-one correspondence between a representation of the path algebra and a classical configuration of bifundamental fields of the quiver gauge theory. Suppose there is a representation M of the path algebra. For each node i ∈ Q 0 , there is a trivial path ei 2 of zero length that begins and ends at i. It is a projection, (ei ) = ei . Since every path starts at some node i and ends at some node j, the sum i ei acts as the identity on the path algebra. Therefore, M = ⊕i∈Q 0 Mi , where Mi = ei M. Let us write Ni = dim Mi . For each path from i to j, one can assign a map from Mi to M j . In particular, there is an Ni × N j matrix for each arrow i → j ∈ Q 1 of the quiver diagram. By identifying this matrix as the bifundamental field associated to the arrow i → j, we obtain a classical configuration of bifundamental fields with the gauge group U (Ni ) at the node i. By reversing the process, we can construct a representation of the path algebra for each configuration of the bifundamental fields. 3.2. F-term constraints and factor algebra A. Let us turn to the F-term constraints. Since the bifundamental fields of the quiver gauge theory is a representation of the path algebra, the F-term equations give relations among generators of the path algebra. It is natural to consider the ideal F generated by the F-term equations and define the factor algebra A = CQ/F. The bifundamental fields obeying the F-term constraints then generate a representation of this factor algebra. Namely, classical configurations of the quiver gauge theory obeying the F-term constraints are in one-to-one correspondence with A-modules. As an example, the algebra A for the conifold geometry contains an idempotent ring C[e1 , e2 ] generated by two elements and is given by the following four generators and relations:6 A = C[e1 , e2 ]a1 , a2 , b1 , b2 / (a1 bi a2 = a2 bi a1 , b1 ai b2 = b2 ai b1 )i=1,2 .
(3.2)
Each A-module for this algebra corresponds to a choice of ranks of the gauge groups and a configuration of the bifundamental fields ai , bi satisfying the F-term constraints. F-term constraints have a nice geometric interpretation on the quiver diagram, which we will find useful in the next section. We observe that each bifundamental field appears exactly twice in the superpotential with different signs of coefficients in the superpotential shown in (2.2). By taking a derivative of the superpotential with respect to a 6 The center Z (A) of this algebra A is generated by x = a b + b a (i, j = 1, 2), and is given by ij i j j i
Z (A) = C[x11 , x12 , x21 , x22 ]/(x11 x22 − x12 x21 ), which is the ring of functions of the conifold singularity.
(3.1)
186
H. Ooguri, M. Yamazaki
Fig. 4. Representation of F-term constraints on the quiver diagram on T2 . In this example, if we write by X AB the bifundamental corresponding to an arrow starting from vertex A and ending at B etc., then the superpotential 2.2 contains a term W = −tr(X AB X BC X C A ) + tr(X AB X B D X D E X E A ), and the F-term condition for X AB (multiplied by X AB ) says that the product of bifundamental fields along the triangle ABC and that along the square AB D E is the same
bifundamental field corresponding to a given arrow, the resulting F-term constraint states that the product of bifundamental fields around a face of the quiver on T2 on one side of the arrow is equal to that around the face on the other side. See Fig. 4 for an example. Therefore, when we have a product of bifundamental fields along a path, any loop on the path can be moved along the path and the resulting product is F-term equivalent to the original one. In [10], it is shown that for any point i, j ∈ Q 0 , we can find a shortest path vi, j from i to j such that any other path a from i to j is F-term equivalent to vi, j ωn with non-negative integer n, where ω is a loop around one face of the quiver diagram. It does not matter where the loop ω is inserted along the path vi, j since different insertions are all F-term equivalent. This means that any path is characterized by the integer n and the shortest path vi, j . In the next subsection, we will impose the D-term constraints on the space of finitely generated left A-modules, mod A. Before doing this, however, it is instructive to discuss topological aspects of mod A by considering its bounded derived category7 D b (mod A). In mathematics, the algebra A gives the so-called “non-commutative crepant resolution” [45]. For singular Calabi-Yau manifolds such as X , the crepant resolution means a resolution that preserves the Calabi-Yau condition.8 For a crepant resolution Y of X , we have the following equivalence of categories9 : D b (coh(Y )) ∼ = D b (mod A),
(3.3)
where D b (coh(Y )) is a bounded derived category of coherent sheaves of crepant resolution Y , and D b (mod A) is the bounded derived categories of finitely generated left A-modules. Equation (3.3) is also interesting from the physics viewpoint. Since D b (coh(Y )) gives a topological classification of A branes on the resolved space Y , the equivalence means that D b (mod A) also classifies D branes, which is consistent with our interpretation above that A-modules are in one-to-one correspondence with a configuration of bifundamental fields obeying the F-term constraints. We should note that the paper [10], which computes the non-commutative DonaldsonThomas invariants for general toric Calabi-Yau manifolds, requires a set of conditions 7 See [44] for an introductory explanation of derived categories in the context of string theory. 8 Mathematically, we mean a resolution f : Y → X such that ω = f ∗ ω , where ω and ω are Y X X Y
canonical bundles of X and Y , respectively. For the class of toric Calabi-Yau threefolds, the existence of crepant resolution is known and different crepant resolutions related by flops are equivalent in derived categories [46]. 9 This is well-known in the case of the conifold (cf. [47]). For general toric Calabi-Yau threefolds, this is not yet mathematically proven as far as the authors are aware of, although there are proofs in several examples [14,48,49].
Crystal Melting and Toric Calabi-Yau Manifolds
187
on brane tilings, namely on the superpotential. We find that most of their conditions (specified in Lemma 3.5 and Conditions 4.12) are automatically satisfied for any quiver gauge theories for D branes on general toric Calabi-Yau manifold. We have not been able to prove that Condition 5.3 also holds in general, but it is satisfied in all the examples we know. 3.3. D-term constraints and θ -stability. We saw that the derived category D b (mod A) of A-modules gives the topological classification of D branes in the toric Calabi-Yau manifold X . To understand the moduli space of D branes, however, we also need to understand implications of the D-term constraints. This is where the θ -stability comes in.10 Let θ ∈ N Q 0 be a vector whose components are real numbers. Consider an A-module M, and recall that this M is decomposed as M = ⊕i∈Q 0 Mi with Mi = ei M. The module M is called θ -stable if θi (dim ei M ) > 0 (3.4) i∈Q 0
for every submodule M of M.11 When > is replaced by ≥, the module M is called θ -semistable. In the language of gauge theory, the stability condition (3.4) is required by the D-term conditions. Some readers might wonder why the D-term conditions, which are equality relations, can be replaced by an inequality as (3.4). In fact, the similar story goes for the Hermitian Yang-Mills equations. There instead of solving the Donaldson-UhlenbeckYau equations, we can consider holomorphic vector bundles with a suitable stability condition, the so-called µ-stability or Mumford-Takemoto stability [52,53]. As we mentioned at the end of Sect. 2, it is known that a configuration of bifundamental fields is mapped to a solution to the D-term equations by a complexified gauge transformation G C if and only if the configuration is θ -stable. Since each A-module M gives a representation of G C = i∈Q 0 G L(Ni , C), where G L(Ni , C) is represented by Mi = ei M at each node, each A-module specifies a particular G C orbit. Thus, finding a θ -stable module is the same as solving the D-term conditions. Up to this point we have not specified the value of θ . Physically, θ ’s correspond to the FI parameters, which are needed to write down D-term equations [50]. Although the Euler number of the space of θ -stable A-modules does not change under infinitesimal deformation of θ , it does change along the walls of marginal stability [13,14]. The noncommutative Donaldson-Thomas invariant defined by [9] is in a particular chamber in the space of θ ’s. Following [10], we hereafter take θ = (0, 0, . . . , 0). We will comment more about this issue in the final section. 3.4. D6 brane and compactification of moduli space. We have found that solutions to the F-term and D-term conditions in the quiver gauge theory are identified with θ -stable A-modules. We want to understand the moduli space of such modules and compute its Euler number. Since D brane charges correspond to the ranks of the gauge groups, we consider moduli space of θ -stable modules with dimension dim Mi = Ni (i ∈ Q 0 ), which we 10 The θ -stability is a special limit of -stability as discussed in [50,51]. 11 In some literature, an additional condition i∈Q 0 θi (dim Mi ) = 0 is imposed for a choice of θ . This is
trivially satisfied for the choice θ = (0, 0, . . . , 0) we choose below.
188
H. Ooguri, M. Yamazaki
denote by M N (A). To compute its Euler number, we need to address the fact that the moduli space of stable A-modules is not always compact. In mathematics literature, the necessary compactification is performed by enlarging the quiver diagram by adding one more node in the following way. Let us fix an arbitrary vertex i 0 , and define a new quiver Qˆ = ( Qˆ 0 , Qˆ 1 ) by Qˆ 0 = Q 0 ∪ {∗},
Qˆ 1 = Q 1 ∪ {a∗ : ∗ → i 0 }.
(3.5)
Namely, we have added one new vertex ∗ and one arrow ∗ → i 0 to obtain the extended ˆ As in the previous case for Q, we can define the path algebra C Q, ˆ the quiver diagram Q. Q 0 ˆ ˆ ˆ ˆ ˆ ideal F generated in CQ by F, and the factor algebra A = C Q/F. Define θ ∈ N +1 by ˆ θˆ = (θ, 1) and define θˆ -stable and semistable A-modules using stability parameter θˆ . It ˆ ˆ is shown in Lemma 2.3 of [10] that θˆ -semistable A-modules are always θ-stable, and the N ˆ ˆ ˆ moduli space Mi0 (A) of θ -stable modules with specified dimension vector N ∈ N Q 0 +1 is compact. Adding the extra-node allows us to compactify the moduli space. In the language of D branes, this corresponds to adding a single D6 brane filling the entire Calabi-Yau manifold, which is necessary to interpret the whole system as a six-dimensional U (1) gauge theory related to the Donaldson-Thomas theory. As we mentioned in Sect. 2, the D6 brane serves as a flavor brane and adds an extra node exactly in the way described in the above paragraph. Note that, in the above paragraph, the ideal Fˆ is generated by the original ideal F. In the quiver gauge theory, this corresponds to the fact that the flavor brane does not introduce a new gauge invariant operator to modify the superpotential. In this way, we arrive at the definition of non-commutative Donaldson-Thomas invariant ˆ N (A)) of cohomologies. With our identification of M ˆ N (A) as the Euler number χ (M i0 i0 ˆ N (A)) with the moduli space of solutions to the F-term and D-term conditions, χ (M i0 computes the Witten index of bound states of D0 and D2 branes bound on a single D6 brane (ignoring the trivial degrees of freedom corresponding to the center of mass of D branes in R1,3 ). We have chosen a specific vertex i 0 to define the non-commutative DonaldsonThomas invariant. The i 0 dependence drops out in simple cases such as C3 and conifold, ˆ N (A)) depends on the choice of the i 0 . We note that the quiver but in general χ (M i0 gauge theory discussed in Sect. 2 also has an apparent dependence on i 0 . There i 0 corresponds to the Q 0 -type domain where the D6 brane is located after the T-duality. Since the fundamental group of the toric Calabi-Yau manifold X is trivial, there is no moduli intrinsic to the D6 brane before taking the T-duality. Thus, we expect that the apparent i 0 dependence in the gauge theory side should disappear with a proper treatment of the T-duality. It would be interesting to study this point further.12 4. Crystal Melting In this section, we define a statistical mechanical model of crystal melting and show that the model reproduces the counting of BPS bound state of D branes. Using the quiver diagram and the superpotential of the gauge theory, we define a natural crystalline structure in three dimensions. We specify a rule to remove atoms from the crystal and show 12 In the example discussed in [14], the result is dependent on the choice of i and a particular vertex should 0 be chosen in order to match with the category of perverse coherent sheaves. We thank Yukinobu Toda for discussions on this point.
Crystal Melting and Toric Calabi-Yau Manifolds
189
that each molten crystal corresponds to a particular BPS bound state of D branes. We use the result of [10] to show that all the relevant BPS states are counted in this way. 4.1. Crystalline structure. Mathematically, the three-dimensional crystal we define here is equivalent to a set of basis for Aei0 , where A is the factor algebra A = CQ/F of the path algebra CQ divided by the ideal F generated by the F-term constraints and ei0 is the path of zero length at the reference node i 0 , which is also the projection operator to the space of paths starting at i 0 . Colloquially, the crystal is a set of paths starting at i 0 modulo the F-term relations. As we shall see later, it corresponds to a BPS state corresponding to a single D6 brane with no D0 and D2 charges. We interpret Aei0 in terms of a three-dimensional crystal as follows. The crystal is composed of atoms piled up on nodes in the universal covering Q˜ on 2 R . By using the projection, π : Q˜ → Q, each atom is assigned with a color corresponding to the node in the original quiver diagram Q. The arrows of the quiver diagram determines the chemical bond between atoms. We start by putting one atom on the top of the reference node i 0 . Next attach an atom at an adjacent node j ∈ Q˜ 0 that is connected i 0 by an arrow going from i 0 to j. The atoms at such nodes are placed lower than the atom at i 0 . In the next step, start with the atoms we just placed and follow arrows emanating from them to attach more atoms at the heads of the arrows. As we repeat this procedure, we may return back to a node where an atom is already placed. In such a case, we use the following rule. As we explained in Sect. 3.2, modulo F-term constraints, any oriented path a starting at i 0 and ending at j can be expressed as vi0 , j ωn , where ω is the loop around a face in the quiver diagram and vi0 , j is one of the shortest paths from i 0 to j. This defines an integer h(a) = n for each path a. The rule of placing atoms is that, if a path a takes i 0 to j and if h(a) = n, we place an atom at the n th place under the first atom on the node j. If there is already an atom at the n th place, we do not place a new atom. By repeating this procedure, we continue to attach atoms and construct a pyramid consisting of infinitely many atoms. Since atoms are placed following paths from i 0 modulo the F-term relations, it is clear that atoms in the crystal are in one-to-one correspondence with basis elements of Aei0 . Note that by construction the crystal has a single peak at the reference node i 0 . This defines a crystalline structure for an arbitrary toric Calabi-Yau manifold. In particular, it reproduces the crystal for C3 discussed in [2,5], and the one for conifold in [9]. See Fig. 5 for the crystalline structure corresponding to the Suspended Pinched Point singularity. In this example, the ridge of the crystal (shown as blue lines in Fig. 5) coincides with the ( p, q)-web of the toric geometry. As we will discuss later, this is a general property of our crystal. 4.2. BPS state and molten crystal. In the forthcoming discussions, the crystal defined above will be identified with a single D6 brane with no D0 and D2 charges. Bound states with non-zero D0 and D2 charges are obtained by removing atoms following the rule specified below. ˆ N (A)) are computed by using the In [9,10], the Donaldson-Thomas invariants χ (M i0 ˆ N corresponding to translational invariance U (1)⊗2 symmetry of the moduli space M i0
of T2 . By the standard localization techniques, the Euler number can be evaluated at the fixed point set of the moduli space under the symmetry. Correspondingly, in the gauge
190
H. Ooguri, M. Yamazaki
Fig. 5. Starting from the universal cover Q˜ of quiver Q shown on the left, we can construct a crystal on the right. Each atom carries a color corresponding to a node in Q, and they are connected by arrows in Q˜ 1 . The green arrows represent arrows on the surfaces of the crystal, whereas the red ones are not. In the case of the Suspended Pinched Point singularity, the atoms come with 3 colors (white, black and gray), corresponding to the 3 nodes of the original quiver diagram Q on T2 shown in Fig. 1
theory side, BPS states counted by the index are those that are invariant under the global U (1)⊗2 symmetry acting on bifundamental fields preserving the F-term constraints since those do not have extra zero modes and do not contribute to the index. We are interested in counting such BPS states. In order for a molten crystal to correspond to U (1)⊗2 invariant θˆ -stable A-modules, we need to impose the following rule to remove atoms from the crystal. Let be a finite set of atoms to be removed from the crystal. The Melting Rule. If aα is in for some a ∈ A, then α should also be in . Since atoms of the crystal correspond to elements of Aei0 , we used the natural action of A on Aei0 to define aα in the above. This means that crystal melting starts at the peak at i 0 and takes place following paths in Aei0 . An example of a molten crystal satisfying this condition is shown in Fig. 6. The melting rule means that a complement I of the vector space spanned by in Aei0 gives an ideal of A. To see this, we just need to take the contraposition of the melting rule. It states: For any β ∈ I and for any a ∈ A, aβ is also in I. Generally speaking, an ideal of an algebra defines a module. To see this, consider a vector |I which is annihilated by all elements of the ideal I. From |I, we can generate a finite dimensional representation of the algebra A by acting elements of A on it. ˆ However, the converse is not always true. Fortunately, when modules are θ-stable and invariant under the U (1)⊗2 symmetry, it was shown in [9,10] that there is a one-to-one correspondence between ideals and modules. It follows that our molten crystal configurations are also in one-to-one correspondence with A-modules and therefore with relevant BPS bound states of D branes. This proves that the statistical model of crystal melting computes the index of D brane bound states.
Crystal Melting and Toric Calabi-Yau Manifolds
191
Fig. 6. Example of a molten crystal and its complement . In this example contains 12 atoms, one hidden behind an atom on the reference point represented by a blue point. It is easy to check that satisfies the melting rule mentioned in the text
It would be instructive to understand explicitly how each molten crystal configuration corresponds to a BPS bound state. Starting from a molten crystal specified by , Prepare a one-dimensional vector space Vα with basis vector eα for each atom α ∈ . ˜ define the action of a on Vα by a(eα ) = eβ when the arrow For each arrow a of Q, a begins from α and ends at another atom β ∈ . Otherwise a(eα ) is defined to be zero. Since an arbitrary path is generated by concatenation of arrows, we have defined an action of a ∈ A on each Vα . By linearly extending the action of a onto the total space M = ⊕α∈ Vα , we obtain a A-module M. There are several special properties about this module M. First, the F-term relations are automatically satisfied. This is because when there exist two different paths a, b ∈ A starting at α and ending at β, a(eα ) and b(eα ) are both defined to be eβ . Second, by construction M is generated by action of the algebra A on a single element ei,0 ∈ Vi,0 . In such a case M is called a cyclic A-module, and by Lemma 2.3 of [10] is also θˆ -stable. Third, by the cyclicity of the module it follows that M is U (1)⊗2 invariant up to gauge transformations. Therefore, M is a U (1)⊗2 invariant θˆ -stable module. It follows from the result of [10] discussed at the beginning of this section that M indeed corresponds to a bound state of D branes contributing to the Witten index. At the beginning of this subsection, we stated without explanation that the original crystal corresponds to a single D6 brane with no D0 and D2 charges, and removing atoms correspond to adding the D brane charges. To understand this statement, let us recall that, in Sect. 2, we started with a configuration of D0 and D2 branes on the toric Calabi-Yau manifold and took a T-duality along the fiber to arrive at the brane configuration. Thus, the number of D2 branes at each node j of the quiver diagram Q is a combination of D0 and D2 charges before T-duality. It is this number that is equal to the rank of the gauge group at j.
192
H. Ooguri, M. Yamazaki
By using the projection π : Q˜ o → Q 0 , the A-module M is decomposed as M = ⊕ j∈Q 0 M j as we saw in Sect. 3, where
Mj = Vα . (4.1) α∈,π(α)= j
In particular, the formula (4.1) means that the rank of the gauge group N j = dimM j at the node j is equal to the number of atoms with the color corresponding to the node j that have been removed from the crystal. Thus, removing an atom at the node j is equivalent to adding D0 and D2 charges carried by the node j. It is interesting to note that each atom in the crystal does not correspond to a single D0 brane or a single D2 brane, but each of them carries a specific combination of D0 and D2 charges. In the crystal melting picture, fundamental constituents are not D0 and D2 branes but the atoms. This reminds us of the quark model of Gell-Mann and Zweig, where the fundamental constituents carry combinations of quantum numbers of hadrons, as opposed to the Sakata model, where existing elementary particles such as the proton, neutron and particle are chosen as fundamental constituents. 4.3. Observations on the crystal melting model. We would like to make a few observations on the statistical model of crystal melting that counts the number of BPS bound states of D branes. We have studied several examples of toric Calabi-Yau manifolds and found that the crystal structure in each case matches with the toric diagram. In particular, the ridges of the crystal, when projected onto the R2 plane, line up with the ( p, q) web of the maximally degenerate toric diagram. This phenomenon is discussed in Appendix. There, we also explain the correspondence between molten crystal configurations and perfect matching of the bipartite graph introduced in Sect. 2.2. So far, we have considered molten crystals that are obtained by removing a few atoms. We may call them low temperature configurations. The high temperature behavior of the model, describing bound states with large D0 and D2 charges, is also interesting. For C3 , it was shown in [2,5] that the high temperature limit of the crystal melting model reproduces the geometric shape of the mirror manifold. Since the high temperature limit of a general statistical model of random perfecting matchings is known to be described by a certain plane algebraic curve [54]; it would be interesting to understand its relation to the mirror of a general toric Calabi-Yau manifold [55]. In the last subsection, we found it useful to describe BPS bound states using ideals of the algebra A. In the case when the toric Calabi-Yau manifold is C3 , ideals are closely related to the quantization of the toric structure as discussed in [2]. The gauge theory for C3 is the dimensional reduction of the N = 4 supersymmetric Yang-Mills theory in four dimensions down to one dimension, and the bifundamental fields are three adjoint fields. The F-term and D-term conditions require that they all commute with each other. Thus chiral ring is generated by three elements x, y, z which commute with each other without any further relation. In this case, any ideal Iπ is characterized by the three-dimensional Young diagram π . Locate each box in the 3d Young diagram π by the Cartesian coordinates (i, j, k) (i, j, k = 1, 2, 3, . . .) of the corner of the box most distant from the origin, and define π to be a set of the 3d Cartesian coordinates (i, j, k) for boxes in π . We can then define the ideal ωπ of the chiral ring by, Iπ = {x i−1 y j−1 w k−1 |(i, j, k) ∈ / π }.
(4.2)
Crystal Melting and Toric Calabi-Yau Manifolds
193
In [2], this description was obtained by quantizing the toric geometry by using its canonical Kähler form and by identifying x i−1 y j−1 w k−1 as states in the Hilbert space. This can be generalized to an arbitrary toric Calabi-Yau manifold X as follows. One starts with the quiver diagram corresponding to X and use the brane tiling to identify the F-term equations. This gives the chiral ring generated by bifundamental fields obeying the F-term and D-term relations. As we saw in Sect. 4.3, each BPS bound state is related to an ideal of the chiral algebra. We expect that such ideals arise from quantization of the toric structure. BPS bound states of D branes emerging from the quantization of background geometry is reminiscent of the bubbling AdS space of [56] and Mathur’s conjecture on black hole microstates [57]. 5. Summary and Discussion In this paper, we established the connection between the counting of BPS bound states of D0 and D2 branes on a single D6 brane to the non-commutative Donaldson-Thomas theory. We studied the moduli space of solutions to the F-term and D-term constraints of the quiver gauge theory which arises as the low energy limit of the brane configuration. We found the direct correspondence between the gauge theory moduli space and the space of modules of the factor algebra of the path algebra for the quiver diagram quotiented by its ideal related to the F-term constraints, subject to a stability condition to enforce the D-term constraints. Using this correspondence, we found a new description of BPS bound states of the D branes in terms of the statistical model of crystal melting. The crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. The relation between the commutative and non-commutative Donaldson-Thomas invariant has been extensively discussed in the recent literature. The degeneracy of D brane bound states changes when the value of θ , used to define the stability condition, jumps along the codimension one subspace, which is called walls of marginal stability. The jump in the degeneracy can be computed by the wall crossing formula [58–60], and if we start from a particular chamber and applying the wall crossing forˆ N (A)) in any chamber we want. In the example of mula, we can obtain the value of χ (M i conifold [13], wall crossing relates non-commutative Donaldson-Thomas invariants to commutative Donaldson-Thomas invariants and to new invariants defined by Pandharipande and Thomas [61]. This story is further generalized by [14] when has no internal lattice point. When the toric diagram contains an internal lattice point, non-commutative Donaldson-Thomas invariant includes D4 branes, since H4 (Y ) = 0. Since (commutative) Donaldson-Thomas invariants do not include D4 brane charges, the above discussion of wall crossing should be modified. It has been proven recently that the topological string theory is equivalent to the commutative Donaldson-Thomas theory for a general toric Calabi-Yau manifold [2,8]. Since the commutative Donaldson-Thomas theory count BPS states for some choice of stability condition, Z BH = Z top ,
(5.1)
is indeed true in some chamber. On the other hand, our result shows that the relation,
Z BH = Z crystal melting ,
(5.2)
194
H. Ooguri, M. Yamazaki
holds in another chamber, where Z BH is the BPS state counting for another choice of the stability condition. Combining these two results, we find that the topological string theory and the statistical model of crystal melting are related by the wall crossing, and we have
Z crystal melting ∼ Z top (modulo wall crossings).
(5.3)
Since there is no wall crossing phenomenon for the Donaldson-Thomas theory on C3 , this result does not contradict with [2], where a direct identification of the topological string theory and the crystal melting is made for C3 . In general, we expect that a proper understanding of the relation between the topological string theory and the crystal melting requires that we take the wall crossing phenomena into account. The OSV formula (1.1) suggests yet another relation between the black hole microstate counting and the topological string theory. According to [62], for a compact Calabi-Yau manifold, the D6/D2/D0 brane system gives rise to a large black hole in four dimensions since it is related to a spinning M theory black hole by the KaluzaKlein reduction. In fact, one can compute the semi-classical Bekenstein-Hawking entropy for such a 4-dimensional black hole and find that it can be made arbitrarily large provided the D2 charges are sufficiently larger than the D0 charge. In this paper, we discussed the D6/D2/D0 system on a non-compact toric Calabi-Yau manifold with an infinite volume. Though it is not obvious that the gravity description in four dimensions is applicable in this case, the OSV formula has been successfully tested for a similar class of non-compact Calabi-Yau manifolds [63–66]. If it is applicable in our case, it would imply the relation between Z crystal melting and |Z top |2 , modulo wall crossings. It would be interesting to find out if such a relation holds. It appears that the crystal melting picture is closely related to the quantization of the toric structure of the Calabi-Yau manifold. It would be interesting to understand the relation better. This could lead to a new insight into quantum geometry, along with the observations in [56] for the bubbling AdS geometry and [57] for black hole microstates. Acknowledgements. We would like to thank Kentaro Nagao, Kazutoshi Ohta, Yukinobu Toda, Kazushi Ueda and Xi Yin for discussions. This work is supported in part by DOE grant DE-FG03-92-ER40701 and by the World Premier International Research Center Initiative of MEXT of Japan. H. O. is also supported in part by a Grant-in-Aid for Scientific Research (C) 20540256 of JSPS and by the Kavli Foundation. M. Y. is also supported in part by the JSPS fellowships for Young Scientists and by the Global COE Program for Physical Sciences Frontier at the University of Tokyo funded by MEXT of Japan.
A. Perfect Matchings In this Appendix we are going to explain the one-to-one correspondence between a molten crystal discussed in the main text and a perfect matching of the bipartite graph. This means that the problem of counting BPS states can also be reformulated as a problem of counting perfect matchings of the bipartite graph, where a perfect matching is a subset of edges of the bipartite graph such that each vertex is contained exactly once. The contents of this appendix is basically a recapitulation of [10]. In Sect. 4.1 we considered a quiver Q˜ = ( Q˜ 0 , Q˜ 1 ), which is a universal cover of the ˜ which we denote by , ˜ can be made bipartite quiver Q on T2 . The dual graph of Q, ˜ and is a universal cover of the bipartite graph on T2 using orientation of arrows of Q, described in Sect. 2.2. What we are going to do is to give an explicit correspondence between a perfect matching of the bipartite graph ˜ and a configuration of molten crystal.
Crystal Melting and Toric Calabi-Yau Manifolds
195
Fig. 7. Given a configuration of a molten crystal, we can construct a perfect matching of the bipartite graph. Each arrow is colored green if it is along the surface of the crystal, and red otherwise. The set of dual of arrows colored red gives a perfect matching of the bipartite graph
We first construct a perfect matching from a molten crystal. Given a molten crystal as shown in Fig. 5, choose all the arrows of Q˜ which are along the surface of the crystal. In the example of Fig. 5, such arrows are colored green in Fig. 5, while the remaining arrows are colored red. Take the set of the dual of edges colored red. It is proven by [10] that such a subset of edges of ˜ is a perfect matching. This is the perfect matching we wanted to construct. In the case when no atoms are removed from the crystal, the perfect matching obtained by this method is called the canonical perfect matching, which we denote by D0 . Since only a finite number of atoms are removed from the crystal, the perfect matching obtained from a molten crystal by the above method coincides with D when sufficiently away from the reference point i. Conversely, given a perfect matching D which coincides with D0 when sufficiently away from the reference point i, we can reproduce a molten crystal. Let us superimpose D with D0 , and we have a finite number of loops, as shown in Fig. 8 in the case of Suspended Pinched Point. Define a height function h D such that (1) h D ( j) = 0 when sufficiently away from i. (2) h D increases by one whenever we cross the loop and go inside it. The example of h D for the case of Suspended Pinched Point is shown in Fig. 8. By removing h D ( j) atoms from each j ∈ Q˜ 0 , we can construct a molten crystal. It was proven in [10] that the set of atoms removed from the crystal so defined satisfies the melting rule of Sect. 4.2. This establishes the one-to-one correspondence between a molten crystal and a perfect matching of the bipartite graph, meaning that BPS states ˜ can also be counted by perfect matchings of the bipartite graph . Finally, let us finish this Appendix by pointing out an interesting connection of the canonical perfect matching D0 with toric geometry. The example of canonical perfect matching D0 for the Suspended Pinched Point is shown in Fig. 9. In this example, the asymptotic form of the bipartite graph has four different patterns. Each of four patterns
196
H. Ooguri, M. Yamazaki
Fig. 8. By superimposing a perfect matching of Fig. 7 with the canonical perfect matching shown later in Fig. 9, we have a set of loops, which defines a height function h D . From this function we can recover a molten crystal
Fig. 9. The canonical perfect matching of the bipartite graph for the Suspended Pinched Point singularity. Asymptotically, the perfect matching corresponds to one of the four perfect matchings of the bipartite graph corresponding to vertices of the toric diagram. The blue borders between different choices of perfect matchings represent the ( p, q)-web
Crystal Melting and Toric Calabi-Yau Manifolds
197
is periodic and therefore can be thought of as a perfect matching of the bipartite graph on T2 . In the brane tiling literature, a perfect matching on the bipartite graph on T2 is known to correspond to one of the lattice points of the toric diagram [25,40].13 We recognize that the four perfect matchings are identified with the four corners of the toric diagram in Fig. 1-(a) and that the borders between different patterns are identified with the blue lines in Fig. 1-(b), which makes the ( p, q)-web of the diagram. In general, for an arbitrary toric Calabi-Yau manifold, we can use the same pattern to construct a perfect matching. Divide the universal covering of the bipartite graph into segments separated by the ( p, q)-web of the toric diagram.14 The perfect matching in each segment is periodic and is identified with one of the perfect matchings of bipartite graphs on T2 , which corresponds to one of the lattice points of the toric diagram and the lattice point in question is precisely the vertex surrounded by the two ( p, q)-webs on T2 .15 This determines a perfect matching. In particular, this means that the ridges of the crystal line up with the ( p, q)-web of the toric diagram. We have examined several other examples as well, and this pattern holds in all cases. Thus, we conjecture that the perfect matching constructed in this way is canonical. We would like to stress again that this conjecture is not needed to construct the crystal melting model. Here we are simply pointing out that, in the examples we have studied, the crystalline structures fit beautifully with the corresponding toric geometries. References 1. Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004) 2. Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. JHEP 0804, 011 (2008) 3. Donaldson, S.K., Thomas, R.P.: Gauge theory in higher dimensions. In: The geometric universe: science, geometry and the work of Roger Penrose. Oxford: Oxford Univ. Press, 1998 4. Thomas, R.P.: A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations. J. Diff. Geom. 54, 367 (2000) 5. Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and classical crystals. http://arXiv.org/ abs/hep-th/0309208v2, 2003 6. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and DonaldsonThomas theory. I. Compos. Math. 142, 1263 (2006) 7. Aganagic, M., Klemm, A., Marino, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005) 8. Maulik, D., Oblomkov, A., Okounkov, A., Pandharipande, R.: Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. http://arxiv.org/abs/0809.3976v1[math.AG], 2008 9. Szendröi, B.: Non-commutative Donaldson-Thomas theory and the conifold. Geom. Topol. 12, 1171 (2008) 10. Mozgovoy, S., Reineke, M.: On the noncommutative Donaldson-Thomas invariants arising from brane tilings. http://arXiv.org/abs/0809.0117v2[math.AG], 2008 11. Young, B.: Computing a pyramid partition generating function with dimer shuffling. http://arXiv.org/abs/ 0709.3079v2[math.CO], 2007 12. Young, B.: with an appendix by J.Bryan, Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds. http://arXiv.org/abs/0802.3948v2[math.CO], 2008 13. Nagao, K., Nakajima, H.: Counting invariant of perverse coherent sheaves and its wall-crossing. http:// arXiv.org/abs/0809.2992v3[math.AG], 2008 14. Nagao, K.: Derived categories of small toric Calabi-Yau 3-folds and counting invariants. http://arXiv. org/abs/0809.2994v3[math.AG], 2008 13 For this correspondence, we consider superimposition of perfect matchings and define a Z2 -valued height function, which is similar to the height function h D defined previously. 14 We choose the diagram that corresponds to the most singular Calabi-Yau manifold. 15 In a consistent quiver gauge theory, it is believed that the multiplicity of perfect matchings at the vertices of the toric diagram is one [28].
198
H. Ooguri, M. Yamazaki
15. Jafferis, D.L., Moore, G.W.: Wall crossing in local Calabi Yau manifolds. http://arxiv.org/abs/0810. 4909v1[hep-th], 2008 16. Chuang, W.Y., Jafferis, D.L.: Wall crossing of BPS states on the Conifold from Seiberg duality and pyramid partitions. http://arxiv.org/abs/0810.5072v2[hep-th], 2008 17. Saulina, N., Vafa, C.: D-branes as defects in the Calabi-Yau crystal. http://arxiv.org/abs/hep-th/ 0404246v1, 2004 18. Katz, S.H.: Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds. http://arxiv.org/abs/math/0408266v2[math.AG], 2004 19. Okuda, T.: Derivation of Calabi-Yau crystals from Chern-Simons gauge theory. JHEP 0503, 047 (2005) 20. Dijkgraaf, R., Vafa, C., Verlinde, E.: M-theory and a topological string duality. http://arxiv.org/abs/hepth/0602087v1, 2006 21. Sulkowski, P.: Crystal model for the closed topological vertex geometry. JHEP 0612, 030 (2006) 22. Dijkgraaf, R., Orlando, D., Reffert, S.: Dimer models, free fermions and super quantum mechanics. http:// arxiv.org/abs/0705.1645v2[hep-th], 2007 23. Jafferis, D.L.: Topological quiver matrix models and quantum foam. http://arxiv.org/abs/0705. 2250v1[hep-th], 2007 24. Heckman, J.J., Vafa, C.: Crystal melting and black holes. JHEP 0709, 011 (2007) 25. Hanany, A., Kennaway, K.D.: Dimer models and toric diagrams. http://arxiv.org/abs/hep-th/0503149v2, 2005 26. Franco, S., Hanany, A., Kennaway, K.D., Vegh, D., Wecht, B.: Brane dimers and quiver gauge theories. JHEP 0601, 096 (2006) 27. Franco, S., Hanany, A., Martelli, D., Sparks, J., Vegh, D., Wecht, B.: Gauge theories from toric geometry and brane tilings. JHEP 0601, 128 (2006) 28. Hanany, A., Vegh, D.: Quivers, tilings, branes and rhombi. JHEP 0710, 029 (2007) 29. Kennaway, K.D.: Brane tilings. Int. J. Mod. Phys. A 22, 2977 (2007) 30. Yamazaki, M.: Brane Tilings and Their Applications. Fortsch. Phys. 56, 555 (2008) 31. Douglas, M.R., Moore, G.W.: D-branes, quivers, and ALE instantons. http://arxiv.org/abs/hep-th/ 9603167v1, 1996 32. Feng, B., He, Y.H., Kennaway, K.D., Vafa, C.: Dimer models from mirror symmetry and quivering amoebae. Adv. Theor. Math. Phys. 12, 3 (2008) 33. Ooguri, H., Vafa, C.: Two-Dimensional black hole and singularities of CY manifolds. Nucl. Phys. B 463, 55 (1996) 34. Imamura, Y.: Anomaly cancellations in brane tilings. JHEP 0606, 011 (2006) 35. Imamura, Y.: Global symmetries and ’t Hooft anomalies in brane tilings. JHEP 0612, 041 (2006) 36. Imamura, Y., Isono, H., Kimura, K., Yamazaki, M.: Exactly marginal deformations of quiver gauge theories as seen from brane tilings. Prog. Theor. Phys. 117, 923 (2007) 37. Klebanov, I.R., Witten, E.: Superconformal field theory on threebranes at a Calabi-Yau singularity. Nucl. Phys. B 536, 199 (1998) 38. Klebanov, I.R., Strassler, M.J.: Supergravity and a confining gauge theory: Duality cascades and χ SBresolution of naked singularities. JHEP 0008, 052 (2000) 39. Imamura, Y., Kimura, K., Yamazaki, M.: Anomalies and O-plane charges in orientifolded brane tilings. JHEP 0803, 058 (2008) 40. Franco, S., Vegh, D.: Moduli spaces of gauge theories from dimer models: Proof of the correspondence. JHEP 0611, 054 (2006) 41. Luty, M.A., Taylor, W.: Varieties of vacua in classical supersymmetric gauge theories. Phys. Rev. D 53, 3399 (1996) 42. King, A.D.: Moduli of representations of finite dimensional algebras. Quart. J. Math. Oxford 45, 515 (1994) 43. Cirafici, M., Sinkovics, A., Szabo, R.J.: Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory. Nucl. Phys. B 809, 452–518 (2009) 44. Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. http://arxiv.org/abs/hep-th/0403166v1, 2004 45. Van den Bergh, M.: Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel, Berlin: Springer, 749 (2004) 46. Bridgeland, T.: Flops and derived categories. Invent. Math. 147, 613 (2002) 47. Van den Bergh, M.: Three-dimensional flops and non-commutative rings. Duke Math. J. 122, 423 (2004) 48. Ueda, K., Yamazaki, M.: Brane tilings for parallelograms with application to homological mirror symmetry. http://arxiv.org/abs/math/0606548v2[math.AG], 2006 49. Ueda, K., Yamazaki, M.: Homological mirror symmetry for toric orbifolds of toric del Pezzo surfaces. http://arxiv.org/abs/math/0703267v1[math.AG], 2007 50. Douglas, M.R., Fiol, B., Romelsberger, C.: Stability and BPS branes. JHEP 0509, 006 (2005) 51. Bridgeland, T.: Stability conditions on triangulated categories. Ann. of Math. 166, 317 (2007)
Crystal Melting and Toric Calabi-Yau Manifolds
199
52. Donaldson, S.K.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. 50, 1 (1985) 53. Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Comm. Pure Appl. Math. 39, 257 (1986) 54. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and Amoebae. http://arxiv.org/abs/math-ph/0311005v1, 2003 55. Ooguri, H., Yamazaki, M.: Work in progress 56. Lin, H., Lunin, O., Maldacena, J.M.: Bubbling AdS space and 1/2 BPS geometries. JHEP 0410, 025 (2004) 57. Mathur, S.D.: The fuzzball proposal for black holes: An elementary review. Fortsch. Phys. 53, 793 (2005) 58. Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. http://arxiv.org/abs/hep-th/ 0702146v2, 2007 59. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. http://arxiv.org/abs/0811.2435v1[math.AG], 2008 60. Gaiotto, D., Moore, G.W., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory. http://arxiv.org/abs/0807.4723v1[hep-th], 2008 61. Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. http://arxiv. org/abs/0707.2348v3[math.AG], 2007 62. Gaiotto, D., Strominger, A., Yin, X.: New connections between 4D and 5D black holes. JHEP 0602, 024 (2006) 63. Vafa, C.: Two dimensional Yang-Mills, black holes and topological strings. http://arxiv.org/abs/hep-th/ 0406058v2, 2004 64. Aganagic, M., Ooguri, H., Saulina, N., Vafa, C.: Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings. Nucl. Phys. B 715, 304 (2005) 65. Dijkgraaf, R., Gopakumar, R., Ooguri, H., Vafa, C.: Baby universes in string theory. Phys. Rev. D 73, 066002 (2006) 66. Aganagic, M., Ooguri, H., Okuda, T.: Quantum entanglement of baby universes. Nucl. Phys. B 778, 36 (2007) Communicated by N. A. Nekrasov
Commun. Math. Phys. 292, 201–215 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0833-1
Communications in
Mathematical Physics
Continuity of Quantum Channel Capacities Debbie Leung1 , Graeme Smith2 1 Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L3G1,
Canada. E-mail:
[email protected]
2 IBM TJ Watson Research Center, 1101 Kitchawan Road, Yorktown Heights,
NY 10598, USA. E-mail:
[email protected] Received: 11 December 2008 / Accepted: 24 February 2009 Published online: 26 May 2009 – © Springer-Verlag 2009
Abstract: We prove that a broad array of capacities of a quantum channel are continuous. That is, two channels that are close with respect to the diamond norm have correspondingly similar communication capabilities. We first show that the classical capacity, quantum capacity, and private classical capacity are continuous, with the variation on arguments apart bounded by a simple function of and the channel’s output dimension. Our main tool is an upper bound of the variation of output entropies of many copies of two nearby channels given the same initial state; the bound is linear in the number of copies. Our second proof is concerned with the quantum capacities in the presence of free backward or two-way public classical communication. These capacities are proved continuous on the interior of the set of non-zero capacity channels by considering mutual simulation between similar channels.
1. Introduction There are several notions of capacity for a noisy quantum communication channel. For example, we may be interested in a channel’s capacity for either classical [1,2], private classical [3], or quantum [3–5] communications. We may have access to auxiliary resources in addition to the channel, such as entanglement, one-way classical communication from the sender to receiver, from the receiver to the sender, or two-way classical communications. In all of these situations, there is a sensible notion of capacity that can be studied. Except when free auxiliary entanglement is available, where the problem is effectively solved [6], the various capacities of even very simple channels are unknown. One property that we would hope for in a capacity is continuity. From a practical point of view there will always be a certain amount of channel uncertainty in real systems. In this setting, if nearby channels had dramatically different capacities, the theory of quantum capacities would be of limited value. However, from a mathematical point of
202
D. Leung, G. Smith
view continuity is not at all obvious—very similar channels can become quite far apart given many copies, and the capacity is operationally defined in terms of an asymptotic number of channel uses. This is not a problem when a single-letter capacity formula is available, in which case we can reason about the formula directly, but when only a multiletter formula is available (or worse, none at all) the problem of continuity becomes a challenge. The continuity of channel capacities has been considered before. For example, in their study of the quantum erasure channel [7], Bennett, DiVincenzo, and Smolin implicitly assumed the continuity of the quantum channel capacity to upper bound the capacity of this channel. For the erasure channel, this assumption was rigorously justified later in [8]. Keyl and Werner explicitly considered continuity of the quantum channel capacity in [9], where it was shown that the capacity is lower semi-continuous. Continuity of the Holevo information (whose regularization gives a multi-letter formula for the classical capacity) was considered in [10], where it was shown to be continuous for finite dimensional outputs and lower semi-continuous in general. A related set of questions concerns the continuity of entropic quantities and entanglement measures, which are functions on quantum states. For example, Fannes [11] found a tight bound on the variation of von Neumann entropy of finite dimensional states. This was subsequently used by Nielsen to study the continuity of entanglement of formation [12]. As another example, Donald and Horodecki proved the continuity of the relative entropy of entanglement [13]. The continuity of asymptotic (i.e., regularized) entanglement measures was studied by Vidal in [14], which were shown to be continuous in any open set of distillable states. More recently, Alicki and Fannes generalized the continuity result in [15] to conditional entropy, and used it to prove the continuity of squashed entanglement [16]. In this work we show the continuity of various communication capacities of quantum channels with finite output dimensions. For the unassisted capacities for classical, private classical, and quantum communication, our tool is an inequality controlling the variation of output entropies of many copies of two nearby channels given the same initial state. By careful use of the Alicki-Fannes inequality [15], this bound is shown to be linear (not quadratic) in the number of copies. For the quantum capacity with two-way classical communication, and the quantum capacity with classical back communication, we also show continuity within an open set of nonzero quantum capacity channels. Our results in this setting build on [14], whose arguments are extended from the distillable entanglement of states to the capacity of channels. The rest of the paper is organized as follows. Section 2 contains various definitions, concepts, and prior results used in this paper. Our main tool, the inequality controlling the variation of output entropies of many copies of two nearby channels given the same initial state is proved in Sec. 3. This is used to show our main results, the continuity of the quantum, classical, and private classical capacity in Sec. 4. For simplicity throughout most of this paper, we focus on channels with finite dimensional inputs and outputs, although the results of Sec. 4 can easily be seen to apply to channels with infinite dimensional inputs and finite outputs. One exception to this focus is in Sec. 5, where we consider a family of pairs of infinite dimensional channels, parameterized by n. As n increases, each pair has decreasing distance, but their capacities differ by at least a constant, thereby showing that finite output dimension is needed for continuity. Continuity for the quantum capacities assisted by backward or two-way classical communication in the interior of the nonzero capacity region is proved in Sec. 6. We make a few concluding remarks in Sec. 7.
Continuity of Quantum Channel Capacities
203
2. Preliminaries In this section, we introduce the concepts, notations, definitions, and background materials, focusing on finite dimensional quantum systems. Notations and discussion in the infinite dimensional case will be deferred to Sec. 5. 2.1. Quantum States and Channels. Let H be a complex Hilbert space, and B(H) be the set of bounded linear operators taking H to itself. A quantum state is represented by a positive semidefinite operator ρ ∈ B(H) with unit trace. Except in Sec. 5, we will be interested in finite-dimensional H. A quantum channel N that takes states from Hin to Hout is a linear map from B(Hin ) to B(Hout ) that is trace-preserving and completely-positive. In particular, when H = Hin = Hout , we denote by I the identity map from B(H) to itself. Recall the definition that N is completely-positive if for any reference system with associated Hilbert space Href , I ⊗ N maps the positive-semidefinite cone in B(Href ⊗ Hin ) to that in B(Href ⊗ Hout ). We also call channels, which are trace-preserving and completely-positive, “TCP maps.” They are exactly the physical operations on a state that are allowed by quantum mechanics. A quantum system is associated with a Hilbert space and its set of bounded operators. We also use the system name loosely. For example, we may say that a channel takes system A to system B, or write N : A → B. We denote the trace, which is a simple example of a TCP map, by Tr [·]. A partial trace on a composite system is simply the trace operation on one component. A pure state is a rank one projector, and is also represented by any vector it projects onto. For a quantum state ρ ∈ B(H), a purification is any pure state |ψψ| ∈ B(H ⊗ H ) such that the partial trace over H gives ρ, and purifications always exist. Any channel N can be represented as a conjugation by an isometry U : Hin → Hout ⊗ Henv , followed by a partial trace: N (ρ) = Tr env UρU † . We sometimes add subscripts to the symbols for quantum states and channels to emphasize what systems they act on, but we may omit these to avoid cluttering. However, for multipartite states, the reduced state on a subset of systems is always subscripted by the subset. Throughout this paper, we use a distance measure between states given by the 1-norm of their difference: ||ρ − σ ||1 = Tr |ρ − σ |. Half of the above is called the trace distance, the quantum analogue of the total variation distance in the classical setting. We use a distance measure between channels (mapping from B(Hin ) to B(Hout )) induced by the diamond norm: ||N1 − N2 || = max{||(N1 − N2 ) ⊗ I(X )||1 : X ∈ B(Hin ⊗ Href ), ||X ||1 = 1}. The maximum can always be attained with X being a pure quantum state. Operationally, the diamond norm on the difference between the two channels characterizes the probability to distinguish them, if one can prepare an optimal state and feed part of it into the channel. The distance measure also has the nice property that, increasing the dimension of the reference system beyond dim(Hin ) does not increase the distinguishability. This gives us control over the trace distance of the output states of different channels given the same input, and subsequently other quantities of interest to be defined in the next subsection.
204
D. Leung, G. Smith
The diamond norm of a channel is closely related to the family of completely bounded norms (cb-norms), and in fact is equal to the usual cb-norm of the adjoint channel as well as a generalized cb-norm of the original channel (for more on cb-norms and their relation to quantum information, see [17,18]). 2.2. Entropic Quantities. For a classical random variable X with Prob(X = x) = px , the Shannon entropy of X is given by H (X ) = − x px log px (or H ({ px })). If X is binary with probabilities p, 1 − p, H (X ) is written as H ( p). Here and throughout this paper, log is in base 2. For a quantum system A prepared in state ρ, the von Neumann entropy is written as S(A)ρ or S(ρ) = − Tr ρ log ρ = H ({λk }), where λk is the k th eigenvalue of ρ. Throughout the paper, subscripts showing states on which entropies and other information theoretic quantities are evaluated are omitted when there is little risk of confusion. For two systems AB in state ρ, we mention a few measures of correlation between A and B: • the quantum mutual information is defined as I (A; B)ρ = S(A) + S(B) − S(AB), where entropies are evaluated on ρ and its partial traces. • The conditional entropy is given by S(A|B) = S(AB) − S(B). • The coherent information I coh (AB)ρ is given by S(B) − S(AB) = −S(A|B). The entropy and conditional entropy, viewed as functions of the underlying states, are both continuous. The following, particularly Theorem 2, will be helpful tools for our task of showing the continuity of capacities. Theorem 1 (Fannes Inequality [11]). For any ρ and σ with ||ρ − σ ||1 ≤ , |S(ρ) − S(σ )| ≤ log d + H (). Theorem 2 (Alicki-Fannes Inequality [15]). For any ρ AB and σ AB with ||ρ − σ ||1 ≤ , S(A|B)ρ − S(A|B)σ ≤ 4 log d A + 2H (). 2.3. Capacities of a quantum channel. Consider a quantum channel N : A → B. The channel N has several different capacities for communication. The following quantities will play crucial roles in the various capacities: • For an input ensemble { px , φx }, let ω = x px |xx| X ⊗ N (φx ) and χ (N ) := max I (X ; B)ω px ,φx
be the optimal Holevo information [19] of the output ensemble (after the channel acts on the input). • For an input state ρ A A , where part of it will be fed into N , let I coh (N , ρ A A ) = I coh (AB)I ⊗N (ρ A A ) be the coherent information generated. Maximizing over the input gives the coherent information of N : I coh (N ) = max I coh (N , ρ A A ). ρ A A
We remark that the maximizing state can be chosen to be pure.
Continuity of Quantum Channel Capacities
205
• For an input ensemble { px , φx }, let ω = x px |xx| X ⊗ (U φx U † ) B E , where U : H A → H B ⊗ H E is an isometric extension of N . Then, I priv (N ) = max (I (X ; B)ω − I (X ; E)ω ) , px ,φx
where the mutual information is evaluated on the reduced states. To give the operational definitions of the different capacities of N for communication, we need to consider n uses of the channel. We will use the shorthands N n , An , B n , and E n to stand for N ⊗n , A⊗n , B ⊗n , and E ⊗n . Definition 1. Classical Capacity. We say that a rate R is -classically-achievable if Kn and a there is an n such that for all n ≥ n there is a classical code {ρk ∈ An }k=1 Kn n n decoding operation Dn : B → {|kk|}k=1 such that ∀k, ||Dn (N (ρk )) − |kk|||1 ≤ with log K n ≥ n R. A rate is classically-achievable if it is -classically achievable for all > 0. The classical capacity of N , C(N ), is the supremum over classically-achievable rates. Theorem 3 (HSW Theorem [1,2]). The classical capacity satisfies C(N ) = lim
n→∞
1 χ (N n ). n
Definition 2. Quantum Capacity. We say that a rate R is -achievable if there is an n such that for all n ≥ n there is a quantum code, Cn ⊂ An and decoding operation Dn : B n → Cn such that for all ψ ∈ B(Cn ), ||Dn (N n (ψ)) − ψ||1 ≤ and log dim HCn ≥ n R. A rate R is achievable if it is -achievable for all > 0. The quantum capacity of N , Q(N ), is the supremum over achievable rates. Theorem 4 (LSD Theorem [3–5]). The quantum capacity satisfies Q(N ) = lim
n→∞
1 coh n I (N ). n
Definition 3. Private Capacity. The private capacity is the capacity of a channel for classical communication with the added requirement that an adversary with access to the environment of the channel is ignorant of the communication. More formally, we say that a rate R is -privately-achievable if there is an n such that ∀n ≥ n Kn there exists a classical code {ρk ∈ An }k=1 with log K n ≥ n R and decoding operation K n such that for all k, Dn : B n → {|kk|}k=1 ||Dn (N n (ρk )) − |kk| ||1 ≤ , and ||ρ Ek n − σ E n ||1 ≤ . n (ρk ), where N (ρ) = Tr B UρU † , with U : H A → H B ⊗ H E an Here ρ Ek n = N isometric extension of N , and σ E n is a fixed state on E n . If R is -privately-achievable for all > 0, it is called privately achievable, and the supremum of privately-achievable rates is called the private capacity. Theorem 5 ([3]). The private capacity satisfies C p (N ) = lim
n→∞
1 priv n I (N ). n
206
D. Leung, G. Smith
The three capacity definitions above are similar in structure, and differing only in the type of information being sent. The corresponding theorems, which give what are called “regularized capacity formulas” also seem to be parallel. In each case, the “regularization”, as the limit over n is called, prevents us from evaluating the capacity of a given channel explicitly, or even numerically. In the case of the quantum capacity [20,21] and the private classical capacity [22] it is known for a while that the regularization cannot be removed in general. More recently, the regularization in the classical capacity was reported to be generally necessary [23]. While very little is known about the capacities above, even less is known about the capacity of a channel for quantum communication assisted by two-way classical communication. To define this capacity, we introduce the notion of an n-use protocol Pn , where n denote the number of times the channel N can be used. Just as in the definition of the unassisted quantum capacity, we consider a system Cn which holds the quantum information to be sent. We use the same symbol to denote Bob’s quantum system which holds the quantum data in his possession at the end of the protocol. Pn is a composition of the following steps (in order of being performed): A0 , M→0 , N , B1 , M←1 , A1 , M→1 , N , B2 , M←2 , · · · An−1 , M→(n−1) , N , Bn , M←n , An . Here, each Ai is performed by the sender Alice on Cn and her auxiliary system after the i th channel use, and each produces an extra system A as an input to the (i+1)th channel use. Each M→i transmits classical communication from Alice to the receiver Bob. Each Bi is performed by Bob on his auxiliary system and all i systems cumulated from the channel uses. Each generates some classical outcome to be sent to Alice in the step M←i . Using the notion of a protocol, we can now define quantum capacity with two-way classical assistance. Definition 4. Quantum Capacity with two-way classical assistance. For any > 0 we say that a rate R is -2-way-achievable if there is an n such that for all n ≥ n there is an n-use protocol Pn such that for any auxiliary reference system A, ψ ∈ Cn ⊗ A, ||Pn ⊗ I(ψ) − ψ||1 ≤ and log dim HCn ≥ n R. In other words, Pn and the identity map on the code space are -close in the diamond norm. A rate is achievable if it is -achievable for all > 0. The quantum capacity of N with two-way classical assistance, Q 2 (N ), is the supremum over achievable rates. Definition 5. Quantum Capacity with back classical assistance Q B (N ). An n-use protocol in this setting is similar to that with two-way assistance, except that M→i are omitted. The rest of the capacity definition is similar to that of Q 2 (N ). Little is known about these assisted capacities. One proven fact [24,25] is that Q 2 (N ) is equal to the entanglement capacity of N (informally, that is the maximum amount of near perfect entanglement generated per use of N , asymptotically). Clearly Q(N ) ≤ Q B (N ) ≤ Q 2 (N ), but beyond that, almost nothing is known about Q B (N ). For instance, there is no known analogue of a connection to entanglement capacity. 3. Continuity of Output Entropy The following theorem is one of our main technical tools. Theorem 6. Let N : A → B and M : A → B be quantum channels and d B be the finite dimension of B. Let A be an auxiliary reference system. If ||N − M|| ≤ , then, for any state φ ∈ B(A An ), S (I ⊗ N n )(φ) − S (I ⊗ Mn )(φ) ≤ n (4 log d B + 2H ()).
Continuity of Quantum Channel Capacities
207
Proof. Let ρ kAB n = I A ⊗ M⊗k ⊗ N ⊗(n−k) (φ A An ). In the above, we have explicitly labeled the auxiliary, the input and the output systems on the states. We omit these subscripts from now on. Setting k = 0 and then n, we have in particular ρ 0AB n = I ⊗ N n (φ) and ρ nAB n = I ⊗ Mn (φ). Since ρ k−1 and ρk differs only in the k th output system, S(AB1 . . . Bk−1 Bk+1 . . . Bn )ρ k−1 = S(AB1 . . . Bk−1 Bk+1 . . . Bn )ρ k .
(1)
The quantity we are interested in is S(AB n )ρ 0 − S(AB n )ρ n , which satisfies S(AB n )ρ 0 − S(AB n )ρ n = ≤
n n n S(AB )ρ k−1 − S(AB )ρ k k=1 n
S(AB n )
ρ k−1
− S(AB n )ρ k .
k=1
Applying Eq.(1) to a single term in this sum, we have S(AB n )
S(AB n )ρ k = S(AB n )ρ k−1 − S(AB1 . . . Bk−1 Bk+1 . . . Bn )ρ k−1 −S(AB n )ρ k + S(AB1 . . . Bk−1 Bk+1 . . . Bn )ρ k = S(Bk |AB1 . . . Bk−1 Bk+1 . . . Bn )ρ k−1 −S(Bk |AB1 . . . Bk−1 Bk+1 . . . Bn ) k .
ρ k−1−
ρ
Because ||N − M|| ≤ , we also have ||ρ k − ρ k−1 ||1 ≤ , so by the Alicki-Fannes inequality, S(Bk |AB1 . . . Bk−1 Bk+1 . . . Bn ) ≤ 4 log d B + 2H ().
ρk
− S(Bk |AB1 . . . Bk−1 Bk+1 . . . Bn )ρ k−1
As a result, we find S(AB n )ρ 0 − S(AB n )ρ n ≤ n(4 log d B + 2H ()), which completes the proof.
208
D. Leung, G. Smith
4. Continuity of Capacities for Channels with Finite Output Dimension We now apply Theorem 6 to show the continuity of C(N ), Q(N ), and C p (N ). Each of these capacities has the form F(N ) = limn→∞ n1 max P (n) f n (N n , P (n) ) for some appropriate family of function { f n } and parameters P (n) to be optimized over. We make repeated use of the following lemma. Lemma 1. If F(N ) = limn→∞ n1 sup P (n) f n (N n , P (n) ) and ∀n, ∀P (n) , | f n (N n , P (n) )− f n (Mn , P (n) )| ≤ nc, then |F(N ) − F(M)| ≤ c. Proof. Let > 0 be arbitrary. Let f n (N n ) = sup P (n) f n (N n , P (n) ). Suppose f n (N n ) (n) (n) and f n (Mn ) are -close to optimal at P1 and P2 . Then, (n)
(n)
f n (N n ) − < f n (N n , P1 ) ≤ f n (Mn , P1 ) + nc ≤ f n (Mn ) + nc, (n)
(n)
f n (Mn ) − < f n (Mn , P2 ) ≤ f n (N n , P2 ) + nc ≤ f n (N n ) + nc. Thus, ∀ > 0, ∀n, | f n (N n ) − f n (Mn )| ≤ nc + . Taking limits → 0, n → ∞, |F(N ) − F(M)| ≤ c. Note that in particular, Lemma 1 holds with sup replaced by max, as needed in the following corollaries. Corollary 1. The classical capacity of a quantum channel with finite-dimensional output is continuous. Quantitatively, if N , M : A → B, where the dimension of B is d B and ||N − M|| ≤ , then |C(N ) − C(M)| ≤ 8 log d B + 4H (). Proof. From the HSW theorem C(N ) = lim
n→∞
1 1 χ (N n ) = lim max I (X ; B n )ω(n) , n→∞ n p ,φ (n) n x x
(n) where ω(n) = x px |xx| X ⊗N ⊗n (φx ). For any N : A → B and M : A → B with ||N − M|| ≤ , and for fixed n and { px , φx(n) }, letting ω = x px |xx| X ⊗ N n (φx(n) ) (n) and ω˜ = x px |xx| X ⊗ Mn (φx ), we have I (X ; B n )ω − I (X ; B n )ω˜ = S(B n )ω − S(B n X )ω − S(B n )ω˜ + S(B n X )ω˜ ≤ S(B n )ω − S(B n )ω˜ + S(B n X )ω˜ − S(B n X )ω ≤ 2n (4 log d B + 2H ()) . Applying Lemma 1 gives the desired result |C(N ) − C(M)| ≤ 8 log d B + 4H (). Corollary 2. The quantum capacity of a quantum channel with finite dimensional output is continuous. Quantitatively, if N , M : A → B, where the dimension of B is d B and ||N − M|| ≤ , then |Q(N ) − Q(M)| ≤ 8 log d B + 4H ().
Continuity of Quantum Channel Capacities
209
Proof. From the LSD Theorem, Q(N ) = lim
n→∞
1 coh n 1 I (N ) = lim max I coh (N n , ρ A An ). n→∞ n n ρ A An
Let ω AB n = I ⊗ N n (ρ A An ) and ω˜ AB n = I ⊗ Mn (ρ A An ). Consider the difference of coherent informations coh n I (N , ρ A An ) − I coh (Mn , ρ A An ) = S(B n )ω − S(AB n )ω − S(B n )ω˜ + S(AB n )ω˜ ≤ S(B n )ω − S(B n )ω˜ + S(AB n )ω − S(AB n )ω˜ ≤ 2n (4 log d B + 2H ()). Applying Lemma 1 gives the result.
Corollary 3. The private classical capacity of a quantum channel with finitedimensional output is continuous. Quantitatively, if N , M : A → B, where the dimension of B is d B and ||N − M|| ≤ , then |C p (N ) − C p (M)| ≤ 16 log d B + 8H (). Proof. Let U and W be the isometric extensions for N and M respectively: 1 priv n 1 I (N ) = lim max I (X ; B n )ω − I (X ; E n )ω , n→∞ n px ,φx n where φx lives in An , ω X B n E n = x px |xx| ⊗ U φx U † and |ω X B n E n G purifies it. Then, C p (N ) = lim
n→∞
I (X ; B n )ω − I (X ; E n )ω = [S(B n ) − S(B n X )]ω − [S(E n ) − S(E n X )]ω = [S(B n ) − S(B n X )]ω − [S(X B n G) − S(B n G)]|ωω| .
(2)
Similarly, define ω˜ X B n E n = x px |xx| ⊗ W φx W † for M. Switching from Eq. (2) to that defined by ω, ˜ the difference can be bounded by applying Theorem 6 to each of the four terms followed by Lemma 1, giving the stated result. 5. Discontinuity of Capacities with Infinite Output Dimension In this section we provide simple examples to show that the classical and quantum capacities of channels with infinite output dimensions are not generally continuous. An earlier demonstration of the discontinuity of the classical capacity for infinite dimensional quantum channel was given by Shirokov [26]. For an infinite dimensional complex Hilbert space H with bounded linear operators B(H), the space of all trace class operators (subset of B(H) with finite trace) is denoted T(H), and its positive semidefinite subset is denoted T+ (H). A quantum state is an element of T+ (H) with unit trace. A quantum channel N from Hin to Hout is a linear map from T(Hin ) to T(Hout ) that is trace-preserving and completely-positive.
210
D. Leung, G. Smith
5.1. Classical Capacity. ∞ , and H = Span{|i}∞ . Consider the channels N Example 1. Let H = Span{|i}i=0 + i=1 and Mn : T(H+ ) → T(H) with
N (|i j|) = Tr(|i j|) |00| and
Mn = 1 −
1 1 N+ idn , log n log n
where idn (|i j|) = |i j| for 1 ≤ i, j ≤ n = Tr(|i j|) |00| otherwise. First of all, we have C(N ) = 0, since N maps every state to |00|. As for the capacity of Mn , an easy lower bound can be obtained by using the codewords |kk| for k = 1, . . . , n, turning Mn to a classical erasure channel in n-dimensions, with erasure probability pe = 1− log1 n . The capacity of the latter is known [7] to be (1− pe ) log n = 1. Thus, C(Mn ) ≥ 1. However,
1 ||N − Mn || = (N − idn ) log n 1 2 ||N − idn || ≤ . = log n log n
5.2. Quantum Capacity. Example 2. Now let N : T(H+ ) → T(H) be defined by N (ρ) =
1 1 Tr(ρ) |00| + ρ. 2 2
That is, N is a 50% erasure channel, so that Q(N ) = 0. Let
1 1 N+ idn . Mn = 1 − log n log n A lower bound of the quantum capacity can be obtained by restricting each input to the span of {|i}i=1,...,n , so that Mn is effectively a quantum erasure channel with 1 n-dimensional inputs and with erasure probability pe = 21 − 2 log n . This quantum erasure channel has capacity[7] (1 − 2 pe ) log n = 1. Therefore, Q(Mn ) ≥ 1. As before, we have ||N − Mn || ≤
2 log n ,
so that Q is also discontinuous.
Continuity of Quantum Channel Capacities
211
6. Two-Way Capacity and Capacity with Back Communication For a general channel, these capacities are not known to have a closed form expression. In this setting, an argument similar to that for continuity of asymptotic entanglement measures in [14] can be used for the interior of the nonzero capacity region. Q 2 and Q B differ in the definition of the n-use protocol, and we will see that this difference does not affect the argument, and we only talk about Q 2 for clarity. For any metric chosen for the space of channels, continuity of Q 2 at N can be stated as ∀ > 0, ∃δ > 0 such that ∀N ∈ B(N , δ), |Q 2 (N ) − Q 2 (N )| ≤ , where B(N , δ) is an open ball of radius δ centered at N (similarly for Q B ). We consider the set of channels taking din to dout dimensions. 6.1. Interior of {Q 2 (N ) > 0}. Let us denote the interior of {Q 2 (N ) > 0} by Q+2 . Suppose N ∈ Q+2 . Using the definition of continuity stated above, we will derive δ as a function of and other relevant parameters, so that ∀ > 0, ∃δ > 0 such that ∀N ∈ B(N , δ), |Q 2 (N ) − Q 2 (N )| ≤ . First, consider B(N , ), where is small enough to ensure B(N , ) ⊂ Q+2 (i.e., Q 2 > 0 on the entire B(N , )). Second, for every M on the boundary of B(N , ), we specify two other channels M1 and M2 so that: M = p1 M1 + (1 − p1 )N , N = p2 M2 + (1 − p2 )M,
(3) (4)
for some p1 , p2 ∈ [0, 1]. M1 , M2 need not be in Q+2 but have to be TCP maps. Such M1 , M2 always exist (for example, we can take them to be M and its antipodal point on B(N , ) respectively). We take M1 , M2 to be on the boundary of the set of channels, as far from N , M as possible to minimize p1 , p2 . The concepts involved in the proof are summarized in the following diagram:
We show how to simulate M by N , from which we derive an upper bound on Q 2 (M), Eq. (5), in terms of Q 2 (N ): (1) We start from the definition of Q 2 (N ). Consider any R1 < Q 2 (N ), with δ1 > 0 such that R1 = Q 2 (N ) − δ1 . For any > 0, ∃n such that ∀n 1 ≥ n , there is a protocol Pn 1 with n 1 uses of N and 2-way classical communication that simulates the identity map on an 2n 1 R1 -dimensional system -close in diamond norm.
212
D. Leung, G. Smith
(2) Any channel can be trivially (and inefficiently) simulated by either one of the two following methods: Alice sends the input noiselessly to Bob who then locally applies the channel, or Alice applies the channel on the input and sends the resulting state to Bob via the noiseless channel. Thus, log d noiseless qubit channels are sufficient for simulating any channel where d = min(din , dout ), in an exact and 1-shot manner. (3) Using the assisting classical communication (only one of the forward or backward direction suffices), Alice and Bob can agree on n biased coins (with probabilities of the two outcomes being p1 , 1− p1 ) and apply the channel N or M1 accordingly. Due to the Chernoff bound, ∀δCh , ∃n Ch such that the probability is less than that it requires more than n( p1 + δCh ) uses of M1 or more than n(1 − p1 + δCh ) uses of N . In this unlikely event, Alice and Bob just run an inaccurate simulation. We now put these 3 steps together. Let n 1 = n (Q 2 (N ) − δ1 )−1 ( p1 + δCh ) log d. We use an n 1 -use protocol of N to simulate n( p1 + δCh ) log d identity channels (it will be -close in diamond norm if n 1 ≥ n ) which in turns simulates n( p1 + δCh ) uses of M1 with the same precision. In addition to the above, we use the coin tosses and n(1 − p1 + δCh ) direct uses of N to simulate n uses of M. This simulation is -close unless an atypical outcome of the coin tosses occurs. If n is large enough, then n 1 ≥ n and n ≥ n Ch , the simulation is 2-close in diamond norm. This takes a total of n(1 − p1 + δCh ) + n (Q 2 (N ) − δ1 )−1 ( p1 + δCh ) log d uses of N . Now, ∀δ2 > 0, R2 = Q 2 (M) − δ2 , ∃m such that ∀n ≥ m , there is a protocol with n uses of M that simulates the identity map on 2n R2 dimensions -close in diamond norm. Substitute these n uses of M by the 2-close simulation above. We have an 3-close simulation of the 2n R2 -dim identity map with n(1 − p1 + δCh ) + n/[Q 2 (N ) − δ1 ] × ( p1 + δCh ) log d uses of N . Letting , δ1 , δ2 , and δCh → 0, we have log d p1 + (1 − p1 ) Q 2 (N ) ≥ Q 2 (M). (5) Q 2 (N ) Running the same argument with N , M reversed and using Eq. (4) instead, we have log d p2 + (1 − p2 ) Q 2 (M) ≥ Q 2 (N ). (6) Q 2 (M) Together, |Q 2 (N ) − Q 2 (M)| ≤ min[ p1 (log d − Q 2 (N )), p2 (log d − Q 2 (M))].
(7)
N
which is colinear with N and M, and is on the boundary of We now consider B(N , δ). We can run the same argument with N in place of M but with the same M1 , M2 . Here, N = δ M + (1 − δ )N . Eliminating M from Eqs. (3) and (4), one can verify that the parameters change as δ ,
δ p2 → q 2 = p2 ·
p1 → q 1 = p1
1 δ
p2
+ (1 − p2 )
≤ p2
1 δ δ ≤ 2 p2 .
1 − p2
In the last inequality, we use the fact that p2 ≤ 1/2 by construction. Using Eq. (5) for
N and substituting p1 , p2 by q1 , q1 , and for δ ≤ 2 log d, |Q 2 (N ) − Q 2 (N )| ≤ min[q1 (log d − Q 2 (N )), q2 (log d − Q 2 (N ))] ≤ .
Continuity of Quantum Channel Capacities
213
Note that δ depends on N ∈ B(N , δ) via the dependence of M and on N . The continuity bound is not as tight as those derived for the unassisted capacities, but it has the merit of being independent of the metric used for the channels. The same argument holds for continuity of Q B in the interior of Q B (N ) > 0 with the only modification in the definition of an n-use protocol. A less -δ-loaded, more concise, and slightly more heuristic derivation in terms of resource inequalities [27] is also possible.1
6.2. Q B of Erasure Channel. The erasure channel of erasure probability p acts on qubit states as follows: E p (ρ) = (1 − p)ρ + p|22|, where |2 can be viewed as an error symbol. Q 2 (E p ) = 1 − p but an expression for Q B (E p ) is unknown, though it is known to be positive for p < 1. Instead of the continuity of Q 2 or Q B at E p , we can ask if these capacities are continuous as a function of p. In other words, we are considering the restriction of these functions to the 1-parameter family of channels E p . 1 We describe the proof in the language of resource inequalities[27]. Each resource inequality (RI) S + 1 S2 · · · ≥ R1 + R2 · · · represents the fact that n units of LHS resources can be used to simulate n units of the RHS resources for asymptotically large n and with sufficient accuracy (say, in diamond norm of the operations involved in the RHS). If the simulation is sufficiently good, manipulating RIs as though they are usual algebraic inequalities can often be justified. In particular, the following can be rigorously proved in many situations. (1) Multiplication by a positive scalar on both sides is allowed. (2) Inequalities can be summed. (3) The inequalities are transitive. (4) Substitution that preserves the inequalities is allowed. Operationally, this requires that simulations are accurate enough to be composable, so that recursive/concatenated simulation is possible. In a related manner, cancellation (or subtraction) is frequently possible. Now, we give an argument for Eq. (11). By the definition of two-way assisted channel capacity,
N + ∞ CC↔ ≥ Q 2 (N ) I,
(8)
where ∞ CC↔ denotes the assistance. Using log d qubit noiseless channels to simulate M1 (see main text), log d I ≥ M1 .
(9)
Together N + ∞ CC↔ ≥ Q 2 (N ) I ≥
Q 2 (N ) M1 . log d
(10)
Q (N )
2 It means that we can use n copies of N to transmit n log d inputs to the receiver, who then applies copies of M1 locally thereby giving a protocol to simulate M1 using N . Equation (3) means that
p1 M1 + (1 − p1 )N + ∞ CC← ≥ M.
(11)
Using free back communication to generate n biased coins (see main text) p1
log d + (1 − p1 ) N + ∞ CC↔ ≥ M. Q 2 (N )
(12)
log d + (1 − p1 ) Q 2 (N ) ≥ Q 2 (M) Q 2 (N )
(13)
This implies p1 as claimed.
214
D. Leung, G. Smith
In this restricted domain, Q 2 (E p ) = 1 − p is clearly continuous. For Q B (E p ), the previous proof now holds on the restricted domain for p < 1. For the point p = 1, continuity still holds because Q B (E p ) ≤ Q 2 (E p ) = 1 − p which is vanishing (converging towards Q B (E1 )) as p → 1. 7. Discussion We have shown that many of the communication capacities of a quantum channel are continuous. For unassisted capacities, such as private, quantum, and classical capacities we proved continuity using Theorem 6. In these cases, we showed variations in capacity of the form A log d + B H () for channels that are distance apart, where the constants A and B depend on the particular capacity under consideration and are typically of order unity. For the more involved case of two-way capacity, we have shown continuity of Q 2 on the interior of {Q(N ) > 0}, and similarly for Q B by making use of an argument of Vidal[14]. In general, application of Theorem 6 will give continuity any time a regularized capacity formula is available. In particular, it can easily be used to show the continuity of the capacity region of multi-user channels such as the multiple access channel [28] and broadcast channels [29,30]. Acknowledgements. We are grateful to Aram Harrow for discussions about continuity of the two-way and back-assisted capacities, and John Smolin for suggesting the example of discontinuity for the classical capacity of infinite-dimensional channels. We thank Bill Rosgen for a careful reading and many helpful corrections on an earlier version of the manuscript. DL was supported by CRC, CFI, ORF, NSERC, CIFAR, MITACS, ARO, and QuantumWorks. GS was supported by the DARPA QUEST program under contract no. HR0011-09-C-0047.
References 1. Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE. Trans. Inf. Theory 44(1), 269–273 (1998) 2. Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997) 3. Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51, 44–55 (2005) 4. Lloyd, S.: Capacity of the noisy quantum channel. Phys. Rev. A 55, 1613–1622 (1997) 5. Shor, P.W.: The quantum channel capacity and coherent information. In: Lecture notes, MSRI Workshop on Quantum Computation, 2002. Available online at http://www.msri.org/publications/ln/msri/2002/ quantumcrypto/shor/1/ 6. Bennett, C.H., Shor, P.W., Smolin, J.A., Thapliyal, A.V.: Entanglement-assisted capacity of a quantum channel and the reverse shannon theorem. IEEE Trans. Inf. Theory 48, 2637–2655 (2002) 7. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A.: Capacities of quantum erasure channels. Phys. Rev. Lett. 78(16), 3217–3220 (1997) 8. Barnum, H., Smolin, J.A., Terhal, B.M.: Quantum capacity is properly defined without encodings. Phys. Rev. A. 58, 3496–3501 (1998) 9. Keyl, M., Werner, R.F.: How to correct small quantum errors. In: Cohereat Evalution in Noisy Environments, Lecture Notes in Physics, 611 Berlin-Heidelberg-New York: Springer Verlag, 2002, pp. 263–286 10. Shirokov, M.E.: The holevo capacity of infinite dimensional channels and the additivity problem. Comm. Math. Phys. 262, 137–159 (2006) 11. Fannes, M.: A continuity property of the entropy density for spin lattice systems. Comm. Math. Phys. 31, 291–294 (1973) 12. Nielsen, M.A.: Continuity bounds for entanglement. Phys. Rev. A 61, 064301 (2000) 13. Donald, M., Horodecki, M.: Continuity of relative entropy of entanglement. Phys. Lett. A 264, 257–260 (1999)
Continuity of Quantum Channel Capacities
215
14. Vidal, G.: On the continuity of asymptotic measures of entanglement. http://arxiv.org/abs/quant-ph/ 0203107vl, 2002 15. Alicki, R., Fannes, M.: Continuity of quantum conditional information. J. Phys. A:Math. Gen. 37, L55–L57 (2004) 16. Christandl, M., Winter, A.: “squashed entanglement” - an additive entanglement measure. J. Math. Phys. 45, 829–840 (2004) 17. Paulsen, V.I.: Completely Bounded Maps and Dilations. John Wiley & Sons, Inc, New York, 1987 18. Devetak, I., Junge, M., King, C., Ruskai, M.B.: Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266, 37–63 (2006) 19. Holevo, A.S.: Statistical problems in quantum physics. In: G. Maruyama, J.V., Prokhorov, eds, Proceedings of the second Japan-USSR Symposium on Probability Theory, Volume 330 of Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1973, pp. 104–119 20. Shor, P.W., Smolin, J.A.: Quantum error-correcting codes need not completely reveal the error syndrome. http://arxiv.org/abs/quant-ph/9604006v2, 1996 21. Smith, G., Smolin, J.A.: Degenerate quantum codes for Pauli channels. Phys. Rev. Lett. 98, 030501 (2007) 22. Smith, G., Renes, J., Smolin, J.A.: Structured codes improve the bennett-brassard-84 quantum key rate. Phys. Rev. Lett. 100, 170502 (2008) 23. Hastings, M.B.: A counterexample to additivity of minimum output entropy. http://arxiv.org/abs/0809. 3972v3quant-ph, 2008 24. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed state entanglement and quantum error correction. Phys. Rev. A. 54, 3824–3851 (1996) 25. Horodecki, M., Horodecki, P., Horodecki, R.: Unified approach to quantum capacities: towards quantum noisy coding theorem. http://arxiv.org/abs/quant-ph/0003040v1, 2000 26. Shirokov, M.: On channels with finite holevo capacity. Theory of Probability and Its Applications 53(4), 732–750 (2008) 27. Devetak, I., Harrow, A., Winter, A.: A family of quantum protocols. Phys. Rev. Lett. 92, 187901 (2004) 28. Yard, J., Devetak, I., Hayden, P.: Capacity theorems for quantum multiple-access channels: Classicalquantum and quantum-quantum capacity regions. IEEE Trans. Inf. Theory 54, 3091–3113 (2008) 29. Yard, J., Hayden, P., Devetak, I.: Quantum broadcast channels. http://arxiv.org/abs/quant-ph/0603098v1, 2006 30. Dupuis, F., Hayden, P.: A father protocol for quantum broadcast channels. http://arxiv.org./abs/quantph/0612155v2, 2006 Communicated by M. B. Ruskai
Commun. Math. Phys. 292, 217–236 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0856-7
Communications in
Mathematical Physics
Invariance of the White Noise for KdV Tadahiro Oh Department of Mathematics, University of Toronto, 40 St. George St, Rm 6290, Toronto, ON M5S 2E4, Canada. E-mail:
[email protected] Received: 11 December 2008 / Accepted: 9 March 2009 Published online: 24 June 2009 – © Springer-Verlag 2009
Abstract: We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space bsp,∞ , sp < −1, contains the support of the white noise. Then, we prove local well-posedness in bsp,∞ for p = 2+, s = − 21 + such that sp < −1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko [21]. 1. Introduction In this paper, we consider the periodic Korteweg-de Vries (KdV) equation: u t + u x x x + uu x = 0 u t=0 = u 0 ,
(1)
where u is a real-valued function on T × R with T = [0, 2π ) and the mean of u 0 is 0. By the conservation of the mean, it follows that the solution u(t) of (1) has the spatial mean 0 for all t ∈ R as long as it exists. Our main goal is to show that the mean 0 white noise dµ = Z −1 exp − 21 u 2 d x du(x), u mean 0 (2) x∈T
is invariant under the flow and that (1) is globally well-posed almost surely on the statistical ensemble (i.e. on the support of µ) without using the complete integrability of the equation. First, we briefly review recent well-posedness results of the periodic KdV (1). In [2], Bourgain introduced a new weighted space-time Sobolev space X s,b whose norm is given by u X s,b (T×R) = ns τ − n 3 b u (n, τ ) L 2 (Z×R) , (3) n,τ
218
T. Oh
where · = 1+|·|. He proved the local well-posedness of (1) in L 2 (T) via the fixed point argument, immediately yielding the global well-posedness in L 2 (T) thanks to the conservation of the L 2 norm. Kenig-Ponce-Vega [14] improved Bourgain’s result and estab1 lished the local well-posedness in H − 2 (T). Colliander-Keel-StaffilaniTakaoka-Tao [9] proved the corresponding global well-posedness result via the I -method. More recently, Kappeler-Topalov [13] proved the global well-posedness of the KdV in H −1 (T), using the complete integrability of the equation. There are also results on the necessary conditions on the regularity with respect to smoothness or uniform continuity of the solution map : u 0 ∈ H s (T) → u(t) ∈ H s (T). Bourgain [3] showed that if the solution map is C 3 , then s ≥ − 21 . Christ-Colliander-Tao [8] proved that if the solution map is uniformly continuous, then s ≥ − 21 . (Also, see Kenig-Ponce-Vega [15].) In [4], Bourgain proved the invariance of the Gibbs measures for the nonlinear Schrödinger equations (NLS). In dealing with the super-cubic nonlinearity, (where only the local well-posedness result was available), he used a probabilistic argument and the approximating finite dimensional ODEs (with the invariant finite dimensional Gibbs measures) to extend the local solutions to the global ones almost surely on the statistical ensemble and showed the invariance of the Gibbs measures. Note that it was crucial that the local well-posedness was obtained with a “good” estimate on the solutions (e.g. via the fixed point argument) for his argument to obtain the uniform convergence of the solutions of the finite dimensional ODEs to those of the full PDE. Also see Burq-Tzvetkov [6], Oh [19], and Tzvetkov [23,24]. In the present paper, we’d like to follow Bourgain’s argument [4]. Unfortunately, it is known (cf. Zhidkov [25]) that the white noise µ in (2) is supported on ∩s<− 1 H s \ 2
1
H − 2 . In view of the results in [3 and 8] described above, we can not hope to have a local-in-time solution via the fixed point argument in H s , s < − 21 . Instead, we will prove a local well-posedness in an appropriate Banach space containing the support of the white noise µ. Define the Besov-type space via the norm ⎛ s p f bsp,∞ := f b p,∞ = sup n f (n) L s
|n|∼2 j
j
= sup ⎝ j
⎞1
p
n | f (n)| sp
p⎠
. (4)
|n|∼2 j
By Hausdorff-Young’s inequality, we have bsp,∞ ⊃ B sp ,∞ for p > 2, where B sp ,∞ is p the usual Besov space with p = p−1 . In Sect. 3, we use the theory of abstract Wiener spaces to show that bsp,∞ contains the full support of the white noise for sp < −1. Now, we’d like to establish the local well-posedness in bsp,∞ for sp < −1. Note 1
that this space is essentially less regular than H − 2 since it contains the support of the bsp,∞ (T). Let X s,b white noise. First, define a variant of the X s,b space adjusted to p be the completion of the Schwartz class S(T × R) under the norm u X s,b = ns τ − n 3 b u (n, τ )bsp,∞ L τp . p
(5)
Then, one of the crucial bilinear estimates that we need to prove is: ∂x (uv)
s,− 21
Xp
u
s, 21
Xp
v
s, 21
Xp
.
(6)
Invariance of White Noise for the Periodic KdV
219
As in [2 and 14], a key ingredient is the algebraic identity n 3 − n 31 − n 32 = 3nn 1 n 2 for n = n 1 + n 2 . However, this is not enough to prove (6) for sp < −1. In establishing the local well-posedness through the usual integral equation, we view the nonlinear problem (1) as a perturbation to the Airy equation u t + u x x x = 0. Noting the Fourier transform of the solution to the Airy equation is a measure supported on {τ = n 3 }, we modify X s,b p with a carefully chosen weight w(n, τ ) in Sect. 4 to treat the resonant cases in (6). (cf. Bejenaru-Tao [1], Kishimoto [17] in the context of NLS.) Theorem 1. Assume the mean 0 condition on u 0 . Let s = − 21 +, p = 2+ such that sp < −1. Then, KdV (1) is locally well-posed in bsp,∞ . Once we prove Theorem 1, we can use the finite dimensional approximation to (1): N u t + u xNx x + P N (u N u xN ) = 0 (7) u N t=0 = u 0N , where PN is the projection onto thefrequencies |n| ≤ N and u N = P N u. Note that (7) is Hamiltonian, and that it preserves (u N )2 d x. Hence, by Liouville’s theorem, the finite dimensional white noise −1 N 2 1 dµ N = Z N exp − 2 (u ) d x du N (x) (8) x∈T
is invariant under the flow of (7). The remaining argument follows just as in [4,6,19,23, 24], and we obtain the a.s. GWP of (1) and the invariance of the white noise µ. Theorem 2. Let {gn (ω)}∞ n=1 be a sequence of i.i.d. standard complex Gaussian random variables on a probability space (, F, P). Consider (1) with initial data u 0 = n =0 gn (ω)einx , where g−n = gn . Then, (1) is globally well-posed almost surely in ω ∈ . Moreover, the mean 0 white noise µ is invariant under the flow. Remark 1.1. This provides an analytical proof of the invariance of the white noise µ. Recently, Quastel-Valko [21] proved the invariance of the white noise under the flow of KdV. Their argument combines the GWP in H −1 (T) via the complete integrability (Kappeler-Topalov [13]), the correspondence between the white noise for KdV and the Gibbs measure (weighted Wiener measure) of mKdV under the (corrected) Miura transform (Cambronero-McKean [7]), and the invariance of the Gibbs measure of mKdV (Bourgain [4].) Their method is not applicable to the general non-integrable coupled KdV system considered in [19], whereas our argument is applicable in the non-integrable case as well. Remark 1.2. Let F L s, p be the space of functions on T defined via the norm f F L s, p = ns f (n) L np . Then, Theorems 1 and 2 can also be established in F L s, p for some s = − 21 +, p = 2+ with sp < −1. See Remark 4.7. This paper is organized as follows: In Sect. 2, we introduce some standard notations. In Sect. 3, we go over the basic theory of Gaussian Hilbert spaces and abstract Wiener spaces. Then, we give the precise mathematical meaning to the white noise µ and show that it is a (countably additive) probability measure on bsp,∞ for sp < −1. In Sect. 4, we introduce the function spaces and linear estimates. Then, we prove Theorem 1 by establishing the crucial bilinear estimate.
220
T. Oh
2. Notation In the periodic setting on T, the spatial Fourier domain is Z. Let dn be the normalized counting measure on Z, and we say f ∈ L p (Z), 1 ≤ p < ∞, if f L p (Z) =
1 | f (n)| dn p
Z
p
:=
1 | f (n)| p 2π
1
p
< ∞.
n∈Z
If p = ∞, we have the obvious definition involving the essential supremum. We often drop 2π for simplicity. If a function depends on both x and t, we use ∧x (and ∧t ) to denote the spatial (and temporal) Fourier transform, respectively. However, when there is no confusion, we simply use ∧ to denote the spatial Fourier transform, the temporal Fourier transform, and the space-time Fourier transform, depending on the context. Given a space X of functions on T×R, we define the local in time restriction X (T× I ) for any time interval I = [t1 , t2 ] ⊂ R, (or simply X [t1 ,t2 ] ) by u X (T×R) : u|I = u . u X I = u X (T×I ) = inf For a Banach space X ⊂ S (T × R), we use X to denote the space of the Fourier trans−1 forms of the functions in X , which is a Banach space with the norm f X = Fn,τ f X , −1 where F denotes the inverse Fourier transform (in n and τ ). Also, for a space Y of to denote the space of the inverse Fourier transforms of the functions on Z, we use Y functions in Y with the norm f Y = F f Y . Now, define bsp,q (T) by the norm s p f bsp,q (T) = f bsp,q (Z) := n f (n) L
|n|∼2 j
q lj
⎛ ⎛ ⎞ q ⎞ q1 p ∞ ⎜ sp p⎠ ⎟ ⎝ =⎝ n | f (n)| ⎠ j=0
(9)
|n|∼2 j
for q < ∞ and by (4) when q = ∞. Lastly, let η ∈ Cc∞ (R) be a smooth cutoff function supported on [−1, 1] with η ≡ 1 on [− 21 , 21 ] and let ηT (t) = η(T −1 t). We use c, C to denote various constants, usually depending only on s and p. If a constant depends on other quantities, we will make it explicit. We use A B to denote an estimate of the form A ≤ C B. Similarly, we use A ∼ B to denote A B and B A and use A B when there is no general constant C such that B ≤ C A. We also use a+ (and a−) to denote a + ε (and a − ε), respectively, for arbitrarily small ε 1. 3. Gaussian Measures in Hilbert Spaces and Abstract Wiener Spaces In this section, we go over the basic theory of Gaussian measures in Hilbert spaces and abstract Wiener spaces to provide the precise meaning of the white noise “dµ = Z −1 exp(− 21 u 2 d x) x∈T du(x)” appearing in Sect. 1. For details, see Zhidokov [25], Gross [12], and Kuo [18].
Invariance of White Noise for the Periodic KdV
221
First, recall (centered) Gaussian measures in Rn . Let n ∈ N and B be a symmetric positive n × n matrix with real entries. The Borel measure µ in Rn with the density 1 −1 1 n dµ(x) = √ exp − B x, x R 2 (2π )n det(B) is called a (nondegenerate centered) Gaussian measure in Rn . Note that µ(Rn ) = 1. Now, we consider the analogous definition of the infinite dimensional (centered) Gaussian measures. Let H be a real separable Hilbert space and B : H → H be a linear positive self-adjoint operator (generally not bounded) with eigenvalues {λn }n∈N and the corresponding eigenvectors {en }n∈N forming an orthonormal basis of H . We call a set M ⊂ H cylindrical if there exists an integer n ≥ 1 and a Borel set F ⊂ Rn such that M = {x ∈ H : (x, e1 H , · · · , x, en H ) ∈ F}.
(10)
For a fixed operator B as above, we denote by A the set of all cylindrical subsets of H . One can easily verify that A is a field. Then, the centered Gaussian measure in H with the correlation operator B is defined as the additive (but not countably additive in general) measure µ defined on the field A via n −1 2 1 n − 21 − n2 µ(M) = (2π ) λj e− 2 j=1 λ j x j d x1 · · · d xn , for M ∈ A as in (10). (11) j=1
F
The following theorem tells us when this Gaussian measure µ is countably additive. Theorem 3.1. The Gaussian measure µ defined in (11) is countably additive on the field A if and only if B is an operator of trace class, i.e. ∞ n=1 λn < ∞. If the latter holds, then the minimal σ -field M containing the field A of all cylindrical sets is the Borel σ -field on H . Consider a sequence of the finite dimensional Gaussian measures {µn }n∈N as follows. For fixed n ∈ N, let Mn be the set of all cylindrical sets in H of the form (10) with this fixed n and arbitrary Borel sets F ⊂ Rn . Clearly, Mn is a σ -field, and setting n −1 2 1 n − 21 − n2 λj e− 2 j=1 λ j x j d x1 · · · d xn µn (M) = (2π ) j=1
F
for M ∈ Mn , we obtain a countably additive measure µn defined on Mn . Then, one can show that each measure µn can be naturally extended onto the whole Borel σ -field M of H by µn (A) := µn (A ∩ span{e1 , · · · , en }) for A ∈ M. Then, we have Proposition 3.2. Let µ in (11) be countably additive. Then, {µn }n∈N constructed above converges weakly to µ as n → ∞. Now, we construct the mean 0 white noise. Let φ = n an einx be a real-valued function on T with mean 0, i.e. we have a0 = 0 and a−n = an . First, define µ N on CN ∼ = R2N with the density N − 12 n=1 |an |2 N (12) dµ N = Z −1 n=1 dan , N e 1 N 2 N where Z N = C N e− 2 n=1 |an | n=1 dan . Note that this measure is the induced probN , where g (ω), n = 1, . . . , N , ability measure on C N under the map ω → {gn (ω)}n=1 n
222
T. Oh
are i.i.d. standard complex Gaussian random variables. Next, define the white noise µ by 1 2 (13) dµ = Z −1 e− 2 n≥1 |an | n≥1 dan , 1 2 where Z = e− 2 n≥1 |an | n≥1 dan . Then, in the above correspondence, we have inx , where {g (ω)} g e φ = n n≥1 are i.i.d. standard complex Gaussian random n =0 n variables and g−n = gn . Let H˙ 0s be the homogeneous Sobolev space restricted to the real-valued mean 0 s inx inx ˙ elements. Let ·, · H˙ s denote the inner product in H0 , i.e. cn e , dn e = H˙ 0s 0 √ 2s 2s −s inx − . Then, the weighted exponentials {|n| e }n =0 n≥1 |n| cn dn . Let Bs = are the eigenvectors of Bs with the eigenvalue |n|2s , forming an orthonormal basis of H˙ 0s . Note that |n|−2s an einx , an einx s = − 21 |an |2 . − 21 B −1 φ, φ H˙ s = − 21 0
n =0
n =0
H˙ 0
n≥1
The right-hand side is exactly the expression appearing in the exponent in (13). By Theorem 3.1, µ is countably additive if and only if B is of trace class, i.e. n =0 |n|2s < ∞. Hence, s<− 1 H s is a natural space to work on. Unfortunately, the results in [3 and 8] 2 state that one can not have a local-in-time solution of (1) via the fixed point argument s in H , s < − 21 . Instead, we propose to work on bsp,∞ (T) defined in (4) for sp < −1 in view of Theorem 1. Since bsp,∞ is not a Hilbert space, we need to go over the basic theory of abstract Wiener spaces. Recall the following definitions [18]: Given a real separable Hilbert space H with norm · , let F denote the set of finite dimensional orthogonal projections P of H . Then, define a cylinder set E by E = {x ∈ H : Px ∈ F}, where P ∈ F and F is a Borel subset of PH , and let R denote the collection of such cylinder sets. Note that R is a field but not a σ -field. Then, the Gauss measure µ on H is defined by x2 − n2 µ(E) = (2π ) e− 2 d x F
for E ∈ R, where n = dimPH and d x is the Lebesgue measure on PH . It is known that µ is finitely additive but not countably additive in R. A seminorm ||| · ||| in H is called measurable if for every ε > 0, there exists P0 ∈ F such that µ(|||Px||| > ε) < ε for P ∈ F orthogonal to P0 . Any measurable seminorm is weaker than the norm of H , and H is not complete with respect to ||| · ||| unless H is finite dimensional. Let B be the completion of H with respect to ||| · ||| and denote by i the inclusion map of H into B. The triple (i, H, B) is called an abstract Wiener space. Now, regarding y ∈ B ∗ as an element of H ∗ ≡ H by restriction, we embed B ∗ in H . Define the extension of µ onto B (which we still denote by µ) as follows. For a Borel set F ⊂ Rn , set µ({x ∈ B : ((x, y1 ), · · · , (x, yn )) ∈ F}) := µ({x ∈ H : (x, y1 H , · · · , x, yn H ) ∈ F}),
Invariance of White Noise for the Periodic KdV
223
where y j ’s are in B ∗ and (·, ·) denote the natural pairing between B and B ∗ . Let R B denote the collection of cylinder sets {x ∈ B : ((x, y1 ), . . . , (x, yn )) ∈ F} in B. Theorem 3.3 (Gross [12]). µ is countably additive in the σ -field generated by R B . In the present context, let H = L 2 (T) and B = bsp,∞ (T) for sp < −1. Then, we have Proposition 3.4. The seminorms · bsp,∞ is measurable for sp < −1. Hence, (i, H, B) = (i, L 2 , bsp,∞ ) is an abstract Wiener space, and µ defined in (13) is countably additive in bsp,∞ . We present the proof of Proposition 3.4 at the end of this section. It seems that the statement in Proposition 3.4 holds true for sp = −1 (cf. Roynette [22] for p = 2.) However, we can choose s and p such that sp < −1 for our application, and thus we will not discuss the endpoint case. It follows from the proof that (i, L 2 , F L s, p ), where F L s, p = bsp, p , is also an abstract Wiener space for sp < −1 (we need a strict inequality in this case.) Given an abstract Wiener space (i, H, B), we have the following integrability result due to Fernique [10]. Theorem 3.5 (Theorem 3.1 in [18]). Let (i, H, B) be an abstract Wiener space. Then, 2 there exists c > 0 such that B ecx B µ(d x) < ∞. Hence, there exists c > 0 such that 2 µ(x B > K ) ≤ e−c K . 2 In our context, if sp < −1, we have µ φ ≥ K , φ mean 0 ≤ e−cK for some s b p,∞ (T) c > 0. With this estimate and Theorem 1, we can follow the argument in [4] to prove Theorem 2. We omit the details. Also, see [6,19,23,24] for the details. Proof of Proposition 3.4 We present the proof only for 2 < p < ∞, which is the relevant case for our application. We just point out that the proof for p ≤ 2 is similar but simpler (where one can use Hölder inequality in place of Lemma 3.6 below.) For p = ∞, see [4,5,19]. It suffices to show that for given ε > 0, there exists large M0 such that µ P>M0 φ bsp,∞ > ε < ε, whereP>M0 is the projection onto the frequencies |n| > M0 . In the following, write φ = n =0 gn einx , where {gn (ω)}∞ n=1 is a sequence of i.i.d. standard complex-valued Gaussian random variables and g−n = gn . First, recall the following lemma. Lemma 3.6. (Lemma 4.7 in [20]) Let {gn } be a sequence of i.i.d standard complexvalued Gaussian random variables. Then, for M dyadic and δ > 0, we have max|n|∼M |gn |2 lim M 1−δ = 0, a.s. 2 M→∞ |n|∼M |gn | Fix K > 1 and δ ∈ (0, 21 ) (to be chosen later.) Then, by Lemma 3.6 and Egoroff’s theorem, there exists a set E such that µ(E c ) < 21 ε and the convergence in Lemma 3.6 is uniform on E, i.e. we can choose dyadic M0 large enough such that {gn (ω)}|n|∼M L ∞ n ≤ M −δ , {gn (ω)}|n|∼M L 2n
(14)
224
T. Oh
for all ω ∈ E and dyadic M > M0 . In the following, we will work only on E and drop ‘∩E’ for notational simplicity. However, it should be understood that all the events are under the intersection with E so that (14) holds. Let {σ j } j≥1 be a sequence of positive numbers such that σ j = 1, and let M j = M0 2 j dyadic. Note that σ j = C2−λj = C M0λ M −λ for some small λ > 0 (to be deterj mined later). Then, we have s µ P>M0 φ > ε ≤ µ {g } > ε s n |n|>M b 0 b p,∞ p,1 ≤
∞
µ {ns gn }|n|∼M j L np > σ j ε ,
(15)
j=0
where bsp,1 is defined in (9). By interpolation and (14), we have p−2
2
p {ns gn }|n|∼M j L np ∼ M sj {gn }|n|∼M j L np ≤ M sj {gn }|n|∼M j Lp 2 {gn }|n|∼M j L ∞
≤ M sj {gn }|n|∼M L 2n
{gn }|n|∼M j L ∞ n
p−2 p
s−δ p−2 p
≤ Mj
{gn }|n|∼M j L 2n
n
n
{gn }|n|∼M j L 2n
a. s. Thus, if we have {ns gn }|n|∼M j L np > σ j ε, then we have {gn }|n|∼M j L 2n R j , −s+δ p−2 p
where R j := σ j εM j
. With p = 2 + 2θ , we have −s + δ p−2 p =
taking δ sufficiently close to −s+δ p−2 p σ j εM j
1 2
−sp+2δθ 2+2θ
>
1 2
by
since −sp > 1. Then, by taking λ > 0 sufficiently small, −s+δ p−2 p −λ
= CεM0λ M j Rj = the polar coordinates, we have µ {gn }|n|∼M j L 2n R j ∼
B c (0,R
j)
e
− 21 |g|2
1
+
CεM0λ M j2 . By a direct computation in |n|∼M j
dgn
∞
1 2
e− 2 r r 2·#{|n|∼M j }−1 dr.
Rj
Note that the implicit constant in the inequality is σ (S 2·#{|n|∼M j }−1 ), a surface measure of n the 2·#{|n| ∼ M j }−1 dimensional unit sphere. We drop it since σ (S n ) = 2π 2 / ( n2 ) −1
2M j 4M j t .
1. By change of variable t = M j 2 r , we have r 2·#{|n|∼M j }−2 r 4M j ∼ M j − 21
Since t > M j
R j = CεM0λ M 0+ j , we have 2M j
Mj
1
1
2
= e2M j ln M j < e 8 M j t and t 4M j < e 8 M j t 1
2
1 2
for M0 sufficiently large. Thus, we have r 2·#{|n|∼M j }−2 < e 4 M j t = e 4 r for r > R. Hence, we have ∞ 2 2 2λ 1+ 2 1 2 µ {gn }|n|∼M j L 2n R j ≤ C e− 4 r r dr ≤ e−c R j = e−cC M0 M j ε . (16) 2
Rj
From (15) and (16), we have ∞ 2 1+2λ+ j+ 2 µ P>M0 φ e−cC M0 2 ε ≤ 21 ε bsp,∞ > ε ≤ j=1
by choosing M0 sufficiently large.
Invariance of White Noise for the Periodic KdV
225
4. Local Well-Posedness in bsp,∞ In this section, we prove Theorem 1 via the fixed point argument. In Subsect. 4.1, we go over the previous local well-posedness theory of KdV to motivate the definition bsp,∞ . Then, we establish the of the Bourgain space W ps,b with the weight, adjusted to basic linear estimates in Subsect. 4.2. Finally, we prove the crucial bilinear estimate in Subsect. 4.3.
4.1. Bourgain Space with a weight. In [14], Kenig-Ponce-Vega proved ∂x (uv)
1
X s,− 2
u
1
X s, 2
v
1
X s, 2
,
(17)
for s ≥ − 21 under the mean 0 assumption on u and v, where X s,b is defined in (3). Their proof is based on proving the equivalent statement: Bs ( f, g) L 2n,τ f L 2n,τ g L 2n,τ ,
(18)
where Bs (·, ·) is defined by Bs ( f, g)(n, τ ) =
1 2π τ
1
− n3 2
n 1 +n 2 =n n 1 =0,n
|n|ns n 1 s n 2 s
f (n 1 , τ1 )g(n 2 , τ2 )dτ1 1
τ1 +τ2 =τ
1
τ1 − n 31 2 τ2 − n 32 2
.
(19) One of the main ingredients is the observation due to Bourgain [2]: n 3 − n 31 − n 32 = 3nn 1 n 2 , for n = n 1 + n 2 ,
(20)
which in turn implies that MAX := max(τ − n 3 , τ1 − n 31 , τ2 − n 32 ) nn 1 n 2
(21)
for n = n 1 + n 2 and τ = τ1 + τ2 with n, n 1 , n 2 = 0. Recall that (21) implies that |n|ns 1 1 |n|ns 1 1 1 1 s s s s 3 3 n 1 n 2 τ − n 3 2 τ1 − n 2 τ2 − n 2 n 1 n 2 MAX 21 1 2
(22)
for s ≥ − 21 . Note that (22) is optimal, for example, when τ − n 3 ∼ 3nn 1 n 2 and τ j −n 3j 3nn 1 n 2 0+ . To exploit this along with the fact the free solution concentrates on the curve {τ = n 3 }, we define the weight w(n, τ ) in the following. For k ∈ Z \ {0}, let 1
Ak = {(n, τ ) : |n| ≥ C, τ − n 3 + 3n(n − k)k n 100 }, for some C > 0. With δ = 0+ (to be determined later), let w(n, τ ) = 1 + min(k, n − k)δ χ Ak . k =0
(23)
226
T. Oh
Note that, for fixed n and τ , there are at most two values of k such that |(n−k)k + τ −n | 0+ 3n 3 1 −n n−1+ 100 . It follows from the definition that w(n, τ ) max(1, τ n ) ≤ τ − 3
n 3 0+ . Now, define the Bourgain space W ps,b with the weight w via the norm uW s,b = u W u + u s,b := w X s,b p
p
p
where ⎧ ⎨ f := ns τ − n 3 b f (n, τ )b0 X s,b
p p,∞ L τ
p
⎩ f Ys,b := p
ns τ
− n 3 b
s,b− 21
p Y
,
(24)
= sup j ns τ − n 3 b f (n, τ ) L p
f (n, τ )b0p,∞ L 1τ = sup j
ns τ
− n 3 b
f (n, τ )
p Lτ |n|∼2 j p L L1 |n|∼2 j τ
,
.
s, 1
For our application, we set b = 21 . Note that Y ps,0 is introduced so that we have W p 2 (T× [−T, T ]) ⊂ C([−T, T ]; bsp,∞ (T)). In the following, we take p > 2. 4.2. Linear Estimates. Let S(t) = e−t∂x and η(t) be a smooth cutoff such that η(t) = 1 on [− 21 , 21 ] and = 0 for |t| ≥ 1. 3
Lemma 4.1. For any s ∈ R, we have η(t)S(t)u 0
s, 21
Wp
u 0 bsp,∞ .
η(τ − Proof. Recall that w(n, τ ) τ − n 3 0+ . Noting that (η(t)S(t)u 0 )∧ (n, τ ) = n 3 )u0 (n), we have 1 η(τ − n 3 ) L τp |u0 (n)| L p η(t)S(t)u 0 s, 1 ≤ sup ns τ − n 3 2 + Wp
2
|n|∼2 j
j
+ sup ns η(τ − n 3 ) L 1τ |u0 (n)| L p
|n|∼2 j
j
1
η(τ ) L τp + η L 1 < ∞. where Cη = τ 2 +
≤ Cη u 0 bsp,∞ ,
Now, we estimate the Duhamel term. By the standard computation [2], we have t ∞ k k i t 3 λ)dλ S(t − t )F(x, t )dt = −i ei(nx+n t) η(λ − n 3 ) F(n, k! 0 n =0 k=1 (1 − η) (λ − n 3 ) iλt einx e F(n, λ)dλ +i λ − n3 n =0 (1 − η) (λ − n 3 ) i(nx+n 3 t) e F(n, λ)dλ +i λ − n3 n =0
=: N1 (F)(x, t) + N2 (F)(x, t) + N3 (F)(x, t).
(25)
Lemma 4.2. For any s ∈ R, we have η(t)N1 (F)
s, 21
Wp
, N2 (F)
s, 21
Wp
, η(t)N3 (F)
s, 21
Wp
F
s,− 21
Wp
.
Invariance of White Noise for the Periodic KdV
227
Proof. Recall that w(n, τ ) τ − n 3 0+ . Let ηk (t) = t k η(t). First, note that |ηk (t)| ≤ |η(t)| since η(t) = 0 for |t| ≥ 1. Moreover, by Hausdorff-Young and Hölder inequal1 ities, we have τ 2 + ηk (τ ) L τp ≤ ηk 1 + ≤ ηk H 1 1 + k. Then, by Minkowski
η(t)N1 (F)
s, 1 Xp 2
t
Ht 2
integral inequality, we have
λ)|dλ ≤ Cη sup ns η(λ − n 3 )| F(n, j
L
p |n|∼2 j
Cη FY s,−1 , p
∞ 1 3 21 + (τ − n 3 ) p ≤ where Cη = supn ∞ k k=1 k! τ − n η k=1 Lτ ∞ 1+k < ∞. Similarly, we have k=1 k! η(t)N1 (F)Y s,0 ≤ p
where Cη = supn
Cη
s λ)|dλ η(λ − n 3 )| F(n, sup n j
∞
1 k (τ k=1 k! η
− p1 −
n
since
1 p +
τ
k!
Cη FY s,−1 ,
p |n|∼2 j
p
− n 3 ) L 1τ . Now, note that
sup ηk (τ − n 3 ) L 1τ ≤ sup τ − n 3 n
L
1
τ 2 + ηk (τ ) L p
1
p Lτ
+
τ − n 3 p ηk (τ − n 3 ) L τp 1 + k,
= 2+. Hence, we have Cη < ∞ as before.
For |τ − n 3 | 1, we have |τ − n 3 | ∼ τ − n 3 . Thus, we have N 2 (F)(n, τ ) 3 −1 τ − n F(n, τ ). Then, by monotonicity (i.e. f s, 1 ≤ g s, 1 for | f | ≤ |g|), we have N2 (F)
s, 1 Wp 2
F
s,− 1 Wp 2
p W
.
2
p W
2
Lastly, by Minkowski integral inequality with w(n, τ ) τ − n 3 0+ , we have η(t)N3 (F)
s, 1 Xp 2
1 = sup ns τ − n 3 2 + η(τ − n 3 ) j
(1 − η)(λ − n 3 ) | F(n, λ)|dλ L p LP 3 λ−n |n|∼2 j τ ≤ Cη FY s,−1 , ×
p
1
1
where Cη = supn τ − n 3 2 + η(τ − n 3 ) L τp = τ 2 + η(τ ) L τp < ∞. Similarly, we have η(t)N3 (F)Y s,0 Cη FY s,−1 , p
η(τ − n 3 ) L 1τ = η L 1τ < ∞. where Cη = supn
p
228
T. Oh
4.3. Bilinear estimate. By expressing (1) in the integral formulation, we see that u is a solution to (1) for |t| ≤ T 1 if and only if u satisfies u(t) : = tu 0 (u) = η(t)S(t)u 0 + η(t)N1 (η2T F(u))(t) + N2 (η2T F(u))(t) + η(t)N3 (η2T F(u))(t), where F(u) = −uu x and η2T (t) = η(t/2T ), i.e. η2T (t) ≡ 1 for |t| ≤ T . In this subsection, we prove the crucial bilinear estimate so that tu 0 (·) defined above is a contraction bsp,∞ (T)) for T sufficiently small. on a ball in W ps (T × [−T, T ]) ⊂ C([−T, T ], Proposition 4.3. Assume that u and v have the spatial means 0 for all t ∈ R. Then, there exist s = − 21 +, p = 2+ with sp < −1, and θ > 0 such that η2T ∂x (uv)
s,− 21
Wp
T θ u
s, 21
Wp
v
s, 21
Wp
.
(26)
Before proving Proposition 4.3, we present some lemmata. Lemma 4.4. (Ginibre-Tsutsumi-Velo [11], Lemma 4.2) Let 0 ≤ α ≤ β and α + β > 21 . Then, we have τ −2α τ − a−2β dτ a−γ , where γ = 2α − [1 − 2β]+ with [x]+ = x if x > 0, = ε > 0 if x = 0, and = 0 if x < 0. Lemma 4.5. For l1 + 2l2 > 1 with l1 , l2 ≥ 0, there exists c > 0 such that for all n = 0 and λ ∈ R, we have 1 1 < c. (27) n 1 l1 λ + n 1 (n − n 1 )l2 n 1 =0,n
Proof. When l1 > 1, (27) is clear. When l2 > 21 , (27) follows from Lemma 5.3 in [16]. Thus, we assume l1 ∈ (0, 1] and l2 ∈ (0, 21 ] in the following. Since l1 + 2l2 > 1, there exists ε > 0 such that l1 + 2l2 − 3ε ≥ 1. If Pn,λ (n 1 ) := λ + n 1 (n − n 1 ) has two real roots, i.e. Pn,λ (n 1 ) = −(n 1 −r1 )(n 1 −r2 ), then there are at most 6 values of n 1 such that |n 1 − r j | ≤ 1. For the remaining values of n 1 , we have Pn,λ (n 1 ) > 14 2j=1 n 1 − r j . Then, (27) follows from Hölder inequality with p = (l1 − ε)−1 and q = (l2 − ε)−1 , we have 1 2 1 p q − pl1 −ql2 n 1 n 1 − r j < c < ∞, LHS of (27) n1
j=1
n1
since pl1 > 1 and ql2 > 1. If Pn,λ (n 1 ) has only one or no real root, then we have |Pn,λ (n 1 )| ≥ (n 1 − 21 n)2 for all n 1 ∈ Z. Then, by Hölder inequality with p = (l1 − ε)−1 and q = (2l2 − 2ε)−1 , we have 1 1 p q − pl1 1 2 −ql2 LHS of (27) ≤ n 1 (n 1 − 2 n) < c < ∞, n1
since pl1 > 1 and 2ql2 =
l2 l2 −ε
n1
> 1.
Lastly, recall the following lemma from [9, (7.50) and Lemma 7.4].
Invariance of White Noise for the Periodic KdV
229
Lemma 4.6. Let 1
(n) = {η ∈ R : η = −3nn 1 n 2 + o(nn 1 n 2 100 ) for some n 1 ∈ Z with n = n 1 + n 2 }. Then, we have
3
τ − n 3 − 4 χ(n) (τ − n 3 )dτ 1.
(28)
Note that (28) is stated with τ − n 3 −1 in [9]. However, by examining the proof of Lemma 7.4 in [9], one immediately sees that (28) is valid with τ − n 3 −α for any 1 α > 23 + 100 . Proof of Proposition 4.3. In the proof, we use (n, τ ), (n 1 , τ1 ), and (n 2 , τ2 ) to denote the Fourier variables for uv, u, and v, respectively, i.e. we have n = n 1 + n 2 and τ = τ1 + τ2 Moreover, by the mean 0 assumption on u and v and by the fact that we have ∂x (uv) on the left-hand side of (26), we assume n, n 1 , n 2 = 0 in the following. First, we prove ∂x (uv)
s,− 21
Wp
u
s, 21
Wp
v
s, 21
Wp
,
(29)
i.e. we first prove (26) with no gain of T θ . Then, it suffices to show B( f, g)(n, τ )
0,− 21
p W
f b0
where B(·, ·) is defined by 1 B( f, g)(n, τ ) = 2π
n 1 +n 2 =n
|n|ns n 1 s n 2 s
p p,∞ L τ
τ1 +τ2 =τ
gb0
p p,∞ L τ
,
(30)
f (n 1 , τ1 )g(n 2 , τ2 )dτ1 . 2 3 21 j=1 w(n j , τ j )τ j − n j
Let MAX := max(τ − n 3 , τ1 − n 31 , τ2 − n 32 ). Then, by (20), we have MAX nn 1 n 2 . 1 0,− 1 p0,− 2 norm on the left-hand side of • Part 1. First, we consider the X p 2 part of the W (30). • Case (1): MAX = τ − n 3 . Without loss of generality, assume |n 1 | ≥ |n 2 |. For fixed 3 n = 0 and τ , let λ = τ −n 3n and define Bn,τ = {n 1 ∈ Z : |n 1 − r j | ≥ 1, j = 1, 2 r j is a real root of Pn,λ (n 1 ) := λ + n 1 (n − n 1 ) 1 or r j = n if no real root}. 2 On Bn,τ , we have τ − n 3 + 3nn 1 n 2 nλ + n 1 (n − n 1 ).
(31)
c . For s > − 1 , we have ◦ Subcase (1.a). On Bn,τ 2
|n|ns 1 1 . 1 n 1 s n 2 s MAX 21 n 2 2 +s
(32)
230
T. Oh
By Lemma 4.4, we have 1
1
τ1 − n 31 − 2 τ2 − n 32 − 2
p L τ1
−1+ p1
τ − n 3 + 3nn 1 n 2
.
c , i.e. the summation Note that for fixed n and τ there are at most four values of n 1 ∈ Bn,τ p over n 1 can be replaced by the L n 1 norm. Then, by Hölder inequality, we have w(n, τ ) f (n 1 , τ1 )g(n 2 , τ2 )dτ1 LHS of (30) sup 1 p +s 3 21 τ − n 3 21 L p 2 j Lτ n τ − n n=n 1 +n 2 2 1 2 1 2 |n|∼2 j τ =τ1 +τ2 1 sup n 2 − 2 −s+δ f (n 1 , ·) L τp g(n 2 , ·) L τp p p . L
j
|n|∼2 j
L n1
Note that w(n, τ ) n 2 δ since |n 1 | ≥ |n 2 |. If |n 1 | |n 2 | and |n| ∼ 2 j , then we have |n 1 | ∼ 2k , where |k − j| ≤ 5: ⎛ ∞ 1 LHS of (30) sup ⎝ n 2 (− 2 −s+δ) p j
|k− j|≤5 |n 1 |∼2k l=0 |n 2 |∼2l p
p
× f (n 1 , ·) L p g(n 2 , ·) L p
∞
τ
1
p
τ
1
2(− 2 −s+δ) p l sup f L p
|n|∼2k
k
l=0
f b0
p p,∞ L τ
gb0
p p,∞ L τ
p
Lτ
sup g L p l
|n|∼2l
p
Lτ
,
by taking δ > 0 sufficiently small such that − 21 − s + δ < 0. Similarly, if |n 1 | ∼ |n 2 | and |n 2 | ∼ 2l , then we have |n 1 | ∼ 2k where |k − l| ≤ 5. ⎛ ∞ 1 n 2 (− 2 −s+δ) p LHS of (30) ⎝ l=0 |k−l|≤5 |n 1 |∼2k |n 2 |∼2l
1
p
×
∞
p p f (n 1 , ·) L p g(n 2 , ·) L p τ τ 1
2(− 2 −s+δ) p l sup f L p
|n|∼2k
k
l=0
f b0
p p,∞ L τ
gb0
p p,∞ L τ
p
Lτ
sup g L p l
|n|∼2l
p
Lτ
.
◦ Subcase (1.b). On Bn,τ . In this case, we have (31). Also, recall that w(n, τ ) τ − n 3 0+ . Moreover, τ − n 3 0+ max(n, n 2 , τ − n 3 + 3nn 1 n 2 )0+ since either τ − n 3 |nn 1 n 2 | or τ − n 3 |nn 1 n 2 | max(n3 , n 2 3 ). In particular, by (31), we have w(n, τ ) (n 2 τ − n 3 + 3nn 1 n 2 )0+ .
(33)
Invariance of White Noise for the Periodic KdV
231
By applying Hölder inequality and proceeding as before, we have LHS of (30) M sup n 2 0− f (n 1 , ·) L τp g(n 2 , ·) L τp L p M f b0
p p,∞ L τ
where
M = sup n,τ
p
|n|∼2 j
j
gb0
p p,∞ L τ
L n1
,
w(n, τ ) 1− p1
1
n 2 2 +s− τ − n 3 + 3nn 1 n 2
p
.
L n1
Thus, it remains to show that M < ∞. By (33), (31), and Lemma 4.5, we have n,τ
1
M p sup
n
p −1−
n2
1 ( 21 +s−) p
n 2
λ + n 1 (n − n 1 ) p −1−
< ∞,
since ( 21 + s−) p + 2( p − 1)− > 1 for p = 2+ < 4 and sp = −1−. Now, assume MAX = τ2 − n 32 . By symmetry, this takes care of the case when MAX = τ1 − n 31 . Note that we have w(n, τ ) τ − n 3 0+ by a crude estimate. Thus, by duality, it suffices to show ∞ |n|ns 1 f (n 1 , τ1 )h(n, τ )dτ 1 1 1 p s n s p − L 3 3 n 3 1 2 2 2 2 L τ2 w(n 2 , τ2 )τ2 − n 2 τ1 − n 1 τ − n n l=0 |n |∼2l 2
sup f L p
∞ p
|n 1 |∼2k
k
L τ1
h
L
j=0
For fixed n 2 = 0 and τ2 , let λ =
p |n|∼2 j
p
Lτ
τ2 −n 32 3n 2
.
(34)
and define
Bn 2 ,τ2 = {n ∈ Z : |n − r j | ≥ 1, j = 1, 2 r j is a real root of Pn 2 ,λ (n) := λ + n(n 2 − n) 1 or r j = n 2 if no real root}. 2 On Bn 2 ,τ2 , we have τ2 − n 32 − 3nn 1 n 2 n 2 λ + n(n 2 − n).
(35)
Also, note that w(n 2 , τ2 ) min(nδ , n 1 δ ) on Bnc2 ,τ2 . 3 • Case (2): MAX = τ2 − n 2 and |n 1 | |n 2 |. In this case, we have |n|ns 1 1 . 1 1 s s n 1 n 2 MAX 2 n 2 2 +s
(36)
1 ◦ Subcase (2.a). On Bnc2 ,τ2 . First, suppose τ2 − n 32 − 3nn 1 n 2 n 2 100 . Thus, by Lemma 4.4, we have 1
1
1
−1 1
τ1 − n 31 − 2 +α τ − n 3 − 2 + L τp τ2 − n 32 − 3nn 1 n 2 − 2 +α+ n 2 100 ( 2 −α)+ (37)
232
T. Oh
for α > 0. Then, by Hölder inequality in τ followed by Young and Hölder inequalities, we have f (n , τ ) −1 1 f (n 1 , τ1 )h(n, τ )dτ 1 1 100 ( 2 −α)+ n h(n, τ ) p 2 1 1 p 3 α − 3 L τ2 ,τ τ1 − n 1 τ1 − n 1 2 τ − n 3 2 L τ2 −1 1
≤ n 2 100 ( 2 −α)+ τ1 − n 31 −α
p−2 p
for fixed n and n 1 . By choosing α >
p p−2
L τ1
f (n 1 , ·) L τp h(n, ·) 1
p
Lτ
= 0+, we have τ1 −n 31 −α
< C < ∞,
p p−2
L τ1
independently of n 1 . Note that if |n| ∼ 2 j and |n 2 | ∼ 2l , then we have |n 1 | ∼ 2k , where |k − j| ≤ 5 or |k − l| ≤ 5, since n = n 1 + n 2 and |n 1 | ≥ |n 2 |. As in Subcase (1.a), for fixed n 2 and τ2 there are at most four values of n ∈ Bnc2 ,τ2 , i.e. the summation over n can be replaced p
by the L n norm. By Hölder inequality in n 2 after switching the order of summations, LHS of (34)
∞ 1 1 ( 2 −α)+ n 2 − 12 −s− 100 f (n 1 , ·)
L τ1 h(n, ·) L τp L p p
l=0
⎛ ∞ ∞ 1 1 1 p (2l )0− sup ⎝ n 2 − 2 −s− 100 ( 2 −α)+ l
l=0
× f (n
p L τ1 |n 2 |∼2l
p p,∞ L τ
h
1 1 ( 2 −α)+ = n 2 − 12 −s− 100 where M
2− 1 1− 100 +
∼
200− 99
j=0 |n|∼2 j
p − n 2 , ·) L p
f 0 M b
for p <
p
|n 2 |∼2l
p
b0p ,1 L τ
p p−2 L n2
h(n, ·)
p
Ln
(38)
p p−2
L n2
1 p
p Lτ
,
< ∞, since
"1 2
+s+
1 1 100 ( 2
# − α)−
p p−2
>1
with sp < −1. Note that we did not make use of w(n 2 , τ2 ) in
this case. 1 Now, suppose τ2 − n 32 − 3nn 1 n 2 n 2 100 . In this case, we can not expect any contribution from τ2 − n 32 − 3nn 1 n 2 in (37). However, as long as we gain a small power of n 2 in the denominator of LHS of (34), we can proceed as before. Note that w(n 2 , τ2 ) ∼ nδ , since |n 1 | |n 2 | implies |n| |n 1 |. If |n 2 | |n|100 , then we have δ w(n 2 , τ2 ) n 2 100 . Otherwise, we have |n 1 | |n 2 | |n|100 . Then, instead of (36), we have |n|ns 1 1 1 1 1 99 1 1 s s n 1 n 2 MAX 2 n 1 ( 2 +s) 100 n 2 2 +s n 2 2 +s+ε for some ε = 0+. Hence, we obtain a small power of n 2 in either case. ◦ Subcase (2.b). On Bn 2 ,τ2 . In this case, we have (35). As in Subcase (2.a), choose small α > p−2 = 0+. By Hölder inequality with (37) and (35), we have p f (n 1 , τ1 )h(n, τ )dτ − 21 +α+ − 21 +α+ f (n 1 , τ1 ) n λ + n(n − n) h(n, τ ) p 2 2 1 1 3 α − 3 Lτ 3 τ − n 2 2 1 τ1 − n 1 τ − n 1
Invariance of White Noise for the Periodic KdV
for fixed n, n 2 , and τ2 with λ = in τ1 , we have
233
τ2 −n 32 3n 2 . Now by (36) and Hölder inequality in n
and then
∞ l 0− −1+α−s+ f (n 1 , τ1 ) LHS of (34) (2 ) M1 n 2 h(n, τ ) p 3 α p L τ2 ,τ L p τ − n 1 Ln 1 l=0 |n 2 |∼2l 1 n 2 −1+α−s+ τ1 − n 31 −α p sup M p−2
l
× f (n 1 , ·) L τp
1
h(n, ·) p Lτ
L τ1
L
p |n 2 |∼2l
p
Ln
,
1 = supn ,τ λ + n(n 2 − n)− 21 +α+ p < ∞ in view of Lemma 4.5 since where M Ln 2 2 2 · ( 21 − α−) p > 1. We also have τ1 − n 31 −α p < C < ∞, independently of n 1 p−2
L τ1
as before. Note that if |n| ∼ 2 j and |n 2 | ∼ 2l , then we have |n 1 | ∼ 2k , where |k − j| ≤ 5 or |k − l| ≤ 5, since n = n 1 + n 2 and |n 1 | |n 2 |. Then, by Hölder inequality in n 2 , we have ⎛ ⎞ 1 p ∞ p p ⎠ ⎝ f (n 1 , ·) L p LHS of (34) M2 sup p h(n, ·) p L l
2 f 0 M b
p p,∞ L τ
2 = n 2 −1+α−s+ where M • Case (3): MAX = have
p p−2
h
τ1
|n 1 |∼2k
j=0 |n|∼2 j
p
b0p ,1 L τ
Lτ
,
p < ∞ since (1 − α + s−) p−2 > 1.
L n2 τ2 − n 32 and
|n 1 | |n 2 |. ⇒ |n 1 | |n 2 | ∼ |n|. In this case, we
|n|ns 1 1 . 1 n 1 s n 2 s MAX 21 n 1 2 +s
(39)
1 ◦ Subcase (3.a). On Bnc2 ,τ2 . If τ2 − n 32 − 3nn 1 n 2 n 2 100 , then we have τ2 − n 32 − 1
3nn 1 n 2 n 1 100 . By repeating the computation in Subcase (2.a), we now have a small negative power of n 1 = n − n 2 in (38), instead of n 2 , which is still summable in p
L np−2 for each fixed n. Note that if |n 2 | ∼ 2l , then we have |n 1 | ∼ 2k and |n| ∼ 2 j , 2 where k = 0, . . . , l and | j − l| ≤ 5. Then, it suffices to see ∞ l=0
n 1
0−
F(n, n 2 )
p
p L Ln |n 2 |∼2l
l ∞ (2k )0− F(n, n − n 1 ) j=0 | j−l|≤5 k=0
∞
sup F(n, n − n 1 )
j=0 k 1
L
p |n|∼2 j
L
p |n 1 |∼2k
L
.
p |n|∼2 j
L
p |n 1 |∼2k
(40)
Now, suppose τ2 − n 32 − 3nn 1 n 2 n 2 100 . Then, we have w(n 2 , τ2 ) ∼ n 1 δ since |n 1 | |n|. This extra gain of n 1 δ in the denominator of (34) lets us proceed as before.
234
T. Oh
◦ Subcase (3.b). On Bn 2 ,τ2 . In this case, we have (35) and we can basically proceed as in Subcase (2.b) with (39) in place of (36). Using (40), the modification is straightforward and we omit the details. 1 p0,−1 part of the W p0,− 2 norm on the left-hand side of • Part 2. Next, we consider the Y (30). Define the bilinear operator Bθ,b (·, ·) by 1 1 Bθ,b ( f, g)(n, τ ) = 2π n=n +n τ − n 3 θ 1
2
τ =τ1 +τ2
|n|ns f (n 1 , τ1 )g(n 2 , τ2 )dτ1 . 2 s s 3 b n 1 n 2 j=1 w(n j , τ j )τ j − n j
If MAX = τ1 − n 31 or τ2 − n 32 , then by Hölder inequality, we have LHS of (30) = sup B−1, 1 ( f, g)(n, τ ) L p 1 ≤ sup τ − n 3 − 2 −ε j
L 1τ
|n|∼2 j
2
j
p Lτ
B− 1 +ε, 1 ( f, g)(n, τ ) L τp L p 2
sup B− 1 +ε, 1 ( f, g)(n, τ ) L p 2
j
2
p
|n|∼2 j
2
|n|∼2 j
Lτ
,
where we choose ε > 0 such that ( 21 + ε) p > 1. For p = 2+, we can take ε = 0+. Then, 1
1
the proof reduces to Cases (2) and (3), where τ − n 3 2 is replaced by τ − n 3 2 −ε . Note that this does not affect the argument in Cases (2) and (3). 1 Now, assume MAX = τ − n 3 . If max(τ1 − n 31 , τ2 − n 32 ) τ − n 3 100 , then by Hölder inequality, we have 1 LHS of (30) ≤ sup τ − n 3 − 2 −ε j
p Lτ
B− 1 , 1 −100ε ( f, g)(n, τ ) L τp L p 2 2
sup B− 1 , 1 −100ε ( f, g)(n, τ ) L p
p
|n|∼2 j
2 2
j
|n|∼2 j
Lτ
.
1
1
Then, the proof reduces to Case (1) with τ j − n 3j 2 replaced by τ j − n 3j 2 −100ε , which does not affect the argument. 1 Lastly, if max(τ1 − n 31 , τ2 − n 32 ) τ − n 3 100 , then by Hölder inequality, we have 1 LHS of (30) ≤ sup τ − n 3 − 2 χ(n) (τ − n 3 ) j
sup B− 1 , 1 ( f, g)(n, τ ) L p j
p
|n|∼2 j
2 2
Lτ
p
Lτ
B− 1 , 1 ( f, g)(n, τ ) L τp L p 2 2
|n|∼2 j
,
where the second inequality follows from Lemma 4.6, since − 21 p = −1+ < − 43 . Once again, the proof reduces to Case (1). • Part 3. In this last part, we discuss how to gain a small power of T in (26) by assuming that u or v are supported locally in time. In Part 1 and 2, we indeed showed ∂x (uv)
s,− 21
Wp
u w v X s,b p
s, 21
Xp
+ w u
s, 21
Xp
v X s,b p
(41)
Invariance of White Noise for the Periodic KdV
235
for some b ∈ (0, 21 ), since we needed the full power of 21 from only one of τ − n 3 , τ1 −n 31 , or τ2 −n 32 , i.e. from the maximum one, and the weight w(n j , τ j ) was needed only when MAX = τ j − n 3j . Thus, (26) follows once we prove η2T u X s,b T θ u p
(42)
s, 21
Xp
for some θ > 0. By interpolation, we have u X s,b uα s,0 u1−α , s, 1 Xp
p
Xp
(43)
2
q η(2T τ ). Hence, we have η$ where α = 1 − 2b ∈ (0, 1). Recall η$ 2T (τ ) = 2T 2T L τ ∼ q−1
q−1
T q η L qτ ∼ T q , i.e. we can gain a positive power of T as long as q > 1. For fixed n, by Young and Hölder inequalities, we have u (n, ·) L τp ≤ η$ η$ 2T ∗ 2T T
p −1 p
p
Lτ
u (n, ·)
1
p
L τ2
1
τ − n 3 − 2 L τp τ − n 3 2 u (n, ·) L τp .
Hence, for p > 2, we have 1
u X s,0 T p u p
s, 21
Xp
.
Then, (42) follows from (43) and (44). This completes the proof.
(44)
Remark 4.7. A simple modification of the proof of Proposition 4.3 can be used to establish the local well-posedness of (1) in F L s, p = bsp, p for some p = 2+, s = − 21 + with sp < −1 as well. Such local solutions can be extended globally a.s. on the statistical ensemble from the discussion in Sect. 3. The modification is straightforward and we omit the details.
References 1. Bejenaru, I., Tao, T.: Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. J. Funct. Anal. 233, 228–259 (2006) 2. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. GAFA. 3, 209–262 (1993) 3. Bourgain, J.: Periodic Korteweg-de Vries equation with measures as initial data. Sel. Math., New Ser. 3, 115–159 (1997) 4. Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166, 1–26 (1994) 5. Bourgain, J.: On the Cauchy and invariant measure problem for the periodic Zakharov system. Duke Math. J. 76, 175–202 (1994) 6. Burq, N., Tzvetkov, N.: Invariant measure for a three dimensional nonlinear wave equation. Int. Math. Res. Not. (2007), no. 22, Art. ID rnm108, 26pp 7. Cambronero, S., McKean, H.P.: The ground state eigenvalue of Hill’s equation with white noise potential. Comm. Pure Appl. Math. 52(10), 1277–1294 (1999) 8. Christ, M., Colliander, J., Tao, T.: Asymptotics, frequency modulation, and low-regularity illposedness of canonical defocusing equations. Amer. J. Math. 125(6), 1235–1293 (2003) 9. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp Global Well-Posedness for KdV and Modified KdV on R and T. J. Amer. Math. Soc. 16(3), 705–749 (2003)
236
T. Oh
10. Fernique, M.X.: Intégrabilité des Vecteurs Gaussiens. Comptes Rendus, Séries A 270, 1698–1699 (1970) 11. Ginibre, J., Tsutsumi, Y., Velo, G.: On the Cauchy Problem for the Zakharov System. J. Funct. Anal. 151, 384–436 (1997) 12. Gross, L.: Abstract Wiener spaces. Proc. 5th Berkeley Sym. Math. Stat. Prob. 2 Berkeley CA: Univ Calif. Press, 1965, 31–42 13. Kappeler, T., Topalov, P.: Global wellposedness of KdV in H −1 (T, R). Duke Math. J. 135(2), 327–360 (2006) 14. Kenig, C., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9(2), 573–603 (1996) 15. Kenig, C., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106(3), 617–633 (2001) 16. Kenig, C., Ponce, G., Vega, L.: Quadratic forms for the 1-D semilinear Schrödinger equation. Trans. Amer. Math. Soc. 348, 3323–3353 (1996) 17. Kishimoto, N.: Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity u 2 . Comm. Pure Appl. Anal. 7(5), 1123–1143 (2008) 18. Kuo, H.: Gaussian Measures in Banach Spaces. Lec. Notes in Math. 463, New York: Springer-Verlag, 1975 19. Oh, T.: Invariant Gibbs measures and a.s. global well-posedness for coupled KdV systems. Diff. Integ. Equ. 22(7–8), 637–668 (2009) 20. Oh, T.: Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system. Preprint, available at http://arXiv.org/abs/0904.2816v1[math.AP], 2009 21. Quastel, J., Valkó, B.: KdV preserves white noise. Commun. Math. Phys. 277(3), 707–714 (2008) 22. Roynette, B.: Mouvement brownien et espaces de Besov. Stochastics Stoch. Rep. 43, 221–260 (1993) 23. Tzvetkov, N.: Invariant measures for the nonlinear Schrödinger equation on the disc. Dyn. Partial Differ. Equ. 3(2), 111–160 (2006) 24. Tzvetkov, N.: Invariant measures for the defocusing Nonlinear Schrödinger equation (Mesures invariantes pour l’équation de Schrödinger non linéaire). Ann. l’Inst. Fourier 58, 2543–2604 (2008) 25. Zhidkov, P.: Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. Lec. Notes in Math. 1756, Berlin-Heidleperg-NewYork:Springer-Verlag, 2001 Communicated by P. Constantin
Commun. Math. Phys. 292, 237–270 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0854-9
Communications in
Mathematical Physics
Stochastically Stable Globally Coupled Maps with Bistable Thermodynamic Limit Jean-Baptiste Bardet1,2 , Gerhard Keller3 , Roland Zweimüller4 1 IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex,
France. E-mail:
[email protected]
2 LMRS, Université de Rouen, Avenue de l’Université, BP.12,
Technopôle du Madrillet, 76801 Saint-Étienne-du-Rouvray, France
3 Department Mathematik, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2,
91054 Erlangen, Germany. E-mail:
[email protected]
4 Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien,
Austria. E-mail:
[email protected] Received: 19 December 2008 / Accepted: 19 March 2009 Published online: 4 July 2009 – © Springer-Verlag 2009
Abstract: We study systems of globally coupled interval maps, where the identical individual maps have two expanding, fractional linear, onto branches, and where the coupling is introduced via a parameter - common to all individual maps - that depends in an analytic way on the mean field of the system. We show: 1) For the range of coupling parameters we consider, finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. 2) For the same range of parameters, the self-consistent Perron-Frobenius operator which captures essential aspects of the corresponding infinite-size system (arising as the limit of the above when the system size tends to infinity), undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium. 1. Introduction Globally coupled maps are collections of individual discrete-time dynamical systems (their units) which act independently on their respective phase spaces, except for the influence (the coupling) of a common parameter that is updated, at each time step, as a function of the mean field of the whole system. Systems of this type have received some attention through the work of Kaneko [10,11] in the early 1990s, who studied systems of N quadratic maps acting on coordinates x1 , . . . , x N ∈ [0, 1], and coupled by a parameter depending in a simple way on x¯ := N −1 (x1 +· · ·+ x N ). His key observation, for huge system size N , was the following: if (x¯ t )t=0,1,2,... denotes the time series of mean field values of the system started in a random configuration (x1 , . . . , x N ), then, for many parameters of the quadratic map, and even for very small coupling strength, pairs (x¯ t , x¯ t+1 ) of consecutive values of the field showed complicated functional dependencies plus some noise of order N −1/2 , whereas for uncoupled systems of the same size the x¯ t , after a while, are constant up to some noise of order N −1/2 . While the latter observation is not surprising for independent units, the complicated dependencies for
238
J.-B. Bardet, G. Keller, R. Zweimüller
weakly coupled systems, a phenomenon Kaneko termed violation of the law of large numbers, called for closer investigation. The rich bifurcation structure of the family of individual quadratic maps may offer some explanations, but since a mathematically rigorous investigation of even a small number of coupled quadratic maps in the chaotic regime still is a formidable task, there seem to be no serious attempts to tackle this problem. A model which is mathematically much√ easier to treat is given by coupled tent maps. Indeed, for tent maps with slope larger than 2 and moderate coupling strength, a system of N mean field coupled units has an ergodic invariant probability density with exponentially decreasing correlations [14]. This is true for all N and for coupling strengths that can be chosen to be the same for all N . Nevertheless, Ershov and Potapov [8] showed numerically that (albeit on a much smaller length scale than in the case of coupled quadratic maps) also mean field coupled tent maps exhibit a violation of the law of large numbers in the aforementioned sense. They also provided a mathematical analysis which demonstrated that the discontinuities of the invariant density of a tent map are at the heart of the problem. Their analysis was not completely rigorous, however, as Chawanya and Morita [3] could show that there are indeed (exceptional) parameters of the system for which there is no violation of the law of large numbers - contrary to the predictions in [8]. On the other hand, references [18,19] contain further simulation results on systems violating the law of large numbers. (But at present, a mathematically rigorous treatment of globally coupled tent maps that is capable of classifying and explaining the diverse dynamical effects that have been observed does not seem to be in sight either.) These studies were complemented by papers by Järvenpää [9] and Keller [13], showing (among other things) that globally coupled systems of smooth expanding circle maps do not display violation of the law of large numbers at small coupling strength, because their invariant densities are smooth. Given this state of knowledge, the present paper investigates specific systems of globally coupled piecewise fractional linear maps on the interval X := [− 21 , 21 ], where each individual map has a smooth invariant density. For small coupling strength, Theorem 4 in [13] extends easily to this setup and proves the absence of a violation of the law of large numbers. For larger coupling strength, however, we are going to show that this phenomenon does occur in the following sense: on Bifurcation. The nonlinear self-consistent Perron-Frobenius operator (PFO) P L 1 (X, λ), which describes the dynamics of the system in its thermodynamic limit, undergoes a supercritical pitchfork bifurcation as the coupling strength increases. (Here and in the sequel λ denotes Lebesgue measure.) Mixing. At the same time, all corresponding finite-size systems have unique absolutely continuous invariant probability measures µ N on their N -dimensional state space, and exhibit exponential decay of correlations under this measure. Stable behaviour. In the stable regime, i.e. for fixed small coupling strength below the bifurcation point of the infinite-size system, the measures µ N converge weakly, as the system size N → ∞, to an infinite product measure (u 0 · λ)N , where u 0 is the unique fixed point of P. Bistable behaviour. In the bistable regime, i.e. for fixed coupling strength above the bifurcation point of the infinite-size system, all possible weak limits of the measures µ N are convex combinations of the three infinite product measures (u r · λ)N , and u ±r∗ are r ∈ {−r∗ , 0, r∗ }, where now u 0 is the unique unstable fixed point of P its two stable fixed points. (We conjecture that the measure (u 0 · λ)N is not charged in the limit.)
Globally Coupled Maps with Bistable Thermodynamic Limit
239
This scenario clearly bears some resemblance to the Curie-Weiss model from statistical mechanics and its dynamical variants. We also stress that a simple modification of our system leads to a variant where, is created at the bifurcation instead of two stable fixed points, one stable two-cycle for P point. This may be viewed as the simplest possible scenario for a violation of the law of large numbers in Kaneko’s original sense. In the next section we describe our model in detail, and formulate the main results. Section 3 contains the proofs for finite-size systems. In Sect. 4 we start the investigation We observe that this operator of the infinite-size system via the self-consistent PFO P. preserves a class of probability densities which can be characterised as derivatives of Herglotz-Pick-Nevanlinna functions. Integral representations of these functions reveal and allows us to describe a hidden order structure, which is respected by the operator P, is extended to arbitrary the pitchfork bifurcation. In Sect. 5 this dynamical picture for P densities. Finally, in Sect. 6, we discuss the situation when some noise is added to the dynamics. 2. Model and Main Results 2.1. The parametrised family of maps. Throughout, all measures are understood to be Borel, and we let P(B) := {probability measures on B}. Lebesgue measure will be denoted by λ. We introduce a 1-parameter family of piecewise fractional-linear transformations Tr on X := [− 21 , 21 ], which will play the role of the local maps. To facilitate manipulation of such maps, we use their standard matrix representation, letting ax + b ab f M (x) := for any real 2 × 2-matrix M = , cd cx + d (x) = (ad − bc)/(cx + d)2 and f ◦ f = f so that f M M N M N . Specifically, we consider the function f Mr , depending on a parameter r ∈ (−2, 2), given by the coefficient matrix r +4r +1 Mr := . 2r 2
One readily checks that f Mr (− 21 ) = − 21 , f Mr ( 21 ) = 23 , f Mr (αr ) = and that (the infimum being attained on ∂ X ) fM (x) = r
4 − r2 2 (r x + 1)
2
2
2 − |r | >0 = inf f M r X 2 + |r |
1 2
for αr := −r/4,
for x ∈ X .
The latter shows that f Mr is uniformly expanding if and only if |r | < 23 , and we define our single-site maps Tr : X → X with parameter r ∈ (−2/3, 2/3) by letting 1 on [− 21 , αr ), f Mr (x) = Tr (x) := f Mr (x) mod Z + f Mr (x) − 1 = f Nr (x) on (αr , 21 ], 2 where
Nr :=
1 −1 0 1
Mr .
240
J.-B. Bardet, G. Keller, R. Zweimüller
2,5 0,4 2 0,2 1,5
y
y
0
1 -0,2 0,5 -0,4 0 -0,4
-0,2
0
0,2
0,4
-0,4
-0,2
x
0
0,2
0,4
x
Fig. 1. The functions Tr (left), and u r (right), for r = − 21
We thus obtain a family (Tr )r ∈(−2/3,2/3) of uniformly expanding, piecewise invertible maps Tr : X → X , each having two increasing covering branches. Note also that this family is symmetric in that (2.1) − Tr (−x) = T−r (x) for r ∈ − 23 , 23 and x ∈ X . According to well-known folklore results, each map Tr , r ∈ (−2/3,2/3), has a unique invariant probability density u r ∈ D := {u ∈ L 1 (X, λ) : u 0, X u dλ = 1}, and Tr is exact (hence ergodic) w.r.t. the corresponding invariant measure. Due to (2.1), we have u −r (x) = u r (−x) mod λ. We denote the Perron-Frobenius operator (PFO), w.r.t. Lebesgue measure λ, of a map T by PT , abbreviating Pr := PTr . In our construction below we will exploit the fact that 2-to-1 fractional linear maps like Tr in fact enable a fairly explicit analysis of their PFOs on a suitable class of densities. In particular, the u r are known explicitly: Remark 1. It is clear that u 0 = 1 X (T0 being piecewise affine). For r ∈ (−2/3, 2/3)\{0} r r we let γr := 1+r and δr := 1−r . Then δr 1 2r 2 (2.2) dy = u r (x) := 2 (r x − (1 − r ))(r x − (1 + r )) γr (1 − x y) is an integrable invariant density for Tr , see [22]. Its normalised version −1 r2 − 4 · u r (x) u r (x) := log 2 9r − 4
(2.3)
is the unique Tr -invariant probability density. The key point in the choice of this family of maps is that for r < 0, Tr is steeper in the positive part of X than in its negative part, hence typical orbits spend more time on the negative part, which is confirmed by the invariant density (see Fig. 1). If r > 0, then Tr favours the positive part. The heuristics of our construction is that for sufficiently strong coupling this effect of “polarisation” is reinforced and gives rise to bistable behaviour.
Globally Coupled Maps with Bistable Thermodynamic Limit
241
2.2. The field and the coupling. For any probability measure Q ∈ P(X ), we denote its mean by φ(Q) := x dQ(x), (2.4) X
and call this the field of Q. With a slight abuse of notation we also write, for u ∈ D, dQ φ(u) := , (2.5) xu(x) d x = φ(Q) if u = dλ X and, for x ∈ X N , φ(x) :=
N N 1 1 xi = φ(Q) if Q = δxi . N N i=1
(2.6)
i=1
To define the system of globally coupled maps (both in the finite- and the infinite-size case) we will, at each step of the iteration, determine the actual parameter as a function of the present field. This is done by means of a feedback function G : X → R := [−0.4, 0.4] which we always assume to be real-analytic 1 and S-shaped in that it satisfies G (x) > 0 and G(−x) = −G(x) for all x ∈ X , while G (x) < 0 if x > 0. The most important single parameter in our model is going to be B := G (0) which quantifies the coupling strength. For the results to follow we shall impose a few additional constraints on the feedback function G, made precise in Assumptions I and II below. Remark 2. The following will be our standard example of a suitable feedback function G: B G(x) := A tanh x . (2.7) A It satisfies Assumptions I and II if 0 < A 0.4 and 0 B 18. 2.3. The finite-size systems. We consider a system T N : X N → X N of N coupled copies of the parametrised map, defined by (T N (x))i = Tr (x) (xi ) with r (x) := G(φ(x)). For the following theorem, which we prove in sect. 3, we need the following assumption (satisfied by the example above): (2.8) Assumption I. G (x) 25 − 50|G(x)| for all x ∈ X . Theorem 1 (Ergodicity and mixing of finite-size systems). Suppose the S-shaped function G satisfies (2.8). Then, for any N ∈ N, the map T N : X N → X N has a unique absolutely continuous invariant probability measure µ N . Its density is strictly positive and real analytic. The systems (T N , µ N ) are exponentially mixing in various strong senses, in particular do Hölder observables have exponentially decreasing correlations. The key to the proof is an estimate ensuring uniform expansion. After establishing the latter in Sect. 3, the theorem follows from “folklore” results whose origins are not so easy to locate in the literature. In a C 2 -setting, existence, uniqueness and exactness of an invariant density were proved essentially by Krzyzewski and Szlenk [16]. Exponential mixing follows from the compactness of the transfer operator first observed by Ruelle [21]. For a result which applies in our situation and entails Theorem 1, we refer to the main theorem of [17]. 1 This is only required to obtain highest regularity of the invariant densities of the finite-size systems in Theorem 1. Everything else remains true if G is merely of class C 2 .
242
J.-B. Bardet, G. Keller, R. Zweimüller
2.4. The self-consistent PFO and the thermodynamic limit of the finite-size systems. Since the coupling we defined is of mean-field type, we can adapt from the probabilistic literature (see for example [5,23]) the classical method of taking the thermodynamic limit of our family of finite-size systems T N , as N → ∞. To do so, consider the set P(X ) of Borel probability measures on X , equipped with the topology of weak convergence : P(X ) → P(X ) by and the resulting Borel σ -algebra on P(X ). Define T (Q):= Q ◦ T −1 , where r (Q) := G(φ(Q)). T r (Q)
(2.9)
. Indeed, if We can then represent the evolution of any finite-size system using T 1 N N (x) := N i=1 δxi is the empirical measure of x = (xi )1i N , then N : X N → ◦ N . P(X ) satisfies N ◦ T N = T Furthermore, when restricted to the set of probability measures absolutely continu is represented by the self-consistent Perron Frobenius operator, ous with respect to λ, T defined as which is the nonlinear positive operator P : L 1 (X, λ) → L 1 (X, λ), P
:= PG(φ(u)) u. Pu
(2.10)
(u · λ) = ( Pu) · λ and preserves the set D of probability Clearly, this map satisfies T densities. Note, however, that it does not contract, i.e. there are u, v ∈ D such that − Pv L 1 (X,λ) > u − v L 1 (X,λ) . Pu on means of Dirac masses, One may finally join these two aspects, the action of T or on absolutely continuous measures, via the following observation: Proposition 1 (Propagation of chaos). Let Q = u · λ ∈ P(X ), with u ∈ D. Then, for Q ⊗N -almost every (xi )i 1 , and any n 0, the empirical measures N (TnN (x1 , . . . , x N )) n u) · λ as N → ∞. converge weakly to ( P represents the This result confirms the point of view that the self-consistent PFO P infinite-size thermodynamic limit N → ∞ of the finite-size systems T N . Its proof is is reasonably simple (easier than for stochastic evolutions). The only difficulty is that T not a continuous map on the whole of P(X ). This can be overcome with the following lemma, which Proposition 1 is a direct consequence of, and whose proof is given in Sect. A.1. at non-atomic measures). Assume that a sequence (Q n )n 1 Lemma 1 (Continuity of T Q n )n 1 converges weakly to in P(X ) converges weakly to some non-atomic Q. Then (T T Q. Here is an immediate consequence of this lemma that will be used below. -invariant Borel probability meaCorollary 1. Assume that a sequence (πn )n 1 of T sures on P(X ) converges weakly to some probability π on P(X ). If there is a Borel set A ⊆ P(X ) with π(A) = 1 which only contains non-atomic measures, then π is also -invariant. T form a π -null set. The mapProof. Due to our condition on A the discontinuities of T ping theorem for weak convergence (e.g. Theorem 5.1 of [2]) thus ensures convergence −1 )n 1 to π ◦ T −1 . But, by assumption, the πn ◦ T −1 = πn converge to π . of (πn ◦ T
Globally Coupled Maps with Bistable Thermodynamic Limit
243
2.5. The long-term behaviour of the infinite-size system. Our goal is to analyse the on D. Some basic features of P can be understood considering the asymptotics of P dynamics of 4 4 H : − 23 , 23 → R := − 10 , 10 , H (r ) := G(φ(u r )), on the densities u r introduced in Sect. 2.1, as which governs the action of P r = PH (r ) u r . Pu
(2.11)
we will always presuppose the following: In studying P, Assumption II. H satisfies the following dichotomy: either h(r ) has a unique fixed point at r = 0 (the stable regime with H (0) 1 and r = 0 stable), or H (r ) has exactly three fixed points − r∗ < 0 < r∗ (the bistable regime with H (0) > 1 and ±r∗ stable).
(2.12)
We will see that H > 0 and H (0) = G (0)/6, so that the stable regime corresponds to the condition G (0) 6. This assumption can be checked numerically for specific feedback functions G. For our example of Remark 2, we check in Sect. A.2 that H is a contraction for B = G (0) 6 and that it is S-shaped for B > 6, which implies Assumption II. By (2.1), H (−r ) = −H (r ). Note, however, that r → φ(u r ) itself is not S-shaped (see Fig. 3 in Sect. A) so that the S-shapedness of G alone is not sufficient for that of H . Observe now that
r =0 (in the stable regime) Pu r = u r iff (2.13) r ∈ {0, ±r∗ } (in the bistable regime) (since u r = u r for r = r , and each Tr is ergodic). We are going to show that the on D comfixed points u 0 = 1 X , and u ±r∗ dominate the long-term behaviour of P pletely, and that they inherit the stability properties of the corresponding parameters −r∗ < 0 < r∗ . Therefore, the stable/bistable terminology for H introduced above also provides an appropriate description of the asymptotic behaviour of P. on D). Consider P : D → D, D equipped with Theorem 2 (Long-term behaviour of P the metric inherited from L 1 (X, λ). Assuming (I) and (II), we have the following: and attracts all densities, that 1) In the stable regime, u 0 is the unique fixed point of P, is, n u = u 0 lim P
n→∞
for all u ∈ D.
Now u 0 is 2) In the bistable regime, {u −r∗ , u 0 , u r∗ } are the only fixed points of P. unstable, while u −r∗ and u r∗ are stable. More precisely: in the sense that their respective basins of a) u ±r∗ are stable fixed points for P attraction are L 1 -open.
244
J.-B. Bardet, G. Keller, R. Zweimüller
b) If u∈ D is not attracted by u −r∗ or u r∗ , then it is attracted by u 0 . c) u 0 is not stable. Indeed, u 0 can be L 1 -approximated by convex analytic densities in the sense made precise from either basin. It is a hyperbolic fixed point of P in Proposition 5 of Sect. 5. Example 1. In case G(x) = A tanh(Bx/A) with 0 < A 0.4 and 0 B 18, both theorems apply. The infinite-size system is stable iff B 6, and bistable otherwise, while all finite-size systems have a unique a.c.i.m. in this parameter region. The theorem summarises the contents of Propositions 3, 4 and 5 of Sect. 5 (which, in fact, provide more detailed information). The proofs rest on the fact that PFOs of maps with full fractional-linear branches leave the class of derivatives of HerglotzPick-Nevanlinna functions invariant. This observation can be used to study the action of in terms of an iterated function system on the interval [−2, 2] with two fractional-linear P branches and place dependent probabilities. In the bistable regime the system is of course not contractive, but it has strong monotonicity properties and special geometric features which allow to prove the theorem. Our third theorem, which is essentially a corollary to the previous ones, describes the passage from finite-size systems to the infinite-size system. Below, weak convergence of the µ N ∈ P(X N ) to some µ ∈ P(X N ) means that ϕ dµ N → ϕ dµ for all contin uous ϕ : X N → R which only depend on finitely many coordinates. (So that ϕ dµ N is defined, in the obvious fashion, for N large enough.) Theorem 3 (From finite to infinite size – the limit as N → ∞). The T N -invariant prob-invariant probability measures ability measures µ N of Theorem 1 correspond to the T −1 -invariant µ N ◦ N on P(X ). All weak accumulation points π of the latter sequence are T probability measures concentrated on the set of measures absolutely continuous w.r.t. λ. Furthermore: −1 1) In the stable regime, the sequence (µ N ◦ N ) N 1 converges weakly to the point mass δu 0 λ . In other words, the sequence (µ N ) N 1 converges weakly to the pure product measure (u 0 λ)N on X N . 2) In the bistable regime, each weak accumulation point π of the sequence (µ N ◦ −1 N ) N 1 is of the form α δu −r∗ λ + (1 − 2α) δu 0 λ + α δur∗ λ for some α ∈ [0, 21 ]. In other words, each weak accumulation point of the sequence (µ N ) N 1 is of the form α(u −r∗ λ)N + (1 − 2α)(u 0 λ)N + α(u r∗ λ)N .
Remark 3. We cannot prove, so far, that α = 21 , which is to be expected because u 0 = 1 X In Sect. 6 we show that α = 1 indeed, if some small is an unstable fixed point of P. 2 noise is added to the system. ◦ N , the T N -invariant probability measures µ N Proof of Theorem 3. As N ◦ T N = T -invariant probability measures µ N ◦ −1 on P(X ). Their of Theorem 1 correspond to T N possible weak accumulation points are all concentrated on sets of measures from P(X ) with density w.r.t. λ, see Theorem 3 in [13]. (The proof of that part of the theorem we refer to does not rely on the continuity of the local maps that is assumed in that paper.) -invariant probTherefore Corollary 1 shows that all these accumulation points are T ability measures concentrated on measures with density w.r.t. λ. In other words, they can be interpreted as P-invariant probability measures on D. Now Theorem 2 implies −1 that the sequence (µ N ◦ N ) N 1 converges weakly to the point mass δu 0 λ in the stable regime, whereas, in the bistable regime, each such limit measure is of the form
Globally Coupled Maps with Bistable Thermodynamic Limit
245
α δu −r∗ λ + (1 − 2α) δu 0 λ + α δur∗ λ for some α ∈ [0, 21 ] (observe the symmetry of the system). Now the corresponding assertions on the measures µ N follow along known lines, for a reference see e.g. [13, Prop. 1]. 3. Proofs: The Finite-Size Systems We assume throughout this section that |G(x)| 0.5
and
G (x) 25 − 50|G(x)| for all |x|
1 . 2
(3.1)
In order to apply the main theorem of Mayer [17] we must check his assumptions (A1) – (A4) for the map T = T N . To that end define F : X N → [− 21 , 23 ] N by (F(x))i = N f Mr (x) (xi ). Obviously T(x) = F(x) mod Z + 21 , and (A1) – (A4) follow readily from the following facts that we are going to prove: Lemma 2. F : X N → [− 21 , 23 ] N is a homeomorphism which extends to a diffeomorphism between open neighbourhoods of X N and [− 21 , 23 ] N . Lemma 3. The inverse F−1 of F is real analytic and can be continued to a holomorphic mapping on a complex δ-neighbourhood of [− 21 , 23 ] N such that F−1 ( ) is contained in a δ -neighbourhood of X N for some 0 < δ < δ. To verify these two lemmas we need the following uniform expansion estimate which we will prove at the end of this section. (Here . denotes the Euclidean norm.) Lemma 4 (Uniform expansion). There is a constant ρ (DF(x))−1 ρ for all N ∈ N and x ∈ X N .
∈
(0, 1) such that
Proof of Lemma 2. Obviously F(X N ) ⊆ [− 21 , 23 ] N . Hence it is sufficient to prove the assertions of the lemma for the map F := 21 (F − ( 21 , . . . , 21 )T ) : X N → X N . As each F extends to an analytic mapping f Mr is differentiable on (−1, 1) (recall that |r | < 23 ), N N from (−1, 1) → R . By Lemma 4, it is locally invertible on ε := (− 21 − ε, 21 + ε) N for each sufficiently small ε 0. (Note that 0 = int(X).) All we need to show is that this implies global invertibility of F| 0 : 0 → 0 , because then the possibility to extend F diffeomorphically to a small open neighbourhood of X N in R N follows again from the local invertibility on ε for some ε > 0. So we prove the global invertibility of F| 0 : 0 → 0 . As each f˜Mr := 21 ( f Mr − 21 ) : X → X is a homeomorphism that leaves fixed the endpoints of the interval X , we have F(∂ X N ) ⊆ ∂ X N and F( 0 ) ⊆ 0 . Observing the simple fact that 0 is a paracompact connected smooth manifold without boundary and with trivial fundamental group, we only need to show that F| 0 : 0 → 0 is proper in order to deduce from [4, Cor. 1] that F| 0 is a diffeomorphism of 0 . So let K be a compact subset of 0 . As F| 0 extends to the continuous map F : X N → X N and as F(∂ X N ) ⊆ ∂ X N , the −1 −1 N N F (K ) ⊂ int(X ) ⊆ X is closed and hence compact. Therefore set F| 0 (K ) = F| 0 : 0 → 0 is indeed proper.
246
J.-B. Bardet, G. Keller, R. Zweimüller
Proof of Lemma 3. As F is real analytic on a real neighbourhood of X N , the real analyticity of F−1 on a real neighbourhood of [− 21 , 23 ] N follows from the real analytic inverse function theorem [15, Theorem 18.1]. It extends to a holomorphic function on a complex δ-neighbourhood of [− 21 , 23 ] N – see e.g. the discussion of complexifications of real analytic maps in[15, pp. 162–163]. If δ > 0 is sufficiently small, Lemma 4 implies that F−1 is a uniform contraction on . Hence the δ in the statement of the lemma can be chosen strictly smaller than δ. Proof of Lemma 4. Recall that (F(x))i = fr (x) (xi ), where r (x) = G(φ(x)), φ(x) = 1 N i=1 x i , and we write fr instead of f Mr . Denote g(x) := G (φ(x)), N 1 (x) := diag( fr (x1 ), . . . , fr (x N )), ∂ fr ∂ fr (x1 ), . . . , (x N )), 2 (x) := diag( ∂r ∂r ⎛ 1 1 ⎞ N ... N ⎜ .. ⎟ , E N := ⎝ ... . ⎠ 1 N
...
1 N
and observe that q(x) := (4 − r 2 ) ∂∂rfr (x)/ fr (x) simplifies to q(x) = 1 − 4x 2 so that 1 (x)−1 2 (x) =
1 1 diag (q(x1 ), . . . , q(x N )) =: 3 (x). 4 − r2 4 − r2
Then the derivative of the coupled map F(x) = ( fr (x) (x1 ), . . . , fr (x) (x N )) is DF(x) = 1 (x) + 2 (x)E N g(x) = 1 (x) 1 +
1 (x)E g(x) , 3 N 4 − r2
with 1 denoting the identity matrix. Letting q = q(x) := (q(x1 ), . . . ., q(x N ))T 1 1 e N := ( , . . . ., )T N N and observing that 3 (x) = diag(q(x)) so that 3 (x)E N = q eTN , the inverse of DF(x) is g(x) T −1 DF(x)−1 = 1 − q(x)e N 1 (x) . 4 − r 2 + g(x)eTN q(x) In order to check conditions under which F is uniformly expanding in all directions, it is sufficient to find conditions under which DF(x)−1 < 1 uniformly in x. Observe 2+|r | 2 first that 1 (x)−1 (inf fr )−1 21 2−|r | < 1 for |r | < 3 . From now on we fix a point x and suppress it as an argument to all functions. Then, if v is any vector in R N that is not perpendicular to q, some scalar multiple of it can be decomposed in a unique way as αv = q − p, where p is perpendicular to q. Denote p¯ = eTN p, q¯ = eTN q, and
Globally Coupled Maps with Bistable Thermodynamic Limit
247
observe that q¯ 0 as q has only nonnegative entries. We estimate the euclidean norm g T of (1 − T q e N )(αv): 2 4−r +ge N q
2 g T 1− q e N (αv) T 2 4 − r + ge N q 2 g p¯ − g q¯ = 1+ q2 + p2 4 − r 2 + g q¯ p¯ − q¯ 2 = 1+ q2 + p2 + q¯ 2 1 + −1 p¯ = q2 + p2 , 1 + −1 q¯
(3.2)
2
−1/2 p and q¯ N −1 q2 (observe that all entries of q where := 4−r g . As p¯ N are bounded by 1), we can continue the above estimate with
q2 + p2 + 2pq
−2 N −1 q2 −1 N −1/2 q 2 + p . (1 + −1 N −1 q2 )2 (1 + −1 N −1 q2 )2
To estimate this expression we abbreviate temporarily t := N −1/2 q. Then 0 t 1, and straightforward maximisation yields: −1 N −1/2 q 9 √ √ , −1 −1 2 2 (1 + N q ) 16 3 −2 −1 2 1 N q . (1 + −1 N −1 q2 )2 4 So we can continue the above estimate by (3.2) (q2 + p2 ) 1 +
9 1 √ √ + 16 3 4
,
where q2 + p2 = αv2 . Hence 1/2 g T 1− 1 + √9 √ + 1 (αv) qe αv 1 − r + geTN q N 16 3 4 and, as the vectors v not perpendicular to q are dense in R N , we conclude DF
−1
(x) ρ := 2
1 2 + |r | · 2 2 − |r |
9 1 1+ √ √ + 4 16 3
1/2 .
Observing = 4−r g one finds numerically that the norm is bounded by 0.99396 uniformly for all x, if −0.5 r 0.5 and 0 g 25−50r . Hence the map F is uniformly expanding in all directions provided (3.1) holds.
248
J.-B. Bardet, G. Keller, R. Zweimüller
4. An Iterated Function System Representation for Smooth Densities 4.1. An invariant class of densities. The PFOs Pr allow a detailed analysis since their action on certain densities has a convenient explicit description: Consider the family (w y ) y∈(−2,2) of probability densities on X given by w y (x) :=
1 − y 2 /4 , (1 − x y)2
x ∈ X.
As pointed out in [22] (using different parametrisations), Perron-Frobenius operators P f M of fractional-linear maps f M act on these densities via their duals f M # , where 01 01 d c ab # M := ·M· = for M = , 10 10 ba cd in that
P f M (1 J · w y ) =
w y dλ · w y J
with y = f M # (y),
(4.1)
for matrices M and intervals J ⊆ X for which f M (J ) = X . (This can also be verified by direct calculation.) Since f (0 1) (x) = x1 , one can compute the duals 10
σr (y) := f Mr# (y) = τr (y) := f Nr# (y) =
1 f Mr ( 1y ) 1 f Nr ( 1y )
=
2(y + r ) (r + 1)y + r + 4
=
σr (y) 2(y + r ) = 1 − σr (y) (r − 1)y − r + 4
and (4.2)
of the individual branches of Tr , then express Pr w y as the convex combination Pr w y = P f Mr (1[− 1 ,αr ) · w y ) + P f Nr (1(αr , 1 ] · w y ) 2
2
= pr (y) · wσr (y) + (1 − pr (y)) · wτr (y) , with weights pr (y) :=
αr
−1/2
w y (x) d x =
r+y 1 1 r+y − and 1 − pr (y) = + . 2 4 + ry 2 4 + ry
(4.3)
(4.4)
It is straightforward to check that for every r ∈ (−2, 2) the functions σr and τr are continuous and strictly increasing on [−2, 2] with images σr ([−2, 2]) = [−2, 2/3] and τr ([−2, 2]) = [−2/3, 2]. From this remark and (4.3) it is clear that the Pr preserve the class of those u ∈ D which are convex combinations u = (−2,2) w y dµ(y) = w• dµ of the special densities w y for some representing measure µ from P(−2, 2). We find that Pr acts on representing measures according to w• dµ = w• d Lr∗ µ with (4.5) Pr (−2,2)
(−2,2)
Lr∗ µ := ( pr · µ) ◦ σr−1 + ((1 − pr ) · µ) ◦ τr−1 , where pr · µ denotes the measure with density pr w.r.t. µ.
Globally Coupled Maps with Bistable Thermodynamic Limit
249
To continue, we need to collect several facts about the dual maps σr and τr . We have σr (y) =
2(4 − r 2 ) (r y + y + r + 4)2
and τr (y) =
2(4 − r 2 ) , (r y − y − r + 4)2
(4.6)
showing that σr and τr are strictly concave, respectively convex, on [−2, 2]. One next gets readily from (4.2) that 1/σr − 1/τr = 1 wherever defined, and σr (y) < τr (y)
for y ∈ [−2, 2]{−r }
while (and this observation will be crucial later on) σr and τr have a common zero zr := −r and 2 2t , and σr (zr + t) = . (4.7) 4 − r2 (r + 1) t + 4 − r 2 4 4 In the following, we restrict our parameter r to the set R = − 10 , 10 . Direct calcu 2 2 lation proves that, letting Y := − 3 , 3 , we have σr (zr ) = τr (zr ) =
σr (Y ) ∪ τr (Y ) ⊆ Y
if r ∈ R,
(4.8)
so that Y is an invariant set for all such σr and τr , and that 2 3 3 2 and sup τr = τr sup σr = σr − 3 4 3 4 Y Y
for r ∈ R,
(4.9)
which provides us with a common contraction rate on Y for the σr and τr from this parameter region. All these features of σr and τr are illustrated by Fig. 2. We denote by w(y) := φ(w y ) the field of the density w y , and find by explicit integration that ∞ y 2k+1 1 + y/2 1 1 1 1 − 2 log + = w(y) = (4.10) 4 y 1 − y/2 y (2k + 1)(2k + 3) 2 k=0
for y ∈ Y . In particular, w(0) = 0 and w (y) 16 > 0, so the field depends monotonically on y. Note also that we have, for all µ ∈ P(Y ), w• dµ = xw y (x) d x dµ(y) = w dµ. (4.11) φ Y
Y
X
Y
We focuson densities u with representing measure µ supported on Y , i.e. on the class D := {u = Y w• dµ : µ ∈ P(Y )}. Writing rµ := G w dµ = G(φ(u)) ∈ R, (4.12) Y
acts on the representing meaand recalling (4.5), we find that our nonlinear operator P sures from P(Y ) via ∗ µ P w• dµ = w• d L with (4.13) Y
Y
∗ µ := Lr∗ µ = ( prµ · µ) ◦ σr−1 + ((1 − prµ ) · µ) ◦ τr−1 . L µ µ µ
250
J.-B. Bardet, G. Keller, R. Zweimüller
0,6
sigma tau
0,4
0,2
0
−0,2
−0,4
−0,6 −0,6
−0,4
−0,2
0
0,2
0,4
0,6
y Fig. 2. The dual maps σr and τr for r = 0.3. The small invariant box has endpoints γr and δr
∗ µ), the support of L ∗ µ, is contained in σr (supp(µ)) ∪ τr (supp(µ)), it is As supp(L immediate from (4.8) that ⊆ D . PD 4 4 For r ∈ R = [− 10 , 10 ] we find that σr and τr each have a unique stable fixed point in Y , given by
σr (γr ) = γr :=
r r +1
and
τr (δr ) = δr :=
−r , r −1
respectively. Note that the interval Yr := [γr , δr ] of width 2r 2 /(1 − r 2 ) between these stable fixed points is invariant under both σr and τr , see the small boxed region in Fig. 2. Furthermore, each of γr and δr is mapped to r under the branch not fixing it, i.e. σr (δr ) = τr (γr ) = r , meaning that, restricted to Yr , σr and τr are the inverse branches of some 2-to-1 piecewise fractional linear map Sr : Yr → Yr . The explicit Tr -invariant densities u r from (2.3) can be represented as u r = Y w• dµr with µr ∈ P(Yr ) ⊆ P(Y ) given by −1 dµr r2 − 4 1Yr (y) (y) = log 2 . dλ 9r − 4 1 − y 2 /4
(4.14)
on D , using its Our goal in this section is to study the asymptotic behaviour of P ∗ . A crucial ingredient of our analysis is an order representation by means of the IFS L
Globally Coupled Maps with Bistable Thermodynamic Limit
251
relation on the space P(Y ) of probability measures µ, ν representing densities from D , defined by µν
:⇔
∀y ∈ Y : µ(y, ∞) ν(y, ∞),
(4.15)
which the IFS will be shown to respect. Recall the fixed points ±r∗ of H defined (in the bistable regime) in § 2.5. We will prove on D ). Take any u ∈ D , u = Proposition 2 (Long-term behaviour of P Y w y dµ(y) for some µ ∈ P(Y ). The following is an exhaustive list of possibilities for the asymptotic ∗n µ)n 0 : behaviour of the sequence (L ∗n µ δ0 for some n 0. Then (L ∗n µ)n 0 converges to µr∗ and hence P n u (1) L converges to u r∗ in L 1 (X, λ). ∗n µ)) contains 0 for all n 0. Then (L ∗n µ)n 0 con(2) The interval conv(supp(L n u converges to u 0 in L 1 (X, λ). In this case also the verges to δ0 and hence P ∗n length of conv(supp(L µ)) tends to 0. ∗n µ)n 0 converges to µ−r∗ and hence P ∗n µ ≺ δ0 for some n 0. Then (L n u (3) L converges to u −r∗ in L 1 (X, λ). In the stable regime, only scenario (2) is possible, so that we always have convergence ∗n µ)n 0 to δ0 in that case. of (L Our arguments will rely on continuity and monotonicity properties of the IFS, that we detail below, before proving Proposition 2 in Sect. 4.4. 4.2. The IFS: continuity. Convergence in P(Y ), lim µn = µ, will always mean weak convergence of measures, Y ρ dµn → Y ρ dµ for bounded continuous ρ : Y → R. Since Y is a bounded interval, this is equivalent to convergence in the Wasserstein-metric dW on P(Y ). If Fµ and Fν are the distribution functions of µ and ν, then ∞ |Fµ (x) − Fν (x)| d x. (4.16) dW (µ, ν) := −∞
The Kantorovich-Rubinstein theorem (e.g. [7, Ch.11]) provides an additional characterisation: dW (µ, ν) = sup ψd(µ − ν) (4.17) ψ: LipY [ψ]1 Y
for any µ, ν ∈ P(Y ). Here, LipY [ψ] := sup y,y ∈Y ;y= y ψ(y) − ψ(y ) / y − y for any ψ : Y → R (and analogously for functions on other domains). We now see that there is a constant K > 0 (the common Lipschitz bound for the functions y → w y (x) on Y , where x ∈ X ) such that w• dµ − w• dν K · dW (µ, ν). (4.18) Y
Y
L 1 (X,λ)
This means that, for densities from D , convergence of the representing measures implies L 1 -convergence of the densities. We will also use the following estimate.
252
J.-B. Bardet, G. Keller, R. Zweimüller
Lemma 5 (Continuity of (r, µ) →Lr∗ µ). There are constants κ1 , κ2 > 0 such that dW (Lr∗ µ, L∗s ν) κ1 dW (µ, ν) + κ2 |r − s| for all µ, ν ∈ P(Y ) and all r, s ∈ R. Proof. For Lipschitz functions ψ on Y we have Y
ψ d Lr∗ µ − L∗s ν LipY [(ψ ◦ σr ) pr + (ψ ◦ τr )(1 − pr )] · dW (µ, ν) + sup Lip R [ψ(σ. (y)) p. (y) + ψ(τ. (y))(1 − p. (y))] · |r − s|.
(4.19)
y∈Y
Suppose that ψ is C 1 , with ψ 1. Then the first Lipschitz constant is bounded by κ1 := sup (τr − σr ) · pr ∞ + σr · pr + τr · (1 − pr )∞ < ∞, r ∈R
and the second one by ∂σr ∂τr ∂ pr κ2 := sup < ∞, · pr + · (1 − pr ) + (τr − σr ) · ∂r ∂r ∂r ∞ r ∈R ∞ ∞ and the lemma follows from the Kantorovich-Rubinstein theorem (4.17), since these C 1 functions ψ uniformly approximate the Lipschitz functions appearing there. ∗ are uniformly Lipschitz-continuous on Corollary 2. The operators Lr∗ , r ∈ R, and L P(Y ) for the Wasserstein metric. ∗ , we recall Proof. The Lipschitz-continuity of Lr∗ is immediate from Lemma 5. For L ∗ µ = Lr∗ µ with rµ = G( w dµ) so that from (2.9) and (4.13) that L Y µ |rµ − rν | Lip(G) Lip(w) dW (µ, ν), which allows to conclude with Lemma 5.
Remark 4. Rigorous numerical bounds give κ1 0.5761 and κ2 0.5334. These estimates can be used to show that not only the individual Lr∗ are uniformly contracting on ∗ is a uniform contracP(Y ), but also (under suitable restrictions on the function G) L ∗ ∗ tion on P(Y ) where Y is a suitable neighbourhood of the support of µr∗ , and µr∗ is the representing measure of u r∗ with r∗ the unique positive fixed point of the equation ∗ , however, does not rely on these estimates, because r = G(φ(u r )). Our treatment of L it is based on monotonicity properties explained below.
Globally Coupled Maps with Bistable Thermodynamic Limit
253
4.3. The IFS: monotonicity. We first collect a few elementary facts about the order relation given by (4.15) (the symbols ≺, and will designate the usual variants of ): µ ν if and only if Y u dµ Y u dν for each bounded and (4.20) non-decreasing u : Y → R. In particular, if µ ν, then Y w dµ Y w dν, and hence rµ rν as well. If µ ν and if ρ1 , ρ2 : Y → Y are non-decreasing and such that ρ1 (y) ρ2 (y) for all y, then µ ◦ ρ1−1 ν ◦ ρ2−1 . If µ ν and if Y u dµ = Y u dν for some strictly increasing u : Y → R, then µ = ν. Let z ∈ Y . Then δz µ if and only if supp(µ) ⊆ [z, ∞).
(4.21) (4.22) (4.23)
We also observe that the representing measures µr of the Tr -invariant densities u r form a linearly ordered subset of P(Y ). Routine calculations based on (4.14) show that µr ≺ µs if r < s.
(4.24)
Our analysis of the asymptotic behaviour of the IFS will crucially depend on the fact ∗ respect this order relation on P(Y ), as made precise in the that the operators Lr∗ and L next lemma: Lemma 6 (Monotonicity of (r, µ) → Lr∗ µ). Let µ, ν ∈ P(Y ) and r, s ∈ R. a) If µ ≺ ν, then Lr∗ µ ≺ Lr∗ ν. b) If r < s, then Lr∗ µ ≺ L∗s µ. ∗ µ ≺ L ∗ ν. c) If µ ≺ ν, then L Proof. Let u : Y → R be bounded and non-decreasing and recall that ∗ u d(Lr µ) = [u ◦ σr · pr + u ◦ τr · (1 − pr )] dµ. Y
(4.25)
Y
a) Let µ ν. In view of (4.20) we can prove L∗ µ L∗ ν by showing that the integrand on the right-hand side of (4.25) is non-decreasing. For this we use the facts that σr and τr are strictly increasing with σr < τr , and that pr is non-increasing, since 4−r 2 pr (y) = − (4+r < 0. One gets then, for x
0
u(σr (x)) pr (y) + u(τr (x))( pr (x) − pr (y)) + u(τr (x))(1 − pr (x)) = u(σr (x)) pr (y) + u(τr (x))(1 − pr (y)) = u(σr (y)) pr (y) + u(τr (y))(1 − pr (y)) − [(u(σr (y)) − u(σr (x))) pr (y) + (u(τr (y)) − u(τr (x))) (1 − pr (y))] . 0
(4.26)
254
J.-B. Bardet, G. Keller, R. Zweimüller
Hence µ ν implies Lr∗ µ Lr∗ ν. Now, if u is strictly increasing, then (4.26) always is a strict inequality, i.e. the integrand on the right-hand side of (4.25) is strictly increasing. Therefore, µ ≺ ν implies Lr∗ µ ≺ Lr∗ ν by (4.22). b) We must show that (4.25) is non-decreasing as a function of r . To this end note first that y2 − 4 ∂ pr (y) = < 0, ∂r (r y + 4)2 ∂σr (y) 8 − 2y 2 = > 0, ∂r (r y + y + r + 4)2 ∂τr (y) 8 − 2y 2 = > 0. ∂r (r y − y − r + 4)2
(4.27) (4.28) (4.29)
Hence, if r < s, then u(σr (x)) pr (x) + u(τr (x))(1 − pr (x)) = u(σr (x)) ps (x) + u(σr (x)) ( pr (x) − ps (x)) +u(τr (x))(1 − pr (x)) >0
u(σs (x)) ps (x) + u(τs (x))( pr (x) − ps (x)) + u(τs (x))(1 − pr (x)) = u(σs (x)) ps (x) + u(τs (x))(1 − ps (x)), and for strictly increasing u we have indeed a strict inequality. c) This follows from a) and b): ∗ µ = Lr∗ µ ≺ Lr∗ ν Lr∗ ν = L ∗ ν L µ µ ν as rµ = G(
Y
w dµ) G(
Y
w dν) = rν by (4.20).
In Sect. 5.2 below, we will also make use of a more precise quantitative version of statement a). It is natural to state and prove it at this point. Lemma 7 (Quantifying the growth of µ → Lr∗ µ). Suppose that α, β > 0 are such that u α and τr , σr β. Then, for µ ν, Y
u d(Lr∗ ν) −
Y
u d(Lr∗ µ) α β
id dµ .
id dν − Y
(4.30)
Y
Proof. Observing that u(σr (y)) − u(σr (x)) and u(τr (y)) − u(τr (x)) are αβ(y − x), we find for the last expression in (4.26) that [(u(σr (y)) − u(σr (x))) pr (y) + (u(τr (y)) − u(τr (x))) (1 − pr (y))] αβy − αβx. This turns (4.26) into a chain of inequalities which shows that the function given by − pr (x)) − αβx is non-decreasing. Hence, by v(x) := u(σr (x)) pr (x) + u(τr (x))(1 (4.20), µ ν entails Y v dµ Y v dν, which is (4.30).
Globally Coupled Maps with Bistable Thermodynamic Limit
255
on D . We are now going to clarify 4.4. Dynamics of the IFS and the asymptotics of P ∗ the asymptotic behaviour of L on P(Y ). In view of (4.18), this also determines the on D , and hence proves Proposition 2. asymptotics of P Our argument depends on monotonicity properties which we can exploit since the topology of weak convergence on P(Y ), conveniently given by the Wasserstein metric, is consistent with the order relation introduced above. Indeed, one easily checks: If (νn ) and (¯νn ) are weakly convergent sequences in P(Y ) with νn ν¯ n for (4.31) all n, then lim νn lim ν¯ n . Recall from Sect. 2.5 that, in the bistable regime, r∗ is the unique positive fixed point of the equation r = G(φ(u r )). For convenience, we now let r∗ := 0 in the stable regime. iff r ∈ {0, ±r∗ }. By Then, in either case, u r with representing measure µr is fixed by P (4.24) we have µ−r∗ µ0 = δ0 µr∗ with strict inequalities in the bistable regime. ∗ µ L ∗2 µ . . ., ∗ µ, then µ L Lemma 8 (Convergence by monotonicity). If µ L ∗n µ)n 0 converges weakly to a measure µr µ with r ∈ {0, ±r∗ }. and the sequence (L The same holds for instead of . ∗n µ)n 0 follows immediately from Proof. The monotonicity of the sequence (L Lemma 6c). Because of (4.31), it implies that the sequence can have at most one ∗ therefore ensure weak accumulation point. Compactness of P(Y ) and continuity of L ∗n ∗ that (L µ)n 0 converges to a fixed point of L , i.e. to one of the measures µr with r ∈ {0, ±r∗ }, and (4.31) entails µr µ. The proof for decreasing sequences is the same. The following lemma strengthens the previous one considerably. It provides uniform control, in terms of the Wasserstein distance (4.16), on the asymptotics of large families of representing measures. Lemma 9 (Convergence by comparison). We have the following: a) In the stable regime, there exists a sequence (εn )n 0 of positive real numbers converging to zero such that ∗n µ, δ0 ) εn d W (L
for µ ∈ P(Y ) and n ∈ N.
b) In the bistable regime, for every y > 0 there exists a sequence (εn )n 0 of positive real numbers converging to zero such that ∗n µ, µr∗ ) εn d W (L
for µ ∈ P(Y ) with µ δ y and n ∈ N.
An analogous assertion holds for measures µ δ−y . Proof. As Y = [− 23 , 23 ], we trivially have δ−2/3 µ δ2/3 for all µ ∈ P(Y ). In ∗ δ2/3 δ2/3 , and Lemma 8 ensures that (L ∗n δ2/3 )n 0 converges. Due to particular, L ∗n ∗n µr∗ L δ2/3 for all n 0, showing, via (4.31), Lemma 6c), we have δ0 µr∗ = L ∗n δ2/3 = µr∗ . In the same way one proves that (L ∗n δ−2/3 )n 0 converges to that lim L µ−r∗ . For the stable regime this means that both sequences converge to δ0 = µ0 .
256
J.-B. Bardet, G. Keller, R. Zweimüller
a) Assume we are in the stable regime. By the above discussion, ∗n δ−2/3 , δ0 ) + dW (L ∗n δ2/3 , δ0 ) εn := dW (L ∗n δ−2/3 L ∗n µ L ∗n δ2/3 for all n 0. tends to zero. For any µ, (4.31) guarantees L ∗n µ, δ0 ) εn . Hence FL∗n δ−2/3 (y) FL∗n µ (y) FL∗n δ2/3 (y) for all y, proving dW (L b) Now consider the bistable regime. Note first that if there is a suitable sequence (εn )n 0 for some y > 0, then it also works for all y > y. Therefore, there is no loss of generality if we assume that y > 0 is so small that 1 G (0) σG(w(y)) (y) = + y + O(y 2 ) > y (4.32) 2 12 (use (4.10), (4.28) and (4.2) to see that this can be achieved). Since Lr∗ δ y = pr (y)δσr (y) + ∗ δ y , (1 − pr (y))δτr (y) and σr (y) < τr (y), we then have δ0 ≺ δ y ≺ L∗G(w(y)) δ y = L ∗n δ y )n 0 converges to µr∗ . In view of recall Lemma 6. Lemma 8 then implies that (L ∗n the initial discussion, (L δ2/3 )n 0 converges to µr∗ as well, so that ∗n δ y , µr∗ ) + dW (L ∗n δ2/3 , µr∗ ) εn := dW (L defines a sequence of reals converging to zero. Now take any µ ∈ P(Y ) with µ δ y , ∗n δ2/3 L ∗n µ L ∗n δ y for all n 0, and dW (L ∗n µ, µr∗ ) εn follows as in then L the proof of a) above. ∗n µ for any µ ∈ P(Y ) This observation enables us to determine the asymptotics of L which is completely supported on the positive half (0, 2/3] of Y (meaning that µ δ0 , cf. (4.23)), or on its negative half [−2/3, 0). Corollary 3. Let µ ∈ P(Y ). ∗n µ)n 0 converges to δ0 . a) In the stable regime, the sequence (L ∗n µ)n 0 converges to µr∗ . If b) In the bistable regime, if µ δ0 , then the sequence (L µ ≺ δ0 , it converges to µ−r∗ . Proof. a) follows immediately from Lemma 9a). We turn to b): Let r := Y w dµ. Then r > 0 because w > 0 on (0, 2/3] and µ δ0 . Therefore σr (0) > σ0 (0) =0. Fix some ∗ µ since σr and y as in Lemma 9b), w.l.o.g. y ∈ (0, σr (0)). Then δ0 ≺ δ y Lr∗ µ = L ∗n µ, µr∗ ) → 0 as n → ∞ by τr map supp(µ) into [σr (0), 2/3], so that indeed dW (L the lemma. ∗n µ)n 0 when none of It remains to investigate the convergence of sequences (L these measures can be compared (in the sense of ≺) to δ0 . To this end let [a0 , b0 ] := Y . Given a sequence of parameters r1 , r2 , . . . ∈ R define an := σrn ◦ . . . ◦ σr1 (a0 ) and bn := τrn ◦ . . . ◦ τr1 (b0 ) for n 1, and, for any µ = µ0 ∈ P(Y ) = P[a0 , b0 ], consider the measures µn := Lr∗n ◦ . . . ◦ Lr∗1 µ. Then supp(µn ) ⊆ supp(Lr∗n µn−1 ) ⊆ σrn ([an−1 , bn−1 ]) ∪ τrn ([an−1 , bn−1 ]) ⊆ [an , bn ] by induction. Write [a, b]ε := [a − ε, b + ε], where ε 0. The next lemma exploits the crucial observation that the two branches σr and τr have tangential contact at their common zero zr , see (4.7) and Fig. 2.
Globally Coupled Maps with Bistable Thermodynamic Limit
257
Lemma 10 (Support intervals close to zeroes). There exists some C ∈ (0, ∞) such that the following holds: Suppose that (rn )n 1 is any given sequence in R. If for some ε 0 and n(ε) ¯ 0 we have zrn+1 ∈ [an , bn ]ε for n n(ε), ¯
(♣ε )
then lim |bn − an | Cε2 ,
(4.33)
n→∞
lim max (|an | , |bn |)
n→∞
3 ε + Cε2 , 4
(4.34)
and, in case ε = 0, ¯ + 1. 0 ∈ [an , bn ] for n n(0)
(4.35)
Proof. Let ε 0 and assume (♣ε ). Note that, for n n¯ = n(ε), ¯ if an > zrn+1 , then 0 < an+1 < 3/4 · ε, if bn < zrn+1 , then − 3/4 · ε < bn+1 < 0, if an zrn+1 bn , then an+1 0 bn+1 . The first implication holds because 0 = σrn+1 (zrn+1 ) < σrn+1 (an ) = an+1 as σrn+1 increases strictly, and since by (♣ε ) we have an ∈ (zrn+1 , zrn+1 + ε], whence an+1 < ε · sup σrn+1 3ε/4 due to (4.9); analogously for the second implication. The third is immediate from monotonicity. Now, as σr and τr share a common zero zr , (4.9) ensures bn+m − an+m 43 (bn+m−1 − ¯ ¯ ¯ an+m−1 ) in case zr ∈ [an+m−1 , bn+m−1 ]. Otherwise, note that zr is ε-close to one of the ¯ ¯ ¯ endpoints, w.l.o.g. to an+m−1 . Since σr and τr are tangent at zr , there is some C > 0 s.t. ¯ 0 τrn+m (an+m−1 ) − σrn+m (an+m−1 ) C4 ε2 in this case, while (4.9) controls the rest of ¯ ¯ ¯ ¯ bn+m − a . In view of diam(Y) = 4/3, we thus obtain, for m 1, ¯ n+m ¯ bn+m − an+m ¯ ¯
3 C 4 (bn+m−1 − an+m−1 ) + ε2 . . . ¯ ¯ 4 4 3
m 3 + Cε2 . 4
Statement (4.33) follows immediately. For the asymptotic estimate (4.34) on max (|an | , |bn |) = max(−an , bn ), use the above inequality plus the observation that, by the first two implications stated in this proof, an+m and −bn+m never exceed 3ε/4. ¯ ¯ Finally, if ε = 0, (4.35) is straightforward from (♣ε ) and the third implication above. While the full strength of this lemma will only be required in the next subsection, the ε = 0 case enables us to now conclude the Proof of Proposition 2. The conclusions of (1) and (3) follow from Corollary 3. If neither of these two cases applies, then the assumption of (2) must be satisfied, and so condition (♣0 ) of Lemma 10 is satisfied with n(0) ¯ = 0. Hence limn→∞ max(|an |, |bn |) = 0 by ∗n µ are supported in [an , bn ], these measures must converge to δ0 . (4.34). As the L
258
J.-B. Bardet, G. Keller, R. Zweimüller
5. Proofs: The Self-Consistent PFO for the Infinite-Size System on D. We are now going to clarify 5.1. Shadowing densities and the asymptotics of P the asymptotics of the self-consistent PFO on the set D of all densities, proving on D). For every u ∈ D, the sequence Proposition 3 (Long-term behaviour of P n u)n 0 converges in L 1 (X, λ), and (P
= u0 in the stable regime, n u lim P n→∞ ∈ {u −r∗ , u 0 , u r∗ } in the bistable regime. n u = u ±r∗ } of the stable fixed points u ±r∗ are L 1 -open. The basins {u ∈ D : limn→∞ P (The set of densities attracted to u 0 in the bistable regime will be discussed in Sect. 5.2 below.) We begin with some notational preparations. Throughout, we fix some u ∈ D. The n−1 u)) (n 1). With this notation, P n u = n u define parameters rn := G(φ( P iterates P Prn . . . Pr1 u. We let π N , N 1, denote the partition of X into monotonicity intervals of Tr N ◦ . . . ◦ Tr1 . Note that each branch of this map is a fractional linear bijection from a member of π N onto X . Since the Tr , r ∈ R, have a common uniform expansion rate, we see that diam(π N ) → 0, and hence, by the standard martingale convergence theorem, E[u σ (π N )] → u in L 1 (X, λ), that is, η N := E[u σ (π N )] − u L 1 (X,λ) −→ 0
as N → ∞.
(5.1)
Write vk(N ) := Pr N +k . . . Pr1 (E[u σ (π N )]) for k 0 and N 1, (N )
and observe that vk ∈ D because it is a weighted sum of images of the constant function 1 under various fractional linear branches (recall (4.1) and (4.8)). (0) For N = 0 we let v0 := u r∗ and write, in analogy to the notation introduced for (0) (0) (0) (0) N 1, vk := Prk . . . Pr1 (v0 ) and η0 := v0 − u L 1 (X,λ) . Obviously, vk ∈ D for all k 0. (N ) (N ) (N ) Hence there are measures µk ∈ P(Y ) such that vk = Y w• dµk . Observe also that (N )
N +k u − v L 1 (X,λ) η N for all k 0 P k
and N 0,
(5.2)
as Pr = 1 for all r , so that in particular N +k u)| η N , |G(φ(v (N ) )) − r N +k+1 | G · η N . |φ(vk(N ) ) − φ( P k ∞ In addition, we need to understand the distances (N ) ,k) n v (N ) L 1 (X,λ) (N := vn+k − P n k
which, in fact, admit some control which is uniform in k:
(5.3)
Globally Coupled Maps with Bistable Thermodynamic Limit
259
Lemma 11 (Shadowing control). There is a non-decreasing sequence (n )n 0 in (0, ∞), not depending on u ∈ D, such that ,k) (N η N · n n (N ,k)
Proof. Let rn
for k, n 0 and N 0.
(5.4)
(N )
n−1 v )), and observe that (4.18) entails := G(φ( P k
,k) (N = Pr N +n+k . . . Pr N +1+k vk(N ) − Pr (N ,k) . . . Pr (N ,k) vk(N ) L 1 (X,λ) n n
K
· dW (Lr∗N +n+k
1
(N ) . . . Lr∗N +1+k µk , L∗(N ,k) r n
(N )
. . . L∗(N ,k) µk ). r1
Applying Lemma 5 repeatedly, we therefore see that ,k) K κ2 (N n
= K κ2
n−1 i=0 n−1
(N ,k)
κ1i |r N +n+k−i − rn−i | N +n+k−i−1 u)) − G(φ( P n−i−1 v (N ) ))| κ1i |G(φ( P k
i=0
K G ∞ κ2 K G ∞ κ2
n−1 i=0 n−1
N +n+k−i−1 u − P n−i−1 v (N ) L 1 (X,λ) κ1i P k (N ,k) κ1i η N + n−i−1 ,
i=0
does not contract on L 1 (X, λ), whence where the last inequality uses (5.2). (Recall that P
n−1 i (N ,k) ,k) (N the need for the n−i−1 -term.) Letting K n := 1 + K G ∞ κ2 i=0 κ1 and := n (N ,k)
max{i
: i = 0, . . . , n − 1}, we thus obtain ,k) (N (N ,k) ) . . . η N · n K nn , K n · (η N + n n−1
which proves our assertion.
(5.5)
We can now complete the Proof of Proposition 3. We begin with the easiest situation: n u − u 0 L 1 (X,λ) = 0. Take any The stable regime. We have to show that limn→∞ P ε > 0. Let (εn )n 0 be the sequence provided by Lemma 9a), and K the constant from (4.18). There is some n (henceforth fixed) for which K εn < ε/3. In view of (5.1), there is some N0 such that (1 + n )η N < 2ε/3 whenever N N0 . We then find, using (5.2), Lemma 11, and (4.18) together with Lemma 9a) that n v (N ) L 1 (X,λ) + K εn N +n u − u 0 L 1 (X,λ) η N + vn(N ) − P P 0 η N + n η N + K εn < ε for N N0 , which completes the proof in this case.
(5.6)
260
J.-B. Bardet, G. Keller, R. Zweimüller
n−1 u)) as before, we let [an , bn ] ⊆ The bistable regime. Given the sequence rn = G(φ( P Y be the sequence of parameter intervals from Lemma 10. Observe that the measures ) representing the vn(N ) satisfy supp(µ(N n ) ⊆ [an , bn ] for all n and N . We now distinguish two cases: First case. For all ε > 0 we have (♣ε ) from Lemma 10. Then, for any ε > 0, the lemma ensures that there is some n (henceforth fixed) with max (|an | , |bn |) < ε/4K , so that (N ) also dW (µn , δ0 ) < ε/2K , whatever N . Due to (5.1), η N < ε/2 for N N0 , and we find, using (5.2) and (4.18), N +n u − u 0 L 1 (X,λ) η N + vn(N ) − u 0 L 1 (X,λ) P ) η N + K dW (µ(N n , δ0 ) < ε for N N0 ,
n u → u 0 . showing that indeed P Second case. There is some ε > 0 s.t. (♣ε ) is violated in that, say, zrn < an−1 − ε
(5.7)
n u → u r∗ . (If (♣ε ) is violated in the for infinitely many n. We show that this implies P n u → u −r∗ then follows by symmetry.) other direction, P ) In view of (5.3), and since (due to µ(N δa N +k , (4.20), and (4.11)) φ(vk(N ) ) k φ(wa N +k ) = w(a N +k ), we have (N )
N +k u)) G(φ(v ) − η N ) r N +k+1 = G(φ( P k (N ) G(φ(vk )) − G ∞ η N G(w(a N +k )) − G ∞ η N , r and hence, observing that ∂σ σ (y) := σG(w(y)) (y) for y ∈ Y , ∂r ∞ 1 and writing σ (a N +k ) − G ∞ η N (5.8) a N +k+1 = σr N +k+1 (a N +k )
∂ for all N and k. Note that σ (0) = σ0 (0)+ ∂r σr (0)|r =0 ·G (0)·w (0) = 21 + 21 ·G (0)· 16 > 1, see (4.28) and (4.10). Therefore, if we fix some ω ∈ (1, σ (0)), there exists some a ∗ > 0 ∗ such that σ (a) ωa forall a ∈ (0, a ]. Without loss of generality, ε/3 < a ∗ . Now fix N such that G ∞ η N < (ω − 1) ε/3, and let N + n + 1 satisfy (5.7). Due to (4.7), we have
a N +n+1 = σr N +n+1 (a N +n ) > σr N +n+1 (zr N +n+1 + ε) > ε/3. Now, if a N +n+1 a ∗ , then, by (5.8),
σ (a N +n+1 ) − G ∞ η N a N +n+2
σ (a ∗ ) − (ω − 1)ε/3 ωa ∗ − (ω − 1)a ∗ = a ∗ > ε/3.
Otherwise, a N +n+1 ∈ (0, a ∗ ), and again
σ (a N +n+1 ) − G ∞ η N a N +n+2 > ωε/3 − (ω − 1) ε/3 = ε/3.
(5.9)
Globally Coupled Maps with Bistable Thermodynamic Limit
261
It follows inductively that lim inf k ak ε/3. More precisely: If N1 and n 1 are integers −1 such that η N1 < := G ∞ (ω − 1) ε/3 and a N1 +n 1 > zr N1 +n1 +1 + ε, then ak > ε/3 for k > N1 + n 1 . In particular, if the initial density u is such that η0 = u r∗ − u L 1 (X,λ) < , then we can take N1 = 0. Next, fix y := ε/6 ∈ (0, ε/3), and choose a sequence (εn )n 0 according to Lemma 9b. (N ) Then 0 < y < ak and hence δ0 ≺ δ y µk for k > N1 + n 1 so that the lemma implies ∗n µ(N ) , µr∗ ) εn . Hence, by (4.18), d W (L k (N )
n v P k
− u r∗ L 1 (X,λ) K · εn for k > N1 + n 1
and all n, N .
We then find, using (5.2) and Lemma 11, n v (N ) L 1 (X,λ) + K · εn N +k+n u − u r∗ L 1 (X,λ) η N + v (N ) − P P k+n k η N + n η N + K · εn
(5.10)
n u − u r∗ L 1 (X,λ) = 0 follows as in the for k > N1 + n 1 and all n, N . Now limn→∞ P stable case. It remains to prove that the basin of attraction of u r∗ is L 1 -open. (Then, by symmetry, is L 1 -continuous, it suffices to show that this basin the same is true for u −r∗ .) As P contains an open L 1 -ball centered at u r∗ . To check the latter condition, first notice that zr∗ < 0 < supp(µr∗ ) so that there is some n 1 > 0 such that σrn∗1 (a0 ) > 0. As we can is L 1 -continuous, assume w.l.o.g. that ε < |zr∗ |, we have σrn∗1 (a0 ) > zr∗ + ε, and as P there is some ∈ (0, ) such that arn1 = σrn1 ◦ · · · ◦ σr1 (a0 ) > zrn1 + ε whenever u − u r∗ L 1 (X,λ) < . Therefore we can continue to argue as in the previous paran u − u r∗ L 1 (X,λ) graph (using the present n 1 and N1 = 0) to conclude that limn→∞ P = 0. Remark 5. We just proved a bit more than what is claimed in Proposition 3: another look is even at Eq. (5.10) reveals that, in the bistable regime, the stable fixed point u r∗ of P Lyapunov-stable (and the same is true for u −r∗ ). Indeed, fix ε > 0, n 1 ∈ N and > 0 as in the preceding paragraph. That choice was completely independent of the particular initial densities investigated there, and the same is true of the choice of the constants K , n and εn occuring in estimate (5.10). Now let δ > 0. Choose n 2 ∈ N such that δ δ εn 2 < 2K and then η := min{, 2(1+ }. Then Eq. (5.10), applied with N = 0, shows n2 ) that for each u ∈ L 1 (X, λ) with η0 = u − u r∗ L 1 (X,λ) < η and for each n 0, n 1 +n 2 +n u − u r∗ L 1 (X,λ) η0 (1 + n 2 ) + K εn 2 < δ. P n u → 5.2. The stable manifold of u 0 in the bistable regime. Let W s (u 0 ) := {u ∈ D : P u 0 } denote the stable manifold of u 0 in the space of all probability densities on X . Clearly, all symmetric densities u (i.e. those satisfying u(−x) = u(x)) belong to W s (u 0 ), because symmetric densities have field φ(u) = 0 so that also the parameter G(φ(u)) = 0, and symmetry is preserved under the operator P0 . However, W s (u 0 ) is not a big set. In the present section we prove
262
J.-B. Bardet, G. Keller, R. Zweimüller
Proposition 4 (The basins of u ±r∗ touch W s (u 0 ) ∩ D ). Each density in W s (u 0 ) ∩ D belongs to the boundaries of the basins of u r∗ and of u −r∗ . n We start by providing on the fields φ( orbits in W s (u 0 )∩D . more information P u) of ∗n n Recall that for u = Y w• dµ ∈ D we have P u = Y w• d(L µ) (n 0). Given ∗n µ, i.e. Rn (u) := such a density, we denote by Rn (u) the “radius” of the support of L ∗n µ). ∗n µ) ⊆ [−ε, ε]}, and let φn (u) := φ( P n u) = w d(L inf{ε > 0 : supp(L Y Lemma 12 (Field versus support radius). In the bistable regime, for each u ∈ W s (u 0 ) ∩ D there exists a constant Cu > 0 such that |φn (u)| Cu · (Rn (u))2 for n 0. w (0)
(5.11) w (0)
Proof. In view of the explicit formula (4.10), we have = and = 0, and therefore see that there is some ε ∈ (0, 13 ) such that for every ε ∈ (0, ε) and all y ∈ [−2ε, 2ε], |y| |y| |w(y)| and |G(y)| > (B − cε)|y|, (5.12) 6 6 − 6ε2 where B := G (0) > 6 and c, too, is a positive constant which only depends on the function G. In addition, elementary calculations based on (4.2) and (4.6) show that letting κ := max(1, B+2 6 ), ε can be chosen such that, for every ε ∈ (0, ε) and r ∈ [0, Bε), also 1 1 |σr (y) − | Bε, |τr (y) − | Bε for |y| ε, 2 2 1 − κε (y + r ) 0 for y ∈ [−r, ε], and Bε τr (y) σr (y) 2 1 + ε (y + r ) > −Bε for y ∈ [−ε, −r ). (5.13) 0 > τr (y) σr (y) 2 1 6
(Recall that σr and τr share at zr = −r .) a zero Finally, note that we can w.l.o.g. take ε 1 B 1 c ¯ so small that B := 2 − ε 1 + 6 − ( 3 + 6 )ε ∈ (1, 3]. (Due to Assumption I we have B 25.) Consider some v = Y w• dν with ν ∈ P(Y ). We claim that for ε ∈ (0, κε ), B¯ · |φ(v)| − ε2 if supp(ν) ⊆ [−ε, ε]. |φ( Pv)|
(5.14)
Denote r := G(φ(v)) which by S-shapedness of G satisfies |r | < Bε. In view of our system’s symmetry, we may assume w.l.o.g. that r 0. According to (4.11) and (4.13) we have ∗ φ( Pv) = φ w• d(L ν) = (w ◦ σr ) · pr dν + (w ◦ τr ) · (1 − pr ) dν Y
Y
Y
so that, due to (5.12) and (5.13), (w ◦ σr ) · pr dν Y 1[−Bε,0) ◦ σr (y) 1[0,Bε] ◦ σr (y) · σr (y) pr (y) dν(y) + 6 − 6ε2 6 Y 1 2 (y) 1[− 2 ,−r ) (y) 1 1 [−r, 3 ] 3 + ε + − κε · (y + r ) pr (y) dν(y). 6 − 6ε2 2 6 2 Y
Globally Coupled Maps with Bistable Thermodynamic Limit
Combining this with the parallel estimate for φ( Pv)
1
2 [− 23 ,−r )
263
Y (w
◦ τr ) · (1 − pr ) dν, we get
1 + ε (y + r ) 2 − κε (y + r ) dν(y) + dν(y). 6 − 6ε2 6 [−r, 23 ]
Continuing, we find that 1 1 +ε 2 − κε 2 − κε · y dν(y) + ·r · y dν(y) + 2 6 6 [− 23 ,−r ) 6 − 6ε [−r, 23 ] 1 1 1 2 +ε 2 − κε 2 − κε · y dν(y) + ·r · y dν(y) + 2 6 6 [− 23 ,0) 6 − 6ε [0, 23 ] 1 − κε · r, K · w(y) dν(y) + K ∗ · w(y) dν(y) + 2 6 [− 23 ,0) [0, 23 ]
φ( Pv)
1 2
where K := ( 21 + ε)/(1 − ε2 ) > K ∗ := ( 21 − κε)(1 − ε2 ). As, because of (5.12), φ(v) = Y w dν 5ε , so that r = G (φ (v)) (B − cε) · φ (v), we conclude
( 21 − κε)(B − cε) 6 1 B 1 −ε + 1+ − φ(v) 2 6 3
φ(v) K ∗ + φ( Pv)
∗
+ (K − K )
[− 23 ,0)
w(y) dν(y)
c ε − ε2 = B¯ · φ(v) − ε2 , 6
since K − K ∗ 3ε and |w(y)| 3ε whenever |y| ε 13 . This proves (5.14). Now take any u ∈ W s (u 0 ) ∩ D . Then φn (u) → 0, and the second alternative of ¯ Proposition 2 applies, so that Rn (u) ε/κ and (1 + 2B Rn (u))2 B+1 2 for all n larger than some n ε . In particular, Rn+1 (u)2 (1 + 2B Rn (u))2 Rn (u)2
B¯ + 1 Rn (u)2 2
(5.15)
n u and for these n in view of (5.13). Applying, for n n ε , the estimate (5.14) to v := P ε := Rn (u), we obtain |φn+1 (u)| B¯ · |φn (u)| − (Rn (u))2 for n n ε . ¯ B−1 2 |φn (u)| for some n > n ε . Then ¯ ¯ ¯ B+1 B−1 2 2 (Rn+1 (u)) B+1 2 (Rn (u)) < 2 2 |φn (u)|
Suppose for a contradiction that (Rn (u))2 < |φn+1 (u)| >
¯ B+1 2 |φn (u)|,
¯ B−1 2 |φn+1 (u)|.
and therefore
< We can thus continue inductively to see that |φn (u)| < |φn+1 (u)| < 2 |φn+2 (u)| < . . . which contradicts φn (u) → 0. Therefore |φn (u)| B−1 (Rn (u))2 for ¯ all n > n ε , and the assertion of our lemma follows. Lemma 13 (W s (u 0) is a thin set for the order ≺). In the bistable regime, if u = Y w• dµ and v = Y w• dν are densities in D with µ ≺ ν, then at most one of u and s v can belong to W (u 0 ).
264
J.-B. Bardet, G. Keller, R. Zweimüller
∗n ν → n v → u r∗ , i.e. L Proof. Suppose that u ∈ W s (u 0 ). We are going to show that P µr∗ as n → ∞. Assume for a contradiction that also v ∈ W s (u 0 ). We denote the parameters obtained ∗(n−1) µ)), and define rn,ν analogously. n−1 u)) = G( w d(L from u by rn,µ := G(φ( P Y Then our assumption implies that limn→∞ rn,µ = limn→∞ rn,ν = 0. 1 In view of (4.10), w 16 , and one checks immediately that inf Y σ0 = 18 49 > 3 1 so that there is n 0 > 0 such that inf Y σrn,µ 3 for all n n 0 . Because of the strict ∗ (Lemma 6) we have L ∗n 0 µ ≺ L ∗n 0 ν, so that (replacing µ and ν by monotonicity of L ∗(n)
these iterates) we can assume w.l.o.g. that n 0 = 0. Denote Lµ := Lr∗n,µ ◦ · · · ◦ Lr∗1,µ ∗n µ = L∗(n) ∗n ν L∗(n) so that L µ µ and (µ → rµ being non-decreasing) L µ ν for n 1. Therefore rn,ν − rn,µ G Y
w d(L∗(n) µ ν)
inf G · X
Y
−G
w d(L∗(n) µ ν) −
Y
Y
w d(L∗(n) µ µ) w d(L∗(n) µ µ)
.
In view of the lower bounds for w and σrn,µ , τrn,µ , repeated application of the estimate (4.30) from Lemma 7 yields rn,ν − rn,µ
inf X G 6 · 3n
id d(ν − µ).
(5.16)
Y
Observe that the last integral is strictly positive because µ ≺ ν, cf. (4.22). On the other hand, due to Proposition 2 there are εn 0 such that ∗n ν) ⊆ [−εn , εn ], ∗n µ) ∪ supp(L supp(L and as σ0 (0) = 21 < 59 and rn,µ , rn,ν → 0 (whence also zrn,µ , zrn,ν → z 0 = 0), there exists a constant C > 0 such that εn C( 59 )n for n n . Hence |φn (u)|, |φn (v)| n max{Cu , Cv } · C 2 ( 25 81 ) for n n by Lemma 12, and as rn,ν −rn,µ sup w · (|φn (u)| + |φn (v)|), this contradicts the previous estimate (5.16). We can now conclude this section with the Proof of Proposition 4. Suppose that u = w• dµ ∈ W s (u 0 ). For t ∈ (0, 1) let u (t) := (t) u, hence u (t) ∈ W s (u ) by the previous 0 Y w• d((1 − t)µ + tδ2/3 ) ∈ D . Then u ∗ , for any t, P n u (t) proposition. Therefore, due to Proposition 2 and monotonicity of L converges to u r∗ u 0 as n → ∞. On the other hand, limt→0 u − u (t) L 1 (X,λ) = 0, so u is in the boundary of the basin of u r∗ . Replacing δ2/3 by δ−2/3 yields the corresponding result for the basin of u −r∗ .
Globally Coupled Maps with Bistable Thermodynamic Limit
265
at C 2 -densities. As P is based on a parametrised family of 5.3. Differentiability of P PFOs where the branches of the underlying map (and not only their weights) depend on the parameter, it is nowhere differentiable, neither as an operator on L 1 (X, λ) nor as an operator on the space BV(X ) of (much more regular) functions of bounded variation on X . On the other hand, as the branches of the map and their parametric dependence are is differentiable as an operator on the space of functions analytic, one can show that P that can be extended holomorphically to some complex neighbourhood of X ⊆ C. Here we will focus on a more general but slightly weaker differentiability statement. at C 2 -densities). Let u ∈ C 2 (X ) be a probability Lemma 14 (Differentiability of P density w.r.t. λ and let g ∈ L 1 (X, λ) have X g dλ = 0. Then ∂ (5.17) P(u + τ g)|τ =0 = Pr (g) + wr (u) · G (φ(u)) φ(g), ∂τ 2 −1 as . If we consider P where r = G(φ(u)), wr (u) := Pr (u vr ) , and vr (x) = 4x 4−r 2 an operator from BV(X ) to L 1 (X, λ), then P is even differentiable at each probability density u ∈ C 2 (X ) ⊂ BV(X ) and u = Pr + G (φ(u)) wr (u) ⊗ φ. D P|
(5.18)
Proof. In order to simplify the notation define a kind of transfer operator L by Lu := u + u ◦ f (1 1) and note that (Lu) = Lu . Observing that f Nr−1 = f Mr−1 ◦ f (1 1) , we have 01
Pr u = L(u ◦ f Mr−1 · f
Mr−1
01
). Define
vr (x) :=
∂ 4x 2 − 1 f Mr−1 ( f Mr (x)) = . ∂r 4 − r2
For a function u ∈ C 2 (X ) denote by U the antiderivative of u. Then u ◦ f Ms−1 · f M −1 − u ◦ f M −1 · f −1 = U ◦ f M −1 − U ◦ f M −1 r s r Mr s ∂ = (s − r ) · (U ◦ f Mr−1 ) + Rs,r , ∂r where
Rs,r (x) := r
As
∂ ∂r (U
s
(s − t)
∂2 (U ( f M −1 (x))) dt. t ∂t 2
∂ ∂r
◦ f Mr−1 ) = u ◦ f Mr−1 · f Mr−1 = (u vr ) ◦ f Mr−1 , we have ∂ (U ◦ f Mr−1 ) = (u vr ) ◦ f Mr−1 · f M −1 . r ∂r
Together with (5.20) this yields Ps u − Pr u = L u ◦ f Ms−1 · f M −1 − u ◦ f M −1 · f −1 r M s r = (s − r ) L (u vr ) ◦ f Mr−1 · f M −1 + L Rs,r r = (s − r ) Pr (u vr ) + L Rs,r
(5.19)
(5.20)
266
J.-B. Bardet, G. Keller, R. Zweimüller
)(x)| C (s −r )2 with a constant that involves only the first two derivatives and |L(Rs,r of u. 2 Now let u ∈ C (X ) be a probability density, and let g ∈ L 1 (X, λ) be such that g dλ = 0. Let r := G(φ(u)) and s := G(φ(u + g)). Then
+ g) − P(u) P(u = (Ps u − Pr u) + Pr g + (Ps g − Pr g) = (s − r ) Pr (u vr ) + Pr (g) + (Ps g − Pr g) + L(Rs,r ).
(5.21)
This implies at once formula (5.17) for the directional derivative, and as Ps g− Pr g1 → 0 (s → r ) uniformly for g in the unit ball of BV(X ), also (5.18) follows at once. In the bistable regime, u ≡ 1 Proposition 5 (u ≡ 1 is a hyperbolic fixed point of P) D∩BV(X ) in the following sense: the derivative of P : is a hyperbolic fixed point of P| D ∩ BV(X ) → L 1 (X, λ) at u ≡ 1 has a one-dimensional unstable subspace and a codimension 1 stable subspace. u≡1 . As G (0) = B and w0 (1) = P0 [2x] = [x], it follows Proof. Let Q := D P| from (5.18) that Q = P0 + B [x] ⊗ φ. (Here [2x] denotes the function x → 2x, etc.) 1 B B Observe now that φ([x]) = 12 . Then Q[x] = P0 [x] + 12 [x] = ( 21 + 12 )[x] so that, B for B > 6, Q has the unstable eigendirection [x] with eigenvalue λ := 21 + 12 > 1. On the other hand, as φ(1) = 0, we have Q1 = P0 1 = 1, so the constant density 1 is a neutral eigendirection, and finally, for f ∈ ker(φ) ∩ ker(λ), we have Q f = P0 f , so Var(Q f ) 21 Var( f ). 6. The Noisy System In Theorem 3 we proved that, in the bistable regime, each weak accumulation point of −1 the sequence (µ N ◦ N ) N 1 is of the form α δu −r∗ λ + (1 − 2α) δu 0 λ + α δur∗ λ for some 1 α ∈ [0, 2 ], i.e. that the stationary states of the finite-size systems approach a mixture of the stationary states of the infinite-size system. It is natural to expect that actually α = 21 , meaning that any limit state thus obtained is a mixture of stable stationary states While we could not prove this for the model discussed so far, we now argue that of P. this conjecture can be verified if we add some noise to the systems. At each step of the dynamics we perturb the parameter of the single-site maps by a small amount. To make this idea more precise, let r (Q, t) = G(φ(Q) + t) for Q ∈ P(X ) and t ∈ R, in particular r (x, t) = G(φ(x) + t) for x ∈ X N and t ∈ R.
(6.1)
Let η1 , η2 , . . . be i.i.d. symmetric real valued random variables with common distribution and |ηn | ε. For n = 1, 2, . . . and x ∈ X N let us define the X N -valued Markov process (ξn )n∈N by ξ0 = x and ξn+1 = Tr (ξn ,ηn+1 ) (ξn ).
(6.2)
Assume now that the distribution of ξn has density h n w.r.t. Lebesgue measure on X N . Then routine calculations show that the distribution of ξn+1 has density R PN ,t h n d(t), where PN ,t is the PFO of the map T N ,t : X N → X N , (T N ,t (x))i = Tr (x,t) (xi ). It is straightforward to check that, for sufficiently small ε, Lemmas 2–4 from Sect. 3 carry
Globally Coupled Maps with Bistable Thermodynamic Limit
267
over to all T N ,t (|t| ε) with uniform bounds, and that X N |PN ,t f − PN ,0 f |dλ N const N · ε · Var( f ) so that the perturbation theorem of [12] guarantees that the process (ξn )n∈N has a unique stationary probability µ N ,ε whose density w.r.t. λ N tends, in L 1 (X N , λ N ), to the unique invariant density of T N as ε → 0. This convergence is not uniform in N , however. Nevertheless, folklore arguments show that there is some ε > 0 such that, for all ε ∈ (0, ε) and all N ∈ N the absolutely continuous stationary measure µ N ,ε is unique so that the symmetry properties of the maps Tr and the random variables ηn guarantee that µ N ,ε is symmetric in the sense that its density h N ,ε satisfies h N ,ε (x) = h N ,ε (−x). On the other hand, for each fixed ε > 0, all weak limit points of the measures −1 as N → ∞ are stationary probabilities for the P(X )-valued Markov process µ N ,ε ◦ N (n )n∈N defined by n+1 = n ◦ Tr−1 (n ,ηn+1 ) ;
(6.3)
: P(X ) → P(X ) in (2.9). The proof is completely analocompare the definition of T gous to the corresponding one for the unperturbed case (see Lemma 1 and Corollary 1). For ε ∈ (0, ε) the symmetry of the µ N ,ε carries over to these limit measures Q in the sense that Q(A) = Q{µˆ : µ ∈ A} for each Borel measurable set A ⊆ P(X ), where µ(U ˆ ) := µ(−U ) for all Borel subsets U ⊆ X . The following proposition then shows that, in the bistable regime and for small ε > 0 and large N , the measures µ N ,ε are weakly close to the mixture 21 (u −r∗ λ)N + (u r∗ λ)N compare also Theorem 3. of the stable states for P; Proposition 6 (Invariant measures for infinite-size noisy systems). Suppose G (0) > 6 so that we are in the bistable regime and recall that the ηn are symmetric random variables. Then, for every δ > 0 there is ε0 > 0 such that for each ε ∈ (0, ε0 ) the stationary distribution Q ε of n on P(X ) is supported on the set of measures u · λ ∈ P(X ) which have density u = Y w• dµ ∈ D with representing measures µ ∈ P(Y ) satisfying dW (µ, 21 (µ−r∗ + µr∗ )) δ. Sketch of the proof. Let Q be a stationary distribution of n that occurs as a weak limit of the measures µ N ,ε . So Q is symmetric. Just as in the proof of Theorem 3, where the is treated, one argues that Q is supported “zero noise limit”, namely the transformation T by the set of measures u · λ, u ∈ D. Arguing as in the derivation of (5.2) one shows that densities in the support of Q can be approximated in L 1 (X, λ) by densities from D , and the stationarity of Q implies that Q is indeed supported by measures with densities ∗ε from D . Therefore the process (n )n 0 can be described by the transfer operator L of an iterated function system on Y just as the self-consistent PFO P is described by ∗ in Eq. (4.13). The only difference is that in this case one first chooses the operator L the parameter r randomly, r = G( Y w dµ + ηn+1 ) and then the branch σr or τr with respective probabilities pr (y) and (1 − pr (y)). Let y > 0 be such that P{ηn > y} > 0. Suppose now that for some realisation of the process (n )n 0 the numbers r (n , ηn+1 ) satisfy condition (♣ε ) of Lemma 10 for all ε > 0. Then it follows, as in the proof of Proposition 2, that limn→∞ rn (n , ηn+1 ) = 0 and the measures n converge weakly to λ so that also limn→∞ r (n , 0) = 0. As ηn > y > 0 for infinitely many n almost surely, both limits cannot be zero at the same
268
J.-B. Bardet, G. Keller, R. Zweimüller
time, and we conclude that almost surely there is some ε > 0 such that (♣ε ) is not satisfied. In particular, there are ε¯ > 0 and n¯ ∈ N such that (♣ε¯ ) is violated for n = n¯ − 1 with some positive probabilityκ. Let n = h n · λ with h n = Y w• dνn . (So h n and νn are random objects.) As in (5.9) we conclude that sup supp(νn¯ ) < −¯ε /3 or inf supp(νn¯ ) > ε¯ /3 in this case. Without loss of generality we assume that the latter happens with probability at least κ2 . Next, as in (4.32) we may choose y ∈ (0, ε¯ /3) so small that 0 < y < y1 := ∗ δ y ). Hence, for reasons of continuity, there is ε1 > 0 such σG(w(y)) (y) inf supp(L that also y inf supp(L∗ε δ y ) if ε ∈ [0, ε1 ). Therefore, in view of the monotonicity of ∗ε , we can conclude that inf supp(νn ) y for all n n¯ with probability the operator L κ at least 2 . Now fix δ > 0. By Lemma 9b there is some (non-random) n 1 ∈ N such that ∗n 1 νn , µr∗ ) δ for all n n¯ with probability at least κ . But then, by continuity d W (L 2 2 reasons again, there is ε0 ∈ (0, ε1 ) such that dW (νn+n 1 , µr∗ ) < δ for all n n¯ with probability at least κ2 . The claim of the proposition follows now, because (n )n is a Markov process and because the stationary distribution Q is symmetric. A. Some Technical and Numerical Results A.1. Proof of Lemma 1. It suffices to prove the convergence for evaluations of any Lipschitz continuous function ϕ defined on X . Let us denote rn = r (Q n ) (resp. r = r (Q)), and αn (resp. α) the discontinuity point of Trn (resp. Tr ). Recall that αn = − r4n (resp. α = − r4 ). Let us fix ε > 0. Q being non-atomic, there exists δ > 0 such that the interval U := [α − δ, α + δ] is of Q-measure smaller than ε. The weak convergence of Q n to Q implies that αn tends to α, and that lim supn→+∞ Q n (U ) Q(U ). Let us choose n 0 such that for all n n 0 , |αn − α| < 2δ and Q n (U ) < ε. One then has ϕ d(T Q) − Q n ) = ϕ ◦ Tr d Q − ϕ d( T ϕ ◦ T d Q rn n X X X X ϕ ◦ Tr d(Q − Q n ) + (ϕ ◦ Trn − ϕ ◦ Tr ) d Q n Uc X + (ϕ ◦ Trn − ϕ ◦ Tr ) d Q n U ϕ ◦ Tr d(Q − Q n ) + Lip(ϕ) sup |Trn − Tr | + 2εϕ∞ . (A.1) Uc
X
Since the application ϕ ◦ Tr has a single discontinuity point, which is of zero Q-measure, the first term converges to zero. The second one also goes to zero since it measures the dependence of Tr on its parameter away from the discontinuity point (one can make an explicit computation). A.2. The fields of the densities u r . We start with some observations on the function ψ(r ) := φ(u r ) that are based on symbolic computations and on numerical evaluations. One finds 4+4 r −3 r 2 1 log 4−4 r −3 r 2 r 7 r 3 461 r 5 4619 r 7 = + ψ(r ) = + + + + ··· . (A.2) 2 r 6 40 2016 13440 log 4−9 r 4−r 2
Globally Coupled Maps with Bistable Thermodynamic Limit
0,08
psi(r) r/6
0,04
Q(r) 189/5*r
14 12
269
0,4 0,2
H(r) r
10 8
0
0
6 −0,04
−0,2
4 2
−0,08 −0,4
−0,2
0
0,2
0,4
0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
r
−0,4 −0,4
−0,2
r
0
0,2
0,4
r
Fig. 3. The functions ψ(r ) := φ(u r ) (left), Q(r ) := ψ (r )2 (centre), and H (r ) = A tanh( BA φ(u r )) with
(ψ (r ))
A = 0.4 and B = 8 (right)
From this numerical evidence (see Fig. 3 for a plot) it is clear that, for r ∈ [0, 0.4], r , and 6 ψ (r ) 189 r 12862 r 3 44487 r 5 346403009 r 7 189 = − + − + ··· r. (ψ (r ))2 5 175 500 4042500 5 ψ(r )
Hence H (r ) = G (ψ(r )) ψ (r ) G ( r6 ) ψ (r ). As G ψ · (ψ )2 · (G ◦ ψ), ◦ψ + H = (G ◦ ψ) = G (ψ )2 H (r ) 0 follows provided filled, if G (x)
G (ψ(r )) G (ψ(r ))
− 189 5 r . Therefore, assumption (2.12) is ful-
G (x) 189 1 or if − · 6x. ψ (6x) G (x) 5
For G(x) = A tanh( BA x), in which case G (x) = B/ cosh( BA x)2 and −2 BA
(A.3) G (x) G (x)
=
this can be checked numerically. (Observe that 0 A 0.4 and distinguish the cases B = G (0) 6 and B > 6.) For an illustration see the rightmost plot of H (r ) in Fig. 3. tanh( BA x),
Acknowledgement. This cooperation was supported by the DFG grant Ke-514/7-1 (Germany). J.-B.B. was also partially supported by CNRS (France). The authors acknowledge the hospitality of the ESI (Austria) where part of this research was done. G.K. thanks Carlangelo Liverani for a discussion that helped to shape the ideas in Section 5.3.
References 1. Bandtlow, O., Jenkinson, O.: Invariant measures for real analytic expanding maps. J. London Math. Soc. 75, 343–368 (2007) 2. Billingsley, P.: Convergence of Probability Measures. New York: Wiley, 1968 3. Chawanya, T., Morita, S.: On the bifurcation structure of the mean-field fluctuation in the globally coupled tent map systems. Physica D 116, 44–70 (1998) 4. Chichilnisky, G.: Topology and invertible maps. Adv. Appl. Math. 21, 113–123 (1998)
270
J.-B. Bardet, G. Keller, R. Zweimüller
5. Dawson, D.A., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20, 247–308 (1987) 6. de la Llave, R.: Invariant manifolds associated to invariant subspaces without invariant complements: a graph transform approach. Mathematical Physics Electronic Journal 9 (2003), Paper 3. (http://www.ma. utexas.edu/mpej/Vol/9/3.ps) 7. Dudley, R.: Real Analysis and Probability. Cambridge: Cambridge Univ. Press, 2002 8. Ershov, S.V., Potapov, A.B.: On mean field fluctuations in globally coupled maps. Physica D 86, 523–558 (1995) 9. Järvenpää, E.: An SRB-measure for globally coupled analytic expanding circle maps. Nonlinearity 10, 1435–1469 (1997) 10. Kaneko, K.: Globally coupled chaos violates the law of large numbers but not the central limit theorem. Phys. Rev. Lett. 65, 1391–1394 (1990) 11. Kaneko, K.: Remarks on the mean field dynamics of networks of chaotic elements. Physica D 86, 158–170 (1995) 12. Keller, G.: Stochastic stability in some chaotic dynamical systems. Monat. Math. 94, 313–333 (1982) 13. Keller, G.: An ergodic theoretic approach to mean field coupled maps. Progress in Probab. 46, 183–208 (2000) 14. Keller, G.: Mixing for finite systems of coupled tent maps. Proc. Steklov Inst. Math. 216, 315–321 (1997) 15. Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Sec. ed., Basel-Boston: Birkhäuser, 1992 16. Krzyzewski, K., Szlenk, W.: On invariant measures for expanding differentiable mappings. Stud. Math. 33, 83–92 (1969) 17. Mayer, D.H.: Approach to equilibrium for locally expanding maps in Rk . Commun. Math. Phys. 95, 1–15 (1984) 18. Nakagawa, N., Komatsu, T.S.: Dominant collective motion in globally coupled tent maps. Phys. Rev. E 57, 1570 (1998) 19. Nakagawa, N., Komatsu, T.S.: Confined chaotic behavior in collective motion for populations of globally coupled chaotic elements. Phys. Rev E 59, 1675–1682 (1999) 20. Pötzsche, C., Siegmund, S.: C m -smoothness of invariant fiber bundles. Topol. Methods Nonlinear Anal. 24, 107–145 (2004) 21. Ruelle, D.: Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34, 231–242 (1976) 22. Schweiger, F.: Invariant measures for piecewise fractional linear maps. J. Austral. Math. Soc. Ser. A 34, 55–59 (1983) 23. Sznitman, A.S.: Topics in propagation of chaos. École d’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464, Berlin: Springer, 1991, pp. 165–251 Communicated by A. Kupiainen
Commun. Math. Phys. 292, 271–284 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0834-0
Communications in
Mathematical Physics
On a Localized Riemannian Penrose Inequality Pengzi Miao School of Mathematical Sciences, Monash University, Victoria 3800, Australia. E-mail: [email protected] Received: 21 January 2009 / Accepted: 9 March 2009 Published online: 19 May 2009 – © Springer-Verlag 2009
Abstract: Let Ω be a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary ∂Ω is the disjoint union of two pieces: Σ H and Σ O , where Σ H consists of the unique closed minimal surfaces in Ω and Σ O is metrically a round sphere. We obtain an inequality relating the area of Σ H to the area and the total mean curvature of Σ O . Such an Ω may be thought of as a region, surrounding the outermost apparent horizons of black holes, in a time-symmetric slice of a space-time in the context of general relativity. The inequality we establish has close ties with the Riemannian Penrose Inequality, proved by Huisken and Ilmanen [9] and by Bray [5]. 1. Introduction Let M be a complete, asymptotically flat 3-manifold with nonnegative scalar curvature. Suppose its boundary ∂ M consists of the outermost minimal surfaces in M. The Riemannian Penrose Inequality, first proved by Huisken and Ilmanen [9] for a connected ∂ M, and then by Bray [5] for ∂ M with any number of components, states that A m AD M (M) ≥ , (1) 16π where m AD M (M) is the ADM mass [1] of M and A is the area of ∂ M. Furthermore, the equality holds if and only if M is isometric to a spatial Schwarzschild manifold outside its horizon. Motivated by the quasi-local mass question in general relativity (see [2,6,7], etc.), we would like to seek a localized statement of the above inequality (1). To be precise, we are interested in a compact, orientable, 3-dimensional Riemannian manifold Ω with boundary. We call Ω a body surrounding horizons if its boundary ∂Ω is the disjoint Research partially supported by the Australian Research Council.
272
P. Miao
union of two pieces: Σ O (the outer boundary) and Σ H (the horizon boundary), and Ω satisfies the following assumptions: (a) Σ O is topologically a 2-sphere. (b) Each component of Σ H is a minimal surface in Ω. (c) There are no other closed minimal surfaces in Ω. Physically, Ω is to be thought of as a finite region in a time-symmetric slice of a spacetime containing black holes and Σ H corresponds to the outermost apparent horizon of the black holes. In such a context, if the spacetime satisfies the dominant energy condition and m Q M (Σ O ) represents some quantity (to be defined) which could measure the quasi-local mass of Σ O , then one would expect m Q M (Σ O ) ≥
A . 16π
(2)
In this paper, we are able to establish an equality of the above form for a special class of body surrounding horizons. Our main result is Theorem 1. Let Ω be a body surrounding horizons whose outer boundary Σ O is metrically a round sphere. Suppose Ω has nonnegative scalar curvature and Σ O has positive mean curvature. Then |Σ H | , (3) m(Σ O ) ≥ 16π where m(Σ O ) is defined by m(Σ O ) =
2 |Σ O | 1 H dσ , 1− 16π 16π |Σ O | Σ O
(4)
where |Σ H |, |Σ O | are the area of Σ H , Σ O , H is the mean curvature of Σ O (with respect to the outward normal) in Ω, and dσ is the surface measure of the induced metric. When equality holds, Σ O is a surface with constant mean curvature. We remark that, assuming (3) in Theorem 1 holds in the first place, one can derive (1) in the Riemannian Penrose Inequality. That is because, by a result of Bray [4], to prove (1), one suffices to prove it for a special asymptotically flat manifold M which, outside some compact set K , is isometric to a spatial Schwarzschild manifold near infinity. On such an M, let Ω be a compact region containing K such that its outer boundary Σ O is a rotationally symmetric sphere in the Schwarzschild region. Applying Theorem 1 to such an Ω and observing that, in this case, the quantity m(Σ O ) coincides with the Hawking quasi-local mass [8] of Σ O , hence agrees with the ADM mass of M, we see that (3) implies (1). On the other hand, our proof of Theorem 1 does make critical use of (1). Therefore, (3) and (1) are equivalent. Besides the Riemannian Penrose Inequality, Theorem 1 is also largely inspired by the following result of Shi and Tam [14]: Theorem 2. (Shi-Tam). Let Ω˜ be a compact, 3-dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose ∂ Ω˜ has finitely many components
On a Localized Riemannian Penrose Inequality
273
Σi so that each Σi has positive Gaussian curvature and positive mean curvature H (with respect to the outward normal), then H dσ ≤ H0 dσ, (5) Σi
Σi
where H0 is the mean curvature of Σi (with respect to the outward normal) when it is isometrically imbedded in R3 . Furthermore, equality holds if and only if ∂ Ω˜ has only one component and Ω˜ is isometric to a domain in R3 . Let Ω˜ be given in Theorem 2. Suppose ∂ Ω˜ has a component Σ which is isometric to a round sphere with area 4π R 2 , then H0 dσ = 8π R, (6) Σ
and (5) yields 1 8π
Σ
H dσ ≤ R.
(7)
Now suppose there is a closed minimal surface Σh in Ω˜ such that Σh and Σ bounds a region Ω which contains no other closed minimal surfaces in Ω˜ (by minimizing area over surfaces homologous to Σ, such a Σh always exists if ∂ Ω˜ has more than one component). Applying Theorem 1 to Ω, we have 2 |Σh | |Σ| 1 ≤ H dσ , (8) 1− 16π 16π 16π |Σ| Σ which can be equivalently written as 1 H dσ ≤ R(R − Rh ), (9) 8π Σ |Σh | where R = |Σ| and R = h 4π 4π . Therefore, Theorem 1 may be viewed as a refinement of Theorem 2 in this special case to include the effect on Σ by the closed minimal surface in Ω˜ that lies “closest” to Σ. In general relativity, Theorem 2 is a statement on the positivity of the Brown-York ˜ [6]. Using the technique of weak inverse mean curvature flow quasi-local mass m BY (∂ Ω) ˜ developed by Huisken and Ilmanen [9], Shi and Tam [15] further proved that m BY (∂ Ω) ˜ is bounded from below by the Hawking quasi-local mass m H (∂ Ω). Suggested by the quantity m(Σ O ) in Theorem 1, we find some new geometric quantities associated to ˜ which are interestingly between m BY (∂ Ω) ˜ and m H (∂ Ω) ˜ (hence providing another ∂ Ω, ˜ ˜ proof of m BY (∂ Ω) ≥ m H (∂ Ω).) We include this discussion at the end of the paper. This paper is organized as follows. In Sect. 2, we review the approach of Shi and Tam in [14] since it plays a key role in our derivation of Theorem 1. The detailed proof of Theorem 1 is given in Sect. 3. In Sect. 3.1, we establish a partially generalized Shi-Tam monotonicity. In Sect. 3.2, we make use of the Riemannian Penrose Inequality. In Sect. 4, we give some discussion on quasi-local mass. In particular, we introduce two quantities motivated by Theorem 1 and compare them with the Brown-York quasi-local mass m BY (Σ) and the Hawking quasi-local mass m H (Σ).
274
P. Miao
2. Review of Shi-Tam’s Approach In [14], Shi and Tam pioneered the idea of using results on asymptotically flat manifolds to study compact manifolds with boundary. We briefly review their approach in this section. Let Ω˜ be given in Theorem 2. For simplicity, we assume ∂ Ω˜ has only one component Σ. Since Σ has positive Gaussian curvature, Σ can be isometrically imbedded in R3 as a strictly convex surface [11]. On the region E exterior to Σ, the Euclidean metric g0 can be written as g0 = dρ 2 + gρ ,
(10)
where gρ is the induced metric on each level set Σρ of the Euclidean distance function ρ to Σ. Motivated by the quasi-spherical metric construction of Bartnik [3], Shi and Tam showed that there exists a positive function u defined on E such that the warped metric gu = u 2 dρ 2 + gρ
(11)
has zero scalar curvature, is asymptotically flat and the mean curvature of Σ in (E, gu ) (with respect to the ∞-pointing normal) agrees with the mean curvature of Σ in Ω. Furthermore, as a key ingredient to prove their result, they showed that the quantity (H0 − Hu ) dσ (12) Σρ
is monotone non-increasing in ρ, and lim (H0 − Hu ) dσ = 8π m AD M (gu ), ρ→∞ Σ ρ
(13)
where H0 , Hu are the mean curvature of Σρ with respect to g0 , gu , and m AD M (gu ) is the ADM mass of gu . Let M be the Riemannian manifold obtained by gluing (E, gum ) to Ω˜ along Σ. The metric on M is asymptotically flat, has nonnegative scalar curvature away from Σ, is Lipschitz near Σ, and the mean curvatures of Σ computed in both sides of Σ in M (with respect to the ∞-pointing normal) are the same. By generalizing Witten’s spinor argument [16], Shi and Tam proved that the positive mass theorem [13,16] remains valid on M (see [10] for a non-spinor proof). Therefore, (H0 − Hu ) dσ ≥ lim (H0 − Hu ) dσ = 8π m AD M (gu ) ≥ 0, (14) ρ→∞ Σ ρ
Σ
with
Σ (H0
− H ) dσ = 0 if and only if H = H0 and Ω˜ is isometric to a domain in R3 .
3. Proof of Theorem 1 We are now in a position to prove Theorem 1. The basic idea is to deform the exterior region of a rotationally symmetric sphere in a spatial Schwarzschild manifold in a similar way as Shi and Tam did on R3 , then attach it to a body surrounding horizons and apply the Riemannian Penrose Inequality to the gluing manifold. The key ingredient in our proof is the discovery of a new monotone quantity associated to the deformed metric. We divide the proof into two subsections.
On a Localized Riemannian Penrose Inequality
275
3.1. A monotonicity property for quasi-spherical metrics on a Schwarzschild background. Consider part of a spatial Schwarzschild manifold
1 (Mrm0 , g m ) = S 2 × [r0 , ∞), dr 2 + r 2 dσ 2 , (15) 1 − 2m r r > 2m, if m ≥ 0 Here m is the ADM mass where r0 is a constant chosen to satisfy 0 r0 > 0, if m < 0. of the Schwarzschild metric g m , r is the radial coordinate on [r0 , ∞), and dσ 2 denotes the standard metric on the unit sphere S 2 ⊂ R3 . Let N be the positive function on Mrm0 defined by 2m . (16) N = 1− r In terms of N , g m takes the form 1 dr 2 + r 2 dσ 2 . (17) N2 The next lemma follows directly from the existence theory established in [14] (see also [3]). gm =
Lemma 1. Let Σ0 be the boundary of (Mrm0 , g m ). Given any positive function φ on Σ0 , there exists a positive function u on Mrm0 such that (i) The metric
u 2 dr 2 + r 2 dσ 2 (18) gum = N has zero scalar curvature and is asymptotically flat. (ii) The mean curvature of Σ0 (with respect to the ∞-pointing normal) in (Mrm0 , gum ) is equal to φ. (iii) The quotient Nu has the asymptotic expansion m0 1 u as r → ∞, (19) =1+ +O N r r2 where m 0 is the ADM mass of gum . Proof. Consider a Euclidean background metric ds 2 = dr 2 + r 2 dσ 2
(20)
on Mrm0 = S 2 × [r0 , ∞). By Theorem 2.1 in [14], there is a unique positive function v on Mrm0 such that gv = v 2 dr 2 + r 2 dσ 2
(21)
has zero scalar curvature, is asymptotically flat and the mean curvature of Σ0 in (Mrm0 , gv ) is given by φ. Furthermore, v has an asymptotic expansion m0 1 v =1+ , (22) +O r r2 where m 0 is the ADM mass of gv . Let u = N v; Lemma 1 is proved.
