Commun. Math. Phys. 227, 1 – 92 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Anderson Transitions for a Family of Almost Periodic Schrödinger Equations in the Adiabatic Case Alexander Fedotov1 , Frédéric Klopp2 1 Department of Mathematical Physics, St. Petersburg State University, 1, Ulianovskaja,
198904 St. Petersburg-Petrodvoretz, Russia. E-mail:
[email protected]
2 Département de Mathématique, Institut Galilée, L.A.G.A., UMR 7539 C.N.R.S, Université de Paris-Nord,
Avenue J.-B. Clément, 93430 Villetaneuse, France. E-mail:
[email protected] Received: 2 July 2001 / Accepted: 13 November 2001
Abstract: This work is devoted to the study of a family of almost periodic one-dimensional Schrödinger equations. Using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit, we prove the existence of an asymptotically sharp Anderson transition in the low energy region. More explicitly, we prove the existence of energy intervals containing only singular spectrum, and of other energy intervals containing absolutely continuous spectrum; the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions. The analysis may be viewed as utilizing a complex WKB method for adiabatic perturbations of periodic Schrödinger equations. The transition energies are interpreted in terms of phase space tunneling. Résumé: Ce travail est consacré à l’étude d’une famille d’équations de Schrödinger quasi-périodiques en dimension 1. Nous définissons une matrice de monodromie pour cette famille. Nous étudions le comportement asymptotique de cette matrice dans le cas adiabatique. A cette fin, nous utilisons une version de la méthode WKB complexe pour l’équation de Schrödinger périodique avec une perturbation adiabatique. L’étude de la matrice de monodromie nous permet de prouver l’existence de transitions d’Anderson. Plus précisément, nous démontrons l’existence d’intervalles d’énergie ne contenant que du spectre singulier, d’autres intervalles contenant surtout du spectre absolument continu; les zones contenant du spectre singulier sont séparées des zones contenant du spectre absolument continu par des seuils de mobilité asymptotiquement ponctuels. Ces seuils de mobilité sont caractérisés grâce à l’effet tunnel dans l’espace des phases.
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A. Fedotov, F. Klopp
1. Introduction In this paper, we study spectral properties of the ergodic family of Schrödinger equations Hφ,ε ψ = −
d2 ψ(x) + (V (x − φ) + α · cos(εx))ψ(x) = Eψ(x), dx 2
x ∈ R. (1.1)
We assume that • V is a locally square integrable, real valued, periodic function, V (x + 1) = V (x),
x ∈ R,
(1.2)
• α > 0 is a coupling constant, • ε is a positive number such that 2π/ε is irrational, • φ ∈ R is a parameter (“indexing the equations of the family”). We note that, under our assumptions, the potential V (x−φ)+α cos(εx) is quasi-periodic. The goal of this paper is to study the family (1.1) at “low” energies. Some of these results were announced in [21, 22]. During the last 20 years, quasi-periodic Schrödinger operators have been the object of many studies both rigorous and numerical (see, e.g. [25, 29, 33, 28, 14] for references). Motivated by the physics of condensed matter, among the main goals was the understanding of their spectra. One of the questions that has drawn a lot of interest is the nature of the spectrum; this is one of the main issues dealt with in this paper. One of the most remarkable phenomena exhibited by the spectrum of these operators is the so-called Anderson transition or “metal-insulator” transition. It was observed numerically (e.g. [27, 28]) that, for certain quasi-periodic Schrödinger operators, the spectrum is split into energy intervals where it is purely absolutely continuous and other energy intervals where it is singular. Roughly speaking, from the physicist’s point of view, the singular spectrum corresponds to a phase where the material is an insulator, and the absolutely continuous spectrum to a phase where the material is a conductor. The transitions from one of these phases to the other one are expected to be sharp and to happen at discrete energies called mobility edges. Such transitions do not always exist; e.g. for highly symmetrical models like the Almost Mathieu equation or for small potentials, it is believed and proved in many cases that no such transitions happen, the spectral type being determined by parameters of this equation ([13, 30]). As for the existence of transitions, in [23], for a model similar to ours, it was proved that the bottom of the spectrum is pure point in the large coupling constant limit. On the other hand, it is known that, for one dimensional quasi-periodic Schrödinger operators with analytic potentials, the spectrum is absolutely continuous for sufficiently high energies ([13]). So, different types of spectra may coexist. In the present paper, we define the monodromy matrix for the family of Eqs. (1.1), compute the asymptotics of a monodromy matrix in the adiabatic limit, i.e. as ε → 0, and use these results to study spectral properties of Eq. (1.1). More precisely, we show that, at low energies, the spectrum of the family (1.1) is located in exponentially small intervals and prove the coexistence of zones where the spectrum is singular and others where most of the spectrum is absolutely continuous. We also prove that the transitions between these regions are asymptotically sharp as ε → 0. The energies where these transitions occur are called asymptotic mobility edges. We give a simple description of these energies in terms of analytic objects naturally associated to Eq. (1.1). To the best
Anderson Transitions for Almost Periodic Schrödinger Equations
3
of our knowledge, this is the first instance of quasi-periodic Schrödinger operators for which such a coexistence has been proved outside of the large coupling regime and sharp transitions have been found. Also, in the present paper, we handle locally square integrable potentials, whereas in previous works analyticity of the potential has always been a crucial assumption. Let us now describe our model more precisely. Consider the periodic Schrödinger operator (H0 ψ)(x) = −
d2 ψ(x) + V (x)ψ(x). dx 2
(1.3)
In this paper, • we assume that the first gap in the spectrum of H0 is open; denote it by (E2 , E3 ); • we restrict our study to a neighborhood of E1 , the bottom of the spectrum, more precisely, to the energies satisfying E − α < E1 and E1 < E + α < E2 .
(1.4)
The techniques that we present here can also be used to study more general adiabatic quasi-periodic perturbations of H0 , i.e. we can replace the cosine by other real analytic potentials. It can also be used to study the situation at higher energies (see [17, 20, 16]). Let us now briefly describe our results, the next section being devoted to the precise mathematical statements. Let E(κ) be the dispersion relation associated to H0 and consider the real and the complex iso-energy curves R and defined by R : :
E(κ) + α · cos(ϕ) = E, κ, ϕ ∈ R, E(κ) + α · cos(ϕ) = E, κ, ϕ ∈ C.
(1.5) (1.6)
The connected components of R are called the real branches of the iso-energy curve (1.6). The real iso-energy curve is 2π -periodic in the κ- and ϕ- (vertical and horizontal) directions. It is symmetric with respect to the ϕ-axis. The role of the real iso-energy curve for adiabatic problems is well known (see, for example, [5]). Under assumption (1.4), the curve has the topology shown in Fig. 1. In this figure, the real branches of R are represented by the full lines, and the dashed lines are complex loops in . Define the actions Sh , Sv and by κdϕ, Sv = i κdϕ and = κdϕ. (1.7) Sh = −i γ2
γ3
γ1
Here, γ1 is a connected component of R , γ2 a loop in connecting γ1 and γ1 − (2π, 0), and, γ3 a loop in connecting γ1 and γ1 + (0, 2π ). We discuss these loops in Sect. 2.6. These loops are oriented so that the integrals in (1.7) be positive. We prove that, on the interval (1.4), the spectrum is situated in exponentially small (in 1/ε) intervals such that the distances between any two neighboring such intervals are of order ε; the “centers” of these intervals are given by the quantization condition cos((E)/ε) = 0. Moreover, if we set S(E) = Sv (E) − Sh (E), then, the subintervals where S(E) < 0 contain only singular spectrum, and the subintervals where S(E) > 0 contain absolutely continuous spectrum; actually, in the latter subintervals, most of the spectrum is absolutely continuous. Finally, we study general properties of the function S(E). For small α, S takes positive values; hence, the bottom of the spectrum of the operator Hφ,ε still contains absolutely continuous spectrum for α small. On the contrary,
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A. Fedotov, F. Klopp
2
|
2
b b
b
'1
3
1
2 |
1
b
'1
0
1 b 2
|
' '1
Fig. 1. The contours on
for large α, the function S becomes negative, and thus, the bottom of the spectrum of the unperturbed operator becomes singular, the absolutely continuous spectrum disappears. For intermediate values of α, there are always one or several asymptotic mobility edges, i.e. points where the function S vanishes. These are the points where the transitions from absolutely continuous to singular spectrum (or vice versa) occur. One can define two tunneling coefficients, a vertical one associated to Sv : tv = exp(− 1ε Sv ), and, a horizontal one associated to Sh : th = exp(− 1ε Sh ). We see that, roughly, if the vertical tunneling is larger than the horizontal one, the spectrum is singular; and if the horizontal is the larger one, the spectrum is absolutely continuous. For Harper’s equation, an heuristic leading to a similar interpretation has been developed in [42].
E 3
S0 c
tr u
m
2
sp
e
1
a
c
0
b
0 1
2
b
3
4
5
6
7
8
1
2
sing. spectrum
3
4
5
6
Fig. 2. The phase diagram
9
Anderson Transitions for Almost Periodic Schrödinger Equations
5
To illustrate our results, we describe an example where we have computed the asymptotic mobility edges numerically, i.e. we have solved S = 0 numerically. We take the simplest example of V , i.e. we take V to be a one-gap potential ([36]) associated to the periodic spectrum [E1 , E2 ] ∪ [E3 , +∞). In our case E1 = 0, E2 = 8. The value of E3 is computed so that the potential V do have period 1. In Fig. 2, we represented the phase diagram, i.e. the curve S0 = {(α, E) : S(E, α) = 0} of the asymptotic mobility edges (for the potential V (· − φ) + α cos(ε·)). The spectrum is represented vertically and the parameter α horizontally. The oblique rectangle delimits the region (α, E) satisfying (1.4). It comprises the bottom of the spectrum of Hφ,ε . For α fixed, the domain marked by “ac spectrum” (resp. “sing. spectrum”) in Fig. 2 contains absolutely continuous (resp. singular) spectrum. α0 is the value of α at which mobility edges start to exist in our energy interval. When one turns the coupling constant α on, the lowest edge of the spectrum starts to “localize” first, though not immediately. When α > α ∗ , all the spectrum in the window we are considering is singular. In a more general case, i.e. if, in (1.1), we replace the cosine by a more general potential, the phase diagram may be quite different (see [17]). Such phase diagrams were already obtained for quasi-periodic finite difference equations using numerical simulations ([27]). Remark 1.1. In the description given above, we used the word “localized” very loosely to denote singular spectrum. 2. The Main Results In this section, we introduce the central object of our study, the monodromy matrix, and present our main results. 2.1. Monodromy equation. We define the monodromy matrix and introduce the monodromy equation. 2.1.1. Monodromy matrix. For any φ fixed, let ψ1,2 (x, φ) be two linearly independent solutions of Eq. (1.1). We say that they form a consistent basis if their Wronskian is independent of φ and if these solutions are 1-periodic in φ, i.e. ψ1,2 (x, φ + 1) = ψ1,2 (x, φ),
∀x, φ.
(2.1)
The existence of a consistent basis is clear: it suffices to take the solutions with canonical Cauchy data at x = 0, i.e. ψ1 (0, φ) = 1 and ψ1 (0, φ) = 0, and ψ2 (0, φ) = 0 and ψ2 (0, φ) = 1. Consider a consistent basis (ψ1,2 ). As cosine is 2π -periodic, the functions ψ1,2 (x + 2π/ε, φ + 2π/ε) are solutions of Eq. (1.1). Therefore, one can write ψ1 (x + 2π/ε, φ + 2π/ε) ψ1 (x, φ) = M(E, φ) , (2.2) ψ2 (x + 2π/ε, φ + 2π/ε) ψ2 (x, φ) where M(E, φ) is a 2 × 2 matrix with coefficients independent of x. The matrix M is called the monodromy matrix corresponding to the consistent basis (ψ1,2 ). Note that if the potential in (1.1) and the solutions ψ1,2 are independent of φ, then our definition becomes the standard definition of the monodromy matrix for the periodic Schrödinger equation (see e.g. [12, 41]). One immediately checks that, for any consistent basis, the monodromy matrix satisfies det M(E, φ) ≡ 1,
M(E, φ + 1) = M(E, φ),
∀φ.
(2.3)
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A. Fedotov, F. Klopp
2.1.2. Monodromy equation. Set h=
2π mod1. ε
(2.4)
Let M be a monodromy matrix corresponding to a consistent basis (ψ1,2 ). The spectral analysis of (1.1) can be reduced to the investigation of the solutions of the monodromy equation Fn+1 = M(E, φ + nh)Fn ,
Fn ∈ C2 ,
∀n ∈ Z.
(2.5)
Going from Eq. (1.1) to the monodromy equation is close to the monodromization transformation introduced in [7] to construct Bloch solutions of difference equations. In our case, it appears that the behavior of solutions of (1.1) for x → ±∞ repeats the behavior of solutions of the monodromy equation for n → ∓∞. And, it is well known that the spectral properties of one dimensional Schrödinger equations can be described in terms of the behavior of its solutions as x → ±∞ (see [24, 31]). Let us formulate the precise statement. In the sequel, identifying the vector solutions of the monodromy Eq. (2.5) with functions F : Z → C2 , we denote the values of F by Fn . One has Theorem 2.1. Let ψj (x, φ), j = 1, 2, be a consistent basis solution of (1.1), locally bounded in (x, φ) together with their derivatives in x. Fix φ = φ0 ∈ R and consider solutions of the monodromy equation. Then, there exists a positive constant C such that, for any vector solution F of Eq. (2.5), there exists a unique solution f of (1.1) satisfying the estimates f (x + 2π n/ε, φ) 1 ≤ CF−n C2 , ∀x ∈ [0, 2π/ε), n ∈ Z. F−n C2 ≤ f (x + 2π n/ε, φ) C2 C (2.6) and reciprocally. Both, for Eqs. (1.1) and (2.5), one can define the Lyapunov exponent (see e.g. [4, 8, 10, 37]). It is defined for almost every φ and independent of φ. Let #(E, φ) be the Lyapunov exponent at energy E for (1.1) and θ (E, φ) be the one for (2.5). Theorem 2.1 then immediately implies Corollary 2.1. The Lyapunov exponents #(E, φ) and θ (E, φ) satisfy the relation #(E, φ) =
ε θ(E, φ). 2π
(2.7)
2.2. The adiabatic limit. We see that the spectral analysis of (1.1) reduces to the analysis of solutions of the monodromy equation (2.5). To compute a monodromy matrix, we use the adiabatic limit ε → 0. In the adiabatic case, the asymptotics of the monodromy matrices take up a very simple form so that one can speak of reducing of (1.1) to some simple model difference equation (i.e. the respective monodromy equation). To describe the reduced model, we first recall some basic facts about one dimensional periodic Schrödinger operators and define some related objects.
Anderson Transitions for Almost Periodic Schrödinger Equations
7
2.3. Periodic Schrödinger operator. Consider the periodic Schrödinger operator (1.3). Its spectrum on L2 (R), denoted by σ (H0 ), is absolutely continuous and consists of intervals [E1 , E2 ], [E3 , E4 ], . . . , [E2n+1 , E2n+2 ], . . . , of the real axis such that E1 < E2 ≤ E3 < E4 . . . E2n ≤ E2n+1 < E2n+2 ≤ . . . , En → +∞, n → +∞. The above intervals are called the spectral bands, and the open intervals (E2 , E3 ), (E4 , E5 ), . . . , (E2n , E2n+1 ), . . . , are called the spectral gaps. Denote the dispersion law associated to (1.3) by E(κ). The inverse function k(E) is called the Bloch quasi-momentum. It is a multi-valued analytic function; its branch points are the edges of the gaps and are of square root type. It is real on the spectral bands; on the spectral gaps, its real part is constant and its imaginary part does not vanish. For more details on this function, we refer to Sect. 5.
2.4. The complex momentum. Consider the function κ(ϕ) defined by the relation (1.6), i.e. κ(ϕ) = k(E − α cos(ϕ)).
(2.8)
We call it the complex momentum. This function is also multi-valued and analytic. The function κ will play a crucial role in the determination of the asymptotics of the monodromy matrix.
2.5. The branch points of κ and the set arccos(R). As the branch points of the Bloch quasi-momentum, all the branch points of κ are of square root type. They are given by the equation E − α cos(ϕ) = En , n ∈ N.
(2.9)
The set of the branch points is 2π -periodic in ϕ. If E ∈ R, it is symmetric with respect to the real line and, moreover, is situated on arccos(R) = R ∪ ∪n∈Z (iR + nπ ), the pre-image of R by the cosine. Under condition (1.4), the complex momentum has a single branch point in (0, π ). Denote it by ϕ1 ; it is defined by E1 = E−α cos(ϕ1 ) (see Fig. 3). The complex momentum also has non-real branch points. Let ϕ2 and ϕ3 be the branch points lying on the line π + iR+ , closest to π, indexed so that Im ϕ2 < Im ϕ3 ; they satisfy (2.9) for n = 2, 3 (see Fig. 3). Consider the map E : ϕ → E − α cos(ϕ). We denote by Z the pre-image of σ (H0 ), the spectrum of H0 and by G the pre-image of the spectral gaps of H0 , i.e. R \ σ (H0 ). Clearly, Z, G ⊂ arccos(R). The connected components of Z and G are separated by branch points of the complex momentum. All the branches of the complex momentum take real values on Z. And, the imaginary part of any branch of the complex momentum is non-zero on G.
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A. Fedotov, F. Klopp
2.6. The phases and tunneling coefficients. We now describe the main analytical objects needed to write out the asymptotics of the coefficients of the monodromy matrix: the phase and the tunneling coefficients th and tv , in terms of contours in the complex plane of ϕ. There is a branch of the complex momentum taking values in iR+ on the interval (−ϕ1 , ϕ1 ), and values in [0, π] on the “cross” [ϕ1 , 2π − ϕ1 ] ∪ [ϕ2 , ϕ2 ] (see Fig. 3 and Sect. 7.1 for more details). Denote this branch by κ∗ . We define the actions ϕ1 ϕ2 Sh = −i κ∗ dϕ and Sv = i (κ∗ − π )dκ, (2.10) −ϕ1
ϕ2
and the phase integral 1 = ε
2π−ϕ1
ϕ1
κ∗ dϕ.
(2.11)
Let J be the interval defined by (2.16). In Sect. 9.1, we prove Lemma 2.1. The actions and phase integrals have the following properties (1) Sh , Sv and take positive values on J ; (2) they are analytic in E in a neighborhood of J ; (3) (E) > 0 for all E ∈ J ; (4) Sv ≤ 2π Im (ϕ2 ) for E ∈ J . We define tunneling coefficients 1 1 th = exp − Sh , tv = exp − Sv . ε ε
(2.12)
Due to point (1) in Lemma 2.1, the tunneling coefficients th and tv are exponentially small in 1/ε. The actions and the phase integral can be written as contour integrals of a branch of κ along closed curves i i 1 Sh = − κdϕ and Sv = κdϕ, and = κdϕ. (2.13) 2 γh 2 γv 2ε γp The loops γh , γv and γp are shown on Fig. 3. We note here that the contours γp , γh and γv can be considered as projections on the complex plane of ϕ, of contours γ1 , γ2 , γ3 located in (see Fig. 1). This follows from the fact that any analytic branch of κ is single valued on each of the curves γp , γh and γv (see Sect. 9.1). This last property is related to the fact that all the branch points of κ are of square root type. 2.7. Asymptotics of the monodromy matrix. As the potential in Eq. (1.1) is real valued, we are able to choose a consistent basis so that the corresponding monodromy matrix has the form a(φ, E) b(φ, E) a b M(φ, E) = (2.14) = ∗ ∗ . b a b(φ, E) a(φ, E)
Anderson Transitions for Almost Periodic Schrödinger Equations
9
'2
v
p
h '1
b 0
b
'1
2
'1
'2
Fig. 3. The branch points and the contours in the complex plane of ϕ
Here and from now on, for f : (z1 , z2 , ...zn ) → f (z1 , z2 , ...zn ) a function of complex variables, f ∗ denotes the function (z1 , z2 , ...zn) → f (z1 , z2 , ..., zn ). The set of a b GL(2, C)-valued functions having the form ∗ ∗ is denoted by M (regardless of b a the variables, these being clear from the context). We introduce another notation that is used throughout the paper; for a : φ → a(φ), a 1-periodic function, we write ˜ a = a0 + a(φ),
(2.15)
where a0 is the zeroth Fourier coefficients of a. Recall that E1 and E2 are the ends of the first spectral interval of the periodic Schrödinger equation (1.3). We assume that E − α ≤ E1 − δ,
E1 + δ < E + α < E2 − δ.
(2.16)
Here and below, δ denotes fixed positive constants independent of ε. This is just a strong version of condition (1.4). In Sect. 8, we prove Theorem 2.2. Pick Y ∈ (Im ϕ2 , Im ϕ3 ). Pick E0 satisfying (2.16). Then, there exists η > 0 and ε0 > 0 such that, for 0 < ε < ε0 , there exists a consistent basis such that M(E, φ), the corresponding monodromy matrix is analytic in V = {|E − E0 | < η} × {|Im φ| ≤ Y /ε}, belongs to M and, in V, its coefficients decomposed according to (2.15) admit the asymptotics 1 i/ε tv e (1 + o(1)), a(φ) ˜ = a(φ, ˜ E) = − e2iπ(φ−φ0 ) (1 + o(1)), th th i tv ˜ ˜ b0 = b0 (E) = ei/ε (1 + o(1)), b(φ) = b(φ, E) = −i e2iπ(φ−φ0 ) (1 + o(1)). th th (2.17) a0 = a0 (E) =
The functions = (E), th = th (E) and tv = tv (E) are the phase and tunneling coefficient defined above. The function φ0 = φ0 (E) is real analytic and satisfies φ0 (E) = O(1). The asymptotics are uniform in (E, φ) in the set V.
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A. Fedotov, F. Klopp
Remark 2.1. The consistent basis solutions are constructed as functions of the variables u = x−φ, ϕ = φ/ε and E. They are analytic in (E, ϕ) ∈ {|E−E0 | < η}×{|Im ϕ| ≤ Y }. The proof of Theorem 2.2 is based on a new asymptotic method developed in [19] to study adiabatic perturbations of periodic Schrödinger operators in dimension 1. This method is based on the analysis of adiabatic asymptotics in the complex plane of ϕ of solutions of equations of the form −ψ (x) + V (x − ϕ, εx)ψ(x) = Eψ(x). In general, this method leads to computations similar to those typical for the standard complex WKB methods (see e.g. [15]). In our case, under condition (2.16), due to some natural symmetries, they remind of the computations made in [6] for the semi-classical analysis Harper equation.
2.8. The spectral results. 2.8.1. The spectrum in the adiabatic limit. We begin with a general observation concerning the location of the spectrum in the adiabatic limit. Recall that, as Hφ,ε is quasiperiodic, its spectrum does not depend on φ (see [1]). Let 6ε denote the spectrum of Hφ,ε . One proves Proposition 2.1. Let 6 = σ (H0 ) + α cos(R) = σ (H0 ) + [−α, α]. Then, one has • ∀ε ≥ 0, 6ε ⊂ 6. • for any K ⊂ 6 compact, there exists a constant C > 0 such that, for ε sufficiently small, one has 1
1
6ε ∩ (E − Cε 2 , E + Cε 2 ) = ∅,
∀E ∈ K.
2.8.2. Location of the spectrum. Let J = J (δ) be the energy interval defined by (2.16). By Lemma 2.1, the function (E) is monotonically increasing on J and its derivative does not vanish there. In J , consider the points (E (l) )l∈N defined by 1 (E (l) ) = π/2 + π l, l ∈ N. ε
(2.18)
The number of these points is of order 1/ε; we denote the minimal and the maximal values of l for which (2.18) admits a solution in J by L1 and L2 . For sufficiently small ε, the distances between the points (E (l) )l∈N satisfy the inequalities c1 ε ≤ E (l) − E (l−1) ≤ c2 ε, l = L1 + 1, . . . , L2 , where c1 and c2 are two positive constant independent of ε. One has Theorem 2.3. There exists a collection of intervals (Il )L1 ≤l≤L2 , Il ⊂ J such that, for ε > 0 sufficiently small, one has • 6ε ∩ J ⊂ ∪L1 ≤l≤L2 Il , • the interval Il lies in an o (ε)-neighborhood of E (l) ,
Anderson Transitions for Almost Periodic Schrödinger Equations
11
• the measure of Il satisfies |Il | = 2
ε(tv (E (l) ) + th (E (l) )) (1 + o(1)). (E (l) )
(2.19)
Moreover, if dNε (E) denotes the density of states measure of Hφ,ε at energy E, then, one has 1 dNε (E) = ε. (2.20) 2π Il Note that the intervals (Il )l are exponentially small and separated by distances of order O(ε). 2.8.3. The nature of the spectrum. Set λ(E) = tv (E)/th (E),
S(E) = ε log λ(E) = Sv (E) − Sh (E).
(2.21)
For δ > 0, define the sets Jδ− = {E ∈ J ; S(E) < −δ} and Jδ+ = {E ∈ J ; S(E) > δ}. If Jδ+ = ∅, then, for sufficiently small ε, the number of intervals Il lying in Jδ+ is of order O(1/ε). We prove Theorem 2.4. Let I ⊆ Jδ+ be an interval, and let λI = exp (− minE∈I S(E) /ε). Pick σ ∈ (0, 1). There exists D ⊂ (0, 1), a set of Diophantine numbers such that • mes (D ∩ (0, ε)) = 1 + O ελσI when ε → 0. ε
(2.22)
• For ε ∈ D sufficiently small, each of the intervals Il ⊂ I contains absolutely continuous spectrum, and for these intervals mes (Il ∩ 6ac ) = 1 + o(1). mes (Il )
(2.23)
Here, 6ac is the absolutely continuous spectrum for the family of Eqs. (1.1). Let us now study the singular spectrum. As before, if Jδ− = ∅, then, for sufficiently small ε, the number of intervals Il lying in Jδ− is of order O(1/ε). We prove Theorem 2.5. For sufficiently small ε, each of the intervals Il ⊂ Jδ− contains only singular spectrum. Moreover, for E ∈ Jδ− , one has 1 (Sh (E) − Sv (E)) + o(ε), 2π where # is the Lyapunov exponent for the family of Eqs. (1.1). #(E) ≥
(2.24)
Remark 2.2. In Theorems 2.4 and 2.5, we have fixed the value of δ independently of ε. The proofs show that one can actually take δ = ε α , 0 < α < 1; in this case, the statements of Theorems 2.4 and 2.5 stay correct except that, in (2.22) and (2.24), the estimates of the error terms have to be modified.
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A. Fedotov, F. Klopp
2.8.4. Heuristics of the proof. Let us outline the basic heuristics guiding the proof of our results. If we omit the factors (1 + o(1)) in (2.17), the monodromy matrix takes the form −ζ −iζ z0 iz0 M = M0 + λM1 , , M1 = λ , M0 = −iz0 z0 iζ −1 −ζ −1 1 i z0 = e ε , ζ = e2iπ(φ−φ0 ) , th where M0 is a matrix with coefficients independent of φ, and M1 is a first order trigonometric polynomial in φ. Now, consider the monodromy equation (2.5) for this matrix. The coefficient λ = λ(E) plays the role of the coupling constant. If λ is small, one expects that the term λM1 can be “omitted” using KAM theory ideas. As a result, in this case, the spectrum should be absolutely continuous. If λ is large, Herman’s idea ([26]) shows that the Lyapunov exponent of the solutions of the monodromy equation is positive. By Corollary 2.1, this implies the positivity of the Lyapunov exponent of the family (1.1). Hence, the spectrum is singular. Finally, since the coefficients of M0 are much bigger than λ, and, since, by (2.3), the monodromy matrix is unimodular,the spectrum has to (E) be located near the zeros of Tr(M0 ), i.e. near the points where cos = 0. ε 2.9. The phase diagram. The transitions, possible in view of Theorems 2.4 and 2.5, actually occur as we shall see now. Therefore, we study the dependence of S(E) = Sv (E) − Sh (E) on E and on α. To underline the dependence on α, let us slightly change our notations and write Sh (α, E) = Sh (E), Sv (α, E) = Sv (E) and S(α, E) = S(E). The potential V is kept fixed. We work in the (α, E) domain S = {(α, E) ∈]0, +∞[×R; E − α ≤ E1 , E1 ≤ E + α ≤ E2 }
(2.25)
(compare with (1.4)). We define S± = {(α, E) ∈ S; S(α, E) ≷ 0}, S0 = {(α, E) ∈ 1 1 S; S(α, E) = 0}, α ∗ := E2 −E and E ∗ := E2 +E 2 2 . The sets S+ , S0 and S− form a partition of S. One has Theorem 2.6. Assume that V satisfy the assumptions described in the introduction. Then, • S0 is a compact analytic curve joining in S the point (α ∗ , E ∗ ) to some point on the half-line E = E1 − α, α > 0. • For any t ∈ [−1, 1], S0 intersects the half-line E = E1 + tα (α > 0) at exactly one point in S. The picture of the phase diagram one gets from Theorem 2.6 is essentially the one given in Fig. 4. Indeed, Theorem 2.6 implies that • S+ is connected and there exists α0 > 0 such that S ∩ {α < α0 } ⊂ S+ . • S− is connected and there exists α1 > α ∗ > 0 such that S ∩ {α > α1 } ⊂ S− . From Theorems 2.4 and 2.6, we deduce that, for α < α0 and for ε sufficiently small, most of the spectrum of Hφ,ε stays absolutely continuous (in the domain S). This can be considered as an analogue of one of the results of [13]; there, it is proved that, for a small d2 quasi-periodic perturbation of − 2 , the spectrum is purely absolutely continuous; here, dx we are dealing with small perturbations of general periodic operators. For α > α1 , by
Anderson Transitions for Almost Periodic Schrödinger Equations
13
E
)
E b
(
E
S+
E2
=
=
E1
E
(
+
S
)
0 b
S0
b
1
b
(
E =
E1 ) Fig. 4. An example of phase diagram
Theorem 2.5, for ε sufficiently small, the spectrum is singular (in the domain S). For α0 < α < α1 , there is at least one mobility edge. At the point (α ∗ , E ∗ ), S0 has the slope dα 1 − ceff = where ceff = dE 1 + ceff
m2 . m1
(2.26)
√ Here, m1,2 are the effective masses defined by |k (Ei + η)| ∼ mi · |η|, i = 1, 2, |η| % 1. By general estimates on the effective mass (see e.g. [32]), one always has ceff < 1. Hence, the picture of the phase diagram that one gets near (α ∗ , E ∗ ) is roughly the same as in Fig. 2. So, we see that, here, the singular spectrum grows with α for α close to α ∗ . Note that if, in (1.1), the cosine is replaced by a more general periodic potential, the dα derivative dE can be negative. In this case, we have singular spectrum “coming from” the “center” of the spectral band of the unperturbed periodic operator.
2.10. Outline of the paper. In the next section, we prove Theorem 2.1 and therefore study the link between solutions of (1.1) and solutions of (2.5). Section 4 is devoted to the proof of the results stated in Sect. 2.8 when assuming Theorem 2.2. Sections 5, 6, 7 and 8 are devoted to the proof of the asymptotics of the monodromy matrix, Theorem 2.2. In Sect. 5, we recall some results on one dimensional periodic Schrödinger operators; in Sect. 6, we describe the adiabatic complex WKB method. Section 7 is devoted to the description of geometrical and analytical objects of the adiabatic complex WKB method when applied to Eq. (1.1). In Sect. 8, we compute the asymptotics of the monodromy matrix. In Sect. 9, we discuss the properties of the phase and action integrals; we also prove the results on the phase diagrams. In Sect. 10, following [40], we adapt Herman’s argument to estimate the Lyapunov exponent in the case when S(E) < 0. And Sect. 11 is devoted to a simple version of KAM needed to study the monodromy equation for small coupling.
14
A. Fedotov, F. Klopp
3. Monodromy Matrices and Monodromy Equation In Sects. 2.1 and 2.1.2, we have introduced the monodromy matrix and the monodromy equation. We now prove Theorem 3.1. Let ψ1,2 be consistent basis solutions of (1.1), and let M be the corresponding monodromy matrix. Fix φ ∈ R. Then, for any solution χ of Eq. (2.5), there exists a unique solution of (1.1), say f , such that f (x + 2π n/ε, φ) (3.1) = A(x, φ − nh) · σ · χ−n , ∀x ∈ R, n ∈ Z, f (x + 2π n/ε, φ) where
A=
ψ1 ψ2
dψ1 dψ2 dx dx
,
σ =
0 −1 , 1 0
h=
2π ε
mod (1).
Reciprocally, for any f solution of (1.1), there exists a unique vector χ solution of (2.5) satisfying (3.1). As the matrix A(x, φ) is periodic in φ and unimodular, Theorem 3.1 immediately implies Theorem 2.1. Proof. Let us recall some elementary facts on difference equations of the form (2.5) where the matrix M is unimodular. Together with vector solutions of (2.5), we also consider matrix solutions, i.e. sequences of 2 × 2-matrices {ϒn }n∈Z such that ϒn+1 = M(φ + nh)ϒn . A matrix solution ϒ of Eq. (2.5) is called fundamental if and only if det ϒn ≡ 1 for all n ∈ Z. To construct a fundamental solution, we let ϒ0 (φ) = I , ∀φ ∈ R, and then, define ϒn for all n just by means of the equation ϒn+1 (φ) = M (φ + nh)ϒn (φ),
ϒn−1 (φ) = M −1 (φ + h(n − 1))ϒn (φ).
(3.2)
This is possible as det M = 0. In result, one obtains a matrix solution of the difference equation (2.5). Since det ϒn+1 (φ) = det M(φ + nh) det ϒn (φ) = det ϒn (φ),
∀n, φ,
this matrix solution is fundamental. Any vector solution of the monodromy equation, say χ , can be represented as χn = ϒn (φ) p,
∀n,
(3.3)
where p is a constant vector. Moreover, for any constant vector p, formula (3.3) gives a vector solution of the monodromy equation. Now, we prove Theorem 3.1. The proof consists of several steps. 1. Let (ψ1 , ψ2 ) be a consistent basis solution of (1.1) and define ψ = (ψ1 , ψ2 )T . Let ϒ(φ) be a fundamental solution of Eq. (2.5). Assuming that n ≥ 0, we compute 2π 2π n ψ (x + 2π n/ε, φ) = M φ − ···M φ − ψ (x, φ − 2π n/ε), ε ε 2π 2π n I = ϒ0 (φ) = M φ − ···M φ − ϒ−n (φ). ε ε
Anderson Transitions for Almost Periodic Schrödinger Equations
15
Hence, as φ → ψ(x, φ) is 1-periodic, we get −1 ψ (x + 2π n/ε, φ) = ϒ−n (φ)ψ (x, φ − nh).
(3.4)
2. Let A be the matrix defined in Theorem 3.1. As detϒl ≡ 1, one has (ϒl−1 )T = −σ ϒl σ ; then, formula (3.4) implies the relation A(x + 2π n/ε, φ) = −A(x, φ − nh) · σ · ϒ−n (φ) · σ.
(3.5)
We have obtained this formula for n ≥ 0; one checks that it remains correct for n < 0. 3. Let χ be a vector solution of (2.5) for a given φ. Represent it in the form (3.3). Denote the components of the vector p in this representation by p1 and p2 . The function f (x, φ) = −p2 ψ1 (x, φ) + p1 ψ2 (x, φ)
(3.6)
satisfies Eq. (1.1). Applying the left- and the right-hand sides of (3.5) to the vector −σp, we get the relation (3.1). This proves the first statement of Theorem 3.1. 4. To prove the second statement, we represent the solution f as a linear combination of the linearly independent solutions ψ1 (x, φ) and ψ2 (x, φ) as in (3.6). Then, we let p be the vector (p1 , p2 )T and remark that χn = ϒn (φ0 )p is a solution of (2.5). Applying (3.5) to the vector −σp, we get (3.1). This completes the proof of Theorem 3.1. & ' 4. The Spectral Study This section is devoted to the proofs of Proposition 2.1, and Theorems 2.3, 2.4 and 2.5 using the asymptotics of the monodromy matrix obtained in Theorem 2.2. 4.1. Convergence to the asymptotic spectrum. We prove Proposition 2.1. The first statement of Proposition 2.1 follows immediately from regular perturbation theory (see e.g. [3, 38]). We turn to the proof of the second statement. The spectrum of Hφ,ε is the same as the d2 spectrum of equation − du 2 ψ(u) + (V (u) + α cos(εu + ϕ))ψ(u) = Eψ(u), ϕ = εφ. Pick ν ∈ (0, 1) and K ⊂ 6 a compact set. Pick E ∈ K. Then, there exists ϕ ∈ [0, 2π ) and E0 ∈ σ (H0 ) such that E = E0 + α cos(ϕ). Hence, equation H0 u = E0 u has a bounded (Bloch) solution, say u, continuous in (x, E0 ) ∈ R × σ (H0 ) together with its derivative in x, see, for example, [41]. Pick χ ∈ C0∞ (R) non negative and not identically vanishing. And put χε (·) = εν/2 χ (ε ν ·). One sets uε = χε u and checks that 2 − d + V − E0 uε ≤ Cεν . 2 2 dx L The constant C is uniform in K. On the other hand, we compute [cos(ε · +ϕ) − cos(ϕ)] uε L2 ≤ Cε 1−ν . So, we obtain u˜ ∈ L2 (R), u ˜ = 1 such that − u˜ + (V + α cos(ε · +ϕ) − E)u ˜ ≤ ν 1−ν C(ε + ε ). Hence, either E ∈ σ (Hφ,ε ) or (Hφ,ε − E)−1 ≥ C/(ε ν + ε 1−ν ). This implies the second statement of Proposition 2.1.
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A. Fedotov, F. Klopp
4.2. Location of the spectrum. Here, we apply Theorem 2.2 to deduce the results of Theorem 2.3 on the location of the spectrum. We proceed as follows. First, we describe conditions guaranteeing that the monodromy equation has two linearly independent solutions rapidly decaying at infinity. Then, we use Theorem 3.1 to construct the corresponding solutions to Eq. (1.1). This describes the complement of the spectrum of (1.1). 4.2.1. Bloch solutions of difference equations. We need some results about solutions of difference equation of the form χ (φ + h) = M (φ) χ (φ),
φ ∈ R,
(4.1)
where h is a positive number and M (φ) ∈ SL (2, C) is a given 1-periodic, matrix-valued function. The results we formulate below can be found in [7]. The set of the vector solutions of (4.1) is a two-dimensional module over the ring of h-periodic functions. For any two vector solutions χ1 and χ2 of (4.1), the determinant of the matrix (χ1 , χ2 ) is h-periodic. The solutions χ1 and χ2 are linearly independent (over the ring of h-periodic functions) if and only if this determinant does not vanish. If χ1 and χ2 are linearly independent solutions, then, any other vector solution χ can be represented as a linear combination of χ1 and χ2 with h-periodic coefficients χ (φ) = a(φ)χ1 (φ) + b(φ)χ2 (φ),
a(φ + h) = a(φ),
b(φ + h) = b(φ),
φ ∈ R.
If φ → χ (φ) is a solution of (4.1), so does φ → χ (φ + 1). Assume, moreover, that χ satisfies χ (φ + 1) = u(φ)χ (φ) for φ ∈ R, where φ → u(φ) is an h-periodic function, then χ is called a Bloch solution, and the function u is its Floquet multiplier. If either |u(φ)| ≥ Const > 1 for all φ ∈ R or |u(φ)| ≤ Const < 1 for all φ ∈ R, the Bloch solution is called monotonous. Remark 4.1. If |u(φ)| ≤ C < 1, then, the solution χ decays exponentially as φ → +∞. Indeed, one has |χ (φ + L)| =
L−1
|u(φ + l)| |χ (φ)| ≤ C L |χ (φ)|.
l=0
If, in addition, χ is bounded on the interval [0, 1[, then |χ (φ)| ≤ C1 e−C2 ·φ with some positive constants C1 and C2 . Similarly, if |u(φ)| ≥ C > 1, then χ decays exponentially as φ → −∞. We finish this section by formulating a simple condition for the existence of the monotonous Bloch solutions. Let M = ((Mij ))1≤i,j ≤2 and set ρ (φ) =
M12 (φ) , M12 (φ − h)
v (φ) = M11 (φ) + ρ (φ) M22 (φ − h).
(4.2)
We also define ρ− = inf |ρ (φ)|, φ∈R
ρ+ = sup |ρ (φ)|, φ∈R
v− = inf |v (φ)|, φ∈R
v+ = sup |v(φ)|. φ∈R
(4.3)
Anderson Transitions for Almost Periodic Schrödinger Equations
17
For f : R → C a periodic continuous non-vanishing function on R, let the index of f , arg f over one period. Clearly, indf is ind f , be the integer equal to the increment of 2π independent of the period chosen to define it. One has Proposition 4.1 ([7]). Assume that M is continuous in φ, and let M12 = 0. If v 2 − , ρ− > 0, ρ+ < 2 ind v = ind ρ = 0,
(4.4) (4.5)
then, Eq. (4.1) has two monotonous Bloch solutions χ ± that satisfy det(χ + , χ − ) = 1 for all φ ∈ R, that are continuous in φ and such that the Floquet multipliers u± of χ ± satisfy the relation u+ (φ)u− (φ) = 1. Remark 4.2. As det(χ + , χ − ) = 1, these solutions are linearly independent. As u+ u− = 1, one of them decays at +∞, and the other one decays at −∞. 4.2.2. Monotonous Bloch solutions and the spectrum of (1.1). Fix an energy E. Let ψ1 and ψ2 form a consistent basis for the family of Eqs. (1.1). Let M(φ) be the corresponding monodromy matrix. Here, instead of the monodromy Eq. (2.5), we consider its continuous analog (4.1). Clearly, if χ (φ) is a vector solution of this equation, then, the vectors χn = χ (φ + nh),
n ∈ Z,
(4.6)
satisfy the monodromy equation (2.5). This and Theorem 2.1 imply Lemma 4.1. Let ψ1,2 be locally bounded in (x, φ) ∈ R2 together with their derivatives in x. If, for a given E, Eq. (4.1) has two linearly independent, locally bounded, monotonous Bloch solutions χ ± , then E is in the resolvent set of Hφ,ε for any φ ∈ R. Proof. Fix φ ∈ R. By formula (4.6), using χ ± , construct two vector solutions χn± of the monodromy equation (2.5). By Theorem 3.1, construct f ± , two solutions of (1.1) satisfying the relation (3.1). Compute the Wronskian of f ± . By (3.1), one has + − w(f + , f − ) = det A(x, φ − nh) det σ det χ−n , χ−n = det χ + (φ − nh), χ − (φ − nh) . As χ ± (x) are linearly independent, the last determinant does not vanish, and f ± are linearly independent solutions of (1.1). As χ ± (x) are monotonous and locally bounded, each of them exponentially decays either at +∞ or at −∞. As they are linearly independent, det(χ + , χ − ) is a nonzero h-periodic function, and, therefore, one of them, say χ + , is exponentially decaying at +∞ and the second one, χ − , is exponentially decaying at −∞. So, we write χ ± (φ) ≤ C1 e∓C2 φ (C1 , C2 > 0). This, and relations (4.6) and (3.1) imply that
hε ±C4 x ± |f (x)| ≤ C3 e , C3 = C1 · sup A(x, φ) , C4 = C2 . 2π x∈[0,2π/ε],φ∈[0,1] (4.7) Here, we have used the fact that ψ1,2 are 1-periodic in φ and locally bounded in (x, φ) together with their derivatives. As (1.1) has two linearly independent solutions f ± satisfying estimates of the form (4.7), the energy E is in the resolvent set of Hφ,ε (see e.g. [9, 41]). & '
18
A. Fedotov, F. Klopp
4.2.3. The functions ρ and v . To study the functions ρ and v defined in (4.3) for the monodromy matrix described in Theorem 2.2, we discuss the coefficients a and b of the monodromy matrix M. Recall that they are 1-periodic and that we have decomposed ˜ them as a = a0 + a(φ) ˜ and b = b0 + b(φ), where a0 and b0 are the zeroth Fourier coefficients of a and b. The asymptotics of a0 , a, ˜ b0 and b˜ are given in Theorem 2.2. One proves Lemma 4.2. For E in a constant neighborhood of E0 , for φ in a constant neighborhood of R and for ε sufficiently small, one has (4.8) ρ(φ) = 1 − e−i/ε tv U (φ) (1 − e−2πih + o (1)), · p(φ, E) + o (λe2π|Im φ0 | ), v(φ) = F (E) − 2λ cos(2π(φ − φ0 − h)) − λ cos ε (4.9) p(φ, E) = 2e−i/ε U (φ)(1 − e−2πih + o (1)),
(4.10)
where F = a0 + a0∗ ,
U (φ) = e2πi(φ−φ0 ) ,
λ = tv /th .
The asymptotics in (4.8), (4.9) and (4.10) are uniform, and the error terms are analytic in E. Remark 4.3. In Lemma 4.2, we used the terminology “constant neighborhood”; here and in the sequel, this means a neighborhood independent of ε. Proof. Using (2.17), we get ˜ 0 = −tv e−i(E)/ε U (φ)(1 + o(1)). b/b Note that this ratio is small in a constant neighborhood of E0 , and so, in this neighborhood, we can write
˜ ˜ − h) ˜ ˜ − h) ˜ b(φ) b(φ b(φ) b(φ 1 + b(φ)/b 0 +o +o . =1+ − ρ= ˜ − h)/b0 b0 b0 b0 b0 1 + b(φ ˜ Substituting into this representation the asymptotics of b(φ)/b 0 , one obtains the asymptotics of ρ. The analyticity in E of the error term follows from the analyticity of the error terms in the monodromy matrix asymptotics. This completes the proof of (4.8). Recall a(φ) ˜ = a(φ) − a0 . As v(φ) = a(φ) + ρ(φ) a ∗ (φ − h), one has v(φ) = F + a(φ) ˜ + (ρ(φ) − 1) a ∗ (φ − h) + a˜ ∗ (φ − h). Using (4.8) and the asymptotics (2.17), we obtain (4.9)–(4.10). This completes the proof of Lemma 4.2. & ' The function F (E) = a0 (E) + a0∗ (E) plays an important role in the spectral analysis of Eq. (1.1). We call it the effective spectral parameter. One has Lemma 4.3. The function F is real analytic in a constant neighborhood of E0 and, in this neighborhood, it admits the following uniform asymptotic representation 1 1 + g1 (E) (E) + g2 (E) , g1,2 = o(1), (4.11) cos F (E) = 2 ε th (E) (E), g (E) = o(1). where g1 and g2 are real analytic, g1,2 1,2
Anderson Transitions for Almost Periodic Schrödinger Equations
19 i
Proof. The function F is real analytic. Theorem 2.2 implies that a0 = e th/ε eg(E) , g being analytic and satisfying the estimate g = o(1) in a constant neighborhood of E0 . Let ¯ g˜ 2 (E) = 1 (g(E)−g(E)). ¯ These two functions are real anag˜ 1 (E) = 21 (g(E)+g(E)), 2i g˜ 1 (E) lytic and admit the estimates g˜ 1,2 = o(1). Clearly, F (E) = 2 eth (E) cos 1ε (E) + g˜ 2 (E) . This implies (4.11). The estimates on the derivatives of g1,2 follow, for example, from the Cauchy estimates for analytic functions. & ' 4.2.4. Location of the spectrum up to exponentially small errors. Now, we are ready to prove the part of Theorem 2.3 concerning the geometry of the spectrum. Our plan is to describe the set R of values of the spectral parameter E for which the monodromy matrix described in Theorem 2.2 satisfies the hypothesis of Proposition 4.1. By Lemma 4.1, the spectrum of Hφ,ε lies in the complement of R. Let V0 be the constant neighborhood of E0 where (2.17) and Lemmas 4.2 and 4.3 are valid. We study the location of the spectrum of (1.1) in J0 = V0 ∩ J . The asymptotics of ρ, v and F imply that the monodromy matrix described by Theorem 2.2 satisfies the assumptions of Proposition 4.1 if cos 1 + g2 (E) ≥ Const(th + tv ), (4.12) ε where g2 is the function from (4.11), and Const is independent of ε. Hence, the values of E satisfying this condition are outside of the spectrum of Hφ,ε . By point (4) of Lemma 2.1, one has 1 C (E) + g2 ≥ , ε ε
E ∈ J0 .
(4.13)
As the tunneling coefficients are exponentially small, inequality (4.13) implies that (4.12) is satisfied outside exponentially small intervals. are located in o (ε) These intervals neighborhoods of the points E (l) defined by cos 1ε (E (l) ) = 0; the distances between these points are of order ε. Near each E (l) , there is exactly one of these intervals. We call these intervals (Il )l∈L . Let us discuss the intervals (Il )l∈L in more detail. Fix an index l ∈ L. Recalling the definitions of tv (E) and th (E), one sees that, up to a factor 1 + o (1), these functions are constant as on Il so in a o(ε)-neighborhood of E (l) . For E ∈ Il , by Lemmas 4.2 and 4.3, we have ρ = 1 + O(tv (E (l) )), v = F (E) − 2λ(E (l) ) cos(2π(φ − φ0 (E (l) ) − h)) + o(λ(E (l) )), 1 + o(1) 1 + g F (E) = 2 cos 2 . ε th (E (l) ) Now we can define the intervals (Il )l∈L more precisely; instead of (4.12), the interval Il situated in the o(ε)-vicinity of E (l) can be described by cos 1 + g2 (E) ≤ (th (E (l) ) + tv (E (l) ))(1 + o(1)), ε
20
A. Fedotov, F. Klopp
where o(1) only depends on ε. This implies that the length of the subinterval situated near E (l) is given by (2.19). Note that any interval Il contains precisely one zero of F (E). We have thus proved the statements of Theorem 2.3 on the location of the spectrum in a constant interval J0 ⊂ R containing E0 . As E0 can be any point in J , this completes the proof of Theorem 2.3 up to the result on the integrated density of states. 4.3. Calculation of the integrated density of states. For ε/π ∈ Q, (1.1) is a metrically transitive family of equations (see e.g. [37]). Recall that 6 is the spectrum of Hφ,ε . 4.3.1. Weyl solutions, Lyapunov exponents and the integrated density of states. We begin with recalling general results on the Lyapunov exponent (see [37, 39]). Consider the family of quasi-periodic equations (1.1). For any fixed value of the spectral parameter E such that Im E ≥ 0, E ∈ / 6, there is a unique solution to (1.1) such that u+ (x) ∈ 2 + L (0, +∞) and u (0) = 1; we call it the Weyl solution of (1.1). For almost all φ ∈ R, the Weyl solution satisfies the estimate |u+ (x, E, φ)|2 + |du+ (x, E, φ)/dx|2 = e−x(#(E)+o (1)) , x → +∞, (4.14) where #(E) is positive, independent of φ and x. #(E) is the Lyapunov exponent of the family (1.1). The Lyapunov exponent is the real part of a function f : E → f (E) analytic in the upper half-plane Im E ≥ 0, E ∈ / 6 and such that the integrated density of states is equal to the limit N (E) = −
1 lim Im f (E + iα), π α→+0
E ∈ R.
4.3.2. More on Bloch solutions of difference equations. Here, following [7], we discuss in more detail the monotonous Bloch solutions described in Proposition 4.1. By Remarks 4.1 and 4.2, in the case of Proposition 4.1, Eq. (4.1) has a continuous Bloch solution χ (φ) satisfying the estimate χ (φ) ≤ C1 eC2 ·φ (C1,2 > 0). This solution can be constructed in the following way. Together with (4.1), consider the finite difference Ricatti equation G(φ) +
ρ = v(φ), G(φ − h)
φ ∈ R,
(4.15)
where v and ρ defined by (4.2). Under condition (4.4), Eq. (4.15) has a continuous, 1-periodic solution that can be represented by the continued fraction G(φ) = v (φ) −
ρ(φ) . ρ(φ − h) v (φ − h) − ρ(φ − 2h) v (φ − 2h) − ···
This continued fraction converges uniformly in φ ∈ R. It satisfies
v− 2 − ρ+ , φ ∈ R, |G(φ) − v(φ)| < v− /2 − 2
(4.16)
(4.17)
Anderson Transitions for Almost Periodic Schrödinger Equations
21
where v− and ρ+ defined by (4.3). For φ ∈ R, define g(φ) = ln G(φ). Condition (4.5) guarantees that the function g can be defined as a 1-periodic continuous function of φ. If v depends analytically on some parameter E in a simply connected domain, then g also is analytic in E. Consider a continuous solution (not necessarily periodic) of the homological equation λ(φ + h) − λ(φ) = g(φ). For any irrational h, one has φ λ(φ) = (θ˜ + o (1)) , h
φ → +∞, where θ˜ =
1
ln G(φ) dφ, Re θ˜ > 0.
(4.18)
0
The function χ1 (φ) = eλ(φ) is the first component of the Bloch solution χ mentioned at the beginning of this section. Note that one has G(φ) =
χ1 (φ + h) . χ1 (φ)
(4.19)
4.3.3. Calculation of the integrated density of states. Let us now come back to Eq. (1.1). 1. We start with a description of the relation between the integrated density of states and the continued fraction G. Let J be a domain in the half plane Im E ≥ 0, J ∩ 6 = ∅. Assume that, on J, one can construct a consistent basis such that the corresponding monodromy matrix is analytic in E ∈ J and satisfies the conditions of Proposition 4.1. Then, for all E ∈ J, one can construct the Bloch solution χ of (4.1) described in Sect. 4.3.2. The vector χ (φ + nh) satisfies the monodromy equation. Clearly, by Theorem 2.1, this solution corresponds to the Weyl solution u+ of (1.1); the Lyapunov exponent # of (1.1) can be calculated by formula (2.7) with θ = Re θ˜ , θ˜ being defined by (4.18). This leads to the formula 1 ε ln G(φ, E) dφ, E ∈ J, (4.20) #(E) = Re 2π 0 where G is the continued fraction (4.16). Since the Lyapunov exponent # is the real part of the analytic function f (E), and ln G is analytic in E ∈ J, one has 1 ε f (E) = ln G(φ, E) dφ + C, E ∈ J, 2π 0 where C is a constant independent of E. Therefore, if γ is a continuous curve in J connecting a ∈ R to b ∈ R, then one has 1 ε N (b) − N (a) = − arg G(φ, E) dφ . (4.21) 2π 2 0 γ Here f |γ denotes the increment of f when going from a to b along γ . We use formula (4.21) to compute the increments of the integrated density of states along the intervals Il , l = L1 , . . . L2 , and, thus, prove (2.20).
22
A. Fedotov, F. Klopp
2. Let us prove Theorem 2.3. Consider one of the intervals Il , L1 ≤ l ≤ L2 . Let E∗ be the zero of F (E) closest to E (l) . By Sect. 4.2.4, we know that E∗ − E (l) = o (ε),
(4.22)
that the interval Il contains E∗ , and that the length of Il is exponentially small, |Il | ≤ e−C/ε . We fix two constants c1 and c2 so that 1 < c2 < c1 and choose J = E ∈ C | Im E ≥ 0, εc1 < |E − E∗ | < εc2 . As Il is exponentially small, one has J ∩ Il = ∅. Moreover, all the other intervals containing some spectrum of Hφ,ε are either to the right or to the left of J. So, J is in the resolvent set of (1.1). Pick a and b in J ∩ R, a < E∗ < b and let γ be the semi-circle connecting a and b in J. To prove the statement of Theorem 2.3 on the integrated density of states, it suffices to check the formula N (b) − N (a) =
ε . 2π
(4.23)
To prove (4.23), we first check Lemma 4.4. For E ∈ J, the monodromy matrix described in Theorem 2.2 satisfies the conditions of Proposition 4.1. So, we can calculate N (b)−N (a) using (4.21). To compute the right-hand side of (4.21), we use Proposition 4.2. For the monodromy matrix from Theorem 2.2, one has
1
arg G(φ, E)dφ = arg F (E)|γ .
(4.24)
γ
0
As F is real analytic, one has arg F (E)|γ ∈ π Z. To get the actual value of arg F (E)|γ , we use Lemma 4.5. Fix c > 0. Then, for ε sufficiently small, one has • (E∗ ) (1 + o(1)), εth (E∗ )
(4.25)
1 , uniformly for |E − E ∗ | ≤ εc . ε 2 th (E∗ )
(4.26)
F (E∗ ) = ±2
•
where the sign ± depends only on Il ; F (E) = O
Anderson Transitions for Almost Periodic Schrödinger Equations
23
By Lemma 2.8, one has ≥ Const > 0 on J ; hence, Lemma 4.5 implies that, on γ , F (E) = F (E∗ )(E − E∗ )(1 + o(1)). Therefore, arg F (E)|γ = −π + o(1). As arg F (E)|γ ∈ π Z, one has arg F (E)|γ = −π . This, (4.21) and Proposition 4.2 imply (4.23). This completes the proof of Theorem 2.3. & ' 3. We now prove Lemmas 4.4, 4.5 and Proposition 4.2. Proof of Lemma 4.5. Formulas (4.25) and (4.26) follow from the representation (4.11). Clearly, for E = E∗ , one has cos(/ε + g2 ) = 0, and sin(/ε + g2 ) = ±1. Therefore, 1 , and as, by Lemma 2.1, (E∗ ) > 0, we one has F (E∗ ) = ±2 1+g th ( /ε + g2 ) E=E∗
get (4.25). To estimate F , one uses Lemma 4.3 and the estimates (E), (E) = O(1), th (E) = O(th /ε) and th (E) = O(th /ε 2 ) which follow from the definitions of the phase integral and the tunneling coefficients. & ' Proof of Lemma 4.4. Therefore, we need the following immediate corollary of Lemma 4.5. Corollary 4.1. There exists C > 0 such that, for ε sufficiently small, 1 ε c1 −1 εc2 −1 ≤ |F | ≤ C , E ∈ J. C th (E∗ ) th (E∗ )
(4.27)
Now, let us check that, for E ∈ J, the monodromy matrix described in Theorem 2.2 satisfies the assumptions of Proposition 4.1. By Lemma 4.2, uniformly in E ∈ J, one has ρ = 1 + O(tv (E∗ )),
v = F (E) + O(λ(E∗ )).
(4.28)
The last formula for v and Corollary 4.1 imply that v−F = O tv (E∗ )ε 1−c1 , F
E ∈ J.
(4.29)
So, as tv is exponentially small, (v − F )/F is small. We check the assumptions of Proposition 4.1: • by (4.28), the index of the periodic function ρ is zero; • by (4.29), one has ind v = ind (F [1 + (v − F )/F ]) = ind (1 + (v − F )/F ) = 0; here, we have used the fact that F is independent of φ; • by (4.28), one has ρ± = 1 + O(tv (E∗ )),
E ∈ J;
(4.30)
by (4.29) and Corollary 4.1, one has v− = |F | min |1 + (v − F )/F | ≥ C φ∈R
therefore, 0 < ρ− and ρ+ <
v− 2 2
ε c1 −1 , th (E∗ )
E ∈ J;
(4.31)
.
So, the assumptions of Proposition 4.1 are fulfilled, and, thus, the solution χ described in Sect. 4.3.2 exists for all E ∈ J. & '
24
A. Fedotov, F. Klopp
Proof of Proposition 4.2. The proof is done in two steps: (1) we check that G/F = 1 + o(1) in J; hence, one can choose a continuous branch of ln G/F so that ln G/F = o(1) in J; 1 (2) for this branch, we prove that 0 ln[G(φ, E)/F (E)]dφ is real for E ∈ J ∩ R. This obviously implies Proposition 4.2. To check step 1, we write G/F = (1 + (v − F )/F ) · (1 + (G − v)/v). We have already seen that (v − F )/F = o(1) in J. The estimate (G − v)/v = o(1) follows from (4.17) and (4.30)–(4.31). This implies point 1. To check step 2, we use the self-adjointness of (1.1). It implies a relation between G(φ, E) and G∗ (φ, E) that we now derive. ∗ Note that,as the monodromy matrix is of the form (2.14), one has σ M σ = M, 01 where σ = . Therefore, if χ is a solution of (4.1), so is σ χ ∗ . Recall that the 10 determinant of the matrix composed of two vector solutions of (4.1) is h-periodic (see Sect. 4.2.1). As both χ and σ χ ∗ decay as φ → −∞, the determinant det (χ (φ), σ χ ∗ (φ)) must vanish identically. Therefore, χ and σ χ ∗ are linearly dependent over the ring of h-periodic functions (see Sect. 4.2.1), i.e. there exists γ , an h-periodic function, such that χ (φ) = γ (φ) σ χ ∗ (φ). Denote the components of the vector χ by χ1 and χ2 . Then, one has χ1 (φ + h)/χ1 (φ) = χ2∗ (φ + h)/χ2∗ (φ).
(4.32)
Moreover, as χ satisfies Eq. (4.1), its first component is related to its second component by the formula χ2 (φ) = (χ1 (φ + h) − a(φ)χ1 (φ))/b(φ), where a and b are the coefficients of the monodromy matrix M. This and (4.19) imply that b(φ) G(φ + h) − a(φ + h) χ2 (φ + h)/χ2 (φ) = G(φ). b(φ + h) G(φ) − a(φ) Substituting this formula and formula (4.19) into (4.32) and applying the operation ∗ , we get the relation G∗ (φ) = G(φ)
b(φ) G(φ + h) − a(φ + h) . b(φ + h) G(φ) − a(φ)
(4.33)
This relation implies point (2). Indeed, as F is real analytic, we can rewrite (4.33) as ∗ G G K(φ + h) 1 − (G(φ) − a(φ))/a ˜ 0 = . (4.34) , K(φ) = ˜ F F K(φ) 1 + b(φ)/b0 Here, the notations a0 , b0 a, ˜ and b˜ are as in (2.15). It is easy to see that K(φ) = 1 + o(1). ˜ Indeed, from the asymptotics of a and b, we get b(φ)/b ˜ 0 = o(1) and a(φ)/a 0 = o(1). Furthermore, from Corollary 4.1, using the asymptotics of a0 , we obtain F /a0 = o(1). As by step 1, G/F = 1 + o(1), this implies that G/a0 = G/F · F /a0 = o(1). Pluging this into the formula for K, see (4.34), we get K(φ) = 1 + o(1). For the branch of ln such that ln 1 = 0, one has ln(G/F )∗ = ln(G/F ) + ln(K(φ + h))−ln K(φ). Integrating this relation in φ from zero to one and taking into account the 1periodicity of ln K, we complete step 2. This also completes the proof of Proposition 4.2. ' &
Anderson Transitions for Almost Periodic Schrödinger Equations
25
4.4. Absolutely continuous spectrum. We now prove Theorem 2.4. Therefore, we assume that the energy E belongs to an interval Il where λ = tv /th is small. We consider Eq. (4.1) where the monodromy matrix M is described in Theorem 2.2. The aim is to put this equation into the form required to apply Proposition 11.1. We proceed as follows. We write the monodromy matrix as a b a˜ b˜ M = M0 + M1 (φ), M0 = 0∗ 0∗ , M1 = ˜ ∗ ∗ , (4.35) b 0 a0 b a˜ where a0 and b0 are zeroth Fourier coefficients of a(φ) and b(φ). The matrix M0 is constant; for φ ∈ R and E ∈ Il , the matrix M1 is of size order λ, see Theorem 2.2. Thus, it is small. We show that, on an essential part of Il , M0 can be diagonalized, M0 = P DP −1 , where P is a constant matrix, and D is a diagonal one. We represent a solution of (4.1) in the form ψ = P A. Then, A satisfies the equation A(φ + h) = (D + λ∗ A)A(φ),
A=
1 −1 P M1 (φ)P λ∗
(4.36)
with λ∗ being a value of λ(E) on Il . We shall check that A is roughly of order O(1). By means of Proposition 11.1, we construct bounded solutions of (4.36); we then use Corollary 2.1 and Ishii-Pastur-Kotani’s Theorem (see e.g. [37, 8, 10]) to complete the proof of Theorem 2.4. Let E0 ∈ Jδ+ , so that λ(E) is small in V0 , a constant neighborhood of E0 . We consider Il ⊂ V0 . Recall that E∗ is the unique zero of the function F (E) located in Il and let λ∗ = λ(E∗ ). 4.4.1. Diagonalization of the monodromy matrix. To use Proposition 11.1, we need to 01 ∗ ∗ . check that D and A belong to M, i.e. σ D σ = D and σ A σ = A, where σ = 10 Here, for matrices M, the mapping M → M ∗ is defined in the same way as for functions (see Sect. 2.7). As M0 and M1 belong to M, it suffices to choose P ∈ M. Assume that ν is an eigenvalue of M0 , and let p be a corresponding eigenvector. Then, σp ∗ is an eigenvector of M0 associated to the eigenvalue ν ∗ . The matrix P = (p, σp ∗ ) belongs to M. We take b0 p= . (4.37) ν − a0 With this choice of p, one checks that det P = −(a0 − ν)(ν − ν ∗ ),
(4.38)
and, if det P = 0, then, the matrix M0 is diagonalizable , and D = Diag (ν, ν ∗ ), A = A1 + A2 , (4.39) 1 1 A1 = Pˇ0 M1 P0 , A2 = −ϒM1 P0 + Pˇ0 M1 ϒ − ϒM1 ϒ , λ∗ det P λ∗ det P (4.40)
26
A. Fedotov, F. Klopp
where b0 −a0∗ P0 = , −a0 b0∗
∗ ∗ b a ˇ P0 = 0 0 , a0 b0
and
0 ν∗ ϒ= . ν 0
4.4.2. The eigenvalues of M0 . Clearly, one has ν 2 − tr M0 ν + det M0 = 0. The trace of M0 is equal to F (E). Let d(E) be the determinant of M0 . Then, one has Lemma 4.6. The function d(E) is real analytic in V , a constant neighborhood of E0 ; moreover, for E ∈ V , one has d(E) = 1 + o(λ(E)2 ),
d (E) = o(λ(E)2 /ε) and d (E) = o(λ(E)2 /ε 2 ). (4.41)
Proof. The real analyticity of d is obvious. As det M ≡ 1, the representation (4.35) implies that d(E) + a˜ a˜ ∗ − b˜ b˜ ∗ = 1, 0
where [f ]0 denotes the zeroth Fourier coefficient of the periodic function f . By (2.17), a˜ ∼ −λU , and b˜ ∼ −iλU . This implies the asymptotics of d. To estimate the first and the second derivatives of d, we note that the asymptotics of d can be rewritten in the form d(E) = 1 + o(1)λ(E)2 , where o(1) is analytic in E. The Cauchy estimates d 2 o(1) imply that, in a (smaller) constant neighborhood of E0 , one has do(1) dE , dE 2 = o(1). This implies (4.41), and completes the proof of Lemma 4.6. & ' To study the eigenvalues of M0 , we use (E) Lemma 4.7. Let f (E) = √Fd(E) , where the branch of the square root is fixed by the √ condition d(E∗ ) ∼ 1. Fix c > 1 and C > 0. One has
(1) for |E − E∗ | ≤ εc , one has f (E) = F (E∗ )(1 + o(1)) when ε → 0; (2) for |w| ≤ C, the equation f (E) = w has only one solution in the domain |E −E∗ | < εc ; w (1 + o(1)) when ε → 0. (3) this solution satisfies E − E∗ = F (E ∗) Proof. Point 3 of Lemma 2.1, Lemmas 4.5 and 4.6 imply that 1 f (E) = F (E∗ )(1 + o(1)) and f (E) = O for |E − E∗ | ≤ εc . (4.42) ε 2 th (E∗ ) The first equality in (4.42) is just point (1) of Lemma 4.7. Assume that |E − E∗ | = εc and |w| ≤ C. Using (4.42) and (4.25), we obtain f (E) − f (E∗ )(E − E∗ ) = O ε c−1 . f (E )(E − E ) − w ∗ ∗ Applying Rouché’s Theorem to the functions f (E) − w and f (E∗ )(E − E∗ ) − w, we see that, in the disc |E − E∗ | ≤ ε c , the function f (E) − w has a single root and that this root is simple. This proves point 2 of Lemma 4.7.
Anderson Transitions for Almost Periodic Schrödinger Equations
27
To get the asymptotics of the root of f (E) − w in the disc |E − E∗ | ≤ εc , we let w Ew = E∗ + f (E and pick α ∈ (0, 1). On the circle |E − Ew | = α|Ew − E∗ |, one has ∗) f (E) − f (E∗ )(E − E∗ ) max|E−E∗ |≤εc |f (E)| |E − E∗ |2 C |E − E∗ |2 ≤ ≤ f (E )(E − E ) − w 2|f (E∗ )| |E − Ew | ε |E − Ew | ∗ ∗ C(1 + α)2 C(1 + α)2 w ≤ |Ew − E∗ | = f (E ) εα εα ∗
th (E∗ ) ≤C . α
Here, C denotes constants that are independent of ε and α. We have used (4.42), (4.25), and Lemma 2.1. If th (E∗ ) = o(α), then, the right-hand side is small; so, by Rouché’s Theorem, the root of f (E) − w in the disc |E − E∗ | ≤ εc is inside the disc |E − Ew | ≤ α|Ew − E∗ |. This gives the following estimate for this root E − E∗ − w = |E − Ew | ≤ α|Ew − E∗ | = α w . f (E∗ ) f (E∗ ) Now, as α can be taken, say, of order of ε, this implies point 3 of Lemma 4.7, and completes the proof of Lemma 4.7. & ' The eigenvalues of M0 are equal to 21 F (E) ±
√
G(E), where G(E) =
F (E) 2
2
−
d(E). As we can rewrite G(E) = d(E) 4 (f (E) − 2)(f (E) + 2), Lemma 4.7 implies that • on the interval [E∗ − ε c , E∗ + ε c ], the function G(E) has two simple roots E∗,± described by the equations f (E∗,± ) = ±2; for the sake of definiteness, we set E∗,− < E∗,+ . • E∗,± admit the asymptotics E∗,± = E∗ ±
2 (1 + o(1)); |F (E∗ )|
(4.43)
• in the interior of I∗ := [E∗,− , E∗,+ ], the function G is negative; • on [E∗ − ε c , E∗ + ε c ], outside of I∗ , the function G is positive. Let us discuss the eigenvalues of M0 on the interval I∗ . The third property of G implies that, in I∗ , they are complex and complex conjugate of each other. So, if ν is one of them, the second is equal to ν ∗ . As νν ∗ = 1, and d is positive on I∗ ⊂ J , one gets (4.44) ν(E) = d(E)eiη , ν ∗ = d(E)e−iη , √ 0, and η = η(E) is real. where √d(E) > −G 4 − f2 As = , for E ∈ I∗ , we can choose F /2 f
4 − f 2 (E) sign f (E∗ ). η(E) = − arctan f (E)
28
A. Fedotov, F. Klopp
Here, the square root is positive, and the branch of arctan is chosen so that, for E ∈ I∗ , one has 0 ≤ η(E) ≤ π. With this choice, η(E) is continuous in E ∈ I∗ . One computes η (E) =
f (E) 4 − f 2 (E)
sign f (E∗ ).
Hence, η(E) is monotonously increasing on I∗ from 0 to π . Finally, we note that the relation F (E) = ν + ν ∗ implies that 2 cos η(E) = f (E).
(4.45)
4.4.3. A new parameterization of the monodromy matrix. As d = 1 + O(λ2 ), on the interval I∗ , the matrix D admits the representation iη(E) e 0 D = D0 + D1 , D0 = , D1 = o(λ2∗ ). (4.46) 0 e−iη(E) Proposition 11.1 is applied to Eq. (11.3) with matrices D and A depending on a parameter η (see (11.2)). So, we shall consider the monodromy matrix as a function of η. As η(E) is monotonous, we can introduce the inverse function E(η) and consider all the objects as functions of η. Let us study E(η). Clearly, it is a monotonous continuous function mapping the interval [0, π ] onto the interval I∗ . Furthermore, (4.45) implies Lemma 4.8. The function E(η) can be analytically continued to V(0,π) , a constant neighborhood of the interval (0, π); and there exists C > 0 such that, in V(0,π) , one has |E(η) − E∗ | ≤ Cε th (E∗ ), and E (η) ∼ −
2 sin η . F (E∗ )
(4.47)
Proof. Fix C > 0. Let DC = {|E − E∗ | ≤ C ε th (E∗ )}. Lemma 4.7 and formula (4.25) imply that • there exists C1 > 0 such that, in the domain DC , one has |f (E)| ≤ C1 for ε sufficiently small; • f bijectively maps DC onto its image; • if C is large enough then, f (DC ) contains the interval (−2, 2); • for E ∈ DC , one has f (E) ∼ F (E∗ ). The function E(η) can be constructed as f −1 (2 cos η), where f −1 is the inverse to f . The information collected on f implies Lemma 4.8. & ' We finish this section with three immediate corollaries of Lemma 4.8. Corollary 4.2. The eigenvalues ν and ν ∗ of the matrix M0 are analytic in η in V(0,π) ; for η ∈ V(0,π) , one has ν = eiη + o(λ2∗ ),
ν ∗ = e−iη + o(λ2∗ ).
Proof. The statement is a consequence of (4.44) and Lemmas 4.8 and 4.6.
' &
Anderson Transitions for Almost Periodic Schrödinger Equations
29
Corollary 4.3. det P and sin η/ det P are analytic in η ∈ V(0,π) ; for E ∈ V(0,π) , one has det P ∼ −2i
ei(E∗ )/ε sin η. th (E∗ )
Proof. Corollary 4.3 is a consequence of formulas (4.38) and (4.44), Lemma 4.8 and ' the asymptotics of a0 . & Corollary 4.4. The matrix D1 = D − Diag eiη , e−iη is analytic in V(0,π) ; in V(0,π) , one has D1 = O(λ2∗ ). Proof. Corollary 4.4 is a consequence of Corollary 4.2.
' &
4.4.4. Estimates of the matrices A1 and A2 . Recall that the matrices A1 and A2 are defined in (4.40). We now estimate them assuming that (φ, η) ∈ R, where R = {|Im φ| ≤ r} × V(0,π) . Here, r is a fixed constant, independent of ε. Trying to estimate A1 , there is one difficulty. Indeed, note that in R, one has |a0 |, |b0 | ∼ 1/th (E∗ ),
˜ ∼ λ∗ . |a|, ˜ |b|
These estimates are a consequence 2.2 and Lemma 4.8. Furthermore, by of Theorem 1 th (E∗ ) ∼ Corollary 4.3, one has 2 sin η . Straightforward norm estimates yield only det P 1 that A1 = O . Thus, the norm of λ∗ A1 is not necessarily small. th (E∗ ) · sin η Let us show that one has 1 A1 = O , (φ, η) ∈ R. (4.48) sin η As A1 ∈ M, it suffices to estimate its coefficients (A1 )11 and (A1 )12 . One computes (A˜ 1 )11 = −a0 a˜ ∗ F + b0 b˜ ∗ F − d a˜ + a0 δ1 , (A˜ 1 )12 = b0∗ a˜ ∗ F − a0∗ b˜ ∗ F + d b˜ ∗ − b0∗ δ1 , where δ1 = a0 a˜ ∗ + a0∗ a˜ − b0 b˜ ∗ − b0∗ b˜
and
A˜ 1 = Pˇ0 M1 P0 .
Note that, by Lemma √ 4.6 and Lemma 4.8, for η ∈ V(0,π) , one has d ∼ 1, and F ∼ 2 cos η (as F = ν+ν ∗ = 2 d(E) cos η). So, to prove (4.48), it suffices to show that δ1 = O(λ∗ ). As M is 1-periodic, the equality det M ≡ 1 implies that all the non constant terms of the Fourier series of det M vanish. Therefore, one has a0 a˜ ∗ + a0∗ a˜ − b0 b˜ ∗ − b0∗ b˜ = a˜ a˜ ∗ − b˜ b˜ ∗ , where {f } denotes the sum of all non constant terms of the Fourier series of a periodic function f . By (2.17), in R, a˜ and b˜ are O(λ∗ ). This implies that δ1 = O(λ∗ ), hence, (4.48).
30
A. Fedotov, F. Klopp
Estimating A2 is straightforward; one just uses norm estimates on the matrices composing A2 to get (A2 ) ≤ O
1 . sin η
(4.49)
So, we have proved Lemma 4.9. The function sin η · A is analytic and bounded in (φ, η) ∈ R. Proof. As A = A1 + A2 , the boundedness follows from (4.48) and (4.49), and the analyticity follows from Corollary 4.3. & ' 4.4.5. Bounded solutions of the monodromy equation. Let us summarize the results obtained in Sect. 4.4.4 to check that one can apply Proposition 11.1 to the “continuous” monodromy equation (4.1) with the monodromy matrix described in Theorem 2.2. We have transformed the “continuous” monodromy equation into λ∗ ˜ A(φ + h) = D0 + A(φ) A(φ) sin η
(4.50)
with D0 = Diag (eiη , e−iη ),
A˜ = sin η (A + D1 /λ∗ ),
(4.51)
where D1 = D − D0 . The matrices D0 and A˜ belong to M. As det M ≡ 1, one has λ∗ ˜ ˜ det(D0 + sin η A) = 1. By Corollary 4.4 and Lemma 4.9, the matrix A is analytic and ˜ r,(0,π) is bounded by a constant uniformly in uniformly bounded in R; so, the norm A ε (for the definition of this norm, see (11.1)). We apply Proposition 11.1 to Eq. (4.50) except near energies E where sin(η(E)) = 0. Therefore, we “cut” the ends of the interval η ∈ (0, π). Fix 0 < α < 1/2 and consider the interval Iα = (λα∗ , π − λα∗ ). On this interval, the Lipschitz norm of the function sin1 η is bounded by Const · λ−2α ∗ , and thus, we can apply Proposition 11.1 with the “effective” coupling constant equal to λ1−2α . As a result, we ∗ get Proposition 4.3. Fix 0 < α < 1/2 and σ < 1. Let σ = (1−2α)σ . If ε is small enough, σ/2 η is outside a subset ∞ of Iα of the measure O(λ∗ ), and h satisfies the Diophantine condition (11.4) with λ = λ∗ , then the monodromy equation (2.5) has bounded solutions. 4.4.6. Back to Eq. (1.1). Let us reformulate the conditions of Proposition 4.3 in terms of the initial equation (1.1). We use the notations λI and S introduced in Theorem 2.4 and in (2.21). We let I = V0 ∩ R. 1. Let us study the set D of values of ε ∈ (0, 1) such that the Diophantine condition (11.4) is satisfied with h = 2π ε mod 1 and λ = λ∗ . Show thatthe set D possesses
Anderson Transitions for Almost Periodic Schrödinger Equations
31
the property (2.22). Recall that λI = exp (−S/ε), where S = minE∈I S(E). So, λ∗ ≤ λI , and one has: ∞ 2π mes ((0, ε) \ D) ≤ dh, (h + n)2 Hσ (n) n=L(ε)
σ Sn
k − 2π / k 3 }, Hσ (n) = ∪∞ k=1 ∪l=0 {h : min |h − l/k| ≤ e
where L(ε) is equal to the integer part of 2π/ε. Therefore as, on I ⊂ J+δ , S > δ > 0, we get ∞ σ nS 1 mes ({(0, ε) \ D}) ≤ C exp − ≤ Cε2 λσI , n2 2π n=L(ε)
where C denotes different positive constants independent of ε. This implies that the measure of D satisfies estimate (2.22). 4 2. By (4.43), the length |I∗ | of the interval I∗ has the asymptotics |I∗ | ∼ |F (E . ∗ )| As λ∗ = tv /th is small, by (2.19), one sees that |Il | has the same asymptotics, i.e. |Il |/|I∗ | ∼ 1 when ε → 0. 3. Let E∞ = E(B), where the set B = (0, λα∗ ) ∪(π − λα∗ , π ) ∪ ∞ , and the function E(η) is the inverse of η(E). Clearly, mes (E∞ ) = B E (η)dη. This and estimates (11.6) σ/2 1 and (4.47) imply that mes (E∞ ) = |F (E · (O(λ∗ ) + O(λ2α ∗ )) = o(|Il |). ∗ )| 4. Let us combine points 1, 2 and 3. We see that, if ε is sufficiently small and belongs to the set D, Eq. (4.1) has bounded solutions for E ∈ I∗ outside some subset of I∗ of measure o(|I∗ |). This means that the monodromy equation itself has bounded solutions for these values of E. Then, Corollary 2.1 implies that, for these energies, the Lyapunov exponent vanishes. Now, applying the Ishii–Pastur–Kotani Theorem, we see that the measure of the absolutely continuous spectrum of (1.1) situated on I∗ has the same asymptotics as the length of I∗ . As Il contains this spectrum, and as |Il |/|I∗ | ∼ 1, we get (2.23). As σ can be made as close to 1 as desired, this completes the proof of Theorem 2.4 for the interval I = V0 ∩R. The theorem clearly remains valid for any fixed subinterval of the interval I and for any finite union of such intervals. This completes the proof of Theorem 2.4. & ' 4.5. Singular spectrum. Here, we prove Theorem 2.5. Therefore, we estimate the Lyapunov exponent of (1.1) on Jδ− using the asymptotics (2.17) of the monodromy matrix described in Theorem 2.2. Let E0 ∈ Jδ− , let V0 be a neighborhood of E0 as in Theorem 2.2. Recall that the asymptotics (2.17) are uniform in E in V0 and in φ in the strip |Im φ| ≤ Y /ε, and that Y > Im ϕ2 (E0 ). 1 Let Y1 = 2π Sv (E). By point 4 of Lemma 2.1, one has Y1 < Im ϕ2 (E) for E ∈ J . Clearly, there is a constant real neighborhood V1 of E0 , where Y − Y1 > δ for some positive constant δ. For E ∈ V1 and for φ in the strip {−Y /ε ≤ Im φ ≤ −(Y − δ)/ε}, the product tv e2πiφ is exponentially large. As takes real values on J , see Lemma 2.1, the last observation and formula (2.17) imply that, for E ∈ V0 ∩ V1 and −Y /ε ≤ Im φ ≤ −(Y − δ)/ε, one has a = −λe2iπ(φ−φ0 ) (1 + o(1)),
b/a = i + o(1),
a ∗ /a = o(1),
b∗ /a = o(1).
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A. Fedotov, F. Klopp
Therefore, M = λe
2iπ(φ−φ0 )
−1 i 0 0
+ o(1) .
Recall that λ is exponentially large on Jδ− . Let E belong to the constant interval J0 = Jδ− ∩V0 ∩V1 . Then, the monodromy matrix satisfies the assumptions of Proposition 10.1, and we get the following estimate for the Lyapunov exponent of the matrix cocycle associated to the pair (M, h): θ (M, h) ≥ log λ + o(1). The Lyapunov exponent #(E) for Eq. (1.1) is related to θ(M, h) by Corollary 2.1. Therefore, ε #(E) ≥ log λ + o(ε). 2π Expressing the coupling constant in terms of the tunneling actions, log λ = 1ε (Sh (E) − Sv (E)) + o(1), we obtain (2.24). On the interval J0 ⊂ Jδ+ , the leading term in (2.24) is positive, hence, so is the Lyapunov exponent for sufficiently small ε. Therefore, by the Ishii–Pastur–Kotani’s Theorem, the spectrum of (1.1) situated on J0 is singular. Since E0 can be taken arbitrary in Jδ+ , this completes the proof of Theorem 2.5. 5. Periodic Schrödinger Operators In this subsection, we discuss the periodic Schrödinger operator (1.3), where V is a 1-periodic, real valued, L2loc -function. We collect all the results needed for this paper. The proofs of these results as well as more details can be found, for example, in [12, 34, 35, 41]. 5.1. Bloch solutions. Let ψ be a solution of the equation H0 ψ = −
d2 ψ (x) + V (x)ψ (x) = Eψ (x), dx 2
x ∈ R,
(5.1)
satisfying the relation ψ (x + 1) = λ ψ (x),
∀x ∈ R,
(5.2)
with λ independent of x. Such a solution is called a Bloch solution, and the number λ is called the Floquet multiplier. Let us discuss the analytic properties of Bloch solutions. As in Sect. 2.3, we denote the spectral bands of the periodic Schrödinger equation by [E1 , E2 ], [E3 , E4 ], . . . , [E2n+1 , E2n+2 ], . . . . Consider G± , two copies of the complex plane E ∈ C cut along the spectral zones. Paste them together to get a Riemann surface with square root branch points. We call this Riemann surface G. One can construct a Bloch solution ψ(x, E) of Eq. (5.1) meromorphic on this Riemann surface. It can be normalized by the condition ψ(1, E) ≡ 1. The poles of this solution are located in the spectral gaps. More precisely, each spectral gap contains exactly one simple pole. It is located either on G+ or on G− . The position of the pole is independent of x. Outside of the edges of the spectrum, the two branches of ψ are linearly independent solutions of (5.1). Finally, we note that, in the spectral gaps, both branches of ψ are real valued functions of x, and, on the spectral bands, they differ only by complex conjugation.
Anderson Transitions for Almost Periodic Schrödinger Equations
33
5.2. The Bloch quasi-momentum. Consider the Bloch solution ψ(x, E). The corresponding Floquet multiplier λ (E) is analytic on G. Represent it in the form λ(E) = exp(ik(E)).
(5.3)
The function k(E) is the Bloch quasi-momentum. 5.2.1. The Bloch quasi-momentum as an analytic multi-valued function. The Bloch quasi-momentum is an analytic multi-valued function of E. It has the same branch points as ψ(x, E). Let D be a simply connected domain containing no branch point of the Bloch quasi-momentum. In D, one can fix an analytic single-valued branch of k, say k0 . All the other single-valued branches of k that are analytic in E ∈ D are related to k0 by the formulae k±,l (E) = ±k0 (E) + 2π l,
l ∈ Z.
(5.4)
The Riemann surface of k is more complicated than the one for ψ. However, on the complex plane cut along the spectral gaps of the periodic operator, one can fix a singlevalued analytic branch of k. 5.2.2. The main branch of the Bloch quasi-momentum. Consider C0 , the complex plane cut along the real line from E1 to +∞. On C0 , one can fix a single valued analytic branch of the quasi-momentum by the condition −ik0 (E) > 0,
E < E1 .
(5.5)
We call k0 the main branch of the Bloch quasi-momentum. The function k0 conformally maps C0 onto the upper half of the complex plane with some vertical slits beginning at the points πl, l ∈ Z, and having finite lengths. It is a bijection. In Fig. 5, we drew two curves in C0 and their images under the transformation E → k0 (E). Consider k0 along the curve γ1 . The quasi-momentum k0 (E) is real and
(E )
1
1
2
3
E1
E2
E3
(k0 (E ))
1 E4
E5 2
2 E1
E2
E3
E4
2
1 0
E5
Fig. 5. The action of mapping k on some curves
2
1 ; 2
3
2
34
A. Fedotov, F. Klopp
monotonically increasing along the spectral zones; along the spectral gaps, it takes complex values and its real part is constant; in particular, we have k0 (E1 ) = 0,
k0 (E2l + i0) = k0 (E2l+1 + i0) = π l,
l = 1, 2, 3 . . . .
(5.6)
If E2n < E2n+1 , one says that the nth gap is open. Along any open gap, Im k0 (E + i0) is a non-constant function with only one non-degenerate maximum. The values of the quasi-momentum k0 on the different sides of the cut E1 < E < +∞ are related to each other by the formula k0 (E + i0) = −k0 (E − i0),
E1 ≤ E.
(5.7)
All the branch point of k0 are of square root type: let El be one√ of the branch points, then, in a sufficiently small neighborhood of El , k0 is analytic in E − El , and k0 (E) − k0 (El ) = cl E − El + O(E − El ), cl = 0. (5.8) The constants ml = |cl |2 /2 are called the effective masses associated to El . 5.3. Periodic components of the Bloch solution. Let D ⊂ C be a simply connected domain that does not contain any branch point of k. On D, we fix an analytic branch k of the Bloch quasi-momentum. There are two disjoint domains in G denoted by D± that project onto D. Define ψ± to be the restrictions of the Bloch solution ψ to D± . The functions ψ± are indexed so that k is the Bloch quasi-momentum of ψ+ . The Bloch solutions ψ± can be represented in the form ψ± (x, E) = e±ik(E)x p± (x, E),
E ∈ D,
(5.9)
where p± (x, E) are functions periodic in x, p± (x + 1, E) = p± (x, E),
∀x ∈ R.
(5.10)
We call p± the periodic components of ψ± with respect to the branch k of the Bloch quasi-momentum. One has 1 1 p+ (t, E)p− (t, E)dt = ψ+ (t, E)ψ− (t, E)dt = −ik (E)w(E), E ∈ D, 0
0
(5.11) where w(E) = w(ψ+ (x, E), ψ− (x, E)), and w(f (x), g(x)) = f (x)g(x)−g (x)f (x). 6. Main Theorem of the Complex WKB Method In this section, we describe the complex WKB method for adiabatically perturbed periodic Schrödinger equations. The reader can find more details and the proofs of the results of this section in [19]. This method was developed for the asymptotic study of equations of the form −
d2 ψ(x) + (V (x) + W (εx))ψ(x) = Eψ(x), dx 2
x ∈ R,
(6.1)
Anderson Transitions for Almost Periodic Schrödinger Equations
35
where V is a real valued 1-periodic function of x, and ε is a small parameter (fixed positive number). This method is designed to study exponentially small effects due to the complex tunneling. Usually, these effects are measured by exponentially small coefficients of certain transition matrices (scattering matrices, monodromy matrices, etc.) relating two distinguished bases of solutions. To calculate such exponentially small terms, one assumes that W is analytic and introduces an additional parameter ϕ into Eq. (6.1) so that it becomes −
d2 ψ(x) + (V (x) + W (εx + ϕ))ψ(x) = Eψ(x), dx 2
x ∈ R.
(6.2)
The idea is that the terms that are exponentially small when ϕ is real may become dominant for complex values of ϕ, and that having computed these terms for complex values of ϕ, one can try to recover their values for ϕ on the real axis. Of course, to realize these ideas, one has to have good enough control of the dependence on ϕ of solutions of Eq. (6.2). However, there is no equation controlling this dependence. But, there is a natural condition which replaces such an equation. We say that (ψ± ) two solutions of (6.2) form a consistent basis if their Wronskian is independent of ϕ and if ψ± (x + 1, ϕ) = ψ± (x, ϕ + ε) ∀ϕ.
(6.3)
Equation (6.3) is called the consistency condition. To clarify this condition, we pass to the variables φ = ϕ/ε and t = x + φ. In terms of these variables, (6.2) takes the form −
d2 ˜ ψ˜ + (V (t − φ) + W (εt))ψ˜ = E ψ, dt 2
(6.4)
and (6.3) just becomes the 1-periodicity ψ˜ in φ. i.e. ψ˜ ± (t, φ + 1) = ψ˜ ± (t, φ).
(6.5)
This condition plays a crucial role for the asymptotic analysis of (6.2). Remark 6.1. Note that if W = α cos(·), then Eq. (6.4) is nothing but Eq. (1.1) and condition (6.5) is precisely the condition (2.1) that we required to define the monodromy matrices. We use the complex WKB method to calculate the Fourier coefficients of the monodromy matrices. Now, following [19], we shall describe the main constructions of the complex WKB method. Below, we assume that V ∈ L2loc , and that W is analytic in a neighborhood D(W ) of the real line. 6.1. Complex momentum. The central analytic object of the complex WKB method is the complex momentum κ(ϕ). It is defined in terms of the Bloch quasi-momentum of the operator (5.1) by the formula κ(ϕ) = k(E − W (ϕ)) in D(W ), the domain of analyticity of the function W .
(6.6)
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A. Fedotov, F. Klopp
The complex momentum κ is a multi-valued analytic function. Its branch points are related to the branch points of the quasi-momentum by the relations El = E − W (ϕ),
l ∈ N,
(6.7)
where El are the ends of the spectral zones of the operator H0 . Let D be a simply connected domain containing no branch points of κ. Then, in D, one can fix an analytic branch κ0 of this function. By (6.10), all the other analytic branches are described by the formulas ± = ±κ0 + 2π m, κm
(6.8)
where ± and m are indexing the branches. 6.2. Canonical domains. The notion of canonical domain is the main geometric notion of the complex WKB method. Throughout the paper, we say that a set is regular if it is in the domain of analyticity of W and contains no branch points of the complex momentum. We use the terms: regular domain, regular curve, regular point. When speaking about regular curves, we suppose in addition that they are smooth and connected. A piecewise smooth, connected curve γ ⊂ C is called vertical if it intersects the lines {Im ϕ = Const} at non-zero angles θ, 0 < θ < π.
(6.9)
Thus, vertical lines are naturally parameterized by Im ϕ. Let γ be a regular, vertical curve. On γ , fix a continuous branch of the momentum κ. The curve γ is said to be canonical if, along γ , ϕ • Im κdϕ is strictly increasing with Imϕ; ϕ • Im (κ − π)dϕ is strictly decreasing with Imϕ. From now on, in this section, ϕ1 and ϕ2 are two regular points such that Im ϕ1 < Im ϕ2 . Definition 6.1. Let K ⊂ D(W ) be a regular, simply connected domain. On K, fix a continuous branch of the quasi-momentum, say κ. The domain K is called canonical if it is the union of curves canonical with respect to κ and connecting ϕ1 and ϕ2 located on ∂K. Note that the boundary of K may contain branch points of the complex momentum. 6.3. Canonical Bloch solutions. Consider the periodic Schrödinger equation d2 ψ(x) + V (x)ψ(x) = Eψ(x), E = E − W (ϕ), x ∈ R. (6.10) dx 2 Here, ϕ plays the role of a parameter. We need two linearly independent Bloch solutions of (6.10) that are analytic in ϕ in a given canonical domain. This property defines these solutions uniquely up to an analytic, non-vanishing factor depending on ϕ. We construct these solutions in terms of ψ(x, E), the Bloch solution of (5.1) meromorphic on the Riemann surface G (see Sect. 5.1) and its Bloch quasi-momentum k(E). Recall that k is defined modulo 2π . Pick ϕ 0 , a regular point such that k (E −W (ϕ0 )) = 0. Begin with a local construction in V 0 , a small enough neighborhood of ϕ 0 . One has, −
Anderson Transitions for Almost Periodic Schrödinger Equations
37
• in V 0 , we fix an analytic branch of κ(ϕ), the complex momentum; • there are two different branches of the function ψ(x, E − W (ϕ)) meromorphic in ϕ ∈ V 0 ; they are linearly independent Bloch solutions of Eq. (6.10); we denote them by ψ± (x, ϕ) so that their respective Bloch quasi-momentum be equal to ±κ(ϕ); • in V 0 , fix an analytic branch of the function (6.11) q(ϕ) = k (E − W (ϕ)). Consider p± (x, ϕ) the periodic components of the Bloch solutions ψ± (x, ϕ). One has ψ± (x, ϕ) = e±iκ(ϕ)x p± (x, ϕ), Let
p± (x + 1, ϕ) = p± (x, ϕ), 1 0
± p∓ (x, ϕ) ∂p ∂ϕ (x, ϕ)dx
0
p+ (x, ϕ)p− (x, ϕ)dx
ω± (ϕ) = − 1 The functions
A± (x, ϕ) = q(ϕ)e
ϕ
ω± (ϕ)dϕ
x ∈ R.
(6.12)
.
(6.13)
ψ± (x, ϕ)
(6.14)
are called the canonical Bloch solutions. In Sect. 3 of [19], we have proved Lemma 6.1. The canonical Bloch solutions are analytic in any regular simply connected domain D containing ϕ 0 . Normalization point. We normalize the canonical Bloch solutions by integrating in (6.14) from a given point ϕ0 , located in D or on its boundary. The point ϕ0 is called the normalization point. Indexing of the canonical Bloch solutions. In D, fix κ1 a continuous branch of the complex momentum. The canonical Bloch solutions A± can be indexed by ± so that κ1 (ϕ) is the Bloch quasi-momentum corresponding to the solution A+ (x, ϕ). The Wronskian of the canonical Bloch solutions. Here, we compute the Wronskian of A± normalized at ϕ0 . We prove Lemma 6.2. One has
w(A+ , A− ) = q 2 (ϕ0 )w(ψ+ (x, ϕ0 ), ψ− (x, ϕ0 )) = i
0
1
ψ+ (x, ϕ0 )ψ− (x, ϕ0 )dx.
The representation of the Wronskian given in Lemma 6.2 shows that, for any regular ϕ0 such that q 2 (ϕ0 ) = k (E − W (ϕ0 )) = 0, the canonical Bloch solutions are linearly independent. Note that, in this case, neither A+ nor A− can be identically zero for any given ϕ ∈ D. The second representation given by Lemma 6.2 allows us to choose as ϕ0 a branch point of the complex momentum. In this case, one has 1 w(A+ , A− ) = i |ψ+ (x, ϕ0 )|2 dx = 0. (6.15) 0
38
A. Fedotov, F. Klopp
Indeed, at the ends of any spectral gap, the branches of the Bloch solution ψ(x, E) coincide and are real valued in x. Proof of Lemma 6.2. The definition of the canonical Bloch solutions and of the functions ω± imply w(A+ , A− ) = q 2 (ϕ) ϕ 1 ∂ exp − p+ (x, ϕ)p− (x, ϕ)dx w(ψ+ (x, ϕ), ψ− (x, ϕ)). log ϕ0 ∂ϕ 0 Relation (5.11) then gives both representations for the Wronskian given in Lemma 6.2. ' &
6.4. The main theorem of the WKB method. Now, we can formulate the main theorem of the complex WKB method for (6.2): Theorem 6.1 ([19]). Fix X > 0. Fix E = E0 ∈ C. Let K be a bounded canonical domain for the family of Eqs. (6.2), and let κ be the branch of the complex momentum with respect to which K is canonical. For sufficiently small positive ε, there exists a consistent basis (f± ) defined for x ∈ R and ϕ ∈ K and having the following properties: • for any fixed x ∈ R, the functions f± (x, ϕ) are analytic in ϕ ∈ K. • For −X ≤ x ≤ X, and ϕ ∈ K, the functions f± (x, ϕ) have the asymptotic representations i ϕ κdϕ (A (x, ϕ) + o(1)), ε → 0. f± (x, ϕ) = e± ε (6.16) ± Here, A± are the canonical Bloch solutions corresponding to the domain K, indexed so that κ(ϕ) is the Bloch quasi-momentum corresponding to the solution A+ (x, ϕ). • The error estimates in (6.16) may be differentiated once in x. Moreover, they are uniform in x ∈ [−X, X] and locally uniform in ϕ in the interior of K. This theorem was proved in [19]. In the sequel, we normalize the solutions f± in (6.14) and in (6.16) by integrating from a point ϕ0 being either in K or on its boundary. When we need to indicate the point ϕ0 explicitly, we write f± (x, ϕ, ϕ0 ). One easily calculates the Wronskian of the solutions f± (x, ϕ, ϕ0 ): w(f+ , f− ) = w(A+ , A− ) + o(1).
(6.17)
By Lemma 6.2, the solutions f± are linearly independent if ϕ0 is regular and k (E − W (ϕ0 )) = 0, or if ϕ0 is one of the branch points of κ. 6.5. Dependence on the spectral parameter and admissible subdomains. To simplify the statement of Theorem 6.1, we have not considered the dependence of the solutions on the spectral parameter E. Therefore, we introduce the notion of admissible subdomains of a canonical domain. Let K be a compact canonical domain and pick δ > 0. Let A be the complementary in K of the δ-neighborhood of ∂K, the boundary of K. The set A is the δ-admissible subdomain of the canonical domain K. One proves
Anderson Transitions for Almost Periodic Schrödinger Equations
39
Proposition 6.1. In the setting of Theorem 6.1, the solutions f± are analytic in E in V0 , a neighborhood of E0 . For A, an admissible subdomain of K, there exists VA ⊂ V0 , a neighborhood of E0 such that the asymptotics (6.16) is uniform in (ϕ, E, x) ∈ A×VA × [−X, X]. It can be once differentiated in x without loosing its uniformity properties. In [19], we have not discussed explicitly the dependence of the solutions f± on the spectral parameter E. In the proof of Theorem 6.1, all the estimates are locally uniform in E. To prove Proposition 6.1, one just has to follow the proof of Theorem 6.1 given in [19], keeping in mind the following additional observation: Lemma 6.3. Let γ be a compact canonical curve for a given value E0 of the spectral parameter E. Then, there exists V1 ⊂ C, a neighborhood of E0 and a constant δ > 0 such that, for any E ∈ V1 , the curve γ remains canonical, and along γ , one has ϕ ϕ d d κdϕ > δ and (κ − π )dϕ < −δ, y = Imϕ, (6.18) Im Im dy dy uniformly in E ∈ V1 . Proof. Since γ is canonical and compact, for E = E0 , along γ , one has ϕ ϕ d d Im κdϕ > δ0 and (κ − π )dϕ < −δ0 , y = Imϕ, Im dy dy for some positive constant δ0 . The branch points of the complex momentum depend continuously on the spectral parameter. So, γ being regular and compact, there exists a closed neighborhood of E0 , say V0 ⊂ C, such that for all E ∈ V0 , the branch points of the complex momentum stay at a non-vanishing distance d of γ . The two derivatives in (6.18) depend continuously on (ϕ, E) ∈ γ × V0 . So, for any constant δ < δ0 , there is V1 ⊂ V0 , a neighborhood of E0 such that, for E ∈ V1 , one obtains the estimates (6.18). This completes the proof of Lemma 6.3. & ' 6.6. How to use the complex WKB method. Let us outline the ideas we use when applying the complex WKB method. Having described canonical domains and the corresponding bases of solutions with the standard asymptotic behavior, one tries to relate the bases corresponding to different canonical domains. Pick two canonical domains K 1,2 and the associated basis of solutions constructed in Theorem 6.1; denote it by (f±1,2 ). Equation (6.2) being linear of second order, there exists a transfer matrix T12 , depending only on ϕ, such that 2 1 f+ f+ = T , ϕ ∈ K 1 ∩ K 2. (ϕ) 12 f−2 f−1 The consistency condition (2.1) implies that T12 is ε-periodic in ϕ. Moreover, (f±1,2 ) being analytic in ϕ, T12 is also analytic. In K 1 ∩K 2 , we can use the asymptotics (6.16) for both bases to obtain the asymptotics of T12 . Using the ε-periodicity of T12 , we see that these asymptotics are valid in a horizontal strip containing K 1 ∩ K 2 . Now, assume that we know the asymptotics of T12 near the boundary of a complex strip containing R. Then, we can easily get the asymptotics of the Fourier coefficients of T12 corresponding to the terms of the Fourier
40
A. Fedotov, F. Klopp
series which are large at this boundary. This enables us to control the Fourier series terms of T12 which are of order exp(−Const/ε) on the real line. The consistency condition directly relates the x-behavior of the solution to its ϕbehavior. This means that one can express the transition matrices appearing in the spectral study of the equation family (6.2) (for example, monodromy matrices) as products of transfer matrices. 6.7. Canonical domains. We finish the section on the complex WKB method by describing a simple general approach to “constructing” canonical domains. We use this approach in this paper. Below, we assume that D is a regular simply connected domain and that κ is a branch of the complex momentum analytic in D. The main elements of the construction are the pre-canonical lines. Pre-canonical lines are made of elementary lines that we discuss now. 6.7.1. Lines of Stokes type. Let γ ⊂ D be a smooth curve. We say that γ is a line of Stokes type with respect to κ if, along γ , either ϕ ϕ Im κdϕ = Const or Im (κ − π )dϕ = Const. 6.7.2. Canonical lines. We identify C and R2 in the usual way. For ϕ ∈ D, denote by S(ϕ) ⊂ C the cone situated between the vectors κ(ϕ) and κ(ϕ)−π and such that, for any vector z ∈ S(ϕ) , one has Im (κ(ϕ) z) > 0, and Im ((κ(ϕ) − π ) z) < 0. The definition of a canonical line implies that a smooth vertical curve γ ∈ D is canonical with respect to κ if and only if for all ϕ ∈ γ , the vector t (ϕ) tangent to γ and oriented upwards belongs to S(ϕ). 6.7.3. Pre-canonical lines. Let γ ⊂ D be a vertical curve. The curve γ is a precanonical line if it consists of a finite union of bounded segments of either canonical lines or lines of Stokes type. One of the main properties of pre-canonical lines is that they can be used to construct canonical lines. Proposition 6.2. Let γ be a pre-canonical curve. Denote the ends of γ by ϕa and ϕb . For V ⊂ D, a neighborhood of γ and Vϕa ⊂ D, a neighborhood of ϕa , there exists a canonical line in V connecting the point ϕb to a point in Vϕa . Proof. The proof of Proposition 6.2 is done in two steps. The pre-canonical curve γ can contain segments of lines of Stokes type. In the first step, we "replace" these segments by nearby segments of canonical curves to prove that, near a given pre-canonical curve, there is a pre-canonical curve consisting only of segments of canonical curves. This curve is not yet canonical, since canonical curves are smooth. So, in the second step, we smooth out the constructed pre-canonical curve to get a canonical one. If the given canonical curve contains no segments of lines of Stokes type, we do not need to make the first step. Otherwise, we use the following lemma: Lemma 6.4. Let β be either a bounded vertical line of Stokes type or a bounded canonical line. Denote the ends of β by ϕa and ϕb . Let V ⊂ D be a neighborhood of β. For Vϕa ⊂ V , a neighborhood of ϕa , there exists Vϕb ⊂ V , a neighborhood of ϕb such that, for each point ϕ2 ∈ Vϕb , there is a point ϕ1 ∈ Vϕa , connected to ϕ2 by a curve canonical with respect to κ and staying in V .
Anderson Transitions for Almost Periodic Schrödinger Equations
41
Proof. If β is a canonical line, the result follows from the fact that any line C 1 -close to βis also canonical. Let us turn to the case of a line of Stokestype. Assume that ϕ ϕ Im κdϕ = Const along β. The case when Im (κ − π )dϕ = Const along β ϕ is treated similarly. On V , define I (ϕ) = ϕa κdϕ, X(ϕ) = Re [I (ϕ)], and Y (ϕ) = Im [I (ϕ)]. Note that along β, Y (ϕ) = 0, and I (β) is an interval [a, b] of R. As V does not contain any branch point of κ, we know that dI /dϕ = κ does not vanish in V . Moreover, along β, X(ϕ) is a strictly monotonous function of Im ϕ. This implies that there exists V ⊂ C a neighborhood of β and U ⊂ C a neighborhood of the interval [a, b] ≡ X(β) ⊂ R such that # : V → U, #(ϕ) = X(ϕ) + iY (ϕ), is a diffeomorphism. Assume that X is increasing along β (the case where X is decreasing is treated similarly). Consider the smooth curves α ⊂ U along which Y is a monotonously increasing function of X, Y = Fα (X), Fα (X) > 0,
X ∈ [a, b] and Fα C 1 ([a,b]) ≤ δ. ϕ Along the curves #−1 (α), the function Im κdϕ is strictly increasing. As ϕδ → 0, the curves #−1 (α) are C 1 -close to β. As β is vertical and as the function Im ϕa κ(ϕ)dϕ is ϕ constant along β, Im ϕa (κ(ϕ) − π )dϕ is decreasing along β. So, if δ is small enough, ϕ (κ − π )dϕ is decreasing along #−1 (α). Thus, the curves #−1 (α) are vertical, and Im −1 the curves # (α) are canonical. This implies the statement of Lemma 6.4. & ' Lemma 6.4 allows us to replace any given pre-canonical line with a neighboring pre-canonical line consisting only of segments of vertical canonical lines. This new precanonical curve begins at ϕa , one of the ends of β, and it connects it in V with a point in any given neighborhood Vb , of ϕb the other end of ϕ. To make the second step of the proof of Proposition 6.2, we have to smooth out the precanonical line βˆ obtained in the first step. In fact, we do it only in small neighborhoods ˆ Let ϕ0 be one such of the common ends of the segments of canonical lines making up β. ˆ points, and let γj ⊂ β, j = 1, 2 be the segments of canonical lines with the common end ϕ0 . As in Sect. 6.7.2 define the cones S(ϕ) for ϕ ∈ V . There exists V0 , a sufficiently small neighborhood of ϕ0 such that S0 ≡ ∩ϕ∈V0 S(ϕ) = ∅ and such that, in V0 , both the tangent vectors to γ1 and the tangent vectors to γ2 belong to S0 . We replace the part of βˆ in V0 by a smooth curve so that the new curve βˆ is smooth and the vectors tangent to this curve in V0 are in S0 . We carry out this smoothening for all the common ends of the ˆ The smoothened curve βˆ is canonical. This segments of canonical lines making up β. completes the proof of Proposition 6.2. & ' By Proposition 6.2, we can construct a canonical curve arbitrarily close to a given pre-canonical curve. But, in general, the ends of these curves can not coincide. Let us single out one case where these curves can have common ends: Lemma 6.5. Let γ be a pre-canonical curve. Assume that it begins with γa and ends with γb , both segments of canonical curves. Denote by ϕa and ϕb those ends of these segments which are internal points of γ ordered so that Im ϕa < Im ϕb . Then, for any positive δ, there is a pre-canonical curve situated in the δ-neighborhood of γ , and containing γa and γb without the δ-neighborhood of ϕa and the δ-neighborhood of ϕb . Proof. If the curve γ consists only of segments of canonical curves, then one proves the lemma just by smoothing out the curve γ as in the proof of Proposition 6.2. Otherwise,
42
A. Fedotov, F. Klopp
we apply Proposition 6.2 to the pre-canonical curve γ , where γ is the part of γ between ϕa and ϕb . This allows us to construct a canonical curve γ arbitrarily close to γ and connecting ϕa to a point ϕb arbitrarily close to ϕb . But, if this point is sufficiently close to ϕb , then we can use the C 1 -stability of canonical curves and slightly deform γb in the δ-neighborhood of ϕb to obtain a new canonical curve γb that ends at ϕb . Now, it suffices to smooth out the pre-canonical curve γa ∪ γ ∪ γb . This completes the proof of Lemma 6.5. & ' 6.8. Enclosing canonical domains. Let γ ⊂ D be a line canonical with respect to κ. Denote by ϕa and ϕb the ends of γ so that Im ϕa < Im ϕb . Let a domain K ⊂ D be a canonical domain corresponding to the triple ϕa , ϕb , κ. If γ ∈ K, then K is called a canonical domain enclosing γ . As any line C 1 -close to γ is canonical, one can always construct a canonical domain enclosing any given canonical curve. We shall call such canonical domains local. In practice, one constructs canonical domains by means of Proposition 6.3. Let γ be a canonical line with respect to κ. Assume that K ⊂ D is a simply connected domain containing γ (without its ends). The domain K is a canonical domain enclosing γ if it is the union of pre-canonical lines each of which is obtained from γ by replacing some of its internal segments by a line pre-canonical with respect to κ. Proof. It suffices to prove that, for any ϕ ∈ K, there is a pre-canonical curve β lying in K, connecting ϕd and ϕu , two internal points of γ , such that Im ϕu > Im ϕd and such that ϕ is an internal point of a segment of β which is a canonical line. Indeed, consider the pre-canonical curve going, first, along γ from ϕa to ϕd , then, along β from ϕd to ϕu , and, at last, along γ from ϕu to ϕb . Consider the segments of this curve from ϕa to ϕ and from ϕ to ϕb . The proof of Proposition 6.3 is then obtained by applying Lemma 6.5 (with δ small enough) to each of these segments. We know that there is a pre-canonical curve β containing ϕ and connecting two internal points of γ . Let us modify this curve to construct the curve β. The point ϕ divides β into two pre-canonical curves βu and βd connecting ϕ respectively to the points ϕu and ϕd of γ . We assume that βd is below βu . Begin with describing the necessary transformation of βu . We can continue βu somewhat beyond the point ϕu so that the new line remain pre-canonical. For this new line, we keep the old notation. By Proposition 6.2, in any neighborhood of βu , there is a canonical line βu beginning at ϕ and ending in any given neighborhood of the other end of βu . We can choose these two neighborhoods sufficiently small so that βu ⊂ K and so that βu intersects γ at, say, ϕu . By construction, one has Im ϕ < Im ϕu . Similarly, starting from βd , we construct a canonical line βd ⊂ K connecting ϕ to a point ϕd ∈ γ situated below ϕ. The precanonical line βd ∪ βu would be the one we need, but ϕ is the common end of the two canonical lines βu and βd . To finish the proof we deform these curves so that ϕ be an internal point of one of them. To this end, we continue βu somewhat beyond ϕ so that the new curve βu remain canonical. Then, using the C 1 -stability of the canonical lines, we slightly deform βd to a new canonical curve βd connecting ϕd to a point of βu situated somewhat below ϕ. The pre-canonical line going along βd from ϕd to βu , and then along βu to ϕu is the one we need. This completes the proof of Proposition 6.3. & '
Anderson Transitions for Almost Periodic Schrödinger Equations
43
7. Constructions of the Complex WKB Method for Eq. (1.1) To use the complex WKB method, we rewrite (1.1) in terms of the variables u = x − φ and ϕ = εφ, −
d2 ψ(u) + (V (u) + α cos(εu + ϕ))ψ(u) = Eψ(u), du2
u ∈ R.
(7.1)
We describe the complex momentum and canonical domains for (7.1); in Sect. 8, these are used to compute the monodromy matrix.
7.1. Complex momentum. Properties of κ depend on the value of the spectral parameter E and on the lengths of the spectral zones and the spectral gaps of the periodic operator H0 . Recall that we assume condition (2.16) to hold. Some general properties of the complex momentum were discussed in Sect. 6.1. In Sect. 2.5, we have stated some properties of the set of branch points of the complex momentum for Eq. (1.1). These properties are obvious consequences of the description of the Bloch quasi-momentum given in Sect. 5.2. Consider the strip {|Re ϕ| < π }. Cut it from −π to −ϕ1 and from ϕ1 to π . Denote the domain thus obtained by D0 . It is regular and simply connected. In D0 , one can fix a single-valued branch of the complex momentum by the conditions Im κ0 (0) > 0,
Re κ0 (0) = 0.
(7.2)
Indeed, the domain D0 is mapped by E : ϕ → E − α cos ϕ onto the upper half of the complex plane. The branch κ0 is related to k0 , the main branch of Bloch quasi-momentum by the formula κ0 (ϕ) = k0 (E − α cos ϕ).
(7.3)
The branch κ0 (defined on D0 ) is called the main branch of the complex momentum. We now describe four properties of the main branch of the complex momentum; they follow from (7.3) and from the properties of k0 described in Sect. 5.2.2. The first two properties are κ0 (−ϕ) = κ0 (ϕ),
(7.4)
κ0 (ϕ) = −κ0 (ϕ).
(7.5)
Note that (7.4) is also valid for complex values of E. The domain {ϕ ∈ D0 ; Re ϕ ≥ 0} is mapped by κ0 onto the upper half plane with finite vertical cuts starting at the points nπ , n ∈ Z (see Fig. 6; in this figure, we have drawn two curves on the complex plane ϕ ∈ C, and their images by κ0 (ϕ)). Finally, we consider the 2π-periodic curve γ going along the real line around the branch points as shown in Fig. 7. Continuing κ0 analytically along this curve, we get κ0 (ϕ + 2π ) = κ0 (ϕ),
ϕ ∈ γ.
(7.6)
44
A. Fedotov, F. Klopp
7
'4
6
'3
5
'2
2
4
'
'1
'4
2
1
'3
2
2
'2
2
3
'
( )
( 0 ( ))
'1
2
3
4
5
6
0
7 2
Fig. 6. The action of the mapping κ
b
'1
0
2
b
'1
2
Fig. 7. The period γ
7.2. The Stokes lines. The definition of the Stokes lines is fairly standard (see e.g. [15]). The integral κ dϕ has the same branch points as the complex momentum. Let ϕ0 be one of them. Consider the curves beginning at ϕ0 described by ϕ (κ (ξ ) − κ (ϕ0 )) dξ = 0. (7.7) Im ϕ0
These curves are the Stokes lines beginning at ϕ0 . It follows from (6.8) that the Stokes line definition is independent of the choice of the branch of κ in (7.7). As the branch points of the complex momentum are of square root type, exactly three Stokes lines begin at any branch point. At this point, the angle between any two nearest neighbor Stokes lines are equal to 2π 3 . These Stokes lines may be finite: they connect pairs of finite branch points; they may also be infinite: they go from finite branch points to infinity. If ϕ0 is a branch point for κ, so is the point ϕ0 + 2π . The Stokes lines starting at ϕ0 + 2π are just the 2π-translates of the Stokes lines starting at ϕ0 . Furthermore, the whole picture of the Stokes lines in the domain 0 ≤ Re ϕ ≤ 2π is symmetric with respect to the real line as well as with respect to the line Re ϕ = π . This is a consequence of the symmetries of the cosine. In Fig. 8, we have shown some of the Stokes lines; they are represented by dotted lines. Let us discuss them briefly. Consider the Stokes lines beginning at the branch point ϕ1 . As κ0 is real on the interval [ϕ1 , π], this interval is a part of the Stokes line beginning at the point ϕ1 . There are two
Anderson Transitions for Almost Periodic Schrödinger Equations
\c"
\b"
45
b'3 b'2
\a" 0
b
2
'1
Fig. 8. The Stokes lines
other Stokes lines beginning at this point. One of them is going upwards. We denote it by “a”. Consider the Stokes lines beginning at the branch point ϕ2 . As κ0 − π is purely imaginary on the segment [ϕ2 , ϕ3 ] of the line π + iR, this segment coincides with the Stokes line connecting the points ϕ2 and ϕ3 . There are two other Stokes lines starting at ϕ2 . We denote by “b” the Stokes line going to the left. As already noted, one of the Stokes lines beginning at ϕ3 coincides with the segment [ϕ2 , ϕ3 ] of the line π + iR. Let “c” be the Stokes line starting at ϕ3 going up to the left. The global behavior of the Stokes lines “a”, “b” and “c” is described by Lemma 7.1. (1) The Stokes line “a” stays vertical; it does not intersect the lines iR and π + iR and stays between them. (2) The Stokes line “b” intersects “a” above ϕ1 ; the segment between its beginning and the intersection is vertical and stays between iR and π + iR. (3) after intersecting “a”, the Stokes line “b” goes downward staying vertical, not intersecting “a” anymore; it intersects either R or the Stokes line “d” symmetric of “a” with respect to iR; (4) The Stokes line “c” stays vertical; it does not intersect the lines “a” and π + iR and stays between these two curves. Proof. First, consider the Stokes line “a”. A Stokes line can become horizontal only at a point where Im κ = 0, i.e. at a point of the pre-image of a spectral band. Therefore, “a” stays vertical as long as it stays (strictly) between iR and π + iR. Assume that “a” goes upward between iR and π + iR and, then, intersects π + iR at ϕa . Denote the segment of “a” between ϕ1 and ϕa by a. ˜ By the definition of Stokes lines and as the interval [ϕ1 , π] is a Stokes line, one has ϕa ϕa κ0 (ζ )dζ = Im κ0 (ζ )dζ = Re κ0 (ζ )dζ. (7.8) 0 = Im a˜
π
π, along π+iR
But, on the line π + iR, one has Re κ0 (ζ ) > 0. So, the right-hand side of (7.8) is nonzero. Thus, the Stokes line “a” does not intersect π + iR and stays vertical if it does not intersect iR. So, now we assume that “a” intersects iR at ϕa . Denote the segment of “a” between ϕ1 and ϕa by a. ˜ Then, as κ0 ∈ iR on iR, 0 0 κ0 (ζ )dζ = Im κ0 (ζ )dζ = Im κ0 (ζ )dζ. 0 = Im a˜
ϕ1
ϕ1 , along R
46
A. Fedotov, F. Klopp
The last integral is non-zero as Im κ0 > 0 on the real line between 0 and ϕ1 . The above two observations prove point 1 of Lemma 7.1. Now, consider the Stokes line “b”. If the lines “b” and “a” do not intersect one another, then “b” intersects either the interval [ϕ1 , π ] or the segment [π, ϕ2 ] of the line π + iR. Assume that there are such intersections. Denote by ϕb the point where it happens for the first time. If ϕb ∈ [π, ϕ2 ], then, as κ0 (ϕ2 ) = π , ϕ2 ϕ2 0 = Im (κ0 (ζ ) − π )dζ = Im (κ0 (ζ ) − π )dζ. ϕb , along "b"
ϕb , along π+iR
As, along [π, ϕ2 ), one has 0 < κ0 < π , the right-hand side in the above formula is non-zero. If ϕb ∈ [ϕ1 , π], one has ϕ2 ϕ2 0 = Im (κ0 (ζ ) − π )dζ = Im (κ0 (ζ ) − π )dζ. ϕb , along “b”
π, along π+iR
As before, we see that the right-hand side in the last formula is non-zero. As a result, we see that “b” must intersect “a” at a point ϕab above R; it stays between “a” and π + iR before the first intersection. This proves point 2 of Lemma 7.1. To prove point 3, we need only to check that “b” can not intersect “a” once more before intersecting either “d” or R. Indeed, assume that it intersects “a”. Denote the intersection point by ζab , then 0 = Im = Im
ϕab
ζab , along “b” ϕab ζab , along “a”
(κ0 (ζ ) − π )dζ (κ0 (ζ ) − π )dζ = −π(ϕab − ζab )
which is impossible. Check the last statement. Using almost the same argument as in the case of “a”, one proves that “c” can not intersect π + iR before intersecting iR. So, it suffices to prove that “c” stays to the right of “a”. Assume that the Stokes lines “a” and “c” intersect. Let ϕac be their first common point. Then, 0 = Im
ϕac ϕ1 , along “a”
κ0 (ζ )dζ = Im
ϕ2 π, along π+iR
κ0 dζ + Im
ϕac ϕ2 , along “c”
κ0 dζ.