276
P. Miao
We note that metrics of the form v 2 dr 2 + r 2 dσ 2 are called (shear free) quasispherical metrics [3]. By the formula (2.26) in [3] (or (1.10) in [14]), the differential equation satisfied by v = u N −1 in Lemma 1 is 2 ∂v (v − v 3 ) v2 , = 2 S2 v + r ∂r r r2
(23)
where S 2 denotes the Laplacian operator of the metric dσ 2 on S 2 . Proposition 1. Let u, gum , m 0 be given in Lemma 1. Let Σr be the radial coordinate sphere in Mrm0 , i.e. Σr = S 2 × {r }. Let HS , Hu be the mean curvature of Σr with respect to the metric g m , gum . Then Σr
N (HS − Hu ) dσ
is monotone non-increasing in r . Furthermore, N (HS − Hu ) dσ = 8π(m 0 − m). lim
(24)
Proof. We have HS = r2 N and Hu = r2 v −1 , where v = u N −1 . Hence 2 (N 2 − N v −1 ) dσ N (HS − Hu ) dσ = Σr Σr r 2r (N 2 − N v −1 ) dω, =
(25)
r →∞ Σ r
S2
where dω = r −2 dσ is the surface measure of dσ 2 on S 2 . As N 2 = 1 − N (HS − Hu ) dσ = (2r − 4m − 2r N v −1 ) dω. Σr
2m r ,
we have (26)
S2
Therefore, d dr
∂ N −1 (2 − 2N v −1 ) − 2r dω v ∂r S2 ∂v + dω. 2r N v −2 2 ∂r S
Σr
N (HS − Hu ) dσ =
(27)
By (23), we have v −2
1 ∂v (v −1 − v) = S2 v + . ∂r 2r 2r
Thus the last term in (27) becomes −2 ∂v 2r N v N S 2 v dω + N (v −1 − v) dω dω = ∂r S2 S2 S2 N (v −1 − v) dω, = S2
(28)
(29)
On a Localized Riemannian Penrose Inequality
277
where we have used the fact that N is a constant on each Σr and S 2 S 2 v dω = 0. Hence the right side of (27) is given by ∂ N −1 (2 − 2N v −1 ) − 2r v + N (v −1 − v) dω. (30) ∂r S2 Replace v by u N −1 , the integrand of (30) becomes 2 − N 2 u −1 − 2r
∂N N u −1 − u. ∂r
(31)
∂N = 1. ∂r
(32)
By (16), we have N 2 + 2r N
Therefore, it follows from (27), (30), (31) and (32) that d N (HS − Hu ) dσ = − u −1 (u − 1)2 dω, 2 dr Σr S
(33)
which proves that Σr N (HS − Hu ) dσ is monotone non-increasing in r . To evaluate limr →∞ Σr N (HS − Hu ) dσ, we have N v −1 = 1 −
(m 0 + m) +O r
1 r2
by (16) and (22). Therefore, by (26) we have N (HS − Hu ) dσ = 2(m 0 − m) dω + O(r −1 ), Σr
(34)
(35)
S2
which implies lim
r →∞ Σ r
Proposition 1 is proved.
N (HS − Hu )dσ = 8π(m 0 − m).
(36)
3.2. Application of the Riemannian Penrose Inequality. In this section, we glue a body surrounding horizons, whose outer boundary is metrically a round sphere, to an asymptotically flat manifold (Mrm0 , gum ) constructed in Lemma 1, and apply the Riemannian Penrose Inequality and Proposition 1 to prove Theorem 1. We start with the following lemma. Lemma 2. Let Ω be a body surrounding horizons. Suppose its outer boundary Σ O has positive mean curvature, then its horizon boundary Σ H strictly minimizes the area among all closed surfaces in Ω that enclose Σ H .
278
P. Miao
Proof. As Ω is compact and the mean curvature vector of Σ O points into Ω, it follows from the standard geometric measure theory that there exist surfaces that minimize the area among all closed surfaces in Ω that enclose Σ H , furthermore none of the minimizers touches Σ O . Let Σ be any such a minimizer. By the Regularity Theorem 1.3 in [9], Σ is a C 1,1 surface, and is C ∞ where it does not touch Σ H ; moreover, the mean curvature of Σ is 0 on Σ\Σ H and equals the mean curvature of Σ H H2 -a.e. on Σ ∩ Σ H . Suppose Σ is not identically Σ H . As Σ H has zero mean curvature, the maximum principle implies that Σ does not touch Σ H . Hence, Σ is a smooth closed minimal surface in the interior of Ω, contradicting the assumption that Ω has no other closed minimal surfaces except Σ H . Therefore, Σ must be identically Σ H . Let Ω be a body surrounding horizons given in Theorem 1. Let R and R H be the area radii of Σ O and Σ H , which are defined by 4π R 2 = |Σ O | and 4π R 2H = |Σ H |.
(37)
It follows from Lemma 2 that R > R H . To proceed, we choose (M m , g m ) to be one-half of a spatial Schwarzschild manifold whose horizon has the same area as Σ H , i.e.
1 m m 2 2 2 2 (M , g ) = S × [R H , ∞), dr + r dσ , (38) 1 − 2m r where m is chosen to satisfy 2m = R H . As R > R H , Σ O can be isometrically imbedded in (M m , g m ) as the coordinate sphere Σ R = {r = R}.
(39)
Henceforth, we identify Σ O with Σ R through this isometric imbedding. Let Mom denote the exterior of Σ O in M m . By Lemma 1 and Proposition 1, there exists a metric gum =
u 2 N
dr 2 + r 2 dσ 2
(40)
on Mom such that gum has zero scalar curvature, is asymptotically flat, and the mean curvature of Σ O (with respect to the ∞-pointing normal) in (Mom , gum ) agrees with H , the mean curvature of Σ O in Ω. Furthermore, the integral N (HS − Hu ) dσ (41) Σr
is monotone non-increasing in r and converges to 8π(m 0 − m) as r → ∞, where m 0 is the ADM mass of gum . Now we attach this asymptotically flat manifold (Mom , gum ) to the compact body Ω along Σ O to get a complete Riemannian manifold M whose boundary is Σ H . The resulting metric g M on M satisfies the properties that it is asymptotically flat, has nonnegative scalar curvature away from Σ O , is Lipschitz near Σ O , and the mean curvatures of Σ O computed in both sides of Σ O in M (with respect to the ∞-pointing normal) agree identically. Lemma 3. The horizon boundary Σ H is strictly outer minimizing in M, i.e Σ H strictly minimizes area among all closed surfaces in M that enclose Σ H .
On a Localized Riemannian Penrose Inequality
279
Proof. By the construction of gum , we know (Mom , gum ) is foliated by {Σr }r ≥R , where each Σr has positive mean curvature. Let Σ be a surface that minimizes area among surfaces in M that encloses Σ H (such a minimizer exists as M is asymptotically flat). We claim that Σ\Ω must be empty, for otherwise Σ\Ω would be a smooth, compact minimal surface in (Mom , gum ) with boundary lying in Σ O , and that would contradict the maximum principle. Therefore, Σ ⊂ Ω. It then follows from Lemma 2 that Σ = Σ H . The next lemma is an application of the “corner smoothing” technique in [10]. Lemma 4. There exists a sequence of smooth asymptotically flat metrics {h k } defined on the background manifold of M such that {h k } converges uniformly to g M in the C 0 topology, each h k has nonnegative scalar curvature, Σ H has zero mean curvature with respect to each h k (in fact Σ H can be made totally geodesic w.r.t h k ), and the ADM mass of h k converges to the ADM mass of g M . Proof. Let M be an exact copy of M. We glue M and M along their common boundary Σ H to get a Riemannian manifold M¯ with two asymptotic ends. Let g M¯ be the resulting
be the copy of Σ in M . Denote by Σ the union of Σ , Σ metric on M¯ and let Σ O O O H
and Σ O , we then know that the mean curvatures of Σ computed in both sides of Σ in ¯ agree. (At Σ O and M¯ (with respect to normal vectors pointing to the same end of M)
, this is guaranteed by the construction of g m , and at Σ , this is provided by the fact ΣO H u that Σ H has zero mean curvature.) Apply Proposition 3.1 in [10] to M¯ at Σ, followed by a conformal deformation as described in Sect. 4.1 in [10], we get a sequence of smooth asymptotically flat met¯ with nonnegative scalar curvature rics {gk }, defined on the background manifold of M, such that {gk } converges uniformly to g M¯ in the C 0 topology and the ADM mass of ¯ Furthermore, as M¯ has a gk converges to the ADM mass of g M¯ on both ends of M. reflection isometry (which maps a point x ∈ M to its copy in M ), detailed checking of the construction in Sect. 3 in [10] shows that {gk } can be produced in such a way that each gk also has the same reflection isometry. (Precisely, this can be achieved by choosing the mollifier φ(t) in Eq. (8) in [10] and the cut-off function σ (t) in Eq. (9) in [10] to be both even functions.) Therefore, if we let M¯ k be the Riemannian manifold ¯ then Σ H remains a surface with zero obtained by replacing the metric g M¯ by gk on M, mean curvature in M¯ k (in fact Σ H is totally geodesic). Define h k to be the restriction of gk to the background manifold of M; Lemma 4 is proved. We continue with the proof of Theorem 1. Let {h k } be the metric approximation of g M provided in Lemma 4. Let Mk be the asymptotically flat manifold obtained by ˜ k , |Σ| ˜ be the area of replacing the metric g M on M by h k . For any surface Σ˜ in M, let |Σ| Σ˜ w.r.t the induced metric from h k , g M respectively.We can not apply the Riemannian
H |k Penrose Inequality directly to claim m AD M (h k ) ≥ |Σ16π . That is because we do not know if Σ H remains to be the outermost minimal surface in Mk . However, since Σ H is a minimal surface in Mk , we know the outermost minimal surface in Mk , denoted by Σk , exists and its area satisfies
˜ k | Σ˜ ∈ S}, |Σk |k = inf{|Σ|
(42)
280
P. Miao
where S is the set of closed surfaces Σ˜ in M that enclose Σ H (see [5,9]). By the Riemannian Penrose Inequality (Theorem 1 in [5]), we have |Σk |k m AD M (h k ) ≥ . (43) 16π Let k approach infinity, we have lim m AD M (h k ) = m AD M (g M ),
(44)
˜ | Σ˜ ∈ S}, lim |Σk |k = inf{|Σ|
(45)
k→∞
and k→∞
where we have used (42) and the fact that {h k } converges uniformly to g M in the C 0 topology. By Lemma 3, we also have ˜ | Σ˜ ∈ S}. |Σ H | = inf{|Σ| Therefore, it follows from (43), (44), (45) and (46) that |Σ H | . m AD M (g M ) ≥ 16π
(46)
(47)
To finish the proof of Theorem 1, we make use of the monotonicity of the integral N (HS − Hu ) dσ. (48) Σr
By Proposition 1, we have N (HS − Hu ) dσ ≥ lim
r →∞ Σ r
ΣO
N (HS − Hu ) dσ
= 8π(m 0 − m).
(49)
On the other hand, we know m 0 = m AD M (gum ) = m AD M (g M ), and 1 m = RH = 2
|Σ H | . 16π
Therefore, it follows from (49), (50), (51) and (47) that N (HS − Hu )dσ ≥ 0. ΣO
Plug in HS =
2 R N,
Hu = H and N = 8π R 1 −
1−
RH R
RH ≥ R
(50)
(51)
(52)
, we then have
ΣO
H dσ.
(53)
On a Localized Riemannian Penrose Inequality
Direct computation shows that (53) is equivalent to (3). Hence, (3) is proved. Finally, when the equality in (3) holds, we have N (HS − Hu )dσ = 0, ∀ r ≥ R. Σr
281
(54)
By the derivative formula (33), u is identically 1 on Mom . Therefore, the metric gum is indeed the Schwarzschild metric g m . Since the mean curvature of Σ O in (Mom , gum ) was arranged to equal H , the mean curvature of Σ O in Ω, we conclude that
1 2 H = R2 1 − RRH , which is a constant. Theorem 1 is proved. Remark. In the above proof, the parameter m was a priori chosen to be 21 R H . If we only require m < 21 R and leave m unspecified, then it follows from (49), (50) and (47) that |Σ H | 1 (55) m+ N (HS − Hu ) dσ ≥ 8π Σ O 16π with N = 1 − 2m R . Minimizing the left side of (55) over m, we have 2 1 R 1 min m + N (HS − Hu ) dσ = H dσ . (56) 1− 8π Σ O 2 16π |Σ O | Σ O m< 12 R Combining (55) and (56), we again obtain (3). Comparing to the equality case in Theorem 2, one would expect that the equality in (3) in Theorem 1 holds if and only if Ω is isometric to a region, in a spatial Schwarzschild manifold, which is bounded by a rotationally symmetric sphere and the Schwarzschild horizon. We believe that this is true, but are not able to prove it at this stage. A confirmation of this expectation seems to require a good knowledge of the behavior of a sequence of asymptotically flat 3-manifolds with controlled C 0 -geometry, on which the equality of the Riemannian Penrose Inequality is nearly satisfied. We leave this as an open question. 4. Some Discussion Let Σ be an arbitrary closed 2-surface in a general 3-manifold M with nonnegative scalar curvature. Consider the quantity 2 |Σ| 1 m(Σ) = H , (57) 1− 16π 16π |Σ| Σ where |Σ| is the area of Σ, H is the mean curvature of Σ in M and we omit the surface measure dσ in the integral. Theorem 1 suggests that, if Σ is metrically a round sphere, m(Σ) may potentially agree with a hidden definition of quasi-local mass of Σ. Such a speculation could be further strengthened by the resemblance between m(Σ) and the Hawking quasi-local mass [8], 1 |Σ| (58) m H (Σ) = 1− H2 . 16π 16π Σ
282
P. Miao
By Hölder’s inequality, we have m(Σ) ≥ m H (Σ)
(59)
for any surface Σ. On the other hand, if Σ is a closed convex surface in the Euclidean space R3 , the classic Minkowski inequality [12] 2 H dσ ≥ 16π |Σ| (60) Σ
implies that m(Σ) ≤ 0 and m(Σ) = 0 if and only if Σ is a round sphere in R3 . Therefore, even though bigger than m H (Σ), m(Σ) shares the same character as m H (Σ) that it is negative on most convex surfaces in R3 . In order to gain positivity and to maintain the same numerical value on metrically round spheres, we propose to modify m(Σ) in a similar way as the Brown-York mass m BY (Σ) [6] is defined. Recall that, for those Σ with positive Gaussian curvature, m BY (Σ) is defined to be 1 m BY (Σ) = H0 dσ − H dσ , (61) 8π Σ Σ where H0 is the mean curvature of Σ when it is isometrically embedded in R3 . Now suppose Σ is metrically a round sphere, then 2 H0 = 16π |Σ|. (62) Σ
In this case, we can re-wriite m(Σ) as either 2 H |Σ| Σ m(Σ) = 1− 16π Σ H0 or 1 m(Σ) = 16π
Σ
2 H Σ H0 . 1− Σ H0
(63)
(64)
This motivates us to consider the following two quantities: Definition 1. For any Σ with positive Gaussian curvature, define 2 H |Σ| Σ m 1 (Σ) = , 1− 16π Σ H0 and 1 m 2 (Σ) = 16π
Σ
2 H Σ H0 , 1− Σ H0
(65)
(66)
where H is the mean curvature of Σ in M and H0 is the mean curvature of Σ when it is isometrically embedded in R3 .
On a Localized Riemannian Penrose Inequality
283
The following result compares m H (Σ), m 1 (Σ), m 2 (Σ) and m BY (Σ). Theorem 3. Suppose Σ is a closed 2-surface with positive Gaussian curvature in a 3-manifold M. Then (i) m 1 (Σ) ≥ m H (Σ), and equality holds if and only if Σ is metrically a round sphere and Σ has constant mean curvature. (ii) m BY (Σ) ≥ m 2 (Σ), and equality holds if and only if Σ H0 dσ = Σ H dσ . (iii) Suppose Σ bounds a domain Ω with nonnegative scalar curvature and the mean curvature of Σ in Ω is positive, then m 2 (Σ) ≥ m 1 (Σ) ≥ 0. Moreover, m 1 (Σ) = 0 if and only if Ω is isometric to a domain in R3 , and m 2 (Σ) = m 1 (Σ) if and only if either Ω is isometric to a domain in R3 in which case m 2 (Σ) = m 1 (Σ) = 0 or Σ is metrically a round sphere. Proof.
(i) Let m(Σ) be defined as in (57). By the Minkowski inequality (60), we have m 1 (Σ) ≥ m(Σ). By (59), we have m(Σ) ≥ m H (Σ). Therefore, m 1 (Σ) ≥ m H (Σ) and equality holds if and only if Σ is metrically a round sphere and the mean curvature of Σ in M is a constant. (ii) This case is elementary. Let a = Σ H and b = Σ H0 . Then (ii) is equivalent to 2 the inequality 1 − ab ≥ 0. (iii) By the result of Shi and Tam [14], i.e. Theorem 2, we have 2 H ≥0 (67) 1 − Σ Σ H0 with equality holding if and only if Ω is isometric to a domain in R3 . (iii) now follows directly from (67) and the Minkowski inequality (60).
Suppose Ω is a compact 3-manifold with boundary with nonnegative scalar curvature and its boundary ∂Ω has positive Gaussian curvature and positive mean curvature. Theorem 3 implies that m BY (∂Ω) ≥ m 2 (∂Ω) ≥ m 1 (∂Ω) ≥ m H (∂Ω)
(68)
with m 1 (∂Ω) ≥ 0 and m BY (∂Ω) = m H (∂Ω) if and only if Ω is isometric to a round ball in R3 . This provides a slight generalization of a previous result of Shi and Tam (Theorem 3.1 (b) in [15]), which showed m BY (∂Ω) ≥ m H (∂Ω). Acknowledgement. The author wants to thank professor Hubert Bray for the helpful discussion leading to Sect. 4. Also the author wishes to thank the referees for their valuable comments and suggestions.
References 1. Arnowitt, R., Deser, S., Misner, C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. (2) 122, 997–1006 (1961) 2. Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62(20), 2346–2348 (1989) 3. Bartnik, R.: Quasi-spherical metrics and prescribed scalar curvature. J. Diff. Geom. 37(1), 31–71 (1993) 4. Bray, H.L.: The Penrose Inequality in General Relativity and Volume Comparison Theorems Involving Scalar Curvature. Stanford University Thesis, 1997
284
P. Miao
5. Bray, H.L.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Diff. Geom. 59(2), 177–267 (2001) 6. Brown, J.D., York, J.W., Jr.: Quasilocal energy in general relativity. In: Mathematical Aspects of Classical Field Theory (Seattle, WA, 1991), Volume 132 of Contemp. Math., Providence, RI: Amer. Math. Soc., 1992, pp. 129–142 7. Christodoulou, D., Yau, S.-T.: Some remarks on the quasi-local mass. In: Mathematics and General Relativity (Santa Cruz, CA, 1986), Volume 71 of Contemp. Math., Providence, RI: Amer. Math. Soc., 1988, pp. 9–14 8. Hawking, S.: Gravitational radiation in an expanding universe. J. Math. Phys. 9, 598–604 (1968) 9. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Diff. Geom. 59(3), 353–437 (2001) 10. Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6), 1163–1182 (2002) 11. Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6, 337–394 (1953) 12. Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, no. 27. Princeton, NJ: Princeton University Press, 1951 13. Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979) 14. Shi, Y., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Diff. Geom. 62(1), 79–125 (2002) 15. Shi, Y., Tam, L.-F.: Quasi-local mass and the existence of horizons. Commun. Math. Phys. 274(2), 277–295 (2007) 16. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981) Communicated by P. T. Chru´sciel
Commun. Math. Phys. 292, 285–301 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0832-2
Communications in
Mathematical Physics
Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions Wu-yen Chuang, Daniel Louis Jafferis NHETC, Department of Physics, Rutgers University, 126 Frelinghuysen Rd, New Jersey 08854, USA. E-mail: [email protected]; [email protected]; [email protected] Received: 24 January 2009 / Accepted: 1 March 2009 Published online: 22 May 2009 – © Springer-Verlag 2009
Abstract: In this paper we study the relation between pyramid partitions with a general empty room configuration (ERC) and the BPS states of D-branes on the resolved conifold. We find that the generating function for pyramid partitions with a length n ERC is exactly the same as the D6/D2/D0 BPS partition function on the resolved conifold in particular Kähler chambers. We define a new type of pyramid partition with a finite ERC that counts the BPS degeneracies in certain other chambers. The D6/D2/D0 partition functions in different chambers were obtained by applying the wall crossing formula. On the other hand, the pyramid partitions describe T 3 fixed points of the moduli space of a quiver quantum mechanics. This quiver arises after we apply Seiberg dualities to the D6/D2/D0 system on the conifold and choose a particular set of FI parameters. The arrow structure of the dual quiver is confirmed by computation of the Ext group between the sheaves. We show that the superpotential and the stability condition of the dual quiver with this choice of the FI parameters give rise to the rules specifying pyramid partitions with length n ERC. 1. Introduction One of the most fruitful areas of overlap between string theory and mathematics has been in the applications of topological field theories and string theories to questions involving integration over various moduli spaces of interesting geometrical objects. One example is that the topological sector of the worldvolume theory of a D6 brane wrapping a Calabi-Yau 3-fold has been identified with Donaldson-Thomas theory. The bound states of D2 and D0 branes to a D6 brane can be regarded as instantons in the topologically twisted N = 2 U (1) Yang-Mills theory in six dimensions. These instantons turn out to correspond to ideal sheaves on the 3-fold. Moreover, Donaldson-Thomas theory involves integration of a virtual fundamental class over the moduli space of such ideal sheaves. There has recently been tremendous progress in understanding the Kähler moduli dependence of the index of BPS bound states of D-branes wrapping cycles in a
286
W.-y. Chuang, D. L. Jafferis
Calabi-Yau 3-fold [1]. On the mathematical side, various generalizations of DonaldsonThomas theory have been proposed [2], in which ideal sheaves are replaced by more general stable objects in the derived category. These invariants will thus also have a dependence on the background Kähler moduli, as encoded in a stability condition. Much work has been done on Donaldson-Thomas theory in toric Calabi-Yau manifolds, which are of necessity noncompact. In particular, using toric localizations, the theory can be solved exactly [3,4]. In this work we will focus on the particular example of the resolved conifold. More recently, there was some extremely interesting work [5,6], in which the Donaldson-Thomas invariants on a noncommutative resolution of the conifold were determined. In that case, the torus fixed points of the moduli space of noncommutative ideal sheaves were identified with pyramid partitions in a length 1 ERC. In [7], these results were reproduced using the physical techniques resulting from the supergravity description of such bound states as multi-centered black holes in IIA string theory compactified on a Calabi-Yau manifold. The Donaldson-Thomas partition function of the commutative and noncommutative resolutions of the conifold were shown to arise as special cases in the moduli space of asymptotic Kähler parameters. Moreover, the D6/D2/D0 partition function was determined in all chambers of a certain real three parameter moduli space that captured the relevant universal behavior as a compact Calabi-Yau threefold degenerated to the noncompact resolved conifold.1 In this work, we will demonstrate an intriguing relationship between the pyramid partitions in a length n ERC and the D6/D2/D0 partition function in various chambers. In other chambers, we will give evidence that the torus fixed points of the moduli space of BPS states are in one to one correspondence with a new type of pyramid partitions in a finite region. It is possible to find a basis of D-brane charges which is both primitive (i.e. it generates the entire lattice) and rigid (i.e. the basis branes have no moduli) for D6/D2/D0 branes in the resolved conifold. Note that this is never possible in compact Calabi-Yau. Thus the BPS bound states are completely described by the topological quiver quantum mechanics, whose fields are the open strings stretched between the basis branes. The quiver that describes D2/D0 in the conifold is in fact the famous U (N ) × U (M) Klebanov-Witten quiver [8], viewed as a 0-dimensional theory in our context. The new ingredient that we introduce is an extended quiver that also includes a U (1) node associated to the D6 brane, in analogy with the case of D6/D0 bound states in C3 studied in [9]. We determine the spectrum of bifundamental strings by computing the appropriate E xt groups. One beautiful feature of this system is that the SU (2) × SU (2) symmetry of the conifold completely fixes the superpotential, SW , up to field redefinition, so we do not need to compute it directly. The moduli space of vacua of this quiver theory depends on the background Kähler moduli through the Fayet-Iliopoulos parameters. This moduli space is obtained by imposing the F-term equations, ∂ W = 0, on the Kähler quotient of the space of fields (here a finite dimensional space of matrices with ranks determined by the D-brane charges) by the U (N )×U (M)×U (1) gauge group. The Kähler quotient is equivalent to the quotient by the complexified gauge group, together with an algebraic stability condition that depends on the FI parameters [10]. In each chamber of the Kähler moduli space of the resolved conifold found in [7], we identify a pair of primitive sheaves carrying D2/D0 charge that become mutually BPS at the boundary of the Kähler cone, in that chamber. In particular, along the boundary 1 These partition functions were also derived first in [18] using a different approach.
Wall Crossing of BPS States on the Conifold
287
−1 of the Kähler cone in the chamber Cn = [Wn−1 Wn−1 ], in the notation of [7] for n > 0, the central charges of OC (−n − 1)[1] and OC (−n) become aligned. In the chamber 1 W 1 ], it is O (−n) and O (−n + 1)[1] which become mutually BPS at the C˜ n = [Wn−1 C C n boundary of the Kähler cone. Therefore in that chamber, we choose those sheaves, together with the pure D6 brane, O X , as our basis objects, and construct the quiver theory, as explained above. Note that all of the above pairs for different n are related to each other by application of Seiberg duality. We will denote the quivers resulting from the choice of basis branes above as Q n and Q˜ n respectively. The fact that the central charges become aligned in that chamber implies that the bifundamental strings between them are massless at tree level, hence the FI parameters, θ , for the two D2/D0 nodes must be equal for those values of the Kähler moduli. It is easy to check that θO X [1] > 0, θOC (−n−1)[1] < 0, and θOC (−n) < 0 is a single chamber, and includes the locus where θOC (−n−1)[1] = θOC (−n) . Therefore we can identify that chamber in the space of FI parameters for the quiver Q n with the chamber Cn in the space of background Kähler moduli. We show that with that choice of FI parameters, the King stability condition becomes equivalent to a simple cyclicity - the quiver representations must be generated by a vector in the C1 associated to the D6 brane U (1) node. The relations obtained from the superpotential are used to see that the torus fixed points of the moduli space of stable representations in this chamber are exactly the pyramid partitions in a length n ERC defined in [5]! Similarly, for the chambers C˜ n , we demonstrate that the torus fixed points are in one to one correspondence with pyramid partitions in a certain finite empty room configuration that we introduce. The generating functions for pyramid partition in a length n ERC was determined in [5,6], and we check that it agrees with the D6/D2/D0 partition function in the resolved conifold in the appropriate chamber found in [7,18,19]. This requires correctly changing variables to take into account the D2/D0 charges of the basis sheaves used in the construction of the quivers. We explicitly check a few examples of the finite type pyramid partitions as well. Thus we have been able to reproduce the D6/D2/D0 partition function on the resolved conifold, in a given chamber in the space of background Kähler moduli, by judiciously choosing a particular Seiberg dual version of the associated three node quiver in which the FI parameters corresponding to the Kähler moduli are of a special simple form. The paper is organized as follows. In Sect. 2 we give a review of wall crossing formulae and the D6/D2/D0 partition function on the resolved conifold based on [1,7]. In Sect. 3, we introduce the relation between the pyramid partition function and the D6/D2/D0 BPS partition function. In Sect. 4 we derive the Seiberg dual quiver and then compute its superpotentials and arrow structures. Next the discussion is on the stability condition and the rules of pyramid partitions. Some future directions will be presented in the conclusion part.
2. Wall Crossing Formula and D6/D2/D0 BPS Partition Function on the Resolved Conifold The index of BPS states with a given total charge is an integer, and thus is a piecewise constant function of the background values of the Kähler moduli. Moreover, the fact that it is a supersymmetric index implies that it can only jump when a state goes to infinity in the moduli space of BPS states, that is when the asymptotics of the potential change. The
288
W.-y. Chuang, D. L. Jafferis
only known way this can happen for the case of BPS bound states of D-branes wrapping a Calabi-Yau manifold is that the physical size of a multi-centered Denef black hole solution diverges at some value of the Kähler parameters [11]. This occurs exactly at (real codimension 1) walls of marginal stability, when the central charges, Z 1 and Z 2 , of the two constituents, 1 and 2 , of the multi-centered supergravity solution become aligned. At such a wall of marginal stability t = tms corresponding to a decay → 1 + 2 , the BPS index will have a discrete jump given by (, t) = (−1)1 ,2 −1 |1 , 2 | (1 , tms )(2 , tms ),
(2.1)
where 1 and 2 are primitive. A semi-primitive wall crossing formula is also given in [1], (1 ) + (1 + N 2 )q N = (1 ) (1 − (−1)k1 ,2 q k )k|1 ,2 |(k2 ) . N
k>0
(2.2) This formula gives a powerful way to construct the D6/D2/D0 BPS generating function on the resolved conifold from the Donaldson-Thomas generating function [7]. The absence of higher genus Gopakumar-Vafa invariants in the resolved conifold implies that only the pure D6 brane exists as a single centered solution. Thus in the core region of the Kähler moduli space, the D6/D2/D0 partition function is just Z = 1. The position of the relevant walls of marginal stability was determined in [7], and using the wall crossing formula for a single D6 bound arbitrary numbers of D2/D0 fragments, the partition function was then computed throughout the moduli space. The wall of marginal stability for = 1 − m β + n d V with h = −m h β + n h d V in a compact Calabi-Yau manifold, X , was shown have a well defined limit as the geometry approached that of the noncompact resolved conifold. Moreover, the walls are independent of m and n , and separate chambers in a real three dimensional space parameterized by the Kähler size, z, of the local P1 , and a real variable ϕ = 13 arg(V ol X ) that characterizes the strength of the B-field along the noncompact directions in units of the Kähler form. The wall of marginal stability for the fragment h was denoted by Wnmhh . The final result for the index of D6/D2/D0 bound states found in [7] was that in the chamber 1 , the generating function is between Wn1 and Wn+1 1 ]) = Z (u, v; [Wn1 Wn+1
n
1 − (−u) j v
j
.
(2.3)
j=1
Similarly, in the chambers where negative D2 charges appear, −2 j k j −1 Z (u, v; [Wn+1 1 − (−u) j 1 − (−u) j v 1 − (−u)k v −1 . Wn−1 ]) = j>0
k>n
(2.4) In the extreme case, n = 0, of the latter they found agreement with the results of Szendr˝oi, who calculated the same partition function at the conifold point using equivariant techniques to find the Euler character of a moduli space of noncommutative sheaves. The n → ∞ limit of (2.4), one obtains the usual “large radius” Donaldson-Thomas theory that was determined in [3], again using equivariant localization.
Wall Crossing of BPS States on the Conifold
289
Fig. 1. This figure illustrates the length 2 ERC of the pyramid partition, from the zeroth layer to the 2nd layer
3. Pyramid Partition and BPS Partition Function on Resolved Conifold In this section we will first discuss the relation between the pyramid partition generating function with length n empty room configuration (ERC) and the D6/D2/D0 BPS state partition function. Afterwards we will discuss how the pyramid partition arises when we look at the torus fixed points on the moduli space Mv of representations of the Calabi-Yau algebra A for the conifold quiver.
3.1. Pyramid partition generating function. The use of pyramid partitions in this context first arose in [5]. Consider the arrangement of stones of two different colors (white and grey) as in Fig. 1. For a generic ERC with length n, there will be n white stones on the zeroth layer.2 On layer 2i, there are (n + i)(1 + i) white stones, while on layer 2i + 1 there are (n + i + 1)(1 + i) grey stones. When we write generating functions, the number of white stones will be counted the power of q0 , and the grey stones by q1 . A finite subset of the ERC is a pyramid partition, if for every stone in , the stones directly above it are also in . Denote as w0 and w1 the number of white and grey stones in the partition. Also denote by Pn the set of all possible pyramid partitions for the ERC of length n. The generating function is defined combinatorially as Z pyramid (n; q0 , q1 ) =
q0w0 (−q1 )w1 .
(3.1)
∈Pn
This function can be computed by some dimer shuffling techniques; we refer the interested reader to [6] for details. Here we just quote the result for the generating function for general n ERC, Z pyramid (n; q0 , q1 ) = M(1, −q0 q1 )2 ×
(1 + q0k (−q1 )k−1 )k+n−1
k≥1
(1 + q0k (−q1 )k+1 )max(k−n+1,0) ,
k≥1 2 We count the layers as the zeroth, 1st, 2nd, and so on.
(3.2)
290
W.-y. Chuang, D. L. Jafferis
Fig. 2. This is the ERC for the fintie type pyramid partition with length 3
where M(x, q) is the MacMahon function M(x, q) =
(
n=1
1 )n . 1 − xq n
(3.3)
Notice that the exponents of the two terms in the product formula start from n and 1 respectively. This function turns out to be exactly the D6/D2/D0 BPS partition function on the resolved conifold found in [7,18,19] in certain chambers after performing the following (n-dependent) parameter identifications: u = −q0 q1 ; v = (−q0 )n−1 q1n .
(3.4)
Now we have Z pyramid (n; q0 , q1 ) = Z D6/D2/D0 (u, v; Cn ) (1 − (−u)k v)k (1 − (−u)k v −1 )k . = M(1, −u) k≥1
(3.5)
k≥n
The upshot is that the pyramid partition for a general empty room configuration counts the number of D6/D2/D0 BPS bound states at a certain value of the background modulus! More precisely, this chamber sits between the conifold point and the large −1 ] in the notation of [7]. radius limit; Cn = [Wn−1 Wn−1 We will explain that this is no coincidence, after we perform a Seiberg duality on the original D6/D2/D0 quiver theory. Moreover, the rules of specifying a pyramid partition encode the stability condition for these BPS states. 1 W 1 ], the D6/D2/D0 BPS states partition function is In the chamber C˜ n = [Wn−1 n given by Z D6/D2/D0 (u, v, C˜ n ) =
n−1
(1 − (−u)−k v)k .
(3.6)
k=1
We conjecture that the partition function in these chambers can be described by some finite type pyramid partitions with length (n − 1) ERC (see Fig. 2 for length 3 example), after a change of variables.
Wall Crossing of BPS States on the Conifold
291
For the finite type pyramid partition with length n, there are n × 1 white stones on the zeroth layer, (n − 1) × 1 grey stones on the first layer, (n − 1) × 2 white stones on the second, (n − 2) × 2 grey stones on the third, and so on until we reach 1 × n. The way of counting is the same. We count the finite subsets of the ERC in which, for every stone in , the stones directly above it are also in . The change of variables is given by u = −q0 q1 , v = −q0n q1n−1 .
(3.7)
Let us consider length 2 ERC as an example to illustrate the correspondence between the BPS states partition function in chambers C˜ n and the finite type pyramid partition, Z n=2
E RC
= 1 + 2q0 + q02 + q02 q1 + 2q03 q1 + q04 q1 = (1 + q02 q1 )(1 + q0 )2 = (1 − (−u)−1 v)(1 − (−u)−2 v)2 = Z (u −1 , v, C˜ 3 ).
(3.8)
3.2. Conifold quiver and pyramid partitions. This section is a review of [5]. Consider the conifold quiver Q = {V, E}, with two vertices V = {0, 1}, and four oriented edges E = {A1 , A2 : 0 → 1, B1 , B2 : 1 → 0}. The F-term relations come from the quartic superpotential W = A1 B1 A2 B2 − A1 B2 A2 B1 .[8] The quiver algebra A contains the idempotent ring C[ f 0 , f 1 ] and can be given by generators and relations as A = C[ f 0 , f 1 ]A1 , A2 , B1 , B2 /B1 Ai B2 − B2 Ai B1 , A1 Bi A2 − A2 Bi A1 , i = 1, 2. (3.9) A is a smooth Calabi-Yau algebra of dimension three [12] and a crepant non-commutative resolution of the singularity Spec (C[x1 , x2 , x3 , x4 ]/(x1 x2 − x3 x4 )). Consider the rank two torus action TW on the moduli space MV of framed cyclic A-modules 3 . It has been shown by Szendr˝oi [5] that the TW -fixed points on the moduli space MV are all isolated and have a one-to-one correspondence with pyramid partitions ∈ P1 of weight (w0 , w1 ). This weight vector is the same as the rank vector of the corresponding quiver. Moreover, given a pyramid partition ∈ P1 , we can obtain the precise framed cyclic module Mπ defined by it from looking at the pyramid partition. First, we draw A1 and A2 fields in the perpendicular direction out of the center of the white stones and draw B1 and B2 fields in the horizontal direction out of grey stones. The superpotential F-term relations require that we get the same result if we follow the arrows of opposite directions of the Ai or Bi fields down to the three lower layers. The cyclicity property of the module turns into the rule that for every stone in , the stones directly above it are also in . The module is generated by the stones on the zeroth layer. We show in Sect. 5 that the cyclicity condition is equivalent to the King stability condition in a particular chamber of the quiver (Fig. 4) obtained by introducing a new node for the D6 brane to the Klebanov-Witten quiver discussed in [5]. In [5], Szendr˝oi generalized the notion of pyramid partitions to length n ERC, and his conjecture for the resulting generating function was proven in [6]. We shall show 3 The torus action fixing the superpotential is a rank three torus T F W , described by (A1 , A2 , B1 , B2 ) → (µa A1 , µb A2 , µc B1 , µ−a−b−c B2 ). And TW is the quotient of TF W by the C ∗ action (µ, µ, µ−1 , µ−1 ).
292
W.-y. Chuang, D. L. Jafferis
Fig. 3. This figure illustrates how to define a framed cyclic module from a pyramid partition
that these partitions arise as torus fixed points of the moduli space of King stable representations of the quiver Q n . For now, note that in the case of general length n ERC, the n stones on the zeroth layer will still play the roles of the framing vectors of the quiver and generate the whole module. There are also new (n − 1) relations for n > 1, if we follow the arrows from the layer zero to layer one, which read: A1 q2 = A2 q1 , A1 q3 = A2 q2 , . . . , A1 qn = A2 qn−1 ,
(3.10)
where q1 . . . qn are the framing nodes on the layer zero. Later we will see that these relations arise from certain cubic terms in the superpotential which are not present for n = 1. 4. Deriving the Quivers via Seiberg Duality Recall that a standard choice of the sheaves representing the conifold quiver is as follows [13]: O X [1], OC , OC (−1)[1].
(4.1)
The arrow structure of this quiver is determined by the Ext group,4 Ext1 (OC , O(−1)[1]) ∼ = Ext1 (O(−1)[1], OC ) ∼ = C2 .
(4.2)
So if we take the rank vector to be (1, M + N , M), the system will have charges (D6, D2, D0) = (1, M, N ). Now we are going to show that this quiver at certain FI parameters leads to the pyramid partition after performing Seiberg dualities. First of all we know that in the pyramid partition there are n marked framing nodes, which are the most top nodes q1 · · · qn and n − 1 relations: A1 q2 = A2 q1 , A1 q3 = A2 q2 , . . . , A1 qn = A2 qn−1 .
(4.3)
This implies that we want to find a quiver representation with the following arrow structure. (See Fig. 4.) For this purpose we choose the basis to be O X , OC (−n − 1), OC (−n)[−1]. 4 We will use Ext and E xt to denote the global and local Ext respectively.
(4.4)
Wall Crossing of BPS States on the Conifold
293
Fig. 4. The n q fields gives n marked points in the quiver, which generates the whole module, and the (n − 1) p fields will give (n − 1) relation via superpotential. The sheaves representing the nodes are to be determined
Fig. 5. This shows the arrow structure of the quiver Q˜ n−1
Or equivalently, by an overall shift,5 O X [1], OC (−n − 1)[1], OC (−n) : Q n .
(4.5)
The quiver with this basis will be called Q n . Now let us try to find the sheaves corresponding to the finite type pyramid partition with length n − 1. There are n − 1 framing nodes on the top and n relations coming from the zeroth layer. So what we have to do is simply to reverse the directions of the p and q fields in Fig. 4. The basis of the quiver is given by O X [1], OC (−n − 1), OC (−n)[1] : Q˜ n−1 .
(4.6)
In the following section we will summarize the brane charges of the Seiberg dual quivers Q n and then confirm the F-term relations imposed by superpotential and the arrow structures of the proposed quivers with Ext group computation. The Ext group computation for the quiver Q˜ n simply follows from the computation for Q n ; therefore we only focus on Q n from now on. 5 In order to make contact with the convention in [13].
294
W.-y. Chuang, D. L. Jafferis
Table 1. Sheaves, ranks, and charges. Here we have taken into account the induced D0 charge of OC . The total charge of the system is M D2 and N D0 Sheaves
Ranks
Charge
FI parameters
O X [1] OC (−n − 1)[1] OC (−n)
1 N + (n − 1)M N + nM
¯ D6 or D6 ¯ n D0 D2, ¯ D2, (n − 1) D0
θ1 θ2 θ3
4.1. Brane charges of the quiver. The Chern character of the charges of the primitive objects in the derived category that we used as a basis in the quiver Q n can be computed as [OC (−n)] = −β + (1 − n)d V,
[OC (−n − 1)[1]] = +β + nd V,
(4.7)
in the conventions of [1]. In the original D6/D2/D0 system, powers u and v count the D0 and D2 charges respectively. Suppose we have a bound state with 1 D6, M D2 and N D0 charges; this will be represented by a quiver with ranks determined by the following computation: N +(n−1)M N +n M q1 .
(−u) N v M = (q0 q1 ) N ((−q0 )n−1 q1n ) M = (−1)n−1 q0
(4.8)
Recall that q0 are the number of white stones. Thus the ranks of the node OC (−n − 1)[1] and OC (−n) are N + (n − 1)M and N + n M respectively. This combination indeed gives the right total charges we are aiming at. We summarize the result in the following table. (See Table 1.) 4.2. O X → OC (−n − 1). We now proceed to determine the number of bifundamental fields that appear in the quiver, by computing the E xt groups between the basis sheaves. First of all, since O X is projective (thus free), we have E xt i (O X , OC ) = 0, i > 0.
(4.9)
And we also have Ext0 (O X , OC (−n − 1)) = 0 when n ≥ 0 since Ext0 (O X , ) is the global section functor. As for Ext1 (O X , OC (−n − 1)), we need to use the following: −m − 1 . (4.10) dim Hn (Pn , O(m)) = −n − m − 1 Therefore, Ext1 (O X , OC (−n − 1)) = H1 (X, OC (−n − 1)) ∼ = Cn , 2 3 Ext (O X , OC (−n − 1)) = Ext (O X , OC (−n − 1)) = 0.
(4.11) (4.12)
4.3. OC (−n)[−1] → O X . According to Chap. 5.3 in [14], the local sheaves E xt k (OC , O X ) are all trivial except for k = 2, E xt 2 (OC , O X ) ∼ = ι∗ KC = ι∗ OC (−2), where KC is the canonical bundle over P 1 .
(4.13)
Wall Crossing of BPS States on the Conifold
295
Fig. 6. The quiver diagram for the pyramid partition with length n ERC
Twisting the sheaf by O X (n), we have E xt 2 (OC (n), O X ) ∼ = ι∗ KC (−n) = ι∗ OC (−2 − n).
(4.14)
There is a local to global spectral sequence which we can apply to get the Ext group. However, if n is large, we can simply apply Property 6.9 in [15] to get the Ext group we want. The property says, if O X (1) is a very ample invertible sheaf and E and F are coherent sheaves on X , there exist an integer n 0 , depending on E, F and i, such that for n > n 0 , Exti (E, F(n)) = (X, E xt i (E, F(n))).
(4.15)
So for n >> 0, we have Ext1 (OC (−n), O X ) = (X, E xt 1 (OC , O X (n))) = 0,
(4.16) n−1 ∼ Ext (OC (−n), O X ) = (X, E xt (OC , O X (n))) = (X, ι∗ OC (n − 2)) = C , (4.17) 1 n−1 ∼ Ext (OC (−n)[−1], O X ) = C . (4.18) 2
2
Now we sum up the computation in a quiver diagram, in which we actually apply an overall shift. (See Fig. 6.)
4.4. Superpotential. In principle, the computation of the superpotential of this quiver quantum mechanics would require evaluating the B-model disk amplitude with boundary conditions determined given by the basis B-branes. Luckily that has already been done for the sheaves OC (n) and OC (m)[−1] by [13], resulting in the Klebanov-Witten superpotential, W = T r (A1 B1 A2 B2 − A1 B2 A2 B1 ). Furthermore, it will turn out that the SU (2) × SU (2) symmetry of the resolved conifold will completely fix the superpotential terms involving the p and q fields, up to field redefinition.
296
W.-y. Chuang, D. L. Jafferis
Recall that the projective coordinates of the resolved conifold, (x1 , x2 , y1 , y2 ) ≡ (λx1 , λx2 , λ−1 y1 , λ−1 y2 ), transform with (x1 , x2 ) in the doublet of SU (2)1 and (y1 , y2 ) in the doublet of SU (2)2 . In the above derivation of the E xt groups, it was clear that the n q and (n − 1) p fields live in various cohomology groups of the P1 . These groups carry an induced action of the SU (2)1 symmetry, which must be realized as a global symmetry group of the quiver quantum mechanics. Therefore we conclude that q and p are in the (n) ¯ and (n − 1) representation of the global SU (2)1 , under which A1 and A2 form a fundamental representation. There is a unique cubic superpotential that is invariant under this SU (2), up to field definition, essentially because there is a single copy of the trivial representation in the tensor product (n) ¯ ⊗ (n − 1) ⊗ (2). We first can construct from A and p a combination which is (n) under SU (2). This is basically the same as constructing angular momentum states |l + 21 , m + 21 from |l, m and | 21 , ± 21 , where 2l + 1 = n − 1. By using the following relation: 1 1 |l + , m + = 2 2
1 1 l +m+1 |l, m| , + 2l + 1 2 2
1 −1 l −m |l, m + 1| , , (4.19) 2l + 1 2 2
we can write down explicitly the form of superpotential (ignoring the color trace and index structure and just focusing on the invariance of the SU (2))
n−2 1 W ∼ p1 A 2 q 1 + ( p2 A 2 + p1 A1 )q2 n−1 n−1 n−3 2 p3 A 2 + p2 A1 )q3 + · · · . +( n−1 n−1
(4.20)
We should perform the following field redefinitions: 1 2 q2 , q˜3 = q3 , q˜1 = q1 , q˜2 = − n−1 (n − 1)(n − 2) 3! = q˜4 − q4 , . . . . (n − 1)(n − 2)(n − 3)
(4.21)
The relations implied by superpotential become A1 q˜2 = A2 q˜1 , A1 q˜3 = A2 q˜2 , . . . , A1 q˜n = A2 q˜n−1 .
(4.22)
Although these field redefinitions are not unitary and will spoil the D-terms, King’s stability condition will not change under these redefinitions. The moduli space of solutions to the F-flatness and D-flatness conditions, modded out by the U (N ) × U (M) gauge symmetry is equivalent to the G L(N , C) × G L(M, C) quotient of the holomorphic F-term constraint, together with the King stability condition. Therefore, we can always bring the superpotential to the form we want, so that (4.3) holds.
Wall Crossing of BPS States on the Conifold
297
4.5. Seiberg duality. Note that given one of the above quivers, all others can be constructed from it simply by repeated application of the rules developed by Berenstein and Douglas [16] for generalized Seiberg dualities. Suppose we begin with quiver Q n . Then dualizing the OC (−n − 1)[1] node reverses the directions of all arrows through that node. In addition, between the other two nodes there will be 2n new mesonic fields Mai = Aa qi for a = 1, 2, i = 1, . . . , n. The superpotential calculated above implies that there is a mass term Mai p j , where the indices are contracted as described above to be consistent with the SU (2) flavor symmetry. This lifts all p j and a corresponding n − 1 of the mesons from the massless spectrum. Therefore, we obtain exactly the field content of the quiver Q n+1 ! This result had to be true, given the previous calculation of the quiver directly from the new basis of objects in the derived category. 5. θ Stability and Cyclicity The moduli space of supersymmetric Higgs branch vacua of the quiver quantum mechanics6 describing the D6/D2/D0 bound states is given by the U (N )×U (M)×U (1) Kähler quotient of the solution of the F-flatness conditions. The background values of the Kähler moduli are encoded in the values of the FI parameters in the Kähler quotient. In general, it is a very difficult problem to determine the Euler character of the resulting moduli space, even by using the toric action to reduce to fixed points. We will find that for a particular choice of FI parameters the situation is dramatically simpler. This motivates us to choose a convenient basis of branes (that is, a particular mutation, Q n , of the quiver) for which the FI parameters are of this simple type in a given chamber in the background Kähler moduli space. It was shown by King [10] that is it possible to replace the D-term equation appearing in the Kähler quotient by a purely holomorphic algebraic condition, called θ -stability. Let (θv )v∈V be the FI parameters, a set of real numbers assigned to the nodes of the quiver, such that θ (N) = Nv θv = 0 for a given dimension vector N. Then a representation R is called θ -stable if for every proper subrepresentation R˜ with dimension ˜ θ (N) ˜ is smaller than θ (N). vector N, Consider the chamber in the space of FI parameters given by θ1 > 0, θ2 < 0, θ3 < 0, for Q n ; θ1 > 0, θ2 < 0, θ3 < 0, for Q˜ n .
(5.1)
Our interest is in bound states with one unit of D6 charge, thus we have that N1 = 1. Then King stability is equivalent to cyclicity, in the sense that the entire representation is generated by a vector in C, the node associated to O X [1]. Firstly, any such representation is King stable for this choice of FI parameters, since any subrepresentation that includes this node must be the entire representation, and thus the proper subrepresentations all ˜ = N˜ 2 θ2 + N˜ 3 θ3 < 0. have θ (N) Moreover, suppose that R is a King stable representation with N1 = 1. Then consider ˜ generated by the vector space C of the D6 node. If it is not all the subrepresentation, R, ˜ = θ1 + N˜ 2 θ2 + N˜ 3 θ3 > 0, and of R, then N˜ 2 < N2 or N˜ 3 < N3 , and one has that θ (R) the representation R must be unstable. 6 Note that our quiver must be understood as a quantum mechanics, describing the BPS configurations of a point-like object in the R 3,1 , rather than a 3 + 1 field theory, as it would then be anomalous. This is obvious from the presence of a Calabi-Yau wrapping brane.
298
W.-y. Chuang, D. L. Jafferis
6. The Big Picture: Connecting the Dots We would like to put together every piece of the story in this section. First of all, we observe that the D6/D2/D0 BPS partition function at a certain chamber, Cn in the Kähler moduli space is the same as the pyramid partition generating function for length n ERC, after the parameter identification (3.4), Z D6/D2/D0 (u, v, Cn ) = Z pyramid (n; q0 , q1 ).
(6.1)
By empirically checking the finite type pyramid partition, we also conjecture that the BPS states partition function is identical to the finite type pyramid partition generating function: Z D6/D2/D0 (u, v, C˜ n ) = Z f inite (n; q0 , q1 ).
(6.2)
Physically, given a set of brane charges, we should be able to use the quiver theory to compute the Euler character of the moduli space. In order to do that, we need to know how to translate the data of chamber Cn or C˜ n into the FI parameters of the corresponding quiver. This, in general, is a very difficult task. In the conifold case, we are in luck because we have the answer from pyramid partition. We showed that pyramid partitions with length n ERC, as well as those of finite type, are torus fixed points in the moduli space of vacua of a certain quiver. Using this answer, we noticed that this quiver is Seiberg dual to the quiver with basis {O X [1], OC , OC (−1)[1]}. And in the Seiberg dual quiver, Q n , we can determine the FI parameters to reproduce the cyclicity property. So we should keep in mind the following relation: Q
Q
Q
Qn
0 Z D6/D2/D0 (u, v, Cn ) = Z quiver (u, v, θi 0 ) n (u, v, θ1 = Z quiver
Z D6/D2/D0 (u, v, C˜ n ) = =
Qn
> 0, θ2
Q0 Q Z quiver (u, v, θ˜i 0 ) Q˜ n Q˜ Z quiver (u, v, θ1 n >
Q˜ n
0, θ2
Qn
< 0, θ3
Q˜ n
< 0, θ3
< 0),
(6.3)
< 0).
(6.4)
The quivers Q n and Q˜ n are Seiberg dual to the quiver Q 0 , which has basis {O X [1], OC , OC (−1)[1]}. Presumably, we should be able to find the mapping: Q0
Cn ↔ θi
Q Q Q˜ ↔ θi n , C˜ n ↔ θ˜i 0 ↔ θi n . Q
Q
(6.5) Q
For Q n , the θ stability condition for {θ1 n > 0, θ2 n < 0, θ3 n < 0} gives exactly the rules for constructing the pyramid partition in length n ERC. On the other hand, ˜ ˜ ˜ the θ stability of the quiver Q˜ n with {θ1Q n > 0, θ2Q n < 0, θ3Q n < 0} gives the rules for constructing the finite type pyramid partition. It is also possible to obtain the mapping (6.5) between the chambers in the space of Kähler moduli and the FI parameters before matching the answers. Consider the cham1 ], which we checked corresponds to the simple choice of FI parameters ber [Wn1 Wn+1 for the quiver Q˜ n . This contains the locus I m(z) = 0, −n − 1 < Re(z) < −n along the
Wall Crossing of BPS States on the Conifold
299
Fig. 7. The quiver Q +n and Q˜ − n
˜ +n Fig. 8. The quiver Q − n and Q
boundary of the Kähler cone for π/3 < ϕ < 2π/3. The D2/D0 branes associated to the sheaves OC (−n − 1)[1] and OC (−n − 2) have charges −β + nd V and +β − (n + 1)d V . When their central charges Z (−β + nd V ; t) = −z − n and Z (+β − (n + 1)d V ; t) = z + n + 1 are aligned, then the bifundamental strings stretched between these two branes must become massless (at tree level) in the quiver quantum mechanics. Referring to the form of the bosonic potential, we see this occurs precisely when the FI parameters for those nodes are equal. Therefore we are in the chamber expected. This provides an a priori derivation of the partition function of D6/D2/D0 bound states in each chamber. One last thing to notice is that we can flip the signs of the θ s and the directions of the arrows of the quiver at the same time, without causing any change to the partition function of the quiver theory. The reason is that in this way we do not change the D-term conditions at all. Therefore, we have: Z Q +n (u, v) = Z Q˜ − (u −1 , v), Z Q −n (u, v) = Z Q˜ + (u −1 , v) , n−1
n−1
(6.6)
where we simplify our notation by specifying the signs of θ1 in the quiver by putting a superscript on the Q n or Q˜ n .
300
W.-y. Chuang, D. L. Jafferis
7. Conclusion In this paper we have studied the relation between the pyramid partition generating function and the D6/D2/D0 BPS state partition function on the resolved conifold. We found that the generating function of the pyramid partition with length n ERC is equivalent to D6/D2/D0 BPS state partition functions in certain chambers in the Kähler moduli space. More precisely, we have the following relation: −1 Z D6/D2/D0 (u, v, Cn ) = Z pyramid (n; q0 , q1 ), Cn = [Wn−1 Wn−1 ],
(7.1)
1 Z D6/D2/D0 (u, v, C˜ n ) = Z f inite (n; q0 , q1 ), C˜ n = [Wn−1 Wn1 ],
(7.2)
−1 1 W 1 ] are defined in [7]. where the chambers [Wn−1 Wn−1 ] and [Wn−1 n From the rules specifying pyramid partitions (of both infinite and finite type), we constructed the corresponding quivers, the θ s parameters, and the superpotentials. We gave the underlying basis of sheaves and verified that they are Seiberg dual to the original D6/D2/D0 systems. The arrow structures of the quivers are also verified by computing E xt groups. The θ parameters in these particular basis are simple and the superpotentials are quartic, so that the rules of pyramid partition emerge. We also noted that the cyclicity condition on quiver representations is the same as the King stability condition in the region of (5.1). It would also be interesting to see if there is a similar story for the BPS partition func−1 1 W 1 ]. We also note that pyramid tion in the chambers other than [Wn−1 Wn−1 ] and [Wn−1 n partitions with more colors have been developed in [17], and we suspect a similar story will emerge in the case of the orbifold Donaldson-Thomas partition function.
Acknowledgements. We would like to thank Emanuel Diaconescu, Greg Moore, Balázs Szendr˝oi and Alessandro Tomasiello for useful conversations. We are also grateful to the Simons Workshop in Mathematics and Physics 2008 for providing a stimulating atmosphere during the final stage of this project. WYC and DJ are supported by DOE grant DE-FG02-96ER40959.
References 1. Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. http://arXiv.org/abs/hep-th/ 0702146v2, 2007 2. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. http://arXiv.org/abs/0811.2435v1[math.AG], 2008 3. Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. JHEP 0804, 011 (2008) 4. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and DonaldsonThomas theory, I. http://arXiv.org/abs/math/0312059v3[math.AG], 2004; Gromov-Witten theory and Donaldson-Thomas theory, II. http://arXiv.org/abs/math/0406092v2[math.AG], 2005 5. Szendr˝oi, B.: Non-commutative Donaldson-Thomas theory and the conifold. Geom. Topol. 12, 1171–1202 (2008) 6. Young, B.: Computing a pyramid partition generating function with dimer shuffling. http://arXiv.org/abs/ 0709.3079v2[math.CO], 2008 7. Jafferis, D.L., Moore, G.W.: Wall crossing in local Calabi Yau manifolds. http://arXiv.org/abs/0810. 4909v1[hep-th], 2008 8. Klebanov, I.R., Witten, E.: Superconformal field theory on threebranes at a Calabi-Yau singularity. Nucl. Phys. B 536, 199 (1998) 9. Jafferis, D.L.: Topological Quiver Matrix Models and Quantum Foam. http://arXiv.org/abs/0705. 2250v1[hep-th], 2007 10. King, A.: Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. 2,45, no. 180, 515–530 (1994)
Wall Crossing of BPS States on the Conifold
301
11. Denef, F.: Supergravity flows and D-brane stability. JHEP 0008:050 (2000) 12. Ginzburg, V.: Calabi-Yau algebras. http://arXiv.org/abs/math/0612139v3[math.AG], 2007 13. Aspinwall, P.S., Katz, S.H.: Computation of superpotentials for D-Branes. Commun. Math. Phys. 264, 227 (2006) 14. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley-Interscience, 1978 15. Hartshorne, R.: Algebraic Geometry. Berlin-Heidelberg-New York: Springer, 1997 16. Berenstein, D., Douglas, M.R.: Seiberg duality for quiver gauge theories. http://arXiv.org/abs/hep-th/ 0207027v1, 2002 17. Young, B., with an appendix by Bryan, J.: Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds. http://arXiv.org/abs/0802.3948v2[math.CO], 2008 18. Nagao, K., Nakajima, H.: Counting invariant of perverse coherent sheaves and its wall-crossing. http:// arXiv.org/abs/0809.2992v3[math.AG], 2008 19. Nagao, K.: Derived categories of small toric Calabi-Yau 3-folds and counting invariants. http://arXiv. org/abs/0809.2994v3[math.AG], 2008 Communicated by N. A. Nekrasov
Commun. Math. Phys. 292, 303–341 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0847-8
Communications in
Mathematical Physics
Mean Field Analysis of Low–Dimensional Systems L. Chayes Department of Mathematics, UCLA, Los Angeles, CA 90059–1555, USA. E-mail: [email protected] Received: 15 August 2007 / Accepted: 15 April 2009 Published online: 21 August 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: For low–dimensional systems, (i.e. 2D and, to a certain extent, 1D) it is proved that mean–field theory can provide an asymptotic guideline to the phase structure of actual systems. In particular, for attractive pair interactions that are sufficiently “spead out” according to an exponential (Yukawa) potential it is shown that the energy, free energy and, in particular, the block magnetization (as defined on scales that are large compared with the lattice spacing but small compared to the range of the interaction) will only take on values near to those predicted by the associated mean–field theory. While this applies for systems in all dimensions, the significant applications are for d = 2 where it is shown: (a) If the mean–field theory has a discontinuous phase transition featuring the breaking of a discrete symmetry then this sort of transition will occur in the actual system. Prominent examples include the two–dimensional q = 3 state Potts model. (b) If the mean–field theory has a discontinuous transition accompanied by the breaking of a continuous symmetry, the thermodynamic discontinuity is preserved even if the symmetry breaking is forbidden in the actual system. E.g. the two–dimensional O(3) nematic liquid crystal. Further it is demonstrated that mean–field behavior in the vicinity of the magnetic transition for layered Ising and XY systems also occurs in actual layered systems (with spread–out interactions) even if genuine magnetic ordering is precluded. 1. Introduction Mean–field theory has traditionally proved to be a reliable guide for predicting, on a coarse level, the behavior in realistic systems. In particular, the location and order of a phase transition may be confidently – if not always accurately – ascertained for a given system by performing the associated mean–field calculation. In recent years some mathematical underpinnings for these trends have been provided in [2,3] (with some ideas therein dating back to [21]). Specifically, the tendency for discontinuous transitions in “realistic” systems was, to a certain extent, elucidated by a comparison to mean–field
304
L. Chayes
theory. The results of [2] may be summarized, roughly, as follows: If H is the Hamiltonian for an nearest neighbor attractive (ferromagnetic) spin–system on Zd and the associated mean–field theory has a discontinuous transition at temperature TMF then, for large dimensions, the actual model has a discontinuous transition at (a normalized) temperature Td → TMF as d → ∞. Further, as d → ∞, the observable characteristics of the system, e.g. latent heat, response functions, etc. approach the corresponding characteristics predicted by the mean–field theory. The nearest neighbor assumption in [2] was to ensure the condition of reflection positivity; d ≥ 3 was required for the convergence of certain (k–space) integrals and, further, d −1 1 was used as a small parameter. In [3], the large d condition was relaxed by the consideration of exponentially decaying interactions but the condition d ≥ 3 was still required. In this note, the obvious step of combining reflection positivity techniques with contour methods will be taken. Thus, at least for models with (exponentially) spread–out interactions, this allows the extensions of the results in [2 and 3] to d = 2 – and, in a weak sense, even to d = 1. In particular, at the foundation of this note, is the result that at length–scales that are “large” – but still small compared with the range of the interaction – the average value of the order parameter is asymptotically close to that predicted by mean–field theory. Foremost, this will be used to demonstrate that in a variety of 2D models with discontinuous transition in their associated mean–field theories, there is an actual discontinuous transition provided that the range parameter of the interaction is sufficiently large. It is remarked that this statement includes 2D models where the breaking of a continuous symmetry, usually associated with an ordered low temperature phase, is necessarily absent. Of course if a discrete symmetry is broken, a magnetized phase is supported in d = 2 and several of the better known transitions of this sort will be discussed as well. As a pertinent example it is established (but not for the first time, cf. the discussion below) that a 2D, three–state Potts model has a first order transition. As for one dimension, since the range of the interaction is ultimately finite, there will be no transitions of any sort. Still, the results concerning the various observables – and the associated thermodynamic potentials – apply. Explicitly, in the vicinity of TMF , there is some sort of pseudo–transition even though all thermodynamic quantities are analytic. While this result is of certain modest æsthetic appeal, it is also pertinent to the study of layered systems. An idealized layered system is a d–dimensional system that is extended L units in the d + 1st direction; the physically relevant cases concern d = 1 and d = 2. While systems of this sort were an important showcase for scaling theory and the renormalization group (see e.g. [24] and, especially, [27] and references therein) it seems that till [10], an honest mean–field theory for systems of this sort had never been derived. In the reference [10], (see also [17]) an issue of seminal importance concerned the transition temperature in the layer. For certain systems, e.g. Ising and XY, (where the spins are unit vectors in n = 1 and n = 2 dimensions respectively and the ordinary dot product is used to define the interaction) the transition temperature was found to deviate from the d + 1–dimensional bulk temperature by an amount t given by |t| ∼ c
π2 , L2
(1.1)
where the constant c is system specific, but explicitly computable, and the asymptotic symbol pertains to the limit L 1. The result in Eq. (1.1) was required to understand the thinning of 4 He layers in the vicinity of the bulk superfluid transition temperature as observed in the cold–temperature experiments of Garcia, Chen and co–workers [15,16].
Mean Field Analysis of Low–Dimensional Systems
305
It would seem that the implicit assumption (or philosophy) behind the analysis is that at the shifted temperature, the layered system actually undergoes a magnetic–type transition. This is false in two–dimensions for XY systems and patently false for any one–dimensional system. However, the overall magnetic – or superfluid – properties of the layer are not, as it turns out, quite so relevant as the free energy per particle within the layer as compared to that of the bulk. For layered systems, by and large, the script follows that which was described above: For a sufficiently spread–out interaction, there is indeed a local ordering that is governed by mean–field theory and the free energetics – which is imminently associated with this local ordering – is accurately described by the layered mean–field theory. Thus, it can be claimed, the mean–field understanding of the layer thinning has some mathematical justification in the context of more realistic systems. It should be emphasized that there is – among others – one severe limitation to this work: The entire approach is contour based and the relevant estimates are enacted via reflection positivity. Hence, to the author’s knowledge, this limits us to attractive pair interactions with only one mechanism for spreading out the interaction, namely using the Yukawa1 (exponential) coupling. (Precise definitions follow in the next section.) By effective contrast: Recently – concurrent with the writing of the present work – a proof of 1st order transitions for some q ≥ 3–state Potts models in d ≥ 2 has been announced [18]. In this approach, which bears certain similarities to the present one, contour estimates are performed with Pirogov–Sinai based methods. Thus, while the work in [18] is ostensibly limited to a single model and, also, a single method to spread– out the interaction, the technique is inherently more flexible. Indeed it seems, albeit with tremendous labor, that these methods might be adapted to a wider variety of interactions and be used to analyze any number of mean–field–type phase transitions associated with the breakdown of discrete symmetries. Let us close this section first, with an informal survey of various results that will be established and then an organizational outline. To start off: for models of a particular type with range parameter µ−1 1, it will be shown that on large blocks the spatially averaged magnetization must, with high probability, be close to a value predicted by the associated mean–field theory. Energy, free energy and other thermodynamic quantities follow suit and the result holds in all dimensions. Thus, if the mean–field theory has a transition at some TMF there is evidently some sort of transition–like behavior in the spread–out system even if all thermodynamic quantities are analytic as in d = 1 or known to be smooth and with no actual magnetic transition like the standard O(n) spin–systems in d = 2. In d ≥ 2 the above considerations allow the proof of first order magnetic transitions in models with discrete symmetries, in particular the Potts models for q ≥ 3 and the cubic models for r ≥ 4. For systems such as the O(n) nematic models with continuous symmetries and first order transitions (for n ≥ 2) in the mean–field theory, the thermodynamic component of the transition, if nothing else, will persist in d ≥ 2, especially d = 2. These transitions will be accompanied by a discontinuity in the local magnetization notwithstanding that other considerations may rule out the possibility of a global magnetization. Models without any particular symmetry can also be treated and a particular example of a tertiary alloy will be discussed. 1 In addition, one can augment or replace the exponential interaction with interactions that decay as a power of the distance; this was a mechanism employed in [3] for treating lower dimensional systems. However, in the unreformed opinion of the author, power law potentials effectively change the dimension of the system and, in any case, cannot be construed as a finite range interactions.
306
L. Chayes
Finally, layered Ising and XY systems will be treated. A mean–field transition temperature, related to the minimal eigenvalue of a certain 1D Laplacian has been calculated in [10] for these systems. By standard methods, it will be shown that this is an upper bound on the actual critical temperature and in the Ising case, for large µ−1 , it is asymptotically a lower bound. More importantly, for both systems it will be proved that the free energy of the actual system is close to that predicted by mean–field theory. The remainder of this paper will be organized as follows: In Sect. 2, first the finite– dimensional (“realistic”) models under consideration will be defined in generality along with some necessary formalism and working notation. Then there will be a subsect. 2.2, devoted to mean–field theory. This will start with some concise definitions and then, within the context of the theory, a definition of (scenario for) a generic 1st order transition. In Subsect. 2.3, layered systems will be described in some limited generality – sufficient to discuss the results derived in [10]. At this point, enough notation will have been established so that by Sect. 3, we are ready for precise statements of theorems. In Sect. 3, all theorems stated will all be of a general nature. The main result will be that if a mean–field model has a generic 1st order scenario then the corresponding “realistic” system will also have this transition d ≥ 2 provided that the range parameter is sufficiently large. A series of propositions and corollaries then follow which cover, in general terms, all items in the above summary. Sect. 4 is devoted to statements about the specific systems mentioned above. Sect. 5 is for proofs. Subsects. 5.1 and 5.2 will be devoted to statements that concern magnetics and energetics respectively; the latter can be omitted without too much loss of continuity. In Subsect. 5.3 proofs of the main general results will be provided and, in Subsect. 5.4 all results concerning specific systems will be established. Finally, Subsect. 5.5 will consist of a brief appendix devoted to some elementary properties of the mean–field theory formalism. 2. Definitions and Setup Here we will fix notation, define briefly a working version of mean–field theory and provide an abbreviated description of layered systems. 2.1. Background. The basic setup will be pretty much the same as in [2] (and [3]). In particular, we will be discussing spin–systems where the spin variables reside in a compact , which is a subset of a finite–dimensional vector space E that is endowed with a positive definite inner product (− · −). Spin variables, generically denoted by an s, are distributed according to some a priori measure denoted by α0 (−). The formal Hamiltonian on Zd is given by −H= Ji, j (si · s j ), (2.1) i, j
where each pair of sites is counted once and the Ji, j ≥ 0. It may, on occasion, be desirable to addan external field to the interaction. Thus, if b ∈ E we may add to −βH the term i (b · si ). However unless the external field represents a parameter of the model that we wish to actively vary, the field term will be implicitly incorporated into the single–spin measure. This work will be exclusively concerned with the so called Yukawa interactions for which the Ji, j are given by Ji, j = K (µ)e−µ|i− j|1 .
(2.2)
Mean Field Analysis of Low–Dimensional Systems
307
Here µ > 0 is the Yukawa parameter – to be considered small – and unless otherwise specified, K (µ) is chosen so that i= j Ji, j = 1. Finite volume Gibbs states and their infinite–volume limits are defined in the standard fashion. Here, for reasons that may already be clear, there will be a vested interest in toroidal measures. In this context, with interactions ranging well beyond nearest neighbor, the convention that will be used, for any given rectangular ⊂ Zd , is to periodically repeat the spin configuration and count all the interactions between spins in and the image spins in c as dictated by the contents of Eqs. (2.1)–(2.2). The Gibbs measures for such “finite volume” spin configurations s at inverse temperature β using the toroidal extension T will be denoted by αβ,T ; that is to say αβ,T (s ) ∝ e−βH (s ) ,
(2.3)
where H (s ) denotes the extended periodic interaction as described above. For most purposes, tori with all linear dimensions the same will be sufficient and the corresponding measures, for tori of scale L, will be denoted by αβ,T L . The normalization constant for the weights in Eq. (2.3) – the partition function – will be denoted by Z ,β or Z L ,β as appropriate. For additional notational continuity, see [2] Sect. 1.2.
2.2. Mean–field theory. Mean–field theory for a Hamiltonian of the form Eq. (2.1) is defined as follows: If β denotes the usual temperature parameter and m ∈ Conv(), the free energy function is defined to be − 21 β(m, m) − S(m) which are, respectively, the energy and entropy terms. The latter will be discussed momentarily, the former will be denoted by − 21 βm 2 . This combination of energy and entropy will be denoted by β (m), the actual mean–field free energy is defined by minimizing β (m): FMF (β) =
inf
β (m) =
m∈Conv()
1 − [ βm 2 + S(m)]. m∈Conv() 2 inf
(2.4)
The entropy is defined, intrinsically, by S(m) = inf [G(h) − (m, h)], h∈E
(2.5)
where e G(h) = α0 (ds)e(s·h) . The entropy is concave which makes the overall combination of β (m) an interesting playoff between a convex and concave piece. We denote by Mβ the set of minimizing magnetizations. It is not difficult to see that Mβ is non–empty and, obviously, confined to the set C = {m ∈ Conv()|S(m) > −∞},
(2.6)
where the entropy is finite. Various convexity/continuity properties will be discussed in a brief appendix; for all intents and purposes we may restrict attention to the interior of C on which m (β) is a continuous function. In mean–field theory, first order transitions come about due to an exchange of minima. The structure of mean–field theory is analytically simple enough so that for first order transitions, the following scenario would seem to be generic: Definition 2.1. Generic MF first order scenario:
308
L. Chayes
(i) At some value of the temperature parameter, βtMF , a degeneracy in the minima of βtMF (m) has occurred. Namely there are two non-empty sets, M I and M I I in Conv() that are separated such that MβtMF = M I ∪ M I I . (These sets need not themselves be connected.) (ii) For all ζ > 0 sufficiently small, there is an interval [β I , β I I ] with βtMF ∈ (β I , β I I ), and separated sets M I and M I I such that (a) M I ⊂ M I and M I I ⊂ M I I . (b) At β = β I , M I contains all the minimizers of β (m) and similarly for M I I at β = β I I . In particular, at β = β I , β I (m) > FMF (β I ) + ζ for all m ∈ M I I and similarly, for M I at β = β I I . (c) Each m ∈ M I ∪M I I if and only if for some β ∈ [β I , β I I ] β (m)− FMF (β) < ζ. (d) Adopting, temporarily, notation for the ζ dependence of the items described in ζ ζ (a) and (b), then, as ζ ↓ 0, we have M I → M I (in the sense ∩ζ >0 M I = M I ) ζ ζ and M I I → M I I while [β I , β I I ] → {βtMF }. Remark 1. It does not seem possible, armed with only the unadorned definitions of this section, to prove a general theorem to the effect that all mean–field first order transitions follow the generic scenario. On the other hand, it is difficult to imagine a mean–field theory of the above type describing a first order transition that is not of this kind. Indeed, as we shall see in the proofs for specific systems, very little is used about the actual systems beyond the occurrence of the first order transition itself. Notwithstanding, some small knowledge of Mβ in the vicinity of βt is inevitably required and therefore one is forced into a case-by-case analysis. Fortunately, much of the difficult work along these lines has already been performed in [2 and 3]. Remark 2. It is further remarked that in the above definition, the temperature parameter has been chosen as the driving parameter for the simple reason that temperature driven transitions are more dramatic and hence better known. In mean–field theory, a first order transition can occur with the variation of other coupling parameters and, with obvious adjustments of notation, a first order scenario can be defined accordingly. Indeed, later on, there will be occasion to use the field driven version of the above first order scenario. 2.3. Layered systems. As mentioned earlier, a general mean–field approach for layered systems has been initiated in [10]. Of course (as discussed in [10]) such systems have been analyzed in the physics literature. But ultimately these analyses rely on independent notions of scaling – all of which turn out to be true. However, as an upshot, they lack in quantitative predictive power (e.g. the coefficient in the shift of the transition temperature for critical layered systems). It should be mentioned that the work in [10] pertains to the analysis of a particular experimental set–up and thus, as far as generalities are concerned, is only of a preliminary nature. Hence, for present purposes, we will be content to discuss an abbreviated version of some ultimate “general theory” for layered systems. In particular, it will be ensured by fiat that the formulation of layered systems fits immediately into the existing framework. The results herein will be sufficient to vindicate the calculations contained in [10] and, it should be mentioned, this was the initial motivation for the current work. The starting point is L copies of Zd which should be regarded, in a natural fashion, as a subset of Zd+1 . For physical applications, one would usually take d equal one or
Mean Field Analysis of Low–Dimensional Systems
309
Fig. 1. First order scenario for the 3–state Potts model: The space may be taken as the vertices of an equilateral triangle. For β < βtMF = 4 log 2 the unique global minimizer is m = 0 which at βtMF becomes degenerate with three secondary minima located a distance of 59 along the axes of the triangle. These three points represent the set M I I , the set M I is simply the origin. Insert shows the mean–field free energy as a function of the scalar magnetization concentrated along one of these axes (going off–axis only increases the free energy). The generic first order scenario follows easily from analytic considerations of [2]. Theorem 4.1 establishes a first order transition in the 2D version of this model with Yukawa couplings at small mass
two. At each site of this lattice, there will be an s ∈ , the position of which will be denoted by a Greek superscript to specify the layer and a Latin subscript to denote the position in Zd . Thus, in certain generality, one may write α δ −H = Ji,α,δ (2.7) j (si · s j ). i, j∈Zd 1≤α,δ≤L
Normally, the interaction in Eq. (2.7) does not connect the top and bottom of the layer; i.e. there is not an L + 1st layer which gets identified with the first layer. For mean–field study, the (finite subsets of) Zd become the complete graph of N −1 sites, that is each Ji,α,δ j becomes independent of i and j and gets scaled by N : Ji,α,δ j →
J α,δ Q . N
(2.8)
The (α, δ) dependence therefore represents a coupling between layers each of which acts as a mean–field system. The simplest non–trivial model, namely ⎧ ⎪ ⎨1; if α = δ α,δ Q = γ ; if |α − δ| = 1 (2.9) ⎪ ⎩0; otherwise already captures most of the essential features (which will be discussed below). For the purposes of this note, it will be assumed that Ji,α,δ j is of the form α,δ Ji,α,δ , j = Ji, j (µ)Q
(2.10)
310
L. Chayes
where Ji, j (µ) is of the Yukawa form that is in Eq. (2.2) and Q α,δ is a symmetric, positive definite matrix. The motivation for the above restriction – more generality is certainly possible – will be clear from the following proposition the proof of which is immediate and will be presented immediately. Proposition 2.2. For single spin–space (which is a subset of a finite–dimensional vector space with positive definite inner product) consider the layered model with Ji,α,δ j = Ji, j (µ)Q α,δ , where Ji, j (µ) is of the Yukawa form (or any other reflection positive pair interaction) and Q is positive definite. Then the interaction for the layered system as described by the Hamiltonian in Eq. (2.7) is reflection positive with respect to all the standard Zd reflections in planes between sites (i.e. in those {1, . . . , L} × Zd−1 hyper–plane segments with normals orthogonal to the layering direction). Proof. The idea is to write the model as a reflection positive model on Zd from which the result follows immediately. To this end, the spin–space will be L and if S and G are “spins” in L with S = (s1 , . . . sL ) and G = (g1 , . . . gL ) we define ((S ◦ G)) = Q α,δ (sα · gδ ). (2.11) α,δ
It is now is enough to demonstrate that ((· ◦ ·)) is a positive definite inner product on L this, we simply write, for any S ∈ L , the relevant expression: ((S ◦ S)) = . To see α,δ (sα · sδ ). Since the original inner product on is positive definite, we may Q α,δ express (sα · sδ ) = n λn cnα cnδ with λn > 0 and the demonstration is completed, by an exchange of the summations, and by noting the positivity of Q. Remark 3. It is remarked that the above sort of grouping is a device that has been employed before, e.g. in [20] – albeit with some extra restrictions. In addition it is noted, without proof, that non–mean–field interactions down the chain can be immediately incorporated into the above formalism by declaring this to be part of the “single– spin measure” on L . Indeed, this will form the basis for some analyses of quantum spin–systems in a future publication. The layered systems of interest have Q of the form in Eq. (2.9) and the generalization that Q α,δ = γ if |α − δ| = with 1 < < L and it may be assumed, for simplicity, that each γ ≥ 0. It is reemphasized that in the layering direction, the coupling is not to be periodically continued. The interaction matrix may be rewritten in the form Q = (1 + 2γ )11 + γ γ with γ = γ (2.12)
and γ the form of a generalized 1D Laplacian. The eigenvalues of this Laplacian are of the form −λk , where 1 k+1 ] (2.13) λk = 2γ [1 − cos π γ L+1
with k = 0, 1, . . . L − 1 and, under most circumstances, k = k. Notice, then that the matrix Q is positive definite if and only if 1 + 2γ cos (k+1)π > 0 for all k; a L+1 condition that shall be henceforth assumed. Let λ0 denote the magnitude of the smallest
Mean Field Analysis of Low–Dimensional Systems
311
eigenvalue. Then, as was shown in [10] there is a magnetic transition in the mean–field layered systems. Indeed, in the Ising system, at temperature parameters greater than that given by (L) −1 βc; I = (1 + 2γ − γ λ0 ) ,
(2.14)
the magnetization profile is not–trivial and satisfies the mean–field equation m α = tanh(β[(1 + 2γ )m α + γ γ m α ]),
(2.15)
where m α magnetization is on the α th layer. The situation for the XY–model is similar with (L)
βc;XY =
1 (1 + 2γ − γ λ0 )−1 2
(2.16)
and the magnetization profile satisfying mα =
I1 (β[(1 + 2γ )m α + γ γ m α ]) I0 (β[(1 + 2γ )m α + γ γ m α ])
,
(2.17)
where the I ’s are modified Bessel functions. It may be presumed that under most circumstances, the smallest eigenvalue corresponds to k = 0. (A sufficient condition, for large L, is that only a finite number of γ ’s are non–zero and that γ1 is large compared to all the others.) Under these (and perhaps other) restrictions, one may compute λ0 ≈
π2 1 γ 2 . L2 γ
(2.18)
It is noted, by a variety of arguments, that the “bulk” transition temperature is simply the L → ∞ limit of the formulas in Eqs. (2.14)–(2.16). Thus, there is a shift in the transition temperature which, written in reduced form, is given by (L)
(∞)
βc;I − βc;I (∞) βc;I
(∞)
=
γβc;I λ0 (∞) 1 − γβc;I λ0
(∞)
≈ γβc;I λ0
(2.19)
and similarly for the XY, where the approximate statement is, e.g. for L large under the assumption that Eq. (2.18) is valid. This is a quantitative version of the qualitative results for the temperature shift found in the aforementioned physics papers. So far – at the level of Eqs. (2.14)–(2.16) – this is exact only in the context of the mean–field theory. Later, in Theorem 4.6 we shall see that these formulas are of some pertinence to the actual systems (with Yukawa style interactions).
312
L. Chayes
3. Statement of Results While all the results of this note concerning phase transitions pertain to Zd with d ≥ 2, most of the results of interest are confined to the 2D cases. Indeed, as mentioned in the Introduction, essentially all of these results can be proved by other methods in d ≥ 3. Moreover, for cases of transitions associated with the breaking of continuous symmetries, some of the results, e.g. concerning local observables, are trivial since these symmetries are also broken in the actual systems when d ≥ 3. Nonetheless, only small additional effort is required for d ≥ 3 so the extra generality will be retained. The main general result of this note is Main Theorem. Let H denote a Yukawa Hamiltonian of the form described in Eqs. (2.1)– (2.2) on Zd with d ≥ 2 and suppose that the associated mean–field theory has a Generic First Order Scenario at temperature parameter βtMF. Then, for µ sufficiently small, the d–dimensional system also has a first order transition at a parameter βt which is near βtMF. In particular, the transition becomes asymptotically close to its mean–field description in the sense that (1) βt → βtMF and (2) on both sides of the transition, the block magnetizations averaged over block regions of scale 0 , where 0 is large compared with unity but small compared with µ−1 can only take on values assymptotically close to the permitted values that are predicted by mean–field theory. In the cases where certain phases of the model may be characterized by a broken symmetry, we have Proposition 3.1. Let H denote a d–dimensional Yukawa Hamiltonian of the form described in Eqs. (2.1)–(2.2) with d ≥ 2 and suppose that the model has a “discrete symmetry” meaning that there is a group, A, of linear maps {A : → |A ∈ A} which are measure preserving bijections and are also isometric (with respect to the inner product). Further suppose, in the context of the associated mean–field theory, that Mβ may be decomposed into k (with k finite) disjoint separated convex sets Mβ = ∪kj=1 M ( j) such that A acts transitively on {M (1) , . . . , M (k) }. Then, for all µ sufficiently small, the model exhibits (as many as) k distinct phases characterized by global magnetizations close to the values in M ( j). In particular, under these conditions, in any shift invariant ergodic Gibbs state derived from this interaction, the magnetization is in the vicinity of one of these sets. In many cases of interest, the first order phase transition may be from a symmetric state into a phase of broken symmetry – sometimes described as a transition featuring the spontaneous breaking of symmetry. For these cases, and certain generalizations, we have Theorem 3.2. Let H denote a d–dimensional Yukawa Hamiltonian of the form described in Eqs. (2.1)–(2.2) with d ≥ 2. Suppose that the associated mean–field theory has a Generic First Order Scenario at some βtMF and that there is a symmetry group A for the model of the sort described in Proposition 3.1. Here is is supposed that MβtMF = M I ∪M I I with M I and M I I separated and each the union of disjoint separated convex sets, e.g. M I = {M I(1) , . . . M I(k I ) }, on which A acts transitively and that this description holds, for the sets M I and M I I , throughout the range [β I , β I I ]. Then, for all µ sufficiently small, at a value of β near βtMF , the model exhibits (as many as) k I + k I I distinct ( j) phases, k I of them characterized by global magnetizations close to the values in M I ,
Mean Field Analysis of Low–Dimensional Systems
313
etc. Furthermore, throughout the above mentioned range of β, in any shift invariant ergodic Gibbs state derived from this interaction, the magnetization is in the vicinity of one of the above mentioned sets. Moreover, and of greater substance: For general, MβtMF = M I ∪ M I I under “only” the hypotheses that for every m I ∈ M I and m I I ∈ M I I , m 2I I ≥ m 2I +
2 E βtMF
for some E > 0, then the model, in fact, has a Generic First Order Scenario. Thus if the detailed structure of M I and M I I is as described in the first paragraph, all of the above conclusions hold along with the obvious necessity that this transition is accompanied by a discontinuity in the energy density. Moreover the gap that is not much smaller than (but perhaps larger than) E . Finally, in cases of a system with degeneracies which are not related by symmetry, the following result for the phase diagram will be established: Theorem 3.3. Let H denote a d–dimensional Yukawa Hamiltonian of the form described in Eqs. (2.1)–(2.2). Suppose that at parameter β, in the associated mean–field theory Mβ consists of k > 1 (non–trivial) interior points {m 1 , . . . m k } of C which, considered as elements of E , are linearly independent. If b ∈ E , we may consider the Hamiltonian augmented by the external field b as described subsequent to Eq. (2.1): (b · si ). (3.1) − βH → −βH + i
Then, for any pair p and q, 1 ≤ p < q ≤ k, there is a one–parameter family of p,q p,q fields bλ , where −1 ≤ λ ≤ +1 with supλ ||bλ || → 0 as µ → 0 such that for some λ ∈ (−1, +1) there is coexistence between two phases with magnetizations near mp and m q respectively. Furthermore for all λ ∈ [−1, +1] at the level of block observables, (on regions that are large compared with unity but small compared with µ−1 ) all the other values of magnetizations are suppressed with high probability. Remark 4. It is remarked that in the context of spin–systems, linear independence of the spin–states and/or the constituent sets of MβtMF is not a common occurrence in the systems that are usually studied. Indeed for a spin–system there is a physical which, perhaps, is endowed with ‘internal’ symmetries that are natural to the problem at hand. The dimension of may be vastly smaller than the actual or effective number of spin–states, but this is where the symmetries come into play. As a consequence of these symmetries, often enough, it is sufficient to align the external field with the desired state to select this state among all others related by symmetry. Then, even for weak fields, this will alter the nature phase diagram and the associated transitions. Well known examples include the q–state Potts model and the r –cubic model where, in the field–temperature plane, generic first order scenarios can be established with phase transitions at non–zero field which are markedly different from those in zero field. However, these sorts of systems – which do not satisfy the linear independence hypotheses of Theorem 3.3 but compensate by having a sufficient degree of internal symmetry – seem difficult to classify under a general principle. Thus, while it is clear that many particular results on in–field transitions can be established with the present methods, here, for the sake of brevity, we
314
L. Chayes
shall refrain from any specific claims. Indeed the Potts model in an external field was analyzed in [3] which, it seems, required a certain degree of effort. In the context of this work, the pertinent cases for Theorem 3.3 are particle systems where the different spin–states represent different particle types. There is no actual in the problem and thus one has to be constructed. Geometrically the simplest – and in a certain sense the most realistic – possibility is mutually orthogonal states in Rn where n corresponds to the number of species, along with some non–diagonal interaction. Here, of course, the external fields represent activities for the various species (albeit with the Euclidean notion of inner product, which is presumably not the same as the inner product defining the particle–particle interactions). Under these circumstances it is most plausible that degenerate minima of β , now representing excesses of various species, will indeed end up linearly independent. 4. Results for Specific Systems In this section, specific examples will be provided for the various phenomena alluded to previously. Foremost: • Discrete spin–systems with symmetry. The best known example is the q–state Potts model where, as is often the convention, each spin si is taken to point to a vertex of a (q − 1)–dimensional hypertetrahedron and the inner product is defined by the usual Euclidean dot product. Hence si · s j is essentially given by a Kronecker delta. On the basis of Proposition 3.1 and Theorem 3.2 along with some analysis of the mean–field theory (most of which was done in [2]) the following is established: Theorem 4.1. Consider the q–state Potts version of the Hamiltonian described in Eqs. (2.1)–(2.2) on Zd with d ≥ 2 and with q ≥ 3. Then for µ−1 large, there is a first order transition at some βt featuring (at least one) high temperature state with small or vanishing magnetization and (at least) q low temperature states characterized by substantial magnetization in the different hypertetrahedral directions. Furthermore, this transition is accompanied by a discontinuity in the energy density. For β > βt , the high temperature state disappears while the low temperature states persist, while for β < βt , the low temperature phases are not present. Finally, the value of βt as well as the free energy, magnetization and energy density at and beyond βt are (at least in some neighborhood of βt ) uniformly close to the appropriate mean–field formulas, e.g. as appear in [34]. Less well known but also of interest are the cubic models in which each spin si points to the face of an r –dimensional cube and one again employs the usual Euclidean inner product. Here the result is Theorem 4.2. Consider the r –cubic version of the Hamiltonian described in Eqs. (2.1)– (2.2) on Zd with d ≥ 2 and with r ≥ 4. Then for µ−1 large, there is a first order transition at some βt , which features coexistence between 2r low temperature states and a high temperature state and a discontinuity in the energy density. The properties of these states are similar, after appropriate modifications, to those described for the Potts model in the statement of Theorem 4.1. • Phase coexistence in models without symmetry. The vast majority of realistic lattice gasses fall into this category – there is no anticipation of symmetries as there would be in a spin–system. Indeed, under these auspices, the range of possible models and their
Mean Field Analysis of Low–Dimensional Systems
315
possible modes of behavior is so vast that the general situation is overwhelming. Thus we shall be content with a single example which, in the opinion of the author, could not easily be treated by other methods. Consider, then, a tertiary alloy – a lattice gas with three species a, b and c. It is assumed that each site i is occupied by one of the three species; thus we have the variables ηia , ηib and ηic each in {0, 1} with ηia + ηib + ηic = 1. It will be stipulated that each species has a pair interaction with strengths Ja > Jb > Jc – so that species b and c will be suppressed, more heavily the latter. As “compensation” there will be repulsion between species a and b and an attraction between b and c. The (formal) Hamiltonian for the model is therefore given by − βH =
[Ji,a j ηia ηaj + Ji,b j ηib ηbj + Ji,c j ηic ηcj − K i,abj ηia ηbj + K i,bcj ηib ηcj ].
(4.1)
i, j
In the above, the notation for couplings has been defined so that all the K ’s and J ’s are non–negative. Here, we have the following: Theorem 4.3. Consider the Hamiltonian defined in Eq. (4.1) with couplings given by the Yukawa form in Eq. (2.2) without the specific normalization condition. Let Ja denote a the sum Ja = i J0,i and similarly for Jb , . . . , K bc . Let us express Ja = J + Da , Jb = J + Db and Jc = J . Then the following holds for all µ sufficiently small: For J sufficiently large, K ab , K bc comparatively (sufficiently) small and Da , Db (sufficiently) , K ) – both components positive – such that smaller still, there is a point K = (K ab bc the Hamiltonian augmented with various (natural) activities exhibits phase coexistences between pairs among three types of phases at certain values of the activities. The three phase types are characterized by dominance of one of the species over the other two. Furthermore, as µ → 0, the requisite activities for coexistence tend to zero. • Low temperature behavior for low–D models with continuous symmetry. Here we shall state the formal results for models with O(n) symmetry; see the paragraph following the proof of Corollary 5.2 for further discussion. Theorem 4.4. Consider the standard O(n) spin–system with n ≥ 2, i.e. each si is an n–dimensional unit vector and the inner product is the usual Euclidean dot product with the Yukawa interaction (Eqs. (2.1)–(2.2)) in d ≥ 1. Let β ≈ βcMF = n. Then, for all µ sufficiently small, there is a scale 0 (which tends to infinity as µ → 0) such that on any compact interval of temperatures, the spatially averaged magnetization at this scale (cf. Eq. (5.1)) is, with high probability, uniformly close to m MF (β)v, ˆ where vˆ is an n– dimensional unit vector and m MF is a solution of the mean–field equation m = mn (βm), where mn (h) is the scalar magnetization function: +1 mn (h) =
hx 2 n−3 2 xd x −1 e (1 − x ) . +1 n−3 hx (1 − x 2 ) 2 d x e −1
(4.2)
Moreover the energy and free energy per spin is uniformly close to the appropriate mean–field formula. The O(n) nematic models are most easily described in the context of the O(n) spin– systems with the pair interaction between spins at sites i and j replaced by (si ·s j )2 . Note that for n = 2 this is, for all intents and purposes, equivalent to an XY spin–system; but
316
L. Chayes
not so for n ≥ 3 which will henceforth be assumed. For a variety of reasons (not all of which are understood by the author) this is always presented via the traceless matrices K p,q = s( p) s(q) −
1 p,q δ , n
(4.3)
where s( p) denotes the p th component of an O(n) spin. Then, the pair interaction between the spins at sites i and j is given by Tr(K i K j ). In any case the device of using symmetric traceless n × n matrices with this notion of inner product and with an a priori measure given by the pullback from the unit sphere in Rn has the advantage that it constitutes the ingredients for a bona fide mean–field theory. Since this mean–field theory was the subject of a good deal of analysis in [2], the relevant results will be summarized briefly: (a) For all β, the minimizing K are orthogonally equivalent to a diagonal matrix of the form 1 1 K = λ(β)diag[1, − ,...,− ], (4.4) n−1 n−1 where λ is a solution of the mean–field equation 1 β λx 2 n−3 (1 − x 2 ) 2 (x 2 − n1 )d x 0 e λ= 1 2 n−3 β λx (1 − x 2 ) 2 d x 0 e
(4.5)
with β = (1 − n1 )β. Indeed all local minima of the free energy function have this property. (b) There is a βtMF such that for β ≤ βtMF , β (K ) is minimized by K ≡ 0 while for β ≥ βtMF , β (K ) is minimized by a non–trivial K as described in (a) with some λ(β) ≥ λ(βtMF ) = λMF t > 0. Thus we see a standard mean–field type of first order transition featuring coexistence of states with differing energy that is accompanied by the breaking of a continuous symmetry. Here we shall prove: Theorem 4.5. Consider an O(n) nematic spin–system in d ≥ 2 with n ≥ 3 as described above (spins K i are n × n symmetric traceless matrices, the inner product given by Tr(K i K j ) and α0 the pullback of Haar measure on the unit n–dimensional sphere; or the simpler description with the usual unit n–dimensional spins and the pair interactions defined by the square of the Euclidean dot product) with couplings as given in Eq. (2.2). Then, for all µ sufficiently small, there is a βt (µ) with βt → βtMF such that at βt (at least) two states coexist; one, a high–temperature state where the energy is small and the other where the energy is substantial. Moreover, in the latter there is a scale 0 , where 0 1 if µ−1 is large such that within blocks of this scale, the spatially averaged nematic–spin variable is, with high probability, of the form in Eq. (4.4) or an orthogonal transformation thereof. In the high temperature state, and for all β < βt the spatial average at scale 0 is close to zero. For β > βt , on any compact interval, there is a µ0 such that for all µ < µ0 the statement concerning the low temperature state holds for β > βt . • Low temperature results for the layered Ising and XY models. Here some preliminary results for the layered systems are presented. More general results for the continuous spin models especially in dimension greater than two are possible but are not of immediate physical relevance and so will be omitted.
Mean Field Analysis of Low–Dimensional Systems
317
Theorem 4.6. Consider layered Ising and XY on Zd × {1, . . . L} as described by the interactions in Eqs. (2.10) and (2.1)–(2.2), where Q is as described just prior to Eq. (2.12) with each γ non–negative. Then for β less than the formulas given in Eqs. (2.14) and (2.16) respectively, the magnetization vanishes. Moreover, for all µ sufficiently small, there is a scale 0 (which can tend to infinity as µ → 0) such that on any compact interval of temperatures, the block averaged magnetization profiles, free energies and energies are uniformly close to those given by the appropriate mean–field formulas. Finally, in the case of the Ising version, the global magnetization profile agrees closely with the block magnetization as just described.
5. Proofs 5.1. General properties: magnetics. Most of the results stated in Sect. 3 are a direct consequence of the following: Lemma 5.1. Consider the spin–system defined by the interaction in Eq. (2.1) with couplings as in Eq. (2.2) at interaction parameter µ defined on the d–dimensional toroidal lattice. For an integer 0 , let 0 denote a cube of side length 0 and let m 0 =
1 si | 0 |
(5.1)
i∈ 0
denote the block magnetization of 0 , and let us assume for simplicity that the linear dimension of the torus, L, is of the form L = 2k 0 . Let m ∈ Conv(), and if > 0 is a 0] real number let N (m) denote the neighborhood ball of radius about m. Let K[ N (m) denote the event 0] K[ N (m) = {m 0 ∈ N (m)}.
Then there is an = ( 0 , µ, ) with the explicit bound 1 < 0 = βω + β 2 + cµ 0 βω2 , 2 where c is a uniform constant of order unity and ω = sup{m|m ∈ Conv()}
(5.2)
such that as L → ∞, N (m)−FMF (β)−) d0
−(β 0] lim αβ,T L (K[ N (m)) ≤ e
,
(5.3)
where, in the above N β (m) =
inf
m ∈N (m)
β (m ).
(5.4)
318
L. Chayes
Proof. By the standard chessboard estimates, ⎡
1 Ld
[ 0 ]
⎤ d0
⎢ Z L ,β (KN (m)) ⎥ 0] αβ,T L (K[ ⎦ , 1 N (m)) ≤ ⎣ Ld Z L ,β
(5.5)
0] where Z L ,β (K[ N (m)) is the partition function which is constrained so that in each translate of 0 by a vector with integer multiples of 0 as components, the translate 0] of the event K[ N (m) occurs. (Arguments of this sort are found in the classic papers on reflection positivity. All relevant results for this work can be found in [30]; the interested reader is also referred to the recent review [1] where these sorts of arguments and some new extensions are well explained. Within the above two reviews, all the necessary referd ences can be found.) Therefore let us seek upper bounds along the lines of ∼ e−β (m)L d on the constrained partition function and lower bounds of the form ∼ e−FMF L on the full partition function. A proof of the latter can be found in the beginning of Theorem 1.1 in [2] although other versions of this result are part of the classic literature on the subject; see, e.g. the book [32] Sect. II.13 – II.14. In any case, we have that
1/L d
Z L ,β ≥ e−FMF +gL
−1
(5.6)
with g a constant. Turning to the necessary upper bound, let us begin with an estimate of the energetic contribution to the partition function under the above mentioned constraints. In particular, we will show that under this constraint, the total energy is approximately − 21 βm 2 times the volume. (Fortunately, it turns out, the ensuing estimate does not depend on the details of how the constraints are satisfied. Moreover the bounds are nearly optimal in the sense that a similar derivation produces a lower bound which does not differ by too much.) Consider two blocks, which are appropriate translates of 0 , that are labeled V p and Vq respectively. Let us define the average coupling J p,q =
1 Ji, j . | 0 |2
(5.7)
i∈V p j∈Vq
For generic i ∈ V p , j ∈ Vq , we may write Ji, j = J p,q (1 + κi, j ), however, it is clear that under the condition µ 0 1, the κi, j are small. Indeed since no two points in V p differ by more than the order of 0 – and similarly for points in Vq – from Eq. (2.2), it is easily seen that |κi, j | ≤ 2cµ 0 ,
(5.8)
where c depends on dimension – but not on p and q – and the two is for convenience. Thus 0] if m p and m q (which satisfy the criterion for the event K[ N (m)) are the magnetizations in their respective blocks, then Ji, j (si · s j ) = | 0 |2 J p,q (m p · m q ) + J p,q κi, j (si · s j ). (5.9) i∈V p j∈Vq
i∈V p j∈Vq
Mean Field Analysis of Low–Dimensional Systems
319
The rightmost term in Eq. (5.9) is relatively small and, further, the quantity (m p · m q ) may be replaced by m˜ 2 for any m˜ ∈ N (m) at cost of only a small additional error. Thus, all in all – for any m˜ ∈ N (m) – Ji, j (si · s j ) ≤ J p,q | 0 |2 [m˜ 2 + 2 ω + 2 + 2cµ 0 ω2 ] i∈V p j∈Vq
≡ J p,q | 0 |2 [m˜ 2 +
2 ]. β
(5.10)
It has been assumed that p = q; a similar–minded argument with corresponding results may be obtained for the “diagonal” terms – or these may be neglected altogether in the small µ limit. In any case, summing Eq. (5.10) over all pairs, (and noting the normalization condition on the Ji, j described after Eq. (2.2)) the anticipated estimate for the energetics has been obtained. Let us turn to the entropic considerations. Since each cube in the torus acts independently, the term to be estimated is simply the appropriately constrained α0 –measure of the spin–configurations in 0 . It is claimed that there is some m ∈ N (m) such that dα0 (s j )][I{m 0 ∈ N (m)} ] ≤ e S(m )| 0 | . (5.11) [ j∈ 0
The derivation is as follows (assuming that the left-hand side is not trivial): Since, conditionally with probability one, m 0 ∈ N (m), then the average of m 0 , denoted by m is also in N (m). Let A 0 , N denote the normalized measure corresponding to the constrained product measure. Then, by Jensen’s inequality, (5.12) d A 0 , N e j (s j ·h) ≥ e| 0 |(h·m ) , i.e.
[
dα0 (s j )e
j∈ 0
≥
[
j [(s j ·h)−(m
·h)]
][I{m 0 ∈ N (m)} ]
dα0 (s j )][I{m 0 ∈ N (m)} ].
(5.13)
j∈ 0
The desired result is obtained by relaxing the constraint on the left and seeking the supremum over h. It is noted that in the preceding, some mild use has been made of the fact that N (m) is a convex set. If it happens that this neighborhood intersects the complement of Conv() the restricted set is still convex. (Or we may stay with the full set and rely on the fact that the measure provides no weight to the complement of Conv() and that outside of Conv(), the free energy is infinite.) 0] To within the stated error tolerances, the upper bound on Z L ,β (K[ N (m)) as it now stands picks a particular point in N (m) to evaluate β . Obviously, this may be replaced with the worst case (perhaps limiting) scenario. As an immediate corollary, we rule out the possibility of any non–mean–field like magnetizations and extend this latter statement to non–toroidal states.
320
L. Chayes
Corollary 5.2. Let η (β) denote the set of magnetizations such that β (m) is within η of FMF (β) and let B[ 0 ],η denote the event that m 0 is in the complementary set, i.e. that the block magnetization in 0 corresponds to a mean–field free energy which is further than η from any minimizer. Then, with apologies for the 3, there is a K (η) such that for all µ 0 sufficiently small, if L → ∞ along power of two multiples of 0 , lim αβ,T L (B[ 0 ],3η ) ≤ K (η)e−η 0 . d
(5.14)
Further, if αβ (−) denotes any shift–invariant infinite–volume Gibbs state corresponding to the Hamiltonian in Eqs. (2.1) – (2.2), then there is a δ that tends to zero with the right-hand side of Eq. (5.14) such that αβ (B[ 0 ],3η ) < δ.
(5.15)
Moreover, if β ranges over a bounded set, then for a given η, the above holds uniformly for any fixed pair (µ, 0 ) – provided µ 0 is sufficiently small. Proof. Once we establish the result in Eq. (5.14), the one in Eq. (5.15) is a direct application of Theorem 2.5 in [5] in the special case of only one “good” event. First let us prove this for fixed β. For m ∈ 3η , let us find a m such that (1) N m (m) ∩ η = ∅, (2) 0 ( 0 , µ, m ) < η. The ability to achieve the former relies on the continuity of β (see the Appendix) and the latter already relies on µ 0 sufficiently small. By compactness, only a finite number, K (η) of these are needed and the result follows immediately. As for the uniformity, let us divide the bounded set of β’s (conveniently thought of as an interval) into pieces each of size no more than β , where 21 β ω2 η. Notice that the free energy at any m ∈ Conv() cannot change by more than this small amount as β varies over the piece. Thus, there is ample space between the union of the η (β)’s and the union of the 3η ’s; items (1) and (2) above can be modified accordingly and the result holds throughout the piece. Since the estimate in Eq. (5.14) depends only on the number, K , of sets used, the maximum can be chosen. Remark 5. There has not been any attempt to provide an optimal scheme for the rate of convergence both here and in the second corollary below. Indeed, it is obvious that the estimates are grossly inefficient. For example, it is clear that the principal contribution to an inequality of the form in Eq. (5.14) should come only from the edge of 3η . This could be existentially rectified by a modification of the second condition to allow bigger neighborhoods in regions far away from the minimizer and/or using the large value. However, the upshot would still be existential so nothing practical would have been gained. Indeed, better estimates, if actually required, can always be obtained in the context of specific models where the particulars of β (m) can be brought into play. The second corollary, namely that as the range of the interaction tends to infinity, the free energy converges to FMF , is also immediately available. It should be remarked that many cases of interest are covered by Theorem II.14.1 in the book [32] (which is in turn based on [29]). Indeed, results of this sort date back to the work of Kac in the early 1960’s.
Mean Field Analysis of Low–Dimensional Systems
321
Corollary 5.3. Consider a spin–system on Zd with interactions as described in 1/|T | Eqs. (2.1)–(2.2). Let Fµ (β) denote the free energy: lim L→∞ Z L ,β L = e−Fµ (β) . Then limµ→0 Fµ (β) = FMF (β). Further, if β ranges over a bounded set, the convergence is uniform. Proof. The lower bound on Z L ,β is already in place. Allowing , 0 , = ( , µ, 0 ), etc. to denote their previous meanings (with fewer restrictions), let K denote the number of neighborhoods required to cover the whole space. Then, by chessboard estimates (see, in particular the “subadditivity lemma”, Lemma 6.3 in [4]) we have |T L | 1 d
Z L ,β ≤ [K ] 0 e−(FMF −)
,
(5.16)
and the result follows first by taking L → ∞ and then µ → 0 which allows 0 → ∞, taking care of the K –term so the (and hence also the < 0 error) can go to zero. Uniformity is established as was done at the end of Corollary 5.2: Most of the above has harmless β dependencies; the interesting term, involving the K , depends on η through , and can be uniformly bounded over the whole range: The term is dispensed with by considering worst case (in β) possibilities. Discussion/Examples. The above results are disquieting, at least at first glance, since the conclusion is that in any system with spread–out interactions, the local magnetization will (more or less) only take on values permitted by mean–field theory. Let us consider the implications in two principal classes of examples: General one–dimensional systems and 2D systems with continuous symmetries. Needless to say, in both cases, the commonly studied mean–field models have phase transitions that are associated with the singular behavior of the magnetic order parameter. This behavior is obviously not possible in a one–dimensional system with Yukawa interactions. (In particular, such interactions are known to satisfy Dobrushin’s criterion for complete analyticity [25].) Notwithstanding, the above tells us that at least locally, but not too locally at length scales 1 ≤ 0 µ−1 , a one–dimensional system will appear to have undergone a phase transition at around a temperature TMF (∝ [βtMF ]−1 ). So, e.g. in cases of magnetic symmetry breaking, once T < TMF , there must be large patches of ordered phase each approximately magnetized according to mean–field theory but, overall, canceling out. A similar picture holds for phase transitions that are not associated with symmetry breaking e.g. that have an m(β) (perhaps with a non–magnetic interpretation) undergoing interesting “discontinuous–like” local behavior at temperatures around TMF . It is remarked that while it is obvious that something along the lines of the above must happen as T → 0, it is now seen, in the large range limit, that this behavior initiates at around TMF and occurs in an understandable and controllable fashion. Indeed, it should be mentioned that for the Ising model [7] and the standard O(2) & O(3) Heisenberg models [6], results along these lines have been obtained previously. However in these instances some particulars of the O(n) with n = 1, 2, 3 were exploited and, moreover, a sustained effort was required. Let us now turn to some interesting 2D cases, namely magnetic systems – such as O(n) systems with n ≥ 2 – that have continuous symmetries. On general principles [19,26] and generally provable by the methods of [12], the symmetry cannot be broken; that is to say in an infinite–volume state, the magnetization will vanish. So in these circumstances, it would appear, the situation is on par with the general one–dimensional systems. However, there are two outstanding exceptions. (1) Cases where the mean–field
322
L. Chayes
model enjoys a discontinuous transition accompanied by a (discontinuous) breaking of the continuous symmetry. Here we will find that the discontinuity persists at the local level signaling a 1st order transition, i.e. phase coexistence. However, at the global level, or even at the level of very large–scale behavior there is no breakdown of the symmetry. (2) The specific case of the 2D–XY (or O(2)) model, has a phase transition, the Kosterlitz Thouless transition, [14,22] with the low temperature phase featuring power law decay of correlations. It is not hard to show, via correlation inequalities, that this must also be the case for the mean–field like version; the question of whether or not this transition actually occurs in the vicinity of TMF is under investigation. 5.2. Further properties: energetics. Till now, our attention has been focused on the block observables m 0 (and their translates) and, as the range of the interaction gets large, we have seen that these concentrate near the values of the magnetization that are dictated by the corresponding mean–field theory. A similar result for the energy is a more ambitious endeavor since, ultimately, the magnetization observables are strictly local whereas the energy observables are more diffuse. In particular, the results of this subsection will be of a more technical nature – e.g. some additional hypotheses concerning the mean–field theory will be required. In fact, not all of this section is strictly necessary when the free energy minima are simply isolated points, which is often enough the case. Indeed, under these circumstances, the magnetization simply drags the energy along with it and a part of the labour of this subsection is rendered unnecessary. Therefore, for some, this section may be read lightly without much loss of continuity. The central result of the subsection amounts to a statement that the actual (µ 1) systems must have “energetics” close to values corresponding to minima or near minima of β . The developments will come about in two stages: The first argument goes via quasilocal energy observables, which holds in full generality. The second part of the argument involves the energy density itself which, if there is coexistence, will require d ≥ 2. The latter is, of course, an absolute necessity since, in d = 1, the energy is continuous and therefore, when the mean–field theory has a discontinuity, the actual system will take on intermediate values. This is brought about by combinations of spatially separated regions which themselves have nearly sharp “allowed” values but are uncorrelated. By contrast, for d ≥ 2 the above mentioned quasilocal energies maintain a coherence and, as a consequence, the global energy is always near some value corresponding to a minimizer or near minimizer of β . This necessitates, above d = 1, energy discontinuities/coexistences in the actual small–µ systems whenever they are exhibited in the corresponding mean–field theory. Let us start with some hypotheses on the energetics corresponding to the set Mβ of minimizers for β (m): Definition 5.4. Regular Energy Hypothesis: Let β (m) denote a mean–field free energy function and η (β) denote the set of magnetizations such that β (m) is less than FMF (β) + η. Then the mean–field theory is said to satisfy the Regular Energy Hypothesis if for all (sufficiently small) η there is a δ(η) with δ(η) → 0 as η → 0 and an R – which does not depend on η such that η (β) may be expressed as the union of R separated sets η =
R r =1
with the properties:
) (r η
(5.17)
Mean Field Analysis of Low–Dimensional Systems
323
(r )
(i) Each m ∈ η has energy (∝ m 2 ) that is within δ of some fixed Er and ) (ii) The sets (r η are decreasing (so the limiting set is either empty or of constant energy Er ). Remark 6. It is noted that the number, R, of such sets is allowed to shrink with η (this is already within the technical leeway of the definition) but not allowed to grow – and especially not without bound. Basically these sets should be thought of as neighborhoods of certain sets of constant energy that minimize the free energy function at or near temperature parameter equal to β. Further, it is underscored that the values Er need not ) themselves be separated, just the magnetizations within the sets (r η that these energies represent. Finally, concurrent with the Generic First Order Scenario, it seems that in all practical circumstances, the Regular Energy Hypothesis holds. However, in contrast to the former, it may well be possible to cook up a model where these hypotheses are violated. Certain Mild Restrictions. Here we shall perform exercises on two scales: the local, 0 which is small compared with µ−1 and the quasilocal, 1 which will be large compared with µ−1 . It will be convenient (not strictly necessary) to assume that 1 is a multiple of 0 and, even more so, that the lattice size L is a power of two multiple of both 0 and 1 . Moreover, in contrast to the magnetic results where no specific details were required concerning how µ 0 → 0, here some mild constraints will come into play; in particular, it will be necessary to ensure that 1 does not go into infinity too fast relative to 0 . The restriction is indeed mild and is easily satisfied if 1 is any superlinear power and 0 any sublinear power of µ−1 . In the forthcoming, often without specifics, all of the above will be referred to as the Mild Restrictions. Definition 5.5. Consider a spin–system described by Eqs. (2.1)–(2.2) and suppose that the corresponding mean–field theory satisfies the Regular Energy Hypothesis. Let 0 and 1 be two length–scales satisfying, if appropriate, the Mild Restrictions. Let m 0 denote, as previously, the block average magnetization in 0 . For a of the form (integer vector)× 0 , let 0 (a) denote the translation of 0 and m 0 (a) the block magnetization in 0 (a). Let θ > 0 and η, etc. denote previous meanings. The block 1 is said to satisfy the Thouroughgood condition of type r (which is actually the (θ , η; r )–Thouroughgood condition) if (r )
(1) For all a such that 0 (a) ⊂ 1 , m 0 (a) ∈ η . (2) The Thouroughgood block energy defined by 1 1 E 1 = − β Ji, j (si · s j ) 2 | 1 | i, j∈ 1
satisfies |Er | − |E 1 | < θ . It noted that if β (m) is minimized by isolated points (and η is sufficiently small while θ is not too small) then condition (2) is trivially satisfied by condition (1). Condition (2) becomes interesting when there is a continuum of minimizing magnetizations which are all of the same “length”. Proposition 5.6. Consider an interaction of the type described in Eqs. (2.1)–(2.2) and suppose that the corresponding β satisfies the Regular Energy Hypothesis. Then, for
324
L. Chayes
any (small) η there is a θ with θ (η) tending to zero such that as for all µ−1 and 0 , 1 sufficiently large satisfying certain Mild Restrictions, the spins in 1 satisfy the (θ, 3η; r )–Thouroughgood condition for at least one value of r with probability tending to one. Proof. Let us begin by ruling out the possibility of not satisfying criterion (1). This can come about in two ways: First, one of the subblocks 0 (a) can satisfy the analogue of B[ 0 ],3η (meaning that m 0 ∈ / 3η (β)). Here we may use Eq. (5.14) and over–count the location. The Mild Condition will force
1 0
d
K (η)e−η 0 → 0 d
as µ → 0. Next there is the possibility that all subblocks are in good shape magnetically but there are specimens that have differing r –phenotype. This, by necessity, will result in a mismatched neighboring pair. Such a possibility can be demonstrated as unlikely by a standard chessboard estimate: Suppose that there are two (particular) neighboring ) (s) blocks with respective block magnetizations in (r 3η and 3η with s = r . Reflecting this event till the torus is covered, it is found that in one direction (along the direction of the pair) there is a dashed pattern and this dash gets extended into the other (d − 1)–directions – stripes, plates, etc. Let us denote the constrained partition function on the torus T L by Z L ,β (r, s). For some fixed –scale, the entropy of Z L ,β (r, s) can be estimated along the lines of Eqs. (5.11)–(5.13) resulting in a factor of
−d
[G˜ r G˜ s ] 0 e 2 [S(m r )+S(m s )+1 ] 1
|T L |
,
where S(m r ), S(m s ) are mean–field entropies representative of the sets r3η and s3η – note that the entropy cannot vary much in these sets – the G˜ (which depend on η) are the appropriate analogs of the K (η) that has appeared before and 1 is a tolerable error – vanishes with η. Let us turn to the energetics of Z L ,β (r, s). Let rr and s denote the sublattice of blocks covered by the two types of events. Following Eqs. (5.7)–(5.10) we may write
Ji, j (si · s j ) ≈ | 0 |2
p,q
i, j
J p,q (m p · m q ) = | 0 |2
(m p · J p,q m q ) p
(5.18)
q
with formal acknowledgment of the small debt from the first step to be made later on. Let us look at the inner summand in the last term – with fixed p – and suppose that p ∈ rr. The terms where q ∈ rr may be replaced, as an upper bound, by β2 (|Er | + δ) – essentially m 2p . Adding and subtracting a “favorable” term, namely m p , for q ∈ s we arrive at (m p ·
q
J p,q m q ) ≤
1 2 (|Er | + δ) + (m p · J p,q [m q − m p ]). | 0 | β q∈s
(5.19)
When this gets (multiplied by | 0 |2 and) summed over p ∈ rr the Er –type term will be half what is expected for an energetic contribution because rr is half of the lattice.
Mean Field Analysis of Low–Dimensional Systems
325
Performing the same routine if p ∈ s and putting additional required factors of β and yields 1 1 1 Ji, j (si · s j ) ≤ β |Er | + |Es | + 2 | L | 2 2 2 i, j
1 − β| 0 |2 J p,q (m p − m q )2 2 p∈rr q∈s
1 2
(5.20)
with 2 another tolerable error term which also accounts for the neglect in Eq. (5.18). The final term cannot be diminished. The minimal squared distance between any pair (r ) (s) magnetizations with one in 3η and the other in 3η will be denoted by β2 Vr,s with Vr,s strictly positive by the Regular Energy Hypothesis. It is noted that as µ → 0, then | L |−1 | 0 |2 times the sum of the J p,q tends to 41 – let us denote a (uniform) lower bound, valid for all µ sufficiently small, by κ; we have arrived at −d 1 −d 1 d Z L ,β (r, s)1/|T L | ≤ [G˜ r ] 0 e 2 (|Er |+S(m r )) [G˜ s ] 0 e 2 (|Es |+S(m s )) e3 ×e− 0 κVr,s (5.21) with all previously discussed errors amalgamated into the final 3 . Aside from terms that are close to unity, the term above in the large square bracket is identified as a negative exponent of the mean–field free energy which, e.g. according to Eq. (5.6), is canceled by the denominator in the chessboard estimate. Thus the probability of a particular mismatched neighboring pair of (otherwise decent) subblock magnetizations is bounded by a quantity that is exponentially small with rate ∝ | 0 |. Accounting for all possible locations and all possible types of mismatches multiplies this by a constant (which depends on R and d) times | 1 |/| 0 | so, overall, is actually more heavily suppressed than the situation where one of the blocks had a “bad magnetization”. In any case, it may be declared that criterion (1) is satisfied with high probability. Let us turn to criterion (2). As may already be obvious, a central reason for the stipulation 0 µ−1 1 is that the total energy of most spins in 1 is accounted for by the pairings with other spins in 1 . Indeed, this reasoning only breaks down for sites that are a distance of order µ−1 from the edge; let us denote Ji, j , (5.22) Q(µ, 1 ) = i∈ 1 j∈c
1
−1 then it is not hard to show, if µ 1 → ∞ with µ → 0, that [µ−1 d−1 1 ] Q(µ, 1 ) tends to (1)\(2) a definitive constant. So, letting B[ ],r denote the event that the spins on 1 satisfy, for 1 energy Er , the Thouroughgood criterion (1) but not criterion (2), let us perform another (1)\(2) chessboard estimate. Let us use Z L ,β (B[ ],r ) to denote the constrained partition func1 tion; it is clear that the entropy is, more or less, |T L |S(m r ) but the energetic contribution −1 is no more than [|Er | − θ + | 1 | Q(µ, 1 ) · 21 βω2 + 4 ]|T L | – all terms understood to be appearing in the exponent – with the ω2 term representing a bound on the largest conceivable energetic contribution to the configuration coming from interactions between spins in differing blocks. Canceling, as before, the mean–field free energy term from
326
L. Chayes
numerator and denominator, taking the appropriate power, namely 1 d /|T L |, we arrive d (1)\(2) at the estimate for αβ,T L (B[ ],r ) of e−[θ−5 ] 1 for yet another j and the proposition is 1 proved. Remark 7. In essence, the above proposition already tells us that in case R = 1, the actual system will have energy close to the mean–field value. If the mean–field theory is sufficiently regular (and let us not pause to axiomatize the concept) it would seem that except at points of a discontinuity, the large µ−1 system will follow the mean–field theory. However, in isolation, this result is not all that illuminating – especially if one considers that so far, there has been no stipulation that we are not in d = 1. Indeed, let us assume that the mean–field theory has a discontinuous transition in the energy at some βt and that the actual system is at some nearby β. Then, in order to ensure that the minimizers on the other side of the transition are excluded from 3η , smaller and smaller η’s have to be chosen as β → βt . This in turn necessitates increasingly larger values of µ−1 in order to bring the results of Proposition 5.6 into play. Thus a non–uniform type of convergence will transpire in the vicinity of a mean–field transition temperature – a result which, after a moments thought, one always anticipates, even in d = 1. Our next result, which definitively requires d ≥ 2, shows that, under reasonable hypotheses, systems with a large enough range parameter are (uniformly) close to some energy corresponding to a near–minimizer of the mean–field theory. This, of course, allows us to keep η fixed at the “expence” of multiple possibilities for the energy. Proposition 5.7. Consider a spin—system on Zd with d ≥ 2 that is described by the interaction in Eqs. (2.1)–(2.2) and suppose that at temperature parameter β, the associated mean–field theory satisfies the Regular Energy Hypothesis. Then, for all µ suffi˜ ciently small, there is an η and a δ(η) and a set of infinite–volume Gibbs states emerging from the αβ,T L such that with probability one, the energy density in any configuration is (r ) within δ˜ of a value Er , r = 1, 2, . . . R associated with . Moreover, if the hypotheses 3η
(r )
hold with a uniform bound on the separations between the various sets 3η while β ranges over a compact set, then with η and δ˜ fixed (and η sufficiently small) the result holds uniformly for all µ below some minimal value. Proof. For a system of the type described above, let η > 0 and let θ (η) denote the quantity described in Proposition 5.6. If R = 1, the argument is somewhat simpler but in any case, let us employ an argument appropriate to R ≥ 2. It is observed that if two distinct “blocks” – translates of 1 by lattice vectors with components (integer)× 1 – are of different Thouroughgood energy type, then these blocks are separated by a closed ∗–connected contour consisting of non–Thouroughgood blocks or Thouroughgood blocks that interface with a Thouroughgood block of a different energy type. The former sort of contour element was, manifestly, estimated in the previous proposition. As we shall see, so has most of what is needed for the latter. Indeed, supposing that the two energy types are r and s, let us neglect all aspects of the interface event save for the fact that there is a row of 1 / 0 boxes which are translates of 0 that have their (r ) magnetization in 3η and this row faces a similar opposing row with magnetizations in (s)
3η . When all this gets reflected to cover the torus, the constrained partition function is exactly Z L ,β (r, s). This time, our estimate will be
Mean Field Analysis of Low–Dimensional Systems
327
[Z −1 L ,β Z L ,β (r, s)]
2 0 d−1 1 |T L |
which is a tremendous – albeit not unexpected – enhancement of the previous run through. Thus it is claimed that with high probability, on T L most blocks are situated exterior to contours. In particular, let us denote by ϑ the estimate for contour elements – the probability that a given block belongs to a contour of length k is bounded above by a constant times ϑ k . Then, it is not difficult to show that if H is any appreciable number (perhaps not too large but certainly in excess of unity) the probability that a fraction larger than ϑ H of all blocks reside inside or on contours tends to zero exponentially at an estimated rate that is a (sublinear) power of the volume. Since results of this sort are well known and the subject of many works in specific systems, let us proceed with a terse, highly non–optimal derivation. d Let us suppose that we desire H ϑ[L −1 1 ] or more non–exterior boxes. We shall say that contours which have between T and 2T elements have done their share if they prod duce at least T1 H ϑ[L −1 1 ] such boxes. Starting at T = 1 and proceeding along powers of 2, it is clear that if none of the groupings have done their share, the event has failed. a0 Let us start (and end) with all T ’s that satisfy T > T0 = [L −1 1 ] , where a0 > 0 is to be determined below. There simply are no such contours with probability greater d T0 than 1 − b1 [L −1 1 ] ϑ , where b1 is a constant of order unity. So it is fairly safe to assume that none of these have done their share. For T ’s that are smaller, let us go to a block lattice with cell size of e.g. 8T 1 , and focusing on a sublattice of 2T 1 , ask if any “site” in this part of the cell belongs to a contour of size between T and 2T . If yes, we surrender the whole cell and relax the criterion of “share” accordingly. Still this d −1 ]d trials with a probability requires T −a1 H1 ϑ[L −1 1 ] successes out of a total of c1 [L T bounded by T a2 H2 ϑ T for each success; the latter is estimated by chessboard methods. In the above, a’s, H ’s etc. are of order unity with H1 numerically large if H is large. The upshot, for a fair share at scales between T and 2T is an upper estimate of the form
T a3 ϑ T −1 H3
c2 T −a1 ϑ H1 L d
with all constants of order unity and both H1 and H3 large if H is large. Clearly the above gets out of hand if we let T get too large but we shall cut off when the above approximately matches our preliminary estimate – which determines the value T0 . For all other T except, perhaps, for the very first few, this will be small due to the ϑ term and the cases T ∼ 1 can rely on large H or (which essentially amounts to the same thing) can be done by hand. With the vast majority of blocks in the exterior of contours, it is indeed the case that the energy content is close to Er for some r – here another estimate using Q(µ, 1 ) is employed. Finally it is noted that all estimates in this and the previous proposition stem from the initial estimate in the first few lines of Proposition 5.6 for which uniformity was established in Corollary 5.2. All subsequent rates, bounds, etc. depend trivially on (r ) the separations between the 3η ’s – which have been deemed to have a minimal value – η, θ and various other parameters can be determined by a worst case scenario on a bounded interval of β’s. Corollary 5.8. Consider a spin–system satisfying the hypotheses of Proposition 5.7. ˜ in Then the conclusion of this proposition holds, perhaps with a slight adjustment of δ, every shift invariant ergodic measure which is a Gibbs state for the interaction.
328
L. Chayes
Proof. The desired result is, almost, an immediate application of Theorem 2.5 and Corollary 2.6 in [5] save for the fact that here (and in various other places throughout this work) the relevant “good” events do not quite satisfy their hypotheses in cases where there are multiple types of goodness. However this can be circumvented by the construction of a superblock: Let us introduce one more length scale, 2 with 2 1 . In contrast to the previous j , this length will not be tied to µ or any other parameters. On the contrary, it is envisioned that 2 → ∞ with all other quantities fixed. However it will be assumed, for connivance, that 1 and L are related to 2 by powers of two. Consider the superblock event, defined on 2 that all but a fraction ϑ of the tiling 1 –sized subblocks satisfy the Thouroughgood condition for the same value of r . If this event is denoted by J (ϑ ), arguments along the lines of those in Proposition 5.7 show that αβ,T L (J ) ≤ e
−K(ϑ )[ 2 ]v 1
(5.23)
for some positive power v and K positive once ϑ is an appreciable multiple of the estimate in the final line of Proposition 5.6. Now let αβ denote any ergodic Gibbs state corresponding to the specified Hamiltonian and suppose that the energy density, E, of αβ is not within the appropriate δ of any Er . Then, with high probability, the αβ –energy per site of a sufficiently large block (i.e. 2 with 2 sufficiently large) is also outside of the anticipated range. In light of the estimate in Eq. (5.23) this is not permitted by the above mentioned theorem in [5]. Corollary 5.9. For spin–systems satisfying the above hypotheses, there is an η 3η such that in any shift invariant ergodic Gibbs state, in almost every configuration the (r) magnetization is in Conv(η ) for some r . Proof. The result follows immediately from the preceding (and continuity of β ). Indeed, in this case, the superblock construction can proceed without the benefit of the intermediate scale. Remark 8. For some systems, e.g. when Mβ consists only of isolated points, the above is in essence the final result. But in others, e.g. the O(n)–systems, this corollary basically provides no information. It is not difficult to imagine that, with the insertion of some further energy hypotheses, we would be in position to directly establish discontinuous transitions in the energy density for “real” systems in d ≥ 2 whenever such transitions occur in the mean–field theory. However, the necessary hypotheses turn out to be slightly nebulous in appearance. Hence we will follow the alternate route of tracking the magnetizations – which in any case are closely tied to the energies – and the results of this subsection will be utilized in a supporting rôle. 5.3. Proofs of main results. Proof of Main Theorem. With what has so far been established, we are in prime position to apply the classic result of Kotecký and Shlosman, which provides a sufficient condition for the occurrence of a 1st order transition. For completeness, let us summarize the hypotheses of [23] Theorem 4 (which have been abbreviated by limiting attention to circumstances where the relevant numerical parameters are small quantities).
Mean Field Analysis of Low–Dimensional Systems
329
Consider a spin–system on Zd , d ≥ 2 belonging to a certain class which includes that defined by H in Eq. (2.1)–(2.2) and suppose there are events A I and A I I defined on the block 0 and an interval of inverse temperature [β I , β I I ] such that for certain small numbers a1 , a2 , b1 and b2 and some L 0 , the following holds for infinitely many L’s that are larger than L 0 : (i) (ii) (iii) (iv)
For all β ∈ [β I , β I I ], αβ,T L ([A I ∪ A I I ]c ) < b1 . The limβ→β I αβ,T L (A I ) ≥ 1 − a1 . The limβ→β I I αβ,T L (A I I ) ≥ 1 − a2 . For all β ∈ [β I , β I I ], if τ j (A I ) denotes the event A I translated to the lattice site j, then for all j, αβ,T L (τ j (A I ) ∩ A I I ) < b2 .
Then there is a βt ∈ (β I , β I I ) such that at β = βt , there are at least two coexisting Gibbs states corresponding to H, denoted by αβI t and αβI tI that are distinguished by αβI t (A I ) ≥ 1 − c ; αβI tI (A I I ) ≥ 1 − c,
(5.24)
where c is small if the a’s and b’s are small. It is clear that for a Generic First Order Scenario, we may utilize the events A I = {m 0 ∈ M I } and similarly for A I I . Using the hypothesis of the scenario and Lemma 5.1, items (i) – (iii) are satisfied; let us turn to (iv). For the latter, we shall adapt some previous notation: If j ∈ Zd , let m 0 ( j) denote the average magnetization in τ j ( 0 ) – thus τ j (A I I ) is the event {m 0 ( j) ∈ M I I }. Now let us define a site i to be good if m 0 (i) ∈ M I ∪ M I I and otherwise bad. It is first noted that A I ∩ A I I = ∅ (and similarly for the translations) since by hypothesis, M I and M I I are separated. Thus there are two types of good sites. Now suppose that the origin is of type I and j is of type II, i.e. the event A I ∩ τ j (A I I ); let us consider the connected component of type I good sites of the origin. We will use the convention that a boundary site is outside the cluster with a neighbor in the cluster. A boundary site could, ostensibly, be a bad site or a site of type II. However, we use the condition 0 1 and the obvious fact that for any lattice vector eˆi , |m 0 ( j + eˆi ) − m 0 ( j)| ≤
ω d−1 0
,
(5.25)
where ω is as big as a spin can get. As a consequence, since M I and M I I are separated, if 0 is large enough, the boundary of any region of type I sites must actually be bad sites. We thus have certain contours and contour events – which will typically be denoted by γ ; these are, technically, ∗–connected contours, that is to say neighbors and next–nearest neighbors are considered connected. It is further remarked that there are actually two types of contours possible depending on “who is separated from whom” plus the possibility of a contour that winds the torus (an SSWC–contour) all of which can be accounted for by doubling the estimate obtained by an a priori infinite sum over contours. Let us focus on the more pertinent issues: Foremost, the events that the individual contour elements (the sites of γ ) represent are actually defined on the larger scale 0 and, even using reflection positivity methods, it is not possible to obtain a tractable Peierls–type estimate without a bit of course–graining. Thus, let us formally consider the lattice T L/ 0 whose “sites” consist of the disjoint blocks that are appropriate translates of 0 . If γ denotes a microscopic ∗–connected contour (or any path) we may associate a cluster, = Q(γ ), on T L/ 0 representing the blocks of scale 0 that were visited by γ . Notice that may itself be only vaguely
330
L. Chayes
contour–like but, at any rate, it is a ∗–connected object. Now if j ∈ T L/ 0 denotes a (block) site, the probability that j belongs to a course grained contour element is exactly the probability that some site in the block is bad. This, in term, may be bounded by the volume of the block times the estimate on the right side of Eq. (5.14). Let us use εC = εC ( 0 , µ, η) to denote this small quantity. A secondary (minor) obstruction occurs for a block contour event associated with a . Indeed we cannot use chessboard methods on each block–element since the relevant events in ∗–neighboring blocks may be entangled. However, disjoint sublattices on T L/ 0 may be considered such that the blocks on each sublattice are devoid of ∗–neighbors in their own sublattice. In d = 2 there are four such sublattices, in general it is 2d . Thus, finally, for each (admissible) cluster , let || denote the maximum of the number of blocks of which reside on the various sublattices. The argument can now be finished along standard lines. The block contour event where the cluster is of size N must be within the distance of the order N of the block containing i or the block at the origin. The number of such clusters is therefore bounded by A(d)N a(d) eκ(d)N with all constants finite and the necessary “double counting” folded into these constants. Therefore, defining N = {∃ γ of bad sites separating 0 from i with |Q(γ )| = N }
(5.26)
αβ,T L ( N ) ≤ AN a eκ N εCN .
(5.27)
we have
Summing from N = 1 the result is small if εC is small and, under the hypotheses concerning 0 , µ, etc. condition (iv) has been verified. Proof of Proposition 3.1. This is, in essence, the 2nd corollary to Proposition 5.7 (Corollary 5.9). First, since the sets M ( j) are convex and separated then small neigh( j) borhoods of these sets – large enough to contain the appropriate η – are convex and separated. Thus the magnetization is always in one of these neighborhoods and there is at least one Gibbs state of the specified form. But now, due to the invariance of the interaction, if there is a Gibbs state associated with one of the M ( j) , then there is a Gibbs state for all the others as well. Proof of Theorem 3.2. Under the hypotheses off a Generic First Order Scenario, the result is established by the Main Theorem (which proves a transition between M I and M I I – like states) and Proposition 3.1 which establishes the nature of the Gibbs states. Alternatively, with the hypothesis of an energy gap between M I and M I I , a Generic First Order Scenario is readily established. Let us start by finding a δm which is small ( j) compared to all separations between the various M J ; explicitly that the δm–neighborhoods of these sets are still separated. Next, let us define an κ which is small enough so that κ (βtMF ) is contained in the union of these neighborhoods. Notice that there is an j unambiguous κI (βtMF ), similarly for I I and also for the various offshoots from the M J . The quantity κ will define both the temperature scale and, for all intents and purposes the (three ×) η. Let [β I , β I I ] be the symmetric interval about βtMF that has, to be definitive, κ = ω2 (β I I − β I )
(5.28)
Mean Field Analysis of Low–Dimensional Systems
331
(and notice that this necessarily implies that ω2 (β I I − β I ) βtMF (m I − m I I )2 for any m J ∈ M J ). Finally 3η will satisfy 3η <
1 βI I − βI E 2 βtMF
(5.29)
so, except for the possibility of some terrible anomaly in the sizes of the m’s in MβtMF , η and κ are comparable. Now define I MI = 3η (β) (5.30) β∈[β I ,β I I ]
and similarly for M I I . Let us demonstrate that M I ∪ M I I is contained in κ – here repeated use will be made of the identity β (m) = β (m) − 21 (β − β )m 2 . Suppose that m ∈ / κ (βtMF ). Then, for β ∈ [β I , β I I ], 1 β (m) ≥ FMF (βtMF ) + κ + (βtMF − β)m 2 2 1 1 ≥ FMF (β) − |βtMF − β|ω2 + κ − |βtMF − β|m 2 2 2 1 1 ≥ FMF (β) − (β I I − β I )ω2 + κ = FMF (β) + κ, 2 2
(5.31)
and since, certainly, 3η < 41 ω2 (β I I − β I ), it is clear that m ∈ / 3η (β). Notice that this also implies that there is a separated M I and M I I which in turn consist of k I and, ( j) respectively, k I I subsets associated with the M J . Let us see that the hypotheses of the Scenario are satisfied. Item (i) is our starting premise. Items (iia) and (iic) have been constructed with the identification of 3η with ζ . Item (iid) is an obvious consequence of continuity. We are left with item (iib) which is to show that at β = β I I , the set M I I contains all the minimizers in the strong sense that FMF (β I I ) falls below β I I (m I ) − 3η for all m I ∈ M I . And we will need the corresponding statement for FMF (β I ). This follows from an argument similar to the above. Let m I ∈ M I . Then for m I I ∈ M I I ⊂ M I I , 1 β I I (m I )−β I I (m I I ) = βtMF (m I )−βtMF (m I I )+ (β I I −β I )(m 2I I −m 2I ). 4
(5.32)
Now βtMF (m I I ) = FMF (βtMF ) and βtMF (m I ) cannot be lower. Meanwhile, (m 2I I − m 2I ) ≥ [2/βtMF ] E ; obviously all the minimizers are in M I I and moreover, the gap is at least 3η. A similar argument holds at the other end of the interval and the proof of a first order transition is complete. The remainder of the statements follow from the first portion of the proof and/or are automatic. Proof of Theorem 3.3. Without loss of generality the treatment shall be confined to the case where m 1 and m 2 are the preferred approximate magnetizations destined for coexistence. Armed with Lemma 5.1 and its corollary, most of the proof amounts to an exercise in linear algebra and analysis. First, by the Gramm–Schmidt procedure (using the inner product defined by the interaction in Eq. (2.1)) let us consider an orthonormal set of fields starting with bˆ1 and bˆ2 covering the span of m 1 and m 2 with, say, bˆ1 ∝ m 1 . The successive fields, bˆ3 , . . . bˆk are now orthogonal to m 1 and m 2 , thus their addition to the
332
L. Chayes
Hamiltonian as described in the statement of this theorem hardly effects the free energy function in the vicinity of these points. The strategy will be to first use these last k − 2 fields to suppress the “unwanted” states and then employ bˆ1 and bˆ2 in tandem to enhance one of {m 1 , m 2 } at the expense of the other. First let ς denote a small quantity and consider the ς –neighborhoods of m j which will be denoted by Nς (m j ) and which may be assumed to be disjoint from one another. Ultimately the applied external field, bλ1,2 , will be small depending on ς and we need not search outside of ∪ j Nς (m j ) for a minimizer of the augmented free energy function. Indeed, letting ϑ denote the minimal surplus outside these regions: ς
inf{β (m)|m ∈ [∪ j M j ]c } = FMF (β) + ϑ,
(5.33)
it is obviously sufficient that ||bλ1,2 ||2 stay bounded by a constant times ϑ, e.g. ϑ/4ω where, it is recalled, ω is the limiting size of the magnetization in all of . Let us start with the construction of the “suppressor fields”; for convenience we shall work with the fields b˜ j ∝ bˆ j that satisfy (b˜ j · m j ) = 1. Let c > 1 denote a constant and let us define coefficients γ3 , . . . γk , γ j ≥ 1 and, say, γ3 = 1 such that γ (b˜ · m j )|. (5.34) γ j = γ j (m j · b˜ j ) ≥ 1 + c| < j
˜ Finally let H = j γ j b j . It is claimed, for all j ≥ 3 that for any m ∈ Nς (m j ) the effect of (H · m) is pretty much of the order unity. Indeed, writing m = m j + δm, γ (b˜ · m j ), (5.35) (H · m) = (H · δm) + γ j + < j
where terms of the form (m j · b˜ ) with > j are absent due to orthogonality. Thus, it is clear, we now have (H · m j ) ≥ 1 − ς ||H ||2 . Now, for λ ∈ [−1, +1], consider the field b˜1,2 (λ) = λ(b˜1 −γ2 b˜2 ), where γ2 is defined along the lines of the above γ ’s: γ2 = γ2 (b˜2 · m 2 ) = 1 + |(b˜1 · m 2 )|.
(5.36)
Obviously if λ = 1, then (b˜1,2 (1) · m 1 ) = 1 and, as is seen, (b˜1,2 (1) · m 2 ) ≤ −1, (b˜1,2 (−1)·m 1 ) = −1 while (b˜1,2 (1)·m 2 ) ≥ 1. Now let ε1 , ε2 > 0 with ε1 ε2 (with the ε’s to be specified with a bit more precision below) and consider bλ1,2 = −ε1 H + ε2 b˜1,2 (λ). For m ∈ Nς (m 1 ), using m = m 1 + δm 1 , we have (m · bλ1,2 ) = ε2 (b˜1,2 (λ) · m 1 ) + ε2 (b˜1,2 (λ) · δm 1 ) − ε1 (H · δm 1 ),
(5.37)
and we see that, at least for |λ| near one, the second term can be neglected relative to the first. Similarly, if we allow ς ε1 small compared with ε2 , the third term may be designated as “unimportant”. Of course the same considerations apply if m ∈ Nς (m 2 ). Meanwhile, if m is in Nς (m j ) with j ≥ 3, then (m · bλ1,2 ) = −ε1 [(H · m)] + ε2 (b˜1,2 (λ) · m),
(5.38)
so the first term is a negative number of order unity times ε1 and, relative to this, the second term may be neglected due to the stipulation concerning the relative sizes of the
Mean Field Analysis of Low–Dimensional Systems
333
ε’s. Thus it is clear that the magnitude of the terms emerging from Nς (m 3 ), . . . Nς (m k ) are always much larger than those from magnetizations inside Nς (m 1 ) and Nς (m 2 ) and, according to the sign of the interaction, these k − 2 regions are ruled out as candidates for the minimizer of the free energy function. Thus the minimum occurs in Nς (m 1 ) ∪ Nς (m 2 ) and it must be the case that the minimizer switches locations for some λ ∈ (−1, +1). Thus for the interaction given by −βH + i (bλ1,2 · si ) it is clear that the associated mean–field theory has a Generic First Order Scenario – albeit field driven; see Remark 2 following Definition 2.1. Indeed, for fixed small ς , and (ε1 , ε2 ) chosen accordingly, it follows from continuity (cf. Theorem 5.11) there is at least one λMF = λMF (ε1 , ε2 ) with λMF ∈ (−1, +1), where the minimum in Nς (m 1 ) coincides with the minimum in Nς (m 2 ). To define M I and M I I we restrict to the subsets of Nς (m 1 ) and Nς (m 2 ) such that hypothesis (iic) is satisfied and then (iia), (iib) and (iid) are easily satisfied. The remains of this proof now follow from the Main Theorem. 5.4. Proofs for specific systems. Let us start with the standard discrete symmetry magnetic transitions: Proof of Theorems 4.1 and 4.2. These systems (as well as a host of others) may be treated together since, in fact, the principal results pertaining to the nature, location etc. of the first order transition are just an application of Theorem (3.2). The secondary result, namely that the high/low temperature states “disappear” on the appropriate side of βt is also, in fact a fairly general feature of these sorts of systems but not really worth abstractifying. Let us start with some basic facts about the mean–field theory which are well known and/or readily derived (and anyway proved in [2], Sect. 4.2 and Sect. 4.3 ). Foremost, for q ≥ 3 and r ≥ 4 there is indeed a first order transition in the mean–field theory; the temperature parameter will in all cases be denoted by βtMF . In both cases the degenerate minima consist of singleton positive magnetization states which are proportional to the values that the spins themselves take as well as a state of zero magnetization. These obviously enact the symmetries of the relevant groups and, needless to say are convex sets. The energy gap is manifest and in addition, it is worth noting that the aforementioned βtMF is the only point of degeneracy between states of differing energy. Thus we apply Theorem 3.2. As for the “disappearance of states”, this follows from elementary considerations. In particular, in the real system, the energy is a monotone function (and so a.e. well defined). Thus, for β < βt there cannot be any states with large magnetization – since that would imply the existence of a substantial energy – and similarly when β > βt there cannot be states with small magnetization. On to the asymmetric situation: To prove the content of Theorem 4.3 it is, by and large, sufficient to establish a triple point in the context of the mean–field theory. The claim, for the mean–field theory, is best summarized in Fig. 2 below and will be proved as a separate lemma. Lemma 5.10. Consider the mean–field theory associated with the Hamiltonian 4.1 which leads to the mean–field free energy function 1 1 1 J (n a , n b , n c ) = − Ja n a2 − Jb n 2b − Jc n 2c − K bc n b n c + K ab n a n b 2 2 2 + n a log n a + n b log n b + n c log n c , (5.39)
334
L. Chayes
Fig. 2. Phase diagram for an asymmetric 3–state model
where n a + n b + n c = 1 and J stands for all the couplings. Using the notations Da , Db and J as described, the following holds for all J > J˜ where J˜ is large (but not unreasonably so): 1. For all Da and Db > 0 and for all K ab and K bc ≥ 0 with the K ’s small compared to J there are three local minima, at least one of which is the global minimum, that are characterized by an abundance of the species a, b and c respectively. These minima will correspondingly be denoted by A, B and C and when they actually minimize they represent the phases. Any other local minima of J are substantially higher. 2. For fixed Da and Db sufficiently small compared to J and (Da − Db ) small compared to Da , there is a finite K˜ such that for K˜ > K ab , and K bc sufficiently small, the A–phase is the minimizer. By contrast, for small values of K ab the B–phase will be prevalent once K bc is sufficiently large. 3. With Da and Db as above, for all K ab < K˜ , there is a transitional point at some value of K bc , where the A and C minima are degenerate. Pertinently, this holds even in the B phase. Furthermore, these points form a “transitional curve” which cuts through the B–phase. The terminal point of this curve in the B–phase is the point K . Proof of Lemma 5.10. Let us start with the situation Da = · · · = K bc = 0 – i.e. the q = 3–state Potts model – with J in excess of some J˜ to be described later. The claim is that there are three minimizers, identical under permutation, with one large and two small populations. While this is of course well known, the forthcoming analysis will demonstrate that these solutions are stable and persistent. Moreover, other local minima (if any) will have substantially higher free energy and/or represent unphysical states. The starting point is, of course, the mean–field equation: n a e−J n a = n b e−J n b = n c e−J n c = λ,
(5.40)
where λ is a Lagrange multiplier adjusted so that n a + n b + n c = 1. A look at the function xe−J x clearly indicates that for λ < (J e)−1 , there are two solutions to x(λ)e x(λ) = λ which, for obvious reasons will be denoted by s(λ) and B(λ). Note that s(λ) is strictly
Mean Field Analysis of Low–Dimensional Systems
335
increasing on [0, (J e)−1 ] while B(λ) is strictly decreasing. The objective, at the level of the q–state Potts model, is to find the value of λ such that (q − 1)s + B = 1. That such a value exists is obvious; as λ → 0, B → ∞ while for the maximum value, λ = (J e)−1 , B = s = 1/J – can be assumed to be way too small. So, by “bringing up” the value of λ, we certainly arrive at a first solution; the claim is that for J in excess of some value J˜ this is (far and away) the only possibility. Direct computation yields (q − 1)s B B˙ + (q − 1)˙s = − , 1 − Js JB −1
(5.41)
where the overdot denotes differentiation with respect to log λ. For q = 2 it may be directly verified that this is negative but not for q ≥ 3. However, this is negative till q J s B ≥ B + (q − 1)s. Now the latter necessarily implies q J s B > B, i.e. J s is already of order unity. But then so is J B; indeed, under the previously mentioned condition, J λ = J se−J s ≥
1 − q1 e , q
(5.42)
1 − q1 e q
(5.43)
thence J λ = J Be−J B ≤
To summarize: For J in excess of some J˜ – not terribly large – there is no hope of a second solution to B + (q − 1)s = 1 because by the time the derivative of B + (q − 1)s gets around to being positive, all B’s and s’s are “hopelessly small”, namely of order J −1 . The only other possibility for minima are two (or more) B–type solutions. However, under these circumstances, it has been shown that the free energy is substantially lowered if, keeping all other n’s fixed, two bigs are exchanged for a small and a (bigger) big; cf. the proof of Lemma 4.4, especially item (i), in [2]. It is clear that the above analysis all goes through with different diagonal couplings, e.g. Ja > Jb > Jc . Let us proceed with the full problem. While we will not use that Da is small compared with J until later, it is conceptually easier to proceed in this vein. The full equations now read n a e−J n a e+K ab n b = n b e−J n b e+K ab n a e−K bc n c = n c e−J n c e−K bc n b = λ;
(5.44)
we are seeking solutions of the form “two small one big”. First off, let us note that there are some restrictions on λ. For example, the third equation certainly requires λ < (J e)−1 and further, for λ comparable to this number, it is easily seen that if there were a solution, it would have (for K ab , K bc J ) n a + n b + n c of the order J −1 . So we shall restrict attention to, say, λ less than λ0 = κ(J e)−1 with some suitably chosen κ of order unity but less than one and proceed. It is not hard to see that there is indeed a unique solution once the big item is specified. Suppose, for example, this is n a . Let us write a facsimile of the first equation, namely Na e−Ja Na e K ab n b = λ, which defines a function Na (n b ). It so happens that this is defined on all of [0, ∞) but not so for Nc (n b ) given from the third equation: Nc e−Jc Nc e K bc n b = λ. However, for λ < λ0 , the quantity n b can safely climb up to the order of J −1 which, as we shall see, is more than ample range. From the middle equation, we can now define a function (n b ) = n b e−Jb n b e+K ab Na (n b ) e−K bc Nc (n b ) ,
(5.45)
336
L. Chayes
and we now wish to solve = λ. Clearly Nb (0) = 0 while, as n b tends to the order of J −1 , the right-hand side will exceed λ0 . (Ignoring the aid from the e+K ab Na (n b ) , Nc does not get any bigger than J −1 so can certainly get almost all the way up to (J e)−1 .) Thus, for all λ of interest there is a solution n a (λ), n b (λ), n c (λ). To see that it is unique (at least for the values of λ that are of interest) we can simply take the derivative: d 1 log (n b ) = − Jb + K ab Na − K bc Nc . dn b nb
(5.46)
Now Na is increasing but so (unfortunately) is Nc . However, in order for K bc Nc = 2 (N −1 − J )−1 to be comparable with n −1 − J , it is obvious that n −1 and J must K bc c b b c b b themselves be comparable which puts in well excess of κλ0 . Notwithstanding, even when n b ≈ J −1 , Nc (n b ) is still small compared with J −1 , and hence given the rest of the range of n b the negative portion of the derivative is not substantial enough to pull the function down below κλ0 . Item 1 has essentially been proved: Having established, e.g. for a dominance over b and c the existence of unambiguous n a (λ), n b (λ) and n c (λ), an argument similar to the K ab = K bc = 0 case shows the existence of a unique λ such that n a + n b + n c = 1. Similarly for the other orderings. Thus, in the region of parameters described, we now have our three well defined “free energies”, A,J , B,J and C,J associated with these three (well separated) local minima. At least one of these functions will represent the actual FMF and all of them are substantially lower than any other value of J outside the vicinity of the minima. Items 2 and 3 are actually not so difficult in light of what has already been established. Indeed, it is observed that the derivatives of the various free energy functions with respect to the couplings admit simple expressions due to the fact that they are already functions evaluated at local minima. For example let us examine A,J expressed in the form of Eq. (5.10) with n c formally eliminated in terms of n a and n b . Then ∂n a ∂n b ∂A,J ∂J ∂J = n a (A)n b (A) + + , (5.47) ∂ K ab ∂n a A ∂ K ab ∂n b A ∂ K ab where the subscripts and arguments of A for various quantities emphasize that the associated functions should be evaluated at the portions of n a and n b (and n c = 1 − n a − n b ) which produce the A–state. However here the relevant partial derivatives vanish because we have a local minimum. Hence ∂A,J /∂ K ab is simply n a n b – as evaluated in the A–state. These derivative arguments will greatly facilitate the proof of all that remains. More pertinent than the above equation is that ∂C,J = −n b (C)n c (C) ∂ K bc
(5.48)
with a formally identical expression for the same derivatives of B,J and A,J but with the right-hand side given by the product of the n’s evaluated in the appropriate states. It is noted that for all K ab , K bc of relevance, n b (A)n c (A) n b (B)n c (B), n b (C)n c (C). For K bc = 0, it is clear that until K ab has become substantial the A–phase is dominant.1 For K ab small, it is clear that once K bc gets large enough, the A and B minima will exchange. Thus, in the vicinity of the origin of the K -space quadrant, the vertical 1 In the absence of additional analysis/analytics, the current argument may represent an unmentioned – and somewhat non–trivial requirement: For large J , we have s(J ) ≈ e−J and since the perturbations must always couple to a subdominant species in order for the K ’s to have impact without themselves becoming unreasonably large it is actually required that Da e J be somewhat small.
Mean Field Analysis of Low–Dimensional Systems
337
axis is enveloped by the A–phase while most of the horizontal axis (and its immediate neighborhood) is dominated by the B–phase. This is item 2. As for item 3, let us start on the vertical axis at a point in the A–phase. Now, we simply compare the derivative in Eq. (5.48) with the counterpart derivative of A : Under the restrictions in the statement of this theorem, the former is always larger in magnitude than the latter. Thus, if Da has been arranged to be suitably small, it is inevitable that C will “catch up” and once it has done so, it will always “stay ahead”. This defines a transitional point which, it is reiterated, may or may not happen within the region of the B–phase. The fact that these points form a curve follows from an elementary argument using (compactness and) the implicit function theorem. Proof of Theorem 4.3. Taking to be the standard positive basis vectors in R3 , as long as the K ’s are not comparable to J , (which is anyway required later) the interaction describes a positive definite inner product. Under the conditions of Lemma 5.10 let us assume, for fixed J ’s, that the K ’s are adjusted so that the mean–field theory is at the point K . The occupation vectors corresponding to the A, B and C phases are manifestly seen to be linearly independent – regarded as vectors in R3 – and are well separated in (Conv()) since each of the vectors has a dominant component. All the conditions of Theorem 3.3 are satisfied; the result follows. Proof of Theorem 4.4. Of course much of the statement of this theorem amounts to a statement about the mean–field theory and this system is well characterized. A brief run through will be provided for completeness. If m ∈ n – the unit sphere in n–dimensions – obvious symmetry considerations reduce all considerations to scalar problems. Thus, e.g. the function mn (h) is given by Eq. (4.2) and once computations are performed, all quantities can be promoted to vectors. Using h in favor of m, (see Proposition 5.12 in the Appendix subsection) the expression for the free energy may be written 1 β = − βm 2 − log G(h) + mh, 2
(5.49)
where all terms involving m are now understood to mean mn (h). Then β = [h −βm]m and noting that m is strictly positive, for all intents and purposes, its presence can be ignored. We are, of course, running through a derivation of the mean–field equation and so far everything is, more or less, general. The specifics for this problem is that mn (h) is a strictly convex function [13,28]. Now it turns out that limh→0 βc mhn (h) = 1 with βc = n. Strict concavity gives us that for positive h, βc mn (h) ≤ h so that if β < βc , the free energy is raised by making h positive, i.e. m(β) = 0. Conversely, if β > βc , raising h away from zero will lower the free energy which continues until the mean–field equation is satisfied. The solution is demonstrably unique by the concavity property and obviously a minimum. The comparison with the actual spin–systems is a direct consequence of Lemma 5.1 and Corollary 5.3, the claims concerning the energy follow from Proposition 5.7; due to the continuous nature of the transition, the Regular Energy Hypothesis is obvious with r = 1 and, finally, the statement concerning the free energy is exactly Corollary 5.3 to Lemma 5.1. Proof of Theorem 4.5. Practically all of what is needed is contained in the second (substantial) half of Theorem 3.2: The appropriate η sets are neighborhoods of the origin 1 and the orbit of λMF t diag[1, − n−1 ] under the action of the full O(n) group. These sets are obviously separated in magnetization and energy. Of course the origin is a singleton – convex – so the “‘magnetic” portions of Theorem 3.2 actually apply which is the
338
L. Chayes
entirety of the claim concerning the high temperature phase. Of course, the low temperature portion of MβtMF is not a convex set. Thus while (at least in the matrix version) the block magnetizations on block scale 0 appear like the mean–field minimizers, there is no reason to expect this sort of coherence on larger scales. A global cooperative effect requires additional ingredients which are present in d > 3 but most definitively absent in d = 2. Proof of Theorem 4.6. The mean–field bound for these systems is actually standard fare: For the Ising system, it follows from an adaptation of a general result to this effect by Sokal [33]. For n–component spins, with any non–negative Ji, j it was shown in [32] that the following inequality holds for n ≥ 2: β (1) (1) si ,h ≤ Ji, j s j ,h + h, (5.50) n j
where the superscripts here denote the first component and −,h denotes the thermal average in system (with certain boundary conditions) at external field h pointing in the direction of the first component. In point of fact, this also holds for n = 1 – at least for h = 0 – where it is the Simon inequality [31] in slightly disguised form. Since it is well known for n = 1 and known [8] for n = 2 how to provide the appropriate boundary conditions for producing the spontaneous magnetization, we might as well take the inequality as it stands with n = 1 or 2, h = 0 and → ∞ replacing thermal averages of spin components by spontaneous magnetizations. In the present context, this reads mα ≤
β α,γ Q mγ , n
(5.51)
where m α (with m α ≥ 0) denotes the spontaneous magnetization in the α th layer. The result now follows pretty easily if we multiply by m α and sum over α (cf. [10] for a more detailed derivation along these lines). All the rest of the claims now follow from previous theorems. For µ small, the block magnetizations are (uniformly on compact intervals of temperature) close to a solution of the mean–field equation by Proposition 5.1 and its corollary. Free energetics and energetics follow from Corollary 5.3 and Proposition 5.7 (where we may use r = 1 because the transition is continuous) and observe that the Regular Energy Hypothesis satisfied. 5.5. Appendix: Continuity properties of β (m). Here are some properties of the free energy function that have been alluded to, or explicitly used in the text. The starting point will be to trim away the inessential portions of E and even , which will later save us the trouble of numerous provisos. Let D denote the set D = {h ∈ E |(s · h) = constant w.p.1}.
(5.52)
Obviously D is a subspace of E and it is seen, after a moments thought, that the non–trivial vectors in D are precisely the ones that are of no interest to the problem at hand. The price of keeping D is that relative topologies must be employed and many statements must be made modulo vectors in D . Thus, without loss of generality, we restrict attention to the essential subspace and, without much apology, continue with the notations , E , etc. But, for future reference it is now noted that (s · h) = 0 constant w.p.1 ⇒ h = 0. The principal result of this section is the continuity of β :
(5.53)
Mean Field Analysis of Low–Dimensional Systems
339
Theorem 5.11. Let , E and α0 be as described with the stipulation in Eq. (5.53) and let C denote the set where the entropy is not −∞. Then the free energy function, 1 β (m) = − βm 2 − (b · m) − S(m), 2
(5.54)
is continuous on Int(C ). As a starting point, for m ∈ E , let us define the entropy functional e(m·h) = e(m·h)−G(h) . (s·h) dα e 0
m =
(5.55)
The object is to maximize m . Obviously C is the set where m is bounded; it is not hard to see that C ⊂ Conv(). For h ∈ E let us use the notation −h for expectation in the tilted measure and define m(h) (= sh ) to be the average magnetization in this measure. In [2] it was proved (Lemma 3.1) that if m ∈ Int(C ), then ∃h ∈ E such that m(h) = m. Here let us prove that this h is unique. Proposition 5.12. Let , E , α0 and C be as described, with the stipulation in Eq. (5.53). Let m ∈ E satisfy m(h) = m for some h ∈ E . Then, in fact, m ∈ C and the h is unique. Proof. The fact that h ∈ C was proved in [2] Lemma 3.1 – but also follows from the argument below which, in fact, is almost exactly the proof of Theorem 2.4 in [9]. In any case, we have, from the above–mentioned lemma in [2] that h maximizes m . Suppose ˜ Then that h˜ also satisfies m = m(h). ˜
m (h) =
˜
˜
e(m·h) e(m·[h−h]) e G(h) ˜ e G(h) e G(h)
˜ = m (h)
˜
e(m·[h−h]) ˜ es·[h−h] h˜
˜
e G(h)
˜ = m (h)
˜
˜ (s·[h−h]) ˜ (s·h) e dα0 e ˜
˜
e(m·[h−h])
˜ (sh˜ ·[h−h]) e(m·[h−h]) = m (h), ˜ ≤ m (h)e
(5.56)
where the inequality is Jensen’s. Evidently h˜ also maximizes the functional. Moreover, ˜ is a.s. a constant according to the since the Jensen inequality has saturated, (s · [h − h]) tilted measure and hence according to α0 . Evidently h = h˜ α0 –a.s. The above proposition allows the definition of an inverse function h(m) defined, at least, on Ran(m). The next result shows that h is continuous: Proposition 5.13. Let , E , α0 and C be as described, with the stipulation in Eq. (5.53). Then Ran(m) = Int(C ) wherein the inverse map h is continuous. Proof. This follows from standard convexity arguments. For example, if m ∈ Ran(m) and it is assumed, with no loss of generality (although, perhaps, some elegance) that by linear transformation the problem has been reduced to n–dimensional Euclidean with standard inner product then the derivative is, explicitly, ∂m a = sa sb h − sa h sb h , ∂h b
(5.57)
340
L. Chayes
where h = h(m). However, the object in Eq. (5.57) is exactly the covariance matrix of the “array” s in the tilted measure. In general this is positive semi–definite but due to the stipulation in Eq. (5.53), it is positive definite. Hence the inverse function is itself differentiable and, moreover, any point in a sufficiently small neighborhood of m can be reached by h. Now by [2] Lemma 3.1 we have that Int(C ) ⊂ Ran(m) ⊂ C but the latter argument tells us that Int(C ) ⊃ Ran(m). As an obvious corollary: Proof of Theorem 5.11. Clearly, it is only necessary to establish continuity of S(m). However, we may now express S(m) = G(h(m)) − (m · h(m)), and the continuity of both portions follows from the continuity of h(m).
(5.58)
Acknowledgements. This research was supported by the NSF under the grant DMS-0306167. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Biskup, M.: Reflection Positivity and Phase Transitions in Lattice Spin Models. Lecture notes from Prague Summer School on Mathematical Statistical Mechanics, 2006 2. Biskup, M., Chayes, L.: Rigorous analysis of discontinuous phase transitions via mean–field bounds. Commun. Math. Phys. 238(1), 53–93 (2003) 3. Biskup, M., Chayes, L., Crawford, N.: Mean-field driven first–order phase transitions in systems with long–range interactions. J. Stat. Phys. 119(6), 1139–1193 (2006) 4. Biskup, M., Chayes, L., Nussinov, Z.: Orbital ordering in transition-metal compounds: I. the 120-degree model. Commun. Math. Phys. 255, 253–292 (2005) 5. Biskup, M., Kotecký, R.: Forbidden gap argument for phase transitions proved by means of chessboard estimates. Commun. Math. Phys. 264(3), 631–656 (2006) 6. Butta, P., Picco, P.: Large-deviation principle for one-dimensional vector spin models with Kac potentials. J. Stat. Phys. 29(1/2), 101–150 (1998) 7. Cassandro, M., Orlandi, E., Presutti, E.: Interfaces and typical Gibbs configurations for one-dimensional Kac potentials. Probab. Theory Relat. Field 96, 57–96 (1993) 8. Chayes, L.: Discontinuity of the spin–wave stiffness in the two–dimensional XY model. Commun. Math. Phys. 197, 623–640 (1998) 9. Chayes, J.T., Chayes, L., Lieb, E.H.: The inverse problem in classical statistical mechanics. Commun. Math. Phys. 93, 57–121 (1984) 10. Zandi, R., Shackell, A., Rudnick, J., Kardar, M., Chayes, L.P.: Thinning of superfluid films below the critical point. Phys. Rev. E 76, 030601 (R) (2007) 11. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. New York: Springer-Verlag Inc., 1998 12. Dobrushin, R.L., Shlosman, S.: Absence of breakdown of continuous symmetry in two–dimensional models of statistical physics. Commun. Math. Phys. 42, 31–40 (1975) 13. Ellis, R.S., Monroe, J.L., Newman, C.M.: The GHS and other correlation inequalities for a class of even ferromagnets. Commun. Math. Phys. 46(2), 167–182 (1976) 14. Frohlich, J., Spencer, T.: The Kosterlitz –Thouless phase transition in two-dimensional abelian spin-systems and the coulomb gas. Commun. Math. Phys. 81, 527–602 (1981) 15. Garcia, R., Chan, M.H.W.: Critical fluctuation-induced thinning of 4 H e films near the superfluid transition. Phys. Rev. Lett. 83(6), 1187–1190 (1998) 16. Ganshin, A., Scheidemantel, S., Garcia, R., Chan, M.H.W.: Critical casimir force in 4 He films: confirmation of finite-size scaling. Phys. Rev. Lett. 97, 075301 (2006) 17. Gambassi, A., Dietrich, S.: Critical dynamics in thin films. J. Stat. Phys. 123(5), 929–1005 (2006)
Mean Field Analysis of Low–Dimensional Systems
341
18. Gobron, T., Merola, I.: First-order phase transition in potts models with finite–range interactions. J. Stat. Phys. 126, 507–583 (2007) 19. Hohenberg, P.C.: Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967) 20. Kennedy, T., Lieb, E.H., Shastry, B.S.: Existence of Néel order in some spin 1/2 Heisenberg antiferromagnets. J. Stat. Phys. 53, 1019–1030 (1988) 21. Kesten, H., Schonmann, R.: Behavior in large dimensions of the potts and Heisenberg models. Rev. Math. Phys. 1, 147–182 (1990) 22. Kosterlitz, J.M., Thouless, D.J.: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973) 23. Kotecký, R., Shlosman, S.B.: First-order phase transitions in large entropy lattice models. Commun. Math. Phys. 83, 493–515 (1982) 24. Lee, D.H., Caflisch, R.G., Joannopoulos, J.D.: Antiferromagnetic classical XY model: A mean–field analysis. Phys. Rev. B 29(5), 2680–2684 (1984) 25. Mayer, D.H.: The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics. Lecture Notes in Physics, Vol. 123, Berlin: Springer Verlag, 1980 26. Mermin, D., Wagner, H.: Absence of ferromagnetism or antiferromagnetism in one – or two–dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966) 27. Nakanishi, H., Fisher, M.E.: Critical Point Shifts in Films. J. Chem. Phys. 78(6), Part I, 15 March 1983 28. Pearce, P.W.: Mean–field bounds on the magnetization for ferromagnetic spin models. J. Stat. Phys. 25(2), 309–320 (1981) 29. Pearce, P.W., Thompson, C.J.: The anisotropic Heisenberg model in the long–range interaction limit. Commun. Math. Phys. 41(2), 191–201 (1975) 30. Shlosman, S.B.: The method of reflection positivity in the mathematical theory of first-order phase transitions. Russ. Math. Surv. 41(3), 83–134 (1986) 31. Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys. 77(2), 111–126 (1980) 32. Simon, B.: The Statistical Mechanics of Lattice Gases. Vol. I., Princeton Series in Physics, Princeton, NJ: Princeton University Press, 1993 33. Sokal, A.D.: Mean-field bounds and correlation inequalities. J. Stat. Phys. 28(3), 431–439 (1982) 34. Wu, F.Y.: The potts model. Rev. Mod. Phys. 54, 235–268 (1982) Communicated by H. Spohn
Commun. Math. Phys. 292, 343–389 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0908-z
Communications in
Mathematical Physics
On Classification of Modular Tensor Categories Eric Rowell1, , Richard Stong2, , Zhenghan Wang3, 1 Department of Mathematics, Texas A&M University, College Station,
TX 77843, U.S.A. E-mail: [email protected]
2 Center for Communications Research, 4320 Westerra Court, San Diego,
CA 92121-1969, U.S.A. E-mail: [email protected] 3 Microsoft Station Q, University of California, CNSI Bldg, Rm 2237, Santa Barbara, CA 93106-6105, U.S.A. E-mail: [email protected]; [email protected] Received: 16 December 2007 / Accepted: 13 June 2009 Published online: 25 August 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Sect. 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Sect. 6. 1. Introduction A modular tensor category (MTC) in the sense of V. Turaev determines uniquely a (2+1)-topological quantum field theory (TQFT) [Tu] (a seemingly different definition appeared in [MS1].) The classification of MTCs is motivated by the application of MTCs to topological quantum computing [F,Ki1,FKW,FLW1,FKLW,P], and by the use of MTCs in developing a physical theory of topological phases of matter [Wil,MR, FNTW,Ki2,Wa,LWe,DFNSS]. G. Moore and N. Seiberg articulated the viewpoint that rational conformal field theory (RCFT) should be treated as a generalization of group theory [MS2]. The algebraic content of both RCFTs and TQFTs is encoded by MTCs. Although two seemingly different definitions of MTCs were used in the two contexts [MS1,Tu], the two notions are essentially equivalent: an MTC in [MS1] consists of essentially the basic data of a TQFT in [Wal]. The theory of MTCs encompasses the most salient feature of quantum mechanics in the tensor product: superposition. Therefore, even without any applications in mind, the classification of MTCs could be pursued as a quantum generalization of the classification of finite groups.
The first author is partially supported by NSA grant H98230-08-1-0020. The second and third authors are partially supported by NSF FRG grant DMS-034772.
344
E. Rowell, R. Stong, Z. Wang
There are two natural ways to organize MTCs: one by fixing a pair (G, λ), where G is a compact Lie group, and λ a cohomology class ∈ H 4 (BG; Z); and the other by fixing the rank of an MTC, i.e. the number of isomorphism classes of simple objects. If a conjecture of E. Witten were true, then every MTC would come from a Chern-Simons-Witten (CSW) TQFT labelled by a pair (G, λ) [Witt,MS1,HRW]. Classification by fixing a compact Lie group G has been carried out successfully for G=finite groups [DW,FQ], G = T n torus [Ma,BM], and G = A, B, C, D simple Lie groups [FK,KW,TW]. In this paper, we will pursue the classification by fixing the rank. This approach is inspired by the study of topological phases of matter and topological quantum computing. Another reason is that we have evidence that there might be exotic (2+1)-TQFTs other than CSW theories [HRW]. Topological phases of matter are like artificial elements. The only known topological phases of matter are fractional quantum Hall liquids: electron systems confined on a disk immersed in a strong perpendicular magnetic field at extremely low temperatures [Wil,DFNSS]. Electrons in the disk, pictured classically as orbiting inside concentric annuli around the origin, organize themselves into some topological order [Wen,WW1,WW2]. Therefore, the classification of topological phases of matter resembles the periodic table of elements. The periodic table does not go on forever, and simpler elements are easier to find. The topological quantum computing project is to find MTCs in Nature, in particular those with non-abelian anyons. Therefore, it is important that we know the simplest MTCs in a certain sense because the chance for their existence is better. There is a hierarchy of structures on a tensor category: rigidity, pivotality, sphericity. We will always assume that our category is a fusion category: a rigid, semi-simple, C-linear monoidal category with finitely many isomorphism classes of simple objects, and the trivial object is simple. It has been conjectured that every fusion category has a pivotal structure [ENO]. Actually, it might be true that every fusion category is spherical. Another important structure on a tensor category is braiding. A tensor category with compatible pivotal and braiding structures is called ribbon. In our case a ribbon category is always pre-modular since we assume it is a fusion category. For each structure, we may study the classification problem. The classification of fusion categories by fixing the rank has been pursued in [O1,O2]. Since an MTC has considerably more structures than a fusion category, the classification is potentially easier, and we will see that this is indeed the case in Sects. 3 and 4. The advantage in the MTC classification is that we can work with the modular S matrix and T matrix to determine the possible fusion rules without first solving the pentagon and hexagon equations. For the classification of MTCs of a given rank, we could start with the infinitely many possible fusion rules, and then try to rule out most of the fusion rules by showing the pentagon equations have no solutions. However, pentagon equations are notoriously hard to solve, and we have no theories to practically determine when a solution exists for a particular set of fusion rules (Tarski’s theorem on the decidability of the first-order theory of real numbers provides a logical solution). So being able to determine all possible fusion rules without solving the pentagon equations greatly simplifies the classification for MTCs. As shown in [HH], all structures on an MTC can be formulated as polynomial equations over Z. Hence the classification of MTC is the same as counting points on certain algebraic varieties up to equivalence. But all the data of an MTC can be presented over a certain finite degree Galois extension of Q, probably over an abelian Galois extension of Q if normalized appropriately. Therefore, the classification problem is closer to number theory than to algebraic geometry. The argument in Sects. 3 and 4 is basically Galois theory
On Classification of Modular Categories
345
plus elementary yet complicated number theory. To complete the classification, we need to solve the pentagons and hexagons given the fusion rules. A significant complication comes from the choices of bases of the Hom spaces when solving the pentagon equations. The choices of basis make the normalization of 6 j symbols into an art: so far no computer programs are available to solve pentagons with a fusion coefficient > 1, but one set of such fusion rules is solved completely [HH]. Currently, there are no theories to count the number of solutions of pentagon equations for a given set of fusion rules without solving the pentagons. For unitary MTCs, there is tension between two desirable normalizations for 6 j symbols: to make the F matrix unitary, or to present all data of the theory in an abelian Galois extension of Q. For the Fibonacci theory, unitarity of the F matrix and abelianess of the Galois extension of Q cannot be achieved simultaneously, but with different F matrices, each can be obtained separately [FW]. This is the reason that we will only define the Galois group of a modular fusion rule and a modular data, but not the Galois group of an MTC. The main result of this paper is the classification of MTCs with rank=2, 3 and unitary MTCs of rank=4. The authors had obtained the classification of all unitary MTCs of rank ≤ 4 in 2004 [Wa]. The delay is related to the open finiteness conjecture: There are only finitely many equivalence classes of MTCs for any given rank. By Ocneanu rigidity the conjecture is equivalent to: There are only finitely many sets of fusion rules for MTCs of a given rank. Our classification of MTCs of rank ≤ 4 supports the conjecture. We also listed all quantum group MTCs up to rank ≤ 12 in Sect. 5. Two well-known constructs of MTCs are the quantum group method, and the quantum double of spherical tensor categories or the Drinfeld center. The quantum double is natural for MTCs from subfactor theory using Ocneanu’s asymptotic inclusions [EK]. It seems that this method might produce exotic MTCs in the sense of [HRW]. Our main technique is Galois theory. Galois theory was introduced into the study of RCFT by J. de Boer and J. Goeree [dBG], who considered the Galois extension K of Q by adjoining all the eigenvalues of the fusion matrices. They made the deep observation that the Galois group of the extension K over Q is always abelian. This result was extended by A. Coste and T. Gannon who used their extension to study the classification of RCFTs [CG]. Fusion rules of an MTC are determined by the modular S-matrix through Verlinde formulas. It follows that the Galois extension K is the same as adjoining to Q all entries of the modular S˜ matrix. When a Galois group element applies to the S˜ matrix entry-wise, this action is a multiplication of S˜ by a signed permutation matrix, which first appeared in [CG]. It follows that the entries of the S˜ matrix are the same up to signs if they are in the same orbit of a Galois group element. For a given rank ˜ ≤ 4, this allows us to determine all possible S-matrices, therefore, all possible fusion rules. Note that the Galois group of a modular data does not change the fusion matrices, but it can change a unitary theory into a non-unitary theory. For example, the Galois conjugate of the Fibonacci theory is the Yang-Lee theory, which is non-unitary. We might expect that for each modular data, one of its Galois conjugates would be realized by a unitary MTC. This is actually false. For example, take a rank=2 modular data with 1 −1 10 S˜ = , and T = . No Galois actions can change the S˜ matrix, hence −1 −1 0i the quantum dimension of the non-trivial simple object from −1 to 1, though the same fusion rules can be realized by a unitary theory: the semion theory. Reference [Ro1] contains a set of fusion rules which has non-unitary MTC realizations, but has no unitary realizations at all.
346
E. Rowell, R. Stong, Z. Wang
Table 1. Unitary MTCs of rank≤ 4 A
2 1
A
4
N
Z2
4 (A1 , 3) 1 2
U A
4
N
Z3
16
N
(A1 , 2)
4 (A1 , 5) 1 2
U A
10 Z2 × Z2
A
8 Z4
N
8 (A1 , 3)
N
4 (A1 , 7) 1
6 Fib × Fib
2
U
N
U
U
The paper is organized as follows. In Sect. 2, we study the implications of the Verlinde formulas using Galois theory. In Sects. 3 and 4, we determine all self-dual modular S˜ matrices of modular symbols of rank=2, 3, and unitary ones for rank=4. Rank=2 is known to experts, and rank=3 fusion rules have been previously classified [CP]. For modular data, Theorems 3.1 and 3.2 can also be deduced from [O1,O2]. In Sect. 5, we determine all UMTCs of rank ≤ 4. In Sect. 6, we discuss some open questions about the structure and application of MTCs. In the Appendix, together with S. Belinschi, we determine all non-self dual unitary modular data of rank ≤ 4. We summarize the classification of all rank ≤ 4 unitary MTCs into Table 1. There are a total of 70 unitary MTCs of rank ≤ 4 (a total of 35 up to ribbon tensor equivalence). The count is done in Sect. 5.4. Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. The 10 non-trivial prime UMTCs are the semion MTC, the Fibonacci MTC or (A1 , 3) 1 , the 2 Z3 MTC, the Ising MTC, the (A1 , 2) MTC, the even half of an SU (2) MTC at level 5 or (A1 , 5) 1 , the Z4 MTC, the toric code MTC, the (D4 , 1) MTC, and the even half 2 of an SU (2) MTC at level 7 or (A1 , 7) 1 . Their explicit data are listed in Sect. 5.3. Out 2 of the 10 non-trivial prime UMTCs, 9 are quantum group categories for a simple Lie group: the semion=SU (2)1 , the Fibonacci=(G 2 )1 , the Z3 =SU (3)1 , the Ising=complex conjugate of (E 8 )2 , the (A1 , 2)=SU (2)2 , the Z4 = SU (4)1 , the toric code= Spin(16)1 , the (D4 , 1) = Spin(8)1 , and the (A1 , 7) 1 =complex conjugate of (G 2 )2 . The Ising MTC 2 and the SU (2)2 MTC have the same fusion rules, but the Frobenius-Schur indicators of the non-abelian anyon σ are +1, −1, respectively. The toric code MTC and the Spin(8)1 MTC have the same fusion rules, but the twists are {1, 1, 1, −1}, and {1, −1, −1, −1}, πi respectively. We choose q = e in the quantum group construction. In the Ising case, πi it is the q = e− theory for E 8 at level=2. For notation and more details, see Sect. 5.3. We do not know how to construct (A1 , 5) 1 by cosets of quantum group categories. 2 The information for each rank is contained in one row of Table 1. Each box contains information of the MTCs with the same fusion rules. The center entry in a box denotes the realization of the fusion by a quantum group category or their products. We also use Fib to denote the Fibonacci category (A1 , 3) 1 . The upper left corner has either A or 2 N , where A means that all anyons are abelian, and N that at least one type of anyons is
On Classification of Modular Categories
347
non-abelian. The right upper corner has a number which is the number of different unitary theories with that fusion rule. If the lower right corner has a U , it means that at least one type of anyons has universal braiding statistics for topological quantum computation. The detailed information about which anyon is abelian or non-abelian, universal or non-universal is given in Sect. 6.3. It is worth noticing that the list of all fusion rules up to rank=4 agrees with the computer search for RCFTs in [GK]. We believe this continues to be true for rank=5. The rank= 6 list in [GK] is not complete. Finally, we comment on the physical realization of UMTCs. The existence of abelian anyons in ν = 13 FQH liquids is established theoretically with experimental support, while non-abelian anyons are believed to exist at the ν = 25 and ν = 12 5 plateaus (see [DFNSS] and the references therein). Current experimental effort is focused on FQH liquids at ν = 25 . But the fermionic nature of electrons complicates direct application of MTCs to FQH liquids because only anyonic properties of bosonic systems can be described fully by MTCs. In other words, we need a refined theory, e.g. a spin MTC, to describe a fermionic system [DW,BM]. 2. Galois Theory of Fusion Rules In this section, we study the implication of Verlinde formulas for fusion rules of MTCs. For more related discussion, see the beautiful survey [G]. Definition 2.1. (1) A rank=n label set is a finite set L of n elements with a distinguished element, denoted by 0, and an involution ˆ : L → L such that 0ˆ = 0. A label i ∈ L is self dual if iˆ = i. The charge conjugation matrix is the n × n matrix C = (δi jˆ ). Note that C is symmetric and C 2 = In , the n × n identity matrix. ˜ where N is a set of n n × n matri(2) A rank=n modular fusion rule is a pair (N ; S), ces Ni = (n i,k j )0≤ j,k≤n−1 , indexed by a rank=n label set L, with n i,k j ∈ Q , and S˜ = (˜si j )0≤i, j≤n−1 is an n × n matrix satisfying the following: (a) s˜00 = 1, s˜i, jˆ = s˜i, j , and all s˜i,0 ’s are non-zero; n−1 2 S˜ (b) If we let D = i=0 s˜i,0 , then S = D is a symmetric, unitary matrix. Furthermore, the matrices Ni in N and S˜ are related by the following: ˜ i Ni S˜ = S
(2.1)
ia for all i ∈ L, where i = (δab λia )n×n is diagonal, and λia = ss˜˜0a . The identities (2.1) or equivalently the Verlinde formulas (2.3) below imply many
ˆ i, j
jˆ i,k
symmetries among n i,k j : n k0, j = δ jk , n i,k j = n kj,i = n kˆ ˆ = n ˆ . The matrix Ni will be called the i th fusion matrix. From identities (2.1), the diagonal entries in i are the eigenvalues of Ni , and the columns of S˜ are the corresponding eigenvectors. The non-zero number D will be called the total quantum order, n−1 2 di = s˜i0 the quantum dimension of the i th label, and D 2 = i=0 di the global quantum dimension. (3) A rank=n modular symbol consists of a triple (N ; S, T ). The pair (N ; sS00 ) is a rank=n modular fusion rule with all n i,k j ∈ N = {0, 1, 2, · · · } (here s00 is the (0,0)entry of the unitary matrix S = (si j )0≤i, j≤n−1 ), and the n×n matrix T = (δab θa )n×n is diagonal, and θ0 = 1. Furthermore, S and T satisfy
348
E. Rowell, R. Stong, Z. Wang
(i) (ST )3 = (D+ s00 )S 2 ; (ii) S 2 = C; (iii) θi ∈ U(1) and θiˆ = θi for each i, n−1 ±1 2 θi di . The following identity can be deduced: where D± = i=0 D+ D− = D 2 .
(2.2)
The complex number θi will be called the twist of the i th label. Note that s00 might be − D1 . A modular symbol is called unitary if each quantum dimension di is the Frobenius-Perron eigenvalue of the corresponding fusion matrix Ni . In particular, the quantum dimensions di ’s are positive real numbers ≥ 1. (4) A modular symbol (N ; S, T ) is called a modular data if there is an MTC whose ˜ fusion rules, modular S-matrix, and T -matrix are given by N , sS00 , T of the modular symbol. (5) Let = {λi j }i, j∈L for a rank=n modular fusion rule, and let K = Q(λi j ), i, j ∈ L be the Galois extension of Q. Then the Galois group G of the Galois field K over Q is called the Galois group of the modular fusion rule. ˜ T ) related in the We are interested in searching for n + 2 tuples (N0 , . . . , Nn−1 ; S, correct fashion. We will index the rows and columns of matrices by 0, 1, . . . , n − 1. ˜ i , the columns of S˜ must be eigenvectors of Ni with eigenvalues λi,0 , Since Ni S˜ = S λi,1 , . . ., and λi,n−1 , respectively. Looking at the first entries of these columns and of ˜ and using the only non-zero 1 of the first row of Ni , we see that λi,0 = di , and Ni S, d j λi, j = s˜i, j . It follows that K is the same as Q(˜si j ), i, j ∈ L. Since S˜ is symmetric, we see that for i = j we have d j λi, j = di λ j,i , and s˜i, j = di λ j,i = d j λi, j for all i and j. Let n i,k j denote the ( j, k) entry of Ni . Since ⎞ ⎛ 0 ··· 0 λi,0 1 ⎜ 0 λi,1 · · · 0 ⎟ ˜† Ni = 2 S˜ ⎝ S , ··· ··· ··· ··· ⎠ D 0 0 · · · λi,n−1 we compute for 0 ≤ j, k ≤ n − 1, n i,k j =
n−1 n−1 s˜i,m s˜ j,m s˜k,m −2 d2 D = λi,m λ j,m λk,m m2 . dm D
m=0
(2.3)
m=0
The fusion matrices can also be described equivalently by fusion algebras. For a rank=n fusion rule, each label i is associated with a variable X i . Then the fusion ring R is the free abelian ring Z[X 0 , . . . , X n−1 ] generated by X i ’s modulo relations (called n−1 k n i, j X k . The fusion algebra will be F = R ⊗ Z K , where fusion rules) X i X j = i=0 K is the Galois field of the fusion rules above. We may replace K by C. If the modular fusion rule is realized by an MTC, then X i is an equivalence class of simple objects, and the multiplication X i X j is just the tensor product. There are modular symbols that are not modular data. Example 2.2. Take the following:
⎛
1 1 √ S= ⎝ 2 2 1
√
2 0 √ − 2
⎞ 1 √ − 2⎠ , 1
On Classification of Modular Categories
349
and T = Diag(1, θ, −1). The fusion matrices Ni are determined by the formulas (2.3), hence are independent of θ . They are the same as those of the Ising MTC in Sect. 5.3. Therefore, for any θ ∈ U(1), we get a modular symbol. But only when θ is a 16th root of unity, do we have modular data. Very likely the modular symbol of an MTC determines the MTC , and we do not know when a modular symbol becomes a modular data. Proposition 2.3. If (N ; S, T ) is a modular data, then we have: (1) θi θ j si j = k n kˆ sk0 θk . ij 4 Ai j Ai j 3 = θi j , (2) j θj j
j
where Ai j = 2n ˆ n ii j + n ii n i ˆ . ii ji θ2 1 i ˆ and is ±1 if k = k. ˆ νk is (3) Let νk = D 2 i, j∈L n k, j di d j θi2 , then νk is 0 if k = k, j
called the Frobenius-Schur indicator of k. πic (4) D+ s00 = e 4 for some c ∈ Q. The rational number c mod 8 is called the topological central charge of the modular data. Proof. For (1), see [BK, Eq. (3.1.2)] on p. 47. For (2), it is [BK, Theorem 3.1.19] found on p. 57. Formula (3) from [Ba] for RCFTs can be generalized to MTCs. (4) follows from Theorem 2.5. Proposition 2.3 (2) implies that the θi are actually roots of unity of finite order, which is often referred to as Vafa’s Theorem. But from Example 2.2, we know that this is not true for general modular symbols, in particular Q(θi ) might not be algebraic for modular symbols. This leads to: Definition 2.4. Given a modular data (N ; S, T ), let K N be the Galois field Q(˜si j , D, θi ), i, j ∈ L. Then the Galois group of K N over Q will be called the Galois group of the modular data. Theorem 2.5. (1) (de Boer-Goeree theorem): The Galois group of a modular fusion rule is abelian. (2) The Galois group of a modular data is abelian. By the Kronecker-Weber theorem, there is an integer m such that K N ⊂ Q(ζm ), 2πi where ζm = e m . The smallest such m for K N is called the conductor of K N , and the order of T always divides N (we intentionally build N into the notation K N ). The Galois group of Q(ζ N ) is the cyclic group of units l such that gcd(l, N ) = 1. Each l acts on K N as the Frobenius map σl : ζ N → ζ Nl . Consequently, σl (T ) = T l and σl (S) = S P˜σ , where the signed permutation matrix P˜σ corresponds to the Galois element σ in the Galois group of the modular fusion rule. It is known that the fusion algebra of a rank=n MTC is isomorphic to the function algebra of n points. A Galois group element σ of the associated modular fusion rule induces an isomorphism of the fusion algebra. It follows that σ determines a permutation of the label set. When we have only a modular fusion rule, the two algebra structures on the fusion algebra a priori might not be isomorphic to each other. But still a Galois group element of the modular fusion determines a permutation of the label set and the
350
E. Rowell, R. Stong, Z. Wang
de Boer-Goeree theorem holds. Actually what we are using in this paper are identities among modular S˜ entries up to some parity signs i,σ = ±1 associated to each Galois element σ . Such parity signs first appeared in [CG] for Galois automorphisms of Q(λi, j , D). First we note the following easy, but very useful fact that the ordered set of eigenvalues of Ni determines the label i uniquely. Proposition 2.6. There do not exist indices j = k such that λi, j = λi,k for all i for any ˜ modular fusion rule (N ; S). Proof. If there were such indices, then the dot product of rows j and k of S˜ would be n−1 |˜si, j |2 > 0, a contradiction. D 2 = i=0 Except (5), the following theorem is contained in [CG]. ˜ Then Theorem 2.7. Let G be the Galois group of a rank=n modular fusion rule (N ; S). (1) the simultaneous action of the Galois group G on the set = {λi j } gives an injective group homomorphism ι : G → Sn , where Sn is the permutation group of n letters; for σ ∈ G, ι(σ )(i) is the associated element in Sn . ˜ is a signed permutation matrix; (2) For any σ ∈ G, the matrix P˜σ = dσ (0) S˜ −1 σ ( S) ˜ furthermore, the map σ → Pσ gives a group homomorphism from G to the signed permutation matrices modulo ±1 which lifts ι. (3) For each σ ∈ G, there are i,σ = ±1 such that 1 σ (k),σ s˜ j,σ (k) . dσ (0)
(2.4)
s˜ j,k = σ ( j),σ k,σ s˜σ ( j),σ −1 (k) ,
(2.5)
σ −1 (k),σ −1 = σ (0),σ 0,σ k,σ .
(2.6)
σ (˜s j,k ) = Moreover,
and
(4) The Galois group G is abelian. n−1 (5) If n is even, then i=0 i,σ = (−1)σ . If n is odd, then D ∈ K , and σ (D) = σ · dσD(0) , n−1 i,σ = σ · (−1)σ . where σ = ±1. We have i=0 We are going to use σ for both the element of the Galois group G and its associated element of Sn . When σ ∈ G applies to a matrix, σ applies entry-wise. Proof. Let K = Q[{λi, j }0≤i, j≤n−1 ] be the Galois extension of Q generated by the eigenvalues of all the Ni and let G be the associated Galois group as above. The action of G on the eigenvalues gives an injection G → Sn × Sn × · · · × Sn , where there are n − 1 factors. Note that we have not assumed the Ni have distinct eigenvalues, therefore this map is not necessarily unique and is not necessarily a group homomorphism. This is not a problem as we will resolve the ambiguity shortly. Just fix one such map for now. Let (σ1 , σ2 , . . . , σn−1 ) denote the image of σ ∈ G under this injection. Note that a priori,
On Classification of Modular Categories
351
there is no relationship between the σi . Let i be the diagonal matrix with diagonal entries λi, j , so ˜ i S˜ −1 . Ni = S Let Pσi = (δi=σi ( j) )0≤i, j≤n−1 be the permutation matrix corresponding to σi . Since i Pσi . Since Ni is rational we have σ (λi, j ) = λi,σi ( j) , we have σ (i ) = Pσ−1 i ˜ σ−1 i Pσi σ ( S) ˜ −1 . ˜ i S˜ −1 = Ni = σ (Ni ) = σ ( S)P S i Rewriting this gives ˜ σ−1 ] = [ S˜ −1 σ ( S)P ˜ σ−1 ]i . i [ S˜ −1 σ ( S)P i i ˜ σ−1 commutes with i . It follows that Bi,σ is block diagonal, Hence Bi,σ = S˜ −1 σ ( S)P i with blocks corresponding to the equal eigenvalues of Ni . In formulas, if the ( j, k) entry ˜ = Bi,σ Pσi = Cσ . Note two facts, if of Bi,σ is nonzero, then λi, j = λi,k . Let S˜ −1 σ ( S) the ( j, k) entry of Cσ is nonzero, then the ( j, σi (k)) entry of Bi,σ is nonzero and hence λi, j = λi,σi (k) . The second fact is that Cσ (as the notation suggests) does not depend on i, only on σ . Suppose Cσ has 2 nonzero entries in column k, say the ( j, k) and (, k) entries. Then λi, j = λi,σi (k) = λi, for all i, contradicting Proposition 2.6 above. If a row or column of Cσ is all zeroes, then det(Cσ ) = 0, a contradiction. Hence Cσ has exactly one nonzero entry in every row and in every column. Thus there is a unique permutation σ ∈ Sn and a diagonal matrix Bσ such that Cσ = Bσ Pσ . Note that we are now using σ for both the element of the Galois group and its associated element of Sn . Note that ˜ = S˜ −1 σ ( S)σ ˜ ( S˜ −1 σ ( S)) ˜ = Cσ σ (Cσ ), Cσ σ = S˜ −1 σ σ ( S) from which it follows that the map G → Sn is a group homomorphism. Thus we have proved that the simultaneous action of the Galois group G on the eigenvalues λi, j of Ni for all i gives an injective group homomorphism G → Sn . n−1 2 di , which must be Note that the squared length of column zero of S˜ is D 2 = i=0 equal to the squared length of column σ (0). Hence
n−1 n−1 2 2 2 D2 = dσ2 (0) λi,σ λi,0 = dσ2 (0) σ (D 2 ). (0) = dσ (0) σ i=0
i=0
Rewriting gives σ
1 D2
=
dσ2 (0) D2
.
It follows that G acts in the same way on the quantities {d j /D 2 }. The Verlinde formulas (2.3) encode the symmetry of the Ni matrices, and give us the complete symmetry under interchanging the last n − 1 Ni and simultaneously reordering the last n − 1 rows and columns of all matrices. Thus n i,k j is invariant under G and hence is necessarily rational if we define it first to be only in R. ˜ = Cσ and inverting this identity gives the two Transposing the identity S˜ −1 σ ( S) −1 T ˜ ˜ equations σ ( S) S = Cσ and ˜ −1 S˜ = Cσ−1 = σ ( S)
D2 ˜ S˜ −1 = d 2 CσT . σ ( S) σ (0) σ (D 2 )
352
E. Rowell, R. Stong, Z. Wang
Hence the matrices dσ (0) Cσ and dσ (0) Bσ are orthogonal. Since Bσ is diagonal it follows that ⎛ ⎞ 0,σ 0 ··· 0 1 ⎜ 0 1,σ · · · 0 ⎟ Bσ = ⎝ ··· ··· ··· ⎠ dσ (0) · · · 0 · · · · · · n−1,σ for some choices of i,σ = ±1. The map σ → dσ (0) Cσ gives a group homomorphism from G to the signed permutation matrices modulo ±1 which lifts the homomorphism ι of (1). ˜ = S˜ Bσ Pσ . Picking out the ( j, k) entry, we have Rewrite the definition of Cσ as σ ( S) σ (˜s j,k ) = dσ1(0) σ (k),σ s˜ j,σ (k) . Moreover, since the left-hand side is symmetric we get ˜ In coordinates this condition becomes s˜ j,k = k σ ( j) s˜σ ( j),σ −1 (k) . S˜ Bσ Pσ = Pσ−1 Bσ S.
Consider the action of G on pairs ( j, k) defined by σ × ( j, k) → (σ ( j), σ −1 (k)). Then we see that |˜s j,k | is constant on orbits of this action. To see identity (2.6), we apply σ −1 to identity (2.4) and compare with identity (2.5). Note that s˜σ −1 (0),σ (0) = σ (0),σ 0,σ by identity (2.5). Given σ1 , σ2 ∈ G, consider first σ2 σ1 (˜s j,k ) = σ2 ( dσ 1(0) σ1 (k),σ1 s˜ j,σ1 (k) ) = σ2 ( dσ 1(0) σ1 (k),σ1 s˜σ1 (k), j ) = 1 Then consider
1
1 dσ2 (0) λσ1 (0),σ2 (0) σ1 (k),σ1 σ2 ( j),σ2 s˜σ1 (k),σ2 ( j) .
1
σ1 σ2 (˜s j,k ) = σ1 σ2 (˜sk, j ) = σ1 σ ( j),σ2 s˜k,σ2 ( j) dσ2 (0) 2 1 1 σ2 ( j),σ2 s˜σ2 ( j),k = σ ( j),σ2 σ1 (k),σ1 s˜σ2 ( j),σ1 (k) . = σ1 dσ2 (0) dσ1 (0) λσ2 (0),σ1 (0) 2 Hence σ1 σ2 = σ2 σ1 using di λ j,i = d j λi, j , i.e. G is abelian. ˜ = ˜ 2 = D 2n , hence det( S) Suppose now that the rank n = 2r is even. Then det( S) 2r ˜ ±D . Since the determinant is a polynomial in the entries of the matrix det(σ ( S)) = ˜ Hence det( S˜ −1 σ ( S)) ˜ = dσ−n ±σ (D 2 )r , with the same sign as det( S). (0) . Since n−1 n−1 −n σ σ det(Cσ ) = dσ (0) (−1) j=0 j,σ , we conclude j=0 j,σ = (−1) . For odd rank ˜ = ±D 2r +1 , hence D ∈ K . Hence σ (D) = σ D/dσ (0) , σ = ±1 and n = 2r + 1, det( S) σ one gets the formula n−1 j=0 j,σ = σ (−1) . Note that the resulting Eqs. (2.5) for the entries s˜ j,k are unchanged if we replace Bσ with −Bσ . We will use this to assume 0 = 1 below. Next we will use the fact that the θi ∈ U(1) to produce a series of twist inequalities ˜ on the entries of S. Theorem 2.8. Given a modular symbol (N ; S, T ) and S is a real matrix, then (1) 2 maxi s˜i,2 j ≤ D|˜s j j | + D 2 for any j. n−1 (2) If j = k, then D ≤ |˜s 1j,k | i=0 |˜si, j s˜i,k |. n−1 σ ( j) s˜ j,σ ( j) (3) i: σ (i)=i θi σ (i) . j=0 θ j θσ ( j) = D−
On Classification of Modular Categories
353
˜ ST ˜ = D+ S. ˜ Then taking the ( j, k) entry of Proof. Rewrite the twist equation as T ST this formula gives θ j θk
n−1
θi s˜i, j s˜i,k = D+ s˜ j,k .
i=0
Since |D+ | = D and |θi | = 1, the largest of the n+1 numbers |˜si, j s˜i,k |, 0 ≤ i ≤ n−1, and D|˜s j,k | must be at most the sum of the other n. If j = k, then i s˜i,2 j = D 2 > D|˜s j j |. Hence this inequality is trivial unless the largest is one of the first n and we get 2 max s˜i,2 j ≤ D|˜s j j | + i
If j = k then
i s˜i, j s˜i,k
n−1
s˜i,2 j .
i=0
= 0 and the nontrivial case is D≤
1
n−1
|˜s j,k |
i=0
|˜si, j s˜i,k |.
We will refer to these as the twist inequalities. Suppose σ ∈ G corresponds to signs i as above. We drop σ for notational easiness. Multiply the identity above by σ ( j) /(θ j θσ ( j) ), set k = σ ( j), and sum over j. The result is n−1
σ ( j)
j=0
n−1
θi s˜i, j s˜i,σ ( j) = D+
i=0
n−1 σ ( j) s˜ j,σ ( j) j=0
θ j θσ ( j)
.
Interchanging the sums and using the fact that s˜i,σ ( j) = σ ( j) σ (i) s˜σ (i), j gives n−1
θi σ (i)
i=0
n−1
s˜i, j s˜σ (i), j = D+
j=0
n−1 σ ( j) s˜ j,σ ( j) j=0
θ j θσ ( j)
.
˜ the innermost sum on the left is zero if i = σ (i) and By orthogonality of the rows of S, D 2 = D+ D− if i = σ (i). Hence n−1 σ ( j) s˜ j,σ ( j) j=0
If σ is fixed point free, then
θ j θσ ( j) n−1 j=0
= D−
σ ( j) s˜ j,σ ( j) θ j θσ ( j)
θi σ (i) .
i: σ (i)=i
= 0.
3. Rank=2 and 3 Modular S Matrices In this section, we determine all possible modular S matrices for rank=2 and 3 modular symbols. The rank=3 case first appeared in [CP], but our proof is new. Theorem 3.1. The only possible rank=2 modular S˜ matrices of some modular symbols are
354
E. Rowell, R. Stong, Z. Wang
(1) 1
, −1
1 ϕ
ϕ , −1
where 2 = 1; (2)
where ϕ 2 = 1 + ϕ. Proof. Since all labels are self-dual, S˜ is a symmetric real unitary matrix of the form 0 1 1 d , so we have d 2 = 1 + md. Simpli. The fusion matrix N1 is of the form 1m d −1 fying D+ D− = D 2 leads√ to θ + θ −1 = 1 − d 2 = −m · d. Since θ ∈ U(1),√so |md| ≤ 2. 2 2 If d > 0, then d = m+ 2m +4 , hence m = 0, 1. If d < 0, then d = m− 2m +4 , hence pπi
[Q(θ + θ −1 ) : Q] ≤ 2. It follows that θ = e q for some ( p, q) = 1, and q is one of {1, 2, 3, 4, 5, 6}. Direct computation shows there are no integral solutions p, q for √ pπ m− m 2 +4 2 cos( q ) = −m · except for q = 2, 5 and m = 0, 1. 2 Theorem 3.2. Then the only possible rank=3 modular S˜ matrices of some modular symbols up to permutations are (1) ⎛
1 ⎝
ω ω2
⎞ ω2 ⎠ , ω
d 0 −d
⎞ 1 −d ⎠ , 1
d1 −d2 1
⎞ d2 1 ⎠, −d1
where 2 = 1, and ω3 = 1, ω = 1. (2) ⎛
1 ⎝d 1 where d 2 = 2. (3) ⎛
1 ⎝d1 d2
where d1 is a real root of x 3 − 2x 2 − x + 1 and d2 = d1 /(d1 − 1) which is a cos(π/7) root of x 3 − x 2 − 2x + 1. The largest d1 = 2 2cos(π/7)−1 = 2.246979604 . . . , and d2 = 2 cos(π/7) = 1.801937736 . . ..
On Classification of Modular Categories
355
Proof. The non-self dual case is given in the Appendix. Hence we assume all fusion rules are self-dual, so S˜ is a real, symmetric, unitary matrix up to the scalar D. It follows that the fusion matrices Ni ’s are commutative, symmetric, integral matrices. One approach to proving the theorem is to analyze case by case for the Galois groups of fusion rules G ∼ = 1, Z2 , Z3 . This strategy will be fully exploited in the rank=4 case in ˜ the next section. Instead we will argue directly from the S-matrix in this section. The fusion matrices N1 , N2 are symmetric, and N1 N2 = N2 N1 . Therefore, they can be written as ⎛ ⎞ 0 1 0 N 1 = ⎝1 m k ⎠ 0 k l and
⎛ 0 N2 = ⎝0 1
0 k l
⎞ 1 l⎠ n
such that 1 + ml + kn = k 2 + l 2 . There characteristic polynomials are p1 (x) = x 3 − ( + m)x 2 + (m − k 2 − 1)x + = 0 and p2 (x) = x 3 − (k + n)x 2 + (nk − 2 − 1)x + k = 0, respectively. Next we turn to the S˜ matrix, which is of the following form: ⎞ ⎛ 1 d1 d2 S˜ = ⎝d1 s˜11 s˜12 ⎠ . d2 s˜12 s˜22 Orthogonality of the columns of the S˜ matrix translates into the equations d1 + d1 s˜11 + d2 s˜12 = 0, d2 + d1 s˜12 + d2 s˜22 = 0, d1 d2 + s˜12 (˜s11 + s˜22 ) = 0. The first two equations give s˜11 = −1 − d2 s˜12 /d1 and s˜22 = −1 − d1 s˜12 /d2 . Plugging these into the third equation gives 2 (d12 + d22 )˜s12 + 2d1 d2 s˜12 − d12 d22 = 0,
hence s˜12 =
d1 d2 . 1± D
356
E. Rowell, R. Stong, Z. Wang
Thus s˜11 = −1 −
d22 , 1± D
s˜22 = −1 −
d12 . 1± D
and
Thus the eigenvalues of N1 are d1 , b =
d22 s˜11 s˜12 d1 1 , and c = = =− − d2 1± D d1 d1 d1 (1 ± D)
and the eigenvalues of N2 are d2 , e =
d12 1 s˜12 d2 , and f = − − . = d1 1± D d2 d2 (1 ± D)
We compute d1 b + d2 f = d1 c + d2 e = bc + e f = −1. Since d1 bc = − and d2 e f = −k, these are equivalent to k k k + = + = + = 1. c e b f d1 d2 Also note that d1 e = d2 b. Let’s deal with the case where = 0 first. Then we have k 2 = kn + 1. Hence k = 1 and n √ = 0. Thus the eigenvalues√ of N2 are 1, 1, and -1 and the eigenvalues of N1 are 2 (m + m + 8)/2, 0, and (m − m 2 + 8)/2. Since N1 has eigenvalues d1 , b, c, and d1 = 0, hence c = 0 which implies m = 0. This gives (k, , m, n) = (1, 0, 0, 0) and ⎛
1 S˜ = ⎝d 1
d 0 −d
⎞ 1 −d ⎠ , 1
where d 2 = 2. The case k = 0 gives essentially the same solution, so we will henceforth assume and k are positive. Since p1 () = −k 2 ≤ 0 and p1 (0) = ≥ 0, we see that the largest root of p1 is > , one of the remaining roots is in (0, ) and the other root is negative. Similarly the largest root of p2 is > k and the other roots are in (0, k) and (−∞, 0).
On Classification of Modular Categories
357
Case 1. The polynomial p1 (x) is reducible. Since d1 > , d1 cannot be an integer. Thus p1 must split into a linear and an irreducible quadratic. Thus Q[d1 , D] is a quadratic extension of Q. Hence Q[d1 , d2 , D] has degree 2 or 4 over Q. Thus p2 is also reducible and also splits into a linear and an irreducible quadratic. Since the /b + k/ f = /c + k/e = 1, the integral roots must be either b and f or c and e. Without loss, we may assume the integer roots are b and f . Let √ √ d2 = α + β s and e = α − β s for rational (in fact integer or half-integer) α and β and integer s. Then since d1 e = d2 b and c is the conjugate of d1 we have √ √ α+β s α−β s d1 = b √ and c = b √ . α−β s α+β s Hence = −d1 bc = −b3 . Since f = −k/(d2 e) = −k/(α 2 − β 2 s) and −b2 = /b = 1 − k/ f = α 2 + 1 − β 2 s. Therefore solving 1 = k/d2 + /d1 for k gives √ √ √ e d2 = α + β s + b3 = α + β s − (α 2 + 1 − β 2 s)(α − β s) d1 b √ = −α(α 2 − β 2 s) + β[α 2 + 2 − β 2 s] s.
k = d2 −
Since k is an integer, this forces α 2 − β 2 s = −2, hence b2 = 1. Since > 0, this √ means = 1 and√b = −1. Also from the equations√above we get k = 2α, d2 = α + √α 2 + 2, e = α − α 2 + 2, f = α, d1 = α 2 + 1 + α α 2 + 2 = d22 /2, c = α 2 + 1 − α α 2 + 2, √ and D = α 2 + 2 + α α 2 + 2. Thus p1 (x) = x 3 − (2α 2 + 1)x 2 − (2α 2 + 1)x + 1 and p2 (x) = x 3 − 3αx 2 + (2α 2 − 2)x + 2α. Thus (k, , m, n) = (2α, 1, 2α 2 , α) and ⎛
1√ S˜ = ⎝α 2 + 1 + α α 2 + 2 √ α + α2 + 2
√ α2 + 1 + α α2 + 2 1 √ −α − α 2 + 2
√ ⎞ α + √α 2 + 2 −α − √α 2 + 2 ⎠ . α2 + α α2 + 2
Note that n = α must be a non-negative integer. Setting α = 0 gives the example found above again. Thus we may assume α ≥ 1. Since d1 = d22 /2, the equation for the θ ’s is |1 + θ2 d22 + (1/4)θ1 d24 | = D = 1 + (1/2)d22 . 2 4 2 If a solution did exist, then 2 − d2 − 1, hence we√would have 1 + (1/2)d2 ≥ (1/4)d√ 2 2 17 ≥ (d2 − 3) or d2 ≤ 3 + 17 = 2.66891 . . .. Since d2 ≥ 1 + 3 = 2.73205 . . ., this cannot occur. Thus these S˜ matrices, for positive α, do not give a modular symbol.
358
E. Rowell, R. Stong, Z. Wang
Case 2. The polynomial p1 (x) is an irreducible cubic. By Case 1, we see that p2 (x) is also an irreducible cubic. Then there must be a Galois symmetry σ with σ (d1 ) = b, σ (b) = c and σ (c) = d1 . Hence σ (d2 ) = f , σ ( f ) = e, and σ (e) = d2 since these roots of p2 pair with the corresponding roots of p1 . Applying σ to the identity d1 e = d2 b, gives d2 b = f c. Thus we must have f c = d1 d2 /(1 ± D). Since fc =
d12 + d22 1 d1 d2 d1 d2 D2 ± D + , + = + 2 2 (1 ± D) d1 d2 (1 ± D) d1 d2 (1 ± D) d1 d2 (1 ± D)
we compute 1± D =
(1 ± D)2 D(1 ± D)2 fc = 1± , d1 d2 d12 d22
and hence
1± D d1 d2
2 = 1.
Since D > 1, we get that s˜12 =
d1 d2 = ±1, 1± D
and hence b = ±1/d2 and e = ±1/d1 . Thus d1 and d2 are units in the ring of algebraic integers. Hence k = = 1 and hence m + n = 1. Without loss we may assume m = 1 and n = 0. Then p1 (x) = x 3 − 2x 2 − x + 1 and p2 (x) = x 3 p1 (1/x) = x 3 − x 2 − 2x + 1. Then one computes d1 =
2 cos(π/7) = 2.246979604 . . . , 2 cos(π/7) − 1
b = 1 − 1/d1 , c = −1/(d1 − 1), d2 = d1 /(d1 − 1) = 1.801937736 . . ., e = 1/d1 , and f = 1 − d1 and ⎛ ⎞ 1 d1 d2 1 ⎠. S˜ = ⎝d1 −d2 d2 1 −d1 4. Rank = 4 Modular S Matrices First we introduce the following notation. For an integer m, define √ m + m2 + 4 φm = , 2 that is, φm is the unique positive root of x 2 − mx − 1 = 0. Note that any algebraic number φ whose only conjugate is −1/φ must be φm for some integer m. Also note the only rational φm is φ0 = 1.
On Classification of Modular Categories
359
Theorem 4.1. The only possible rank=4 modular S˜ matrices of unitary modular symbols up to permutations are (1)
⎛ 1 ⎜1 ⎝1 1
1 1 −1 −1
1 −1 ω ω¯
⎞ 1 −1⎟ , ω¯ ⎠ ω
⎛ 1 ⎜1 ⎝1 1
1 1 −1 −1
1 −1 1 −1
⎞ 1 −1⎟ ; −1⎠ 1
⎛ 1 ⎜1 ⎝1 1
1 −1 1 −1
1 1 −1 −1
⎞ 1 −1⎟ ; −1⎠ 1
ϕ −1 ϕ −1
1 ϕ −1 −ϕ
⎞ ϕ −1 ⎟ , −ϕ ⎠ 1
ϕ ϕ2 −1 −ϕ
⎞ ϕ2 −ϕ ⎟ ⎟; −ϕ ⎠ 1
where ω = ±i; (2)
(3)
(4)
⎛
1 ⎜ϕ ⎝1 ϕ where ϕ = (5)
√ 1+ 5 2
is the golden ratio; ⎛
1 ⎜ϕ ⎜ ⎝ϕ ϕ2 (6)
⎛
1 ⎜d 2 − 1 ⎜ ⎝ d +1 d
ϕ −1 ϕ2 −ϕ
d2 − 1 0 −d 2 + 1 d2 − 1
d +1 −d 2 + 1 d −1
⎞ d d2 − 1 ⎟ ⎟, −1 ⎠ −d − 1
where d is the largest real root of x 3 − 3x − 1. Proof. The non-self dual case is treated in the Appendix, so we will assume that S˜ is real in the following. Since the fusion coefficients n i,k j are totally symmetric in i, j and k for self-dual categories, we will instead write n i, j,k in what follows. For notational easiness, when the Galois group element σ is clear from the context, we simply write i,σ as i . Identities for i and s˜ jk that are not referenced are all from Theorem 2.7. All the twist inequalities are from Theorem 2.8.
360
E. Rowell, R. Stong, Z. Wang
Case 1. G contains a 4-cycle. By symmetry we may assume σ = (0 1 2 3) ∈ G. The conditions s˜ j,k = k σ ( j) s˜σ ( j),σ −1 (k) and 1 2 3 = −1 give ⎛
1 d ⎜ S˜ = ⎝ 1 d2 d3
d1 1 2 d2 − 2 d3 1
d2 − 2 d3 −1 2 d1
⎞ d3 1 ⎟ . 2 d1 ⎠ − 1 2 d2
By symmetry under interchanging N1 and N3 , we may assume 2 = +1. Note that σ 2 (d2 ) = λ2,2 = −1/d2 = d2 . Hence the characteristic polynomial p2 of N2 is irreducible. Since σ 2 (d1 ) = −d3 /d2 < 0. Hence σ 2 (d1 ) = d1 . Thus p1 is irreducible. Since 1 /d3 is a root of p1 , it follows that p3 is also irreducible. We see that λ1,1 = 1 d2 /d1 , λ1,2 = −d3 /d2 , and λ1,3 = 1 /d3 . In particular d1 λ1,1 λ1,2 λ1,3 = −1. Orthogonality of the rows of S˜ is equivalent to
d1 + 1 d1 d2 − d2 d3 + 1 d3 = 0, or 1 1 + λ1,3 = λ1,1 + . d1 λ1,2
Write p1 (x) = x 4 −c1 x 3 +c2 x 2 +c3 x −1. Then p4 (x) = x 4 − 1 c3 x 3 −c2 x 2 + 1 c1 x −1. Note that c1 = Trace(N1 ) ≥ 0 and 1 c3 = Trace(N3 ) ≥ 0. Multiplying together the orthogonality condition above and five of its formal conjugates gives
128 + (c32 − c12 )2 − 16c1 c3 + 12(c32 − c12 )c2 + 32c22 = 0. This equation forces c1 and c3 to be even. Let = c1 −c3 and = c1 +c3 (hence and are even and congruent mod 4). Then solving the quadratic equation√above for c2 we 3±
( 2 −32)(2 +32)
. see that we must have ( 2 −32)(2 +32) to be a square and c2 = 16 It follows that || ≥ 6. If and are multiples of 4, then we see they are multiples of 8 and either sign gives an integral c2 . If and are both 2 mod 4, then there is a unique choice of the sign for which c2 is integral. The Galois group of p1 must be Z/4Z, otherwise it would contain the (0 1)(2 3). Applying this to the orthogonality identity above gives 1/λ1,3 + d1 = λ1,2 + 1/λ1,1 . Multiplying this by the original identity gives d1 λ11,3 + d1 λ1,3 = λ1,11λ1,2 + λ1,1 λ1,2 . Hence d1 λ1,3 = (λ1,1 λ1,2 )±1 , either of which contradicts the product of all four roots being −1. In particular, p1 cannot have complex roots, since complex conjugation would give a transposition in the Galois group. Applying σ to the orthogonality identity gives d1 + 1/λ1,1 = λ1,2 + 1/λ1,3 .
On Classification of Modular Categories
361
We know from the preliminary discussionthat all three of the resulting Ni matrices
will be rational. Define P =
16c2 −3 2 +32
n 1,1,1 = n 1,1,2 = n 1,1,3 = n 1,2,2 = n 1,2,3 = n 1,3,3 = n 2,2,2 = n 2,2,3 = n 2,3,3 = n 3,3,3 =
=±
2 −32 , 2 +32
then we compute
c1 − c3 5c1 − 3c3 − P, 8 8 1 (P − 1), c1 − c3 c1 + c3 − P , 1 8 8 c1 + c3 c1 − c3 + P, 4 4 P, c1 + c3 c1 − c3 − P, 8
8 c32 + c12 − 2P , 1 2c2 − 4 c1 + c3 c1 − c3 + P , 1 4 4 1 (P + 1), and 5c3 − 3c1 c1 − c3 − P . 1 8 8
Recall that the n i, j,k must be nonnegative integers. This restricts the ci . First looking at n 1,2,3 , we see that P must be a positive integer. Hence c2 must be given by the upper sign. This condition in fact guarantees integrality of all the n i, j,k . (The additional factors of 2 in the denominator cancel out if c2 is integral and can be ignored.) Integrality of P severely restricts , since it requires all odd prime factors of 2 + 32 to be congruent to 1 mod 8 (since 2 and −2 are both squares mod any such prime). In particular either = 0 or || ≥ 6. Since P = 0, we see that 2 ≥ 2 + 64, hence || > ||. Thus c3 must be positive and > 0. Since we saw above 1 c3 ≥ 0, we see that 1 = +1. Thus rewriting the orthogonality relation gives d1 /d3 = (d2 − 1)/(d2 + 1). The twist inequality coming from the (0, 3) entry reads d1 (1 + d2 ). D ≤ 1+ d3 Plugging in the preceding identity simplifies this to D ≤ 2d2 . Rearranging gives 3d22 ≥ d12 + d32 + 1 > d12 + d32 and plugging in the identity d1 = d3 (d2 − 1)/(d2 + 1) yields 2 3d22 d3 3 < < . 2 d2 + 1 2 2(d2 + 1) To see why this is helpful, expand the equations Trace(Ni ) = ci for i = 1, 3 and use the identity above to eliminate d1 . The result is d22 − 2d2 − 1 d 2 + 2d2 − 1 1 d3 + 2 · , and d2 (d2 + 1) d2 − 1 d3 d22 + 2d2 − 1 d22 − 2d2 − 1 1 d3 − · . c3 = d2 (d2 + 1) d2 − 1 d3
c1 =
362
E. Rowell, R. Stong, Z. Wang
Subtracting these gives = c1 − c3 = −4
d2 + 1 d3 +2 . d2 + 1 d2 d3
√ 3 > −4 3/2 > −4.9. However, we saw above that either = 0 Hence > −4 d2d+1 or || ≥ 6. It follows that ≥ 0. Since ≥ 6 and c2 ≥ 3/16, it follows that c2 > = c1 − c3 . Thus p3 (1) = c1 − c2 − c3 < 0. Thus d3 > 1. Hence we see < 2(d2 + 1)/(d2 d3 ) < 4. It follows that = 0, i.e., c1 = c3 . Since = 0, = 2c1 is a multiple of 8 and 2 c 2 P 1 −2 = −1. 32 4 In this case the characteristic polynomials become p1 (x) = x 4 − c1 x 3 + 2 2(c1 /4)2 − 1x 2 + c1 x − 1, p2 (x) = x 4 − 4 2(c1 /4)2 − 1x 3 − 6x 2 + 4 2(c1 /4)2 − 1x + 1, and p3 (x) = x 4 − c1 x 3 − 2 2(c1 /4)2 − 1x 2 + c1 x − 1. In particular p1 (x) > 0 for x ≥ c1 . Hence d1 < c1 . We have p2 (x) = (x 2 − t1 x − 1)(x2 − t2 x − 1), where t1 > 0 > t2 are the two roots of t 2 − 4 2(c1 /4)2 − 1t − 4 = 0. Since the larger root of x 2 − t x − 1 is an increasing function of t, d2 must correspond to t1 . Hence 1 4 2 d2 = t1 + > t1 = 4 2(c1 /4) − 1 + > 4 2(c1 /4)2 − 1. d2 t1 In particular, d2 > 4 since the square root above is integral. Finally the twist inequality coming from the (0, 2) entry reads d1 d3 . D ≤2 1+ d2 Squaring and using the identity d3 = d1 (d2 + 1)/(d2 − 1) to eliminate d3 gives the inequality 4(d2 + 1)2 d14 − 2d2 (d23 − 4d22 + d2 + 4)d12 − d22 (d2 − 1)2 (d22 − 3) ≥ 0. Dividing through by 4(d2 + 1)2 d12 and rearranging gives d12 ≥
d22 (d2 − 1)2 (d22 − 3) d2 3 2 (d − 4d + d + 4) + . 2 2 2 2(d2 + 1)2 4(d2 + 1)2 d12
The right-hand side of this inequality is an increasing function of d2 for d2 > 4 and a decreasing function of d1 , hence we may replace d1 by its upper bound c1 and d2 by the lower bound above. The result is 4560 − 2138c12 + 264c14 − 8c16 + (608 − 276c12 + 32c14 ) 2c12 − 16 ≥ 0.
On Classification of Modular Categories
363
Since the coefficient of the square root is nonnegative it follows that √ 4560 − 2138c12 + 264c14 − 8c16 + (608 − 276c12 + 32c14 )c1 2 ≥ 0. √ This polynomial in c1 is negative for c1 > 6 2, hence c1 ≤ 8. The only multiple of 4 in the range 3 ≤ c1 ≤ 8 for which the Pell equation above is satisfiable is c1 = 4. This gives c1 = c3 = 4, c2 = 2, P = 32. However plugging in shows that the twist inequality D ≤ 2(1 + d1 d3 /d2 ) does not actually hold in this case. Thus there are no solutions in this case. Case 2. G is the Klein 4-group. Let σ1 , σ2 , and σ3 be the elements of G which correspond to (0 1)(2 3), (0 2)(1 3), and (0 3)(1 2), respectively. Let Bσ1 correspond to signs i with 1 2 3 = 1 and Bσ2 correspond to signs δi with δ1 δ2 δ3 = 1. Then using the usual identities gives ⎛
1 d ⎜ S˜ = ⎝ 1 d2 d3
d1 1 2 d3 1 2 d2
d2 2 d3 s˜2,2 s˜2,3
⎞ ⎛ 1 d3 1 2 d2 ⎟ ⎜d1 = s˜2,3 ⎠ ⎝d2 1 s˜2,2 d3
d1 s˜1,1 δ1 d3 s˜1,3
d2 δ1 d3 δ2 δ1 δ2 d1
⎞ d3 s˜1,3 ⎟ . δ1 δ2 d1 ⎠ δ2 s˜1,1
Comparing these we see 2 = δ1 , s˜1,1 = 1 , and s˜2,2 = δ2 , hence ⎛
1 d1 ˜S = ⎜ ⎝d 2 d3
d1 1 2 d3 1 2 d2
d2 2 d3 δ2 2 δ2 d1
⎞ d3 1 2 d2 ⎟ . 2 δ2 d1 ⎠ 1 δ2
Orthogonality of the rows of S˜ gives the three conditions (1 + 1 )(d1 + 2 d2 d3 ) = (1 + δ2 )(d2 + 2 d1 d3 ) = (1 + 1 δ2 )(d3 + 1 2 d1 d2 ) = 0. Suppose 1 = +1, then we see d1 = − 2 d2 d3 , hence 2 = −1 and the remaining orthogonality relations become d2 (1 + δ2 )(1 − d3 ) = d3 (1 + δ2 )(1 − d2 ). We cannot have d2 = d3 = 1, since this would make d1 = 1, hence δ2 = −1. This gives ⎛
1 d d ⎜ 2 S˜ = ⎝ 3 d2 d3
d2 d3 1 −d3 −d2
d2 −d3 −1 d2 d3
⎞ d3 −d2 ⎟ . d2 d3 ⎠ −1
The eigenvalues of N2 are d2 and −1/d2 each with multiplicity 2. Hence d2 = φm for some integer m. The eigenvalues of N3 are d3 and −1/d3 each with multiplicity 2, hence d3 = φn for some integer n. So ⎛
1 φ φn ⎜ m S˜ = ⎝ φm φn
φm φn 1 −φn −φm
φm −φn −1 φm φn
⎞ φn −φm ⎟ . φm φn ⎠ −1
364
E. Rowell, R. Stong, Z. Wang
The resulting Ni matrices are necessarily rational, but in this case they are all integral, namely, ⎞ 0 1 0 0 ⎜1 mn m n ⎟ , =⎝ 0 m 0 1⎠ 0 n 1 0 ⎞ ⎛ 0 0 1⎟ ⎜0 , and N3 = ⎝ 0⎠ 0 0 1 ⎛
N1 = N2 N3 = N3 N2 ⎛ N2
=
0 ⎜0 ⎝1 0
0 m 0 1
1 0 m 0
0 n 1 0
0 1 0 0
⎞ 1 0⎟ . 0⎠ n
Note that nonnegativity of the entries forces m, n ≥ 0 and hence φm , φn ≥ 1. The strongest twist inequalities are the (0, 1) and (2, 3) cases which give D ≤ 4 or (φm2 + 1)(φn2 +1) ≤ 16. This gives, up to symmetry, the solutions (m, n) = (0, 0), (0, 1), (0, 2), or (1, 1). These are all excluded since the resulting Galois group G is at most Z/2Z. (These examples will return when we look at smaller Galois groups.) Case 3. G contains a 3-cycle. Since we can exclude Cases 1 and 2 above, the image of G in S4 cannot be transitive. It follows that G must fix the point j not on the 3-cycle. Thus λi, j is rational (hence integral) for every i. Up to symmetry there are two cases for the 3-cycle. We could have σ = (1 2 3) or σ = (0 1 2). If σ = (1 2 3), then the di are integral. The identities s˜ j,k = σ ( j) k s˜σ ( j),σ −1 (k) and 1 2 3 = 1 give i = 1 for all i (since di = s˜0,i = i s˜0,i+1 = i di+1 for 1 ≤ i ≤ 2) and ⎛
1 d ⎜ S˜ = ⎝ 1 d1 d1
d1 s˜1,1 s˜3,3 s˜2,2
d1 s˜3,3 s˜2,2 s˜1,1
⎞ d1 s˜2,2 ⎟ . s˜1,1 ⎠ s˜3,3
Orthogonality of the columns of S˜ gives s˜1,1 + s˜2,2 + s˜3,3 = −1 and s˜1,1 s˜2,2 + s˜2,2 s˜3,3 + s˜3,3 s˜1,1 = −d12 . The first of these gives −1/d1 = λ1,1 + λ1,2 + λ1,3 from which we see −1/d1 is an algebraic integer. Hence d1 = 1 and λ1,i < 1. The second equation gives λ1,1 λ1,2 + λ1,2 λ1,3 + λ1,3 λ1,1 = −1. Hence λ1,1 , λ1,2 , and λ1,3 are the three roots of g(x) = x 3 + x 2 − x + n for some integer n. This cubic must be irreducible and have three real roots all less than 1. Irreducibility excludes n = 0 and n = −1. For the roots of g to be less than 1, we must have g(1) > 0 or n + 1 > 0. Hence n ≥ 1. However, this results in complex roots. Thus this case gives no solutions. Thus we must have σ = (0 1 2) and λi,3 is integral for all i. The identities for s˜ j,k give ⎛
1 d ⎜ S˜ = ⎝ 1 d2 d3
d1 1 2 d2 1 2 d3
d2 1 2 d1 1 2 d3
⎞ d3 2 d3 ⎟ . 1 2 d3 ⎠ s˜3,3
Since σ (d3 ) = λ3,1 = 2 d3 /d1 and σ 2 (d3 ) = 1 2 d3 /d2 , we must have σ (d3 ) = d3 . (Otherwise 1 = 2 = d1 = d2 = 1 which fails.) Thus d3 is a root of an irreducible cubic g(x) = x 3 − c1 x 2 + c2 x − c3 and 2 d1 and 1 2 d2 are ratios of roots of g. If g had
On Classification of Modular Categories
365
Galois group S3 , then the ratios of the roots of g would be roots of an irreducible sextic. Thus g has Galois group Z/3Z and G = {1, σ, σ 2 }. Note that d3 (d1 d2 + 2 d2 + 1 2 d1 ), d1 d2 d2 c2 = d3 λ3,1 + λ3,1 λ3,2 + λ3,2 d3 = 3 ( 2 d2 + 1 + 1 2 d1 ), and d1 d2 3 d c3 = d3 λ3,1 λ3,2 = 1 3 . d1 d2 c1 = d3 + λ3,1 + λ3,2 =
Orthogonality of the columns of S˜ gives c1 2 d1 + 1 2 d2 + 1 d1 d2 = = −1, and c3 d32 c2 1 + 2 d1 + 1 2 d2 s˜3,3 = =− = −λ3,3 ∈ Z. c3 d3 d3 Thus g(x) = x 3 − cx 2 + ncx + c for integer n, c. Since g has Galois group Z/3Z, δ2 =
1 discr(g) = (n 2 + 4)c2 − 2n(2n 2 + 9)c − 27 c2
must be a square. The resulting n i, j,k are 2 2 δ (δ − nc − 1) − 2 , 2 n +3 1 2 (−δ + nc − 2n 2 + 3), = 2(n 2 + 3) 1 (−nδ + (n 2 + 2)c + 3n), = 2(n 2 + 3) 2 (δ + nc − 2n 2 + 3), = 2(n 2 + 3) 1 (3n − c), = 2 n +3 2 (δ − nc + 2n 2 − 3), = 2(n 2 + 3) 1 2 δ 1 2 (δ + nc + 1) + 2 , =− 2 n +3 1 (nδ + (n 2 + 2)c + 3n), = 2 2(n + 3) 1 2 (−δ − nc + 2n 2 − 3), = 2(n 2 + 3) c + n3 . = 2 n +3
n 1,1,1 = n 1,1,2 n 1,1,3 n 1,2,2 n 1,2,3 n 1,3,3 n 2,2,2 n 2,2,3 n 2,3,3 n 3,3,3
Integrality of n 1,2,3 requires c ≡ 3n (mod n 2 + 3). If we write c = 3n + a(n 2 + 3) for integer a, then we compute δ 2 = (n 2 + 3)2 (a 2 (n 2 + 4) + 2an − 3). Hence δ = (n 2 + 3)β, where β is integral and β 2 = a 2 (n 2 + 4) + 2an − 3. Note that in particular this forces
366
E. Rowell, R. Stong, Z. Wang
a = 0. Rewriting it as β 2 = (an + 1)2 + 4a 2 − 4, we see that β ≡ an + 1 (mod 2). Thus we compute 2 2 ((n + 1)β − 3n 2 − an(n 2 + 3) − 1), 2 1 2 (−β + an + 1), = 2 1 = (−nβ + 3n + a(n 2 + 2)), 2 2 = (β + an + 1), 2 = − 1 a, 2 = (β − an − 1), 2 1 2 2 ((n + 1)β + 3n 2 + an(n 2 + 3) + 1), =− 2 1 = (nβ + 3n + a(n 2 + 2)), 2 1 2 (−β − an − 1), = 2 = a + n,
n 1,1,1 = n 1,1,2 n 1,1,3 n 1,2,2 n 1,2,3 n 1,3,3 n 2,2,2 n 2,2,3 n 2,3,3 n 3,3,3
and these are all integral. Nonnegativity of these entries gives further restrictions on the parameters. Looking at n 1,2,2 + n 1,3,3 = 2 β, we see that 2 is the sign of β (or β = 0, but this gives a = ±1, n = −a, c = a and g(x) = (x − a)(x 2 − 1) which is reducible). Looking at n 1,1,2 + n 2,3,3 = − 1 2 β, we see that 1 = −1. Looking at n 1,2,3 we see that a > 0. Nonnegativity provides additional constraints on the parameters, but instead we look at the twist inequalities. We saw above that a > 0, hence c = 3n + a(n 2 + 3) ≥ n 2 + 3n + 3 > 0. Thus two of the roots d3 , λ3,1 and λ3,2 of g must be positive. By symmetry, we may assume d3 > λ3,1 > 0 > λ3,2 . Then 2 = 1 and we have ⎛
1 d /λ ⎜ 3 3,1 S˜ = ⎝ −d3 /λ3,2 d3
d3 /λ3,1 d3 /λ3,2 −1 d3
−d3 /λ3,2 −1 d3 /λ3,1 −d3
⎞ d3 d3 ⎟ . −d3 ⎠ nd3
Let M = max(1/λ3,1 , 1/|λ3,2 |) so that Md3 = max(d1 , d2 ). Since D 2 = (n 2 + 3)d32 and 1 1 1 + + = n 2 + 2, d32 λ23,1 λ23,2 the diagonal twist inequality coming from the (0, 0) entry gives √ n2 + 3 2 2 2M ≤ n + 3 + . d3 This inequality allows only finitely many choices of the parameters.
On Classification of Modular Categories
367
If n > 0, then g(−1/φn ) = −φn−3 < 0 and therefore λ3,2 > −1/φn . Thus M > φn and we get √ 2φn2
n2 + 3 . d3
Further since d3 λ3,1 = Mc and d3 > λ3,1 , we have d3 >
φn c ≥ φn (n 2 + 3n + 3).
For n ≥ 2, we get a contradiction by noting that φn > n, hence these equations force 1 2n 2 < n 2 + 3 + √ , n a contradiction. For n = 1, plugging in gives a contradiction. If n = 0, then β 2 = 4a 2 −3, hence a = 1 and g(x) = x 3 −3x 2 +3. Since s˜3,3 = 0, the (3, 3) entry of the twist equation gives θ32 d32 (1 + θ1 + θ2 ) = 0, hence θ1 = θ¯2 = e±2πi/3 . The (0, 3) and (1, 3) entries give θ3 (1 + d1 θ1 − d2 θ2 ) = D+ = θ1 θ3 (d1 − d2 θ1 + θ2 ). This case is realized by (A1 , 7) 1 . 2 If n < 0, then 1/M = λ3,1 . One easily checks that M and d3 are increasing functions of c, therefore it suffices to check that the inequality 2M 2 ≤ n 2 + 3 + (n 2 + 3)1/2 d3−1 fails for a = 1 and hence c = n 2 + 3n + 3. The inequality fails for n ≤ −2. (To see this simply compute both sides for n = −2. For n ≤ −3, note that g(−1/n) = −((n + 1)/n)3 < 0 and g((n + 1)2 ) = −(n + 1)4 + n + 2 < 0. Therefore M < −n and d3 > (n + 1)2 . Hence we have 2M 2 > 2n 2 > n 2 + 4 and (n 2 + 3)1/2 /d3 < 1, but these combine to contradict the inequality.) For n = −1, a = 1, the inequality holds, but the resulting polynomial g(x) = x 3 − x 2 − x + 1 is reducible. Moving up to the next case n = −1, a = 3, the inequality fails. Thus there are no solutions in this case. With the cases above completed, we consider G which is not transitive and contains no 3-cycle. Up to symmetry, it follows that G must be a subgroup of Z/2Z × Z/2Z = (0 1), (2 3). Case 4. G contains the transposition σ = (2 3). In this case the parity condition gives 1 2 3 = −1. Three instances of the usual identity give d2 = s˜0,2 = 2 s˜0,3 = 2 d3 , d3 = s˜0,3 = 3 s˜0,2 = 3 d2 , and d1 = s˜0,1 = 1 s˜0,1 = 1 d1 . Since the di are positive we conclude 1 = 2 = 3 = 1, a contradiction. Thus we are left with only three possibilities. Either G = Z/2Z = (0 1)(2 3), G = Z/2Z = (0 1), or G is trivial. Case 5. G contains σ = (0 1)(2 3). Using the identities s˜ j,k = σ ( j) k s˜σ ( j),σ −1 (k) and 1 2 3 = 1 gives ⎛
1 d ⎜ S˜ = ⎝ 1 d2 d3
d1 1 2 d3 1 2 d2
d2 2 d3 s˜2,2 s˜2,3
⎞ d3 1 2 d2 ⎟ . s˜2,3 ⎠ 1 s˜2,2
368
E. Rowell, R. Stong, Z. Wang
Suppose first that 1 = 1. Then, orthogonality of the first two columns of S˜ forces 2 = −1 and d1 = d2 d3 . Orthogonality of the last column with the first three gives the equations s˜2,2 d2 + s˜2,3 d3 = d2 (d32 − 1), s˜2,2 d3 + s˜2,3 d2 = d3 (d22 − 1), and s˜2,2 s˜2,3 = −d2 d3 . Looking at the cases d2 = d3 and d2 = d3 separately, these solve to give s˜2,2 = −1 and s˜2,3 = d1 = d2 d3 . Hence ⎛
1 d d ⎜ S˜ = ⎝ 2 3 d2 d3
d2 d3 1 −d3 −d2
d2 −d3 −1 d2 d3
⎞ d3 −d2 ⎟ . d2 d3 ⎠ −1
This is exactly the S˜ matrix of Case 2 above. Exactly as in that case, we get d2 = φm , d3 = φn , and d1 = d2 d3 . The Ni matrices and the twist inequalities are the same, hence we conclude (m, n) = (0, 1), (0, 2), or (1, 1). (Here we exclude m = n = 0 since it gives G trivial.) In the first case, (m, n) = (0, 1), the possible twist matrices are given by θ3 = e±4πi/5 , θ2 = ±i, and θ1 = θ2 θ3 . In the second case, (m, n) = (0, 2), no twist matrix exists. (To see this, note that the (1, 1) and (3, 3) entries in the twist equation give θ12 (φ22 + θ1 + θ2 φ22 + θ3 ) = D+ , −θ32 (φ22 + θ1 + θ2 φ22 + θ3 ) = D+ . Thus θ1 = ±iθ3 . The (0, 1) and (0, 3) entries give θ1 (1 + θ1 − θ2 − θ3 ) = D+ , θ3 (1 − θ1 + θ2 − θ3 ) = D+ . Subtracting these and using the result above gives θ2 = ±i. Plugging these equations into 1 + θ1 φ22 + θ2 + θ3 φ22 = D+ , gives D+ = (1 ± i)(1 + θ3 φ22 ). Equating squared norms √ gives θ3 + θ¯3 = 1 − φ22 . However 1 − φ22 = −2 − 2 2 < −2, so this is impossible.) In the third case, (m, n) = (1, 1), the possible twist matrices are given by θ2 = e±4πi/5 , θ3 = e±4πi/5 , and θ1 = θ2 θ3 . Next consider the case 1 = −1 so ⎛ 1 d1 d −1 ⎜ S˜ = ⎝ 1 d2 2 d3 d3 − 2 d2
d2 2 d3 s˜2,2 s˜2,3
⎞ d3 − 2 d2 ⎟ . s˜2,3 ⎠ −˜s2,2
By symmetry under interchanging N2 and N3 , we may assume 2 = +1. Since σ is the only nontrivial element of the Galois group, we conclude that d1 −
1 d3 d2 = n, d2 + = r, and d3 − =s d1 d1 d1
On Classification of Modular Categories
369
are integers. Hence d1 = φn . If n < 0, then d1 < 1 and since it is the largest root d3 < d2 . But then it follows that trace(N1 ) = d1 − 1/d1 + d3 /d2 − d2 /d3 < 0, an impossibility. Further if n = 0, then the same argument shows d1 = 1 and d3 = d2 . However, looking at the eigenvalues of N2 shows σ (d2 ) = d3 /d1 > 0 and looking at N3 shows σ (d3 ) = −d2 /d1 < 0. Thus n ≥ 1 and d1 is irrational. Since G = Z/2Z, it follows that K = Q[d1 ]. Hence we can write d2 = ad1 + b and d3 = ad ˜ 1 + b˜ for rational ˜ Hence a, a, ˜ b, b. r = d2 +
d3 ˜ 1 + b + a˜ − n b, ˜ and s = d3 − d2 = (a˜ − b)d1 + b˜ − a + nb. = (a + b)d d1 d1
Hence b˜ = −a, a˜ = b, r = na + 2b and s = nb − 2a. Note that in complex terms this gives d2 + id3 = (a + ib)(d1 − i) and r + is = (a + ib)(n − 2i). In particular D 2 = 1 + d12 + d22 + d32 = (1 + d12 )(n 2 + r 2 + s 2 + 4)/(n 2 + 4). 2 + s˜ 2 = 1 + d 2 . Since s˜ /d is an Since the columns of S˜ are of equal length s˜2,2 2,2 2 2,3 1 eigenvalue of N2 , (˜s2,2 /d2 )σ (˜s2,2 /d2 ) = s˜2,2 s˜2,3 /(d2 d3 ) is an integer. Further s˜2,2 = 0, since s˜2,2 = 0 would force σ (˜s2,2 ) = 0 and hence s˜2,3 = 0. Thus 1 + d12 |˜s2,2 s˜2,3 | ≥ ≥ 1. 2d2 d3 d2 d3 The twist inequality coming from the (0, 1) entry of S˜ gives d2 d3 d1 2d2 d3 1 + d12 ≤ 2+ · d1 1 + d12
(1 + d12 )1/2 (1 + a 2 + b2 )1/2 = D ≤ 2 + 2
≤ 2+
1 + d12 (1 + d1 )2 = . d1 d1
Rewriting this gives using r 2 + s 2 ≤ (n 2 + 4)
4d13 + 5d12 + 4d1 + 1 d12 (1 + d12 )
.
The twist inequality coming from the (0, 0) entry of S˜ is 2d12 ≤ D 2 + D, hence 2d12 ≤
(1 + d1 )4 (1 + d1 )2 + , or d1 d12
d14 ≤ 5d13 + 8d12 + 5d1 + 1. It follows that d1 < 7, hence 1 ≤ n ≤ 6. Together with the bound on r 2 + s 2 above, this leaves only finitely many possibilities (110 of them, insisting that d2 and d3 be positive, but finite).
370
E. Rowell, R. Stong, Z. Wang
Since G = Z/2Z, the quantities d2 d3 d1 d3 d2 − d2 d3 s˜2,2 s˜2,3 + d2 d3 s˜2,3 s˜2,2 − d2 d3 s˜2,2 s˜2,3 d2 d3
= = = = =
r 2 + nr s − s 2 , n2 + 4 (n 2 + 4)r s n+ 2 , r + nr s − s 2 (n 2 − 4)r 3 − 12nr 2 s − 3(n 2 − 4)r s 2 + 4ns 3 , (r 2 + s 2 )(r 2 + nr s − s 2 ) 4nr 3 + 3(n 2 − 4)r 2 s − 12nr s 2 − (n 2 − 4)s 3 , (r 2 + s 2 )(r 2 + nr s − s 2 ) (4 − 3n 2 )(r 4 − 6r 2 s 2 + s 4 ) − 2n(n 2 − 12)r s(r 2 − s 2 ) (r 2 + s 2 )2 (r 2 + nr s − s 2 )
are all integers. Only 6 of the 110 examples pass these integrality conditions. These are (n, r, s) = (1, 2, 1), (1, 3, −1), (2, 2, 2), (3, 2, 3), (4, 2, 4), and (4, 3, 1). The cases with n = s and r = 2 can be ignored since they give a = 0 and b = 1, hence d2 = 1, d3 = d1 , and s˜2,2 = −1. Thus s˜3,3 = 1. Invoking the symmetry under interchanging N1 and N3 puts us back in the case 1 = 1. These are just the (0, n) examples discussed above. The remaining two examples fail to give integral Ni matrices and also fail the twist inequalities. Thus there are no new examples in this case. Case 6. G contains the transposition σ = (0 1). Since we can exclude the cases above, σ must be the only non-trivial element of G. Up to symmetry there are two cases. The parity condition gives 1 2 3 = −1. Since d2 = s˜0,2 = 1 2 s˜1,2 and s˜1,2 = 2 s˜0,2 = 2 d2 , we conclude 1 = 1 and 3 = − 2 . Then s˜2,3 = 2 3 s˜3,2 and the fact that S˜ is symmetric forces s˜2,3 = s˜3,2 = 0. By symmetry we may assume 2 = 1 and 3 = −1. Thus we get ⎛ ⎞ 1 d1 d2 d3 1 d2 −d3 ⎟ ⎜d S˜ = ⎝ 1 . d2 d2 s˜2,2 0 ⎠ d3 −d3 0 s˜3,3 Note that λ2,2 = s˜2,2 /d2 and λ3,3 = s˜3,3 /d3 are integers. Note that orthogonality of the third column of S˜ with the first forces λ2,2 < 0. Since d1 ≥ σ (d1 ) = 1/d1 , we conclude d1 ≥ 1. Thus orthogonality of the first and fourth columns of S˜ gives λ3,3 ≥ 0. It is straightforward, though somewhat tedious, to build the Ni matrices in this case and worry about their integrality and nonnegativity; however, there is an easier approach. Since s˜2,3 = s˜3,2 = 0, the twist equation for the (2, 3) entry becomes θ2 θ3 (d2 d3 − θ1 d2 d3 ) = 0. Thus we conclude θ1 = 1. Using this fact the (2, 2) entry becomes 2 θ22 (2d22 + θ2 s˜2,2 ) = D+ s˜2,2 .
Since s˜2,2 /d2 = λ2,2 < 0 is integral and |D+ |2 = D 2 = (λ22,2 + 2)d22 , equating the squared norms of the sides of this equation gives θ2 + θ¯2 = 1 −
2 . λ22,2
On Classification of Modular Categories
371
The left-hand side is an algebraic integer, hence we conclude λ2,2 = −1 and θ2 = exp(±2πi/3). Similarly, the (3, 3) entry gives λ3,3 = 1 and θ3 = exp(±2πi/3). Equating the squared lengths of the last two columns of S˜ now gives d22 = d32 , hence d2 = d3 . This is a contradiction, since σ (d2 ) = λ2,1 = d2 /d1 but σ (d3 ) = λ3,1 = −d3 /d1 . Case 7. G is trivial. This case is also contained in [CZ]. In this case all the di and λi, j are integral. By symmetry, we may assume 1 ≤ d1 ≤ d2 ≤ d3 . Since every column of S˜ must have squared length D 2 , we see that D 2 must be a multiple of di2 for all i. If d1 = d2 = d3 , then d12 must divide D 2 = 3d12 + 1. Hence d1 = d2 = d3 = 1. Up to symmetry orthogonality of the columns of S˜ forces ⎛ ⎞ 1 1 1 1 ⎜1 −1 −1 1 ⎟ S˜ = ⎝ , 1 −1 1 −1⎠ 1 1 −1 −1 or ⎞ ⎛ 1 1 1 1 ⎜1 1 −1 −1⎟ . S˜ = ⎝ 1 −1 1 −1⎠ 1 −1 −1 1 For the first ⎛ 0 ⎜1 N1 = ⎝ 0 0
S˜ matrix, this gives integral Ni ⎞ ⎛ 1 0 0 0 0 0 0 0⎟ ⎜0 0 , N2 = ⎝ 0 0 1⎠ 1 0 0 1 0 0 1
matrices ⎞ ⎛ 1 0 0 0 1⎟ ⎜0 , and N3 = ⎝ 0 0⎠ 0 0 0 1
The possibilities for the corresponding twist matrix are ⎛ ⎞ ⎛ 1 0 0 0 1 0 ⎜0 i 0 0 ⎟ ⎜0 i T =⎝ , 0 0 1 0 ⎠ ⎝0 0 0 0 0 −i 0 0
0 0 −1 0
0 0 1 0
0 1 0 0
⎞ 1 0⎟ . 0⎠ 0
⎞ 0 0⎟ , 0⎠ i
or their complex conjugates. (Note that this is the case m = n = 0 of the form found in Cases 2 and 5.) For the second S˜ matrix, the compatible T matrices are listed in Table 2. If the di are not all equal, then d1 < d3 , hence 1 + d12 < d32 and d32 < D 2 < 3d32 . Thus we must have D 2 = 2d32 , i.e., two of the s˜i,3 are zero and the other two are ±d3 . Of course s˜0,3 = d3 . Suppose s˜ j,3 = ±d3 is the other nonzero entry in the last column. Orthogonality of the first and last columns gives d3 (1 ± d j ) = 0, hence the lower sign is correct and d j = 1. Thus using symmetry we may assume j = 1. Orthogonality of the remaining columns with the last column gives s˜1,1 = d1 = 1 and s˜2,1 = d2 . Orthogonality of the third column and the first two gives s˜2,2 = −2 and ⎛ ⎞ 1 1 d2 d3 1 d2 −d3 ⎟ ⎜1 S˜ = ⎝ . d2 d2 −2 0 ⎠ d3 −d3 0 0 However equality of the squared lengths of the last two columns now gives D 2 = 2d32 = 2d22 + 4 or d32 = d22 + 2, an impossibility.
372
E. Rowell, R. Stong, Z. Wang
5. Realization of Fusion Rules and Classification of MTCs ˜ in Theorems 3.1, 3.2, 4.1, we will In this section, for each modular fusion rule (N ; S) first determine all modular symbols with this fusion rule which also satisfy (1), (2) of Proposition 2.3, then classify all MTCs realizing each such modular symbol. ˜ there are two choices of compatible S matriFor each modular fusion rule (N ; S), 1 ˜ 1 ˜ ces: S = D S or − D S. When the two modular symbols are realized by (2+1)-TQFTs, respectively, one TQFT is obtained from the other by tensoring the trivial theory with S = (−1). The quantum invariant of the 3-sphere will be D1 or − D1 , respectively. Also the topological central charge c of the two theories will differ by 4. Another symmetry for modular symbols is complex conjugation: to change (N ; S, T ) to (N ; S † , T † ). Complex conjugation of a modular symbol gives rise to a different modular symbol if one of the S, T is not a real matrix. Given an S˜ matrix, we can obtain all fusion matrices by using the Verlinde formulas. Instead of listing fusion matrices, we will present them as fusion rules. In the following, we will not list trivial fusion rules such as 1 ⊗ x = x and those that can be obtained from obvious identities such as x ⊗ y = y ⊗ x. We will also write x ⊗ y as x y sometimes. Then we use relations (3)(i)-(iv) of Definition 2.1 together with (1) (2) of Proposition 2.3 to determine the possible T -matrices. As Example 2.2 illustrates, Proposition 2.3 is necessary to get finitely many solutions in some cases. We find that there are finitely many modular symbols (N ; S, T ) of rank≤ 4 satisfying Proposition 2.3. Modulo the symme2πi try S → −S, these modular symbols are classified in Table 2. In the table, ζm = e m . The labels will be {1, X } for rank=2, {1, X, Y } for rank=3, and {1, X, Y, Z } for rank=4. They will correspond to rows 1, 2, 3, 4 of the S˜ matrices. The # is the number of modular symbols satisfying Proposition 2.3 modulo the symmetry S → −S. The column P stands for primality, and the column G is the Galois group of the modular fusion rule. With modular symbols determined, we turn to realizing each of them with MTCs. ˜ First let us consider the S-matrices corresponding to Theorem 4.1(3)-(5). In each of these cases there is a rank=2 tensor subcategory corresponding to the objects labelling columns 1 and 3 of the S˜ matrix. Further inspection shows that the submatrix of S˜ corresponding to rows and columns 1 and 3 is invertible. It is obvious that the tensor subcategory generated by the trivial object and the object labelling column 3 is a ˜ modular subcategory. For the S-matrices of Theorem 4.1(3),(4) these rank=2 modular subcategories are equivalent to the UMTCs corresponding to Theorem 3.1(1), while the ˜ modular subcategory corresponding to the S-matrix of Theorem 4.1(5) is equivalent to (one of) the UMTCs coming from Theorem 3.1(2). By [M2, Theorem 4.2] this implies that the MTCs corresponding to these S˜ matrices are direct products of rank=2 MTCs. For this reason we will not write down realizations or complete data for these MTCs as they can be deduced from their product structure. MTCs realizing the remaining 8 nontrivial modular symbols are prime, i.e. they do not have non-trivial modular subcategories. To complete the classification, we need to solve the pentagon and hexagon equations for all 8 modular symbols. The solutions of the pentagon equations are organized into the F-matrices whose entries are called 6 j symbols. The solutions of hexagons are given by the braiding eigenvalues. 2 5.1. F-matrices. Given an MTC C, a 4-punctured sphere Sa,b,c,d , where the 4 punctures are labelled by a, b, c, d, can be divided into two pairs of pants (=3-punctured spheres) in two different ways. In the following figure, the 4-punctured sphere is the boundary of a thickened neighborhood of the graph in either side, and the two graphs encode the
On Classification of Modular Categories
373
Table 2. Rank≤ 4 unitary modular symbols S˜ matrix S˜ = (1)
Fusion rules
Thm 3.1(1)
X2 = 1
Thm 3.1(2)
X2 = 1 + X
Thm 3.2(1)
X 2 = X ∗ , X X ∗ = 1, (X ∗ )2 = X
Thm 3.2(2)
X2 = 1 + Y , XY = X, Y 2 = 1
Thm 3.2(3)
X2 = 1 + X + Y , XY = X + Y , Y2 = 1 + X
T matrix
#s
P
G
(1)
1
Yes
1
Diag(1, ±i)
2
Yes
1
2
Yes
Diag(1, ζ3±1 , ζ3±1 )
Z2
2
Yes
Z2
Diag(1, (ζ16 )2k+1 , −1)
8
Yes
Z2
Diag(1, (ζ7 )±5 , (ζ7 )±1 )
2
Yes
Z3
Diag(1, (ζ5 )±2 )
Thm 4.1(1)
X 2 = Y = (X ∗ )2 , X X ∗ = 1 = Y 2, X Y = X ∗, X ∗Y = X
Diag(1, −1, (ζ8 )±m , (ζ8 )±m ), m=1,3
4
Yes
Z2
Thm 4.1(2)
X 2 = 1, X Y = Z , X Z = Y , Y 2 = 1, Y Z = X, Z2 = 1
Diag(1, −1, 1 , 1 ), 12 = 1
2
Yes
1
Thm 4.1(3)
X 2 = 1, X Y = Z , X Z = Y , Y 2 = 1, Y Z = X, Z2 = 1
Diag(1, θ1 , θ2 , θ1 θ2 ), θi2 = −1
3
No
1
Thm 4.1(4)
X2 = 1 + X, XY = Z, X Z = Y + Z , Y 2 = 1, Y Z = X, Z2 = 1 + X
Diag(1, θ1 , θ2 , θ1 θ2 ), θ1 = (ζ5 )±2 , θ2 = ±i
4
No
Z2
Thm 4.1(5)
X2 = 1 + X, XY = Z, X Z = Y + Z, Y2 = 1 + Y, Y Z = X + Z, Z2 = 1 + X + Y + Z
Diag(1, θ1 , θ2 , θ1 θ2 ), θ1 = (ζ5 )±2 , θ2 = (ζ5 )±2
3
No
Z2
Thm 4.1(6)
X2 = 1 + X + Y , XY = X + Y + Z, X Z = Y + Z, Y 2 = 1 + X + Y + Z, Y Z = X + Y, Z2 = 1 + X
Diag(1, ζ9±2 , (ζ9 )±6 , (ζ9 )∓6 )
2
Yes
Z3
two different pants-decompositions of the 4-punctured sphere. The F-move is just the change of the two pants-decompositions. When bases of all pair of pants spaces Hom(a ⊗ b, c) are chosen, then the two pants 2 decompositions of Sa,b,c,d determine bases of the vector spaces Hom((a ⊗ b) ⊗ c, d), and Hom(a ⊗ (b ⊗ c), d), respectively. Therefore the F-move induces a matrix Fda,b,c : Hom((a ⊗ b) ⊗ c, d) → Hom(a ⊗ (b ⊗ c), d), which are called the F-matrices. Consistency of the F matrices are given by the pentagon equations. For each quadruple (a, b, c, d), we have an F-matrix whose entries are indexed by a pair of triples ((m, s, t), (n, u, v)), where m, n are the labels for the internal edges, and s, t, u, v are indices for a basis of the Hom(x ⊗ y, z) spaces with dim > 1. For the MTCs in our paper, none of the Hom(a ⊗ b, c) has dim > 1, so we will drop the s, t, u, v from our notation. If one of the a, b, c in Fda,b,c is the trivial label, then we may assume Fda,b,c is the identity matrix. But we cannot always do so if d is the trivial label.
374
E. Rowell, R. Stong, Z. Wang
In the following, any unlisted F matrix is the identity. a2 b c a b c 22 22 abc Fd 22 22 n m 22 22 2 d d 5.2. Braidings and twists. The twist of the simple type X i will be denoted by θi , and it is defined by the following positive twist:
θi
The braiding eigenvalues are defined by the following diagram: a
b
Rcab c
a' b '' '' '' '
c
The consistency equations of the braidings are given by two independent families of hexagon equations. If c is the trivial label, and label a is self-dual, then R1aa = νa θa−1 , where νa is the Frobenius-Schur indicator of a. 5.3. Explicit data. In this section, we give the explicit data for at least one realization of each prime modular fusion rule. Since each modular symbol can have up to 4 MTC realizations, we will present the complete data for only one of them. We choose one with the following properties: (1) The (0, 0) entry of the S matrix is D1 , where D is the total quantum order. (2) In a category with a generating non-abelian simple object X , we choose a theory with the positive exponent of the twist θ X being the smallest. This is inspired by anyon theory that the simple object with the smallest exponent is the most relevant in physical experiments. If a modular symbol (N ; S, T ) is realized by an MTC, then the modular symbol (N ; −S, T ) is also realized by an MTC. The modular symbol (N ; S˜ † , T † ) is realized by complex conjugating all F matrices and braidings of (N ; S, T ). So in the following each group of data will be for 4 MTCs if any of S, T, F and braidings are not real; otherwise there will be two. We choose the F matrices to be unitary, and real if possible. In anyon theory, labels will be called anyon types. The smallest positive exponent of a twist θi will be called the topological spin of the anyon type i. Topological spins are the
On Classification of Modular Categories
375
conformal dimensions modulo integers of the corresponding primary field if the MTC has a corresponding RCFT. The last line of the data lists all quantum group realizations of the same theory. We did not list the Frobenius-Schur indicators of anyons because they can be calculated by the formula in Proposition 2.3. In the following data, only the semion s and the (A1 , 2) non-abelian anyon σ have Frobenius-Schur indicator=−1. 5.3.1. Semion MTC We will use s to denote the non-trivial label. Anyon types: {1, s} Fusion rules: s 2 = 1 Quantum dimensions: {1, 1} Twists: θ1 = 1, θs = i √ Total quantum order: D = 2 Topological central charge: c = 1 Braidings: R1ss = i 1 1 1 √ S-matrix: S = 2 1 −1 F-matrices: Fss,s,s = (−1) Realizations: (A1 , 1), (E 7 , 1) 5.3.2. Fibonacci MTC We will use ϕ to denote the golden ratio ϕ = non-trivial label. Anyon types: {1, τ } Fusion rules: τ 2 = 1 + τ Quantum dimensions: {1, ϕ} 4πi Twists: θ1 = 1, θτ = e 5 √ π Total quantum order: D = 2 cos( 10 ) = 2 sin(5π ) Topological central charge: c = 4πi
14 5
√ 1+ 5 2
and τ the
5
3πi
Braidings: R1τ τ = e−5 , Rτττ = e 5 1 ϕ 1 S-matrix: S = √2+ϕ ϕ −1 −1 −1/2 ϕ ϕ F-matrices: Fττ,τ,τ = ϕ −1/2 −ϕ −1 Realizations: (A1 , 3) 1 , (G 2 , 1), complex conjugate of (F4 , 1) 2
5.3.3. Z3 MTC We will use ω for both a non-trivial label and the root of unity ω = e2πi/3 . No confusions should arise. Anyon types: {1, ω, ω∗ } Fusion rules: ω2 = ω∗ , ωω∗ = 1, (ω∗ )2 = ω Quantum dimensions: {1, 1, 1} 2πi Twists: θ1 = 1, θω = θω∗ =√e 3 Total quantum order: D = 3 Topological central charge: c = 2 ∗ 2πi ω∗ ,ω ω∗ ,ω∗ = e− 4πi 3 =⎞e− 3 , Rωω,ω Braidings: R1ω,ω = ∗ = Rω ⎛ R1 1 1 1 S-matrix: S = √1 ⎝1 ω ω2 ⎠ 3 1 ω2 ω
376
E. Rowell, R. Stong, Z. Wang
F-matrices: Fda,b,c = (1) for any a, b, c, d Realizations: (A2 , 1), (E 6 , 1) 5.3.4. Ising MTC We will use 1, σ, ψ to denote the non-trivial labels. Anyon types: {1, σ, ψ} σ ψ = ψσ = σ, ψ 2 = 1 Fusion rules: σ 2 = 1 + ψ,√ Quantum dimensions: {1, 2, 1} πi Twists: θ1 = 1, θσ = e 8 , θψ = −1 Total quantum order: D = 2 Topological central charge: c = 21 πi
ψψ
ψσ
σψ
Braidings: R1σ σ = e− 8 , R1 = −1, Rσ = Rσ = −i, Rψσ σ = e √ ⎛ ⎞ 1 2 √ 1 √ S-matrix: S = 21 ⎝ 2 √ 0 − 2⎠ 1 − 2 1 1 1 ψ,σ,ψ σ,ψ,σ σ,σ,σ F-matrices: Fσ ,Fσ = √1 = (−1),Fψ = (−1) 2 1 −1 Realizations: complex conjugate of (E 8 , 2)
3πi 8
5.3.5. (A1 , 2) MTC We will use 1, σ, ψ to denote the non-trivial labels again. Anyon types: {1, σ, ψ} Fusion rules: σ 2 = 1 + ψ,√ σ ψ = ψσ = σ, ψ 2 = 1 Quantum dimensions: {1, 2, 1} 3πi Twists: θ1 = 1, θσ = e 8 , θψ = −1 Total quantum order: D = 2 Topological central charge: c = 23 πi
ψψ
ψσ
σψ
πi
Braidings: R1σ σ = −e− 8 , R1 = −1, Rσ = Rσ = i, Rψσ σ = e 8 √ ⎛ ⎞ 1 2 √ 1 √ S-matrix: S = 21 ⎝ 2 √ 0 − 2⎠ 1 − 2 1 1 1 ψ,σ,ψ σ,ψ,σ F-matrices: Fσσ,σ,σ = − √1 ,Fσ = (−1),Fψ = (−1) 2 1 −1 Realizations: (A1 , 2)
5.3.6. (A1 , 5) 1 MTC We will use 1, α, β to denote the non-trivial labels. Note that 2 1, α, β are special labels for 1, Y, X in Theorem 3.2(3) of Table 2. Anyon types: {1, α, β} Fusion rules: α 2 = 1 + β, αβ = α + β, β 2 = 1 + α + β Quantum dimensions: {1, d, d 2 − 1}, where d = 2 cos( π7 ) Twists: θ1 = 1, θα = e
2πi 7
, θβ √= e
10πi 7
Total quantum order: D =
7 2 sin( π7 ) Topological central charge: c = 48 7 2πi 10πi ββ Braidings: R1αα = e− 7 , R1 = e− 7 2πi 9πi ββ αβ βα Rα = e − 7 , Rα = Rα = e 7 5πi 6πi ββ αβ βα Rβ = e− 7 , Rβ = Rβ = e 7 , Rβαα
= e−
4πi 7
On Classification of Modular Categories
377
⎛
⎞ 1 d d2 − 1 S-matrix: S = D1 ⎝ d −d 2 + 1 1 ⎠ d2 − 1 1 −d F-matrices: see the end of this subsection Realizations: (A1 , 5) 1 2
5.3.7. Z4 MTC We will use 1, , σ, σ ∗ to denote the non-trivial labels. Anyon types: {1, , σ, σ ∗ } Fusion rule: 2 = σ σ ∗ = 1, σ 2 = (σ ∗ )2 = , σ = σ ∗ , σ ∗ = σ Quantum dimensions: {1, 1, 1, 1} πi Twists: θ1 = 1, θ = −1, θσ = θσ ∗ = e 4 Total quantum order: D = 2 Topological central charge: c = 1 ∗ ∗ πi πi ∗ Braidings: R1 , = −1, R σ,σ = R σ ,σ ∗ = e 4 , R1σ,σ = R1σ ,σ = e− 4 , ,σ σ ∗ , = R ,σ ∗ = −i Rσσ, ∗ = Rσ ∗ = Rσ σ ⎛ ⎞ 1 1 1 1 ⎜1 1 −1 −1⎟ S-matrix: S = 21 ⎝ 1 −1 −i i ⎠ 1 −1 i −i ∗ , ∗ ∗ ∗ ∗ ∗ = Fσσ ,σ ,σ = Fσ ,σ, = Fσ ,σ = F σ, ,σ = F σ , ,σ = (−1) F-matrices: Fσσ,σ,σ ∗ ∗ Realizations: (A3 , 1), (D9 , 1) 5.3.8. Toric code MTC The fusion rules are the same as Z2 × Z2 , but the theory is not a direct product. We will use 1, e, m, to denote the non-trivial labels. Anyon types: {1, e, m, } Fusion rules: e2 = m 2 = 2 = 1, em = , e = m, m = e Quantum dimensions: {1, 1, 1, 1} Twists: θ1 = θe = θm = 1, θ = −1 Total quantum order: D = 2 Topological central charge: c = 0 Braidings: R1 , = −1, R e,m = 1, R m,e = −1, R1e,e = R1m,m = 1, ,m e, ,e Re = 1, Rem, = −1, −1 ⎛ Rm = 1, Rm = ⎞ 1 1 1 1 ⎜1 1 −1 −1⎟ S-matrix: S = 21 ⎝ 1 −1 1 −1⎠ 1 −1 −1 1 F-matrices: Fda,b,c = (1) for all a, b, c, d Realizations: (D8 , 1), D(Z2 )—quantum double of Z2 5.3.9. (D4 , 1) MTC The fusion rules are the same as Z2 × Z2 , but the theory is not a direct product. We will use 1, e, m, to denote the non-trivial labels again. Anyon types: {1, e, m, } Fusion rules: e2 = m 2 = 2 = 1, em = , e = m, m = e Quantum dimensions: {1, 1, 1, 1} Twists: θ1 = 1, θe = θm = θ = −1 Total quantum order: D = 2 Topological central charge: c = 4
378
E. Rowell, R. Stong, Z. Wang
Braidings: R1 , = −1, R e,m = −1, R m,e = 1, R1e,e = R1m,m = −1, Re ,m = e, ,e 1, Rem, = −1, Rm = 1, Rm = −1 ⎛ ⎞ 1 1 1 1 ⎜1 1 −1 −1⎟ S-matrix: S = 21 ⎝ 1 −1 1 −1⎠ 1 −1 −1 1 a,b,c = (1) for all a, b, c, d F-matrices: Fd Realizations: (D4 , 1) 5.3.10. (A1 , 7) 1 MTC We will use 1, α, ω, ρ to denote the non-trivial labels. Note that 2 1, α, ω, ρ are special labels for 1, Z , Y, X in Theorem 4.1(6) of Table 2. Anyon types: {1, α, ω, ρ} Fusion rules: α 2 = 1 + ω, αω = α + ρ, αρ = ω + ρ, ω2 = 1 + ω + ρ ωρ = α + ω + ρ, ρ 2 = 1 + α + ω + ρ Quantum dimensions: {1, d, d 2 − 1, d + 1}, where d = 2 cos( π9 ) and d 3 = 3d + 1 2πi
Twists: θ1 = 1, θα = e 3 , θω = e 3 Total quantum order: D = 2 sin( π ) Topological central charge: c = Braidings: R1αα = e 7πi
− 2πi 3
4πi 9
4πi 3
9
10 3
, R1ωω = e−
ωρ
, θρ = e
ρω
4πi 9
ρρ
, R1 = e −
4πi
4πi 3
ρρ
Rααω = Rαωα = e 9 , Rα = Rα = e 9 , Rα = −1 2πi 2πi 5πi πi 7πi αρ ρα ωρ ρω ρρ Rω = Rω = e 9 , Rω = Rω = e− 3 , Rωαα = e 9 , Rω = e− 9 , Rωωω = e 9 8πi πi 7πi 2πi αρ ρα ρω ωρ Rραω = Rρωα = e− 9 , Rρ = Rρ = e− 3 , Rρ = Rρ = e 9 , Rρωω = e 9 , ρρ
Rρ = e −
2πi 3
⎛
⎞ 1 d d2 − 1 d + 1 ⎜ d −d − 1 d 2 − 1 −1 ⎟ ⎟ S-matrix: S = D1 ⎜ ⎝d 2 − 1 d 2 − 1 0 −d 2 + 1⎠ d +1 −1 −d 2 + 1 d F-matrices: see below Realizations: (A1 , 7) 1 , complex conjugate of (G 2 , 2) 2 The list of all F matrices for an MTC can occupy many pages. But they are needed for the computation of quantum invariants using graph recouplings, the Hamiltonian formulation of MTCs as in [LWe] or the study of anyon chains [FTL]. For the MTCs (A1 , k) 1 with odd k, all the data of the theory can be obtained from [KL]. For k = 5, 2
2πi
choose A = ie− 28 , the label set is L = {0, 2, 4} in [KL] and 0 = 1, 4 = α, 2 = β. 2πi For k = 7, set A = ie 36 , the label set is L = {0, 2, 4, 6} in [KL] and 0 = 1, 6 = α, 2 = ω, 4 = ρ. The twist is given by θa = (−1)a Aa(a+2) , and the braiding Rcab = a+b−c
a(a+2)+b(b+2)−c(c+2)
2 . The formulas for 6 j symbols can be found in Chap. 10 of (−1) 2 A− [KL]. The F matrices from [KL] are not unitary, but the complete data can be presented over an abelian Galois extension of Q. To have unitary F matrices, we need to normalize the θ symbols as θ (i, j, k) = di d j dk . The (A1 , k) 1 , k odd, MTCs have peculiar properties regarding the relation between 2 the bulk (2 + 1)-TQFTs and the boundary RCFTs. To realize (A1 , k) 1 , k odd, using the 2πi ± 4(k+2)
2
Kauffman bracket formalism, we set A = ie . In order to follow the convention 2πi 2πi above, we choose A = ie− 4(k+2) if k = 1 mod 4, and A = ie 4(k+2) if k = −1 mod 4.
On Classification of Modular Categories
379
Note that in both cases A is a 2(k + 2)th root of unity. We have (A1 , k) = (A1 , k) 1 × the semion,
(5.1)
(A1 , k) = (A1 , k) 1 × the semion,
(5.2)
2
if k = 1 mod 4, and 2
3k if k = −1 mod 4. The central charge of (A1 , k) is k+2 , which implies that the central 3k 3k if k = −1 mod 4. charge of (A1 , k) 1 is ck = 1 − k+2 if k = 1 mod 4, and ck = 1 + k+2 2 In Table 3, we list all unitary quantum groups categories of rank≤ 12 from the standard construction. For notation, see [HRW].
Remark 5.1. The following serves as a guide to Table 3. (1) In general we will list these categories as (X r , k) for the category obtained from a quantum group of type X r at level k. Observe that the corresponding root of unity is of order = mk + h, where m = 1 for X = A, D, or E; m = 2 for X = B, C or F and m = 3 for X = G, and h is the dual Coxeter number. (2) The category (Ar , k) has a modular subcategory (Ar , k) 1 generated by the objects r +1 with integer weights provided gcd(r +1, k) = 1. These are found on line 5 of Table 3 where L = {(1, 2s + 1), (2s, 2), (2, 4), (2, 5), (2, 7), (3, 3), (4, 3), (6, 3) : 1 ≤ s ≤ 11}. (3) We include the examples of pseudo-unitary categories coming from low-rank coincidences for quantum groups of types F4 and G 2 at roots of unity of order coprime to 2 and 3 respectively. (4) This list includes different realizations of equivalent categories. We eliminate those coincidences that occur because of Lie algebra isomorphisms such as sp4 ∼ = so5 etc., and do not include the trivial rank= 1 category. (5) NSD means the category contains non-self-dual objects. (6) “c.f. (X r , k)” means the categories in question have the same fusion rules as those of (X r , k). (7) We include the three categories coming from doubles of finite groups with rank≤12, although they are not strictly speaking of quantum group type.
5.4. Classification. In this section, we explain Table 1. We identify MTCs whose label sets differ by permutations. For the trivial MTCs, the two MTCs are distinguished by the S matrices: S = (±1). For the Z2 fusion rule, unitary MTCs are the semion MTC and those from the two symmetries S → −S and complex conjugate. For the Fibonacci fusion rule, unitary MTCs are the Fibonacci MTC and those from the two symmetries S → −S and complex conjugate. For the Z3 fusion rule, all unitary MTCs are the one listed in the last subsection and those from the two symmetries S → −S and complex conjugate. For the Ising fusion rule, there are a total of 16 theories divided into two groups according to the Frobenius-Schur indicator of the non-abelian anyon X, X 2 = 1 + Y . There are 8 unitary MTCs with Frobenius-Schur indicator=1. Their twists are given
380
E. Rowell, R. Stong, Z. Wang Table 3. Unitary Quantum Group Categories of rank ≤ 12
(X r , k)
Rank
Notes
(Ar , 1), r ≤ 11
r +1
r ≥ 2 NSD, abelian
r +2
NSD
5
(A1 , k), k ≤ 11
k+1
(A2 , 2)
6
(A2 , 3), (A3 , 2) (Ar , k) (Br , 1)
1 r +1
, (r, k) ∈ L
10
k+2
1 k+r r +1 k
NSD
6
r ≥ 2 NSD
k +r +1 4r
3
c.f. (A1 , 2)
(Br , 2), r ≤ 8
r +4
finite braid image?
(B2 , 3)
10
(Cr , 1) r ≤ 11
r +1
(C3 , 2)
10
(D2r , 1)
4
r even c.f. D ω (Z2 )
4r − 1
(D2r +1 , 1)
4
c.f. (A3 , 1)
4r + 1
(Dr , 2), r = 4, 5
11, 12
r = 5 NSD
8, 10
(E 6 , k), k = 1, 2
3, 9
NSD
13, 14
(E 7 , k), 1 ≤ k ≤ 3
2, 6, 11
19, 20, 21
(E 8 , k), 2 ≤ k ≤ 4
3, 5, 10
32, 33, 34
(F4 , k), 1 ≤ k ≤ 3
2, 5, 9
20, 22, 24
(G 2 , k), 1 ≤ k ≤ 5
2, 4, 6, 9, 12
F4
10
G2
5, 8, 10
D ω (Z2 ) D ω (Z3 ) D ω (S3 )
mπi
4r + 2 12
c.f. (A1 , r )
2(r + 2) 12
15, 18, 21, 24, 27 c.f. (E 8 , 4)
17 11, 13, 14
4
prime
9
prime
8
c.f. (B4 , 2)
by θ X = e 8 for m = 1, 7, 9, 15. The Ising MTC is the simplest one with m = 1 and central charge c = 21 . The theory m = 1, m = 15 are complex conjugate of each other, so are the m = 7, 9. The other 4 MTCs are obtained by choosing −S. There are mπi 8 unitary MTCs with Frobenius-Schur indicator=−1. Their twists are θ X = e 8 for m = 3, 5, 11, 13. The SU (2) at level k = 2 is the simplest one with m = 3 and central charge c = 23 . The MTCs m = 3 and m = 13 are complex conjugate, so are m = 5, 11. The other 4 are those with −S. The Ising MTC is not an SU (2) theory. It can, however, be obtained as a quantum group category as the complex conjugate of E 8 at level=2. Note that the F matrices in each group of 8 are the same, but their braidings are different. The SU (2) level=2 theory has FXX X X = −Fσσ σ σ with the other F matrices the same as the Ising theory. For the (A1 , 5) 1 fusion rule, all unitary MTCs are the one listed in the last subsection 2 and those from the two symmetries S → −S and complex conjugate. For the Z2 × Z2 fusion rules, there are two groups of theories depending on whether or not the theory is a product. There are 4 theories which are not direct products, and 6 product theories. The toric code MTC has another version, which could also be called the toric code: it has θe = θm = −1. All F matrices are 1. The braidings R1ee = R1mm = R em = −1, R me = 1, and others are the same as the toric code. Another two are the −S versions. The product theories are the products of the semion MTC and its complex
On Classification of Modular Categories
381
conjugate. There are 4 possible theories, but two of them are the same: semion × complex conjugate is the same as complex conjugate × semion. Hence there are 3 theories here. With the −S versions, we have 6 product theories. For the Z4 fusion rule, the Galois group action of the MTC listed above is Z4 . Its mπi actions give rise to 4 theories with θ X = θ X ∗ = e 4 for m = 1, 3, 5, 7. They all have the same F matrices. The −S versions give a total of 8. For the (A1 , 3) fusion rule, this is the product of the semion fusion rule with the Fibonacci fusion rule. There are 4 product theories from semion, Fibonacci and their complex conjugates. These 4 theories are different, and the other 4 come from their −S versions. Let us choose the product of the semion with the Fibonacci as a representative theory, then we have 4 anyons, 1, ϕ, τ, s, where τ is the Fibonacci anyon, and ϕ is the same as τ tensoring the semion s. For the (A1 , 7) 1 fusion rule, all unitary MTCs are the ones listed in last subsection 2 and those from the two symmetries S → −S and complex conjugate. The analysis of the Fibonacci × Fibonacci fusion rule is the same as that of the semion × semion fusion rule. 6. Conjectures and Further Results In this section we briefly discuss several conjectures concerning the structure and application of MTCs. 6.1. Fusion rules and the finiteness conjecture. Since topological phases of matter are discrete in the space of theories, therefore, MTCs, encoding the universal properties of topological phases of matter, should also be discrete. It is conjectured [Wa]: Conjecture 6.1. If the rank of MTCs is fixed, then there are only finitely many equivalence classes of MTCs. By Ocneanu rigidity, this is equivalent to there are only finitely many modular fusion rules realizing by MTCs of a fixed rank. Proposition 6.2. There are only finitely many equivalence classes of unitary MTCs with total quantum order D ≤ c, where c is any given universal constant. √ Proof. For a unitary rank=n MTC, all quantum dimensions dr ≥ 1, r ∈ L. So D ≥ n. ∗ sir s jr skr If D ≤ c, then n ≤ c2 . By Verlinde formula 2.3, we have n i,k j = | rn−1 =0 s0r | ≤ 1 3 D rn−1 =0 dr ≤ n D ≤ c for any i, j, k. Therefore, there are only finitely many possible fusion rules. By Ocneanu rigidity, there are only finitely many possible MTCs. 6.2. Topological qubit liquids and the fault-tolerance conjecture. Topological phases of matter are quantum liquids such as the electron liquids exhibiting the FQHE, whose topological properties emerged from microscopic degrees of freedom. This inspires the following discussion. Let be a triangulation of a closed surface , be its dual triangulation: vertices are centers of the triangles in , and two vertices are connected by an edge if and only
382
E. Rowell, R. Stong, Z. Wang
if the corresponding triangles of share an edge. The dual triangulation of is a celluation of whose 1-skeleton is a tri-valent graph. It is well-known that any two triangulations of the same surface can be transformed from one to the other by a finite sequence of two moves and their inverses: the subdivision of a triangle into 3 new triangles; and the diagonal flip of two adjacent triangles that share an edge (=the diagonal). Dualizing the triangulations into celluations, the two moves become the inflation of a vertex to a triangle and the F move. Definition 6.3. (1) Given an integer k > 0, a k-local, or just local, qubit model on (, ) is a pair (H , H ), where H is the Hilbert space ⊗e∈ C2 , and H is a k local Hamiltonian in the following sense: H is a sum of Hermitian operators of the form id ⊗ · · · ⊗ id ⊗ Ok ⊗ id ⊗ · · · ⊗ id, where Ok acts on ≤ k qubits. (2) A modular functor V is realized by a topological qubit liquid if there is a sequence ∞ of whose meshes → 0 as i → ∞, an integer k, and of triangulations {i }i=1 uniform local qubit models on (, i ) such that (i) the groundstates manifold of each Hi is canonically isomorphic to the modular functor V () as Hilbert spaces; (ii) the mapping class group acts as unitary transformations compatibly; (iii) there is a spectral gap in the following sense: if the eigenvalues of the Hamiltonians Hi are normalized such that 0 = λi0 < λi1 < · · ·, then λi1 ≥ c for all i, where c > 0 is some universal constant. The scheme for the local qubit models should be independent of the geometry of the surface , and have a uniform local description. The modular functor determines a unique topological inner product on V (). We require that the restricted inner products from Hi to the groundstates of Hi agree with the topological inner product on V (). To identify the Hilbert space Hi of one triangulation with another, we consider the two basic moves: F move and inflation of a vertex. The F move does not change the number of qubits, so the two Hilbert spaces Hi have the same number of qubits. We require that the identification be an isometric. For the inflation of a vertex, the inflated celluation has 3 new qubits, so we need to choose a homothetic embedding with a universal homothecy constant. The action of the mapping class group is defined as follows: consider the moduli space of all triangulations of that two triangulations are equivalent if there dual graphs are isomorphic as abstract graphs. By a sequence of diagonal flips, we can realize a Dehn twist. Each diagonal flip is an F move, and their composition is the unitary transformation associated to the Dehn twist. Conjecture 6.4. (1) Every doubled MTC C can be realized as a topological qubit liquid. (2) The groundstates V () ∼ = Hi ⊂ Hi form an error-correction code for each triangulation i . 6.3. Topological quantum compiling and the universality conjecture. Every unitary MTC gives rise to anyonic models of quantum computers as in [FKLW]. Quantum gates are realized by the braiding matrices of anyons, i.e. the afforded representations of the braid groups. Topological quantum compiling is the question of realizing desired unitary transformations by braiding matrices in quantum algorithms, in particular for those algorithms which are first described in the quantum circuit model such as Shor’s famous factoring algorithm.
On Classification of Modular Categories
383
To choose a computational subspace, we will use the so-called conformal block basis for the Hilbert space V (D 2 , ai ; a∞ ) of a punctured disk, where a∞ labels the boundary. Conformal block basis is in one-one correspondence to admissible labelings m, n, . . . , p of the internal edges of the following graph subject to the fusion rules at each trivalent vertex. As explained in Sect. 5.1, the tri-valent vertices also need to be indexed if multiplicities n i,k j > 1. am aJ1 a2 a3 JJ JJ JJ m JJJ n JJJJ JJ J p JJJ J a∞ The braiding of two anyons ai , ai+1 in a conformal block basis state is represented by stacking the braid on top of the above graph at i, i + 1 positions. Definition 6.5. An MTC C has property F if for every object X in C and every m the representation ρ Xm of Bm on V (D 2 , X, · · · , X ; a∞ ) factors over a finite group for any a∞ ∈ L. The following is conjectured by the first author (see [NR]): Conjecture 6.6. Let C be an MTC. (a) If C is unitary, then it has property F if and only if (di )2 ∈ N for each simple object X i or, equivalently, if and only if the global quantum dimension D 2 ∈ N. (b) In general, C has property F if and only if (FPdim(X i ))2 ∈ N for each simple object X i , where FPdim is the Frobenius-Perron dimension, i.e. the Frobenius-Perron eigenvalue of the fusion matrix Ni . The verification of this conjecture for UMTCs of rank≤ 4 is summarized in Table 4. Theorem 6.7. The following anyons are universal in the sense of [FKLW]: the Fibonacci anyon τ , the (A1 , 5) 1 anyons α, β, the (A1 , 7) 1 anyons α, ω, ρ, the 2 2 two anyons ϕ, τ in (A1 , 3) (see 5.4 for notation), and the two τ ’s in Fib × Fib. Table 4. Unitary prime MTCs rank≤4 Realization
PSL(2, Z), Relations
Property F?
Vect C
1, S = T = 1
Yes
(A1 , 1)
PSL(2, 3), T 4 = I
Yes
(A1 , 3) 1
PSL(2, 5), T 5 = I
No
(A2 , 1)
Yes
(A1 , 5) 1
PSL(2, 3), T 3 = I PSL(2, 8), T 16 = (T 2 ST )3 = I PSL(2, 7), T 7 = (T 4 ST 4 S)2 = I
(A3 , 1)
PSL(2, 8), T 8 = (T 2 ST )3 = I
Yes
D(Z2 )
PSL(2, 2), T 2 = I
Yes
(A1 , 7) 1
PSL(2, 9), T 9 = (T 4 ST 5 S)2 = I
No
2
(A1 , 2) 2
2
Universal Anyons
τ
Yes No
α,β
α, ω, ρ
384
E. Rowell, R. Stong, Z. Wang
Universal anyonic quantum computation can also be achieved with the anyon τ × τ in Fib × Fib, but images of the representations of the braid groups from this anyon are not as large as possible. Anyons that correspond to ϕ, τ, α, β, ω, ρ in other versions are also universal. Proof. We deduce the proof from [FLW2,LRW,LWa]. Universality of ϕ and τ is given in [FLW2]. The anyons α are both the fundamental representations of (A1 , k) up to abelian anyons. The universality of fundamental representation anyons are established in [FLW2]. Therefore, both α’s are universal. To prove that β, ω, ρ are universal, we first show that their braid representations are irreducible. By inspecting the braiding eigenvalues in Section 5.3, we see that they satisfy the conditions of [TW][Lemma 5.5] [HRW][Proposition 6.1]. It follows that the braid representations are irreducible. Universality now can be proved following [FLW2 or LRW]. Appendix. Non-Self Dual Rank≤ 4 MTCs with S. Belinschi Every rank=1, 2 MTC is self-dual, so we will start with rank=3. A.1. Nonselfdual rank=3. The three labels will be 0, 1, 2 such that 0ˆ = 0, 1ˆ = 2, 2ˆ = 1. The modular S˜ matrix is of the form: ⎛
1 ⎝d d
d x x¯
⎞ d x¯ ⎠ . x
s˜22 = s˜11 , s˜12 = s˜11 follows from s˜i, ˆ j = s˜i, j . Unitarity of S implies 1 + d 2 = 2|x|2 , d 2 + x 2 + x¯ 2 = 0, 1 + x + x¯ = 0. The fusion matrix N1 has eigenvalues d, dx , dx¯ . Their sum d + |x|2
1+d 2
(A.1) (A.2) (A.3) x+x¯ d
= d−
1 d
is an
1 1 2(d
+ d) is also an integer. Therefore, d is an integer. Their product d = 2d = integral multiple of 21 , so d is an integer. Let θ be the twist of label 1, hence of label 2. Using identity (2.2), we get 1 − 2d 2 + θ + θ −1 = 0.
(A.4)
Therefore, 2d 2 ≤ 3. Since d = 0, the only possible integers are d 2 = 1, hence |x| = 1. 2πi Then 1 + x + x¯ = 0 leads to x = e± 3 . A.2. Nonselfdual rank=4. Now we turn to the non-self dual rank=4 case. The 4 labels will be denoted as 1, Y, X, X ∗ , where Y is self dual and X, X ∗ dual to each other. Taking into account all symmetries among n i,k j , we can write the non-trivial fusion matrices as:
On Classification of Modular Categories
NY
NX
NX∗
⎛ 0 ⎜1 = ⎝ 0 0 ⎛ 0 ⎜0 = ⎝ 0 1 ⎛ 0 ⎜0 = ⎝ 1 0
385
1 n1 n2 n2
0 n2 n3 n4
0 n2 n4 n3
1 n3 n5 n7
0 n2 n3 n4
0 n4 n7 n6
⎞ 0 n2⎟ ; n4⎠ n3 ⎞ 0 n4⎟ ; n6⎠ n7 ⎞ 1 n3⎟ . n7⎠ n5
The modular S˜ matrix is⎞of the form: ⎛ 1 d1 d2 d2 x y y⎟ ⎜d S˜ = ⎝ 1 , where x, y are real, and z is not real. d2 y z z¯ ⎠ d2 y z¯ z We will work on unitary modular symbols, so d1 ≥ 1, d2 ≥ 1. The argument for the general case should have only minor changes. The identity N X NY = NY N X leads to the identities: 1 + n 1 n 3 + n 2 (n 5 + n 7 ) = n 22 + n 23 + n 24 , n 1 n 4 + n 2 (n 6 + n 7 ) = n 1 n 4 + n 2 (n 5 + n 6 ) =
n 22 n 22
(A.5)
+ 2n 3 n 4 ,
(A.6)
+ 2n 3 n 4 .
(A.7)
NY N X ∗ = N X ∗ NY gives no new identities. But N X N X ∗ = N X ∗ N X gives us: n2n4 + n4n6 = n2n3 + n4n5, n5 = n7, 2 n 4 + n 26 = 1 + n 23 + n 27 .
(A.8) (A.9) (A.10)
Case 1. n 4 = 0. If n 4 = 0, then n 2 n 3 = 0. First if n 2 = 0, then 1 + n 1 n 3 = n 23 which implies n 3 = 1, n 1 = 0. It follows that n 1 = n 2 = n 4 = 0, n 3 = 1. This leads to n 26 = 2 + n 27 which has no solutions. Secondly if n 3 = 0, then n 26 = 1 + n 27 which implies n 6 = 1, n 7 = 0. Hence n 3 = n 4 = n 5 = n 7 = 0, n 6 = 1. This leads to n 2 = 1, and n 1 is arbitrary. To rule out this case, notice that the labels 1, X, X ∗ have exactly the same fusion rules as the rank=3 non-self dual theory. Therefore, it is a pre-modular category with the same fusion rules, which is necessarily modular by [Br]: Suppose otherwise, then (d2 , z, z¯ ) would be a d2 times (1, d2 , d2 ) as vectors, contradicting z is not real. It follows d2 = 1, z = ω for some ω3 = 1. Comparing the squared lengths of row 1 and row 3 of the S˜ matrix, we see that y 2 = d12 . Also note that d12 = 3 + n 1 d1 . Equality of the squared lengths of row 1 and row 2 implies x 2 + 2d12 = 3. Since x is real, this does not hold if d1 > 0. Case 2. n 4 = 0. If n 2 = 0, then n 1 = 2n 3 , 1 + n 1 n 3 = n 23 + n 24 . Hence 1 + n 23 = n 24 which implies n 4 = 1, n 3 = 0. So we have n 1 = n 2 = n 3 = 0, n 4 = 1, n 5 = n 6 = n 7 . The labels 1, Y form a subcategory the same as the Z2 theory, hence d12 = 1, x 2 = 1. If x = −1, then y = 0, and d1 d2 = 0 which is a contradiction. If x = 1, then y 2 = d22 .
386
E. Rowell, R. Stong, Z. Wang
So d22 = d12 = 1. Using d22 = 1 + n 3 d1 + 2n 5 d2 below, we see that n 3 , n 5 , hence n 6 = n 7 = 0. So we have n 1 = n 2 = n 3 = n 5 = n 6 = n 7 = 0, n 4 = 1, which is the Z4 fusion rule. Suppose n 4 = 0, n 2 = 0. The fusion rules in Table 4 gives us the following identities: d12 = 1 + n 1 d1 + 2n 2 d2 , d1 d2 = n 2 d1 + (n 3 + n 4 )d2 , d22 = n 4 d1 + (n 5 + n 6 )d2 ,
(A.11) (A.12) (A.13)
d22 = 1 + n 3 d1 + 2n 5 d2 .
(A.14)
Combining equations, we have (n 4 − n 3 )d1 + (n 6 − n 5 )d2 = 1.
(A.15)
If n 4 = n 3 , then n 6 = 1 + n 27 which implies n 6 = 1, n 7 = 0. Hence n 5 = 0. By Eq. (A.15), n 5 = n 6 which is a contradiction. If n 5 = n 6 , then n 24 = 1 + n 23 which implies n 3 = 0, n 4 = 1. Solving all equations, we get n 1 = n 2 = n 3 = 0, n 4 = 1, n 5 = n 6 = n 7 , which is the Z4 fusion rule. So we may assume from now on n 2 = 0, n 4 = 0, n 4 = n 3 , n 5 = n 6 . By Eq. (A.8), we have n 4 (n 5 − n 6 ) = n 2 (n 4 − n 3 ).
(A.16)
n4 n4 . d1 − n2 n 2 (n 4 − n 3 )
(A.17)
Hence we have d2 =
Plugging into (A.15) and simplifying, we have d12 = (n 1 + 2n 4 )d1 −
n3 . n4 − n3
(A.18)
The orthogonality of S˜ gives us: x 2 + 2y 2 = 1 + 2d22 ,
(A.19)
y + 2|z| = (1 + x)d1 + 2yd2 = 0, yd1 + (1 + z + z¯ )d2 = 0, d1 d2 + (x + z + z¯ )y = 0, d22 + y 2 + z 2 + z¯ 2 = 0. 2
2
1 + d12
+ d22 ,
(A.20) (A.21) (A.22) (A.23) (A.24)
Note that y cannot be 0. Suppose otherwise, then x = −1, so d2 = 0, a contradiction. 1 The eigenvalues of NY are d1 , dx1 , dy2 , dy2 . Their sum d1 + dx1 − (1+x)d = d1 − dd12 + d2 2
2
z ( d11 − dd12 )x is an integer. The eigenvalues of N X are d2 , dy1 , dz2 , dz¯2 . Their sum d2 + dy1 + z+¯ d2 2
is an integer.
On Classification of Modular Categories
If
1 d1
−
d1 d22
387
= 0, then d12 = d22 . By Eq. (A.12), ±d1 = n 2 + n 3 + n 4 , then d1 , d2 are
integers. But the sum of the eigenvalues of NY d1 − d11 is also an integer, so d1 = ±1. It follows that ±1 = n 2 + n 3 + n 4 , but n 2 , n 4 are both = 0, a contradiction. If d11 − dd12 = 0, then x and subsequently all y, z + z¯ , |z|2 are in Q(d1 , d2 ). So all 2
x, y, z + z¯ , |z|2 , z 2 + z¯ 2 are in Q(d1 , d2 ). By Eq. A.15, Q(d1 , d2 ) is a degree≤ 2 Galois extension of Q. Therefore, the Galois group of the characteristic polynomial p1 (t) of NY is either trivial or Z2 . If it is trivial, then all eigenvalues d1 , dx1 , dy2 , dy2 and d2 are integers. So d1 , d2 , x, y are all integers. From the unitary assumption, d1 , d2 ≥ 1. Since dx1 , dy2 are integers, |x| ≥ d1 , |y| ≥ d2 . Equation (A.19) implies x = ±1, y = ±d2 . Since dx1 is an integer, |x| = d1 = 1.
Then 2yd2 = −2 implies d2 = 1, contradicting
1 d1
−
d1 d22
= 0.
Therefore the Galois group of p1 (t) is Z2 . Since p1 (t) has a pair of repeated roots, then p1 (t) is (t − m)2 q1 (t) for some irreducible quadratics q1 (t) and integer m or (q1 (t))2 . Assume q1 (t) = t 2 + bt + c, where b, c are integers. Note that d1 has to be an irrational root of p1 (t). If p1 (t) has integral roots m, then dy2 = m, so y 2 ≥ d22 . x = d1 dx1 = c implies |x| ≥ 1. By Eq. (A.19), y 2 ≥ d22 implies x 2 ≤ 1, hence |x| = 1, y 2 = d22 . It follows from Eq. (A.21) that d1 = d22 . By Eq. (A.13), (n 4 − 1)d1 + (n 5 + n 6 )d2 = 0. Since n 4 ≥ 1, it follows that n 4 = 1, n 5 = n 6 = 0, contradicting n 5 = n 6 . Hence p1 (t) = q1 (t)2 , and d1 = dx1 , i.e. x = d12 ≥ 1, and y 2 ≤ d22 . So the roots of p1 (t) are d1 , d1 , dy2 , dy2 . Then d1 + d1 d2 y
+ x = −(z + z¯ ) =
yd1 d2
y d2
and
are both integers. By Eqs. A.22,A.23,
+ x = x( dd12y + 1), √ + 1 = 0. Then d1 = s, which is also
+ 1 is an integer. On the other hand,
so x = d12 would be a rational number s if √ −b± b2 −4c , 2
d1 y d2
d2 d1 y
d1 d2 y
d2 d1 y + 1 = 0, d2 d12 = 2 d22 − 1. By 1
but not a rational number, hence b = 0, a contradiction. If
then y = − dd21 . Substituting this and x = d12 into Eq. (A.21), we get d2 d1
∈ Q, hence d12 would be a rational number s again, a contradiction. Putting everything together, we have the only desired modular S˜ matrix.
Eq. (A.15),
Acknowledgement. ZW thanks Nick Read for his insightful Comments on earlier versions. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References [Ba] [BM] [BK] [Br] [CG] [CP]
Bantay, P.: The Frobenius-Schur indicator in conformal field theory. Phys. Lett. B 394(1–2), 87– 88 (1997) Belov, D., Moore, G.: Classification of spin abelian Chern-Simons theories. http://arxiv.org/abs/ hep-th/0505235v1, 2005 Bakalov, B., Kirillov, A. Jr.: Lectures on Tensor Categories and Modular Functors. University Lecture Series, Vol. 21, Providence, RI: Amer. Math. Soc., 2001 Bruguières, A.: Catégories prémodulaires, modularisations et invariants des variétés de dimension 3. (French) Math. Ann. 316(2), 215–236 (2000) Coste, A., Gannon, T.: Remarks on Galois symmetry in rational conformal field theories. Phys. Lett. B 323(3–4), 316–321 (1994) Caselle, M., Ponzano, G.: Analyticity, modular invariance and the classification of three operator fusion algebras. Phys. Lett. B 242(1), 52–58 (1990)
388
[CZ] [dBG] [DFNSS] [DW] [ENO] [EK] [F] [FKLW] [FKW] [FLW1] [FLW2] [FW] [FQ] [FK] [FNTW] [FTL] [G] [GK] [HH] [HRW] [Ki1] [Ki2] [KL] [KW] [LRW] [LWa] [LWe] [Ma] [M1] [M2] [MR]
E. Rowell, R. Stong, Z. Wang
Cuntz, M.: Integral modular data and congruences. J. Alg. Comb. 29(3), 357–387 (2009) de Boer, J., Goeree, J.: Markov traces and II1 factors in conformal field theory. Commun. Math. Phys. 139(2), 267–304 (1991) Das Sarma, S., Freedman, M., Nayak, C., Simon, S.H., Stern, A.: Non-Abelian Anyons and Topological Quantum Computation. Rev. Mod. Phys. 80, 1083 (2008) Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Commun. Math. Phys. 129(2), 393–429 (1990) Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. of Math. (2) 162(2), 581–642 (2005) Evans, D., Kawahigashi, Y.: Quantum Symmetries on Operator Algebras. Oxford Mathematical Monographs. New York: Oxford Science Publications. The Clarendon Press/Oxford University Press, 1998 Freedman, M.H.: P/NP, and the quantum field computer. Proc. Natl. Acad. Sci. USA 95(1), 98–101 (1998) Freedman, M., Kitaev, A., Larsen, M., Wang, Z.: Topological quantum computation. Bull. Amer. Math. Soc. (N.S.) 40(1), 31–38 (2003) Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227(3), 587–603 (2002) Freedman, M.H., Larsen, M.J., Wang, Z.: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227(3), 605–622 (2002) Freedman, M.H., Larsen, M.J., Wang, Z.: The two-eigenvalue problem and density of Jones representation of braid groups. Commun. Math. Phys. 228, 177–199 (2002) Freedman, M.H., Wang, Z.: Large quantum Fourier transforms are never exactly realized by braiding conformal blocks. Phys. Rev. A (3) 75(3), 032322 (2007) Freed, D., Quinn, F.: Chern-Simons theory with finite gauge group. Commun. Math. Phys. 156(3), 435–472 (1993) Frohlich, J., Kerler, T.: Quantum Groups, Quantum Categories and Quantum Field Theory. Lecture Notes in Mathematics, 1542. Berlin: Springer-Verlag, 1993 Fradkin, E., Nayak, C., Tsvelik, A., Wilczek, F.: A Chern-Simons effective field theory for the Pfaffian quantum Hall state. Nucl. Phys. B516, 704–718 (1998) Feiguin, A., Trebst, S., Ludwig, A.W.W., Troyer, M., Kitaev, A., Wang, Z., Freedman, M.: Interacting anyons in topological quantum liquids: The golden chain. Phys. Rev. Lett. 98, 160409 (2007) Gannon, T.: Modular data: the algebraic combinatorics of conformal field theory. J. Alg. Comb. 22(2), 211–250 (2005) Gepner, D., Kapustin, A.: On the classification of fusion rings. Phys. Lett. B 349(1–2), 71–75 (1995) Hagge, T., Hong, S.: Some non-braided fusion categories of rank 3. http://arxiv.org/abs/0704. 0208v2[math.GT], 2007 Hong, S.-M., Rowell, E.C., Wang, Z.: On exotic modular tensor categories. Commun. Contemp. Math. 10(Suppl. 1), 1049–1074 (2008) Kitaev, A.: Fault-tolerant quantum computation by anyons. Ann. Physics 303(1), 2–30 (2003) Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Physics 321(1), 2–111 (2006) Kauffman, L., Lins, S.: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Ann. Math. Studies, 134. Princeton, NJ: Princeton University Press, 1994 Kazhdan, D., Wenzl, H.: Reconstructing monoidal categories. In: I. M. Gelfand Seminar, Adv. Soviet Math. 16, Part 2, Providence, RI: Amer. Math. Soc., 1993, pp. 111–136 Larsen, M.J., Rowell, E.C., Wang, Z.: The N -eigenvalue problem and two applications. Int. Math. Res. Not. 2005(64), 3987–4018 (2005) Larsen, M., Wang, Z.: Density of the SO(3) TQFT representation of mapping class groups. Commun. Math. Phys. 260(3), 641–658 (2005) Levin, M., Wen, X.-G.: String-net condensation: A physical mechanism for topological phases. Phys. Rev. B71, 045110 (2005) Manoliu, M.: Abelian Chern-Simons theory. I. a topological quantum field theory. J. Math. Phys. 39(1), 170–206 (1998) Müger, M.: From subfactor to categories and topology, II. J. Pure Appl. Alg. 180(1–2), 159–219 (2003) Müger, M.: On the structure of modular categories. Proc. London Math. Soc. (3) 87(2), 291–308 (2003) Moore, G., Read, N.: Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360(2–3), 362–396 (1991)
On Classification of Modular Categories
[MS1] [MS2] [NR] [O1] [O2] [P] [Ro1] [Tu] [TV] [TW] [Wa] [Wal] [Wen] [WW1] [WW2] [Wenz] [Wil] [Witt]
389
Moore, G., Seiberg, N.: Lectures on RCFT. Superstrings ’89 (Trieste, 1989), River Edge, NJ: World Sci. Publ., 1990 pp. 1–129 Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–254 (1989) Naidu, D., Rowell, E.C.: A finiteness property for braided fusion categories. http://arxiv.org/abs/ 0903.4157v1[math.QA], 2009 Ostrik, V.: Fusion categories of rank 2. Math. Res. Lett. 10(2–3), 177–183 (2003) Ostrik, V.: Pre-modular categories of rank 3. Mosc. Math. J. 8(1), 111–118 (2008) Preskill, J.: Chapter 9 at http://www.theory.caltech.edu/~preskill/ph229/ Rowell, E.C.: From quantum groups to unitary modular tensor categories. In: Contemp. Math. 413, 215–230 (2006) Turaev, V.: Quantum Invariants of Knots and 3-Manifolds. De Gruyter Studies in Mathematics, Berlin: Walter de Gruyter, 1994 Turaev, V., Viro, O.: State sum invariants of 3-manifolds and quantum 6 j-symbols. Topology 31(4), 865–902 (1992) Tuba, I., Wenzl, H.: On braided tensor categories of type BC D. J. Reine Angew. Math. 581, 31–69 (2005) Wang, Z.: Topologization of electron liquids with Chern-Simons theory and quantum computation. In: Differential Geometry and Physics, Nankai Tracts Math., 10, Hackensack, NJ: World Sci. Publ., 2006 pp. 106–120 Walker, K.: On Witten’s 3-manifold Invariants. 1991 notes at http://canyon23.net/math/ Wen, X.-G.: Topological orders and edge excitations in fractional quantum Hall states. Adv. in Phys. 44, 405 (1995) Wen, X.-G., Wang, Z.: A classification of symmetric polynomials of infinite variables- a construction of Abelian and non-Abelian quantum Hall states. Phys. Rev. B 77, 235108 (2008) Wen, X.-G., Wang, Z.: Topological properties of Abelian and non-Abelian quantum Hall states from the pattern of zeros. Phys. Rev. B 78, 155109 (2008) Wenzl, H.: C ∗ tensor categories from quantum groups. J. Amer. Math. Soc. 11(2), 261–282 (1998) Wilczek, F.: Fractional Statistics and Anyon Superconductivity, Singapore: World Scientific Pub. Co. Inc., 1990 Witten, E. (1989) The search for higher symmetry in string theory. In: Physics and Mathematics of Strings. Philos. Trans. Roy. Soc. London Ser. A 329(1605), 349–357 (1989)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 292, 391–415 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0907-0
Communications in
Mathematical Physics
Conformal Mappings and Dispersionless Toda Hierarchy Lee-Peng Teo Faculty of Information Technology, Multimedia University, Jalan Multimedia, Cyberjaya, 63100, Selangor, Malaysia. E-mail: [email protected] Received: 5 August 2008 / Accepted: 18 May 2009 Published online: 20 August 2009 – © Springer-Verlag 2009
Abstract: Let D be the space consists of pairs ( f, g), where f is a univalent function on the unit disc with f (0) = 0, g is a univalent function on the exterior of the unit disc with g(∞) = ∞ and f (0)g (∞) = 1. In this article, we define the time variables tn , n ∈ Z, on D which are holomorphic with respect to the natural complex structure on D and can serve as local complex coordinates for D. We show that the evolutions of the pair ( f, g) with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting D to the subspace consists of pairs where f (w) = 1/g(1/w), ¯ we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin [31]. Since every C 1 homeomorphism γ of the unit circle corresponds uniquely to an element ( f, g) of D under the conformal welding γ = g −1 ◦ f , the space HomeoC (S 1 ) can be naturally identified as a subspace of D characterized by f (S 1 ) = g(S 1 ). We show that we can naturally define complexified vector fields ∂n , n ∈ Z on HomeoC (S 1 ) so that the evolutions of ( f, g) on HomeoC (S 1 ) with respect to ∂n satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings ( f −1 , g −1 ). Moreover, in the latter case, the time variables are Fourier coefficients of γ and 1/γ −1 . 1. Introduction Introduced in [24,25] as the dispersionless limit of the well-known Toda lattice hierarchy [30], dispersionless Toda hierarchy can also be interpreted as describing the evolutions ˜ with respect to a set of formal time of the coefficients of two formal power series (L, L) ˜ variables tn , n ∈ Z. Here L(w) = w+ (lower power terms), and L(w) = w+ higher power terms. Under certain analytic conditions, L is a function univalent in a neighborhood of ∞, and L˜ is a function univalent in a neighborhood of the origin. It is therefore natural to link up evolutions of conformal mappings with dispersionless Toda hierarchy. Starting from the work of Wiegmann and Zabrodin [31], the integrable structure
392
L.-P. Teo
of conformal mappings has aroused considerable interest [1–6,13–18,20,22,26,33]. Wiegmann and Zabrodin [31] defined a set of time variables tn , n ≥ 0, on the space of conformal mappings g that map the exterior of the unit disc onto the exterior of a simply connected domain that contains the origin. They showed that the evolutions of the conformal mappings (g(w), 1/g(1/w)) ¯ with respect to (. . . , −t¯2 , −t¯1 , t0 , t1 , t2 , . . .) satisfy the dispersionless Toda hierarchy. They also defined the notion of tau function for analytic curves, which is the tau function for the hierarchy. Later it was revealed that this problem is closely related to the Dirichlet boundary problem and two dimensional inverse potential problem [20,32], and it can be put under the framework of conformal field theory [26]. By a straightforward modification, it was shown in [20] that the deformation of the conformal mapping f of the interior domain can also be described by the dispersionless Toda hierarchy. However, the evolutions of the interior mappings f and the evolutions of the exterior mappings g are treated using different time coordinates. In this paper, we consider a more general conformal mappings problem. Denote by D the space that consists of pairs of conformal mappings ( f, g), where f is a univalent function on the unit disc D and g is a univalent function on the exterior of the unit disc D∗ , normalized so that f (0) = 0, g(∞) = ∞ and f (0)g (∞) = 1. We also assume that both f and g can be extended to C 1 homeomorphisms of the extended complex plane. Moreover, the interior domain +1 = f (D) does not contain ∞ and the exterior domain ∗ − 2 = g(D ) does not contain the origin. A set of complex time variables tn , n ∈ Z, are defined on D so that the coefficients of f and g depend holomorphically on tn . We construct a tau function on the space D and use it to show that the evolutions of the conformal mappings (g, f ) with respect to the set of time variables tn , n ∈ Z, satisfy the dispersionless Toda hierarchy, with L = g and L˜ = f . In the language of Takasaki and Takebe [24], the solution to the dispersionless Toda hierarchy we considered is the solution to the Riemann-Hilbert problem ˜ M = M, ˜ LM−1 = L,
(1.1)
which has been considered in the formal level in [23] in relation to two dimensional ˜ are the Orlov-Shulman functions. A consequence of (1.1) string theory. Here M and M is that L and L˜ satisfy the string equation L, L˜ −1 = 1, or more precisely, T
g(w), f (w)−1
T
=w
∂g(w) ∂ f (w)−1 ∂ f (w)−1 ∂g(w) −w = 1. ∂w ∂t0 ∂w ∂t0
Let be the subspace of D that consists of ( f, g) with f (w) = 1/g(1/w). ¯ We show that is characterized by t¯n = −t−n . The restriction of the dispersionless Toda flows to the subspace is the integrable structure of conformal mappings considered by Wiegmann and Zabrodin [31]. Another interesting subspace of D is the space characterized by f (S 1 ) = g(S 1 ), which is equivalent to g −1 ◦ f being a C 1 homeomorphism of the unit circle. In fact, for every C 1 homeomorphism γ of the unit circle, there is a unique element ( f, g) of D such that γ = g −1 ◦ f . Therefore, we can identify the space HomeoC (S 1 ) of C 1 homeomorphisms of the unit circle as a subspace of D containing ( f, g) with f (S 1 ) = g(S 1 ). Assume that this subspace is locally defined by the equations t¯n = Zn (tm ), then we can define vector fields ∂/∂tn , n ∈ Z, as a restriction of ∂/∂tn , n ∈ Z, to HomeoC (S 1 ). It is easy to deduce from the results on D that the evolutions of (g, f ) ∈ HomeoC (S 1 ) with respect to tn , n ∈ Z, are also governed by the dispersionless Toda flows.
Conformal Mappings and Dispersionless Toda Hierarchy
393
The inverse functions f −1 , g −1 are Riemann mappings of the respective domains and − 2 . It is also interesting to study their evolutions under the context of integrable hierarchies. We show that restricted to the space HomeoC (S 1 ), we can define time variables tn , n ∈ Z, which are some Fourier coefficients of γ and 1/γ −1 . There are complexified vector fields ∂n on HomeoC (S 1 ) whose action on tn , n ∈ Z, is given by ∂n tm = δn,m . Therefore, we can identify ∂/∂tn on HomeoC (S 1 ) with ∂n . We construct a tau function on HomeoC (S 1 ) and show that the evolutions of (g −1 , f −1 ) with respect to tn , n ∈ Z, satisfy the dispersionless Toda hierarchy. The layout of this paper is as follows. In Sect. 2, we review some facts we need about generalized Grunsky coefficients, generalized Faber polynomials and dispersionless Toda hierarchy. In Sect. 3, we prove that there is an integrable structure on the space D of pairs of conformal mappings. In Sect. 4, we discuss the Riemann-Hilbert data associated to our solution to the dispersionless Toda hierarchy. In Sect. 5, we discuss the relation of our work with the work of Wiegmann and Zabrodin. In Sect. 6, we consider the restriction of the integrable hierarchy to conformal mappings ( f, g) satisfying f (S 1 ) = g(S 1 ). In Sect. 7, we consider the corresponding problem for Riemann mappings. +1
2. Background Materials 2.1. Grunsky coefficients and Faber polynomials. We review some concepts we need about univalent functions. For details, see [8,21,28,29]. Let F(z) = α1 z + α2 z 2 + · · · be a function univalent in a neighborhood of the origin and G(z) = βz + β0 + β1 z −1 + · · ·, β = α1−1 , be a function univalent in a neighborhood of ∞. We define the generalized Grunsky coefficients bm,n , m, n ∈ Z and Faber polynomials Pn and Q n by the following formal power series expansion: ∞
∞
G(z) − G(ζ ) = log β − bmn z −m ζ −n , z−ζ
log
G(z) − F(ζ ) = log β − z
log
log
F(z) − F(ζ ) =− z−ζ
m=1 n=1 ∞ ∞
bm,−n z −m ζ n ,
m=1 n=0
∞ ∞
b−m,−n z m ζ n ,
m=0 n=0 ∞
Pn (w) G(z) − w =− z −n , log bz n n=1
∞
log
F(z) Q n (w) n w − F(z) = log − z ; w α1 z n n=1
and for m ≥ 0, n ≥ 1, b−m,n = bn,m . By definition, the Grunsky coefficients are symmetric, i.e., bm,n = bn,m for all m, n ∈ Z. The coefficient b0,0 is given explicitly by − log α1 = log β, where α1 = F (0) and β = g (∞). Pn (w) is a polynomial of degree n in w and Q n (w) is a polynomial of degree n in 1/w. More precisely, 1 −1 n . Pn (w) = (G (w) )≥0 , Q n (w) = F−1 (w)n ≤0
394
L.-P. Teo
n Here when S is a subset of integers and A(w) = nnAn w is a (formal) power series, we denote by (A(w)) S the truncated sum n∈S An w . The functions log(G(z)/z), P ◦ G and Q ◦ G are meromorphic in a neighborhood of ∞ and the functions log(F(z)/z), Pn ◦ F and Q n ◦ F are meromorphic in a neighborhood of the origin. Their power series expansions are given by ∞
log
∞
G(z) = log β − b0,m z −m , z
log
m=1
Pn (G(z)) = z n + n
∞
F(z) = log α1 − b0,−m z m z m=1 ∞
bnm z −m ,
Pn (F(z)) = nbn,0 + n
m=1
Q n (G(z)) = −nb−n,0 + n
bn,−m z m ,
m=1 ∞
b−n,m z −m ,
Q n (F(z)) = z −n + n
m=1
∞
b−n,−m z m .
m=1
(2.1)
2.2. Dispersionless Toda hierarchy. The dispersionless Toda hierarchy is a hierarchy of equations describing the evolutions of the coefficients of a pair of formal power series ˜ where (L, L), L(w) = r (t)w +
∞
u n+1 (t)w −n ,
n=0 ∞
−1 ˜ (L(w)) = r (t)w −1 +
u˜ n+1 (t)w n .
n=0
Here r (t), u n (t) are functions of tn , n ∈ Z, which we denote collectively by t; w is a formal variable independent of t. The evolution of the coefficients u n are encoded in the following Lax equations: ∂L = {Bn , L}T , ∂tn ∂ L˜ ˜ T, = {Bn , L} ∂tn
∂L = { B˜ n , L}T , ∂t−n ∂ L˜ ˜ T. = { B˜ n , L} ∂t−n
Here {·, ·}T is the Poisson bracket { f, g}T = w
∂ f ∂g ∂ f ∂g , −w ∂w ∂t0 ∂t0 ∂w
and 1 Bn = (Ln )>0 + (Ln )0 , 2
1 B˜ n = (L˜ n )<0 + (L˜ n )0 . 2
(2.2)
Conformal Mappings and Dispersionless Toda Hierarchy
395
Proposition 3.1 in [28] can be reformulated as Proposition 2.1. If there exists a function τ of t such that log τ generates the generalized Grunsky coefficients of a pair (F, G) of formal power series, namely ⎧ ⎪−|mn|bm,n (t), if m = 0, n = 0 2 ∂ logτ (t) ⎨ = |m|bm,0 (t), if m = 0, n = 0 ⎪ ∂tm ∂tn ⎩−2b (t), if m = n = 0, 0,0 then the pair of formal power series (G−1 , F−1 ) satisfies the dispersionless Toda hierarchy. Here G−1 and F−1 are the inverse functions of G and F respectively. 3. Pairs of Conformal Mappings and Dispersionless Toda Hierarchy Let D be the unit disc and D∗ its exterior. In this section, we consider the space of pairs of conformal mappings and show that it has an integrable structure modeled by dispersionless Toda hierarchy. First we define the spaces: S0 = f : D → C univalent f (w) = a1 w + a2 w 2 + · · · ; a1 = 0; ˆ , ∞∈ / f (D); f is extendable to a C 1 homeomorphism of C. S∞ = g : D∗ → C univalent g(w) = bw + b0 + b1 w −1 + · · · ; b = 0; ˆ , 0∈ / g(D∗ ); g is extendable to a C 1 homeomorphism of C.
D = ( f, g) f ∈ S0 , g ∈ S∞ ; f (0)g (∞) = a1 b = 1. . − + ∗ Let +1 = f (D) and − 1 its exterior, 2 = g(D ) and 2 its exterior. C1 and C2 denotes 1 1 1 the C curves C1 = f (S ) and C2 = g(S ) respectively. Notice that f −1 is a Riemann mapping of +1 and g −1 is a Riemann mapping of − 2 . The functions tn and vn are functions on the space D defined in the following way: For n ≥ 1, 1 g(w)−n z −n 1 dg(w) = dz, tn = 2πin S 1 f (w) 2πin C2 f ◦ g −1 (z) −1 −1 t−n = g(w) f (w)n−2 d f (w) = z n−2 g ◦ f −1 (z)dz, 2πin S 1 2πin C1 (3.1) 1 g(w)n zn 1 dg(w) = dz, vn = 2πi S 1 f (w) 2πi C2 f ◦ g −1 (z) −1 −1 v−n = g(w) f (w)−n−2 d f (w) = z −n−2 g ◦ f −1 (z)dz. 2πi S 1 2πi C1
For n = 0,
1 1 g(w) 1 dg(w) = d f (w), 2πi S 1 f (w) 2πi S 1 f (w)2 1 1 1 dz = z −2 g ◦ f −1 (z)dz, (3.2) = 2πi C2 f ◦ g −1 (z) 2πi C1 g(w) g (w) f (w) f (w) 1 1 g(w) log − log g(w) dw. − v0 = 2πi S 1 w f (w) w f (w)2 w f (w) t0 =
396
L.-P. Teo
It is obvious that tn and vn depend analytically on the coefficients of the functions f and g. We also define the following functions: S± (z) =
1 2πi
1 S˜± (z) = 2πi
S1
S1
1 (1/ f )(w)g (w) 1/( f ◦ g −1 )(ζ ) dw = dζ, z ∈ ± 2, g(w) − z 2πi C2 ζ −z (3.3) 1 g(w) f (w) g ◦ f −1 (ζ ) ± dw = dζ, z ∈ 1 . f (w)2 ( f (w) − z) 2πi C1 ζ 2 (ζ − z)
They are meromorphic functions in the respective domains. In a neighborhood of the origin or ∞, they have the series expansion S+ (z) =
S˜+ (z) = −
ntn z n−1 ,
n=1
S− (z) = −t0 z
∞
v−n z n−1 ,
n=1 −1
−
∞
vn z
−n−1
S˜− (z) = −t0 z
,
n=1
−1
+
∞
(3.4) nt−n z
−n−1
.
n=1
Moreover, restricted to the curve C2 , ∞
1 n−1 −1 = S (z) − S (z) = nt z + t z + vn z −n−1 ; + − n 0 f ◦ g −1 (z) n=1
(3.5)
n=1
whereas restricted to the curve C1 , ∞
∞
n=1
n=1
g ◦ f −1 (z) = S˜+ (z) − S˜− (z) = − nt−n z −n−1 + t0 z −1 − v−n z n−1 . (3.6) 2 z We would like to show that {tn , n ∈ Z} is a complete set of local complex coordinates on the space D and therefore the partial derivatives ∂t∂n , ∂∂t¯ are well-defined. First we n have the following lemmas: Lemma 3.1. Let t → ( f t , gt ) be any smooth curve on D, then ∂ 1 (∂(gt ◦ f t−1 )/∂t) ◦ f t (w) f t (w) = ◦ gt (w)gt (w), w ∈ S 1 . f t (w)2 ∂t f t ◦ gt−1 Proof. It is straightforward to verify that ⎛ ⎞ − ddtf + gf dg d 1 dt ⎠ (z) = ⎝ ◦ g −1 (z), dt f ◦ g −1 f2 ⎛ ⎞ dg g d f −1 − 1 d(g ◦ f ) dt f dt ⎠ (z) = ⎝ ◦ f −1 (z). z2 dt f2 The assertion follows immediately.
(3.7)
Conformal Mappings and Dispersionless Toda Hierarchy
397
Lemma 3.2. Let Ct , t ∈ (−ε, ε) be a smooth family of C 1 curves and let h(z, z¯ , t) be a smooth family of C 1 functions defined in a neighborhood of Ct . Then ∂h ∂wt ∂wt dh d (z, z¯ , t)dz + (z, z¯ , t) dz − d z¯ . h(z, z¯ , t)dz = t=0 dt t=0 Ct ∂ z¯ ∂t ∂t C dt (3.8) Here wt (z, z¯ ) is a family of C 1 functions such that w0 (z, z¯ ) = id, wt (Ct ) = C0 . The proof of this lemma is straightforward. In the following, we are mostly dealing with function h such that h z¯ = 0 on C. In this case the second term in (3.8) vanishes. With these lemmas, we can show that Proposition 3.3. If ( f t , gt ), t ∈ (−ε, ε) ⊆ R, is a smooth curve on D such that dtn /dt = 0 for all n, then ( f t , gt ) = ( f 0 , g0 ) for all t. Proof. If ( f t , gt ) is a one parameter family in D such that dtn /dt = 0 for all n, then (3.4) gives d S+ (z) = 0, dt
d S˜− (z) = 0, dt
and d S− = O(z −2 ) as z → ∞. dt We conclude from (3.5) that d((1/ f )◦g −1 )/dt is the boundary value of the holomorphic ∗ function d S− /dt on − 2 , which vanish at ∞. Since g is holomorphic on D , 1 d ◦ g(w)g (w), w ∈ S 1 (3.9) dt f ◦ g −1 is the boundary value of a holomorphic function on D∗ . Similarly, we conclude that d(g ◦ f −1 )/dt ◦ f (w) f (w), w ∈ S 1 (3.10) ( f (w))2 is the boundary value of a holomorphic function on D. Lemma 3.1 implies that the function holomorphic in D∗ which is equal to (3.9) on S 1 and the function holomorphic in D which is equal to (3.10) on S 1 agree on S 1 . Therefore, we have a holomorphic ˆ which vanishes at ∞. This function must be identically zero. Therefore function on C −1 d(g ◦ f )/dt = 0 and working this formula out explicitly, we have 1 d f (w) 1 dg(w) = , w ∈ S1. g (w) dt f (w) dt However, restricted to S 1 , d log b 1 dg(w) = w + lower order terms in w. g (w) dt dt 1 d f (w) d log a1 = w + higher order terms in w. f (w) dt dt Comparing coefficients and using the fact that a1 = b−1 , we conclude that dg/dt = d f /dt = 0. This implies that ( f t , gt ) = ( f 0 , g0 ) for all t.
398
L.-P. Teo
Proposition 3.3 shows that the set {tn , n ∈ Z} of variables on D is complete, i.e., any nontrivial vector field on D will change at least one of tn . To show that this set of variables are independent, we first introduce some notations. Let κm,n and Pn , Qn be the generalized Grunsky coefficients and Faber polynomials of the pair of univalent functions ( f −1 , g −1 ). We have in particular κ0,0 = − log( f −1 ) (0)) = log f (0) = log a1 = − log b. The following proposition shows that the variables tn , n ∈ Z, are independent. Proposition 3.4. There are variations ∂n , n ∈ Z, of ( f, g) on D so that ∂n tm = δn,m . Proof. For n ≥ 1 and w ∈ S 1 , let u n (w) = u −n (w) =
∞ Pn (w) f (w)2 = u n;m w m , f (w)g (w) m=−∞ ∞ Qn (w) f (w)2 = u −n;m w m , f (w)g (w) m=−∞
and let u 0 (w) =
∞ f (w)2 = u 0;m w m . w f (w)g (w) m=−∞
Notice that since Pn (w) is a polynomial in w of degree n − 1 and Q n (w) is a polynomial of degree n +1 in w−1 without w −1 and w 0 terms, the Z×Z matrix {u n;m } is nonsingular. Define independent variations ∂n , n ∈ Z on D so that ∞ 1 m u n;1 w + u n;m w ∂n f (w) = − f (w) , 2 m=2 ∞ 1 u n;1 w + u n;−m w −m . ∂n g(w) = g (w) 2 m=0
It follows from (3.7) that for n ≥ 1 and z ∈ C2 , 1 un f (z) = ◦ g −1 (z) = Pn ◦ g −1 (z)(g −1 ) (z), ∂n f ◦ g −1 f2 1 u −n f (z) = ◦ g −1 (z) = Qn ◦ g −1 (z)(g −1 ) (z), ∂−n f ◦ g −1 f2 1 u0 f (g −1 ) (z) −1 (z) = ◦ g ∂0 (z) = . f ◦ g −1 f2 g −1 (z) For z ∈ C1 ,
u n g 1 −1 ◦ g −1 (z) = Pn ◦ f −1 (z)( f −1 ) (z), (z) = ∂n g ◦ f z2 f2 1 u −n g −1 ◦ g −1 (z) = Qn ◦ f −1 (z)( f −1 ) (z), (z) = ∂−n g ◦ f z2 f2 u 0 g ( f −1 ) (z) 1 −1 −1 ◦ g . g ◦ f (z) = ∂ (z) = 0 z2 f2 f −1 (z)
Conformal Mappings and Dispersionless Toda Hierarchy
399
As follows from (2.1), in a neighbourhood of z = ∞, Pn ◦ g −1 (z)(g −1 ) (z) = nz n−1 − n
∞
mκnm z −m−1 ,
m=1 ∞
Qn ◦ g −1 (z)(g −1 ) (z) = −n (g −1 ) (z) g −1 (z)
=
mκ−n,m z −m−1 ,
m=1 ∞
1 + z
mκ0,m z −m−1 .
m=1
In a neighbourhood of z = 0, Pn ◦ f −1 (z)( f −1 ) (z) = n
∞
mκn,−m z m−1 ,
m=1
Qn ◦ f −1 (z)( f −1 ) (z) = −nz −n−1 + n
∞
mκ−n,−m z m−1 ,
m=1
( f −1 ) (z) f −1 (z)
=
1 − z
∞
mκ0,−m z m−1 .
m=1
The definitions (3.1) and (3.2) of tn , n ∈ Z, and Lemma 3.2 then shows that ∂n tm = δn,m . As follows from this proposition, {tn , n ∈ Z} are good local coordinates on D. Therefore the partial derivatives ∂t∂n and ∂∂t¯ are well defined and ∂t∂n coincides with the n variation ∂n defined in the proposition above. We then have Proposition 3.5. The variation of the functions vm , m ∈ Z, with respect to the coordinates tn , n ∈ Z, is given by the following. For m = 0, ∂vm = −|mn|κn,m , n = 0, ∂tn
∂vm = |m|κ0,m , ∂t0
(3.11)
and for m = 0, ∂v0 = |n|κn,0 , n = 0, ∂tn
∂v0 = −2κ0,0 . ∂t0
(3.12)
Moreover, ∂vm = 0, for all m, n. ∂ t¯n
(3.13)
Proof. Equation (3.13) follows immediately since both tn , n ∈ Z, and vm , m ∈ Z, depend on the coefficients of f and g holomorphically. Equation (3.11) follows from
400
L.-P. Teo
the definition (3.1) of vm and the proof of Proposition 3.4. For (3.12), a straightforward computation gives g(w) g (w) f (w) f (w) ∂ 1 g(w) log − log g(w) − ∂tn w f (w) w f (w)2 w f (w) g(w) g (w) ∂ f (w) 1 g(w) ∂ ∂g(w) 1 ∂g(w) g (w) − log + log = f (w) ∂tn g(w) f (w) w ∂w ∂tn w f 2 (w) ∂tn f (w) ∂ f (w) f (w) ∂g(w) f (w) −g(w) − log 3 f (w) ∂tn w ∂tn f (w)2 ∂ 1 ∂ f (w) f (w) g(w) − log w ∂w f (w)2 ∂tn g(w) ∂ f (w) 1 ∂g(w) + − w f (w) ∂tn w f (w)2 ∂tn 1 ∂g(w) f (w) g(w) ∂ f (w) ∂ g(w) − log = log ∂w f (w) ∂tn w w f (w)2 ∂tn f (w) g(w) 1 ∂ f (w) ∂g(w) log . − log g (w) + − f (w) f (w)2 w w ∂tn ∂tn The definition (3.2) of v0 then implies that ∂v0 f (w) g(w) 1 1 ∂ f (w) ∂g(w) log dw. − log g = (w) − f (w) ∂tn 2πi S 1 f (w)2 w w ∂tn ∂tn We obtain from the proof of Proposition 3.4 that ∂v0 f (w) g(w) 1 1 log − log dw =− ∂t0 2πi S 1 w w w = − log a1 + log b = 2 log b = −2κ0,0 , and for n ≥ 1, 1 ∂v0 =− ∂tn 2πi
S1
Pn (w)
f (w) g(w) 1 g(w) log P (w) log − log dw = dw w w 2πi S 1 n w
g −1 (z) 1 Pn ◦ g(z)(g −1 ) (z) log dz = nκn,0 , 2πi C2 z 1 f (w) f (w) 1 g(w) =− Qn (w) log Qn (w) log − log dw = − dw 1 1 2πi S w w 2πi S w f −1 (z) 1 Qn ◦ f (z)( f −1 ) (z) log dz = nκ−n,0 . = 2πi C1 z =−
∂v0 ∂t−n
Since the Grunsky coefficients κm,n are symmetric, Proposition 3.5 shows that there should formally exist a function F on D such that ∂F/∂tn = vn . Let (z) be a holomorphic function on +1 which has a series expansion
(z) =
∞ v−n n=1
n
zn
Conformal Mappings and Dispersionless Toda Hierarchy
401
in a neighbourhood of z = 0; and let (z) be a meromorphic function on − 2 which has a series expansion (z) =
∞ vn n=1
n
z −n
in a neighbourhood of z = ∞. Notice that (z) = − S˜+ (z) and (z) = S− (z) + t0 /z. From Proposition 3.5 and (2.1), we have Lemma 3.6. The variations of the functions and with respect to tn , n ∈ Z are given by ∂ (z) = −Pn ( f −1 (z)) + nκn,0 , ∂tn ∂ f −1 (z) − log a1 , (z) = − log ∂t0 z ∂ (z) = −Qn ( f −1 (z)) + z −n , ∂t−n
∂ (z) = −Pn (g −1 (z)) + z n , ∂tn ∂ g −1 (z) − log b, (z) = − log ∂t0 z ∂ (z) = −Qn (g −1 (z)) − nκ−n,0 . ∂t−n
Now define the tau function τ by ¯ τ = |T|2 = TT,
(3.14)
where T is a holomorphic function on D defined by log T =
t2 1 g (w) t0 v0 − 0 + g(w) (g(w)) + 2(g(w)) dw 2 4 8πi S 1 f (w) g(w) f (w) 1 f (w) + ( f (w)) − 2 ( f (w)) dw. (3.15) 8πi S 1 f (w)2
When the sum converges absolutely, log T can be written explicitly as ∞
log T =
t2 1 t0 v0 − 0 − (n − 2)(tn vn + t−n v−n ). 2 4 4 n=1
By definition, log τ is a harmonic function on D. The partial derivatives of log τ with respect to tn are given by the following proposition: Proposition 3.7. For all n ∈ Z, we have ∂ log τ = vn , ∂tn
∂ log τ = v¯n . ∂ t¯n
Proof. Since log τ = log T + log T and T is holomorphic, it is sufficient to prove that ∂ log T = vn . ∂tn
402
L.-P. Teo
We consider only the case n ≥ 1. The case where n ≤ 0 is similar. From (3.15), we have ∂ ∂ log T ∂v0 1 1 z 4 = 2t0 + (z) + 2(z) dz ∂tn ∂tn 2πi C2 ∂tn f ◦ g −1 (z) ∂ ∂(z) 1 1 ∂(z) dz z + +2 2πi C2 f ◦ g −1 (z) ∂z ∂tn ∂tn 1 1 ∂(g ◦ f −1 )(z) + z (z) − 2 (z) dz 2πi C1 z 2 ∂tn 1 ∂ ∂ (z) g ◦ f −1 (z) ∂ (z) + z dz. (3.16) −2 2πi C1 z2 ∂z ∂tn ∂tn Now
∂v0 ∂tn
= nκn,0 ,
∂ 1 1 z (z) + 2(z) dz 2πi C2 ∂tn f ◦ g −1 (z) ∂ S+ (z) 1 z (z) + 2(z) dz = (−n + 2)vn , = 2πi C2 ∂tn ∂ ∂(z) 1 1 ∂(z) dz z +2 2πi C2 f ◦ g −1 (z) ∂z ∂tn ∂tn 1 1 n −1 −1 −1 = (n + 2)z − zP (g (z))(g ) (z) − 2P (g (z)) dz n n 2πi C2 f ◦ g −1 (z) 1 1 (n + 2)g(w)n g (w) − g(w)Pn (w) − 2Pn (w)g (w) dw = 2πi S 1 f (w) 1 1 g(w)Pn (w) + 2Pn (w)g (w) dw, = (n + 2)vn − 2πi S 1 f (w) 1 1 ∂(g ◦ f −1 )(z) z (z) − 2 (z) dz 2πi C1 z 2 ∂tn ∂ S˜− (z) 1 z (z) − 2 (z) dz = 0, = 2πi C1 ∂tn 1 ∂ ∂ (z) g ◦ f −1 (z) ∂ (z) z dz −2 2πi C1 z2 ∂z ∂tn ∂tn g(w) 1 = −2nt0 κn,0 − f (w)P (w) − 2P (w) f (w) dw. n n 2πi S 1 f (w)2 Adding the terms, we find that 4
∂ log T 1 = 4vn − ∂tn πi
which is the desired assertion.
S1
d g(w)Pn (w) dw = 4vn , dw f (w)
Conformal Mappings and Dispersionless Toda Hierarchy
403
Using this proposition and Proposition (3.5), we find that Corollary 3.8. For the function τ defined by (3.15), we have ⎧ ⎪ ⎨−|mn|κm,n , if m = 0, n = 0, 2 ∂ log τ = |m|κm,0 , if m = 0, n = 0, ⎪ ∂tm ∂tn ⎩−2κ , if m = n = 0, 0,0 where κm,n is the generalized Grunsky coefficients of the pair of univalent functions ( f −1 , g −1 ). It follows from Proposition 2.1 that Theorem 3.9. The evolutions of the pair of conformal mappings (g, f ) with respect to tn , n ∈ Z, satisfy the dispersionless Toda hierarchy (2.2). Since the coefficients of f and g depend holomorphically on the variables tn , n ∈ Z, and log τ is a harmonic function on D, it follows that we have another solution of the dispersionless Toda hierarchy if we take t¯n as the time variables. More precisely, let f¯ and g¯ be conformal mappings defined respectively by f¯(z) = f (¯z ) and g(z) ¯ = g(¯z ). Then Theorem 3.10. The evolutions of the pair of conformal mappings (g, ¯ f¯) with respect to t¯n , n ∈ Z, satisfy the dispersionless Toda hierarchy (2.2). An interesting thing to note is that on the subspace of D defined by f = f¯ and g = g, ¯ the coefficients of f and g are real. Therefore tn are real variables. It follows that restricted to this totally real submanifold of D, we have the usual dispersionless Toda flows where the time variables tn , n ∈ Z, are real. 4. Symplectic Structure and Riemann Hilbert Data In the language of Takasaki and Takebe (see [25] and the references therein), to every solution of the dispersionless Toda hierarchy, one can associate a Riemann Hilbert data ˜ (or called the twistor data). Namely there exist two pairs of functions (r, h) and (˜r , h) of the variables w and t0 such that ˜ T = r˜ , {˜r , h} ˜ M), ˜ ˜ M). ˜ ˜ L, r (L, M) = r˜ (L, h(L, M) = h( {r, h}T = r,
(4.1)
˜ are the Orlov-Schulman functions. They are defined so that they can be Here M and M written as M=
∞
ntn Ln + t0 +
n=1 ∞
˜ =− M
n=1
∞
vn L−n ,
n=1
nt−n L˜ −n + t0 −
∞ n=1
v−n L˜ n ;
404
L.-P. Teo
˜ i.e. and they form a canonical pair with L and L, ∂L ∂M ∂M ∂L −w = L, ∂w ∂t0 ∂w ∂t0 ˜ ˜ ˜ ˜ ˜ M} ˜ T = w ∂ L ∂ M − w ∂ M ∂ L = L. ˜ {L, ∂w ∂t0 ∂w ∂t0
{L, M}T = w
(4.2)
˜ are formal power series of Conversely, Takasaki and Takebe also showed that if (L, L) ˜ are formal functions of the form (4.2), and there exist (r, h), (˜r , h) ˜ the form (2.2), M, M ˜ is a solution to the dispersionless Toda hierarchy. The proof satisfying (4.1), then (L, L) is formal. The main technique is comparing powers of w. However nothing is assumed about the convergence of the series. For the conformal mapping problems we study in ˜ as (4.2), then compare to the previous section, L = g, L˜ = f . If we define M and M 1 (3.5) and (3.6), we find that restricted to the unit circle S , g(w) , f (w) g(w) ˜ M(w) = f (w) S˜+ ( f (w)) − S˜− ( f (w)) = . f (w)
(4.3)
˜ M = M.
(4.4)
M(w) = g(w) (S+ (g(w)) − S− (g(w))) =
In other words,
Moreover, ˜ LM =L
or
˜ = L. L˜ M
(4.5)
To prove the identities (4.2) in our context, notice that (4.2) shows that for w ∈ S 1 , ∞ 1 1 {L, M}T = {g(w), t0 }T + {g(w), vn }T g(w)−n−1 L g(w) n=1 ∞ g (w) ∂vn −n−1 + =w g (w) g(w) . g(w) ∂t0
(4.6)
n=1
Using (3.11) and (2.1), this gives ∞ ∂ 1 −n {L, M}T = w κn,0 g(w) log g(w) − L ∂w n=1 g −1 (g(w)) ∂ log g(w) + log − log b−1 = 1. =w ∂w g(w) ˜ M ˜ The identity L, = L˜ can be proved in a similar way, or by observing that (4.4) T and (4.5) give 1˜ ˜ 1 ˜ ˜ ˜ 1 L, M = LM, M = {L, M}T = 1. ˜ T T L L˜ L˜ M
Conformal Mappings and Dispersionless Toda Hierarchy
405
It is interesting to note that the identity L1 {L, M}T = 1 can also be written as 1 = L, L−1 M = L, L˜ −1 = g(w), f (w)−1 , T
T
T
which is known as the string equation. We can read from (4.4) and (4.5) that the Riemann-Hilbert data for our problem is r (w, t0 ) = w, h(w, t0 ) = t0 ,
r˜ (w, t0 ) = pt0 , ˜ h(w, t0 ) = t0 .
(4.7)
On the other hand, we can transform the pair of Eqs. (4.4) and (4.5) to ˜ L = L˜ M,
L˜ −1 = L−1 M,
which is the Riemann Hilbert problem studied by Takasaki [23] in connection to string theory. 5. Integrable Structure of Conformal Mappings à la Wiegmann-Zabrodin In this section, we would like to discuss the relation between the integrable structure on pairs of conformal mappings discussed in Sect. 3 and the integrable structure of conformal mappings observed by Wiegmann and Zabrodin [17,20,31]. In the approach of Wiegman and Zabrodin, the coordinates tn are defined as harmonic moments. Given a domain − containing ∞, bounded by an analytic curve C and with complement + , define 1 1 tn = z −n z¯ dz, vn = z n z¯ dz, n ≥ 1; 2πi C 2πi C (5.1) 1 1 2 2 z¯ dz, v0 = log |z| d z. t0 = 2πi C π + For n ≥ 1, let t−n = −t¯n and v−n = −v¯n . Denote by g the conformal map mapping the exterior disc to the domain − normalized so that g(∞) = ∞, g (∞) > 0. Let ˜ L(w) = g(w), and L(w) =
1 g(1/w) ¯
.
(5.2)
˜ is a solution of the dispersionless Toda hierWiegmann and Zabrodin show that (L, L) archy with respect to the time variables tn , n ∈ Z. Consider the transformation on the space D defined by 1 1 , g(w) ˜ = . (5.3) ( f, g) → f˜(w) = g(1/w) ¯ f (1/w) ¯ Under this transformation, it is easy to check that t˜n = −t¯−n for all n = 0, v˜n = −v¯−n for all n = 0,
and and
t˜0 = t¯0 , v˜0 = v¯0 .
406
L.-P. Teo
In other words, in terms of the coordinates tn , n ∈ Z, the transformation (5.3) is furnished by the automorphism −t¯−n , if n = 0, {tn } → tn , where tn = ¯t0 , if n = 0, of D. The invariant subspace of this automorphism is the space defined by the equations t¯n = −t−n for n = 0 and t0 = t¯0 , which is the space containing all pairs of conformal mappings of the form (1/g(1/w), ¯ g(w)). It is straightforward to check that on the subspace , the definitions of the variables tn and vn (3.1) reduce to (5.1). Notice that is not a complex manifold because of the one extra dimension furnished by the real variable t0 . We can take the variables t0 , Re tn , Im tn , n ≥ 1 as coordinates on the real manifold . To make some distinctions, we denote by tn the variables tn restricted to so that for n = 1, t−n = −¯tn . Any function FD(tn , t¯n ) on D restricted to the function F (tn ) = FD (tn , −t−n ) on . Therefore, the partial derivatives ∂t∂n on can be defined in terms of the partial derivatives ∂t∂n and ∂∂t¯ by n
∂ ∂ ∂ ∂ ∂ ∂ = − , for n = 0, and = + . ∂tn ∂tn ∂ t¯−n ∂t0 ∂t0 ∂ t¯0 Notice that ∂t∂n are well-defined (complex) vector fields on since they annihilate the defining functions Z n (tm , t¯m ) = t¯n + t−n , n ∈ Z, of . Now since the functions T (3.15) and vn (3.1) on D are holomorphic, we find immediately from the results in Sect. 3 that their restrictions to satisfy ⎧ ⎪ if mn = 0 ⎨−|mn|κm,n , 2 ∂ log T ∂ log T = vn , = |m|κm,0 , (5.4) if m = 0, n = 0, ⎪ ∂tn ∂tm ∂tn ⎩−2κ , if m = n = 0. 0,0 ¯ g(w)) is a solution of the dispersionless Proposition 2.1 then shows that (1/g(1/w), Toda hierarchy with respect to the time variables tn , n ∈ Z, with restriction t−n = −¯tn . This is precisely the result of Wiegmann and Zabrodin [17,20,31]. The corresponding tau function is the restriction of T (3.15) to . We left it as an exercise for the reader to show that restricted to , the function T is given by 1 1 1 log T = − 2 log − d 2 ζ d 2 z, π − − z ζ which is a real-valued function. Therefore, restricted to , τ = T2 . As a result, the restriction of the tau function τ on D to is not the corresponding tau function on . 6. Conformal Weldings and Dispersionless Toda Hierarchy In this section, we review the concept of conformal weldings and discuss their evolutions under the dispersionless Toda flow. For details about conformal weldings, one can see [9,11,12,19].
Conformal Mappings and Dispersionless Toda Hierarchy
407
Let HomeoC (S 1 ) be the space of all C 1 homeomorphisms on the unit circle S 1 . Notice that a C 1 homeomorphism γ ∈ HomeoC (S 1 ) is also a quasi-symmetric homeomorphism, i.e., γ (eiθ ) satisfies the inequality 1 γ (ei(θ+ω) ) − γ (eiθ ) ≤ M, ≤ M γ (eiθ ) − γ (ei(θ−ω) )
∀ θ, ω ∈ R, 0 < ω <
π , 2
(6.1)
for some constant M > 1. Therefore, according to the theory of quasiconformal mapˆ = C ∪ {∞}, pings, γ can be extended to be a C 1 map on the extended complex plane C which is also denoted by γ , and satisfies 1 1 = , ∀z ∈ C. γ z¯ γ (z) ˆ 1. Moreover, γ is real analytic on C\S In [29], we used the theory of quasiconformal mappings to show that given a quasi-symmetric homeomorphism with its quasiconformal extension γ , there exist quasiconformal mappings f˜ and g˜ such that γ = g˜ −1 ◦ f˜, and f˜ D and g˜ D∗ are univalent functions. Moreover, f˜ and g˜ are unique if we impose the conditions f˜(0) = 0, ˜ = ∞. Define f = r ◦ f˜ and g = r ◦ g, ˜ where r is a complex number f˜ (0) = 1 and g(∞) 2 −1 so that r = 1/g˜ (∞); we find that γ = g ◦ f and f (0)g (∞) = r 2 f˜ (0)g˜ (∞) = 1. 1 In other words, we have shown that given γ ∈ HomeoC (S 1 ), there exists two C −1 homeomorphisms f and g of the plane, such that γ = g ◦ f , and f D and g D∗ are the unique univalent functions satisfying f (0) = 0, g(∞) = ∞ and f (0)g (∞) = 1. The decomposition of γ as g −1 ◦ f is known as conformal welding or sewing.1 Given γ ∈ HomeoC (S 1 ) with conformal welding γ = g −1 ◦ f , we can associate γ with the simply connected domain + = f (D) = g(D), its exterior − = f (D∗ ) = g(D∗ ) and their common boundary C = f (S 1 ) = g(S 1 ), a C 1 curve. However, such an association is not one-to-one. If γ1 = g1−1 ◦ f 1 and γ2 = g2−1 ◦ f 2 are associated to the same domain, then f 1 (D) = f 2 (D) implies that f 1−1 ◦ f 2 is a univalent function on D mapping the unit disc back to itself. Therefore, f 1−1 ◦ f 2 is a linear fractional transformation of z+a the form eiθ 1+ az ¯ for some a ∈ D and θ ∈ R. However, the condition f 1 (0) = f 2 (0) = 0 forces a = 0. Therefore, we are left with the possibility f 2 (z) = f 1 (eiθ z). Similar argument shows that g1 (D∗ ) = g2 (D∗ ) implies that g2 (z) = g1 (eiω z) for some ω ∈ R. The condition f j (0)g j (∞) = 1, j = 1, 2, then forces eiθ = eiω . On the other hand, it is easy to show that given γ ∈ HomeoC (S 1 ) with conformal welding γ = g −1 ◦ f and given r ∈ S 1 , the conformal welding of r −1 ◦ γ ◦ r ∈ HomeoC (S 1 ) is given by γ = (g ◦ r )−1 ◦ ( f ◦ r ) and therefore γ and r −1 ◦ γ ◦ r are both associated to the domain + = f (D) = f ◦ r (D). As a conclusion, γ1 and γ2 are associated with the same domain + if and only if γ2 = r −1 ◦ γ1 ◦ r for some r ∈ S 1 . Now return to our discussion on the evolutions of conformal mappings; we see from the unique decomposition HomeoC (S 1 ) γ = g −1 ◦ f that we can identify HomeoC (S 1 ) as a subspace of D containing the pairs ( f, g) with f (S 1 ) = g(S 1 ). Unlike the subspace which can be easily identified as the subspace of D defined by t¯n = −t−n , the characterization of the space HomeoC (S 1 ) is a highly nontrivial issue. Assume that the subspace HomeoC (S 1 ) can be defined locally by t¯n = Zn (tm ). Then 1 The conformal welding of γ ∈ S 1 \Diff (S 1 ), where Diff (S 1 ) is the space of diffeomorphisms on the + + unit circle, was first discussed in [11].
408
L.-P. Teo
we can take tn = tn as a set of local parameters on HomeoC (S 1 ) so that any function FD(tn , t¯n ) on D restricted to the function FHomeoC (S 1 ) (tn ) = FD (tn , Zn (tm )) on HomeoC (S 1 ). The (complex) vector fields ∞ ∂ ∂Zm ∂ ∂ = + , n ∈ Z, ∂tn ∂tn m=−∞ ∂tn ∂ t¯m
(6.2)
are then well-defined vector fields on the subspace HomeoC (S 1 ). Now using the same reasoning as in Sect. 5, one can prove that (5.4) still holds. It follows that with L = g and L˜ = f , their evolutions with respect to tn satisfy the dispersionless Toda hierarchy (2.2). One should take note that the tn -flow on HomeoC (S 1 ) is different from the tn flow on D. We would also like to remark that now the variables tn cannot be treated as local coordinates on HomeoC (S 1 ) since they are complex variables and their complex conjugates satisfy some nontrivial relations t¯n = Zn (tm ) on HomeoC (S 1 ). 7. Riemann Mappings and Dispersionless Toda Hierarchy Since the functions f −1 and g −1 are respectively the Riemann mappings of the domains +1 and − 2 , it is natural to ask whether we can describe the evolutions of the Riemann mappings (g −1 , f −1 ) by dispersionless Toda flows. Here we are not going to explore all the possibilities. We restrict our consideration to the solutions governed by the same Riemann-Hilbert data (4.7). Formally, one can just replace all the f and g in the definitions and proofs above by f −1 and g −1 and get the desired results. However, analytically this is not feasible. Tracing from the beginning the definitions of tn and vn , we immediately bumped into the problem that f −1 ◦ g and g −1 ◦ f are not well defined for general f and g. To make f −1 ◦ g and g −1 ◦ f well defined, we have to restrict our consideration to the space HomeoC (S 1 ), where f (S 1 ) = g(S 1 ). g −1 ◦ f and f −1 ◦ g are then C 1 homeomorphisms of the unit circle. In this case, the functions tn and vn defined as in (3.1) can be considered as Fourier coefficients. More precisely, given a C 1 homeomorphism γ of the unit circle, the functions tn , vn , n < 0 , and t0 on HomeoC (S 1 ) are the coefficients of the absolutely convergent Fourier series expansion of γ = g −1 ◦ f on S 1 : γ (w) = −
∞
nt−n w
−n+1
+ t0 w −
n=1
∞
v−n w n+1 , w = eiθ .
(7.1)
n=1
For n > 0, the functions tn , vn are the coefficients of the Fourier series expansion of 1/γ −1 = (1/ f −1 ) ◦ g on S 1 : ∞
∞
n=1
n=1
1 n−1 −1 = nt w + c w + vn w −n−1 , w = eiθ . n 0 γ −1 (w) For the coefficient c0 , it is easy to check that it coincides with t0 : c0 =
1 2πi
S1
1 1 dw = −1 γ (w) 2πi
S1
1 1 dγ (w) = w 2πi
S1
γ (w) dw = t0 . w2
(7.2)
Conformal Mappings and Dispersionless Toda Hierarchy
409
Finally, similar to (3.2), the function v0 is defined as −1 f (w) γ (w) 1 1 g (z) dz 1 g(w) log dw− . − log v0 = 2 −1 2πi S 1 w w w γ (w) 2πi C f −1 (z) z Heuristically, it is equal to 1 v0 = − 2πi
S1
γ (w) log w (log w) 2 − −1 dw. w γ (w)
One should take note that as in Sect. 5, the condition f (S 1 ) = g(S 1 ) implies some nontrivial relations between the variables tn , ¯tn , n ∈ Z. As in Sect. 5, assume that locally, we can regard HomeoC (S 1 ) as a submanifold of a complex manifold defined by the zeros of the functions t¯n − Zn (tm ), then we can define the vector fields ∂t∂n by (6.2). Alternatively, one can regard ∂t∂n as complexified vector fields on HomeoC (S 1 ) as shown by the following proposition. Proposition 7.1. There are complexified vector fields ∂n on HomeoC (S 1 ) such that ∂n tm = δn,m . local coordinate chart is given by (u 0 , Re u n , Proof. At every point γ ∈ HomeoC (S 1 ), a inθ and u Im u n ) → ei(θ+u(θ)) ◦ γ , where u(θ ) = −n = u¯ n . Equivalently, n∈Z u n e we can also use (u n )n∈Z as local coordinates and a complexified vector field ∂ on HomeoC (S 1 ) can be written as ∂=
∞
cn
n=−∞
where cn ∈ C. Its action on γ is
∞
∂γ (w) = i
∂ , ∂u n
cn w
n+1
◦ γ (w).
n=−∞
Let bm,n be the generalized Grunsky coefficients of ( f, g) and Pn (z), Q n (z) the associated generalized Faber polynomials. Consider the complexified vector fields ∂n whose action on γ is given by ∂n γ (w) =
∞
mnb−m,n w m+1 = w 2 Pn ( f (w)) f (w),
m=1
∂−n γ (w) = −nw
−n+1
+
∞
(7.3) mnb−m,−n w
m+1
=w
2
Q n ( f (w)) f (w),
m=1
for n ≥ 1 and ∂0 γ (w) = w −
∞ m=1
mbm,0 w m+1 = w 2
f (w) . f (w)
(7.4)
As in the proof of Proposition 3.4, one can show that ∂n , n ∈ Z, give rise to independent variations of γ .
410
L.-P. Teo
From the definitions (7.1) and (7.2), we find that for n ∈ Z, ∂n γ (w) = −
∞
m[∂n t−m ]w −m+1 + [∂n t0 ]w −
m=1
∞
[∂n v−m ]w m+1 ,
(7.5)
m=1
and ∞ ∞ (γ −1 ) (w)(∂n γ ) ◦ γ −1 (w) m−1 −1 = m[∂ t ]w + [∂ t ]w + [∂n vm ]w −m−1 . n m n 0 γ −1 (w)2 m=1
m=1
(7.6) Using the definition of ∂n γ (w) given by (7.3) and (7.4), we find from (7.5) that ∞
mnb−m,n w m+1 = −
m=1
∞
m[∂n t−m ]w −m+1 + [∂n t0 ]w −
m=1
−nw −n+1 +
∞
∞
[∂n v−m ]w m+1 ,
(7.7)
m=1
mnb−m,−n w m+1
m=1
=−
∞
m[∂−n t−m ]w −m+1 + [∂−n t0 ]w −
m=1
∞
[∂−n v−m ]w m+1 ,
(7.8)
m=1
for n ≥ 1, and w−
∞
mbm,0 w
m+1
=−
m=1
∞
m[∂0 t−m ]w
−m+1
+ [∂0 t0 ]w −
m=1
∞
[∂0 v−m ]w m+1 .
m=1
(7.9) On the other hand, using (7.6) and f ∞
◦ γ −1
m[∂n tm ]w m−1 + [∂n t0 ]w −1 +
m=1
= g, we have ∞
[∂n vm ]w −m−1
m=1
(γ −1 ) (w)(w 2 Pn ◦ f ) ◦ γ −1 (w) f ◦ γ −1 (w) = = Pn (g(w))g (w) γ −1 (w)2 ∞ n−1 − nmbnm w −m−1 ; (7.10) = nw m=1 ∞
m[∂−n tm ]w m−1 + [∂−n t0 ]w −1 +
m=1
∞
[∂−n vm ]w −m−1
m=1
= Q n (g(w))g (w) = −
∞
nmbm,−n w −m−1
(7.11)
m=1
for n ≥ 1; and ∞ m=1
m[∂0 tm ]w m−1 + [∂0 t0 ]w −1 +
∞ m=1
[∂0 vm ]w −m−1 =
∞ g (w) mbm,0 w −m−1 . = g(w) m=1
(7.12)
Conformal Mappings and Dispersionless Toda Hierarchy
411
We can then read from (7.7), (7.8), (7.9), (7.10), (7.11) and (7.12) that ∂n tm = δn,m for all n, m ∈ Z. It follows from this proposition that we can identify the vector field defined in the proof. One can also trace from the proof that
∂ ∂tn
with ∂n
Proposition 7.2. Let bm,n be the generalized Grunsky coefficients of the pair of univalent functions ( f, g). The variation of vm , m ∈ Z, with respect to tn , n ∈ Z, is given by the following: ∂vm ∂vm = −|mn|bn,m , n = 0, and = |m|b0,m . ∂tn ∂t0 For the function v0 , we have Proposition 7.3. The variation of v0 with respect to tn , n ∈ Z, is given by ∂v0 ∂vn = |n|bn,0 = , ∂tn ∂t0
n = 0,
∂v0 = −2b0,0 . ∂t0
Proof. We have 1 ∂v0 = ∂tn 2πi
f (w) (∂γ /∂tn )(w) ∂ 1 g(w) log (w)dw. − log w w2 w ∂tn γ −1 S1
Using the series expansion for each term gives the desired result.
For the tau function, we define it as t2 t0 v0 1 1 log τ = − 0 + wφ (w) + 2φ(w) dw −1 2 4 8πi S 1 γ (w) γ (w) 1 + wψ (w) − 2ψ(w) dw, 2 8πi S 1 w where ψ(w) =
∞ v−n n=1
n
w n , w ∈ D, φ(w) =
∞ vn n=1
n
w −n , w ∈ D∗ .
From Proposition 7.2 and the identities in (2.1), we find that the variations of ψ(w) and φ(w) with respect to tn , n ∈ Z, are given by: Lemma 7.4. The variations of the functions ψ and φ with respect to tn , t−n , n ≥ 1, and t0 are given by ∂ψ (w) = −Pn ( f (w)) + nbn,0 , ∂tn ∂ψ f (w) (w) = − log + log a1 , ∂t0 w ∂ψ (w) = −Q n ( f (w)) + w −n , ∂t−n
∂φ (w) = −Pn (g(w)) + w n , ∂tn ∂φ g(w) (w) = − log + log b, ∂t0 w ∂φ (w) = −Q n (g(w)) − nb−n,0 . ∂t−n
412
L.-P. Teo
From this, we can prove as in Proposition 3.7 that Proposition 7.5. The tau function generates the functions vn , namely ∂ log τ = vn ∂tn for all n ∈ Z. Combining this proposition with Proposition 7.2 and Proposition 7.3, we have ⎧ ⎪ if m = 0, n = 0 ⎨−|mn|bm,n , 2 ∂ log τ = |m|bm,0 , (7.13) if m = 0, n = 0 ⎪ ∂tm ∂tn ⎩−2b , if m = n = 0. 0,0 Therefore, we conclude by Proposition 2.1 that Theorem 7.6. The evolutions of the Riemann mappings (g −1 , f −1 ) with respect to tn , n ∈ Z, satisfy the dispersionless Toda hierarchy (2.2). Acknowledgements. The author would like to thank A. Zabrodin and L. Takhtajan for helpful comments. We would also like to thank the anonymous referee for the illuminating suggestions which have greatly improved the presentation of this article. This project is funded by Ministry of Science, Technology and Innovation of Malaysia under eScienceFund 06-02-01-SF0021.
Appendix A. The Subgroup of Linear Fractional Transformations In this section, we consider the subspace of D containing those ( f, g) where f and g are linear fractional transformations. The conditions f (0) = 0, g(∞), f (0)g (∞) = 1, ∞∈ / f (D) and 0 ∈ / g(D∗ ) imply that f and g have the following forms: c 1 w f (w) = , |a| < 1, g(w) = bw + c, b = 0, < 1. b 1 + aw b Notice that here we have three complex parameters a, b and c. It is straightforward to compute from the definition (3.1) of tn and vn that t−1 = −c, t0 = b2 , t1 = ab, tn = 0 for all |n| ≥ 2; for n ≥ 1,
(A.1) vn = b2 cn , v−n = −bn+2 a n , and v0 = b2 log b2 − b2 + abc.
Therefore, we see that the subspace of linear fractional transformations is characterized by tn = 0 for all |n| ≥ 2. As a function of t−1 , t0 and t1 , we have n , vn = (−1)n t0 t−1
v−n = −t0 t1n ,
v0 = t0 log t0 − t0 − t−1 t1 .
A straightforward computation shows that the τ function (3.14) is given by 2 t02 3 2 2 2 log t0 − t0 − t−1 t0 t1 . τ = |T| = exp 4 4 From this, it is easy to verify that ∂ log τ = v−1 , ∂t−1
∂ log τ = v0 , ∂t0
∂ log τ = v1 . ∂t1
(A.2)
Conformal Mappings and Dispersionless Toda Hierarchy
413
Now we consider the restriction of ( f, g) considered above to the subspace and HomeoC (S 1 ). Restricted to , 1 w 1 w 1 . = = f (w) = b 1 + aw b¯ 1 + c¯¯ w g(1/w) ¯ b Therefore, b = b¯
c¯ = a. b¯
and
Equation (A.1) then implies that t0 is real and t¯1 = −t−1 . Therefore v¯n = t0 t1n = −v−n
for n = 1,
and the tau function T is
v0 = t0 log t0 − t0 + |t1 |2 ,
and
t02 3 2 2 2 log t0 − t0 + t0 |t1 | . T = exp 4 4
Again, one can show that (A.2) holds. For the restriction to HomeoC (S 1 ), the condition f (S 1 ) = g(S 1 ) is equivalent to γ = g −1 ◦ f is a linear fractional transformation mapping S 1 to itself. Equivalently, γ ∈ PSL(2, R). This implies that iα
b=
iα
e2
(1 − |a|2 )
, c = −
ae ¯ −2
(1 − |a|2 )
,
and
γ (w) = e−iα
w + a¯ . 1 + aw
Substituting into (A.1) gives iα
t−1
ae ¯ −2 = , 1 − |a|2
eiα t0 = , 1 − |a|2
iα
ae 2
t1 = . 1 − |a|2
Therefore, t¯−1 = t1 ,
t¯0 =
(1 + t1 t−1 )2 , t0
t¯1 = t−1 .
As a function of t−1 , t0 , t1 , the tau function is given by t02 3 2 2 log t0 − t0 − t−1 t0 t1 . T = exp 4 4
(A.3)
Equation (A.2) still holds. Next we consider the time variables tn and the functions vn for the evolutions of the Riemann mappings (g −1 , f −1 ). Since γ (w) = ae ¯ −iα + 1 γ −1
(w) = −a +
∞
(−1)n e−iα a n (1 − |a|2 )w n+1 ,
n=0 ∞ −i(n+1)α
e
n=0
(1 − |a|2 )a¯ n w −n−1 ,
414
L.-P. Teo
we find that the variables tn and vn are given by t1 = −a, vn = e
t0 = e−iα (1 − |a|2 ),
−i(n+1)α
(1 − |a| )a¯ , 2
t−1 = −ae ¯ −iα ,
v−n = (−1)
n
tn = 0,
n−1 −iα n
a (1 − |a| ),
e
2
∀|n| ≥ 2, n ≥ 1.
On the other hand, we also have v0 = t0 log t0 − t0 − t1 t−1 . The local coordinates α, a, a¯ of HomeoC (S 1 ) can be expressed in terms of t−1 , t0 and t1 by a = −t1 ,
e−iα = t0 + t1 t−1 ,
a¯ = −
t−1 , t0 + t1 t−1
and the functions vn , n ∈ Z can be written as functions of t−1 , t0 and t1 by n , vn = (−1)n t0 t−1
v−n = −t0 t1n ,
v0 = t0 log t0 − t0 − t1 t−1 .
In terms of t−1 , t0 and t1 , we have ¯t−1 =
t1 , t0 + t1 t−1
¯t0 =
t0 , (t0 + t1 t−1 )2
¯t1 =
t−1 . t0 + t1 t−1
The tau function is given by
t02 3 2 2 log t0 − t0 − t−1 t0 t1 . τ = exp 4 4 Its dependence on t−1 , t0 , t1 is the same as (A.3). References 1. Alonso, L.M.: Genus-zero Whitham hierarchies in conformal-map dynamics. Phys. Lett. B 641, 466–473 (2006) 2. Alonso, L.M., Medina, E.: Solutions of the dispersionless Toda hierarchy constrained by string equations. J. Phys. A: Math. Gen. 37, 12005–12017 (2004) 3. Alonso, L.M., Medina, E.: Exact solutions of integrable 2D, contour dynamics. Phys. Lett. B 610, 277–282 (2005) 4. Alonso, L.M., Medina, E., Manas, M.: String equations in Whitham hierarchies: tau-functions and Virasoro constraints. J. Math. Phys. 47, 083512 (2006) 5. Bauer, M., Bernard, D.: 2D growth processes: SLE and Loewner chains. Phys. Rep. 432, 115–221 (2006) 6. Boyarsky, A., Marshakov, A., Ruchayskiy, O., Wiegmann, P., Zabrodin, A.: Associativity equations in dispersionless integrable hierarchies. Phys. Lett. B 515, 483–492 (2001) 7. Crowdy, D.: The Benney hierarchy and the Dirichlet boundary problem in two dimensions. Phys. Lett. A 343, 319–329 (2005) 8. Duren, P.L.: Univalent functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 259, New York: Springer-Verlag, 1983 9. Gardiner, F.P., Lakic, N.: Quasiconformal Teichmüller theory. Mathematical Surveys and Monographs, Vol. 76, Providence, RI: American Mathematical Society, 2000 10. Gardiner, F.P., Sullivan, D.P.: Symmetric structures on a closed curve. Amer. J. Math. 114(4), 683–736 (1992) 11. Kirillov, A.A.: Kähler structure on the K -orbits of a group of diffeomorphisms of the circle. Funkt. Anal. i Pril. 21(2), 42–45 (1987) 12. Kirillov, A.A., Yuriev, D.V.: Kähler geometry of the infinite-dimensional homogeneous space M = diff + (S 1 )/rot(S 1 ). Funkt. Anal. i Pril. 21(4), 35–46 (1987)
Conformal Mappings and Dispersionless Toda Hierarchy
415
13. Konopelchenko, B., Alonso, L.M., Ragnisco, O.: The partial derivative-approach to the dispersionless KP hierarchy. J. Phys. A: Math. Gen. 34, 10209–10217 (2001) 14. Konopelchenko, B., Alonso, L.M.: Dispersionless scalar integrable hierarchies, Whitham hierarchy, and ¯ the quasiclassical ∂–dressing method. J. Math. Phys. 43, 3807–3823 (2002) 15. Konopelchenko, B., Alonso, L.M.: Nonlinear dynamics on the plane and integrable hierarchies of infinitesimal deformations. Stud. Appl. Math. 109, 313–336 (2002) 16. Kostov, I.K.: String equation for string theory on a circle. Nucl. Phys. B 624, 146–162 (2002) 17. Kostov, I.K., Krichever, I.M., Mineev-Weinstein, M., Zabrodin, A., Wiegmann, P.B.: The τ -function for analytic curves. In: Random matrix models and their applications, Math. Sci. Res. Inst. Publ., Vol. 40, Cambridge: Cambridge Univ. Press, 2001, pp. 285–299 18. Krichever, I., Marshakov, A., Zabrodin, A.: Integrable structure of the dirichlet boundary problem in multiply-connected domains. Commun. Math. Phys. 259, 1–44 (2005) 19. Lehto, O.: Univalent functions and Teichmüller spaces. Graduate Texts in Mathematics, Vol. 109, New York: Springer-Verlag, 1987 20. Marshakov, A., Wiegmann, P., Zabrodin, A.: Integrable structure of the Dirichlet boundary problem in two dimensions. Commun. Math. Phys. 227(1), 131–153 (2002) 21. Pommerenke, C.: Univalent functions. Göttingen, Vandenhoeck & Ruprecht: 1975; with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV 22. Prokhorov, D., Vasil’ev, A.: Univalent functions and integrable systems. Commun. Math. Phys. 262, 393– 410 (2006) 23. Takasaki, K.: Dispersionless Toda hierarchy and two-dimensional string theory. Commun. Math. Phys. 170(1), 101–116 (1995) 24. Takasaki, K., Takebe, T.: SDiff(2) Toda equation—hierarchy, tau function, and symmetries. Lett. Math. Phys. 23(3), 205–214 (1991) 25. Takasaki, K., Takebe, T.: Integrable hierarchies and dispersionless limit. Rev. Math. Phys. 7(5), 743–808 (1995) 26. Takhtajan, L.A.: Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves. Current correlation functions. Lett. Math. Phys. 56 (3), 181–228 (2001), (EuroConférence Moshé Flato 2000, Part III (Dijon)) 27. Takhtajan, L.A., Teo, L.P.: Weil-Petersson metric on the universal Teichmuller space. Mem. Amer. Math. Soc. 183(861), (2006) 28. Teo, L.P.: Analytic functions and integrable hierarchies—characterization of tau functions. Lett. Math. Phys. 64(1), 75–92 (2003) 29. Teo, L.P.: The Velling-Kirillov metric on the universal Teichmüller curve. J. Anal. Math. 93, 271–307 (2004) 30. Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 4, Amsterdam: North-Holland, 1984, pp. 1–95 31. Wiegmann, P.B., Zabrodin, A.: Conformal maps and integrable hierarchies. Commun. Math. Phys. 213(3), 523–538 (2000) 32. Zabrodin, A.V.: The dispersionless limit of the Hirota equations in some problems of complex analysis. Teoret. Mat. Fiz. 129(2), 239–257 (2001) 33. Zabrodin, A.V.: Growth processes related to the dispersionless Lax equations. Physica D 235, 101–108 (2007) Communicated by L. Takhtajan
Commun. Math. Phys. 292, 417–429 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0904-3
Communications in
Mathematical Physics
On the Existence of Traveling Waves in the 3D Boussinesq System Marta Lewicka1 , Piotr B. Mucha2 1 Department of Mathematics, University of Minnesota,
127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, USA. E-mail: [email protected] 2 Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02097 Warszawa, Poland. E-mail: [email protected] Received: 22 August 2008 / Accepted: 1 July 2009 Published online: 22 August 2009 – © Springer-Verlag 2009
Abstract: We extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. For threedimensional channels not aligned with the gravity direction and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected. 1. Introduction The Boussinesq-type system of reactive flows is a physical model in the description of flame propagation in a gravitationally stratified medium [24]. It is given as the reaction-advection-diffusion equation for the reaction progress T (which can be interpreted as temperature), coupled to the fluid motion through the advection velocity, and the Navier-Stokes equations for the incompressible flow u driven by the temperature-dependent force term. After passing to non-dimensional variables [3,19], the Boussinesq system for flames takes the form: Tt + u · ∇T − T = f (T ), u t + u · ∇u − νu + ∇ p = T ρ, div u = 0.
(1.1)
Here, ν > 0 is the Prandtl number, that is the ratio of the kinematic and thermal diffusivities (inverse proportional to the Reynolds number). The vector ρ = ρ g corresponds to the non-dimensional gravity g scaled by the Rayleigh number ρ > 0. The reaction rate is given by a nonnegative ’ignition type’ Lipschitz function f of the temperature, this last one normalized to satisfy: 0 ≤ T ≤ 1. The above model can be derived from a more complete system under the assumption that the Lewis number equals 1.
418
M. Lewicka, P. B. Mucha
We study the system (1.1) in an infinite cylinder D ⊂ R3 with a smooth, connected crossection ⊂ R2 . Recent numerical results, motivated by the astrophysical context [19,20], suggest that the initial perturbation in T either quenches or develops a curved front, which eventually stabilizes and propagates as a traveling wave. On the other hand, existence of non-planar traveling waves for the single reaction-advection-diffusion equation in a prescribed flow has been a subject of active study in the last decade [2,14,23]. For system (1.1), existence of traveling waves has been considered under the no-stress or Dirichlet boundary conditions in u, in channels of various inclinations and dimensions [3–5,10,17]. The main difference presents itself at the orientation of D with respect to g; when they are aligned there are no non-planar fronts at small Rayleigh numbers [4], while in the other case a traveling front, necessarily non-planar, is expected to exist at any range of parameters. This has been rigorously proven: in [3] for n = 2 dimensional channels D and under no-stress boundary conditions, in [5] for n = 2 and the more physical no-slip conditions and in [10] for the same boundary conditions and arbitrary dimension n, but for a simplified system (corresponding to the infinite Prandtl number ν = ∞) when the Navier-Stokes part of (1.1) is replaced by the Stokes system. The purpose of this paper is to remove this last assumption, for three dimensional channels. Namely, we will investigate the model supplied by the Navier-Stokes system. We assume that ρ is not parallel to the unbounded direction of D, which after an elementary change of variables [3] amounts to studying: D = (−∞, ∞) × = {(x, x); ˜ x ∈ R, x˜ ∈ } and ρ · e3 = 0. We will prove the existence of a traveling wave solution to (1.1): T (x−ct, x), ˜ u(x−ct, x), ˜ with the speed c to be determined and under the boundary conditions: ∂T =0 ∂ n
and
u = 0 on ∂ D,
(1.2)
where n is the unit normal to ∂ D. Such a front satisfies: − cTx − T + u · ∇T = f (T ), −cu x + du · ∇u − νu + ∇ p = T ρ, div u = 0.
(1.3)
We set constant d to be 0 or 1. For the simplified system, when the advection in u has been neglected and d = 0, the theorem below states existence of a non-planar traveling wave, for any crossection , Prandtl number ν and Rayleigh number ρ. For the full system when d = 1, we need to assume the following relative thinness condition, involving ν, ρ, the area ||, and the Poincaré and the Poincaré-Wirtinger constants C P , C P W of : 1/2 √ CP 1/2 2 14 √ || |ρ · (0, x)| ˜ |ρ|C PW + < 1. (1.4) ν πν This condition is essential in our analysis and it is not clear if the below existence result holds without it. Recall that C P is determined by the thinness of , and hence
Traveling Waves in the 3D Boussinesq System
419
(1.4) admits domains with large area which are sufficiently thin. Respectively, C P W depends on the maximum of (inner) distances between points in . On the other hand, the quantities relating to smoothness of ∂ have no direct influence on (1.4). The nonlinear Lipschitz continuous function f is assumed to be of ignition type: f (T ) = 0 on (−∞, θ0 ] ∪ [1, ∞),
f (T ) > 0 on (θ0 , 1)
for some ignition temperature θ0 ∈ (0, 1). The following is our main result: Theorem 1.1. Assume that either d = 0 or d = 1 and (1.4) holds. Then there exist 1,α c > 0, T ∈ C 2,α (D) with ∇T ∈ L 2 (D), u ∈ H 3 ∩ C 2,α (D), p ∈ Cloc (D) satisfying (1.3) and (1.2) together with: lim ||u(x, ·)||C 2 () = lim ||∇T (x, ·)|| L ∞ () = 0.
x→±∞
x→±∞
(1.5)
Moreover T (D) ⊂ [0, 1], maxx≥0,y∈ T (x, y) = θ0 , and there is a nonzero reaction: ˆ f (T ) ∈ (0, ∞). D
The limits of T satisfy: lim ||T (x, ·)|| L ∞ () = 0,
x→+∞
lim ||T (x, ·) − θ− || L ∞ () = 0
x→−∞
for some: θ− ∈ (0, θ0 ] ∪ {1}. The following sections are devoted to the proof of Theorem 1.1. In Sect. 2 we formulate some auxiliary results, of an independent interest. In particular, we prove a weak version of Xie’s conjecture [22] for the Stokes operator (established in [21] for the Laplacian). Based on results in [11], we then derive an a priori estimate valid in any channel D, whose cross-section fulfills the geometrical constraint (1.4). This allows us to obtain uniform bounds on the quantities involved in the fixed point argument (Theorem 4.2) in Sect. 3 and 4; in particular the bounds are independent of length of the compactified domains Ra = [−a, a] × . The set-up for the Leray-Schauder degree is different than in [5,10]: we solve the flow equations in the full unbounded channel D, while the reaction equation is solved in Ra . Once the uniform bounds are established, we refer to [3,10] for further details of the proofs. In Sect. 5 we improve a sufficient condition from [10] for the left limit θ− of the temperature profile T obtained in Theorem 1.1 to be equal to 1. We remark that the a priori estimates we derive do not preclude the solutions (T, u) to have arbitrary large norms. Indeed, the main chain of estimates eventually leads to inequality (4.4), whose right hand side has a linear growth in terms of the left-hand side, and thanks to condition (1.4), the main bound on u L ∞ does not restrict the magnitude of this quantity. Similar estimates are known also for solutions to the Navier-Stokes equations for the 2d and cylindrical symmetric systems [9,18,12,13], and in the presence of a special geometrical constraint on the domain [12,13]. We will always calculate all numerical constants at the leading order terms explicitly. By convention, the norms of a vector field u on D are given as: u L ∞ (D) = 1/2 1/2 3 3 i 2 i 2
u and
u =
u . 2 ∞ 2 L (D) i=1 i=1 L (D) L (D)
420
M. Lewicka, P. B. Mucha
2. Auxiliary Results In the sequel, we will need a uniform estimate for the supremum of the solution to Stokes 1/2 1/2 system in D. The known proofs of the inequality u L ∞ ≤ C ∇u L 2 Pu L 2 , P being the Helmholtz projection, are based on the a-priori estimates in [1], which hold for smooth domains. Therefore the constant C depends strongly on the boundary curvature, and becomes unbounded as tends to any domain √ with a reentrant corner. It has been conjectured by Xie [22] that actually C = 1/ 3π . To our knowledge, this is still an open question. Below we prove its weaker version, sufficient to our purpose and involving lower order terms. Theorem 2.1. Let g ∈ L 2 (D). Then the solution u ∈ H 2 ∩ H01 (D) to the Stokes system: − νu + ∇ p = g,
div u = 0 in D
(2.1)
satisfies the bound: 2 1/2 1/2
u L ∞ (D) ≤ √
∇u L 2 (D) g L 2 (D) + C ∇u L 2 (D) , 2π ν where constant C depends only on the crossection . Proof. 1. We first quote two results, whose combination will yield the proof. The first one is Xie’s inequality [21] for the Laplace operator in a 3d domain. Namely, for any u ∈ H 2 ∩ H01 (D) there holds: 1 1/2 1/2
u L ∞ (D) ≤ √ u L 2 (D) ∇u L 2 (D) . 2π
(2.2)
√ The crucial information in the above estimate is that the constant 1/ 2π is good for all open subsets of R3 . The next result is a recent commutator estimate by Liu, Liu and Pego [11]. Recall first [15] that for any vector field u ∈ L 2 (D) there exists the unique decomposition u = Pu + ∇q with div(Pu) = 0, and q solving in the sense of distributions: q = div u
in D,
∂q =0 ∂ n
on ∂ D.
This Helmholtz projection satisfies: Pu L 2 (D) ≤ u L 2 (D) . In this setting, it has been proved in [11] that for every > 0 there exists C , > 0 such that: ˆ ˆ ˆ 1 2 1 2 2 + ∀u ∈ H ∩ H0 (D) |(P − P)u| ≤ |u| + C , |∇u|2 . 2 D D D (2.3) The proof in [11], written for bounded domains, can be directly used also for the case of cylindrical domains D with smooth boundary (since the covering number for the partition of unity on ∂ D is finite). 2. Applying the Helmholtz decomposition to (2.1) we arrive at: −νPu = P g, which can be restated as: 1 −u = (P − P)u + P g, ν
Traveling Waves in the 3D Boussinesq System
421
since Pu = u. Using (2.3) we obtain:
u L 2 (D) ≤
3 1
u L 2 (D) + C ∇u L 2 (D) + P g L 2 (D) , 4 ν
which yields:
u L 2 (D) ≤
4
g L 2 (D) + C ∇u L 2 (D) . ν
Now combining (2.4) and (2.2) proves the result.
(2.4)
We will also need an extension result for divergence free vector fields. Define a compactified domain Ra = [−a, a] × . Theorem 2.2. For any a > 0 and any > 0 there exists a linear continuous extension operator E : C 1,α (Ra ) −→ C 1,α (D), such that for every u ∈ C 1,α (Ra ) there holds: (i) (Eu)|Ra = u, (ii) if div u = 0 in Ra , then div (Eu) = 0 in D, (iii) Eu L ∞ (D) ≤ (1 + ) u L ∞ (Ra ) . Proof. Given a vector field u ∈ C 1,α ([−a, 0] × ) we shall construct its extension u˜ ∈ C 1,α ([−a, ∞) × ) such that (ii) holds together with:
u ˜ L ∞ ([−a,∞)×) ≤ (1 + ) u L ∞ ([−a,0)×) .
(2.5)
This construction, being linear and continuous with respect to the C 1,α norm, will be enough to establish the lemma. Fix a large n > 0. For x ∈ [−a, a/2n 2 ] and x˜ ∈ , define the vector v(x, x) ˜ with components: 1 u (x, x) ˜ for x ∈ [−a, 0] 1 v (x, x) ˜ = ˜ + λ2 u 1 (−nx, x) ˜ + λ3 u 1 (−n 2 x, x) ˜ for x ∈ [0, a/2n 2 ], λ1 u 1 (0, x) for i = 2, 3 : i u (x, x) ˜ for x ∈ [−a, 0] i ˜ = v (x, x) i 2 i 2 −nλ2 u (−nx, x) ˜ − n λ3 u (−n x, x) ˜ for x ∈ [0, a/2n 2 ], where: λ1 =
(1 + n)(1 + n 2 ) 1 + n2 1+n , λ3 = 3 . , λ = − 2 n3 n 2 (n − 1) n (n − 1)
3 Since we have: i=1 λi = 1, −nλ2 − n 2 λ3 = 1 and n 2 λ2 + n 4 λ3 = 1, it follows 1,α 2 that v ∈ C ([−a, a/2n ] × ). Also, by an explicit calculation, we see that div u = 0 implies div v = 0. Let now φ ∈ C ∞ (R, [0, 1]) be a non-increasing cut-off function such that φ(x) = 1 for x < 0 and φ(x) = 0 for x > a/3n 2 . Define: ˆ x u(x, ˜ x) ˜ = φ(x)v(x, x) ˜ + φ (s)v 1 (s, x) ˜ ds · e1 . 0
422
M. Lewicka, P. B. Mucha
Clearly, u˜ ∈ C 1,α ([−a, ∞) × ) and div u˜ = 0 if div u = 0. Further: |u˜ 1 (x, x)| ˜ ≤ (|λ2 | + |λ3 |) u 1 L ∞ + |λ1 | · φ(x)u 1 (0, x) ˜ + (1 − φ(x)) v 1 L ∞ ≤ (|λ2 | + |λ3 |) u 1 L ∞ + |λ1 |(|λ1 | + |λ2 | + |λ3 |) u 1 L ∞ , |u˜ i (x, x)| ˜ ≤ (n|λ2 | + n 2 |λ3 |) u i L ∞ for i = 2, 3. Since λ1 → 1, |λ2 |, λ3 → 0, n|λ2 | → 1 and n 2 λ3 → 0 as n → ∞, the estimate (2.5) holds if only n is sufficiently large, which ends the proof. We remark that the norm of the operator E blows up when → 0. Indeed, one cannot have = 0 in (iii) and keep the norm of E bounded. The following elementary fact will be often used: Lemma 2.3. For any u ∈ L 2 ∩ C 0,α (D) there holds: lim x→±∞ u(x, ·) L ∞ () = 0. 3. The Bound on u L ∞ ( D) In this section, given c ∈ R, τ ∈ [0, 1], a divergence-free vector field v˜ ∈ C 1,α (D) and a boundedly supported T˜ ∈ C 1,α (D), we consider the following problem: −cu x − νu + τ d v˜ · ∇u + ∇ p = τ T˜ ρ in D, div u = 0 in D, u = 0 on ∂ D and lim x→±∞ u(x, ·) C 1 () = 0.
(3.1)
Theorem 3.1. There exists the unique u ∈ H 3 ∩ C 2,α (D) solving (3.1) with some 2 (D). Moreover: p ∈ Hloc (i) when d = 0 then u satisfies: u L ∞ (D) ≤ C ∇ T˜ L 2 (Ra ) , (ii) when d = 1 then we have: 1/2 2C P 1/2 2 |ρ · (0, x)| ˜ |ρ|C PW +
u L ∞ (D) ≤ √
v ˜ L ∞ (D) ∇ T˜ L 2 (D) ν πν + C ∇ T˜ L 2 (D) . Above, C P and C P W denote, respectively, the Poincaré and the Poincaré-Wirtinger constants of , while the constant C is independent of c, τ, a, T˜ or v. ˜ Proof. 1. The bound by ∇ T˜ . Define the quantity: 1/2 2 L= |ρ · (0, x)| ˜
and consider the following vector field with boundedly supported gradient: ˆ x q(x, x) ˜ = ρ · e1 T˜ (s, ·) ds + ρ · (0, x) ˜ T˜ (x, ·). 0
By an easy calculation we see that: T˜ (x, x) ˜ ρ − ∇q(x, x) ˜ = T˜ (x, x) ˜ −
T˜ (x, ·) ρ − ρ · (0, x) ˜
∂ ˜ T (x, ·)e1 ∂x
Traveling Waves in the 3D Boussinesq System
423
and therefore:
T˜ ρ − ∇q L 2 (D) ≤ (|ρ|C P W + L) ∇ T˜ L 2 (D) .
(3.2)
Recall that the Poincaré-Wirtinger constant C P W on is the inverse of the first nonzero eigenvalue of the related Neumann problem. By a mollification argument, we may also assume that q ∈ C 2 (D) and that (3.2) is still satisfied. 2. Existence of a weak solution. Following the Galerkin method, define:
V = cl H 1 (D) u ∈ Cc∞ (D, R3 ), div u = 0 . ´ Clearly, V is a Hilbert space with the scalar product u, wV = D ∇u : ∇w. The 1/2 norms u V := u, uV and u H 1 (D) are equivalent in V , by virtue of the Poincaré inequality in , which yields: u L 2 (D) ≤ C P u V . Since V is a subspace of H 1 (D), it is also separable and hence it admits a Hilbert ∞ (orthonormal) basis {ψn }∞ n=1 ∈ Cc (D) ∩ V . For each n, let Vn = span {ψ1 . . . ψn } and let Pn : Vn −→ Vn be given by:
ˆ ˆ ˆ n ˜ Pn (u) = νu, ψi V − c u x ψi + τ d (v˜ · ∇u)ψi − τ (T ρ − ∇q)ψi ψi . D
i=1
D
D
The operator Pn is continuous and it satisfies: ˆ ˆ ˆ Pn (u), uV = ν u 2V − c u x u + τ d (v˜ · ∇u)u − τ (T˜ ρ − ∇q)u D
D
D
≥ ν u 2V − C P T˜ ρ − ∇q L 2 (D) u V > 0 2C P ˜ when u V =
T ρ − ∇q L 2 (D) , ν ´ ´ where we used 2 D u x u = D (|u|2 )x = 0 and the nullity of the trilinear term. By Lemma 2.1.4 in [16], there exists u n ∈ Vn , bounded in V by the above quantity and solving: Pn (u n ) = 0. Since V is reflexive, it follows that {u n } converges weakly (up to a subsequence) to some u ∈ V such that: ˆ ˆ ˆ ˆ ∀w ∈ V ν ∇u : ∇w − c u x w + τ d (v˜ · ∇u)w − τ T˜ ρw = 0. (3.3) D
D
D
D
This identity follows first for w = ψn , and then by the density of the linear combinations of {ψn } in V . Taking w = u and using (3.2) we obtain:
u V ≤
CP P W + L) ∇ T˜ L 2 (D) . (|ρ|C ν
(3.4)
3. Regularity. Recall that by de Rham’s theorem (see, for example, Proposition 1.1.1 in [16]), a necessary and sufficient condition for a distribution field v ∈ D that v = ∇ p for some p ∈ D is that v, wV = 0 for all w ∈ V . Hence, (3.3) implies the first equality in (3.1) in the weak sense. By the standard regularity theory [1,16] and in view of (3.4) we may deduce now that the same equality holds in the classical sense and that u ∈ H 3 (D), ∇ p ∈ H 1 (D) (since ∇ T˜ ∈ L 2 (D)). Therefore u ∈ C 1,α (D) (for α < 1/4) and the boundary conditions in (3.1) follow, together with the asymptotic conditions as
424
M. Lewicka, P. B. Mucha
|x| → ∞, in view of Lemma 2.3. Next, recalling that T˜ ∈ C 1,α (D), the potential theory [8] employed to the localized problem and the classical Schauder estimates give that u ∈ C 2,α (D). 4. The bound on cu x . Since ∇q has a bounded support and is C 1 , thus ∇( p−q) ∈ H 1 (D). Consequently: ˆ ˆ ˆ u x ∇( p − q) = (u∇( p − q))x − div (u( p − q)x ) = 0, D
D
D
because in view of u ∈ H 3 (D) and ∇( p − q) ∈ H 1 (D) one has: ˆ lim
|x|→∞
ˆ |u∇( p − q)|(x, ·) +
|u ( p − q)x |(x, ·) = 0. 1
Integrating the first equality in (3.1) against cu x on D we obtain: ˆ ˆ ˆ 2 ˜ ∇u : ∇u x − cu x (τ T ρ − τ ∇q) + τ d cu x (v˜ · ∇u)
cu x L 2 (D) = νc D D D ≤ cu x L 2 (D) T˜ ρ − ∇q L 2 (D) + d v˜ · ∇u L 2 (D) , where we have once more used that u ∈ H 3 (D). Therefore, by (3.2): P W + L) ∇ T˜ L 2 (D) + d v˜ · ∇u L 2 (D) .
cu x L 2 (D) ≤ (|ρ|C
(3.5)
5. The bound and uniqueness for d = 0. It is now easy to conclude the proof, when d = 0. Using the standard elliptic estimates for the Stokes system (2.1), following from the theory in [1], and the Sobolev interpolation inequality, it follows that:
u L ∞ (D) ≤ C u H 2 (D) u H 1 (D) ≤ C ∇ T˜ L 2 (D) , 1/2
1/2
in virtue of (3.2), (3.4) and (3.5). The constant C is uniform and depends only on the geometry of , and the constants |ρ| and ν. Uniqueness of u also follows from the above bound. 6. The case of d = 1. Denote: g = cu x − τ v˜ · ∇u + (τ T˜ ρ − τ ∇q). By (3.2) and (3.5) we obtain that:
g L 2 (D) ≤ 2 v ˜ L ∞ (D) ∇u L 2 (D) + 2 (|ρ|C P W + L) ∇ T˜ L 2 (D) . Therefore, Theorem 2.1 implies: 2 1/2
u L ∞ (D) ≤ √ v ˜ L ∞ (D) ∇u L 2 (D) πν 2 1/2 1/2 +√ P W + L)1/2 ∇ T˜ L 2 (D) ∇u L 2 (D) + C ∇u L 2 (D) , (|ρ|C πν which by (3.4) establishes the result.
Traveling Waves in the 3D Boussinesq System
425
4. The Uniform Bounds and Existence of Traveling Waves In this section we prove the uniform bounds on solutions to system (1.3), and then establish existence of a traveling wave in (1.1) by a Leray-Schauder degree argument. Given c ∈ R, τ ∈ [0, 1], a divergence-free vector field v ∈ C 1,α (Ra ) and Z ∈ C 1,α (Ra ) consider first the reaction-advection-diffusion problem: − cTx − T + τ v · ∇T = τ f (Z )
in Ra ,
T (−a, x) ˜ = 1, T (a, x) ˜ = 0 for x˜ ∈ , ∂T (x, x) ˜ = 0 for x ∈ [−a, a] and x˜ ∈ ∂, ∂ n
(4.1)
together with the following normalization condition, whose eventual role is to single out a correct approximation of the traveling wave in T , in the moving frame which chooses to have f (T (x, ·)) = 0 for x ≥ 0: (4.2) max T (x, x); ˜ x ∈ [0, a], x˜ ∈ = θ0 . We now recall the bounds on solutions to the above problems, proved in [3] and used in [5,10]. The right-hand sides of (iii), (iv) and (v) follow by re-examining the proofs. Theorem 4.1. Let T = Z ∈ C 1,α (Ra ) satisfy (4.1) and (4.2). Then one has: T (x, x) ˜ ∈ [0, 1] for all (x, x) ˜ ∈ Ra , T (x, x) ˜ ≤ θ0 for all x > 0, x˜ ∈ , 1/2 |c| ≤ v L ∞ (Ra ) + 2 f L ∞ ([0,1]) ,
∇T 2L 2 (R ) ≤ || 27 c − v 1 L ∞ (Ra ) + a1 , a ´ (v) Ra f (T ) ≤ || 4 c − v 1 L ∞ (Ra ) + a1 .
(i) (ii) (iii) (iv)
Given T ∈ C 1,α (Ra ) satisfying boundary conditions as in (4.1), we will consider its boundedly supported C 1,α (D) extension: ⎧ ˜ ⎨ T (x, x) for x ∈ [−a, a], ˜ for x < −a, (4.3) T˜ (x, x) ˜ = φ(|x| − a) · 2 − T (−2a − x, x) ⎩ −φ(|x| − a) · T (2a − x, x) ˜ for x > a. Here φ ∈ C ∞ (R, [0, 1]) satisfies φ(x) = 1 for x < 1/3, φ(x) = 0 for x > 2/3 and
∇φ L ∞ ≤ 4. Also, for a divergence-free v ∈ C 1,α (Ra ), let v˜ ∈ C 1,α (D) be its divergence-free extension, given in Theorem 2.2. Theorem 4.2. Let c ∈ R, τ ∈ [0, 1], T = Z ∈ C 1,α (Ra ), v ∈ C 1,α (Ra ) and u ∈ C 1,α (D) satisfy (3.1), (4.1), (4.2), with T˜ and v˜ defined as above. Moreover, let u |Ra = v and assume that either d = 0, or d = 1 and (1.4) holds. Then, for large a: ˆ |c| + T C 1,α (Ra ) + u C 2,α (Ra ) + ∇T L 2 (Ra ) + u H 3 (Ra ) + f (T ) ≤ C, Ra
where C is a numeric constant independent on a, τ and the estimated quantities.
426
M. Lewicka, P. B. Mucha
Proof. By Theorem 4.1 (iii) and (iv) we obtain: || 1/2
∇T 2L 2 (R ) ≤ 7|| u L ∞ (D) + f L ∞ ([0,1]) + . a a On the other hand, the boundary conditions for T imply that, for large a: √ √ 1/2 1/4
∇ T˜ L 2 (D) ≤ 2 ∇T L 2 (Ra ) + 8 ≤ 14||1/2 u L ∞ (D) + f L ∞ ([0,1]) + 9. Consequently, for d = 0 the uniform bound on u L ∞ (D) follows by Theorem 3.1 (i). When d = 1 then Theorem 3.1 (ii) yields the same bound, under condition (1.4). Indeed,√ let > 0 be such that the quantity in the left hand side of (1.4) is strictly smaller than 1/ 1 + . Then: 1/2 √ 2C P 2
u L ∞ (D) ≤ 1 + √ |ρ · (0, x)| ˜ |ρ|C PW + ν πν √ 1/2 1/4 1/2 × 14||1/2 u L ∞ (D) + f L ∞ ([0,1]) + C u L ∞ (D) 1/2 1/4 + C u L ∞ (D) + f L ∞ ([0,1]) + 1 1/2 1/4 (4.4) ≤ q u L ∞ (D) + C u L ∞ (D) + 1 f L ∞ ([0,1]) + 1 , 1/2 for some q ∈ (0, 1). Hence u L ∞ (D) ≤ C f L ∞ ([0,1]) + 1 and so, recalling (3.4) and Theorem 3.1, we have: ˆ 1/2 2 ∞ f (T ) ≤ C f L ∞ ([0,1]) + 1 . (4.5) |c| + u L (D) + u H 1 (D) + Ra
In (4.4) and (4.5) the constant C is independent of a, τ , the nonlinearity f and the estimated quantities. ´ The uniform bounds on u H 2 (D) , |c|, ∇T L 2 (Ra ) and Ra f (T ) follow by Theorem 4.1, (3.4) and (2.4). Now, the standard local elliptic estimates for the Stokes system (2.1) (see [1], also compare [10]) imply that:
u H 3 (D) ≤ C( ∇g L 2 (D) + u H 1 (D) ). we obtain the uniform bound on u H 3 (D) . The Taking g = cu x − τ v˜ · ∇u + τ T˜ ρ, bounds on u C 2,α (D) and T C 1,α (Ra ) follow by Hölder’s estimates for system (3.1) and (4.1) [1,7]. The proof is done. We remark that our result does not imply smallness of C in the uniform bound. In particular, C depends on the constant C , from Theorem [11], which can be arbitrarily large. We finally have: Proof of Theorem 1.1. For every sufficiently large a > 0, consider an operator: K a : R × C 1,α (Ra ) × Cd1,α (Ra ) × [0, 1] −→ R × C 1,α (Ra ) × Cd1,α (Ra ), where Cd1,α (Ra ) stands for the Banach space of the divergence-free, C 1,α regular vector fields on the compact domain Ra . Define: ˜ x ∈ [0, a], x˜ ∈ }, T, u |Ra , K a (c, Z , v, τ ) := c − θ0 + max{T (x, x);
Traveling Waves in the 3D Boussinesq System
427
where T is the solution to (4.1), and u solves (3.1) are known also with T˜ and v˜ defined as in (4.3) and Theorem 2.2. The operator K a is continuous, compact [7] and all its fixed points (c, T, v) such that K a (c, T, v, τ ) = (c, T, v) for some τ ∈ [0, 1] are uniformly bounded, in view of Theorem 4.2. We may now employ the Leray-Schauder degree theory, as in [3,5,10], to obtain the existence of a fixed point of K a (·, ·, ·, 1), since the degree of the map K a (·, ·, ·, 0) is nonzero. This fixed point (ca , T a , v a ) again satisfies the bounds in Theorem 4.2. By a bootstrap argument we moreover obtain the uniform bound on T a C 2,α (Ra−1 ) . One may thus choose a sequence an → ∞ such that cn := can converges to some 2,α c ∈ R, and Tn := T an , vn := v an converge in Cloc (D) to some T, u ∈ C 2,α (D). 2 2,α Further, u ∈ H (D) ∩ C (D) and hence the first convergence in (1.5) follows. Since ∇T ∈ L 2 ∩ C 0,α (D), we obtain the other convergence in view of Lemma 2.3. The positivity of the propagation speed c and the existence of the right and left limits of T , together with the statement in (ii) follow exactly as in [10]. 5. A Sufficient Condition for θ− = 1 The following lemma improves on the result in [10], where a sufficient condition for the left limit θ− of T to be 1 required a cubic bound: f (T ) ≤ k[(T − θ0 )+ ]3 . Lemma 5.1. In the setting of Theorem 1.1, if moreover the nonlinearity satisfies: f (T ) ≤ k[(T − θ0 )+ ]2 and k 1 + f L ∞ ([0,1]) + f L ∞ ([0,1]) ≤ C , (5.1) then θ− = 1. Here C > 0 is a constant, depending only on ν, ρ and . Proof of Theorem 1.1. 1. In the course of the proof, C will denote any positive constant depending only on ν, ρ and . Integrating the temperature equation in (1.3) against T and on D yields, respectively: ˆ ˆ 1
∇T 2L 2 (D) = f (T )T − cθ−2 || ≤ f (T ), (5.2) 2 D D
T L 2 (D) ≤ f (T ) L 2 (D) + u · ∇T L 2 (D) , (5.3) ´ where in (5.3) we used that D Tx T = 0. The interpolation, Hölder and Sobolev inequalities imply that: 1/2
1/2
u · ∇T L 2 (D) ≤ u L 6 (D) ∇T L 3 (D) ≤ u L 6 (D) ∇T L 2 (D) ∇T L 6 (D) ≤
C
u 2H 1 (D) ∇T L 2 (D) + ∇T H 1 (D) .
Now, taking above sufficiently small and introducing (5.3) and (5.4) into:
∇T H 1 (D) ≤ C ∇T L 2 (D) + T L 2 (D) we obtain, in view of (5.2):
∇T H 1 (D) ≤ C ∇T L 2 (D) + f (T ) L 2 (D) + u 2H 1 (D) ∇T L 2 (D) 1/2 ˆ 1/2 ≤ C 1 + f L ∞ ([0,1]) + u 2H 1 (D) f (T ) . D
(5.4)
428
M. Lewicka, P. B. Mucha
By (4.5) and convergences established in the proof of Theorem 1.1, this implies: 1/2 ˆ 1/2 1/2 f (T ) . (5.5)
∇T H 1 (D) ≤ C 1 + f L ∞ + f L ∞ D
2. Now, for every x ∈ R denote M(x) = max x∈ T (x, x), ˜ m(x) = min x∈ T (x, x) ˜ ˜ ˜ and notice that m(x) is non-increasing. This can be proved for each Tn on Rn , using the maximum principle. Passing with n to ∞, one obtains the same result in the limit. We now argue by contradiction. If θ− ≤ θ0 then m(x) ≤ θ0 for every x ∈ R and: ˆ ˆ +∞ 2 [(T − θ0 )+ ] ≤ || |M(x) − m(x)|2 dx D −∞ ˆ +∞ ≤ 2||
T (x, ·) − T (x, ·) L ∞ () dx ≤ C ∇T 2H 1 (D) . −∞
Together with (5.5) and the assumption in (5.1) the above yields: ˆ ˆ 2 [(T − θ0 )+ ] ≤ C 1 + f L ∞ + f L ∞ k [(T − θ0 )+ ]2 , D
D
which by the assumption on k implies that both sides above must be zero. Consequently, f (T ) ≡ 0 and one can deduce (see [3,10]) that T ≡ 0 as well, contradicting the results of Theorem 1.1. The condition (5.1) seems to be artificial and we believe that it can be further relaxed or even omitted altogether, for the wave (T, u) obtained in the limiting procedure of Theorem 1.1. Acknowledgements. M.L. was partially supported by the NSF grant DMS-0707275 and by the Center for Nonlinear Analysis (CNA) under the NSF grants 0405343 and 0635983. P.B.M. has been supported by MNiSW grant No. N N201 268935 and by ECFP6 M. Curie ToK program SPADE2, MTKD-CT-2004-014508 and SPB-M.
References 1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35–92 (1964) 2. Berestycki, H.: The influence of advection on the propagation of fronts in reaction-diffusion equations. Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H. Berestycki, Y. Pomeau, eds, Dordrecht: Kluwer 2003 3. Berestycki, H., Constantin, P., Ryzhik, L.: Non-planar fronts in Boussinesq reactive flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(4), 407–437 (2006) 4. Constantin, P., Kiselev, A., Ryzhik, L.: Fronts in reactive convection: bounds, stability and instability. Comm. Pure Appl. Math. 56, 1781–1803 (2003) 5. Constantin, P., Lewicka, M., Ryzhik, L.: Traveling waves in 2D reactive Boussinesq systems with no-slip boundary conditions. Nonlinearity 19, 2605–2615 (2006) 6. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Springer Tracts in Natural Philosophy, 38. 39, Berlin-Heidelberg-NewYork: Springer, 1998 7. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Berlin: Springer-Verlag, 2001 8. Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, London-NewYork: Academic Press, 1968
Traveling Waves in the 3D Boussinesq System
429
9. Ladyzhenskaya O.A.: On unique solvability of three-dimensional Cauchy problem for the Navier–Stokes equations under the axial symmetry. Za. Nauchn. Sem. LOMI 7, 155–177 (1968) (in Russian) 10. Lewicka, M.: Existence of traveling waves in the Stokes-Boussinesq system for reactive flows. J. Diff. Eq. 237(2), 343–371 (2007) 11. Liu, J.-G., Liu, J., Pego, R.L.: Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate. Comm. Pure Appl. Math. 60(10), 1443–1487 (2007) 12. Mucha, P.B.: On a problem for the Navier-Stokes equations with the infinite Dirichlet integral. Z. Angew. Math. Phys. 56(3), 439–452 (2005) 13. Mucha, P.B.: On cylindrical symmetric flows through pipe-like domains. J. Diff. Eqs. 201(2), 304–323 (2004) 14. Roquejoffre, J.M.: Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders. Ann. Inst. Henri Poincare 14(4), 499–552 (1997) 15. Sohr, H.: The Navier-Stokes equations. An elementary functional analytic approach. Basel: Birkhäuser Verlag, 2001 16. Temam, R.: Navier-Stokes equations. Theory and numerical analysis. AMS Chelsea Publishing, Providence, RI, Amer. Math. Soc., 2001 17. Texier-Picard, R., Volpert, V.: Problemes de reaction-diffusion-convection dans des cylindres non bornes. C. R. Acad. Sci. Paris Sr. I Math. 333, 1077–1082 (2001) 18. Ukhovskij, M.R., Yudovich, V.I. Axially symmetric motions of prefect and viscous fluids filling all space. Prihl. Math. Mekh. 32, 59–69 (1968) (in Russian) 19. Vladimirova, N., Rosner, R.: Model flames in the Boussinesq limit: the efects of feedback. Phys. Rev. E. 67, 066305 (2003) 20. Vladimirova, N., Rosner, R.: Model flames in the Boussinesq limit: the case of pulsating fronts. Phys. Rev. E. 71, 067303 (2005) 21. Xie, W.: A sharp pointwise bound for functions with L 2 -Laplacians and zero boundary values of arbitrary three-dimensional domains. Indiana Univ. Math. J. 40(4), 1185–1192 (1991) 22. Xie, W.: On a three-norm inequality for the Stokes operator in nonsmooth domains. In: The NavierStokes equations II—theory and numerical methods, Springer Lecture Notes in Math. Vol. 1530, BerlinHeidelberg-Newyork: Springer, 1992, pp. 310–315 23. Xin, J.: Front propagation in heterogeneous media. SIAM Review 42(2), 161–230 (2000) 24. Zeldovich, Ya.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M.: The Mathematical Theory of Combustion and Explosions. New York: Consultants Bureau, 1985 Communicated by P. Constantin
Commun. Math. Phys. 292, 431–456 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0896-z
Communications in
Mathematical Physics
The Third Order Helicity of Magnetic Fields via Link Maps R. Komendarczyk Department of Mathematics, Univ. of Pennsylvania, Philadelphia, PA 19104, USA. E-mail: [email protected], URL: www.math.upenn.edu/~rako Received: 5 September 2008 / Accepted: 4 June 2009 Published online: 23 August 2009 – © Springer-Verlag 2009
Abstract: We introduce an alternative approach to the third order helicity of a volume preserving vector field B, which leads us to a lower bound for the L 2 -energy of B. The proposed approach exploits correspondence between the Milnor µ¯ 123 -invariant for 3-component links and the homotopy invariants of maps to configuration spaces, and we provide a simple geometric proof of this fact in the case of Borromean links. Based on these connections we develop a formulation for the third order helicity of B on invariant unlinked domains of B, and provide Arnold’s style ergodic interpretation of this invariant as an average asymptotic µ¯ 123 -invariant of orbits of B. 1. Introduction A purpose of this paper is to develop a particular formula for the third order helicity on certain invariant sets of a volume preserving vector field B. The third order helicity, [3], is an invariant of B under the action of volumorphisms isotopic to the identity (denoted here by SDiff0 (M)). Importance of such invariants stems from the basic fact that the evolution of the vorticity in the ideal hydrodynamics or of the magnetic field B0 in the ideal magnetohydrodynamics (MHD), occurs along a path t −→ g(t) ∈ SDiff0 (M), [3, p. 176]. Namely, B(t) = g∗ (t)B0 which is a direct consequence of Euler’s equations: d B + [v, B] = 0, dt
d g(t) = v. dt
(1.1)
One often says that the magnetic field B is frozen in the velocity field v of plasma, and the action by SDiff0 (M) is frequently referred to as frozen-in-field deformations. A fundamental example of such an invariant defined for a general class in SVect(M) is the helicity H12 (B1 , B2 ) defined for a pair of vector fields B1 and B2 on M = S 3 or a homology sphere. Helicity has been first introduced by Woltjer, [39], in the context of This project is supported by DARPA, #FA9550-08-1-0386.
432
R. Komendarczyk
magnetic fields, and is a measure of how orbits of B1 , and B2 link with each other. This topological interpretation of helicity has been made precise by Arnold, who introduced the concept of the average asymptotic linking number of a volume preserving vector field B on M, [2]. The subject has been further investigated in [1,5,25,36] (see [3] for additional references), where authors approach higher helicities via the Massey products under various assumptions about the vector fields or their domains (work in [16,35,37] concerns yet other approaches to the problem). Extensions of the helicity concept to higher dimensional foliations can be found in [18,23,33] and recently in [10]. In this paper we present an alternative to these approaches, which is a natural extension of the notion of the linking number as a degree of a map, and exploits relations to the homotopy theory of certain maps associated to the link. The paper has two parts. In the first part we show how the Milnor µ¯ 123 -invariant: µ¯ 123 (L) of a parametrized Borromean link L in S 3 can be obtained as a Hopf degree of an associated map to the configuration space of three points in S 3 . The presented approach to link homotopy invariants of 3-component links has been proposed in [20] for n-component links in R3 , and conveniently simplified for 3-component links in S 3 in the joint work [12], where the full correspondence between µ¯ 123 (L) and the Pontryagin-Hopf degree is proved. In Sect. 2 we present the original proof of this correspondence in the Borromean case, which is sufficient for our purposes. The second part of the paper discusses a new definition of the third order helicity denoted here by H123 (B; T ), where B is a volume preserving vector field having an invariant unlinked domain T ⊂ S 3 . The simplest of such domains are three invariant handlebodies in S 3 which have pairwise unlinked cycles in the first homology. This includes the case of Borromean flux tubes already investigated in [5,25]. In Sect. 5 we develop an ergodic formulation of H123 (B; T ) as an average asymptotic µ¯ 123 -invariant, in the spirit of Arnold’s average asymptotic linking number, which allows us to extend the definition of the invariant to topologically more complicated unlinked domains. We also derive, in Sect. 7, a lower bound for the L 2 -energy of B in terms of H123 (B; T ).
2. The Milnor µ123 -invariant and the Hopf degree The µ-invariants ¯ of n-component links in S 3 have been introduced by Milnor in [28,29] as invariants of links up to link homotopy. Recall that the link homotopy is a deformation of a link in S 3 which allows each component to pass through itself but not through a different component. Clearly, this is a weaker equivalence than the equivalence of links up to isotopy where components are not allowed to pass through themselves at all. The fundamental example of a µ-invariant ¯ is the linking number (denoted by µ¯ 12 ) which is a complete invariant of the 2-component links up to link homotopy. In the realm of 3-component links the relevant invariants are the pairwise linking numbers µ¯ 12 , µ¯ 23 , µ¯ 32 , and the third invariant µ¯ 123 in Zgcd(µ¯ 12 ,µ¯ 23 ,µ¯ 32 ) , which is a well defined integer, if and only if, µ¯ 12 = µ¯ 23 = µ¯ 32 = 0. In the second part of the paper we will interpret this statement as a topological condition on the invariant set of a vector field. A precise definition of µ-invariants ¯ is algebraic and involves the Magnus expansion of the lower central series of the fundamental group: π1 (S 3 − L) of the link complement. We refer the interested reader to the works in [28,29]. In the remaining part of this section we will prove that µ¯ 123 (L) is a Hopf degree for an appropriate map associated to the link L, provided that the link is Borromean, i.e. the pairwise linking numbers are zero (note that the Borromean links are more general then Brunnian links, [28]).
Third Order Helicity of Magnetic Fields via Link Maps
433
Let us review basic facts about the Hopf degree H ( f ) of a map f : S 3 −→ S 2 , (see e.g. [6]). A well known property of the Hopf degree is that H : f −→ H ( f ) provides an isomorphism between π3 (S 2 ) and Z. Recall that up to a constant multiple we may express H ( f ) as (M = S 3 ), α ∧ f ∗ν = α∧ω = α ∧ dα, (2.1) H (f) = M
M
M
where ν is the area 2-form on S 2 , and α satisfies ω = f ∗ ν = dα. Notice that f ∗ ν is always exact since the cohomology of S 3 in dimension 2 vanishes. We may also interpret H ( f ) as an intersection number, [6,30]. Namely, consider two regular values p1 and p2 ∈ S 2 of the map f , then l1 = f −1 ( p1 ) and l2 = f −1 ( p2 ) form a link in S 3 , and the integral formula (2.1) can be interpreted as the intersection number of l1 with the Seifert surface spanning l2 : H ( f ) = lk(l1 , l2 ).
(2.2)
If we replace S 3 with an arbitrary closed compact orientable 3-dimensional manifold M, we may still obtain an invariant of f : M −→ S 2 this way, provided that the condition f ∗ ν = dα holds. Proposition 2.1. Let M be a closed Riemannian manifold, and ν ∈ 2 (S 2 ) the area form on S 2 . The formula (2.1) provides a homotopy invariant for a map f : M −→ S 2 , if the 2-form f ∗ ν is exact. Up to a constant multiple this invariant can be calculated as an intersection number defined in (2.2), where l1 = f −1 ( p1 ) and l2 = f −1 ( p2 ) form a link in M, where both l1 and l2 are null-homologous. Proof. Given a homotopy F : I × M → S 2 , f 1 = F(1, · ), f 0 = F(0, · ), we define ωˆ = F ∗ ν. We have ωˆ = F ∗ ν = d α, ˆ
ω1 = f 1∗ ν = i 1∗ F ∗ ν = d α1 ,
ω0 = f 0∗ ν = i 0∗ F ∗ ν = d α0 ,
where i 0 : M → M × I , i 0 (x) = (x, 0), i 1 : M → M × I , i 1 (x) = (x, 1) are appropriate inclusions. Potentials: αˆ {0}×M , α0 , and αˆ {1}×M , α1 differ by a closed form αˆ {0}×M −α0 = β0 , αˆ {1}×M −α1 = β1 , dβ0 = dβ1 = 0,
therefore the Stokes Theorem immediately implies that Formula (2.1) is independent of the choice of the potential. For the proof of invariance under homotopies we revoke the standard argument in [6, p. 228] ν∧ν = ωˆ ∧ ωˆ = ωˆ ∧ d αˆ 0= F(M×I ) M×I M×I = d(ωˆ ∧ α) ˆ = ωˆ ∧ αˆ = ω1 ∧ α1 − ω0 ∧ α0 M×I ∂(M×I ) M M = dα1 ∧ α1 − dα0 ∧ α0 = H ( f 1 ) − H ( f 0 ). M
M
The interpretation of H ( f ) as the intersection number (2.2) is the same as in [6, p. 230].
434
R. Komendarczyk
Given a 3-component parametrized link L = {L 1 , L 2 , L 3 } in S 3 we wish to associate a certain map FL : S 1 × S 1 × S 1 → S 2 to it, and interpret its Hopf degree as the Milnor µ¯ 123 -invariant. Recall the definition of the configuration space of k points in M: Confk (M) := {(x1 , x2 , . . . , xk ) ∈ (M)k | xi = x j , for i = j}. As an introduction to the method we review the Gauss formula for the linking number of a 2-component link L = {L 1 , L 2 } in R3 . Denote parameterizations of components by L 1 = {x(s)}, L 2 = {y(t)} and consider the map r
L
FL : S 1 × S 1 −→ Conf2 (R3 ) −→ S 2 ,
L(s, t) = (x(s), y(t)),
x−y where r (x, y) = x−y
is a retraction of Conf2 (R3 ) onto S 2 . It yields the classical Gauss linking number formula: µ¯ 12 (L) = lk(L 1 , L 2 ) = deg(FL ), deg(FL ) = FL∗ (ν), S 1 ×S 1
where ν ∈ 2 (S 2 ) is the area form on S 2 . Consequently, the linking number lk(L 1 , L 2 ), also known as the Milnor µ¯ 12 -invariant, can be obtained as the homotopy invariant of the map FL associated to L. Observe that homotopy classes [S 1 × S 1 , S 2 ] are isomorphic to Z and deg : F → deg(F) provides the isomorphism; we also point out that as sets: [S k , Confn (R3 )] = πk (Confn (R 3 )), and [S k , Confn (S 3 )] = πk (Confn (S 3 )) (cf. [17, p. 421]). Thus considering the based homotopies, in the context of the link homotopy of Borromean links, and base point free homotopies is equivalent in this setting. In [20,21], the authors consider a natural extension of this approach to n-component parame-trized links L in R3 by considering maps FL : S 1 × . . . × S 1 −→ Confn (R3 ) and their homotopy classes, we refer to this type of maps loosely as link maps, (cf. [21]). In particular, Kohno [20] proposed specific representatives of cohomology classes of the based loop space of Confn (R3 ) as candidates for appropriate link homotopy invariants of L. It has been observed, in [12], that in the 3-component case it is beneficial to consider Conf3 (S 3 ), and L ⊂ S 3 , since the topology of Conf3 (S 3 ) simplifies dramatically (in comparison to Conf3 (R3 )). We review this simplification in the following paragraph as it is essential for the proof of the main theorem in this section. Consider a 3-component link L = {L 1 , L 2 , L 3 } in S 3 parametrized by {x(s), y(t), z(u)} and the following map: H
L
FL : S 1 × S 1 × S 1 −→ Conf3 (S 3 ) −→ S 2 ,
L(s, t, u) = (x(s), y(t), z(u)), (2.3)
where we denote by H : S 3 × R3 × (R3 \ {0}) → S 2 the projection on the S 2 factor, first concluding that Conf3 (S 3 ) ⊂ S 3 × S 3 × S 3 is diffeomorphic to S 3 × Conf2 (R3 ) = S 3 × R3 × (R3 \ {0}), and consequently deformation retracts onto S 3 × S 2 . Considering S 3 as unit quaternions, the map H can be expressed explicitly by the formula, [12]: H
Conf3 (S 3 ) (x, y, z) −→
pr(x−1 · y) − pr(x−1 · z) ∈ S2,
pr(x−1 · y) − pr(x−1 · z)
(2.4)
Third Order Helicity of Magnetic Fields via Link Maps
435
where · stands for the quaternionic multiplication, −1 is the quaternionic inverse, and pr : S 3 −→ R3 the stereographic projection from 1. As a result one has the following particular expression for FL : FL (s, t, u) =
pr(x(s)−1 · z(u)) − pr(x(s)−1 · y(t)) .
pr(x(s)−1 · z(u)) − pr(x(s)−1 · y(t))
(2.5)
At this point we note that one has freedom in choosing the deformation retraction H in (2.4), but the above particular formula makes the proof of the main theorem of this section possible. Let T = S 1 × S 1 × S 1 denote the domain of FL ; notice that, thanks to (2.5), restricting FL to the subtorus T23 in the second and third coordinate (t, u) of T, we obtain the usual Gauss map of the 2-component link {x−1 · L 2 , x−1 · L 3 }. Since the diffeomorphism x−1 · of S 3 is orientation preserving we conclude deg(FL |T23 ) = lk(L 2 , L 3 ). We claim that for any 2-component sublink {L i , L j } of L: deg(FL |Ti j ) = ±lk(L i , L j ),
1 ≤ i < j ≤ 3,
(2.6)
where i, j index the coordinates of T. Indeed, since it is already true for i = 2 and j = 3, the general case follows by applying a permutation σ ∈ 3 of coordinate factors in Conf3 (S 3 ) ⊂ (S 3 )3 . Notice that σ is a diffeomorphism of Conf3 (S 3 ) ⊂ (S 3 )3 either preserving or reversing the orientation (which explains the sign in (2.6)). We infer (2.6) because σ induces an isomorphism on homotopy groups of Conf3 (S 3 ). The main theorem of this section is Theorem 2.2. Let L = {L 1 , L 2 , L 3 } be a 3-component Borromean link in S 3 , consider the associated map FL defined in (2.3). The Hopf degree of this map satisfies H (FL ) = ±2 µ¯ 123 ,
(2.7)
where the sign depends on the choice of orientations of components of L. Proof. By the Borromean rings we understand any 3-component link with µ¯ 123 = ±1. Every such link is link homotopic to the diagram presented on Fig. 1 (where the sign can be determined from the orientation of components). The proof of the theorem can be reduced to the case of Borromean rings L Borr , as follows: if L and L are link-homotopic then FL and FL are homotopic maps. By the Milnor classification of 3-component links up to link homotopy (see [28]), every 3-component link L with zero pairwise linking numbers and µ¯ 123 = ±n is represented by the right diagram on Fig. 1. Consequently, up to homotopy, the associated map FL can be obtained from FL Borr by covering one of the S 1 factors in T, n-times. Therefore, in order to prove the claim it suffices to show H (FL Borr ) = ±2.
(2.8)
According to Proposition 2.1, H (FL ) is well defined for a link L ⊂ S 3 provided FL∗ ν ∈ 2 (T) is trivial in H 2 (T), which is true thanks to (2.6) and because the pairwise linking numbers of L are zero. The method of proof relies on a direct calculation of H (FL Bor ), for a carefully chosen parametrization of L Bor in S 3 . This calculation is achieved by visualization of the link l S,N = l S ∪ l N in T, and application of Formula (2.2), where l S := FL−1 Bor (S),
l N := FL−1 Bor (N )
436
R. Komendarczyk
Fig. 1. left: µ¯ 123 = ±1 and right: µ¯ 123 = ±n
Fig. 2. The model of L Borr parametrized by {L 1 = pr(x(s)), L 2 = pr(y(t)), L 3 = pr(z(u))}. Arc A1 is a part of the unit circle on xy-plane, A2 is a part of the circle of radius 1 +
are preimages of the North pole N = (0, 0, 1) and South pole S = (0, 0, −1) in S 2 ⊂ R3 . Notice that [FL∗ ν] = 0 in H 2 (T), if and only if, [l S ] = 0 and [l N ] = 0 in H1 (T). We begin by identifying S 3 with the set of unit quaternions in R4 with standard coordinates: (w, x, y, z) = w + x i + y j + z k, and choosing a specific parametrization of the Borromean rings L Bor in S 3 . That is, define the L 1 component of L Bor to be the great circle in S 3 through 1 and k, parametrized as x(s) = cos(s) + sin(s) k, x(s)−1 = cos(s) − sin(s) k . Observe that pr(x(s)) parameterizes the z-axis in R3 . Figure 2 shows how to define the second and the third component {L 2 , L 3 } of the Borromean rings L Borr in R3 considered as an image of S 3 − {1} under the stereographic projection pr : S 3 ⊂ R4 −→ R3 from 1 ∈ S 3 . L 2 will bound the annuli with a rounded wedge removed, i.e. an arc A1 of the circle of radius 1. The arc A2 belongs to the circle of radius r = (1 + ) in the (x, y)-plane. The component L 3 is chosen to be a vertical ellipse linking with L 2 . Next we focus on the Formula (2.5); observe that multiplication by x(s)−1 has an effect of a rotation by angle s in the (w, z)-plane and (x, y)-plane of R4 , which can be
Third Order Helicity of Magnetic Fields via Link Maps
437
Fig. 3. Four positions corresponding to angles: s = 0, π2 , π, 3π 2 , small arrows next to N and S indicate a motion as s
directly calculated: x(s)−1 · (w, x, y, z) = (cos(s)w + sin(s)z, cos(s)x + sin(s)y, cos(s)y − sin(s)x, cos(s)z − sin(s)w) . The flow defined by this S 1 -action is tangent to the great circles of S 3 , thus the projected flow on R3 , via the stereographic projection pr, presents the standard picture of the Hopf fibration. Let us call an invariant Hopf torus an r -torus, if and only if, it contains a circle of radius r in the (x, y)-plane. Without loss of generality we assume that L 2 on Fig. 2 belongs to the r /2 -Hopf torus. Every point on a r -torus traces a (1, 1)-curve under the S 1 -action. For sufficiently small , this motion can be regarded as a composition of the rotation by angle s in both the direction of the meridian and the longitude of a r -torus. Therefore, for different values of s the S 1 -action “rotates” the components L 2 and L 3 , by sliding along the Hopf tori by angle s in the meridian and the longitudinal direction. We denote resulting link components by L s2 (t) = pr(x−1 (s) · y(t)),
and
L s3 (u) = pr(x−1 (s) · z(u)).
The unit circle on the (x, y)-plane is left invariant under this action and therefore can be considered as the “axis of the rotation”. This justifies the choice of the particular shape of L Borr pictured on Fig. 2. Next, we seek to visualize the projection of l S,N = l S ∪ l N on the su-face and st-face of the domain T of FL Bor parameterized by (s, t, u) ∈ T, (it is convenient to think about T as a cube in (s, t, u)-coordinates, see Fig. 4). For example when s = 0, (x(0) = 1), a point (0, t0 , u 0 ) belongs to l N , if and only if, the vector v0 = pr(y(t0 )) − pr(z(u 0 )) points in the direction of N = (0, 0, 1), the analogous condition holds for direction S = (0, 0, −1), and l S . In order to determine a diagram of l S,N , we must keep track of the “head” and “tail” of the vector vs = L s2 (t) − L s3 (u), for various values of s and record values of t and u for which vs points “North” and “South” (Fig. 3). This reads as the following condition: (s, t, u) ∈ l S,N ,
if and only if,
vs S or N .
(2.9)
Without loss of generality we assume that L s2 is parametrized by the unit t-interval, and L s3 is parametrized by the unit u-interval. The process of recording values of u and t such that (2.9) holds is self-explanatory and is shown on Fig. 2 for values s = 0, π2 , π, 3π 2 , which is sufficient to draw projections of l S and l N on st- and tu-faces of T. Collecting the information on Fig. 3, we draw the projection of l S,N on the su-face of T represented by square (A) in Fig. 4. Analogously, the projection of l S,N on the st-face of T is obtained
438
R. Komendarczyk
(A)
(B)
(C)
(D)
Fig. 4. Projection of l S,N on the su-face and st-face of T. The strands l S (solid line) and l N (dashed line) are oppositely oriented since l S and l N are null homologous in T
and pictured in square (B). In order to obtain the diagram of l S,N we resolve the double points of Diagram (A) into crossings. For example, let us resolve the “circled” double point on (A), which occurs at s = π2 in the left two stands of l S,N . It suffices to determine the value of the t-coordinate at this point. Diagram (B) tells us that l S is below l N , because T is oriented so that the t-axis points above the su-face (see Fig. 4). Resolving the remaining crossings in a similar fashion leads to a diagrams of l S,N presented in squares (C) and (D). Clearly, the linking number of l S and l N is equal to ±2 in Diagram (C), (as the intersection number of e.g. l S with the obvious annulus on Diagram (C)). This justifies (2.8), and ends the proof. Results of Theorem 2.2 and Proposition 2.1 combined with Formula (2.1) allow us to express µ¯ 123 (L) of a Borromean link L as µ¯ 123 (L) = for
FL∗ ν
T FL∗ ν
∧α =
T
L ∗ ω ∧ α,
= dα, ω = H ∗ ν ∈ 2 (Conf3 (S 3 )),
(2.10)
where ν is the area form on S 2 , and H : Conf3 (S 3 ) −→ S 2 is the deformation retraction (as in e.g. (2.4)). Alternatively, we may view ω as a 2-form on (S 3 )3 which is singular along the diagonals ⊂ (S 3 )3 , and the singularity is of order O(r 2 ), where r is a distance to . Consequently, ω is integrable but not square integrable on (S 3 )3 . Remark 2.3. Notice that the integral formula (2.10) exhibits the following property of µ¯ 123 : µ¯ 123 (L 1 , L 2 , L 3 ) = sign(σ )µ¯ 123 (L σ (1) , L σ (2) , L σ (3) ),
σ ∈ 3 .
Third Order Helicity of Magnetic Fields via Link Maps
439
3. Invariants of Volume Preserving Flows. Helicities Given finitely many volume preserving vector fields B1 , B2 , . . . Bk ∈ SVect(M) on M = S 3 or a homology 3-sphere one seeks quantities I(B1 , B2 , . . . , Bk ) invariant under the action of volumorphisms isotopic to the identity g ∈ SDiff0 (M), commonly known as helicities or higher helicities: I(B1 , B2 , . . . , Bk ) = I(g∗ B1 , g∗ B2 , . . . , g∗ Bk ), for all g ∈ SDiff0 (M),
(3.1)
where g∗ is a push-forward by a diffeomorphism g. To distinguish the case of a single vector field B (i.e. B = B1 = . . . = Bk ) we often refer to I(B) = I(B, B, . . . , B) as self helicity. We elucidated in the introduction a fundamental example of such an invariant is the ordinary helicity H(B1 , B2 ) of a pair of vector fields. In the remaining part of this section we review well known formulations of the helicity, which will later help us to point out analogies to the proposed formulation of the 3rd order helicity. Let T = T1 ∪T2 represent two invariant subdomains (not necessarily disjoint) under flows of B1 and B2 in S 3 and let T = T1 × T2 ⊂ Conf2 (M) ⊂ M × M. Recall that the formula for H(B1 , B2 ), from [19,38], specialized to invariant subdomains T = T1 ∪ T2 may be expressed as ω ∧ ι B1 µ ∧ ι B2 µ, (3.2) H12 (B1 , B2 ) = T1 × T 2
where ω is known as the linking form on M × M. When T = M × M this formula is equivalent to a more commonly known expression: H(B1 , B2 ) = M ι B1 µ ∧ d −1 (ι B2 µ) (because ω also represents the integral kernel of d −1 ). Philosophically, H(B1 , B2 ) can be derived (cf. [2]) from the linking number of a pair of closed curves, which is expressed by Arnold’s Helicity Theorem. For orbits {O 1 (x), O 2 (y)} of B1 and B2 through x, y ∈ M, we introduce the following notation for the long pieces of closed up orbits: OTBi (x) = {it (x) | 0 ≤ t ≤ T } ⊂ O i (x), O¯ i (x) := O i (x) ∪ σ (x, i (x, T )), T
i = 1, 2,
(3.3)
T
where σ (x, y) denotes a short path, [38], connecting x and y in M (see Sect. 5). Paraphrasing [2] we state (for the proof also see [38]), Theorem 3.1 (Arnold’s Helicity Theorem, [2]). Given B1 , B2 ∈ SVect(M), the following limit exists almost everywhere on M × M: m¯ B1 B2 (x, y) = lim
T →∞
1 lk(O¯T1 (x) × O¯T2 (y)). T2
Moreover, m¯ B1 B2 is in L 1 (M × M), and m¯ B1 B2 (x, y) µ(x) ∧ µ(y). H12 (B1 , B2 ) =
(3.4)
(3.5)
M×M
The function m¯ B1 B2 represents an asymptotic linking number of orbits {O 1 (x), O 2 (y)}, and the identity (3.5) tells us that the helicity H12 (B1 , B2 ) is equal to the average asymptotic linking number. In the following paragraphs, we will demonstrate how this philosophy is applied to obtain the asymptotic µ¯ 123 -invariant for 3-component links and the third order helicity.
440
R. Komendarczyk
4. Definition of “µ123 -Helicity” on Invariant Unlinked Handlebodies In this section we apply the formulation of the µ¯ 123 -invariant for the 3-component links in S 3 , obtained in Sect. 2, to define the third order helicity of a volume preserving vector field B on certain invariant sets T of B in S 3 . In the following paragraphs as a “warmup” to a more general case treated in Sect. 5, we consider the case of three disjoint unlinked handlebodies: T = T1 ∪ T2 ∪ T3 in S 3 each of genus g(Ti ). Henceforth, we use “unlinked” to mean “with pairwise unlinked connected components”. When T represents three unlinked tubes (also known as flux tubes [3]) in R3 , the third order helicity has been developed by several authors [5,27,25] via Massey product formula for the µ¯ 123 -invariant, we compare our approach to these known works in Sect. 8. Assume Ti s have smooth boundary and B to be tangent to ∂Ti ; we set i = 1, 2, 3, Bi := B T , i
and denote the flow of B on S 3 by , and flows of restrictions Bi by i . Clearly, such T is an invariant set of B. Given any domain T with three connected components {Ti } we may always associate a product domain in Conf3 (S 3 ) as follows: T := T1 × T2 × T3 ⊂ Conf3 (S 3 ) ⊂ S 3 × S 3 × S 3 .
(4.1)
Notice that T is a domain with corners in Conf3 (S 3 ), and we use the same notation for the product of Ti as for the union in S 3 . Wherever needed, we also assume that (S 3 )3 is equipped with a product Riemannian metric. Let a domain T defined in (4.1), where Ti ∩ T j =Ø, i = j and each Ti is a handlebody in S 3 be called unlinked handlebody, if and only if, the 2-form ω ∈ 2 (Conf3 (S 3 )) defined in Eq. (2.10) is exact on T , i.e. ω has a local potential αω ∈ 1 (T ): ω = dαω .
(4.2)
Denote a volume preserving vector field B and an unlinked handlebody T as a pair (B; T ). Remark 4.1. Since ω is a dual cohomology class to the S 2 factor in Conf3 (S 3 ) ∼ = S3 × S2, ω does not admit a global potential. Because each handlebody Ti has a homotopy type of a bouquet of circles, there is a natural choice of the basis for H1 (Ti ) which consists of cycles {L ik }k=1,...,g(∂ Ti ) corresponding to the circles. We have the following practical characterization of unlinked handlebodies: Lemma 4.2. T is an unlinked handlebody, if and only if, lk(L ik , L rj ) = 0,
for all i, j,
i = j.
(4.3)
Proof. The standard integral pairing, [6], H 2 (T ) × H2 (T ) −→ R implies that a closed k-form is exact, if and only if, it evaluates to zero on all k-cycles of the domain. By the K¨unneth formula H2 (T ) = H2 (T1 × T2 × T3 ) is generated by L ik ⊗ L rj , and by (2.6): lk(L ik , L rj ) = ω i = j. L ik ×L rj
Therefore the condition (4.3) is necessary and sufficient for ω to be exact on T .
Third Order Helicity of Magnetic Fields via Link Maps
441
We define the µ¯ 123 -helicity of (B; T ) denoted by H123 (B; T ) or H123 (B1 , B2 , B3 ) as follows: def. H123 (B; T ) = H123 (B1 , B2 , B3 ) = (αω ∧ dαω ) ∧ ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 , T
(4.4) where µi denotes the pull-back of the volume form µ on S 3 under the projection πi : S 3 × S 3 × S 3 −→ S 3 ,
πi (x1 , x2 , x3 ) = xi ,
(4.5)
and ι Bi is a contraction by a vector field Bi . Our notational convention is to denote by ι Bi µi both the forms on the base of πi and the pullbacks: πi∗ ι Bi µi . Notice that µ = µ1 ∧ µ2 ∧ µ3 is a volume form on the product: S 3 × S 3 × S 3 . There are obvious analogies between Formula (4.4) above, Formula (3.2) for H12 (B; T ), and the integral formula (2.10) for the µ¯ 123 -invariant. The 3-form: γω := αω ∧ dαω = αω ∧ ω, plays a role of the linking form as ω in Formula (3.2). The main motivation behind definition (4.4) is the ergodic interpretation of H123 (B; T ) as an average asymptotic µ¯ 123 -invariant of orbits of B, which will become apparent in Sect. 5. Formula (4.4) can be also regarded as the third order helicity of three distinct vector fields Bi , supported on the handlebodies Ti . In Sect. 7, we indicate how to construct the potential αω from the basic elliptic theory of differential forms. Theorem 4.3 (Helicity Invariance Theorem). On every unlinked invariant handlebody T in S 3 , H123 (B; T ) is (i) independent of a choice of the potential αω , (ii) invariant under the action of SDiff0 (S 3 ), i.e. for every g ∈ SDiff0 (S 3 ): H123 (B; T ) = H123 (g∗ B; g(T )).
(4.6)
Proof. To prove (i) observe for every other potential αω of ω, the difference β = αω −αω is a closed 1-form on T (since ω = dαω = dαω ). Therefore,
(β ∧ ω) ∧ ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 H123 (B; T ) − H123 (B; T ) = T (1) = d β ∧ αω ∧ ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 T (2) = β ∧ αω ∧ ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 = 0, ∂T
where in (1) we applied d(ι Bi µi ) = 0 (since Bi ’s are divergence free), and in (2): ι Bi µi ∂ T = 0, i
(because each vector field Bi is tangent to the boundary ∂Ti ), where ∂T = (∂T1 × T2 × T3 ) ∪ (T1 × ∂T2 × T3 ) ∪ (T1 × T2 × ∂T3 ) .
442
R. Komendarczyk
The proof of (ii) is in the style of [5,27], but adapted to our setting. For any given g ∈ SDiff(S 3 ), by definition, there exists a path t −→ g(t) ∈ SDiff0 (S 3 ), such that g(0) = id S 3 ,
g(1) = g.
Denote by V the divergence free vector field on S 3 , given by V (x) = g(t) is a flow of V , and push-forward fields Bi by
d dt g(t, x)|t=0 ,
i.e.
Bit := g(t)∗ Bi . It is well known (see Appendix A, or [15, p. 224]) that 2-forms: ι B t µ are frozen in the i flow of V , i.e. d g(t)∗ ι B t µ = (∂t + LV )ι B t µ = 0. (4.7) i i dt We also have a path g(t) ˆ = (g(t), g(t), g(t)) in SDiff0 (S 3 × S 3 × S 3 ), which analogously leads to the vector field Vˆ = (V, V, V ). (Recall that a tangent bundle T (S 3 )3 has a natural product structure.) Equation (4.7) implies (∂t + LVˆ ) πi∗ ι B t µ = (∂t + LVˆ )ι B t µi = 0. (4.8) i
i
(In the second equation we merely revoke our notational conventions: ι B t µi ≡ i πi∗ (ιg(t)∗ Bi µ).) d Let T (t) = g(t)(T ˆ (0)) ⊂ Conf3 (S 3 ); we must show dt H123 (B1t , B2t , B3t ) = 0. Notice that for small enough and t ∈ (t0 − , t0 + ) we can assume, by (i), that αω is a time independent potential obtained from the slightly bigger domain T which deformation retracts on T (t0 ), and satisfies T (t) ⊂ T,
for t ∈ (t0 − , t0 + ).
ˆ = id(S 3 )3 ; at t0 we calculate: Without loss of generality set t0 = 0, and g(0) d d H123 (g(t)∗ B1 , g(t)∗ B2 , g(t)∗ B3 ) = dt dt =
T (t)
αω ∧ dαω ∧ ι B t µ1 ∧ ι B t µ2 ∧ ι B t µ3 1
2
3
d gˆ (t)∗ αω ∧ dαω ∧ ι B t µ1 ∧ ι B t µ2 ∧ ι B t µ3 1 2 3 T (0) dt = L∂ +Vˆ (αω ∧ dαω ) ∧ ι B t µ1 ∧ ι B t µ2 ∧ ι B t µ3 , t T (0)
1
2
3
where in the last identity we applied (4.8) and the product rule for the Lie derivative. Now because ω ∧ αω is time independent (for t ∈ (t0 − , t0 + )), the Cartan magic formula yields L∂t +Vˆ (αω ∧ dαω ) = LVˆ (αω ∧ dαω ) = ιVˆ d(αω ∧ dαω ) + d(ιVˆ (αω ∧ dαω )) = d(ιVˆ (αω ∧ dαω )),
where d(αω ∧ dαω ) = ω ∧ ω = 0. Since Bit are tangent to the boundary of Ti (t), the same argument as in the proof of (i) shows that the right-hand side of the previous equation vanishes.
Third Order Helicity of Magnetic Fields via Link Maps
443
Remark 4.4. Notice that the above argument indicates that if we replace αω ∧ dαω by virtually any closed 3-form η on Conf3 (S 3 ) (or (S 3 )3 ) we obtain some invariant under frozen-in-field deformations. If η is exact we obtain trivial invariants, therefore the only sensible candidates here are cohomology classes of Conf3 (S 3 ) ∼ = S 3 × S 2 . In dimension 3 it leaves us with a dual to the S 3 factor in Conf3 (S 3 ). Based on the considerations in Sect. 2 one may argue that an invariant obtained this way is trivial. Indeed, the cohomology class η evaluated on any 3-torus obtained from a 3-component link in S 3 via the map L in (2.3) is zero. Therefore, one could apply the ergodic approach of Sect. 5 to show that η ∧ ι B µ1 ∧ ι B µ2 ∧ ι B µ3 defines a trivial invariant. The crucial obstacle in extending the formula in (4.4) to encompass the whole (S 3 )3 is the fact that the potential αω cannot be globally defined on Conf3 (S 3 ). 5. The Ergodic Interpretation of H123 (B;T ) The following statement is often seen in the literature [8,9]: Helicity measures the extent to which vector fields twist and coil around each other. A beauty of Arnold’s ergodic approach to the helicity H12 (B) is that it makes this statement precise, by interpreting H12 (B) as an average asymptotic linking number of orbits of B. But, it also has a practical application as it allows us to extend our approach to certain invariant sets of B. In this section we apply this philosophy to our newly defined invariant H123 (B; T ), and interpret it as the average asymptotic µ¯ 123 -invariant of orbits of B in T . Moreover, this ergodic interpretation leads us to an alternative, more intuitive proof of the Helicity Invariance Theorem 4.3. We begin by observing that given a volume preserving vector field B on M and its flow t , we may regard B as three vector fields on (M)3 . Thus, (, , ) induces a natural R3 action defined as follows:
: R × (M)3 −→ (M)3 , ((s, t, u), x, y, z) −→ ((s, x), (t, y), (u, z)). 3
(5.1)
Observe that is a volume preserving action on (M)3 . Our analysis is rooted in techniques developed in [2,24,25,38], the main tool is the following Theorem 5.1 (Multi-parameter Ergodic Theorem, [4]). For any real valued L 1 -function F on (M)3 , the time averages under the action in (5.1): ¯ F(x, y, z) = lim
T →∞
1 T3
T 0
0
T
T
F((x, s), (y, t), (z, u)) ds dt du
0
converge almost everywhere. In addition, the limit function F¯ satisfies ¯ L 1 ((M)3 ) ≤ F L 1 ((M)3 ) , (i) F
(ii) F¯ is invariant under the -action, (iii) if (M)3 is of finite volume then (M)3
F¯ =
(M)3
F.
(5.2)
444
R. Komendarczyk
Definition 5.2. Define the invariant unlinked domain T of B as an arbitrary -invariant set, with topological closure T which belongs to a larger product of open sets T = T1 × T2 × T3 in Conf3 (S 3 ), satisfying the following: (A) T admits a short path system S, (B) Equation (4.2) holds on T, where by a system of short paths on T, [38,3], we understand a collection of curves S = {σi (x, y)} on each open set Ti such that (a) for every pair of points x, y ∈ Ti there is a connecting curve σi (x, y) : I → Ti in S, σi (0) = x and σi (1) = y, (b) the lengths of paths in S are uniformly bounded above by a common constant. Topologically, every -invariant set is a union of products of orbits of B in (S 3 )3 . It is often convenient to think of the orbits -action as a foliation of (S 3 )3 . Then -invariant sets are just the union of leaves of this foliation. A fundamental example of an invariant unlinked domain is the case of -invariant set T contained in the product T = T1 × T2 × T3 of disjoint open unlinked handlebodies Ti . Note that in this case we do not require B to be tangent to ∂ Ti , and T always admits a short path system as we describe in the following: Remark 5.3. In [38], Vogel shows that on a closed manifold M, geodesics always provide a short path system. When T is contained in the product of unlinked handlebodies T we may easily construct such a system on T as follows: because Ti are proper subsets of S 3 we generally do not want to use ambient geodesics from S 3 as they may not lie entirely in Ti . To obtain S one puts an artificial Riemannian metric on each Ti which makes ∂ Ti totally geodesic, and choose S to be geodesics on such a Riemannian manifold. Observe that applying a diffeomorphism g ∈ Diff(T) to S results in the system gS on g(T). The following result is an analog of Arnold’s Helicity Theorem in our setting: Theorem 5.4 (Ergodic interpretation of H123 (B; T )). Given (B; T ), the following limit (asymptotic µ¯ 123 -invariant of orbits) exists for almost all (x, y, z) ∈ T : 1 (5.3) m¯ B (x, y, z) = lim 3 µ¯ 123 O¯TB1 (x), O¯TB2 (y), O¯TB3 (y) . T →∞ T Moreover,
H123 (B; T ) =
T
m¯ B (x, y, z) µ(x) ∧ µ(y) ∧ µ(z).
(5.4)
Proof. The proof is similar to the one in e.g. [38]. Before we start, we must point out the following identity (valid for any 3-form β on M × M × M and vector fields B1 , B2 , B3 on M) (ι B3 ι B2 ι B1 β) ∧ µ1 ∧ µ2 ∧ µ3 = β(B1 , B2 , B3 ) µ1 ∧ µ2 ∧ µ3 = β ∧ ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 .
(5.5)
The first equation follows from the definition, the second one is a consequence of the fact that ι B is an antiderivation, i.e. ι B (α ∧ β) = (ι B α) ∧ β + (−1)|α| α ∧ (ι B β),
(5.6)
Third Order Helicity of Magnetic Fields via Link Maps
445
and ι Bi µ j = 0, for i = j (see Appendix A). As a result, H123 (B1 , B2 , B3 ) = (αω ∧ dαω ) ∧ ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 T ι B3 ι B2 ι B1 (αω ∧ dαω ) µ1 ∧ µ2 ∧ µ3 = T = m B1 ,B2 ,B3 (x, y, z) µ1 ∧ µ2 ∧ µ3 , T
where m B1 ,B2 ,B3 := αω ∧ dαω (B1 , B2 , B3 ). For convenience, let us set (see (3.3)): ¯ O(x, y, z; T ) := O¯TB1 (x) × O¯TB2 (y) × O¯TB3 (y). ¯ Observe that if the orbit O(x, y, z; T ) is nondegenerate, it represents a Borromean link and we may apply Formula (2.10) to get µ¯ 123 O¯TB1 (x), O¯TB2 (y), O¯TB3 (y) = αω ∧ dαω O¯ (x,y,z;T ) = αω ∧ dαω + (I ), O (x,y,z;T )
where the term (I ) involves integrals over short paths in S (see Appendix B). For degenerate orbits (such as fixed points etc.), the above formula still makes sense because µ¯ 123 is a homotopy invariant of the associated map as proven in Sect. 2. For every (x, y, z): D(x,y,z) [∂i ] = Bi , i = 1, 2, 3, where ∂1 := ∂s ,∂2 := ∂t , ∂3 := ∂u , thus we obtain ∗(x,y,z) (αω ∧ dαω ) = (αω ∧ dαω (B1 , B2 , B3 )(1 (x, s), 2 (y, t), 3 (z, u)) ds ∧ dt ∧ du = m B1 ,B2 ,B3 (1 (x, s), 2 (y, t), 3 (z, u)) ds ∧ dt ∧ du.
Therefore, O (x,y,z;T )
αω ∧ dαω =
T
0
T 0
0
T
m B1 ,B2 ,B3 (1 (x, s),
2 (y, t), 3 (z, u)) ds ∧ dt ∧ du. Function m B1 ,B2 ,B3 is smooth bounded on T and hence L 1 . Because short paths do not contribute to the time average (see Appendix B): lim
T →∞
1 (I ) = 0. T3
Theorem 5.1 applied to the function m B1 ,B2 ,B3 yields almost everywhere existence of the limit (5.3). Hence we obtain the invariant L 1 -function m¯ B := m¯ B1 ,B2 ,B3 on T . The identity (5.4) follows from (iii) of Theorem 5.1. Theorem 5.5 (Helicity Invariance Theorem-ergodic version). On every unlinked domain T in S 3 , (i) and (ii) of Theorem 4.3 hold for H123 (B; T ).
446
R. Komendarczyk
Proof. The proof of (i) immediately follows from independence of the limit (5.3) of the choice of the potential αω . For the proof of (ii) we must show the following: m B1 ,B2 ,B3 µ1 ∧ µ2 ∧ µ3 = m g∗ B1 ,g∗ B2 ,g∗ B3 µ1 ∧ µ2 ∧ µ3 . T
g(T )
Theorem 5.1 tells us that m B1 ,B2 ,B3 and m g∗ B1 ,g∗ B2 ,g∗ B3 admit L 1 -averages m¯ B1 ,B2 ,B3 and m¯ g∗ B1 ,g∗ B2 ,g∗ B3 , under actions of Bi and g∗ Bi respectively. It suffices to show the following identity: m¯ B1 ,B2 ,B3 (x, y, z) = m¯ g∗ B1 ,g∗ B2 ,g∗ B3 (g(x), g(y), g(z)),
a.e.,
(5.7)
then (4.6) is an immediate consequence of Eq. (5.2), change of variables for integrals, and the fact that g preserves volume (i.e. µi = g ∗ µi ). Borrowing notation from the previous theorem set ¯ g O(x, y, z; T ) := g(O¯TB1 (x)) × g(O¯TB2 (y)) × g(O¯TB3 (z)) g B g B g B = O¯T∗ 1 (g(x)) × O¯T∗ 2 (g(y)) × O¯T∗ 3 (g(z)),
where the second identity is a consequence of the fact that the flow of g∗ Bi is obtained ¯ as a composition of g and the flow of Bi . Since g is isotopic to the identity, O(x, y, z; T ) ¯ and g O(x, y, z; T ) are homotopic as link maps (for nondegenerate orbits they are in fact isotopic Borromean links in S 3 ) and by theorems of Sect. 2, we have ¯ ¯ y, z; T )) = µ¯ 123 (g O(x, y, z; T )). µ¯ 123 (O(x, As a result of the above identity and (5.3) we derive a.e. 1 ¯ µ¯ 123 (O(x, y, z; T )) T →∞ T 3 1 ¯ y, z; T )) = m¯ g∗ B1 ,g∗ B2 ,g∗ B3 (g(x), g(y), g(z)). = lim 3 µ¯ 123 (g O(x, T →∞ T
m¯ B1 ,B2 ,B3 (x, y, z) = lim
Notice that in the last equation we used the “pushed forward” short paths system: gS. Since the lengths of paths in gS are bounded as well, they do not contribute to the limit. This proves the identity (5.7), and consequently (4.6). Notice that the above argument does not require the Stokes Theorem, and as such may lead to further generalizations. Clearly, for H123 (B; T ) to be nontrivial T must be of nonzero measure. 6. Flux Formula for H123 (B;T ) The following formula is a well known property of the ordinary helicity H12 (B; T ) of the flux tubes T modeled on a 2-component link L = {L 1 , L 2 } (see e.g. [24,7]) H12 (B; T ) = H12 (B1 , B2 ) = lk(L 1 , L 2 ) Flux(B1 ) Flux(B2 ).
(6.1)
Here we show an analogous property for H123 (B; T ), when T is an invariant unlinked handlebody. Recall that {L ik }k=1,...,g(∂ Ti ) denotes the basis of H1 (T ) defined in Lemma 4.2.
Third Order Helicity of Magnetic Fields via Link Maps
447
Fig. 5. The simplest unlinked handlebodies: flux tubes T Borr modeled on the Borromean rings (left), and unlinked genus 2 handlebodies (right)
Proposition 6.1. H123 (B1 , B2 , B3 ) on invariant unlinked handlebodies T satisfies the following formula: j H123 (B1 , B2 , B3 ) = µ¯ 123 (L i1 , L 2 , L k3 ) Flux i (B1 ) Flux j (B2 ) Flux k (B3 ), (6.2) i, j,k j
where {L i1 ⊗ L 2 ⊗ L k3 } is a basis of H3 (T ), Flux(Bi ) stands for the flux of Bi through a cross sectional surface k of Ti , which represents the homology Poincar´e dual of L ik in H2 (Ti , ∂Ti ). Proof. Recall that the flux Flux k (Bi ) of a vector field Bi though a cross-sectional surface k in Ti is given by: Flux k (Bi ) = ι Bi µ = h k ∧ ι Bi µ Ti k = ι Bi h k ∧ µ = h k (Bi ), (6.3) Ti
Ti
where 1-forms h k represent cohomology Poincar´e duals of k , and we applied (5.6) in the third equation. For every closed curve γ ⊂ T h k satisfies h k = #(γ , k ) = deg(γ , L k ), γ
j
j
where deg(γ , L k ) measures how many times γ “wraps around” the cycle L k . For simplicity, we first assume that T is modeled on a Borromean link L = {L 1 , L 2 , L 3 } (such as T Borr on Fig. 5). Then, h = h 1 ∧h 2 ∧h 3 ∈ 3 (T ) is a cohomology class dual to the cycle L 1 ⊗ L 2 ⊗ L 3 in H3 (T ). Define H := ι B3 ι B2 ι B1 h = h(B1 , B2 , B3 ), which is a smooth function, and let H¯ be the time average of H as in Theorem 5.1. It suffices to show m¯ B1 ,B2 ,B3 = µ¯ 123 (L) H¯ ,
a.e.
(6.4)
448
R. Komendarczyk
Analogously, as in the proof of Theorem 4.3, Eq. (6.4) immediately implies Formula ¯ (6.2). Assume the notation of Theorem 4.3, for given T consider O(x, y, z; T ). Thanks to Theorem 2.2 (see the first paragraph of the proof) we have ¯ y, z; T )) = µ¯ 123 (L) deg(O¯TB1 (x), L 1 ) deg(O¯TB2 (y), L 2 ) deg(O¯TB3 (x), L 3 ) µ¯ 123 (O(x, = µ¯ 123 (L) h. O¯ (x,y,z;T )
Therefore, 1 ¯ µ¯ 123 (O(x, y, z; T )) T3 1 H = µ¯ 123 (L) lim 3 T →∞ T O¯ (x,y,z;T ) = µ¯ 123 (L) H¯ (x, y, z),
m¯ B1 ,B2 ,B3 (x, y, z) = lim
T →∞
where the last equality is again the consequence of short paths not contributing to the limit. From the product structure of T and (iii) of Theorem 5.1 we get
¯ H= h 1 (B1 ) h 2 (B2 ) h 3 (B3 ) , H= T
T
T1
T2
T3
which combined with Eq. (6.3) for fluxes concludes the proof in the case of Borromean flux tubes. The proof in the case of a general handlebody is analogous, once we show the following: Lemma 6.2. Let O = {O1 , O2 , O3 } be a 3-component link in S 3 such that Oi ⊂ Ti for each i. Then O is Borromean and j j µ¯ 123 (L i1 , L 2 , L k3 ) deg(O1 , L i1 ) deg(O2 , L 2 ) deg(O3 , L k1 ). (6.5) µ¯ 123 (O) = i, j,k
Proof. Thanks to the interpretation of the µ¯ 123 -invariant in Sect. 2, it is not only a link homotopy invariant, but also a homotopy invariant of the associated map FO defined in (2.3). Observe that each component Oi can be homotoped inside of its handlebody Ti i ∼ i is a to become a bouquet of circles O = S 1 ∨ S 1 ∨ . . . ∨ S 1 so that each factor in O j multiple of the cycle represented by {L i }in H1 (Ti ). As a result, we obtain the associated map FO , and 1 H (FO ). 2 Interpreting H (FO ) as the intersection number and summing up intersection numbers we conclude (6.5). µ¯ 123 (O) =
In the case of Borromean flux tubes, Formula (6.1) reduces to H123 (B1 , B2 , B3 ) = µ¯ 123 (L) Flux 1 (B1 )Flux 2 (B2 )Flux 3 (B3 ),
(6.6)
where i denotes homology Poincar´e duals to L i in H2 (Ti , ∂Ti ). Since the fluxes are invariant under frozen-in-field deformations, Formula (6.1) is yet another proof of Theorem 4.3 in the setting of invariant unlinked handlebodies. In [25] the authors develop the same formula for the Borromean flux tubes. This clearly must be the case, as we work with the same topological invariants of links via a different approach. Additional advantage of our formulation is that we do not have to separately deal with null points of vector fields as in [25].
Third Order Helicity of Magnetic Fields via Link Maps
449
7. Energy Bound In this section we indicate how the quantity H123 (B; T ) invariant under frozen-in-field deformations provides a lower bound for the L 2 -energy E 2 (B) of a volume preserving field B on M = S 3 . We restrict our considerations to the case of an invariant unlinked handlebody T , defined in Sect. 4. For the notation used in this section see Appendix C. Recall the definition E 2 (B) = M
|B|2 = B 2L 2 (M) .
(7.1)
The ordinary helicity H12 (B) provides a well known lower bound (see [3, p. 123]): 1 |H12 (B)| ≤ E 2 (B), λ1 where λ1 is the first eigenvalue of the elliptic self adjoint operator ∗ d : 1 (M) → 1 (M) (known as the curl operator), ∗ denotes the Hodge star. Importance of such lower energy bounds stems from an area of interest in the ideal magnetohydrodynamics, [32], as this constrains the phenomenon of “magnetic relaxation”, [14]. A need for higher helicities can be justified by the fact that one may easily produce examples of vector fields B for which H12 (B) vanishes, but the energy of the field B still cannot be relaxed. For example, consider a classical case of Borromean flux tubes T Borr with B smooth and vanishing outside the tubes. Furthermore, assume that orbits of B are just “parallel circles” inside each tube. By taking B = B1 + B2 + B3 , bilinearity of H12 ( · , · ) on SVect(S 3 ) and disjoint supports of Bi yield H12 (B) = H12 (B, B) =
3 i=1
H12 (Bi ) +
H12 (Bi , B j ) = 0,
i = j
where “cross-helicities” H12 (Bi , B j ), i = j vanish by Formula (6.1), and self helicities H12 (Bi ) vanish because the average linking number of orbits is zero (orbits are just parallel circles). Nevertheless, Formula (6.2) tells us H123 (B; T Borr ) = 0; as a result we may regard H123 (B; T ) as a possible “higher obstruction” to the energy relaxation or the third order cross-helicity of B on T . To obtain a lower bound for E 2 (B) in such situations we notice that H123 (B; T ) is the L 2 -inner product of the 6-forms: ∗(ω ∧ αω ) and ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 . Indeed, from (4.4) we have H123 (B; T ) = ∗(αω ∧ ω), ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 L 2 6 ((S 3 )3 ) . Let CT = αω ∧ ω L 2 , for a fixed Riemannian product metric on (S 3 )3 ; this constant depends on the domain T in (S 3 )3 . We estimate using the Cauchy-Schwarz inequality,
450
R. Komendarczyk
|H123 (B; T )| ≤ αω ∧ ω L 2 ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 L 2 1
2 = CT (ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 ) ∧ ∗(ι B1 µ1 ∧ ι B2 µ2 ∧ ι B3 µ3 ) T
(1)
= CT
(2)
T
1 2 |B1 |2 |B2 |2 |B3 |2 µ
= CT
T1
1 |B1 |2
2
T2
1 |B2 |2
2
T3
1 |B3 |2
2
r
3
(7.2) ≤ CT E 2 (B) 2 , where E 2 (B) = S 3 |B|2 . To observe (1) first note that for any pair of forms: α ∈ k (M), and β ∈ j (N ), on Riemannian manifolds M and N , on the product M × N we have ∗ ∗ (α) ∧ π N∗ (β)) = (∗ M π M α) ∧ (∗ N π N∗ β), ∗ M×N (π M
(7.3)
where π M : M × N −→ M and π N : M × N −→ N are the natural projections (the proof is a simple calculation in an orthogonal frame of the product and is left to the reader). Now, step (1) in (7.2) follows by applying (7.3) to the integrand, and observing in the coframe {ηki }: ι Bi µi ∧ ∗ι Bi µi = (a1 η2i ∧ η3i − a2 η1i ∧ η3i + a3 η1i ∧ η2i ) ∧ (a1 η1i − a2 η2i + a3 η3i ) = (a12 + a22 + a32 ) η1i ∧ η2i ∧ η3i = |Bi |2 η1i ∧ η2i ∧ η3i = |Bi |2 µi , where Bi = (a1 , a2 , a3 ). Step (2) in (7.2) follows from the Fubini Theorem. Next, we aim to provide an estimate for CT . For this purpose we review some basic L 2 -theory of the operator d −1 (i.e. inverse of the exterior derivative d). The main goal is to estimate an L 2 -norm of the potential αω of ω in (4.2). Following the standard elliptic theory of differential forms, [34], the potential αω in (4.2) can be obtained via a solution to the Neumann problem for 2-forms on T (see Appendix C)
φ N = ω, in T n φ N = n dφ N = 0,
on ∂ T,
(7.4)
where n stands for the normal component of a differential form, and = dδ + δd, δ = ± ∗ d∗, (cf. [34]). As T is a domain with corners we replace it by a slightly larger domain T in (S 3 )3 with the same topology (i.e. T is a deformation retract of T) but with smooth boundary ∂ T. (One may argue that it is not really necessary, since T is Lipschitz and elliptic problems, such as (7.4) are well posed on Lipschitz domains, [31]). Because of (i) in Theorem 4.3, we may use the restriction of αω = δφ N ,
(7.5)
to T (see Appendix C for justification of (7.5)). Associated to (7.4) is the Neumann Laplacian N : H 2 2N (T) −→ L 2 2 (T),
Third Order Helicity of Magnetic Fields via Link Maps
451
which has a discrete positive spectrum {λi,N } and eigenvalues satisfy the variational principle called Rayleigh-Ritz quotient, [11]. The first (principal) eigenvalue λ1,N may be expressed as λ1,N = inf
dϕ 2L 2 + δϕ 2L 2 1 2 2 ⊥ . ϕ ∈ H N (M) ∩ H N (M)
ϕ 2L 2
We denote the inverse of N by G N : L 2 2 (T) −→ H 2 2N (T), which restricts to a compact, self-adjoint operator on L 2 . As a result the spectrum of G N is discrete and given as {1/λi,N }. Note that based on these considerations we may define d −1 := δG N . Theorem 7.1 (Energy bound). For every volume preserving vector field B which has an invariant unlinked domain T , the L 2 -energy of B on S 3 is bounded below by the third order helicity H123 (B; T ), as follows: 3
|H123 (B; T )| ≤ CT (E 2 (B)) 2 .
(7.6)
Also, we may estimate the constant CT : CT ≤
1
ω 2L ∞ 2 (T ) (Vol(S 3 ))3 , λ1,N
(7.7)
where λ1,N is the first eigenvalue of the Neumann Laplacian on 2 (T). Proof. We estimate
αω 2L 2 = δφ N 2L 2 ≤ dφ N 2L 2 + δφ N 2L 2 = |(φ N , φ N )| ≤ ω L 2 φ N L 2 , where we used Green’s formula [34, p. 60] and boundary conditions of (7.4) in the sec1 ond identity. Now, because φ N = G N ω, and it is a well known fact that G N L 2 = λ1,N , (G N is compact self adjoint on L 2 ) we obtain: 1
αω L 2 ≤
ω L 2 . λ1,N As a result we estimate CT : CT = αω ∧ ω L 2 ≤ ω L ∞ αω L 2 1 ≤
ω L 2 ω L ∞ . λ1,N
452
R. Komendarczyk
Notably, the best energy estimate so far has been obtained by Freedman and He, [15], for the L 3/2 -energy of B, in the case when B admits an invariant domain T modeled on an n-component link L = {L 1 , . . . , L n } in R3 . Their estimate is based on the asymptotic crossing number and reads n π 1/4 ac(L k , L)|Flux(Bk )| · min {|Flux(Bk )|}, (7.8) E 3/2 (B) ≥ 1≤k≤n 16 k=1
where the asymptotic crossing numbers ac(L k , L) for Borromean links can be estimated below by a smallest genus among surfaces in R3 \{L 1 ∪. . .∪ Lˆ k ∪. . .∪ L n } with a single boundary component L i . Since L 3/2 -energy of B bounds the L 2 -energy, inequality (7.8) leads to a lower estimate purely in terms of fluxes and topological data. It is not clear to the author if this approach can be extended to the case of invariant handlebodies considered in Sect. 4. A different, more optimal estimate, has been obtained by Laurence and Stredulinsky, via the Massey product formula, in [26], but the proof is provided only in a special case of the vector field B. Contrary to these lower bounds, which are given in terms of topological data, the estimate in (7.6) depends on the geometry of the domain T , and also ω L ∞ . Unfortunately, ω blows up on the diagonals ⊂ (S 3 )3 , and as a result the estimate is meaningless when the handlebodies Ti get close to each other during the evolution of the magnetic field B. At this point, we need an assumption for Ti to stay 1cm apart during the evolution. Another drawback is that λ1,N is a geometric constant which is altered during the evolution as well. If we consider the situation in which the boundaries of Ti are invariant during the evolution, the estimate may be useful. Under such assumption, which occurs whenever the velocity field v of plasma in (1.1) is tangent to ∂Ti , the bound in (7.7) stays constant. 8. Comparison to the Known Approaches via Massey Products In several prior works [5,13,25,27] helicities were developed via the Massey product formula for µ¯ 123 . These approaches are equivalent to the one presented here in the sense that invariants obtained this way measure the same topological information. Most notably the work [25] provides an explicit expression for the third order helicity of the Borromean flux tubes, where the ergodic interpretation in the style of Arnold’s asymptotic linking number is also provided. In [27] one finds the following formula for the third order helicity: M123 (B1 , B2 , B3 ) = A1 ∧ A2 ∧ A3 , (8.1) M
where Ai = d −1 (ι Bi µ). This formula is valid for three distinct vector fields Bi on a closed manifold M. For invariant domains with boundary, (8.1) defines an invariant pro vided Ai ∂ M = 0, but this only happens in certain situations (e.g. M is simply connected, and Ai ’s are appropriately chosen). The most commonly known formula directly related to the Massey products was developed by Berger [5], in the case of Borromean flux tubes T = T1 ∪ T2 ∪ T3 , M123 (B1 , B2 , B3 ) = A1 ∧ F23 + F12 ∧ A3 . (8.2) ∂ T1
Third Order Helicity of Magnetic Fields via Link Maps
453
In [5] it is expressed as a volume integral over T by applying gauge fixing. When Ti are topologically solid tori there exists a single Massey product < a1 , a2 , a3 > in the complement S 3 \T , represented by the 2-form A1 ∧ F23 + F12 ∧ A3 . When Ti are handlebodies there are multiple Massey products, but the formula (8.2) should still be valid. So far, such extensions have not been considered in the literature and the volume integrals over T may be harder to obtain in such a case. One may also point out that ergodic interpretations of Massey products are more involved [25] compared to the approach presented in Sect. 5. Acknowledgements. The inspiration for the presented approach to the link homotopy invariants comes from the paper of Toshitake Kohno [20], and I am grateful to him for the valuable e-mail correspondence. I have enjoyed conversations with many colleagues at the University of Pennsylvania, who have influenced this work. I wish to thank Herman Gluck for weekly meetings and his interest in this project, Frederic Cohen, Dennis DeTurck, Charlie Epstein, Paul Melvin, Tristan Rivi`ere, Clay Shonkwiler, Jim Stasheff, and David Shea Vick for the valuable input. I am also grateful to my advisor Robert Ghrist who introduced me to the subject a long time ago. After posting recent joint results in [12] we were informed by Paul Kirk about related works of Urlich Koschorke in [21,22], on homotopy invariants of link maps and Milnor µ-invariants. ¯ The author acknowledges financial support of DARPA, #FA9550-08-1-0386.
Appendices A. Equations for the Frozen-in-Field Forms Given a volume preserving vector field B on M and a path t −→ g(t) ∈ SDiff(M), let B t := g∗ (t)B. Then for B 0 = B, by definition d t B t=0 = −LV B = −[V, B], B˙ t = dt
(A.1)
and as a result d ι B t µ|t=0 = −ι[V,B] µ. dt Next, calculate (∂t + LV ≡ L∂t +V ), (∂t + LV )ι B t µ = ∂t (ι B t µ) + LV ι B t µ = ι B˙t µ + ι B t (LV µ) + ι[V,B t ] µ = ι B˙t +[V,B t ] µ, where in the second identity we applied the general formula: ι[A,B] = L A ι B − ι B L A , and in the third equation the fact that V is volume preserving i.e. LV µ = 0. As a result of (A.1) we obtain (∂t + LV )ι B t µ = 0. Next, we justify Formula (5.5). First use (5.6) to calculate ι Bi (µ j ∧ µk ) = (ι Bi µ j ) ∧ µk − µ j ∧ (ι Bi µk ), ι Bi (µ1 ∧ µ2 ∧ µ3 ) = (ι Bi µ1 ) ∧ µ2 ∧ µ3 − µ1 ∧ ι Bi (µ2 ∧ µ3 ) (A.2) = (ι Bi µ1 ) ∧ µ2 ∧ µ3 − µ1 ∧ ι Bi µ2 ∧ µ3 + µ1 ∧ µ2 ∧ ι Bi µ3 ,
454
R. Komendarczyk
since ι Bi µ j = 0 for i = j only one term in the above expressions remains for each i. Set α := ι B2 ι B1 β, β ∈ 3 ((S 3 )3 ); since α ∧ µ1 ∧ µ2 ∧ µ3 = 0 and α is a 1-form we obtain 0 = ι B3 (α ∧ µ1 ∧ µ2 ∧ µ3 ) = (ι B3 α) ∧ µ1 ∧ µ2 ∧ µ3 − α ∧ µ1 ∧ µ2 ∧ ι B3 µ3 , where in the last step we used (A.2). Therefore (ι B3 ι B2 ι B1 β) ∧ µ1 ∧ µ2 ∧ µ3 = (ι B2 ι B1 β) ∧ µ1 ∧ µ2 ∧ ι B3 µ3 . Analogously, (ι B2 ι B1 β) ∧ µ1 ∧ µ2 = (ι B1 β) ∧ µ1 ∧ ι B2 µ2 and ι B1 β ∧ µ1 = β ∧ ι B1 µ1 which justifies Eq. (5.5). B. Zero Contribution of Short Paths to the Time Average It is clear when β ∈ 3 (T) is at least a C 1 on T ⊂ S 3 × S 3 × S 3 , then f = β(B1 , B2 , B3 ) is continuous on T and f O¯T1 (x) O¯T2 (y) O¯T3 (z)
=
OT1 (x)
+
σ (1 (x,T ),x)
OT2 (y)
+
σ (2 (y,T ),y)
OT3 (z)
+
σ (3 (z,T ),z)
f .
After expanding, it is obvious that we must show the following (for all choices), when T → ∞: 1 f −→ 0, (B.1) T 3 σ (1 (x,T ),x) OT2 (y) OT3 (z) 1 f −→ 0, (B.2) T 3 σ (1 (x,T ),x) σ (2 (y,T ),y) OT3 (z) 1 f −→ 0. (B.3) T 3 σ (1 (x,T ),x) σ (2 (y,T ),y) σ (3 (z,T ),z) Since the lengths of the short paths in S are bounded by a common constant d, (B.1)-(B.3) follow immediately, e.g. for (B.1) we have 1 1 f ≤ 3 d(T + d)(T + d) f ∞ −→ 0. 3 T σ (1 (x,T ),x) O 2 (y) O 3 (z) T T T C. Notation in Section 7 We adopt notation from the elegant exposition in [34]. Let M be an orientable manifold with smooth boundary. k (M) = C ∞ (M, k ), smooth differential forms on M. kN (M) = {φ ∈ k (M) | nφ = 0, ndφ = 0} the subspace satisfying the Neumann boundary conditions, (n denotes a normal component of a form along ∂ M).
Third Order Helicity of Magnetic Fields via Link Maps
455
The L 2 -inner product on k (M) is defined as ω ∧ ∗η, (ω, η) L 2 = M
L 2 k (M), L 2 -differential forms on M. HkN (M) = {λ ∈ H 1 k (M) | dλ = δλ = 0, nλ = 0} the subspace of the Neumann harmonic fields. Next we justify (7.5), first observe that for any γ ∈ H 1 k−1 (M): (dγ , λ) L 2 = tγ ∧ ∗nλ = 0, ∀λ ∈ HkN (M), ∂M
(C.1)
where t and n stands for respectively tangent and normal to ∂ M components of the form. As a result, if ω ∈ HkN (M)⊥ we obtain a solution φ to the Neumann problem: δdφ + dδφ = ω, ⇒
ω − dδφ = δdφ.
(C.2)
Formula (C.1) implies: (ω − dδφ) ∈ HkN (M)⊥ , moreover n(ω − dδφ) = n(δdφ) = δ n(dφ) = 0, by the boundary condition in (7.4). If ω is a closed form, ω − dδφ is also closed, and clearly coclosed by (C.2). Thus ω − dδφ is a harmonic field with zero normal component, and therefore it has to be in HkN (M), and therefore the zero form. This yields ω = dδφ. As a result we obtain necessary and sufficient conditions for ω to be exact: (i) dω = 0, (ii) (ω, λ) L 2 = 0, for all λ ∈ HkN (M).
References 1. Akhmetiev, P.: On a new integral formula for an invariant of 3-component oriented links. J. Geom. Phys. 53(2), 180–196 (2005) 2. Arnold, V.: The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5(4), 327–345 (1986) 3. Arnold, V., Khesin, B.: Topological methods in hydrodynamics. Vol 125 of Applied Mathematical Sciences. New York: Springer-Verlag, 1998 4. Becker, M.E.: Multiparameter groups of measure-preserving transformations: a simple proof of Wiener’s ergodic theorem. Ann. Probab. 9(3), 504–509 (1981) 5. Berger, M.: Third-order link integrals. J. Phys. A 23(13), 2787–2793 (1990) 6. Bott, R., Tu, L.: Differential forms in algebraic topology. Volume 82 of Graduate Texts in Mathematics. New York: Springer-Verlag, 1982 7. Cantarella, J.: A general mutual helicity formula. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2003), 2771–2779 (2000) 8. Cantarella, J., DeTurck, D., Gluck, H.: The Biot-Savart operator for application to knot theory, fluid dynamics, and plasma physics. J. Math. Phys. 42(2), 876–905 (2001) 9. Cantarella, J., DeTurck, D., Gluck, D., Teytel, M.: Isoperimetric problems for the helicity of vector fields and the Biot-Savart and curl operators. J. Math. Phys. 41(8), 5615–5641 (2000) 10. Cantarella, J., Parsley, J.: A new cohomological formula for helicity in R 2k+1 reveals the effect of a diffeomorphism on helicity. http://arxiv.org/abs/0903.1465v1[math,GT], 2009 11. Chavel, I.: Eigenvalues in Riemannian geometry. Includes a chapter by Burton Randol. With an appendix by Jozef Dodziuk, Volume 115 of Pure and Applied Mathematics. Orlando, FL: Academic Press Inc., 1984
456
R. Komendarczyk
12. DeTurck, D., Gluck, H., Komendarczyk, R., Melvin, P., Shonkwiler, C., Vela-Vick, D.: Triple linking numbers, Hopf invariants and Integral formulas for three-component links. http://arxiv.org/abs/0901. 1612v1[math,GT], 2009 13. Evans, N.W., Berger, M.A.: A hierarchy of linking integrals. In: Topological aspects of the dynamics of fluids and plasmas (Santa Barbara, CA, 1991), Volume 218 of NATO Adv. Sci. Inst. Ser. E Appl. Sci., Dordrecht: Kluwer Acad. Publ., 1992, pp. 237–248 14. Freedman, M.: Zeldovich’s neutron star and the prediction of magnetic froth. In: The Arnoldfest (Toronto, ON, 1997), Volume 24 of Fields Inst. Commun., Providence, RI: Amer. Math. Soc., 1999, pp. 165–172 15. Freedman, M., He, Z.: Divergence-free fields: energy and asymptotic crossing number. Ann. of Math. (2) 134(1), 189–229 (1991) 16. Gambaudo, J.-M., Ghys, É.: Enlacements asymptotiques. Topology 36(6), 1355–1379 (1997) 17. Hatcher, A.: Algebraic topology. Cambridge: Cambridge University Press, 2002 18. Khesin, B.: Ergodic interpretation of integral hydrodynamic invariants. J. Geom. Phys. 9(1), 101– 110 (1992) 19. Khesin, B.: Topological fluid dynamics. Notices Amer. Math. Soc. 52(1), 9–19 (2005) 20. Kohno, T.: Loop spaces of configuration spaces and finite type invariants. In: Invariants of knots and 3-manifolds (Kyoto, 2001), Volume 4 of Geom. Topol. Monogr. Coventry: Geom. Topol. Publ., 2002, pp. 143–160 (electronic) 21. Koschorke, U.: A generalization of Milnor’s µ-invariants to higher-dimensional link maps. Topology 36(2), 301–324 (1997) 22. Koschorke, U.: Link homotopy in S n × Rm−n and higher order µ-invariants. J. Knot Theory Ramifications 13(7), 917–938 (2004) 23. Kotschick, D., Vogel, T.: Linking numbers of measured foliations. Ergodic Theory Dynam. Systems 23(2), 541–558 (2003) 24. Laurence, P., Avellaneda, M.: A Moffatt-Arnold formula for the mutual helicity of linked flux tubes. Geophys. Astrophys. Fluid Dynam. 69(1–4), 243–256 (1993) 25. Laurence, P., Stredulinsky, E.: Asymptotic Massey products, induced currents and Borromean torus links. J. Math. Phys. 41(5), 3170–3191 (2000) 26. Laurence, P., Stredulinsky, E.: A lower bound for the energy of magnetic fields supported in linked tori. C. R. Acad. Sci. Paris Sér. I Math. 331(3), 201–206 (2000) 27. Mayer, C.: Topological link invariants of magnetic fields. Ph.D. thesis, 2003 28. Milnor, J.: Link groups. Ann. of Math. (2) 59, 177–195 (1954) 29. Milnor, J.: Isotopy of links. In: R. Fox, editor, Algebraic Geometry and Topology, Princeton, NJ: Princeton University Press, 1957, pp. 280–306 30. Milnor, J.: Topology from the Differentiable Viewpoint. Chapter 7 in Princeton Landmarks in Mathematics and Physics. Princeton, NJ: Princeton University Press, 1997 (Revised reprint of 1965 original) 31. Mitrea, D., Mitrea, M., Taylor, M.: Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds. Mem. Amer. Math. Soc. 150, no. 713, Providence, RI: Amer. Math. Soc., 2001 32. Priest, E.: Solar Magnetohydrodynamics. Dordrecit: D. Rediel Publishing Comp., 1984 33. Rivière, T.: High-dimensional helicities and rigidity of linked foliations. Asian J. Math. 6(3), 505– 533 (2002) 34. Schwarz, G.: Hodge decomposition—a method for solving boundary value problems. Vol. 1607 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1995 35. Spera, M.: A survey on the differential and symplectic geometry of linking numbers. Milan J. Math. 74, 139–197 (2006) 36. Bodecker, H.v., Hornig, G.: Link invariants of electromagnetic fields. Phys. Rev. Lett. 92(3), 030406, 4 (2004) 37. Verjovsky, A., Vila Freyer, R.F.: The Jones-Witten invariant for flows on a 3-dimensional manifold. Commun. Math. Phys. 163(1), 73–88 (1994) 38. Vogel, T.: On the asymptotic linking number. Proc. Amer. Math. Soc. 131(7), 2289–2297 (2003) (electronic) 39. Woltjer, L.: A theorem on force-free magnetic fields. Proc. Nat. Acad. Sci. U.S.A. 44, 489–491 (1958) Communicated by P.T. Chru´sciel
Commun. Math. Phys. 292, 457–477 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0898-x
Communications in
Mathematical Physics
Convergence to SPDEs in Stratonovich Form Guillaume Bal Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA. E-mail: [email protected] Received: 8 September 2008 / Accepted: 28 April 2009 Published online: 15 August 2009 – © Springer-Verlag 2009
Abstract: We consider the perturbation of parabolic operators of the form ∂t + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems. 1. Introduction We consider the parabolic equation ∂u ε 1 x + P(x, D)u ε − d q u ε = 0, ∂t ε ε2 u ε (0, x) = u 0 (x),
(1)
where P(x, D) is an elliptic pseudo-differential operator with principal symbol of order m > d and x ∈ Rd . The initial condition u 0 (x) is assumed to belong to L 1 (Rd )∩L 2 (Rd ). We assume that q(x) is a mean zero, Gaussian, stationary field defined on a probability space (, F, P) with integrable correlation function R(x) = E{q(0)q(x)}.
458
G. Bal
The main objective of this paper is to construct a solution to the above equation in L 2 ( × Rd ) uniformly in time on bounded intervals (see Theorem 3 below) and to show that the solution converges in distribution as ε → 0 to the unique mild solution of the following stochastic partial differential equation (SPDE): ∂u + P(x, D)u − σ u ◦ W˙ = 0, ∂t u(0, x) = u 0 (x),
(2)
where W˙ denotes spatial white noise, ◦ denotes the Stratonovich product and sigma is defined as ˆ σ 2 := (2π )d R(0) E{q(0)q(x)}d x. (3) = Rd
We denote by G(t, x; y) the Green’s function associated to the above unperturbed operator. In other words, G(t, x; y) is the distribution kernel of the operator e−t P(x,D) . Our main assumptions on the unperturbed problem are that G(t, x; y) = G(t, y; x) is continuous and satisfies the following regularity conditions: sup |G(t, x; y)|d x + sup t α |G(t, x; y)|2 d x + sup t α |G(t, x; y)| < ∞. t,y
Rd
t,y
Rd
t,x,y
(4) d . Note that 0 < α < 1. Note that the L 2 bound is a Here, we have defined α := m consequence of the L 1 and L ∞ bounds. Such regularity assumptions may be verified e.g. for parabolic equations with m = 2 and d = 1 or more generally for equations with m = 2n an even number and d < m. The convergence of the random solution to the solution of the SPDE is obtained under the additional continuity constraint d . sup s γ |G(s, x, ζ ) − G(s, x + y, ζ )|d x → 0 as y → 0 for γ = 2 1 − m Rd s∈(0,T ),ζ
(5) Such a constraint may also be verified for Green’s functions of parabolic equations with m m = 2n and d < m as well as for operators of the form P(x, D) = (− ) 2 ; see Lemma 4.1 below. We look for mild solutions of (2), which we recast as t u(t, x) = e−t P(D) u 0 (x) + G(t − s, x; y)u(s, y) ◦ σ dW (y)ds. (6) 0
Rd
Here, dW is the standard Wiener measure on Rd and ◦ means that the integral is defined as a (anticipative) Stratonovich integral. In Sect. 2, we define the Stratonovich integral for an appropriate class of random variables and construct a solution to the above equation in L 2 ( × Rd ) uniformly in time on bounded intervals by the method of Duhamel expansion; see Theorem 1 below. In Sect. 3, we show that the solution to the above equation is unique in an adapted functional setting. The convergence of the solution u ε (t) to its limit u(t) is addressed in Sect. 4; see Theorem 4 below. The analysis of stochastic partial differential equations of the form (2) with m = 2 and with the Stratonovich product replaced by an Itô (Skorohod) product or a Wick product
Convergence to SPDEs in Stratonovich Form
459
and the white noise in space replaced by a white noise in space time is well developed; we refer the reader to e.g. [6,9,12,15,16,22]. The case of space white noise with Itô product is analyzed in e.g. [10]. One of the salient features obtained in these references is that solutions to stochastic equations of the form (2) are found to be square-integrable for sufficiently small spatial dimensions d and to be elements in larger distributional spaces for larger spatial dimensions; see in particular [6] for sharp criteria on the existence of locally mean square random processes solution to stochastic equations. The theory presented in this paper shows that the solution to (2) may indeed be seen as the ε → 0 limit of solutions to a parabolic equation (1) with highly oscillatory coefficient when the spatial dimension is sufficiently small. In larger spatial dimensions, the behavior observed in [2] is different. The solution to (1) with a properly scaled potential m (of amplitude proportional to ε− 2 for m < d) converges to the deterministic solution of a homogenized equation, at least for sufficiently small times. The solution to a stochastic model of the form (2) with multiplicative noise interpreted either as an Itô product or a Stratonovich product, when it exists, no longer represents the asymptotic behavior of the solution to an equation of the form (1) with highly oscillatory random coefficients. The justification for the stochastic models of the form (2) is then more difficult. The analysis of equations with highly oscillatory random coefficients of the form (1) has also been performed in other similar contexts. We refer the reader to [18] for a recent analysis of the case m = 2 and d = 1 with much more general potentials than the Gaussian potentials considered in this paper. When the potential has smaller amplitude, then the limiting solution as ε → 0 is given by the unperturbed solution of the parabolic equation where q has been set to 0. The analysis of the random fluctuations beyond the unperturbed solution was addressed in e.g. [1,8]. Equation (1) may also be seen as a continuous version of the parabolic Anderson problem; see e.g. [5].