Now, using the definition of the Stokes line “c”, we transform the right hand side into
ϕ2
π, along π+iR
Re κ0 dIm ζ + π Im (ϕac − ϕ2 ).
This expression is positive. So, “a” and “c” can not intersect. This implies point 4 and completes the proof of Lemma 7.1. & '
7.3. Canonical domains. Now we describe the canonical domains used in the next section to compute the monodromy matrix.
Anderson Transitions for Almost Periodic Schrödinger Equations
47
∞ 7.3.1. The domain K0 . Consider the simply connected ϕ regular domain K0 correspondκ0 dζ = Const are represented by ing to Fig. 9, part A. In this figure, the lines Im ϕ continuous curves, and the lines Im (κ0 − π )dζ = Const are shown as dotted curves. The boundary of K0∞ consists of Stokes lines. This domain exists for all E on the interval J defined by (1.4). On K0∞ , we fix an analytic branch κ0 of the complex momentum as the analytic continuation of the main branch of the complex momentum. Consider its subdomain K0 corresponding to Fig. 9, part B. Its boundary contains the lines of Stokes type (with respect to κ0 ) passing by ζ1 and ζ2 (as shown in the same figure). Assuming that such a domain exists, we define the horizontal strip |Im ϕ| ≤ Y (K0 ), where K0 and K0∞ coincide. One has
b2 b '3 b
b '2 '1
b
b '3
b
b
'1
0
b
b
b
b
b
b '2 '1
b
b
0
b
b
'1
b
b
b
A
B
1 b
Fig. 9. Canonical domains: K0∞ and K0
Proposition 7.1. One can choose the points ζ1 and ζ2 so that the domain K0 exist and so that Y (K0 ) > Im ϕ3 is as large as desired. The domain K0 is canonical with respect to the branch κ0 and the points ζ1 and ζ2 . ϕ Proof. As usual, we identify R2 and C. The lines of Stokes type Im κ0 dζ = Const are ϕ integral curves of the vector field κ0 (ϕ), and the lines of Stokes type Im (κ0 −π )dζ = Const are integral curves of the vector field κ0 (ϕ) − π . First, choosing the points ζ1 and ζ2 in K0∞ properly, we show that there exists a domain K0 bounded by the lines of Stokes type as shown in Fig. 9, and that Y (K0 ) can be made as large as desired. The proof is split into a few steps. ϕ 1. Pick a point A on the Stokes line “c”, see Fig. 10. Recall that along “c”, Im ϕ3 (κ0 − ϕ π)dζ = 0. Consider γA , the line of Stokes type Im A κ0 dζ = 0, passing through A. It is transversal to the Stokes line “c” at A. Show that, above A, it is vertical and stays between “a” and “c”. At A, γA is tangent to the vector κ0 (A); in a sufficiently small neighborhood of A, above A, it goes up and stays to the left of “c” and to the right of “a”. Above A, it can not intersect “a” (at least, without intersecting “c”) as “a” is also a line of Stokes type ϕ along which Im κ0 dζ = const. Assumes that γA intersects “c” above A. Denote the
48
A. Fedotov, F. Klopp
B
B
b
2
b
\d"
\a"
b b
\c" A b
0 b
b
0
'1
A
b'3
\b" b'2 b
'1
b b
b
b
b1 Fig. 10. The construction of K0
intersection point by ζi . Then, ζi 0 = Im κ0 dζ A, along γA ζi
= Im
A, along “c”
κ0 dζ = π Im (ζi − A),
which is impossible. So, γA stays between “a” and “c” which also implies that it is vertical above A. ϕ 2. We pick ζ2 on γA above A. Consider γB , the line of Stokes type Im ζ2 (κ0 −π )dζ = 0 passing by ζ2 . At ζ2 , it is transversal to γA and vertical. One shows that, below ζ2 , • γB goes down staying to the left of the lines γA , “c”, the segment [ϕ2 , ϕ3 ] and the line “b”, • then, it intersects either R or “d”, the Stokes line symmetric to “a” with respect iR, • it stays vertical between ζ2 and B, the intersection point with either R or “d”. Indeed, being tangent to the vector κ0 (ζ2 ) − π at ζ2 , γB goes down from ζ2 and, in a sufficiently small neighborhood of ζ2 , stays to the left of γA and to the right of “d”. Then, it stays vertical and goes down at least while between “d” and γA . Let R be the “rectangle” bounded by “d” , the line Im ϕ = Im ζ2 , γA and the line Im ϕ = Im A. If γB leaves R via “d”, the second step is completed. If it leaves R via ζi , a point of γA , then ζi ζi πIm (ζi − ζ2 ) = Im κ0 dζ = Im κ0 dζ = 0, ζ2 , along γB
ζ2 , along γA
which is impossible. So, we are left with the case where it leaves R through a point of the line Im ϕ = Im A staying between “d” and γA . Now, consider “P”, the “polygon” bounded by the lines “d”, Im ϕ = Im A, “c”, the segment [ϕ2 , ϕ3 ] of the line π + iR, the line “b” and, possibly, R. Inside P , γB stays vertical and goes down. If γB leaves P via “d” or R, the second step is completed. This
Anderson Transitions for Almost Periodic Schrödinger Equations
49
is the case as it can not intersect the other lines ϕforming the boundary of P since all of them are lines of Stokes type along which Im (κ0 − π )dζ = const. Note that, if B is real, then it is located between −ϕ1 , the starting point of “d”, and ϕ1 , the starting point of “a”. 3. Show that, by choosing Im ζ2 large enough, one can make B belong to “d” and Im B as large as desired. First assume that B ∈ R. Using the definitions of the lines of Stokes type, one computes πIm (ζ2 ) = Im
ζ2
B, along γB
κ0 dζ = Im
ϕ1 B, along R
κ0 dζ + Im
A ϕ1 , in K0∞
κ0 dζ.
The second term in the right-hand side is constant, and the first one increases with Im ζ2 . So, if Im ζ2 is large enough, then B becomes a point of “d”. Assume B ∈“d”. Along ϕ both “d” and γA , Im κ0 dζ = Const; so, one gets π Im (ζ2 − B) = Im
A −ϕ1 , in K0∞
κ0 dζ.
So, Im B linearly increases with Im ζ2 . This completes the third step. 4. One chooses the point ζ1 symmetric to ζ2 with respect to 0. As the cosine, hence, κ0 ϕ ϕ are even (see (7.4)), the families of lines Im (κ0 − π )dζ = Const and Im κ0 dζ = Const are symmetric with respect to ϕ = 0. As a result, we see that the domain K0 exists and that Y (K0 ), the size of the horizontal strip, where K0 coincides with K0∞ , can be made as large as wanted. By means of Proposition 6.3, we show that K0 is canonical. The proof is again split into a few steps. 5. We say that the lines of a family (lv )v∈U fibrate a domain D if D is the disjoint ϕ union of the lines (lv )v∈U . We use the fact that the families lines of Stokes type Im κ0 dζ = ϕ Const and Im (κ0 − π)dζ = Const both fibrate K0 . Indeed, consider, for example, the first family. Pick a point ζ0 in K0 . If there are two lines of Stokes type passing through ϕ ζ0 , then ζ0 is a critical point for the harmonic function Im 0 κ0 dζ . Then κ0 (ζ0 ) = 0. This is possible only at a branch point of κ0 and there are no branch points inside of K0 . 6. We construct a line γ ∈ K0 connecting ζ1 and ζ2 and canonical with respect to κ0 . Therefore, we use Lemma 6.5 and make some preparations that we present now. First, in a neighborhood of ζ2 , pick a segment γ2 of a vertical, say, straight line starting at ζ2 and going down between γA and γB . At ζ2 , the last two lines are tangent respectively to the vectors κ0 (ζ2 ) and κ0 (ζ2 ) − π . So, at least, the part of γ2 situated in a small enough neighborhood of ζ2 is a canonical line. To keep the notations simple, we denote it also by γ2 . The line γ1 starting at ζ1 and symmetric to γ2 with respect to 0 is also canonical. Describe a pre-canonical line β ⊂ K0 connecting a1 , an internal point of γ1 to a2 , an internal point aof γ2 . Consider σ2 , the line of Stokes type containing ϕ a2 and satisfying the relation Im ϕ 2 (κ0 − π)dζ = 0. As the lines of Stokes type Im (κ0 − π )dζ = const fibrate K0 , choosing a2 close enough to γB , we get that (1) σ2 stays in the part of K0 where Im κ0 > 0 and, so is vertical in K0 ; (2) the segment of σ2 going down from a2 intersects “d”. In particular, together with γB , σ2 intersects the line iR at a point b2 having a positive imaginary part. The pre-canonical line β is symmetric with respect to 0. It begins at a2 , then, along σ2 , it goes down to its intersection with iR, and then goes down along iR to the origin.
50
A. Fedotov, F. Klopp
The line β˜ = γ1 ∪β ∪γ2 is pre-canonical. The pre-canonical line β˜ found, Lemma 6.5 implies that, as close to β as desired, there exists γ , a canonical line connecting ζ1 to ζ2 so that, in a sufficiently small neighborhood of ζ1 , γ coincides with γ1 , and, in a sufficiently small neighborhood of ζ2 , it coincides with γ2 . We choose γ so that it belongs to K0 . 7. By means of Proposition 6.3, we show that K0 is a canonical domain enclosing γ (actually, it is the maximal canonical domain, but, we do not need this fact). Pick a point ζ0 in K0 . We need only to check that, in K0 , there is a pre-canonical curve β connecting two internal points of γ and containing ζ0 . We assume that Im ζ0 ≥ 0. Due to the symmetry of K0 , β and γ1,2 with respect to the origin, for Im ζ0 < 0, the analysis is similar. There are four cases to be considered. 7a. First, we assume that ζ0 is to the left of “a”. Consider σ0 , a line of Stokes type ϕ Im ζ0 κ0 dζ = 0, containing ζ0 . As ζ0 is between “d” and “a”, and as the lines of Stokes type fibrate K0 , the line σ0 stays between the lines “a” and “d”, and also between “a” and “d”, the lines symmetric to “a” and “d” with respect to the real line. So, it stays vertical and has to intersect one of the lines γA and γB and one of the lines −γA and −γB , the symmetrics of γA and γB with respect to the origin. In fact, as both γA and −γA belong to the same family of lines of Stokes type as σ0 , it intersects γB and −γB . To construct the needed pre-canonical curve, we pick c1 and c2 , two internal points of σ0 , so that Im c1 < Im ζ0 < Im c2 . Consider α2 , the line of Stokes type from the same family as γB and beginning at c2 . As the lines of Stokes type fibrate K0 , choosing c2 close enough to γB , we get that (1) α2 stays in the part of K0 where Im κ0 > 0, and, thus, is vertical; (2) the segment of α2 beginning at c2 and going up intersects γA . As γA is a part of the right boundary of K0 , this segment intersects also γ . Similarly, by properly choosing c1 , one constructs α1 , a vertical line of Stokes type in K0 , connecting the point c1 to an internal point of γ situated below c1 . The needed pre-canonical line β goes from this point of γ along α1 to σ0 , then, along σ0 , to α2 , and, then along α2 to an internal point of γ . The construction of β is illustrated by Fig. 10. 7b. Assume that ζ0 is situated to the right of or on “a” and above “b”. The construction of the pre-canonical curve corresponds to Fig. 11, part B. 7c. If ζ0 is to the left of the line π + iR and either below “b” or on “b”, we construct β as in Fig. 11, part C. 7d. If ζ0 is either to the right of the line π + iR or on this line, then, we first construct a line β˜ as shown in Fig. 11, part D. This line contains δ1 and δ2 , two segments of the line π + iR. As, on the line π + iR, 0 < κ0 < π , this line and its segments δ1 and δ2 are canonical. But, the line β˜ is not yet pre-canonical : as, on π + iR, κ0 is real, the lines b2
b2
b0 b
b
b
'1
0
'1
b b
b '3
b'3
b '2
b
b
b
b
b
B
Æ2
'2
b
b
b
b b0
'1
0
'1
b b
1
b2
b
b
b
b
b
1
C
Fig. 11. The pre-canonical line β
'2
b b
'1
0
'1
b b
b '3
~
Æ1
b
b
1
D
b
b
0
Anderson Transitions for Almost Periodic Schrödinger Equations
51
of Stokes type are horizontal (i.e. are not vertical) at the points of π + iR. To correct this, we use the C 1 -stability of the canonical lines. We slightly deform the canonical line ˜ π + iR so as to get a pre-canonical line from its deformed segments δ1 and δ2 and β. We have shown that, for any point ζ0 ∈ K0 , there exists a pre-canonical line β containing ζ0 as an internal point and connecting two internal points of the canonical ' line γ . As explained, this implies that K0 is canonical. This completes the proof. & We finish this section by noting that the relations (7.6) and (7.5) imply that any domain obtained from K0 by means of 2π -translations and/or the reflection with respect to the real line is canonical. 7.3.2. The domain K1 . Consider the simply connected regular domain K1∞ corresponding to Fig. 12, part A. The boundary of K1∞ consists of Stokes lines. This domain exists for all E in the interval J defined by (1.4). Let κ1 be the analytic continuation of κ0 from K0∞ to K1∞ through their common part. Consider the subdomain K1 ⊂ K1∞ corresponding to Fig. 12, part B. Its boundary contains the lines of Stokes type (with respect to κ1 ) passing by ζ1 and ζ2 (as shown in Fig. 12). Consider the horizontal strip |Im ϕ| ≤ Y (K1 ), where K1 and K1∞ coincide. One has Proposition 7.2. One can choose the points ζ1 and ζ2 so that the domain K1 exists and so that Y (K1 ) > Im ϕ3 is as large as desired. The domain K1 is canonical with respect to the branch κ1 and the points ζ1 and ζ2 . The proof of this statement being analogous to the one of Proposition 7.1, we omit it.
0
b'1
'3 b
'3 b
'2 b
'2 b
b
b
2
'2
b 2
'3
2
0
'1
b
b
A
b
b
2
1
b
2
'2
b 2
'3
2
B
Fig. 12. Canonical domains: K1∞ and K1
We finish this section by noting that κ1 satisfies the following symmetry relations: κ1 (2π − ϕ) = κ1 (ϕ),
(7.9)
κ1 (ϕ) = κ1 (ϕ).
(7.10)
52
A. Fedotov, F. Klopp
1 the canonical Bloch These relations follow from the definition of κ1 . Consider A± solutions constructed for the domain K1 and indexed so that κ1 is the quasi-momentum 1 . Discuss the corresponding functions ψ (x, ϕ) = ψ 1 (x, ϕ) and ω = ω1 (see of A+ ± ± ± ± Sect. 6.3). One has 1 1 (x, ϕ), ψ− (x, ϕ) = ψ+
1 1 ψ± (x, 2π − ϕ) = ψ± (x, ϕ).
(7.11)
Indeed, the first symmetry follows from the discussion at the end of Subsect. 5.1 as the interval [ϕ1 , 2π − ϕ1 ] is a connected component of the pre-image of a spectral band. The second symmetry holds as cos(2π − x) = cos(x). Furthermore, the definitions of ω± , see (6.13), and relations (7.11) imply that 1 (ϕ) = ω1 (ϕ), ω+ −
1 1 ω± (2π − ϕ) = −ω± (ϕ).
(7.12)
7.3.3. Bloch solutions along periodic curves. We need the following simple observations. Consider the two branches of ψ(x, E − α cos ϕ), the Bloch solution of (6.10) with W (ϕ) = α cos ϕ. Consider them on the domain D0 where the main branch κ0 was 0 (x, ϕ) so that their Bloch quasi-momenta are defined. We index these branches by ψ± equal to ±κ0 (ϕ). 0 analytically along the periodic curve γ described in Fig. 7. Then, Continue ψ± along γ , 0 0 ψ± (x, ϕ + 2π ) = ψ± (x, ϕ),
(7.13)
0 0 ω± (ϕ + 2π ) = ω± (ϕ),
(7.14)
0 are constructed by (6.13) in terms of ψ 0 . Equation (7.14) follows from the where ω± ± definition of the functions ω± and from (7.13). And (7.13) holds as (1) there are only two branches of the function ψ(x, E − α cos ϕ); (2) κ0 is the Bloch quasi-momentum 0 ; (3) κ is 2π -periodic along γ . of ψ+ 0 Recall that the derivative k0 (E) can vanish only inside the spectral bands of the periodic operator (1.3) (see Subsect. 5.2.2). So, on γ , we can fix an analytic branch of
q0 =
k0 (E − α cos ϕ). Recall that k0 (E) ∈ −iR+ for E < E1 (see Subsect. 5.2.2).
We fix the branch of q0 so that q0 ∈ e−iπ/4 R+ between ϕ1 and −ϕ1 . One has q0 (ϕ + 2π ) = q0 (ϕ),
ϕ ∈ γ.
(7.15)
This relation easily follows from the analytic properties of the main branch of the Bloch quasi-momentum k0 . 7.3.4. The domain K2 . Define the domain K2 by K2 = K0 + 2π ; by the remark concluding Sect. 7.3.1, the domain K2 is canonical. Let us discuss the solutions constructed 2 (x, ϕ), ω2 and q as the analytic by Theorem 6.1 on this domain. In K2 , define κ2 , ψ± 2 ± 0 0 and q to K along the pecontinuations of the functions κ0 , ψ± (x, E − α cos ϕ), ω± 0 2 riodic curve γ corresponding to Fig. 13. Recall that the analytic continuations of κ0 , 0 (x, E − α cos ϕ), ω0 and q along γ are 2π -periodic. This implies ψ± 0 ± Lemma 7.2. The domain K2 is canonical with respect to κ2 . Moreover, if f±0 are the consistent basis solutions constructed for K0 by Theorem 6.1 and normalized at the point ϕ = 0, then, on K2 , the functions f±2 (ϕ) = f±0 (ϕ − 2π ) have the same asymptotics as the consistent basis solutions constructed for K2 by Theorem 6.1 and normalized at the point ϕ = 2π .
Anderson Transitions for Almost Periodic Schrödinger Equations
53
'3 '2
0
'1
2
K2 : K1 : K0 :
Fig. 13. The three canonical domains
8. The Proof of Theorem 2.2 This section is devoted to the proof of Theorem 2.2. We begin with an observation on the analyticity in ϕ of the solutions constructed in Theorem 6.1. They can be analytically continued outside of K. Indeed, let S(Y1 , Y2 ) = {Y1 < Im ϕ < Y2 } be the smallest strip containing the domain K. Fix 0 < ν < (Y2 − Y1 )/2. Consider the domain Kν = {ζ ∈ K; Y1 + ν < Im ϕ < Y2 − ν}, and its horizontal width, w = sup |ϕ − ϕ |. Clearly, ϕ,ϕ ∈Kν Im ϕ=Im ϕ
w > 0. Assume that ε < w. Then, the functions f± , being defined for all x ∈ R and analytic in ϕ ∈ K, can be analytically continued in the whole strip {Y1 + ν < Im ϕ < Y2 − ν} using the consistency condition (6.3). To simplify the notations below, when speaking about the solutions f± constructed for a canonical domain K outside of this domain, we shall assume that ϕ ∈ {Y1 + ν < Im ϕ < Y2 − ν}.
8.1. The monodromy matrix. Let (f±0 ) be the solutions of (7.1) constructed by Theorem 6.1 for the canonical domain K0 . To this basis, we associate the matrix M˜ defined by 0 ˜ F 0 (u, ϕ + 2π ) = M(ϕ)F (u, ϕ),
f+0 . f−0
F0 =
(8.1)
In (8.1), changing the variables (u, ϕ) to the variables (x, φ) of the input Eq. (1.1), ˜ we see that M˜ is related to the monodromy matrix M by the formula M(ϕ) = M(φ), ϕ = εφ. The matrix M˜ is unimodular and ε-periodic (compare with (2.3)). We define ˜ the coefficients of the matrix M(ϕ) by m11 (ϕ) m12 (ϕ) ˜ M(ϕ) = . m21 (ϕ) m22 (ϕ)
54
A. Fedotov, F. Klopp
We prove Proposition 8.1. Under the assumptions of Theorem 2.2, there is a positive η such that M˜ is analytic in V = {|E − E0 | < η} × {|Im φ| ≤ Y /ε} and, in V, its coefficients admit the uniform asymptotics m11 (ϕ) = th · ei/ε (1 + o(1)), 2iπ m12 (ϕ) = iG · ei/ε (1 + o(1)) − tv · e ε (ϕ−ϕ(0) ) (1 + o(1)) , 2iπ m21 (ϕ) = −iG−1 · ei/ε (1 + o(1)) − tv · e− ε (ϕ−ϕ(0) ) (1 + o(1)) , m22 (ϕ) =
1 −i/ε 1 e (1 + o(1)) + ei/ε (1 + o(1)) th th 2π tv i 2π (ϕ−ϕ(0) ) e ε − (1 + o(1)) + e−i ε (ϕ−ϕ(0) ) (1 + o(1)) . th
Here, the phase , the coefficients th , tv are defined in (2.10) and (2.11), and ϕ1 iε ϕ2 (ω+ − ω− )dϕ , ϕ(0) = − (ω− − ω+ )dϕ − π, G = exp 2π ϕ1 0
(8.2)
where the integrals are taken along curves situated in K0 , and the functions ω± are the ones defined for K0 in 7.3.1. The function ϕ(0) is real analytic. We deduce Theorem 2.2 from Proposition 8.1 by changing the consistent basis; this is done in Sect. 8.6. The next sections are devoted to the proof of Proposition 8.1. 8.2. The canonical domains K0 , K1 , K2 and the associated transition matrices. It would be easy to compute the asymptotics of M˜ in a subset of K0 where we know the asymptotic behavior of both F 0 (u, ϕ + 2π) and F 0 (u, ϕ). Unfortunately, a quick look at Fig. 9, part B, shows us that K0 ∩ (K0 + 2π ) = ∅. Therefore, we need to introduce additional canonical domains. The first domain we need is K1 (see Fig. 12 B and Subsect. 7.3.2). Let (f±1 ) be the consistent basis constructed by Theorem 6.1 in K1 . As (f±0 ) and (f±1 ) are both bases of solutions of (7.1), we can write 1 f (8.3) F 0 (u, ϕ) = T1 (ϕ)F 1 (u, ϕ), F 1 = +1 , f− where the matrix T1 is independent of u. As (f±0 ) and (f±1 ) satisfy the consistency condition 6.3, the matrix T1 is ε-periodic. The second canonical domain we need is K2 , see Sect. 7.3.4. Define f±2 (ϕ) = f±0 (ϕ − 2π). It is a consistent basis of solutions with standard asymptotic behavior on K2 . As (f±2 ) and (f±1 ) are two bases of solutions of (7.1), we have 2 f F 1 (u, ϕ) = T2 (ϕ)F 2 (u, ϕ), F 2 = +2 , (8.4) f− where the matrix T2 is independent of u and ε-periodic in ϕ. Putting (8.3) and (8.4) together, we get F 0 (u, ϕ) = T1 (ϕ)T2 (ϕ)F 2 (u, ϕ) = T1 (ϕ)T2 (ϕ)F 0 (u, ϕ − 2π ).
Anderson Transitions for Almost Periodic Schrödinger Equations
55
Hence, the monodromy matrix reads ˜ M(ϕ) = T1 (ϕ + 2π )T2 (ϕ + 2π ).
(8.5)
To prove Proposition 8.1, we study the matrices T1 and T2 . We denote their coefficients as follows a (ϕ) b1 (ϕ) a (ϕ) b2 (ϕ) T1 (ϕ) = 1 , T2 (ϕ) = 2 . (8.6) c1 (ϕ) d1 (ϕ) c2 (ϕ) d2 (ϕ) Below, when describing the asymptotics of the matrices T1 and T2 and when computing j these asymptotics, the contour integrals of κj and ω± , j = 0, 1, 2, are taken along curves in Kj in all the cases when no other choice is described explicitly. One proves Proposition 8.2. Under the assumptions of Theorem 2.2, there is a positive η such that T1 is analytic in V = {|E − E0 | < η} × {|Im φ| ≤ Y /ε} and, in V, its coefficients admit the uniform asymptotics i
a1 (ϕ) = e ε
π
π
κ0 dϕ+
0 dϕ ω+
(1 + o(1)),
(8.7)
· o(e−δ1 /ε ), π ϕ 1 i π 1 2i κ dϕ ϕ1 0 c1 (ϕ) = −ie− ε 0 κ0 dϕ e ε ϕ1 1 e 0 ω− dϕ− π ω+ dϕ (1 + o(1))
(8.8)
b1 (ϕ) = e
i ε
0
π 0
0
κ0 dϕ
i
−e−2 ε d1 (ϕ) = e
− εi
π 0
ϕ π
e
ϕ2 0
ϕ
0 dϕ− ω−
π
0 dϕ −2π i (ϕ−π) ω+ ε
2
e
(1 + o(1)) , (8.9)
π
κ0 dϕ+
2 (κ −π)dϕ 0
0 0 ω− dϕ
(1 + o(1)).
(8.10)
In (8.8), δ1 is a positive constant. and Proposition 8.3. Under the assumptions of Theorem 2.2, there is a positive η such that T2 is analytic in V = {|E − E0 | < η} × {|Im φ| ≤ Y /ε} and, in V, its coefficients admit the uniform asymptotics i
a2 (ϕ) = e ε b2 (ϕ) = ie
2π π
− εi
2π
κ2 dϕ+
2π
κ2 dϕ
2 dϕ ω+
π
e
2 εi
(1 + o(1)),
2π −ϕ
1
κ1 dϕ
(8.11) 1
1 dϕ− ω+
2π −ϕ1
2 dϕ ω−
(1 + o(1)) π 1 2π 2 i ϕ¯2 ω dϕ− ϕ¯ ω+ dϕ 2π i (ϕ−π) 2 − e2 ε π (κ1 −π)dϕ e ϕ¯2 − e ε (1 + o(1)) , i
c2 (ϕ) = e ε d2 (ϕ) = e
π
2π
− εi
π
κ2 dϕ
2π π
π
e
2π −ϕ π
· o(e−δ2 /ε ), 2π
κ2 dϕ+
π
2 dϕ ω−
(1 + o(1)).
2π
(8.12)
(8.13) (8.14)
In (8.13), δ2 is a positive constant. The next sections are devoted to the proof of Propositions 8.2 and 8.3. As it is similar to the proof of Proposition 8.2, we do not give a detailed proof of Proposition 8.3; we just describe the starting points and partial results. From now on, C denotes different positive constants independent of ϕ and ε.
56
A. Fedotov, F. Klopp
8.3. The asymptotics of T1 . By definition, we have 1 w(f+0 , f−1 ), w1 1 c1 (ϕ) = w(f−0 , f−1 ), w1
1 w(f+0 , f+1 ), w1 1 d1 (ϕ) = − w(f−0 , f+1 ), w1 d d w1 = w(f+1 , f−1 ) = f−1 f+1 − f+1 f−1 . du du
a1 (ϕ) =
b1 (ϕ) = −
(8.15)
j
The solutions (f± ) are normalized at the points νj , j = 0, 1 where ν0 = 0,
ν1 = π. j
For convenience, we recall the asymptotics of f± j
f± (u, ϕ) = e
± εi
ϕ
νj
κj dϕ
j
(A± (u, ϕ) + o(1)),
j
A± (x, ϕ) = qj (ϕ)e
ϕ
νj
j
ω± dϕ
ϕ ∈ Kj ,
(8.16)
j
ψ± (x, ϕ).
(8.17)
1 , ω1 and q are the analytic continuations of respectively κ , ψ 0 , ω0 Recall that κ1 , ψ± 1 0 ± ± ± and q0 along the periodic curve γ from K0 to K1 (see Sect. 7.3.3). Note that, in view of Lemma 6.2, one has j
j
j
j
wj = w(f+ , f− ) = qj2 w(ψ+ , ψ− )|ϕ=νj + o(1).
(8.18)
These Wronskians are non-zero. Indeed, j
j
• as νj are not branch points of the complex momentum, ψ+ and ψ− are linearly independent; • q 2 = k (E − α cos ϕ) only vanishes at points in the pre-image of the spectral gaps ∪∞ l=1 (E2l , E2l+1 ) with respect to the mapping E : ϕ → E − α cos ϕ (see Sect. 5.2.2), and (νj )j =1,2 do not belong to this pre-image. 8.3.1. Three substrips. We assume that K0 and K1 are chosen so that Y (K0 ), Y (K1 ) > Im ϕ3 , the definitions and properties of Y (K0 ) and Y (K1 ) are described in Sects. 7.3.1 and 7.3.2. We study the transition matrix T1 in the strip −Y0 < Im ϕ < Y0 ,
Y0 = Im ϕ3 .
When computing the asymptotics of T1 , we divide this strip into three different smaller substrips called (I), (II) and (III) (see Fig. 14). Each of these strips requires a different type of computation.
Anderson Transitions for Almost Periodic Schrödinger Equations
57
8.3.2. Properties of the analytic objects of the complex WKB method in the substrips. In j j the section, we compare κj , ψ± , ω± and qj for j = 0 and j = 1 in each of the different strips (I), (II) and (III). In the strip (II), we have a non-empty intersection for K0 and K1 . In the intersection, by definition, we have κ1 = κ0 ,
1 0 ψ± = ψ± ,
1 0 ω± = ω± ,
q1 = q0 .
and
(8.19)
In the strip (I), consider the common boundary of K0 and K1 (see Fig. 14). It is the Stokes line beginning at ϕ = ϕ1 and going downwards. Along this line, we get κ1 (ϕ + 0) = −κ0 (ϕ − 0), 1 0 ψ± (u, ϕ + 0) = ψ∓ (u, ϕ − 0), 1 0 ω± (ϕ + 0) = ω∓ (ϕ − 0),
(8.20) (8.21) (8.22) (8.23)
q1 (ϕ + 0) = iq0 (ϕ − 0),
where ϕ + 0 (resp. ϕ − 0) denotes the limit taken from the right (resp. left). These formulae hold as ϕ1 is a pre-image of E1 (the infimum of the first band of the periodic Schrödinger operator) by the mapping E : ϕ → E − α cos ϕ. Indeed, formula (8.20) holds as ϕ1 is a branch point of the complex momentum, as ϕ1 is of j square root type and as κ0 (ϕ1 ) = 0. Formula (8.21) holds as ψ± are just branches of the Bloch solution ψ(x, E), E = E − α cos ϕ, which has only two different branches, and j as E1 is a branch point of this function. The relation for ω± follows from (8.21) and the definition of ω± . The relation for q follows from the definition of q as E1 is a square root branch point of the Bloch quasi-momentum. We also notice that, as the imaginary part of the main branch κ0 is positive in K0 along the common boundary of K0 and K1 , the relation (8.20) implies that, along this boundary, one has Im κ1 (ϕ + 0) = −Im κ0 (ϕ − 0) < 0.
'3 '2
(8.24)
b (III)
b
(II) 0
b
'1
2
b
b
2
'2
b2
'3
(I)
K1
:
K0
:
Fig. 14. Going from K0 to K1
58
A. Fedotov, F. Klopp
In the strip (III), the common boundary of K0 and K1 is the interval [ϕ2 , ϕ3 ], that is the Stokes line joining ϕ2 to ϕ3 . Along [ϕ2 , ϕ3 ], we have κ1 (ϕ + 0) = 2π − κ0 (ϕ − 0), 1 ψ± (u, ϕ 1 ω± (ϕ
+ 0) =
(8.25)
0 ψ∓ (u, ϕ − 0), 0 ω∓ (ϕ − 0),
(8.26)
+ 0) = q1 (ϕ + 0) = −iq0 (ϕ − 0),
(8.27) (8.28)
Im κ1 (ϕ + 0) = −Im κ0 (ϕ − 0) < 0.
(8.29)
and
8.3.3. A continuation lemma. In the strips (I) and (III), the δ-admissible sub-domains of canonical domains K0 and K1 are separated by a distance 2δ. To get uniform asymptotics for f±0 in K1 , we use Lemma 8.1. Let ϕ− , ϕ+ , ϕ0 be fixed points (see Figs. 16 and 17) such that • Im ϕ− = Im ϕ+ ; • there is no branch point of ϕ → κ(ϕ) on the interval [ϕ− , ϕ+ ]; • for ϕ0 ∈ (ϕ− , ϕ+ ), q(ϕ0 ) = 0. Fix a continuous branch of κ on [ϕ− , ϕ+ ]. Let f (u, ϕ), f± (u, ϕ) be solutions of (7.1) for ϕ ∈ [ϕ− , ϕ+ ] and u ∈ [−U, U ] satisfying condition (6.3), and such that: i
ϕ
κdϕ
(1) f (u, ϕ) = e ε ϕ0 (A+ (u, ϕ) + o(1)) for ϕ ∈ [ϕ− , ϕ0 ] when ε → 0, and the asymptotic is differentiable in u; ±i
ϕ
κdϕ
(A± (u, ϕ) + o(1)) for ϕ ∈ [ϕ− , ϕ+ ] when ε → 0, and the (2) f± (u, ϕ) = e ε ϕ0 asymptotic is differentiable in u. Here, (A± ) are canonical Bloch solutions with the Bloch quasi-momenta ±κ. Then, • if Im(κ(ϕ)) > 0 for all ϕ ∈ [ϕ− , ϕ+ ], there exists C > 0 such that, for ε > 0 small enough, 1 ϕ df (u, ϕ) + |f (u, ϕ)| ≤ Ce ε ϕ0 |Imκ|dϕ , ϕ ∈ [ϕ0 , ϕ+ ]; (8.30) du • if Im(κ(ϕ)) < 0 for all ϕ ∈ [ϕ− , ϕ+ ], then i
f (u, ϕ) = e ε
ϕ
ϕ0
κdϕ
(A+ (u, ϕ) + o(1)),
ϕ ∈ [ϕ0 , ϕ+ ],
(8.31)
and the asymptotic is differentiable in u. Remark 8.1. This lemma reflects a heuristic that says that the WKB asymptotics of a solution stays valid as long as its leading term is exponentially increasing. A similar statement for a class of difference equations can be found in [6].
Anderson Transitions for Almost Periodic Schrödinger Equations
59
Proof. Note that w(f+ , f− ) = w(A+ , A− )|ϕ=ϕ0 + o(1). As q(ϕ0 ) = 0, by Lemma 6.2, the leading term in this formula is a non-zero constant, and f± are linearly independent. So, we can write f = a(ϕ)f+ + b(ϕ)f− ,
(8.32)
where a(ϕ) =
w(f, f− ) w(f+ , f− )
w(f+ , f ) . w(f+ , f− )
b(ϕ) =
and
By (6.3), a and b are ε-periodic in ϕ. As, on [ϕ− , ϕ0 ], the solutions f and f+ have the same asymptotics, one computes a(ϕ) = 1 + o(1).
(8.33)
The coefficient a is ε-periodic in ϕ so that (8.33) holds on [ϕ− , ϕ+ ]. Let us now estimate the coefficient b. We start with the case when Im(κ(ϕ)) > 0 for all ϕ ∈ [ϕ− , ϕ+ ]. Then, on [ϕ− , ϕ0 ], we have 2i ϕ 2 ϕ0 κdϕ Im(κ)dϕ |w(f+ , f )| ≤ C e ε ϕ0 . ≤ Ce ε ϕ So, for ϕ ∈ [ϕ0 − ε, ϕ0 ], we get that |b(ϕ)| ≤ C; hence, by ε-periodicity, we have |b(ϕ)| ≤ C for ϕ ∈ [ϕ− , ϕ+ ]. On the other hand, on [ϕ0 , ϕ+ ], we have |f+ | ≤ Ce
− 1ε
ϕ
ϕ0
Im(κ)dϕ
1
, |f− | ≤ Ce ε
ϕ
ϕ0
Im(κ)dϕ
.
So that, by (8.32), we get (8.30) on [ϕ 02i, ϕϕ+ ]. IfIm(κ(ϕ)) < 0 for all ϕ ∈ [ϕ− , ϕ+ ] then, κdϕ for ϕ ∈ [ϕ− , ϕ0 ], we get |b(ϕ)| ≤ C e ε ϕ0 . Using this estimate for ϕ ∈ [ϕ− , ϕ− +ε] and the ε-periodicity, for ϕ ∈ [ϕ− , ϕ+ ], we get |b(ϕ)| ≤ Ce
− 2ε
ϕ
0 ϕ−
|Im(κ)|dϕ
.
Hence, by (8.32) and (8.33), on [ϕ0 , ϕ+ ], we have f = a(ϕ)f+ + b(ϕ)f− ϕ ϕ i ϕ κdϕ − 2iε ϕ κdϕ − 2ε ϕ 0 |Im(κ)|dϕ ϕ ε − 0 0 A+ (u, ϕ) + O e + o(1) =e e i
= eε
ϕ
ϕ0
κdϕ
(A+ (u, ϕ) + o(1)) .
This proves (8.31) and completes the proof of Lemma 8.1.
' &
Remark 8.2. Assume that, for E = E0 , (1) the functions f± are two solutions constructed by Theorem 6.1 on some canonical domain K , (2) the interval [ϕ− , ϕ+ ] is in the δ-admissible subdomain of K , (3) the function f is a solution constructed by Theorem 6.1 on a canonical domain K , (4) the interval [ϕ− , ϕ0 ] is located in its δ-admissible subdomain of K . Then, the proof of Lemma 8.1 shows the asymptotics and estimates of f obtained by Lemma 8.1 are uniform in a constant neighborhood of E0 .
60
A. Fedotov, F. Klopp
8.3.4. Complex values of E and admissible subdomains. Let E0 ∈ J be the point fixed in Theorem 2.2. And let K0 and K1 be the canonical domains corresponding to this value of E. Fix δ > 0. We describe the precise choice of δ later. To get asymptotics of T1 uniformly in E, we do not work with K0 and K1 , but j with their δ-admissible subdomains. More precisely, we use the asymptotics of f± , j = 1, 2, given by Theorem 6.1 only for ϕ in these admissible subdomains. Then, in view of Proposition 6.1, these asymptotics are valid and uniform in E in a constant complex neighborhood of the point E0 fixed in Theorem 2.2. This guarantees that the asymptotics of the coefficients of T1 are valid and uniform in this complex neighborhood. Let Y be chosen as in Theorem 6.1. In the sequel, we fix δ sufficiently small so that δ < Y0 − Y = Im ϕ3 − Y . In the sequel, if no other choice is indicated explicitly, K0 and K1 denote the δ-admissible subdomains of K0 and K1 .
8.3.5. Auxiliary canonical domains. To use the Continuation Lemma, Lemma 8.1, “to bridge” the gap between the admissible subdomains of K1 and K0 in the strip (III), we need two linearly independent solutions of Eq. (1.1) with standard asymptotic behavior in the strip (III) between these admissible subdomains. So, we introduce an auxiliary canonical domain K˜ 0 corresponding to Fig. 15, part A. Let κ˜ 0 be the analytic continuation of κ0 through the interval (ϕ2 , ϕ3 ). The domain K˜ 0 is canonical with respect to κ˜ 0 . We construct K˜ 0 by means of Proposition 6.3. This construction being similar to (and much simpler than) the one of K0 done in Sect. 7.3.1, we only illustrate it in parts B and C of Fig. 15. Here, the dotted lines are lines of Stokes type Im (κ˜ 0 − π )dϕ = Const, the thin continuous lines of Stokes type Im κ˜ 0 dϕ = Const, and the thick lines are canonical. Along the continuous lines Im ϕ decays as Re ϕ increases. Note that the domain K˜ 0 can be constructed so that it comprises any given compact subinterval of (ϕ2 , ϕ3 ).
2 b
b'3
b
~
'2 b
b1
b'3
B
K0
A
2 b
b
C '2 b
b1
Fig. 15. The local canonical domain
Also, we need to apply Lemma 8.1 to bridge the gap between the admissible subdomains of K0 and K1 in the strip (I). In this case, the auxiliary canonical domain is K0 , the symmetric of K0 with respect to the real line. We are now ready to start with the computations of the coefficients of T1 .
Anderson Transitions for Almost Periodic Schrödinger Equations
61
8.3.6. The asymptotics of d1 (ϕ). By (8.15), we need to compute the asymptotics of w(f−0 , f+1 ) when ε → 0. Strip (I). In view of (8.24), Lemma 8.1 allows us to “continue” the asymptotics of f−0 from K0 to K1 across the common boundary of the canonical domains in the strip (I). The consistent basis used to apply Lemma 8.1 is the basis corresponding to the auxiliary canonical domain K0 . Let the interval [ϕ− , ϕ+ ] be as in Fig. 16, i.e. the points ϕ− and ϕ0 are in the admissible subdomain of K0 , the point ϕ+ is situated in the admissible subdomain of K1 , and the point ϕb is on the common boundary of the canonical domains. Lemma 8.1 implies that the asymptotics (8.16) of f−0 stays valid on [ϕb , ϕ+ ]. We rewrite it in the form f−0 = e
− εi
ϕ1 0
κ0 dϕ− εi
ϕ
b ϕ1
κ0 dϕ − εi
e
ϕ
κ0 dϕ
ϕb
0 (A− + o(1)),
where the last integral is taken along [ϕ− , ϕ+ ]. By (8.16) and as κ1 = κ0 in the strip (II), and κ1 = −κ0 in the strip (I), for ϕ ∈ [ϕ− , ϕ+ ] ∩ K1 , we can write i
f+1 = e ε
ϕ π
1
κ0 dϕ− εi
ϕ
b ϕ1
κ0 dϕ − εi
e
ϕ
ϕb
κ0 dϕ
1 (A+ + o(1)),
where the last integral is taken along [ϕ− , ϕ+ ]. So, i
|w (f−0 , f+1 )| ≤ C|e− ε
π 0
κ0 dϕ
| · |e
− 2iε
As the points ϕ1 and ϕb are on the Stokes line Im 1.
ϕ
b ϕ1
κ0 dϕ
ϕ
ϕ1 κ0 dϕ
K0
| · |e
− 2iε
ϕ
ϕb
κ0 dϕ
= 0, one has |e
|.
− εi
ϕ
b ϕ1
κ0 dϕ
|=
'1
'
'b '0
K1
' '+
Fig. 16. Continuation below ϕ1
Now, as ϕ has to be in an admissible sub-domain of K1 , we know that Re (ϕ−ϕb ) ≥ δ. Assuming that δ < Re (ϕ − ϕb ) ≤ δ + ε, we get i
|w(f−0 , f+1 )| ≤ C|e− ε
π 0
κ0 dϕ
|eCδ/ε .
(8.34)
By the ε-periodicity of the Wronskian, it holds everywhere on the line {Im (ϕ) = Im (ϕb )}. The estimate (8.34) is valid as long as the interval [ϕ− , ϕ+ ] stays off the δ-neighborhood of the branch points, and as long as its ends are in the admissible subdomains of K0 and K1 . Hence, for −Y0 + δ ≤ Im ϕ ≤ −δ, we have i
|d1 (ϕ)| ≤ C|e− ε
π 0
κ0 dϕ
|eCδ/ε .
(8.35)
62
A. Fedotov, F. Klopp
This estimate is uniform along horizontal lines Im ϕ = C. Strip (II). In this case, by (8.19), for ϕ ∈ K0 ∩ K1 , we have i
f−0 = e− ε
π 0
π
κ0 dϕ+
0
ϕ
0 dϕ − i ω− ε
e
κ0 dϕ
π
1 (A− (u, ϕ) + o(1)).
Comparing this with the formula ϕ
i
f+1 = e ε
κ0 dϕ
π
1 (A+ (u, ϕ) + o(1)),
one easily obtains i
w(f−0 , f+1 ) = −e− ε
π 0
κ0 dϕ
e
π 0
0 dϕ ω−
(w(f+1 , f−1 ) + o(1)).
Hence, i
d1 (ϕ) = e− ε
π 0
κ0 dϕ
e
π 0
0 dϕ ω−
(1 + o(1)).
(8.36)
By the ε-periodicity of the Wronskian, (8.36) is uniform along horizontal lines in the strip δ ≤ Im ϕ ≤ Im ϕ2 − δ. Strip (III). In this region, we satisfy ourselves with an estimate on w(f−0 , f+1 ). In view of (8.29), by Lemma 8.1, we can continue the asymptotics of f−0 from [ϕ− , ϕ0 ] to the whole interval [ϕ− , ϕ+ ], see Fig. 17. The consistent basis needed to apply Lemma 8.1 is obtained by applying Theorem 6.1 for the auxiliary canonical domain K˜ 0 . This domain comprises a part of the common boundary of the canonical domains K0 and K1 , i.e. of the segment (ϕ2 , ϕ3 ) (see Figs. 14, 15 and 17). On the segment [ϕb , ϕ+ ], we have f−0 = e
− εi
ϕ2 0
κ0 dϕ− εi
ϕ
b ϕ2
κ0 dϕ − εi
e
ϕ
ϕb
κ0 dϕ
0 (A− + o(1)),
(8.37)
where the last integral is taken along [ϕ− , ϕ+ ]. In view of (8.19) and (8.25), we also have i
f+1 = e ε
ϕ π
2
κ0 dϕ− εi
ϕ
ϕ2
κ0 dϕ
i
1 e ε (ϕ−ϕ2 )2π (A+ + o(1)),
ϕ ∈ [ϕ− , ϕ+ ] ∩ K1 ,
(8.38)
where, in the last integral, we integrate the analytic continuation of κ0 from K0 to K1 through (ϕ2 , ϕ3 ), and this integral is taken along a curve in K1 . Representations (8.37) and (8.38) imply i π −2 i ϕ (κ −π)dϕ |d1 (ϕ)| ≤ C e− ε 0 κ0 dϕ · e ε ϕ2 0 .
(III)
K0
'
'3
'0 '2
K1
'b ' '+
Fig. 17. Continuation through [ϕ2 , ϕ3 ]
Anderson Transitions for Almost Periodic Schrödinger Equations
63
Recall that, for ϕ ∈ [ϕ2 , ϕ3 ], one has κ0 (ϕ) = π + it, where t ≥ 0. Therefore, if |ϕ − ϕb | is of order δ, we obtain i
|d1 (ϕ)| ≤ C|e− ε
π 0
κ0 dϕ
|eCδ/ε .
(8.39)
Since d1 is ε-periodic, this estimate is valid uniformly along horizontal lines in the strip {Im ϕ2 + δ ≤ Im ϕ ≤ Y0 − δ} (recall that Y0 = Im ϕ3 ). Uniform asymptotics. As d1 (ϕ) is ε-periodic in ϕ and analytic in a strip {−Y0 ≤ Imϕ ≤ Y0 }, we can expand d1 in a Fourier series with exponentially decreasing coefficients ϕ 1 ϕ0 +ε 2iπn ϕε δn e where δn = d1 (ϕ)e−2iπn ε dϕ (8.40) d1 (ϕ) = ε ϕ0 n∈Z
for any ϕ0 ∈ {−Y0 ≤ Imϕ ≤ Y0 }. For n = 0, we get i
|δn | ≤ C|e− ε
π
|e−2π|n|(Y0 −δ)/ε eCδ/ε .
κ0 dϕ
0
(8.41)
To prove this estimate for n < 0, one uses (8.39) and (8.40) with Im ϕ0 = Y0 − δ. In the case of n > 0, one uses estimate (8.35) and (8.40) with Im ϕ0 = −Y0 + δ. Furthermore, by means of the estimate (8.36) and (8.40) with Im ϕ0 = δ, we get i
δ0 = e− ε
π 0
κ0 dϕ
e
π
0 dϕ ω−
0
(1 + o(1)).
(8.42)
Choose δ < min{2π/C, Y0 − Y }. Then, (8.41) and (8.42) imply i
d1 (ϕ) = e− ε
π 0
π
κ0 dϕ+
0 dϕ ω−
0
(1 + o(1)),
uniformly along horizontal lines in the strip {|Im ϕ| ≤ Y }. 8.3.7. An estimate for b1 (ϕ). We only estimate b1 when ε → 0. Strip (I). Let γ be the Stokes line beginning at ϕ = ϕ1 and bordering K0 and K1 . In view of (8.24), Lemma 8.1 only gives us an estimation on f+0 when we cross γ along [ϕ− , ϕ+ ] (see Fig. 16). This estimation is i
|f+0 | ≤ C|e ε
ϕ1 0
κ0 dϕ+ εi
ϕ
b ϕ1
κ0 dϕ+ εi
ϕ
0 ϕb
κ0 dϕ
ϕ
1
|e ε
ϕ0
|Imκ0 |dϕ
, ϕ ∈ [ϕ− , ϕ+ ] ∩ K1 .
Recall that κ0 = κ1 in the strip (II) and that, in the strip (I), one has κ0 = −κ1 . Therefore, i
|f+1 | ≤ C|e ε
ϕ π
1
κ0 dϕ− εi
ϕ
b ϕ1
κ0 dϕ− εi
ϕ
0 ϕb
κ0 dϕ
1
|e ε
ϕ
ϕ0
|Im κ0 |dϕ
, ϕ ∈ [ϕ− , ϕ+ ] ∩ K1 .
Note that the first derivatives with respect to x of the solutions satisfy the same estimates. Therefore, for ϕ on [ϕb , ϕ+ ] and inside the admissible subdomain of K1 , we obtain i
|w(f+0 , f+1 )| ≤ C|e ε
ϕ1 0
κ0 dϕ
2
|e ε
ϕ
ϕ0
|Imκ0 |dϕ
.
We have used the fact that κ0 is real on [ϕ1 , π ]. Now, the only restriction we have on ϕ0 is that it has to be in the admissible subdomain of K0 ; hence, we choose |ϕ0 − ϕ| ∼ 2δ, where 2δ is the distance between the admissible subdomains of K0 and K1 . So, for −Y0 + δ ≤ Im ϕ ≤ −δ, uniformly along horizontal lines, we have δ
i
|b1 (ϕ)| ≤ CeC ε |e ε
ϕ1 0
κ0 dϕ
δ
i
| = eC ε |e ε
π 0
κ0 dϕ
|.
(8.43)
64
A. Fedotov, F. Klopp
Strip (II). We don’t need to study b1 here. Strip (III). In view of (8.29), Lemma 8.1 gives us only an estimate on f+0 in K1 . Using the notations of Fig. 17, we get i
|f+0 | ≤ C|e ε
π 0
κ0 dϕ+ εi
ϕ π
2
κ0 dϕ+ εi
ϕ
b ϕ2
κ0 dϕ
i
eε
ϕ
0 ϕb
κ0 dϕ
1
|e ε
ϕ
|Im κ0 |dϕ
, ϕ ∈ [ϕ− , ϕ+ ] ∩ K1 , ϕ0
(8.44)
where the last integral is taken along [ϕ− , ϕ+ ]. We also have i
|f+1 | ≤ C|e ε
ϕ π
2
κ1 dϕ+ εi
ϕ
b ϕ2
ϕ
κ1 dϕ+ εi
0 ϕb
κ1 dϕ
ϕ
1
|e ε
ϕ0
|Im κ1 |dϕ
, ϕ ∈ [ϕ− , ϕ+ ] ∩ K1 . (8.45)
In the last two integrals, κ1 is the analytic continuation of the branch κ1 through [ϕ2 , ϕ3 ]. The derivatives of f+0 and f+1 also satisfy (8.44) and (8.45). This and the relations (8.19) and (8.25) imply that i
|w(f+0 , f+1 )| ≤ C|e ε
π 0
κ0 dϕ
i
||e2 ε
ϕ π
2
κ0 dϕ+ 2πε i (ϕ0 −ϕ2 )
2
|e ε
ϕ
ϕ0
|Imκ0 |dϕ
.
ϕ The branch κ0 is positive along [π, ϕ2 ] and Im(ϕ0 −ϕ2 ) > 0. Hence, Im ( π 2 κ0 dϕ+ ϕmain π ϕ20 dϕ) > 0. So, there exists δ1 > 0 (independent of ε and of δ) such that δ
i
|b1 (ϕ)| ≤ CeC ε |e ε
π
κ0 dϕ
0
|e−δ1 /ε ,
uniformly along horizontal lines in the strip Im ϕ2 + δ ≤ Im ϕ ≤ Y0 − δ. Uniform asymptotics. Using the same method as for d1 , in the strip {|Imϕ| ≤ Y }, we get i
b1 (ϕ) = e ε
π 0
κ0 dϕ
· o(e−δ1 /ε ).
8.3.8. The asymptotics of a1 (ϕ). By (8.15), we need to compute the asymptotics of w(f+0 , f−1 ) when ε → 0. The computations being similar to those made for b1 and d1 , we give only the results. Strip (I). For −Y0 + δ ≤ Im ϕ ≤ −δ, δ
i
|a1 (ϕ)| ≤ CeC ε |e ε
π 0
κ0 dϕ
|.
Strip (II). For 0 < δ ≤ Im ϕ ≤ Im ϕ2 − δ, we get i
a1 (ϕ) = e ε
π 0
κ0 dϕ
e
π 0
0 dϕ ω+
(1 + o(1)).
Strip (III). For Im ϕ2 + δ ≤ Im ϕ ≤ Y0 − δ, we get δ
i
|a1 (ϕ)| ≤ CeC ε |e ε
π 0
κ0 dϕ
|.
Uniform asymptotics. Estimating the Fourier coefficients of a1 , uniformly in the strip {|Imϕ| ≤ Y }, we get that i
a1 (ϕ) = e ε
π 0
π
κ0 dϕ+
0
0 dϕ ω+
(1 + o(1)).
Anderson Transitions for Almost Periodic Schrödinger Equations
65
8.3.9. The asymptotics of c1 (ϕ). Now, we need to compute the asymptotics of w(f−0 , f−1 ) when ε → 0. Strip (I). In view of (8.20), we can use Lemma 8.1 to “continue” the asymptotic expansion for f−0 from K0 to K1 along a horizontal line (see Fig. 16). This allows us to get an asymptotic of c1 . We use formulae (8.20)–(8.23). The asymptotic expansion for f−0 on [ϕb , ϕ+ ] is ϕ 0 i ϕ 0 f−0 (u, ϕ) = e− ε 0 κ0 dϕ · q0 e 0 ω− dϕ (ψ− + o(1)) . Here, we integrate along a curve going from K0 to K1 below the point ϕ1 ; the functions 0 and q are the analytic continuations of these functions along this curve. We κ0 , ω− 0 use (8.20) and compute ϕ1 ϕ ϕ κ0 dϕ = κ0 dϕ − κ1 dϕ 0 0 ϕ1 π ϕ ϕ1 (8.46) κ0 dϕ − κ0 dϕ − κ1 dϕ. = ϕ1
0
Similarly, by (8.22), one obtains ϕ1 ϕ 0 0 ω− dϕ = ω− dϕ − 0
0
ϕ1 π
π
0 ω+ dϕ +
ϕ π
1 ω+ dϕ,
(8.47)
Using (8.46), (8.47), (8.21) and (8.23), we get f−0 (u, ϕ) = −ie
− εi
ϕ1 0
κ0 dϕ+ εi
ϕ1
π
κ0 dϕ+
ϕ1
0
ϕ
0 dϕ− ω−
π
1
0 dϕ ω+
i
eε
ϕ π
κ1 dϕ
1 (A+ (u, ϕ) + o(1)) .
On the other hand, we know that ϕ
i
f−1 (u, ϕ) = e− ε
π
κ1 dϕ
1 (A− (u, ϕ) + o(1)).
Therefore, one has i
w(f−0 , f−1 ) = −i · e− ε
π 0
i κ0 dϕ 2 ε
e
π
ϕ1
κ0 dϕ
e
ϕ1 0
ϕ
0 dϕ− ω−
π
1
0 dϕ ω+
(w1 + o(1)),
that is i
c1 (ϕ) = −i · e− ε
π 0
i κ0 dϕ 2 ε
e
π
ϕ1
κ0 dϕ
e
ϕ1 0
ϕ
0 dϕ− ω−
π
1
0 dϕ ω+
(1 + o(1)).
(8.48)
This holds in the strip {−Y0 + δ ≤ Im ϕ ≤ −δ}, uniformly along horizontal lines. Strip (II). We do not need any information on c1 in this region. Strip (III). We use Lemma 8.1 to compute the asymptotics of c1 . Let the segment [ϕ− , ϕ+ ] be as in Fig. 17. In view of (8.29), for ϕ ∈ [ϕ− , ϕ+ ], Lemma 8.1 implies that ϕ 0 i ϕ 0 f−0 = e− ε 0 κ0 dϕ q0 e 0 ω− dϕ ψ− + o(1) . (8.49)
66
A. Fedotov, F. Klopp
In (8.49), the integration contour first goes from 0 to ϕ− in K0 , then to ϕ along [ϕ− , ϕ+ ]. Using (8.25), we rewrite the first integral in (8.49) in the form ϕ π ϕ2 ϕ κ0 dϕ = κ0 dϕ + 2 κ0 dϕ + 2π(ϕ − ϕ2 ) − κ1 dϕ. 0
π
0
Also, by (8.27), we get ϕ 0 ω− dϕ = 0
π 0
π
0 ω− dϕ
+
ϕ2 π
0 (ω−
0 − ω+ )dϕ
+
ϕ π
1 ω+ dϕ.
Combining these formulae with (8.26) and (8.28), we obtain i
f−0 = ie− ε
π 0
κ0 dϕ−2 εi
ϕ π
2 (κ −π)dϕ 0
i
e−2π ε (ϕ−π) e
ϕ2 0
ϕ
0 dϕ− ω−
π
2
0 dϕ ω+
i
eε
ϕ
κ1 dϕ
π
1 (A+ + o(1)) .
Using this in conjunction with i
f−1 = e− ε
ϕ
κ1 dϕ
π
1 (A− + o(1)),
we get i
c1 (ϕ) = ie− ε
π 0
κ0 dϕ−2 εi
ϕ π
2 (κ −π)dϕ 0
i
e−2π ε (ϕ−π) e
ϕ
ϕ2
0 dϕ− ω−
0
π
2
0 dϕ ω+
(1 + o(1)). (8.50)
This asymptotic is valid for Im ϕ2 + δ ≤ ϕ ≤ Y0 − δ uniformly along horizontal lines. Uniform asymptotics. To get global information on c1 , we let ϕ−π π −ϕ 1 ϕ0 +ε c1 (ϕ) = γn e2iπn ε , γn = c1 (ϕ)e2iπn ε dϕ. (8.51) ε ϕ0 n∈Z
To study γn with n > 0, we use (8.48) and (8.51) with Im ϕ0 = −Y0 + δ. This gives i
|γn | ≤ C|e− ε
π 0
κ0 dϕ
|e−
2π ε |n|Y0
,
In the case of n = 0, by (8.48) and (8.51) with Im ϕ0 = −δ, we get γ0 = −ie
− εi
π 0
κ0 dϕ+2 εi
π
ϕ1
κ0 dϕ
e
ϕ1 0
ϕ
0 dϕ− ω−
π
1
0 dϕ ω+
(1 + o(1)).
For n < 0, we use the asymptotic obtained in the strip (III) and (8.51) with Im ϕ0 = Y0 − δ. We obtain π −ϕ 1 ϕ0 +ε γ−1 = c1 (ϕ)e−2iπ ε dϕ ε ϕ0 i
= ie− ε
π 0
κ0 dϕ−2 εi
ϕ π
2 (κ −π)dϕ 0
e
ϕ2 0
ϕ
0 dϕ− ω−
π
2
0 dϕ ω+
(1 + o(1)),
and, for n < −1, one has ϕ0 +ε 1 i π 2i ϕ2 2π 2inπ π −ϕ ε |γn | = c1 (ϕ)e dϕ ≤ C|e− ε 0 κ0 dϕ− ε π (κ0 −π)dϕ |e− ε |n+1|(Y0 −δ) . ε ϕ0
Anderson Transitions for Almost Periodic Schrödinger Equations
67
Putting all this together, we get for ϕ in the strip {|Imz| ≤ Y } (recall that Y < Y0 − δ): c1 (ϕ) = − ie
− εi i
−e−2 ε
π 0
ϕ π
κ0 dϕ
e
2 εi
2 (κ −π)dϕ 0
e
π
ϕ1
ϕ2 0
κ0 dϕ
e
ϕ1 0
ϕ
0 dϕ− ω−
π
ϕ
0 dϕ− ω− 2
0 dϕ ω+
1
0 dϕ ω+
(1 + o(1)) i e−2π ε (ϕ−π) (1 + o(1)) ,
π
where o(1) is uniform in the strip {|Imz| ≤ Y }. This ends the proof of Proposition 8.2.
8.4. The asymptotics of T2 . One has 1 w(f+1 , f−2 ), w2 1 c2 (ϕ) = w(f−1 , f−2 ), w2
1 w(f+1 , f+2 ), w2 1 d2 (ϕ) = − w(f−1 , f+2 ), w2 d d w2 = w(f+2 , f−2 ) = f−2 f+2 − f+2 f−2 . du du
a2 (ϕ) =
b2 (ϕ) = −
(8.52) (8.53) (8.54)
These Wronskians are ε-periodic and, together with the solutions, analytic in ϕ. 8.4.1. Three substrips. The coefficients of T2 are computed in the strip −Y0 + δ ≤ Im ϕ ≤ Y0 − δ. Here, Y0 is the same as before, and δ is an arbitrarily small positive constant that is independent of ε. Again we divide this strip into three smaller strips denoted respectively by (I), (II) and (III), see Fig. 18.
K1
:
K2
:
'3 '2
b b
(I) 0
b
'1
b
2
(II)
b (III)
2
'2
b
Fig. 18. Going from K1 to K2
'1
2
68
A. Fedotov, F. Klopp
2 , ω2 and q are 8.4.2. Properties of analytic objects. Recall that the branches κ2 , ψ± 2 ± 1 , ω1 and q from K to K through their defined as analytic continuations of κ1 , ψ± 1 1 2 ± common part (or along the periodic curve described in Fig. 7 and Sect. 7.3.4). In the strip (II), K1 and K2 do intersect. In K1 ∩ K2 , we have
κ1 = κ2 ,
1 2 ψ± = ψ± ,
1 2 ω± = ω± ,
q2 = q1 .
Consider the strip (I). Consider the common boundary of K2 and K1 in this strip (see Fig. 18). It is the Stokes lines beginning at 2π − ϕ1 and going upwards. Along this line, one has κ1 (ϕ − 0) = −κ2 (ϕ + 0),
1 2 ψ± (u, ϕ − 0) = ψ∓ (u, ϕ + 0),
1 2 ω± (ϕ − 0) = ω∓ (ϕ + 0),
q1 (ϕ − 0) = iq2 (ϕ + 0),
and Im κ2 (ϕ + 0) = −Im κ1 (ϕ − 0) > 0. In the strip (III), the common boundary of K2 and K1 is a part of the interval [ϕ2 , ϕ3 ]. This interval is the Stokes line joining ϕ2 to ϕ3 . Along this line, one has κ1 (ϕ − 0) = 2π − κ2 (ϕ + 0),
1 2 ψ± (u, ϕ − 0) = ψ∓ (u, ϕ + 0),
1 2 ω± (ϕ − 0) = ω∓ (ϕ + 0),
q1 (ϕ − 0) = −iq2 (ϕ + 0)
and Im κ2 (ϕ + 0) = −Im κ1 (ϕ − 0) > 0. 8.4.3. Auxiliary canonical domains. When applying Lemma 8.1 to compute T2 , we use two auxiliary canonical domains: (1) the domain symmetric to K˜ 0 with respect to the point π ; (2) the domain K2 which is symmetric to K2 with respect to the real line and symmetric to the auxiliary canonical domain K0 with respect to the point π. 8.4.4. Partial results. Now, we list the asymptotics of the coefficients of T2 in the strips (I)–(III). As before, δ is a sufficiently small, fixed, positive constant such that δ < Y0 −Y , and Y is as in Theorem 2.2. Coefficient a2 . As in the strip δ ≤ Im ϕ ≤ Y0 − δ so in the strip −Y0 + δ ≤ Im ϕ ≤ −Im ϕ2 − δ, we get the estimate i
|a2 (ϕ)| ≤ C|e ε
2π
2π −ϕ1
κ2 dϕ
|eCδ/ε .
It is uniform along horizontal lines. We get also the uniform asymptotics 2π
i
a2 (ϕ) = e ε
π
κ2 dϕ
e
2π π
2 dϕ ω+
(1 + o(1)),
−Im ϕ2 + δ ≤ Im ϕ ≤ −Im δ.
Coefficient b2 . We get two asymptotic formulae i
b2 (ϕ) = ie ε
2π −ϕ π
1
κ1 dϕ+ εi
2π −ϕ1 2π
2π −ϕ
κ2 dϕ+
π
1
2π −ϕ1
1 dϕ− ω+
2π
2 dϕ ω−
(1 + o(1)),
Anderson Transitions for Almost Periodic Schrödinger Equations
69
as δ ≤ Im ϕ ≤ Y0 − δ, and 2i
b2 (ϕ) = −ie ε
ϕ π
2 (κ −π)dϕ− i 1 ε
2π π
ϕ
κ2 dϕ+
π
2π
1 dϕ+ ω+
2
ϕ2
2 dϕ ω−
e
2π i ε (ϕ−π)
(1 + o(1)),
as −Y0 + δ < Im ϕ < −ϕ2 − δ. These asymptotics are uniform along horizontal lines. Coefficient c2 . We get only estimates uniform along the horizontal lines. If δ ≤ Im ϕ ≤ Y0 − δ, we have δ
2π
i
|c2 (ϕ)| ≤ CeC ε |e ε
π
κ2 dϕ
|,
and, if −Y0 + δ ≤ Im ϕ ≤ −Im ϕ2 − δ, we have i
|c2 (ϕ)| ≤ C|e ε
2π π
κ2 dϕ
|e−δ2 /ε .
Here, δ2 is a positive constant independent of ε and δ. Coefficient d2 . We get the estimates i
|d2 (ϕ)| ≤ C|e− ε
2π π
κ2 dϕ
|eCδ/ε .
They are uniform along horizontal lines as in the strip δ ≤ Im ϕ ≤ Y0 − δ so in the strip −Y0 + δ ≤ Im ϕ ≤ −Im ϕ2 − δ}. And, in the strip −Im ϕ2 + δ ≤ Im ϕ ≤ −δ, we get the uniform asymptotics i
d2 (ϕ) = e− ε
2π π
κ2 dϕ
e
2π π
2 dϕ ω−
(1 + o(1)).
Uniform asymptotics for the coefficients of T2 are obtained by analyzing their Fourier series. This leads to the formulae announced in Proposition 8.3. ˜ Let K := K0 ∪ K1 ∪ K2 . This domain is represented on 8.5. The computation of M. Fig. 19. Note that it is cut along the Stokes lines shown by the bold lines. It is simply connected and regular. Continue analytically the main branch of the complex momentum from K0 to K. Denote the analytic continuation simply by κ. By definition, the functions κj , j = 0, 1, 2, are the restrictions of κ to Kj , i.e. κ|Kj = κj . Similarly, we introduce the functions ω± single valued and analytic on K so that j
ω± |Kj = ω± . These functions κ and ω± have the following symmetry properties: κ(2π − ϕ) = κ(ϕ)
and
ω± (2π − ϕ) = −ω± (ϕ).
(8.55)
1 , see (7.9) and (7.12). Equation (8.55) follows from the analogous properties of κ1 and ω± To get the asymptotics of the matrix M˜ described in Proposition 8.1, we proceed as follows: j
• in the asymptotic formulae for T1 and T2 , we replace κj and ω± , j = 1, 2 by κ and ω respectively;
70
A. Fedotov, F. Klopp
'2 b
0
b
'1
2
b
b'
b
'
2
2
b
2
'1
'2
Fig. 19. The domain K
• we compute the product T1 (ϕ + 2π )T2 (ϕ + 2π ) (compare with (8.5)); • in the formulae thus obtained, we simplify the integrals by means of the relations (8.55). In result, after a rather long, but elementary calculation, we get i i ˜ ˜ m11 (ϕ) = t˜h · e ε 1 + o(1) + o e− ε e−(c1 +c2 )/ε , i i 2π i ˜ ˜ m12 (ϕ) = iG e ε 1 + o(1) + o e− ε e−c1 /ε − t˜v e−iθ + ε (ϕ+π) (1 + o(1)) , i 3i 2π i ˜ ˜ m21 (ϕ) = −iG−1 e ε 1 + o(1) + o e− 2ε e−c2 /ε − t˜v eiθ− ε (ϕ+π) (1 + o(1)) , i ˜ i ˜ 1 i ˜ m22 (ϕ) = e ε (1 + o(1)) + e− ε (1 + o(1)) + o t˜v2 e− ε − th 2π tv −iθ+ 2π i (ϕ+π) ε (1 + o(1)) + eiθ−i ε (ϕ+π) (1 + o(1)) . e th Here, c1 , c2 > 0 and we have set 2i
t˜h = e ε
ϕ1 0
κdϕ
G = exp
,
ϕ1
0
˜ =
2π−ϕ1
ϕ1
i
− t˜v = e ε
κdϕ,
(ω+ − ω− )dϕ ,
θ =i
ϕ2 ϕ1
ϕ2 ϕ2
(κ−π)dϕ
, (8.56)
(ω+ − ω− )dϕ.
All the integrals are taken along curves in K. We note that κ is real along the interval ˜ is real for real E, and all the terms of the form o (e... ) are in fact [ϕ1 , 2π − ϕ1 ]. So, o(1) in a sufficiently small constant neighborhood of E0 . This implies the asymptotic ˜ = ; this is done formulae of Proposition 8.1 if one checks that t˜h = th , t˜v = tv and in Sects. 9.1.1 and 9.1.2. 8.5.1. Completing the proof of Proposition 8.1. To finish the proof of ϕProposition 8.1, we check that ϕ(0) is real for E ∈ R. Therefore, we prove that θ = i ϕ12 (ω+ − ω− )dϕ is real. One has ϕ2 ϕ2 θ = −i (ω+ − ω− )dϕ = −i (ω+ (ϕ) ¯ − ω− (ϕ))dϕ. ¯ ϕ1
ϕ1
Anderson Transitions for Almost Periodic Schrödinger Equations
71
ϕ As ω+ (ϕ) = ω− (ϕ) (see (7.12)), and ϕ22 ω± dϕ = 0 (see (8.55)), one has θ = ϕ ' i ϕ12 (ω+ − ω− )dϕ = θ . This completes the proof of Proposition 8.1. & 8.6. Getting a monodromy matrix of the form (2.14). To get such a monodromy matrix, we change the consistent basis. Recall that, using Theorem 6.1, we have constructed the solution f+0 (u, ϕ) with the “standard” asymptotics in K0 . Denote it by f0 . Consider the function f0∗ (u, ϕ) = f0 (u, ϕ). It satisfies Eq. (7.1) and the consistency condition (6.3). First, we compute T0 (ϕ), the matrix of (f0 , f0∗ ) in the basis (f+0 , f−0 ). This matrix is ε-periodic and analytic. Then, we check that the Wronskian of f0 and f0∗ is a nonvanishing constant. So, the basis (f0 , f0∗ ) is consistent; the monodromy matrix for this basis is the one described in Theorem 2.2. It is given by −1 ˜ M(ϕ) = T0 (ϕ + 2π )M(ϕ)T 0 (ϕ) ,
(8.57)
where M˜ is the monodromy matrix for the basis (f+0 , f−0 ). To get the asymptotics of M, ˜ we use the asymptotics of T0 and M. As f0 = f+0 , the matrix T0 has the form 1 0 T0 (ϕ) = , (8.58) a0 b0 where a0 and b0 are defined f0∗ = a0 f+0 + b0 f−0 and, thus, are given by the formulae: a0 =
w(f−0 , f0∗ ) w(f−0 , f+0 )
and b0 =
w(f0∗ , f+0 ) w(f−0 , f+0 )
=
w(f0∗ , f0 ) w(f−0 , f+0 )
.
(8.59)
When computing a0 and b0 , as in Sect. 8.5, we use the functions κ, ω± , ψ± and q 0 , ψ 0 and q from K to K = K ∪ K ∪ K . defined by analytic continuation of κ0 , ω± 0 0 0 1 2 ± As before, δ is a sufficiently small (but independent of ε) positive number. Let us first compute a0 . First, note that, K0 ∩ K0 contains the rectangle R1 = {ϕ ∈ K0 ; |Im ϕ| ≤ Im ϕ3 − δ, |Re ϕ| ≤ δ}. In this rectangle, we know the asymptotic of f0 . To get the one of f0∗ , we note that ψ+ (u, ϕ) = ψ+ (u, ϕ),
κ(ϕ) = −κ(ϕ), ω+ (u, ϕ) = ω+ (u, ϕ),
q(ϕ) = iq(ϕ),
ϕ ∈ R1 .
(8.60)
The first of these relations follows as κ is purely imaginary on (0, ϕ1 ). The second is valid as the branches of the Bloch solution ψ(x, E) are real for E < E1 . The third one follows from the second one and the definitions of ω± . The last one follows from the choice of the branch of q0 : q0 ∈ e−iπ/4 R+ on (0, ϕ1 ), see Sect. 7.3.3. By (8.16), (8.17) and (8.60), we get i
f0∗ (u, ϕ) = ie ε
ϕ 0
κdϕ
0 (A+ (u, ϕ) + o(1)).
(8.61)
Comparing this result with the asymptotics of f0 = f+0 , see (8.16), we obtain the uniform formula a0 = i + o(1),
ϕ ∈ R1 .
(8.62)
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A. Fedotov, F. Klopp
From the ε-periodicity of a0 , we deduce that this asymptotics is valid and uniform in the whole strip {|Im ϕ| ≤ Im ϕ3 − δ}. We now turn to the asymptotics of b0 . Therefore, we compute the asymptotics of f0 in K1 ∩ {−Im ϕ3 + δ ≤ Im ϕ ≤ −δ}. By definition of T1 and (8.6), we know that f0 = a1 f+1 + b1 f−1 . The asymptotics of a1 , b1 and of (f+1 , f−1 ) are known in K1 ∩ {−Im ϕ3 + δ ≤ Im ϕ ≤ −δ} (see (8.16) and Proposition 8.2). So, we get that i
f0 (u, ϕ) = e ε
π 0
κdϕ
e i
π 0
ω+ dϕ
π
i
ϕ
i
ϕ
eε
π
κdϕ
1 (A+ (u, ϕ) + o(1))
1 + o(1)e ε 0 κdϕ e− ε π κdϕ (A− (u, ϕ) + o(1)) i ϕ 2i ϕ κdϕ 0 1 A+ (u, ϕ) + o(1) + o(1)e− ε π κdϕ (A− (u, ϕ) + o(1)) = eε 0 i
= eε
ϕ 0
κdϕ
0 (A+ (u, ϕ) + o(1)).
(8.63) ϕ
Here, we have used the fact that the coefficient c = e− ε π κdϕ is (exponentially) small. Indeed, the domain K1 is canonical. So, there is a canonical curve connecting ϕ with the uppermost point of K1 . As Im ϕ < 0, it intersects the interval (ϕ1 , 2π − ϕ1 ) which is a Stokes line. So, in the formula for c, we can integrate from ϕ to this interval along the canonical line, and, then, along this interval to π . Now, the definitions of the canonical lines and the Stokes lines imply the smallness of c. As a result of the computation (8.63), we see that the leading term of the asymptotics of f0 in K1 ∩ {−Im ϕ3 + δ ≤ Im ϕ ≤ −δ} is obtained by analytic continuation from K0 across the common part of K0 and K1 . Consider the “rectangle” R2 = {ϕ1 + δ ≤ Re (ϕ) ≤ π −δ, δ ≤ |Im ϕ| ≤ Im ϕ3 −δ}. We can assume that it is contained in K0 ∪K1 . We have 2i
i
f0 = e ε
ϕ 0
κdϕ
0 (A+ (u, ϕ) + o(1)),
ϕ ∈ R2 .
(8.64)
In R2 ∩{Im ϕ < 0}, this follows from (8.63), and for Im ϕ > 0, this is just the asymptotics from (8.16). Now, we get from (8.64), the asymptotics of f0∗ in R2 . We note that κ(ϕ) = κ(ϕ),
ψ− (u, ϕ) = ψ+ (u, ϕ),
ω− (u, ϕ) = ω+ (u, ϕ),
q(ϕ) = q(ϕ),
ϕ ∈ R2 .
(8.65)
The first of these relations follows as κ is real on (ϕ1 , 2π − ϕ1 ). The second is valid as the branches of the Bloch solution ψ(x, E) differ by the complex conjugation on the spectral bands. The third one follows from the second one and the definitions of ω± . The last one follows from the choice of the branch of q0 , see Sect. 7.3.3. The relations (8.65) and (8.60), and the asymptotics (8.64) imply that i
f0∗ (u, ϕ) = e ε 2i
=eε
ϕ1 0
i κdϕ − ε
ϕ1 0
e
κdϕ
e
ϕ
ϕ1 0
ϕ1
κdϕ
e
ϕ1 0
(ω+ −ω− )dϕ
(ω+ −ω− )dϕ
i
e− ε
ϕ 0
0 (A− (u, ϕ) + o(1))
κdϕ
0 (A− (u, ϕ) + o(1)) .
The asymptotics is uniform for ϕ ∈ R2 . Now, using the asymptotics of f0 and f0∗ , formula (8.2) defining G and formula (9.3) for Sh , we get w(f0∗ , f0 ) = th Gw(f−0 , f+0 )(1 + o(1)).
(8.66)
Anderson Transitions for Almost Periodic Schrödinger Equations
73
As the Wronskian is analytic and periodic in ϕ, this asymptotic is valid and uniform in the strip {|Im ϕ| ≤ Im ϕ3 − δ}. We see that the leading term of the Wronskian is independent of ϕ. The factor 1+o(1) can depend on ϕ. Recall that in order to define the monodromy matrix, we need that the Wronskian be constant. To “correct” the situation, we slightly modify f0 . Denote the factor 1 + o(1) from (8.66) by g. The factor g is ε-periodic in ϕ as the Wronskians are (the coefficients th and G are constant). Formula (8.66) shows that g is real analytic up to a constant factor g1 = 1 + o(1) (as w(f0∗ , f0 ) is real analytic, th and G are real, and th Gw(f−0 , f+0 ) is independent of ϕ). So, we can (and we do) redefine f0 √ √ by dividing it by the real analytic and ε-periodic factor g/g1 = 1 + o(1) so that w(f0∗ , f0 ) = th w(f−0 , f+0 )g1 be constant. The new functions f0 and f0∗ form a consistent basis and still have the "old" asymptotics in R1 and R2 . So, for the new solutions, we get the "old" formulas for a0 and b0 . Now, combining (8.58) with the asymptotics obtained for a0 and b0 , we get
1 0 T0 (ϕ) = , i + o(1) th G(1 + o(1))
|Im ϕ| ≤ Im ϕ3 − δ.
Being obtained by means of the asymptotics of f±0 (constructed by Theorem 6.1) and the asymptotics of a1 and b1 given by Proposition 8.2, the asymptotics of T0 stay uniform in a constant neighborhood of E = E0 ∈ J . As the monodromy matrix M is associated to the basis (f0 , f0∗ ), it takes the form
a(ϕ) b(ϕ) M(ϕ) = ∗ . b (ϕ) a ∗ (ϕ) Using the asymptotics of T0 , of the monodromy matrix M˜ obtained in Theorem 8.1 and formula (8.57), one computes the coefficients a and b to obtain 2iπ 1 i/ε e (1 + o(1)) − tv e ε (ϕ−ϕ(0) ) (1 + o(1) , th 2iπ i i/ε b(ϕ) = (1 + o(1)) − tv e ε (ϕ−ϕ(0) ) (1 + o(1) . e th
a(ϕ) =
(8.67)
The asymptotics are valid uniformly in the strip |Im (ϕ)| ≤ Y if δ < Im (ϕ3 ) − Y (Y is as in Theorem 2.2). Now, to get the statement of Theorem 2.2, one passes to the variable φ = ϕ/ε. The asymptotic representations for a0 , b0 , a˜ and b˜ described in Theorem 2.2 follow from (8.67) and standard estimates of the Fourier coefficients. & '
9. Properties of the Phase and Actions Integrals. The Phase Diagrams First, we study properties of the phase integral and the tunneling actions Sv and Sh . Then, we investigate the location of the asymptotic mobility edges and prove Theorem 2.6 and formula (2.26).
74
A. Fedotov, F. Klopp
9.1. Properties of the action integrals. 9.1.1. Properties of the phase integral and tunneling coefficients. We begin with discussing the phase integral . We prove its properties described by Lemma 2.1 and the representation (2.13) of by a contour integral. The branch κ∗ chosen in Sect. 2.6 to define the phase integral is actually the branch κ defined in Sect. 8.5. Indeed, (1) κ coincides with κ0 on K0 and, in particular, on (−ϕ1 , ϕ1 ). By (7.4) and as the main branch κ0 ∈ iR+ on (0, ϕ1 ), we see that κ ∈ iR+ on (−ϕ1 , ϕ1 ). (2) κ coincides with κ1 on K1 and, in particular, on the cross (ϕ1 , 2π − ϕ1 ) ∪ (ϕ2 , ϕ2 ). By (7.9), as κ1 coincides with the main branch κ0 on the part of this cross (ϕ1 , π ] ∪ [π, ϕ2 ), and as κ0 ∈ (0, π) there, we see that κ ∈ (0, π ) on the cross. Now, let us prove the properties of described in Lemma 2.1. The positivity of is obvious. Check that (E) > 0 on J . Note that 2 π κ0 dϕ, (9.1) = ε ϕ1 where κ0 is the main branch of the complex momentum, and κ0 (ϕ) = k0 (E − α cos(ϕ)); here, k0 is the main branch of the Bloch quasi-momentum. Therefore, 2 π k (E − α cos(ϕ))dϕ. (9.2) (E) = ε ϕ1 0 Note that, since k0 (E) has square root branch points at the ends of the spectral zones, the integral in (9.2) converges. As k0 (E) > 0 inside the first spectral zone, Eq. (9.2) implies the positivity of inside J . The analyticity of in E will be checked by means of the representation (2.13) of by a contour integral taken along a closed curve γp corresponding to Fig. 3. Let us describe the choice of the branch of the complex momentum in (2.13) and prove (2.13). It suffices to assume that E is real and to consider contours γp symmetric with respect to the real axis and to the line π + iR. The branch of the complex momentum in (2.13) coincides with the main one on the part of the contour γp situated in the domain {0 < Re ϕ < π ; Im ϕ > 0}. We have to show that it can be analytically continued to a branch continuous on γp . Denote by a the right point of the intersection of γp and R. As we can continue κ on γp without this point, we need only to prove that the branch thus obtained is continuous at a. This follows from the relation κ(2π − ϕ) = κ(ϕ), ϕ ∈ γp . This relation itself follows from the fact that the main branch κ0 is real on the segments (π, ϕ2 ) and (π, ϕ2 ) of the line π + iR. Note that κ(ϕ) = −κ(ϕ), ϕ ∈ γp ; it follows from (7.5). Now, it suffices to deform γp so that it goes around the interval [ϕ1 , 2π − ϕ1 ] just along this interval. As the main branch is real along this interval, the last relation implies (2.13). The representation (2.13) implies the analyticity of in E as γp can be a closed curve going around [ϕ1 , 2π − ϕ1 ] and staying at a positive distance from the branch points. We have completed the analysis of . The analysis of the tunneling actions Sv and Sh is done in the same way. In particular, one has ϕ2 ϕ1 (κ0 (ϕ) − π )dϕ, Sh (E) = −2i κ0 (ϕ)dϕ, (9.3) Sv (E) = 2i π
0
Anderson Transitions for Almost Periodic Schrödinger Equations
75
where, in the first integral, we integrate along the segment [π, ϕ2 ] of the line π + iR. Point 4 of Lemma 2.1 follows from the representation for Sv in (9.3) as, on the integration contour, 0 < κ0 < π. ˜ t˜v and t˜h ˜ t˜v and t˜h . In view of (9.1) and (9.3), the functions , 9.1.2. Functions , defined by (8.56) satisfy the relations ˜ = ,
t˜v = tv ,
t˜h = th ,
needed to complete the computations of the monodromy matrix in Sect. 8.5. 9.1.3. Dependence of the action integrals on α. We now study the actions Sh and Sv defined in (2.10) as functions of α and E. It will be convenient to change variables (α, E) → (α, t) so that E = E1 + tα, and to keep the notations Sh (α, t) and Sv (α, t) for the functions obtained from Sh (α, E) and Sv (α, E) by this change of variables. Representations (9.3) imply that ϕ2 Sv (α, t) = 2i (k0 (E1 + α(t − cos(ϕ))) − π )dϕ, π ϕ1 (9.4) Sh (α, t) = −2i k0 (E1 + α(t − cos(ϕ)))dϕ, 0
where k0 is the main branch of the Bloch quasi-momentum, see Sect. 5.2.2. The domain S defined by (2.25) becomes K := {(α, t); −1 ≤ t ≤ 1, 0 < α, (1 + t)α ≤ E2 − E1 } . The formulae (9.4) show that the actions Sh (α, t) and Sv (α, t) are real analytic in (α, t) in int (K), the interior of K, and continuous at its boundary. One proves Lemma 9.1. In int (K), one has ∂Sv < 0, ∂α
∂Sh ∂S ∂(Sv − Sh ) > 0, and = < 0. ∂α ∂α ∂α
t 1
K
2
0 3
1
1
Fig. 20. The domain K
(9.5)
76
A. Fedotov, F. Klopp
Proof of Lemma 9.1. Taking (9.4) into account, we compute ϕ1 ∂Sh k0 (E1 + α(t − cos(ϕ)))(t − cos(ϕ))dϕ, = −2i ∂α Sh = ∂α 0 ϕ2 ∂Sv ∂ α Sv = k0 (E1 + α(t − cos(ϕ)))(t − cos(ϕ))dϕ. = 2i ∂α π We notice that • for ϕ ∈ (0, ϕ1 ) ⊂ R, we have E−α cos(ϕ) < E1 ; hence, −ik0 (E−α cos(ϕ)) < 0 (see Fig. 5). Furthermore, the inequality E−α cos(ϕ) < E1 is equivalent to t −cos(ϕ) < 0. Hence, ∂α Sh ≥ 0. Moreover, ∂α Sh = 0 if and only if ϕ1 = 0, i.e. if and only if t = 1. • for ϕ ∈ (π, ϕ2 ) ⊂ {π + iR}, we have E1 < E − α cos(ϕ) < E2 ; hence, k0 (E − α cos(ϕ)) > 0 (see Fig. 5). On the other hand, E1 < E − α cos(ϕ) means that t − cos(ϕ) > 0. Hence, ∂α Sv ≤ 0. Moreover, ∂α Sv = 0 if and only if ϕ2 = π , i.e. if and only if (1 + t)α = E2 − E1 . This completes the proof of Lemma 9.1.
' &
9.2. The phase diagram. Now, we study the asymptotic mobility edges: we prove Theorem 2.6 and formula (2.26). We use the notations of Sect. 9.1.3 9.2.1. The curve S = 0. Lemma 9.1 implies that, for every t ∈ [−1, 1], there exists at most one α such that (α, t) ∈ K and Sh (α, t) = 0. Let us study the behavior of S on the boundary of K. The boundary ∂K is made of four curves (1) (2) (3) (4)
0 1 2 3
: α = 0, −1 ≤ t ≤ 1; : t = −1, 0 < α; 1 : t = 1, 0 < α < α ∗ , α ∗ = E2 −E 2 ; ∗ : α ≤ α, (1 + t)α = E2 − E1 .
Note that in terms of α and t, the relations defining ϕ1 and ϕ2 have the form: cos ϕ1 = t,
cos ϕ2 = t −
E 2 − E1 . α
(9.6)
Now, we can study S on the boundary of K. • On 3 , one has ϕ2 = π, ϕ1 ≥ 0, and ϕ1 = 0 only at (α ∗ , 1); hence, by (9.4), S < 0 except at the point (α ∗ , 1), where S = 0. • Consider Sv and Sh near 0 i.e. for α small and t ∈ [−1, 1]. As usual, one has ϕ1 ∈ [0, π ], and, by (9.4), Sh (α, t) stays bounded. On the other hand, uniformly for t ∈ [−1, 1], one has Sv (t, α) → +∞ as α → 0. Indeed, pick 0 < β < E2 − E1 . For sufficiently small α, t − β/α < −1, and we can define ϕβ ∈ π + iR+ by cos ϕβ = t − β/α. Then, on the interval Iβ = {ϕ ∈ π + iR; 0 ≤ Im ϕ ≤ Im ϕβ }, one has k0 (E1 + α(t − cos ϕ)) − π ≤ k0 (E1 + β) − π < 0. Hence, by (9.4), one has Sv (t, α) ≥ 2(π − k0 (E1 + β))Im ϕβ . This implies that Sv (t, α) → +∞ as α → 0. Hence, for α sufficiently small, S is positive. • The statements of Lemma 9.1 remain valid on 1 . • On 1 , one has ϕ1 = π; and, when α → +∞, one has ϕ2 → π , hence, Sv → 0; on the other hand, Sh → +∞; so S is negative for α large enough. • On 2 , one has ϕ1 = 0 and Im ϕ2 > 0. So, on 1 , S > 0.
Anderson Transitions for Almost Periodic Schrödinger Equations
77
As a conclusion of this discussion, we see that, for every t ∈ [−1, 1], there exists a unique α(t) such that S(α(t), t) = 0 and (α(t), t) ∈ K. For t = 1, α(t) = α ∗ ; for t = −1, 0 < α(t) < ∞; for −1 < t < 1, (α(t), t) ∈ int (K). The analyticity of the curve t → α(t) is a consequence of the analyticity of S, Lemma 9.1 and the Local Inversion Theorem. This completes the proof of Theorem 2.6. & ' 9.2.2. Mobility edges near the point (α ∗ , 1). Consider the asymptotic mobility edge S in a neighborhood of the point (α ∗ , 1). When proving Theorem 2.6, we have seen that Sh (α ∗ , 1) = Sv (α ∗ , 1) = 0. Let the (mi )i=1,2 be the effective masses corresponding to (Ei )i=1,2 the ends of the first spectral band of the periodic Schrödinger operator (see Sect. 5.2.2). One can obtain the Taylor formula for Sh and Sv at (α ∗ , 1):
π 2m2 ∗ 2(α − α) + α ∗ (t − 1) and Sh (α, t) ∼ 2 α∗ π ∗ Sv (α, t) ∼ 2α m1 (1 − t), α ∼ α ∗ , t ∼ 1, 2 where we have omitted the standard error terms. As S = Sv − Sh , this implies (2.26). 10. Positive Lyapunov Exponent Let (M(φ, ε))0<ε<1 be a family of SL(2, C)-valued, sufficiently regular, 1-periodic functions. Let h be a positive number. Define the matrix cocycle PN (φ, ε) = M(φ + N h, ε) · M(φ + (N − 1)h, ε) · · · M(φ + h, ε) · M(φ, ε). Then, γ (φ), the Lyapunov exponent of this matrix cocycle at the point φ, is defined by γ (φ) =
lim
N→+∞
1 log PN (φ, ε). N
(10.1)
It is well known (see [8, 37, 40] and references therein) that, if h is irrational, then this limit exists for almost all φ, is independent of φ, and satisfies 1 1 γ = lim log PN (θ, ε)dθ. (10.2) N→+∞ N 0 We prove Proposition 10.1. Pick ε0 > 0. Assume that there exist y0 and y1 satisfying the inequalities 0 < y0 < y1 < ∞ and such that, for any ε ∈ (0, ε0 ) one has • the function φ → M(φ, ε) is analytic in the strip S = {φ ∈ C; 0 ≤ Im φ ≤ y1 /ε}; • in the strip S1 = {φ ∈ C; y0 /ε ≤ Im φ ≤ y1 /ε} ⊂ S, M(φ, ε) admits the representation M(φ, ε) = λ(ε)ei2πn0 φ · (M0 (ε) + M1 (φ, ε)) ,
n0 ∈ N \ {0},
with a positive constant λ and a matrix M0 independent of φ; 1 β(ε) • M0 (ε) = ; 0 α(ε)
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A. Fedotov, F. Klopp
• there exist constants β > 0 and α ∈ (0, 1) independent of ε and such that |α(ε)| ≤ α and |β(ε)| ≤ β; • m(ε) = sup M1 (φ, ε) → 0 as ε → 0. Here, . is the matrix norm associated to φ∈S1
the vector norm for v = (v1 , v2 )T ∈ C2 .
v = |v1 | + |v2 |,
Then, there exist C > 0 and ε1 > 0 such that, if 0 < ε < ε1 , one has γ > log λ(ε) − Cm(ε).
(10.3)
Proof. To prove this result, we follow the ideas of Herman ([26]) as extended by Sorets and Spencer ([40]). One computes
1 0 1 · U (ε)−1 , where U (ε) = 0 α(ε) 0
M0 (ε) = U (ε) ·
−β(ε) 1−α(ε) .
1
Under our assumptions on M0 , the family (U (ε))ε∈(0,ε0 ) is uniformly bounded. Let n0 be as in Proposition 10.1. Put ˜ M(φ, ε) = e−i2πn0 φ · U (ε)−1 · M(φ, ε) · U (ε).
(10.4)
Note that, in S1 , ˜ M(φ, ε) = λ(ε)
1 0 ˜ + M1 (φ, ε) , 0 α(ε)
sup M˜ 1 ≤ const · m(ε).
φ∈S1
(10.5)
˜ h), Denote by P˜N (φ, ε) the matrix cocycle associated to the pair (M, ˜ + N h, ε) · · · M(φ ˜ + h, ε) · M(φ, ˜ P˜N (φ, ε) = M(φ ε). Hence, by (10.4) and (10.2), one has γ =
1 N→+∞ N
lim
0
1
log P˜N (θ, ε)dθ.
(10.6)
Define the function gN by gN (φ, ε) =
1 1 , P˜N (φ, ε) , 0 0
where the angular brackets denote the scalar product in R2 . Then, 0
1
log P˜N (θ, ε)dθ ≥
1 0
log |gN (θ, ε)|dθ.
(10.7)
Anderson Transitions for Almost Periodic Schrödinger Equations
79
Let us now recall a version of Jensen’s formula proved in [40]. Lemma 10.1 ([40]). Pick 0 < ρ < 1. Let f be an analytic function in the annulus A = {z ∈ C; ρ ≤ |z| ≤ 1} and continuous on A. Let f = 0 on the boundary of A. Let (rj )j denote the roots of f inside A. Then, one has 1 1 log |f (ei2πθ )|dθ = log |f (ρei2πθ )|dθ 0 0 (10.8) f (z) 1 − log |rj | − dz log ρ. 2iπ |z|=ρ f (z) rj ∈A
We let z = e2πiφ . As gN is analytic and 1-periodic in φ ∈ S, the relation fN (z, ε) = gN (φ, ε) defines a function analytic in the annulus 1 ≥ |z| ≥ e−2πy1 /ε . We take ρ = e−2πy/ε , where y satisfies y0 < y < y1 , and apply Lemma 10.1 to fN (z, ε). Taking into account the fact that the contribution of the zeroes of fN to (10.8) is non-positive, we get 1 1 fN (ζ, ε) y dζ . (10.9) log |gN (θ, ε)|dθ ≥ log(fN (ρei2πθ ), ε)dθ + ε |z|=ρ fN (ζ, ε) 0 0 We are now left with estimating fN and fN on the circles Cy = {|z| = e−2πy/ε }, y0 < y < y1 . Let M˜ 1 be as in (10.5). We define inductively {ak , bk }∞ k=0 so that a0 = b0 = 0, 1 0 1 1 = + M˜ 1 (φ + h(k − 1), ε) , (1 + bk ) 0 α(ε) ak ak−1
k ≥ 1.
(10.10)
Then, one checks that fN (z, ε) = λ(ε)N (1 + bN+1 ) · · · (1 + b1 ). We use Lemma 10.2. For ε sufficiently small and z ∈ Cy , y0 ≤ y ≤ y1 , one has |ak | ≤ m(ε) and |bk | ≤ 2m(ε), k ∈ N.
(10.11)
Proof. Lemma 10.2 is proved inductively. Obviously, (10.11) holds for k = 0. Let us assume that it holds up to rank k. Equation (10.10) implies 1 1 1 + bk+1 = 1 + , , M˜ 1 (φ + hk, ε) 0 ak 0 1 (1 + bk+1 )ak+1 = α(ε)ak + . , M˜ 1 (φ + hk, ε) 1 ak Using the assumptions on α(ε) and m(ε), one gets 1 [α|ak | + m(ε)(1 + |ak |)] and |bk+1 | ≤ m(ε)(1 + |ak |). (10.12) |ak+1 | ≤ 1 + bk+1 Using the assumption on m(ε) and the induction assumption, for ε sufficiently small, one gets |bk+1 | ≤ 2m(ε)
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uniformly in k. Plugging this into the first inequality from (10.12), for ε sufficiently small, one obtains |ak+1 | ≤
1 α m(ε) + m(ε)(1 + m(ε)) ≤ m(ε), 1 − 2m(ε)
uniformly in k. By induction, this completes the proof of Lemma 10.2.
' &
As an immediate consequence of (10.10) and Lemma 10.2, we get that, for any k, the functions ak and bk are analytic in φ in the strip {y0 /ε ≤ Im φ ≤ y1 /ε}. Hence, by the estimation (10.11) for bk and by the Cauchy estimates, the derivative of bk satisfies the estimate |bk | ≤ Const · ε · m(ε) in some smaller strip {y˜0 /ε ≤ Im φ ≤ y˜1 /ε}. As bk is 1-periodic in φ, we can also consider it as an analytic function of z = e2πiφ . Then 1 dbk dbk = . So, there exists C > 0 such that, if ρ = e−2πy/ε for y˜0 < y < y˜1 , dz 2πiz dφ one has N fN (z, ε) 1 bk (z) 1 dz ≤ dz ≤ CN m(ε). ε |z|=ρ fN (z, ε) ε |z|=ρ 1 + bk (z) k=1
and 0
1
log |fN (ρe
i2πθ
, ε)|dθ = N log λ(ε) +
N k=1 0
1
log |1 + bk (ρei2πθ )|dθ
≥ N log λ(ε) − CN m(ε). Plugging this into Eq. (10.9), one completes the proof of Proposition 10.1 by means of (10.7) and (10.6). & ' 11. A KAM Theory Construction 11.1. The model equation. Let φ∈ C. Recall that a matrix function M(φ) ∈ GL(2, C) a b belongs to M if it is of the form ∗ ∗ . b a We consider 1-periodic functions in M depending on a parameter η in B, a Borel subset of R. For functions M that are analytic in the strip {|Im φ| ≤ r} and that depend on η ∈ B, we consider their Lipschitz norms Mr,B ≡
sup
|Im φ|≤r,η∈B
M(φ, η) +
sup
|Im φ|≤r, η,η ∈B, η =η
M(φ, η) − M(φ, η ) . |η − η | (11.1)
Here, . is the matrix norm associated to the vector norm v = |v1 | + |v2 |,
for
v = (v1 , v2 )T ∈ C2 .
If M does not depend on φ, we write MB instead of Mr,B .
Anderson Transitions for Almost Periodic Schrödinger Equations
81
Let I be a bounded real interval. Let D and A in M be two matrix-valued functions of φ ∈ C such that (1) the matrices A and D depend on a parameter η ∈ I , D = diag (exp(iη), exp(−iη)),
A = A(φ, η);
(11.2)
(2) the matrix A(φ, η) is analytic in φ in a strip |Im φ| ≤ r and Ar,I ≤ 1. Let λ be a parameter taking positive values, and let h be a fixed positive number. Consider the equation A(φ + h) = (D + λA)A(φ),
φ ∈ R.
(11.3)
Our aim is to study matrix solutions of Eq. (11.3) for small values of λ. Therefore, we use standard KAM theory ideas avoiding small neighborhoods of the KAM resonances (see [11, 2]). This allows us to construct solutions of (11.3) outside of some set of η of small measure. As usual for KAM methods, we impose a Diophantine condition on the number h. We fix σ ∈ (0, 1) and define l λσ Hσ = h ∈ (0, 1); min h − ≥ 3 , k = 1, 2, 3 . . . . (11.4) l∈N k k We will assume that h ∈ Hσ . Remark 11.1. Consider the complement of Hσ in (0, 1). For λ < 1, it is contained in the union of open intervals Il,k , k ∞
R \ Hσ ⊂
Ik,l ,
k=1 l=0
centered at the points l/k and of length
2λσ k3
mes ( (0, 1) \ Hσ ) ≤
. Hence, one gets
∞ 2λσ k=1
k3
(k + 1) = C · λσ .
(11.5)
In the sequel, C denotes a constant depending only on I , σ and r (but not on η, φ, h, λ or A). We prove Proposition 11.1. Let h, A and D be chosen as above; assume that det(D + λA) ≡ 1. Then, there exists λ0 = λ0 (r, σ, I ) such that, for 0 < λ < λ0 , there is a Borel set ∞ ⊂ I , ∞ = ∞ (r, σ, I ), mes ∞ ≤ Cλσ/2 ,
(11.6)
such that, for η ∈ B∞ = I \ ∞ , Eq. (11.3) has a solution of the form iη ·φ/ h e ∞ 0 A(φ, η) = U (φ, η) , 0 e−iη∞ ·φ/ h where • U is defined and analytic for |Im φ| < r/2 and satisfies U − 1r/2, B∞ ≤ Cλ1−σ ; • η∞ is a real valued Lipschitz function of η that satisfies η∞ (η) − ηB∞ ≤ Cλ.
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α β , we let 11.2. The proof of Proposition 11.1. For a 1-periodic matrix M(φ) = β ∗ α∗ α0 0 Diag M = , where α0 is the mean value of the periodic function α. Let τh be 0 α0∗ the shift operator defined by τh f (φ) = f (φ + h). 11.2.1. The induction procedure. One proves Proposition 11.1 inductively. First, let us explain the heuristics guiding the KAM induction. The aim is to transform equation (11.3) with small matrix λA into an equation A∞ (φ + h) = D∞ A∞ (φ) with a constant diagonal matrix D∞ . Therefore, we search for a solution of (11.3) in the form A0 = (I + U1 (φ))A1 , where U1 is 1-periodic and small. This leads to the equation A1 (φ + h) = (I + τh U1 (φ))−1 (D + λA)(I + U1 (φ))A1 (φ). As U1 and λA are small, one has (I + τh U1 )−1 (D + λA)(I + U1 ) = D + (DU1 − τh U1 D + λA) + . . . , where the dots denote lower order terms. So, if DU1 − τh U1 D + λA = 0, then (11.3) is replaced by a new equation of the same form, namely, A1 (φ + h) = (D + A1 (φ))A1 (φ). The new matrix A1 is smaller than λA. We prove that one can find a periodic solution U of the equation DU − τh U D + A = 0 if Diag A = 0. Let us turn to the induction. First, we let D0 = D + λDiag A,
A0 = λA − λDiag A.
At the k th step of the induction, we define two matrix-valued functions Uk and Bk so that (I + τh Uk (φ))−1 (Dk−1 + Ak−1 (φ))(I + Uk (φ)) = Dk−1 + Bk (φ),
(11.7)
and set Dk = Dk−1 + Diag Bk ,
Ak (φ) = Bk (φ) − Diag Bk .
(11.8)
As Uk we take a periodic solution of the homological equation τh Uk (φ)Dk−1 − Dk−1 Uk (φ) = Ak−1 (φ).
(11.9)
Then, by (11.7), one has Bk (φ) = (I + τh Uk (φ))−1 Ak−1 (φ) Uk (φ).
(11.10)
We check that Uk and Ak quickly converge to zero and that the product P (φ) = X∞ k=1 (I + Uk (φ)) = (I + U1 (φ)) (I + U2 (φ)) (I + U3 (φ)) . . .
(11.11)
converges to an invertible matrix. As a result, we obtain P −1 (φ + h)(D + λA(φ))P (φ) = D∞ ,
(11.12)
where D∞ is a constant diagonal matrix. This immediately implies that (11.3) has a solution of the form A(φ) = P (φ) exp(φD∞ / h).
(11.13)
We prove that this solution has all the properties announced in Proposition 11.1.
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83
11.2.2. Homological equation and small denominators. Consider the homological equation (11.9). At each step of the induction procedure, Ak is a 1-periodic in φ. Therefore, we look for 1-periodic solutions of (11.9). This leads to a small denominator problem. We write the homological equation as ˜ (φ) = A(φ). ˜ τh U (φ)D˜ − DU
(11.14)
We prove Lemma 11.1. Let A˜ and D˜ belong to M, and let ˜ d˜ ∗ ), where d˜ does not depend on φ; • D˜ be diagonal, D˜ = diag (d, • Diag A˜ ≡ 0; • A˜ be analytic in a strip |Im φ| ≤ r˜ ; • A˜ and D˜ be Lipschitz functions of a parameter η ∈ B (B, a Borel subset of R) satisfying ˜ r˜ ,B < ∞, A
˜ B < ∞, D
˜ > 0. and inf η∈B |d| If the number h satisfies the Diophantine condition (11.4), and the parameter η is outside of the set ˜ =
k,l∈Z
λσ/2 η ∈ B : |α(η) − hk − l| < , max2 {1, k}
α(η) =
1 ˜ arg d(η), π (11.15)
then, there exists C0 > 0, a universal constant, and U ∈ M, a solution of the homological equation (11.14) analytic in the strip |Im φ| < r˜ and satisfying the estimate U r˜ −ρ, B\˜
˜ r˜ ,B D ˜ B A ≤ C0 ˜2 λσ ρ 5 inf η∈B |d|
˜ 2 D B ρ + (1 + ρ ) 1 + ˜2 inf η∈B |d| 2
5
∀ρ ∈ (0, r˜ ).
, (11.16)
Proof. We are looking for U = ((Uij ))1≤i,j ≤2 ∈ M. Let u = U11 and v = U12 . Then, Eq. (11.14) is equivalent to the system of equations ˜ τh u − u = a/d, e
−2iπα
˜ τh v − v = b/d,
(11.17) (11.18)
where a = A˜ 11 , b = A˜ 12 are the coefficients of A˜ = ((A˜ ij ))1≤i,j ≤2 . To solve the first equation, consider the Fourier series of a, i.e. a = k =0 ak e2iπkφ . The zeroth term vanishes as Diag A˜ = 0. The Lipschitz norm of ak is estimated by ˜ r˜ ,B . ak B ≤ e−2π r˜ |k| A
(11.19)
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A. Fedotov, F. Klopp
The 1-periodic solution of (11.17) is given by u=
1 ak e2πikφ . e2πikh − 1 d˜ k =0
This representation and (11.19) imply that e−2πρ|k| ˜ r˜ ,B D ˜ B e−2πρ|k| 1 A ˜ ur˜ −ρ,B ≤ ≤ . d˜ Ar˜ ,B 2πikh ˜2 |e − 1| |e2πikh − 1| inf η∈B |d| B k =0 k =0 (11.20) 1
As
|e2πikh
− 1|
≤ Cλ−σ k 2 and
k 2 e−2πρ|k| ≤ C/ρ 3 , 0 < ρ, there exists C > 0, a
k =0
universal constant, such that ur˜ −ρ,B ≤ C
˜ B ˜ r˜ ,B D A , where 0 < ρ < r˜ . ˜2 λσ ρ 3 inf η∈B |d|
(11.21)
In terms of the Fourier coefficients of the function b, the 1-periodic solution of Eq. (11.18), can be written as v=
1 bk e2πikφ ˜ r˜ ,B . , and bk B ≤ e−2π r˜ |k| A e2πi(kh−α(η)) − 1 d˜ k∈Z
The function v is estimated by 1 1 −2πρ|k| ˜ vr˜ −ρ,B\˜ ≤ Ar˜ ,B e . 2πi(kh−α(η)) ˜ e − 1 B\˜ d B k∈Z On the other hand, one has 2 1 1 2πi(hk−α(η)) ≤ sup − 1B , e2πi(kh−α(η)) − 1 ˜ e2πi(kh−α(η)) − 1 e ˜ B\ η∈B\ and ˜ B ≤1+ e2πi(hk−α(η)) − 1B = e2πihk d˜ ∗ /d˜ − 1B ≤ 1 + d˜ ∗ /d
By (11.15), one also has 2 1 ≤ Cλ−σ max{1, k 4 }, sup 2πi(kh−α(η)) e − 1 ˜ η∈B\ k 4 e−2πρ|k| ≤ C/ρ 5 , 0 < ρ. k>0
and
˜ 2 D B . ˜2 inf η∈B |d|
Anderson Transitions for Almost Periodic Schrödinger Equations
Finally, one obtains vr˜ −ρ,B\˜
˜ r˜ ,B D ˜ B A ≤C ˜2 λσ ρ 5 inf η∈B |d|
85
˜ 2 D B 1+ ˜2 inf η∈B |d|
(1 + ρ 5 ),
0 < ρ < r˜ . (11.22)
Estimates (11.21) and (11.22) imply (11.16) with a universal constant. This completes the proof of Lemma 11.1. & ' 11.2.3. Induction procedure: Estimates for Ak , Uk and Dk . At each step of the induction procedure, we use Lemma 11.2 to construct Uk , a solution of the homological equation (11.9) for the matrices Ak−1 and Dk−1 . Then, (11.10) and (11.8) yield the matrices Ak and Dk . The price of the use of Lemma 11.2 is twofold: • first, we can construct Uk only outside some set of values of η; let Bk−1 be the set denoted by B in Lemma 11.1 when A˜ = Ak and D˜ = Dk . The set of “bad” values of η “thrown away” at the kth step is described by (11.15) with d˜ = (Dk−1 )11 ; denote this set by k ; then, one has λσ/2 k = η ∈ Bk−1 : |αk−1 (η) − hl − m| < , max(l 2 , 1) l,m∈Z
Bk−1 = I \
k−1
j ,
B0 = I,
(11.23)
j =1
where
αk (η) =
1 arg dk (η), π
dk = (Dk )11 ;
(11.24)
note that, for all k ≥ 0, one has Bk+1 ⊂ Bk . • second, by formula (11.16) we can control only U r˜ −ρ, B\˜ , but not U r˜ , B\˜ ; so, if Ak−1 was constructed in a strip |Im φ| ≤ rk−1 , then to get a good enough norm control, we have to consider Uk in a smaller strip |Im φ| ≤ rk , 0 < rk < rk−1 ; we choose rk = r −
k
ρl ,
r0 = r,
ρl =
l=1
r ; 2l+1
note that, one has r∞ ≡ r −
∞
ρl = r/2.
(11.25)
l=1
We prove Lemma 11.2. Let k ≥ 0. Assume that Bk+1 is nonempty. Let C0 be the constant obtained in Lemma 11.1 and define S(k) =
k 2+l l=0
2l
,
S = lim Sk , k→∞
Q = 25 ,
C(r) = 24C0 (10 + r 2 + r 5 )/r 5 .
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Assume λ < 1/4 and λ1−σ 2C(r)QS < 1/2.
(11.26)
Then, for k ≥ 0, one has Bk+1 rk+1 ,Bk+1 ≤ Uk+1 rk+1 ,Bk+1
1 k λσ Q2 S(k) F (k + 1), 2C(r) 1 k−1 ≤ Qk/2+1+2 S(k) F (k). 4
(11.27) (11.28)
2k where F (k) = 2C(r)λ1−σ . Proof. Equations (11.27) and (11.28) are proved inductively. To simplify the notations, we write Ak , Dk , Uk and Bk instead of Ak Bk ,rk , Dk Bk , Uk Bk ,rk and Bk Bk ,rk . Let us check (11.27) – (11.28) for k = 0. One has D0 ≤ 1 + λ < 5/4,
inf |d0 | ≥ 1 − λ > 3/4,
η∈B0
A0 ≤ 2λ.
Using this and Lemma 11.1 with A˜ = A0 , D˜ = D0 and ρ = ρ1 , one gets a very rough estimate 2 + 1 + 25 (1 + ρ 5 ) 5 ρ 1 9 A0 r 1 20 U1 < C0 σ 9 λ ρ15 r5 (11.29) % & 2 + 10 + r 5 20 r 1 < C0 A0 λ−σ Q2 < Q2 C(r)λ1−σ . 9 2 r5 So (11.28) is proved for k = 0. To estimate B1 , we compare (11.29) with the second condition on λ; this implies that U1 < 1/8 (since Q2 < QS ). Therefore, (11.10) implies that B1 < 87 U1 A0 . And using (11.29) and A0 ≤ 2λ, we see that B1 < 2C(r)Q2 λ2−σ . This proves (11.27) for k = 0. Now, assume that (11.27) and (11.28) are valid for 1 ≤ l ≤ k. We check that (11.27) and (11.28) then hold for k + 1. Begin with checking that Dk ≤ 3/2,
inf |dk | ≥ 1/2.
η∈Bk
By (11.8), one has Dk ≤ Dk − Dk−1 + Dk−1 − Dk−2 + · · · + D1 − D0 + D0 ≤
k−1 j =0
Bj +1 + D0 .
(11.30)
Anderson Transitions for Almost Periodic Schrödinger Equations
87
Using (11.27) to estimate this last sum, we obtain k−1
Bj +1 ≤
j =0
k−1 1 j λσ Q2 S(j ) F (j + 1). 2C(r)
(11.31)
j =0
By (11.26), the terms in the right hand sum above are super exponentially decaying. So, it is roughly of order of the first term, and we need only a very rough estimate: 2j S(j )
Q
F (j + 1) ≤
2j +1 2C(r)λ1−σ QS
Q2
jS
2C(r)λ1−σ QS ≤ QS
2(j +1)
2C(r)λ1−σ QS ≤ QS
2 ·
1 , 22j
where, in the last step, we have used (11.26). This and (11.31) imply that k−1
Bj +1 <
j =0
2 1 σ S 1−σ λ C(r)λ = 4QS C(r)λ2−σ . 2Q C(r)QS
Since D0 ≤ 1 + λ, we see that Dk ≤ 1 + λ(1 + 4QS C(r)λ1−σ ). This with (11.26) implies that Dk satisfies (11.30). The estimate for dk is obtained in a similar way; one has dk − eiη ≤ dk − dk−1 + · · · + d1 − d0 + d0 − eiη ≤ Dk − Dk−1 + · · · + D1 − D0 + d0 − eiη ≤
k−1
S
Bj +1 + λ < λ(1 + 4Q C(r)λ
1−σ
(11.32)
).
j =0
As 2QS C(r)λ1−σ < 1/2, |dk | satisfies (11.30). Lemma 11.1 with D˜ = Dk , A˜ = Ak and ρ = ρk , and the estimates (11.30) yields Uk+1 ≤
6C0 (10 + r 2 + r 5 ) −σ r 5 1 λ Ak = C(r)Qk+2 Ak λ−σ . 5 4 r5 ρk+1
By (11.8), one has Ak ≤ 2Bk . Using (11.27) for Bk , we get Uk+1 ≤
1 k+2+2k−1 S(k−1) 1 k−1 Q F (k) = Qk/2+1+2 S(k) F (k). 4 4
So, Uk+1 satisfies (11.28). To prove that Bk+1 satisfies (11.27), we first get a very rough estimate Uk+1 ≤ 21 . It follows from
1 1 2k−1 k+2 k−1 +S(k) 2k F (k) Uk+1 ≤ Qk/2+1+2 S(k) F (k) = Q 4 4 2k 1 k 1 ≤ Q2 S F (k) = 2C(r)QS λ1−σ 4 4
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A. Fedotov, F. Klopp
and (11.26). The rough estimate and (11.10) give Bk+1 ≤ 2Ak Uk+1 ≤ 4Bk Uk+1 . Using (11.27) for Bk and (11.28) for Uk+1 , we get Bk+1 ≤
1 k−1 k−1 λσ Q2 S(k−1)+k/2+1+2 S(k) F (k + 1). 2C(r)
Using the equality 2k−1 S(k − 1) + k/2 + 1 + 2k−1 S(k) = 2k S(k), we see that Bk+1 satisfies (11.27). This completes the proof of Lemma 11.2. & ' To complete the proof of Proposition 11.1, we also need the following rough estimates Corollary 11.1. In the case of Lemma 11.2, there exists C > 0 depending only on r such that, for k ≥ 0, one has dk+1 − dk Bk+1 ≤ Cλσ (Cλ1−σ )k+1 , iη
dk − e Bk ≤ 2λ.
(11.33) (11.34)
Proof. Estimate (11.33) follows immediately from (11.8) and (11.27). To prove (11.34), we note that, by (11.32), dk − eiη Bk ≤ λ(1 + 4QS C(r)λ1−σ ).
(11.35)
By the second estimate from (11.26), this implies that dk −eiη Bk ≤ 2λ. This completes the proof of Corollary 11.1. & ' 11.2.4. Construction of the solutions. Let us for a while assume that the set B∞ = ∩k Bk = ∅; we prove it later. Then, as the sequences (Bk )k and (rk )k are decreasing, for η ∈ B∞ and for all k ≥ 1, Bk and Uk are analytic in the band |Im φ| ≤ r∞ , and one has 2k+1 1 λσ 2C(r)QS λ1−σ , 2C(r) 2k Uk+1 r∞ ,B∞ ≤ Uk+1 rk+1 ,Bk+1 ≤ 2C(r)QS λ1−σ .
Bk+1 r∞ ,B∞ ≤ Bk+1 rk+1 ,Bk+1 ≤
Hence, if λ satisfies (11.26), the product P (φ) defined by (11.11) converges to a 1periodic, GL(2, C)-valued function that is analytic in the strip |Im φ| ≤ r∞ . Moreover, one has I − P (φ)r∞ ,B∞ ≤ Cλ1−σ . As Ak r∞ ,B∞ ≤ 2Bk r∞ ,B∞ → 0, we get (11.12) where the constant diagonal matrix D∞ , satisfies the estimate D∞ B∞ < ∞. By construction, all the matrices Uk belong to M; thus, (11.12) implies that D∞ belongs to M. Moreover, as we assumed det(D + λA) ≡ 1, (11.12) gives 1 ln | det D∞ | = dφ(ln | det P (φ)| − ln | det P (φ + h)|) = 0. 0
These two observations imply that D∞ has the form described in Proposition 11.1 with η∞ ∈ R. Letting in (11.34) k → +∞, we see that d∞ − dB∞ ≤ 2λ. Hence, η∞ can be defined as a Lipschitz function of η, and η∞ (η) − η ≤ Cλ. So, we have proved that (11.13) is a solution to (11.3) with all the properties described in Proposition 11.1. To complete the proof of Proposition 11.1, we are left with estimating the measure of B∞ .
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89
11.2.5. The set B∞ . To study the set B∞ , we study its complement and the sets k defined in (11.2.3); therefore, we first study the functions η → αk (η) defined by (11.24). Since dk is continuous on Bk , and, by (11.34), has no zeros, the function η → αk (η) can be taken continuous. By Corollary 11.1, the function dk is close to eiη for λ small; hence, the function αk can be taken with values close to η/π . More precisely, we get Lemma 11.3. There exists C > 0 (depending on r, σ , I ) such that, for sufficiently small λ, one has αk+1 − αk Bk+1 ≤ Cλσ (Cλ1−σ )k+1 , αk − η/π Bk ≤ Cλ,
k ≥ 0,
k ≥ 0.
(11.36) (11.37)
Proof. We write . instead of .Bk+1 . Note that αk+1 − αk =
1 arg(1 + wk ), π
wk =
dk+1 − dk . dk
(11.38)
By (11.30) and (11.33), for sufficiently small λ, we get that |wk (η)| ≤ 1/2,
∀η ∈ Bk+1 ,
∀k.
This and (11.38) gives αk+1 − αk ≤ Cwk for some C > 0. Using (11.34) and (11.38), we get αk+1 − αk ≤ Cdk+1 − dk . The bound (11.33) implies (11.36). Estimate (11.37) follows from (11.36). Indeed, αk − η/π ≤ αk − αk−1 + · · · + α1 − α0 + α0 − η/π ≤ λσ (Cλ1−σ )k + · · · + λσ (Cλ1−σ ) + Cλ ≤ Cλ. This completes the proof of Lemma 11.3.
(11.39)
' &
The function αk is defined only on Bk . It is more convenient to work with functions defined on I = B0 . Therefore, we extend αk to I in the following inductive way. For α0 , we have nothing to do. Suppose that, for 0 ≤ l ≤ k, αl have been extended to I . Consider the function αk+1 . We have to define it on I \ Bk+1 which is a countable set of open intervals. On each of these intervals, we define αk+1 so that the function αk+1 − αk is linear, and the function αk+1 remain continuous. To keep notations short, the extended functions are still called αk . Clearly, the continuation procedure does not change neither the Lipschitz norms of the differences αk+1 −αk , i.e. αk+1 − αk Bk+1 = αk+1 − αk B0 , nor the sets k (see (11.2.3)). Moreover, one checks that the estimate (11.37) remains valid with . Bk+1 replaced by . B0 . This follows from the fact that, in (11.39), we can change the sets Bk+1 , Bk , . . . , B1 by B0 = I . The idea of the estimate of ∞ is the following. As α0 is close to η/π , the measure of 0 is small. As long as αk is close to α0 , k almost coincides with 0 . When these sets are quite different, the measure of k becomes very small. We prove Lemma 11.4. Let fi , i = 1, 2, be two Lipschitz functions defined on an interval I such that, for some positive δ and δ1 , 0 < δ1 < 1, |f1 (η) − f2 (η)| ≤ δ ∀η ∈ I,
and
f1,2 − I dI ≤ δ1 < 1.