2. Stratonovich Integrals and Duhamel Solutions The analysis of (6) requires that we define the multi-parameter Stratonovich integral used in the construction of a solution to the SPDE. The construction of Stratonovich integrals and their relationships to Itô integrals is well-studied. We refer the reader to e.g. [7,11,13,17,20]. The construction that we use below closely follows the functional setting presented in [14]. The convergence of processes to multiple Stratonovich integrals may be found in e.g. [3,4]. Let f (x1 , . . . , xn ) be a function of n variables in Rd . We want to define the iterated Stratonovich integral In ( f ). Let us first assume that f separates as a product of n functions defined on Rd , i.e., f (x1 , . . . , xn ) = nk=1 f k (xk ). Then we define In
n
f k (xk ) =
k=1
n
I1 ( f k (xk )),
(7)
k=1
where I1 ( f ) = Rd f (x)dW (x) is the usual multi-parameter Wiener integral. It then remains to extend this definition of the integral to more general functions f (x). We define the symmetrized function f s(x1 , . . . , xn ) =
1 f (xs(1) , . . . , xs(n) ), n! s∈Sn
(8)
460
G. Bal
where the sum is taken over the n! permutations of the variables x1 , . . . , xn . We then define In ( f ) = In ( f s) and thus now consider functions that are symmetric in their arguments. For the rest of the paper, we write Stratonovich integrals using the notation dW rather than ◦ dW . For the Itô convention of integration, we use the notation δW . Let f and g be two functions of n variables. We formally define the inner product
f, gn = E f (x)dW (x1 ) . . . dW (xn ) g(x)dW (x1 ) . . . dW (xn ) nd Rnd R f (x)g(x )E {dW (x1 ) . . . dW (xn )dW (xn+1 ) . . . dW (x2n )} , (9) = R2nd
since the latter has to hold for functions defined as in (7). Here, x = (xn+1 , . . . , x2n ). We need to expand the moment of order 2n of Gaussian random variables. The moment is defined as follows: 2n E dW (xk ) = δ(xk − xl(k) )d xk d xl(k) . (10) p∈P k∈A0 (p)
k=1
Here, p runs over all possible pairings of 2n variables. There are card(P) = cn =
(2n − 1)! (2n)! = = (2n − 1)!! n−1 (n − 1)!2 n!2n
(11)
such pairings. Each pairing is defined by a map l = l(p) constructed as follows. The domain of definition of l is the subset A0 = A0 (p) of {1, . . . , 2n} and the image of l is B0 = B0 (p) = l(A0 ) defined as the complement of A0 in {1, . . . , 2n}. The cardinality of A0 and B0 is thus n and there are cn choices of the function l such that l(k) ≥ k + 1. The formula (10) thus generalizes the case n = 1, where E{dW (x)dW (y)} = δ(x − y)d xd y. We extend by density the iterated Stratonovich integral defined in (7) to the Banach space Bn of functions f that are bounded for the norm ⎛
f n = ⎝
2nd p∈P R
| f ⊗ f |(x1 , . . . , x2n )
⎞1 2
δ(xk − xl(k) )d xk d xl(k) ⎠ .
k∈A0 (p)
(12) The above Banach space may be constructed as the completion of smooth functions with compact support for the above norm [19]. Since the sum of product of functions of one d−dimensional variable are dense in the space of continuous functions, they are dense in the above Banach space and the Stratonovich integral is thus defined for such integrands f (x). A more explicit expression may be obtained for the above norm for functions f (x) that are symmetric in their arguments. Since we do not use the explicit expression in this paper, we shall not derive it explicitly. We note however that
f 2n = E{In+n (| f ⊗ f |)},
(13)
since In ( f )In ( f ) = I2n ( f ⊗ f ). Note that the above space is a Banach subspace of the Hilbert space of square integrable functions since the L 2 norm of f appears for the pairing k δ(xk − xk+n ). Note also
Convergence to SPDEs in Stratonovich Form
461
that the above space is dense in L 2 (Rnd ) for its natural norm. Indeed, let f be a square integrable function. We can construct a sequence of functions f k that vanish on a set of measure k −1 in the vicinity of the sets of Lebesgue measure 0 where the distributions δ(xk − xl ), 1 ≤ k, l ≤ n, are supported and equal to f outside of this set. For such functions, we verify that f k n is the L 2 (Rnd ) norm of f k . Moreover, f k converges to f as k → ∞ as an application of the dominated Lebesgue convergence theorem so that Bn is dense in L 2 (Rnd ). Note finally that the above expression still defines a norm for functions that are not necessarily symmetric in their arguments. This norm applied to non-symmetric functions is not optimal as far as the definition of iterated Stratonovich integrals is concerned since many cancellations may happen by symmetrization (8). However, the above norm is sufficient in the construction of a Duhamel expansion solution to the SPDE. Duhamel solution. Let us define formally the integral Hu(t, x) = σ
t 0
Rd
G(t − s, x; y)u(s, y)dW (y)ds,
(14)
where we recall that dW means an integral in the Stratonovich sense. The Duhamel solution is defined formally as u(t, x) =
∞
u n (t, x), u n+1 (t, x) = Hu n (t, x), u 0 (t, x) = e−t P(x,D) [u 0 (x)],
n=0
(15) where u 0 (x) is the initial conditions of the stochastic equation, which we assume is integrable. The above solution is thus defined formally as a sum of iterated Stratonovich integrals u n (t, x) = In ( f n (t, x, ·)). The main result of this section is the following. Theorem 1. Let u(t, x) be the function defined in (15). The iterated integrals u n (t, x) = In ( f n (t, x, ·)) are defined in L 2 (Rd ; Bn ) uniformly in time t ∈ (0, T ) for all T > 0 and n ≥ 1. When the initial condition u 0 (x) ∈ L 1 (Rd ) ∩ L 2 (Rd ), then u(t, x) is a mild solution to the SPDE in L 2 (Rd × ) uniformly in time t ∈ (0, T ) for all T > 0. When u 0 (x) ∈ L 1 (Rd ), then the deterministic component u 0 (t, x) in u(t, x) satisfies d t 2m u 0 (t, x) ∈ L 2 (Rd ) uniformly in time. Proof. The L 2 norm of u(t, x) is defined by E{u 2 (t, x)}d x = E{In+m ( f n (t, x, ·) ⊗ f m (t, x, ·))}d x Rd
d n,m≥0 R
≤
d n,m≥0 R
E{In+m (| f n (t, x, ·) ⊗ f m (t, x, ·)|)}d x.
We now prove that the latter is bounded uniformly in time on compact intervals. The proof shows that f n (t, ·) is also uniformly bounded in L 2 (Rd ; Bn ) so that the iterated integrals u n (t, x) are indeed well defined.
462
G. Bal
Note that n + m = 2n¯ for otherwise the above integral vanishes. Then, using the notation t0 = s0 = t, we have In,m (t) =
Rd
=
E{In+m (| f n (t, x) ⊗ f m (t, x)|)}d x n−1 tk
Rd k=0 0
×
×
Rd
n−1
Rdn k=0
n G(tn , xn ; ξ )u 0 (ξ )dξ dtk k=1
m−1 sl l=0
|G|(tk − tk+1 , xk ; xk+1 )
0
m−1
Rdm l=0
|G|(sl − sl+1 , yl ; yl+1 )
×δ(x0 − x)δ(y0 − x)σ n+m E
n
dW (xk )
k=1
Rd
m
m G(sm , ym ; ζ )u 0 (ζ )dζ dsl l=1
dW (yl ) d x.
l=1
Using the fact that 2ab ≤ a 2 + b2 with a and b the Green’s functions involving x and d
the fact that τ m G 2 (τ, x, y)d x is uniformly bounded, we bound the integral in x by a d constant. Let us define φ(s) = |t − s|− m . As a consequence, we obtain that In,m (t)
t
φ(t1 )
n−1 tk
|G|(tk − tk+1 , xk ; xk+1 )
Rd(n−1) k=1
k=1 0
0
n−1
n × G(tn , xn ; ξ )u 0 (ξ )dξ dtk Rd
×
k=1
t m−1 sl 0 l=1
0
m−1
Rd(m−1) l=1
|G|(sl − sl+1 , yl ; yl+1 )
m × G(sm , ym ; ζ )u 0 (ζ )dζ dsl d R
l=1
×σ
n+m
E
n k=1
dW (xk )
m
dW (yl ) .
l=1
Here a b means that a ≤ Cb for some constant C > 0. In the above term φ(t1 ) should be replaced by φ(t1 ) + φ(s1 ). The second contribution involving φ(s1 ) is treated in the same manner as that involving φ(t1 ), which we now analyze.
Convergence to SPDEs in Stratonovich Form
463
Let us re-label xn+l = yl and tn+l = sl for 1 ≤ l ≤ m. We also define x = (x0 , . . . , xn+m+1 ). Then we find that 2n¯ n+m Hn,m (t, x) E dW (xk ) , In,m (t) ≤ σ ¯ R2nd
t
Hn,m (t, x) =
φ(t1 )
0
n−1 tk
0 l=1
k=1 0
×
Rd
k=1 t m−1 tn+1+l n+m−1 0
G(tn , xn ; ξ )u 0 (ξ )dξ
|G|(tk − tk+1 , xk ; xk+1 )
k=1,k =n
Rd
n+m G(tn+m , xn+m ; ζ )u 0 (ζ )dζ dtk . k=1
We now recall the pairings introduced in (10) and replace n by n¯ there. Let us introduce the notation ⎧ ⎨ xk+1 k = n, n + m k = n, n + m t k=n τk = k+1 yk = ξ 0 k = n, n + m, ⎩ζ k = n + m, so that Hn,m (t, x) is bounded by
R2d
t
φ(t1 )
0
n−1 tk
k=1 0
2n¯ t m−1 tn+l
0 l=1
0
n+m
|G|(tk − τk , xk ; yk )|u 0 (ξ )| |u 0 (ζ )| dξ dζ
k=1
dtk .
k=1
Now, we have for each pairing p ∈ P, 2n¯
|G|(tk − τk , xk ; yk ) =
k=1
|G|(tk − τk , xk ; yk )|G|(tl(k) − τl(k) , xl(k) ; yl(k) ),
k∈A0
and as a consequence, using the delta functions appearing in (10), In,m (t) ≤ σ n+m × ≤
2d p∈P R
p∈P 0
φ(t1 )
n−1 tk k=1 0
0
t m−1 tn+l 0 l=1
|u 0 (ξ )| |u 0 (ζ )|
0
n+m |G|(tk −τk , xk ; yk )|G|(tl(k) −τl(k) , xk ; yl(k) )d xk dξ dζ dtk
¯ Rnd k∈A0
t
t
φ(t1 )
k=1 n−1 k=1 0
tk
t m−1
0 l=1
tn+l 0
k∈A0
C 2
u
dtk , 0 1 L (tk(k) − τk(k) )α n+m
k=1
(16) for some positive constant C in which we absorb σ 2 . On the second line above, the yl(k) are evaluated at xl(k) = xk . The function k → k(k) for k ∈ A0 is at the moment an arbitrary function such that k(k) = k or k(k) = l(k). The last line is obtained iteratively in increasing values of k in A0 by using that one of the Green’s function is integrable in xk uniformly in the other variables and that the other Green’s function is bounded independent of the spatial variables by a constant times the time variable to the power −α,
464
G. Bal
d where we recall that α = m . We have used here assumption (4). It then remains to integrate in the variables ξ and ζ and we use the initial condition u 0 (x) for this. Let us now choose the map k(k). It is constructed as follows. When both k and l(k) belong to {1, . . . , n} or both belong to {n + 1, n + m}, then we set k(k) = k. When k ∈ {1, . . . , n} and l(k) ∈ {n + 1, n + m} (i.e., when there is a crossing from the n first variables to the m last variables), then we choose k(k) = k for half of these crossings and k(k) = l(k) for the other half. When the number of crossings is odd, the last crossing is chosen with k(k) = k. Let us define A10 = k(A0 ) ∩ {0, . . . , n} and A20 = k(A0 )\A10 . Let n 0 = n 0 (p) be the number of elements in A10 and m 0 = m 0 (p) be the number of elements in A20 such that n 0 + m 0 = n. ¯ Let p = p(p) be the number of crossings in p. Then, by construction of m, we have p+1 n−p m − p p n0 = + , m0 = + , (17) 2 2 2 2 p+1 p p+1 p where [ p+1 2 ] = 2 if p is odd and 2 if p is even, with [ 2 ] + [ 2 ] = p. Thus, n 0 is n+1 m bounded by 2 and m 0 by 2 . We thus obtain that ⎡ ⎤ n n−1 t ⎢ k ⎥ 1 In,m (t) ≤ C n¯ u 0 2L 1 φ(t1 ) dtk ⎦ ⎣ α (tk − tk+1 ) 0 1
⎡
p∈P
m−1 sl
⎢ ×⎣ l=0
0
k=0
n+l∈A20
k∈A0
⎤
k=1
m 1 ⎥ dsl ⎦, (sl − sl+1 )α l=1
with the convention that t0 = s0 = t, tn+1 = 0 and sm+1 = 0. It remains to estimate the time integrals, which are very small, and sum over a very large number of them. It turns out that these integrals admit explicit expressions. The construction of the mapping k(k) ensures that the number of singular terms of the form τ −α is not too large in the integrals over the t and the s variables. Let αk for 0 ≤ k ≤ n be defined such that α0 = α, αk = α for k ∈ A10 and αk = 0 otherwise. Still with the convention that tn+1 = 0, we thus want to estimate In = In (p) =
n−1 tk
n
k=0 0
k=0
n 1 dtk . (tk − tk+1 )αk
(18)
k=1
The integrals are calculated as follows. Let us consider the last integral: 0
tn−1
1 1 1−β dtn = tn−1 n−1 (tn−1 − tn )αn−1 tnαn
0
1
1 du, (1 − u)αn−1 u αn
where we define βn = αn and βm = βm+1 + αm for 0 ≤ m ≤ n − 1. The latter integral is thus given by 1−β
tn−1 n−1 B(1 − βn , 1 − αn−1 ),
Convergence to SPDEs in Stratonovich Form
465
where B(x, y) = (x)(y) (x+y) is the Beta function and (x) the Gamma function equal to (x − 1)! for x ∈ N∗ . The integration in tn−2 then yields
tn−2
1−β
tn−1 n−1
2−β
(tn−2 − tn−1 )αn−2
0
dtn−2 = tn−2 n−2 B(2 − βn−1 , 1 − αn−2 ).
By induction, we thus obtain that n−β0
I n = t0
n−1
n−1
n−β0
B(n − k − βk+1 , 1 − αk ) = t0
k=0
k=0
(n − k − βk+1 )(1 − αk ) . (n − k + 1 − βk+1 − αk ) (19)
Since βk+1 + αk = βk , we obtain by telescopic cancellations that n−β0
I n = t0
n−1 (1 − βn ) (1 − αk ). (n + 1 − β0 ) k=0
Then with our explicit choices for the coefficients αk above, we find that β0 = (n 0 + 1)α so that n−(n 0 +1)α (1 − αn )
I n = t0
n 0 (1 − α)
(n + 1 − (n 0 + 1)α)
.
For a fixed p, we see that the contribution of the time integrals in In,m (t) is bounded by a constant (since (1 − α) is bounded as α < 1) times ¯ ¯ t0n+m−α n−α t0n+m−α n−α n¯ (1 − α) n¯ (1 − α) ≤ (n + 1 − (n 0 + 1)α)(m + 1 − m 0 α) ((n + 1)(1 − α2 ) − α)((m + 1)(1 − α2 )) 1
2 z z based on the values of n 0 and m 0 . Using Stirling’s formula (z) ∼ ( 2π z ) ( e ) so that z z (z) is bounded from below by ( C ) for C < e, we find that the latter term is bounded by
¯ t0n+m−α n−α C n+m α
α
n n(1− 2 ) m m(1− 2 )
,
(20)
for some positive constant C. The latter bound holds for each p ∈ P. Using the Stirling formula again, we observe that the number of graphs in P is bounded by ( 2en¯ )n¯ . As a consequence, we have n¯ 2n¯ CnCm ¯ Im,n ≤ t0n+m−α n−α (21) α α . n(1− e 2 ) m m(1− 2 ) n Using the concavity of the log function, we have n+m 2 n + m2 n + m n+m nn m m ≥ ≥ , n+m 2 so that n
m
n¯ n¯ ≤ C n C m n 2 m 2 .
466
G. Bal
As a consequence, we have (n+m)(1− α2 )−α
In,m ≤ Jn,m (t) := t0
CnCm
1 n
n 2 (1−α)
m
m 2 (1−α)
.
(22)
The bound with n = m shows that for n ≥ 1, u n (t, x) belongs to L 2 (Rd ; Bn ) uniformly in time on compact intervals since 2(1−α) > 0. Now the deterministic component u 0 (t, x) corresponding to n = m = 0 is in L 2 (Rd ) uniformly in time when u 0 (x) ∈ L 2 (Rd ), α while t 2 u 0 (t, x) is in L 2 (Rd ) uniformly in time when u 0 (x) ∈ L 1 (Rd ). Upon summing the above bound over n and m, we indeed deduce that u(t, x) belongs to L 2 ( × Rd ) uniformly in time on compact intervals when u 0 ∈ L 2 (Rd ). The above uniform convergence shows that Hu(t, x) is $ well defined in L 2 ( × Rd ) uniformly in time. Moreover, we verify that Hu(t, x) = n≥1 u n (t, x) = u(t, x) − u 0 (t, x). This shows that u(t, x) is a mild solution of the stochastic partial differential equation and concludes the proof of the theorem. 3. Uniqueness of the SPDE Solution Let us assume that two solutions exist in a linear vector space M. Then their difference, which we call u, solves the equation u = Hu = H p u, for all p ≥ 0. The space M is defined so that H p u is well-defined and is constructed as follows. We construct u ∈ M as a sum of iterated Stratonovich integrals In ( f n (t, x, ·)). u(t, x) = n≥0
Because the iterated Stratonovich integrals do not form an orthogonal basis of random variables in L 2 (), the above sum is formal and needs to be defined carefully. We need to ensure that the sum converges in an appropriate sense and that M is closed under the application of H. One way to do so is to construct u(t, x) using the classical Wiener-Itô chaos expansion Im (gm (t, x, ·)), u(t, x) = m≥0
where Im is the iterated Itô integral, and to show that the above series is well defined. We then also impose that the chaos expansion of H p u is also well-defined. We first need a calculus to change variables from a definition in terms of iterated Stratonovich integrals to one in terms of iterated Itô integrals. This is done by using the Hu-Meyer formulas. We re-derive this expression as follows. We denote by δW an Itô integral and by dW a Stratonovich integral. We project Stratonovich integrals onto the orthogonal basis of Itô integrals as follows E{In ( f n )Im (φm )} = E{Im (gm )Im (φm )} = m! gm φm d x, Rmd
Convergence to SPDEs in Stratonovich Form
467
where φm is a test function. We find that E{In ( f n )Im (φm )} is equal to f n (x1 , . . . , xn )φm (y1 , . . . , ym )E{dW (x1 ) . . . dW (xn )δW (y1 ) . . . δW (ym )}. R(n+m)d
The moment of product of Gaussian variables is handled as in (10) with the exception that E{δW (yk )δW (yl )} = 0 for k = l by renormalization of the Itô-Skorohod integral. The functions f n and φm are symmetric in their arguments (i.e., invariant by permutation of their variables). We observe that the variables y need be paired with m variables x. There are mn ways of pairing the y variables. There remain n − m = 2k variables that need be paired, for a possible number of pairings equal to (2k − 1)! . (k − 1)!2k−1 The above term is thus given by k m + 2k (2k − 1)! d xl f m+2k (y1 , . . . ym , x1 , x1 , . . . xk , xk ) m (k − 1)!2k−1 l=1
×φm (y1 , . . . ym )
m
dy p .
p=1
This shows that gm is given by (m + 2k)! gm (x1 , . . . , xm ) = m!k!2k
f m+2k (x1 , . . . , xm , y1⊗2 , . . . , yk⊗2 )
k
dyk .
p=1
Here y ⊗2 ≡ (y, y). The coefficients gm are therefore obtained by integrating n − m factors pairwise in the coefficients f n . This allows us to write the iterated Stratonovich integral as a sum of iterated Itô integrals as follows: n
In ( f n ) =
[2] k=0
n! In−2k (n − 2k)!k!2k
Rkd
f n (xn−2k , y ⊗2 )dy .
This is the Hu-Meyer formula. More interesting for us is the reverse change of coordinates. Let us define formally f = In ( f n ) = Im (gm ). n≥0
Then we find that gm (x) =
m≥0
(m + 2k)! k≥0
m!k!2k
Rkd
f m+2k (x, y ⊗2 )dy.
The square integrability of the coefficients gm is a necessary condition for the random variables f to be square integrable, and more generally, to be in larger spaces of distributions [9]. The above formula provides the type of constraints we need to impose on
468
G. Bal
the traces of the coefficients f n . For square integrable variables, we consider the normed vector space M f of random variables f = In ( f n ), n≥0
where the coefficients { f n } are bounded for the norm ⎛ ⎜
f M f = ⎝ m!
m≥0
⎛ ⎝
(m + 2k)! k≥0
m!k!2k
⎞2
Rkd
| f m+2k |(x, y
⊗2
⎞1 2
⎟ )dy ⎠ d x ⎠ < ∞.
Note that the above defines a norm as the triangle inequality is clearly satisfied and for k = 0, we find that the L 2 norm of each f m has to vanish, so that f m ≡ 0 for all m when the norm vanishes. Note also that M f is a dense subset of L 2 () as any square integrable function gm may be approximated by a function f mk , which vanishes in a set of Lebesgue measure at most k −1 in the vicinity of the measure 0 set of diagonals given by k the support of the distributions δ(xk − xl ). For such functions, we verify that f mk = gm k converges so that the Itô and Stratonovich iterated integrals agree. We also have that gm to gm by density. Since every square integrable random variable may be approximated by a finite number of terms in the chaos expansion, this concludes our proof that M f is dense in L 2 () equipped with its natural metric. Let us now move to the analysis of the stochastic integral H. It turns out that M f is not stable under H nor is it in any natural generalization of M f . Let us define u(t, x) = In ( f n (t, x, ·)), Hu(t, x) = In ((H f )n (t, x, ·)). n≥0
n≥0
We then observe that
H f n+1 (t, x, y) = σ s
t
G(t − s, x, y1 ) f n (s, y)ds ,
0
where s is the symmetrization with respect to the d(n + 1)−dimensional y variables. Let us consider H2 f n+2 , which depends only on f n . Let cm,k = (m+2k)! be the coefficient m!k!2k that appears in the definition of gm . Then, for H2 f n+2 relative to f n , the coefficients indexed by k are essentially replaced by coefficients indexed by k + 1. Since cm,k+1 is not bounded by a multiple of cm,k uniformly, the integral operator H2 cannot be bounded in M f . The reason why solutions to the stochastic equation may still be found is because the integrations in time after n iterations of the integral H provide a factor inversely proportional to n!. This factor allows us to stabilize the growth in the traces that appears by going from cm,k to cm,k+1 . Uniqueness of the solution may thus only be obtained in a space where the factor n! appears, at least implicitly. A suitable functional space is constructed as follows. Let gm be the chaos expansion coefficients associated to the coefficients | f n | and gm, p the chaos expansion coefficients associated to the coefficients |H p f n |. Then we impose that the coefficients { f n } be bounded for the norm sup sup
1 2 2 sup < ∞, c p gm, p (t, x, y)d xd y
m≥0 p≥0 t∈(0,T )
(23)
Convergence to SPDEs in Stratonovich Form
469
where c p is an increasing series such that c p → ∞ as p → ∞. Here T is a fixed (arbitrary) positive time. We denote by M = M(T ) the normed vector space of random fields u(t, x) for which the decomposition in iterated Stratonovich integrals satisfies the above constraint. We are now ready to state the main result of this section. Theorem 2. Let T > 0 be an arbitrary time and u 0 (x) ∈ L 1 (Rd )∩ L 2 (Rd ). The solution constructed in Theorem 1 is the unique mild solution to the stochastic partial differential equation (2) in the space M = M(T ) for an appropriate sequence of terms c p → ∞ as p → ∞. Proof. Let us first prove uniqueness in M. We have u = H p u for all p ≥ 0. This implies that gm (t, x, ·) = gm, p (t, x·). The latter converges to 0 in the L 2 sense as p → ∞. This implies that gm (t, x, ·) uniformly vanishes for all m so that u ≡ 0. Let now u(t, x) be given by the following Duhamel expansion: u(t, x)= u n (t, x), u n+1 (t, x) = Hu n (t, x)=Hn+1 u 0 (t, x), u 0 (t, x) = e−t P u 0 (x). n≥0
We thus verify that Hk u(t, x) =
u n (t, x).
n≥k
The proof of construction of the Duhamel solution shows that k 2 E{(H u) (t, x)}d x ≤ E In+m (| f n |⊗| f m |) (t, x)d x n,m≥k
:=
m!
2 gm,k (t, x, y)d xd y
m≥0
converges to 0 and is bounded by a constant we call ck−1 u 0 2L 1 (Rd ) for k ≥ 1. Here, ck
may be chosen independently of u 0 ∈ L 1 (Rd ). For such a sequence of terms ck → ∞ as k → ∞, (23) is clearly satisfied. This shows that u belongs to M so constructed. The same theory holds when the supremum in m is replaced by a sum with weight m! so that M becomes a subspace of L 2 . In some sense, the subspace created above is the largest we can consider that is stable under application of H. When u 0 (x) ∈ L 1 (Rd ) not necessarily in L 2 (Rd ), then the deterministic component u 0 (t, x) is not square integrable uniformly in time. The space M may then be replaced by a different space α where c p in (23) is replaced by t 2 c p . 4. Convergence Result Let us now come back to the solution of the equation with random coefficients (1). The theory of existence for such an equation is very similar to that for the stochastic limit. We define formally the integral t Hε u(t, x) = G(t − s, x; y)u(s, y)qε (y)dyds, (24) 0
Rd
470
G. Bal d
where we have defined qε (y) = ε− 2 q( εy ). The Duhamel solution is defined formally as u ε (t, x) =
∞
u n,ε (t, x), u n+1,ε (t, x) = Hε u n,ε (t, x), u 0 (t, x) = e−t P(x,D) [u 0 (x)],
n=0
(25) where u 0 is the initial condition of the stochastic equation, which we assume is integrable and square integrable. We have the first result: Theorem 3. The function u ε (t, x) defined in (25) solves u ε (t, x) = Hε u ε (t, x) + e−t P(x,D) [u 0 (x)],
(26)
and is in L 2 (Rd × ) uniformly in time t ∈ (0, T ) for all T > 0. Proof. The proof goes along the same lines as that of Theorem 1. The L 2 norm of u ε (t, x) is defined by E{u 2ε (t, x)}d x = E{u n,ε (t, x)u m,ε (t, x)}d x Rd
d n,m≥0 R
≤
d n,m≥0 R
E{|u n,ε |(t, x)|u m,ε |(t, x)}d x ≤ Im,n,ε (t),
where In,m,ε (t) =
n−1 tk
Rd k=0 0
× ×
Rd
n−1
Rdn k=0
|G|(tk − tk+1 , xk ; xk+1 )
n G(tn , xn ; ξ )u 0 (ξ )dξ dtk
m−1 s l l=0
0
k=1
m−1
Rdm l=0
|G|(sl−sl+1 , yl ; yl+1 )
×δ(x0 − x)δ(y0 − x)E{
n
qε (xk )d xk
k=1
m
Rd
m G(sm , ym ; ζ )u 0 (ζ )dζ dsl l=1
qε (yl )dyl }d x.
l=1
Following the proof of Theorem 1, we obtain In,m,ε (t) ≤
¯ R2nd
Hn,m (t, x) E
2n¯
qε (xk )d xk .
k=1
Statement (10) now becomes 2n¯ xk − xl(k) −d d xk d xl(k) , qε (xk )d xk = ε R E ε k=1
p∈P k∈A0 (p)
(27)
Convergence to SPDEs in Stratonovich Form
471
where we recall that R(x) = E{q(0)q(x)} is the correlation function of the Gaussian field q. As in the construction of the Duhamel solution for (2), this yields n−1 t tk t m−1 tn+l φ(t1 ) |u 0 (ξ )| |u 0 (ζ )| In,m,ε (t) ≤ 2d p∈P R
k=1 0
0
0 l=1
¯ Rnd k∈A0
0
× |G|(tk − τk , xk ; yk )|G|(tl(k) − τl(k) , xl(k) ; yl(k) )ε−d n+m xk − xl(k) d x dξ dζ × R d x dtk . k l(k) ε k=1
For each k ∈ A0 considered iteratively with increasing order, the term between parentheses is bounded by the L 1 norm of the Green’s function integrated in xk(k) times the integral of the correlation function in the variable xk (k) , with (k(k), k (k)) = (k, l(k)), which gives a σ 2 contribution, thanks to definition (3), times the L ∞ norm of the Green’s function in the variable xk (k) . Using (4) and the integrability of the correlation function R(x), this shows that n−1 n+m tk t m−1 tn+l t C 2 In,m,ε (t) ≤ φ(t1 )
u
dtk , 0 1 L α (tk(k) − τk(k) ) 0 0 0 0 p∈P
k=1
l=1
k∈A0
k=1
as in the proof of Theorem 1. The rest of the proof is therefore as in Theorem 1 and shows that each u n,ε (t, x) is well defined in L 2 (Rd × ) uniformly in time and that the series defining u(t, x) converges uniformly in the same sense. Mollification and convergence result. We now have defined a sequence of solutions u ε (t, x) and a limiting solution u(t, x). When qε and the white noise W used in the construction of u(t, x) are independent, then the best we can hope for is that u ε converges in distribution to u. The convergence is in fact much stronger by constructing qε d x as a ˆ ) be the power spectrum of q, which is defined as the Fourier mollifier of dW . Let R(ξ transform of R(x). By Bochner’s theorem, the power spectrum is non-negative and we ˆ )) 21 . Let ρ(x) be the inverse Fourier transform of ρ. may define ρ(ξ ˆ ) = ( R(ξ ˆ We may then define q(x) ˜ = ρ(x − y)dW (y), (28) Rd
and obtain a stationary Gaussian process q(x). ˜ This process is mean-zero and its correlation function is given by ˜ ρ(x − y)ρ(y)dy = R(x), R(x) = Rd
by inverse Fourier transform of a product. As a consequence, q(x) and q(x) ˜ have the same law since they are mean zero and their correlation functions agree. The corresponding Duhamel solutions u ε and u˜ ε also have the same law by inspection. It thus obviously remains to understand the limiting law of u˜ ε to obtain that of u ε . It turns out that u˜ ε may be interpreted as a mollifier of u(t, x), the solution constructed in Theorem 1, and as such converges strongly to its limit. In addition to the assumptions on the Green’s function in (4) and (5), we also assume that ρ(x) ∈ L 1 (Rd ). Then we have
472
G. Bal
Theorem 4. Let u ε (t, x) be the solution constructed in Theorem 3 and u(t, x) the solution constructed in Theorem 1. Then we have that u ε (t, x) converges in distribution to u(t, x) as ε → 0. More precisely, let u˜ ε (t, x) be the Duhamel solution corresponding to the random potential q˜ in (28). Then we have that
u˜ ε (t) − u(t) L 2 (Rd ×) → 0,
ε → 0,
(29)
uniformly in time over compact intervals. Proof. Let us drop the upper˜to simplify notation. We have E{(u(t) − u ε (t))2 }d x = δ Iε,n,m (t), δ Iε (t) = δ Iε,n,m (t) =
Rd
Rd
n,m
E{(u n (t) − u n,ε (t))(u m (t) − u m,ε (t))}d x.
Following the proofs of Theorems 1 and 3, we observe that δ Iε,n,m (t) n−1 = ×
tk
Rd k=0 0 m−1 sl 0
l=0
×δ(y0−x)E
n−1
Rdn k=0 m−1
Rdm l=0 n
G(tk − tk+1 , xk ; xk+1 )
Rd
G(tn , xn ; ξ )u 0 (ξ )dξ
Rd
n σ dW (xk )− qε (xk )d xk
k=1
G(sm , ym ; ζ )u 0 (ζ )dζ m
k=1
dtk
k=1
G(sl−sl+1 , yl ; yl+1 )
n
m
dsl δ(x0−x)
l=1 m σ dW (yk )− qε (yl )dyl
l=1
d x.
l=1
Here, we have again that t0 = s0 = t. The integration in x is handled as in the proof of Theorem 1 so that (with φ(t1 ) + φ(s1 ) replaced by φ(t1 ) as above) |δ Iε,n,m (t)| n−1 n−1 n t tk G(tk−tk+1 , xk ; xk+1 ) G(tn , xn ; ξ )u 0 (ξ )dξ dtk φ(t1 ) 0 0 Rd(n−1) Rd k=1 t m−1 sl
×
0 l=1 0 n
×E
k=1 m−1
Rd(m−1) l=1
σ dW (xk ) −
k=1
G(sl − sl+1 , yl ; yl+1 )
n
qε (xk )d xk
k=1
m
Rd
G(sm , ym ; ζ )u 0 (ζ )dζ
σ dW (yk ) −
l=1
k=1 m
dsl
l=1
m l=1
qε (yl )dyl .
The main difference with respect to previous proofs is that we cannot bound the Green’s functions by their absolute values just yet. The moment of Gaussian variables is handled as follows. We recast it as n m n σ dW (xk ) − qε (xk )d xk qε (yl )dyl k=1
k=1
l=1
Convergence to SPDEs in Stratonovich Form
473
1 plus a second contribution that is handled similarly. We denote by δ Iε,n,m (t) the cor2 1 responding contribution in δ Iε,n,m (t) and by δ Iε,n,m (t) = δ Iε,n,m (t) − δ Iε,n,m (t). The above contribution is recast as n q−1
n+m σ dW (xk ) σ dW (xq ) − qε (xq )d xq qε (x p )d x p ,
q=1 p=1
(30)
p=q+1
where we have defined xn+l = yl for 1 ≤ l ≤ m. We have therefore n (or more precisely n ∧ m by decomposing the product over m variables when m < n) terms of the form ⎧ ⎫ 2n¯ ⎨q−1 ⎬ n+m σ dW (xk ) σ dW (xq )−qε (xq )d xq qε (x p )d x p := E ak,ε (d xk ) , E ⎩ ⎭ p=1
p=q+1
k=1
where each measure ak,ε (d xk ) is Gaussian. Then, (10) is replaced in this context by 2n¯ * + ak,ε (d xk ) = E ak,ε (d xk )al(k),ε (d xl(k) ) E p∈P k∈A0 (p)
k=1
:=
h ε,k (xk − xl(k) )d xk d xl(k) .
(31)
p∈P k∈A0 (p)
The functions h ε,k (xk − xl(k) ) come in five different forms according to: E{dW (x)dW (y)} = σ 2 δ(x − y)d xd y, 1 x−y d xd y, E{dW (x)qε (y)dy} = σ d ρ ε ε 1 x−y d xd y, (32) E{qε (x)d xqε (y)dy} = d R ε ε 1 x−y d xd y, E{(dW (x) − qε (x)d x)dW (y)} = σ 2 δ(x − y) − σ d ρ ε ε 1 1 x−y x−y − dR d xd y. E{(dW (x) − qε (x)d x)qε (y)dy} = σ d ρ ε ε ε ε At this point, we have obtained that 1 |δ Iε,n,m (t)| p∈P
n−1 n−1 t tk × φ(t1 ) G(tk−tk+1 , xk ; xk+1 ) 0 Rd(n−1) k=1 k=1 0 t m−1 n sl G(tn , xn ; ξ )u 0 (ξ )dξ dtk × Rd
k=1
0 l=1
0
m−1
Rd(m−1) l=1
G(sl − sl+1 , yl ; yl+1 )
× G(sm , ym ; ζ )u 0 (ζ )dζ dsl h ε,k (xk − xl(k) )d xk d xl(k) . Rd l=1 k∈A0 (p)
m
474
G. Bal
Using the notation as in the proof of Theorem 1, we obtain that n−1 tk t m−1 tn+l t 1 φ(t1 ) u 0 (ξ )u 0 (ζ ) |δ Iε,n,m (t)| 2d ¯ Rnd 0 0 l=1 0 k=1 0 p∈P R
k∈A0 (p)
n+m × G(tk−τk , xk ; yk )G(tl(k)−τl(k) , xl(k) ; yl(k) )h ε,k (xk−xl(k) )d xk d xl(k) dξ dζ dtk k=1 n−1 t tk t m−1 tn+l φ(t1 ) |u 0 (ξ )||u 0 (ζ )| 2d 0 ¯ 0 l=1 0 Rnd k=1 0 k∈A0 (p) p∈P R n+m × G(tk−τk , xk ; yk )G(tl(k)−τl(k) , xl(k) ; yl(k) )h ε,k (xk−xl(k) )d xk d xl(k) dξ dζ dtk . k=1
It remains to handle the multiple integral between absolute values. For k ∈ A0 (p) for which h ε,k is of the form given in the last two lines of (32), we observe that the corresponding term between parentheses in the above expression is of the form G(s, x; ζ )G(τ, y; ξ )h ε (x − y)d xd y R2d 1 x−y G(s, x, ζ ) g = x; ξ ) − G(τ, y; ξ )) dy dx (G(τ, d ε Rd Rd ε G(s, x, ζ ) g(y) (G(τ, x; ξ ) − G(τ, x + εy; ξ )) dy d x, (33) = Rd
Rd
where the function g(x) is given by either g(x) = ±σρ(x)
or
g(x) = ±(R(x) − σρ(x)).
(34)
This is because ρ averages to σ while R averages to σ 2 . Let k0 be the index for which h ε,k0 is in the form of a difference as above. This yields, with g = g[k0 ] as above, 1 (t)| |δ Iε,n,m
2d p∈P R
t
φ(t1 )
n−1 tk k=1 0
0
t m−1 tn+l
0 l=1
|u 0 (ξ )||u 0 (ζ )|
0
¯ Rnd k0 =k∈A0 (p)
× G(tk − τk , xk ; yk )G(tl(k) − τl(k) , xl(k) ; yl(k) )h ε,k (xk − xl(k) ) d xk d xl(k) ×|G(tk(k0 ) − τk(k0 ) , xk(k0 ) ; yk(k0 ) )||g(xk (k0 ) )|dξ dζ
n+m
dtk
k−1
× G(·, ·, ·) − G(·, · + εxk (k0 ) , ·) (tk (k0 ) − τk (k0 ) , xk(k0 ) ; yk (k0 ) ) d xk0 d xl(k0 ) . The above term is now handled as in the proof of Theorem 1. For k = k0 , the bounds are obtained as before because ρ and R are integrable functions by hypothesis. The Green’s function |G(tk(k0 ) −τk(k0 ) , xk(k0 ) ; yk(k0 ) )| is bounded by a constant times |tk(k0 ) −τk(k0 ) |−α . The integration d xk0 d xl(k0 ) = d xk(k0 ) d xk (k0 ) then yields a contribution bounded by |tk (k0 ) − τk (k0 ) |−γ times sup τγ |g(y)| |G(τ, x; ξ ) − G(τ, x + εy; ξ )| d xd y. (35) Mε = τ ∈(0,T ),ξ ∈Rd
R2d
Convergence to SPDEs in Stratonovich Form
475
The presence of the factor γ is sufficient to ensure that Mε converges to 0 as ε → 0. As a consequence, as in the derivation of (16), we obtain that |δ Iε,n,m (t)|
t n−1 tk t m−1 tn+l
p∈P 0 k=1 0
0 l=1
0
n+m 2n Mε φ(t1 ) C dtk . d γ |tk (k0 )−τk (k0 ) | k∈A0 (tk(k) −τk(k) ) m k=1
The factor 2n comes from twice the summed contributions in (30). The presence of the factor γ increases the time integrals as follows. Assume that k0 ≤ n for concreteness; the case k0 ≥ n + 1 is handled similarly. Then β0 in the proof of Theorem 1 should be replaced by β0 +γ . This does not significantly modify the analysis of the functions and the contribution of each graph is still bounded by a term of the form (20). The behavior in time, however, is modified by the presence of the contribution γ and we find that (n+m)(1− α2 )−α−γ
|δ Iε,n,m (t)| ≤ Cn Mε t0
CnCm
1 n
n 2 (1−α)
m
m 2 (1−α)
.
The above bound is of interest for n + m ≥ 2 since the case n = m = 0 corresponds to the ballistic component u 0 (t, x), which is the same for u ε (t, x) and u(t, x) so that δ Iε,0,0 = 0. By choosing γ = 2(1 − α) > 0, we observe that 2(1 − α2 ) − α − γ ≥ 0 for n + m ≥ 2 so that |δ Iε,n,m (t)| is bounded uniformly in time. The new factor n may be absorbed into C n so that after summation over n and m, we get
u ε (t) − u(t) 2L 2 (Rd ×) ≤ C Mε .
(36)
By assumption (5), the integrand in (35) converges point-wise to 0 and an application of the dominated Lebesgue convergence theorem shows that Mε → 0. This concludes the proof of the convergence result. A continuity lemma. We conclude this paper by showing that the constraints (4) and (5) imposed on the Green’s functions of the unperturbed problem throughout the paper are satisfied for a natural class of parabolic operators. m
Lemma 4.1. Let G(t, x) be defined as the Fourier transform of e−t|ξ | , i.e., the Green’s m function of the operator ∂t + (− ) 2 for m > d. Then the conditions in (4) and (5) are satisfied. Moreover, when m is an even number, then Mε in (35) satisfies the bound Mε ε β ,
β = 2(m − d) ∧ 1. d
1
m
Proof. By scaling invariance, we find that G(t, x) = t − m G(1, t − m x). Since |ξ | p e−|ξ | is integrable for all p, we obtain that G(1, x) belongs to C ∞ (Rd ). Since G(1, x) is d bounded, then so is t m G(t, x) uniformly in t and x. By the above scaling, G(t, x) belongs to L 1 (Rd ) uniformly in time if and only if m G(1, x) does. When m in an even integer, then e−|ξ | belongs to S(Rd ), the space of Schwartz functions, so that G(1, x) ∈ S(Rd ) as well. It is therefore integrable and has an integrable gradient. When m is not an even integer, we have m
e−|ξ | − 1 =
∞ (−1)k k=1
k!
|ξ |k m.
476
G. Bal
The Fourier transform of the homogeneous function |ξ |k m is given by [21] c(k)|x|−k m−d ,
c(k) = Cd 2k m
( 21 (km + d)) (− 21 km )
, m
where Cd is a normalization constant independent of k. The Fourier transform of e−|ξ | may then be written as a constant times |x|−(d+m) plus a smoother contribution that converges faster to 0 (for instance because it belongs to some H s (Rd ) with s > d2 + k sufficiently large so that k derivatives of this contribution are integrable). It is therefore integrable for m > 0. The L 2 bound follows from the L 1 and L ∞ bounds. We obtain by scaling invariance and from the definition of G(t, x) that 1 tγ |G(t, x) − G(t, x + εy)|d x = t γ |G(1, x) − G(1, x + t − m εy)|d x. Rd
Rd
The above derivation shows that the gradient of G is also integrable for m > 1 so we 1 may bound the above quantity by t γ (1 ∧ t − m ε|y|). Now, sup(t γ ∧ t γ − m ε|y|) (ε|y|)γ m ∨ ε|y| = (ε|y|)2(m−d) ∨ ε|y|, 1
t
according to γ m < 1 or γ m ≥ 1. We thus obtain (5) by sending εy → 0. When g(y) is sufficiently regular, then we obtain the more precise bound 2(m−d) 2(m−d) Mε ε |g(y)||y| dy ∨ ε |g(y)||y|dy, Rd
provided that the latter integrals are well-defined.
Rd
Acknowledgement. This work was supported in part by NSF Grants DMS-0239097 and DMS-0804696.
References 1. Bal, G.: Central limits and homogenization in random media. Multiscale Model. Simul. 7(2), 677–702 (2008) 2. Bal, G.: Homogenization with large spatial random potential, Submitted, 2009. available at http://www. columbia.edu/~gb2030/PAPERS/large-potential-homogenization.pdf, 2009 3. Bardina, X., Jolis, M.: Weak convergence to the multiple Stratonovich integral. Stochastic Processes Appl. 90, 277–300 (2000) 4. Budhiraja, A., Kallianpur, G.: Two results on multiple Stratonovich integrals. Statistica Sinica 7, 907– 922 (1997) 5. Carmona, R.A., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108(518), 125 (1994) 6. Dalang, R.C.: Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E.’s. Electron. J. Probab. 26, 1–29 (1999) 7. Delgado, R., Sanz, M.: The Hu-Meyer formula for non deterministic kernels. Stochastics Stochastics Rep. 38, 149–158 (1992) 8. Figari, R., Orlandi, E., Papanicolaou, G.: Mean field and Gaussian approximation for partial differential equations with random coefficients. SIAM J. Appl. Math. 42, 1069–1077 (1982) 9. Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic partial differential equations. A modeling, white noise functional approach. Prob. Appl., Boston, MA: Birkhäuser, 1996 10. Hu, Y.: Chaos expansion of heat equations with white noise potentials. Potential Anal. 16, 45–66 (2002) 11. Hu, Y.Z., Meyer, P.-A.: Chaos de Wiener et intégrale de Feynman. In: Séminaire de Probabilités, XXII, Vol. 1321 of Lecture Notes in Math.; Berlin: Springer, pp. 51–71, 1988
Convergence to SPDEs in Stratonovich Form
477
12. Itô, K.: Foundations of stochastic differential equations in infinite-dimensional spaces. Vol. 47 of CBMSNSF Regional Conference Series in Applied Mathematics, Philadelphia, PA:SIAM, 1984 13. Johnson, G.W., Kallianpur, G.: Homogeneous chaos, p-forms, scaling and the feynman integral. Trans. Amer. Math. Soc. 340, 503–548 (1993) 14. Jolis, M.: On a multiple Stratonovich-type integral for some Gaussian processes. J. Theoret. Probab. 19, 121–133 (2006) 15. Nualart, D., Rozovskii, B.: Weighted stochastic sobolev spaces and bilinear spdes driven by space-time white noise. J. Funct. Anal. 149, 200–225 (1997) 16. Nualart, D., Zakai, M.: Generalized Brownian functionals and the solution to a stochastic partial differential equation. J. Funct. Anal. 84, 279–296 (1989) 17. Nualart, D., Zakai, M.: On the relation between the stratonovich and ogawa integrals. Ann. Probab 17, 1536–1540 (1989) 18. Pardoux, E., Piatnitski, A.: Homogenization of a singular random one dimensional PDE. GAKUTO Internat. Ser. Math. Sci. Appl. 24, 291–303 (2006) 19. Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. 2nd ed. New York: Academic Press, Inc., 1980 20. Solé, J.L., Utzet, F.: Stratonovich integral and trace. Stochastics Stochastics Rep. 2, 203–220 (1990) 21. Taylor, M.E.: Partial Differential Equations I. New York: Springer Verlag, 1997 22. Walsh, J.B.: An introduction to stochastic partial differential equations. École d’été de probabilités de Saint-Flour, XIV—1984, Vol. 1180 of Lecture Notes in Math., Berlin: Springer, pp. 265–439, 1986 Communicated by H.-T. Yau
Commun. Math. Phys. 292, 479–510 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0895-0
Communications in
Mathematical Physics
Limit of Fluctuations of Solutions of Wigner Equation Tomasz Komorowski1 , Szymon Peszat2 , Lenya Ryzhik3 1 Institute of Mathematics, UMCS, pl. Marii Curie-Skłodowskiej 1, Lublin 20-031,
´ Poland, and Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-950 Warsaw, Poland. E-mail: [email protected] 2 Institute of Mathematics, Polish Academy of Sciences, Sw. Tomasza 30/7, 31-027 Cracow, Poland. E-mail: [email protected] 3 Department of Mathematics, University of Chicago, Chicago, IL 60637, USA. E-mail: [email protected] Received: 13 October 2008 / Accepted: 25 May 2009 Published online: 14 August 2009 – © Springer-Verlag 2009
Abstract: We consider fluctuations of the solution Wε (t, x, k) of the Wigner equation, which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of Wε (t, x, k) converges as ε → 0 to W¯ (t, x, k) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, Wε (0, x, k) = δ(x) f (k) and f ∈ S(Rd ), then the laws of the rescaled fluctuation Z ε (t) := ε−1/2 [Wε (t, x, k) − W¯ (t, x, k)] converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered. 1. Introduction A weak random potential in the Schrödinger equation i
∂φ 1 + x φ − δV (t, x)φ = 0 ∂t 2
(1.1)
with the parameter δ 1, strongly effects the behavior of solutions on times scales of the order t ∼ O(δ −2 ) and larger. The corresponding rescaled problem for φε (t, x) := ε−d/2 φ(t/ε, x/ε) reads √ ∂φε ε2 t x + x φε − εV , φε = 0, iε (1.2) ∂t 2 ε ε with ε = δ 2 . Physically, on this time scale waves undergo multiple scattering and propagate in all directions. In such regimes behavior of the spatial energy density E ε (t, x) = |φε (t, x)|2 is best described in terms of the Wigner transform [15,21,29] defined as εy ∗ εy dy φε t, x + Wε (t, x, k) = eik·y φε t, x − . 2 2 (2π )d
480
T. Komorowski, S. Peszat, L. Ryzhik
Here and everywhere below a ∗ denotes the complex conjugate of a ∈ C. Note that Wε∗ (t, x, k) = Wε (t, x, k). Since the wave energy density can be decomposed as E ε (t, x) = Wε (t, x, k)dk, it is customary to interpret the Wigner transform as a “phase space resolved” wave energy density. It is well known that, provided that φε L 2 ≤ C, in the limit ε → 0 the Wigner transform converges (possibly after passing to a subsequence) to a limit W (t, x, k) which is a non-negative measure. The Wigner transform of a solution of (1.2) satisfies an evolution equation ∂ Wε i p p Vˆ (t/ε, d p) + k · ∇ x Wε = √ ei p·x/ε [Wε (k − ) − Wε (k + )] . (1.3) ∂t 2 2 (2π )d ε When V (t, x) is a spatially and temporally statistically homogeneous random process with sufficiently fast decaying correlations, the expected value of the limit, W¯ (t, x, k) = EW (t, x, k) satisfies the radiative transport equation [2,8,9,14,22,29,30] 2 p2 dp k ¯ ¯ ˆ − , k − p [W¯ (t, x, p) − W¯ (t, x, k)] Wt + k · ∇ W = R , (1.4) 2 2 (2π )d where R(t, x) = E[V (s, y)V (t +s, x +y)] is the two-point correlation function of the proˆ cess V (t, x) and R(ω, k) is its power spectrum, that is, the Fourier transform of R(t, x). Moreover, it has been shown in [3,9,24] that in many situations the limit is actually selfaveraging, or, more precisely, that for any test function θ ∈ S(R2d ) the process Wε , θ converges, as ε → 0+, to the deterministic limit W¯ , θ in probability. However, this result was generally established only when the initial data for the Wigner transform is sufficiently regular, which may be achieved, for instance, by consideration of mixtures of states. The restriction on the regularity of the initial data is not simply technical: sufficiently singular initial data W0 (x, k) does lead to the absence of self-averaging. In particular, it has been shown in [1] that when the initial data is localized both in space and wave vectors: W0 (x, k) = δ(x)δ(k −k0 ) then the Wigner transform is not self-averaging. The problem of the self-avaraging of the energy density has a practical aspect as recent inversion techniques based on the kinetic limits for wave propagation in random media have been developed and even tested in physical experiments [4–6]. This makes important understanding of the statistics of the fluctuation Wε − W¯ , as well as its dependence on the regularity of the initial data. The first step in this direction was made recently in [7], where the limit of the scintillation of Wε , θ was analyzed in the simplest case when the random process V (t, x) is a white noise in time. The purpose of the present paper is to obtain the weak limit of the full fluctuation process Z ε (t, x, k) =
Wε (t, x, k) − W¯ (t, x, k) √ ε
(1.5)
in the case of the initial data of the form W (0, x, k) = δ(x) f (k). This Cauchy data corresponds to the physically perhaps most interesting case of a point source in a random medium with a smooth distribution of energy over wave vectors. One consequence of [7] is that the size of Wε − W¯ should depend on the regularity of the initial data: if W0 (x, k) is smooth then Wε − W¯ is of √ the size O(εd/2 ) while for a spatially localized initial data as above the fluctuation is O( ε) independent of the dimension. We choose the random
Limit of Fluctuations of Solutions of Wigner Equation
481
potential to be still of the white noise type to simplify the analysis but the main asymptotic results should hold for a much larger class of the random potentials, in particular, for Gaussian and Markovian fields [20]. Let us explain our main result informally. We consider the solution of the random Wigner equation dWε (t, x, k) = −k · ∇x Wε (t, x, k)dt B(d S t, d p) p p +i ei p·x/ε Wε (t, x, k − ) − Wε (t, x, k + ) , d 2 2 (2π )d R (1.6) with the initial data Wε (0, x, k) = δ(x) f (k). Equation (1.6) is a formal analog of the Wigner equation (1.3) when the random potential is a white noise in time, or a limit of (1.3) when V (t, x) is oscillating faster in time than in space. The stochastic integral in the right side is understood in the Stratonovich sense and {B(t), t ≥ 0} is a spatially homogeneous Wiener process with the spectral measure µ. That is, it can be written as B(t) = Bn (t)F (en µ), t ≥ 0, n
where en is a basis of L 2 space over the spectral measure µ and F, or denote the Fourier transform. Then the expectation W¯ (t) = EWε (t) does not depend on ε and satisfies the kinetic equation, also known as the radiative transport equation
∂ W¯ ¯ + k · ∇x W = W¯ (t, x, k − p) − W¯ (t, x, k) µ(dp), ∂t (1.7) ¯ W (0) = δ(x) f (k). ˆ ˆ p) and Note that in case when V (t, x) in (1.1) is a white noise in t then R(ω, p) = R( ˆ p)d p. Eq. (1.7) is a special case of (1.4) with µ(d p) = (2π )−d R( We show that the correction Z ε defined by (1.5) converges weakly to a random limit Z (t, x, k) which satisfies the same kinetic equation (1.7) but with a random initial data: ∂Z + k · ∇x Z = [Z (t, x, k − p) − Z (t, x, k)] µ(d p), ∂t (1.8) Z (0) = δ(x)X (k). Here X (k) is a distribution valued Gaussian random variable given by +∞ σ dBn (s) ei p·(k+σ p/2)s f (k + σ p/2)en ( p)µ(d p). (1.9) X (k) := −i n σ =±1
0
Rd
From a physical standpoint the fact that Z satisfies a deterministic problem with a random initial data is quite natural, the reason being once again the fact that the size of the fluctuation depends on the regularity of the initial data. Solution of the radiative transport equation W¯ (t, x, k) is less singular for t > 0 than the initial data W0 (x, k) √ = δ(x) f (k) – hence, the fluctuation that is produced at positive times is smaller than O( ε), and the main random contribution to Z ε comes from an initial time layer when W¯ (t, x, k) still has a spatially localized singularity. Hence, the stochastic nature of Z ε (t, x, k) manifests
482
T. Komorowski, S. Peszat, L. Ryzhik
itself, in the leading order, only as the initial data for the limiting kinetic equation. Let us repeat that we believe this fact is not related to the delta-correlation of the random potential in time. The exact form of the angular distribution X (k) may be deduced formally from the initial layer problem for the fluctuation. In the fast variables s = t/ε, y = x/ε Eq. (1.3) may be re-written as √ ∂ Wε ε σp + k · ∇ y Wε = −i ), σ ei p·y Vˆ (s, d p)Wε (s, y, k + d ∂s (2π )d 2 R σ =±1 where Wε (s, y, k) := Wε (εs, εy, k). We introduce a formal asymptotic expansion Wε (t, x, k) = W¯ (t, x, k) +
√
ε Z (t, x, k) + . . . ,
then, Wε (s, y, k) = W¯ (s, y, k) +
√
ε Z (s, y, k) + . . . .
The leading order term satisfies the homogeneous transport equation W¯ s + k · ∇ y W¯ = 0, W¯ (0, y, k) = ε−d δ(y) f (k), and is, therefore, given by W¯ (s, y, k) = ε−d δ(y −ks) f (k). The equation for Z (s, y, k) is σp ∂s Z (s, y, k) + k · ∇ y Z (s, y, k) = −i σ ei p·y Vˆ (s, d p)W¯ s, y, k + 2 Rd σ =±1
with the initial data Z (0, y, k) = 0. For a random potential of the form (1.7) this gives an explicit formula for Z (s, y, k): s Z (s, y, k) = −i σ ei p·(y−k(s−τ )) W¯ (τ, y − k(s − τ ), n σ =±1
0
Rd
σp )en ( p)µ(d p)d Bn (τ ) 2 s −d = −iε σ ei p·(k+σ p/2)τ k+
Rd
0
n σ =±1
σp σp δ(y − ks − τ )en ( p)µ(d p)dBn (τ ). × f k+ 2 2 We obtain therefore: Z (t, x, k) = Z (t/ε, x/ε, k) = −iε−d σ n σ =±1
× δ(ε
−1
0
t/ε
σp ei p·(k+σ p/2)τ f k + 2 Rd
(x − kt + εσ pτ/2))en ( p)µ(d p)dBn (τ ),
Limit of Fluctuations of Solutions of Wigner Equation
483
and since ε−d δ(z/ε) = δ(z) we obtain that for small t 1 the quantity εσ pτ ≤ pt 1 can be neglected, thus ∞ σp Z (0, x, k) ≈ −i δ(x)en ( p)µ(d p)dBn (τ ). σ ei p·(k+σ p/2)τ f k + 2 Rd 0 n σ =±1
The paper is organized as follows. We first recall some basic facts about homogeneous Wiener processes in Sect. 2. The basic existence theory for the Wigner equation with a white-noise potential is described in Sect. 3. We recall that when the initial data for the Wigner equation (1.6) is the Wigner transform of the initial data for the Schrödinger equation (1.2) then existence of the solution of the Wigner equation can be deduced from the respective property of the Schrödinger equation with a white-noise potential [11]. However, to the best of our knowledge, such a theory for the Wigner equation with an arbitrary initial data is not available in the literature. The principal result of this section is Theorem 3.2. Sect. 4 contains the main result of this paper, Theorem 4.1, which describes the asymptotics of the fluctuation process. 2. Preliminaries Basic notation. We denote by S(Rd ) and S(Rd ; C) the spaces of rapidly decreasing functions of the Schwartz class and by S (Rd ) and S (Rd ; C) the corresponding spaces of tempered distributions. The value of a distribution ξ on a test function ψ will be denoted by ξ, ψ . Let τx ψ(·) := ψ(x + ·), x ∈ Rd be the group of translations on S(Rd ). It can be extended to S (Rd ) by setting τx ξ, ψ := ξ, τ−x ψ . We denote by ( p) := Fψ( p) = ψ e−i p·x ψ(x)dx Rd
the Fourier transform of a function ψ(x). Also, we use the notation F1 ( f )(q, k) := e−iq·x f (x, k)dx, F2 ( f )(x, y) := e−iy·k f (x, k)dk Rd
Rd
for the partial Fourier transform in just one of the variables. Given s, u ∈ R we denote by H s,u the mixed Sobolev space with the norm (1 + |q|2 )s/2 (1 + |y|2 )u/2 | fˆ(q, y)|2 dqdy, f ∈ S(R2d ). f 2H s,u := R2d
In the ensuing notation we shall also write H1s := H s,0 and H2u := H 0,u . Given p1 , p2 ∈ [1, +∞) we denote by A p1 , p2 , B p1 , p2 the Banach spaces that are the completions of S(R2d ) under the norms
p1 / p2 p1 p2 φ p1 , p2 := |F1 (φ)(q, k)| dk dq, Rd
and (φ(pB1 ,)p2 ) p1
Rd
:=
Rd
Rd
ˆ |φ(q, y)| p2 dy
p1 / p2 dq,
respectively. The definition can be easily extended to cover the case when one, or both of the indices equal +∞.
484
T. Komorowski, S. Peszat, L. Ryzhik
Some functional spaces formed over the spectral measure. Given a function ψ( p), p ∈ Rd , we set ψ(s) ( p) := ψ ∗ (− p) and say that ψ is even if ψ = ψ(s) . Assume that µ is a finite Borel measure on Rd that is symmetric, that is, µ( ) = µ(− ) for all sets
∈ B(Rd ). The real Hilbert space L 2(s) (µ) consists of all functions ψ ∈ L 2C (µ) that are even. Note that ψ1 ( p)ψ2∗ ( p)µ(d p), ∀ ψ1 , ψ2 ∈ L 2C (µ)
ψ1 , ψ2 µ := Rd
is a real valued scalar product on L 2(s) (µ), provided µ is symmetric. We will need the following proposition. Proposition 2.1. Let {en } be an orthonormal basis of L 2(s) (µ). Then for any ψ1 , ψ2 ∈ L 2C (µ) we have
ψ1 , en µ ψ2 , en µ = ψ1 ( p)ψ2 (− p)µ(d p). (2.1) Rd
n
Proof. Given ψ ∈ L 2C (µ), consider its symmetrization S[ψ] ∈ L 2(s) (µ), 1
S[ψ]( p) := ψ( p) + ψ ∗ (− p) . 2 For any φ ∈ L 2(s) (µ) and ψ1 ∈ L 2C (µ) we have, using the symmetry of µ: 1
S[ψ1 ], φ µ = ψ1 ( p)φ ∗ ( p) + ψ1∗ (− p)φ ∗ ( p) µ(d p) 2 Rd 1 =
ψ1 , φ µ + φ, ψ1 µ = Re ψ1 , φ µ , 2 and thus
S[iψ1 ], φ µ = Re iψ1 , φ µ = −Im ψ1 , φ µ . Therefore, for all ψ1 ∈ L 2C (µ) and φ ∈ L 2(s) (µ), we have
ψ1 , φ µ = S[ψ1 ], φ µ − i S[iψ1 ], φ µ , which implies that I :=
ψ1 , en µ ψ2 , en µ n
S[ψ1 ], en µ − i S[iψ1 ], en µ S[ψ2 ], en µ − i S[iψ2 ], en µ = n
= S[ψ1 ], S[ψ2 ] µ − i S[iψ1 ], S[ψ2 ] µ − i S[ψ1 ], S[iψ2 ] µ − S[iψ1 ], S[iψ2 ] µ = S[ψ1 ] − iS[iψ1 ], S[ψ2 ] µ − iS[ψ1 ] + S[iψ1 ], S[iψ2 ] µ = ψ1 , S[ψ2 ] µ − i ψ1 , S[iψ2 ] µ , since ψ = S[ψ] − iS[iψ]. It follows that I = ψ1 , S[ψ2 ] µ − iψ1 , S[iψ2 ] µ = ψ1 , S[ψ2 ] + iS[iψ2 ] µ = ψ1 ( p)ψ2 (− p)µ(d p), Rd
which is (2.1).
Limit of Fluctuations of Solutions of Wigner Equation
485
This result can be further generalized by a standard density argument leading to Corollary 2.2. Let ν be the Borel measure on R2d defined by δ( p +q)µ(d p)dq. Suppose that ∈ L 2C (ν), then ( p, q)en (q)en ( p)µ(d p)µ(dq) = ( p, − p)µ(d p). n
R2d
Rd
Spatially homogeneous Wiener process. Let µ be a non-negative symmetric, Borel measure on Rd . Recall that an S (Rd ) -valued, Gaussian process {B(t), t ≥ 0} is called a spatially homogeneous Wiener process on Rd if it has the following properties, see e.g. [25–27]: (i) for any ψ ∈ S(Rd ), { B(t), ψ , t ≥ 0} is a real-valued Wiener process, (ii) for any t ≥d0, the law of B(t) is invariant with respect to the group of translations τx , x ∈ R acting on S (Rd ). Equivalently, one can prove, see e.g. [25], that {B(t), t ≥ 0} is Gaussian and its covariance is of the form E [ B(t), ψ1 B(s), ψ2 ] = ψˆ 1 , ψˆ 2 µ (t ∧ s),
ψ1 , ψ2 ∈ S(Rd )
(2.2)
for some Borel measure µ. Since B(t) takes values in the space of tempered distributions, there is an n ≥ 0 such that −n 1 + | p|2 µ(d p) < +∞. Rd
The measure µ is called the spectral measure of {B(t), t ≥ 0}. It is known that if µ is finite then B is a Gaussian random field on [0, ∞) × Rd satisfying E [ B(t, x)B(s, y)] = µ(x − y)(t ∧ s), Let
x, y ∈ Rd , t, s ≥ 0.
Hµ := F (ψµ) : ψ ∈ L 2(s) (µ) ⊂ S (Rd )
be the real Hilbert space equipped with the scalar product induced from L 2(s) (µ) by F, that is, for all ψ1 , ψ2 ∈ L 2(s) (µ) ∩ S(Rd ; C) we have
F (ψ1 µ) , F (ψ2 µ) Hµ = ψ1 , ψ2 µ = F (ψ1 µ) , F (ψ2 ) . According to [25], the reproducing kernel Hilbert space of {B(t), t ≥ 0} can be identified with Hµ , that is, {B(t), t ≥ 0} is the cylindrical Wiener process on Hµ . The above property is expressed by the following. Proposition 2.3. For any orthonormal basis {en } of L 2(s) (µ) there is a sequence of independent standard real-valued Wiener processes {Bn (t), t ≥ 0} such that B(t) = Bn (t)F (en µ), t ≥ 0, (2.3) n
where the series converges in the L 2 sense and P-a.s in any Hilbert space H such that the embedding Hµ → H is Hilbert–Schmidt.
486
T. Komorowski, S. Peszat, L. Ryzhik
3. The Wigner Equation with a Spatially Homogeneous Random Potential The Wigner equation. Let {B(t), t ≥ 0} be a spatially homogeneous Wiener process with spectral measure µ and let ε > 0. We are concerned with the initial problem for the following SPDE, called the Wigner equation, dWε (t, x, k) = −k · ∇x Wε (t, x, k)dt B(d S t, d p) p p +i ei p·x/ε Wε (t, x, k − ) − Wε (t, x, k + ) , (3.1) d 2 2 (2π )d R Wε (0, x, k) = W0 (x, k). The stochastic integral above is understood in the Stratonovich sense. We give a rigorous definition of the solution to (3.1) in an appropriate functional space that shall be specified later on. For any φ ∈ S(s) (Rd ; C) - the space of Schwartz class functions that are complex even (i.e. φ(− p) = φ ∗ ( p)) - we have
B(t), φ =
F(F(en µ)), φ Bn (t), t ≥ 0, n
while
F(F(en µ)), φ = (2π )2d en , F −1 (F −1 φ) µ = (2π )d en ( p)φ ∗ (− p)µ(d p) = (2π )d Rd
Rd
It follows that
B(t), φ = (2π )d
en ( p)φ( p)dµ.
(3.2)
Bn (t)
n
Rd
en ( p)φ( p)dµ, t ≥ 0, φ ∈ S(s) (Rd ; C). (3.3)
Taking into account (2.3) and (3.2) we can rewrite (3.1) into the following form (recall that Wε (t, x, k) is real valued): Cε [Wε ]en d S Bn , dWε = AWε dt + Cε [Wε ]d S B = AWε dt + (3.4) n Wε (0, x, k) = W0 (x, k), where A : S(R2d ) → S(R2d ) is given by Aψ(x, k) := −k · ∇x ψ(x, k),
(3.5)
and the operator Cε : S(R2d ) → L(L 2(s) (µ), C ∞ (R2d )) is given by Cε [ψ]ϕ(x, k) := −i
σ =±1
S(R2d )
σ
σp ϕ( p)µ(d p) ei( p·x)/ε ψ x, k + 2 Rd
(3.6)
for ψ ∈ and ϕ ∈ Here L(X, Y ) denotes the space of continuous linear operators between linear topological spaces X and Y . We shall further specify the above operators later on when we define the notion of a solution to (3.4). L 2(s) (µ).
Limit of Fluctuations of Solutions of Wigner Equation
487
The Wigner equation in the Itô form. Equation (3.1) can be rewritten in the Itô form 1 dWε = AWε + L ε Wε dt + Cε [Wε ]en dBn , 2 n (3.7) Wε (0) = W0 , where L ε ψ :=
Cε [Cε [ψ]en ]en .
n
We have, more explicitly: L ε ψ(x, k) = −i
σ
n σ =±1
=−
n
Rd
Rd
σp µ(d p) ei( p·x)/ε Cε [ψ]en x, k + 2 Rd
ei( p+q)·x/ε (x, k, p, q)en (q)en ( p)µ(d p)µ(dq),
where (x, k, p, q) :=
σ,σ =±1
σ p σ q σ σ ψ x, k + + . 2 2
By Proposition 2.2, the definition of L ε does not in fact depend on ε. We shall drop therefore ε, from this point on, from its notation. The explicit expression for this operator is given by (x, k, p, − p)µ(d p) Lψ(x, k) = − d R = [ψ(x, k − p) − 2ψ(x, k) + ψ(x, k + p)] µ(d p) Rd
= 2Mψ(x, k) − 2ψ(x, k), where
(3.8)
Mψ(x, k) :=
Rd
ψ(x, k + p)µ(d p)
(3.9)
and := µ(Rd ) < +∞.
(3.10)
The last equality in (3.8) follows from the fact that µ(−d p) = µ(d p). A simple consequence of (3.7) is that W¯ (t) := EWε (t) does not depend on ε and it satisfies the linear kinetic equation 1 d W¯ (t) = A + L W¯ (t), (3.11) dt 2 W¯ (0) = W0 .
488
T. Komorowski, S. Peszat, L. Ryzhik
The probabilistic representation of a solution of the kinetic equation. We now recall a probabilistic formula for the solution to (3.11) treating it as the solution of Kolmogorov’s equation for a certain Markov jump process. The results of this section are standard and their proofs can be found, for instance, in Appendix 2 of [19]. Define the probability measure ν(A) := −1 µ(A), where is given by (3.10) and A is a Borel set. Let {L i , i ≥ 0} be a sequence of i.i.d. random variables (momenta i th jump) distributed according to ν and set K 0 := 0, K n :=
n−1
L i , n ≥ 1.
i=0
Let σ0 , σ1 , . . . be i.i.d. random variables (times between the jumps), independent of L 0 , L 1 , . . . such that σ0 is exponentially distributed with the intensity parameter . Consider the “jump times” t0 := 0, tn :=
n−1
σi , n ≥ 1.
i=0
The jump process K (t) is defined as K (t) := K n , for t ∈ [tn , tn+1 ). For any function φ ∈ L ∞ (Rd ) we have Eφ(k + K (t)) = e−t φ(k) +
+∞
φn (t, k),
(3.12)
n=1
where φn (t, k) := e−t
(t)n Eφ (K n + k), n!
n ≥ 1.
Since the laws of K 1 and −K 1 are identical we have E[ψ1 (k + K (t))]ψ2 (k)dk = E[ψ2 (k + K (t))]ψ1 (k)dk Rd
Rd
(3.13)
∞ d for any pair of functions t ψi ∈ L (R ), i = 1, 2. Let X (t) := kt + 0 K (s)ds. The process {(−X (t), K (t)), t ≥ 0} is Markovian with the generator A + 1/2L. Thus, the solution of (3.11) can be written as
W¯ (t, x, k) = E {W0 (x − X (t), k + K (t))}.
(3.14)
Using (3.12) we obtain therefore W¯ (t, x, k) = e−t W0 (x − kt, k) +
+∞
Wn (t, x, k),
n=1
where W0 (t, x, k) := e−t W0 (x − kt, k), E W0 (x − kt − Xn , k + K n ) dτ (n) Wn (t, x, k) := e−t n n (t)
(3.15)
Limit of Fluctuations of Solutions of Wigner Equation
489
for n ≥ 1, and n (t) := [(τ0 , . . . , τn−1 ) : t ≥
n−1
τi , τi ≥ 0],
τn := t −
i=0
dτ (n) := dτ0 . . . dτn−1 ,
Xn :=
n
n−1
τi ,
i=0
K i τi .
i=1
We shall introduce a semigroup of operators given by W¯ (t) := S(t)W0 , t ≥ 0. Proposition 3.1. The family {S(t), t ≥ 0} extends to a C0 -semigroup of contractions on spaces A p1 , p2 , B p1 , p2 for all p1 , p2 ∈ [1, +∞) and H s,u for all s, u ∈ R. Proof. Note that for any W0 ∈ A p1 , p2 we have
p
S(t)W0 p11 , p2 =
Rd
|F1 (S(t)W0 )(q, k)| p2 dk
Rd
p1 / p2 dq.
Using formula (3.14) we obtain that the right-hand side equals
p2 p1 / p2 t E exp iq · kt + i q · K (s)ds F1 (W0 )(q, k + K (t)) dk dq Rd Rd 0 p1 / p2 p ≤ E |F1 (W0 )(q, k + K (t))| p2 dk dq = W0 p11 , p2 .
Rd
Rd
The proofs for B p1 , p2 and H s,u are similar. Continuity easily follows from contractivity of the semigroup and the fact that the property in question holds on S(Rd ). For any θ belonging to A p1 , p2 (or B p1 , p2 , or H s,u ) define S ∗ (t)θ (x, k) = E {θ (x + X (t), k + K (t))}.
(3.16)
Using integration by parts we easily conclude that for W0 ∈ A p1 , p2 and θ ∈ A p1 , p2 we have the following duality relation
S(t)W0 , θ = W0 , S ∗ (t)θ .
(3.17)
A similar statement holds if A p1 , p2 is replaced by B p1 , p2 , or by H s,u and their dual counterparts.
Existence and uniqueness result for solutions to the Wigner equation. Note that for ψ ∈ S(R2d ) the Hilbert-Schmidt norm of the operator Cε [ψ] equals Cε [ψ]en 2H −s Cε [ψ]2L (L 2 (µ),H −s ) = (H S)
(s)
1
n
=
n
R2d
1
|F1 (Cε [ψ]en ) (q, k)|2
dkdq . (1 + |q|2 )s/2
490
T. Komorowski, S. Peszat, L. Ryzhik
Here F1 denotes the Fourier transform performed with respect to the first variable. The sum inside the integral is 2 |F1 (Cε [ψ]en ) (q, k)| = σ ei(q+ p/ε)·x 2d R n n σ =±1 2 2 σp × ψ(x, k + Φ( p, q, k)en ( p)µ(d p) , )en ( p)µ(d p)dx = d 2 R n where Φ( p, q, k) :=
σ =±1
We have
σ ei(q+ p/ε)·x ψ(x, k −
Rd
σp )dx. 2
n
2 Φ( p, q, k)en ( p)µ(d p) Rd = Φ( p, q, k)en ( p)µ(d p) =
n
Rd
n
R2d
Rd
Φ ∗ ( p , q, k)en (− p )µ(d p )
Φ( p, q, k)Φ ∗ (− p , q, k)en ( p)en ( p )µ(d p)µ(d p ).
Therefore, by Corollary 2.2, we obtain 2 Φ( p, q, k)en ( p)µ(d p) = n
Rd
Rd
(3.18)
|Φ( p, q, k)|2 µ(d p),
and, consequently, Cε [ψ]2L
−s 2 (H S) (L (s) (µ),H1 )
Now, write
=
Φ± ( p, q, k) := ±
Rd
R2d
|Φ( p, q, k)|2
e−i(q+ p/ε)·x ψ(x, k ±
so that Φ = Φ− + Φ+ , and, moreover, 2 2 |Φ− ( p, q, k)| dk = |Φ+ ( p, q, k)| dk = Rd
Rd
µ(d p)dq . (1 + |q|2 )s/2
Rd
Rd
e
p )dx, 2
ix·(q+ p/ε)
2 ψ(x, k)dx dk.
Hence, the Hilbert-Schmidt norm of the operator Cε [ψ] is bounded as 2 µ(d p)dkdq ix·(q+ p/ε) Cε [ψ]2L (L 2 (µ),H −s ) ≤ 2 e ψ(x, k)dx (1 + |q|2 )s/2 (H S) (s) 1 R3d Rd =2
p 2 µ(d p)dkdq ≤ aε ψ2H −s , F1 (ψ) q + , k 1 ε (1 + |q|2 )s/2 R3d
(3.19)
Limit of Fluctuations of Solutions of Wigner Equation
491
where aε :=2 sup
q∈Rd
s/2 p 2 −s/2 1 + q + 1 + |q|2 µ(d p) < +∞. ε Rd
(3.20)
We have shown that for each ε fixed, the operator Cε : H1−s → L (H S) (L 2(s) (µ), H1−s )) is bounded. In addition, we show in Proposition 3.1 below that the operator A − (1/2)L generates a C0 -semigroup on H1−s . We have shown therefore the following existence and uniqueness result, see Theorem 9.7, of [27]: Theorem 3.2. Under condition (3.20) for any Wε (0) ∈ H1−s and ε > 0, there exists a unique solution to (3.4) starting from Wε (0) and such that Wε (·) ∈ C([0, +∞), H1−s ) a.s. Moreover, (3.4) defines a Markov family on H1−s satisfying the Feller property. Remark. Observe that when s > 0 a sufficient condition for (3.20) is s/2 1 + | p|2 µ(d p) < +∞. Rd
(3.21)
Indeed, suppose with no loss of generality that ε = 1 in (3.20). Then for |q| ≥ 2| p| we have 1 + |q + p|2 ≥ 1 + (|q| − | p|)2 ≥ 1 + (|q|/2)2 . Hence, there exists C > 0 such that 1 + |q|2 ≤ C(1 + | p + q|2 )(1 + | p|2 ). This of course, in light of (3.21), implies (3.20). One can easily show an example of a s/2 µ(d p) = +∞, for which condition (3.20) fails. measure µ such that Rd 1 + | p|2 An a priori estimate. Suppose that { f n (t), t ≥ 0}, n ≥ 1 are H1−s ∩ A p1 , p2 -valued processes that together with {Wε (t), t ≥ 0} are adapted with respect to the filtration corresponding to the Brownian motions {Bn (t), t ≥ 0}, n ≥ 0. Suppose also that Uε (t) is an H1−s -valued process that satisfies
Uε (t), θ = S(t)W0 , θ + +
∞
t
∞ n=1 0
t
S(t − s)Cε [Uε (s)]en , θ dBn (s)
S(t − s) f n , θ dBn (s)
(3.22)
n=1 0
for all θ ∈ S(R2d ). We formulate here a certain a priori estimate for E Uε (t), θ 2 , that will be useful in what follows. Recall that pi := pi /( pi − 1) when pi > 1, or pi := +∞ if pi = 1 for i = 1, 2.
492
T. Komorowski, S. Peszat, L. Ryzhik
Proposition 3.3. Suppose that W0 ∈ H1−s ∩ A p1 , p2 , for some p1 , p2 ≥ 1. Then, E[ Uε (t), θ ] ≤ 3e 2
sup
θ p1 , p2 ≤1
3t
W0 2p , p + 1
2
sup
t
θ p1 , p2 ≤1 0
E f n (s), θ ds 2
n
(3.23) (B )
for all t > 0. A similar result holds also when the norm · p1 , p2 is replaced by · | p1 , p2 . Proof. Suppose that θ p1 , p2 ≤ 1. Observe that from Lemma 2.1 we have
E S(t − s)Cε [Uε (s)]en , θ 2 n
=
σσ E
R4d
σ,σ =±1
exp iε−1 p · (x − x ) S ∗ (t − s)θ (x, k)S ∗ (t − s)θ (x , k )
× Uε (s, x, k + σ p/2)Uε (s, x , k + σ p/2)dxdx dkdk µ(d p) E Uε (s), θεσ, p (t − s) Uε (s), (θεσ , p (t − s))∗ µ(d p), = d σ,σ =±1 R
(3.24)
where θεσ, p (s, x, k) := σ exp iε−1 p · x (S ∗ (s)θ )(x, k + σ p/2).
(3.25)
Note that p
p1 / p2 p |F1 (S ∗ (s)θ ) q ∓ , k + σ p/2 | p2 dk dq ε Rd Rd p p = S ∗ (s)θ p11 , p2 ≤ S ∗ (s)θ p11 , p2 ,
θεσ, p (s) p11 , p2 =
by virtue of Proposition 3.1. Hence, using the above we obtain from (3.22),
E[ Uε (t), θ ] ≤ 3 W0 2p , p + 2
1
+
sup
sup
0 θ p1 , p2 ≤1
2
θ p1 , p2 ≤1 n
t
t
E[ Uε (s), θ 2 ]ds
E[ f n (s), θ 2 ]ds .
(3.26)
0
Taking the supremum over θ p1 , p2 ≤ 1 on the left hand side and using Gronwall’s inequality we conclude the proof of the proposition.
Limit of Fluctuations of Solutions of Wigner Equation
493
4. Asymptotics of the Fluctuations Assumptions on the spectral measure. We shall assume that the spectral measure µ satisfies the following condition: 1 1+ µ(d p) < +∞. (4.1) sup | p + q| q∈Rd We will actually require a more refined version of (4.1) around p = 0: µ(d p)
( f ) := lim sup εγ −1 < +∞ γ ε→0+ [| p|≤ε ] | p|
(4.2)
for some γ ∈ (0, 1). Next, set |det(x, y)| := {|x|2 |y|2 − (x · y)2 }1/2 . We shall assume that sup |det( p + q, p + p1 + q)|−1 µ(d p)µ(d p1 ) < +∞.
(4.3)
q
The main result. Define the rescaled fluctuation Z ε (t) := ε−1/2 [Wε (t) − W¯ (t)], where Wε (·) satisfies the Wigner equation (3.7) and W¯ (·) is the solution of the kinetic Eq. (3.11). Then Z ε (·) satisfies 1 Cε [Z ε ]en dBn + ε−1/2 Cε [W¯ ]en dBn , dZ ε = A + L Z ε dt + 2 (4.4) n n Z ε (0) = 0. We will consider the initial data for the Wigner equation of the form W0 (x, k) = δ(x) f (k). For simplicity we assume that the angular distribution f (k) ≥ 0 is a Schwartz class function: f ∈ S(Rd ). This assumption may be greatly relaxed at the expense of more technicalities which we avoid to keep the presentation as simple as possible. Suppose that { Z¯ (t), t ≥ 0} is a unique, H −s,−u -valued, solution of equation d Z¯ (t) = A + 21 L Z¯ (t), dt Z¯ (0) = δ ⊗ X.
(4.5)
Here X is a Gaussian, random S (Rd )-valued element given by +∞ X (k) := −i σ dBn (s) ei p·(k+σ p/2)s f (k + σ p/2)en ( p)µ(d p). n σ =±1
0
Rd
( Z ε (t1 ), θ1 , . . . , Z ε (t N ), θ N ), as ε → 0+, to the law of ( Z¯ (t1 ), θ1 , . . . ,
Z¯ (t N ), θ N ) for arbitrary t1 ≤ . . . ≤ t N and θ1 , . . . , θ N ∈ H −s,−u , where s, u > d. Our principal result can be now stated as follows. Theorem 4.1. Assume that (4.1)–(4.3) hold. Then, for any t0 > 0 the laws of {Z ε (t), t ≥ 0} over C([t0 , +∞)S (Rd )) converge, as ε → 0+, to the law of the solution of (4.5).
494
T. Komorowski, S. Peszat, L. Ryzhik
Note that we need an initial time layer after which the weak convergence could be claimed. The reason is that the non-zero initial data for the limit Z¯ (t) cannot be a weak limit of Z ε (0) = 0, as ε → 0+. We will actually show that after a short time t = o(1) the process Z ε (t) is no longer small and thus the initial angular distribution of the limit Z¯ (t) is the limit of the outgoing distributon of Z ε (t) after a short initial time layer. This of course precludes the claim of the weak convergence on the entire [0, +∞). The initial angular distribution. We may describe X (k) as a real distribution-valued, mean zero, random field and covariance function
(4.6) E X, θ X, θ = C(θ, θ ), where
C(θ, θ ) :=
σσ
gσ,σ (q, p) := θ, θ
R2d
+∞
µ(d p)
Rd
σ,σ =±1
and
eis(σ +σ
0
)| p|2 /2
gσ,σ ( ps, p)ds
eiq·(k−k ) f (k + σ p/2) f k − σ p/2 θ (k)θ (k )dkdk ,
with ∈ In fact, the law of X is supported in any Sobolev space H −u for u > d equipped with Borel σ algebra generated by the weak topology. To see that consider the approximants T N X N ,T (k) := −i σ dBn (s) ei p·(k+σ p/2)s f (k + σ p/2)en ( p)µ(d p). S(Rd ).
Rd
0
n=1 σ =±1
We obtain EX N ,T 2H −u = −
N
σσ
T
ds 0
n=1 σ,σ =±1
R5d
× f (k + σ p/2) f (k + σ p /2)en ( p)en ( p )µ(d p)µ(d p ) =−
N
σσ
n=1 σ,σ =±1
T
ds 0
e−iq·k+iq·k ei p·(k+σ p/2)s ei p ·(k +σ
dkdk dq (1 + |q|2 )u/2
R5d
∞
T
ds
n=1 0
R3d
e−iq·k+iq·k e−i p·(k−σ p/2)s e−i p ·(k −σ
× f (k − σ p/2) f (k − σ p /2)en∗ ( p)en∗ ( p )µ(d p)µ(d p ) =
p /2)s
X N (s, q, p)Y N (s, q, p )en∗ ( p)en∗ ( p )
p /2)s
dkdk dq (1 + |q|2 )u/2
µ(d p)µ(d p )dq , (1 + |q|2 )u/2
where X N and Y N are the orthogonal projections in L 2(s) (µ) (in the p-variable) of σ e−iq·k e−i p·(k−σ p/2)s f (k − σ p/2)dk X (s, q, p) = i σ =±1
=i
σ =±1
Rd
σ e−iσ q· p/2 fˆ(q + ps)
Limit of Fluctuations of Solutions of Wigner Equation
and Y(s, q, p) = i
σ
σ =±1
Now, Corollary 2.2 implies that T 2 ds EX N ,T H −u = while Y(s, q, − p) = X N∗ (s, q, p) =
0 ∗ X (s, q,
Rd
R2d
=
eiq·k e−i p·(k−σ p/2)s f (k − σ p/2)dk.
X N (s, q, p)Y N (s, q, − p)
N N
X , en ∗µ en∗ ( p) =
d n=1 R
N d n=1 R N d n=1 R
µ(d p)dq , (1 + |q|2 )u/2
p) and thus
n=1
=
495
en ( p )X ∗ (s, q, p )en (− p)µ(d p )
en (− p )Y(s, q, − p )en (− p)µ(d p ) en∗ ( p )Y(s, q, p )en (− p)µ(d p )
= Y N (s, q, − p). Therefore, we have EX N ,T 2H −u
T
=
ds
0 +∞
≤ 0
≤2
Rd
Rd
X N (s, q)2L 2
dq
(s) (µ) (1 + |q|2 )u/2
dq ds X (s, q)2L 2 (µ) 2 )u/2 d (1 + |q| (s) R+∞ 2 ˆ | f (q + ps)| µ(d p)ds 0
Rd
dq < +∞ (1 + |q|2 )u/2
S(Rd )
for f ∈ and u > d, under assumption (4.1). Taking T = M, with M ∈ N, we conclude that the sequence of laws of {X N ,M ; N , M ∈ N} is tight, thus also weakly pre-compact by the results of [17], in the weak topology of H −u . The existence of the limit can be established by verifying that, for all θ, θ ∈ S(Rd ), lim E[ X N ,M , θ X N ,M , θ ] = C(θ, θ ).
N ,M
We leave this as an exercise to the reader. 5. The Proof of Theorem 4.1 An auxiliary Gaussian process. As the first step we will approximate Z ε (t) by the solution of (4.4) but without the middle term on the right side. That is, suppose that { Z¯ ε (t), t ≥ 0} is the solution of equation 1 d Z¯ ε = A + L Z¯ ε dt + ε−1/2 Cε [W¯ ]en dBn , 2 (5.1) n ¯ Z ε (0) = 0.
496
T. Komorowski, S. Peszat, L. Ryzhik
It is Gaussian given explicitly by a stochastic convolution t −1/2 ¯ S(t − s)Cε [W¯ (s)]en dBn (s). Z ε (t) = ε n
(5.2)
0
This is simply the same kinetic equation satisfied by W¯ (t) but with an additional random forcing which depends on W¯ (t). The following lemma is crucial in estimating the difference between Z ε (t) and Z¯ ε (t). Lemma 5.1. Suppose that µ satisfies the assumptions of Theorem 4.1 and s, u > d. Then, there exists C > 0 such that for any T > 0 we have 2 T E
Cε [ Z¯ ε (t)]en , θ dBn (t) ≤ Cεθ 2H s,u , ∀ ε ∈ (0, 1], θ ∈ H s,u . (5.3) 0
n
Proof. According to Lemma 2.1, the expectation appearing on the left side of (5.3) equals T 2 E
Cε [ Z¯ ε (t)]en , θ dt n
=
0
σ,σ =±1
σσ
0
T
R5d
ei p·(x−x )/ε (θ ⊗ θ )(x, k, x , k )
σp ¯ σp ¯ ×E Z ε t, x, k + Z ε t, x , k + dtdxdx dkdk µ(d p). 2 2
(5.4)
Using (5.2), the definition of Z¯ ε (t), we can re-write the expectation of the expression appearing in the right side of (5.4) as t ε,n σp σp ε,n −1 ds, K t,s ⊗ K t,s x, k + ε ,x ,k + 2 2 0 n where ε,n K t,s (x, k) := S(t − s)[Cε [W¯ (s)]en ] (x, k) .
Substituting into (5.4) we conclude that the expression on its right hand side equals T t 1 σp σp θ x , k − σσ ds ei p·(x−x )/ε θ x, k − Iε = ε n 2 2 R5d 0 0 σ,σ =±1 × S(t − s)[Cε [W¯ (s)]en ] (x, k) S(t −s)[Cε [W¯ (s)]en ] x , k dxdx dkdk µ(d p) T t 1 σ p =− σ σ σ1 σ1 ds S ∗ (t − s) ei p·x/ε θ x, k − ε n 2 R7d 0 0 σ,σ =±1 σ1 ,σ1 =±1 σp eiq·x/ε eiq1 ·x /ε ×S ∗ (t − s) e−i p·x /ε θ x , k − 2 σ q1 σ1 q ¯ Ws x , k + 1 × W¯ s x, k + 2 2
Limit of Fluctuations of Solutions of Wigner Equation
497
×en (q)en (q1 )dxdx dkdk µ(d p)µ(dq)dµ(dq1 ) T t 1 σ σ σ1 σ1 dt ds = ε 0 0 σ,σ =±1 σ p × eiq·x/ε e−iq·x /ε S ∗ (t − s) ei p·x/ε θ x, k − 2 R6d p σ1 q σ ×S ∗ (t − s) e−i p·x /ε θ x , k − W¯ s x, k + 2 2 q σ ×W¯ s x , k + 1 dxdx dkdk µ(d p)µ(dq). 2
This can be written more succinctly as Iε := ε
−1
σσ
σ1 σ1
T
t
0
σ,σ ,σ1 ,σ1 =±1
0
R6d
σ1 p1 σ1 p1 ,x ,k + × W¯ s ⊗ W¯ s x, k + 2 2 σ p, p,ε
×L t,s
ei p1 ·(x−x )/ε
σ p, p,ε ∗
) (x, k, x , k )dtdsdxdx dkdk µ(d p)µ(d p1 ). (5.5)
⊗ (L t,s
Here W¯ s := W¯ (s), Tl f (x, k) := f (x, k + l) and q, p,ε
L t,s
(x, k) := S ∗ (t − s)T−q/2 θ˜ε (x, k; p),
with θ˜ε (x, k; p) := ei p·x/ε θ (x, k). Using (3.15) we can represent W¯ s as a series W¯ s = n≥0 Ws(n) , where Ws(n) = Wn (s) is given by (3.15). Likewise, we can write q, p,ε,m q, p,ε L t,s (x, k) = L t,s (x, k), m≥0
where
q q (t − s), k − ; p , (x, k) := e−(t−s) θ˜ε x + k − 2 2 q q, p,ε,n n −(t−s) L t,s (t − s) + Xn , (x, k) = e E θ˜ε x + k − 2 q, p,ε,0
L t,s
n (t−s)
q k − + K n ; p dτ (n). 2 Here we have maintained the notation introduced in (3.15). Therefore, the expression in (5.5) can be represented accordingly as Iε = Iεn,n ,m,m , n,n ,m,m ≥0
where
Iεn,n ,m,m :=
1 ε
T
dt 0
0
t
ds e−2t
R2d
Wm,n Wm∗ ,n µ(d p)µ(d p1 )
498
T. Komorowski, S. Peszat, L. Ryzhik
and
σ1 p1 σ p, p,ε,m L t,s exp {i p1 · x/ε} Ws(n) x, k + (x, k)dxdk 2 R2d σ,σ =±1 = m+n σσ dτ dρ exp {i( p + p1 ) · x/ε}
Wm,n :=
σσ
m (t−s)
σ,σ =±1
R2d
n (s)
σp /ε × exp i (t − s) p · k − 2 σp σp (t − s) + Xm , k − + Km ×E exp {i p · Xm /ε} θ x + k − 2 2 σ1 p 1 σ1 p1 s − Yn , k + + L n dxdk. (5.6) × W0 x − k + 2 2
Here Xm := mj=0 K j τ j and Yn := nj=0 L j ρ j are the random variables arising from the probabilistic interpretation for the kinetic equation. All variables K j , L j are i.i.d., each with the law ν(·). We can further rewrite the right-hand side of (5.6) using the law of the random variables representing momentum and obtain
Wm,n =
σ σ1
σ,σ1 =±1
ˆ m (t−s)
dτ
ˆ n (s)
dρ
R(m+n)d+2
dxdk
m
µ(dk j )
j=1
n
µ(dl j )
j=1
⎧ ⎫ m ⎨ ⎬ σp /ε × exp i p · ( × exp i( p + p1 ) · x/ε + i(t − s) p · k − k j τ j )/ε ⎩ ⎭ 2 j=1 ⎞ ⎛ m m σ p σ p (t − s) + kjτj, k − kj⎠ + × θ ⎝x + k − 2 2 j=1 j=1 ⎛ ⎞ n n σ σ p p 1 1 1 1 s− + ljρj, k + l j ⎠, ×W0 ⎝x − k + 2 2 j=1
j=1
ˆ n (s) := [(s1 , . . . , sn ) : s ≥ sn ≥ . . . s1 ≥ 0]. Thanks to symmetry we can where rewrite the above expression as
1 Wm,n = n!m!
σ,σ1 =±1
σ σ1
m (t−s)
dτ
n (s)
dρ
R(m+n)d+2
dxdk
m
µ(dk j )
j=1
n j=1
µ(dl j )
⎧ ⎫ m ⎨ ⎬ σp /ε exp i p · ( k j τ j )/ε × exp i( p + p1 ) · x/ε + i(t − s) p · k − ⎩ ⎭ 2 j=1
Limit of Fluctuations of Solutions of Wigner Equation
⎛
499
⎞ m m σ p σ p (t − s) + + × θ ⎝x + k − kjτj, k − kj⎠ 2 2
j=1
j=1
⎛
⎞ n n p p σ σ 1 1 1 1 × W0 ⎝ x − k + s− + ljρj, k + lj⎠ 2 2 j=1
j=1
with n (s) := [(s1 , . . . , sn ) : si ∈ [0, s], i = 1, . . . , n]. Using the fact that W0 (x, k) = δ(x) f (k) and performing the Fourier transform of both f (k) and θ (x, k) we conclude that Wm,n
1 = n!m!
σ σ1
σ,σ1 =±1
m (t−s)
×
R(m+n)d+4
dkdydqdz
m
dτ
n (s)
µ(dk j )
j=1
dρ
n
µ(dl j )
j=1
σ1 p1 ×θˆ (q, y) fˆ (z) exp i[ε−1 ( p + p1 + εq)s + z] · k + 2 σp × exp i[ε−1 ( p + εq)(t − s) + y] · k − 2 ⎫ ⎧ ⎫ ⎧ m n ⎬ ⎨ ⎬ ⎨ −1 −1 (ε ( p + εq)τ j + y) · k j exp i [ε ( p + p1 + εq)ρ j + z] · l j × exp i ⎭ ⎩ ⎭ ⎩ j=1
=
εm+n m!n!
R4d
× m
j=1
σ1 p1 dkdydqdz θˆ (q, y) fˆ (z) exp i[ε−1 ( p + p1 + εq)s + z] · k + 2
s t −s , p + εq, y n , p + p1 + εq, z ε ε
σ p . × exp i[ε−1 ( p + εq)(t − s) + y] · k − 2
Here we have set
t
(t, k, x) =
dτ 0
Rd
ei p·(τ k+x) µ( p)d p.
(5.7)
Finally, we change variables t := t/ε and s := s/ε. We have shown that the expression in (5.5) equals Iε = εI˜ ε , where I˜ ε =
T /ε 0
e−2εt dt
t
ds 0
R2d
|Fε |2 µ(d p)µ(d p1 ),
(5.8)
500
T. Komorowski, S. Peszat, L. Ryzhik
and Fε :=
σ σ1
R4d
σ,σ1 =±1
θˆ (q, y) fˆ (z) exp {ε[(t −s, p+εq, y)+(s, p + p1 +εq, z)]}
σ1 p1 × exp i[( p + p1 + εq)s + z] · k + 2 σ p dkdydqdz × exp i[( p + εq)(t − s) + y] · k − 2 = θˆ (q, y) fˆ (−y − t ( p + εq) − sp1 ) σ σ1 R2d
σ,σ1 =±1
× exp {ε[(t − s, p + εq, y) + (s, p + p1 + εq, −y − t ( p + εq) − sp1 )]} × exp {(i/2) [( p + εq)(t − s) + y] · (σ1 p1 + σ p)]} dqdy. Changing variables u := t − s, s := s we obtain that |Fε |2 µ(d p)µ(d p1 ), I˜ ε = e−ε(u+s) duds R2d
0≤s,u,s+u≤T /ε
and, as f is real so that fˆ(y) is complex-even in y, we have θˆ (q, y) fˆ∗ (y + (s + u)( p + εq) + sp1 ) σ σ1 Fε = σ,σ1 =±1
R2d
× exp {ε[(u, p + εq, y) + (s, p + p1 + εq, −y − u( p + εq) −s( p + p1 + εq))]} exp {(i/2) [( p + εq)(t − s) + y] · (σ1 p1 + σ p)]} dqdy. (5.9) Note that directly from the definition (5.7) we have (u, p, y) ≥ 0 and it can be estimated as follows u eik·(up) − 1 ik·y ik·(τ p+y) (u, p, y) = e µ(dk) e dτ µ(dk) = Rd Rd i(k · p) 0 ik·(up) − 1 e (5.10) ≤u µ(dk) ≤ u. Rd k · (up) Thus, the expression in the exponent in (5.9) can be bounded as (u, p + εq, y) + (s, p + p1 + εq, y + u( p + εq) − s( p + p1 + εq)) ≤ (u + s) ≤ T /ε. Therefore, expression in (5.8) may be estimated by +∞ +∞ |I˜ ε | ≤ 4eT duds µ(d p)µ(d p1 ) 0
R2d
0
2 ˆ ˆ |θ (q, y) || f (y + u( p + εq) + s( p + p1 + εq)) |dqdy × R2d +∞ +∞ (B ) T ≤ 4e θ 1,1 duds µ(d p)µ(d p1 )dqdy|θˆ (q, y) |
0
0
R4d
×| fˆ (y + u( p + εq) + s( p + p1 + εq)) |2 .
Limit of Fluctuations of Solutions of Wigner Equation
501
The integral in u and s may be treated as +∞ +∞ | fˆ (y + u( p + εq) + s( p + p1 + εq)) |2 duds 0
0
≤ C( f )|det( p + εq, p + p1 + εq)|−1 . Taking this into account we can estimate (B ) |det( p + q, p + p1 + q)|−1 µ(d p)µ(d p1 ). |I˜ ε | ≤ 4eT C( f )(θ 1,1 )2 sup q
Finally, to get (5.3) it suffices only to recall assumption (4.3) and observe that when (B ) s, u > d/2 there exists C > 0 such that θ 1,1 ≤ Cθ H s,u for all θ ∈ S(R2d ). Approximating Z ε by Z¯ ε . We now use Lemma 5.1 to estimate the difference between the true corrector Z ε and Z¯ ε . The error Uε (t) := Z ε (t) − Z¯ ε (t) satisfies the equation 1 Cε [Uε ]en dBn + Cε [ Z¯ ε ]en dBn , dUε = A + L Uε dt + 2 (5.11) n n Uε (0) = 0. We have the following estimate: Lemma 5.2. For any t > 0 there exists C > 0 such that for all θ ∈ S(R2d ) we have (B ) E Uε (t), θ 2 ≤ Cε(θ 1,1 )2 . Proof. Using estimate (3.23), with Wε (0) = 0, we obtain 2 t (B ) 2 2 ¯ E[ Uε (t), θ ] ≤ 3(θ 1,1 ) sup E
Cε [ Z ε (s)]en , θ dBn (s) . (B)
θ1,1 ≤1
The result then follows from Lemma 5.1.
n
0
Tightness of Z¯ ε (t), as ε → 0+. The asymptotics of Z ε (t), as ε → 0+, is therefore the same as that of Z¯ ε (t). Using decomposition of generator L as in the last line of (3.8) we can write that Cε [W¯ ]en dBn , d Z¯ ε = (A − + M) Z¯ ε dt + ε−1/2 n
Z¯ ε (0) = 0, where the operator M is given by (3.9). Therefore by Duhamel’s formula we have t t Z¯ ε (t) = S0 (t − s)M Z¯ ε (s)ds + ε−1/2 S0 (t − s)Cε [W¯ (s)]en dBn (s). (5.12) 0
n
0
Here S0 f (t) := e−t f (x − kt, k) for an appropriate f and t ∈ R.
(5.13)
502
T. Komorowski, S. Peszat, L. Ryzhik
Suppose we are given a family of Borel probability measures {Pε , ε > 0} defined over a certain topological space. We say that the family is weakly pre-compact, as ε → 0+, if for any sequence εn → 0, as n → +∞ one can choose a subsequence from {Pεn , n ≥ 1} that is weakly convergent. Proposition 5.3. Suppose that s, u > d, t0 > 0 and the space H −s,−u is equipped with the weak topology. Then, the family of laws of the processes { Z¯ ε (t), t ≥ 0} considered in C([t0 , +∞), H −s,−u ) is weakly pre-compact when ε → 0+. Proof. According to [17], Theorem 3.1, p. 276, to show weak pre-compactness of the laws in D([t0 , +∞), H −s,−u ) it suffices only to show that for each δ > 0, T1 ≥ t0 there exists K > 0 such that P sup Z¯ ε (t)−s,−u ≤ K ≥ 1 − δ (5.14) t∈[t0 ,T1 ]
and that for any test function θ ∈ H s,u , the laws of { Z¯ ε (t), θ , t ∈ [t0 , T ]}, ε ∈ (0, 1] are tight in C[t0 , T ]. H −s,−u )
(5.15)
H −s,−u ),
Since C([t0 , +∞), is a closed subset of D([t0 , +∞), see Proposition 1.6, p. 267 of [17], this implies weak pre-compactness of the laws in C([t0 , +∞), H −s,−u ). In order to conclude (5.14) it is a actually enough to prove that T −1 sup ε (T − t)−2α Cε [W¯ (t)]2L (L 2 (µ),H −s,−u ) dt < +∞ (5.16) ε∈(0,1],T ∈[t0 ,T1 ]
(H S)
0
(s)
for α ∈ (0, 1/2). Using Lemma 7.2, p. 182 of [12] and estimates (7.11) and (7.12), p. 184 of ibid. we would be able then to conclude that 2 E sup Z¯ ε (t)−s,−u < +∞, (5.17) t∈[t0 ,T1 ]
which in particular implies (5.14). Hence, we will now show that (5.16) holds. Note that, by (3.19), the expression under the supremum in (5.16) can be bounded from above by µ(d p)dydq 2 T −2α (T − t) dt Jε := 2 )s/2 (1 + |y|2 )u/2 3d ε 0 (1 + |q| R 2 × σ e−ix·(q+ p/ε) e−iy·k W¯ (t, x, k + σ p/2)dxdk . (5.18) 2d R σ =±1
Using probabilistic representation of W¯ (t, x, k) in a similar way as it has been done in the proof of Lemma 5.1 we obtain that e−ix·(q+ p/ε) e−iy·k W¯ (t, x, k + σ p/2)dxdk R2d
=
e−t n
n!
R2d
dxdk
n (t)
⎛
dτ
n
(Rd )n j=1
µ(dk j )
⎞ n n σ p σ p t− + kjτj, k + kj⎠ . × e−ix·(q+ p/ε) e−iy·k W0 ⎝x − k + 2 2 j=1
j=1
Limit of Fluctuations of Solutions of Wigner Equation
503
Taking into account the fact that W0 (x, k) = δ(x) f (k) and substituting into (5.18) we obtain µ(d p)dydq 2 T −2t −2α e dt Jε = (T − t) 2 s/2 2 u/2 ε 0 R3d (1 + |q| ) (1 + |y| ) σp − iy · k × σ exp −i(t/ε)( p + εq) · k + 2d 2 R σ =±1 2 σp + iz · k + + ε(t/ε, q, z) fˆ(z)dkdz . 2 Recall, see (5.10), that ε(t/ε, q, z) ≤ T for t ∈ [0, T ]. Let ω( p, q) := ( p + εq)| p + εq|−1 and gu (y) := (1 + |y|2 )−u/2 . Integrating out the k and z variables and replacing t := t/ε we obtain −2α 2 C T T /ε T Jε ≤ 2α −t dt gu (y)gs (q) fˆ(y + t ( p + εq)) µ(d p)dydq ε ε R3d 0 gu (y)gs (q) ≤ CT µ(d p)dydq R3d | p + εq| −2α 2 T | p + εq| 2α T | p+εq|/ε T | p + εq| ˆ × −t f (y + tω( p, q)) dt ε ε 0 = CT gs (q)| p + εq|−1 µ(d p)dq × sup S 2α
R2d S
× 0
S>0,ω∈Sd−1
2 (S − t)−2α gu ∗ fˆ (tω)dt ≤ C( f, T ) < +∞
for a function f ∈ S(Rd ), with a constant C( f, T ) that does not depend on ε ∈ (0, 1). Hence, (5.16) holds. Next, we establish (5.15). Suppose first that θ ∈ S(R2d ). For each ε ∈ (0, 1] the real valued process { Z¯ ε (t), θ , t ≥ 0} is Gaussian. In order to prove its tightness we will show that its covariance Rε (t, s) satisfies |Rε (t, s) − R(s, s)| + |Rε (t, t) − Rε (t, s)| ≤ C(t0 , T ; θ )(t − s)
(5.19)
for all t > s, ε ∈ (0, 1), and t0 ≤ t, s ≤ T . For t > s the covariance Rε (t, s) of the process Z¯ ε (t), θ equals s 1 σσ Rε (t, s) = ei(x−x )· p/ε W¯ (u, x, k + σ p/2)W¯ (u, x , k + σ p/2) ε 0 σ,σ =±1
× S ∗ (t − u)θ (x, k)S ∗ (s − u)θ (x , k )duµ(d p)dxdx dkdk . Hence, we have Rε (t, s) − Rε (s, s) =
1 ε ×
σσ
s
t
du 0
σ,σ =±1
du
s
ei(x−x )· p/ε W¯ (u, x, k + σ p/2)W¯ (u, x , k + σ p/2)
×S ∗ (u − u)θ A,L (x, k)S ∗ (s − u)θ (x , k )µ(d p)dxdx dkdk ,
504
T. Komorowski, S. Peszat, L. Ryzhik
where θ A,L (x, k) := (−A + 21 L)θ (x, k). Using the same argument as in the proof of Lemma 5.1 we obtain t s 1 Rε (t, s) − Rε (s, s) = σσ du du e−(u +s) ε 0 s m,m ,n,n σ,σ =±1 A,L ˜ m,n,σ ˜ m ,n ,σ µ(d p), W W × (5.20) Rd
where A,L ˜ m,n,σ W =
m n 1 dτ dρ dxdk µ(dk j ) µ(dl j )ei x· p/ε m!n! m (u −u) n (u) j=1 j=1 ⎛ ⎞ m m × θ A,L ⎝x + k(u − u) + kjτj, k + k j⎠ j=1
⎛
× W0 ⎝x − k + and ˜ m ,n ,σ = (m !n !)−1 W ×
σp 2
u−
j=1
n
ljρj, k +
j=1
m (s−u)
dx dk ⎛
dτ
m
l j⎠ ,
j=1
n (u)
dρ
µ(dk j )
j=1
× θ ⎝x + k (s − u) + ⎛
⎞
σp + 2
n
n
µ(dl j )e−i x · p/ε
j=1 m
k j τ j , k +
j=1
m
⎞ k j⎠
j=1
⎞ n n p p σ σ × W0 ⎝x − k + u− + l j ρ j , k + l j⎠. 2 2 j=1
j=1
As before, using the specific form of the initial data W0 (x, k) = δ(x) f (k) and performing the Fourier transform of θ (x, k) and f (k) we obtain u 1 A,L ˜ , p + εq, z Wm,n,σ = ε n n m!n! ε pu + qu + y + z) · k × m u − u, q, y exp i( ε u × exp i(σ p/2) · ( p + εq) + z θˆA,L (q, y) fˆ(z)dkdqdydz ε u pu εn , p + εq, − − qu − y m (u − u, q, y) = n m!n! ε ε
pu × exp i(σ p/2) · (u − u )q − y θˆA,L (q, y) fˆ(− − qu − y)dqdy. ε
Limit of Fluctuations of Solutions of Wigner Equation
Likewise, we have
pu − q s − y m (u, q , y ) ε ε pu ˆ − q s − y )dq dy . × exp −i(σ p/2) · y θ q , y fˆ( ε In consequence, (5.20) becomes t s 1 −(u +s) Rε (t, s) − Rε (s, s) = σσ du du e µ(d p)dqdq dydy ε 0 s σ,σ =±1 u pu , p + εq, − − qu − y + (u − u, q, y) × exp ε ε uε −1 , − p + εq , ε pu − q s − y + (u, q , y ) θˆA,L (q, y) θˆ q , y × exp ε ε
pu pu − qu − y) fˆ( − q s − y ). × exp i(σ p/2) · (u − u )q − y − y fˆ(− ε ε Changing variable u new := u/ε we obtain that ˜ m ,n ,σ = W
εn m !n !
n
u
505
, − p + εq ,
(B )
(B )
|Rε (t, s) − Rε (s, s)| ≤ C(T, f )θ A,L 1,1 θ 1,1 (t − s)
(5.21)
for all ε ∈ (0, 1], t, s ∈ [0, T ]. This estimates the first term in (5.19). On the other hand, for t > s we also have t s 1 σσ du du ei(x−x )· p/ε Rε (t, t) − Rε (t, s) = ε 0 s σ,σ =±1
× W¯ (u, x, k + σ p/2)W¯ (u, x , k + σ p/2) ×S ∗ (u − u)θ A,L (x, k)S ∗ (t − u)θ (x , k )µ(d p)dxdx dkdk t 1 + σσ du ei(x−x )· p/ε W¯ (u, x, k + σ p/2) ε s σ,σ =±1 ¯
× W (u, x , k + σ p/2)S ∗ (t − u)θ (x, k) × S ∗ (t − u)θ (x , k )µ(d p)dxdx dkdk .
(5.22)
Denote the first and the second terms on the right-hand side of (5.22) by R1 and R2 respectively. The first term R1 can be estimated exactly in the same way as |Rε (t, s) − Rε (s, s)| and we obtain (B )
(B )
|R1 | ≤ Cθ A,L 1,1 θ 1,1 (t − s) for a certain constant independent of ε > 0 and θ . On the other hand, the term R2 equals t 1 −2t R2 = σσ e du µ(d p)dqdq dydy ε s σ,σ =±1 × exp −i(σ p/2) · y + y θˆ (q, y) θˆ q , y u pu × exp ε , p + εq, − − q(t − u) − y + (t − u, q, y) ε uε pu × exp ε , p + εq , − − q(t − u) − y + (t − u, q , y ) ε ε pu pu ˆ − q(t − u) − y) fˆ( − q (t − u) − y ). × f (− ε ε
506
T. Komorowski, S. Peszat, L. Ryzhik
We can further decompose R2 as R2 = R21 + R22 , where the terms R21 , R22 correspond to integration with respect to the p variable over the regions [| p| ≤ εγ ] and [| p| > εγ ] with some γ ∈ (0, 1): µ(d p) (B ) |R21 | ≤ C T (t − s) fˆ2∞ (θ 1,1 )2 εγ −1 . γ [| p|≤ε ] | p| On the other hand, for R22 we note that CT t |R22 | ≤ du µ(d p) dqdq dydy |θˆ (q, y) ||θˆ q , y | ε s [| p|>εγ ] pu pu ×| fˆ(− − q(t − u) − y)|| fˆ( − q (t − u) − y )|. ε ε Let c > 0 be a fixed constant. We can split the region of integration over q and y variables over the region A consisting of those (q, y, q , y ), for which at least one of these variables is greater than cεγ −1 , and its complement Ac . Denote the respective terms by R22 and R22 . Note that since s ≥ t0 > 0 in the latter case we can find an appropriate c > 0 such that |R22 | ≤
CT (B ) 2 (t − s)(θ 1,1 ) sup | fˆ(z)|2 ≤ C(t − s), ε [|z|≥cεγ −1 ]
since f ∈ S(Rd ). Finally if one of the variables (q, y, q , y ) is greater than cεγ −1 we can use the fact that θ ∈ S(R2d ) to obtain that |R22 | ≤ C(t − s) for some constant C > 0 independent of ε > 0. We conclude that for any T1 > t0 > 0 there exists a constant, independent of ε > 0, such that |Rε (t, t) − Rε (t, s)| ≤ C(t − s)
(5.23)
for any t > s belonging to [t0 , T1 ]. Combining (5.21) with (5.23) and using Gaussianity of { Z¯ ε (t), t ≥ 0} we deduce (5.19) and hence tightness of the laws of { Z¯ ε (t), θ , t ≥ 0}, ε ∈ (0, 1] in C[t0 , +∞) when θ ∈ S(R2d ) and t0 > 0 is fixed. To show tightness for an arbitrary θ ∈ H s,u it suffices only to use density of S(R2d ) in H s,u and boundedness estimate (5.14). Convergence of the initial data. Finally, we prove that for any t0 > 0 the laws of the processes t −1/2 S0 (−s)Cε [W¯ (s)]en dBn (s), Gε (t) := ε n
0
which appear in the right side of (5.12), converge weakly, over C([t0 , +∞); H −s,−u ), to the law of a constant process G(t) ≡ X . As we have pointed out the law of the latter is supported in this space, provided that s, u > d. The proof of tightness essen t tially follows the same argument as the one for tightness of Z¯ ε (t) = ε−1/2 n 0 S(t − s)Cε [W¯ (s)]en dBn (s). We focus therefore on the limit identification. Thanks to Gaussianity of {Gε (t), t ≥ 0} it suffices only to calculate the limit of covariance
Cε (t, s; θ, θ ) := E Gε (t), θ Gε (s), θ
Limit of Fluctuations of Solutions of Wigner Equation
507
as ε → 0+ for t > s and θ, θ ∈ S(R2d ). A simple calculation shows that s 1 Cε (t, s; θ, θ ) = σσ ei(x−x )· p/ε W¯ (u, x, k + σ p/2)W¯ (u, x , k +σ p/2) 5d ε 0 R
=
σ,σ =±1 ×S0∗ (−u)θ (x, k)S0∗ (−u)θ (x , k )duµ(d p)dxdx dkdk s 1 ˜ nW ˜ µ(d p), σσ du W n ε 0 n,n σ,σ =±1
where S0∗ (−u) is the adjoint of S0 (−u) defined in (5.13) and n 1 ˜ Wn := dρ dxdk µ(dl j ) n! n (u) R(n+2)d j=1 ⎛ ⎞ n n σp σp u− + l j ρ j ,k + l j ⎠. ei x· p/ε θ (x − ku, k) W0 ⎝x − k + 2 2 j=1
j=1
˜ is similar, except θ , n, k, x are replaced by θ , n , k , x and ei x· p/ε The formula for W n by e−i x · p/ε . Using the same approach as in the proof of Lemma 5.1 we obtain n ˜n = ε W F1 (θ ) (q, k) fˆ(z)n n! R2d × (u/ε, p + εq, z) exp {i(k + σ p/2) · [(u/ε)( p + εq) + z]} dqdz. Hence, changing variable u new := u/ε, we obtain that the covariance is s/ε Cε (t, s; θ, θ ) = σσ du dqdzdq dz µ(d p) exp i p σ z + σ z /2 σ,σ =±1
R5d
0
× exp {ε [ (u, p + εq, −z − u( p + εq)) + u, − p + εq , −z − (− p + εq )u θˆ (q, −z) × fˆ(−z − ( p + εq)u)θˆ q , −z fˆ(−z − (− p + εq )u). Passing to the limit ε → 0+, which can easily be justified via Lebesgue dominated convergence theorem, we obtain +∞ lim Cε (t, s; θ, θ ) = σσ du dzdz µ(d p) exp i p σ z + σ z /2 ε→0+
0
σ,σ =±1
R3d
× F2 θ (0, z) fˆ(z − pu)F2 θ 0, z fˆ(z + pu) = E X, θ X, θ ,
cf. formula (4.6). Suppose that {( Z¯ (t), G(t)), t ≥ 0} is a limiting point of {( Z¯ ε (t), Gε (t)) t ≥ 0}, ε ∈ (0, 1]. Then, for any θ ∈ S(R2d ) we have t
Z¯ (t), θ =
S0 (t − s)M Z¯ (s), θ ds + G(t), θ , 0
which is equivalent to (4.5). Thus, Theorem 4.1 follows. Thanks to Lemma 5.2 this concludes the proof of the convergence of finite dimensional distributions ( Z ε (t1 ), θ1 , . . . ,
Z ε (t N ), θ N ), as ε → 0+, to the law of ( Z¯ (t1 ), θ1 , . . . , Z¯ (t N ), θ N ) for arbitrary t1 ≤ . . . ≤ t N and θ1 , . . . , θ N ∈ H −s,−u where s, u > d.
508
T. Komorowski, S. Peszat, L. Ryzhik
Tightness of {Z ε (t), t ≥ 0}, as ε → 0+. According to the results of [23] it suffices only to show that for any θ ∈ S(Rd ) the family of processes { Z ε (t), θ , t ≥ 0} is tight in C[t0 , +∞), as ε → 0+ for any t0 > 0. From our previous consideration we know that { Z¯ ε (t), θ , t ≥ t0 } is tight, as (θ) ε → 0+. Since Z ε (t) = Uε (t) + Z¯ ε (t) it suffices therefore to prove that {Uε (t) :=
Uε (t), θ , t ≥ t0 } is tight, as ε → 0+. We start with the following modification of Lemma 5.2. Proposition 5.4. Suppose that T > 0. Then, there exists a constant C depending on , T such that sup
θ H s,u ≤1 t∈[0,T ]
E[ Uε (t), θ 4 ] ≤ Cε2
(5.24)
for all ε ∈ (0, 1]. Proof. Suppose, that θ H s,u ≤ 1. Using (3.22), with initial data vanishing, and Burkholder-Davis-Gundy inequality, see e.g. Corollary 4.2, p. 161 of [28], we obtain ⎧ ⎡ ⎤2 ⎪ ⎨ t
S(t − s)Cε [Uε (s)](en ), θ 2 ds ⎦ E[ Uε (t), θ 4 ] ≤ C E ⎣ ⎪ 0 ⎩ n≥0 2 ⎫ ⎬ t +E
S(t − s)Cε [ Z¯ ε (s)](en ), θ 2 ds . (5.25) ⎭ 0 n σ , p
From (3.24) (notation for θε (s) is the same as in (3.25)) we obtain that the first term on the right hand side equals ⎧ ⎫2 ⎨ t ⎬ E
Uε (s), θεσ, p (t − s) Uε (s), (θεσ , p (t − s))∗ µ(d p)ds ⎩ ⎭ d σ,σ =±1 0 R t ≤ 4t 2 sup E Uε (s), θ 4 ds. 0 θ H s,u ≤1
To deal with the second term we observe that t
S(t − s)Cε [ Z¯ ε (s)](en ), θ 2 ds 0
n
belongs to the space of second degree polynomials formed over the Gaussian Hilbert space corresponding to the Gaussian stochastic process { Z¯ ε (s), s ∈ [0, t]}, see Sect. 1.3 of [18] for the definition of Gaussian Hilbert spaces and Definition 2.1 of ibid. for the definition of polynomials. According to Theorem 5.10, p. 62 of ibid. all L p norms on the space of given degree polynomials are equivalent. Therefore the term in question can be estimated by , 2 t 2 ¯ C E
S(t − s)Cε [ Z ε (s)](en ), θ ds ≤ C ε2 , n
0
thanks to the argument contained in the proof of Lemma 5.1 (recall that θ H s,u ≤ 1). Estimate (5.24) follows then upon an application of Gronwall’s inequality.
Limit of Fluctuations of Solutions of Wigner Equation
509
(θ)
To verify tightness of {Uε (t), t ≥ 0} it suffices to establish, see e.g. [10] Theorem 12.3, p. 95, that for each T > t0 > 0 there exists a constant C independent of ε ∈ (0, 1] and such that E[Uε(θ) (t) − Uε(θ) (s)]4 ≤ C(t − s)2 , ∀ t0 ≤ s < t ≤ T.
(5.26)
From (5.11) and Hölder inequality we obtain that the left hand side of (5.26) can be bounded by ⎧ ⎡ ⎤4 ⎪ t ⎨
S(t − τ )Cε [Uε (τ )]en , θ dBn ⎦ C E⎣ ⎪ s n≥0 ⎩ ⎤4 ⎫ ⎡ ⎪ t ⎬ +E⎣
S(t − τ )Cε [ Z¯ ε (τ )]en , θ dBn ⎦ (5.27) ⎪ s n≥0 ⎭ for some constant C > 0. Using the Burkholder-Davis-Gundy inequality we can estimate the first term above by ⎧ ⎫2 ⎨ t ⎬ CE
Uε (τ ), θεσ, p (t − τ ) Uε (τ ), (θεσ , p (t − τ ))∗ µ(d p)dτ .(5.28) ⎩ ⎭ Rd s σ,σ =±1
From (5.24) and Cauchy-Schwartz inequality we obtain that the expression in (5.28) is bounded by C(t − s)2 . To deal with the second term in (5.27) we again use equivalence of L p norms on Gaussian spaces and obtain that the term in question is estimated by ⎫2 ⎧ ⎬ ⎨ t E Z¯ ε (τ ), θεσ, p (t − τ ) Z¯ ε (τ ), (θεσ , p (t − τ ))∗ µ(d p)dτ .(5.29) C ⎭ ⎩ Rd s σ,σ =±1
Using (5.17) we again estmate (5.29) by C(t − s)2 , thus (5.26) follows. Acknowledgements. This work has been partly supported by Polish Ministry of Science and Higher Education Grants N 20104531 (T.K.), PO3A03429 (Sz.P.). In addition T.K. and Sz.P. acknowledge the support of EC FP6 Marie Curie ToK programme SPADE2, MTKD-CT-2004-014508 and Polish MNiSW SPB-M. The work of L.R. has been supported by NSF grant DMS-0604687 and ONR.
References 1. Bal, G.: On the self-averaging of wave energy in random media. SIAM Multiscale Model. Simul. 2, 398–420 (2004) 2. Bal, G.: Kinetics of scalar wave fields in random media. Wave Motion 43, 132–157 (2005) 3. Bal, G., Komorowski, T., Ryzhik, L.: Self-averaging of the Wigner transform in random media. Commun. Math. Phys. 242, 81–135 (2003) 4. Bal, G., Carin, L., Liu, D., Ren, K.: Experimental validation of a transport-based imaging method in highly scattering environments. Inverse Problems 26, 2527–2539 (2007) 5. Bal, G., Liu, D., Vasudevan, S., Krolik, J., Carin, L.: Electromagnetic Time-Reversal Imaging in Changing Media: Experiment and Analysis. IEEE Trans. Anten. and Prop. 55, 344–354 (2007) 6. Bal, G., Pinaud, O.: Kinetic models for imaging in random media. Multiscale Model. Simul. 6, 792–819 (2007)
510
T. Komorowski, S. Peszat, L. Ryzhik
7. Bal, G., Pinaud, O.: Self-averaging of kinetic models for waves in random media. Kinet. Relat. Models 1, 85–100 (2008) 8. Bal, G., Papanicolaou, G., Ryzhik, L.: Radiative transport limit for the random Schroedinger equation. Nonlinearity 15, 513–529 (2002) 9. Bal, G., Papanicolaou, G., Ryzhik, L.: Self-averaging in time reversal for the parabolic wave equation. Stoch. Dyn. 2, 507–531 (2002) 10. Billingsley, P.: Convergence of probability measures. New York: Wiley & Sons, 1968 11. Dawson, D., Papanicolaou, G.: A random wave process. Appl. Math. Optim. 12, 97–114 (1984) 12. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press, 1992 13. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge: Cambridge University Press, 1996 14. Erdös, L., Yau, H.T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger Equation. Comm. Pure Appl. Math. 53, 667–735 (2000) 15. Gérard, P., Markowich, P.A., Mauser, N.J., Poupaud, F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50, 323–380 (1997) 16. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. GroningenAmsterdam: North-Holland, 1981 17. Jakubowski, A.: On the Skorochod topology. Ann. de l’I.H.P., Sect. B 22, 263–285 (1986) 18. Janson, S.: Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, Cambridge: Cambridge University. press, 1997 19. Kipnis, C., Landim C.: Scaling limits of interacting particle systems. Berlin-Heidelberg-New York: Springer-Verlag, 1999 20. Komorowski, T., Ryzhik, L.: Fluctuations of the Wigner transform in Markovian potentials. In preparation 21. Lions, P.-L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9, 553–618 (1993) 22. Lukkarinen, J., Spohn, H.: Kinetic limit for wave propagation in a random medium. Arch. Ration. Mech. Anal. 183, 93–162 (2007) 23. Mitoma, I.: Tightness of Probabilities On C([0, 1]; S) and D([0, 1]; S). The Annals of Probability 11, 989–999 (1983) 24. Papanicolaou, G., Ryzhik, L., Solna, K.: Self-averaging from lateral diversity in the Itô-Schrödinger equation. SIAM MMS 6, 468–492 (2007) 25. Peszat, S., Zabczyk, J.: Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Processes Appl. 72, 187–204 (1997) 26. Peszat, S., Zabczyk, J.: Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116, 421–443 (2000) 27. Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise (an Evolution Equation Approach). Cambridge: Cambridge University Press, 2007 28. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. 3rd edition, Berlin-Heidelberg-New York: Springer-Verlag, 1999 29. Ryzhik, L., Papanicolaou, G., Keller, J.B.: Transport equations for elastic and other waves in random media. Wave Motion 24, 327–370 (1996) 30. Spohn, H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17, 385–412 (1977) Communicated by H. Spohn
Commun. Math. Phys. 292, 511–528 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0892-3
Communications in
Mathematical Physics
On the Behavior of Eisenstein Series Through Elliptic Degeneration D. Garbin1, , A.-M. v. Pippich2, 1 Mathematics Ph.D. Program, The Graduate Center of CUNY, 365 Fifth Avenue,
New York, NY, U.S.A. E-mail: [email protected]
2 Mathematics Ph.D. Program, Humboldt-Universität zu Berlin, Institut für Mathematik,
Rudower Chaussee 25, 12489 Berlin, Germany. E-mail: [email protected] Received: 15 October 2008 / Accepted: 5 May 2009 Published online: 20 August 2009 – © Springer-Verlag 2009
Abstract: Let be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = \H be the associated finite volume hyperbolic Riemann surface. If γ is a primitive parabolic, hyperbolic, resp. elliptic element of , there is an associated parabolic, hyperbolic, resp. elliptic Eisenstein series. In this article, we study the limiting behavior of these Eisenstein series on an elliptically degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. The elliptic Eisenstein series associated to a degenerating elliptic element converges up to a factor to the parabolic Eisenstein series associated to the parabolic element which fixes the newly developed cusp on the limit surface. 1. Introduction 1.1. Eisenstein series. Let ⊆ PSL2 (R) denote a Fuchsian group of the first kind acting by fractional linear transformations on the upper half-plane H := {z ∈ C | z = x + i y, y > 0}. Let M := \H, which is a Riemann surface of finite volume with respect to the natural hyperbolic metric induced from H. Associated to each cusp of M, or equivalently, associated to any -inconjugate, primitive parabolic element γ ∈ , there is a classically studied non-holomorphic Eisenstein series Epar;M,γ (z, s), which is defined by a generalized Dirichlet series that converges for Re(s) > 1 and which admits a meromorphic continuation to the whole s-plane. We will refer to the series Epar;M,γ (z, s) as parabolic Eisenstein series. The theory of these Eisenstein series plays a prominent role in the theory of automorphic functions and automorphic forms, and is, by now, a classical part of mathematical literature. For example, the significance of the Eisenstein series Epar;M,γ (z, s) in the spectral theory of automorphic functions for a The first author acknowledges support from the PSC–CUNY grant 69288-00-38.
The second author acknowledges support from the DFG Graduate School Berlin Mathematical School
and the DFG Research Training Group Arithmetic and Geometry.
512
D. Garbin, A.-M. v. Pippich
Fuchsian group relies on the fact that these series are eigenfunctions of the hyperbolic Laplacian hyp for the continuous spectrum. Further, associated to any primitive hyperbolic element γ ∈ , one can define a hyperbolic Eisenstein series Ehyp;M,γ (z, s), which is the scalar-valued analog to the formvalued hyperbolic Eisenstein series introduced by S. Kudla and J. Millson in [KM 79]. The hyperbolic Eisenstein series has been recently studied in [Ri 04,Fa 07,GJM 08], and [JKVP]. Following these ideas, J. Jorgenson and J. Kramer were lead to consider the so-called elliptic Eisenstein series Eell;M,γ (z, s) associated to any primitive elliptic element γ ∈ . In fact, these series have been introduced in the unpublished paper [JK 03] in order to derive optimal sup-norm bounds for cusp forms of weight 2 for the group . For further aspects of the elliptic Eisenstein series, the reader is referred to [vP 05,KVP], and to the second author’s thesis project [vP], where the meromorphic continuation of the elliptic Eisenstein series Eell;M,γ (z, s) is proven, various expansions of the series are computed and a Kronecker limit type formula is established there.
1.2. The main results. This article continues in the direction of the existing papers [Fa 07,GJM 08, and Ob 08]. The article [Fa 07] studies the asymptotic behavior of hyperbolic Eisenstein series when considering a hyperbolically degenerating family of finite volume hyperbolic Riemann surfaces, in short, a family parametrized by a finite number of closed geodesics with hyperbolic lengths converging to 0 and developing new cusps on the limit surface. In brief, the main result in [Fa 07] is that the limit of the (properly scaled) hyperbolic Eisenstein series associated to the primitive hyperbolic element which fixes the pinching geodesic is equal to the parabolic Eisenstein series associated to the newly developed cusp on the limit surface. The main results in [Fa 07] are reproved in [GJM 08] using different methods, namely counting function arguments and the Stieltjes integral representations of the considered Eisenstein series. In this article, we study the asymptotic behavior of Eisenstein series when considering an elliptically degenerating family of finite volume hyperbolic Riemann surfaces, in short, a family parametrized by a finite number of elliptic fixed points with orders converging to infinity, resp. with hyperbolic cone angles converging to zero, and developing new cusps on the limit surface (see [GJ]). Since it is of general interest (see, for example, the article [IO 08], where parabolic, hyperbolic, and elliptic Poincaré series are investigated), we study the complete picture of Eisenstein series, i.e parabolic, hyperbolic, and elliptic Eisenstein series. Moreover, we give the missing statement for the elliptic Eisenstein series in the case of a hyperbolically degenerating family of finite volume hyperbolic Riemann surfaces. This completes the picture of the asymptotic behavior of Eisenstein series in the context of degenerating families of finite volume hyperbolic Riemann surfaces. Precise definitions and references to all concepts will be given in Sect. 2 below. However, with these comments made, we are able to state the main result of this article. Main Theorem. Let {Mq }q=(q1 ,...,qm ) be an elliptically degenerating family of finite volume hyperbolic Riemann surfaces with limit surface M∞ . For j = 1, . . . , m, let γq j ∈ q be a degenerating primitive elliptic element of order n q j (n q j ∈ N, n q j > 1) and hyperbolic cone angle αq j = 2π/n q j , and let γw j ∈ ∞ denote the primitive parabolic element of width w j which fixes the newly developed cusp Pw j on the limit surface M∞ . Then, the following assertions hold true:
On the Behavior of Eisenstein Series Through Elliptic Degeneration
513
(i) For a primitive parabolic, hyperbolic, resp. non-degenerating elliptic element γ ∈ q , we have lim E∗;Mq ,γ (z, s) = E∗;M∞ ,γ (z, s),
q→∞
where ∗ stands for parabolic, hyperbolic, elliptic, respectively. (ii) For a degenerating primitive elliptic element γq j ∈ q ( j = 1, . . . , m), we have lim α −s Eell;Mq ,γq j (z, s) q→∞ q j
= Epar;M∞ ,γw j (z, s).
In all instances, the convergence is uniform on compact subsets of M∞ bounded away from the developing cusps and elliptic cones, and in half-planes of the form Re(s) ≥ 1+δ for δ > 0. 1.3. Related studies. For quite some time, the behavior of the Selberg zeta function on families of hyperbolically degenerating Riemann surfaces has been of interest to physicists in the field of string theory. Namely the computations of the closed bosonic as well as fermionic strings amount to evaluating a certain integral over the moduli space of compact hyperbolic Riemann surfaces of genus g, where the integrand involves some special values of the Selberg zeta function (see pp. 2–3 of [He 90]). In connection with string theory, D. Hejhal ([He 90]) and S. Wolpert ([Wo 87]) had been motivated to study the asymptotic behavior of the Selberg zeta function on families of compact hyperbolic Riemann surfaces in the setting of hyperbolic degeneration. The Selberg zeta function can be viewed as a spectral invariant. More generally, in a series of papers ([JL 95,JL 97a,JL 97b,HJL 97]) other spectral invariants such as the regularized trace of the hyperbolic heat kernel, the spectral determinant, or the spectral zeta function, have been identified and their behavior through hyperbolic degeneration has been investigated. In [DJ 98], the authors consider such invariants on families of degenerating 3-manifolds. More recently ([GJM 08]), it has been shown that the Eisenstein series mentioned in Sect. 1.1 are also invariant objects through hyperbolic degeneration. In [GJ], the authors consider a new type of degeneration of hyperbolic Riemann surfaces (elliptic degeneration) and investigate the behavior of such spectral invariants. 1.4. Outline of the paper. The paper is organized as follows. In Sect. 2, we recall and summarize basic notations, definitions and known results used in this article. Of particular importance is the Stieltjes integral representation of the parabolic, hyperbolic, resp. elliptic Eisenstein series employing parabolic, hyperbolic, elliptic counting functions, respectively. In Sect. 3, we study the limiting behavior of these counting functions through elliptic degeneration. We conclude by proving the Main Theorem in Sect. 4. Our proofs closely follow the method of proof in [GJM 08]. However, for the sake of completeness and convenience of the reader we include such proofs here. 2. Background Material 2.1. Basic notation. As mentioned in the Introduction, we let ⊆ PSL2 (R) denote a Fuchsian group of the first kind acting by fractional linear transformations on the upper half-plane H := {z ∈ C | z = x +i y, y > 0}. We let M := \H, which is a finite volume
514
D. Garbin, A.-M. v. Pippich
hyperbolic Riemann surface, and denote by p : H −→ M the natural projection. The 2 , resp. the hyperbolic Laplacian hyperbolic line element dshyp hyp , are given as 2 ∂ d x 2 + dy 2 ∂2 2 2 . dshyp := , resp. := −y + hyp y2 ∂ x 2 ∂ y2 By dhyp (z, w) we denote the hyperbolic distance from z ∈ H to w ∈ H. Under the change of coordinates x := eρ cos(θ ) and y := eρ sin(θ ), the hyperbolic line element, resp. the hyperbolic Laplacian, are rewritten as 2 dρ 2 + dθ 2 ∂ ∂2 2 2 dshyp . = = − sin (θ ) + , resp. hyp ∂ρ 2 ∂θ 2 sin2 (θ ) For z = x + i y ∈ H, we define the hyperbolic polar coordinates = (z), ϑ = ϑ(z) centered at i ∈ H by (z) := dH (i, z), ϑ(z) := ( L, Tz ), where L := {z ∈ H | x = Re(z) = 0} denotes the positive y-axis and Tz is the euclidean tangent at the unique geodesic passing through i and z at the point i. In terms of the hyperbolic polar coordinates, the hyperbolic line element, resp. the hyperbolic Laplacian take the form 2 dshyp = sinh2 ( )dϑ 2 + d 2 , resp. hyp = −
∂ 1 ∂2 1 ∂2 − − . ∂ 2 tanh( ) ∂ sinh2 ( ) ∂ϑ 2
In a slight abuse of notation, we will at times identify M with a fundamental domain F in H and identify points on M with their preimages in F . 2.2. Parabolic Eisenstein series. Let γ be a primitive parabolic element of . Hence there is an element σ˜ = σ˜ (γ ) ∈ PSL2 (R) such that 1 w −1 , σ˜ γ σ˜ = 0 1 where w = w(γ ) (w ∈ N, w > 1)denotes the is fixed by width of the cusp Pγ which √ w 0 −1 1 1 γ . With the scaling-matrix σ := σ˜ 0 1/√w , we get σ γ σ = 0 1 . We note that γ := Stab Pγ = γ .
For z ∈ H and s ∈ C, the parabolic Eisenstein series Epar;M,γ (z, s) associated to γ ∈ is given by Epar;M,γ (z, s) := Im(σ −1 ηz)s . (1) η∈γ \
Referring to [He 83,Iw 02 or Ku 73], e.g., where detailed proofs are provided, we recall that the series (1) converges absolutely and locally uniformly for any z ∈ H and s ∈ C with Re(s) > 1, and that it is invariant with respect to . Moreover, the series (1) is an eigenfunction of hyp , i.e. it satisfies the differential equation hyp − s(1 − s) Epar;M,γ (z, s) = 0. (2)
On the Behavior of Eisenstein Series Through Elliptic Degeneration
515
2.3. Hyperbolic Eisenstein series. Let γ be a primitive hyperbolic element of . Hence there is a scaling-matrix σ = σ (γ ) ∈ PSL2 (R) such that /2 0 e −1 , σ γσ = 0 e−/2 where = (γ ) denotes the hyperbolic length of the closed geodesic L γ on M in the homotopy class determined by γ . We note that L γ ), L γ = p( where L γ := σ L and L is the positive y-axis, and that γ := Stab L γ = γ . Using the coordinates ρ = ρ(z) and θ = θ (z) introduced in Subsect. 2.1, the hyperbolic Eisenstein series Ehyp;M,γ (z, s) associated to γ ∈ is defined by s −s Ehyp;M,γ (z, s) := sin θ (σ −1 ηz) = cosh dhyp (ηz, Lγ ) , (3) η∈γ \
η∈γ \
L) from z to the geodesic line L is characrecalling that the hyperbolic distance dhyp (z, terized by the formula L) = 1. sin (θ (z)) · cosh dhyp (z, Referring to [Fa 07,GJM 08,vP 05, or Ri 04], where detailed proofs are provided, we recall that the series (3) converges absolutely and locally uniformly for any z ∈ H and s ∈ C with Re(s) > 1, and that it is invariant with respect to . A straightforward computation shows that the series (3) satisfies the differential equation (4) hyp − s(1 − s) Ehyp;M,γ (z, s) = s 2 Ehyp;M,γ (z, s + 2). 2.4. Elliptic Eisenstein series. Let γ be a primitive elliptic element of . Hence there is a scaling-matrix σ = σ (γ ) ∈ PSL2 (R) such that cos(π/n) sin(π/n) , σ −1 γ σ = − sin(π/n) cos(π/n) where n = n(γ ) (n ∈ N, n > 1) denotes the order of the elliptic fixed point E γ which is fixed by γ and αn := 2π/n denotes the hyperbolic cone angle. Note that γ := Stab E γ = γ . Using the hyperbolic polar coordinates = (z) and ϑ = ϑ(z) introduced in Subsect. 2.1, the elliptic Eisenstein series Eell;M,γ (z, s) associated to γ ∈ is defined by −s Eell;M,γ (z, s) := sinh (σ −1 ηz) . (5) η∈γ \
Referring to [vP 05], where a detailed proof is provided, we recall that the series (5) converges absolutely and locally uniformly for z ∈ H with z = ηE γ for any η ∈ , and for s ∈ C with Re(s) > 1, and that it is invariant with respect to . A straightforward computation shows that the series (3) satisfies the differential equation hyp − s(1 − s) Eell;M,γ (z, s) = −s 2 Eell;M,γ (z, s + 2). (6)
516
D. Garbin, A.-M. v. Pippich
2.5. Parabolic, hyperbolic, and elliptic counting functions. Using the notations of Subsect. 2.2, we fix ε = ε(z, ) ∈ R sufficiently small such that 1/ε > Im(σ −1 ηz) for all η ∈ , and we define the parabolic counting function Npar;M,γ (T ; z, ε) as
ε ) < T , Npar;M,γ (T ; z, ε) := card η ∈ γ \ dhyp (σ −1 ηz, L ε := {z ∈ H | y = Im(z) = 1/ε}. Equivalently, the ε denotes the horocycle L where L function Npar;M,γ (T ; z, ε) counts the number of geodesic paths, i.e. preimages of geodesics on H, each from a different homotopy class, from z ∈ M to the horocycle Lε := ε ) on M of length less than T . Since dhyp (σ −1 ηz, L ε ) = log 1/ε Im(σ −1 ηz) , p(σ L we have ε ) . Im(σ −1 ηz)s = ε−s exp −s · dhyp (σ −1 ηz, L With this observation, we can express the parabolic Eisenstein series (1) as a Stieltjes integral, namely we have ∞ e−su d Npar;M,γ (u; z, ε). (7) Epar;M,γ (z, s) = ε−s 0
Observe that the integral in (7) depends on the choice of ε through the parabolic counting function; however, after multiplying by ε−s the product itself is independent of ε. Using the notations of Subsect. 2.3, we define the hyperbolic counting function Nhyp;M,γ (T ; z) as
L) < T . Nhyp;M,γ (T ; z) := card η ∈ γ \ dhyp (σ −1 ηz, Equivalently, the function Nhyp;M,γ (T ; z) counts the number of geodesic paths, i.e. preimages of geodesics on H, each from a different homotopy class, from z ∈ M to the closed geodesic L γ on M of length less than T . Using the counting function Nhyp;M,γ (T ; z) we can express the hyperbolic Eisenstein series (3) as a Stieltjes integral, namely we have ∞ Ehyp;M,γ (z, s) = cosh(u)−s d Nhyp;M,γ (u; z). (8) 0
Using the notations of Subsect. 2.4, we define the elliptic counting function Nell;M,γ (T ; z) as
Nell;M,γ (T ; z) := card η ∈ γ \ dhyp (σ −1 ηz, i) < T . Equivalently, the function Nell;M,γ (T ; z) counts the number of geodesic paths, i.e. preimages of geodesics on H, each from a different homotopy class, from z ∈ M to the elliptic fixed point p(E γ ) on M of length less than T . Using the counting function Nell;M,γ (T ; z) we can express the elliptic Eisenstein series (5) as a Stieltjes integral, namely we have ∞ ∞ Eell;M,γ (z, s) = sinh(u)−s d Nell;M,γ (u; z) = sinh(u)−s d Nell;M,γ (u; z), 0
u 0 (z)
(9)
On the Behavior of Eisenstein Series Through Elliptic Degeneration
517
where u 0 (z) := minη∈γ \ dhyp (σ −1 ηz, i). Observe that u 0 (z) > 0, since z ∈ H with z = ηE γ for any η ∈ . By following the method of proof in Lemma 3 of [JL 95] (see also [Lu 93 and GJM 08]) which simply utilizes elementary hyperbolic geometric considerations, we can establish the following bounds. For any point z ∈ M with injectivity radius r , and any u ≥ T0 > r , we have 2 T0 −r sinh2 u+r − sinh 2 2 Npar;M,γ (u; z, ε) ≤ Npar;M,γ (T0 ; z, ε) + , (10) sinh2 r2 2 T0 −r sinh2 u+r − sinh 2 2 r Nhyp;M,γ (u; z) ≤ Nhyp;M,γ (T0 ; z) + , (11) 2 sinh 2 sinh2 u+r − sinh2 T02−r 2 Nell;M,γ (u; z) ≤ Nell;M,γ (T0 ; z) + . (12) sinh2 r2 2.6. Elliptically degenerating families of Riemann surfaces. For the convenience of the reader, we recall the notion of an elliptically degenerating family of finite volume hyperbolic Riemann surfaces. For further details, we refer to [Ab 77,Be 83,Fa 73,Ra 79], as well as [Ju 93,Ju 95, and Ju 98]. For m ∈ N>0 and q := (q1 , . . . , qm ), let q ⊆ PSL2 (R) denote a Fuchsian group of the first kind with primitive elliptic elements γq1 , . . . , γqm of order n q1 , . . . , n qm (n q j ∈ N, n q j > 1) with fixed points E q1 , . . . , E qm and scaling-matrices σq1 , . . . , σqm , respectively. By Mq := q \H we denote the associated finite volume hyperbolic Riemann surface and by pq : H → Mq the natural projection. We call a primitive elliptic element γ ∈ q degenerating, if γ = γq j for some j = 1, . . . , m, and non-degenerating, otherwise. For j = 1, . . . , m, let Cq j denote the infinite hyperbolic cone of order n q j . We identify Cq j with its fundamental domain Fq j in H given by
Fq j = z ∈ H 0 ≤ ϑ(z) < 2π/n q j with hyperbolic metric induced from H. For ε > 0, let Cq j ,ε ⊆ Cq j denote the submanifold of Cq j of hyperbolic volume ε given under the identification of Cq j with Fq j by the fundamental domain
Fq j ,ε = z ∈ Fq j 0 ≤ (z) < arccosh(1 + εn q j /2π ) . The boundary ∂Cq j ,ε of Cq j ,ε is identified with ∂Fq j ,ε := {z ∈ Fq j | (z) = arccosh(1+ εn q j /2π )}. Further, let C∞ denote the infinite cusp of width 1. We identify C∞ with its fundamental domain F∞ in H given by
F∞ = z ∈ H 0 ≤ Re(z) < 1 with hyperbolic metric induced from H. For ε > 0, let C∞,ε ⊆ C∞ denote the submanifold of C∞ of hyperbolic volume ε given under the identification of C∞ with F∞ by the fundamental domain
F∞,ε = z ∈ F∞ Im(z) > 1/ε . The boundary ∂C∞,ε of C∞,ε is identified with ∂F∞,ε := {z ∈ F∞ | Im(z) = 1/ε}.
518
D. Garbin, A.-M. v. Pippich
Fig. 1. Elliptic degeneration of q1 and q2
For notational convenience, we write q → ∞ for |q| := inf j=1,...,m n q j → ∞ and q > q for |q| > |q |. An elliptically degenerating family {Mq } of finite volume hyperbolic Riemann surfaces with limit surface M∞ is a family indexed by q = (q1 , . . . , qm ) with q → ∞ satisfying the following conditions: (i) As q → ∞, the finite volume hyperbolic Riemann surface Mq converges to a finite volume hyperbolic Riemann surfaces M∞ which can be realized as M∞ = ∞ \H for a Fuchsian group ∞ ⊆ PSL2 (R) of the first kind with primitive parabolic elements γw1 , . . . , γwm of width w1 , . . . , wm with fixed points Pw1 , . . . , Pwm and scaling- matrices σw1 , . . . , σwm , respectively. The points p∞ (Pw1 ), . . . , p∞ (Pwm ) ∈ M∞ , simply denoted by Pw1 , . . . , Pwm , are the cusps which developed from degeneration (see Fig. 1) and will be called newly developed cusps. (ii) For ε > 0 sufficiently small, the submanifold Cq j ,ε ⊆ Cq j embeds isometrically into Mq for j = 1, . . . , m, and the submanifold C∞,ε ⊆ C∞ embeds m-times isometrically into M∞ . More precisely, for ε > 0 sufficiently small, we have σq j Cq j ,ε ⊆ Mq and σw j C∞,ε ⊆ M∞ , and, as q → ∞, the submanifold σq j Cq j ,ε ⊆ Mq converges to the submanifold σw j C∞,ε ⊆ M∞ ( j = 1, . . . , m). (iii) The hyperbolic metric on Mq converges to the hyperbolic metric on M∞ . Moreover, the convergence is uniform on compact subsets K of M∞ bounded away from the developing cusps, i.e. for compact subsets K ⊆ M∞ \ ∪mj=1 σw j C∞,ε for some ε > 0 sufficiently small. For every q, it is possible to identify points z(q) and w(q) on Mq \ ∪mj=1 σq j Cq,ε such that limq→∞ dq (z(q), w(q)) = d∞ (z(∞), w(∞)). Henceforth, we shall suppress the q-dependence of points which are identified during degeneration and simply write z and w. Suppose that M is a finite volume hyperbolic Riemann surface with p cusps. Then, for 0 < m ≤ p, there exists an elliptically degenerating family {Mq } of finite volume hyperbolic Riemann surfaces Mq with p −m cusps and m degenerating elliptic elements with limit surface M∞ = M (see [Ju 98]). Let us describe how one can convert a cusp to a cone. The discussion follows [Ju 98] (see also Sect. 2 of [Ju 95]), which for reasons of transparency we decide to include here. First consider the infinite cylinder S1 × (0, ∞) equipped with some smooth metric. Denote by (t) the length of the circle S1 × {t} and suppose that (t) tends to 0 as t
On the Behavior of Eisenstein Series Through Elliptic Degeneration
519
tends to ∞. Denote by d(s, t) the signed distance between the two circles S1 × {s} and S1 × {t}. Then we can define the cone angle α as (s) − (t) , α = lim lim t→∞ s→t d(s, t) provided that the limit exists. In such case, we call α the angle of the cone with apex at ∞. To model a cone with angle α, α = 0, and apex at ∞, we consider (S1 ×(0, ∞), m α ), where α m α = ρα (y)2 (d x 2 + dy 2 ) with ρα (y) = . sinh(αy) First we observe that eucl log(ρα (y)) = −ρα (y)2 , where eucl = −(∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 ) denotes the Euclidean Laplacian. This in turn shows that the metric m α has constant curvature equal to −1. Furthermore, it also follows that the length of the circle and the distance between two circles are given by
csch(αy2 ) + coth(αy2 )
α
. and dα (y1 , y2 ) = log
α (y) = sinh(αy) csch(αy1 ) + coth(αy1 )
This makes (S1 × (0, ∞), m α ) a hyperbolic cone of angle α with apex at ∞. Note that as α approaches 0, we see that ρα (y)2 approaches y −2 . To model a cusp, we consider (S1 ×(0, ∞), m 0 ) with metric given by m 0 = y −2 (d x 2 + 2 dy ). It easily follows that (y) = 1/y and d(y1 , y2 ) = | log(y1 /y2 )|, hence α = 0 in this case. Furthermore, m 0 has constant curvature equal to −1, thus (S1 × (0, ∞), m 0 ) is a hyperbolic cone with angle 0 and apex at ∞, i.e. a hyperbolic cusp. Now suppose that M is a hyperbolic Riemann surface with conical ends. For simplicity, let us assume that M has exactly one such end E ⊂ M. Consider a hyperbolic metric g on M such that (E, g) is a cone as described above. In [Ju 93], the author shows that (E, g) embeds isometrically into (M, g). The main theorem in [Ju 98] states the following. Let M be a surface with one end E. Let [g] be a pointwise conformal class of hyperbolic metrics on M. For each α ∈ [0, 2π(1 − χ (M))), let gα be the unique hyperbolic metric on M such that (E, gα ) is a cone of angle α. Define the conformal factor wα by gα = ewα g0 . Then the map that sends α to the smooth factor wα is real analytic. In other words, a hyperbolic surface with cusps belongs to a unique conformal deformation of hyperbolic surfaces with cones and this deformation is real-analytically parametrized by the angles of the cones. 2.7. A Stieltjes integral inequality. Referring to [JK 01] for a detailed proof, we state the following inequality of Stieltjes integrals, which is a key component in our analysis. Let F be a real valued, non-negative, smooth, decreasing function defined for u > 0 and let g1 , g2 be real valued, non decreasing functions defined for u ≥ a > 0 and satisfying g1 (u) ≤ g2 (u) for u ≥ a. Then, the following inequality of Stieltjes integrals ∞ ∞ F(u) dg1 (u) + F(a) g1 (a) ≤ F(u) dg2 (u) + F(a) g2 (a) a
holds, provided both integrals
a
exist.1
1 The authors would like to thank Jürgen Elstrodt for pointing out that the function F must be non-negative, for otherwise the statement does not hold.
520
D. Garbin, A.-M. v. Pippich
3. Convergence of Counting Functions In this section we establish the limiting behavior of the parabolic, hyperbolic, and elliptic counting functions introduced in Subsect. 2.5 on an elliptically degenerating family of finite volume hyperbolic Riemann surfaces. Lemma 3.1. With notations as in Sect. 2, let {Mq } be an elliptically degenerating family of finite volume hyperbolic Riemann surfaces with limit surface M∞ . For T > 0, the following assertions hold true: (a) For a primitive parabolic element γ ∈ q , we have lim Npar;Mq ,γ (T ; z, ε) = Npar;M∞ ,γ (T ; z, ε).
q→∞
(b) For a primitive hyperbolic element γ ∈ q , we have lim Nhyp;Mq ,γ (T ; z) = Nhyp;M∞ ,γ (T ; z).
q→∞
(c) For a non-degenerating primitive elliptic element γ ∈ q , we have lim Nell;Mq ,γ (T ; z) = Nell;M∞ ,γ (T ; z).
q→∞
In all instances, the convergence is uniform on compact subsets of M∞ bounded away from the developing cusps. Proof. For j = 1, . . . , m, let γq j ∈ q be a degenerating elliptic element of order n q j . To prove (c), let γ ∈ q denote a non- degenerating primitive elliptic element with fixed point E γ ∈ Mq . Fix z ∈ Mq and let ε1 > 0 be sufficiently small such that σq j Cq j ,ε1 ⊆ Mq for j = 1, . . . , m and z ∈ Mq \∪mj=1 σq j Cq j ,ε1 for all q. Choose ε0 > 0, ε0 < ε1 sufficiently small, such that dhyp (σq j ∂Cq j ,ε0 , σq j ∂Cq j ,ε1 ) > T . But then, any geodesic path from z ∈ Mq \ ∪mj=1 σq j Cq j ,ε1 to the elliptic fixed point E γ on Mq with length less than T necessarily lies entirely in Mq \ ∪mj=1 σq j Cq j ,ε0 . Since the hyperbolic metric on Mq converges uniformly on compact subsets of M∞ bounded away from the developing cusps (see Subsect. 2.6), this completes the proof of (c). The remaining two cases follow by an analoguous argument. Lemma 3.2. With notations as in Sect. 2, let {Mq } be an elliptically degenerating family of finite volume hyperbolic Riemann surfaces with limit surface M∞ . Let γq j ∈ q ( j = 1, . . . , m) be a degenerating elliptic element of order n q j and let γw j ∈ ∞ be the parabolic element which fixes the newly developed cusp Pw j on the limit surface M∞ . (a) For ε > 0 sufficiently small such that Cq j ,ε embeds isometrically into Mq , we define
Nell;Mq ,∂Cq j ,ε (T ; z) := card η ∈ γq j \q dhyp (σq−1 ηz, ∂Cq j ,ε ) < T . j Then, for T > 0 and for z ∈ Mq \ ∪mj=1 σq j Cq j ,ε , we have Nell;Mq ,γq j (T + g(ε, q j ); z) = Nell;Mq ,∂Cq j ,ε (T ; z), where
εn q j 1 + 1 + 4π/(εn q j ) . g(ε, q j ) := log 1 + 2π
(13)
On the Behavior of Eisenstein Series Through Elliptic Degeneration
521
Fig. 2. Convergence of σq j ∂Cq j ,ε to Lw j ,ε
(b) For T > 0, we have lim Nell;Mq ,γq j (T + g(ε, q j ); z) = Npar;M∞ ,γw j (T ; z, ε)
q→∞
uniformly on compact subsets of M∞ bounded away from the developing cusps. Proof. Fix q and let z ∈ Mq \∪mj=1 σq j Cq j ,ε . For j = 1, . . . , m, lift z to a point z ∈ H and choose the representative η in γq j \q such that ηz ∈ σq j Fq j . Then, z ∈ Mq \σq j Cq j ,ε implies ηz ∈ σq j (Fq j \Fq j ,ε ) which allows us to write dhyp (σq−1 ηz, i) = dhyp (σq−1 ηz, ∂Fq j ,ε ) + dhyp (∂Fq j ,ε , i). j j Since
εn q j εn q j = log 1 + 1 + 1 + 4π/(εn q j ) dhyp (∂Fq j ,ε , i) = arccosh 1 + , 2π 2π
we find ηz, i) = dhyp (σq−1 ηz, ∂Cq j ,ε ) + g(ε, q j ). dhyp (σq−1 j j This proves (a). Now, as q → ∞, the boundary σq j ∂Cq j ,ε of σq j Cq j ,ε converges to the ε ) boundary σw j ∂C∞,ε of σw j C∞,ε which can be identified with Lw j ,ε := p∞ (σw j L (see Fig. 2). Further, the hyperbolic metric on Mq converges to the hyperbolic metric on M∞ away from the developing cusps (see Subsect. 2.6). Hence, we obtain lim Nell;Mq ,γq j (T + g(ε, q j ); z) = lim Nell;Mq ,∂Cq j ,ε (T ; z) q→∞
ε ) < T = Npar;M∞ ,γw (T ; z, ε). = card η ∈ γw j \∞ dhyp (σw−1j ηz, L j
q→∞
This completes the proof of the lemma.
4. Convergence of Eisenstein Series In this section, we prove the Main Theorem stated in Subsect. 1.2. In brief, our proof uses the convergence of the counting functions for fixed T > 0, the uniform bounds for the counting functions, and the Stieltjes integral inequality.
522
D. Garbin, A.-M. v. Pippich
4.1. Proof of Main Theorem. Using the notation of Sect. 2, let {Mq } be an elliptically degenerating family of hyperbolic Riemann surfaces of finite volume with limit surface M∞ . Let γq j ∈ q denote the degenerating elliptic element with elliptic fixed point E q j ( j = 1, . . . , m). Further, let s ∈ C with Re(s) ≥ 1 + δ for some fixed δ > 0 and let z ∈ Mq be bounded away from the developing cusps and elliptic cones. Denote by r = r (q) the injectivity radius of Mq at z. Proof of part (i). In order to prove the statement of convergence in the elliptic case, let γ ∈ q be a non-degenerating primitive elliptic element with elliptic fixed point E γ . We have to show that for every > 0, there exists a q0 = q0 (), q0 > 0, such that for all q > q0 the bound
Eell;M ,γ (z, s) − Eell;M ,γ (z, s) < q ∞ holds. Using (9), we write for T0 > r , T0 −s Eell;Mq ,γ (z, s) = sinh(u) d Nell;Mq ,γ (u; z)+ u 0 (z)
∞ T0
sinh(u)−s d Nell;Mq ,γ (u; z) (14)
and proceed analoguously for Eell;M∞ ,γ (z, s). Then,
Eell;M ,γ (z, s) − Eell;M ,γ (z, s)
q ∞
T0
T0
≤
sinh(u)−s d Nell;Mq ,γ (u; z) − sinh(u)−s d Nell;M∞ ,γ (u; z)
u (z) u 0 (z) 0∞ ∞ + sinh(u)−(1+δ) d Nell;Mq ,γ (u; z) + sinh(u)−(1+δ) d Nell;M∞ ,γ (u; z). (15) T0
T0
To bound the first term on the right hand side of (15) let us choose T0 such that T0 is a point of continuity of Nell;M∞ ,γ (T ; z), meaning there is no geodesic path from z to the conical point E γ associated to γ on M∞ with length equal to T0 . Then, with T0 chosen, there is an integer N and a q0 sufficiently large such that for q > q0 , we have N = Nell;Mq ,γ (T0 ; z) = Nell;M∞ ,γ (T0 ; z). Let {dk,Mq } ⊂ [u 0 (z), T0 ] be the set of lengths on Mq such that for any C > 0 we have Nell;Mq ,γ (dk,Mq − C; z) < Nell;Mq ,γ (dk,Mq + C; z). For simplicity, we count the elements in the set {dk,Mq } with multiplicities so that we have
T0
u 0 (z)
sinh(u)−s d Nell;Mq ,γ (u; z) =
N
sinh(dk,Mq )−s .
k=1
Further, by Lemma 3.1, which we apply for T ∈ [u 0 (z), T 0 ], for := δ /N with δ = δ () > 0, there is a q0 such that for q > q0 , we have dk,Mq − dk,M∞ < for k = 1, . . . , N , so then, N
dk,M − dk,M < δ . q ∞ k=1
(16)
On the Behavior of Eisenstein Series Through Elliptic Degeneration
523
With this, for > 0 and for q > q0 := max(q0 , q0 ), we arrive at the bound
T0
u 0 (z)
≤
−s
sinh(u) d Nell;Mq ,γ (u; z) −
N
T0
u 0 (z)
sinh(u) d Nell;M∞ ,γ (u; z)
−s
sinh(dk,M )−s − sinh(dk,M )−s < , q ∞
(17)
k=1
where the last bound follows from (16) and the absolute continuity of sinh(u)−s on [u 0 (z), T0 ]. To bound the second term on the right hand side of (15), we show the following bound: Given any > 0, there is a T0 = T0 (, δ, r ) such that for all q, we have
∞ T0
sinh(u)−(1+δ) d Nell;Mq ,γ (u; z) < .
(18)
In the notation of Subsect. 2.7, let F be the real valued, non-negative, smooth, decreasing function defined for u > 0 by F(u) := sinh(u)−(1+δ) , and let g1 , g2 be the real valued, non decreasing functions defined for u ≥ T0 > 0 by g1 (u) := Nell;Mq ,γ (u; z), g2 (u) := Nell;Mq ,γ (T0 ; z) +
sinh2
u+r 2
− sinh2 sinh2 r2
T0 −r 2
.
Using inequality (12), we have that g1 (u) ≤ g2 (u) for u ≥ T0 > r . With all this, the Stieltjes integral inequality from Subsect. 2.7 gives the bound
∞ T0
sinh(u)−(1+δ) d Nell;Mq ,γ (u; z)
≤
∞
sinh(u)−(1+δ) dg2 (u) + sinh(T0 )−(1+δ) ·
T0
1 · = 2 sinh2 r2
∞ T0
sinh2
sinh(u)−(1+δ) sinh(u + r )du +
T0 +r 2
− sinh2 sinh2 r2
T0 −r 2
sinh(r ) sinh(T0 )−δ , sinh2 r2
(19)
where for the last equality we used the following identities proven by elementary calculations using trigonometric identities: sinh(u + r ) du, 2 sinh2 r2 T0 + r T0 − r − sinh2 = sinh(r ) sinh(T0 ). sinh2 2 2
dg2 (u) =
524
D. Garbin, A.-M. v. Pippich
√ By the trivial bound sinh(u) ≤ eu /2 and the bound sinh(u) ≥ eu /4 for u ≥ log( 2), we obtain from (19) the estimates ∞ ∞ 4δ e r e−δT0 −δu sinh(u)−(1+δ) d Nell;Mq ,γ (u; z) ≤ e du + · 2 sinh2 r2 T0 T0 δ r 4 e 1 1 r · (20) = e−δT0 · + 2 δ 2 sinh 2 which clearly can be made smaller than any > 0, namely, by taking 1 1 4δ e r 1 T0 > + − log() + log . δ δ 2 sinh2 r2
(21)
Therefore, we have proved the bound asserted in (18). To put all this together, taking T0 a point of continuity of Nell;M∞ ,γ (T ; z) satisfying the inequalities T0 > u 0 (z) , T0 > r , and (21), the first term on the right hand side of (15) is arbitrarily small by (17), and the second and the third term on the right hand side of (15) are arbitrarily small by (20). This completes the proof of part (i) of the Main Theorem in the elliptic case. The analoguos statement in the parabolic, resp. hyperbolic case, is proven starting with the Stieltjes integral representation (7), resp. (8). Then, we use similar arguments as in the elliptic case with a slight modification. Namely, we define F(u) := e−su , resp. F(u) := cosh(u)−s and g1 (u), g2 (u) as suggested by the inequality (10), resp. (11). This completes the proof of part (i) of the Main Theorem. Proof of part (ii). In this part, we show the convergence of the (properly scaled) elliptic Eisenstein series Eell;Mq ,γq j (z, s) associated to the degenerating elliptic element γq j ∈ q to the parabolic Eisenstein Epar;M∞ ,γw j (z, s) associated to the parabolic element γw j ∈ ∞ ( j = 1, . . . , m). Let ε > 0 be as in Lemma 3.2 sufficiently small such that z ∈ Mq \ ∪mj=1 σq j Cq j ,ε . Using (9), we write for T0 + g(ε, q j ) > r , T0 +g(ε,q j ) Eell;Mq ,γq j (z, s) = sinh(u)−s d Nell;Mq ,γq j (u; z) u 0 (z) ∞ sinh(u)−s d Nell;Mq ,γq j (u; z), (22) + T0 +g(ε,q j )
with g(ε, q j ) defined by (13) in Lemma 3.2. We shall multiply both sides of (22) by 2−s esg(ε,q j ) and let q approach infinity. For the second integral in (22), we use part (a) of Lemma 3.2 to write ∞ ∞ −s sinh(u)−s d Nell;Mq ,γq j (u; z) = sinh u + g(ε, q j ) d Nell;Mq ,∂Cq j ,ε (u; z). T0 +g(ε,q j )
T0
The geometric argument from [Lu 93] which produced (10), (11) and (12) immediately extends to give the bound 2 T0 −r sinh2 u+r − sinh 2 2 r Nell;Mq ,∂Cq j ,ε (u; z) ≤ Nell;Mq ,∂Cq j ,ε (T0 ; z) + , 2 sinh 2
On the Behavior of Eisenstein Series Through Elliptic Degeneration
525
for u ≥ T0 > r . Following the computations in (20), we arrive at the estimate
∞
2 δ er 1
−s sg(ε,q j )
−s −δT0 · +1 . sinh(u) d Nell;Mq ,γq j (u; z) ≤ e ·
2 e
δ sinh2 r2 T0 +g(ε,q j ) (23) By choosing 1 1 2 δ er T0 > +1 − log() + log , δ δ sinh2 r2
(24)
we have that the upper bound in (23) is less than . For the first integral in (22), we observe the following elementary result: For fixed x > 0 and s ∈ C with Re(s) > 0, we have lim 2−s esr (sinh(x + r ))−s = e−sx .
r →∞
(25)
Furthermore, the limit (25) is uniform for all x > 0 and Re(s) ≥ 1 + δ. Let f (s, q j ) := 2−s ε−s esg(ε,q j ) . Then, by Lemma 3.2 and the argument yielding (17), we have, for T0 as in (21), the limit
T0 +g(ε,q j )
lim f (s, q j ) sinh(u)−s d Nell;Mq ,γq j (u; z) q→∞ u 0 (z) ∞ e−su d Npar;M∞ ,γw j (u; z, ε). = ε−s u 0 (z)
(26)
Now using (23) and (26) and the triangle inequality, as in (15), we obtain lim f (s, q j )Eell;Mq ,γq j (z, s) = Epar;M∞ ,γw j (z, s).
q→∞
(27)
To complete the proof of part (ii), it remains to evaluate f (s, q j ). From the definition (13), i.e. εn q j 1 + 1 + 4π/(εn q j ) g(ε, q j ) = log 1 + , 2π we immediately derive the relation f (s, q j ) ∼ 2−s ε−s
εn q j π
s
=
nq j
s
2π
= αq−sj
as q → ∞. Substituting (28) into (27), we obtain lim α −s Eell;Mq ,γq j (z, s) q→∞ q j
= Epar;M∞ ,γw j (z, s).
This completes the proof of part (ii) of the Main Theorem.
(28)
526
D. Garbin, A.-M. v. Pippich
Remark 4.2. Since every degenerating elliptic element γq j is parametrized by its hyperbolic cone angle αq j = 2π/n q j which approaches zero as the order n q j runs off to infinity, the statement of part (ii) of the Main Theorem can be rewritten as lim αq−sj Eell;Mq ,γq j (z, s) = Epar;M∞ ,γw j (z, s)
αq j →0
for j = 1, . . . , m, z ∈ Mq bounded away from the developing cusps, and s ∈ C with Re(s) > 1. Remark 4.3. Consider the differential equation (6) satisfied by Eell;Mq ,γq j (z, s) which, after multiplying by the hyperbolic cone angle αq−sj and taking the limit as αq j → 0, gives lim
αq j →0
hyp − s(1 − s) αq−sj Eell;Mq ,γq j (z, s)
= − lim (sαq j )2 αq−(s+2) Eell;γq j (z, s + 2). j αq j →0
(29)
By Remark 4.2, this implies hyp − s(1 − s) Epar;M∞ ,γw j (z, s) = 0 for j = 1, . . . , m, z ∈ Mq bounded away from the developing cusps, and s ∈ C with Re(s) > 1. The point here is that the term on the right hand side of (29) vanishes through elliptic degeneration. This shows that in the setting of part (ii) of the Main Theorem, the differential equation for the elliptic Eisenstein series limits to the differential equation for the parabolic Eisenstein series. Remark 4.4. As mentioned in the Introduction, one can study the asymptotic behavior of Eisenstein series through hyperbolic degeneration. Using the method of proof as in [GJM 08], where parabolic and hyperbolic Eisenstein series are considered, one can obtain the following analogous result for the elliptic Eisenstein series. Let {M } be a hyperbolically degenerating family of finite volume hyperbolic Riemann surfaces with limit surface M0 . Let γ ∈ be a degenerating primitive hyperbolic element of length . For a primitive elliptic element γ ∈ , we have lim Eell;M ,γ (z, s) = Eell;M0 ,γ (z, s).
→0
The convergence is uniform on compact subsets of M0 bounded away from the developing cusps, and in half-planes of the form Re(s) ≥ 1 + δ for δ > 0. Acknowledgement. The authors would like to thank their advisors Jay Jorgenson (for the first named author) and Jürg Kramer (for the second named author) for their support in the process of writing this paper. The first named author would also like to express his gratitude to Yiannis Petridis and Józef Dodziuk for their mathematical guidance as well as financial assistance.
On the Behavior of Eisenstein Series Through Elliptic Degeneration
527
References [Ab 77] [Be 83]
Abikoff, W.: Degenerating families of riemann surfaces. Ann. of Math. (2) 105(1), 29–44 (1977) Beardon, A.F.: The geometry of discrete groups. Graduate Texts in Mathematics 91, New York: Springer-Verlag, 1983 [DP 86] D’Hoker, E., Phong, D.H.: On determinants of laplacians on riemann surfaces. Commun. Math. Phys. 104(4), 537–545 (1986) [DJ 98] Dodziuk, J., Jorgenson, J.: Spectral asymptotics on degenerating hyperbolic 3-manifolds. Mem. Amer. Math. Soc. 135(643) (1998) [Fa 07] Falliero, T.: Dégénérescence de séries d’eisenstein hyperboliques. Math. Ann. 339(2), 341–375 (2007) [Fa 73] Fay, J.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics 352, Berlin: Springer-Verlag, 1973 [GJ] Garbin, D., Jorgenson, J.: Spectral convergence of elliptically degenerating Riemann surfaces. In preparation [GJM 08] Garbin, D., Jorgenson, J., Munn, M.: On the appearance of eisenstein series through degeneration. Comment. Math. Helv. 83(4), 701–721 (2008) [He 83] Hejhal, D.: The Selberg trace formula for PSL(2, R). Vol. 2. Lecture Notes in Mathematics 1001, Berlin: Springer-Verlag, 1983 [He 90] Hejhal, D.: Regular b-groups, degenerating Riemann surfaces, and spectral theory. Mem. Amer. Math. Soc. 88, no. 437 (1990) [HJL 97] Huntley, J., Jorgenson, J., Lundelius, R.: On the asymptotic behavior of counting functions associated to degenerating hyperbolic riemann surfaces. J. Funct. Anal. 149(1), 58–82 (1997) [IO 08] Imamoglu, Ö., O’Sullivan, C.: Parabolic, hyperbolic, and elliptic Poincaré series. http://arxiv. org/abs/0806.4398v1[math.NT], 2008 [Iw 02] Iwaniec, H.: Spectral methods of automorphic forms. Graduate Studies in Mathematics 53, Providence, RI: Amer. Math. Soc., 2002 [JZ 93] Ji, L., Zworski, M.: The remainder estimate in spectral accumulation for degenerating hyperbolic surfaces. J. Funct. Anal. 114(2), 412–420 (1993) [JK 01] Jorgenson, J., Kramer, J.: Bounds for special values of selberg zeta functions of riemann surfaces. J. Reine Angew. Math. 541, 1–28 (2001) [JK 03] Jorgenson, J., Kramer, J.: Canonical metrics, hyperbolic metrics, and Eisenstein series for PSL2 (R). Preprint, 2003 [JKVP] Jorgenson, J., Kramer, J., v. Pippich, A.-M.: On the spectral expansion of hyperbolic Eisenstein series. Submitted [JL 95] Jorgenson, J., Lundelius, R.: Convergence of the heat kernel and the resolvent kernel on degenerating hyperbolic riemann surfaces of finite volume. Quaestiones Math. 18(4), 345–363 (1995) [JL 97a] Jorgenson, J., Lundelius, R.: A regularized heat trace for hyperbolic riemann surfaces of finite volume. Comment. Math. Helv. 72(4), 636–659 (1997) [JL 97b] Jorgenson, J., Lundelius, R.: Convergence of the normalized spectral counting function on degenerating hyperbolic riemann surfaces of finite volume. J. Funct. Anal. 149(1), 25–57 (1997) [Ju 93] Judge, C.: The Laplace spectrum of surfaces with cone points. Ph.D. thesis, Univ. of Maryland College Park, 1993 [Ju 95] Judge, C.: On the existence of maass cusp forms on hyperbolic surfaces with cone points. J. Amer. Math. Soc. 8(3), 715–759 (1995) [Ju 98] Judge, C.: Conformally converting cusps to cones. Conform. Geom. Dyn. 2, 107–113 (1998) (electronic) [KVP] Kramer, J., v. Pippich, A.-M.: Elliptic Eisenstein series for PSL2 (Z). Submitted [Ku 73] Kubota, T.: Elementary theory of Eisenstein series. Tokyo: Kodansha Ltd., 1973 [KM 79] Kudla, S.S., Millson, J.J.: Harmonic differentials and closed geodesics on a riemann surface. Invent. Math. 54(3), 193–211 (1979) [Lu 93] Lundelius, R.: Asymptotics of the determinant of the laplacian on hyperbolic surfaces of finite volume. Duke Math. J. 71(1), 211–242 (1993) [Ob 08] Obitsu, K.: Asymptotics of degenerating Eisenstein series. http://arxiv.org/abs/0801. 3691v3[math.cv], 2008 [vP 05] v. Pippich, A.-M.: Elliptische Eisensteinreihen. Diplomarbeit, Humboldt-Universität zu Berlin, 2005 [vP] v. Pippich, A.-M.: The arithmetic of elliptic Eisenstein series. Ph.D. thesis, Humboldt-Universität zu Berlin. In preparation [Po 81] Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207–210 (1981) [Ra 79] Randol, B.: Cylinders in riemann surfaces. Comment. Math. Helv. 54(1), 1–5 (1979)
528
[Ri 04] [Sa 87] [Wo 87]
D. Garbin, A.-M. v. Pippich
Risager, M.S.: On the distribution of modular symbols for compact surfaces. Int. Math. Res. Not. 41(41), 2125–2146 (2004) Sarnak, P.: Determinants of laplacians. Commun. Math. Phys. 110(1), 113–120 (1987) Wolpert, S.A.: Asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces. Commun. Math. Phys. 112(2), 283–315 (1987)
Communicated by L. Takhtajan
Commun. Math. Phys. 292, 529–568 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0894-1
Communications in
Mathematical Physics
Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory Yuji Kodama1, , Virgil U. Pierce2, 1 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA.
E-mail: [email protected]
2 Department of Mathematics, University of Texas – Pan American,
Edinburg, TX 78539, USA. E-mail: [email protected] Received: 3 November 2008 / Accepted: 18 May 2009 Published online: 14 August 2009 – © Springer-Verlag 2009
Abstract: It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ -function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ -functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function Fnm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the 1 dispersionless Toda lattice hierarchy, and nm Fnm are the Grunsky coefficients of the Faber polynomials. Contents 1.
Introduction and Background . . . . . . . . . . . . . . . . . . 1.1 The Toda lattice hierarchy . . . . . . . . . . . . . . . . . . 1.2 Unitary ensembles of random matrices . . . . . . . . . . . 1.3 The Pfaff lattice hierarchy . . . . . . . . . . . . . . . . . . 1.4 Orthogonal and symplectic ensembles of random matrices Both authors are partially supported by NSF grant DMS0806219.
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
530 530 534 537 538
530
Y. Kodama, V. U. Pierce
1.5 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . The Dispersionless Toda Hierarchy . . . . . . . . . . . . . . . . . . 2.1 The dToda curve and its deformation with the dToda hierarchy . 2.2 The dToda hierarchy implies the dKP hierarchy . . . . . . . . . 3. Combintorial Results for the dToda Hierarchy . . . . . . . . . . . . 3.1 Explicit formulae for Fnm (1; 0) . . . . . . . . . . . . . . . . . . 3.2 Graph enumeration on the sphere . . . . . . . . . . . . . . . . . 4. The Dispersionless Pfaff Hierarchy . . . . . . . . . . . . . . . . . . 4.1 The dispersionless limits of the symplectic Pfaff lattice hierarchy 4.2 The dPfaff curve and the rescaled dPfaff hierarchy . . . . . . . . 5. Universality of the dToda Hierarchy . . . . . . . . . . . . . . . . . . 5.1 Universality in random matrix theory . . . . . . . . . . . . . . . 5.2 The dToda hierarchy implies the dPfaff hierarchy . . . . . . . . 6. Appendix A: The dKP Hierarchy . . . . . . . . . . . . . . . . . . . 7. Appendix B: Computation of C0 (T0 ) . . . . . . . . . . . . . . . . . 7.1 GUE case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 GOE case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 GSE case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
541 541 541 544 548 548 552 556 556 557 559 560 561 562 564 564 565 566 567
1. Introduction and Background We begin with a brief review of the dispersionless limits of the Toda and Pfaff lattice hiearchies (see for examples [2,22,32,33]). Here the Toda hierarchy is the 1-dimensional one with the parameters {t0 , t1 , t2 , . . .}. These limits are found from the equations for the τ -functions of the lattice hierarchies by subsituting an asymptotic expansion for the logarithm of the τ -functions in the small limit. We also describe the connection of the τ -functions of these hierarchies with the partition functions of the unitary, orthogonal, and symplectic ensembles of random matrix models with the matrix size N (or 2N ) related to N = 1 (see for examples [15,17,29]). In particular the asymptotic expansions of the free energy, the logarithm of the partition functions, as the size of the matrices approaches infinity satisfy the underlying assumption for the asymptotic expansion of the logarithm of the τ -functions of the Toda and Pfaff lattice hierarchies. Then the leading order terms of the asymptotic expansions satisfy the dispersionless Toda and Pfaff lattice hierarchies, which are written in terms of the two-point functions denoted by Fnm (see for examples [8,9,33]). There is a rich literature dealing with dispersionless integrable systems including the KP and Toda lattice hierarchies, which mainly involve studying algebraic and complex analytic aspects of the equations and/or discussing applications to 2-dimensional topological field theories (see, for examples, [4,14,24,26,34–36]). However, it seems that there is no paper which connects directly the dispersionless integrable systems with combinatorial problems associated with random matrix ensembles. The present paper is to deal with this connection for combinatorial problems such as counting ribbon graphs on a compact surface of genus zero, and give a universality result for the dispersionless integrable systems which is an analogue of that for the random matrix ensembles.
1.1. The Toda lattice hierarchy. The Toda lattice equation (the first member of the Toda lattice hierarchy) is given by
Dispersionless Integrable Systems and Universality in Random Matrix Theory
⎧ ∂an ⎪ ⎪ = an (bn+1 − bn ), ⎨ ∂t1 ⎪ ∂b ⎪ ⎩ n = an − an−1 . ∂t1
531
n = 1, 2, . . . .
There exists a sequence {τn : n ≥ 0} of τ -functions with τ0 = 1 which generate the an , bn by the formulas τn+1 τn−1 ∂ τn . an = , bn = log τn2 ∂t1 τn−1 We may then write the Toda lattice equation in the Hirota bilinear form, D12 τn · τn = 2τn+1 τn−1 .
(1.1)
Here D1 is the usual Hirota derivative, i.e. for a variable tk which is a flow-parameter for the k th member of the Toda lattice hierarchy, Dk is defined by ∂ ∂ − f (tk )g(tk ) . Dk f · g := tk =tk ∂tk ∂tk Note that the Hirota equation (1.1) of the Toda lattice gives an =
∂2 τn+1 τn−1 log τn = . τn2 ∂t12
(1.2)
The Toda lattice is expressed in the Lax representation with a tridiagonal semi-infinite matrix L by ∂L = [B1 , L], ∂t1
B1 = [L]≥0 ,
where [L]≥0 is the upper triangular part of the matrix L given by ⎛ b1 ⎜a1 ⎜0 L=⎜ ⎜0 ⎝ .. .
1 b2 a2 0 .. .
0 1 b3 a3 .. .
0 0 1 b4 .. .
⎞ ··· · · ·⎟ · · ·⎟ ⎟. · · ·⎟ ⎠ .. .
∂ The Lax matrix is also written in terms of a shift operator := exp( ∂n ), that is, with the eigenvector φ = (φ1 , φ2 , . . .), we have
(Lφ)n = φn+1 + bn φn + an−1 φn−1 = ( + bn + an−1 −1 )φn = λφn . The hierarchy of the Toda lattice is defined by ∂L = [Bk , L], ∂tk
Bk = [L k ]≥0 ,
k = 1, 2, 3, . . . .
532
Y. Kodama, V. U. Pierce
The τ -functions are then the functions of infinite variables, i.e. τ (t) with t = (t1 , t2 , . . .). Then it is well-known that each τn -function also satisfies the KP hierarchy [2,6]: ˆ − 1 D1 Dk τn · τn = 0, h k+1 (D) k = 3, 4, 5, . . . , (1.3) 2 ˆ = (D1 , 1 D2 , 1 D3 , . . .) and h n (x) with x = D ˆ are the elementary symmetric where D 2 3 functions of x = (x1 , x2 , . . .) defined by ∞ ∞ xn z n = h n (x)z n . exp n=1
n=0
In particular, the first equation with k = 3 gives (−4D1 D3 + D14 + 3D22 )τn · τn = 0, which is the KP equation, that is, the function u = 2 ∂∂x 2 log τn for each n satisfies 2
∂ ∂x
∂u ∂ 3 u ∂ 2u ∂u −4 + 3 + 6u + 3 2 = 0, ∂t ∂ x ∂x ∂y
(1.4)
with x = t1 , y = t2 and t = t3 . It is also well-known that the τn -functions of the Toda lattice hierarchy satisfy the following set of equations (see for example [1]): ˆ τn+1 · τn = 0, Dk − h k (D) k = 2, 3, 4, . . . . (1.5) The first equation with k = 2 gives (D2 − D12 )τn+1 · τn = 0.
(1.6)
This equation with (1.1) gives the nonlinear Schrödinger equation (with the change t2 → it2 ), i.e. i
∂ψ ∂ 2 ψ + 2 + 2ψ 2 ψ¯ = 0, ∂t2 ∂t1
with ψ = ττn+1 and ψ¯ = τn−1 τn . Thus, the nonlinear Schrödinger equation is the second n member of the Toda lattice hierarchy. We now briefly summarize the dispersionless limit of the Toda hierarchy: The key ingredient in the dispersionless limit is to introduce a free energy, denoted F, 1 −1 τn (t; ) = exp F(T , T) + O( ) , (1.7) 0 2 where is a small parameter, and T = (T1 , T2 , . . .) represents the slow variables with Tk = tk for k ≥ 1. Note in particular that the limit → 0 gives a continuous limit of the lattice structure, that is, the lattice spacing has the order O() and the limit introduces the continuous variable T0 , n −→ T0 .
Dispersionless Integrable Systems and Universality in Random Matrix Theory
533
The free energy F(T0 , T) is calculated from the limit, F(T0 , T) = lim 2 log τn −1 T; . →0
Then the dispersionless Toda hierarchy can be written in terms of the second derivatives, Fmn =
∂2 F , ∂ Tm ∂ Tn
m, n ≥ 0.
The Fmn play an important role for the dispersionless integrable systems, and they are sometimes referred to as the two point functions of the corresponding topological field theory (see for examples [14,24,4]). In the limit → 0, the an , bn variables in the Toda lattice become ⎧ F00 ⎪ ⎨ an (t) = exp (log τn+1 − 2 log τn + log τn−1 ) −→ e , ∂ ⎪ (log τn − log τn−1 ) −→ F01 . ⎩ bn (t) = ∂t1 Then the Hirota form of the Toda lattice equation (1.1) becomes F11 = e F00 ,
(1.8)
which is called the dispersionless Toda (dToda) equation. Likewise the KP equation (1.4) gives 2 + 3F22 = 0, −4F13 + 6F11
which is the dispersionless KP (dKP) equation. Also the dispersionless limit of (1.6) is 2 = 0. F02 − 2F11 − F01
One should also note that in the dispersioless limit, the spectral problem Lφ = λφ with the Lax operator and the eigenvector φ = (φ1 , φ2 , . . .)T leads to an algebraic equation (the spectral curve of the dToda equation), λ = p(λ) + F01 +
e F00 , p(λ)
(1.9)
where p(λ) is a quasi-momentum given by lim→0 φ/φ = exp(∂ S/∂ T0 ). This is a spectral curve of genus 0, and the dToda hierarchy defines an integrable deformation of the curve (see Lemma 2.1). Here the eigenvector φ is assumed to be in the WKB form as → 0, 1 S(T0 , T) + O(1) . φn (t; ) = exp Although the function S plays a key role for the dispersionless theory, we will not use it in this paper. We just mention that the S-function can be expressed in the form (see for example [32]), S(T0 , T) =
∞ n=1
λn Tn + T0 log λ −
∞ 1 F (T , T). k k 0 k=1 kλ
(1.10)
534
Y. Kodama, V. U. Pierce
The dToda hierarchy may be formulated by the form, ⎧ λe−D(λ)F0 − µe−D(µ)F0 ⎪ ⎪ ⎨ e D(λ)D(µ)F = λ−µ , e F00 (D(λ)+D(µ))F0 ⎪ −D(λ)D(µ)F ⎪ ⎩e e =1− λµ
(1.11)
where D(λ) =
∞ 1 ∂ . n ∂T nλ n n=1
This is obtained by the reduction from the 2-dimesional dToda hierarchy in [8,32] with the constraints t¯k = −tk for all k, and the Toda hierarchy with those constraints is referred to as the 1-dimensional Toda hierarchy. Then one can show that the dToda hierarchy (1.11) leads to the curve (1.9) (see Lemma 2.1 below). We will further discuss the detailed structure of the hierarchy (1.11) in Sect. 2, where we show that (1.11) indeed gives the dispersionless limits of (1.3) and (1.5).
1.2. Unitary ensembles of random matrices. A special class of solutions to the Toda lattice equations defines the partition functions of the unitary ensembles of random matrices. Explicitly the τ -functions are taken to be n 1 (2) 1 2 τn (t) = Z n (V0 (λ); t) = dλ1 · · · dλn |λi − λ j | exp − Vt (λk ) , n! n! R R k=1 i< j j where Vt (λk ) = V0 (λk ) − ∞ j=1 λk t j and V0 (λ) is a polynomial of even degree (e.g. V0 (λ) = 21 λ2 for the Gaussian ensemble). This function represents integration over the eigenvalues of the random matrices of the unitary ensemble, and its Taylor expansion in t contains the moments of traces of powers of the matrix (see for example a common reference [27]). A ribbon graph is a one complex embedded on an oriented genus g surface, such that the complement of the complex is a disjoint collection of sets homeomorphic to discs. We may think of a ribbon graph as a collection of discs (vertices) and ribbons (edges) such that the ribbons are glued together preserving the local orientations of the discs, in other words the ribbons lay flat on the oriented surface. Our ribbon graphs will be labeled in the following way: The vertices are labeled so that they are distinct, and the edges emanating from each vertex are labeled as distinct as well. We also define the degree of the vertex as the number of edges attached to the vertex. We represent ribbon graphs for this paper as graphs drawn on an oriented surface (we only consider a compact surface of genus zero (i.e. a sphere) in this paper, and refer to a ribbon graph on the sphere as a ribbon graph of genus zero). In Fig. 1, we illustrate one vertex of degree 8 and two vertices of degrees of 3 and 5 on a sphere. A fundamental result of random matrix theory is that for a quadratic potential V0 (λ), (2) log(Z N ) possesses an asymptotic expansion in even powers of N , whose terms give generating functions partitioning ribbon graphs by genus:
Dispersionless Integrable Systems and Universality in Random Matrix Theory
1
8
b1
a1
2 7
b5
a3 3
6 5
a2
4
535
b2 b4
b3
Fig. 1. A (ribbon) graph with one oriented vertex of degree 8 on a sphere (left) and a graph with two oriented vertices of degrees 3 and 5 on a sphere (right). Examples of complete connected graphs are shown. The edges drawn are to represent ribbons laying flat on the sphere
Theorem 1.1 (Bessis-Itzykson-Zuber [5], Ercolani-McLaughlin [15]). For the Gaussian unitary ensemble (GUE), we have (2) N 2 (2) N 2 λ ; NT λ ;0 log Z N ZN = eg (T)N 2−2g , 2 2 g≥0 where j
j
T 1T 2 · · · Tj eg (T) = κg ( j1 , j2 , . . . ) 1 2 = κg (j) . j1 ! j2 ! · · · j! 0≤ j1 , j2 ,... j The coefficient κg (j) gives the number of the connected ribbon graphs with jk labeled vertices of degree k for k = 1, 2, . . . on a compact surface of genus g. Note that Theorem 1.1 does not contain our T0 = Nn = n variable in the large N limit. The T0 -variable can be inserted naturally by taking the τ -function in the form with t = N T (or T = t), N 2 1 λ ; NT . τn (t; ) = Z n(2) n! 2 Then the free energy is given by N 2 N 2 1 (2) N 2 log τn (−1 T; ) = log Z n(2) λ ; N T Z n(2) λ ; 0 +log Zn λ ;0 . 2 2 n! 2 √ We note here that the scaling λ → T0 λ leads to n n2 n 2 n (2) N 2 (2) Zn λ ; N T = Zn λ2 ; n Tˆ , λ ; T = T0 2 Z n(2) 2 2T0 T0 2 where Tˆ j = T0 T j (which is the so called Penner scaling). This expression then has the asymptotic expansion including T0 , ˆ 2−2g + C(T0 ; N ) log τn (−1 T; ) = eg (T)n j/2−1
g≥0
=
g≥0
2−2g
T0
ˆ N 2−2g + C(T0 ; N ), eg (T)
536
Y. Kodama, V. U. Pierce (2)
1 where C(T0 ; N ) = log[ n! Z n ( N2 λ2 ; 0)] which can be computed directly (see Appendix B). Note that this asymptotic expansion agrees (at leading order) with the assumption (1.7); and thus gives a class of solutions of the Toda lattice satisfying this assumption. Then the leading order of this asymptotic expansion gives a solution of the dispersionless Toda equation, 1 1 (2) N 2 F(T0 , T) = lim Z λ log ; N T N →∞ N 2 n! n 2 2 ˆ = T0 e0 (T) + C0 (T0 ),
where C0 (T0 ) is the leading order of C(T0 ; N ) for the large limit of N , and as shown in Appendix B, it is given by 1 2 3 . (1.12) C0 (T0 ) = T0 log T0 − 2 2 This formula has also been found directly from the dToda equations in a connection to the topological field theory (see (8.6) of p. 209 in [4]). In particular, Theorem 1.1 implies that the two point function Fnm (1; 0) for mn = 0 represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere (g = 0), i.e. ∂2 F (1; 0) ∂ Tn ∂ Tm ∂ 2 e0 = (0) = κ0 (0, . . . , 1, 0, . . . , 1, 0, . . .), ∂ Tn ∂ Tm
Fnm (1; 0) =
n, m ≥ 1,
where 1’s in κ0 (j) are at the n- and m th places. We will give an explicit formula for Fnm (1; 0) in Sect. 3. We also attach an enumerative meaning to derivatives with respect to T0 : Corollary 1.1. Each derivative F0,2k for k ≥ 1, corresponds to counting the number of connected ribbon graphs with a vertex of degree 2k and a marked face on a sphere. Proof. We start with our relation ˆ + C0 (T0 ) F(T0 ; T) = T02 e0 (T) Tˆ j = κ0 (j)T02 + C0 (T0 ) j! j 1 2− ji + 2 i ji j T i + C0 (T0 ). κ0 (j)T0 i = j! j
(1.13)
One then notes that the number of vertices of a ribbon graph with vertices of type j = ( j1 , j2 , . . . ) is v = i ji and the number of edges is e = 21 i i ji . If the ribbon graph is of genus 0, then by the Euler characteristic formula, the number of faces is f =2−v+e =2−
i
ji +
1 ji i 2 i
Dispersionless Integrable Systems and Universality in Random Matrix Theory
537
so that by inserting this equation into (1.13), we find F(T0 ; T) =
j
f
κ0 (j)T0
Tj + C0 (T0 ), j!
(1.14)
from which one sees that differentiating with respect to T0 is related to counting ribbon graphs with marked faces in the enumerative part (i.e. except for the contributions coming from C0 (T0 )). For example: F0,2k (1; 0) with k = 0 is the number of ribbon graphs with a single vertex of degree 2k and one of the f = 1 + k faces marked. These numbers will also be found in Theorem 3.1 as a consequence of the dToda hierarchy. 1.3. The Pfaff lattice hierarchy. The Pfaff lattice hierarchy is normally referred to as the DKP hierarchy. (The “D” stands for the D-type Lie group which is the symmetry group for the hierarchy [20]). The first member of the DKP hierarchy, called the DKP equation, is given by the set of equations ⎧ ∂ ∂u ∂2 + − ∂u ∂ 3 u ∂ 2u ⎪ ⎪ ⎪ + + 12u = 12 (v v ) −4 + 3 ⎪ ⎨ ∂t1 ∂t3 ∂t13 ∂t1 ∂t22 ∂t12 (1.15) 2 ± t1 ⎪ ± ⎪ ∂v ± ∂ 3 v ± v ∂u ∂v ∂ ⎪ ⎪ ⎩2 + + 6u ∓3 + 2v ± dt1 = 0. ∂t3 ∂t1 ∂t1 ∂t2 ∂t2 ∂t13 The left-hand side of the first equation is just the KP equation, and the right-hand side gives a coupling term with the field v ± . Because of this, the DKP equation is sometimes called a coupled KP equation [18]. In terms of the τ -functions, u and v ± are expressed as u=
∂2 log τn , ∂t12
v± =
τn±1 . τn
The DKP equation is then given by (−4D1 D3 + D14 + 3D22 )τn · τn = 24τn+1 τn−1 (2D3 + D13 ∓ 3D1 D2 )τn±1 · τn = 0.
(1.16)
(1.17)
It is known that the τ -functions are given by Pfaffians [3,18,22]. Since the equation with the set of τ -functions has a lattice structure (i.e. τn is determined by the previous τ -functions with τ0 = 1), we call it the Pfaff lattice in this paper. This choice of name is also convenient for this paper as we wish to compare the Pfaff lattice, as given in [3,22], and the Toda lattice. The Pfaff lattice hierarchy is then given by [2,32] ˜ − 1 D1 Dk+3 τn · τn = h k (D) ˜ τn+1 · τn−1 , h k+4 (D) 2 k = 0, 1, 2, . . . . ˜ τn±1 · τn = 0, h k+3 (∓D) In a similar manner as in the Toda lattice case, one can also consider the dispersionless limit of the Pfaff lattice equation. The dispersionless Pfaff (dPfaff) lattice equation is then given by 2 −4F13 + 6F11 + 3F22 = 12e F00 (1.18) 3 2F03 + F01 + 6F01 F11 − 3F01 F02 − 6F12 = 0.
538
Y. Kodama, V. U. Pierce
Here we have used the limits, v ± = exp(log τn±1 − log τn ) → exp(±−1 F0 ) and v + v − = exp(log τn+1 − 2 log τn + log τn−1 ) → exp(F00 ) as → 0. In [32], Takasaki formulates the dispersionless DKP (Pfaff lattice) hierarchy by taking the dispersionless limit in the differential Fay identities of the Pfaff lattice hierarchy: ⎧ λ2 e−D(λ)F0 − µ2 e−D(µ)F0 ⎪ ⎪ , ⎨ e D(λ)D(µ)F (λ + µ − (D(λ) + D(µ))F1 − F01 ) = λ−µ (1.19) F e 00 (D(λ) − D(µ))F1 ⎪ ⎪ ⎩ e−D(λ)D(µ)F 1 − = 1 − 2 2 e(D(λ)+D(µ))F0 . λ−µ λ µ This equation is somewhat similar to the dToda hierarchy (1.11), and in fact, this similarity is one of our motivations to consider the universality between dPfaff and dToda hierarchies. 1.4. Orthogonal and symplectic ensembles of random matrices. Two special classes of solutions of the Pfaff lattice hierarchy are given by the partition functions of the orthogonal and symplectic ensembles of random matrices (see for examples [2,21,22]). We denote those partition functions by Z (β) with β = 1 for the orthogonal ensembles and β = 4 for the symplectic ensembles. With this notation, the partition function (or τ -function) for the Toda hierarchy is denoted by Z (2) , i.e. β = 2. 1.4.1. Orthogonal ensemble. In this case, the partition function is taken to be 2n (1) Z 2n (V0 (λ); t) = dλ1 · · · dλ2n |λi − λ j | exp − Vt (λk ) , R
R
∞
i< j
k=1
j
where Vt (λk ) = V0 (λk ) − j=1 λk t j and V0 (λ) is a polynomial of even degree. This integral represents integration over the eigenvalues of the random matrices in the orthogonal ensemble, and as in the case of the unitary ensemble, its Taylor expansion in t gives the moments of traces of powers of the matrix. This partition function is also related to the combinatorial problem of counting ribbon graphs on a compact surface. In this case we will consider Möbius graphs, which are defined to be one-complexes embedded in an unoriented surface of Euler characteristic χ such that the complement of the complex is a collection of sets homeomorphic to discs. We may think of a Möbius graph as a ribbon graph where the ribbons are allowed to be glued together with a twist, i.e. reversing the local orientation of the vertices. Our Möbius graphs are labeled in the same manner as the ribbon graphs. We introduce Möbius graphs here for completeness sake. However in this paper we consider only those Möbius graphs of Euler characteristic 2, such graphs are equivalent, up to a choice of local orientations at the vertices, to ribbon graphs embedded in the sphere (see Lemma 5.1), so we will not encumber our discussion here with examples of Möbius graphs. One may see [29] for a discussion of Möbius graphs with Euler characteristics less than 2. (1) For a quadratic potential V0 (λ), we have an asymptotic expansion for log(Z 2n ) of the same basic character as that of the unitary ensembles: Theorem 1.2 (Goulden-Harer-Jackson [17]). For the Gaussian orthogonal ensemble (GOE), we have (1) (1) N 2 (1) N 2 λ ; 2N T λ ;0 log Z 2N Z 2N = E χ (T)(2N )χ , 2 2 χ ≤2
Dispersionless Integrable Systems and Universality in Random Matrix Theory
539
where E χ(1) (T) =
j
Kχ (j)
Tj , j!
and Kχ (j) is the number of connected Möbius graphs with jk labeled vertices of degree k for k = 1, 2, . . . , on a compact surface of Euler characteristic χ . We then take our τ -function in the form with T0 = Nn = n, 1 (1) N 2 1 2 n (1) n 2 λ ; N T = n (2T0 )n + 2 Z 2n λ ; 2n Tˆ , τn (−1 T; ) = n Z 2n 2 n! 4 2 n! 2 j
where Tˆ is given by the Penner scaling, i.e. Tˆ = (Tˆ1 , Tˆ2 , . . .) with Tˆ j = (2T0 ) 2 −1 T j . This expression then has the asymptotic expansion (1) χ ˆ log τn (−1 T; ) = E χ (T)(2n) + C (1) (T0 ; N ) χ ≤2
=
χ ≤2
ˆ χ + C (1) (T0 ; N ), (2T0 )χ E χ(1) (T)N
(1)
where C (1) (T0 ; N ) = log[ 2n1n! Z 2n ( N4 λ2 ; 0)]. Note that the form of this asymptotic expansion agrees (at leading order) with the assumption (1.7). For future reference, for this initial condition, the leading order of the expansion gives a solution of the dPfaff hierarchy, 1 1 (1) N 2 (1) F (T0 , T) = lim λ ; NT log n Z 2n N →∞ N 2 2 n! 2 ˆ + C (1) (T0 ), = 4T02 E (1) (T) 2
0
(1)
where C0 (T0 ) is the leading order of C (1) (T0 ; N ). In Appendix B, we explicitly calculate C0(1) (T0 ), and it gives 3 1 C0(1) (T0 ) = T02 log(2T0 ) − T02 = C0(2) (2T0 ), 2 2 where C0(2) (T0 ) = C0 (T0 ) given in (1.12). The relation with C0(2) (T0 ) of the dToda hierarchy is a consequence of the universality, which will be discussed in Sect. 5. j With Tˆ j = (2T0 ) 2 −1 T j , we have the expansion formula for the solution of the dPfaff hierarchy corresponding to the Gaussian orthogonal ensemble, (1)
(1)
ˆ + C (T0 ) F (1) (T0 , T) = (2T0 )2 E 2 (T) 0 = (2T0 )2
K2 (j)(2T0 )e−v
j
=
j
K2 (j)(2T0 ) f
Tj (1) + C0 (T0 ) j!
Tj (1) + C0 (T0 ). j!
(1.20)
This formula then gives the combinatorial meaning of the solution of the dPfaff hierarchy (also notice the similarity with the formula (1.14) for the dToda hierarchy.
540
Y. Kodama, V. U. Pierce
1.4.2. Symplectic ensemble. In this case, the partition function is given by n (4) 4 Z n (V0 (λ); t) = dλ1 · · · dλn |λi − λ j | exp − 2Vt (λk ) , R
R
k=1
i< j
j where Vt (λk ) = V0 (λk ) − ∞ j=1 λk t j and V0 (λ) is a polynomial of even degree. This function represents integration over the eigenvalues of the random matrices in the symplectic ensemble, and the moments of traces of powers of the matrix are obtained in its Taylor expansion in t. Then the corresponding combinatorial result is give by the following theorem: Theorem 1.3 (Mulase-Waldron [29]). For the Gaussian symplectic ensemble (GSE), we have (1) (4) N 2 (4) N 2 log Z N λ ; 2N T λ ;0 ZN = E χ (T)(−2N )χ , 2 2 χ ≤2 (1)
where E χ (T) is defined as in Theorem 1.2. This theorem can also be found from Theorem 4.1 of [7], in particular the 1/(4N )n−m appearing in that theorem leads to the 2N T in the above formula of Theorem 1.3. Remark 1.4. There is a duality in the Gaussian orthogonal and symplectic ensembles. Theorems 1.2 and 1.3 give the N → −N duality between the partition functions of the Gaussian orthogonal and symplectic ensembles of random matrices. In general the even terms of the asymptotic expansions agree, while the odd terms have opposite signs. See the discussions in [7,29]. The τ -function in this case is defined by n 1 (4) N 2 1 2 n −1 τn T; = n Z n λ ; N T = n (2T0 )n − 2 Z n(4) λ2 ; 2n Tˆ , 2 n! 4 2 n! 2 where Tˆ is the Penner scaling defined in the same form as in the case of orthogoj nal ensemble, i.e. Tˆ = (Tˆ1 , Tˆ2 , . . .) with Tˆ j = (2T0 ) 2 −1 T j . Then the free energy is given by −1 (4) N 2 (4) N 2 log τn T; = log Z n λ ; NT λ ; 0 + C (4) (T0 ; N ) Zn 4 4 n n λ2 ; 2n Tˆ λ2 ; 2n Tˆ + C (4) (T0 ; N ) Z n(4) = log Z n(4) 2 2 ˆ χ + C (4) (T0 ; N ), (−2T0 )χ E χ(1) (T)N = χ ≤2
(4)
where C (4) (T0 ; N ) = log[ 2n1n! Z n ( N4 λ2 ; 0)]. In the limit N → ∞, the leading order of the free energy is given by 1 1 (4) N 2 F (4) (T0 , T) = lim log ; N T Z λ N →∞ N 2 2n n! n 4 (4) 2 (1) ˆ (1.21) = (2T0 ) E (T) + C (T0 ), 2
0
Dispersionless Integrable Systems and Universality in Random Matrix Theory
541
(4)
where C0 (T0 ) is the leading order of C (4) (T0 ; N ), and in Appendix B it is found to (1) be the same as C0 (T0 ). This implies that with (1.20) we have the duality of the free energies, i.e. F (1) (T0 , T) = F (4) (T0 , T). In Sect. 5, we further discuss the universality for those free energies corresponding to the GUE, GOE and GSE random matrix theories. 1.5. Outline of the paper. The paper is organized as follows: In Sect. 2, we first show that the dToda hierarchy is determined by variables F00 and F01 (Proposition 2.1). Then we discuss some properties of Fmn (Proposition 2.4). In particular, we show that F1n are related to the Catalan numbers, which gives a combinatorial meaning of the dToda hierarchy (Proposition 2.2). In Sect. 3, we first derive the explicit formulae for the two point functions Fnm for the dToda hierarchy under the conditions of F01 = 0 and F11 = 1 (Theorem 3.1), and then discuss the combinatorial description of Fmn as the enumeration of ribbon graphs on a sphere (Proposition 3.1). We also show that the combinatorial meaning of Fmn can be directly obtained from the dToda hierarchy (Theorem 3.2). In Sect. 4, we describe the dPfaff hierarchy. In particular, we show that under the assumption of the special dependency of the free energy F in the variables (T0 , T), the dPfaff hierarchy can be reduced to the dToda hierarchy (Proposition 4.1). This corresponds to the dispersionless limit of the Pfaff lattice hierarchy restricted to symplectic matrices [23]. In Sect. 5, we first show the universality among the free energies for the GUE, GOE and GSE in the leading order of large N expansions (Proposition 5.1). Then we show in general that the solution of the dToda hierarchy also satisfies the dPfaff hierarchy under the rescaling of the two point functions Fmn (Theorem 5.2). This means that the dPfaff hierarchy contains the dToda hierarchy, and shows the universality of the dToda hierarchy at the leading orders among the unitary, orthogonal and symplectic ensembles in random matrix theory. In Appendix A, we give a brief introduction of the dKP theory, and derive the dKP hierarchy (2.3) which plays an important role in this paper. We also (β) give in Appendix B explicit computations of the terms C0 (T0 ) for β = 1, 2 and 4. 2. The Dispersionless Toda Hierarchy In this section, we first show that the dToda hierarchy (1.11) defines a deformation of the algebraic curve given by (1.9), and as a result we show that the two point functions Fnm are determined by F01 and F00 . We also discuss a connection of the dToda hierarchy with the dKP hierarchy, and define the Faber polynomials as the generators of the higher members of the dKP hierarchy. These results will then be used in Sect. 3 to derive explicit formulae of Fmn solving the two-vertex problem. 2.1. The dToda curve and its deformation with the dToda hierarchy. Let us first define the function p(λ), p(λ) := λ exp (−D(λ)F0 ) . This corresponds to the equation p(λ) = exp ∂∂TS0 with the S-function given by (1.10). Then the dToda hierarchy is defined on the dToda curve (1.9):
542
Y. Kodama, V. U. Pierce
Lemma 2.1. The dToda hierarchy (1.11) gives a deformation of the algebraic curve of genus 0 given by e F00 . p
λ = p + F01 +
Proof. Eliminating e D(λ)D(µ)F from two equations in (1.11), we obtain e F00 p(λ) − p(µ) 1− . 1= λ−µ p(λ) p(µ) We then separate the variables, λ − p(λ) −
e F00 e F00 = µ − p(µ) − . p(λ) p(µ)
With the asymptotic condition, λ − p(λ) → F01 + O(λ−1 ), we obtain the curve. With this Lemma 2.1, we define in this paper the dispersionless Toda (dToda) hierarchy in the form, ⎧ e F00 ⎪ ⎪ = e−D(λ)D(µ)F, ⎨1 − p(λ) p(µ) (2.1) e F00 ⎪ ⎪ ⎩ λ = p(λ) + F01 + with p(λ) = λe−D(λ)F0 . p(λ) In particular one should note through the paper that the dToda curve (the second equation) on (λ, p)-plane plays a central role for the combinatorial description of the dToda hierarchy. We now note that the first equation in (2.1) implies: Lemma 2.2. Each Fmn with m, n ≥ 1 can be determined by the set {F0k : 0 ≤ k ≤ m + n}. Proof. The first equation with p(λ) = λ exp(−D(λ)F0 ) gives Fmn 1 e F00 (D(λ)+D(µ))F0 . e D(λ)D(µ)F = = − log 1 − m n λµ m≥1 n≥1 mn λ µ The right hand side depends only on F0k for k ≥ 0. Also, setting λ = µ and expanding for large λ, the first equation gives 1−
∞ 1 e F00 e F00 = 1 − h n (2Fˆ 0 ) p(λ)2 λ2 n=0 λn ∞ 1 2 ˆ h n (−Z), = e−D(λ) F = n=0 n
where Fˆ 0 = ( Fˆ01 , Fˆ02 , . . .) with Fˆ0k = n ≥ 2 defined by Zˆ n =
1 k F0k ,
Fkl k+l=n kl
and Zˆ = (0, Zˆ 2 , Zˆ 3 , . . .) with Zˆ n for
(k, l ≥ 1).
Dispersionless Integrable Systems and Universality in Random Matrix Theory
543
Comparing the coefficients of the power λ−n , we obtain ˆ + e F00 h n (2Fˆ 0 ) = 0 h n+2 (−Z)
n ≥ 0.
(2.2)
This set of equations is called the dipersionless Hirota equation for the dToda hierarchy, which gives a half set of the dispersionless Hirota equations (see for example [8], and the other half is given by (2.4) below). The first equation with n = 0 then gives the dispersionless Toda equation, F11 = e F00 . For n = 1, we obtain F12 = 2e F00 F01 = 2F01 F11 . Thus F11 and F12 are determined by only F00 and F01 . In fact, we have from the second equation (dToda curve) that Proposition 2.1. The two point functions Fnm of the dToda hierarchy are determined by F00 and F01 only. Proof. The curve gives 1 = e−D(λ)F0 +
F01 e F00 D(λ)F0 + 2 e . λ λ
Expanding this with large λ, we have h n (−Fˆ 0 ) + e F00 h n−2 (Fˆ 0 ) = 0 n ≥ 2. Then one can see that F0n is determined by the previous F0k for 0 ≤ k < n. For examples, we have for n = 2 and 3, 2 2 + 2e F00 = F01 + 2F11 , F02 = F01 3 1 3 3 F03 = F01 F02 − F01 + 3F01 e F00 = F01 + 6F01 F11 . 2 2
Then from Lemma 2.2, we conclude that all Fmn for m, n ≥ 1 are determined by F00 and F01 (or F01 and F11 = exp F00 ). From the dToda curve, p(λ) can be explicitly calculated as follows: Proposition 2.2. The inverse of the dToda curve for the case p(λ) → λ as λ → ∞ is give by p(λ) = λ − F01 −
∞
F1,n+1 , (n + 1)λn+1 n=0
where [ n2 ] n F1,n+1 k+1 F n−2k F11 Ck , = n+1 2k 01 k=0
2k 1 . with Ck = k+1 k
Note here that F11 = e F00 and Ck is the k th Catalan number with C0 = 1.
544
Y. Kodama, V. U. Pierce
Proof. From the curve, first we have p 2 − (λ − F01 ) p + F11 = 0. Solving this for p with the asymptotic condition p(λ) → λ as λ → ∞, we have 1 λ − F01 + (λ − F01 )2 − 4F11 p= 2 1 λ − F01 − (λ − F01 )2 − 4F11 . = λ − F01 − 2 Noting the last term can be expressed as √ ∞ F11 F11 1 − 1 − 4x = Ck x k λ − F01 2x λ − F01 k=0
with x =
F11 , (λ − F01 )2
thus we have ∞
k+1 ∞ F11 F1,n+1 = C . k n+1 (λ − F01 )2k+1 n=0 (n + 1)λ k=0
Then the coefficients Pn can be found by ∞ F1,n+1 k+1 = Ck F11 n+1 k=0
Using the expansion (λ + F01 )n =
λ=∞
dλ (λ + F01 )n . 2πi λ2k+1
n j n− j j=0 j λ F01 , we obtain the formula.
n
The appearance of the Catalan numbers indicates some connections of combinatorial problems to the dToda hierarchy. This is one of the main motivations of the present study. 2.2. The dToda hierarchy implies the dKP hierarchy. Here we consider the second equation of (1.11), that is, in terms of p(λ), p(λ) − p(µ) = e D(λ)D(µ)F . λ−µ
(2.3)
This equation is also known as the dKP hierarchy, and p(λ) in the dKP hierarchy is given by p(λ) = λ − D(λ)F1 (see Appendix A, and also for examples [8,9]). For the dToda hierarchy, p(λ) = λ exp(−D(λ)F0 ) can also be expressed in a similar form. This is obtained from an interesting connection with the Faber polynomials: Definition 2.1. Let λ be a Laurent series in p given by λ = p + u1 +
∞ u k+1 . k p k=1
Then the Faber polynomials n ( p) of degree n in p are defined by ! n ( p) = λ( p)n ≥0 n = 0, 1, 2, . . . ,
Dispersionless Integrable Systems and Universality in Random Matrix Theory
545
where [λ( p)n ]≥0 represents the polynomial part of λ( p)n in p. The polynomial n ( p(λ)) has the Laurent series in λ, n ( p(λ)) = λn −
∞ 1 Q , m nm m=1 λ
where Q nm are the functions of u = (u 1 , u 2 , . . .). Those Q nm are related to the Grunsky coefficients cnm = n1 Q nm . Then (2.3) gives the generating function of the 1 Faber polynomials with cnm = nm Fnm (see Remark 2.2 below for a further discussion). In the dKP theory, the Faber polynomials give the generators of the flows in the hierarchy (see Appendix A). Then with (2.3), we have Proposition 2.3. The Faber polynomials can be expressed as n ( p(λ)) = λn − D(λ)Fn
n ≥ 1,
and 0 ( p) = 1. Proof. We first note that log
p(λ) − p(µ) λ−µ
= D(λ)D(µ)F.
Taking the derivative with respect to λ, we have ∞ 1 ∂ p(λ) 1 1 = − D(µ)Fn . n+1 p(λ) − p(µ) ∂λ λ − µ n=1 λ
Then for n ≥ 1, we have
λn ∂ p(λ) dλ = µn − D(µ)Fn λ=∞ 2πi p(λ) − p(µ) ∂λ ! dp λ( p)n = λ( p(µ))n ≥0 . = p=∞ 2πi p − p(µ)
This completes a proof. We now derive the other set of the dispersionless Hirota equations: Taking the limit λ → µ of the equation in (2.3), we have ∂ p(λ) 2 = e D(λ) F . ∂λ With p(λ) = λ exp(−D(λ)F0 ), we have −λ
∂ 2 D(λ)F0 = e D(λ) F+D(λ)F0 − 1 ∂λ ∞ ∞ 1 1 − 1 = = exp Z h (Z), n n n n n=1 λ n=1 λ
546
Y. Kodama, V. U. Pierce
where Z = (Z 1 , Z 2 , . . .) with Z k defined by Z 1 = F01 ,
Zn =
Fkl F0n + . n k+l=n kl
This then gives the other half set of the dispersionless Hirota equation for the dToda hierarchy (the first half set is given by (2.2)), F0n = h n (Z) n ≥ 2,
(2.4)
which gives the dispersionless limit of the hierarchy (1.5). The first equation of (2.4) with n = 2 gives 2 F02 = F01 + 2F11 .
With Proposition 2.3 with the case n = 1, i.e. 1 ( p) = p + F01 , the function p(λ) can also be expressed by p(λ) = λ − F01 − D(λ)F1 . Then note that that the dHirota equation of the dKP hierarchy is derived from the same 2 equation ∂ p/∂λ = e D(λ) F with p(λ) = λ − F01 − D(λ)F, ˆ F1n = h n+1 (Z)
n ≥ 3,
which is the dispersionless limit of (1.3). Here recall that Zˆ = (0, Zˆ 2 , Zˆ 3 , . . .) with Zˆ k given by Zˆ k = Z k − F0k /k. The first equation with n = 3 gives the dKP equation, 2 − 4F13 + 6F11 + 3F22 = 0.
(2.5)
We also mention the positivity of Fmn : Proposition 2.4. The coefficients Fmn satisfy the following properties: (a) If F1m ≥ 0 for all m ≥ 1, then Fmn ≥ 0 for all m, n ≥ 1. (b) If F1,2k = 0 for all k ≥ 1 and all others F1,m ≥ 0, then Fmn = 0 for all m, n with m + n = odd. Proof. With p(λ) = λ − F01 − D(λ)F1 , we have p(λ) − p(µ) (D(λ) − D(µ))F =1− λ−µ λ−µ ∞ F1m 1 1 1 = 1− − λ − µ m=1 m λm µm m ∞ 1 1 F1m . = 1+ λµ m=1 m k=1 λm−k µk−1
e D(λ)D(µ)F =
Then taking the logarithm and the differentiation with respect to λ of this equation, we have m Fmn m−k+1 1 −D(λ)D(µ)F F1m . =e m−k+2 µk−1 n λm+1 µn m n m m k=1 λ This shows the property (a), i.e. ∀F1m ≥ 0 implies ∀Fmn ≥ 0.
Dispersionless Integrable Systems and Universality in Random Matrix Theory
547 2
The second property (b) can be obtained from ∂ p(λ)/∂λ = e D(λ) F . This equation gives ∞ 1 Fkl ∂ D(λ)F1 D(λ)2 F = = log 1 − ∂λ n n=2 λ k+l=n kl ∞ F1k , = log 1 + λk+1 k=1
which shows that if all F1,2k = 0, then n+m=odd
Fmn = 0. mn
Then from (a), we conclude Fmn = 0 if m + n =odd. Remark 2.2. The Faber polynomials n ( p) appear in a classical complex function theory: The Bieberbach conjecture states that if f (z) is a univalent function on |z| < 1 with the expansion f (z) = z + a2 z 2 + a3 z 3 + · · ·, then the coefficients an satisfy |an | ≤ n. Let g(z) be the function defined by 1 ,
g(z) = f
1 z
which is univalent for |z| > 1. Then the Faber polynomials n ( p) appear in an expansion, ∞ ( p) g(z) − p n =− . log n z nz n=1 With the expansion of n (g(z)) for large z, n (g(z)) = z n + n
∞ c nm . m z m=1
(2.6)
Grunsky defined the Grunsky coefficients cnm . Then Grunsky showed that his coefficients should satisfy a sequence of inequalities in order that g is univalent on |z| > 1. In particular, an explicit formula for the Grunsky coefficient was found in the paper by Schur in terms of the coefficients an of f [30]. In Sect. 3.1, we find an explicit formula of Fnm for the case where the coefficients are given by the Catalan numbers and discuss their combinatorial meaning (our derivation is independent from the calculation used in [30]). From (2.3) and (2.6), one can see that the Grunsky coefficients cnm are related to 1 our two point functions, that is, cnm = nm Fnm . The dToda hierarchy then defines an integrable deformation with an infinite number of parameters T = (T1 , T2 , . . .) for the univalent function f (z), so that the Grunsky coefficients are given by the second derivatives of a function F, the free energy, i.e. cnm =
1 ∂2 F . nm ∂ Tn ∂ Tm
(2.7)
Thus a most important consequence of the dToda hierarchy is to show the existence of a unique function F, the free energy.
548
Y. Kodama, V. U. Pierce
3. Combintorial Results for the dToda Hierarchy The appearance of the Catalan numbers in the dToda hierarchy (see Proposition 2.2) directly demonstrates a combinatorial meaning of the functions Fnm . In particular, if F01 (1; 0) = 0 and F11 (1; 0) = 1, then we have, from Proposition 2.2, p(λ) = λ −
∞ C k , 2k+1 k=0 λ
which gives F1,2k+1 = (2k + 1)Ck .
(3.1)
In general, the two point functions Fnm have the following combinatorial interpretation: One sees from the connection to random matrices discussed in Sect. 1.2 that for the ˆ particular choice of F(T0 , T) = T02 e0 (T)+C 0 (T0 ) the two point function with n, m ≥ 1, ∂ 2 F ∂2 2 ˆ Fnm (T0 = 1; T = 0) = = T e ( T) , 0 0 ∂ Tn ∂ Tm T0 =1;T=0 ∂ Tn ∂ Tm ˆ ˆ T0 =1;T=0 is the number of genus 0 connected ribbon graphs with a vertex of degree n and a vertex of degree m (see Theorem 1.1). Here the degree of the vertex represents the number of edges (ribbons) attached to the vertex. For example, the F1,2k+1 gives the number of connected graphs with a vertex with a single edge and a vertex with 2k + 1 edges on a sphere (recall that the number of ways to connect 2k edges of a single vertex is given by the Catalan number Ck ). In this section we compute a remarkable closed formula for the Fn,m (1; 0) in the case with F01 (1; 0) = 0 and F11 (1; 0) = 1. Also recall that differentiation with respect to T0 has the meaning of counting ribbon graphs with a marked face. 3.1. Explicit formulae for Fnm (1; 0). Let us first recall Proposition 2.4 (b): if F1,2k = 0, ∀k > 0 and F1,m ≥ 0, ∀m > 0, we have Fnm = 0
if n + m = odd.
Also from Proposition 2.2, if F10 = 0 and F11 = 1, we have F1,2k = 0 and F1,2k+1 = (2k + 1)Ck . So we calculate only Fnm with m + n =even and positive. Then we state the following theorem which gives an explicit formula of Fnm for those cases of m and n. The method to find the formula is based on the dispersionless Toda hierarchy. Theorem 3.1. The two point functions Fnm = ∂ Tn∂∂ Tm F for the dToda hierarchy (2.1) with F01 = 0 and F11 = 1 (i.e. F00 = 0) are given by ⎧ F0,2k = (k + 1)Ck , k = 1, 2, . . . , ⎪ ⎪ ⎪ ⎪ ( j+1)(k+1) ⎨F j, k = 0, 1, 2, . . . , 2 j+1,2k+1 = (2 j + 1)(2k + 1) j+k+1 C j C k , ⎪ ⎪ F2 j,2k = jk ( j+1)(k+1) C j Ck , j, k = 1, 2, . . . , ⎪ j+k ⎪ ⎩ Fnm = 0, otherwise, 2
where C j is the j th Catalan number.
Dispersionless Integrable Systems and Universality in Random Matrix Theory
549
Proof. We first derive the formula F0,2k for k = 0: We start with the definition of p, p(λ) = λe−D(λ)F0 , which implies ∞ F 0,2k 1 = log D(λ)F0 = 2k k=1 2k λ
λ . p(λ)
Therefore we may compute F0,2k by the contour integral 1 F0,2k = 2k
λ=∞
dλ 2k−1 λ log 2πi
λ . p(λ)
Using the dToda curve, 1 1 λ( p) = p + = p 1 + 2 , p p we write the integral with respect to p, and then integrate by parts, 1 1 d dp 1 2k log 1 + 2 F0,2k = λ( p) 2k dp p p=∞ 2πi 2k 2k−1 1 dp 1 2k−3 1 2k − 1 1+ 2 p . = = p k k−1 p=∞ 2πi k In the last step, we have used the expansion, (1 + we obtain
1 2k−1 ) p2
=
2k−1 2k−1 j=0
j
p −2 j . Thus
F0,2k = (k + 1)Ck . We now derive the formula Fnm for nm = 0: We use the Faber polynomials in Proposition 2.3, i.e. n ( p) = [λ( p)n ]≥0 = λn − D(λ)Fn . This implies Fmn =− m
λ=∞
dλ m−1 λ n ( p(λ)). 2πi
Now changing variables from λ to p, we have d dp λ( p)m n ( p) Fmn = − dp p=∞ 2πi 1 n d 1 m dp p+ p+ =− . p dp p p=∞ 2πi ≤0
550
Y. Kodama, V. U. Pierce
Here we have noticed that n ( p) can be replaced by λ( p)n with d([λ( p)m ]<0 ]) for d(λ( p)m ). The notation [ f ( p)]<0 implies the part of the Laurent series f ( p) with the inverse powers of p, i.e. [ f ( p)]<0 = f ( p) − [ f ( p)]≥0 . Thus we have Fmn
!
1 n 2k−m−1 dp m p+ = (m − 2k) p . p k p=∞ 2πi k=0 m
2
Then from the expansion ( p + 1p )n = p=∞
n n−2l , l=0 l p
n
(3.2)
we have the residue integral
dp n−m+2(k−l)−1 p , 2πi
which implies n − m + 2(k − l) = 0 in the non-zero case. Therefore, Fmn = 0 only if n − m is even (recall Proposition 2.4). So we have two cases: (a) m = 2 j + 1, n = 2k + 1, and (b) m = 2 j, n = 2k. With the residue calculation for the case (a), we obtain F2 j+1,2k+1 =
min( j,k)
(2l + 1)
l=0
2j + 1 j −l
2k + 1 . k −l
We consider the case (a) (the case (b) can be shown in the similar way): Let us assume k ≥ j (because of the symmetry in j, k, this case is enough to show). Also confirm that the case of j = 0 gives (3.1). Then from (3.2), we have F2 j+1,2k+1 =
j l=0
2j + 1 2k + 1 . l k − j +l
(2 j − 2l + 1)
We note the following identity which is a key equation in the proof, 2j + 1 2j 2j (2 j − 2l + 1) = (2 j + 1) − . l l l −1
(3.3)
Also we write
2k + 1 k(k − 1) · · · (k − j + l + 1) 1 = Ck . 2k + 1 k − j + l (k + 2)(k + 3) · · · (k + j − l + 1)
Then we have the following formula with the common denominator (k + 2)(k + 3) · · · (k + j + 1), 2j 2j k(k − 1) · · · (k − j + l + 1) Ck − (k + 2)(k + 3) · · · (k + j − l + 1) l l −1 Ck G( j, k), = (k + 2)(k + 3) · · · (k + j + 1)
j F2 j+1,2k+1 = (2 j + 1)(2k + 1) l=0
where G( j, k) is the polynomial of degree j in k, G( j, k) :=
j−l−1 l−1 2j 2j − (k − α) × (k + j + 1 − α). l l −1 l=0 j
α=0
α=0
(3.4)
Dispersionless Integrable Systems and Universality in Random Matrix Theory
551
" Here the products −1 α=0 (· · · ) are understood to give 1 when l = 0 and l = j. We then claim that the polynomial G( j, k) can be expressed by the simple form,
2j G( j, k) = (k + 1)(k + 2) · · · (k + j) j = ( j + 1)C j (k + 1)(k + 2) · · · (k + j).
(3.5)
j 2 j . This then gives the desired Here note that the coefficient 2jj is just l=0 2l j − l−1 formula, that is, this equation for G( j, k) leads to the formula F2 j+1,2k+1 . To prove the claim, we write (3.4) in the sum of the terms with the coefficients 2l j (like a telescoping sum): G( j, k) = −2
j−1
( j − l)
l=0
j−l−2 l−1 2j (k − α) × (k + j + 1 − α) l α=0
α=0
2j + (k + 2)(k + 3) . . . (k + j + 1), j
(3.6)
where the last term in (3.6) is the remainder from the telescoping. We then observe that this reduces the degree of the polynomials of both sides of (3.5), that is (3.5) becomes the equation
2
j−1
( j −l)
l=0
2j l
j−l−2
(k −α) ×
α=0
l−1
(k + j +1−α) = j
α=0
2j (k + 2)(k + 3) · · · (k + j), j (3.7)
where the right-hand side of this equation is the difference between the remainder in (3.6) and the right hand side of (3.5). We then note that the coefficients in the sum can be written in the form similar to (3.3), 2j 2j − 1 2j − 1 ( j − l) = j − . l l l −1 Here we have used the relations (the second one is the Pascal rule), 2j 2j − 1 l = 2j l l −1
and
2j 2j − 1 2j − 1 = + . l l l −1
Then we can write (3.7) in a similar form to (3.5), j−l−2 l−1 2j − 1 2j − 1 − (k − α) × (k + j + 1 − α) l l −1 l=0 α=0 α=0 2j − 1 = (k + 2)(k + 3) · · · (k + j). j −1 j−1
552
Y. Kodama, V. U. Pierce
Again we write the sum in the terms with the coefficients to the right-hand side to arrive at the equation j−2 l=0
(2 j − 2l − 1)
2 j−1 l
and move the remainder
j−l−3 l−1 2j − 1 (k − α) × (k + j + 1 − α) l α=0
2j − 1 = (k + 3)(k + 4) · · · (k + j). j −1
α=0
Both sides are now polynomials of degree j − 2 in k. In the similar way with the identity, 2j − 1 2j − 2 2j − 2 (2 j − 2l − 1) = (2 j − 1) − , l l l −1 the above equation takes the similar form as before, i.e. j−l−3 l−1 2j − 2 2j − 2 − (k − α) × (k + j + 1 − α) l l −1 l=0 α=0 α=0 2j − 2 = (k + 3)(k + 4) · · · (k + j). j −2 j−2
Then we can further reduce the degree of the polynomials in the same way. Continuing this process, we complete the proof. 3.2. Graph enumeration on the sphere. We recall that Fnm has a combinatorial meaning of enumeration of connected (ribbon) graphs on a sphere. The presentation of such combinatoric problems in terms of the function F(T0 ; T) arising from the Gaussian unitary ensemble example is not a new idea. However viewing the genus zero solutions as arising purely from the dToda hierarchy is a new approach to solving the problem, although a similar idea was used in the case of a single even time for the computations of [16]. Surprisingly we have found a closed form for the numbers Fnm with the conditions F01 = 0 and F11 = 1 for general n and m. The solution of the two-vertex problem given by Theorem 3.1 is new, previously the problem has been solved only in the so-called “dipole” case when the vertices are of the same degree (i.e. Fnn ) [19,25]. These authors consider the general genus problem of two vertices of the same degree on a genus g surface. Similar formulae were also found for other low genus problems in [5,16], but again the vertices must all have the same degree. Let us start by giving a combinatoric argument for the F0,2k formula of Theorem 3.1, that is, F0,2k gives the number of genus 0 ribbon graphs with a vertex of degree 2k and a marked face: As shown above counting graphs with marked faces in the case of fixed genus is equivalent to just counting the number of graphs, the number of faces being given by the vertex structure. So the problem is then to count the number of genus 0 ribbon graphs with a single vertex of degree 2k. It is easy to see that this is equivalent to counting the number of ways to make pair-wise connections with 2k points on a circle without crossing and the number of sections divided by the chords giving the connections of the vertices. The first number is well-known and is given by the k th Catalan number Ck . Then the sphere is divided up to k + 1 sections. So we have (k + 1)Ck which is F0,2k (see Theorem 3.1).
Dispersionless Integrable Systems and Universality in Random Matrix Theory
553
Now we consider the two vertices problem, and show that Fnm satisfy the following recursion relations as a direct consequence of the enumeration of connected ribbon graphs appearing in the problem: Proposition 3.1. The two point functions Fmn with nm = 0 for the two-vertex problem satisfy the recursion relations, ⎧ j ⎪ ⎪ ⎪ C j−l F2l−1,2k+1 + (2k + 1)C j+k , j, k = 0, 1, 2, . . . , ⎨ F2 j+1,2k+1 = 2 l=1
j ⎪ ⎪ ⎪ C j−l F2l−2,2k + 2kC j+k−1 , ⎩ F2 j,2k = 2
j, k = 1, 2, . . . .
l=2
Proof. We consider the case of F2 j+1,2k+1 (the case of F2 j,2k can be shown in the same way). We will use Va and Vb to refer to the vertex of degree 2 j +1 and 2k +1 respectively. We start with the graph, in which a specific edge, say 1, of Va is glued to an edge of the vertex Vb . The number of ways to accomplish this gluing is 2k + 1 (the number of edges at Vb ). Having glued the edge 1 of Va with an edge of Vb , we have the situation of one-vertex problem of degree 2 j + 2k. So the number of connected graphs for each situation is given by the Catalan number C j+k . Thus the total number of the edge 1 gluing is (2k + 1)C j+k , which is the last term in the equation. Now we calculate the other cases where the edge 1 is glued to another edge of Va . Let the edge 1 be glued with the edge 2s + 2 (s = 0, 1, . . . , j − 1). This divides the edges of Va into two sets, one of which is cut off the rest of the graph (see Fig. 2). This group of edges is marked by {2, 3, . . . , 2s + a}, and these 2s edges of Va must be glued together within this group. This is the one-vertex problem of degree 2s, and the number of gluings is Cs . Now we are left with the two vertices problem with the vertex of degree 2 j −2s −1 for Va and the vertex of degree 2k + 1. The number of connected graphs of this problem is, of course, F2 j−2s−1,2k+1 . This situation with a one-vertex problem of degree 2s and a two-vertex problem of degrees 2 j − 2s − 1 and 2k + 1 appears also in the case where the edge 1 is glued with the edge 2 j − 2s + 1. Here the group of 2s edges in the one-vertex problem is {2 j − 2s + 2, 2 j − 2s + 3, . . . , 2 j + 1}. So we have the total number of connected graphs is F2 j+1,2k+1 = 2
j−1
Cs F2 j−2s−1,2k+1 + (2k + 1)C j+k .
s=0
Then changing the variables in the sum with s = j − l, we have the equation. We also note that these relations in Proposition 3.1 define a recursion relation of the Faber polynomials on the dToda curve: We consider only the first equation (the second 1 equation can be shown in the same manner). Multiplying (2k+1)λ 2k+1 to the first equation in Proposition 3.1 and summing up for k = 0, 1, 2 . . ., we obtain, after rearranging the terms, 2 j+1 − 2
j l=1
C j−l 2l−1 = λ2 j 1 −
j
C j−l λ2l−1 .
(3.8)
l=1
∞ 1 1 (Recall that n = λn − ∞ m=1 mλm Fnm and 1 = p = λ − k=0 λ2k+1 C k .) This can be considered as a recursion relation of the Faber polynomials defined on the dToda curve λ = p + 1p , that is, with (3.8), one can define 2 j+1 from 1 = p. Conversely, starting
554
Y. Kodama, V. U. Pierce
2s 2
Va
2k+1
1 2
2s+1
1
2s+2
2j+1
2s+3
Vb
2j-2s-1 Fig. 2. Two-vertex problem with degrees 2 j + 1 and 2k + 1. The gluing of the edge 1 of Va to the edge 2s + 2 of the same vertex divides into the two sets of edges; the set of 2s edges marked by {2, 3, . . . , 2s − 1} is of the one-vertex problem, since those edges can not cross the glued ribbon between the edges 1 and 2s + 2 on a sphere. The gluing of the 2 j − 2s − 1 edges of the vertex Va to 2k + 1 edges of Vb gives the two-vertex problem
from this recursion, one can derive the first equation of proposition 3.1. This implies that we have an alternative proof of this Proposition using only the Faber polynomials, that is, the explicit form of two point functions Fmn obtained in Theorem 3.1 satisfy the recursion relations of Proposition 3.1. Let us now show that the relation (3.8) can be directly derived from the dToda curve. We first note that the left-hand side is a polynomial in p. It is indeed easy to see that the projection of the right hand side on the polynomial part gives the polynomial in the right hand side. This means that the non-polynomial part in the right hand side must vanish, when (λ, p) is on the dToda curve, i.e. λ = p + 1p . So we want to prove # $ j 2j 2l−1 λ( p) p = C j−l λ( p) , (3.9) <0
l=1
<0
where [ f ( p)]<0 is the non-polynomial part of the Laurent series f ( p). With the dToda curve, this leads to j−1 j−1 2 j −2( j−k)+1 j−l−1 2( j − l) − 1 −2( j−k)+2l+1 p p = Cl . k k=0 k l=0 k=0 Then from the coefficients of the power p −2( j−k)+1 and using the Pascal relation, 2kj = 2 j−1 2 j−1 + k−1 , we have the following lemma which proves Eq. (3.9): k Lemma 3.1. For j > k, we have k 2j − 1 2( j − l) − 1 = . Cl k−1 k −l l=1 Proof. Let us first note that we may write the second binomial coefficient as 2n − 1 2( j − l) − 1 = , ( j − l) − ( j − k) n−i where we denoted n = j − l and i = j − k. We also note 2n − 1 (n + i) i 2n = . n−i 2i n n − i
(3.10)
Dispersionless Integrable Systems and Universality in Random Matrix Theory
555
Now let z(t) be a generating function of the Catalan numbers, z=
∞
Cjt j,
j=0
which satisfies 1 = z − t z 2 with z = 1 at t = 0. Then one can show that ∞ i 2n tn. (z − 1) = n=i n n − i i
This result can be found by computing the contour integral, using t = 2−z dz, z3
t=0
dt 1 (z − 1)i = 2πi t n+1
z=1
z−1 z2
and dt =
z 2n−1 dz (2 − z) 2πi (z − 1)n−i+1
du (u + 1)2n−1 (1 − u) u n−i+1 u=0 2πi 2n 2n − 1 2n − 1 i . = − = n n−i n−i n−i −1
=
Then from (3.10), we see that
2n − 1 = n−i t=0 = t=0 = t=0
dt 1 1 d i t (z − 1)i n+i 2πi t 2i dt 1 dt (z − 1)i−1 z − 1 + t z n+1 2πi 2t z 1 dt i (z − 1) 1 + 2πi 2t n+1 2−z
t=0
dt 1 (z − 1)i . 2πi t n+1 2 − z
= This implies that
∞ 2n − 1 (z − 1)i tn. = n−i 2−z n=i Also with C0 = 1, we have z − 1 = (z − 1)
∞
l=1 Cl t
l,
so that we see by setting i = j − k,
∞ ∞ 2n − 1 (z − 1)i t n+l Cl = 2−z n − ( j − k) l=1 n= j−k m− ∞ j+k 2(m − l) − 1 tm. = Cl (m − l) − ( j − k) m= j−k+1 l=1
556
Y. Kodama, V. U. Pierce
Then the sum in the lemma is given by the coefficient of t j , i.e. k 2( j − l) − 1 dt 1 (z − 1) j−k = Cl (z − 1) j+1 k −l 2−z t=0 2πi t l=1 2 j−1 dz z du (u + 1)2 j−1 = = k uk z=1 2πi (z − 1) u=0 2πi 2j − 1 = . k−1 This lemma proves the relation (3.9) which leads to the recursion relation of the Faber polynomials (3.8) defined on the dToda curve. The point here is that the combinatorial meaning of the two point functions Fmn given in Proposition 3.1 can be directly obtained from the structure of the Faber polynomials defined on the dToda curve (i.e. independent from the random matrix description of the two point functions). We summarize this as the following theorem: 1 Theorem 3.2. The Grunsky coefficients cnm = nm Fnm of the Faber polynomials defined 1 on the dToda curve λ = p + p satisfy the recursion relations given in Proposition 3.1, that is, they have the combinatorial meaning of counting the connected ribbon graphs for the two-vertex problem on a sphere.
4. The Dispersionless Pfaff Hierarchy In this section, we consider some basic properties of the dPfaff hierarchy. In particular, we show that the dPfaff hierarchy can be reduced to the dToda hierarchy by choosing a particular section of the solution space which depends only on the even times of the hierarchy (i.e. the symplectic Pfaff lattice hierarchy studied in [23]). We also discuss a rescaling of the variables based on the structure of the τ -functions of the Toda and Pfaff lattice hierarchies. That structure is that one is given by determinants and the other by Pfaffians. 4.1. The dispersionless limits of the symplectic Pfaff lattice hierarchy. Let us first note that the dPfaff hierarchy (1.19) reduces to the dToda hierarchy as a special choice of the variables. F Proposition 4.1. Let α = λ2 and β = µ2 . Suppose F2m+1 = ∂t∂2m+1 = 0 for all m ≥ 0, and let sn = 2t2n for n ≥ 1 and s0 = t0 . Then the dPfaff hierarchy (1.19) is reduced to the dToda hierarchy (1.11), ⎧ ˜ ˜ ˜ ˜ ⎪ αe− D(α) F0 − βe− D(β) F0 ⎪ ˜ ˜ D(β) F˜ ⎪ e D(α) = ⎨ α−β F˜00 ⎪ e ⎪ − D(α) ˜ ˜ ˜ ˜ ˜ D(β) F˜ ⎪ e( D(α)+ D(β)) F0 , =1− ⎩e αβ 1 ∂ ˜ ˜ 0 , s1 , . . .) = F(t0 , t1 , . . .) and D(α) = ∞ where F(s k=1 kα k ∂sk .
Dispersionless Integrable Systems and Universality in Random Matrix Theory
557
Proof. The proof is based on the following observation: With α = λ2 and F2m+1 = 0 (∀m ≥ 0), ∞
1 ∂F 2m ∂t2m m=1 2mλ ∞ 1 ∂ F˜ ˜ ˜ = = D(α) F. m m=1 mα ∂sm
D(λ)F =
It is then immediate to see that (1.19) with F1 = 0 gives (1.11). This can be considered as the dispersionless limit of the Pfaff lattice restricted to symplectic matrices (a continuous version of the SR algorithm [23]). The dPfaff equation is then given by F22 = 4e F00 which is
˜ F˜11 = e F00 .
The second equation in the dPfaff (dDKP) equation (1.18) becomes trivial. 4.2. The dPfaff curve and the rescaled dPfaff hierarchy. We consider the following scalings: 1 F. (4.1) 2 The scaling in the time variables may be considered as the size of the matrix models, i.e. the matrices we consider for the orthogonal ensembles are twice the size of matrices for the unitary ensembles. The scaling in the free energy function is due to the structure of the τ -functions, i.e. the determinant for the Toda lattice and the Pfaffian for the Pfaff lattices (a Pfaffian is a square root of a determinant). We will show in Sect. 5 that those scalings appear in the particular choices of our τ -functions corresponding to the Gaussian ensembles discussed in the Introduction. With the scalings (4.1), Fmn is scaled by 2Fmn for all m, n ≥ 0, and we redefine the dPfaff hierarchy (1.19) in the following form similar to the dToda hierarchy, ⎧ e2F00 ⎪ −2D(λ)D(µ)F q(λ) − q(µ) ⎪ = e ⎨ 1− p(λ)2 p(µ)2 λ−µ (4.2) 2F00 e ⎪ ⎪ ⎩ q(λ)2 = p(λ)2 + G 2 + , p(λ)2 T0 → 2T0 , T → 2T, and F →
with p(λ) = λe−D(λ)F0 , q(λ) = λ − F01 − 2D(λ)F1 , and 2 G 2 = F02 − F01 − 4F11 .
We call the algebraic equation of ( p, q) in (4.2) the dPfaff curve. The first equation is the scaled version of the second one in (1.19), and the first one in (1.19) is derived from this definition (4.2): Lemma 4.1. From (4.2), we have p(λ)2 − p(µ)2 = e2D(λ)D(µ)F (q(λ) + q(µ)), λ−µ which is the first equation in (1.19) with the scaling Fmn → 2Fmn .
558
Y. Kodama, V. U. Pierce
Proof. From the dPfaff curve, we have 2 2 2 2 1− q(λ) − q(µ) = p(λ) − p(µ)
e2F00 p(λ)2 p(µ)2 q(λ) − q(µ) , = p(λ)2 − p(µ)2 e−2D(λ)D(µ)F λ−µ
which leads to the equation in the lemma. This shows the equivalence between (1.19) and (4.2). Let us now discuss the functional relations among Fmn , i.e. the dispersionless Hirota equations for the dPfaff hierarchy: We first note Lemma 4.2. Each Fmn for m, n ≥ 1 can be determined by the set {F0k : 0 ≤ k ≤ m + n} ∪ {F11 }. Proof. From the dDKP curve, q 2 = p 2 + G 2 + e2F00 / p 2 , we can see that each F1,m for m ≥ 2 can be determined by {F0k : 0 ≤ k ≤ m + 1} ∪ {F11 }. Then from the first equation, we have (D(λ) − D(µ))F1 e2F00 + log 1 − 2 . 2D(λ)D(µ)F = − log 1 − p(λ)2 p(µ)2 λ−µ This implies the assertion. In the similar way as in the dToda hierarchy, the dispersionless Hirota equations can be derived as follows: From the dPfaff curve, we have for n ≥ 1, h n+2 (−2Fˆ 0 ) + e2F00 h n−2 (2Fˆ 0 ) F1 j F1k 4 4 F1,n+1 + F01 F1n + 4 , =− n+1 n k j+k=n j
(4.3)
F where Fˆ 0 = ( Fˆ01 , Fˆ02 , . . .) with Fˆ0 j = 0j j . This gives the first half of the dispersionless Hirota equations for the dPfaff hierarchy, which shows that F1n are determined by {F0m : 0 ≤ m ≤ n + 1}. The first nontrivial equation with n = 1 is given by
1 2 3 F03 − F01 F02 + 2F01 F11 + F01 . 3 3 This is the second equation of the dPfaff lattice equation (1.18) with the scaling Fnm → 2Fnm . The second half of the dispersionless Hirota equations is given by taking λ = µ in the first equation, i.e. F12 =
1−
e2F00 2 dq . = e−2D(λ) F p(λ)4 dλ
Expanding for large λ, this gives ∞ 1 e2F00 h n (4Fˆ 0 ) λ4 n=0 λn ∞ ∞ 1 1 ˆ = . h (−2Z)) 1+2 F n n n+1 1n n=0 λ n=1 λ
1−
Dispersionless Integrable Systems and Universality in Random Matrix Theory
559
Then we have for n ≥ 4, ˆ + h n (−2Z)
ˆ = −e2F00 h n−4 (4Fˆ 0 ), 2F1k h j (−2Z)
(4.4)
k+ j=n−1
where Zˆ = (0, Zˆ 2 , Zˆ 3 , . . .) with Zˆ n =
F jk j+k=n jk . The first equation with n
= 4 gives
1 2 2 + F22 = e2F00 . − F13 + 2F11 3 2 This is the first equation of the dispersionless limit of the Pfaff lattice equation (1.18) with the scaling Fmn → 2Fmn . Remark 4.1. The dPfaff curve in (4.2) has the structure on (P, Q) ∈ C2 with P = p 2 and Q = q 2 , Q = P + G2 +
e2F00 , P
which is the same as the dToda curve representing a sphere. The curve has also only two parameters of the deformation. However, in terms of the coordinates (λ, p) ∈ C2 , the curve has an infinite number of parameters F0k , unlike the dToda case. In ( p, q)coordinates, the dPfaff curve has an asymptotic property q = p+
∞
G 2m , 2m−1 m=1 2mp
where G 2m are determined by G 2 and F00 . Then one can consider the Faber polynomials associated with the function q( p). However it seems that there is no role for those polynomials in the dPfaff theory.
5. Universality of the dToda Hierarchy In this section we first show that the combinatoric descriptions for the functions e0 (T) and E 2(1) (T) in the leading order terms of the free energies (see Theorems 1.1 and 1.2) are intimately related. The fundamental fact underlying this is that the Euler characteristic two Möbius graphs are equivalent to genus 0 ribbon graphs together with some data specifying local orientations at the vertices. The relationship between the leading (β) orders of the asymptotic expansions of the free energies log(Z N ) in the GUE (β = 2), GOE (β = 1) and GSE (β = 4) random matrix settings implies that for the special initial condition corresponding to these examples the solutions of the dToda and dPfaff hierarchies agree (after a rescaling with proper choice of initial conditions). We will then show that this is merely an example of a more general fact: the solution of the dToda hierarchy is in fact always a solution of the rescaled dPfaff hierarchy, or in other words the dToda hierarchy implies the dPfaff hierarchy. However the two hierarchies are not equivalent, the family of solutions to the dPfaff hierarchy being much larger (see [22]).
560
Y. Kodama, V. U. Pierce
5.1. Universality in random matrix theory. In broad terms universality in random matrix theory states that the limiting behavior of eigenvalues of the unitary ensembles of random matrices is independent of the external potential V0 (λ) in certain scaling regimes. There is also a universality for the limiting behavior of the eigenvalues of the random matrices across the orthogonal, unitary, and symplectic ensembles in some regions and choices of scalings, see [12,13] and [31]. In the Gaussian case, when V0 (λ) is quadratic, we will illustrate universality explicitly by giving a simple relationship between the leading order expressions of the free energy F (β) (T0 , T) for the GUE and GOE random matrix models (recall that F (1) = F (4) , that is, the GOE and GSE cases give the same leading order). A related result is that there is a universality between integrals over the unitary and orthogonal groups (not the same expressions as our Z n(2) and Z n(1) ) after appropriate scalings [10]. Let us first note the following lemma relating the leading order combinatoric interpretation given by Theorems 1.1 and 1.2. Lemma 5.1. Möbius graphs of Euler characteristic 2 are in 1-1 correspondence with pairs made up of a ribbon graph of genus 0 together with one of the 2v−1 possible configurations for the local orientations of the vertices, where v is the number of vertices of the graph Proof. It is easy to see that a ribbon graph of genus 0 together with a choice of local orientations of the v vertices gives a Möbius graph of Euler characteristic 2. For a Möbius graph to have Euler characteristic 2 there must be a choice of the local orientations of the vertices so that the ribbons are untwisted (as the graph lays flat on a sphere), thus giving the pair of a ribbon graph of genus 0 and a choice of local orientations. From this lemma, we see that (for leading order only) there is the following relation between the coefficients κ0 (j) and K2 (j) of the expansions of the free energies (1.14) and (1.20), where v =
K2 (j) = 2v−1 κ0 (j),
i ji
is the number of vertices. We now have:
Proposition 5.1. The free energies F (β) (T0 , T) have the following relations: F (1) (T0 , T) = F (4) (T0 , T) =
1 (2) F (2T0 , 2T). 2
Proof. The duality between F (1) and F (4) has already been shown in Sect. 1.4. The relation between F (1) and F (2) is a direct consequence of the relation between κ0 (j) and K2 (j) above. Namely from (1.20), we have
Tj (1) + C0 (T0 ) j! j Tj (1) + C0 (T0 ) = κ0 (j)2v−1 (2T0 ) f j! j (2T)j 1 (2) 1 + C0 (2T0 ) = κ0 (j)(2T0 ) f 2 j j! 2 1 (2) = F (2T0 , 2T). 2
F (1) (T0 , T) =
K2 (j)(2T0 ) f
Dispersionless Integrable Systems and Universality in Random Matrix Theory
561
This proposition implies that the free energy of the dToda hierarchy gives an universality over those Gaussian ensembles in the leading order with the scalings given in (4.1). Also notice that the scalings in (4.1) explicitly appear in this result. Remark 5.1. It is interesting to note that the scalings in Proposition 5.1, which give universality between the dToda and dPfaff hierarchies, agree with the scalings found in [10] giving the universality at leading order between the matrix integrals over the orthogonal and unitary groups (see Theorems 7.1 and 9.1 of [10]). In our case the partition functions are derived from integrals over the spaces of symmetric and hermitian matrices.
5.2. The dToda hierarchy implies the dPfaff hierarchy. We now show that a solution of the dToda hierarchy (2.1) gives a solution of the dPfaff hierarchy (4.2). This extends the previous results for special choices of the free energies associated with the Gaussian ensembles, and it implies that the universality holds for the general choice of the potential V0 (λ) or the measure in the random matrix theory. Theorem 5.2. The free energy F of the dToda hierarchy (2.1) also satisfies the dPfaff hierarchy (4.2). Proof. We first calculate the square of the first equation of the dToda hierarchy, e F00 e2F00 1 = e2D(λ)D(µ)F 1 − 2 + p(λ) p(µ) p(λ)2 p(µ)2 e F00 e2F00 − 1− = e2D(λ)D(µ)F 2 1 − p(λ) p(µ) p(λ)2 p(µ)2 e2F00 D(λ)D(µ)F 2D(λ)D(µ)F 1− . −e = 2e p(λ)2 p(µ)2 Then from Lemma 2.3 of the kernel formula of p(λ), we have e2F00 p(λ) − p(µ) e2D(λ)D(µ)F 1 − =2 − 1. p(λ)2 p(µ)2 λ−µ Now recall that p(λ) for the dToda hierarchy is expressed by p(λ) = λ − F01 − D(λ)F1 . Then noting that 2 p(λ) = q(λ) + λ − F01 , we obtain the first equation of the dPfaff hierarchy (4.2). Now we show that the dPfaff curve q 2 = p 2 + G 2 + e2F00 p −2 is reduced to the dToda curve. If Fmn are the solutions of the dToda hierarchy, then the G in the curve is given by 2 + F02 = −2F11 = −2e F00 . G 2 = −4F11 − F01 2 + 2F and F = e F00 (see Sect. 2.1). This implies Here we have used F02 = F01 11 11
q 2 = p 2 − 2e F00 +
e2F00 = p2
p−
e F00 p
2 .
With q = λ − F01 − 2D(λ)F1 = 2 p − λ + F01 , this equation leads to the dToda curve, F00 which is equivalent to the choice, q = p − e p .
562
Y. Kodama, V. U. Pierce
Finally, we remark that this universality can be seen in the dPfaff equation with the rescaling and the proper normalization: The dPfaff equation is given by 2 2 2 + F22 = e2F00, − F13 + 2F11 3 4 which can be split to 1 1 2 1 2 2 − F13 + F11 + F22 = −F11 + e2F00 . 3 2 4 Then if F satisfies the dToda hierarchy, then both sides of this equation are zero. Notice that the left-hand side is the dKP equation and the other side is the dToda equation. This splitting can be also seen in the Pfaff lattice equation (1.15), i.e. the first equation in (1.15) can be written in the form, ∂ ∂u ∂2 ∂u ∂ 3 u ∂ 2u + 3 + 6u −4 + 3 2 = −3 2 (u 2 − 4v + v − ). ∂t1 ∂t3 ∂t1 ∂t1 ∂t2 ∂t1 Then one has a KP solution for u which also satisfies the Toda lattice equation given in the right hand side (see e.g. [6]). 6. Appendix A: The dKP Hierarchy The purpose of this Appendix is to show how one can get the kernel formula (2.3) for the KP hierarchy, which is one of the most important formulae in this paper. We begin by giving a quick introduction to the dKP hierarchy: Let λ be a Laurent series in p given by λ= p+
u2 u3 + 2 + · · ·, p p
where u k are the functions of an infinite variables (T1 , T2 , . . .). Then the dKP hierarchy is defined by the set of equations, ∂λ = { n , λ} ∂ Tn
n = 1, 2, . . . ,
where n (λ( p)) is the Faber polynomial given by n (λ( p)) = [λ( p)n ]≥0 . Here [ f ( p)]≥0 is a polynomial part of the Laurent series f ( p) in p. In particular, 1 (λ( p)) = p, and we denote sometime T1 = X . The bracket { f, g} is the Poisson bracket defined by { f, g} =
∂ f ∂g ∂ f ∂g − . ∂p ∂ X ∂ X ∂p
The Faber polynomial can be written in the Laurent series (see Definition 2.1), n (λ) = λn −
∞ 1 Q . m nm m=1 λ
Dispersionless Integrable Systems and Universality in Random Matrix Theory
563
In particular, we have 1 (λ( p)) = p(λ) = λn −
∞ 1 Q . m 1m m=1 λ
Then, following the computation in [9] (see also [8]), we first multiply λn−1 ∂∂λp to p(λ), i.e. p(λ)λn−1
∞ ∂λ ∂λ ∂λ = λn − Q 1m λn−m−1 . ∂p ∂ p m=1 ∂p
Taking the projection of this equation on the polynomial part, n−1 ∂ 1 ∂ 1 1 ∂ p n = n+1 − n−m . Q 1m ∂p n ∂p n + 1 ∂p n − m m=1 Now multiply by µ−n and sum over n ≥ 1, we have ∞ 1 ∂ ∞ 1 ∂ ∞ n−1 Q 1m 1 ∂ n−m n n+1 = − p n n m n−m ∂ p n − m n n=1 µ ∂ p n=1 µ ∂ p n + 1 n=1 m=1 mµ µ ∞ 1 ∂ ∞ Q ∞ j n 1m 1 ∂ =µ − 1 − . n m j n j n=1 µ ∂ p m=1 µ j=1 µ ∂ p This then gives,
∞ ∞ Q 1 ∂ n 1,m 1 = µ− p− m n n m=1 µ n=1 µ ∂ p ∞ 1 ∂ n . = ( p(µ) − p) n n n=1 µ ∂ p
Thus we have ∞ 1 ∂ 1 = p(µ) − p n=1 µn ∂ p
n (λ( p)) . n
Integrating this with respect to p, and fixing the boundary condition at µ = ∞, we obtain ∞ 1 µ = exp (λ) n p(µ) − p(λ) n=1 nµ ∞ ∞ 1 1 n λ − Q = exp n m nm n=1 nµ m=1 λ ∞ ∞ 1 Q nm 1 . = exp − m n n 1 − µλ n=1 m=1 λ µ Then expressing the Grunsky coefficients in terms of the free energy, i.e. cnm = 1 nm Fnm , we obtain the dKP hierarchy (2.3), p(µ) − p(λ) = e D(λ)D(µ)F . µ−λ
1 n Q nm
=
564
Y. Kodama, V. U. Pierce
One should note here that the dispersionless Hirota equation for Fnm is completely deter u m+1 mined by the function λ = p + ∞ (spectral curve). Then the dKP hierarchy m m=1 p defines an integrable deformation of the function λ( p), and as a result, the Grunsky coefficients can be expressed by the second derivatives of the free energy with the deformation parameters. We also remark that in [32] (see also [33]), Takasaki derives this formula as a classical limit of the differential Fay identity, which can be obtained from the bilinear identity [11], dz ξ(t −t,z) τ t − [z −1 ] τ t + [z −1 ] = 0, e z=∞ 2πi ∞ n where ξ(t, z) = n=1 z tn and [z −1 ] is given by 1 1 1 , 2, 3, ... . [z −1 ] = z 2z 3z Takasaki also derives the dPfaff hierarchy from the classical limit of the differential Fay identity obtained from the bilinear identity for the DKP hierarchy [20]. This is remarkable, since there seems to be no proper classical limit for the Pfaff lattice hierarchy in the matrix representation given in [2,22] (this is contrary to the Toda lattice hierarchy). 7. Appendix B: Computation of C0 (T0 ) (β)
Here we give the computation of C0 (T0 ) for β = 1, 2 and 4. We use formula (3.3.10) given in [27], reproduced here: With the integral, n β β (β) β 2 2 λi − λ j exp − Zn λ ;0 = dλ1 . . . dλn λ , 2 2 j=1 j R R i< j
one finds the following formula, which we call Mehta’s formula: n n β −n βj (β) β 2 − n2 −β n(n−1) 2 4 1+ λ ; 0 = (2π ) β . Zn 1+ 2 2 2 j=1
7.1. GUE case. For β = 2, we have C
(2)
1 (2) Z (T0 ; N ) = log n! n
N 2 λ ;0 2
.
From Mehta’s formula, we have n
Z n(2) (λ2 ; 0) = (2π ) 2 2−
n2 2
n
j!.
j=1
A straightforward change of variables gives n n2 n (2) N 2 λ ; 0 = N − 2 (2π ) 2 Zn j!. 2 j=1
Dispersionless Integrable Systems and Universality in Random Matrix Theory
565
Inserting this into our formula for C (2) (T0 ; N ) we find: n n n n2 λ2 ; 0 = − log N + log(2π ) + log Z n(2) log( j!). 2 2 2 j=1 So the problem is to find the leading order behavior of log( j!). We compute this in a generalization of the Riemann sums approach to deriving Sterlings approximation. We start by rewriting the expression log( j!) as a double summation which then simplifies: n
log( j!) =
j=1
=
j n
log k =
j=1 k=1 n
n n
log k
k=1 j=k
(n − k + 1) log k.
k=1
We then approximate this sum by an integral using the trapezoid rule n n 1 log( j!) = log n + (n − x + 1) log x d x + E, 2 1 j=1 where the error in the approximation can be computed using the Euler-MacLaurin formula. For our purposes it is enough to show that E = O(n log n), in truth it performs much better. The integral then gives n
3 1 log( j!) = − n 2 + n 2 log n + O(n log n). 4 2 j=1 Inserting this back into our expression for C (2) (T0 ; N ) we find 1 2 1 3 n log n − n 2 log N + n 2 + O(n log n). 2 2 4 Then in the limit, we have with n = N T0 , C (2) (T0 ; N ) =
(2)
C0 (T0 ) := lim
N →∞
1 1 3 C(T0 ; N ) = T02 log(T0 ) − T02 . N2 2 4
7.2. GOE case. In the case of β = 1, we have 1 (1) N 2 C (1) (T0 ; N ) = log n Z 2n λ ;0 . 2 n! 4 Mehta’s formula then gives −2n 2n 3 j (1) 1 2 n λ ; 0 = (2π ) 1+ Z 2n 2 2 2 j=1
= 23n
n
j!
j=1 n
= 22n π 2 n!
n−1 j=0
n−1 j=1
√ π (2 j + 1)! 22 j+1 j!
2−2 j (2 j + 1)!.
566
Y. Kodama, V. U. Pierce
Likewise in this case we find that 1 n 2 1 (1) N 2 (1) 1 2 (1) 2 C (T0 ; N ) = log n Z 2n = n + λ ;0 log + log n Z 2n λ ;0 2 n! 4 2 N 2 n! 2 n n−1 2 j + log[(2 j + 1)!] + O(n log n). = n 2 log + log(n!) − 2 log 2 N j=1 j=1 Stirlings approximation implies that log(n!) =
n
log k = O(n log n).
k=1
So we only need to find the behavior of log((2 j + 1)!) for large n: as above we follow the basic method of Stirlings approximation and write the sum as the approximation of an integral plus an error term. For large n the error term is of order O(1); n−1
log((2 j + 1)!) =
j=1
j+1 n−1 2
log k =
j=1 k=1
=
n−1
2n−1 n−1
log k
k=1 j= k 2
! (n − l) log(2l) + (n − l) log(2l + 1)
l=1
n−1 1 = log((2n − 2)(2n − 1)) + (n − x) log(2x(2x + 1))d x + E 2 1 3 = − n 2 + n 2 log(2n) + O(n log n). 2 Then we find 3 C (1) (T0 ; N ) = n 2 log(2n) − n 2 log N − n 2 O(n log n), 2 which gives (1)
C0 (T0 ) := lim
N →∞
1 (1) 3 C (T0 ; N ) = T02 log(2T0 ) − T02 . 2 N 2
7.3. GSE case. For β = 4, we have C
(4)
1 (T0 ; N ) = log n Z n(4) 2 n!
N 2 λ ;0 4
.
Mehta’s formula with β = 4 gives n
Z n(4) (λ2 ; 0) = π 2 2−2n
2+ n 2
n
(2 j)!,
j=1
which together with a change of variables gives n n π 2 −n 2 + n (4) N 2 2 λ ;0 = Zn N (2 j)!. 4 2 j=1
Dispersionless Integrable Systems and Universality in Random Matrix Theory
567
Therefore C (4) (T0 ; N ) = −n 2 log N +
n
log(2 j)! + O(n log n).
j=1
So we must compute the asymptotic behavior of log(2 j)!; it is not suprising that it is identical to the asymptotic behavior of log(2 j + 1)! found in the β = 1 case. This is in fact yet more evidence of the duality that exists between β = 1 and 4. Inserting this back into our expression for C (4) (T0 ; N ) we find 3 C (4) (T0 ; N ) = −n 2 log N − n 2 + n 2 log(2n) + O(n log n), 2 from which we conclude that 3 (4) C0 (T0 ) = T02 log(2T0 ) − T02 . 2 Note added in proof: After submitting this paper, we were informed (1/23/09) that the explicit formulas Fmn in Theorem 3.1. also appeared in [28] (Eq. (11) in p.27) where the Fmn provide the numbers of non-crossing chord diagrams in an annulus. We thank Prof. Drew Armstrong for this information. References 1. Adler, M., van Moerbeke, P.: Vertex operator solutions of the discrete KP-hierarchy. Commun. Math. Phys. 203, 185–210 (1999) 2. Adler, M., van Moerbeke, P.: Toda versus Pfaff lattice and related polynomials. Duke Math. J. 112, 1–58 (2002) 3. Adler, M., Shiota, T., van Moerbeke, P.: Pfaff τ -functions. Math. Ann. 322, 423–476 (2002) 4. Aoyama, S., Kodama, Y.: Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy. Commun. Math. Phys. 182, 185–219 (1996) 5. Bessis, D., Itzykson, C., Zuber, J.B.: Quantum field theory techniques in graphical enumeration. Adv. Applied Math. 1, 109–157 (1980) 6. Biondini, G., Kodama, Y.: On a family of solutions of the Kadomtsev-Petviashvili equation which also satisfy the Toda lattice hierarchy. J. Phys. A: Math. Gen. 36, 10519–10536 (2003) 7. Bryc, W., Pierce, V.U.: Duality of real and quaternionic random matrices. Electron. J. Probab. 14, 452– 476 (2009) 8. Chen, Y.-T., Tu, M.-H.: On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy. J. Math. Phys. 47, 102702 (1–19) (2006) 9. Caroll, R., Kodama, Y.: Solution of the dispersionless Hirota equations. J. Phys. A: Math. Gen. 28, 6373– 6387 (1995) 10. Collins, B., Guionnet, A., Maurel-Segala, E.: Asymptotics of unitary and orthogonal matrix integrals. http://arxiv.org/abs/math/0608193v3[math.PR], 2008 11. Date, E., Kashiwara, M., Jimbo, M., Miwa, T.: Transformation groups for soliton equations. In: Nonlinear Integrable Systems - Classical Theory and Quantum Theory, Singapore: World Scientific, 1983, pp. 39–119 12. Deift, P., Gioev, D.: Universality in random matrix theory for orthogonal and symplectic ensembles. IMRP 2, 116 (2007) 13. Deift, P., Gioev, D.: Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Comm. Pure Appl. Math. 60, 867–910 (2007) 14. Dubrovin, B.A.: Integrable systems and classification of 2-dimensional topological field theories. In: Integrable Systems, Prog. Math. 115, Boston: Birkhäuser, 1993, pp. 313–359 15. Ercolani, N.M., McLaughlin, K.T.-R.: Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration. IMRN 14, 755–820 (2003) 16. Ercolani, N.M., McLaughlin, K.T.-R., Pierce, V.U.: Random matrices, graphical enumeration and the continuum limit of Toda lattices. Commun. Math. Phys. 278, 31–81 (2008)
568
Y. Kodama, V. U. Pierce
17. Goulden, I.P., Harer, J.L., Jackson, D.M.: A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves. Trans. Amer. Math. Soc. 353, 4405– 4427 (2001) 18. Hirota, R., Ohta, Y.: Hierarchies of coupled soliton equations. I. J. Phys. Soc. Japan 60, 798–809 (1991) 19. Jackson, D.M.: On an integral representation for the genus series for 2-cell embeddings. Trans. Amer. Math. Soc. 344, 755–772 (1993) 20. Jimbo, M., Miwa, T.: Soliton equations and infinite dimensional Lie algebras. Publ. RIMS, Kyoto University, 19, 943–1001 (1983) 21. Kakei, S.: Orthogonal and symplectic matrix integrals and coupled KP hierarchy. J. Phys. Soc. Japan 68, 2875–2877 (1999) 22. Kodama, Y., Pierce, V.U.: Geometry of the Pfaff lattices. Inter. Math. Res. Notes, 2007, art.Drnm120, doi:10.1093/imrn/rnm120 (1–55), 2007 23. Kodama, Y., Pierce, V.U.: The Pfaff lattice on symplectic matrices. http://arxiv.org/abs/[nlin.SI], 2009 24. Krichever, I.M.: The τ -function of the unversal Whitham hierarchy, matrix models and topological field theories. Comm. Pure Appl. Math. 47, 437–475 (1994) 25. Kwak, J.H., Lee, J.: Genus polynomials of dipoles. Kyungpook Math. J. 33, 115–125 (1993) 26. Marshakov, A., Wiegmann, P., Zabrodin, A.: Integrable structure of the Dirichlet boundary proble in two dimensions. Commun. Math. Phys. 227, 131–153 (2002) 27. Mehta, M.L.: Random Matrices 3rd ed. Amsterdam: Elsevier, (2004) 28. Mingo, J.A., Speicher, R., Tian, E.: Second order cumulants of products. (arXiv:0708.0586) 29. Mulase, M., Waldron, A.: Duality of orthogonal and symplectic matrix integrals and quaternionic Feynman graphs. Comm. Math. Phys. 240, 553–586 (2003) 30. Schur, I.: On Faber polynomials. Amer. J. Math. 67, 33–41 (1945) 31. Shcherbina, M.: On the universality for orthogonal ensembles of random matrice. Commun. Math. Phys. 285, 257–974 (2009) 32. Takasaki, K.: Differential Fay identities and auxiliary linear problem of integrable hierarchies. http:// arxiv.org/abs/0710.5356v4[nlin.SI], 20 33. Takasaki, K., Takebe, T.: Integrable hierarchies and dispersionless limit. Rev. Mod. Pfys. 7, 743–808 (1995) 34. Teo, L.-P.: Analytic functions and integrable hierarchies - Characterization of tau functions. Lett. Math. Phys. 64, 75–92 (2003) 35. Wiegmann, P.B., Zabrodin, A.: Conformal maps and integrable hierarchies. Commun. Math. Phys. 213, 523–538 (2000) 36. Zabrodin, A.: The dispersionless limit of the Hirota equations in some problems of complex analysis. Teoret. Mat. Fiz. 129, 239–257 (2001) Communicated by L. Takhtajan
Commun. Math. Phys. 292, 569–605 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0888-z
Communications in
Mathematical Physics
Extended Scaling Relations for Planar Lattice Models G. Benfatto1 , P. Falco2 , V. Mastropietro1 1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica,
00133 Roma, Italy. E-mail: [email protected]
2 Mathematics Department, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Received: 18 November 2008 / Accepted: 20 May 2009 Published online: 14 August 2009 – © Springer-Verlag 2009
Abstract: It is widely believed that the critical properties of several planar lattice systems, like the Eight Vertex or the Ashkin-Teller models, are well described by an effective continuum fermionic theory obtained as a formal scaling limit. On the basis of this assumption several extended scaling relations among their indices were conjectured. We prove the validity of some of them, among which the ones predicted by Kadanoff (Phys Rev Lett 39:903–905, 1977) and by Luther and Peschel (Phys Rev B 12:3908–3917, 1975).
1. Introduction and Main Results Integrable models in statistical mechanics, like the Ising or the Eight vertex (8V) models in two dimensions, provide conceptual laboratories for the understanding of phase transitions. Integrability is however a rather delicate property requiring very special features, and it is usually lost in more realistic models. The principle of universality, phenomenologically quite well verified, says that the singularities for second order phase transitions should be insensitive to the specific details of the model. From the theoretical side, a mathematical justification of universality in planar lattice models is rather difficult to provide. Only very recently Pinson and Spencer established, see [29,32], a form of universality for the 2D Ising model; they added to the Ising Hamiltonian a perturbation breaking the integrability and showed that the indices they can compute are exactly the same as the Ising model ones. While the critical indices of the Ising model are expressed by pure numbers, there are other lattice models in which some of the critical exponents vary continuously with the parameters appearing in the Hamiltonian. A celebrated example is provided by the Eight vertex model, solved by Baxter in [2]; even though it can be mapped in two Ising models coupled by a quartic interaction, its critical indices are different from the Ising ones.
570
G. Benfatto, P. Falco, V. Mastropietro
Several authors, starting from Kadanoff and collaborators [16–18] and Luther and Peschel [20], have argued that many models, like the Askhin-Teller (AT) model and several others, belong to the class of universality of the 8V model. The notion of universality in this case is much more subtle; it does not mean that the indices are the same for all the models in the same class (on the contrary, the indices depend on all the details of the Hamiltonian), rather it means that there are scaling relations between them, such that all the indices of a single model can be expressed in terms of one of the indices of the same model. The notion of universality for models with continuously varying indices has been deeply investigated over the years, see for instance [18,27,28,33]; it has been pointed out that such models are well described in the scaling limit by an effective continuum fermionic theory, and on the basis of this assumption several extended scaling relations between their indices were derived. While the assumption of a continuum scaling limit description of planar lattice models is very powerful, it is well known that a mathematical justification of it is very difficult, see e.g. [31]. The aim of this paper is to provide a mathematical proof of some of the exact scaling relations derived in the literature for planar lattice models. We will focus mainly on the 8V and AT models, but, as we shall explain after the main theorem below, our result can be extended to several other models. We start from the well known (see [3]) Ising formulation of the 8V and the AT mod2 els. Let be a square subset of Z of side L; if x = (x0 , x) ∈ and e0 = (1, 0), e1 = (0, 1), we consider two independent configurations of spins, {σx = ±1}x∈ and {σx = ±1}x∈ and the Hamiltonian H (σ, σ ) = H J (σ ) + H J (σ ) − J4 V (σ, σ ),
(1.1)
where J > 0 and J > 0 are two parameters, H J is the (ferromagnetic) Ising Hamiltonian in the lattice , H J (σ ) = −J σx σx+e j , (1.2) j=0,1 x∈
V is the quartic interaction and −J4 is the coupling. In the AT model, J and J can be different (in which case the model is called anisotropic) and V = V AT , with (see Fig. 1) V AT (σ, σ ) = σx σx+e j σx σx+e . (1.3) j j=0,1 x∈
In the 8V model J = J and V = V8V , with (see Fig. 1) V8V (σ, σ ) = σx+ j (e0 +e1 ) σx+e0 σx+ j (e0 +e1 ) σx+e1 .
(1.4)
j=0,1 x∈
In this paper we will focus our attention on two observables, σx σx+e j + ε σx σx+e , ε = ±, Oxε = j j=0,1
(1.5)
j=0,1
and their truncated correlations in the thermodynamic limit G ε (x − y) = lim Oxε Oyε − Oxε Oyε , ε = ±, →∞
(1.6)
Extended Scaling Relations
571
8V
AT
Fig. 1. The quartic interaction in the 8V and in the AT case. The gray and the black square are the same square of the lattice
where · is the average over all configurations of the spins with statistical weight e−β H (σ,σ ) . In the AT model, Ox+ is called the energy, while Ox− is called the crossover; and viceversa in the 8V model, see e.g. [27]. Despite their similarity, an exact solution exists for the 8V model but not for the AT model. In recent times the methods of fermionic Renormalization Group (RG) (introduced in [7]; see e.g. [25] for an updated introduction) has been applied to such models (for J4 small), using the well known representation of such correlations in terms of Grassmann integrals, see e.g. [30]. It was proved in [21,22] that both the 8V and the isotropic (J = J ) AT systems have a nonzero critical temperature, Tc , such that, if T = Tc , G ε (x − y) decays faster than any power of ξ −1 |x − y|, with ξ −1 ∼ C |T − Tc |ν , as T → Tc . Moreover, for T → Tc , there are two constants Cε , ε = ±, such that Cε , as |x − y| → ∞, G ε (x − y) ∼ |x − y|2xε
(1.7)
(1.8)
where x± are critical indices expressed by convergent series in J4 . The analysis in [22] allows to compute the indices ν, x± with arbitrary precision (by an explicit computation of the lowest orders and a rigorous bound on the remainder); however, the complexity of such expansions makes it essentially impossible to see directly from them the scaling relations. In the case of the anisotropic AT model, it was proven in [13,14] that there are two critical temperatures, T1,c and T2,c , and the corresponding critical indices are the same as those of the Ising model. However as J − J → 0: |T1,c − T2,c | ∼ |J − J |x T ,
(1.9)
with a transition index, x T , different from 1 if J4 = 0. In this paper we will prove the following theorem: Theorem 1.1. If the coupling is small enough, the critical indices of the 8V or AT models verify 1 , x+ 1 ν= ; 2 − x+ and, in the case of the anisotropic AT model, x− =
xT =
2 − x+ . 2 − x−
(1.10) (1.11)
(1.12)
572
G. Benfatto, P. Falco, V. Mastropietro
Remarks. 1. Equation (1.10) is the extended scaling law first conjectured by Kadanoff for the AT and 8V models (see Eq. (13b) and (15b) of [16]). Equation (1.11) was conjectured by Luther and Peschel in [20] (see Eq. (16) and Table I of that paper). 2. The scaling relation (1.12) was never conjectured before. 3. All the critical indices we consider can be expressed as simple functions of one of them, in agreement with the general belief. 4. The exact value of the index ν for the Eight Vertex model has been obtained by Baxter, see (10.12.23b) of [3]; hence the values of x± , which cannot be computed from the exact solution, can be obtained from (1.10) and (1.11). 5. We have considered just the 8V or the AT models for definiteness, but our theorem is valid for any Hamiltonian of the form (1.1), if the quartic interaction is small enough and verifies some symmetry conditions, listed in App. O of [22]; in particular, our result is valid for a generic interaction of the form V (σ, σ ) = v(x − y)σx σx+e j σy σy+e , (1.13) j j=0,1 x,y∈
with v(x − y) a rotational invariant short range potential (|v(x − y)| ≤ Ce−κ|x−y| , C, κ positive constants). 6. Our analysis can be extended to a large class of (integrable or non integrable) quantum spin chains or fermion models known as Luttinger liquids [15]. For instance in the case of the X Y Z spin chain with a magnetic field, analyzed in [9], calling x+ the index appearing in the oscillating part of the spin-spin correlation along the z direction (see (1.20) of [9]), and α the index appearing in the decay rate (see (1.19) of [9]), it is a simple corollary of the proof of Theorem 1.1 that ν = (2 − x+−1 )−1 . 7. Several other relations are conjectured in the literature, concerning critical indices, which are much more difficult to study with our methods, like the indices of the polarization correlations. New ideas seems to be required to treat such cases. The main steps of the proof are the following. (i) The correlations of the spin models are written in terms of interacting one dimensional fermion models, whose correlations can be computed in terms of Grassmann integrals; (ii) the critical indices are expressed in terms of convergent expansions by a Renormalization Group analysis of the functional integrals; (iii) the original theory is shown to be equivalent (in the sense that their critical indices coincide) to an effective continuum fermionic model, defined as the formal scaling limit of the original one, provided that the coupling λ∞ of the equivalent model is chosen properly as a function of J4 ; (iv) The effective model is expressed in terms of Grassmann integrals which are identical to the ones appearing in certain Quantum Field Theory (QFT) models; we take advantage of the Gauge symmetry and of a property called anomaly non-renormalization, see [23], to exactly compute the correlations and the critical indices. It turns out that the dependence of the critical indices on λ∞ is so simple that we can check that the extended scaling relations are verified. The proof relies on several results derived in detail in previous papers, in particular [6,22,23]; here the attention is mostly focused on the new technical points which are required for the proof, and the use of earlier published results is through precise statements proved elsewhere. The proof of (1.11) is at the end of §2.3, while (1.12) is proved in §2.4; (1.10), the main result of this paper, is proved in §4.2.
Extended Scaling Relations
573
2. RG Analysis of Spin Models In this section we summarize the analysis given in [22] for the 8V or isotropic AT model and in [14] for the anisotropic AT model. The correlations of the AT or 8V models are written in terms of Grassmann integrals and are analyzed using RG methods. The critical indices x+ , x− , ν and x T are written, in the small coupling region, as model independent convergent series of a single parameter, λ−∞ , as we shall call the asymptotic limit of the effective coupling on large scale. Notice that λ−∞ is in turn a convergent series, depending on all the details of the lattice model, of the coupling J4 . Such expansions allow in principle to compute the indices with arbitrary precision, but the complexity of such expansions makes it essentially impossible to see directly from them the extended scaling relations. 2.1. Fermionic representation of the spin models. The partition function Z (I ) of the Ising model with external sources A j,x , and periodic conditions at the boundary of is ⎤ ⎡ ⎥ ⎢ (2.1) exp ⎣ I j,x σx σx+e j ⎦ , Z (I ) = σ
j=0,1 x∈
where I j,x = A j,x + β J . It is well known, see [30] or App. A1 of [14], that Z (I ) can be written as sum of Grassman integrals. Let γ = (ε0 , ε1 ), with ε0 , ε1 = ± and let {Hx , H¯ x , Vx , V¯x }x∈ be a family of Grassmann variables verifying the γ -boundary conditions, namely H¯ x+(L ,0) = ε0 H¯ x ,
H¯ x+(0,L) = ε1 H¯ x ,
Hx+(L ,0) = ε0 Hx ,
Hx+(0,L) = ε1 Hx ,
(2.2)
and similar relations for V, V¯ (we are skipping the γ dependence in the H ’s and V ’s). Then we consider the Grassmann functional integral Z γ = d H d V e S(t) , (2.3) where the action S(t) is the following function of the parameters t = {t j,x } the Grassmann variables with γ −boundary condition:
t0,x H¯ x Hx+e0 + t1,x V¯x Vx+e1 S(t) = x∈
+
H¯ x Hx + V¯x Vx + V¯x H¯ x + Vx Hx + Vx H¯ x + Hx V¯x .
x∈ j=0,1
and of
(2.4)
x∈
If we choose t j,x = tanh I j,x and put c j,x = cosh I j,x , the partition function (2.1) can be written in the following way: ⎛ ⎞
(−1)δγ Zγ Z (I ) = (−1)|| 2|| ⎝ c j,x ⎠ (2.5) 2 γ j,x
where δγ = 1 for γ = (+, +), and δγ = 0 otherwise.
574
G. Benfatto, P. Falco, V. Mastropietro
By using (2.4), the correlation functions (1.6) of the spin model (1.1) can be written as (see §A.1 for details) 2 ln Z¯ ( A) ¯ ∂ T Oxε ; Oyε = + SL (x, y), (2.6) ε ε ¯ ¯ ∂ Ax ∂ Ay A≡0 ¯ where SL (x, y) is a correction term, vanishing in the thermodynamic limit (see App. 1), and ¯ ¯ ¯ (2.7) Z ( A) = d H d V d H d V e S(s)+S(s )+2λV +B( A) , s, s and λ being parameters independent of j and x (see (A.9) below), such that s = tanh(β J )+O(β J4 ), s = tanh(β J )+O(β J4 ) and λ = O(β J4 ). Moreover, the (H, V ) and (H , V ) variables verify antiperiodic boundary conditions and V is a quartic interaction that, in the AT case, is given by V AT =
H¯ x Hx+e0 H¯ x Hx+e , + V¯x Vx+e1 V¯x Vx+e 0 1
(2.8)
x∈
while, in the 8V case, is given by
V8V =
H¯ x Hx+e0 V¯x Vx+e . + V¯x+e0 Vx+e0 +e1 H¯ x+e H 1 1 x+e1 +e0
(2.9)
x∈
¯ is an interaction with external sources A¯ εx , given, in the AT case, by Finally B( A) ¯ = B( A)
+ V¯x Vx+e A¯ εx qε H¯ x Hx+e0 + V¯x Vx+e1 + qε H¯ x Hx+e 0 1
x∈ ε=±
+
+ V¯x Vx+e1 V¯x Vx+e , A¯ εx pε H¯ x Hx+e0 H¯ x Hx+e 0 1
(2.10)
x∈ ε=±
while, in the 8V case, it is given by
¯ = B( A)
A¯ εx qε H¯ x Hx+e0 + V¯x+e0 Vx+e0 +e1
x∈,ε=± H +qε H¯ x+e 1 x+e1 +e0
+
+ V¯x Vx+e 1
, + V¯x+e0 Vx+e0 +e1 H¯ x+e H A¯ εx pε H¯ x Hx+e0 V¯x Vx+e 1 1 x+e1 +e0
(2.11)
x∈ ε=±
with qε , qε and pε parameters independent of j and x, such that qε = 1 − tanh(β J ) + O(β J4 ), qε = ε[1 − tanh(β J )] + O(β J4 ) and pε = O(β J4 ). Notice the crucial difference between (2.1) and (2.7); in the second case also quartic monomials appear in the exponent, while in the first case only quadratic terms appear. In other words, the Ising model correspond to a non-interacting fermionic theory, while the model (1.1) is mapped into an interacting fermionic system.
Extended Scaling Relations
575
2.2. The case J = J . We consider first the case J = J . The Grassmann integral (2.7) is not very convenient for the RG analysis, but it can be properly rewritten, by a suitable change of variables in the Grassmann algebra, in a much better form (see [22,14] and App. A for details), such that the analogy of the above functional integral with a fermionic (Euclidean) Quantum Field Model is clearer. 1 2π 1 Let D be the set of k’s such that k0 = 2π L (n 0 + 2 ) and k1 = L (n 1 + 2 ), for n 0 , n 1 = − L2 , . . . , L2 − 1, and L an even integer. Then the functional integral (2.7) can be written as (see §A.2 for details) 1 ¯ ¯ ¯ Z ( A) = (2.12) P(dψ)Pχ (dχ ) eQ(ψ,χ )+V (ψ,χ )+B( A) , N where N is a normalization constant, P(dψ) is the (Grassmannian) Gaussian measure with propagator g(x) =
1 −ikx −1 e T (k), L2
(2.13)
k∈D
with
⎛ T (k) = u ⎝
⎞
i sin k0 + sin k1
−iµ(k)
iµ(k)
i sin k0 − sin k1
⎠,
(2.14)
u = tanh(β J ) + O(β J4 ), µ(k) = (cos k0 + cos k1 − 2) + 2
1−
√ u
2+u
,
(2.15)
Pχ (dχ ) is the Gaussian measure with propagator gχ (x), which is obtained from g(x) by replacing T (k) with T χ (k), T χ (k) being the matrix obtained from T (k) by substituting µ(k) with √ 1+ 2+u χ . (2.16) µ (k) = (cos k0 + cos k1 − 2) + 2 u The interaction with the external source is given by − − + + − + + − ¯ =i B( A) q+ A¯ +x [ψx,+ ψx,− − ψx,− ψx,+ + χx,+ χx,− − χx,− χx,+ ] x∈
+i
− + + − + + − − ¯ q− A¯ − x [ψx,+ ψx,− + ψx,+ ψx,− + χx,+ χx,− + χx,+ χx,− ] + R B ( A),
x∈
(2.17) ¯ contains terms either quartic in the fields or quadratic with derivatives, where R B ( A) whose explicit form (not essential for our purposes) can be obtained by performing the changes of variables explained in App. A. Moreover Q(ψ, χ ) = −
1 + − + − k,ω χ ψk,ω ]Q ω,ω (k), [ψ k,ω + χ k,ω || k∈D ω,ω
(2.18)
576
G. Benfatto, P. Falco, V. Mastropietro
where Q(k) is a matrix which vanishes at k = 0. Finally, the quartic self interaction is given by
− + + − + + − − V(ψ, χ ) = λ ψx,+ ψx,− ψx,+ ψx,− + χx,+ χx,− χx,+ χx,− x∈
+ v(ψ, χ ) + R V (ψ, χ ),
(2.19)
where v(ψ, χ ) is a quartic interaction depending both on ψ and χ , which has a different expression in the AT and 8V models, and R V (ψ, χ ) is the sum of quartic monomials with at least a derivative; the explicit form for such expressions can again be obtained by performing the changes of variables explained in App. A. If J > 0 and J4 is any real number, u is a strictly increasing function of tanh(β J ) and has range (0, 1), as one can check by using the definition of s, see (A.9). On the other hand, det √ T (k) = 0 only if k = 0 and µ(k) = 0; hence, g(x) has a singularity at u = u c = 2 − 1, which is an allowed value; moreover, if β|J4 | 1 (as we shall suppose in the following), u = tanh(β J ) + O(β J4 ). Since we expect that the interaction will move this singularity, it is convenient to modify the interaction by adding a − + ψ finite counterterm iν1 L12 ω,k ωψ k,ω k,−ω , which is compensated by replacing, in the matrix T (k), µ(k) with √ u∗ , u ∗ = 2 − 1 − ν1 . µ1 (k) = (cos k0 + cos k1 − 2) + 2 1 − (2.20) u Let us call T1 (k) the new matrix and P1 (dψ) the corresponding measure; we get 1 (1) ¯ ¯ ¯ (2.21) P1 (dψ)Pχ (dχ ) eQ(ψ,χ )+V (ψ,χ )+B( A) , Z ( A) = N1 where V (1) (ψ, χ ) = iν1
1 + − k,ω ψk,−ω + V(ψ, χ ), ωψ L2
(2.22)
ω,k
and ν1 has to be determined so that the interacting propagator has an infrared singularity at u = u ∗ ; the critical temperature is uniquely determined by the value of u ∗ . Let us now remark that det T χ (k) is strictly positive for any k, as one can easily see by using the fact that u ∈ (0, 1). Hence, if we define + = ψ + QTχ−1 , ψ − = Tχ−1 Qψ − , ψ
(2.23)
+ , χ − → χ − + ψ − , allows us to rewrite (2.21) in the change of variables χ + → χ + + ψ the form 1 (1) ¯ ¯ ¯ (2.24) Z ( A) = PZ 1 ,µ1 (dψ)Pχ (dχ ) eV (ψ,χ −ψ )+ B( A) , N ¯ is the functional obtained from B( A) ¯ by replacing χ with χ − ψ and where B( A) PZ 1 ,µ1 (dψ) is the Gaussian measure with propagator g(x) =
1 −ikx (1) −1 e (T ) (k), L2 k∈D
(2.25)
Extended Scaling Relations
577
where T (1) (k) = T (k) − Q(k)Tχ−1 Q(k). It is also convenient to perform the trivial change of variables + k,ω + , ψ − → ψ − , k = (k0 , k1 ), ψ → −iωψ k = (k1 , k0 ). k,ω k,ω k,ω
(2.26)
Hence, by an explicit calculation of Q(k) and using the identity u ∗ /u = 1 − µ1 (0)/2, one can see that T (1) (k) is the matrix −µ1 − µ+,− (k) −i sin k0 + sin k1 + µ+,+ (k) (2.27) Z 1 C1 (k) −µ1 − µ+,− (k) −i sin k0 − sin k1 + µ−,− (k) with C1 (k) = 1, µ1 = 2µ1 (0)/(2 − µ1 (0)) and Z 1 = u ∗ ; moreover µ+,+ (k) = −µ−,− (k)∗ is an odd function of k of the form µ+,+ (k) = 2µ1 (0)(−i sin k0 +sin k1 )/(4− 2µ1 (0)) + O(|k|3 ), while µ+,− (k) is a real even function, of order |k|2 , which vanishes only at k = 0. Finally, det T (1) (k) ≥ C(2 − cos k0 − cos k1 ), so that PZ 1 ,µ1 (dψ) has the same type of infrared singularity as P1 (dψ). The fact that det Tχ (k) is strictly positive implies that gχ (x) is an exponential decaying function; hence, we can safely perform the integration over the field χ in (2.24). The result can be written in the following form (see Lemma 1 of [22]) 2 (1) ¯ (1) (1) ¯ ¯ S ( A) ¯ ¯ Z ( A) ≡ e = PZ 1 ,µ1 (dψ)e L N +V (Z 1 ψ)+B ( A) , (2.28) where N (1) is a constant and the effective potential V¯ (1) (ψ) can be represented as 2n V¯ (1) = Wω,α,ε,2n (x1 , . . . , x2n )∂ α1 ψxε11,ω1 . . . ∂ α2n ψxε2n ,ω2n . (2.29) n≥1 α,ω,ε x1 ,...,xn
The kernels Wω,α,ε,2n in the previous expansions are analytic functions of λ and ν1 near the origin; if we suppose that ν1 = O(λ), their Fourier transforms satisfy, for any n ≥ 1, the bounds, see [22], α,ω,ε,2n (k1 , . . . , k2n−1 )| ≤ L 2 C n |λ|n . |W
(2.30)
¯ A similar representation can be written for the functional of the external field B (1) ( A). As explained in detail in [22], the symmetries of the two models we are considering imply that, in the r.h.s. of (2.29), there are no local terms quadratic in the field, which are relevant or marginal, except those which are already present in the free measure.
2.3. Multiscale analysis. We briefly recall here the analysis in [22] (see also [7,8]). The integration in (2.28) can be done by iteratively integrating the fields with decreasing momentum scale and by moving to the free measure all the marginal terms quadratic in the field. We introduce a scaling parameter γ = 2, a decomposition of the unity 1 = f 1 + 0h=−∞ f h (k), with f h (k) a smooth function with support {γ h−1 π/4 ≤ |k| ≤ γ h+1 π/4}, and the corresponding decomposition of the field ψ = 1j=−∞ ψ ( j) . If the fields ψ (1) , . . . , ψ (h+1) are integrated, we get (h) ¯ (h) √ (≤h) (h) √ (≤h) ¯ ¯ (2.31) eS ( A) = e S ( A) PZ¯ h ,µh (dψ (≤h) )eV ( Z h ψ )+B ( Z h ψ , A) ,
578
G. Benfatto, P. Falco, V. Mastropietro
h ( j) and P where ψ (≤h) = j=−∞ ψ Z¯ h ,µh (dψ) is the Gaussian fermionic measure with the propagator obtained from (2.25) by replacing in (2.27) C1 (k) with C h (k) = h f h (k)]−1 , µ1 with µh , Z 1 with the function Z¯ h (k) (to be defined below) and [ k=−∞ the functions µσ,σ (k) with similar functions µ(h) σ,σ (k); finally, the constant Z h , which ¯ rescale the field, is given by Z h = Z h (0). The effective interaction V (h) (ψ) is a sum over monomials in the Grassmann variables and we define a localization operator (see e.g. §4 of [22]) as (h)
LV (h) (ψ) = (−m h + γ h n h )Fν(h) + lh Fλ − z h Fz(h) ,
(2.32)
where m h , n h , z h and lh are suitable real numbers, 1 (≤h)+ (≤h)− ψ (2.33) k,ω ψk,−ω , L2 ω k 1 (≤h)− , (≤h)+ ψ = 2 (−i sin k0 + ω sin k1 )ψ (2.34) k,ω k,−ω L ω k 1 (≤h)+ ψ (≤h)+ ψ (≤h)− ψ (≤h)− δ(k1 − k2 + k3 − k4 ). ψ = 8 k1 ,+ k3 ,− k2 ,+ k4 ,− L
Fν(h) = Fz(h) Fλ(≤h)
Moreover we define
k1 ,...,k4
(ε) ¯ = LB(h) ( Z h ψ (≤h) , A) Z h A¯ εx Ox(≤h)ε ,
(2.35)
ε,x
where (≤h)+ (≤h)− (≤h)+ (≤h)− Ox(≤h)+ = ψx,+ ψx,− + ψx,− ψx,+ , (≤h)+
Ox(≤h)− = i[ψx,+
(≤h)+
ψx,−
(≤h)−
+ ψx,+
(≤h)−
ψx,−
].
(2.36)
We now move to the fermionic measure the terms proportional to m h and z h in (2.32) and we rescale the fields so that (h) ¯ 2 ¯ eS ( A) = e S ( A)+L th PZ¯ h−1 ,µh−1 (dψ (≤h) ) ¯ (h) (√ Z h−1 ψ (≤h) )+B¯ (h) (√ Z h−1 ψ (≤h) , A) ¯
eV
,
(2.37)
with th a normalization constant and Z¯ h−1 (k) = Z h (1 + z h C h−1 (k)), µh−1 =
Zh [µh (k) + m h C h−1 (k)]. (2.38) ¯ Z h−1 (k)
The renormalized potential V¯ (h) (ψ) can be written as V¯ (h) (ψ) = γ h νh Fν(h) + λh Fλ(h) + R (h) (ψ),
(2.39)
with νh = n h (Z h /Z h−1 ) and λh = lh (Z h /Z h−1 )2 ; R (h) (ψ) is a√sum over monomi¯ = als similar to (2.29), with 2n + α1 + .. + α2n > 4. Finally, B¯ (h) ( Z h−1 ψ (≤h) , A) √ ¯ The field ψ (h) is integrated and the procedure can be iterated. The B (h) ( Z h ψ (≤h) , A). above integration procedure is done till the scale h ∗ defined as the maximal j such that
Extended Scaling Relations
579 ∗
γ j ≤ |µ j |, and the integration of the fields ψ (≤h ) can be done in a single step. Roughly speaking, h ∗ defines the momentum scale of the mass. Notice that the propagator of the field ψ (≤h) can be written, for h ≤ 0, as g (≤h) (x, y) = gT(≤h) (x, y) + r (≤h) (x, y),
(2.40)
where 1 −ik(x−y) 1 −1 e T (k), L2 Zh h k∈D −ik0 + k1 −µh , Th (k) = C h (k) µh −ik0 − k1 (≤h)
gT
(x, y) =
(2.41) (2.42)
and, for any positive integer M, |r (≤h) (x, y)| ≤ C M
γ 2h . 1 + (γ h |x − y| M )
(2.43)
(h)
The propagator gT (x, y) verifies a similar bound with γ h replacing γ 2h . A similar decomposition can be done for g (h) (x, y). The definition of the localization operator L selects in V (h) , B (h) the terms with positive or vanishing scaling dimension, which is given, for the monomials with n ψfields and m A-fields, by D =2−
n − m. 2
(2.44)
In the RG language, the terms with positive or vanishing dimension are called relevant or marginal terms, respectively. Notice that a priori many other possible local marginal or relevant terms could be generated in the RG integration, with respect to the one listed in (2.32); however these terms are absent, thanks to the symmetries of the problem, as proved in [22], App. F (see also [14], §A2.2). The outcome of the above procedure is that the kernels in V¯ ( j) and B¯ ( j) are analytic functions of the running coupling constants λk , νk , k ≥ j, provided that supk≥ j (|λk | + |νk |) is small enough, see [22] and §3 of [9]. The running couplings λ j (which, by construction, are the same in the massless µ = 0 or in the massive µ = 0 case, see [13]), satisfy a recursive equation of the form ( j)
( j)
λ j−1 = λ j + βλ (λ j , . . . , λ0 ) + β¯λ (λ j , ν j ; . . . ; λ0 , ν0 ), ( j)
(2.45)
( j)
where βλ , β¯λ are µ-independent and expressed by a convergent expansion in λ j , ( j) ν j . . . , λ0 , ν0 ; moreover β¯λ vanishes if at least one of the νk is zero. The running coupling λ j stays close to λ for any j as a consequence of the following property, called vanishing of the Beta function, which was proved in Theorem 2 of [11]; for suitable positive constants C and ϑ < 1: ( j)
|βλ (λ j , . . . , λ j )| ≤ C|λ j |2 γ ϑ j .
(2.46)
Indeed, it is possible to prove that, for a suitable choice of ν1 = O(λ), ν j = O(γ ϑ j λ¯ j ), if λ¯ j = supk≥ j |λk |, and this implies, by the short memory property (see for instance
580
G. Benfatto, P. Falco, V. Mastropietro
( j) A4.6 of [13]), β¯λ = O(γ ϑ j λ¯ 2j ) so that the sequence λ j converges, as j → −∞, to a smooth function λ−∞ (λ) = λ + O(λ2 ), such that
|λ j − λ−∞ | ≤ Cλ2 γ ϑ j .
(2.47)
Z j−1 ( j) ( j) = 1 + βz (λ j , . . . , λ0 ) + β¯z (λ j , ν j ; . . . , λ0 , ν0 ), Zj
(2.48)
Moreover
( j) with β¯z vanishing if at least one of the νk is zero so that, by using the bound ν j = ( j) O(γ ϑ j λ¯ j ) and the short memory property, β¯z = O(λ j γ ϑ j ). Finally ( j)
βz (λ j , . . . , λ0 ) = βz (λ−∞ ) + O(λγ ϑ j ),
(2.49)
where the last identity follows from (2.47) and the short memory property. An important point is that the function βz (λ−∞ ) is model independent. Similar equations hold for Z h(±) , µh , with leading terms again model independent, that is ( j)
β± (λ j , . . . , λ0 ) = β± (λ−∞ ) + O(λγ ϑ j ).
(2.50)
By an explicit computation and (2.49), (2.50), there exist η+ (λ−∞ ) = c1 λ−∞ + O(λ2−∞ ), η− (λ−∞ ) = −c1 λ−∞ + O(λ2−∞ ), ηµ (λ−∞ ) = c1 λ−∞ + O(λ2−∞ ) and ηz (λ−∞ ) = c2 λ2−∞ + O(λ3−∞ ), with c1 and c2 strictly positive, such that, for any j ≤ 0, | logγ (Z j−1 /Z j ) − ηz (λ−∞ )| ≤ Cλ2 γ ϑ j , | logγ (µ j−1 /µ j ) − ηµ (λ−∞ )| ≤ C|λ|γ ϑ j , (±) (±) | logγ (Z j−1 /Z j ) − η± (λ−∞ )|
(2.51)
≤ Cλ2 γ ϑ j .
The critical indices are functions of λ−∞ only, as it is clear from (2.49); moreover from (6.28) and (5.4) of [22], the indices x± appearing in (1.8) are such that x ± = 1 − η± + η z , ηµ = η+ − η z = 1 − x + .
(2.52)
When the limit µ → 0 is taken (after the limit L → ∞, so that all the Z γ ,γ have the same limit), the multiscale integration procedure implies the power law decay of the correlations given by (1.8). If µ = 0 (that is, if the temperature is not the critical one), the correlations decay faster than any power with rate proportional to µh ∗ , where, if [x] denotes the largest integer ≤ x, h ∗ is given by ∗
h =
logγ |µ| 1 + ηµ
,
which implies, together with (2.52), the identity (1.11) of Theorem 1.1.
(2.53)
Extended Scaling Relations
581
2.4. The anisotropic Ashkin-Teller model. In order to derive (1.12), we briefly recall the analysis of the anisotropic Ashkin-Teller model in [13,14] with J = J . We still obtain an expression similar to (2.21), the main difference being that (see (A.21) below) ε(≤h) ε(≤h) P1 (dψ) contains in the exponents also terms of the form ψx,ω ψx,−ω , and the same is true for Pχ (dχ ). The integration procedure is similar to the one in §2.3, but we have to substitute the Grassmann integration PZ h ,µh (dψ (≤h) ) in (2.31) with a new measure PZ h ,µh ,σh (dψ (≤h) ), where µh and σh are the constants multiplying, respectively, the quadratic mass terms 2
(≤h)−
(≤h)+ ψx,ω ψx,−ω
and
− 2i
ω=±
(≤h)ε
(≤h)ε
ψx,+ ψx,− .
(2.54)
ε=±
One can see that | logγ (µ j−1 /µ j ) − ηµ (λ−∞ )| ≤ Cλ2 γ ϑ j , | logγ (σ j−1 /σ j ) − ησ (λ−∞ )| ≤ Cλ2 γ ϑ j .
(2.55)
Hence, since the two mass terms are clearly proportional, respectively, to the operators O + and O − , we find that ηµ = η+ − η z ,
ησ = η− − η z .
(2.56)
It turns out that the difference of the critical temperatures scales as |J − J |x T , where x T , see (5.26) of [13] (where the indices are defined with a different sign and the definitions of µh and σh are exchanged), is given by xT =
1 + ηµ , 1 + ησ
(2.57)
which implies (1.12), since ηµ = 1 − x+ and ησ = 1 − x− .
3. Equivalence with a Continuum Model In this section we show that the spin model (1.1) is equivalent, for the purpose of computing the long distance behavior of the correlations we are considering, to a fermionic theory defined as the formal scaling limit of the original one plus an ultraviolet regularization; more exactly, we prove that the critical indices x+ , x− , ν and x T of the spin model (1.1) are equal to the indices of a fermionic theory provided that the bare coupling λ∞ of the new theory is properly chosen as a suitable function of the parameters of the 8V or AT models. The new fermionic theory has correlations expressed by Grassmann integrals which are identical to the ones appearing in certain Quantum Field Theory models; in particular it verifies extra Gauge symmetries with respect to the original spin Hamiltonian (1.1).
582
G. Benfatto, P. Falco, V. Mastropietro
3.1. The model. The continuum (or QFT) model is defined as the limit N → ∞, followed by the limit −l → ∞, to be called the removed cutoff limit, of a model with an infrared γ l and an ultraviolet γ N momentum cut-off, −l, N 0. This model is expressed in terms of the following Grassmann integral: W N (A,J,η) [l,N ] (N ) [l,N ] = P(dψ ) exp V (ψ )+ e dx Aεx Oε,x +
ε
dx
[l,N ]+ [l,N ]− ψx,ω [Jx,ω ψx,ω
+[l,N ] − + ψx,ω ηx,ω
+ [l,N ]− + ηx,ω ψx,ω ]
, (3.1)
ω
, a square subset of R2 of size | | ≤ γ −2l , Ox+ and Ox− are defined in (2.36) where x ∈ and P(dψ [l,N ] ) is a Gaussian measure with propagator gT[l,N ] (x, y) given by (2.41) with N −1 µh = 0, Z h = 1 and C h−1 (k) replaced by Cl,N (k) = k=l f k (k); moreover ηx± are ε external fermionic fields and Ax , Jx,ω are external bosonic fields. The interaction is λ∞ − + + − V (N ) (ψ) = ψy,−ω ψx,ω ψy,−ω , (3.2) dx dyv K (x − y)ψx,ω 2 ω where K < N and v K (x − y) is given by v K (x − y) =
1 χ0 (γ −K p)eip(x−y) , L2 p
(3.3)
χ0 (p) being a smooth function with support in {|p| ≤ 2} and equal to 1 for {|p| ≤ 1}. The correlation functions are found by making suitable derivatives with respect to the external fields Ax , Jx , ηx and setting them equal to zero. We consider K fixed, for example K = 0, so that no ultraviolet regularization is needed, as we shall see, when we take the limit N → ∞. We shall study the functional W N (A, J, η) by performing a multiscale integration of (3.1); we have to distinguish two different regimes: the first regime, called ultraviolet, contains the scales h ∈ [K + 1, N ], while the second one contains the scales h ≤ K , and is called infrared. 3.2. The ultraviolet integration. We describe how to control the integration of the ultraviolet scales, in order to remove the ultraviolet cut-off N → ∞. For simplicity, we shall only consider the case A = η = 0, but the result is valid for the full problem (see also the remark at the end of this section). Assume that the fields ψ (N ) , ψ (N −1) , . . . , ψ (h+1) are integrated so that (h) eW N (0,J,0) = eS (J ) P(dψ [l,h] ) exp V (h) (ψ [l,h] ) + B (h) (ψ [l,h] , J ) , (3.4) where V (h) +B (h) is the sum of integrated monomials in m ψx+i ,ωi variables, i = 1, . . . , m, m ψy−i ,ωi variables and n Jz j ,ωj external fields, j = 1, . . . , n, multiplied by suitable
kernels Wω(n;2m)(h) (z; x, y). The scaling dimension is again (2.44), and, as in §2.3, we ;ω define a localization operator on the terms with positive or vanishing scaling dimensions which, as in the previous case, are the terms with (2m, n) = (2, 0) or (4, 0) or (2, 1).
Extended Scaling Relations
583
Notice however that in this case the localization operation is defined as the identity on the (0;2)(h) (1;2)(h) (0;4)(h) relevant or marginal terms, that is Wω , Wω ;ω and Wω,ω , while it annihilates, as always, all the other contributions to the effective potential. (n;2m)(h) These kernels Wω ;ω (z; x, y) are represented as power expansions in the running (0;2)(k)
(1;2)(k)
(0;4)(k)
coupling functions Wω , Wω ;ω and Wω,ω , k ≥ h, whose size is estimated by 1 the L norm, as well as the kernels themselves. Of course, since the kernels may contain delta functions, we extend, as usual, the definition of the L 1 norm, by treating the delta as a positive funcion. Hence, we define (n;2m)(k) (n;2m)(k) de f 1 (3.5) dzdxdy Wω ;ω = (z; x, y) Wω ;ω | | and we prove the following theorem. Theorem 3.1. If λ∞ is small enough, there exist two constants C1 > 1 and C2 , such that, if K ≤ h ≤ N , the relevant or marginal contributions to the effective potential satisfy the bounds: Wω(0;2)(h) ≤ C1 |λ∞ |γ h γ −2(h−K ) ,
(1;2)(h) Wω ;ω (0;4)(h) Wω,ω
− δ2 δω,ω ≤ C2 |λ∞ |γ
−(h−K )
(3.6) ,
− λ∞ vδ4 δω,−ω ≤ C2 |λ∞ |2 γ −(h−K ) ,
(3.7) (3.8)
where δ2 (z; x, y) ≡ δ(z − x)δ( z − y) and vδ4 (x1 , x2 , y1 , y2 ) ≡ δ(x1 − y1 )v K (x1 − x2 )δ(x2 − y2 ). Before proving the theorem, notice that, as for the multiscale analysis in §2.3, the fact that the running coupling functions are small for λ∞ small enough (as it follows by the bounds (3.6), (3.7), (3.8)) implies the following standard “dimensional” bound for all other kernels with negative scaling dimension, for λ∞ small enough, see e.g. App. A of [23]: (n;2m)(k)
Wω ;ω
≤ C n+dn,m |C1 λ∞ |dn,m γ k(2−n−m) ,
(3.9)
where dn,m = max{m − 1, 0}, if n > 0, and dn,m = max{m − 1, 1}, if n = 0, and C is (n;2m)(k) a suitable constant larger, at least, than γ . Indeed, in the tree expansion for Wω ;ω defined in [23], all the vertices of the tree have negative scaling dimension and there (0;2)(h) (1;2)(h) (0;4)(h) are three types of endpoints (see [23]), associated to Wω , Wω ;ω , Wω,ω , which contribute (up to dimensional factors and for λ∞ small enough) a factor C1 |λ∞ |, 1 + C2 |λ∞ | ≤ C and |λ∞ |[1 + C2 |λ∞ |] ≤ C1 |λ∞ |, respectively. Notice that the condition C > γ comes from the bound of the trivial tree (that with only one endpoint) (0;2)(k) contributing to the tree expansion of Wω . The bounds (3.6), (3.7), (3.8) follow from a “power counting improvement”, similar to the one used in [19] for the Yukawa model, in which the non-locality of the interaction plays an essential role. Proof of Theorem 3.1. The proof is by induction: we assume that the bounds (3.6)–(3.8) hold for h : k + 1 ≤ h ≤ N (for h = N they are true with C1 = C2 = 0) and we prove them for h = k.
584
G. Benfatto, P. Falco, V. Mastropietro
(0;2)(k)
(1;2)(k) and W−ω;ω , the [k+1,N ] dotted lines the external fermionic lines, the paired line is the fermionic propagator gω and the wiggly line is the interaction v K
Fig. 2. Graphical representation of (3.10); the gray blobs represent the kernels Wω
The inductive assumption implies the validity of (3.9) and we need to improve such bound when 2 − n − m ≥ 0. We can write, by using the properties of the fermionic trun(1;0) cated expectations and the fact that, by the oddness of the free propagator, Wω (k) = 0, Wω(0;2)(k) (x, y) (1;2)(k) = λ∞ dwdw v K (x − w)gω[k+1,N ] (x − w )W−ω;ω (w; w , y),
(3.10)
which can be bounded, by using (3.9), as (Fig. 2) (1;2)(k)
Wω(0;2)(k) ≤ |λ∞ |v K L ∞ W−ω;ω
N
gω( j) L 1
j=k+1
c1 ≤ γ 2K C|λ∞ |γ −k ≤ C1 |λ∞ |γ k γ −2(k−K ) , 1 − γ −1
(3.11)
c1 where, for example, C1 = max{2, 1−γ −1 C}; hence (3.6) is proved. Notice that the condition C1 ≥ 2 is introduced only because C1 is the same constant appearing in (3.9). (1;2)(k)
Let us now consider Wω ;ω (z; x, y) and notice that it can be decomposed as the sum of the five terms in Fig. 3. The term denoted by (a) in Fig. 3 can be bounded as (1;2)(k)
(2;2)(k)
W(a);ω ;ω ≤ |λ∞ |v K L ∞ Wω ,−ω;ω
N
gω( j) L 1 ≤ CC1 |λ∞ |γ −2(k−K ) . (3.12)
j=k+1 (0;2)(k)
The bounds for the graphs (c) and (d) are an easy consequence of the bound for Wω . In order to obtain an improved bound also for the graph (b) of Fig. 3, we need to (2;0)(k) further expand Wω,ω as done in Fig. 4, if we define the graph (b2) so that the vertex u can be either on the fermion line joining w with w (as in the figure) or on the other fermion line ending in w. The bound for the graph (b2) can be done by using the previous arguments. We can write (1;2)(k) 2 W(b2)ω dwdu dz v K (x − w)v K (u − z ) ;ω (z; x, y) = λ∞ δ(x − y) · dudw gω[k+1,N ] (w − u)gω[k+1,N ] (u − w)gω[k+1,N ] (w − u ) (2;2)(k)
· Wω ,ω;−ω (z, z ; w , u).
(3.13)
Extended Scaling Relations
585
(a)
(b)
(c)
(d)
(1;2)(k)
Fig. 3. Graphical representation of Wω ;ω (z; x, y); the external wiggly line represents the external field J , while the internal wiggly line is the interaction v K , as before
(b)
(b1)
(b2)
(b3)
Fig. 4. Graphical representation of the term (b) in Fig. 3
In order to get the right bound, it is convenient to decompose the three propagators gω into scales and then bound by the L ∞ norm the propagator of lowest scale, while the two others are used to control the integration over the inner space variables through their L 1 norm. Hence we get: (1;2)(k)
(2;2)(k)
W(b2)ω ;ω ≤ |λ∞ |2 v K L ∞ v K L 1 Wω ,−ω;ω · 3! gω( j) L 1 gω(i) L 1 gω(i ) L ∞ ≤ C3 |λ∞ |2 γ −2(k−K )
(3.14) (3.15)
k+1≤i ≤ j≤i≤N
for some constant C3 . The bound of (b1) and (b3) requires a new argument, based on a cancelation following from the particular form of the free propagator. Let us consider, for instance, (b1): (1;2)(k)
W(b1)ω ;ω (z; x, y) = λ∞ δω ,−ω δ(x − y)
[k+1,N ] dw v K (x − w) g−ω (w − z)
2
.
(3.16)
586
G. Benfatto, P. Falco, V. Mastropietro
Since the cutoff function Ck,N (k) is symmetric in the exchange between k0 and k1 , it is easy to see that gω[k,N ] (x0 , x1 ) = −iωgω[k,N ] (x1 , −x0 ); hence 2 [k+1,N ] (u) = 0. (3.17) du g−ω It follows, by using (3.17) and the identity
v K (x − w) = v K (x − z) +
(z j − w j )
ds ∂ j v K (x − z + s(z − w)),
1
(3.18)
0
j=0,1
that we can write (1;2)(k)
W(b1)ω ;ω (z; x, y) = λ∞ δω ,−ω δ(x − y) · 1 [k+1,N ] ds dw ∂ j v K (x − z + s(z − w)) (z j − w j ) g−ω (w − z) ·
2
.
j=0,1 0
(3.19) Hence, (1;2)(k) W(b1)ω ;ω ≤ 4|λ∞ |
·
N i
( j)
g−ω L ∞
dx (∂ j v K )(x)
(3.20)
i=k j=k (i)
dw |w j ||g−ω (w)| ≤ C4 |λ∞ |γ −(k−K ) .
(3.21)
By summing all the bounds, we see that there is a constant C2 such that (1;2)(k)
Wω ;ω
− δω,ω δ2 ≤ C2 |λ∞ |γ −(k−K ) ,
(3.22)
which proves (3.7). The bound (3.8) for W (0;4)(k) follows from similar arguments.
Remark. In the presence of the A fields the above analysis can be repeated, with the only difference being that in the analogue of Fig. (4) the (b1) and (b3) terms are missing.
3.3. Equivalence of the spin and the QFT models. As a consequence of the integration of the ultraviolet scales discussed in the previous section, we can write the removed cutoffs limit of (3.1), with η = 0 and with the choice K = 0, as (0) (≤0) (0) (≤0) lim lim (3.23) Pµ0 ,Z 0 (dψ (≤0) )eV (ψ )+B (ψ ,A,J ) , l→−∞ N →∞
where the propagator of the integration measure in (3.23) coincides with gT(≤0) (x, y), (0) defined in (2.41), LV (0) = λ0 Fλ and LB (0) , when J = 0, defined as in (2.37); from the analysis of the previous section it follows that λ0 is a smooth function of λ∞ , such that λ0 = λ∞ + O(λ2∞ ).
Extended Scaling Relations
587
The multiscale integration for the negative scales can be done exactly as described in §2.3, with the only difference being that, by the oddness of the free propagator, ν j = 0 and ( j)
(λ j , . . . λ0 ), λ j−1 = λ j + β λ
(3.24)
where, by (2.40) and the short memory property, ( j) (λ j , . . . λ0 ) = β ( j) (λ j , . . . λ0 ) + O(λ¯ 2j γ ϑ j ), β λ λ
(3.25)
( j)
βλ (λ j , . . . λ j ) being the function appearing in the bound (2.46), so that we can prove in the usual way that λ−∞ = λ0 + O(λ20 ); since λ0 = λ∞ + O(λ2∞ ), we have λ−∞ = h(λ∞ ) = λ∞ + O(λ2∞ ),
(3.26)
for some analytic function h(λ∞ ), invertible for λ∞ small enough. Moreover, by using (2.40), Z± j−1 Z± j
( j) ± =1+β (λ j , . . . , λ0 ),
(3.27)
with ( j) ( j) ± β (λ j , . . . , λ0 ) = β± (λ j , . . . λ0 ) + O(λ¯ 2j γ ϑ j ),
(3.28)
( j)
β± being the functions appearing in (2.50). This implies that η± = logγ [1 + β± (λ−∞ )],
(3.29)
that is the critical indices in the AT or 8V or in the model (3.1) are the same as functions of λ−∞ . Of course λ−∞ is a rather complex function of all the details of the models. However, if we call λj (λ) the effective couplings of the lattice model of the previous sections, the invertibility of h(λ∞ ) implies that we can choose λ∞ so that h(λ∞ ) = λ−∞ (λ).
(3.30)
With this choice of λ∞ (λ), the critical indices are the same, as they depend only on λ−∞ ; the rest of this chapter is devoted to the proof that the critical indices have, as functions of λ∞ , simple expressions, which imply the scaling relations in the main theorem. 4. Ward Identities and Schwinger-Dyson Equation In this section we prove that the Gauge symmetry of the equivalent QFT model implies closed equations for the correlations, from which an explicit expression of the correlations and the indices as a function of λ∞ can be derived; such expressions are so simple that the validity of the extended scaling relation can be checked. Such a simplicity follows from the fact that the Ward Identities for the equivalent QFT model, from which the closed equations are derived, verify a property called anomaly non-renormalization.
588
G. Benfatto, P. Falco, V. Mastropietro
4.1. Schwinger-Dyson equations and Ward Identities. The Schwinger-Dyson equations for the model (3.1) are generated by the identity, see [6], Dω (k)
∂eW N − W N (0,η) (0, η) = χl,N (k) ηk,ω e + ∂ ηk,ω ! ∂ 2 eW N dp − λ∞ v K (p) (0, η) , + (2π )2 ∂ Jp,−ω ∂ ηk+p,ω
(4.1)
where Dω (k) = −ik0 + ωk and we have shortened the notation of W N (0, J, η) into W N (J, η). ± → e±iαx,ω ψ ± , we obtain another On the other hand, by the change of variables ψx,ω x,ω identity: ∂W N ∂W N (0, η) − τ v K (p)D−ω (p) (0, η) ∂ Jp,ω ∂ Jp,−ω " ! ∂W N ∂W N dk ∂WA − + (0, η) ηk,ω + (0, 0, η), = η + (0, η) − − (2π )2 k+p,ω ∂ ηk,ω ∂ αp,ω ∂ ηk+p,ω
Dω (p)
where τ is a constant to be chosen later, # (N ) [l,N ] eWA (α,J,η) = P(dψ [l,N ] )eV (ψ )+ ω dx ·e
# ω
[l,N ]+ − + ψ [l,N ]− ] dx[ψx,ω ηx,ω +ηx,ω x,ω
[l,N ]+ [l,N ]− Jx,ω ψx,ω ψx,ω
e[A0 −τ A− ]
α,ψ [l,N ]
,
dq dp + − q,ω ψp,ω , A0 (α, ψ) = Cω (q, p) αq−p,ω ψ 4 (2π ) ω=± dq dp de f − + q,−ω p,−ω A− (α, ψ) = ψ D−ω (p − q) v K (p − q) αq−p,ω ψ , 4 (2π ) ω=± de f
(4.2)
(4.3) (4.4) (4.5)
−1 −1 (p) − 1]Dω (p) − [χl,N (q) − 1]Dω (q), (4.6) Cω (q, p) = [χl,N N and χl,N (k) = k=l f k (k). An explicit derivation of (4.2) can be found in §2.2 of [10]; (4.2) is obtained by introε (k) never vanishing for all values of k = 0 and equivalent ducing a cut-off function χl,N to χl,N (k) as far as the scaling properties of the theory are concerned; ε is a small paramε (k) = χ eter and limε→0+ χl,N l,N (k). This further regularization (to be removed before ε )−1 (k) − 1]χ ε (k) = taking the removed cutoffs limit) ensures that the identity [(χl,N l,N ε 1 − χl,N (k) is satisfied for all k = 0. When this further regularization is removed, all the quantities we shall study have a well defined expression and, by the change of variables ± → e±iαx,ω ψ ± , we get the Ward identity (WI): ψx,ω x,ω
∂W N (0, η) ∂ Jp,ω " ! ∂W N dk ∂W N ∂WA − + = (0, η) − (0, η) η (0, 0, η) η + . (4.7) k,ω + − τ =0 (2π )2 k+p,ω ∂ ηk,ω ∂ αp,ω ∂ ηk+p,ω
Dω (p)
Extended Scaling Relations
589
Equation (4.2) is obtained by just adding and subtracting the second term in the first line of (4.2). The last term in (4.7) is a correction to the formal WI, due to the presence of the ultraviolet cut-off function χl,N . The two equations obtained from (4.2) by putting ω = ±1 can be solved w.r.t. ∂eW N /∂ Jp,ω and, if we define 1 1 , a(p) ¯ = , 1−τ v K (p) 1+τ v K (p) a(p) + εa(p) ¯ , Aε (p) = 2
a(p) =
(4.8)
we obtain the identity Aωω (p) ∂eWA ∂eW N (0, η) − (0, 0, η) Dω (p) ∂ αp,ω ∂ Jp,ω ω " ! WN WN Aωω (p) dk ∂e ∂e − + = (0, η) ηk,ω . ηk+p,ω + (0, η) − − Dω (p) (2π )2 ∂ ηk,ω ∂ ηk+p,ω ω
(4.9)
By using (4.9) and (4.1), we easily get: ∂eW N − W N (0,η) (0, η) = χl,N (k) ηk,ω e + ∂ ηk,ω dp A−ωω (p) − λ∞ v (p) 2 K (2π ) D−ω (p) ω " ! ∂ 2 eW N ∂ 2 eW N dq − + (0, η) − + (0, η) ηq,ω · ηq+p,ω + + − (2π )2 ∂ ηq,ω ∂ ηk+p,ω ∂ ηk+p,ω ∂ ηq+p,ω dp ∂ 2 eWA A−ωω (p) − λ∞ v K (p) (0, 0, η) , (4.10) + (2π )2 D−ω (p) ∂ αp,ω ∂ ηk+p,ω
Dω (k)
ω
where we have used that, by simple parity arguments, λ∞
A−ωω (p) ∂eW N dp v (p) K + (0, η) = 0. (2π )2 D−ω (p) ∂ ηk,ω
(4.11)
4.2. Closed equation. If we make an arbitrary number of functional derivatives with respect to the η external fields in (4.10), then we set η = 0 and perform the Fourier transform, we obtain a set of differential equations. We will prove in the last section the following crucial result: Theorem 4.1. If λ∞ is small enough and we put τ=
λ∞ , 4π
(4.12)
590
G. Benfatto, P. Falco, V. Mastropietro
then the Fourier transforms of the correlation functions generated by setting η = 0 after deriving w.r.t. η the functional dp A−ωω (p) ∂ 2 eWA (0,0,η) v (p) , (4.13) K + (2π )2 D−ω (p) ∂ αp,ω ∂ ηk+p,ω ω
vanish at distinct points in the removed cutoff limit (defined at the beginning of §3.1). This theorem will be proved in §4.3. We want now to show how to use it to prove the identity (1.10), so completing the proof of Theorem 1.1. By using Theorem 4.1 and some regularity property of the Schwinger functions (for details, see §A.1 in [6]) we get, in the removed cutoff limit, a set of closed equation for the Schwinger functions. In particular, if we define de f
− + ψx,ω ψy,ω = Sω (x − y), de f
− − + + ψy,−ω ψu,−ω ψv,ω = G ω (x, y, u, v), ψx,ω
(4.14) (4.15)
we get
where ∂ω = ∂x0
(∂ω Sω ) (x) − λ∞ FK ,− (x)Sω (x) = δ(x), # K ,− (−p), with + iω∂x1 and FK ,− (x) = dp/(2π )2 e−ipx F
(4.16)
f v K (p)Aε (p) K ,ε (p) de . F = D−ω (p)
(4.17)
Sω (x) = eλ∞ − (x,0) gω (x),
(4.18)
The solution of (4.16) is:
having defined ε (x, z) =
dk e−ikx − e−ikz FK ,ε (−k). (2π )2 Dω (k)
(4.19)
Notice that, for large |x|, thanks to (4.8), ε (x, 0) ∼ −
a(0) + εa(0) ¯ Aε (0) ln |x| = − ln |x|, 2π 4π
(4.20)
which implies, in particular, that the critical index ηz , defined in (2.51) is given by ηz =
2τ 2 λ∞ [a(0) − a(0)] ¯ = . 4π 1 − τ2
(4.21)
− − + Moreover, if we take in (4.10) three derivatives w.r.t. ηq,−ω , ηk+q−s,ω and ηs,−ω , we find, after Fourier transforming and in the cutoffs limit, x ∂ω G ω (x, y, u, v) = δ(x − v)S−ω (y − u)
(4.22) + λ∞ FK ,+ (x − y) − FK ,+ (x − u) − FK ,− (x − v) G ω (x, y, u, v),
Extended Scaling Relations
591
which is solved, by using (4.18), by G ω (x, y, u, v) = e−λ∞ [+ (x−y,v−y)−+ (x−u,v−u)] · Sω (x − v)S−ω (y − u).
(4.23)
The r.h.s. of (4.23) is well defined for x = u and y = v, if x = y, or x = y and + ψ− u = v, if x = u. This is a consequence of the fact that the operators ψx,ω x,−ω and + + ψx,ω ψx,−ω are well defined even in the limit N → ∞, thanks to the non locality of the
∂ WN N interaction. Hence, one expects that ∂∂A+W + and ∂ A+ ∂ A+ can be calculated by simply x ∂ Ay x y using Eqs. (4.23), (4.18). A rigorous proof of this statement could be done by a simple (1) extension of Lemma 4.1 of [6] (with Z¯ N = c1 = 1 + O(λ∞ )), where a similar (more difficult) problem is considered. If we put (4.23) x = u and y = v, we find, using also (4.15) and (4.20), that 2
2
− − + + − + + − ψx,ω ψx,−ω ψy,−ω ψy,ω = ψx,ω ψx,−ω ψy,−ω ψy,ω 0 e−2λ∞ [+ (x−y,0)−− (x−y,0)]
∼
|x−y|→∞
C |x
¯ ∞ /2π )] − y|2[1−a(0)(λ
.
(4.24)
If we put instead x = y and u = v, we get − − + + − + + − ψx,ω ψx,−ω ψu,−ω ψu,ω = ψx,ω ψx,−ω ψu,−ω ψu,ω 0 e2λ∞ [+ (x−u,0)+− (x−u,0)]
∼
|x−u|→∞
C . |x − u|2[1+a(0)(λ∞ /2π )]
(4.25)
Let us now choose λ∞ , so that (3.30) is satisfied. Then, by using (2.36), (4.8), (4.12) and the definition (1.8) of x± we get the identities x+ =
1−τ 1+τ , x− = , 1+τ 1−τ
(4.26)
which imply the identity (1.10). Remark. The proof of the relation x− x+ = 1 follows from two main ingredients, namely the linearity of τ as a function of λ∞ , see (4.12), and the vanishing of the last term in (4.10). The validity of such properties is due to our choice of the equivalent continuum model; it is indeed known, as proved in [5], that in other QFT models, still equivalent to the spin model, such properties are not true so that they do not allow to derive the relation x− x+ = 1. The linearity of τ as a function of λ∞ corresponds to a property called in the physical literature non renormalization of the anomaly or Adler-Bardeen theorem, see [23–25]. 4.3. Proof of Theorem 4.1. We start with the multiscale integration of the Grassmann integral WA (α, 0, η) (4.3) appearing on the WI (4.2). Notice # that W+A (α,−0, η) (4.3) is very similar to W N (J, η), see (3.1), the difference being that Jx,ω ψx,ω ψx,ω is replaced by A0 − τ A− . A crucial role in the analysis is played by the function Cω (p, q) appearing in the definition of A0 ; this function is very singular, but it appears in the various equations relating the correlation functions only through the regular function f ω(i, j) (q + p, q) de = χ N (p)Cω (q + p, q) gω(i) (q + p) gω( j) (q), U
(4.27)
592
G. Benfatto, P. Falco, V. Mastropietro
where χ N (p) is a smooth function, with support in the set {|p| ≤ 3γ N +1 } and equal to 1 in the set {|p| ≤ 2γ N +1 }; we can add freely this factor in the definition, since ω(i, j) (q + p, q) will only be used for values of p such that χ N (p) = 1, thanks to the U (i, j) support properties of the propagator. It is easy to see that Uω vanishes if neither j nor i equals N or l; this has the effect that at least one of the fields in A0 has to be integrated at the N or l scale. As a matter of fact, the terms in which at least one field is integrated at scale l are much easier to analyze, see the considerations after (4.61) below. In order to study the (i, j) others, it is convenient to introduce the function Sω,ω such that ¯ ω(i, j) (q + p, q) = U
ω¯
(i, j) Dω¯ (p) Sω,ω (q + p, q). ¯
One can show that, if we define dp dq −ip(x−z) iq(y−z)(i, j) (i, j) Sω,ω (z; x, y) = e e (p, q), Sω,ω ¯ ¯ (2π )4
(4.28)
(4.29)
then, given any positive integer M, there exists a constant C M such that, if j > l, (N , j)
|Sω,ω (z; x, y)| ≤ C M ¯
γN γj , 1 + [γ N |x − z|] M 1 + [γ j |y − z|] M
(4.30)
a bound which is used to control the renormalization of the marginal terms containing a vertex of type A0 . We choose τ as given by τ = λ∞
N i, j=l+1
dq (i, j) (q, q) ; S (2π )2 −ω,ω
(4.31)
by an explicit calculation one can see that, for any l < 0 and N > 0, τ satisfies (4.12). We remark that, to get this result, it is important to exclude from the sum in the r.h.s. of (4.31) the couples (i, j) with one of the indices equal to l; without this restriction, τ would be equal to 0, for any N > 0. We will proceed as in the analysis of W N (J, η), by integrating first the ultraviolet scales N , N − 1, . . . , h + 1, h ≥ K , and following a procedure very similar to the one described in §3.2; we have new marginal terms with one α field and two ψ fields and we have to prove the analogue of (3.7) for them. The marginal terms such that only one of these two fields is contracted are proportional to W (0;2)(k) , so that one can use (3.6) to bound them. Hence, we shall consider in detail only the terms such that both fields of (n;2m)(k) A0 or A1 are contracted and we shall call K the corresponding kernels of the ;ω;ω monomials with 2m ψ-fields and n α-fields. In the case n = 1, we decompose them as follows: (1;2m)(k) (p; k), (1;2m)(k) Dσ ω (p)W (4.32) K ;ω;ω (p; k) = ;σ,ω;ω σ
where p is the momentum flowing along the external α-field. As in §3.2, we have to (1;2)(k) improve the dimensional bound of W;σ,ω;ω . We can write the following identity, which
Extended Scaling Relations
593
(a)
(b)
(c)
(d)
(e)
(1;2)(k)
Fig. 5. Graphical representation of W;−1,ω;ω ; the triangular vertex represents Cω (q, p) given by (4.5)
is represented by the first line of Fig. 5 in the case σ = −1: N (1;2)(k) (0;4)(k) j) W;σ,ω;ω (z; x, y) = (z; u, w)Wω,ω (u, w, x, y) dudw Sσ(i,ω,ω i, j=k
−τ δ−1,σ
(1;2)(k)
dw v K (z − w)W−ω;ω (w; x, y).
(4.33)
(1;2)(k)
We can further decompose W;−1,ω;ω as in the last three lines of Fig. 5. The term (c) can be written as N (i, j) λ∞ dudu dwdw S−ω,ω (z; u, w)gω[k,N ] (u − u )v K (u − w ) i, j=k (1;4)(k)
· W−ω;ω,ω (w ; u , w, x, y).
(4.34) (i, j)
de f
Hence, if we put b j (x) = γ j /(1 + [γ j |x|]3 ), we recall that S−ω,ω is different from 0 only if either i or j is equal to N , and we use the bound (4.30), we see that the norm of (c) is bounded by N∗ (1;4)(k) C3 |λ∞ |v K L ∞ dxdu dwdw |W−ω;ω,ω (w ; u , w, x, y)| ·
i, j,m=k
dzdu bi (z − w)b j (z − u)|gω(m) (u − u )|,
(4.35)
594
G. Benfatto, P. Falco, V. Mastropietro
where ∗ reminds us that max{i, j} = N . Since the L 1 and the L ∞ norm of b j satisfy a ( j) bound similar to analogous bounds of gω , we can proceed as in the previous section to # bound dzdu bi (z − w)b j (z − u)|gω(m) (u − u )|, by taking the L ∞ norm for the factor with the smaller index and the L 1 norm for the other two. By also using (3.9), we get the bound Cϑ |λ∞ |2 γ −2(k−K ) γ −ϑ(N −k) ,
(4.36)
for any 0 < ϑ < 1 (Cϑ is divergent for ϑ → 1). With respect to the analogous bound in §3.2 ((b2) in Fig.4), there is an improvement of a factor γ −ϑ(N −k) . The term (d) can be bounded by N ∗
C|λ∞ |v K L ∞
bi L 1 b j L 1 ≤ C|λ∞ |γ −(k−K ) γ −(N −k) ;
i, j=k
for the term (e) we get the bound C|λ∞ |2 γ −3(k−K ) γ −(N −k) . By putting together all the previous bounds, we get (c) + (d) + (e) ≤ Cϑ |λ∞ |γ −(k−K ) γ −ϑ(N −k) .
(4.37)
We consider now the terms (a) and (b), whose sum can be written as ⎡ ⎤ N (i, j) du ⎣λ∞ S−ω,ω (z; u, u) − τ δ(z − u)⎦ ·
i, j=k (1;2)(k) dw v K (u − w)W−ω;ω (w; x, y).
(4.38)
By using the identity (3.18), (4.38) can be written also as ⎤ ⎡ N (i, j) (1;2),(k) ⎣λ∞ (w; x, y) du S−ω,ω (z; u, u) − τ ⎦ dw v K (z − w)W−ω;ω i, j=k
+ λ∞
N
(i, j)
du S−ω,ω (z; u, u)(u p − z p )
p=0,1 i, j=k
1
·
ds 0
(1;2),(k) dw (∂ p v K )(z − w + s(u − z))W−ω;ω (w; x, y).
(4.39)
The latter term is again irrelevant and vanishing for N − k → +∞; in fact, its norm can be bounded by (1;2),(k)
2|λ∞ |W−ω;ω ∂v K L 1
N ∗
dz bi (z − u)b j (z − u)|u − z p |
i, j=k
≤ C|λ∞ |γ −(k−K ) γ −(N −k) .
(4.40)
Contrary to what happened for the graph (b1) of Fig. 4, the contribution of the graph (a) to the first term in the r.h. side of (4.39) is not zero (that is, the fermionic bubble is not
Extended Scaling Relations
595
vanishing); however, in this case its value is compensated by the graph (b), thanks to the explicit choice we made for τ . Indeed we have λ∞
N
(i, j)
du S−ω,ω (z; u, u) − τ = −2λ∞
i, j=k
k−1
(N , j)
du S−ω,ω (z; u, u), (4.41)
j=l+1
that easily implies that the norm of the first term in the r.h. side of (4.39) is bounded by C|λ∞ |γ −(N −k) . (1;2)(k) Let us finally consider W;+1,ω;ω , for which we can use a graph expansion similar to that of Fig. 5, the only differences being that τ is replaced by 0 and the indices −ω are replaced by ω. Hence a bound can be obtained with the same arguments used above, with only one important difference: the contribution that in the previous analysis was compensated by the graph (b) now is zero by symmetry reasons. Indeed, if we call k∗ the (i, j) ∗ ∗ (i, j) (k , p ) = −ωω¯ Sω,ω (k, p), which vector k rotated by π/2, it is easy to see that Sω,ω ¯ ¯ implies that N
du
(i, j) (z; u, u) Sω,ω
=
i, j=k
N i, j=k
dk (i, j) (k, −k) = 0. S (2π )2 ω,ω
(4.42)
We have then proved that (1;2)(k)
W;σ,ω;ω ≤ C|λ∞ |γ −ϑ(N −k) ,
(4.43)
which implies, as in the proof of Theorem 3.1, that, for K ≤ k ≤ N , (1;2m)(k) m (1−m)k −ϑ(N −k) γ . W;σ,ω;ω ≤ (C|λ∞ |) γ
(4.44)
With respect to the bounds appearing in Theorem 3.1, there is an extra factor γ −ϑ(N −k) , implying that such kernels vanish at fixed k in the N → ∞ limit. However, this is not sufficient to prove Theorem 4.1, as in (4.13) the derivatives of WA (α, 0, η) with respect to the external fields are integrated over p. It is convenient to write (4.13) as ω
∂WT,ε A−ωω (p) ∂ 2 eWA (0,0,η) dp v K (p) = (0, η), + 2 k,ω (2π ) D−ω (p) ∂ αp,ω ∂ ηk+p,ω ∂β ε=±
(4.45)
where eWT,ε (β,η) =
P(dψ [l,N ] )eV
·e
(ε)
(ε)
T1 −τ T−
(N ) (ψ [l,N ] )+
ψ l,N ,β
ω
#
[l,N ]+ − + ψ [l,N ]− ] dx[ψx,ω ηx,ω +ηx,ω x,ω
(4.46)
596
G. Benfatto, P. Falco, V. Mastropietro
and (ε)
T1 (ψ, β) =
dk dp dq (ε) C−εω (q + p, q) v K (p) (2π )4 D−ω (p) ω k,ω ψ − ψ + − ·β k+p,ω q+p,−εω ψq,−εω ,
(ε)
T− (ψ, β) =
dk dp dq (ε) − ψ + − k,ω ψ u K (p)β k+p,ω q+p,εω ψq,εω , 4 (2π ) ω
(4.47)
(4.48)
where Dεω (p) de f (ε) (ε) (ε) ε (p), . u K (p) = v K (p) v K (p) v K (p) = v K (p) A D−ω (p) (±)
(4.49)
(−)
Notice that v K (x) and u K (x) are smooth functions of fast decay, hence they are equiv(+) alent to v K (x) in the bounds. This is not true for u K (x), whose Fourier transform is bounded but discontinuous in p = 0. However, in the following we shall only need to (+) (+) (+) know that u K L ∞ ≤ Cγ 2K and that | u K (p)| ≤ | v K (p) v K (p)|, which are easy to prove. As in §4.1, we now perform a multiscale integration for the ultraviolet scales N , N − 1, . . . , k + 1, k ≥ K , very similar to the one described in §3.2, the main difference being that there appear in the effective potential new monomials in the external field β and in ψ. Again, as in Theorem 3.1, one has to produce an improved bound only on the terms with positive or vanishing dimension, so that one has to analyze the kernels of the monomials with a β field and one or three ψ fields. We consider first the terms contributing to WT,ε (β, η), in which at least one of the (ε) two ψ-fields in T1 (ψ, β) with momentum q + p or q is contracted at scale N . We shall (1;2m−1) call WT,ε;ω;ω the corresponding kernels of the monomials with 2m − 1 ψ-fields and 1 α-field. We can write (1;1)(k) (1;1)(k) (1;1)(k) = W(a)T,ε;ω,ω + W(b)T,ε;ω,ω , WT,ε;ω,ω
(4.50)
where (1;1)(k) (ε) a) W(a)T,ε;ω,ω is the sum over the terms such that the field β belongs only to a T1 -ver+ q+p,−εω − tex, whose ψ-field ψ either is contracted with ψ k+p,ω (this can happen only for (0;2)(k) ω ε = −1) or is connected to it through a kernel W (q + p). (1;1)(k)
b) W(b)T,ε;ω,ω is the sum over the remaining terms. −1 (k) − 1 = 0; hence Let us consider the first term. Given k, for N large enough, χl,N we can write:
v (−1) dp (1;1)(k) K (p) W(a)T,ε;ω,ω (k) = δε,−1 [χ−∞,N (p + k) − 1] (2π )2 D−ω (p) ω(0;2)(k) (p + k) 1 + ω(0;2)(k) (k) . · 1 + gω[k+1,N ] (p + k)W gω[k+1,N ] (k)W
(4.51)
Extended Scaling Relations
597
(1;1)(k)
Fig. 6. Graphical representation of W(b)T,ε;ω,ω ; the dotted line coming from x represents the external field β (−1)
(−1)
Moreover, since v K (p) = 0 for |p| ≥ 2γ K , then χ−∞,N (p+k)−1 = 0, if vK and N is large enough. It follows that, given a fixed k, for N large enough,
(p) = 0
(1;1)(k) (k) = 0. W (a)T,ε;ω,ω
(4.52)
(1;1)(k)
Let us now consider W(b)T,ε;ω,ω (x − y), which can be decomposed as in Fig. 6. By using (4.33), it can be written as (ε) (1;2)(k) dz u K (x − z)gω[k,N ] (x − w)W;σ,−εω;ω (z; y, w), (4.53) σ
hence its norm, by using (4.43), can be bounded by ∞ u (ε) K L
N
(1;2)(k) |gω( j) | L 1 W;σ,−εω;ω ≤ C|λ∞ |γ k γ −2(k−K ) γ −ϑ(N −k) ,
(4.54)
j=k
so that (1;1)(k)
WT,ε;ω,ω ≤ C|λ∞ |γ k γ −ϑ(N −k) γ −2(k−K ) .
(4.55)
(1;3)(k) (1;3)(k) (1;3)(k) WT,ε;ω;ω = W(a)T,ε;ω;ω + W(b)T,ε;ω;ω ,
(4.56)
Moreover
(1;3)(k) k+p,ω of T1 and T− is not where W(a)T,ε;ω;ω contains the terms in which the field ψ (0;2)(k) ω contracted or is linked to a kernel W , while the other terms are collected in (1;3)(k) W(b)T,ε;ω;ω . (1;3)(k)
Let us consider first W(b)T,ε;ω;ω , which can be represented as in Fig. 7. We can write (1;3)(k)
W(b)T,ε;ω,ω (x, y, u, v) (ε) (1;4)(k) = dzdw u K (x − z)gω[k,N ] (x − w)W,ε;ω,ω (z; w, y, u, v), (1;4)(k)
(4.57) (ε)
so that, by the bounds (4.43), W,ε;ω,ω ≤ C|λ∞ |γ −k γ −ϑ(N −k) and u K L ∞ ≤ Cγ 2K , we get: (1;3)(k)
W(b)T,ε;ω;ω ≤ C|λ∞ |γ −2(k−K ) γ −ϑ(N −k) .
(4.58)
598
G. Benfatto, P. Falco, V. Mastropietro
(b1)
(b2)
(b3)
(b4)
(1;3)(k)
Fig. 7. Graphical representation of W(b)T,ε;ω;ω (1;3)(k)
Let us now consider W(a)T,ε;ω;ω ; its Fourier transform, if we call k+ and k− the momenta
of the two fields connected to the line u (ε) K , can be written as (notice that ω is of the form (ω, ω , ω )): (1;3)(k) (k; k+ , k− ) = 1 + ω(0;2)(k) (k + k+ − k− ) W gω[k+1,N ] (k + k+ − k− )W (a)T,ε;ω;ω (1;2)(k) (ε) − + − − W · u (k+ − k− ) (k + k − k , k ). K
σ
;σ,−εω,ω
(4.59) (−1)
Then, if ε = −1, since v K
L 1 ≤ C, by using the bounds (4.43) and (3.6), we find
(1;3)(k) −ϑ(N −k) W(a)T,−1;ω;ω . ≤ C|λ∞ |γ
(4.60)
Using the general bound (2.58) of [5], we get that the contributions to the derivatives of WT,ε with respect to η at distinct space points, coming from trees containing one (1;1)(k) (1;3)(k) (1;3)(k) endpoint associated with one of the kernels WT,−1;ω;ω , W(a)T,−1;ω;ω , W(a)T,−1;ω;ω , −N
are bounded by C k k!4 λk∞ δ −2k ( γ δ )ϑ , with 0 < ϑ < 1 and δ the minimal distance between the external points; hence they are vanishing in the removed cutoff limit. A similar conclusion is true for the contributions to the derivatives of WT,ε with respect to η at distinct points, coming from trees containing one endpoint associated (1;3)(k) (+) | ≤ γ −2l , it is easy to with W(a)T,+;ω;ω , even if u K (x) is not integrable. In fact, since | # (+1) −l show that dx|v K (x)| ≤ Cγ , so that the previous bound has to be multiplied by −l γ ; however, we take the limit −l → ∞ after the limit N → ∞, hence the conclusion is the same. Finally we have to consider the contributions to the correlation functions such that one of the ψ-fields in T1(ε) (ψ, β) with momentum q + p or q, see (4.47), is contracted at scale l. In such a case we can use the bound (i,l) U γ −l−i ω (q + p, q) , if |p| ≥ 2γ l+1 , (4.61) ≤ Cγ −(i−l) Dω (p) Z i−1
Extended Scaling Relations
599
and the factor γ −(i−l) in the r.h.s. of this bound makes negative the scaling dimension of marginal terms with an external β line (there are no relevant terms), see §4.8 of [11] for details. Again by the bound (2.58) of [5] we get, for this kind of contributions to the derivatives of WT,ε with respect to η at distinct points, the bound C k k!4 λk∞ δ −2k (γ l /δ)ϑ , with 0 < ϑ < 1, so they are vanishing as l → −∞. This completes the proof of Theorem 4.1. Acknowledgements. P.F. is indebted to David Brydges for stimulating his interest in the topic with the request of a review seminar on the papers [29] and [22].
A. Fermionic Representation of the Partition Function A.1. Proof of (2.6). Since σx , σx = ±1, % $ = cosh(α) + σx σx+e j σy σy+e exp ασx σx+e j σy σy+e sinh(α), j j so that the partition function for the AT or 8V model with external fields is given by: Z (I, I ) = [cosh(β J4 )]2|| "
· 1 + tanh(β J4 ) j=0,1 x∈
∂2 j,x A ∂A j,x
! Z (I )Z (I ),
(A.1)
j,x = A j,x and A = A , while, in where I j,x = Aj,x + β J and, in the AT case, A j,x j,x 0,x = A0,x , A = A , A 1,x = A1,x+e0 , A = A the 8V case, A . 0,x 1,x 1,x 0,x+e1 Let us call t j,x , c j,x the expressions obtained from t j,x , c j,x by substituting A j,x with j,x ; in a similar way we define t j,x , cj,x . Let us now define: A t j,x f j,x = 1 + tanh(β J4 ) t j,x , g j,x = h j,x =
t j,x tanh(β J4 ) t j,x tanh(β J4 ) , g j,x = 2 , ( c j,x )2 f j,x ( c j,x ) f j,x 1 ( cj,x )2 ( c j,x )2
(A.2)
tanh(β J4 ) − g j,x g j,x . f j,x
we can write the partition function j,x and A By explicitly taking the derivatives w.r.t. A j,x (A.1) as ⎛ ⎞
Z (I, I ) = 4|| [cosh(β J4 )]2|| ⎝ f j,x c j,x cj,x ⎠ j,x
·
(−1) γ ,γ
δγ +δγ
4
Z γ ,γ (I, I ),
where Z γ ,γ (I, I ) is the Grassmannian functional integral Z γ ,γ (I, I ) = d H d V d H d V e S(t+g)+ S (t +g )+V (h) ,
(A.3)
(A.4)
600
G. Benfatto, P. Falco, V. Mastropietro
with boundary conditions γ = (ε0 , ε1 ) and γ = (ε0 , ε1 ) on the variables H , V and H , V , respectively. Moreover S(t) and S (t) have a definition which depends on the model. S(t) is equal to S(t) in the AT model, while, in the 8V model, it is the function which is obtained from S(t), by substituting, in the first line of (2.4), V¯x Vx+e1 with V¯x+e0 Vx+e0 +e1 . S (t), in the AT case, is obtained from S(t), by simply replacing H, V with H , V , while, in the 8V case, we also have to substitute H¯ x Hx+e with V¯x Vx+e and V¯x Vx+e 0 1 1 with H¯ x+e1 Hx+e1 +e0 . V (h) is a quartic interaction that, in the AT case, is given by V AT (h) =
+ h 1,x V¯x Vx+e1 V¯x Vx+e h 0,x H¯ x Hx+e0 H¯ x Hx+e , 0 1
(A.5)
x∈
while, in the 8V case, is given by V8V (h) =
+ h 1,x V¯x+e0 Vx+e0 +e1 H¯ x+e H h 0,x H¯ x Hx+e0 V¯x Vx+e . (A.6) 1 1 x+e1 +e0
x∈
We remark that g j,x , g j,x , h j,x = O(β J4 ). The truncated correlations of the quadratic observables are obtained by taking two derivatives of ln Z (I, I ) w.r.t. the external sources in two different points, and putting such external sources to zero. The addends 2|| ln[2 cosh(β J4 )] and j,x (ln f j,x + ln c j,x + ln cj,x ) do not contribute when we take two derivatives in the A variables of two different points. If we define ∂ εj,x = ∂/∂ A j,x + ε∂/∂ Aj,x , we get: T Oxε ; Oyε =
i, j
⎤ δγ +δγ ε ε ⎣ ⎦ ∂i,x ∂ j,y ln (−1) Z γ ,γ (I, I ) γ ,γ ⎡
,
(A.7)
A≡0
so that we get, by some simple algebraic calculations: ∂ 2 Z¯ γ ,γ (A) 1 (−1)δγ +δγ Z γ ,γ ∂ A¯ εx ∂ A¯ εy A=0 ¯ ¯ 1 δγ +δγ ∂ Z γ ,γ (A) − (−1) 2 ε ( Z) ∂ A¯ x ¯
T Oxε ; Oyε =
γ ,γ
(−1)δγ +δγ
A=0 γ ,γ
∂ Z¯ γ ,γ (A) ∂ A¯ εy ¯
,
A=0
(A.8) with Z γ ,γ defined as in (2.7), with (γ , γ )-boundary conditions (instead of antiperiodic in all variables) and Z = γ ,γ (−1)δγ +δγ Z¯ γ ,γ (0, 0); the parameters s, s and λ are given by s = t j,x + g j,x A≡0 = tanh(β J ) + O(β J4 ), s = t j,x + g j,x = tanh(β J ) + O(β J4 ), A≡0 2λ = h j,x A≡0 = O(β J4 ),
(A.9)
Extended Scaling Relations
601
and the parameters appearing in (2.10) and (2.11) are given by & ' ∂ ∂ +ε , qε = { t, g → t , g }, qε = ( ti,x + gi,x ) ∂ A j,x ∂ A j,x i A≡0 & ' ∂h i,x ∂h j,x +ε . (A.10) pε = ∂ A j,x ∂ A j,x i
A≡0
In order to prove (2.6) we note that, as proved in App. G of [22], if we put Z¯ γ = Z γ | A=0 , the quantities Z¯ γ ,γ (0)/ Z¯ γ Z¯ γ are exponentially insensitive to boundary conditions in the thermodynamic limit, away from the critical temperature; this implies that Z coincides, in the thermodynamic limit, with ( Z¯ γ¯ ,γ¯ (0)/ Z¯ γ¯ Z¯ γ¯ )( Z¯ )2 with γ¯ = (−, −) and Z¯ = γ (−1)δγ Z¯ γ . Notice that Z¯ is non vanishing; indeed, as proved in §4 of [26], away from the critical temperature | Z¯ γ | is exponentially insensitive to boundary conditions and below the critical temperature Z γ is positive for any γ while above is negative if γ = (+, +) and positive in all other cases. Moreover, as proved in App. G of [22], ∂ Z¯ γ ,γ (A) ∂ 2 Z¯ γ ,γ (A) 1 and Z¯ γ ,γ (0) ∂ A¯ εx A=0 Z¯ γ ,γ (0) ∂ 2 A¯ εx ∂ A¯ εy A=0 ¯ ¯ 1
(A.11)
are exponentially insensitive to boundary conditions, so that the r.h.s. of (A.8) coincides, in the thermodynamic limit, with
¯ ∂ 2 log Z¯ γ¯ ,γ¯ ( A) | A=0 ¯ . ∂ 2 A¯ εx ∂ A¯ εy
A.2. Proof of (2.12) . In order to make more evident the analogy of the above functional integral with the action of a fermionic (Euclidean) Quantum Field Model, it is convenient to make a change of variables in the Grassmann algebra. The new Grassmannian variables will be denoted by ψx , ψ¯ x , χx and χ¯ x and are related to the old ones by the equations: π
H¯ x + i Hx = ei 4 (ψx − χx ) , V¯x + i Vx = ψx + χx , π H¯ x − i Hx = e−i 4 ψ¯ x − χ¯ x , V¯x − i Vx = ψ¯ x + χ¯ x .
(A.12)
A similar transformation is done for the primed variables. After a straightforward computation, we see that the action (2.4), calculated at t j,x = s, ∀ j, x, can be written in terms of the Majorana fields as S(s) = A(ψ, m s ) + A(χ , Ms ) + Q(ψ, χ ), where m s = 1 −
√
2 + s, Ms = 1 +
A(ψ, m) =
√ 2 + s and, if we define ∂ i ψx = ψx+ei − ψx ,
% s $ 0 ψ¯ x ψx ψx ∂ − i∂ 1 ψx + c.c. − im 4 x∈ x∈ % s $ 0 1 + ψ¯ x −i∂ − i∂ ψx + c.c. , 4 x∈
(A.13)
(A.14)
602
G. Benfatto, P. Falco, V. Mastropietro
% s $ 0 ψx ∂ + i∂ 1 χx + {ψ ↔ χ } + c.c. 4 x∈ % s $ χ¯ x −i∂ 0 + i∂ 1 ψx + {ψ ↔ χ } + c.c. , − 4
Q(ψ, χ ) = −
(A.15)
x∈
where, in agreement with (A.12), we are calling complex conjugation (c.c.) the operation on the Grassmann algebra which amounts to exchange ψx with ψ¯ x , χx with χ¯ x and i with −i. The quartic interaction of the AT model becomes:
ψ¯ x ψx ψ¯ x ψx + ψ¯ x ψx χ¯ x χx + {ψ ↔ χ } V AT = −λ x∈
−λ
χ¯ x ψx χ¯ x ψx + χ¯ x ψx ψ¯ x χx + {ψ ↔ χ } + R V ,
(A.16)
x∈
where R V is the sum of quartic terms with at least one (discrete) derivative. In the case of the 8V model, the second square bracket has +λ in front, rather than −λ. If we set bε = (qε + εqε )/2 and dε = (qε − εqε )/2, the interaction with the external field is given by
¯ = −i bε A¯ εx ψ¯ x ψx + εψ¯ x ψx + χ¯ x χx + εχ¯ x χx B( A) x∈ ε=±
−i
dε A¯ εx ψ¯ x ψx − εψ¯ x ψx + χ¯ x χx − εχ¯ x χx + R B ,
x∈ ε=±
where R B is the sum of monomials quartic in the fields or quadratic with derivatives. We remark that, if J = J , then dε = 0, while bε = 1 − tanh(β J ) + O(β J4 ). We now make another change of variables, defined by the relations ε ψx,+ =
ψ¯ x − εi ψ¯ x ψx − εiψx ε = , ψx,− , ε = ±, √ √ 2 2
(A.17)
and the similar ones for the χ -variables. This change of variables is analogous in the euclidean theories of the transformation from Majorana fermions to Dirac fermions in real time QFT. If we put u = (s+s )/2, v = (s−s )/2 (s, s defined in (A.9)) and m ε = (m s +εm s )/2 (m s and m s defined after (A.13)), we get A(ψ, m s ) + A(ψ , m s ) $ % $ % u + − − + ψx,+ ∂ 0 − i∂ 1 ψx,+ ∂ 0 − i∂ 1 ψx,+ + ψx,+ + c.c. = 4 x∈ % $ % u − $ 0 − + + i∂ + i∂ 1 ψx,− i∂ 0 + i∂ 1 ψx,− + ψx,+ + c.c. + ψx,+ 4 $ % $ % v + + − − ∂ 0 − i∂ 1 ψx,+ ∂ 0 − i∂ 1 ψx,+ + ψx,+ + c.c. + ψx,+ 4 $ % $ % v − − + + i∂ 0 + i∂ 1 ψx,− i∂ 0 + i∂ 1 ψx,− + ψx,+ + c.c. + ψx,+ 4
( − − + − + − + + −im + ψx,− + im − ψx,+ , (A.18) ψx,+ − ψx,+ ψx,− ψx,− + ψx,+ ψx,−
Extended Scaling Relations
603
ε with ψ −ε and i with −i. In where now the c.c. operation amounts to exchange ψx,ω x,−ω the new variables the interaction with the external source is given by − + + − ¯ =i B( A) (b+ A¯ +x + d− A¯ − x )[ψx,+ ψx,− − ψx,− ψx,+ x∈ − + + − + χx,+ χx,− − χx,− χx,+ ]+i
¯+ (b− A¯ − x + d+ Ax )
x∈ − + + − + + − − ¯ · [ψx,+ ψx,− + ψx,+ ψx,− + χx,+ χx,− + χx,+ χx,− ] + R B ( A),
(A.19)
and V(ψ, χ ) is given by (2.19). 1 2π 1 Let D be the set of k’s such that k0 = 2π L (n 0 + 2 ) and k1 = L (n 1 + 2 ), for L L n 0 , n 1 = − 2 , . . . , 2 − 1, and L an even integer. Then, the Fourier transform for the fermions with antiperiodic boundary condition is defined by de f
ε ψx,ω =
1 iεkx ε k,ω . ψ e ||
(A.20)
k∈D
Therefore (A.18) can be written as A(ψ, m s ) + A(ψ , m s ) =
u + k S(k)k , 2||
(A.21)
k∈D
where + + − , ψ − , ψ −k,+ −k,− k = (ψ ,ψ ), k,+ k,− − − + + + ,ψ ), k = (ψk,+ , ψk,− , ψ −k,+
(A.22)
−k,−
and , if we define µ(k) as in (2.15) and ω (k) = −i sin k0 + ω sin k1 , D v v σ (k) = (cos k0 + cos k1 − 2) + 2 , u u the matrix S(k) is given by ⎛ − (k) D ⎜ ⎜ ⎜ −iµ(k) ⎜ S(k) = ⎜ ⎜v ⎜ D ⎜ u − (k) ⎝ −iµ(k)
iµ(k)
v u
− (k) D
+ (k) D
−iσ (k)
+iµ(k)
− (k) D
+ (k) D
−iσ (k)
v u
iσ (k)
⎞
⎟ ⎟ D+ (k)⎟ ⎟ ⎟. ⎟ iσ (k) ⎟ ⎟ ⎠ D+ (k)
v u
(A.23)
In the case J = J we have v = 0 and σ (k) ≡ 0, so that we get the much simpler equation A(ψ, m s ) + A(ψ , m s ) = −
1 + k,ω ψ − Tω,ω (k), ψ k,ω || k∈D ω,ω
(A.24)
604
G. Benfatto, P. Falco, V. Mastropietro
with Tω,ω (k) given by (2.14) and A(χ , Ms ) + A(χ , Ms ) = −
1 + − χ χ k,ω χ k,ω Tω,ω (k), ||
(A.25)
k∈D ω,ω
T χ (k) being the matrix obtained from T (k) by substituting µ(k) with (2.16). Moreover, ¯ can be written as in (2.17). εqε = qε , so that dε = 0 and bε = qε ; it follows that B( A) This completes the proof of (2.12).
References 1. Ashkin, J., Teller, E.: Statistics of two-dimensional lattices with four components. Phys. Rev. 64, 178– 184 (1943) 2. Baxter, R.J.: Eight-vertex model in lattice statistics. Phys. Rev. Lett. 26, 832–833 (1971) 3. Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press, Inc., 1989 4. Barber, M., Baxter, R.J.: On the spontaneous order of the eight-vertex model. J. Phys. C 6, 2913– 2921 (1973) 5. Benfatto, G., Falco, P., Mastropietro, V.: Functional integral construction of the massive thirring model: verification of axioms and massless limit. Commun. Math. Phys. 273, 67–118 (2007) 6. Benfatto, G., Falco, P., Mastropietro, V.: Massless Sine-Gordon and massive thirring models: proof of the Coleman’s equivalence. Commun. Math. Phys. 285, 713–762 (2008) 7. Benfatto, G., Gallavotti, G.: Perturbation Theory and the Fermi surface in a quantum liquid. A general quasi-particle formalism and one dimensional systems. J. Stat. Phys. 59, 541–664 (1990) 8. Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta functions and schwinger functions for a many fermions system in One dimension. Commun. Math. Phys. 160, 93–171 (1994) 9. Benfatto, G., Mastropietro, V.: Renormalization group, hidden symmetries and approximate ward identities in the XYZ model. Rev. Math. Phys. 13, 1323–1435 (2001) 10. Benfatto, G., Mastropietro, V.: On the density-density critical indices in interacting fermi systems. Commun. Math. Phys. 231, 97–134 (2002) 11. Benfatto, G., Mastropietro, V.: Ward identities and chiral anomaly in the luttinger liquid. Commun. Math. Phys. 258, 609–655 (2005) 12. Falco, P., Mastropietro, V.: Renormalization group and asymtotic spin–charge separation for chiral luttinger liquid. J. Stat. Phys. 131, 79–116 (2008) 13. Giuliani, A., Mastropietro, V.: Anomalous critical exponents in the anisotropic Ashkin-Teller model. Phys. Rev. Lett. 93, 190603–07 (2004) 14. Giuliani, A., Mastropietro, V.: Anomalous universality in the anisotropic AshkinâTeller model. Commun. Math. Phys. 256, 681–725 (2005) 15. Haldane, D.M.: General relation of correlarion exponents and spectral properties of one dimensional Fermi systems: application to the anisotropic S = 1/2 Heisenberg chain. Phys. Rev. Lett. 45, 1358–1362 (1980) 16. Kadanoff, L.P.: Connections between the critical behavior of the planar model and that of the eight-vertex model. Phys. Rev. Lett. 39, 903–905 (1977) 17. Kadanoff, L.P., Brown, A.C.: Correlation functions on the critical lines of the baxter and ashkin-teller models. Ann. Phys. 121, 318–345 (1979) 18. Kadanoff, L.P., Wegner, F.J.: Some critical properties of the eight-vertex model. Phys. Rev. B 4, 3989– 3993 (1971) 19. Lesniewski, A.: Effective action for the Yukawa2 quantum field theory. Commun. Math. Phys. 108, 437– 467 (1987) 20. Luther, A., Peschel, I.: Calculations of critical exponents in two dimension from quantum field theory in one dimension. Phys. Rev. B 12, 3908–3917 (1975) 21. Mastropietro, V.: Non-universality in ising models with four spin interaction. J. Stat. Phys. 111, 201–259 (2003) 22. Mastropietro, V.: Ising models with four spin interaction at criticality. Commun. Math. Phys. 244, 595–642 (2004) 23. Mastropietro, V.: Nonperturbative Adler-Bardeen theorem. J. Math. Phys 48, 022302 (2007) 24. Mastropietro, V.: Non-perturbative aspects of chiral anomalies. J. Phys. A 40, 10349–10365 (2007) 25. Mastropietro, V.: Non-perturbative Renormalization. River Edge, NJ: World Scientific, 2008 26. McCoy, B., Wu, T.: The two dimensional Ising model. Cambridge, MA: Harvard Univ. Press, 1973 27. den Nijs, M.P.M.: Derivation of extended scaling relations between critical exponents in two dimensional models from the one dimensional Luttinger model. Phys. Rev. B 23, 6111–6125 (1981)
Extended Scaling Relations
605
28. Pruisken, A.M.M., Brown, A.C.: Universality for the critical lines of the eight vertex, Ashkin-Teller and Gaussian models. Phys. Rev. B 23, 1459–1468 (1981) 29. Pinson, H., Spencer, T.: Unpublished 30. Samuel, S.: The use of anticommuting variable integrals in statistical mechanics. I. The computation of partition functions. J. Math. Phys. 21, 2806 (1980) 31. Smirnov, S.: Towards conformal invariance of 2D lattice models. Proceedings Madrid ICM, Zürich Europ. Math. Soc, 2006 32. Spencer, T.: A mathematical approach to universality in two dimensions. Physica A 279, 250–259 (2000) 33. Zamolodchikov, A.B., Zamolodchikov, Al.B.: Conformal field theory and 2D critical phenomena, part 1. Soviet Sci. Rev. A 10, 269 (1989) Communicated by G. Gallavotti
Commun. Math. Phys. 292, 607–636 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0926-x
Communications in
Mathematical Physics
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras David Pask1 , Adam Rennie2,3 , Aidan Sims1 1 School of Mathematics and Applied Statistics, University of Wollongong,
Wollongong, NSW, Australia. E-mail: [email protected]; [email protected] 2 Institute for Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark 3 Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia. E-mail: [email protected]
Received: 19 January 2007 / Accepted: 29 July 2009 Published online: 26 September 2009 – © Springer-Verlag 2009
Abstract: In [PRen] we constructed smooth (1, ∞)-summable semifinite spectral triples for graph algebras with a faithful trace, and in [PRS] we constructed (k, ∞)-summable semifinite spectral triples for k-graph algebras. In this paper we identify classes of graphs and k-graphs which satisfy a version of Connes’ conditions for noncommutative manifolds. Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . Background on Graph C ∗ -Algebras and Spectral Triples 2.1 The C ∗ -algebras of graphs . . . . . . . . . . . . . 2.2 Semifinite spectral triples . . . . . . . . . . . . . . 2.3 Summability . . . . . . . . . . . . . . . . . . . . . 2.4 The gauge spectral triple for a graph C ∗ -algebra . . 3. Conditions for Locally Compact Semifinite Manifolds . 3.1 The analytic conditions . . . . . . . . . . . . . . . 3.2 The orientation and closedness conditions . . . . . 3.3 The bimodule conditions . . . . . . . . . . . . . . 4. k-Graph Manifolds . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
607 608 608 610 612 614 616 616 617 622 628 635
1. Introduction The object of this paper is to address the general definition of noncommutative manifolds using new examples. The phrase ‘noncommutative manifold’ is one which is still open to some degree of interpretation. Broadly speaking, a noncommutative manifold is a spectral triple (A, H, D) satisfying some additional conditions, such as those originally
608
D. Pask, A. Rennie, A. Sims
proposed by Connes, [C1]. However, the conditions presented in [C1] only make sense when A is unital (that is, a compact noncommutative space). Moreover, the proof of Connes’ spin manifold theorem in [C2] uses a version of Connes’ conditions which do not have an immediate generalisation to the noncommutative case, since the volume form is assumed antisymmetric. We suspect that supplementing the conditions in [C2] with the closedness condition discussed at length in [RV] will resolve this issue. In this paper we consider the set of conditions presented in [RV], and show that there is a natural generalisation of each to noncommutative, nonunital and semifinite spectral triples. In the process, we show that for certain graphs and k-graphs, the spectral triples constructed in [PRen,PRS] satisfy these conditions, making them reasonable candidates for the title of noncommutative manifolds. Conditions for noncompact noncommutative manifolds have previously been considered in [R1,R2,GGISV]. We have made an effort to generalise the conditions from [RV] in as minimal and stringent a way as possible. Nevertheless, our conditions must be regarded as provisional. Additional examples and applications are required to determine the ‘correct’ conditions characterising noncommutative manifolds. The vast majority of examples of noncommutative manifolds in this paper come from nonunital algebras (see [PRen,PRS]), so our conditions must address aspects of ‘noncompact noncommutative manifolds.’ Moreover, most of our examples are semifinite, in that the trace employed is not the operator trace on Hilbert space; it is a faithful normal semifinite trace on a different von Neumann algebra. This is not to say that the C ∗ -algebras arising in our examples do not admit type I spectral triples. By considering traces which reflect the geometry of the underlying graph (or k-graph), we are naturally led to semifinite spectral triples. For simplicity we discuss only graph algebras (i.e. algebras of 1-graphs) in detail; in a final section we summarise the k-graph situation, since it is largely similar. The main result for k-graph algebras that is novel is the construction of a Hochschild k-cycle representing the volume form. While the proof of the main result of [RV] is incorrect, several important results still hold. For instance, the conditions introduced in Sect. 3 do not employ Poincaré Duality in K -theory, but rather the Morita equivalence condition characterising spinc structures, [P]. The equivalence of these two conditions in the compact commutative case (in the presence of the other conditions) follows from [C2] and [RV, Sect. 7]. In addition, we do not consider the metric condition, since this has been shown to be redundant in the compact case [RV, App. B]. 2. Background on Graph C ∗-Algebras and Spectral Triples 2.1. The C ∗ -algebras of graphs. For a more detailed introduction to graph C ∗ -algebras we refer the reader to [BPRS,KPR,R] and the references therein. A directed graph E = (E 0 , E 1 , r, s) consists of countable sets E 0 of vertices and E 1 of edges, and maps r, s : E 1 → E 0 identifying the range and source of each edge. The graph is row-finite if each vertex emits at most finitely many edges and locally finite if it is rowfinite and each vertex receives at most finitely many edges. We write E n for the set of paths µ = µ1 µ2 · · · µn of length |µ| := n; that is, sequencesof edges µi such that r (µi ) = s(µi+1 ) for 1 ≤ i < n. The maps r, s extend to E ∗ := n≥0 E n in an obvious way. A loop in E is a path L ∈ E ∗ with s(L) = r (L), we say that a loop L has an exit if there is v = s(L i ) for some i which emits more than one edge. If V ⊆ E 0 then we write
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
609
V ≥ w if there is a path µ ∈ E ∗ with s(µ) ∈ V and r (µ) = w. A sink is a vertex v ∈ E 0 with s −1 (v) = ∅, a source is a vertex w ∈ E 0 with r −1 (w) = ∅. The infinite path space of E is denoted E ∞ ; if E is row-finite then E ∞ becomes a σ -compact topological space in the topology inherited from the product topology on E 1 . For a row-finite graph E, a Cuntz-Krieger E-family in a C ∗ -algebra B consists of mutually orthogonal projections { pv : v ∈ E 0 } and partial isometries {Se : e ∈ E 1 } satisfying the Cuntz-Krieger relations Se∗ Se = pr (e) for e ∈ E 1 and pv = Se Se∗ whenever v is not a sink. {e:s(e)=v}
Theorem 1.2 of [KPR] shows that there is a universal C ∗ -algebra C ∗ (E) generated by a universal Cuntz-Krieger E-family {Se , pv }. A product Sµ := Sµ1 Sµ2 . . . Sµn is non-zero precisely when µ = µ1 µ2 · · · µn is a path in E n . Since the Cuntz-Krieger relations imply that the projections Se Se∗ are also mutually orthogonal, we have Se∗ S f = 0 unless e = f , and words in {Se , S ∗f } collapse to sums of products of the form Sµ Sν∗ for µ, ν ∈ E ∗ satisfying r (µ) = r (ν) (cf. [KPR, Lemma 1.1]). Indeed, because the family {Sµ Sν∗ } is closed under multiplication and involution, we have C ∗ (E) = span{Sµ Sν∗ : µ, ν ∈ E ∗ and r (µ) = r (ν)}.
(1)
The algebraic relations and the density of span{Sµ Sν∗ } in C ∗ (E) play a critical role. The universal property of C ∗ (E) gives rise to a strongly continuous action γ , called the gauge action of S 1 on C ∗ (E) satisfying γz (Se ) = zSe for all e ∈ E 1 and γz (Pv ) = Pv for all v ∈ E 0 . Because S 1 is compact, averaging over γ with respect to normalised Haar measure gives an expectation of C ∗ (E) onto the fixed-point algebra C ∗ (E)γ : 1 (a) := γz (a) dθ for a ∈ C ∗ (E), z = eiθ . 2π S 1 The map is positive, has norm 1, and is faithful in the sense that (a ∗ a) = 0 implies a = 0. From Equation (1), it is easy to see that a graph C ∗ -algebra is unital if and only if the underlying graph is finite. When we consider infinite graphs, formulas which involve sums of projections may contain infinite sums. To interpret these, we use strict convergence in the multiplier algebra of C ∗ (E). The following is proved in [PR]. Lemma 2.1. Let E be a row-finite graph, let A be a C ∗ -algebra generated by a CuntzKrieger E-family {Te , qv }, and let { pn } be a sequence of projections in A. If pn Tµ Tν∗ converges for every µ, ν ∈ E ∗ , then { pn } converges strictly to a projection p ∈ M(A). To build spectral triples we will require traces. The following two results from [PRen] characterise the ‘nice’ traces on graph algebras. Lemma 2.2. Let E be a row-finite directed graph. (i) If C ∗ (E) has a faithful, semifinite trace then no loop can have an exit. (ii) If C ∗ (E) has a faithful, semifinite, lower semicontinuous, gauge-invariant trace τ then τ ◦ = τ and τ (Sµ Sν∗ ) = δµ,ν τ ( pr (µ) ).
610
D. Pask, A. Rennie, A. Sims
This result was sharpened using the notion of a graph trace, which we also require in this paper. Definition 2.3 (See [T]). If E is a row-finite directed graph, then a graph trace on E is a function g : E 0 → R+ such that for any v ∈ E 0 which is not a sink we have g(r (e)). (2) g(v) = s(e)=v
Proposition 2.4. Let E be a row-finite directed graph. Then there is a one-to-one correspondence between faithful graph traces on E and faithful, semifinite, lower semicontinuous, gauge invariant traces on C ∗ (E). Another graph theoretic concept useful for the graphs we will be dealing with is the following. Definition 2.5. Let E be a row-finite directed graph. An end will mean either a sink or the shift-tail equivalence class of an infinite path with no exits. For more information about shift-tail equivalence see [KPRR, §2]. Lemma 2.6. Let A = C ∗ (E) be a graph C ∗ -algebra such that no loop in the locally finite graph E has an exit. Then, K 0 (C ∗ (E)) = Z#ends ,
K 1 (C ∗ (E)) = Z#loops .
In particular, K ∗ (C ∗ (E)) is finitely generated if there are finitely many ends in E. 2.2. Semifinite spectral triples. We begin with some semifinite versions of standard definitions and results. Let τ be a fixed faithful, normal, semifinite trace on the von Neumann algebra N . Let KN be the τ -compact operators in N (that is the norm closed ideal generated by the projections E ∈ N with τ (E) < ∞). Definition 2.7. A semifinite spectral triple (A, H, D) consists of a Hilbert space H, a ∗-algebra A ⊂ N , where N is a semifinite von Neumann algebra acting on H, and a densely defined unbounded self-adjoint operator D affiliated to N such that 1) [D, a] is densely defined and extends to a bounded operator in N for all a ∈ A, and 2) a(λ − D)−1 ∈ KN for all λ ∈ R and all a ∈ A. The triple is said to be even if there is ∈ N such that ∗ = , 2 = 1, a = a for all a ∈ A and D + D = 0. Otherwise it is odd. Definition 2.8. A semifinite spectral triple (A, H, D) is QC k for k ≥ 1 (Q for quantum) if for all a ∈ A the operators a and [D, a] are in the domain of δ k , where δ(T ) = [|D|, T ] is the partial derivation on N defined by |D|. We say that (A, H, D) is QC ∞ if it is QC k for all k ≥ 1. Note. The notation is meant to be analogous to the classical case, but we introduce the Q so that there is no confusion between quantum differentiability of a ∈ A and classical differentiability of functions. Remarks concerning derivations and commutators. By partial derivation we mean that δ is defined on some subalgebra of N which need not be (weakly) dense in N . More
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
611
precisely, dom δ = {T ∈ N : δ(T ) is bounded}. We also note that if T ∈ N , one can show that [|D|, T ] is bounded if and only if [(1 + D2 )1/2 , T ] is bounded, by using the functional calculus to show that |D| − (1 + D2 )1/2 extends to a bounded operator in N . In fact, writing |D|1 = (1 + D2 )1/2 and δ1 (T ) = [|D|1 , T ] we have dom δ n = dom δ1n
for all n.
We also observe that if T ∈ N and [D, T ] is bounded, then [D, T ] ∈ N . Similar comments apply to [|D|, T ], [(1 + D2 )1/2 , T ]. The proofs of these statements can be found in [CPRS2]. Definition 2.9. A ∗-algebra A is smooth if it is Fréchet and ∗-isomorphic to a proper dense subalgebra i(A) of a C ∗ -algebra A which is stable under the holomorphic functional calculus. Thus saying that A is smooth means that A is Fréchet and a pre-C ∗ -algebra. Asking for i(A) to be a proper dense subalgebra of A immediately implies that the Fréchet topology of A is finer than the C ∗ -topology of A (since Fréchet means locally convex, metrizable and complete.) It has been shown that if A is smooth in A then Mn (A) is smooth in Mn (A), [GVF,S]. This ensures that the K -theories of A and A are isomorphic, the isomorphism being induced by the inclusion map. Stability under the holomorphic functional calculus extends to nonunital algebras, since the spectrum of an element in a nonunital algebra is defined to be the spectrum of this element in the ‘one-point’ unitization, though we must of course restrict to functions satisfying f (0) = 0. Likewise, the definition of a Fréchet algebra does not require a unit. The point of contact between smooth algebras and QC ∞ spectral triples is the following lemma, proved in [R1]. Lemma 2.10. If (A, H, D) is a QC ∞ spectral triple, then (Aδ , H, D) is also a QC ∞ spectral triple, where Aδ is the completion of A in the locally convex topology determined by the seminorms qn,i (a) = δ n d i (a) , n ≥ 0, i = 0, 1, where d(a) = [D, a]. Moreover, Aδ is a smooth algebra. We call the topology on A determined by the seminorms qni of Lemma 2.10 the δ-topology. Whilst smoothness does not depend on whether A is unital or not, many analytical problems arise because of the lack of a unit. As in [R1,R2,GGISV], we make two definitions to address these issues. Definition 2.11. An algebra A has local units if for every finite subset of elements n {ai }i=1 ⊂ A, there exists φ ∈ A such that for each i, φai = ai φ = ai . Definition 2.12. Let A be a Fréchet algebra and Ac ⊆ A be a dense subalgebra with local units. Then we call A a quasi-local algebra (when Ac is understood). If Ac is a dense ideal with local units, we call Ac ⊂ A local. Separable quasi-local algebras have an approximate unit {φn }n≥1 ⊂ Ac such that φn+1 φn = φn , [R1].
612
D. Pask, A. Rennie, A. Sims
Example. For a graph C ∗ -algebra A = C ∗ (E), Eq. (1) shows that Ac = span{Sµ Sν∗ : µ, ν ∈ E ∗ and r (µ) = r (ν)} is a dense subalgebra. It has local units because Sµ Sν∗ v = s(µ) ∗ . pv Sµ Sν = 0 otherwise Similar comments apply to right multiplication by ps(ν) . By summing the source and range projections (without repetitions) of all Sµi Sν∗i appearing in a finite sum a=
cµi ,νi Sµi Sν∗i ,
i
we obtain a local unit for a ∈ Ac . By repeating this process for any finite collection of such a ∈ Ac we see that Ac has local units. We also require that when we have a spectral triple the operator D is compatible with the quasi-local structure of the algebra, in the following sense. Definition 2.13. If (A, H, D) is a spectral triple, then we define CD (A) to be the algebra generated by A and [D, A]. Definition 2.14. A local spectral triple (A, H, D) is a spectral triple with A quasi-local such that there exists an approximate unit {φn } ⊂ Ac for A satisfying CD (A)n , CD (Ac ) = n
where CD (A)n = {ω ∈ CD (A) : φn ω = ωφn = ω}. Remark. A local spectral triple has a local approximate unit {φn }n≥1 ⊂ Ac such that φn+1 φn = φn φn+1 = φn and φn+1 [D, φn ] = [D, φn ]φn+1 = [D, φn ], [R2]. This is the crucial property we require to prove most of our summability results, to which we now turn. 2.3. Summability. In the following, let N be a semifinite von Neumann algebra with faithful normal trace τ . Recall from [FK] that if S ∈ N , the t th generalized singular value of S for each real t > 0 is given by µt (S) = inf{||S E|| : E is a projection in N with τ (1 − E) ≤ t}. The ideal L1 (N √ ) consists of those operators T ∈ N such that T 1 := τ (|T |) < ∞ where |T | = T ∗ T . In the Type I setting this is the usual trace class ideal. We will simply write L1 for this ideal in order to simplify the notation, and denote the norm on L1 by · 1 . An alternative definition in terms of singular values is that T ∈ L1 if ∞ T 1 := 0 µt (T )dt < ∞.
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
613
Note that in the case where N = B(H), L1 is not necessarily complete in this norm but it is complete in the norm ||.||1 + ||.||∞ . (where ||.||∞ is the uniform norm). Another important ideal for us is the domain of the Dixmier trace: (1,∞)
L
(N ) = T ∈ N : T L(1,∞)
1 := sup log(1 + t) t>0
t
µs (T )ds < ∞ .
0
We will suppress the (N ) in our notation for these ideals, as N will always be clear from context. The reader should note that L(1,∞) is often taken to mean an ideal in the algebra of τ -measurable operators affiliated to N . Our notation is however consistent with N that of [C] in the special case N = B(H). With this convention the ideal of τ -compact ) such that operators, K(N ), consists of those T ∈ N (as opposed to N µ∞ (T ) := lim µt (T ) = 0. t→∞
Definition 2.15. A semifinite local spectral triple is (1, ∞)-summable if a(D − λ)−1 ∈ L(1,∞) for all a ∈ Ac , λ ∈ C \ R. Remark. If A is unital, ker D is τ -finite dimensional. Note that the summability requirements are only for a ∈ Ac . We do not assume that elements of the algebra A are all integrable in the nonunital case. Strictly speaking, this definition describes local (1, ∞)summability, however we use the terminology (1, ∞)-summable to be consistent with the unital case. We need to briefly discuss the Dixmier trace, but fortunately we will usually be applying it in reasonably simple situations. For more information on semifinite Dixmier traces, see [CPS2]. For T ∈ L(1,∞) , T ≥ 0, the function FT : t →
1 log(1 + t)
t
µs (T )ds
0
is bounded. For certain generalised limits ω ∈ L ∞ (R∗+ )∗ , we obtain a positive functional on L(1,∞) by setting τω (T ) = ω(FT ). This is the Dixmier trace associated to the semifinite normal trace τ , denoted τω , and we extend it to all of L(1,∞) by linearity, where of course it is a trace. The Dixmier trace τω is defined on the ideal L(1,∞) , and vanishes on the ideal of trace class operators. Whenever the function FT has a limit at infinity, all Dixmier traces return the value of the limit. We denote the common value of all Dixmier traces on measurable operators by −. So if T ∈ L(1,∞) is measurable, for any allowed functional ω ∈ L ∞ (R∗+ )∗ we have τω (T ) = ω(FT ) = − T.
614
D. Pask, A. Rennie, A. Sims
d Example. Let D = 1i dθ act on L 2 (S 1 ). Then it is well known that the spectrum of D consists of eigenvalues {n ∈ Z}, each with multiplicity one. So, using the standard operator trace, Trace, the function F(1+D2 )−1/2 is N 1 (1 + n 2 )−1/2 , log 2N n=−N
and this is bounded. Hence (1 + D2 )−1/2 ∈ L(1,∞) and Traceω ((1 + D2 )−1/2 ) = − (1 + D2 )−1/2 = 2.
(3)
In [R1,R2] we proved numerous properties of local algebras. The introduction of quasilocal algebras in [GGISV] led us to review the validity of many of these results for quasi-local algebras. Most of the summability results of [R2] are valid in the quasi-local setting. In addition, the summability results of [R2] are also valid for general semifinite spectral triples since they rely only on properties of the ideals L( p,∞) , p ≥ 1, [C,CPS2], and the trace property. We quote the version of the summability results from [R2] that we require below. Proposition 2.16 ([R2]). Let (A, H, D) be a QC ∞ , local (1, ∞)-summable semifinite spectral triple relative to (N , τ ). Let T ∈ N satisfy T φ = φT = T for some φ ∈ Ac . Then T (1 + D2 )−1/2 ∈ L(1,∞) . For Re(s) > 1, T (1 + D2 )−s/2 is trace class. If the limit lim + (s − 1/2)τ (T (1 + D2 )−s ) exists, then it is equal to
s→1/2
1 − T (1 + D2 )−1/2 . (4) 2
In addition, for any Dixmier trace τω , the function a → τω (a(1 + D2 )−1/2 ) defines a trace on Ac ⊂ A. 2.4. The gauge spectral triple for a graph C ∗ -algebra. In this section we summarise the construction of a Kasparov module and a semifinite spectral triple for locally finite directed graphs with no sources. This material is based on [PRen]. We begin by constructing a Kasparov module. For E a row finite directed graph, we set A = C ∗ (E), F = C ∗ (E)γ , the fixed point algebra for the S 1 gauge action. The algebras Ac , Fc are defined as the finite linear span of the generators. Right multiplication makes A into a right F-module, and similarly Ac is a right module over Fc . We define an F-valued inner product (·|·) R on both these modules by (a|b) R := (a ∗ b), where is the expectation A → F. Completing A in the norm x 2X := (x|x) R F = (x ∗ x) F gives us a right C ∗ -F-module denoted X . The algebra A acting by multiplication on the left of X provides a representation of A as adjointable operators on X .
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
615
We let X c be the copy of Ac ⊂ X . For each k ∈ Z, define an adjointable endomorphism k on X by 1 k (x) = z −k γz (x)dθ, z = eiθ , x ∈ X, so 2π S 1 Sα Sβ∗ |α| − |β| = k ∗ . (5) k (Sα Sβ ) = 0 |α| − |β| = k Proposition 2.17 ([PRen]). Let X be the right C ∗ -F-module defined above. Let k 2 (xk |xk ) R < ∞}, X D = {x ∈ X : k∈Z
and define D : X D → X by D and regular.
k∈Z x k
=
k∈Z kx k .
Then D is closed, self-adjoint
Theorem 2.18. Suppose that the graph E is locally finite and has no sources. Let V = D(1 + D2 )−1/2 . Then (X, V ) is an odd Kasparov module for A-F and so defines an element of K K 1 (A, F). To construct a semifinite spectral triple, we suppose that our graph C ∗ -algebra also has a faithful gauge invariant trace τ : A → C. Using τ , we define a C-valued inner product ·, · on X c by x, y := τ ((x|y) R ) = τ ((x ∗ y)) = τ (x ∗ y), the last equality following from the gauge invariance of τ . Denote the Hilbert space completion of X c by H = L 2 (X, τ ). The operator D extends to a self-adjoint operator on H, [PRen, Lemma 5.5], and for all a ∈ Ac the commutator [D, a] extends to a bounded operator on H. Lemma 2.19. The algebra Ac and the linear space [D, Ac ] are contained in the smooth domain of the derivation δ, where for T ∈ B(H), δ(T ) = [|D|, T ]. So the completion of Ac in the δ-topology, which we denote by A, is a Fréchet pre-C ∗ -algebra. Moreover A is a quasi-local algebra with dense subalgebra Ac , and CD (Ac ) ⊂ CD (A) is also quasi-local. The last piece of information we require is the von Neumann algebra and trace which give us a semifinite spectral triple. Let End F00 (X c ) be the finite rank endomorphisms of the pre-C ∗ -module X c . Proposition 2.20. Let N = (End F00 (X c )) . Then there exists a faithful, normal, semifinite trace τ˜ : N → C such that for all rank one endomorphisms x,y of X c we have τ˜ (x,y ) = τ ((y|x) R ), x, y ∈ X c . Moreover, for all a ∈ A and λ ∈ C \ R, the operator a(λ − D)−1 lies in KN . Hence we obtain a semifinite spectral triple. However, more is true.
616
D. Pask, A. Rennie, A. Sims
Theorem 2.21. Let E be a locally finite graph with no sources, and let τ be a faithful, semifinite, norm lower-semicontinuous, gauge invariant trace on C ∗ (E). Then (A, H, D) is a QC ∞ (1, ∞)-summable odd local semifinite spectral triple (relative to (N , τ˜ )). For all a ∈ A the operator a(1 + D2 )−1/2 is not trace class. For any v ∈ E 0 which does not connect to a sink we have τ˜ω ( pv (1 + D2 )−1/2 ) = 2τ ( pv ), where τ˜ω is any Dixmier trace associated to τ˜ . The main point is that for v ∈ E 0 such that v does not connect to a sink, and for k ∈ Z we have τ˜ ( pv k ) = τ ( pv ). This is the spectral triple we will be working with for the rest of the paper, and we refer to it as the gauge spectral triple of the directed graph E (or algebra C ∗ (E)). We remind the reader that the existence of this spectral triple depends only on the graph E being locally finite with no sources, and the existence of a faithful, semifinite, gauge invariant, norm lower-semicontinuous trace τ : A → C. The latter is a nontrivial condition. 3. Conditions for Locally Compact Semifinite Manifolds We now review in turn the conditions for noncommutative manifolds as presented in [RV]. We will consider natural generalisations of these conditions to the semifinite and nonunital setting and consider when the gauge spectral triple satisfies these generalisations. We will present each condition as stated for the type I and unital case, where (N , τ ) = (B(H), Trace) and (1 + D2 )−1/2 ∈ K(H), and then present the necessary modification of the condition, if it requires modification. When dealing with these generalisations we will suppose that (A, H, D) is a local semifinite spectral triple relative to N , τ˜ . We will not suppose that A is unital, but will suppose that Ac ⊂ A gives us a quasi-local algebra. When considering the conditions as applied to graph algebras, we will suppose that E is a locally finite directed graph with no sources and possessing a faithful graph trace g. We will let (A, H, D) be the gauge spectral triple of E described in the previous section. The conditions are somewhat interdependent, and we have found it is difficult to present them in a logical fashion. It seems that this difficulty is greatly eased if we assume at the outset that the Hilbert space H carries commuting representations π : A → B(H) and π op : Aop → B(H). The former representation actually has π(A) ⊂ N ⊂ B(H), but we do not assume this for the latter representation. We will explicitly state this bimodule requirement again when we look at the first order condition, but it will be apparent that several of our conditions require a bimodule structure for their statement. In all the following, we identify a ∈ A with π(a) ∈ N unless stated otherwise. 3.1. The analytic conditions. Old Condition 1 (Dimension). The type I unital spectral triple (A, H, D) is ( p, ∞)summable for a fixed positive integer p, for which Traceω ((1 + D 2 )− p/2 ) > 0 for all Dixmier limits ω.
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
617
To generalise this condition we evidently need to replace the operator trace, Trace, by the trace τ˜ : N → C which determines the compactness and summability requirements of our spectral triple. We also need to restate the requirement, since in general for a nonunital spectral triple, even type I, we will not have (1 + D2 )− p/2 ∈ L(1,∞) , [R2]. So we have a simultaneous generalisation to the nonunital and semifinite case. New Condition 1 (Semifinite Nonunital Dimension). The local semifinite spectral triple (A, H, D) is ( p, ∞)-summable for a fixed positive integer p, for which τ˜ω (a(1 + D 2 )− p/2 ) > 0 for all ω and all 0 = a ∈ Ac with a ≥ 0. Remark. In the type I setting, we also have the condition of Absolute Continuity which states: For all nonzero a ∈ A with a ≥ 0, and for any ω-limit, the following Dixmier trace is positive: Traceω (a(1 + D2 )− p/2 ) > 0. This is half of Connes’ finiteness and absolute continuity condition, [C1,C2,GVF], the other half being finiteness discussed in Sect. 3.3 below; see also [RV]. Here we have demanded positivity only for positive elements of Ac , but this extends to positive elements of A, provided we allow the value +∞. Of course our reformulation of the dimension condition already subsumes a semifinite version of absolute continuity, so the natural generalisation of the absolute continuity condition is already satisfied by our gauge spectral triples. This shows that even in the unital case it makes sense to combine the dimension and absolute continuity conditions, as mentioned in [RV]. Thus our formulation of the conditions has rendered the absolute continuity condition redundant. This generalisation of the dimension condition is satisfied by the gauge spectral triple of a directed graph with p = 1. Provided the graph E has no sinks this follows from Theorem 2.21 since the Dixmier trace of a ∗ a, 0 = a ∈ Ac , is given by τ˜ω (a ∗ a(1 + D2 )−1/2 ) = 2τ (a ∗ a) > 0. Even if the graph has sinks, the proof of Theorem 2.21 in [PRen] shows that we still have positivity. Old Condition 2 (Regularity). The spectral triple (A, H, D) is QC ∞ . Without loss of generality, we assume that A is complete in the δ-topology and so is a Fréchet pre-C ∗ -algebra. It follows from Lemma 2.19 that this condition is satisfied with no need to modify it at all. New Condition 2 (Regularity). The spectral triple (A, H, D) is QC ∞ . Without loss of generality, we assume that A is complete in the δ-topology and so is a Fréchet pre-C ∗ -algebra.
3.2. The orientation and closedness conditions. This section examines the orientation and finiteness conditions. The orientability condition for spectral triples with unital algebra A is Old Condition 3 (Orientability). Let p be the metric dimension of (A, H, D). We require that the spectral triple be even, with Z2 -grading , if and only if p is even.
618
D. Pask, A. Rennie, A. Sims
For convenience, we take = I dH when p is odd. We say the spectral triple (A, H, D) is orientable if there exists a Hochschild p-cycle c=
n α=1
aα0 ⊗ bαop ⊗ aα1 ⊗ · · · ⊗ aαp ∈ Z p (A, A ⊗ Aop )
(6a)
whose Hochschild class [c] ∈ H H p (A, A ⊗ Aop ) may be called the “orientation” of (A, H, D), such that aα0 bαop [D, aα1 ] . . . [D, aαp ] = . (6b) πD (c) := α
Here A ⊗ Aop is a bimodule for A via a · (x ⊗ y op ) = ax ⊗ y op , (x ⊗ y op ) · a = xa ⊗ y op , a, x, y ∈ A. Now, typically, we have a nonunital algebra, and require a different formulation. We adopt the attitude that we should have a locally finite but possibly infinite cycle, as would be the case for a volume form on a noncompact manifold. New Condition 3 (Nonunital Orientability). Let p be the metric dimension of (A, H, D). We require that the spectral triple be even, with Z2 -grading , if and only if p is even. For convenience, we take = I dH when p is odd. We say the spectral triple (A, H, D) is orientable if there exists a Hochschild p-cycle c=
∞ α=1
aα0 ⊗ bαop ⊗ aα1 ⊗ · · · ⊗ aαp
(7a)
whose Hochschild class [c] may be called the “orientation” of (A, H, D), such that aα0 bαop [D, aα1 ] . . . [D, aαp ] = , (7b) πD (c) := α
where the sum in (7b) converges strongly. Remark. We have deliberately omitted any mention of the homology groups that c should belong to, there being many possibilities and few examples to guide us. We offer one possible candidate, without examining the subject in detail. op
Let Cn (Ac , Ac ⊗ Ac ) be the linear space of algebraic Hochschild n-chains for Ac . Suppose A is the completion of Ac in the topology determined by the seminorms qk , let op {qk }k∈Nn be a corresponding family of seminorms on Cn (Ac , Ac ⊗ Ac ) and let {φ j } be ∞ op a local approximate unit for A, [R1]. Define Cn (A, A ⊗ A ) to be the completion of op Cn (Ac , Ac ⊗ Ac ) for the topology determined by the family of seminorms qk, j (a 0 ⊗ bop ⊗ a 1 ⊗ · · · ⊗ a n ) := qk (φ j a 0 ⊗ (φ j b)op ⊗ φ j a 1 ⊗ · · · ⊗ φ j a n ). This should be viewed as similar to uniform convergence of all derivatives on compacta, and so analogous to a C ∞ topology. Ultimately more nonunital examples are required to clarify this issue; for more comments see [GGISV,R1,R2]. We leave these homological questions for future investigation.
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
619
For the case of graph algebras, we consider the sum over all edges in the graph c= Se∗ ⊗ Se . (8) e∈E 1
Before worrying about the convergence of this sum (in the multiplier algebra), we apply the Hochschild boundary b to find b(c) = (Se∗ Se − Se Se∗ ) = pr (e) − pv , e
v not sink
e
where we have used the Cuntz-Krieger relation to obtain the second sum on the righthand side. Thus if there are no sinks, the second sum on the right-hand side converges to the identity (in the multiplier algebra or the ‘one-point’ unitization). The first sum on the right-hand side contains each vertex projection pv with multiplicity equal to the number of edges entering it, which we denote by |v|1 . Thus b(c) = (|v|1 pv − pv ) + |v|1 pv . v a sink
v∈E 0 , v not sink
In particular, if each vertex has precisely one edge entering it, and no vertex is a sink, b(c) = 0. We say that such a graph E has no sinks, and satisfies the single entry condition. Observe that the single entry condition (together with the requirement that no loop has an exit) rules out loops except for the case where the (connected) graph comprises a single loop. The C ∗ -algebra of a graph consisting of a simple loop on n vertices is isomorphic to Mn (C(S 1 )). For a one-edge loop, the Hochschild cycle c is z −1 ⊗ z, the usual volume form for the circle. The single entry condition also rules out sources, so unless our (connected) graph comprises a single loop, it is an infinite directed tree with no sources or sinks, in which case the C ∗ -algebra is AF [KPR,
Theorem 2.4]. If E satisfies the single entry condition then we claim that Se∗ ⊗ Se converges to a partial isometry in the multiplier algebra of C ∗ (E) ⊗ C ∗ (E). Let X e = Se∗ ⊗ Se , then it is clear that X e is a partial isometry in C ∗ (E) ⊗ C ∗ (E) with X e X e∗ = (Se∗ ⊗ Se )(Se ⊗ Se∗ ) = Pr (e) ⊗ Se Se∗ , X e∗ X e = (Se ⊗ Se∗ )(Se∗ ⊗ Se ) = Se Se∗ ⊗ Pr (e) . By the relations in C ∗ (E) the Se Se∗ are mutually orthogonal, and then by the single entry hypothesis the Pr (e) are too. Hence the X e have mutually orthogonal ranges, and a standard argument (see [PR2, Lemma 1.1] or [BPRS, Lemma 1.1]) finishes off the claim. Using the single-entry condition, we see that the Hochschild cycle defined in (8) is represented by πD (c) = Se∗ [D, Se ] = Se∗ Se = pr (e) = I dH , (9) e
e
e
showing that the new condition of orientation is satisfied for this cycle. The sums in (9) converge in the strict topology as an operator on the C ∗ -module X , and also converge strongly on H. It may well be possible that there is a Hochschild cycle for a more general family of graphs, and we are not claiming that the above conditions are necessary for the orientability condition to hold, only sufficient.
620
D. Pask, A. Rennie, A. Sims
From now on we suppose that E has no sinks and satisfies the single entry condition. As noted above, it follows that the algebra C ∗ (E) is then AF unless it is Mn (C(S 1 )). In the AF case, E is a directed tree. We record the following lemma describing the fixed-point algebra of the directed tree examples. Lemma 3.1. Suppose that E is a directed tree with no sinks satisfying the single entry condition and having finitely many ends. Then F = C ∗ (E)γ is an abelian algebra, isomorphic to the continuous functions on the infinite path space E ∞ of E. Letting N denote the number of ends, each f ∈ Fc can be represented as N
f =
v∈E 0
∗ cv,n Sv,n Sv,n , cv,n ∈ C,
n=1
where (v, n) denotes a path with source v and range in the n th tail. The C ∗ -norm of such an f is f 2F = sup |cv,n |2 . Proof. The assertion that F ∼ = C0 (E ∞ ) follows from an argument along the lines of [KPRR, Lemma 4.3]. To see that it is possible to write f ∈ Fc in the above form, consider a path α with range r (α) a vertex emitting one edge e. Then ∗ Sαe Sαe = Sα Se Se∗ Sα∗ = Sα ps(e) Sα∗ = Sα pr (α) Sα∗ = Sα Sα∗ .
So any Sα Sα∗ is equal to Sβ Sβ∗ where β is an extension of α not passing through a vertex emitting more than one edge. If α is a path with range a vertex emitting, say, k edges, e1 , . . . , ek , then Sα Sα∗
=
Sα pr (α) Sα∗
=
k
Sα Sei Se∗i Sα∗ ,
i=1
and this can be subsequently extended until the next vertices emitting more than one edge. This process terminates after finitely many steps because there are finitely many ∗ are mutually orthogonal, so ends. The Sv,n Sv,n f∗f =
v
∗ |cv,n |2 Sv,n Sv,n ,
n
and f 2F = sup |cv,n |2 . The next condition is closedness, which, in its original form, is basically Stoke’s theorem for the Dixmier trace applied to elements of A ⊗ Aop . The original formulation for ( p, ∞)-summable triples using the operator trace Trace is Old Condition 4 (Closedness). The ( p, ∞)-summable spectral triple (A, H, D) is closed if for any a1 , . . . , a p ∈ A ⊗ Aop , the operator [D, a1 ] · · · [D, a p ](1 + D2 )− p/2 has vanishing Dixmier trace; thus, for any Dixmier trace Traceω , Traceω [D, a1 ] · · · [D, a p ](1 + D2 )− p/2 = 0. (10)
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
621
Remark. By setting φ(a0 , . . . , a p ) := Traceω a0 [D, a1 ] · · · [D, a p ](1 + D2 )− p/2 , Eq. (10) may be rewritten [C, VI.2] as B0 φ = 0, where B0 is defined on (k + 1)-linear functionals by (B0 φ)(a1 , . . . , ak ) := φ(1, a1 , . . . , ak ) + (−1)k φ(a1 , . . . , ak , 1). To see the utility of this condition, we introduce some notation so that we can quote Lemma 3 of [C, VI.4.γ ]. Let ∗ (A) be the universal differential algebra of A, [C, II.1.α]. Then πD : ∗ (A) → CD (A) defined by πD (a0 δa1 . . . δan ) = a0 [D, a1 ] · · · [D, an ] is a ∗-algebra representation. Denote by ∗D (A) the graded differential algebra we obtain by quotienting CD (A) by the differential ideal πD (δ(ker πD )), where δ is the universal derivation on ∗ (A). We denote by d the derivation on ∗D (A). See [C, Chap VI] for more information. Finally, let Z k (A, A∗ ) denote the Hochschild cocycles. Lemma 3.2. Let (A, H, D) be ( p, ∞)-summable and satisfy Old Condition 5 (first order). Then for each k = 0, 1, . . . , p and η ∈ k A, a Hochschild cocycle Cη ∈ Z p−k (Aop , (Aop )∗ ) is defined by Cη (a 0 , . . . , a p−k ) := Traceω πD (η) a 0 [D, a 1 ] . . . [D, a p−k ] (1 + D2 )− p/2 , a 0 , . . . , a p−k ∈ Aop . Moreover, if Old Condition 4 (closedness) also holds, then Cη depends only on the class of πD (η) in kD A, and B0 Cη = (−1)k Cdη . Thus the first order condition together with closedness give us tools to study the Hochschild and cyclic homology of the algebra A. More information can be found in [C, VI.4.γ ]. The difficulty we face is that we have a Dixmier trace defined on N ⊃ A which we can not apply to A ⊗ Aop . As we discuss in the next section, we do not believe having a spectral triple for A ⊗ Aop is of central importance. Nevertheless, the utility of Lemma 3.2 is greatly reduced by our new formulation. New Condition 4 (Semifinite Closedness). The ( p, ∞)-summable local semifinite spectral triple (A, H, D) is closed if for any Dixmier trace τ˜ω we have (11) τ˜ω [D, a1 ] · · · [D, a p ](1 + D2 )− p/2 = 0 for all a1 , . . . , a p ∈ A. It would seem that this formulation does not give us tools to study the Hochschild and cyclic cohomology of A as in the type I case described above, [C, VI.4.γ ]. More examples are required to understand the proper extension of this condition to the semifinite setting. For the gauge spectral triple of a graph algebra and generator Sµ Sν∗ ∈ A, [PRen, Theorem 5.8], τ˜ω ([D, Sµ Sν∗ ](1 + D2 )−1/2 ) = (|µ| − |ν|)τ˜ω (Sµ Sν∗ (1 + D2 )−1/2 ) = 2(|µ| − |ν|)τ (Sµ Sν∗ ).
622
D. Pask, A. Rennie, A. Sims
The gauge invariance of the trace says that τ (Sµ Sν∗ ) is non-zero only if |µ| = |ν|, whence the whole expression always vanishes. Hence the new closedness condition holds for the gauge spectral triple.
3.3. The bimodule conditions. This section is concerned with the relation between the bimodule structure of the Hilbert space and the spectral triple. First we have the first order condition which specifies the bimodule structure. In the original type I setting we have Old Condition 5 (First Order). There are commuting representations π : A → B(H) and π op : Aop → B(H) of the opposite algebra Aop (or equivalently, an antirepresentation of A). Writing a for π(a), and bop for π op (b), we ask that [a, bop ] = 0. In addition, the bounded operators in [D, A] commute with Aop ; in other words, [[D, a], bop ] = 0 for all a, b ∈ A.
(12)
In the type I setting the first order condition gives us a spectral triple for A ⊗ Aop , but we believe this is not essential, and just an artefact of the type I setting. Rather we focus on the fact that in the type I setting the algebra CD (A) is contained in the endomorphism algebra of the right A module H∞ . The finiteness condition (below) asks that H∞ = m≥1 dom Dm be a finite projecR (H ), tive (right) A module. The first order condition then says that CD (A) ⊆ EndA ∞ where R is for right. One would expect this finite projective condition to be symmetric in some sense, but this is an extra requirement. If H∞ is also a finite projective left L (H ), L for left. Typically however, these two A-module, then CD (Aop ) ⊆ EndA ∞ algebras of endomorphisms, one left and one right, will not commute with each other. They do for the gauge spectral triple of a graph algebra, but this is a one-dimensional phenomenon (see also [GGISV]). A moment’s thought shows that regarding the (sections of the) spinor bundle of a spin manifold M as a C ∞ (M) bimodule, the two collections of endomorphisms we obtain do not commute, since both algebras of endomorphisms are the same Clifford algebra. These arguments show that the most important aspect of the first order condition is that the algebra CD (A) acts as endomorphisms of a noncommutative bundle, and that the ‘symbol’ of D is such an endomorphism. Moreover, in the semifinite setting we begin with a representation π : A → N ⊂ B(H). The von Neumann algebra N is thus required to contain A and the spectral projections of D, and these are the only requirements. So typically, π op (Aop ) ⊂ N , and this is the case for the gauge spectral triple. In particular, Aop need not lie in the domain of the trace we employ, and even supposing we have a version of the first order condition, we will not obtain a spectral triple for A ⊗ Aop . We therefore change the first-order condition only very slightly as follows: New Condition 5 (Semifinite First Order). There are commuting representations π : A → N and π op : Aop → B(H) of the opposite algebra Aop . Writing a for π(a), and bop for π op (b), we ask that [a, bop ] = 0. In addition, the bounded operators in [D, A] commute with Aop ; in other words, [[D, a], bop ] = 0 for all a, b ∈ A.
(13)
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
623
For the gauge spectral triple of a directed graph, the Hilbert space naturally carries commuting representations of A and Aop . The first order condition [[D, A], Aop ] = 0 follows since [D, A] ⊂ A, and the left and right actions of A on the Hilbert space commute. The condition of finiteness in the unital case is Old Condition 6 (Finiteness). The dense subspace of H which is the smooth domain of D, H∞ := dom Dm , m≥1
is a finitely generated projective right A-module. Thus H∞ q Am , where q ∈ Mm (A) is an idempotent. Without loss of generality, we may suppose q = q ∗ also, so that, without further hypotheses, H∞ carries
∗ q b when an A-valued Hermitian pairing, namely, that given by (ξ, η) := a j,k j jk k
m , η = ( q b )m . ξ = ( j qi j a j )i=1 k ik k i=1 In the nonunital case, this is necessarily more subtle as the elements of H∞ = ∩m domDm must also satisfy integrability conditions. In [R1], the notion of smooth module was introduced for nonunital algebras which are local. As we are dealing with quasi-local algebras, most of the results on smooth modules in [R1] are not applicable. We take the attitude that: Point 1) H∞ should be a continuous A-module, Point 2) H∞ should embed continuously as a dense subspace in the C ∗ -A-module X A = H∞ , Point 3) X should be the completion of q A N for some N and some projection q in M N (Ab ), where Ab is a unitization of A, Point 4) the Hermitian product H∞ x, y → (x|y) should have range in A (acting on the right). Point 1) is implied by the condition of regularity. Proof. For x ∈ H∞ and a ∈ A we have n n δ n− j (a op )|D| j (x) D (xa) H = |D| (xa) H = j n
n
≤
j=0 n j=0
n j
H
δ n− j (a op ) |D| j (x) H .
(14)
The continuity of the action of A on H∞ now follows easily. Point 2 above is included to ensure that we can recover the ‘module of continuous sections vanishing at infinity’ from H∞ , and it is a nontrivial condition as we shall see. Once we have a continuous embedding, the image will be dense for our graph algebra examples, since Ac ⊂ H∞ .
624
D. Pask, A. Rennie, A. Sims
Once we can recover the module X , we demand that it be ‘finitely generated and projective’ in the sense of 3): see also [R1, Theorem 8]. The examples arising from graph algebras have A dense in X , so taking N = 1 and q = id Ab in any unitization Ab of A shows that 3) is always satisfied for the gauge spectral triple of a graph algebra. All four points are satisfied in the unital case, so we will ignore the case of a single loop in the following, focussing attention instead on the directed trees. Roughly speaking, without points 2) and 4), H can contain many ‘functions’ on the graph which are unbounded, and so are not in the algebra A or the module X . Modules of unbounded ‘functions’ are not terrible per se, but we prefer to remain close to the C ∗ -theory. Example. Let E be the ‘dyadic directed tree’.
•
•
• 1
··· • •
.........................
............. ........................ ................... ................... ............................. ................................... .........
•
2
•
.......... .......................... .......................
•.................................................................
•
• •
....................
.. ......................... ....................... .. ....................... ....................... ....................... .. ........................ ...
•
···
•
........ .................................. ............................
•........................................................ ....
........... .................... ............... ............... ....................... ........... ........... ........... ............... ........... .. ....... 1 ................... ...... . ..... . . . . 2 .... . . . . .... . . . . .... . . . . . ...... ..... ..... ...... ............................................................................ ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ............... ...... ........ ............... ........... .............. ........... . . . . . . . . . . .... ....................... ............... ............... 1 ..................... ..........
•
Define a faithful trace as follows. If v is a vertex before the first split, let τ ( pv ) = 1. If v occurs after n splits and before n + 1 splits, define τ ( pv ) = 2−n . Finally define τ (Sµ Sν∗ ) = δµ,ν τ ( pr (µ) ). Then the Hilbert space H = L 2 (X, τ ) contains a = lim
N →∞
N
2i/4 pi ,
(15)
i=1
where τ ( pi ) = 2−i , and the pi are mutually orthogonal. The element a ∈ H in Eq. (15) does not lie in the C ∗ -module X , as the limit does not exist in the norm · X . New Condition 6 (Nonunital Finiteness). The dense subspace of H which is the smooth domain of D, H∞ := dom Dm m≥1
has a right inner product A-module structure. Moreover, H∞ embeds as a dense subspace of a C ∗ -A-module which is finitely generated and projective over some unitization Ab of A.
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
625
Having identified a working generalisation of the finiteness condition, we identify the restrictions it places on a graph C ∗ -algebra. So to check that New Condition 6 holds, we must verify points 2) and 4). Proposition 3.3. Suppose that the locally finite directed graph E has no sinks, no loops and satisfies the single entry condition. The A-module H∞ satisfies 2) if and only if the K -theory of A is finitely generated. In this case the Hilbert space H also satisfies point 2). If the K -theory of A is finitely generated then point 4) holds. Remark. Thus for the directed tree examples, the finiteness condition is satisfied if and only if the K -theory of A is finitely generated. Proof. We begin with condition 2) for our directed trees. First of all we must have ker D ∩ H∞ = L 2 (Fc , τ ) ⊂ X . Thus we require a C > 0 such that f 2X = f ∗ f F ≤ Cτ ( f ∗ f )1/2 = C f 2H , 1/2
for all f ∈ L 2 (Fc , τ ). In particular, we require for all v ∈ E 0 that 1 = pv F ≤ Cτ ( pv )1/2 . Hence τ ( pv ) must be bounded below, which implies, by the definition of a graph trace, the faithfulness of τ , and since E is connected and row-finite, that there exist at most finitely many ends, and so K 0 (A) is finitely generated. Thus the condition is necessary. Conversely, suppose that K 0 (A) is finitely generated, and let rank(K 0 (A)) = N < ∞ be the number of ends. Observe that having finitely many ends implies that any faithful graph trace is bounded from below. Then if f ∈ Fc , Lemma 3.1 allows us to write f =
N
∗ cv,n Sv,n Sv,n , cv,n ∈ C,
v∈E 0 n=1
where (v, n) denotes a path with source v and range in the n th end. We have f 2F = sup |cv,n |2 . Now suppose that f ∈ H, so that f 2H = τ ( f ∗ f ) =
v
|cv,n |2 τ ( pn ) < ∞,
n
where pn is any projection in the n th end. Then f 2H =
v
|cv,n |2 τ ( pn ) ≥ min{τ ( pn )}
|cv,n |2
v
n
≥ min{τ ( pn )} sup |cv,n | = 2
v,n
n min{τ ( pn )} f 2F
= min{τ ( pn )} f 2X .
626
D. Pask, A. Rennie, A. Sims
Hence f ∈ X . Finally, suppose that x ∈ H, so x = As f k := xk∗ xk ∈ F and is positive, we have x 2H =
τ (xk∗ xk ) =
k
k∈Z x k
f k H ≥ (min{τ ( pn )})1/2
and
k
= (min{τ ( pn )})1/2
k
xk∗ xk X = (min{τ ( pn )})1/2
k
= (min{τ ( pn )})
1/2
∗ k∈Z τ (x k x k )
< ∞.
fk X
(xk∗ xk )2 F
1/2
k
xk∗ xk F
= (min{τ ( pn )})
1/2
k
sup xk∗ xk F k
= (min{τ ( pn )})1/2 x 2X . This proves that the finite generation of K 0 (A) is necessary and sufficient for the second point. For point 4), we assume that K 0 (A) is finitely generated. We observe that if x, y ∈ X c = Ac ⊂ H∞ we have x ∗ y ∈ Ac ⊂ A. In particular, x ∗ y is in the smooth domain of the derivation δ = [|D|, ·]. Thus for x, y ∈ X c we have, by Lemma 2.19,
δ m (x ∗ y) 2 ≤
|k − l|2m xk∗ yl 2A ,
k,l∈Z
where the sum over k, l is finite and we have used δ m ((x ∗ y)∗op ) = δ m (x ∗ y) for a ∈ Ac to avoid writing op throughout the following calculation. Now xk∗ yl 2A = yl∗ xk xk∗ yl A ≤ yl∗ yl A xk xk∗ A ≤ C 2 τ (yl∗ yl )τ (xk∗ xk ), the last inequality using the finite generation of K 0 (A). So we have the inequality δ m (x ∗ y) ≤ C 2
|k − l|2m τ (yl∗ yl )τ (xk∗ xk )
k,l∈Z
≤C
2
(|k| + |l|)2m τ (yl∗ yl )τ (xk∗ xk )
k,l∈Z
= C2
2m 2m |k|2m− j |l| j τ (yl∗ yl )τ (xk∗ xk ) j
k,l∈Z j=0
2m 2m τ ((|D| j/2 yl )∗ (|D| j/2 yl )) =C j 2
k,l∈Z j=0
= C2
2m j=0
×τ ((|D|(2m− j)/2 xk )∗ (|D|(2m− j)/2 xk )) 2m |D| j/2 y 2H |D|(2m− j)/2 x 2H . j
(16)
So suppose that {x i } ⊂ X c is a sequence converging to x ∈ H∞ in the topology determined by the seminorms x → |D|m x H , m ≥ 0, and similarly y i → y.
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
627
The estimate (16) shows that δ m (x ∗j y j − xi∗ yi ) 2A = δ m (x ∗j y j − x ∗j yi + x ∗j yi − xi∗ yi ) 2A 2m 2m |D| j/2 (y j − yi ) 2H |D|(2m− j)/2 (x j ) 2H ≤ C2 j j=0
2m 2m |D| j/2 (yi ) 2H |D|(2m− j)/2 (x j − xi ) 2H , +C j 2
j=0
and this goes to zero. Hence the sequence x ∗j y j is Cauchy in A, and so for the limits x, y ∈ H∞ , the inner product (x|y) A = x ∗ y is in the completion of Ac for the δ-topology, and so x ∗ y ∈ A. Remark. We also note that Connes stipulates that when we restrict the Hilbert space inner product to H∞ we should have x, y = − (x|y)A (1 + D2 )−1/2 , where the Hermitian product is the A-valued one: (x|y)A = x ∗ y. However, the trace satisfies τ = τ ◦ , so τ ((x|y)A ) = τ (x ∗ y) = τ ((x ∗ y)) = τ ((x|y) R ), and the inner product does indeed satisfy this formula, up to a factor of 2; see Eq. (3). The factor of 2 also occurs in the type I case, and is simply a matter of normalisation of the inner product, and does not affect the Hilbert space; see [R2, Sect. 5] for the constants in the commutative case. The following condition describes a spinc structure for the noncommutative manifold, [P]. Old Condition 7 (Spinc ). The C ∗ -A-module completion of H∞ is a Morita equivalence bimodule between A and the norm completion of the algebra CD (A) generated by A and [D, A]. Since for graph algebras the A-bimodule A is contained in X , we have a natural Morita equivalence bimodule between A and A. As the norm closed algebra generated by A and [D, A] is just A in the case of the gauge spectral triple, the Morita equivalence follows. Thus there is no need to alter the spinc condition to deal with semifiniteness or lack of a unit (at least for graph algebras). New Condition 7 (Spinc ). The C ∗ -A-module completion of H∞ is a Morita equivalence bimodule between A and the norm completion of the algebra CD (A) generated by A and [D, A]. In the case where A = C ∞ (M), M a manifold, the spinc condition (together with orientability) provides a spinc structure for M, [P]. Given a spinc manifold M, M is spin if and only if at least one (oriented) Morita equivalence bimodule admits a bijective antilinear map satisfying the requirements of the reality condition, [GVF, Theorem 9.6]. Thus the reality condition below, in conjunction with the spinc condition, may be regarded as a noncommutative spin structure.
628
D. Pask, A. Rennie, A. Sims
Old Condition 8 (Reality). There is an antiunitary operator J : H → H such that J a ∗ J −1 = a op for all a ∈ A; and moreover, J 2 = ±1, J D J −1 = ±D and also J J −1 = ± in the even case, according to the following table of signs depending only on p mod 8: p mod 8
0 2 4 6
J 2 = ±1
+−− +
J D J −1 = ±D + + + + J J −1 = ± + − + −
p mod 8
1 3 5 7
J 2 = ±1
+ −−+
(17)
J D J −1 = ±D − + − +
For the origin of this sign table in K R-homology, we refer to [GVF, §9.5]. For the gauge spectral triple, the operator J : L 2 (X, τ ) → L 2 (X, τ ), J (x) = x ∗ satisfies the reality condition for p = 1, namely, J 2 = 1, J a ∗ J = a op and J D J = −D, so the bimodule and spectral triple are real. This can be directly verified with ease. For this reason we retain the reality condition in its original form. New Condition 8 (Reality). There is an antiunitary operator J : H → H such that J a ∗ J −1 = a op for all a ∈ A; and moreover, J 2 = ±1, J D J −1 = ±D and also J J −1 = ± in the even case, according to the table (17) of signs. For the type I case connectedness of the underlying noncommutative space is formulated in the following condition. Old Condition 9 (Irreducibility). The spectral triple (A, H, D) is irreducible: that is, the only operators in B(H) commuting with D and all a ∈ A are the scalars. In a von Neumann algebra context it is clear what we should replace this condition with. New Condition 9 (Semifinite Irreducibility). The semifinite spectral triple (A, H, D) is irreducible: that is, the only operators in N commuting with D and all a ∈ A are the scalars. For our algebra A, only the fixed-point subalgebra F commutes with D. For graphs satisfying the single entry condition, F is abelian. A graph-theoretic argument shows that if E is connected, then no nontrivial element of F can commute with all of A. We summarise our results for graph algebras. Theorem 3.4. Let E be a connected locally finite graph with no sinks, admitting a faithful graph trace, satisfying the single entry condition and having finitely generated K -theory. Then the gauge spectral triple (A, H, D) of E satisfies the new (semifinite, nonunital) conditions 1 to 9. If E is not a single loop, the gauge spectral triple is both nonunital (noncompact) and semifinite. 4. k-Graph Manifolds In [PRS] we adapted the construction of [PRen] described earlier to construct a Kasparov module and semifinite spectral triple for suitable k-graph algebras. This was accomplished by ‘pushing forward’ the Dirac operator (of the simplest spin structure) on the k-torus, using the canonical Tk action on a k-graph algebra.
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
629
We will not go into the details of these constructions as we did for graph algebras, noting only that they are essentially analogous to the graph case. We also omit a general discussion of k-graph algebras, as this is lengthy. We will adopt the definitions, notations and conventions of [PRS], and refer the reader to this work for an introduction to k-graph algebras adapted to this context. We do require several notational reminders so that we can state our results here with the minimum of ambiguity. In particular: Warning In this section we reverse our conventions regarding range and source of edges. This means that sinks and sources play opposite roles, the single entry condition becomes the single exit condition, and so on. This is in keeping with the notation employed in [PRS]. Briefly, a k-graph is a set of paths with a degree map d : → Nk . For n ∈ Nk , we write n for d −1 (n), and regard 0 as the set of vertices. Paths have the unique factorisation property: if d(λ) = m + n, then there are unique paths µ ∈ m and ν ∈ n such that λ = µν. In particular, if m ≤ n ≤ l = d(λ), then there is a unique factorisation λ = λ(0, m)λ(m, n)λ(n, l), where d(λ(0, m)) = m and so forth. It also follows that each path λ has a unique range r (λ) ∈ 0 such that r (λ)λ = λ; likewise for sources. With this in mind, we write vn for r −1 (v) ∩ n and n v for s −1 (v) ∩ n for n ∈ Nk and v ∈ 0 . The C ∗ -algebra C ∗ () of a k-graph is the universal C ∗ -algebra generated by a set {Sλ : λ ∈ } of partial isometries satisfying Cuntz-Krieger type relations [KP]. For the remainder of this section, ‘k-graph’ shall be an abbreviation for ‘locally convex, locally finite k-graph without sinks, which possesses a faithful k-graph trace’. All the conditions below refer to the general semifinite nonunital versions discussed for graph algebras (with appropriate changes to the dimensions involved where necessary). The gauge spectral triples (A, H, D) for k-graph algebras satisfy the new dimension, regularity (smoothness) and absolute continuity conditions, with dimension k. All this is proved in [PRS]. The new first order condition is satisfied just as in the graph case, and the new irreducibility condition is also satisfied if the k-graph is connected. The remaining conditions which we need to verify are the new finiteness, orientability, closedness, Morita equivalence (spinc ) and reality conditions. In order to do this, we will need to assume that our k-graphs are row-finite with no sources (0 < |vn | < ∞ for v ∈ 0 and n ∈ Nk ), and satisfy the single exit condition (|n v| = 1 for each v ∈ 0 and n ∈ Nk ). Finiteness and Morita Equivalence. The proof of our rather strict finiteness condition for k-graphs is almost identical to the proof for the 1-graph case. In fact, once we have the following result it is virtually identical. Suppose that is a row-finite k-graph with no sources satisfying the single-exit condition. We claim there is an isomorphism of the fixed point algebra C ∗ ()γ onto C0 (∞ ) (where the infinite path space ∞ is endowed with the topology generated by the cylinder sets λ∞ , λ ∈ ). The isomorphism takes Sλ Sλ∗ to the characteristic function χλ∞ for each λ ∈ . To see this, first recall from [FPS] (see also [KP]) that for an arbitrary row-finite k-graph with no sources, there is an isomorphism of the diagonal D() := span{Sλ Sλ∗ : λ ∈ } onto C0 (∞ ) which takes Sλ Sλ∗ to χλ∞ . We also know that C ∗ ()γ = span{Sµ Sν∗ : d(µ) = d(ν)}, but the single-exit condition ensures that whenever Sµ Sν∗ = 0 and d(µ) = d(ν), we have µ = ν. Hence C ∗ ()γ = D() when satisfies the single-exit condition, and this establishes the claim.
630
D. Pask, A. Rennie, A. Sims
In particular, it is not hard to deduce from this an exact analogue of Lemma 3.1: if is row-finite, satisfies the single exit condition, and has finitely many (say N ) ends, then each element a of Fc can be expressed as a :=
N v∈0
∗ b(v,i) S(v,i) S(v,i) ,
(18)
i=1
where (v, i) is a path from the i th end to v. As in the graph case, there is almost nothing to prove when the algebra is unital. This follows since then the trace of the identity is finite, and we can compare the Hilbert space and C ∗ -module norms easily. For the nonunital case we have the following. Proposition 4.1. Suppose that the locally finite, locally convex k-graph (, d) has no sources and satisfies the single exit condition. The A-module H∞ embeds continuously in the C ∗ -A-module completion if and only if the K -theory of A is finitely generated; in this case the Hilbert space H does also. If K ∗ (A) is finitely generated then the C ∗ -inner product restricted to H∞ takes values in A. Apart from the above result describing the fixed-point algebra of C ∗ (), we also require the K -theory computations of [PRS] which show that in the situation of Proposition 4.1 the K -theory is finitely generated if and only if there are finitely many ends. With these results in hand, our corresponding proof for 1-graphs can be applied with minor modifications. The Morita equivalence condition is now simple, since CD (A) ∼ = Cli f f (Rk ) ⊗ A [k/2] (or Cli f f + (Rk ) in odd dimensions) and H∞ = A2 . So the gauge spectral triple of a k-graph is spinc . [k/2] Orientation. As noted above, H∞ = A2 , and the operator D acts on generators k k j j x ∈ X c ⊂ H with degree d(x) = n ∈ N by Dx = j=1 γ (in j ) x, where the γ are constant matrices generating the complex Clifford algebra of Rk . In what follows we write 1k for (1, . . . , 1) ∈ Nk . We let k be the group of permutations of {1, . . . , k}. Fix a k-graph and a path µ ∈ 1k . Given a permutation σ ∈ k , the factorisation property guarantees that there is a unique factorisation µ = µσ1 µσ2 . . . µσk such that µiσ ∈ eσ (i) for 1 ≤ i ≤ k. For example, let k = 2 and let µ = e f = ab be a commuting square, so that d(e) = d(b) = e1 and d( f ) = d(a) = e2 . There are two elements of 2 , namely the (1,2) (1,2) flip (1, 2) and the identity id. We have µ1 = a and µ2 = b whilst µid 1 = e and id µ2 = f . We use the notation (−1)σ for the canonical homomorphism σ → (−1)σ from k to {−1, 1} which takes the 2-cycles (i, j) to −1. Proposition 4.2. Let be a row-finite k-graph with no sources, and suppose that for every v ∈ 0 and 1 ≤ i ≤ k we have |ei v| = 1 (single exit). Define ck := i
k+1 2
1 (−1)σ Sµ∗ ⊗ Sµσ1 ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk . k! 1
µ∈
k
(19)
σ ∈k
Then b(ck ) = 0, where b is the Hochschild boundary operator, and πD (ck ) = , where
is the grading for k even, and the identity for k odd.
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
631
Proof. We begin by establishing that πD (ck ) = because this is the easier of the two calculations. To see this, we just calculate (here, the γ j are the generators of Cli f f (Rk )): π D (ck ) = i = i = i
k+1 2
k+1 2
k+1 2
1 (−1)σ Sµ∗ [D, Sµσ1 ][D, Sµσ2 ] . . . [D, Sµσk ] k! 1
µ∈
k
µ∈
k
σ ∈k
1 (−1)σ Sµ∗ Sµσ1 γ σ (1) Sµσ2 γ σ (2) . . . Sµσk γ σ (k) k! 1 σ ∈k
1 Sµ∗ Sµσ1 Sµσ2 . . . Sµσk = ωC ps(µ) , k! 1 1
γ1 ···γk
µ∈
k
σ ∈k
µ∈
k
where ωC is the complex volume form in Cli f f (Rk ). The single exit assumption ensures that the sum of vertex projections in the last line has exactly one term for each vertex of , and hence converges to the identity in the multiplier algebra of C ∗ (), establishing that πD (ck ) = . Now we need to establish that b(ck ) = 0. To begin with, fix µ ∈ 1k . We claim that ⎞ ⎛ (−1)σ Sµ∗ ⊗ Sµσ ⊗Sµσ ⊗ · · · ⊗ Sµσ ⎠ b⎝ 1
σ ∈k
2
k
=
σ ∈k
(−1)σ Sµ∗ σ ...µσ ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk 2
(20)
k
+(−1)k Sµσk Sµ∗ σ Sµ∗ σ ...µσ 1
k
k−1
⊗ Sµσ1 ⊗ · · · ⊗ Sµσk−1 .
To see this, we apply the definition of the Hochschild boundary b to obtain ⎛ ⎞ b⎝ (−1)σ Sµ∗ ⊗ Sµσ ⊗Sµσ ⊗ · · · ⊗ Sµσ ⎠ 1
σ ∈k
2
=
k
σ ∈k
+
(−1)σ Sµ∗ σ ...µσ ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk 2
k
k−1 (−1) j Sµ∗ ⊗ Sµσ1 ⊗ · · · ⊗ Sµσj Sµσj+1 ⊗ · · · ⊗ Sµσk j=1
+(−1)k Sµσk Sµ∗ σ Sµ∗ σ ...µσ k
1
k−1
⊗ Sµσ1 ⊗ · · · ⊗ Sµσk−1 .
To establish (20), it therefore suffices to show that for 1 ≤ j ≤ k − 1, we have σ ∈k
(−1)σ Sµ∗ ⊗ Sµσ1 ⊗ · · · ⊗ Sµσj Sµσj+1 ⊗ · · · ⊗ Sµσk = 0.
To see this, we fix 1 ≤ j ≤ k − 1, and note that we may partition k as k = A j B j where A j := {σ ∈ k : σ ( j) < σ ( j + 1)} and B j := {σ ∈ k : σ ( j) > σ ( j + 1)}. Let t j ∈ k be the transposition ( j, j + 1). Then σ → σ ◦ t j is a bijection from A j to B j .
632
D. Pask, A. Rennie, A. Sims
Hence (−1)σ Sµ∗ ⊗ Sµσ1 ⊗ · · · ⊗ Sµσj Sµσj+1 ⊗ · · · ⊗ Sµσk
σ ∈k
=
(−1)σ Sµ∗ ⊗ Sµσ1 ⊗ · · · ⊗ Sµσj Sµσj+1 ⊗ · · · ⊗ Sµσk
σ ∈A j
+(−1)σ ◦t j Sµ∗
⊗S
σ ◦t j
µ1
⊗ ··· ⊗ S
σ ◦t j
µj
S
σ ◦t j
µ j+1
⊗ ··· ⊗ S
.
σ ◦t j
µk
The definition of t j guarantees that (−1)σ + (−1)σ ◦t j = 0 for all σ ∈ A j , and we will therefore have established (20) if we can show that for fixed 1 ≤ j ≤ k − 1 and fixed σ ∈ A j , we have ∗ ⊗ S σ ⊗ ··· ⊗ S σ S σ Sµ µ µ µ 1
j
j+1
∗⊗ S ⊗ · · · ⊗ Sµσ = Sµ
σ ◦t j
µ1
k
⊗ ··· ⊗ S
σ ◦t j
µj
S
σ ◦t j
µ j+1
⊗ ··· ⊗ S
σ ◦t j
µk
.
(21) σ ◦t
By definition of t j we have µiσ = µi j whenever i = j, j + 1. If we set m :=
j−1 k i=1 eσ (i) ∈ N , then the factorisation property in ensures that σ ◦t j
µσj µσj+1 = µ(m, m + eσ ( j) + eσ ( j+1) ) = µ(m, m + eσ ◦t j ( j) + eσ ◦t j ( j+1) ) = µ j
σ ◦t
µ j+1j .
It follows that corresponding terms in the elementary tensors on either side of (21) are identical. This establishes (20). We must now show that if we sum the right-hand side of (20) over all µ ∈ 1k , we obtain zero. Fix, for the time being, µ ∈ 1k and σ ∈ k . Consider the expression (−1)σ Sµ∗ σ ...µσ ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk 2
(22)
k
appearing as a summand in the first term on the right-hand side of (20). Let λ := µσ2 µσ3 . . . µσk , so that µ = µσ1 λ. Let ψk ∈ k be the permutation defined by ψk (i) = i +1 for i ≤ k − 1 and ψk (k) = 1. Fix α ∈ s(λ)eσ (1) . Then λα ∈ 1k . Consider the expression x(λ, σ ◦ ψk , α) := (−1)σ ◦ψk (−1)k S(λα)σ ◦ψk S ∗ k
σ ◦ψk
(λα)k
S∗
σ ◦ψk
(λα)1
σ ◦ψ
...(λα)k−1 k
⊗ S(λα)σ ◦ψk ⊗ · · · ⊗ S(λα)σ ◦ψk 1
k−1
which appears in the second term on the right-hand side of (20) for λα ∈ 1k and σ ◦ ψk ∈ k . We have (−1)ψk = (−1)k−1 , and hence (−1)σ ◦ψk (−1)k = −(−1)σ . By σ ◦ψ σ ◦ψ definition of ψk , we have (λα)k k = α, and (λα) j k = µσj+1 for 1 ≤ j ≤ k − 1. Hence, we may rewrite x(λ, σ ◦ ψk , α) = −(−1)σ Sα Sα∗ Sµ∗ σ ...µσ ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk . 2
k
By the Cuntz-Krieger relation, we have Sα Sα∗ = ps(λ) = ps(µσk ) , α∈s(λ)
eσ (1)
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
and hence Sµ∗ σ ...µσ ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk − 2
k
α∈s(λ)
x(µσ2 . . . µσk , σ ◦ ψk , α) = 0.
633
(23)
eσ (1)
The single-exit condition and the unique factorisation property guarantee that each Sµ∗ σ ...µσ ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk 2
k
occurs exactly once in the first summand of the right-hand side of (20) as µ ranges over 1k and σ ranges over k . The factorisation property shows that for fixed µ and σ , a term x(λ, σ , α) is of the form x ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk for some x ∈ C ∗ () only if σ = σ ◦ ψk , λ = µσ2 . . . µσk and α ∈ s(λ)eσ (1) . Hence we may formally rewrite b(ck ) = (−1)σ Sµ∗ σ ...µσ ⊗ Sµσ2 ⊗ · · · ⊗ Sµσk 2
µ∈1k σ ∈k
k
x(µσ2 . . . µσk , σ ◦ ψk , α) ,
−
α∈s(µσk )
eσ (1)
which formally collapses to zero by (23).
One can check relatively easily, using the approximate identity µ∈1k Sµ Sµ∗ for C ∗ (), that the infinite sums involved in the definition of ck and the formal calculations in this proof make sense in the multiplier algebra of the k +1-fold tensor power of C ∗ (). Closedness. To show that for all a1 , . . . , ak ∈ A we have τ˜ω ( [D, a1 ] · · · [D, ak ](1 + D2 )−k/2 ) = 0, it suffices to prove the result for generators of the algebra. So let Tµ j ,ν j = Sµ j Sν∗j , j = 1, . . . , k, be generators. Then [D, Tµ j ,ν j ] = γ (id j ) = i
k
γ m n m, j Tµ j ,ν j ,
m=1
where d j = (n 1, j , . . . , n k, j ) is the degree of Tµ j ,ν j . With this notation we have τ˜ω ( [D, Tµ1 ,ν1 ] · · · [D, Tµk ,νk ](1 + D2 )−k/2 ) ⎛ ⎞ γ j1 n j1 ,1 ) · · · ( γ jk n jk ,k )Tµ1 ,ν1 · · · Tµk ,νk (1 + D2 )−k/2 ⎠ . = i p τ˜ω ⎝ ( j1
(24)
jk
Now = ωC ⊗ 1, where ωC is the (representation of) the complex volume form in the Clifford algebra. Since the only products of generators of the Clifford algebra with nonzero trace are multiples of the identity, the only surviving terms on the right-hand side
634
D. Pask, A. Rennie, A. Sims
of Eq. (24) when we expand the products are those with precisely one of each generator γ j . Thus τ˜ω ( [D, Tµ1 ,ν1 ] · · · [D, Tµk ,νk ](1 + D2 )−k/2 ) ⎛ ⎞ = i p τ˜ω ⎝
γ σ (1) n σ (1),1 · · · γ σ (k) n σ (k),k )Tµ1 ,ν1 · · · Tµk ,νk (1 + D2 )−k/2 ⎠ σ ∈Sk
⎛
= i p−[( p+1)/2] τ˜ω ⎝
⎞ (−1)σ n σ (1),1 · · · n σ (k),k )Tµ1 ,ν1 · · · Tµk ,νk (1 + D2 )−k/2 ⎠
σ ∈Sk
=i
p−[( p+1)/2]
det(n j,k )τ˜ω Tµ1 ,ν1 · · · Tµk ,νk (1 + D2 )−k/2 .
Now, the trace τ˜ω (Tµ1 ,ν1 · · · Tµk ,νk (1 + D2 )−k/2 ) = τ (Tµ1 ,ν1 · · · Tµk ,νk ) is zero unless Tµ1 ,ν1 · · · Tµk ,νk ∈ F, since τ is gauge invariant. This is equivalent to k j=1
dj = 0 ⇔ ∀l
k
nl,m = 0.
m=1
Hence the first, say, column of the matrix (n j,k ) is a linear combination of the other columns, and det(n j,k ) = 0. Hence for any generators Tµ j ,ν j = Sµ j Sν∗j , we have τ˜ω ( [D, Tµ1 ,ν1 ] · · · [D, Tµk ,νk ](1 + D2 )−k/2 ) = 0. Reality. We take the complex Clifford algebra Cli f f k to be generated by k elements γ j , j = 1, . . . , k such that (γ j )∗ = −γ j and γ j γ l + γ l γ j = −2δl, j I d. We make some further specifications on the generators consistent with these conventions. Denote by j the antilinear operator on X such that ⎞ ⎛ ∗ ⎞ ⎛ x1 x1 ⎜ .. ⎟ ⎜ .. ⎟ jx = j ⎝ . ⎠ = ⎝ . ⎠ . x2∗[k/2] x2[k/2] Let s(k) = [ k2 ](k + 1) − k and label the generators of the Clifford algebra so that (−1)s(k) γ j j odd . γ j = jγ j j = (−1)s(k)+1 γ j j even Observe that s(k) is even only when k = 4n, so except for these dimensions the odd generators have complex entries and are invariant under transpose, while the even generators have real entries and are antisymmetric. In dimensions 4n the situation is of course reversed. Let χ = γ 2 γ 4 · · · γ 2[k/2] be the product of the even generators (take χ = 1 when k = 1). Since the entries of χ are real for all k (if k = 4n there are 2n factors in χ and so the entries of χ are real) we have χ¯ = χ .
Noncommutative Manifolds from Graph and k-Graph C ∗ -Algebras
635
Using (γ j )∗ = −γ j we find χ ∗ = (−1)[k/2]([k/2]+1)/2 χ . We then define J := χ ◦ j = j ◦ χ . Lemma 4.3. The operator J satisfies J 2 = , J D = D J and for k even J = J , where , , are given in the table in Eq. (17). Proof. To check the sign , one needs only J ∗ J = 1 (which is straightforward) and J ∗ = j∗ ◦ χ ∗ = (−1)[k/2]([k/2]+1)/2 j ◦ χ = (−1)[k/2]([k/2]+1)/2 J. The sign can now be easily checked. The sign , in even dimensions, arises because j preserves the ±1 eigenspace decomposition of ωC , and so commutes with ωC , while ωC χ = (−1)k/2 χ ωC . For this is more subtle. We require the straightforward identity jn j = −n which may be checked on generators. Then we compute iγ j n j )n J ∗ = χ (−i jγ j jn j )jn J ∗ J Dn J ∗ = J ( j
⎛
= −iχ ⎝
j
(−1)s(k) γ j n j +
j odd
= −i(−1)[(k+1)/2](k+2)
⎞
(−1)s(k)+1 γ j n j ⎠ jn J ∗
(25)
j even
γ j n j J n J ∗ = (−1)[(k+1)/2](k+2) D−n .
j
Using the orthogonality of the n , for any x ∈ DomD we have J Dn J ∗ x = (−1)[(k+1)/2](k+2) D−n x = (−1)[(k+1)/2](k+2) Dx. J D J ∗x = n∈Zk
n∈Zk
The reader will check that the sign appearing here agrees with the values of in the table above. Theorem 4.4. Let (, d) be a connected, locally convex, locally finite k-graph with no sources, a faithful k-graph trace, satisfying the single exit condition and having finitely generated K -theory. Then the gauge spectral triple (A, H, D) of satisfies the (semifinite, nonunital) Conditions 1 to 9. Acknowledgements We thank J. Varilly for useful discussions. This work was supported by the ARC and the Danish Research Council.
References [BPRS] [CPS2] [CPRS1] [CPRS2]
Bates, T., Pask, D., Raeburn, I., Szyma´nski, W.: The C ∗ -algebras of row-finite graphs. New York J. Math 6, 307–324 (2000) Carey, A., Phillips, J., Sukochev, F.: Spectral flow and Dixmier traces. Adv. in Math. 173, 68–113 (2003) Carey, A., Phillips, J., Rennie, A., Sukochev, F.: The Hochschild class of the Chern character of semifinite spectral triples. J. Funct. Anal. 213, 111–153 (2004) Carey, A., Phillips, J., Rennie, A., Sukochev, F.: The local index theorem in semifinite von Neumann algebras I: spectral flow. Adv. in Math. 202, 451–516 (2006)
636
[C] [C1] [C2] [D] [FK] [FPS] [GGISV] [GVF] [KP] [KPR] [KPRR] [L] [M] [P] [PR] [PRen] [PRS] [R] [RW] [RS] [RSz] [R1] [R2] [RV] [RLL] [S] [T]
D. Pask, A. Rennie, A. Sims
Connes, A.: Noncommutative Geometry. London-New York: Academic Press, 1994 Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996) Connes, A.: On the spectral characterization of manifolds. http://arxiv.org/abs/0810. 2088v1[math.OA], 2008 Dixmier, J.: von Neumann Algebras. Amsterdam: North-Holland, 1981 Fack, T., Kosaki, H.: Generalised s-numbers of τ -measurable operators. Pacific J. Math. 123, 269–300 (1986) Farthing, C., Pask, D., Sims, A.: Crossed products by Zl as higher rank graph C ∗ -algebras. Houston J. Math. (to appear) Gayral, V., Gracia-Bondía, J.M., Iochum, B., Schücker, T., Varilly, J.C.: Moyal planes are spectral triples. Commun. Math. Phys. 246, 569–623 (2004) Gracia-Bondía, J.M., Varilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Boston: Birkhauser, 2001 Kumjian, A., Pask, D.: Higher rank graph C ∗ -algebras. New York J. Math. 6, 1–20 (2000) Kumjian, A., Pask, D., Raeburn, I.: Cuntz-Krieger algebras of directed graphs. Pacific J. Math. 184, 161–174 (1998) Kumjian, A., Pask, D., Raeburn, I., Renault, J.: Graphs, groupoids and Cuntz-Krieger algebras. J. Funct. Anal. 144, 505–541 (1997) Lance, E.C.: Hilbert C ∗ -Modules. Cambridge: Cambridge University Press, 1995 Mallios, A.: Topological Algebras, Selected Topics. London: Elsevier Science Publishers B.V., 1986 Plymen, R.J.: Strong Morita equivalence, spinors and symplectic spinors. J. Operator Th. 16, 305–324 (1986) Pask, D., Raeburn, I.: On the K-theory of Cuntz-Krieger algebras. Publ. RIMS, Kyoto Univ. 32, 415–443 (1996) Pask, D., Rennie, A.: The noncommutative geometry of graph C ∗ -algebras I: the index theorem. J. Funct. Anal. 233, 92–134 (2006) Pask, D., Rennie, A., Sims, A.: The noncommutative geometry of k-graph C ∗ -algebras. J. K-Theory 1, 259–304 (2008) Raeburn, I.: Graph Algebras, CBMS Regional Conference Series in Mathematics, Vol. 103, Providence, RI: Amer. Math. Soc., 2005 Raeburn, I., Williams, D.P.: Morita Equivalence and Continuous-Trace C ∗ -Algebras, Math. Surveys & Monographs, vol. 60, Providence, RI: Amer. Math. Soc., 1998 Reed, M., Simon, B.: Volume I: Functional Analysis, Volume II: Fourier Analysis, Self-Adjointness. New York: Academic Press, 1980 Raeburn, I., Szymanski, W.: Cuntz-Krieger algebras of infinite graphs and matrices. Trans. Amer. Math. Soc. 356, 39–59 (2004) Rennie, A.: Smoothness and locality for nonunital spectral triples. K-Theory 28, 127–165 (2003) Rennie, A.: Summability for nonunital spectral triples. K-Theory 31, 71–100 (2004) Rennie, A., Varilly, J.: Reconstruction of manifolds in noncommutative geometry. http://arxiv. org/abs/math/0610418v4[math.OA], 2008 Rørdam, M., Larsen, F., Laustsen, N.J.: An Introduction to K -Theory and C ∗ -Algebras. LMS Student Texts, 49, Cambridge: Cambridgr Univ. Press, 2000 Schweitzer, L.B.: A short proof that Mn (A) is local if A is local and Fréchet. Int. J. Math. 3, 581–589 Tomforde, M.: The ordered K 0 -group of a graph C ∗ -algebra. C.R. Math. Acad. Sci. Soc. R. Can 25, 19–25 (2003)
Communicated by A. Connes
Commun. Math. Phys. 292, 637–666 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0891-4
Communications in
Mathematical Physics
Spectral Gap and Transience for Ruelle Operators on Countable Markov Shifts Van Cyr, Omri Sarig∗ Mathematics Department, The Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected], [email protected] Received: 1 December 2008 / Accepted: 30 March 2009 Published online: 25 August 2009 – © Springer-Verlag 2009
Abstract: We find a necessary and sufficient condition for the Ruelle operator of a weakly Hölder continuous potential on a topologically mixing countable Markov shift to act with spectral gap on some rich Banach space. We show that the set of potentials satisfying this condition is open and dense for a variety of topologies. We then analyze the complement of this set (in a finer topology) and show that among the three known obstructions to spectral gap (weak positive recurrence, null recurrence, transience), transience is open and dense, and null recurrence and weak positive recurrence have empty interior.
1. Introduction 1.1. Overview. Thermodynamic formalism is a branch of ergodic theory which studies, for a given dynamical system T : X → X and a given function φ : X → R, the existence andproperties of invariant probability measures µφ which maximize the quantity h µ (T ) + φdµ (“equilibrium measures”). The key tool is the Ruelle operator, (L φ f )(x) =
eφ(y) f (y).
(1.1)
T y=x
Under fairly mild conditions, if L φ acts with spectral gap on some sufficiently rich Banach space L, then µφ exists, and quite a lot can be said about its properties (see the books [B,HH,PP,R], or Theorem 1.1). O.S. was partially supported by an NSF grant DMS-0652966 and by an Alfred P. Sloan Research Fellowship.
638
V. Cyr, O. Sarig
Here we ask how large is the set of functions φ : X → R for which such a space L can be found. We study this question within the cadre of countable Markov shifts, and weakly Hölder continuous functions φ : X → R (see below). We (a) identify a necessary and sufficient condition on φ for the existence of a Banach space on which L φ acts with spectral gap; (b) analyze the topological structure of the set of functions φ which satisfy this condition; (c) compare the topological properties of the various obstructions to this condition, and figure out which obstruction is the most important. 1.2. Setting. Let S be a countable set, and A = (ti j )S ×S be a matrix of zeroes and ones. The countable Markov shift (CMS) with set of states S and transition matrix A is the dynamical system T : X → X , where X := {(x0 , x1 , . . . , ) ∈ S N∪{0} : txi xi+1 = 1 for all i}, and T (x)i := xi+1 . We think of X as of the collection of one sided infinite admissible paths on a directed graph with vertices v ∈ S, and edges v1 → v2 (v1 , v2 ∈ S , tv1 v2 = 1). We equip X with the metric d(x, y) = 2−t (x,y) , t (x, y) := inf{k : xk = yk } (where inf ∅ := ∞). The resulting topology is generated by the cylinder sets [a0 , . . . , an−1 ] := {x ∈ X : xi = ai , i = 0, . . . , n − 1} (a0 , . . . , an−1 ∈ S, n ≥ 1). A word a ∈ S n is called admissible if the cylinder it defines is non-empty. The length of an admissible word a = (a0 , . . . , an−1 ) is |a| := n. We assume throughout that T : X → X is topologically mixing. This is the case when for any two states a, b there is an N (a, b) such that for all n ≥ N (a, b) there is an admissible word of length n which starts at a and ends at b. Next we consider real valued functions φ : X → R. We define the variations of a function φ : X → R to be the numbers var n (φ) := sup |φ(x) − φ(y)| : x0n−1 = y0n−1 , n where here and throughout z m := (z m , . . . , z n ). We say that φ has summable variations, if n≥2 var n φ < ∞. We say that φ is θ –weakly Hölder continuous for 0 < θ < 1, if there exists Aφ > 0 such that var n (φ) ≤ Aφ θ n for all n ≥ 2. A weakly Hölder continuous function is Hölder (with respect to the metric defined above) iff it is bounded. A bounded θ –weakly Hölder function is called θ –Hölder. k The Birkhoff sums of a function φ are denoted by φn := n−1 k=0 φ ◦ T . Suppose φ has summable variations and X is topologically mixing. The Gurevich pressure of φ is the limit
1 log Z n (φ, a), where Z n (φ, a) = eφn (x) 1[a] (x), and a ∈ S. n→∞ n n
PG (φ) = lim
T x=x
This limit is independent of a, and if sup φ < ∞, then it is equal to sup{h µ (T ) + φdµ}, where the supremum ranges over all invariant probability measures such that the sum is not of the form ∞ − ∞ [S1].
Spectral Gap and Transience for Ruelle Operators
639
1.3. The spectral gap property. Recall that the Ruelle operator associated with φ is the operator (L φ f )(x) := T y=x eφ(y) f (y). This is well defined for functions f such that the sum converges for all x ∈ X . Let dom(L φ ) denote the collection of such functions. Definition 1.1. Suppose φ is θ –weakly Hölder continuous, and PG (φ) < ∞. We say that φ has the spectral gap property (SGP) if there is a Banach space of continuous functions L s.t. L ⊂ dom(L φ ) and L ⊇ {1[a] : a ∈ S n , n ∈ N}; f ∈ L ⇒ | f | ∈ L, | f | L ≤ f L ; L–convergence implies uniform convergence on cylinders; L φ (L) ⊆ L, and L φ : L → L is bounded; L φ = λP + N , where λ = exp PG (φ), and P N = N P = 0, P 2 = P, dim Im P = 1, and the spectral radius of N is less than λ; (f) If g is θ –Hölder, then L φ+zg : L → L is bounded, and z → L φ+zg is analytic on some complex neighborhood of zero.
(a) (b) (c) (d) (e)
The motivation is the following (compare with [R,HH,Li,PP,BS,GH,AD]). Suppose X is a topologically mixing CMS, and φ : X → R is a weakly Hölder continuous potential with finite Gurevich pressure, finite supremum, and the SGP. Write L φ = λP + N as above, then Theorem 1.1. P takes the form P f = h f dν, where h ∈ L is positive, and ν is a measure which is finite and positive on all cylinders. The measure dm φ = hdν is a T –invariant probability measure with the following properties: (a) If m φ has finite entropy, then m φ is the unique equilibrium measure of φ. (b) There is a 0 < κ < 1 s.t. for all g ∈ L ∞ (m φ ) and f bounded Hölder continuous, ∃C( f, g) > 0 s.t. | Covm φ ( f, g ◦ T n )| ≤ C( f, g)κ n . (Cov = covariance.) (c) Suppose ψ is a bounded Hölder continuous function such that Em φ [ψ] = 0. If √ ψ = ϕ − ϕ ◦ T with ϕ continuous, then ∃σ > 0 s.t. ψn / n converges in distribution (w.r.t. m φ ) to the normal distribution with mean zero and standard deviation σ. (d) Suppose ψ is a bounded Hölder continuous function, then t → PG (φ + tψ) is real analytic on a neighborhood of zero. We remark that the assumption that m φ has finite entropy is satisfied trivially for all CMS with finite Gurevich entropy PG (0) < ∞. Versions of Theorem 1.1 were shown in a variety of contexts by many people [R,GH,HH,AD,Li,BS,G1] (this is a partial list). The proof in our context is given in Appendix A. 1.4. The problem. When does a potential φ satisfy the SGP? How common is this phenomenon? What are the most important obstructions? If |S| < ∞ then every (weakly) Hölder continuous function has SGP (Ruelle [R]), but this is not the case when |S| = ∞ because of the phenomena of null recurrence, transience [S2], and positive recurrence with sub-exponential decay of correlations [S4] or non–analytic pressure function [S5,Lo,PrS]. Doeblin and Fortet have given sufficient conditions for spectral gap for potentials φ associated to a class of countable Markov chains [DF]. Aaronson & Denker had constructed Banach spaces with spectral gap for potentials associated with Gibbs–Markov
640
V. Cyr, O. Sarig
measures [AD]. The underlying CMS must satisfy a certain combinatorial condition (the “big images” property). Young had constructed Banach spaces with spectral gap for certain functions φ on CMS satisfying a different combinatorial condition (“tower structure”), see [Y]. 1.5. Notational convention. a = c ± ε means c − ε < a < c + ε, a = B ±1 c means B −1 ≤ a/c ≤ B, and an cn means that ∃B s.t. an = B ±1 cn for all large n. 2. Summary of Results 2.1. A necessary and sufficient condition for SGP. The condition is in terms of the discriminant, a notion which was introduced in [S3]. We recall the definition, and refer the reader to Appendix A for further information. If one induces a CMS on one of its states a ∈ S, then the result is a full shift. It is useful to fix the following notation: (a) S := {[a] = [a, ξ1 , . . . , ξn−1 ] : n ≥ 1, ξi = a, [a, ξ , a] = ∅}; N∪{0}
(b) X := S , viewed as a countable Markov shift with set of states S; (c) π : X → [a]; π([a 0 ], [a 1 ], . . .) = (a 0 , a 1 , . . .). This is a conjugacy between the left shift on X , and the induced (=first return) map on [a]. Every function φ : X → R has an “induced version” φ : X → R given by ⎞ ⎛ ϕ a −1 φ := ⎝ φ ◦ T k ⎠ ◦ π, where ϕa (x) := 1[a] (x) inf{n ≥ 1 : T n (x) ∈ [a]}. k=0
It is easy to see that if φ is weakly Hölder continuous on X , then φ is weakly Hölder continuous on X (moreover, var 1 ψ < ∞ even when var 1 ψ = ∞). The a–discriminant of φ is the (possibly infinite) quantity
a [φ] := sup{PG (φ + p) : p ∈ R s.t. PG (φ + p) < ∞}. The sign of this number has meaning [S3], see Appendix A. A weakly Hölder continuous function φ on a topologically mixing countable Markov shift is called strongly positive recurrent, (SPR), if it has finite Gurevich pressure and if there is a state a s.t. a [φ] > 0. Strong positive recurrence is a generalization of the notion of stable positive recurrence for positive infinite matrices due to Gurevich and Savchenko [GS]. It has its roots in the classical work of Vere-Jones on the problem of geometric ergodicity for Markov chains [VJ]. Theorem 2.1. Suppose X is a topologically mixing CMS, and φ : X → R is weakly Hölder continuous with finite Gurevich pressure, then φ has the spectral gap property iff φ is strongly positive recurrent. That SGP implies SPR is fairly routine, given the results of [S3]. The main part of the theorem is the other direction. It is perhaps useful at this point to explain how to check strong positive recurrence. Define Z n∗ (φ, a) := eφn (x) 1[ϕa =n] (x), T n x=x
Spectral Gap and Transience for Ruelle Operators
641
and let R denote the of rφ (x) := n≥1 x n Z n∗ (φ, a), then [S3]
radius of convergence
∞ proves that either
a [φ] − log rφ (R) ≤ n=2 var n φ or a [φ] = log rφ (R) = ∞. In particular, if rφ (·) diverges at its radius of converges, then φ is SPR. It is easy to construct examples of φ with SGP on any topologically mixing CMS: Start with any weakly Hölder continuous φ : X → R with finite pressure, and fix some state a. One checks that rφ+t1[a] (x) = et rφ (x), thus φ +t1[a] is strongly positive recurrent for any t large enough. 2.2. SGP is open and dense. Let denote the collection of weakly Hölder continuous functions φ : X → R with finite Gurevich pressure. There are many different useful topologies on . To list them concisely, fix an infinite sequence ω = (ωn )n≥1 , 0 ≤ ωn ≤ ∞ and define for a function f : X → R,
f ω := sup | f | +
∞
ωn var n ( f ), where 0 · ∞ := 0,
n=1
V (φ, ε) := {φ ∈ : φ − φ ω < ε}. The ω–topology is the topology generated by V (φ, ε), (ε > 0, φ ∈ ). The choice ω = (0, 0, . . .) is useful for the study of perturbations in the sup norm. Other important choices are ω = (0, . . . , 0, ∞, ∞, . . .) (finite memory), ω = (0, 1, 1, . . .) (summable variations), and ω = (0, θ −1 , θ −2 , . . .) (Hölder). Theorem 2.2. The set of φ ∈ with the spectral gap property is open and dense in with respect to the ω–topology, for any ω = (ωn )n≥1 . In particular, the spectral gap property is stable under perturbations in with sufficiently small sup norm (ω = (0, 0, . . .)); and any φ ∈ can be perturbed to be strongly positive recurrent using a perturbation of arbitrarily small Hölder norm, or even finite memory of length one (ω = (∞, ∞, . . .)). This means that there is an open and dense set of φ ∈ which satisfy the conclusion of Theorem 1.1. Loosely speaking these are potentials whose thermodynamic formalism is similar to the behavior of thermodynamic systems at equilibrium without a phase transition. The following works contain related results: (1) Gurevich and Savchenko showed in [GS] that if φ ∈ is “stably positive recurrent” and φ is Markovian (i.e. var 2 φ = 0), then there is an ε > 0 s.t. any Markovian φ ∈ s.t. φ − φ ∞ < ε is positive recurrent (cf. Appendix A). For Markovian potentials, “stable positive recurrence” can be easily seen to be equivalent to strong positive recurrence. (2) Gallavotti & Miracle–Sole considered in [GM] multi-dimensional lattice gas models, and showed that in a certain topology there is a dense G δ –set of interaction potentials whose pressure functional is differentiable. Next we consider the larger set SV of all φ : X → R with summable variations and finite Gurevich pressure. Again, we can define the ω–topology on SV as the topology generated by {φ ∈ SV : φ − φ ω < ε} for all ε > 0, φ ∈ SV . Theorem 2.2 . Let SV denote the collection of all φ : X → R with summable variations and finite Gurevich pressure, then {φ ∈ SV : φ is strongly positive recurrent} is open and dense in SV for every ω–topology.
642
V. Cyr, O. Sarig
Obstructions to the SGP. If a potential φ ∈ does not have the spectral gap property, then by Theorem 2.1 it is not strongly positive recurrent, and a [φ] ≤ 0. Potentials with strictly negative discriminant are called transient. Potentials with zero discriminant are divided into two groups: null recurrent, and weakly positive recurrent (see Appendix A for a summary of the definitions and properties of the various modes of recurrence – in particular see Theorem 7.3 to equate the above definition of transience with that in Definition 7.1). We ask whether one of these obstructions is more common, in some sense, than the others. The ω–topologies are too weak to detect the difference between transience, null recurrence, and weak positive recurrence (they are all nowhere dense), so we need to use a stronger topology. The topologies of perturbations of finite support are sufficient for this purpose. To N define these topologies, fix a (nonempty) finite collection of states B = i=1 [ai ]. The uniform topology localized at B (or just the “B–uniform topology”) is the topology generated by the basis U (φ; ε, B) := {φ ∈ : φ − φ ∞ < ε, φ | X \B = φ| X \B } (ε > 0, φ ∈ ). Denote the resulting topology by LU(B). Theorem 2.3. Let (Tr) := {φ ∈ : φ is transient}. With respect to LU(B), (Tr) is open in , and open and dense in {φ ∈ : φ does not have SGP}. As a corollary of this theorem and its proof we have the following topological description of the various modes of recurrence in each of the LU(B)–topologies: (a) (b) (c)
strong positive recurrence: open; transience: open; weak positive recurrence and null recurrence: empty interior, contained in the boundaries of the first two sets.
In other words, transience is the most common obstruction to spectral gap. 3. Proof of Theorem 2.1 3.1. Strong Positive Recurrence implies Spectral Gap. Assume w.l.o.g. that PG (φ) = 0 (otherwise pass to φ − PG (φ), cf. §7.1). Fix some state a ∈ S s.t. a [φ] > 0. By the discriminant theorem (Appendix A, Theorem 7.3), PG (φ) = 0, where the over bar indicates induction on [a]. Therefore, by strong positive recurrence, there exists εa such that 0 < PG (φ + 2εa ) < ∞. This εa must be positive, because p(t) := PG (φ + t) is an increasing function. The function φ is by assumption weakly Hölder so there exists 0 < θ < 1 and Aφ > 0 such that var n φ ≤ Aφ θ n for all n ≥ 2. Make εa so small that 0 < θ e p < 1, where p := PG (φ + εa ).
(3.1)
This is possible to do, because p(t) := PG (φ + t) is continuous (being convex and finite) on (−∞, 2εa ) (see (7.4) in Appendix A). Define ψ := φ + εa − p1[a] , then using the properties of PG (·) listed in Appendix A §7.1 it readily follows that (1) PG (ψ) = 0, becausePG (ψ) = PG (φ + εa − p) = PG (φ + εa ) − p = 0;
Spectral Gap and Transience for Ruelle Operators
643
(2) ψ is strongly positive recurrent, because PG (ψ + εa ) ≤ PG (φ + 2εa ) < ∞, and PG (ψ + εa ) = PG (ψ + εa ϕa ) ≥ PG (ψ) + εa = εa > 0, so a [ψ] > 0; (3) PG (ψ) = 0, because PG (ψ) = 0 and ψ is (strongly positive) recurrent, see Appendix A, Theorem 7.3, part (1). Since ψ is SPR, it is positive recurrent (Appendix A, Theorem 7.3). By the generalized Ruelle Perron Frobenius theorem (Appendix A, Theorem 7.2) and the assumption that PG (ψ) = 0, there exists a Borel measure ν0 , finite and positive on cylinders, and a positive continuous function h 0 : X → R such that L ∗ψ ν0 = ν0 , L ψ h 0 = h 0 , and h 0 dν0 = 1. Moreover, var 1 [log h 0 ] ≤ ≥2 var φ. Setting C0 := exp ≥2 var φ, we see that for every x, h 0 (x) = C0±1 h 0 [x0 ], where h 0 [x0 ] := sup[x0 ] h 0 . Define for x, y ∈ X , t (x, y) := min{n : xn = yn }, where min ∅ = ∞, sa (x, y) := #{0 ≤ i ≤ t (x, y) − 1 : xi = yi = a} (compare with the notion of “separation time” due to L.-S. Young [Y]). Let L denote the collection of continuous functions f : X → C for which
1 sa (x,y)
f L := sup : x, y ∈ [b],x = y < ∞. sup | f (x)|+sup | f (x)− f (y)|/θ b∈S h 0 [b] x∈[b] It is clear that (L, · L ) is a Banach space. We show that L φ (L) ⊆ L, and that L φ : L → L is a bounded operator with spectral gap. The proof uses the strengthening of the Ionsecu-Tulcea & Marinecu theorem due to Hennion ([HH], Theorem II.5). Suppose there exists a continuous semi-norm · C on L with the following properties: (A) There is a constant M > 0 s.t. L φ f C ≤ M f C for all f ∈ L; (B) Let ρ(L φ ) denote the spectral radius of L φ : L → L. There are constants n 0 ∈ N, 0 < r < ρ(L φ ), and R > 0 such that
L nφ0 f L ≤ r n 0 f L + R f C ;
(3.2)
(C) Every sequence { f n }n≥1 ∈ L s.t. sup f n L ≤ 1 has a subsequence { f n k }k≥1 s.t.
L φ f n k − g C −−−→ 0 for some g ∈ L. k→∞
Hennion’s theorem then says that L = F ⊕ N , where F, N are L φ –invariant subspaces such that dim(F) < ∞, ρ(L φ |N ) < ρ(L φ ), and such that every eigenvalue of L φ |F is of modulus ρ(L φ ). As we shall see below, the theory of equilibrium measures on topologically mixing CMS implies that ρ(L φ ) = 1, that the only eigenvalue on the unit circle is one, and that this eigenvalue is simple. This gives the spectral gap property with λ = 1, P the eigenprojection of one, and N := L φ (I − P). We will apply Hennion’s theorem to L φ : L → L. The semi–norm we use is
· C := · L 1 (ν0 ) .
644
V. Cyr, O. Sarig
Step 1. (A) holds: · C is a continuous semi-norm on L, and there is a constant M such that L φ f C ≤ M f C for all f ∈ L. Proof. To see that · C is continuous, suppose that f n − f L → 0. Then f n → f pointwise, and | f n (x) − f (x)| ≤ f n − f L h 0 [x0 ] ≤ C0 f n − f L h 0 (x) at every point. Since h 0 ∈ L 1 (ν0 ), f n − f C = | f n − f |dν0 → 0. Next fix f ∈ L. Then | f | ≤ C0 f L h 0 . The identity φ = ψ − εa + p1[a] ≤ ψ + p − εa shows that |L φ f | ≤ e p−εa L ψ (C0 f L h 0 ) = C0 e p−εa f L h 0 . Integrating w.r.t ν0 , we get L φ f C ≤ C0 e p−εa f L and the step follows with M := C0 exp( p − εa ). Step 2. Proof of (3.2). Proof. We need some notation. For every b ∈ S, set P n (b) := { p = ( p0 , . . . , pn−1 ) : ( p, b) is admissible}. For every p = ( p0 , . . . , pn−1 ) admissible, let n( p) := #{0 ≤ i ≤ n − 1 : pi = a}, and set Pkn (b) := { p ∈ P n (b) : n( p) ≥ k + 1}. In what follows we fix k (to be determined later), and estimate L nφ f L for arbitrary f ∈ L and n ≥ 1. Part 1. Analysis of supx∈[b] |(L nφ f )(x)| (b ∈ S). Suppose x ∈ [b]. Since φ = ψ − εa + p1[a] and | f | ≤ C0 f L h 0 , |(L nφ f )(x)| ≤ eφn ( px) | f ( px)| = eψn ( px)−nεa + pn( p) | f ( px)| p∈P n (b)
≤
p∈P n (b)
eψn ( px)−nεa + pn( p) | f ( px)|
p∈Pkn (b)
+ C0 ekp−nεa f L
eψn ( px) h 0 ( px)
p∈P n (b)\Pkn (b)
≤
eψn ( px)−nεa + pn( p) | f ( px)| + C0 ekp−nεa f L h 0 [b],
p∈Pkn (b)
because the last sum is bounded by (L nψ h 0 )(x) = h 0 (x) ≤ h 0 [b]. Every p ∈ Pkn (b) admits a unique decomposition p = (α, β, γ ) with α ∈ Ak , β ∈ B and γ ∈ C, where: Ak := {α : n(α) = k, and (α, a) is admissible}, B := {β : β starts at a, and (β, a) is admissible} ∪ {empty word}, C := {γ : γ contains exactly one a, at its beginning, and (γ , b) is admissible}. Conversely, every triplet (α, β, γ ) ∈ Ak × B × C such that |α| + |β| + |γ | = n (where |w| := length of w) gives rise to an element of Pkn (b). Thus
Spectral Gap and Transience for Ruelle Operators
645
|(L nφ f )(x)| ≤ C0 ekp−nεa f L h 0 [b] ⎧ ⎨ eψγ (γ x)−γ εa + ⎩ α+β+γ =n
×
γ ∈C,|γ |=γ
eψβ (β γ x)−βεa + p[n(β γ )+k]
β∈B,|β|=β
⎫ ⎬ eψα (α β γ x)−αεa | f (α β γ x)| , ⎭
α∈Ak ,|α|=α
(3.3) with the convention that ψ0 ≡ 0. We estimate the innermost sum. Since n(α) = k, | f (α β γ x)| ≤ inf | f | + f L θ k h 0 [α0 ] ≤ inf | f | + C0 f L θ k h 0 (α β γ x). [α,a]
[α,a]
Since vari φ = vari ψ for all i ≥ 1, eψα (α β γ x) ≤ C0 inf eψα . [α,a]
We can thus estimate the inner sum by C0 inf eψα −αεa inf | f |+C0 f L θ k e−αεa α∈Ak ,|α|=α
≤ C0
[α,a]
[α,a]
eψα (α β γ x) h 0 (α β γ x)
α∈Ak ,|α|=α
inf eψα −αεa | f | + C0 f L θ k e−αεa h 0 [a] (∵ (β γ )0 = a)
α∈Ak ,|α|=α
[α,a]
≤ C0 e−αεa
α∈Ak ,|α|=α
≤ C0 e−αεa
α∈Ak ,|α|=α
C0 e−αεa = ν0 [a]
1 ν0 [a] 1 ν0 [a]
α∈Ak ,|α|=α
[a]
eψα (α y) | f (α y)|dν0 (y) + C0 f L θ k e−αεa h 0 [a]
L αψ (1[α,a] | f |)dν0 + C0 f L θ k e−αεa h 0 [a]
1[α,a] | f |dν0 + C0 f L θ k e−αεa h 0 [a] (∵ L ∗ψ ν0 = ν0 )
e−αεa
C0
f C + C0 h 0 [a]θ k e−αεa f L ν0 [a] 1 + h 0 [a] . ≤ C1 e−αεa f C + θ k f L , where C1 := C0 ν0 [a] ≤
Substituting this estimate in (3.3), we see that |(L nφ f )(x)| ≤ C0 ekp−nεa f L h 0 [b] + ⎡ ×⎣
eψγ (γ x)−γ εa
α+β+γ =n γ ∈C,|γ |=γ
⎤ eψβ (β γ x)−βεa + p[n(β γ )+k] C1 e−αεa f C + θ k f L ⎦ .
β∈B,|β|=β
(3.4)
646
V. Cyr, O. Sarig
By construction n(β γ ) = n(β) + 1 and ψβ (β γ x) = φβ (β γ x) + βεa − pn(β), so the sum in the square brackets is eφβ (β γ x)+ p(k+1) C1 e−αεa f C + θ k f L β∈B,|β|=β
= C1 e−αεa f C + θ k f L · e p(k+1)
eφβ (β γ x)
β∈B,|β|=β
≤ C1 e
p(k+1)−αεa
f C + θ k f L · C0 Z β (φ, a), where Z β (φ, a):= eφβ (z) 1[a] (z), T β z=z
because β, γ start with a. We claim that supβ Z β (φ, a) ≤ 2C0 : Had there been a β with Z β (φ, a) > 2C0 , then we would have had Z nβ (φ, a) ≥ [ C10 Z β (φ, a)]n ≥ 2n , in contradiction to the assumption that n1 log Z n (φ, a) −−−→ PG (φ) = 0. Setting C2 := 2C0 , n→∞ we obtain that the sum in the square brackets in (3.4) is bounded by C0 C1 C2 e(k+1) p−αεa f C + θ k f L . Substituting this in (3.4), gives |(L nφ f )(x)| ≤ eψγ (γ x)−γ εa C0 C1 C2 e(k+1) p−αεa α+β+γ =n γ ∈C,|γ |=γ
× f C + θ k f L + C0 ekp−nεa f L h 0 [b] C e−(α+γ )εa 0 ≤ C0 C1 C2 e(k+1) p f C + θ k f L eψγ (γ x) h 0 (γ x) h 0 [a] α+β+γ =n
f L h 0 [b] ≤ C02 C1 C2 e(k+1) p f C + θ k f L + C0 e
γ ∈C,|γ |=γ
kp−nεa
α+β+γ =n
e−(α+γ )εa h 0 [b] h 0 [a]
f L h 0 [b]. (∵ L ψ h 0 =h 0 ) It is easy to check that supn∈N α+β+γ =n e−(α+γ )εa −εa 2 , then for all x ∈ [b], C3 = 1 + ≥0 e + C0 e
kp−nεa
|(L nφ f )(x)| ≤ e(k+1) p
≤
≥0 e
−εa 2 .
C02 C1 C2 C3
f C + (θ k + e−nεa ) f L h 0 [b]. h 0 [a]
Let
(3.5)
Part 2. Analysis of the Lipschitz constant of L nφ f on [b]. Suppose x, y ∈ [b]. |(L nφ f )(x) − (L nφ f )(y)|
eφn ( px) 1 − eφn ( py)−φn ( px) | f ( px)| + eφn ( py) | f ( px) − f ( py)| ≤ p∈P n (b)
≤
eφn ( px) C4 θ t (x,y) | f ( px)| +
p∈P n (b)
p∈P n (b)
eφn ( py) C0 h 0 ( py) f L θ n( p)+sa (x,y) ,
p∈P n (b)
Spectral Gap and Transience for Ruelle Operators
! Aφ where C4 := max 1, 1−θ sup |(L nφ
f )(x)−(L nφ
f )(y)| ≤ θ
Aφ |δ|≤ 1−θ
⎡
647
δ "
1−e
|φ(x)−φ(y)|
δ , and Aφ := sup θ t (x,y) . Thus
⎤
C4 sup L nφ | f | + C0 f L eφn ( py) h 0 ( py)θ n( p) ⎦ [b] p∈P n (b)
sa (x,y)⎣
=: θ sa (x,y) [I + II],
(3.6)
where I := C4 sup L nφ | f | ≤ e(k+1) p [b]
by (3.5), and II := C0 f L
C02 C1 C2 C3 C4
f C + (θ k + e−nεa ) f L h 0 [b] h 0 [a]
eφn ( py) θ n( p) h 0 ( py)
p∈P n (b)
= C0 f L
eψn ( py)−nεa + pn( p) θ n( p) h 0 ( py)
p∈P n (b)
= C0 f L e−nεa
eψn ( py) (e p θ )n( p) h 0 ( py)
p∈P n (b)
≤ C0 f L e−nεa
eψn ( py) h 0 ( py), because e p θ < 1 by (3.1)
p∈P n (b)
≤ C02 f L e−nεa h 0 [b], because L ψ h 0 = h 0 and y ∈ [b] ≤ e(k+1) p
C02 C1 C2 C3 C4 −nεa e
f L h 0 [b], because p > 0 and C1 C2 C3 C4 > h 0 [a]. h 0 [a]
Substituting the estimates for I and II in (3.6), we see that for all x, y ∈ [b], 2 |(L nφ f )(x) − (L nφ f )(y)| (k+1) p C 0 C 1 C 2 C 3 C 4 k −nεa
f
h 0 [b]. ≤ e + (θ + 2e ) f
C L h 0 [a] θ sa (x,y) (3.7) Part 3. Putting everything together to obtain (3.2). Equations (3.5) and (3.7), together with the fact that C4 ≥ 1 give C 2 C1 C2 C3 C4
L nφ f L ≤ 3e(k+1) p 0
f C + (θ k + e−nεa ) f L h 0 [a] 2C C C C C 1 2 3 4 ≤ 3e p 0 ekp f C + ((e p θ )k + ekp−nεa ) f L (3.8) h 0 [a] At this stage, it is probably useful to recall the definition of the constants Ci : C0 := exp
∞
var φ,
=2
C3 := 1 +
#∞ =0
$2 e
−εa
,
C1 := C0 (h 0 [a] + 1/ν0 [a]) , C2 := 2C0 , ⎫ ⎧ ⎪ ⎬
δ ⎪ ⎨ Aφ
C4 := max 1, 1−θ sup 1−e
δ ⎪. ⎪ A ⎭ ⎩ |δ|≤ φ 1−θ
648
V. Cyr, O. Sarig
These constants do not depend on k or n. Using (3.1), it is no problem to choose first k and then n 0 so that 1 3C0 e(k+1) p &
L nφ f L ≤ R f C + f L for all n ≥ n 0 , where R := Ci . 2 h 0 [a] 4
(3.9)
i=0
In the particular case n = n 0 , we get (3.2) with r := 2−1/n 0 . In the next step we shall see that r < ρ(L φ ). Step 3. L φ is a bounded operator on L and ρ(L φ ) = 1, thus (B) holds. Proof.
· C ≤ C0 · L on L, because for every f ∈ L, | f | ≤ C0 f L h 0 , and h 0 dν0 = 1. Thus (3.8) implies that L φ < ∞, and (3.9) says that sup L nφ < ∞. It follows that L φ is bounded, and that its spectral radius is not larger than one. We claim that the spectral radius is equal to one. Otherwise, there is some κ < 1 such that L nφ = O(κ n ), and then |L nφ 1[a] | = O(κ n ) uniformly on [a]. Now L nφ 1[a] Z n (φ, a) uniformly on [a] (Appendix A, Remark 7.1), so this means that 0 = PG (φ) = limn→∞ n1 log Z n (φ, a) ≤ log κ < 0, a contradiction. Step 4. Every sequence { f n }n≥1 in C such that sup f n L < ∞ has a subsequence which converges w.r.t. · C to some element of L. Since L φ < ∞, (C) holds. Proof. Let X 0 denote the subset of X consisting of all sequences which contain the symbol a infinitely many times. This is a subset of ν0 –full measure, because ν0 is an ergodic conservative measure which charges every partition set. The function δ(x, y) := θ sa (x,y) is a metric on X 0 , and (X 0 , δ) is a complete separable metric space. The family { f n }n≥1 is uniformly Lipschitz on partition sets with respect to this metric. By the Arzela–Ascoli theorem, there is a subsequence { f n k }k≥1 which converges pointwise on X 0 to some function g0 : X 0 → C. Since | f n k (x)| ≤ C0 (sup f n L )h 0 (x), and h 0 dν0 < ∞, X 0 | f n k − g0 |dν0 → 0. We show that ∃g ∈ L such that g| X 0 = g0 . Choose points y b ∈ [b] ∩ X 0 , (b ∈ S), and define a map ϑ : X → X 0 by ⎧ x ⎪ ∃i s.t. xi = a, ⎨y 0 a a ϑ(x) := (x0 , . . . , xk , y1 , y2 , . . .) ∃i s.t. xi = a, k := max{i : xi = a} < ∞, ⎪ ⎩x ∃ infinitely many i s.t. xi = a. We claim that for all x, y ∈ X , sa (ϑ(x), y) ≥ sa (x, y). If sa (x, y) = 0 or ϑa (x) = x, then there is nothing to prove. Otherwise, x has finitely many coordinates equal to a. Let k := max{i : xi = a, x0i = y0i } and k := max{i : xi = a, ϑ(x)i0 = y0i }, then sa (x, y) = #{0 ≤ j ≤ k : y j = a} and sa (ϑ(x), y) = #{0 ≤ j ≤ k : y j = a}. By construction, ϑ(x)k0 = x0k = y0k , therefore k ≥ k and sa (ϑ(x), y) ≥ sa (x, y). Now set g := g0 ◦ ϑ. Since ϑ| X 0 = id, g| X 0 = g0 . If x ∈ [b], then ϑ(x) ∈ [b], so |g(x)| = |g0 (ϑ(x))| ≤ sup | f n (ϑ(x))| ≤ h 0 [b] supn≥1 f n L . If x, y ∈ [b], then |g(x) − g(y)| ≤ |g0 (ϑ(x)) − g0 (ϑ(y))| ≤ sup | f n (ϑ(x)) − f n (ϑ(y))| ≤ sup f n L · h 0 [b]θ n
We conclude that g ∈ L, and that
X
n sa (ϑ(x),ϑ(y))
| f n k − g|dν0 =
≤ sup f n L · h 0 [b]θ sa (x,y) . n
X0
| f n k − g0 |dν0 → 0.
Spectral Gap and Transience for Ruelle Operators
649
Step 5. L φ : L → L satisfies parts (a)–(e) of the spectral gap property. Proof. It is clear that every element of L is continuous, and that L contains all indicators of cylinder sets. Parts (a) and (d) of the spectral gap property were shown in Step 3. Parts (b) and (c) are obvious from the definition of · L . We prove part (e). The previous steps show that the conditions of Hennion’s theorem are satisfied and that ρ(L φ ) = 1. It follows that L = F ⊕ N , where L φ (F) ⊆ F, L φ (N ) ⊆ N , F is a finite dimensional space, the eigenvalues of L φ |F are all of modulus one, and the spectral radius of L φ |N is strictly less than one. We show that F = span{h} for some function h s.t. L φ h = h. Once this is done, we let P : L → F denote the eigenprojection of the eigenvalue 1, and N := L φ (I − P). It is clear that L φ = P + N , P 2 = P, P N = N P = 0, and dim Im P = dim F = 1. To see that ρ(N ) < 1, we use the facts ρ(L φ |N ) < 1 and L φ |F = id to see that L nφ = P + N n → P, whence N = { f ∈ L : L nφ f −−−→ 0}. n→∞
It follows that N = ker P. Thus N = L φ (I − P) is equal to zero on F and equal to L φ on N . Since F and N are L φ –invariant and ρ(L φ |N ) < 1, ρ(N ) < 1 and (e) is proved. Step 5.1. 1 is an eigenvalue of L φ |F : F → F. We construct an eigenfunction. Recall that φ is (strongly) positive recurrent with pressure zero. By the generalized RPF theorem (Appendix A, Theorem 7.2) there is a positive continuous function h and a Borel measure ν such that L φ h = h, L ∗φ ν = ν, hdν = 1. The measure dµ = hdν is known to be an exact invariant probability measure, and for every cylinder [a], L nφ 1[a] −−−→ hν[a] n→∞ pointwise [S2]. We claim that h ∈ L. By (3.8), supn≥1 L nφ 1[a] L < ∞. By Step 4, ∃n k → ∞ such L 1 (ν0 )
that L nφk 1[a] −−−→ g ∈ L. The limit must agree with the pointwise limit of L nφk 1[a] , n→∞
whence with h. Thus h = (1/ν[a])g ν0 –almost everywhere, whence by continuity — everywhere. Thus h ∈ L. We claim that h ∈ F. Write h = h 1 +h 2 , where h 1 ∈ F and h 2 ∈ N . Since L φ h = h, h = L nφ h 1 + L nφ h 2 . The first summand stays inside F, and the second summand tends to zero in norm, because ρ(L φ |N ) < 1. It follows that h ∈ F. But dim F < ∞ so F = F. Thus h ∈ F. Since h ∈ F and L φ h = h, 1 is an eigenvalue of L φ : F → F. Step 5.2. 1 is the only eigenvalue of L φ |F : F → F. This eigenvalue is simple. By the definition of F, all the eigenvalues of L φ |F : F → F have modulus one. Suppose f ∈ F \ {0} and L φ f = eiθ f , we show that eiθ = 1 and f = const h. We claim that f ∈ L 1 (ν). Since L φ is positive, L φ | f | ≥ |L φ f | = | f |, whence N n=1
L nφ [L φ | f | − | f |] ≤ L φN +1 | f | ≤ C0 sup L nφ f L h 0 for all N . n
650
V. Cyr, O. Sarig
But ν is conservative and ergodic and L ∗φ ν = ν, so for every F ≥ 0 such that Fdν = 0, n L φ F = ∞ ([A], Proposition 1.3.2). Thus L φ | f | = | f | ν–almost everywhere. It follows that | f | is an absolutely continuous invariant density of ν. An ergodic conservative measure can have at most one invariant density, so | f | = const h ν–a.e., whence f ∈ L 1 (ν). We claim that f is proportional to h and eiθ = 1. Let dµ := hdν. Since f ∈ L 1 (ν), 'ψ = 1 L φ (hψ).1 Since µ is exact, f / h ∈ L 1 (µ). The transfer operator of µ is T h n ' 'n ( f / h) = einθ ( f / h),
T ( f / h) − ( f / h)dµ L 1 (µ) → 0 ([A], Theorem 1.3.3). But T so this can only happen if eiθ = 1 and f / h = const almost everywhere. Since f, h are continuous and ν has global support, f / h = const. Step 5.3. dim F = 1. Since the spectrum of L φ |F consists of a single simple eigenvalue equal to one, and since (by construction) dim F < ∞, F has a basis with respect to which L φ : F → F is represented by dim(F) × dim(F) Jordan block with ones on the diagonal. The iterates of such a matrix diverge when dim(F) > 1 (the (1, 2)–entry escapes to infinity). This cannot be the case, because sup L nφ < ∞ by (3.8). The conclusion is that dim(F) = 1. We conclude that F = span{h}, where L φ h = h. By the discussion above, part (e) of SGP is proved. Step 6. Proof of part (f) of SGP. Part (f) of SGP says that if f ∈ F is θ –Hölder, then z → L φ+z f is analytic at zero. Write for every θ –Hölder continuous function g,
g θ := sup |g| + sup{|g(x) − g(y)|/θ t (x,y) : x, y ∈ X }. It is easy to verify that g f L ≤ g θ f L for all f ∈ L. It follows that the operator Mn : f → L φ (g n f ) is bounded, and that Mn ≤ zn
L φ
g nθ . Thus the series ∞ n=0 n! Mn converges absolutely in the operator norm for all zn |z| < 1/ g θ . As a result, L φ+zg ≡ ∞ n=0 n! Mn is analytic on {z ∈ C : |z| < 1/ g θ }. This shows (f), and completes the proof of SGP. 3.2. Spectral Gap implies Strong Positive Recurrence. Suppose φ has the spectral gap property, and write L φ = λP + N with λ = exp PG (φ) and P, N as above. Since P N = N P = 0 and P 2 = P, L nφ = λn P + N n . Since the spectral radius of N is less than λ, λ−n N n = O(κ n ) where 0 < κ < 1. Thus for (any) fixed x ∈ [a], λ−n Z n (φ, a) λ−n (L nφ 1[a] )(x) = P1[a] (x) + O(κ n ) (Appendix A, Remark (7.1)). It is impossible for P1[a] (x) to vanish, because this would imply that Z n (φ, a) = O((κλ)n ), whereas n1 log Z n (φ, a) → log λ and κ < 1. Thus P1[a] (x) = 0. According to the theory of analytic perturbations of linear operators [K], there exists ε > 0 s.t. every L : L → L which satisfies L − L φ < ε can be written in the form L = λ(L)P(L) + N (L), where P(L), N (L) are bounded linear operators s.t. P(L)2 = P(L), dim Im P(L) = 1, N (L)P(L) = P(L)N (L) = 0, and such that the spectral radius of N (L) is smaller 1 The transfer operator of a measure µ s.t. µ ◦ T −1 µ is the operator T 1 ' 1 : L (µ) → L (µ) whose value ' f dµ = ϕ ◦ T · f dµ for all test functions on a function f ∈ L 1 (µ) is determined by the condition ϕ T ϕ ∈ L ∞ (µ).