(11.40)
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Fix c ∈ I . Let > 0. Define F1,2 = η ∈ I : |f1,2 (η) − c| ≤ ,
and δ0 = 2
1 − δ1 . 1 + δ1
Then, one has mes (F1,2 ) ≤ 2/(1 − δ1 ). Moreover, if δ < δ0 , then mes (F2 \ F1 ) ≤ 2δ/(1 − δ1 ), and if δ ≥ δ0 , then mes (F2 \ F1 ) ≤ 2/(1 − δ1 ). Proof. Let us extend f1 and f2 continuously outside of I so that they keep the properties (11.40). It suffices to prove the statements for these “new” f1 and f2 , and I = R. First, we note that (η − η )(1 − δ1 ) ≤ fj (η) − fj (η ) ≤ (η − η )(1 + δ1 ),
j = 1, 2.
So, fj are monotonous and Fj are intervals. Denote the left (resp. right) end of these − + intervals by η1,2 (resp. η1,2 ). Obviously, one has 2 2 + − ≤ η1,2 − η1,2 ≤ , 1 + δ1 1 − δ1
f1 (η1± ) = f2 (η2± ) = c±.
So, one has |η1± − η2± | ≤
|f1 (η1± ) − f1 (η2± )| |f2 (η2± ) − f1 (η2± )| δ = ≤ . 1 − δ1 1 − δ1 1 − δ1
Now, if δ < δ0 then 2/(1 + δ1 ) > δ/(1 − δ1 ); hence, the intersection of F1 and F2 is not empty, and mes (F2 \ F1 ) can be estimated by the sum |η1+ − η2+ | + |η1− − η2− |, that is, by 2δ/(1 − δ1 ). If δ ≥ δ0 , we use the trivial estimate mes (F2 \ F1 ) ≤ mes (F2 ) ≤ 2/(1 − δ1 ). This completes the proof of Lemma 11.4. & ' Now, let us discuss the sets k+1 . Each of these sets is contained in a countable union of open intervals Il,m (k), l, m ∈ Z defined by λσ/2 Il,m (k) = η ∈ I : |αk (η) − hl − m| < . max(1, l 2 ) In view of Lemma 11.4 and (11.37), if λ is sufficiently small, the length of the interval Il,m (k) can be bounded by Cλσ/2 / max{1, l 2 } uniformly in m and k. Fix k and l. On any Il,m (k), one has −1 − lh + αk ≤ m ≤ 1 − lh + αk ; so, the number of Il,m (k) that are not empty does not exceed M = maxη∈B0 αk − minη∈B0 αk + 3. In view of (11.37), we see that M is bounded uniformly in k and l. Note that this implies in particular that, for some C > 0, one has mes (1 ) ≤ Cλσ/2 . We can represent the set ∞ as ∞ = 1 ∪
(k+1 \ k ).
k≥1
Let us estimate the mes (k+1 \ k ). One has mes Il,m (k + 1) \ Il,m (k) . mes (k+1 \ k ) ≤ l∈Z m∈Z
Anderson Transitions for Almost Periodic Schrödinger Equations
91
To estimate mes Il,m (k + 1) \ Il,m (k) , we apply Lemma 11.4 with f1 = π αk and f2 = παk+1 . In view of (11.36)–(11.37), one then has δ = δ(k) = Cλσ (Cλ1−σ )k+1 , δ1 = Cλ. λσ/2 We choose = λσ/2 / max{1, l 2 }. Then, δ0 (l) = 2 1−Cλ 1+Cλ max{1,l 2 } . For sufficiently small λ, one has • mes Il,m (k + 1) \ Il,m (k) ≤ Cλσ (Cλ1−σ )k+1 if λσ (Cλ1−σ )k+1 < Cλσ/2 / l 2 , i.e. −σ/4 (Cλ1−σ )−(k+1)/2 , if l ≤ L, L = Cλ • mes Il,m (k + 1) \ Il,m (k) ≤ Cλσ/2 l −2 otherwise. This implies mes (k+1 \ k ) ≤ CLλσ (Cλ1−σ )k+1 +
C σ/2 ≤ Cλ3σ/4 (Cλ1−σ )(k+1)/2 . λ L
Summing the exponentially convergent series in k, we see that ∞ satisfies (11.6) and, hence, that B∞ is not empty. This completes the proof of Proposition 11.1. & ' Acknowledgements. This work was done while A.F. held a PAST professorship at Université Paris 13. F.K. gratefully acknowledges support of the European TMR network ERBFMRXCT960001. While working on these results, we had the pleasure of many interesting and helpful discussions with many friends and colleagues. In particular, we would like to thank M. Aizenman, S. Jitomirskaya, L. Pastur, B. Simon and Y. Sinaï.
References 1. Avron, J. and Simon, B.: Almost periodic Schrödinger operators, II. The integrated density of states. Duke Math. J., 50, 369–391, (1983) 2. Bellissard, J., Lima, R. and Testard, D.: Metal-insulator transition for theAlmost Mathieu model. Commun. Math. Phys. 88, 207–234 (1983) 3. Birman, M.Sh. and Solomjak, M.Z.: Spectral theory of selfadjoint operators in Hilbert space. Dordrecht: D. Reidel Publishing Co., 1987 Translated from the 1980 Russian original by S. Khrushchëv and V. Peller 4. Bougerol, P. and Lacroix, J.: Products of random matrices with applications to Schrödinger operators. Boston, MA: Birkhäuser Boston Inc., 1985 5. Buslaev, V.: Adiabatic perturbation of a periodic potential. Teor. Mat. Fiz. 58, 223–243 (1984) (in Russian) 6. Buslaev, V. and Fedotov, A.: The complex WKB method for Harper’s equation. Preprint, Mittag-Leffler Institute, Stockholm, 1993 7. Buslaev, V. and Fedotov, A.: Bloch solutions of difference equations. St Petersburg Math. J. 7, 561–594 (1996) 8. Carmona, R. and Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Basel: Birkhäuser, 1990 9. Coddington, E. and Levinson, N.: Theory of ordinary differential equations. New-York: McGraw-Hill, 1955 10. Cycon, H.L., Froese, R.G., Kirsch, W. and Simon, B.: Schrödinger Operators. Berlin: Springer Verlag, 1987 11. Dinaburg, E.I. and Sina˘ı, Ja.G.: The one-dimensional Schrödinger equation with quasiperiodic potential. Funk. Anal. i Priložen. 9(4), 8–21 (1975) 12. Eastham, M.: The spectral theory of periodic differential operators. Edinburgh: Scottish Academic Press, 1973 13. Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992) 14. Eliasson, L.H.: Reducibility and point spectrum for linear quasi-periodic skew products. In: Proceedings of the ICM 1998,Berlin, Volume II, pp. 779–787, 1998 15. Fedoryuk, M.: Asymptotic analysis. Berlin: Springer Verlag, 1st edition, 1993 16. Fedotov, A. and Klopp., F.: On the singular spectrum of one dimensional quasi-periodic Schrödinger operators in the adiabatic limit. Preprint Universität Potsdam 17. Fedotov, A. and Klopp, F.: The spectrum of adiabatic quasi-periodic Schrödinger operators on the real line. In progress 18. Fedotov, A. and Klopp, F.: The monodromy matrix for a family of almost periodic equations in the adiabatic case. Preprint, Fields Institute, Toronto, 1997
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19. Fedotov, A. and Klopp, F.: A complex WKB analysis for adiabatic problems. Asymptotic Anal. 27, 219–264 (2001) 20. Fedotov, A. and Klopp, F.: On the absolutely continuous spectrum of one dimensional quasi-periodic Schrödinger operators in the adiabatic limit. Preprint, Université Paris-Nord, 2001 21. Fedotov, A. and Klopp, F.: Coexistence of different spectral types for almost periodic Schrödinger equations in dimension one. In: Mathematical results in quantum mechanics (Prague, 1998). Basel: Birkhäuser, 1999, pp. 243–251 22. Fedotov, A. and Klopp, F.: Transitions d’Anderson pour des opérateurs de Schrödinger quasi-périodiques en dimension 1. In Seminaire: Équations aux Dérivées Partielles, 1998–1999. Palaiseau: École Polytech., 1999, pp. Exp. No. IV, 15 23. Fröhlich, J., Spencer, T. and Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132, 5–25 (1990) 24. Gilbert, D. and Pearson, D.: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. and its Appl. 128, 30–56 (1987) 25. Helffer, B., Kerdelhué, P. and Sjöstrand, J.: Le papillon de Hofstadter revisité. Mém. Soc. Math. France (N.S.) 43, 87 (1990) 26. Herman, M.-R.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol’d et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58 (3), 453–502 (1983) 27. Hiramoto, H. and Kohmoto, M.: Electronic spectral and wavefunction properties of one-dimensional quasi-periodic systems: A scaling approach. Int. J. Mod. Phys. B, 164 (3–4), 281–320 (1992) 28. Janssen, T.: Aperiodic Schrödinger operators. In: R. Moody, ed., The Mathematics of Long-Range Aperiodic Order. Dordrecht: Kluwer, 1997, pp. 269–306 29. Jitomirskaya, S.: Almost everything about the almost Mathieu operator. II. In: XIth International Congress of Mathematical Physics (Paris, 1994), Cambridge: Internat. Press 1995, pp. 373–382 30. Jitomirskaya, S.Ya.: Metal-insulator transition for the almost Mathieu operator. Ann. of Math. (2) 150 (3), 1159–1175 (1999) 31. Jitomirskaya, S.Ya. and Last, Y.: Power law subordinacy and singular spectra. II. Line operators. Comm. Math. Phys. 211 (3), 643–658 (2000) 32. Kargaev, P. and Korotyaev, E.: Effective masses and conformal mappings. Commun. Math. Phys. 169, 597–625 (1995) 33. Last, Y.: Almost everything about the almost Mathieu operator. I. In: XIth International Congress of Mathematical Physics (Paris, 1994), Cambridge: Internat. Press, 1995, pp. 366–372 34. Marchenko, V. and Ostrovskii, I.: A characterization of the spectrum of Hill’s equation. Math. USSR Sbornik 26, 493–554 (1975) 35. McKean, H. and Trubowitz, E.: The spectrum of Hill’s equation. Invent. Math. 30, 217–274 (1975) 36. McKean, H. and van Moerbeke, P.: Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Comm. Pure Appl. Math. 29, (2), 143–226 (1976) 37. Pastur, L. and Figotin, A.: Spectra of Random and Almost-Periodic Operators. Berlin: Springer-Verlag, 1992 38. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol I: Functional Analysis. New York: Academic Press, 1980 39. Simon, B.: Almost periodic Schrödinger operators: a review. Advances in Applied Mathematics 3, 463–490 (1982) 40. Sorets, E. and Spencer, T.: Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials. Commun. Math. Phys. 142 (3), 543–566 (1991) 41. Titschmarch, E.C.: Eigenfunction expansions associated with second-order differential equations. Part II. Oxford: Clarendon Press, 1958 42. Wilkinson, M.: Tunnelling between tori in phase space. Phys. D 21 (2–3), 341–354 (1986) Communicated by M. Aizenman
Commun. Math. Phys. 227, 93 – 118 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
The Perturbation of the Quantum Calogero–Moser–Sutherland System and Related Results Yasushi Komori, Kouichi Takemura 1 Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan.
E-mail:
[email protected]
2 Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku,
Yokohama 236-0027, Japan. E-mail:
[email protected] Received: 30 May 2001 / Accepted: 27 November 2001
Abstract: The Hamiltonian of the trigonometric Calogero–Sutherland model coincides with a certain limit of the Hamiltonian of the elliptic Calogero–Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero–Moser model and the regularity (convergence) of the perturbation for the arbitrary root system. We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x1 , . . . , xN ) for the AN−1 -case. As a result, the algebraic calculation of the perturbation is justified. 1. Introduction The Hamiltonian of the elliptic Calogero–Moser model is given as follows ([8]), H := −
N
i=1
1≤i<j ≤N
1 ∂2 + β(β − 1) 2 ∂xi2
℘ (xi − xj ),
(1.1)
where β is the coupling constant. This Hamiltonian reduces to that of the trigonometric Calogero–Sutherland model √ by setting τ → −1∞, where τ is the ratio of two basic periods of the elliptic function. As for the trigonometric Calogero–Sutherland model, it is well-known by specialists that their eigenstates are given by the Jack polynomials (or the AN−1 -Jacobi polynomials). So far, many researchers have studied the Jack polynomials and its q-deformed version, the Macdonald polynomials, and clarified various properties such as the orthogonality, the norms, the Pieri formula, the Cauchy formula, and the evaluations at (1, . . . , 1). The Calogero–Sutherland model is extended to those associated with simple Lie algebras. From this point of view the Hamiltonian (1.1) is called the AN−1 -type. Studies of these models are being developed by using their algebraic structures.
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In contrast with the trigonometric models, the elliptic models are less investigated and the spectrum or the eigenfunctions are not sufficiently analyzed. There is, however, some important progress due to Felder andVarchenko. They clarify that the BetheAnsatz works well for the AN−1 -type elliptic Calogero–Moser model ([1]). Although this method may have applications to the spectral problem and indeed some partial results are obtained in the articles ([10, 11]), we will employ another approach in the present article. In this article, we will add some knowledge of the elliptic Calogero–Moser model based on the analysis of the trigonometric model, which we will explain below. One topic is the essential self-adjointness of the elliptic Calogero–Moser model for the arbitrary root system. Firstly we will establish it for the trigonometric model by taking the Jacobi polynomials as its domain in the space of square integrable functions. We will obtain the elliptic version by perturbation. A second topic is to obtain the eigenvalues and the eigenfunctions of the elliptic Calogero–Moser model for arbitrary root systems. There are at least two ways to perform it. One is to use the Bethe Ansatz method, which is valid only for the AN−1 case. From this viewpoint some results are obtained in ([10, 11]). The other is to use the well-developed perturbation theory, which we will consider in this article. We regard the Hamiltonian of the elliptic Calogero–Moser model as the perturbed operator of √ the Calogero–Sutherland model by the parameter p = exp(2π τ −1). We have such abundant knowledge about the eigenvalues and the eigenfunctions of the Calogero– Sutherland model that we can apply the perturbation method. Then we will obtain the eigenvalues and the eigenfunctions as a formal power series of p. In general, such formal d2 2 4 power series do not converge. For example, consider the operator H := − dx 2 +x +αx , then the formal power series of the eigenvalues and eigenfunctions diverge for any α = 0. However, in our cases, the formal power series converges if p is sufficiently small. The convergence is assured by the functional analytic method introduced by Kato and Rellich. We mean the convergence of the eigenfunctions in the L2 -norm sense. The other topics are the holomorphy of the eigenfunctions, the relationship with the higher commuting operators, and giving the elliptic analogue of the Jacobi polynomials, which are valid for the AN−1 case. The Kato–Rellich method does not give the holomorphy a priori. We will obtain the holomorphy by using several properties of the Jack (or the AN−1 -Jacobi) polynomials. Thanks to the holomorphy, the eigenspaces of the secondorder Hamiltonian are compatible with the higher commuting operators. By considering the joint eigenfunctions of the Hamiltonian and the higher commuting operators, we see the well-definedness of an elliptic analogue of the AN−1 -Jacobi polynomials. We remark that Langmann obtained the algorithm for constructing the eigenfunctions and eigenvalues as the formal power series of p ([5]) His algorithm would be closely related to ours which is explained in Sect. 4.3. There are some merits for comparing the perturbation method to the Bethe Ansatz method. The calculation of the perturbation does not essentially depend on the coupling constant β though the calculation of the Bethe Ansatz method strongly depends on β. In addition, the Bethe Ansatz method is applied to the AN−1 type and β ∈ Z>1 cases, but the perturbation method may be valid for all types and the coupling constant does not need to be an integer.
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2. Jacobi Polynomials and Self-Adjointness The Hamiltonians of the trigonometric and the elliptic Calogero–Sutherland models are respectively given by HT := − +
kα (kα − 1)|α|2
α∈R+
HE := − +
1 1 − , 4 sin2 ( α, h /2) 12
kα (kα − 1)|α|2 ℘ ( α, h ),
(2.1) (2.2)
α∈R+
where the coupling constant kα is real and invariant under the action of the Weyl group kα = kwα , and ℘ (x) = ℘ (x; π, π τ ) is the Weierstrass ℘ function. For our later convenience, we have subtracted a constant term from the original trigonometric Hamiltonian. √ is the Laplacian on T := hR /2π Q∨ . By using the variable p = exp(2τ π −1), we often write HE (p) = HE in order to emphasize the dependency of p. In this notation, we have HT = HE (0). We first show that the Hamiltonian of the trigonometric model is defined on a dense subspace of L2 (T , dµ)W , where µ is the normalized Haar measure, and is essentially self-adjoint with respect to the inner product (f, g) := f · g. (2.3) T
We denote by · := (· , ·)1/2 the norm in L2 (T , dµ). We define HT and HE on C 2 (T )W ∩ D(V ), where D(V ) denotes the domain of the multiplication operators in (2.1) and (2.2). Then we see that these operators are symmetric. If kα ≥ 2, HT has a C 2 -class W -invariant eigenfunction HT = E0 ,
(2.4)
where E0 = (ρ(k)|ρ(k)) − e0 , 1 kα (kα − 1)|α|2 , e0 = 12
(2.5) (2.6)
α∈R+
1 kα α, 2 α∈R+ | sin( α, h /2)|kα . =
ρ(k) =
(2.7) (2.8)
α∈R+
Let C[P ] be the polynomial ring of the weight lattice P . For each λ, let eλ denote the corresponding element, so that eλ eµ = eλ+µ and e0 = 1. We also regard the element eλ √ λ −1 λ,h ˙ := e as a function on T by the rule e (h) , where h˙ ∈ T is the image of h ∈ hR . Let mλ for λ ∈ P+ be the monomial symmetric functions mλ := |Wλ |−1 ewλ = eµ , (2.9) w∈W
µ∈W λ
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where Wλ denotes the stabilizer of λ in W . The set {mλ |λ ∈ P+ } forms a basis of C[P ]W . Define the partial order ≺ in P by ν µ ⇔ µ − ν ∈ Q+ .
(2.10)
Let · and (· , ·) denote the norm and the inner product in L2 (T , 2 dµ) respectively. Definition 1 (Heckman-Opdam). There exists a family of polynomials {Jµ |µ ∈ P+ } which consists of a basis of C[P ]W satisfying the following conditions: Jµ = mµ + uµν mν , (2.11) ν≺µ
(Jµ , Jν ) = 0, −1 HT .
Let H0 := its eigenfunctions. Proposition 2.1.
if µ = ν.
(2.12)
Then these polynomials are characterized by the operator H0 as
H0 Jµ = Eµ Jµ ,
(2.13)
where Eµ = (µ + ρ(k)|µ + ρ(k)) − e0 . It is well known that the normalized Jacobi polynomials J˜λ (λ ∈ P+ ) form a complete orthonormal system in the space L2 (T , 2 dµ)W with the inner product (· , ·) if kα ≥ 0. It follows that Lemma 2.2. Assume kα ≥ 0. Then P := C[P ]W is a dense subspace in L2 (T , dµ)W . Theorem 2.3. Assume kα ≥ 2. Then HT is essentially self-adjoint on P. Proof. From Proposition 2.1 and P ⊂ C 2 (T )W ∩ D(V ) we see that Jλ are the eigenfunctions of HT . Then the theorem is obtained from Lemma 2.2 since it implies that the range of (HT ± i) is dense. If 0 < kα < 2, then ∈ C 2 (T )W ∪ D(V ) and P is not an appropriate domain for HT . However Theorem 2.3 is generalized in the following sense in terms of the adjoint operator HT∗ : Theorem 2.4. Assume kα ≥ 0. Then HT∗ |P is essentially self-adjoint on P. We rewrite HE (p) = W(p)+HT , where W(p) = (HE (p)−HT ) is a multiplication operator with 1 1 + W(p)(h) := kα (kα − 1)|α|2 ℘ ( α, h ) − . (2.14) 4 sin2 ( α, h /2) 12 α∈R+
By the formula (A.13), we see that W(p)u ≤ W(p)max u,
(2.15)
since the function W(p)(h) is a continuous function on T . This implies that W(p) is ∗ (p) = W(p)∗ + H∗ . bounded. Hence we have HE T ∗ (p)| is essentially selfTheorem 2.5 ([4]). Let −1 < p < 1 and kα ≥ 0. Then HE P adjoint on P. Proof. The symmetry of the operator W(p) is trivial. Then we deduce that W(p)+HT∗ |P is essentially self-adjoint on P. ∗ (p)| . In the next section, we abuse the symbols HT and HE (p) for HT∗ |P and HE P
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3. Perturbation in the L2 -Space In this section, we employ the variable p with |p| < 1 instead of τ as a parameter of perturbation and treat mainly the gauge-transformed Hamiltonian defined below with kα > 0. For a linear operator T , we denote by D(T ) its domain and by R(T ) its range respectively. 3.1. The resolvent in the L2 space. For a bounded linear operator A, we denote by A the operator norm, i.e., A := supv =1 Av . We set W (p) := −1 (HE (p) − HT )(= W(p)), −1
T (p) :=
HE (p).
(3.1) (3.2)
Then T (p) = H0 + W (p) is a closable operator on L2 (T , 2 dµ)W with D(T (p)) = C[P ], and particularly if −1 < p < 1, T (p) is an essentially self-adjoint operator. Here W (p) is a bounded operator on L2 (T , 2 dµ)W with an upper bound, W (p) ≤ Wmax (p) := 4
∞ n|p|n · kα (kα − 1)|α|2 , 1 − |p|n n=1
(3.3)
α∈R+
which is monotonous with respect to |p| and tends to 0 as p → 0. Let T˜ denote the closure of a closable operator T . Then T˜ (p) for p ∈ (−1, 1) is the unique extension of T (p) to the self-adjoint operator. In particular H˜ 0 = T˜ (0) is the self-adjoint extension of H0 . T˜ (p) is a self-adjoint holomorphic family [3]. Notice that the spectrum of the operator H˜ 0 is discrete. Let σ (H˜ 0 ) be the set of the spectrum and let ρ(H˜ 0 ) be the resolvent set of the operator H˜ 0 . We have σ (H˜ 0 ) = {(λ + ρ(k)|λ + ρ(k)) − e0 |λ ∈ P+ }.
(3.4)
The following proposition is obvious. Proposition 3.1. For each a ∈ σ (H˜ 0 ), the corresponding eigenspace {v ∈ L2 (T , 2 dµ)W | H˜ 0 v = av} is finite dimensional. For ζ ∈ ρ(H˜ 0 ), the resolvent (H˜ 0 − ζ )−1 is compact and (H˜ 0 − ζ )−1 = (dist(ζ, σ (H˜ 0 )))−1 . We have (H˜ 0 − ζ )−1
c λ Jλ =
λ
(Eλ − ζ )−1 cλ Jλ , λ
(3.5)
where λ cλ Jλ ∈ L2 (T , 2 dµ)W and H˜ 0 Jλ = Eλ Jλ . The proof of the Kato-Rellich theorem also implies the compactness of the resolvent of T˜ (p) for −1 < p < 1. If (H˜ 0 − ζ )−1 W (p) < 1, then (H˜ 0 − ζ )−1 (T˜ (p) − ζ ) = 1 + (H˜ 0 − ζ )−1 W (p) has a bounded inverse by Neumann series and thus T˜ (p) − ζ has also a bounded inverse
−1 (T˜ (p) − ζ )−1 = 1 + (H˜ 0 − ζ )−1 W (p) (H˜ 0 − ζ )−1 (3.6) =
∞
j −(H˜ 0 − ζ )−1 W (p) (H˜ 0 − ζ )−1 .
j =0
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In particular, if (H˜ 0 − ζ )−1 < W (p)−1 , then the bounded inverse of T˜ (p) − ζ exists. The right-hand side of this expression implies that the resolvent of T˜ (p) is also compact. By the equality (H˜ 0 −ζ )−1 = (dist(ζ, σ (H˜ 0 )))−1 and Eq. (3.4), the resolvent set ρ(T˜ (p)) is included outside the union of the closed disks dist(ζ, σ (H˜ 0 )) ≤ W (p). Proposition 3.2. Let T be a closed operator with the resolvent set ρ(T ). Let 41 , 42 be circles which are contained in ρ(T ) and whose interiors are disjoint. We set 1 (T − ζ )−1 dζ, (i = 1, 2). Pi := − √ 2π −1 4i Then we have Pi2 = Pi and P1 P2 = P2 P1 = 0. Proposition 3.3. Let P , Q be bounded operators subject to P 2 = P , Q2 = Q and P − Q < 1. Then we have rank P = rank Q. Proof. For u ∈ R(Q), we have P u = u + P u − Qu due to Qu = u. Then P u ≥ (1 − P − Q)u,
(3.7)
which implies that P |R(Q) : R(Q) → R(P Q) ⊂ R(P ) is one-to-one and rank P ≥ rank Q. Similarly we have rank P ≤ rank Q and hence rank P = rank Q. Let 4 ⊂ ρ(H˜ 0 ) be a circle and let r = dist(4, σ (H˜ 0 )). Then there exists p0 > 0 such that for all |p| < p0 , W (p) < r and thus 4 ⊂ ρ(T˜ (p)). Notice (H˜ 0 − ζ )−1 ≤ r −1 on 4. Let 1 P4 (p) := − √ (T˜ (p) − ζ )−1 dζ. 2π −1 4 Then we have 1 P4 (p) − P4 (0) ≤ 2π ≤
1 2π
4
(T˜ (p) − ζ )−1 − (H˜ 0 − ζ )−1 |dζ |
∞ 4 j =1
(H˜ 0 − ζ )−1 j +1 W (p)j |dζ |
1 r −2 W (p) < . |dζ | 2π 4 1 − r −1 W (p)
(3.8) (3.9)
(3.10)
Fix ai ∈ σ (H˜ 0 ). Since the set σ (H˜ 0 ) is discrete, we can choose a circle 4i and 0 < pi such that 4i contains only one element ai inside it and Pi (p) = P4i (p) satisfying Pi (p) − Pi (0) < 1 for |p| < pi . By Propositions 3.2, 3.3, we see that rank Pi (p) = rank Pi (0) and in particular, Pi (p) is a degenerate operator. By the proof of Proposition 3.3, we see that Vi (p) := R(Pi (p)) is spanned by the image of Vi (0) = R(Pi (0)), i.e., the eigenspace of H˜ 0 with the eigenvalue ai . We choose a basis of Vi (p) as {Pi (p)J˜λ | λ such that H˜ 0 J˜λ = ai J˜λ }, where J˜λ is the normalized Jacobi polynomial. One sees that Vi (p) is a finite dimensional invariant subspace of T˜ (p) due to the commutativity of Pi (p) and T˜ (p). Lemma 3.4. The matrix elements of T˜ (p)|Vi (p) : Vi (p) → Vi (p) with respect to Pi (p)J˜λ are real-holomorphic functions of p.
Perturbation of Calogero–Moser–Sutherland System µ
99 µ
Proof. We define the functions cλ (p) and dλ (p) by T˜ (p)Pi (p)J˜λ =
µ
Pi (0)Pi (p)J˜λ =
µ
µ
cλ (p)Pi (p)J˜µ , µ
dλ (p)J˜µ .
(3.11) (3.12)
Then we see that Pi (0)T˜ (p)Pi (p)|Vi (0) : Vi (0) →µVi (0) and Pi (0)Piµ(p)|Vi (0) : Vi (0) → Vi (0) are real-holomorphic. Equivalently, µ cλ (p)dµν (p) and dλ (p) are real-holoµ morphic. By Proposition 3.3, Pi (0)Pi (p)|Vi (0) or the matrix dλ (p) is invertible, which µ implies cλ (p) is real-holomorphic. µ
The matrix c(p) = (cλ (p)) is symmetric. It is known that if all the matrix elements of the symmetric operator on the finite dimensional vector space are real-holomorphic, then its eigenvalues and eigenvectors are real-holomorphic (see [3]). Hence we have Proposition 3.5. The eigenvalues of T˜ (p) are on Vi (p) real-holomorphic and coincide with ai when p = 0. The eigenfunctions are also real-holomorphic. Summarizing, we obtain the following theorem. Theorem 3.6. For each ai ∈ σ (H˜ 0 ), there exists pi > 0 such that for −pi < p < pi , the dimension of the eigenspace whose eigenvalues are included in |ζ − ai | < Wmax (p) is equal to the dimension of the eigenspace of eigenvalue ai . Moreover the eigenfunctions and the eigenvalues depend on p real-holomorphically. If the coupling constants kα (> 0) are all rational numbers, we can estimate the eigenvalues uniformly. We will explain this below. Suppose kα are all rational. Let kα = kα,num /kden be such that kα,num are integers and kden is a positive integer. Let n be the minimal positive integer such that (P |P ) ⊂ Z/n. Then we see that the spectrum of H˜ 0 is uniformly separated. To be more precise, if a, b ∈ σ (H˜ 0 ) and a = b, we have |a − b| ≥ 1/nkden . Hence if we take p0 as Wmax (p0 ) = 1/4nkden ,
(3.13)
then there exists a set of circles 4i such that for |p| < p0 , each 4i ⊂ ρ(T˜ (p)) contains only one element ai ∈ σ (H˜ 0 ) inside it, any two circles never cross, Pi (p)−Pi (0) < 1, and every element of σ (T˜ (p)) is contained inside some circle 4i . Therefore we have Theorem 3.7. Suppose kα ∈ Q>0 and let p0 be defined in (3.13). Then Theorem 3.6 holds for pi = p0 . All eigenvalues of T˜ (p) on the L2 (T , 2 dµ)W space are contained in ∪a∈σ (H˜ 0 ) {ζ | |ζ − a| < Wmax (p)} for −p0 < p < p0 . All eigenfunctions are realholomorphically connected to the eigenfunction of H˜ 0 as p → 0.
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4. AN−1 -Cases 4.1. In Sect. 3.1, we considered the spectrum problem of the gauge-transformed Hamiltonian T˜ (p) in the L2 (T , 2 dµ)W space and show that the perturbation is holomorphic by use of the theory of Kato and Rellich. On the other hand, it is known that there is a commuting family of differential operators (e.g. (4.2) for the AN−1 case) which commute with the Hamiltonian ([8, 7, 2]). In this section, we will investigate the relationship between the functions obtained by applying the projections Pi (p) and the commuting family of differential operators. As a result, we will prove that the perturbation series which is obtained by the algorithmic calculation is not only square-integrable but also holomorphic w.r.t. the variables of the coordinate. For this purpose, we will consider the spectrum problem in the C ω (T )W -space. In this section, we consider the AN−1 cases. 4.2. We introduce some known result for the AN−1 cases. We realize the AN−1 root system in RN . Let {8i }i=1,... ,N be an orthonormal basis. N The space h∗ is defined by h∗ := {h = N i=1 hi 8i | i=1 hi = 0}. The simple roots are {8i − 8i+1 |i = 1, . . . , N − 1}. We set xi = (h|8i ). Let us recall the Hamiltonian of the elliptic Calogero–Moser model, H := −
N
i=1
1≤i<j ≤N
1 ∂2 + β(β − 1) 2 ∂xi2
℘ (xi − xj ).
This system is integrable, i.e., there exists sufficiently many commuting operators. The existence and the explicit expressions are known in ([8, 7, 2]), etc. Here, we exhibit Hasegawa’s expression which will be used in the proof of Proposition 4.1. Later we will discuss the relationship between the expression of Ochiai–Oshima–Sekiguchi ([7]) and the one of Hasegawa ([2]). Following ([2]), we set ∂ β j ∈J ∂xj :(x) ∂ ˆ Hi := , (4.1) :(x) ∂xj |I |=i J ⊂I
j ∈I \J
Hi := :(x) Hˆ i :(x)−β (1 ≤ i ≤ N ), (4.2) where :(x) := 1≤i<j ≤N θ ((xi − xj )/2π ), θ(x) is the theta function defined in Sect. A.2. The operators Hˆ i , Hi , H act on the space of functions which are meromorphic except for the branches along xj − xk ∈ 2π(Z + Zτ ) (j = k). On this space, we have [Hˆ i1 , Hˆ i2 ] = [Hi1 , Hi2 ] = 0 (1 ≤ i1 , i2 ≤ N ) and [Hi , H] = 0 (1 ≤ i ≤ N ). ˜ := 1≤j
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˜ −1 H ˜ coincide with the one of the operator (1 ≤ i ≤ N ). The action of the operator T˜ (p) for the smooth functions on the real domain except for xj = xk (j = k). From ˜ −1 H. ˜ The operators T (p) and H (i) (p) this reason, we adopt the notation T (p) = act on the space of meromorphic functions on the complex domain. The operator T (p) is expressed by some combinations of H (1) (p) and H (2) (p). Proposition 4.1. The operators T (p) and H (i) (p) preserve the space C ω (T )W , where T is the torus hR /2πQ∨ ; (Q∨ (% Q): the coroot lattice of type AN−1 ). Proof. Since the operator T (p) is expressed in terms of H (1) (p) and H (2) (p), it is enough to show the H (i) (p) (1 ≤ i ≤ N ) cases. The Weyl group of type AN−1 acts on the space of the function on hR by the permutation of the variable. We denote the action of σ on f (x) by f (x σ ). Let us recall the definition of C ω (T )W , i.e., f (x σ ) = f (x) (∀σ ∈ W ), C ω (T )W = f (x) ∈ C ω (RN ) f (x + u N . 8 ) = f (x) (∀u ∈ R), i i=1 f (x + 2π(8i − 8j )) = f (x) (1 ≤ i, j ≤ N ) (4.3) Let f (x) be a function in C ω (T )W . From the definition of the operators H (i) (p) (1 ≤ i ≤ N ), the function f˜(x) := H (i) (p)f (x) satisfies the relations f˜(σ x) = f˜(x) (∀σ ∈ W ), ˜ ˜ ˜ f˜(x + u N j =1 8j ) = f (x) (∀u ∈ R), f (x + 2π(8j1 − 8j2 )) = f (x) (1 ≤ j1 , j2 ≤ N ). It remains to show the holomorphy of the function f˜(x) on RN . The function (θ (x)/ sin πx)β is a non-zero holomorphic function in R, because the function θ (x)/ sin πx is non-zero on R and does not admit any branching points on R. From the definition of the operators Hˆ i (1 ≤ i ≤ N ), the function H (i) (p)f (x) does not have poles except for xj1 − xj2 + 2π k = 0 (1 ≤ j1 = j2 ≤ N, k ∈ Z) on RN . If the function H (i) (p)f (x) has a pole along xj1 − xj2 = 0, the order of the pole is one, but it contradicts the Weyl group invariance of the function H (i) (p)f (x). Therefore the function H (i) (p)f (x) is holomorphic along xj1 − xj2 = 0. The holomorphy along xj1 −xj2 +2π k = 0 (k ∈ Z\{0}) follows from the periodicity of H (i) (p)f (x). From the commutativity of H and Hi we have [T (p), H (i) (p)] = [H (i) (p), H (j ) (p)] = 0, (1 ≤ i, j ≤ N ),
(4.4)
on the space C ω (T )W . (p) (1 ≤ By using the formula (A.12), we can expand the operators T (p) and H (i) √ i ≤ N ) as the formal power series of operators w.r.t. the parameter p(= e2π −1τ ), T (p) = T (0) +
∞
T {j } (0)p j ,
j =1
H (i) (p) = H (i) (0) +
∞ j =1
H (i),{j } (0)p j .
(4.5)
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Y. Komori, K. Takemura √
We set zi = e −1xi . The operators T (0), H (i) (0), T {j } (0), and H (i),{j } (0) are expressed as the combination of the rational functions of z1 , z2 , . . . , zN and the polynomials of ∂ ∂ ∂z1 , . . . , ∂zN . Proposition 4.2. Let f be an element of C[P ]W . Then the functions T (0)f , T {j } (0)f , H (i) (0)f , and H (i),{j } (0)f are elements of C[P ]W for j ∈ Z≥1 and 1 ≤ i ≤ N . Proof. We prove H (i),{j } (0)f ∈ C[P ]W . For the other cases, the proofs are similar. From the definition, the function H (i),{j } (0)f is symmetric and rational with respect to the variables z1 , . . . , zN . The possible poles of the rational function H (i),{j } (0)f are zk − zk ' = 0 (1 ≤ k < k ' ≤ N ) and the degree of each pole is one, but it contradicts the Weyl group invariance of the function H (i),{j } (0)f . Therefore the rational function H (i),{j } (0)f does not have any poles, and we have (i),{j } (0) ∈ C[P ]W . H We notice that the operator T (0) = A1 H˜ 0 + A2
(4.6)
is the gauge-transformed Hamiltonian of the trigonometric Calogero–Sutherland model up to constants A1 , A2 , where H˜ 0 =
N i=1
2
∂ zi ∂zi
zi + z j ∂ ∂ zi . +β − zj zi − z j ∂zi ∂zj
(4.7)
i<j
The joint eigenfunction of the operators H (i) (0) (1 ≤ i ≤ N ) is the Jacobi polynomial (i) (i) Jλ (λ ∈ P+ ). We denote the joint eigenvalue Eλ by H (i) (0)Jλ = Eλ Jλ . (i) (i) Suppose β > 0. Let λ, µ ∈ P+ . It is known that the condition Eλ = Eµ for all i ∈ {1, . . . , N} is equivalent to λ = µ. In other words, the joint eigenvalue is nondegenerate. From now on we will discuss the symmetry (self-adjointness) of the higher commuting Hamiltonians. For this purpose, we will discuss the relationship between the expressions of the higher commuting Hamiltonians in ([7]) and the ones in ([2]). Following ([7, 9]), we introduce the operators Ik =
0≤j ≤[ 2k ]
1 − 2j )!
2j j !(k
· · · u(x2j −1 − x2j )
σ
(u(x1 − x2 )u(x3 − x4 ) · · ·
σ ∈W
∂
∂
∂x2j +1 ∂x2j +2
...
∂ ∂xk
(4.8)
,
where k = 1, . . . , N, W is the Weyl group of AN−1 -type (N th symmetric group), 1 , . . . , xN )) = f (xσ −1 (1) , . . . , xσ −1 (N) ) for σ ∈ W , and u(x) = β(β − 1)℘ (x). The domains of the operators Ik (k = 1, . . . , N ) are the same as the ones of Hk (k = 1, . . . , N). By a straightforward calculation, we have H3 = I3 + CI1 for some constant C. Applying Theorem 5.2. in ([9]), we obtain that the operators Hk (k = 1, . . . , N ) are expressed as the polynomial of I1 , I2 , . . . , IN . σ (f (x
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103
Let R[I1 , I2 , . . . , IN ] be a polynomial ring generated by I1 , I2 , . . . , IN and ς be an involution on R[I1 , I2 , . . . , IN ] such that ς F (x1 , . . . , xN ) = F (−x1 , . . . , −xN ). Then ς I = (−1)k I and ς H = (−1)k H . Hence H admit the expansion, k k k k k Hk = cj1 ,... ,jm Ij1 · · · Ijm , (4.9) j1 ≤···≤jm
where cj1 ,... ,jm ∈ R and if k − (j1 + · · · + jm ) ∈ 2Z≥0 then cj1 ,... ,jm = 0. From a similar discussion, the operators Ik admit the expansion, Ik = c˜j1 ,... ,jm Hj1 · · · Hjm ,
(4.10)
j1 ≤···≤jm
where c˜j1 ,... ,jm ∈ R and if k − (j1 + · · · + jm ) ∈ 2Z≥0 , then c˜j1 ,... ,jm = 0. Lemma 4.3. We suppose β > N . For f, g ∈ C ω (T )W , we have (H (k) (p)f, g) = (−1)k (f, H (k) (p)g) (1 ≤ k ≤ N ). ˜ 2 dµ and H (k) (p) = ˜ −1 Hk , ˜ it is enough Proof. Since (f, g) = T f (x)g(x)|| k ˜ ˜ ˜ Hk (g(x)||) ˜ to show T Hk (f (x)||) g(x)||dµ = (−1) T f (x)|| dµ. We have the equality T h(x)dµ = A 0≤x1 ,... ,xN ≤2πN h(x)dx1 · · · dxN for some non-zero constant A, which follows from the correspondence between the integration of the sln invariant function and the one of the gln . From this equality, the property (4.9), and the commutativity [Ij1 , Ij2 ] = 0 (1 ≤ j1 , j2 ≤ N ), if we show ˜ ˜ ˜ Ik (g(x)||) ˜ Ik (f (x)||) g(x)||dx = (−1)k f (x)|| dx, (4.11) D
D
RN |0
where D = {(x1 , . . . , xN ) ∈ ≤ x1 , . . . , xN ≤ 2π N } and dx = dx1 dx2 · · · dxN , then we obtain Lemma 4.3. ˜ g(x)|| ˜ are C N -class. If β > N then the functions f (x)||, ω W From the definition of C (T ) (4.3), we have the periodicity f (x1 , . . . , xl + 2πN, . . . , xN ) = f (x1 , . . . , xl , . . . , xN ) (1 ≤ l ≤ N ) for f (x1 , . . . , xN ) ∈ C ω (T )W . ˜ and g(x) ˜ The functions f˜(x), g(x) We set f˜(x) = f (x)|| ˜ = g(x)||. ˜ are smooth on RN except for xi − xj + 2π k = 0 (1 ≤ i = j ≤ N, k ∈ Z). The behaviors of the ˜ functions f˜(x), g(x) ˜ around xi − xj + 2π k = 0 are O(|xi − xj |β ), i.e. f (x) β and g(x) ˜ |xi −xj |β
|xi −xj |
are bounded around xi − xj + 2π k = 0. From the expression of Ik (4.8), if we show ∂ ∂ ˜ u(x1 − x2 )u(x3 − x4 ) · · · u(x2j −1 − x2j ) ··· f (x) g(x)dx ˜ (4.12) ∂x2j +1 ∂xk D ∂ ∂ ˜ = f (x) u(x1 − x2 )u(x3 − x4 ) · · · u(x2j −1 − x2j ) ··· g(x) ˜ dx, ∂x2j +1 ∂xk D
for all j s.t. 0 ≤ j ≤ [ 2k ], then we obtain (4.11) and Lemma 4.3. The number β satisfies β > N ≥ 2. Though the function u(x2l−1 − x2l ) = β(β − 1)℘ (x2l−1 − x2l ) (l = 1, . . . , j ) has a double pole along x2l−1 − x2l + 2π k = 0 (k ∈ Z),
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the integrands of (4.12) are bounded around x2l−1 − x2l + 2π k = 0 from the properties f˜(x) = O(|x2l−1 − x2l |2 ) and g(x) ˜ = O(|x2l−1 − x2l |2 ) around x2l−1 − x2l + 2π k = 0. Hence the singularities along x2l−1 − x2l + 2π k = 0 (l = 1, . . . , j, k ∈ Z) do not affect the integration. Since the integrands of (4.12) are continuous, we can replace the range of integration of both sides of (4.12) with D ' , where D ' = {(x1 , . . . xN ) ∈ D|x2l−1 − x2l + 2π k = 0 (l = 1, . . . , j, k ∈ Z)}. It is obvious that ∂ ∂ ˜ u(x1 − x2 )u(x3 − x4 ) · · · u(x2j −1 − x2j ) f (x) g(x)dx ˜ (4.13) ··· ∂x2j +1 ∂xk D' ∂ ∂ = ··· u(x1 − x2 )u(x3 − x4 ) · · · u(x2j −1 − x2j )f˜(x) g(x)dx. ˜ ∂xk D ' ∂x2j +1 By applying the integration by parts repeatedly, we find that the r.h.s. of (4.13) is equal to ∂ ∂ (−1)k−2j u(x1 − x2 )u(x3 − x4 ) · · · u(x2j −1 − x2j )f˜(x) ··· g(x)dx. ˜ ' ∂x ∂x 2j +1 k D Here we used the periodicities on xl → xl + 2π N (l = 2j + 1, . . . , k). Hence we obtain (4.12) and Lemma 4.3. Proposition 4.4. We suppose β ≥ 0. For f, g ∈ C ω (T )W , we have√ (H (k) (p)f, g) = k (k) (−1) (f, H (p)g) (1 ≤ k ≤ N ). In other words, the operators ( −1)k H (k) (p) are symmetric on the space C ω (T )W . Proof. It is trivial for the β = 0 case. We assume β > 0. Let f (x) ∈ C ω (T )W . Then H (k) (p)f (x) is a polynomial in the parameter β of degree at most k and H (k) (p)f (x) ∈ C ω (T )W . k j ω W (0 ≤ j ≤ k), We set H (k) (p)f (x) = j =0 fj (x)β . Then fj (x) ∈ C (T ) (x) = because H (k) (p)f (x) ∈ C ω (T )W for all β. For f (x) and H (k) (p)f (x), we set f k j (k) f (x) and H (p)f (x) = fj (x)β . j =0
We fix the functions f (x), g(x) ∈ C ω (T )W . It is enough to show that the equations (H (k) (p)f, g) − (−1)k (f, H (k) (p)g) = 0 hold for β > 0 and 1 ≤ k ≤ N . We set T ' = {(x1 , . . . , xN ) ∈ RN |
N
xi = 0, 0 ≤ xi − xj ≤ 2π (1 ≤ i < j ≤ N )},
i=1 ◦
T ' = {(x1 , . . . , xN ) ∈ RN |
N i=1
xi = 0, 0 < xi − xj < 2π (1 ≤ i < j ≤ N )},
∗ ˜ 2 − (−1)k f (x) H (k) (p)g(x) || ˜ 2, h (x) = H (k) (p)f (x) g(x)|| ˜ 2 − (−1)k f ˜ 2. (x) H (k) (p)g(x) (p)f (x) g(x) h(x) = H (k)
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∗ Then the equation (H (k) (p)f, g) − (−1)k (f, H (k) (p)g) = 0 is equivalent to T h ∗ ∗ 1 (x)dµ = 0, where T = hR /2π Q∨ . From the equation N! T h (x)dµ = T ' h (x)dµ, it ∗ is sufficient to show T ' h (x)dµ = 0. ◦
∗
◦
We have h(x) =h (x) on the domain T ' , because sin((xi − xj )/2) > 0 on T ' for i < j and the branch of the function sinβ ((xi − xj )/2) is chosen to be a positive real number. For β ∈ C, the branch of the function sinβ ((xi − xj )/2) (i < j ) is canonically chosen by the relation a β = exp(β log a) for a = sin((xi − xj )/2) > 0. Hence it is sufficient to show the equation T ' h(x)dµ = 0 for β > 0. ∗ From Lemma 4.3, T ' h(x)dµ = T ' h (x)dµ = 0 holds for β > N . (k) ω W ω W From Proposition 4.1, we have H (p)f (x) ∈ C (T ) when f (x) ∈ C (T ) and β ∈ C. Hence the integral T ' h(x)dµ is well-defined if Reβ > 0. We fix β0 (Reβ0 > 0). Since the function h(x) is holomorphic in β and the functions ∂ h(x) and ∂β h(x) are uniformly bounded in (x, β) ∈ T ' × {β ' | |β ' − β0 | < 8} for some 8 ∈ R>0 , the integral T ' h(x)dµ is also holomorphic at β = β0 (Reβ0 > 0) by Lebesgue’s theorem. By the identity theorem, the equation T ' h(x)dµ = 0 holds for β s.t. Reβ > 0. Therefore we obtain the proposition. 4.3. Perturbation. We start with the general proposition related to the perturbation method. Proposition 4.5. Let {v1 , v2 , . . . } be linearly independent vectors in a vector space {k} V over R. Let Hi (k ∈ Z≥0 , i = 1, . . . , N ) be linear operators on V such that {k} {k},i {0},i H i vj = for all i, j, k. We assume that there exists Ej ∈ R j ' :finite dj,j ' vj ' {0}
{0},i
such that Hi vj = Ej
vj for all i, j and if j1 = j2 then there exists some i such {0},i that = Ej . Let ( , ) be an inner product on V such that (vi , vj ) = δi,j . Let ∞ 2 {k} k Hi (p) := k=0 Hi p be formal power series of the linear operators and assume [Hi1 (p), Hi2 (p)] = 0 for all i1 , i2 ∈ {1, . . . , N} as the formal power series of p. Then {0},i Ej1
there exists formal power series of vectors vj (p) = vj +
∞ k=1
j ' :finite
{k}
cj,j ' vj ' p k ,
(4.14) {0},i
such that Hi (p)vj (p) = Eji (p)vj (p) and (vj (p), vj (p)) = 1, where Eji (p) = Ej + ∞ {k},i k p is a formal power series on p and the equalities hold as the formal power k=1 Ej series of p. For each j , the normalized formal power series of the joint eigenfunction of the form (4.14) is unique. Proof. We introduce variables w1 , . . . , wN and set H (w, p) :=
N i=1
wi Hi (p),
j'
{k}
dj,j ' (w)vj ' :=
N i=1
{k}
wi H i vj ,
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vj (p) := vj +
∞ k=1 j '
Ej (w, p) :=
∞ k=0
{k}
{k}
Ej (w)p k =
N i=1
{k}
cj,j ' vj ' p k ,
{0},i
Ej
wi +
N ∞ k=1 i=1
{k},i
Ej
wi p k .
{0},i
are given in advance. We will investigate the condiThe numbers dj,j ' (w) and Ej tions for the coefficients of the formal power series vj (p) and Ej (w, p) satisfying the following relations: H (w, p)vj (p) = Ej (w, p)vj (p), (vj (p), vj (p)) = 1.
(4.15)
{0}
We set cj,j ' = δj,j ' By comparing the coefficients of vj ' p k , we obtain that the conditions (4.15) are equivalent to the following relations: k {k−k ' } {k ' } k−1 {k−k ' } {k ' } k ' =1 ( j '' cj,j '' dj '' ,j ' (w)) − k ' =1 cj,j ' Ej (w) {k} , (j ' = j ), (4.16) cj,j ' = {0} {0} Ej (w) − Ej ' (w) k−1 ' ' 1 {k} {k } {k−k } (4.17) cj,j ' cj,j ' , cj,j = − 2 ' ' k =1 j
{k}
Ej (w) =
k
k ' =1
j'
{k−k ' } {k ' } dj ' ,j (w) −
cj,j '
k−1 k ' =1
{k−k ' }
cj,j
{k ' }
Ej (w).
(4.18)
We remark that the denominator of (4.16) is non–zero by the non–degeneracy condition. {k} {k} The numbers cj,j ' and Ej (w) are determined recursively and they exist uniquely. We {k}
have recursively that for each j and k, #{j ' | cj,j ' = 0} is finite and the summations in (4.16–4.18) on the parameters j ' and j '' are indeed the finite summations. {k} At this stage, the apparent expression of the coefficients cj,j ' may depend on w. We {k}
will show that the coefficients cj,j ' do not depend on w. We denote vj (p) by vj (w, p). From the commutativity of H (w, p) and H (w' , p) we have H (w, p)(H (w' , p)vj (w, p)) = Ej (w, p)(H (w ' , p)vj (w, p)). Since the vector H (w' , p)vj (w, p) admits the expansion H (w ' , p)vj (w, p) = {0} Ej (w ' )vj + O(p), we obtain the following relation from the uniqueness of the formal eigenvector. H (w ' , p)vj (w, p) = vj (w, p), f (w, w ' , p)
where f (w, w' , p) := (H (w' , p)vj (w, p), H (w' , p)vj (w, p)) and 1/ f (w, w ' , p) is k −1/2 = regarded power series on p from the formula (a02 + ∞ k=1 ak p ) as a formal n −1/2 ∞ k a0−1 a0−2 ∞ . Therefore we have H (w ' , p)vj (w, p) = n=0 k=1 ak p n
Perturbation of Calogero–Moser–Sutherland System
107
f (w, w ' , p)vj (w, p). On the other hand we have H (w ' , p)vj (w ' , p) = Ej (w ' , p)vj (w ' , p). By the uniqueness of the formal eigenvector whose leading term is {k} vj , we have vj (w, p) = vj (w ' , p). Therefore the coefficients cj,j ' do not depend on w. From (4.18), we obtain recursively that the coefficients of the formal eigenvalue {k} {k},i (k ∈ Z≥1 ) are deterEj (w) are linear in w1 , . . . , wN . Therefore the numbers Ej mined appropriately. Proposition 4.6. Proposition 4.5 is applicable for the AN−1 -type elliptic Calogero– Moser model by the following correspondence: Hi (p) ⇔ The commuting differential operator H (i) (p), vj ⇔ The normalized Jacobi polynomial J˜λ . {k},i {k} Proof. The finiteness of the summation Hi vj = j ' dj,j ' vj ' follows from Proposition 4.2 and the fact that the Jacobi polynomial forms a basis of C[P ]W . {0},i The non-degeneracy of the joint eigenvalues Ej follows from the non-degeneracy of the joint eigenvalue of the Jacobi polynomial. Summarizing, we have the algorithm of computing the “formal” eigenvalues and “formal” eigenfunctions of the elliptic Calogero–Moser model of AN−1 -type by using the Jacobi polynomial. In the next subsection, we will discuss the convergence.
4.4. Analyticity and the higher commuting operators. In this subsection, we will consider the spectral problem in the C ω (T )W -space for the AN−1 elliptic Calogero–Moser model. We assume β > 1. Since T is compact, we have C ω (T )W ⊂ L2 (T , 2 dµ)W . We will show the holomorphy of the eigenfunctions which we have found on the L2 (T , 2 dµ)W space in Sect. 3. After having the holomorphy of the eigenfunctions, we will justify the convergence and the holomorphy of the joint eigenfunctions of the higher commuting operators obtained by the algorithmic calculation, which we have explained in Sect. 4.3. For this purpose, we need the following propositions. Proposition 4.7. For each eigenvalue ai ∈ σ (H˜ 0 ) and eigenfunction J˜λ of the Hamiltonian H˜ 0 of the trigonometric model such that H˜ 0 J˜λ = ai J˜λ , there exists a positive number pi such that the function Pi (p)J˜λ is holomorphic in (x1 , . . . , xN , p) on the set Bpi , where the operator Pi (p) is a projection on the Hilbert space L2 (T , 2 dµ)W which was defined in Sect. 3.1 and B8 = {(x1 , . . . , xN , p) ∈ CN × R| |Im xj | < 8 (j = 1, . . . , N ), −8 < p < 8}. (4.19) Proposition 4.8. For all eigenvalue ai ∈ σ (H˜ 0 ) and the Jacobi polynomial J˜λ , we have H (j ) (p)Pi (p)J˜λ = Pi (p)H (j ) (p)J˜λ , (j = 1, . . . , N ), when |p| is sufficiently small. We will prove Propositions 4.7 and 4.8 in the next section.
(4.20)
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Remark. For the A1 and β ∈ Z>1 cases, and the A2 and β = 2 case, Proposition 4.7 is obvious from the construction of the eigenfunctions via the Bethe Ansatz method ([11]). We fix the eigenvalue ai ∈ σ (H˜ 0 ). From Propositions 3.1, 4.7, and 4.8, if |p| is sufficiently small then the operators H (j ) (p) act on the finite dimensional space Vi (p), where CPi (p)J˜λ , (4.21) Vi (p) = λ|H˜ 0 J˜λ =ai J˜λ
and we have Vi (p) ⊂ C ω (T )W . √ From Proposition 4.4, the higher commuting operators ( −1)j H (j ) (p) (j = 1, . . . , N ) are symmetric both on the space C ω (T )W and the finite dimensional space Vi (p). √ The joint eigenvalues are real-holomorphic w.r.t the parameter p and the operators ( −1)j H (j ) (p) are simultaneously diagonalizable in the space Vi (p) if |p| is sufficiently small and p ∈ R. The joint eigenfunctions are holomorphic on the domain B8 for sufficiently small 8 ∈ R>0 . Therefore the joint eigenfunction of H (1) (p), . . . , H (N) (p) admits the holomorphic expansion in the variable p. Since the joint eigenvalues of H (1) (0), . . . , H (N) (0) are distinct, the expansion is unique up to the normalization (see Sect. 4.3.) Hence the perturbation series which is obtained by the method introduced in Sect. 4.3 converges holomorphically and coincides with the eigenvalue and eigenfunction which is obtained by diagonalizing the finite dimensional space Vi (p). Summarizing, we have Theorem 4.9. For the AN−1 and β > 1 cases, the perturbation expansion of the commuting operators H (1) (p), . . . , H (N) (p) which is performed in Sect. 4.3 converges holomorphically and defines the eigenfunction which is holomorphic when |I mxj | (j = 1, . . . , N ) and |p| (p ∈ R) are sufficiently small. The joint eigenvalue is holomorphic in the parameter p(+ 1). Remark. It was pointed out by Prof. T. Oshima that the real-holomorphy of the squareintegrable eigenfunction ψ(x) (i.e. T (p)ψ(x) = E(p)ψ(x), ψ(x) ∈ L2 (T , 2 dµ)W ) is also obtained by the following argument. From the ellipticity of the operator T (p) and Weyl’s lemma, we have the realholomorphy of the eigenfunction ψ(x) on the domain T˙ = T \ T ' , where T ' := {(x1 , . . . , xN ) ∈ T |∃(i = j ), xi = xj }. Next we consider the analytic continuation of the function ψ(x). The equation T (p)ψ(x) = E(p)ψ(x) has regular singularities along xi = xj (i = j ), and the exponents at the singularity are (0, −2β − 1). It follows that the function ψ(x) is holomorphic along xi − xj = 0 from the property ψ(x) ∈ L2 . Hence we have the real-holomorphy of ψ(x) on T . 5. Proof of Propositions 4.7 and 4.8 In this section, we assume that the root system is of the AN−1 -type. For λ ∈ P+ and j = 1, . . . , N − 1, we set mλ = µ∈W λ e µ,h and eFj = mFj , where Fj is the j th fundamental weight. For λ = lj =1 Fij (l ∈ Z≥0 , ij ∈ {1, . . . , N − l 1} (j = 1, . . . , l)), we set e˜λ = j =1 eFij . Then we have e˜λ = eλ' on h∗ , where eµ is the elementary symmetric function for the partition µ defined in Macdonald’s book ([6], p. 20) and λ' is the conjugate of λ.
Perturbation of Calogero–Moser–Sutherland System
We set t (x, p) :=
∞
tk (x)p k := ℘ (x) −
k=1
109
1 4 sin2 (x/2)
+
1 , 12
which converges uniformly on a strip around R × [−8, 8] for 0 ≤ 8 < 1. From the formula (A.13), we have tk (x) = −2 j (cos j x − 1).
(5.1)
j |k
Here, j |k means that the positive integer j is a divisor of k. Lemma 5.1. For a real number c such that c > 1, there exists a positive number a ' such that |tk (x)| < a ' ck for all x ∈ R. formula (5.1), we have |tk (x)| ≤ 4tk < Proof. Let tk be the sum of all divisors of k. By the 4k 2 . Since the convergence radius of the series k 2 p k is equal to 1, the convergence radius of the series tk p k is equal to or less than 1. Therefore we have the lemma. The W (p) defined in (3.1) has an expansion in terms of p given by W (p) = ∞operator (k) p k , where T (k) is the operator of multiplication by the function T (k) (h) := T k=1 ∞ (k) p k converges α∈R+ β(β − 1)tk ( α, h ). For each p ∈ (1, −1), the series k=1 T uniformly on hR . Proposition 5.2. For a real number c such that c > 1, there exists a positive number a such that T (k) ≤ ack . Proof. It follows from Lemma 5.1 and the inequality |T (k) f |2 2 dµ ≤ sup |T (k) (h)|2 |f |2 2 dµ. T
h∈T
T
(5.2)
Lemma 5.3. The function α∈+ tk ( α, h ) admits the expansion, tk ( α, h ) = cµ m µ . α∈+
(5.3)
√ µ∈Q∩P+ ,|µ|≤ 2k
Proof. From formula (5.1), we have tk ( α, h ) = − ( j (ej α,h + e−j α,h − 2) = − 2j (mj θ − 1), α∈+
α∈+ j |k
j |k
where θ is the highest root of the root system AN−1 . Since |θ | = we have the lemma.
√
Sublemma 5.4. Let Jλ be the AN−1 -Jacobi polynomial. We have Jλ eFr = c¯ν Jν , ν∈P+ ,ν−λ∈{wFr |w∈W }
for some constants c¯ν .
2 and j θ ∈ Q ∩ P+ ,
(5.4)
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Proof. This follows from the Pieri formula ([6], p. 332 and Sect. VI.10.).
Sublemma 5.5. Let l be a positive integer. Assume ij ∈ {1, . . . , N − 1} and wj ∈ W , (j = 1, . . . , l). We have | lj =1 wj (Fij )| ≤ | lj =1 Fij |. Proof. It is sufficient to show (λ + µ, λ + µ) ≥ (λ + w(µ), λ + w(µ)) for λ, µ ∈ P+ and w ∈ W . This inequality is equivalent to (λ, µ − w(µ)) ≥ 0. From the property µ − w(µ) ∈ Q+ , we have (λ, µ − w(µ)) ≥ 0. Sublemma 5.6. If λ, µ ∈ P+ and λ − µ ∈ Q+ , then we have |λ| ≥ |µ|. Proof. Immediate from the equality (λ, λ) − (µ, µ) = (λ − µ, λ + µ).
Sublemma 5.7 ([6], p. 20). The monomial symmetric function mλ has the expansion
mλ = e˜λ +
cˇν e˜ν ,
(5.5)
ν∈P+ ,λ−ν∈Q+ \{0}
for some constants cˇν . Lemma 5.8. We have the expansion,
J λ mµ =
c¯ν Jν ,
(5.6)
ν∈P+ ,|ν−λ|≤|µ|
for some constants c¯ν . Proof. First, we expand mµ by using Sublemma 5.7. Then Jλ mµ is expressed as the linear combination of Jλ e˜ν , where ν ∈ P+ and µ − ν ∈ Q+ . We set ν = lj =1 Fij (l ∈ Z≥0 , ij ∈ {1, . . . , N − 1} (j = 1, . . . , l)) We repeatedly apply Sublemma 5.4 for Jλ e˜ν . Then Jλ e˜ν is expressed as the linear combination of Jν ' , where ν ' = λ + lj =1 wj (Fij ) for some wj ∈ W (j = 1, . . . , l). From Sublemma 5.5, we have |ν ' − λ| ≤ | lj =1 Fij | = |ν|. Applying Sublemma 5.6 for µ and ν, we obtain Lemma 5.8. Proposition 5.9. Let |p| < 1 and λ ∈ P+ . Write ∞ k=1
T (k) p k J˜λ =
t˜λ,µ J˜µ ,
µ∈P+ ,λ−µ∈Q
where J˜λ is the normalized AN−1 -Jacobi polynomial. For each C such that C > 1 and √ C|p| < 1, there exists a number C '' ∈ R>0 such that |t˜λ,µ | ≤ C '' (C|p|)(|λ−µ|+1)/2 2 for all µ ∈ P+ . Proof. Since the normalized Jacobi polynomials form the complete orthonormal system (k) p k )J˜ , J˜ ) . with respect to the inner product ( , ) , we have t˜λ,µ = (( ∞ λ µ k=1 T
Perturbation of Calogero–Moser–Sutherland System
111
We √ fix λ, µ ∈ P+ . Let m be the smallest integer which is greater or equal to |λ − µ|/ 2. If k < m, then we have (T (k) p k J˜λ , J˜µ ) = 0 by Lemmas 5.3, 5.8 and the orthogonality. Therefore we have ∞ (k) k |t˜λ,µ | = (J˜µ , ( T p )J˜λ ) k=1 ∞ = (J˜µ , ( T (k) p k )J˜λ ) k=m ∞ k ˜ ˜ 2 = tk ( α, h )p Jλ Jµ dµ T k=m α∈+ ∞ k ≤ sup tk ( α, h )p |J˜λ J˜µ 2 |dµ h∈T k=m α∈ T + kα (kα − 1)N (N − 1) k ≤ tk |p| · |J˜λ |2 dµ |J˜µ |2 dµ 2 T T k≥m
kα (kα − 1)N (N − 1) ≤ tk |p|k . 2 k≥m
k Similarly, we have |t˜λ,λ | ≤ kα (kα −1)N(N−1) 2 k≥1 ntk |p| . Since the convergence radius of the series n tn p is equal to 1, we obtain that there √ exists a number C '' ∈ R>0 such that |t˜λ,λ | ≤ C '' (C|p|) and |t˜λ,µ | ≤ C '' (C|p|)|λ−µ|/ 2 for λ = µ. Hence we have the proposition. Proposition 5.10. Let suppose dist(ζ, σ (H˜ 0 )) ≥ D. Write D be a positive number. We −1 −1 ˜ ˜ ˜ ˜ (T (p)−ζ ) Jλ = µ tλ,µ Jµ , where (T (p)−ζ ) is defined in (3.6). For each λ ∈ P+ and C ∈ R>1 , there exists C ' ∈ R>0 and p0 ∈ R>0 which do not depend on ζ (but depend on D) such that tλ,µ satisfies |λ−µ| √ 2
|tλ,µ | ≤ C ' (C|p|) 2N
,
(5.7)
for all p, µ s.t. |p| < p0 and µ ∈ P+ . Proof. Let us recall that the operator (T˜ (p) − ζ )−1 is defined by the Neumann series (3.6). (k) p k ). From the We fix the number D(∈ R>0 ) and set X := (ζ − H˜ 0 )−1 ( ∞ k=1 T expansion (3.6) and Proposition 5.2, there exists a number p1 ∈ R>0 such that the inequality X < 1/2 holds for p (|p| < p1 ) and ζ (dist(ζ, σ (H˜ 0 )) > D). In this case, i )(H ˜ 0 − ζ )−1 = (T˜ (p) − ζ )−1 . We write X J˜λ = ˇ ˜ we have ( ∞ µ tλ,µ Jµ . i=0 X ' −1 ' ˜ For the series µ cµ J˜µ , write µ cµ J˜µ = (H˜ 0 − ζ ) µ cµ Jµ . We have |cµ | ≤ D −1 |cµ | for each µ. Combining with Proposition 5.9, we obtain that for each C such '' ∈ R that C > 1 and C+1 >0 which does not depend on ζ but D 2 p1 < 1, there exists C √ C+1 '' (|λ−µ|+1)/2 2 ˇ if |p| < p1 . such that |tλ,µ | ≤ C ( 2 |p|) To obtain Proposition 5.10, we use the method of majorants.
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Y. Komori, K. Takemura
Z N We introduce the symbol eλ (λ ∈ ( N ) ) to avoid inaccuracies. We remark that Z N P+ ( N ) . We will apply the method of majorants for the formal series µ∈( Z )N cµ eµ N instead of µ∈P+ cµ eµ . ˜ by the rule For the formal series µ∈( Z )N cµ eµ , we define the partial ordering ≤ N (1) (2) (1) (2) ˜ cµ eµ ≤ cµ eµ ⇔ ∀µ, |cµ | ≤ |cµ |. Z N µ∈( N )
Z N µ∈( N )
(i)
Z N We will later consider the case that each cµ (µ ∈ ( N ) , i = 1, 2) is expressed as (2) the infinite sum. If one shows the absolute convergence of cµ for each µ, one has the (1) absolute convergence of cµ for each µ by the majorant. ˜ ˇ ˇ We set Xeλ = µ tλ,µ eµ , where the coefficients tλ,µ were defined by X Jλ = ˜ ˇ µ tλ,µ Jµ . i −1 ˜ ˜ Our goal is to show (5.7) for tλ,µ s.t. µ tλ,µ J˜µ = ( ∞ i=0 X )(H0 − ζ ) Jλ . Since (H˜ 0 − ζ )−1 J˜λ = (Eλ − ζ )−1 J˜λ and |(Eλ − ζ )−1 | ≤ D −1 , it is enough to show that there exists C I ∈ R>0 and p 0 ∈ R>0 which do not depend on ζ (but depend on D) such ∞ I is well-defined by I k that tλ,µ k=0 X eλ = tλ,µ eµ and satisfies |λ−µ| √ 2
I | ≤ C I (C|p|) 2N |tλ,µ
for all p, µ s.t. |p| < p0 and µ ∈ P+ . We set Y eλ := yλ,µ eµ := µ,λ−µ∈ZN
µ,λ−µ∈ZN
Zeλ :=
zλ,µ eµ :=
µ,λ−µ∈ZN
C
C
''
''
µ,λ−µ∈ZN
,
(5.8)
C+1 p 2 C+1 p 2
(|µ−λ|+1)/2√2
Ni=1 (|µi −λ√i |+1) 2N 2
eµ ,
eµ .
We have the inequality
˜ eλ ≤Ze ˜ λ. Xeλ ≤Y k Let k ∈ Z≥1 . If the coefficients of Z eλ w.r.t the basis {eµ } converge absolutely, then the coefficients of the series Xk eλ and Y k eλ are well-defined and we have ˜ k eλ ≤Z ˜ k eλ . X k eλ ≤Y From the equality Z k eλ = ν (1) ,... ,ν (k−1) zλ,ν (1) zν (1) ,ν (2) · · · zν (k−1) ,µ eµ and the property zλ,µ = z0,µ−λ , we have 1 Z k eλ = ··· √ (2π −1)N |s1 |=1 |sN |=1 µ,λ−µ∈ZN k λ −µ −1 νN λ1 −µ1 −1 z0,ν s1ν1 · · · sN s1 · · · sNN N ds1 · · · dsN eµ ν=(ν1 ,... ,νN )∈ZN
=
µ,λ−µ∈ZN
1 √ (2π −1)N
N i=1 |si |=1
ν∈Z
(C '' ) N 1
C +1 p 2
(|ν|+1) √ 2N 2
k λ −µi −1
siν si i
dsi eµ .
Perturbation of Calogero–Moser–Sutherland System 1√ 2
2N Set p˜ = ( C+1 2 p)
∞
113
, we have
Z k eλ
k=1
k N 1 λ −µ −1 '' N1 (|ν|+1) ν = (C ) p˜ si si i i dsi eµ √ N (2π −1) |s |=1 i k=1 µ,λ−µ∈ZN i=1 ν∈Z k N ∞ 1 1 λ −µ −1 '' (|ν|+1) ν ˜ (C ) N p˜ si si i i dsi eµ ≤ √ N (2π −1) |s |=1 i i=1 k=1 ν∈Z µ,λ−µ∈ZN = Zλ,µ eµ , ∞
µ,λ−µ∈ZN
where Zλ,µ =
1 N λ −µ −1 (C '' ) N (p˜ − p˜ 3 )si i i dsi 1 . √ 1 (2π −1)N i=1 |si |=1 (1 − ps ˜ i )(1 − ps ˜ i−1 ) − (C '' ) N (p˜ − p˜ 3 )
(5.9)
N N ∞ k k Remark that we used the inequality ∞ k=1 i=1 (ai ) ≤ i=1 k=1 (ai ) for 0 < q−q 3 |n|+1 n x = (1−qx)(1−qx −1 ) . The equality (5.9) a1 , . . . , aN < 1 and the formula n∈Z q makes sense for p˜ < p2 , where p2 is the positive number satisfying the inequalities 1
(C '' ) N |p2 −p23 | (1−p2 )2
1
< 1 and (C '' ) N p2 < 1. k Therefore each coefficient of ∞ k=1 Z eλ w.r.t the basis {eµ } converges absolutely. Hence the following inequality makes sense:
p2 < 1,
∞ k=0
˜ λ+ X k eλ ≤e
∞
˜ λ+ Z k eλ ≤e
k=1
Zλ,µ eµ .
µ,λ−µ∈ZN 1
˜ be the solution of the equation (1− ps)(1− ˜ ˜ −1 )−(C '' ) N (p˜ − p˜ 3 ) = 0 on Let s(p) ps ˜ < 1. Then s(p) ˜ is holomorphic in p˜ near 0 and admits the expansion s satisfying |s(p)| ˜ = p˜ + c2 p˜ 2 + · · · . We have s(p) 1 √
(2π −1)
1
(C '' ) N (p˜ − p˜ 3 )s n−1 ds |s|=1
1
˜ ˜ −1 ) − (C '' ) N (p˜ − p˜ 3 ) (1 − ps)(1 − ps
˜ (p)s( ˜ p) ˜ |n| , = pf
(5.10)
˜ is a holomorphic function defined near p˜ = 0. For the n ≥ 0 case, we have where f (p) ˜ For the n < 0 case, we the relation (5.10) by calculating the residue around s = s(p). ˜ need to change the variable s → s −1 and calculate the residue around s = s(p). k The coefficient of eµ on the series ∞ k=0 X eλ satisfying λ − µ ∈ Q has to be zero ˜ |ν1 |+···+|νN | ≤ |s(p)| ˜ from the definition of X. By the inequality |s(p)|
2 ν12 +···+νN
, we
114
Y. Komori, K. Takemura
have ∞
˜ λ+ X k eλ ≤e
k=0
µ,λ−µ∈Q
N
˜ λ ˜ (p))s( ˜ ˜ |λi −µi | eµ ≤e (pf p)
i=1
+
˜ (p)) ˜ N s(p) ˜ |λ−µ| eµ . (pf
µ,λ−µ∈Q
˜ < p3 and p3 : a sufficiently small positive number. for |p|
1√ 2N 2 , the inequality C+1 < C, and the expanCombining the relation p˜ = C+1 2 p 2 2 ˜ = p˜ + c2 p˜ + . . . , we obtain (5.8) and the proposition. sion s(p) Proposition 5.11. Let ai ∈ σ (H˜ 0 ) and 4i be a circle in C which contains only one element ai of σ (H˜ 0 ) inside it. Let λ ∈ P+ satisfying H˜ 0 J˜λ = ai J˜λ . We set Pi (p) = − 2π √1 −1 4i (T˜ (p) − ζ )−1 dζ and write Pi (p)J˜λ = µ sλ,µ J˜µ . For each C ∈ R>1 , there exists C ' ∈ R>0 and p∗ ∈ R>0 such that sλ,µ satisfies |λ−µ| √ 2
|sλ,µ | ≤ C ' (C|p|) 2N
,
(5.11)
for all p, µ s.t. |p| < p∗ and µ ∈ P+ . Proof. Since the spectrum σ (H˜ 0 ) is discrete, there exists a positive number D such that inf ζ ∈4i dist(ζ, σ (H˜ 0 )) ≥ D. We write (T˜ (p) − ζ )−1 J˜λ = µ tλ,µ (ζ )J˜µ . From Proposition 5.10, we obtain that for each C ∈ R>1 , there exists C∗ ∈ R>0 and p∗ ∈ R>0 which does not depend on |λ−µ| √
ζ (∈ 4i ) such that |tλ,µ (ζ )| ≤ C∗ (C|p|) 2N 2 for all p, µ s.t. |p| < p∗ and µ ∈ P+ . Let L be the length of the circle 4i and write − 2π √1 −1 4i (T˜ (p) − ζ )−1 dζ J˜λ = ˜ ˜ µ sλ,µ Jµ . By integrating µ tλ,µ (ζ )Jµ over the circle 4i , we have |sλ,µ | ≤ |λ−µ|
√ L 2N 2 2π C∗ (C|p|)
for all p, µ s.t. |p| < p∗ and µ ∈ P+ . Therefore we have Proposition 5.11.
Proposition 5.12. Let µ ∈ P+ and cµ be a number satisfying |cµ | < a|p|b|µ| (|µ| > M) for some a, b > 0 and M ∈ Z. The function µ cµ J˜µ is holomorphic when |I mxj | (j = 1, . . . , N ) and |p| are sufficiently small. √ Proof. Since zi = e −1xi , it is enough to show that the function µ cµ J˜µ is holomorphic when |p| is sufficiently small and 1/2 < |zj | < 2 (j = 1, . . . , N ). We count roughly the number of the elements of P+ of a given length. The rough estimate is given by #{λ ∈ P+ | (λ|λ) = l} ≤ (2lN )N . We will use this in the inequality (5.12). In the proof, we will use the notations and the results written in Sect. A.1. In Sect. A.1, there are parameters r and C0 . We fix r = 2 and C0 = 2. There is another number A defined in Sect. A.1.
Perturbation of Calogero–Moser–Sutherland System
115
We have ≤ ˜ J c (z , . . . , z ) a|p|b|µ| |J˜µ (z1 , . . . , zN )| µ µ 1 N µ∈P+ ,|µ|≥M µ∈P+ ,|µ|≥M 1/2≤|zi |≤2 1/2≤|zi |≤2 √ √ √ ≤ a2(N−1) 2(µ|µ) 2 N(µ|µ) |p|b (µ|µ) µ∈P+ ,|µ|≥M
≤
√ √ 2+ N
aA(2nN )N (2(N−1)
|p|b )
√
(5.12)
n
n≥M,n∈Z/N
by the formulae (A.2), (A.11). √ √ (N−1) 2+ N |p|b < 1 then the bottom part of the inequality converges. We If 2 √ √ choose a positive number p0 which satisfies 2(N−1) 2+ N p0b < 1. Then the series | µ∈P+ ,|µ|≥M cµ J˜µ (z1 , . . . , zN )| is uniformly bounded and uniformly absolutely converges for |p| < p0 and 1/2 ≤ |zi | ≤ 2 (i = 1, . . . , N ). Since the functions J˜µ (z1 , . . . , zN ) are holomorphic, we have the holomorphy of the function ˜ µ cµ Jµ (z1 , . . . , zN ) by Weierstrass’s theorem. Combining Propositions 5.11 and 5.12, we have Proposition 4.7. From Propositions 5.10 and 5.12, the function (T˜ (p) − ζ )−1 J˜λ is real-holomorphic on (x1 , . . . , xN ) if |p| is sufficiently small. From Proposition 4.1, the operators H (j ) (p) (j = 1, . . . , N ) act well-definedly on the function (T˜ (p) − ζ )−1 J˜λ and we have H (j ) (p)(T˜ (p) − ζ )−1 J˜λ ∈ C ω (T )W . It follows from the commutativity of the operators T˜ (p) and H (j ) (p) (4.4) that H (j ) (p)J˜λ = H (j ) (p)(T˜ (p) − ζ )(T˜ (p) − ζ )−1 J˜λ = (T˜ (p) − ζ )H (j ) (p)(T˜ (p) − ζ )−1 J˜λ . Hence we have H (j ) (p)(T˜ (p) − ζ )−1 J˜λ = (T˜ (p) − ζ )−1 H (j ) (p)J˜λ . By integrating it on the variable ζ over the circle 4i , we have H (j ) (p)Pi (p)J˜λ = Pi (p)H (j ) (p)J˜λ . Therefore we have Proposition 4.8. A. Jack Polynomial and Special Functions A.1. Jack polynomial and AN−1 -Jacobi polynomial. We will see the relationship between the Jack polynomial and the AN−1 -Jacobi polynomial. Let MN be the set of partitions with at most N parts, i.e., MN := {λ = (λ1 , . . . , λN ) | λi − λi+1 ∈ Z≥0 , (i = 1, . . . , N − 1), λN ∈ Z≥0 }. We set M0N := {λ = (λ1 , . . . , λN ) ∈ MN | λN = 0}. The Jack polynomial JλI (z1 , . . . , zN ) (λ ∈ MN ) is a symmetric polynomial of variables (z1 , . . . , zN ) which is an eigenfunction of the gauge-transformed Hamiltonian H˜ 0 (4.6). Let mIλ be the monomial symmetric polynomial. The Jack polynomial admits the following expansion: uλµ mIµ , (A.1) JλI = mIλ + µ≺λ
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where the dominant ordering of MN is given by λ µ ⇔ ij =1 λj ≤ ij =1 µj (i = N 1, . . . , N − 1), N j =1 λj = j =1 µj . We see the correspondence between the Jack polynomial and the AN−1 -Jacobi poly nomial. Let JλI (z1 , . . . , zN ) (λ ∈ MN ) be a Jack polynomial. We set |λ|I = N i=1 λi N and λ = i=1 (λi −|λ|I /N )8i . Then λ ∈ P+ , where P+ is the set of dominant weights of type AN−1 . The function (z1 · · · zN )−|λ|I /N JλI (z1 , . . . , zN ) is precisely the AN−1 -Jacobi polynomial Jλ . By this correspondence, the Jack polynomial JλI (z1 , . . . , zN ) (λ ∈ M0N ) corresponds with the AN−1 -Jacobi polynomial Jλ (λ ∈ P+ ) one-to-one. Let λ be an element in M0N and λ be the corresponding element in P+ . Since (λ|λ) ≥ (λ1 − |λ|I /N )2 + (|λ|I /N )2 ≥ (λ1 )2 /2 ≥
|λ|2I , 2(N−1)2
we have
|λ|I ≤ (N − 1) 2(λ|λ). Let us recall the Cauchy formula for the Jack polynomial. (1 − κXi Yj )−β = κ |λ|I JλI (X)JλI (Y )jλ−1 ,
(A.2)
(A.3)
λ∈MN
1≤i,j ≤N
where 0 ≤ jλ =
a(s) + βl(s) + 1 ≤ 1, a(s) + βl(s) + β
(A.4)
s∈λ
due to β ≥ 1. a(s) is the arm-length and l(s) is the leg-length. Since p. 379 of Macdonald’s book ([6]), we have JλI = mIλ + µ≺λ uλµ mIµ with uλµ > 0 if β > 0. Hence we have uλµ mµ , (A.5) Jλ = mλ + λ−µ∈Q+
with uλµ > 0. Let r be a real number greater than 1. √ N If 1/r < |zi | < r for all i then |mλ (z1 , . . . , zN )| ≤ r i=1 |(λ|8i )| mλ (1) ≤ r N(λ|λ) mλ (1). Therefore we have 0 ≤ |Jλ (z)| ≤ r
√ N(λ|λ)
Jλ (1)
on 1/r < |zi | < r for all i. By setting Xi = Yj = 1 in (A.3), we have 2 (1 − κ)−βN = κ n cn = κ |λ|I JλI (1)2 jλ−1 , n∈Z≥0
where cn = A such that
|λ|I =n
(A.7)
λ∈MN
4(βN 2 +n+1) . For each β ≥ 1 and C0 4(βN 2 +1)4(n+1) 2n 2 cn < A C0 for all n ∈ Z≥0 . Thus
JλI (1)2 ≤
(A.6)
> 1, there exists a positive number
JλI (1)2 jλ−1 < A2 C02n .
(A.8)
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By the inequality (A.2), we have √ (N−1) 2(λ|λ)
|Jλ (1)| < AC0
.
(A.9)
The square of the norm of JλI is 4(ξi − ξj + β)4(ξi − ξj − β + 1) , 4(ξi − ξj )4(ξi − ξj + 1)
JλI 2 =
(A.10)
i<j
where ξi = λi + β(N − i). (See ([6], p. 383)) If β ≥ 1 then we have JλI 2 ≥ 1 because of the convexity of the function log 4(x). Therefore we have Jλ 2 ≥ 1. Generally we have for r > 1, max
1/r≤|zi |≤r
|J˜λ (z)| ≤
max
√ √ (N−1) 2(λ|λ) N(λ|λ)
1/r≤|zi |≤r
|Jλ (z)| ≤ AC0
r
,
(A.11)
where J˜λ (z) is the normalized AN−1 -Jacobi polynomial.
A.2. Special functions. We define some functions needed in this article: θ1 (x) := 2
∞
√ (−1)n−1 exp(τ π −1(n − 1/2)2 ) sin(2n − 1)π x,
(A.12)
n=1
θ(x) :=
℘ (x; ω1 , ω3 ) :=
1 + z2
(m,n)∈Z2 \{(0,0)}
θ1 (x) , θ1' (0) 1
(z + 2mω1 + 2nω3 )2
−
1 (2mω1 + 2nω3 )2
,
℘ (x) := ℘ (x; π, π τ ). We have ℘ (x) =
∞ 1 1 np n − − 2 (cos nx − 1), 2 1 − pn 4 sin (x/2) 12 n=1
(A.13)
√ where p = exp(2τ π −1). Acknowledgements. The authors would like to thank Prof. M. Kashiwara and Prof. T. Miwa for discussions and support. Thanks are also due to Dr. T. Koike and Prof. T. Oshima. They thank the referee for valuable comments. One of the authors (YK) is a Research Fellow of the Japan Society for the Promotion of Science.
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References 1. Felder, G., Varchenko, A.: Three formulae for eigenfunctions of integrable Schrödinger operator. Comp. Math. 107, no. 2, 143–175 (1997) 2. Hasegawa, K.: Ruijsenaars’ commuting difference operators as commuting transfer matrices. Commun. Math. Phys. 187, no. 2, 289–325 (1997) 3. Kato, T.: Perturbation theory for linear operators, Corrected printing of second ed., Berlin–Heidelberg: Springer-Verlag, 1980 4. Komori, Y.: Algebraic analysis of one-dimensional quantum many-body systems. Ph.D thesis, Univ. of Tokyo (2000) 5. Langmann, E.: Anyons and the elliptic Calogero–Sutherland model. Lett. Math. Phys. 54, no. 4, 279–289 6. Macdonald, I.G.: Symmetric functions and Hall polynomials. Second edition. New York: Oxford Science Publications, The Clarendon Press, Oxford University Press, 1995 7. Ochiai, H., Oshima, T., Sekiguchi, H.: Commuting families of symmetric differential operators. Proc. Japan Acad. Ser. A Math. Sci. 70, no. 2, 62–66 (1994) 8. Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep. 94, no. 6, 313–404 (1983) 9. Oshima, T., Sekiguchi, H.: Commuting families of differential operators invariant under the action of a Weylgroup. J. Math. Sci. Univ. Tokyo 2, no. 1, 1–75 (1995) 10. Ruijsenaars, S.N.M: Generalized Lamé functions. I. The elliptic cases. J. Math. Phys. 40, no. 3, 1595–1626 (1999) 11. Takemura, K.: On the eigenstates of the elliptic Calogero–Moser model. Lett. Math. Phys. 53, no. 3, 181–194 (2000) Communicated by L. Takhtajan
Commun. Math. Phys. 227, 119 – 130 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Singular Spectrum of Lebesgue Measure Zero for One-Dimensional Quasicrystals Daniel Lenz1,2, 1 Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel 2 Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, 60054 Frankfurt, Germany.
E-mail:
[email protected] Received: 3 July 2001 / Accepted: 11 December 2001
Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. 1. Introduction This article is concerned with discrete random Schrödinger operators associated to minimal topological dynamical systems. This means we consider a family (Hω )ω∈ of operators acting on 2 (Z) by (Hω u)(n) ≡ u(n + 1) + u(n − 1) + f (T n ω)u(n),
(1)
where is a compact metric space, T : −→ is a homeomorphism and f : −→ R is continuous. The dynamical system (, T ) is called minimal if every orbit is dense. For minimal (, T ), there exists a set ⊂ R s.t. σ (Hω ) = , for all ω ∈ ,
(2)
where we denote the spectrum of the operator H by σ (H ) (cf. [6, 36]). This research was supported in part by THE ISRAEL SCIENCE FOUNDATION (Grant no. 447/99) and by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
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We will be particularly interested in the case that (, T ) is a subshift over a finite alphabet A ⊂ R. In this case is a closed subset of AZ , invariant under the shift operator T : AZ −→ AZ given by (T a)(n) ≡ a(n + 1) and f is given by f : −→ A ⊂ R, f (ω) ≡ ω(0). Here, A carries the discrete topology and AZ is given the product topology. Operators associated to subshifts arise in the quantum mechanical treatment of quasicrystals (cf. [3, 40] for background on quasicrystals). Various examples of such operators have been studied in recent years. The main examples can be divided in two classes. These classes are given by primitive substitution operators (cf. e.g. [4, 5, 7, 11, 41, 42]) and Sturmian operators respectively more generally circle map operators (cf. e.g. [6, 12, 15, 16, 26, 27, 30]). A recent survey can be found in [14]. For these classes and in fact for arbitrary operators associated to subshifts satisfying suitable ergodicity and aperiodicity conditions, one expects the following features: (S) Purely singular spectrum; (A) absence of eigenvalues; (Z) Cantor spectrum of Lebesgue measure zero. Note that (S) combined with (A) implies purely singular continuous spectrum and note also that (S) is a consequence of (Z). Let us mention that (S) is by now completely established for all relevant subshifts due to recent results of Last–Simon [34] in combination with earlier results of Kotani [32]. For discussion of (A) and further details we refer the reader to the cited literature. The aim of this article is to investigate (Z) and to relate it to ergodic properties of the underlying subshifts. The property (Z) has been investigated for several models by a number of authors: Following work by Bellissard–Bovier–Ghez [5], the most general result for primitive substitutions so far has been obtained by Bovier/Ghez [7]. They can treat a large class of substitutions which is given by an algorithmically accessible condition. The Rudin– Shapiro substitution does not belong to this class. For arbitrary Sturmian operators, Bellissard–Iochum–Scoppola–Testard established (Z) [6], thereby extending the work of Süt˝o in the golden mean case [41, 42]. A different approach, which recovers some of these results, is given in [13, 19]. A canonical starting point in the investigation of (Z) for subshifts is the fundamental result of Kotani [32] that the set {E ∈ R : γ (E) = 0} has Lebesgue measure zero if (, T ) is an aperiodic subshift. Here, γ denotes the Lyapunov exponent (precise definition given below). This reduces the problem (Z) to establishing the equality
= {E ∈ R : γ (E) = 0}.
(3)
As do all other investigations of (Z) so far, our approach starts from (3). Unlike the earlier treatments mentioned above our approach does not rely on the so called trace maps. Instead, we present a new method, the cornerstones of which are the following: (1) A strong type of Oseledec theorem by A. Furman [21]. (2) A uniform ergodic theorem for a large class of subshifts by the author [37]. This new setting allows us (∗)
to characterize validity of (3) for arbitrary strictly ergodic dynamical systems by an essentially ergodic property viz by uniform existence of the Lyapunov exponent (Theorem 1), (∗∗) to present a large class of subshifts satisfying this property (Theorem 2). Here, (∗) gives the new conceptual point of view of our treatment and (∗∗) gives a large class of examples. Put together (∗) and (∗∗) provide a soft argument for (Z) for a large class of examples which contains, among other examples, all primitive substitutions.
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The paper is organized as follows. In Sect. 2 we present the subshifts we will be interested in, introduce some notation and state our results. In Sect. 3, we recall results of Furman [21] and of the author [37] and adopt them to our setting. Section 4 is devoted to a proof of our results. Finally, in Sect. 5 we provide some further comments and discuss a variant of our main result.
2. Notation and Results In this section we discuss basic material concerning topological dynamical systems and the associated operators and state our results. As usual a dynamical system is said to be strictly ergodic if it is uniquely ergodic (i.e. there exists only one invariant probability measure) and minimal. A minimal dynamical system is called aperiodic if there does not exist an n ∈ Z, n = 0, and ω ∈ with T n ω = ω. As mentioned already, our main focus will be the case that (, T ) is a subshift over the finite alphabet A ⊂ R . We will then consider the elements of (, T ) as double sided infinite words and use notation and concepts from the theory of words. In particular, we then associate to the set W of words associated to consisting of all finite subwords of elements of . The length |x| of a word x ≡ x1 . . . xn with xj ∈ A, j = 1, . . . , n, is defined by |x| ≡ n. The number of occurrences of v ∈ W in x ∈ W is denoted by v (x). We can now introduce the class of subshifts we will be dealing with. They are those satisfying uniform positivity of weights (PW) given as follows: (PW) There exists a C > 0 with lim inf |x|→∞
v (x) |x| |v|
≥ C for every v ∈ W.
One might think of (PW) as a strong type of minimality condition. Indeed, minimality can easily be seen to be equivalent to lim inf |x|→∞ |x|−1 v (x)|v| > 0 for every v ∈ W [39]. The condition (PW) implies strict ergodicity [37]. The class of subshifts satisfying (PW) is rather large. By [37], it contains all linearly repetitive subshifts (see [20, 33] for definition and thorough study of linearly repetitive systems). Thus, it contains, in particular, all subshifts arising from primitive substitutions as well as all those Sturmian dynamical systems whose rotation number has bounded continued fraction expansion [20, 33, 38]. In our setting the class of subshifts satisfying (PW) appears naturally as it is exactly the class of subshifts admitting a strong form of the uniform ergodic theorem [37]. Such a theorem in turn is needed to apply Furmans results (s. below for details). After this discussion of background from dynamical systems we are now heading towards introducing key tools in spectral theoretic considerations viz transfer matrices and Lyapunov exponents. The operator norm · on the set of 2 × 2-matrices induces a topology on GL(2, R) and SL(2, R). For a continuous function A : −→ GL(2, R), ω ∈ , and n ∈ Z, we define the cocycle A(n, ω) by
A(T n−1 ω) · · · A(ω) : n > 0 Id : n = 0 A(n, ω) ≡ −1 n A (T ω) · · · A−1 (T −1 ω) : n < 0.
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By Kingman’s subadditive ergodic theorem (cf. e.g. [31]), there exists (A) ∈ R with 1 (A) = lim log A(n, ω) n→∞ n for µ a. e. ω ∈ if (, T ) is uniquely ergodic with invariant probability measure µ. Following [21], we introduce the following definition. Definition 1. Let (, T ) be strictly ergodic. The continuous function A : (, T ) −→ GL(2, R) is called uniform if the limit (A) = limn→∞ n1 log A(n, ω) exists for all ω ∈ and the convergence is uniform on . Remark 1. It is possible to show that uniform existence of the limit in the definition already implies uniform convergence. The author learned this from Furstenberg and Weiss [22]. They actually have a more general result. Namely, they consider a continuous subadditive cocycle (fn )n∈N on a minimal (, T ) (i.e. fn are continuous real-valued functions on with fn+m (ω) ≤ fn (ω) + fm (T n ω) for all n, m ∈ N and ω ∈ ). Their result then gives that existence of φ(ω) = limn→∞ n−1 fn (ω) for all ω ∈ implies constancy of φ as well as uniform convergence. For spectral theoretic investigations a special type of SL(2, R)-valued function is relevant. Namely, for E ∈ R, we define the continuous function M E : −→ SL(2, R) by E − f (T ω) −1 E . (4) M (ω) ≡ 1 0 It is easy to see that a sequence u is a solution of the difference equation u(n + 1) + u(n − 1) + (f (T n ω) − E)u(n) = 0
(5)
if and only if
u(n + 1) u(n)
= M E (n, ω)
u(1) , n ∈ Z. u(0)
(6)
By the above considerations, M E gives rise to the average γ (E) ≡ (M E ). This average is called the Lyapunov exponent for the energy E. It measures the rate of exponential growth of solutions of (5). Our main result now reads as follows. Theorem 1. Let (, T ) be strictly ergodic. Then the following are equivalent: (i) The function M E is uniform for every E ∈ R. (ii) = {E ∈ R : γ (E) = 0}. In this case the Lyapunov exponent γ : R −→ [0, ∞) is continuous. Remark 2. (a) As will be seen later on, M E is always uniform for E with γ (E) = 0 and for E ∈ R \ . From this point of view, the theorem essentially states that M E can not be uniform for E ∈ with γ (E) > 0.
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(b) Continuity of the Lyapunov exponent can easily be inferred from (ii) (though this does not seem to be in the literature). More precisely, continuity of γ on {E ∈ R : γ (E) = 0} is a consequence of subharmonicity. Continuity of γ on R \ follows from the Thouless formula (see e. g. [10] for discussion of subharmonicity and the Thouless formula). Below, we will show that continuity of γ follows from (i) and this will be crucial in our proof of (i) ⇒ (ii). Having studied (∗) of the introduction in the above theorem, we will now state our result on (∗∗). Theorem 2. If (, T ) is a subshift satisfying (PW), then the function M E is uniform for each E ∈ R. Remark 3. (a) Uniformity of M E is rather unusual. This is, of course, clear from Theorem 1. Alternatively, it is not hard to see directly that it already fails for discrete almost periodic operators. More precisely, the Almost–Mathieu-Operator with coupling bigger than 2 has uniform positive Lyapunov exponent [24]. By a deterministic version of the theorem of Oseledec (cf. Theorem 8.1 of [34] for example), this would force pure point spectrum for all these operators, if M E were uniform on the spectrum. However, there are examples of such Almost–Mathieu Operators without point spectrum [2, 29]. (b) The above theorem generalizes [18, 35], which in turn unified the work of Hof [25] on primitive substitutions and of Damanik and the author [17] on certain Sturmian subshifts. (c) The theorem is a rather direct consequence of the subadditive theorem of [37]. The two theorems yield some interesting conclusions. We start with the following consequence of Theorem 1 concerning (Z). A proof is given in Sect. 4. Corollary 2.1. Let (, T ) be an aperiodic strictly ergodic subshift. If M E is uniform for every E ∈ R, then the spectrum is a Cantor set of Lebesgue measure zero. As = {E : γ (E) = 0} holds for arbitrary Sturmian dynamical subshifts [6, 41] (cf. [19] as well), Theorem 1 immediately implies the following corollary. Corollary 2.2. Let ((α), T ) be a Sturmian dynamical system with rotation number α. Then M E is uniform for every E ∈ R. Remark 4. So far uniformity of M E for Sturmian systems could only be established for rotation numbers with bounded continued fraction expansion [17]. Moreover, the corollary is remarkable as a general type of uniform ergodic theorem actually fails as soon as the continued fraction expansion of α is unbounded [37, 38]. Theorem 1, Theorem 2 and Corollary 2.1 directly yield the following corollary. Corollary 2.3. Let (, T ) be a subshift sastisfying (PW). Then = {E ∈ R : γ (E) = 0}. If (, T ) is furthermore aperiodic, then is a Cantor set of Lebesgue measure zero. Remark 5. For aperiodic (, T ) satisfying (PW), this gives an alternative proof of (S). As discussed above primitive substitutions satisfy (PW). As validity of (Z) for primitive substitutions has been a special focus of earlier investigations (cf. the discussion in Sect. 1 and Sect. 5), we explicitly state the following consequence of the foregoing corollary. Corollary 2.4. Let (, T ) be aperiodic and associated to a primitive substitution, then
is a Cantor set of Lebesgue measure zero.
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3. Key Results In this section, we present (consequences of) results of Furman [21] and of the author [37]. We start with some simple facts concerning uniquely ergodic systems. Define for a continuous b : −→ R and n ∈ Z the averaged function An (b) : −→ R by k n−1 n−1 k=0 b(T ω) : n > 0 0 : n=0 An (b)(ω) ≡ (7) −1 |n| −k |n| b(T ω) : n < 0. k=1 Moreover, for a continuous b as above and a finite measure µ on we set µ(b) ≡ b(ω) dµ(ω). The following proposition is well known, see e.g. [43]. Proposition 3.1. Let (, T ) be uniquely ergodic with invariant probability measure µ. Let b be a continuous function on . Then the averaged functions An (b) converge uniformly towards the constant function with value µ(b) for |n| tending to infinity. The following consequence of a result by A. Furman is crucial to our approach. Lemma 3.2. Let (, T ) be strictly ergodic with invariant probability measure µ. Let B : −→ SL(2, R) be uniform with (B) > 0. Then, for arbitrary U ∈ C2 \ {0} and ω ∈ , there exist constants D, κ > 0 such that B(n, ω)U ≥ D exp(κ|n|) holds for all n ≥ 0 or for all n ≤ 0. Here, · denotes the standard norm on C2 . Proof. Theorem 4 of [21] states that uniformity of B implies that (in the notation of [21]) either (B) = 0 or B is continuously diagonalizable. As we have (B) > 0, we infer that B is continuously diagonalizable. This means that there exist continuous functions C : −→ GL(2, R) and a, d : −→ R with 0 −1 exp(a(ω)) B(1, ω) = C(T ω) C(ω). 0 exp(d(ω)) By multiplication and inversion, this immediately gives exp(nAn (a)(ω)) 0 B(n, ω) = C(T n ω)−1 C(ω), n ∈ Z. 0 exp(nAn (d)(ω))
(8)
As C : −→ GL(2, R) is continuous on the compact space , there exists a constant ρ > 0 with 0 < ρ ≤ C(ω), | det C(ω)|, C −1 (ω), | det C −1 (ω)| ≤
1 < ∞, for all ω ∈ . ρ (9)
In view of (8) and (9), exponential growth of terms as B(n, ω)U will follow from suitable upper and lower bounds on An (a)(ω) and An (d)(ω) for large |n|. To obtain these bounds we proceed as follows. Assume without loss of generality µ(a) ≥ µ(d). By (9), (8) and Proposition 3.1, we then have exp(a(·)) 0 0 < (B) = (C(T ·)−1 BC) = ( = µ(a). (10) 0 exp(d(·))
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Moreover, det B(ω) = 1 implies det B(n, ω) = 1 for all n ∈ Z. Thus, taking determinants, logarithms and averaging with n1 in (8), we infer 0 = An (a)(ω) + An (d)(ω) +
1 log | det(C(T n ω)−1 C(ω))|. n
Taking the limit n → ∞ in this equation and invoking (9) as well as Proposition 3.1, we obtain µ(a) = −µ(d). As µ(a) > 0 by (10), Proposition 3.1 then shows that there exists κ > 0, e.g. κ = 21 µ(a), s.t. for large |n|, we have An (a)(ω) > κ, and An (d)(ω) < −κ for all ω ∈ . Now, the statement of the lemma is a direct consequence of (8) and (9).
Lemma 3.3. Let (, T ) be strictly ergodic. Let A : −→ SL(2, R) be uniform. Let (An ) be a sequence of continuous SL(2, R)-valued functions converging to A in the sense that d(An , A) ≡ supω∈ {An (ω)−A(ω)} −→ 0, n −→ ∞. Then, (An ) −→ (A), n −→ ∞. Proof. This is essentially a result of [21]. More precisely, Theorem 5 of [21] shows that (An ) converges to (A) whenever the following holds: A is a uniform GL(2, R)−1 valued function and d(An , A) −→ 0 and d(A−1 n , A ) −→ 0, n −→ ∞. Now, for −1 functions An , A with values in SL(2, R), it is easy to see that d(A−1 n , A ) −→ 0, n −→ ∞ if d(An , A) −→ 0, n −→ ∞. The proof of the lemma is finished. Lemma 3.4. Let (, T ) be uniquely ergodic. Let A : −→ GL(2, R) be continuous. Then, the inequality lim supn→∞ n−1 log A(n, ω) ≤ (A) holds uniformly on . Proof. This follows from Corollary 2 of [21] (cf. Theorem 1 of [21] as well).
Finally, we need the following lemma providing a large supply of uniform functions if (, T ) is a subshift satisfying (PW). Lemma 3.5. Let (, T ) be a subshift satisfying (PW). Let F : W −→ R satisfy F (xy) ≤ (x) exists. F (x) + F (y) (i.e. F is subadditive). Then, the limit lim|x|→∞ F|x| Proof. This is just Proposition 4.2 of [37].
4. Proofs of the Main Results In this section, we use the results of the foregoing section to prove the theorems stated in Sect. 2. We start with some lemmas needed for the proof of Theorem 1. Lemma 4.1. Let (, T ) be strictly ergodic. If M E is uniform for every E ∈ R then
= {E ∈ R : γ (E) = 0} and the Lyapunov exponent γ : R −→ [0, ∞) is continuous.
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Proof. We start by showing continuity of the Lyapunov exponent. Consider a sequence (En ) in R converging to E ∈ R. As the function M E is uniform by assumption, by Lemma 3.3, it suffices to show that d(M En , M E ) → 0, n → ∞. This is clear from the definition of M E in (4). Now, set / ≡ {E ∈ R : γ (E) = 0}. The inclusion / ⊂ follows from general principles (cf. e.g. [10]). Thus, it suffices to show the opposite inclusion ⊂ /. By (2), it suffices to show σ (Hω ) ⊂ / for a fixed ω ∈ . Assume the contrary. Then there exists spectrum of Hω in the complement / c ≡ R \ / of / in R. As γ is continuous, the set / c is open. Thus, the spectrum of Hω can only exist in / c , if spectral measures of Hω actually give weight to / c . By standard results on the generalized eigenfunction expansion [8], there exists then an E ∈ / c admitting a polynomially bounded solution u = 0 of (5). By (6), this solution satisfies (u(n + 1), u(n))t = M E (n, ω)(u(1), u(0))t , n ∈ Z, where v t denotes the transpose of v. By E ∈ / c , we have (M E ) ≡ γ (E) > 0. As M E is uniform by assumption, we can thus apply Lemma 3.2 to M E to obtain that (u(n+1), u(n))t is, at least, exponentially growing for large values of n or large values of −n. This contradicts the fact that u is polynomially bounded and the proof is finished. Lemma 4.2. If (, T ) is uniquely ergodic, M E is uniform for each E ∈ R with γ (E) = 0. Proof. By det M E (ω) = 1, we have 1 ≤ M E (n, ω) and therefore 0 ≤ lim inf n→∞ n−1 log M E (n, ω) ≤ lim supn→∞ n−1 log M E (n, ω). Now, the statement follows from Lemma 3.4. The following lemma is probably well known. However, as we could not find it in the literature, we include a proof. Lemma 4.3. If (, T ) is strictly ergodic, M E is uniform with γ (E) > 0 for each E ∈ R \ . Proof. Let E ∈ R \ be given. The proof will be split in four steps. Recall that is the spectrum of Hω for every ω ∈ by (2) and thus E belongs to the resolvent of Hω for all ω ∈ . Step 1. For every ω ∈ , there exist unique (up to a sign) normalized U (ω), V (ω) ∈ R2 such that M E (n, ω)U (ω) is exponentially decaying for n −→ ∞ and M E (n, ω)V (ω) is exponentially decaying for n −→ −∞. The vectors U (ω), V (ω) are linearly independent. For fixed ω ∈ they can be chosen to be continuous in a neighborhood of ω. Step 2. Define the matrix C(ω) by C(ω) ≡ (U (ω), V (ω)). Then C(ω) is invertible and there exist functions a, b : −→ R \ {0} such that a(ω) 0 C(T ω)−1 M E (ω)C(ω) = . (11) 0 b(ω) Step 3. The functions |a|, |b|, C, C −1 : −→ R are continuous. Step 4. M E is uniform with γ (E) > 0. Ad Step 1. This can be seen by standard arguments. Here is a sketch of the construction. Fix ω ∈ and set u0 (n) ≡ (Hω − E)−1 δ0 (n) and u−1 (n) ≡ (Hω − E)−1 δ−1 (n), where δk , k ∈ Z, is given by δk (k) = 1 and δk (n) = 0, k = n. By Combes–Thomas
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arguments, see e.g. [10], the initial conditions (u0 (0), u0 (1)) and (u−1 (0), u−1 (1)) give rise to solutions of (5) which decay exponentially for n → ∞. It is easy to see that not both of these solutions can vanish identically. Thus, after normalizing, we find a vector U (ω) with the desired properties. The continuity statement follows easily from continuity of ω → (Hω − E)−1 x, for x ∈ 2 (Z). The construction for V (ω) is similar. Uniqueness follows by standard arguments from constancy of the Wronskian. Linear independence is clear as E is not an eigenvalue of Hω . Ad Step 2. The matrix C is invertible by linear independence of U and V . The uniqueness statements of Step 1, show that there exist functions a, b : −→ R with M E (ω)U (ω) = a(ω)U (T ω) and M E (ω)V (ω) = b(ω)V (T ω). This easily yields (11). As the left hand side of this equation is invertible, the right hand side is invertible as well. This shows that a and b do not vanish anywhere. Ad Step 3. Direct calculations show that the functions in question do not change if U (ω) or V (ω) or both are replaced by −U (ω) resp. −V (ω). By Step 1, such a replacement can be used to provide a version of V and U continuous around an arbitrary ω ∈ . This gives the desired continuity. Ad Step 4. As C and C −1 are continuous by Step 3 and is compact, there exists a constant κ > 0 with κ ≤ C(ω), C −1 (T ω) ≤ κ −1 for every ω ∈ . Thus, uniformity of M E will follow from uniformity of ω → C −1 (T ω)M E (ω)C(ω), which in turn will follow by Step 2 from uniformity of |a|(ω) 0 ω → D(ω) ≡ . 0 |b|(ω) As |a| and |b| are continuous by Step 3 and do not vanish by Step 2, the functions ln |a|, ln |b| : −→ R are continuous. The desired uniformity of D follows now by Proposition 3.1 (see proof of Lemma 3.2 for similar reasoning). Positivity of γ (E) is immediate from Step 1. A simple but crucial step in the proof of Theorem 2 is to relate the transfer matrices to subadditive functions. This will allow us to use Lemma 3.5 to show that the uniformity assumption of Lemma 3.2 and Lemma 3.3 holds for subshifts satisfying (PW). We proceed as follows. Let (, T ) be a strictly ergodic subshift and let E ∈ R be given. To the matrix valued function M E we associate the function F E : W −→ R by setting F E (x) ≡ log M E (|x|, ω), where ω ∈ is arbitrary with ω(1) · · · ω(|x|) = x. It is not hard to see that this is well defined. Moreover, by submultiplicativity of the norm · , we infer that F E satisfies F E (xy) ≤ F E (x) + F E (y). Proposition 4.4. M E is uniform if and only if the limit lim|x|→∞
F E (x) |x|
exists.
Proof. This is straightforward. Now, we can prove the results stated in Sect. 2. Proof of Theorem 1. The implication (i)⇒(ii) is an immediate consequence of Lemma 4.1. This lemma also shows continuity of the Lyapunov exponent. The implication (ii)⇒(i) follows from Lemma 4.2 and Lemma 4.3.
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Proof of Corollary 2.1. As is closed and has no discrete points by general principles on random operators, the Cantor property will follow if has measure zero. But this follows from the assumption and Theorem 1, as the set {E ∈ R : γ (E) = 0} has measure zero by the results of Kotani theory discussed in the introduction. Proof of Theorem 2. This is immediate from Lemma 3.5 and Proposition 4.4.
5. Further Discussion In this section we will present some comments on the results proven in the previous sections. As shown in the introduction and the proof of Theorem 1, the problem (Z) for subshifts can essentially be reduced to establishing the inclusion ⊂ {E ∈ R : γ (E) = 0}. This has been investigated for various models by various authors [5–7,13,19,42]. All these proofs rely on the same tool viz trace maps (see [1, 9] for study of trace maps as well). Trace maps are very powerful as they capture the underlying hierarchical structures. Besides being applicable in the investigation of (Z), trace maps are extremely useful because • trace map bounds are an important tool to prove absence of eigenvalues. Actually, most of the cited literature studies both (A) and (Z). In fact, (Z) can even be shown to follow from a strong version of (A) [19] (cf. [13] as well). While this makes the trace map approach to (Z) very attractive, it has two drawbacks: • The analysis of the actual trace maps may be quite hard or even impossible. • The trace map formalism only applies to substitution-like subshifts. Thus, trace map methods can not be expected to establish zero-measure spectrum in a generality comparable to the validity of the underlying Kotani result. Let us now compare this with the method presented above. Essentially, our method has a complementary profile: It does not seem to give information concerning absence of eigenvalues. But on the other hand it only requires a weak ergodic type condition. This condition is met by subshifts satisfying (PW) and this class of subshifts contains all primitive substitutions. In particular, it gives information on the Rudin-Shapiro substitution which so far had been unattainable. Moreover, quite likely, the condition (PW) will be satisfied for certain circle maps, where (Z) could not be proven by other means. All the same, it seems worthwhile pointing out that (PW) does not contain the class of Sturmian systems whose rotation number has an unbounded continued fraction expansion. This is in fact the only class known to satisfy (Z) (and much more [6, 12, 15–17, 27, 28, 41]) not covered by (PW). For this class, one can use the implication (ii) ⇒ (i) of Theorem 1, to conclude uniform existence of the Lyapunov exponent as done in Corollary 2.2. Still it seems desirable to give a direct proof of uniform existence of the Lyapunov exponent for these systems. Finally, let us give the following strengthening of (the proof of) Theorem 1. It may be of interest whenever the strictly ergodic system is not a subshift. Theorem 3. Let (, T ) be strictly ergodic. Then,
= {E ∈ R : γ (E) = 0} ∪ {E ∈ R : M E is not uniform}, where the union is disjoint.
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Proof. The union is disjoint by Lemma 4.2. The inclusion “⊃” follows from Lemma 4.3. To prove the inclusion “⊂”, let E ∈ R with M E uniform and γ (E) > 0 be given. By Lemma 3.3, we infer positivity of the Lyapunov exponent for all F ∈ R close to E. Moreover, by Theorem 4 of [21], for F ∈ R with γ (F ) > 0, uniformity of M F is equivalent to existence of an n ∈ N and a continuous C : −→ GL(2, R) such that all entries of C(T n ω)−1 M F (n, ω)C(ω) are positive for all ω ∈ . By uniformity of M E this latter condition holds for M E . By continuity of (F, ω) → C(T n ω)−1 M F (n, ω)C(ω) and compactness of , it must then hold for M F as well whenever F is sufficiently close to E. These considerations prove existence of an open interval I ⊂ R containing E on which uniformity of the transfer matrices and positivity of the Lyapunov exponent hold (cf. top of p. 811 of [21] for related arguments). Now, replacing / c with I , one can easily adopt the proof of Lemma 4.1 to obtain the desired inclusion. Note added. After this work was completed, we learned about the very recent preprint “Measure Zero Spectrum of a Class of Schrödinger Operators” by Liu–Tan–Wen–Wu (mp-arc 01-189). They present a detailed and thorough analysis of trace maps for primitive substitutions. Based on this analysis, they establish (Z) for all primitive substitutions thereby extending the approach developed in [5, 7, 9, 41]. Acknowledgements. This work was done while the author was visiting The Hebrew University, Jerusalem. The author would like to thank Y. Last for hospitality as well as for many stimulating conversations on a wide range of topics including those considered above. Enlightening discussions with B. Weiss are also gratefully acknowledged. Special thanks are due to H. Furstenberg for most valuable discussions and for bringing the work of A. Furman [21] to the authors attention. The author would also like to thank D. Damanik for an earlier collaboration on the topic of zero measure spectrum [19].
References 1. Allouche, J.-P., Peyière, J.: Sur une formule de récurrence sur le s traces de produits de matrices associés à certaines substitutions. C.R. Acad. Sci. Paris 302, 1135–1136 (1986) 2. J. Avron, B. Simon, Almost periodic Schrödinger Operators, II. The integrated density of states. Duke Math. J. 50, 369–391 (1983) 3. Baake, M.: A guide to mathematical quasicrystals. In: Quasicrystals, eds. J.-B. Suck, M. Schreiber, P. Häussler, Berlin: Springer, 1999 4. Bellissard, J.: Spectral properties of Schrödinger operators with a Thue-Morse potential. In: Number theory and physics, eds. J.-M. Luck, P. Moussa, M. Waldschmidt, Proceedings in Physics 47, Berlin: Springer, 140–150 (1989) 5. Bellissard, J.,Bovier, A., Ghez, J.-M.: Spectral properties of a tight binding Hamiltonian with period doubling potential. Commun. Math. Phys. 135, 379–399 (1991) 6. Bellissard, J., Iochum, B., Scoppola, E. and Testard, D.: Spectral properties of one-dimensional quasicrystals. Commun. Math. Phys. 125, 527–543 (1989) 7. Bovier, A., Ghez, J.-M.: Spectral Properties of One-Dimensional Schrödinger Operators with Potentials Generated by Substitutions, Commun. Math. Phys. 158, 45–66 (1993); Erratum: Commun. Math. Phys. 166, 431–432 (1994) 8. Berezanskii, J.M.: Expansions in eigenfunctions of self-adjoint operators. Transl. Math. Monographs 17, Am. Math. Soc. Providence, R.I. (1968) 9. Casdagli, M.: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Commun. Math. Phys. 107, 295–318 (1986) 10. Carmona, R., Lacroix, J.: Spectral theory of Random Schrödinger Operators. Boston: Birkhäuser (1990) 11. Damanik, D.: Singular continuous spectrum for a class of substitution Hamiltonians. Lett. Math. Phys. 46, 303–311 (1998) 12. Damanik, D.: α-continuity properties of one-dimensional quasicrystals. Commun. Math. Phys. 192, 169– 182 (1998)
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13. Damanik, D.: Substitution Hamiltonians with bounded trace map orbits. J. Math.Anal.Appl. 249, 393–411 (2000) 14. Damanik, D.: Gordon-type arguments in the spectral theory of one-dimensional quasicrystals. In: Directions in Mathematical Quasicrystals, eds. M. Baake, R.V. Moody, CRM Monograph Series 13, Providence, RI: AMS, 2000, 277–305 15. Damanik, D., Killip, R. and Lenz,D.: Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity. Commun. Math. Phys. 212, 191–204 (2000) 16. Damanik, D., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues. Commun. Math. Phys. 207, 687–696 (1999) 17. Damanik, D., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent. Lett. Math. Phys. 50, 245–257 (1999) 18. Damanik, D., Lenz, D.: Linear repetitivity I., Subadditive ergodic theorems. To appear in Discr. Comput. Geom. 19. Damanik, D., Lenz, D.: Half-line eigenfunctions estimates and singular continuous spectrum of zero Lebesgue measure. Preprint 20. Durand, F.: Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20, 1061–1078 (2000) 21. Furman, A.: On the multiplicative ergodic theorem for uniquely ergodic ergodic systems. Ann. Inst. Henri Poincaré Probab. Statist. 33, 797–815 (1997) 22. Furstenberg, H., Weiss, B.: Private communication 23. Geerse, C., Hof, A.: Lattice gas models on self-similar aperiodic tilings. Rev. Math. Phys. 3, 163–221 (1991) 24. Herman, M.-R.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant the caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv 58, 4453–502 (1983) 25. Hof, A.: Some Remarks on Aperiodic Schrödinger Operators. J. Stat. Phys. 72, 1353–1374 (1993) 26. Hof, A., Knill, O., Simon, B.: Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys. 174, 149–159 (1995) 27. Jitomirskaya, S., Last, Y.: Power law subordinacy and singular spectra. I. Half-line operators. Acta Math. 183, 171–189 (1999) 28. Jitomirskaya, S., Last, Y.: Power law subordinacy and singular spectra. II. Line Operators. Commun. Math. Phys. 211, 643–658 (2000) 29. Jitomirskaya, S., Simon, B.: Operators with singular continuous spectrum. III.Almost perodic Schrödinger operators. Commun. Math. Phys. 165, 201–205 (1994) 30. Kaminaga, M.: Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential. Forum Math. 8, 63–69 (1996) 31. Katznelson, Z.,Weiss, B.: A simple proof of some ergodic theorems. Israel J. Math. 34, 291–296 (1982) 32. Kotani, S.: Jacobi matrices with random potentials taking finitely many values. Rev. Math. Phys. 1, 129–133 (1989) 33. Lagarias, J.C., Pleasants, P.A.B.: Repetitive Delone Sets and Quasicrystals, To appear in Ergod. Th. & Dynam. Sys. 34. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum for onedimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999) 35. Lenz, D.: Aperiodische Ordnung und gleichmässige spektrale Eigenschaften von Quasikristallen, Dissertation, Frankfurt/Main, Logos, Berlin (2000) 36. Lenz, D.: Random operators and crossed products. Mathematical Physics, Analysis and Geometry 2, 197–220 (1999) 37. Lenz, D.: Uniform ergodic theorems on subshifts over a finite alphabet. Ergod. Th. & Dynam. Syst. 22, 245–255 (2002) 38. Lenz, D.: Hierarchical structures in Sturmian dynamical systems. Preprint 39. Queffélec, M.: Substitution Dynamical Systems – Spectral Analysis. Lecture Notes in Mathematics, Vol. 1284. Berlin–Heidelberg–New York: Springer, 1987 40. Senechal, M.: Quasicrystals and geometry. Cambridge. Cambridge University Press, 1995 41. Süt˝o, A.: The spectrum of a quasiperiodic Schrödinger operator. Commun. Math. Phys. 111, 409–415 (1987) 42. Süt˝o, A.: Singular continuous spectrum on a Cantor set of zero Lebesgue measure. J. Stat. Phys. 56, 525–531 (1989) 43. Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics, 79, Berlin: Springer, 1982 Communicated by B. Simon
Commun. Math. Phys. 227, 131 – 153 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions A. Marshakov1,2 , P. Wiegmann3,4 , A. Zabrodin5,2 1 Theory Department, Lebedev Physics Institute, Leninsky pr. 53, 117924 Moscow, Russia 2 ITEP, Bol. Cheremushkinskaya str. 25, 117259 Moscow, Russia 3 James Franck Institute and Enrico Fermi Institute of the University of Chicago, 5640 S.Ellis Avenue,
Chicago, IL 60637, USA
4 Landau Institute for Theoretical Physics, Moscow, Russia 5 Institute of Biochemical Physics, Kosygina str. 4, 119991 Moscow, Russia
Received: 18 September 2001 / Accepted: 18 December 2001
Abstract: We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the taufunction is identical to the partition function of the planar large N limit of the Hermitian one-matrix model. 1. Introduction The subject of the Dirichlet boundary problem in two dimensions [1] is a harmonic function in a domain of the complex plane bounded by a closed curve with a given value on the boundary and continuous up to the boundary. The question we address in this paper is how the harmonic function in the bulk varies under a small deformation of the the shape of the domain. Remarkably, this standard problem of complex analysis possesses an integrable structure [2, 3] which we intend to clarify further in this paper. It is described by a particular solution of an integrable hierarchy of partial differential equations known in the literature as dispersionless Toda (dToda) hierarchy. Moreover, related integrable hierarchies arise in the context of 2D topological theories and just the same solution to the dToda hierarchy emerges in the study of 2D quantum gravity [4,5] (we do not elaborate these relations in this paper). Let D be a simply connected domain in the complex plane bounded by a smooth simple curve γ . The Dirichlet problem is to find a harmonic function u(z) in D such that it is continuous up to the boundary and equals a given function u0 (z) on the boundary.
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The problem has a unique solution written in terms of the Green function G(z1 , z2 ) of the Dirichlet boundary problem: 1 u(z) = − u0 (ξ )∂n G(z, ξ )|dξ | , (1.1) 2π γ where ∂n is the normal derivative on the boundary with respect to the second variable, and the normal vector n always looks inside the domain, where the Dirichlet problem 1 is posed. Equivalently, the solution is represented as u(z) = u0 (ξ )∂ξ G(z, ξ )dξ , π i ∂D where ∂D is understood as γ runs anticlockwise with respect to the domain. The main object to study is, therefore, the Dirichlet Green function. It is uniquely determined by the following properties [1]: (G1) The function G(z1 , z2 ) − log |z1 − z2 | is symmetric, bounded and harmonic everywhere in D in both arguments; (G2) G(z1 , z2 ) = 0 if any one of the variables belongs to the boundary. The definition implies that G(z1 , z2 ) is real and negative in D. The Green function can be written explicitly through a conformal map of the domain D onto some “reference” domain for which the Green function is known. A convenient choice is the unit disk. Let f (z) be any bijective conformal map of D onto the unit disk (or its complement), then f (z1 ) − f (z2 ) , G(z1 , z2 ) = log (1.2) f (z1 )f (z2 ) − 1 where bar means complex conjugation. Such a map exists by virtue of the Riemann mapping theorem [1]. It thus suffices to study variations of the conformal map f (z) under deformations of the boundary. This problem was discussed in [2, 3], where it was shown that evolution of the conformal map under changing harmonic moments of the domain is given by the dToda integrable hierarchy. (A relation between conformal maps of slit domains and special solutions to some integrable equations of hydrodynamic type was earlier observed by Gibbons and Tsarev [6].) The study of the Dirichlet problem approaches this subject from another angle. Our starting point is the Hadamard variational formula [7]. It gives the variation of the Green function under small deformations of the domain in terms of the Green function itself: 1 δG(z1 , z2 ) = ∂n G(z1 , ξ )∂n G(z2 , ξ )δh(ξ )|dξ |. (1.3) 2π γ Here δh(ξ ) is the thickness between the curve γ and the deformed curve, counted along the normal vector at the point ξ ∈ γ . We show that already this remarkable formula reflects all integrable properties of the Dirichlet problem. A smooth closed curve γ (for simplicity, we may assume it to be analytic in order to have an easy sufficient justification of some arguments below) divides the complex plane into two parts having a common boundary: a compact interior domain Dint , and an exterior domain Dext containing ∞. Correspondingly, one recognizes interior and exterior Dirichlet problems. The main contents of the paper is common for both of them. To stress this, we try to keep the notation uniform calling the domain simply D. We will
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show that (logarithms of) the tau-functions, introduced in [2,3] and further studied in [8], for the interior and exterior problems are related to each other by a Legendre transform. The exterior Dirichlet problem makes sense when the interior domain degenerates into a segment (a plane with a gap). We will show that in this case a deformation problem is described by the dispersionless limit of the Toda chain hierarchy and discuss its relation to the planar limit of the Hermitian matrix model. 2. Deformations of the Boundary Let D be a simply-connected domain in the extended complex plane bounded by a smooth simple curve γ . Consider a basis ψk (z), k ≥ 1, of holomorphic functions in D such that ψk (z0 ) = 0 for some point z0 ∈ D. We call z0 the normalization point. The basis is assumed to be fixed and independent of the domain. For example, in case of the interior problem one may assume, without loss of generality, that the origin is in D and set z0 = 0, ψk (z) = zk /k, while a natural choice for the exterior problem is z0 = ∞ and ψk (z) = z−k /k. Throughout the paper, these bases for interior and exterior problems are referred to as natural ones. Let tk be moments of the domain D defined with respect to the basis ψk : 1 tk = κ ψk (z) d 2 z, k = 1, 2, . . . , (2.1) π D where κ = ± for the interior (exterior) problem. We also assume that the functions ψk for domains containing ∞ are integrable, or the integrals are properly regularized (see below). Besides, we denote by t0 the area (divided by π ) of the domain D in the case of the interior problem and that of the complementary (compact) domain in the case of the exterior problem: 1 d 2 z for compact domains π D . t0 = 1 2 d z for non-compact domains π C\D Let us note that the moments (except for t0 ) are in general complex. We call the quantities tk , t¯k and t0 harmonic moments of the domain D. The Stokes formula represents the harmonic moments as contour integrals 1 tk = ψk (z)¯zdz, k = 0, 1, 2, . . . 2π i γ (where it is set ψ0 (z) = 1) providing, in particular, a regularization of possibly divergent integrals (2.1) in case of the exterior problem. Throughout the paper the contour in γ is run in the anticlockwise direction both for interior and exterior problems. The basic fact of the theory of deformations of closed analytic curves is that the (in general complex) moments tk supplemented by the real variable t0 form a set of local coordinates in the space of smooth closed curves [9] (see also [10]). This means that under any small deformation of the domain the set {t0 , t1 , . . . } is subject to a small change and vice versa. More precisely, let γ (t) be a family of curves such that ∂t tk = 0 in some neighborhood of t = 0, then all the curves γ (t) coincide with γ = γ (0) in this neighborhood.
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γ
ξ
ξ
.
γ
.
Dext
Dint
Fig. 1. Action of the operator ∇(ξ ) in the case of interior Dint (left) and exterior Dext (right) domains. In our convention bump always looks outside the (interior or exterior) domain
The family of differential operators
ψk (z)∂tk + ψk (z)∂t¯k ∇(z) = ∂t0 +
(2.2)
k≥1
span the complexified tangent space to the space of curves. They are invariant under change of variables in the following sense: let t˜k be harmonic moments defined with ˜ respect to another basis, ψ˜ k , of holomorphic functions in D; then ∇(z) = ∇(z). Note that ∇(z0 ) = ∂t0 since ψk (z0 ) = 0. The operator ∇(z) has a clear geometrical meaning described below. Let us consider a special deformation of the domain obtained by adding to it an infinitesimal smooth bump (of an arbitrary form) with area located at the point ξ ∈ γ . Our convention is that > 0 if the bump looks outside the domain in which the Dirichlet problem is posed, as is shown in Fig. 1. Let A be any functional of a domain that depends on the harmonic moments only. The variation of such a functional in the leading order in , is given by ∇(ξ )A, ξ ∈ γ. π (∂tk Aδtk + ∂t¯k Aδ t¯k ) and Indeed, combining δA = ∂t0 Aδt0 + δ(ξ ) A = κ
δtk = κ
1 π
(2.3)
k≥1
bump
ψk (z)d 2 z = κ
ψk (ξ ) π
we obtain (2.3). So, the result of the action of the operator ∇(ξ ) with ξ ∈ γ on A is proportional to the variation of the functional under attaching a bump at the point ξ . To put it differently, we can say that the boundary value of the function ∇(z)A is given by the l.h.s. of (2.3). For functionals A such that the series ∇(z)A converges everywhere in D up to the boundary, this remark gives a usable method to find the function ∇(z)A everywhere in the domain. This function is harmonic in D with the boundary value determined from (2.3). It is given by (1.1): 1 |dξ |∂n G(z, ξ )δ(ξ ) A. (2.4) ∇(z)A = κ 2π γ π This gives the result of the action of the operator ∇(z), when the argument is anywhere in D.
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For example, given any regular function f in a domain containing the interior domain D, set Af = D f (z)d 2 z. We have: δ(ξ ) Af = ∇(ξ )Af = f (ξ ), ξ ∈ γ . If the π function f is harmonic in D, then ∇(z)Af = πf (z) for any z ∈ D. The subject of the deformation theory of the Dirichlet boundary problem is to compute ∇(z)G(z1 , z2 ) through the conformal map or the Green function of the original domain. In the next section, we do this using the Hadamard variational formula. 3. Hadamard Variational Formula and Dispersionless Integrable Hierarchy 3.1. The Hadamard integrability condition. Variation of the Green function under small deformations of the domain is known due to Hadamard [7], see Eq. (1.3). Being specified to the particular case of attaching a bump of the area , it reads: δ(ξ ) G(z1 , z2 ) = − ∂n G(z1 , ξ )∂n G(z2 , ξ ), ξ ∈ γ. (3.1) 2π To find how the Green function changes under a variation of the harmonic moments, we use (2.4) to employ the harmonic continuation procedure explained in the previous section. The harmonic function in D with a boundary value given by the Hadamard formula is 1 ∇(z3 )G(z1 , z2 ) = κ ∂n G(z1 , ξ )∂n G(z2 , ξ )∂n G(z3 , ξ )|dξ |. (3.2) 4π γ It is obvious from the r.h.s. of (3.2) that the result of the action of the operator ∇(z) on the Green function is harmonic and symmetric in all three arguments, i.e., ∇(z3 )G(z1 , z2 ) = ∇(z1 )G(z2 , z3 ).
(3.3)
This is our basic relation. It has the form of the integrability condition. In the rest of the paper we will draw consequences of this symmetry and underlying algebraic structures. Note also that despite the fact that the Green function vanishes on the boundary, its derivative (the l.h.s. of Eq. (3.3)) with respect to the deformation of the domain does not. The basic equation (3.3) is a compressed form of an integrable hierarchy. To unfold it, let us separate holomorphic and antiholomorphic parts of this equation. Let E be the exterior to the unit disk. Given a point a ∈ D, consider a bijective conformal map fa : D → E such that fa (a) = ∞. The Dirichlet Green function then is G(a, z) = − log |fa (z)|.
(3.4)
Under a proper normalization of the map the integrability condition (3.3) becomes holomorphic: ∇(b) log fa (z) = ∇(a) log fb (z)
(3.5)
for all a, b, z ∈ D. The following normalization will be convenient: the overall phase is chosen to be argfa (z0 ) = π − arg
(z0 − a) if a = z0 , where z0 ∈ D is the normalization point. If a = z0 we set lim (z − z0 )2 f (z) to be real and negative. Under these conditions z→z0
fa (z) =
(a¯ − z¯ 0 ) f (a) (a − z0 ) f (a)
1/2
f (z)f (a) − 1 f (a) − f (z)
(3.6)
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(for a, z0 = ∞). In the vicinity of the point a (a = ∞) ra
fa (z) = eiω(a,z0 ) pk (a)(z−a)k , 1+ z−a
(3.7)
k≥1
where the real constant ra is called the conformal radius of the domain [11] with respect to the point a, and ω is a phase determined from the normalization condition. In particular, they read ω(z0 , z0 ) = 0. Similarly, the map fa (z) can be defined in the case when either a or z0 lies at infinity. To verify (3.5), we note that the holomorphic function ∇(b) log fa (z)−∇(a) log fb (z) is also antiholomorphic (in z) by virtue of (3.3), and thus must be a constant. Setting z = z0 we find that the latter is zero: ∇(b) log |fa (z0 )| − ∇(a) log |fb (z0 )| + i∇(b) arg fa (z0 ) − i∇(a) arg fb (z0 ) = 0 (the first line vanishes due to (3.3), the second one vanishes because the normalization does not depend on the shape of the domain). 3.2. Harmonic moments as commuting flows. Equation (3.5) suggests to treat log fa (z) as a generating function of commuting flows with respect to spectral parameter a. The expansion
log fa (z) = H0 (z) + (3.8) ψk (a) Hk (z) − ψk (a) H˜ k (z) k≥1
defines generators Hk , H˜ k of the commuting flows. Clearly, H0 (z) = log f (z). It implies evolution equations for f (z), ∂ log f (z) ∂Hk (z) = , ∂tk ∂t0
∂ log f (z) ∂ H˜ k (z) =− ¯ ∂ tk ∂t0
and integrability conditions: ∂Hj (z) ∂Hk (z) = , ∂tj ∂tk
∂ H˜ j (z) ∂ H˜ k (z) = , ∂ t¯j ∂ t¯k
∂ H˜ j (z) ∂Hk (z) =− . ∂ t¯j ∂tk
(3.9)
The real part of (3.8) vanishes on the boundary (as it is the Dirichlet Green function), therefore, the boundary values of H˜ k and Hk are complex conjugated: H˜ k (z) = Hk (z) ,
z ∈ γ.
(3.10)
The structure of integrable hierarchy becomes explicit if instead of functions of z one passes to functions of its image w under the map fz0 : w = fz0 (z) ≡ f (z). Using the chain rule, one can write ∇(a) log fb (z) = ∇(a) log fb (z(w)) + (∇(a) log f (z)) w∂w log fb . w
In the last term we observe that ∇(a) log f (z) = ∂t0 log fa (z) (using (3.5) at b = z0 ). Subtracting the same equality with a, b interchanged, we come to the equation of zerocurvature type: ∇(a) log fb − ∇(b) log fa − {log fa , log fb } = 0,
Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions
where the Poisson brackets are defined as {f, g} ≡ w
137
∂f ∂g ∂g ∂f −w and tk -deriva∂w ∂t0 ∂w ∂t0
tives are taken now at fixed w. Let z(w) be the map inverse to w = f (z). Equation (3.5) at b = z0 , being rewritten for the inverse map, has the form of a one-parametric family of evolution equations of the Lax type labeled by the spectral parameter a. They are ∇(a)z(w) = {log fa (z(w)), z(w)}.
(3.11)
We refer to them as to deformation equations. The zero-curvature conditions ensure that these equations are consistent, i.e. flows with different values of spectral parameter commute. The integrability conditions (3.9) in the new variable acquire the form of the zerocurvature equations ∂tj Hi (w) − ∂ti Hj (w) + {Hi (w), Hj (w)} = 0, ∂t¯j H˜ i (w) − ∂t¯i H˜ j (w) + {H˜ j (w), H˜ i (w)} = 0,
(3.12)
∂tj H˜ i (w) + ∂t¯i Hj (w) + {H˜ i (w), Hj (w)} = 0. From (3.6) it is easy to see that Hk are polynomials in w while H˜ k are polynomials in w−1 . Furthermore, for any basis such that ψk (z) = O( (z−z0 )k ) these polynomials are of degree k. In other words, the generators are meromorphic functions on the Riemann sphere with two marked points at w = 0 and w = ∞. This is a particular case of the universal Whitham hierarchy [12], known as the dispersionless Toda lattice. In [13], it is proved that the full set of zero-curvature conditions (3.12), together with the polynomial structure of the generators, already imply existence of the Lax function and Lax-Sato equations. For a particular choice of the basis, the Lax function can be expressed through the inverse conformal map. Consider the interior problem, set z0 = 0 ∈ D and fix the basis of holomorphic functions in D to be the natural one: ψk (z) = zk /k. From (3.7) and (3.8) it follows that H˜ k are holomorphic in D while Hk are meromorphic with the k th order pole at 0 so that Hk = z−k + O(1) as z → 0. Combining these properties with the polynomial structure of the generators as functions of w, and taking into account (3.10), one gets
1 −k Hk = z−k (w) z (w) , + >0 0 2
1 −k −1 −k + H˜ k = z¯ (w ) z¯ (w −1 ) , <0 0 2
(3.13)
where z(w) is the map inverse to f (z) and z¯ (w) = z(w) ¯ is its Schwarz double. The symbols (f (w))>0 , (f (w))<0 mean truncated Laurent series, where only terms with strictly positive (negative) powers of w are kept, while (f (w))0 is the free term (w 0 ) of the series. The free terms in (3.13) are found with the help of an easily proved identity w(z) 1 −k log = (z (w))0 zk valid for any series of the form w(z) = z + w1 z2 + . . . . z k k≥1
Similar formulas hold for the exterior problem with the choice z0 = ∞, ψk (z) = z−k /k.
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3.3. Deformation equations and dispersionless Toda hierarchy. Expanding the deformation equations (3.11) in spectral parameter we obtain the Lax representation of the dToda hierarchy. To see this, we recall that the dToda hierarchy is an infinite set of evolution (Lax-Sato) equations for two Lax functions L(w) = rw + uk w −k , k≥0
˜ L(w) = rw−1 +
u˜ k w k .
k≥0
The equations are ∂L(w) ∂L(w) = {Hk (w), L(w)} , = {L(w), H˜ k (w)} ∂tk ∂ t¯k ˜ and the same for L(w). Here the Poisson brackets with respect to w and t0 are defined as above, and
1 k Hk (w) = Lk (w) + L (w) , >0 0 2
1 k k H˜ k (w) = L˜ (w) + L˜ (w) <0 0 2 are generators of the flows. One may also introduce H0 (w) = log w. They obey the dispersionless zero-curvature conditions (3.12) which express consistency of the Lax equations and generate an infinite set of partial differential equations for coefficients of the Lax functions. To summarize, we have the following identification of the Lax functions with conformal maps: • For the interior problem (z0 = 0 with the natural basis): 1 1 ˜ L(w) = , L(w) = , z(w) z¯ (w −1 ) where z(w) is the function inverse to w = f0 (z); • For the exterior problem (z0 = ∞ with the natural basis):
(3.14)
˜ L(w) = z¯ (w−1 ) ,
(3.15)
L(w) = z(w) ,
where z(w) is the function inverse to w = f∞ (z). So the inverse map, z(w), and its Schwarz double, z¯ (w −1 ), both obey the Lax equations. Let us comment on another choice of basis. Suppose λ−1 is a local parameter at the normalization point z0 , i.e., 1 λ= + c0 + c1 (z−z0 ) + O( (z−z0 )2 ), z0 = ∞, z−z0 λ = z + c0 +
c1 + O(z−2 ) , z
z0 = ∞.
with some domain-independent coefficients cj . Assuming that λ−1 (z) is a well defined local parameter in a domain that contains D, we choose the basis of holomorphic functions in D to be ψk (z) = λ−k (z)/k. This results in a linear change of times with the help of a triangular matrix. Repeating the above arguments, one identifies the Lax function with λ(z): L(w) = λ(z(w)). This Lax function provides a solution to an equivalent hierarchy in the sense of [14].
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3.4. String equations. A customary way to fix a solution to a dispersionless hierarchy is to impose an additional constraint on the Lax functions (sometimes called string equation [15]). We are going to show that not only the deformation equations but the string equation, too, is an easy consequence of the Hadamard formula and our basic relation (3.3). By r = rz0 denote the conformal radius of the curve γ with respect to the normalization point z0 . Let us calculate δ(ξ ) log r in two different ways. The first one is to use the Hadamard formula (3.1) when both arguments tend to the normalization point: δ(ξ ) log r =
2 |∂ξ G(z0 , ξ )|2 = |∂ξ f (ξ )|2 π 2π
(recall that |f (ξ )| = 1 for ξ ∈ γ ). The second one is obtained from (3.3) in the limit z1 → ξ ∈ γ , z2 , z3 → z0 : δ(ξ ) log r = −
∂t log |f (ξ )| , π 0
where we have used (2.3). Combining the results, we obtain the relation ∂t0 log |f (z)|2 = −|∂z f (z)|2 ,
z ∈ γ.
Passing to the variable w = f (z), one rewrites this equation as ∂z(w) ∂z(w) 2Re w = 1, |w| = 1 , ∂w ∂t0 where z(w) is the inverse map, as before. Being analytically continued from the unit circle, it reads {z(w), z¯ (w−1 )} = 1. This is the customary form of the semiclassical string equation. Similar arguments in case of the exterior problem lead to the same equation. For the natural basis, let us rewrite the string equation in terms of the Lax functions ˜ Using the identifications (3.14) and (3.15), we have: L, L. {L−1 (w), L˜ −1 (w)} = −1 for the interior problem, ˜ {L(w), L(w)} = 1 for the exterior problem.
(3.16)
So, although the string equation in terms of z(w) is the same, the interior and exterior problems correspond to two different solutions of the same (dToda) integrable hierarchy (cf. [16]). 4. Tau-Function and Dispersionless Hirota Equations 4.1. Tau-function. The symmetry relation (3.3) implies that there exists a real-valued function of harmonic moments F (t0 , t, t¯) = F (t0 ; t1 , t2 , . . . ; t¯1 , t¯2 , . . . ) such that 1 G(z1 , z2 ) = g0 (z1 , z2 ) + ∇(z1 )∇(z2 )F, 2
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where the function g0 does not depend on the domain, i.e., on moments (the coefficient 1 ¯ 2 is set for future convenience). Note that adding to F a quadratic form in tk , tk amounts to changing the function g0 only. From the definition of the Green function it follows that g0 (z1 , z2 ) is symmetric and harmonic in D in both arguments with the only singularity log |z1 − z2 | as z1 → z2 , or, in terms of a local parameter λ−1 , log |λ−1 (z1 ) − λ−1 (z2 )|. In the case of the interior problem, D can be any domain not containing the point of infinity. The function g0 must be the same for all such domains. Therefore, singularities of g0 in the extended complex plane, other than the logarithmic singularity at merging points, may occur only at infinity. In other words, g0 is of the form g0 (z1 , z2 ) = log |λ−1 (z1 ) − λ−1 (z2 )| + h0 (z1 , z2 ), where h0 is regular and harmonic in both arguments everywhere in the complex plane but at infinity. The function h0 is in fact a matter of definition of the tau-function. For some particular basis we are free to choose it to be zero. Hence we obtain the following important relation: 1 G(z1 , z2 ) = log |λ−1 (z1 ) − λ−1 (z2 )| + ∇(z1 )∇(z2 )F , 2
z1 , z2 ∈ D.
(4.1)
Changing the normalization point or passing to another local parameter results in modifying the function g0 . As is already pointed out, this is equivalent to adding to F a quadratic form in times. Taking into account the remark at the end of Sect. 3.4, one sees that this agrees with the relation between tau-functions of equivalent hierarchies discussed in [17]. It is easy to see that in this form the relation holds true for the exterior problem as well. For clarity, let us specify the above equation for the natural basis. Instead of the differential operator ∇(z) defined for an arbitrary basis, it is convenient to use its specialization for the natural basis z−k z¯ −k D(z) = ∂t0 + (4.2) ∂t + ∂¯ , k k k tk k≥1
so that ∇(z) = D(z) for the exterior problem and ∇(z) = D(z−1 ) for the interior one. Equation (4.1) is then rewritten in the form 1 log |z1 − z2 | + D(z1−1 )D(z2−1 )F for the interior problem 2 G(z1 , z2 ) = (4.3) log |z−1 − z−1 | + 1 D(z1 )D(z2 )F for the exterior problem 1 2 2 The function F plays a central role in what follows. It is the tau-function of curves introduced and studied in [2, 3, 8]. It is dispersionless limit of the logarithm of the taufunction of an integrable hierarchy or dispersionless tau-function [12]. For brevity we refer to it simply as the tau-function. The tau-function has been related to variations of conformal maps in [2, 3, 8]. The relation between the tau-function and the Green function has been observed by L.Takhtajan. See [10] for a rigorous proof and discussion. In Sect. 5 we give a few explicit integral representations for the tau-function.
4.2. Conformal maps and the tau-function. In this section, we use the natural basis. To unify formulas for the interior and exterior cases, we write ψk (z) = λ−k (z)/k, where
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141
λ(z) = z−1 for the interior problem and λ(z) = z for the exterior one. (Correspondingly, z0 is either 0 or ∞.) We also employ the notation f (zj ) = fj , λ(zj ) = λj , D(z) =
z−k ∂ , k ∂tk
¯ z) = D(¯
k≥1
z¯ −k ∂ , k ∂ t¯k
(4.4)
k≥1
¯ z). so that D(z) = ∂t0 + D(z) + D(¯ Representations of the conformal map through the tau-function can be obtained from (4.3) by separating holomorphic and antiholomorphic parts in z1 and z2 of the Green function. Holomorphic parts in both variables yield log
f1 − f2 1 = D(λ1 )D(λ2 )F − ∂t20 F. λ1 − λ 2 2
Terms holomorphic in z1 and antiholomorphic in z2 yield
1 ¯ 2 )F. = −D(λ1 )D(λ log 1 − f1 f2
(4.5)
(4.6)
Various specifications of (4.3), (4.5) and (4.6) lead to representations of the conformal maps through the tau function. Tending, for example, z2 → z0 in (4.3) and extracting holomorphic parts in z1 , we get the following expressions for the conformal maps through the tau-functions:
1 f (z) = λ(z) exp − ∂t20 F − ∂t0 D(λ(z))F . (4.7) 2 The leading coefficients as z → z0 give formulas for the conformal radius: 2 log r = −κ∂t20 F
(4.8)
(recall that the sign factor κ is introduced in Sect. 2 to distinguish between interior and exterior problems). Limits z2 → z0 in (4.5), (4.6) give other representations for the conformal maps,
1 2 f (z) = e− 2 ∂t0 F λ(z) − ∂t20 t1 F − D(λ(z))∂t1 F , 1 2 1 = e− 2 ∂t0 F D(λ(z))∂t¯1 F , f (z)
(4.9)
and another representation of the conformal radius: r −2κ = ∂t21 t¯1 F.
(4.10)
These formulas agree with (4.8) provided F satisfies the dispersionless Toda equation ∂t21 t¯1 F = e∂t0 F . 2
(4.11)
In fact all coefficients of the Taylor expansion of the conformal map around the normalization point can be easily read from (4.9).
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Merging the points z1 and z2 in (4.5), (4.6), we get formulas for the derivative and the modules of the conformal maps: f (z) = λ (z)eD (λ(z))F − 2 ∂t0 F ,
¯ . 1 − |f (z)|−2 = exp −D(λ(z))D(λ(z))F 1 2
2
(4.12)
It is easy to see that the generators of commuting flows introduced in (3.8) are expressed through the tau-function as follows:
1 ∂t0 + D(λ(z)) ∂tk F , Hk (z) = λk (z) − 2
1 H˜ k (z) = ∂t0 + D(λ(z)) ∂t¯k F , 2
1 H0 (z) = log λ(z) − ∂t0 + D(λ(z)) ∂t0 F . 2 As is seen from (4.3), coefficients of the Laurent series of the regular part of the Green function are second order derivatives of the tau-function. The formula d log fz1 (z) d log fz2 (z) 1 2(z1 ∂z1 + z2 ∂z2 )G(z1 , z2 ) = 2π i ∂D d log z can be easily proved by substituting (3.4) and evaluating residues at the poles z1 , z2 . The left hand side of this formula could be thought of as a dispersionless analog of the Gelfand-Dikii resolvent. Expanding both sides in power series in z1 , z2 using (4.3) and (3.8), we obtain ∂ 2F dHn dHm 1 = , ∂tn ∂tm 2πi(n+m) γ d log z dHn dHm 1 ∂ 2F , =− ∂tn ∂ t¯m 2πi n γ d log z
n ≥ 0,
m ≥ 1,
n, m ≥ 1
(in the last formula we used (3.10)). In addition to the relations above, we point out that the tau-function provides new representations for some classical objects of complex analysis. Consider, for example,
2 (z) (z) the Schwarz derivative S(f, z) = ff (z) − 23 ff (z) of the conformal map f . A nonsingular part of lim ∂z ∂ζ G(z, ζ ) is ζ →z
1 12 S(f, z),
so Eq. (4.3), with z0 = ∞ and λ(z) = z,
yields [3]
where ∂z D(z) = −
S(f, z) = 6(∂z D(z))2 F, z−k−1 ∂tk . The reproducing kernel [11] of the (exterior) domain
k≥1
D, K(z, ζ¯ ) = 2∂z ∂ζ¯ G(z, ζ ), is given by ¯ ζ¯ )F. K(z, ζ¯ ) = ∂z D(z)∂ζ¯ D(
(4.13)
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4.3. Dispersionless Hirota equations. The tau-function obeys an infinite set of nonlinear differential equations. They can be obtained by excluding f from the identities of the type (4.5, 4.6) by means of Eq. (4.7). Equations obtained in this way are dispersionless limits of the Hirota bilinear relations [18] obeyed by the tau-functions of integrable hierarchies. For example, consider Eq. (4.5) for pairs of points (z1 , z2 ), (z2 , z3 ) and (z3 , z1 ). Exponentiating and summing the equations one gets the identity (z1 − z2 )eD(z1 )D(z2 )F + (z2 − z3 )eD(z2 )D(z3 )F + (z3 − z1 )eD(z3 )D(z1 )F = 0. (4.14) obeyed by the tau-function F for both interior and exterior problems. This is the symmetric form of Hirota’s equation for the dKP hierarchy. It encodes algebraic relations between the second order derivatives of the tau-function. These relations are obtained on expanding the Hirota equation in powers of z1 , z2 , z3 and comparing coefficients1 . The ¯ z) are defined in (4.4). Equation (4.14), as well as other Hirota operators D(z) and D(¯ equations given below, are the same for interior and exterior problems. More general equations obtained in a similar way include derivatives with respect to t0 and t¯k . These are equations of the dToda hierarchy: (z1 − z2 )eD(z1 )D(z2 )F = z1 e−∂t0 D(z1 )F − z2 e−∂t0 D(z2 )F , ¯
1 − e−D(z1 )D(¯z2 )F =
1 ∂t (∂t +D(z1 )+D(¯ ¯ z2 ))F e 0 0 . z1 z¯ 2
(4.15)
(4.16)
Note that Eq. (4.15) can be obtained from (4.14) in the formal limit z3 → 0 if to understand limz→0 D(z) as −∂t0 . These equations allow one to express the second order derivatives ∂t2m tn F , ∂t2 t¯ F with m, n ≥ 1 as certain functions of the derivatives ∂t20 tk F , m n ∂t2 t¯ F . The dispersionless Toda equation (4.11) is the limit of (4.16) as z1 , z2 → ∞. 0 k Each side of this equation is the squared conformal radius as one can see from (4.8, 4.10). In the same manner one can derive other equivalent forms of the Hirota equations: z1 (z3 − z2 )e−D(z3 )(D(z1 )−D(z3 ))F − z2 (z3 − z1 )e−D(z3 )(D(z2 )−D(z3 ))F = z3 (z1 − z2 )e(D(z1 )−D(z3 ))(D(z2 )−D(z3 ))F , ¯
(4.17)
¯
1 − e−(D(z1 )−D(z3 ))(D(¯z2 )−D(¯z3 ))F ¯
¯
= (z1−1 − z3−1 )(¯z2−1 − z¯ 3−1 )eD(z3 )(D(z3 )+D(z1 )−D(z3 )+D(¯z2 )−D(¯z3 ))F ,
(4.18)
where the operator D(z) is defined in (4.2). Equations (4.17), (4.18) may be interpreted as analogs of Eqs. (4.15, 4.16) with the normalization point moved to z3 , in which case ∂t0 should be substituted by D(z3 ). In the limit z3 → ∞ they convert into Eqs. (4.15), (4.16) respectively. In (4.18) z¯ 3 can be regarded as an independent formal variable, not necessarily complex conjugate to z3 . We stress that the full set of differential equations for F obtained by expanding the Hirota equations (4.15, 4.16) or (4.17,4.18) is the same. 1 In [19, 20], the Hirota equation for the dKP hierarchy was obtained in an equivalent but less symmetric form which follows from (4.14) in the limit z3 → ∞.
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The special case of (4.18) z2 = z1 is worth mentioning: ¯
¯ ¯
1 − e−(D(z)−D(ζ ))(D(¯z)−D(ζ ))F = |z−1 − ζ −1 |2 eD(ζ )D(z)F . Further specialization ζ → z yields ¯ z)F = eD |z|4 ∂z D(z)∂z¯ D(¯
2 (z)F
.
(4.19)
This equation is especially remarkable. On the one hand, it looks like the dispersionless ¯ z) reToda equation (4.11) where ∂t0 , ∂t1 , ∂t¯1 are replaced by D(z), z2 ∂z D(z), z¯ 2 ∂z¯ D(¯ spectively (moreover, (4.19) becomes (4.11) as z → ∞). On the other hand, (4.19) is equivalent to the Liouville equation. Indeed, Eq. (4.19) tells us that the field χ (z) = D2 (z)F − log |z|4 obeys the Liouville equation
∂z ∂z¯ χ = 2eχ
for z ∈ D. (Here we imply that D contains ∞, but similar equations can be written for the interior problem, too.) By virtue of (4.3), (4.12), the solution to this equation can be written as follows: χ (z) = 2 lim ) − log |z − z | G(z, z z →z
or eχ(z) =
|f (z)|2 . (|f (z)|2 − 1)2
Note that ds 2 = eχ dzd z¯ is the pull back of the Poincaré metric with constant negative curvature Rds 2 = −2e−χ ∂z ∂z¯ χ = −4 in D. We also note that the Liouville field χ equals the value of the reproducing kernel [11] of the domain D at merging points: χ (z) = log K(z, z¯ ), where K(z, ζ¯ ) is the reproducing kernel (4.13). 4.4. Residue formulas. In this section we present formulas for third order derivatives of F . Formulas of this type are known in the theory of dispersionless integrable hierarchies and are referred to as residue formulas [12]. They are used, in particular, to prove the associativity equations for tau-functions of Whitham hierarchies [21–23]. The basic relation to derive the residue formulas is (3.2). By virtue of (4.1), its left hand side is a generating function for third order derivatives of the tau-function. Let us first rewrite this formula in holomorphic terms. To do that we note that ∂n G(z, ξ ) = −2|∂ξ G(z, ξ )| for ξ on the boundary (the vector grad G is normal to the boundary). Further, since G vanishes on the boundary, ∂ξ G(z, ξ )dξ + ∂ξ¯ G(z, ξ )d ξ¯ = 0. Therefore, |∂ξ Gdξ | = −κi∂ξ Gdξ holds on the boundary. Taking all this into account, we rewrite (3.2) as 4 ∂z G(a, z)∂z G(b, z)∂z G(c, z) ∇(a)∇(b)∇(c)F = (dz)3 . πi γ dzd z¯ Finally, using G(a, z) = − log |fa (z)|, where fa (z) is a holomorphic function of z (see (3.6)), we get 1 d log fa (z) d log fb (z) d log fc (z) ∇(a)∇(b)∇(c)F = − . 2π i γ dzd z¯
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Let us expand both sides in the basis ψk in each argument using (2.2), (3.8). Comparing the coefficients, we come to the residue formulas for third order derivatives of the taufunction: ∂ 3F 1 dHl dHm dHn = − , l, m, n ≥ 0, ∂tl ∂tm ∂tn 2π i γ dzd z¯ (4.20) 1 dHl dHm dHn ∂ 3F = , l, m ≥ 0 , n ≥ 1. ∂tl ∂tm ∂ t¯n 2π i γ dzd z¯ The formulas (4.20) were first obtained by I. Krichever [9] by other means. 5. Integral Representations of the Tau-Function In this section we discuss integral representations of the tau-functions and the connection between the tau-functions for the interior and exterior Dirichlet problems. We employ the special notation Dint for the interior domain bounded by the curve γ and Dext for the exterior one. We use the natural basis, i.e., the normalization points are 0 ∈ Dint and ∞ ∈ Dext , and the basis functions are zk /k and z−k /k respectively. At last, the tau-functions are denoted as F int , F ext . 5.1. Electrostatic potentials. Let us begin with the exterior problem. Our starting point is Eq. (4.3) which we rewrite here introducing the auxiliary function (potential) 4ext (z) = D(z)F ext .
(5.1)
Equation (4.3) then reads D(z)4ext (ξ ) = 2G(z, ξ ) − 2 log z−1 − ξ −1 ¯ z).) This function for both z and ξ being in Dext . (We recall that D(z) = ∂t0 + D(z) + D(¯ admits an electrostatic interpretation explained below. When one of the points reaches the boundary, the Green function vanishes and one has D(ξ )4ext (z) = −2 log |z−1 − ξ −1 |,
ξ ∈ γ.
The last formula makes sense of the variation of the function 4ext under the bump deformation of the domain (see Sect. 2). According to (2.3), the variation is δ(ξ ) 4ext (z) =
2 log |z−1 − ξ −1 | π
and, therefore, one may write the integral representation 2 d 2 ζ log z−1 − ζ −1 4ext (z) = − π Dint
(5.2)
up to a harmonic function that does not depend on shape of the domain. Adding to (5.2) harmonic functions independent of times amounts to redefinition of the tau-function by terms linear in times, which contribute neither to the Green functions nor to the
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formulas for conformal maps (4.7), (4.9), and leave Hirota equations unchanged. We set all such terms to vanish. In a similar way, for the interior problem we get δ(ξ ) 4int (z) = 2 log |z − ξ | and π 2 4int (z) = d 2 ζ log |z − ζ | − c, π Dext where the constant c is necessary for regularization of the divergent integral over Dext . Let us cut off the integral at a big circle of radius R and set c = c(R) = 2 d 2 ζ log |ζ | = R 2 log R 2 − R 2 . Then we get π |ζ |
2 int 2 d ζ log |z − ζ | − c(R) 4 (z) = lim R→∞ π Dext 2 = d 2 ζ log |1 − zζ −1 | − v0 , (5.3) π Dext where we introduced v0 =
1 π
Dint
log |z|2 d 2 z.
Formulas (5.2), (5.3) present the potentials 4ext , 4int in the form of an integral over the complementary domain, since by definition (5.1) the functions 4ext , 4int are harmonic in Dext , Dint respectively. One may use these formulas to extend the functions 4 to the whole complex plane. For example, (5.2) defines a function 4ext which is harmonic in the exterior domain and satisfies the Poisson equation −∂z ∂z¯ 4ext (z) = 1 − π t0 δ(z) in Dint . This function is the potential generated by a uniformly distributed charge in Dint and a compensating point-like charge at the origin. The expansions of 4ext at small and large z read: 4ext (z) = −|z|2 + 2t0 log |z| + (tkext zk + t¯kext z¯ k ) , z → 0, k>0 (5.4) (tkint z−k + t¯kint z¯ −k ) , z → ∞, 4ext (z) = v0 + k>0
where the additional superscripts are set to distinguish between exterior and interior moments. Under our assumptions, these series converge in Dint and Dext respectively. The functions 4ext and ∂z 4ext are continuous at the boundary. 5.2. Integral formulas for the tau-function. Using the same strategy, we set z in (5.1) to the boundary and interpret this formula as a result of the bump deformation of the domain. It is easy to check that the variation of 1 1 1 1 F ext = d 2 z4ext (z) = − 2 d 2z d 2 ζ log − (5.5) 2π Dint π Dint z ζ Dint is δ(ξ ) F ext =
ext 1 4 (ξ ) + d 2 zδ(ξ ) 4ext (z) 2π 2π Dint ext = 4 (ξ ) − 2 d 2 z log |z−1 − ξ −1 | = 4ext (ξ ). 2π π Dint π
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Equation (5.5) presents the tau-function for the exterior Dirichlet problem as a double integral over the domain complement to the Dext [8]. Similar arguments give the tau-function of the interior problem:
1 F int = lim − 2 d 2z d 2 ζ log |z − ζ | + C(R) − c(R)t0 R→∞ π Dext Dext with c(R) as in (5.3), and 1 1 2 C(R) = 2 d z d 2 ζ log |z − ζ | = R 4 (2 log R 2 − 1). π |z|
(5.6)
In fact, the tau-functions admit other useful integral representations. Let us mention some of them. In terms of the potential 4ext , extended to the whole complex plane as explained above, the tau-function of the exterior problem may be represented as a (regularized) integral over the whole complex plane:
1 ∂z 4ext 2 d 2 z + t 2 log ε , F ext = lim (5.7) 0 ε→0 2π C\Dε where Dε is a small disk of radius ε centered at the origin. In the electrostatic interpretation, the tau-function is basically the energy of the system of charges mentioned above. Stokes formula gives an integral representation through a contour integral: 1 2 1 z¯ dz − zd z¯ ext F = − t0 + . (5.8) 4ext (z) 4 2π γ 4i Similar formulas can be written for the interior problem. 5.3. Legendre transform F int ↔ F ext . The tau-functions for the interior and exterior problems are connected by a Legendre transform. The former is the function of the interior moments tkint and the area t0 , while the latter is the function of the exterior moments tkext and the area. From (5.1) we read that first order derivatives of the taufunction for the Dirichlet problem in Dint or Dext with respect to the harmonic moments are (up to the factor k) harmonic moments of the complimentary domain: ∂F int = ktkext , ∂tkint
∂F ext = ktkint ∂tkext
and
∂F ext ∂F int =− = v0 . ∂t0 ∂t0 Using (5.6), we obtain the relation F int (t0 , tint , t¯int ) =
∞
k=1
k tkint tkext + k t¯kint t¯kext − F ext (t0 , text , t¯ext ).
(5.9)
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By virtue of (5.9), it means that the functions F int and F ext are related by Legendre transforms with respect to all variables but t0 : F int =
∞ k=1
F
ext
=
∞
tkext
k=1
tkint
ext ∂F ext ext ∂F ¯ − F ext , + t k ∂tkext ∂ t¯kext ∂F int ∂F int int − F int . + t¯k ∂tkint ∂ t¯kint
(5.10)
5.4. Homogeneity properties of the tau-function. Expansion of the potential 4ext (z) around the origin (5.4) allows one to prove another important property of the taufunctions. Integrating both sides of it over Dint and using (5.9), we obtain a quasihomogeneity condition for F ext . A similar condition for F int is most easily derived from (5.10). They are: 4F int = t02 + 2t0 ∂t0 F int +
k>0
4F
ext
=
−t02
+ 2t0 ∂t0 F
ext
+
(2 + k)(tk ∂tk F int + t¯k ∂t¯k F int ),
k>0
(2 − k)(tk ∂tk F ext + t¯k ∂t¯k F ext ).
(5.11)
These formulas reflect the scaling of moments as z → λz with real λ: tkext → λ2+k tkext , tkint → λ2−k tkint . The logarithmic moment v0 , under the same rescaling, exhibits a more complicated behaviour: v0 → λ2 v0 + t0 λ2 log λ2 . To get rid of the “anomaly term” t02 one may modify the tau-function by subtracting 41 t02 log t02 . 6. Dirichlet Problem on the Plane with a Gap Consider the case when the domain Dint shrinks to a segment of the real axis. Then the interior Dirichlet problem does not seem to make sense anymore but the exterior one is still well-posed: find a bounded harmonic function in the complex plane such that it equals a given function on the segment. The problem admits an explicit solution (see e.g. [24]). Possible variations of the data are a variation of the function on the segment and the endpoints of the segment. Solution of this problem as well as its integrable structure may be obtained from the formulas for a smooth domain as a result of a singular limit when a smooth domain shrinks to the segment. The tau-function, obtained in this way, is a partition function of the Hermitian onematrix model for the one-cut solution in the planar large N limit [25].
6.1. Shrinking the domain: The limiting procedure. Let us consider a family of curves γ (ε) obtained from a given curve γ by rescaling of the y-axis as y → εy. (If the curve γ is given by an equation P (x, y) = 0, then γ (ε) is given by P (x, y/ε) = 0.) We are interested in the limit ε → 0, γ (0) being a segment of the real axis. We denote the endpoints by α, β. Let =y(x) be the thickness of the domain bounded by the curve γ (ε) at the point x (see Fig. 2).
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∆ y(x) x
ρ (x) β
a
x
Fig. 2. A thin domain stretched along the real axis with thickness =y(x) shrinks into a segment with density ρ(x)
We introduce the function ρ(x) = lim
ε→0
=y(x) . ε
It is easy to see that in case of general position this function can be represented as (6.1) ρ(x) = (x − α)(β − x) M(x), where M(x) is a smooth function regular at the edges. So, instead of the space of contours γ we have the space of real positive functions ρ(x) of the form (6.1) with a finite support [α, β] (the endpoints of which are not fixed but may vary). Let us use the first equation in (5.4) to define times as coefficients of the expansion of the potential 4(x) = 4ext (x) generated by the thin domain with a uniform charge density (and with a point-like charge at the origin). In the leading order in ε the rescaled potential is
1 1 β 1 2 4(x) φ(x) ≡ lim =− − = T0 log x 2 + dx ρ(x ) log Tk x k . ε→0 ε π α x x k≥1
(6.2) Comparing this with (5.4), we get T0 = lim ε −1 t0 , ε→0
Tk = lim ε −1 (tkext + t¯kext ), ε→0
k ≥ 1, k = 2,
(6.3)
T2 = lim ε −1 (t2ext + t¯2ext − 1).
ε→0
β
1 ρ(x)dx but similar integral representations for other times, Tk = Note that T0 = π α β 2 ρ(x)x −k dx, which formally follow from (6.2), are ill-defined. On the other hand, πk α
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harmonic moments of the interior behave, in the scaling limit, as tkint = εk −1 µk +O(ε 2 ), where µk are well-defined moments of the function ρ on the segment: 1 β ρ(x)x k dx . (6.4) µk = π α Using the integral formula (5.5), it is now straightforward to find the scaling limit of the tau-function F ext for the exterior problem. Taking into account (6.3), we introduce the function F cut as follows: F ext (εt0 ; εt1 ,
1 1 + εt2 , εt3 , . . . ; ε t¯1 , + ε t¯2 , ε t¯3 , . . . ) 2 2 = ε2 F cut (T0 ; T1 , T2 , T3 , . . . ) + O(ε 3 ),
(6.5)
where T0 = t0 , Tk = tk + t¯k . Since the second order derivatives of F ext are invariant under the rescaling (6.5), the function F cut obeys the Hirota equations (4.14)–(4.16) (as F ext does) where one has to set ∂tk = ∂t¯k = ∂Tk . The latter means that F cut is a solution of the reduced dToda hierarchy (see e.g. [12]). This sort of reduction is usually referred to as the dispersionless Toda chain. We conjecture that other types of reduction correspond, in the same way, to shrinking of Dint to slit domains of a more complicated form. An integral representation for the F cut (obtained as a limit of (5.5)) reads: β β 1 ρ(x1 )ρ(x2 ) log |x1−1 − x2−1 |dx1 dx2 . F cut = − 2 π α α β 1 1 ρ(x)φ(x)dx = µk Tk , where µk are defined in Represent this as F cut = 2π α 2 k≥0
(6.4) and µ0 =
1 π
β α
ρ(x) log x 2 dx .
It is clear from the limit of (5.9) that µk = ∂Tk F cut for k ≥ 0. Therefore, we obtain the relation 2F cut = Tk ∂Tk F cut , (6.6) k≥0
which means that F cut is a homogeneous function of degree 2. This formula also means that the tau-function F cut is “self-dual” under the Legendre transform with respect to T0 , T1 , T2 , . . . . However, the analog of the Legendre transform (5.10) we discussed in Sect. 5.3 does not include T0 , so the analog of the function F int is the function E cut =
k≥1
∂F cut ∂T0 β β 1 =− 2 ρ(x1 )ρ(x2 ) log |x1 − x2 |dx1 dx2 , π α α
Tk µk − F cut = F cut − T0
which is the electrostatic energy of the segment with the charge density ρ on it, regarded as a function of the variables T0 , µ1 , µ2 , . . . . Properties of this function and its possible relation to the Hamburger 1D moment problem are to be further investigated.
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6.2. Conformal maps. The conformal map f : C \ [α, β] −→ E from the plane with the gap onto the exterior E of the unit disk is given by the explicit formula √ 2z − α − β + 2 (z − α)(z − β) . f (z) = β −α All formulas which connect conformal maps and the tau-function remain true in this case, 4z 2(α + β) too. In particular, expanding f (z) as z → ∞, f (z) = − + O(z−1 ), β −α β −α we read from (4.9) formulas for the endpoints of the cut:
1 β − α = 4 exp ∂T20 F cut , β + α = 2∂T20 T1 F cut . 2 The Green function is expressed through the f (z) by the same formula (1.2). Set w = f (z), then the inverse map is z(w) =
β −α β +α (w + w −1 ) + . 4 2
˜ Clearly, z(w) = z¯ (w−1 ), so the constraint on the Lax functions is now L(w) = L(w) which signifies the reduction to the dispersionless Toda chain.
6.3. Relation to matrix models. In analogy with variations of the data of the Dirichlet problem discussed in Sect. 3, we can consider the following problem: given a set of Tk , k ≥ 0, to find endpoints α, β of the segment and the density ρ(x) as functions of these parameters. This problem has appeared in studies of the planar N → ∞ limit of the Hermitian matrix model. In this case the function ρ(x) is a density of eigenvalues [25]. For completeness, we recall the basic points. Taking the derivative of (6.2), we get the equation which connects the set of Tk with α, β and ρ(x): β 2 ρ(x )dx v.p. = V (x) , π x − x α where V (x) = φ(x) − T0 log x 2 =
Tk x k .
(6.7)
k≥1
Consider the function W (z) =
1 π
α
β
ρ(x)dx . z−x
It is analytic in the complex plane with the cut [α, β]. As z → ∞, it behaves as W (z) = T0 z−1 + O(z−2 ). On the cut, its boundary values are W (x ± i0) = − 21 V (x) ∓ iρ(x). The function W (z) is uniquely defined by these analytic properties. So, to find ρ one should find the holomorphic function from its mean value on the cut. The result is given by the explicit formula √ 1 dζ V (ζ ) (z − α)(z − β) W (z) = , (6.8) √ 4π i ζ −z (ζ − α)(ζ − β)
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where the contour encircles the cut but not the point z. The endpoints of the cut are fixed by comparing the leading terms of (6.8) with the required asymptote of the W (z) as z → ∞. This leads to the hodograph-like formulas 1 V (z)dz zV (z)dz 1 = 0, = −2T0 , √ √ 2πi 2π i (z − α)(z − β) (z − α)(z − β) which implicitly determine α, β as functions of Tk . It follows from the above that F cut coincides with the one-cut free energy of the Hermitian one-matrix model with potential (6.7) in the planar large N limit. (As a matter of fact, it was shown in [26] that the partition function of the matrix model at finite N is the tau-function of the Toda chain hierarchy with dispersion.) Combining (6.6) with the ε → 0 limit of the second relation in (5.11), we obtain kTk ∂Tk F cut + T02 = 0. k≥1
This identity is known as the Virasoro L0 -constraint on the dispersionless tau-function F cut [26, 12]. Other Virasoro constraints can be obtained in the ε → 0 limit from the W1+∞ -constraints on the tau-function F ext . We do not discuss them here. 6.4. An example: Gaussian matrix model. If only the first three variables are nonzero (i.e., T0 , T1 , T2 = 0) the function F cut can be found explicitly. Consider a family of ellipses with axes l and εs centered at some point x0 on real axis: y2 1 (x − x0 )2 + = . l2 ε2 s 2 4 The harmonic moments, as ε → 0, are (see Appendix in ref. [3]): t0 = 41 εsl, t1ext = 2εx0 sl −1 + O(ε 2 ), t2ext = 21 − εsl −1 + O(ε2 ) and all other moments of the complement to the ellipse vanish. From (6.3) we read values of the rescaled variables: T0 =
1 (β − α)s , 4
T1 = 2
β +α s, β −α
T2 = −
2 s, β −α
and all the rest are zero. Here we have expressed times through the endpoints α = x0 − 21 l, β = x0 + 21 l of the segment and the extra parameter s. The density function is ρ(x) = −T2 (β − x)(x − α). (Note that T2 < 0.) Using the explicit form of the tau-function for an ellipse [3, 8], t1 t¯1 + t12 t¯2 + t¯12 t2 1 t0 3 F ext ellipse = t02 log − t02 + t0 , 2 1 − 4t2 t¯2 4 1 − 4t2 t¯2 and the scaling procedure (6.5), we find F cut (T0 , T1 , T2 , 0, 0, . . . ) =
T0 T12 3 1 2 T0 T0 log − T02 − . 2 −2T2 4 4T2
This is the expression for the free energy of the Gaussian matrix model in the planar large N limit [25]. The same result can be obtained from the integral formula for F cut .
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Acknowledgements. We acknowledge useful discussions with A. Boyarsky, L. Chekhov, B. Dubrovin, P. Di Francesco, M. Mineev-Weinstein, V. Kazakov, A. A. Kirillov, I. Kostov, A. Polyakov, O. Ruchayskiy, and especially with I. Krichever and L. Takhtajan. P.W. also acknowledges a discussion with L. Takhtajan who suggested to use the Hadamard formula to prove the relation between the tau-function and the Dirichlet Green function. The work of A.M. and A.Z. was partially supported by CRDF grant RP1-2102 and RFBR grant No. 00-02-16477, P.W. and A.Z. have been supported by grants NSF DMR 9971332 and MRSEC NSF DMR 9808595. A.M. was partially supported by INTAS grant 97-0103 and grant for support of scientific schools No. 00-15-96566. A.Z. was partially supported by grant INTAS-99-0590 and grant for support of scientific schools No. 00-15-96557.
References 1. Hurwitz, A. and Courant, R.: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie. Berlin– Heidelberg–New York: Springer-Verlag, 1964 (Russian translation, adapted by M.A. Evgrafov: Theory of functions, Moscow: Nauka, 1968) 2. Mineev-Weinstein, Wiegmann, M.P.B. and Zabrodin, A.: Phys. Rev. Lett. 84, 5106 (2000), e-print archive: nlin.SI/0001007 3. Wiegmann, P.B. and Zabrodin,A.: Commun. Math. Phys. 213, 523 (2000), e-print archive: hep-th/9909147 4. Hanany, A., Oz, Y. and Plesser, R.: Nucl. Phys. B425, 150–172 (1994); Takasaki, K.: Commun. Math. Phys. 170,101–116 (1995); Eguchi, T. and Kanno, H.: Phys. Lett. 331B, 330 (1994) 5. Daul, J.M., Kazakov V.A. and Kostov, I.K.: Nucl. Phys. B409, 311-338 (1993); Bonora, L. and Xiong, C.S.: Phys. Lett. B347, 41–48 (1995) 6. Gibbons, J. and Tsarev, S.P.: Phys. Lett. 211A, 19–24 (1996); ibid 258A, 263 (1999) 7. Hadamard, J.: Mém. présentés par divers savants à l’Acad. sci., 33 (1908) 8. Kostov, I.K., Krichever, I.M.,Mineev-Weinstein, M., Wiegmann, P.B. and Zabrodin, A.: τ -function for analytic curves. In: Random matrices and their applications, MSRI publications, Vol. 40, Cambridge: Academic Press, 2001, e-print archive: hep-th/0005259 9. Krichever, I.: Unpublished 10. Takhtajan, L.: Lett. Math. Phys. 56, 181-228 (2001). e-print archive: math.QA/0102164. 11. Hille, E.: Analytic function theory, V. II: Ginn and Company, 1962 12. Krichever, I.M.: Funct. Anal Appl. 22, 200–213 (1989); Commun. Pure. Appl. Math. 47, 437 (1992), e-print archive: hep-th/9205110 13. Takebe, T.: Adv. Series in Math. Phys. 16 (1992), Proceedings of RIMS Research Project 1991, pp 923– 940. 14. Shiota, T.: Invent. Math. 83, 333–382 (1986) 15. Douglas, M.: Phys. Lett. B238, 176 (1990) 16. Mineev-Weinstein, M. and Zabrodin, A.: Proceedings of the Workshop NEEDS 99 (Crete, Greece, June 1999), e-print archive: solv-int/9912012 17. Kharchev, S., Marshakov, A., Mironov, A. and Morozov, A.: Mod. Phys. Lett. A8, 1047–1061 (1993), e-print archive: hep-th/9208046 18. Sato, M.: Soliton Equations and Universal Grassmann Manifold Math. Lect. Notes Ser., Vol. 18, Sophia University, Tokyo (1984); E. Date, M. Jimbo, M.Kashiwara and T. Miwa, Transformation groups for soliton equations. In: Nonlinear Integrable Systems, eds. M. Jimbo and T. Miwa, Singapore: World Scientific, 1983 19. Gibbons, J. and Kodama, Y.: Proceedings of NATO ASI “Singular Limits of Dispersive Waves”, ed. N. Ercolani, London–New York: Plenum, 1994; Carroll, R. and Kodama, Y.: J. Phys. A: Math. Gen. A28, 6373 (1995) 20. Takasaki, K. and Takebe, T.: Rev. Math. Phys. 7, 743-808 (1995) 21. Dijkgraaf, R., Verlinde, E. and Verlinde, H.: Nucl. Phys. B352, 59 (1991) 22. Marshakov, A., Mironov, A. and Morozov, A.: Phys. Lett. B389, 43–52 (1996), e-print archive: hepth/9607109. 23. Boyarsky, A., Marshakov, A., Ruchayskiy, O., Wiegmann, P. and Zabrodin, A.: Phys. Lett. B515, 483–492 (2001) e-print archive: hep-th/0105260. 24. Gakhov, F.: Boundary problems, Moscow: Nauka, 1977 (in Russian); Bitsadze, A.: Foundations of the theory of analytic functions of a complex variable, Moscow: Nauka, 1984 (in Russian) 25. Brézin, E., Itzykson, C., Parisi, G. and Zuber, J.-B.: Commun. Math. Phys. 59, 35 (1978) 26. Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A. and Orlov, A.: Nucl. Phys. B357, 565–618 (1991) Communicated by L. Takhtajan
Commun. Math. Phys. 227, 155 – 190 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
The Canonical Solutions of the Q-Systems and the Kirillov–Reshetikhin Conjecture Atsuo Kuniba1 , Tomoki Nakanishi2 , Zengo Tsuboi3 1 Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan. E-mail:
[email protected] 2 Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan.
E-mail:
[email protected]
3 Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan.
E-mail:
[email protected] Received: 2 August 2001 / Accepted: 27 December 2001
Abstract: We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite Q-systems according to the number of functions and equations. The finite Qsystems appear as the thermal equilibrium conditions (the Sutherland–Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov–Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.
1. Introduction
In the series of works [K1, K2, KR], Kirillov and Reshetikhin studied the formal counting problem (the formal completeness) of the Bethe vectors of the XXX-type integrable spin chains, and they empirically reached a remarkable conjectural formula on the characters of a certain family of finite-dimensional modules of the Yangian Y (g). Let us formulate it in the following way. Conjecture 1.1. Let g be a complex simple Lie algebra of rank n. We set y = (ya )na=1 , (a) ya = e−αa for the simple roots αa of g. Let Qm (y) be the normalized g-character of (a) the KR module Wm (u) (a = 1, . . . , n; m = 1, 2, . . . ; u ∈ C) of the Yangian Y (g); and
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Qν (y) :=
(a) (a) νm . (a,m) (Qm (y))
Qν (y)
Then, the formula
(1 − e−α ) =
α∈+
Pm(a) (ν, N ) =
Pm(a) (ν, N ) + Nm(a)
(a) N=(Nm ) (a,m)
∞ k=1
(a)
νk min(k, m) −
(b,k)
(a)
(ya )mNm ,
(a) Nm
(b)
Nk da Aab min
m k , db da
(1.1)
(1.2)
holds. Here, A = (Aab ) is the Cartan matrix of g, da are coprime positive integers such that (da Aab ) is symmetric, + is the set of all the positive roots of g, and ab = Γ (a + 1)/Γ (a − b + 1)Γ (b + 1). Remark 1.2. Due to the Weyl character formula, the series in the RHS of (1.1) should be a polynomial of y, and its coefficients are identified with the multiplicities of the (a) (a) (a) (a) g-irreducible components of the tensor product (a,m) Wm (um )⊗νm , where um are arbitrary. Remark 1.3. There are actually two versions of Conjecture 1.1. The above one is the version in [HKOTY] which followed [K1, K2]. In the version in [KR], the binomial coefficients ab are set to be 0 if a < b; furthermore, the equality is claimed, not for the entire series in both sides of (1.1), but only for their coefficients of the powers y M “in the fundamental Weyl chamber”; namely, M = (Ma )na=1 satisfies (a,m)
(a) νm m a −
n
Ma αa ∈ P+ ,
(1.3)
a=1
where a are the fundamental weights and P+ is the set of the dominant integral weights of g. So far, it is not proved that the two conjectures are equivalent. Both conjectures are naturally translated into ones for the untwisted quantum affine algebras, which are extendable to the twisted quantum affine algebras [HKOTT]. In this paper, we refer to all these conjectures as the Kirillov–Reshetikhin conjecture. More comments and the current status of the conjecture will be given in Sect. 5.7. (a)
In [KR, K3], it was claimed that the Qm (y)’s satisfy a system of equations (a)
(a)
2 (Q(a) m (y)) = Qm−1 (y)Qm+1 (y) 2 + (ya )m (Q(a) m (y))
(b)
(Qk (y))Gam,bk .
(1.4)
(b,k)
(a)
Here, Q0 (y) = 1, and Gam,bk are the integers defined as −Aba (δm,2k−1 + 2δm,2k + δm,2k+1 ) db /da = 2, −A (δ db /da = 3, ba m,3k−2 + 2δm,3k−1 + 3δm,3k Gam,bk = +2δ + δ ) m,3k+1 m,3k+2 −A δ otherwise. ab da m,db k
(1.5)
See (4.22) for the original form of (1.4) in [KR, K3]. The relations (1.4) and (4.22) are often called the Q-system. The importance of the role of the Q-system to the formula
Canonical Solutions of the Q-Systems
157
(1.1) was recognized in [K1, K2, KR], and more explicitly exhibited in [HKOTY, KN2]. In this paper we proceed one step further in this direction; we study Eq. (1.4) in a more general point of view, and give a characterization of the special power series solution in (1.1). For this purpose, we introduce finite and infinite Q-systems, where the former (resp. the latter) is a finite (resp. infinite) system of equations for a finite (resp. infinite) family of power series of the variable with finite (resp. infinite) components. Equation (1.4), which is an infinite system of equations with the variable with finite components, is regarded as an infinite Q-system with the specialization of the variable (a specialized Qsystem). We show that every finite Q-system has a unique solution which has the same type of power series formula as (1.1) (Theorem 2.4). In contrast, infinite Q-systems and their specializations, in general, admit more than one solution. However, every infinite Q-system, or its specialization, has a unique canonical solution (Theorems 3.7 and 4.2), whose definition is given in Definition 3.5. The formula (1.1) turns out to be exactly the power series formula for the canonical solution of (1.4) (Theorem 4.3 and Proposition 4.9). Therefore, one can rephrase Conjecture 1.1 in a more intrinsic way as (a) follows (Conjecture 5.5): The family (Qm (y)) of the normalized g-characters of the KR modules is characterized as the canonical solution of (1.4). This is the main statement of the paper. Interestingly, the finite Q-systems also appear in other types of integrable statistical mechanical systems. Namely, they appear as the thermal equilibrium condition (the Sutherland-Wu equation) for the Calogero-Sutherland model [S], as well as the one for the ideal gas of the Haldane exclusion statistics [W]. The property of the solution of the finite Q-systems are studied in [A,AI, IA] from the point of view of the quasihypergeometric functions. We expect that the study of the Q-system and its variations and extensions will be useful for the representation theory of the quantum groups, and for the understanding of the nature of the integrable models as well. 2. Finite Q-Systems A considerable part of the results in this section can be found in the work by Aomoto and Iguchi [A, IA]. We present here a more direct approach. More detailed remarks will be given in Sect. 2.4. 2.1. Finite Q-systems. Throughout Sect. 2, let H denote a finite index set. Let w = (wi )i∈H and v = (vi )i∈H be complex multivariables, and let G = (Gij )i,j ∈H be a given complex square matrix of size |H |. We consider a holomorphic map D → CH , v → w(v) with wi (v) = vi (1 − vj )−Gij , (2.1) j ∈H
where D is some neighborhood of v = 0 in CH . The Jacobian (∂w/∂v)(v) is 1 at v = 0, so that the map w(v) is bijective around v = w = 0. Let v(w) be the inverse map around v = w = 0. Inverting (2.1), we obtain the following functional equation for vi (w)’s: vi (w) = wi (1 − vj (w))Gij . (2.2) j ∈H
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By introducing new functions Qi (w) = 1 − vi (w),
(2.3)
Eq. (2.2) is written as Qi (w) + wi
(Qj (w))Gij = 1.
(2.4)
j ∈H
From now on, we regard (2.4) as a system of equations for a family (Qi (w))i∈H of power series of w = (wi )i∈H with the unit constant terms (i.e., the constant terms are 1). Here, for any power series f (w) with the unit constant term and any complex number α, we mean by (f (w))α ∈ C[[w]] the α th power of f (w) with the unit constant term. We can easily reverse the procedure from (2.1) to (2.4), and we have Proposition 2.1. The power series expansion of Qi (w) in (2.3) gives the unique family (Qi (w))i∈H of power series of w with the unit constant terms which satisfies (2.4). Definition 2.2. The following system of equations for a family (Qi (w))i∈H of power series of w with the unit constant terms is called a (finite) Q-system: For each i ∈ H ,
(Qj (w))Dij + wi
j ∈H
(Qj (w))Gij = 1,
(2.5)
j ∈H
where D = (Dij )i,j ∈H and G = (Gij )i,j ∈H are arbitrary complex matrices with det D = 0. Equation (2.4), which is the special case of (2.5) with D = I (I : the identity matrix), is called a standard Q-system. It is easy to see that there is a one-to-one correspondence between the solutions of the Q-system (2.5) and the solutions of the standard Q-system Qi (w) + wi
j ∈H
(Qj (w))Gij = 1,
G = GD −1 ,
(2.6)
where the correspondence is given by Qi (w) = Qi (w) =
(Qj (w))Dij ,
(2.7)
j ∈H
j ∈H
(Qj (w))(D
−1 ) ij
.
(2.8)
Therefore, from Proposition 2.1, we immediately have Theorem 2.3. There exists a unique solution of the Q-system (2.5), which is given by (2.8), where (Qi (w))i∈H is the unique solution of the standard Q-system (2.6).
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2.2. Power series formulae. In what follows, we use the binomial coefficient in the following sense: For a ∈ C and b ∈ Z≥0 , a Γ (a + 1) , (2.9) = Γ (a − b + 1)Γ (b + 1) b where the RHS means the limit value for the singularities. We set N := (Z≥0 )H . For D, G in (2.5) and ν = (νi )i∈H ∈ CH , we define two power series of w, N ν KD,G (w) = K(D, G; ν, N )wN , wN = wi i , (2.10) N ∈N
ν (w) RD,G
=
i∈H
R(D, G; ν, N )w
N
(2.11)
N ∈N
with the coefficients
Pi + Ni , Ni
K(D, G; ν, N ) =
i∈H (N)
R(D, G; ν, N ) =
det Fij
H (N)
i∈H (N)
(2.12)
1 Pi + N i − 1 , Ni − 1 Ni
where we set H (N) = { i ∈ H | Ni = 0 } for each N ∈ N , Pi = Pi (D, G; ν, N ) := − νj (D −1 )j i − Nj (GD −1 )j i , j ∈H
(2.13)
(2.14)
j ∈H
Fij = Fij (D, G; ν, N ) := δij Pj + (GD −1 )ij Nj ,
(2.15)
and notation for det i,j ∈H (N) . In (2.12) and (2.13), det ∅ and detH (N) is a shorthand ν ν ∅ mean 1; namely, KD,G (w) and RD,G (w) are power series with the unit constant γ terms. It is easy to check that both series converge for |wi | < |γi i /(γi + 1)γi +1 |, where −1 z γi = −(GD )ii and z = exp(z log z) with the principal branch −π < Im(log z) ≤ π chosen. Now we state our main results in this section. Theorem 2.4 (Power series formulae). Let (Qi (w))i∈H be the unique solution of (2.5). For ν ∈ CH , let QνD,G (w) := i∈H (Qi (w))νi . Then, ν 0 (w)/KD,G (w), QνD,G (w) = KD,G
QνD,G (w)
=
ν RD,G (w).
(2.16) (2.17)
The power series formulae for Qi (w) are obtained as special cases of (2.16) and (2.17) by setting ν = (νj )j ∈H as νj = δij . One may recognize that the first formula (2.16) is analogous to the formula (1.1), 0 (w) in (2.16) corresponds to the Weyl denominator in the where the denominator KD,G LHS of (1.1). As mentioned in Sect. 1, the formula (1.1) is interpreted as the formal completeness of the XXX-type Bethe vectors. In the same sense, the second formula (2.17) is analogous to the formal completeness of the XXZ-type Bethe vectors in [KN1, KN2]. See Sect. 2.4 for more remarks.
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Example 2.5. Let |H | = 1. Then, (2.5) is an equation for a single power series Q(w), (Q(w))D + w(Q(w))G = 1,
(2.18)
where D = 0 and G are complex numbers, and the series (2.11) reads as ν (w) = RD,G
∞ ν Γ ((ν + N G)/D)(−w)N . D Γ ((ν + N G)/D − N + 1)N!
(2.19)
N=0
Equation (2.18) and the power series formula (2.19) are well known and have a very long history since Lambert (e.g. [B, pp. 306–307]). Example 2.6. Consider the case G = O in (2.5), (Qj (w))Dij + wi = 1.
(2.20)
j ∈H
This is easily solved as
Qi (w) =
(1 − wj )(D
−1 ) ij
,
(2.21)
j ∈H
and, therefore, QνD,O (w) =
(1 − wi )
j ∈H
νj (D −1 )j i
=
i∈H
(1 − wi )−Pi (D,O;ν,N) ,
(2.22)
i∈H
where N ∈ N is arbitrary. Using the binomial theorem ∞ β +N N −β−1 x , = (1 − x) N
(2.23)
N=0
one can directly check that QνD,O (w)
=
Pi − 1 + Ni N ν wi i = RD,O (w), Ni
N∈N i∈H (N)
QνD,O (w)
=
j ∈H (1 − wi )
j ∈H
νj (D −1 )j i −1
j ∈H (1 − wi )
=
−1
ν (w) KD,O 0 (w) KD,O
.
(2.24)
(2.25)
2.3. Proof of Theorem 2.4 and basic formulae. Theorem 2.4 is regarded as a particularly nice example of the multivariable Lagrange inversion formula (e.g. [G]) where all the explicit calculations can be carried through. Here, we present the most direct calculation based on the multivariable residue formula (the Jacobi formula in [G, Theorem 3]). We first remark that Lemma 2.7. Let G = GD −1 . For each ν ∈ CH , let ν ∈ CH with νi = j ∈H νj (D −1 )j i . Then,
QνD,G (w) = QνI,G (w), ν KD,G (w)
=
ν KI,G (w),
(2.26) ν RD,G (w)
=
ν RI,G (w).
(2.27)
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Proof. The equality (2.26) is due to Theorem 2.3. The ones (2.27) follow from the fact Pi (D, G; ν, N ) = Pi (I, G ; ν , N ). By Lemma 2.7, we have only to prove Theorem 2.4 for the standard case D = I . Recall that (Proposition 2.1) QνI,G (w) = i∈H (1 − vi (w))νi , where v = v(w) is the inverse map of (2.1). Thus, Theorem 2.4 follows from Proposition 2.8 (Basic formulae). Let v = v(w) be the inverse map of (2.1). Then, the power series expansions det H
w ∂v j i ν (w) (1 − vi (w))νi −1 = KI,G (w), vi ∂wj i∈H ν (1 − vi (w))νi = RI,G (w)
(2.28) (2.29)
i∈H
hold around w = 0. Proof. The first formula (2.28). We evaluate the coefficient for w N in the LHS of (2.28) as follows: ∂v (1 − vi (w))νi −1 (vi (w))−1 (wi )1−Ni −1 dw (w) w=0 ∂w i∈H
−Ni = Res (1 − vj )−Gij (1 − vi )νi −1 (vi )−1 vi dv Res
v=0
= Res v=0
i∈H
j ∈H
(1 − vi )−Pi (I,G;ν,N)−1 (vi )−Ni −1 dv
i∈H
Pi (I, G; ν, N ) + Ni
=
i∈H
Ni
= K(I, G; ν, N ),
where we used (2.23) to get the last line. Thus, (2.28) is proved. The second formula (2.29). By a simple calculation, we have det H
v ∂w
j i (v) (1 − vi ) = det δij + (−δij + Gij )vi H wi ∂vj i∈H = dJ vi , J ⊂H
(2.30)
i∈J
where dJ := detJ (−δij + Gij ), and the sum is taken over all the subsets J of H . Therefore, the LHS of (2.29) is written as (θ(true) = 1 and θ(false) = 0) det H
w ∂v j i (1 − vi (w))νi −1 vi (w)θ(i∈J ) . (w) dJ vi ∂wj J ⊂H
i∈H
(2.31)
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By a similar residue calculation as above, the coefficient for w N of (2.31) is evaluated as (1 − vi )−Pi (I,G;ν,N)−1 (vi )−Ni +θ(i∈J )−1 dv dJ Res J ⊂H
v=0
i∈H
Pi (I, G; ν, N ) + Ni − θ(i ∈ J ) = dJ Ni − θ(i ∈ J ) J ⊂H (N) i∈H (N) 1 Pi + N i − 1 = dJ Ni (Pi + Ni ) Ni Ni − 1 i∈J J ⊂H (N) i∈H (N)\J i∈H (N)
1 P + N − 1 i i = det δij (Pj + Nj ) + (−δij + Gij )Nj H (N) Ni Ni − 1
i∈H (N)
= R(I, G; ν, N). Thus, (2.29) is proved.
This completes the proof of Theorem 2.4. Example 2.9. We say that the map w(v) in (2.1) is lower-triangular if the matrix Gij is strictly lower-triangular with respect to a certain total order ≺ in H (i.e., Gij = 0 for i j ). Let w(v) be a lower-triangular map. Then,
v ∂w
Gij vj j i (v) = det δij + = 1. (2.32) det H wi ∂vj H 1 − vj Thus, the formula (2.28) is simplified as ν (1 − vi (w))νi −1 = KI,G (w).
(2.33)
i∈H
This type of formulae has appeared in [K1, K2, HKOTY]. Let us isolate the case ν = 0 from (2.28), together with the formula (2.30), for later use: Corollary 2.10 (Denominator formulae).
w ∂v j i 0 (w) = det (w) (1 − vi (w))−1 , KI,G H vi ∂wj i∈H −1 0 (w) = det δij (1 − vi (w)) + Gij vi (w) . KI,G H
(2.34) (2.35)
From (2.35) and the first formula of Theorem 2.4, we obtain Corollary 2.11. QνI,G (w) = gJ :=
J ⊂H
ν+δJ gJ KI,G (w),
J ⊂H |J |=|J |
JJ sgn JJ
(2.36)
det
i∈J,j ∈J
δij − Gij
det
i∈J ,j ∈J
Gij ,
where δJ = (θi )i∈H , θi = 1 if i ∈ J and 0 otherwise, and J = H \ J .
(2.37)
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163
From Corollary 2.11, one can easily reproduce the second formula of Theorem 2.4. We leave it as an exercise for the reader.
2.4. Remarks on related works. i) The formal completeness of the Bethe vectors. In [K1, K2, HKOTY, KN1, KN2, KNT], the formal completeness of the XXX/XXZ-type Bethe vectors are studied. In the course of their analysis, several power series formulae in this section appeared in specialized/implicit forms. For example, Lemma 1 in [K1] is a special case of (2.33), Theorem 4.7 in [KN2] is a special case of Proposition 2.8, etc. From the current point of view, however, the relation between these power series formulae and the underlying finite Q-systems was not clearly recognized therein. As a result, these power series formulae and the infinite Q-systems were somewhat abruptly combined in the limiting procedure to obtain the power series formula for the infinite Qsystems. We are going to straighten out this logical entanglement, and make the logical structure more transparent by Theorem 2.4 and the forthcoming Theorems 3.10, 4.3, Proposition 4.9, and Conjecture 5.5. ii) The ideal gas with Haldane statistics and the Sutherland–Wu equation. The series ν (w) has an interpretation of the grand partition function of the ideal gas with the KD,G Haldane exclusion statistics [W]. The finite Q-system appeared in [W] as the thermal equilibrium condition for the distribution functions of the same system. See also [IA] for another interpretation. The one variable case (2.18) also appeared in [S] as the thermal equilibrium condition for the distribution function of the Calogero–Sutherland model. As an application of our second formula in Theorem 2.4, we can quickly reproduce the “cluster expansion formula” in [I, Eq. (129)], which was originally calculated by the Lagrange inversion formula, as follows: log Qi (w) = =
∂ ν RI,G (w) ν=0 ∂νi
det Fj k (I, G; 0, N )
H (N) N∈N j,k =i
j ∈H (N)
1 Pj (I, G; 0, N ) + Nj − 1 N w , Nj Nj − 1
(2.38)
where {Qi (w)}i∈H is the solution of (2.4). The Sutherland-Wu equation also plays an important role for the conformal field theory spectra. (See [BS] and the references therein.) ν (w) is a special example of iii) Quasi-hypergeometric functions. The series KD,G the quasi-hypergeometric functions by Aomoto and Iguchi [AI]; when Gij are all integers, it reduces to a general hypergeometric function of Barnes–Mellin type. A quasihypergeometric function satisfies a system of fractional differential equations and a system of difference-differential equations [AI]. It also admits an integral representation ν (w) reduces to a simple form ([A, [A]. In particular, the integral representation for KI,G Eq. (2.30)], [IA, Eq. (89)]); in our notation, 1 = tiνi −1 fi (w, t)−1 dt, √ (2π −1)|H | i∈H Gij fi (w, t) := ti − 1 + wi tj ,
ν KI,G (w)
j ∈H
(2.39) (2.40)
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where the integration is along a circle around ti = 1 starting from ti = 0 for each ti . We see that fi (w, t) = 0 is the standard Q-system (2.4). The integral (2.39) is easily evaluated by the Cauchy theorem as [A, Eq. (2.32)] ν KI,G (w) = QνI,G (w)/ det(δij Qi (w) + Gij (1 − Qj (w))), H
(2.41)
where {Qi (w)}i∈H is the solution of (2.4). The formula (2.41) reproduces a version of the Lagrange inversion formula (the Good formula [G, Theorem 2]), and it is equivalent to the formulae (2.16), (2.30), and (2.34). 3. Infinite Q-Systems 3.1. Infinite Q-systems. Throughout Sect. 3, let H be a countable index set. We fix an increasing sequence of finite subsets of H , H1 ⊂ H2 ⊂ · · · ⊂ H such that lim HL = H . − → The result below does not depend on the choice of the sequence {HL }∞ L=1 . A natural choice is H = N and HL = { 1, . . . , L }. However, we introduce this generality to accommodate the situation we encounter in Sect. 4 (cf. (4.1)). Let w = (wi )i∈H be a multivariable with infinitely many components. For each L ∈ N, let wL = (wi )i∈HL be the submultivariable of w. The field C[[wL ]] of the power series of wL over C is equipped with the standard XL -adic topology, where XL is the ideal of C[[wL ]] generated by wi ’s (i ∈ HL ). For L < L , there is a natural projection pLL : C[[wL ]] → C[[wL ]] such that pLL (wi ) = wi if i ∈ HL and 0 if i ∈ HL \ HL . A power series f (w) of w is an element of the projective limit C[[w]] = lim C[[wL ]] ← − of the projective system C[[w1 ]] ← C[[w2 ]] ← C[[w3 ]] ← · · ·
(3.1)
with the induced topology. Let pL be the canonical projection pL : C[[w]] → C[[wL ]], and fL (wL ) be the Lth projection image of f (w) ∈ C[[w]]; namely, fL (wL ) = pL (f (w)) and f (w) = (fL (wL ))∞ L=1 . Here are some basic properties of power series which we use below: (i) We also present a power series f (w) as a formal sum f (w) = aN w N , aN ∈ C, (3.2) N∈N
N = { N = (Ni )i∈H | Ni ∈ Z≥0 , all but finitely many Ni are zero },
(3.3)
(the definition of N is reset here for the infinite index set H ) whose Lth projection image is fL (wL ) = aN w N , (3.4) N∈NL
NL = { N ∈ N | Ni = 0 for i ∈ / HL }.
(3.5)
(ii) For any power series f (w) with the unit constant term and any complex number α, the α th power (f (w))α := ((fL (wL ))α )∞ L=1 ∈ C[[w]] is uniquely defined and has the unit constant term again. (iii) Let fi (w) (i ∈ H ) be a family of power series and fi,L (wL ) be their Lth projections. If their infinite product exists irrespective ofthe order of the product, we write it as i∈H fi (w). i∈H fi (w) exists if and only if i∈H fi,L (wL ) exists for each L; furthermore, if they exist, the latter is the Lth projection of the former.
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Definition 3.1. The following system of equations for a family (Qi (w))i∈H of power series of w with the unit constant terms is called an (infinite) Q-system: For each i ∈ H, (Qj (w))Dij + wi (Qj (w))Gij = 1. (3.6) j ∈H
j ∈H
Here, D = (Dij )i,j ∈H and G = (Gij )i,j ∈H are arbitrary infinite-size complex matrices satisfying the following two conditions: (D) The matrix D is invertible, i.e., there exists a matrix D −1 such that DD −1 = D −1 D = I . (G’) The matrix product G = GD −1 is well-defined. When D = I , Eq. (3.6) is called a standard Q-system. Remark 3.2. The condition (G’) is rephrased as “for each i and k, all but finitely many Gij (D −1 )j k (j ∈ H ) are zero”. Similarly, the condition (D) implies that, for each i and k, all but finitely many Dij (D −1 )j k , (D −1 )ij Dj k (j ∈ H ) are zero. For the standard case, (D) is trivially satisfied, and (G’) is satisfied for any complex matrix G. Unlike the finite Q-systems, the uniqueness of the solution does not hold for the infinite Q-systems, in general. For instance, the following example admits infinitely many solutions. Example 3.3. Let H = Z, and consider a Q-system, Qi−1 (w)Qi+1 (w) + wi = 1, (Qi (w))2
(3.7)
where Q0 (w) = 1. This can be easily solved as Qi (w) = (Q1 (w))
i
i−1
(1 − wj )i−j ,
(3.8)
j =1
where Q1 (w) is an arbitrary series of w with the unit constant term. 3.2. Canonical solution. 3.2.1. Solution of standard Q-system. First, we consider the standard case Qi (w) + wi (Qj (w))Gij = 1.
(3.9)
j ∈H
Let Qi,L (wL ) := pL (Qi (w)) be the Lth projection image of Qi (w). Then, (3.9) is equivalent to a series of equations (L = 1, 2, . . . ), Qi,L (wL ) + pL (wi ) (Qj,L (wL ))Gij = 1, (3.10) j ∈H
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which are further equivalent to
Qi,L (wL ) + wi
Qi,L (wL ) = 1
i∈ / HL ,
(3.11)
(Qj,L (wL ))Gij = 1
i ∈ HL .
(3.12)
j ∈HL
Namely, a standard infinite Q-system is an infinite family of standard finite Q-systems which is compatible with the projections (3.1). By Proposition 2.1, (3.12) uniquely determines Qi,L (wL ) for i ∈ HL . Furthermore, so determined (Qi,L (wL ))∞ L=1 belongs to C[[w]], again because of the uniqueness of the solution of (3.12). Therefore, Proposition 3.4. There exists a unique solution (Qi (w))i∈H of the standard Q-system (3.9), whose Lth projections Qi,L (wL ) := pL (Qi (w)) are determined by (3.11) and (3.12). 3.2.2. Canonical solution. As we have seen in Example 3.3, the uniqueness property does not hold for a general infinite Q-system (3.6). This is because, unlike the standard case, the Lth projection of (3.6) is not necessarily a finite Q-system. The non-uniqueness property also implies that, unlike the finite case, (3.6) does not always reduce to the standard one, Qi (w) + wi
j ∈H
(Qj (w))Gij = 1,
G = GD −1 .
(3.13)
In fact, the relations (2.7) and (2.8) are no longer equivalent due to the infinite products therein. However, the construction of a solution of a general Q-system from a standard one in Theorem 2.3 still works. We call the so obtained solution as canonical solution. Let us give a more intrinsic definition, however. Definition 3.5. We say that a solution (Qi (w))i∈H of the Q-system (3.6) is canonical if it satisfies the following condition: (Inversion property): For any i ∈ H ,
(Qk (w))(D
−1 ) D ij j k
= Qi (w).
(3.14)
j ∈H k∈H
Remark 3.6. The condition (3.14) is not trivial, because, in general, one cannot freely exchange the order of the infinite double product therein. Theorem 3.7. There exists a unique canonical solution of the Q-system (3.6), which is given by Qi (w) =
j ∈H
(Qj (w))(D
−1 ) ij
,
where (Qi (w))i∈H is the unique solution of the standard Q-system (3.13).
(3.15)
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167
Proof. First, we remark that the infinite product (3.15) exists, because its Lth projection image reduces to the finite product −1 (Qj,L (wL ))(D )ij (3.16) Qi,L (wL ) = j ∈HL
due to (3.11). Let us show that the family (Qi (w))i∈H in (3.15) is a solution of the Q-system (3.6). With the substitution of (3.16), the Lth projection image of the first term in the LHS of (3.6) is −1 (Qj,L (wL ))Dij = (Qk,L (wL ))Dij (D )j k j ∈H
j ∈H k∈HL
Qi,L (wL ) i ∈ HL = 1 = Qi,L (wL ) i ∈ / HL .
(3.17)
In the second equality above, we exchanged the order of the products. It is allowed because the double product is a finite one (cf. Remark 3.2). The second term in the LHS of (3.6) can be calculated in a similar way as follows: −1 (Qj,L (wL ))Gij = (Qk,L (wL ))Gij (D )j k j ∈H
j ∈H k∈HL
=
k∈HL
(Qk,L (wL ))Gik .
(3.18)
From (3.17) and (3.18), we conclude that (3.6) reduces to (3.13). Furthermore, by (3.17), we have (Qj (w))Dij = Qi (w). (3.19) j ∈H
Then, substituting (3.19) in (3.15), we obtain (3.14). Therefore, (Qi (w))i∈H is a canonical solution of (3.6). Next, we show the uniqueness. Suppose that (Qi (w))i∈H is a canonical solution of (3.6). We define Qi (w) as Qi (w) =
(Qj (w))Dij .
(3.20)
j ∈H
Then, by the inversion property (3.14), we have −1 Qi (w) = (Qj (w))(D )ij .
(3.21)
j ∈H
Also, by (3.6), Qi,L (wL ) = 1,
i∈ / HL .
(3.22)
With (3.21) and (3.22), the same calculation as (3.18) shows that (Qi (w))i∈H is the (unique) solution of (3.13). Therefore, by (3.21), Qi (w) is unique.
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Example 3.8. Let us find the canonical solution of the Q-system (3.7) in Example 3.3. We have Dij = −2δij + δi,j −1 + δi,j +1 ,
(D −1 )ij = − min(i, j ).
(3.23)
Let HL = {1, . . . , L}. By (3.20) and (3.22), the Lth projection of the LHS of (3.14) equals +1 L j
(Qk,L (wL ))(D
−1 ) D ij j k
j =1 k=j −1
=
L k+1
(Qk,L (wL ))(D
−1 ) D ij j k
(3.24)
k=1 j =k−1
× (QL+1,L (wL ))(D =
L
−1 ) D iL L,L+1
(QL,L (wL ))−(D
−1 ) i,L+1 DL+1,L
(Qk,L (wL ))δik (QL+1,L (wL ))− min(i,L) (QL,L (wL ))min(i,L+1) .
k=1
Therefore, condition (3.14) reads Qi,L (wL )(QL,L (wL )/QL+1,L (wL ))i Qi,L (wL ) = QL,L (wL )(QL,L (wL )/QL+1,L (wL ))L
i≤L i ≥ L + 1.
(3.25)
This is equivalent to Qi,L (wL ) = QL,L (wL ),
i ≥ L + 1.
(3.26)
Using (3.8) and (3.26), one can easily obtain Q1 (w) =
∞
(1 − wj )−1 .
(3.27)
j =1
Therefore, the canonical solution of (3.7) is given by Qi (w) =
∞
(1 − wj )− min(i,j ) .
(3.28)
j =1
3.3. Power series formula. Let (Qi (w))i∈H be the canonical solution of (3.6), and (Qi (w))i∈H be the unique solution of the standard Q-system (3.13). For the matrix D in (3.6), let ν(D) be the set of all ν = (νi )i∈H such that νi ∈ C and, for each i, the sum j ∈H νj (D −1 )j i exists (i.e., all but finitely many νj (D −1 )j i (j ∈ H ) are zero). For each ν ∈ ν(D), we define −1 QνD,G (w) := (3.29) (Qi (w))νi = (Qj (w))νi (D )ij . i∈H
i∈H j ∈H
Canonical Solutions of the Q-Systems
169
The last infinite product exists, because its Lth projection image reduces to a finite product due to (3.11) and the definition of ν(D). For each ν ∈ ν(D), let ν = (νi ) ∈ ν(I ), νi = j ∈H νj (D −1 )j i . Then, by (3.29), we have
QνD,G (w) = QνI,G (w),
G = GD −1 .
(3.30)
It follows from (3.11) and (3.30) that Lemma 3.9. ν
pL (QνD,G (w)) = QI L,G (wL ), L
(3.31)
L
where the RHS is for the solution of the finite Q-system with the finite index set HL , and IL = (δij )i,j ∈HL , GL = (Gij )i,j ∈HL , νL = (νi )i∈HL are the HL -truncations of I , G , ν , respectively. ν (w) and R ν For D, G in (3.6) and ν ∈ ν(D), we define the power series KD,G D,G (w) by the superficially identical formulae (2.10)–(2.15) with D, G, ν, N , etc., therein being replaced by the ones for the infinite index set H .
Theorem 3.10 (Power series formulae). For the canonical solution (Qi (w))i∈H of (3.6) and ν ∈ ν(D), let QνD,G (w) be the series in (3.29). Then, ν 0 ν QνD,G (w) = KD,G (w)/KD,G (w) = RD,G (w).
(3.32)
Proof. By Theorem 2.4 and Lemma 3.9, it is enough to show that ν
ν (w)) = KI L,G (wL ), pL (KD,G L
L
ν
ν pL (RD,G (w)) = RI L,G (wL ). L
L
(3.33)
By (3.2)–(3.5), (3.33) further reduces to the following equality: Pi (D, G; ν, N ) = Pi (IL , GL ; νL , NL ), where NL = (Ni )i∈HL is the HL -truncation of N .
N ∈ NL , i ∈ HL ,
(3.34)
4. Q-Systems of KR Type In this section, we introduce a class of infinite Q-systems which we call the Q-systems of KR type. This is a preliminary step towards the reformulation of Conjecture 1.1. 4.1. Specialized Q-systems. Throughout the section, we take the countable index set as H = {1, . . . , n} × N
(4.1)
for a given natural number n. We choose the increasing sequence H1 ⊂ H2 ⊂ · · · ⊂ H with lim HL = H as HL = {1, . . . , n} × {1, . . . , L}. Let y = (ya )na=1 be a multivariable − → with n components.
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Definition 4.1. The following system of equations for a family (Qm (y))(a,m)∈H of power series of y with the unit constant terms is called a specialized (infinite) Q-system: For each (a, m) ∈ H ,
(b)
(b,k)∈H
(Qk (y))Dam,bk + (ya )m
(b)
(b,k)∈H
(Qk (y))Gam,bk = 1,
(4.2)
where the infinite-size complex matrices D = (Dam,bk )(a,m),(b,k)∈H and G = (Gam,bk )(a,m),(b,k)∈H satisfy the same conditions (D) and (G’) as in Definition 3.1. A solution of (4.2) is called canonical if it satisfies the condition
(c)
(b,k)∈H (c,j )∈H
(Qj (y))(D
−1 ) am,bk Dbk,cj
= Q(a) m (y).
(4.3)
Let C[[y]] be the field of power series of y with the standard topology, JL be the ideal of C[[y]] generated by (ya )L+1 ’s (a = 1, . . . , n), and C[[y]]L be the quotient C[[y]]/JL . We can identify C[[y]] with the projective limit of the projective system, C[[y]]1 ← C[[y]]2 ← C[[y]]3 ← · · · .
(4.4)
(a)
Let w = (wm )(a,m)∈H be a multivariable, and let w(y) be the map with (a) wm (y) = (ya )m .
(4.5)
The map (4.5) induces the maps ψL and ψ such that C[[wL ]] ← C[[w]] ψL ↓ ψ↓ C[[y]]L ← C[[y]].
(4.6)
We call the image ψ(f (w)) ∈ C[[y]] the specialization of f (w), and write it as f (w(y)). Explicitly, for f (w) in (3.2), f (w(y)) =
∞ M1 ,...,Mn =0
aN
n
(ya )Ma .
(4.7)
a=1
N∈N ∞ (a) m=1 mNm =Ma
Theorem 4.2. There exists a unique canonical solution of the specialized Q-system (4.2), (a) (a) which is given by the specialization Qm (y) = Qm (w(y)) of the canonical solution (a) (Qm (w))(a,m)∈H of the following Q-system: (b,k)∈H
(b)
(a) (Qk (w))Dam,bk + wm
(b,k)∈H
(b)
(Qk (w))Gam,bk = 1.
(4.8)
Proof. Since the map ψ is continuous, it preserves the infinite product. Therefore, the specialization of the canonical solution of (4.8) gives a canonical solution of (4.2). Let us show the uniqueness. By repeating the same proof for Theorem 3.7, the uniqueness
Canonical Solutions of the Q-Systems
171
is reduced to the one for the standard case D = I . Let us write (4.2) for D = I as (L = 1, 2, . . . ) m Q(a) m (y) + (ya )
Q(a) m (y) ≡1
(b,k)∈HL
(b) (Qk (y))Gam,bk
≡1
mod JL
(a, m) ∈ / HL ,
(4.9)
mod JL
(a, m) ∈ HL .
(4.10)
(a)
(a)
These equations uniquely determine Qm (y) mod JL . Since L is arbitrary, Qm (y) is unique. By the specialization of Theorem 3.10, we immediately obtain (a)
Theorem 4.3 (Power series formulae). Let (Qm (y))(a,m)∈H be the canonical solution (a) (a) of the Q-system (4.2). Let QνD,G (y) = (a,m)∈H (Qm (w))νm , ν ∈ ν(D). Then, ν 0 QνD,G (y) = KD,G (y)/KD,G (y) = RνD,G (y), ν KD,G (y)
ν (w(y)) KD,G
where the series = and cializations of the series in Theorem 3.10.
RνD,G (y)
=
(4.11)
ν RD,G (w(y))
are the spe-
4.2. Convergence property. Let us consider the special case of the specialized Q-system (4.2) where the matrix D and its inverse D −1 are given by Dam,bk = −δab (2δmk − δm,k+1 − δm,k−1 ), (D
−1
(4.12)
)am,bk = −δab min(m, k).
(4.13)
(a)
Then, (4.2) is written in the form (Q0 (y) = 1) (a)
(a)
2 (Q(a) m (y)) = Qm−1 (y)Qm+1 (y) 2 + (ya )m (Q(a) m (y))
(b,k)∈H
(b)
(Qk (y))Gam,bk .
(4.14)
(a)
Proposition 4.4. A solution (Qm (y)) of the specialized Q-system (4.14) is canonical if and only if it satisfies the following condition: (Convergence property): For each a, the limit lim Q(a) m (y) exists in C[[y]]. m→∞
(4.15)
(a)
Proof. Let (Qm (y))(a,m)∈H be a solution of (4.14). The same calculation as (3.24) in Example 3.8 shows that (4.3) is equivalent to the following equality for each L (cf. (3.26)): (a)
Q(a) m (y) ≡ QL (y)
mod JL ,
m ≥ L + 1.
(4.16)
Clearly, condition (4.15) follows from condition (4.16). Conversely, assume condition (4.15). By (4.14), we have (a)
(a)
(a) Q(a) m (y)/Qm−1 (y) ≡ Qm+1 (y)/Qm (y)
mod JL ,
(m ≥ L + 1). (a)
(4.17) (a)
Because of (4.15), both sides of (4.17) are 1 mod JL . Thus, we have Qm (y) ≡ Qm−1 (y) mod JL (m ≥ L + 1). Therefore, (4.16) holds.
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4.3. Q-system of KR type and denominator formula. Definition 4.5. A specialized Q-system (4.2) is called a Q-system of KR (Kirillov– Reshetikhin) type if the matrices D and G further satisfy the following conditions: (KR-I) The matrix D and its inverse D −1 are given by (4.12) and (4.13). (KR-II) There exists a well-order ≺ in H such that G = GD −1 has the form Gam,bk = gab m for (a, m) (b, k),
(4.18)
where gab (a, b = 1, . . . , n) are integers with det 1≤a,b≤n gab = 0. Example 4.6. Let ta > 0 and hab (a, b = 1, . . . , n) be real numbers such that gab := hab tb are integers and det hab = 0. We define a well-order ≺ in H as follows: (a, m) ≺ (b, k) if tb m < ta k, or if tb m = ta k and a < b. Then, Gam,bk = hab min(tb m, ta k)
(4.19)
satisfies the condition (KR-II) with gab = hab tb . Let x = (xa )na=1 be a multivariable with n components, and y(x) be the map ya (x) =
n
(xb )−gab ,
(4.20)
b=1
where gab are the integers in (4.18). We set m (a) Q(a) m (x) := (xa ) Qm (y(x)),
(4.21)
which are Laurent series of x. (a)
(a)
Proposition 4.7. The family (Qm (x))(a,m)∈H satisfies a system of equations (Q0 (x) = 1), (a) (a) (b) 2 (a) 2 (Qk (x))Gam,bk . (Q(a) m (x)) = Qm−1 (x)Qm+1 (x) + (Qm (x)) (4.22) (b,k)∈H
Proof. By comparing (4.14) and (4.22), it is enough to prove the equality ∞
Gam,bk (−k) = gab m.
(4.23)
k=1
Due to the condition (KR-II), for given (a, m) and b, there is some number L such that Gam,bk = gab m holds for any k ≥ L. Then, for k > L, we have Gam,bk =
∞ j =1
Gam,bj Dbj,bk = gab m(−2 + 1 + 1) = 0.
(4.24)
Therefore, the LHS of (4.23) is evaluated as ∞ L k=1 j =1
Gam,bj Dbj,bk (−k) = (L + 1)Gam,bL − LGam,bL+1 = gab m.
(4.25)
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173
Remark 4.8. The relation (4.22) is the original form of the Q-system in [K2, K3, KR], where the matrix G is taken as (1.5). See also (5.14) and (5.16). Note that, in the second (a) term of the RHS in (4.22), the factor (Qm (y))2 is cancelled by the factor in the product for (b, k) = (a, m), because Gam,am = −2. (a)
Proposition 4.9 (Denominator formula). Let (Qm (y))(a,m)∈H be the canonical solution 0 0 0 (x) := KD,G (y(x)), where KD,G (y) is the of the Q-system of KR type (4.14). Let KD,G power series in (4.11). Then, the formula
∂Q(a) 0 1 (x) = det (x) (4.26) KD,G 1≤a,b≤n ∂xb holds. A proof of Proposition 4.9 is given in Appendix A. In Conjecture 5.7, Proposition 4.9 0 will be used to identify KD,G (x) for some G with the Weyl denominators of the simple Lie algebras. 5. Q-Systems and the Kirillov–Reshetikhin Conjecture In this section, we reformulate Conjecture 1.1 in terms of the canonical solutions of certain Q-systems of KR type (Conjecture 5.5). Then, we present several character formulae, all of which are equivalent to Conjecture 5.5. 5.1. Quantum affine algebras. We formulate Conjecture 1.1 in the following setting: Firstly, we translate the conjecture for the KR modules of the (untwisted) quantum affine (1) algebra Uq (Xn ), based on the widely-believed correspondence between the finite(1) dimensional modules of Y (Xn ) and Uq (Xn ) (for the simply-laced case, see [V]). Secondly, we also include the twisted quantum affine algebra case, following [HKOTT]. First, we introduce some notations. Let g = XN be a complex simple Lie algebra of rank N. We fix a Dynkin diagram automorphism σ of g with r = ord σ . Let g0 be the σ -invariant subalgebra of g; namely, g r g0
Xn 1 Xn
A2n 2 Bn
A2n−1 2 Cn
Dn+1 2 Bn
E6 2 F4
D4 3 G2
(5.1)
See Fig. 1. Let A = (Aij ) (i, j ∈ I ) and A = (Aij ) (i, j ∈ Iσ ) be the Cartan matrices of g and g0 , respectively, where Iσ is the set of the σ -orbits on I . We define the numbers di , di , Ci , Ci (i ∈ I ) as follows: di (i ∈ I ) are coprime positive integers such that (di Aij ) is symmetric; di (i ∈ Iσ ) are coprime positive integers such that (di Aij ) is symmetric, and we set di = dπ(i) (i ∈ I ), where π : I → Iσ is the canonical projection; Ci = r if σ (i) = i, and 1 otherwise; Ci = 2 if Aiσ (i) < 0, and 1 otherwise. It immediately follows (r)
(2)
that di = di and Ci = 1 if r = 1; di = 1 if r > 1; Ci = 1 if XN = A2n . It is easy to (r) (2) check the following relations: Set κ0 = 2 if XN = A2n , and 1 otherwise. Then, κ0 di
r s=1
Aiσ s (j ) = di Aπ(i)π(j ) , κ0 Ci di = Ci di .
(5.2) (5.3)
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A. Kuniba, T. Nakanishi, Z. Tsuboi (XN , r)
g0
(A2n , 2)
❞
❞
(A2n−1 , 2)
t
t
t
t ❞
❞
❞
t
t
Bn Cn
❞
❞
❞> t
t
t
t< ❞
❞
❞
❞> t
t (Dn+1 , 2)
❞
❞
❞
t
Bn
❞ t
(E6 , 2)
t
❞
t
t
❞
F4
❞> t
t
t t
(D4 , 3)
❞
t
❞> t
G2
Fig. 1. The Dynkin diagrams of XN and g0 for r > 1. The filled circles in XN correspond to the ones in g0 which are short roots of g0
For q ∈ C× , we set qi = q κ0 di , qi = q di , and [n]q = (q n − q −n )/(q − q −1 ).
(r)
We use the “second realization” of the quantum affine algebra Uq = Uq (XN ) [D2, ± (i ∈ I, k ∈ Z), Hik (i ∈ I, k ∈ Z \ {0}), Ki±1 (i ∈ I ), and the J] with the generators Xik ±1/2 central elements c . As far as finite-dimensional Uq -modules are concerned, we can ±1/2 set c = 1. Some of the defining relations in the quotient (the quantum loop algebra) Uq /(c±1/2 − 1) are presented below to fix notations (here we follow the convention in [CP2, CP3]): ± Xσ±(i)k = ωk Xik ,
Ki Xj±k Ki−1 = q
r
±κ0 di
[Hik , Xj±l ] = ± + [Xik , Xj−l ] =
s=1 Aiσ s (j )
r 1 k
r
s=1
±1 Kσ±1 (i)k = Kik ,
Hσ (i)k = ωk Hik , Xj±k ,
(5.5)
± [kκ0 di Aiσ s (j ) ]q ωsk Xj,k+l ,
δσ s (i),j ωsl
s=1
(5.4)
+ − − Hi,k+l Hi,k+l
qi − qi−1
(5.6)
,
(5.7)
± where ω = exp(2πi/r), and Hik (i ∈ I, k ∈ Z) are defined by ∞ k=0
± Hi,±k uk
=
Ki±1 exp
±(qi − qi−1 )
∞
Hi,±l u
l
(5.8)
l=1
± with Hik = 0 (±k < 0).
Remark 5.1. In [CP3], there are some misprints which are relevant here. Namely, the relation [Hik , Xj±l ] should read (5.6) here; in Proposition 2.2 and Theorem 3.1 (ii), q should read qi for such i that σ (i) = i and aiσ (i) = 0 therein. We thank V. Chari for the correspondence concerning these points.
Canonical Solutions of the Q-Systems
175
Let V (ψ ± ) denote the irreducible Uq -module with a highest weight vector v and the ± highest weight ψ ± = (ψik ), namely, + v = 0, Xik
± v Hik
(5.9)
± ψik v,
=
± ψik
∈ C.
(5.10)
The following theorem gives the classification of the finite-dimensional Uq -modules: (r)
Theorem 5.2 (Theorem 3.3 [CP2], Theorem 3.1 [CP3]). The Uq (XN )-module V (ψ ± ) is finite-dimensional if and only if there exist N -tuple of polynomials (Pi (u))i∈I with the unit constant terms such that ∞ k=0
+ k ψik u
=
∞ k=0
− ψi,−k u−k
=
qi Ci deg Pi
Pi (qi −2Ci uCi )
Pi (uCi )
,
(5.11)
where the first two terms are the Laurent expansions of the third term about u = 0 and u = ∞, respectively. The polynomials (Pi (u))i∈I are called the Drinfeld polynomials of V (ψ ± ). It follows from (5.3), (5.10), and (5.11) that
±Ci deg Pi
Ki±1 v = qi ±Ci deg Pi v = qi
v.
(5.12)
5.2. The KR modules. We take an inclusion ι : Iσ K→ I such that π ◦ ι = id, and regard Iσ as a subset of I . Let us label the set Iσ with {1, . . . , n}. The Drinfeld polynomials (5.11) satisfy the relation Pσ (i) (u) = Pi (ωu) (σ (i) = i) by (5.4) and (5.8). Therefore, it is enough to specify the polynomials Pi (u) only for those i ∈ {1, . . . , n} ⊂ I . We set H = {1, . . . , n} × N as in (4.1). (a)
Definition 5.3. For each (a, m) ∈ H and ζ ∈ C× , let Wm (ζ ) be the finite-dimensional irreducible Uq -module whose Drinfeld polynomials Pb (u) (b = 1, . . . , n) are specified as follows: Pb (u) = 1 for b = a, and Pa (u) =
m
(1 − ζ qa Ca (m+2−2k) u).
(5.13)
k=1 (a)
We call Wm (ζ ) a KR (Kirillov–Reshetikhin) module. ± and Ka±1 (a = 1, . . . , n) generate the subalgebra By (5.2) and (5.5), we see that Xa0 (a) Uq (g0 ). It is well known that all Wm (ζ ) (ζ ∈ C× ) share the same Uq (g0 )-module ± structure. If we set Ka±1 = qa±Ha and take the limit q → 1, Xa0 and Ha (a = 1, . . . , n) (a) generate the Lie algebra g0 . Accordingly, Wm (ζ ) is equipped with the g0 -module struc(a) ture. We call its g0 -character the g0 -character of Wm (ζ ). The g0 -highest weight of (a) Wm (ζ ), in the same sense as above, is mCa a by (5.12) and (5.13).
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A. Kuniba, T. Nakanishi, Z. Tsuboi
5.3. The Kirillov–Reshetikhin conjecture. We define the matrix G = (Gam,bk )(a,m),(b,k)∈H with the entry r
m k db Gam,bk = (5.14) Abσ s (a) min , db da C s=1 b db Aba min( dmb , dka ) r = 1 = 1 (5.15) r > 1. Cb Aba min(m, k) It follows from (5.15) and Example 4.6 that G satisfies the condition (KR-II) in Definition 4.5 with gab = Aba /Cb . Below, we consider the Q-system of KR type with the matrix G := G D, where D is the matrix in (4.12). By using (A.6) of [KN2]), the entry of G is explicitly written as 1 r>1 − Cb Aba δm,k db /da = 2 −Aba (δm,2k−1 + 2δm,2k + δm,2k+1 ) Gam,bk = −Aba (δm,3k−2 + 2δm,3k−1 + 3δm,3k (5.16) db /da = 3 +2δm,3k+1 + δm,3k+2 ) −Aab δda m,db k otherwise. Let αa and a (a = 1, . . . , n) be the simple roots and the fundamental weights of g0 . We set xa = eCa a ,
ya = e−αa .
(5.17)
Then, they satisfy the relation (4.20) for the above gab ; namely, ya =
n b=1
−Aba /Cb
xb
.
(5.18)
(a)
Definition 5.4. Let Qm (x) be the Laurent polynomial of x = (xa )na=1 representing (a) (a) (a) the g0 -character of the KR module Wm (ζ ). Then, Qm (y) := (xa )−m Qm (x)|x=x(y) , n where x(y) is the inverse map of (5.18), is a polynomial of y = (ya )a=1 with the unit (a) (a) constant term. We call Qm (y) the normalized g0 -character of Wm (ζ ). Now we present a reformulation of Conjecture 1.1. This is the main statement of the paper. (a)
(a)
ν 0 (y)/KD,G (y) = RνD,G (y) Qν (y) = KD,G
(5.19)
Conjecture 5.5. Let Qm (y) be the normalized g0 -character of the KR module Wm (ζ ) (r) (a) of Uq (XN ). Then, the family (Qm (y))(a,m)∈H is characterized as the canonical solution of the Q-system of KR type (4.14) with G given in (5.16). (a) (a) Let Qν (y) = (a,m) (Qm (y))νm for ν ∈ ν(D). By Theorem 4.3, Conjecture 5.5 is equivalent to Conjecture 5.6 ([KN2]). The formulae ν (y) KD,G
and RνD,G (y) are the power series in (4.11) with D in (4.12) and hold, where G in (5.16). Therefore, RνD,G (y) is a polynomial of y, and its coefficients are identified (a) (a) (a) with the g0 -weight multiplicities of the tensor product (a,m)∈H Wm (ζm )⊗νm , where (a) ζm are arbitrary.
Canonical Solutions of the Q-Systems
177 g
5.4. Equivalence to Conjecture 1.1. Let + denote the set of all the positive roots of g. (r) (2) Originally, Conjecture 5.5 is formulated for XN = A2n as follows (cf. Conjecture 1.1): (r)
(2)
Conjecture 5.7 ([K1, K2, HKOTY, HKOTT]). For XN = A2n , the formula ν (y) KD,G Qν (y) = (1 − e−α )
(5.20)
g
α∈+0 ν (y) is the power series in (4.11) with D in (4.12) and G in (5.16). holds, where KD,G ν Therefore, KD,G (y) is a polynomial of y, and its coefficients are identified with the multi (a) (a) (a) plicities of the g0 -irreducible components of the tensor product (a,m)∈H Wm (ζm )⊗νm , (a)
where ζm are arbitrary. (r)
(2)
Proof of the equivalence between Conjectures 5.6 and 5.7 for XN = A2n . Suppose that Conjecture 5.7 holds. Then, setting ν = 0 in (5.20), we have 0 KD,G (y) =
(1 − e−α ).
(5.21)
g α∈+0
ν 0 Therefore, Qν (y) = KD,G (y)/KD,G (y) holds. Conversely, suppose that the family of (a)
the normalized g0 -characters (Qm (y))(a,m)∈H is the canonical solution of (4.14). Then, the equality (5.21) follows from Proposition 4.9 and the lemma below. Lemma 5.8. Let g be a complex simple Lie algebra of rank n, and αa and a be the simple roots and the fundamental weights of g. We set xa = e a , ya = e−αa /ka , where ka (a = 1, . . . , n) are 1 or 2. Suppose that fa (y) (a = 1, . . . , n) are polynomials of y with the unit constant terms such that fa (x) = xa fa (y(x)) are invariant under the action of the Weyl group of g. Then,
det
∂f
1≤a,b≤n
a
∂xb
(x) = (1 − e−α ).
Proof. The proof is the same as the one for Lemma 8.6 in [HKOTY]. (2)
(5.22)
g α∈+
In the case A2n , (5.21) does not hold under Conjecture 5.6, because the assumption (2) in Lemma 5.8 is not satisfied by (5.17). We treat the case A2n separately below.
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A. Kuniba, T. Nakanishi, Z. Tsuboi
0
n−1 n ❞> ❞ 1 0
1
❞> ❞
n n−1
(2) Fig. 2. The Dynkin diagram of A2n . The upper and lower labels respect the subalgebra Bn and Cn , respectively
(2)
5.5. The A2n case. (2)
5.5.1. The Bn -character. For A2n , g0 = Bn . Let {1, . . . , n} label Iσ as the upper label in Fig. 2. Accordingly, Ca = 1 for a = 1, . . . , n − 1, and 2 for a = n. We continue to set ya = e−αa as in Sect. 5.3. We will show later, in (5.34) and (5.36), that under Conjecture 5.5 the following formula holds instead of the formula (5.21): 0 KD,G (y) =
n
n
a=1
k=a
1+
(r)
yk
(1 − e−α ).
(5.23)
n α∈B +
(2)
Therefore, Conjecture 5.5 for XN = A2n is equivalent to (r)
(2)
Conjecture 5.9. For XN = A2n , the formula ν
Q (y) =
ν (y) KD,G
n
a=1
1+
n
(1 − e
k=a yk −α
−1
)
(5.24)
n α∈B +
holds for the normalized Bn -characters of the KR-modules. (2)
5.5.2. The Cn -character. As is well-known, Uq (A2n ) has a realization with the “Chevalley generators” Xa± and Ka±1 (a = 0, . . . , n) (e.g. [CP3, Proposition 1.1]). Among them, ± Xa± and Ka±1 (a = 1, . . . , n) are identified with Xa0 , Ka±1 in (5.4)–(5.8), and generate ± the subalgebra Uq (Bn ). On the other hand, Xa and Ka±1 (a = 0, . . . , n − 1) generate the subalgebra Uq 2 (Cn ). See Fig. 2. If we set Ka = qaHa (a = 0, . . . , n − 1), where q0 = q d0 , d0 = 4, then Xa± and Ha (a = 0, . . . , n − 1) generate the Lie algebra Cn (a) in the limit q → 1. This provides Wm (ζ ) with the Cn -module structure, by which the (a) Cn -character of Wm (ζ ) is defined. ˙ Let α˙ a and a (a = 1, . . . , n) be the simple roots and the fundamental weights labeled with the lower label in Fig. 2. By looking at the same Uq -module as Bn and Cn -modules as above, a linear bijection φ : h∗ → h˙ ∗ is induced, where h∗ and h˙ ∗ are the duals of the Cartan subalgebras of Bn and Cn , respectively. ˙ 0 = 0): Lemma 5.10. Under the bijection φ, we have the correspondence ( ˙ n−a − ˙ n, Ca a → α˙ n−a a = 1, . . . , n − 1 αa → −(α˙ 1 + · · · + α˙ n−1 + 21 α˙ n ) a = n.
(5.25) (5.26)
Canonical Solutions of the Q-Systems
179 (2)
Proof. It is obtained from the relations among Hi and αi for A2n [Kac]: 0=c=
n i=0
ai∨ Hi ,
0=δ=
n
a i αi ,
(5.27)
i=0
where (a0∨ , . . . , an∨ ) = (2, . . . , 2, 1) and (a0 , . . . , an ) = (1, 2, . . . , 2) for the upper label in Fig. 2. Let W(Xn ) denote the Weyl group of Xn . Lemma 5.11. There is an element s ∈ W(Cn ) which acts on h˙ ∗ as follows: ˙ a (a = 1, . . . , n), φ(Ca a ) → 1 φ(αa ) → α˙ a (a = 1, . . . , n). Ca
(5.28) (5.29)
Proof. We take the standard orthonormal basis εa of h˙ ∗ . Let s be the element such that s : εa → −εn−a+1 . Then, ˙ n = −(εn−a+1 + · · · + εn ) → ε1 + · · · + εa = ˙ a, ˙ n−a − α˙ n−a = εn−a − εn−a+1 → εa − εa+1 = α˙ a (a = 1, . . . , n − 1), 1 1 − (α˙ 1 + · · · + α˙ n−1 + α˙ n ) = −ε1 → εn = α˙ n . 2 2
(5.30) (5.31) (5.32)
According to (5.30)–(5.32), we set ˙
xa = e a ,
ya = e−α˙ a /Ca .
(5.33)
Then, the relation (5.18) is preserved, since φ and s above are linear. Lemma 5.11 assures that the following definition is well-defined. (a)
Definition 5.12. Let Qm (x) be the Laurent polynomial of x = (xa )na=1 representing the (a) (a) (a) Cn -character of the KR module Wm (ζ ). Then, Qm (y) := (xa )−m Qm (x)|x=x(y) is a (a) polynomial of y = (ya )na=1 with the unit constant term. We call Qm (y) the normalized (a) Cn -character of Wm (ζ ). (a)
Moreover, by Lemma 5.11 and the W(Cn )-invariance of the Cn -character of Wm (ζ ), we have Proposition 5.13. The normalized Bn -character and the normalized Cn -character of (a) (2) Wm (ζ ) of Uq (A2n ) coincide as polynomials of y. (2)
Thus, Conjecture 5.5 for the normalized Bn -characters of A2n is applied for the normalized Cn -characters as well. Furthermore, in contrast to the Bn case, Lemma 5.8 is now applicable for (5.33). Therefore, under Conjecture 5.5, we have 0 KD,G (y) = (1 − e−α ). (5.34) n α∈C +
(r)
(2)
Hence, we conclude that Conjecture 5.5 for XN = A2n is also equivalent to
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A. Kuniba, T. Nakanishi, Z. Tsuboi (r)
(2)
Conjecture 5.14 ([HKOTT]). For XN = A2n , the formula ν (y) KD,G Qν (y) = (1 − e−α )
(5.35)
n α∈C +
holds for the normalized Cn -characters of the KR-modules, where y is specified as (5.33). The following relation is easily derived from the explicit expressions of the Weyl denominators of Bn and Cn (e.g. [FH]):
(1 − e−α ) =
n α∈C +
n
n
a=1
k=a
1+
yk
(1 − e−α ),
(5.36)
n α∈B +
where the equality holds under the following identifications: ya = e−α˙ a /Ca for the LHS and ya = e−αa for the RHS under the label in Fig. 2. From (5.34) and (5.36), we obtain (5.23).
5.6. Characters for the rank n subalgebras. The procedure to deduce the Cn -characters (2) from the Bn -characters for A2n in Sect. 5.5 is also applicable to the g˙ -characters for any (r) rank n subalgebra g˙ = g0 of XN . (The characters of the lower rank subalgebras are obtained by their specializations.) Let us demonstrate how it works in two examples: (r) (1) Case I. XN = Bn , g0 = Bn , g˙ = Dn .
(r) (2) Case II. XN = A2n−1 , g0 = Cn , g˙ = Dn . ˙ a ) (a = 1, . . . , n) be the simple roots and the fundaLet αa and a (resp. α˙ a and mental weights of g0 (resp. g˙ ) labeled with the upper (resp. lower) label in Fig. 3. As in Sect. 5.5, a linear bijection φ : h∗ → h˙ ∗ is induced, where h∗ and h˙ ∗ are the duals of the Cartan subalgebras of g0 and g˙ , respectively.
0
0
❞ 2 n❅ ❅❞ 1
❞
n−1 n ❞> ❞ 1 0
n−2
n−1
❞
n−1 n ❞< ❞ 1 0
2 n❅ ❅ 1
❞
❞
n−2
n−1 (1)
(2)
Bn (1)
Fig. 3. The Dynkin diagrams of Bn g˙ , respectively
A2n−1 (2)
and A2n−1 . The upper and lower labels respect the subalgebra g0 and
Canonical Solutions of the Q-Systems
181
Doing a similar calculation to Lemmas 5.10 and 5.11, we have ˙ 0 = 0): Lemma 5.15. Under the bijection φ, we have the correspondence ( Case I. ˙ n−a − ˙ n a = 1, n a → ˙ ˙ n a = 2, . . . , n − 1, n−a − 2 α˙ n−a a = 1, . . . , n − 1 αa → − 21 (2α˙ 1 + · · · + 2α˙ n−2 + α˙ n−1 + α˙ n ) a = n. Case II.
(5.37)
(5.38)
˙ n−1 − ˙n a =1 ˙ ˙ n a = 2, . . . , n, n−a − 2
(5.39)
α˙ n−a a = 1, . . . , n − 1 −(2α˙ 1 + · · · + 2α˙ n−2 + α˙ n−1 + α˙ n ) a = n.
(5.40)
a → αa →
Lemma 5.16. There is an element s ∈ W(Dn ) which acts on h˙ ∗ as follows: Case I. ˙a a = 1, . . . , n − 2, n φ( a ) → ˙ ˙ n a = n − 1, n−1 + α˙ a a = 1, . . . , n − 1 φ(αa ) → 1 ˙ n−1 + α˙ n ) a = n. 2 (−α Case II.
˙a ˙ n−1 + ˙n φ( a ) → 2 ˙n α˙ a φ(αa ) → −α˙ n−1 + α˙ n
(5.41)
(5.42)
a = 1, . . . , n − 2 a =n−1 a = n,
(5.43)
a = 1, . . . , n − 1 a = n.
(5.44)
Accordingly, we set Case I. ˙
˙
˙
xa = e a (a = 1, . . . , n − 2, n), e n−1 + n (a = n − 1), ya = e
−α˙ a
(a = 1, . . . , n − 1), e
(α˙ n−1 −α˙ n )/2
(a = n).
(5.45) (5.46)
Case II. ˙
˙
˙
˙
xa = e a (a = 1, . . . , n − 2), e n−1 + n (a = n − 1), e2 n (a = n), ya = e
−α˙ a
(a = 1, . . . , n − 1), e
α˙ n−1 −α˙ n
(a = n).