Spectral Gap and Transience for Ruelle Operators
651
than |λ(L)|. Moreover, if ε > 0 is sufficiently small, then L → λ(L), P(L), N (L) are analytic on {L : L − L φ < ε}. Since g := 1[a] is Hölder continuous, t → L φ+tg is real analytic, whence continuous, at zero. So ∃δ > 0 such that if |t| < δ, then L φ+tg − L φ < ε and L φ+tg = λt Pt + Nt , where λt := λ(L φ+tg ), Pt := P(L φ+tg ), Nt := N (L φ+tg ). Since L–convergence implies pointwise convergence, Pt 1[a] (x) −−→ P1[a] (x). We t→0
saw above that for any x ∈ [a], P1[a] (x) = 0. Choosing our δ sufficiently small, we can ensure that (Pt 1[a] )(x) = 0 for all |t| < δ for some x ∈ [a]. We now repeat the argument above for φ + tg and see that for all t real such that |t| < δ, |λt |−n Z n (φ + tg, a) |λt |−n (L nφ+tg 1[a] )(x) = |(Pt 1[a] )(x) + o(1)|, whence |λt |−n Z n (φ + tg, a) 1. This implies that for all |t| < δ, |λt | = exp PG (φ + tg) and φ + tg is recurrent. By the discriminant theorem, a [φ + tg] ≥ 0 for all |t| < δ. But a [φ + tg] = a [φ + t1[a] ] = a [φ] + t (Appendix A, Lemma 7.1). If this is non-negative for all |t| < δ, then it must be the case that a [φ] > 0. 4. Strong Positive Recurrence is Open and Dense The material in this section relies on the theory of modes of recurrence, which we summarized for the convenience of the reader in Appendix A. Main Lemma. As we shall see below, it is fairly easy to approximate a recurrent potential by a strongly positive recurrent potential. Here we show that every potential can be approximated by a recurrent potential. Lemma 4.1. If φ ∈ is a transient potential, a ∈ S, and ψ is a non-positive bounded weakly Hölder function s.t. supp ψ ⊂ [a], then φ + ψ ∈ , φ + ψ is transient, and PG (φ + ψ) = PG (φ). Proof. Since φ + ψ ≤ φ we have PG (φ + ψ) ≤ PG (φ). To see the other inequality, we note that since φ is transient, 1 log Z n∗ (φ, a) (Appendix A, (7.6)) n n→∞ 1 = lim sup log Z n∗ (φ + ψ, a) (∵ supp φ ⊂ [a] and sup |ψ| < ∞) n→∞ n ≤ PG (φ + ψ). (∵ (7.5))
PG (φ) = lim sup
This shows that PG (φ) = PG (φ + ψ). Using the transience of φ and the non-positivity of ψ, we see that ∞
e−n PG (φ+ψ) Z n (φ + ψ, a) =
n=0
∞ n=0
≤
∞ n=0
so φ + ψ is transient.
e−n PG (φ) Z n (φ + ψ, a) e−n PG (φ) Z n (φ, a) < ∞,
652
V. Cyr, O. Sarig
Lemma 4.2 (Main Lemma). Suppose φ ∈ is transient, then for any ε > 0 there exists a recurrent ϕ ∈ so that ϕ − φ ∞ ≤ ε and var 1 [ϕ − φ] = 0. k
Proof. Recall that S denotes the set of states. We write a − → b for a, b ∈ S if there is an admissible word with k + 1 symbols which starts with a and ends with b. Fix ε > 0 and b ∈ S. We construct finite sets of states {c1k , . . . , crkk } (k ≥ 0) by induction as follows. When k = 0, let r0 := 1, and c10 := b. Now suppose we have carried the construction for each < k. Let b1k , b2k , b3k , . . . be the list of all different
states c for which b − → c for ≤ k. If this collection is finite, let rk be its size, and set {c1k , . . . , crkk } := {b1k , . . . , brkk }. If it is infinite, observe that # Z n∗
φ+ε
∞ i=1
$ 1[bk ] , b ≥ enε Z n∗ (φ, b) (1 ≤ n ≤ k), i
since for any x with T n x = x and x0 = b we have added an extra factor of ε to the potential at states x0 , x1 , . . . , xn−1 . Therefore we can find sk ∈ N such that $ # sk ε ∗ Zn φ + ε 1[bk ] , b ≥ en· 2 Z n∗ (φ, b) (1 ≤ n ≤ k). (4.1) i=1
i
We let {c1k , . . . , crkk } be the set {c1k−1 , . . . , crk−1 } ∪ {b1k , . . . , bskk } where, in this case, rk k−1 k is the number of different states ci so defined. Set φ[0] := φ, and define for k ≥ 1, φ[k] = φ + ε
rk
1[ck ] ,
i=1
i
We interpolate these potentials. Observe that for all k ≥ 1, φ[k] = φ[k − 1] + ε
mk i=1
1[d k ] , where {d1k . . . , dmk k } := {c1k . . . , crkk } \ {c1k−1 , . . . , crk−1 }, k−1 i
with m k defined by the above identity. Define for k ≥ 1 and 0 ≤ i ≤ m k , φ[k, i] := φ[k − 1] + ε
i j=1
1[d k ] . j
Then φ[k, 0] = φ[k − 1], and φ[k, m k ] = φ[k]. We claim that there must be some k, i such that φ[k, i] is recurrent. Assume by way of contradiction that this is not the case: φ[k, i] is transient for all k, i. In this case, the sequence φ[k] = φ[k, m k ] ≥ φ[k, m k − 1] ≥ · · · ≥ φ[k, 1] ≥ φ[k − 1, m k−1 ] ≥ · · · is a decreasing sequence of transient potentials where each term is equal to its predecessor minus ε times the indicator of some partition set. By Lemma 4.1, all terms
Spectral Gap and Transience for Ruelle Operators
653
in the sequence have the same Gurevich pressure. Since the sequence terminates after finitely many steps at φ[0] = φ, PG (φ[k]) = PG (φ) for all k.
(4.2)
Consider now the power series tk (x) := 1 +
∞
Z i (φ[k], b)x i ,
i=1
rk (x) :=
∞
Z i∗ (φ[k], b)x i .
i=1
Both have radius of convergence exp[−PG (φ)]: the first by the definition of the Gurevich pressure and (4.2), and the second because of the assumption that φ[k] is transient (Appendix A, (7.6)). They are related by the following inequality for all 0 < x < exp[−PG (φ)] (Appendix A, (7.2)): ∞
1 [tk (x) − 1] ≤ tk (x)rk (x) ≤ B 2 [tk (x) − 1], where B := exp var n φ. (4.3) 2 B n=2
By (4.3), rk (x) ≤ B 2 for all 0 < x < exp[−PG (φ)] and k ≥ 1. But this cannot be the case, because for exp[−PG (φ) − 2ε ] < x < exp[−PG (φ)], rk (x) ≥
k
Z n∗ (φ[k], b)x n ≥
n=1
−−−→ k→∞
k
ε
en· 2 Z n∗ (φ, b)x n (by (4.1))
n=1 ∞
Z n∗ (φ, b)(eε/2 x)n = ∞.
n=1
This contradiction shows that there must be some k0 , i 0 for which ϕ := φ[k0 , i 0 ] is recurrent. By construction ϕ ∈ , var 1 [ϕ − φ] = 0, and ϕ − φ ∞ = ε. Proof of Theorem 2.2. The proof has two parts: (a) If φ ∈ , then for every ε > 0 there is a strongly positive recurrent potential ϕ ∈ s.t. ϕ − φ ∞ < and var 1 [ϕ − φ] = 0. (b) The set of strongly positive recurrent potentials is open w.r.t the sup norm on . The first part shows that the set of strongly positive recurrent potentials is dense in the strongest possible ω–topology; the second step shows that it is open in the weakest possible ω–topology. Part 1. Approximating general potentials by strongly positive recurrent potentials. Fix φ ∈ and ε > 0. By Lemma 4.2, there exists a recurrent ψ ∈ such that φ − ψ ∞ < ε/2 and var 1 [φ − ψ] = 0. We now appeal to the discriminant theorem (Appendix A, Theorem 7.3): Fix some a ∈ S, then the recurrence of ψ implies that a [φ] ≥ 0. If ϕ := ψ + 2ε · 1[a] , then
a [ϕ] = a [ψ] +
ε (Appendix A, Lemma 7.1), 2
654
V. Cyr, O. Sarig
so ϕ is strongly positive recurrent. It is obvious that φ − ϕ ∞ < ε and var 1 [ϕ − φ] = var 1 [ψ − φ] = 0. Part 2. For every strongly positive recurrent φ ∈ there exists a δ > 0 such that if ϕ ∈ and ϕ − φ ∞ < δ, then ϕ is strongly positive recurrent. We fix some a ∈ S and work with the induced system on [a], X , as defined in §2.1. By the definition of the discriminant, if φ ∈ is strongly positive recurrent then there exists p ∈ R such that 0 < PG (φ + p) < ∞. W.l.o.g. PG (φ + p ) < ∞ for some p > p. The map x → PG (φ + x) is convex and finite on (−∞, p ), whence continuous on (−∞, p] (Appendix A (7.4)). It is also strictly increasing (because φ + x + h ≥ (φ + x) + h for all h > 0). Hence, there exist numbers p1 < p2 s.t. 0 < PG (φ + p1 ) < PG (φ + p2 ) < ∞. Take p0 := ( p1 + p2 )/2 and δ := ( p2 − p1 )/2. If ϕ ∈ and ϕ − φ ∞ < δ, then φ + p1 ≤ ϕ + p0 ≤ φ + p2 so 0 < PG (φ + p1 ) < PG (ϕ + p0 ) < PG (φ + p2 ) < ∞, proving that a [ϕ] > 0. This shows that the set of strongly positive recurrent potentials is · ∞ -open. Proof of Theorem 2.2 . The proof is identical to the proof of Theorem 2.2 with the words “weakly Hölder” replaced by “summable variations”. 5. Transience is Open and Dense in the Set of Non-strongly Positive Recurrent Potentials The reader is referred to Appendix A for the definition and properties of transient, null recurrent, and weakly positive recurrent potentials. Proof of Theorem 2.3. Lemma 7.1 in Appendix A says that for every a ∈ S and t ∈ R,
a [φ + t · 1[a] ] = (
a [φ] + t. Suppose B = ri=1 [ai ], and φ ∈ is transient. Then a1 [φ] < 0. Find ε1 > 0 s.t. φ (1) := φ + ε1 · 1[a1 ] satisfies a1 [φ (1) ] < 0. Then φ (1) is transient. The transience of φ (1) means that a2 [φ (1) ] < 0, so we can find ε2 > 0 s.t. φ (2) := φ (1) + ε2 · 1[a2 ] satisfies a2 [φ (2) ] < 0. So φ (2) is also transient. Continuing in this manner, we obtain ε1 , . . . , εr > 0 s.t. ψ := φ (r ) = φ +
r
εi · 1[ai ] is transient.
i=1
Take δ := min{ε1 , . . . , εr }. We claim that every ϕ ∈ such that ϕ − φ ∞ < δ and φ| X \B = ϕ| X \B is transient. To see this, we observe that ϕ can be obtained from ψ by subtracting the r non-negative functions (ψ − ϕ)1[ai ] . By Lemma 4.1 each subtraction preserves transience, so the end result ϕ is transient. This proves that the set of transient potentials is LU(B)–open. We claim that it is dense in the complement of the strongly positive recurrent potentials. To see this, it is enough to show that every φ ∈ s.t. a1 [φ] = 0 can be approximated in LU(B) by a transient potential. Take φ + t · 1[a1 ] with t → 0− .
Spectral Gap and Transience for Ruelle Operators
655
6. More on Transience The previous arguments suggest the following new characterization of transience: Theorem 6.1. φ ∈ is transient if and only if there exists ψ ∈ such that ψ ≥ φ, ψ ≡ φ, and PG (ψ) = PG (φ). Proof. If φ is transient, then for any a ∈ S, ψ := φ + t · 1[a] is transient for all t > 0 sufficiently small (Theorem 2.3). By Lemma 4.1, PG (ψ − s1[a] ) = PG (ψ) for all s > 0. In the particular case s = t we get PG (ψ) = PG (φ), and ψ is as required. We will show that if φ is recurrent then no such ψ can exist. Suppose by way of contradiction that ∃ψ ∈ such that ψ ≡ φ, ψ ≥ φ, and PG (ψ) = PG (φ). Find some word [a] := [a1 , . . . , an ] such that ψ − φ > α on [a] for some α > 0. Since φ ≤ φ + α · 1[a] ≤ ψ and PG (·) is increasing, PG (φ + α · 1[a] ) = PG (φ). The potential ϕ := φ + α · 1[a] must be recurrent, because ∞
Z n (ϕ, a)e
−n PG (ϕ)
n=1
=
∞ n=1
Z n (ϕ, a)e
−n PG (φ)
≥
∞
Z n (φ, a)e−n PG (φ) = ∞,
n=1
by the recurrence of φ. Therefore there exists a positive continuous function h such that L ϕ h = e PG (ϕ) h = e PG (φ) h (Appendix A, Theorem 7.2). This and φ ≤ ϕ implies that L φ h ≤ e PG (φ) h, and it is easy to see that this inequality is strict on T [a]. Now consider the non-negative function f := h − e−PG (φ) L φ h. This is a non-negative continuous function, not everywhere equal to zero, such that ∞
e−k PG (φ) L kφ f ≤ h < ∞ everywhere.
k=0
In particular the series on the left (all of whose summands are non-negative) converges almost surely. But this is impossible: φ is recurrent, so L φ has a conservative ergodic eigenmeasure ν, L ∗φ ν = e PG (φ) ν. Since L ∗φ ν = e PG (φ) ν, e−PG (φ) L φ is the transfer operator of k ν, whence L φ f = ∞ ν–almost everywhere (cf. [A], Proposition 1.3.2) , whence at least at one point. This contradiction shows that ψ cannot exist. The result should be compared with the results of S. Ruette [Rt] on the transience of φ ≡ 0. Acknowledgements. The authors would like to thank the referees for their careful reading of the paper and for many valuable suggestions.
7. Appendix A: The Discriminant and the Three Modes of Recurrence The purpose of this section is to summarize the results of [S1,S2 and S3] concerning the thermodynamic formalism of countable Markov shifts (CMS). Throughout this section we assume that X is a topologically mixing CMS with set of states S and transition matrix A, which we think of as the set of one sided admissible paths on a directed graph G. We use the notation introduced in §1.2.
656
V. Cyr, O. Sarig
7.1. Gurevich pressure. Suppose φ has summable variations, and define as always φn := φ + φ ◦ T + · · · + φ ◦ T n−1 . The Gurevich pressure of φ is 1 log Z n (φ, a), where Z n (φ, a) := eφn (x) 1[a] (x). n→∞ n n
PG (φ) = lim
T x=x
The limit exists, is independent of the choice of a, and satisfies [S1]: (a) For every constant c, PG (φ + c) = PG (φ) + c; (b) φ ≤ ψ ⇒ PG (φ) ≤ PG (ψ); (c) if φ, ψ have summable variations, then PG (tφ+(1−t)ψ) ≤ t PG (φ)+(1−t)PG (ψ) for all t ∈ [0, 1]. Theorem 7.1 (Variational Principle [S1]). If sup φ < ∞ and φ has summable vari ations, then PG (φ) = sup{h µ (T ) + φdµ}, where the supremum ranges over all T –invariant Borel probability measures such that (h µ (T ), φdµ) = (∞, −∞). Remark 7.1. If X is topologically mixing and φ has summable variations, then L nφ 1[a] Z n (φ, a) uniformly on [a]. 7.2. Modes of recurrence. Recall that ϕa (x) := 1[a] (x) inf{n ≥ 1 : T n (x) ∈ [a]}, and set Z n∗ (φ, a) = eφn (x) 1[ϕa =n] (x). T n x=x ∗ (φ, a) are related by the following “approximate renewal equation”: Z n (φ, a) and Z n set B := exp 2 ∞ n=2 varn (φ) , then
∗ Z n (φ, a) = B ±1 Z n−1 (φ, a)Z 1∗ (φ, a)+· · ·+ Z 1 (φ, a)Z n−1 (φ, a) + Z n∗ (φ, a) . (7.1) Passing to the generating functions, tφ (x) = 1 +
∞
Z n (φ, a)x
n
n=1
and rφ (x) =
∞
Z n∗ (φ, a)x n ,
n=1
we obtain 1 tφ (x)rφ (x) ≤ tφ (x) − 1 ≤ B 2 tφ (x)rφ (x) B2 for every x ∈ [0, R), where R = e−PG (φ) is the radius of convergence of tφ (·). Definition 7.1. Set λ = e PG (φ) . We call φ • transient, if tφ (λ−1 ) < ∞; • positive recurrent, if tφ (λ−1 ) = ∞ but rφ (λ−1 ) < ∞; • null recurrent, if tφ (λ−1 ) = ∞ and rφ (λ−1 ) = ∞.
(7.2)
Spectral Gap and Transience for Ruelle Operators
657
We have the following [S2, Theorem 1]: Theorem 7.2 (Generalized Ruelle-Perron-Frobenius Theorem [S2]). φ is recurrent iff there exist λ > 0, a conservative measure ν, finite and positive on cylinders, and a positive continuous function h such that L ∗φ ν = λν and L φ h = λh. In this case λ = e PG (φ) and ∃ an ∞ such that for every cylinder [a] and x ∈ X , n 1 −k k λ (L φ 1[a] )(x) −−−→ h(x)ν[a], n→∞ an k=1
where {an }n satisfies an ∼ ( more,
[a] hdν)
−1
n
k=1 λ
−k Z
k (φ, a)
for every a ∈ S. Further-
(1) if φ is positive recurrent then ν(h) < ∞, an ∼ n ·const and for every [a],λ−n L nφ 1[a] −−−→ hν[a]/ν(h) uniformly on compacts; n→∞
(2) if φ is null recurrent then ν(h) = ∞, an = o(n) and for every cylinder [a], λ−n L nφ 1[a] −−−→ 0 uniformly on compacts. n→∞
It is not difficult to see, using the representation of h as the limit above, that var 1 [log h] ≤ n≥2 var n φ. 7.3. The discriminant. Fix a state a ∈ S, and recall the operation of passing from the pair (X, φ) to (X , φ) as explained in §2.1. Define pa∗ [φ] := sup{ p | PG (φ + p) < ∞} (the bar means that we induce on the state a). This number can be calculated by the formula [S3] pa∗ [φ] = − lim sup n→∞
1 log Z n∗ (φ, a). n
(7.3)
Moreover, the map p(t) = PG (φ + t)
(7.4)
is convex, strictly increasing and continuous on {t : t ≤ pa∗ [φ]} ([S3, Prop. 3]). The discriminant of φ at a ∈ S is defined to be
a [φ] := sup{PG (φ + p) | p < pa∗ [φ]}. The following is frequently useful so we state it as a lemma. Lemma 7.1. If X is topologically mixing and φ has summable variations and finite pressure then a [φ + t · 1[a] ] = a [φ] + t. Proof. PG (φ + t · 1[a] + p) = PG (φ + p+t) = PG (φ + p)+t, so pa∗ [φ+t ·1[a] ] = pa∗ [φ] and a [φ + t · 1[a] ] = a [φ] + t. The discriminant detects modes of recurrence: Theorem 7.3 (Discriminant Theorem [S3]). Let X be a topologically mixing countable Markov shift and let φ : X → R be some function with summable variations and finite Gurevich pressure. Let a ∈ S be some arbitrary fixed state.
658
V. Cyr, O. Sarig
(1) The equation PG (φ + p) = 0 has a unique solution p(φ) if a [φ] ≥ 0 and no solution if a [φ] < 0. The Gurevich pressure of φ is given by ! − p(φ) i f a [φ] ≥ 0 ; PG (φ) = − pa∗ [φ] i f a [φ] < 0 (2) φ is positive recurrent if a [φ] > 0 and transient if a [φ] < 0. In the case
a [φ] = 0, φ is either positive recurrent or null recurrent. In particular, strong positive recurrence implies positive recurrence. Definition 7.2. We say that φ is weakly positive recurrent if it is positive recurrent but not strongly positive recurrent. Corollary 7.1. Suppose X is topologically mixing and φ has summable variations and finite Gurevich pressure. If φ is recurrent then PG (φ) ≥ lim sup
1 log Z n∗ (φ, a) n
(7.5)
PG (φ) = lim sup
1 log Z n∗ (φ, a). n
(7.6)
n→∞
and if φ is transient then n→∞
The first equation is by definition of the pressure and Z n (φ, a) ≥ Z n∗ (φ, a). The second equation is the discriminant theorem and (7.3). 8. Appendix B: Proof of Theorem 1.1 Throughout this section, assume that T : X → X is a topologically mixing countable Markov shift, and that φ ∈ . We use the thermodynamic formalism for CMS as summarized in Appendix A. 8.1. Some technical implications of SGP.
Lemma 8.1. If φ has SGP, then the P in Definition 1.1 has the form Pg = h gdν for all g ∈ L, where h ∈ L is positive and bounded away from zero on cylinders, ν is finite and positive on cylinders, and L φ h = λh, L ∗φ ν = λν, hdν = 1. Proof. We show that φ is positive recurrent (Appendix A, Definition 7.1). The idea is to fix a ∈ S and show that λ−n Z n (φ, a) 1, where Z n (φ, a) = T n x=x eφn (x) 1[a] (x). This implies recurrence by definition, and rules out null recurrence because if φ were null recurrent, then λ−n Z n (φ, a) λ−n L nφ 1[a] −−−→ 0 on [a] because of Theorem 7.2, n→∞
part (2), which contradicts λ−n Z n (φ, a) 1. Write L φ = λP + N with λ = exp PG (φ) and P, N as in Definition 1.1. Since P N = N P = 0 and P 2 = P, L nφ = λn P + N n . Since the spectral radius of N is less than λ, λ−n N n = O(κ n ), where 0 < κ < 1. We have for (any) fixed x ∈ [a], λ−n Z n (φ, a) λ−n (L nφ 1[a] )(x) = P1[a] (x) + O(κ n ) (see (7.1) in Appendix A). It is impossible for P1[a] (x) to vanish, because this would imply that Z n (φ, a) =
Spectral Gap and Transience for Ruelle Operators
659
O((κλ)n ), whereas n1 log Z n (φ, a) → log λ and κ < 1. Thus P1[a] (x) = 0. It follows that λ−n Z n (φ, a) 1, whence the positive recurrence of φ. By the generalized RPF theorem (Appendix A, Theorem 7.2), ∃h positive, continuous, and bounded away from zero on cylinders, and ∃ν positive and finite on cylinders such that L φ h = λh, L ∗φ ν = λh, hdν = 1. Moreover, λ−n L nφ 1[a] −−−→ ν[a]h n→∞
pointwise. But λ−n L nφ 1[a] − P1[a] L ≤ λ−n N n 1[a] L → 0, so λ−n L nφ 1[a] → P1[a] pointwise. We see that P1[a] = ν[a]h. Since P(L) ⊆ L, h ∈ L. Since dim Im P = 1, there exists ϕ ∈ L∗ s.t. Pg = ϕ(g)h for all g ∈ L. We show that ϕ(g) = gdν for all g ∈ L. Let m φ := hdν. The relations L ∗φ ν = λν and L φ h = λh can be used to see that m φ is T –invariant measure. The methods of [ADU] show that it is mixing (even exact). 1 g ∈ L ∩ L 1 (ν), then gh −1 Suppose ∈ L (m φ ), and the mixing of m φ implies that −1 n (gh )1[a] ◦ T dm φ −−−→ m φ [a] gdν. On the other hand n→∞ (gh −1 )1[a] ◦ T n dm φ = g1[a] ◦ T n dν = λ−n L nφ (g1[a] ◦ T n )dν ) * −n n Pg + λ−n N n g dν −−−→ ϕ(g)m φ [a], = λ L φ gdν = [a]
n→∞
[a]
because λ−n N n g L → 0, whence λ−n N n g → 0 uniformly on [a]. Comparing the limits we see that ϕ(g) = gdν for all g ∈ L ∩ L 1 (ν). It remains to see that L ⊂ L 1(ν). Otherwise there exists f ∈ L s.t. | f |dν = ∞. Since f ∈ L, g := | f | ∈ L, and gh −1 dm φ = ∞. The mixing of m φ implies that (gh −1 )1[a] ◦ T n dm φ −−−→ ∞ n→∞
(bound gh −1 from below by a bounded function with large integral). But g ∈ L, so we can write as before (gh −1 )1[a] ◦ T n dm φ = [a] λ−n L nφ gdν −−−→ ϕ(g)m φ [a]. This n→∞ limit is finite, so we arrive at a contradiction. Lemma 8.2. Let ν be as in the previous lemma, then there exists some constant C0 s.t.
· L 1 (ν) ≤ C0 · L . Proof. Suppose f ∈ L. By assumption,L has the lattice property: f ∈ L ⇒ | f | ∈ L. By the previous lemma, P| f | = h | f |dν, so f L 1 (ν) = P| f | L / h L ≤
P
P
h L | f | L ≤ h L f L . So take C 0 := P / h L . Lemma 8.3. Suppose φ ∈ has the SGP. If ψ is (bounded and) Hölder continuous, then ψ f ∈ dom(L φ ) and L φ (ψ f ) ∈ L for all f ∈ L. The operator f → L φ (ψ f ) is a bounded linear operator on L. Proof. If f ∈ L, then | f | ∈ L. Since L ⊆ dom(L φ ), | f | ∈ dom(L φ ). If ψ is bounded, then |ψ f | < C| f | for some C, so |ψ f | ∈ dom(L φ ), whence ψ f ∈ dom(L φ ). Next, by assumption, t → L φ+tψ is a real analytic Hom(L, L)–valued map on a neighborhood of zero. This means that for every f ∈ L, t → L φ+tψ f is a real analytic L–valued map on a neighborhood of zero. Differentiating at zero, we see that there exists g ∈ L s.t. 1 L [L φ+tψ f − L φ f ] −−→ g ∈ L. t→0 t
660
V. Cyr, O. Sarig
Since L-convergence implies pointwise convergence, 1 g(x) = lim [L φ+tψ f − L φ f ](x) = L φ (ψ f )(x) + lim L φ t→0 t t→0
etψ − 1 −ψ t
f (x)
for every x ∈ X . Since ψ is bounded and | f | ∈ L ⊂ dom(L φ ),
tψ
τt
e − 1 e −1
L φ
− ψ f ≤ sup
− 1
ψ ∞ (L φ | f |)(x) −−→ 0.
t→0 t tτ |τ |≤ ψ ∞ Thus g(x) = L φ (ψ f )(x) for all x, whence L φ (ψ f ) = g ∈ L. We estimate L φ (ψ f ) L . We just saw that L φ (ψ f ) is the derivative at zero of the L–valued function t → L φ+tψ f . By SGP, this function extends to a holomorphic function z → L φ+zψ on some complex neighborhood U of the origin. Let C be a circle with center zero and radius r so small that C ⊂ U , then for every f ∈ L: + + + 1 , 1 + L f dz
L φ (ψ f ) L = + 2πi + ≤ r1 maxz∈C L φ+zψ · f L . φ+zψ 2 C z L
It follows that f → L φ (ψ f ) is a bounded operator.
8.2. Equilibrium measures. It was proved in [BS] that if a weakly Hölder continuous function φ with finite pressure and supremum has an equilibrium measure, then this measure is of the form hdν with h > 0 continuous and ν s.t. L φ h = λh, L ∗φ ν = λν, hdν = 1. Here we show the converse: If h, ν are as above, and dm = hdν has finite entropy, then it is an equilibrium measure (by [BS] the unique one). Let α := {[a] : a ∈ S} denote the natural generator. Lemma 8.4 (Rokhlin). Let µ be a shift invariant measure on a CMS X , and let α be the natural generator. Then h µ (T ) ≥ Hµ (α|α1∞ ), with equality when Hµ (α) < ∞. Proof. The equality when Hµ (α) < ∞ is standard, so we focus on the case when Hµ (α) = ∞. We use the following notational conventions for partitions. Suppose γ is a measurable partition of X , then σ (γ ) :=the sigma algebra generated by γ ; γmn := n -n −k γ =the smallest partition s.t. σ (γ n ) ⊇ −k γ ); and γ ∞ :=the m k=m T k=m σ (T 1 n smallest sigma–algebra which contains n≥1 σ (γ1 ). Take an increasing sequence of finite partitions β (n) such that σ (β (n) ) ↑ σ (α). For every fixed n, since Hµ (β (n) ) < ∞, k−1 . −1/ 1 1 (n) (n) H β = lim (β −H ) h µ T, β (n) = lim Hµ (β (n) )k−1 µ µ 0 0 k→∞ k k→∞ k 0 =1
1 k→∞ k
= lim
k−1 =1
k−1 1 Hµ β (n) |(β (n) )1 ≥ lim Hµ β (n) |α1 , k→∞ k
(8.1)
=1
because σ (α) ⊃ σ (β (n) ). We claim that Hµ β (n) |α1 −−−→ Hµ β (n) |α1∞ . →∞
(8.2)
Spectral Gap and Transience for Ruelle Operators
661
This is because (a) Iµ β (n) |α1 −−−→ Iµ β (n) |α1∞ µ–a.e. (Martingale Convergence Theorem) →∞ (b) sup≥1 Iµ β (n) |α1 dµ < ∞, by the Chung–Neveu Lemma ([P], Lemma 2.1); (c) the Dominated Convergence Theorem. By (8.1) and (8.2), for all n, h µ T, β (n) ≥ Hµ β (n) |α1∞ ≡ Iµ β (n) |α1∞ dµ. Now Iµ β (n) |α1∞ ↑ Iµ (α|α1∞ ), because β (n) increase to α (see e.g. [P], Theorem 2.2 (ii)). By the Monotone Convergence Theorem Hµ β (n) |α1∞ ↑ Hµ α|α1∞ , and we con clude that h µ T, β (n) ≥ Hµ β (n) |α1∞ −−−→ Hµ α|α1∞ . Since h µ (T )≥h µ T, β (n) , n→∞ the proof is completed. Proposition 8.1. Suppose φ has summable variations, has finite Gurevich pressure, and sup φ < ∞. Suppose further that h > 0 ispositive continuous, ν is positive and finite on cylinders, L φ h = λh, L ∗φ ν = λν, and hdν = 1. If dµ = hdν has finite entropy, then it is an equilibrium measure of φ. dµ ≡ −[φ + ln h − ln h ◦ T − Proof. One can show, as in [L], that Iµ α|α1∞ = − ln dµ◦T PG (φ)], so (Iµ (α|α1∞ ) + φ + ln h − ln h ◦ T )dµ = PG (φ). By Lemma 8.4, Iµ α|α1∞ dµ = Hµ α|α1∞ ≤ h µ (T ) < ∞, so Iµ is absolutely integrable (it is a non-negative function). Since φ + ln h − ln h ◦ T is bounded from above (by PG (φ)), it is also absolutely integrable, and h µ (T ) + [φ + ln h − ln h ◦ T ]dµ ≥ [Iµ + φ + ln h − ln h ◦ T ]dµ = PG (φ). (8.3)
We claim that φ ∈ L 1 (µ), and φdµ = [φ + ln h − ln h ◦ T ]dµ. The following holds for almost every x ∈ X : (a) φn (x)/n −−−→ φdµ (because sup φ < ∞ and µ is ergodic); n→∞ ) * (b) φn (x) + ln h(x) − ln h (T n x) /n −−−→ [φ + ln h − ln h ◦ T ]dµ (because n→∞
φ + ln h − ln h ◦ T ∈ L 1 (µ)); (c) ∃n k (x) ↑ ∞ s.t. | ln h(x) − ln h(T n k (x) x)| ≤ 1 (because of the Poincaré recurrence theorem, and the continuity of h). Choose one such x, then 1 1 φn k (x) = lim φn k (x) + ln h(x) − ln h(T n k (x) x) φdµ = lim k→∞ n k (x) k→∞ n k (x) 1 φn + ln h(x) − ln h(T n x) = (φ + ln h − ln h ◦ T ) dµ. = lim n→∞ n By (8.3) h µ (T ) + φdµ ≥ PG (φ). The proposition now follows from the variational principle (Appendix A, Theorem 7.1).
662
V. Cyr, O. Sarig
8.3. Proof of Theorem 1.1. Suppose that X is topologically mixing, and φ ∈ has the SGP and satisfies sup φ < ∞. Let λ, P, N be as in Definition 1.1. Proof of (a). Lemma 8.1 says that P is of the form P f = h f dν, where L φ h = λh, L ∗φ ν = λν, and hdν = 1. Proposition 8.1 says that if dm φ = hdν has finite entropy, then m φ is an equilibrium measure for φ. By [BS], there is at most one such measure, so m φ is unique. Proof of (b). Let ρ(N ) denote the spectral radius of N : L → L. By the SGP, ∃κ ∈ (ρ(N )/λ, 1). If f is (bounded and) Hölder continuous, then L φ ( f h) ∈ L (Lemma 8.3). If g ∈ L ∞ (m φ ), then the identities dm φ = hdν, L ∗φ ν = λν, and P L φ ( f h) = λh f dm φ imply
−n n n−1
f · g ◦ T n dm φ −
λ = gdm f dm [L ( f h) − λ P L ( f h)]gdν φ φ
φ φ
+ + + + + + + + ≤ g ∞ +λ−n N n−1 L φ ( f h)+ 1 ≤ C0 g ∞ +λ−n N n−1 L φ ( f h)+ ≤ C0 λ
−1
g ∞ λ
−(n−1)
N
L (ν) n−1
L
L φ ( f h) L ≤ const g ∞ L φ ( f h) L κ n .
Part (c). We assume without loss of generality that λ = 1, Em φ [ψ] = 0. To arrange this, replace φ by φ − log λ and ψ by ψ − Em φ [ψ]. Part (e) of the SGP is stable under perturbation in Hom(L, L) [K]: There exists a neighborhood U of L φ in Hom(L, L) and analytic maps P, N : U → Hom(L, L), λ : U → C such that for all L ∈ U , L = λ(L)P(L) + N (L), P(L)N (L) = N (L)P(L) = 0, P(L)2 = P(L), dim Im P(L) = 1. If U is sufficiently small, then there is some ε0 > 0 s.t. for all L ∈ U , the spectral radius of N (L) is less than 1 − 2ε0 and the spectral radius of L (equal to |λ(L)|) is more than 1 − ε0 . By the SGP, t → L t := L φ+itψ is analytic on a neighborhood of zero. The maps λt = P(L t ), Pt = P(L t ), Nt = N (L t ) must also be analytic in t on a small neighborhood I of zero. Recall that there is a constant C0 s.t. · L 1 (ν) ≤ C0 · L . For t in I , Em φ eitψn = λ−n L nφ eitψn h dν = L nt hdν (∵ λ = 1) * ) n = λnt Pt h + λ−n t Nt h dν = λnt [1 ± C0 Pt − P + |λt |−n Ntn h L ]. The spectral radius of Nt is less than 1 − 2ε0 , and |λt | ≥ 1 − ε0 , so this gives Em φ [eitψn ] = λnt [1 + εn (t)] for all n, where εn (t) −−−−−−→ 0. t→0,n→∞
Later we will see that if ψ is not cohomologous to a constant, then λt = 1 −
σ2 2 t + o(t 2 ) as t → 0, 2
(8.4)
Spectral Gap and Transience for Ruelle Operators
663
√ where σ > 0. It will then follow that Em φ [exp(itψn / n)] −−−→ exp(−σ 2 t 2 /2), which n→∞
means that √1n ψn converges in distribution (w.r.t m φ ) to a normal law with mean zero and standard deviation σ . To prove (8.4), we expand λt as in [GH]. Define for this purpose h t := Pt 1/ Pt 1dν (the denominator approaches 1 as t → 0 so it is not zero for all |t| sufficiently small), and write L := L 0 = L φ . Then L t h t = λt h t and so λt = L t h t dν = (L t − L)(h t − h)dν + (L t − L)hdν + Lh t dν = (L t − L)(h t − h)dν + Eν [L((eitψ − 1)h)] + h t dν (∵ L ∗ ν = λν = ν) t2 (8.5) = (L t − L)(h t − h)dν − ψ 2 hdν + o(t 2 ) + 1, 2 2
where we have used the fact that ψ is bounded to expand eitψ = 1 + itψ − t2 ψ 2 + o(t 2 ), and the assumption that Em φ [ψ] = 0 to note that ψhdν = 0. (The assumption that ψ is bounded is an overkill.) The analyticity of t → L t , Pt and the estimate · L 1 (ν) ≤ C0 · L can be used to show that (L t − L)(h t − h)dν = o(t) as t → ∞. Thus λt = 1 + o(t). Next we study the difference h t − h, as in [G1]. In what follows, o(1) means an element of L whose L–norm is o(1): ht − h λt h t − h = + o(1) (because λt = 1 + o(t) and h t L is bounded near zero) t t ht − h L t h t − Lh ht + o(1) = (L t − L) + L + o(1). = t t t Subtracting the second summand from both sides, we obtain itψ ht − h = (L t − L) htt + o(1) = L e t −1 h t + o(1). (1 − L) t
(8.6)
The left side of (8.6) converges in L, whence in L 1 (ν), to (1 − L)a, where
d
ht . a := dt t=0 The right side of (8.6) converges in L 1 (ν) to i L(ψh). To see this, note the following: (a) ψ is bounded, so ∃M s.t. |(eitψ − 1)/t| ≤ M for all |t| < 1; itψ L 1 (ν) e −1 h −−−→ iψh, because of the dominated convergence theorem and the (b) t t→0
≤ C+0 h L < ∞; bound + itψ h L 1 (ν) + + e −1 (h t − h)+ 1 ≤ C0 M h t − h L −−→ 0. (c) + t t→0 L (ν) itψ Thus e t −1 h t −−→ iψh in L 1 (ν). t→0
Now L extends to a bounded operator on L 1 (ν) s.t. L ≤ 1 (the transfer operator itψ L 1 (ν) of ν), so L e t −1 h t −−−→ i L(ψh). t→0
664
V. Cyr, O. Sarig
Passing to the limit t → 0 in (8.6), we see that (1 − L)a = i L(ψh) ν–a.e. Since all elements of L are assumed to be continuous, and since ν is globally supported, (1 − L)a = i L(ψh). Apply L k to both sides: L k a − L k+1 a = i L k (ψh). The norm of the right hand side is summable:
L k (ψh) L = P(L(ψh)) + N k−1 L(ψh) L = N k
L(ψh) L (∵ P[L(ψh)] = h
L[ψh]dν = h
ψdm φ = 0),
k and N k < ∞. Summing over k ≥ 0, we obtain a = i ∞ k=1 L (ψh). Returning to the expansion (8.5) of λt , we see that t2 λt = (L t − L)(at + o(t))dν − ψ 2 hdν + o(t 2 ) + 1 2 itψ e −1 t2 (a + o(1))dν − = t2 ψ 2 hdν + o(t 2 ) + 1 t 2 ∞ t2 L k (ψh)dν + o(t 2 ), = 1− ψ 2 hdν − t 2 ψ 2 k=1
and we obtain (8.4) with
σ := 2
∞ 2 k ψ + ψ L (ψh) dm φ . h 2
k=1
But it is not yet clear that σ 2
is strictly positive. Tok see this we follow [G1] and rewrite the integrand in terms of the function u := ∞ k=0 L (ψh), noting that ψh = u − Lu: 1 1 2 2 2 (ψh) (u − Lu) σ = + 2ψh(u − ψh) dm = + 2(u − Lu)Lu dm φ φ h2 h2
2 1 2 1 2 2 L(h · u/ h) u dm = − (Lu) = (u/ h) − dm φ . φ h2 h ' : v → h −1 L(hv) preserves m φ : T '∗ m φ = m φ (it is the transfer The operator T operator of m φ = hdν). Thus we get '[(u/ h)2 ] − (T '(u/ h))2 dm φ . T σ2 = ' takes the form T ' f = T y=x g(y) f (y), where g = eφ It is not difficult to see that T '[(u/ h)2 ] ≥ (T '(u/ h))2 , h/ h ◦ T . We have T y=x g(y) ≡ 1. Since t → t 2 is convex, T '[(u/ h)2 ] = [T '(u/ h)]2 m φ –a.e. and we get that σ 2 ≥ 0, with equality iff T 2 By the strict convexity of t → t , u/ h must be constant on {y : T y = x} for a.e. x. Since m φ ∼ m φ ◦ T (∵ dm/dm ◦ T = eφ h/ h ◦ T > 0), this means that there is a function ϕ s.t. u/ h = ϕ ◦ T almost everywhere. Thus ψ=
1 1 (u − Lu) = ϕ ◦ T − L(hϕ ◦ T ) = ϕ ◦ T − ϕ a.e. h h
Spectral Gap and Transience for Ruelle Operators
665
It follows that ψ is an almost everywhere coboundary w.r.t m φ . By the Livsic theorem of Gouëzel [G2], ψ is a coboundary with a continuous transfer function. But part (c) assumes that ψ is not like that. Part (d). Suppose ψ is a (bounded) Hölder continuous function, and let L t , λt , Pt , Nt be as above. We saw that t → λt , Pt , Nt are analytic on some complex neighborhood of 0, and that for all |t| sufficiently small, ρ(Nt ) < |λt |. We claim that λt = exp PG (φ + tψ) on some real neighborhood of t = 0. This is because of the estimates Z n (φ + tψ, a) L nt 1[a] (x) = λnt Pt 1[a] (x) + Ntn 1[a] (x) λnt , which hold uniformly in x on [a] provided t is small enough that ρ(Nt ) < |λt | (see Appendix A, Remark 7.1). In particular λ0 = exp PG (φ) = 0, and PG (φ + tψ) = log λt is real analytic on a neighborhood of zero. Note added in proof. The first author has recently found a combinatorial characterization of the topologically mixing CMS for which {φ ∈ : φ does not have SGP} is not empty. As it turns out, “most” infinite state CMS are like that, e.g. all CMS whose associated transition graph G contains an infinite ray. Complete statements and detailed proofs will appear elsewhere. References [ADU] Aaronson, J., Denker, M., Urbanski, F.: Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337(2), 495–548 (1993) [A] Aaronson, J.: An introduction to infinite ergodic theory. Math. Surv. and Monog. 50, Providence, RI: Amer. Math. Soc. 1997 [AD] Aaronson, J., Denker, M.: Local limit theorems for Gibbs–Markov maps. Stochastics Dyn. 1, 193– 237 (2001) [B] Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics 16, Singapore: World Scientific, 2000 [BS] Buzzi, J., Sarig, O.: Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Erg. Thy. Dynam. Sys. 23, 1383–1400 (2003) [DF] Doeblin, W., Fortet, R.: Sur des chaînes à liasions complètes. Bull. Soc. Math. France 65, 132– 148 (1937) [GM] Gallavotti, G., Miracle-Sole, S.: Statistical mechanics of lattice systems. Commun. Math. Phys. 5(5), 317–323 (1967) [G1] Gouëzel, S.: Central limit theorem and stable laws for intermittent maps. Probab. Theory Rel. Fields 128(1), 82–122 (2004) [G2] Gouëzel, S.: Regularity of coboundaries for nonuniformly expanding Markov maps. Proc. Amer. Math. Soc. 134(2), 391–401 (2006) [GH] Guivarc’h, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. (In French, English summary) [Limit theorems for a class of Markov chains and applications to Anosov diffeomorphisms] Ann. Inst. H. Poincaré Probab. Statist. 24(1), 73–98 (1988) [GS] Gurevich, B.M., Savchenko, S.V.: Thermodynamics formalism for countable Markov chains. Usp Mat. Nauk 53, 2, 3–106 (1998). Engl. transl. in Russ. Math. Surv. 53:2 3–106 (1998) [HH] Hennion, H., Hervé, L.: Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi–compactness, LNM 1766, Berlin Heidelberg-New York: Springer, 2001 [K] Kato, T.: Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics. Berlin: Springer-Verlag, 1995 [L] Ledrappier, F.: Principe variationnel et systemes dynamiques symboliques. Z. Wahrs. Verb. Geb. 30, 185–202 (1974)
666
V. Cyr, O. Sarig
[Li]
Liverani, C.: Central limit theorem for deterministic systems. International Conference on Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. Ser. 362, Harlow: Longman, 1996, pp. 56–75 Lopes, A.: The zeta function, nondifferentiability of pressure, and the critical exponent of transition. Adv. Math. 101(2), 133–165 (1993) Parry, W.: Entropy and generators in ergodic theory. New York: W.A. Benjamin Inc., 1969 Parry, W.: Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–8 (1990) Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66(1–2), 503–514 (1992) Ruelle, D.: Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics. Second Edition, Cambridge: Cambridge Univ. Press, 2004 Ruette, S.: On the Vere-Jones classification and existence of maximal measures for countable topological Markov chains. Pacific J. Math. 209(2), 365–380 (2003) Sarig, O.: Thermodynamic formalism for countable Markov shifts. Erg. Th. Dynam. Syst. 19, 1565– 1593 (1999) Sarig, O.: Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121, 285–311 (2001) Sarig, O.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217, 555–577 (2001) Sarig, O.: Subexponential decay of correlations. Invent. Math. 150, 629–653 (2002) Sarig, O.: Critical exponents for dynamical systems. Commun. Math. Phys. 267, 631–667 (2006) Vere–Jones, D.: Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford 13(2), 7–28 (1962) Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(2), 585–650 (1998)
[Lo] [P] [PP] [PrS] [R] [Rt] [S1] [S2] [S3] [S4] [S5] [VJ] [Y]
Communicated by G. Gallavotti
Commun. Math. Phys. 292, 667–719 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0886-1
Communications in
Mathematical Physics
Lie Conformal Algebra Cohomology and the Variational Complex Alberto De Sole1 , Victor G. Kac2 1 Dipartimento di Matematica, Universitá di Roma “La Sapienza”,
Cittá Universitaria, 00185 Rome, Italy. E-mail: [email protected]
2 Department of Mathematics, MIT, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA. E-mail: [email protected] Received: 22 December 2008 / Accepted: 11 June 2009 Published online: 14 August 2009 – © Springer-Verlag 2009
Dedicated to Corrado De Concini on his 60th birthday Abstract: We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a g-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators. 1. Introduction Lie conformal algebras encode the properties of operator product expansions in conformal field theory, and, at the same time, of local Poisson brackets in the theory of integrable evolution equations. Recall [K] that a Lie conformal algebra over a field F is an F[∂]-module A, endowed with a λ-bracket, that is an F-linear map A ⊗ A → F[λ] ⊗ A, denoted by a ⊗ b → [aλ b], satisfying the two sesquilinearity properties [∂aλ b] = −λ[aλ b],
[aλ ∂b] = (∂ + λ)[aλ b],
(1)
such that the skew-symmetry [aλ b] = −[b−∂−λ a]
(2)
[aλ [bµ c]] − [bµ [aλ c]] = [[aλ b]λ+µ c]
(3)
and the Jacobi identity
hold for any a, b, c ∈ A. It is assumed in (2) that ∂ is moved to the left. A module over a Lie conformal algebra A is an F[∂]-module M, endowed with a λ-action, that is an F-linear map A ⊗ M → F[λ] ⊗ M, denoted by a ⊗ b → aλ b, such
668
A. De Sole, V. G. Kac
that sesquilinearity (1) holds for a ∈ A, b ∈ M and the Jacobi identity (3) holds for a, b ∈ A, c ∈ M. A cohomology theory for Lie conformal algebras was developed in [BKV]. Given a Lie conformal algebra Aand an A-module M, one first defines the basic cohomology complex Γ• (A, M) = k∈Z+ Γk , where Γk consists of F-linear maps γ : A⊗k → F[λ1 , . . . , λk ] ⊗ M, satisfying certain sesquilinearity and skew-symmetry properties, and endows this complex with a differential δ : Γk → Γk+1 , such that δ 2 = 0. This complex is isomorphic to the Lie algebra cohomology complex for the annihilation Lie algebra g− of A with coefficients in the g− -module M [BKV, Theorem 6.1]. Next, one endows Γ• (A, M) with a structure of a F[∂]-module, such that ∂ commutes with δ, which allows one to define the reduced cohomology complex Γ • (A, M) = Γ• (A, M)/∂ Γ• (A, M), and this is the Lie conformal algebra cohomology complex, introduced in [BKV]. Our first contribution to this theory is a more explicit construction of the reduced cohomology complex. Namely, we introduce a new cohomology complex C • (A, M) = ⊕k∈Z+ C k , where C 0 = M/∂ M, C 1 = HomF[∂] (A, M), and for k ≥ 2, C k consists of poly λ-brackets, namely of F-linear maps c : A⊗k → F[λ1 , . . . , λk−1 ] ⊗ M, satisfying certain sesquilinearity and skew-symmetry conditions, and we endow C • (A, M) with a square zero differential d. We construct embeddings of complexes: Γ • (A, M) ⊂ C¯ • (A, M) ⊂ C • (A, M),
(4)
C¯ • (A,
where M) consists of cocycles which vanish if one of the arguments is a torsion element of A. In fact, C¯ k = C k , unless k = 1. We show that Γ • (A, M) = C¯ • (A, M), provided that, as an F[∂]-module, A is isomorphic to a direct sum of its torsion and a free F[∂]-module (which is always the case if A is a finitely generated F[∂]-module). Our opinion is that the slightly larger complex C • (A, M) is a more correct Lie conformal algebra cohomology complex than the complex Γ • (A, M) of [BKV]. This is illustrated by our Theorem 3.1(c), which says that the F[∂]-split abelian extensions of A by M are parameterized by H 2 (A, M) for the complex C • (A, M). This holds for the cohomology theory of [BKV] only if A is a free F[∂]-module. Following [BKV], we also consider the superspace of basic chains Γ• (A, M) and its subspace of reduced chains Γ• (A, M) (they are not complexes in general). Corresponding to the embeddings of complexes (4), we introduce the vector superspaces of chains C• (A, M) and C¯ • (A, M), and the maps: C• (A, M) C¯ • (A, M) → Γ• (A, M).
(5)
We develop the theory further in the important case for the calculus of variations, when the A-module M is endowed with a commutative associative product, such that ∂ and aλ for all a ∈ A are derivations of this product. In this case one can endow the superspace Γ• (A, M) with a commutative associative product [BKV]. Furthermore, we introduce a Lie algebra bracket on the space g := Π Γ1 (A, M) (Π , as usual, stands for reversing of the parity). Let g = ηg ⊕ g ⊕ F∂η be a Z-graded Lie superalgebra extension of g, where η is an odd indeterminate, η2 = 0. We endow Γ• (A, M) with a structure of a g-complex, which is a Z-grading preserving Lie superalgebra homomorphism ϕ : g → EndF Γ• (A, M), such that ϕ(∂η ) = δ. We also show that ϕ( g) lies in the • subalgebra of derivations of the superalgebra Γ (A, M). For each X ∈ g we thus have the Lie derivative L X = ϕ(X ) and the contraction operator ι X = ϕ(ηX ), satisfying all the usual relations, in particular, the Cartan formula L X = ι X δ + δι X .
Lie Conformal Algebra Cohomology and the Variational Complex
669
Denoting by g∂ the centralizer of ∂ in g, we obtain the induced structure of a g∂ -complex for Γ • (A, M), which we, furthermore, extend to the larger complex C • (A, M). Namely, we introduce a canonical Lie algebra bracket on all spaces of 1-chains with reversed parity (see (5)), so that all the maps ΠC1 Π C¯ 1 → Π Γ1 → Π Γ1 are Lie algebra homomorphisms, and the embeddings (4) are morphisms of complexes, endowed with a corresponding Lie algebra structure. What does it all have to do with the calculus of variations? In order to explain this, introduce the notion of an algebra of differential functions (in variables). This is a differential algebra, i.e., a unital commutative associative algebra V with a derivation ∂, endowed with commuting derivations ∂(n) , i ∈ I = {1, . . . , }, n ∈ Z+ , such that only a finite number of rules with ∂ hold:
∂f (n) ∂u i
∂u i
are non-zero for each f ∈ V, and the following commutation
∂ (n) ∂u i
,∂ =
∂
(the RHS is 0 if n = 0) .
(n−1) ∂u i
(6) (n)
An important example is the algebra of differential polynomials F[u i |i ∈ I , n ∈ Z+ ] (n) (n+1) , n ∈ Z+ , i ∈ I . Other examples include any localization by a with ∂(u i ) = u i multiplicative subset or any algebraic extension of this algebra. • = Ω • (V) over V is defined as an exterior superalgeThe basic de Rham complex Ω 1 = i∈I, n∈Z Vδu (n) on generators δu (n) with odd parity. bra over the free V-module Ω i i + k , where Ω 1 . This Z-graded superalgebra 0 = V, Ω k = Λk Ω • = k∈Z Ω We have: Ω V + (n) f is endowed by an odd derivation δ of degree 1, such that δ f = i∈I, n∈Z+ ∂ (n) δu i for ∂u i
0 and δ(δu (n) ) = 0. One easily checks that δ 2 = 0, so that Ω • is a cohomology f ∈Ω i complex. Let g be the Lie algebra of derivations of the algebra V of the form X=
i∈I, n∈Z+
Pi,n
∂ (n)
∂u i
,
where Pi,n ∈ V.
(7)
To any such derivation X we associate an even derivation L X (Lie derivative) and an odd • by letting L X |V = X, L X (δu (n) ) = derivation ι X (contraction) of the superalgebra Ω i (n) • with a structure of a g-complex, δ Pi,n , ι X |V = 0, ι X (δu i ) = Pi,n . This provides Ω by letting ϕ(X ) = L X and ϕ(ηX ) = ι X . Also, the derivation ∂ extends to an (even) • by letting ∂(δu (n) ) = δu (n+1) . derivation of Ω i i It is easy to check, using (6), that ∂ and δ commute, hence we can consider the reduced complex • (V)/∂ Ω • (V), Ω • (V) = Ω which is called the variational complex. This is, of course, a g∂ -complex. Our main observation is the interpretation of the variational complex Ω • (V) in terms of Lie conformal algebra cohomology, given by Theorem 1 below. Let R = i∈I F[∂]u i be a free F[∂]-module of rank , endowed with the trivial λ-bracket [aλ b] = 0 for all a, b ∈ R. Let V be an algebra of differential functions.
670
A. De Sole, V. G. Kac
We endow V with the structure of an R-module by letting u iλ f =
n∈Z+
λn
∂f ∂u i(n)
, i ∈ I,
and extending to R by sesquilinearity. Let g be the Lie algebra of derivations of V of the form (7), and let g∂ be the subalgebra of g, consisting of derivations commuting with ∂. Theorem 1. The g∂ -complexes C • (R, V) and Ω • (V) are isomorphic. As a result, we obtain the following interpretation of the complex Ω • (V), which explains the name “calculus of variations”. of Ω 0 are called We have: Ω 0 = V/∂V, Ω 1 = HomF[∂] (R, V) = V ⊕ . Elements 0 local functionals and the image of f ∈ V in Ω is denoted by f . Elements of Ω 1 are called local 1-forms. differential δ : Ω 0 → Ω 1 is identified with the variational
The δ f = δδuf , where derivative: δ f = δu i i∈I
δf ∂f = (−∂)n (n) . δu i ∂u i n∈Z+
(8)
Furthermore, the space of 2-cochains C 2 is identified with the space of skew-adjoint differential operators by associating to a λ-bracket {· λ ·} : R ⊗2 → F[λ] ⊗ V the × matrix Si j (∂) = {u j ∂ u i }→ , where the arrow means that ∂ is moved to the right. The differential δ : Ω 1 → Ω 2 is expressed in terms of the Frechet derivative D F (∂)i j =
∂ Fi (n)
n∈Z+ ∂u j
∂n,
i, j ∈ I,
(9)
which defines an F-linear map: V → V ⊕ . Namely: δ F = D F (∂) − D F (∂)∗ . The subspace of closed 2-cochains in C 2 is identified with the space of symplectic differential operators. A 2-cochain, which is a skew-adjoint differential operator Si j (∂), can be identified with the corresponding F-linear map (V )2 → V/∂V, of “differential type”, given by Q i Si j (∂)P j . S(P, Q) = i, j∈I
Skew-adjointness of S translates to the skew-symmetry condition S(P, Q) = −S(Q, P). More generally, the space of k-cochains C k for k ≥ 2 is identified with the space of all skew-symmetric F-linear maps S : (V )k → V/∂V, of “differential type”: 1 ,...,n k f in1 ,...,i (∂ n 1 Pi11 ) · · · (∂ n k Pikk ), S(P 1 , . . . , P k ) = k i 1 ,...,i k ∈I n 1 ,...,n k ∈Z+
1 ,...,n k where f in1 ,...,i ∈ V. The skew-symmetry condition is simply S(P 1 , . . . , P k ) = k sign(σ )S(P σ (1) , . . . , P σ (k) ), for every σ ∈ Sk . The subspace of closed k-cochains for k ≥ 2 is the subspace of “symplectic” k-differential operators.
Lie Conformal Algebra Cohomology and the Variational Complex
671
We prove in [BDK] that the cohomology H j of the complex Ω • (V) is zero for j ≥ 1 f and H 0 = C/(C ∩ ∂V), where C := { f ∈ V | ∂ (n) = 0 ∀i ∈ I, n ∈ Z+ }, provided ∂u i
that V is normal, as defined in Sect. 5.6. (Any algebra of differential functions can be δ included in a normal one.) As a corollary, we obtain (cf. [D]) that Ker δu = ∂V + C, and δ F ∈ Im δu iff D F (∂) is a self-adjoint differential operator, provided that V is normal. The first result can be found in [D] (see also [Di] and [Vi], where it is proved under stronger conditions on V), but it is certainly much older. The second result, at least under stronger conditions on V, goes back to [H], [V]. We also obtain the classification of symplectic differential operators (cf. [D]) and of symplectic poly-differential operators for normal V, which seems to be a new result. Thus, the interaction between the Lie conformal algebra cohomology and the variational calculus has led to progress in both theories. On the one hand, the variational calculus motivated some of our constructions in the Lie conformal algebra cohomology. On the other hand, the Lie conformal algebra cohomology interpretation of the variational complex has led to a better understanding of this complex and to a classification of symplectic differential operators. The ground field is an arbitrary field F of characteristic 0. We wish to thank Bojko Bakalov for very valuable comments, in particular, for the observation that our complex C • (A, M) is isomorphic to the complex in [BDAK, Sect. 15.1], in the case when the Hopf algebra H is F[∂]. 2. Lie Conformal Algebra Cohomology Complexes 2.1. The basic cohomology complex Γ• and the reduced cohomology complex Γ • . Let us review, following [BKV], the definition of the basic and reduced cohomology complexes associated to a Lie conformal algebra A and an A-module M. A k-cochain of A with coefficients in M is an F-linear map γ : A⊗k → F[λ1 , . . . , λk ] ⊗ M, a1 ⊗ · · · ⊗ ak → γλ1 ,...,λk (a1 , . . . , ak ), satisfying the following two conditions: A1. γ λ1 ,...,λk (a1 , . . . , ∂ai , . . . , ak ) = −λi γλ1 ,...,λk (a1 , . . . , ak ) for all i, A2. γ is skew-symmetric w.r.t. simultaneous permutations of the ai ’s and the λi ’s. Remark 1. Note that Condition A1 implies that γλ1 ,...,λk (a1 , . . . , ak ) is zero if one of the elements ai is a torsion element of the F[∂]-module A. k • • k We letk Γ = Γ (A, M) be the space of all k-cochains, and Γ = Γ (A, M) = γ is defined by the following formula: k≥0 Γ . The differential δ of a k-cochain (δ γ )λ1 ,...,λk+1 (a1 , . . . , ak+1 ) = +
k+1 i, j=1 i< j
k+1
(−1)i+1 ai λi
γ i
i j
λ1 ,··· ˇ ···,λ ˇ k+1 ,λi +λ j
i
λ1 ,···,λ ˇ k+1
i=1
(−1)k+i+ j+1 γ
i
j
(a1 , · ˇ· ·, ak+1 ) (10)
(a1 , · ˇ· ·· ˇ· ·, ak+1 , [ai λi a j ]).
One checks that δ maps Γk to Γk+1 , and that δ 2 = 0. The Z-graded space Γ• (A, M) with the differential δ is called the basic cohomology complex associated to A and M.
672
A. De Sole, V. G. Kac
Define the structure of an F[∂]-module on Γ• by letting (∂ γ )λ1 ,...,λk (a1 , . . . , ak ) = (∂ M + λ1 + · · · + λk ) γ λ1 ,...,λk (a1 , . . . , ak ) ,
(11)
where ∂ M denotes the action of ∂ on M. One checks that δ and ∂ commute, and therefore ∂ Γ• ⊂ Γ• is a subcomplex. We can consider the reduced cohomology com plex Γ • (A, M) = Γ• (A, M)/∂ Γ• (A, M) = k∈Z+ Γ k (A, M). For example, Γ 0 = M/∂ M M, and we denote, as in the calculus of variations, by m the image of m ∈ M in M/∂ M M. As before we let, for brevity, Γ • = Γ • (A, M) and Γ k = Γ k (A, M), k ∈ Z+ . In the following sections we will find a simpler construction of the reduced cohomology complex Γ • , in terms of poly λ-brackets.
2.2. Poly λ-brackets. Let A and M be F[∂]-modules, and, as before, denote by ∂ M the action of ∂ on M. For k ≥ 1, a k-λ-bracket on A with coefficients in M is, by definition, an F-linear map c : A⊗k → F[λ1 , . . . , λk−1 ] ⊗ M, denoted by a1 ⊗ · · · ⊗ ak → {a1λ1 · · · ak−1 λk−1 ak }c , satisfying the following conditions: B1. {a1λ1 · · · (∂ai )λi · · · ak−1 λk−1 ak }c = −λi {a1λ1 · · · ak−1 λk−1 ak }c , for 1 ≤ i ≤ k − 1; B2. {a1λ1 · · · ak−1 λk−1 (∂ak )}c = (λ1 + · · · + λk−1 + ∂ M ){a1λ1 · · · ak−1 λk−1 ak }c ; B3. c is skew-symmetric with respect to simultaneous permutations of the ai ’s and the λi ’s in the sense that, for every permutation σ of the indices {1, . . . , k}, we have: {a1λ1 · · · ak−1 λk−1 ak }c = sign(σ ){aσ (1) λσ (1) · · · aσ (k−1) λσ (k−1) aσ (k) }c The notation in the RHS means that λk is replaced by λ†k = − occurs, and ∂ M is moved to the left.
k−1
j=1 λ j
λk →λ†k
.
− ∂ M , if it
Remark 2. A structure of a Lie conformal algebra on A is a 2-λ-bracket on A with coefficients in A, satisfying the Jacobi identity (3). We let C 0 = M/∂ M M and, for k ≥ 1, we denote by C k = C k (A, M) the space 1 of all k-λ-brackets on A with coefficients in M. For example, C is the space of all F[∂]-module homomorphisms c : A → M. We let C • = k∈Z+ C k , the space of all poly λ-brackets. We also define C¯ • = k∈Z+ C¯ k , where C¯ 0 = C 0 = M/∂ M M, and C¯ k ⊂ C k is the subspace of k-λ-brackets c with the following additional property: {a1λ1 · · · ak−1 λk−1 ak }c is zero if one of the elements ai is a torsion element in A. Clearly, C¯ 1 needs not be equal to C 1 . On the other hand, it is easy to check, using the sesquilinearity Conditions B1 and B2, that C¯ k = C k for k ≥ 2.
Lie Conformal Algebra Cohomology and the Variational Complex
673
2.3. The complex of poly λ-brackets. We next define a differential d on the space C • of poly λ-brackets such that d(C k ) ⊂ C k+1 and d 2 = 0, thus making C • a cohomology complex. For m ∈ C 0 = M/∂ M M, we let d m ∈ C 1 be the following F[∂]-module homomorphism:
(12) d m (a) = {a}d m := a−∂ M m. This is well defined since, if m ∈ ∂ M M, the RHS is zero due to sesquilinearity. For c ∈ C k , with k ≥ 1, we let dc ∈ C k+1 be the following poly λ-bracket: {a1λ1 · · · ak λk ak+1 }dc := k
+
i=1 i
j
c
(−1)k+i+ j+1 a1λ1 · ˇ· ·· ˇ· · ak λk ak+1 λ† [ai λi a j ] k+1
i, j=1 i< j
+(−1)k ak+1 λ†
k+1
+
k i (−1)i+1 ai λi a1λ1 · ˇ· · ak λk ak+1
a1λ1 · · · ak−1 λk−1 ak
c
c
i (−1)i a1λ1 · ˇ· · ak λk [ai λi ak+1 ] ,
k
(13)
c
i=1
where, as before, λ†k+1 = − kj=1 λ j − ∂ M , and ∂ M is moved to the left. For example, for an F[∂]-module homomorphism c : A → M, we have {aλ b}dc = aλ c(b) − b−λ−∂ c(a) − c([aλ b]).
(14)
Proposition 1. (a) For c ∈ C k , we have d(c) ∈ C k+1 and d 2 (c) = 0. This makes (C • , d) a cohomology complex. (b) d(C¯ k ) ⊂ C¯ k+1 for all k ≥ 0. Hence (C¯ • , d) is a cohomology subcomplex of (C • , d). Proof. We prove part (b) Mfirst. For k ≥ 1 there is nothing to prove. For k = 0 just notice that, if m ∈ M/∂ M and a ∈ A is a torsion element, then, by (12), we have d m (a) = 0, since torsion elements of A act trivially in any module [K]. Hence d m ∈ C¯ 1 . In order to prove part (a) we have to check that, if c ∈ C k , then dc, defined by (12) and (13), satisfies Conditions B1, B2, B3, and d(dc) = 0. To simplify the arguments, we rewrite (13) in a concise form: {a1λ1 · · · ak λk ak+1 }dc := +
k+1 i, j=1 i< j
k+1
(−1)k+i+ j+1 a1λ1
(−1)
i+1
ai λi
i a1λ1 · ˇ· · ak λk ak+1
i=1
i j · ˇ· ·· ˇ· · ak+1λk+1 [ai λi a j ] c
c
λk+1 =λ†k+1
,
(15)
where the RHS is evaluated at λk+1 = λ†k+1 = − kj=1 λ j − ∂ M , with ∂ M acting from the left. The above equation should be interpreted by saying that, in the first term in
674
A. De Sole, V. G. Kac
the RHS, for i = k + 1, the last index λk does not appear in the poly λ-bracket. Let us replace ah by ∂ah in Eq. (15). It is not hard to check, using Conditions B1 and B2 for c and the sesquilinearity of the λ-action of A on M, that, for 1 ≤ h ≤ k, each term in the RHS of (15) gets multiplied by −λh , while, for h = k + 1, each term gets multiplied by −λ†k+1 = kj=1 λ j + ∂ M . Hence dc satisfies Conditions B1 and B2. In order to prove B3., let σ be a permutation of the set {1, . . . , k + 1}. A basic observation σ (k+1)
is that, if we first replace λσ (k+1) by λ†σ (k+1) = −λ1 − · ˇ· · −λk+1 − ∂ M , and then λk+1
by λ†k+1 = −λ1 · · · − λk − ∂ M , as a result λσ (k+1) stays unchanged. Notice, moreover, σ (i)
i
that, for 1 ≤ i ≤ k + 1, {σ (1), · ˇ· ·, σ (k + 1)} is a permutation of {1, · ˇ· ·, k + 1}, and its sign is (−1)i+σ (i) sign(σ ). Hence, using the assumption B3 on c, we get i aσ (i) λσ (i) aσ (1) λσ (1) · ˇ· · aσ (k) λσ (k) aσ (k+1) † c λk+1 =λk+1 σ (i) i+σ (i) = sign(σ )(−1) aσ (i) λσ (i) a1λ1 · ˇ· · ak λk ak+1 . (16) † λk+1 =λk+1
c j
i
Similarly, for the second term in (15), we notice that {σ (1), · ˇ· ·· ˇ· ·, σ (k + 1)} is a perσ (i)σ ( j)
mutation of {1, · ˇ· · · ˇ· · , k + 1}, and its sign is (−1)i+ j+σ (i)+σ ( j) sign(σ ) if σ (i) < σ ( j), and it is (−1)i+ j+σ (i)+σ ( j)+1 sign(σ ) if σ (i) > σ ( j). Hence, for σ (i) < σ ( j) we have i j aσ (1) λσ (1) · ˇ· ·· ˇ· · aσ (k+1) λσ (k+1) [aσ (i) λσ (i) aσ ( j) ] † λk+1 =λk+1
c
= sign(σ )(−1)i+ j+σ (i)+σ ( j)
σ (i)σ ( j)
× a1λ1 · ˇ· · · ˇ· · ak+1λk+1 [aσ (i) λσ (i) aσ ( j) ] c
λk+1 =λ†k+1
,
(17)
while for σ (i) > σ ( j) we have, by the skew-symmetry of the λ-bracket in A, i j aσ (1) λσ (1) · ˇ· ·· ˇ· · aσ (k+1) λσ (k+1) [aσ (i) λσ (i) aσ ( j) ] † c
λk+1 =λk+1
= sign(σ )(−1)i+ j+σ (i)+σ ( j)
σ ( j)σ (i)
× a1λ1 · ˇ· · · ˇ· · aσ (k+1) λk+1 [aσ ( j) −λσ (i) −∂ aσ (i) ] c
= sign(σ)(−1)i+ j+σ (i)+σ ( j) σ ( j)σ (i)
× a1λ1 · ˇ· · · ˇ· · aσ (k+1) λk+1 [aσ ( j) λσ ( j) aσ (i) ] c
λk+1 =λ†k+1
λk+1 =λ†k+1
.
(18)
In the last identity we used the assumption that c satisfies condition B2. Clearly, Eqs. (16), (17) and (18), together with the definition (15) of dc, imply that dc satisfies condition B3. We are left to prove that d 2 c = 0. We have, by (15),
Lie Conformal Algebra Cohomology and the Variational Complex
{a1λ1 · · · ak+1λk+1 ak+2 }d 2 c = +
k+2
675
k+2 i (−1)i+1 ai λi a1λ1 · ˇ· · ak+1λk+1 ak+2 i=1
j
i
dc
(−1)k+i+ j a1λ1 · ˇ· ·· ˇ· · ak+2λk+2 [ai λi a j ]
i, j=1 i< j
λk+2 =λ†k+2
dc
,
(19)
M M is moved to where, in the RHS, we replace λk+2 by λ†k+2 = − k+1 j=1 λ j − ∂ , and ∂ the left. Again by (15) and by sesquilinearity of the λ-action of A on M, the first term in the RHS of (19) is ⎛
k+2 ⎜ ⎜ (−1)i+ j (i, j)a j λ j ⎝
i
j
a1λ1 · ˇ· ·· ˇ· · ak+1λk+1 ak+2
ai λi
i, j=1 i = j
c
k+2
+
(−1)k+i+ j+h (i, h)( j, h)
i, j,h=1 i< j i, j =h
j
i
⎞
h
⎟ ⎟ ⎠
× ah λh a1λ1 · ˇ· ·· ˇ· ·· ˇ· · ak+2λk+2 [ai λi a j ]
λk+2 =λ†k+2
c
,
(20)
where (i, j) is +1 if i < j and −1 if i > j. Similarly, by (13) the second term in the RHS of (19) is ⎛ ⎜ k+2 ⎜ h i j ⎜ (−1)k+i+ j+h+1 (h, i)(h, j)ah λh a1λ1 · ˇ· ·· ˇ· ·· ˇ· · ak+2λk+2 [ai λi a j ] ⎜ ⎜ c ⎝i, j,h=1 i< j i, j =h
+
k+2
(−1)
i+ j
[ai λi a j ]λi +λ j
i
j
a1λ1 · ˇ· ·· ˇ· · ak+1λk+1 ak+2
i, j=1 i< j
+
c
k+2
(−1)i+ j+ p+q ( p, i)( p, j)(q, i)(q, j)
i, j, p,q=1 i< j, p
×
p q
i
j
(21)
a1λ1 · ˇ· ·· ˇ· ·· ˇ· ·· ˇ· · ak+2λk+2 [ai λi a j ]λi +λ j [a p λ p aq ] c
676
A. De Sole, V. G. Kac
+
k+2
(−1)k+i+ j+h (h, i)(h, j)
i, j,h=1 i< j i, j =h
⎞ × a1λ1
⎟ ⎟ ⎟ · ˇ· ·· ˇ· ·· ˇ· · ak+2λk+2 [ah λh [ai λi a j ]] ⎟ . † ⎟ c ⎠ λk+2 =λk+2 h
i
j
Notice that the first term in (20) is the negative of the second term in (21), and the second term in (20) is the negative of the first term in (21). Moreover, it is not hard to check, using the Jacobi identity for the λ-bracket on A, that the last term in (21) is identically zero, and, using the skew-symmetry Condition B3 on c, that also the third term in (21) is zero. In conclusion, d 2 c = 0, as we wanted. In the next section we shall embed the cohomology complex Γ • , introduced in Sect. 1.1, in the cohomology complex C¯ • , and we shall prove that, if the F[∂]-module A decomposes as a direct sum of the torsion and a free submodule, then this embedding is an isomorphism. We believe that the (slightly) bigger cohomology complex C • is a more natural and a more correct definition for the Lie conformal algebra cohomolgy complex. This will be clear when interpreting in Sect. 3 the cohomology H (C • , d) in terms of abelian Lie conformal algebra extensions of A by the module M. 2.4. Isomorphism of the cohomology complexes Γ • and C¯ • . We define, for k ≥ 1, an F-linear map ψ k : Γk → C k , as follows. Given γ ∈ Γk , we define ψ k ( γ ) : A⊗k → F[λ1 , . . . , λk−1 ] ⊗ M, by: γλ {a1λ1 · · · ak−1 λk−1 ak }ψ k ( γ) = where, as before, λ†k = −
k−1
† 1 ,...,λk−1 ,λk
j=1 λ j
(a1 , . . . , ak ),
(22)
− ∂ M , and ∂ M is moved to the left.
γ ) ∈ C¯ k . Lemma 1. (a) For γ ∈ Γk , we have ψ k ( (b) We have Ker (ψ k ) = ∂ Γk . Hence ψ k induces an injective F-linear map ψ k : Γ k = Γk /∂ Γk → C¯ k ⊂ C k . (c) Suppose that the Lie conformal algebra A decomposes, as F[∂]-module, as A = T ⊕ (F[∂] ⊗ U ) ,
(23)
where T is the torsion of A and A¯ = F[∂] ⊗ U is a complementary free submodule. ∼ Then ψ k (Γk ) = C¯ k , hence ψ k induces a bijective F-linear map ψ k : Γ k → C¯ k . Proof. Let γ ∈ Γk , and consider c = ψ k ( γ ). We want to prove that c ∈ C¯ k . It is clear that c satisfies Conditions B1 and B2. Let us check that it also satisfies Condition B3. Let σ be a permutation of the set {1, . . . , k}, and let i = σ (k). Since γ satisfies the skew-symmetry Condition A2, we have {aσ (1) λσ (1) · · · aσ (k−1) λσ (k−1) aσ (k) }c = γλ
† σ (1) ,...,λσ (k−1) ,λσ (k)
= sign(σ ) γλ
† 1 ,...,λi ,...,λk
(a1 , . . . , ak ).
(aσ (1) , . . . , aσ (k) ) (24)
Lie Conformal Algebra Cohomology and the Variational Complex
677
If we then replace λk by λ†k , as prescribed by Condition B3, we get i
λi† → −λ1 − · ˇ· · −λk−1 − λ†k − ∂ M = λi .
(25)
Therefore the RHS of (24) becomes sign(σ ){a1λ1 · · · ak−1 λk−1 ak }c , as required. It is also clear that c vanishes on the torsion of A, thanks to Remark 1, so that c ∈ C¯ k . This proves part (a). By the definition (11) of the action of ∂ on Γk , and the definition (22) of ψ k , we have {a1λ1 · · · ak−1 λk−1 ak }ψ k ∂ γ )λ γ = (∂
† 1 ,...,λk−1 ,λk
(a1 , . . . , ak ) = 0,
since −λ1 − · · · − λk−1 − λ†k − ∂ M = 0. Hence ∂ Γk ⊂ Ker (ψ k ). For the opposite incluγλ ,...,λ ,λ† (a1 , . . . , ak ) = 0. By Taylor expanding in sion, let γ ∈ Ker (ψ k ). Namely, 1
k−1
λ†k − λk , we have
k
∞ 1 dn γλ ,...,λk (a1 , . . . , ak ) = 0, (−Λ − ∂ M )n n n! dλk 1
(26)
n=0
where Λ = kj=1 λ j . We denote by ϑ : A⊗k → F[λ1 , . . . , λk ] ⊗ M the following F-linear map ϑλ1 ,...,λk (a1 , . . . , ak ) =
∞ 1 dn (−Λ − ∂ M )n−1 n γλ ,...,λk (a1 , . . . , ak ). n! dλk 1 n=1
Equation (26) can then be rewritten as γλ1 ,...,λk (a1 , . . . , ak ) = (∂ M + λ1 + · · · + λk ) ϑλ1 ,...,λk (a1 , . . . , ak ).
(27)
It follows from Eq. (27) that ϑ satisfies Conditions A1 and A2, since γ does. Hence ϑ ∈ Γk . Equation (27) then implies that γ = ∂ ϑ ∈ ∂ Γk , thus proving (b). Assume next that A decomposes as in (23). We need to prove that, for c ∈ C¯ k , we can find γ ∈ Γk such that γ ) = c. ψ k (
(28)
Such a k-cochain can be constructed as follows. For u 1 , . . . , u k ∈ U , we let γλ1 ,...,λk (u 1 , . . . , u k ) = {u 1
λ1 − Λ+∂k
M
· · · u k−1
λk−1 − Λ+∂k
M
u k }c ,
(29)
k−1 where Λ = i=0 λi , and we extend it to (F[∂] ⊗ U )⊗k by the sesquilinearity Condition ⊗k A1, and to A letting it be zero if one of the arguments is in the torsion T . We need to check that γ satisfies Conditions A1, A2 and (28). Condition A1 is obvious. It suffices to check Condition A2 for elements ai = u i ∈ U, i = 1, . . . , k. Let σ be a permutation of the indices {1, . . . , k}. We have, γλσ (1) ,...,λσ (k) (u σ (1) , . . . , u σ (k) ) = {u σ (1) Λ+∂ M · · · u σ (k−1) λσ (1) −
k
λσ (k−1) − Λ+∂k
M
u σ (k) }c .
(30)
678
A. De Sole, V. G. Kac
We then observe that
†
k−1 Λ + ∂M Λ + ∂M Λ + ∂M λk − λi − − ∂ M = λk − =− . k k k i=1
Hence, since c satisfies Condition B3, the RHS of (30) is equal to sign(σ ){u 1
λ1 − Λ+∂k
M
· · · u k−1
λk−1 − Λ+∂k
M
u k }c = sign(σ ) γλ1 ,...,λk (u 1 , . . . , u k ).
Finally, we prove that (28) holds. We have, for u 1 , . . . , u k ∈ U , γλ {u 1λ1 · · · u k−1 λk−1 u k }ψ k ( γ) =
† 1 ,...,λk−1 ,λk
(u 1 , . . . , u k ).
(31)
Note that, if we replace λk by λ†k , Λ + ∂ M becomes 0. Hence, by the definition (29) of γ , the RHS of (31) is equal to {u 1λ1 · · · u k−1 λk−1 u k }c . This proves that (28) holds for elements of U . Clearly both sides of (28) are zero if one of the elements ai is in T . Since both ψ k ( γ ) and c satisfy the sesquilinearity Conditions B1 and B2, we conclude that (28) holds for every ai ∈ A. Theorem 2. The identity map on M/∂ M and the maps ψ k , k ≥ 1, induce an embedding of cohomology complexes Γ • → C¯ • . If, moreover, the Lie conformal algebra A decomposes, as F[∂]-module, in a direct sum of a free module and the torsion, this map is an isomorphism of complexes: Γ • C¯ • . Proof. By Lemma 1 we already know that ψ k factors through an injective F-linear map ψ k : Γ k → C¯ k , and that, if A decomposes as in (23), this map is bijective. Hence, in order to prove the theorem, we only have to prove that the following diagrams are commutative: C¯ 1 v; O v v d vv ψ1 vv vv / Γ1 M/∂ M M δ
C¯O k
d
/ C¯ k+1 O
ψk
,
Γk
(32)
ψ k+1 δ
/ Γ k+1
, ∀k ≥ 1.
First, given m ∈ M/∂ M M, we have (δm)λ (a) = aλ m, so that ψ 1 δm (a) = a−∂ M m = d m (a), namely the first diagram in (32) is indeed commutative. Next, given k ≥ 1, let γ ∈ Γk be a representative of γ ∈ Γ k . We need to prove that dψ k ( γ ) = ψ k+1 (δ γ ).
(33)
From (13) and (22), we have {a1λ1 · · · ak λk ak+1 }dψ k ( γ) =
k
i (−1)i+1 ai λi a1λ1 · ˇ· · ak λk ak+1
ψ k ( γ)
i=1
+ (−1)k ak+1 λ†
k+1
a1λ1 · · · ak−1 λk−1 ak
ψ k ( γ)
Lie Conformal Algebra Cohomology and the Variational Complex k
+
(−1)
i
679
j
a1λ1 · ˇ· ·· ˇ· · ak λk ak+1 λ† [ai λi a j ]
k+i+ j+1
k+1
i, j=1 i< j
ψ k ( γ)
k i i + (−1) a1λ1 · ˇ· · ak λk [ai λi ak+1 ]
ψ k ( γ)
i=1
=
k (−1)i+1 ai λi γ
i
i
λ1 ,···,λ ˇ k ,λ†k+1
i=1
(a1 , · ˇ· ·, ak+1 )
+ (−1)k ak+1 λ† γλ1 ,...,λk (a1 , . . . , ak ) k+1
+
k
(−1)k+i+ j+1 γ
λ1 ,··· ˇ ···,λ ˇ k ,λ†k+1 ,λi +λ j
i, j=1 i< j
+
k (−1)i γ i=1
i
i j
j
(a1 , · ˇ· ·· ˇ· ·, ak+1 , [ai λi a j ])
i
i
λ1 ,···,λ ˇ k ,λi +λ†k+1
(a1 , · ˇ· ·, ak , [ai λi ak+1 ]).
(34)
In the last equality we used the sesquilinearity of the λ-action of A on M. Clearly, the RHS of (34) is the same as {a1λ1 · · · ak λk ak+1 }ψ k+1 (δ γ ). This proves Eq. (33) and the theorem. 2.5. Exterior multiplication on Γ• . To complete the section, we review the definition of the wedge product on the basic Lie conformal algebra cohomology complex Γ• (A, M) (cf. [BKV]). We assume that A is a Lie conformal algebra and M is an A-module endowed with a commutative, associative product such that ∂ M : M → M, and aλM : M → F[λ] ⊗ M, are (ordinary) derivations of this product. Consider the basic Lie conformal algebra cohomology complex Γ• (A, M) intro ∈ Γk , we define their exterior duced in Sect. 2.1. Given two cochains α ∈ Γh and β k+h multiplication α∧β ∈Γ by the following formula: )λ1 ,...,λh+k (a1 , . . . , ah+k ) = ( α∧β
sign(σ ) αλσ (1) ,...,λσ (h) (aσ (1) , . . . , aσ (h) ) h!k!
σ ∈Sh+k
λσ (h+1) ,...,λσ (h+k) (aσ (h+1) , . . . , aσ (h+k) ), β where the sum is over the set Sh+k of all permutations of {1, . . . , h + k}. Proposition 2. (a) The exterior multiplication (35) makes Γ• into a Z-graded commutative associative superalgebra, generated by Γ0 ⊕ Γ1 , M = Γ0 being an even subalgebra. (b) The operator ∂, acting on Γ• by (11), is an even derivation of the superalgebra Γ• . (c) The differential δ, defined by (10), is an odd derivation of the superalgebra Γ• . Proof. It follows from the analogous well known results for the Lie algebra cohomology, using the isomorphism of the complex Γ• to the Lie algebra cohomology complex for the annihilation Lie algebra of A (see [BKV, Theorem 5.1]).
680
A. De Sole, V. G. Kac
3. Cohomology and Extensions In this section we interpret the cohomology of the complex (C • , d) in terms of extensions of the Lie conformal algebra A and its modules. We start by reviewing the notions of extensions of a module over a Lie conformal algebra and of a Lie conformal algebra. Let A be a Lie conformal algebra and let M, N be A-modules. We denote by ∂ M and ∂ N the F[∂]-module structure on M and N respectively, and by aλM and aλN the λ-action of a ∈ A on M and N respectively. An extension of M by N is, by definition, an A-module E together with a short exact sequence of A-modules 0 → N → E → M → 0. We can fix a splitting E = M ⊕ N as F-vector spaces. This space is an A-module extension of M by N if it is endowed with: 1. an endomorphism ∂ E of M ⊕ N , such that ∂ E | N = ∂ N and ∂ E m − ∂ M m ∈ N for every m ∈ M, which makes M ⊕ N into an F[∂]-module; 2. a λ-action of A on M ⊕ N such that aλE | N = aλN for every a ∈ A and aλE m − aλM m ∈ F[λ] ⊗ N for every a ∈ A and m ∈ M, which makes M ⊕ N into an A-module. In this setting, two structures of A-module extensions E and E on M ⊕ N are isomorphic if there is an A-module isomorphism σ : E → E such that σ | N = 1I N and σ (m) − m ∈ N for every m ∈ M. An extension E is split if it is isomorphic to M ⊕ N as an A-module, and it is said to be F[∂]-split if it is isomorphic to M ⊕ N as an F[∂]module, namely if we can choose the F-vector space splitting E = M ⊕ N such that ∂E = ∂M ⊕ ∂N. We can also talk about extensions of a Lie conformal algebra. Let A, B be two Lie conformal algebras, and assume that the F[∂]-module B is endowed with a structure of an A-module. We denote by ∂ A and ∂ B the F[∂]-module structure on A and B respectively, and by [· λ ·] A and [· λ ·] B the λ-brackets on A and B respectively. An extension of A by B is, by definition, a Lie conformal algebra E together with a short exact sequence of Lie conformal algebras 0 → B → E → A → 0. In other words, if we fix a splitting E = A ⊕ B as F-vector spaces, the structure of a Lie conformal algebra extension on E consists of: 1. an endomorphism ∂ E of A ⊕ B, such that ∂ E | B = ∂ B and ∂ E a − ∂ A a ∈ B for every a ∈ A, which makes A ⊕ B into an F[∂]-module, 2. a λ-bracket on A ⊕ B such that [· λ ·] E | B = [· λ ·] B , [aλ b] E = aλ b for every a ∈ A and b ∈ B, and [aλ a ] E − [aλ a ] A ∈ F[λ] ⊗ B for every a, a ∈ A, which makes A ⊕ B into a Lie conformal algebra. As before, two structures of Lie conformal algebra extensions E and E on A ⊕ B are isomorphic if there is a Lie conformal algebra isomorphism σ : E → E such that σ | B = 1I B and σ (a) − a ∈ B for every a ∈ A. E is a split extension if it is isomorphic, as a Lie conformal algebra, to the semi-direct sum of A and B, and it is said to be F[∂]-split if it is isomorphic to A ⊕ B as an F[∂]-module, namely if we can choose the F-vector space splitting E = A ⊕ B such that ∂ E = ∂ A ⊕ ∂ B .
Lie Conformal Algebra Cohomology and the Variational Complex
681
We next review the construction of the module Chom(M, N ) of conformal homomorphisms [K]. A conformal homomorphism from the F[∂]-module M to the F[∂]-module N is an F-linear map ϕλ : M → F[λ] ⊗ N such that ϕλ (∂ M m) = (∂ N + λ)ϕλ (m). We denote by Chom(M, N ) the space of all conformal homomorphisms from M to N . It has the structure of an F[∂]-module given by (∂ϕ)λ = −λϕλ . If, moreover, M and N are modules over the Lie conformal algebra A, then Chom(M, N ) has the structure of an A-module, given by (aλ ϕ)µ = aλN ◦ ϕµ−λ − ϕµ−λ ◦ aλM . In the following theorem, we denote by H • (A, M) = k∈Z+ H k (A, M) the cohomology of the complex (C • , d) associated to the Lie conformal algebra A and the A-module M (see Sect. 2.3). Theorem 3. (a) H 0 (A, M) is naturally identified with the set of isomorphism classes of extensions of F, considered as A-module with trivial action of ∂ and trivial λ-action of A, by the A-module M. (b) H 1 (A, Chom(M, N )) is identified with the set of isomorphism classes of F[∂]-split extensions of the A-module M by the A-module N . (c) H 2 (A, M) is naturally identified with the set of isomorphism classes of F[∂]-split extensions of the Lie conformal algebra A by the A-module M, viewed as a Lie conformal algebra with the zero λ-bracket. Proof. By definition, H 0 (A, M) consists of elements m ∈ M/∂ M M in the kernel of d, namely such that a−∂ M m = 0 for every a ∈ A. In other words, H 0 (A, M) = m ∈ M | a−∂ M m = 0, ∀a ∈ A /∂ M M. (35) On the other hand, as discussed above, a structure of an A-module extension E of F by M on the space M ⊕ F is uniquely defined by an element ∂ E 1 = m ∈ M such that aλM m ∈ (∂ M + λ)F[λ] ⊗ M (or, equivalently, such that a−∂ M m = 0) for every a ∈ M. Indeed, the corresponding of aλE 1 ∈ M[λ], is then uniquely defined by the E λ-action M M equation (∂ + λ) aλ 1 = aλ m, imposed by sesquilinearity. It is immediate to check that this construction makes E = M ⊕F into an A-module. Furthermore, let m, m ∈ M be such that a−∂ M m = a−∂ M m = 0 for every a ∈ A, and consider the corresponding structures of A-module extensions E and E on M ⊕ F. An isomorphism of A-module extensions σ : E → E is completely defined by an element σ (1) − 1 = n ∈ M, such that ∂ E σ (1) = σ (∂ E 1), or, equivalently, m = m + ∂n. Hence, m and m correspond to isomorphic extensions if and only if they differ by an element of ∂ M. This proves part (a). By definition, C 1 (A, Chom(M, N )) is the space of F[∂]-linear maps c : A → Chom(M, N ). It can be identified, letting c(a)λ (m) = aλc m, with the space of F-linear maps A ⊗ M → F[λ] ⊗ N , satisfying the following sesquilinearity conditions (for a ∈ A, m ∈ M): (∂a)cλ m = −λaλc m,
aλc (∂m) = (λ + ∂ N )(aλc m).
(36)
682
A. De Sole, V. G. Kac
The equation dc = 0 for c to be closed can then be rewritten, recalling (13) and using the above notation, as follows (a, b ∈ A, m ∈ M): c c m) + aλc (bµM m) − bµN (aλc m) − bµ (aλM m) − [aλ b]cλ+µ m = 0. aλN (bµ
(37)
Notice that, if ϕλ ∈ Chom(M, N ), then ϕ0 is an F[∂]-linear map from M to N , and conversely, any F[∂]-linear map ϕ : M → N can be thought of as an element of Chom(M, N ) which is independent of λ. It follows that any element in d(C 0 (A, Chom(M, N ))), when written in the above notation, is of the form (dϕ)
aλ
m = aλN ϕ(m) − ϕ(aλM m),
(38)
for an F[∂]-linear map ϕ : M → N . In conclusion, H 1 (A, ! Chom(M, N )) {c of the form (38)} . = c : A ⊗ M → F[λ] ⊗ N (36) − (37) hold
(39)
On the other hand, as discussed at the beginning of the section, a structure of F[∂]-split extension E of M by N on the space M ⊕ N is uniquely determined by the elements aλE m −aλM m =: aλc m ∈ F[λ]⊗ N , and the requirement that E = M ⊕ N is an A-module exactly says that aλc m satisfies conditions (36) and (37). Furthermore, let E and E be two such extensions, associated to the closed elements c and c respectively. An isomorphism σ : E → E is uniquely determined by the elements σ (m) − m =: ϕ(m) ∈ N . The condition that σ commutes with the action of ∂ = ∂ M ⊕∂ N , i.e. σ (∂ M m) = (∂ M ⊕∂ N )σ (m), is equivalent to ϕ(∂ M m) = ∂ N ϕ(m), namely ϕ : M → N is an F[∂]-linear map. The condition that σ commutes with the λ-action of A, i.e. σ (aλE m) = aλE σ (m), is equivalent to
aλc m + ϕ(aλM m) = aλc m + aλN ϕ(m), which means that c and c differ by an exact element. This proves part (b). We are left to prove part (c). The space C 2 (A, M) consists of F-linear maps c : ⊗2 A → F[λ] ⊗ M, denoted by a ⊗ b → {aλ b}c , satisfying the conditions of sesquilinearity {∂aλ b}c = −λ{aλ b}c ,
{aλ ∂b}c = (λ + ∂ M ){aλ b}c ,
(40)
and skew-symmetry {bλ a}c = −{a−λ−∂ M b}c .
(41)
Recalling the definition (13) of d and using the skew-symmetry of the λ-bracket on A, the equation dc = 0 for c to be closed can be written as follows: aλ {bµ z}c − bµ {aλ z}c + z −λ−µ−∂ M {aλ b}c +{aλ [bµ z]}c − {bµ [aλ z]}c + {z −λ−µ−∂ M [aλ b]}c = 0,
(42)
for every a, b, z ∈ A. Recall that C 1 (A, M) consists of F[∂]-linear maps ϕ : A → M. Hence exact elements c = dϕ are of the form {aλ b}dϕ = aλ ϕ(b) − b−λ−∂ M ϕ(a) − ϕ([aλ b]).
(43)
Lie Conformal Algebra Cohomology and the Variational Complex
683
We thus have
! {c of the form (43)} . H 2 (A, M) = c : A⊗2 → F[λ] ⊗ M (40) − (42) hold (44)
Once we fix an F[∂]-splitting E = M ⊕ A, an “abelian” extension E of the Lie conformal algebra A by the A-module M is determined by a λ-bracket [· λ ·] E : (M ⊕ A)⊗2 → F[λ] ⊗ (M ⊕ A), satisfying the axioms of a Lie conformal algebra, and such that [m λ m ] E = 0 for m, m ∈ M, [aλ m] E = aλ m for a ∈ A and m ∈ M, and [aλ b] E − [aλ b] ∈ F[λ] ⊗ M. Let {aλ b}c := [aλ b] E − [aλ b] ∈ F[λ] ⊗ M. It is not hard to check that the axioms of sesquilinearity, skew-symmetry and Jacobi identity for [· λ ·] E become Eqs. (40), (41) and (42) respectively. Namely [· λ ·] E defines a structure of Lie conformal algebra extension on E if and only if c is a closed element of C 2 (A, M). Let E and E be two such extensions, associated to the closed elements c and c respectively. An isomorphism σ : E → E is uniquely determined by the elements σ (a) − a =: ϕ(a) ∈ M. It is easy to check that σ commutes with the action of ∂ = ∂ M ⊕ ∂ A if and only if ϕ(∂a) = ∂ M ϕ(a), namely ϕ : A → M is an F[∂]linear map. Finally, σ defines a Lie conformal algebra isomorphism, i.e. σ ([aλ b] E ) = [σ (a)λ σ (b)] E , if and only if {aλ b}c + ϕ([aλ b]) = {aλ b}c + aλ ϕ(b) − b−λ−∂ M ϕ(a), which means that c and c differ by an exact element.
Remark 3. Part (a) of Theorem 3 is the same statement as [BKV, Theorem 3.1-2]. Part (b) is equivalent to [BKV, Theorem 3.1-3]. This is due to Theorem 2 and the fact that Chom(M, N ) is free as an F[∂]-module, hence C • (A, Chom(M, N )) = C¯ • (A, Chom(M, N )). However, [BKV, Theorem 3.1-4] is false, unless A is free as an F[∂]module. Part (c) of Theorem 3 is the corrected version of it. This lends some support to our opinion that the cohomology complex (C • , d) is a more correct definition of a Lie conformal algebra cohomology complex. Moreover, as it appears from the proof of Theorem 3, the identification of the cohomology of the complex (C • , d) with the extensions of Lie conformal algebras and their representations is more direct and natural than for the complex (Γ • , δ). As B. Bakalov pointed out, Theorem 3 is a special case of Theorem 5.1 from [BDAK], valid for any Lie pseudoalgebra. Example 1. Consider the centerless Virasoro Lie conformal algebra Vir 0 = F[∂]L, with λ-bracket given by [L λ L]0 = (∂ + 2λ)L. We have C 1 (Vir 0 , F) = Fa, where a : Vir 0 → ⊗2 F[λ] is determined by a(L) = 1, and C 2 (Vir 0 , F) = Fα ⊕ Fβ, where α, β : Vir 0 → 3 F[λ] are determined by {L λ L}α = λ and {L λ L}β = λ . In particular da = 2α. Therefore H 2 (Vir 0 , F) is one-dimensional, meaning that, up to isomorphism, there is a unique 1-dimensional central extension of Vir 0 , namely Vir = F[∂]L ⊕ FF, with C central and 3 [L λ L] = (∂ + 2λ)L + λ12 C. Note that, since Vir 0 is free as an F[∂]-module, this is the same answer that we get if we consider the cohomology complex Γ • C¯ • . On the other hand, we have C 1 (Vir, F) = Fa ⊕ Fb, where a, b : Vir → F are determined by a(L) = 1, a(C) = 0, b(L) = 0, b(C) = 1, which is strictly bigger than C¯ 1 (Vir, F) = Fa. The 2-cochains are as before: C 2 (Vir, F) = Fα ⊕ Fβ, with α and β
684
A. De Sole, V. G. Kac
1 determined by {L λ L}α = λ and {L λ L}β = λ3 . In particular da = 2α and db = 12 β. 2 Therefore H (Vir, F) = 0, which corresponds to the fact that there are no non-trivial 1-dimensional central extensions of Vir. On the contrary, the second cohomology of the complex Γ • C¯ • is one-dimensional.
Remark 4. Since C¯ k = C k for k = 1, the corresponding cohomologies H n (C¯ • ) and H n (C • ) are isomorphic unless n = 1 or 2. In particular, it follows from [BKV, Theorem 7.1], that for the complex C • (Vir, F) we have: H n (Vir, F) = F for n = 0 or 3, and H n (Vir, F) = 0 otherwise. 4. The Space of k-Chains, Contractions and Lie Derivatives 4.1. g-complexes. Recall that a cohomology complex is a Z-graded vector space B • , endowed with an endomorphism d, such that d(B k ) ⊂ B k+1 and d 2 = 0. We view B • as a vector superspace, where elements of B k have the same parity as k in Z/2Z, so that d is an odd operator. Let g be a Lie algebra, and let g = ηg ⊕ g ⊕ F∂η , where η is odd such that η2 = 0, be the associated Z-graded Lie superalgebra. A g-structure on the complex B • is a Z-grading preserving Lie algebra homomorphism ϕ : g → End B • , such that ϕ(∂η ) = d. A complex with a given g-structure is called a g-complex. Given X ∈ g, the operator ι X = ϕ(ηX ) on B • is called the contraction, and the operator L X = ϕ(X ) is called the Lie derivative (along X ). Note that we have Cartan’s formula L X = [d, ι X ],
(45)
and the commutation relations [d, L X ] = 0, [ι X , ιY ] = 0, [L X , ιY ] = [ι X , L Y ] = ι[X,Y ] , [L X , L Y ] = L [X,Y ] . (46) Remark 5. In order to construct a g-structure on a complex (B • , d), it suffices to construct commuting odd operators ι X on B • , depending linearly on X , such that ι X (B k ) ⊂ B k−1 , and [[d, ι X ], ιY ] = ι[X,Y ] , ∀X, Y ∈ g.
(47)
Indeed, if we define L X by (45), all commutation relations (46) hold. Let ∂ be an endomorphism of the complex (B • , d), i.e. such that ∂(B k ) ⊂ B k and [d, ∂] = 0. Let g∂ = {X ∈ g | [ι X , ∂] = 0} ⊂ g. Notice that [L X , ∂] = 0 for all X ∈ g∂ . It follows that g∂ is a Lie subalgebra of g, and that (∂ B • , d) is a subcomplex of (B • , d) with a g∂ -structure. The corresponding quotient complex (B • /∂ B • , d),
Lie Conformal Algebra Cohomology and the Variational Complex
685
has an induced g∂ -structure, and it is called the reduced g∂ -complex. A morphism of a g-complex (B • , ϕ) to an h-complex (C • , ψ) is a Lie algebra homomorphism π : g → h and a Z-grading preserving linear map ρ : B • → C • , such that ρ(ϕ(g)b) = ψ(π(g))ρ(b), for all b ∈ B • and g ∈ g, where π is extended to a Lie superalgebra homomorphism g → h by letting π(ηX ) = ηπ(X ) and π(∂η ) = ∂η . Such a morphism is an isomorphism if both π and ρ are isomorphisms. 4.2. The basic and reduced spaces of chains Γ• and Γ• . The definitions of the basic and reduced spaces of k-chains are obtained, following [BKV], by dualizing, respectively, the definitions of the spaces Γk and Γ k introduced in Sect. 2.1. In particular, the basic space Γk (A, M) of k-chains of the Lie conformal algebra A with coefficients in the A-module M is, by definition, the quotient of the space A⊗k ⊗ Hom(F[λ1 , . . . , λk ], M), where Hom(F[λ1 , . . . , λk ], M) is the space of F-linear maps from F[λ1 , . . . , λk ] to M, by the following relations: C1. a1 ⊗ · · · ∂ai · · · ⊗ ak ⊗ φ = −a1 ⊗ · · · ⊗ ak ⊗ (λi∗ φ), where we denote λi∗ φ ∈ Hom(F[λ1 , . . . , λk ], M) is defined by (λi∗ φ)( f (λ1 , . . . , λk )) = φ(λi f (λ1 , . . . , λk )) ;
(48)
C2. aσ (1) ⊗ · · · ⊗ aσ (k) ⊗ (σ ∗ φ) = sign(σ )a1 ⊗ · · · ⊗ ak ⊗ φ, for every permutation σ ∈ Sk , where σ ∗ φ ∈ Hom(F[λ1 , . . . , λk ], M) is defined by (σ ∗ φ)( f (λ1 , . . . , λk )) = φ( f (λσ (1) , . . . , λσ (k) )).
(49)
We let, for brevity, Γk = Γk (A, M) and Γ• = Γ• (A, M) = k∈Z+ Γk . The following statement is the analogue of Remark 1 for the space of k-chains. Lemma 2. If one of the elements ai is a torsion element of the F[∂]-module A, we have a1 ⊗ · · · ⊗ ak ⊗ φ = 0 in Γk . In particular, Γk can be identified with the quotient of the space A¯ ⊗k ⊗ Hom(F[λ1 , . . . , λk ], M) by the relations C1 and C2 above, where A¯ = A/ Tor A denotes the quotient of the F[∂]-module A by its torsion. Proof. If P(∂)ai = 0 for some polynomial P, we have, by the relation C1., 0 = a1 ⊗ · · · (P(∂)ai ) · · · ⊗ ak ⊗ φ ≡ a1 ⊗ · · · ai · · · ⊗ ak ⊗ (P(−λi∗ )φ). To conclude the lemma we are left to prove that the linear endomorphism P(−λi∗ ) of the space Hom(F[λ1 , . . . , λk ], M) is surjective. For this, consider the subspace P(−λi )F[λ1 , . . . , λk ] ⊂ F[λ1 , . . . , λk ], and fix a complementary subspace U ⊂ F[λ1 , . . . , λk ], so that F[λ1 , . . . , λk ] = P(−λi )F[λ1 , . . . , λk ] ⊕ U . Given φ ∈ Hom(F[λ1 , . . . , λk ], M), we define the linear map ψ : F[λ1 , . . . , λk ] → M by letting ψ|U = 0 and ψ(P(−λi ) f (λ1 , . . . , λk )) = φ( f (λ1 , . . . , λk )) for every f ∈ F[λ1 , . . . , λk ]. Clearly, P(−λi∗ )ψ = φ.
686
A. De Sole, V. G. Kac
The space Γ• is endowed with a structure of a Z-graded F[∂]-module, with the action of ∂ induced by the natural action on A⊗k ⊗ Hom(F[λ1 , . . . , λk ], M): ∂ (a1 ⊗ · · · ⊗ ak ⊗ φ) =
k
a1 ⊗ · · · (∂ai ) · · · ⊗ ak ⊗ φ + a1 ⊗ · · · ⊗ ak ⊗ (∂φ)
i=1
= a1 ⊗ · · · ⊗ ak ⊗ (−λ∗1 − · · · − λ∗k + ∂)φ ,
(50)
, . . . , λk ], M) is defined by (∂φ)( f ) = ∂ M (φ( f )). The reduced where ∂φ ∈ Hom(F[λ1 space of chains Γ• = k∈Z+ Γk is, by definition, the subspace of ∂-invariant chains: Γk = {ξ ∈ Γk | ∂ξ = 0} ⊂ Γk . For example, for k = 0 we have Γ0 = M and Γ0 = {m ∈ M | ∂m = 0}. Next, consider the case k = 1 and suppose that the F[∂]-module A admits the decomposition (23), as a direct sum of Tor A and a complementary free submodule F[∂] ⊗ U . We already pointed out in Lemma 2 that a ⊗ φ = 0 in Γ1 if a ∈ Tor(A). Moreover, by the sesquilinearity Condition C1 we have (P(∂)u) ⊗ φ = u ⊗ (P(−λ∗ )φ) in Γ1 , for every u ∈ U, φ ∈ Hom(F[λ], M) and every polynomial P. Hence we can identify Γ1 U ⊗ Hom(F[λ], M). Under this identification, an element u ⊗ φ ∈ Γ1 is annihilated by ∂ if and only if the map φ : F[λ] → M, satisfies the equation (−λ∗ + ∂)φ = 0, namely if φ(λn ) = ∂ n φ(1) for every n ∈ Z+ . Clearly, there is a bijective correspondence between such maps and the elements of M, given by φ → φ(1) ∈ M. In conclusion, we have an isomorphism Γ1 U ⊗ M. Remark 6. Apparently, there is no natural way to define a differential δ on k-chains, making Γ• and Γ• homology complexes. The one given in [BKV, Sect. 4] is divergent, unless any m ∈ M is annihilated by a power of ∂ M . 4.3. Contraction operators acting on Γ• and Γ • . Assume, as in Sect. 2.5, that A is a Lie conformal algebra and M is an A-module endowed with a commutative, associative product µ : M ⊗ M → M, such that ∂ M : M → M, and aλ : M → C[λ] ⊗ M, satisfy the Leibniz rule. Given an h-chain ξ ∈ Γh , we define the contraction operator ιξ : Γk → Γk−h , k ≥ h, as follows. If a1 ⊗ · · · ⊗ ah ⊗ φ ∈ A⊗h ⊗ Hom(F[λ1 , . . . , λh ], M) is a representative of ξ ∈ Γh , and γ ∈ Γk , we let (ιξ λ1 ,...,λk (a1 , . . . , ak ) , γ )λh+1 ,...,λk (ah+1 , . . . , ak ) = φ µ γ
(51)
where, in the RHS, φ µ denotes the composition of the maps, commuting with λh+1 , . . . , λk , φ⊗1I
µ
F[λ1 , . . . , λh ] ⊗ M −→ M ⊗ M −→ M.
(52)
γ) = We extend the definition of ιξ to all elements ξ ∈ Γh by linearity on ξ , and we let ιξ ( 0 if k < h. We also define the Lie derivative L ξ by Cartan’s formula: L ξ = [δ, ιξ ]. It is immediate to check, using the sesquilinearity and skew-symmetry Conditions A1 and A2 for γ (cf. Sect. 2.1), that the RHS in (51) does not depend on the choice of the representative for ξ in A⊗h ⊗ Hom(F[λ1 , . . . , λh ], M). Moreover, if γ ∈ Γk , it follows that ιξ ( γ ) satisfies both Conditions A1 and A2, namely ιξ ( γ ) ∈ Γk−h .
Lie Conformal Algebra Cohomology and the Variational Complex
687
Proposition 3. The contraction operators on the superspace Γ• commute, i.e. for ξ ∈ Γh and ζ ∈ Γj we have ιξ ιζ = (−1)h j ιζ ιξ . Proof. Let a1 ⊗ · · · ⊗ ah ⊗ φ ∈ A⊗h ⊗ Hom(F[λ1 , . . . , λh ], M) be a representative for ξ ∈ Γh , b1 ⊗ · · · ⊗ b j ⊗ ψ ∈ A⊗ j ⊗ Hom(F[µ1 , . . . , µ j ], M) be a representative for γ ∈ Γk . By the definition (51) of the contraction operators, we have ζ ∈ Γj , and let (ιζ ιξ γ )ν1 ,...,νk−h− j (c1 , . . . , ck−h− j ) = ψ µ φµ γλ1 ,...,λh ,µ1 ··· ,µ j ,ν1 ,...,νk−h− j (a1 , . . . , ah , b1 , . . . , b j , c1 , . . . , ck−h− j ) . Since obviously φ µ and ψ µ commute, the proposition follows from Condition A2 for . γ Proposition 4. For every basic h-chain ξ ∈ Γh , we have [∂, ιξ ] = ∂ ◦ ιξ − ιξ ◦ ∂ = ι∂ξ .
(53)
In particular, if ξ ∈ Γh is a reduced h-chain, then ιξ commutes with ∂, and it induces a well-defined contraction operator on the reduced cohomology complex: ιξ : Γ k → Γ k−h . γ ∈ Γk . By the Proof. Let a1 ⊗ · · · ⊗ ah ⊗ φ be a representative of ξ ∈ Γh , and let k definition (11) of the action of ∂ on Γ , we have ∂ιξ γ λ ,...,λ (ah+1 , . . . , ak ) h+1 k (54) γλ1 ,...,λk (a1 , . . . , ak ) , = (∂ M + λh+1 + · · · + λk )φ µ and, similarly,
γ λ ,...,λ (ah+1 , . . . , ak ) ιξ ∂ k h+1 γλ1 ,...,λk (a1 , . . . , ak ) . = φ µ (∂ M + λ1 + · · · + λk )
(55)
On the other hand, by the definition (50) of the action of ∂ on Γh , we have γλ1 ,...,λk (a1 , . . . , ak ) γ λ ,...,λ (ah+1 , . . . , ak ) = (∂φ)µ ι∂ξ h+1 k − φ µ (λ1 + · · · + λh ) γλ1 ,...,λk (a1 , . . . , ak ) .
(56)
Equation (53) then follows by (54), (55), (56), and by the following result. Lemma 3. For every linear map φ : F[λ1 , . . . , λh ] → M, we have [∂ M , φ µ ] = ∂ M ◦ φ µ − φ µ ◦ (id ⊗ ∂ M ) = (∂φ)µ ,
(57)
where φ µ : F[λ1 , . . . , λh ] ⊗ M → M is defined in (52). Proof. Given f ⊗ m ∈ F[λ1 , . . . , λh ] ⊗ M we have
∂ M ◦ φ µ ( f ⊗ m) = ∂ M (φ( f ) · m) ,
φ µ ◦ (1I ⊗ ∂ M ) ( f ⊗ m) = φ( f ) · (∂ M m), (∂φ)µ ( f ⊗ m) = (∂ M φ( f )) · m. Equation (57) follows since, by assumption, ∂ M is a derivation of M.
688
A. De Sole, V. G. Kac
For example, for h = 0, the contraction by m ∈ M = Γ0 is given by the commutative associative product in M, namely we have, (ιm γ )λ1 ,...,λk (a1 , . . . , ak ) = m γλ1 ,...,λk (a1 , . . . , ak ) (which is the same as the exterior multiplication by m ∈ Γ0 = M). If, moreover, m ∈ M is such that ∂m = 0, we have ιm ∂ γ = ∂ιm γ , so that ιm induces a well-defined map Γ k → Γ k . Next, consider the case h = 1. Recall from the previous section that, if A decomposes as in (23), we have Γ1 U ⊗ Hom(F[λ], M). The contraction operator associated to ξ = u ⊗ φ ∈ Γ1 is given by (ιξ γ )λ2 ,...,λk (a2 , . . . , ak ) = φµ γλ,λ2 ,...,λk (u, a2 , . . . , ak ) . Moreover, we have Γ1 U ⊗ M, and the contraction operator associated to ξ = u ⊗ m is given by (ιξ γ )λ2 ,...,λk (a2 , . . . , ak ) = γ∂ M ,λ2 ,...,λk (u, a2 , . . . , ak )→ m,
(58)
where the arrow in the RHS means that ∂ M should be moved to the right. Clearly, ιξ ∂ γ = ∂ιξ γ , and ιξ induces a well-defined map Γ k → Γ k−1 . 4.4. The Lie algebra structure on g = Π Γ1 and the g-structure on the complex (Γ• , δ). In this section we want to define a Lie algebra structure on the space of 1-chains Γ1 , thus making Γ• a Π Γ1 -complex (recall the definition in Sect. 4.1), where Π means that we take opposite parity, namely we consider Γ1 as an even vector space. We start by describing the space of 1-chains in a slightly different form. Recall that Γ1 is the quotient of the space A ⊗ Hom(F[λ], M) by the image of the operator ∂ ⊗ 1 + 1 ⊗ λ∗ . We shall identify Hom(F[λ], M) with M[[x]] via the map 1 φ → φ(λn )x n . n! n∈Z+
It is immediate to check that, under this identification, the action of ∂ on Hom(F[λ], M) corresponds to the natural action of ∂ on M[[x]], while the operator λ∗ acting on Hom(F[λ], M) corresponds to the operator ∂x = ddx on M[[x]]. Thus, the space of 1-chains is " Γ1 = (A ⊗ M[[x]]) (∂ ⊗ 1 + 1 ⊗ ∂x )(A ⊗ M[[x]]). Recalling (50), the corresponding action of ∂ on Γ1 is given by ∂(a ⊗ m(x)) = a ⊗ (∂ − ∂x )m(x),
(59)
and the reduced space of 1-chains is Γ1 = {ξ = a ⊗ m(x) ∈ Γ1 | ∂ξ = 0}. In particular, if A admits a decomposition (23) as a direct sum of Tor A and a complementary free submodule F[∂] ⊗ U , we have Γ1 U ⊗ M[[x]], and the reduced subspace Γ1 ⊂ Γ1 consists of elements of the form ξ = u ⊗ (e x∂ m), u ∈ U, m ∈ M.
(60)
Given ξ ∈ Γ1 , we can write the action of the contraction operator ιξ : Γk → Γk−1 , defined by (51). Consider the pairing M[[x]] ⊗ F[λ] → M given by x m , λn = n! δm,n , m, n ∈ Z+ .
(61)
It induces a pairing , : M[[x]] ⊗ (F[λ] ⊗ M) → M, given by , ⊗1I
µ
M[[x]] ⊗ F[λ] ⊗ M −→ M ⊗ M −→ M,
(62)
Lie Conformal Algebra Cohomology and the Variational Complex
689
where µ in the last step denotes the commutative associative product on M. Then, if a1 ⊗ m(x1 ) ∈ A ⊗ M[[x1 ]] is a representative of ξ ∈ Γ1 , the contraction operator ιξ : Γk → Γk−1 acts as follows: # $ (ιξ γ )λ2 ,...,λk (a2 , . . . , ak ) = m(x1 ), γλ1 ,λ2 ,...,λk (a1 , a2 , . . . , ak ) , (63) where, in the RHS, , denotes the contraction of x1 with λ1 defined in (62). Clearly, if ξ is as in (60), Eq. (63) reduces to (58). We can also write down the formula for the Lie derivative L ξ = δ ◦ ιξ + ιξ ◦ δ. Let a1 ⊗ m(x1 ) ∈ A ⊗ M[[x1 ]] be a representative of ξ ∈ Γ1 . Recalling the expression (10) of the differential δ, we have (διξ γ )λ2 ,...,λk+1 (a2 , . . . , ak+1 ) % & i k+1 γ (a , a , · ˇ · ·, a ) = i=2 (−1)i ai λi m(x1 ), i 1 2 k+1 λ1 ,λ2 ,···,λ ˇ k+1 ( ' i j k+1 k+i+ j + i, j=2 (−1) γ (a1 , a2 , · ˇ· ·· ˇ· ·, ak+1 , [ai λi a j ]) , m(x1 ), i j λ1 ,λ2 ,··· ˇ ···,λ ˇ k+1 ,λi +λ j
i< j
and γ )λ2 ,...,λk+1 (a2 , . . . , ak+1 ) (ιξ δ %
& i k+1 γ i (−1)i+1 m(x1 ), ai λi (a1 , · ˇ· ·, ak+1 ) = i=1 λ1 ,···,λ ˇ k+1 ( ' i j k+1 k+i+ j+1 + i, j=1 (−1) γ i j (a1 , · ˇ· ·· ˇ· ·, ak+1 , [ai λi a j ]) . m(x1 ), λ1 ,··· ˇ ···,λ ˇ k+1 ,λi +λ j
i< j
We then use the assumption that the λ-action of A on M is by derivations of the commutative associative product of M, to get, from the above two equations, # $ (L ξ γλ2 ,...,λk+1 (a2 , . . . , ak+1 ) γ )λ2 ,...,λk+1 (a2 , . . . , ak+1 ) = m(x1 ), a1λ1 % & k+1 i + (−1)i ai λi m(x1 ) , γ (a1 , a2 , · ˇ· ·, ak+1 ) i λ1 ,λ2 ,···,λ ˇ k+1
i=2
−
k+1 #
$ m(x1 ), γλ2 ,...,λ1 +λ j ,...,λk+1 (a2 , . . . , [a1λ1 a j ], . . . , ak+1 ) .
(64)
j=2
We next introduce a Lie algebra structure on g = Π Γ1 and the corresponding g-structure on the complex (Γ• , δ). Define the following bracket on the space A ⊗ M[[x]]: [a ⊗ m(x), b ⊗ n(x)] = [a∂x1 b] ⊗ m(x1 )n(x) x =x 1 # $ # $ −a ⊗ n(x1 ), bλ1 m(x) + b ⊗ m(x1 ), aλ1 n(x) , (65) where, as before, , in the RHS denotes the contraction of x1 with λ1 defined in (62). Lemma 4. (a) The bracket (65) on A ⊗ M[[x]] induces a well-defined Lie algebra bracket on the space g = Π Γ1 . (b) The operator ∂ ⊗ 1 + 1 ⊗ ∂ on A ⊗ M[[x]] is a derivation of the bracket (65). In particular, ∂ defined in (59) is a derivation of the Lie algebra g = Π Γ1 , and g∂ = Π Γ1 ⊂ g is a Lie subalgebra.
690
A. De Sole, V. G. Kac
Proof. Notice that, by the definition (62) of the inner product , , f (x1 ), λ1 g(x1 ) = ∂x1 f (x1 ), g(λ1 ).
(66)
Hence, by (65) and the sesquilinearity conditions, we have # $ [(∂ ⊗ 1 + 1 ⊗ ∂x ) (a ⊗ m(x)) , b ⊗ n(x)] = −(∂ ⊗ 1 + 1 ⊗ ∂x ) a ⊗ n(x1 ), bλ1 m(x) , and [a ⊗ m(x), (∂ ⊗ 1 + 1 ⊗ ∂x ) (b ⊗ n(x))] = (∂ ⊗ 1 + 1 ⊗ ∂x ) [a∂x1 b] ⊗ m(x1 )n(x) x
1 =x
# $
+ b ⊗ m(x1 ), aλ1 n(x) .
It follows that (∂ ⊗ 1 + 1 ⊗ ∂x ) is a derivation of the bracket " (65), and that (65) induces a well-defined bracket on the quotient Γ1 = A ⊗ M[[x]] (∂ ⊗ 1 + 1 ⊗ ∂x )(A ⊗ M[[x]]). Next, let us prove skew-symmetry. We have [a ⊗ m(x), b ⊗ n(x)] + [b ⊗ n(x), a ⊗m(x)] , = [a∂x1 b] + [b∂x a] ⊗ m(x1 )n(x) x1 =x
and the RHS belongs to (∂ ⊗ 1 + 1 ⊗ ∂x )(A ⊗ M[[x]]), due to the skew-symmetry of the λ-bracket on A. For part (a), we are left to prove the Jacobi identity. Applying twice (65), we have [a ⊗ m(x), [b ⊗ n(x), c ⊗ p(x)]]
= [a∂x1 [b∂x2 c]] ⊗ m(x1 )n(x2 ) p(x) x1 =x2 =x # $ − [a∂x1 b] ⊗ m(x1 ) p(x2 ), cλ2 n(x) x =x # $ 1 + [a∂x1 c] ⊗ m(x1 ) n(x2 ), bλ2 p(x) x =x # $ 1 + [b∂x2 c] ⊗ m(x1 ), aλ1 (n(x2 ) p(x)) x =x ## $$ 2 − a ⊗ n(x1 ) p(x2 ), [bλ1 c]λ1 +λ2 m(x) $ $ ## + a ⊗ p(x2 ), cλ2 n(x1 ) , bλ1 m(x) $ $ ## − a ⊗ n(x2 ), bλ2 p(x1 ) , cλ1 m(x) # $$ # − b ⊗ m(x1 ), aλ1 p(x2 ), cλ2 n(x) # $$ # + c ⊗ m(x1 ), aλ1 n(x2 ), bλ2 p(x) .
For the fifth term in the RHS we used (66) and the following obvious identity: f (x1 )g(x2 )x =x , h(λ2 ) = f (x1 )g(x2 ), h(λ1 + λ2 ), 1
2
(67)
(68)
where, in the RHS, we denote by , the pairing of F[[x1 , x2 ]] and F[λ1 , λ2 ], defined by contracting x1 with λ1 and x2 with λ2 , as in (61). Similarly, we have [b ⊗ n(x), [a ⊗ m(x), c ⊗ p(x)]]
= [b∂x2 [a∂x1 c]] ⊗ m(x1 )n(x2 ) p(x) x1 =x2 =x # $ − [b∂ a] ⊗ n(x1 ) p(x2 ), cλ m(x) x1
2
x1 =x
Lie Conformal Algebra Cohomology and the Variational Complex
# $ + [b∂x2 c] ⊗ n(x2 ) m(x1 ), aλ1 p(x) x =x # $ 2 + [a∂x1 c] ⊗ n(x2 ), bλ2 (m(x1 ) p(x)) x =x ## $$ 1 − b ⊗ m(x1 ) p(x2 ), [aλ1 c]λ1 +λ2 n(x) $ $ ## + b ⊗ p(x2 ), cλ2 m(x1 ) , aλ1 n(x) $ $ ## − b ⊗ m(x1 ), aλ1 p(x2 ) , cλ2 n(x) # $$ # − a ⊗ n(x1 ), bλ1 p(x2 ), cλ2 m(x) $$ # # + c ⊗ n(x2 ), bλ2 m(x1 ), aλ1 p(x) ,
691
(69)
and, for the third term of Jacobi identity, [[a ⊗ m(x), b ⊗ n(x)], c ⊗ p(x)] = [[a∂x1 b]∂x1 +∂x2 c] ⊗ m(x1 )n(x2 ) p(x) x =x =x # $ 1 2 − [a∂x1 b] ⊗ p(x2 ), cλ2 (m(x1 )n(x)) x =x 1 # $ − [a∂x1 c] ⊗ n(x2 ), bλ2 m(x1 ) p(x) x =x 1 # $ + [b∂x2 c] ⊗ m(x1 ), aλ1 n(x2 ) p(x) x =x ## $$2 + c ⊗ m(x1 )n(x2 ), [aλ1 b]λ1 +λ2 p(x) $ $ ## − c ⊗ n(x2 ), bλ2 m(x1 ) , aλ1 p(x) $ $ ## + c ⊗ m(x1 ), aλ1 n(x2 ) , bλ2 p(x) $$ # # + a ⊗ p(x2 ), cλ2 n(x1 ), bλ1 m(x) # $$ # − b ⊗ p(x2 ), cλ2 m(x1 ), aλ1 n(x) .
(70)
We now combine Eqs. (67), (69) and (70), to get the Jacobi identity. In particular, the first terms in the RHS of (67), (69) and (70) combine to zero, due to the Jacobi identity for the λ-bracket on A. For the second terms in the RHS of (67), (69) and (70), we use the skew-symmetry of the λ-bracket on A and the Leibniz rule for the λ-action of A on M, to conclude that their combination belongs to (∂ ⊗ 1 + 1 ⊗ ∂x )(A ⊗ M[[x]]). The third term in the RHS of (67) combines with the fourth term in the RHS of (69) and the third term in the RHS of (70) to give zero, and similarly for the combination of the fourth term in the RHS of (67), the third term in the RHS of (69) and the fourth term in the RHS of (70). Furthermore, the combination of the fifth, sixth and seventh terms in the RHS of (67), the eighth term in the RHS of (69) and the eighth term in the RHS of (70) give ## $$ a ⊗ n(x1 ) p(x2 ), −[bλ1 c]λ1 +λ2 m(x) + bλ1 cλ2 m(x) − cλ2 bλ1 m(x) , which is zero due to the Jacobi identity for the λ-action of A on M, and similarly for the remaining terms in (67), (69) and (70). We are left to prove part (b). We have [(∂ ⊗ 1 + 1 ⊗ ∂)(a ⊗ m(x)), b ⊗ n(x)] = [∂a∂x1 b] ⊗ m(x1 )n(x) x =x + [a∂x1 b] ⊗ (∂m(x1 ))n(x) x =x 1 1 # $ # $ −(∂a) ⊗ n(x1 ), bλ1 m(x) − a ⊗ n(x1 ), bλ1 (∂m(x)) $ # $ # + b ⊗ m(x1 ), (∂a)λ1 n(x) + b ⊗ (∂m(x1 )), aλ1 n(x) ,
(71)
692
A. De Sole, V. G. Kac
and [a ⊗ m(x), (∂ ⊗ 1 + 1 ⊗ ∂)(b ⊗ n(x))] = [a∂x1 ∂b] ⊗ m(x1 )n(x) x =x + [a∂x1 b] ⊗ m(x1 )(∂n(x)) x =x 1 # $ # $ 1 − a ⊗ n(x1 ), (∂b)λ1 m(x) − a ⊗ (∂n(x1 )), bλ1 m(x) $ # $ # + (∂b) ⊗ m(x1 ), aλ1 n(x) + b ⊗ m(x1 ), aλ1 (∂n(x)) .
(72)
Putting Eqs. (71) and (72) together we get (∂ ⊗ 1 + 1 ⊗ ∂) [a ⊗ m(x), b ⊗ n(x)] = [(∂ ⊗ 1 + 1 ⊗ ∂)(a ⊗ m(x)), b ⊗ n(x)] + [a ⊗ m(x), (∂ ⊗ 1 + 1 ⊗ ∂)(b ⊗ n(x))] . This completes the proof of the lemma.
Proposition 5. The basic cohomoloy complex (Γ• , δ) admits a g-structure, ϕ : g→ End Γ• , where g = Π Γ1 is the Lie algebra with the Lie bracket induced by (65), given by ϕ(∂η ) = δ, ϕ(ηξ ) = ιξ , ϕ(ξ ) = L ξ , ξ ∈ Γ1 . The corresponding reduced (by ∂) g∂ -complex is (Γ • , δ). Proof. In view of Remark 5 and Proposition 3, we only have to check that [L ξ1 , ιξ2 ] = ι[ξ1 ,ξ2 ] ,
(73)
where, for ξ ∈ Γ1 , ιξ is given by (63) and L ξ is given by (64). For i = 1, 2, let then ai ⊗ m i (x) ∈ A ⊗ M[[x]] be a representative of ξi ∈ Γ1 . We have γ )λ3 ,...,λk+1 (a3 , . . . , ak+1 ) (L ξ1 ιξ2 # $$ # γλ2 ,λ3 ,...,λk+1 (a2 , a3 , . . . , ak+1 ) = m 1 (x1 ), a1λ1 m 2 (x2 ), % % && k+1 i + (−1)i (ai λi m 1 (x1 )), m 2 (x2 ), γ (a , a , a , · ˇ · ·, a ) i 1 2 3 k+1 λ1 ,λ2 ,λ3 ,···,λ ˇ k+1
i=3
−
k+1
#
# $$ m 1 (x1 ), m 2 (x2 ), γλ2 ,...,λ1 +λ j ,...,λk+1 (a2 , . . . , [a1λ1 a j ], . . . , ak+1 ) , (74)
j=3
where, as in (61), with , we contract x1 with λ1 and x2 with λ2 . Similarly, we have (ιξ2 L ξ1 γ )λ3 ,··· ,λk+1 (a3 , . . . , ak+1 ) # $$ # γλ2 ,...,λk+1 (a2 , . . . , ak+1 ) = m 2 (x2 ), m 1 (x1 ), a1λ1 % % && k+1 i i + (−1) m 2 (x2 ), (ai λi m 1 (x1 )), γ (a1 , a2 , · ˇ· ·, ak+1 ) i λ1 ,λ2 ,···,λ ˇ k+1
i=2
−
k+1 #
# $$ m 2 (x2 ), m 1 (x1 ), γλ2 ,...,λ1 +λ j ,...,λk+1 (a2 , . . . , [a1λ1 a j ], . . . , ak+1 ) .(75)
j=2
Lie Conformal Algebra Cohomology and the Variational Complex
693
Combining Eqs. (74) and (75), we get ([L ξ1 , ιξ2 ] γ )λ3 ,··· ,λk+1 (a3 , . . . , ak+1 ) $ $ ## γλ2 ,λ3 ,...,λk+1 (a2 , a3 , . . . , ak+1 ) = m 1 (x1 ), (a1λ1 m 2 (x2 )) , $ $ ## γλ1 ,λ3 ,...,λk+1 (a1 , a3 , . . . , ak+1 ) − m 2 (x2 ), (a2λ2 m 1 (x1 )) , $$ ## γλ1 +λ2 ,λ3 ,...,λk+1 ([a1λ1 a2 ], a3 , . . . , ak+1 ) , + m 1 (x1 )m 2 (x2 ),
(76)
where, for the first term, we used the fact that the λ-action of A on M is by derivations of the product on M. To conclude, we use Eqs. (66) and (68) to rewrite the RHS of (76) as (ιξ γ )λ3 ,...,λk+1 (a3 , . . . , ak+1 ), where # $ # $ ξ = a2 ⊗ m 1 (x1 ), (a1λ1 m 2 (x)) − a1 ⊗ m 2 (x2 ), (a2λ2 m 1 (x)) = [ξ1 , ξ2 ]. + [a1∂ a2 ] ⊗ m 1 (x1 )m 2 (x) x1
x1 =x
4.5. The space of chains C• . Recall from Theorem 2 that the cohomology complex Γ • is a subcomplex of the cohomology complex C • defined in Sect. 2.3. One may ask whether, for a reduced h-chain ξ ∈ Γh , there is a natural way to extend the definition of the contraction operator ιξ to the complex C • . In order to formulate the statement, in Theorem 4 below, we first define a new space of chains, obtained by dualizing the definition of the complex C • . We let C• = k∈Z+ Ck , where C0 = {m ∈ M | ∂m = 0} (= Γ0 ), and, for k ≥ 1, we define the space Ck of k-chains of A with coefficients in M as the quotient of the space A⊗k ⊗ Hom(F[λ1 , . . . , λk−1 ], M) by the following relations: D1. a1 ⊗ · · · ∂ai · · · ⊗ ak ⊗ φ ≡ −a1 ⊗ · · · ⊗ ak ⊗ (λi∗ φ), for every 1 ≤ i ≤ k − 1; D2. a1 ⊗ · · · ⊗ ak−1 ⊗ (∂ak ) ⊗ φ ≡ a1 ⊗ · · · ⊗ ak ⊗ ((λ∗1 + · · · + λ∗k−1 − ∂)φ); D3. aσ (1) ⊗ · · · ⊗ aσ (k) ⊗ (σ ∗ φ) ≡ sign(σ )a1 ⊗ · · · ⊗ ak ⊗ φ, for every permutation σ ∈ Sk , where σ ∗ φ ∈ Hom(F[λ1 , . . . , λk−1 ], M) is defined by
(77) (σ ∗ φ)( f (λ1 , . . . , λk−1 )) = φ f (λσ (1) , . . . , λσ (k−1) )λ →λ , k
k†
where in the RHS we have to replace λk by λk † = −λ1 − · · · − λk−1 + ∂ M and move ∂ M to the left of φ. For example, C1 = (A ⊗ M)/∂(A ⊗ M). In particular, in C1 it is not necessarily true that a ⊗ m is equivalent to zero for every torsion element a of the F[∂]-module A. On the other hand the analogue of Lemma 2 holds for k ≥ 2: Lemma 5. If k ≥ 2 and ai ∈ Tor A for some i, we have a1 ⊗ · · · ⊗ ak ⊗ φ = 0 in Ck . Proof. For 1 ≤ i ≤ k − 1, relation D1 is the same as relation C1, hence the same argument as in the proof of Lemma 2 works. Similarly, for i = k, if P(∂)ak = 0, we have by the relation D2, 0 = a1 ⊗· · ·⊗ak−1 ⊗(P(∂)ak )⊗φ = a1 ⊗· · ·⊗ak ⊗(P(λ∗1 + · · · + λ∗k−1 − ∂)φ), and to conclude the lemma we need to prove that the linear endomorphism P(λ∗1 + · · · + λ∗k−1 − ∂) of Hom(F[λ1 , . . . , λk−1 ], M) is surjective. In other words, given an element
694
A. De Sole, V. G. Kac
φ ∈ Hom(F[λ1 , . . . , λk−1 ], M), we want to find an element ψ ∈ Hom(F[λ1 , . . . , λk−1 ], M) such that P(λ∗1 + · · · + λ∗k−1 − ∂)ψ = φ. Suppose, for simplicity, that the polynomial P is monic of degree N . Hence P(λ∗1 + · · · + λ∗k−1 − ∂) = (λ∗1 ) N +
N
∂ n Rn (λ∗1 , . . . , λ∗k−1 ),
n=0
where the polynomials Rn ∈ F[λ∗1 , . . . , λ∗k−1 ], considered as polynomials in λ1 , have degree strictly less than N . Then ψ can be constructed recursively by saying that ψ(λn1 1 λn2 2 · · · λn−1 k−1 ) = 0 for n 1 < N , and ψ(λ1N +n 1 λn2 2 . . . λn−1 k−1 ) = φ(λn1 1 · · · λn−1 k−1 ) −
n1 n−1 M n ψ R (λ , . . . , λ ∂ )λ · · · λ n 1 k−1 n=0 1 k−1 .
N
m−1 1 m2 Since the RHS only depends on ψ(λm 1 λ2 · · · λk−1 ) with m 1 < N + n 1 , the above equation defines ψ by induction on n 1 . Clearly, P(λ∗1 + · · · + λ∗k−1 − ∂)ψ = φ. In analogy with the notation used in Sect. 2.2, we introduce the space C¯ • = k∈Z+ C¯ k , by taking the quotient of the space C• by the torsion of A. More precisely, let C¯ 0 = {m ∈ M | ∂m = 0} = C0 and, for k ≥ 1, C¯ k is the quotient of the space A¯ ⊗k ⊗ Hom(F[λ1 , . . . , λk−1 ], M), where A¯ = A/ Tor A, by the relations D1, D2 and D3 above. In particular, by Lemma 5, C¯ k = Ck for k = 1, and there is a natural surjective map C1 C¯ 1 . We next want to describe the relation between the spaces Ck and Γk . In particular, we are going to define a canonical map χk : Ck → Γk , and we will prove in Proposition 7 that, if the F[∂]-module A decomposes as a direct sum of its torsion and a free submodule, χk factors through an isomorphism C¯ k Γk . For k ≥ 1, let ρk : Hom(F[λ1 , . . . , λk ], M) Hom(F[λ1 , . . . , λk−1 ], M), be the restriction map associated to the inclusion F[λ1 , . . . , λk−1 ] ⊂ F[λ1 , . . . , λk ]. Let
χk : Hom(F[λ1 , . . . , λk−1 ], M) → Hom(F[λ1 , . . . , λk ], M), be the injective linear map defined by
(χk φ) ( f (λ1 , . . . , λk )) = φ f (λ1 , . . . , λk−1 , λk † ) , M M where in the RHS we let λk † = − k−1 j=1 λ j + ∂ and we move ∂ to the left.
(78)
Lemma 6. (a) We have ρk ◦ χk = 1I on Hom(F[λ1 , . . . , λk−1 ], M). Hence χk ◦ ρk is a projection operator on Hom(F[λ1 , . . . , λk ], M), whose image is naturally isomorphic to Hom(F[λ1 , . . . , λk−1 ], M). (b) The image of χk consists of the elements φ ∈ Hom(F[λ1 , . . . , λk ], M) such that (λ∗1 + · · · + λ∗k )φ = ∂φ.
(79)
(c) We have the commutation relations λi∗ ◦ χk = χk ◦ λi∗ ∀1 ≤ i ≤ k − 1, λ∗k ◦ χk = χk ◦ (−λ∗1 − · · · − λ∗k−1 + ∂), (80) where λi∗ is the linear endomorphism of Hom(F[λ1 , . . . , λk ], M) defined by (48).
Lie Conformal Algebra Cohomology and the Variational Complex
695
(d) For every permutation σ ∈ Sk we have σ ∗ ◦ χ k = χk ◦ σ ∗ ,
(81)
where σ ∗ in the LHS denotes the endomorphism of Hom(F[λ1 , . . . , λk ], M) defined by Eq. (49), while in the RHS it denotes the endomorphism of Hom(F[λ1 , . . . , λk−1 ], M) defined by (77). Proof. Part (a) is obvious. Given φ ∈ Hom(F[λ1 , . . . , λk−1 ], M), we have, by the definition (78) of χk , ∗ (λ1 + · · · + λ∗k − ∂)χk φ ( f (λ1 , . . . , λk )) = (χk φ) (λ1 + · · · + λk − ∂ M ) f (λ1 , . . . , λk ) = 0, namely χk φ satisfies Eq. (79). Conversely, if φ ∈ Hom(F[λ1 , . . . , λk ], M) solves Eq. (79), we have, by Taylor expanding in λk † − λk = −λ1 − · · · − λk + ∂ M , (χk ρk φ) ( f (λ1 , . . . , λk )) = φ f (λ1 · · · , λk−1 , λk † ) 1 (−λ∗1 − · · · − λ∗k + ∂)n φ (∂λnk f )(λ1 , . . . , λk ) = φ ( f (λ1 , . . . , λk )) . = n! n∈Z+
Hence, φ is in the image of χk , as we wanted. This proves part (b). For part (c), the first equation in (80) is clear. The second equation follows by part (b). We are left to prove part (d). Given φ ∈ Hom(F[λ1 , . . . , λk−1 ], M) we have, for every permutation σ ∈ Sk ,
(σ ∗ χk φ) ( f (λ1 , . . . , λk )) = φ f (λσ (1) , . . . , λσ (k) )λ →λ , k
k†
and (χk σ ∗ φ) ( f (λ1 , . . . , λk )) =φ
f (λσ (1) , . . . , λσ (k−1) , −λσ (1) − · · · − λσ (k−1)
+ ∂M)
λk →λk †
.
Equation (81) follows by the fact that, for σ (k) = k, when we replace λk by λk † = −λ1 − · · · − λk−1 + ∂ M , the expression −λσ (1) − · · · − λσ (k−1) + ∂ M is replaced by λσ (k) . We extend χk to an injective map χk : A⊗k ⊗ Hom(F[λ1 , . . . , λk−1 ], M) → A⊗k ⊗ Hom(F[λ1 , . . . , λk ], M), given by χk (a1 ⊗ · · · ⊗ ak ⊗ φ) = a1 ⊗ · · · ⊗ ak ⊗ χk (φ).
(82)
Moreover, we denote by C1, C2 ⊂ A⊗k ⊗ Hom(F[λ1 , . . . , λk ], M) the subspace generated by the relations C1 and C2 from Sect. 4.2, and by D1, D2, D3 ⊂ A⊗k ⊗ Hom(F[λ1 , . . . , λk−1 ], M) the subspace generated by the relations D1, D2 and D3. Proposition 6. (a) χk (D1, D2, D3) ⊂ C1, C2. (b) For every x ∈ A⊗k ⊗ Hom(F[λ1 , . . . , λk−1 ], M), we have ∂χk (x) ∈ C1, C2. (c) χk induces a well-defined linear map χk : Ck → Γk .
696
A. De Sole, V. G. Kac
Proof. For 1 ≤ i ≤ k − 1, we have χk a1 ⊗ · · · (∂ai ) · · · ⊗ ak ⊗ φ + a1 ⊗ · · · ⊗ ak ⊗ (λi∗ φ)
= a1 ⊗ · · · (∂ai ) · · · ⊗ ak ⊗ χk (φ) + a1 ⊗ · · · ⊗ ak ⊗ χk (λi∗ φ) ,
and this is in C1, C2 thanks to Lemma 6(c). Similarly, by the second equation in (80), χk a1 ⊗ · · · ⊗ ak−1 ⊗ (∂ak ) ⊗ φ − a1 ⊗ · · · ⊗ ak ⊗ (λ∗1 + · · · + λ∗k−1 − ∂)φ) = a1 ⊗ · · · ⊗ ak−1 ⊗ (∂ak ) ⊗ χk (φ) + a1 ⊗ · · · ⊗ ak ⊗ λ∗k χk (φ) ∈ C1, C2. Furthermore, by Lemma 6(d), we have, for every permutation σ ∈ Sk , χk a1 ⊗ · · · ⊗ ak ⊗ φ − sign(σ )aσ (1) ⊗ · · · ⊗ aσ (k) ⊗ (σ ∗ φ) = a1 ⊗ · · · ⊗ ak ⊗ χk (φ) − sign(σ )aσ (1) ⊗ · · · ⊗ aσ (k) ⊗ (σ ∗ χk (φ)) ∈ C1, C2. This proves part (a). From (50) and Lemma 6(b), we have ∂χk (a1 ⊗ · · · ⊗ ak ⊗ φ) ≡ a1 ⊗ · · · ⊗ ak ⊗ (−λ∗1 − · · · − λ∗k + ∂)χk (φ) = 0, thus proving (b). Part (c) follows from (a) and (b).
Proposition 7. If the Lie conformal algebra A decomposes, as an F[∂]-module, in a direct sum of the torsion and a complementary free F[∂]-submodule, the identity map on C0 = {m ∈ M | ∂m = 0} and the maps χk : Ck → Γk , k ≥ 1, factor through a bijective map C¯ • Γ• . Proof. Suppose that the F[∂]-module A decomposes as in (23). By definition, in the space C¯ k we have that a1 ⊗ · · · ⊗ ak ⊗ φ ≡ 0 if one of the elements ai is in T = Tor A. The same is true in the space Γk by Lemma 2. It follows that χk induces a well-defined map χk : C¯ k → Γk ⊂ Γk .
(83)
Moreover, in the space Γk we have, using relation C1, that (P1 (∂)u 1 ) ⊗ · · · ⊗ (Pk (∂)u k ) ⊗ φ ≡ u 1 ⊗ · · · ⊗ u k ⊗ P1 (−λ∗1 ) · · · Pk (−λ∗k )φ , for every u i ∈ U and φ ∈ Hom(F[λ1 , . . . , λk ], M). Hence, we can identify the space Γk with the quotient of the space U ⊗k ⊗ Hom(F[λ1 , . . . , λk ], M) by the relation C2. Similarly, in the space C¯ k we have, using the relations D1 and D2, that (P1 (∂)u 1 ) ⊗ · · · ⊗ (Pk (∂)u k ) ⊗ φ ≡ u 1 ⊗ · · · ⊗ u k ⊗ P1 (−λ∗1 ) · · · Pk−1 (−λ∗k−1 )Pk (λ∗1 + · · · + λ∗k−1 − ∂)φ , for every u i ∈ U and φ ∈ Hom(F[λ1 , . . . , λk−1 ], M). Hence, we can identify the space C¯ k with the quotient of the space U ⊗k ⊗ Hom(F[λ1 , . . . , λk−1 ], M) by the relation D3. The map χk in (83) is then induced by the map U ⊗k ⊗ Hom(F[λ1 , . . . , λk−1 ], M) → U ⊗k ⊗ Hom(F[λ1 , . . . , λk ], M), given by u 1 ⊗ · · · ⊗ u k ⊗ φ → u 1 ⊗ · · · ⊗ u k ⊗ χk (φ),
Lie Conformal Algebra Cohomology and the Variational Complex
697
for every u i ∈ U and φ ∈ Hom(F[λ1 , . . . , λk−1 ], M). Recalling (50), the action of ∂ on Γk is induced by the map U ⊗k ⊗Hom(F[λ1 , . . . , λk ], M) → U ⊗k ⊗Hom(F[λ1 , . . . , λk ], M), given by u 1 ⊗ · · · ⊗ u k ⊗ φ → u 1 ⊗ · · · ⊗ u k ⊗ (−λ∗1 − · · · − λ∗k )φ . Hence, the subspace Γk ⊂ Γk is spanned by elements of the form u 1 ⊗ · · · ⊗ u k ⊗ φ, such that (−λ∗1 − · · · − λ∗k )φ = 0. By Lemma 6(b), this is the same as the image of χk . Therefore the map (83) is surjective. Finally, injectiveness of (83) is clear since, by Lemma (6)(d), relation D3 corresponds, via χk , to relation C2. 4.6. Contraction operators acting on C • . Assume, as in Sect. 4.3, that A is a Lie conformal algebra and M is an A-module endowed with a commutative, associative product µ : M ⊗ M → M, such that ∂ M : M → M, and aλ : M → C[λ] ⊗ M, satisfy the Leibniz rule. Given an h-chain x ∈ C h , we define the contraction operator ιx : C k → C k−h , k ≥ h, in the same way as we defined, in Sect. 4.3, the contraction operator associated to an element of Γh . If a1 ⊗ · · · ⊗ ah ⊗ φ ∈ A⊗h ⊗ Hom(F[λ1 , . . . , λh−1 ], M) is a representative of x ∈ C h , and c ∈ C k , we let, for h < k, {ah+1λh+1 · · · ak−1 λk−1 ak }ιx c
= (χh φ)µ {a1λ1 · · · ah λh ah+1λh+1 · · · ak−1 λk−1 ak }c ,
(84)
where, in the RHS, φ µ is defined by (52) and χh is given by (78). For h = k, equation (84) has to be modified as follows:
ιx c = φ µ {a1λ1 · · · ak−1 λk−1 ak }c ∈ M/∂ M M = C 0 . (85) Lemma 7. (a) For c ∈ C k , the RHS of (84) does not depend on the choice of the representative for x in A⊗h ⊗Hom(F[λ1 , . . . , λh−1 ], M). Hence the contraction operator ιx is well defined for x ∈ C h . (b) For c ∈ C k , the RHS of (84) satisfies Conditions B1, B2 and B3. Hence ιx c ∈ C k−h . Proof. If x = a1 ⊗ · · · (∂ai ) · · · ⊗ ah ⊗ φ + a1 · · · ⊗ ah ⊗ (λi∗ φ), for 1 ≤ i ≤ h − 1, we have {ah+1λh+1 · · · ak−1 λk−1 ak }ιx c = (χh φ)µ {a1λ1 · · · (∂ai )λi · · · ak−1 λk−1 ak }c
+ λi {a1λ1 · · · ak−1 λk−1 ak }c , and this is zero since, by assumption, c satisfies Condition B1. Similarly, if x = a1 ⊗ · · · ⊗ ah−1 ⊗ (∂ah ) ⊗ φ − a1 · · · ⊗ ah ⊗ ((λ∗1 + · · · + λ∗h−1 − ∂)φ), we have {ah+1λh+1 · · · ak−1 λk−1 ak }ιx c = (χh φ)µ {a1λ1 · · · (∂ah )λh · · · ak−1 λk−1 ak }c
−(λ1 + · · · + λh−1 ){a1λ1 · · · ak−1 λk−1 ak }c
+ (χh ∂φ)µ {a1λ1 · · · ak−1 λk−1 ak }c .
(86)
698
A. De Sole, V. G. Kac
Using Condition B1 for c, we can rewrite the RHS of (86) as
(χh φ)µ −(λ1 + · · · + λh−1 + λh ){a1λ1 · · · ak−1 λk−1 ak }c
+(χh ∂φ)µ {a1λ1 · · · ak−1 λk−1 ak }c , which is zero thanks to Lemma 6(c). Furthermore, if x = a1 ⊗ · · · ⊗ ah ⊗ φ − sign(σ )aσ (1) ⊗ · · · ⊗ aσ (h) ⊗ (σ ∗ φ), for a permutation σ ∈ Sh , we have {ah+1λh+1 · · · ak−1 λk−1 ak }ιx c
= (χh φ)µ {a1λ1 · · · ak−1 λk−1 ak }c
− sign(σ )(χh σ ∗ φ)µ {a1λ1 · · · ak−1 λk−1 ak }c = (χh φ)µ {a1λ1 · · · ak−1 λk−1 ak }c
− sign(σ ){aσ (1) λσ (1) · · · aσ (h) λσ (h) ah+1λh+1 · · · ak−1 λk−1 ak }c ,
(87)
where, in the second equality, we used Lemma (6)(d) and the definition (49) of σ ∗ acting on Hom(F[λ1 , . . . , λh ], M). Clearly, the RHS of (87) is zero since, by assumption, c satisfies Condition B3. This proves part (a). For part (b), Condition B1 for ιx c follows immediately from the same condition on c. We have
{ah+1λh+1 · · · ak−1 λk−1 (∂ak )}ιx c = (χh φ)µ {a1λ1 · · · ak−1 λk−1 (∂ak )}c
= (χh φ)µ (λ1 + · · · + λk−1 + ∂ M ){a1λ1 · · · ak−1 λk−1 ak }c . (88) By Lemmas 3 and 6(c), the RHS of (88) is the same as
(λh+1 + · · · + λk−1 + ∂ M )(χh φ)µ {a1λ1 · · · ak−1 λk−1 ak }c = (λh+1 + · · · + λk−1 + ∂ M ){ah+1λh+1 · · · ak−1 λk−1 ak }ιx c , namely ιx c satisfies Condition B2. Similarly, for Condition B3, let σ be a permutation of the set {h + 1, . . . , k}. We have {aσ (h+1) λσ (h+1) · · · aσ (k−1) λσ (k−1) aσ (k) }ιx c
= (χh φ)µ {a1λ1 · · · ah λh aσ (h+1) λσ (h+1) · · · aσ (k−1) λσ (k−1) aσ (k) }c .
(89)
M M We then observe that, replacing in the above equation λk by − k−1 j=h+1 λ j − ∂ , ∂ µ acting from the left, is the same as replacing it, inside the argument of (χh φ) in the M RHS, by − k−1 j=1 λ j − ∂ . For this we use Lemmas 3 and 6(c). After this substitution, the RHS of (89) becomes, using Condition B3 for c,
sign(σ )(χh φ)µ {a1λ1 · · · ak−1 λk−1 ak }c = {ah+1λh+1 · · · ak−1 λk−1 ak }ιx c . Proposition 8. The contraction operators on the superspace C • commute, i.e. for x ∈ C h and y ∈ C j we have ιx ι y = (−1)h j ι y ιx .
Lie Conformal Algebra Cohomology and the Variational Complex
699
Proof. Let a1 ⊗ · · · ⊗ ah ⊗ φ ∈ A⊗h ⊗ Hom(F[λ1 , . . . , λh−1 ], M) be a representative for x ∈ C h , b1 ⊗ · · · ⊗ b j ⊗ ψ ∈ A⊗ j ⊗ Hom(F[µ1 , . . . , µ j−1 ], M) be a representative for y ∈ C j , and let c ∈ C k . For k > h + j, the proof is similar to that of Proposition 3. Thus we only have to consider the case k = h + j. Recalling (84) and (85), we have
ι y (ιx c) = ψ µ (χh φ)µ {a1λ1 · · · ah−1 λh−1 ah λh b1µ1 · · · b j−1 µ j−1 b j }c . Applying the skew-symmetry Condition B3 for c and using the definition (78) of χh , we get, after integration by parts, that the RHS is
(−1)h j (χ j ψ)µ φ µ {b1µ1 · · · b j−1 µ j−1 b j µ j a1λ1 · · · ah−1 λh−1 ah }c , which is the same as (−1)h j ιx (ι y c). For example, given an element m ∈ C0 = {m ∈ M | ∂m = 0}, we have {a1λ1 · · · ak−1 λk−1 ak }ιm c = m{a1λ1 · · · ak−1 λk−1 ak }c . Recall also that C1 = A ⊗ M/∂(A ⊗ M). The contraction operators associated to 1-chains are given by the following formulas: if c ∈ C 1 = HomF[∂] (A, M), then ιa⊗m c = mc(a), (90) while if c ∈ C k , with k ≥ 2, then {a2λ2 · · · ak−1 λk−1 ak }ιa1 ⊗m c = {a1 ∂ M a2λ2 · · · ak−1 λk−1 ak }c → m,
(91)
where the arrow in the RHS means, as usual, that ∂ M should be moved to the right. Also we have the following formulas for the Lie derivative L x = [d, ιx ] by a 1-chain x ∈ C1 acting on C 0 = M/∂ M M and C 1 = HomF[∂] (A, M): (92) L a⊗m n = (a∂ M n)→ m, (L a⊗m c)(b) = a∂ M c(b) → m +← (b−∂ M m)c(a) − c [a∂ M b] → m, where the left arrow in the RHS means, as usual, that ∂ M should be moved to the left. The definitions of the contraction operators associated to elements of Γ• and C• are “compatible”. This is stated in the following: Theorem 4. For x ∈ C h and γ ∈ Γ k , with k ≥ h, we have ιx (ψ k (γ )) = ψ k−h (ιχh (x) (γ )), where ψ k : Γ k → C k , denotes the injective linear map defined in Theorem 2, and χh : C h → Γh , denotes the linear map defined in Proposition 6. In other words, there is a commutative diagram of linear maps: CO k
ιx
/
ψk
C k−h O ,
? Γk
ψ k−h ιξ
/
? Γ k−h
provided that ξ ∈ Γh and x ∈ C h are related by ξ = χh (x).
(93)
700
A. De Sole, V. G. Kac
Proof. Let γ ∈ Γk be a representative of γ ∈ Γ k , and let a1 ⊗ · · · ⊗ ah ⊗ φ ∈ ⊗h A ⊗ Hom(F[λ1 , . . . , λh−1 ], M) be a representative of x ∈ C h . Recalling the definition (22) of ψ k and the definition (84) of ιx , we have
µ γ = (χ φ) (a , . . . , a ) , (94) {ah+1λh+1 · · · ak−1 λk−1 ak }ιx ψ k ( † h 1 k γ) λ ,...,λ ,λ 1
where, in the RHS, λ†k stands for −
k−1
k
k−1
j=1 λ j
− ∂ M , with ∂ M acting on the argument of M (χh φ)µ . By Lemmas 3 and (6)(c), we can replace λ†k by − k−1 j=h+1 λ j − ∂ , where now M µ ∂ is moved to the left of (χh φ) . Hence, the RHS of (94) is the same as (χh φ)µ γλ1 ,λ2 ,...,λk (a1 , . . . , ak ) λ →λ† k k = {ah+1λh+1 · · · ak−1 λk−1 ak }ψ k−h (ιχ (x) ( γ )) , h
thus completing the proof of the theorem.
4.7. Lie conformal algebroids. A Lie conformal algebroid is an analogue of a Lie algebroid. Definition 1. A Lie conformal algebroid is a pair (A, M), where A is a Lie conformal algebra, M is a commutative associative differential algebra with derivative ∂ M , such that A is a left M-module and M is a left A-module, satisfying the following compatibility conditions (a, b ∈ A, m, n ∈ M): L1. ∂(ma) = (∂ M m)a + m(∂a), L2. aλ (mn) = (aλ m)n + m(aλ n), L3. [aλ mb] = (aλ m)b + m[aλ b]. It follows from Condition M L3
and skew-symmetry (2) of the λ-bracket, that ∂ ∂ λ m [aλ b] + (aλ+∂ m)→ b, L3’. [maλ b] = e ∞ 1 M i where the first term in the RHS is i=0 i! (λ + ∂ ) m (a(i) b), and in the second term the arrow means, as usual, that ∂ should be moved to the right, acting on b. We next give two examples analogous to those in the Lie algebroid case. Let M be, as above, a commutative associative differential algebra. Recall from Sect. 3 that a conformal endomorphism on M is an F-linear map ϕ(= ϕλ ) : M → F[λ] ⊗ M satisfying ϕλ (∂ M m) = (∂ M + λ)ϕλ (m). The space Cend(M) of conformal endomorphism is then a Lie conformal algebra with the F[∂]-module structure given by (∂ϕ)λ = −λϕλ , and the λ-bracket given by [ϕλ ψ]µ = ϕλ ◦ ψµ−λ − ψµ−λ ◦ ϕλ . Example 2. Let Cder(M) be the subalgebra of the Lie conformal algebra Cend(M) consisting of all conformal derivations on M, namely of the the conformal endomorphisms satisfying the Leibniz rule: ϕλ (mn) = ϕλ (m)n +mϕλ (n). Then the pair (Cder(M), M) is a Lie conformal algebroid, where M carries the tautological Cder(M)-module structure, and Cder(M) carries the following M-module structure: M
(mϕ)λ = e∂ ∂λ m ϕλ . (95)
Lie Conformal Algebra Cohomology and the Variational Complex M
701 M
M
This is indeed an M-module, since e x∂ (mn) = (e x∂ m)(e x∂ n). Furthermore, ConM M dition L1 holds thanks to the obvious identity e∂ ∂λ λ = (λ + ∂ M )e∂ ∂λ . Condition L2. holds by definition. Finally, for Condition L3 we have [ϕλ mψ]µ (n) = ϕλ (mψ)µ−λ (n) − (mψ)µ−λ (ϕλ (n)) M
M
= ϕλ e∂ ∂µ m ψµ−λ (n) − e∂ ∂µ m ψµ−λ (ϕλ (n))
M
M = e(λ+∂ )∂µ ϕλ (m) ψµ−λ (n) + e∂ ∂µ m ϕλ ψµ−λ (n) − ψµ−λ (ϕλ (n)) M
M
= e∂ ∂µ ϕλ (m) ψµ (n) + e∂ ∂µ m [ϕλ ψ]µ (n) = (ϕλ (m)ψ + m[ϕλ ψ])µ (n). Example 3. Assume, as in Sect. 4.3, that A is a Lie conformal algebra and M is an Amodule endowed with a commutative, associative product, such that ∂ M : M → M, and aλ : M → C[λ] ⊗ M, for a ∈ A, satisfy the Leibniz rule. The space M ⊗ A has a natural structure of F[∂]-module, where ∂ acts as ∂(m ⊗ a) = (∂ M m) ⊗ a + m ⊗ (∂a).
(96)
Clearly, M ⊗ A is a left M-module via multiplication on the first factor. We define a left λ-action of M ⊗ A on M by M
(m ⊗ a)λ n = e∂ ∂λ m (aλ n), (97) and a λ-bracket on M ⊗ A by [(m ⊗ a)λ (n ⊗ b)] M
= e∂ ∂λ m n ⊗ [aλ b] + ((m ⊗ a)λ n) ⊗ b − e∂∂λ ((n ⊗ b)−λ m ⊗ a) .
(98)
We claim that (96) and (98) make M ⊗ A a Lie conformal algebra, (97) makes M an M ⊗ A-module, and the pair (M ⊗ A, M) is a Lie conformal algebroid. This will be proved in Proposition 9, using Lemmas 8 and 9. Lemma 8. (a) The following λ-bracket defines a Lie conformal algebra structure on the C[∂]-module M ⊗ A: M
e∂ ∂λ m n ⊗ [aλ b]. (99) [(m ⊗ a) λ (n ⊗ b)]0 = (b) For x, y ∈ M ⊗ A and m ∈ M, we have M
[mxλ y]0 = e∂ ∂λ m [xλ y]0 , [xλ my]0 = m[xλ y]0 .
(100)
Proof. For the first sesquilinearity condition, we have M
M
) * ∂(m ⊗ a) λ (n ⊗ b) 0 = e∂ ∂λ ∂ M m n ⊗ [aλ b] − e∂ ∂λ m n ⊗ λ[aλ b] = −λ [(m ⊗ a) λ (n ⊗ b)]0 . The second sesquilinearity condition and skew-symmetry can be proved in a similar way, and they are left to the reader. Let us check the Jacobi identity. We have M
M
) ) * * (m ⊗ a) λ (n ⊗ b) µ ( p ⊗ c) 0 0 = e∂ ∂λ m e∂ ∂µ n p ⊗ [aλ [bµ c]].
702
A. De Sole, V. G. Kac
Exchanging a ⊗ m with b ⊗ n and λ with µ, we get M
M
) * (n ⊗ b) µ [(m ⊗ a) λ ( p ⊗ c)]0 0 = e∂ ∂λ m e∂ ∂µ n p ⊗ [bµ [aλ c]]. Furthermore, we have
M M
[[m ⊗ a λ n ⊗ b]0 ν p ⊗ c]0 = e∂ ∂ν e∂ ∂λ m n p ⊗ [[aλ b]ν c].
Putting ν = λ + µ, the RHS becomes M
M
e∂ ∂λ m e∂ ∂µ n p ⊗ [[aλ b]λ+µ c]. Hence, the Jacobi identity for the λ-bracket (99) follows immediately from the Jacobi identity for the λ-bracket on A. This proves part (a). Part (b) is immediate. We define another λ-product on M ⊗ A: (m ⊗ a)λ (n ⊗ b) = ((m ⊗ a)λ n) ⊗ b.
(101)
Notice that the λ-bracket (98) can be nicely written in terms of the λ-bracket (99) and the λ-product (101): [xλ y] = [xλ y]0 + xλ y − y−λ−∂ x.
(102)
Lemma 9. (a) The λ-product (101) satisfies both sesquilinearity conditions (for x, y ∈ M ⊗ A): ( ∂ x)λ y = −λ xλ y, xλ ( ∂ y) = (λ + ∂)(xλ y).
(103)
(b) For x ∈ M ⊗ A, m ∈ M and y either in M ⊗ A or in M, we have M
(mx)λ y = e∂ ∂λ m xλ y, xλ (my) = (xλ m)y + m(xλ y).
(104)
(c) We have the following identity for x, y, z ∈ M ⊗ A: xλ [yµ z]0 = [(xλ y)λ+µ z]0 + [yµ (xλ z)]0 .
(105)
(d) We have the following identity for x, y ∈ M ⊗ A and z either in M or in M ⊗ A: xλ (yµ z) − yµ (xλ z) = [xλ y]λ+µ z.
(106)
Proof. We have M
( ∂(m ⊗ a))λ (n ⊗ b) = e∂ ∂λ (∂ M − λ)m (aλ n) ⊗ b. The first sesquilinearity condition follows from the obvious identity e∂ ∂λ (∂ M − λ) = M −λe∂ ∂λ . The second sesquilinearity condition can be proved in a similar way. This proves part (a). Part (b) is immediate. For part (c) and (d), let x = a ⊗m, y = b ⊗n, z = c ⊗ p ∈ A ⊗ M. We have M
M
xλ [yµ z]0 = e∂ ∂λ m aλ e∂ ∂µ n p ⊗ [bµ c]. (107) M
Lie Conformal Algebra Cohomology and the Variational Complex
Similarly,
M M
[(xλ y)ν z]0 = e∂ ∂ν e∂ ∂λ m (aλ n) p ⊗ [bν c].
703
(108)
Hence, if we put ν = λ + µ, the RHS becomes M
M
M
M
e∂ ∂λ m e∂ ∂µ (aλ n) p ⊗ [bλ+µ c] = e∂ ∂λ m aλ e∂ ∂µ n p ⊗ [bµ c], (109) where we used the sesquilinearity of the λ-bracket on A. Furthermore, we have M
M
[yµ (xλ z)]0 = e∂ ∂µ n e∂ ∂λ m (aλ p) ⊗ [bµ c]. (110) Combining Eqs. (107), (109) and (110), we immediately get (105), thanks to the assumption that the λ-action of A on M is a derivation of the commutative associative product on M. We are left to prove part (d). We have M
M
xλ (yµ p) = e∂ ∂λ m aλ e∂ ∂µ n (bµ p) M
M
M
M
= e∂ ∂λ m e∂ ∂µ n aλ (bµ p) + e∂ ∂λ m e∂ ∂µ (aλ n) (bλ+µ p). (111) For the second equality, we used the Leibniz rule and the sesquilinearity condition for the λ-action of A on M. Exchanging x with y and λ with µ, we have M
M
M
M
yµ (xλ p) = e∂ ∂λ m e∂ ∂µ n bµ (aλ p) + e∂ ∂µ n e∂ ∂λ (bµ m) (aλ+µ p). (112) By similar computations, we get M
M
(xλ y)λ+µ z = e∂ ∂λ m e∂ ∂µ (aλ n) (bλ+µ p), and
M
M
(y−λ−∂ x)λ+µ p = e∂ ∂µ n e∂ ∂λ (bµ m) (aλ+µ p).
Finally, it follows by a straightforward computation that M
M
[xλ y]0 λ+µ z = e∂ ∂λ m e∂ ∂µ n [aλ b]λ+µ p. Equation (106) is obtained combining Eqs. (111), (112), (113), (114) and (115).
(113)
(114)
(115)
Proposition 9. (a) The λ-bracket (98) defines a Lie conformal algebra structure on the F[∂]-module M ⊗ A. (b) The λ-action (97) defines a structure of a M ⊗ A-module on M. (c) The pair (M ⊗ A, M) is a Lie conformal algebroid. (d) We have a Lie conformal algebroid homomorphism (M ⊗ A,M)→(Cder(M), M), given by the identity map on M and the following Lie conformal algebra homomorphism from M ⊗ A to Cder(M): M
m ⊗ a → e∂ ∂λ m aλ .
704
A. De Sole, V. G. Kac
Proof. It immediately follows from Lemma 8 and Lemma 9(a) that the λ-bracket (102) satisfies sesquilinearity and skew-symmetry. Furthermore, the Jacobi identity for the λbracket (98) follows from Lemma 8 and Eqs. (105) and (106). This proves part (a). Part (b) is Lemma 103(c), in the case z ∈ M. For part (c) we need to check Conditions L1, L2 and L3. The first two conditions are immediate. The last one follows from Eqs. (100) and (104). Finally, part (d) is straightforward and is left to the reader. 4.8. The Lie algebra structure on ΠC1 and the ΠC1 -structure on the complex (C • , d). Recall that the space of 1-chains of the complex (C • , d) is C1 = (A ⊗ M)/∂(A ⊗ M) with odd parity. We want to define a Lie algebra structure on ΠC1 , where, as usual, Π denotes parity reversing, making C • into a ΠC1 -complex. By Proposition 9(a), we have a Lie conformal algebra structure on M ⊗ A. Hence, if we identify M ⊗ A with A ⊗ M by exchanging the two factors, we get a structure of a Lie algebra on the quotient space (A ⊗ M)/∂(A ⊗ M), induced by the λ-bracket at λ = 0 [K]. Explicitly, we get the following well-defined Lie algebra bracket on ΠC1 = (A ⊗ M)/∂(A ⊗ M): [a ⊗ m, b ⊗ n] = [a∂ M b]→ ⊗ mn + b ⊗ a∂ M n → m − a ⊗ b∂ M m → n, (116) 1
where in the RHS, as usual, the right arrow means that ∂ M should be moved to the right, and in the first summand ∂1M denotes ∂ M acting only on the first factor m. " Recall from Sect. 4.4 that Γ1 = (A ⊗ M[[x]]) (∂ ⊗ 1 + 1 ⊗ ∂x )(A ⊗ M[[x]]), and Γ1 = ξ ∈ Γ1 | ∂ξ = 0 , where the action of ∂ on Γ1 is given by (59). Under this identification, the map χ1 : C1 → Γ1 defined by (78) and (82) is given by M
χ1 (a ⊗ m) = a ⊗ e x∂ m.
(117)
Proposition 10. The map χ1 : C1 → Γ1 is a Lie algebra homomorphism, which factors through a Lie algebra isomorphism χ1 : C¯ 1 → Γ1 , provided that A decomposes as in (23). Proof. We have, by (116) and (117) that M M
χ1 ([a ⊗ m, b ⊗ n]) = [a∂ M b]→ ⊗ e x∂ m e x∂ n 1 M M x∂ +b ⊗ e (a∂ M n)→ m − a ⊗ e x∂ (b∂ M m)→ n . Recalling formula (65) for the Lie bracket on Γ1 , we have
M M [χ1 (a ⊗ m), χ1 (b ⊗ n)]) = [a∂x1 b] ⊗ e x1 ∂ m e x∂ n x1 =x # $ # $ +b ⊗ m(x1 ), aλ1 n(x) − a ⊗ n(x1 ), bλ1 m(x) .
(118)
(119)
Clearly, the first term in the RHS of (118) is the same as the first term in the RHS of (119). Recalling the definition (62) of the pairing , , and using the sesquilinearity of the λ-action of A on M, we have that the second term in the RHS of (118) is the same as the second term in the RHS of (119), and similarly for the third terms. The last statement follows from Proposition 7.
Lie Conformal Algebra Cohomology and the Variational Complex
705
1 → End C • , given Proposition 11. The complex (C • , d) has a ΠC1 -structure ϕ : ΠC by ϕ(∂η ) = d, ϕ(ηx) = ιx , ϕ(x) = L x = [d, ιx ]. Moreover, (C¯ • , d) is a ΠC1 -subcomplex. Proof. Due to Remark 5 and Proposition 8, we only need to check that, for x, y ∈ ΠC1 , we have [L x , ι y ] = ι[x,y] .
(120)
This follows from a long but straightforward computation, using the explicit formulas (13) and (91) for the differential and the contraction operators. It is left to the reader. Notice though that, in the special case when A decomposes as in (23), Eq. (120) is a corollary of Proposition 5, Theorem 2 and Theorem 4 for h = 1. Indeed, due to these results, it suffices to check that both sides of (120) coincide when acting on C 1 = HomF[∂] (A, M). In the latter case, using Eqs. (12), (14), (90), (91), (92) and (116), we have, for c ∈ C 1 , L a⊗m (ιb⊗n c) = c(b) a∂ M n → m + n a∂ M c(b) → m, ιb⊗n (L a⊗m c) = n a∂ M c(b) → m + c(a) b∂ M m → n − nc [a∂ M b] → m, ι[a⊗m,b⊗n] c = nc [a∂ M b] → m + c(b) a∂ M n → m − c(a) b∂ M m → n. It follows that (120) holds when applied to elements of C 1 .
The above results imply the following Theorem 5. The maps ψ • : Γ • → C¯ • ⊂ C • and χ1 : C1 → Γ1 define a homomorphism of g-complexes. Provided that A decomposes as in (23), we obtain an isomorphism ∼ of ΠC1 Π Γ1 -complexes ψ • : Γ • → C¯ • . Proof. It follows from Theorem 2, Proposition 7, Theorem 4 and Proposition 10.
4.9. Pairings between 1-chains and 1-cochains. Recall that Γ0 = M. Hence, the contraction operators of 1-chains, restricted to the space of 1-cochains, define a natural pairing Γ1 × Γ1 → M, which, to ξ ∈ Γ1 and γ ∈ Γ1 , associates γ = φ µ ( γλ (a)) ∈ M, ιξ
(121)
where a ⊗ φ ∈ A ⊗ Hom(F[λ], M) is a representative of ξ . When we consider the reduced spaces, we have Γ 0 = M/∂ M, and the above map induces a natural pairing Γ1 × Γ 1 → M/∂ M, which, to ξ ∈ Γ1 and γ ∈ Γ 1 , associates γλ (a)) ∈ M/∂ M, (122) ιξ γ = φ µ ( where again a ⊗ φ ∈ A ⊗ Hom(F[λ], M) is a representative of ξ , and γ ∈ Γ1 is a representative of γ . A similar pairing can be defined for 1-chains in C1 and 1-cochains in C 1 . Recall that C 0 = M/∂ M, C 1 is the space of F[∂]-module homomorphisms c : A → M, and C1 = A ⊗ M/∂(A ⊗ M). The corresponding pairing C1 × C 1 → M/∂ M, is obtained as follows. To x ∈ C1 and c ∈ C 1 , we associate, recalling (85), (123) ιx (c) = m · c(a) ∈ M/∂ M, where a ⊗ m ∈ A ⊗ M is a representative of x. Recalling Theorems 2 and 4, the above pairings (122) and (123) are compatible in the sense that ιx (c) = ιξ (γ ), provided that γ ∈ Γ 1 and c ∈ C 1 are related by c = ψ 1 (γ ), and ξ ∈ Γ1 and x ∈ C1 are related by ξ = χ1 (x).
706
A. De Sole, V. G. Kac
4.10. Contraction by a 1-chain as an odd derivation of Γ• . Recall that, if the A-module M has a commutative associative product, and ∂ M and aλM are even derivations of it, then the basic cohomology complex Γ• is a Z-graded commutative associative superalgebra with respect to the exterior product (35), and the differential δ is an odd derivation of degree +1. Proposition 12. The contraction operator ιξ , associated to a 1-chain ξ ∈ Γ1 , is an odd derivation of the superalgebra Γ• of degree -1. Proof. Let a1 ⊗ φ, with a1 ∈ A and φ ∈ Hom(F[λ1 ], M), be a representative of ξ ∈ Γ1 . By the definition (35) of the exterior product, we have ))λ2 ,...,λh+k (a2 , . . . , ah+k ) (ιξ ( α∧β sign(σ ) αλσ (1) ,...,λσ (h) (aσ (1) , . . . , aσ (h) )× = φµ h!k! σ ∈Sh+k λσ (h+1) ,...,λσ (h+k) (aσ (h+1) , . . . , aσ (h+k) ) . ×β
(124)
, we can rewrite the RHS of (124) as By the skew-symmetry condition A2 for α and β h i sign(σ ) i+1 µ (−1) φ (a1 , aσ (1) , · ˇ· ·, aσ (h) ) α i h!k! λ1 ,λσ (1) ,···,λ ˇ σ (h) i=1 σ | σ (i)=1
λσ (h+1) ,...,λσ (h+k) (aσ (h+1) , . . . , aσ (h+k) ) ×β +
h+k
sign(σ ) (−i)i−h+1 αλσ (1) ,...,λσ (h) (aσ (1) , . . . , aσ (h) ) × h!k! i=h+1 σ | σ (i)=1 ×φ µ β
i
i
λ1 ,λσ (h+1) ,···,λ ˇ σ (h+k)
(a1 , aσ (h+1) , · ˇ· ·, aσ (h+k) ) .
(125)
The set of all permutations σ ∈ Sh+k such that σ (i) = 1, is naturally in bijection with the set of all permutations τ of {2, . . . , h + k}, and the correspondence between the signs is sign(τ ) = (−1)i+1 sign(σ ). Hence, (125) can be rewritten as sign(τ ) τ
h(ιξ α )λτ (2) ,...,λτ (h) (aτ (2) , . . . , aτ (h) )× h!k! λτ (h+1) ,...,λτ (h+k) (aτ (h+1) , . . . , aτ (h+k) ) ×β
αλτ (2) ,...,λτ (h+1) (aτ (2) , . . . , aτ (h+1) )× + k(−1)h )λτ (h+2) ,...,λτ (h+k) (aτ (h+2) , . . . , aτ (h+k) ) × (ιξ β )λ2 ,...,λh+k (a2 , . . . , ah+k ) α) ∧ β = (ιξ ( h ))λ2 ,...,λh+k (a2 , . . . , ah+k ). + (−1) ( α ∧ ιξ (β Remark 7. One can show that the g-structure of all our complexes Γ• , Γ • and C • can be extended to a structure of a calculus algebra, as defined in [DTT]. Namely, one can extend the Lie algebra bracket from the space of 1-chains to the whole space of chains (with reverse parity), and define there a commutative superalgebra structure, which extends our g-structure and satisfies all the properties of a calculus algebra.
Lie Conformal Algebra Cohomology and the Variational Complex
707
5. The Complex of Variational Calculus as a Lie Conformal Algebra Cohomology Complex 5.1. Algebras of differential functions. An algebra of differential functions V in the variables u i , indexed by a finite set I = {1, . . . , }, is, by definition, a differential algebra (i.e. a unital commutative associative algebra with a derivation ∂), endowed with commuting derivations ∂(n) : V → V, for all i ∈ I and n ∈ Z+ , such that, given f ∈ V,
∂ (n) ∂u i
∂u i
f = 0 for all but finitely many i ∈ I and n ∈ Z+ , and the following
commutation rules with ∂ hold:
∂ (n) ∂u i
,∂ =
∂ (n−1) ∂u i
,
(126)
where theRHS is considered to be zero if n = 0. As in the previous sections, we denote by f → f the canonical quotient map V → V/∂V. Denote by C ⊂ V the subspace of constant functions, i.e. ∂f C= f ∈V = 0 ∀i ∈ I, n ∈ Z+ . (127) (n) ∂u i It follows from (126) by downward induction that Ker (∂) ⊂ C.
(128)
Also, clearly, ∂C ⊂ C. Typical examples of algebras of differential functions are: the ring of polynomials (n)
R = F[u i | i ∈ I, n ∈ Z+ ], (n)
(129)
(n+1)
, any localization of it by some multiplicative subset S ⊂ R, where ∂(u i ) = u i (n) such as the whole field of fractions Q = F(u i | i ∈ I, n ∈ Z+ ), or any algebraic extension of the algebra R or of the field Q obtained by adding a solution of a certain polynomial equation. In all these examples the action of ∂ : V → V is given by ∂ ∂= u i(n+1) (n) . Another example of an algebra of differential functions is the ∂u i i∈I,n∈Z+ ∂ (n) (n+1) ∂ + ring R [x] = F[x, u i | i ∈ I, n ∈ Z+ ], where ∂ = ui . (n) ∂x ∂u The variational derivative
δ δu
i∈I,n∈Z+
i
: V → V ⊕ is defined by
δf ∂f := (−∂)n (n) . δu i ∂u i n∈Z+
(130)
It follows immediately from (126) that δ (∂ f ) = 0, δu i for every i ∈ I and f ∈ V, namely, ∂V ⊂ Ker
δ δu .
(131)
708
A. De Sole, V. G. Kac
A vector field is, by definition, a derivation of V of the form X=
Pi,n
i∈I,n∈Z+
∂ ∂u i(n)
,
Pi,n ∈ V.
(132)
We let g be the Lie algebra of all vector fields. The subalgebra of evolutionary vector fields is g∂ ⊂ g, consisting of the vector fields commuting with ∂. By (126), a vector field X is evolutionary if and only if it has the form XP =
(∂ n Pi )
i∈I,n∈Z+
∂ (n)
∂u i
,
where P = (Pi )i∈I ∈ V .
(133)
5.2. Normal algebras of differential functions. Let V be an algebra of differential functions in the variables u i , i ∈ I = {1, . . . , }. For i ∈ I and n ∈ Z+ we let ∂f Vn,i := f ∈ V (m) = 0 if (m, j) > (n, i) in lexicographic order . (134) ∂u j We also let Vn,0 = Vn−1, . (n) A natural assumption on V is to contain elements u i , for i ∈ I, n ∈ Z+ , such that (n)
∂u i
(m)
∂u j
= δi j δmn .
(135)
Clearly, such elements are uniquely defined up to adding constant functions. Moreover, choosing these constants appropriately, we can assume that ∂u i(n) = u i(n+1) . Thus, under this assumption V is an algebra of differential functions extension of the algebra R in (129). Lemma 10. Let V be an algebra of differential functions extension of the algebra R . Then: (a) We have ∂ = ∂ R + ∂ , where ∂R =
i∈I,n∈Z+
(n+1)
ui
∂ ∂u i(n)
and ∂ is a derivation of V which commutes with all
, ∂ (n) ∂u i
(136)
and which vanishes on
R ⊂ V. In particular, ∂ Vn,i ⊂ Vn,i . (b) If f ∈ Vn,i \Vn,i−1 , then ∂ f ∈ Vn+1,i \Vn+1,i−1 , and it has the form (n+1) h ju j + r, ∂f = j≤i
where h j ∈ Vn,i for all j ≤ i, r ∈ Vn,i , and h i = 0. (c) For f ∈ V, f g = 0 for every g ∈ V if and only if f = 0.
(137)
Lie Conformal Algebra Cohomology and the Variational Complex
709
f Proof. Part (a) is clear. By part (a), we have that ∂ f is as in (137), where h j = ∂ (n) ∈ ∂u j (m) ∂ f Vn,i , and r = j∈I,m≤n u j (m−1) + ∂ f ∈ Vn,i . We are left to prove part (c). Suppose ∂u j f = 0 is such that f g = 0 for every g ∈ V. By taking g = 1, we have that f ∈ ∂V. (n+1) Hence f has the form (137) for some i ∈ I and n ∈ Z+ . But then u i f does not have (n+1) this form, so that u i f = 0.
Definition 2. The algebra of differential functions V is called normal if we have (n) ∂ du i f ∈ Vn,i (n) Vn,i = Vn,i for all i ∈ I, n ∈ Z+ . Given f ∈ Vn,i , we denote by ∂u i
a preimage of f under the map Vn,i−1 .
∂ (n) . This integral is defined up to adding elements from ∂u i
Proposition 13. Any normal algebra of differential functions V is an extension of R . (n)
Proof. As pointed out above, we need to find elements u i ∈ V, for i ∈ I, n ∈ Z+ , such that (135) holds. By the normality assumption, there exists vin ∈ Vn,i such that ∂vin
∂v n
∂v n
= 1. Note that ∂(n) (n)i = ∂1(n) = 0, hence (n)i ∈ Vn,i−1 . If we then replace ∂u ∂u i−1 ∂u i−1 ∂u i−1 ni ∂vin ∂win ∂wn n n n vi by wi = vi − du i−1 (n) , we have that (n) = 1 and (n)i = 0. Proceeding by (n) ∂u i
∂u i−1
∂u i
(n)
downward induction, we obtained the desired element u i .
∂u i−1
Clearly, the algebra R is normal. Moreover, any extension V of R can be further extended to a normal algebra, by adding missing integrals. For example, the localization of R1 = F[u (n) | n ∈ Z+ ] by u is not a normal algebra, since it doesn’t contain du u . Note that any differential algebra (A, ∂) can be viewed as a trivial algebra of differential functions with ∂(n) = 0. Such an algebra does not contain R , hence it is not ∂u i
normal.
5.3. The complex of variational calculus. Let V be an algebra of differential functions. • = Ω • (V) is defined as the free commutative superalThe basic de Rham complex Ω (n) • consists of gebra over V with odd generators δu i , i ∈ I, n ∈ Z+ . In other words Ω finite sums of the form (m ) (m ) 1 ···m k 1 ···m k f im1 ···i δu i1 1 ∧ · · · ∧ δu ik k , f im1 ···i ∈ V, (138) ω= k k ir ∈I,m r ∈Z+
and it has a (super)commutative product given by the wedge product ∧. We have a nat k defined by saying that elements in V have degree • = ural Z+ -grading Ω Ω k∈Z+ (n) k is a free module over V with 0, while the generators δu have degree 1. Hence Ω i
(m 1 )
(m k )
, with (m 1 , i 1 ) > · · · > (m k , i k ) (with 0 = V and Ω 1 = i∈I,n∈Z Vδu (n) . respect to the lexicographic order). In particular Ω i + 1 Notice that there
is a natural V-linear pairing Ω × g → V defined on generators by (m) 1 × g by V-bilinearity. δu i , ∂(n) = δi, j δm,n , and extended to Ω
basis given by the elements δu i1
∂u j
∧ · · · ∧ δu ik
710
A. De Sole, V. G. Kac
• , such that δ f = We let δ be an odd derivation of degree 1 of the complex Ω (n) (n) ∂f for f ∈ V, and δ(δu i ) = 0. It is immediate to check that δ 2 = 0 (n) δu i i∈I, n∈Z+
∂u i
k as in (138), we have and that, for ω∈Ω δ( ω) =
1 ···m k ∂ f im1 ···i k
ir ∈I,m r ∈Z+ j∈I,n∈Z+
∂u (n) j
(n)
(m 1 )
δu j ∧ δu i1
(m k )
∧ · · · ∧ δu ik
.
(139)
• → Ω • , as an odd derivation For X ∈ g we define the contraction operator ι X : Ω (n) • of degree -1, such that ι X ( f ) = 0 for f ∈ V, and ι X (δu ) = X (u (n) ). If X ∈ g of Ω i i k is as in (138), we have is as in (132) and ω∈Ω ω) = ι X (
k
ir ∈I,m r ∈Z+ q=1
q
1) 1 ···m k k) (−1)q+1 f im1 ···i Piq ,m q δu i(m ∧ · ˇ· · ∧δu i(m . 1 k k
(140)
In particular, for f ∈ V we have ι X (δ f ) = X ( f ).
(141)
It is easy to check that the operators ι X , X ∈ g, form an abelian (purely odd) subalgebra • , namely of the Lie superalgebra Der Ω [ι X , ιY ] = ι X ◦ ιY + ιY ◦ ι X = 0.
(142)
The Lie derivative L X along X ∈ g is defined as a degree 0 derivation of the super• , commuting with δ, and such that algebra Ω LX( f ) = X( f )
0 . for f ∈ Ω
(143)
One can easily check (on generators) Cartan’s formula (cf. (45)): L X = [δ, ι X ] = δ ◦ ι X + ι X ◦ δ.
(144)
We next prove the following: [ι X , L Y ] = ι X ◦ L Y − L Y ◦ ι X = ι[X,Y ] .
(145)
0 = V. It is clear by degree considerations that both sides of (145) act as zero on Ω Moreover, it follows by (141) that [ι X , L Y ](δ f ) = ι X διY δ f − ιY δι X δ f = X (Y ( f )) − Y (X ( f )) = [X, Y ]( f ) = ι[X,Y ] (δ f ) for every f ∈ V. Equation (145) then follows Finally, as by the fact that both sides are even derivations of the wedge product in Ω. immediate consequence of Eq. (145), we get that [L X , L Y ] = L X ◦ L Y − L Y ◦ L X = L [X,Y ] .
(146)
• is a g-complex, • by derivations. Thus, Ω g acting on Ω • , such that Note that the action of ∂ on V extends to a degree 0 derivation of Ω (n)
(n+1)
∂(δu i ) = δu i
, i ∈ I, n ∈ Z+ .
(147)
Lie Conformal Algebra Cohomology and the Variational Complex
711
This derivation commutes with δ, hence we can consider the corresponding reduced de Rham complex Ω • = Ω • (V), usually called the complex of variational calculus: + k /∂ Ω k , Ω• = Ωk, Ωk = Ω k∈Z+
with the induced action of δ. With an abuse of notation, we denote by δ and, for X ∈ g∂ , by ι X , L X , the maps induced on the quotient space Ω k by the corresponding maps on k . Obviously, Ω • is a g∂ -complex. Ω 5.4. Isomorphism of the cohomology g∂ -complexes Ω • and Γ • . Proposition 14. Let V be an algebra of differential functions. Consider the Lie conformal algebra A = ⊕i∈I F[∂]u i with the zero λ-bracket. Then V is a module over the Lie conformal algebra A, with the λ-action given by ∂f λn (n) . (148) ui λ f = ∂u i n∈Z+ Moreover, the λ-action of A on V is by derivations of the associative product in V. Proof. The fact that V is an A-module follows from the definition of an algebra of differential functions. The second statement is clear as well. Let Γ• = Γ• (A, V) and Γ • = Γ • (A, V) be the basic and reduced Lie conformal algebra cohomology complexes for the A-module V, defined in Proposition 14. Thus, to every algebra of differential functions V we can associate two apparently unrelated types of cohomology complexes: the basic and reduced de Rham cohomology com• (V) and Ω • (V), defined in Sect. 5.3, and the basic and reduced Lie conformal plexes, Ω • • algebra cohomology complexes Γ (A, V) and Γ (A, V), defined in Sect. 2.1, for the Lie conformal algebra A = i∈I F[∂]u i , with the zero λ-bracket, acting on V, with the λ-action given by (148). We are going to prove that, in fact, these complexes are isomorphic, and all the related structures (such as exterior products, contraction operators, Lie derivatives,...) correspond via this isomorphism. We denote, as in Sect. 4.2, by Γ• = Γ• (A, V) (resp. Γ• = Γ• (A, V)) the basic (resp. reduced) space of chains of A with coefficients in V. Recall from Sect. 4.4 that Π Γ1 is " identified with the space (A ⊗ V[[x]]) (∂ ⊗ 1 + 1 ⊗ ∂x )(A ⊗ V[[x]]), and it carries a Lie algebra structure given (65), which in this case takes the form, by the1 Lie bracket 1 for i, j ∈ I and P(x) = m∈Z+ m! Pm x m , Q(x) = n∈Z+ n! Q n x n ∈ V[[x]]: [u i ⊗ P(x), u j ⊗ Q(x)] = −u i ⊗
n∈Z+
Qn
∂ P(x) (n) ∂u j
+uj ⊗
m∈Z+
Pm
∂ Q(x) (m)
∂u i
.
(149)
Moreover, ∂ acts on Γ1 by (59). Its kernel Π Γ1 consists of elements of the form u i ⊗ e x∂ Pi , where Pi ∈ V, (150) i∈I
and it is a Lie subalgebra of Π Γ1 . We also denote, as in Sect. 5.1, by g the Lie algebra of all vector fields (132) acting on V, and by g∂ ⊂ g the Lie subalgebra of evolutionary vector fields (133).
712
A. De Sole, V. G. Kac
Proposition 15. The map Φ1 : Π Γ1 → g, which maps ξ=
u i ⊗ Pi (x) =
i∈I,n∈Z+
i∈I
1 u i ⊗ Pi,n x n ∈ Γ1 , n!
(151)
to
Φ1 (ξ ) =
Pi,n
i∈I, n∈Z+
∂ (n)
∂u i
,
(152)
is a Lie algebra isomorphism. Moreover, the image of the space of reduced 1-chains via Φ1 is the space of evolutionary vector fields. Hence we have the induced Lie algebra ∼ isomorphism Φ1 : Π Γ1 → g∂ . Proof. Clearly, Φ1 is a bijective map, and, by (150), Φ1 (Γ1 ) = g∂ . Hence we only need to check Φ1 is a Lie algebra homomorphism. This is immediate from Eq. (149). • , such that Φ 0 = 1I|V and, for k ≥ 1, Φ k : Γk → Theorem 6. The map Φ • : Γ• → Ω k is given by Ω Φ k ( γ) =
1 k!
(m 1 )
ir ∈I,m r ∈Z+
1 ···m k f im1 ···i δu i1 k
(m k )
∧ · · · ∧ δu ik
,
(153)
mk 1 ···m k 1 where f im1 ···i ∈ V is the coefficient of λm γλ1 ,...,λk (u i1 , . . . , u ik ), is an iso1 · · · λk in k morphism of superalgebras, and an isomorphism of g-complexes, (once we identify the Lie algebras g and Π Γ1 via Φ1 , as in Proposition 15). Moreover, Φ • commutes with the action of ∂, hence it induces an isomorphism of the ∼ corresponding reduced g∂ -complexes: Φ • : Γ • → Ω • .
Proof. Since I is a finite index set, the RHS of (153) is a finite sum, so that Φ k (Γk ) ⊂ k . By the sesquilinearity and skew-symmetry Conditions A1 and A2 in Sect. 2.1, Ω elements γ ∈ Γk areuniquely determined by the collection of polynomials 1 ···m k m 1 k γλ1 ,...,λk (u i1 , . . . , u ik ) = m r ∈Z+ f im1 ···i λ1 · · · λm k , which are skew-symmetric with k respect to simultaneous permutation of the variables λr and the indices ir . We want to k . In fact, denote by Ψ k : Ω k → Γk check that Φ k is a bijective linear map from Γk to Ω k the linear map which to ω as in (138) associates the k-cochain Ψ ( ω), such that 1 ···m k m 1 k ω)λ1 ,...,λk (u i1 , . . . , u ik ) = f im1 ···i λ1 · · · λm Ψ k ( k , k m r ∈Z+
where f denotes the skew-symmetrization of f : m σ (1) ···m σ (k) 1 ···m k = sign(σ ) f iσ (1) f im1 ···i ···i σ (k) , k σ
and Ψ k ( ω) is extended to A⊗k by the sesquilinearity Condition A1. It is straightforward to check that Ψ k ( ω) is indeed a k-cochain, and that the maps Φ k and Ψ k are inverse to each other. This proves that Φ • is a bijective map. Next, let us prove that Φ • is an associative superalgebra homomorphism. Let α ∈ h ∈ Γk and let α m 1 ,...,m h be the coefficient of λm 1 · · · λm h in the polynomial Γ ,β 1 i 1 ,...,i h h λ1 ,...,λk αλ1 ,...,λh (u i1 , . . . , u i h ), and let β n 1 ,...,n k be the coefficient of λn 1 · · · λn k in β j1 ,..., jk
1
k
Lie Conformal Algebra Cohomology and the Variational Complex
713
m h+k 1 )λ1 ,...,λh+k (u i1 , . . . , u i h+k ) (u j1 , . . . , u jk ). By (35), the coefficient of λm α ∧β 1 · · · λh+k in ( is sign(σ ) m σ (1) ,...,m σ (h) m σ (h+1) ,...,m σ (h+k) α β . h!k! iσ (1) ,...,iσ (h) iσ (h+1) ,··· ,iσ (h+k) σ ∈Sh+k
) = Φ h ( ) follows by the definition (153) of Φ k . The identity Φ h+k ( α∧β α ) ∧ Φ k (β m ···m mk 1 k 1 k Let γ ∈ Γ , and denote by f i1 ···ik ∈ V the coefficient of λm γλ1 ,...,λk 1 · · · λk in k+1 k (u i1 , . . . , u ik ). We want to prove that Φ (δ γ ) = δΦ ( γ ). By assumption, the λ-bracket
on A is zero, and the λ-action of A on V is given by (148). Hence, recalling (10), the m k+1 1 coefficient of λm γ )λ1 ,...,λk+1 (u i1 , . . . , u ik+1 ) is 1 · · · λk+1 in the polynomial (δ r
k+1
ˇ k+1 ∂ f mr1 ···m
(−1)r +1
r =1
i 1 ···i ˇ k+1 (m ) ∂u ir r
.
It follows that q
Φ k+1 (δ γ) = =
1 (k + 1)! 1 k!
k+1 (−1)q+1
ir ∈I,m r ∈Z+ q=1 1 ···m k ∂ f im1 ···i k (m 0 ) ∂u i0 ir ∈I,m r ∈Z+
ˇ k+1 ∂ f mq1 ···m i 1 ···i ˇ k+1 (m ) ∂u iq q
1) k+1 ) δu i(m ∧ · · · ∧ δu i(m 1 k+1
0) k) δu i(m ∧ · · · ∧ δu i(m = δΦ k ( γ ), 0 k
thus proving the claim. mk 1 ···m k 1 Similarly, the coefficient of λm γ )λ1 ,...,λk (u i1 , . . . , u ik ) is ∂ M f im1 ···i 1 · · · λk in (∂ k k 1 ···m r −1···m k + r =1 f im1 ···i , so that k ⎛ 1 k ⎝∂ M f m 1 ···m k δu (m 1 ) ∧ · · · ∧ δu (m k ) γ) = Φ (∂ i1 ik i 1 ···i k k! ir ∈I,m r ∈Z+ ⎞ k (m q +1) (m 1 ) (m k ) ⎠ m 1 ···m k = ∂Φ k ( + f i1 ···ik δu i1 ∧ · · · ∧ δu iq ∧ · · · ∧ δu ik γ ). q=1
Φ•
This proves that is compatible with the action of ∂. Finally, we prove that Φ • is compatible with the contraction operators. Let γ ∈ Γk be as in the statement of the theorem, and let ξ ∈ Γ1 be as in (151). By Eq. (63), we have the following formula for the contraction operator ιξ , # $ (ιξ Pi1 (x1 ), γ )λ2 ,...,λk (u i2 , . . . , u ik ) = γλ1 ,λ2 ,...,λk (u i1 , u i2 , . . . , u ik ) , i 1 ∈I
where , denotes the contraction of x1 with λ1 defined in (62). Hence, the coefficient mk 2 of λm γ )λ2 ,...,λk (u i2 , . . . , u ik ) is 2 · · · λk in (ιξ Pi1 ,m 1 f im1 i12m···i2 k···m k . i 1 ∈I,m 1 ∈Z+
714
A. De Sole, V. G. Kac
It follows that γ )) = Φ k−1 (ιξ (
1 (k − 1)!
ir ∈I,m r ∈Z+
(m 2 )
Pi1 ,m 1 f im1 i12m···i2 k···m k δu i2
(m k )
∧ · · · ∧ δu ik
,
γ )). This completes the proof which, recalling (140) and (152), is the same as ιΦ1 (ξ ) (Φ k ( of the theorem. Let V be an alge5.5. An explicit construction of the g∂ -complex of variational calculus. bra of differential functions in the variables {u i }i∈I , let A = i∈I F[∂]u i be the free F[∂]-module of rank , considered as a Lie conformal algebra with the zero λ-bracket, and consider the A-module structure on V, with the λ-action given by (148). By Theorem 6, the g∂ -complex of variational calculus Ω • (V) is isomorphic to the Π Γ1 -complex Γ • (A, V). Furthermore, due to Theorems 2 and 4, the Π Γ1 -complex Γ • (A, V) is iso• morphic to the ΠC1 -complex C (A, V) = k∈Z+ C k , which is explicitly described in Sects. 2.3 and 4.6. In this section we use this isomorphism to describe explicitly the ΠC1 g∂ -complex of variational calculus C • (A, V) Ω • (V), both in terms of “poly-symbols”, and in terms of skew-symmetric “poly-differential operators”. We shall identify these two complexes via this isomorphism. We start by describing all vector spaces Ω k and the maps d : Ω k → Ω k+1 , k ∈ Z+ . First, we have Ω 0 = V/∂V.
(154)
Next, Ω 1 = HomF[∂] (A, V), hence we have a canonical identification Ω 1 = V ⊕ .
(155)
Comparing (12) and (148), we see that d : Ω 0 → Ω 1 is given by the variational derivative: δf . d f = δu
(156)
For arbitrary k ≥ 1, the space Ω k can be identified with the space of k-symbols in u i , i ∈ I . By definition, a k-symbol is a collection of expressions of the form u i1 λ1 u i2 λ2 · · · u ik−1 λk−1 u ik ∈ F[λ1 , . . . , λk−1 ] ⊗ V, (157) where i 1 , . . . , i k ∈ I , satisfying the following skew-symmetry property: u i1 λ1 u i2 λ2 · · · u ik−1 λk−1 u ik = sign(σ ) u iσ (1) λ · · · u iσ (k−1) λ u iσ (k) , σ (1)
σ (k−1)
(158)
for every permutation σ ∈ Sk , where λk is replaced, if it occurs in the RHS, by λ†k = − k−1 j=1 λ j − ∂, with ∂ acting from the left. Clearly, by sesquilinearity, for k ≥ 1, the space Ω k = C k of k-λ-brackets is one-to-one correspondence with the space of k-symbols.
Lie Conformal Algebra Cohomology and the Variational Complex
715
⊕ For example, the space of 1-symbols is the same as V . A 2-symbol is a collection of elements u i λ u j ∈ F[λ] ⊗ V, for i, j ∈ I , such that u i λ u j = − u j −λ−∂ u i . A 3-symbol is a collection of elements u i λ u j µ u k ∈ F[λ, µ] ⊗ V, for i, j, k ∈ I , such that u i λ u j µ u k = − u j µ u i λ u k = − u i λ u k −λ−µ−∂ u j ,
and similarly for k > 3. Comparing (13) and (148) we see that, if F ∈ V ⊕ , its differential d F corresponds to the following 2-symbol: n ∂ Fj n ∂ Fi ui λu j = − (−λ − ∂) λ = (D F ) ji (λ) − (D ∗F ) ji (λ), (159) (n) (n) ∂u ∂u i j n∈Z+ where D F is the Frechet derivative defined by (9). More generally, the differential of a k-symbol for k ≥ 1 is given by the following formula:
d {u i1 λ1 · · · u ik−1 λk−1 u ik } i 1 ,...,i k ∈I ⎛ k s s+1 n ∂ ⎝ = (−1) λs (n) u i1 λ1 · ˇ· · u ik λk u ik+1 (160) ∂u is n∈Z+ s=1 ⎛ ⎞ ⎞n k ∂ ⎠ ⎝− u + (−1)k λ j − ∂⎠ · · · u u . i i i 1 k−1 k λ λk−1 1 (n) ∂u j=1 i k+1 n∈Z+ i 1 ,...,i k+1 ∈I
Provided that V is an algebra of differential functions extension of R , an equivalent language is that of skew-symmetric poly-differential operators. By definition, a k-differential operator is an F-linear map S : (V )k → V/∂V, of the form ,...,n k n 1 1 f in1 1,...,i (∂ Pi1 ) · · · (∂ n k Pikk ). (161) S(P 1 , . . . , P k ) = k n 1 ,...,n k ∈Z+ i 1 ,...,i k ∈I
The operator S is called skew-symmetric if S(P 1 , . . . , P k ) = sign(σ )S(P σ (1) , . . . , P σ (k) ), for every P 1 , . . . , P k ∈ V and every permutation σ ∈ Sk . Given a k-symbol n ,...,n n k−1 1 k−1 f i1 ,...,i λn 1 · · · λk−1 , i 1 , . . . , i k ∈ I, u i1 λ1 · · · u ik−1 λk−1 u ik = k−1 ,i k 1
(162)
n 1 ,...,n k−1 ∈Z+
n ,...,n
1 k−1 where f i1 ,...,i ∈ V, we associate to it the following poly-differential operator: k k S : (V ) → V/∂V, is n 1 ,...,n k−1 1 k S(P , . . . , P ) = f i1 ,...,i (∂ n 1 Pi11 ) · · · (∂ n k−1 Pik−1 )Pikk . (163) k−1 ,i k k−1
n 1 ,...,n k−1 ∈Z+ i 1 ,...,i k ∈I
716
A. De Sole, V. G. Kac
Clearly, the skew-symmetry property of the k-symbol is translated to the skew-symmetry of the poly-differential operator. Conversely, integrating by parts, any k-differential operator can be written in the form (163). Thus we have a surjective map Ξ from the space of k-symbols to the space of skew-symmetric k-differential operators. Provided that V is an algebra of differential functions extension of R , by Lemma 10(c), the k-differential operator S can be written uniquely in the form (163). Hence, the map Ξ is an isomorphism. Note that the space of 1-differential operators S : V → V/∂V can be canonically identified space Ω 1 = V ⊕ . Explicitly, to the 1-differential operator with the n ∂ n P , we associate: S(P) = f i i∈I,n∈Z+ i ⎛ ⎝
⎞
(−∂)n f in ⎠
n∈Z+
∈ V ⊕ .
(164)
i∈I
We can write down the expression of the differential d : Ω k → Ω k+1 in terms of poly-differential operators. First, if F ∈ Ω 1 = V ⊕ , the 2-differential operator corresponding to d F ∈ Ω 2 is obtained by looking at Eq. (159): (d F)(P, i) i∈I Q i X P (Fi ) − Pi X Q (F Q) = (165) = i, j∈I Q i D F (∂)i j P j − Pi D F (∂)i j Q j , where X P denotes the evolutionary vector field associated to P ∈ V , defined in (133), and D F (∂) is the Frechet derivative (9). Next, if S : (V )k → V/∂V is a skew-symmetric k-differential operator, its differential d S, obtained by looking at (160), is the following k + 1-differential operator: (d S)(P 1 , . . . , P k+1 ) =
k+1 s (−1)s+1 (X P s S) (P 1 , · ˇ· ·, P k+1 ).
(166)
s=1
In the above formula, if S is as in (161), X P S denotes the k-differential operator obtained ,...,n k 1 ,...,n k from S by replacing the coefficients f in1 ,...,i by X P ( f in1 1,...,i ). k k Remark 8. For k ≥ 2, a k-differential operator can also be understood as a map S : (V )k−1 → V ⊕ of the following form: n ,...,n k−1 n1 1 n k−1 k−1 S(P 1 , . . . , P k−1 )ik = f i1 1,...,ik−1 Pik−1 ). (167) ,i k (∂ Pi 1 ) · · · (∂ n 1 ,...,n k−1 ∈Z+ i 1 ,...,i k−1 ∈I
This corresponds to the k-symbol (162) in the obvious way. With this notation, the differential d S is the following map (V )k → V ⊕ : (d S)(P 1 , . . . , P k )i =
k s (−1)s+1 (X P s S)(P 1 , · ˇ· ·, P k )i s=1
+ (−1)
k
j∈I,n∈Z+
(−∂)
n
P jk
∂S (n)
∂u i
(P , . . . , P 1
k−1
)j .
(168)
Lie Conformal Algebra Cohomology and the Variational Complex
717
Recall that the Lie algebra g∂ ΠC1 is identified with the space V via the map P → X P , defined in (133). Given P ∈ V , we want to describe explicitly the action of the corresponding contraction operator ι P and the Lie derivative L P = [d, ι P ]. First, for F ∈ V ⊕ = Ω 1 , we have (cf. (90)): ι P (F) = Pi Fi ∈ V/∂V = Ω 0 . (169) i∈I
Next, the contraction of a k-symbol for k ≥ 2 is given by the following formula (cf. (91)):
ι P u i1 λ1 · · · u ik−1 λk−1 u ik i 1 ,...,i k ∈I ⎞ ⎛ u i1 ∂ u i2 λ2 · · · u ik−1 λk−1 u ik = ⎝ Pi1 ⎠ , (170) →
i 1 ∈I
i 2 ,...,i k ∈I
where, as usual, the arrow in the RHS means that ∂ is moved to the right. For k = 2, the above formula becomes ⎞ ⎛ ι P u i λ u j i, j∈I = ⎝ u j ∂ ui → Pj ⎠ ∈ V ⊕ = Ω 1 . (171) j∈I
i∈I
We can write the above formulas in the language of poly-differential operators. For a k-differential operator S, we have (ι P 1 S)(P 2 , . . . , P k ) = S(P 1 , P 2 , . . . , P k ).
(172)
For k = 2 ι P 1 S is a 1-differential operator which, by (164), is the same as an element of V ⊕ = Ω 1 . Remark 9. In the interpretation (167) of a k-differential operator, the action of the contraction operator is given by (ι P 1 S)(P 2 , . . . , P k−1 )ik = S(P 1 , P 2 , . . . , P k−1 )ik . Next, we write the formula for the Lie derivative L Q : Ω k → Ω k , associated to Q ∈ V g∂ , using Cartan’s formula L Q = [ι Q , d]. Recalling (156) and (169), after integration by parts we obtain, for f ∈ Ω 0 = V/∂V: f = X Q ( f ), (173) LQ where X Q is the evolutionary vector field corresponding to Q (cf. (133)). Similarly, recalling (159) and (171), we obtain, for F ∈ Ω 1 = V ⊕ : dι Q (F) = D F (∂)∗ Q + D Q (∂)∗ F, ι Q d(F) = D F (∂)Q − D F (∂)∗ Q, where D F (∂) denotes the Frechet derivative (9), and D F (∂)∗ is the adjoint differential operator. Putting the above formulas together, we get: L Q F = D F (∂)Q + D Q (∂)∗ F.
(174)
718
A. De Sole, V. G. Kac
For k ≥ 2, L Q acts on a k-symbol in Ω k by the following formula, which can be derived from (160) and (170): L Q {u i1 λ1 · · · u ik−1 λk−1 u ik } = X Q {u i1 λ1 · · · u ik−1 λk−1 u ik } k−1 s + (−1)s+1 {u j λs +∂ u i1 λ1 · ˇ· · u ik−1 λk−1 u ik }→ D Q (λs ) jis s=1
j∈I
+ (−1)
k+1
{u j λ† +∂ u i1 λ1 · · · u ik−2 λk−2 u ik−1 }→ D Q (λ†k ) jik . j∈I
k
In the RHS the evolutionary vector field X Q is applied to the coefficients of the k-symbol, in the last two terms the arrow means, as usual, that we move ∂ to the right, D Q (λ) denotes the Frechet derivative (9) considered as a polynomial in λ, and, in the last term, λ†k = −λ1 − · · · − λk−1 − ∂, where ∂ is moved to the left. This formula takes a much nicer form in the language of k-differential operators. Namely we have: (L Q S)(P 1 , . . . , P k ) = (X Q S)(P 1 , . . . , P k ) +
k
S(P 1 , . . . , X Q P s , . . . , P k ).
s=1
(175) Here X Q S has the same meaning as in Eq. (166). This formula can be obtained from the previous one by integration by parts. 5.6. An application to the classification of symplectic differential operators. Recall that C ⊂ V denotes the subspace (127) of constant functions. In [BDK] we prove the following: Theorem 7. If V is normal, then H k (Ω • , d) = δk,0 C/(C ∩ ∂V). Recall that a symplectic differential operator (cf. [D] and [BDK]) is a skew-adjoint differential operator S(∂) = Si, j (∂) i, j∈I : V → V ⊕ , which is closed, namely the following condition holds (cf. (168)): u i λ Sk j (µ) − u j µ Ski (λ) − u k −λ−µ−∂ S ji (λ) = 0,
(176)
where the λ-action of u i on V is defined by (148). We have the following corollary of Theorem 7. Corollary 1. If V is a normal algebra of differential functions, then any symplectic differential operator is of the form: S F (∂) = D F (∂) − D F (∂)∗ , for some F ∈ V ⊕ . Moreover, S F = SG if and only if F − G = δδuf for some f ∈ V. A skew-symmetric k-differential operator S : (V )k → V/∂V is called symplectic if it is closed, i.e. k+1
s
(−1)s+1 (X P s S) (P 1 , · ˇ· ·, P k+1 ) = 0.
s=1
The following corollary of Theorem 7 is a generalization of Corollary 1 and uses Proposition 13
Lie Conformal Algebra Cohomology and the Variational Complex
719
Corollary 2. If V is a normal algebra of differential functions, then any symplectic k-differential operator, for k ≥ 1, is of the form: S(P 1 , . . . , P k ) =
k s (−1)s+1 (X P s T ) (P 1 , · ˇ· ·, P k ), s=1
for some skew-symmetric k − 1-differential operator T . Moreover, T is defined up to adding a symplectic k − 1-differential operator. Remark 10. It follows from the proof of Theorem 7 that, Corollaries 1 and 2 hold in any algebra of differential functions V, provided that we are allowed to take F and T respectively in an extension of V, obtained by adding finitely many integrals of elements (n) of V (an integral of an element f ∈ Vn,i is a preimage du i f of ∂(n) independent on (m) uj
∂u i
with (m, j) > (n, i)).
Remark 11. The map Ξ defined in Sect. 5.5 may have a non-zero kernel if V is not an extension of the algebra R , but, of course, for any V the image of Ξ is a g∂ -complex. The 0th term of this complex is V/∂V and the k th term, for k ≥ 1, is the space of skew-symmetric k-differential operators S : (V )k → V/∂V. Remark 12. Throughout this section we assumed that the number of variables u i is finite, but this assumption is not essential, and our arguments go through with minor modifications. This is the reason for distinguishing V from V ⊕ , in order to accommodate the case = ∞. References [BKV] [BDAK] [BDK] [D] [Di] [DTT] [H] [K] [Vi] [V]
Bakalov, B., Kac, V.G., Voronov, A.A.: Cohomology of conformal algebras. Commun. Math. Phys. 200, 561–598 (1999) Bakalov, B., D’Andrea, A., Kac, V.G.: Theory of finite pseudoalgebras. Adv. Math. 162(1), 1–140 (2001) Barakat, A., De Sole, A., Kac, V.G.: Poisson vertex algebras in the theory of Hamiltonian equations. http://arXiv.org/abs/0907.1275, 2009 Dorfman, I.: Dirac structures and integrability of non-linear evolution equations. New York: John Wiley and Sons, 1993 Dickey, L.A.: Soliton equations and Hamiltonian systems. Advanced Ser. Math. Phys. 26, Second ed., Singapore: World Sci., 2003 Dolgushev, V., Tamarkin, D., Tsygan, B.: Formality of the homotopy calculus algebra of Hochschild (co)chains. http://arXiv.org/abs/0807.5117v1[math.KT], 2008 Helmholtz, H.: Uber der physikalische bedentung des princips der klinstein wirkung. J. Reine Angen Math 100, 137–166 (1887) Kac, V.G.: Vertex algebras for beginners. Univ. Lecture Ser., Vol 10, 1996, Second edition, Providence, RI: Amer. Math. Soc., 1998 Vinogradov, A.M.: On the algebra-geometric foundations of lagrangian field theory. Sov. Math. Dokl. 18, 1200–1204 (1977) Volterra, V.: Leçons sur les Fonctions de Lignes. Paris: Gauthier-Villar, 1913
Communicated by Y. Kawahigashi
Commun. Math. Phys. 292, 721–759 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0884-3
Communications in
Mathematical Physics
Abelian Sandpiles and the Harmonic Model Klaus Schmidt1,2 , Evgeny Verbitskiy3,4 1 Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria.
E-mail: [email protected]
2 Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria 3 Philips Research, High Tech Campus 36 (M/S 2), 5656 AE, Eindhoven, The Netherlands.
E-mail: [email protected]
4 Department of Mathematics, University of Groningen, PO Box 407, 9700 AK, Groningen, The Netherlands
Received: 15 January 2009 / Accepted: 14 April 2009 Published online: 15 August 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: We present a construction of an entropy-preserving equivariant surjective map from the d-dimensional critical sandpile model to a certain closed, shift-invariant d subgroup of TZ (the ‘harmonic model’). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.
Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . 1.1 Four models . . . . . . . . . . . . . . . . . 1.2 Outline of the paper . . . . . . . . . . . . . 2. A Potential Function and its 1 -Multipliers . . . 3. The Harmonic Model . . . . . . . . . . . . . . 3.1 Linearization . . . . . . . . . . . . . . . . 3.2 Homoclinic points . . . . . . . . . . . . . . 3.3 Symbolic covers of the harmonic model . . 3.4 Kernels of covering maps . . . . . . . . . . 4. The Abelian Sandpile Model . . . . . . . . . . 5. The Critical Sandpile Model . . . . . . . . . . . 5.1 Surjectivity of the maps ξg : R∞ −→ X f (d) 5.2 Properties of the maps ξg , g ∈ I˜d . . . . . . 6. The Dissipative Sandpile Model . . . . . . . . . 6.1 The dissipative harmonic model . . . . . . (γ ) 6.2 The covering map ξ (γ ) : R∞ −→ X f (d,γ ) . 7. Conclusions and Final Remarks . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
722 722 723 723 733 734 734 735 739 744 747 747
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
754 755 755 756 758 759
722
K. Schmidt, E. Verbitskiy
1. Introduction For any integer d ≥ 2 let hd = 0
1
··· 0
1
log 2d − 2
d
cos(2π xi )
d x1 · · · d xd ,
(1.1)
i=1
h 2 = 1.166, h 3 = 1.673, etc. It turns out that for d ≥ 2, h d is the topological entropy of three different d-dimensional models in mathematical physics, probability theory, and dynamical systems. For d = 2, there is even a fourth model with the same entropy h d . 1.1. Four models. The d-dimensional abelian sandpile model was introduced by Bak, Tang and Wiesenfeld in [3,4] and attracted a lot of attention after the discovery of the Abelian property by Dhar in [8]. The set of infinite allowed configurations of the sandpile d model is the shift-invariant subset R∞ ⊂ {0, . . . , 2d − 1}Z defined in (4.4) and discussed in Sect. 4.1 In [10], Dhar showed that the topological entropy of the shift-action σR∞ on R∞ is also given by (3.4), which implies that every shift-invariant measure µ of maximal entropy on R∞ has entropy (1.1). Shift-invariant measures on R∞ were studied in some detail by Athreya and Jarai in [1,2], Jarai and Redig in [13]; however, the question of uniqueness of the measure of maximal entropy is still unresolved. Spanning trees of finite graphs are classical objects in combinatorics and graph theory. In 1991, Pemantle in his seminal paper [17] addressed the question of constructing uniform probability measures on the set Td of infinite spanning trees on Zd — i.e., on the set of spanning subgraphs of Zd without loops. This work was continued in 1993 by Burton and Pemantle [5], where the authors observed that the topological entropy of the set of all spanning trees in Zd is also given by the formula (1.1). Another problem discussed in [5] is the uniqueness of the shift-invariant measure of maximal entropy on Td (the proof in [5] is not complete, but Sheffield has recently completed the proof in [22]. This coincidence of entropies raised the question about the relation between these models. A partial answer to this question was given in 1998 by R. Solomyak in [24]: she constructed injective mappings from the set of rooted spanning trees on finite regions of Zd into X f (d) such that the images are sufficiently separated. In particular, this provided a direct proof of coincidence of the topological entropies of α f (d) and σTd without making use of formula (1.1). In dimension 2, spanning trees are related not only to the sandpile models (cf. e.g., [19] for a detailed account) and, by [24], to the harmonic model, but also to a dimer model (more precisely, to the even shift-action on the two-dimensional dimer model) by [5]. However, the connections between the abelian sandpiles and spanning trees (as well as dimers in dimension 2), are non-local: they are obtained by restricting the models to finite regions in Zd (or Z2 ) and constructing maps between these restrictions, but these maps are not consistent as the finite regions increase to Zd . In this paper we study the relation between the infinite abelian sandpile models and the algebraic dynamical systems called the harmonic models. The purpose of this paper is to define a shift-equivariant, surjective local mapping between these models: from the d
1 In the physics literature it is more customary to view the sandpile model as a subset of {1, . . . , d}Z by adding 1 to each coordinate.
Abelian Sandpiles and the Harmonic Model
723
infinite critical sandpile model R∞ to the harmonic model. Although we are not able to prove that this mapping is almost one-to-one it has the property that it sends every shift-invariant measure of maximal entropy on R∞ to Haar measure on X f (d) . Moreover, it sheds some light on the somewhat elusive group structure of R∞ . Firstly, the dual group of X f (d) is the group Gd = Rd /( f (d) ), ± where Rd = Z[u ± 1 , . . . , u d ] is the ring of Laurent polynomials with integer coefficients in the variables u 1 , . . . , u d , and ( f (d) ) is the principal ideal in Rd generated by d (u i + u i−1 ). The group Gd is the correct infinite analogue of the groups f (d) = 2d − i=1 of addition operators defined on finite volumes, see [9,19] (cf. Sect. 7). Secondly, the map ξ Id constructed in this paper gives rise to an equivalence relation ∼ on R∞ with
x ∼ y ⇐⇒ x − y ∈ ker(ξ Id ), such that R∞ /∼ is a compact abelian group. Moreover, R∞ /∼ , viewed as a dynamical system under the natural shift-action of Zd , has the topological entropy (1.1). This extends the result of [16], obtained in the case of dissipative sandpile model, to the critical sandpile model. Finally, we also identify an algebraic dynamical system isomorphic to the dissipative sandpile model. This allows an easy extension of the results in [16]: namely, the uniqueness of the measure of maximal entropy on the set of infinite recurrent configurations in the dissipative case. Unfortunately, we are not yet able to establish the analogous uniqueness result in the critical case. 1.2. Outline of the paper. Sect. 2 investigates certain multipliers of the potential function (or Green’s function) of the simple random walk on Zd . In Sect. 3 these results are used to describe the homoclinic points of the harmonic model. These points are then used to define shift-equivariant maps from the space ∞ (Zd , Z) of all bounded d-parameter sequences of integers to X f (d) . In Sect. 4 we introduce the critical and dissipative sandpile models. In Sect. 5 we show that the maps found in Sect. 3 send the critical sandpile model R∞ onto X f (d) , preserve topological entropy, and map every measure of maximal entropy on R∞ to Haar measure on the harmonic model. After a brief discussion of further properties of these maps in Subsect. 5.2, we turn to dissipative sandpile models in Sect. 6 and define an analogous map to another closed, shift-invariant subgroup of d TZ . The main result in [16] shows that this map is almost one-to-one, which implies that the measure of maximal entropy on the dissipative sandpile model is unique and Bernoulli. 2. A Potential Function and its 1 -Multipliers Let d ≥ 1. For every i = 1, . . . , d we write e(i) = (0, . . . , 0, 1, 0, . . . , 0) for the i th unit vector in Zd , and we set 0 = (0, . . . , 0) ∈ Zd . d We identify the cartesian product Wd = RZ with the set of formal real power series ±1 ±1 in the variables u 1 , . . . , u d by viewing each w = (wn ) ∈ Wd as the power series wn u n (2.1) n∈Zd
724
K. Schmidt, E. Verbitskiy
with wn ∈ R and u n = u n1 2 · · · u nd d for every n = (n 1 , . . . , n d ) ∈ Zd . The involution w → w ∗ on Wd is defined by wn∗ = w−n , n ∈ Zd .
(2.2)
For E ⊂ Zd we denote by π E : Wd −→ R E the projection onto the coordinates in E. For every p ≥ 1 we regard p (Zd ) as the set of all w ∈ Wd with ⎛ ⎞1/ p |wn | p ⎠ < ∞. w p = ⎝ n∈Zd
Similarly we view ∞ (Zd ) as the set of all bounded elements in Wd , equipped with the ±1 1 d supremum norm · ∞ . Finally we denote by Rd = Z[u ±1 1 , . . . , u d ] ⊂ (Z ) ⊂ Wd the ring of Laurent polynomialswith integer coefficients. Every h in any of these spaces will be written as h = (h n ) = n∈Zd h n u n with h n ∈ R (resp. h n ∈ Z for h ∈ Rd ). The map (m, w) → u m ·w with (u m ·w)n = wn−m is a Zd -action by automorphisms of the additive group Wd which extends linearly to an Rd -action on Wd given by h·w = hnun · w (2.3) n∈Zd
for every h ∈ Rd and w ∈ Wd . If w also lies in Rd this definition is consistent with the usual product in Rd . For the following discussion we assume that d ≥ 2 and consider the irreducible Laurent polynomial d f (d) = 2d − (u i + u i−1 ) ∈ Rd . (2.4) i=1
The equation
f (d) · w = 1
(2.5)
with w ∈ Wd admits a multitude of However, there is a distinguished (or fundamental) solution w (d) of (2.5) which has a deep probabilistic meaning: it is a certain multiple of the lattice Green’s function of the symmetric nearest-neighbour random walk on Zd (cf. [6,12,25,27]). solutions.2
Definition 2.1. For every n = (n 1 , . . . , n d ) ∈ Zd and t = (t1 , . . . , td ) ∈ Td we set n, t = dj=1 n j t j ∈ T. We denote by F (d) (t) =
f n(d) e2πi n,t = 2d − 2 ·
d
n∈Zd
cos(2π t j ), t = (t1 , . . . , td ) ∈ Td ,
(2.6)
j=1
the Fourier transform of f (d) . 2 Under the obvious embedding of R → ∞ (Zd , Z), the constant polynomial 1 ∈ R corresponds to the d d element δ (0) ∈ ∞ (Zd , Z) given by
(0) δn =
1 if n = 0, 0 otherwise.
Abelian Sandpiles and the Harmonic Model
(1) For d = 2, wn(2)
:=
725
e−2πi n,t − 1 dt for every n ∈ Z2 . F (2) (t) T2
(2) For d ≥ 3, wn(d) :=
e−2πi n,t dt for every n ∈ Zd . (d) Td F (t)
The difference in these definitions for d = 2 and d > 2 is a consequence of the fact that the simple random walk on Z2 recurrent, while on higher dimensional lattices it is transient. Theorem 2.2. ([6,12,25,27]) We write · for the Euclidean norm on Zd . (i) For every d ≥ 2, w(d) satisfies (2.5). (ii) For d = 2, ⎧ ⎨ 0 if n = 0, 1 4 +n 4 )− 3 wn(2) = (n ⎩− 1 log n − κ2 − c2 n 4 1 2 2 4 + O( n −4 ) if n = 0, 8π n (2)
where κ2 > 0 and c2 > 0. In particular, w0 Moreover, 4 · wn(2) =
∞
(2.7)
(2)
= 0 and wn < 0 for all n = 0.
(P(X k = n|X 0 = 0) − P(X k = 0|X 0 = 0)),
k=1
where (X k ) is the symmetric nearest-neighbour random walk on Z2 . (iii) For d ≥ 3, n
d−2
wn(d)
= κd + cd
1 n 4
d
4 i=1 n i n 2
−
3 d+2
+ O( n −4 )
(2.8)
as n → ∞, where κd > 0, cd > 0. Moreover, 2d · wn(d) =
∞
P(X k = n|X 0 = 0) > 0 for every n ∈ Zd ,
k=0
where (X k ) is again the symmetric nearest-neighbour random walk on Zd . Definition 2.3. Let w(d) ∈ Wd be the point appearing in Definition 2.1. We set Id = g ∈ Rd : g · w (d) ∈ 1 (Zd ) ⊃ ( f (d) ),
(2.9)
(d) where ( f (d) ) = f (d) · Rd is the principal ideal generated by f (d) . Since wn(d) = w−n ∗ d ∗ for every n ∈ Z it is clear that Id = Id = {g : g ∈ Id }.
726
K. Schmidt, E. Verbitskiy
Theorem 2.4. The ideal Id is of the form Id = ( f (d) ) + I3d ,
(2.10)
Id = {h ∈ Rd : h(1) = 0} = (1 − u 1 ) · Rd + · · · + (1 − u d ) · Rd
(2.11)
where
with 1 = (1, . . . , 1). For the proof of Theorem 2.4 we need several lemmas. We set Jd = ( f (d) ) + I3d ⊂ Rd .
(2.12)
Lemma 2.5. Let g = k∈Zd gk u k ∈ Rd . Then g ∈ Jd if and only if it satisfies the following conditions (2.13)–(2.16). gk = 0, (2.13)
k∈Zd
gk ki = 0
for i = 1, . . . , d,
(2.14)
gk ki k j = 0
for 1 ≤ i = j ≤ d,
(2.15)
gk (ki2 − k 2j ) = 0
for 1 ≤ i = j ≤ d.
(2.16)
k=(k1 ,...,kd )∈Zd
k=(k1 ,...,kd
)∈Zd
k=(k1 ,...,kd )∈Zd
Proof. Condition (2.13) is equivalent to saying that g ∈ Id . In conjunction with (2.13), (2.14) is equivalent to saying that g ∈ I2d : indeed, if g ∈ Id , then it is of the form g= with ai ∈ Rd for i = 1, . . . , d. Then ∂g = ∂u j
d
(1 − u i ) · ai
(2.17)
i=1
k −1
gk k j · u k11 · · · u j j
· · · u kdd = −a j +
k=(k1 ,...,kd )∈Zd
d
(1 − u i ) ·
i=1
∂ai , ∂u j
∂g and ∂u (1) = 0 if and only if a j ∈ Id . j If g ∈ Id is of the form (2.17) and satisfies (2.14) we set
aj =
d (1 − u i ) · bi, j i=1
with bi, j ∈ Rd . Condition (2.15) is satisfied if and only if ∂a j ∂2g ∂ai (1) = − − = bi, j (1) + b j,i (1) = 0 ∂u i ∂u j ∂u j ∂u i for 1 ≤ i = j ≤ d.
(2.18)
Abelian Sandpiles and the Harmonic Model
727
Finally, if g satisfies (2.13)–(2.14) and is of the form (2.17)–(2.18) with bi, j ∈ Rd for all i, j, then (2.16) is equivalent to the existence of a constant c ∈ R with
gk ki2 = −2
k=(k1 ,...,kd )∈Zd
∂ai (1) = 2bi,i (1) = c ∂u i
for i = 1, . . . , d. The last equation shows that bi,i − b1,1 ∈ Id for i = 2, . . . , d. By combining all these observations we have proved that g satisfies (2.13)–(2.16) if and only if it is of the form g = h1 ·
d (1 − u i )2 + h 2
(2.19)
i=1
with c ∈ Z, h 1 ∈ Rd and h 2 ∈ I3d . The set of all such g ∈ Rd is an ideal which we d denote by J˜. Clearly, I3d ⊂ J˜ and i=1 (1 − u i )2 ∈ J˜. Since (1 − u i )2 · (1 − u i−1 ) ∈ I3d for i = 1, . . . , d as well, we conclude that f (d) =
d d (1 − u i )2 − (1 − u i−1 ) · (1 − u i )2 ∈ J˜. i=1
(2.20)
i=1
This shows that J˜ ⊂ Jd , and the reverse inclusion also follows from (2.20) and (2.19). Lemma 2.6. Id ⊂ Jd . (d) Proof. We assume that g ∈ Id and set v = g · w . In order to verify (2.13) we argue by contradiction and assume that k gk = 0. If d = 2 then gk log n + l.o.t., vn = − k 2π
for large n . If d ≥ 3, then vn =
κd k gk + l.o.t. n d−2
for large n . In both cases it is evident that v ∈ 1 (Zd ). By taking (2.13) into account one gets that, for every d ≥ 2, (d) vn = (g · w (d) )n = gk wn−k k
=
Td
e
−2πi n,t
2d
2πi k,t k gk e d − 2 j=1 cos(2π t j )
Hence v = (vn ) is the sequence of Fourier coefficients of the function 2πi k,t k gk e . H (t) = d 2d − 2 j=1 cos(2π t j )
dt.
728
K. Schmidt, E. Verbitskiy
If v ∈ 1 (Zd ), then H must be a continuous function Td . Since t = 0 is the only on2πi k,· (d) d zero of F on T (cf. (2.6)), the numerator G = k gk e must compensate for this singularity. Consider the Taylor series expansion of G at t = 0: G(t) =
gk + 2πi
k
d
tj
j=1
gk k j − 2π 2
d j=1
k
t 2j
gk k 2j − 4π 2
ti t j
i= j
k
gk ki k j
k
+h.o.t. The Taylor series expansion of F (d) at t = 0 is given by F (d) (t) = 4π 2
d
t 2j + h.o.t.
j=1
Suppose that h(t) =
a0 +
d
j=1 b j t j
d
2 j=1 c j t j + i= j 2 2 t1 + · · · + td + h.o.t
+
di, j ti t j + h.o.t
is continuous at t = 0. Then a0 = 0, b j = 0 for all j, c j = c for all j, di j = 0 for all i = j, and for some constant c. If any of these conditions is violated, then one easily produces examples of sequences t(m) → 0 as m → ∞ with distinct limits limm→∞ h(t(m) ). By applying this to H we obtain (2.13)–(2.16), so that g ∈ Jd by Lemma 2.5. To establish the inclusion Jd ⊆ Id , we have to show that for any g ∈ Jd , g·u ∈ 1 (Zd ), where u ∈ Wd of the form d n4 1 ωn = i=1d+4i , or ωn = with γ ≥ d − 2. n n γ For d = 2, we also have to treat the case ωn = log n . These results are obtained in the following three lemmas. Lemma 2.7. Suppose that d ≥ 2 and that ω ∈ Wd is given by
0 if n = 0, d ωn = i=1 n i4 if n = 0. n d+4 If g ∈ Rd satisfies (2.13), then g · ω ∈ 1 (Zd ). Proof. Let M = max{ k : gk = 0}, and suppose that n > M. Then d d 4 3 4 i=1 n i + O( n ) i=1 (n i − ki ) (g · ω)n = gk = g k n − k d+4 n d+4 (1 + O( n −1 )) k k d 4 1 1 i=1 n i =O . = gk + O n d+4 n d+1 n d+1 Therefore,
k
n |(g
· ω)n | < ∞.
Abelian Sandpiles and the Harmonic Model
729
For the reverse inclusion Jd ⊂ Id we need different arguments for d = 2 and for d ≥ 3. We start with the case d = 2. k ∈ R satisfies (2.13). We set S = Lemma 2.8. Suppose that g = 2 + k∈Z2 gk u {k : gk > 0} and S− = {k : gk < 0}. Put Mg = 2 gk = 2 |gk | k∈S+
k∈S−
and define two polynomials in the variables (n 1 , n 2 ): gk (n 1 − k1 )2 + (n 2 − k2 )2 = n − k 2gk , P+ (n 1 , n 2 ) = k∈S+
P− (n 1 , n 2 ) =
(n 1 − k1 )2 + (n 2 − k2 )2
|gk |
k∈S+
=
k∈S−
n − k 2|gk | .
(2.21)
k∈S−
Let m g be the degree of P = P+ − P− . If Mg − m g ≥ 3, then g · ω ∈ 1 (Z2 ), where
ωn =
Proof. Since
k∈Z2
(2.22)
0 if n = (0, 0), log n if n = (0, 0).
gk = 0 by (2.13), Mg = deg P+ = deg P− and
m g = deg P < max(deg P+ , deg P− ) = Mg . Let v = g · ω. Hence, for all n with n > max{ k : k ∈ S+ ∪ S− }, one has P+ (n 1 , n 2 ) 1 P+ (n 1 , n 2 ) − P− (n 1 , n 2 ) 1 = log 1 + |(g · ω)n | = log . 2 P− (n 1 , n 2 ) 2 P− (n 1 , n 2 ) There exist constants C, N such that mg P+ (n 1 , n 2 ) − P− (n 1 , n 2 ) C 1 ≤ C n = < P− (n 1 , n 2 ) 2 n Mg n Mg −m g for n ≥ N . Hence we can find another constant C˜ such that |(g · ω)n | ≤
C˜ n Mg −m g
for all sufficiently large n . Since Mg −m g ≥ 3, we finally conclude that g·ω ∈ 1 (Z2 ). Lemma 2.9. Suppose that g ∈ Jd (cf. (2.13)–(2.16)), and that ω ∈ Wd is given by
0 if n = 0, ωn = 1 if n = 0, n γ for some integer γ ≥ d − 2. Then g · ω ∈ 1 (Zd ).
730
K. Schmidt, E. Verbitskiy
Proof. Let Sg = {k ∈ Zd : gk = 0}, M = max{ k : k ∈ Sg }, and note that Sg ⊂ Bd = {y ∈ Rd : y ≤ M},
(2.23)
where · is the Euclidean norm on Zd ⊂ Rd . We fix n ∈ Zd with n > M and set h (n) (k) = n − k −γ =
d
−γ /2 (n i − ki )2
.
(2.24)
i=1
In calculating the Taylor expansion of h (n) as a function of the variables k1 , . . . , kd we use the notation I ! = i 1 ! · · · i d !, |I | = i 1 + · · · + i d and
∂ |I | h (n) ∂ i1 +···+id h (n) = , ∂k I ∂k1i1 · · · ∂knin
(2.25)
for I = (i 1 , . . . , i d ) ∈ Zd+ , k = (k1 , . . . , kd ) ∈ Zd , where Z+ = {n ∈ Z : n ≥ 0}. Then the Taylor expansion of h (n) for k ≤ M is given by 1 ∂ |I | h (n) (n) (0) k I + RI kI , I I ! ∂k |I |≤2 |I |=3 1 ∂ |I | h (n) (n) |R I | ≤ sup (y) I y∈Bd I ! ∂k
h (n) (k) = where
(cf. (2.23)). The first and second order derivatives of h (n) have the following form. ∂h (n) (k) = γ · (n i − ki ) · n − k −γ −2 for i = 1, . . . , d, ∂ki ∂ 2 h (n) (k) = γ · (γ + 2) · (n i − ki ) · (n j − k j ) · n − k −γ −4 ∂ki ∂k j for i, j = 1, . . . , d, i = j, 2 (n) ∂ h (k) = γ · (γ + 2) · (n i − ki )2 · n − k −γ −4 − γ · n − k −γ −2 ∂ki2 for i = 1, . . . , d. It follows that h (n) (0) = n −γ , ∂h (n) (0) = γ · n i · n −γ −2 , ∂ki ∂ 2 h (n) (0) = γ · (γ + 2) · n i · n j · n −γ −4 , i = j, ∂ki ∂k j ∂ 2 h (n) (0) = γ · (γ + 2) · n i2 · n −γ −4 − γ · n −γ −2 . ∂ki2
Abelian Sandpiles and the Harmonic Model
731
For I = (i 1 , . . . , i d ) ∈ Zd+ and y ∈ Rd , ∂ |I | h (n) (y) = PI (n 1 , . . . , n d ) · n − y −γ −2|I | , ∂k I where PI is a polynomial of degree at most |I | in the variables n 1 , . . . , n d . Therefore, for every I ∈ Zd+ with |I | = 3, −γ −3 |R (n) ). I | ≤ O( n
(2.26)
By using the Taylor series expansion of h (n) above we obtain that, for all n with sufficiently large norm, |(g · ω)n | = gk h (n) (k) k∈Sg ⎛ ⎞ d (n) (0) (n) ∂h ⎝ ≤ h (0) gk + gk ki ⎠ i=1 ∂ki k∈Sg k∈Sg ⎛ ⎞ 2 (n) ∂ h (0) ⎝ + gk ki k j ⎠ i= j ∂ki ∂k j k∈Sg ⎛ ⎞ d 1 ∂ 2 h (n) (0) ⎝ 2 ⎠ −(γ +3) + g k ). (2.27) k i + O( n 2 2 ∂k i i=1 k∈S g
The first three terms on the right-hand side of the above inequality vanish because of (2.13), (2.14), and (2.15). The fourth term is estimated as follows: (2.16) implies that gk ki2 = const for all i = 1, . . . , d, k∈Sg
and we denote by C this common value. Then ⎛ ⎞ d ∂ 2 h (n) (0) ⎝ gk ki2 ⎠ 2 ∂k i i=1 k∈S g
=
d γ (γ + 2) · n i2 · n −γ −4 −γ · n −γ −2 C = γ (γ + 2) − γ d C||n||−γ −2 . i=1
Therefore, if γ = d − 2, then the fourth term vanishes. If γ > d − 2, i.e., if γ ≥ d − 1, then the fourth term is of the order O( n −(d+1) ), and is thus summable over Zd . The remainder term in (2.27) is always summable since γ + 3 ≥ d + 1. Proof of Theorem 2.4. We start with the case d ≥ 3. Recall that for n = 0, d 4 κd 3cd 1 i=1 n i (d) +c − +O( n −(d+2) ) =: ωn(1) + ωn(2) + ωn(3) + rn . wn = d d−2 d+4 n n d + 2 n d
732
K. Schmidt, E. Verbitskiy
Applying g, we conclude that g · w ∈ 1 (Zd ), because g · ω(1) , g · ω(3) ∈ 1 (Zd ) by Lemma 2.9 for γ = d − 2 and γ = d, respectively; g · ω(2) ∈ 1 (Zd ) by Lemma 2.7; (g · r )n = O( n −(d+2) ), and hence g · r ∈ 1 (Zd ) as well. Now consider the case d = 2. Then wn(2) = −
n4 + n4 1 3 1 log n −κ2 − c2 1 4+22 − +O( n −4 ) = ωn(1) + ωn(2) + ωn(3) + rn . 8π n 4 n 2
For any g ∈ J2 , g · ω(2) , g · ω(3) , g · r ∈ 1 (Z2 )
(2.28)
by the results of Lemmas 2.7 and 2.9. The remaining term g · ω(1) has to be treated slightly differently. First of all, note that since J2 = ( f ) + (u 1 − 1)3 · R2 + (u 1 − 1)2 (u 2 − 1) · R2 + (u 1 − 1)(u 2 − 1)2 · R2 + (u 2 − 1)3 · R2 , it is sufficient to check that g · ω(1) ∈ 1 (Z2 ) only for the set of generators, i.e., for g = f (2) , (u 1 − 1)3 , (u 1 − 1)2 (u 2 − 1), (u 1 − 1)(u 2 − 1)2 , (u 2 − 1)3 . For g = f (2) , f (2) · w (2) = δ (0) ∈ 1 (Z2 ) (cf. (2.5) and Footnote 2 on page 724), and hence, given (2.28), f · ω(1) ∈ 1 (Z2 ) as well. For g = (u 1 − 1)3 ∈ R2 we apply Lemma 2.8. Note that S+ = {(1, 0), (3, 0)}, S− = {(0, 0), (2, 0)}, P+ = ((n 1 − 3)2 + n 22 )((n 1 − 1)2 + n 22 )3 , P− = ((n 1 − 2)2 + n 22 )3 (n 21 + n 22 ) and P+ − Pi = 9 − 60n 1 + 108n 21 − 84n 31 + 30n 41 − 4n 51 − 36n 22 + 60n 1 n 22 −36n 21 n 22 + 8n 31 n 22 − 18n 42 + 12n 1 n 42 . Hence Mg = deg P+ = deg P− = 8, m g = deg P = 5, Mg − m g = 3. Therefore, by Lemma 2.8, |(g · ω(1) )n | = O( n −3 ), and hence g · ω(1) ∈ 1 (Z2 ), which is equivalent to g ∈ I2 . The same calculation shows that (u 2 − 1)3 ∈ I2 . Furthermore, since f (2) ∈ I2 and 3 (2) 2 u −1 = −u −1 1 (u 1 − 1) + f 2 (u 1 − 1)(u 2 − 1) ,
we obtain that (u 1 − 1)(u 2 − 1)2 ∈ I2 and, by symmetry, that (u 1 − 1)2 (u 2 − 1) ∈ I2 . This proves that J2 ⊂ I2 , and Lemma 2.6 yields that J2 = I2 .