(5.47) (5.48)
(a)
Then, the relation (5.18) is preserved. Define the g˙ -characters of Wm (ζ ) in the same way as Definition 5.12. Then, the normalized g0 -character and the normalized g˙ -character
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A. Kuniba, T. Nakanishi, Z. Tsuboi (a)
of Wm (ζ ) coincide as polynomials of y. Thus, Conjecture 5.5 for the normalized g0 characters is applied for the normalized g˙ -characters as well. So far, the situation is (2) parallel to the Cn case for A2n . From now on, the situation is parallel to the Bn case (2) for A2n . The following relations are easily derived from the explicit expressions of the Weyl denominators for Bn , Cn , Dn :
(1 − e−α ) =
n α∈B +
(1 − e−α ) =
n α∈C +
n
n
a=1
k=a
1−
yk
(5.49)
n α∈D +
n
n
a=1
k=a
1 − yn−1
(1 − e−α ),
yk2
(1 − e−α ),
(5.50)
n α∈D +
where the equalities hold under the following identifications: (5.17) for the LHSs, (5.46) for the RHS of (5.49), (5.48) for the RHS of (5.50) under the label in Fig. 3. We conclude (1) (2) that Conjecture 5.5 for Bn and A2n−1 is equivalent to (1)
Conjecture 5.17. (i) For Bn , the formula −1 ν (y) na=1 1 − nk=a yk KD,G ν Q (y) = (1 − e−α )
(5.51)
n α∈D +
holds for the Dn -characters of the KR-modules, where y is specified as (5.46). (2) (ii) For A2n−1 , the formula −1 ν (y) na=1 1 − yn−1 nk=a yk2 KD,G ν (5.52) Q (y) = (1 − e−α ) n α∈D +
holds for the Dn -characters of the KR-modules, where y is specified as (5.48). The manifest polynomial expressions of the numerators in the RHSs of (5.24), (5.51), (a) and (5.52) for Qm (y) are available in [HKOTT] with some other examples. 5.7. Related works. Below we list the related works on Conjectures 1.1 and 5.5–5.7 ν mostly chronologically. However, the list is by no means complete. The series KD,G (y) in (5.20) admits a natural q-analogue called the fermionic formula. This is another fascinating subject, but we do not cover it here. See [BS, HKOTY, HKOTT] and references therein. It is convenient to refer to the formula (5.20) with the binomial coefficient (2.9) as type I, and the ones with the binomial coefficient in Remark 1.3 as type II. (In the (a) (a) context of the XXX-type integrable spin chains, Nm and Pm represent the numbers of (a) m-strings and m-holes of color a, respectively. Therefore one must demand Pm ≥ 0, which implies that the relevant formulae are necessarily of type II.) The manifest ex(a) pression of the decomposition of Qm such as (2)
Q1 = χ ( 2 ) + χ ( 5 )
(5.53)
Canonical Solutions of the Q-Systems
183
is referred to as type III, where χ (λ) is the character of the irreducible Xn -module V (λ) with highest weight λ. Since there is no essential distinction between these conjectured (1) formulae for Y (Xn ) and Uq (Xn ), we simply refer to both cases as Xn below. At this moment, however, the proofs should be separately given for the nonsimply-laced case [V]. 0 [Be]. Bethe solved the XXX spin chain of length N by inventing what is later known as the Bethe ansatz and the string hypothesis. As a check of the completeness of his eigenvectors for the XXX Hamiltonian, he proved, in our terminology, the type II (1) formula of Qν (y) with νm = N δm1 for A1 . See [F, FT] for a readable exposition in the framework of the quantum inverse scattering method. 1 [K1, K2]. Kirillov proposed and proved the type I formula of the irreducible modules V (m a ) for A1 [K1] and An [K2]. The idea of the use of the generating function and the Q-system, which is extended in the present paper, originates in this work. 2 [KKR]. Kerov et al. proposed and proved the type II formula for An by the combinatorial method, where the bijection between the Littlewood-Richardson tableaux and the rigged configurations was constructed. 3 [D1]. Drinfeld claimed that V (m a ) can be lifted to a Y (Xn )-module, if the Kac (1) label for αa in Xn is 1. These modules are often called the evaluation modules, and (1) identified with some KR-modules. A method of proof is given in [C] for Uq (Xn ). (a) Therefore, the type III formula Qm = χ (m a ) holds for those a. Some of the corresponding R-matrices for the classical algebras, Xn = An , Bn , Cn , Dn , were obtained earlier in [KRS, R] by the reproduction scheme (also known as the fusion procedure) in the context of the algebraic Bethe ansatz method. (a) 4 [OW]. Ogievetsky and Wiegmann proposed the type III formula of Q1 for some a for the exceptional algebras from the reproduction scheme. 5 [KR, K3]. Kirillov and Reshetikhin formulated the type II formula for any simple Lie algebra Xn . For that purpose, they vaguely introduced a family of Y (Xn )-modules, which we identify with the KR modules here. They proposed the type II formula for any Xn , and the Q-system and the type III formula for Xn = Bn , Cn , Dn . The Q-system for exceptional algebras Xn was also proposed in [K3]. Due to the long-term absence of the proofs of the announced results by the authors, we regard these statements as conjectures at our discretion in this paper. See Remark 5.18 for the further remark. (a)
(a)
Remark 5.18. Let Xn = Bn , Cn , Dn . Let Qm and Qm be the Xn -character and the normalized Xn -character of the “KR module” proposed in [KR]. Then, one can organize the conjectures in [KR] as follows: (a)
(i) All Qm ’s are given by the type III formula in [KR]. (a) (a) (ii) The family (Qm )(a,m)∈H satisfies the Q-system (4.22) for Xn , and Q1 ’s (a = 1, . . . , n) are given by the type III formula in [KR]. (Note that the Q-system (a) (4.22), or equivalently (1.4), recursively determines all Qm ’s from the initial data (a) n (Q1 )a=1 .) (iii) Any power Qν is given by the type II formula. As stated in [KR], one can certainly show the equivalence between (i) and (ii) without referring of the KR-modules themselves. See [HKOTY]. One can also confirm the equivalence between (i) and a weak version of (iii), (a)
(iii’) All Qm ’s are given by the type II formula.
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A. Kuniba, T. Nakanishi, Z. Tsuboi (a)
See [Kl] and Appendix A in [HKOTY]. The family (Qm )(a,m)∈H given by (i) satisfies the convergence property (4.15). Thus, (i), (ii), and (iii’) are all equivalent to (a)
(iv) The family (Qm )(a,m)∈H is the canonical solution of the Q-system (1.4). Therefore, as shown in Section 5.4 (also [KN2]), they are also equivalent to (v) Any power Qν is given by the type I formula (1.1). This is why we call Conjecture 1.1 the Kirillov-Reshetikhin conjecture. The equivalence between (iii) and the others has not been proved yet as we mentioned in Remark 1.3. (a)
6 [CP1, CP2]. Chari and Pressley proved the type III formula of Q1 in most cases (1) for Y (Xn ) [CP1], and for Uq (Xn ) [CP2], where the list is complete except for E7 and E8 . (a) 7 [Ku]. The type III formula of Qm was proposed for some a for the exceptional algebras. 8 [Kl]. Kleber analyzed a combinatorial structure of the type II formula for the (a) simply-laced algebras. In particular, it was proved that the type III formula of Qm and the corresponding type II formula are equivalent for An and Dn . 9 [HKOTY, HKOTT]. Hatayama et al. gave a characterization of the type I formula as the solution of the Q-system which are C-linear combinations of the Xn -characters with the property equivalent to the convergence property (4.15). Using it, the equivalence of (a) the type III formula of Qm and the type I formula of Qν (y) for the classical algebras was shown [HKOTY]. In [HKOTT], the type I and type II formulae, and the Q-systems (r) (a) (2) for the twisted algebras Uq (XN ) were proposed. The type III formula of Qm for A2n , (2) (2) (3) A2n−1 , Dn+1 , D4 was also proposed, and the equivalence to the type I formula was shown in a similar way to the untwisted case. 10 [KN1, KN2]. The second formula in Conjecture 5.6 was proposed and proved for A1 [KN1] from the formal completeness of the XXZ-type Bethe vectors. The same formula was proposed for Xn , and the equivalence to the type I formula was proved [KN2]. The type I formula is formulated in the form (5.19), and the characterization of type I formula in [HKOTY] was simplified as the solution of the Q-system with the convergence property (4.15). (a) (1) 11 [C]. Chari proved the type III formula of Qm for Uq (Xn ) for any a for the classical algebras, and for some a for the exceptional algebras. 12 [OSS]. Okado et al. constructed bijections between the rigged configurations and (a) the crystals (resp. virtual crystals) corresponding to Qν (y), with νm = 0 for m > 1, (1) (2) (2) for Cn and A2n (resp. Dn+1 ). As a corollary, the type II formula of those Qν (y) was (1)
(2)
proved for Cn and A2n . Assembling all the above results and the indications to each other, let us summarize the current status of the Kirillov-Reshetikhin conjectures into the following theorem. (r) Here, we mention the results only for the quantum affine algebra Uq (XN ) case. Also, we exclude the isolated results only valid for small m. (1)
Theorem 5.19. (i) Conjecture 5.5 and the type I formula of Qν (y) are valid for An , (1) (1) (1) Bn , Cn , Dn .
Canonical Solutions of the Q-Systems
185 (1)
(a)
(ii) The type II formula of Qν (y) is valid for An and valid for those ν with νm = 0 (1) (2) (a) for m > 1 for Cn and A2n . The type II formula of Qm (y) is valid for the following a (1) (1) (1) (1) (1) in [C]: any a for Bn , Cn , Dn ; a = 1, 6 for E6 ; a = 7 for E7 . (a) (1) (1) (1) (1) (iii) The type III formula of Qm is valid for all a for An , Bn , Cn , Dn , and for (1) (1) (1) (1) (1) those a listed in [C] for E6 , E7 , E8 , F4 , G2 . The formula is found in [C]. A. The Denominator Formulae We give a proof of Proposition 4.9. The proof is divided into three steps. A.1. Step 1. Reduction of the denominator formula. In Steps 1 and 2, we consider the unspecialized infinite Q-system (4.8), and we assume that D and G satisfy the condition (KR-II) in Definition 4.5. For a given positive integer L, let HL = {1, . . . , n} × {1, . . . , L} be the finite subset (a) (a) of H in Sect. 4.1. With multivariables vL = (vm )(a,m)∈HL , wL = (wm )(a,m)∈HL , (a) zL = (zm )(a,m)∈HL , we define the bijection vL → wL around v = w = 0 (cf. (2.1)) by (b) (a) (a) wm (vL ) = vm (1 − vk )−Gam,bk , (A.1) (b,k)∈HL
and the bijection vL → zL around v = z = 0 by (b) (a) (a) zm (vL ) = wm (vL ) (1 − vk )gab m
(A.2)
(b,k)∈HL
(a) = vm
(b,k)∈HL
(b)
(1 − vk )−Gam,bk +gab m ,
(A.3)
where gab is the one in (KR-II). Let us factorize the bijection wL → vL as wL → zL → vL . The map wL → zL is described as (a) (a) zm (wL ) = wm
n
(Qb (wL ))−gab m ,
Qb (wL ) :=
b=1
L
(b)
(1 − vk (wL ))−1 .
(A.4)
k=1
By the assumption (KR-II) and the expression (A.3), the map vL → zL is lowertriangular in the sense of Example 2.9. Therefore, the following equality holds: det HL
w (b) ∂z(a) m k (w ) = det (w ) L L , (b) HL z(a) ∂w (b) ∂wk m k
w (b) ∂v (a) m k (a)
vm
(A.5)
where detHL means the abbreviation of det (a,m),(b,k)∈HL . We now simultaneously specialize the variables wL and zL with the variables y = (ya )na=1 and u = (ua )na=1 as (cf. (4.5)) (a) (a) wm = wm (y) = (ya )m ,
(a) (a) zm = zm (u) = (ua )m .
(A.6)
This specialization is compatible with (A.4) and the map y → u, ua (y) = ya
n b=1
(qb (y))−gab ,
qb (y) := Qb (wL (y)).
(A.7)
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Proposition A.1. Let GL = (Gam,bk )(a,m),(b,k)∈HL be the HL -truncation of G , KI0 ,G (wL ) be the one in (2.34), and KI0 ,G (y) := KI0 ,G (wL (y)) be its specialL L L L L L ization by (A.6). Then, the formula (2.34) reduces to KI0L ,G (y) = L
n
y ∂u b a (y) qa (y). 1≤a,b≤n ua ∂yb
det
(A.8)
a=1
Proof. Because of (A.5), it is enough to prove the equality det HL
w (b) ∂z(a) m k
(w (y)) = L (b)
(a)
zm ∂wk
y ∂u b a (y) . 1≤a,b≤n ua ∂yb det
(A.9)
We remark that ya
L ∂ (a) ∂ = mwm , (a) ∂ya ∂wm
(A.10)
m=1
det(δam,bk + mαabk ) = HL
δab +
det
1≤a,b≤n
L
kαabk ,
(A.11)
k=1
where αabk are arbitrary constants depending on a, b, k. Set Fa (wL ) =
n
(Qb (wL ))−gab .
b=1
Then, (A.9) is obtained as
(b) (LHS) = det δam,bk + mwk HL
=
δab +
det
1≤a,b≤n
L k=1
∂ (b)
∂wk
log Fa (wL (y))
∂
(b)
kwk
(b)
∂wk
log Fa (wL (y))
∂ log Fa (wL (y)) 1≤a,b≤n ∂yb
y ∂u b a = det (y) , 1≤a,b≤n ua ∂yb =
det
δab + yb
where we used (A.4), (A.11), (A.10), and (A.7).
A.2. Step 2. Change of variables. We introduce the change of the variables y and u in (A.6) to x = (xa )na=1 and q = (qa )na=1 as ya (x) =
n b=1
(xb )−gab ,
ua (q) =
n
(qb )−gab .
b=1
(A.12)
Canonical Solutions of the Q-Systems
187
Thus, if f (y) is a power series of y, then f (y(x)) is a Laurent series of x because of the assumption in (KR-II) that gab ’s are integers. This specialization is compatible with (A.7) and the map x → q, qa (x) = xa qa (y(x)).
(A.13)
Let us summarize all the maps and variables in a diagram: (A.2)
(A.4)
v ←→ z ←→ w (A.6)↑ ↑ (A.6) (A.7)
u ←→ y (A.12)↑ ↑(A.12)
(A.14)
(A.13)
q ←→ x With these changes of variables, (A.8) becomes the Jacobian of q(x): Proposition A.2. Let KI0
L ,GL
KI0
L ,GL
(y) be the one in Proposition A.1, and let KI0
L ,GL
(y(x)). Then, the formula KI0L ,G (x) = L
∂q
det
1≤a,b≤n
a
∂xb
(x) :=
(x)
(A.15)
holds. Proof. By (A.12), we have
q ∂u
x ∂y b a b a det = det = det (−gab ) = 0. 1≤a,b≤n ua ∂qb 1≤a,b≤n ya ∂xb 1≤a,b≤n
(A.16)
Using Proposition A.1, (A.13), and (A.16), we obtain KI0L ,G (x) = L
n
y ∂u b a (y(x)) qa (y(x)) 1≤a,b≤n ua ∂yb
det
a=1
n
x ∂q
∂q b a a = det (x) qa (y(x)) = det (x) . 1≤a,b≤n qa ∂xb 1≤a,b≤n ∂xb
(A.17)
a=1
A.3. Step 3. Denominator formula for the Q-systems for KR type. Now we are ready to prove Proposition 4.9; namely, 0 0 0 Proposition A.3. Let KD,G (x) := KD,G (y(x)), where KD,G (y) is the denominator in (4.11) for the Q-system of KR type (4.14). Then, the formula 0 (x) = KD,G (a)
(a)
det
∂Q(a)
1≤a,b≤n
1
∂xb
(x)
(A.18) (a)
holds, where we set Q1 (x) := xa Q1 (y(x)) for the canonical solution (Qm (y))(a,m)∈H of (4.14).
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Proof. We recall the following four facts: Fact 1: By (3.33) and (4.6), we have 0 KD,G (y) ≡ KI0L ,G (y) L
mod JL .
(A.19) (a)
Fact 2: By Theorem 3.7 and the proof therein, the canonical solution (Qm (y))(a,m)∈H (a) of (4.14) and the solution (Qm (y))(a,m)∈H of the corresponding standard Q-system are related as Q(a) m (y) =
(a)
(a)
Qm+1 (y)Qm−1 (y) (a)
(Qm (y))2
.
(A.20)
Fact 3: By Propositions 2.1, 3.4, and (4.6), the series qb (y) in (A.7) satisfies qa (y) ≡
L m=1
−1 (Q(a) m (y))
mod JL ,
(A.21)
(a)
where Qm (y) is the one in Fact 2. Note that qb (y) depends on L. Fact 4: By the proof of Proposition 4.4, it holds that (a)
(a)
QL (y) ≡ QL+1 (y)
mod JL .
(A.22) (a)
Combining Facts 2–4, we immediately have qa (y) ≡ Q1 (y) mod JL . Thus, (a) limL→∞ qa (y) = Q1 (y) holds. Therefore, taking the limit L → ∞ of (A.8) with the help of Fact 1, we obtain 0 KD,G (y) =
n
y ∂U b a (a) (y) Q1 (y), 1≤a,b≤n Ua ∂yb
Ua (y) := ya
det
n
(A.23)
a=1
(b)
(Q1 (y))−gab .
(A.24)
b=1
The equality (A.18) is obtained from (A.23) in the same way as the proof of Proposition A.2. Acknowledgement. We would like to thank V. Chari, G. Hatayama, A. N. Kirillov, M. Noumi, M. Okado, T. Takagi, and Y. Yamada for very useful discussions. We especially thank K. Aomoto for the discussion where we recognize the very close relation between the present work and his work, and also for pointing out the reference [G] to us.
References [A] [AI] [B] [Be]
Aomoto, K.: Integral representations of quasi hypergeometric functions. In: Proc. of the International Workshop on Special Functions, Hong Kong, 1999, C. Dunkl et al. (eds), Singapore: World Scientific, pp. 1–15 Aomoto, K. and Iguchi, K.: On quasi-hypergeometric functions. Methods and Appli. of Anal. 6, 55–66 (1999) Berndt, B.C.: Ramanujan’s Notebooks, Part I. Berlin: Springer-Verlag Bethe, H.A.: Zur Theorie der Metalle, I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Physik 71, 205–231 (1931)
Canonical Solutions of the Q-Systems [BS]
189
Bouwknegt, P. and Schoutens, K.: Exclusion statistics in conformal field theory – generalized fermions and spinons for level-1 WZW theories. Nucl. Phys. B547, 501–537 (1999) [C] Chari, V.: On the fermionic formula and the Kirillov–Reshetikhin conjecture. Intern. Math. Res. Notices 12, 629–654 (2001) [CP1] Chari, V. and Pressley, A.: Fundamental representations of Yangians and singularities of Rmatrices. J. reine angew. Math. 417, 87–128 (1991) [CP2] Chari, V. and Pressley, A.: Quantum affine algebras and their representations, Canadian Math. Soc. Conf. Proc. 16, 59–78 (1995) [CP3] Chari, V. and Pressley, A.: Twisted quantum affine algebras. Commun. Math. Phys. 196, 461–476 (1998) [D1] Drinfeld, V.: Hopf algebras and the quantumYang-Baxter equations. Sov. Math. Dokl. 32, 264–268 (1985) [D2] Drinfeld, V.: A new realization of Yangians and quantum affine algebras. Sov. Math. Dokl. 36, 212–216 (1988) [F] Faddeev, L.D.: Lectures on quantum inverse scattering method. In: Integrable systems (Tianjin, 1987), Nankai Lectures Math. Phys., Teaneck, NJ: World Sci. Publishing, 1990, pp. 23–70 [FT] Faddeev, L.D. and Takhtadzhyan, L.A.: The spectrum and scattering of excitations in the onedimensional isotropic Heisenberg model. J. Sov. Math. 24, 241–246 (1984) [FH] W. Fulton and J. Harris, Representation theory: a first course, Springer-Verlag, New York, 1991 [G] Gessel, I.M.: A combinatorial proof of the multivariable Lagrange inversion formula. J. Combin. Theory Ser. A 45, 178–195 (1987) [HKOTT] Hatayama, G., Kuniba, A., Okado, M., Takagi, T. and Tsuboi, Z.: Paths, crystals and fermionic formulae, math.QA/0102113. To appear in Progr. in Math. Phys. [HKOTY] Hatayama, G., Kuniba, A., Okado, M., Takagi, T. and Yamada, Y.: Remarks on fermionic formula. Contemporary Math. 248, 243–291 (1999) [I] Iguchi, K.: Generalized Lagrange theorem and thermodynamics of a multispecies quasiparticle gas with mutual fractional exclusion statistics. Phys. Rev. B58, 6892–6911 (1998); Erratum: Phys. Rev. B 59, 10370 (1999) [IA] Iguchi, K. and Aomoto, K.: Integral representation for the grand partition function in quantum statistical mechanics of exclusion statistics. Int. J. Mod. Phys.: B14, 485–506 (2000) [J] Jing, N.-H.: On Drinfeld realization of quantum affine algebras. Ohio State Univ. Math. Res. Inst. Publ. 7, 195–206 (1998) [Kac] Kac, V.G.: Infinite dimensional Lie algebras, 3rd edition. Cambridge: Cambridge University Press, 1990 [KKR] Kerov, S.V., Kirillov, A.N. and Reshetikhin, N.Yu.: Combinatorics, the Bethe ansatz and representations of the symmetric group. J. Sov. Math. 41, 916–924 (1988) [K1] Kirillov, A.N.: Combinatorial identities and completeness of states for the Heisenberg magnet. J. Sov. Math. 30, 2298–3310 (1985) [K2] Kirillov, A.N.: Completeness of states of the generalized Heisenberg magnet. J. Sov. Math. 36, 115–128 (1987) [K3] Kirillov, A.N.: Identities for the Rogers dilogarithm function connected with simple Lie algebras. J. Sov. Math. 47, 2450–2459 (1989) [KR] Kirillov, A.N. and Reshetikhin, N.Yu.: Representations ofYangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras. J. Sov. Math. 52, 3156–3164 (1990) [Kl] Kleber, M.: Combinatorial structure of finite-dimensional representations ofYangians: the simplylaced case. Int. Math. Res. Note 7, 187–201 (1997) [KRS] Kulish, P., Reshetikhin, N.Yu. and Sklyanin, E.: Yang–Baxter equations and representation theory I. Lett. Math. Phys. 5, 393–403 (1981) (1) [Ku] Kuniba, A.: Thermodynamics of the Uq (Xr ) Bethe ansatz system with q a root of unity. Nucl. Phys. B 389, 209–244 (1993) [KN1] Kuniba, A. and Nakanishi, T.: The Bethe equation at q = 0, the Möbius inversion formula, and weight multiplicities: I. The sl(2) case. Prog. in Math. 191, 185–216 (2000) [KN2] Kuniba, A. and Nakanishi, T.: The Bethe equation at q = 0, the Möbius inversion formula, and weight multiplicities: II. The Xn case. math.QA/0008047. To appear in J. Algebra [KNT] Kuniba, A., Nakanishi, T. and Tsuboi, Z.: The Bethe equation at q = 0, the Möbius inversion (r) formula, and weight multiplicities: III. The XN case. Lett. Math. Phys. 59, 19–31 (2002) [OW] Ogievetsky, E. and Wiegmann, P.: Factorized S-matrix and the Bethe ansatz for simple Lie groups. Phys. Lett. B 168, 360–366 (1986) [OSS] Okado, M., Schilling, A. and Shimozono, M.: Virtual crystals and fermionic formulas of type (2) (2) (1) Dn+1 , A2n , and Cn . math.QA/0105017
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A. Kuniba, T. Nakanishi, Z. Tsuboi
[R]
Reshetikhin, N.Yu.: Integrable models of quantum 1-dimensional magnetics with O(n) and Sp(2k)-symmetry. Theoret. Math. Phys. 63, 555–569 (1985) Sutherland, B.: Quantum many-body problem in one dimension: Thermodynamics. J. Math. Phys. 12, 251–256 (1971) Varagnolo, M.: Quiver Varieties and Yangians. Lett. Math. Phys. 53, 273–283 (2000) Wu, Y.-S.: Statistical distribution for generalized ideal gas of fractional statistical particles. Phys. Rev. Lett. 73, 922–925 (1994)
[S] [V] [W]
Communicated by L. Takhtajan
Commun. Math. Phys. 227, 191 – 209 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Destruction of the Beating Effect for a Non-Linear Schrödinger Equation Vincenzo Grecchi1 , André Martinez1 , Andrea Sacchetti2 1 Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna,
Italy. E-mail:
[email protected],
[email protected]
2 Dipartimento di Matematica, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/B,
41100 Modena, Italy. E-mail:
[email protected] Received: 31 May 2001 / Accepted: 23 January 2002
Abstract: We consider a non-linear perturbation of a symmetric double-well potential as a model for molecular localization. In the semiclassical limit, we prove the existence of a critical value of the perturbation parameter giving the destruction of the beating effect. This value is twice the one corresponding to the first bifurcation of the fundamental state. Here we make use of a particular projection operator introduced by G. Nenciu in order to extend to an infinite dimensional space some known results for a two-level system. 1. Introduction As it is well known, quantum double-well problems exhibit some caracteristic features such as “splitting of the energy levels”, “delocalization” and “beating effect”. It is also known that certain molecules, e.g. the ammonia one NH3 , are such that one of the nuclei (the nitrogen nucleus N in the case of ammonia), in the Born–Oppenheimer approximation, moves according to a double-well effective potential. The beating effect for such molecules, related to the periodic motion of a state passing from localization at one of the wells to localization at the other one, appears as an “inversion line” on the spectrum. For non-isolated molecules we have the “red shift” of the “inversion line”, and, if the ammonia gas is at a pressure large enough (about 2 atmospheres) the inversion line disappears, the N nucleus becomes localized: the well known pyramidal shape of the molecule (molecular structure) appears. Thus, we see classical behaviors of microscopical systems. The cause of this phenomenon should be the polarity of the pyramidal molecule which polarizes the environement, so that the reaction field stabilizes the molecular structure. We consider a standard model for molecular structure: a symmetric double-well potential with a non-linear perturbation [5]. In previous research [6, 7] a critical value of the perturbation parameter has been found giving a bifurcation of the fundamental state and new asymmetrical states.
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The present research shows that for such a value of the parameter the dynamics is not qualitatively changed with respect to the unperturbed case; in particular the beating effect is unchanged. On the other side, here we found another critical value of the parameter at which we have the destruction of the beating effect. In particular, beating motion exists for any value of the parameter smaller than the critical one, at the limit of the critical value, the period of this motion diverges, and for larger values of the parameter the beating effect is absent (see Theorem 2 and Corollary 1). Curiously enough, this second critical value of the parameter is nearly (exactly in the limit considered) twice the previous one. The factor 2 between the critical values of the parameter is explained by the similar role played by two different “energy” invariants belonging to the original problem and to the linearized one respectively. Our work is based on the reduction of the problem to a bi-dimensional space in the semiclassical limit and it makes use of the known results about the dynamics of the reduced two level problem [13, 17]. The paper is organized in the following way. In Sect. 2 we describe the model and we give the main results. In Sect. 3 we prove the theorems. In particular in Sect. 3.2 the reduction of the time-dependent problem into a bi-dimensional space in the semiclassical limit is given. In Sect. 3.3 we recall some known results about the bi-dimensional problem, concerning the trajectories and the frequencies of the motion for different values of the parameter. Finally, in Sect. 3.4 we prove the stability result and the existence of the critical parameter in the full problem. 2. Description of the Model and Main Results We consider here the time-dependent non-linear Schrödinger equation ∂ψ h¯ 2 i h¯ ∂t = H0 ψ + f (x, ψ)ψ, H0 ψ = − 2m ψ + V (x)ψ, ψ(t, x)|t=0 = ψ 0 (x)
(1)
where V (x), x ∈ Rn , is a double-well symmetric potential: V (x , −xn ) = V (x),
x = (x , xn ) ∈ Rn ,
x = (x1 , . . . , xn−1 ),
and f (x, ψ) = ψ, W ψ W (x),
(2)
where is a real parameter and W ∈ C(Rn ) is a given real-valued, bounded, odd function: W (x , −xn ) = −W (x), x = (x , xn ).
(3)
In such a case W locally represents the position operator xn and Eq. (1) would describe the effect of the spontaneous symmetry breaking for a symmetric molecule [4–7, 12]. It is well known [2] that when the nonlinear term has a form given by (2) then we have the conservation of the energy defined below: 1 1 H = H0 ψ, ψ + ψ, W ψ 2 = H0 ψ 0 , ψ 0 + ψ 0 , W ψ 0 2 . 2 2 Hereafter, · , · and · respectively denote the scalar product and the norm in the Hilbert space L2 (Rn ).
Destruction of Beating Effect for Non-Linear Schrödinger Equation
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Remark 1. If we consider the locally linear problem defined as i h¯
∂ψ = H lin ψ, ∂t
where H lin ψ = H0 ψ + f (x, ψ 0 )ψ, f (x, ψ 0 ) = ψ 0 , W ψ 0 W, then we have the conservation of the energy defined as E = ψ 0 , H lin ψ 0 = H0 ψ 0 , ψ 0 + ψ 0 , W ψ 0 2 . Let σ (H0 ) be the spectrum of the self-adjoint realization of H0 on the Hilbert space L2 (Rn , dx). We assume that the discrete spectrum of H0 is not empty and let E+ < E− be the two lowest eigenvalues of H0 , with associated normalized eigenvectors ϕ+ and ϕ− . It is well known [8, 14, 15] that, under very general conditions on V , the splitting between the first two eigenvalues, defined as ω = E− − E+ , satisfies to the following asymptotic behavior: ω ∼ e−C/h¯ , as h¯ → 0,
(4)
for some positive constant C (hereafter C denotes any generic positive constant). In the same limit we also have 1 ϕ± (x) ∼ √ [ϕ0 (x) ± ϕ0 (−x)] , as h¯ → 0, 2 where ϕ0 (x) is a function localized within one well, for instance the right-hand one corresponding to positive values of xn . We also assume that (5) dist {E+ , E− }, σ (H0 ) \ {E+ , E− } ∼ C h, ¯ as h¯ → 0, for some positive C. Now, let 1 1 ϕR = √ (ϕ+ + ϕ− ) and ϕL = √ (ϕ+ − ϕ− ) , 2 2 they are normalized functions such that ϕR (x) ∼ ϕ0 (x) and ϕL (x) ∼ ϕ0 (−x), as h¯ → 0.
(6)
That is ϕR , the so-called right-hand well wave-function, is localized within the righthand well and ϕL , the so-called left-hand well wave-function, is localized within the other well. The solution of Eq. (1) can be written in the form ψ(t, x) = aR (t)ϕR (x) + aL (t)ϕL (x) + ψc (t, x), aR,L (t) ∈ C,
(7)
where ψc = !c ψ is the projection of ψ on the eigenspace orthogonal to the twodimensional space spanned by ϕR and ϕL ; that is: !c = 1 − · , ϕR ϕR − · , ϕL ϕL .
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It is well known that when the perturbation term f is absent in Eq. (1) then a state, initially prepared on the two lowest states, that is ψc0 ≡ 0, generically makes experience of a beating motion between the two wells with period 4π h/ω ¯ and the expectation value W t = ψ(t, ·), W (·)ψ(t, ·) has an oscillating behavior within the interval [−|w|, |w|], w = ϕR , W ϕR . Now, we are going to consider the effect of the perturbation f on these beating motions in the semiclassical limit. In the following we assume that the perturbation strength is of the same order of the splitting and we introduce the non-linearity parameter defined as µ=
c = O(1), as h¯ → 0, ω
(8)
where c = 2w2 = 2ρ02 , w = ϕR , W ϕR = ϕ+ , W ϕ− = ρ0 , the choice of ϕ± can be made such that ρ0 ∈ R − {0}. We state our main results: 0 2 n 1 Theorem 1. For any ψ ∈ H (R ), Eq. (1) admits a unique solution ψ ∈ C Rt ; L2 (Rn ) ∩ C 0 Rt ; H 2 (Rn ) such that ψ(0, x) = ψ 0 (x). Moreover, for all t ∈ R we have that
ψ(t, ·) = ψ 0 (·) .
(9)
Theorem 2. If ψc0 = !c ψ 0 ≡ 0 and if 2|H − &| > δ, & = 1 (E+ + E− ), − 1 ω 2 for some δ > 0 fixed and any h¯ small enough; then there exists τB and a positive constant C independent of h¯ and such that for any α < 1 ψ t + 2τB , · − ψ(t, ·) = O(ω˜ α ), ∀t ∈ [0, t , ], ω˜ for h¯ small enough, where t, =
τ, 1 ln and τ , = (α − 1)/C. ω˜ ω˜
In particular, the expectation value W t is, up to an error of order O(ω˜ α ), a periodic function with pseudo-period T = 2τB /ω˜ and: (i) if 2|H − &| <1−δ ω
(10)
for some δ > 0 and any h¯ small enough, then there exists t0 > 0 such that for any K ∈ N and η > 0 fixed then W t > 0, t0 + η + kT < t < t0 + (k + 1/2) T − η,
Destruction of Beating Effect for Non-Linear Schrödinger Equation
195
and W t < 0, t0 + (k + 1/2) T + η < t < t0 + (k + 1)T − η for any k = 0, 1, . . . , K and h¯ small enough; (ii) in contrast, if 2|H − &| >1+δ ω
(11)
for some δ > 0 and any h¯ small enough, then W t = 0, ∀t ∈ [0, t , ]. Remark 2. Condition (10) implies that H ∈ (E+ , E− ) and condition (11) implies that H∈ / [E+ , E− ]. Remark 3. For an expression of the pseudo-period we refer to Sect. 3.3; in particular, τB is given by Eq. (45) in the case (10) and τB is given by Eq. (46) in the case (11). For what concerns the dynamics of a state initially prepared on one well, e.g. the right-hand one, we have that: Corollary 1 (Beating Destruction: The critical parameter). Let ψ 0 = ψR and µ∞ = ±2, where µ = µ∞ +o(1), as h¯ → 0. Then the state returns near to the initial condition after a pseudo-period T of order 1/ω; ˜ that is for any α < 1 and any K ∈ N fixed then: ψ(kT , ·) − ψR (·) = O(ω˜ α ), for any k = 1, 2, . . . , K. Moreover, if: (i) |µ∞ | < 2, then ψ ((k + 1/2)T , ·) − ψL (·) = O(ω˜ α ), k = 1, 2, . . . , K, and we have the beating motion between the two wells as in the unperturbed case; (ii) |µ∞ | > 2, then W t > 0, ∀t ∈ [0, t , ], that is the state ψ is localized within the right-hand well. 3. Proof of the Theorems 3.1. Proof of Theorem 1. We denote ψ˜ = eitH0 /h¯ ψ and W˜ = eitH0 /h¯ W e−itH0 /h¯ . Then Eq. (1) is equivalent to: i h¯
∂ ψ˜ ˜ = F (ψ), ∂t
where ˜ = ψ, ˜ W˜ ψ ˜ W˜ ψ˜ F (ψ)
(12)
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satisfies to the following Lipschitz-type estimate: for any ψ˜ 1 , ψ˜ 2 ∈ L2 (Rn ) we have that
(13) F (ψ˜ 1 ) − F (ψ˜ 2 ) ≤ C ψ˜ 1 2 + ψ˜ 2 2 ψ˜ 1 − ψ˜ 2 for some positive constant C. Therefore, for small enough, a local existence and unicity result follows from Cauchy’s theorem. Moreover, for any solution ψ˜ of (12) we have that
˜ 2 ∂ ψ ∂ ψ˜ ˜ ψ ˜ = 0; ˜ = 2 , ψ = 2h¯ −1 F (ψ), ∂t ∂t ∂ ψ˜ ˜ is constant with respect to t. As a consequence, hence, ψ ∂t remains uniformly bounded on any open interval of time where it is defined and thus the global existence in time follows from standard arguments. ifψ 0 ∈ H 2 (Rn ) onealso has that Finally, n n ∞ 2 1 2 ˜ ψ ∈ C (Rt ; H (R )) and thus ψ ∈ C Rt ; L (R ) ∩ C 0 Rt ; H 2 (Rn ) . 3.2. Reduction to a two-level system. Here, we prove a stability result which allows us to reduce the analysis of Eq. (1) to a bi-dimensional space. To this purpose we make use of some ideas contained in [10, 11] and [16]. Now, let ω˜ = where
ω˜ h¯
ω 1 and H1 = H0 , h¯ h¯
(14)
= O(1) as h¯ → 0. We treat ω˜ as a new semiclassical parameter. We have that:
Theorem 3. Let ψ(t, x) = aR (t)ϕR (x) + aL (t)ϕL (x) + ψc (t, x), aR,L (t) ∈ C, ψc = !c ψ, be the solution of Eq. (1) satisfying the initial condition ψc0 ≡ 0. Then there exists a positive constant C such that
and
˜ ˜ ˜ C ωt , ψc ≤ C ωe ˜ C ωt H0 ψc ≤ C ωe
(15)
˜ ˜ C ωt aR,L (t) − e−i(E+ +E− )t/2h¯ AR,L (tω/2h¯ ) ≤ C ωe
(16)
for h¯ small enough and any t ∈ R+ , where AR,L (τ ) are the solutions of the non-linear system iAR = −AL + 2ν0 ρ0 AR AR,L (0) = aR,L (0) , (17) , |AR (τ )|2 + |AL (τ )|2 = 1 iAL = −AR − 2ν0 ρ0 AL where means the derivative with respect to τ and ν0 = ν0 (τ ) =
ρ0 (|AR (τ )|2 − |AL (τ )|2 ), ρ0 = ϕ+ , W ϕ− . h¯ ω˜
In particular, for any α ∈ (0, 1), then H0 ψc ≤ C ω˜ α , ψc ≤ C ω˜ α
(18)
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and
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aR,L (t) − e−i(E+ +E− )t/2h¯ AR,L (tω/2h¯ ) ≤ C ω˜ α
for any t ∈ [0, t , ], t , = (τ , /ω) ˜ ln(1/ω), ˜ τ , = (α − 1)/C. Proof. In order to prove the theorem we investigate the solution ψ of (1) with initial data: 0 0 0 2 0 2 ϕ+ + a − ϕ− , |a+ | + |a− | = 1. ψ 0 = a+
(19)
In order to do that, we make the change of time scale: t →τ =
ωt ωt ˜ = 2h¯ 2
which transform (1) into (for the sake of simplicity ψ still denotes the solution of the new equation): i ω˜ ∂ψ = H1 ψ + ψ, W ψ W ψ. h¯ 2 ∂τ
(20)
Our first aim is to construct an approximation of ψ as ω˜ → 0+ . Let us define χ 0 = χ , χ ∈ L2 (Rn ), and χ 1 = H˜ 1 χ , χ ∈ D(H˜ 1 ), where H˜ 1 = H1 + c1 1, c1 is such that H˜ 1 ≥ 1,
(21)
and therefore χ 0 ≤ χ 1 for any χ ∈ D(H˜ 1 ). We start by proving the following lemma. Lemma 1. Let ψ be the solution of Eq. (20) with initial data (19). Let j = 0 or j = 1, ϕ ∈ C 1 (Rτ ; L2 (Rn )) ∩ C 0 (Rτ ; H 2 (Rn )) be such that ϕ(τ, ·) ≤ C for some C > 0 and any τ , ϕ(0, ·) − ψ 0 (·) j = O(ω) ˜ and
i ω˜ ∂ φ= − + H1 + ϕ, W ϕ W ϕ h¯ 2 ∂τ
(22)
φ(τ, ·) j = O(ω˜ 2 )
(23)
be such that
uniformly for τ ≥ 0 and ω˜ small enough. Then, there exists C > 0 such that: ϕ(τ, ·) − ψ(τ, ·) j ≤ C ωe ˜ Cτ , ∀τ ≥ 0.
(24)
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Proof. In order to prove this lemma, first consider j = 0. Let us denote ϕ˜ = e2iτ H1 /ω˜ ϕ, ψ˜ = e2iτ H1 /ω˜ ψ, φ˜ = e2iτ H1 /ω˜ φ, u = ϕ˜ − ψ˜ and W˜ = e2iτ H1 /ω˜ W e−2iτ H1 /ω˜ . We have i ω˜
ϕ, W ϕ W˜ ϕ˜ − ψ, W ψ W˜ ψ˜ − φ˜ u = h¯ 2 and therefore ∂ u 2 ∂τ = 2| u , u |
1 ˜ u = 4 ϕ, W ϕ W˜ ϕ˜ − ψ, W ψ W˜ ψ˜ − φ, h¯ ω˜ 2ω˜
≤ C u 2 + ω u ˜ ≤ C u 2 + ω˜ 2
(25)
for any τ ≥ 0 and for some constant C > 0 since (8), (23) and ab ≤ 21 a 2 + 21 b2 for any a, b > 0. As a result it follows that ∂ −Cτ u 2 ≤ Ce−Cτ ω˜ 2 , e ∂τ ˜ and thus, since u|τ =0 = O(ω): e−Cτ u 2 ≤ C ω˜ 2
(26)
for some C > 0. Then (24) immediately follows. Moreover, we have that (24) is still true when we replace the usual norm χ by ˜ χ 1 = H˜ 1 χ , χ ∈ D(H1 ), where H˜ 1 = H1 + c1 1 ≥ 1 for some c1 . Indeed, let ϕ, ˜ ψ, ˜ ˜ W and φ as above and let now ˜ u1 = H˜ 1 (ϕ˜ − ψ). Then i ω˜
ϕ, W ϕ H˜ 1 W˜ ϕ˜ − ψ, W ψ H˜ 1 W˜ ψ˜ − H˜ 1 φ˜ u1 = h¯ 2 and
∂ u1 2 = 4 ϕ, W ϕ H˜ 1 W˜ ϕ˜ − ψ, W ψ H˜ 1 W˜ ψ˜ − 1 H˜ 1 φ, ˜ u1 ∂τ h¯ ω˜ 2ω˜
≤ C u1 2 + ω u ˜ 1 ≤ C u1 2 + ω˜ 2
for some constant C > 0 since (8), (23) and H˜ 1 W H˜ 1−1 and H˜ 1−1 are bounded operators, uniformly with respect to ω. ˜ As above, it follows that (26) is true, from which (24) follows. !
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Now, in order to prove Theorem 3 we explicitly construct a solution ϕ satisfying the assumptions of Lemma 1. We re-write Eq. (22) as: i ω˜ ∂ − ˜ ϕ = φ, (27) + H1 + ωνW 2 ∂τ where ν = ν(τ ) =
ϕ, W ϕ , = O(1), as h¯ → 0, h¯ ω˜ h¯ ω˜
and where ϕ and φ must satisfy the conditions 0 ˜ ϕ(0, ·) − ψ (·) 1 ≤ C ω, ϕ(τ, ·) ≤ C, ∀τ ≥ 0, φ(τ, ·) 1 ≤ C ω˜ 2 , ∀τ ≥ 0,
(28)
for some C > 0, ψ 0 is given by (19). We denote by !0 = 1 − !c the orthogonal projection onto Cϕ+ ⊕ Cϕ− , that is: 1 !0 = (ζ − H1 )−1 dζ, 2π i γ where γ is a simple complex loop encircling h1¯ E+ , h1¯ E− , leaving the rest of σ (H1 ) in its exterior and such that (see (5)) dist (γ , σ (H1 )) ≥ C, for some constant C > 0. We also set: 1 !1 = (ζ − H1 )−1 W (ζ − H1 )−1 dζ 2π i γ
(29)
and, for any ν ∈ C 1 (R), ˜ )!1 . !ν(τ ) = !0 + ων(τ
(30)
From the definition, from (5) and (21) and since 1 H1 !1 = −W !0 + ζ (ζ − H1 )−1 W (ζ − H1 )−1 dζ, 2π i γ then it follows that !1 χ 1 ≤ C χ 1 , for any χ ∈ D(H1 ). We look for a solution ϕ of the linear equation (27) of the form ϕ(τ ) = !ν(τ ) b+ (τ )ϕ+ + b− (τ )ϕ− . For such a choice of ϕ and from the definition of ν we have ν(τ ) = ϕ, W ϕ = ν0 + ωα(τ ˜ )ν(τ ) + ω˜ 2 β(τ )ν 2 (τ ) , h¯ ω˜ h¯ ω˜
(31)
(32)
(33)
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where 2
ν0 =
b¯s(>) bs(> ) ϕs(> ) , W ϕs(>) = 2 b+ b¯− ρ0 ,
>, > =1
s(1) = + and s(2) = −, since (3), and α(τ ) and β(τ ) are functions independent of ω˜ given by: α=
2
bs(> ) b¯s(>) αs(> ),s(>) , β =
>, > =1
2
bs(> ) b¯s(>) βs(> ),s(>) ,
>, > =1
where α±,± = ϕ± , W !1 ϕ± + !1 ϕ± , W ϕ± , β±,± = !1 ϕ± , W !1 ϕ± . From this fact and since following behavior:
h¯ ω˜
= O(1), it follows that ϕ(τ, ·) ≤ C and ν satisfies the
ν, ν = O(1), uniformly w.r. to ω˜ > 0 small enough and τ ≥ 0,
(34)
provided that the unknown functions b± and their first derivative are bounded uniformly with respect to τ and ω. ˜ Now, observing that: i ω˜ ∂ ˜ !ν = K, + H1 + ωνW, − 2 ∂τ where K=−
i ω˜ 2 ˜ ([H1 , !1 ] + [W, !0 ]) + ω˜ 2 ν 2 [W, !1 ] ν !1 + ων 2
is such that Kχ 1 ≤ C ω˜ 2 χ 1 since (34), [H1 , !1 ] + [W, !0 ] = 0, by definition of !1 , and since H˜ 1 W H˜ 1−1 is a bounded operator. By inserting (32) into (27) we obtain that b+ (τ ) and b− (τ ) must satisfy to the following equation: i ω˜ E± !ν(τ ) {c+ ϕ+ + c− ϕ− } = φ, c± = − b± + ωνW ˜ b± , + (35) h¯ 2 where
and ν =
b± (0) = a± + O(ω), φ = −K(b+ ϕ+ + b− ϕ− ), φ(τ, ·) 1 ≤ C ω˜ 2 , ∀τ ≥ 0,
h¯ ω˜ ϕ, W ϕ
= ν0 + O(ω) ˜ has to satisfy (34).
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Now, we have that !ν(τ ) (W ϕ± ) = !0 W ϕ± + ων(τ ˜ )!1 W ϕ± , where !0 W ϕ± = ϕ+ , W ϕ± ϕ+ + ϕ− , W ϕ± ϕ− and ων(τ ˜ )!1 W ϕ± 1 ≤ C ω W ˜ ϕ± 1 ≤ C ω, ˜ since (31). Moreover, let (ζ − H1 )−1 K1 = (ζ − H1 − ωνW ˜ ˜ )−1 − (ζ − H1 )−1 − (ζ − H1 )−1 ωνW (ζ − H1 )−1 where
−1 ˜ − 1 − ωνW ˜ K1 = 1 − ωνW (ζ − H1 )−1 (ζ − H1 )−1 −1 = ω˜ 2 ν 2 (ζ − H1 )−1 W (ζ − H1 )−1 W 1 − ωνW ˜ (ζ − H1 )−1
is such that for any ζ ∈ γ then K1 χ 1 ≤ C ω˜ 2 χ 1 . From this it follows that !ν(τ ) =
1 2π i
γ
˜ )−1 dζ + K2 , (ζ − H1 − ωνW
where K2 χ 1 ≤ C ω˜ 2 χ 1 ; hence we can write that: !2ν(τ ) = !ν(τ ) + K3 , K3 χ 1 ≤ C ω˜ 2 χ 1 . Therefore:
i ω˜ E+ !ν(τ ) c+ ϕ+ = !ν(τ ) − b+ + + ωνW ˜ b + ϕ+ h¯ 2 i ω˜ E+ + ωνW ˜ b + ϕ+ + φ1 = !2ν(τ ) − b+ + h¯ 2 i ω˜ 1 = !ν(τ ) − b+ + E+ b+ ϕ+ + ωνb ˜ + ϕ− , W ϕ+ ϕ− + φ2 , h¯ 2
where φ> 1 ≤ C ω˜ 2 , > = 1, 2, and ϕ+ , W ϕ+ = 0. Therefore, (35) can be re-written as: !ν(τ ) {d+ ϕ+ + d− ϕ− } = φ3 , d± = −
i ω˜ 1 ˜ 0 b∓ , b + E± b± + ωνρ 2 ± h¯
where 0 b± (0) = a± + O(ω), ˜ φ3 (τ, ·) 1 = O(ω˜ 2 ), ∀τ ≥ 0,
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with ρ0 = ϕ+ , W ϕ− ∈ R. As a result it is enough to find b± , bounded together with their first derivative for any τ , such that ˜ 0 ρ 0 b− = 0 − i ω˜ b + 1 E b + ων 2 + h¯ + + + 1 E b + ων (36) ˜ 0 ρ 0 b+ = 0 . − i2ω˜ b− h¯ − − b (0) = a 0 , ν = 2 (b b¯ ρ ) ± 0 + − 0 ± h¯ ω˜
Setting: 1 1 aR = √ (b+ + b− ) and aL = √ (b+ − b− ) 2 2 the system (36) becomes + aR + 21 ωa ˜ L − ων ˜ 0 ρ0 aR − i2ω˜ aR = − E−2+E h¯ E +E i ω ˜ 1 − a = − − + aL + ωa ˜ 0 ρ0 aL 2 L 2h¯ 2 ˜ R + ων , 2 − |a |2 ) ν = ρ (|a | 0 0 R L h ω ˜ ¯ aR (0) = √1 (a 0 + a 0 ) and aL (0) = √1 (a 0 − a 0 ) + − + − 2
(37)
2
and we look for a solution of the form: aR (τ ) = AR (τ )e−i(E+ +E− )τ/h¯ ω˜ , aL (τ ) = AL (τ )e−i(E+ +E− )τ/h¯ ω˜ with AR and AL independent of ω. Then (37) is transformed into the correspondent system: iAR = −AL + 2ν0 ρ0 AR , (38) iAL = −AR − 2ν0 ρ0 AL where ν0 =
ρ0 (|AR |2 − |AL |2 ), AR,L (0) = aR,L (0). h¯ ω˜
It easy to verify that |AL (τ )|2 +|AR (τ )|2 = 1 since ρ0 ∈ R; hence, the solutions AR,L (τ ) exist for any τ and they are bounded, together with their first derivative, uniformly with respect to τ and ω˜ small enough since h¯ω˜ = O(1). Then (34) will be satisfied uniformly with respect to ω˜ (actually (38) is independent of ω). ˜ From these facts and by (30), (32) and Lemma 1 then the solution of (19)–(20) satisfies the estimates (15) and (16). Theorem 3 is proved. ! 3.3. Dynamics of the two-level system. In order to study the system of Eqs. (17) we re-write it in the form iAR = −AL + 2µ|AR |2 AR AR,L (0) = aR,L (0) , (39) , iAL = −AR + 2µ|AL |2 AL |AR (τ )|2 + |AL (τ )|2 = 1 where denotes the derivative with respect to τ , c = 2ρ02 , µ = c ω plays the role of the parameter of non-linearity and we re-define AR,L (τ ) up to a phase factor, i.e. AR,L (τ ) → AR,L (τ )ei2cτ .
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Remark 4 (Gross–Pitaevskii equation). If the perturbation term has the form f (x, ψ) = |ψ(x)|2 W (x),
(40)
where W (x) is a given real-valued even function: W (x , −xn ) = W (x), then Eq. (1) takes the form of the Gross–Pitaevskii equation [1] and we have the conservation of the energy defined below: 1 H = H0 ψ, ψ + ψ 2 , W ψ 2 . 2 In particular, the same arguments given above prove that the two-level system for the Gross–Pitaevskii equation takes the form (39), where c = ϕR , W |ϕR |2 ϕR and where the function W is such that this scalar product is defined. In discussing two-level systems we have characterized the states in terms of AR (τ ) = p(τ )eiα(τ ) and AL (τ ) = q(τ )eiβ(τ ) ,
(41)
where p, q, α and β are real-valued functions, 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1. From the redundancy of the common phase factor we have that the state can be described now by means of a vector in an abstract Euclidean three-dimensional space with components (p, q cos(β−α), q sin(β−α)). In particular, from the normalization condition p 2 +q 2 = 1, it belongs, in such an Euclidean space, to the surface of the sphere. Hence, in order to study the solution of the two-level system (39) we represent the surface of the sphere by means of a Mercator-type chart; that is by means of two real coordinates (P , z), where P = p2 ∈ [0, 1] is the square of the modulus of AR and z = α − β ∈ T = R/2π Z = [0, 2π) belongs to the one-dimensional torus and represents the difference between the phases of AR and AL (see Ch. 13, [9]). We underline that this representation is singular at P = 0 and P = 1; in fact, for P = 0 (respectively P = 1) and any z we have localization on the left-hand (respectively right-hand) well. If the non-linear term is absent in Eq. (39), then P (τ ) is a periodic function with period π and, if initially P (0) = 0 or P (0) = 1, then P (τ ) periodically assumes the values 0 and 1. The system of equations (39) has been recently studied [13]. Here, we recall the most relevant results. Lemma 2. et P (τ ) = p 2 (τ ) and z(τ ) = α(τ ) − β(τ ); then P (τ ) and z(τ ) satisfy the following system of ordinary differential equations: √ √ P = 2 P 1− P sin z . (42) cos z z = (1 − 2P ) 2µ + √ √1 P 1−P
Equations (42) have four stationary solutions (I) (P = 1/2, z = 0) , (II) (P = 1/2, z = π ) , 1 + 1 − 1/µ2 ,z = (III) P = 2 1 − 1 − 1/µ2 (IV) P = ,z = 2
π 1+ 2 π 1+ 2
|µ| µ |µ| µ
, if |µ| ≥ 1, , if |µ| ≥ 1,
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where, for |µ| < 1, (I) and (II) are center points while, for µ > 1 (respectively µ < −1), the stationary solutions (I) (respectively (II)), (III) and (IV) are center points and the stationary solution (II) (respectively (I)) is a saddle point. Moreover, the function √ √ √ √ I = I (P , z, µ) = P 1 − P µ P 1 − P + cos z (43) is an integral of motion and the dynamics of the two-level system, with initial condition (P0 , z0 ), could be described by means of the integral path defined by the implicit equation I (P , z, µ) = I (P0 , z0 , µ). In particular, we consider the following two behaviors: [C1] P (τ ) is a periodic continuous function, with given period τB , such that P (τ ) = 21 , for τ = τ˜ , τ˜ + 21 τB , for some τ˜ , and P (τ ) < 21 and P (τ + 21 τB ) > 21 for any τ ∈ (τ˜ , τ˜ + 21 τB ). [C2] P (τ ) is a periodic continuous function such that P (τ ) = 21 for any τ . We have that: Lemma 3. Let (P0 , z0 ) ∈ [0, 1] × T be the initial state in the two-level representation. We have: (i) if |µ| ≤ 1, then P (τ ) has a time behavior of type C1 for any initial condition (P0 , z0 ), but the ones corresponding to the stationary solutions (I) and (II) (see Fig. 1);
1
0.8
0.6
P 0.4
0.2
0 -3
-2
-1
0
1
2
3
z
Fig. 1. Integral paths of the equation I (P , z, µ) = I˜ for some values of I˜ and for µ = − 21 fixed. The bold line represents the integral path of the beating motion, that is the transition from localization on a well to localization on the other one. Localization on the right-hand (respectively left-hand) well occurs at P = 1 (respectively P = 0) for any z
Destruction of Beating Effect for Non-Linear Schrödinger Equation
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1
0.8
0.6
P 0.4
0.2
0 -3
-2
-1
0
1
2
3
Fig. 2. Integral paths of equation I (P , z, µ) = I˜ for some values of I˜ and for µ = − 23 fixed. We observe the stability of the beating motion (bold line) despite the appearance of the bifurcation of one fixed point. Broken lines represent the two sepratrices; inside the region enclosed by these lines we have closed paths, around the asymmetrical stationary state originated from the bifurcation of the fundamentals state, representing periodic oscillations within only one well
(ii) if |µ| > 1, let D = D(µ) be the bounded open set enclosed by the path with equation π |µ| 1 + 2µP (1 − P ) − µ/2 z= 1+ ± arccos (44) √ √ 2 µ 2 P 1−P and containing the stationary solutions (III) and (IV); then for any (P0 , z0 ) ∈ D, (P0 , z0 ) different from the stationary solutions (III) and (IV), P (τ ) has a behavior of ¯ where D¯ denotes the closure of D, and (P0 , z0 ) type C2; in contrast, if (P0 , z0 ) ∈ / D, is different from the stationary solution (I), then P (τ ) has a behavior of type C1 (see Figs. 2 and 3). Remark 5. Let (P0 , z0 ) be such that P0 = 0 or P0 = 1. Then I (P0 , z0 , µ) = 0 and ¯ if |µ| < 2. Hence, for |µ| < 2 we observe (P0 , z0 ) ∈ D, if |µ| > 2, and (P0 , z0 ) ∈ / D, the beating motion, such that P (τ ) periodically assumes the values 0 and 1 (see the bold line in Fig. 2). The beating motion corresponds to the path with equation I (P , z, µ) = 0; that is: √ √ π |µ| zf b = 1− ± arccos |µ| P 1 − P . 2 µ In contrast, for 2 < |µ| we have that the beating motion between the two wells is not possible (see the bold line in Fig. 3, where µ = − 25 ); in particular, if initially P (0) = 1
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1
0.8
0.6
P 0.4
0.2
0 -3
-2
-1
0
1
2
3
z
Fig. 3. Integral paths of equation I (P , z, µ) = I˜ for some values of I˜ and for µ = − 25 fixed. We observe the destruction of the beating motion. The trajectory (bold line) starting from the localization point corresponding to P = 1 (respectively P = 0) stays in the region P > 21 (respectively P < 21 ) and it encircles one asymmetrical stationary state originated from the bifurcation of the fundamental state
(respectively P (0) = 0) then during the motion we have that P (τ ) > P (τ ) < 21 ) for any τ .
1 2
(respectively
As a result of Lemma 3, it follows that we generically observe a periodic motion with period τB that depends on the parameter µ and on the initial condition (P0 , z0 ). In particular: ¯ where the set D is defined in Lemma 3, then the beating / D, Lemma 4. If (P0 , z0 ) ∈ motion between the two wells has period given by
√ (µ−4I +2)(µ−4I −2) √ EK µ µ 2 −(1+ 1+4µI )2 τB = τB (I, µ) = 4 , (45) √ (1 + 1 + 4µI )2 − µ2 where I = I (P0 , z0 ; µ) and EK is the complete elliptic integral of the first kind. We close this section with the following remarks. Remark 6. If (P0 , z0 ) ∈ D then we have a periodic motion within one well with period: √ √ x2 µ x1 x2 x2 τB = −2i √ EF − EK , (46) , √ √ µ x1 x2 µ x1 µ x1 where EF denotes the incomplete elliptic integral of the first kind and where x1 = µ2 − 4 − 8µI + 16I 2 and x2 = µ2 − (1 + 4µI + 1)2 .
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Remark 7. The frequency ν f b = 1/τ f b of the beating motion, corresponding to the value I = 0 of the integral of motion, depends on |µ| and monotonically decreases and vanishes at |µ| = 2; indeed, we have that 4 − µ2 fb
ν = 4EK iµ/ 4 − µ2 which is a monotone decreasing function as µ ∈ [0, 2). From formulas (106.02) and (112.01) [3], it follows that νf b ∼
1 1 as |µ| → 2− . 2 ln(8/ 4 − µ2 )
We remark also that the range of frequencies is given by (νmin , νmax ], where 1√ 1 − |µ|, if |µ| < 1 νmin = π 0, if |µ| ≥ 1 and νmax =
1 1 + |µ|, for any µ. π
In particular we observe that the interval (νmin , νmax ] broadens as µ increase. 3.4. Beating destruction for large non-linearity. Now, we complete the proof of Theorem 2 and of the corollary. To this end we remark that the energy has the form 1 H = ψ, H0 ψ + ψ, W ψ 2 , 2 where, in order to take into account the contribution due to the term ψc , we observe that
W t = ψ, W ψ = |aR |2 − |aL |2 ϕR , W ϕR + R1 + R2 , where R1 = aR a¯ L ϕR , W ϕL + a¯ R aL ϕL , W ϕR +|aL |2 (ϕL , W ϕL + ϕR , W ϕR ) , R2 = ψc , W ψc + aR ϕR , W ψc + a¯ R ψc , W ϕR +aL ϕL , W ψc + a¯ L ψc , W ϕL , and 1 ψ, H0 ψ = &(|aR |2 + |aL |2 ) − ω(aR a¯ L + aL a¯ R ) + R3 , 2 where R3 = ψc , H0 ψc .
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From Theorem 3 we have that aR (t) and aL (t) are such that for any α < 1,
1 − |aR (t)|2 + |aL (t)|2 = ψc (t, ·) 2 = O(ω˜ 2α ) for any t ∈ [0, (τ , /ω) ˜ ln(1/ω)], ˜ for some fixed τ , , and sup ˜ |aR,L (t)| − |AR,L (t ω/2)| = O(ω˜ α ), t∈[0,(τ , /ω) ˜ ln(1/ω)] ˜
where AR,L (τ ) are computed in Sects. 3.3. From (3) and since the wave-functions ϕR,L are localized on just one well [8], it follows that R1 = O(ω), as h¯ → 0,
(47)
for any t ≥ 0. Moreover, making use of Theorem 3, we have that R2 = O(ω˜ α ) and R3 = O(ω˜ 2α )
(48)
for any τ ∈ [0, (τ , /ω) ˜ ln(1/ω)] ˜ and for some τ , > 0. From these facts and from (41) then it follows that for any t ∈ [0, (τ , /ω) ˜ ln(1/ω)] ˜ we have that W t = (2P − 1) ϕR , W ϕR + O(ω˜ α ), where P = |AR |2 is the periodic solution given in Lemma 3. Then W t is, up to an error of order O(ω˜ α ), a periodic function with period T given in Lemma 4. If we remark that 1 H = & + ωµ − ωI (P , z, µ) + O(ω˜ 2α ), 4 ¯ From this where we choose α > 21 , then we have that (11) implies that (P0 , z0 ) ∈ / D. fact and from the stability result the beating motion between the two wells follows. In contrast, (10) implies that (P0 , z0 ) ∈ D; hence, the beating motion disappears. In particular, we observe that the energy corresponding to the beating motion with initial condition P0 = 0 (or P0 = 1) is such that Hf b ≈ & + 41 µω. Hence, the beating motion disappears for |µ| > 2. Theorem 2 and the corollary are proved. Acknowledgement. This work is partially supported by the Italian MURST and INDAM-GNFM. V.G. and A.M. are supported by the University of Bologna (funds for selected research topics). V.G. is also supported by the INFN.
References 1. D’Agosta, R., Malomed, B.A., Presilla, C.: Stationary solutions of the Gross–Pitaevskii equation with linear counterpart. Phys. Lett. A 275, 424–434 (2000) 2. Bourgain, J.: Global solutions of nonlinear Schrödinger equations. AMS - Coll. Publ. 46 (1999) 3. Byrd, P.F., Friedman, M.D.: Handbook of elliptic integrals for engineers and physicists. Berlin: SpringerVerlag, 1954 4. Claviere, P., Jona Lasinio, G.: Instability of tunneling and the concept of molecular structure in quantum mechanics: the case of pyramidal molecules and the enantiomer problem. Phys. Rev. A 33, 2245–2253 (1986) 5. Davies, E.B.: Symmetry breaking for a non-linear Schrödinger operator. Commun. Math. Phys. 64, 191–210 (1979) 6. E.B. Davies: Nonlinear Schrödinger operators and molecular structure. J. Phys. A: Math. and Gen. 28, 4025-4041 (1995)
Destruction of Beating Effect for Non-Linear Schrödinger Equation
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7. Grecchi, V., Martinez, A.: Non-linear Stark effect and molecular localization. Commun. Math. Phys. 166, 533–548 (1995) 8. Helffer, B., Sjöstrand, J.: Multiple wells in the semi-classical limit I. Comm. P.D.E. 9, 337–408 (1984) 9. Merzbacher, E.: Quantum Mechanics. New York: Wiley Int. Ed., 2nd edition, 1970 10. Nenciu, G.: Adiabatic theorem and spectral concentration. Commun. Math. Phys. 82, 121–135 (1981/82) 11. Nenciu, G.: Linear adiabatic theory, exponential estimates. Commun. Math. Phys. 152, 479–496 (1993) 12. Pratt, R.F.: Spontaneous deformation of hydrogen atom shape in an isotropic environment. J. Phys. France 49, 635–641 (1988) 13. Raghavan, S., Smerzi, A., Fantoni, S., Shenoy, S.R.: Coherent oscillations between two weakly coupled Bose–Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping. Phys. Rev. A 59, 620–633 (1999) 14. Simon, B.: Semi-classical analysis of low lying eigenvalues I. Ann. I.H.P. 38, 295–307 (1983) 15. Simon, B.: Semi-classical analysis of low lying eigenvalues II: tunneling. Ann. of Math. 120, 89–118 (1984) 16. Sjöstrand, J.: Projecteurs adiabatique du point de vue pseudo-différentiel. C.R. Acad. Sci. Paris 317, Série I, 217–220 (1993) 17. Vardi, A.: On the role of intermolecular interactions in establishing chiral stability. J. Chem. Phys. 112, 8743–46 (2000) Communicated by A. Kupiainen
Commun. Math. Phys. 227, 211 – 241 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Weighted Bergman Kernels and Quantization Miroslav Engliš ˇ Žitná 25, 11567 Prague 1, Czech Republic. E-mail:
[email protected] MÚ AV CR, Received: 29 December 2000 / Accepted: 14 December 2001
Abstract: Let be a bounded pseudoconvex domain in CN , φ, ψ two positive functions on such that − log ψ, − log φ are plurisubharmonic, and z ∈ a point at which − log φ is smooth and strictly plurisubharmonic. We show that as k → ∞, the Bergman kernels with respect to the weights φ k ψ have an asymptotic expansion ∞
kN bj (x, y) k −j , Kφ k ψ (x, y) = N π φ(x, y)k ψ(x, y) j =0
for x, y near z, where φ(x, y) is an almost-analytic extension of φ(x) = φ(x, x) and similarly for ψ. Further, b0 (x, x) = det[−∂ 2 log φ(x)/∂xj ∂x k ]. If in addition is of finite type, φ, ψ behave reasonably at the boundary, and − log φ, − log ψ are strictly plurisubharmonic on , we obtain also an analogous asymptotic expansion for the Berezin transform and give applications to the Berezin quantization. Finally, for smoothly bounded and strictly pseudoconvex and φ a smooth strictly plurisubharmonic defining function for , we also obtain results on the Berezin–Toeplitz quantization. Introduction Let be a domain in CN , ρ a positive continuous function on , and Kρ the reproducing kernel of the weighted Bergman space A2 (, ρ) of all holomorphic functions on square-integrable with respect to the measure ρ(z) dz, dz being the Euclidean volume element in CN ; we call Kρ the weighted Bergman kernel corresponding to ρ, and for ρ ≡ 1 we will speak simply of the Bergman kernel K of . The Berezin transform Bρ is the integral operator defined by |Kρ (x, y)|2 Bρ f (y) = f (x) ρ(x) dx (1) Kρ (y, y) The author’s research was supported by GA CR ˇ grant no. 201/00/0208 and GA AV CR ˇ grant A1019005.
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for all y for which Kρ (y, y) = 0. In terms of the operator Mf of multiplication by f on the space L2 (, ρ dz) this can be rewritten as Bρ f (y) =
Mf Kρ (·, y), Kρ (·, y) , Kρ (·, y)2
from which it is immediate that the integral (1) converges, for instance, for any bounded measurable function f . The Berezin transform was first introduced by F. A. Berezin [Ber] in the context of quantization of Kähler manifolds. More specifically, let φ be a positive function on such that − log φ is strictly plurisubharmonic, and set gj k = ∂ 2 (− log φ)/∂zj ∂zk
(2)
and χ = det(gj k ) (so that ds 2 = gj k dzj dzk is the Kähler metric with potential − log φ and χ the corresponding volume density). For a bounded symmetric domain in CN and φ(z) = 1/K (z, z) (so that ds 2 is the Bergman metric), Berezin showed that for all m ≥ 1 it holds that Kφ m χ (x, y) = p(m)φ(x, y)−m ,
(3)
where φ(x, y) = 1/K (x, y) is a function on × holomorphic in x, y such that φ(x, x) = φ(x), and p is a polynomial of degree N which depends only on ; and that 1 1 ˜ (y) + O Bφ m χ f (y) = f (y) + f (4) m m2 ˜ is the Laplace-Beltrami operator of the metric ds 2 on . Using (4), as m → ∞, where he was then able to construct a nice quantization procedure for mechanical systems whose phase-space is with the Bergman metric. Later the present author showed that to get (4) it suffices that (3) holds only asymptotically as m → ∞ in a certain sense and used this to extend the range of applicability of Berezin’s original procedure to all plane domains with the Poincaré metric [E1], to some complete Reinhardt domains in C2 with natural rotation-invariant Kähler metrics [E2], and finally to any strictly pseudoconvex domain with real-analytic boundary and φ a real-analytic defining function for such that − log φ is strictly plurisubharmonic [E6]. In fact, [E6] even dealt with the more general setting of weights of the form ρ = φ m ψ M with −φ, −ψ two C ω defining functions of a strictly pseudoconvex domain such that − log φ, − log ψ are plurisubharmonic, M fixed and m → ∞. Then (3), with φ m ψ M in place of φ m χ , holds asymptotically for (x, y) near the diagonal, and (4) holds for any f ∈ L∞ () which is smooth in a neighbourhood of y. The aim of the present paper is to improve these results by relaxing the hypotheses of real-analyticity and of φ, ψ being defining functions. For a function f on a domain in Cn , we say that f is almost analytic at x = a if ∂f/∂x j , j = 1, . . . , n vanish at a together with their partial derivatives of all orders. It is known that any C ∞ function φ(x) possesses a (non-unique) almost analytic extension φ(x, y) such that φ(x, y) is almost-analytic in x and y at all points of the diagonal x = y, and φ(x, x) = φ(x). Further, if φ(x) is real-valued, then the extension may be chosen so that φ(y, x) = φ(x, y) (just replace φ(x, y) by 21 (φ(x, y) + φ(y, x))); in the sequel, we will always assume that an extension with this additional property has been chosen for a real-valued φ(x). We now have the following results.
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Theorem 1. Let be a bounded pseudoconvex domain in CN , φ, ψ two bounded positive continuous functions on such that − log φ, − log ψ are plurisubharmonic, and let x0 ∈ be a point in a neighbourhood of which φ and ψ are C ∞ and − log φ is strictly plurisubharmonic. Fix an integer M ≥ 0. Then there is a smaller neighbourhood U of x0 such that the asymptotic expansion Kφ k ψ M (x, y) =
∞ kN · bj (x, y)k −j π N φ(x, y)k ψ(x, y)M
(5)
j =0
holds uniformly for all x, y ∈ U as k → ∞, in the sense that for each m > 0, sup φ(x)k/2 φ(y)k/2 Kφ k ψ M (x, y) x,y∈U
k N φ(x)k/2 φ(y)k/2 − N π φ(x, y)k ψ(x, y)M
N+m−1
bj (x, y)k
j =0
−m = O(k )
(6)
−j
as k → ∞. Here φ(x, y), ψ(x, y) are fixed almost-analytic extensions of φ(x) and ψ(x) to U × U, respectively. The coefficients bj (x, y) ∈ C ∞ (U × U) are almost-analytic at x = y, and their jets at a point (x, x) on the diagonal depend only on the jets of φ and ψ at x. In particular, 1 . (7) b0 (x, x) = det ∂∂ log φ(x) In the situation of the last theorem, consider the domain = {(z1 , z2 , z3 ) ∈ × CM × C :
|z3 |2 |z2 |2 + < 1}. φ(z1 ) ψ(z1 )
(8)
Recall that for domain D in Cn , a boundary point z ∈ ∂D is called smooth if in some neighbourhood of z, ∂D is a C ∞ -submanifold of Cn ; the domain D is called smoothly bounded if it is bounded and all its boundary points are smooth. A smooth boundary point z is said to be of finite type ≤ m if there is no complex analytic variety passing through z which has order of contact with ∂D at z bigger than m. (Thus, for instance, a strictly pseudoconvex boundary point is of type 2.) Finally, a smoothly bounded domain is said to be of finite type if all its boundary points are of finite type. Theorem 2. Assume that the hypotheses of Theorem 1 are fulfilled, and that in addition is smoothly bounded and of finite type. (This implies, in particular, that φ, ψ ∈ C ∞ ().1 ) Then for any f ∈ L∞ () which is C ∞ in a neighbourhood of x0 , there is an asymptotic expansion Bφ k ψ M f (y) =
∞
Qj f (y) · k −j ,
(9)
j =0
uniformly for all y in a neighbourhood of x0 , where Qj are linear differential operators whose coefficients involve only the derivatives of φ, ψ at y and Q0 is the identity operator. 1 But not necessarily C ∞ ()!
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We remark that in the applications to the Berezin quantization, − log φ is the potential of the Kähler metric, and thus is automatically strictly plurisubharmonic on all of . Strictly speaking, the nonnegative integer M in Theorem 1 is redundant (one can rechristen ψ M to ψ and take M = 1); we preferred to keep it since the same weights φ k ψ M appear in Theorem 2, where M already seems to be essential (we do not know whether the hypotheses of Theorem 2 are also satisfied for the pair ψ M , 1 if they are satisfied for ψ, M). We should also remark that the hypotheses of smooth boundedness and finite type of are still not completely satisfactory – although they cover a lot of situations, they still exclude some important cases like, for instance, the Berezin quantization of a general (i.e. non-symmetric) domain with the Bergman or the Cheng–Yau metric; cf. Remark (2) after Theorem 11 below. The whole approach can also be adapted to arbitrary Kähler manifolds in place of domains in CN [Pe], and sections of line bundles in place of functions. The function φ then defines the metric structure of the line bundle, and ∂∂ log φ is the corresponding curvature form. For compact Kähler manifolds and − log φ strictly plurisubharmonic on all of (i.e. the line bundle of strictly negative curvature) the analogue of Theorem 1 has been obtained independently by Zelditch [Ze] for x = y and by Catlin [Ca] for general x, y; and the analogue of Theorem 2 in this setting was established by Karabegov and Schlichenmaier [KS]. In [BMS] and [Sch] the authors also obtain (still in the context of compact manifolds) somewhat stronger results concerning the Berezin–Toeplitz quantization, and we finish by observing that the same results can also be obtained in our noncompact situation. (ρ) Recall that, quite generally, the Toeplitz operator TF with symbol F ∈ L∞ () is the operator on A2 (, ρ) given by the recipe (ρ) f (x)F (x)Kρ (y, x)ρ(x) dx, TF f (y) =
or, equivalently, (ρ)
TF f = Pρ (Ff ), where Pρ is the orthogonal projection of L2 (, ρ) onto A2 (, ρ). For simplicity, we state our result on the Berezin–Toeplitz quantization only for the weights which are of most interest to us, viz. ρ = φ m χ with χ = det[−∂∂ log φ]. Theorem 3. Let be a smoothly bounded strictly pseudoconvex domain in CN and −φ a smooth defining function for = {φ > 0} such that − log φ is strictly plurisubharmonic. Then: (φ m χ)
→ f ∞ as m → ∞; (i) for any f ∈ C ∞ (), Tf (ii) there exist bilinear operators Cj : C ∞ () × C ∞ () → C ∞ () (j = 0, 1, 2, . . . ) such that for any f, g ∈ C ∞ () and any integer k, k m (φ χ) (φ m χ) −j (φ m χ) T − m TCj (f,g) = O(m−k−1 ) Tf g
(10)
j =0
as m → ∞. Further, C0 (f, g) = f g and C1 (f, g) − C1 (g, f ) = i{f, g}, the Poisson bracket of f and g with respect to the metric (2).
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The result of the kind appearing in Theorem 3 was first obtained for a domain in C with the Poincaré metric by Klimek and Lesniewski [KL], using uniformization techniques, and for a bounded symmetric domain with the invariant (Bergman) metric and = Cn with the Euclidean metric by Borthwick, Lesniewski and Upmeier [BLU] and Coburn [Co], respectively, in both cases with the aid of the computational machinery available thanks to the specific nature of the domain and metric. For a compact manifold with an arbitrary Kähler metric, Theorem 3 was proved by Bordemann, Meinrenken and Schlichenmaier [BMS]. In our case (i) is a fairly straightforward consequence of Theorem 2, while (ii) follows, as in [BMS] and [Sch], from the Boutet de Monvel–Guillemin calculus of generalized Toeplitz operators [BG]; see also [Gu]. It turns out that Cj are, in fact, differential operators (see Corollary 15); for compact Kähler manifolds, this was proved in [KS]. As in [E6], our method of proof of Theorem 1 is based on the analysis of the Bergman of the Forelli-Rudin domain (8) over ; (5) is then obtained from Fefferman’s kernel K near the boundary. This is done in Sect. 2, after establishing asymptotic expansion of K some localization theorems for the Bergman kernel in Sect. 1. Theorem 2 is proved in Sect. 3, and its applications to quantization are described in Sect. 4. The Berezin– Toeplitz quantization is discussed in Sect. 5. At the end of each section we provide various remarks, comments on related developments, open problems, etc. It is perhaps appropriate to point out briefly what are the new ingredients in Sects. 1– 3 here against [E6]. In [E6], we proved a stronger assertion than (6) (recalled in (24) below) under stronger hypotheses on , φ and ψ. Here, by enhancing the treatment of the technical matters (cf. Lemmas 7 and 8 in Sect. 2), we prove the weaker assertion (6) under weaker hypotheses, and then show that (6) is still sufficient to yield the conclusion of Theorem 2 under the additional assumption of smooth boundedness and finite type . of Throughout the paper, “psh” is an abbreviation for “plurisubharmonic”. 1. Preliminaries Our starting point is the following proposition, reproduced here from [E6] (see also [BFS]), which relates the weighted Bergman kernels Kφ k ψ M on to the unweighted . Bergman kernel of the domain Proposition 4. Let be an arbitrary domain in CN (it need not be bounded), φ, ψ two the domain defined positive continuous functions on , M a nonnegative integer, and by (8). Then the Bergman kernel K := K of is given by t) = K(z;
∞ (k + l + M + 1)! Kψ l+M φ k+1 (z1 , t1 ) z2 , t2 l (z3 t 3 )k . k! l! π M+1
(11)
k,l=0
. The series converges uniformly on compact subsets of is pseudoconvex if and Note that by the familiar criterion for Hartogs domains, only if is pseudoconvex and − log φ, − log ψ are psh. Proof. Arguing as in [Lig], Proposition 0 shows that β β t) = K(z; Kwαβ (z1 , t1 ) z2 t 2 z3α t α3 , α,β
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where the summation is over all multiindices α ∈ N, β ∈ NM , N = {0, 1, 2, . . . }, and
wαβ (z1 ) =
(z2 , z3 ) :
= Since
|α|=k
|z2 |2 ψ(z1 )
+
|z3 |2 φ(z1 )
<1
|zβ |2 |z3α |2 dz2 dz3 2
π M+1 α! β! φ(z1 )|α|+1 ψ(z1 )|β|+M . (|α| + |β| + M + 1)!
x α y α /α! = x, y k /k!, the required assertion follows.
The construction similar to (8) was first used by Forelli and Rudin [For, FR, Rud]; for other applications, see [Lig, KLR] and the references therein. Let us recall the asymptotic formula for the boundary behaviour of the Bergman kernel due to Fefferman [Fef] and Boutet de Monvel–Sjöstrand [BS]. Let be a strictly pseudoconvex domain in Cn with smooth boundary and −φ a C ∞ defining function for , i.e. = {z : φ(z) > 0}, φ is C ∞ in a neighbourhood of , ∇φ = 0 on ∂, and the Levi matrix (−∂ 2 φ/∂zj ∂zk ) is positive definite on the complex tangent space (the last condition is equivalent to the Monge-Ampére matrix in (13) below having n positive and 1 negative eigenvalue, for any z ∈ ∂). Then there exist functions a(x, y), b(x, y), φ(x, y) ∈ C ∞ (Cn × Cn ) such that (a) a a(x, y), b(x, y), φ(x, y) are almost-analytic in x, y in the sense that ∂φ(x, y)/∂x and ∂φ(x, y)/∂y have a zero of infinite order at x = y, and similarly for a(x, y) and b(x, y); (b) φ(x, x) = φ(x); (c) for x ∈ ∂, a(x, x) =
n! J [φ](x) > 0, πn
(12)
where J [φ] is the Monge-Ampére determinant
−φ −∂φ/∂zk J [φ] = − det −∂φ/∂zj −∂ 2 φ/∂zj ∂zk
(13)
whose positivity follows from the strong pseudoconvexity of ∂; (d) the Bergman kernel of is given by the formula K(x, y) =
a(x, y) + b(x, y) log φ(x, y) φ(x, y)n+1
(14)
for (x, y) ∈ 4 = {|x − y| < 4, dist(x, ∂) < 4}, where 4 > 0 is sufficiently small; (e) outside any 4 the Bergman kernel is C ∞ up to the boundary of × ; (f) if the boundary ∂ is even real-analytic, then the functions a(x, y), b(x, y) and φ(x, y) can in fact be chosen to be holomorphic in x, y in a neighbourhood of the boundary diagonal {(x, x); x ∈ ∂} in Cn , and outside any 4 the Bergman kernel is holomorphic in x, y in a neighbourhood of × .
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The original proofs in [Fef] and [BS] deal only with (a)–(e); part (f) is due to Kashiwara [Kas] and Bell [Be1]. Observe that if φ (x, y) is another function satisfying (a) and (b), then h = (φ /φ)−1 vanishes at x = y to an infinite order; thus (14) remains in force with φ and a = (1 + h)n+1 a + φ n+1 b log(1 + h) in the place of φ and a. It follows that even for any function φ(x, y) satisfying (a) and (b) there exist a(x, y), b(x, y) such that the conclusions (a)–(d) hold. This allows us to work with a convenient φ(x, y) in concrete situations later on: for instance, if φ(x) is of the form |x1 |2 + (a function of x2 , . . . , xn ), we can take φ(x, y) = x1 y 1 + (a function of x2 , . . . , xn , y2 , . . . , yn ). We will find convenient the following two (probably well-known) “localization lemmas”, which can be used to obtain a local variant of Fefferman’s theorem (see [E6]). Lemma 5. Let 1 ⊂ be two bounded pseudoconvex domains and U a neighbourhood of a point x0 ∈ ∂ such that U ∩ ∂1 = U ∩ ∂ and the piece of common boundary U ∩ ∂ is smooth and strictly pseudoconvex. Then the difference K1 (x, y) − K (x, y) is C ∞ on (U ∩ 1 ) × (U ∩ 1 ). Lemma 6. Let be a pseudoconvex domain (possibly unbounded) and x0 ∈ ∂ a strictly pseudoconvex point of its boundary. Then there exists a bounded strictly pseudoconvex domain 1 ⊂ such that ∂ and ∂1 coincide in a neighbourhood of x0 . Further, if x0 is a smooth boundary point, then 1 can be chosen to be smoothly bounded. Proof of Lemma 5. For , 1 smoothly bounded and strictly pseudoconvex, this is the content of Lemma 1 on p. 6 in [Fef]. The local version given here follows in the same way by J. J. Kohn’s local regularity theorems for the ∂-operator and subelliptic estimates at x0 ([Ko], Theorems 1.13 and 1.16) by the argument as on p. 469 in [Be2], cf. in particular the formula (2.1) there. Proof of Lemma 6. Let u be a defining function for = {u < 0} strictly-psh in a neighbourhood B(x0 , δ) of x0 (see e.g. [Krn], Proposition 3.2.1). Choose a C ∞ function θ : [0, 1) → R+ such that θ ≡ 0 on [0, 1/2], θ ≥ 0 on [1/2, 1) and θ(1−) = +∞. Set 1 = {x : u(x) + θ (|x − x0 |2 /δ 2 ) < 0}. Then 1 ⊂ ∩ B(x0 , δ), ∂1 coincides with ∂ in B(x0 , δ/2), and as θ ≥ 0, θ(|x − x0 |2 /δ 2 ) is psh, so 1 is strictly pseudoconvex. Finally, if u is C ∞ in B(x0 , δ), then 1 is smoothly bounded. It turns out that the boundedness hypothesis on in Lemma 5 is, in fact, unnecessary: see [E7], Sect. 4, where also the full details of the proof can be found. The conclusion of the lemma fails, however, if U ∩∂ = U ∩∂1 is only assumed to be weakly pseudoconvex: for instance, take = {max(|z1 |, |z2 |) < 1} ⊂ C2 , 1 = {max(|z1 |, 2|z2 |) < 1}, and x0 = (1, 0). Similarly, the hypothesis that be pseudoconvex cannot be dispensed with: an example is 1 = {z ∈ C2 : |z| < 2}, = 1 ∪ {|z1 | < 3, 1 < |z2 | < 3}, x0 = (2, 0). On the other hand, the hypothesis that 1 be pseudoconvex is not needed in the proof and can be omitted (but we won’t have any use for this refinement in the sequel). A similar construction as in Lemma 6 was used by Bell [Be2] (cf. also the references therein). Remark. There is also a “local version” of part (f) of Fefferman’s theorem: namely, if is bounded pseudoconvex and z ∈ ∂ is a strictly pseudoconvex and real-analytic boundary point (i.e. ∂ is a C ω -submanifold of Cn in some neighbourhood of z), then there exists a neighbourhood U of z and functions a(x, y), b(x, y) and φ(x, y) on U ×U ,
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holomorphic in x, y, such that −φ(x, x) is a local defining function for on U and (12) and (14) hold. See [Kan], §9, in particular the Theorem on p. 94. (The author is obliged to M. Kashiwara and Gen Komatsu for this information.) 2. Weighted Bergman Kernels We now use Fefferman’s asymptotic expansion together with Proposition 4 to determine the asymptotics of Kφ m ψ M (z, z) as z and M are fixed and m → ∞. Let us start with two technical lemmas. Let φ(x, y) be a function in C ∞ ( × ) almost-analytic in x, y on the diagonal and such that φ(x, y) = φ(y, x), φ(x, x) =: φ(x) > 0, and − log φ(x) is strictly psh at some point x0 . The last condition implies that there exists c > 0 and a small ball U centered at x0 such that φ(x)φ(y) ≤ 1 − c|x − y|2 |φ(x, y)|2
∀x, y ∈ U.
Denote D = {(x, y, τ ) ∈ U × U × C : |τ |2 <
(15)
φ(x)φ(y) }. |φ(x, y)|2
Then |x − y|2 <
2 |1 − τ | c
∀(x, y, τ ) ∈ D.
(16)
Let G m and G be the set of all functions in C m (D) and C ∞ (D), respectively, which are almost-analytic in x, y, τ at all points (x, x, 1), x ∈ U . Denote (1 − τ )k (k < 0), uk (τ ) = 1 (1 − τ )k log (k ≥ 0). 1−τ Lemma 7. Let f (x, y, τ ) be a function in G which is in fact holomorphic in τ on D. Assume that m f (x, y, τ ) = aj (x, y)uj (τ ) + g(x, y, τ ), j =−n−1
where g ∈ G m . Then the Taylor coefficients fk (x, y) of f with respect to τ satisfy −1 n m (−1)j j +k k fk (x, y) − a (x, y) − a (x, y) −j −1 j k j +1 j +1 j =0 j =0 ∂τm g∞ |φ(x, y)| k ≤ √ k(k − 1) · · · (k − m) φ(x)φ(y) as k → ∞. (Here and elsewhere ∂τm , etc., is a shorthand for ∂ m /∂τ m , etc.)
Bergman Kernels
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Proof. This is immediate from the formulas (j ≥ 0) (1 − τ )
−j −1
=
∞ k+j k
k=0
τ k,
∞ (−1)j j ! 1 (1 − τ ) log = τ j +1 − τ (τ − 1)j + τ k, 1−τ k(k − 1) · · · (k − j )
(17)
j
k=j +1
and the Cauchy estimates for the Taylor coefficients of g r =
√ φ(x)φ(y) : |φ(x,y)|
1 2π ∂ m g iθ −ikθ (x, y, re ) e dθ |(k + 1) · · · (k + m)gk+m (x, y)r k+m | = 2π 0 ∂τ m m ∂ g ≤ ∂τ m . ∞ Lemma 8. Each G ∈ G admits a decomposition G(x, y, τ ) = G0 (x, y) + (τ − 1)g1 (x, y, τ )
(18)
with G0 ∈ C ∞ (U × U ) almost-analytic in x, y on the diagonal, and g1 ∈ G. Proof. Extend G to a C ∞ function on U × U × C (still denoted by G) and set G0 (x, y) := G(x, y, 1),
g1 (x, y, τ ) :=
G(x, y, τ ) − G(x, y, 1) , τ −1
(x, y, τ ) ∈ D.
Clearly the relation (18) is satisfied, and G0 (x, y) is almost-analytic on the diagonal. Thus we only need to show that for any multiindices a, b, c, d and nonnegative integers j, k, the function b d k ∂xa ∂ x ∂yc ∂ y ∂τj ∂ τ g1 (x, y, τ ) extends continuously to the points where τ = 1, and vanishes there unless |b| = |c| = k = 0. For brevity, let us introduce the shorthand b
d
k
Habcdj k := ∂xa ∂ x ∂yc ∂ y ∂τj ∂ τ G. We have G(x, y, τ ) − G(x, y, 1) = 0
1
(τ − 1)∂τ G(x, y, τt ) + (τ − 1)∂ τ G(x, y, τt ) dt
where τt = 1 + (τ − 1)t. Hence 1 τ −1 ∂τ G(x, y, τt ) + ∂ τ G(x, y, τt ) dt. g1 (x, y, τ ) = τ −1 0 b
d j k
Applying ∂xa ∂ x ∂yc ∂ y ∂τ ∂ τ to both sides, the first term in the integrand becomes just t j +k Ha,b,c,d,j +1,k (x, y, τt ),
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while the second one yields an expression of the form j k
t i+l cil
i=0 l=0
(τ − 1)αil Ha,b,c,d,i,l+1 (x, y, τt ), (τ − 1)βil
(19)
with cil ∈ C, αil ∈ {0, 1}, and βil ∈ {1, . . . , j + 1}. Consider the function γ (x, y, τ ) := Ha,b,c,d,i,l+1 (x, y, τ ). By the Taylor formula, for any integer n ≥ 0 and (x, y, τ ) ∈ U × U × C, γ (x, y, τ ) =
n x − y y − x 1 m ∇ γ( x+y , x+y ,1) , ,τ − 1 2 2 m! 2 2 m=0 1 x−y y−x + ∇ n+1 γXθ , ,τ − 1 , (n + 1)! 2 2
x+y x−y y−x m where Xθ := ( x+y 2 , 2 , 1) + θ ( 2 , 2 , τ − 1), for some θ ∈ [0, 1]. Here ∇ γX Y th stands for the m - total differential of γ at the point X evaluated on the m-vector (Y, . . . , Y ). But in view of the almost-analyticity of G, we have ∇ m γ(z,z,1) ≡ 0 for all m and z; hence
|γ (x, y, τ )| ≤
|x − y|2 n+1 1 2 ∇ n+1 γX · , sup + |1 − τ |2 (n + 1)! X∈U ×U ×[τ,1] 2
where [τ, 1] denotes the line segment in C with endpoints τ and 1. Replacing τ by τt , noting that |1 − τt | = t|1 − τ |, and letting i and l vary, it follows that the expression (19) is bounded in modulus by Cabcdj kn
|x − y|n+1 + |1 − τ |n+1 , |1 − τ |j +1
uniformly for x, y ∈ U , t ∈ [0, 1], and τ in compact subsets of C. However, for (x, y, τ ) ∈ D we have the estimate (16), hence the last quantity is in that case estimated by Cabcdj kn |1 − τ |
n+1 2 −j −1
.
Consequently, a b c d j k ∂ ∂ ∂ ∂ ∂ g (x, y, τ ) − ∂x x y y τ τ 1
1 0
n−2j −1 Ha,b,c,d,j +1,k (x, y, τt ) dt ≤ Cabcdj kn |1 − τ | 2
∀(x, y, τ ) ∈ D. As n is arbitrary, it follows that the limit b
lim
D"(x,y,τ )→(z,z,1)
d
k
∂xa ∂ x ∂yc ∂ y ∂τj ∂ τ g1 (x, y, τ )
exists for each z ∈ U , and equals Ha,b,c,d,j +1,k (z, z, 1), which vanishes unless |b| = |c| = k = 0. This completes the proof of the lemma.
Bergman Kernels
221
is a bounded pseudoconvex domain Proof of Theorem 1. The hypotheses ensure that with z1 = x0 and z2 = 0 are smooth strictly pseuin CN+M+1 , the points z ∈ ∂ doconvex boundary points, and u(z) = |z3 |2 + g(z1 )|z2 |2 − φ(z1 ), with g = φ/ψ, is a smooth local defining function near any such point. By continuity, there is δ > 0 with |z1 − x0 |2 + |z2 |2 < δ are strictly pseudoconvex. such that all points of ∂ We may assume that δ < dist(x0 , ∂). Choose a C ∞ function θ on [0, 1) such that θ ≥ 0, θ ≡ 0 on [0, 2/3] and θ(t) = − log(1 − t) on (3/4, 1), and define u (z) = = {u < 0}. A similar |z3 |2 + g(z1 )|z2 |2 − exp[−θ ((|z1 − x0 |2 + |z2 |2 )/δ)]φ(z1 ) and argument as in the proof of Lemma 6 shows that is a strictly pseudoconvex domain and that and coincide in a neighbourhood of C = {z : |z1 −x0 |2 +|z2 |2 ≤ δ/2}. ⊂ , and also by Thus the conclusions (a)–(e) of Fefferman’s theorem are applicable to is C ∞ on C ∩ is C ∞ on C ∩ ×C ∩ . It follows that K ×C ∩ Lemma 5 K − K minus the boundary diagonal S = {(z, z) : z ∈ C ∩ ∂ }, while near S it is of the form t) = K(z,
a(z, t) + b(z, t) log[−u(z, t)], [−u(z, t)]N+M+2
(20)
with some almost-analytic C ∞ functions a, b and u satisfying u(z, z) = u(z) and a(z, z) =
(N + M + 1)! J [−u](z), π N+M+1
. z ∈ C ∩ ∂
.) Let us now specialize to (Here and in (20) we have used the fact that u = u on C ∩ z, t ∈ C ∩ with z2 = t2 = 0, i.e. to points of the form (x, 0, z3 ) with (x, z3 ) ∈ D := {|x − x0 |2 < δ/2, |z3 |2 < φ(x)} ⊂ CN+1 . It follows that the function 0, z3 ; y, 0, t3 ) g(x, z3 ; y, t3 ) = K(x, is C ∞ on√D × D minus the boundary diagonal = {(x, z3 ; y, t3 ) : x = y, z3 = t3 , |z3 | = φ(x)}, while near it is of the form g(x, z3 ; y, t3 ) =
a (x, z3 ; y, t3 ) + b (x, z3 ; y, t3 ) log[φ(x, y) − z3 t 3 ], [φ(x, y) − z3 t 3 ]N+M+2
where a , b – the restrictions of a, b to z2 = t2 = 0 – are holomorphic in x, y, z3 , t 3 on . Switching to the variable z3 t 3 τ= φ(x, y) we thus see that g(x, z3 ; y, t3 ) = F (x, y, τ ) for a function F on the domain D1 = {(x, y, τ ) : x, y ∈ U, |τ |2 <
φ(x)φ(y) }, |φ(x, y)|2
where U = {x : |x − x0 |2 < δ/2}, such that F is C ∞ on D1 minus the set 1 = {(x, y, τ ) : x = y, τ = 1}, while near 1 it is of the form F (x, y, τ ) =
G(x, y, τ ) + H (x, y, τ ) log(1 − τ ), (1 − τ )N+M+2
(21)
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M. Engliš
where H (x, y, τ ) = b (x, y, τ φ(x, y)) and a (x, y, τ φ(x, y)) +(1−τ )N+M+2 b (x, y, τ φ(x, y)) log[φ(x, y)N+M+2 ] φ(x, y)N+M+2
G(x, y, τ ) =
are functions in C ∞ (D1 ) almost-analytic in x, y, τ on 1 . Shrinking U if necessary so that (15) is satisfied, we are in a position to apply Lemma 8; thus there exist functions G0 , G1 , G2 , . . . in C ∞ (U × U ), almost-analytic on the diagonal, such that for each integer m ≥ 0, G(x, y, τ ) =
m
Gj (x, y) · (1 − τ )j + (1 − τ )m+1 gm (x, y, τ )
j =0
with gm ∈ G, and similarly for H . From (21) we therefore obtain, for any m ≥ 0, F (x, y, τ ) =
N+M+1 j =0
m
Gj (x, y) + Hj (x, y)(1 − τ )j log(1 − τ ) + Rm (x, y, τ ), j +1 (1 − τ ) j =0
where the function Rm ∈ G m is holomorphic in τ on D. Thus Lemma 7 can be applied to F (x, y, τ ), with n = N + M + 1, m any integer ≥ 0, and al−N−M−2 (x, y) = Gl (x, y), al (x, y) = Hl (x, y),
l = 0, . . . , N + M + 1, l ≥ 0.
As the Taylor coefficients of F (x, y, ·) are, in view of Proposition 4, given by fk (x, y) =
(k + M + 1)! Kφ k+1 ψ M (x, y) φ(x, y)k , π M+1 k!
we thus conclude that N+M+1 k + j (k + M + 1)! k GM+N+1−j (x, y) π M+1 k! Kφ k+1 ψ M (x, y) φ(x, y) − j j =0 m Cm (−1)j |φ(x, y)| k − k Hj (x, y) ≤ m √ k φ(x)φ(y) j =0 (j + 1) j +1 ∼ (k + 1)j and as k → ∞, for any m ≥ 0, uniformly for x, y ∈ U . As k+j j k j +1 for each fixed j , we thus obtain (6) and (5). The claim concerning the j +1 ∼ (k +1) local character of the coefficients bj (x, y) follows from the fact that the jet of the function , are known b and the (N + d + M)-jet of the function a in (20) at a point (z, z), z ∈ ∂ √ to depend only on the jet of the boundary ∂ at the point z, hence for z = (x, 0, φ(x)) only on the jets of φ and ψ at x (see e.g. [BFG], p. 312); and, consequently, so do, in
Bergman Kernels
223
turn, a and b , a and b , G and H , Gj and Hj , and bj . In particular, for x = y the leading order term is k N bN (x, x) k N π M+1 G0 (x, x) k N π M+1 a (x, x, φ(x)) = = π N φ(x)k+1 ψ(x)M φ(x)k+1 (N + M + 1)! (N + M + 1)!φ(x)N+M+k+2 √ √ √ √ k N π M+1 a (x, φ(x); x, φ(x)) k N π M+1 a(x, 0, φ(x); x, 0, φ(x)) = = (N + M + 1)!φ(x)N+M+k+2 (N + M + 1)!φ(x)N+M+k+2 √ k N J [−u](x, 0, φ(x)) = . π N φ(x)N+M+k+2 Standard matrix manipulations show that J [−u](z) = so
φ N+M+1 |z2 |2 1 1 |z2 |2 det 1 − log log · ∂∂ + · ∂∂ , ψM ψ φ ψ ψ bN (x, x) N π φ(x)k+1 ψ(x)M
=
det[−∂∂ log φ(x)] , π N φ(x)k+1 ψ(x)M
whence (7) follows. This completes the proof of Theorem 1.
Corollary 9. Let be a domain in CN and φ a positive function on . Assume that • is bounded and pseudoconvex, • φ and J [φ] are bounded, and − log φ is psh on , 1 log J [φ] is psh on , and • there exists an integer M ≥ 0 such that − log φ − M • − log φ is smooth and strictly psh at a point x0 ∈ . Let χ = det[−∂∂ log φ]. Then as k → ∞, there is an asymptotic expansion m k N−m−1 kN −j K k (x, y) − ≤ C β (x, y) k j m,U φχ π N φ(x, y)k φ(x)k/2 φ(y)k/2 j =0
for x, y in a small neighbourhood U of x0 and any m ≥ 0, where the coefficients βj (x, y) ∈ C ∞ (U × U ) are almost analytic on the diagonal, their jets at (x0 , x0 ) depend only on the jet of φ(x) at x0 , and β0 = 1. Proof. Apply the previous theorem with ψ = φ · J [φ]1/M , observe that χ= and replace k + M + N + 1 by k.
J [φ] , φ N+1
(22)
Remarks. (1) The boundedness assumptions on , φ, and ψ or J [φ] in Theorem 1 ) are in fact and Corollary 9 (which are equivalent to the boundedness of the domain unnecessary, cf. the remark after Lemma 5. (2) In the applications to the Berezin quantization, one takes for − log φ the Kähler potential of the metric (2); hence − log φ is automatically smooth and strictly plurisubharmonic at all points of . The third condition in Corollary 9 can in this context be
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M. Engliš
rephrased in terms of the Ricci tensor: by (22), it is equivalent to the plurisubharmonicity of −(M + N + 1) log φ − log χ ; but ∂ 2 log χ = Ricj k ∂zj ∂zk is the Ricci tensor of the metric gj k . Thus the condition says that Ricj k ≤ (M + N + 1)gj k at all points of , in the sense that the difference of the right-hand and the left-hand side is positive semidefinite. (3) Theorem 1 fails if − log φ is only assumed to be psh but not strictly psh at x0 . Indeed, examples given at the end of Sect. 2 in [E6] suggest that in that case one gets in (5) an asymptotic expansion not in the negative powers of k, but instead in negative√powers of k 2/m , where m is the type (assumed to be finite) of the boundary point (x0 , 0, φ(x0 )) ∈ ; and similarly it seems that ∂ β0 (x, x) = lim
k→∞
π N k
φ(x)k Kφ k χ (x, x) =
2 , m
with the right-hand side interpreted as zero for points of infinite type. (4) Theorem 1 fails even more drastically if − log φ is not even psh at x0 . Indeed, it is an immediate consequence of (6) that lim Kφ k ψ M (x0 , x0 )1/k = 1/φ(x0 ).
k→∞
(23)
On the other hand, for arbitrary positive lower-semicontinuous functions φ, ψ on (not necessarily such that − log φ or − log ψ are psh), it follows from Proposition 4, the formula for the radius of convergence, and the fact that the domain of convergence of a power series is always a log-convex complete Reinhardt domain, that lim sup Kφ k ψ M (x, x)1/k = 1/φ ∗ (x), k→∞
where log(1/φ ∗ ) is the greatest psh minorant of log(1/φ); in particular, (23) cannot hold if − log φ is not psh. See [E3] and the references therein for related matters. (5) If φ, ψ are assumed to be not only C ∞ but real-analytic (C ω ) near x0 , the assertion (6) of Theorem 1 can be substantially sharpened, namely to sup φ(x, y)k Kφ k ψ M (x, y) −
x,y∈U
kN N π ψ(x, y)M
N+m−1
bj (x, y) k −j = O(k −m ). (24)
j =0
Also, the coefficients bj (x, y) are not merely almost-analytic, but holomorphic in x, y on U × U . For x = y, the estimate (24) is better than (6) by an exponential factor, cf. (15). For strictly pseudoconvex with C ω boundary and φ, ψ two C ω defining functions for , (24) was proved in [E6], Theorem 11; the local version stated above follows exactly in the same way from the corresponding local variant of part (f) of Fefferman’s theorem mentioned in the end of Sect. 1.
Bergman Kernels
225
(6) A consequence of (24) is that Kφ k ψ M (x, y) is zero-free on U × U as soon as k is sufficiently large, and Kφ k ψ M (x, y)−1/k → φ(x, y)
(25)
on U × U as k → ∞ when the holomorphic branches of the roots are chosen appropriately. Note that for φ, ψ not C ω but merely C ∞ , (6) is too weak to yield this (unless x = y); on the other hand, it can be shown that the sequence of the absolute values |Kφ k ψ M (x, y)|1/k
(26)
is always locally uniformly bounded, on all of × (!), for any positive lower semicontinuous functions φ, ψ (see [E3]). Almost nothing seems to be known about the limiting behaviour of this sequence, however. For plane domains with − log φ the Kähler potential of the Poincaré metric and M = 0, this problem was studied by the present author in [E4]; it turns out that in that case the limit (25) exists for all (x, y) not in the cut locus of (i.e. if there is a unique shortest geodesic connecting x to y), while for (x, y) in the cut locus the sequence (26) can display an oscillatory behaviour as k → ∞. Understanding the limiting behaviour of (26) in the general case seems quite intriguing. (7) For a compact manifold, Catlin [Ca] has obtained the estimates (6) even for the derivatives of Kφ k ψ M with respect to the x and y variables. Our methods can also be used to give a similar result in the present context as well, as it is easily seen that the formula (11) in Proposition 4 can be differentiated termwise with respect to x, y any number of times. We omit the details. (8) The asymptotics as k → ∞ of the integral Ik (f ) =
f (x)Kφ k (x, x) φ(x)k dx
are of interest in the study of Pauli operators (quantum Hamiltonians) in magnetic fields; see [Erd, Rai].
3. The Berezin Transform We begin by recalling the following generalization of part (e) of Fefferman’s theorem (that part is actually due to Kerzman [Ke]). Proposition 10. Let D be a bounded pseudoconvex domain in Cn and x, y ∈ ∂D two smooth boundary points of finite type. Then there exist neighbourhoods U, V of x and y, respectively, in Cn such that the (unweighted) Bergman kernel K(x, y) of D extends to a C ∞ function on (D ∩ U ) × (D ∩ V ). The proof of Proposition 10 can be found in [Be2] and [Bo]; again, the boundedness hypothesis can be dropped [E7].
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M. Engliš
Proof of Theorem 2. By Theorem 1, there exists a neighbourhood U of x0 such that the asymptotic expansion (6) holds for x, y in a neighbourhood of U . This implies, first of all, that Kφ k ψ M (y, y) = 0 as soon as k is large enough; thus the Berezin transform Bφ k ψ M f (y) =
f (x)
|Kφ k ψ M (x, y)|2 Kφ k ψ M (y, y)
φ(x)k ψ(x)M dx
(27)
is always defined for all y ∈ U as soon as k is sufficiently large. Split the integration in (27) into integration over U and over \ U . Consider the function 0, s; y, 0, φ(y)). f (x, y, s) = K(x, Let V be another neighbourhood of x0 whose closure is contained in U . By Proposition 10 and the finite type hypothesis, f is a C ∞ function on the set compact W = {(x, y, s) : x ∈ \ U, y ∈ V , |s|2 ≤ φ(x)}. Thus for any integer j ≥ 0, ∂j f j = cj < +∞. (x,y,s)∈W ∂s sup
On the other hand, By Proposition 4, ∞ (k + M + 1)! f (x, y, s) = Kφ k+1 ψ M (x, y) φ(y)k/2 s k . k! π M+1
(28)
k=0
Applying Cauchy estimates to the function s ' → ∂ j f (x, y, s)/∂s j , holomorphic in the √ disc |s| < φ(x), we thus obtain |Kφ k+1 ψ M (x, y)φ(x)k/2 φ(y)k/2 | ≤
(k − j )! π M+1 cj , (k + M + 1)!
for all x ∈ \ U , y ∈ V and k ≥ j ; that is, |Kφ k ψ M (x, y)|2 φ(x)k φ(y)k ≤ cj k −2(M+j +1)
∀x ∈ \ U, y ∈ V , k ≥ j + 1,
and, upon invoking (6) with x = y, |Kφ k ψ M (x, y)|2 Kφ k ψ M (y, y)
φ(x)k ψ(x)M ≤
cj ψ(x)M
k N+2j +2M+2
for all x ∈ \ U , y ∈ V and k ≥ j + 1. It follows that the integral over \ U is O(k −j ), uniformly as y ∈ V , for any j . It remains to deal with the integral over U . In that case, by (6) we have the asymptotic formula Kφ k ψ M (x, y) · φ(x)
k/2
φ(y)
k/2
m+N k N φ(x)k/2 φ(y)k/2 = N bj (x, y) k −j π φ(x, y)k ψ(x, y)M j =0
+k
−m−1
Cm (x, y, k)
Bergman Kernels
227
for m any integer ≥ 0, with supx,y∈U,k≥1 |Cm (x, y, k)| < ∞. Combining this with the similar estimates with x and y interchanged and with x = y, respectively, we arrive at the asymptotic expansion |Kφ k ψ M (x, y)|2 Kφ k ψ M (y, y) =
φ(x)k ψ(x)M
m+N k N φ(x)k φ(y)k ψ(x)M ψ(y)M |b0 (x, y)|2 γj (x, y) k −j π N |φ(x, y)|2k |ψ(x, y)|2M b0 (y) j =0
+ k N−m−1 Cm (x, y, k) (x, y, k)| < ∞, and with γ = 1. We thus see that as k → ∞ the with supx,y∈U,k≥1 |Cm 0 integral over U has the same asymptotic expansion as ∞ k N
π
k
−j
U
j =0
f (x) γj (x, y)
ψ(x)M ψ(y)M |b0 (x, y)|2 φ(x)φ(y) k dx. (29) · |ψ(x, y)|2M b0 (y) |φ(x, y)|2
Finally, recall the familiar formula for the asymptotics of Laplace integrals: if D is a bounded region in Rn , F a complex-valued and S a real-valued function in C ∞ (D), and S peaks at a single point x∗ ∈ D, then as λ → +∞, D
F (x)eλS(x) dx =
2π n/2 λ
√
∞ eλS(x∗ ) aj λ−j , | Hess S(x∗ )| j =0
(30)
where the coefficients aj depend only on the derivatives of F and S at x∗ , a0 = F (x∗ ), and ∂ 2S Hess S(x∗ ) = det (x∗ ) . ∂xj ∂xk Moreover, if F and S (and x∗ ) depend in addition smoothly on some additional parameter y ∈ D ⊂ Rn , then the asymptotic expansion (30) holds uniformly as y ranges over a compact subset of D . (See [Fed], Theorems II.4.1 and II.4.4, or [BH], Sect. 8.3.) As we have already observed in (15), owing to the strict plurisubharmonicity of − log φ, φ(x)φ(y) the function x ' → has a strict local maximum at x = y. Diminishing U if |φ(x, y)|2 necessary, we may thus assume that the function S(x) = log
φ(x)φ(y) |φ(x, y)|2
(31)
peaks only at x = y on U × U ; shrinking U further if needed we may likewise assume that f is C ∞ on U . Consequently, the formula (30) can be applied to the integrals in (29); and since a short computation reveals that Hess S(y) = 4N (det[−∂∂ log φ])2 = 4N b0 (y)2 , the assertion of the theorem follows.
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M. Engliš
Remarks. (1) Though the coefficients aj in (30) and the operators Qj in (9) can in principle be evaluated explicitly, the required computations are extremely cumbersome (cf. the discussion after Theorem 12 in [E6]). In the case which occurs in the Berezin quantization, namely for Bφ k χ with χ = det[−∂∂ log φ], the following general description is therefore of interest. It also shows that the operators Qj are very natural objects from the point of view of Kähler geometry. Theorem 11. The operators Qm are finite sums of differential operators of the form C i1 ···ik j 1 ···j l f/j 1 ···j l i1 ···ik f '→ i1 ,... ,ik ,j1 ,... ,jl
with k, l ≤ m, where the slash stands for the covariant differentiation and C i1 ···ik j 1 ···j l are tensor fields on , symmetric in i1 , . . . , ik and in j 1 , . . . , j l , which are contractions of tensor products of the contravariant metric tensor g j k , the curvature tensor Rij kl , and the latter’s covariant derivatives. In particular, Q0 = I (the identity operator), ˜ ≡ Q1 =
gj k
j,k
∂2 , ∂zj ∂zk
the Laplace-Beltrami operator corresponding to the metric (2), and Q2 =
∂2 1 2 1 ˜ + Ricj k , 2 2 ∂zj ∂zk j,k
with Ricj k the contravariant Ricci tensor, i.e. Ricj k = χ = det[gj k ] and g j k is the inverse matrix to gj k .
j l mk l,m g g
∂ 2 log χ , where ∂zm ∂zl
For the proof, see [E5], Theorem 4. (The inequalities k, l ≤ m are not mentioned there explicitly, but they follow easily from the formula (2.22) there.) In [E6] it is also shown how to evaluate the coefficients bj (x, y) in (5), for instance, √ χ (x) ˜ b1 (x, x) = log . ψ(x)M in Theorem 2 (2) The condition of finite type, and even of smooth boundedness, of is rather unsatisfactory, since none of the standard metrics associated with a domain – the Bergman and the Cheng–Yau metric, for instance – satisfy it in general. Indeed, the Bergman metric corresponds to the choice φ(z) = 1/K (z, z), with K (z, z) the unweighted Bergman kernel of , which is almost never C ∞ up to the boundary, even for strictly pseudoconvex, owing to the presence of the logarithmic term in (14). Similarly, for the Cheng–Yau metric on a strictly pseudoconvex domain (the unique Kähler–Einstein metric for which Ricj k = −gj k ) the potential has a similar logarithmic singularity at the boundary, by a result of Lee and Melrose [LM]. Thus in both cases, . fails to be C ∞ on the “equator” {(x, 0) : x ∈ ∂} ⊂ ∂ (3) What prevents us from getting Theorem 2 without the smooth boundedness and finite type hypotheses is only Proposition 10, which was needed in the proof to estimate
Bergman Kernels
229
the “tail” of the integral (27) defining the Berezin transform. However, examples suggest that even though Proposition 10 obviously fails in the absence of the above-mentioned hypotheses, the required estimate for the “tail” need not. Hence, Theorem 2 could be extended to all pseudoconvex domains and positive functions φ, ψ ∈ C ∞ () such that − log φ, − log ψ are psh if we had a direct proof of the following assertion: 2 Problem. If , φ, ψ are as above and M is a nonnegative integer, then for any x ∈ and any neighbourhood U of x there exists δ > 0 and an integer m0 such that
sup
y∈\U
|Kφ m ψ M (x, y)|2
1/m
Kφ m ψ M (x, x)Kφ m ψ M (y, y)
≤1−δ
∀m ≥ m0 .
(In view of (5), one may replace 1/Kφ m ψ M (x, x) by φ m (x), and similarly for y.) Obviously, this problem is related to the problem of understanding the asymptotic behaviour of |Kφ m ψ M (x, y)|1/m as m → ∞ away from the diagonal x = y, mentioned in Remark (6) in Sect. 2 above. 4. The Berezin Star-Product As this seems not to be done in full anywhere in the literature, we now describe briefly how to construct a star product from Theorem 2. For the sake of brevity, denote h = 1/m and Ah = A2 (, φ m ψ M ), Kh = Kφ m ψ M , Bh = Bφ m ψ M (m = 1, 2, . . . ). For T ∈ Bh , the Banach algebra of all bounded linear operators on Ah , define the function T (x, y) – the covariant symbol of T – on × by T (x, y) =
T Kh (·, y), Kh (·, x) Ah , Kh (x, y)
x, y ∈ .
(32)
Then T is representable as the integral operator on with kernel T (x, y)Kh (x, y) with respect to the measure φ(x)m ψ(x)M dx, and for T1 , T2 ∈ Bh , Kh (x, z)Kh (z, y) (T1 T2 )(x, y) = T1 (x, z)T2 (z, y) (33) φ(z)m ψ(z)M dz. Kh (x, y) Set T˜ (x) := T (x, x) and let Ah be the vector space of all functions of the form T˜ (x) with T ∈ Bh . Since the correspondence T ' → T˜ is one-to-one, one can transfer the algebraic operations and operator involution from Bh into Ah , which endows Ah with the structure of an involutive complex algebra, with complex conjugation as the involution, and with an (associative, but not commutative) product which we denote by ∗h . See [Ber], or [E2], p. 415, for the details. Let A be the algebraic direct sum of all Ah , h = 1, 1/2, . . . , and let A˜ ⊂ A be the subset of all elements of the form f (h; x) = T˜h (x), where Th ∈ Bh for each h, for which there exist functions fj (x, y) on × (j = 1, 2, . . . ), holomorphic in x and y, such that for each N ∈ N, Th (x, y) = f0 (x, y) + hf1 (x, y) + · · · + hN fN (x, y) + hN+1 FN (h; x, y), 2 Here, again, it suffices to take M = 1 (replacing ψ M by ψ).
(34)
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where sup |fj (x, y)| < ∞,
sup |fj (y, x)| < ∞,
x
x
sup |FN (h; x, y)| < ∞, x,h
∀y ∈ , j = 0, 1, . . . , N, h = 1, 1/2, 1/3, . . . .
(35)
Clearly A˜ is closed under addition, scalar multiplication and complex conjugation. We denote the product in A by ∗. The next theorem, in combination with a certain “approximation” result (based, roughly speaking, on the observation that functions of the form f g, with f, g bounded holomorphic, are in each Ah ), enable us to extend the product ∗ to all formal power series in h with C ∞ ()-coefficients. At the end of the next section we describe how this product relates to the Berezin–Toeplitz star-product considered therein. ˜ Then for each z ∈ , we have an asymptotic expansion as Theorem 12. Let f, g ∈ A. h → 0, (f ∗ g)(h; z) = Ck (Rfi , Rgj ) hi+j +k , (36) i,j,k≥0
C ∞ ( × )
C ∞ ()
where R : → is the operator of restriction to the diagonal x = y, and Ck are bilinear differential operators of the form αβ T(i,j,k) u/α v/β , u, v ∈ C ∞ (), (37) Ck (u, v) = 0≤i,j ≤k |α|=i |β|=j
where α, β are multiindices, the slash stands for covariant differentiation with respect αβ to the metric (2), and T(i,j,k) are, for each fixed i, j, k ≥ 0, i, j ≤ k, tensor fields on , symmetric in the entries of α and of β, of the same form as in Theorem 11. In particular, 1 j k lm C0 (u, v) = uv, C1 (u, v) = g j k u/j v/k , C2 (u, v) = g g u/j l v/km , 2 j,k
j,k,l,m
where as before g j k is the inverse matrix to gj k = −∂∂ log φ. Proof. Denote u(h; x) = f (h; z, x)g(h; x, z), where f (h; z, x) = Th (z, x) and similarly for g. Owing to (33) we then have |Kh (x, z)|2 (f ∗ g)(h; z) = u(h; x) φ(x)1/ h ψ(x)M dx = (Bh u(h; ·))(z). Kh (z, z) In view of (34), for each N ∈ N, u(h; x) = u0 (x) + hu1 (x) + · · · + hN uN (x) + hN+1 UN (h; x), where uk (x) =
fi (z, x)gj (x, z),
k = 0, 1, . . . , N,
i+j =k
UN (h; x) =
hi+j −N−1 fi (z, x)gj (x, z)
i+j >N, i,j ≤N
+ f0 (z, x)GN (h; x, z) + FN (h; z, x)g0 (x, z).
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231
The hypotheses on A˜ ensure that u1 , . . . , uN and UN (h; ·) are bounded on , the last uniformly in h. By Theorem 2, we therefore get (Bh u(h; ·))(z) =
N
hk Bh uk (z) + hN+1 O(UN (h; ·)∞ )
k=0
=
N
hk
k=0
=
∞ l=0
hl+k Ql uk (z) + O(hN+1 )
l,k≥0 l+k≤N
=
hl Ql uk (z) + hN+1 O(1)
l,i,j ≥0 l+i+j ≤N
hl+i+j Ql [fi (z, x)gj (x, z)]x=z + O(hN+1 ),
where in the last expression the operators Ql act on the x variable. As N is arbitrary, we thus obtain an asymptotic expansion (f ∗ g)(h; z) = hi+j +l Kl (fi , gj )(z), i,j,l≥0
with
Kk (fi , gj )(z) := Qk [fi (z, ·)gj (·, z)]z .
We have to show that this coincides, in fact, with Ck (Rfi , Rgj )(z) for certain differential operators Ck of the form (37). Consider thus, quite generally, two functions f, g on × holomorphic in the first variable and anti-holomorphic in the second, fix a point z ∈ , and let us show that Qk [f (z, ·)g(·, z)], evaluated at z, comes as a sum k i,j =0 |α|=i |β|=j
αβ
S(i,j ) (Rf )/α (Rg)/β
evaluated at z,
αβ
where S(i,j ) , i, j = 0, . . . , k, are tensor fields on of the form described in the theorem. For brevity, denote F = f (z, ·) and G = g(·, z); thus G is holomorphic and F is antiholomorphic on . In view of Theorem 11, it is sufficient to show that each covariant derivative of the product F G, evaluated at z, is a sum of expressions of the form S αβ F/α G/β z , (38) with S αβ being components of tensor fields involving only the metric tensor, the curvature tensor, and the latter’s covariant derivatives; the assertion will then follow since, as G is holomorphic, G/β (z) = (Rg)/β (z), and similarly for F/α . Now by the product rule, (F G)/... comes as a sum of products F/... G/... of covariant derivatives of F and G. Using the Ricci formula, which exhibits the commutator (T...... )/pq − (T...... )/qp of two successive covariant differentiations of a
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M. Engliš
tensor T...... as a sum of contractions of T...... against the curvature tensor (see e.g. [E5], formula (2.17)), we further reduce to showing that F/γ α G/δβ |z is of the form (38), for any multiindices α, β, γ and δ. However, in view of the holomorphy of F and G, we have F/ l = G/l = 0 for any l, hence the last expression vanishes unless |γ | = |δ| = 0. Thus the required assertion follows. ˜ we have K0 (f, g)(z) = F (z)G(z) and Finally, recalling that Q0 = I and Q1 = , j k K1 (f, g)(z) = j,k g F/j G/k |z , hence in terms of u := Rf and v := Rg we get C0 (u, v) = uv and C1 (u, v) = j,k g j k u/j v/k . The formula for C2 follows similarly from the formula for Q2 in Theorem 11. This completes the proof. Consider now the ring (over the field of complex numbers) C ∞ ()[[h]] of all formal power series ∞ f (h; x) = fj (x)hj , fj ∈ C ∞ () ∀j, j =0
with the usual algebraic operations, and endow it with the C[[h]]-linear product ∗ given by fi h i ∗ gj hj := Ck (fi , gj ) hi+j +k (39) i≥0
j ≥0
i,j,k≥0
with the operators Ck from (36). Theorem 13. The product ∗ is a star-product, i.e. it is associative and satisfies (f ∗ g)0 = f0 g0 (the pointwise product in C ∞ ()), and f ∗ g − g ∗ f h
0
= i{f, g}
(40)
(41)
(the Poisson bracket with respect to the metric (2)). ˜ this is immediate from the preceding theorem; as the Proof. For f (h; x), g(h; x) ∈ A, α term at hm in (39) – namely, i+j +k=m Ck (fi , gj ) – involves only ∂ fi and ∂ β gj with ∞ i, j, |α|, |β| ≤ m, it therefore suffices to check that for each f ∈ C ()[[h]], z ∈ and N ∈ N, there exists f ∈ A˜ such that α
α
∂ fj (z) = ∂ fj (z)
∀ j, |α| ≤ N.
(42)
Indeed, one then has Ck (f, g)(z) = Ck (f , g )(z) ∀k = 0, 1, . . . , N, so (f ∗ g)(h; z) = (f ∗ g )(h; z) mod hN+1 , and the validity of (40) and (41) for f, g follows from its validity for f , g ; similarly for the associativity. We will search for such an f ∈ A˜ in the form f (h; x) =
N j =0
pj (x) hj
Bergman Kernels
233
with pj bounded holomorphic functions on . Then fj (x, y) = pj (y), so (34) and (35) α
hold, and (42) is equivalent to ∂ α pj (z) = ∂ fj (z) for all |α|, j ≤ N . Clearly, such 1 α functions pj exist (for instance, take the polynomials pj (x) = |α|≤N α! ∂ f j (z)x α ). It remains to show that f (h; x) = T˜h (x) for some family of operators Th ∈ Bh (h = 1, 1/2, . . . ). However, observe that for T the operator on Ah of multiplication by a bounded analytic function p, one has from (32), pKh (·, x), Kh (·, x) T˜ (x) = Kh (x, x) p(x)Kh (x, x) = = p(x), Kh (x, x) by the reproducing property of Kh . It follows that T∗ = p. Further, T is bounded if p is. Thus the operators N ∗ Th = multiplication by hj pj on Ah j =0
do the job we need.
Remark. It follows from (37) that the Berezin star product is an example of a “deformation quantization with separation of variables” in the sense of Karabegov [Kar], i.e. f ∗ g = f g (pointwise product) if f or g is analytic. It is also easy to show that f ∗ g = g ∗ f , i.e. ∗ preserves complex conjugation. 5. Berezin–Toeplitz Quantization In this section we restrict our attention to the case of smoothly bounded and strictly pseudoconvex, −φ a smooth defining function for (i.e. φ is C ∞ in a neighbourhood of , φ > 0 on , φ < 0 on the complement of , and φ = 0, ∇φ = 0 on ∂) such that − log φ is strictly psh, and to the weights φ k χ , χ = det[−∂∂ log φ]. For the proof of part (ii) of Theorem 3, we need to use the theory of Boutet de Monvel– Guillemin Toeplitz operators [BG], in the same way as in [BMS] and [Sch] for compact manifolds; we include a sketch of the proof below for convenience. The point we wish to make here is that part (i) of Theorem 3 is an easy consequence (not using the Boutet de Monvel–Guillemin theory in any way) of Theorem 2 and the following elementary observation. Lemma 14. Let be a strictly pseudoconvex domain in CN and −φ a smooth defining function for such that − log φ is strictly psh. Then for any positive continuous function g on there exists an integer d > 0 such that − log φ − d1 log g is strictly psh on . Proof. Observe that
0 −φ −φk 1 0 −1 1 φk /φ = φ , −φj −φj k − (φ/d)(log g)j k φj /φ 1 0 (− log φ − d1 log g)j k 0 1
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where for brevity we have used the subscripts j, k to denote differentiations by zj , zk . Thus − log φ − (1/d) log g is strictly psh at z ∈ if and only if the square matrix
1 1 0 −φ −φk 0 −φj −φj k − d 0 (log g)j k φ ≡ A(z) − d B(z) has 1 negative and N positive eigenvalues. However, in view of the hypotheses on φ and g, both matrix-valued functions A(z) and B(z) are continuous on and A(z) has the required signature (N, 0, 1) for each z ∈ ; thus supz∈ B(z) = b < ∞ and K = {A(z), z ∈ } is a compact subset of the open set N of all Hermitian matrices of signature (N, 0, 1). Taking d so large that b/d < dist(K, ∂N ), the assertion follows. Proof of part (i) of Theorem 3. Let J [φ] be the Monge-Ampére determinant (13). As χ = J [φ]/φ N+1 , the strict plurisubharmonicity of − log φ implies that J [φ] > 0 on ; on the other hand, the fact that φ is a smooth defining function of the strictly pseudoconvex domain implies that J [φ] ∈ C ∞ () and J [φ] > 0 on ∂. Thus the preceding lemma applies to g = J [φ]. Let d be the constant provided by the lemma, and apply Theorem 2 to , φ, M = d, and ψ = J [φ]1/d φ (thus φ m ψ M = φ m+d+N+1 χ ). This gives Bφ m χ f (x) = f (x) + O(m−1 )
as m → ∞,
(43)
for any f ∈ L∞ () ∩ C ∞ (), uniformly for x in compact subsets. On the other hand, let Kφ m χ (·, x) kx(m) = Kφ m χ (x, x)1/2 be the unit vector in the direction of Kφ m χ (·, x) (the coherent state). Then by the definition of the Toeplitz operator and of the Berezin transform, (φ m χ) (m) (m) kx , kx L2 (,φ m χ) .
Bφ m χ f (x) = Tf It follows that
(φ m χ)
|Bφ m χ f (x)| ≤ Tf
(φ m χ)
kx(m) 2 = Tf
(φ m χ)
Combining this with (43), we see that lim inf m→∞ Tf hand, for any weight ρ,
(44)
.
≥ f ∞ . On the other
(ρ)
Tf h = Pρ (f h) ≤ f h ≤ f ∞ h, (ρ)
whence Tf ≤ f ∞ . Thus the desired assertion follows.
Proof of part (ii) of Theorem 3. Let us now consider the domain (8) with M = 0: = {(z, w) ∈ × C : |w|2 < φ(z)}.
(45)
is a smoothly bounded, strictly pseudoconvex The hypotheses then mean precisely that domain in Cn , n = N + 1, and r(z, w) := |w|2 − φ(z) is a smooth defining function . The boundary X = ∂ is a compact manifold, and we denote by α the restriction for to X of the 1-form Im ∂r = (∂r − ∂r)/2i. Then α is a contact form, i.e. α ∧ (dα)n−1 determines a nonvanishing volume form on X. Following [BG], let L2 be the Lebesgue
Bergman Kernels
235
space on X with respect to this measure, and H 2 the closure in L2 of the subspace of . For each (z, w) ∈ , the all functions in C ∞ (X) which extend holomorphically into evaluation functional f ' → f (z, w) turns out to be continuous on H 2 , hence is given by the scalar product with a certain element k(z,w) ∈ H 2 . The function Szegö (z1 , w1 ; z2 , w2 ) := k(z2 ,w2 ) , k(z1 ,w1 ) H 2 K × minus the × is called the Szegö kernel; it extends smoothly to all of on Szegö on X is the Szegö boundary diagonal, and the integral operator determined by K projection π, i.e. the orthogonal projection in L2 onto H 2 . For F ∈ L∞ (X), the (global) Toeplitz operator TF is the operator on H 2 defined by TF f = π(Ff ). Let H(m) ⊂ H 2 (m = 0, 1, 2, . . . ) be the subspace of all functions of the form f (z, w) = f (z) wm
((z, w) ∈ X).
A routine computation shows that (see e.g. [Ran], p. 291) α ∧ (dα)n−1 = (n − 1)!
J [r] dS, ∂r
dS being the surface element (i.e. (2n − 1)-dimensional Hausdorff measure) on X. Introducing the coordinates (z, w) = (z, eiθ φ(z))
(z ∈ , θ ∈ [0, 2π ])
on X, we have dS = φ + ∂φ2 dz dθ; and as φ + ∂φ2 = ∂r and J [r] = J [φ] = φ N+1 χ , we thus have X
|f (z)w m |2 α ∧ (dα)n−1 = (n − 1)!
2π
0
= 2π (n − 1)!
|f (z)|2 φ(z)m J [r] dz dθ
|f |2 φ m+N+1 χ dz.
It follows that H(m) is isometrically (up to the immaterial factor (n−1)!2π) isomorphic to the Bergman space A2 (, φ m+N+1 χ ). As the subspaces H(m) are pairwise orthogonal and span all of H 2 , we therefore arrive at the following analogue of Proposition 4 (cf. [Lig, BFS, KLR]): Szegö (z1 , w1 ; z2 , w2 ) = K
∞ 1 Kφ m+N +1 χ (z1 , z2 ) · (w1 w 2 )m . 2π N !
(46)
m=0
For f ∈ C ∞ (), let fˆ ∈ C ∞ (X) be defined by fˆ(z, w) = f (z)
(47)
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M. Engliš
and consider the Toeplitz operator Tfˆ . Then all the subspaces H(m) are invariant under Tfˆ , and the restriction of Tfˆ to H(m) is (up to the unitary equivalence above) precisely (φ m+N +1 χ)
the Toeplitz operator Tf
. That is,
Tfˆ *
∞
T (φ
m χ)
.
(48)
m=N+1
Finally, a generalized Toeplitz operator on H 2 is the operator TP given by TP = π P π, where P is a pseudodifferential operator on the compact manifold X. The order ord(TP ) and the symbol σ (TP ) of TP are defined as the order of P and the restriction of the symbol σ (P ) of P to the submanifold N := {(x, ξ ) : ξ = tαx , t > 0} of the cotangent bundle of X, respectively. It was then shown in [BG] that these two definitions are unambiguous, and (P1) (P2) (P3) (P4) (P5)
the generalized Toeplitz operators form an algebra under composition, ord(T1 T2 ) = ord(T1 ) + ord(T2 ), σ (T1 T2 ) = σ (T1 )σ (T2 ), σ ([T1 , T2 ]) = {σ (T1 ), σ (T2 )} (the Poisson bracket), if ord(T ) = 0, then T is a bounded operator on H 2 , and if ord(T1 ) = ord(T2 ) = k and σ (T1 ) = σ (T2 ), then ord(T1 − T2 ) ≤ k − 1.
Let T be the subalgebra of all generalized Toeplitz operators which commute with the circle action Uθ : f (z, w) ' → f (z, eiθ w)
((z, w) ∈ X, θ ∈ R)
on H 2 ; clearly, the operators Tfˆ with fˆ as in (47) belong to T . Let D be the infinitesimal generator of the semigroup Uθ ; one has Dh = imh
∀h ∈ H(m) ,
(49)
and D = T∂/∂θ is a generalized Toeplitz operator of order 1. Using (P1)–(P5) it was then shown in [BMS] (see also [Sch] and [Gu]) that if T ∈ T is of order 0, then T = Tfˆ + D −1 R for some (uniquely determined) f ∈ C ∞ () and R ∈ T of order 0. Repeated application of this formula reveals that, for each k ≥ 0, T =
k j =0
D −j Tfˆj + D −k−1 Rk ,
with fj ∈ C ∞ () and Rk ∈ T of order 0. Invoking the boundedness of zeroth order operators, it follows that k D k+1 T − D −j Tfˆj j =0
Bergman Kernels
237
is a bounded operator on H 2 , that is, by (49) and (48), k (φ m χ) m−j Tfj T |H(m) − = O(m−k−1 ). j =0
Taking for T the product Tfˆ Tgˆ with f, g ∈ C ∞ (), we obtain (10). Finally, the assertions concerning C0 and C1 follow from the properties (P2) and (P3) of the symbol; see the references mentioned above for the details. Remarks. (1) The same comment as in Remark (2) at the end of Sect. 3 applies here as well: namely, the machinery of [BG] makes heavy use of the compactness of the , so the last proof does not easily generalize to unbounded domains . manifold X = ∂ The problem is that, first, zeroth order pseudodifferential operators no longer need to be bounded, and second, that the various smoothing operators which arise as error terms need not be bounded either. The former can be coped with by means of the Calderon– Vaillancourt theorem (by replacing C ∞ () by the space of all functions on all of whose derivatives up to a certain order are continuous and uniformly bounded); the latter has so far been dealt with successfully only in the case of = Cn with the Euclidean metric (hence with the potential − log φ(z) = |z|2 , giving rise to the Segal–Bargmann spaces 2 of entire functions square-integrable with respect to the Gaussian measures e−m|z| dz), see [Bw]. be smooth prevents us from dealing with Similarly, the requirement that X = ∂ the case of φ not being a defining function (or a power of one), thus excluding the most interesting cases such as the Bergman and the Cheng–Yau (Kähler–Einstein) metric. Apparently, this difficulty will probably be even harder to overcome than the previous one. We remark that, on the other hand, to some extent the above approach can be generalized to some non-Kähler compact manifolds as well; see [BU]. (2) On a purely formal level, the Berezin–Toeplitz quantization has been carried out on any Kähler manifold by Reshetikhin and Takhtajan [RT]; however, it is not clear whether these arguments (involving a formal application of the stationary-phase method) can be made rigorous. Cf. also the similar formal expansion by Cornalba and Taylor [CT]. For bounded symmetric domains in Cn (Hermitian symmetric spaces) with the invariant metric, a similar line of attack – namely, a study of the asymptotics of the weighted Bergman projections Pφ m : L2 (, φ m ) → A2 (, φ m ) as m tends to infinity – has been initiated by Arazy and Ørsted [AO]. (3) The proofs in Sections 2 and 3 feature an interplay between two subjects: the ∂-techniques due to Kohn, Catlin, and others, on the one hand, and the theory of Fourier integral operators on the other hand. The latter is more powerful for the applications we have – in particular, the Boutet de Monvel–Guillemin theory of generalized Toeplitz operators in this section relies on it completely, and it is also hidden in Boutet de Monvel’s and Sjöstrand’s proof of the Fefferman theorem which was our departing point for most of the developments in Sect. 2. On the other hand, the former lend themselves much more easily to localization – they work in a neighbourhood of a smooth strictly pseudoconvex point even if the boundary is bad (nonsmooth, weakly pseudoconvex, even unbounded) away from it. From this point of view, it would be very desirable to have at least the following “noncompact version” of the Boutet de Monvel–Guillemin theory. Let be a bounded pseudoconvex domain in CN and φ a function on which is positive and C ∞ on , vanishes at ∂ (but it is not assumed to be C ∞ up to the
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M. Engliš
as in (45), boundary), and such that − log φ is strictly psh on . Consider the domain which lies over . Then X and let X = {(z, w) ∈ ×C : |w|2 = φ(z)} be the part of ∂ is a noncompact strictly pseudoconvex CR-manifold; the form α ∧ (dα)N is, as before, a nonvanishing surface element on X, and we define H 2 as the closure in L2 of the subspace of functions that extend holomorphically to × C. The corresponding Szegö kernel will then still satisfy (46), and one can again introduce the global Toeplitz operators TF , F ∈ L∞ (X), and Tfˆ , f ∈ L∞ (), and the generalized Toeplitz operators TP , with P a pseudodifferential operator on X which we now assume in addition to be bounded on L2 . Let again T be the set of all generalized Toeplitz operators commuting with the circle action Uθ . By the Calderon–Vaillancourt theorem, Tfˆ ∈ T if f ∈ BC ∞ (), the subspace in C ∞ () of functions all partial derivatives of which belong to L∞ (). Problem. Do the properties (P1)–(P5) prevail in this setting? As has already been mentioned above, this was settled in the affirmative in the case of = CN and φ(z) = |z|2 in [Bw]. (4) We conclude by exhibiting how the Berezin–Toeplitz star product implicit in Theorem 3, namely, f ∗ g :=
∞
hj Cj (f, g),
f, g ∈ C ∞ ()[[h]],
(50)
j =0
with Cj given by (10) and extended C[[h]]-linearly from C ∞ () to C ∞ ()[[h]], is related to the Berezin star product constructed in Sect. 4. Let us for the moment write ∗BT for the former and ∗B for the latter, and as in Sect. 4 set again h = 1/m and (φ m χ) (h) (h) (m) Tf = Tf , kx = kx , etc. For f, g ∈ C ∞ (), we thus have (Bh f ∗B Bh g)(x) =
∞ k=0
CkB (Bh f, Bh g)(x) hk ,
where CkB are the operators Ck from (39), extended to C ∞ ()[[h]] by C[[h]]-linearity, and we regard Bh f =
∞
Qj f hj
(51)
j =0
(h) as an element of C ∞ ()[[h]]. On the other hand, since Bh f = Tf and, by the definition (h) (h) (h) (h) of ∗B , Tf ∗B Tg = (Tf Tg )˜(cf. (33) and (44)), we have (h)
(Bh f ∗B Bh g)(x) = Tf Tg(h) kx(h) , kx(h) Ah . But in view of (44) again and (10), the last expression is equal to Bh
∞ k=0
CkBT (f, g) hk ,
Bergman Kernels
239
where CkBT are the operators Ck from (50). Thus the two star products are related by ∞ k=0
Bh CkBT (f, g) hk =
∞ k=0
CkB (Bh f, Bh g) hk .
(52)
As Q0 = I , it follows from (51) that Bh , regarded as a formal power series in h with differential operators as coefficients, has an inverse Bh−1 ; one can thus rewrite (52) as ∞ k=0
CkBT (f, g) hk = Bh−1
∞ k=0
CkB (Bh f, Bh g) hk ,
or, regarding Bh as a C[[h]]-linear operator on C ∞ ()[[h]] (i.e. as a formal differential operator), Bh (f ∗BT g) = Bh f ∗B Bh g.
(53)
Thus the two star products differ only by a “change of ordering” f ↔ Bh f . In the terminology of [Kar], (53) says that the star products ∗B and ∗BT are duals of one another. (5) Using Theorems 11 and 12 and the formulas for Q1 , Q2 , C1B and C2B there, we obtain from (52) the following corollary. Corollary 15. The operators CjBT are bilinear differential operators, with coefficients of the same form as in Theorem 11. In particular, C1BT (u, v) = − g j k u/k v/j , j,k
C2BT (u, v) =
1 j k lm g g u/km v/j l − Ricj k u/k v/j . 2 j,k,l,m
j,k
(6) Again, the Berezin–Toeplitz star product is an example of a deformation quantization with the separation of variables, but this time with the role of holomorphic and anti-holomorphic variables swapped (i.e. f ∗BT g = f g if g is holomorphic or f is anti-holomorphic); see [KS] for the identification of ∗BT within this scheme. Acknowledgement. Part of this work was done while the author was visiting Erwin Schrödinger Institute for Mathematical Physics in Vienna in November 1999. It is a pleasure for the author to acknowledge its support, and to thank E. Straube for helpful discussions. The author also thanks the referee for very useful comments.
References [AO]
Arazy, J., Ørsted, B.: Asymptotic expansions of Berezin transforms. Indiana Univ. Math. J. 49, 7–30 (2000) [BFG] Beals, M., Fefferman, C., Grossman, R.: Strictly pseudoconvex domains in Cn . Bull. Amer. Math. Soc. 8, 125–326 (1983) [Be1] Bell, S.: Extendibility of the Bergman kernel function. In: Complex Analysis II. College Park 1985–86, Lecture Notes in Math. 1276. Berlin–New York: Springer-Verlag 1987, pp. 33–41 [Be2] Bell, S.: Differentiability of the Bergman kernel and pseudo-local estimates. Math. Z. 192, 467–472 (1986)
240
[Ber] [BH] [Bo]
M. Engliš
Berezin, F.A.: Quantization. Math. USSR Izvestiya 8, 1109–1163 (1974) Bleistein, N., Handelsman, R.A.: Asymptotic expansions of integrals. New York: Dover, 1986 Boas, H.: Extension of Kerzman’s theorem on differentiability of the Bergman kernel function. Indiana Univ. Math. J. 36, 495–499 (1987) [BFS] Boas, H.P., Fu, S., Straube, E.: The Bergman kernel function: explicit formulas and zeros. Proc. Amer. Math. Soc. 127, 805–811 (1999) [BMS] Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler manifolds and gl(n), n → ∞ limits. Commun. Math. Phys. 165, 281–296 (1994) [Bw] Borthwick, D.: Microlocal techniques for semiclassical problems in geometric quantization. In: Perspectives on quantization. South Hadley 1996, Contemp. Math. 214, Providence, RI: AMS 1998, pp. 23–27 [BLU] Borthwick, D., Lesniewski, A., Upmeier, H.: Non-perturbative deformation quantization on Cartan domains. J. Funct. Anal. 113, 153–176 (1993) [BU] Borthwick, D., Uribe, A.: Almost complex structures and geometric quantization. Preprint math/9608006, Math. Res. Lett. 3, 845–861 (1996) [BG] Boutet de Monvel, L., Guillemin, V.: The spectral theory of Toeplitz operators. Ann. Math. Studies 99. Princeton, NJ: Princeton University Press, 1981 [BS] Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. Astérisque 34–35, 123–164 (1976) [Ca] Catlin, D.: The Bergman kernel and a theorem of Tian. In: Analysis and geometry in several complex variables, Katata 1997, Trends in Math, Boston, NY: Birkhäuser, 1999, pp. 1–23 [Co] Coburn, L.A.: Deformation estimates for the Berezin–Toeplitz quantization, Commun. Math. Phys. 149, 415–424 (1992) [CT] Cornalba, L., Taylor IV, W.: Holomorphic curves from matrices. Preprint hep-th/9807060. Nuclear Phys. B 536, 513–552 (1999) [E1] Engliš, M.: Asymptotics of the Berezin transform and quantization on planar domains. Duke Math. J. 79, 57–76 (1995) [E2] Engliš, M.: Berezin quantization and reproducing kernels on complex domains. Trans. Amer. Math. Soc. 348, 411–479 (1996) [E3] Engliš, M.: Asymptotic behaviour of reproducing kernels of weighted Bergman spaces, Trans. Amer. Math. Soc. 349, 3717–3735 (1997) [E4] Engliš, M.: Asymptotic behaviour of reproducing kernels, Berezin quantization and mean-value theorems. In: S. Saitoh, D. Alpay, J.A. Ball, T. Ohsawa (eds.), Reproducing kernels and their applications, International Society for Analysis, Applications and Computation, 3, Dordrecht: Kluwer, 1999, pp. 53–64 [E5] Engliš, M.: The asymptotics of a Laplace integral on a Kähler manifold. J. reine angew. Math. 528, 1–39 (2000) [E6] Engliš, M.: A Forelli-Rudin construction and asymptotics of weighted Bergman kernels. J. Funct. Anal. 177, 257–281 (2000) [E7] Engliš, M.: Pseudolocal estimates for ∂ on general pseudoconvex domains. Indiana Univ. Math. J. (2001), to appear [Erd] Erd˝os, L.: Ground state density of the Pauli operator in the large field limit. Lett. Math. Phys. 29, 219–240 (1993) [Fed] Fedoryuk, M.V.: Asymptotics, integrals, series. Moscow: Nauka, 1987 [Fef] Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Inv. Math. 26, 1–65 (1974) [For] Forelli, F.: Measures whose Poisson integrals are pluriharmonic. Illinois J. Math. 18, 373–388 (1974) [FR] Forelli, F., Rudin, W.: Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J. 24, 593–602 (1974) [Gu] Guillemin, V.: Star products on pre-quantizable symplectic manifolds. Lett. Math. Phys. 35, 85–89 (1995) [Kan] Kaneko, A.: Introduction to Kashiwara’s microlocal analysis for the Bergman kernel. Preprint JTokyo-Math 89-03, Univ. of Tokyo, Dept. of Mathematics, 1989 [Kar] Karabegov, A.V.: Deformation quantization with separation of variables on a Kähler manifold. Commun. Math. Phys. 180, 745–755 (1996) [KS] Karabegov, A.V., Schlichenmaier, M.: Identification of Berezin–Toeplitz deformation quantization. Preprint math/0006063 (2000) [Kas] Kashiwara, M.: Analyse microlocale du noyau de Bergman. Séminaire Goulaouic–Schwartz 1976– 1977, exposé no. 8. Palaiseau: École Polytechnique, 1977 [Ke] Kerzman, N.: The Bergman kernel function. Differentiability at the boundary. Math. Ann. 195, 149– 158 (1972) [KL] Klimek, S., Lesniewski, A.: Quantum Riemann surfaces, I: The unit disc. Commun. Math. Phys. 146, 103–122 (1992); II: The discrete series. Letters in Math. Phys. 24, 125–139 (1992)
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Kohn, J.J.: Subellipticity of the ∂-Neumann problem on pseudo-convex domains: sufficient conditions. Acta Math. 142, 79–122 (1979) [Krn] Krantz, S.G.: Function theory of several complex variables. Pacific Grove: Wadsworth & Brooks/Cole, 1992 [KLR] Krantz, S.G., Li, S.-Y., Rochberg, R.:Analysis of some function spaces associated to Hankel operators. Illinois J. Math. 41, 398–411 (1997) [LM] Lee, J., Melrose, R.: Boundary behavior of the complex Monge-Ampére equation. Acta Math. 148, 159–192 (1982) [Lig] Ligocka, E.: On the Forelli–Rudin construction and weighted Bergman projections. Studia Math. 94, 257–272 (1989) [Pe] Peetre, J.: The Berezin transform and Ha-plitz operators. J. Operator Theory 24, 165–186 (1990) [Rai] Raikov, G.D.: Eigenvalue asymptotics for the Pauli operator in strong non-constant magnetic fields. Ann. Inst. Fourier 49, 1603–1636 (1999) [Ran] Range, R.M.: Holomorphic functions and integral representations in several complex variables. Berlin: Springer, 1986 [RT] Reshetikhin, N., Takhtajan, L.: Deformation quantization of Kähler manifolds. Preprint math/9907171, 1999 [Rud] Rudin, W.: Function theory in the unit ball of Cn . Berlin–Heidelberg–New York: Springer, 1980 [Sch] Schlichenmaier, M.: Deformation quantization of compact Kähler manifolds by Berezin–Toeplitz quantization. Preprint math.QA/9910137, 1999 [Ze] Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998) [Ko]
Communicated by A. Connes
Commun. Math. Phys. 227, 243 – 279 (2002)
Communications in
Mathematical Physics
Segregation in the Falicov–Kimball Model James K. Freericks1, , Elliott H. Lieb2, , Daniel Ueltschi3,,† 1 Department of Physics, Georgetown University, Washington, DC 20057, USA.
E-mail:
[email protected]
2 Departments of Mathematics and Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA.
E-mail:
[email protected]
3 Department of Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA.
E-mail:
[email protected] Received: 21 June 2001 / Accepted: 4 January 2002
Abstract: The Falicov–Kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on-site repulsion; alternatively, it is a model of itinerant electrons and fixed nuclei. It can be seen as a simplification of the Hubbard model; by neglecting the kinetic (hopping) energy of the spin up particles, one gets the Falicov–Kimball model. We show that away from half-filling, i.e. if the sum of the densities of both kinds of particles differs from 1, the particles segregate at zero temperature and for large enough repulsion. In the language of the Hubbard model, this means creating two regions with a positive and a negative magnetization. Our key mathematical results are lower and upper bounds for the sum of the lowest eigenvalues of the discrete Laplace operator in an arbitrary domain, with Dirichlet boundary conditions. The lower bound consists of a bulk term, independent of the shape of the domain, and of a term proportional to the boundary. Therefore, one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary. For the Falicov- Kimball model, this corresponds to having a single “compact” domain that has no heavy particles. Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . The Discrete Laplace Operator in a Finite Domain Lower Bound Involving the Boundary . . . . . . . Finite U . . . . . . . . . . . . . . . . . . . . . .
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© 2001 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
J. K. F. is partially supported by the Office of Naval Research under grant N00014-99-1-0328. E. H. L. and D. U. are partially supported by the National Science Foundation under grant PHY-98 20650. † Present address: Department of Mathematics, University of California, Davis, CA 95616, USA.
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Upper Bound . . . . . . . . . . Positive Electronic Temperature Conclusion . . . . . . . . . . . Appendix . . . . . . . . . . . .
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1. Introduction 1.1. The Falicov–Kimball model. Introduced thirty years ago to describe the semiconductor-metal transition in SmB6 and related materials [FK], the Falicov–Kimball model is a simple lattice model with rich and interesting properties. The system consists of two species of spinless electrons with different effective masses: one species has infinite mass (so the particles do not move – we call them “classical particles”), while the second species represents itinerant spinless electrons whose kinetic energy is represented by a hopping matrix. The Hamiltonian in a finite domain ⊂ Zd is cx† cy + 2d nx + U wx nx . (1.1) HU ({wx }) = − x,y∈ |x−y|=1
x∈
x∈
Here, cx† , cx , denote creation, annihilation operators of an electron at site x; nx = cx† cx ; wx = 0, 1 is the number of classical particles (“heavy electrons”) at x, and U 0 is an on-site repulsion between the two species of particles. HU ({wx }) represents the energy of the electrons under a potential U wx . The term 2d nx in (1.1) is for convenience only. It makes HU positive, and this term only adds 2d times the electron number, N. At zero temperature, one is typically interested in the configurations of classical particles that minimize the ground state energy of the electrons. The model was reinvented in [KL] as a simplification of the Hubbard model, by neglecting the hoppings of electrons of spin ↑, say. This simplification changes the nature of the model somewhat, mainly because the continuous SU(2) symmetry is lost. Connections between the two models are therefore not immediate; however, the greater knowledge obtained for the Falicov–Kimball model may help in understanding the Hubbard model. Rigorous results in [KL] include a proof that equilibrium states display long-range order of the chessboard type when both species of particles have density 1/2; this holds for all dimensions greater than 1 and for all U = 0 (including U < 0), provided the temperature is low enough. The model is reflection positive under a suitable magnetic field, or when the electrons are replaced by hard-core bosons; this property can be used to establish long-range order [MP]. Perturbative methods allow for an extension of these results for large U and small temperature, see [LM, MM, DFF]. Absence of long-range order when the inverse temperature β is small, or βU is small, was also established in [KL]. One may increase the density of one species and decrease the density of the other species while maintaining the half-filling condition, namely that the total density is 1. (However, as was shown in [KL], the lowest energy is achieved when both species have density 1/2.) The one-dimensional case was considered in [Lem]; if classical particles and electrons have respective densities pq and 1 − pq , the ground state is the “most homogeneous configuration” for U large enough; this configuration is periodic with a period no greater than q. Away from half-filling the particles segregate: classical particles occupy one side of the chain, leaving room for electrons on the other side. There are
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several results in 2D. Gruber et. al. [GJL] performed a 1/U expansion and found periodic phases when the density of the classical particles is 1/3, 1/4, 1/5. This was made rigorous by Kennedy [Ken]. These results are reviewed in [GM]. The knowledge of the 2D phase diagram for large U was further extended in [Ken2, Hal, HK]. New ground states for various rational densities were uncovered; for some densities they are periodic, but there are also densities where coexistence of configurations with different periods have minimum energy. The results are summarized in Fig. 1 in [HK]. Finally, it was understood in [DMN] that the 111 interface is stable, due to the effective interactions between the particles.
1.2. Away from half-filling. The purpose of our study is to explore the situation away from half-filling; i.e., we take the total density to differ from 1. For any density away from half-filling we prove that the ground state is segregated for U large enough. When U = ∞, the ground state is segregated for all densities (at half filling all configurations have the same energy, including segregated and periodic ones). Hole-particle symmetries for both species of particles [KL] imply that the results for positive U and densities (ne , nc ) of electrons and heavy (classical) particles, transpose to (a) positive U and densities (1 − ne , 1 − nc ), and (b) negative U and densities (ne , 1 − nc ) or (1 − ne , nc ). For simplicity, we take the total density ne + nc to be strictly less than 1. We start our study by taking the limit U → ∞. Electrons are described by wave functions that vanish on sites occupied by the classical particles, and the question is to find the arrangement of classical particles that minimizes the energy of the electrons. This amounts to minimizing the sum of the lowest eigenvalues of the discrete Laplace operator with Dirichlet boundary conditions. This is explained in Sect. 2, where it is shown that the energy per site of N electrons in a finite domain ⊂ Zd with volume ||, is bounded below by the energy per site of the electrons in the infinite lattice with density n = N/||. One can refine this lower bound by including a term proportional to |∂|, the volume of the boundary ∂ of (Sect. 3). This implies that the configuration of the heavy, fixed electrons that minimizes the ground state energy of the movable electrons has, more or less, one large hole with relatively small perimeter. Thus the movable particles are separated from the fixed ones. This behavior was conjectured in [FF] and is opposite to the checkerboard configuration, in which both kinds of particles are inextricably mixed. Segregation was shown to occur in the ground state of the d = 1 model in [Lem], and of the d = ∞ model in [FGM]. The present paper proves that this holds for all dimensions, and in particular for the relevant physical situations d = 2 and d = 3. Segregation is more difficult to understand on a heuristic level than the chessboard phase. The latter is a local phenomenon that results from effective interactions between nearest neighbor sites, while the former is a global phenomenon involving extended wave functions. This remark should also apply to the Hubbard model, for which antiferromagnetism is much better understood than ferromagnetism. The fact that the sum of the lowest N eigenvalues of the Laplacian in a domain of volume || is bounded below by the infinite volume value at the same density is not unexpected and holds also in the continuum. Indeed, the original idea, due to Li and Yau [LY] (see also [LL] Sects. 12.3 and 12.11), was demonstrated in the continuum, and we only adapted it to the lattice context. However, the fact that the error term is proportional to |∂|, the area of the boundary, is a completely different story. Its proof, at least the one given here, is complicated. More to the point, such a bound does not hold in the
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continuum. One can easily construct a continuum domain with finite volume ||, but with |∂| = ∞, and for which all eigenvalues are finite. Taking U large instead of infinite decreases the energy, but the gain is at most proportional to |∂|, as explained in Sect. 4. Therefore the infinite–U segregation effect outweighs this gain and the particles are still separated (at least when the heavy particle density is far enough from 1/2). Finally, an upper bound derived in Sect. 5 shows that the energy of electrons in really consists of a bulk term independent of the shape of , plus a term of the order of the boundary. These results are summarized in Theorem 1.1 below. One can use them to discuss the electronic free energy at inverse temperature β, for a fixed configuration of classical particles, see Theorem 1.2. The conclusion of this paper involves a discussion of firstorder phase transitions at finite temperature, of what happens when classical particles have a small hopping term, and of the possible links with ferromagnetism in systems of interacting electrons with spins. In order to present the main result of this paper, we need a few definitions. For k ∈ (−π, π]d , we set εk = 2d − 2
d
cos ki .
(1.2)
i=1
The energy per site e(n) of a density n of free electrons in the infinite volume Zd is 1 e(n) = εk dk, (1.3) (2π )d εk <εF where the Fermi level εF = εF (n) is defined by the equation 1 n= dk. (2π )d εk <εF
(1.4)
We can specify the configuration (wx )x∈Zd of classical particles by the domain ⊂ / and Zd consisting of those sites without particles (holes), that is, w(x) = 1 if x ∈ w(x) = 0 if x ∈ . Let hU denote the one-particle Hamiltonian whose action on a square summable, complex function ϕ on Zd is [hU ϕ(y) + 2dϕ(x) + U χ c (x)ϕ(x). (1.5) ϕ](x) = − y,|y−x|=1
Here, χ c (x) is the characteristic function that is 1 if x belongs to the complement c U of , and is 0 if x ∈ / c . We define E,N to be the ground state energy of N electrons for the configurations defined by , i.e. U = E,N
inf
{ϕ1 ,...,ϕN }
N i=1
(ϕi , hU ϕi ),
(1.6)
where the infimum is taken over N orthonormal functions, i.e. (ϕi , ϕj ) = δij . There exist normalized minimizers if the Fermi level is below U ; they are not identically zero inside , and decay exponentially outside. U Notice that E,N is increasing in U , since (ϕ, hU ϕ) is increasing in U for any ϕ.
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We define the boundary by ∂ = {x ∈ : dist (x, c ) = 1}, where c is the complement of , i.e., the points in Zd not in . The following theorem summarizes the results obtained in this paper. It contains upper and lower bounds for the ground state energy. We set n = N/||. Theorem 1.1. There are functions α(n) > 0 and γ (U ) with limU →∞ U γ (U ) = 8d 2 , such that for all finite domains ,
U 2dn − e(n) |∂| E,N − ||e(n) α(n) − γ (U ) |∂|.
Furthermore, α(n) = α(1 − n), and for n |Sd |/(4π )d , it can be chosen as α(n) =
2d−3 π d d 3 |S
2/d d|
2
n1+ d .
Here, |Sd | is the volume of the unit sphere in d dimensions. An explicit expression for γ (U ) can be found in Proposition 4.1. The lower bound α(n) vanishes when n = 0 (no itinerant electrons) or n = 1 (fully occupied lower band). Theorem 1.1 is relevant only when α(n) > γ (U ), that is, sufficiently away from half-filling (depending on U ). The theorem states that the “good” configurations for which electrons have low energy must have small boundaries. As a consequence, the system displays phase separation in the ground state. The upper bound is symmetric under the transformation n → 1−n due to a symmetry of the Hamiltonian, and it is saturated for U = ∞ by configurations with isolated holes. Indeed, in this case the eigenstates consist of δ functions on the holes, with eigenvalues equal to 2d, and ∂ = . The lower bound is first explained in Sect. 2 for U = ∞ and without the term involving the boundary. The latter requires more effort and is derived using Lemmas 3.2–3.5 in Sect. 3. Proposition 4.1 then extends it to the case of finite U . The upper bound is proved in Sect. 5. Theorem 1.1 is described in [FLU], which also reviews the rigorous results obtained so far for the Falicov–Kimball model.
1.3. Electrons at low temperature. It is natural to consider the situation at positive temperature. The relevant object is the Gibbs state obtained by averaging over the configurations of classical particles, and by taking the trace of the Gibbs operator exp −βHU ({wx }) . We expect the system to display a first-order phase transition in the grand-canonical ensemble; densities of both types of particles should have discontinuities as functions of the chemical potentials. But a rigorous treatment of this phase transition is beyond reach at present. However, we do obtain some properties of the system when the configuration of the classical particles is fixed, and the electrons are at positive temperature. Namely, one can extend the estimates of the ground state energy to estimates of the electronic free energy. The results are described in this section, and their derivation can be found in Sect. 6. Let us consider a box with periodic boundary conditions. The configuration of classical particles is specified by the set of holes ⊂ (later, in Corollary 1.3, we shall
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average over ). With µ being the chemical potential, the grand-canonical electronic free energy (equal to −||/β times the pressure) is U F, (β, µ) = −
1 U + βµN . log Tr exp −βH, β
(1.7)
U Here, H, = HU ({wx }) as defined in (1.1), N = x∈ nx is the number of electrons in , and the trace is in the Fock space of antisymmetric wave functions on . U is obtained by considering independent electrons, which A simple “guess” for F, are either in or else in c . In the latter case the effective chemical potential is µ − U . Our “guess” would then be U F, (β, µ) ≈ ||f (β, µ) + (|| − ||)f (β, µ − U ),
where f (β, µ) is the free energy per site for free electrons: 1 1 f (β, µ) = − dk log 1 + e−β(εk −µ) . d d β (2π ) [−π,π]
(1.8)
(1.9)
Formula (1.8) is, indeed, correct when U is large – in the sense that the error is proportional only to |∂|. More precisely, Theorem 1.2. There are functions α(β, ¯ µ) > 0 with limβ→∞ α(β, ¯ µ) > 0 if 0 < µ < 4d, and γ¯ (U ) with limU →∞ U γ¯ (U ) = 16d 2 + 2d+3 d 4 , such that for all finite domains and ⊂ , 1 U ||1− d F, (β, µ) − ||f (β, µ) + (|| − ||)f (β, µ − U ) Cd,µ |∂| + Cd,µ α(β, ¯ µ) − γ¯ (U ) |∂|, with √ 4π d 1 = + 2d(2d + 1) , |Sd |1/d 1 + e−βµ √ 1 4π d = . 1/d −β(µ−U ) |Sd | 1 + e
Cd,µ Cd,µ 1
The term ||1− d on the left side is not exactly proportional to |∂|. However, we 1 have in mind that || and || are comparable, in which case ||1− d is no greater than |∂| (up to a factor). Notice that the upper bound vanishes as µ → −∞, i.e. when the density tends to 0. In the limit U → ∞, Theorem 1.2 takes a simpler form, namely U =∞ Cd,µ |∂| F, (β, µ) − ||f (β, µ) α(β, ¯ µ)|∂|.
(1.10)
This extension of Theorem 1.1 to the case of positive (electronic) temperatures is explained in Sect. 6. The lower bound follows from Propositions 6.1 and 6.2, while the upper bound is stated in Proposition 6.3.
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Our next step is to find upper and lower bounds for the total grand-canonical “free energy” by averaging over (i.e., averaging over the positions of the classical particles). This can be done with the aid of the Ising model free energy fIsing (β, h), fIsing (β, h) = −
1 1 lim exp −β sx sy − βh sx , β ||→∞ || {sx =±1}
{x,y}⊂ |x−y|=1
(1.11)
x∈
where the sum is over configurations of classical spins on . Corollary 1.3. If U is large enough (so that α(β, ¯ µ)− γ¯ (U ) > 0), we have Ising bounds for the full free energy, 1 1 1 α¯ ¯ + 4d fIsing 4d αβ, ¯ 2d 2 [f (β, µ) + f (β, µ − U )] + 4 α α¯ [f (β, µ) − f (β, µ − U )] −βF U (β,µ) 1 1 , − e lim log β ||→∞ ||
⊂ 1 1 2 [f (β, µ) + f (β, µ − U )] + 2 dCd,µ 1 + 21 Cd,µ fIsing 21 Cd,µ β, Cd,µ [f (β, µ) − f (β, µ − U )] ,
where α¯ = α(β, ¯ µ) − γ¯ (U ). The proof can be found at the end of Sect. 6. Another consequence of Theorem 1.2 concerns the equilibrium state; namely, it allows for a precise meaning of segregation. We consider the probability that sites x and y are both occupied by classical particles, or both are unoccupied. Namely, we consider U ⊂:wx =wy exp −βF, (β, µ) , δwx ,wy = (1.12) U ⊂ exp −βF, (β, µ) where the sums are over subsets of such that || = [(1 − nc )||] ([z] denotes the integer part of z ∈ R). The restriction wx = wy means that either both x and y belong to , or both belong to the complement of . Segregation means that up to a small fraction of sites that are close to the boundary between classical particles and empty sites, any two sites at finite distance are either both hosts of a classical particle, or are both empty. The fraction of sites close to the boundary vanishes in the thermodynamic limit. Hence we expect that lim
lim
lim δwx ,wy = 1,
β→∞ |x−y|→∞ ||→∞
(1.13)
but we are unable to prove it. Notice that using Theorem 1.1, one can conclude that lim
lim
lim δwx ,wy = 1.
|x−y|→∞ ||→∞ β→∞
(1.14)
Indeed, taking the limit of zero temperature at finite volume, the sum over becomes restricted to the ground state configuration(s), whose boundary fraction |∂|/|| tends to zero in the thermodynamic limit. We can however take advantage of Theorem 1.2 to obtain a result that is better than (1.14):
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Corollary 1.4. If U is large enough (depending on µ and d only), the ground state of the Falicov–Kimball model displays segregation, in the sense that lim
lim
lim δwx ,wy = 1.
|x−y|→∞ β→∞ ||→∞
The proof of this corollary can be found at the end of Sect. 6. 2. The Discrete Laplace Operator in a Finite Domain 2.1. Basic properties. We start our investigations by taking the limit U → ∞. Let us =∞ , the corresponding Hamiltonian, which acts on functions ϕ ∈ L2 () denote h ≡ hU as follows: if x ∈ , ϕ(y) + 2d ϕ(x). (2.1) [h ϕ](x) = − y∈,|x−y|=1
Some observations can readily be made that will be useful in the sequel. For ϕ ∈ L2 (), one has the following formula, ϕ(x) − ϕ(y) 2 , (ϕ, h ϕ) = (2.2) {x,y}:|x−y|=1
where the sum is over all x, y ∈ Zd , with the understanding that ϕ(x) = 0 for x ∈ / . Equation (2.2) takes a simple form because of the diagonal term in h . The effect of the Dirichlet boundary conditions appears through a term x∈∂ |ϕ(x)|2 , that is due to pairs {x, y} with x ∈ and y ∈ / . The matrix h is self-adjoint, and (2.2) shows that h 0. Its spectrum has a symmetry. Let ϕ be an eigenvector with eigenvalue e, and ϕ¯ be the function ϕ(x) ¯ = (−1)|x| ϕ(x); one easily checks that ϕ¯ is also an eigenvector, with eigenvalue (4d − e). The spectrum is therefore contained in the interval [0, 4d], and is symmetric around 2d. Furthermore, one has E,||−N = 2d || (1 − 2n) + E,N , e(1 − n) = 2d (1 − 2n) + e(n).
(2.3)
This allows to restrict ourselves to the case n 21 ; indeed, existence of a lower bound for a density n implies a lower bound for the density 1 − n. This symmetry holds only for U = ∞. 2.2. The bulk term. We are looking for a lower bound for the sum E,N of the first N eigenvalues of h . This problem was considered by Li and Yau [LY] for the Laplace operator in the continuum. Let ⊂ Rd be a bounded domain. They prove that the sum SN of the first N eigenvalues of the Laplace operator with Dirichlet boundary conditions is bounded below, SN > (2π)2
2 2 2 d |Sd |− d N 1+ d ||− d , d +2
(2.4)
where |Sd | and || are the volumes of respectively the d-dimensional sphere and of .
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The corresponding inequality in the discrete case – our Theorem 1.1 without the boundary correction – constitutes the heart of this paper, and we explain below the proof of Li and Yau; see also [LL], Theorem 12.3. It is useful to introduce the Fourier transform for functions in L2 (Zd ), that is, on the whole lattice. A function ϕ ∈ L2 () can be considered an element of L2 (Zd ) by setting ϕ(x) = 0 outside . This Fourier transform will lead to the electronic energy density for the infinite lattice, which is the bulk term for the energy of electrons in . The Fourier transform of a function ϕ is defined by ϕ(k) ˆ = ϕ(x)eikx , k ∈ [−π, π ]d , (2.5) x∈Zd
and the inverse transform is ϕ(x) =
1 (2π )d
[−π,π]d
−ikx dk ϕ(k)e ˆ .
(2.6)
Using the Fourier transform, a little thought shows that the energy of a particle in a state ϕ in L2 () is 1 2 (ϕ, h ϕ) = dk |ϕ(k)| ˆ εk , (2.7) (2π )d [−π,π]d / in (2.5). with εk defined in (1.2) and with ϕ(x) = 0 if x ∈ Let us consider N orthonormal functions ϕ1 , . . . , ϕN , and let E (ϕ1 , . . . , ϕN ) be their energy. We have 1 E (ϕ1 , . . . , ϕN ) = dk ρ(k)εk , (2.8) (2π )d [−π,π]d with ρ(k) =
N
|ϕˆj (k)|2 .
(2.9)
j =1
The function ρ(k) satisfies the following equations: 0 ρ(k) ||, 1 dk ρ(k) = N. (2π )d [−π,π]d
(2.10a) (2.10b)
Indeed, positivity of ρ is immediate and the last equation is Plancherel’s identity. The upper bound (2.10a) for ρ(k) can be seen by writing ρ(k) = (f, P f ),
(2.11)
where P is the projector onto {ϕj }N j =1 , Px,y =
N j =1
ϕj (x)ϕj∗ (y),
(2.12)
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and f is the vector fx = e−ikx χ (x).
(2.13)
Then, since P 1l, we have ρ(k) f 2 = ||. Clearly, we have the lower bound 1 E,N inf dk ρ(k) εk . ρ: 0ρ|| (2π )d [−π,π]d
(2π)−d
(2.14)
ρ=N
We can use the bathtub principle ([LL], Theorem 1.14) to find the infimum: it is given by the function || if εk εF ρmin (k) = (2.15) 0 otherwise,
1 where the Fermi level εF is given by the relation (2π) d εk <εF dk = N/||. Thus the right side of (2.14) is precisely equal to || e(N/||). 3. Lower Bound Involving the Boundary In the previous section, we showed that E,N is bounded below by its bulk term. Now we strengthen this inequality and prove that E,N also includes a term proportional to the boundary of . This can be checked for d = 1 by explicit computation, but higher dimensions require more elaborate treatment. We start with a lemma that applies when the density n is small enough (or, by the symmetry for h , when it is close to 1). Lemma 3.1. If n |Sd |/(4π)d , we have 2
E,N
2d−3 n1+ d ||e(n) + d 3 |∂|. π d |Sd |2/d
Proof. Recall that E,N =
1 (2π )d
[−π,π]d
dk ρ(k)εk
(3.1)
2 . We want to show that ρ(k) cannot be too close to ρ with ρ(k) = N ˆ min (k) j =1 |ϕ(k)| in (2.15). By completeness of the set of eigenvectors {ϕj }, we have ρ(k) = || −
||
2 |ϕ(k)| ˆ .
(3.2)
j =N+1
We now use the Schrödinger equation. We have h ϕj (x) = ej ϕj (x) for x ∈ . Let us take ϕj (x) = 0 for x ∈ / ; then the following equation holds true for all x ∈ Zd : − ϕj (x + e) + χ c (x) ϕj (x + e) + 2dϕj (x) = ej ϕj (x). (3.3) e
e:x+e∈
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The middle term of the left side is necessary for the equation to hold at sites outside , that are neighbors of sites in . Taking the Fourier transform, we get εk ϕˆj (k) + (bk , ϕj ) = ej ϕˆj (k), where bk is a “boundary vector”,
bk (x) = χ ∂ (x)e−ikx
(3.4)
e−ike .
(3.5)
e:x+e∈ /
Notice that |∂| bk 2 (2d)2 |∂| if |k|∞
π 3.
From (3.4), we have
|(bk , ϕj )|2 1 |(bk , ϕj )|2 . (3.6) 2 (εk − ej ) (4d)2
The electronic energy in is given by ρ(k)εk . We saw in Sect. 2 that 0 ρ(k) ||. By (3.5) and (3.6), this can be strengthened to |ϕˆj (k)|2 =
const P− bk 2 ρ(k) || − const P+ bk 2 ,
(3.7)
where P− (resp. P+ ) is the projector onto the subspace spanned by (ϕ1 , . . . , ϕN ) (resp. (ϕN+1 , . . . , ϕ|| )). See Fig. 3.1 for intuition. We show below that P+ bk 2 const |∂|,
(3.8)
and this will straightforwardly lead to the lower bound.
||
const P+ bk 2 ρmin (k) ρ(k)
εk
εF const P− bk 2 −π
0
π
k
Fig. 3.1. Illustration of the expression (2.8) for E,N ; ρ(k) satisfies more stringent estimates than those stated in (2.10a), and this plays an important role in deriving the lower bound
In order to see that the boundary vector has a projection in the subspace of the eigenvectors with large eigenvalues, we first remark that (bk , h bk ) = |bk (x) − bk (y)|2 bk 2 . (3.9) {x,y}:|x−y|=1
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We used the fact that each site x of ∂ has at least one neighbor y outside of , and we obtained an inequality by restricting the sum over such pairs. Let us introduce N such that eN 21 and eN +1 > 21 . We first consider the situation where N N . Using first ej 4d and then the previous inequality, we have 4d
||
||
|(bk , ϕj )| 2
j =N +1
|(bk , ϕj )|2 ej
j =N +1
bk − 2
N j =1
For |k|∞
π 3,
|(bk , ϕj )|2 ej 21 bk 2 .
(3.10)
(3.6) and (3.10) imply ρ(k) || −
|∂| . 2(4d)3
(3.11)
We can write a lower bound by proceeding as in Sect. 2, but using the bound (3.11) for ρ(k), instead of ||. The bathtub principle then gives 1 |∂| dk ε || − E,N − ||e(n) k (2π )d εF <εk <εF 2(4d)3 1 |∂| − dk εk , (3.12) d (2π ) εk <εF 2(4d)3 where we introduce εF such that N=
|∂| 1 || − (2π)d 2(4d)3
εk <εF
dk.
(3.13)
Notice that for n |Sd |/(4π)d , we have εF < 21 , so that εk < εF implies |k|∞ < π3 . This justifies the use of (3.11). We bound the first integral of (3.12) using εk > εF , and we obtain 1 |∂| dk(εF − εk ). (3.14) E,N − ||e(n) 2(4d)3 (2π )d εk <εF One can derive a more explicit expression for the lower bound. First, dk(εF − εk ) 21 εF dk. εk <εF
Second we use 1 −
θ2 2
εk 21 εF
cos θ 1 −
4 2 θ , π2
to get
8 |k|2 εk |k|2 . π2 One can use the upper bound of (3.16) to get dk |Sd |( 21 εF )d/2 . εk < 21 εF
(3.15)
(3.16)
(3.17)
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Recall that |Sd | is the volume of the unit sphere in d dimensions. The lower bound of (3.16) allows to write 25 n2/d . |Sd |2/d
εF
(3.18)
Then one gets the bound
2
24+2d n1+ d dk(εF − εk ) . |Sd |2/d εk <εF
(3.19)
Hence the boundary correction to E,N is bounded below by α(n)|∂| with α(n) =
2d−3 π d d 3 |S
2/d d|
2
n1+ d .
(3.20)
Recall that we supposed N N , where N is the index of the largest eigenvalue that is smaller than 21 . Were it not the case, we can write, with n = N /||,
E,N =
N j =1
ej +
N j =N +1
ej ||e(n ) + α(n )|∂| + 21 ||(n − n ).
We used the previous inequality to bound the first sum, and ej This is greater than ||e(n) + α(n)|∂| provided e(n ) + α(n )
1 2
(3.21)
for the second sum.
|∂| 1 |∂| + 2 (n − n ) e(n) + α(n) . || ||
(3.22)
A sufficient condition is that 21 n − e(n) − α(n) is an increasing function of n. Computing the derivative (the derivative of e(n) is εF (n), that is smaller than (2π )2 n2/d /|Sd |2/d using (3.16)), and requiring it to be positive leads to the condition n
{2(2π )2
+
The right side is greater than |Sd |/(4π )d .
|Sd | . 2d−2 (1 + d2 )}d/2 π d d3
(3.23)
It may seem obvious that the extra energy due to the presence of the boundary increases as n increases, until it reaches 21 . But we can provide no proof for this, and hence we need a new derivation for the lower bound with higher densities. We proceed in two steps. First we give a lemma that works when the boundary has few nearest neighbors; the proof is similar to that of the previous lemma. Then we give three lemmas, with more intricate demonstrations, and that establish the lower bound for boundaries where at least a density of sites have nearest neighbors. We need some notation to characterize the configuration around a site x of the boundary. Let e, e be unit vectors in Zd ; the notation e i means that e is parallel to the i th direction; equivalently, the components of e are given by ek = ±δik . We introduce integers qx,i and qx,ij ; for x ∈ ∂, we set qx,i = #{e i : x + e ∈ / }, qx,ij = #{(e, e ) : e i, e j,
x + e ∈ ∂,
x + e + e ∈ / }.
(3.24)
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Notice that 0 qx,i 2 and 0 qx,ij 4. Also, / qx,ii = #{e i, x + e ∈ ∂, x + 2e ∈ }, and 0 qx,ii 2. We also define qx = i qx,i . The following lemma applies to domains where most boundary sites x satisfy qx,ij ≡ 0, in which case x has no neighbors that belong to the boundary. Here, qx,ij ≡ 0 means that, at x, qx,ij = 0 for all 1 i, j d. Lemma 3.2. For all ⊂ Zd with #{x ∈ ∂ : qx,ij ≡ 0}
1 |∂|, 32d 4
there exists α(n) > 0 such that E,N ||e(n) + α(n)|∂|. Remark. limn→0 α(n) = 0 and α(1 − n) = α(n) by the symmetry for h . Proof. We can suppose N || 2 . The definition of ρ(k) involves a sum over the first N eigenvectors (more precisely, of their Fourier transforms). In case of degenerate eigenvalues one is free to choose any eigenvectors. For the proof of Lemma 3.2 it turns out that the possible degeneracy of 2d brings some burden, and it is useful to redefine ρ(k) by averaging over eigenvectors with eigenvalue 2d: N if N N˜ j =1 |ϕˆj (k)|2 ρ(k) = ˜ (3.25) 2 if N > N N |ϕˆj (k)|2 + N−N˜ ˜; j :ej =2d |ϕˆ j (k)| j =1 ˜ ||−2N
2d. The degeneracy of 2d is || − 2N˜ (which here, N˜ is such that eN˜ < 2d and eN+1 ˜ may be zero). Of course, E,N is still given as the integral of ρ(k) multiplied by εk . || The goal is to prove that ρ(k) cannot approach ρmin in (2.15). Since j =1 |ϕˆj (k)|2 = ||, we have ρ(k) || −
||
|ϕˆj (k)|2 +
j =||−N˜ +1
1 |ϕˆj (k)|2 . 2
(3.26)
j :ej =2d
We introduce S(k) =
|| ˜ j =||−N+1
|(bk , ϕj )|2 +
1 |(bk , ϕj )|2 , 2
(3.27)
j :ej =2d
with bk the boundary vector defined in (3.5). By the inequality (3.6), it is enough to show that S(k) is bounded below by a quantity of the order of |∂|. We have (3.28) S(k) = bk , P+ bk + 21 (bk , P0 bk ), where P+ is the projector onto the subspace spanned by all ϕj with ej > 2d, and P0 is the projector corresponding to the eigenvalue 2d. We want to show that S(k) const|∂| for small |k|. This amounts to prove that the vector bk cannot lie entirely in the subspace spanned by {ϕj }1j N .
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Notice that if x ∈ ∂ and qx,ij ≡ 0, then x has no neighbors in ∂. Using the assumption of Lemma 3.2, as well as |bk (x)| 2d and |∂| bk 2 , we get (bk , h bk ) = |bk (x) − bk (y)|2 {x,y}:|x−y|=1
2d
|bk (x)|2 − 2d
x∈∂
|bk (x)|2
(3.29)
x∈∂ qx,ij ≡0
1 2d − bk 2 . 4d The last inequality uses the assumption of Lemma 3.2, and the fact that |bk (x)| is at most 2d and at least 1. Next we consider (h − 2d)bk 2 . We have, for x ∈ , (h − 2d)bk (x) = − bk (x + e), (3.30) e π 3,
and therefore, if |k|∞
(h − 2d)bk 2
|bk (x + e)|2 =
x∈ e
(2d − qx )|bk (x)|2 .
(3.31)
x
We write bk = b + b , with b (x) = bk (x) if qx = 2d, 0 otherwise. Notice that b ⊥ b . Clearly, P0 b = b , and therefore S(k) = (b , P+ b ) + 21 (b , P0 b ) + 21 b 2 .
(3.32)
Furthermore, from (3.29) and (3.31), b satisfies (b , (h − 2d)b ) −
1 bk 2 , 4d
(b , (h − 2d)2 b ) b 2 .
(3.33)
Because |ej − 2d| 2d, the last inequality implies |(ϕj , b )|2 (ej − 2d) + 2 |(ϕj , b )|2 (ej − 2d) − j
j :ej >2d
=
|(ϕj , b )|2 |ej − 2d|
j
b 2 . 2d
(3.34)
With the first inequality in (3.33), this yields
|(ϕj , b )|2 (ej − 2d)
j :ej >2d
bk 2 b 2 − , 4d 8d
(3.35)
hence j :ej >2d
|(ϕj , b )|2
b 2 bk 2 − . 8d 2 16d 2
(3.36)
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Back to (3.32), we obtain S(k)
b 2 bk 2 bk 2 + 21 b 2 − . 2 2 8d 16d 16d 2
(3.37)
We can combine this bound with (3.26) and (3.6); we have then for all |k|∞ < ρ(k) || −
|∂| . (4d)4
We introduce ε˜ F such that 1 || |∂| dk + dk = N, || − (2π)d (4d)4 εk <˜εF ,|k|∞ < π3 (2π )d εk <˜εF ,|k|∞ > π3
π 3,
(3.38)
(3.39)
and we have N E,N − ||e( || ) 1 |∂| || − dk εk (2π)d (4d)4 εF <εk <˜εF ,|k|∞ < π3 || 1 |∂| + dk ε − dk εk . k (2π)d εF <εk <˜εF ,|k|∞ > π3 (2π )d (4d)4 εk <εF ,|k|∞ < π3
(3.40)
We bound the first two integrals using εk > εF ; from the definitions of εF and ε˜ F we have |∂| || dk = dk. (3.41) (2π)d εk <˜εF ,|k|∞ < π3 (2π )d εF <εk <˜εF As a result, we obtain the bound we were looking for, |∂| 1 N E,N − ||e( || ) dk εF − εk . (4d)4 (2π )d εk <εF ,|k|∞ < π3
(3.42)
We present now another lemma that claims the lower bound for E,N , and that involves a new assumption. We shall see below in Lemmas 3.4 and 3.5 that for all volumes, at least one of these lemmas applies. Lemma 3.3. Let δ > 0 and n |Sd |/(4π )d . We assume that (h − eN )bk0 2 δ|∂|, for some k0 belonging to the Fermi surface, i.e. εk0 = εF , where εF is the Fermi energy N for density n = || . Then we have E,N − ||e(n) η |∂| with η = |Sd |5 δ 30d+2 /(2271d+23 π 10d+2 d 130d+9 ).
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259
The constant η that appears as a lower bound seems ridiculously small, but we cannot do better. Notice that this lower bound is much smaller than the one obtained in Lemma 3.1 at low density, with n = |Sd |/(4π )2 . We expect however that the lower bound is an increasing function of n for 0 n 21 , although we cannot prove it. Proof. We have E,N where
1 − ||e(n) = (2π )d
[−π,π]d
dk 2− (k)εk − 2+ (k)εk ,
(3.43)
2+ (k) = || − ρ(k) χ εk < εF , 2− (k) = ρ(k) χ εk > εF .
χ · is 1 if · is We used here another convention for the characteristic
function, namely true, and is 0 otherwise. Notice that dk 2− (k) = dk 2+ (k). Then we both have
1 d dk 2− (k)(εk − εF ) (2π) E,N − ||e(n) (3.44)
1 d dk 2+ (k)(εF − εk ). (2π) And by Hölder, this implies 5 1 4 1 1/5 − 41 E,N − ||e(n) dk [2 dk |ε (k)] − ε | . ± k F (2π)d (2π )d (3.45) One shows in Lemma A.1 (a) that the integral of |εk − εF |− 4 is bounded by 2. Recall that {ϕj }1j || are the eigenvectors of h . Let P− , resp. P+ , be the projectors onto the first N eigenvectors, resp. the last || − N eigenvectors. By (3.6), one has inequalities 1
1 P+ bk 2 (4d)2 1 2− (k) P− bk 2 (4d)2
2+ (k)
if εk < εF , if εk > εF .
Let us introduce sets A and A by A = k : εk < εF and |k − k0 | < A = k : εk > εF and |k − k0 | <
δ3 225 d 25/2 δ3 225 d 25/2
(3.46)
,
.
(3.47)
We obtain a lower bound by substituting (3.46) into (3.45), and restricting the integrals to A and A . Namely, 5 1 1 2/5 E,N − ||e(n) 8 2 dkP b , + k 2 d (2π )d A 5 1 1 E,N − ||e(n) 8 2 dkP− bk 2/5 . d 2 d (2π ) A
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Let b˜k = bk /bk . From the assumption of the lemma, and using h − ε 4d and Lemma A.1 (d), we have that for all k ∈ A ∪ A , bk 2 δ 5 2. |∂| 2 d
(3.48)
Extracting a factor |∂|, and using the above inequality, we can write 5 δ ˜k 2/5 , E,N − ||e(n) |∂| 13 4 b dk P + 2 d (2π )5d A 5 δ ˜k 2/5 . E,N − ||e(n) |∂| 13 4 dk P b − 2 d (2π )5d A
(3.49)
Consider k ∈ A. The assumption of the lemma for k0 , together with the bound for the gradient in Lemma A.1 (e), implies (bk , (h − eN )2 bk ) δ . |∂| 2
(3.50)
Therefore (b˜k , (h − eN )2 b˜k )
δ . 8d 2
(3.51)
This can be rewritten as ||
δ |(ϕj , b˜k )|2 (ej − eN )2 2 , 8d
(3.52)
δ 2 |(ϕj , b˜k )|2 (ej2 + eN ) 2 + 2eN (b˜k , h b˜k ). 8d
(3.53)
j =1
that is, || j =1
Hence (b˜k , h b˜k ) eN +
|| j =1
|(ϕj , b˜k )|2
ej2 2eN
−
δ eN − 5 3. 2 2 d
The quantity in the brackets is negative for j N . Observing that eN 1/2d+1 π 2 (because n |Sd |/(4π)d by 2d+4 π 2 d 2 . Therefore,
(3.54) e(|Sd |/(4π)d ) |Sd |/(4π)d
and using Lemma A.1 (c)), the bracket is bounded
(b˜k , h b˜k ) eN + 2d+4 π 2 d 2 P+ b˜k 2 −
δ . 25 d 3
(3.55)
On the other hand, for k ∈ A , (b˜k , h b˜k )
|| j =N+1
|(ϕj , b˜k )|2 ej eN+1 − 4dP− b˜k 2 .
(3.56)
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261
Since |k − k | δ 3 /224 d 25/2 , we have from Lemma A.1 (g) and (3.48), (b˜k , h b˜k ) − (b˜k , h b˜k )
δ 26 d 3
.
(3.57)
Therefore eN+1 − 4dP− b˜k 2 eN + 2d+4 π 2 d 2 P+ b˜k 2 −
δ 25 d 3
+
δ . 26 d 3
(3.58)
Clearly, eN eN+1 ; then P+ b˜k 2 +
1 P− b˜k 2 2d+2 π 2 d
δ 2d+10 π 2 d 5
.
(3.59)
We use now (3.49). The worst situation happens when P+ b˜k 2 is equal to the right side of the previous equation. Using Lemma A.1 (b) we finally get the lower bound of Lemma 3.3. Now we show that we can use Lemma 3.3 for all such that Lemma 3.2 does not apply. Let ax = (2d − ε)qx,i 1id and Qx = (1 + δij )qx,ij 1i,j d . More generally, we let a denote a vector with entries (2d − ε)qx,i , and Q a matrix with entries 2qx,ii in the diagonal and qx,ij off the diagonal, that correspond to a possible configuration around x. With c = (cos ki )1id , we introduce F (c; a, Q) = (a, c) + 21 Tr Q − (c, Qc).
(3.60)
This function appears when establishing a lower bound for (h − ε)bk 2 . Let Q be the set of all matrices Q (for which there exists some compatible configuration); we introduce Q = {Q ∈ Q : Qii ≡ 2 and Qij + Qj i = 4 for all i = j }
(3.61)
Q = {Q ∈ / Q : Qij ≡ 0}.
(3.62)
and The reason behind the definition of Q is that we can provide a lower bound only if F (c; a, Q) is not uniformly zero when k moves along the Fermi surface (i.e. with εk = εF ); and we can show that F (c; a, Q) is not uniformly zero only for Q ∈ Q , see Lemma 3.5 below. For given εF , we define µ(εF ) = min
min
max |F (c; a, Q)|.
a,Q∈Q ε∈[0,2d] c:εk =εF
(3.63)
We state a lower bound involving µ(εF ), and check below in Lemma 3.5 that µ(εF ) is strictly positive for εF > 0. Lemma 3.4. Let d 2. For all finite satisfying #{x ∈ ∂ : qx,ij ≡ 0} we have max (h − ε)bk 2
k:εk =εF
1 |∂|, 32d 4 µ(εF )
2 6 d 5 5d
2
|∂|.
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1 The factor 32d 4 is arbitrary here, and has been taken such in order to complement the condition of Lemma 3.2.
Proof. Let us introduce qk (x) = χ ∂ (x)
e−ike .
(3.64)
e:x+e∈ /
By the definition of the discrete Laplacian,
(h − ε)bk (x) = e−ikx (2d − ε)qk (x) − e−ike qk (x + e) .
(3.65)
e
Let us denote by rk (x) the quantity inside the brackets above. Clearly, (h − ε)bk 2 = rk 2 . Let Ra , a = 1, . . . , 2d , represent all combinations of inversions of some coordinates. We have the following inequality: 2d 2d 2 1 1 2 r r . R k R k a a 2d 2d a=1
a=1
Indeed, starting from the RHS, we have in essence (with 0 ai 1 and ai v i , ai v i = ai aj (v i , v j ) i
i
(3.66)
i
ai = 1)
i,j
2 √ √ ai ai v i
(3.67)
i
1/2 1/2 2 ai ai v i 2 i
i
which is the LHS of (3.66). The RHS of (3.66) is clearly smaller than maxk:εk =εF (h − ε)bk 2 . One computes d now 2a=1 rRa k (x) for x ∈ ∂. First, d
d 2 1 (2d − ε)q (x) = (2d − ε) qx,i cos ki . R k a 2d a=1
(3.68)
i=1
Second, d
2 1 − d 2
e−iRa k e
=−
qx,ii cos(2ki ) −
i=1
= −2
e−iRa k e
e :x+e+e ∈ /
a=1 e:x+e∈∂ d
d i=1
1 qx,ij cos(ki + kj ) + cos(ki − kj ) 2 i,j :i =j
qx,ii cos2 ki +
d i=1
qx,ii −
i,j :i =j
qx,ij cos ki cos kj .
(3.69)
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263
We used cos(2ki ) = 2 cos2 ki −1, and the bracket in the second line is [·] = 2 cos ki cos kj . Gathering (3.68) and (3.69), we obtain d
2 1 rRa k (x) = F (c; ax , Qx ). 2d
(3.70)
a=1
/ Q and differs from 0, there exists a neighbor y One can check that whenever Qx ∈ that belongs to Q . Then the condition of the lemma implies that #{x ∈ ∂ : Qx ∈ Q }
1 26 d 5
|∂|.
(3.71)
Furthermore, Q has less than 5d elements since 0 Qij 4; then for any that satisfies the assumption of the lemma there exists Q ∈ Q such that 2
#{x ∈ ∂ : Qx = Q}
1 26 d 5 5d
2
|∂|.
(3.72)
We get a lower bound for (h − ε)bk 2 by considering only those sites, i.e.
max (h − ε)bk 2
k:εk =εF
uniformly in ε ∈ [0, 2d].
x∈∂:Qx =Q
max |F (c; ax , Qx )|
c:εk =εF
µ(εF ) 26 d 5 5d
2
|∂|
(3.73)
There remains to be checked that µ(εF ) differs from 0. Lemma 3.5. For all εF > 0, we have µ(εF ) = 0. Proof. We proceed ab absurdo and explore ways where F (c; a, Q) could be uniformly zero. Let u be the vector such that ui ≡ 1. The constraint εk = εF takes a simple form, namely (u, c) = d − 21 εF . Furthermore, c satisfies |c|∞ 1; if εF = 0, we can find δc such that |c + δc|∞ 1 and (u, c + δc) = d − 21 εF – in which case δc must be perpendicular to u. The condition F (c + δc; a, Q) = F (c; a, Q) for all δc ⊥ u implies that a − 2Qc u. This should also be true when c is replaced with c + δc, hence Qδc u for all δc ⊥ u. Now take (δc)8 = δi8 − δj 8 . We have (Qδc)i = Qii − Qij , (Qδc)j = −Qjj + Qj i ,
(3.74)
and these two components must be equal, since Qδc is parallel to u. Hence Qii + Qjj = Qij + Qj i , or 2qx,ii + 2qx,jj = qx,ij + qx,j i .
(3.75)
In this case F (c; a, Q) takes the form F (c; a, Q) = (2d − ε)
d i=1
qx,i ci − (2d − εF )
d i=1
qx,ii ci +
d i=1
qx,ii .
(3.76)
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J.K. Freericks, E.H. Lieb, D. Ueltschi
Since Q ∈ Q we have qx,ij ≡ 0; if (u, c) = 0, one can take c = 0, and F (c; a, Q) is strictly positive, so we can suppose (u, c) = 0. Let s = i qx,ii /(u, c), and v be the vector with components vi = (2d − ε)qx,i − (2d − εF )qx,ii + s.
(3.77)
Then F (c; a, Q) = (v, c). If we require this to be zero for c u, then we need v ⊥ u. But we also require (v, c + δc) = (v, c) for all δc ⊥ u, hence v u. So v must be zero, i.e. (2d − ε)qx,i − (2d − εF )qx,ii + s = 0
(3.78)
for all 1 i d. We also have qx,i + qx,ii 2, and qx,ii cannot be always equal to 2. If s = 0, one checks that necessarily qx,ii ≡ 1, which is impossible because Q ∈ Q . Hence F (c; a, Q) cannot be uniformly zero when moving along the Fermi surface. 4. Finite U We consider now the Falicov–Kimball model with finite repulsion U , and establish a lower bound for the ground state energy of N electrons in a configuration specified by . More precisely, we show that when decreasing the repulsion U , one does not lower the energy more than const · |∂|/U . For any , the spectrum of hU , is included in [0, 4d] ∪ [U, U + 4d]. When U > 4d, eigenstates with energy in [0, 4d] show exponential decay outside of ; and eigenstates with energy in [U, U + 4d] show exponential decay inside . Hence and \ are essentially decorrelated, and the situation is close to that with U = ∞. The following proposition compares the energies of electrons with finite and infinite U . It is useful to introduce η(U ), η(U ) =
2d 2 (U − 2d)2 j (U − 2d)2d = − 1. U − 2d U (U − 4d) (U (U − 4d))d d
(4.1)
j =1
Notice that limU →∞ U 2 η(U ) = 4d 3 , as it easily comes out from the middle expression. Proposition 4.1. If U > 4d, we have U E,N − γ (U )|∂|, E,N
with γ (U ) =
8d 2 + d 2d+2 η(U ). U − 2d
Proof. First, we remark that eigenvectors of hU , with eigenvalue smaller than 4d have exponential decay outside of . Indeed, for x ∈ / the Schrödinger equation can be written ϕj (x + e) ϕj (x) = e . (4.2) U + 2d − ej
Segregation in the Falicov–Kimball Model
265
If ej 4d, we have ||
||
j =1 2d
|ϕj (x)| 2
e
|ϕj (x + e)|2
(U − 2d)2
j =1
.
(4.3)
Using this inequality, we can proceed by induction on the distance between x and . The induction hypothesis is that the following holds true: ||
|ϕj (x)|2
j =1
2d 2n U − 2d
(4.4)
for any x such that dist (x, ) n. As a result, we have ||
2d 2 dist (x,) . U − 2d
(4.5)
χ (x) ϕj∗ (x) ϕj (y) χ (y).
(4.6)
|ϕj (x)|2
j =1
Let us introduce ρ˜xy =
N j =1
U We show that E,N is bounded below by Tr ρh ˜ , up to a contribution no greater than const |∂|/U . Recall that h is the Hamiltonian with infinite repulsions. If P is the projector onto the domain , let ϕ˜j = P ϕj , U E,N =
N
|ϕj (x) − ϕj (y)|2 + U
|ϕj (x) − ϕj (y)|2 −
|ϕj (x)|2
x∈,y ∈ / |x−y|=1
{x,y} ⊂ |x−y|=1
Tr ρh ˜ −2
|ϕ˜j (x) − ϕ˜j (y)|2
j =1 {x,y}:|x−y|=1
+
|ϕj (x)|2
x ∈ /
j =1 {x,y}:|x−y|=1 N
N
|ϕj (x)| |ϕj (y)|.
(4.7)
/ j =1 x∈,y ∈ |x−y|=1
By the Schwarz inequality, the last term is smaller than 2
N
x∈,y ∈ / j =1 |x−y|=1
|ϕj (x)|2
1/2
N
x∈,y ∈ / j =1 |x−y|=1
|ϕj (y)|2
1/2
8d 2 |∂|. U − 2d
(4.8)
We used (4.5) with dist (x, ) being respectively 0 and 1, in order to control the quantities in both brackets.
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Recall that ej denotes the j th eigenvalue of the Hamiltonian h ; that is, with infinite repulsions. Let us introduce the projector Pj onto the corresponding eigenstate. Then Tr ρh ˜ =
||
ej Tr ρP ˜ j ≡
j =1
where the nj satisfy 0 nj 1, and obtain the lower bound
||
ej n j ,
(4.9)
j =1
j
nj = Tr ρ. ˜ By the bathtub principle [LL], we
Tr ρh ˜
Tr ρ˜
ej .
(4.10)
j =1
There remains to show that Tr ρ˜ is close to N . We have N − Tr ρ˜ =
N
|ϕj (x)|2
x ∈ / j =1 ∞
|∂|
#{x : dist (x, ) = n}
n=1
2d
n=1
We bounded #{·} 2d
∞
n+d−1 d−1
2d 2n = 2d η(U )|∂|. U − 2d
n+d−1 d−1
2d 2n U − 2d
|∂|. Since ej 4d, we obtain the proposition.
(4.11)
5. Upper Bound We establish now an upper bound for the sum of the first N eigenvalues in a finite domain U , for the case of infinite repulsion. The bound carries over to finite U , since E,N is increasing in U . The strategy is to average h over a huge box. The “strength” of the averaged Hamiltonian depends on the number of bonds in , which is roughly 2d|| − |∂|. The averaged Hamiltonian is, up to a factor, the hopping matrix in the huge box, and its ground state energy is easy to compute in the thermodynamic limit. This can be compared to E,N by concavity of the sum of lowest eigenvalues of self-adjoint operators. The result is Proposition 5.1. The sum of the first N eigenvalues of the Laplace operator in a domain with Dirichlet boundary conditions, satisfies the upper bound E,N ||e(n) + |∂|(2dn − e(n)). Proof. Let L be a multiple of ||, and NL be such that NL /Ld = N/||. We consider a box {1, . . . , L}d . We introduce ε˜ = 21 (eN + eN+1 ). Let Ra , a = 1, . . . , Ld d!, represent a translation possibly followed by an axis permutation. We define the averaged Hamiltonian h¯ L,
Ld d! 1 = d hRa − ε˜ 1lRa . L d! a=1
(5.1)
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267
Then d
L d! 1 SNL hRa − ε˜ 1lRa SNL (h¯ L, ) d L d! a=1
= SN (h − ε˜ 1l ).
(5.2)
Indeed, all summands in the above equation are equal, and the Hamiltonian h − ε˜ 1l has no more than N negative eigenvalues, and at least Ld − || zero eigenvalues. The RHS is equal to E,N − N ε˜ . Let Ki be the number of sites in that have i neighbors in . We have || = 2d i=0 Ki 1 2d and |∂| = 2d−1 K ; and the number of bonds in is iK . Then the averaged i i i=0 i=0 2 Hamiltonian is
h¯ L,
xy
=−
t || δ|x−y|=1 + (2d − ε˜ ) d δxy , Ld L
(5.3)
with t=
2d 1 iKi . 2d
(5.4)
i=0
Let K =
2d
i=0
2d−i 2d Ki ;
then t = || − K and K |∂|. One easily checks that
|| K || h¯ L, = d h{1,...,L}d + d 2d1l{1,...,L}d − h{1,...,L}d − ε˜ d 1l{1,...,L}d . L L L
(5.5)
Notice that all operators commute. In (5.2), the terms involving ε˜ cancel, since = N . Now, as L → ∞, SNL (1l{1,...,L}d ) = NL , and NL || Ld 1 SN (h d ) → e(n). Ld L {1,...,L}
(5.6)
||e(n) + K(2dn − e(n)) E,N .
(5.7)
Therefore (5.2) implies
6. Positive Electronic Temperature This section considers the electronic free energy at positive temperature, for a fixed configuration of classical particles. We will see that the inequalities satisfied by the sums over lowest eigenvalues have an extension to free energies.
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J.K. Freericks, E.H. Lieb, D. Ueltschi
6.1. Lower bound for U = ∞. We start with U → ∞. Let F (β, µ) = − β1 log Tr e−βH , where the trace is taken in the Fock space of antisymmetric wave functions U =∞ is the second quantized form of the one-particle Hamiltonian on , and H ≡ H, h defined by (2.1). Proposition 6.1. For all finite , one has the lower bound F (β, µ) − ||f (β, µ) α(β, ¯ µ)|∂|, ¯ µ) > 0 if 0 < µ < 4d. where α(β, ¯ µ) > 0 satisfies limβ→∞ α(β, Proof. The fermionic free energy F (β, µ) can be expressed in terms of the eigenvalues of h , F (β, µ) = −
|| 1 log 1 + e−β(ej −µ) . β
(6.1)
j =1
In order to compare this with the corresponding infinite-volume expression (1.9), we partition the Brillouin zone [−π, π ]d according to the level sets of the function εk ; more precisely, we define measures µj , 1 j ||, by || −1 j χ εF ( j|| ) < ε(k) < εF ( || ) dk. (6.2) d (2π)
1 || dk ∗ Notice that dµj (k) = 1 and || j =1 dµj (k) = (2π)d . Next we introduce ej , that are equal to εk averaged over µj : (6.3) ej∗ = dµj (k) εk . dµj (k) =
The ground state energy (1.3) of a density N/|| of electrons in Zd can then be written as N
e(N/||) =
1 ∗ ej . ||
(6.4)
j =1
From the lower bound without a boundary term, we have N j =1
ej >
N j =1
ej∗ ,
(6.5)
for all N < ||, and equality when N = ||. Actually, inequality (6.5) can be strengthened by introducing a term depending on the boundary of . In Theorem 1.1, α(n) can be taken to be increasing in n for n 21 . Also, α(1 − n) = α(n). Therefore there exists a function a(ε), with a(ε) > 0 for 0 < ε < 2d, a(4d − ε) = −a(ε), and 1 α(n) = dk a(εk ). (6.6) (2π )d εk <εF
Segregation in the Falicov–Kimball Model
Next we define ej =
269
dµj (k) εk +
|∂| || a(εk ) ;
(6.7)
then the following is stronger than (6.5) and holds true, N
ej
j =1
N j =1
ej .
(6.8)
With a(ε) chosen appropriately both sequences (ej ) and (ej ) are increasing, and the inequality above is an equality when N = ||. The sequence (ej ) is said to ‘majorize’ (ej ). We can apply an inequality due to Hardy, Littlewood and Pólya (and independently found by Karamata); see [Mit] p. 164. For any concave function g, we have ||
g(ej )
j =1
|| j =1
g(ej ).
(6.9)
(Conversely, if (6.9) holds for all concave g, then (ej ) majorizes (ej ).) We use this inequality with g(e) = −
1 log(1 + e−β(e−µ) ), β
(6.10)
which is concave. We get F (β, µ)
|| j =1
g(ej )
|| (2π )d
[−π,π]d
dk g εk +
|∂| || a(εk ) ,
where the last step is Jensen’s inequality. Then 1 1 dk g εk + |∂| a(εk ) − g(εk ) . F (β, µ) − f (β, µ) || d || (2π ) In the limit β → ∞, we have e − µ if e < µ g(e) = 0 if e µ.
(6.11)
(6.12)
(6.13)
As a result, for all 0 < µ < 4d we get a lower bound for large β that is uniform in the limit β → ∞. One also gets a lower bound by using concavity of g, that holds for all temperatures, but that is not uniform in β: εk + |∂| a(εk ) || 1 1 dk de g (e) F (β, µ) − f (β, µ) d (2π ) || εk
1 |∂| 2 = dk a(ε )g (ε ) − O(a(ε ) ) k k k || (2π )d |∂| 1 = dka(εk ) || (2π )d εk <2d
× g (εk ) − g (4d − εk ) − O(a(εk )) . (6.14)
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J.K. Freericks, E.H. Lieb, D. Ueltschi
The integrand in the last line is strictly positive if a(εk ) is small enough, and chosen to vanish appropriately as εk → 2d. 6.2. Lower bound with finite U . We extend now the results of the previous section to the case of finite repulsion U . As we noted in Sect. 4, when U > 4d all eigenstates have exponential decay, either in or in \ . We show that the total free energy in is equal to a term involving \ only, plus a term involving only, up to a correction of order |∂|/U . Proposition 6.2. U F, (β, µ) F (β, µ) + F\ (β, µ − U ) − γ¯ (U )|∂|
with
(4d)2 , U − 2d and η(U ) is defined in (4.1). Notice that limU →∞ U γ¯ (U ) = 16d 2 + 8d 4 2d . γ¯ (U ) = (2dU + 4d + 8d 2 )2d η(U ) +
Proof. Let us introduce ϕj (x) if 1 j || and x ∈ , or if || < j || and x ∈ / ϕ˜j (x) = 0 otherwise. (6.15) We assume N > || (otherwise, replace || by N in the next expressions, and ignore the sums whose initial number is greater than the final one). Then N j =1
ejU =
|| j =1
|ϕ˜j (x) − ϕ˜j (y)|2 +
{x,y} |x−y|=1
|ϕj (x) − ϕj (y)|2
{x,y} ⊂ |x−y|=1
−
|ϕj (x)|2 + U
x∈,y ∈ / |x−y|=1
+
N j =||+1
|ϕ˜j (x) − ϕ˜j (y)|2 +
{x,y} |x−y|=1
x ∈ /
|ϕj (x) − ϕj (y)|2
{x,y} ⊂c |x−y|=1
−
|ϕj (x)|2
|ϕj (x)|2 + U
x ∈,y∈ / |x−y|=1
(6.16)
|ϕj (x)|2 .
x ∈ /
We proceed as in Sect. 4 and define ρ˜xy =
|| j =1
ρ˜xy =
χ (x)ϕj∗ (x)ϕj (y)χ (y),
N j =||+1
χ c (x)ϕj∗ (x)ϕj (y)χ c (y).
(6.17)
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271
Then N j =1
ejU Tr ρh ˜ + Tr ρ˜ [h\ + U ] − 2
N
|ϕj (x)| |ϕj (y)|.
(6.18)
|ϕj (x)| |ϕj (y)|.
(6.19)
/ j =1 x∈,y ∈ |x−y|=1
The inequality (4.10) is still valid, for both ρ˜ and ρ˜ . Hence N j =1
ejU
Tr ρ˜
ej +
j =1
||+1+Tr ρ˜
(e¯j + U ) − 2
N
/ j =1 x∈,y ∈ |x−y|=1
j =||+1
Here, e¯j , || < j || are the eigenvalues of the operator h\ . We define 2 if 1 j || x∈,y ∈ / |ϕj (x)| |ϕj (y)| 4d x ∈ / |ϕj (x)| + 2 |x−y|=1 δj = 2 / |ϕj (x)| |ϕj (y)| if || < j ||. (4d + U ) x∈ |ϕj (x)| + 2 x∈,y ∈ |x−y|=1
(6.20) Then (6.19) takes the simpler form N j =1
ejU
|| j =1
N
(ej − δj ) +
(e¯j + U − δj ).
(6.21)
j =||+1
The sequence in the RHS is not necessarily increasing, but one gets a lower bound by rearranging the terms. Hence one can apply Hardy, Littlewood, Pólya inequality. Indeed, it also works when the total sum over elements of the sequences are not equal, provided the concave function is increasing – which is the case with g(e). One obtains || j =1
g(ejU )
||
g(ej − δj ) +
j =1
||
g(e¯j + U − δj ).
(6.22)
j =||+1
We use now g(e − δ) g(e) − δ, and we find U F, (β, µ)
F (β, µ) + F\ (β, µ − U ) −
||
δj .
(6.23)
j =1
The remaining effort consists in estimating the sum of δj , using exponential decay of eigenfunctions ϕj either in or in \ . Retracing (4.11) and (4.8), we get || j =1 || j =||+1
δj 4d 2d η(U )|∂| +
8d 2 |∂|, U − 2d
δj (U + 4d) 2d η(U )|∂( \ )| +
8d 2 |∂|. U − 2d
(6.24)
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J.K. Freericks, E.H. Lieb, D. Ueltschi
Notice that the last term can be written with |∂| instead of |∂( \ )|, as can be seen from (4.8). We use ∂( \ )| 2d|∂|, and we finally obtain ||
δj (2dU + 4d + 8d 2 ) 2d η(U )|∂| +
j =1
(4d)2 |∂|. U − 2d
(6.25)
6.3. Upper bound. We turn to the upper bound for the electronic free energy. We first notice that the free energy is raised when one decorrelates the domain occupied by the classical particles, from the empty domain. The following proposition applies to all finite subsets of Zd , and it also applies when is a finite d-dimensional torus. Proposition 6.3. We have the upper bounds U (β, µ) F (β, µ) + F • F, \ (β, µ − U ). 4π √d d−1 || d + 2d|∂| . • F (β, µ) ||f (β, µ) + 1+e1−βµ |S |1/d d
d−1
Notice that the isoperimetric inequality implies that for all finite ⊂ Zd , || d |∂|. This does not hold, however, when is e.g. a box with periodic boundary conditions. Proof. The Peierls inequality allows us to write −β(ψ ,[H U −µN ]ψ ) U j j , Tr e−β(H, −µN ) e ,
(6.26)
j
for any set of orthonormal functions {ψj } (in the Fock space of antisymmetric wave functions on ). We can choose the ψj to be eigenfunctions of H and H\ — decorrelating and \ . In \ , the free electrons experience a uniform potential U ; the energy levels are given by the spectrum of h\ plus U . This only shifts the chemical potential, so that we obtain the first claim of the proposition. Now we estimate F (β, µ). Let us introduce |∂| ∗ |∂| e˜j = (1 − || )ej + 2d || = dµj (k) εk + (2d − εk ) |∂| (6.27) || ; || || then e˜j e˜j +1 , j =1 e˜j = j =1 ej∗ , and the upper bound for the ground state energy can be cast in the form N j =1
ej
N
e˜j .
(6.28)
j =1
This allows us to summon again the Hardy, Littlewood, Pólya inequality, and we get || F (β, µ) . g dµj (k) εk + (2d − εk ) |∂| || j =1
(6.29)
Segregation in the Falicov–Kimball Model
273
The derivative of g(e) satisfies 0 < g (e)
1 1 + e−βµ
(6.30)
(recall that e 0). Since the measure µj is concentrated on those k where εk lies −1 j between εF ( j|| ) and εF ( || ), we can bound (6.29) by F (β, µ)
|| (2π)d
1 j j−1 ε ( )−ε ( ) + 2d|∂| . F F || || 1 + e−βµ ||
[−π,π ]d
dk g(εk ) +
j =1
(6.31) j −1 We need a bound for εF ( || ) − εF ( j|| ); since ∇εk = 2(sin k1 , . . . , sin kd ), we √ −1 have ∇εk 2 d. Let us take k such that εk = εF ( j|| ), and δk k such that
j εk+δk = εF ( || ). Then
√ j −1 ) − εF ( j|| ) 2 d δk. εF ( ||
(6.32)
If δkmin is chosen so as to minimize the norm of such δk, we have 1 1 1 dk δmin d |Sd |. = d j −1 j (2π) εF ( || )<εk <εF ( || ) (2π )d ||
(6.33)
Combining this inequality with (6.32), we get j −1 εF ( || ) − εF ( j|| )
√ 4π d ||−1/d . |Sd |1/d
This leads to the upper bound of Proposition 6.3.
(6.34)
6.4. Proofs of the corollaries. Proof of Corollary 1.3. Let e− = −d + h and e+ = −d − h be the energies per site of the all − and all + Ising configurations. A configuration can be specified by the set of − spins. Let B() be the set of bonds connecting and \ . Notice that 1 2d |B()| |∂| |B()|. The partition function of the Ising model can be written as − + ZI, = e−β[||e +|\|e ] e−2β|B()| . (6.35) ⊂ U (β, µ) implies that the partition function of the Falicov– Now the upper bound for F, Kimball model is bounded below by
Z
e−β[||f (β,µ)+|\|f (β,µ−U )] e−βCd,µ |∂| e−βCd,µ ||
d−1 d
.
(6.36)
⊂
The last factor vanishes in the thermodynamic limit. One then makes the connection with Ising by multiplying Z by
exp β|| d2 Cd,µ + 21 f (β, µ) + f (β, µ − U ) ,
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J.K. Freericks, E.H. Lieb, D. Ueltschi
and by choosing the temperature to be 21 Cd,µ β, and the magnetic field to be h=
1 [f (β, µ) − f (β, µ − U )] Cd,µ
(6.37)
(the magnetic field is negative). The other bound is similar, simply replace Cd,µ by α/2d. ¯ Proof of Corollary 1.4. Because was assigned periodic boundary conditions, we have 1 δwx+z ,wy+z . (6.38) δwx ,wy = || z∈
It is not hard to check that for any configuration w specified by ⊂ , one has δwx+z ,wy+z || − |∂| |x − y|1 . (6.39) z∈
Then δwx ,wy 1 − |x − y|1
|∂| ||
.
(6.40)
We need a bound for the last term. The fact is that typical configurations of classical 2 log 2 particles cannot have too much boundary: |∂| . Indeed, || is smaller than r = β α(β,µ) ¯
⊂
U χ |∂| > r|| e−βF, (β,µ) U (β,µ) −βF, ⊂ e
2|| e−β||f (β,µ)−β(||−||)f (β,µ−U ) e−(2 log 2)||
e−β||f (β,µ)−β(||−||)f (β,µ−U ) e−βCd,µ (nc ||)
2−|| eβ(Cd,µ +Cd,µ )||
d−1 d
d−1 d
e−βCd,µ ||
d−1 d
(6.41)
.
Therefore δwx ,wy 1 −
d−1 2 log 2 d |x − y|1 − 2−|| eβ(Cd,µ +Cd,µ )|| . β α(β, ¯ µ)
(6.42)
The last term vanishes in the limit ( Zd , and the term involving |x − y|1 vanishes when β → ∞.
7. Conclusion Our analysis of the Falicov–Kimball model away from half-filling allows some extrapolations. We expect segregation to survive at small temperature, when both the classical particles and the electrons are described by the grand-canonical ensemble, at inverse temperature β and with chemical potentials µc and µe . Segregation is a manifestation of coexistence between a phase with many classical particles and few electrons, and a phase with many electrons and few classical particles. It is therefore natural to conjecture the following, for d 2:
Segregation in the Falicov–Kimball Model
275
A first order phase transition occurs at low temperature, when varying the chemical potentials. The transition from the chessboard state at half-filling for the itinerant and heavy electrons (and large U ) to the segregated state is still not clear. A heuristic analysis suggests that these states could coexist, hence there could be another first-order phase transition. Alternate possibilities include mixtures between other periodic phases and the empty or full lattice before the segregation sets in. One interest of the Falicov–Kimball model is its possible relevance in understanding the Hubbard model, a notoriously difficult task. See e.g. [Lieb2] and [Tas2] for reviews of rigorous results on the Hubbard model. The relationship between the Falicov–Kimball model and the Hubbard model is like the one between the Ising and Heisenberg models for magnetism. The former does not possess the continuous symmetry of the latter, and therefore the approximation is a crude one. Still, the two models share many similarities; for instance, the Falicov–Kimball model displays long-range order of the chessboard type at half-filling and at low temperature [KL], and the ground state of the Hubbard model is a spin singlet [Lieb]. Ferromagnetism in the Hubbard model depends on the dimension, on the filling, and on the geometry: it has been shown to occur on special lattices such as “line-graphs” [MT,Tas, Mie,Tas2]. Does ferromagnetism take place in the Hubbard model on Z3 , for large repulsions and away from half-filling? Returning to Falicov–Kimball, let us walk on the road that leads to Hubbard. We consider the asymmetric Hubbard model that describes spin 21 electrons with hoppings depending on the spins (this interpretation is more convenient than physical). Its Hamiltonian is † † Ht = − cx↑ cy↑ − t cx↓ cy↓ + U nx↑ nx↓ . (7.1) x,y:|x−y|=1
x,y:|x−y|=1
x
Notice that H0 is the Falicov–Kimball model, while H1 is the usual Hubbard model. Although we did not prove it, it is rather clear that segregation still takes place for very small t. Furthermore, the density of the phase with classical particles, in the ground state, should still be exactly 1 – indeed, the electrons exert a sort of “pressure” that packs the classical particles together, and the tendency of the latter to delocalize is not strong enough to overcome this pressure. This is summarized in the following conjecture: For t t0 , segregation occurs in the ground state, at large U and away from half-filling, in the form of a coexistence between a phase of classical particles with density 1, and a phase of electrons with smaller density. This should also hold at positive temperature, although the density of the phase of the classical particles will be reduced, due to the presence of some holes. If we increase t, assuming that segregation remains, we should reach a critical value tc < 1 where the region of classical particles starts to grow. The density of the phase of particles with smaller hoppings is now strictly less than 1. A major question is whether segregation survives all the way to the point where t reaches 1 – this would imply the existence of a ferromagnetic phase in the Hubbard model. We note, however, that while it is conceivable that there is a segregated (i.e., ferromagnetic) ground state at t = 1, it cannot be true that every ground state (for equal number of up and down spins) is segregated. This follows from the SU(2) symmetry. If @ is a saturated ferromagnetic ground state with 2N up electrons, we can construct A = (S− )N @, which is also
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a ground state, with N up and N down electrons. However, A has the up and down electrons inextricably mixed, which is the opposite of a segregated state. Indeed, the SU(2) symmetry is restored precisely at t=1, and the ground states have at least the degeneracy due to this symmetry. The Hubbard model is a rich and complicated model that poses difficult challenges. The Falicov–Kimball model can be of some help, for instance in checking scenarios that should apply to both models. This discussion of ferromagnetism illustrates however that the links between them are subtle. A. Appendix We derive in the sequel various expressions that are too intricate to appear in the main body of this paper. Lemma A.1.
1 (a) (2π) dk|εk − εF |−1/4 < 2. d (b) Assume that α 2
√ 16 2πd 1/d n ; |Sd |1/d
then for all k such that εk = εF ,
α 2 d dk χ |k − k| < α |Sd | . 8π d εk <εF
2
2
9 d 1+ d (c) e(n) 12( 10 ) n /|Sd | d .
5/2 . k (d) ∇ b |∂| 8d 2
−ε)bk (e) ∇ (h|∂| 29 d 11/2 . 2
,h bk ) 32d 7/2 . (f) ∇ (bk|∂|
(g) Assume that bk 2 /|∂| η. Then if η 1, ∇ (bkb,h2bk ) η−2 28 d 11/2 . d
k
Proof of Lemma A.1 (a). Setting Y = 2d − εF − 2 i=2 cos ki , and making the change of variables ξ = cos k1 , one gets dk|εk − εF |−1/4 π 1 =2 dk2 · · · dkd dk1 d−1 |Y − 2 cos k1 |1/4 [−π,π] 0 1 1 1 =2 dk2 · · · dkd dξ 1 − ξ 2 |Y − 2ξ |1/4 [−π,π]d−1 −1 1 2/3 1 1/3 1 −3/4 2 dk2 · · · dkd 2 dξ dξ |Y − 2ξ | (1 − ξ 2 )3/4 [−π,π]d−1 0 −1 1 1/3 1 2/3 1 −3/4 −3/4 2 2 dk2 · · · dkd 2 dζ √ dξ |ξ | . ζ (1 − ζ )3/4 [−π,π]d−1 0 −1 The integral over ζ can be split into one running from 0 to 21 , and one running from 21 1 to 1. For the first part we bound √ζ (1−ζ 23/4 √1ζ , while the bound for the second )3/4
Segregation in the Falicov–Kimball Model
part can be chosen to be
√
277
2 (1−ζ1 )3/4 . Everything can now be computed explicitly, and
we find 231/12 32/3 (2π)d−1 < 2(2π )d .
Proof of Lemma A.1 (b). Let us introduce a map γ (ξ ) such that 1 − 21 γ 2 (ξ ) = cos ξ ; precisely, √ 2(1 − cos ξ ) if ξ ∈ [0, π ] γ (ξ ) = √ − 2(1 − cos ξ ) if ξ ∈ [−π, 0]. The condition εk < εF becomes
d
i=1 |γ (ki )|
2
(A.1)
< εF . The derivative of γ is
| sin ξ | dγ =√ . dξ 2(1 − cos ξ )
(A.2)
We check now that |γ (ξ ) − γ (ξ )| > |ξ − ξ |2 /4π . Let us assume that γ (ξ ) > γ (ξ ). Then ξ | sin λ| 1 ξ γ (ξ ) − γ (ξ ) = dλ √ dλ| sin λ| (A.3) 2 ξ 2(1 − cos λ) ξ 1 ξ dλ|λ| |ξ − ξ |2 /4π. (A.4) π ξ
Then we can write
εk <εF
dk χ |k − k| < α
εk <εF
dk χ |ki − ki | <
dγ1 . . . dγd χ
d i=1
√α d
∀i
|γi |2 < εF χ |γi − γi | <
α2 4πd
.
One gets a lower bound by replacing the last characteristic function by the condition d α2 32 2 2/d ; the assumption of the lemma i=1 |γi − γi | < 4πd . Recall that εF |Sd |2/d n √ α2 implies that εF > 4πd ; as a consequence, a lower bound is the volume of the sphere of radius
α2 8πd .
Proof of Lemma A.1 (c). By (3.16), d 8d|Sd | 8 +1 εF2 . dk|k|2 = 2 e(n) 2 d d π (2π) |k|2 <εF π (2π ) (d + 2)
(A.5)
The lower bound then follows from εF
32 n2/d . |Sd |2/d
(A.6)
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J.K. Freericks, E.H. Lieb, D. Ueltschi
Proof of Lemma A.1 (d)–(g). Since bk =
eik(e−e ) ,
(A.7)
/ x∈∂ e:x+e∈ / e :x+e ∈
we have
b 1 k = i(ej − ej )eik(e−e ) . ∇ |∂| j |∂|
(A.8)
x∈∂ e,e
This is less than 2(2d)2 , and we obtain the bound (d). We consider now (h − ε)bk 2 , e−ike − eikx [(h − ε)bk ](x) = (2d − ε) e :x+e ∈ /
=
e−ik(e+e )
e:x+e∈∂ / e :x+e+e ∈
e−ik(e+e ) (2d − ε) χ |e| = 0 − χ |e| = 1 .
e:x+e∈∂,|e|=0,1 / e :x+e+e ∈
(A.9) In the last line, e is allowed to be 0. Let ξ(e) = (2d − ε) χ |e| = 0 − χ |e| = 1 . Then (h − ε)bk (x) 2 = eik(e+e −e −e ) ξ(e)ξ(e ).
e:x+e∈∂,|e|=0,1 e :x+e ∈∂,|e |=0,1 / e :x+e+e ∈ / e :x+e +e ∈
(A.10) component of the gradient; it involves a term ej +ej −ej −ej there are sums over e , e , with less than (2d)2 terms; the sum
One computes now the j th
that is smaller than 4; e |ξ(e)| is bounded by 4d; finally, the number of sites where (h − ε)bk differs from 0 is bounded by 2d|∂|. As a result, the j th component of the gradient is bounded by 1 5 2 (4d) , and we obtain (e). We estimate now the gradient of (bk , h bk ). One easily checks that eik(e−e −e ) . (A.11) (bk , h bk ) = bk 2 − / e :x+e ∈∂ e :x+e +e ∈ x∈∂ e:x+e∈ /
We can use the bound (d) for the gradient of bk 2 . The gradient of the last term is less than 3(2d)3 |∂|, so we can write (b , h b ) k k (A.12) 8d 5/2 + 24d 7/2 32d 7/2 . ∇ |∂| Finally, one easily checks that (b , h b ) 2 |∂| 2 (b , h b ) 2 k k k k 2 ∇ ∇ bk 2 bk 2 |∂| (b , h b ) 2 |∂| 4 b 2 2 k k k +2 ∇ . |∂| bk 2 |∂| Using (d) and (f), as well as (bk , h bk )/|∂| 2(2d)3 , one gets (g).
(A.13)
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Acknowledgements. It is a pleasure to thank Michael Loss for several valuable discussions.
References [DFF]
Datta, N., Fernández, R. and Fröhlich, J.: Effective Hamiltonians and phase diagrams for tightbinding models. J. Stat. Phys. 96, 545–611 (1999) [DMN] Datta, N., Messager, A. and Nachtergaele, B.: Rigidity of interfaces in the Falicov–Kimball model. J. Stat. Phys. 99, 461–555 (2000) [FK] Falicov, L.M. and Kimball, J.C.: Simple model for semiconductor-metal transitions: SmB6 and transition-metal oxides. Phys. Rev. Lett. 22, 997–999 (1969) [FF] Freericks, J.K. and Falicov, L.M.: Two-state one-dimensional spinless Fermi gas. Phys. Rev. B 41, 2163–2172 (1990) [FGM] Freericks, J.K., Gruber, Ch. and Macris, N.: Phase separation and the segregation principle in the infinite-U spinless Falicov–Kimball model. Phys. Rev. B 60, 1617–1626 (1999) [FLU] Freericks, J.K., Lieb, E.H. and Ueltschi, D.: Phase separation due to quantum mechanical correlations. Phys. Rev. Lett. 88, 106401 1–4 (2002) [GJL] Gruber, Ch., J¸edrzejewski, J. and Lemberger, P.: Ground states of the spinless Falicov–Kimball model II. J. Stat. Phys. 66, 913–938 (1992) [GM] Gruber, Ch. and Macris, N.: The Falicov–Kimball model: A review of exact results and extensions. Helv. Phys. Acta 69, 850–907 (1996) [Hal] Haller, K.: Ground state properties of the neutral Falicov–Kimball model. Commun. Math. Phys. 210, 703–731 (2000) [HK] Haller, K. and Kennedy, T.: Periodic ground states in the neutral Falicov–Kimball model in two dimensions. J. Stat. Phys. 102, 15–34 (2001) [Ken] Kennedy, T.: Some rigorous results on the ground states of the Falicov–Kimball model. Rev. Math. Phys. 6, 901–925 (1994) [Ken2] Kennedy, T.: Phase separation in the neutral Falicov–Kimball model. J. Stat. Phys. 91, 829–843 (1998) [KL] Kennedy, T. and Lieb, E.H.: An itinerant electron model with crystalline or magnetic long range order. Physica A 138, 320–358 (1986) [LM] Lebowitz, J.L., and Macris, N.: Long range order in the Falicov–Kimball model: Extension of Kennedy-Lieb theorem. Rev. Math. Phys. 6, 927–946 (1994) [Lem] Lemberger, P.: Segregation in the Falicov–Kimball model. J. Phys. A 25, 715–733 (1992) [LY] Li, P. and Yau, S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983) [Lieb] Lieb, E.H.: Two theorems on the Hubbard model. Phys. Rev. Lett. 62, 1201–1204; Errata 62, 1927 (1989) [Lieb2] Lieb, E.H.: The Hubbard model: Some rigorous results and open problems. XIth International Congress of Mathematical Physics (Paris, 1994), Cambridge, MA: Internat. Press, 1995, 392–412 [LL] Lieb, E.H. and Loss, M.: Analysis. 2nd edition, Providence, RI: Amer. Math. Soc., 2001 [MP] Macris, N. and Piguet, C.-A.: Long-range orders in models of itinerant electrons interacting with heavy quantum fields. J. Stat. Phys. 105, 909–935 (2001) [MM] Messager, A. and Miracle-Solé, S.: Low temperature states in the Falicov–Kimball model. Rev. Math. Phys. 8, 271–299 (1996) [Mie] Mielke, A.: Ferromagnetism in single-band Hubbard models with a partially flat band. Phys. Rev. Lett. 82, 4312–4315 (1999) [MT] Mielke, A. and Tasaki, H.: Ferromagnetism in the Hubbard model. Examples from models with degenerate single-electron ground states. Commun. Math. Phys. 158, 341–371 (1993) [Mit] Mitrinovi´c, D.S.: Analytic Inequalities. Grundlehren der mathematischen Wissenschaften, Berlin– Heidelberg–New York: Springer, 1970 [Tas] Tasaki, H.: Ferromagnetism in Hubbard models. Phys. Rev. Lett. 75, 4678–4681 (1995) [Tas2] Tasaki, H.: From Nagaoka’s ferromagnetism to flat-band ferromagnetism and beyond – An introduction to ferromagnetism in the Hubbard model. Prog. Theor. Phys. 99, 489–548 (1998) Communicated by J. L. Lebowitz
Commun. Math. Phys. 227, 281 – 302 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Super-Diffusivity in a Shear Flow Model from Perpetual Homogenization Gérard Ben Arous1 , Houman Owhadi2 1 DMA, EPFL, 1015 Lausanne, Switzerland. E-mail:
[email protected] 2 William Davidson Faculty (Bloomfield), Technion, 32000 Haifa, Israel.
E-mail:
[email protected] Received: 1 June 2001 / Accepted: 11 January 2002
Abstract: This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dyt = dωt − ∇(yt )dt, y0 = 0 and d = 2. is a 2 × 2 skewsymmetric matrix associated to a shear flow characterized by an infinite number of spatial n (x /R ), where hn are smooth scales 12 = −21 = h(x1 ), with h(x1 ) = ∞ γ h n 1 n n=0 functions of period 1, hn (0) = 0, γn and Rn grow exponentially fast with n. We can show that yt has an anomalous fast behavior (E[|yt |2 ] ∼ t 1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . The Model . . . . . . . . . Main Results . . . . . . . . Proofs Under Hypothesis 1 Proofs Under Hypothesis 2
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281 283 284 288 295
1. Introduction Turbulent incompressible flows are characterized by multiple scales of mixing length and convection rolls. It is heuristically known and expected that a diffusive transport in such media will be super-diffusive. The first known observation of this anomaly is attributed to Richardson [27] who analyzed available experimental data on diffusion in air, varying on about 12 orders of magnitude. On that basis, he empirically conjectured that the diffusion coefficient Dλ in turbulent air depends on the scale length λ of the measurement. The Richardson law, 4
Dλ ∝ λ 3
(1)
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G. Ben Arous, H. Owhadi 1
was related to Kolmogorov–Obukhov turbulence spectrum, v ∝ λ 3 , by Batchelor [4]. The super-diffusive law of the root-mean-square relative displacement λ(t) of advected particles 1
3
λ(t) ∝ (Dλ(t) t) 2 ∝ t 2
(2)
was derived by Obukhov [22] from a dimensional analysis similar to the one that led 1 Kolmogorov [18] to the λ 3 velocity spectrum. More recently physicists and mathematicians have started to investigate the superdiffusive phenomenon (from both heuristic and rigorous points of view) by using the tools of homogenization or renormalization; we refer to M. Avellaneda and A. Majda [3, 2], J. Glimm et al. [12, 13, 15], J. Glimm and Q. Zhang [16], Q. Zhang [28], M.B. Isichenko and J. Kalda [17], G. Gaudron [14]. It is now well known that homogenization over a periodic or ergodic divergence free drift has the property to enhance the diffusion [10, 11, 19]. It is also expected that several spatial scales of eddies should give rise to anomalous diffusion between proper time scales outside the homogenization regime or when the bigger scale has not yet been homogenized. We refer to M. Avellaneda [1]; A. Fannjiang [8]; Rabi Bhattacharya [6] (see also [7] by Bhattacharya, Denker and Goswami); A. Fannjiang and T. Komorowski [9], and this panorama is certainly not complete. The purpose of this paper is to implement rigorously on a shear flow model the idea that the key to anomalous fast diffusion in turbulent flows is an unfinished homogenization process over a large number of scales of eddies without a sharp separation between them. We will assume that the ratios between the spatial scales are bounded. The underlying phenomenon is similar to the one related to anomalous slow diffusion from perpetual homogenization on an infinite number of scales of gradient drifts [25, 5], the main difference lies in the asymptotic behavior of the multi-scale effective diffusivities D(n) associated with n spatial scales, i.e. D(n) diverge towards ∞ or converge towards 0 with exponential rate depending on the nature of the scales: eddies or obstacles. Note that the shear-layer model is exactly solvable ([3, 14]). When the geometrically divergent scales are recast into the Fourier setting with a power-law spectrum, superdiffusivity has already been proven in the limit t → ∞. Our purpose in this paper is to show that never-ending homogenization can be used as a tool to obtain a quantitative control on the anomaly for finite times, not just an asymptotic result and without any self-similarity assumption. Moreover it will be shown that the mean-squared displacement E[yt2 ] of the diffusion in the shear flow behaves like D(n(t))t (see (24)). In this formula, n(t) has a logarithmic growth and corresponds to the number of scales that can be considered as homogenized at time t, casting into light the role of never-ending homogenization in the anomalous fast behavior of a diffusion process in a shear-flow model. Moreover it will be shown in [26] that the strategy associated to never-ending homogenization can be extended to higher dimensions (and non shear flow models of turbulence). We would like to refer the reader to an interesting and related recent preprint by S. Olla and T. Komorowski [29] on “the superdiffusive behavior of passive tracer with a Gaussian drift”.
Super-Diffusivity in a Shear Flow Model from Perpetual Homogenization
283
2. The Model Let us consider in dimension two a Brownian motion with a drift given by the divergence of a shear flow stream matrix, i.e. the solution of the stochastic differential equation: dyt = dωt − ∇(yt )dt,
y0 = 0,
where is a skew-symmetric 2 × 2 shear flow matrix, 0 h(x1 ) (x1 , x2 ) = . −h(x1 ) 0
(3)
(4)
The function (x1 , x2 ) → h(x1 ) is given by a sum of infinitely many periodic functions with (geometrically) increasing periods h(x1 ) =
∞
γn h n
x
n=0
1
Rn
,
(5)
where hn are smooth functions of period 1. We will assume that hn (0) = 0.
(6)
We will normalize the functions hn by the choosing their variance equal to one: 1 1 Var(hn ) = (hn (x) − hn (y)dy)2 dx = 1. 0
(7)
0
Rn and γn grow exponentially fast with n, i.e. Rn =
n
rk ,
(8)
k=0
where rn are integers, r0 = 1, ρmin = inf∗ rn ≥ 2 n∈N
and
ρmax = sup rn < ∞.
and
γmax = sup(γn+1 /γn ) < ∞.
n∈N∗
(9)
We choose γ0 = 1 and γmin = inf (γn+1 /γn ) > 1 n∈N
n∈N
(10)
It is assumed that the first derivate of the potentials hn are uniformly bounded. (Osc(h) stands for sup h − inf h) K0 = sup Osc(hn ) < ∞, n∈N
K1 = sup hn ∞ < ∞. n∈N
(11)
In this paper we shall distinguish two hypotheses Hypothesis 1. ρmin > γmax .
(12)
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G. Ben Arous, H. Owhadi
Hypothesis 2. 1/2 . ρmin > γmax
(13)
hn (0) = 0
(14)
K2 = sup ∂12 hn ∞ < ∞.
(15)
For all n ∈ N,
and n∈N
Let us observe that under Hypotheses 1, (6), (9), (10) and (11) h is a well defined C 1 function on Rd and |h(x)| ≤ K1 |x|(1 − γmax /ρmin )−1
|h (x)| ≤ K1 (1 − γmax /ρmin )−1 .
(16)
Thus under Hypothesis 1, is a well defined Lipschitz stream matrix and the solution of the stochastic differential equation (3) exists; is unique up to sets of measure 0 with respect to the Wiener measure and is a strong Markov continuous Feller process. Under Hypothesis 2, (6), (9), (10) and (11), is no more Lipschitz continuous but h is still a well defined C 2 function on Rd and 2 )−1 |h(x)| ≤ K2 |x|2 (1 − γmax /ρmin
2 |h (x)| ≤ K2 (1 − γmax /ρmin )−1 |x|.
(17)
3. Main Results 3.1. Under Hypothesis 1. Our objective is to show that the solution (3) is abnormally fast and the asymptotic sub-diffusivity will be characterized as an anomalous behavior of the variance at time t, i.e. E0 [yt2 ] ∼ t 1+ν as t → ∞. More precisely there exists a constant ρ0 (γmin , γmax , K0 , K1 ) and a time t0 (γmin , γmax , R1 , K0 , K1 ) such that Theorem 1. If ρmin > ρ0 and yt is a solution of (3) then for t > t0 , E0 [|yt .e2 |2 ] = t 1+ν(t)
(18)
with ln γmin ln ρmax + ln
γmax γmin
−
C1 ln γmax ≤ ν(t) ≤ ln t ln ρmin + ln
γmin γmax
+
C2 , ln t
(19)
where the constants C1 and C2 depend on ρmin , γmin , γmax , ρmax , K0 , K1 . We remark that if γmax = γmin = γ and ρmax = ρmin = ρ then ν(t) ∼ ln γ / ln ρ. The key of the fast asymptotic behavior of the variance of the solution of (3) is the geometric rate of divergence towards ∞ of the multi-scale effective matrices associated to a finite number of scales. More precisely, for k, p ∈ N, k ≤ p we will write H k,p =
p n=k
γn hn (x/Rn )
(20)
Super-Diffusivity in a Shear Flow Model from Perpetual Homogenization
285
k,p
and k,p the skew-symmetric matrix given by 1,2 (x1 , x2 ) = H k,p (x1 ). Let D( 0,p ) be the effective diffusivity associated to homogenization of the periodic operator L 0,p = 1/2# − ∇ 0,p ∇. Then it is easy to see that D( 0,p ) =
1 0 0 D( 0,p )22
(21)
and it will be shown that Theorem 2. For
−1 $ = 4K1 ρmin (γmin − 1) < 1,
1 + 4(1 − $)
p k=0
γk2 ≤ D( 0,p )22 ≤ 1 + 4(1 + $)
(22) p k=0
γk2 .
(23)
The super-diffusive behavior can be explained and controlled by a perpetual homogenization process taking place over the infinite number of scales 0, . . . , n, . . . . The idea of the proof of Theorem 1 is to distinguish, when one tries to estimate (18), the smaller scales which have already been homogenized (0, . . . , nef called effective scales), the bigger scales which have not had a visible influence on the diffusion (ndri , . . . , ∞ called drift scales because they will be replaced by a constant drift in the proof) and some intermediate scales that manifest their particular shapes in the behavior of the diffusion (nef + 1, . . . , ndri − 1 = nef + nper called perturbation scales because they will enter in the proof as a perturbation of the homogenization process over the smaller scales). The number of effective scales is fixed by the mean squared displacement of yt .e1 . Writing nef (t) = inf{n : t ≤ Rn2 } one proves that E[(yt .e2 )2 ] ∼ D( 0,nef (t) )t.
(24)
Assume for instance Rn = ρ n and γn = γ n then nef (t) ∼ ln t/(2 ln ρ) and ln γ
E[(yt .e2 )2 ] ∼ t 1+ ln ρ . We remark that the quantitative control is sharper than the one associated to a perpetual homogenization on a gradient drift [25]; this is explained by the fact that the number of perturbation scales is limited to only one scale with a divergence free drift. Nevertheless the main difficulty is to control the influence of this intermediate scale and the core of that control is based on the following mixing stochastic inequality (we write TdR = RTd the torus of dimension d and side R). 1 Proposition 1. Let R > 0 and f, G ∈ (H 1 (T1R ))2 such that 0 f (y)dy = 0 and 1 0 G(y)dy = 0, let r > 0 and t > 0, t ∂1 f (rbs ) ds ≤ f L2 (T1 ) G L2 (T1 ) 2r −1 . E G(bt ) 0
R
R
(25)
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3.2. Physical interpretation. We want to emphasize that to obtain a super-diffusive behavior of the solution of (3) of the shape E[yt2 ] ∼ t 1+ν with ν > 0 it is necessary to assume the exponential rate of growth of the parameters γn ; this has a clear meaning when the flow is compared on a heuristic point of view to a real turbulent flow. First, note that our model starts with the dissipation scale and expresses the inertial range of scales as geometrically divergent. Next, the parameters γn hn ∞ /Rn represents the amplitude of the pulsations of the eddies of size Rn . It is a well known characteristic of turbulence ([20] p. 129) that the amplitude of the pulsations increase with the scale, since for all scales hn ∞ ≤ K1 one should have lim γn /Rn → ∞ to reflect that image. In our model, it is sufficient to assume the exponential rate of divergence of γn to obtain a super-diffusive behavior. Let us also notice on a heuristic point of view (our model is not isotropic and does not depend on the time) that the energy dissipated per unit time and unit volume in the eddies of scale n is of order of $n ∝
γn2 2 K . Rn4 2
(26)
So saying that the energy is dissipated mainly in the small eddies is equivalent to saying that γn /Rn2 → 0 as n → ∞ or if Rn = ρ n and γn = ρ αn , this equivalent to say that α < 2, which is included in Hypothesis 2. The Kolmogorov–Obukhov law is equivalent to say that K2 < ∞, for all n hn (0) = 0 and 4
γn ∝ Rn3 ,
(27)
or if Rn = ρ n and γn = ρ αn , this is equivalent to say that α = 43 corresponding to the Kolmogorov spectrum, which violates the hypothesis 1, ρ > γ but not Hypothesis 2 1 (ρ > γ 2 ) which will be addressed below. Overlapping ratios. The super-diffusive behavior in Theorem 1 requires a minimal separation between scales, i.e. ρmin > ρ0 and this condition is necessary. Assume for instance Rn = ρ n and γn = γ n , then if hn (x1 ) = g(x1 ) − γ p g(a p x1 ) (with g ∈ C 1 (T1 ) where T1 is the torus R/Z and a ∈ N∗ ) it is easy to see that for ρ = a, is bounded and yt has a normal behavior by Norris’s Aronson type estimates [21]. Thus, as for a gradient drift [25], it is easy to see for simple examples that when ρ ≤ ρ0 the solution of (3) may have a normal or a super-diffusive behavior depending on the value of ρ and the shapes of hn and ratios of normal behavior may be surrounded by ratios of anomalous behavior. This phenomenon is created by a strong overlap between spatial ratios between scales. Thus the hypothesis (12) is necessary not only to ensure that one has a well defined C 1 stream matrix but also that its associated diffusion may show an anomalous fast behavior. Indeed with hn (x) = sin(2π x) − 3 sin(6π x), γn = ρn = 3n one has h∞ < ∞, which leads to a normal behavior of yt . The case ρmin ≤ γmax will be addressed with Hypothesis 2. Let us observe that to investigate this case we had to add further information on the higher derivate of hn to 2 ensure that h is well defined: hn (0) = 0, hn ∞ ≤ K2 under the constraint ρmin > γmax (which includes the Kolmogorov case γ = ρ 4/3 ).