Abelian Sandpiles and the Harmonic Model
733
3. The Harmonic Model Let d > 1. We define the shift-action α of Zd on TZ by d
(α m x)n = xm+n
(3.1)
for every m, n ∈ Zd and x = (xn ) ∈ TZ and consider, for every h ∈ Rd , the group homomorphism d d h(α) = h m α m : TZ −→ TZ . (3.2) d
m∈Zd
Since Rd is an integral domain, Pontryagin duality implies that h(α) is surjective for d ∼ every nonzero h ∈ Rd (it is dual to the injective homomorphism from Rd = TZ to itself consisting of multiplication by h). d Let f (d) ∈ Rd be given by (2.4) and let X f (d) ⊂ TZ be the closed, connected, shift-invariant subgroup ⎧ d ⎨ d (xn+e( j) + xn−e( j) ) = 0 X f (d) = ker f (d) (α) = x = (xn ) ∈ TZ : 2d xn − ⎩ j=1 ⎫ (3.3) ⎬ for every n ∈ Zd . ⎭ We denote by α f (d) the restriction of α to X f (d) . Since every α m , m ∈ Zd , is a continuf (d) ous automorphism of X f (d) , the Zd -action α f (d) preserves the normalized Haar measure λ X f (d) of X f (d) .
The Laurent polynomial f (d) can be viewed as a Laplacian on Zd and every x = (xn ) ∈ X f (d) is harmonic (mod 1) in the sense that, for every n ∈ Zd , 2d · xn is the sum of its 2d neighbouring coordinates (mod 1). This is the reason for calling (X f (d) , α f (d) ) the d-dimensional harmonic model. According to [21, Theorem 18.1] and [21, Theorem 19.5], the metric entropy of α f (d) with respect to λ X f (d) coincides with the topological entropy of α f (d) and is given by hλX
f (d)
(α f (d) ) = h top (α f (d) ) 1 1 = ··· log f (d) (2πit1 , . . . , 2πitd ) dt1 · · · dtd < ∞. 0
(3.4)
0
Furthermore, α f (d) is Bernoulli with respect to λ X f (d) (cf. [21]). Since every constant element of TZ lies in X f (d) , α f (d) has uncountably many fixed points and is therefore nonexpansive: for every ε > 0 there exists a nonzero point x = (xn ) in X f (d) with d
| xn| < ε for every n ∈ Zd , where | t (mod 1)|| = min {|t − n| : n ∈ Z}, t ∈ R.
(3.5)
734
K. Schmidt, E. Verbitskiy
3.1. Linearization. Consider the surjective map ρ : Wd = RZ −→ TZ given by d
d
ρ(w)n = wn (mod 1)
(3.6)
for every n ∈ Zd and w = (wn ) ∈ Wd . We write σ for the shift action (σ m w)n = (u −m · w)n = wm+n of Zd on Wd (cf. (2.3)). As in (3.2) we set, for every g = n 1 d n∈Zd h n u ∈ (Z ), h(σ ) = h n σ n : Wd −→ Wd .
(3.7)
n∈Zd
gn u n ∈ Rd , h =
(3.8)
n∈Zd
Then h(σ )(w) = h ∗ · w, g(α)(ρ(w)) = ρ(g ∗ · w)
(3.9)
for every w ∈ Wd (cf. (2.2) and (2.3)). d We set Wd (Z) = ZZ ⊂ Wd . According to (3.3), W f (d) := ρ −1 (X f (d) ) = {w ∈ Wd : ρ(w) ∈ X f (d) } = f (d) (σ )−1 (Wd (Z)) = {w ∈ Wd : f (d) · w ∈ Wd (Z)}.
(3.10)
For later use we denote by d R ⊂ Wd , Z ⊂ Wd (Z), T ⊂ TZ
(3.11)
the set of constant elements. If c is an element of R, Z or T we denote by c˜ the corresponding constant element of R, Z or T. Equation (3.10) allows us to view W f (d) as the linearization of X f (d) . 3.2. Homoclinic points. Let β be an algebraic Zd -action on a compact abelian group Y , i.e., a Zd -action by continuous group automorphisms of Y . An element y ∈ Y is homoclinic for β (or β-homoclinic to 0) if limn→∞ β n y = 0. The set of all homoclinic points of β is a subgroup of Y , denoted by β (Y ). If β is an expansive algebraic Zd -action on a compact abelian group Y then β (Y ) is countable, and β (Y ) = {0} if and only if β has positive entropy with respect to the Haar measure λY (or, equivalently, positive topological entropy). Furthermore, β (Y ) is dense in Y if and only if β has completely positive entropy w.r.t. λY . Finally, if β is expansive, then β n x → 0 exponentially fast (in an appropriate metric) as n → ∞. All these results can be found in [14]. If β is nonexpansive on Y , then there is no guarantee that β (Y ) = {0} even if β has completely positive entropy. Furthermore, β-homoclinic points y may have the property that β n y → 0 very slowly as n → ∞. The Zd -action α f (d) on X f (d) is nonexpansive and the investigation of its homoclinic points therefore requires a little more care. In particular we shall have to restrict our attention to α f (d) -homoclinic points x for which α nf (d) x → 0 sufficiently fast as n → ∞. For this reason we set
Abelian Sandpiles and the Harmonic Model
(1) α (X f (d) ) =
⎧ ⎨ ⎩
735
x ∈ α (X f (d) ) :
⎫ ⎬ | x n| < ∞ , ⎭ d
(3.12)
n∈Z
where | · | is defined in (3.5). In order to describe the homoclinic groups α (X f (d) ) and (1) α (X f (d) ) we set x = ρ(w (d) ) ∈ X f (d) .
(3.13)
The fact that x ∈ X f (d) is a consequence of Theorem 2.2 (1) and (3.10). Proposition 3.1. Let α f (d) be the algebraic Zd -action on the compact abelian group X f (d) defined in (3.3). Then every homoclinic point z ∈ α (X f (d) ) is of the form z = ρ(h · w(d) ) for some h ∈ Rd . Furthermore, (d) (X ) = ρ {h · w : h ∈ I } (3.14) (1) (d) d f α (cf. Theorem 2.2, (2.9) and (3.12)). Proof. If z ∈ α (X f (d) ), then we choose w ∈ ∞ (Zd ) with limn→∞ wn = 0 and ρ(w) = z. From (3.10) we know that f (d) · w ∈ Wd (Z), and the smallness of (most of) the coordinates of w guarantees that h = f (d) · w ∈ Rd = 1 (Zd ) ∩ ∞ (Zd , Z), where ∞ (Zd , Z) = {w = (wn ) ∈ ∞ (Zd ) : wn ∈ Z for every n ∈ Zd }. If we multiply the last identity by w(d) we get that w (d) · f (d) · w = w = w (d) · h = h · w (d) for some h ∈ Rd . (1) If z ∈ α (X f (d) ) then w ∈ 1 (Zd ) and hence, by definition, h ∈ Id . Conversely, if (1)
h ∈ Id , then z = ρ(h · w (d) ) ∈ α (X f (d) ).
Remark 3.2. A homoclinic point z of an algebraic Zd -action β on a compact abelian group Y is fundamental if its homoclinic group β (Y ) is the countable group generated by the orbit {β n z : n ∈ Zd } (cf. [14]). Proposition 3.1 shows that x = ρ(w (d) ) also has the property that its orbit under (1) α f (d) generates the homoclinic groups α (X f (d) ) and α (X f (d) ), although x itself may not be homoclinic (e.g., when d = 2). 3.3. Symbolic covers of the harmonic model. We construct, for every homoclinic point ∞ d z ∈ (1) α (X f (d) ), a shift-equivariant group homomorphism from (Z , Z) to X f (d) which we subsequently use to find symbolic covers of α f (d) . (1)
According to Proposition 3.1, every homoclinic point z ∈ α (X f d ) is of the form z = g(α)(x ) = ρ(g ∗ · w (d) ) for some g ∈ Id . We define group homomorphisms d ξ¯g : ∞ (Zd ) −→ ∞ (Zd ) and ξg : ∞ (Zd ) −→ TZ by ξ¯g (w) = (g · w (d) )(σ )(w) = (g ∗ · w (d) ) · w and ξg (w) = (ρ ◦ ξ¯g )(w).
(3.15)
736
K. Schmidt, E. Verbitskiy
These maps are well-defined, since ξ¯g (w)n =
wn−k · (g ∗ · w (d) )k
k∈Zd
converges for every n, and equivariant in the sense that ξ¯g ◦ σ n = σ n ◦ ξ¯g , ξg ◦ σ n = α n ◦ ξg , ξ¯g ◦ h(σ ) = h(σ ) ◦ ξ¯g , ξg ◦ h(σ ) = h(α) ◦ ξg ,
(3.16)
for every n ∈ Zd , g ∈ Id and h ∈ Rd . We also note that vn α −n g(α)(x ) ξg (v) = for every v = (vn ) ∈ ∞ (Zd , Z). Proposition 3.3. For every g ∈ Id , ∞
ξg ( (Z , Z)) = d
n∈Zd
{0} if g ∈ ( f (d) ), X f (d) if g ∈ I˜d := Id ( f (d) ),
(3.17)
(cf. (2.9) and (3.15)–(3.16)). We begin the proof of Proposition 3.3 with two lemmas. Lemma 3.4. For every w ∈ ∞ (Zd ) and g ∈ Id , ( f (d) (σ ) ◦ ξ¯g )(w) = f (d) · (g ∗ · w (d) ) · w = g ∗ · ( f (d) · w (d) ) · w = g ∗ · w = g(σ )(w). (3.18) Furthermore, ξg (∞ (Zd , Z)) ⊂ X f (d) . Proof. For every h, v ∈ Rd , Theorem 2.2 (1) implies that f (d) · h ∗ · w (d) · v = h ∗ · f (d) · w (d) · v = h ∗ · v.
(3.19)
Zd
Fix g ∈ Id and let K ≥ 1 and VK = {−K + 1, . . . , K − 1} ⊂ ∞ (Zd , Z). Then VK is shift-invariant and compact in the topology of pointwise convergence, and the set VK ⊂ VK of points with only finitely many nonzero coordinates is dense in VK . For v ∈ VK ⊂ Rd , ξ¯g (v) = (g ∗ · w (d) ) · v
(3.20)
( f (d) (σ ) ◦ ξ¯g )(v) = f (d) · g ∗ · w (d) · v = g ∗ · f (d) · w (d) · v = g ∗ · v
(3.21)
and by (3.15) and (3.19). Since both ξ¯g and multiplication by g ∗ are continuous on VK , (3.21) holds for every v ∈ VK . By letting K → ∞ we obtain (3.21) for every v ∈ ∞ (Zd , Z), hence for every v ∈ M1 ∞ (Zd , Z) with M ≥ 1, and finally, again by coordinatewise convergence, for every w ∈ ∞ (Zd ), as claimed in (3.18). For the last assertion of the lemma we note that ξg (v) = ρ((g ∗ · w (d) ) · v) = (g · v ∗ )(α)(x ) ∈ X f (d)
(3.22)
for every v ∈ VK (cf.(3.13)). The continuity argument above yields that ξg (v) ∈ X f (d) for every v ∈ ∞ (Zd , Z).
Abelian Sandpiles and the Harmonic Model
737
Lemma 3.5. If g ∈ I˜d then ξg (∞ (Zd , Z)) = X f (d) . In fact, ξg (2d ) = X f (d) , where m = {0, . . . , m − 1}Z ⊂ ∞ (Zd , Z) for every m ≥ 1. Furthermore, the restriction of ξg to 2d (or to any other closed, bounded, shift-invariant subset of ∞ (Zd , Z)) is continuous in the product topology on that space. d
Proof. We fix x ∈ X f (d) and define w ∈ W f (d) by demanding that ρ(w) = x and 0 ≤ wn < 1 for every n ∈ Zd . If v = f (d) (σ )(w) then −2d + 1 ≤ vn ≤ 2d − 1 for every n ∈ Zd . Since ξ¯g commutes with f (d) (σ ) by (3.16), (3.21) shows that ξg (v) = (ρ ◦ ξ¯g )(v) = g(α)(x).
(3.23)
X f (d) ⊃ ξg (∞ (Zd , Z)) ⊃ ξg (V2d ) ⊃ g(α)(X f (d) ),
(3.24)
Hence
where VK = {−K + 1, . . . , K − 1}Z ⊂ ∞ (Zd , Z). We claim that d
g(α)(X f (d) ) = X f (d) .
(3.25)
Indeed, consider the exact sequence g(α)
{0} −→ ker g(α) ∩ X f (d) −→ X f (d) −→ X f (d) −→ {0}, set Y = ker g(α)∩ X f (d) , Z = g(α)(X f (d) ) ⊂ X f (d) , write αY and α Z for the restrictions of α to Y and Z , and denote by α the Zd -action induced by α on X f (d) /Z . Yuzvinskii’s addition formula ([21, (14.1)]) implies that h top (α f (d) ) = h top (αY ) + h top (α Z ) = h top (α ) + h top (α Z ), where we are using the fact that the topological entropies of these actions coincide with their metric entropies with respect to Haar measure. Since the polynomials f (d) and g have no common factors, h top (αY ) = 0 by [21, Corollary 18.5], hence h top (α f (d) ) = h top (α Z ) is given by (3.4) and 0 < h top (α f (d) ) < ∞. Since the Haar measure λ X f (d) of X f (d) is the unique measure of maximal entropy for α f (d) we conclude that λ X f (d) (g(α) (X f (d) )) = 1 and g(α)(X f (d) ) = X f (d) , as claimed in (3.25). We have proved that ξg (V2d ) = X f (d) . If v ∈ ∞ (Zd , Z) satisfies that vn = 2d − 1 for every n ∈ Zd , then v + V2d = 4d−1 , and (3.24) implies that ξg (4d−1 ) = ξg (V2d ) + ξg (v ) = X f (d) + ξg (v ) = X f (d) . We still have to show that ξg (2d ) = X f (d) . Fix M ≥ 1 for the moment and put Q M = {−M, . . . , M}d ⊂ Zd . Let ∞ (Zd , Z+ ) = {v ∈ ∞ (Zd , Z) : vn ≥ 0 for every n ∈ Zd }.
(3.26)
738
K. Schmidt, E. Verbitskiy
For every v ∈ ∞ (Zd , Z+ ) and n ∈ Zd we set
u n · f (d) if vn ≥ 2d (v,n) = h 0 otherwise, and we put H (v,M) = h (v,n) , T (v) = v − H (v,M) . n∈Q M
If
D M (v) =
vn · n 2max ,
(3.27)
n∈Q M
where · max is the maximum norm on Rd , then T (v) = v if and only if vn < 2d for every n ∈ Q M , and D M (T (v)) ≥ D M (v) + 2
(3.28)
otherwise. We define inductively T n (v) = T (T n−1 (v)), n ≥ 2, and conclude from (3.28) that there exists, for every v ∈ ∞ (Zd , Z+ ), an integer K M (v) ≥ 0 with v˜ (M) = T k (v) for every k ≥ K M (v). For v ∈ 4d−1 and any M ≥ 1, the corresponding
v˜ (M)
(3.29)
satisfies
0 ≤ v˜n(M) ≤ 2d − 1 if n ∈ Q M , v˜n(M) ≥ vn if n max = M + 1, v˜n(M) − vn ≤ (2d − 1) · (2M + 1)d ,
(3.30)
{n: n max =M+1}
v˜n(M) = vn if n max > M + 1, where · max is the maximum norm on Rd . Let V˜ (M) = {v˜ (M) : v ∈ 4d−1 }. Since v − v˜ (M) ∈ ( f (d) ) it is clear that ξg (v) = ξg (v˜ (M) ) for every v ∈ 4d−1 and g ∈ I˜d . Since g ∈ I˜d , Theorem 2.4 implies that there exists a constant C > 0 with −d−1 |(g ∗ · w (d) )n | ≤ C · n max for every nonzero n ∈ Zd .
Hence |ξ¯g (v˜ (M) )0 − ξ¯g (v¯ (M) )0 | < 4d · (2M + 1)d · C · (M + 1)−d−1 → 0 as M → ∞, where v¯n(M) It follows that
(M) if n ∈ Q M , v˜ = n vn otherwise.
lim ξ¯g (v − v¯ (M) ) = 0
M→∞
in the topology of coordinate-wise convergence. Since v¯ (M) ∈ {v ∈ 4d−1 : 0 ≤ vn < 2d for every n ∈ Q M } for every v ∈ 4d−1 and M ≥ 1, we conclude that ξg (2d ) is dense in X f (d) . As ξg (2d ) is also closed, this implies that ξg (2d ) = X f (d) , as claimed.
Abelian Sandpiles and the Harmonic Model
739
Remark 3.6. Although we have not yet introduced sandpiles and their stabilization (this will happen in Sect. 4), the second part of the proof of Lemma 3.5 is effectively a ‘sandpile’ argument, and v˜ (M) is a stabilization of v in Q M . Proof of Proposition 3.3. If g lies in I˜d , Lemma 3.5 shows that ξg (2d ) = ξg (∞ (Zd , Z)) = X f (d) . On the other hand, if g = h· f (d) for some h ∈ Rd , then g ∗ ·w (d) ∈ Rd , and hence ξ¯g (v)n ∈ Z for every n ∈ Zd and v ∈ ∞ (Zd , Z), implying that ξg (v) = 0. 3.4. Kernels of covering maps. Having found compact shift-invariant subsets V ⊂ ∞ (Zd , Z) such that the restrictions of ξg to V are surjective for every g ∈ I˜d (cf. Lemma 3.5), we turn to the problem of determining the kernels of the group homomorphisms ξg : ∞ (Zd , Z) −→ X f (d) , g ∈ Id (cf. (3.15)). We shall see below that ker(ξg ) depends on g and that ker ξgh ker(ξg ) for g ∈ Id and 0 = h ∈ Rd . In view of this it is desirable to characterize the set Kd = ker(ξg ) (3.31) g∈Id
of all v ∈ ∞ (Zd , Z) which are sent to 0 by every ξg , g ∈ Id . In the following discussion we set, for every ideal J ⊂ Rd , X J = {x ∈ TZ : g(α)(x) = 0 for every g ∈ J } = d
ker g(α),
(3.32)
g∈J
and put /( f (d) ) = X f (d) / X Id . X˜ f (d) = Id
(3.33)
In order to explain (3.33) we note that the dual group of X˜ f (d) is a subgroup of X f (d) = (d) Rd /( f ), hence X˜ f (d) is a quotient of X f (d) by a closed, shift-invariant subgroup, which is the annihilator of Id /( f (d) ) and hence equal to X Id . The Zd -action α f (d) on X f (d) induces a Zd -action α˜ f (d) on X˜ f (d) . Note that α˜ nf (d) is dual to multiplication by u n
on Id /( f (d) ). With this notation we have the following result.
Theorem 3.7. There exists a surjective group homomorphism η : ∞ (Zd , Z) −→ X˜ f (d) with the following properties: (1) The homomorphism η is equivariant in the sense that η ◦ σ n = α˜ nf (d) ◦ η for every n ∈ Zd ; (2) ker(η) = K d ; (3) The topological entropy of α˜ f (d) coincides with that of α f (d) (cf. (3.4)). For the proof of Theorem 3.7 we choose and fix a set of generators G d = {g (1) , . . . , of Id (for d = 2 we may take, for example, G 2 = {g (1) , g (2) , g (3) } with g (1) = (1 − u 1 )2 · (1 − u 2 ), g (2) = (1 − u 1 ) · (1 − u 2 )2 and g (3) = (1 − u 1 )2 + (1 − u 2 )2 ); for d ≥ 3 we can use the set of generators G d = { f (d) } ∪ {(u i − 1) · (u j − 1) · (u k − 1) : i, j, k = 1, . . . , d}). With such a choice of G d we define a map g (m) }
ξ Id : ∞ (Zd , Z) −→ X mf(d)
(3.34)
740
K. Schmidt, E. Verbitskiy
by setting ξ Id (v) = (ξg(1) (v), . . . , ξg(m) (v))
(3.35)
for every v ∈ ∞ (Zd , Z). Lemma 3.8. There exists a continuous shift-equivariant group isomorphism θd : ξ Id (∞ (Zd , Z)) −→ X˜ f (d) .
(3.36)
Proof. We define a continuous group homomorphism θ : X f (d) −→ X mf(d) by setting
θ (x) = (g (1) (α)(x), . . . , g (m) (α)(x)) for every x ∈ X f (d) . According to (3.15) and (3.16),
ξg ◦ h(σ )(v) = ρ(g ∗ · w (d) · h ∗ · v) = g(α) ◦ ξh (v) for every g, h ∈ I˜d and v ∈ Rd , and hence, by continuity, for every g, h ∈ I˜d and v ∈ ∞ (Zd , Z). Since ξh (∞ (Zd , Z)) = X f (d) by Lemma 3.5 we conclude that ξ Id (∞ (Zd , Z)) ⊃ ξ Id ◦ h(σ )(∞ (Zd , Z)) = θ (X f (d) ). On the other hand, ξ Id (v) = (g (1) (α) ◦ v ∗ (α)(x ), . . . , g (m) (α) ◦ v ∗ (α)(x )) ∈ θ (X f (d) ) for every v ∈ Rd and hence, again by continuity, for every v ∈ ∞ (Zd , Z). We have proved that ξ Id (∞ (Zd , Z))) = θ (X f (d) ). The homomorphism θ has kernel X Id and induces a group isomorphism θ : X˜ f (d) −→ θ (X f (d) ). The proof is completed by setting θd = (θ )−1 . Proof of Theorem 3.7. We set η = θd ◦ ξ Id (cf. (3.34)–(3.36)). By definition, K d = ker(ξ Id ) = ker(η). The equivariance of η is obvious. Furthermore, h top (α) ˜ ≤ h top (α f (d) ), since X˜ f (d) is an equivariant quotient of X f (d) . On the other hand, X˜ f (d) ∼ = ξ Id (∞ (Zd , Z)), and the ∞ d first coordinate projection π1 : ξ Id ( (Z , Z) −→ X f (d) is surjective by Lemma 3.5. This implies that h top (β1 ) = h top (α) ˜ ≥ h top (α f (d) ), so that these entropies have to coincide. In order to characterize the kernel K d of η further we need a lemma and a definition. Lemma 3.9. For every y ∈ ∞ (Zd ) with ρ(y) ∈ X I3 there exists a unique c(y) ∈ [0, 1) d ˜ ∈ ∞ (Zd , Z), where c(y) ˜ denotes the element of R with c(y) ˜ n = c(y) with f (d) · y + c(y) for every n ∈ Zd .
Abelian Sandpiles and the Harmonic Model
741
Proof. Let x ∈ X I3 and y ∈ ∞ (Zd ) with ρ(y) = x. According to the definition of X I3 d d this means that g(α)(x) = ρ(g ∗ · y) = 0 for every g ∈ I3d . Since g j = (u j − 1) · f (d) ∈ I3d for j = 1, . . . , d, g j (α)(x) = ρ(g ∗j · y) = 0 for j = 1, . . . , d, which implies that f (d) (α)(x) is a fixed point of the Zd -action α on X I3 . ˜ ∈ ∞ (Zd , Z). Hence there exists a unique constant c(y) ∈ [0, 1) with f (d) · y + c(y)
d
Definition 3.10. We call points v ∈ ∞ (Zd ) and x ∈ TZ periodic if their orbits (under σ and α, respectively) are finite. If ⊂ Zd is a subgroup of finite index we denote by (Zd )() , ∞ (Zd , Z)() and () K d the sets of all -invariant elements in the respective spaces. d
Theorem 3.11. (1) For every y ∈ ∞ (Zd ) with ρ(y) ∈ X I3 , d ˜ + m˜ ∈ K d ⊂ ∞ (Zd , Z) v = f (d) · y + c(y)
(3.37)
for every m˜ ∈ Z (cf. (2.10), (3.11), (3.32) and Lemma 3.9). (2) Let ⊂ Zd be a subgroup of finite index. An element v ∈ ∞ (Zd , Z)() lies in K d if and only if it is of the form (3.37) with y ∈ ∞ (Zd )() , ρ(y) ∈ X I3 and m˜ ∈ Z. d
We start the proof of Theorem 3.11 with two lemmas. Lemma 3.12. For every g ∈ Id and every constant element m˜ ∈ ∞ (Zd , Z), ξg (m) ˜ = 0. In other words, Z ⊂ Kd . Proof. We know that g ∈ Id if and only if it satisfies (2.13)–(2.16). We fix g = 2 ∈ Z (note that k ∈ I , put v = g ∗ · w (d) ∈ 1 (Zd ), and set c = g u g k d d d k∈Z k k∈Z k j this value is independent of j ∈ {1, . . . , d} by (2.16)). For every n ∈ Zd , −2πi k,t (d) ∗ (d) −2πi n,t k gk e vn = (g · w )n = dt. gk wn+k = e d 2d − 2 j=1 cos(2π t j ) Td k∈Zd Hence v = (vn ) is the sequence of Fourier coefficients of the function −2πi k,t k gk e Hg (t) = . d 2d − 2 j=1 cos(2π t j ) Since these Fourier coefficients are absolutely summable by assumption, we get that vn = Hg (0). (3.38) n∈Zd
742
K. Schmidt, E. Verbitskiy
On the other hand, given the Taylor series expansion of Hg at t = 0, we have 2 + h.o.t −2π 2 dj=1 t 2j g k k −2π 2 c dj=1 t 2j + h.o.t k j Hg (t) = = , 4π 2 dj=1 t 2j + h.o.t 4π 2 dj=1 t 2j + h.o.t and hence Hg (0) = −c/2.
We are going to show that Hg (0) ∈ Z. Indeed, since k gk k j = 0 for all j by (2.14), we have that k j (k j − 1) 1 1 gk k 2j = − gk k j (k j − 1) = − gk Hg (0) = − ∈ Z. (3.39) 2 2 2 k
k
k
Z, we have Finally, for any g ∈ Id and m˜ ∈ ξ¯g (m) ˜ =m· vn = m Hg (0) ∈ Z ˜ =0∈X by (3.38), and hence ξg (m)
n∈Zd f (d) .
(3.40)
Lemma 3.13. For every g ∈ I3d , Hg (0) = 0 (cf. (3.38)). Proof. Every element of I3d is of the form h · g with h ∈ Rd and g = (u i − 1) · (u j − 1) · (u k − 1) for set v = g ∗ · w (d) and obtain from some i, j, k ∈ {1, . . . , d}. We (3.39) that ∗ · w (d) = h ∗ · v, then H (0) = H (0) = v = 0. If w = (hg) hg n∈Zd wn = g n∈Zd n h v = 0. d d k k∈Z n∈Z n−k Proof of Theorem 3.11. Let x ∈ X I3 , y ∈ ∞ (Zd ) with ρ(y) = x, m˜ ∈ Z, and v = d
f (d) · y + c(y) ˜ + m˜ ∈ ∞ (Zd , Z) (cf. Lemma 3.9). Then g(α)(x) = ρ(g ∗ · y) = 0
for every g ∈ I3d . We set w = g ∗ · w (d) and obtain from (3.16), (3.18) and Lemma 3.12, that ˜ + m) ˜ = ξg ( f (d) · y + c(y)) ˜ ξg (v) = ξg ( f (d) · y + c(y) = ρ(g ∗ · w (d) · f (d) · y + g ∗ · w (d) · c(y)) ˜ = ρ(g ∗ · y + w · c(y)) ˜ ∗ = ρ(g · y) = 0,
since n∈Zd wn = 0 by Lemma 3.13. This proves that every v ∈ ∞ (Zd , Z) of the form (3.37) lies in K d . For (2) we assume that ⊂ Zd is a subgroup of finite index. In view of (1) we only have to verify that every v ∈ ∞ (Zd , Z)() ∩ K d has the form (3.37). Assume therefore that v ∈ ∞ (Zd , Z)() ∩ K d . We choose a set C ⊂ Zd which () intersects each coset of in Zd in a single point and set 0 = {w ∈ ∞ (Zd )() : () (d) (σ )) = R there exists, for n∈C wn = 0}. As 0 is finite-dimensional and ker( f (d) · y = y. every y ∈ () , a unique y ∈ () 0 0 with f d ˜ then Put a˜ = n∈C vn /|Z / |, regarded as an element of R. If v = v − a, ()
v ∈ 0
()
and f (d) · y = v for some y ∈ 0 .
Abelian Sandpiles and the Harmonic Model
743
Since v ∈ K d , ξg (v) = 0 for every g ∈ Id . For g ∈ I3d , Lemma 3.13 shows that ξ¯g (v) = g ∗ · w (d) · v = g ∗ · w (d) · v + g ∗ · w (d) · a˜ = g ∗ · y + g ∗ · w (d) · a˜ = g ∗ · y ∈ ∞ (Zd , Z). Hence ρ(g ∗ · y) = g(α)(ρ(y)) = 0 for all g ∈ I3d , so that ρ(y) ∈ X I3 . d We obtain that v = f (d) · y + a˜ R, which completes the proof for some y ∈ ∞ (Zd ) with ρ(y) ∈ X I3 and some a˜ ∈ d of (2). Theorem 3.11 implies that there exist nonconstant elements v ∈ K d f (d) (σ )(∞ ∈ ∞ (Zd , Z) differ in only finitely many coordinates, then they get identified under ξ Id (i.e., their difference lies in K d ) if and only if they differ by an element in ( f (d) ) ⊂ ∞ (Zd , Z). This is a consequence of the following assertion:
(Zd , Z)). However, if two elements v, v
Proposition 3.14. For every g ∈ I˜d , ker(ξg ) ∩ Rd = ( f (d) ) = f (d) · Rd . Proof. Suppose that h ∈ Rd ∩ ker(ξg ). Then v := ξ¯g (h) = g ∗ · w (d) · h ∈ ∞ (Zd , Z).
(3.41)
Since g ∈ Id , g ∗ · w (d) ∈ 1 (Zd ) and hence v ∈ Rd = 1 (Zd ) ∩ ∞ (Zd , Z). If we multiply both sides of (3.41) by f (d) we get that f (d) · v = g · h. As Rd has unique factorization this implies that h ∈ f (d) · Rd .
Remarks 3.15. (1) One can show that the periodic points are dense in K d , so that every v ∈ K d is a coordinate-wise limit of elements of the form (3.37) in Theorem 3.11. (2) Theorem 3.11 (1) gives a ‘lower bound’ for the kernel K d of the maps ξg , g ∈ Id . There is also a straightforward ‘upper bound’ for that kernel: an element v ∈ ∞ (Zd , Z) lies in K d if and only if ξ¯g (v) = g ∗ · w (d) · v =: wg ∈ ∞ (Zd , Z) for every g ∈ Id . By multiplying this equation with f (d) we obtain that K d ⊂ {v ∈ ∞ (Zd , Z) : g · v ∈ f (d) · ∞ (Zd , Z) for every g ∈ Id } =: K¯ d . (3.42) It is not very difficult to see that the inclusion in (3.42) is strict. In fact, K¯ d /K d turns out to be isomorphic to Td . (3) In [18], the kernel K¯ d of ξ Id was studied using methods of commutative algebra.
744
K. Schmidt, E. Verbitskiy
4. The Abelian Sandpile Model Let d ≥ 2, γ ≥ 2d, and let E ⊂ Zd be a nonempty set. For every n ∈ E we denote by N E (n) the number of neighbours of n in E, i.e., (4.1) N E (n) = E ∩ {n ± e(i) : 1 = 1, . . . , d} , where e(i) is the i th unit vector in Zd . We set γ = {0, . . . , γ − 1}Z
d
(4.2)
(cf. Lemma 3.5) and put (γ )
P E = {v ∈ {0, . . . , γ − 1} E : vn ≥ N E (n) for at least one n ∈ E}, (γ ) PF . RE =
(4.3)
∅= F⊂E 0<|F|<∞
(γ )
In the literature the set R E is called the set of recurrent configurations on E. A configuration v ∈ {0, . . . , γ − 1} E is recurrent if and only if it passes the burning test, which is described as follows: given v ∈ {0, . . . , γ − 1} E , delete (or burn) all sites n ∈ E such that vn ≥ N E (n), (1)
thereby obtaining a configuration v ∈ {0, . . . , γ − 1} E with E (1) ⊂ E. We repeat the process and obtain a sequence E ⊃ E (1) ⊃ · · · ⊃ E (k) ⊃ · · ·. If at some stage E (k) = E (k+1) = ∅ we say that v fails the burning test, and v is a forbidden (or nonrecurrent) configuration. The closed, shift-invariant subset (γ )
(γ )
R∞ = RZd ⊂ γ ⊂ ∞ (Zd , Z)
(4.4) (2d)
is called the d-dimensional sandpile model with parameter γ . For γ = 2d, R∞ = R∞ (γ ) is called the critical sandpile model, and for γ > 2d, the model R∞ is said to be dissipative. In order to motivate this terminology we assume that E ⊂ Zd is a nonempty set. An element v ∈ Z+E is called stable if yn < γ for every n ∈ E. If v ∈ Z+E is unstable at some n ∈ E, i.e., if vn ≥ γ , then v topples at this site: the result is a configuration Tn (v) with ⎧ ⎪ ⎨vn − γ if k = n, Tn (v)k = vk + 1 if n − k max = 1, ⎪ ⎩v otherwise. k If vm , vm ≥ γ for some m, n ∈ E, m = n, then Tn (Tn (v)) = Tm (Tn (v)), i.e., toppling operators commute. A stable configuration v˜ ∈ {0, . . . , γ − 1} E is the result of toppling of v, if there exist n(1) , . . . , n(k) ∈ E such that k Tn(i) (v). v˜ = i=1
Abelian Sandpiles and the Harmonic Model
745
If the set E is finite (in symbols: E Zd ), then every v ∈ Z+E will lead to a stable configuration v˜ by repeated topplings. However, if E is infinite, then repeated toppling of a configuration v ∈ Z+E will, in general, lead to a stable configuration v˜ ∈ {0, . . . , γ − 1} E only if γ > 2d, i.e., in the dissipative case.3 (γ ) We denote by σ = σR(γ ) the shift-action of Zd on R∞ ⊂ ∞ (Zd , Z) ⊂ Wd ∞ (cf. (3.7)). For the following discussion we introduce the Laurent polynomial f (d,γ ) = γ −
d ± (u i + u i−1 ) ∈ Rd = Z[u ± 1 , . . . , u d ].
(4.5)
i=1
For γ = 2d, f (d,γ ) = f (d) (cf. (2.4)). Proposition 4.1. Let d ≥ 2 and γ ≥ 2d. The following conditions are equivalent for every v ∈ γ : (γ )
(1) v ∈ R∞ ; (2) For every nonzero h ∈ Rd with h n ∈ {0, 1} for every n ∈ Zd , ( f (d,γ ) · h)n + vn ≥ γ for at least one n ∈ supp(h) = {m ∈ Zd : h m = 0}. (3) For every h ∈ Rd with h n > 0 for some n ∈ Zd , ( f (d,γ ) · h)n + vn ≥ γ for at least one n ∈ {m ∈ Zd : h m > 0}. (γ )
Furthermore, if v, v ∈ R∞ and 0 = v − v ∈ Rd , then v − v ∈ / f (d,γ ) · Rd . Proof. Fix an element v ∈ γ . If h ∈ Rd with h n ∈ {0, 1} for every n ∈ Zd and E = supp(h), then ( f (d,γ ) · h)n + vn ∈ {0, . . . , γ − 1} for every n ∈ E if and only if (γ ) vn ≤ N E (n) − 1 for every n ∈ E, in which case π E (v) ∈ / P E and v ∈ / R∞ (cf. (4.3)). This proves the equivalence of (1) and (2). Now suppose that h ∈ ∞ (Zd , Z) with Mh = maxm∈Zd h m > 0, and that f (d,γ ) · h + v ∈ γ . We set Smax (h) = {n ∈ Zd : h n = Mh }
(4.6)
and observe that vn + ( f (d,γ ) · h)n ≥ vn + Mh · (γ − NSmax (h) ) < γ for every n ∈ Smax (h), so that vn ≤ NSmax (h) − 1 for every n ∈ Smax (h).
(4.7)
If h ∈ Rd , then Smax (h) is finite and (4.7) yields a contradiction to the definition of (γ ) R∞ . This proves the implication (1) ⇒ (3), and the reverse implication (3) ⇒ (2) is obvious. The last assertion of this proposition is a consequence of (3). The proof of Proposition 4.1 has the following corollary. 3 Even in the dissipative case stable configurations will, in general, only arise as coordinate-wise limits of infinite sequences of topplings of v.
746
K. Schmidt, E. Verbitskiy (γ )
Corollary 4.2. If v ∈ R∞ , and if h ∈ ∞ (Zd , Z) satisfies that maxm∈Zd h m > 0 and (γ ) v + f (d,γ ) · h ∈ R∞ , then every connected 4 component of Smax (h) is infinite (cf. (4.6)). Proof. If Smax (h) has a finite connected component C then (4.7) shows that ¯ n + vn = γ − NC (n) ( f (d,γ ) · h)n + vn ≥ ( f (d,γ ) · h) for every n ∈ C, where
¯h n = h n if n ∈ C, 0 otherwise.
As in (4.7) we obtain a contradiction to (4.3).
(γ )
(γ )
Remark 4.3. Proposition 4.1 implies that ( f (d,γ ) (σ )(h) + R∞ ) ∩ R∞ = ∅ for every d nonzero h ∈ Rd . However, if h ∈ {0, 1}Z satisfies that the set S(h) = {n ∈ Zd : h n = (γ ) 1} is infinite and connected, then one checks easily that there exists a v ∈ R∞ with (γ ) f (d) (σ )(h) + v ∈ R∞ . In spite of this the following result holds. Proposition 4.4. The set (γ )
(γ )
V = {v ∈ R∞ : v + w ∈ / R∞ for every nonzero w ∈ f (d,γ ) (σ )(∞ (Zd , Z))}
(4.8)
(γ )
is a countable intesection of dense open sets (i.e., a dense G δ -set) in R∞ . (γ )
Proof. Let v ∈ R∞ and h ∈ ∞ (Zd , Z) be such that maxn∈Zd h n ≥ 0, f (d,γ ) · h = 0 (γ ) and v + f (d,γ ) · h ∈ R∞ . We set Mh = maxm∈Zd h m , define Smax (h) ⊂ Zd as in (4.6), and put ∂Smax (h) = {n ∈ Smax (h) : m − n max = 1 for some m ∈ Zd Smax (h)}. As ( f (d,γ ) · h)n > 0 for every n ∈ ∂Smax (h), the set ∂Smax (h) must have empty intersection with F(v) = {n ∈ Zd : vn = γ − 1}. (γ )
Now suppose that v ∈ R∞ has the following properties: (a) The set F(v) is connected. (b) Every connected component of Zd F(v) is finite. (c) minn∈Zd vn = 0. According to Corollary 4.2, every connected component C of Smax (h) is infinite. If C = Zd , then the hypotheses (a)–(b) above guarantee that the boundary ∂C = C ∩ ∂Smax (h) of C is a union of finite sets, each of which is contained in one of the connected components of Zd F(v). Let C and D be connected components of Smax (h) and Zd F(v), respectively, with D ∩ ∂C = ∅. Since C is infinite and connected and F(v) is connected, we must have that h m = Mh = 0 for every m ∈ F(v). 4 A set S ⊂ Zd is connected if we can find, for any two coordinates m and n in S, a ‘path’ p(0) = m, p(1), . . . , p(k) = n in S with p( j) − p( j − 1) max = 1 for every j = 1, . . . , k.
Abelian Sandpiles and the Harmonic Model
747
Define h˜ by
h n if n ∈ D h˜ n = 0 otherwise. ˜ n = ( f (d,γ ) · h)n for every n ∈ D, and 0 ≤ ( f (d,γ ) · h) ˜ n ≤ ( f (d,γ ) · h)n Then ( f (d,γ ) · h) d (d,γ ) ˜ for every n ∈ F(v). For n ∈ Z (F(v) ∪ D), ( f · h)n = 0. By combining these (γ ) statements we see that v + f (d,γ ) · h˜ ∈ R∞ . Since 0 = h˜ ∈ Rd we obtain a contradiction to Proposition 4.1. (γ ) (γ ) This shows that v + f (d,γ ) · h ∈ / R∞ for every v ∈ R∞ satisfying conditions (a)–(b) above and every nonzero h ∈ ∞ (Zd , Z) with maxn∈Zd h n ≥ 0. If γ = 2d and h ∈ ∞ (Zd , Z) satisfies that f (d) · h = 0, then we may add a constant to h, if necessary, to ensure that maxn∈Zd h n ≥ 0. Since such an addition will not affect (γ ) / R∞ = R∞ for every v ∈ R∞ satisfying conditions f (d) ·h, we obtain that v + f (d) ·h ∈ (a)–(c) above and every nonconstant h ∈ ∞ (Zd , Z). If γ > 2d and h ∈ ∞ (Zd , Z) satisfies that maxn∈Zd h n < 0, then ( f (d,γ ) · h)n < 0 (γ ) (γ ) for every n ∈ Zd , and v + f (d,γ ) · h ∈ / R∞ for every v ∈ R∞ satisfying condition (c) above. (γ ) Let V ⊂ R∞ be the set of all points satisfying conditions (a)–(c) above. This set is clearly dense and (γ )
(γ )
V ⊂ V = {v ∈ R∞ : v + w ∈ / R∞ for every nonzero w ∈ f (d) (σ )(∞ (Zd , Z))}. (4.9) The set V is therefore dense, and it is obviously shift-invariant. In order to verify that V is a G δ we write its complement as an Fσ of the form (γ ) π˜ {(v, h) ∈ R∞ × B N (∞ (Zd , Z)) :
(γ )
R∞ V = M≥1 N ≥1 0=c∈Z Q M
(γ ) v + f (d) · h ∈ R∞ and π Q M ( f (d,γ ) · h) = c} ,
where B N (∞ (Zd , Z)) = {h ∈ ∞ (Zd , Z) : h ∞ ≤ N }, Q M appears in (3.26) and (γ ) (γ ) π˜ : R∞ × ∞ (Zd , Z) −→ R∞ is the first coordinate projection. 5. The Critical Sandpile Model (2d)
Throughout this section we assume that d ≥ 2 and γ = 2d. We write R∞ = R∞ for d the critical abelian sandpile model, define the harmonic model X f (d) ⊂ TZ by (2.4) and (3.3), and use the notation of Sect. 3. 5.1. Surjectivity of the maps ξg : R∞ −→ X f (d) . For every g ∈ I˜d (cf. (3.17)) we define the map ξg : ∞ (Zd , Z) −→ X f (d) by (2.9) and (3.15). We shall prove the following results.
748
K. Schmidt, E. Verbitskiy
Theorem 5.1. For every g ∈ I˜d , ξg (R∞ ) = X f (d) . Furthermore, the shift-action σR∞ of Zd on R∞ has topological entropy, 1 log π Q N (R∞ ) N →∞ |Q N | 1 1 = ··· log f (d) (e2πit1 , . . . , e2πitd ) dt1 · · · dtd = h(α f (d) ).
h top (σR∞ ) = lim
0
(5.1)
0
For the proof of this result we need a bit of notation and several lemmas. For every Q ⊂ Zd and v ∈ Wd we set S (Q) (v) = {v ∈ Wd : πZd Q (v ) = πZd Q (v)}.
(5.2)
If V ⊂ Wd is a subset we set SV(Q) (v) = S (Q) (v) ∩ V . We fix g ∈ I˜d . Let ε with 0 < ε < 1/4d. Since g ∗ · w (d) ∈ 1 (Zd ) we can find K ≥ 1 with |ξ¯g (v)0 − ξ¯g (v )0 | < ε for every v, v ∈ 2d with π Q K (v) = π Q K (v )
(5.3)
(cf. (3.26)) (Q)
Lemma 5.2. Let v ∈ 2d , Q ⊂ Zd be a finite set and v ∈ S2d (v) (cf. (4.2) and (5.2)). (1) ξg (v ) = ξg (v) if and only if v − v ∈ ( f (d) ). (2) If ξg (v ) = ξg (v), then | ξg (v )n − ξg (v)n| ≥ 1/4d for some n ∈ Q + Q K = {m + k : m ∈ Q, k ∈ Q K }, where K is defined in (5.3), Q K in (3.26) and | · | in (3.5). Proof. We put y = ξ¯g (v), x = ρ(y) = ξg (v), y = ξ¯g (v ) and x = ξg (v). Assume that | xn − xn| < 1/4d
(5.4)
for every n ∈ Q + Q K . Since (5.4) holds automatically for n ∈ Zd (Q + Q K ) by (5.3), it holds for every n ∈ Zd . We choose z ∈ W f (d) with ρ(z) = x − x and z n ∞ < 1/4d (cf. (3.10)). Then f (d) · z ∈ ∞ (Zd , Z), and the smallness of the coordinates of z implies that f (d) · z = 0. Since ρ(z) = ρ(y − y) we obtain that z − (y − y) ∈ ∞ (Zd , Z). As the coordinates of z are small and limn→∞ |y − y| = limn→∞ |ξ¯g (v − v)| = 0, due to the continuity of ξ¯g , we conclude that h = z − (y − y) ∈ Rd . According to (3.18), f (d) · (z − (y − y)) = f (d) · h = g ∗ · (v − v). As Rd has unique factorization and g ∗ is not divisible by f (d) , v − v must lie in the ideal ( f (d) ) ⊂ Rd . Theorem 2.2 (i) and (3.15) together imply that ξg (v ) = x = x = ξg (v).
Abelian Sandpiles and the Harmonic Model
749
If ε > 0 and Q ⊂ Zd we call a subset Y ⊂ X f (d) (Q, ε )-separated if there exists, for every pair of distinct points x, x ∈ Y , an n ∈ Q with | xn − xn | ≥ ε . The set Y is (Q, ε )-spanning if there exists, for every x ∈ X f (d) , an x ∈ Y with | xn − xn | < ε for every n ∈ Q. (Q+Q K )
Lemma 5.3. Let Q ⊂ Zd be a finite set and v ∈ 2d . Then the set ξg (S2d (Q, ε)-spanning.
(v)) is
Proof. According to Lemma 3.5, ξg (2d ) = X f (d) . If we fix w ∈ 2d and set
vn if n ∈ Q + Q K , wn = wn otherwise, (Q+Q K ) then w ∈ S (v) and | ξg (w)n − ξg (w )n| < ε for every n ∈ Q by (5.3). 2d
Lemma 5.4. For every finite set Q ⊂ Zd and every w ∈ R∞ , the restriction of ξg to (Q) (Q) SR∞ (w) is injective and the set ξg (SR∞ (w)) is (Q + Q K , 1/4d)-separated. (Q) (w), then Proposition 4.1 and Lemma 5.2 show Proof. If v, v are distinct points in SR ∞ that | ξg (v)n − ξg (v )n| ≥ 1/4d for some n ∈ Q + Q K . We write every h ∈ Rd as h = n∈Zd h n u n and set supp(h) = {n ∈ Zd : h n = 0}. For Q ⊂ Zd we put
R(Q) = {h ∈ Rd : supp(h) ⊂ Q}, R + (Q) = {h ∈ R(Q) : h n ≥ 0 for every n ∈ Zd },
(5.5)
S + (Q) = {h ∈ R(Q) : h n ∈ {0, 1} for every n ∈ Zd }. For L ≥ 1, v ∈ 2d and q ≥ 0 we set Yv (q) = {w ∈ S (Q L+K +1 ) (v) : for every n ∈ Zd , 0 ≤ wn < 2d if n max = L + K + 1 and − q ≤ wn < 2d if n max = L + K + 1}, (5.6) Yv (q) = {w ∈ Yv (q) : π Q L+K (w) ∈ π Q L+K (R∞ )}. Lemma 5.5. Let L ≥ 1, q ≥ 0 and v ∈ 2d . Then Yv (q) = Yv (q)
(Yv (q + 1) − h · f (d) ).
(5.7)
0=h∈S + (Q L+K )
Proof. Suppose that v ∈ Yv (q). According to the proof of Proposition 4.1 there exists, for every nonzero h ∈ S + (Q L+K ), an n ∈ supp(h) ⊂ Q L+K with (v+h · f (d) )n > 2d −1. In particular, v + h · f (d) (q) ∈ / Yv (q + 1) and v ∈ / Yv (q + 1) − h · f (d) . This shows that Yv (q) ⊂ Yv (q)
(Yv (q + 1) − h · f (d) ). 0=h∈S + (Q
!
L+K )
Conversely, if v ∈ Yv (q) 0=h∈S + (Q L+K ) (Yv (q + 1) − h · f (d) ), but v ∈ / Yv (q), + then the proof of Proposition 4.1 allows us to find a nonzero h ∈ S (Q L+K ) with
750
K. Schmidt, E. Verbitskiy
(v+h · f (d) )n < 2d for every n ∈ supp(h). If (v+h · f (d) )n < 0 for some n ∈ Q L+K , then n∈ / supp(h) and −2d ≤ (v+h· f (d) )n < 0. We replace h by h = h+u n ∈ S + (Q L+K ) and obtain that 0 ≤ (v +h · f (d) )n < 2d for every n ∈ supp(h ). By repeating this process we can find h ∈ S + (Q L+K ) with supp(h ) ⊃ supp(h) such that 0 ≤ (v+h · f (d) )n ≤ 2d−1 for every n ∈ Q L+K . Since 0 ≥ (h · f (d) )n ≥ −1 if n max = L + K + 1 and (h · f (d) )n = 0 outside Q L+K +1 , we see that v + h · f (d) ∈ Yv (q + 1). This contradicts our choice of v and proves (5.7). Lemma 5.6. For every v ∈ 2d and L ≥ 1 there exists an h ∈ R + (Q L ) with v = v + h · f (d) ∈ Yv ((2d − 1) · (2L + 1)d ). Proof. For every v ∈ ∞ (Zd , Z) we define D Q L+1 (v) by (3.27). Since D Q L+1 (v + u n · f (d) ) ≤ D Q L+1 (v) − 2 for every n ∈ Q L , D Q L+1 (v + h · f (d) ) ≤ D Q L+1 (v) − 2 h 1 for every h ∈ S + (Q L ). Suppose that v ∈ 2d . If w ∈ / Yv (0) then (5.7) shows that we can find a nonzero (1) + (1) (1) h ∈ S (Q L ) with v = v + h · f (d) ∈ Yv (1), and the first paragraph of this proof shows that D Q L+1 (v (1) ) ≤ D Q L+1 (v) − 2 h (1) 1 . If v (1) ∈ / Yv (1) we can repeat this argument and find a nonzero h (2) ∈ S + (Q L ) with (2) (1) v = v + h (2) · f (d) ∈ Yv (2) and D Q L+1 (v (2) ) ≤ D Q L+1 (v) − 2 h (1) 1 − 2 h (2) 1 . Proceeding by induction, we choose nonzero elements h (1) , . . . , h (m) ∈ S + (Q L ) with (k) v = v + (h (1) + · · · + h (k) ) · f (d) ∈ Yv (m) for every k = 1, . . . , m. We claim that v (k) ∈ Yv ((2d − 1) · (2L + 1)d ) for every k ≥ 1, and that this process has to stop, i.e., that v = v (m) = v + (h (1) + · · · + h (m) ) · f (d) ∈ Yv ((2d − 1) · (2L + 1)d )
(5.8)
for some m ≥ 1. In order to verify this we assume that we have found h (1) , . . . , h (k) ∈ S + (L) with (k) v (k) = v + (h (1) + · · · + h (k) ) · f (d) ∈ Yv (k). Since n∈Q L+1 vn = n∈Q L+1 vn , (k)
(k)
(k)
0 ≤ vn ≤ 2d − 1 for n ∈ Q L , vn ≤ vn if n max = L + 1 and vn = vn for every n∈ / Q L+1 , we know that vn ≥ vn(k) (2d − 1) · 2d · (2L + 1)d−1 ≥ {n: n max =L+1}
≥
{n: n max =L+1}
{n: n max =L+1}
vn −
vn(k)
(5.9)
n∈Q L
≥ −(2d − 1) · (2L + 1)d , so that v (k) ∈ Yv ((2d − 1) · (2L + 1)d ) for every k ≥ 1. Furthermore, k D Q L+1 (v (k) ) = D Q L+1 (v) − 2 h ( j) 1 ≤ D Q L+1 (v) − 2k j=1
< (L + 1) · (2d − 1) · (2L + 3)d − 2k 2
and D Q L+1 (v (k) ) ≥ −(L + 1)2 · (2d − 1) · (2L + 1)d · |Q L+1 Q L | for every k, so that the integer k has to remain bounded. This shows that our inductive process has to terminate, which proves (5.8).
Abelian Sandpiles and the Harmonic Model
751
Before we complete the proof of Theorem 5.1 we state another consequence of Lemmas 5.5 and 5.6. Proposition 5.7. Let v ∈ ∞ (Zd , Z) and M ≥ 1. Then there exists a unique h ∈ Rd with the following properties: (1) (2) (3) (4)
supp(h) = {m ∈ Zd : h m = 0} ⊂ Q M ; If v = v + h · f (d) , then π Q M (v ) ∈ π Q M (R∞ ); for every m ∈ Zd with m vm = vm max > M + 1; | ≤ (2M + 3)d · v . |v ∞ {n: n max =M+1} n
Proof. The proof of Lemma 5.5 allows us to find a polynomial h − ∈ Rd with nonnegative coefficients and supp(h − ) ⊂ Q M such that (v −h − · f (d) )n < 2d for every n ∈ Q M . Next we proceed as in the proof of Lemma 5.6 and choose a polynomial h + ∈ Rd with nonnegative coefficients and supp(h + ) ⊂ Q M such that v = v +(h + −h − )· f (d) satisfies (2). Condition (3) holds obviously, and (4) follows from the fact that n∈Q M+1 vn = n∈Q M+1 vn . In order to verify the uniqueness of h = h + − h − we assume that h ∈ Rd is another polynomial with supp(h ) ⊂ Q M such that v = v + h · f (d) satisfies Condition (2) above. We assume without loss in generality that h m > h m for some m ∈ Q M and set g = h − h and
v if n ∈ Q M , wn = n 2d otherwise. Then w ∈ R∞ and (w + g · g (d) )n = vn < 2d for every n ∈ Q M . Since supp(g) ⊂ Q M and gn > 0 for some n ∈ Q M this contradicts Proposition 4.1. Proof of Theorem 5.1. We fix ε > 0 and choose K according to (5.3). Lemma 5.6 and (5.9) show that X f (d) = ξg (2d ) = ξg (2d (L + K + 1, (2d − 1) · (2L + 2K + 1)d )), where 2d (M, q) = v ∈ ∞ (Zd , Z) : vm < 2d for every n ∈ Zd , vn ≥ 0 for every n ∈ Zd with n max > M + 1, vn ≥ −q and π Q M (v) ∈ π Q M (R∞ ) . d {n∈Z : n max =M+1}
(5.10)
Exactly the same argument as in the proof of Lemma 3.5 shows that ξg (R∞ ) = X f (d) . Since ξg (R∞ ) = X f (d) we know that h top (σR∞ ) ≥ h top (α f (d) )=
0
1
1
···
log f (d) (e2πis1 , . . . , e2πisd ) ds1 · · · dsd
(5.11)
0
(cf. [15] or [21, Theorem 18.1]). (Q ) In order to prove the reverse inequality we note that ξg is injective on SR∞L (v) for
(Q L ) (v)) is a (Q L+K , 1/4d)-separated subset of every v ∈ R∞ and L ≥ 1 and that ξg (SR ∞ X f (d) , by Proposition 4.1 and Lemma 5.2. In particular, if v¯ ∈ R∞ is given by
752
K. Schmidt, E. Verbitskiy
v¯n = 2d − 1 for every n ∈ Zd ,
(5.12) (Q L ) then π Q L (SR (v)) ¯ = π Q L (R∞ ) for every L ≥ 1. ∞ For every L ≥ 0 we denote by n(L + K ) the maximal size of a (Q L+K , 1/4d)separated set in X f (d) . From the definition of topological entropy we obtain that 1 1 (Q ) log π Q L (R∞ ) = lim log SR∞L (v) ¯ L→∞ |Q L | L→∞ |Q L | 1 1 (Q ) = lim log ξg (SR∞L (v)) log n(L + K ) ¯ ≤ lim L→∞ |Q L | L→∞ |Q L | 1 (d) = lim log n(L + K ) = h top (α f ), L→∞ |Q L+K |
h top (σR∞ ) = lim
which completes the proof of the theorem.
(5.13)
Remark 5.8. The expression (3.4) for the topological entropy of σR∞ can be found in [10, p. 56]. By using the fact that α f (d) and σR∞ have the same topological entropy one can prove Theorem 5.1 a little more directly: Lemmas 5.3 and 5.4 imply that the restriction of α to the closed, shift-invariant subset ξg (R∞ ) ⊂ X f (d) has the same topological entropy as α f (d) . Since the Haar measure λ X f (d) is the unique measure of maximal entropy for α f (d) by [15], ξg (R∞ ) has to coincide with X f (d) , as claimed in Theorem 5.1. (w)
Theorem 5.9. For every w ∈ R∞ and L ≥ 1 we denote by ν L the equidistributed (Q ) probability measure on the set SR∞L (w) in (5.2). Fix w ∈ R∞ and let µ(w) be any limit point of the sequence of probability measures (w)
µL
=
1 k (w) σ∗ ν L |Q L | k∈Q L
as L → ∞. Then µ(w) is a measure of maximal entropy on R∞ and (ξg )∗ µ(w) = λ X f (d) for every g ∈ I˜d . In fact, if µ is any shift-invariant probability measure of maximal entropy on R∞ , then (ξg )∗ µ = λ X f (d) for every g ∈ I˜d . (w)
Proof. We fix w ∈ R∞ . Let L ≥ 1 and let ν˜ L = (ξg )∗ ν L be the equidistributed (Q L ) probability measure on the (Q L+K , 1/4d)-separated set ξg (SR (w)) of cardinality ∞ ≥ π Q L−1 (R∞ ). (w) (w) 1 k We set µ˜ (w) k∈Q L (α f (d) )∗ ν˜ L . By choosing a suitable subseL = (ξg )∗ µ L = |Q L |
(w) quence (L k , k ≥ 1) of the natural numbers we may assume that limk→∞ µ(w) Lk = µ (w)
and limk→∞ µ˜ L k = µ˜ (w) = (ξg )∗ µ(w) . We denote by µ = (π{0} )∗ µ˜ (w) the projection of µ˜ (w) onto the zero coordinate in X f (d) and choose a partition {I1 , . . . , I8d } of T into half-open intervals of length 1/8d such that the endpoints of these intervals all have µ -measure zero. For i = 1, . . . , 8d we set Ai = {x ∈ X f (d) : x0 ∈ Ii } and observe that µ˜ (w) (∂ Ai ) = 0. We write ζ = {A1 , . . . , A8d } for the resulting partition of X f (d) .
Abelian Sandpiles and the Harmonic Model
For every L ≥ 1 we set ζ L =
753
" k∈Q L+K
(w)
α −k (ζ ). Since each atom of ζ L conf (d)
tains at most one atom of ν˜ L (by Lemma 5.4) and all these atoms have equal mass, (Q L ) Hν˜ (w) (ζ L ) = log |SR (w)|. ∞ L Exactly the same argument as in the proof of the inequality (∗) in [28, Theorem 8.6] shows that, for every M, L ≥ 1 with 2M + 2K < L, |Q M | (Q L ) log |SR (w)| = Hν˜ (w) (ζ M ) ≤ Hµ˜ (w) (ζ M ) ∞ L L |Q L | |Q M+K | · (|Q L+K | − |Q L−M−K | + · log(8d). |Q L | By setting L = L k and letting k → ∞ we obtain from (5.13) that |Q M | · h top (α f (d) ) ≤ lim Hµ˜ (w) (ζ M ) = Hµ˜ (w) (ζ M ) k→∞
Lk
for every M ≥ 1, and hence that h top (α f (d) ) ≤ lim
M→∞
1 |Q M+K |
· Hµ˜ (w) (ζ M ) = h µ˜ (w) (α f (d) ).
Since λ X f (d) is the unique measure of maximal entropy on X f (d) , µ˜ (w) coincides with
λ X f (d) , and µ(w) is a measure of maximal entropy on R∞ . In order to complete the proof of Theorem 5.9 we assume that µ is an arbitrary ergo(γ ) dic shift-invariant probability measure with maximal entropy on R∞ . We let M ≥ 5, put F = π Q M (R∞ ) and set, for every z ∈ F, Oz = {v ∈ R∞ : π Q M (v) = z}. Fix z ∈ F with c = µ(Oz ) > 0. The ergodic theorem guarantees that 1 lim 1O (σ 3Mm v) = c N →∞ |Q N |
(5.14)
m∈Q N
for µ-a.e. v ∈ R∞ . Let z ∈ F be given by
2d − 1 if n max = M, zn = zn if n ∈ Q M−1 . We claim that µ(Oz ) > 0. In order to see this we assume that µ(Oz ) = 0 (which implies, of course, that z = z ). If v ∈ R∞ is fixed for the moment, and if Sv = {n ∈ Zd : σ 3Mn v ∈ Oz }, then we can replace the coordinates of σ 3Mm v in Q M by those of z for every m ∈ Sv , and we can do so independently at every m ∈ Sv . The resulting points v will always lie in R∞ . An elementary entropy argument shows that we could increase the entropy of µ under the Zd -action n → σ 3Mn by making all these points v equally likely, which would violate the maximality of the entropy of µ (a more formal argument should be given in terms of conditional measures). Exactly the same kind of argument as in the preceding paragraph allows us to conclude that the cylinder sets Oz with z n ∈ F and z n = 2d − 1 for every n ∈ Q M with n max = M, all have equal measure. A slight modification of the proof of the first part of this theorem now shows that h((ξg )∗ µ) = h(λ X f (d) ), i.e., that (ξg )∗ µ = λ X f (d) .
754
K. Schmidt, E. Verbitskiy
5.2. Properties of the maps ξg , g ∈ I˜d . 5.2.1. The ‘group structure’ of R∞ In (3.4) we saw that σR∞ and α f (d) have the same topological entropy. If µ is a shift-invariant measure of maximal entropy on R∞ , then the dynamical system (R∞ , µ, σR∞ ) has a Bernoulli factor of full entropy (cf. [23]). As (X f (d) , λ X f (d) , α f (d) ) is Bernoulli by [20], the full entropy Bernoulli factor of (R∞ , µ, σR∞ ) is measurably conjugate to (X f (d) , λ X f (d) , α f (d) ). In particular, there exists a µ-a.e. defined measurable map φ : R∞ −→ X f (d) with φ∗ µ = λ X f (d) and φ ◦ σR∞ = α f (d) ◦ φ µ-a.e. What distinguishes the maps ξg , g ∈ I˜d , from these abstract factor maps φ : R∞ −→ X f (d) is that the ξg are not only continuous and surjective, but that they also reflect the somewhat elusive group structure of R∞ in the following sense. It is well known that the set R E of recurrent sandpile configurations on a finite set E ⊂ Zd in (4.3) is a group (cf. [8–10]). However, the group operation does not extend in any immediate way to the infinite sandpile model R∞ . Fix g ∈ I˜d and suppose that v, v ∈ R∞ , and that w = v + v ∈ 4d−1 (with coordinate-wise addition). Proposition 5.7 shows that there exists, for every M ≥ 1, an element w (M) ∈ ∞ (Zd , Z) satisfying conditions (1)–(4) there. Since w − w(M) ∈ ( f (d) ) for every M ≥ 1, ξg (w (M) ) = ξg (w) for every M ≥ 1. Exactly as in the proof of the lemma we observe that any coordinate-wise limit w˜ ∈ R∞ of the sequence (w (M) , M ≥ 1) still satisfies that ξg (w) ˜ = ξg (w) = ξg (v) + ξg (v ). The ‘sum’ w˜ of v and v is, of course, not uniquely defined, but any two versions of this sum are identified under ξg . Moreover, if ∼ is the equivalence relation on R∞ defined by v ∼ v if and only if v − v ∈ ker(ξ Id ) = K d (cf. (3.31)), then R∞ /∼ is a compact abelian group isomorphic to X˜ f (d) = X f (d) / X Id (cf. Lemma 3.8): if [v] is the equivalence class of v ∈ R∞ , then the map θd ◦ ξ Id : R∞ −→ X˜ f (d) in (3.36) sends [v] to θd ◦ ξ Id (v) and maps the group operation [v] ⊕ [v ] := [v + v ] on R∞ /∼ to that on X˜ f (d) . 5.2.2. The problem of injectivity In Subsect. 5.2.1 we saw that R∞ has a natural group structure modulo elements in the kernel of ξg . Another problem which depends on the intersection of Rd with the cosets of ker ξ Id is the question of ‘pulling back’ to R∞ dynamical properties of α f (d) , such as uniqueness or the Bernoulli property of the measure of maximal entropy of R∞ . It is clear that the map ξ Id (and hence all the maps ξg , g ∈ I˜d ) must be noninjective on R∞ , since these maps are continuous, Rd is zero-dimensional, and the groups X f (d) and X¯ f (d) are connected. The following lemma shows that some of the maps ξg , g ∈ I˜d , are ‘more injective’ than others and is the reason for determining the ideal Id precisely in Sect. 2. Lemma 5.10. Let g ∈ I˜d and h ∈ Rd . For every v, w ∈ R∞ with ξg (w) ∈ ξg (v) + ker h(α), ξg·h (v) = ξg·h (w). It follows that |{w ∈ R∞ : ξg·h (w) = ξg·h (v)}| = | ker h(α f (d) )| for every v ∈ R∞ .
(5.15)
Abelian Sandpiles and the Harmonic Model
755
Proof. If x = ξg (v), y ∈ ker h(α f (d) ) and w ∈ R∞ satisfies that ξg (w) = x + y (cf. Theorem 5.1), then ξg·h (w) = h(α)(ξg (w)) = h(α)(x + y) = h(α)(x) = ξg·h (v).
6. The Dissipative Sandpile Model In this section we fix d ≥ 2 and γ > 2d, and consider the dissipative sandpile model γ R∞ ⊂ γ described in Sect. 4 and investigated in [26,7,16]. 6.1. The dissipative harmonic model. Consider the Laurent polynomial f (d,γ ) ∈ Rd defined in (4.5) and the corresponding compact abelian group d # d X f (d,γ ) = ker f (d,γ ) (α) = x = (xn )n∈Zd ∈ TZ : γ xn − (xn+e(i) + xn−e(i) ) = 0
$
i=1
for every n ∈ Z . d
(6.1)
We write α X f (d,γ ) for the shift-action (3.1) of Zd on X f (d,γ ) ⊂ TZ . d
Lemma 6.1. The shift-action α of Zd on X f (d,γ ) is expansive, i.e., there exists an > 0 such that sup | xn − xn | >
n∈Zd
for every x, x ∈ X f (d,γ ) with x = x . The entropy of α f (d,γ ) is given by h top (α f (d,γ ) ) = h λ X
f
(α f (d,γ ) ) = (d,γ )
1 0
1
···
log f (d,γ ) (e2πit1 , . . . , e2πitd ) dt1 · · · dtd ,
0
and the Haar measure λ X f (d,γ ) is the unique shift-invariant measure of maximal entropy on X f (d,γ ) . Proof. Since f (d,γ ) has no zeros in Sd = (z 1 , . . . , z d ) ∈ Cd : |z i | = 1 for i = 1, . . . , d , α f (d,γ ) is expansive by [21, Theorem 6.5]. The last two statements follow from [21, Theorems 19.5, 20.8 and 20.15].
756
K. Schmidt, E. Verbitskiy (γ )
6.2. The covering map ξ (γ ) : R∞ −→ X f (d,γ ) . Since α f (d,γ ) is expansive and has completely positive entropy, the equation f (d,γ ) · w = 1
(6.2)
has a unique solution w = w(d,γ ) ∈ 1 (Zd ), given by (d,γ )
wn
1
= 0
··· 0
1
e−2πi n,t dt1 · · · dtd , d γ − 2 · i=1 cos(2π ti )
where t = (t1 , . . . , td ) (cf. (2.5), [14] and [6]). Since w (d,γ ) ∈ 1 (Zd ), we can proceed (γ ) as in (3.15) and define a homomorphism ξ (γ ) : R∞ → X f (γ ,d) by ξ¯ (γ ) (v)n = (w (d,γ ) · v)n = (γ )
for every v ∈ R∞ , and by
(d,γ )
vn−k wk
n∈Zd
ξ (γ ) = ρ ◦ ξ¯ (γ ) . Proposition 6.2. The map ξ (γ ) has the following properties: (γ )
(a) ξ (γ ) (R∞ ) = X f (d,γ ) ; (γ )
(b) For v, v ∈ R∞ , ξ (γ ) (v) = ξ (γ ) (v ) if and only if v = v + f (d,γ ) · h
(6.3)
for some h ∈ ∞ (Zd , Z); (γ ) (c) ξ (γ ) (v) = ξ (γ ) (v) for all v, v ∈ R∞ with v − v ∈ Rd . Furthermore, the topological entropies of the shift-actions α f (d,γ ) on X f (d,γ ) and σR(γ ) ∞ (γ ) on R∞ coincide. Proof. The proofs are completely analogous to (but simpler than) those of the corresponding results in the critical case. Corollary 6.3. For every v ∈ ∞ (Zd , Z) there exists a h ∈ ∞ (Zd , Z) such that w = (γ ) v + f (d,γ ) · h ∈ R∞ . Proof. This follows from Proposition 6.2 (a)–(b).
Remark 6.4. The element w in Corollary 6.3 can be constructed explicitly by using the method described in the proofs of Lemma 3.5, Theorem 4.1 and Subsect. 5.2.1. In [16], two elements v, v ∈ ∞ (Zd , Z) are called equivalent (denoted by v ∼ v ) if they satisfy (6.3) for some h ∈ ∞ (Zd , Z).5 We write [v] ⊂ ∞ (Zd , Z) for the equivalence class of v in this relation. The following theorem summarizes the results of [16]. 5 Definition 3.2 in [16, p. 404] contains a misprint: the requirement that h ∈ ∞ (Zd , Z) is omitted, although it is used subsequently.
Abelian Sandpiles and the Harmonic Model
757
(γ )
(γ )
Theorem 6.5. The quotient R∞ /∼ is a compact space. Moreover, (R∞ /∼ , ⊕) is a compact abelian group, where [y] ⊕ [ y˜ ] = [y + y˜ ]. (γ )
Furthermore, there exists a shift-invariant measure of maximal entropy on R∞ , denoted by µ, such that (γ ) (γ ) µ y ∈ R∞ : [y] ∩ R∞ is a singleton = 1. (6.4) Proof. The first two statements are the results of [16, Prop. 3.2 and Th. 3.1]. Furthermore, the main result of [16], Theorem 3.2, states that, if µV is the uniform measure on (γ ) πV R Q(N ) , where V ⊂ Zd is a rectangle, then the set of limit points of sequences µV , V Zd , is a singleton. Denote by µ this unique limit point. We claim that µ is a (γ ) shift-invariant measure on R∞ , which moreover, has maximal entropy. The invariance follows immediately from the uniqueness of the weak limit point. (γ ) (γ ) Denote by σ the Zd -shift action on R∞ . For every Borel set A ⊆ R∞ , every n ∈ Zd , and any sequence of rectangles E k Zd : µ(σ n A) = lim µ E k (σ n A) = lim µ E k +n (A) = µ(A). k→∞
k→∞
Using the methods of [28, Chap. 8] (see also the proof of Theorem 5.9 above), one can show that 1 1 (γ ) log |R E | µ([y E ]) log µ([y E ]) = lim h µ (σR(γ ) ) = lim − d d ∞ |E| |E| E→Z E→ Z (γ ) y E ∈R E
= h top (σR(γ ) ), ∞
(γ )
where σR(γ ) is the restriction of σ to R∞ . Finally, (6.4) is the result of [16, Prop. 3.3]. ∞ We are now able to extend the results of [16] further. (γ )
Theorem 6.6. Let d ≥ 2, γ > 2d, and let R∞ be the dissipative sandpile model (4.4). (γ ) (γ ) (γ ) (i) The set C = y ∈ R∞ : [y] ∩ R∞ is a singleton is a dense G δ -subset of R∞ ; (γ )
(ii) The group (R∞ /∼ , ⊕) is isomorphic to X f (d,γ ) ; (γ )
(iii) The subshift R∞ admits a unique measure µ of maximal entropy. (γ ) (iv) The shift action of Zd on (R∞ , µ) is Bernoulli. Proof. The first statement is proved in Proposition 4.4. Using the properties of ξ γ : (γ ) R∞ → X f (d,γ ) (Lemma 6.2), the second statement is immediate. The same proof as (γ )
in Theorem 5.9 shows that h top (σR(γ ) ) = h top (X f (d,γ ) ), and that ξ∗ ν = λ X f (d,γ ) for ∞
(γ )
every shift-invariant probability measure ν of maximal entropy on R∞ . (γ ) Since the restriction of the continuous map ξ (γ ) : R∞ −→ X f (d,γ ) to C is injective, ξ (γ ) (C) is a Borel subset of X f (d,γ ) with full Haar measure.
758
K. Schmidt, E. Verbitskiy (γ )
(γ )
If ν is a shift-invariant probability measure of maximal entropy on R∞ , then ξ∗ ν = λ X f (d,γ ) . Hence ν(C) = 1, and the injectiveness of ξ (γ ) on C implies that ν = µ, where µ is the measure appearing in Theorem 6.5. This proves (iii). (γ ) The Bernoulli property of the shift-action of Zd on (R∞ , µ) follows from the corresponding property of α f (d,γ ) on (X f (d,γ ) , λ X f (d,γ ) ) proved in [20], since the two systems are measurably conjugate. 7. Conclusions and Final Remarks (1) In [11], toppling invariants have been constructed for the abelian sandpile model in finite volume. These are functions which are linear in height variables and are invariant under the topplings. It is also obvious that the definition [11, Eq. (3.3)] cannot be extended to the infinite volume. The underlying problem (non-summability of the lattice potential function) is precisely the problem overcome by the introduction of 1 -homoclinic points {v = g · w (d) : g ∈ Id }. The inevitable drawback is a larger kernel ξg f (d) · ∞ (Zd , Z). Nevertheless, we are tempted to conjecture that for d ≥ 2, the set {v ∈ R∞ : there exists v˜ ∈ R∞ : v˜ = v and ξ Id (v) = ξ Id (v)} ˜ has measure 0 with respect to any measure of maximal entropy. As in the dissipative case, this would imply that R∞ carries a unique measure of maximal entropy. (2) In the present paper we did not address the properties of the infinite volume sandpile dynamics, see e.g. [13]. We note that the sandpile dynamics takes a particularly simple form in the image space, the harmonic model X f (d) or its factor group X˜ f (d) . Namely, given any initial configuration v, suppose one grain of sand is added at site n. For every g ∈ I˜d = Id ( f (d) ), ξg (v + δ (n) ) = ξg (v) + ξg (δ (n) ) = ξg (v) + ρ(α −n z (g) ), where δ (n) = σ −n δ (0) (cf. Footnote 2) and z (g) = ρ(g ∗ · w (d) ) ∈ (1) α (X f (d) ) is the homoclinic point appearing in (3.14). It might be interesting to understand whether any statistical properties of the harmonic model can be used to draw any conclusions on the distribution of avalanches and other dynamically relevant notions in R∞ . Finally, as already mentioned in the Introduction, the group Gd = Rd /( f (d) ) is the appropriate infinite analogue of the groups of addition operators in finite volumes: on the sandpile model, Gd can be viewed as the abelian group generated by the elementary addition operators {an : n ∈ Zd } satisfying the basic relations an2d = ak k: k−n max =1
for all n ∈ Zd . These addition operators are well-defined on R E , E Zd , but for the infinite volume limit R∞ these operators are not defined everywhere. Under the maps ξg : R∞ −→ X f (d) , g ∈ Id , or ξ Id : R∞ −→ X˜ f (d) = X f (d) / X Id , the addition operator an is sent to addition of the homoclinic points ξg (δ (n) ) = ρ(g ∗ · w (d) ) = g(α)(x ) (on X f (d) ) and ξ Id (δ (n) ) (on X˜ f (d) ), respectively. These additions are defined everywhere on X f (d) and X˜ f (d) , and the isomorphism between X˜ f (d) and R∞ /∼ implies that the addition operators an , n ∈ Zd , are defined everywhere on R∞ /∼ (cf. Subsect. 5.2.1).
Abelian Sandpiles and the Harmonic Model
759
Acknowledgement. E.V. would like to acknowledge the hospitality of the Erwin Schrödinger Institute (Vienna), where part of this work was done. E.V. is also grateful to Frank Redig, Marius van der Put and Thomas Tsang for illuminating discussions. K.S. would like to thank EURANDOM (Eindhoven) and MSRI (Berkeley), for hospitality and support during part of this work. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Athreya, S.R., Járai, A.A.: Infinite volume limit for the stationary distribution of abelian sandpile models. Commun. Math. Phys. 249(1), 197–213 (2004) 2. Athreya, S.R., Járai, A.A.: Erratum: Infinite volume limit for the stationary distribution of abelian sandpile models. Commun. Math. Phys. 264(3), 843 (2006) 3. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/ f noise. Phys. Rev. Lett. 59, 381–384 (1987) 4. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988) 5. Burton, R., Pemantle, R.: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21, 1329–1371 (1993) 6. de Boor, C., Höllig, K., Riemenschneider, S.: Fundamental solutions for multivariate difference equations. Amer. J. Math. 111, 403–415 (1989) 7. Daerden, F., Vanderzande, C.: Dissipative abelian sandpiles and random walks. Phys. Rev. E 63, 30301– 30304 (2001) 8. Dhar, D.: Self organized critical state of Sandpile Automaton models. Phys. Rev. Lett. 64, 1613– 1616 (1990) 9. Dhar, D.: The abelian sandpiles and related models. Phys. A 263, 4–25 (1999) 10. Dhar, D.: Theoretical studies of self-organized criticality. Phys. A 369, 29–70 (2006) 11. Dhar, D., Ruelle, P., Sen, S., Verma, D.-N.: Algebraic aspects of abelian sandpile models. J. Phys. A 28, 805–831 (1995) 12. Fukai, Y., Uchiyama, K.: Potential kernel for the two-dimensional random walk. Ann. Probab. 24, 1979– 1992 (1996) 13. Járai, A., Redig, F.: Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3. Probab. Theor. Relat. Fields 141, 181–212 (2008) 14. Lind, D., Schmidt, K.: Homoclinic points of algebraic Zd -actions. J. Amer. Math. Soc. 12, 953–980 (1999) 15. Lind, D., Schmidt, K., Ward, T.: Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101, 593–629 (1990) 16. Maes, C., Redig, F., Saada, E.: The infinite volume limit of dissipative abelian sandpiles. Commun. Math. Phys. 244, 395–417 (2004) 17. Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19, 1559–1574 (1991) 18. van der Put, M., Tsang, F.L.: Discrete Systems and Abelian Sandpiles. J. Alg. 322(1), 153–161 (2009) 19. Redig, F.: Mathematical aspects of the abelian sandpile model. Mathematical Statistical Physics, Volume Session LXXXIII: Lecture Notes of the Les Houches Summer School 2005 (Les Houches), Bovier, A., Dunlop, F., den Hollander, F., van Enter, A., Dalibard, J. (eds.), Amsterdam: Elsevier, (2006), pp. 657–728 20. Rudolph, D.J., Schmidt, K.: Almost block independence and Bernoullicity of Zd -actions by automorphisms of compact groups. Invent. Math. 120, 455–488 (1995) 21. Schmidt, K.: Dynamical Systems of Algebraic Origin. Basel-Berlin-Boston: Birkhäuser Verlag, 1995 22. Sheffield, S.: Uniqueness of maximal entropy measure on essential spanning forests. Ann. Probab. 34, 857–864 (2006) 23. Sinai, Ya.G.: On a weak isomorphism of transformations with invariant measure. Mat. Sb. 63(105), 23–42 (1964) 24. Solomyak, R.: On coincidence of entropies for two classes of dynamical systems. Ergod. Th. & Dynam. Sys. 18, 731–738 (1998) 25. Spitzer, F.: Principles of random walks. New York: van Nostrand Reinhold, 1964 26. Tsuchiya, T., Tomori, M.: Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model. Phys. Rev. E 61, 1183–1188 (2000) 27. Uchiyama, K.: Green’s function for random walks on Z N . Proc. London Math. Soc. 77, 215–240 (1998) 28. Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics, Vol. 79, BerlinHeidelberg-New York: Springer Verlag, 1982 Communicated by G. Gallavotti
Commun. Math. Phys. 292, 761–795 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0853-x
Communications in
Mathematical Physics
Quantum Inequalities from Operator Product Expansions Henning Bostelmann1,2, , Christopher J. Fewster2 1 Dipartimento di Matematica, Universita di Roma “Tor Vergata”,
Via della Ricerca Scientifica, 00133 Roma, Italy
2 Department of Mathematics, University of York, Heslington,
York, YO10 5DD, United Kingdom. E-mail: [email protected]; [email protected] Received: 29 January 2009 / Accepted: 3 April 2009 Published online: 9 July 2009 – © Springer-Verlag 2009
Dedicated to the memory of Bernd Kuckert Abstract: Quantum inequalities are lower bounds for local averages of quantum observables that have positive classical counterparts, such as the energy density or the Wick square. We establish such inequalities in general (possibly interacting) quantum field theories on Minkowski space, using nonperturbative techniques. Our main tool is a rigorous version of the operator product expansion. 1. Introduction The principal qualitative difference between classical and quantum physics lies in the fundamentally unsharp nature of the latter, quantitatively expressed by the uncertainty principle. This distinction becomes particularly acute when one seeks analogues in quantum theory of quantities that are classically positive. In quantum mechanics, for example, one replaces a probability distribution over classical phase space by the Wigner function, which is pointwise positive only for Gaussian states [1]. Consequently, Weyl quantization of classically positive observables does not generally yield positive operators. Similarly, a positive (local) quadratic form in a classical field and its derivatives, such as the energy density of a free minimally coupled scalar field, would not be expected to have a positive analogue in quantum field theory, owing to the subtractions necessary to renormalize products of fields at a point. Nonetheless, positivity is not completely destroyed in quantization. The sharp Gårding inequalities [2] show that classically positive symbols have Weyl quantizations that are positive modulo corrections of lower order; that is, operators corresponding to a lower rate of growth in momentum. The aim of this paper is to establish analogous results for quantum field theory in a model independent and nonperturbative setting. The key to our approach is a recently-developed microscopic phase space condition [3] that controls the degrees of freedom available to the theory at small scales and bounded energy, and Supported by the EU network “Noncommutative Geometry” (MRTN-CT-2006-0031962).
762
H. Bostelmann, C. J. Fewster
guarantees the existence of a rigorous operator product expansion (OPE) [4]. In any theory obeying this condition (along with other standard criteria set out in Sect. 3) we identify a class of ‘classically positive’ operator products and show how this classical positivity is reflected in estimates on suitable smearings of the composite fields appearing in the corresponding OPEs. If there is a distinguished normal product associated with the underlying classically positive expression the picture is closely analogous to that emerging from the Gårding inequalities: suitable smearings of the normal product are positive modulo corrections of a lower order. As we will describe, our results significantly generalize the quantum (energy) inequalities, developed over recent years, that provide lower bounds on smearings of quadratic normal ordered quantities in free field theories. In the following subsections, we will describe the background and motivation for our study. 1.1. Quantum inequalities. It has been known for many years that expectation values of quantities such as the Wick square or energy density of a free scalar field may assume negative values and are pointwise unbounded from below as the quantum state is varied. Indeed, no local observable (other than the zero operator) can be both positive and have a vanishing vacuum expectation value [5]. Thirty years ago, Ford made the key observation that, as unrestricted negative energy densities or fluxes could produce macroscopic violations of the second law of thermodynamics, it was to be expected that QFT itself places strict limits on such departures from positivity [6]. Subsequently, Ford and Roman were able to derive lower bounds, called quantum inequalities (QIs), on averaged energy densities for scalar fields in Minkowski space [7–9]; these results were generalized to static curved spacetimes by Pfenning and Ford [10]. In the results just mentioned, the averaging is performed along a timelike geodesic with respect to a Lorentzian weight. With Eveson, one of us (CJF) obtained similar results for general weight functions [11]. As an example, the renormalized energy density :ρ: of the field of mass m in four-dimensional Minkowski space obeys the inequality ∞ 1 2 2 dt ω(:ρ: (t, 0))|g(t)| ≥ −Q[g] := − du u 4 |g(u)| ˜ (1.1) 16π 3 m for any g ∈ D(R) and all Hadamard states ω (this is slightly weaker than the bound of [11]). Here g˜ denotes the Fourier transform. Similar bounds are obeyed by any classically positive field of form i :(Pi φ)2 :, where the Pi are partial differential operators with smooth real coefficients. We will understand the term ‘quantum inequality’ to apply to any bound of this type, and not just those relating to the energy density (for which the more specific term ‘quantum energy inequality’ (QEI) is also used). The basic technique of [11] generalizes straightforwardly to static spacetimes [12] and the electromagnetic field [13]. It also underlies the general and rigorous results of [14], which give QIs for averaging with arbitrary weights along arbitrary timelike curves in arbitrary globally hyperbolic spacetimes, valid for all Hadamard states. (The bound in [14] is expressed using a reference state; see [15] for analogous results with a purely local geometric bound.) Similar results hold for spin-1 fields [16]. We note that averaging in timelike directions is essential for establishing inequalities; while averaging over spacetime volumes also yields lower bounds (see, e.g., [15]), purely spatial [17] or lightlike [18] averaged energy densities do not generally obey quantum inequalities. An important feature of the lower bound in (1.1) is that it is independent of the state ω, and can be rewritten as an operator inequality :ρ : (|g|2 ) ≥ −Q(g)1. One cannot
Quantum Inequalities from Operator Product Expansions
763
expect bounds of this type for general interacting theories [19] (although they do hold for conformal field theories in two dimensions [20]; see also [21,22] for precursors). Indeed, the nonminimally coupled scalar field provides an example of a free field theory in which averaged energy densities are unbounded from below [23]. The best that can be expected, in general, is an inequality of the form :ρ: (|g|2 ) ≥ −Q(g), where Q(g) is now permitted to be an operator. As noted in [24], this would be a rather empty notion without some constraints on Q(g) (for example, Q(g) = − :ρ : (|g|2 ) gives a trivial inequality of this type). To qualify as a nontrivial inequality, Q should be of ‘lower order’ than :ρ: in a defined sense. For example, the nonminimally coupled scalar field obeys bounds of the form :ρ: (|g|2 ) ≥ −Q1 (g)1 + 2ξ :φ 2: (g˙ 2 )
(1.2)
in four-dimensional Minkowski space for coupling ξ ∈ [0, 1/4] [23]. Crucially, the right-hand side is bounded relative to (1 + H ) p for any p > 2, while the left-hand side is not bounded relative to any (1 + H )q with q < 3, where H is the Hamiltonian. In the present paper we will weaken the criterion of nontriviality slightly owing to the approximate nature of OPEs. As we explain in outline in Sect. 2 and in detail in Sect. 6, we permit bounds containing a remainder term that is of higher order in energetic terms than the field of interest, but which is vanishing in the small distance limit. All the results mentioned so far rely on positivity of an underlying classical expression, namely, a sum of squares of fields and their derivatives; this is also the focus of the present work. However, it is important to recall that the energy density of a Dirac field is not expressed in this way; accordingly different techniques are required to obtain quantum energy inequalities in this case (see [25–28] for spin-1/2 and [29,30] for spin-3/2).
1.2. Perturbative versus nonperturbative approaches. While QIs were first studied for free fields on Minkowski space, it is now known – as mentioned above – that the concept is compatible at least with some simple types of interaction, specifically the coupling to an external gravitational field and those in conformal field theories. However, on the technical side, the existing results typically rely on the rather simple structure of linear quantum fields fulfilling c-number commutation relations. For dealing with general, possibly self-interacting quantum fields, this is far too restrictive. Instead, our aim here is to derive inequalities from general principles of quantum field theory that are not restricted to linear fields. A fully rigorous construction of self-interacting quantum field theories remains a challenge, and has to date been completed in low-dimensional models only [31–33]. In physical spacetime, interacting theories have generally been established in a perturbative setting, usually without any control on the convergence of the perturbation series. It would therefore seem natural to investigate QIs in a perturbative context. However, severe conceptual difficulties arise here. In order to give any reasonable meaning to quantum inequalities theory, we need to determine when a formal power in perturbation k series, say P[g] = ∞ k=0 ck g with ck ∈ C, and with the formal variable g being interpreted as a “coupling constant”, should be considered positive. Understanding the set of formal power series as a ∗-algebra, the natural notion of positivity is as follows [34]: P is considered positive if and only if P[g] = Q ∗ [g]Q[g] for some formal power series Q.
(1.3)
764
H. Bostelmann, C. J. Fewster
It turns out that this condition is equivalent to the following one: P[g] = g 2n
∞
dk g k with n ∈ N0 , dk ∈ R, d0 > 0.
(1.4)
k=0
(Here (1.3) ⇒ (1.4) is immediate; the converse follows by inserting x = (d0−1 g −2n P[g]− √ 1) into the power series of 1 + x around x = 0.) Now Eq. (1.4) shows that this notion of positivity is not useful in our context, since it roughly says that positivity of P is determined by its lowest-order coefficient. (See [35] for a slight variant.) The order-0 coefficient however is supposed to be the contribution from free field theory. So—with this definition—QIs would hold at finite coupling if and only if they hold at coupling g = 0; the effects of interaction on inequalities cannot be captured in this approach. Let us illustrate these difficulties in a simple example: should one consider the following formal power series positive? P[g] =
∞ (−1)k k=0
(2k)!
g 2k .
(1.5)
Considering P as a convergent series, it would be positive for small, but not for all g. Forgetting all convergence properties, the only information that remains is positivity at g = 0, i.e., of the zero-order coefficient. The question of interest, however, would be √ whether the physical value of g falls into the convergence radius of P; this question is not accessible in formal perturbation theory. It is therefore necessary to conduct our investigation in a nonperturbative formulation of quantum field theory, such as the Wightman setting [36] or the C ∗ algebraic formulation of Haag and Kastler [37]. (We shall actually use a combination of both; the technical details will be recalled in Sect. 3.) This is, in a way, a very strong assumption to start with, since we assume that our QFT models have been fully constructed and are under complete topological control. Indeed, as mentioned above, the rigorous construction of interacting models in physical space-time remains an open challenge. The virtues of our axiomatic approach, however, are of a different nature: within the framework of algebraic quantum field theory, we can formulate physically motivated, qualitative properties of quantum field theories, which can explicitly be verified in simple models such as free field theory, but which appear general enough to be postulated for the interacting situation. We can then show how observable consequences, such as quantum inequalities, follow from these postulated properties. 1.3. Phase space conditions. The specific qualitative properties we will employ are known as phase space conditions. Semi-classical considerations (originating with Bohr and Sommerfeld) suggest that only finitely many independent states (or, dually, observables) are required to describe a quantum system which is restricted to a finite volume in phase space e.g., by cut-offs in configuration space and energy. In quantum field theory, this picture can certainly persist only qualitatively and in an approximate sense. However, it is possible to give a precise meaning to the aforementioned concepts, expressed as the compactness or nuclearity of certain maps; see e.g. [38–40]. These phase space conditions have physically interesting consequences: for example, they imply the existence of thermal equilibrium states [41] and are important for the particle interpretation of quantum field theories [42].
Quantum Inequalities from Operator Product Expansions
765
The role of phase space conditions for QIs has been partially investigated before. Even in the free field situation described above, one may see the need for some restrictions on the phase space behaviour of the theory [43]: instead of one field of mass m, consider an infinite number of fields with masses m j (for simplicity, in four-dimensional Minkowski space). The total energy density will obey a QI, ∞ 1 2 dt ω(:ρ: (t, 0))|g(t)|2 ≥ − du u 4 N (u)|g(u)| ˜ , (1.6) 16π 3 0 where N (u) =
ϑ(u − m j )
(1.7)
j
counts the number of species with masses below energy u. If N grows no faster than polynomially with u, the lower bound is finite for all g ∈ C0∞ (R); the same condition is known to guarantee that this theory obeys nuclearity in the sense of Buchholz and Wichmann [39]. Other ideas concerning the relationship between QEIs and nuclearity conditions are discussed in [44], while connections with thermodynamic stability are described in [45,46]. However, the results presented here are the first in which QIs have been derived as a consequence of phase space criteria. For our purposes, we will use a microscopic phase space condition recently introduced by one of us (HB) in [3]; we shall recall its formulation and consequences in Sect. 3. Compared with other similar conditions, it is specifically sensitive in the short-distance regime, the realm which is of most interest for QIs. Indeed, one heuristically expects [47] that at short distances and finite energies, the theory may be well-approximated in terms of finitely many observables corresponding to pointlike quantum fields. This approximation of bounded observables by quantum fields can indeed be made precise [3] and plays a central role in our approach. Its use is twofold. First, it tells us how our primary objects—local algebras of bounded operators—relate to the quantum fields for which inequalities are formulated. Second, it serves to establish an additional structure for the quantum fields, namely a rigorous version of the operator product expansion [4]. We can understand this OPE, which describes the “structure constants” of the “improper algebra” of quantum fields, as containing all relevant information about the interaction, and in this sense as a replacement for the Lagrangian [48]. In fact, it is the OPE from which our inequalities will be computed. In particular, the OPE allows us to generalize the notion of normal ordering that has a key role for QIs of linear fields, replacing it with normal products in the sense of Zimmermann [49]. The remainder of the paper is organized as follows. We start with a non-technical account of our main methods and results in Sect. 2. Then, in Sect. 3, we introduce the framework of nonperturbative quantum field theory that we work in, including the phase space condition mentioned above. Section 4 presents some technical preliminaries from distribution theory. In Sect. 5, we establish the rigorous operator product expansion in the variant that we require. This expansion will be the base of our quantum inequalities, derived in Sect. 6. Dilation covariance as a special case is covered in Sect. 7. We end with a brief outlook in Sect. 8. 2. Overview We now give a non-technical overview of our main techniques and results, postponing rigorous arguments to later sections. Throughout, we work in Minkowski space of
766
H. Bostelmann, C. J. Fewster
dimension 2 + 1 or more (possible generalizations are discussed in Sect. 8). For simplicity, we shall always pick a fixed Lorentz frame, and hence a fixed time axis; all quantum fields φ(t) = φ(t, x = 0) will be restricted to this time axis, and smeared expressions φ( f ) = dt f (t) φ(t) will refer to one-dimensional integration only. This is sufficient for regularizing Wightman fields [50]; due to the symmetry properties of Minkowski space, it covers the essential features of the inequalities we wish to consider. To illustrate our approach we begin by sketching the derivation of a QI for the Wick square of the free real scalar field, essentially following the argument of [14] but in a form which is amenable to our generalization. We will then indicate which changes are necessary to deal with the general situation. Accordingly, let φ denote the free field and let σ be a normal state in the vacuum sector with sufficiently regular high-energy behaviour such that the expectation values in the following are finite. The distributional integration kernel F(t, t ) = σ (φ(t)φ(t ))
(2.1)
is positive-definite, in the sense that for any test function g, we have
dt dt F(t, t )g(t)g(t ) = σ (φ(g)∗ φ(g)) ≥ 0.
(2.2)
Then, also F(t, t )/ıπ(t − t − ı0) is positive-definite; namely we have by Fourier analysis,
dp dt dt F(t, t )g(t)g(t ) ϑ( p)e−ı p(t−t ) π ∞ dp = dt dt F(t, t )eı pt g(t)eı pt g(t ) ≥ 0. π 0 (2.3)
g(t)g(t ) dt dt F(t, t ) = ıπ(t − t − ı0)
We now use Wick ordering and introduce new variables s = (t + t )/2, s = t − t in order to rewrite the kernel F: F(t, t ) = σ (:φ 2: (s)) + + (s ) + σ (R(s, s )),
(2.4)
where + (t −t ) = ω(φ(t)φ(t )) is the vacuum two-point function of φ, and the remainder R is given by
t + t 2
2 :
R(s, s ) = :φ(t)φ(t ): − :φ = U (s) :φ(s /2)φ(−s /2) − φ 2 (0): U (s)∗ ,
(2.5)
with U (s) being the time translation unitaries. Note that R(s, s ) is a smooth function when evaluated in σ . Inserting this into Eq. (2.3), we obtain σ (:φ 2: ( f ) + cg 1) ≥ −Rσ,g ,
(2.6)
Quantum Inequalities from Operator Product Expansions
767
where
g(s + s /2)g(s − s /2) , ıπ(s − ı0) g(s + s /2)g(s − s /2) , cg := ds ds + (s ) ıπ(s − ı0) g(s + s /2)g(s − s /2) . Rσ,g := ds ds σ (R(s, s )) ıπ(s − ı0) f (s) :=
ds
(2.7) (2.8) (2.9)
It seems plausible that Rσ,g becomes small as supp g shrinks to a point. We will give more quantitative estimates in that respect later. Here, let us consider the special case where g is real-valued. Then both g(s + s /2)g(s − s /2) and R(s, s ) are even functions in s . Hence, in Eqs. (2.7) and (2.9), we can replace the factor (s − ı0)−1 with its even part, 1 −1 1 + (2.10) = ıπ δ(s ). 2 s − ı0 s + ı0 Since R(s, 0) = 0, this results in Rσ,g = 0 and f (s) = g(s)2 . Thus Eq. (2.6) gives the more usual inequality for the Wick square, :φ 2: (g 2 ) ≥ −cg 1.
(2.11)
We now aim at a generalization beyond free field theory. So let φ be a general, possibly self-interacting local quantum field. (The term “quantum field” is used here in a generic fashion, and may include derivatives of fields as well as composite fields or suitably defined powers of fields.) The main difficulty we face in applying the above construction is that no concept of normal ordering is available; we cannot use Wick ordering to split the product into higher-order and lower-order terms, as in Eq. (2.4). Instead, we shall use an operator product expansion for the product φ(t)∗ φ(t ), φ ∗ (t)φ(t ) =
n
C j (t − t )φ j ((t + t )/2) + Rn (t, t ).
(2.12)
j=1
Here Rn is a remainder term, which is “small” if t and t are close, while the φ j are composite fields. Smearing against g(t)g(t ), where g ∈ D(R), the left-hand side is then a positive operator, and this remains true if we multiply (2.12) with any positive-type kernel K (t − t ), which takes the role of K (t − t ) = 1/ıπ(t − t − ı0) above. In other words, Eqs. (2.2) and (2.3) remain valid. We can then rearrange and obtain as analogue to Eq. (2.6), n
φj( f j) ≥ −
dt dt g(t)g(t )K (t − t )Rn (t, t ),
(2.13)
j=1
where the test functions f j are given in terms of g, K , and the OPE coefficients C j by f j (s) = ds K (s ) C j (s ) g(s + s /2)g(s − s /2). (2.14)
768
H. Bostelmann, C. J. Fewster
Note that there is no guarantee that these functions are necessarily pointwise positive (the issues here are related to Hudson’s theorem [1] and the ‘choice of basis’ invoked in the OPE). We will return to this below. In order to establish our results rigorously, the main task is to establish the OPE and to control the remainder term on the right-hand side. We will show in Theorem 6.1 that, given α ≥ 0, one may find n, m and so that for all d > 0 and g ∈ D(−d, d), n
φ j ( f j ) ≥ − (d)g2d,m (1 + H )2 ,
(2.15)
j=1
where H is the Hamiltonian, (d) = o(d α ) as d → 0 and · d,m is equivalent to the Sobolev norm on W0m,1 (−d, d). (Of course, finite sums of field products can be, and are, accommodated by our result.) The relationship with the QIs described in Sect. 1.1 is most apparent in the case where one of the composite fields, say φ1 , is of higher order than the others, in the sense that there exists for which (1 + H )− φ j ( f )(1 + H )− is bounded for j ≥ 2, while (1 + H )− φ1 ( f )(1 + H )− is unbounded. Then we may rearrange to write φ1 ( f 1 ) ≥ −
n
φ j ( f j ) − (d)g2d,m (1 + H )2 .
(2.16)
j=2
In cases where the remainder term vanishes (2.16) is then a nontrivial QI in the sense of [24]: namely, one cannot find constants C, C such that |σ (φ1 ( f 1 ))| ≤ C
n
|σ (φ j ( f j ))| + C
(2.17)
j=2
for all (sufficiently regular) states σ because φ1 is of higher energetic order than the fields on the right-hand side. Examples include the QI (2.11), where the only composite field on the right-hand side is the identity, and the QEI (1.2) on the nonminimally coupled field, where both the identity and Wick square appear on the right-hand side. This simple situation does not persist in general, however. First, it does not seem guaranteed that a unique choice of a highest-order field φ1 exists. For interacting fields, one would expect φ1 to be the normal product of φ ∗ φ in the sense of Zimmermann [49]; but there are indications from perturbation theory that in some cases, this normal product might not be unique [51]. Second, the remainder term cannot be expected to vanish in general–this reflects that the OPE is a controlled approximation, rather than an exact formula. Third, the remainder term is not of lower energetic order than the fields: in fact, is chosen so that each (1 + H )− φ j ( f j )(1 + H )− is bounded. Although the inequalities in Eq. (2.15) remain valid, it is necessary to adapt the criterion of nontriviality to our setting. Our approach is to focus on the short-distance behaviour, in which the remainder term vanishes as o(d α ). By contrast, we will show in Sect. 6.2 that, for the bounds we obtain, n −2 − − (2.18) sup gd,m (1 + H ) φ j ( f j )(1 + H ) g∈D (−d,d) j=1
Quantum Inequalities from Operator Product Expansions
769
is not o(d α ) as d → 0. Thus the remainder term cannot dominate the contribution of the composite fields in the small. In this context, it turns out to be crucial to formulate the OPE, and correspondingly the inequalities, in a “basis-independent” fashion, that is, in a way that is independent of a possible arbitrariness in the choice of composite fields. For practical purposes, it is still important to understand the more specific question as to whether there is a normal product of strictly higher energetic order than the other fields in the OPE. At present it seems to us that this must be discussed in the light of particular examples. Last but not least, one would like to gain more insight in the properties of the sampling functions f j , given by Eq. (2.14), in particular for the function f 1 corresponding to a “highest-order” composite field, which generalizes f 1 (s) = g(s)2 from the free field case. In general, it is certainly not expected that f 1 depends on g in a simple pointwise fashion. However, one may ask whether φ1 can be chosen so that f 1 retains other properties that are apparent in the free-field situation, for example whether f 1 ≥ 0, either pointwise or in an averaged sense. This will depend crucially on the form of the OPE coefficients C j , which are however unknown in general. We will give two approaches to this problem. The first, in Sect. 6.3, indicates conditions under which one may simultaneously tune the leading sampling function to a given positive form, while also reducing the remainder term. These conditions are broadly met under the assumption that the OPE coefficients have scaling limits in the sense of [52]. Our argument here is essentially to form a Riemann sum of QIs over small distance scales, in which the remainder term is suppressed. This approach is, however, tied to basis representations of the OPE; it would appear to be most useful in the context of particular models. Second, in Sect. 7, we discuss the particular case of dilation covariant theories. This is of interest since we can expect that our theory is approximated by a dilation covariant “scaling limit theory” in the ultraviolet. In this restricted situation, we will derive explicit criteria on g that guarantee positivity of f 1 . Since positivity of f 1 also fixes the sign of the composite field φ1 , this gives us a means of distinguishing the positive “normal square” of φ from its negative.
3. Algebras of Observables and Pointlike Quantum Fields As a mathematical basis of quantum field theory, we adopt the framework of local quantum physics [37]. Specifically, for describing pointlike quantum fields, we use the methods set forth in [3]. For the convenience of the reader, we will collect the relevant notions and results below, and introduce some notations that are useful in our context. We set out from a local net of algebras, O → A(O), in the vacuum sector. That is, for each bounded open region O of Minkowski space, we have an algebra A(O) of bounded operators; we take these to be von Neumann algebras acting on a common Hilbert space H. Further, we have a strongly continuous unitary representation (x, ) → U (x, ) of the proper orthochronous Poincaré group on H, with a common invariant unit vector ∈ H. We write the translation subgroup as U (x, 1) = exp ı Pµ x µ . Together these objects are supposed to fulfil the following axioms: (i) Isotony: A(O1 ) ⊂ A(O2 ) if O1 ⊂ O2 . (ii) Locality: [A1 , A2 ] = 0 if O1 , O2 are two spacelike separated regions, and Ai ∈ A(Oi ). (iii) Covariance: U (x, )A(O)U (x, )∗ = A(O + x) for all Poincaré transformations (x, ).
770
H. Bostelmann, C. J. Fewster
(iv) Positivity of energy: The joint spectrum of the Pµ falls into the closed forward light cone. (v) Uniqueness of the vacuum: is unique (up to a phase) as an invariant vector for all U (x, 1). We are primarily interested in the algebras associated with standard double cones Or of radius r centred at the origin, and use A(r ) as shorthand for A(Or ). Also, for most parts we only use the time-translation subgroup of U (x, ), which we denote as t → U (t), with positive generator H = P0 ≥ 0. We write the spectral projectors of H for the interval [0, E] as P(E). Let be the set of ultraweakly continuous functionals on B(H). We consider for > 0 the subspaces
C () = σ ∈ σ ( ) := σ (1 + H ) · (1 + H ) < ∞ , (3.1) which are Banach spaces in the norm · ( ) . Their duals C ()∗ consist of linear forms φ for which the dual norm φ(− ) = (1 + H )− φ(1 + H )− is finite. (More precisely, φ are quadratic forms on a dense subspace of H × H, for which the form (1 + H )− φ(1 + H )− , with the multiplication defined in the weak sense, is bounded.) We also introduce the space of smooth functionals, C ∞ () = ∩ >0 C (), and equip it with the Fréchet topology induced by all norms · ( ) . The dual space C ∞ ()∗ is then given by ∪ >0 C ()∗ , and will be considered with the weak∗ topology. Further, we define for E > 0 the set of energy-bounded functionals, (E) = {σ (P(E) · P(E)) | σ ∈ }. Then ∪ E>0 (E) is dense in C ∞ () and weakly dense in . Each space C () is invariant under the natural action of hermitian conjugation, i.e. σ ∗ (A) = σ (A∗ ), and this structure transfers to the dual spaces; so we can speak of hermitian elements in C ∞ () and C ∞ ()∗ . With respect to pointlike fields, we assume that the theory fulfils a specific type of phase space condition [3], sensitive in the ultraviolet. To formulate this, consider the inclusion map : C ∞ () → . We assume that can be approximated with finite-rank maps in the following sense. Definition 3.1 (Microscopic phase space condition). A net O → A(O) is said to satisfy the microscopic phase space condition if for every γ ≥ 0, there exists a continuous linear map ψ : C ∞ () → of finite rank such that for sufficiently large > 0, r −γ ( − ψ)A(r )( ) → 0 as r → 0. Here the restriction A(r ) is applied to the image points of the maps, which are functionals in . This phase space condition is known to be fulfilled in free field theory in at least 3 + 1 space-time dimensions, for massive free fields also in 2 + 1 dimensions [53]. The consequences of this condition are as follows [3]. While the maps ψ are not uniquely fixed by the property above, the image of their dual maps, img ψ ∗ =: γ , is actually unique at fixed γ , provided that the rank of ψ is chosen minimal. These finitedimensional spaces γ form an increasing sequence 0 ⊂ 1 ⊂ 2 . . ., and their union ∪γ γ = FH is precisely the field content of the theory as defined by Fredenhagen and Hertel [54]. After smearing with test functions, the elements φ ∈ γ are local Wightman fields. Actually it suffices for regularizing φ to smearit along the time axis; that is, for f ∈ S(R) and φ ∈ FH , the quadratic form φ( f ) = dt f (t) U (t) φ U (t)∗ can be continued to an unbounded, but closable operator on the dense invariant domain C ∞ (H) = ∩ >0 (1 + H )− H. Further, φ ∈ γ can be approximated with bounded operators in a controlled way; cf. [3, Lemma 3.5] and the remark following it:
Quantum Inequalities from Operator Product Expansions
771
Theorem 3.2. Let φ ∈ FH . One can find constants > 0, k > 0 and operators Ar ∈ A(r ) for each r > 0 such that, as r → 0, Ar − φ(− ) = O(r ), Ar (− ) = O(1), dn Ar = O(r −k ), ∀n ∈ N : n U (t)Ar U (t)∗ = O(r −k−n ). dt
φ(− ) < ∞,
Moreover, the spaces γ are related to the approximation of bounded operators in the short distance limit; see [3, Eq. (4.4)]: Theorem 3.3. Let pγ : C ∞ ()∗ → γ ⊂ C ∞ ()∗ be a continuous projection onto γ . Then, for sufficiently large > 0, (pγ ∗ − )A(r )( ) = o(r γ ). Here pγ ∗ : C ∞ () → C ∞ () is the pre-dual map to pγ , which always exists due to its finite rank. Of course, such projections pγ exist in abundance. Since the spaces γ are invariant under conjugation, it is possible to choose pγ hermitian, i.e., such that pγ (A∗ ) = pγ (A)∗ . It was shown in [4] that due to the properties explained above, operator product expansions exist in a rigorous sense. In fact, [4] established the expansion of φ(x)φ (y) for spacelike separated points x and y. A similar scheme can be applied for arbitrary x and y, in the sense of distributions, as sketched in [4] and worked out in more detail in [53, Ch. 5.5]. (See also [55, Sect. 4].) For our purposes, we will need a specific variant of this product expansion, which will be established in Sect. 5.2. 4. Distributions as Boundary Values of Analytic Functions If σ ∈ is energy-bounded and φ a Wightman field with sufficiently regular high-energy behaviour, then the distribution1 σ (φ ∗ (t)φ(t )) is the boundary value of an analytic function in the half plane Im (t − t ) < 0. If further σ is positive, then the distribution is positive-definite. These types of distributions have certain well-known characterizations [36,56]. Since we will need specific quantitative estimates in our context, we will repeat some of those arguments in detail. First of all, for each d > 0 and m ∈ N we define a norm on D(−d, d) by f d,m := max d n f (n) 1 . 0≤n≤m
(4.1)
This norm is equivalent to the Sobolev norm defining the space W0m,1 (−d, d) [57], but it is convenient to use the above norms owing to their behaviour under scaling. Namely, if f ∈ D(−d, d) and λ > 0, and we set f λ (t) = λ−1 f (t/λ), then f λ ∈ D(−λd, λd) and ∀m ∈ N :
f λ λd,m = f d,m .
(4.2)
Let us now define the class of analytic functions that is of interest. 1 Throughout the paper, we will write distributions in terms of their formal integration kernels, such as K (x) f (x)d x for the evaluation of a distribution K ∈ S(R) on a test function f ∈ S(R), even if K does not arise from an integrable function or measure. This is merely a notational convention.
772
H. Bostelmann, C. J. Fewster
Definition 4.1. We say that an analytic function F : R − ıR+ → C is regular at the boundary if there exists > 0 such that F(− ) :=
sup
−1≤Im z<0
|F(z)| |Im z|
is finite. The space of all such functions for given is denoted as K ; and K := ∪ >0 K . As the name suggests, functions in K have distributional boundary values on the real line. Proposition 4.2. Let F ∈ K . Then the limit limy→0+ F(x − ıy) exists as a tempered distribution in x. The limit distribution F(x − ı0) satisfies the following estimate for f ∈ D(−d, d), d > 0: f (x) F(x − ı0) d x ≤ 4 +2 ( + 3)(1 + d − −2 ) F(− ) f d, +2 . Proof. For fixed f ∈ S(R), consider the function g(y) := f (x)F(x − ıy)d x, 0 < y ≤ 1.
(4.3)
Since F is analytic in z = x − ıy, we can obtain the derivatives of g using integration by parts: j d jg j d j ∀ j ∈ N0 : f ( j) (x)F(x − ıy)d x. (y) = f (x)(−ı) F(x − ıy)d x = ı dy j dz j (4.4) Thus we have the estimate ∀ j ∈ N0 :
j d g ≤ y − F(− ) f ( j) 1 . (y) dy j
(4.5)
We now want to deduce the following improved estimate: j +2 d g ≤ 4 − j+2 (1 + y 3/2− j ) F(− ) (y) f (k) 1 for j ∈ {0, . . . , + 2}. dy j
(4.6)
k=0
In fact, for j = + 2, this directly follows from Eq. (4.5). Now suppose that Eq. (4.6) holds for j + 1 in place of j. We compute: j j+1 1 d g d jg g d dy j+1 (y ) dy j (y) ≤ dy j (1) + dy y 1 +2 ≤ F(− ) f ( j) 1 + 4 − j+1 F(− ) f (k) 1 dy (1 + (y )1/2− j ) k=0
≤ 4 − j+1 F(− )
+2 k=0
f (k) 1 (4 + 2y 3/2− j ).
y
(4.7)
Quantum Inequalities from Operator Product Expansions
773
This proves Eq. (4.6). In particular, the case j = 1 shows that dg/dy is bounded as y → 0; thus g(y) converges in this limit. Setting j = 0 in Eq. (4.6) then shows that g(0+) =: f (x)F(x − ı0)d x defines a tempered distribution. Also, if f ∈ D(−d, d), we can combine the estimate m f (k) 1 ≤ (m + 1) max{1, d −m } f d,m (4.8) k=0
with Eq. (4.6), where j = 0 and m = + 2, in order to show the proposed estimate for the limit distribution. As is well known, the analytic function F is uniquely determined by its boundary distribution F( · − ı0). Further, for two functions which are regular at the boundary, it follows from Definition 4.1 that their product inherits this property. More explicitly, for F ∈ K and G ∈ Km , we have F G(− −m) ≤ F(− ) G(−m) .
(4.9)
Thus the product of the boundary distributions is well-defined by multiplying the corresponding analytic functions. On the other hand, the Fourier transforms of the boundary distributions have support in [0, ∞). This allows for an alternative definition of the distribution product by convolution in Fourier space. The two definitions are in fact equivalent [56, Ch. IX.10, Example 4]. Apart from our distributions being boundary values of analytic functions, we also need to consider questions of positivity. We remind the reader of the definitions (the terminology is not completely consistent in the literature). For g1 , g2 ∈ D(R), we introduce the abbreviation g1 g2 (s, s ) := g1 (s + s /2)g2 (s − s /2). Definition K ∈ S(R2 ) is called positive-definite if for all g ∈ S(R), 4.3. A distribution one has ds ds K (s, s ) g¯ g(s, s ) ≥ 0. If here K depends on the second variable only, so K ∈ S(R) , it is called a distribution of positive type. With K+ ⊂ K we denote the subset of positive type distributions. The Bochner-Schwartz Theorem asserts that distributions of positive type are precisely the Fourier transforms of positive, polynomially bounded measures. We now show that the product of distributions, as discussed further above, preserves positivity if both factors are positive, at least in a special situation that is of interest to us. Proposition 4.4. Let F ∈ K+ . Let G : R × (R − ıR+ ) → C such that G(s, · ) ∈ K for some and every fixed s, where the map R → K , s → G(s, · ) is bounded and continuous in · (− ) . Suppose further that G(s, s −ı0) is positive-definite. Then the product distribution P(s, s ) = F(s − ı0)G(s, s − ı0) is continuous in s and positive-definite. Proof. First, due to Prop. 4.2, the boundedness and continuity of s → G(s, · ) implies that P(s, s ) f (s )ds is continuous and bounded in s; in particular P ∈ S(R2 ) is well-defined. Now let µ be the positive measure that arises by Fourier transform of F(x − ı0). Since F(x − ı0) is a boundary value, we know supp µ ⊂ [0, ∞). Therefore we have for g ∈ S(R), ds ds P(s, s )g¯ g(s, s ) ∞ = lim dµ( p) ds ds e−ı p(s −ı ) G(s, s − ı )g¯ g(s, s ) . (4.10)
→0+ 0
=:I ( , p)
774
H. Bostelmann, C. J. Fewster
Supposing for a moment that the integrand I ( , p) has an integrable bound in p, uniform in , we can apply the dominated convergence theorem and obtain ∞ dµ( p) ds ds G(s, s − ı0)g¯ p g p (s, s ), ds ds P(s, s )g¯ g(s, s ) = 0
(4.11) where g p (t) = eı pt g(t). This is clearly non-negative, since G is positive-definite. It remains to prove appropriate bounds for I ( , p). To that end, choose n ∈ N so large that dµ( p)(1 + p)−n < ∞. We use integration by parts in s to obtain I ( , p) = (1 + p)−n ds ds (1 + p)n e−ı(s −ı ) p G(s, s − ı )g¯ g(s, s ) ∂ n = (1 + p)−n ds ds e−ı(s −ı ) p 1 − ı G(s, s − ı )g¯ g(s, s ). (4.12) ∂s Via the Leibniz rule, we can distribute the derivatives ∂/∂s to G(s, s ) and to the test function. Now note that with G, also the derivatives ∂ k G/∂z k fulfil polynomial bounds when Im z → 0−; namely, we can use the Cauchy integral formula for a circle of radius |Im z/2| around z in order to obtain the estimate k ∂ G(s, z) 1 k G(x, · )(− ) |Im z|− −k for − ≤ Im z < 0. (4.13) ∂z k ≤ 2 k! sup 2 x This implies that e−ı pz/2 ∂ k /∂z k G(s, z/2) belongs to K +k with norm uniform in s and p. Applying Proposition 4.2, we can then obtain finite bounds on the integral in (4.12) as → 0, so |I ( , p)| ≤ c (1 + p)−n for small ,
(4.14)
with a constant c depending on G and g. This is a bound of the required form. 5. Products Our next aim is to describe products of quantum fields that are of interest to us, and derive an operator product expansion for them. Specifically, we are interested in the products of two quantum fields φ, φ , displaced to different points t, t on the time axis; this product then exists as a distribution in the difference variable s = t − t . In addition, we wish to multiply this distribution with a c-number distributional kernel in t − t , and also consider sums of such expressions. The operator product expansion we use is derived by means of techniques described in [4]; however, we need to generalise the construction both to include the weighting factors and also to obtain more detailed estimates on OPEs at timelike-separated points. We can formally describe the products of interest as elements of the algebraic tensor product space prod := K ⊗ C ∞ ()∗ ⊗ C ∞ ()∗ . Any element ∈ prod has the form of a finite sum, K j ⊗ φ j ⊗ φ j , K j ∈ K, φ j , φ j ∈ C ∞ ()∗ . (5.1) = j
Quantum Inequalities from Operator Product Expansions
775
For > 0, we set prod = K ⊗C ()∗ ⊗C ()∗ ⊂ prod ; clearly, prod = ∪ >0 prod . Further we consider the subspace prod,loc = K ⊗ FH ⊗ FH ⊂ prod , the space of products of pointlike fields. To each product ∈ prod , we can associate a distribution T , heuristically given by t + t ∗ t + t T (t − t )U = K j (t − t − ı0)φ j (t)φ j (t ). (5.2) U 2 2 j
We shall first discuss in which sense these product distributions exist, before deriving an operator product expansion for them, in the case where φ j and φ j are local fields. Then we will introduce certain convolutions of these distributions with test functions, generalize the OPE for them, and single out a minimal set of composite fields that will be of use to us. 5.1. Operator products. Before considering our operator products, let us first define the set of distributions of interest. Definition 5.1. A C ∞()∗-valued distribution is a linear map T : D(R) → C ∞ ()∗ such that there exist constants > 0 and m ∈ N0 , and, for each d > 0, a constant cd , with the property that ∀ f ∈ D(−d, d) :
T ( f )(− ) ≤ cd f d,m .
Equivalently, we might say that T D(−d, d) extends to a map of W0m,1 (−d, d) to (− ) C ()∗ , with finite norm T d,m ≤ cd . In more standard terms, T might be called a distribution of finite order, but since we will not use other distributions in this context, we drop the extra qualifier. As before, we shall denote these distributions using their formal kernels: T ( f ) = d x T (x) f (x). Their expectation values σ (T (x)), for fixed σ ∈ C (), are then distributions in D(R) in the usual sense. We shall call a C ∞ ()∗ -valued distribution skew-hermitian if T (x)∗ = T (−x). We shall now clarify in which precise sense the distributions T in Eq. (5.2) exist. Proposition 5.2. Let > 0. To each = j K j ⊗ φ j ⊗ φ j ∈ prod , there exists a unique C ∞ ()∗ -valued distribution T such that for any σ ∈ ∪ E>0 (E), σ (T ( f )) = ds f (s )K j (s − ı0)σ φ j (s /2 − ı0)φ j (−s /2 + ı0) . j
The map → T is linear. Further, there is a constant c > 0 such that for any d ≤ 1, (−5 −1) K j (− ) φ j (− ) φ j (− ) . T d,3 +2 ≤ c d −(3 +2) j
Proof. Without loss of generality, we can assume that is of the form = K ⊗ φ ⊗ φ . Let σ ∈ (E), where E > 0 is fixed for the moment. Then, due to the spectrum condition, the distribution σ (φ(s /2)φ (−s /2)) is indeed the boundary value of an analytic function, namely of F(s − ıs ) := σ e(s +ıs )H/2 φ e−(s +ıs )H φ e(s +ıs )H/2 , s > 0. (5.3)
776
H. Bostelmann, C. J. Fewster
This function fulfils the bounds
|F(s − ıs )| ≤ σ e Es (1 + E)2 φ(− ) φ (− ) sup e−λs (1 + λ)2 (− )
≤ σ φ
(− )
φ
λ>0 2 (1+E)s
(1 + E) e
(2 )2 (s )−2 .
(5.4)
So F is regular at the boundary in the sense of Definition 4.1. Rescaling its argument, we explicitly have F(
z )(−2 ) ≤ cσ φ(− ) φ (− ) (1 + E)4 , 1+ E
(5.5)
where the constant c depends on only. The distributional product K (s − ı0)F(s − ı0) therefore exists. Rescaling also K , and applying Proposition 4.2 and Eq. (4.9), we obtain for any g ∈ D(−d, d) and with another constant c , ds g(s )K ( s − ı0 )F( s − ı0 ) 1+ E 1+ E ≤ c σ K (− ) φ(− ) φ (− ) (1 + E)5 (1 + d −3 −2 )gd,3 +2 .
(5.6)
Now let f ∈ D(−d, d), d ≤ 1. We set g(s ) = (1 + E)−1 f (s /(1 + E)) ∈ D(−(1 + E)d, (1+ E)d) and obtain using Eq. (5.6) with (1+ E)d in place of d, ds f (s )K (s − ı0)F(s − ı0) ≤ c σ K (− ) φ(− ) φ (− ) (1 + E)5 d −3 −2 f d,3 +2 .
(5.7)
This serves to define T ( f ) on (E) for any E. Using [55, Lemma 2.6], we can extend this linear form to C 5 +1 (), and obtain another constant c such that (−5 −1)
T d,3 +2 ≤ c K (− ) φ(− ) φ (− ) d −3 −2 .
(5.8)
The extension is unique by density. It is also clear by construction that T ( f ) is linear in f and in , i.e. multilinear in φ, φ , and K . Then, the estimate (5.8) shows that T is a C ∞ ()∗ -valued distribution in the sense of Def. 5.1.
5.2. Product expansions. We now prove the operator product expansion for a product of pointlike fields, ∈ prod,loc , in the following form. Theorem 5.3. Let ∈ prod,loc , and let α ≥ 0. There exist > 0, m ∈ N, γ ≥ 0, and a hermitian projector pγ : C ∞ ()∗ → γ onto γ such that (− ) T − pγ T d,m = o(d α ) as d → 0.
This is a variant of [4, Theorem 3.2]. Note that the approximation emerges into a more familiar form of operator product expansion if pγ is written in a basis.
Quantum Inequalities from Operator Product Expansions
777
Proof. Again, we can assume = K ⊗ φ ⊗ φ ∈ prod for some > 0, where now φ, φ ∈ FH . Further, after possibly increasing , we choose k > 0 and approximating sequences Ar , Ar for φ, φ as in Theorem 3.2. Set Br = K ⊗ Ar ⊗ Ar . We define m := 3 + 2 and γ := (2k + + 3)(α + m + 1), and choose a hermitian projector pγ onto ˆ γ . Now we estimate for an as yet unspecified , ˆ (− )
ˆ (− )
ˆ (− )
T − pγ T d,m ≤ T − TBr d,m + TBr − pγ TBr d,m ˆ ˆ
ˆ
) + pγ (− , ) T(−Br ) (− d,m .
(5.9)
ˆ ˆ
Here pγ (− , ) is a constant independent of r and d, finite if ˆ is large. We will show ˆ below that for large , ˆ (− )
T(−Br ) d,m = O(r d −m ), ˆ
) −2k− −2 − −2 TBr − pγ TBr (− d (r + d)γ ). d,m = O(r
(5.10) (5.11)
Setting r (d) = d α+m+1 , and using γ = (2k + + 3)(α + m + 1), both terms above are of order O(d α+1 ), from which the theorem follows. To show Eq. (5.10), we write − Br = K ⊗ (φ − Ar ) ⊗ φ + K ⊗ Ar ⊗ (φ − Ar ).
(5.12)
For the first summand, we estimate by Proposition 5.2: (−5 −1)
TK ⊗(φ−Ar )⊗φ d,m
≤ O(d −m ) K (− ) φ − Ar (− ) φ (− ) = O(r d −m ), (5.13)
as proposed. The second summand of Eq. (5.12) has a similar estimate, which combined gives Eq. (5.10). For Eq. (5.11), we use the short-distance approximation of Theorem 3.3 on the operator ArP (s ) := Ar (s /2)Ar (−s /2) ∈ A(Or +d ), where |s | ≤ d, and on its derivatives in s . Using the estimates on the derivatives of Ar (t) and Ar (t) provided by Theorem 3.2, ˆ this entails that for large , n ˆ d (− ) P P A (s ) − p A (s ) = O((r + d)γ )O(r −2k−n ). γ r r (ds )n
(5.14)
Now we compute TBr − pγ TBr , first on a fixed test function f ∈ D(−d, d), d ≤ 1, and on a fixed functional σ ∈ C ∞ (). By Prop. 5.2, we have σ (TBr ( f ) − pγ TBr ( f )) = ds h(s )K (s − ı0), where h(s ) = f (s )g(s ), g(s ) = σ ArP (s ) − pγ ArP (s ) . (5.15) (Note that here h is smooth, the only divergent factor is K . Therefore, also, sharp energybounds of σ do not play a role.) Using Proposition 4.2, it follows that |σ (TBr ( f ) − pγ TBr ( f ))| ≤ cK (− ) d − −2 hd, +2
(5.16)
778
H. Bostelmann, C. J. Fewster
with a numerical constant c. For the Sobolev norm of h, we can derive the following estimate by the Leibniz formula: hd, +2 = f gd, +2 ≤ 2 +2 f d, +2
max d n
0≤n≤ +2
sup |g (n) (t)|.
(5.17)
t∈[−d,d]
The derivatives of g can be estimated by Eq. (5.14). For t ∈ [−d, d] one has ˆ
|g (n) (t)| ≤ σ ( ) O((r + d)γ )O(r −2k−n ),
(5.18)
where the O(. . .) estimates are uniform in σ . Combining Eqs. (5.16)–(5.18), we obtain ˆ
) − −2 TBr − pγ TBr (− )O((r + d)γ )O(r −2k− −2 ), d, +2 ≤ O(d
(5.19)
which gives Eq. (5.11). The bounds established are certainly not strict, in particular regarding the value of γ (i.e., the number of approximation terms needed in the OPE). They might be improved at the price of extra computational effort, but this is not relevant for our purposes. Note however that the kernels K introduce an extra divergence that might make more OPE terms necessary than in the “ordinary” OPE version with K = 1. 5.3. Convolutions. In order to establish the existence of quantum inequalities, we need to analyse distributions evaluated on certain convolutions of test functions, similar to Eqs. (2.7)–(2.9) in the free field case. Let us define them, and establish their welldefinedness. We remind the reader of the abbreviation g1 g2 (s, s ) = g1 (s +s /2)g2 (s − s /2), and of the notion of skew-hermitian C ∞ ()∗ -valued distributions, which fulfil T (s )∗ = T (−s ). Lemma 5.4. Let T be a C ∞ ()∗ -valued distribution. Then the bilinear map κ0 [T ] : D(R) × D(R) → D(R, C ∞ ()∗ ), κ0 [T ](g1 , g2 )(s) = ds g1 g2 (s, s )T (s ) is well-defined; indeed, if g1 , g2 ∈ D(−d, d), then supp κ0 [T ](g1 , g2 ) ⊂ (−d, d). Further, κ[T ] : D(R) × D(R) → C ∞ ()∗ , κ[T ](g1 , g2 ) = ds U (s) (κ0 [T ](g1 , g2 )(s)) U (s)∗ is well-defined as a weak integral. Both κ0 [T ] and κ[T ] are linear in T . If T is skewhermitian, then κ[T ](g, ¯ g) is hermitian for arbitrary g ∈ D(R). For any m ∈ N and d > 0, one has the estimate (− )
(− )
κ[T ]d,m ≤ 2m+1 T 2d,m . The Sobolev norms of the bilinear maps are understood here with respect to a product of identical Sobolev norms on the two arguments.
Quantum Inequalities from Operator Product Expansions
779
Proof. First, κ0 [T ](g1 , g2 )(s) is well-defined since g1 g2 (s, · ) lies in D(−2d, 2d) for each fixed s; and it is (weakly) smooth in s since s → g1 g2 (s, · ) is smooth in the D(R) topology. The support properties are clear. Further, one sees that (− )
κ0 [T ](g1 , g2 )(s)(− ) ≤ T 2d,m g1 g2 (s, · )2d,m , which is locally bounded in s. Therefore, for each σ ∈ C ∞ (), the map R → C, s → σ U (s) κ0 [T ](g1 , g2 )(s) U (s)∗
(5.20)
(5.21)
is continuous. Hence κ[T ] is well-defined as a weak integral. Using the Leibniz rule and a change of variables, one finds ds (g1 g2 )(s, ·)2d,m ≤
m n n (r ) (n−r ) d r g1 1 d n−r g2 1 r n=0 r =0
≤ (2m+1 − 1)g1 d,m g2 d,m .
(5.22)
Together with Eq. (5.20), this yields the estimate (− ) (− ) κ[T ]d,m ≤ 2m+1 T 2d,m ,
(5.23)
as proposed. Also, it is clear in matrix elements that both κ0 [T ](g) and κ[T ](g) are linear in T . If T is skew-hermitian, one uses the identity g¯ g(s, s ) = gg(s, ¯ −s ) to conclude ∗ ∗ κ0 [T ](g, ¯ g)(s) = κ0 [T ](g, ¯ g)(s) and, in consequence, κ[T ](g, ¯ g) = κ[T ](g, ¯ g). The estimates above show that our operator product expansion for T , as established in Theorem 5.3, can be transferred to κ[T ]. This is in fact the form of OPE we shall use for establishing quantum inequalities. Corollary 5.5. Let ∈ prod,loc , and let α ≥ 0. There exist > 0, m ∈ N, γ ≥ 0, and a hermitian projector pγ : C ∞ ()∗ → γ onto γ such that (− )
κ[T − pγ T ]d,m = o(d α ) as d → 0. 5.4. Minimal approximating projectors. The operator product expansion allows us to approximate a given product with a finite number of composite fields. It is important for our applications to choose the minimal number of composite fields needed, so that none of the approximation terms can be considered “redundant”. Let us introduce that notion of approximation by finitely many terms more abstractly. This is similar, but not identical to the analysis of normal products in [4, Sect. IV]. Definition 5.6. Let ∈ prod,loc , and α ≥ 0. A hermitian projector2 p in C ∞ ()∗ with finite-dimensional image in FH is called α-approximating for if there are constants > 0 and m ∈ N such that (− )
κ[T − pT ]d,m = o(d α ) as d → 0. 2 Projectors in this space will always be assumed as continuous.
780
H. Bostelmann, C. J. Fewster
The operator product expansion in Corollary 5.5 tells us that for any given α, we can choose γ large enough such that any hermitian projector p onto γ is α-approximating for . However, this is in a way an “upper estimate” to the OPE, since γ may contain elements that are not actually needed for approximating the given product. We will therefore minimize the approximating projector in a well-defined sense. This is done as follows. On the family of all α-approximating projectors for a given product , we introduce a partial order by p1 ≤ p2
:⇔
(img p1 ⊂ img p2 ) ∧ (ker p1 ⊃ ker p2 ).
(5.24)
Minimal elements with respect to this partially ordered set will be called minimal α-approximating projectors. By dimensional arguments, any decreasing sequence in the set must eventually become constant; so minimal elements certainly exist, and can be constructed below each given α-approximating projector. However, there seems to be no reason why they should be unique. This is in contrast to the situation for normal product spaces [4, Sect. IV], where the approximation property depends on img p only, i.e., any other projector onto the same space would also be α-approximating. In that case, one finds a unique minimal approximating space of fields. In our situation, these stronger results do not seem to follow, the main difficulty being that the convolution κ[ · ] does not commute with projectors. This turns out not to be a problem however: each minimal α-approximating projector will give us a nontrivial quantum inequality. Let us summarize the main point of the above discussion: Proposition 5.7. Let α ≥ 0 and ∈ prod,loc . There exists at least one minimal α-approximating projector p for . 6. Quantum Inequalities We now establish quantum inequalities as a consequence of the operator product expansion above, and prove that they are nontrivial as discussed in Sect. 2. 6.1. Existence of inequalities. In order to establish inequalities, we define a set of products pos ⊂ prod,loc which are “classically positive”, namely a finite sum of absolute squares with positive-type coefficients:
pos := (6.1) K j ⊗ φ ∗j ⊗ φ j K j ∈ K+ , φ j ∈ FH . j
For any ∈ pos , the distribution T is then skew-hermitian. (One verifies this in matrix elements by the integral formula in Prop. 5.2, using the relation K j (z) = K j (−¯z ) for the positive-type kernels K j .) Products from pos now give rise to quantum inequalities. To formulate these, we use the abbreviation R := (1 + H )−1 . Theorem 6.1. Let ∈ pos and α ≥ 0. Let p be an α-approximating projector for . There exist > 0, m ∈ N, and a function : R+ → R+ of order (d) = o(d α ) such that the following inequality between bounded operators holds. ∀d > 0, g ∈ D(−d, d) :
R κ[ pT ](g, ¯ g)R ≥ − (d)(gd,m )2 1.
Quantum Inequalities from Operator Product Expansions
781
Proof. By Def. 5.6, there exist , m and (d) = o(d α ) such that κ[ pT − T ](g, ¯ g)(− ) ≤ (d) (gd,m )2 .
∀d > 0, g ∈ D(−d, d) :
(6.2)
¯ g) is guaranteed to be hermitian Note here that, since T is skew-hermitian, κ[T ](g, by Lemma 5.4. Since p is hermitian, the same is true for κ[ pT ](g, ¯ g). The expectation values of these expressions in positive functionals are therefore real. Thus, for any ρ ∈ ∪ E (E), ρ ≥ 0, we obtain from Eq. (6.2), ρ(κ[ pT ](g, ¯ g)) − ρ(κ[T ](g, ¯ g)) ≥ −ρ( ) (d) (gd,m )2 . (6.3) Now ∈ pos is of the form = j K j ⊗ φ ∗j ⊗ φ j . Due to energy-boundedness of ρ, we have by Prop. 5.2 and Lemma 5.4, ρ(κ[T ](g, ds ds g¯ g(s, s ) K j (s − ı0) ¯ g)) = j
Here
×ρ U (s) φ ∗j (s /2) φ j (−s /2) U (s)∗ .
(6.4)
G j (s, s − ı0) := ρ U (s)φ ∗j (s /2)φ j (−s /2)U (s)∗ = ρ φ ∗j (s + s /2)φ j (s − s /2)
(6.5)
are positive-definite distributions, as they give ρ(φ j (g)∗ φ j (g)) when integrated with g¯ g. Also, a similar estimate as in Eq. (5.4) shows that s → G j (s, · ) is uniformly bounded in · (− ) , and also continuous since the energy-bounded state ρ is analytic for U (s). Thus the products of G j with the positive-type kernels K j are positive-definite as well (Prop. 4.4). Therefore, the expression in Eq. (6.4) is non-negative. Setting ρˆ = ρ(R − · R − ), we can thus reduce Eq. (6.3) to ρ(R ˆ κ[ pT ](g, ¯ g) R ) ≥ − (d) (gd,m )2 ρ(1). ˆ
(6.6)
¯ g)R can be extended to a bounded operator by Eq. (6.2). Since ρˆ Here R κ[ pT ](g, can be chosen from a dense subset in the set of all positive functionals, the theorem now follows. The connection of the theorem with more usual forms of quantum inequalities becomes clear when we write the projector p in a basis: p=
n
σ j ( · )φ j , where σ j ∈ C ∞ (), φ j ∈ FH , σ j (φk ) = δ jk .
(6.7)
j=1
Here we choose φ j and σ j hermitian, which is possible since p is hermitian. Then, the inequality in the theorem can be rewritten as n j=1
R φ j ( f j )R ≥ − (d) (gd,m )2 1,
(6.8)
782
H. Bostelmann, C. J. Fewster
where the functions f 1 , . . . , f n are given by f j (s) = ds g¯ g(s, s ) σ j (T (s )).
(6.9)
These f j are actually of compact support, namely supp f j ⊂ (−d, d) if g ∈ D(−d, d), see Lemma 5.4. They are also smooth, since s → g¯ g(s, · ) is differentiable in the S-topology; so they are indeed proper test functions in D(R). Further, the f j are realvalued, which follows from hermiticity of σ j and skew-hermiticity of T . The inequality (6.8) is of an asymptotic nature, inasmuch as only the asymptotic behaviour of the remainder, (d) = o(d α ), is known. For the sake of concreteness, we may choose a fixed test function g ∈ D(−1, 1), and define a family of scaled functions gd (t) = d −1 g(t/d). For these, gd d,m is independent of d, so that the right-hand side of Eq. (6.8) simplifies; the inequality is then valid as the parameter d of the family goes to 0. While the functions f j are real-valued, they are not guaranteed to be pointwise positive, in contrast to the free field situation [11]. To see this, we note that (6.9) has a strong analogy to Weyl quantization. Letting C˜ j be the Fourier transform of C j (s ) = σ j (T (s )), one has dp ˜ dp ˜ f j (s) = C j ( p) ds eı ps g¯ g(s, s ) = C j ( p)Wg (s, p), (6.10) 2π 2π where Wg is the Wigner function associated with the “state” g, Wg (s, p) = ds eı ps g(s + s /2)g(s − s /2).
(6.11)
Now the Wigner function cannot be pointwise positive for compactly supported g [1], so positivity of f j can only be expected in special situations; see e.g., Prop. 7.3. Note that Eq. (6.8) is a far-reaching generalization of the usual inequalities for squares of fields in free field theory. In particular, the estimate will in general not be restricted to two fields, such as the Wick square and the identity in Eq. (2.11), but will involve a possibly large number of fields smeared with different sampling functions. One of the φ j will typically be the identity operator, and another φ j will typically be a normal product in the sense of Zimmermann [4,49]. This term will usually be distinguished as a highest-order field, relating e.g. to scaling dimensions. But there seems to be no guarantee that such a highest-order field exists uniquely, and even less that only two fields φ1 , φ2 appear in the inequality. Compared with the usual free-field situation, we also encounter a remainder term (d) which seems unavoidable in this context, but is of negligible order compared with the contributions of the field operators, as we shall see below. 6.2. Nontriviality. While Thm. 6.1 asserts that our construction yields a large variety of valid quantum inequalities, there remains the concern that they could be trivial in the sense that the lower bound could also serve as an upper bound, cf. [24]. In particular, an inequality for a bounded operator A of the form A ≥ −A1 would be considered trivial. Since the exponent in Eq. (6.8) is so large that all R φ j R are bounded, we might well encounter this situation: the left-hand side of Eq. (6.8) might be dominated in norm by the remainder (d). More generally, since Thm. 6.1 puts no further restrictions
Quantum Inequalities from Operator Product Expansions
783
on the projector p, it might also be possible that pT contains single “redundant terms” that are individually dominated by (d), and are thus essentially irrelevant. We shall show now that if the approximating projector is chosen minimal, these problems do not occur, and in this sense the inequality is nontrivial. Theorem 6.2. Let ∈ pos and α ≥ 0. Let p be a minimal α-approximating projector for . Let V := img p. For sufficiently large > 0 and m ∈ N, and for any hermitian projector q : V → V , q = 0, it holds that d −α
κ[qpT ](g, ¯ g)(− ) → 0 as d → 0. (gd,m )2 g∈D (−d,d) sup
Proof. Suppose that m, and a hermitian projector q : V → V are given such that d −α
sup
g∈D (−d,d)
(gd,m )−2 κ[qpT ](g, ¯ g)(− ) → 0 as d → 0.
(6.12)
We will show q = 0. First, we can use the polarization identity for the quadratic form κ[qpT ](g1 , g2 ) in order to show (− )
d −α κ[qpT ]d,m → 0.
(6.13)
The triangle inequality then yields (− ) d −α κ[(1 − q) pT − T ]d,m (− ) (− ) ≤ d −α κ[ pT − T ]d,m + d −α κ[qpT ]d,m → 0,
(6.14)
since p is α-approximating; we suppose here that m, are sufficiently large. Now (6.14) shows that (1 − q) p is also α-approximating for . It is clear that (1 − q) p ≤ p. Since however p is minimal, this implies (1 − q) p = p. Thus q = 0. Again, let us illustrate the content of the theorem by passing to a basis representation of p, as in Eq. (6.7). For the case q = 1V , the theorem precisely shows that the left-hand side of Eq. (6.8) does not vanish in norm as fast as (d) = o(d α ). Further, choose q specifically as q = σk φk with fixed k. Then one obtains κ[qpT ](g, ¯ g) = φk ( f k ), with f k as in Eq. (6.9). Thus, Thm. 6.2 provides us with a null sequence (di )i∈N , a constant c > 0, and a sequence of functions g (i) ∈ D(−di , di ) with g (i) di ,m = 1 such that (i)
R φk ( f k ) R ≥ c (di )α for all i ∈ N, where (i)
f k (s) =
ds g¯ (i) g (i) (s, s ) σk (T (s )).
(6.15)
(6.16)
So the field φk in the inequality (6.8) gives a contribution that is large compared to the remainder (d). Theorem 6.2, in full generality, shows that this conclusion is true independent of the choice of basis. We have argued in Prop. 5.7 that minimal α-approximating projectors p exist for any product , and any α ≥ 0. So we always obtain nontrivial quantum inequalities in the sense above. One might suspect here that the minimization of the approximating projector p could lead to p = 0, which might again be seen as trivial. While this is not the case even in a simple free field example, we shall give a general argument that shows that p = 0 cannot occur, under a mild extra assumption.
784
H. Bostelmann, C. J. Fewster
Theorem 6.3. Let α ≥ 0, and ∈ pos \{0}. Suppose that the vacuum vector is separating for the smeared fields φ( f ), with φ ∈ FH and f ∈ D(R). If p is an α-approximating projector for , then p = 0. We note that the condition of a separating vacuum vector is indeed a rather weak one. It would suffice, for example, that there exists a wedge region W such that is cyclic for A(W). Proof. Suppose that α and are given such that p = 0 is α-approximating for . We will show = 0. To that end, we choose and m sufficiently large, and pick a fixed positive test function g ∈ D(−1, 1). Then gd := d −1 g(d −1 · ) lies in D(−d, d), and gd d,m = g1,m . Employing Def. 5.6, we obtain κ[T ](g¯d , gd )(− ) → 0 as d → 0.
(6.17)
Evaluating the convolution integral in the vacuum state ω yields due to translation invariance, (6.18) ds ds g¯d gd (s, s ) ω(T (s )) → 0 as d → 0. As argued in the proof of Thm. 6.1, the distribution ω(T (s )) is of positive type. Hence it is the Fourier transform of a polynomially bounded positive measure µ. With this information, we can rewrite Eq. (6.18) as dµ( p) |g˜d ( p)|2 → 0 as d → 0. (6.19) 2 > 0 locally uniformly. Since µ However, as d → 0, we have |g˜d ( p)|2 → |g(0)| ˜ is positive, we can conclude here that µ is the zero measure. So ω(T (s )) = 0 as a distribution. Using ∈ pos , we have a representation
0 = ω(T (s )) =
n
K j (s − ı0) ω(φ ∗j e−ı(s −ı0)H φ j ) with K j ∈ K+ , φ j ∈ FH .
j=1
(6.20) Since all summands are of positive type, each of them must vanish individually; and clearly, also their analytic continuations must vanish. Thus, for any j, we have either K j = 0 or ω(φ ∗j U (−s )φ j ) = 0. But the latter implies φ j ( f ) = 0 for any f of compact support; thus φ j ( f ) = 0 by assumption, and ultimately φ j = 0 by passing to a delta sequence. In total, this means = 0. One might also be concerned that p might project only onto multiples of the identity. Again, this does not occur in the simple example of the Wick square of the free field, as discussed in Sect. 2. In general, we conjecture, but have not proved, that in this case all fields appearing in the product must be multiples of the identity. At the very least, one can show that the projector may be taken to be of the form p = ω( · )1, where ω is the vacuum state. If this p is indeed α-approximating for , the normal product of can be defined by point splitting, and vanishes identically. So this does not seem to be a case of great interest.
Quantum Inequalities from Operator Product Expansions
785
6.3. Mesoscopic bounds. The inequalities derived above involve a remainder term that vanishes in the small distance limit. Here, we discuss how the remainder can be reduced for test functions of fixed supports, essentially by forming a Riemann integral of the bounds at short distance. Let χ ∈ D(−1, 1) and f ∈ D(−d, d) be fixed nonnegative functions. We set χλ (s) = λ−1 χ (s/λ) for λ ∈ (0, 1]. As in Thm. 6.1, we suppose p to be an α-approximating projector for ∈ pos , with α ≥ 0. The basic inequality of Thm. 6.1, applied to χλ , entails R κ[ pT ](χλ , χλ )R ≥ − (λ)(χλ λ,m )2 1 = − (λ)(χ 1,m )2 1
(6.21)
for suitable > 0 and m ∈ N, where (λ) = o(λα ). Applying a time-translation through λk, multiplying by λ f (λk) and summing, we find λ f (λk)U (λk)κ[ pT ](χλ , χλ )U (λk)∗ ≥ − (λ)(χ 1,m )2 λ f (λk)R −2 k∈Z
k∈Z
≥ − (λ)(χ 1,m )2 ( f 1 + λ f 1 )R −2 . (6.22) Passing to a basis representation, we may rewrite this inequality in the form n
φ j (F j,λ ) ≥ −2 (λ)(χ 1,m )2 f d,1 R −2
(6.23)
j=1
for λ ≤ d where F j,λ (s) =
λ f (λk)
ds σ j (T (s ))χλ χλ (s − λk, s ).
(6.24)
k∈Z
Owing to the support properties of χλ , at most two terms contribute to the sum on k for each fixed s; moreover, F j,λ ∈ D(−d, d). In any fixed state in C ∞ () the expectation value of the right-hand side of (6.23) can be made arbitrarily small by reducing λ, while the behaviour of the terms on the right-hand side is determined by the asymptotic behaviour of the F j,λ , regarded as compactly supported distributions. In the unlikely event that each F j,λ converged to a limit in the weak-∗ topology on E (R), we would have established a quantum inequality without remainder term. It may be useful to give two examples. If the OPE coefficient σ j (T (s )) is smooth, then convergence does occur, with F j,λ → σ j (T (0))(χ 1 )2 f
in E (R) as λ → 0.
(6.25)
(To see this, one integrates against u(s) and observes that the k th summand is subject to only an O(λ2 ) error if u(s)σ j (T (s )) is replaced by u(λk)σ j (T (0)); as there are at most O(λ−1 ) nonzero summands the result follows by a simple calculation.) On the other hand, if σ j (T (s )) = (ıπ )−1 /(s − ı0), we find λF j,λ → (χ 2 )2 f
in E (R) as λ → 0.
(6.26)
(Note that it is the L 2 -norm that appears here, in contrast to the first example.) In general, therefore, it cannot be expected that all of the F j,λ converge as λ → 0. Nonetheless, as in the second example, its leading order behaviour in λ can be identified as follows.
786
H. Bostelmann, C. J. Fewster
Proposition 6.4. Let q be the order of the germ of σ j (T (s )) at s = 0 and define (6.27) η j (λ) = ds σ j (T (s ))(χλ ∗ χˆ λ )(s ), where χ(s ˆ ) = χ (−s ). If λ−q η j (λ)−1 = o(1) as λ → 0 then in E (R) as λ → 0.
F j,λ /η j (λ) → f
(6.28)
In particular, this is satisfied if σ j (T (s )) has a scaling limit of degree β < 0 and q = −1 − β. Here, the order of the germ of σ (T (s )) q ∈ N0 for which at s = 0 is the minimal q there are λ0 > 0 and C > 0 such that | ds σ (T (s ))u(s )| ≤ C r =0 sup |u (r ) | for all u ∈ D(−λ0 , λ0 ). The notion of scaling limit is taken from [52]: namely, the scaling limit exists if there exists a monotone positive function N (λ) for which (6.29) N (λ) ds σ j (T (s ))u λ (s ) → S(u)
for all u ∈ D(R), with a nonzero limit for at least one u. Under these circumstances, S is a homogeneous distribution, i.e., S(u λ ) = λβ S(u), with degree β ∈ R determined by lim
λ →0
N (λ ) = λβ . N (λλ )
(6.30)
(Our definition of the degree coincides with that of [58, Ch. I Sect. 1.6.] and differs from [52].) If β < 0, for example, the distribution (s − i0)β (log s − i0)γ has a scaling limit of degree β and (germ) order −1 − β, and therefore meets the criteria stated. Proof of Prop. 6.4. We choose λ0 ∈ (0, 1] sufficiently small that σ (T (s )) has order q on (−2λ0 , 2λ0 ), and assume henceforth that 0 < λ < λ0 . As in the second example above, we integrate F j,λ against u ∈ E(R) and approximate u(s) by u(λk) in the k th summand, to obtain λ f (λk) ds χλ χλ (s, s )u(s + λk) ds F j,λ (s)u(s) = ds σ j (T (s )) = η j (λ)
k∈Z
λ f (λk)u(λk) + R j,λ ,
(6.31)
k∈Z
where R j,λ =
ds σ j (T (s ))
λ f (λk)
ds χλ χλ (s, s )[u(s + λk) − u(λk)].
k∈Z
(6.32) Now R j,λ is, at worst, of order O(λ−q ) as λ → 0, as is easily seen using the estimate C sup ds σ j (T (s ))χλ χλ (s, s ) ≤ q+2 ; (6.33) λ s
Quantum Inequalities from Operator Product Expansions
787
and the facts that (i) the sum contains at most O(λ−1 ) nonzero terms; (ii) the s-integral extends over the region [−λ, λ]. This establishes ds F j,λ (s)u(s) = η j (λ) (6.34) ds f (s)u(s) + O(λ) + O(λ−q ) as λ → 0, from which (6.28) follows immediately. Now suppose that σ j (T (s )) has a scaling limit of degree β < 0. It is easy to see that (6.29) implies N (λ)η j (λ) → S(χ ∗ χ). ˆ The spectrum condition entails that S = C(ı(· − ı0))β , where the nonzero constant C is real owing to hermiticity (cf. the proof of Prop. 7.2 below). As β < 0, we may verify directly that S(χ ∗ χ) ˆ = 0, that N (λ) is necessarily monotone decreasing and vanishing as λ → 0. Thus η j (λ) → ±∞ depending on the sign of C. Moreover, Eqs. (6.29) and (6.30) entail (λ )q η j (λ ) = λ−β−q . λ →0 (λλ )q η j (λλ ) lim
(6.35)
By hypothesis, σ j (T (s )) has order q = −1 − β (as does S). Thus −β − q > 0 and we deduce that λ−q η j (λ)−1 → 0 as λ → 0. The significance of this result becomes clear in the situation where one of the composite fields, say φ1 , is identified as a field of particular interest, e.g., the normal product. By hermiticity of the projection p, η1 is real-valued; the hypothesis of Prop. 6.4 requires that |η1 (λ)| → ∞ as λ → 0. If, in fact, η1 (λ) → +∞, we may divide the quantum inequality (6.23) by this factor to obtain a bound 1 2 (λ) (χ 1,m )2 f d,1 R −2 φ j (F j,λ ) ≥ − η1 (λ) η1 (λ) n
φ1 (F1,λ /η1 (λ)) +
(6.36)
j=2
for λ < d. (If η1 (λ) → −∞ we simply reverse the sign of φ1 and hence σ1 (T (s )) and η1 to obtain the same result; the possibility that η1 oscillates in sign as λ → 0 can be excluded if the scaling limit exists.) In this form, it is clear that the remainder term may be diminished by reducing λ, at the possible cost of increasing the magnitude of the terms in composite fields with j ≥ 2 (if η j (λ) grows more rapidly than η1 (λ)). Moreover, the expectation value of the first term tends to that of φ1 ( f ) as λ → 0 for any state in C ∞ (). Further progress is only possible with more detailed information regarding the (germs of the) OPE coefficient distributions. Nonetheless, we expect that the results presented here will be of use in the context of particular models. 7. Scaling Limits and Dilation Covariance For a concrete interpretation of our quantum inequalities, it is of particular interest to investigate the detailed structure of the sampling functions with which the composite fields are smeared, e.g. the functions f j in Eq. (6.9). For example, one is interested whether they are pointwise positive, or at least “mostly positive” in a well-defined sense. Of course, these properties depend crucially on the structure of the OPE coefficients involved, about which little is known in the general case. The most reasonable approach therefore seems to investigate those properties under more restrictions on the theory.
788
H. Bostelmann, C. J. Fewster
In the preceding section, our approach was to approximate a given sampling function with a Riemann sum; this relied on some assumptions on the behavior of the OPE coefficients in the small, and was tied to a choice of basis in the field spaces. In the following, we want to take a different approach: we investigate the structure of sampling functions in a restricted class of quantum field theories, namely in the presence of dilation symmetries. While for a realistic description of microphysics, one would not consider dilation covariant quantum field theories, this case is still important as an idealization at short scales. Namely, in the short-distance regime, quantum field theories should be approximated by a scaling limit theory, which indeed possesses a dilation symmetry. Let us briefly sketch how the scaling limit of quantum field theories fits into our context. It has been shown by Buchholz and Verch [59] that scaling limits can be formulated very naturally on the level of local algebras. Every quantum field theory possesses a scaling limit in this sense, although it might not be unique. The limit theory is, under a suitable choice of limit states, covariant under a strongly continuous unitary representation of the dilation group [55]. However, the structure of these dilation unitaries may be very intricate, acting on a nonseparable Hilbert space. (See also [60].) In [55], it was shown that this picture is compatible with the usual notion of field renormalization: if the original algebraic theory fulfils a slightly sharpened version of Def. 3.1, then the limit theory fulfils Def. 3.1 too; and pointlike fields in the original theory converge, under a multiplicative renormalization scheme, to pointlike fields in the limit theory. In a certain sense, the projectors pγ onto γ converge to corresponding (0) projectors pγ in the limit theory. Also, this scheme is compatible with products of pointlike fields and operator product expansions. Thus one can expect that the structures exhibited in Sect. 6 properly converge in the scaling limit, and yield quantum inequalities in the limit theory. Our aim here is neither to describe this passage to the limit theory in detail, nor to treat all possible cases of dilation group representations that may appear in the limit. Rather, we take the above as a motivation to investigate quantum inequalities in dilation covariant theories, and to show in certain simple cases that stricter classification results on the form of quantum inequalities can be achieved. In the remainder of this section, we will therefore assume that our theory A has a dilation symmetry; i.e., that there exists a strongly continuous unitary representation λ → U (λ) of the dilation group on H, which is compatible with the Poincaré group representation, and acts on the local algebras in the usual geometric way. The adjoint action of U (λ) can then be extended to C ∞ ()∗ , where we write δλ φ = U (λ)φU (λ)∗ in the weak sense. The spaces γ are invariant under δλ [3, Sect. IV]. We shall now consider the action of δλ on the structures considered so far, and introduce some definitions for convenience. Definition 7.1. A quadratic form φ ∈ C ∞ ()∗ is called dilation covariant if, with some β ∈ R, δλ φ = λβ φ for all λ > 0. A product ∈ prod is called dilation covariant if, with some β ∈ R, δλ T (s) = λβ T (λs) for all λ > 0, in the sense of distributions. A projector p in C ∞ ()∗ is called dilation covariant if δ1/λ ◦ p ◦ δλ A(O1 ) = pA(O1 ) for all 0 < λ ≤ 1, where O1 is the standard double cone of radius 1.
Quantum Inequalities from Operator Product Expansions
789
Note that the restriction to A(O1 ) in the definition of dilation covariant projectors is unavoidable if we want p to be norm-bounded on B(H). Namely, suppose that δ1/λ ◦ p ◦ δλ (A) = p(A) for all A ∈ B(H) and 0 < λ ≤ 1, and hence for all λ by the group relation. Since δλ acts as a norm isomorphism on B(H), norm-boundedness of p would lead to δλ being uniformly bounded on the finite dimensional space img p, both for λ → 0 and for λ → ∞, which would exclude that img p contains fields with nonzero scaling dimension. Dilation covariant products can easily be constructed, e.g. by choosing dilation covar iant fields φ1 , φ2 , and setting = (ı z)−β ⊗ φ1 ⊗ φ2 with some β ≥ 0. If and p are both dilation covariant, Def. 7.1 implies that δλ p T (s ) = λβ p T (λs ) for 0 < λ ≤ 1 and for s ∈ [−1, 1];
(7.1)
that is, the equation holds when evaluated on test functions with support in [−1, 1]. This follows by approximating T with sequences of bounded local operators, as in the proof of Thm. 5.3. We will now consider the form of quantum inequalities in our case, that is, investigate the structure of minimal approximating projectors p and their subprojectors. We shall restrict here to the simplest case, where one deals with one-dimensional subrepresentations of δλ . In this case, we can find a full classification of our quantum inequality terms. Proposition 7.2. Let ∈ pos be dilation covariant. Let p be a one-dimensional dilation covariant projector in C ∞ ()∗ . Then, there exist a dilation covariant field φ ∈ FH and β ∈ R such that pT (s ) = (ı(s − ı0))β φ on the interval (−1, 1). Proof. We choose φ ∈ FH and σ ∈ C ∞ () such that p = σ ( · )φ. Since σ (φ) = 1, and since φ can be approximated by bounded operators as in Thm. 3.2, we can find A ∈ A(O1 ) such that σ (A) = 1. Using that p is dilation covariant, we obtain σ (δλ A) δ1/λ φ = σ (A)φ = φ for all 0 < λ ≤ 1,
(7.2)
δλ φ = σ (δλ A)φ = c(λ)φ for all 0 < λ ≤ 1.
(7.3)
and thus
Here the C-valued function c(λ) is continuous in λ and fulfils c(1) = 1, c(λ)c(λ ) = c(λλ ) if λ, λ ∈ (0, 1]. This suffices to conclude that there exists a β1 ∈ C such that c(λ) = λβ1 for all 0 < λ ≤ 1.
(7.4)
Due to the group relation, we then obtain for all λ ∈ R+ , δλ φ = λβ1 φ.
(7.5)
Splitting φ = φ R + ıφ I into real and imaginary parts, we note that δλ preserves this splitting, which means that β1 must be real. So φ is dilation covariant. Inserting into Eq. (7.1), we arrive at σ (T (s )) = λβ2 −β1 σ (T (λs )) in the sense of D(−1, 1) ,
(7.6)
790
H. Bostelmann, C. J. Fewster
where β2 ∈ R is the exponent relating to . Using the right-hand side as a definition for |s | > 1, we can construct a homogeneous distribution3 D ∈ D(R) of degree β := β1 − β2 such that D(s ) = σ (T (s )) in the sense of D(−1, 1) .
(7.7)
The homogeneous distributions of one variable are however fully classified (cf. [58, Ch. I Sect. 3.11.]): they are of the form D(s ) = c+ (s + ı0)β + c− (s − ı0)β with c± ∈ C.
(7.8)
We can further restrict the possible form of D. Since σ can be approximated by energybounded functionals σ E , and σ E (T (s )) has an analytic continuation to the lower halfplane, the only singular direction (in the sense of wave front sets) of σ (T (s )) at 0 can be the positive half-line. Since the wave front set is determined locally, Eq. (7.7) entails that c+ = 0. Absorbing a factor ı −β c− into the field φ, we finally obtain β p T (s ) = ı(s − ı0) φ on the interval (−1, 1),
(7.9)
as proposed. Now in the above situation, we can easily describe the quantum inequality terms that arise. One finds for any g ∈ D(−1, 1), κ[ pT ](g, ¯ g) = φ( f ) with f (s) =
ds (ı(s − ı0))β g¯ g(s, s ).
(7.10)
This expression would not represent the entire quantum inequality, as approximating projectors will typically not be one-dimensional. Rather, (7.10) would represent one of the summands of the inequality in Eq. (6.8). In typical cases, one may expect that there exists a distinguished highest-order term in the operator product expansion, which corresponds to the “normal product” part of , and which is described by a one-dimensional dilation covariant projector as above. Note that Prop. 7.2 determines the field φ uniquely. In particular, for β ≤ 0, requiring the distributional factor to be of positive type fixes the phase factor of φ. While other conditions might be used to restrict this phase factor, such as demanding that φ be hermitian, the quantum inequalities give a stronger restriction that even fixes a ± sign in φ. In this sense, our quantum inequalities can be used to distinguish the normal square of a field from its negative; squares of fields retain certain aspects of positivity in the quantum case. Let us further investigate the structure of the smearing function f obtained in Eq. (7.10). We assume for a moment that g is real-valued, and thus g¯ g(s, s ) is symmetric in s . By a standard computation [58, Ch. I Sect. 3 Nr. 8], one obtains the 3 As mentioned in Sect. 6.3, alternative conditions that force a distribution in the scaling limit to be homogeneous are discussed in [52].
Quantum Inequalities from Operator Product Expansions
791
following simplified expressions in terms of convergent integrals: βπ ∞ β ds (s ) g g(s, s ) for β > −1, (7.11) f (s) = 2 cos 2 0 ∞ βπ ds (s )β g g(s, s ) f (s) = 2 cos 2 0 [(−β−1)/2] 1 ∂ 2k g g 2k =0 s − for β < −1, |β| ∈ 2N + 1, (7.12) s (2k)! (∂s )2k k=0 (−1)k π ∂ 2k g g for β = −2k − 1, k ∈ N0 . (7.13) f (s) = (2k)! (∂s )2k s =0 Using these explicit characterizations, we can directly investigate the positivity properties of the function f . For reasons of simple interpretation, it would be convenient if the f (s) are positive at each s. We can give some sufficient conditions to this end. Proposition 7.3. Let g ∈ D(R), β ∈ R, and f be given as in Eq. (7.10). If any of the following conditions is fulfilled, it follows that f (s) ≥ 0 for all s ∈ R. (i) −1 < β ≤ 1, and g(t) ≥ 0 for all t ∈ R. (ii) β = −1, and g is real-valued. (iii) −3 < β < −1, supp g is a connected interval I , and g is logarithmically concave within I . Proof. The case (i) follows immediately from Eq. (7.11). In case (ii) , we obtain f (s) = π g(s)2 from Eq. (7.13), which yields the result. For (iii) , observe that in this case Eq. (7.12) reads βπ ∞ β 2 g(s) f (s) = 2 cos ds (s ) − g(s + s /2)g(s − s /2) . (7.14) 2 0
Now the concavity of t → log g(t) precisely implies that g(s)2 ≥ g(s + s /2)g(s − s /2) for any s and s . The case β = −1 corresponds to the leading order of the OPE in the Wick square of a massless free field theory, as discussed in Sect. 2. Our main interest is therefore in the case where β is near −1, which might be expected in asymptotically free theories. This realm is covered in the above proposition. In models, it might be possible to exploit the choice of positive-type kernels K j in the definition of in order to arrive at precisely the case β = −1, so that the function f = g 2 has a simple interpretation. We do however not investigate this possibility in detail here. In more generality, for any β ≤ 0, we can at least state the following more qualitative result: since (ı(s − ı0))β is of positive type, one finds ds f (s) ≥ 0, (7.15) so f has at least a non-negative average, regardless of the choice of g. A bit more generally, one can deduce Gårding inequalities for f , similar to those familiar from quantum mechanics [61]: for suitable test functions χ , one has (7.16) χ (s) f (s)ds ≥ −cχ (g2 )2 . Thus positivity of the test function f is preserved at least in a generalized sense.
792
H. Bostelmann, C. J. Fewster
8. Conclusions and Outlook We have shown that quantum field theories obeying the microscopic phase space condition of [3] admit a large class of nontrivial quantum inequalities: to every classically positive expression, i.e., a sum of absolute squares, we find a combination of composite fields that is positive up to an error obeying defined estimates and vanishing in the short distance limit. The composite fields appearing in such QIs are smeared with test functions derived from OPE coefficients as well as a choice of test function g. In the free field case, these smearing functions bore a simple relationship to g, at least for the normal product; here, the relationship is less direct, although we have succeeded in classifying their structure under simplifying assumptions within dilation covariant theories. Our inequalities are primarily valid in the short-distance limit, when the support of the test functions shrinks to a point. However, we also discussed how to obtain inequalities for smearing functions with extended (mesoscopic) support, in which the remainder term can be reduced at the expense of increasing the contributions from other composite fields. To conclude we mention a number of open questions and avenues for further investigation. First, more progress can be made in understanding the sampling functions arising. For example, in the dilation covariant setting, one could also allow general finite-dimensional irreducible representations of the dilation group. Second, it would probably not be hard to generalise our bounds from smearing along a fixed timelike inertial curve to smearing along arbitrary smooth timelike curves in Minkowski space. The structure of inequalities is not expected to change significantly under this generalization. Third, one would also like to establish OPE-based quantum inequalities in curved spacetime. Here, the situation is complicated by the lack of a global Hamiltonian to specify scales of spaces of states and fields. A replacement for the topologies thus induced might be found in the detailed microlocal structure of n-point functions, for example, using wave-front sets modulo Sobolev regularity (see, e.g., [62]). An alternative approach would be to use the stress-energy tensor as the basis for estimates of high-energy behaviour. Hollands has recently established an OPE on curved spacetime for perturbatively constructed theories [63]; however, the generalization of the nonperturbative methods used here presently remains a challenging problem. Fourth, it would be desirable to obtain results that directly constrain the energy density of a quantum field theory, returning to the original motivation for quantum inequalities. One may heuristically expect from perturbation theory that the energy density in purely bosonic theories does arise from such a sum of squares (although a generalization would be needed to cater for theories with fermionic fields) and would therefore be amenable to our approach. However, more direct connections to the energy density are unknown at present; in fact, the very concept of energy density is not well established in a nonperturbative context in purely Minkowski space quantum field theory. More generally, no general nonperturbative version of the Noether theorem has been found to date. In the Wightman framework, only very few results about pointlike Noether currents are available [64,65], in particular an existence proof is missing. In the algebraic framework, partial results have been achieved [66] on the base of the so-called split property of the local algebras [67,68]. In effect, it is possible to construct “local” energy operators HO,Oˆ , which are associated with the observable algebra A(O) of a bounded region O and ˆ for a slightly smaller region Oˆ ⊂⊂ O. These act like the global Hamiltonian on A(O) operators fulfil HO,Oˆ ≥ 0, which may be interpreted as a very weak form of energy inequality: starting from local integrals of the energy density, it seems always possible
Quantum Inequalities from Operator Product Expansions
793
to add appropriate “boundary terms”, associated with O ∩ Oˆ , such that the resulting operator HO,Oˆ is positive. However, there is no explicit control on these boundary terms, not even a means of separating them from a “main term”, so that this approach does not yet lead to a meaningful interpretation in terms of quantum energy inequalities. In curved spacetime, however, the situation is better. Brunetti, Fredenhagen and Verch have shown the existence of a stress-tensor in locally covariant quantum field theories obeying the time-slice axiom [69]. This stress-energy tensor is obtained by functional differentiation with respect to metric perturbations. This prevents an immediate identification of the energy density as a sum of absolute squares of basic fields. Nevertheless, this may serve as a starting point for future study. Acknowledgements. This work was initiated during the programme ‘Mathematical and Physical Aspects of Perturbative Approaches to Quantum Field Theory’ at the Erwin Schrödinger Institute, Vienna, and the authors thank the organisers of the programme and the ESI for financial support. CJF also thanks the Fakultät für Mathematik, University of Vienna for hospitality and financial support at various stages of the work. It is a pleasure to thank Stefan Hollands for valuable discussions in the early phases of this work. HB also thanks the II. Institut für Theoretische Physik, Hamburg, for hospitality and Klaus Fredenhagen for helpful remarks. The discussion of positivity of formal power series in Sect. 1.2 arose from conversations between HB and Bernd Kuckert, to whose memory this paper is dedicated.
References 1. Hudson, R.L.: When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6, 249–252 (1974) 2. Fefferman, C., Phong, D.H.: The uncertainty principle and sharp Gårding inequalities. Comm. Pure Appl. Math. 34, 285 (1981) 3. Bostelmann, H.: Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301 (2005) 4. Bostelmann, H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005) 5. Epstein, H., Glaser, V., Jaffe, A.: Nonpositivity of the energy density in quantized field theories. Nuovo Cimento 36, 1016–1022 (1965) 6. Ford, L.H.: Quantum coherence effects and the second law of thermodynamics. Proc. Roy. Soc. London A 364, 227–236 (1978) 7. Ford, L.H.: Constraints on negative-energy fluxes. Phys. Rev. D 43, 3972–3978 (1991) 8. Ford, L.H., Roman, T.A.: Averaged energy conditions and quantum inequalities. Phys. Rev. D 51, 4277–4286 (1995) 9. Ford, L.H., Roman, T.A.: Restrictions on negative energy density in flat spacetime. Phys. Rev. D 55, 2082–2089 (1997) 10. Pfenning, M.J., Ford, L.H.: Scalar field quantum inequalities in static spacetimes. Phys. Rev. D 57, 3489–3502 (1998) 11. Fewster, C.J., Eveson, S.P.: Bounds on negative energy densities in flat spacetime. Phys. Rev. D 58, 084010 (1998) 12. Fewster, C.J., Teo, E.: Bounds on negative energy densities in static space-times. Phys. Rev. D 59, 104016, 10 (1999) 13. Pfenning, M.J.: Quantum inequalities for the electromagnetic field. Phys. Rev. D 65, 024009, 13 (2002) 14. Fewster, C.J.: A general worldline quantum inequality. Class. Quant. Grav. 17, 1897–1911 (2000) 15. Fewster, C.J., Smith, C.J.: Absolute quantum energy inequalities in curved spacetime. Ann. Henri Poincaré 9, 425–455 (2008) 16. Fewster, C.J., Pfenning, M.J.: A quantum weak energy inequality for spin-one fields in curved space-time. J. Math. Phys. 44, 4480–4513 (2003) 17. Ford, L.H., Helfer, A.D., Roman, T.A.: Spatially averaged quantum inequalities do not exist in fourdimensional spacetime. Phys. Rev. D 66, 124012 (2002) 18. Fewster C.J., Roman T.A.: Null energy conditions in quantum field theory. Phys. Rev. D 67, 044003, 11 (2003) 19. Olum, K.D., Graham, N.: Static negative energies near a domain wall. Phys. Lett. B 554, 175–179 (2003)
794
H. Bostelmann, C. J. Fewster
20. Fewster, C.J., Hollands, S.: Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17, 577–612 (2005) 21. Flanagan, É.É.: Quantum inequalities in two-dimensional Minkowski spacetime. Phys. Rev. D 56, 4922– 4926 (1997) 22. Vollick, D.N.: Quantum inequalities in curved two-dimensional spacetimes. Phys. Rev. D 61, 084022, 5 (2000) 23. Fewster, C.J., Osterbrink, L.W.: Quantum energy inequalities for the non-minimally coupled scalar field. J. Phys. A 41, 025402 (2008) 24. Fewster, C.J.: Quantum energy inequalities and local covariance. II: Categorical formulation. Gen. Rel. Grav. 39, 1855–1890 (2007) 25. Fewster, C.J., Verch, R.: A quantum weak energy inequality for Dirac fields in curved spacetime. Commun. Math. Phys. 225, 331–359 (2002) 26. Fewster, C.J., Mistry, B.: Quantum weak energy inequalities for the Dirac field in flat spacetime. Phys. Rev. D 68, 105010, 6 (2003) 27. Dawson, S.P., Fewster, C.J.: An explicit quantum weak energy inequality for Dirac fields in curved spacetimes. Class. Quant. Grav. 23, 6659–6681 (2006) 28. Smith, C.J.: An absolute quantum energy inequality for the Dirac field in curved spacetime. Class. Quant. Grav. 24, 4733–4750 (2007) 29. Yu, H., Wu, P.: Quantum inequalities for the free Rarita-Schwinger fields in flat spacetime. Phys. Rev. D 69, 064008 (2004) 30. Hu, B., Ling, Y., Zhang, H.: Quantum inequalities for massless spin-3/2 field in Minkowski spacetime. Phys. Rev. D 73, 045015 (2006) 31. Glimm, J., Jaffe, A.: Quantum Physics–A Functional Integral Point of View. 2nd edition, New York: Springer, 1987 32. Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton, NJ: Princeton University Press, 1991 33. Lechner, G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008) 34. Dütsch, M., Fredenhagen, K.: A local (perturbative) construction of observables in gauge theories: The example of QED. Commun. Math. Phys. 203, 71–105 (1999) 35. Bordemann, M., Waldmann, S.: Formal GNS construction and states in deformation quantization. Commun. Math. Phys. 195, 549–583 (1998) 36. Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. New York: Benjamin, 1964 37. Haag, R.: Local Quantum Physics. 2nd edition. Berlin: Springer, 1996 38. Haag, R., Swieca, J.A.: When does a quantum field theory describe particles? Commun. Math. Phys. 1, 308–320 (1965) 39. Buchholz, D., Wichmann, E.H.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106, 321–344 (1986) 40. Buchholz, D., Porrmann, M.: How small is the phase space in quantum field theory? Ann. Inst. H. Poincaré 52, 237–257 (1990) 41. Buchholz, D., Junglas, P.: On the existence of equilibrum states in local quantum field theory. Commun. Math. Phys. 121, 255–270 (1989) 42. Porrmann, M.: Particle weights and their disintegration II. Commun. Math. Phys. 248, 305–333 (2004) 43. Fewster, C.J.: Quantum energy inequalities and stability conditions in quantum field theory. In: A. Boutet de Monvel, D. Buchholz, D. Iagolnitzer, U. Moschella, eds., Rigorous Quantum Field Theory: A Festschrift for Jacques Bros, Volume 251 of Progress in Mathematics, Boston: Birkhäuser, 2006, pp. 95–111 44. Fewster, C.J., Ojima, I., Porrmann, M.: p-nuclearity in a new perspective. Lett. Math. Phys. 73, 1–15 (2005) 45. Fewster, C.J., Verch, R.: Stability of quantum systems at three scales: Passivity, quantum weak energy inequalities and the microlocal spectrum condition. Commun. Math. Phys. 240, 329–375 (2003) 46. Schlemmer, J., Verch, R.: Local thermal equilibrium states and quantum energy inequalities. Annales Henri Poincare 9, 945–978 (2008) 47. Haag, R., Ojima, I.: On the problem of defining a specific theory within the frame of local quantum physics. Ann. Inst. H. Poincaré 64, 385–393 (1996) 48. Wilson, K.G.: Non-Lagrangian models of current algebra. Phys. Rev. 179, 1499–1512 (1969) 49. Zimmermann, W.: Local operator products and renormalization in quantum field theory. In: S. Deser, M. Grisaru, H. Pendleton, eds., Lectures on Elementary Particles and Quantum Field Theory, Volume 1. Cambridge, MA: MIT Press, 1970 50. Borchers, H.J.: Field operators as C ∞ functions in spacelike directions. Nuovo Cimento (10) 33, 1600–1613 (1964)
Quantum Inequalities from Operator Product Expansions
795
51. Johnson, K.: Solution of the equations for the Green’s functions of a two dimensional relativistic field theory. Nuovo Cimento 20, 773–790 (1961) 52. Fredenhagen, K., Haag, R.: Generally covariant quantum field theory and scaling limits. Commun. Math. Phys. 108, 91–115 (1987) 53. Bostelmann, H.: Lokale Algebren und Operatorprodukte am Punkt. Thesis, Universität Göttingen, 2000. Available online at http://webdoc.sub.gwdg.de/diss/2000/bostelmann/ 54. Fredenhagen, K., Hertel, J.: Local algebras of observables and pointlike localized fields. Commun. Math. Phys. 80, 555–561 (1981) 55. Bostelmann, H., D’Antoni, C., Morsella, G.: Scaling algebras and pointlike fields. A nonperturbative approach to renormalization. Commun. Math. Phys. 285, 763–798 (2009) 56. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Volume II: Fourier Analysis, SelfAdjointness. San Diego, CA: Academic Press, 1975 57. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Volume 140 of Pure and Applied Mathematics. 2nd edition, London-NewYork: Academic Press, 2003 58. Gelfand, I.M., Shilov, G.E.: Generalized Functions. Volume 1, New York: Academic Press, 1968 59. Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. Rev. Math. Phys. 7, 1195–1239 (1995) 60. Bostelmann, H., D’Antoni, C., Morsella, G.: On dilation symmetries arising from scaling limits. http://arxiv.org/abs/0812.4762v1 [math.ph], 2008, to appear in Commun. Math. Phys. 61. Eveson, S.P., Fewster, C.J., Verch, R.: Quantum inequalities in quantum mechanics. Ann. Henri Poincaré 6, 1–30 (2005) 62. Junker, W., Schrohe, E.: Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties. Ann. Henri Poincaré 3, 1113–1181 (2002) 63. Hollands, S.: The operator product expansion for perturbative quantum field theory in curved spacetime. Commun. Math. Phys 273, 1–36 (2007) 64. Orzalesi, C.A.: Charges and generators of symmetry transformations in quantum field theory. Rev. Mod. Phys. 42, 381–408 (1970) 65. Lopuszanski, J.: An Introduction to Symmetry and Supersymmetry in Quantum Field Theory. Singapore: World Scientific, 1991 66. Buchholz, D., Doplicher, S., Longo, R.: On Noether’s theorem in quantum field theory. Ann. Phys. (N.Y.) 170, 1 (1986) 67. Doplicher, S.: Local aspects of superselection rules. Commun. Math. Phys. 85, 73–86 (1982) 68. Doplicher, S., Longo, R.: Local aspects of superselection rules II. Commun. Math. Phys. 88, 399–409 (1983) 69. Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle - A new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003) Communicated by Y. Kawahigashi
Commun. Math. Phys. 292, 797–810 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0851-z
Communications in
Mathematical Physics
Nonlinear Instability for the Critically Dissipative Quasi-Geostrophic Equation Susan Friedlander1 , Nataša Pavlovi´c2 , Vlad Vicol1 1 Department of Mathematics, University of Southern California,
3620 South Vermont Ave., KAP 108, Los Angeles, CA 90089, USA. E-mail: [email protected]; [email protected] 2 Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, USA. E-mail: [email protected] Received: 31 January 2009 / Accepted: 27 April 2009 Published online: 28 June 2009 – © Springer-Verlag 2009
Abstract: We prove that linear instability implies non-linear instability in the energy norm for the critically dissipative quasi-geostrophic equation. 1. Introduction A fundamental equation in oceanography and meteorology is the 3 dimensional Navier-Stokes equation in the context of a rapidly rotating, density stratified, viscous, incompressible fluid. Both the forces of rotation and stratification impose a tendency toward 2 dimensionality on the 3 dimensional fluid motion, and this leads to approximate and simpler mathematical models. Important non-dimensional parameters are the Ekman number (the strength of the viscous term relative to rotation) and the Rossby number (the strength of the nonlinearity relative to rotation). In many geophysical problems these parameters are very small. A set of approximations based on asymptotic expansions in powers of these small parameters yields an approximate equation for the 3 dimensional pressure known as the general quasi-geostrophic equation with appropriate boundary conditions. Further simplifying assumptions reduce the problem to the study of a 2 dimensional equation which describes the evolution of the temperature field on a surface that bounds the fluid. In the geophysical fluids literature this equation is known as the surface quasi-geostrophic equation. A derivation of this equation and a discussion of its physical relevance can be found, for example, in Pedlosky [Pe], Salmon [S], Held at al [HPGS]. The effects of viscosity are incorporated via a boundary layer analysis and a mechanism known as Ekman layer pumping which produces the dissipative term in the 2 dimensional quasi-geostrophic equation. In the mathematical literature this 2 dimensional equation is often called the dissipative quasi-geostrophic equations (QG equation) with the word surface being omitted since the equation is 2 dimensional. This equation, for an unknown active scalar (x, t) representing the temperature on the boundary surface, is given by ∂t + U · ∇ + (−)β = f,
(1.1)
798
S. Friedlander, N. Pavlovi´c, V. Vicol
where U (x, t) is the velocity vector and f (x) is a given external force. The velocity is coupled with the temperature via a stream function (x, t): = (−)1/2 = ,
(1.2)
U = ∇ ⊥ = (∂x2 , −∂x1 ) = (R2 , −R1 ),
(1.3)
and
where Ri is the i th Riesz transform. Our analysis of (1.1) – (1.3) considers x in the 2 dimensional torus [0, 2π ]2 = T2 and t ∈ [0, ∞). Both the non-dissipative and the dissipative QG equations have received much attention following the seminal article of Constantin et al [CMT]. They observed a number of similar features between the full 3 dimensional Euler and Navier-Stokes equations and the much simpler QG equations in terms of possible formation of singularities. Recent results concerning the dissipative QG equations include [CC,CCW,CV,CW,DD,DP, J1,J2,KN,KNV,M,W] and references therein. The appropriate power β of the negative Laplacian in the derivation from the general 3D viscous quasi-geostrophic models and Ekman boundary layer analysis is β = 1/2. Dimensionally the 2D QG equation with β = 1/2 is the analogue of the 3D NavierStokes equation. β = 1/2 is called the critical case. The first results concerning regularity of solutions to the dissipative QG equation were given in the simpler (but non-physical) subcritical case where β > 1/2: see, for example, Constantin and Wu [CW]. In the critical case, β = 1/2, Constantin, Cordoba and Wu [CCW] proved existence of a unique global solution evolving from any initial data that are small in L ∞ . Very recently, the smallness assumption was removed independently in breakthrough works of Caffarelli and Vasseur [CV] and Kiselev, Nazarov and Volberg [KNV]. In particular, Caffarelli and Vasseur [CV] used harmonic extension to establish regularity of the Leray-Hopf weak solution. On the other hand, Kiselev et al [KNV] proved the global well posedness of the critical dissipative QG equations with periodic C ∞ data. Their argument is based on a certain non-local maximum principle for a suitably chosen modulus of continuity. In the present article we consider the question of nonlinear instability of a steady solution of the forced critical QG equations. We note that the above mentioned references concern the case f = 0, but in order to ensure the existence of a large class of steady states we must consider the nontrivially forced problem. In particular, we need to reprove certain results that are known to hold for the unforced equations but not in the forced context, namely the nonlocal maximum principle of Kiselev et al [KNV]. The main result of this paper is that linear instability implies nonlinear Lyapunov instability for , and hence U , in the function space L 2 . Such results connecting linear and nonlinear instability have been proven under certain restrictions for the 2D Euler equations, see Bardos et al [BGS], Friedlander and Vishik [VF], and Lin [L]. There the methods utilize a bootstrap technique where closure relies on the special property of conservation of vorticity which is valid for 2D Euler but not for 3D Euler, where the equivalent instability result is still unproven. This property cannot be utilized for the QG equation because the relation between the temperature and the stream function is not equivalent to the relation between the vorticity and the stream function in the 2D Euler equations. In fact this is one reason why it is conjectured that the QG equations might mimic possible singularity development in the 3D fluid equations. The result that linear instability implies nonlinear instability in L 2 for the Navier-Stokes equations in any dimension was proved in Friedlander et al [FPS] (see also the seminal text of Yudovich [Y]). In this case the special ingredient that permits
Nonlinear Instability for the Critical QG
799
the bootstrap argument to close is the smoothing property of the Laplacian with respect to the nonlinear term. The arguments in [FPS] carry over directly to the subcritical dissipative QG equation (i.e. β > 1/2) because the dissipative term again smooths the nonlinear term in (1.1) - (1.3). However the case of the critical QG equation is more subtle because the critical dissipative term (β = 1/2) and the nonlinear term are now of the same order. Hence to prove that linear instability implies nonlinear instability in L 2 for the critical dissipative QG equations via the bootstrap argument requires a different special ingredient. The one we use in this article is the existence of a global bound on ∇(t) L ∞ . This result for the unforced critically dissipative QG was proved in [KNV] and a recent preprint of Kiselev and Nazarov [KN] shows that the result also holds for the equation augmented by a dispersion term. The existence of this global bound for the forced equations is proven in Sect. 5. We note that the fairly general abstract theorem of Friedlander et al [FSV] can be applied to the critical QG equations. Since the spectrum of the linearized QG operator is discrete (see Sect. 3), the spectral gap condition of the abstract theorem is satisfied. It then follows from [FSV] that linear instability implies nonlinear instability in H s , with s > 2. The novel result of this present paper is to prove instability in the “physically natural” energy space L 2 .
Organization of the paper. In Sect. 2 we formulate the stability problem in terms of the temperature (x, t) perturbed about a steady state θ0 (x) ∈ C ∞ . Also in the same section we define nonlinear stability/instability and we state the main instability result, Theorem 2.1. In Sect. 3 we study the linear operator L for the dissipative QG equations in perturbation form. This operator is elliptic of order 1, with compact resolvent, and hence its spectrum is purely discrete for x ∈ T2 . We prove certain properties of L that will be used in the bootstrap argument. Then in Sect. 4 we use this argument to prove Theorem 2.1. In Sect. 5 we prove, in the spirit of [KNV], that the forced equation has a global C ∞ solution and that supt≥0 ∇(t) L ∞ < ∞. This result in used in the bootstrap argument that proves the main theorem. 2. Notation and Formulation of the Result Let θ0 be the temperature of a smooth steady 2D flow with velocity q0 , and smooth force f , that is we have q0 · ∇θ0 + θ0 = f, q0 = (R2 θ0 , −R1 θ0 ).
(2.1) (2.2)
Here we consider θ0 , q0 , f ∈ C ∞ (T2 ). We linearize (1.1) about the steady state (θ0 , q0 ) by writing (x, t) = θ0 (x) + θ (x, t) and U (x, t) = q0 (x) + q(x, t). In such a way we obtain an equation that governs the perturbation θ : ∂t θ = Lθ + N (θ ),
(2.3)
where the linear operator L is defined by Lθ = −q0 · ∇θ − q · ∇θ0 − θ,
(2.4)
800
S. Friedlander, N. Pavlovi´c, V. Vicol
the velocity is coupled with the temperature via q = (R2 θ, −R1 θ )
(2.5)
N (θ ) = −q · ∇θ.
(2.6)
and
zero mean on the torus, and in the For simplicity of the presentation we let θ0 , f, θ have following we shall denote H s = {v ∈ H s (T2 ) : T2 vd x = 0}, for all s ≥ 0. We define a suitable version of stability (the same definition was used, e.g. in [FPS,VF]). Definition. Let (X, Z ) be a pair of Banach spaces. A solution θ0 of (2.1)-(2.2) is called (X, Z ) nonlinearly stable if for any ρ > 0, there exists ρ > 0 so that if θ (0) ∈ X and θ (0) Z < ρ , then we have (i) there exists a global in time solution to (2.3) such that θ (t) ∈ C([0, ∞); X ); (ii) θ (t) Z < ρ for a.e. t ∈ [0, ∞). An equilibrium θ0 that is not stable (in the above sense) is called Lyapunov unstable. The Banach space X is the space where a local existence theorem for the nonlinear equations is available, while Z is the space where the spectrum of the linear operator is analyzed, and where the instability is measured. In the case of the critical dissipative QG we let X be the critical Sobolev space H 1 (cf. [CC,CW,DD,J1,J2,M]), while the growth of the perturbation is considered in the energy space Z = L 2 . Now we are ready to formulate the main result of the present paper. Theorem 2.1. Suppose that θ0 is a smooth mean-free steady state solution of the critical dissipative QG, i.e., it solves (2.1)–(2.2). If the associated linear operator L, as defined in (2.4), has spectrum in the unstable region, then the steady state is (H 1 , L 2 ) Lyapunov nonlinearly unstable. 3. Linearized Dissipative QG The linear operator L defined in (2.4) via Lθ = −q0 · ∇θ − q · ∇θ0 − θ is a nonlocal operator with principal symbol a(x, k) = −|k| + iq0 (x) · k, which does not vanish on T2 × Z2 \ {0}. Therefore L is elliptic of order 1. Since q0 , ∇θ0 ∈ C ∞ , for large enough α > 0, we have that (L − α I )−1 is a bounded operator from L 2 into H 1 . Moreover, the domain of L D(L) = {v ∈ H 1 (T2 ), vd x = 0} ⊂ L 2 (T2 ) (3.1) T2
is compactly embedded in L 2 by Rellich’s theorem, so that resolvent (L − α I )−1 is a compact operator. Thus L has discrete spectrum.
Nonlinear Instability for the Critical QG
801
Let µ be the eigenvalue of L with maximal positive real part over L 2 . Let λ = Re µ and φ ∈ L 2 be the corresponding eigenfunction1 . For a fixed 0 < δ < Cλ , where Cλ > 0 is a constant depending on λ to be determined later, we denote by L δ , L δ = L − (λ + δ)I.
(3.2)
The shift ensures that L δ generates a bounded C0 -semigroup over L 2 and that the resolvent set of L δ contains the right half plane. The following lemma shows that L δ generates an analytic semigroup over L 2 . Lemma 3.1. Over L 2 the operator L δ generates an analytic semigroup. The proof of the lemma modifies the proof of [P, Theorem 7.2.7], which shows the analyticity of a strongly elliptic operator of order 2m over L 2 , to the case of the linearized QG operator, which is elliptic of order 1. Proof. Define the operator G via Gv = v + q0 · ∇v + R(v) · ∇θ0 + 2βv = −Lv + 2βv,
(3.3)
where we have denoted R(v) = (R2 v, −R1 v) and β = ∇θ0 L ∞ . Since q0 is divergence-free we have that G satisfies Gärding’s inequality Re (Gv, v) ≥ 1/2 v2L 2 + βv2L 2 .
(3.4)
In the above estimate we also used R(v) L 2 ≤ v L 2 . Similarly, for every v ∈ D(G), we have |Im (Gv, v)| ≤ |(Gv, v)| ≤ 1/2 v2L 2 + 3βv2L 2 .
(3.5)
Since v is a scalar, it follows from (3.4) and (3.5) that the numerical range S(G) (cf. [P, pp. 12]) is contained in the set Sϑ0 = {λ ∈ C : −ϑ0 < arg λ < ϑ0 },
(3.6)
where ϑ0 = arctan(3) < π/2. Choosing ϑ0 < ϑ < π/2 and defining ϑ = {z ∈ C : | arg z| > ϑ}, we have that there is a constant C = C(ϑ, ϑ0 ) > 0 such that dist(z, S(G)) ≥ C|z|, for all z ∈ ϑ .
(3.7)
We now claim that all real x < 0 are in the resolvent set ρ(G) of the operator G. Recall that G = −L + 2β I , and moreover that the spectrum of the operator L is contained in the half plane {z ∈ C : Re z ≤ λ}, where 0 < λ = Re µ, and µ is the eigenvalue of L with largest real part with associated eigenfunction φ. Since q0 is divergence free we also have that µφ2L 2 = (Lφ, φ) = −1/2 φ2L 2 − (R(φ) · ∇θ0 , φ),
(3.8)
and by taking real parts this implies that λ ≤ ∇θ0 L ∞ = β; hence the spectrum of G is contained in the right half plane, proving the claim. 1 The steady flow q = (sin mx , 0) gives an example for which the operator L has unstable eigenvalues 0 2 over L 2 . This follows from an extension of the analysis in Friedlander and Shvydkoy [FS] to the dissipative equations (see also Meshalkin and Sinai [MS]).
802
S. Friedlander, N. Pavlovi´c, V. Vicol
We have hence proven that ϑ is contained in the complement of S(G) and has non-empty intersection with ρ(G); by [P, Theorem 1.3.9] we have that ϑ ⊂ ρ(G) and for every z ∈ ϑ we have the resolvent estimate R(z : G) L 2 →L 2 ≤
1 dist(z : S(G))
≤
1 . C|z|
(3.9)
Therefore −G is the infinitesimal generator of an analytic semigroup (cf. [P, Theorem 2.5.2]) and so L δ = −G + (2β − λ − δ)I generates an analytic semigroup on L 2 , since it is a bounded perturbation of −G.
Now we state and prove the lemma that will be used in the proof of our main result, Theorem 2.1. Lemma 3.2. For 0 ≤ γ ≤ 1 there exists a constant C > 0 such that e L δ t v L 2 →L 2 ≤
C 1−γ γ v L 2 −1 v L 2 , tγ
(3.10)
for all smooth functions v ∈ L 2 , where C = C(γ , δ, α, θ0 ). Proof. Since q0 is divergence free, it is convenient to use the operator Aα , defined via Aα v = −q0 · ∇v − v − αv = L δ v + R(v) · ∇θ0 − (α − λ − δ)v,
(3.11)
where α > max{λ + δ, Cθ0 2H 2+ }, > 0, and C is a sufficiently large dimensional constant. We treat L δ as a bounded perturbation of Aα . The operator Aα is also elliptic and 2 has discrete spectrum, so by possibly choosing a different α, we have that A−1 α ∈ L(L ). First, we claim that A−1 α v L 2 ≤ Cv L 2 ,
(3.12)
for all smooth v ∈ L 2 with zero mean. In order prove this, denote h = A−1 α v, which also has zero mean, and observe that (3.12) is equivalent to h L 2 ≤ C−1 Aα h L 2 .
(3.13)
The definition of Aα implies that (−1 Aα h, h) = −(−1 (q0 · ∇h), h) − h2L 2 − α−1/2 h2L 2 , and therefore h2L 2 + α−1/2 h2L 2 ≤ −1 Aα h L 2 h L 2 + |(q0 · ∇h, −1 h)|.
(3.14)
Note that (q0 · ∇−1/2 h, −1/2 h) = 0 since div q0 = 0. Using Plancherel’s theorem, we write this inner product in terms of Fourier coefficients (cf. [KV] and references therein) (q0 · ∇h, −1 h) = (q0 · ∇h, −1 h) − (q0 · ∇−1/2 h, −1/2 h) = i(2π )2 qˆ0 j · k |l|−1/2 − |k|−1/2 hˆ k |l|−1/2 hˆ l . j+k+l=0
(3.15)
Nonlinear Instability for the Critical QG
803
In the above summation, the Fourier frequencies j, k, l ∈ Z2 \ {0} because q0 and h are mean free, and hˆ k denotes the k th Fourier coefficient of h. Since |l| = | j + k| the triangle inequality gives ||l| − |k|| ≤ | j|, and therefore 1 ||l|1/2 − |k|1/2 | 1 | j||k| ≤ 1/2 1/2 1/2 |k| 1/2 − 1/2 ≤ |k| ≤ | j|. 1/2 1/2 |l| |k| |l| |k| |l| |k| (|l| + |k|1/2 ) Therefore, by (3.15) and the Cauchy-Schwartz inequality we have that | j||qˆ0 j ||hˆ k ||l|−1/2 |hˆ l | |(q0 · ∇h, −1 h)| ≤ C j+k+l=0
≤C
| j||qˆ0 j |
j∈Z2 \{0}
|hˆ − j−l ||l|−1/2 |hˆ l |
l∈Z2 \{0,− j}
≤ Ch L 2 −1/2 h L 2
| j|2+ |qˆ0 j || j|−1−
j∈Z2 \{0} −1/2
≤ Ch L 2
h L 2 2+ θ0 L 2 .
We insert the above estimate into (3.14) and apply the Young’s inequality ab ≤ thus obtaining
(3.16) a2 4
+ b2 ,
1 h2L 2 + (α − C2+ θ0 2L 2 )−1/2 h2L 2 ≤ −1 Aα h2L 2 . 2 Since α > Cθ0 2H 2+ , the above estimate proves (3.13). Now we prove that for smooth v ∈ L 2 we have L −1 δ Aα v L 2 ≤ Cv L 2 ,
(3.17)
for a sufficiently large constant C > 0. The inequality (3.17) follows by writing −1 −1 L −1 δ Aα v = v + L δ (R(v) · ∇θ0 ) − (α − δ − λ)L δ v,
L −1 δ
(3.18)
L2
is bounded on (cf. [P, Lemma 2.6.3]). Together with and noting that the operator the boundedness of the Riesz-transforms on L 2 , (3.18) implies ∞ L −1 δ Aα v L 2 ≤ v L 2 (1 + C(∇θ0 L + α − δ − λ)),
L −1 δ Aα
(3.19)
L(L 2 ).
∈ which proves (3.17) and therefore In order to conclude the proof of the lemma we use the fact that L δ generates an analytic semigroup (cf. Lemma 3.1) and therefore (cf. [P, Theorem 2.6.13]) we have that C −γ γ −γ e L δ t v L 2 →L 2 = L δ e L δ t L δ v L 2 →L 2 ≤ γ L δ v L 2 . (3.20) t −γ
Now we bound L δ v L 2 by interpolating (cf. [P, Theorem 2.6.10]) as follows: −γ
1−γ
L δ v L 2 = L δ
γ
−1 (L −1 δ v) L 2 ≤ Cv L 2 L δ v L 2 1−γ
γ
−1 −1 ≤ Cv L 2 (L −1 δ Aα )(Aα )( v) L 2 1−γ
γ
≤ Cv L 2 −1 v L 2 , 1−γ
(3.21)
where in order to obtain (3.21) we used (3.17) and (3.12). We conclude the proof of the lemma by combining (3.20) and (3.21).
804
S. Friedlander, N. Pavlovi´c, V. Vicol
4. Proof of Theorem 2.1 Here we prove Theorem 2.1. In order to do this it is sufficient to show that the trivial solution θ = 0 of (2.3) is (H 1 , L 2 ) Lyapunov unstable. With this goal in mind, we consider a family of solutions θ ε to ∂t θ ε = Lθ ε + N (θ ε ), θ ε |t=0 = εφ,
(4.1) (4.2)
where φ is as above an eigenfunction of L associated with the eigenvalue with maximal positive real part λ. We will prove the following proposition that clearly implies the desired Lyapunov instability result. Proposition 4.1. There exist positive constants C¯ and ε¯ ≤ 1 such that for every ε ∈ ¯ (0, ε¯ ), there exists Tε > 0 such that θ ε (Tε ) L 2 ≥ C. We remark that if θ ε (x, t) solves (4.1)–(4.2), then the function ε (x, t) = θ ε (x, t) + θ0 (x) solves the forced QG equations (5.1)–(5.3), with initial data ε (x, 0) = θ0 (x) + εφ(x) ∈ C ∞ (T2 ). Moreover, in Lemma 5.1 of Sect. 5 we prove that the global smooth solution of the forced QG equations satisfies ∇ε (t) L ∞ ≤ C0ε for all t ≥ 0, where the constant C0ε depends solely on the L ∞ and W 1,∞ norms of the initial data and the force. For ε ∈ (0, 1], we have ε (0) L ∞ ≤ θ0 L ∞ + φ L ∞ , and similarly ∇ε (0) L ∞ ≤ ∇θ0 L ∞ + ∇φ L ∞ , which are independent of ε, and therefore there exists a fixed C0 > 0 such that ∇ε (t) L ∞ ≤ C0 , for all ε ∈ (0, 1] and for all t ≥ 0. We refer the reader to the proof of Lemma 5.1 for further details. The triangle inequality then implies that by possibly increasing C0 we have sup ∇θ ε (t) L ∞ ≤ C0
(4.3)
t≥0
for all ε ∈ (0, 1]. We will henceforth denote θ ε simply as θ and will use the analogous notation for q. All constants in the following are ε-independent. Proof of Proposition 4.1. For R > Cφ := φ L 2 to be chosen later, let T = T (R, ε) be the maximal time such that θ (t) L 2 ≤ ε Reλt ,
for t ∈ [0, T ].
(4.4)
Clearly T ∈ (0, ∞] due to the strong continuity in L 2 of t → θ (t) and the chosen initial condition. Using Duhamel’s formula we write the solution of (4.1)–(4.2) as θ (t) = e Lt εφ + B(t),
(4.5)
where
t
B(t) =
e L(t−s) N (θ )(s) ds.
(4.6)
1+γ /2 B(t) L 2 ≤ C1 ε Reλt ,
(4.7)
0
First, we shall prove that
Nonlinear Instability for the Critical QG
805
where γ ∈ (0, 1) and C1 = C(C0 , λ, δ, γ ) > 0 are constants. To show (4.7), we rewrite the operator B and then use Lemma 3.2 as follows: t B(t) L 2 = e(λ+δ)(t−s) e L δ (t−s) N (θ (s)) ds L 2 0 t ≤ e(λ+δ)(t−s) e L δ (t−s) N (θ (s)) L 2 →L 2 ds 0 t 1 1−γ γ ≤C e(λ+δ)(t−s) N (θ (s)) L 2 −1 N (θ (s)) L 2 ds, (4.8) γ (t − s) 0 where γ ∈ (0, 1) is arbitrary, and C > 0. In order to bound the factor −1 N (θ (s)) L 2 we recall the explicit representation (cf. [CC]) of the nonlinear term −1 (R(θ ) · ∇θ ) = Cn (R1 (θ R2 (θ )) − R2 (θ R1 (θ ))) ,
(4.9)
for some dimensional constant Cn , and the fact that the Riesz transforms are bounded on L 2 and L 4 , to obtain that −1 N (θ (s)) L 2 ≤ Cθ Ri θ L 2 ≤ Cθ 2L 4 .
(4.10)
By interpolating, we have 1/3
2/3
θ L 4 ≤ Cθ L 2 θ L 8 .
(4.11)
On the other hand by the Gagliardo-Nirenberg inequality and the Hölder inequality we have that (cf. [N]) 3/8
3/8
θ L 8 = θ 8/3 L 3 ≤ C∇(θ 8/3 ) L 6/5 3/8
5/8
3/8
≤ Cθ 5/3 ∇θ L 6/5 ≤ Cθ L 2 ∇θ L ∞ .
(4.12)
By combining (4.10) with (4.11) and (4.12) we obtain γ
3γ /2
γ /2
−1 N (θ (s)) L 2 ≤ Cθ L 2 ∇θ L ∞ .
(4.13)
On the other hand, by Hölder’s inequality, and the boundedness of the Riesz transforms on L 2 , we have 1−γ
1−γ
1−γ
N (θ ) L 2 ≤ θ L 2 ∇θ L ∞ .
(4.14)
Recall that by (4.3) we have ∇θ (t) L ∞ ≤ C0 , for all t ≥ 0. Using assumption (4.4) and the fact that 0 < δ < Cλ = λγ /2, we substitute the bounds (4.13) and (4.14) into (4.8), to conclude 1+γ /2 , B(t) L 2 ≤ C1 ε Reλt
(4.15)
for some positive constant C1 = C(C0 , λ, δ, q, γ ), proving (4.7). The Duhamel formula (4.5) and the bound (4.7) imply 1+γ /2 θ (t) L 2 ≤ Cφ εeλt + C1 ε Reλt .
(4.16)
806
S. Friedlander, N. Pavlovi´c, V. Vicol
Observing that R was chosen such that R > Cφ , it follows that we have the following estimate on the maximal time T :
R − Cφ 2/γ λT =: C2 > 0, (4.17) εe ≥ C1 R 1+γ /2 which clearly holds if T = ∞. On the other hand, if T is finite, (4.17) is obtained by combining the continuity of t → θ (t) L 2 , (4.4) and (4.16) to obtain γ /2 , ε ReλT ≤ Cφ εeλT + C1 R 1+γ /2 εeλT εeλT which, in turn, implies (4.17). Therefore we have T ≥ Tε , where we defined Tε =
1 C2 ln . λ ε
(4.18)
To conclude the proof we must find a lower bound on θ (Tε ) L 2 . We use Duhamel’s formula (4.5), the triangle inequality, and (4.15) to obtain 1+γ /2 θ (Tε ) L 2 ≥ Cφ εeλTε − C1 ε ReλTε .
(4.19)
Using (4.18), with C2 given by (4.17), the lower bound (4.19) implies R − Cφ ) C1 R 1+γ /2 = C2 (2Cφ − R) := C¯ > 0,
θ (Tε ) L 2 ≥ C2 (Cφ − C1 R 1+γ /2
by choosing Cφ < R < 2Cφ . This concludes the proof of the proposition which, in turn, implies Theorem 2.1.
5. Global Well-Posedness for the Forced QG Equation In this section, by modifying the argument of Kiselev et al [KNV], we prove that the forced QG equation has a unique global smooth solution. More precisely, we prove the following: Lemma 5.1. Assume that 0 , f ∈ C ∞ are T2 -periodic functions with zero mean. Then there exists a unique global in time smooth solution of ∂t + U · ∇ + = f, U = R() = (R2 , −R1 ), (0) = 0 .
(5.1) (5.2) (5.3)
Moreover for all t ≥ 0 we have ∇(t) L ∞ ≤ C0 , where C0 = C0 (0 L ∞ , ∇0 L ∞ , f L ∞ , ∇ f L ∞ ) is a positive constant.
(5.4)
Nonlinear Instability for the Critical QG
807
The proof of the lemma is in the spirit of [KNV], but we additionally need to treat the force term, which a-priori could cause growth of the solution. Since (t) is mean free, it can be shown a-priori that (t) L p , with 2 ≤ p ≤ ∞, remains bounded for all time. However the same methods do not work for the subcritical quantity ∇(t) L ∞ , and therefore we need to prove the nonlocal maximum principle of [KNV] for the forced QG equation (5.1)–(5.3). This is achieved by suitably choosing a scaling parameter B and making use of the fact that due to periodicity we do not need to consider arbitrarily large length scales. We note that the scaling parameter B is used only in the modulus of continuity, whereas the solutions to (5.1)–(5.3) are not space-time rescaled. Proof of Lemma 5.1. We recall that a continuous, increasing, unbounded, concave function ω : [0, ∞) → [0, ∞), with ω(0) = 0 is a modulus of continuity for a function f if | f (x) − f (y)| ≤ ω(|x − y|),
(5.5)
for all x, y ∈ R2 . The modulus is strict if the strict inequality holds in (5.5). We consider a modulus of continuity that also satisfies ω (0) < ∞, and limξ →0+ ω
(ξ ) = −∞, namely, as in [KNV] we let ξ − ξ 3/2 , 0 ≤ ξ ≤ δ, (5.6) ω(ξ ) = δ − δ 3/2 + γ log(1 + 41 log(ξ/δ)), ξ > δ, where δ > γ > 0 are sufficiently small fixed constants. Since 0 ∈ C ∞ , there exists a sufficiently large B > 0 such that 0 has strict modulus of continuity ω B (ξ ) = ω(Bξ ). The scaling parameter B may be chosen as B = C∇0 L ∞ exp(exp(C0 L ∞ )),
(5.7)
where C is a sufficiently large positive constant. Moreover, since ω is unbounded, by possibly increasing B we may ensure that AB 2 ≥ ∇ f L ∞ ,
(5.8)
where the fixed dimensional constant A is as in [KNV, Lemma], and also ω B (d) ≥ 4π f L ∞ , d
(5.9)
√ where d = diam(T2 ) = 2π 2 will be fixed throughout this section. We fix a B that satisfies (5.7)–(5.9) and note that the modulus of continuity is now given by 0 ≤ ξ ≤ Bδ , Bξ − (Bξ )3/2 , ω B (ξ ) = ω(Bξ ) = (5.10) Bξ 1 3/2 δ − δ + γ log(1 + 4 log( δ )), ξ > Bδ . Denote ω B (ξ ) = Bω (Bξ ) and ω B
(ξ ) = B 2 ω
(Bξ ). We claim that ω B (ξ ) is preserved by the evolution (2.3), so that is a global solution. We extend , U, 0 , f to T2 -periodic functions on R2 . The first step of the proof is to show that if (t) has strict modulus of continuity ω B for t ∈ [0, T ], then there exists τ > 0 such that (t) has strict modulus of continuity ω B on t ∈ [0, T + τ ). Since ∇(t) L ∞ < ω B (0), we have that (t) ∈ C ∞ for all
808
S. Friedlander, N. Pavlovi´c, V. Vicol
t ∈ [0, T ], and by the local regularity theorem (cf. [CC,J1,J2,M]) for some time τ > 0 beyond T . We must show that by possibly shrinking τ , we have that |(x, t) − (y, t)| < ω B (|x − y|)
(5.11)
for all t ∈ (T, T + τ ) and x = y ∈ R2 . Define the compact set K = [−2π, 2π ]2 × [−2π, 2π ]2 ⊂ R4 . Since is T2 -space periodic, we have that for any (x, y) ∈ R4 , with x = y, there exist (x , y ) ∈ K , with x = y , such that |x − y | ≤ |x − y|, (x, t) = (x , t), and (y, t) = (y , t). Because ω B is increasing, if (5.11) holds for all (x , y ) ∈ K with x = y , then we have that for all x = y ∈ R2 , |(x, t) − (y, t)| = |(x , t) − (y , t)| < ω B (|x − y |) ≤ ω B (|x − y|). Therefore it is sufficient to prove that there exists τ > 0 such that (5.11) holds for x = y, with (x, y) ∈ K . By assumption, there exists > 0 such that ∇(T ) L ∞ < ω B (0) − 2, and by continuity, for small enough τ we have that ∇(t) L ∞ < ω B (0) − for all t ∈ [T, T + τ ). Therefore for (x, y) ∈ K , with 0 < |x − y| = ξ < ρ, where ρ ≤ min(δ/B, 2 /B 3 ), we have |(x, t) − (y, t)| ≤ ξ ∇(t) L ∞ < ξ(B − ) ≤ Bξ − (Bξ )3/2 = ω B (ξ ), for all t ∈ [T, T + τ ). On the other hand, due to the continuity in time of |(x, t) − (y, t)|, the compactness of the set {(x, y) ∈ K : |x − y| ≥ ρ}, and and the fact that (5.11) holds at t = T , we have that there is a sufficiently small τ > 0 such that (5.11) holds for all (x, y) ∈ K , x = y, and t ∈ [T, T + τ ). The second part is to rule out the case in which there exists T > 0 and x = y ∈ R2 such that (x, T ) − (y, T ) = ω B (|x − y|) (cf. [KNV]). Note that by the periodicity of , for such x = y ∈ R2 fixed, there exist x , y ∈ T2 such that ω B (|x − y|) = (x, T ) − (y, T ) = (x , T ) − (y , T ) ≤ ω B (|x − y |) ≤ ω B (d), and since ω B is increasing, we must have 0 < ξ = |x − y| ≤ d = diam(T2 ). We conclude by showing that d dt ((x, t) − (y, t))|t=T
< 0,
(5.12)
which contradicts the fact that the strict modulus of continuity is lost at t = T . In the following we suppress the time dependence of and U , since we work at t = T fixed. Since has modulus of continuity ω B (ξ ), we know (cf. [KNV, Lemma]) that U has modulus of continuity B (ξ ), where we defined
ξ ∞ ω B (η) ω B (η) dη + ξ B (ξ ) = A dη , η η2 0 ξ for some positive constant A. Then as in [KNV, Sect. 4] we have that ω B (ξ + h|U (x) − U (y)|) − ω B (ξ ) |(U · ∇)(x) − (U · ∇)(y)| ≤ lim+ h→0 h
≤ |U (x) − U (y)|ω B (ξ ) ≤ B (ξ )ω B (ξ ). (5.13)
Nonlinear Instability for the Critical QG
809
The dissipative terms are estimated as in [KNV, Sect. 5], namely by the negative quantity 1 ξ/2 ω B (ξ + 2η) + ω B (ξ − 2η) − 2ω B (ξ ) M B (ξ ) = dη π 0 η2 1 ∞ ω B (2η + ξ ) − ω B (2η − ξ ) − 2ω B (ξ ) + dη. (5.14) π ξ/2 η2 Lastly, the force term is estimated using the mean value theorem | f (x) − f (y)| ≤ FB (ξ ) =
ξ ∇ f L ∞ , 0 ≤ ξ ≤ ξ > Bδ . 2 f L ∞ ,
δ B,
(5.15)
Thus, in order to conclude the proof of (5.12), we must show that for all 0 < ξ ≤ d, we have B (ξ )ω B (ξ ) + FB (ξ ) + M B (ξ ) < 0.
(5.16)
First we treat the case 0 < ξ ≤ δ/B. By keeping track of B, and using condition (5.8), similar arguments as in [KNV, Sect. 7] show that δ ξ ) + ξ ∇ f L ∞ + ω B
(ξ ) B (ξ )ω B (ξ ) + FB (ξ ) + M B (ξ ) ≤ AB 2 ξ(3 + log Bξ π
δ 3 2 −1/2 . ≤ B ξ A(4 + log )− (Bξ ) Bξ 4π Since we have 0 < Bξ ≤ δ, the above quantity is strictly negative if δ is sufficiently small. Note that δ does not depend on B. For the case δ/B ≤ ξ ≤ d, we follow the estimates in [KNV, Sect. 8] to conclude that if γ and δ are sufficiently small, independent of B, then B (ξ )ω B (ξ ) + FB (ξ ) + M B (ξ ) ≤ Aγ ω Bξ(ξ ) + 2 f L ∞ −
1 ω B (ξ ) π ξ .
But B was chosen so that (5.9) is satisfied, i.e. 2 f L ∞ ≤ ω B (d)/2π d. Because on [δ/B, ∞) the function ω B (ξ )/ξ is decreasing, for any ξ ∈ [δ/B, d] we have that 2 f L ∞ ≤ ω B (d)/2π d ≤ ω B (ξ )/2π ξ . Thus
1 1 ω B (ξ ) ω B (ξ ) 1 ω B (ξ ) + 2 f L ∞ − ≤ Aγ + − < 0, Aγ ξ π ξ 2π π ξ if γ is sufficiently small, independent of B. Therefore (5.16) holds for all 0 < ξ ≤ d, and so (5.12) is proven. Therefore the solution (t) exists for all time and has strict modulus of continuity ω B , which implies that ∇(t) L ∞ < ω B (0) = B for all t ≥ 0, concluding the proof of the lemma.
Remark 5.2. We note that it is also possible to adapt the De Giorgi-type techniques used by Caffarelli and Vasseur [CV] to treat the forced QG equation. First one proves boundedness of the solution in L 2 using energy estimates, and then similarly to [CV, Sect. 2] one obtains boundedness (not decay) for all time of (t) in L ∞ and of U (t) in B M O. The second step is to show that the solution is actually Hölder and that it remains bounded in this space for all t ≥ 0. Adding a smooth force does not create additional difficulties. Since this is already subcritical regularity, in the third step it is standard to bootstrap to higher regularity and prove that the W 1,∞ norm of (t) is bounded in time.
810
S. Friedlander, N. Pavlovi´c, V. Vicol
Acknowledgements. We thank Hongjie Dong, Alexander Kiselev, Anna Mazzucato, Roman Shvydkoy and Alexis Vasseur for very helpful discussions. The work of S.F. is supported by NSF grant DMS 0803268. The work of N.P. is supported by NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.
References [BGS] [CC] [CCW] [CMT] [CV] [CW] [DD] [DP] [FPS] [FS] [FSV] [HPGS] [J1] [J2] [KN] [KNV] [KV] [L] [M] [MS] [N] [P] [Pe] [S] [VF] [W] [Y]
Bardos, C., Guo, Y., Strauss, W.: Stable and unstable ideal plane flows. Dedicated to the Memory of Jacques-Lious Lions, Chinese Ann. Math. Ser B. 23(2), 149–164 (2002) Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004) Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equation. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000), Indiana Univ. Math. J. 50 (2001), Special Issue, 97–107 (2001) Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994) Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasigeostrophic equation. To appear in Annals of Math, available at http://pjm.math.berkeley.edu/ editorial/uploads/annals/accepted/090120-caffarelli/090120-caffarelli-v1.pdf Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30(5), 937–948 (1999) Dong, H., Du, D.: Global well-posedness and a decay estimate for the critical dissipative quasigeostrophic equation in the whole space. Discrete Contin. Dyn. Syst. 21(4), 1095–1101 (2008) Dong, H., Pavlovi´c, N.: Regularity criteria for the dissipative quasi-geostrophic equations in Hölder spaces. Commun. Math. Phys. doi:10.1007/s00220-009-0756-x Friedlander, S., Pavlovi´c, N., Shvydkoy, R.: Nonlinear instability for the Navier-Stokes equations. Commun. Math. Phys. 264, 335–347 (2006) Friedlander, S., Shvydkoy, R.: The unstable spectrum of the surface quasi-geostropic equation. J. Math. Fluid Mech. 7(suppl. 1), S81–S93 (2005) Friedlander, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2), 187–209 (1997) Held, I.M., Pierrehumbert, R.T., Garner, S.T., Swanson, K.L.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995) Ju, N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the sobolev space. Commun. Math. Phys. 251(2), 365–376 (2004) Ju, N.: Dissipative quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions. Indiana Univ. Math. J. 56(1), 187–206 (2007) Kiselev, A., Nazarov, F.: Global regularity for the critical dispersive dissipative surface quasigeostrophic equation. Preprint Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasigeostrophic equation. Invent. Math. 167(3), 445–453 (2007) Kukavica, I., Vicol, V.: On the radius of analyticity of solutions to the three-dimensional euler equations. Proc. Amer. Math. Soc. 137, 669–677 (2009) Lin, Z.: Nonlinear instability of ideal plane flows. Int. Math. Res. Not. 41, 2147–2178 (2004) Miura, H.: Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space. Commun. Math. Phys. 267(1), 141–157 (2006) Meshalkin, L., Sinai, Y.: Investigation of stability for a system of equations descibing plane motion of a viscous incompressible fluid. Appl. Math. Mech. 25, 1140–1143 (1961) Nirenberg, L.: On elliptic partial differential equations. Annali Della Scuola Normale Superiore di Pisa - Classe di Scienze Sér. 3(13(2), 115–162 (1959) Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematics Sciences v. 44, New York: Springer-Verlag, 1983 Pedlosky, J.: Geophysical Fluid Dynamcs. New York: Springer-Verlag, 1987 Salmon, R.: Lectures on Geophysical Fluid Dynamics. New York: Oxford University Press, USA, 1998 Vishik, M., Friedlander, S.: Nonlinear instability in two dimensional ideal fluids: the case of a dominant eigenvalue. Commun. Math. Phys. 243(2), 261–273 (2003) Wu, J.: The 2d dissipative quasi-geostrophic equation. Appl. Math. Lett. 15(8), 925–930 (2002) Yudovich, V.I.: The Linearization Method in Hydrodynamical Stability Theory. Transactions of Mathematical Monographs, Vol. 74, Providence, RI: Amer. Math. Soc. 1989
Communicated by P. Constantin
Commun. Math. Phys. 292, 811–827 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0852-y
Communications in
Mathematical Physics
The Navier-Stokes Equations in the Critical Lebesgue Space Hongjie Dong1, , Dapeng Du2, 1 Division of Applied Mathematics, Brown University, 182 George Street, Box F,
Providence, RI 02912, USA. E-mail: [email protected]
2 School of Mathematical Sciences, Fudan University, Shanghai 200433,
P. R. China. E-mail: [email protected] Received: 5 February 2009 / Accepted: 23 April 2009 Published online: 7 July 2009 – © Springer-Verlag 2009
Abstract: We study regularity criteria for the d-dimensional incompressible Navier-Stokes equations. We prove in this paper that if u ∈ L t∞ L dx ((0, T ) × Rd ) is a Leray-Hopf weak solution, then u is smooth and unique in (0, T ) × Rd . This generalizes a result by Escauriaza, Seregin and Šverák [5]. Additionally, we show that if T = ∞ then u goes to zero as t goes to infinity. 1. Introduction In this paper we consider the incompressible Navier-Stokes equations in d spatial dimensions with unit viscosity and zero external force: ∂t u + u · ∇u − u + ∇ p = 0, div u = 0
(1.1)
for x ∈ Rd and t ≥ 0 with the initial condition u(0, x) = a(x), x ∈ Rd .
(1.2)
Here u is the velocity and p is the pressure. For sufficiently regular data a, the local strong solvability of such problems is well known (see, for example, [9,13,33 and 14]). The solution is unique and locally smooth in both spatial and time variables. On the other hand, the global in time strong solvability is an outstanding open problem for d ≥ 3. Another important type of solutions are called Leray-Hopf weak solutions (see Sect. 2.1 for the notation and definition). In the pioneering works of Leray [20] and Hopf [12], it is shown that for any divergence-free vector field a ∈ L 2 , there exists at Hongjie Dong was partially supported by the National Science Foundation under agreement No. DMS0111298 and DMS-0800129. Dapeng Du was partially supported by China Postdoctor Science Fund CPSF 20070410683.
812
H. Dong, D. Du
least one Leray-Hopf weak solution of the Cauchy problem (1.1)-(1.2) on (0, ∞) × Rd . Although the problems of uniqueness and regularity of Leray-Hopf weak solutions are still open, since the seminal work of Leray there is an extensive literature on conditional results under various criteria. The most well-known condition is so-called Ladyzhenskaya-Prodi-Serrin condition, that is for some T > 0, u ∈ L rt L qx (Rd+1 T ),
(1.3)
where the pair (r, q) satisfies 2 d + ≤ 1, q ∈ (d, ∞]. r q Under the condition (1.3), the uniqueness of Leray-Hopf weak solutions was proved by Prodi [22] and Serrin [28], and the smoothness was obtained by Ladyzhenskaya [15]. For further results, we refer the reader to [8,30,31] and the recent preprint [2], and references therein. The borderline case (r, q) = (∞, d) is much more subtle since the result cannot be proved by the usual methods using the local smallness of certain norms of u which are invariant under the natural scaling u(t, x) → λu(λ2 t, λx),
p(t, x) → λ2 p(λ2 t, λx).
(1.4)
For d = 3, this case was studied recently by Escauriaza, Seregin and Šverák in a remarkable paper [5]. The main result of [5] is the following theorem. Theorem 1.1 (Escauriaza, Seregin and Šverák). Let d = 3. Suppose that u is a LerayHopf weak solution of the Cauchy problem (1.1)-(1.2) in (0, T ) × R3 and u satisfies the condition (1.3) with (r, q) = (∞, 3). Then u ∈ L 5 ((0, T ) × R3 ), and hence it is smooth and unique in (0, T ) × R3 . Before we give a description of Theorem 1.1, we shall recall another important concept, the partial regularity of weak solutions. The study of partial regularity of the Navier-Stokes equations was originated by Scheffer in a series of papers [23–25]. In three space dimensions, he established various partial regularity results for weak solutions satisfying the so-called local energy inequality. For d = 3, the notion of suitable weak solutions was introduced in a celebrated paper [1] by Caffarelli, Kohn and Nirenberg. They called a pair (u, p) a suitable weak solution if u has finite energy norm, p belongs to the Lebesgue space L 5/4 , u and p are weak solutions to the Navier-Stokes equations and satisfy a local energy inequality. It is proved that, for any suitable weak solution (u, p), there is an open subset in which the velocity field u is Hölder continuous, and the complement of it has zero 1-D Hausdorff measure. In [21], with zero external force, Lin gave a more direct and sketched proof of Caffarelli, Kohn and Nirenberg’s result. A detailed treatment was then later given by Ladyzhenskaya and Seregin in [18]. For other results in this direction, we refer the reader to [3,11,31] and references therein. The proofs in [5] are highly nontrivial and rely on certain regularity criteria in the light of [1,21 and 18]. That is, roughly speaking, if some scaling invariant quantities are small then the solution is locally regular. Another main ingredient of the proof is a backward uniqueness theorem of heat equations with bounded coefficients of lower order terms in the half space (see also [6]). Under an additional assumption on the pressure, there are some extensions of Theorem 1.1 to the half space case and the bounded domain case; we refer the reader to [26] and [19] for some results in this direction. Another interesting open problem is the extension to the higher dimensional Navier-Stokes equations. It seems
The Navier-Stokes Equations
813
to us that the argument in [5] breaks down in several places when d ≥ 4. In particular, the regularity criterion, Theorem 2.2 [5], is unknown for the higher dimensional NavierStokes equations. We now state the main results of the article. Theorem 1.2. Let d ≥ 3, K > 0 and T ∈ (0, ∞). Suppose that u is a Leray-Hopf weak solution of the Cauchy problem (1.1)-(1.2) in (0, T ) × Rd and u satisfies the condition u ∈ L t∞ L dx ((0, T ) × Rd ), u L t∞ L x ((0,T )×Rd ) ≤ K . d
(1.5)
Then u ∈ L d+2 ((0, T ) × Rd ), and hence it is smooth and unique in (0, T ) × Rd . Theorem 1.3. Let d ≥ 3 and K > 0. Suppose that u is a Leray-Hopf weak solution of the Cauchy problem (1.1)-(1.2) in (0, ∞) × Rd and u satisfies the condition u ∈ L t∞ L dx ((0, ∞) × Rd ), u L t∞ L x ((0,∞)×Rd ) ≤ K . d
(1.6)
Then u is smooth and unique in (0, ∞) × Rd . Moreover, we have lim u(t, ·) L ∞ = 0.
t→∞
(1.7)
We give a brief description of our argument. As in [5] we prove by contradiction and blow up the solution near a singular point at the first blow-up time to obtain a sequence of solutions {u k }. The limiting function u ∞ of this sequence is a suitable weak solution of the Navier-Stokes equations. Note that the solutions u k are smooth before the first blow-up time. As we mentioned before, we are not able to establish a regularity criterion similar to Theorem 2.2 [5], which says if certain scaling invariant quantities are small then the solution is locally Hölder continuous. Instead we use a modified one. Roughly speaking, we show that if the solutions are smooth, the L t∞ L dx norm is bounded and some scaling invariant quantities are small, then we have a priori L ∞ bound for the solutions on a much smaller ball. Here the point is the a priori L ∞ bound only depends on the L t∞ L dx norm and the dimension. This regularity criterion together with the L p -convergence of u k yields the local boundedness of u ∞ outside a large cylinder. The local boundedness implies the local smoothness of u ∞ . Then we use the backward uniqueness proved in [5] to see that u ∞ is equivalent to zero outside a large cylinder, which further implies that u ∞ ≡ 0 by using the spatial analyticity of strong solutions and the weak-strong uniqueness of the Navier-Stokes equations. This means the sequence u k converges to zero in L p on any compact set. Going back to the original solution u we see that the modified regularity criterion applies, which gives a contradiction and proves Theorem 1.2. To prove Theorem 1.3, we notice that u is in L 4 ((0, ∞) × Rd ), which implies the smallness of its L 4 norm in (T, ∞) × Rd for large T . Then we use the modified regularity criterion again and the scaling (1.4). We remark that a decay result similar to that of Theorem 1.3 was obtained in [7] by using a completely different method. The remaining part of the article is organized as follows. We give a few definitions and prove several preliminary results in the next section. In Sect. 3, we prove a key estimate (Proposition 3.1) about the scaling invariant quantities and construct a sequence of solutions by blowing up the solution at a singular point. Section 4 is devoted to the proof of a local boundedness estimate (Theorem 4.1). We finish the proof of Theorem 1.2 and 1.3 in Sect. 5.
814
H. Dong, D. Du
2. Preliminaries We make a few preparations in this section. We use the notation in [18]. Let ω be a domain in some finite-dimensional space. Denote L p (ω; Rn ) and W pk (ω; Rn ) to be the usual Lebesgue and Sobolev spaces of functions from ω into Rn . Denote the norm of the spaces L p (ω; Rn ) and W pk (ω; Rn ) by · L p (ω) and · W pk (ω) respectively. As usual, for any measurable function u = u(x, t) and any p, q ∈ [1, +∞], we define u(x, t) L tp L qx := u(x, t) L qx t . Lp
For summable functions p, u = (u i ) and τ = (τi j ), we use the following differential operators: ∂u ∂u , u ,i = , ∇ p = ( p,i ), ∇u = (u i, j ), ∂t ∂ xi div u = u i,i , div τ = (τi j, j ), u = div∇u, ∂t u = u t =
which are understood in the sense of distributions. We use the notation of spheres, balls and parabolic cylinders, S(x0 , r ) = {x ∈ R4 | |x − x0 | = r }, S(r ) = S(0, r ), S = S(1); B(x0 , r ) = {x ∈ R4 | |x − x0 | < r }, B(r ) = B(0, r ), B = B(1); Q(z 0 , r ) = B(x0 , r ) × (t0 − r 2 , t0 ), Q(r ) = Q(0, r ), Q = Q(1). Also we denote mean values of summable functions as follows: 1 u(x, t) d x, [u]x0 ,r (t) = |B(r )| B(x0 ,r ) 1 (u)z 0 ,r = u dz. |Q(r )| Q(z 0 ,r ) We recall the following well-known interpolation inequality. Lemma 2.1. For any functions u ∈ W21 (Rd ) and any q ∈ [2, 2d/(d − 2)] and r > 0, d(q/4−1/2)
|u|q d x ≤ N (q) Br
q/2−d(q/4−1/2)
|∇u|2 d x Br
+ r −d(q−2)/2
|u|2 d x Br
q/2
|u|2 d x
.
(2.1)
Br
2.1. Leray-Hopf weak solutions. We denote C˙ 0∞ the space of all divergence-free infinitely differentiable vector fields with compact support in Rd . Let J˙ and J˙21 be the closure of C˙ 0∞ in the spaces L 2 and W21 , respectively. For any T ∈ (0, ∞], denote d Rd+1 T = (0, T ) × R .
By a Leray-Hopf weak solution of (1.1)-(1.2) in Rd+1 T , we mean a vector field u such that:
The Navier-Stokes Equations
815
i) u ∈ L ∞ (0, T ; J˙) ∩ L 2 (0, T ; J˙21 ); ii) The function t → Rd u(t, x) · w(x) d x is continuous on [0, T ] for any w ∈ L 2 ; iii) Equation (1.1) holds weakly in the sense that for any w ∈ C˙ 0∞ (Rd+1 T ), (−u · ∂t w − u ⊗ u : ∇w + ∇u : ∇w) d x dt = 0; (2.2) Rd+1 T
iv) The energy inequality: 1 1 |u(t, x)|2 d x + |∇u|2 d x ds ≤ |a(x)|2 d x 2 Rd 2 Rd Rdt holds for any t ∈ [0, T ], and we have u(t, ·) − a(·) L 2 → 0 as t → 0. It is well known that for any a ∈ J˙, there exists at least one Leray-Hopf weak solution of the Cauchy problem (1.1)-(1.2) on (0, ∞) × Rd (see [20 and 12]). 2.2. Suitable weak solutions. The definition of suitable weak solutions was introduced in [1] (see also [21 and 18]). Let ω be an open set in Rd . By a suitable weak solution of the Navier-Stokes equations on the set (0, T ) × ω, we mean a pair (u, p) such that i) u ∈ L ∞ (0, T ; J˙) ∩ L 2 (0, T ; J˙21 ) and p ∈ L d/2 ((0, T ) × ω); ii) u and p satisfy Eq. (1.1) in the sense of distributions (2.2). iii) For any t ∈ (0, T ) and for any nonnegative function ψ ∈ C0∞ (Rd ) vanishing in a neighborhood of the parabolic boundary {t = 0} × ω ∪ [0, T ] × ∂ω, we have the local energy inequality ess sup0<s≤t |u(s, x)|2 ψ(s, x) d x + 2 |∇u|2 ψ d x ds ω (0,t)×ω 2 2 ≤ {|u| (ψt + ψ) + (|u| + 2 p)u · ∇ψ} d x ds. (2.3) (0,t)×ω
2.3. Scaling invariant quantities. The following notation will be used throughout the article: 1 A(r ) = A(r, z 0 ) = ess supt0 −r 2 ≤t≤t0 d−2 |u(x, t)|2 d x, r B(x0 ,r ) 1 E(r ) = E(r, z 0 ) = d−2 |∇u|2 dz, r Q(z 0 ,r ) 1 C(r ) = C(r, z 0 ) = d−4/(d+1) |u|2(d+3)/(d+1) dz, r Q(z 0 ,r ) 1 D(r ) = D(r, z 0 ) = d−4/(d+1) | p|(d+3)/(d+1) dz. r Q(z 0 ,r ) We notice that these quantities are all invariant under the natural scaling (1.4). We shall use the following two lemmas involving these quantities.
816
H. Dong, D. Du
Lemma 2.2. Let ρ > 0, ε > 0 be constants and a pair (u, p) be a suitable weak solution of (1.1). Suppose Q(z 0 , ρ) ⊂ Rd+1 T and C(ρ) + D(ρ) ≤ ε2(d+3)/(d+1) . Then under the condition (1.5), we have A(ρ/2) + E(ρ/2) ≤ N ε2 . Proof. By a scaling argument, we may assume without loss of generality that ρ = 1. In the energy inequality (2.3), we put t = t0 and choose a suitable smooth cut-off function ψ such that d+1 ψ ≡ 0 in Rd+1 t0 \ Q(z 0 , 1), 0 ≤ ψ ≤ 1 in RT ,
ψ ≡ 1 in Q(z 0 , 1/2), |∇ψ| < N , |∂t ψ| + |∇ 2 ψ| < N in Rd+1 t0 . By using (2.3), we get A(1/2) + 2E(1/2) ≤ N
|u| dz + N 2
Q(z 0 ,1)
Q(z 0 ,1)
(|u|2 + 2| p|)|u| dz.
Due to Hölder’s inequality, one can obtain |u|2 dz ≤ N (C(1))(d+1)/(d+3) ≤ N ε2 , Q(z 0 ,1)
and
Q(z 0 ,1)
(|u|2 + 2| p|)|u| dz
≤N ≤N
Q(z 0 ,1)
Q(z 0 ,1)
|u|
d+3 2
dz 1
|u|d dz
d
2 d+3
Q(z 0 ,1)
|u|
2(d+3) d+1
+ | p|
d+3 d+1
d+1 d+3
dz
(C(1) + D(1))(d+1)/(d+3)
≤ N ε2 , where in the last inequality we used (1.5). The conclusion of Lemma 2.2 follows immediately. Lemma 2.3. Suppose γ ∈ (0, 1/2], ρ > 0 are constants and Q(z 0 , ρ) ∈ Rd+1 T . Then we have D(γρ) ≤ N γ −d+4/(d+1) C(ρ) + γ 4/(d+1) D(ρ) . (2.4) Proof. Let η(x) be a smooth function on Rd supported in the unit ball B(1), 0 ≤ η ≤ 1 ¯ and η ≡ 1 on B(2/3). It is known that for a.e. t ∈ (t0 −ρ 2 , t0 ), in the sense of distribution, one has
(2.5) p = Di j u i u j .
The Navier-Stokes Equations
817
For these t, we consider the decomposition p = px0 ,ρ + h x0 ,ρ in B(x0 , ρ), where px0 ,ρ is the Newtonian potential of
Di j u i u j η((x − x0 )/ρ). Then h x0 ,ρ is harmonic in B(x0 , 2ρ/3). Denote r = γρ. By using the Calderón-Zygmund estimate, one has | px0 ,ρ (x, t)|(d+3)/(d+1) dz Q(z 0 ,r ) ≤ | px0 ,ρ (x, t)|(d+3)/(d+1) dz Q(z ,ρ) 0 |u|2(d+3)/(d+1) dz. ≤ Q(z 0 ,ρ)
(2.6)
Since h x0 ,ρ is harmonic in B(x0 , 2ρ/3), any Sobolev norm of h x0 ,ρ in a smaller ball can be estimated by any of its L p norm in B(x0 , 2ρ/3). Thus, one obtains |h x0 ,ρ |(d+3)/(d+1) d x B(x0 ,r )
≤ Nr d sup |h x0 ,ρ |(d+3)/(d+1) d x B(x0 ,r )
d −d
≤ Nr ρ
B(x0 ,ρ)
|h x0 ,ρ |(d+3)/(d+1) d x.
Integrating (2.7) in t ∈ (t0 − r 2 , t0 ), we obtain |h x0 ,ρ |(d+3)/(d+1) dz Q(z 0 ,r ) ≤ Nr d ρ −d |h x0 ,ρ |(d+3)/(d+1) dz Q(z ,ρ) 0 d −d | p|(d+3)/(d+1) + | px0 ,ρ |(d+3)/(d+1) dz ≤ Nr ρ Q(z ,ρ) 0 d −d | p|(d+3)/(d+1) dz + N |u|2(d+3)/(d+1) dz, ≤ Nr ρ Q(z 0 ,ρ)
Q(z 0 ,ρ)
(2.7)
(2.8)
where we used (2.6) in the last inequality. By combining (2.6) and (2.8) we reach (2.4). The lemma is proved. 2.4. Strong solutions and spatial analyticity. We recall the following local strong solvability of (1.1)-(1.2) (see, for example, [9,13,33 and 14]), and the spatial analyticity of strong solutions (see, for example, [10 and 4]). Proposition 2.4. For any divergence-free initial data a ∈ L p (Rd ), p ≥ d, the Cauchy problem (1.1)-(1.2) has a unique strong solution u ∈ C([0, δ); L p (Rd )) for some δ > 0. Moreover, u is infinitely differentiable and spatial analytic for t ∈ (0, δ).
818
H. Dong, D. Du
3. A Blowup Procedure We begin this section by proving the following key estimate, which shows if the quantities C and D are sufficiently small in a cylinder, then they must be also small in any sub-cylinder. Proposition 3.1. Let (u, p) be a pair of suitable weak solution of (1.1). Suppose that Q(z 0 , ρ) ⊂ Rd+1 and the condition (1.5) holds. Then for any ε0 > 0 there exists an T ε∗ > 0 depending only on ε0 and d such that if C(ρ, z 0 ) + D(ρ, z 0 ) ≤ ε∗ , then we have C(r, z 1 ) + D(r, z 1 ) ≤ ε0 for any z 1 ∈ Q(z 0 , ρ/2) and r ∈ (0, ρ/2). Proposition 3.1 follows immediately from the next lemma by using a covering argument and an iteration. Lemma 3.2. Let (u, p) be a suitable weak solution of (1.1). Suppose that Q(z 0 , ρ) ⊂ Rd+1 and condition (1.5) holds. Then there exist universal constants ε∗ > 0 and γ ∈ T (0, 1/4] such that for any ε ∈ (0, ε∗ ] if C(ρ, z 0 ) + D(ρ, z 0 ) ≤ ε, then we have C(γρ, z 0 ) + D(γρ, z 0 ) ≤ ε. Proof. As before, one may assume ρ = 1. We prove by contradiction. Let γ ∈ (0, 1/4] be a constant to be specified later. Suppose there exist a decreasing sequence {εk } converging to 0, and a sequence of pairs of suitable weak solutions (u k , pk ) such that 2(d+3)/(d+1)
C(1, z 0 , u k , pk ) + D(1, z 0 , u k , pk ) ≤ εk C(γ , z 0 , u k , pk ) + D(γ , z 0 , u k , pk ) >
,
(3.1)
2(d+3)/(d+1) εk .
(3.2)
By Lemma 2.2, one also has A(1/2, z 0 , u k , pk ) + B(1/2, z 0 , u k , pk ) ≤ N εk2 ,
(3.3)
where the constant N is independent of k. We define (vk , qk ) = (u k /εk , qk /εk ). Then (vk , qk ) is a suitable weak solution of ∂t vk + εk vk · ∇vk − vk + ∇qk = 0, div vk = 0.
(3.4)
From (3.1), (3.2) and (3.3), we get C(1, z 0 , vk , qk ) + D(1, z 0 , vk , qk ) ≤ 1, C(γ , z 0 , vk , qk ) + D(γ , z 0 , vk , qk ) > 1, A(1/2, z 0 , vk , qk ) + B(1/2, z 0 , vk , qk ) ≤ N .
(3.5) (3.6) (3.7)
The Navier-Stokes Equations
819
By using (3.7), applying the interpolation inequality (2.1) with q = 2(d + 2)/d and integrating in t, we bound vk L 2(d+2)/d (Q(z 0 ,1/2)) by N . Thus by Hölder’s inequality, vk · ∇vk L (d+2)/(d+1) (Q(z 0 ,1/2)) ≤ N . Due to the coercive estimate for the Stokes system (see, for instance, [29]) with a suitable cut-off function, we reach
2(d+2) d+2 d+2 d+2 |vk | d + |∂t vk | d+1 + |D 2 vk | d+1 + |∇qk | d+1 dz ≤ N , Q(z 0 ,1/3)
where the constant N is independent of k. Thanks to the compact embedding theorem and (3.5), there exist v ∈ L 2(d+3)/(d+1) (Q(z 0 , 1/3)), q ∈ L (d+3)/(d+1) (Q(z 0 , 1/3)), and a subsequence, which is still denoted by (vk , qk ) such that vk → v in L 2(d+3)/(d+1) (Q(z 0 , 1/3)), qk q in L (d+3)/(d+1) (Q(z 0 , 1/3)).
(3.8)
This together with (3.4) implies ∂t v − v + ∇q = 0, div v = 0.
(3.9)
Moreover, v L 2(d+3)/(d+1) (Q(z 0 ,1/3)) + q L (d+3)/(d+1) (Q(z 0 ,1/3)) ≤ N . By the classical estimate of the Stokes system, one has sup
Q(z 0 ,1/4)
|v| ≤ N ,
which gives C(γ , z 0 , v, q) ≤ N γ 4/(d+1) . This contradicts (3.6) and (3.8), if we choose γ sufficiently small. The lemma is proved. Lemma 3.3. Under the assumptions of Theorem 1.2, we have u(t, ·) L d (Rd ) ≤ N ,
(3.10)
2 d u ∈ L 4 (Rd+1 T ), ∂t u, D u, ∇ p ∈ L 4/3 ((δ, T ) × R ),
(3.11)
for each t ∈ [0, T ], and
for any δ ∈ (0, T ). Moreover, (u, p) is a suitable weak solution of (1.1) in Rd+1 T .