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3.3. Under Hypothesis 2. In this subsection we will give theorems putting into evidence the anomalous fast behavior of the solutions of (3) under Hypothesis 2 which includes the Kolmogorov spectrum. Theorem 3. Under Hypothesis (2), if there exists a constant z > 2 such that γmin ≥ z then there exists a constant C0 depending on z, K0 , K2 such that if 2 −1 ρmin γmax > C0
(28)
then there exist constants C1 > 0 depending on z, C2 > 0 depending on z, K0 , K2 , γmax , C3 > 0 on z, K0 and C4 > 0 on z, K0 , ρmax such that if yt is a solution of (3) and t ≥ C3 , then
2 2 C1 tγp(t) ≤ E (yt .e2 )2 ≤ C2 tγp(t) (29) and
C4 t 1+ν(t) ≤ E (yt .e2 )2 ≤ C2 t 1+ν(t)
(30)
−2 p(t) := sup{n ∈ N : 16(1 + K02 ) 1 − (γmin − 1)−1 Rn2 < t}
(31)
ν(t) := ln γp(t) ln Rp(t) .
(32)
with
and
Remark. It is important to note that Eq. (29) emphasizes the role of never-ending homogenization in the fast behavior of the diffusion and its proof is also based on the separation of the infinite number of scales into smaller ones which act through their effective properties, larger ones which will be bounded by a constant drift and an intermediate one that one has to control. Equation (30) gives quantitative control of the anomaly without any self-similarity assumption; ν(t) is not a constant if the multi-scale velocity distribution associated to (3) is not self-similar but one has ln γmin ln ρmax ≤ ν(t) ≤ ln γmax ln ρmin . (33) Definition 1. The Stochastic Differential Equation (3) is said to have a self-similar velocity distribution if and only if ρmin = ρmax = ρ
(34)
γmin = γmax = γ .
(35)
α = ln γ / ln ρ.
(36)
and
Then we will write
From Theorem 3 one easily deduces the following theorem.
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G. Ben Arous, H. Owhadi
Theorem 4. Assume that the SDE (3) has a self-similar velocity distribution with 0 < α < 2. Under Hypothesis (2), there exists a constant C0 depending on α, K0 , K2 such that if ρ > C0
(37)
then there exist constants C1 > 0 depending on K0 , ρ, C2 > 0 on K0 , K2 , ρ, α and C3 on K0 such that if yt is a solution of (3) and t ≥ C3 , then
C1 t 1+α ≤ E (yt .e2 )2 ≤ C2 t 1+α . (38) Remark. Observe that all the velocity distribution 0 < α < 2 is covered by Theorem 4. The condition (37) is still needed to avoid overlapping ratios and cocycles. It is important to note that even if Theorem 3 allows to cover a larger spectrum of velocity distribution than Theorem 1, its proof is based on the regularity of the drift associated to the diffusion (K2 < ∞). Whereas in Theorem 1, the drift −∇ associated to the stochastic differential equation (3) can be non-Holder-continuous (only the regularity of is needed to prove an anomalous fast behavior). 3.4. Remark. Fast separation between scales. When γmin > 1 and γmax < ∞, the feature that distinguishes a strong anomalous behavior from a weak one is the rate at which spatial scales do separate. Indeed one can follow the proof of Theorem 1, changing α the condition ρmax < ∞ into Rn = Rn−1 [ρ n /Rn−1 ] (ρ, α > 1) and γmax = γmin = γ to obtain Theorem 5. For t > t0 (γ2 , R2 , K0 , K1 ), C1 tγ β(t) ≤ E0 [|yt .e2 |2 ] ≤ C2 tγ β(t) ,
(39)
β(t) = 2(2 ln ρ)− α (ln t) α ,
(40)
with
1
1
where the constants C1 and C2 depend on ρ, γ , α, K1 . And as α ↓ 1 the behavior of the solution of (3) pass from weakly anomalous to strongly anomalous. 4. Proofs Under Hypothesis 1 4.1. Notations and proof of Theorem 2. In this subsection we will prove Theorem 2 using the explicit formula of the effective diffusivity. We will first introduce the basic notations that will also be used to prove Theorem 1. Let J be smooth T12 periodic 2 × 2 skew-symmetric matrix such that J12 (x1 , x2 ) = j (x1 ) and consider the periodic operator LJ = 1/2# − ∇J ∇. We call χl the solution of the cell problem associated to LJ , i.e. the T12 periodic solution of LJ (χl − l.x) = 0 with χl (0) = 0. One easily obtains that χl (x1 , x2 ) = −2l2
0
x1
1
j (y)dy − x1 0
j (y)dy .
(41)
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289
The cell problem allows to compute the effective diffusivity t lD(J )l = solution of the 2 T12 |l − ∇χl (x)| dx that leads to 1 0 D(J ) = . (42) 0 1 + 4 Var(j ) For a R periodic function f we will write R Var(f ) = 1/R (f (x) − 0
R
f (y)dy)2 dx.
(43)
0
Using the notation (20), from Eq. (42) we obtain (21) with D( 0,p )2,2 = 1 + 4 Var(H p ).
(44)
Now we will prove by induction on p that (for $ given by (22)) (1 − $)
p k=0
γk2
p
≤ Var(H ) ≤ (1 + $)
p k=0
γk2 .
(45)
Then Eq. (23) of theorem will follow by (44). Equation (45) is trivially true for p = 0. From the explicit formula of H p we will show in paragraph 4.1 that Var(H p ) − Var(H p−1 ) − γ 2 ≤ 2γp K1 r −1 Var(H p−1 ). (46) p p Assuming (45) is true at the rank p one obtains p 1 1 2 p 2 Var(H ) ≤ (1 + $) γp+1 (γk /γp+1 )2
(47)
k=0
≤ (1 + $) γp+1 (γmin − 1)−1 , 1 2
1
and it is easy to see that the condition (22) implies that $ ≥ 2K1 (1 + $) 2 (ρmin (γmin − 1))−1 ; combining this with (46) and (47) one obtains that Var(H p+1 ) − Var(H p ) − γ 2 ≤ $γ 2 (48) p+1 p+1 which proves the induction and henceforth the theorem. 4.1.1. From the equation 1 1 p−1 Var(H p ) = (H (Rp x) − H p−1 (Rp y)dy) + γp (hp (x) − 0
one obtains
0
Var(H p ) − Var(H p−1 ) − γ 2 ≤ 2γp |E| p
with
1 0
2 hp (y)dy) dx
(49)
E = Cov(SRp H p−1 , hp ), 1 1 where Cov stands for the covariance: Cov(f, g) = 0 (f (x) − 0 f (y)dy)(g(x) − 1 0 g(y)dy)dx and SR is the scaling operator SR f (x) = f (Rx). We will use the following mixing lemma whose proof is an easy exercise
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G. Ben Arous, H. Owhadi
2 Lemma 1. Let (g, f ) ∈ C 1 (T d and R ∈ N∗ then
T1d
g(x)SR f (x)dx −
T1d
g(x)dx
T1d
f (x)dx ≤ (g ∞ /R)
T1d
f dx.
(50)
By Lemma 1 and the Cauchy–Schwartz inequality, |E| ≤
∇hp ∞ Rp−1 Rp
1 0
|SRp H p−1 (x)|dx ≤
K1 Var(H p−1 ). rp
(51)
Combining this with (49) one obtains (46).
4.2. Anomalous mean squared displacement: Theorem 1. 4.2.1. Anomalous behavior from perpetual homogenization. Let yt be the solution of (3). Define 2 nef (t) = inf{n ∈ N : t ≤ Rn+1 (γn−1 /γn+1 )2 2−3 K1−2 (1 − γmax /ρmin )2 }.
(52)
nef (t) will be the number of effective scales that one can consider homogenized in the estimation of the mean squared displacement at the time t. Indeed, we will show in Subsect. 4.2.2 that for ρmin > C1,K1 ,γmin ,γmax and t > C2,K0 ,γmin ,γ1 ,R1 one has
(1/4)tγn2ef (t)−1 ≤ E (yt .e2 )2 ≤ C3,K1 ,γmin γn2ef (t)+1 t.
(53)
Combining this with the bounds (9) and (10) on Rn and γn one easily obtains Theorem 1. 4.2.2. Distinction between effective and drift scales. Proof of Eq. (53). By the Ito formula one has t yt .e2 = ωt .e2 + ∂1 h(ωs .e1 ) ds. (54) 0
And by the independence of ωt .e2 from ωt .e1 one obtains 2 t
E (yt .e2 )2 = t + E ∂1 h(ωs .e1 )ds .
(55)
0
Thus for all p ∈ N∗ , using h = H p + H p+1,∞ one easily obtains (writing H p = H 0,p ) 2 2 t
1 t E (yt .e2 )2 ≥ t + E ∂1 H p (ωs .e1 )ds ∂1 H p+1,∞ (ωs .e1 )ds −E 2 0 0 t 2 2 t ≤ t + 2E ∂1 H p (ωs .e1 )ds ∂1 H p+1,∞ (ωs .e1 )ds + 2E . 0
0
(56)
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Now we will bound the larger scales ∂1 H p+1,∞ as drift scales, i.e. bound them by a
−1 constant drift using ∂1 H p+1,∞ ∞ ≤ K1 γp+1 /Rp+1 )(1 − γmax /ρmin , 2 2
tK1 γp+1 1 t 2 E (yt .e2 ) ≥ t + E − ∂1 H p (ωs .e1 )ds 2 Rp+1 (1 − γmax /ρmin ) 0 t 2 2 (57) tK1 γp+1 +2 ≤ t + 2E ∂1 H p (ωs .e1 )ds . Rp+1 (1 − γmax /ρmin ) 0
2 t Write Ip = E 0 ∂1 H p (ωs .e1 )ds . Since H p is periodic, for t large enough, Ip should behave like tγp2 . Nevertheless since the ratios between the scales are bounded, to control the asymptotic lower bound of the mean squared displacement in (57) we will need a quantitative control of Ip that is sharp enough to show that the influence of the effective scales is not destroyed by the larger ones. This control is based on stochastic mixing inequalities and it will be shown in Subsect. 4.2.3 that for ρmin > 8K1 /(γmin −1) one has 2
Ip ≤ t23 γp−1 (1 − 1/γmin )−2 + K0 K1 γp−1 (γp /rp )(1 − 1/γmin )−1 √ + t 2 K12 (γp2 /Rp2 ) + t68γp−1 γp Rp−1 K02 (1 − 1/γmin )−1 (58) 2 2 + 60K02 (1 − 1/γmin )−1 γp−1 γp (Rp2 /rp ) + 16K02 γp−1 Rp−1
and Ip ≥ tγp−1 (γp−1 − 16K1 γp /rp ) −
√
− 60K02 (1 − 1/γmin )−1 γp−1 γp
t68γp−1 γp Rp−1 K02 (1 − 1/γmin )−1
Rp2 rp
2 2 − 8K02 γp−1 Rp−1 (1 − 1/γmin )−2 .
(59)
Choosing p = nef (t) given by (52) one obtains (53) from (58), (59) and (57) by straightforward computation under the assumption ρmin > CK0 ,K1 ,γmax ,γmin . 4.2.3. Influence of the intermediate scale on the effective scales: Proof of the inequalities (59) and (58). In this subsection we will prove the inequalities (59) and (58) by distinguishing the scale p as a perturbation scale, i.e. controlling its influence on the homogenization process over the scales 0, . . . , p − 1. More precisely, writing bt the Brownian motion ωt .e1 one has t 2 Ip = Ip−1 + E ∂1 H p,p (bs )ds (60) + 2Jp 0
with
t t p,p Jp = E ∂1 H (bs )ds ∂1 H p−1 (bs )ds . 0
We will then control (60) by bounding ∂1 H p,p by a constant drift to obtain t 2 E ≤ t 2 K12 γp2 /Rp2 . ∂1 H p,p (bs )ds 0
(61)
0
(62)
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G. Ben Arous, H. Owhadi
In Subsect. 4.2.4 we will control the homogenization process over the scales 0, . . . , p−1 and use our estimates on D( 0,p−1 ) to obtain that for ρmin > 8K1 /(γmin − 1), 2 2 2 20(1 − 1/γmin )2 + Rp−1 γp−1 16K02 (1 − 1/γmin )2 , Ip−1 ≤ tγp−1
Ip−1 ≥
2 tγp−1 2(1 − 1/γmin )2
2 2 − Rp−1 γp−1 8K02 (1 − 1/γmin )2 .
(63) (64)
In Subsect. 4.2.5 we will bound Jp is an error term by mixing stochastic inequalities to obtain √ |Jp | ≤ γp−1 γp (1 − 1/γmin )−1 tRp−1 34K02 + (t/rp )8K1 + Rp2 rp−1 30K02 . (65) Combining (60), (62), (63), (64) and (65) one obtains (58) and (59).
4.2.4. Control of the homogenization process over the effective scales: Proof of Eqs. (63) and (64). Writing for m ≤ p, Rp H m,p (y)dy (66) κ m,p = 1/Rp 0
and
κp
0
=
κ 0,p
one obtains by the Ito formula bt t t ∂1 H p−1 (bs )ds = 2 (H p−1 (y) − κ p−1 )dy − 2 (H p−1 (bs ) − κ p−1 )dbs . 0
0
(67) Using the periodicity of H p : x
H p−1 (y) − κ p−1 dy ≤ Rp−1 K0 γp−1 (1 − 1/γmin )−1 .
(68)
0
Now we will show that for ρmin > 8K1 /(γmin − 1) and p ∈ N∗ , t
E[ (H p (bs ) − κ p )2 ds] ≤ (1 − 1/γmin )−2 tγp2 5 + Rp2 γp2 4K02 0
≥
(69)
tγp2 2 − Rp2 γp2 4K02 (1 − 1/γmin )−2 .
Combining (67), (68) and (69) one obtains (63) and (64) by straightforward computation. The proof of (69) is based on standard homogenization theory: writing x p
2 (70) H (y) − κ p − Var(H p )x, f (x) = 0
and g(x) = 2
0
x
f (y) dy − (x/Rp )
Rp
f (y) dy .
(71)
0
One obtains by the Ito formula t E[ (H p (bs ) − κ p )2 ds] = Var(H p )t + E[g(bt )].
(72)
0
Using the periodicity of g, one has g∞ ≤ 4K02 γp2 Rp2 (1 − 1/γmin )−2 . Combining this with the estimate (45) on Var(H p ) one obtains (69) from (72).
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293
4.2.5. Control of the influence of the perturbation scale: Proof of Eq. (65). From Eqs. (61), (67), x
H p−1 (y) − κ p−1 dy ≤ Rp−1 γp−1 K0 (1 − 1/γmin )−1 (73) 0
and t 0
∂1 H p,p (ωs .e1 )ds = 2
bt
(H p,p (y) − κ p,p )dy − 2
0
t 0
(H p,p (bs ) − κ p,p ) dbs . (74)
One obtains by using straightforward computation (using the Cauchy–Schwartz inequality) that √ (75) |Jp | ≤2γp−1 γp Rp−1 K02 (1 − 1/γmin )−1 t + 4|Jp,2 | + 2|Jp,3 | with
t Jp,2 = E (H p−1 (bs ) − κ p−1 )(H p,p (bs ) − κ p,p ) ds
(76)
0
and Jp,3
t p−1 =E ∂1 H (bs )ds
bt
(H p,p (y) − κ p,p )dy .
(77)
0
0
We will show in Subsect. 4.2.6 that the ratio rp allows a stochastic separation between the scales 0, . . . , p − 1 and p reflected by the following inequality: √
Jp,2 ≤ γp−1 γp K0 (1 − 1/γmin )−1 8K0 Rp−1 t + 2tK1 /rp . (78) x Using Proposition 1 (that we will prove Subsect. 4.2.7) with G(x) = 0 (H p,p (y) − κ p,p )dy, r = rp and f (rp x) = H p−1 (x) − k p−1 that Jp,3 ≤ γp−1 γp R 2 r −1 15K 2 (1 − 1/γmin )−1 . (79) p p 0 Combining (75), (78) and (79) one obtains (65).
4.2.6. Stochastic separation between scales: Proof of Eq. (78). Writing for x ∈ R, x g(x) = (H p−1 (y) − κ p−1 )(H p,p (y) − κ p ) dy, (80) 0
one obtains by Ito formula 2E[ 0
bs
g(y)dy] = Jp,2 .
Using the functional mixing Lemma 1 one obtains easily that
|g(x)| ≤ 2K0 γp−1 (1 − 1/γmin )−1 4γp K0 Rp−1 + |x|K1 (γp /rp ) . Combining this with (81) one obtains (78).
(81)
(82)
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4.2.7. Stochastic mixing: Proof of Proposition 1. By the scaling law of the Brownian motion t t R2 E G(bt /R) (1/R)∂1 f (rbs /R) ds = RE G(b t ) ∂1 f (rbs ) ds R2
0
0
and it is sufficient to prove the proposition assuming R = 1. Let us write t I = E G(bt ) ∂1 f (rbs ) ds .
(83)
0
We will prove Proposition 1 by expanding (83) on the the Fourier decompositions of f and G (written fk and Gk ) and controlling trigonometric functions, f (x) = fk eik2πx . (84) k∈Z
Write for k, m ∈ Z,
Jk,m =
t
E eikr2πbs eim2πbt ds.
(85)
0
By straightforward computation t 2 (kr+m)2 2 m2 Jk,m = e−(2π) 2 s−(2π) 2 (t−s) 0
=e
2
−(2π)2 m2 t
2 ( (kr)2 +krm)t 2
1 − e−(2π)
(86)
(2π )2 ( (kr) 2 + krm) 2
(in last fraction of the above equation, if the denominator is equal to 0, we consider it as a limit to obtain the exact value t). Now I= Jkm ik2πfk Gm . (87) k,m∈Z2
Thus since f0 = G0 = 0, by the Cauchy–Schwartz inequality |I | ≤
|fk |
k∈Z∗
(2π ) m |Gm | 2
2
m∈Z
2
21
1
Jk2 .
(88)
with 1 1 − e−(2π)2 ( (kr)2 +krm)t 2 2 2 Jk = e−(2π) m t . 2r2 2 k 2 m ( + krm)(2π ) ∗ 2
m∈Z
(89)
2
Using (1 − e−tx )/x ≤ 3t for x > 0 and the fact that the minimum of k 2 r/2 + km is reached for m0 ∼ kr/2, we obtain 4t 2 1 1 2 2 2 Jk ≤ e−(2π) m t + 2 k2 r 2 m2 krm(2π )2 ∗ ∗ (90) m∈Z 2
≤4/r .
m∈Z
Super-Diffusivity in a Shear Flow Model from Perpetual Homogenization
Observing that G 2L2 (T1 ) = (2π )2
m∈Z m
2 |G
m|
|I | ≤ G L2 (T1 ) (2/r)
2
295
one obtains from (88) and (90)
|fk |,
(91)
|I | ≤ G L2 (T1 ) f L2 (T1 ) 2/r.
(92)
k∈Z∗
which, using the Cauchy–Schwartz inequality leads to
5. Proofs Under Hypothesis 2 In this section we will prove Theorem 3 under Hypothesis 2. Observe that it is sufficient to prove Eq. (29) under Hypothesis 2, (28) and (31). We will use the same notations as in Sect. 4. Let us observe that from Eq. 55 one obtains
E (yt .e2 )2 = t + E X2 (b, 0, t) + E Y 2 (b, 0, t) + 2E X(b, 0, t)Y (b, 0, t) (93) with for p ∈ N and bs = ωs .e1 ,
t
∂1 H 0,p (bs ) ds
(94)
∂1 H p+1,∞ (bs ) ds.
(95)
X(b, 0, t) = 0
and
t
Y (b, 0, t) = 0
We will prove in Subsect. 5.1 the following lemma (which is the core of the proof of Theorem 3) which gives quantitative control on decorrelation between the drifts associated with the small scales and those associated to the larger ones. Lemma 2. For t > Rp2 ,
E[X(b, 0, t)Y (b, 0, t)] ≤ t 3/2 R 2 R −2 + 8tRp R −2 Rp γp γp+1 p+1 p+2 p+1
−1 −1 −1 2 1 − γmax /ρmin 12K0 K2 1 − γmin .
(96)
Now by standard homogenization as has been done in Subsect. 4.2.4 it is easy to prove that −1 −1 E[X(b, 0, t)2 ] ≤ (Rp2 + t)γp2 8K02 1 − γmin (97) and
−1 −1 . E[X(b, 0, t)2 ] ≥ 2t Var(H p ) − 8Rp2 γp2 K02 1 − γmin
(98)
Now we will give in Lemma 3 (proven in Subsect. 5.2) the exponential rate of divergence of Var(H p ). Lemma 3. For γmin > 2 one has
2 −1 −2 γp2 1 − (γmin − 1)−1 ≤ Var(H p ) ≤ γp2 1 − γmin .
(99)
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The proof of Lemma 3 is different from that of Theorem 2 and is based on the domination of the influence of the biggest scale over the smaller ones (which is ensured by γmin > 2, which is also necessary in the case ρmin > γmax to avoid cocycles). Combining (99) with (98) one obtains that
2 −1 −1 E[X(b, 0, t)2 ] ≥ 2tγp2 1 − (γmin − 1)−1 − 8Rp2 γp2 K02 1 − γmin . (100) Now let us observe that from (93) and the Cauchy–Schwartz inequality one obtains
E (yt .e2 )2 ≤ t + 2E X 2 (b, 0, t) + 2E Y 2 (b, 0, t)
≥ t + E X2 (b, 0, t) − 2 E[X(b, 0, t)Y (b, 0, t)] .
Combining (100), (97), (96), (101) and bounding E Y 2 (b, 0, t) by
E Y 2 (b, 0, t) ≤ ∂12 H p+1,∞ 2∞ t 2 E[bt2 ] 2 −4 −1 −1 2 2 K2 (1 − γmin t 3 Rp+1 ) (1 − γmax /ρmin )−1 , ≤ γp+1
(101)
(102)
2 >γ one obtains that for t > Rp2 , γmin > 2 and ρmin max that
2 −4 2 2 E[(yt .e2 )2 ] ≤ 32(Rp2 + t)γp2 K02 + 8γp+1 K2 (1 − γmax /ρmin t 3 Rp+1 )−1
(103)
and
2 E[(yt .e2 )2 ] ≥ 2tγp2 1 − (γmin − 1)−1 − 16Rp2 γp2 K02
−2 −2 2 2 − t 3/2 Rp+1 Rp+2 + 8tRp Rp+1 Rp γp γp+1 48K0 K2 (1 − γmax /ρmin )−1 . (104)
It follows (104) and (103) that for
−2 16(1 + K02 ) 1 − (γmin − 1)−1 Rp2 < t
−2 2 . ≤ 16(1 + K02 ) 1 − (γmin − 1)−1 Rp+1 (105)
One has
2 −2 2 400K0 K2 (1 − γmax /ρmin )−1 E[(yt .e2 )2 ] ≥ tγp2 1 − (γmin − 1)−1 − γmax ρmin
−1 × 1 + (1 + K0 )ρmin (γmin − 1)(γmin − 2)−1 (106)
and E[(yt .e2 )2 ] ≤ tγp2 (1 + K02 )
−4 2 2 × 1 + 2050γmax )−2 , 1 − (γmin − 1)−1 K22 (1 − γmax /ρmin (107) which proves Eq. (29) under Hypothesis 2, (28) and (31) and thus Theorem 3.
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5.1. Proof of Lemma 2. Let us introduce Z a random variable idependent from bs and taking its values in [0, Rp ] uniformly with respect to the Lebesgue measure. First, let us observe that by the periodicity of H p , E[X(b + Z, 0, t)Y (b, 0, t)] = 0.
(108)
Next, define τ = inf{s > 0 : |bs + Z| = 0 or |bs + Z| = Rp }. Now we will decompose X(b + Z, 0, t) and Y (b, 0, t) as follows: X(b + Z, 0, t) = X(b + Z, 0, τ ) + X(b + Z, τ, t), Y (b, 0, t) = Y (b, 0, τ ) + Y (b, τ, t).
(109) (110)
Let b be a BM independent from b and Z with the same law as b. Combining the decompositions (109) and (110) with the strong Markov property and the periodicity of H p one obtains that E[X(b + Z, 0, t)Y (b, 0, t)] = E[X(b + Z, 0, τ ∧ t)Y (b, 0, τ ∧ t)] + E[X(b , 0, (t − τ )+ )Y (b + bτ , 0, (t − τ )+ )] + E[X(b + Z, 0, τ ∧ t)Y (b, τ ∧ t, t)] + E[X(b + Z, τ ∧ t, t)Y (b, 0, τ ∧ t)], (111) where we have used E[X(b , 0, (t − τ )+ )Y (b + bτ , 0, (t − τ )+ )] = E[X(b + Z, τ ∧ t, t)Y (b, τ ∧ t, t)]. Using a similar decomposition for E[X(b, 0, t)Y (b, 0, t)] and subtracting (111) combined with (108) one obtains that E[X(b, 0, t)Y (b, 0, t)] = I1 + I2 + I3 + I4 + I5 + I6 + I7 .
(112)
with
I1 = E X(b , 0, (t − τ )+ ) Y (b , 0, (t − τ )+ ) − Y (b + bτ , 0, (t − τ )+ ) ,
I2 = E X(b , (t − τ )+ , t) Y (b , (t − τ )+ , t) ,
I3 = −E X(b + Z, 0, τ ∧ t)Y (b, 0, τ ∧ t) ,
I4 = −E X(b + Z, 0, τ ∧ t)Y (b, τ ∧ t, t) ,
I5 = −E X(b + Z, τ ∧ t, t)Y (b, 0, τ ∧ t) ,
I6 = E X(b , 0, (t − τ )+ )Y (b , (t − τ )+ , t) ,
I7 = E Y (b , 0, (t − τ )+ )X(b , (t − τ )+ , t) .
(113) (114) (115) (116) (117) (118) (119)
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We shall use standard homogenization techniques to estimate the influence of the effective scales in those terms. Let us observe that by the Ito formula one has a.s. for all t > 0, t p
(120) H (bs ) − κ p dbs X(b, 0, t) = χ p (bt ) − 2 0
with (using the notation introduced in (66)) x H p (y)dy − x2κ p . χ p (x) = 2
(121)
0
Next we will control the larger scales by bounding the growth of their drift. Indeed using the uniform control on the second derivate of the functions hn in Hypothesis 2 one has −2 2 (1 − γmax /ρmin )−1 . ∂12 H p+1,∞ ∞ ≤ K2 γp+1 Rp+1
(122)
Using the Cauchy–Schwartz inequality in (114), (120) and (122) one obtains that 21 21 |I2 | ≤ E X(b , 0, (t − τ )+ )2 E (Y (b , (t − τ )+ , t))2
1 21 −1 −1 ≤ (1 − γmin ) 4K02 γp2 Rp2 + 4K02 γp2 t 2 ∂12 H p+1,∞ ∞ E |bt |2 τ 2 . Thus for t > Rp2 , |I2 | ≤
√ −2 −1 −1 2 tγp γp+1 Rp3 Rp+1 12K0 K2 (1 − γmin ) (1 − γmax /ρmin )−1 .
(123) (124)
Similarly using (120) one obtains that −2 −1 −1 2 12K0 K2 (1 − γmin ) (1 − γmax /ρmin )−1 . |I3 | ≤ γp γp+1 Rp4 Rp+1
To estimate I4 we use the conditional independence
I4 = − E 1τ ∧t
− E 1τ ∧t=t E X(b + Z, 0, τ ∧ t) τ ∧ t E Y (b, τ ∧ t, t) τ ∧ t .
(125)
(126)
Since Y (b, t, t) = 0 the second term in (126) is null. Moreover conditionally on the chances 1/2 is equalto 0 or Rp , so using (120) one event τ < t, bτ ∧t + Z with equal
obtains that on {τ ∧ t < t}, E X(b + Z, 0, τ ∧ t) τ ∧ t = 0, which leads to I4 = 0.
(127)
Similarly, using conditional independence (120) and (122) one easily obtains −2 −1 −1 2 12K0 K2 (1 − γmin ) (1 − γmax /ρmin )−1 , |I5 | ≤ γp γp+1 Rp4 Rp+1
|I6 | ≤
√
−2 −1 −1 2 tγp γp+1 Rp3 Rp+1 12K0 K2 (1 − γmin ) (1 − γmax /ρmin )−1 .
(128) (129)
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To estimate I7 let us observe that from (120) and the spectral gap associated to the Laplacian on the torus (which is equivalent to the Poincaré inequality, see [23]) one has |E[X(b , (t − τ )+ , t)|t − τ ]| ≤ |χ p (bt ) − χ p (b(t−τ )+ )| −2 /2(t−τ ) +
≤ 12e−Rp
−1 −1 K0 (1 − γmin ) γp R p .
(130)
Thus using conditional independence and distinguishing the events τ > t/2 (whose probability is bounded using the spectral gap) and τ ≤ t/2 , one obtains −2 /4
|I7 | ≤ t 3/2 e−tRp
−1 −1 2 γp γp+1 16Rp Rp−2 K0 K2 (1 − γmin ) (1 − γmax /ρmin )−1
−1 −1 2 ≤ γp γp+1 16Rp4 Rp−2 K0 K2 (1 − γmin ) (1 − γmax /ρmin )−1 .
(131)
To estimate I1 we will distinguish the scale p + 1 as an intermediate scale. First observe that Y (b , 0, (t − τ )+ ) − Y (b + bτ , 0, (t − τ )+ ) = 2γp+1 J1 + 2J2 with
J2 =
(t−τ )+
0
and
J1 =
∂1 H p+2,∞ (bs ) − ∂1 H p+2,∞ (bs + bτ ) ds
(t−τ )+
∂1 hp+1 (bs ) − ∂1 hp+1 (bs + bτ ) ds.
0
(132)
(133)
(134)
Using |bτ | ≤ Rp and (122), one obtains that for t > Rp , E X(b , 0, (t − τ )+ )J2 −2 −1 −1 2 ≤ t 3/2 Rp γp γp+1 Rp+2 2K0 K2 (1 − γmin ) (1 − γmax /ρmin )−1 .
Now using the Ito formula one has bτ hp+1 (x)dx − J1 = 0
(t−τ )+
− 0
b(t−τ ) +bτ +
b(t−τ )+
(135)
hp+1 (x)dx
hp+1 (bs ) − hp+1 (bs
(136)
+ bτ )
dbs
Thus observing that (τ, bτ ) has the same law as (τ, −bτ ) one obtains that E X(b , 0, (t − τ )+ )J1 bτ
= E X(b , 0, (t − τ )+ ) hp+1 (x) − hp+1 (−x) /2 dx 0
+ E X(b , 0, (t − τ )+ )
b(t−τ ) +bτ +
b(t−τ )
− E X(b , 0, (t − τ )+ )
0
+
(t−τ )+
hp+1 (x) − hp+1 (x − bτ ) /2 dx
hp+1 (bs ) − hp+1 (bs + bτ ) dbs .
(137)
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It follows from (137) that for t > Rp , E X(b , 0, (t − τ )+ )γp+1 J1
−1 −2 −1 −1 2 1 − γmax /ρmin K2 1 − γmin . ≤ 6K0 γp tRp2 γp+1 Rp+1
(138)
Thus from (138) and (135) we deduce that
−1 −2 −2
−1 −1 2 + tRp Rp+1 . Rp γp γp+1 12K0 K2 1 − γmin 1 − γmax /ρmin |I1 | ≤ t 3/2 Rp+2 (139) Combining (139), (124), (125), (127), (128), (129), (131) and (96) we have obtained that for t > Rp2 , Eq. (96) is valid.
5.2. Proof of Lemma 3. Observe that (using the notation introduced in (66)) Var(H p ) = Var(H p−1 ) + γp2 + 2Rp−1
Rp 0
H p−1 (x) − κ 0,p−1 hp (x) − κ p,p . (140)
It follows by the Cauchy–Schwartz inequality and the normalization condition (7) that Var(H p−1 ) + γp2 + 2γp Var(H p−1 ) 2 ≥ Var(H p ) 1
≥ Var(H p−1 ) + γp2 − 2γp Var(H p−1 ) 2 , 1
from which we deduce Var(H p ) 2 ≤ γp + Var(H p−1 ) 2
(141)
Var(H p ) 2 ≥ γp − Var(H p−1 ) 2 .
(142)
1
1
and 1
1
From (141) one obtains by induction that −1 −1 Var(H p ) 2 ≤ γp (1 − γmin ) . 1
(143)
Combining (142) with (143) one obtains that
1 Var(H p ) 2 ≥ γp 1 − (γmin − 1)−1 , and observe that 1 − (γmin − 1)−1 > 0 for γmin > 2.
(144)
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Acknowledgements. The authors would like to thank the referees for useful comments. Part of this work was supported by the Aly Kaufman fellowship.
References 1. Avellaneda, M.: Homogenization and renormalization, the mathematics of multi-scale random media and turbulent diffusion. Lectures in Applied Mathematics, Volume 31, 1996, pp. 251–268 2. Avellaneda, M., Majda, A.: Homogenization and renormalization of multiple-scattering expansions for green functions in turbulent transport. In: Composite Media and Homogenization Theory, Volume 5 of Progress in Nonlinear Differential Equations and Their Applications, 1987, pp. 13–35 3. Avellaneda, M., Majda, A.: Mathematical models with exact renormalization for turbulent transport. Commun. Math. Phys. 131, 381–429 (1990) 4. Batchelor, G.: Diffusion in a field of homogeneous turbulence ii. The relative motion of particles. Proc. Cambridge Philos. Soc. 48, 345 (1952) 5. Ben Arous, G., Owhadi, H.: Multi-scale homogenization with bounded ratios and anomalous slow diffusion. Preprint available at http://dmawww.epfl.ch/∼owhadi/ (2001) 6. Bhattacharya, R.: Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media. Ann. Appl. Probab. 9(4), 951–1020 (1999) 7. Bhattacharya, R., Denker, M., Goswami, A.: Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales. Stoch. Processes and their Appl. 80, 55–86 (1999) 8. Fannjiang, A.: Anomalous diffusion in random flows. In: Mathematics of Multiscale Materials – The IMA volumes in mathematics and its applications, Volume 99, 1999, pp. 81–99 9. Fannjiang, A., Komorowski, T.: Fractional brownian motion limit for motions in turbulence. Ann. of Appl. Prob. 10(4) (2001) 10. Fannjiang, A., Papanicolaou, G.: Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54, 333–408 (1994) 11. Fannjiang, A., Papanicolaou, G.: Diffusion in turbulence. Probab. Theory Related Fields 105(3), 279–334 (1996) 12. Furtado, F., Glimm, J., Lindquist, B., Pereira, F.: Multiple length scale calculus of mixing lenght growth in tracer floods. Kovarik, F., ed., Proceeding of the emerging technologies conference, Houston TX, Institute for Improved Oil Recovery, 1990, pp. 251–259. 13. Furtado, F., Glimm, J., Lindquist, B., Pereira, F., Zhang, Q.: Time dependent anomalous diffusion for flow in multi-fractal porous media. Verheggan, T., editor, Proceeding of the workshop on numerical methods for simulation of multiphase and complex flow. New York: Springer Verlag, 1991, pp. 251–259. 14. Gaudron, G.: Scaling laws and convergence for the advection-diffusion equation. Ann. of Appl. Prob. 8, 649–663 (1998) 15. Glimm, J., Lindquist, B., Pereira, F., Peierls, R.: The multi-fractal hypothesis and anomalous diffusion. Mat. Apl. Comput. 11(2), 189–207 (1992) 16. Glimm, J., Zhang, Q.: Inertial range scaling of laminar shear flow as a model of turbulent transport. Commun. Math. Phys. 146, 217–229 (1992) 17. Isichenko, M., Kalda, J.: Statistical topography. ii. Two-dimensional transport of a passive scalar. J. Nonlinear Sci. 1, 375–396 (1991) 18. Kolmogorov, A.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl.Akad.Nauk SSSR 4(30), 9–13 (1941) 19. Komorowski, T., Olla, S.: On homogenization of time dependent random flows. To appear in Probability Theory and Related Fields (2000) 20. Landau, L., Lifshitz, E. Fluid Mechanics, 2nd ed. Moscow: MIR, 1984 21. Norris, J.: Long-time behaviour of heat flow: Global estimates and exact asymptotics. Arch. Rational Mech. Anal. 140, 161–195 (1997) 22. Obukhov, A.: On the distribution of energy in the spectrum of turbulent flow. Izv. Akad. Nauk SSSR Ser. Georg. Geofiz. 5, 453 (1941) 23. Olla, S.: Homogenization of Diffusion Processes in Random Fields. Ecole Polytechnique (1994). Cours Ecole Polytechnique 24. Owhadi, H.: Anomalous diffusion and homogenization on an infinite number of scales. Ph.D. thesis, EPFL – Swiss Federal Institute of Technology (2001). Thesis no 2340, January 2001. Available at http://dmawww.epfl.ch/∼owhadi/ 25. Owhadi, H.: Anomalous slow diffusion from perpetual homogenization. Submitted (2001). Preprint available at http://dmawww.epfl.ch/∼owhadi/
302
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26. Owhadi, H.: Richardson law linked with Kolmogorov law in turbulent diffusion. In preparation (2001) 27. Richardson, L. F.: Atmosphere diffusion shown on a distance-neighbour graph. Proc. R. Soc. London, Ser A 110, 709 (1926) 28. Zhang, Q.: A multi-scale theory of the anomalous mixing length growth for tracer flow in heterogeneous porous media. J. Stat. Phys. 505, 485–501 (1992) 29. Komorovski, T., Olla, S.: On the superdiffusive behavior of passive tracer with a Gaussian drift. Preprint available at http://www.cmap.polytechnique.fr/olla/pubolla.html (December 2001) Communicated by P. Constantin
Commun. Math. Phys. 227, 303 – 348 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Generating Functional in CFT on Riemann Surfaces II: Homological Aspects Ettore Aldrovandi1, , Leon A. Takhtajan2 1 S.I.S.S.A./International School for Advanced Studies, Via Beirut 2/4, 34013 Trieste, Italy.
E-mail:
[email protected]
2 Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA.
E-mail:
[email protected] Received: 29 June 2000 / Accepted: 16 January 2002
Dedicated to the memory of Han Sah Abstract: We revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne cohomology class over the fundamental class of the underlying topological surface. This Deligne ˇ class is constructed by applying a descent procedure with respect to a Cech resolution of any covering map of a Riemann surface. Detailed calculations are presented in the ˇ two cases of an ordinary Cech cover, and of the universal covering map, which was used in our previous approach. We also establish a dictionary that allows to use the same formalism for different covering morphisms. The Deligne cohomology class we obtain depends on a point in the Earle–Eells fibration over the Teichmüller space, and on a smooth coboundary for the Schwarzian cocycle associated to the base-point Riemann surface. From it, we obtain a variational characterization of Hubbard’s universal family of projective structures, showing that the locus of critical points for the chiral action under fiberwise variation along the Earle– Eells fibration is naturally identified with the universal projective structure. Contents 1. 2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . Preliminaries and Notations . . . . . . . . . . . . 2.1 Quasi-conformal maps and deformations . 2.2 Sheaves and Deligne complexes . . . . . . ˇ 2.3 Cech formalism for generalized coverings 2.4 Evaluation over the fundamental class . . . Construction of the Action . . . . . . . . . . . . .
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Current address: Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA. E-mail:
[email protected]
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A.
E. Aldrovandi, L.A. Takhtajan
3.1 General remarks . . . . . . . . . ˇ 3.2 Setup for regular Cech coverings 3.3 The local Lagrangian cocycle . . 3.4 Other coverings – a dictionary . . Variation and Projective Structures . . . 4.1 Variation . . . . . . . . . . . . . 4.2 Relative projective structures . . 4.3 Geometry of the vertical variation Appendix . . . . . . . . . . . . . . . . . A.1 Cones . . . . . . . . . . . . . . A.2 Fundamental class . . . . . . . .
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1. Introduction This paper is a follow-up to our previous paper [2], where we presented an algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. The aim of the present work is two-fold. First, in light of the renewed interest for Classical Field Theory [13], we present a case study for an action functional whose construction exhibits non-trivial algebraic properties – the action is actually the evaluation of a certain Deligne class. The functional is non-topological, which should be contrasted with cases where methods of homological algebra and algebraic topology were used to construct topological terms [18, 20, 3]. Furthermore, in the recent development of String Theory, there appear dynamical fields of a new geometric content, such as, for example, the B-field. It is very important to find adequate geometric structures to describe these fields and to devise suitable action functionals [19]. Some attempts have been made at introducing the language of gerbes as the proper geometric structure, at least in the lower degrees (where the language itself ˇ makes sense). In this approach, one usually settles for a Cech description relative to some open covering of the underlying manifold. Therefore an added motivation to our work, although we mention gerbes only in passing, was to show the universal nature ˇ of the Cech paradigm for constructing action functionals. By this we mean to develop ˇ a method which works for general Cech resolutions and cohomology with respect to arbitrary coverings, and not just the standard open cover, and which allows to freely change among the coverings. This brings us to the second goal: to describe explicitly the dependence of the chiral action functional on various default choices, which is necessary in order to make our construction in [2] work for arbitrary coverings. In particular, this calls for the following: 1. A detailed analysis of the descent equations with respect to the nerve of the cover, where the use of Deligne complexes becomes crucial. 2. An analysis of the dependence of the chiral action on the choice of the projective structure on the Riemann surface. Recall that the choice of the universal cover for a Riemann surface, made in [2], yields a default choice for the projective structure: the Fuchsian projective structure, provided by the uniformization map. Since the universal Conformal Ward Identity (CWI) determines the chiral action only up to a holomorphic projective connection, the dependence of the chiral action functional on the choice of a projective structure should be compatible with it. Indeed, we prove this for the chiral action “on shell”, i.e., for solutions of the classical equations of motion.
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In order to describe the content of this paper in more detail, we briefly recall the main results in [2]. Let µ be a Beltrami coefficient on C – a smooth bounded function µ with the property µ∞ = supz∈C |µ(z)| < 1 – and let f be a solution of the Beltrami equation fz¯ = µ fz , a self-map f : C → C, unique up to post-composition with a Möbius transformation. The Euclidean version of Polyakov’s action functional for two-dimensional quantum gravity [35] has the form fzz S[f ] = 2π i µz dz ∧ d z¯ , C fz and solves the universal Conformal Ward Identity (∂¯ − µ ∂ − 2 µz )
δW c = µzzz , δµ(z) 12π
where W [µ] is the generating functional for the vacuum chiral conformal block, and c W [µ] = − S[f ] . 96π 2 Here c is the central charge of the theory, and we denoted by δ the variational operator. In [2], we extended Polyakov’s ansatz from C to a compact Riemann surface X of genus g > 1, using the following construction. Consider the universal cover H → X, where H is the upper half-plane, and let µ be a Beltrami coefficient on H, which is a pull-back of a Beltrami coefficient on X (see 2.1 and [2], and also [1, 34] for details). Depending on the extension of µ into the lower half-plane, there exists a unique solution f to the Beltrami equation on H. It is a map f : H → D with the following intertwining property: f ◦ = ˜ ◦ f, where is a Fuchsian group uniformizing the Riemann surface X (it is isomorphic to π1 (X) as an abstract group), and → ˜ is an isomorphism onto a discrete subgroup of PSL2 (C). The domain D = f (H) is diffeomorphic to H and can be made equal to H by choosing an appropriate extension of µ. In this way one gets a deformation map ∼ ˜ f :X∼ = \H → \D = X˜ (which is also denoted by f ) onto a new Riemann surface ˜ X. The de Rham complex on H is a complex of -modules for the obvious pull-back action. The basic 2-form of Polyakov’s ansatz ω[f ] =
fzz µz dz ∧ d z¯ fz
on H is manifestly not invariant under the action of ; this means that regarding ω[f ] as a 0-cochain for with values in 2-forms, its group coboundary is not zero. Nevertheless, ω[f ] can be extended to a cocycle [f ] of total degree 2 living in the double complex Cp,q = C q (, Ap (H)), whose total cohomology coincides with the de Rham cohomology of X. Simple integration for the genus zero case is replaced by the evaluation over a suitable representative of the fundamental class [X] of X, defining S[f ] = [f ], .
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This construction [2] extends the definition of the chiral action to a higher genus Riemann surface X, and the functional S[f ] has the same variational properties as Polyakov’s action on the complex plane. In particular, it solves the universal CWI, the general solution being the sum of W [µ] = −c/96π 2 S[f ] and an arbitrary quadratic differential, holomorphic with respect to the new complex structure on X determined by the Beltrami differential µ. The main advantage of working with the universal cover H is that one can use formulas from the genus zero case and simply “push them onto” the double complex Cp,q = C q (, Ap (H)).1 However, working with the universal cover uses several default choices, as follows: – The groups and ˜ are discrete subgroups of PSL2 (R) and PSL2 (C) respectively, so that local sections to the covering maps H → X and D → X˜ are projective structures ˜ respectively. These projective subordinated to the complex structures of X and X, structures are inherent in the choice of H as a cover, and they do not appear explicitly in the expression for the total cocycle [f ]. – H 3 (X, C) = 0 has to be invoked to close the descent equations leading from ω to the total cocycle . This fact can be interpreted as the vanishing of an obstruction or, in other words, as an integrability property for the problem of choosing integration constants to the last descent equation. An element of arbitrariness is introduced in the explicit computation of by choosing a shift of a C-valued 3-cochain in this equation ˇ to turn it into Cech coboundary. – A specific choice of logarithm branches was made in [2]. The analysis of this construction shows that what we have used were not some specific features of the universal cover H → X, but rather its algebraic properties relative to the double complex Cp,q : the facts that H is contractible, and that is cohomologically trivial with respect to modules of smooth forms on H. These are precisely the properties ˇ of a “good” cover [7], one for which the Cech–de Rham double complex computes cohomology groups for both theories. ˜ defined, say, as the solution As in [2], start with the deformation map f : X → X, of the Beltrami equation on X. It is natural to ask whether it is possible to carry out the same scheme as with H with respect to a different cover of X, for example an ordinary open cover UX = {Ui }i∈I of X, with the requirement that it should allow for a change of covering morphism without changing the formalism. This is achieved by considering, for a given covering map U → X and a sheaf F , or complex of sheaves F • ˇ • (U → X; F • ), ˇ on X, its Cech cohomology Hˇ • (U → X; F ), or hypercohomology H respectively. The framework of the universal cover is retrieved from the observation that ˇ group cohomology for is Cech cohomology for the covering H → X. Our main difference from [2] is the use of the Deligne complex instead of the simpler de Rham complex. In particular, introducing the smooth de Rham sheaves A•X , we work ı
d
d
with the Deligne complex of length 3: Z(3)•D : Z(3) → A0X → A1X → A2X , where def
Z(3) = (2πi)3 Z, and apply the same procedure as before. Namely, we form the double p complex Cp,q = Cˇ q (U → X; Z(3)D ), localize the Polyakov’s 2-form ω to U as an element of degree (3, 0) in this complex2 , and perform the usual descent calculations. The 1 Another procedure would be to find a covariant version of everything on the base X (cf. [27, 40]), but this introduces additional “background” structures with no direct bearing to the complex and algebro-topological structures of X. 2 There is a degree shift caused by the insertion of the integers at degree zero in the Deligne complex.
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latter procedure was first introduced into mathematical physics in [17]. Specifically, we solve for elements θ and $ of degree (2, 1) and (1, 2), respectively, satisfying equations ˇ ˇ = dθ and δθ ˇ = d$, with δ$ ˇ ∈ Z(3), where δˇ is the Cech δω coboundary operator. It is crucial that these equations are solvable due to the vanishing of the tame symbol T X, T X in holomorphic Deligne cohomology. As a result, starting from Polyakov’s 2-form ω[f ] we obtain a cocycle [f ] of total degree 3 in the total complex Tot C•,• . This constitutes the first result of the paper, Proposition 1. Note that it is convenient, for a regular open cover UX , to consider the most general form of the bulk term for the Polyakov’s action, given by adding a smooth projective connection h to the local basic 2-form for genus 0: ω[f ] =
fzz µz dz ∧ d z¯ + 2 µ h dz ∧ d z¯ . fz
Here z is a local coordinate for U ∈ UX , and h a representative in U of a smooth projective connection on X – a smooth coboundary for the usual Schwarzian cocycle relative to the cover UX . The space Q(X) of all such coboundaries is an affine space over the vector space of smooth quadratic differentials on X. On H, the pull-back of a projective connection is a quadratic differential. See Sects. 2.2, 2.3 and 3.2, 3.3 for details. ˇ In Sect. 3.4, we translated the generalized Cech formalism for the universal cover ∼ H → X into group cohomology for = π1 (X), so that Proposition 1 translates into Proposition 3, thus refining the corresponding results in [2]. For the construction of the action functional, we need to evaluate the cocycle [f ] against the fundamental class [X] of a Riemann surface X, which we represent as a cycle in a homological double complex Sp,q = Sp (Nq U ) of singular p-simplices in the q + 1-fold product of U with itself. Using the pairing , between Deligne cocycles and cycles, which is well-defined because dim X = 2 = 3 − 1, we can define S[f ] = [f ], , where is the shift of the cycle so that it has total homological degree 3. Due to the insertion of integers into the Deligne complex, the pairing , is well defined only modulo Z(3), so that the action functional S[f ] is well-defined only modulo Z(3). Using the exponential map z → exp{z/(2π i)2 }, that identifies C/Z(3) with C∗ , one can d log
d
replace the complex Z(3)•D with A∗X −−→ A1X → A2X and resets all degrees by one, so that cocycle would correspond to a cocycle ) of total degree 2. The corresponding pairing , m will be now multiplicative and single-valued, with values in C∗ . As a result, the single-valued functional A[f ] = )[f ], m = exp{S[f ]/(2π i)2 } is the exponential of the action, which is quite natural since we are dealing with an effective action in QFT. Details of this construction are presented in Sects. 2.2 and 2.4. In Sect. 3.3.4 we prove the independence of the functional A[f ] from the choices of logarithm branches, establish its relations with Bloch dilogarithms, and show that it can be considered as C∗ -torsor. The second result of the paper should be understood from the viewpoint of Classical Field Theory. Let B(X) → T (X) be the Earle–Eells principal fibration over the Techmüller space T (X). The total space B(X) of this fibration is the unit ball in the L∞ norm in the space of all smooth Beltrami differentials on X. To every µ ∈ B(X) there
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˜ a solution of the Beltrami equation on corresponds a deformation map f (µ) : X → X, X, uniquely determined by the condition that when pulled back to the universal cover H, it gives a Fuchsian deformation, i.e. f (H) = H. This allows to consider the functional A[f ] as a map A : Q(X) × B(X) → C∗ . When studying the variational problem for the functionals S[f ] and A[f ], we consider the deformation map f as the dynamical field, and the projective connection h as an external field, with the problem to compute the variation with respect to f . Geometrically, these variations are tangent vectors to B(X), and are of two types, depending on whether they deform the complex structure of X or not, i.e., whether the associated Kodaira-Spencer cocycle (see Sect. 4.1) is holomorphically trivial or not. In the former case, the variations correspond to vertical tangent vectors to the Earle–Eells fibration B(X) → T (X), and here we consider only these variations. One needs to show that this variational problem is well-defined even though the action itself is not expressed in terms of a simple integration over X of a 2-form. In “physical” terminology, the bulk term given by the 2-form ω is a multi-valued one, and we prove in Theorem 1 that the variation of the action depends solely on the variation of the bulk term and is a well-defined 2-form on X. We give two proofs of this result. The first one is based on a careful analysis of the descent equations for the variations of all components of the Deligne cocycle [f ]. The second proof, albeit in a sketchy form, shows that this result is, in fact, more general, and depends only on descent properties of the variational bicomplex. Takens’ results [37, 13, 41] are essential in this context. We plan to return to this result with more details in a more general situation, not limited to dimension 2, elsewhere. However, this result holds only thanks to the good gluing properties of the variations, which follow from the triviality of the Kodaira-Spencer cocycle, and this formalism can not be directly applied to the case of general variations. In this respect, we point out that there was an error in the computation of general variation in the universal cover formalism [2]. While a brute-force calculation would achieve the goal, we prefer to defer it until the development of the proper treatment of the variational formalism for multi-valued actions, where variational bicomplex(es) glue in a more complicated way due to the non-vanishing of the deformation class. Returning to the present paper, we also give a geometric interpretation of Theorem 1. It states that at critical points under vertical variations of the dynamical field f , the external field – the smooth projective connection h – is holomorphic with respect to the complex structure on X defined by the deformation map f . In Sect. 4.2, we reformulate this by saying that the space of critical points coincides with the pull-back to B(X) of Hubbard’s universal projective structure P(X) → T (X), studied in [25, 34]. The paper is organized as follows. In Sect. 2 we set up some necessary tools. In ˇ particular, we give a brief tour of Deligne complexes and explain the Cech formalism with respect to a covering U → X. We also present the minimum amount of formulas necessary to perform the evaluation over representatives of the fundamental class [X]. A more in-depth presentation would have led us through a rather long detour from the main line of the paper, therefore we provide it in the appendix, in A.2. Sections 3 and 4 comprise the main body of the paper. After some general remarks in 3.2 and 3.3, we ˇ construct the representative cocycle [f ], using Cech formalism with respect to an open cover. We analyze the changes under redefinition of the logarithm branches and of the trivializing coboundary for the tame symbol TX , TX in 3.3.5. In 3.4, we present our construction in the form suitable for coverings U → X other than the open one UX , and in particular translate everything in terms of U = H. Finally, in 4.1 we discuss the
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variation of the action. After a brief reminder of some basic notions about families of projective structures in 4.2, we present in 4.3 a geometric interpretation of the vertical variation of the action. 2. Preliminaries and Notations 2.1. Quasi-conformal maps and deformations. Let X be compact Riemann surface of genus g > 1. A Riemann surface is called marked, if a system of standard generators of its fundamental group π1 (X) is chosen (up to an inner automorphism). Let T (X) be the Teichmüller space of marked compact Riemann surfaces of genus g, with base point the Riemann surface X. It is defined as the set of equivalence classes of orientation preserving diffeomorphisms ˜ f : X −→ X, where the triples [X, f1 , X˜ 1 ] and [X, f2 , X˜ 2 ] are said to be equivalent if the map f2 ◦f1−1 is homotopic to a conformal mapping of X˜ 1 onto X˜ 2 . It is well-known (see, e.g., [34]), that T (X) is a smooth manifold of real dimension 6g − 6, and it admits a complex structure. ˜ let µ = µ(f ) be the Beltrami differential For any quasi-conformal map f : X → X, for X associated to f . It is a section of T X ⊗ T¯ X ∗ , where T X is the holomorphic tangent bundle of X, satisfying the Beltrami equation ¯ = µ∂f, ∂f where ∂ = ∂/∂z, ∂¯ = ∂/∂ z¯ . Conversely, if a C ∞ Beltrami differential µ has L∞ -norm less than one, µ∞ < 1, then the Beltrami equation is solvable and its solution f is a diffeomorphism. ∗ Denote by A−1,1 (X) = (X, T X ⊗ T¯ X ) the vector space of all smooth Beltrami differentials for X, and by B(X) the open unit ball in A−1,1 (X) with respect to the L∞ norm. It is known that B(X) is the total space of a smooth infinite-dimensional principal fibration over T (X) with structure group G(X), the group of all orientation preserving diffeomorphisms of X isotopic to the identity [14, 34]. Briefly, for every µ ∈ B(X) we lift it to the universal cover H and consider the solution f (µ) of the Beltrami equation on H with the condition that f (H) = H. Such an f exists and is unique up to a postdef
composition with Möbius automorphism of H. If g ∈ G(X), then µg = µ(f ◦ g). This provides an identification between the description of the Teichmüller space as ˜ with fixed X, and as the quotient the space of equivalence classes of the triples [X, f, X] of B(X) by G(X). For any µ ∈ B(X) denote by [µ] the corresponding element in T (X) and by f (µ) : X → Xµ the resulting deformation of X. Though actually Xµ depends only on the class [µ], we suppress this in the notation, and whenever the element µ is fixed, or clear from ˜ as above. the context, we denote Xµ by X, ∗ ⊗q p,q ∗ ) be the space of C ∞ tensors of weight Let A (X) = (X, T X ⊗p ⊗ T¯ X (p, q), with the proviso that we take the tangent bundle whenever either p or q is p,q negative (like A−1,1 (X) for Beltrami differentials). Denote by AX the corresponding sheaves of sections. It is well-known that the operator k,1 ∂¯ µ = ∂¯ − µ∂ − k∂µ : Ak,0 X → AX
(2.1.1)
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¯ is the ∂-operator for the complex structure determined by µ – the pull-back by f of the complex structure on Xµ . This gives rise to the exact sequence ∂¯ µ
0 → A−1,0 (X) −→ A−1,1 (X) → H 1 (Xµ , $µ ) → 0, where $µ is the tangent sheaf of Xµ , which is isomorphic to 0 → Tµ (B(X)/T (X)) → Tµ (B(X)) → T[µ] (T (X)) → 0 , and provides the canonical identification T[µ] (T (X)) = H 1 (Xµ , $µ ) (see, e.g. [34]). p
2.2. Sheaves and Deligne complexes. For any smooth manifold M, we denote by AM the sheaf of smooth complex-valued p-forms on M, and by Ap (M) the corresponding spaces of global sections. Then A0M ≡ AM , the sheaf of smooth complex-valued funcp tions. When M is complex, we denote by M the sheaves of holomorphic p-forms. In 0 particular, M ≡ OM , the structure sheaf. Recall that the hypercohomology groups Hp (M, F • ) of a complex of sheaves F • : F 0 −→ F 1 −→ · · · are defined as the cohomology groups of the total complex of a suitable resolution I •,• ˇ of the complex F • . In practice, one usually takes a Cech resolution relative to some (sufficiently fine) cover UM of M and considers the double complex def Cp,q = Cˇ q (UM , F p ) .
The hypercohomology Hp (M, F • ) is computed by taking H p (Tot C•,• ), with the conˇ vention that the total differential D in degree (p, q) is given by D = d + (−1)p δ, ˇ where d is the differential in the complex F • and δˇ is the differential in the Cech direction. Furthermore, two complexes F • and G• are said to be quasi-isomorphic if there is a morphism F • → G• inducing an isomorphism of their cohomology sheaves: ∼ → H • (G). The standard (spectral sequence) argument implies that their hyperH • (F ) − cohomology groups are the same. We will apply this machinery to the case when the complex F • is a smooth Deligne complex. The use of Deligne complexes is nowadays fairly common, so we just recall the notations and a few basic facts needed in the sequel. It is convenient to use the “algebraic def
geometers’ twist” and set Z(p) = (2π i)p Z. Following [16, 9] we have: Definition 1. Let M be a smooth manifold. The following complex of sheaves ı
d
d
d
p−1
Z(p)•D : Z(p)M −→ AM −→ A1M −→ · · · −→ AM
is called the smooth Deligne complex. The smooth Deligne cohomology groups of M q – denoted by HD (M, Z(p)) – are the hypercohomology groups Hq (M, Z(p)•D ). Remark 1. Z(p) is placed in degree zero and the degree of each term ArM in Z(p)•D is r + 1. The first differential is just the inclusion ı of Z(p) in AX , while d is the usual de Rham differential. The complex is truncated to zero after degree p. An equivalent definition of the Deligne complex is presented in the appendix, cf. A.1.
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The exponential map e : AM → A∗M , f → exp(f/(2π i)p−1 ), induces a quasiisomorphism d log d d p−1 Z(p)•D ∼ = (A∗M −−→ A1M −→ · · · −→ AM )[−1],
where [−1] denotes the operation of shifting a complex one step to the right. Namely, for a complex F • the complex F • [−1] is defined as F [−1]k = F k−1 , with d[−1] = −d. To prove this quasi-isomorphism, observe that the non zero cohomology sheaves p−1 p−2 of the complex Z(p)•D are CM /Z(p)M and AM /dAM , located in degree 1 and p, i
respectively. Next, consider the standard exponential exact sequence 0 −→ Z(p)M −→ e AM −→ A∗M −→ 1, implying the following commutative diagram: ι
d
Z(p)M −−−−→ AM −−−−→ e
d
d
p−1
−d
−d
p−1
A1M −−−−→ · · · −−−−→ AM
−d log
A∗M −−−−→ A1M −−−−→ · · · −−−−→ AM
where the first vertical arrow on the left is the exponential map, and the others are given by multiplication by (−1)k−1 /(2π i)p−1 in degree k. Now it is obvious that the two complexes have the same cohomology sheaves (by identifying C/Z(p) ∼ = C∗ through the exponential map) and therefore have the same hypercohomology groups, up to an q p−1 index shift: HD (M, Z(p)) ∼ = Hq−1 (M, A∗M → A1M → · · · → AM ). Remark 2. In general, the truncation of the Deligne complex Z(p)•D after degree p is fundamental. However, when dim M = p − 1, this truncation is irrelevant. In other • becomes words, when the length of the complex coincides with the dimension, Z(p)D • an augmented de Rham complex: Z(p)M → AM [15]. Therefore the only non trivial cohomology sheaf occurs in degree 1, and Z(p)•D becomes quasi-isomorphic to CM /Z(p)M [−1]. As a result, q HD (M, Z(p)) ∼ = H q−1 (M, C/Z(p)) ∼ = H q−1 (M, C∗ ),
where the latter isomorphism is given by the exponential map. Working out explicitly the first cohomology groups, one gets the following isomor1 (M, Z(1)) ∼ H 0 (M, A∗ ) – the multiplicative group of global invertible phisms: HD = M 2 functions – HD (M, Z(1)) ∼ = H 1 (M, A∗M ) – the group of isomorphism classes of smooth 2 (M, Z(2)) ∼ H1 (M, A∗ → A1 ) – the group of isomorphism line bundles – and HD = M M classes of line bundles with connection. Higher Deligne cohomology groups describe more complicated higher geometric structures – e.g., gerbes and 2-gerbes. When M is complex, there is an entirely analogous definition for the holomorphic Deligne complex: ı
d
d
d
p−1
Z(p)•D,hol : Z(p)M −→ M −→ 1M −→ · · · −→ M , • with the holomorphic Deligne cohomology groups HD ,hol (M, Z(p)) being the hypercohomology groups of the complex Z(p)•D,hol .
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Many of the formal properties of the smooth Deligne complex are also valid in the holomorphic category. In particular, there is the exponential quasi-isomorphism d log d d p−1 Z(p)•D,hol ∼ = (∗M −−→ 1M −→ · · · −→ M )[−1],
since non-trivial cohomology sheaves of these complexes occur only in degrees 1 and p and coincide, which implies the isomorphism in the hypercohomology, so that q p−1 HD,hol (M, Z(p)) ∼ = Hq−1 (M, ∗M → 1M → · · · → M ). When dimC M = p − 1 • the truncation becomes irrelevant and Z(p)D,hol is just Z(p)M → •M . Therefore, thanks to the exactness of the holomorphic de Rham complex, Z(p)•D,hol is also quasiisomorphic to CM /Z(p)M [−1], and we have Hq (M, Z(p)•D,hol ) ∼ = H q−1 (M, C/Z(p)) ∼ = H q−1 (M, C∗ ). In particular, when M is a Riemann surface X and p = 2 we have, for obvious dimensional reasons H3 (X, Z(2)•D,hol ) ∼ = H 2 (X, C∗ ) ∼ = C∗ and H4 (X, Z(2)•D,hol ) ∼ = H 3 (X, C∗ ) = 0. These elementary facts will play a major role in the constructions in Sect. 3. There is a cup product ∪ : Z(p)•D ⊗ Z(q)•D → Z(p + q)•D given by [16, 9]: deg f = 0, f · g f ∪ g = f ∧ dg deg f ≥ 0 and deg g = q, 0 otherwise, r (M, Z(p)) ⊗ H s (M, Z(q)) → H r+s and induced product in cohomology: ∪ : HD D D (M, Z(p + q)). Note that since Deligne cohomology is defined using resolutions of complexes of sheaves, one has to take into account the appropriate sign rules. That is, for two complexes F • and G• one forms the double complexes
Cp,q (F ) = Cˇ q (UX , F p )
and
Cr,s (G) = Cˇ s (UX , Gr )
and defines the cup product ∪ : Cp,q (F ) ⊗ Cr,s (G) −→ Cˇ q+s (UX , F p ⊗ Gr ) ⊂ Cp+r,q+s (F ⊗ G) of two elements {fi0 ,...,iq } ∈ Cp,q (F ) and {gj0 ,...,js } ∈ Cr,s (G) by (−1)qr fi0 ,...,iq ⊗ giq ,iq+1 ,...,iq+s .
(2.2.1)
In this formula, one could replace the ⊗ by any other product F • ⊗ G• → (F • ∪ G• ), in particular by the cup product for Deligne complexes, introduced above. Brylinski and McLaughlin [11] spell out several cup products for the first few degrees representing interesting symbol maps. We will use one of them later, so here we recall its construction. 2 (M, Z(1)) corresponds to the group of smooth line bundles As already observed, HD ˇ ˇ on M. Working out details of the Cech resolution relative to the Cech cover UM =
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2 (M, Z(1)) is represented by the cocycle (f , m ), {Ui }i∈I , one finds that a class in HD ij ij k where fij ∈ (Ui ∩ Uj , A0M ) and mij k ∈ (Ui ∩ Uj ∩ Uk , Z(1)M ) are subject to the relations:
mj kl
fj k − fik + fij = mij k , − mikl − mij l + mij k = 0.
ˇ 1-cocycle with values in invertible functions, as expected. Thus gij = exp fij is a Cech Consider now two line bundles L and L! over M, represented by cocycles (fij , mij k ) and (fij! , m!ij k ), respectively. Their cup product, to be denoted by the “tame” symbol 4 (M, Z(2)), represented by the cocycle L, L! (see, e.g., [11]), is an element of HD (−fij dfj! k , mij k fkl! , mij k m!klp ). A similar interpretation holds for the holomorphic Deligne cohomology. In particular, 2 of holomorphic line bundles on M, and the HD ,hol (M, Z(1)) corresponds to the group 4 cup product of two such line bundles is L, L! ∈ HD ,hol (M, Z(2)). When dim C M = 1, according to the previous remark, the cup product of two holomorphic line bundles is a trivial cocycle: L, L! = 0. In this paper our main emphasis will be on smooth Deligne cohomology in degree 3 (M, Z(3)) is represented by ˇ three. With respect to the Cech cover UM , a class in HD the total cocycle (ωi , aij , fij k , mij kl ), where ωi ∈ (Ui , A2M ), aij ∈ (Ui ∩ Uj , A1M ), fij k ∈ (Ui ∩ Uj ∩ Uk , A0M ), and mij kl ∈ (Ui ∩ Uj ∩ Uk ∩ Ul , Z(3)M ) are subject to the relations: ωj − ωi = d aij , aj k − aik + aij = −dfij k , ˇ ij klp = 0 . ˇ ij kl = mij kl , δm δf
(2.2.2)
3 (M, Z(3)) is the group of isomorphism classes of gerbes on According to [9, 10], HD M, equipped with connective structure described by {aij }, and with curving described by {ωi }.
ˇ 2.3. Cech formalism for generalized coverings. In this section, we provide the necessary ˇ machinery to translate statements and computations carried out in a conventional Cech covering by open sets to other kinds of coverings, such as the universal cover, that will allow to merge results from our previous approach [2] into the present one. This formalism is not yet part of a mathematical physics curriculum, so here we present ˇ the prerequisites necessary for computing Cech cohomology, referring to the standard sources [4, 32, 5] where the theoretical background is explained. Let M be a smooth manifold or topological space. The general idea is to pass from inclusions U 7→ M to general local homeomorphisms U → M which are not necessarily injective. Technically, one fixes a category CM whose objects are spaces étale over M, morphisms are the covering maps, and which is closed with respect to the fiber product of the maps over M, with M being the terminal object in CM . The coverings are surjective families of local
homeomorphism in CM , namely families {fi : Ui → U } of M-maps such that U = i fi (Ui ). In practice, we shall restrict our attention to covering maps of M itself. The key observation is that if Ui 7→ M and Uj 7→ M are inclusions, then Ui ∩ Uj ≡ Ui ×M Uj – the fiber product of maps Ui 7→ M and Uj 7→ M – so
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that the notion of fiber product for covering maps replaces the notion of intersection of open sets. For a covering U → M in CM we obtain an augmented simplicial object [6] Mo
π
o U o
o U ×M U oo
o U ×M U ×M U ooo
...
by considering the nerve N• (U → M). Specifically, for any integer q ≥ 0 we define Nq (U → M) = U ×M · · · ×M U (q+1)-times
where for i = 0, . . . , q the arrows are the maps di : Nq (U → M) → Nq−1 (U → M), forgetting the i th factor in the product. ˇ complex relative to For an abelian sheaf F on M (more precisely, on CM ) the Cech a covering U → M in CM is defined by setting for any q ≥ 0, Cˇ q (U ; F ) = (Nq (U → M), F )
with
δˇ =
q i=0
(−1)i di∗ .
ˇ The ordinary Cech formalism is recovered by considering an open cover UM = {Ui }i∈I of M and the covering i∈I Ui → M, so that in degree q we just get the disjoint union of all q-fold intersections Nq (UM ) = Ui0 ∩ · · · ∩ Uiq , i0 ,...,iq
ˇ and the resulting Cech complex is the standard one. At the other extreme, let U → M be a regular covering map and G = Deck(U/M) the corresponding group of deck transformations acting properly on U on the right. One immediately verifies that · · × G, U × ··· × U ∼ =U ×G × · M M (q+1)−times
q−times
and under this isomorphism the maps di : Nq (U → M) → Nq−1 (U → M) become i=0 (x · g1 , g2 , . . . , gq ) di (x, g1 , . . . , gq ) = (x, g1 , . . . , gi gi+1 , . . . , gq ) i = 1, . . . , q − 1 (x, g , . . . , g ) i = q. 1 q−1 ˇ Hence, the Cech complex with respect to U → M becomes the usual EilenbergMacLane cochain complex on G with values in the G-module F (U ): Cˇ q (U ; F ) ∼ = C q (G; F (U )). ˇ Thus the Cech cohomology of this complex is just the group cohomology of G with values in the G-module F (U ), where the module structure is given by the pull-back action. A particular case of special interest for us is when U is the universal cover of M, so that G = π1 (M).
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The formalism clearly extends to the case where we consider a complex A• of sheaves on M – typically, the de Rham complex. The hypercohomology with respect to a covering U → M will be the cohomology of the total complex of Cˇ q (U ; Ap ). In some favorable cases, one or both spectral sequences associated to the double complex above will degenerate at the first level. Degeneration at the first level of the first spectral sequence, that is, the one associated to the filtration on p, is equivalent to Hˇ q (U → M; Ap ) = 0
for all q > 0.
Since each Ap is assumed to be a sheaf, that is, Ap (M) is the kernel Ap (M)
/ Ap (U )
// Ap (U × U ) , M
p the cohomology of the total complex Cˇ q (U ; Ap ) equals HdR (M). On the other hand, the degeneration of the other spectral sequence (at the same level) means the complex A• is a resolution of some sheaf F , so that the total cohomology equals Hˇ p (U → M; F ). Therefore, when both of these cases are realized, we have a ˇ Cech-de Rham type situation [7], that is p
Hp (U → M; A• ) ∼ = Hˇ p (U → M; F ) ∼ = HdR (M). ˇ The obvious example ofthis situation is the Cech–de Rham double complex relative to the ordinary cover i∈I Ui , where the above isomorphism gives the usual de p Rham theorem: Hˇ p (M, C) ∼ = HdR (M). Another example of utmost importance is the universal cover H → X of a Riemann surface X of genus g > 1. Since there exist π1 (X)p p equivariant partitions of unity [26], the sheaves AH are acyclic: H q (π1 (X), AH ) = 0 for q > 0 and all p. Moreover, since H is contractible, the de Rham complex A• (H) is obviously acyclic in dimension greater than zero, and as a result we have the isomorphism3 p
H p (π1 (X), C) ∼ = HdR (M). 2.4. Evaluation over the fundamental class. For the construction of the action functional we need to evaluate Deligne cohomology classes against the fundamental class [X] of X, which we need to represent as a cycle in a suitable homological double complex – in ˇ a way analogous to the use of Cech resolutions to compute the hypercohomology. The aim of this section is to introduce the minimum set of tools necessary to describe the homological (double) complex and to perform the evaluation, relegating all technical details to the appendix. There, we construct an explicit representative of [X] with respect to a covering U → X by mirroring the cohomology computations done in 3.3. The computations are explicit enough that the reader who is only interested in the formulas for can read A.2 directly. Also, the reader interested only in the construction of the local action cocycle can safely proceed to Sect. 3. As usual, whenever we mention facts that are not specific to X being a Riemann or topological surface, we use the notation M to denote a general smooth manifold or topological space with covering U → M. 3 See also [2] for a simple-minded proof without spectral sequences.
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2.4.1. Consider the double complex Sp,q = Sp (Nq (U → M)), where N• (U → M) is the nerve of the covering U → M and S• is the singular simplices functor, i.e., Sp (M) is the set of continuous maps 9p → M, where 9p is the standard simplex. For every p ≥ 0, the covering map U → M induces a corresponding map : : Sp,0 = Sp (U ) → Sp (M) between simplices – the augmentation map. The double complex S•,• has two boundary operators: the usual boundary operator on singular chains, ∂ ! : Sp,q → Sp−1,q , and the boundary operator ∂ !! : Sp,q → Sp,q−1 induced by the face maps of the nerve: ∂ !! = (−1)i di , where di : Nq (U ) → Nq−1 (U ) and di is the induced map on singular chains. As usual, we have the simple complex Tot S with total differential ∂ = ∂ ! + (−1)p ∂ !! on Sp,q . ˇ If U is the ordinary Cech covering UM = i∈I Ui , then Sp (Ui0 ∩ · · · ∩ Uiq ). Sp (Nq (UM )) = i0 ,...,iq
If, on the other hand, U is a regular covering space with G as a group of deck transformations, then Sp (U ) is a G-module with G-action given by translation of simplices. It follows that Sp (Nq (U )), for q > 0, consists of simplices into U parameterized by q-tuples of elements in G. Taking into account the expression for the face maps di , computed in 2.3, we get Sp (Nq (U )) = Sp (U ) ⊗ZG Bq (G), where B• (G) is the bar resolution [28] and ZG is the integral group ring of G. Hence, for any p, the ∂ !! -homology is just the group homology Hq (Sp (N• (U ))) = Hq (G; Sp (U )). We are interested in the situation when S•,• has no homology with respect to the second index, except in degree zero, namely we want ∼ Sp (M) q = 0 Hq (Sp (N• (U → M))) = 0 q > 0, for the ∂ !! homology. In this case we say that Sp,• resolves Sp (M) and one has the isomorphism H• (M, Z) ≡ H• (S• (M)) ∼ = H• (Tot S•,• ). This isomorphism is induced by the augmentation map : : Tot S → S• (M), which assigns to any chain of total degree n in Tot S the chain :( n,0 ), where n,0 is the component in Sn,0 . It is easy to see that this map is a chain map, it sends cycles into cycles and induces the above isomorphism. Details can be found, e.g., in [28]. 4 ˇ Observe that this situation is realized for both the examples of an open Cech cover and of a regular covering U → M (cf. the appendix). For completeness, in the appendix we briefly analyze the implications of the requirement that the double complex S•,• is acyclic with respect to the first index, and their relations with good covers. 4 A detailed calculation along these lines can also be found in the appendix of [2].
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2.4.2. For a topological manifold M of dimension n, we need to represent [M] with a total cycle of degree n in Tot S•,• . It has the form = 0 +
n
(−1)
k−1
l=0 (n−l)
k ,
k=1
where k ∈ Sn−k,k and ∂ ! 0 = ∂ !! 1 , . . . , ∂ ! k−1 = ∂ !! k , . . . , ∂ ! n = 0. The choice of signs ensures ∂ = 0, where ∂ is the total differential in Tot S•,• . By definition, is a “lift” of M considered as a chain in Sn (M), i.e. :( 0 ) = M, where M = i σi for a suitable collection of singular simplices σi ∈ Sn (M). The existence of the elements 1 , . . . , n follows from the ∂ !! -exactness assumption and the fact that 0 lifts M. Indeed, we have 0 = ∂M = ∂:( 0 ) = :(∂ ! 0 ), so that there exists 1 ∈ Sn−1,1 such that ∂ ! 0 = ∂ !! 1 , and so on. Specializing to the case when M ≡ X is a Riemann surface, the representative of the fundamental class [X] is the cycle = 0 + 1 − 2 , with components k ∈ S2−k,k satisfying ∂ ! 0 = ∂ !! 1 , ∂ ! 1 = ∂ !! 2 , and ∂ !! 0 = ∂ ! 2 = 0. This cycle is explicitly ˇ constructed in the appendix for the case of an ordinary Cech cover U = UX and in [2] for the case of the universal cover H → X. ˇ Here we present the basic formulas for the Cech case, which also gives the flavor of the general procedure which carries over to the other coverings unchanged. Following [21, 38], introduce the symbol 9i0 ,...,iq to denote the (q + 1)-fold intersection thought of as a generator in Sp,q , so that a generic element can be written in the form: σ = σi0 ...iq · 9i0 ...iq , i0 ...iq
where σi0 ...iq are are p-simplices for Ui0 ∩ · · · ∩ Uiq , i.e. continuous maps 9p → Ui0 ∩ · · · ∩ Uiq . It is immediate to verify that !!
∂ 9i0 ...iq =
q j =0
(−1)j 9i0 ,...,ij ,...,iq ,
where thesign denotes omission. Then q
∂ !! σ =
(−1)k σi0 ,..., j
,...,iq
↑ k-th
i0 ,...,iq−1 k=0 j
· 9i0 ,...,iq−1 ,
where the summation goes over ordered sets of indices (it is assumed that I is an ordered set). Thus with the convention that i0 ,...,iq is the sum over sets of indices {i0 , . . . , iq } with i0 ≤ · · · ≤ iq , we can rewrite the last equation as !!
∂ σ =
q k=0
(−1)k
i0 ,..., j ,...,iq ↑ k-th
σi0 ,...,j,...,iq · 9i0 ,...,iq−1 .
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Now, consider the problem of constructing the total cycle = 0 + 1 − 2 representing [X]. Representing the components i as: σi · 9 i , 1 = σij · 9ij , 2 = σij k · 9ij k , 0 = i
ij
ij k
we first construct 0 as follows. Starting from the nerve of the cover UX consider a triangulation of X by U-small simplices, i.e. each simplex comprising the triangulation has support in some open set Ui belonging to the cover (cf. the appendix for the detailed procedure). Then X = i σi , where each chain σi is a sum of simplices whose support is contained in Ui for each i, and one immediately writes 0 = i σi · 9i . The other components are determined by the ∂ !! -exactness condition of the complex. Namely, from the above expression and ∂ ! 0 = ∂ !! 1 , ∂ ! 1 = ∂ !! 2 one gets the equations ! ∂ σi · 9 i = σj i − σij · 9i , i
i,j
∂ ! σij · 9ij =
i,j
k,i,j
i,j
σkij −
i,k,j
σikj +
σij k · 9ij k
i,j,k
for components σij and σij k . Explicit expression for these components in terms of the barycentric decomposition is given in the appendix. 2.4.3. In order to discuss the evaluation pairing, we need to address the issue of the index shift in the Deligne complex. One way is to explicitly use the exponential map described in 2.2 to revert the indexing to the familiar form without a shift, at the cost of introducing an explicit multiplicative structure via the exponential. Another way is to introduce an ad hoc index shift in homology to mirror the one in the Deligne complex, i.e. to consider singular q-simplices to be of homological degree q + 1. The resulting pairing will be additive, but only defined mod Z(p). The two approaches are in the end the same. We start with the second approach. Let (K• , ∂) be a homological complex. The def
canonical way to shift it is to introduce K[1]• = K•−1 , with ∂[1] = −∂, cf. [28]. We require instead that the new differential be simply ∂, while retaining the index shift. Thus we replace Sr,s = Sr (Ns (U → M)) by the new double complex
Sr,s = Sr−1 (Ns (U → M)),
with differential ∂ = ∂ ! + (−1)r ∂ !! , where ∂ ! is the usual singular boundary, as before. If = ( 0 , . . . , q ), with k ∈ Sq−k,k , is a q-chain in Tot S•,• , then it maps to the (q + 1)-chain = ((−1)q 0 , . . . , (−1)q−k k , . . . , q , 0) in Tot S•,• , and is a cycle if and only if is a cycle. ˇ Let C•,• be a Cech resolution of the Deligne complex Z(p)•D with respect to the covering U → M. The pairing between Cr,s and Sr,s is defined as follows (cf. [21, 38]). To the pair (φ, σ ), where φ is a collection {φ i0 ,...,is } of (r−1)-forms on Ns (U → M) for r > 0, or integers Z(p) for r = 0 and σ = σi0 ,...,is · 9i0 ,...,is ∈ Sr,s we assign i0 ,...,is σi0 ,...,is φi0 ,...,is r > 0 (2.4.1) φ, σ = 0 r = 0.
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To extend this pairing to Tot C•,• and Tot S•,• , let σ = (σ0 , σ1 , · · · , σn−1 , 0), with σk ∈ Sn−k,k , and > = (φ0 , φ1 , . . . , φn ), with φk ∈ Cn−k,k . Then we define >, σ =
n−1
φk , σk ,
(2.4.2)
k=0
where, φk = 0 for all k < n − p, if, of course, n > p. Note that so far the pairing was defined to have values in C. However, the fundamental fact is that away from the truncation degree, i.e. when the total degree n is strictly less than p, and therefore the form degree is strictly less than p − 1, the total differentials D and ∂ are transpose to each other modulo Z(p): D>, = >, ∂
mod Z(p)
(2.4.3)
for all > ∈ Tot C•,• , ∈ Tot S•,• . This readily follows from the very definition of the Deligne complex. Equation (2.4.3) means that the pairing , , considered modulo Z(p), defines a pairing between H • (Tot C•,• ) and H• (Tot S•,• ) away from the truncation degree p − 1. Formula (2.4.3) would not hold for degrees bigger than or equal to p − 1, unless dim M = p − 1 – the case where the truncation becomes unimportant. This is the situation we will be interested in in Sect. 3. Therefore in this case the pairing (2.4.2) descends to the corresponding homology and cohomology groups and is non degenerate. It defines a pairing between H • (Tot C•,• ) and H• (Tot S•,• ) which we continue to denote by , . Let us show how these formulas work in the case of a Riemann surface X and a Deligne cocycle = (ωi , aij , fij k , mij kl ) of total degree 3. (Recall that the individual elements are subject to the relations (2.2.2).) Let = ( 0 , 1 , − 2 ) be a representative in Tot S•,• of the fundamental class [X]. Then the corresponding element in the shifted complex will be
= ( 0 , − 1 , − 2 , 0).
Omitting the indices, the evaluation of the class of over [X] will be computed by the expression , = ω, 0 − a, 1 − f, 2 ,
(2.4.4)
where each term in (2.4.4) should be expanded according to (2.4.1). This evaluation takes its values in C/Z(3) and does not depend on the representative cocycle of the 3 (M, Z(3)). Deligne cohomology class [] ∈ HD Another way to define the pairing is to use explicitly the quasi-isomorphism d log d d p−1 Z(p)•D ∼ = A∗M −−→ A1M −→ · · · −→ AM [−1], induced by the exponential map (see 2.2). In this way a cocycle representing a class of k (M, Z(p)) becomes a cocycle of degree k − 1 in the double complex degree k in HD p−1 Cˇ • U → M, A∗M → A1M → · · · → AM .
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In particular, if = (ωi , aij , fij k , mij kl ), subject to the relations (2.2.2), is a cocycle of total degree 3 in Cˇ • (U → M, Z(3)•D ) representing a Deligne class of total degree 3, the element )=
fij k 1 1 ω , − a , exp i ij (2πi)2 (2π i)2 (2π i)2
is the corresponding cocycle of total degree 2. As in the previous discussion, we will consider only the case when dim M = p − 1, where p is the length of the Deligne complex, so that the truncation becomes irrelevant. ˜ •,• the double complex Cˇ • (U → M, A∗ → A1 → · · · → An ), with Denote by C M M M n = p − 1. ˜ r,s and Sr,s which assigns to the pair Then there exists a natural pairing between C (ψ, c) the evaluation of the r-form ψ over a chain c = σi0 ,...,is ·9i0 ,...,is ∈ Sr (Ns (U → M)): ψ, c =
σi0 ,...,is
ψi0 ,...,is ,
with the understanding that for r = 0 this is just the pointwise evaluation of an invertible function, defined through the exponential map. To define a multiplicative pair˜ •,• and Tot S•,• , let C = (c0 , c1 , . . . , cn ), with ci ∈ Sn−i,i , and ing between Tot C ˜ n−i,i . Then we define ) = (ψ0 , ψ1 , . . . , ψn ), with ψi ∈ C ), Cm =
n−1
exp(ψi , ci ) · ψn , cn ∈ C∗ .
(2.4.5)
i=0
˜ •,• and S•,• , the total differentials By the very construction of the double complexes C D and ∂ are transpose to each other, namely D), C m = ), ∂Cm ˜ •,• , C ∈ S•,• . The pairing (2.4.5) descends to the corresponding homology for all ) ∈ C ˜ •,• ) and cohomology groups and is non-degenerate. It defines a pairing between H • (Tot C and H• (Tot S•,• ) which we continue to denote by , m . It is easy to describe the relation between the multiplicative pairing , m and the C/Z(p)-valued additive pairing introduced earlier. Namely, let > ∈ Tot C•,• ), C ∈ S•,• ˜ •,• . Then we have and let ) be the corresponding element in C ), C m = exp{>, C /(2π i)p−1 }. It what follows we will use freely both forms of the pairing, multiplicative and additive, depending on the context.
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3. Construction of the Action 3.1. General remarks. The next sections will be devoted to the detailed construction of the action functional – or rather its exponential – by specifying the following: (a) A resolution of the Deligne complex Z(3)•D . 3 (X, Z(3)) that “starts” from a collection {ω [f ]} (b) A representative for a class in HD i i∈I of “local Lagrangians densities” – top forms on X – defined with respect to a given covering UX = {Ui }i∈I of X. The latter data come from Polyakov’s ansatz, with dynamical field given by a deformation map f : X → X˜ and with external field given by a smooth projective connection of X. Before doing so, we make some remarks of general character. – The Deligne complex Z(3)•D is especially convenient for treating various logarithmic terms produced in descent calculations, while keeping additivity. – The “local Lagrangian” [f ] appears as a total cocycle of total degree 3 in the Deligne complex Z(3)•D , and we define the action functional by evaluating this cocycle over the representative of the fundamental class of the Riemann surface X, S[f ] = [f ], , described in 2.4. According to 2.4, S[f ] ∈ C/Z(3), so that the functional A[f ] = [f ], m = exp{S[f ]/(2π i)2 } is the exponential of the action. – A similar approach was taken in [3, 20] in order to describe certain topological terms arising in two-dimensional quantum field theories. In our case the field is a deformation f : X → X˜ and the procedure differs in that we construct the whole representing cocycle starting from one end of the descent staircase. – According to [18, 13] the exponentials of action functionals should be more properly regarded as C∗ -torsors rather than numbers. This is most apparent when dealing with manifolds with boundaries. A similar situation occurs in our case, when X is a compact Riemann surface: the definition of the local Lagrangian cocycle [f ] depends on the trivialization of the tame symbol (T X, T X], described by an (f -independent) element of H 2 (X, C∗ ) ∼ = C∗ . As a result, the multiplicative action functional A[f ] is a C∗ torsor. – The action functional A[f ], defined through hypercohomology admits the following 3 (X, Z(3)) classifies geometric interpretation. According to Sect. 2.2, the group HD isomorphism classes of gerbes equipped with connective structure and curving [10, 9]. Since dim X = 2, these are necessarily flat, therefore they are classified by their 3 (X, Z(3)) ∼ H 2 (X, C∗ ). Thus A[f ] can be interholonomy via the isomorphism HD = preted as the holonomy of an appropriate higher algebraic structure. ˇ 3.2. Setup for regular Cech coverings. Let UX = {Ui }i∈I be an open cover of X, which we assume to be a good cover, i.e. all nonempty intersections Ui0 ,...,ip = Ui0 ∩ · · · ∩ Uip ˇ are contractible. Therefore, we are in a Cech–de Rham situation [7, 38], and the double p,q def ˇ q p • complex CD = C (UX , Z(3)D ) computes HD (X, Z(3)). Let {zi : Ui → C}i∈I be holomorphic coordinates for the complex structure of X, and let zij : zj (Ui ∩ Uj ) → zi (Ui ∩ Uj ) be coordinate change functions: zi = zij ◦ zj on Ui ∩ Uj .
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Remark 3. One could also use coordinate functions with P1 instead of C. More generally, for Ui0 ,...,iq there are holomorphic coordinates zi0 , . . . , ziq with p ˇ zik = zik ik+1 (zik+1 ), k = 0, . . . , q − 1. If φ ∈ Cˇ q−1 (UX , AX ) is a Cech cochain, i.e. φ = {φi0 ,...,iq−1 }, where the components φi0 ,...,iq−1 are p forms on Ui0 ,...,iq−1 its ˇ Cech differential is defined as ˇ i0 ,...,iq = δφ
q−1 k=0
(−1)k φi0 ,...,iˆk ,...,iq + (−1)q (ziq−1 iq )∗ φi0 ,...,iq−1 .
ˇ It is understood that each component φi0 ,...,iq−1 of a Cech cochain φ is expressed in the coordinate ziq−1 , i.e. the one determined by the last index, and we will use this convention throughout the paper. ˜ denote by VX = {Vi }i∈I , where Vi = Given a quasi-conformal map f : X → X, ˜ Let {wi : Vi → C}i∈I be holomorphic f (Ui ), the corresponding good open cover for X. ˜ and let wij : wj (Vi ∩ Vj ) → wi (Vi ∩ Vj ) coordinates for the complex structure of X, be the corresponding coordinate change functions: wi = wij ◦ wj on Vi ∩ Vj . Let fi = wi ◦ f |Ui ◦ zi−1 , i ∈ I , be local representatives of the map f , satisfying the transformation law fi ◦ zij = wij ◦ fj .
(3.2.1)
def ¯ i def Denote ∂fi = ∂fi /∂zi and ∂¯ i fi ≡ ∂f = ∂fi /∂ z¯ i , and introduce local representa¯ i /∂fi . tives of the Beltrami differential µ by µi = ∂f It follows from (3.2.1) that ! ! ∂fi ◦ zij · zij = wij ◦ fj · ∂fj ,
¯ i ◦ zij · z! ∂f ij
=
! wij
¯ j, ◦ fj · ∂f
(3.2.2) (3.2.3)
and µi ◦ zij ·
! zij ! zij
= µj .
(3.2.4)
def
! ◦ z = dz /dz are transition functions for the holomorphic tanSince ξij = zij j i j def ! ◦ w = dw /dw are the transition functions for gent bundle T X, and ξ˜ij = wij j i j ˜ or ˜ T X, it follows from (3.2.2) that ∂f is a section of the bundle T ∗ X ⊗ f −1 T X, 1,0 −1 −1 ˜ ˜ ˜ ∂f ∈ A (f T X). Here f T X is the pull-back of the tangent bundle over X by f . ¯ ∈ A0,1 (f −1 T X). ˜ Similarly, ∂f In the C ∞ category f −1 T X˜ ∼ = T X, so that T ∗ X⊗f −1 T X˜ is isomorphic to the trivial bundle. This is also implied directly by the transition formula (3.2.2), since ∂fi % = 0, f ˜ being a diffeomorphism. Thus ∂f is an explicit trivializing section for T ∗ X ⊗ f −1 T X, ∗ −1 ˜ that establishes the isomorphism between T X ⊗ f T X and the trivial line bundle. Introducing representatives cij k and c˜ij k for the first Chern classes c1 (T X) = c1 (T˜ X), we have ˇ cij k = δ({log z··! })ij k , (3.2.5a)
ˇ w··! ◦ f· })ij k , c˜ij k = δ({log bij =
! log wij
! ◦ fj − log zij
(3.2.5b) − log ∂fi ◦ zij + log ∂fj ,
(3.2.5c)
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ˇ ·· })ij k = c˜ij k − cij k . All the numbers bij , cij k and c˜ij k are in Z(1). and, obviously, δ({b Although one can get c˜ij k = cij k and bij = 0 through a suitable redefinition of the logarithm branches, there is no additional complication (except, perhaps, the notation) in keeping the general situation. 3.3. The local Lagrangian cocycle. In order to construct the action functional, one needs an ansatz for its top degree part. Following [2], we promote the standard Polyakov’s chiral action5 , ωi =
∂ 2 fi ∂µi dzi ∧ d z¯ i + 2µi hi dzi ∧ d z¯ i , ∂fi
(3.3.1)
√ ∞ to an element {2π −1ωi }i∈I ∈ C3,0 D . Here h = {hi }i∈I is a C coboundary for the Schwarzian cocycle 2 d 3 zi 3 d 2 zi {zi , zj } = − , 2 dzj2 dzj3 relative to the cover UX (see [23]). In other words, it satisfies the following transformation law: ! 2 {zi , zj } = hj − hi ◦ zij · (zij )
(3.3.2)
on Ui ∩ Uj . Clearly, such an h exists, since the Schwarzian cocycle is already zero in the holomorphic category [23]. The space Q(X) of all such h includes the holomorphic ⊗2 projective connections, and is an affine space over the vector space H 0 (X, (A1,0 X ) ). Let us call such an h a smooth projective connection (even though that we do not relate it to projective structures). Following the usual strategy [17] of descending the staircase in the double complex C•,• D , starting with the 0-cochain {ωi } of 2-forms on X, we find a 1-cochain of 1-forms {θij } and a 2-cochain of functions {$ij k } satisfying ˇ · )ij = dθij , δ(ω ˇ ·· )ij k = d$ij k . δ(θ ˇ = 0 mod Z(2) ensures that the total element Imposing the condition δ$ √ def = 2π −1 {ωi }, {θij }, {−$ij k }, {−mij kl } , ˇ is a cocycle in the total complex. where m = δ$, Solvability of the descent equations is proved in the standard way using the acyclic ˇ property of the good cover UX and Poincaré lemma on differential forms. Namely, δdω = ˇ = dθ and 0 = δdθ ˇ = d δθ ˇ implies δθ ˇ = d$. Finally, from δd$ ˇ ˇ = 0 implies δω = d δ$ p ˇ ∈ Zˇ 3 (UX , CX ). From de Rham theorem Hˇ p (X, C) ∼ 0 one concludes δ$ = HdR (X) it ˇ = 0, after possible rescaling of constants. follows for dimensional reasons that δ$ The foregoing shows that one can get a “minimal” cocycle with the condition mij kl = 0, albeit not in explicit form. However, our goal is to have a cocycle [f ] with “good” 5 More precisely, Polyakov’s chiral action has no second term in (3.3.1), which, in fact, is not necessary in genus zero.
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dependence on the dynamical field f (i.e. with the same variational properties as in the genus zero case). It is most remarkable that such cocycle [f ] can in fact be computed ˇ = 0 mod Z(2). This explicitly, allowing for a geometric interpretation as to why δ$ computation is accomplished in the following steps. ˇ = dθ. We find, using the transformation rules (3.2.2)–(3.2.4), 3.3.1. δω ˇ ij = ωj − ωi δω !! zij ! ! ! ! = d 2 µj ! d z¯ j − log(wij ◦ fj ) + log zij ◦ fj ) d log zij d log ∂fj + log(wij zij + 2µj hj dzj ∧ d z¯ j − 2µi hi dzi ∧ d z¯ i − 2µj {zi , zj } dzj ∧ d z¯ j . (3.3.3) In light of (3.3.2), Eq. (3.3.3) reads ˇ = dθ, δω with θ given by the first two terms on the RHS of (3.3.3), that is, θij = 2µj
!! zij ! zij
! ! d z¯ j − log(wij ◦ fj ) + log zij d log ∂fj ! + log(wij
◦ fj ) d
(3.3.4)
! log zij .
ˇ ˇ . The first term on the RHS of (3.3.4) is a cocycle, as it has the Cech 3.3.2. δθ cup product of two terms which are cocycles themselves. We can ignore it from now on. The term on the second line of (3.3.4) is also cup product, so its coboundary is computed by ˇ ∪ b) = δ(a) ˇ ˇ applying δ(a ∪ b + (−1)deg a a ∪ δ(b). For the remaining term the cocycle is computed directly. The final result is ! ! δˇ θ ij k = − log wij d log wj! k + log zij d log zj! k ! (3.3.5) − (c˜ij k + cij k ) d log ∂fk − d log zij log wj! k ! , + c˜ij k d log zik
where we suppressed the f -dependence. To restore it, notice that on the triple intersection ! ◦ Ui ∩ Uj ∩ Uk everything is evaluated with respect to the coordinate zk , so that log wij ! ! fj |Uk = log wij ◦ fj ◦ zj k ≡ log wij ◦ wj k ◦ fk . We shall use this convention in the sequel, in order to keep some of the expressions less cumbersome. ˇ = d$. Here we are using Deligne tame symbols in holomorphic category, 3.3.3. δθ ˇ = d$ and to check that introduced in 2.2 in order to find $ satisfying the equation δθ ˇδ$ = 0 mod Z(2). ˇ Consider the tame symbol T X, T X , which is represented in Cech cohomology by the element ! ! − log zij d log zj! k , cij k log zkl , cij k cklm ∈ Cˇ 2 (1X ) ⊕ Cˇ 3 (OX ) ⊕ Cˇ 4 (Z(2)X ),
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where {cij k } represents the first Chern class of T X. As we mentioned in Sect. 2.2, H4 (X, Z(2)•D,hol ) ∼ = H 3 (X, C∗ ) = 0, so that the total cocycle representing T X, T X is a coboundary: ! ! d log zj! k , cij k log zkl , cij k cklm = D τij , φij k , nij kl , − log zij where (τij ) ∈ Cˇ 1 (UX , 1X ), (φij k ) ∈ Cˇ 2 (UX , OX ) and (nij kl ) ∈ Cˇ 3 (UX , Z(2)X ). Computing the RHS yields the relations ! ˇ )ij k + dφij k , d log zj! k = (δτ − log zij ! ˇ ij kl + nij kl , cij k log zkl = −(δφ)
cij k cklm
ˇ ij klm . = (δn)
(3.3.6a) (3.3.6b) (3.3.6c)
There is an entirely similar situation for the deformed Riemann Surface X˜ and the ˜ T X˜ , for which we introduce the corresponding objects corresponding symbol T X, ˇ as τ˜ij , φ˜ ij k and n˜ ij kl . Using these results we rewrite δθ ˇ ∗ (τ˜ ))ij k + df ∗ (φ˜ ij k ) − δ(τ ˇ )ij k − dφij k δˇ θ ij k = δ(f ! − (c˜ij k + cij k ) d log ∂fk − d log zij log wj! k ! , + c˜ij k d log zik
where f ∗ (τ˜ij ) and f ∗ (φ˜ ij k ) are pull-backs of forms τ˜ij and φ˜ ij k on X. Now, perform the shift: def θij → θˆij = θij − f ∗ (τ˜ij ) + τij .
This is possible since τij and τ˜ij are holomorphic relative to the respective complex structures, implying dτij = 0 and df ∗ (τ˜ij ) = 0, so that ˇ ij , d θˆij = dθij = δ(ω) without affecting the 2-form part of the action. From now on we assume that θij has been redefined in this way, that is θij = θijold − f ∗ (τ˜ij ) + τij ,
(3.3.7)
where θijold is given by formula (3.3.4), and we can finally put ˇ = d$, δθ with $ij k = f ∗ (φ˜ ij k ) − φij k − (c˜ij k + cij k ) log ∂fk ! ! − log zij log wj! k + c˜ij k log zik .
(3.3.8)
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ˇ 3.3.4. δ$. Using the relations (3.3.6) we compute: ˇ ij kl = n˜ ij kl − nij kl − (c˜ij k + cij k )bkl + cij l c˜j kl − cikl c˜ij k , δ$ ˇ ∈ Cˇ 3 (UX , Z(2)X ). Setting so that δ$ def ˇ mij kl = (δ$) ij kl ,
we can summarize the foregoing in the following Proposition 1. The total cochain √ def = 2π −1 ωi , θij , −$ij k , −mij kl , 3 (X, Z(3)). with ωi given by the Polyakov form (3.3.1), represents a class in HD
Proof. All the preceding computations amount to show that √ ˇ ij k − d$ij k , (δ$) ˇ ij kl − mij kl , (δm) ˇ ij klp D = 2π −1 −ωj + ωi + dθij , (δθ) = 0. Then represents a class since the double complex C•,• D computes the hypercohomology. ) ( Now that we have constructed the Lagrangian cocycle from the Polyakov top form in (3.3.1), we can finally give the Definition 2. Let µ ∈ B(X) be a Beltrami coefficient, f be the associated deformation map, and [f ] be the local Lagrangian cocycle constructed from (3.3.1). The Polyakov action functional on X is given by the evaluation def
S[f ] = [f ], ,
(3.3.9)
over the representative of the fundamental class of X given in 2.4 and in the appendix. Remark 4. As it follows from the definition, Polyakov’s action is well-defined modulo Z(3), so that only its exponential A[f ] = exp{S[f ]/(2π i)2 } is well-defined. It also follows from the definition of the pairing in Sect. 2.4 that the functional A[f ] actually 3 (X, Z(3)) of the local Lagrangian cocycle depends only on the cohomology class in HD [f ]. By construction, the cocycle [f ] depends also on a smooth projective connection h ∈ Q(X), so that the exponential of the action defines the map A : Q(X) × B(X) −→ C∗ , where the dependence on the first factor is that of an external field. Here we analyze the dependence of the action functionals S[f ] and A[f ] on the choice of branches. We also study the trivializing coboundary for the tame the logarithm symbol T X, T X , analyze the dependence of the action on this trivialization, and show that A[f ] should be in fact considered as taking its values in a C∗ -torsor.
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3.3.5. Dependency on logs. Here we prove the following Proposition 2. The functional A[f ] is independent of the choice of the logarithm branches in (3.2.5). Proof. It is sufficient to show that changing the definition of the various logarithm branches in amounts to change it by a coboundary. First, we change these branches, ! ! −→ log zij + kij , log zij ! ! log w −→ log w + k˜ij , ij
ij
log ∂fi −→ log ∂fi + pi , where kij , k˜ij , pi ∈ Z(1). The effect of these changes on the representatives of the Chern classes of T X and T X˜ is bij −→ bij + k˜ij − kij + pj − pi , ˇ ij k , cij k −→ cij k + δ(k) ˜ ij k . ˇ k) c˜ij k −→ c˜ij k + δ( While the term ωi is obviously invariant under these changes, θij and $ij k , by descent theory, transform as follows: θij −→ θij + dψij , ˇ − rij k , $ −→ $ + δψ where ψ ∈ Cˇ 1 (UX , A0X ) and r ∈ Cˇ 2 (UX , C). Note that if rij k ∈ Z(2) for any ij k, then −→ + Dλ, where λ = (0, ψij , rij k ), and we are done. To prove that r ∈ Cˇ 2 (UX , Z(2)), we actually compute the shift for $. First, we explicitly determine ! ψij = −(k˜ij + kij ) log ∂fj + k˜ij log zij .
Next, we explicitly compute the shift of the total cocycle representing T X, T X . This is a straightforward calculation, using relations (3.3.6), with the result: τij −→ τij , φij k −→ φij k − kij log zj! k , ˇ ij k kkl . nij kl −→ nij kl + kij cj kl + cij k kkl + (δk) ˜ T X˜ . Putting everything together, we Similar formulas are valid for the shift of T X, get ˜ ij k pk ˇ ij k + (δˇ k) rij k = (k˜ij + kij )bj k + c˜ij k + cij k + (δk) ˜ ij k kik + k˜j k cij k + k˜ij cij k ∈ Z(2). + kij k˜j k − c˜ij k kik − (δˇ k) (3.3.10) ) (
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3.3.6. A more detailed analysis of the vanishing tame symbol. Here we analyze the condition T X, T X = 0 as an element of H4 (X, Z(2)•D,hol ) in more detail. In particular, we investigate the possibility of putting the trivializing cochains (τij , φij k , nij kl ) and (τ˜ij , φ˜ ij k , n˜ ij kl ) into some specific forms. This analysis is based on the relations (3.3.6), which we rewrite here: ! ˇ )ij k + dφij k , d log zj! k = (δτ − log zij ! ˇ ij kl + nij kl , cij k log zkl = −(δφ)
ˇ ij klm . cij k cklm = (δn) The first equation above calls for the differential equation ! − log zij ◦ zj k d log zj! k = dLij k .
(3.3.11)
Its! solution Lij k (zk ) can be considered as a Bloch dilogarithm associated to the symbol zij , zj! k , which is the cup-product in Deligne cohomology of the two invertible func! and z! and is a trivial element of H 2 (U , Z(2)) (see [16] for more details). tions zij jk D ij k The consistency condition on quadruple intersections Uij kl is obtained by applying the ˇ Cech coboundary to the differential equation satisfied by Lij k . One gets ! ˇ ij kl + αij kl , cij k log zkl = −(δL)
ˇ coboundwhere αij kl is a C-valued cochain – an integration constant. By taking the Cech ary of the last relation we get ˇ ij klm . cij k cklm = (δα) Therefore, ˇ − n) = 0, δ(α that is, the element α−n is a 3-cocycle. By dimensional reasons, it must be a coboundary, ˇ α = n + δβ, with β being a 2-cochain with values in C. It follows that ! ˇ − β)ij kl + nij kl . = −δ(L cij k log zkl
As a result, we effectively obtained a trivializing cocycle for the tame symbol (T X, T X] which does not include a 1-form: ! ! − log zij d log zj! k , cij k log zkl , cij k cklm = D 0, Lij k , nij kl , where we relabeled L − β → L.
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3.3.7. Relation with C∗ -torsors. Notice that the trivialization of the tame symbol T X, T X is defined up to a cocycle representing an element in H3 (X, Z(2)•D,hol ) ∼ = 2 2 ∗ ∗ ∗ ∼ ∼ H (X, C/Z(2)) = H (X, C ) = C . Thus there is a C -action on the functional A[f ] which simply is the shift of the total trivializing cochain (τij , φij k , nij kl ) by a cocycle representing a class in H3 (X, Z(2)•D,hol ). From this it is clear that, keeping f fixed, the functional A[f ] does not simply take its values in C∗ , but rather in a C∗ -torsor T . From this perspective, choosing a specific total cochain to trivialize the symbol T X, T X ∼
amounts to choosing an isomorphism T → C. The C∗ -action can be described explicitly if we make use ! of the cocycle (0, Lij k , nij kl ), obtained by choosing a dilogarithm Lij k for the symbol zij , zj! k . Namely, as it follows from the discussion in the previous section, we can add to Lij k a cocycle (βij k , pij kl ) representing an element in ı H3 (X, Z(2) → C) ∼ = H 2 (X, C∗ ).
ˇ = p ∈ Z(2). Note that, by definition, δβ Since the action functional is defined using trivialization of two tame symbols, ˜ T X˜ , the above argument should be applied to both cochains T X, T X and T X, (τij , φij k , nij kl ) and (τ˜ij , φ˜ ij k , n˜ ij kl ), so that we have in fact two C∗ -actions. From a Teichmüller theory point of view, these two actions refer to very different structures. One is defined in terms of the complex structure X which is fixed throughout (a base point in Teichmüller space), while the other is relative to the f -dependent complex ˜ The latter action depends on the dynamical field f . structure X. Thus it is appropriate to speak of a (C∗ , C∗ )-action, in the sense that the space T where the action takes its values carries two simultaneous (and compatible) C∗ -actions. 3.4. Other coverings – a dictionary. In this section we set up a dictionary connecting ˇ the generalized Cech formalism developed in 2.3 and 2.4 with the formalism used in [2] for the universal cover of X. Besides comparing the two formalisms, by applying the dictionary to the formulas in 3.3, we also clarify the explicit form of the Lagrangian cocycle constructed in [2]. Specifically, we treat the “integration constants” arising from solving the descent equations via Deligne complexes and analyze explicit dependence of the action functional on background projective structures. 3.4.1. Start from the universal cover U → X, which we specify as the upper half-plane H. Then Deck(H/X) ∼ = π1 (X) ∼ = , a finitely-generated, purely hyperbolic Fuchsian group (a discrete subgroup of PSL2 (R)), uniformizing the Riemann surface X. The group acts on H by Möbius transformations. Geometric objects on X correspond to -equivariant objects on H: a tensor φ ∈ Ap,q (X) corresponds to an automorphic form φ for of weight (2p, 2q), i.e. a function (indicated by the same name) φ : H → C such that φ ◦ γ · (γ ! )p (γ ! )q = φ, γ ∈ . Clearly, an automorphic form is just a zero cocycle on with values in Ap,q (H). Examples of automorphic forms of geometric origin are provided by Beltrami differentials on X, that correspond to forms of weight (−2, 2), by abelian differentials on X – global sections of X – that correspond to holomorphic forms of weight (2, 0), and by quadratic
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differentials on X – global sections of ⊗2 X – that correspond to holomorphic forms of weight (4, 0). The deformation map f is realized as a quasi-conformal map f : H −→ D satisfying on H the Beltrami equation fz¯ = µfz ,
(3.4.1)
where µ is a Beltrami differential for on H such that µ∞ < 1. The Beltrami equation on H should be supplemented by boundary conditions that guarantee the following: 1. D = f (H) is a quasi-disk, i.e. a domain in P1 bounded by a closed Jordan curve and analytically isomorphic to H; 2. ˜ = f ◦ ◦ f −1 ⊂ PSL2 (C) is a discrete subgroup, isomorphic to as an abstract group, acting on D, i.e. a so-called quasi-fuchsian group. The isomorphism → ˜ intertwines f . These boundary conditions are specified by extending µ to the whole complex plane, where the Beltrami equation has a unique solution up to a post-composition with Möbius transformation [1, 34]. The following two types are of particular importance. def
¯ µ(z) = µ(¯z) for z ∈ H. (a) Extension of µ by reflection to the lower half plane H: Then D = H and ˜ is also a Fuchsian group. (b) Extension of µ by setting µ(z) = 0 for z ∈ H. In this case D is a quasi-disc and the dependence of the mapping f on µ is holomorphic. The formalism developed below will be independent of a particular boundary condition chosen. 3.4.2. Here we address a minor normalization problem caused by the fact that the action of PSL2 (R) – and therefore of and ˜ – by Möbius transformations is on the left instead of on the right, as we assumed in 2.3. Assuming a right action yields all the standard formulas in group cohomology. On the other hand, a left action of is more convenient in view of the fact that H itself is the quotient of a principal fibration: H ∼ = PSL2 (R)/ SO(2).6 As a result, the surface itself is presented as a double coset space: X ∼ = \ PSL2 (R)/ SO(2). For a left action G × U → U for a G-space U → M there is the isomorphism U ×M · · · ×M U ∼ = Gq × U, q+1
sending the q-tuple (x0 , . . . , xq ) to the tuple (g1 , . . . , gq , x) such that (x0 , . . . , xq ) = (g1 . . . gq x, g2 . . . gq x, . . . , gq x, x). This arrangement makes the face maps di appear in backward order, that is d0 (g1 , . . . , gq , x) = (g2 , . . . , gq , x) . . . dq (g1 , . . . , gq , x) = (g1 , . . . , gq−1 , gq x). 6 In general, we prefer to consider right principal fibrations.
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As a result, the action on the coefficients would be on the right and the coboundary operator δ in group cohomology should actually be read from right to left, as in (δφ)g1 ,...,gq = φg2 ,...,gq +
q−1 i=1
(−1)i φg1 ,...,gi gi+1 ,...,gq q
(3.4.2)
∗
+ (−1) gq φg1 ,...,gq−1 , for φ a (q − 1) cochain. Observe that the pull-back action on the coefficients is a right one. The familiar formulas in group cohomology can be retrieved by turning the left action into a right one using the standard trick x · g = g −1 x, def
g ∈ G, x ∈ U,
which at the level of nerves amounts to performing the swap (g1 , . . . , gq , x) → (x, gq−1 , . . . , g1−1 ) in degree q. It follows that one has to evaluate all cochains over inverses of group elements. This is the convention we followed in [2]. On the other hand, given the action of on H as a left one, keeping the non stanˇ dard form (3.4.2) parallels more closely the Cech framework if we consider the pair (γ (z), z) ∈ H×X H, for z ∈ H and γ ∈ , as a change of coordinates, much like the pair (zi , zj ) ∈ Ui ×X Uj ≡ Ui ∩ Uj with zi = zij (zj ). More generally, for Ui0 ∩ · · · ∩ Uiq there are coordinates zi0 , . . . , ziq with zik = zik ik+1 (zik+1 ), k = 0, . . . , q − 1, and if φ ∈ Cˇ q−1 (UX , Ap ) then we have ˇ i0 ,...,iq = δφ
q−1 k=0
(−1)k φi0 ,...,iˆk ,...,iq + (−1)q (ziq−1 iq )∗ φi0 ,...,iq−1 ,
(3.4.3)
where the convention is that each component is expressed in the coordinate determined by the last index. This is the formula we used when performing explicit computations ˇ with Cech cochains for the calculation of the local Lagrangian cocycle. Thus (3.4.3) becomes formally equal to (3.4.2) when we interpret the last pull-back by gq as the restriction isomorphism expressing everything in terms of the last coordinate. 3.4.3. The translation of the constructions in 3.2 and 3.3 to the upper-half plane is now done according to the following table: ˇ Cech: UX Ui0 ∩ · · · ∩ Uin zi0 , . . . , zin zik = zik ik+1 (zik+1 ), k = 0, . . . , n − 1 p q φi0 ,...,in (zin )dzin d z¯ in (3.4.3)
zk−1
Upper-half plane H n × H γ1 , . . . , γn , z = γk (zk ), k = 1, . . . , n, zn = z φγ1 ,...,γn (z)dzp d z¯ q (3.4.2)
Similar provisions of course relate the deformed coordinates wi and elements of the ˜ Any construction explicitly involving the map f must take into deformed group . account the equivariance property f ◦ γ = γ˜ ◦ f for any γ ∈ , where γ˜ is the
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˜ We have relations entirely similar to corresponding element in the deformed group . (3.2.2) and (3.2.3) which can be found in [2]; for example γ˜ ! ◦ f fz = fz ◦ γ . γ!
(3.4.4)
ˇ In order to handle the logarithm of (3.4.4) in the same way as we just did in the Cech case ! ! (see (3.2.5)) we depart from [2]. The problem is to relate log(γ1 γ2 ) and log γ1 ◦γ2 +log γ2! ˜ Instead of directly analyzing the branch-cuts for any γ1 , γ2 ∈ , and similarly for . (thus introducing an element of explicit dependence on the choice of the branches) we set cγ1 ,γ2 = log γ2! − log(γ1 γ2 )! + log γ1! ◦ γ2 ,
(3.4.5a)
c˜γ˜1 ,γ˜2 =
(3.4.5b)
log γ˜2! !
− log(γ˜1 γ˜2 )
!
!
+ log γ˜1!
◦ γ˜2 ,
bγ = log γ˜ ◦ f − log γ − log fz ◦ γ + log fz .
(3.4.5c)
The numbers cγ1 ,γ2 , c˜γ˜1 ,γ˜2 and bγ belong to Z(1), and c, c˜ are cocycles with c˜ = c + δb. Since γ ! is the automorphy factor for T X, the geometric interpretation is that again c represents c1 (T X) [24]. Alternatively, c represents the Euler class of the S 1 -bundle \ PSL2 (R) → X ([33, 39, 8], see also [30]). Indeed, the first of Eqs. (3.4.5) can be written in terms of rotation numbers: cγ1 ,γ2 = −2 w(γ2 ) − w(γ1 γ2 ) + w(γ1 ) ◦ γ2 , where w ac db = arg(cz + d). More precisely, this is the Euler class of the RP1 -bundle obtained by letting PSL2 (R) act on the real projective line realized as the boundary of H (see [39] for details). Again, a similar discussion holds for ˜ with the obvious changes. ˇ As was shown in Sect. 2.3, Cech cohomology with respect to the cover H → X is the same as group cohomology of π1 (X) ∼ = with values in the appropriate coefficients. Also it was noted there that H → X is a good covering acyclic for fine sheaves, so p that H p (π1 (X), C) ∼ = HdR (X) ∼ = H p (X, C). Similar arguments show that the double p • (X, Z(3)). q complex C (, Z(3)D ) computes HD The choice of the covering H → X – or, more generally, D → X – contains more information than simply using an abstract universal covering map U → X: it includes the choice of a projective structure. Indeed, since the Schwarzian derivative of any Möbius transformation vanishes, any local section of the canonical projection would precisely be a system of projective charts for it. It follows that when working with H → X the explicit inclusion of projective connections becomes – strictly speaking – unnecessary. Indeed, these were not considered in [2]. However, it is known [2, 40] that the effective action (that is, the class of the local Lagrangian cocycle) in higher genus is determined – say, by the Universal Ward Identity – only up to holomorphic quadratic differentials. Interpreting the latter as lifts of projective connections, the precise statement is that the effective action is determined up to the choice of a projective structure. In light of this observation, and also to keep a ˇ strict parallel with the Cech formulation, we make this dependence on a generic projective connection explicit.7 In this way we obtain a unified formalism consistent with the treatment of variations in Sect. 4, where conditions on the projective connections will be enforced by the variational process. 7 Here the term projective connection is to be understood in the same way as in 3.3, i.e. as not necessarily holomorphic one.