820
H. Dong, D. Du
Proof. The first assertion is due to (1.5) and the weak continuity of Leray-Hopf weak solutions. By using Lemma 2.1 with q = 2d/(d − 2) and r = ∞, we have u L t L x
d+1 2 2d/(d−2) (RT )
≤ N,
which together with (3.10) and Hölder’s inequality yields u L 4 (Rd+1 ) ≤ N , u · ∇u L 4/3 (Rd+1 ) ≤ N . T
T
Thus, (3.11) follows from the coercive estimate for the Stokes system. Finally, due to x (Rd+1 ). the pressure equation (2.5) and the Calderón-Zygmund estimate, p ∈ L t∞ L d/2 T Therefore, it is clear that (u, p) is a suitable weak solution. The lemma is proved. Remark 3.4. From (3.11), one can infer that u ∈ C((0, T ]; L 4/3 (B R )) for any R > 0. For this combined with (3.10) and Hölder’s inequality, we get u ∈ C((0, T ]; L p (B R )) for any p ∈ [1, d). Because of the local strong solvability and the weak-strong uniqueness (see, for instance, [32]), we know that u is regular for t ∈ (0, T0 ) for some T0 ∈ (0, T ]. Suppose T0 is the first blowup time of u, and Z 0 = (T0 , X 0 ) is a singular point. Take a decreasing sequence {λk } converging to 0. We rescale the pair (u, p) at time T0 and define u k (t, x) = λk u(T0 + λ2k t, X 0 + λk x), pk (t, x) = λ2k p(T0 + λ2k t, X 0 + λk x). Then for each k = 1, 2, . . ., (u k , pk ) is a suitable weak solution of (1.1) and u k is smooth for t ∈ (−λ−2 k T0 , 0). We finish this section by constructing a limiting solution. Proposition 3.5. i) There is a subsequence of {(u k , pk )}, which is still denoted by {(u k , pk )}, such that u k → u ∞ in C([t0 − 1, t0 ]; L q1 (B(x0 , 1))), pk p∞ in
x L qt 2 L d/2 (Q(z 0 , 1))
(3.12) (3.13)
for any z 0 ∈ (−∞, 0] × Rd , q1 ∈ [1, d) and q2 ∈ [1, ∞). ii) Furthermore, (u ∞ , p∞ ) is a suitable weak solution of (1.1) in (−∞, 0) × Rd , and u ∞ ∈ L qt 2 L dx ((−T1 , 0) × Rd ),
x p∞ ∈ L qt 2 L d/2 ((−T1 , 0) × Rd )
for any T1 > 0 and q2 ∈ [1, ∞). Proof. First we fix a z 0 ∈ (−∞, 0] × Rd . Since pk , k = 1, 2, . . . have a uniform bound x ((t − 1, t ) × Rd ) norm, and consequently a uniform bound of their of the L t∞ L d/2 0 0 x t L q2 L d/2 (Q(z 0 , 1)) norms, there is a subsequence, which is still denoted by { pk }, such that (3.13) holds. Similarly, u k L t∞ L x (Q(z 0 ,3)) ≤ u k L t∞ L x ((t0 −9,t0 )×Rd ) ≤ N , d
d
where N is independent of k. By Lemma 2.2, we have A(2, z 0 , u k , pk ) + B(2, z 0 , u k , pk ) ≤ N .
(3.14)
The Navier-Stokes Equations
821
Now following the proof of Lemma 3.3, we deduce u k ∈ L 4 (Q(z 0 , 3/2)), ∂t u k , D 2 u k , ∇ pk ∈ L 4/3 (Q(z 0 , 3/2)) with uniform norms. Therefore, we can find a subsequence still denoted by {u k } such that u k → u ∞ in C([t0 − 1, t0 ]; L 4/3 (B(x0 , 1))). This together with (3.14) gives (3.12) by using the Hölder’s inequality. To finish the proof of Part i), it suffices to use a Cauchy diagonal argument. Part ii) then follows from Part i) and (3.14). 4. Schoen’s Trick The objective of this section is to establish the following regularity criterion. Theorem 4.1. Suppose u is a regular solution of (1.1) in Q(z 0 , ρ1 ). Then for any K > 0 there exists an ε1 = ε1 (d, K ) > 0 such that the following is true. If any z 1 ∈ Q(z 0 , ρ1 /2), ρ ∈ (0, ρ1 /2) we have C(ρ, z 1 ) ≤ ε1 , p L t∞ L x
d/2 (Q(z 1 ,ρ))
≤ K,
(4.1)
then sup
Q(z 0 ,ρ1 /4)
|u(z)| < N (ρ1 , d).
Proof. We prove the theorem by using Schoen’s trick. Let δ ∈ (0, ρ12 /4) be a number and denote d(z) = (t0 + ρ12 /4 − t)1/2 ,
Mδ =
max
¯ 0 ,ρ1 /2)∩{t≤t0 −δ} Q(z
d(z)|u(z)|.
If for all δ ∈ (0, ρ12 /4) we have Mδ ≤ 2, then there’s nothing to prove. Otherwise, ¯ 0 , ρ1 /2) ∩ {t ≤ t0 − δ}, suppose for some δ and z 1 ∈ Q(z M := Mδ = |u(z 1 )|d(z 1 ) > 2. Let r1 = d(z 1 )/M < d(z 1 )/2. We make the scaling as follows: u(y, ¯ s) = r1 u(r12 s + t1 , r1 y + x1 ), p(y, ¯ s) = r1 p(r12 s + t1 , r1 y + x1 ). The pair (u, ¯ p) ¯ satisfies (1.1) in Q(0, 1) and u¯ is smooth. Obviously, sup |u| ¯ ≤ 2, |u(0, ¯ 0)| = 1.
(4.2)
Q(0,1)
By the scaling-invariant property of the quantity C, in what follows we view it as the object associated to (u, ¯ p) ¯ at the origin. For any ρ ∈ (0, 1], from (4.1) we have C(ρ) ≤ ε1 , p ¯ L t∞ L x (Q(1)) ≤ K . d/2
(4.3) (4.4)
822
H. Dong, D. Du
We decompose p¯ as in the proof of Lemma 2.3: p¯ = p¯ 0,1 + h¯ 0,1 . Because of (4.2), we have | p¯ 0,1 |4(d+2) dz ≤ N .
(4.5)
Q(0,1)
Since h¯ 0,1 (t, ·) is harmonic in B(2/3) for a.e. t ∈ (−1, 0), it holds that |h¯ 0,1 |4(d+2) dz Q(0,1/2)
≤N ≤N
0
sup |h¯ 0,1 (t, ·)|4(d+2) dt
−1/4 B(1/2) 0
−1/4
≤N
|h¯ 0,1 |d/2 d x
8(d+2)/d
B(2/3)
| p¯ 0,1 |
4(d+2)
dt
8(d+2)/d | p(t, ¯ ·)|
dz + sup
Q(0,1)
t∈(0,1)
d/2
dx
B(0,1)
≤ N,
(4.6)
where in the last inequality we used (4.4) and (4.5). Thus, we deduce from (4.5) and (4.6) that | p| ¯ 4(d+2) dz ≤ N . (4.7) Q(0,1/2)
Now we note that (u, ¯ p) ¯ satisfies the equation ¯ − ∇( p) ¯ ∂t u¯ − u¯ = div(u¯ ⊗ u) in Q(0, 1). Owing to (4.2), (4.7) and the classical Sobolev space theory of parabolic equations, we have 1,1/2
u¯ ∈ W4(d+2) (Q(0, 1/3)), u ¯ W 1,1/2
4(d+2) (Q(0,1/3))
≤ N.
By the Sobolev embedding theorem (see [16]), we obtain ¯ C 1/4 (Q(0,1/4)) ≤ N , u¯ ∈ C 1/4 (Q(0, 1/4)), u where N is a universal constant depending only on d and K . Therefore, we can find δ1 < 1/5 independent of ε1 such that |u(x, ¯ t)| ≥ 1/2 in Q(0, δ1 ).
(4.8)
Now we choose ε1 small enough which makes (4.8) and (4.3) a contradiction. The theorem is proved.
The Navier-Stokes Equations
823
5. Proof of Theorem 1.2 and 1.3 We finish the proof of Theorem 1.2 in this section. Let u k , pk , u ∞ and p∞ be the functions constructed in Sect. 3. First we verify that the assumptions of Theorem 4.1 hold for (u k , pk ) when k is sufficiently large and the parabolic cylinder is far away from the origin. Lemma 5.1. For any ε2 > 0 and T1 ≥ 1, we can find R ≥ 1 such that, for any z 0 ∈ (−T1 − 1, 0] × (Rd \ B R+1 ), lim sup C(1, z 0 , u k , p∞ ) ≤ ε2 .
(5.1)
k→∞
Proof. Due to Proposition 3.5 ii), we can find R large such that |u ∞ |d dz (−T1 −2,0)×(Rd \B R )
is sufficiently small. This together with Proposition 3.5 i) proves the lemma.
Lemma 5.2. For any ε3 > 0 and T1 ≥ 1, we can find R ≥ 1 and ρ3 ∈ (0, 1/2] such that, for any ρ ∈ (0, ρ3 ] and z 0 ∈ (−T1 − 1, 0] × (Rd \ B R+2 ), lim sup (C(ρ, z 0 , u k , pk ) + D(ρ, z 0 , u k , pk )) ≤ ε3 .
(5.2)
k→∞
Proof. The lemma is a consequence of Lemma 5.1, 2.3 and Proposition 3.1. Indeed, since D(1, z 0 , u k , pk ) has a uniform bound, for any ε > 0, we can choose γ small in (2.4), then ε2 small in (5.1) and R large such that lim sup (C(γ , z 0 , u k , pk ) + D(γ , z 0 , u k , pk )) ≤ ε k→∞
holds for any z 0 ∈ (−T1 − 1, 0] × (Rd \ B R+1 ). Now it suffices to choose ε small depending on ε3 and apply Proposition 3.1. We finish the proof by setting ρ3 = γ /2. Next we show that u ∞ is identically equal to zero. Proposition 5.3. Under the assumptions of Theorem 1.2, let (u ∞ , p∞ ) be the suitable weak solution constructed in Sect. 3. Then, u ∞ (t, ·) ≡ 0 ∀t ∈ (−∞, 0). Proof. Let ε1 be the constant in Theorem 4.1. Let T1 ≥ 1 be a number. Owing to Lemma 5.2, we can find R ≥ 1 and ρ3 ∈ (0, 1/2] such that, for any ρ ∈ (0, ρ3 ] and z 0 ∈ (−T1 − 1, 0] × (Rd \ B R+2 ) estimate (5.2) holds with ε1 /2 in place of ε3 . Moreover, we recall that for each K , pk L t∞ L x
d/2 ((−∞,0)×R
d)
≤ N (d)K .
Thus Theorem 4.1 yields that lim sup
sup
k→∞ Q(z 0 ,ρ3 /4)
|u k (z 0 )| ≤ N (d, ρ3 )
824
H. Dong, D. Du
for any z 0 ∈ [−T1 − 1, 0) × (Rd \ B R+2 ). Now by Proposition 3.5, we obtain |u ∞ (z)| ≤ N (d, ρ3 ) for a.e. z ∈ [−T1 − 1, 0) × (Rd \ B R+2 ). Upon using the regularity results for linear Stokes systems, one can estimate higher derivatives |D j u ∞ (z)| ≤ N (d, j, ρ3 )
(5.3)
for any j ≥ 1 and a.e. z ∈ [−T1 , 0) × (Rd \ B R+3 ). We now claim that u ∞ (0, ·) ≡ 0 by adapting the argument in the proof of Theorem 1.4 [5]. For any x0 ∈ Rd , by using (3.12), |u ∞ (x, 0)| d x B(x0 ,1) ≤ |u k (x, 0) − u ∞ (x, 0)| d x + |u k (x, 0)| d x B(x0 ,1)
B(x0 ,1)
≤ u k − uC([−1,1];L 1 (B(x0 ,1))) + N (d) = u k − uC([−1,1];L 1 (B(x0 ,1))) + N (d)
1/d |u k (x, 0)| d x d
B(x0 ,1)
B(λk x0 ,λk )
1/d |u(y, 0)|d dy
.
The right-hand side of the above inequality goes to zero as k → ∞, which proves the claim. Because of (5.3), the vorticity ω = curl u ∞ satisfies the differential inequality |∂t ω − ω| ≤ N (|ω| + |∇ω|) on (−T1 , 0] × (Rd \ B R+3 ). Thanks to the backward uniqueness theorem proved in [5] (see also [6]), we reach ω(z) = 0 on (−T1 , 0] × (Rd \ B R+3 ).
(5.4)
Now we fix a t0 ∈ (−T1 , 0). Take an increasing sequence {tk } ⊂ (−T1 , 0) converging to t0 . For each k, we consider Eq. (1.1) with initial data u ∞ (tk , ·). By Proposition 2.4, one can locally find a strong solution vk ∈ C([tk , tk + δk ); L d (Rd )) for some small δk , and vk (t, ·) is spatial analytic for t ∈ (tk , tk + δk ). By the weak-strong uniqueness, vk ≡ u ∞ for t ∈ [tk , tk + δk ). Therefore, ω(t, ·) is also spatial analytic for t ∈ (tk , tk + δk ). Because of (5.4), we get ω(z) = 0 on (tk , tk + δk ) × Rd , which implies that u ∞ ≡ 0 in the same region. In particular, there exists a sequence {sk } converging to t0 such that tk < sk ≤ t0 , u ∞ (sk , ·) ≡ 0. This together with the weak continuity of u ∞ yields that u ∞ (t0 , ·) ≡ 0. Since t0 ∈ (−T1 , 0) is arbitrary and T1 ≥ 1 is also arbitrary, we complete the proof of the theorem. We are ready to prove Theorem 1.2.
The Navier-Stokes Equations
825
Proof of Theorem 1.2. We prove the theorem in three steps. Step 1. First we show that u is regular for t ∈ (0, T ]. Thanks to Proposition 3.5 and 5.3, u k → 0 in C([−3, 0]; L 2(d+3)/(d+1) (B(3))). Also recall that D(1, z 0 , u k , pk ) has a uniform bound. Following the proof of Lemma 5.2 we have: for any ε4 > 0, there is a ρ4 ∈ (0, 1/2] and a positive integer k0 such that, for any ρ ∈ (0, ρ3 ] and z 0 ∈ (−2, 0] × B(2), C(ρ, z 0 , u k0 , pk0 ) + D(ρ, z 0 , u k0 , pk0 ) ≤ ε4 . We choose ε4 sufficiently small and apply Theorem 4.1 to get sup
(−1,0)×B(1)
|u k0 | < ∞,
which implies that sup
Q(Z 0 ,λk0 )
|u| < ∞.
This contradicts the assumption that (T0 , X 0 ) is a blowup point. Therefore, u is regular for t ∈ (0, T ]. Step 2. We bound the sup norm of u in this step. Fix a δ ∈ (0, T ). Since u L t∞ L x ((0,T )×Rd ) ≤ N , p L t∞ L x
d/2 ((0,T )×R
d
d)
≤ N,
by the same reasoning as at the beginning of the proof of Proposition 5.3, we see that there exists a large R ≥ 1 such that sup [δ,T )×(Rd \B(R))
|u| ≤ N .
(5.5)
¯ Next we estimate the sup norm of u in [δ, T )× B(R). Fix a z 0 = (t0 , x0 ) in [δ, T ]× B(R). In the construction of u k , we replace (T0 , X 0 ) by (t0 , x0 ). By the same reasoning as in the first step, for some ε = ε(T0 , X 0 ) > 0, we have sup |u| < ∞.
Q(z 0 ,ε)
¯ By the compactness of [δ, T ] × B(R), it holds that sup
¯ [δ,T )× B(R)
|u| ≤ N .
This together with (5.5) yields sup [δ,T )×Rd
|u| ≤ N .
Step 3. Finally we prove the uniqueness. Owing to the local strong solvability of (1.1), we have u ∈ L d+2 (Rd+1 T1 ) for some T1 ∈ (0, T ). On the other hand, for t ∈ [T1 , T ] the solution is uniformly bounded and belongs to L t∞ L dx ((T1 , T ) × Rd ), thus u ∈ L d+2 (Rd+1 T ). The uniqueness then follows.
826
H. Dong, D. Du
Now we give Proof of Theorem 1.3. Thanks to Theorem 1.2, it remains to prove (1.7). Let λ > 0 be a constant to be specified later. We define u λ (t, x) = λu(λ2 t, λx), pλ (t, x) = λ2 p(λ2 t, λx). Then (u λ , pλ ) is also a Leray-Hopf weak solution of (1.1) in (0, ∞) × Rd , and u λ satisfies (1.6) with the same constant K due to the scaling invariant property. By the proof of Lemma 3.3, we have u λ ∈ L 4 ((0, ∞) × Rd ). Thus for any ε > 0, there is a T > 0 such that u λ L 4 ((T,∞)×Rd ) ≤ ε. Let ε1 be the constant in Theorem 4.1. Upon using Lemma 2.3 and Proposition 3.1, we can find a large T = Tλ such that C(ρ, z 0 , u λ , pλ ) + D(ρ, z 0 , u λ , pλ ) ≤ ε1 , for any ρ ∈ (0, 1/2] and z 0 ∈ [T, ∞) × Rd . Owing to Theorem 4.1, we conclude sup
Q(z 0 ,1/4)
|u λ (z)| < N ,
where N = N (d, K ) is independent of λ. Therefore, sup t≥λ2 T,x∈Rd
|u(t, x)| < N /λ.
Sending λ → ∞ yields the desired result. The theorem is proved.
Acknowledgement. The authors would like to express their sincere gratitude to Vladmir Šverák for very helpful comments and suggestions. The authors are also grateful to Gabriel Koch and the referee for useful comments on a previous version of the manuscript.
References 1. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982) 2. Cheskidov, A., Shvydkoy, R.: On the regularity of weak solutions of the 3D Navier-Stokes equations in −1 B∞,∞ . http://arxiv.org/abs/0708.3067v2[math:AP], 2007 3. Dong, H., Du, D.: Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time. Commun. Math. Phys. 273(3), 785–801 (2007) 4. Dong, H., Li, D.: Optimal local smoothing and analyticity rate estimates for the generalized Navier-Stokes equations. Commun. Math. Sci. 7(1), 67–80 (2009) 5. Escauriaza, L., Seregin, G., Šverák, V.: L 3,∞ -solutions of Navier-Stokes equations and backward uniqueness (In Russian). Usp. Mat. Nauk 58(2)(350), 3–44 (2003); translation in Russ. Math. Surv. 58(2), 211–250 (2003) 6. Escauriaza, L., Seregin, G., Šverák, V.: Backward uniqueness for parabolic equations. Arch. Rat. Mech. Anal. 169(2), 147–157 (2003) 7. Gallagher, I., Iftimie, D., Planchon, F.: Asymptotics and stability for global solutions to the Navier-Stokes equations. Ann. Inst. Fourier (Grenoble) 53(5), 1387–1424 (2003) 8. Giga, Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differ. Eq. 62, 186–212 (1986) 9. Giga, Y., Miyakawa, T.: Solution in L r of the Navier-Stokes initial value problem. Arch. Rat. Mech. Anal. 89, 267–281 (1985) 10. Giga, Y., Sawada, O.: On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem. In: Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday. 1,2, Dordrecht: Kluwer Acad. Publ., 2003, pp. 549–562
The Navier-Stokes Equations
827
11. Gustafson, S., Kang, K., Tsai, T.: Interior regularity criteria for suitable weak solutions of the NavierStokes equations. Commun. Math. Phys. 273(1), 161–176 (2007) 12. Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951) 13. Kato, T.: Strong L p -solutions of the Navier-Stokes equation in Rm with applications to weak solutions. Math. Z. 187, 471–480 (1984) 14. Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001) 15. Ladyzhenskaya, O.: On the uniqueness and smoothness of generalized solutions to the Navier-Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5, 169–185 (1967); English transl.: Sem. Math. V. A. Steklov Math. Inst. Leningrad 5, 60–66 (1969) 16. Ladyzhenskaya, O., Solonnikov, V., Ural’tseva, N.: Linear and quasi-Linear equations of parabolic type. Moscow: Nauka, 1967 (in Russian); English translation: Providence, RI: Amer. Math. Soc., 1968 17. Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flows. 2nd edition, London: Gordon and Breach, 1969 18. Ladyzhenskaya, O., Seregin, G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999) 19. Mikhailov, A., Shilkin, T.: L 3,∞ -solutions to the 3D-Navier-Stokes system in the domain with a curved boundary (English, Russian summary). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 336, 133–152 (2006); translation in J. Math. Sci. (N. Y.) 143(2), 2924–2935 (2007) 20. Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933) 21. Lin, F.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51, 241–257 (1998) 22. Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959) 23. Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66, 535–552 (1976) 24. Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55, 97–112 (1977) 25. Scheffer, V.: The Navier-Stokes equations on a bounded domain. Commun. Math. Phys. 73, 1–42 (1980) 26. Seregin, G.: On smoothness of L 3,∞ -solutions to the Navier-Stokes equations up to boundary. Math. Ann. 332(1), 219–238 (2005) 27. Seregin, G., Sverak, V.: On smoothness of suitable weak solutions to the Navier-Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 306, 186–198 (2003); translation in J. Math. Sci. (N. Y.) 130(4), 4884–4892 (2005) 28. Serrin, J.: On the interior regularity of weak solutions of Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962) 29. Maremonti, P., Solonnikov, V.: On estimates for the solutions of the nonstationary Stokes problem in S. L. Sobolev anisotropic spaces with a mixed norm. (Russian. English, Russian summary) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222, 124–150 (1995); translation in J. Math. Sci. (New York) 87(5), 3859–3877 (1997) 30. Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear Problems, R. Langer, ed., Madison, WI: Univ. of Wisconsin Press, 1963, 69–98 31. Struwe, M.: On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math. 41, 437–458 (1988) 32. von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Braunschweig: Vieweg, 1985 33. Taylor, M.: Analysis on Morrey spaces and applications to Navier-Stokes equation. Comm. Part. Differ. Eqs. 17, 1407–1456 (1992) Communicated by P. Constantin
Commun. Math. Phys. 292, 829–870 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0868-3
Communications in
Mathematical Physics
An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field Fanny Delebecque-Fendt, Florian Méhats IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. E-mail: [email protected]; [email protected] Received: 6 February 2009 / Accepted: 28 April 2009 Published online: 24 July 2009 – © Springer-Verlag 2009
Abstract: We study the limiting behavior of a singularly perturbed Schrödinger-Poisson system describing a 3-dimensional electron gas strongly confined in the vicinity of a plane (x, y) and subject to a strong uniform magnetic field in the plane of the gas. The coupled effects of the confinement and of the magnetic field induce fast oscillations in time that need to be averaged out. We obtain at the limit a system of 2-dimensional Schrödinger equations in the plane (x, y), coupled through an effective selfconsistent electrical potential. In the direction perpendicular to the magnetic field, the electron mass is modified by the field, as the result of an averaging of the cyclotron motion. The main tools of the analysis are the adaptation of the second order long-time averaging theory of ODEs to our PDEs context, and the use of a Sobolev scale adapted to the confinement operator. 1. Introduction 1.1. The singularly perturbed problem. Many electronic devices are based on the quantum transport of a bidimensional electron gas (2DEG) artificially confined in heterostructures at nanometer scales, see e.g. [2,4,20,31]. In this article, we derive an asymptotic model for the quantum transport of a 2DEG subject to a strong uniform magnetic field which is parallel to the plane of the gas. The aim of this paper is to understand how the cyclotron motion competes with the effects of the potential confining the electrons and the nonlinear effects of the selfconsistent Poisson potential. Our tool is an asymptotic analysis from a singularly perturbed Schrödinger-Poisson system towards a reduced model of bidimensional quantum transport. In particular, we generalize in this context the notion of cyclotron effective mass, usually explicitly calculated in the simplified situation of a harmonic confinement potential [20,28]. Our starting model is thus the 3D Schrödinger-Poisson system, singularly perturbed by a confinement potential and the strong magnetic field. The three-dimensional space variables are denoted by (x, y, z) and the associated canonical basis of R3 is denoted by
830
F. Delebecque-Fendt, F. Méhats
(ex , ey , ez ). The particles are subject to three effects: a confinement potential depending on the z variable, a uniform magnetic field applied to the gas along the ey axis, and the selfconsistent Poisson potential. Given a small parameter ε > 0, which is the typical extension of the 2DEG in the z direction, our starting model is the following dimensionless Schrödinger-Poisson system: i∂t ε =
1 2 1 2 2 −∂ + B z + V (z) ε − 2i Bz∂x ε − x,y ε + V ε ε , c z 2 ε ε ε (0, x, y, z) = 0 (x, y, z), 1 V ε (t, x, y, z) = ∗ | ε |2 , 4πr ε
(1.1) (1.2) (1.3)
where we have denoted r ε (x, y, z) =
x2 + y 2 + ε2 z 2 .
(1.4)
The scaling is discussed in the next subsection. This system describes the transport of electrons under the action of: – The applied confinement potential ε12 Vc (z), nonnegative, such that Vc (z) → +∞ as |z| → +∞. The precise assumptions of this potential are made below in Assumptions 1.1 and 1.2. – The applied uniform magnetic field Bε ey (with B > 0 fixed), which derives from the magnetic potential 1ε Bzex . We have chosen to work in the Landau gauge. – The Poisson selfconsistent potential V ε . Note that (1.1) is equivalent to 1 2 Bz 2 ε ε i∂t = 2 −∂z + Vc (z) + i∂x − − ∂y2 ε + V ε ε . ε ε ε
(1.5)
The goal of this work is to exhibit an asymptotic system for (1.1), (1.2), (1.3) as ε → 0. Let us end this subsection with short bibliographical notes. In a linear setting, quantum motion constraint on a manifold has been studied for a long time by several authors, see [15,18,21,30] and references therein. Nonlinear situations were studied more recently. The approximation of the Schrödinger-Poisson system with no magnetic field was studied when the electron gas is constrained in the vicinity of a plane in [7,25] and when the gas is constrained on a line in [5]. When the nonlinearity depends locally on the density, as for the Gross-Pitaevskii equation, asymptotic models for confined quantum systems were studied in [8,6,12]. In the classical setting, collisional models in situations of strong confinement have been studied in [17]. Finally, let us draw a parallel with the problem of homogenization of the Schrödinger equation in a large periodic potential, studied in [1 and 29]. At the limit ε → 0, as noted above, we will obtain an homogenized system which takes the form of bidimensional Schrödinger equations with an effective mass in the x direction. However, this phenomenon is due to an averaging of the cyclotron motion induced by a strong magnetic field, and is not exactly the same notion as the usual effective mass for the transport in a lattice or in a crystal. Nevertheless, it is interesting to observe that the scaling used in [1,29] in the case of a strong periodic potential is similar to the strong confinement scaling used in the present paper.
2DEG in a Strong Magnetic Field
831
1.2. The physical scaling. In order to clarify the physical assumptions underlying our singularly perturbed system, let us derive (1.1), (1.2), (1.3) from the Schrödinger-Poisson system written in physical variables. This system reads as follows: 2 eB i∇ − zex + eVc + eV, c e ∗ ||2 . V= 4π x2 + y2 + z2
i∂t =
1 2m
(1.6) (1.7)
Each dimensionless quantity in (1.1), (1.2), (1.3) is the associated physical quantity normalized by a typical scale: x=
x y z ||2 Vc V , y = , z = , | ε |2 = , Vε = , , Vc = x y z N Vc V
B=
B B
. (1.8)
Now we introduce two energy scales in this problem: a strong energy E con f , which will be the energy of the confinement in z and of the magnetic effects, and a transport energy E transp , which will be the typical energy of the longitudinal transport in (x, y) and also of the selfconsistent effects. We introduce the following small dimensionless parameter: ε=
E transp E con f
1/2 1.
(1.9)
Then our scaling assumptions are the following. We set to the scale E con f the confinement potential, the magnetic energy and the kinetic energy along z: E con f
1 := eVc = m 2
eB mc
2 z2 =
2 2mz 2
,
(1.10)
and we set to the scale E transp the selfconsistent potential energy, the kinetic energies along x and y, and we finally choose a time scale adapted to this energy: E transp := eV =
2 e2 N x z 2 = = = . 2 t 2mx 2my 2
(1.11)
By inserting (1.8) in (1.6), (1.7), then by using (1.9), (1.10) and (1.11), we obtain directly our singularly perturbed problem (1.1), (1.3). Note that (1.10) and (1.11) imply that ε is also the ratio between the transversal and the longitudinal space scales: ε=
z z = . x y
1.3. Heuristics in a simplified case. In this section, we analyze a very simplified situation where analytic calculations can be directly done. We assume here that Vc is a harmonic confinement potential and we neglect the Poisson potential V ε . We formally analyze the heuristics in this simplified case, that will be further compared to our result obtained in the general case.
832
F. Delebecque-Fendt, F. Méhats
We thus consider here a new system, similar to (1.1) where we prescribe Vc (z) = α 2 z 2 , α > 0, and where the Poisson potential V ε is replaced by 0: 1 1 i∂t ε = 2 −∂z2 + (α 2 + B 2 )z 2 ε − 2i Bz∂x ε − x,y ε , (1.12) ε ε ε (0, x, y, z) = 0 (x, y, z). (1.13) In this situation, there is a trick which enables us to transform the equation. Indeed, by remarking that −∂z2 + (α 2 + B 2 )z 2 − 2i Bεz∂x − ε2 ∂x2 2 B α2 2 2 2 = −∂z + (α + B ) z − 2 iε∂ − ε2 ∂x2 , x α + B2 α2 + B 2 we obtain that (1.12) is equivalent to 2 1 B α2 ε i∂t ε = 2 −∂z2 + (α 2 + B 2 ) z − 2 iε∂ − ∂ 2 ε − ∂y2 ε . x ε α + B2 α2 + B 2 x (1.14) Introduce now the following operator: for a function u ∈ L 2 (R3 ), we set B ε −1 ( u)(x, y, z) = Fx εξ ) , Fx u(ξ, y, z + 2 α + B2 where Fx denotes the Fourier transform in the x variable. Note that this operator ε is unitary on L 2 (R3 ) and commutes with ∂x and ∂y . Hence, we deduce from (1.14) and by direct calculations that the function u ε = ε ε satisfies the following system: i∂t u ε =
1 ε α2 u − ∂ 2 u ε − ∂y2 u ε , u ε (t = 0) = ε 0 , H z ε2 α2 + B 2 x
where
z = −∂z2 + (α 2 + B 2 )z 2 . H Let us now filter out the oscillations by introducing the new unknown
z /ε2 )u ε .
ε = exp(it H
z /ε2 ) commutes with ∂x , ∂y and, finally, the following equaAgain, the operator exp(it H tion is equivalent to (1.12): i∂t ε = −
α2
α2 ∂ 2 ε − ∂y2 ε , ε (t = 0) = ε 0 . + B2 x
(1.15)
As ε → 0, it is not difficult to see that, for sufficiently smooth initial data, we have ε 0 → 0 . Therefore, one can show that, in adapted functional spaces, we have
ε → as ε → 0, with solution of the limit system: i∂t = −
α2 ∂ 2 − ∂y2 , (t = 0) = 0 . α2 + B 2 x
(1.16)
2DEG in a Strong Magnetic Field
833
This equation is a bidimensional Schrödinger equation with an anisotropic operator that can be interpreted as follows. Whereas, as expected, the dynamics in the y is not perturbed by the magnetic field (since it is parallel to y), in the x direction the electrons are 2 2 transported as if their mass was augmented by a factor α α+B > 1. This coefficient is 2 called the (dimensionless) electron cyclotron mass [20,28]. In this article, the model that we want to treat is the nonlinear system (1.1), (1.3), with a general confinement potential Vc instead of α 2 z 2 and the selfconsistent Poisson potential. Consequently, it is not possible to simplify the equation (1.1) by the above trick. Moreover, the potential V ε depends on the z variable and on the function ε itself. Therefore, one has to be careful for instance when filtering out the fast oscillations by
z /ε2 ), since in this nonlinear framework some interferapplying the operator exp(it H ence effects between the elementary waves might appear. In this article, we present a general strategy that enables to overcome these difficulties. The strategy will be inspired from [6] where the nonlinear Schrödinger equation under strong partial confinement was analyzed. Two main differences appear here. First, the Poisson nonlinearity is nonlocal, which requires specific estimates. Observe that, at the limit ε → 0, the nonlinearity in the present paper reads 4π1|x| ∗ |ψ|2 dz and does not depend on z. This makes an important difference with the case of [6], in particular no resonance effects due to the nonlinearity will appear. Second, the magnetic field induces in (1.1) a singular term at an intermediate scale 1ε between the confinement operator (at the scale ε12 ) and the nonlinearity (at the scale ε10 ). Hence, compared to [6], the average techniques have to be pushed to the order two and resonance effects will finally appear here due to this magnetic term. 1.4. Main result. Consider the system (1.1), (1.2), (1.3). We assume that the confinement potential Vc satisfies two assumptions. The first one concerns the behavior of this function at infinity. Assumption 1.1. The potential Vc is a C ∞ nonnegative even function such that a 2 |z|2 ≤ Vc (z) ≤ C|z| M for |z| ≥ 1,
(1.17)
where a > 0, M > 0, and |∂z Vc (z)| = O |z|−M , Vc (z)
|∂zk Vc (z)| = O(1) for all k ∈ N∗ , Vc (z)
(1.18)
as |z| → +∞, where M > 0. Note that a smooth even potential of the form Vc (z) = C|z|s for |z| ≥ |z 0 |, with C > 0, s ≥ 2, satisfies these assumptions. In particular the harmonic potential Vc = a 2 z 2 fits these conditions. Let us discuss the assumptions. The assumption that the function Vc (z) is even is important in our analysis, see e.g. Step 4 in Subsect. 1.5. The left inequality in the first condition (1.17) implies that Vc tends to +∞ as |z| → +∞. The fact that Vc (z) ≥ a 2 z 2 is not essential in our analysis but simplifies it (see below, it allows to give a simple characterization of the energy space related to our system). As it is well-known [26], the spectrum of operator Hz defined by Hz = −∂z2 + B 2 z 2 + Vc (z)
(1.19)
834
F. Delebecque-Fendt, F. Méhats
is discrete, when Hz is considered as a linear, unbounded operator over L 2 (R), with domain D(Hz ) = {u ∈ L 2 (R), Hz u ∈ L 2 (R)}. The complete sequence of eigenvalues of Hz will be denoted by (E p ) p∈N , taken strictly increasing with p (recall indeed that in dimension 1 the eigenvalues are simple), and the associated Hilbert basis of real-valued eigenfunctions will be denoted by (χ p (z)) p∈N . The right inequality in (1.17) and the second condition (1.18) are more technical and are here to simplify the use of a Sobolev scale based on the operator Hz , which is well adapted to our problem. More precisely, these assumptions are used in Lemma 2.3. The second assumption on Vc concerns the spectrum of the confinement operator Hz . Assumption 1.2. The eigenvalues of the operator Hz defined by (1.19) satisfy the following property: there exists C > 0 and n 0 ∈ N such that ∀ p ∈ N, E p+1 − E p ≥ C(1 + p)−n 0 . The most simple situation where (1.2) is satisfied is when there exists a uniform gap between the eigenvalues: for all p ∈ N∗ , E p+1 − E p ≥ C0 > 0. Note that in this case we have n 0 = 0. This property is true in the following examples: √ – If Vc (z) = a 2 z 2 + V1 (z), with V1 L ∞√< 2 a 2 + B 2 . Indeed, in this case the perturbation theory gives |E p − (2 p + 1) a 2 + B 2 | < V1 L ∞ . – If Vc (z) ∼ a|z|s as |z| → +∞, with s > 2. Indeed, in this case the Weyl asymptotics 2s [19] gives E p ∼ C p s+2 , so E p+1 − E p → +∞ as p → +∞. Let us now give a few indications on the Cauchy problem for (1.1), (1.2), (1.3). This system benefits from two conservation laws, the mass and energy conservations: ∀t ≥ 0, ε (t) 2L 2 = 0 2L 2 , E( ε (t)) = E(0 ),
(1.20)
where the total energy of the wavefunction ε is defined by E( ε ) =
1 1 1 ε 2 ∂ + Vc ε 2L 2 + 2 (ε∂x + i Bz) ε 2L 2 z 2 L ε2 ε2 ε 1 √ + ∂y ε 2L 2 + V ε ε 2L 2 . 2
(1.21)
For fixed ε > 0, the Cauchy theory for the Schrödinger-Poisson with a constant uniform magnetic field was solved in [14,16] in the energy space. It is not difficult to adapt these proofs (see also the reference book [13]) to our case where an additional confinement potential is applied. The energy space in our situation is the set of functions u such that E(u) is finite: B 1 = u ∈ L 2 (R3 ) : ∂z u ∈ L 2 (R3 ), Vc u ∈ L 2 (R3 ), ∂y u ∈ L 2 (R3 )
i Bz u ∈ L 2 (R3 ) . and ∂x + ε
2DEG in a Strong Magnetic Field
835
This space seems to depend on ε, which would not be convenient for our asymptotic analysis. In fact, it does not. Indeed, thanks to our assumption (1.17) on the confinement potential, one has zu L 2 ≤
1 Vc u L 2 , a
so u ∈ B 1 implies that zu ∈ L 2 and thus ∂x u ∈ L 2 . Hence one has B 1 = u ∈ H 1 (R3 ) : Vc u ∈ L 2 (R3 ) , and, on this space, we will use the following norm independent of ε: u 2B 1 = (I − x,y + Hz )1/2 u 2L 2 = u 2L 2 + (−x,y )1/2 u 2L 2 + (Hz )1/2 u 2L 2 = u 2H 1 + Vc u 2L 2 + B 2 zu 2L 2 ,
(1.22)
where we used the selfadjointness and the positivity of −x,y and of the operator Hz defined by (1.19), and where I denotes the identity operator. In this paper, we will assume that the initial datum 0 in (1.2) belongs to this space B 1 . Then, for all ε > 0, the system (1.1), (1.2), (1.3) admits a unique global solution ε ∈ C 0 ([0, +∞), B 1 ). Our aim is to analyze the asymptotic behavior of ε as ε → 0. We are now in position to state our main results. Here and throughout this paper, we will use the notation ∀u ∈ L 1z (R), u = u(z) dz. (1.23) R
Let us introduce the limit system. First define the following coefficients: 2Bzχ p χq 2 ∀ p ∈ N, α p = 1 − , Eq − E p
(1.24)
q= p
where we recall that (E p , χ p ) p∈N is the complete sequence of eigenvalues and eigenfunctions of the operator Hz defined by (1.19). Then, we introduce the following infinite dimensional, nonlinear and coupled differential system on the functions φ p (t, x, y): ∀ p ∈ N, i∂t φ p = −α p ∂x2 φ p − ∂y2 φ p + W φ p , φ p (t = 0) = 0 χ p , (1.25) ⎛ ⎞ 1 W = ∗⎝ |φ p |2 ⎠ . (1.26) 4π x2 + y 2 p∈N Note that the convolution in (1.26) holds on the variables (x, y) ∈ R2 . Equation (1.26) is nothing but the Poisson equation for a measure valued distribution of mass whose support is constrained to the plane z = 0: ⎡ ⎛ ⎞⎤ 1 W (t, x, y) = ⎣ ∗⎝ |φ p (t, x, y)|2 δz=0 ⎠⎦ . 4π x2 + y 2 + z 2 p∈N z=0
836
F. Delebecque-Fendt, F. Méhats
In order to compare with ε , we introduce the following functions:
(t, x, y, z) = φ p (t, x, y) χ p (z), p∈N
ε app (t, x, y, z) =
e−it E p /ε φ p (t, x, y) χ p (z). 2
(1.27)
p∈N ε can be deduced from through the application of the operator eit Hz /ε , Remark that app unitary on B 1 : 2
ε = e−it Hz /ε . app 2
This explicit relation is the only dependency in ε of the limit system (1.25), (1.26), (1.27). Our main result is the following theorem. Theorem 1.3. Assume that Vc satisfies Assumptions 1.1 and 1.2 and let 0 ∈ B 1 . For all ε ∈ (0, 1], denote by ε ∈ C 0 ([0, +∞), B 1 ) the unique global solution of the initial system (1.1), (1.2), (1.3). Then the following holds true: ε ∈ (i) The limit system (1.25), (1.26), (1.27) admits a unique maximal solution app 0 1 C ([0, Tmax ), B ), where Tmax ∈ (0, +∞] is independent of ε. If Tmax < +∞ ε (t, ·) then app B 1 → +∞ as t → Tmax . (ii) For all T ∈ (0, Tmax ), we have ε = 0. lim ε − app 0 1 ε→0
C ([0,T ],B )
Comments on Theorem 1.3. 1. The cyclotron effective mass. Theorem 1.3 thus states that, on all time intervals where the limit system (1.27), (1.25), (1.26) is well-posed, the solution ε of the singularly ε . As expected, the dynamics in the y perturbed system (1.1), (1.2), (1.3) is close to app direction, i.e. parallel to the magnetic field, is not affected by the magnetic field, since the operator is still −∂y2 . On the other hand, the situation is different in the direction x and the averaging of the cyclotron motion results in a multiplication of the operator −∂x2 by the factor α p which only depends on Vc and B. The coefficient α1p plays in (1.25) the role of an effective mass in the direction perpendicular to the magnetic field. We find that the effective mass in the Schrödinger equation for the mode p depends on the index p of this mode. We do not know whether these coefficients are positive for a general Vc . Notice that the effective mass could be predicted heuristically by the following argument. Denoting by kx , ky the wavevectors of the 2DEG in the plane (x, y), the electron dispersion relation E p (kx , ky ) in the transversal subbands can be written from (1.1) by computing the eigenvalues of the operator 1 d2 2 2 2 2 2 2 − 2 + B z + Vc (z) + 2ε Bzkx + ε kx + ε ky . ε2 dz Since ε is small, an approximation of E p (kx , ky ) can be computed thanks to perturbation theory, which gives the following parabolic band approximation: 2Bzχ p χq 2 Ep 2 E p (kx , ky ) = 2 − kx + kx2 + ky2 + o(1). ε Eq − E p q= p
2DEG in a Strong Magnetic Field
837
We can read on this formula that the effective mass is 1 in the y direction and is α −1 p according to (1.24) in the x direction. Note that the specific case of the harmonic potential is treated below (see Comment 3). 2. Conservation of the energy for the limit system. Let us write the conservation of the energy for the limit system. The total energy for this system can be splitted into a confinement energy Econ f ( ) and a transport energy Etr ( ) defined by Econ f ( ) = Etr ( ) =
p∈N
p∈N
E p φ p 2L 2 ,
α p ∂x φ p 2L 2 +
p∈N
(1.28)
∂y φ p 2L 2
1 1 + |φ p (x, y)|2 |φq (x , y )|2 dxdydx dy . (1.29) 2 p,q R4 4π |x − x |2 + |y − y |2 An interesting property is that these two quantities are separately conserved by the limit ε solves (1.25), (1.26), (1.27), then, for all t ∈ [0, T ], we have system. If app ε ε ε ε (t)) = Econ f (app (0)) and Etr (app (t)) = Etr (app (0)). (1.30) Econ f (app
In particular, by summing up the two equalities in (1.30), we obtain the following conservation property: ε ε ε ε Econ f (app (t)) + Etr (app (t)) = Econ f (app (0)) + Etr (app (0)).
(1.31)
Note that, in the general case, we do not know whether the energy defined by (1.29) is the sum of nonnegative terms. This point is related to the fact that the well-posedness for t ∈ [0, +∞) of the Cauchy problem for the nonlinear system (1.25), (1.26) is an open issue. Nevertheless, when the α p are such that the energy is coercive on B 1 , i.e. when we have ∀ ∈ B 1 , C0 2B 1 ≤ Econ f ( ) + Etr ( ) ≤ C1 2B 1 + C2 4B 1 ,
(1.32)
with a constant C0 > 0 independent of ε, then the maximal solution of (1.25), (1.26) is globally defined: Tmax = +∞. Corollary 1.4 (Global in time convergence). Under the assumptions of Theorem 1.3, assume moreover that there exists 0 < α < α such that the coefficients α p defined by (1.24) satisfy the following condition: ∀ p ∈ N,
α ≤ α p ≤ α.
(1.33)
ε ∈ C0 Then the system (1.27), (1.25), (1.26) admits a unique global solution app ([0, +∞), B 1 ) and, for all T > 0, we have ε = 0, lim ε − app 0 1 ε→0
C ([0,T ],B )
where ε ∈ C 0 ([0, +∞), B 1 ) denotes the solution of (1.1), (1.2), (1.3).
838
F. Delebecque-Fendt, F. Méhats
The proof of this corollary is immediate and will not be detailed in this paper. Indeed, ε (t) of (1.25), (1.26) remarking that (1.33) implies (1.32), we obtain that the solution app satisfies the following uniform bound: ε ε ε app (t) 2B 1 ≤ C E con f (app (t)) + E tr (app (t)) = C E con f (0 ) + E tr ( 0 ) , where the quantity in the right-hand side is finite as soon as 0 ∈ B 1 . 3. Case of harmonic confinement. In the special case of a harmonic confinement potential Vc (z) = a 2 z 2 , the eigenvalues and eigenfunctions of Hz = −∂z2 + (a 2 + B 2 )z 2 can be computed explicitly and one has E p = (2 p + 1) a 2 + B 2 , χ p (z) = (a 2 + B 2 )1/8 u p (a 2 + B 2 )1/4 z , where (u p ) p∈N are the normalized Hermite functions defined e.g. in [24, Appendix B-8] and satisfying −u + z 2 u p = (2 p + 1)u p . The properties of the Hermite functions give √ √ 2( p + 1) 2p χ p+1 + 2 χ p−1 , 2zχ p = 2 (a + B 2 )1/4 (a + B 2 )1/4 and one can compute explicitly the coefficients α p = 1 − B2
2zχ p χ p+1 2 2zχ p χ p−1 2 a2 + B2 = 2 . E p+1 − E p E p − E p−1 a + B2
We thus recover here the coefficient found in Subsect. 1.3 in the simplified situation. Note that, in this case, condition (1.33) is satisfied and the convergence result holds on an arbitrary time interval. It is reasonable to conjecture that this condition (1.33) holds again when Vc (z) = a 2 z 2 + V1 (z), where V1 is a small perturbation. 4. Towards a more realistic model. Since we aim at describing the transport of electrons, which are fermions, our model should not be restricted to a pure quantum state. The following model describes the transport of an electron gas in a mixed quantum state and is more realistic: 1 1 i∂t εj = 2 −∂z2 + B 2 z 2 + Vc (z) εj − 2i Bz∂x εj − x,y εj + V ε εj , ∀ j, ε ε (1.34) ε j (0, x, y, z) = j,0 (x, y, z), ∀ j, (1.35) 1 V ε (t, x, z) = ∗ ρε, ρε = λ j | εj |2 , (1.36) 4πr ε j
where λ j , the occupation factor of the state εj , takes into account the statistics of the electron ensemble and is fixed once and for all at the initial time. Note that the Schrödinger equations (1.34) are only coupled through the selfconsistent Poisson potential. Therefore, we claim that our main Theorem 1.3, which has been given for the sake of simplicity in the case of the pure quantum state, can be extended to this system (1.34), (1.35), (1.36), with appropriate assumptions on the initial data ( j,0 ). Similarly, a given smooth external potential could be incorporated in the initial system. We also claim that our result can be easily adapted if we add in the right-hand side of (1.1) a term of the form Vext (t, x, y, εz) ε (which is coherent with our scaling), and the result does not change qualitatively.
2DEG in a Strong Magnetic Field
839
1.5. Scheme of the proof. In this section, we sketch the main steps of the proof of the main theorem. Step 1: A priori estimates. The first task is to obtain uniform in ε a priori estimates for the solution of (1.1), (1.2), (1.3), which are of course crucial in the subsequent nonlinear analysis. Due to the presence of the singular ε12 and 1ε terms in (1.1), this task is not obvious here. In Subsect. 2.1, we introduce a well adapted functional framework: a Sobolev scale based on the operators −x,y and Hz . More precisely, for all m ∈ N, we introduce the Hilbert space m/2 B m = u : u 2B m = u 2L 2 (R3 ) + (−x,y )m/2 u 2L 2 (R3 ) + Hz u 2L 2 (R3 ) < +∞ . (1.37) In Subsect. 2.1, we give some equivalent norms which are easier to handle here. Then in Subsect. 2.2 we take advantage of this functional framework and derive some a priori estimates for (1.1), (1.2), (1.3). Step 2: The filtered system. In [3,6], the asymptotics of NLS equations under the form i∂t u ε =
1 Hz u ε − x,y u ε + F(|u ε |2 )u ε , ε2
(1.38)
such as the Gross-Pitaevskii equation, was analyzed. In (1.38), F : R+ → R is a given function and the nonlinearity depends locally on the density |u ε |. It appeared in [6] that a fruitful strategy is to filter out the oscillations in time induced by the term ε12 Hz , without projecting on the eigenmodes of Hz . Indeed, projecting (1.38) on the Hilbert basis χ p leads to difficult problems of series summations and of small denominators in oscillating phases. Introducing the new unknown: v ε (t, x, z) = exp it Hz /ε2 u ε (t, x, z), the filtered system associated to (1.38) reads 2 2 2 2 i∂t v ε = −x,y v ε + eit Hz /ε F e−it Hz /ε v ε e−it Hz /ε v ε ,
(1.39)
where we used the fact that Hz , thus eit Hz , commutes with ∂x and ∂y . Then, the analysis of the limit ε → 0 amounts to prove that it is possible to define an average of the nonlinearity in (1.39) with respect to the fast variable t/ε2 . Let us adapt this strategy to our problem. Introduce
ε (t, x, z) = exp it Hz /ε2 ε (t, x, z). One deduces from (1.1), (1.2), (1.3) the following equation for ε : 2B it Hz /ε2 −it Hz /ε2 t ε ε ε ε e (i∂x ) − x,y + F ze , (t) , (1.40) i∂t = − ε ε2 where we introduced the nonlinear function 1 −iτ Hz 2 −iτ Hz iτ Hz ∗ e u e u, (τ, u) → F (τ, u) = e 4πr ε and where r ε is still defined by (1.4).
(1.41)
840
F. Delebecque-Fendt, F. Méhats
Step 3: Approximation by an intermediate system. Before performing the limit ε → 0 in (1.40), we remark that (1.41) can be approximated in order to get rid of the fast time variable t/ε2 in the nonlinear term of (1.40). By writing formally 1 1 = + o(1), 2 2 2 2 2 x +y +ε z x + y2 we remark that
(1.42)
! 1 1 −iτ Hz −iτ Hz 2 = ∗ u ∗ u e + o(1) e rε x2 + y 2 # " 1 = ∗ |u|2 + o(1), x2 + y 2
where the symbol ∗ denotes here a convolution in the (x, y) variables only, and where we used the fact that eiτ Hz is unitary on L 2z (R). Hence, inserting this Ansatz in (1.41) yields # " 1 iτ Hz 2 F (τ, u) = e ∗ |u| e−iτ Hz u + o(1) 2 2 4π x + y # " 1 2 = ∗ |u| u + o(1). 4π x2 + y 2 Denoting
F0 (u) =
"
1
∗ |u| 4π x2 + y 2
2
#
u,
$ε of the following intermediate system: and introducing the solution
2B it Hz /ε2 −it Hz /ε2 $ε = − $ε ) − x,y
$ε + F0 ε (t) , e (i∂x
ze i∂t
ε
(1.43)
(1.44)
we expect that the solution ε of (1.40) satisfies $ε + o(1).
ε =
(1.45)
Subsection 2.3 is devoted to the rigorous proof of this heuristics. We give sense to the o(1) in Lemma 2.7 and we prove that the solutions of the two nonlinear equations (1.40) and (1.44) are close together and that (1.45) holds true in the sense of the B 1 norm. This statement is given in Proposition 2.1. Step 4: Second order averaging of oscillating systems. Thanks to Step 3, we can consider the simplest system (1.44) instead of (1.40). We are now left with the analysis of the asymptotics of this intermediate system as ε → 0. Note that (1.44) is under the general form t 1 u(t) + g(u(t)) (1.46) i∂t u = f ε ε2 with f (τ ) = −2Beiτ Hz ze−iτ Hz i∂x
and
g(u) = −x,y u + F0 (u).
2DEG in a Strong Magnetic Field
841
At this point, a critical fact has to be noticed. Equations under the form i∂t u = f
t ε2
u(t) + g(u(t))
(1.47)
can be averaged when, due to some ergodicity property, one can give a sense to the time average 1 f = lim T →+∞ T
T
0
f (τ ) dτ.
(1.48)
0
Indeed, under rather general assumptions, the techniques of averaging of dynamical systems – see the reference book on the topic by Sanders and Verhulst [27]– enable us to show that (1.47) is well approximated by the averaged equation i∂t u = f 0 u(t) + g(u(t)). Yet, the oscillating term in (1.46), compared to the same term in (1.47), is multiplied by 1ε . Therefore, a necessary condition in order to perform the averaging of (1.46) is that the average f 0 of f is zero. In our case, the integral kernel of the operator eiτ Hz ze−iτ Hz , defined by ∀u,
e
iτ Hz
ze
−iτ Hz
u=
R
G(τ, z, z )u(z )dz ,
is given by G(τ, z, z ) =
p∈N q∈N
=
eiτ (E p −Eq ) zχ p χq χ p (z)χq (z ) eiτ (E p −Eq ) zχ p χq χ p (z)χq (z ).
p∈N q= p
In the last inequality, we used the fact that, by Assumption 1.1, Vc is even. Indeed, this property implies that, for all p, (χ p )2 is also even, thus z(χ p )2 = 0. Consequently, since p = q implies E p = E q , the kernel G(τ, z, z ) is a series of functions which all have a vanishing average in time. We thus expect that the operator-valued function f (τ ) has the same property: f 0 = lim
T →+∞
1 T
T
f (τ ) dτ = 0.
0
In such a situation, the theory of averaging has to be pushed to the second order [27] in order to obtain the limit of (1.46) as ε → 0. Section 3 is devoted to this question of second order averaging, which leads to the limit system (1.25), (1.26). The main result of Sect. 3 is Proposition 3.2. In the short last Sect. 4, we prove our main Theorem 1.3 by just gathering the results proved in the previous sections.
842
F. Delebecque-Fendt, F. Méhats
2. The Nonlinear Analysis In this section, we obtain some a priori estimates uniform in ε for the initial system (1.1), (1.2), (1.3) and we prove that it can be approximated by an intermediate system, where we regularize the initial data and where we replace the Poisson nonlinearity by its formal limit given in (1.43). This intermediate system takes the form $ε = i∂t
1 1 $ε − 2i Bz∂x $ε − x,y $ε + W ε $ε , Hz ε2 ε $ε (0, x, y, z) = $0 (x, y, z),
W ε (t, x, z) =
1
4π x2 + y 2
# " $ε |2 . ∗ |
(2.1)
(2.2)
(2.3)
Notice that (2.3) is nothing but the Poisson equation (1.3) where we replace $0 in (2.2) r ε = x2 + y 2 + ε2 z 2 by r 0 = x2 + y 2 . Moreover, the initial datum m will be chosen as a regularization in B of the initial datum 0 . Recall the definition (1.37) of the space B m . The main result of this section is the following proposition. Proposition 2.1 (Approximation of the initial system). Assume that Vc satisfies Assumptions 1.1, 1.2 and that 0 ∈ B 1 . For all ε ∈ (0, 1], denote by ε ∈ C 0 (R+ , B 1 ) the unique global solution of the initial system (1.1), (1.2), (1.3). Then the following holds true: (i) There exists a maximal positive time such that ε is bounded uniformly in ε : the quantity % & T0 := sup T ≥ 0 : sup ε C 0 ([0,T ],B 1 ) < +∞ ε∈(0,1]
(2.4)
satisfies T0 ∈ (0, +∞]. If T0 < +∞ then lim sup ε C 0 ([0,T0 ],B 1 ) = +∞. ε→0
(ii) For all T ∈ (0, T0 ), where T0 is defined by (2.4), for all δ > 0 and for all integers $0 ∈ B m and εδ such that the following holds true. For all m ≥ 2, there exist ε ∈ (0, εδ ], the intermediate system (2.1), (2.2), (2.3) admits a unique solution $ε ∈ C 0 ([0, T ], B m ) satisfying the following uniform estimates: $ε C 0 ([0,T ],B 1 ) ≤ δ, ε − ∀ε ≤ εδ $ε C 0 ([0,T ],B m ) ≤ C( 0 B 1 ) $0 B m .
(2.5) (2.6)
Remark 2.2. It is a priori not excluded that T0 < +∞. Indeed, although we are in a repulsive case, the energy conservation does not enable us to obtain ε-independent a priori estimates in B 1 (see the proof of Lemma 2.6). This may be linked to the possible formation of caustics, as for the nonlinear Schrödinger equation in semiclassical regime, see e.g. [11].
2DEG in a Strong Magnetic Field
843
2.1. Preliminaries. As we explained in Subsect. 1.5, our nonlinear analysis will deeply rely on the use of the functional spaces B m defined by (1.37) and adapted to the operators Hz and −x,y . The following result was proved in [6] by using an appropriate Weyl-Hörmander pseudodifferential calculus, inspired by [9,22]: Lemma 2.3 ([6]). Under Assumption 1.1, consider the Hilbert space B m defined by (1.37) for m ∈ N. Then the norm · B m in (1.37) is equivalent to the following norm: u H m (R3 ) + Vc (z)m/2 u L 2 (R3 ) .
(2.7)
Moreover, for all u ∈ B m+1 , we have
1/2 Hz u B m + ∂x u B m + ∂y u B m + ∂z u B m + Vc u B m u B m+1 .
(2.8)
The operator x,y commutes with the rapidly oscillating operator e±it Hz /ε and with the operator i z∂x . This will enable us to obtain uniform bounds for the solution of (1.1) by simply applying x,y to this equation. Unfortunately, the operator Hz does not satisfy this property. For this reason, we introduce the following operator: 2
Hε = Hz − 2iε Bz∂x − ε2 ∂x2 = −∂z2 + Vc (z) + (iε∂x − Bz)2 .
(2.9)
This operator enables us to define another norm equivalent to the B m norm. The following lemma is proved in Appendix A. Lemma 2.4. The operator Hε defined by (2.9) on L 2 (R3 ) with domain B 2 is self-adjoint and nonnegative. There exists a constant C1 > 0 such that, for all ε ∈ (0, 1] and for all u ∈ B 1 , we have 1 u 2B 1 ≤ u 2L 2 (R3 ) + (−x,y )1/2 u 2L 2 (R3 ) + Hε1/2 u 2L 2 (R3 ) ≤ C1 u 2B 1 . (2.10) C1 Moreover, for all integers m ≥ 2, there exists εm ∈ (0, 1] such that, for all ε ∈ (0, εm ], for all u ∈ B m , we have 1 u 2B m ≤ u 2L 2 (R3 ) + (−x,y )m/2 u 2L 2 (R3 ) + Hεm/2 u 2L 2 (R3 ) ≤ 2 u 2B m . (2.11) 2 2.2. A priori estimates. In this subsection, we obtain an a priori estimate uniform in ε for the initial Schrödinger-Poisson model (1.1), (1.3) and the intermediate model (2.1), (2.2), (2.3). Remark first that these two models can be considered in a unified way. For all u ∈ B 1 and for α ∈ {0, 1}, denote 1 2 Fα (u) = ∗ |u| u, (2.12) 4π x2 + y 2 + αε2 z 2 where the convolution holds on the three variables (x, y, z) ∈ R3 . Remark that for α = 0, this definition coincides with the definition (1.43). We shall consider for ε ∈ (0, 1] and α ∈ {0, 1} the nonlinear equation 1 Hε u ε − ∂y2 u ε + Fα (u ε ), ε2 u ε (0, x, y, z) = u 0 (x, y, z),
i∂t u ε =
(2.13) (2.14)
844
F. Delebecque-Fendt, F. Méhats
where the operator Hε was defined by (2.9). Note that for u 0 = 0 and α = 1, (2.13), $0 and α = 0, (2.13), (2.14) is the initial system (1.1), (1.2), (1.3), and that for u 0 = (2.14) is the intermediate system (2.1), (2.2), (2.3). Let us first state a technical lemma concerning the nonlinearities F1 and F0 , which is proved in Appendix B. Lemma 2.5. There exists a constant C > 0 such that, for all ε ∈ (0, 1], for α = 0 or 1, we have ∀u, v ∈ B 1 , Fα (u) − Fα (v) B 1 ≤ C u 2B 1 + v 2B 1 u − v B 1 , (2.15) where Fα is defined by (2.12). Moreover, for all m ∈ N∗ , there exists Cm > 0 such that we have the tame estimate ∀ε ∈ (0, 1], ∀α ∈ {0, 1}, ∀u ∈ B m , Fα (u) B m ≤ Cm u 2B 1 u B m .
(2.16)
Now we are able to derive uniform a priori estimates for the solution of (2.13), (2.14). Lemma 2.6. Let ε ∈ (0, 1], α ∈ {0, 1} and u 0 ∈ B 1 . Then the solution u ε of Eq. (2.13), (2.14) exists and is unique in C 0 ([0, +∞), B 1 ) and the following uniform in ε estimates hold true: (i) For all M > 0, there exist T > 0, only depending on M and u 0 B 1 , such that, for all ε ∈ (0, 1], we have u ε C 0 ([0,T ],B 1 ) ≤ (1 + M) u 0 B 1 .
(2.17)
> 0, we have (ii) Let m ≥ 2 be an integer and assume that u 0 ∈ B m . Then, for all T the estimate
u ε 2 0 , (2.18) ∀ε ∈ (0, εm ], u ε C 0 ([0,T ],B m ) ≤ C u 0 B m exp C T 1 C ([0,T ],B ) where εm > 0 is as in Lemma 2.4. Proof. Step 1. The Cauchy problem and the conservation laws. For any given ε > 0, the existence and uniqueness of a maximal solution u ε ∈ C 0 ([0, T ), B 1 ) can be obtained by standard techniques [13]. We leave this first part of the proof to the reader. This solution satisfies both L 2 and energy conservation laws: ∀t ≥ 0, u ε (t) L 2 = u 0 L 2 and Eα (u ε (t)) = Eα (u 0 ),
(2.19)
where the energy Eα is defined by 1 1 (Hε u, u) L 2 + ∂y u 2L 2 + (Fα (u), u) L 2 2 ε 2 1 1 1 1 2 2 = 2 ∂z u L 2 + 2 Vc u L 2 + 2 (ε∂x + i Bz)u 2L 2 + ∂y u 2L 2 + (Fα (u), u) L 2 . ε ε ε 2
Eα (u) =
We recall that the operator Hε is defined by (2.9). These conservation laws show that the solution u ε is global, i.e. that T = +∞. Unfortunately, due to the ε12 terms in this expression, one cannot use the energy conservation to get uniform in ε estimates. Instead, we will directly write the equations satisfied by ∂x u ε , ∂y u ε or (Hε )1/2 u ε and use the standard L 2 -estimates for these equations and the fact that the self-adjoint operators Hε , ∂x and ∂y commute together.
2DEG in a Strong Magnetic Field
845
Step 2. B 1 estimate. This yields i∂t (∇x,y u ε )(t) = and
1 Hε (∇x,y u ε ) − ∂y2 (∇x,y u ε ) + ∇x,y Fα (u ε ) 2 ε
1 i∂t Hε1/2 u ε (t) = 2 Hε (Hε1/2 u ε ) − ∂y2 (Hε1/2 u ε ) + Hε1/2 Fα (u ε ) . ε
Hence, u ε (t) L 2 + ∇x,y u ε (t) L 2 + Hε1/2 u ε (t) L 2 ≤ u 0 L 2 + ∇x,y u 0 L 2 + Hε1/2 u 0 L 2 t ∇x,y Fα (u ε (s)) L 2 + Hε1/2 Fα (u ε (s)) L 2 ds +C 0
and, for ε ∈ (0, 1], the equivalence of norms given in Lemma 2.4, yields t ε u (t) B 1 ≤ C u 0 B 1 + C Fα (u ε (s)) B 1 ds 0 t ≤ C u 0 B 1 + C u ε (s) 3B 1 ds,
(2.20)
0
where we used (2.15) with v = 0 to estimate Fα (u ε (s)). Hence, by applying the Gronwall lemma to the integral inequality (2.20), we prove Item (i) of the lemma. Step 3. B m estimate. Let T > 0, m ≥ 2, u 0 ∈ B m and let ε ∈ (0, εm ], where 0 < εm ≤ 1 m/2 as in Lemma 2.4. Since the operators Hε and x,y commute together, Hε u ε satifies the following equation: 1 i∂t Hεm/2 u ε (t) = 2 Hε (Hεm/2 u ε ) − ∂y2 (Hεm/2 u ε ) + Hεm/2 Fα (u ε ) , ε thus, for all t ∈ [0, T ], m/2 ε u (t)
Hε
L2
t ≤ Hεm/2 u 0 L 2 + Hεm/2 Fα (u ε (s)) L 2 ds 0 t ≤ C u 0 B m + C Fα (u ε (s)) B m ds 0 t 2 ≤ C u 0 B m + C u ε C u ε (s) B m ds, 0 ([0,T ],B 1 )
(2.21)
0
where we used Lemma 2.4 and the tame estimate (2.16). Similarly, −x,y u ε satisfies the following equation: 1 i∂t (−x,y u ε )(t) = 2 Hε (−x,y u ε ) − ∂y2 (−x,y u ε ) − x,y Fα (u ε) ε and, using the definition of B m (1.37) and (2.16) yields: t (−x,y )m/2 u ε (t) L 2 ≤ (−x,y )m/2 u 0 L 2 + (−x,y )m/2 Fα (u ε (s)) L 2 ds, 0 t ε 2 ≤ C u 0 B m + C u C 0 ([0,T ],B 1 ) u ε (s) B m ds. (2.22) 0
846
F. Delebecque-Fendt, F. Méhats
Therefore, by using again the equivalence of norms given by Lemma 2.4 and the L 2 conservation law in (2.19), we deduce from (2.21) and (2.22) that, for t ≤ T , we have t ε ε 2 u (t) B m ≤ C u 0 B m + C u C 0 ([0,T ],B 1 ) u ε (s) B m ds, 0
and the Gronwall lemma gives (2.18).
2.3. Proof of Proposition 2.1. In this subsection, we prove Proposition 2.1, ie we show that this solution can be uniformly approximated by a regular solution of the intermediate system. We first state a technical lemma on the Poisson kernels, which is proved in Appendix 4. Lemma 2.7. There exists a constant C > 0 such that, for all ε ∈ (0, 1], we have ∀u ∈ B 2 ,
F1 (u) − F0 (u) B 1 ≤ C ε1/3 u 3B 2 ,
(2.23)
where F0 and F1 are defined by (2.12). We are now ready to prove the main result of this section. Proof of Proposition 2.1. Let 0 ∈ B 1 , let an integer m ≥ 2 be fixed, and define the $0 by regularized initial datum $0 = I − ηx,y −m/2 (I + ηHz )−m/2 0 ,
(2.24)
where η > 0 is a small parameter that will be fixed further and where I denotes the identity operator. Denote by ε the solution of the initial system (1.1), (1.2), (1.3) and $ε the solution of the intermediate system (2.1), (2.2), (2.3) with the initial datum by $ε . (2.24). We shall estimate the difference ε − Step 1. Uniform bounds for ε . Let 0 < ε ≤ 1. From Lemma 2.6 (i), we first deduce that there exists T1 > 0 only depending on 0 B 1 such that, for all ε ∈ (0, 1], ε C 0 ([0,T1 ],B 1 ) ≤ 2 0 B 1 . This implies that T0 defined by (2.4) satisfies T0 ≥ T1 > 0. Clearly, if T0 < +∞, we have lim sup ε C 0 ([0,T0 ],B 1 ) = +∞, ε→0
otherwise by reiterating the above procedure we could find a uniform bound on [0, T2 ] with T2 > T0 . Now we fix T ∈ (0, T0 ) and δ > 0 for the sequel of this proof. Definition (2.4) of T0 implies that ε C 0 ([0,T ],B 1 ) ≤ C 0 B 1 , independent of ε ∈ (0, 1]. (2.25) $0 . First, we deduce from (2.24) that Step 2. Bounds for the initial datum $0 = (I − ηx,y )−m/2 (I + ηHz )−m/2 (I − x,y + Hz )1/2 0 , (I − x,y + Hz )1/2
2DEG in a Strong Magnetic Field
847
hence $0 L 2 (I − x,y + Hz )1/2 ≤ (I − ηx,y )−m/2 (I + ηHz )−m/2 (I − x,y + Hz )1/2 0 L 2 ≤ (I − x,y + Hz )1/2 0 L 2 , where we used the fact that the operators (I − ηx,y )−m/2 and (I + ηHz )−m/2 are bounded on L 2 , with bounds equal to 1. Therefore, using (1.22), we obtain $0 B 1 ≤ 0 B 1 ,
(2.26)
where we recall that the right-hand side is independent of ε. Next, we get from (2.24) the two following identities: for all integer ≤ m, $0 = (−x,y )/2 (I − ηx,y )−/2 (−x,y )/2+1/2 ×(I − ηx,y )/2−m/2 (I + ηHz )−m/2 (−x,y )1/2 0 , and /2+1/2 $ 0
Hz
/2
= Hz (I + ηHz )−/2 (I + ηHz )/2−m/2 (I − ηx,y )−m/2 Hz 0 . 1/2
Thus, from the bound ∀λ ∈ R+ , λ/2 (1 + ηλ)−/2 ≤ Cη−/2 , /2
we deduce that both operators (−x,y )/2 (I − ηx,y )−/2 and Hz (I + ηHz )−/2 are bounded on L 2 , with bounds equal to Cη−/2 , and thus ∀ ≤ m,
$0 B +1 ≤ Cη−/2 0 B 1 ,
(2.27)
where we recall the definition (1.37) of the B m norms. Finally, we obtain also from (2.24) that $0 ) = I − (I − ηx,y )−m/2 (I + ηHz )−m/2 (I − x,y + Hz )1/2 (0 − ×(I − x,y + Hz )1/2 0 . Decompose v = (I − x,y + Hz )1/2 0 on the Hilbert basis (χ p ) p∈N of eigenmodes of Hz : v(x, y, z) = v p (x, y) χ p (z), p∈N
and denote by v'p (ξ ), ξ ∈ R2 , the Fourier transform of v p (x, y). By (1.22), we have 2 2 $ 1 − (1 + η|ξ |2 )−m/2 (1 + ηE p )−m/2 |v'p (ξ )|2 dξ. 0 − 0 B 1 = p∈N
Hence, using that
R2
R2 p∈N
|v'p (ξ )|2 dξ = 0 2B 1 < +∞
(2.28)
848
F. Delebecque-Fendt, F. Méhats
and that
lim 1 − (1 + η|ξ |2 )−m/2 (1 + ηE p )−m/2 = 0,
∀ξ ∈ R2 , ∀ p ∈ N,
η→0
we deduce from Lebesgue’s dominated convergence theorem and from the convergence of the series in (2.28) that
0 B 1 = 0. lim 0 −
(2.29)
η→0
$ε . Consider Step 3. Uniform a priori estimates for $ε C 0 ([0,T ],B 1 ) ≤ 2 ε C 0 ([0,T ],B 1 ) }. (2.30) Tη := sup{τ ∈ (0, T ] : ∀ε ∈ (0, 1], η Note that, from (2.26) and Lemma 2.6 (i), we know that Tη ∈ (0, T ] is well-defined. Then, from Lemma 2.6 (ii), we deduce the following estimate: $0 B +1 $ε C 0 ([0,T ],B +1 ) ≤ C $ε C 0 ([0,T ],B 1 ) ∀ε ∈ (0, εm ], ∀ ≤ m, η η ε $0 B +1 ≤ C C 0 ([0,T ],B 1 ) $0 B +1 , ≤ C 0 B 1 (2.31) where we used (2.30) and (2.25).
ε . Using the notations defined in (2.9) and Step 4. Estimate of the difference ε − $ε satisfy (2.13),(2.14) with α = 1, u 0 = 0 and α = 0, u 0 = $0 (2.12), ε and respectively. The Duhamel formulation of these equations read respectively t 2 2 ε (t) = e−it (Hε −∂y ) 0 + e−i(t−s)(Hε −∂y ) F1 ( ε (s)) ds, 0
$0 + $ε (t) = e−it (Hε −∂y ) 2
t
$ε (s)) ds. e−i(t−s)(Hε −∂y ) F0 ( 2
0
Hence, for all t ∈ [0, Tη ] and ε ∈ (0, εm ], t $ε (t) B 1 ≤ 0 − $0 B 1 + $ε (s)) B 1 ds ε (t) − F1 ( ε (s)) − F1 ( 0 t $ε (s)) − F0 ( $ε (s)) B 1 ds + F1 ( 0
$0 B 1 + C ≤ 0 −
t 0
$ε (s) B 1 ds + C ε1/3 η−3/2 , ε (s) −
where we used (2.15), (2.25), (2.30), (2.23) and (2.31) with = 1, coupled to (2.27). Here C denotes a generic constant depending only on T and 0 B 1 . Hence, by the Gronwall Lemma, we get, for all t ∈ [0, Tη ], $ε (t) B 1 ≤ 0 − $0 B 1 + C ε1/3 η−3/2 eC T . ε (t) − (2.32)
2DEG in a Strong Magnetic Field
849
Now, according to (2.29), we fix η such that δ 1 ε CT $ , C 0 ([0,T ],B 1 ) 0 − 0 B 1 e ≤ min 2 3 and, in a second step, we fix εδ ∈ (0, εm ] such that δ 1 ε 1/3 −3/2 C T , C 0 ([0,T ],B 1 ) . e ≤ min C εδ η 2 3 From (2.32), we deduce that 2 $ε (t) B 1 ≤ min δ, ε C 0 ([0,T ],B 1 ) . ∀t ∈ [0, Tη ], ∀ε ∈ (0, εδ ], ε (t) − 3 (2.33) Therefore, we have $ε C 0 ([0,T ],B 1 ) ≤ ε C 0 ([0,T ],B 1 ) + ε − $ε C 0 ([0,T ],B 1 ) η η η ≤
5 ε C 0 ([0,T ],B 1 ) . 3
(2.34)
We claim that Tη = T . Indeed, if Tη < T , then, applying again Lemma 2.6 at Tη and using (2.34) enables us to find τ > 0 such that, for all ε ∈ (0, 1), $ε C 0 ([T ,T +τ ],B 1 ) ≤ 2 ε C 0 ([0,T ],B 1 ) , η η which, together with (2.34), contradicts the definition (2.30) of Tη . Finally, (2.33) gives (2.5) and (2.31) with = m − 1 gives (2.6). The proof of Proposition 2.1 is complete. 3. Second Order Averaging In this section, we focus on the intermediate system (2.1), (2.2), (2.3) as ε goes to zero. As we explained in Subsect. 1.5, it is interesting to consider the filtered version of this $0 ∈ B m be a given initial data, let $ε be the corresponding solution of equation. Let (2.1), (2.2), (2.3) and set $ε (t, ·) = exp it Hz /ε2 $ε (t, ·).
(3.1) This function satisfies the system 2B it Hz /ε2 −it Hz /ε2 $ε = − $ε ) − x,y
$ε + F0 ε (t) , e i∂t
(i∂x
ze ε $ε (t = 0) = $0 ,
(3.2)
where F0 is defined by (1.43). The advantage of this intermediate system, compared to $ε ) has no dependence in the fast variable t2 . (1.40) is that the nonlinearity F0 (
ε We will analyze the filtered system (3.2) in the framework of second order averaging of fast oscillating ODEs under the form (1.46) –see [27]– that we adapt here to our context of nonlinear PDEs. Recall that (E p ) p∈N , (χ p ) p∈N are the complete families
850
F. Delebecque-Fendt, F. Méhats
of eigenvalues and eigenfunctions of the operator Hz and denote by p the spectral projector on χ p : ∀ ∈ L 2 (R3 ),
p = χ p χ p .
Introduce now the following unbounded operator on L 2 (R3 ): A0 =
−∂x2
αp p
2Bzχ p χq 2 with α p = 1 − . Eq − E p
(3.3)
q= p
p≥0
With this notation, the limit system (1.25), (1.26), (1.27) can be rewritten in a more compact form as i∂t = A0 − ∂y2 + F0 ( ),
(t = 0) = 0 .
(3.4)
We state the main results of this section in the following two propositions. Proposition 3.1. Assume that Vc satisfies Assumptions 1.1 and 1.2. Then the following properties hold true. (i) The unbounded operator A0 defined by (3.3) on L 2 (R3 ) with the domain D(A0 ) = { ∈ L 2 (R3 ) : ∂x2 α p p ∈ L 2 (R3 )} p≥0
is selfadjoint. Moreover, the operator A0 satisfies ∀ ≥ 0, ∀u ∈ B 2n 0 +4+ ,
A0 u B ≤ C u B 2n0 +4+ ,
(3.5)
where n 0 is as in Assumption 1.2. (ii) Let 0 ∈ B 1 . The limit system (3.4) admits a unique maximal solution
∈ C 0 ([0, Tmax ), B 1 ). If Tmax < +∞ then (t) B 1 → +∞ as t → Tmax . Proposition 3.2 (Averaging of the intermediate system). Assume that Vc satisfies Assumptions 1.1 and 1.2. Then there exists an integer m ≥ 2 such that the following holds true. $0 ∈ B m , we consider the solution
$ε ∈ C 0 ([0, +∞), B m ) of (3.2) and the maximal For 0 1 $0 as initial data: solution ∈ C ([0, Tmax ), B ) of the limit system with t 2 2 $0 − i
(t) = e−it (A0 −∂y ) e−i(t−s)(A0 −∂y ) F0 ( (s))ds. (3.6) 0
We assume that there exist T ∈ (0, Tmax ), ε0 > 0 such that $ε C 0 ([0,T ],B m ) < +∞. M := sup
(3.7)
$ε − C 0 ([0,T ],B 1 ) ≤ ε C M ,
(3.8)
ε∈(0,ε0 ]
Then we have
where C M is independent of ε.
2DEG in a Strong Magnetic Field
851
3.1. Well-posedness of the limit system. In this section, we prove Proposition 3.1. Step 1. Basic properties of the operator A0 . First, from Vc (z) ≥ a 2 z 2 , we deduce that the pth eigenvalue of Hz is larger than the p th eigenvalue of the harmonic oscillator d2 2 2 2 − dz 2 + (a + B )z : ∀ p ∈ N, E p ≥ a 2 + B 2 (2 p + 1). (3.9) From Assumption 1.2, we deduce that the coefficients α p in (3.3) satisfy 2 2Bzχ p χq = 1 + C(1 + p)n 0 Bzχ p 2L 2 |α p | ≤ 1 + C(1 + p)n 0 q≥0
≤
C E np0 +1 , 1/2
where we used (3.9) and that Bzχ p L 2 ≤ E p . Now, consider a nonnegative integer and u in B 2n 0 +4+ . Let n 0 be defined as in Assumption 1.2, and decompose u over the χ p family which is orthogonal in L 2 : A0 u 2B = α 2p ∂x2 p u 2B p≥0
≤C
0 +2 u 2 E 2n ≤C p p B +2
p≥0
≤C
Hzn 0 +1 p u 2B +2
p≥0
p u 2B 2n0 +4+
=
C u 2B 2n0 +4+ ,
p≥0
where we used Lemma 2.3. This proves (3.5). Furthermore, by passing to the limit as N → +∞ in the identity ∀ , ∈ D(A),
N
α p (∂x2 p , p ) L 2 =
p=0
N
α p ( p , ∂x2 p ) L 2 ,
p=0
we obtain that the operator A0 is symmetric. Moreover, the equation A0 + i = f admits a solution ∈ D(A0 ) for all f ∈ L 2 (R3 ). Indeed, the projection of this equation on χ p reads −α p ∂x2 φ p + iφ p = f p , and this elliptic equation can obviously be solved for all f p ∈ L 2 (R2 ). Therefore, by the standard criterion for selfadjointness [26], the operator A0 is selfadjoint. We have proved the first part of Proposition 3.1. Step 2. Well-posedness and stability of the limit system. The operator A0 being selfadjoint, the Stone theorem can be applied and the operator −i A0 generates a unitary group of continuous operators e−i A0 t on L 2 and also on B 1 . The Duhamel formulation of (3.4) reads t 2 2
(t) = e−it (A0 −∂y ) 0 − i e−i(t−s)(A0 −∂y ) F0 ( (s))ds (3.10) 0
∂y2
(recall that A0 and commute together). Since, by (2.15), the application F0 is locally Lipschitz continuous on B 1 , it is easy to prove by a standard fixed point technique that
852
F. Delebecque-Fendt, F. Méhats
(3.10) admits a unique maximal solution ∈ C 0 ([0, Tmax ), B 1 ). The details are left to the reader. Note that, if Tmax < +∞, then (t) B 1 → +∞ as t → Tmax . Item (ii) of Proposition 3.1 is proved. Remark 3.3. In fact, this strategy of proof by a fixed point mapping leads to a stability result. For all η > 0 and for all T ∈ (0, Tmax ), there exists δη,T > 0 such that the
0 satisfying following holds true. For all
0 B 1 ≤ δη,T , 0 − Eq. (3.6),
0 − i
(t) = e−it (A0 −∂y ) 2
t
e−i(t−s)(A0 −∂y ) F0 ( (s))ds, 2
0
admits a unique solution ∈ C 0 ([0, T ], B 1 ) and we have sup (t) − (t) B 1 ≤ η.
t∈[0,T ]
(3.11)
3.2. Proof of Proposition 3.2. This subsection is devoted to the proof of Proposition 3.2, which relies on a reformulation of the Duhamel formula for (3.2). Step 1. Reformulation of the Duhamel formula. Introduce the following family of unbounded self-adjoint operators on L 2 (R3 ): ∀τ ∈ R,
a(τ ) = −2Beiτ Hz ze−iτ Hz i∂x
(3.12)
with domain B 2 . Note that, from (1.17) and Lemma 2.3, we deduce that, for all ∈ N, ∀u ∈ B 2 , ∀τ ∈ R,
a(τ )u L 2 ≤ C u B 2 .
(3.13)
The Duhamel representation of (3.2) reads $ε (t) = $0 −
i ε
t 0
t s ε $ε (s) + F0 ε (s) ds. (3.14) $ −x,y
a 2 (s)ds − i ε 0
Introduce the primitive of a: ∀u ∈ B 2 , ∀τ ∈ R,
τ
A(τ )u =
a(s)u ds,
(3.15)
0
which is well-defined as a Riemann integral, thanks to (3.13), and is such that ∀u ∈ B 2 , ∀τ ∈ R,
A(τ )u L 2 ≤ Cτ u B 2 .
(3.16)
$ε ∈ $ε ∈ C 0 ([0, T ], B 4 ), then by (3.2) we have that ∂t
Now, we notice that if
C 0 ([0, T ], B 2 ). Hence one can integrate by parts in the first integral of (3.14) and,
2DEG in a Strong Magnetic Field
853
if m ≥ 4, the following expression holds true for all t ∈ [0, T ], in the sense of functions in C 0 ([0, T ], L 2 ): t i t s ε s t $ε (t) $ε (s)ds − iε A $ (s)ds = iε
− a 2
A 2 ∂t
ε 0 ε ε ε2 0 t s s t $ε (t) $ε (s)ds − iε A
= A 2 a 2
ε ε ε2 0 t s $ε (s) + F0 ε (s) ds, +ε A 2 −x,y
ε 0 $ε . Finally, the Duhamel formula (3.14) becomes where we used (3.2) to evaluate i∂t
$ε (t)
s s t ε $ε (t) $ (s)ds − iε A $0 +
= A 2 a 2
ε ε ε2 0 t s $ε (s) + F0 ε (s) ds +ε A 2 −x,y
ε 0 t $ε (s) + F0 ε (s) ds. −i −x,y
t
(3.17)
0
Step 2. Approximation of the Duhamel formula. Denote t (ε (t) =
$ε (t) $ε (t) + iε A
ε2 and rewrite (3.17) as follows: t t s s 2
ε (s)ds − i $ε (s) + F0 ε (s) ds. $ A 2 a + i∂ −∂y2
x 2 ε ε 0 0 t ε s $ε (s) + F0 (s) ds. +ε A 2 −x,y
(3.18) ε 0
(ε (t) = $0 +
In this step, we prove that $ε (t) −
(ε (t) B 1 ≤ ε C M sup
t∈[0,T ]
(3.19)
and that (ε (t) = $0 − i
t 0
(ε (s) − ∂y2
(ε (s) + F0
(ε (s) + ε f ε (s) ds, A0
(3.20)
with sup f ε B 1 ≤ C M .
t∈[0,T ]
(3.21)
In order to prove this claim, we state two technical lemmas which are proved in Appendix 4 so that the proof would be more readable.
854
F. Delebecque-Fendt, F. Méhats
Lemma 3.4. Let Vc satisfy Assumptions 1.1 and 1.2. Then, for all integer , the operator A(τ ) defined by (3.15) satisfies t 0 2n 0 ++8 ∀u ∈ C ([0, T ], B ), sup A 2 u(t) ≤ C u C 0 ([0,T ],B 2n0 ++8 ) , ε t∈[0,T ] B (3.22) where n 0 is as in Assumption 1.2 and C is independent of ε. Lemma 3.5. Let Vc satisfy Assumptions 1.1 and 1.2. Let T > 0 and m = 4n 0 + 17. Let u ∈ C 0 ([0, T ], B m ) such that ∂t u ∈ C 0 ([0, T ], B m−2 ). Then we have, for all ε ∈ (0, 1], t t s s 2 2 A 2 a 2 + i∂x u(s)ds + i A0 u(s)ds sup 1 ≤ Cε u , (3.23) ε ε 0 0 t∈[0,T ] B where A0 , a and A are respectively defined by (3.3), (3.12) and (3.15) and where u denotes shortly u C 0 ([0,T ],B m ) + ∂t u C 0 ([0,T ],B m−2 ) . In order to apply these lemmas, we need some bounds for ε and ∂t ε . Let us fix m = 4n 0 + 17, where n 0 is as in Assumption 1.2 and assume that we have the uniform estimate (3.7). By (2.8), we deduce that 2 2 $ε x,y ε C 0 ([0,T ],B m−2 ) + eit Hz /ε ze−it Hz /ε ∂x
≤ C M . (3.24) 0 m−2 C ([0,T ],B
)
Moreover, from (2.16), we deduce that F0 ( ε ) C 0 ([0,T ],B m ) ≤ C M .
(3.25)
Hence, from (3.2), (3.24) and (3.25), we get ∂t ε C 0 ([0,T ],B m−2 ) ≤
CM . ε
(3.26)
Therefore, applying Lemmas 3.4 and 3.5 and using (3.7), (3.24), (3.25) and (3.26) yield t ε $ sup A 2 (t) (3.27) 2n +5 ≤ C M , ε t∈[0,T ] B 0 ε t ε $
ε sup A 2 −x,y (t) + F0 (t) ε t∈[0,T ]
B1
≤ ε CM ,
(3.28)
and
t t s s 2 $ε ε (s)ds $
sup A a + i∂
(s)ds + i A 0 x 1 ≤ ε C M , (3.29) 2 2 ε ε 0 0 t∈[0,T ] B
where we used that m ≥ 4n 0 + 17, thus in particular m ≥ 4n 0 + 13 and m ≥ 2n 0 + 11. Hence, from (3.27), we deduce (3.19) and $ε − F0
(ε 0 $ε −
(ε ) C 0 ([0,T ],B 1 ) + F0
∂y2 (
1 ≤ ε C M , (3.30) C ([0,T ],B )
2DEG in a Strong Magnetic Field
855
where we also used the estimate (2.15). Moreover, from (3.5) and (3.27), we get $ε −
(ε ) C 0 ([0,T ],B 1 ) ≤ ε C M . A0 (
(3.31)
Finally, inserting (3.28), (3.29), (3.30), (3.31) in (3.18) yields (3.20) with the estimate (3.21). (ε Step 3. A stability result for the limit system. First notice that (3.20) implies that
satisfies in the strong sense the equation (ε = A0
(ε − ∂y2
(ε + F0 (
(ε ) + ε f ε , i∂t
$0 ,
(t = 0) =
which has the following mild formulation: t 2 −it (A0 −∂y2 ) $ ε ( (ε (s)) + ε f ε ds.
(t) = e 0 − i e−i(t−s)(A0 −∂y ) F0 (
(3.32)
0
$0 as initial data: there exists a maximal soluApply now Proposition 3.1 (ii) with 0 1 tion ∈ C ([0, Tmax ), B ) to Eq. (3.6). Assume that T is such that 0 < T < Tmax . Subtracting (3.6) to (3.32) leads, for all t ≤ T , to t F0 (
(ε (t) − (t) B 1 ≤ (ε (s)) − F0 ( (s)) 1 ds + ε f ε C 0 ([0,T ],B 1 )
B 0 t
(ε (s) − (s) 1 ds , ≤ CM ε + B 0
(ε C 0 ([0,T ],B 1 ) ≤ C M . Therefore, the Gronwall where we used (2.15), (3.21) and
Lemma gives the estimate (3.8) and the proof of Proposition 3.2 is complete. 4. Proof of the Main Theorem This section is devoted to the proof of the main Theorem 1.3. Remark that the statement (i) is already proved in Proposition 3.1. Let us prove the statement (ii) of Theorem 1.3. Let 0 ∈ B 1 . Denote by ε ∈ C 0 ([0, +∞), B 1 ) the solution of (1.1), (1.2), (1.3) and let T0 ∈ (0, +∞] be the maximal time given by Proposition 2.1 (i). We also introduce the maximal solution ∈ C 0 ([0, Tmax ), B 1 ) of the limit system (3.4), given by Proposition 3.1. Pick T such that 0 < T < min(T0 , Tmax ) and let η > 0. Since T < Tmax , according to Remark 3.3, one can define δη/3,T > 0 such that the
0 satisfying following holds true. For all
0 B 1 ≤ δη/3,T , 0 −
∈ C 0 ([0, T ], B 1 ) and we have (3.11): Eq. (3.6) admits a unique solution sup (t) − (t) B 1 ≤ η/3.
t∈[0,T ]
856
F. Delebecque-Fendt, F. Méhats
Next, we fix m ≥ 2 according to Proposition 3.2 and δ > 0 by δ = min
η 3
, δη/3,T .
(4.1)
$0 ∈ B m and εδ such that the Since T < T0 , Proposition 2.1 (ii) enables to choose ε
corresponding solution of the intermediate system (2.1), (2.2), (2.3) satisfies (2.5) and (2.6) for all ε ≤ εδ :
ε C 0 ([0,T ],B 1 ) ≤ δ ≤ ε −
η , 3
(4.2)
ε is bounded in C 0 ([0, T ], B m ) uniformly with respect to ε. and
0 satisfies Now, we remark that by (4.2) this initial data
0 B 1 ≤ δ ≤ δη/3,T . 0 − Hence, Remark 3.3 gives that the solution of Eq. (3.6) satisfies − C 0 ([0,T ],B 1 ) ≤
η , 3
or, equivalently, 2 2 e−it Hz /ε − e−it Hz /ε C 0 ([0,T ],B 1 ) ≤
η . 3
(4.3)
ε in C 0 ([0, T ], B m ) enables us to apply Proposition Moreover, the uniform bound of $ε satisfies 3.2, which gives that the function $ε − e−it Hz /ε2 C 0 ([0,T ],B 1 ) ≤ δ ≤
η , 3
(4.4)
for ε ≤ εδ , where solves (3.6). Finally, (4.2), (4.3) and (4.4) yield the existence of ε0 such that, for all ε ∈ (0, ε0 ] we have ε − e−it Hz /ε C 0 ([0,T ],B 1 ) ≤ η. 2
(4.5)
To conclude, it remains to remark that T0 ≥ Tmax . Indeed, if T0 < Tmax , then we have, by Proposition 2.1 (i), lim sup ε C 0 ([0,T0 ],B 1 ) = +∞, ε→0
which implies by (4.5) that lim (T ) B 1 = +∞.
T →T0
This contradicts T0 < Tmax . The proof of Theorem 1.3 is complete.
2DEG in a Strong Magnetic Field
857
Appendix A. Proof of Lemma 2.4 First, by integrating by parts and applying Cauchy-Schwarz, we obtain Bz∂x u 2L 2 = B 2 z 2 |∂x u|2 dxdydz = (B 2 z 2 u)(−∂x2 u)dxdydz ≤ u 2B 2 . R3
R3
Hence, the first properties stated in the lemma are obvious from the definition (2.9), and we shall only detail the proof of the equivalence of norms. Step 1. The case m = 1. From the definition (2.9) and the assumption (1.17) on Vc , we deduce that Hε1/2 u 2L 2 = ((−∂z2 + Vc )u, u) L 2 + (ε∂x + i Bz)u 2L 2 = ((−∂z2 + Vc )u, u) L 2 + B 2 zu 2L 2 + ε2 ∂x u 2L 2 − 2ε BIm (zu, ∂x u) L 2 1 a2 ((−∂z2 + Vc )u, u) L 2 + ( + B 2 ) zu 2L 2 + ε2 ∂x u 2L 2 − 2ε B zu L 2 ∂x u 2L 2 2 2 2 1 a a2 ≥ ((−∂z2 + Vc )u, u) L 2 + zu 2L 2 + 2 ε2 ∂x u 2L 2 2 4 a + 4B 2 1/2 ≥ C Hz u 2L 2 + Cε2 ∂x u 2L 2 . ≥
Conversely, from (1.22) and (2.9), we estimate directly (Hε u, u) L 2 ≤ C Hz u 2L 2 + C ε2 ∂x u 2L 2 . 1/2
For all ε ∈ (0, 1], this yields the equivalence of norms (2.10). For m ≥ 2, we will proceed by induction. For the clarity of the proof, let us introduce two notations. For m ∈ N, we denote by (Pm ) the property (Pm ): There exists εm > 0 such that, for all ε ∈ (0, εm ] and for all u ∈ B m , we have 1 2 m/2 2 u 2B m ≤ u 2L 2 (R3 ) + m/2 u L 2 (R3 ) ≤ 2 u 2B m , x,y u L 2 (R3 ) + Hε 2 and by (Qm ) the property (Qm ): There exists Cm > 0 such that, for all u ∈ B m and ε ∈ (0, 1], the operator Am = 1ε (Hεm − Hzm ) satisfies |(Am u, u) L 2 | ≤ Cm u 2B m . Note that the lemma will proved if we show that (Pm ) holds true for all m ≥ 0. Note also that, up to a possible modification of the sequence (εm )m∈N , this sequence can be chosen nonincreasing. Step 2. (Qm ) implies (Pm ). Let m ≥ 0 be fixed. From (Qm ), we deduce that Hεm/2 u 2L 2 = (Hεm u, u) L 2 = (Hzm u, u) L 2 + ε(Am u, u) L 2 m/2
= Hz
u 2L 2 + ε(Am u, u) L 2 ,
thus m/2
Hz
m/2
u 2L 2 − εCm u 2B m ≤ Hεm/2 u 2L 2 ≤ Hz
Setting εm =
1 , 2Cm
u 2L 2 + εCm u 2B m .
(A.1)
858
F. Delebecque-Fendt, F. Méhats
we deduce directly from (1.37) and (A.1) that, for ε ≤ εm , 1 2 m/2 2 u 2B m ≤ u 2L 2 (R3 ) + m/2 u L 2 (R3 ) ≤ 2 u 2B m . x,y u L 2 (R3 ) + Hε 2 We have proved (Pm ). Step 3. Proof of (Qm ) for m = 0 and 1. For m = 0, choose A0 = 0 and (Q0 ) is obvious. Let us prove (Q1 ). From (2.9), we have Hε = Hz + ε A1 , with A1 = −2i Bz∂x − ε∂x2 .
(A.2)
For all u ∈ B 1 , we have |(A1 u, u) L 2 | = | − 2i B(∂x u, zu) L 2 + ε ∂x u 2L 2 | ≤ C( zu 2L 2 + ∂x u 2L 2 ) ≤ C1 u 2B 1 , where we applied Cauchy-Schwarz and Lemma 2.3. We have proved (Q1 ). Step 4. Proof of (Qm ) for m ≥ 2. We shall now proceed by induction. Let m ≥ 2 and assume that (Qm−2 ) and (Qm−1 ) hold true. Let us prove (Qm ). We compute Hεm = (Hz + ε A1 )Hεm−2 (Hz + ε A1 ) = Hz Hεm−2 Hz + ε A1 Hεm−1 + ε Hεm−1 A1 = Hzm + ε Hz Am−2 Hz + ε A1 Hεm−1 + ε Hεm−1 A1 , where we have applied (Qm−2 ). Hence, denoting Am = Hz Am−2 Hz + A1 Hεm−1 + Hεm−1 A1 ,
(A.3)
we obtain Hεm = Hzm + ε Am and, for all u ∈ B m , we get from the definition (A.3) that |(Am u, u) L 2 | ≤ |(Hz Am−2 Hz u, u) L 2 | + 2|(Hεm−1 u, A1 u)| L 2 , where we used that Hεm−1 and the operator A1 defined by (A.2) are selfadjoint. It remains to estimate the two terms in the right-hand side of this inequality. The first one can be estimated as follows: |(Hz Am−2 Hz u, u) L 2 | = |(Am−2 Hz u, Hz u) L 2 | ≤ Cm−2 Hz u 2B m−2 ≤ Cm−2 u 2B m , where we used (Qm−2 ) and (2.8). The second one can be estimated as follows: |(Hεm−1 u, A1 u)| L 2 = Hεm−1 u, (i∂x )(−2Bzu + i∂x u) 2 L m−1 m−1 2 2 = Hε (i∂x u), Hε (−2Bzu + i∂x u) L2 m−1 m−1 2 2 ≤ Hε (i∂x u) Hε (−2Bzu + i∂x u) L2
≤ C ∂x u B m−1 zu B m−1
L2
+ C ∂x u 2B m−1
≤ C u 2B m , where we used that Hε commutes with ∂x , the Cauchy-Schwarz inequality, the property (Pm−1 ) and, at the last step, (2.8). Therefore, we have proved that |(Am u, u) L 2 | ≤ Cm u 2B m , which proves (Qm ). The proof of the lemma is complete.
2DEG in a Strong Magnetic Field
859
Appendix B. Proof of Lemma 2.5 For readability, we introduce in this Appendix the following notation: ∀(x, y, z) ∈ R3 , ∀α ∈ {0, 1}, ∀ε ∈ (0, 1), rαε (x, y, z) = x2 + y 2 + αε2 z 2 . With this notation, for all u ∈ B 1 , and α ∈ {0, 1}, the nonlinearity Fα (u) defined in (2.12) reads 1 2 Fα (u) = ∗ (|u| ) u. 4πrαε In order to prove the estimates stated in Lemma 2.5, we prove the following technical lemma on the Poisson nonlinearity. Lemma B.1. The following estimates hold: (i) There exists a positive constant C that does not depend on ε ∈ (0, 1] or α ∈ {0, 1} such that 1 ∀u, v ∈ H 1 (R3 ), ∗ (B.1) (uv) rε ∞ ≤ C u H 1 v H 1 . α L (ii) There exists a positive constant C that does not depend on ε ∈ (0, 1] or α ∈ {0, 1} such that, if D denotes a derivative with respect to x, y or z, 1 D ∀u, v ∈ H 1 (R3 ), ∀v ∈ H 1 (R3 ), ∗ (uv) 3 ∞ ≤ C u H 1 v H 1 . ε rα L L x,y
z
(B.2) (iii) For any integer k, let β = (βx , βy , βz ) ∈ N3 be a multiinteger of length |β| = β β β βx + βy + βz = k and let D β = ∂x x ∂y y ∂z z be the associated derivative. Then there exists a positive constant Ck depending only on k such that β 1 2 D ∀u ∈ H k , ∗ |u| (B.3) 3 ∞ ≤ Ck u H 1 u H k . rαε L L x,y
z
Proof. Noting that, for all (x, y) ∈ R2 , 1 1 (x, y, ·) ∗ ≤ (uv) ∞ rε (x − x )2 + (y − y )2 R2 α L (R ) × uv(x , y , ·) 1 dx dy , L (R )
we only need estimates for the convolution with √
1 x2 +y 2
(B.4)
in R2 . Here, we refer the reader
to Lemma B.1 of [7] where it was shown that for any f ∈ L p (R2 ) ∩ L 1 (R2 ) with 2 < p ≤ ∞, the following bound holds: 1 ∗ f ≤ C p f θL p (R2 ) f 1−θ , (B.5) L 1 (R 2 ) x2 + y 2 ∞ 2 L (R )
860
F. Delebecque-Fendt, F. Méhats
where θ = p/(2 p − 2). Moreover, from Cauchy-Schwarz and Sobolev embeddings, we deduce that for all p ∈ [1, +∞), uv(x, y, ·) L 1 (R) p 2 ≤ u(x, y, ·) L 2 (R) v(x, y, ·) L 2 (R) p 2 L (R ) L (R ) ≤ u L 2 p L 2 v L 2 p L 2 ≤ u H 1 (R3 ) v H 1 (R3 ) . x,y
x,y
z
z
Combined with (B.4) and (B.5), this proves Item (i). In order to prove Item (ii), consider a first order derivative D with respect to x, y or z and let u, v ∈ H 1 (R3 ). The usual properties of the convolution give D
1 1 1 = ε ∗ D (uv) = ε ∗ (D(u)v + u D(v)) . ∗ (uv) ε rα rα rα
Using (B.4) combined with the generalized Young formula gives 1 1 ∗ (D(u)v + u D(v)) ∗ D(u)v + u D(v) L 1z rε 3 ∞ ≤ 2 2 3 x +y α L x,y L z L x,y ≤ C D(u)v + u D(v) L 1z 6/5 , (B.6) L x,y
since the function x → √
1 x2 +y 2
belongs to L 2w (R2 ). We end the proof of Item (ii) noting
that, thanks to Sobolev embeddings, D(u)v + u D(v) L 1z 6/5 ≤ C D(u) L 2 v L 3x,y L 2z + C D(v) L 2 u L 3x,y L 2z L x,y
≤ C u H 1 v H 1 . In order to prove Item (iii), we follow the same lines with derivatives of higher orders. Consider the derivative D β where β = (βx , βy , βz ) ∈ N3 is a multiinteger of length |β| = βx + βy + βz = k. The usual properties of the convolution give β 1 2 D ∗ |u| rε α
L 3x,y L ∞ z
1 β 2 = ε ∗ D |u| r α
L 3x,y L ∞ z
.
Again, using (B.4) combined to the generalized Young’s formula lead to: 1 1 β 2 ∗ D β |u|2 D |u| ≤ ∗ 3 ∞ rε L 1z (R) x2 + y 2 α L x,y L z L 3x,y β ≤ C D |u|2 6/5 1 . L x,y L z
We now write D β (uu) =
β ≤β
Cβ D β (u)D β−β (u),
(B.7)
2DEG in a Strong Magnetic Field
861
where the sum is over the set of multiintegers β = (βx , βy , βz ) such that βx ≤ βx , βy ≤ βy and βz ≤ βz . Thus, combining (B.7) with Sobolev embeddings gives as above β 1 2 D ∗ |u| ≤C D β (u) L 2 u L 3x,y L 2z ε rα L 3x,y L ∞ z |β |=k +C D β u L 3x,y L 2z D β−β ( u ) L 2 |β |=, 0≤
≤ C u H k u H 1 + C
k−1
u H +1 u H k− .
=0
We conclude noting that, by interpolation, for all ≤ k − 1, u H +1 u H k− ≤ u H k u H 1 .
(B.8)
Proof of Lemma 2.5. We first prove (2.15). In that view, let us fix u and v in B 1 , ε ∈ (0, 1) and α ∈ {0, 1}, and note that 1 1 2 Fα (u) − Fα (v) = ∗ [(|u| + |v|)(|u| − |v|)] u + ∗ |v| (u − v) . 4πrαε 4πrαε (B.9) Hence (2.15) is a straightforward consequence of the following claim. There exists a positive constant C such that for all u 1 , u 2 and u 3 ∈ B 1 , 1 ∗ (u u ) u3 (B.10) 1 2 1 ≤ C u 1 B 1 u 2 B 1 u 3 B 1 . rε α B Proof of the claim (B.10). According to Lemma 2.3 we have 1 1 1 r ε ∗ (u 1 u 2 ) u 3 1 ≤ C r ε ∗ (u 1 u 2 ) u 3 1 + C Vc r ε ∗ (u 1 u 2 ) u 3 2 . α α α B H L (B.11) First, applying (B.1) and then Lemma 2.3, 1 1 r ε ∗ (u 1 u 2 ) u 3 2 ≤ r ε ∗ (u 1 u 2 ) ∞ u 3 L 2 ≤ C u 1 B 1 u 2 B 1 u 3 B 1 . α α L L (B.12) Similarly, we have 1 Vc 1 ∗ (u 1 u 2 ) u 3 ≤ ε ∗ (u 1 u 2 ) Vc u 3 L 2 ε rα rα L2 L∞ ≤ C u 1 B 1 u 2 B 1 u 3 B 1 .
(B.13)
Moreover, if D denotes any differential operator of order 1, 1 1 1 D ∗ (u u ) u ∗ (u u ) u + ∗ (u u ) D(u 3 ). (B.14) = D 1 2 3 1 2 3 1 2 rαε rαε rαε
862
F. Delebecque-Fendt, F. Méhats
Applying the Hölder inequality leads to D 1 ∗ (u 1 u 2 ) u 3 ≤ D 1 ∗ (u 1 u 2 ) 3 ε ε rα rα L2 L
∞ x,y L z
u 3 L 6x,y L 2z
≤ C u 1 B 1 u 2 B 1 u 3 B 1 , where we used (B.2). Finally, using (B.1), 1 1 D(u 3 ) L 2 ∗ (u u ) D(u ) ≤ ∗ (u u ) 1 2 3 1 2 rε ε rα α L2 L∞ ≤ C u 1 B 1 u 2 B 1 u 3 B 1 .
(B.15)
(B.16)
We deduce the claim (B.10) by combining (B.11) with (B.12), (B.13), (B.14), (B.15) and (B.16). In order to prove (2.16), consider a positive integer m and fix u ∈ B m . According m/2 to Lemma 2.3, we only need to estimate Fα (u) H m and Vc Fα (u) L 2 . In that view, we readily have 1 m/2 m/2 2 ∗ |u| (1 + Vc )Fα (u) 2 3 ≤ ∞ 3 (1 + Vc )u L 2 (R3 ) ε L (R ) 4πrα L (R ) ≤ C u 2B 1 (R3 ) u B m ,
(B.17)
where we applied (B.1) and Lemma 2.3. Now, let D β denote any derivative of length m and write D β (Fα (u)) =
Cβ D β
β ≤β
1 2 D β−β (u). ∗ |u| 4πrαε
Hence, 1 2 β ≤C 4πr ε ∗ |u| ∞ D u L 2 L 2 (R 3 ) α L |β |=m 1 β−β Dβ +C ∗ |u|2 (u) L 6x,y L 2z 3 ∞ D ε 4πr α L L
β D Fα (u)
1≤|β |≤m
x,y
≤ C u 2H 1 u H m + C
m
z
u H 1 u H u H m−+1 ,
(B.18)
=1
where we applied (B.1), (B.3) and Sobolev embeddings. Using the interpolation estimate (B.8) gives m D Fα (u)
L 2 (R 3 )
≤ C u 2H 1 u H m ≤ C u 2B 1 u B m .
(B.19)
We conclude the proof of (2.16) combining (B.17) and (B.19). This ends the proof of Lemma 2.5.
2DEG in a Strong Magnetic Field
863
Appendix C. Proof of Lemma 2.7 In this section, we set for simplicity X = (x, y) ∈ R2 . In order to prove estimate (2.23), we first study the difference between both convolution kernels. First Step. Difference between the convolution kernels. Let u, v be two functions of B 2 . Denote 1 1 δ(u, v)(X, z) = − |X − X | |X − X |2 + ε2 (z − z )2 R2 R ×u(X , z )v(X , z )dz d X .
(C.1)
We split the integral as follows:
δ(u, v)(X, z) = δ + (u, v)(X, z) + δ − (u, v)(X, z) =
X ∈+
z ∈R
+
X ∈−
z ∈R
,
where + = {X ∈ R2 , |X − X | > ε},
− = {X ∈ R2 , |X − X | < ε}.
For all η, µ ∈ R, and X = X , we have
1 |X − X |2 + ε2 η2
−
1 |X −
X |2
+ ε2 µ2
=
εη
εµ
−ξ 3/2 dξ (C.2) |X − X |2 + ξ 2
and 1 1 2 . − ≤ 2 2 2 2 2 2 |X − X | |X − X | + ε η |X − X | + ε µ
(C.3)
Besides, a simple study gives ∀X = X, ∀ξ ∈ R,
|ξ | |X −
X |2
3/2 + ξ2
1 2 ≤ √ . |X − X |2 3 3
(C.4)
Equation (C.2), combined with (C.3) and (C.4) allows us to claim that for all θ ∈ (0, 1), 1 1 1 − . (C.5) ≤ Cεθ |η − µ|θ 2 2 2 |X − X |2 + ε2 η2 |X − X |1+θ |X − X | + ε µ Now, applying (C.5) with η = z − z , µ = z and θ = 3/8 leads to 1 1 − u(X , z )v(X , z )dz d X 2 2 2 2 2 2 + R |X − X | + ε (z − z ) |X − X | + ε z 1 ≤ Cε3/8 |z|3/8 u(X , ·)v(X , ·) L 1 d X |11/8 + R |X − X 1 ≤ Cε3/8 |z|3/8 1/24 u L 6 L 2 v L 6 L 2 ≤ Cε1/3 |z|3/8 u B 1 v B 1 , X z X z ε
864
F. Delebecque-Fendt, F. Méhats
where we used the Hölder inequality and Sobolev embeddings. Similarly, applying (C.5) with η = z , µ = 0 and θ = 3/4 leads to 1 1 − , z )||v(X , z )|dz d X |u(X + R |X − X | |X − X |2 + ε2 z 2 1 ≤ Cε3/4 |z |3/4 u(X , ·)v(X , ·) L 1 d X R + |X − X |7/4 1 ≤ Cε3/4 5/12 z 3/8 u L 6 L 2 v L 6 L 2 X z X z ε ≤ Cε1/3 u B 2 v B 1 . We have proved that |δ + (u, v)(X, z)| ≤ Cε1/3 (1 + |z|3/8 ) u B 2 v B 1 .
(C.6)
Consider now δ − . Using (C.2) again leads to |ξ | |u(X , z )||v(X , z )|dξ dz d X . |δ − (u, v)(X, z)| ≤ 2 2 3/2 − R R (|X − X | + ξ ) (C.7) Moreover, a simple computation gives |ξ | 2 . dξ = |2 + ξ 2 )3/2 (|X − X |X − X | R Hence, (C.7) gives
1 |u(X , z )||v(X , z )|dz d X − R |X − X | ≤ Cε1/3 u L 6 L 2 v L 6 L 2
|δ − (u, v)(X, z)| ≤ C
X
z
X
z
≤ Cε1/3 u B 1 v B 1 .
(C.8)
Combining (C.6) and (C.8) allows to conclude that (C.9) |δ(u, v)(X, z)| ≤ Cε1/3 (1 + Vc (z)) u B 2 v B 1 , √ where we have used z 3/8 ≤ C(1 + Vc (z)), deduced from (1.17). Step 2: Difference between the nonlinearities. In order to prove Lemma 2.7, we need to estimate the following quantity in B 1 : F1 (u) − F0 (u) = δ(u, u) u,
(C.10)
where u ∈ B 2 is given. According to Lemma 2.3, we have F1 (u) − F0 (u) B 1 ≤ C Vc (F1 (u) − F0 (u)) L 2 + C F1 (u) − F0 (u) H 1 . First, we deduce from (C.9) that (1 + Vc )δ(u, u) u L 2 ≤ Cε1/3 (1 + Vc )u L 2 u B 2 v B 1 ≤ Cε1/3 u 3B 2 , (C.11)
2DEG in a Strong Magnetic Field
865
where we used Lemma 2.3. Let now D denote a first order derivative with respect to x, y or z. We clearly have D (F1 (u) − F0 (u)) L 2 1 1 ≤ − ∗ (D(u)u + u D(u)) u + δ(u, u)D(u) L 2 2 2 2 2 |X | |X | + ε z L ≤ 2|δ(u, D(u))u L 2 + δ(u, u)D(u) L 2 . (C.12) According to (C.9), we have δ(u, D(u))u L 2 ≤ Cε1/3 (1 + and δ(u, u)D(u) L 2 ≤ Cε1/3 (1 +
Vc )u L 2 u B 2 D(u) B 1 ≤ Cε1/3 u 3B 2
(C.13)
Vc )D(u) L 2 u B 2 u B 1 ≤ Cε1/3 u 3B 2 , (C.14)
where we used again Lemma 2.3. Combining (C.10), (C.11), (C.12),(C.13) and (C.14) gives (2.23). The proof of Lemma 2.7 is complete. Appendix D. Proof of the Technical Lemmas 3.4 and 3.5 Let us develop the operators a and A defined by (3.12) and (3.15) on the eigenbasis χ p . We have a(τ )u = − eiτ (E p −Eq ) a pq i∂x u q χ p , p≥0 q≥0
where we have introduced the coefficients a pq = 2Bzχ p χq .
(D.1)
Recall that, by Assumption 1.1, the potential Vc is even, so for all p, the function (χ p (z))2 is even. Therefore, we have # " ∀ p ∈ N, a pp = 2Bzχ p2 = 0, thus a(τ )u = −
eiτ (E p −Eq ) a pq i∂x u q χ p .
(D.2)
p≥0 q= p
Let us now integrate this formula in order to compute the operator A defined by (3.15): A(τ )u = i
eiτ (E p −Eq ) − 1 a pq i∂x u q χ p . E p − Eq
(D.3)
p≥0 q= p
Before proving Lemmas 3.4 and 3.5, let us give a useful estimate on coefficients a pq . For all p ∈ N, q ∈ N, k ∈ N we have (k+1)/2
|a pq | ≤ C
Eq
k/2
Ep
.
(D.4)
866
F. Delebecque-Fendt, F. Méhats
Indeed, we have k/2 k/2 k/2 E p a pq = 2B Hz χ p , zχq 2 = 2B χ p , Hz (zχq ) 2 L
L
k/2 2B Hz (zχq ) L 2
≤ ≤ 2B zχq B k
(k+1)/2
≤ C χq B k+1 ≤ C E q
,
where we applied Lemma 2.3. Proof of Lemma 3.4. Let n 0 be as in Assumption 1.2, let ∈ N and u ∈ C 0 ([0, T ], B 2n 0 +8+ ). Denoting u p = uχ p ,
2 µ2p = u p χ p C 0 ([0,T ],B 2n 0 +8+ ) ,
(D.5)
we have 2 u C 0 ([0,T ],B 2n 0 +8+ ) =
µ2p < +∞.
(D.6)
p≥0
From (D.3), we obtain
t A 2 ε
u(t) C 0 ([0,T ],B ) ≤ C
p≥0 q= p
(1 + q)n 0 |a pq | u q χ p C 0 ([0,T ],B +1 ) ,
where we used Assumption 1.2. Besides, applying Lemma 2.3 gives u q C 0 ([0,T ],H s (R2 )) = ≤
1 n +4+(−s)/2 Hz 0 (I n 0 +4+(−s)/2 Eq s/2 Eq C n +4+/2 µq , Eq 0
+ (−x,y )s/2 )(u q χq ) C 0 ([0,T ],L 2 )
(D.7)
for all s ≤ 2n 0 + 8 + . Hence, from the definition (1.37), we get (+1)/2
u q χ p C 0 ([0,T ],B +1 ) ≤ C E p
u q C 0 ([0,T ],L 2 (R2 )) + C u q C 0 ([0,T ]H +1 (R2 ))
(+1)/2
≤C
Ep
(+1)/2
+ Eq
n +4+/2
Eq 0
µq ,
and, by using (D.4) and (3.9), (+5)/2
(1 + q)n 0 |a pq | u q χ p C 0 ([0,T ],B +1 ) ≤ C ≤C
Ep E qn 0 |a pq | 2 Ep 1 µq . E 2p E q
(+1)/2
+ Eq
n +4+/2
Eq 0
E 2p
µq
2DEG in a Strong Magnetic Field
867
Therefore,
t A 2 ε
⎛
u(t) C 0 ([0,T ],B )
⎞⎛ ⎞ 1 µq ⎠⎝ ⎠ ≤C⎝ E 2p Eq p≥0 q≥0 ⎛ ⎞3/2 ⎛ ⎞1/2 1 ⎠ ⎝ ≤C⎝ µq2 ⎠ E 2p p≥0
q≥0
by Cauchy-Schwarz. To conclude, it suffices to use (3.9) and (D.6): the series converge and we obtain the desired estimate (3.22). Proof of Lemma 3.5. Let m = 4n 0 + 17 and let u ∈ C 0 ([0, T ], B m ) such that ∂t u ∈ C 0 ([0, T ], B m−2 ). Denoting now 2 2 u p = uχ p , (D.8) ν 2p = u p χ p C 0 ([0,T ],B m ) + ∂t u p χ p C 0 ([0,T ],B m−2 ) , we have 2 2 u C 0 ([0,T ],B m ) + ∂t u C 0 ([0,T ],B m−2 ) =
ν 2p < +∞.
(D.9)
p≥0
Applying Lemma 2.3 as above yields (m−s)/2
Ep
(m−2−s)/2
u p C 0 ([0,T ],H s (R2 )) + E p
∂t u p C 0 ([0,T ],H s (R2 )) ≤ Cν p (D.10)
for all s ≤ m. By composing the expressions (D.3) and (D.2) for A and a, we obtain A(τ )a(τ )u = i
eiτ (E p −Eq ) − 1 eiτ (Eq −E n ) a pq aqn ∂x2 u n χ p E p − Eq p≥0 q= p n=q
1 − eiτ (Eq −E p ) =i (a pq )2 ∂x2 u p χ p E p − Eq p≥0 q= p
+i
eiτ (E p −E n ) − eiτ (Eq −E n ) a pq aqn ∂x2 u n χ p . E p − Eq p≥0 q= p n = q
n = p
Now, remark that, by (1.24) and (D.1), we have for all p ∈ N the identity 1+
(a pq )2 = αp. E p − Eq
q= p
Therefore we get, using the definition (3.3), A(τ )a(τ ) + i∂x2 u = −i A0 u (a pq )2 2 −i eiτ (Eq −E p ) ∂ u p χp E p − Eq x p≥0 q= p a a pq qn eiτ (E p −E n ) − eiτ (Eq −E n ) +i ∂ 2un χ p , E p − Eq x p≥0 q= p n = q
n = p
868
F. Delebecque-Fendt, F. Méhats
and, integrating, t t s s A 2 + i∂x2 a 2 u(s)ds + i A0 u(s)ds ε ε 0 0 t (a pq )2 2 χp eis(Eq −E p )/ε ∂x2 u p (s) ds = −i E p − Eq 0 p≥0 q= p t a pq aqn 2 2 eis(E p −E n )/ε − eis(Eq −E n )/ε ∂x2 u n (s) ds. χp +i E p − Eq 0 p≥0 q= p n = q
n = p
(D.11) In order to estimate the right-hand side of this identity, we claim that, for all p ∈ N, p ∈ N and λ = 0, we have t iλs/ε2 2 χ p (z) e ∂x u q (s, x, y) ds
1/2
C 0 ([0,T ],B 1 )
0
1/2
ε2 E p + Eq ≤ CT νq , (D.12) |λ| E q(m−4)/2
where C T only depends on T and νn is defined by (D.8). This claim is proved below. As a consequence, we can estimate (D.11) as follows: t t s s 2 A 2 a 2 + i∂x u(s)ds + i A0 u(s)ds 0 ε ε 0 0 C ([0,T ],B 1 ) (a pq )2 1 ≤ Cε2 νp 2 (m−5)/2 |E p − E q | E p p≥0 q= p
+Cε2
1/2 1/2 |a pq ||aqn | E p + En 1 1 + νn (m−4)/2 |E p − E q | |E p − E n | |E q − E n | En p≥0 q= p n = q n = p
E 3p (1 + p)2n 0 ≤ Cε2 νp E q2 E (m−5)/2 p p≥0 q≥0
n +11/2 En 0 1 (1 + q)n 0 (1 + n)n 0 ν n 0 +2 (m−4)/2 n 2 E p Eq En p≥0 q≥0 n≥0 νp 1 νn 1 1 2 , ≤ Cε2 + Cε 2 3 2 2 (1 + q ) 1 + p (1 + p ) (1 + q ) 1 + n
+Cε2
p≥0 q≥0
p≥0 q≥0 n≥0
where we used Assumption 1.2, (D.4), (3.9) and recall that m = 4n 0 + 17. Hence we deduce (3.23) by using Cauchy-Schwarz and (D.9). It remains to prove the claim. Proof of the claim (D.12). Let
t
v(t, x, y, z) = χ p (z) 0
2
eiλs/ε ∂x2 u q (s, x, y) ds,
(D.13)
2DEG in a Strong Magnetic Field
869
for p ∈ N, q ∈ N and λ = 0. An integration by parts in (D.13) yields t ε2 2 2 v(t, x, y, z) = i χ p eiλs/ε ∂x2 ∂t u q (s, x, y) ds + eiλt/ε ∂x2 u q (t, x, y) λ 0 −∂x2 u q (0, x, y) . Hence, by using (D.10), we obtain 1/2
v C 0 ([0,T ],B 1 ) ≤ C T
1/2
ε2 E p + Eq νq , |λ| E q(m−4)/2
where C T only depends on T . This concludes the proof of (D.12). The proof of Lemma 3.5 is complete. Acknowledgement. The authors were supported by the Agence Nationale de la Recherche, ANR project QUATRAIN. They wish to thank N. Ben Abdallah and F. Castella for fruitful discussions.
References 1. Allaire, G., Piatnitski, A.: Homogenization of the Schrödinger equation and effective mass theorems. Commun. Math. Phys. 258(1), 1–22 (2005) 2. Ando, T., Fowler, B., Stern, F.: Electronic properties of two-dimensional systems. Rev. Mod. Phys. 54, 437–672 (1982) 3. Bao, W., Markowich, P.A., Schmeiser, C., Weishäupl, R.: On the Gross-Pitaevski equation with strongly anisotropic confinement: formal asymptotics and numerical experiments. Math. Models Meth. Appl. Sci. 15(5), 767–782 (2005) 4. Bastard, G.: Wave Mechanics Applied to Semi-conductor Heterostructures. Les Éditions de Physique, Les Ulis: EDP Sciences, 1992 5. Ben Abdallah, N., Castella, F., Delebecque-Fendt, F., Méhats, F.: The strongly confined Schrödinger-Poisson system for the transport of electrons in a nanowire. SIAM J. Appl. Math. 69(4), 1162–1173 (2009) 6. Ben Abdallah, N., Castella, F., Méhats, F.: Time averaging for the strongly confined nonlinear Schrödinger equation, using almost periodicity. J. Diff. Eq. 245(1), 154–200 (2008) 7. Ben Abdallah, N., Méhats, F., Pinaud, O.: Adiabatic approximation of the Schrödinger-Poisson system with a partial confinement. SIAM J. Math. Anal 36, 986–1013 (2005) 8. Ben Abdallah, N., Méhats, F., Schmeiser, C., Weishäupl, R.M.: The nonlinear Schrödinger equation with strong anisotropic harmonic potential. SIAM J. Math. Anal. 37(1), 189–199 (2005) 9. Bony, J.-M., Chemin, J.-Y.: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. France 122(1), 77–118 (1994) 10. Brezzi, F., Markowich, P.A.: The three dimensional Wigner -Poisson Problem : existence, uniqueness and approximation. Math. Meth. Appl. Sci. 14(1), 35–61 (1991) 11. Carles, R.: Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential. Commun. Contemp. Math. 7(4), 483–508 (2005) 12. Carles, R., Markowich, P.A., Sparber, C.: On the Gross-Pitaevskii equation for trapped dipolar quantum gases. Nonlinearity 21(11), 2569–2590 (2008) 13. Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes 10, Providence, RI: Amer. Math. Soc., 2003 14. Cazenave, T., Esteban, M.J.: On the stability of stationary states for nonlinear Schrödinger equations with an external magnetic field. Mat. Apl. Comput. 7, 155–168 (1988) 15. da Costa, R.C.T.: Quantum mechanics for a constraint particle. Phys. Rev. A 23(4), 1982–1987 (1981) 16. de Bouard, A.: Nonlinear Schrödinger equations with magnetic fields. Diffel. Int. Eqs. 4(1), 73–88 (1991) 17. Degond, P., Parzani, C., Vignal, M.-H.: A Boltzmann model for trapped particles in a surface potential. Multiscale Modeling & Simulation, SIAM 5(2), 364–392 (2006) 18. Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7(1), 73–102 (1995)
870
F. Delebecque-Fendt, F. Méhats
19. Egorov, Yu. V., Shubin, M.A.: Partial Differential Equations. I. Encyclopaedia Math. Sci., 30, Berlin: Springer, 1992 20. Ferry, D.K., Goodnick, S.M.: Transport in Nanostructures. Cambridge: Cambridge University Press, 1997 21. Froese, R., Herbst, I.: Realizing holonomic constraints in classical and quantum mechanics. Commun. Math. Phys. 220(3), 489–535 (2001) 22. Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Lalacians. Berlin-Heidelberg-NewYork: Springer, 2005 23. Illner, R., Zweifel, P.F., Lange, H.: Global Existence, Uniqueness and Asymptotic Behaviour of Solutions of the Wigner-Poisson and Schrödinger-Poisson Systems. Math. Meth. Appl. Sci. 17(5), 349–376 (1994) 24. Messiah, A.: Mécanique Quantique, Tome 1. Paris: Dunod, 2003 25. Pinaud, O.: Adiabatic approximation of the Schrödinger-Poisson system with a partial confinement: the stationary case. J. Math. Phys. 45(5), 2029–2050 (2004) 26. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. 1–4, New York, San FranciscoLondon: Academic Press, 1972–1979 27. Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Appl. Math. Sci. vol. 59, New York-Heidelberg-Tokio: Springer-Verlag, 1985 28. Smrˇcka, L., Jungwirth, T.: In-plane magnetic-field-induced anisotropy of 2D Fermi contours and the field-dependent cyclotron mass. J. Phys. Conds. Matter 6, 55–64 (1994) 29. Sparber, C.: Effective mass theorems for nonlinear Schrödinger equations. SIAM J. Appl. Math. 66(3), 820–842 (2006) 30. Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Lecture Notes in Mathematics 1821, Berlin-Heidelberg-New York: Springer-Verlag, 2003 31. Vinter, B., Weisbuch, C.: Quantum Semiconductor Structures: Fundamentals & Applications. LondonNewYork: Academic Press, 1991 Communicated by P. Constantin
Commun. Math. Phys. 292, 871–912 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0901-6
Communications in
Mathematical Physics
Cardy Algebras and Sewing Constraints, I Liang Kong1,2 , Ingo Runkel3 1 Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany.
E-mail: [email protected]
2 Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 45, 53115 Bonn, Germany 3 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom.
E-mail: [email protected] Received: 28 March 2009 / Accepted: 7 June 2009 Published online: 13 August 2009 – © Springer-Verlag 2009
Abstract: This is part one of a two-part work that relates two different approaches to two-dimensional open-closed rational conformal field theory. In part one we review the definition of a Cardy algebra, which captures the necessary consistency conditions of the theory at genus 0 and 1. We investigate the properties of these algebras and prove uniqueness and existence theorems. One implication is that under certain natural assumptions, every rational closed CFT is extendable to an open-closed CFT. The relation of Cardy algebras to the solutions of the sewing constraints is the topic of part two. Contents 1. 2.
Introduction and Summary . . . . . . . . . . . . . Preliminaries on Tensor Categories . . . . . . . . 2.1 Tensor categories and (co)lax tensor functors . 2.2 Algebras in tensor categories . . . . . . . . . 2.3 Modular tensor categories . . . . . . . . . . . 2.4 The functors T and R . . . . . . . . . . . . . 3. Cardy Algebras . . . . . . . . . . . . . . . . . . 3.1 Modular invariance . . . . . . . . . . . . . . 3.2 Two definitions . . . . . . . . . . . . . . . . 3.3 Uniqueness and existence theorems . . . . . . A. Appendix . . . . . . . . . . . . . . . . . . . . . . A.1 Proof of Lemma 2.7 . . . . . . . . . . . . . A.2 Proof of Lemma 3.20 . . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
871 875 875 878 884 885 891 891 895 899 905 905 906
1. Introduction and Summary This is part I of a two-part work which relates two different approaches to two-dimensional open-closed rational conformal field theory (CFT).
872
L. Kong, I. Runkel
The first approach uses a three-dimensional topological field theory to express correlators of the open-closed CFT [Fe,FRS,Fj]. Here one starts from a modular tensor category, which defines a three-dimensional topological field theory [RT,T], and from a special symmetric Frobenius algebra in this modular tensor category. To each openclosed world sheet X one assigns a 3-bordism M X with embedded ribbon graph constructed from this Frobenius algebra. To the boundary of M X the topological field theory assigns a vector space B(X ) and to M X itself a vector C X ∈ B(X ). One proves that this collection of vectors C X provides a so-called solution to the sewing constraints [Fj]. If the modular tensor category is the category of representations of a suitable vertex operator algebra, the spaces B(X ) are spaces of conformal blocks, and the C X are the correlators of an open-closed CFT. In this approach one thus makes an ansatz for the correlators on all world sheets simultaneously and then proves that they obey the necessary consistency conditions. The relation to CFT rests on convergence and factorisation properties of higher genus conformal blocks, and the precise list of conditions the vertex operator algebra has to fulfill for these properties to hold is not known. However, from a physical perspective one expects that interesting classes of models [W,FK] will have all the necessary properties. The second approach uses the theory of vertex operator algebras to construct directly the correlators of the genus 0 and genus 1 open-closed CFT [HK1,HK2,K3]. More precisely, in this approach one uses a notion of CFT defined in [K3, Sect. 1] (and called partial CFT1 there), where one glues Riemann surfaces around punctures with local coordinates as in [V,H1] instead of gluing around parametrised circles as in [Se]. This approach is based on the precise relation between genus-0 CFT and vertex operator algebras [H1], and on the fact that the category of modules over a rational vertex operator algebra is a modular tensor category [HL,H2]. Let us call a vertex operator algebra rational if it satisfies the conditions in [H2, Sect. 1]. If one analyses the consistency conditions of a genus-0,1 open-closed CFT, one arrives at a structure called Cardy CV |CV ⊗V - algebra in [K3]. It is formulated in purely categorical terms in the categories CV and CV ⊗V of modules over the rational vertex operator algebras V and V ⊗ V , respectively. Cardy algebras (Definition 3.7) are the central objects in part I of this work, and we will describe their relation to CFT in slightly more detail below. The data in a Cardy algebra amounts to an open-closed CFT on a generating set of world sheets, from which the entire CFT can be obtained by repeated gluing. The conditions on this data are necessary for this procedure to give a consistent genus-0,1 open-closed CFT. The two approaches just outlined start at opposite ends of the same problem. In both cases the difficulty to obtain a complete answer lies in the lack of control over the properties of higher genus conformal blocks. Nonetheless, both approaches give rise to notions formulated in entirely categorical terms, and we can compare the structures one finds. In part II we will come to the satisfying conclusion that giving a solution to the sewing constraints is essentially equivalent, in a sense made precise in part II, to giving a Cardy algebra. To motivate the notion of a Cardy algebra and our interest in it, we would like to outline how it emerges when formulating closed CFT and open-closed CFT at genus-0,1 in the language of vertex operator algebras. The next one and a half pages, together with a few
1 The qualifier ‘partial’ refers to the fact that the gluing of punctures is only defined if the coordinates ζ , 1 ζ2 around two punctures can be analytically extended to a large enough region containing no other punctures, so that the identification ζ1 ∼ 1/ζ2 is well-defined. That is, if ζ1 can be extended to a disc of radius r , then ζ2 must be defined on a disc of radius greater than 1/r . Both discs must not contain further punctures.
Cardy Algebras and Sewing Constraints, I
873
remarks in the main text, are the only places where we make reference to vertex operator algebras. The reader who is not familiar with this structure is invited to skip ahead. All types of field algebras occurring below are called self-dual if they are endowed with non-degenerate invariant bilinear forms. A genus-0 closed CFT is equivalent to an algebra over a partial dioperad consisting of spheres with arbitrary in-coming and out-going punctures. The dioperad structure allows to compose one in-going and one out-going puncture of distinct spheres, so that the result is again a sphere. Such an algebra with additional natural properties is canonically equivalent to a so-called self-dual conformal full field algebra [HK2,K1]. A conformal full field algebra contains chiral and anti-chiral parts, the easiest nontrivial example is given by V ⊗V , where V is a vertex operator algebra. A conformal full field algebra containing V ⊗ V as a subalgebra is called a conformal full field algebra over V ⊗ V . When V is rational, the category of self-dual conformal full field algebras over V ⊗ V is isomorphic to the category of commutative symmetric Frobenius algebras in CV ⊗V [K1, Thm. 4.15]. Similarly, a genus-0 open CFT is an algebra over a partial dioperad consisting of disks with an arbitrary number of in-coming and out-going boundary punctures. Such an algebra with additional natural properties is canonically equivalent to a self-dual open-string vertex operator algebra as defined in [HK1]. A vertex operator algebra V is naturally an open-string vertex operator algebra. An open-string vertex operator algebra containing V as a subalgebra in its meromorphic centre is called an open-string vertex operator algebra over V . When V is rational, the category of self-dual open-string vertex operator algebras over V is isomorphic to the category of symmetric Frobenius algebras in CV , see [HK1, Thm. 4.3] and [K3, Thm. 6.10]. Finally, a genus-0 open-closed CFT is an algebra over the Swiss-cheese partial dioperad, which consists of disks with both interior punctures and boundary punctures, and is equipped with an action of the partial spheres dioperad. Such an algebra can be constructed from a so-called self-dual open-closed field algebra [K2]. It consists of a self-dual conformal full field algebra Acl , a self-dual open-string vertex operator algebra Aop , and interactions between Acl and Aop satisfying certain compatibility conditions. Namely, if Acl is defined over V ⊗ V and Aop over V , one requires that the boundary condition on a disc is V -invariant in the sense that both the chiral copy V ⊗ 1 and the anti-chiral copy 1 ⊗ V of V in Acl give the copy of V in Aop in the limit of the insertion point approaching a point on the boundary of the disc [K2, Def. 1.25]. An open-closed field algebra with V -invariant boundary condition is called an open-closed field algebra over V . When V is rational, the category of self-dual open-closed field algebras over V is isomorphic to the category of triples (Aop |Acl , ι˜cl-op ), where Acl is a commutative symmetric Frobenius CV ⊗V -algebra, Aop a symmetric Frobenius CV -algebra, and ι˜cl-op an algebra homomorphism T (Acl ) → Aop satisfying a centre condition (given in (3.20) below), see [K2, Thm. 3.14] and [K3, Sect. 6.2]. Here T : CV ⊗V → CV is the Huang-Lepowsky tensor product functor [HL]. The genus-1 theory does not provide new data as it is determined by taking traces of genus-0 correlators, but it does provide two additional consistency conditions: the modular invariance condition for one-point correlators on the torus [So], and the Cardy condition for boundary-two-point correlators on the annulus [C2,Lw]. Their categorical formulations have been worked out in [HK3,K3]. Adding them to the axioms of a self-dual open-closed field algebra over V finally results in the notion of a Cardy CV |CV ⊗V -algebra. One can prove that the category of self-dual open-closed field algebras over a rational vertex operator algebra V satisfying the two genus-1 consistency conditions is isomorphic to the category of Cardy CV |CV ⊗V -algebras [K3, Thm. 6.15].
874
L. Kong, I. Runkel
If V is rational, then so is V ⊗ V [DMZ,HK2]. Thus both CV and CV ⊗V are modular tensor categories. In fact, CV ⊗V ∼ = CV (CV )− (see [FHL, Thm. 4.7.4] and [DMZ, Thm 2.7]), where the minus sign relates to the particular braiding used for CV ⊗V . Namely, for a given modular tensor category D, we denote by D− the modular tensor category obtained from D by inverting braiding and twist. We will also sometimes write D+ for D. The product amounts to taking direct sums of pairs of objects and tensor products of morphisms spaces. The definition of a Cardy algebra can be stated in a way that no longer makes reference to the vertex operator algebra V , and therefore makes 2 ≡ C C , this leads sense in an arbitrary modular tensor category C. Abbreviating C± + − 2 to the definition of a Cardy C|C± -algebra. The relation to genus-0,1 open-closed CFT outlined above is the main motivation for 2 -algebras. In part I of this work we investigate how much one our interest in Cardy C|C± can learn about Cardy algebras in the categorical setting, and without the assumption that the modular tensor category C is given by CV for some V . We briefly summarise our approach and results below. In Sect. 2.1–2.3, we recall some basic notions we will need, such as (co)lax tensor functors, Frobenius functors, and modular tensor categories. In Sect. 2.4, we study the 2 → C, which is defined by the tensor product on C via T (⊕ A × B ) = functor T : C± i i i ⊕i Ai ⊗ Bi for Ai , Bi ∈ C. Using the braiding of C one can turn T into a tensor functor. A tensor functor is automatically also a Frobenius functor, and so takes a Frobenius algebra A in its domain category to a Frobenius algebra F(A) in its target category. 2 , also defined in Sect. 2.4. An important object in this work is the functor R : C → C± We show that R is left and right adjoint to T . As a consequence, R is automatically a lax and colax tensor functor, but it is in general not a tensor functor. However, we will show that it is still a Frobenius functor, and so takes Frobenius algebras in C to Frobenius 2 . In fact, it also preserves the properties simple, special and symmetric algebras in C± of a Frobenius algebra. In the case C = CV the functor R : CV → CV ⊗V was first constructed in [Li1,Li2] using techniques from vertex operator algebras. This motivated the present construction and notation. The functor R was also considered in a slightly different context in [ENO2]. The above results imply that R and T form an ambidextrous adjunction, and we will 2 . For example, use this adjunction to transport algebraic structures between C and C± the algebra homomorphism ι˜cl-op : T (Acl ) → Aop in C is transported to an algebra 2 . This gives rise to an alternative definition homomorphism ιcl-op : Acl → R(Aop ) in C± 2 of a Cardy C|C± -algebra as a triple (Aop |Acl , ιcl-op ). To prepare the definition of a Cardy algebra, in Sect. 3.1 we discuss the so-called 2 (Definition 3.1 below). We show that modular invariance condition for algebras in C± when Acl is simple, the modular invariance condition can be replaced by an easier condition on the quantum dimension of Acl (namely, the dimension of Acl has to be that of the modular tensor category C), see Theorem 3.4. In Sect. 3.2 we give the two definitions of a Cardy algebra and prove their equivalence. Section 3.3 contains our main results. We first show that for each special symmetric Frobenius algebra A in C (see Sect. 2.2 for the definition of special) one obtains a Cardy algebra (A|Z (A), e), where Z (A) is the full centre of A (Theorem 3.18). The full centre [Fj, Def. 4.9] is a subobject of R(A) and e : Z (A) → R(A) is the canonical embedding. Next we prove a uniqueness theorem (Theorem 3.21), which states that if (Aop |Acl , ιcl-op ) is a Cardy algebra such that dim Aop = 0 and Acl is simple, then Aop is special and (Aop |Acl , ιcl-op ) is isomorphic to (Aop |Z (Aop ), e). When combined with part II of this work, this result amounts to [Fj, Thm. 4.26] and provides an alternative
Cardy Algebras and Sewing Constraints, I
875
(and shorter) proof. Finally we show that for every simple modular invariant commu2 there exists a simple special symmetric tative symmetric Frobenius algebra Acl in C± Frobenius algebra Aop and an algebra homomorphism ιcl-op : Acl → R(Aop ) such that (Aop , Acl , ιcl-op ) is a Cardy algebra (Theorem 3.22). This theorem is closely related to a result announced in [Mü2] and provides an independent proof in the framework of Cardy algebras. In physical terms these two theorems mean that a rational open-closed CFT with a unique closed vacuum state can be uniquely reconstructed from its correlators involving only discs with boundary punctures, and that every closed CFT with unique vacuum and left/right rational chiral algebra V ⊗ V occurs as part of such an open-closed CFT. 2. Preliminaries on Tensor Categories In this section, we review some basic facts of tensor categories and fix our conventions and notations along the way. 2.1. Tensor categories and (co)lax tensor functors. In a tensor (or monoidal) category C with tensor product bifunctor ⊗ and unit object 1, for U, V, W ∈ C, we denote the ∼ =
associator U ⊗ (V ⊗ W ) − → (U ⊗ V ) ⊗ W by αU,V,W , the left unit isomorphism ∼ =
∼ =
→ U by lU , and the right unit isomorphism U ⊗ 1 − → U by rU . If C is braided, 1⊗U − for U, V ∈ C we write the braiding isomorphism as cU,V : U ⊗ V → V ⊗ U . Let C1 and C2 be two tensor categories with units 11 and 12 respectively. For simplicity, we will often write ⊗, α, l, r for the data of both C1 and C2 . Lax and colax tensor functors are defined as follows, see e.g. [Y, Ch. I.3] or [Ln, Ch. I.1.2]. Definition 2.1. A lax tensor functor G : C1 → C2 is a functor equipped with a morphism φ0G : 12 → G(11 ) in C2 and a natural transformation φ2G : ⊗◦(G × G) → G ◦⊗ such that the following three diagrams commute: α
G(A) ⊗ (G(B) ⊗ G(C)) id G(A) ⊗φ2G
φ2G ⊗id G(C)
G(A) ⊗ G(B ⊗ C)
G(A ⊗ B) ⊗ G(C),
φ2G
l G(A)
G(11 ) ⊗ G(A)
G(α)
/ G(A) G(l −1 A )
φ0G ⊗id G(A)
φ2G
(2.1)
φ2G
G(A ⊗ (B ⊗ C)) 12 ⊗ G(A)
/ (G(A) ⊗ G(B)) ⊗ G(C)
/ G(11 ⊗ A)
,
/ G((A ⊗ B) ⊗ C) G(A) ⊗ 12
r G(A)
G(r A−1 )
id G(A) ⊗φ0G
G(A) ⊗ G(11 )
/ G(A)
φ2G
.
/ G(A ⊗ 11 )
(2.2) Definition 2.2. A colax tensor functor is a functor F : C1 → C2 equipped with a morphism ψ0F : F(11 ) → 12 in C2 , and a natural transformation ψ2F : F ◦⊗ → ⊗◦(F × F) such that the following three diagrams commute:
876
L. Kong, I. Runkel α
F(A) ⊗ (F(B) O ⊗ F(C))
/ (F(A) ⊗ F(B)) ⊗ F(C) O
id F(A) ⊗ψ2F
ψ2F ⊗id F(C)
F(A) ⊗ F(B ⊗ C) O
F(A ⊗ B)O ⊗ F(C),
ψ2F F(α)
F(A ⊗ (B ⊗ C))
12 ⊗ O F(A) o
l −1 F(A)
F(11 ) ⊗ F(A) o
ψ2F
/ F((A ⊗ B) ⊗ C)
F(A)O ⊗ 12 o
F(A) O F(l A ),
ψ0F ⊗id F(A)
(2.3)
ψ2F
−1 r F(A)
F(A) O F(r A )
id F(A) ⊗ψ0F
F(A) ⊗ F(11 ) o
F(11 ⊗ A)
ψ2F
F(A ⊗ 11 ). (2.4)
We denote a lax tensor functor by (G, φ2G , φ0G ) or just G, and a colax tensor functor by (F, ψ2F , ψ0F ) or F. Definition 2.3. A tensor functor T : C1 → C2 is a lax tensor functor (T, φ2T , φ0T ) such that φ0T , φ2T are both isomorphisms. A tensor functor (T, φ2T , φ0T ) is automatically a colax tensor functor (T, ψ2T , ψ0T ) with ψ0T = (φ0T )−1 and ψ2T = (φ2T )−1 . In the next section we will discuss algebras in tensor categories. The defining properties (2.1) and (2.2) of a lax tensor functor are analogues of the associativity, the left-unit, and the right-unit properties of an algebra. Indeed, a lax tensor functor G : C1 → C2 maps a C1 -algebra to a C2 -algebra. Similarly, (2.3) and (2.4) are analogues of the coassociativity, the left-counit and the right-counit properties of a coalgebra, and a colax tensor functor F : C1 → C2 maps a C1 -coalgebra to a C2 -coalgebra. We will later make use of functors that take Frobenius algebras to Frobenius algebras. This requires a stronger condition than being lax and colax and leads to the notion of a ‘functor with Frobenius structure’ or ‘Frobenius monoidal functor’ [Sz,DP,P], which we will simply refer to as Frobenius functor. Definition 2.4. A Frobenius functor F : C1 → C2 is a tuple F ≡ (F, φ2F , φ0F , ψ2F , ψ0F ) such that (F, φ2F , φ0F ) is a lax tensor functor, (F, ψ2F , ψ0F ) is a colax tensor functor, and such that the following two diagrams commute: F(A) ⊗ (F(B) O ⊗ F(C))
α
id F(A) ⊗ψ2F
φ2F ⊗id F(C)
F(A ⊗ B)O ⊗ F(C)
F(A) ⊗ F(B ⊗ C) φ2F
F(A ⊗ (B ⊗ C))
/ (F(A) ⊗ F(B)) ⊗ F(C)
ψ2F F(α)
/ F((A ⊗ B) ⊗ C)
(2.5)
Cardy Algebras and Sewing Constraints, I
877 α −1
F(A) ⊗ (F(B) ⊗ F(C)) o
(F(A) ⊗ F(B)) ⊗ F(C) O
id F(A) ⊗φ2F
ψ2F ⊗id F(C)
F(A) ⊗ F(B ⊗ C) O
F(A ⊗ B) ⊗ F(C)
ψ2F
F(A ⊗ (B ⊗ C)) o
F(α −1 )
(2.6)
φ2F
F((A ⊗ B) ⊗ C)
Proposition 2.5. If (F, φ2F , φ0F ) is a tensor functor, then F is a Frobenius functor with ψ0F = (φ0F )−1 and ψ2F = (φ2F )−1 . Proof. Since F is a tensor functor, it is lax and colax. If we replace ψ2F by (φ2F )−1 in (2.5) and (2.6), both commuting diagrams are equivalent to (2.1), which holds because F is lax. Thus F is a Frobenius functor.
The converse statement does not hold. For example, the functor R which we define in Sect. 2.4 is Frobenius but not tensor. Let us recall the notion of adjunctions and adjoint functors [Ma, Ch. IV.1]. Definition 2.6. An adjunction from C1 to C2 is a triple F, G, χ , where F and G are functors F : C1 → C2 , G : C2 → C1 , and χ is a natural isomorphism which assigns to each pair of objects A1 ∈ C1 , A2 ∈ C2 a bijective map ∼ =
χ A1 ,A2 : HomC2 (F(A1 ), A2 ) −−→ HomC1 (A1 , G(A2 )), which is natural in both A1 and A2 . F is called a left-adjoint of G and G is called a right-adjoint of F. For simplicity, we will often abbreviate χ A1 ,A2 as χ . Associated to each adjunction ρ
δ
→ G F and F G − → idC2 , where F, G, χ , there are two natural transformations idC1 − idC1 and idC2 are identity functors, given by δ A1 = χ (id F(A1 ) ),
ρ A2 = χ −1 (id G(A2 ) )
(2.7)
for Ai ∈ Ci , i = 1, 2. They satisfy the following two identities: δG
Gρ
id G
G −→ G F G −→ G = G −−→ G,
Fδ
ρF
id F
F −→ F G F −→ F = F −−→ F.
(2.8)
We have, for g : F(A1 ) → A2 and f : A1 → G(A2 ), χ (g) = G(g) ◦ δ A1 ,
χ −1 ( f ) = ρ A2 ◦ F( f ).
(2.9)
For simplicity, δ A1 and ρ A2 are often abbreviated as δ and ρ, respectively. Let F, G, χ be an adjunction from a tensor category C1 to a tensor category C2 and (F, ψ2F , ψ0F ) a colax tensor functor from C1 to C2 . We can define a morphism
878
L. Kong, I. Runkel
φ0G : 11 → G(12 ) and a natural transformation φ2G : ⊗ ◦ (G × G) → G ◦ ⊗ by, for A, B ∈ C2 , δ11
G(ψ0F )
φ0G = χ (ψ0F ) = 11 −→ G F(11 ) −−−−→ G(12 ), δ
→ G F (G(A) ⊗ G(B)) φ2G = χ ((ρ A ⊗ ρ B ) ◦ ψ2F ) = G(A) ⊗ G(B) − G(ψ2F )
G(ρ A ⊗ρ B )
−−−−→ G (F G(A) ⊗ F G(B)) −−−−−−→ G(A ⊗ B),
(2.10)
φ2G
where we have used the first identity in (2.9). Notice that is natural because it is a composition of natural transformations. One can easily show that ψ0F and ψ2F can be re-obtained from φ0G and φ2G as follows: Fφ0G
ρ
→ 12 , ψ0F = χ −1 (φ0G ) = F(11 ) −−→ F G(12 ) − F(δ⊗δ)
ψ2F = χ −1 (φ2G ◦ (δ ⊗ δ)) = F(U ⊗ V ) −−−−→ F (G F(U ) ⊗ G F(V )) Fφ2G
ρ
−−→ F G (F(U ) ⊗ F(V )) − → F(U ) ⊗ F(V ).
(2.11)
for U, V ∈ C1 . The following result is standard; for the sake of completeness, we give a proof in Appendix A.1. Lemma 2.7. (F, ψ2F , ψ0F ) is a colax tensor functor iff (G, φ2G , φ0G ) is a lax tensor functor. 2.2. Algebras in tensor categories. An algebra in a tensor category C, or a C-algebra, is a triple A = (A, m, η), where A is an object of C, m (the multiplication) is a morphism A ⊗ A → A such that m ◦ (m ⊗ id A ) ◦ α A,A,A = m ◦ (id A ⊗ m), and η (the unit) is a morphism 1 → A such that m ◦ (id A ⊗ η) = id A ◦ r A and m ◦ (η ⊗ id A ) = id A ◦ l A . If C is braided and m ◦ c A,A = m, then A is called commutative. A left A-module is a pair (M, m M ), where M ∈ C and m M is a morphism A⊗M → M such that m M ◦(id A ⊗m M ) = m M ◦(m A ⊗id M )◦α A,A,M and m M ◦(η A ⊗id M ) = id M ◦l M . Right A-modules and A-bimodules are defined similarly. Definition 2.8. Let C be a tensor category and let A be an algebra in C. (i) A is called simple iff it is simple as a bimodule over itself. Let C be in addition k-linear, for k a field. (ii) A is called absolutely simple iff the space of A-bimodule maps from A to itself is one-dimensional, dimk Hom A|A (A, A) = 1. (iii) A is called haploid iff dimk Hom(1, A) = 1 [FS, Def. 4.3]. In the following we will assume that all tensor categories are strict to avoid spelling out associators and unit constraints. A C-coalgebra A = (A, , ε) is defined analogously to a C-algebra, i.e. : A → A ⊗ A and ε : A → 1 obey coassociativity and counit conditions. If C is braided and if A and B are C-algebras, there are two in general non-isomorphic algebra structures on A ⊗ B. We choose A ⊗ B to be the C-algebra with multiplication m A⊗B = (m A ⊗m B )◦(id A ⊗c−1 A,B ⊗id B ) and unit η A⊗B = η A ⊗η B . Similarly, if A and B are C-coalgebras, then A ⊗ B becomes a C-coalgebra if we choose the comultiplication A⊗B = (id A ⊗ c A,B ⊗ id B ) ◦ ( A ⊗ B ) and the counit ε A⊗B = ε A ⊗ ε B .
Cardy Algebras and Sewing Constraints, I
879
Definition 2.9. A Frobenius algebra A = (A, m, η, , ε) is an algebra and a coalgebra such that the coproduct is an intertwiner of A-bimodules, (id A ⊗ m) ◦ ( ⊗ id A ) = ⊗ m = (m ⊗ id A ) ◦ (id A ⊗ ). We will use the following graphical representation for the morphisms of a Frobenius algebra, A
A
A
A
m=
, η=
, =
, ε=
.
(2.12)
A A
A
A
A Frobenius algebra A in a k-linear tensor category, for k a field, is called special iff m ◦ = ζ id A and ε ◦ η = ξ id1 for nonzero constants ζ , ξ ∈ k. If ζ = 1 we call A normalised-special. A Frobenius algebra homomorphism between two Frobenius algebras is both an algebra homomorphism and a coalgebra homomorphism. A (strictly) sovereign tensor category is a tensor category equipped with a left and a right duality which agree on objects and morphisms (see e.g. [Bi,FS] for more details). We will write the dualities as = d˜U : U ⊗ U ∨ → 1,
= dU : U ∨ ⊗ U → 1,
U∨
U
U
U∨
U
U∨
U∨
U
(2.13)
= b˜U : 1 → U ∨ ⊗ U.
= bU : 1 → U ⊗ U ∨ ,
In terms of these we define the left and right dimension of an object U as diml U = dU ◦ b˜U
, dimr U = d˜U ◦ bU ,
(2.14)
both of which are elements of Hom(1, 1). Let now C be a sovereign tensor category. For a Frobenius algebra A in C, we define two morphisms: A∨
A =
A∨
,
A
A =
.
(2.15)
A
Definition 2.10. A Frobenius algebra A is symmetric iff A = A . The following lemma shows that under certain conditions we do not need to distinguish the various notions of simplicity in Definition 2.8. Lemma 2.11. Let A be a commutative symmetric Frobenius algebra in a C-linear semisimple sovereign braided tensor category C and suppose that diml A = 0. Then the following are equivalent.
880
L. Kong, I. Runkel
(i) A is simple. (ii) A is absolutely simple. (iii) A is haploid. Proof. (ii)⇔(iii): A is haploid iff it is absolutely simple as a left module over itself [FS, Eq. (4.17)]. Furthermore, for a commutative algebra we have Hom A (A, A) = Hom A|A (A, A), and so A is haploid iff it is absolutely simple. (i)⇒(ii): If A is simple, then every nonzero element of Hom A|A (A, A) is invertible. Hence this space forms a division algebra over C, and is therefore isomorphic to C. (iii)⇒(i): Since C is semi-simple and A is haploid, also Hom(A, 1) is one-dimensional. The counit ε is a nonzero element in this space, and so gives a basis. This implies firstly, that ε ◦ η = 0, and secondly, that there is a constant β ∈ C such that β · ε = d A ◦ (id A∨ ⊗ m) ◦ (b˜ A ⊗ id A ).
(2.16)
Composing with η from the right yields β ε◦η = diml A. The right-hand side is nonzero, and so β = 0. By [FRS, Lem. 3.11], A is special. We have already proved (ii)⇔(iii), and so A is absolutely simple. A special Frobenius algebra in a semi-simple category has a semi-simple category of bimodules (apply [FS, Prop. 5.24] to the algebra tensored with its opposite algebra). For semi-simple C-linear categories, simple and absolutely simple are equivalent.2 Thus A is simple.
Remark 2.12. For a Frobenius algebra A the morphisms (2.15) are invertible, and hence A∼ = A∨ . In this case one has diml A = dimr A [FS, Rem. 3.6.3] and so we could have stated the above lemma equivalently with the condition dimr A = 0. Let F : C1 → C2 be a lax tensor functor between two tensor categories C1 , C2 and m F(A)
let (A, m A , η A ) be an algebra in C1 . Define morphisms F(A) ⊗ F(A) −−−→ F(A) and η F(A)
12 −−−→ F(A) as m F(A) = F(m A ) ◦ φ2F ,
η F(A) = F(η A ) ◦ φ0F .
(2.17)
Then (F(A), m F(A) , η F(A) ) is an algebra in C2 [JS, Prop. 5.5]. If f : A → B is an algebra homomorphism between two algebras A, B ∈ C1 , then F( f ) : F(A) → F(B) is also an algebra homomorphism. If (M, m M ) is a left (or right) A-module in C1 , then (F(M), F(m M )◦φ2F ) is a left (or right) F(A)-module; if M has a A-bimodule structure, then F(M) naturally has a F(A)-bimodule structure. Similarly, if (A, A , ε A ) is a coalgebra in C1 and F : C1 → C2 is a colax tensor func F(A)
ε F(A)
tor, then F(A) with coproduct F(A) −−−→ F(A) ⊗ F(A) and counit F(A) −−−→ 12 given by F(A) = ψ2F ◦ F( A ),
ε F(A) = ψ0F ◦ F(ε A ),
(2.18)
is a coalgebra in C2 . If f : A → B is a coalgebra homomorphism between two coalgebras A, B ∈ C1 , then F( f ) : F(A) → F(B) is also a coalgebra homomorphism. 2 To see this note that if U is simple, then the C-vector space Hom(U, U ) is a division algebra, and hence Hom(U, U ) = C idU . Conversely, if U is not simple, then U = U1 ⊕ U2 and Hom(U, U ) contains at least two linearly independent elements, namely idU1 and idU2 .
Cardy Algebras and Sewing Constraints, I
881
Proposition 2.13. 3 If F : C1 → C2 is a Frobenius functor and (A, m A , η A , A , ε A ) a Frobenius algebra in C1 , then (F(A), m F(A) , η F(A) , F(A) , ε F(A) ) is a Frobenius algebra in C2 . Proof. One Frobenius property, (m F(A) ⊗ id F(A) ) ◦ (id F(A) ⊗ F(A) ) = F(A) ◦ m F(A) , follows from the commutativity of the following diagram (we spell out the associativity isomorphisms): F(A) ⊗ F(A)
id F(A) ⊗F( A )
/ F(A) ⊗ F(A ⊗ A)
φ2F
F(id A ⊗ A )
F(A ⊗ A)
F( A )
F(A)
α F(A),F(A),F(A)
/ F(A ⊗ (A ⊗ A))
(F(A) ⊗ F(A)) ⊗ F(A)
F(α A,A,A )
/ F(A) ⊗ (F(A) ⊗ F(A))
φ2F
F((A ⊗ A) ⊗ A)
F(m A )
id F(A) ⊗ψ2F
ψ2F
/ F(A ⊗ A) ⊗ F(A)
ψ2F
F(m A ⊗id A )
/ F(A ⊗ A)
(2.19)
φ2F ⊗id F(A)
F(m A )⊗id F(A)
/ F(A) ⊗ F(A)
The commutativity of the upper-left subdiagram follows from the naturalness of φ2F , that of the upper-right subdiagram follows from (2.5), that of the lower-left subdiagram follows from the Frobenius properties of A, and that of the lower-right subdiagram follows from the naturalness of ψ2F . The proof of the other Frobenius property is similar.
Proposition 2.14. If F : C1 → C2 is a tensor functor and A a Frobenius algebra in C1 , then: (i) F(A) has a natural structure of Frobenius algebra as given in Proposition 2.13; (ii) If A is (normalised-)special, so is F(A). Proof. Part (i) follows from Propositions 2.5 and 2.13. Part (ii) is a straightforward
verification of the definition, using ψ2F = (φ2F )−1 and ψ0F = (φ0F )−1 . Let C1 , C2 be sovereign tensor categories and F : C1 → C2 a Frobenius functor. We ∨ ∨ define two morphisms I F(A∨ ) , I F(A ∨ ) : F(A ) → F(A) , for a Frobenius algebra A in C1 , as follows: I F(A∨ ) = ((ψ0F ◦ F(d A ) ◦ φ2F ) ⊗ id F(A)∨ ) ◦ (id F(A∨ ) ⊗ b F(A) ) , I ∨ = (id F(A)∨ ⊗ (ψ0F ◦ F(d˜ A ) ◦ φ2F )) ◦ (b˜ F(A) ⊗ id F(A∨ ) ).
(2.20)
F(A )
It is easy to see that these are isomorphisms. Lemma 2.15. If F : C1 → C2 is a Frobenius functor and A a Frobenius algebra in C1 , then F(A) = I F(A∨ ) ◦ F( A ),
F(A) = I F(A ∨ ) ◦ F( A ).
(2.21)
3 After the preprint of the present paper appeared we noticed that this proposition is also proved in [DP, Cor. 5].
882
L. Kong, I. Runkel
Proof. We only prove the first equality, the second one can be seen in the same way. By definition, we have I F(A∨ ) ◦ F( A ) = ((ψ0F ◦ F(d A ) ◦ φ2F ) ⊗ id F(A)∨ ) ◦ (id F(A∨ ) ⊗ b F(A) ) ◦ F( A ) = (ψ0F ◦ F(d A ) ◦ φ2F ) ◦ (F( A ) ⊗ id F(A) ) ⊗ id F(A)∨ ◦ (id F(A) ⊗ b F(A) ). For the term inside the square brackets we find ψ0F ◦ F(d A ) ◦ φ2F ◦ (F( A ) ⊗ id F(A) ) = ψ0F ◦ F(d A ) ◦ F( A ⊗ id A ) ◦ φ2F = ψ0F ◦ F(d A ◦ ( A ⊗ id A )) ◦ φ2F = ψ0F ◦ F(ε A ◦ m A ) ◦ φ2F .
(2.22)
On the other hand, by definition, F(A) = [((ψ0F ◦ F(ε A ) ◦ (F(m A ) ◦ φ2F )) ⊗ id F(A)∨ ] ◦ (id F(A) ⊗ b F(A) ). This demonstrates the first equality in (2.21).
Proposition 2.16. Let F : C1 → C2 be a tensor functor, G : C2 → C1 a functor, F, G, χ an adjunction, A a C1 -algebra, and B a C2 -algebra. Then f : A → G(B) is an algebra homomorphism if and only if f˜ = χ −1 ( f ) : F(A) → B is an algebra homomorphism. Proof. We need to show that m G(B) ◦ ( f ⊗ f ) = f ◦ m A and f ◦ η A = ηG(B) ,
(2.23)
m B ◦ ( f˜ ⊗ f˜) = f˜ ◦ m F(A) and f˜ ◦ η F(A) = η B s.
(2.24)
is equivalent to
We first prove that the first identity in (2.23) is equivalent to the first identity in (2.24). For the left-hand side of the first identity in (2.23) we have the following equalities: (1)
m G(B) ◦ ( f ⊗ f ) = G(m B ) ◦ φ2G ◦ ( f ⊗ f ) (2)
= G(m B ) ◦ G(ρ ⊗ ρ) ◦ G(ψ2F ) ◦ δ ◦ ( f ⊗ f )
(3)
= G(m B ) ◦ G(ρ ⊗ ρ) ◦ G(ψ2F ) ◦ G F( f ⊗ f ) ◦ δ
(4)
= G(m B ) ◦ G(ρ ⊗ ρ) ◦ G(F( f ) ⊗ F( f )) ◦ G(ψ2F ) ◦ δ (5) = G m B ◦ (ρ ⊗ ρ) ◦ (F( f ) ⊗ F( f )) ◦ ψ2F ◦ δ (6) (2.25) = χ m B ◦ (ρ ⊗ ρ) ◦ (F( f ) ⊗ F( f )) ◦ ψ2F ,
where (1) is the definition of m G(B) in (2.17), (2) is the second identity in (2.10), (3) and (4) are naturality of δ and ψ2F , respectively, step (5) is functoriality of G and finally step (6) is (2.9). For the right hand side of the first identity in (2.23) we get (1)
f ◦ m A = Gρ ◦ δG ◦ ( f ◦ m A ) (2)
= Gρ ◦ G F( f ◦ m A ) ◦ δ
(3)
= G(ρ ◦ F( f ◦ m A )) ◦ δ
(4)
= χ (ρ ◦ F( f ) ◦ F(m A )),
(2.26)
Cardy Algebras and Sewing Constraints, I
883
where (1) is the adjunction property (2.8), (2) is naturality of δ, (3) functoriality of G, and (4) amounts to (2.9) and functoriality of F. On the other hand, we see that the first equality in (2.24) is equivalent to m B ◦ (ρ ⊗ ρ) ◦ (F( f ) ⊗ F( f )) = ρ ◦ F( f ) ◦ F(m A ) ◦ φ2F .
(2.27)
Using that φ2F is invertible with inverse (φ2F )−1 = ψ2F and that χ is an isomorphism, it follows that the statement that (2.25) is equal to (2.26) is equivalent to the identity (2.27). Now we prove that the second identity in (2.23) is equivalent to the second identity in (2.24). Using (2.17) and (2.10) we can write ηG(B) = G(η B )◦φ0F = G(η B )◦G(ψ0F )◦δ1 . Together with (2.9) this shows that the second identity in (2.23) is equivalent to f ◦ η A = χ (η B ◦ ψ0F ).
(2.28)
On the other hand, the second identity in (2.24) is equivalent to ρ ◦ F( f ) ◦ F(η A ) ◦ φ0F = η B ,
(2.29)
which, by φ0F = (ψ0F )−1 and (2.9), is further equivalent to (2.28).
Definition 2.17. Let (A, m A , η A , A , ε A ) and (B, m B , η B , B , ε B ) be two Frobenius algebras in a tensor category C. For f : A → B, we define f ∗ : B → A by f ∗ = ((ε B ◦ m B ) ⊗ id A ) ◦ (id B ⊗ f ⊗ id A ) ◦ (id B ⊗ ( A ◦ η A )).
(2.30)
The following lemma is immediate from the definition of (·)∗ and the properties of Frobenius algebras. We omit the proof. Lemma 2.18. Let C be a tensor category, let A, B, C be Frobenius algebras in C, and let f : A → B and g : B → C be morphisms. (g ◦ f )∗ = f ∗ ◦ g ∗ . f is a monomorphism iff f ∗ is an epimorphism. f is an algebra map iff f ∗ is a coalgebra map. If f is a homomorphism of Frobenius algebras, then f ∗ ◦ f = id A and f ◦ f ∗ = id B . (v) If C is sovereign and if A and B are symmetric, then f ∗∗ = f .
(i) (ii) (iii) (iv)
Let C and D be tensor categories and let F : C → D be a Frobenius functor. Given Frobenius algebras A, B in C and a morphism f : A → B, the next lemma shows how (·)∗ behaves under F. Lemma 2.19. F( f ∗ ) = F( f )∗ . Proof. The definition of the structure morphisms of the Frobenius algebra F(A) is given in (2.17) and (2.18). Substituting these definitions gives F( f )∗ = ψ0F ◦ F(ε B ) ◦ F(m B ) ◦ φ2F ⊗ id F(A) ◦ id F(B) ⊗ F( f ) ⊗ id F(A) ◦ id F(B) ⊗ ψ2F ◦ F( A ) ◦ F(η A ) ◦ φ0F = (ψ0F ⊗ id F(A) ) ◦ F (ε B ◦ m B ◦ (id B ⊗ f )) ⊗ id F(A) ◦(φ2F ⊗ id F(A) ) ◦ (id F(B) ⊗ ψ2F ) ◦ id F(B) ⊗ F( A ◦ η A ) ◦ (id F(B) ⊗ φ0F ).
(2.31)
884
L. Kong, I. Runkel
In the middle line of the last expression we can use the defining property (2.5) of F, namely we substitute (φ2F ⊗ id F(A) ) ◦ (id F(B) ⊗ ψ2F ) = ψ2F ◦ φ2F . Then ψ2F can be moved to the left, and φ2F to the right, until they can be omitted against ψ0F and φ0F , respectively, using (2.2) and (2.4). This results in F( f )∗ = F ((ε B ◦ m B ◦ (id B ⊗ f )) ⊗ id A ) ◦ F (id B ⊗ ( A ◦ η A )) ,
(2.32)
which is nothing but F( f ∗ ).
2.3. Modular tensor categories. Let C be a modular tensor category [T,BK], i.e. an abelian semi-simple finite C-linear ribbon category with simple tensor unit 1 and a nondegeneracy condition on the braiding (to be stated in a moment). We denote the set of equivalence classes of simple objects in C by I, elements in I by i, j, k ∈ I and their representatives by Ui , U j , Uk . We also set U0 = 1 and for an index k ∈ I we define k¯ by Uk¯ ∼ = Uk∨ . Since the tensor unit is simple, we shall for modular tensor categories identify Hom(1, 1) ∼ = C (cf. footnote 2). Define numbers si, j ∈ C by4
si, j =
Uj
.
Ui
(2.33)
They obey si, j = s j,i and s0,i = dim Ui , see e.g. [BK, Sect. 3.1]. (In a ribbon category the left and right dimension (2.14) of Ui coincide and are denoted by dim Ui .) The non-degeneracy condition on the braiding of a modular tensor category is that the |I|×|I|-matrix s should be invertible. In fact [BK, Thm. 3.1.7], sik sk j = Dim C δi,j¯ , (2.34) k∈I
where Dim C = i∈I (dim Ui )2 . One can show (even in the weaker context of fusion categories over C) that Dim C √ ≥ 1 [ENO1, Thm. 2.3]. In particular, Dim C = 0. We fix once and for all a square root Dim C of Dim C. Nk
k (i, j)k Ni j }α=1
ij in HomC (Ui ⊗ U j , Uk ) and the dual basis {ϒα Let us fix a basis {λα(i, j)k }α=1
(i, j)k
in HomC (Uk , Ui ⊗U j ). The duality of the bases means that λα(i, j)k ◦ ϒβ = δα,β idUk . We also fix λ(0,i)i = λ(i,0)i = idUi . We denote the basis vectors graphically as follows: Ui
Uk
λα(i, j)k =
α
Ui
Uj
,
ϒα(i, j)k =
Uj α
.
(2.35)
Uk
4 In the graphical notation used below, we have given an orientation to the ribbons indicated by the arrows. For example, it is understood that this orientation determines which of the duality morphisms in (2.13) to use.
Cardy Algebras and Sewing Constraints, I
885 (i;α)
For V ∈ C we also choose a basis {bV
V } of HomC (V, Ui ) and the dual basis {b(i;β) }
(i;α)
of HomC (Ui , V ) for i ∈ I such that bV notation
V ◦ b(i;β) = δαβ idUi . We use the graphical
Ui
b(i;α) = V
V α
,
V b(i;α) =
V
α
.
(2.36)
Ui
Given two modular tensor categories C and D, by C D we mean the tensor product of additive categories over C [BK, Def. 1.1.15], i.e. the category whose objects are direct sums of pairs V × W of objects V ∈ C and W ∈ D and whose morphism spaces are HomCD (V × W, V × W ) = HomC (V, V ) ⊗C HomC (W, W )
(2.37)
for pairs, and direct sums of these if the objects are direct sums of pairs. If we replace the braiding and the twist in C by the antibraiding c−1 and the antitwist −1 θ respectively, we obtain another ribbon category structure on C. In order to distinguish these two distinct structures, we denote (C, c, θ ) and (C, c−1 , θ −1 ) by C+ and C− respectively. As in the Introduction, we will abbreviate 2 C± = C+ C− .
(2.38)
2 is given by U × U for Note that a set of representatives of the simple objects in C± i j i, j ∈ I. For the remainder of Sect. 2 we fix a modular tensor category C.
2.4. The functors T and R. The tensor product bifunctor ⊗ can be naturally extended 2 → C. Namely, T (⊕ N V × W ) = ⊕ N V ⊗ W for all V , W ∈ C to a functor T : C± i i i i i=1 i i=1 i and N ∈ N. The functor T becomes a tensor functor as follows. For φ0T : 1 → T (1 × 1) take φ0T = id1 (or l1−1 in the non-strict case). Next notice that, for U, V, W, X ∈ C, T (U × V ) ⊗ T (W × X )= (U ⊗ V ) ⊗ (W ⊗ X ), T ((U × V ) ⊗ (W × X ))= (U ⊗ W ) ⊗ (V ⊗ X ).
(2.39)
We define φ2T : T (U × V ) ⊗ T (W × X ) → T ((U × V ) ⊗ (W × X )) by −1 φ2T = idU ⊗ cW V ⊗ id X .
(2.40)
(In the non-strict case the appropriate associators have to be added.) The above definition of φ2T can be naturally extended to a morphism φ2T : T (M1 ⊗ M2 ) → T (M1 ) ⊗ T (M2 ) 2 . The following result can be checked by direct for any pair of objects M1 , M2 in C± calculation [JS, Prop. 5.2]. Lemma 2.20. The triple (T, φ2T , φ0T ) gives a tensor functor.
886
L. Kong, I. Runkel
In particular, (T, φ2T , φ0T , ψ2T , ψ0T ), where ψ2T = (φ2T )−1 and ψ0T = (φ0T )−1 , gives a Frobenius functor. 2 as follows: for A ∈ C and f ∈ Hom (A, B), Define the functor R : C → C± C (A ⊗ Ui∨ ) × Ui , R( f ) = ( f ⊗ idUi∨ ) × idUi . (2.41) R(A) = i∈I
i∈I
This functor was also considered in a slightly different context in [ENO2, Prop. 2.3]. DimC ∨ The family of isomorphisms γ AR = ⊕i∈I dim Ui id(A⊗Ui )×Ui ∈ Aut(R(A)) defines a R natural isomorphism γ : R → R. Our next aim is to show that R is left and right adjoint to T , in other words R and T form an ambidextrous adjunction (see e.g. [Ld] for a discussion of ambidextrous 2, adjunctions). To this end we introduce two linear isomorphisms, for A ∈ C and M ∈ C± χˆ : HomC (T (M), A) −→ HomC 2 (M, R(A)), ±
(2.42)
χˇ : HomC (A, T (M)) −→ HomC 2 (R(A), M). ±
N M l × M r , then χˆ and χˇ are given by If we decompose M as M = ⊕n=1 n n A
χˆ :
N
→
fn
N n=1 i∈I
n=1
Mnl
Ui∨
A
fn
×
Mnr
α
Ui
α
(2.43)
α
Mnr
Mnl
Mnr
and Mnl
χˇ :
Mnr
N n=1
gn
A
Mnl
→
N n=1 i∈I
α
Mnr α Mnr
×
gn
A
Ui∨
α
DimC . dim Ui
(2.44)
Ui
Notice that χˆ and χˇ are independent of the choice of basis. Theorem 2.21. T, R, χˆ and R, T, χˇ −1 are adjunctions, i.e. R is both left and right adjoint of T . N M l × M r . The isomorphism χˆ amounts to the following Proof. Write M as M = ⊕n=1 n n composition of natural isomorphisms,
HomC (T (M), A) = ⊕n HomC (Mnl ⊗ Mnr , A) ∼ = ⊕n,i HomC (Mnl ⊗ Ui , A) ⊗ HomC (Mnr , Ui ) ∼ = ⊕n,i HomC (Mnl , A ⊗ Ui∨ ) ⊗ HomC (Mnr , Ui ) = HomC±2 (M, R(A)). (2.45)
Cardy Algebras and Sewing Constraints, I
887
Thus χˆ is natural. Let (γ AR )∗ : HomC 2 (R(A), M) → HomC 2 (R(A), M) denote the ± ± pull-back of γ AR . The isomorphism χˇ is equal to the composition of (γ AR )∗ and the following sequence of natural isomorphisms: HomC (A, T (M)) = ⊕n HomC (A, Mnl ⊗ Mnr ) ∼ = ⊕n,i HomC (A, M l ⊗ Ui ) ⊗ HomC (Ui , M r ) n
n
∼ = ⊕n,i HomC (A ⊗ Ui∨ , Mnl ) ⊗ HomC (Ui , Mnr ) = HomC±2 (R(A), M). (2.46)
We have proved that both χˆ and χˇ are natural isomorphisms.
There are four natural transformations associated to χˆ and χ, ˇ namely δˆ
δˇ
ρˇ
idC 2 − → RT − → idC 2 ±
ρˆ
and idC − →TR − → idC ,
±
(2.47)
2, defined by, for A ∈ C, M ∈ C±
δˆ M = χ(id ˆ T (M) ), ρˆ A = χˆ −1 (id R(A) ), ρˇ M = χˇ (id T (M) ), δˇ A = χˇ −1 (id R(A) ).
(2.48)
N Ml × Mr , They can be expressed graphically as follows, with M = ⊕n=1 n n
δˆ M =
n,i
Mnl
Mnr
Ui∨
α
α
Ui
×
Mnl
α
,
ρˆ A =
A
,
i∈I
Mnr
A
Ui∨
Ui
(2.49)
Mnl
ρˇ M =
n,i
Mnr
×
α
α
α Mnl
Mnr
Ui∨
A
Ui
DimC dim Ui
, δˇ A =
Ui∨
Ui dim Ui DimC .
i∈I A
Note that ρˇ M ◦ δˆ M = Dim C · id M and ρˆ A ◦ δˇ A = id A .
(2.50)
Lemma 2.22. The functors T and R as maps on the sets of morphisms have left inverses, and thus are injective. Proof. Let f : A → B be a morphism in C. We define a map Q R : HomC 2 (R(A), ± R(B)) → HomC (A, B) by f → ρˆ B ◦ T ( f ) ◦ δˇ A . Then we have Q R ◦ R( f ) = ρˆ B ◦ T R( f ) ◦ δˇ A = ρˆ B ◦ δˇ B ◦ f = f,
(2.51)
888
L. Kong, I. Runkel
where we used naturality of δˇ and (2.50) in the second and third equalities, respectively. So Q R is a left inverse of R on morphisms. Thus R is injective on morphisms. Similarly, 2 . We define a map Q : Hom (T (M), T (N )) → let g : M → N be a morphism in C± T C −1 HomC 2 (M, N ) by g → (Dim C) · ρˇ N ◦ R(g ) ◦ δˆ M . Then we have ±
Q T ◦ T (g) = (Dim C)−1 · ρˇ N ◦ RT (g) ◦ δˆ M = (Dim C)−1 · ρˇ N ◦ δˆ N ◦ g = g. (2.52) So Q T is a left inverse of T on morphisms. Thus T is injective on morphisms.
Using (2.9) and (2.50), one can express the two inverse maps χˆ −1 , χˇ −1 as follows, for f ∈ HomC 2 (M, R(A)) and g ∈ HomC 2 (R(A), M), ±
±
ˇ χˆ −1 ( f ) = ρˆ ◦ T ( f ), χˇ −1 (g) = T (g) ◦ δ.
(2.53)
By Proposition 2.5 and Lemma 2.7, R is both a lax and colax tensor functor. In particular, φ0R : 1 × 1 → R(1) is given by φ0R = χˆ (ψ0T ) = R(ψ0T ) ◦ δˆ1 × 1 = id1×1
(2.54)
ˆ which can and φ2R : R(A) ⊗ R(B) → R(A ⊗ B) by φ2R = R(ρˆ A ⊗ ρˆ B ) ◦ R(ψ2T ) ◦ δ, be expressed graphically as A
φ2R =
ψ0R
Uk
×
i, j,k∈I α
α
α
A
Similarly,
Uk∨
B
Ui∨
B
U∨ j
Ui
. (2.55)
Uj
: R(1) → 1 × 1 is given by ψ0R = ρˇ1 ◦ R(φ0T ) = Dim C id1×1
(2.56)
and ψ2R : R(A ⊗ B) → R(A) ⊗ R(B) by ψ2R = ρˇ ◦ R(φ2T ) ◦ R(δˇ A ⊗ δˇ B ), which in graphical notation reads A
ψ2R =
Ui∨
B
U∨ j
Ui
Uj
α
×
α
i, j,k∈I α
A
B
Uk∨
dim Ui dim U j . dim Uk Dim C
Uk
(2.57) If C has more than one simple object, then R does not take the tensor unit of C to the 2 and so is clearly not a tensor functor. However, we will show that R is tensor unit of C± still a Frobenius functor. This will imply that if A is a Frobenius algebra in C, then R(A) = (R(A), m R(A) , η R(A) , R(A) , ε R(A) )
(2.58)
Cardy Algebras and Sewing Constraints, I
889
2 , where the structure morphisms were given in (2.17) and is a Frobenius algebra in C± (2.18). In the case A = 1 it was proved in [Mü1, Prop. 4.1] (see also [Fr, Lem. 6.19] and [K1, Thm. 5.2]) that (2.58) is a commutative simple symmetric normalised-special 2 . In fact, given a Frobenius algebra A in C, it is straightforward Frobenius algebra in C± to verify that the structure morphisms in (2.58) are precisely those of (A × 1) ⊗ R(1), cf. Sect. 2.2.
Proposition 2.23. (R, φ2R , φ0R , ψ2R , ψ0R ) is a Frobenius functor. Proof. Using the explicit graphical expression of φ2R , φ0R , ψ2R , ψ0R , it is easy to see that the commutativity of the diagrams (2.5) and (2.6) are equivalent to the statement that 2 . The latter R(1) with structure morphisms as in (2.58) is a Frobenius algebra in C± statement is true by [Mü1, Prop. 4.1].
From Lemma 2.20 and Proposition 2.23 we see that T and R take Frobenius algebras to Frobenius algebras. The following two propositions show how the properties of Frobenius algebras are transported. 2 . Then T (A) is a Frobenius algebra Proposition 2.24. Let A be a Frobenius algebra in C± in C and
(i) A is symmetric iff T (A) is symmetric. (ii) A is (normalised-)special iff T (A) is (normalised-)special. N Al × Ar . Then the maps I Proof. For part (i) write A as a direct sum ⊕n=1 T (A) , I T (A) : n n ∨ ∨ T (A ) → T (A) defined in (2.20) are given by: N c(Aln )∨ ,(Arn )∨ . IT (A) = IT (A) = ⊕n=1
(2.59)
Therefore, by (2.21), T (A) = T (A) is equivalent to T ( A ) = T (A ). Since by Lemma 2.22, T is injective on morphisms, this proves part (i). Part (ii) can be checked in the same way, for example the condition m T (A) ◦ T (A) = ζ id T (A) is easily checked to be equivalent to T (m A ◦ A ) = ζ T (id A ).
Proposition 2.25. Let A be a Frobenius algebra A in C. Then R(A) is a Frobenius 2 and algebra in C± (i) A is symmetric iff R(A) is symmetric. (ii) A is (normalised-)special iff R(A) is (normalised-)special. Proof. Recall that the structure morphisms of the Frobenius algebra R(A) are equal to those of (A × 1) ⊗ R(1). Using this equality, part (i) and (ii) follow because R(1) is symmetric and normalised-special. For example, m R(A) ◦ R(A) = [(m A ◦ A ) × id1 ] ⊗ id R(1) = R(m A ◦ A ),
(2.60)
so that m R(A) ◦ R(A) = ζ id R(A) is equivalent to R(m A ◦ A ) = ζ R(id A ), which by Lemma 2.22 is equivalent to m A ◦ A = ζ id A .
The functor R has one additional property not shared by T , namely R takes absolutely simple algebras to absolutely simple algebras. We will see explicitly in Sect. 3.3 that this is not true for T .
890
L. Kong, I. Runkel
Lemma 2.26. For a C-algebra A, the map R : Hom A|A (A, A) → Hom R(A)|R(A) (R(A), R(A))
(2.61)
given by f → R( f ) is well-defined and an isomorphism. Proof. Since R is a lax tensor functor, R(A) is naturally a R(A)-bimodule. It is easy to see that R in (2.61) is a well-defined map. R(A) is also naturally a R(1)-bimodule, which can be identified with the induced R(1)-bimodule structure on (A × 1) ⊗ R(1), where the left R(1) action on (A ×1)⊗ R(1) is given by (id A×1 ⊗m R(1) )◦(c−1 A×1,R(1) ⊗id R(1) ). We have the following natural isomorphisms: χˆ −1
∼ =
→ HomC 2 (A × 1, R(A)) −−→ HomC (A, A). Hom R(1)|R(1) (R(A), R(A)) − ±
(2.62)
which, by (2.53), are given by, for f ∈ Hom R(1)|R(1) (R(A), R(A)), f → f = f ◦ (id A×1 ⊗ η R(1) ) → f = ρˆ ◦ T ( f ),
(2.63)
and its inverse is given by, for g ∈ HomC (A, A), g → g = R(g) ◦ δˆ → g = (id A×1 ⊗ m R(1) ) ◦ (g ⊗ id R(1) ),
(2.64)
where g is indeed a R(1)-bimodule map due to the commutativity of R(1). It is easy to check that g = R(g) in (2.64). Therefore R gives an isomorphism from HomC (A, A) to Hom R(1)|R(1) (R(A), R(A)). Moreover, one verifies that R(g) is an R(A)-bimodule map iff g is an A-bimodule map. In other words, R : g → R(g) gives an isomorphism ∼ =
Hom A|A (A, A) − → Hom R(A)|R(A) (R(A), R(A)).
Corollary 2.27. Let A be a C-algebra. (i) A is absolutely simple iff R(A) is absolutely simple. (ii) Let A be in addition Frobenius. Then A is simple and special iff R(A) is simple and special. Proof. Part (i) immediately follows from Lemma 2.26. The statement of part (ii) without the qualifier ‘simple’ is proved in Proposition 2.25. But, as in the proof of (iii)⇒(i) in Lemma 2.11, a special Frobenius algebra in a semi-simple category has a semi-simple category of bimodules, and for a semi-simple C-linear category, simple and absolutely simple are equivalent. Part (ii) then follows from part (i).
The following lemma will be needed in Sect. 3.2 below to discuss the properties of Cardy algebras. Lemma 2.28. Let A be a Frobenius algebra in C. Then (δˇ A )∗ = ρˆ A . Proof. Recall from (2.50) that δˇ A is a morphism A → T R(A). Since T and R are both Frobenius functors, T R(A) is a Frobenius algebra in C. Substituting the definitions, after a short calculation one finds
Cardy Algebras and Sewing Constraints, I
εT R(A) ◦ m T R(A) = Dim(C)
891
(2.65)
i∈I A
Ui∨
Ui
A
Uı¯∨
Uı¯
Substituting this in the definition of (δˇ A )∗ gives, again after a short calculation, the morphism ρˆ A . At an intermediate step one uses that the part of the morphism (δˇ A )∗ , which is made up of Ui and Uı¯ ribbons, their duals, and the basis morphisms λ(i,¯ı )0 and ϒ (i,¯ı )0 , can be replaced by dim1 Ui · dUi .
3. Cardy Algebras In this section we start by investigating the properties of Frobenius algebras which satisfy the so-called modular invariance condition. We then give two definitions of a Cardy algebra and prove their equivalence. Finally, in Sect. 3.3, we study the properties of these algebras and state our main results. 2 is an abbreviation for C C . We fix a modular tensor category C. Recall that C± + −
2 , we define the object K and the morphism ω : K → K 3.1. Modular invariance. In C± as
K =
Ui × U j ,
ω=
i, j∈I
dim Ui dim U j idUi ×U j . Dim C
(3.1)
i, j∈I
They have the property (see e.g. [BK, Cor. 3.1.11])
(Uk × Ul )∨
Ui × U j
K
= δi,k δ j,l
ω
(Uk × Ul )∨
Ui × U j
Dim C b˜Ui ×U j ◦ dUi ×U j . dim Ui dim U j
(3.2)
892
L. Kong, I. Runkel
Definition 3.1. 2 . A morphism f : A ⊗ B → B is called S-invariant iff (i) Let A, B be objects of C± W
W K
B B f
f
=
A
W
ω
(3.3)
A
W
2. C±
holds for all for W ∈ 2 -algebra (A , m , η ) is called modular invariant iff θ (ii) A C± cl cl cl Acl = id Acl and m cl is S-invariant. Lemma 3.2. The morphism f : A ⊗ B → B is S-invariant if and only if Ui × U j
Ui × U j
B
α
f
=
B
Dim C dim Ui dim U j α
(3.4)
f B α
A
Ui × U j
A
Ui × U j
holds for all i, j ∈ I. Proof. Condition (3.3) holds for all W iff it holds for all W = Ui × U j , i, j ∈ I, so it is enough to show that the right-hand side of (3.3) with W = Ui × U j is equal to the right-hand side of (3.4). Recall the notation for basis morphisms in (2.36). Starting from (3.3), write ⎛ ⎞ ∨ (k×l;α) ∨ B ⎠ ⊗ idUi ×U j , bB ◦ b(k×l;α) (3.5) id B ∨ ⊗ idUi ×U j = ⎝ k,l,α
and then apply (3.2). The graphical representation of the resulting morphism can be deformed to give (3.4).
2 -algeRemark 3.3. As shown in [K3, Sect. 6.1], the modular invariance condition of a C± bra exactly coincides with the modular invariance condition for torus 1-point correlation functions of a genus-0,1 closed CFT. In particular, the condition θ Acl = id Acl is equivalent to invariance under the modular transformation T : τ → τ + 1, and the condition (3.4) with f = m cl is equivalent to invariance under S : τ → − τ1 . Combining the modular invariance condition with the genus zero properties of a genus-0,1 closed CFT 2. results in a modular invariant commutative symmetric Frobenius algebra in C±
Cardy Algebras and Sewing Constraints, I
893
2 -algebra. Evaluating (3.4) for f = m , composing Let Acl be a modular invariant C± cl it with ηcl ⊗ idUi ×U j and taking the trace implies the following identity:
Zi j =
1 DimC
Ui ×U j
Acl
where
(3.6) Z i j = dimC HomC 2 (Ui ×U j , Acl ). ±
Decomposing Acl into simple objects, this gives Zi j =
Sik Z kl Sl−1 j
√ where Si j = si, j / DimC,
(3.7)
k,l∈I
which in CFT terms is of course nothing but the invariance of the torus partition function under the modular S-transformation. The following theorem gives a simple criterion for modular invariance. 2. Theorem 3.4. Let Acl be a haploid commutative symmetric Frobenius algebra in C±
(i) If Acl is modular invariant, then dim Acl = Dim C. (ii) If dim Acl = Dim C, then Acl is special and modular invariant. Proof. Part (i): Since Acl is haploid, for i = j = 0, Equation (3.6) reduces to 1 = dim Acl /DimC. Part (ii): By the same reasoning as in the proof of (iii)⇒(i) in Lemma 2.11 one shows that Acl is special. Thus m cl ◦ cl = ζcl id Acl for some ζcl = 0. 2 )loc of local A -modules is again a modBy [KO, Thm. 4.5], the category (C± cl Acl 2 )loc = (Dim C/ dim A )2 (see [Fr, Prop. 3.21 & ular tensor category and Dim (C± cl Acl Rem. 3.23] for the same statement in the notation used here). Thus by assumption we 2 )loc = 1. It then follows from [ENO1, Thm. 2.3] that up to isomorphism, have Dim(C± Acl loc 2 (C± ) Acl has a unique simple object (namely the tensor unit). In other words, every simple local Acl -module is isomorphic to Acl (seen as a left-module over itself). We have the following isomorphisms between morphism spaces [FS, Prop. 4.7 & 4.11], Hom Acl (Acl ⊗ (Ui × U j ), Acl ) ∼ = HomC±2 (Ui × U j , Acl ), Hom A (Acl , Acl ⊗ (Ui × U j )) ∼ = Hom 2 (Acl , Ui × U j ). C±
cl
Using these to transport the bases (2.36) from the right to the left, we obtain bases {b(iα j) }α (i j)
of Hom Acl (Acl ⊗ (Ui × U j ), Acl ) and {bβ }β of Hom Acl (Acl , Acl ⊗ (Ui × U j )). These can be expressed graphically as
894
L. Kong, I. Runkel Acl
Ui × U j
Acl
β
b(iα j) =
Acl
(i j)
,
bβ
α
Acl
=
1 DimC dim Ui dim U j ζcl
Ui × U j
Acl ,
(3.8)
Acl
(i j)
(i j)
where the nonzero factor in bβ is included for convenience. Notice that b(iα j) ◦ bβ a left Acl -module map. Since Acl is simple as a left module over itself, we have (i j)
b(iα j) ◦ bβ
= λαβ id Acl
is
(3.9)
(i j)
for some λαβ ∈ C. By computing tr(b(iα j) ◦ bβ ), it is easy to verify that λαβ = δαβ . We will now prove the following identity: Acl
Ui × U j
Acl
Ui × U j
α Acl
1 ζcl
=
α
DimC 1 dim Ui dim U j ζcl
Acl
.
Acl
(3.10)
α
Acl
Ui × U j
Acl
Ui × U j
One checks that the left-hand side of this equation is an idempotent, which we denote by ζcl−1 PAl cl (Ui ×U j ), cf. [Fr, Sect. 3.1]. By [Fr, Prop. 4.1] the image Im(ζcl−1 PAl cl (Ui ×U j )) is a local Acl -module, and hence isomorphic to A⊕N cl for some N ∈ Z≥0 . All left-module morphisms from Acl to Acl ⊗(Ui ×U j ) are linear combinations of the (i j) (i j) (i j) (i j) bβ . Furthermore one verifies that ζcl−1 PAl cl (Ui ×U j ) ◦ bβ = bβ . Therefore, the bβ
(i j) describe precisely the image of the idempotent, i.e. ζcl−1 PAl cl (Ui ×U j ) = α bα ◦b(iα j) , which is nothing but (3.10). Composing (3.10) with ζcl · εcl ⊗ idUi ×U j from the left (i.e. from the top) produces (3.4). In addition, since Acl is commutative symmetric Frobenius it satisfies θ Acl = id Acl [Fr, Prop. 2.25]. Altogether, this shows that Acl is modular invariant.
Remark 3.5. As we were writing this paper, we heard that the results in Theorem 3.4 were obtained independently by Kitaev and Müger [Ki]. Remark 3.6. Setting i = j = 0 in (3.6) gives the identity dim Acl = Z 00 Dim C [KR, Prop. 2.3]. Combining this with Theorem 3.4 (ii) one may wonder if a general modular 2 is isomorphic to a direct invariant commutative symmetric Frobenius algebra Acl in C± sum of simple such algebras. However, this is not so. For example, one can take the commutative symmetric Frobenius algebra Acl = C[x]/x2 in the category of vector spaces equipped with the non-degenerate trace ε(ax + b) = a. In this case the modular invariance condition holds automatically, but Acl is clearly not a direct sum of two algebras.
Cardy Algebras and Sewing Constraints, I
895
For a general modular tensor category C, the algebra C[x]/x2 R(1), understood as an 2 via the braided monoidal isomorphism Vect (C) C 2 → C 2 , provides algebra in C± f ± ± another counter-example. 3.2. Two definitions. Define a morphism PAl : A → A for a Frobenius algebra A in C 2 as follows [Fr, Sect. 2.4], or C± A A
PAl =
.
m
(3.11)
A
If A is also commutative and obeys m A ◦ A = ζ A id A , we have PAl = ζ A id A . In particular, this holds if A is commutative and special. Using the fact that the Frobenius algebra l R(1) is commutative and normalised-special, one can check that PR(A) : R(A) → R(A) takes the following form: Ui∨
A
l PR(A) =
i∈I
Ui A
×
m
.
(3.12)
Ui∨
A
Ui
2 -algebra, With these ingredients, we can now give the first definition of a Cardy C|C± which was introduced in [K3, Def. 5.14], cf. Remark 3.15 below. 2 -algebra I). A Cardy C|C 2 -algebra is a triple (A |A , ι Definition 3.7 (Cardy C|C± op cl cl-op ), ± where (Acl , m cl , ηcl , cl , εcl ) is a modular invariant commutative symmetric Frobe2 -algebra, (A , m , η , , ε ) is a symmetric Frobenius C-algebra, and nius C± op op op op op ιcl-op : Acl → R(Aop ) an algebra homomorphism, such that the following conditions are satisfied:
(i) Centre condition: R(Aop )
R(Aop ) m R(Aop ) m R(Aop )
=
R(Aop )
ιcl-op
ιcl-op
Acl
.
R(Aop )
R(Aop )
Acl
R(Aop )
(3.13)
896
L. Kong, I. Runkel
(ii) Cardy condition: R(Aop ) R(Aop )
ιcl-op ◦ ι∗cl-op =
.
(3.14)
R(Aop )
Remark 3.8. 2 -algebra” in Definition 3.7 was chosen because many of (i) The name “Cardy C|C± the important ingredients were first studied by Cardy: the modular invariance of the closed theory [C1], the consistency of the annulus amplitude [C2], and the bulk-boundary OPE [CL]. On the other hand, the boundary-boundary OPE and the OPE analogue of the centre condition were first considered in [Lw]. (ii) One can easily see that in the special case that C is the category Vect f (C) of finite2 -algebra gives exactly the algebraic dimensional C-vector spaces, a Cardy C|C± formulation of two-dimensional open-closed topological field theory over C (cf. Remark 6.14 in [K3]), see [Lz, Sect. 4.8], [Mo, Thm. 1.1], [AN, Thm. 4.5], [LP, Cor. 4.3], [MS, Sect. 2.2]. When passing to a general modular tensor category C there are two important differences to the two-dimensional topological field theory. Firstly, the algebras Acl and Aop now live in different categories, which in particular affects the formulation of the centre condition and the Cardy condition. Secondly, the modular invariance condition has to be imposed on Acl . In the case C = Vect f (C), modular invariance holds automatically. (1)
(1)
(1)
(2)
2 -algebras (A |A , ι Definition 3.9. A homomorphism of Cardy C|C± op cl cl-op ) → (Aop (2)
(2)
(1)
(2)
|Acl , ιcl-op ) is a pair ( f op , f cl ) of Frobenius algebra homomorphisms f op : Aop → Aop (2) and f cl : A(1) cl → Acl such that the diagram
A(1) cl
f cl
/ A(2) cl
(1)
(2)
ιcl-op
(1) R(Aop )
ιcl-op
(3.15)
R( f op ) / R(A(2) ) op
commutes. Remark 3.10. Since a homomorphism of Frobenius algebras is invertible (cf. Lemma 2.18 (iv)), a homomorphism of Cardy algebras is always an isomorphism. 2 -algebras, using the commutativity of For a homomorphism ( f op , f cl ) of Cardy C|C± (3.15) and the fact that f cl and f op are both algebra and coalgebra homomorphisms, it is easy to show that (3.15) commutes iff (1)
R(Aop )
R( f op )
/ R(A(2) ) op
(1)
(ιcl-op )∗
(2)
(1)
Acl
f cl
(ιcl-op )∗
/ A(2) cl
(3.16)
Cardy Algebras and Sewing Constraints, I
897
commutes. 2 -algebra. Define the morphism Let (Aop |Acl , ιcl-op ) be a Cardy C|C± ι˜cl-op = χˆ −1 (ιcl-op ) : T (Acl ) −→ Aop .
(3.17) (n)
N C l × C r such that C l × C r = η (1 × 1). We use ι Decompose Acl as Acl = ⊕n=1 cl n n 1 cl-op 1
to denote the restriction of ιcl-op to Cnl × Cnr and ι˜(n) cl-op to denote the restriction of ι˜cl-op l r to Cn ⊗ Cn . We introduce the following graphical notation: Aop
(n)
ι˜cl-op =
. Cnl
(3.18)
Cnr
By (2.43), ιcl-op can be expressed in terms of ι˜cl-op as follows: Ui∨
Aop
ιcl-op =
N n=1 i∈I
Ui
×
α
α
α
Cnr
Cnl
.
(3.19)
Cnr
Lemma 3.11. The centre condition (3.13) is equivalent to the following condition in C: Aop
Aop
Aop
=
Cnl
Cnr
Aop
Aop
Cnl
for n = 1, . . . , N .
Cnr
(3.20)
Aop
Proof. First, insert (3.19) and the definition (2.17) of m R(Aop ) into (3.13). Then apply the commutativity of R(1) to the left hand side of (3.13). The equivalence between (3.13) and (3.20) follows immediately.
Remark 3.12. The centre condition (3.20) is very natural from the open-closed conformal field theory point of view. Correlators on the upper half plane are expressed in terms of conformal blocks on the full complex plane. The objects Cnl and Cnr are associated to the field insertion at a point z in the upper half plane and at the complex conjugate point z¯ in the lower half plane, respectively. The object Aop corresponds to a field inserted at a point r on the real axis. The centre condition (3.20) simply says that the correlation functions in the disjoint domains |z| > r > 0 and r > |z| > 0 are analytic continuations of each other, see [K2, Prop. 1.18].
898
L. Kong, I. Runkel
Recall that we define ι˜∗cl-op : Aop → T (Acl ) as in (2.30). We introduce the graphical notation Cnl
ι˜∗cl-op =
Cnr
Cnl
N
N
=
Aop
Cnr T (−1 A ), cl
n=1
(3.21)
n=1 Aop
Aop
where the second equality follows from (2.15), (2.21) and (2.59). Lemma 3.13. The Cardy condition (3.14) is equivalent to the following identity in C: Ui∨
Aop
Aop
Ui∨
Cnr
N DimC dim Ui α n=1
α
Aop
=
Cnl α
op
Cnr
Aop Aop
for all i ∈ I . (3.22)
m op
Ui∨
Ui∨
Proof. By (2.44), (3.12) and (3.19), it is easy to see that (3.22) is equivalent to the following identity: l ˇ ι∗cl-op ) = PR(A . ιcl-op ◦ χ(˜ op )
(3.23)
Therefore, it is enough to show that χ(˜ ˇ ι∗cl-op ) = ι∗cl-op .
(3.24)
We have ∗ ∗ (4) (1) (2) (3) χˇ −1 (ι∗cl-op ) = T (ι∗cl-op ) ◦ δˇ = T (ιcl-op )∗ ◦ δˇ∗∗ = ρˆ ◦ T (ιcl-op ) = ι˜cl-op . (3.25) In step (1) we use the expression (2.53) for χˇ −1 , step (2) follows from Lemma 2.18 (v) and Lemma 2.19. Step (3) is Lemma 2.18 (i) and Lemma 2.28, and finally step (4) amounts to substituting (2.53) and (3.17). Acting with χˇ on both sides of the above equality produces (3.24).
Combining Lemmas 3.11 and 3.13, and Proposition 2.16, we obtain the following 2 -algebra (recall the graphical notation (3.18) for equivalent definition of Cardy C|C± ι˜cl-op and (3.21) for ι˜∗cl-op ).
Cardy Algebras and Sewing Constraints, I
899
2 -algebra II). A Cardy C-algebra is a triple (A |A , ι˜ Definition 3.14 (Cardy C|C± op cl cl-op ), 2 -algebra satisfying property (3.4) where Acl is a commutative symmetric Frobenius C± with f = m cl , Aop is a symmetric Frobenius C-algebra, and ι˜cl-op : T (Acl ) → Aop is an algebra homomorphism satisfying the conditions (3.20) and (3.22).
Remark 3.15. Up to a choice of normalisation, Definition 3.14 is the same as the original one in [K3, Def. 6.13]. The difference between the two definitions is the factor Dim C/ dim Ui on the left hand side of (3.22), which in [K3, Def. 6.13] is given by √ Dim C/ dim Ui . The √ two definitions √ are related by rescaling the coproduct cl and counit εcl of Acl by 1/ Dim C and Dim C, respectively. We chose the convention in (3.22) to remove all dimension factors from the expression (3.14) for the Cardy condition.
3.3. Uniqueness and existence theorems. In this subsection we investigate the structure of Cardy algebras. We start with the following proposition, which, when combined with the results of part II, provides an alternative proof of [Fj, Prop. 4.22]. 2 -algebra. If A is simple and Proposition 3.16. Let (Aop |Acl , ιcl-op ) be a Cardy C|C± cl dim Aop = 0, then Aop is simple and special.
Proof. By Remark 3.6, we have dim Acl = Z 00 Dim C = 0, and by Lemma 2.11, Acl is therefore haploid. Restricting the Cardy condition (3.22) to the case Ui = 1 and composing both sides with εop from the left, we see that εop kills all terms associated to U j × 1 ∈ Acl in the sum except for a single 1 × 1 term. Thus we obtain the following identity: β εop = d˜ Aop ◦ (m op ⊗ id A∨op ) ◦ (id Aop ⊗ b Aop ),
(3.26)
where β ∈ C. Composing with ηop from the right in turn implies that βεop ◦ ηop = dim Aop , which is nonzero by assumption. Thus also β = 0 and εop is a nonzero multiple of the morphism on the right hand side of (3.26). By [FRS, Lem. 3.11], Aop is special. Since Aop is a special Frobenius algebra, Aop is semi-simple as an Aop -bimodule (apply [FS, Prop. 5.24] to Aop tensored with its opposite algebra). Suppose Aop is not (1) (2) (1) (2) simple, so that we can write Aop = Aop ⊕ Aop for nonzero Aop -bimodules Aop and Aop . We denote the canonical embeddings and projections associated to this decomposition as ι1,2 and π1,2 . We have the identities m op ◦ (ι1 ⊗ ι2 ) = 0,
εop ◦ ηop =
2
εop ◦ ιi ◦ πi ◦ ηop .
(3.27)
i=1
The first identity follows since π1 ◦ m op ◦ (ι1 ⊗ ι2 ) = 0 (as m op gives the left action of (2) Aop on Aop and hence it preserves Aop ), and similarly π2 ◦ m op ◦ (ι1 ⊗ ι2 ) = 0. The second identity is just the completeness of ι1,2 , π1,2 . Since εop ◦ ηop = 0, without losing generality we can assume εop ◦ ι1 ◦ π1 ◦ ηop = 0. Using that π2 is a bimodule map we compute π2 ◦ [LHS of (3.22)]Ui =1 ◦ ι1 ◦ π1 ◦ ηop = π2 ◦ [RHS of (3.22)]Ui =1 ◦ ι1 ◦ π1 ◦ ηop = P l (2) ◦ π2 ◦ ι1 ◦ π1 ◦ ηop = 0. Aop
(3.28)
900
L. Kong, I. Runkel
On the other hand, using that Acl is haploid, that ι˜cl-op is an algebra map, and that ∗ ι˜cl-op is a coalgebra map, one can check that the left-hand side of (3.28) is equal to ι1 ◦ π1 ◦ ηop ) π2 ◦ ηop for some λ = 0. This implies that π2 ◦ ηop = 0. Thus λ(εop ◦
ηop = i ιi ◦ πi ◦ ηop = ι1 ◦ π1 ◦ ηop . Hence, we have 0 = π2 ◦ ι2 = π2 ◦ m op ◦ (ηop ⊗ ι2 ) = π2 ◦ m op ◦ ((ι1 ◦ π1 ◦ ηop ) ⊗ ι2 ). (3.29) However, the right-hand side is zero by (3.27). This is a contradiction and hence Aop must be simple.
To formulate the next theorem we need the notion of the full centre of an algebra [Fj, Def. 4.9]. Recall that an algebra A in a braided tensor category has a left centre and a right centre [VZ,O], both of which are sub-algebras of A. Of these two, we will only need the left centre. The following definition is [Fr, Def. 2.31], which in our setting is equivalent to that of [VZ,O]. Definition 3.17. Let A be a symmetric special Frobenius algebra such that m A ◦ A = ζ A id A . (i) The left centre Cl (A) of A is the image of the idempotent ζ A−1 PAl . (ii) The full centre Z (A) is Cl (R(A)). That ζ A−1 PAl is an idempotent follows from [FRS, Lem. 5.2] when keeping track of the factors ζ A ([FRS] assumes normalised-special, i.e. ζ A = 1). Note that Cl (A) is again 2 . Let el : C (A) → A be the embedding an object of C, while Z (A) is an object of C± l A of Cl (A) into A. The left centre is in fact the maximal subobject of A such that m A ◦ c A,A ◦ (elA ⊗ id A ) = m A ,
(3.30)
see [Fr, Lem. 2.32]. This observation explains the name left centre and also makes the connection to [O, Def. 15]. l The full centre is by definition the image of the idempotent ζ A−1 PR(A) : R(A) → 2 R(A). Since C± is abelian, the idempotent splits and we obtain the embedding and restriction morphisms e : Z (A) → R(A) and r : R(A) Z (A)
(3.31)
l ζ A−1 PR(A) .
It follows from Proposition 2.25 and which obey r ◦ e = id Z (A) and e ◦ r = 2 with [Fr, Prop. 2.37] that Z (A) is a commutative symmetric Frobenius algebra in C± 5 structure morphisms m Z (A) = r ◦ m R(A) ◦ (e ⊗ e), η Z (A) = r ◦ η R(A) , Z (A) = ζ A · (r ⊗ r ) ◦ R(A) ◦ e, ε Z (A) = ζ A−1 · ε R(A) ◦ e.
(3.32)
Moreover, if A is simple then Z (A) is simple, and if A is simple and dim A = 0, then Z (A) is simple and special. The normalisation of the counit is such that ε Z (A) ◦ η Z (A) = ζ A−2 dim A DimC.
(3.33)
5 The normalisation of product and unit is the standard one. The factors in the coproduct and counit have to be included in order for (A|Z (A), e) to be a Cardy algebra, see Theorem 3.18 below. The normalisation of the counit enters the Cardy condition (3.14) through the definition of ( · )∗ .
Cardy Algebras and Sewing Constraints, I
901
Theorem 3.18. Let A be a special symmetric Frobenius C-algebra. Then (A|Z (A), e) 2 -algebra. is a Cardy C|C± The proof of this theorem makes use the following two lemmas. Lemma 3.19. e : Z (A) → R(A) is an algebra map, and e∗ = ζ A · r . Proof. It follows from [Fr, Lem. 2.29] (or by direct calculation, using in particular m R(A) ◦ R(A) = ζ A id R(A) ) that l m R(A) ◦ (e ⊗ e) = ζ A−1 · PR(A) ◦ m R(A) ◦ (e ⊗ e).
(3.34)
l shows that e is compatible with multiplication. For the Substituting e ◦ r = ζ A−1 PR(A) unit one finds l e ◦ η Z (A) = e ◦ r ◦ η R(A) = ζ A−1 PR(A) ◦ η R(A) = η R(A) .
(3.35)
Thus e is an algebra map. For the second statement one computes Z (A)
Z (A)
R(A)
r
e
(1)
e∗ = ζ A
Z (A)
R(A)
R(A)
(2)
=
r R(A)
r R(A)
R(A)
e R(A)
R(A)
Z (A) r R(A)
(3)
(4)
=
= ζ A · r,
(3.36)
R(A)
where in (1) the definitions (2.30) and (3.32) have been substituted, step (2) is e ◦ l r = ζ A−1 PR(A) , step (3) uses that R(A) is symmetric Frobenius, and step (4) is again l e ◦ r = ζ A−1 PR(A) .
Lemma 3.20. Let A be a symmetric Frobenius algebra in C. The morphism l l l PR(A) : R(A) ⊗ R(A) −→ R(A) ◦ m R(A) ◦ PR(A) ⊗ PR(A)
(3.37)
is S-invariant. The proof of this lemma is a slightly lengthy explicit calculation and has been deferred to Appendix A.2.
902
L. Kong, I. Runkel
Proof of Theorem 3.18. That e is an algebra map was proved in Lemma 3.19. The centre condition (3.13) holds by property (3.30) of the left centre. The Cardy condition (3.14) also is an immediate consequence of Lemma 3.19, l ιcl-op ◦ ι∗cl-op = e ◦ (ζ A r ) = PR(A) .
(3.38)
The full centre Z (A) is a commutative symmetric Frobenius algebra. It remains to prove modular invariance. That θ Z (A) = id Z (A) is implied by commutativity and symmetry of Z (A) [Fr, Prop. 2.25]. The S-invariance condition (3.3) follows from Lemma 3.20: l In (3.37) substitute PR(A) = ζ A e ◦ r and then put the resulting morphism into (3.3). Compose the resulting equation with e ⊗ id W from the right (i.e. from the bottom) and substitute the definition (3.32) of m Z (A) . This results in the statement that m Z (A) is S-invariant.
The following theorem is analogous to [LR, Prop. 2.9] and [Fj, Thm. 4.26], which, roughly speaking, answer the question under which circumstances the restriction of a two-dimensional conformal field theory to the boundary already determines the entire conformal field theory. The first work is set in Minkowski space and uses operator algebras and subfactors, while the second work is set in Euclidean space and uses modular tensor categories. 2 -algebra such that dim A = 0 and Theorem 3.21. Let (A|Acl , ιcl-op ) be a Cardy C|C± Acl is simple. Then A is special and (A|Acl , ιcl-op ) ∼ = (A|Z (A), e) as Cardy algebras.
Proof. By Proposition 3.16, A is simple and special. Since Acl is simple, the algebra map ιcl-op : Acl → R(A) is either zero or a monomorphism. But ιcl-op ◦ ηcl = η R(A) , and so ιcl-op = 0. Thus ιcl-op is monic. By Lemma 2.18 (ii), ζ A−1 ι∗cl-op is epi. The Cardy condition (3.14) implies l = e ◦ r. ιcl-op ◦ ζ A−1 ι∗cl-op = ζ A−1 PR(A)
(3.39)
Composing this with e ◦ r from the left yields e ◦ r ◦ ιcl-op ◦ ζ A−1 ι∗cl-op = e ◦ r = ιcl-op ◦ ζ A−1 ι∗cl-op . Since ζ A−1 ι∗cl-op is epi, we have
e ◦ r ◦ ιcl-op = ιcl-op .
(3.40)
Actually, (3.40) also follows from (3.13) and specialness of R(A). We will prove that ( f op , f cl ) : (A|Acl , ιcl-op ) −→ (A|Z (A), e) where f op = id A , f cl = r ◦ ιcl-op , (3.41) is an isomorphism of Cardy algebras. f cl is an algebra map: Compatibility with the units follows since ιcl-op is an algebra map, f cl ◦ ηcl = r ◦ ιcl-op ◦ ηcl = r ◦ η R(A) = η Z (A) .
(3.42)
Compatibility with the multiplication also follows since ιcl-op is an algebra map, m Z (A) ◦ ( f cl ⊗ f cl ) = r ◦ m R(A) ◦ (e ⊗ e) ◦ (r ⊗ r ) ◦ (ιcl-op ⊗ ιcl-op ) = r ◦ m R(A) ◦ (ιcl-op ⊗ ιcl-op ) = r ◦ ιcl-op ◦ m cl = f cl ◦ m cl , where in the second step we used (3.40).
(3.43)
Cardy Algebras and Sewing Constraints, I
903
f cl is an isomorphism: As above, since f cl is an algebra map and since Acl is simple, f cl has to be monic. By Lemma 3.19, r ∗ = ζ A−1 e. Thus f cl∗ = ι∗cl-op ◦ r ∗ = ζ A−1 ι∗cl-op ◦ e and f cl ◦ f cl∗ = r ◦ ιcl-op ◦ ζ A−1 ι∗cl-op ◦ e = r ◦ e ◦ r ◦ e = id Z (A) ,
(3.44)
and so f cl is also epi, and hence iso. f cl is a coalgebra map: Since f cl is an algebra map, so is f cl−1 . By (3.44), f cl−1 = f cl∗ and by Lemma 2.18 (iii) this implies that f cl is a also coalgebra map. The diagram (3.15) commutes: Commutativity of (3.15) is equivalent to e ◦ f cl = ιcl-op , which holds by (3.40).
Let A be a special symmetric Frobenius algebra. So far we have seen that (A|Z (A), e) is a Cardy algebra, and that all Cardy algebras with Aop = A and simple Acl are of this form. It is now natural to ask if every simple Acl does occur as part of a Cardy algebra. The following theorem provides an affirmative answer. Recall that for an A-left module M, the object M ∨ ⊗ A M is an algebra (see e.g. [KR, Lem. 4.2]). Theorem 3.22. If Acl is a simple modular invariant commutative symmetric Frobenius 2 -algebra, then there exist a simple special symmetric Frobenius C-algebra A and a C± morphism ιcl-op : Acl → R(A) such that ∼ Z (A) as Frobenius algebras; (i) Acl = 2 -algebra; (ii) (A|Acl , ιcl-op ) is a Cardy C|C± ∨ ∼ (iii) T (Acl ) = ⊕κ∈J Mκ ⊗ A Mκ as algebras, where {Mκ }κ∈J is a set of representatives of the isomorphism classes of simple A-left modules. Proof. By Remark 3.6, we have dim Acl = Z 00 Dim C = 0, and by Lemma 2.11, Acl is haploid. It then follows from Theorem 3.4 that Acl is special. By Proposition 2.24, T (Acl ) is a special symmetric Frobenius algebra in C. Thus T (Acl ) = ⊕i Ai , where the Ai are simple symmetric Frobenius algebras. We will show that at least one of the Ai is special. Since T (Acl ) is special, we have m T (Acl ) ◦ T (Acl ) = ζ id T (Acl ) for some ζ ∈ C× . Restricting this to the summand Ai shows m i ◦ i = ζ id
Ai . Furthermore, εT (Acl ) ◦ ηT (Acl ) = ξ id1 for some ξ ∈ C× . But εT (Acl ) ◦ ηT (Acl ) = i εi ◦ ηi , and so at least one of the εi ◦ ηi has to be nonzero. Therefore, at least one of the Ai is special; let A ≡ Ai be this summand. We denote the embedding A → T (Acl ) by e0 and the restriction T (Acl ) A by r0 . Notice that r0 is an algebra homomorphism. Define ιcl-op = χˆ (r0 ) : Acl −→ R(A).
(3.45)
By Proposition 2.16, ιcl-op is an algebra homomorphism. Next we verify the centre condition (3.13), or rather its equivalent form (3.20). By substituting the definitions, one 2 implies the can convince oneself that the commutativity m cl ◦ c Acl ,Acl = m cl of Acl in C± condition m T (Acl ) ◦ = m T (Acl ) in C, see [K2, Prop. 3.6]. Here, : T (Acl ) ⊗ T (Acl ) → T (Acl ) ⊗ T (Acl ) is given by idCnl ⊗ cCml ,Cnr ⊗ idCmr ◦ cCml ,Cnl ⊗ cC−1r ,C r ◦ idCml ⊗ cC−1l ,C r ⊗ idCnr , = n
m,n
m
n
m
(3.46) and we decomposed Acl as Acl =
⊕n Cnl
× Cnr . As a consequence
we obtain the identity
r0 ◦ m T (Acl ) ◦ ◦ (id Acl ⊗ e0 ) = r0 ◦ m T (Acl ) ◦ (id Acl ⊗ e0 ).
(3.47)
904
L. Kong, I. Runkel
Using that r0 is an algebra map, and that by definition ι˜cl-op = r0 , we obtain (3.20). In order to show that (A|Acl , ιcl-op ) is a Cardy algebra, it remains to show that the Cardy condition (3.14) is satisfied. We will demonstrate this via a detour by first proving that Acl ∼ = Z (A) as Frobenius algebras. Recall the notations e and r given in (3.31). Using the centre condition (3.20) one l can check that PR(A) ◦ ιcl-op = m R(A) ◦ R(A) ◦ ιcl-op . By specialness of A we have l m A ◦ A = ζ A id A and so together with e ◦ r = ζ A−1 PR(A) we get,
e ◦ r ◦ ιcl-op = ιcl-op .
(3.48)
f cl = r ◦ ιcl-op : Acl −→ Z (A).
(3.49)
Next, consider the morphism
By the same derivation as in (3.42) and (3.43) one sees that f cl is an algebra map. In particular, f cl ◦ ηcl = η Z (A) = 0 and so f cl = 0. Since Acl is simple, f cl has to be a monomorphism. By the same argument as used in the proof of Theorem 3.4 (ii), up to isomorphism Acl is the unique simple local Acl -(left-)module. The algebra monomorphism f cl turns Z (A) into an Acl -module. Since Z (A) is commutative, it is local as an Acl -module, and so Z (A) ∼ = A⊕N cl for some N ≥ 1. By construction, A is a simple special symmetric Frobenius algebra. Proposition 2.25 and Corollary 2.27 show that R(A) inherits all these properties, and thus Z (A) is simple (see the comment below Eq. (3.32)). By Theorem 3.18, Z (A) is modular invariant, and then by Theorem 3.4 (i), dim Z (A) = Dim C. This implies that N = 1 in Z (A) ∼ = A⊕N cl , and so f cl is in fact an isomorphism. Since Acl and Z (A) are both haploid, we have ε Z (A) ◦ f cl = ξ εcl for some ξ ∈ C× . The counit uniquely determines the Frobenius structure on Acl and Z (A) (see e.g. [FRS, Lemma 3.7]), so that f cl is a coalgebra isomorphism iff ξ = 1. To compute ξ we compose the above identity with ηcl from the right. Defining ζcl via εcl ◦ ηcl = ζcl−1 Dim C · id1 and using (3.33) gives ξ = dim A ζcl /ζ A2 . By rescaling the comultiplication and the counit of A, and consequently changing ζ A , we can always achieve ξ = 1. This proves part (i) of the theorem. Equation (3.48) implies that ιcl-op = e ◦ f cl . Since f cl is an isomorphism of Frobenius algebras, by Lemmas 2.18 and 3.19 we have l ιcl-op ◦ ι∗cl-op = e ◦ f cl ◦ f cl∗ ◦ e∗ = ζ A e ◦ r = PR(A) .
(3.50)
Thus (A|Acl , ιcl-op ) is a Cardy algebra. This proves part (ii) of the theorem. Part (iii) can be seen as follows. By [KR, Prop. 4.3], T Z (A) ∼ = ⊕κ∈J Mκ∨ ⊗ A Mκ as algebras. Together with the observation that T ( f cl ) : T (Acl ) → T Z (A) is an isomorphism of algebras, this proves part (iii).
Remark 3.23. Part (i) of Theorem 3.22 was announced by Müger [Mü2]. We provide an independent proof in the setting of Cardy algebras The above theorem, together with Lemma 2.11 and Theorem 3.4, shows that a sim2 -algebra A with dim A = Dim C is always ple commutative symmetric Frobenius C± cl cl part of a Cardy algebra (Aop |Acl , ιcl-op ) for some simple special symmetric Frobenius algebra Aop in C. However, the above proof also illustrates that Aop is not unique. This raises the question how two Cardy algebras with a given Acl can differ. This question is answered by [KR, Thm. 1.1], which in the present framework can be restated as follows.
Cardy Algebras and Sewing Constraints, I (i)
(i)
905
(i)
2 -algebras such that Theorem 3.24. If (Aop |Acl , ιcl-op ), i = 1, 2 are two Cardy C|C± (i) (i) (1) ∼ (2) A is simple and dim Aop = 0 for i = 1, 2, then A = A as algebras if and only cl
(1)
cl
(2)
cl
if Aop and Aop are Morita equivalent. (i) Proof. Theorem 1.1 in [KR] is stated for A(i) op being non-degenerate algebras and Acl = (i) (i) Z (Aop ) for i = 1, 2. By Proposition 3.16, Aop are simple and special for i = 1, 2. (i) Then by [KR, Lem. 2.1], Aop are non-degenerate algebras. By Theorem 3.21, we have (i) ∼ (i) (1) (2) Acl = Z (Aop ) as Frobenius algebras. Finally, by [KR, Thm. 1.1], Z (Aop ) ∼ = Z (Aop ) (1) (2) as algebras iff Aop and Aop are Morita equivalent.
2 ) be the set of equivalence classes [B] of simple modular invariant Let Cmax (C± 2 . Two such algebras B and B are commutative symmetric Frobenius algebras B in C± equivalent if B and B are isomorphic as algebras (but not necessarily as Frobenius algebras). Let Msimp (C) be the set of Morita classes of simple special symmetric Frobenius 2 ) by z : {A} → [Z (A)], where algebras in C. Define the map z : Msimp (C) → Cmax (C± {A} denotes the Morita class of A. From Theorem 3.22 (i) and [KR, Thm. 1.1] we learn: 2 ) is a bijection. Corollary 3.25. The map z : Msimp (C) → Cmax (C±
A. Appendix A.1. Proof of Lemma 2.7 . We will show that if (F, ψ2F , ψ0F ) is a colax tensor functor from C1 to C2 , then (G, φ2G , φ0G ) is a lax tensor functor from C2 to C1 . Applying this result to the opposed categories then gives the converse statement. We need to show that φ0G and φ2G make the diagrams (2.1) and (2.2) commute. We first prove the commutativity of (2.1). Consider the following diagram: id G(A) ⊗φ2G
G(A) ⊗ (G(B) ⊗ G(C))
/ G(A) ⊗ G(B ⊗ C)
Gψ2F ◦δ
G(F G(A) ⊗ F(G(B) ⊗ G(C)))
G(F(id G(A) )⊗F(φ2G ))
Gψ2F ◦δ
/ G(F G(A) ⊗ F G(B ⊗ C)) G(ρ A ⊗ρ B⊗C )
G(id F G(A) ⊗ψ2F )
G(F G(A) ⊗ (F G(B) ⊗ F G(C)))
G(ρ A ⊗(ρ B ⊗ρC ))
/ G(A ⊗ (B ⊗ C)).
(A.1) The top subdiagram is commutative because of the naturality of ◦ δ. The commutativity of the bottom subdiagram follows from the following identities: Gψ2F
(ρ B ⊗ ρC ) ◦ ψ2F = (ρ B ⊗ ρC ) ◦ ψ2F ◦ ρ F ◦ Fδ = ρ B⊗C ◦ F G(ρ B ⊗ ρC ) ◦ F G(ψ2F ) ◦ Fδ = ρ B⊗C ◦ F(φ2G )
(A.2)
as a map F(G(B) ⊗ G(C)) → B ⊗ C. The commutativity of (A.1) implies that the composition of maps in the left column in (2.1) can be replaced by (A.3) G(ρ A ⊗ (ρ B ⊗ ρC )) ◦ G (id F G(A) ⊗ ψ2F ) ◦ ψ2F ◦ δ.
906
L. Kong, I. Runkel
Similarly, we can show that the composition of maps in the right column in (2.1) can be replaced by (A.4) G((ρ A ⊗ ρ B ) ⊗ ρC ) ◦ G (ψ2F ⊗ id F G(C) ) ◦ ψ2F ◦ δ. Using the commutativity of (2.3), it is easy to see that (2.1) with the left and right columns of (2.1) replaced by (A.3) and (A.4) respectively is commutative. Hence (2.1) is commutative. Now we prove the commutativity of the first diagram in (2.2). φ2G ◦ (φ0G ⊗ id G(A) )
= G(ρ12 ⊗ ρ A ) ◦ Gψ2F ◦ δ ◦ (Gψ0F ◦ δ11 ) ⊗ id G(A)
(1)
(2)
= G(ρ12 ⊗ ρ A ) ◦ Gψ2F ◦ G F(Gψ0 ⊗ id G(A) ) ◦ G F(δ11 ⊗ id G(A) ) ◦ δ
(3)
= G(ρ12 ⊗ ρ A ) ◦ G(F G(ψ0F ) ⊗ id F G(A) ) ◦ Gψ2F ◦ G F(δ11 ⊗ id G(A) ) ◦ δ (4) = G(id12 ⊗ ρ A ) ◦ G [ψ0F ◦ ρ F(11 ) ◦ (Fδ)11 ] ⊗ id F G(A) ◦ Gψ2F ◦ δ (5)
= G(id12 ⊗ ρ A ) ◦ G(ψ0F ⊗ id F G(A) ) ◦ Gψ2F ◦ δ
(6)
= G(id12 ⊗ ρ A ) ◦ G(l −1 F G(A) ) ◦ G F(l G(A) ) ◦ δ
(7)
= G(l −1 A ) ◦ Gρ A ◦ δG ◦ l G(A)
(8)
= G(l −1 A ) ◦ l G(A) ,
(A.5)
where in step (1) we substituted the definition of φ0G , φ2G given in (2.10); in step (2) we used the naturality of δ; in step (3) we used the naturality of Gψ2F ; in step (4) we switched the position between Gρ12 and G F G(ψ0F ) and the position between Gψ2 and G F(δ12 ⊗ id G(A) ) using the naturality of ρ and Fψ2G respectively; in step (5) we applied the second identity in (2.8); in step (6) we used (2.4); in step (7) we used the naturality of l −1 and δ; in step (8) we used the first identity in (2.8). The proof of the commutativity of the second diagram in (2.2) is similar. Thus we have shown that G is a lax tensor functor.
A.2. Proof of Lemma 3.20. To prepare the proof, recall that for a given object B ∈ C, the modular group P S L(2, Z) acts on the space ⊕i HomC (B ⊗ Ui , Ui ), see e.g. [BK, Sect 3.1] and [K3, Eq. (4.55)]. We will only need the action of S and S −1 . Let f ∈ ⊕i HomC (B ⊗ Ui , Ui ). Then Uj Ui Ui
S :
f
−→
i∈I B
Ui
dim U j √ DimC i∈I j∈I
f
B
,
Uj
(A.6)
Cardy Algebras and Sewing Constraints, I
907
Uj Ui Ui
S −1 :
f
−→
i∈I B
dim U j √ DimC i∈I j∈I
f
Ui
B
.
(A.7)
Uj
By Lemma 3.2, to establish that (3.37) is S-invariant, it is enough to prove the identity (3.4) when f is given by (3.37). Using (A.6) and (A.7), we can see that Eq. (3.4) simply says that ⊕i, j [RHS of (3.4)] is invariant under the action of S × S. Consider the element ∨ g of ⊕ j,k∈I HomC 2 (R(A) ⊗ (U ∨ j × Uk ), U j × Uk ) given by ±
U∨ j × Uk α
R(A)
g
=
R(A) R(A)
j,k∈I
.
(A.8)
α α U∨ j × Uk
R(A)
By the above arguments, proving S-invariance of (3.37) is equivalent to proving invariance of g under the action of S × S. For i ∈ I, we denote by gi the component of g in ∨ ⊕ j∈I HomC (A ⊗ Ui∨ ⊗ U ∨ j , U j ) ⊗ (⊕k∈I Hom C (Ui ⊗ Uk , Uk )) . We view the second Hom-space in the above tensor product as a Hom-space in C+ instead of C− . It is enough to show that gi is invariant under the action of S × S −1 . Note that the action of S −1 in C+ is equivalent to that of S in C− . The morphism gi can be canonically identified with a bilinear pairing ⎛ ⎞ ∨ ∨ ⎠ ( · , · )i : ⎝ HomC (U ∨ j , A ⊗ Ui ⊗ U j ) ×
j∈I
HomC (Uk , Ui ⊗ Uk )
−→ C
(A.9)
k∈I ∨ ∨ as follows. For h 1 ∈ HomC (U ∨ j , A ⊗ Ui ⊗ U j ) and h 2 ∈ Hom C (Uk , Ui ⊗ Uk ) we set
(h 1 , h 2 )i = (dim U j dim Uk )−1 trU ∨j ×Uk [gi ◦ (h 1 × h 2 )] .
(A.10)
908
L. Kong, I. Runkel
When substituting the explicit form of the product m R(A) of R(A) = (A × 1) ⊗ R(1), after a short calculation one finds
A A
α A Uk
(h 1 , h 2 )i =
α
1 dim U j
Uj
Ui Uk A
A
A
h2
.
(A.11)
α
U∨ j
A h1
l Here the top morphism PR(A) has been simplified with the help of the identity
PR(A) ◦ m R(A) ◦ (PR(A) ⊗ PR(A) ) = ((m A ◦ A ) ⊗ idUi∨ ) × idUi ◦ m R(A) ◦ (PR(A) ⊗ PR(A) ),
(A.12)
i∈I
which can be checked by direct calculation along the same lines as in the proof of [Fr, Lem 3.10]. The action of the modular transformation S on ⊕i∈I (B ⊗ Ui , Ui ) for B ∈ C naturally induces an action on ⊕i∈I (Ui , B ⊗ Ui ) [K3, Prop. 5.14], which we denote by ∨ ∨ S ∗ . In the present case we get an action of S ∗ on ⊕ j∈I HomC (U ∨ j , A ⊗ Ui ⊗ U j ) and ⊕k∈I HomC (Uk , Ui ⊗ Uk ). Then to show gi is invariant under the action of S × S −1 amounts to showing that
(h 1 , h 2 )i = ((S −1 )∗ h 1 , S ∗ h 2 )i ,
(A.13)
Cardy Algebras and Sewing Constraints, I
909
∨ ∨ for all h 1 ∈ HomC (U ∨ j , A ⊗ Ui ⊗ U j ) and h 2 ∈ Hom C (Uk , Ui ⊗ Uk ). We have
A A
α A
Ui
dim Un ((S −1 )∗ h 1 , S ∗ h 2 )i = DimC m,n,α
Um
Un
A
A
.
h2 Uk
A
α
Uj
A h1
(A.14)
Now drag the upper vertex indexed by α in the above graph along its Um∨ -leg until it meets the lower vertex also indexed by α, then sum over α and m. This gives
A A
A Ui
((S −1 )∗ h 1 , S ∗ h 2 )i =
dim Un n
A
A
DimC
h2 Uk
A Uj
A
.
Un
h1
(A.15)
910
L. Kong, I. Runkel
If we just look at the neighbourhood of the Un -loop in the above graph, we see the following subgraphs: A
A ∨ Uk
Uk∨
dim Un n
Un
DimC
U∨ j
A∨
A
=
α
Uk∨
Uk
1 dim U j
, α
α
U∨ j
Uj
Uj
(A.16)
where we have applied [BK, Cor. 3.1.11]. Substituting this subgraph back to the original graph in (A.15), we obtain
α A
A
A
A
Ui
((S
−1 ∗
∗
) h 1 , S h 2 )i =
α
Uk
1 dim U j
A
mA
Uj
.
A h2 Uk
A A
α U∨ j
A h1
(A.17) The graph in (A.17) is equal to that in (A.11). In order to see this, we first drag the “bubble” (m A ◦ A ) along A lines and through the m A vertex (because m A ◦ A is a bimodule map) until it reaches the lower-left leg of the upper vertex indexed by α. Then drag the m A vertex along the (red) dotted line in the above graph. Finally, we apply the associativity of A, (A.12), and [Fr, Lem 3.11]. Then we see that the graph in (A.17) exactly matches with the one in (A.11).
Acknowledgements. We would like to thank the organisers of the Oberwolfach Arbeitsgemeinschaft “Algebraic structures in conformal field theories” (April 2007), where this work was started, for an inspiring meeting. We would further like to thank the Hausdorff Institute for Mathematics in Bonn and the organisers of the stimulating meeting “Geometry and Physics” (May 2008). We are indebted to Alexei Davydov, Jens Fjelstad, Jürgen Fuchs, Yi-Zhi Huang, Alexei Kitaev, Urs Schreiber, Christoph Schweigert, Stephan Stolz and Peter Teichner for helpful discussions and/or comments on a draft of this paper. The research of IR was partially supported by the EPSRC First Grant EP/E005047/1, the PPARC rolling grant PP/C507145/1 and the Marie Curie network ‘Superstring Theory’ (MRTN-CT-2004-512194).
Cardy Algebras and Sewing Constraints, I
911
References [AN] [Bi] [BK] [C1] [C2] [CL] [DMZ] [DP] [ENO1] [ENO2] [Fe] [FHL] [Fj] [FK] [Fr] [FRS] [FS] [H1] [H2] [HK1] [HK2] [HK3] [HL] [JS] [K1] [K2] [K3] [Ki] [KO] [KR] [Ld] [Li1] [Li2]
Alexeevski, A., Natanzon, S.M.: Noncommutative two-dimensional topological field theories and hurwitz numbers for real algebraic curves. Sel. Math., New Ser. 12, 307–377 (2006) Bichon, J.: Cosovereign hopf algebras. J. Pure Appl. Alg. 157, 121–133 (2001) Bakalov, B., Kirillov, A.A.: Lectures on Tensor Categories and Modular Functors. Providence, RI: Amer. Math. Soc., 2001 Cardy, J.L.: Operator content of two-dimensional conformal invariant theories. Nucl. Phys. B 270, 186–204 (1986) Cardy, J.L.: Boundary conditions, fusion rules and the verlinde formula. Nucl. Phys. B 324, 581–596 (1989) Cardy, J.L., Lewellen, D.C.: Bulk and boundary operators in conformal field theory. Phys. Lett. B 259, 274–278 (1991) Dong, C.-Y., Mason, G., Zhu, Y.-C.: Discrete series of the Virasoro algebra and the moonshine module. In: Algebraic Groups and Their Generalizations: Quantum and infinite-dimensional Methods, Proc. Symp. Pure Math. 56, Part 2, Providence, RI: Amer. Math.Soc., 1994, pp. 295–316 Day, B., Pastro, C.: Note on frobenius monoidal functors. New York J. Math. 14, 733–742 (2008) Etingof, P.I., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. 162, 581–642 (2005) Etingof, P.I., Nikshych, D., Ostrik, V.: An analogue of radford’s s 4 formula for finite tensor categories. Int. Math. Research Notices 54, 2915–2933 (2004) Felder, G., Fröhlich, J., Fuchs, J., Schweigert, C.: Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology. Comp. Math. 131, 189–237 (2002) Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993) Fjelstad, J., Fuchs, J., Runkel, I., Schweigert, C.: Uniqueness of open/closed rational cft with given algebra of open states. Adv. Theor. Math. Phys. 12, 1283–1375 (2008) Fröhlich, J., King, C.: The chern-simons theory and knot polynomials. Commun. Math. Phys. 126, 167–199 (1989) Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Correspondences of ribbon categories. Adv. Math. 199, 192–329 (2006) Fuchs, J., Runkel, I., Schweigert, C.: Tft construction of rcft correlators. I: partition functions. Nucl. Phys. B 646, 353–497 (2002) Fuchs, J., Schweigert, C.: Category theory for conformal boundary conditions. Fields Inst. Commun. 39, 25–71 (2003) Huang, Y.-Z.: Two-dimensional conformal geometry and vertex operator algebras. Progress in Mathematics, Vol. 148, Boston: Birkhäuser, 1997 Huang, Y.-Z.: Rigidity and modularity of vertex tensor categories. Comm. Contemp. Math. 10, 871–911 (2008) Huang, Y.-Z., Kong, L.: Open-string vertex algebra, category and operads. Commun. Math. Phys. 250, 433–471 (2004) Huang, Y.-Z., Kong, L.: Full field algebras. Commun. Math. Phys. 272, 345–396 (2007) Huang, Y.-Z., Kong, L.: Modular invariance for conformal full field algebras. http://arxiv.org/abs/ math/0609570v2[math.QA], 2006 Huang, Y.-Z., Lepowsky, J.: Tensor products of modules for a vertex operator algebra and vertex tensor categories, In: Lie Theory and Geometry, in honor of Bertram Kostant, ed. R. Brylinski, J.-L. Brylinski, V. Guillemin, V. Kac, Boston: Birkhäuser, 1994, pp. 349–383 Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993) Kong, L.: Full field algebras, operads and tensor categories. Adv. Math. 213, 271–340 (2007) Kong, L.: Open-closed field algebras. Commun. Math. Phys. 280, 207–261 (2008) Kong, L.: Cardy condition for open-closed field algebras. Commun. Math. Phys. 283, 25–92 (2008) Kitaev, A.: Private communication Kirillov, A.A., Ostrik, V.: On q-analog of mckay correspondence and ade classification of sl(2) conformal field theories. Adv. Math. 171, 183–227 (2002) Kong, L., Runkel, I.: Morita classes of algebras in modular tensor categories. Adv. Math. 219, 1548–1576 (2008) Lauda, A.D.: Frobenius algebras and ambidextrous adjunctions. Theo. Appl. Cat. 16, 84– 122 (2006) Li, H.-S.: Regular representations of vertex operator algebras. Commun. Contemp. Math. 4, 639–683 (2002) Li, H.-S.: Regular representations and huang-lepowsky tensor functors for vertex operator algebras. J. Alge. 255, 423–462 (2002)
912
[Ln] [LP] [LR] [Lw] [Lz] [Ma] [Mo] [MS] [Mü1] [Mü2] [O] [P] [RT] [Se] [So] [Sz] [T] [V] [VZ] [W] [Y]
L. Kong, I. Runkel
Leinster, T.: Higher operads, higher categories. London Mathematical Society Lecture Note Series 298. Cambridge: Cambridge University Press, 2004 Lauda, A., Pfeiffer, H.: Open-closed strings: two-dimensional extended tqfts and frobenius algebras. Topology Appl. 155, 623–666 (2008) Longo, R., Rehren, K.H.: Local fields in boundary conformal qft. Rev. Math. Phys. 16, 909 (2004) Lewellen, D.C.: Sewing constraints for conformal field theories on surfaces with boundaries. Nucl. Phys. B 372, 654–682 (1992) Lazaroiu, C.I.: On the structure of open-closed topological field theory in two dimensions. Nucl. Phys. B 603, 497–530 (2001) Mac Lane, S.: Categories for the working mathematician. Brelin-Heidelberg-NewYork: Springer, 1998 Moore, G.: Some comments on branes, G-flux, and K -theory. Int. J. Mod. Phys. A16, 936– 944 (2001) Moore, G., Segal, G.: D-branes and K-theory in 2D topological field theory. http://arxiv.org/abs/ hep-th/0609042v1, 2006 Müger, M.: From subfactors to categories and topology II. The Quantum Double of Tensor Categories and Subfactors. J. Pure Appl. Alg. 180, 159–219 (2003) Müger, M.: Talk at workshop ‘Quantum Structures’ (Leipzig, 28. June 2007), Preprint in preparation Ostrik, V.: Module categories, weak hopf algebras and modular invariants. Transform. Groups 8, 177–206 (2003) Pfeiffer, H.: Finitely semisimple spherical categories and modular categories are self-dual. Adv. Math. 221, 1608–1652 (2009) Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Inv. Math. 103, 547–597 (1991) Segal, G.: The definition of conformal field theory. Preprint 1988; also in: U. Tillmann (ed.), Topology, geometry and quantum field theory, London Math. Soc. Lect. Note Ser. 308, Cambridge: Cambridge Univ. Press, 2004, pp. 421–577 Sonoda, H.: Sewing conformal field theories II. Nucl. Phys. B 311, 417–432 (1988) Szlachányi, K.: Adjointable monoidal functors and quantum groupoids. In: Hopf algebras in noncommutative geometry and physics, Caenepeel, S., Oystaeyen, F.V. (eds.) Lecture Notes in Pure and Applied Mathematics 239 Boca Raton, FL: CRC Press, 2004, pp. 297–307 Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. New York: de Gruyter, 1994 Vafa, C.: Conformal theories and punctured surfaces. Phys. Lett. B 199, 195–202 (1987) Van Oystaeyen, F., Zhang, Y.H.: The brauer group of a braided monoidal category. J. Algebra 202, 96–128 (1998) Witten, E.: Quantum field theory and the jones polynomial. Commun. Math. Phys. 121, 351–399 (1989) Yetter, D.N.: Functorial knot theory. Categories of tangles, coherence, categorical deformations, and topological invariants. Series on Knots and Everything 26, River Edge, NJ: World Scientific, 2001
Communicated by Y. Kawahigashi