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Now we set out to write the correspondence: (3, 0) (2, 1) (1, 2) (0, 3)
UX ωi (zi ) θij (zj ) $ij k (zk ) mij kl
H ω(z) θγ (z) $γ1 ,γ2 (z) mγ1 ,γ2 ,γ3
For the first two lines we start by translating (3.3.1) and (3.3.4), respectively: ∂ 2f ∂µ dz ∧ d z¯ + 2 µ h dz ∧ d z¯ , (3.4.6) ∂f γ !! θγ (z) = 2µ ! d z¯ − log(γ˜ ! ◦ f ) + log γ ! d log ∂f + log(γ˜ ! ◦ f ) d log γ ! , (3.4.7) γ ωγ (z) =
where h is a smooth quadratic differential. In this way the last term of (3.4.6) is automorphic of weight (1, 1), hence it is killed by the coboundary operator. This would be consistent with a translation of (3.3.3). We stress (3.4.7) is a direct translation of the expression for the (2, 1) component prior to the computation of δθ = d$. As before, the existence of $γ1 ,γ2 is guaranteed by the vanishing of the analog of the symbol T X, T X in holomorphic Deligne cohomology. This time, the tame symbol is represented by the cocycle − log γ1! ◦ γ2 d log γ2! , cγ1 ,γ2 log γ3! , cγ1 ,γ2 cγ3 ,γ4 ∈ C 2 (, 1 (H)) ⊕ C 3 (, O(H)) ⊕ C 4 (, Z(2)). Since H → X is a good cover, the quasi-isomorphism d log ∼ ∼ Z(2)•D,hol → O∗ (H) −−→ 1 (H) [−1] → C/Z(2) ∼ = C∗ is still in place by holomorphic Poincaré lemma on H. Hence H4 (, Z(2)•D,hol ) ∼ = H 3 (, C∗ ) = 0, again, by obvious dimensional reasons. It follows that we can still introduce (τγ ) ∈ C 1 (, 1 (H)), (φγ1 ,γ2 ) ∈ C 2 (, O(H)) and (nγ1 ,γ2 ,γ2 ) ∈ C 3 (, Z(2)) such that − log γ1! ◦ γ2 d log γ2! , cγ1 ,γ2 log γ3! , cγ1 ,γ2 cγ3 ,γ4 = D τγ , φγ1 ,γ2 , nγ1 ,γ2 ,γ3 , where various γi ’s are used as place-holders for added clarity. Obviously, the treatment for the corresponding quantities depending on ˜ is entirely similar. As a result, we can either compute the coboundary of (3.4.7) or simply translate (3.3.5) and repeat step by step what was done in Sect. 3.3 to arrive at θγ = θγold − τ˜γ + τγ ,
(3.4.8)
with θγold given by (3.4.7) and, finally: $γ1 ,γ2 = φ˜ γ1 ,γ2 − φγ1 ,γ2 − (c˜γ1 ,γ2 + cγ1 ,γ2 ) log ∂f − log γ1! ◦ γ2 log γ˜2! + c˜γ1 ,γ2 log(γ1 ◦ γ2 )! mγ1 ,γ2 ,γ3 = n˜ γ1 ,γ2 ,γ3 − nγ1 ,γ2 ,γ3 − (c˜γ1 ,γ2 + cγ1 ,γ2 ) bγ3 + cγ1 ,γ2 ◦γ3 c˜γ2 ,γ3 − cγ1 ◦γ2 ,γ3 c˜γ1 ,γ2 . Therefore the analog of Proposition 1 holds
(3.4.9) (3.4.10)
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Proposition 3. The total cochain √ def = 2π −1 ω, θγ , −$γ1 ,γ2 , −mγ1 ,γ2 ,γ3 , 3 (X, Z(3)). with ω given by the Polyakov form (3.4.6), represents a class in HD
The action functional S[f ] is computed by evaluating [f ] over the appropriate representative of [X], which in this case would be a total cocycle in Sp (H) ⊗Z Bq () whose (2, 0) component can be taken as a fundamental domain F for in the form of a standard 4g-gon, as detailed in [2]. 4. Variation and Projective Structures 4.1. Variation. Here we compute the variation of the action functional S[f ] with respect to the dynamical field f , i.e. we compute its differential in field space. We denote by δ the variational operator – the exterior differential in field space [37, 41, 13] – and we will ˇ use coordinates with respect to a good Cech cover UX whenever a local computation is required. Since the dynamical field f is a deformation map on X, we can either choose to allow variations that effectively deform the complex structure or restrict ourselves to the “trivial” ones – deformations corresponding to vertical tangent vectors in the Earle–Eells fibration over the Teichmüller space. From (3.2.1) we get f ∗ (κij ) =
δfj δfi ! −1 ◦ zij · (zij ) − , ∂fi ∂fj
(4.1.1)
where def δwij κ = κij = ! wij is the standard Kodaira–Spencer deformation cocycle, and f ∗ (κij ) = κij ◦ fj /∂fj is ˜ $), ˜ ˜ where $ ˜ is the tangent sheaf of X, its pull-back. The condition [κ] = 0 in H 1 (X, selects variations that leave the complex structure X fixed. Specifically, if [κ] = 0 then it follows from (4.1.1) that δfi /∂fi represents a smooth (1, 0)-vector field on X – possibly after redefining it by a holomorphic coboundary for f ∗ (κij ). Furthermore, the variation δµ of the corresponding Beltrami differential as a tangent vector to B(X) at µ is δf δµ = ∂¯ µ , ∂f (−1,1)
(4.1.2)
so the class [δµ] ∈ H∂¯ (X) corresponds to [κ] under the Dolbeault isomorphism. µ In the sequel we shall confine ourselves to vertical variations, that is, to those with i [κ] = 0. Then δf ∂fi defines a smooth vector field on X. We start to compute the variation of the Lagrangian cocycle with respect to f . From a purely formal point of view, the calculation for the variation of the top form part proceeds as usual, where in each coordinate patch we have δωi = ai + dηi
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with (ai ) ∈ Cˇ 0 (UX , A2X ) and (ηi ) ∈ Cˇ 0 (UX , A1X ), where δfi dzi ∧ d z¯ i . ai (f, δf ) = −2∂¯ µ hi − {fi , zi } ∂fi Using the well-known identity ∂¯ µ {f, z} = ∂ 3 µ, where µ = µ(f ) and z is a local coordinate on X (the index i is omitted here), we get ∂¯ µ ({f, z} − h) = (∂¯ − µ ∂ − 2 ∂µ)({f, z} − h) = ∂ 3 µ − (∂¯ − µ ∂ − 2 ∂µ)h ¯ = ∂ 3 µ + 2h ∂µ + ∂h µ − ∂h ¯ = Dh µ − ∂h. Here, for any smooth projective connection h ∈ Q(X), Dh is the following third order differential operator: Dh = ∂ 3 + 2h∂ + ∂h.
(4.1.3)
It is well-known (see, e.g., [22]) that it has the property Dh : A−1,l −→ A2,l X X , for all l; in particular, Dh maps global forms of weight (−1, l) to global forms of weight (2, l). Thus the final expression for the variation of the top form term is, ¯ i − Dh µi δfi dzi ∧ d z¯ i . ai (f, δf ) = −2 ∂h ∂fi
(4.1.4)
Thanks to (3.3.2) and to the fact that Dh is a well defined map, ai (f, δf ) is a well defined global 2-form on X. The 1-form ηi has the expression ηi = δ log ∂fi d log ∂fi + 2∂(log ∂fi )dzi δµi − 2(hi − {fi , zi })
δfi dzi + µi d z¯ i , ∂fi (4.1.5)
where is the interior product between 1-forms and vectors. The main point is that the term (4.1.4) alone constitutes the variation of the whole Lagrangian cocycle. Namely, we have √ Theorem 1. The variation of the total cocycle [f ] = 2π −1 ωi , θij , −$ij k , −mij kl under vertical variation is given by the 2-form (4.1.4) up to a total coboundary in the Deligne complex. The variation of the action functional S[f ] is √ δS[f ] = 2π −1 a(f, δf ), X
giving the following Euler-Lagrange equation ¯ = 0. Dh µ − ∂h
(4.1.6)
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We can give two different proofs of this theorem. One is more in keeping with the spirit of this work and uses the explicit form of . The other is based only on Takens’ acyclicity theorem [37] for the variational bicomplex and the formal machinery of descent equations. Although we present both, the second one will only be sketched here, as providing details for it would lead us far afield.8 Proof (First proof). The procedure is to compute the variation of the various components ˇ ij , that can be computed in two of by applying δ to the descent equations. Start with δδω ˇ different ways: from equation δω = dθ, and from the variational relation δω = a + dη. Since ai = aj , we have ˇ ij ) = 0, d(δθij − δη and we deduce, using the Poincaré Lemma, that ˇ ij = dλij , δθij − δη for (λij ) ∈ Cˇ 1 (UX , AX ). An explicit calculation using (3.3.7) and (4.1.5) confirms this relation with λij = 2
!! wij
! wij
! ! δ log ∂fj − τ˜ij ◦ fj δfj . ◦ fj δfj − log wij ◦ fj + log zij
(4.1.7)
The last term in this formula is obtained by varying the difference f ∗ (τ˜ij ) − τij , that enters Eq. (3.3.7). Clearly, the variation of τij is zero and for the variation of f ∗ (τ˜ij ) we have δf ∗ (τ˜ij ) = δ τ˜ij ◦ fj dfj = δ(τ˜ij ◦ fj ) dfj + τ˜ij ◦ fj δdfj = τ˜ij! ◦ fj δfj dfj + τ˜ij ◦ fj dδfj = d τ˜ij ◦ fj δfj , since τ˜ij ∈ 1 (U˜ i ∩ U˜ j ) (see Sect. 3.3.3). Computing the coboundary of (4.1.7) yields ˇ ij k = −(c˜ij k + cij k )δ log ∂fk δλ ! ! − (log wij ◦ fj + log zij ) δ log wj! k ◦ fk − (δˇ τ˜ )ij k ◦ fk δfk .
On the other hand, the variation of (3.3.8) gives ! δ log wj! k ◦ fk δ$ij k = φ˜ ij! k ◦ fk δfk − (c˜ij k + cij k )δ log ∂fk − log zij ! ˇ ij k + log wij = δλ ◦ fj δ(log wj! k ◦ fk ) + φ˜ ij! k ◦ fk δfk + (δˇ τ˜ )ij k ◦ fk δfk .
Using the first equation in (3.3.6): ! ˇ )ij k = − log wij d log wj! k , d φ˜ ij k + (δτ
we get ˇ ij k . δ$ij k = δλ 8 We plan to return to the topic from a more general point of view elsewhere.
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Finally, putting it all together, we obtain δ = (ai ) + D(η + λ), as wanted. Proof (Second proof). The 2-form ai in the relation δωi = ai + dηi is a source form [41], hence it is uniquely determined by the de Rham class of ωi . Moreover, given a specific ωi , the form dηi is also determined (so ηi is determined up to an exact form). Since ωj = ωi + dθij , we must have ai = aj as both ai and aj are source forms for the same Lagrangian problem. Here the requirement that the variation be vertical is crucial in order to ensure that δf/∂f glue as a geometric object – a vector field on X. Therefore, ˇ ij = δδω ˇ ij , we get from δ δω ˇ ij + dλij δθij = δη by the Poincaré lemma. Proceeding in the same fashion we also get ˇ ij k ) = 0. d(δ$ij k − δλ ˇ ij k are forms of degree one in the field direction, i.e. they contain Now, both δ$ij k and δλ one variation. Takens’ acyclicity theorem [37, 41, 13] asserts the variational bicomplex is acyclic in all degrees except the top one in the de Rham direction, provided the degree in the variational direction is at least one. Hence, ˇ ij k , δ$ij k = δλ and we reach the same conclusion as in the previous proof.
) (
4.2. Relative projective structures. Here we interpret of the Euler–Lagrange equation from the previous section through the principal G(X)-bundle over the universal family of projective structures. First, we reformulate Theorem 1 as follows Theorem 2. The Euler–Lagrange equation ∂¯ hi = Dh µi for the vertical variational problem is the condition that the push-forward of the projective connection {hi } onto X˜ by the map f is holomorphic. Proof. Indeed, the push-forward of h is f∗ (h) = {h˜ ◦ fi−1 · (∂fi−1 /∂wi )2 }, where h˜ i = hi − {fi , zi }. It is a projective connection on X˜ because of the transformation law ! 2 ) = {wi , wj } ◦ fj (∂fj )2 . h˜ j − h˜ i ◦ zij (zij
The Euler–Lagrange equation is equivalent to the equation ∂¯ µ h˜ i = 0, which is precisely ˜ ( the condition that the projective connection f∗ (h) is holomorphic on X. )
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It is well-known (see, e.g. [23]) that a holomorphic projective connection on X determines a projective structure on X, and vice versa. The space of all projective structures on X is an affine space modeled over H 0 (X, ⊗2 X ) – the vector space of holomorphic quadratic differentials on X. For any holomorphic family C → S of Riemann surfaces parameterized by a complex manifold S, there is the holomorphic family PS (C) → S of relative projective structures on C [25]. The fiber over s ∈ S is the affine space of all (holomorphic) projective structures for Cs . We will be interested in the universal case S = T (X) and denote by P(X) the universal family of relative projective structures. Following [34], consider the following pullback diagram: S(X) −−−−→ B(X)
(4.2.1)
P(X) −−−−→ T (X) where the vertical arrows are principal G(X)-bundles, and the horizontal ones are affine bundles with spaces affine over H 0 (Xµ , ⊗2 Xµ ) as fibers, µ ∈ B(X). (The curve Xµ depends only on the class of µ modulo G(X) and so do its holomorphic objects.) Here S(X) is the space of all projective structures on X holomorphic with respect to some complex structure determined by µ ∈ B(X), without considering the quotient by G(X). Since every projective structure determines a complex structure, there is an obvious projection S(X) → B(X). As it follows from Theorem 2, ¯ S(X) = {(h, µ) ∈ Q(X) × B(X)|Dh µ = ∂h}, so that S(X) is the critical manifolds for the mapping A : Q(X) × B(X) → C∗ (as well as for the map S : Q(X) × B(X) → C/Z(3)). The projection S(X) → B(X) is now just the projection on the second factor, and every fiber over µ in S(X) is indeed an affine space over the vector space H 0 (Xµ , ⊗2 Xµ ) (or rather its pull-back by f ). Indeed, if (h, µ) and (h! , µ) are two projective structures subordinated to µ, then we have ¯ Dh µ = ∂h
and
¯ !, Dh! µ = ∂h
¯ = ∂¯ µ ({f, z} − h) imply that which using the identity Dh µ − ∂h ∂¯ µ (h! − h) = 0, concluding that h − h! is a µ-holomorphic quadratic differential. ¯ – is The local meaning of the Euler–Lagrange equation – the condition Dh µ = ∂h the following. Lemma 1. The operators ∂¯ µ and Dh commute if and only if the Euler-Lagrange equation is satisfied. Proof. Let v be a local section of A−1,0 X . As a result of a direct calculation we have (omitting the coordinate index i) ¯ Dh ∂¯ µ v − ∂¯ µ Dh v = Lv (Dh µ − ∂h),
(4.2.2)
where Lv = v ∂ + 2 ∂v is the Lie derivative operator on A2,1 X . Thus the “if” part is clear. For the “only if” part, assume the RHS of (4.2.2) is zero for all v. Therefore, if ¯ = 0, implying we consider f v for any local f , then we must have v(f ) · (Dh µ − ∂h) (4.1.6). ( )
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We conclude that S(X) is the geometric locus where the commutativity condition Dh ∂¯ µ = ∂¯ µ Dh is satisfied. Then we can consider Dh as a map between two augmented Dolbeault complexes (where $µ and ⊗2 Xµ are actually pull-backs of the corresponding sheaves from Xµ to X by the map f (µ)): ∂¯ µ
ı
0 −−−−→ $µ −−−−→ A−1,0 −−−−→ A−1,1 −−−−→ 0 X X Dh Dh Dh
(4.2.3)
0 −−−−→ ⊗2 −−−→ A2,0 −−−−→ A2,1 −−−−→ 0 Xµ − X X ı
∂¯ µ
Dh
where the morphism $µ −→ ⊗2 Xµ is now the usual third order µ-holomorphic operator [22, 25], also familiar from the theory of the KdV equation [29]. It fits into the exact sequence ı
Dh
−−−→ 0, 0 −−−−→ V X (h) −−−−→ $µ −−−−→ ⊗2 Xµ −
(4.2.4)
where V X (h) is a rank three local system depending on the projective structure h – a locally constant sheaf on X. Actually, it is the sheaf of polynomial vector fields of degree not greater than two in the coordinates adapted to (h, µ). Passing to cohomology, we get: 1 1 0 → H 0 (Xµ , ⊗2 Xµ ) → H (X, V X (h)) → H (Xµ , $µ ) → 0.
(4.2.5)
According to the theorem of Hubbard [25], sequence (4.2.5) is isomorphic to the tangent bundle sequence for the relative projective structure P(X) → T (X) at (h, µ). Furthermore, the usual machinery of local systems shows that H 1 (X, V X (h)) is isomorphic to the Eichler cohomology group H 1 (π1 (X, p), V (h)p ), where V (h)p is the stalk of V X (h) over the point p. The proof that this coincides with the classical Eichler cohomology (see [26]), can be obtained by lifting everything to the universal cover H of X and using factors of automorphy (see [25] for further details). On the other hand, from our description of S(X) we have ˙ ˙ µ) T(h,µ) S(X) = {(h, ˙ ∈ A2,0 (X) × A−1,1 (X)|Dh µ˙ = ∂¯ µ h}
(4.2.6)
and the RHS can be written as the fiber product A2,0 (X) ×A2,1 (X) A−1,1 (X) with respect to the pair of maps ∂¯ µ and Dh . For vertical – along the fiber of S(X) → B(X) – tangent vectors to S(X) at (h, µ) we have ˙ µ) (h, ˙ = (Dh v, ∂¯ µ v), where v ∈ A−1,0 (X) is the infinitesimal generator. This pair clearly satisfies the condition in (4.2.6), since S(X) is the geometric locus of the commutativity condition. Thus the map sending v → (Dh v, ∂¯ µ v) describes the vertical tangent bundle of S(X) → B(X).
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Therefore, if [h, µ] denotes the class of (h, µ), we have for the vertical tangent space to P(X) at [h, µ]: TV ,[h,µ] P(X) ∼ (4.2.7) = A2,0 (X) ×A2,1 (X) A−1,1 (X) /(Dh , ∂¯ µ )(A−1,0 (X)), which obviously projects onto H∂¯−1,1 (X) ∼ = H 1 (Xµ , $µ ). Now, this is just the C ∞ µ image of the Eichler cohomology description of the tangent sheaf to the relative projective structure P(X) → T (X) and we have the following Proposition 4. The differential geometric description of the tangent space to P(X) at the class of (h, µ) as given by (4.2.7) coincides with the algebraic description given by the Eichler cohomology group H 1 (X, V X (h)). Proof. Consider the cone of Dh : A−1,• → A2,• X X : ∂¯ µ ⊕Dh
Dh −∂¯ µ
2,1 −−−−→ A−1,1 ⊕ A2,0 C •X : 0 −→ A−1,0 X X X −−−−→ AX −→ 0.
Its cohomology sheaf complex equals V X (h), thus by standard homological algebra arguments (see, e.g. [28]) one has H1 (X, C •X ) = H 1 (X, V X (h)) and from the canonical sequence −1,• • 0 −→ A2,• −→ 0 X [−1] −→ C X −→ AX
one gets (4.2.5). On the other hand, the RHS of (4.2.7) is the first cohomology group of the complex ∂¯ µ ⊕Dh
Dh −∂¯ µ
0 −→ A−1,0 (X) −−−−→ A−1,1 (X) ⊕ A2,0 (X) −−−−→ A2,1 (X) −→ 0 which is equal to the first term
p,q E1
C p (X) p q ∼ ˇ = H (X, C X ) = 0
of the spectral sequence computing H• (X, C •X ).
q = 0, q > 0,
) (
4.3. Geometry of the vertical variation. Here we consider the functional A[f ] as as map A : Q(X) × B(X) −→ C∗ , where Q(X) is the affine space of all C ∞ projective connections on X and B(X) is the total space of the Earle–Eells fibration. By Theorem 2, the critical manifold for A[f ] coincides with S(X). Considering critical values of A (”on shell” condition) leads to the function A : S(X) −→ C∗ , where A(h, µ) = [h, µ], m . Since S(X) is a principal G(X)-bundle over P(X), it is interesting to analyze the behavior of A under the G(X)-action. It is given by the following Lemma 2. The directional derivative of the action functional A for the vertical tangent vector (Dh v, ∂¯ µ v) to S(X) at (h, µ), where v ∈ A−1,0 (X), is given by √ 4π −1 µ Dh v · A. (4.3.1) X
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Proof. We just repeat the computation of the vertical variation with the additional term ¯ − Dh µ) v dz ∧ d z¯ 2 µ δh dz ∧ d z¯ , where δh = Dh v. Since the main term, given by 2 (∂h vanishes “on shell”, this proves the result. ( ) ∗ Formula (4.3.1) defines a function c : S(X) −→ Lie G(X) by assigning to the pair (h, µ) a linear functional on Lie G(X) ∼ = A−1,0 (X) as follows: µDh v. (4.3.2) v → 2 X
Equivalently, c is a 1-cochain over Lie G(X) with values in functions over S(X) with left Lie G(X)-action. Proposition 5. The 1-cochain c is a 1-cocycle. Proof. For v, w ∈ A−1,0 (X) ∼ = Lie G(X) we have δc(v, w) = v · c(w) − w · c(v) − c([v, w]), where c(u) : S(X) → C is the function c(u)(h, µ) = 2
X
µDh u.
Using the infinitesimal action, v · c(w)(h, µ) = 2 we get
(δc)(h, µ) = 2
X
∂¯ µ vDh w + µ Lv (Dh w ,
X
∂¯ µ vDh w + µLv (Dh w
−∂¯ µ wDh v − µLw (Dh v) − µ Dh Lv w ,
where Lv = v∂ + 2∂v is the Lie derivative on A2 (X), and the Lie bracket in A−1,0 (X) is the usual vector field Lie bracket: [v, w] = Lv w = (v ∂w − w ∂v). Using the identity Lv (Dh w) − Lw (Dh v) − Dh Lv (w) = 0, we are left with (δc)(v, w)(h, µ) = 2
X
=2 = 0,
X
∂¯ µ vDh w − ∂¯ µ w Dh v
v Dh ∂¯ µ w − ∂¯ µ Dh w
because of the commutativity condition and the skew-symmetry of the operator Dh .
) (
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A. Appendix A.1. Cones. Recall [28] that for a map u : A• → B• the cone C•u of u is the complex: C•u = A• [1] ⊕ B• with differential d(a, b) = (−da, u(a) + db). The cone fits into the exact sequence: 0 −→ B• −→ C•u −→ A• [1] −→ 0. If the map u is injective, this is the same as the cokernel of u (up to a shift in the resulting exact cohomology sequence). For the Deligne complex, we often find that the equivalent definition [15, 13] of Z(p)•D is ı− Z(p)•D = Cone Z(p) ⊕ F p (A)•M −−→ A•M [−1],
(A.1.1)
where : F p (A)•M → A•M is the Hodge-Deligne filtration (filtration bête), that is, the nth sheaf of F p (A)•M is AnM if n ≥ p, and zero otherwise. Briefly, the equivalence is shown as follows. The cone in (A.1.1) is equal to Cone Z(p) −→ Cone(F p (A)•M −→ A•M ) [−1]. The inner cone can clearly be replaced by the cokernel of the inclusion map, namely the (sharp) truncation τ ≤p−1 A•M of the de Rham complex. Thus we have Cone Z(p) −→ τ ≤p−1 A•M [−1], which equals Z(p)•D as defined in the main text. A.2. Fundamental class. We want to collect here some technical facts and computations related to the construction of a representative of the fundamental class [M], that are not strictly necessary in the main body of this paper. Recall that we work with the double complex Sp,q = Sp (Nq (U → X)), where N• (U → X) is the nerve of the covering U → X, and S• is the singular simplices functor. A.2.1. We saw in the main text, Sect. 2.4, that when Sp,• resolves Sp (M) for any fixed p, the total homology of S•,• is equal to H• (M, Z). By definition, this condition is that H0 (Sp (N• (U ))) ∼ = Sp (M) and Hq (Sp (N• (U ))) = 0 for q > 0. Then the isomorphism H• (M, Z) ∼ = H• (Tot S) can be easily obtained by carefully lifting a cocycle in S• (M) 1 = H (S (N (U ))) = 0 for to a total cocycle in S•,• .9 More concisely, we have Ep,q q p • 9 See, e.g., [28]. Details for this calculation can be found in the appendix of [2].
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2 = E∞ ∼ q > 0 (the spectral sequence collapses) and at the next step one has Ep,0 p,0 = Hp (S• (M)) = Hp (M, Z), as wanted. ˇ These requirements are met for a Cech covering UM , where a contracting homotopy for Sp (N• (UM )) can be constructed explicitly [38] (see also [21], the appendix on the de Rham theorem). Indeed, one can easily show that H0 (Sp (N• (UM )) ∼ = Sp (M) by applying S (−) to the sequence · · · N (U ) ⇒ N (U ) → M. The resulting maps p 1 M 0 M are i σi → i σi and ij σij → i ( j (σj i − σij )), so the composition is zero. Moreover, if i σi = 0, for any pair of indices ij , we must have σi |Uij + σj |Uij = 0, so that σi |Uij σj |Uij = σi |Uij −σi |Uij = ∂ !! σi |Uij , proving the claim. Similarly, if U → M is a regular covering with G = Deck(U/M) acting on the right on U , then Sp,0 = Sp (N0 (U )) ≡ Sp (U ) is a free (right) G-module [28], so that Sp (N• (U )) ∼ = Sp (U ) ⊗ZG B• (G) resolves Sp (U ) ⊗ZG Z ∼ = Sp (M), hence ∼ Sp (M) q = 0 Hq (Sp (N• (U → M))) = 0 q > 0,
as wanted. A.2.2. Since S•,• is a double complex, it is well known that its associated total complex can be filtered in two ways – with respect to either p or q. Filtering over the second index of Sp,q = Sp (Nq (U )) yields the second spectral sequence with
! 1 ∼ Ep,q = Hp∂ (S•,q ) ≡ Hp (S• (Nq (U )).
Although not required in the following it is interesting to see when and whether this latter sequence also degenerates, like the other one. In other words, we want to consider the case when for fixed q the complex S•,q is acyclic in degree > 0. Assumption. The covering U → M is good, that is, each Nq (U ) = U ×M · · · ×M U is contractible, hence is acyclic for the singular simplices functor. Remark 5. The assumption on U → M guarantees the de Rham complex is a resolution of C, so the second cohomological spectral sequence H p (Cˇ q (U ; A• )) degenerates and the total cohomology equals Hˇ q (U ; C). By virtue of the assumption, E 1 is computed as 1 ∼ Z < Nq R U > Eq,p = 0
p=0 p > 0,
where Nq R U is the set of connected components of Nq (U ) and Z < Nq R U > is the abelian group generated by Nq R U . This follows from the fact that H0 gives us a factor Z for every connected component of Nq (U ). These connected components arrange into a simplicial set N• R U , where the face maps are induced by the face maps of the nerve N• (U ), specifying where every component goes. Thus N• R U expresses the pure combinatorics of the covering. Since the spectral sequence collapses, the total homology is equal to
2 ∞ ∼ Eq,0 = Eq,0 = Hq (Z < N• RU >)
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and (see [31]) Hq (Z < N• RU >) ∼ = Hq (|N• RU |), where |·| is the geometric realization of Nq R U , namely, the CW-complex obtained by putting in a standard q-simplex 9q for each element in Nq R U and gluing them together according to the face maps. Therefore, for a good covering the three homologies are equal: Hq (Tot S•,• ) ∼ = Hq (M, Z) ∼ = Hq (|N• R U |). ˇ In our concrete examples, an ordinary Cech covering is good if all Ui0 ∩ · · · ∩ Uiq ! ∂ are contractible. In this case, to compute H0 (S• (Nq UM )) we must assign a Z factor to each Ui0 ∩ · · · ∩ Uiq . Following [38, 21], denote Ui0 ∩ · · · ∩ Uiq as a generator in this group by the symbol 9i0 ,...,iq , so that Z < Nq R U >= i0 ,...,iq Z · 9i0 ,...,iq and Nq R U = {9i0 ,...,iq }. Therefore, N• R U represents the abstract nerve of the open cover and |N• R U | is the CW-complex obtained by replacing each 9i0 ,...,iq – in other words, each non-void intersection – by a standard q-simplex and gluing them according to the face maps of N• R U . On the other hand, if U → M is a G-covering, then according to 2.3 Nq (U ) = U × Gq , and it is good if U is contractible. Thus Z < N• R U >∼ = Z ⊗ZG B• (G), so that Hq (Z < N• R U >) ∼ = Hq (G; Z) ∼ = Hq (BG, Z), where BG = |N• R U | is the classifying space of G, where in this case Nq R U = Gq for q ≥ 1 and N0 R U = point. A.2.3. Let us return to the main problem of representing the fundamental class of X as a total cycle in the double complex Sp,q . If the sequence 0 ← Sp (X) ← Sp (N• (U )) is exact, then there exists a splitting τ : Sp (X) → Sp (N• (U )), i.e. the map τ satisfies : ◦ τ = idSp (X) . In other words, τ is the first step of an explicit contracting homotopy for Sp (N• (U )). Then a cycle representing [X] can be produced by lifting X via τ and completing τ (X) to a total cycle using the standard descent argument. In the concrete examples we have been looking at, this can be done as follows. The case where U is a regular G-covering can be handled by starting from a fundamental domain F for the action of G on U , where we regard F as an element of degree (p, 0) in Sp,0 ∼ = Sp (U ). Full details are spelled out in [2]. If U comes = Sp (U ) ⊗ZG B0 (G) ∼ ˇ from an ordinary Cech covering UX , we first replace Sp (X) by UX -small simplices: 0 ←− SpU (X) ←− Sp (N• UX ), those whose support is contained in the open cover where the UX -small simplices are UX = {Ui }. Second, write X = i σi , where all σi are UX -small, and set τ (X) = def i σi · 9i = 0 ∈ Sp,0 . Since :(∂ ! 0 ) = ∂ ! : 0 = ∂ ! :τ (X) = ∂ ! X ≡ 0, by the standard argument there exist 1 , 2 , . . . , p , with k ∈ Sp−k,k , k = 1, . . . , p, such that ∂ ! 0 = ∂ !! 1 , . . . , ∂ ! q−1 = ∂ !! q , . . . , ∂ ! p = 0.
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j
345
i
ij ijk jk
ik
k
Fig. 1. Intersections and their nerve
This schema can be implemented in a fairly explicit way using a map h : N• R U → S•U (X) constructed in [7] (Th. 13.4, proof) to realize the nerve of a covering. Of course, our case of interest here is p = 2. ˜ U of N• R U In order to describe h we shall need the barycentric decomposition N• R (see [36] for a more complete explanation). For any finite subset τ of the index set I denote Uτ = ∩i∈τ Ui , and let: ˜U = {Uτ }, N0 R ˜U = N1 R
τ ⊂I
{Uτ1 ⊂ Uτ0 },
τ0 ⊂τ1 ⊂I
˜U = Nq R
··· {Uτq ⊂ · · · ⊂ Uτ0 }.
τ0 ⊂···⊂τq ⊂I
In order to construct the mapping h, assign to each Uτ a point vτ ∈ Uτ , to any inclusion Uτ1 ⊂ Uτ0 a path from vτ0 to vτ1 , and to Uτ2 ⊂ Uτ1 ⊂ Uτ0 the cone from vτ0 to the path from vτ1 to vτ2 , which is of course a 2-simplex. Denote by 9(vτ0 ), 9(vτ0 , vτ1 ) and 9(vτ0 , vτ1 , vτ2 ) the 0, 1 and 2-simplices so obtained. Observe how the simplices constructed in this way inherit an orientation from the natural one on the barycentric ˜ U ; this is the main reason for using N• R ˜ U in place of N• R U . So, decomposition N• R for example, 9(vτ0 , vτ1 , vτ2 ) has the orientation induced by the order vτ0 ≤ vτ1 ≤ vτ2 associated to the inclusion τ0 ⊂ τ1 ⊂ τ2 . The typical situation for the indices i, j, k looks as in Fig. 1: to the index sets i, ij and ij k correspond the points vi , vij and vij k in Ui , Uij and Uij k , respectively. Then 9(vi , vij ) is the 1-simplex joining vi and vij , 9(vij , vij k ) the one joining vij and vij k , and so on. After these preparations, define an element 0 in S2,0 as st(vi ) · 9i , 0 = i∈I
where st(vi ) =
j,k:9ij k %=0
:ij k 9(vi , vij , vij k ) − 9(vi , vik , vij k )
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is the star of the vertex vi , and :ij k = ±1 according to whether the order of the triple i, j, k agrees with the orientation or not, namely whether the order ij k is the same as the cyclic (counterclockwise) order around the vertex vij k . Recall that 9τ is the symbol corresponding to Uτ , when considered as a generator in the abelian group generated by the nerve, as in A.2. Rewriting 0 as
st(vi ) · 9i =
i∈I
where
:ij k 9(vi , vij , vij k ) − 9(vi , vik , vij k ) · 9i
i,j,k
means the sum over triples of indices in I , its first differential is:
i,j,k
∂ ! 0 =
− 9(vj , vij , vij k ) − 9(vj , vj k , vij k ) · 9j + 9(vk , vik , vij k ) − 9(vk , vj k , vij k ) · 9k ,
:ij k 9(vik , vij k ) · (9k − 9i )
i,j,k
+
− 9(vij , vij k ) · (9j − 9i ) − 9(vj k , vij k ) · (9k − 9j ) :ij k 9(vi , vij ) · 9i − 9(vi , vik ) · 9i − 9(vj , vij ) · 9j
i,j,k
+ 9(vj , vj k ) · 9j + 9(vk , vik ) · 9k − 9(vk , vj k ) · 9k . The last sum is easily seen to be zero, while the first can be rewritten as ∂ !! 1 for the following element in S1,1 : 1 =
:ij k 9(vik , vij k ) · 9ik − 9(vij , vij k ) · 9ij − 9(vj k , vij k ) · 9j k .
i,j,k
Again, computing the first differential gives ∂ ! 1 =
:ij k vij k · (9ik − 9ij − 9j k )
i,j,k
+
:ij k vij · 9ij − vik · 9ik + vj k · 9j k ,
i,j,k
with the last sum being identically zero. The first term can be rewritten as ∂ !! 2 , where 2 = −
i,j,k
:ij k vij k · 9ij k .
Generating Functional in CFT on Riemann Surfaces II: Homological Aspects
347
Finally, the total chain ≡ 0 + 1 − 2 is a cycle, ∂ = 0, and we have the following expression for the representative of the fundamental class of X in the double complex: :ij k 9(vi , vij , vij k ) − 9(vi , vik , vij k ) · 9i = i,j,k
− 9(vj , vij , vij k ) − 9(vj , vj k , vij k ) · 9j + 9(vk , vik , vij k ) − 9(vk , vj k , vij k ) · 9k :ij k 9(vik , vij k ) · 9ik − 9(vij , vij k ) · 9ij − 9(vj k , vij k ) · 9j k + i,j,k
+
:ij k vij k · 9ij k .
i,j,k
Remark 6. By taking the second augmentation, the total cycle maps to: :ij k 9ij k , i,j,k
which is the 2-cycle in the CW complex representing the combinatorics of the cover U, and therefore the homology of X, in degree p = 2. Acknowledgements. At the early stage of this work we appreciated useful discussions with J.L. Dupont and especially C.-H. Sah, who passed away in July 1997. His generosity of mind and enthusiasm made all our discussions special. He is deeply missed. The work of L.T. was partially supported by the NSF grant DMS-98-02574.
References 1. Ahlfors, L.V.: Lectures on Quasiconformal Mappings. Toronto–New York–London: Van Nostrand, 1966 2. Aldrovandi, E., Takhtajan, L.A.: Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces. Commun. Math. Phys. 188, 29–67 (1997) 3. Alvarez, O.: Quantization and Cohomology. Commun. Math. Phys. 100, 279–309 (1985) 4. Artin, M.: Grothendieck Topologies. Harvard Univ. Math. Dept. Lecture Notes, 1962 5. Artin, M., Grothendieck, A. and Verdier, J.-L.: Théorie des Topos et Cohomologie étale des Schemas. Lecture Notes in Mathematics 269, 270, 305, Berlin–Heidelberg–NewYork: Springer-Verlag, 1972–1973 6. Artin, M., Mazur, B.: Étale homotopy. Springer Lecture Notes in Mathematics 100, Berlin–Heidelberg– New York: Springer-Verlag, 1969 7. Bott, R., Tu, L.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics 82, Berlin– Heidelberg–New York: Springer-Verlag, 1982 8. Brooks, R., Goldman, W.: The Godbillon-Vey invariant of a transversely homogeneous foliation. Trans. AMS 286, 651–664 (1984) 9. Brylinsky, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Prog. in Math. 107, Basel–Boston: Birkhäuser, 1993 10. Brylinsky, J.-L., McLaughlin, D.: The geometry of degree four characteristic classes and of line bundles on loop spaces I. Duke J. Math. 75, 603–632 (1994) 11. Brylinsky, J.-L., McLaughlin, D.: The geometry of degree four characteristic classes and of line bundles on loop spaces II. Duke J. Math. 83, 105–139 (1996) 12. Brylinski, J.-L.: Geometric construction of Quillen line bundles. In: Advances in Geometry J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan, P. Xu, eds. Prog. in Math. 172, Boston: Birkhäuser, 1999 13. Deligne, P., Freed, D.: Classical Field Theory. In: Quantum Fields and Strings: A Course for Mathematicians, Vol. 1. P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison, E. Witten, eds., Providence, RI: AMS and IAS, 1999 14. Earle, C. J., Eells, J.: A fibre bundle description of Teichmüller theory. J. Diff. Geom. 3, 19–43 (1969)
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15. Esnault, H.: Characteristic classes of flat bundles. Topology 27, 323–352 (1988) 16. Esnault, H., Viehweg, E.: Deligne–Beilinson cohomology. In: Beilinson’s Conjectures on Special values of L-Functions. M. Rapoport, N. Shappacher and P. Schneider, eds. Perspective in Math., New York: Academic Press, 1988 17. Faddeev, L., Shatashvili, S.,: Realization of the Schwinger term in the Gauss law and the possibility of correct quantization of a theory with anomalies. Phys. Lett. B167, 225–228 (1986) 18. Freed, D.: Higher Algebraic Structures and Quantization. Commun. Math. Phys. 159, 343–398 (1994) 19. Freed, D., Witten, E.: Anomalies in String Theory with D-Branes. arXiv:hep-th/9907189 20. Gaw¸edzki, K.: Topological actions in two-dimensional quantum field theory. In: Nonperturbative Quantum Field Theories. G. ’t Hooft, A. Jaffe, G. Mack, P.K. Mitter, R. Stora, eds. NATO Series Vol. 185, New York: Plenum Press, 1988, pp. 101–142 21. Goldberg, S.I.: Curvature and Homology. Revised reprint of the 1970 edition. Mineola: Dover, 1998 22. Gunning, R.C.: Special coordinate coverings of Riemann surfaces. Math. Ann. 170, 67–86 (1967) 23. Gunning, R.C.: Lectures on Riemann Surfaces. Princeton, NJ: Princeton Univ. Press, 1966 24. Gunning, R.C.: Riemann Surfaces and Generalized Theta Functons. Berlin–Heidelberg–New York: Springer-Verlag, 1976 25. Hubbard, J.: The monodromy of projective structures. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Sony Brook Conference. I. Kra, B. Maskit, eds., Princeton, NJ: Princeton Univ. Press, 1980 26. Kra, I.: Automorphic forms and Kleinian groups. New York: Benjamin, 1972 27. Lazzarini, S.: Doctoral Thesis, LAPP Annecy-le-Vieux (1990) and references therein 28. Mac Lane, S.: Homology. Berlin–Heidelberg–New York: Springer-Verlag, 1975. Weibel, C.A.: An introduction to homomological algebra. Cambridge: Cambridge Univ. Press, 1994 29. Magri, F.: A simple model for the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978) 30. Matsumoto, S.: Some remarks on foliated S 1 bundles. Inv. Math. 90, 334–358 (1987) 31. May, J.P.: Simplicial objects in Algebraic Topology. Reprint of the 1967 original. Chicago Lectures in Mathematics, Chicago: Univ. of Chicago Press, 1992 32. Milne, J.S.: Étale Cohomology. Princeton Mathematical Series 33 Princeton, NJ: Princeton Univ. Press, 1980 33. Milnor, J.: On the existence of a connection with curvature zero. Comment. Math. Helv. 32, 215–223 (1958) 34. Nag, S.: The complex analytic theory of Teichmüller spaces. New York: Wiley Intersc., 1988 35. Polyakov, A. M.: Quantum gravity in two dimensions. Mod. Phys. Lett. A 2, 893–898 (1987) 36. Segal, G.: Classifying Spaces and Spectral Sequences. Publ. IHES 34, 105–112 (1968) 37. Takens, F.: A global version of the inverse problem of the calculus of variations. J. Diff. Geom. 14, 543–562 (1979) 38. Weil, A.: Sur les théorèmes de de Rham. Comment. Math. Helv. 26, 119–145 (1952) 39. Wood, J.W.: Bundles with totally disconnected structure group. Comment. Math. Helv. 46, 257–273 (1971) 40. Zucchini, R.: A Polyakov action on Riemann surfaces. II. Commun. Math. Phys. 152, 269–298 (1993) 41. Zuckerman, G.: Action principles and global geometry. In: Mathematical Aspects of String Theory. S.T. Yau, ed., Singapore: World Scientific Publishing, 1987 Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 227, 349 – 384 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Twisted Sectors for Tensor Product Vertex Operator Algebras Associated to Permutation Groups Katrina Barron, , Chongying Dong , Geoffrey Mason† Department of Mathematics, University of California, Santa Cruz, CA 95064, USA. E-mail:
[email protected];
[email protected] Received: 20 April 2000 / Accepted: 20 January 2002
Abstract: Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V ⊗k . We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V ⊗k are isomorphic to the categories of weak, weak admissible and ordinary V -modules, respectively. The main result is an explicit construction of the weak g-twisted V ⊗k -modules from weak V -modules. For an arbitrary permutation automorphism g of V ⊗k the category of weak admissible g-twisted modules for V ⊗k is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γ g-twisted V ⊗k -modules for γ a general automorphism of V acting diagonally on V ⊗k and g a permutation automorphism of V ⊗k . 1. Introduction Orbifold theory and coset construction theory [GKO] are two important ways of constructing a new conformal field theory from a given one. The first orbifold conformal field theory was introduced in [FLM1] and the theory of orbifold conformal fields was subsequently developed, for example, in [DHVW1, DHVW2, FLM3] and [DVVV]. The first critical step in orbifold conformal field theory is to construct twisted sectors. In [Le1] and [FLM2], twisted sectors for finite automorphisms of even lattice vertex operator algebras were first constructed – the twisted vertex operators were constructed, The first author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship and a University of California President’s Postdoctoral Fellowship. Present address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail:
[email protected] The second author was supported by NSF grant DMS-9700923 and a research grant from the Committee on Research, UC Santa Cruz. † The third author was supported by NSF grant DMS-9700909 and a research grant from the Committee on Research, UC Santa Cruz.
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and in [Le2] and [DL2], the twisted Jacobi identity was formulated and shown to hold for these operators. These results led to the introduction of the notion of a g-twisted V -module [FFR], [D], for V a vertex operator algebra and g an automorphism of V . This notion records the properties obtained in [Le1], Sect. 3.3 of [FLM2], and [Le2], and provides an axiomatic definition of twisted sectors. The focus of this paper is the construction of twisted sectors for permutation orbifold theory. Let V be a vertex operator algebra, and for a fixed positive integer k, consider the tensor product vertex operator algebra V ⊗k (see [FLM3, FHL]). Any element g of the symmetric group Sk acts on V ⊗k in the obvious way, and thus it is appropriate to consider g-twisted V ⊗k -modules. This is the setting for permutation orbifolds. In the case of V a lattice vertex operator algebra, this becomes a special case of the more general result of [Le1, Le2, FLM2] and [DL2], mentioned above. Permutation orbifold theory has also been studied in the physics literature, e.g., in [KS] and [FKS], and in [BHS], the twisted vertex operators for the generators were given for affine, Virasoro, super-Virasoro and W3 cyclic permutation orbifolds. The characters and modular properties of permutation orbifolds are presented in [Ba]. However, the construction of twisted sectors for general permutation orbifold theory and for arbitrary orbifold theory have been open problems. The main result of this paper is the explicit construction of twisted sectors for general permutation orbifold theory. In addition, for g a k-cycle, we show that the categories of weak, weak admissible and ordinary g-twisted V ⊗k -modules are isomorphic to the categories of weak, weak admissible and ordinary V -modules, respectively. (The definitions of weak, weak admissible and ordinary twisted modules are given in Sect. 3.) Our proof includes an explicit construction of the g-twisted V ⊗k -modules given a V -module. We show that for an arbitrary permutation automorphism g of V ⊗k the category of weak admissible g-twisted modules for V ⊗k is semisimple and the simple objects are determined if V is rational. In addition, we extend our results to the more general setting of γ g-twisted V ⊗k -modules for γ a general automorphism of V acting diagonally on V ⊗k and g an arbitrary permutation automorphism of V ⊗k . One can use our constructions to calculate the characters and perform modular transformations. Moreover, we expect that the methods introduced in this paper can be extended to construct twisted sectors for arbitrary orbifold theory. We next point out some recent results and conjectures in the theory of vertex operator algebras with which we hope to put the results of this paper into perspective and show some of the motivation which led to our results. Let V be a vertex operator algebra and g an automorphism of V . In [DM] it is shown that given a weak V -module (M, Y ), one can define a new weak V -module g ◦M such that g ◦M = M as the underlying space and the vertex operator associated to v is given by Y (gv, z). Then M is called g-stable if g ◦ M and M are isomorphic as weak V -modules. It is a well-known conjecture that if V is rational and g is of finite order, then the number of isomorphism classes of irreducible gstable V -modules is finite and equal to the number of isomorphism classes of irreducible g-twisted V -modules. It is proved in [DLM3] that if V is rational and satisfies the C2 condition, and g is of finite order, then the number of isomorphism classes of irreducible g-twisted modules is finite and less than or equal to the number of isomorphism classes of irreducible g-stable V -modules. Moreover, if V is also assumed to be g-rational, then the number of isomorphism classes of irreducible g-twisted V -modules is equal to the number of isomorphism classes of irreducible g-stable V -modules. Now consider the tensor product vertex operator algebra V ⊗k as discussed above with g a k-cycle. From Proposition 4.7.2 and Theorem 4.7.4 in [FHL], it follows that the number of isomorphism classes of irreducible g-stable V ⊗k -modules is equal to the
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number of isomorphism classes of irreducible V -modules. Combining this result with the conjecture above, we have the conjecture that if V is rational and g is a k-cycle, then the number of isomorphism classes of irreducible g-twisted V ⊗k -modules is equal to the number of isomorphism classes of irreducible V -modules. Thus we see that the main result in this paper is actually a stronger result than this conjecture, namely that the categories of weak, weak admissible and ordinary g-twisted V ⊗k -modules are in fact isomorphic to the categories of weak, weak admissible and ordinary V -modules, respectively, even without the assumption that V is rational. To construct the isomorphism between the category of weak g-twisted V ⊗k -modules and the category of weak V -modules for g a k-cycle, we define a weak g-twisted V ⊗k module structure on any weak V -module. Our construction has been motivated by the following two results. The first motivating result is the modular invariance of the trace function in orbifold theory [DLM3]. If V is a holomorphic vertex operator algebra (see [DLM3]), then V is the only irreducible module for itself. Thus by Proposition 4.7.2 and Theorem 4.7.4 in [FHL], V ⊗k is the only irreducible g-stable V ⊗k -module. Following the notation and results of [DLM3], consider the symmetric group Sk as an automorphism group of V ⊗k and denote by Z(x, y, τ ) the y-trace on the unique x-twisted V ⊗k -module V ⊗k (x) where x, y ∈ Sk commute. Then the span of Z(x, y, τ ) is modular invariant. Using the modular invariance result of [DLM3], one can show that the graded dimension of irreducible g-twisted V ⊗k -modules is exactly the graded dimension of V except that Vn is graded by n/k (plus a uniform shift) instead of n for n ∈ Z. (This fact has also been observed and used in [DMVV] to study elliptic genera of symmetric products and second quantized strings.) This leads one to expect a g-twisted V ⊗k -module structure on V with the new gradation. The second motivating result is the construction in [Li2] of certain g-twisted V modules for g a certain automorphism of V . Let h ∈ V1 such that the zero-mode operator h0 for Y (h, z) acts semisimply on V with only finitely many eigenvalues and such that these eigenvalues are rational. Then e2πh0 i is an automorphism of V . Li’s construction defines a new action on any V -module and this action gives a e2πh0 i -twisted V -module structure on the original V -module. The main feature in the new action is an exponential operator (z) built up from the component operators hn of Y (h, z) for n ≥ 0. This kind of operator first appeared in the construction of twisted sectors for lattice vertex operator algebras [FLM2, FLM3]. Now let M be a weak admissible V -module and let g be the k-cycle g = (12 · · · k). As suggested above from our first motivating result, one expects a weak g-twisted V ⊗k module structure on M with some vertex operator Yg (v, z) for v ∈ V ⊗k . From the twisted Jacobi identity one sees that the component operators of Yg (u ⊗ 1 ⊗ · · · ⊗ 1, z) for u ∈ V form a Lie algebra. In addition, the fact that Yg (gv, z) = limz1/k →η−1 z1/k Yg (v, z), for η = e−2πi/k , indicates that all the vertex operators Yg (v, z) for v ∈ V ⊗k are generated by Yg (u ⊗ 1 ⊗ · · · ⊗ 1, z) for u ∈ V . Therefore, the key point is to define Yg (u ⊗ 1 ⊗ · · · ⊗ 1, z) which is expected to be Y (k (z)u, z1/k ) for some operator k (z) ∈ (End V )[[z1/k , z−1/k ]] due to our second motivating result above. In [FLM2, FLM3] and [Li2] an operator (z) was introduced by using vertex operators associated to certain Heisenberg algebras. But for an arbitrary vertex operator algebra, we do not have a Heisenberg algebra available. So we must find another way to construct (z). Note that such a construction should also work if V is the vertex operator algebra associated to the highest weight modules for the Virasoro algebra. Therefore in general, the only
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available operators with which we can construct (z) are L(n) for n ∈ Z. And in fact, our operator k (z) is built up from L(n) for n ≥ 0 (see Sect. 2). The paper is organized as follows. In Sect. 2, we define the operator k (z) on V and prove several important properties of k (z) which are needed in subsequent sections. The main ideas for the proofs of these identities come from [H]. In Sect. 3, we define a weak g-twisted V ⊗k -module structure on any weak V -module M for g = (12 · · · k) by using the operator k (z). As a result we construct a functor Tgk from the category of weak V -modules to the category of weak g-twisted V ⊗k -modules such that Tgk maps weak admissible (resp., ordinary) V -modules into weak admissible (resp., ordinary) g-twisted V ⊗k -modules. In addition, Tgk preserves irreducible objects. In Sect. 4, we define a weak V -module structure on any weak g-twisted V ⊗k -module. In so doing, we construct a functor Ugk from the category of weak g-twisted V ⊗k -modules to the category of weak V -modules such that Tgk ◦ Ugk = id and Ugk ◦ Tgk = id. In Sect. 5, we give the extension of the results of Sect. 3 and 4 for g a general k-cycle. Section 6 is devoted to twisted modules for an arbitrary permutation g ∈ Sk . In particular we prove that if V is rational, then V ⊗k is g-rational. We also construct irreducible g-twisted V ⊗k -modules from irreducible V -modules. In Sect. 7, we study various twisted modules for an automorphism of V ⊗k which is a product of a permutation and an automorphism of V . Here the automorphisms of V act on V ⊗k diagonally. For γ an automorphism of V , g the k-cycle g = (12 · · · k), and γ g an automorphism of V ⊗k , we show that the category of weak γ g-twisted V ⊗k -modules is isomorphic to the category of weak γ k -twisted V -modules and that this isomorphism preserves admissible, ordinary and irreducible objects. Finally we construct γ g-twisted V ⊗k -modules from γ -twisted V -modules for g an arbitrary permutation and show that if V is γ -rational, then V ⊗k is γ g-rational. 2. The Operator k (z) In this section we define an operator k (z) = Vk (z) on a vertex operator algebra V for a fixed positive integer k. In Sect. 3, we will use k (z) to construct a g-twisted V ⊗k -module from a V -module where g is a certain k-cycle. Let Z+ denote the positive integers. Let x, y, z, z0 , and αj for j ∈ Z+ be formal variables commuting with each other. Consider the polynomial 1 1 (1 + x)k − ∈ xC[x]. k k
By Proposition 2.1.1 in [H], for any formal power series j ∈Z+ cj x j ∈ xC[[x]] there exist unique aj ∈ C for j ∈ Z+ such that ∂ c1 exp − ·x = aj x j +1 cj x j . ∂x j ∈Z+
j ∈Z+
Thus for k ∈ Z+ , we can define aj ∈ C for j ∈ Z+ , by ∂ 1 1 exp − aj x j +1 · x = (1 + x)k − . ∂x k k j ∈Z+
For example, a1 = (1 − k)/2 and a2 = (k 2 − 1)/12.
Permutation Twisted Tensor Product VOAs
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Let R be a ring, and let O be an invertible linear operator on R[x, x −1 ]. We define ∂ another linear operator O x ∂x by ∂
O x ∂x · x n = O n x n for any n ∈ Z. For example, since z1/k can be thought of as an invertible linear mul∂ tiplication operator on C[x, x −1 ], we have the operator z(1/k)x ∂x from C[x, x −1 ] to ∂ C[z1/k , z−1/k ][x, x −1 ]. Note that z(1/k)x ∂x can be extended to a linear operator on C[[x, x −1 ]] in the obvious way. Let ∂ f (x) = z1/k exp − ·x aj x j +1 ∂x j ∈Z+ ∂ j +1 ∂ = exp − aj x · z(1/k)x ∂x · x ∂x j ∈Z+
=
z1/k k
(1 + x)k −
z1/k k
∈ z1/k xC[x].
Then the compositional inverse of f (x) in xC[z−1/k , z1/k ][[x]] is given by ∂ ∂ f −1 (x) = z−(1/k)x ∂x exp aj x j +1 ·x ∂x j ∈Z+ −1/k −j/k j +1 ∂ =z ·x exp aj z x ∂x j ∈Z+
= (1 + kz
−1/k
x)1/k − 1,
where the last line is considered as a formal power series in z−1/k xC[z−1/k ][[x]], i.e., we are expanding about x = 0 taking 11/k = 1. Let fα (x) denote the formal power series 1/k j +1 ∂ fα (x) = z exp − αj x · x ∈ z1/k x + z1/k x 2 C[α1 , α2 , ...][[x]]. ∂x j ∈Z+
Then fα−1 (x) ∈ z−1/k x + x 2 C[x][z−1/k ][[α1 , α2 , ...]], and fα (fα−1 (x) + z−1/k y) − x ∈ C[x][z1/k , z−1/k ][[α1 , α2 , ...]][[y]]. Furthermore, the coefficient of the monomial y in fα (fα−1 (x) + z−1/k y) − x is in 1 + x 2 C[x][z1/k , z−1/k ][[α1 , α2 , ...]]. Therefore, following [H], we can define *j = *j ({−αn }n∈Z+ , z1/k , x) ∈ C[x][z1/k , z−1/k ][[α1 , α2 , ...]] for j ∈ N by e
*0
exp
j ∈Z+
*j y
j +1
∂ y = fα (fα−1 (x) + z−1/k y) − x. ∂y
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Proposition 2.1. *j ({−an }n∈Z+ , z1/k , k1 z1/k−1 z0 ) for j ∈ N is well defined in C[z0 ][[z−1/k ]]. Furthermore 1 *j {−an }n∈Z+ , z1/k , z1/k−1 z0 = −aj (z + z0 )−j/k (2.1) k for j ∈ Z+ , and 1 exp *0 ({−an }n∈Z+ , z1/k , z1/k−1 z0 ) = z1/k−1 (z + z0 )−1/k+1 , k
(2.2)
where (z + z0 )−j/k is understood to be expanded in nonnegative integral powers of z0 . Proof. By Lemma 4.3.4 in [H], the formal series *j ({−αn }n∈Z+ , z1/k , x) for j ∈ N, are actually in C[x][α1 , α2 , ...][[z−1/k ]]. Therefore *j ({−an }n∈Z+ , z1/k , x) is well defined in C[x][[z−1/k ]], for j ∈ N, and the first statement of the proposition follows. In yC[z0 ][[z−1/k ]][[y]], we have ∂ z1/k−1 (z + z0 )−1/k+1 exp − aj (z + z0 )−j/k y j +1 ·y ∂y j ∈Z+ ∂ −(1/k)y ∂y 1/k−1 −1/k+1 1/k j +1 ∂ =z (z + z0 ) (z + z0 ) (z + z0 ) exp − aj y ·y ∂y j ∈Z+
∂ −(1/k)y ∂y
−1
f (y) = z (z + z0 )(z + z0 ) −1 −1/k = z (z + z0 )f ((z + z0 ) y) z1/k z1/k = z−1 (z + z0 ) (1 + (z + z0 )−1/k y)k − k k k z1/k z1/k −1/k y + z−1/k (z + kz1−1/k x)1/k − = z − x k k x= k1 z1/k−1 z0 = f (z−1/k (z + kz1−1/k x)1/k − 1 + z−1/k y) − x 1 1/k−1 x= k z z0 −1 −1/k = f (f (x) + z y) − x 1 1/k−1 x= k z z0 = exp *0 ({−an }n∈Z+ , z1/k , x) ∂ · exp · y *j ({−an }n∈Z+ , z1/k , x)y j +1 . ∂y j ∈Z+
x= k1 z1/k−1 z0
Equations (2.1) and (2.2) follow. Let V = (V , Y, 1, ω) be a vertex operator algebra. In (End V )[[z1/k , z−1/k ]], define V −j/k k (z) = exp aj z L(j ) k −L(0) z(1/k−1)L(0) . j ∈Z+
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Proposition 2.2. In (End V )[[z1/k , z−1/k ]], we have Vk (z)Y (u, z0 )Vk (z)−1 = Y (Vk (z + z0 )u, (z + z0 )1/k − z1/k ), for all u ∈ V . Proof. By Eq. (5.4.10)1 in [H] and Proposition 2.1 above, we have Vk (z)Y (u, z0 )Vk (z)−1 1 −j/k = exp aj z L(j ) Y (k −L(0) z(1/k−1)L(0) u, z1/k−1 z0 ) · k j ∈Z+ · exp − aj z−j/k L(j ) j ∈Z+
1 = z1/kL(0) exp aj L(j ) z−(1/k)L(0) Y (k −L(0) z(1/k−1)L(0) u, z1/k−1 z0 ) k j ∈Z+ · z(1/k)L(0) exp − aj L(j ) z−(1/k)L(0)
j ∈Z+
1 = z(1/k)L(0) Y z−(1/k)L(0) exp − *j ({−an }n∈Z+ , z1/k , z1/k−1 z0 )L(j ) k j ∈Z+ 1 · exp −*0 ({−an }n∈Z+ , z1/k , z1/k−1 z0 )L(0) · k −L(0) z(1/k−1)L(0) u, k 1 f −1 ( z1/k−1 z0 ) z−(1/k)L(0) k = z(1/k)L(0) Y z−(1/k)L(0) exp aj (z + z0 )−j/k L(j ) z−(1/k−1)L(0) j ∈Z+
1/k · (z + z0 )(1/k−1)L(0) · k −L(0) z(1/k−1)L(0) u, 1 + z−1 z0 − 1 z−(1/k)L(0) 1/k = z(1/k)L(0) Y z−(1/k)L(0) Vk (z + z0 )u, 1 + z−1 z0 − 1 z−(1/k)L(0) = Y Vk (z + z0 )u, (z + z0 )1/k − z1/k as desired.
Define xk (z) ∈ (End C[x, x −1 ])[[z1/k , z−1/k ]] by ∂ ∂ ∂ aj z−j/k x j +1 k x ∂x z(−1/k+1)x ∂x . xk (z) = exp − ∂x j ∈Z+
1 There is a typo in this equation in [H]. A(0) in the first line of Eq. (5.4.10) should be A(1) which is the (1) (1) (1) infinite series {Aj }j ∈Z+ , where Aj ∈ C. In our case, Aj = −aj .
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Proposition 2.3. In (End C[x, x −1 ])[[z1/k , z−1/k ]], we have 1 ∂ ∂ ∂ x + z1/k−1 xk (z) = (z), ∂x k ∂x ∂z k ∂ ∂ ∂ −xk (z)−1 + kz−1/k+1 xk (z)−1 = kz−1/k+1 xk (z)−1 . ∂x ∂x ∂z −xk (z)
(2.3) (2.4)
Proof. In C[x, x −1 ][[z1/k , z−1/k ]], we have ∂ 1 ∂ · x + z1/k−1 xk (z) · x ∂x k ∂x ∂ 1 1/k−1 ∂ −1/k+1 −(1/k)x ∂x 1/k j +1 ∂ = −1 + z z z exp − aj x kz ·x k ∂x ∂x j ∈Z+ 1 ∂ −1/k+1 −(1/k)x ∂ ∂x f (x) = −1 + z1/k−1 kz z k ∂x 1 1/k−1 −1/k+1 ∂ = −1 + z kz f (z−1/k x) k ∂x = −1 + z−1/k f (z−1/k x) = −1 + (1 + z−1/k x)k−1 = −1 + (1 + z−1/k x)k − z−1/k x(1 + z−1/k x)k−1 1/k z1/k −1/k z −1/k k = (−1 + k)z x) − (1 + z k k 1/k 1/k ∂ z z (1 + z−1/k x)k − + kz−1/k+1 ∂z k k 1 ∂ = k(− + 1)z−1/k f (z−1/k x) + kz−1/k+1 f (z−1/k x) k ∂z ∂ −1/k+1 = f (z−1/k x) kz ∂z ∂ −1/k+1 −(1/k)x ∂ ∂x f (x) = z kz ∂z ∂ ∂ −1/k+1 −(1/k)x ∂x 1/k j +1 ∂ = z z exp − aj x kz ·x ∂z ∂x j ∈Z+ ∂ −1/k+1 −j/k j +1 ∂ = exp − aj z x kz ·x ∂z ∂x
−xk (z)
j ∈Z+
∂ x = (z) · x. ∂z k Since xk (z) · x n is well defined in C[x, x −1 ][[z1/k , z−1/k ]] for all n ∈ Z, by Proposition 2.1.7 in [H], we have xk (z2 ) · x n = (xk (z2 ) · x)n
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for all n ∈ Z. Therefore ∂ ∂ 1 · x n + z1/k−1 xk (z) · x n −xk (z) ∂x k ∂x n ∂ ∂ x 1 · x + z1/k−1 k (z) · x = −nxk (z) · x n−1 ∂x k ∂x ∂ = −n xk (z) · x n−1 xk (z) ·x ∂x n−1 ∂ x 1 1/k−1 x +n z k (z) · x k (z) · x k ∂x x n−1 ∂ 1 ∂ x x −k (z) = n k (z) · x · x + z1/k−1 k (z) · x ∂x k ∂x x n−1 ∂ x = n k (z) · x k (z) · x ∂z n ∂ x = k (z) · x ∂z ∂ x = k (z) · x n ∂z for all n ∈ Z. Equation (2.3) follows by linearity. Similarly, in C[x, x −1 ][[z1/k , z−1/k ]], we have ∂ ∂ · x + kz−1/k+1 xk (z)−1 · x ∂x ∂x ∂ ∂ ∂ ∂ = −1 + kz−1/k+1 aj z−j/k x j +1 z(1/k−1)x ∂x k −x ∂x exp ·x ∂x ∂x j ∈Z+ ∂ (1/k−1)x ∂ −x ∂ 1/k −1 ∂x k ∂x z = −1 + kz−1/k+1 f (x) z ∂x ∂ −1 1/k−1 −1 = −1 + kz k x) f (z ∂x ∂ = −1 + kz (1 + z−1 x)1/k − 1 ∂x = −1 + (1 + z−1 x)1/k−1 = (1 + z−1 x)1/k − 1 − z−1 x(1 + z−1 x)1/k−1 ∂ 1/k −1/k+1 −1 1/k 1/k ∂ −1 1/k = kz (1 + z x) − 1 + z z (1 + z x) − 1 ∂z ∂z ∂ 1/k −1 1/k−1 −1 = kz−1/k+1 k x) z f (z ∂z ∂ (1/k−1)x ∂ −x ∂ 1/k −1 ∂x k ∂x z = kz−1/k+1 f (x) z ∂z ∂ ∂ −1/k+1 ∂ (1/k−1)x ∂x −x ∂x −j/k j +1 ∂ k exp aj z x = kz z ·x ∂z ∂x
−xk (z)−1
j ∈Z+
∂ = kz−1/k+1 xk (z)−1 · x. ∂z
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The proof of identity (2.4) on x n for n ∈ Z is analogous to the proof of identity (2.3) on x n for n ∈ Z. Identity (2.4) then follows by linearity. Let L be the Virasoro algebra with basis Lj , j ∈ Z, and central charge d ∈ C. The above identity can be thought of as an identity for the representation of the Virasoro ∂ algebra on C[x, x −1 ] given by Lj → −x j +1 ∂x , for j ∈ Z, with central charge equal to zero. We want to prove the corresponding identity for certain other representations of the Virasoro algebra, in particular for vertex operator algebras. We do this by following the method of proof used in Chapter 4 of [H]. Letting κ be another formal variable commuting with z and L, we first prove the identity in U. (L)[[z1/k , z−1/k ]][[κ, κ −1 ]], where U. (L) is a certain extension of the universal enveloping algebra for the Virasoro algebra, and then letting κ = k, the identity will follow in (End V )[[z1/k , z−1/k ]], where V is a certain type of module for the Virasoro algebra. We want to construct an extension of U(L), the universal enveloping algebra for the Virasoro algebra, in which κ −L0 and z(1/k−1)L0 can be defined. Let V. be a vector space over C with basis {Pj : j ∈ Z}. Let T (L ⊕ V. ) be the tensor algebra generated by the direct sum of L and V. , and let I be the ideal of T (L ⊕ V. ) generated by
Li ⊗ Lj − Lj ⊗ Li − [Li , Lj ], Li ⊗ d − d ⊗ Li , Pi ⊗ Pj − δij Pi , Pi ⊗ d − d ⊗ Pi , Pi ⊗ Lj − Lj ⊗ Pi+j : i, j ∈ Z . Define U. (L) = T (L ⊕ V. )/I. For any formal variable z and for n ∈ Z, we define Pj znj ∈ U. (L)[[z, z−1 ]]. znL0 = j ∈Z
Note κ −L0 and z(1/k−1)L0 are well-defined elements of U. (L)[[z1/k , z−1/k ]][[κ, κ −1 ]]. In U. (L)[[z1/k , z−1/k ]][[κ, κ −1 ]], define L −j/k aj z Lj κ −L0 z(1/k−1)L0 . k (z) = exp j ∈Z+
Proposition 2.4. In U. (L)[[z1/k , z−1/k ]][[κ, κ −1 ]], we have 1 1/k−1 ∂ L L−1 L z (z), k (z) = κ ∂z k ∂ −1 −1/k+1 −1 L L−1 L = kz−1/k+1 L (z)−1 . k (z) L−1 − κz k (z) ∂z k L k (z)L−1 −
(2.5) (2.6)
Proof. In U. (L)[[z1/k , z−1/k ]][[κ, κ −1 ]], we have 1 1/k−1 L−1 L z k (z) κ
1 a z−j/k Lj = z1/k−1 e j ∈Z+ j , L−1 κ −L0 z(1/k−1)L0 κ 1 1 = z1/k−1 aj1 · · · ajn z−(j1 +···+jn )/k κ n! n∈Z+ j1 ,... ,jn ∈Z+ · Lj1 Lj2 · · · Lji−1 [Lji , L−1 ]Lji+1 · · · Ljn κ −L0 z(1/k−1)L0 ,
L k (z)L−1 −
i=1,... ,n
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which is a well-defined element of U. (L)[[z1/k , z−1/k ]][[κ, κ −1 ]] involving only elements Lj with j ∈ N. The right-hand side of (2.5) also involves only Lj for j ∈ N. Thus ∂ comparing with the identity (2.3) for the representation Lj → −x j +1 ∂x , the identity (2.5) must hold. The proof of (2.6) is analogous. Let V be a module for the Virasoro algebra satisfying V = n∈Z Vn . For j ∈ Z, let L(j ) ∈ End V and c ∈ C be the representation images of Lj and d, respectively, for the Virasoro algebra. Assume that for v ∈ Vn , we have L(0)v = nv. For any formal variable z, define zj L(0) ∈ (End V )[[z, z−1 ]] by zj L(0) v = zj n v for v ∈ Vn . Or equivalently, let P (n) : V → Vn be the projection from V to the homogeneous subspace of weight n for n ∈ Z. Then zj L(0) v =
zj n P (n)v
n∈Z
for v ∈ V . The elements P (n) ∈ End V can be thought of as the representation images of Pn in the algebra U. (L). Note that for k a positive integer, k −L(0) is a well-defined element of End V and (1/k−1)L(0) z is a well-defined element of (End V )[[z1/k , z−1/k ]]. In (End V )[[z1/k , z−1/k ]], define Vk (z) = exp aj z−j/k L(j ) k −L(0) z(1/k−1)L(0) . (2.7) j ∈Z+
From Proposition 4.1.1 in [H] and Proposition 2.4, we obtain the following corollary. Corollary 2.5. In (End V )[[z1/k , z−1/k ]], we have 1 ∂ Vk (z)L(−1) − z1/k−1 L(−1)Vk (z) = Vk (z), k ∂z ∂ V −1 −1/k+1 V −1 k (z) L(−1) − kz L(−1)k (z) = kz−1/k+1 Vk (z)−1 . ∂z
(2.8) (2.9)
In particular, the identities hold for V being any vertex operator algebra. 3. The Twisted Sector for g = (12 · · · k) We first review the definitions of weak, weak admissible and ordinary g-twisted modules for a vertex operator algebra V and an automorphism g of V of finite order k (cf. [DLM1]– [DLM3]). Let (V , Y, 1, ω) be a vertex operator algebra. A weak g-twisted V = (V , YM )module is a C-linear space M equipped with a linear map V → (EndM)[[z1/k , z−1/k ]], −n−1 , such that for u, v ∈ V and w ∈ M given by v → YM (v, z) = n∈Q vn z the following hold: (1) vm w = 0 if m is sufficiently large; (2) YM (1, z) = 1; (3) YM (v, z) = n∈r/k+Z vn z−n−1 for v ∈ V r , where V r = {v ∈ V |gv = e−2πir/k v}; (4)
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the twisted Jacobi identity holds: for u ∈ V r , z1 − z2 z2 − z1 YM (u, z1 )YM (v, z2 ) − z0−1 δ YM (v, z2 )YM (u, z1 ) z0−1 δ z0 −z0
=
z2−1
z1 − z 0 z2
−r/k z1 − z0 YM (Y (u, z0 )v, z2 ). δ z2
(3.1)
It can be shown (cf. Lemma 2.2 of [DLM1], and [DLM2]) that YM (ω, z) has component operators which satisfy the Virasoro algebra relations and YM (L(−1)u, z) = d dz YM (u, z). If we take g = 1, then we obtain a weak V -module. A weak admissible g-twisted V -module is a weak g-twisted V -module M which carries a k1 Z+ -grading M = ⊕n∈ 1 Z+ M(n) k
(3.2)
such that vm M(n) ⊆ M(n + wt v − m − 1) for homogeneous v ∈ V . We may assume that M(0) = 0 if M = 0. If g = 1, we have a weak admissible V -module. Remark 3.1. Above we used the term “weak admissible g-twisted module” whereas in most of the literature (cf. [DLM1, Z]) the term “admissible g-twisted module” is used for this notion. We used the qualifier “weak” to stress that these are indeed only weak modules and in general are not ordinary modules. However, for the sake of brevity, we will now drop the qualifier “weak”. An (ordinary) g-twisted V -module is a weak g-twisted V -module M graded by C induced by the spectrum of L(0). That is, we have M= Mλ , (3.3) λ∈C
where Mλ = {w ∈ M|L(0)w = λw}. Moreover we require that dim Mλ is finite and Mn/k+λ = 0 for fixed λ and for all sufficiently small integers n. If g = 1 we have an ordinary V -module. The vertex operator algebra V is called g-rational if every admissible g-twisted V module is completely reducible, i.e., a direct sum of irreducible admissible g-twisted modules. It was proved in [DLM2] that if V is g-rational then: (1) every irreducible admissible g-twisted V -module is an ordinary g-twisted V -module; and (2) V has only finitely many isomorphism classes of irreducible admissible g-twisted modules. Now we turn our attention to tensor product vertex operator algebras. Let V = (V , Y, 1, ω) be a vertex operator algebra and k a fixed positive integer as in Sect. 2. Then V ⊗k is also a vertex operator algebra (see [FHL]), and the permutation group Sk acts naturally on V ⊗k as automorphisms. Let g = (12 · · · k). In this section we construct a functor Tgk from the category of weak V -modules to the category of weak g-twisted modules for V ⊗k . We do this by first defining g-twisted vertex operators on a weak V module M for a set of generators which are mutually local (see [Li2]). These g-twisted vertex operators generate a local system which is a vertex algebra. We then construct a homomorphism of vertex algebras from V ⊗k to this local system which thus gives a weak g-twisted V ⊗k -module structure on M.
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For v ∈ V denote by v j ∈ V ⊗k the vector whose j th tensor factor is v and whose other tensor factors are 1. Then gv j = v j +1 for j = 1, . . . , k, where k + 1 is understood to be 1. Let W be a weak g-twisted V ⊗k -module, and let η = e−2πi/k . Then it follows immediately from the definition of twisted module that the g-twisted vertex operators on W satisfy Yg (v j +1 , z) = lim Yg (v 1 , z). z1/k →η−j z1/k
Since V ⊗k is generated by v j for v ∈ V and j = 1, . . . , k, the vertex operators Yg (v 1 , z) for v ∈ V determine all the vertex operators Yg (u, z) on W for any u ∈ V ⊗k . This observation is very important in our construction of twisted sectors. Let u, v ∈ V . Then by (3.1) the twisted Jacobi identity for Yg (u1 , z1 ) and Yg (v 1 , z2 ) is z1 − z2 z2 − z1 −1 −1 1 1 Yg (u , z1 )Yg (v , z2 ) − z0 δ Yg (v 1 , z2 )Yg (u1 , z1 ) z0 δ z0 −z0 =
k−1 1 −1 (z1 − z0 )1/k Yg (Y (g j u1 , z0 )v 1 , z2 ) δ ηj z2 1/k k z j =0
(cf. [Le2, D]). Since g j u1
(3.4)
2
uj +1 , we see that Y (g j u1 , z0 )v 1
= only involves nonnegative integer powers of z0 unless j = 0 (mod k). Thus 1 −1 (z1 − z0 )1/k 1 1 Yg (Y (u1 , z0 )v 1 , z2 ). (3.5) [Yg (u , z1 ), Yg (v , z2 )] = Resz0 z2 δ 1/k k z2
This shows that the component operators of Yg (u1 , z) for u ∈ V on W form a Lie algebra. Now let M = (M, Y ) be a weak V -module. For u ∈ V and k (z) = Vk (z) given by (2.7), define Y¯ (u, z) = Y (k (z)u, z1/k ). When we put a weak g-twisted V ⊗k -module structure on M this Y¯ (u, z) will be the twisted vertex operator acting on M associated to u1 . Here we give several examples of Y¯ (u, z). If u ∈ Vn is a highest weight vector then k (z)u = k −n z(1/k−1)n u and Y¯ (u, z) = k −n z(1/k−1)n Y (u, z1/k ). In particular, if n = 1 we have Y¯ (u, z) = k −1 z1/k−1 Y (u, z1/k ). This case is important in the study of symmetric orbifold theory for the vertex operator algebras associated to affine Lie algebras. Now we take u = ω. Recall that a2 = (k 2 − 1)/12. Thus c −2/k z2(1/k−1) ω + a k (z)ω = 2 z k2 2 z2(1/k−1) (k 2 − 1)c −2/k = ω+ , z k2 24
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where c is the central charge. Therefore z2(1/k−1) (k 2 − 1)c −2 Y¯ (ω, z) = Y (ω, z1/k ) + z . 2 k 24k 2
(3.6)
We next study the properties of the operators Y¯ (u, z). Lemma 3.2. For u ∈ V , d ¯ Y (u, z). Y¯ (L(−1)u, z) = dz Proof. By Corollary 2.5, we have Y¯ (L(−1)u, z) = Y (k (z)L(−1)u, z1/k ) d 1 =Y k (z)u, z1/k + z1/k−1 Y (L(−1)k (z)u, z1/k ) dz k d 1 1/k 1/k−1 d =Y + z k (z)u, z Y (k (z)u, x) dz k dx x=z1/k d d =Y k (z)u, z1/k + Y (k (z)u, x 1/k ) dz dx x=z
d = Y (k (z)u, z1/k ) dz d ¯ = Y (u, z) dz as desired.
In the proof of the following lemma and again later on, we will need some properties of the δ-function. We first note that from Proposition 8.8.22 of [FLM3], for p ∈ Z we have −p/k p/k z1 − z0 z2 + z0 −1 z1 − z0 −1 z2 + z0 = z1 , (3.7) δ δ z2 z2 z2 z1 z1 and it is easy to see that k−1 z1 − z0 p/k p=0
z2
z2−1 δ
z1 − z0 z2
=
z2−1 δ
(z1 − z0 )1/k 1/k
z2
.
Therefore, we have the δ-function identity (z1 − z0 )1/k (z2 + z0 )1/k −1 = z . δ z2−1 δ 1 1/k 1/k z2 z1 Lemma 3.3. For u, v ∈ V ,
1 (z1 − z0 )1/k ¯ [Y¯ (u, z1 ), Y¯ (v, z2 )] = Resz0 z2−1 δ Y (Y (u, z0 )v, z2 ). 1/k k z2
(3.8)
(3.9)
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363 1/k
1/k
Proof. Replacing Y (u, z1 ) and Y (v, z2 ) by Y (k (z1 )u, z1 ) and Y (k (z2 )v, z2 ), respectively, in the commutator formula z1 − x −1 [Y (u, z1 ), Y (v, z2 )] = Resx z2 δ Y (Y (u, x)v, z2 ) (3.10) z2 which is a consequence of the Jacobi identity on M, we have −1/k [Y¯ (u, z1 ), Y¯ (v, z2 )] = Resx z2 δ
1/k
z1
−x
1/k
z2
1/k
Y (Y (k (z1 )u, x)k (z2 )v, z2 ). (3.11) 1/k
1/k
We want to make the change of variable x = z1 − (z1 − z0 )1/k , where by z1 − (z1 − z0 )1/k we mean the power series expansion in positive powers of z0 . In this case, we note that for n ∈ Z, 1/k (z1 − x)n 1/k x=z1 −(z1 −z0 )1/k m n 1/k 1/k−l n/k−m/k (−1)m z1 − z1 = (−1)l z0l m l m∈N l∈Z+ 1/k −z0 l m n m n/k (−1) z1 − = m l z1 m∈N l∈Z+ 1/k −z0 l n n/k 1+ = z1 l z1 l∈Z+ z0 n/k n/k =z1 1 − z1 n/k =(z1 − z0 ) . 1/k
− (z1 − z0 )1/k into
1/k
−x
Thus substituting x = z1 −1/k
z2
δ
z1
1/k z2
1/k
Y (Y (k (z1 )u, x)k (z2 )v, z2 )
we have a well-defined power series given by (z1 − z0 )1/k 1/k 1/k δ Y (Y (k (z1 )u, z1 − (z1 − z0 )1/k )k (z2 )v, z2 ). 1/k z2 Let f (z1 , z2 , x) be a complex analytic function in z1 , z2 , and x, and let h(z1 , z2 , z0 ) be a complex analytic function in z1 , z2 , and z0 . Then if f (z1 , z2 , h(z1 , z2 , z0 )) is well defined, and thinking of z1 and z2 as fixed, i.e., considering f (z1 , z2 , h(z1 , z2 , z0 )) as a Laurent series in z0 , by the residue theorem of complex analysis, we have ∂ Resx f (z1 , z2 , x) = Resz0 h(z1 , z2 , z0 ) f (z1 , z2 , h(z1 , z2 , z0 )) (3.12) ∂z0
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which of course remains true for f and h formal power series in their respective variables. 1/k Thus making the change of variable x = h(z1 , z2 , z0 ) = z1 −(z1 −z0 )1/k , using (3.11), (3.12), the δ-function identity (3.9) and Proposition 2.2, we obtain [Y¯ (u, z1 ), Y¯ (v, z2 )] 1 −1/k (z1 − z0 )1/k = Resz0 z2 (z1 − z0 )1/k−1 δ 1/k k z 2
1/k · Y (Y (k (z1 )u, z1
1/k
− (z1 − z0 ) )k (z2 )v, z2 ) 1 −1 (z1 − z0 )1/k 1/k 1/k = Resz0 z2 δ Y (Y (k (z1 )u, z1 − (z1 − z0 )1/k )k (z2 )v, z2 ) 1/k k z2 1 −1 (z2 + z0 )1/k 1/k 1/k = Resz0 z1 δ Y (Y (k (z1 )u, z1 − (z1 − z0 )1/k )k (z2 )v, z2 ) 1/k k z1 1 −1 (z1 − z0 )1/k = Resz0 z2 δ 1/k k z2 1/k
1/k
1/k
· Y (Y (k (z2 + z0 )u, (z2 + z0 )1/k − z2 )k (z2 )v, z2 ) 1 (z1 − z0 )1/k = Resz0 z2−1 δ Y (k (z2 )Y (u, z0 )v, z2 ) 1/k k z2 1 (z1 − z0 )1/k ¯ = Resz0 z2−1 δ Y (Y (u, z0 )v, z2 ), 1/k k z2 as desired.
We are now in a position to put a weak g-twisted V ⊗k -module structure on M. For u ∈ V set Yg (u1 , z) = Y¯ (u, z) Note that Yg (uj , z) =
and
Yg (uj +1 , z) =
k−1
p j p=0 Yg (u , z),
p
lim
z1/k →η−j z1/k
where Yg (uj , z) =
Yg (u1 , z).
(3.13)
j −n−1 . n∈p/k+Z un z
Lemma 3.4. Let u, v ∈ V . Then
1 −1 ηj −i (z1 − z0 )1/k Yg ((Y (u, z0 )v)j , z2 ), [Yg (u , z1 ), Yg (v , z2 )] = Resz0 z2 δ 1/k k z2 (3.14) where (Y (u, z0 )v)j = n∈Z (un v)j z0−n−1 , and i
j
p
[Yg (ui , z1 ), Yg (v j , z2 )] 1 = Resz0 z2−1 η(j −i)p k
z1 − z 0 z2
−p/k z1 − z0 Yg ((Y (u, z0 )v)j , z2 ). δ z2 1/k
(3.15) 1/k
Proof. By Lemma 3.3, Eq. (3.14) holds if i = j = 1. Replacing z1 and z2 by 1/k 1/k η−i+1 z1 and η−j +1 z2 , respectively, we obtain Eq. (3.14) for any i, j = 1, . . . , k. Equation (3.15) is a direct consequence of (3.14).
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By Lemma 3.4 for u, v ∈ V , there exists a positive integer N such that [Yg (ui , z1 ), Yg (v j , z2 )](z1 − z2 )N = 0.
(3.16)
Letting z1/k go to η−i+1 z1/k in Lemma 3.2, we have Yg (L(−1)ui , z) =
d Yg (ui , z). dz
Thus the operators Yg (ui , z) for u ∈ V , and i = 1, . . . , k are mutually local and generate a local system A in the sense of [Li2]. Let σ be a map from A to A such that σ Yg (ui , z) = Yg (ui+1 , z) for u ∈ V and i = 1, . . . , k. By Theorem 3.14 of [Li2]2 , the local system A generates a vertex algebra we denote by (A, YA ), and σ extends to an automorphism of A of order k such that M is a natural weak σ -twisted A-module in the sense that Y (α(z), z1 ) = α(z1 ) for α(z) ∈ A are σ -twisted vertex operators on M. Remark 3.5. σ is given by σ a(z) =
lim
z1/k →η−1 z1/k
a(z)
for a(z) ∈ A (see [Li2]). Let Ai = {c(z) ∈ A|σ c(z) = ηi c(z)} and a(z) ∈ Ai . For any integer n and b(z) ∈ A, the operator a(z)n b(z) is an element of A given by a(z)n b(z) = Resz1 Resz0
z1 − z 0 z
i/k
z0n · X,
(3.17)
where X = z0−1 δ
z1 − z z − z1 a(z1 )b(z) − z0−1 δ b(z)a(z1 ). z0 −z0
Or, equivalently, a(z)n b(z) is defined by:
(a(z)n b(z)) z0−n−1 = Resz1
n∈Z
z1 − z 0 z
i/k
· X.
(3.18)
Thus following [Li2], for a(z) ∈ Ai , we define YA (a(z), x) by setting YA (a(z), z0 )b(z) equal to (3.18). Lemma 3.6. We have [YA (Yg (ui , z), z1 ), YA (Yg (v j , z), z2 )] = 0. 2 There is a typo in the statement of Theorem 3.14 in [Li2]. The V in the theorem should be A. That is, the main result of the theorem is that the local system A of the theorem has the structure of a vertex superalgebra.
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Proof. From the vertex algebra structure of A, we have [YA (Yg (ui , z), z1 ), YA (Yg (v j , z), z2 )] z1 − z0 −1 = Resz0 z2 δ YA (YA (Yg (ui , z), z0 )Yg (v j , z), z2 ). z2 So we need to compute YA (Yg (ui , z), z0 )Yg (v j , z). p Note that Yg (ui , z) = n∈p/k+Z uin z−n−1 is an eigenvector for σ with eigenvalue ηp . Set Xp =
z0−1 δ
x−z z−x p i p −1 j Yg (u , x)Yg (v , z) − z0 δ Yg (v j , z)Yg (ui , x). z0 −z0
Then by (3.18),
i
j
YA (Yg (u , z), z0 )Yg (v , z) =
k−1
Resx
p=0
x − z0 z
p/k
Xp .
Using Lemma 3.4 we compute Resz0 z2−1 δ
z1 − z0 z2
= Resz0 Resx
k−1
YA (Yg (ui , z), z0 )Yg (v j , z)
z2−1 δ
p=0
= Resz0 Resx
∞ k−1
z1 − z0 z2
z2−1 δ
z1 − z0 z2
x − z0 z
p/k
x − z0 z
p=0 n=0 −1 p i · z0 [Yg (u , x), Yg (v j , z)](x − z)n z0−n ∞ k−1 x z1 − z0 −1 Resz0 Resx z2 δ z2 p=0 n=0
Xp
p/k
− z0 p/k −1 = z0 (x − z)n z0−n z 1 x − y −p/k x−y · Resy z−1 η(j −i)p δ Yg ((Y (u, y)v)j , z) k z z k−1 ∞ x − z0 p/k z1 − z0 = Resz0 Resx Resy z2−1 δ z2 z p=0 n=0 1 x − y −p/k x−y · z0−1 y n z0−n z−1 η(j −i)p δ Yg ((Y (u, y)v)j , z). k z z
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Thus using the δ-function identity (3.7), we have x − z0 p/k x − y −p/k x−y δ Resx z−1 z z z p/k z + y p/k z+y −1 x − z0 = Resx x δ z x x z z +y 0 p/k −p/k −1 p/k = Resx x (1 − ) z (z + y) δ x x z z+y 0 −1 p/k −p/k p/k = Resx x (1 − (z + y) δ ) z z+y x = z−p/k (z + y − z0 )p/k . Finally, we have Resz0 z2−1 δ
z1 − z0 z2
= Resy Resz0
YA (Yg (ui , z), z0 )Yg (v j , z)
k−1 1 −1 z1 − z0 1 z2 δ k z2 z0 − y p=0
(z + y − z0 )p/k Yg ((Y (u, y)v)j , z) k−1 1 −1 z1 − (z0 + y) 1 = Resy Resz0 z2 δ k z2 z0 ·η
(j −i)p −p/k
z
p=0
(z − z0 )p/k Yg ((Y (u, y)v)j , z) k−1 1 −1 z1 − y = Resy η(j −i)p Yg ((Y (u, y)v)j , z) z2 δ k z2 ·η
(j −i)p −p/k
z
p=0
= 0, as desired. Lemma 3.7. For u1 , . . . , uk ∈ V , we have YA (Yg (ukk , z)−1 · · · Yg (u22 , z)−1 Yg (u11 , z), x)
= YA (Yg (ukk , z), x) · · · YA (Yg (u22 , z), x)YA (Yg (u11 , z), x),
where Yg (uii , z)−1 is the component vertex operator of YA (Y (uii , z), x). Proof. The lemma follows from Lemma 3.6 above and formula (13.26) of [DL1]. Define the map f : V ⊗k → A by f : V ⊗k → A, u1 ⊗ · · · ⊗ uk = (ukk )−1 · · · (u22 )−1 u11 → Yg (ukk , z)−1 · · · Yg (u22 , z)−1 Yg (u11 , z) for u1 , . . . , uk ∈ V . Then f (ui ) = Yg (ui , z).
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Lemma 3.8. f is a homomorphism of vertex algebras. Proof. We need to show that f Y (u1 ⊗ · · · ⊗ uk , x) = YA (Yg (ukk , z)−1 · · · Yg (u22 , z)−1 Yg (u11 , z), x)f for ui ∈ V . Take vi ∈ V for i = 1, . . . , k. Then f Y (u1 ⊗ · · · ⊗ uk , x)(v1 ⊗ · · · ⊗ vk ) = f (Y (u1 , x)v1 ⊗ · · · ⊗ Y (uk , x)vk ) = Yg (Y (ukk , x)vkk , z)−1 · · · Yg (Y (u22 , x)v22 , z)−1 Yg (Y (u11 , x)v11 , z). By Lemma 3.7, we have YA (Yg (ukk , z)−1 · · · Yg (u22 , z)−1 Yg (u11 , z), x)f (v 1 ⊗ · · · ⊗ v k )
= YA (Yg (ukk , z), x) · · · YA (Yg (u22 , z), x)YA (Yg (u11 , z), x)Yg (vkk , z)−1 · · · Yg (v22 , z)−1 Yg (v11 , z).
By Lemma 3.6, it is enough to show that Yg (Y (ui , x)v i , z) = YA (Yg (ui , z), x)Yg (v i , z) for u, v ∈ V and i = 1, . . . , k. In fact, in view of the relation between Y (u1 , z) and Y (ui , z) for u ∈ V , we only need to prove the case i = 1. By Proposition 2.2, 1/k
Yg (Y (u1 , z0 )v 1 , z2 ) = Y (k (z2 )Y (u, z0 )v, z2 ) 1/k
1/k
= Y (Y (k (z2 + z0 )u, (z2 + z0 )1/k − z2 )k (z2 )v, z2 ). On the other hand, YA (Yg (u , z2 ), z0 )Yg (v , z2 ) = 1
1
k−1 p=0
where
Resz1
z1 − z 0 z2
p/k
X,
z1 − z2 z2 − z1 −1 1 1 Yg (u , z1 )Yg (v , z2 ) − z0 δ Yg (v 1 , z2 )Yg (u1 , z1 ). X= z0 −z0 By Eq. (3.16), there exists a positive integer N such that z0−1 δ
(z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 ) = (z1 − z2 )N Yg (v 1 , z2 )Yg (u1 , z1 ). Thus
z1 − z2 Yg (u1 , z1 )Yg (v 1 , z2 ) z0 z2 − z1 −N − z0−1 δ z0 (z1 − z2 )N Yg (v 1 , z2 )Yg (u1 , z1 ) −z0 z1 − z2 −N = z0−1 δ z0 (z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 ) z0 z2 − z1 −N − z0−1 δ z0 (z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 ) −z0 z1 − z0 = z2−1 z0−N δ (z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 ) , z2
X = z0−1 δ
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where we have used the elementary δ-function relation z0−1 δ
z1 − z2 z0
− z0−1 δ
z2 − z1 −z0
= z2−1 δ
z1 − z0 z2
(3.19)
(cf. [FLM3]). Therefore using the δ-function relation (3.8), we have YA (Yg (u1 , z2 ), z0 )Yg (v 1 , z2 ) = Resz1 z0−N z2−1 δ
(z1 − z0 )1/k
1/k
z2
(z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 ) .
Let x be a new formal variable which commutes with z0 , z1 , z2 . Then −1/k z2 δ
1/k
−x
z1
1/k
z2
= x −1 δ
1/k
z1
(z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 ) 1/k
− z2 x
(z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 )
1/k + z1 (z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 ) x 1/k 1/k z − z2 1/k 1/k = (z1 − z2 )N x −1 δ 1 Y (k (z1 )u, z1 )Y (k (z2 )v, z2 ) x 1/k 1/k −z2 + z1 1/k 1/k − (z1 − z2 )N x −1 δ Y (k (z2 )v, z2 )Y (k (z1 )u, z1 ) x 1/k z −x −1/k 1/k = (z1 − z2 )N z2 δ 1 1/k Y (Y (k (z1 )u, x)k (z2 )v, z2 ). z2
− x −1 δ
1/k
−z2
Note that the first term in the above formula is well defined when x is replaced by 1/k z1 −(z1 −z0 )1/k , and therefore the last term is also well defined under this substitution. Thus −1/k
z0−N z2
−1/k
= z2 =
δ δ
−1/k z2 δ
(z1 − z0 )1/k 1/k z2
(z1 − z0 )1/k 1/k z2
(z1 − z0 )1/k 1/k z2
(z1 − z2 )N Yg (u1 , z1 )Yg (v 1 , z2 )
1/k
Y (Y (k (z1 )u, z1
1/k
− (z1 − z0 )1/k )k (z2 )v, z2 )
1/k
1/k
Y (Y (k (z2 + z0 )u, (z2 + z0 )1/k − z2 )k (z2 )v, z2 ).
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K. Barron, C. Dong, G. Mason
Finally we have YA (Yg (u1 , z2 ), z0 )Yg (v 1 , z2 ) (z1 − z0 )1/k 1/k −1 = Resz1 z2 δ Y (Y (k (z2 + z0 )u, (z2 + z0 )1/k − z2 ) 1/k z2 1/k
· k (z2 )v, z2 ) 1/k
1/k
= Y (Y (k (z2 + z0 )u, (z2 + z0 )1/k − z2 )k (z2 )v, z2 ) = Yg (Y (u1 , z0 )v 1 , z2 ), as desired.
Let (M, Y ) be a weak V -module. Define Tgk (M, Y ) = (Tgk (M), Yg ) = (M, Yg ). That is Tgk (M, Y ) is M as the underlying vector space and the vertex operator Yg is given by (3.13). Now we state our first main theorem of the paper. Theorem 3.9. (Tgk (M), Yg ) is a weak g-twisted V ⊗k -module such that Tgk (M) = M, and Yg , defined by (3.13), is the linear map from V ⊗k to (End Tgk (M))[[z1/k , z−1/k ]], defining the twisted module structure. Moreover, (1) (M, Y ) is an irreducible weak V -module if and only if (Tgk (M), Yg ) is an irreducible weak g-twisted V ⊗k -module. (2) M is an admissible V -module if and only if Tgk (M) is an admissible g-twisted V ⊗k module. (3) M is an ordinary V -module if and only if Tgk (M) is an ordinary g-twisted V ⊗k module. Proof. It is immediate from Lemma 3.8 that Tgk (M) = M is a weak g-twisted V ⊗k module with Yg (u1 , z) = Y¯ (u, z). Note that Yg ((k (z)−1 u)1 , z) = Y¯ (k (z)−1 u, z) = Y (u, z1/k ) and that all twisted vertex operators Yg (v, z) for v ∈ V ⊗k can be generated from Yg (u1 , z) for u ∈ V . It is clear now that M is an irreducible weak V -module if and only if Tgk (M) is an irreducible weak g-twisted V ⊗k -module. So (1) has been proved. For (2) since M is an admissible V -module, we have M = ⊕n∈N M(n) such that for m ∈ k1 Z, the component operator um satisfies um M(n) ⊂ M(wt u − m − 1 + n) if u ∈ V is of homogeneous weight. Define a k1 Z+ -gradation on Tgk (M) such that Tgk (M)(n/k) = M(n) for n ∈ Z. Recall that Yg (v, z) = m∈ 1 Z vn z−m−1 for v ∈ V ⊗k . k
We have to show that vm Tgk (M)(n) ⊂ Tgk (M)(wt v − m − 1 + n) for m, n ∈ k1 Z. As before, since all twisted vertex operators Yg (v, z) for v ∈ V ⊗k can be generated from Yg (u1 , z) for u ∈ V , it is enough to show u1m Tgk (M)(n) ⊂ Tgk (M)(wt u − m − 1 + n).
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Let u ∈ Vp for p ∈ Z. Then k (z)u =
∞
u(i)z1/k−p−i/k ,
i=0
where u(i) ∈ Vp−i . Thus Yg (u , z) = Y (k (z)u, z 1
1/k
)=
∞
Y (u(i), z1/k ))z1/k−p−i/k ,
i=0
and for m ∈ k1 Z, u1m =
∞
u(i)(1−k)p−i−1+km+k .
i=0
Since the weight of u(i)(1−k)p−i−1+km+k is k(p − m − 1), we see that u1m Tgk (M)(n) = u1m M(kn) ⊂ M(k(p − m − 1 + n)). That is u1m Tgk (M)(n) ⊂ Tgk (M)(p − m − 1 + n), showing that Tgk (M) is an admissible g-twisted V ⊗k -module. Similarly, one can show that if Tgk (M) is an admissible g-twisted V ⊗k -module, then M is an admissible V -module with M(n) = Tgk (M)(n/k) for n ∈ Z. ¯ z) = n∈Z Lg (n)z−n−2 , where ω¯ = kj =1 ωj . In order to prove (3) we write Yg (ω, We have ¯ z) = Yg (ω,
k−1 i=0
lim
Yg (ω1 , z).
(k 2 −1)c 24k .
This immediately implies (3).
z1/k →η−i z1/k
It follows from (3.6) that Lg (0) = k1 L(0) +
Let V be an arbitrary vertex operator algebra and g an automorphism of V of finite order. We denote the categories of weak, admissible and ordinary g-twisted V -modules g g by Cw (V ), Ca (V ) and C g (V ), respectively. If g = 1, we habitually remove the index g. Now again consider the vertex operator V ⊗k and the k-cycle g = (12 · · · k). Define g (V ⊗k ) Tgk : Cw (V ) −→ Cw
(M, Y ) → (Tgk (M), Yg ) = (M, Yg ) f → Tgk (f ) = f for (M, Y ) an object and f a morphism in Cw (V ). The following corollary to Theorem 3.9 is obvious. g
Corollary 3.10. Tgk is a functor from the category Cw (V ) to the category Cw (V ⊗k ) such that: (1) Tgk preserves irreducible objects; (2) The restrictions of Tgk to Ca (V ) and C(V ) g are functors from Ca (V ) and C(V ) to Ca (V ⊗k ) and C g (V ⊗k ), respectively. g
In the next section we will construct a functor Ugk from the category Cw (V ⊗k ) to the category Cw (V ) such that Ugk ◦ Tgk = idCw (V ) and Tgk ◦ Ugk = idCwg (V ⊗k ) .
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Remark 3.11. In constructing the weak g-twisted V ⊗k -module structure on the weak V -module (M, Y ), we chose to define Yg (u1 , z) by (3.13) which then generate Yg (v, z) for all v ∈ V ⊗k . But our choice of u1 for u ∈ V as generators is not canonical. We could j just as well have chosen to define Yg (uj , z) = Y¯ (u, z) for j = 2, . . . , k, and generated j g-twisted operators Yg (v, z) for v ∈ V ⊗k from these rather than from Yg1 (u1 , z). These j
new g-twisted operators Yg (v, z) are related to the old ones Yg = Yg1 by j +1
Yg
(v, z) =
lim
z1/k →ηj z1/k
Yg (v, z)
for j = 1, . . . , k − 1. This is a reflection of the fact that for any vertex operator algebra (V , Y, 1, ω) and any automorphism g of V , if (M, Yg ) is a weak g-twisted V -module, then so is (M, Y˜g ), where Y˜g (v, z) =
lim
z1/k →ηj z1/k
Yg (v, z).
Furthermore, if (M, Yg ) is admissible, then (M, Yg ) and (M, Y˜g ) are isomorphic as g-twisted V -modules via f : (M, Yg ) → (M, Y˜g ) w → ηj kn w for w ∈ M(n). (Note that here we have used the fact that ηk(wt v) = 1 for all v ∈ V of homogeneous weight wt v.) 4. Constructing a Weak V -Module Structure on a Weak g = (12 · · · k)-Twisted V ⊗k -Module For k ∈ Z+ and g = (12 · · · k), let M = (M, Yg ) be a weak g-twisted V ⊗k -module. Motivated by the construction of weak g-twisted V ⊗k -modules from weak V -modules in Sect. 3, we consider Yg ((k (zk )−1 u)1 , zk )
(4.1)
for u ∈ V , where k (z)−1 = Vk (z)−1 is given by (2.7), i.e., k (z)−1 = z−(1/k−1)L(0) k L(0) exp − aj z−j/k L(j ) . j ∈Z+
Note that (4.1) is multivalued since Yg ((k (z)−1 u)1 , z) ∈ (EndM)[[z1/k , z−1/k ]]. We thus define YU (u, z) = Yg ((k (zk )−1 u)1 , zk ) to be the unique formal Laurent series in (EndM)[[z, z−1 ]] given by taking (zk )1/k = z. Our goal in this section is to construct g a functor Ugk : Cw (V ⊗k ) → Cw (V ) with Ugk (M, Y ) = (Ugk (M), YU ) = (M, YU ). If we instead define YU by taking (zk )1/k = ηj z for η = e−2πi/k with j = 1, . . . , k − 1, then (M, YU ) will not be a weak V -module. Further note that this implies that if we allow z to be a complex number and if we define z1/k using the principal branch of the logarithm, then much of our work in this section is valid if and only if −π/k < arg z < π/k.
Permutation Twisted Tensor Product VOAs
Lemma 4.1. For u ∈ V , we have
373
d d ((zk )1/k ) YU (u, z) dz dz d = YU (u, z) dz
YU (L(−1)u, z) =
on Ugk (M) = M. Thus the L(−1)-derivative property holds for YU . Proof. The proof is similar to that of Lemma 3.2. By Corollary 2.5 we have k (z)−1 L(−1) − kz−1/k+1 L(−1)k (z)−1 = kz−1/k+1
d k (z)−1 . dz
Making the change of variable z → zk gives k (zk )−1 L(−1) − k(zk )−1/k zk L(−1)k (zk )−1 = (zk )−1/k z
d k (zk )−1 . dz
Thus if (zk )1/k = ηj z, we have d Yg ((k (zk )−1 u)1 , zk ) dz 1 d d = Yg k (zk )−1 u , zk + Yg ((k (zk )−1 u)1 , x k )|x=z dz dx d 1 = Yg k (zk )−1 u , zk + kzk−1 Yg (L(−1)(k (zk )−1 u)1 , zk ) dz d 1 = Yg k (zk )−1 u , zk + kzk−1 Yg ((L(−1)k (zk )−1 u)1 , zk ) dz j = η Yg ((k (zk )−1 L(−1)u)1 , zk ). Since by definition YU (u, z) = Yg ((k (zk )−1 u)1 , zk ) with (zk )1/k = z, the result follows. Lemma 4.2. Let u, v ∈ V . Then in Ugk (M) = M, z1 − z0 YU (Y (u, z0 )v, z2 ). [YU (u, z1 ), YU (v, z2 )] = Resz0 z2−1 δ z2 Proof. The proof is similar to the proof of Lemma 3.3. From the twisted Jacobi identity, we have 1 (z1 − z0 )1/k Yg (Y (u1 , z0 )v 1 , z2 ). (4.2) [Yg (u1 , z1 ), Yg (v 1 , z2 )] = Resz0 z2−1 δ 1/k k z2 Therefore, [YU (u, z1 ), YU (v, z2 )] = [Yg ((k (z1k )−1 u)1 , z1k ), Yg ((k (z2k )−1 v)1 , z2k )] (z1k − x)1/k 1 −k = Resx z2 δ Yg (Y ((k (z1k )−1 u)1 , x)(k (z2k )−1 v)1 , z2k ). k z2
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We want to make the change of variable x = z1k − (z1 − z0 )k , where we choose z0 such that ((z1 − z0 )k )1/k = z1 − z0 . Then noting that (z1k − x)n/k |x=zk −(z1 −z0 )k = (z1 − z0 )n 1 for all n ∈ Z, and using (3.12), we have [YU (u, z1 ), YU (v, z2 )] = Resz0 z2−k (z1 − z0 )k−1 δ
z1 − z0 z2
Yg (Y ((k (z1k )−1 u)1 , z1k − (z1 − z0 )k )
· (k (z2k )−1 v)1 , z2k ) z1 − z0 = Resz0 z2−1 δ Yg (Y ((k (z1k )−1 u)1 , z1k − (z1 − z0 )k )(k (z2k )−1 v)1 , z2k ) z2 z1 − z0 = Resz0 z2−1 δ Yg ((Y (k ((z2 + z0 )k )−1 u, (z2 + z0 )k − z2k ) z2 · k (z2k )−1 v)1 , z2k ). Thus the proof is reduced to proving Y (k ((z2 + z0 )k )−1 u, (z2 + z0 )k − z2k )k (z2k )−1 = k (z2k )−1 Y (u, z0 ) , i.e., proving k (z2k )Y (k ((z2 + z0 )k )−1 u, (z2 + z0 )k − z2k )k (z2k )−1 = Y (u, z0 ) .
(4.3)
In Proposition 2.2, substituting u, z and z0 by k ((z2 + z0 )k )−1 u, z2k and (z2 + z0 )k − z2k , respectively, gives Eq. (4.3). Theorem 4.3. With the notations as above, Ugk (M, Yg ) = (Ugk (M), YU ) = (M, YU ) is a weak V -module. Proof. Since the L(−1)-derivation property has been proved for YU in Lemma 4.1, we only need to prove the Jacobi identity which is equivalent to the commutator formula given by Lemma 4.2 and the associator formula which states that for u, v ∈ V and w ∈ Ugk (M) there exists a positive integer n such that (z0 + z2 )n YU (u, z0 + z2 )YU (v, z2 )w = (z2 + z0 )n YU (Y (u, z0 )v, z2 )w. 1 1 i 1 Write u1 = k−1 i=0 u(i) where gu(i) = η u(i) . Then from the twisted Jacobi identity, we have the following associator: there exists a positive integer m such that for n ≥ m, (z0 + z2 )i/k+n Yg (u1(i) , z0 + z2 )Yg (v 1 , z2 )w = (z2 + z0 )i/k+n Yg (Y (u1(i) , z0 )v 1 , z2 )w for i = 0, . . . , k − 1. Replacing z2 by z2k and z0 by (z0 + z2 )k − z2k gives (z0 + z2 )i+kn Yg (u1(i) , (z0 + z2 )k )Yg (v 1 , z2k )w = (z2 + z0 )i+kn Yg (Y (u1(i) , (z2 + z0 )k − z2k )v 1 , z2k )w. Note that if a ∈ V ⊗k such that ga = ηi a, then Yg (a, z) = l∈i/k+Z an z−l−1 . Thus there exists a positive integer mi such that if ni ≥ mi , then (z0 + z2 )ni Yg (u1(i) , (z0 + z2 )k )Yg (v 1 , z2k )w = (z2 + z0 )ni Yg (Y (u1(i) , (z2 + z0 )k − z2k )v 1 , z2k )w
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for i = 0, . . . , k − 1. As a result we see that there exists a positive integer m such that if n ≥ m, then (z0 + z2 )n Yg (u1 , (z0 + z2 )k )Yg (v 1 , z2k )w = (z2 + z0 )n Yg (Y (u1 , (z2 + z0 )k − z2k )v 1 , z2k )w. Now write k ((z0 +z2 )k )−1 u =
j ∈N uj (z0 +z2 )
sj
for some uj ∈ V and integers sj , t and note that this is a finite sum. Similarly we have a finite sum k (z2k )−1 v = j ∈N vj z2j for some vj ∈ V and tj ∈ Z. Thus there exists a positive integer m such that if n ≥ m, then (z0 + z2 )n+si Yg (u1i , (z0 + z2 )k )Yg (vj1 , z2k )w = (z2 + z0 )n+si Yg (Y (u1i , (z2 + z0 )k − z2k )vj1 , z2k )w for all i, j ∈ N. Finally we have for n ≥ m, (z0 + z2 )n YU (u, z0 + z2 )YU (v, z2 )w = (z2 + z0 )n Yg ((k ((z0 + z2 )k )−1 u)1 , (z0 + z2 )k )Yg ((k (z2k )−1 v)1 , z2k )w t = (z0 + z2 )n+si z2j Yg (u1i , (z0 + z2 )k )Yg (vj1 , z2k )w i,j ≥0
=
i,j ≥0
t
(z2 + z0 )n+si z2j Yg (Y (u1i , (z2 + z0 )k − z2k )vj1 , z2k )w
= (z2 + z0 )n Yg (Y (k ((z2 + z0 )k )−1 u)1 , (z2 + z0 )k − z2k )(k (z2k )−1 v)1 , z2k )w = (z2 + z0 )n Yg ((k (z2k )−1 Y (u, z0 )v)1 , z2k )w = (z2 + z0 )n YU (Y (u, z0 )v, z2 )w,
where we have used Eq. (4.3), completing the proof. g
Theorem 4.4. Ugk is a functor from the category Cw (V ⊗k ) of weak g-twisted V ⊗k modules to the category Cw (V ) of weak V -modules such that Tgk ◦ Ugk = idCwg (V ⊗k ) and g Ugk ◦ Tgk = idCw (V ) . In particular, the categories Cw (V ⊗k ) and Cw (V ) are isomorphic. Moreover, (1) The restrictions of Tgk and Ugk to the category of admissible V -modules Ca (V ) and g to the category of admissible g-twisted V ⊗k -modules Ca (V ⊗k ), respectively, give category isomorphisms. In particular, V is rational if and only if V ⊗k is g-rational. (2) The restrictions of Tgk and Ugk to the category of ordinary V -modules C(V ) and to the category of ordinary g-twisted V ⊗k -modules C g (V ⊗k ), respectively, give category isomorphisms. Proof. It is trivial to verify Tgk ◦ Ugk = idCwg (V ⊗k ) and Ugk ◦ Tgk = idCw (V ) from the definitions of the functors Tgk and Ugk . Parts 1 and 2 follow from Theorem 3.9.
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5. Twisted Sectors for an Arbitrary k-Cycle In this section we extend Theorem 4.4 to the category of weak g-twisted V ⊗k -modules, where g is an arbitrary k-cycle. But we first recall some general results following, for example, [DLM3]. Let V be any vertex operator algebra, and let g, h ∈ Aut(V ). There is an isomorphism g from the category of weak g-twisted V -modules Cw (V ) to the category of weak hgh−1 hgh−1
twisted V -modules Cw
(V ) given by −1
g hgh (V ) −→ Cw (V ) h : Cw (M, Yg ) → (M, Yhgh−1 ),
where Yhgh−1 (u, z) = Yg (hu, z) for u ∈ V . Moreover it is clear that (M, Yg ) is irreducible, admissible, or ordinary if and only if (M, Yhgh−1 ) is irreducible, admissible, or ordinary, respectively. Now suppose g is an arbitrary k-cycle and let g = (12 · · · k). Then there exists h ∈ Sk such that g = hgh−1 . However, this h is unique only up to multiplication on the right by powers of g. Thus given g , we can specify h uniquely by requiring that h leave 1 fixed. Denote this by h1 . Then we have the following corollary to Theorem 4.4. Corollary 5.1. Let g ∈ Sk be a k-cycle, let g = (12 · · · k), and let h1 be the unique element of Sk that fixes 1 and satisfies g = h1 gh−1 1 . Then we have the following isomorphism of categories
g (V ⊗k ) Tgk = h1 ◦ Tgk : Cw (V ) −→ Cw
(M, Y ) → (Tgk (M), Yg ) = (M, Yh1 gh−1 ), 1
where Yg (v, z) = Yh1 gh−1 (v, z) = Yg (h1 v, z) 1
for v ∈
V ⊗k ,
and Yg is uniquely determined by Yg (u1 , z) = Y¯ (u, z) = Y (k (z)u, z1/k ), Yg (uj +1 , z) =
lim
z1/k →η−j z1/k
Yg (u1 , z)
for u ∈ V . Moreover, Tgk preserves irreducible, admissible and ordinary objects. Remark 5.2. In the corollary above, we could just have easily defined Tgk by Tgk = hl ◦Tgk , where hl , rather than fixing 1, is the unique element of Sk that takes 1 to l for l = 2, . . . , k and satisfies g = hl gh−1 l . Then these new g -twisted operators defined on M denoted l by Yg would differ from those defined in Corollary 5.1, denoted by Yg , by 1 Ygl+1 (u , z) =
lim
z1/k →ηl z1/k
Yg (u1 , z)
⊗k is determined for l = 1, . . . , k − 1 and u ∈ V , where of course Ygl+1 (v, z) for v ∈ V l+1 1 j +1 , z) = lim by Ygl+1 (u z1/k →η−j z1/k Yg (u , z).
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6. Twisted Sectors for Arbitrary Permutations In order to determine the various twisted module categories for an arbitrary permutation, we need to study twisted modules for tensor product vertex operator algebras in general. For s ∈ Z+ and i = 1, ...s, let V i = (V i , Yi , 1i , ωi ) be a vertex operator algebra and gi an automorphism of V i of finite order. Then g = g1 ⊗ · · · ⊗ gs is an automorphism of the tensor product vertex operator algebra V 1 ⊗ · · · ⊗ V s of finite order. If W i are weak gi -twisted V i -modules for i = 1, . . . , s, then the tensor product W 1 ⊗ · · · ⊗ W s = W is a weak g-twisted V 1 ⊗ · · · ⊗ V s -module in the obvious way. Thus from Proposition 4.7.2 in [FHL], we have the following lemma. Lemma 6.1. If W i are irreducible weak gi -twisted V i -modules for i = 1, . . . , s, then W = W 1 ⊗ · · · ⊗ W s is an irreducible weak g-twisted V 1 ⊗ · · · ⊗ V s -module. Proposition 6.2. The notation is the same as before. If V i is gi -rational for all i then V 1 ⊗· · ·⊗V s is g-rational and each irreducible V 1 ⊗· · ·⊗V s -module W is isomorphic to W 1 ⊗ · · · ⊗ W s for some irreducible gi -twisted V i -modules W i . Proof. Let W = ⊕n∈Q+ W (n) be an admissible g-twisted V 1 ⊗ · · · ⊗ V s -module. We need to show that W is completely reducible. It is sufficient to prove that a submodule generated by any vector w ∈ W (n) is completely reducible, and thus we can assume that W is generated by w. Identify V i with the vertex operator algebra 11 ⊗ · · · 1i−1 ⊗ V i ⊗ 1i+1 ⊗ · · · ⊗ 1s . (This is “almost” a subalgebra of V 1 ⊗ · · · ⊗ V s , failing to be a subalgebra due to the fact that 11 ⊗ · · · 1i−1 ⊗ V i ⊗ 1i+1 ⊗ · · · ⊗ 1s has a different Virasoro element from that of V 1 ⊗ · · · ⊗ V s .) With this identification, the submodule W generated by w is an admissible gi -twisted V i -module. By [DM] and [Li1], W is spanned 1 · · · v s w|v i ∈ V i , m ∈ Q}. Set W i to be the span of {v w|v ∈ V i , m ∈ Q}. by {vm i m ms 1 Since V i is gi -rational, W i is an admissible gi -twisted V i -module which is a direct sum of irreducible gi -twisted V i -modules. Note that the map f from W 1 ⊗ · · · ⊗ W s to W 1 w ⊗· · ·⊗v s w to v 1 · · · v s w for v i ∈ V i , m ∈ Q is a V 1 ⊗· · ·⊗V s which sends vm i ms m1 ms 1 homomorphism. By Lemma 6.1, W 1 ⊗ · · · ⊗ W s is completely reducible. Thus W as a homomorphic image of W 1 ⊗ · · · ⊗ W s is completely reducible. In particular, if W is irreducible then W is isomorphic to a tensor product of irreducible gi -twisted V i modules. Remark 6.3. Proposition 6.2 is a generalization of a special case of a result proved in [FHL] which states that in the case gi = 1 for all i, any ordinary irreducible V 1 ⊗ · · · ⊗ V s -module W on whose lowest weight space the operators Li (0) have only rational eigenvalues is isomorphic to the tensor product of some irreducible V i -modules. In our case Li (0) is the component operator of Yi (ωi , z) of weight 0, the gi are general automorphisms of V i of finite order, but we have restricted to the case where each V i is gi -rational. Let k be a fixed positive integer, and let g ∈ Sk . Then g can be written as a product of disjoint cycles g = g1 · · · gp , where the order of gi is ki such that i ki = k. (Note that we are including 1-cycles.) Furthermore, there exists h ∈ Sk satisfying g = hg1 · · · gp h−1 such that gi is a ki -cycle which permutes the numbers ( i−1 j =1 kj ) + i i−1 1, ( j =1 kj ) + 2, ..., j =1 kj . In Sects. 3, 4 and 5 we have determined various gi twisted V ⊗ki -module categories in terms of corresponding V -module categories via the functor Tgkii for gi a ki -cycle. Thus we have the following:
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Theorem 6.4. Let Wi be a weak V -module for i = 1, . . . , p. Given g ∈ Sk and a k decomposition g = hg1 · · · gp h−1 as above, h ◦ (Tgk1 (W1 ) ⊗ · · · ⊗ Tg p (Wp )) is a weak p
1
g-twisted V ⊗k -module. Moreover,
k
(1) If each Wi is irreducible then h ◦ (Tgk1 (W1 ) ⊗ · · · ⊗ Tg p (Wp )) is irreducible. 1
p
(2) If each Wi is an admissible (resp., ordinary) V -module, then h ◦ (Tgk1 (W1 ) ⊗ · · · ⊗ k
Tg p (Wp )) is an admissible (resp., ordinary) g-twisted V ⊗k -module.
1
p
(3) If V is a rational vertex operator algebra, then V ⊗k is g-rational and all irreducible k g-twisted modules are given by h ◦ (Tgk1 (W1 ) ⊗ · · · ⊗ Tg p (Wp )) where Wi are p 1 irreducible V -modules. Proof. Parts (1) and (2) follow from Corollary 5.1 and Lemma 6.1. Part (3) follows from Corollary 5.1 and Proposition 6.2. Remark 6.5. Note that in the theorem above, we have not specified a unique decomposik tion g = hg1 · · · gp h−1 but rather have given the functor h◦(Tgk1 ⊗· · ·⊗Tg p ) for a given 1
p
such (non-unique) decomposition g = hg1 · · · gp h−1 . However, for any decomposition k
g = hg1 · · · gp h−1 , the resulting g-twisted V ⊗k -module h◦(Tgk1 (W1 )⊗· · ·⊗Tg p (Wp )) p 1 is isomorphic to that obtained from any other decomposition. Similar to the symmetries discussed in Remarks 3.11 and 5.2 above, the isomorphism consists of transformations lim 1/ki −ji 1/ki of each i th tensor factor in the g-twisted vertex operator where ηki =
z →ηk z i −2πi/k i and j e
i
= 1, . . . , ki − 1.
7. Twisted Sectors for the Product of a Permutation with an Automorphism of V In this section we give a slight generalization of our previous results to a broader class of automorphisms. Note that Aut(V ) acts on V ⊗k diagonally and this action commutes with the action of Sk . Let γ ∈ Aut(V ) and g ∈ Sk . In this section we determine various γ g-twisted V ⊗k -module categories. (Although the diagonal action of γ on V ⊗k is more appropriately denoted by γ ⊗k , to simplify notation, we will write this as γ with the diagonal action being understood.) First, we assume that g = (12 · · · k) and show that suitable variants of all the arguments and results in Sects. 3 and 4 remain valid for the case of γ g-twisted V ⊗k -module categories. Then we generalize the permutation g as we did in Sects. 5 and 6 and show that again suitable variants of our arguments remain valid. We proceed to formulate these results. Let g = (12 · · · k). For each positive integer n, set ηn = e−2πi/n . Then ηn/m = ηnm . Let o(γ ) = l, and set o(γ ) = l = sd, where d is the greatest common divisor of l and k. Furthermore, if we let o(γ g) = n, then o(γ g) = n = sk. Let (W, Y ) be a weak γ g-twisted V ⊗k -module. Then the analogue of Eq. (3.4) is z1 − z2 z2 − z1 −1 −1 1 1 Y (u , z1 )Y (v , z2 ) − z0 δ Y (v 1 , z2 )Y (u1 , z1 ) z0 δ z0 −z0 n−1 1/n 1 −1 j (z1 − z0 ) = z2 Y (Y (γ j g j u1 , z0 )v 1 , z2 ) δ ηn (7.1) 1/n n z j =0 2
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for u, v ∈ V . Let u be an eigenvector for γ with eigenvalue ηlr for some r = 1, . . . , l. rj Then γ j g j u1 = ηl uj +1 , where uj +k = uj . Thus similar to the situation in Sect. 3, j j 1 1 Y (γ g u , z0 )v only involves nonnegative integer powers of z0 if j = 0 (mod k), and [Y (u1 , z1 ), Y (v 1 , z2 )] s−1 1/n 1 kq (z1 − z0 ) rkq = Y (Y (ηl u1 , z0 )v 1 , z2 ) Resz0 z2−1 δ ηn 1/n n z q=0
=
s−1 q=0
=
1/n 1 −1 q (z1 − z0 ) rqk/d ηs Resz0 z2 δ ηs Y (Y (u1 , z0 )v 1 , z2 ) 1/n n z
n−1 s−1 q=0 p=0
1 qp Resz0 z2−1 ηs n
· Y (Y (u1 , z0 )v 1 , z2 ) =
2
k s−1 q=0 t=1
Resz0
1 −1 z sk 2
2
z1 − z 0 z2
z1 − z 0 z2
p/n z1 − z0 rqk/d ηs δ z2
(ts−rk/d)/n z1 − z0 δ z2
· Y (Y (u1 , z0 )v 1 , z2 ) k z1 − z0 t/k z1 − z0 −r/ l 1 z1 − z0 = Resz0 z2−1 δ k z2 z2 z2 t=1
· Y (Y (u1 , z0 )v 1 , z2 ) z1 − z0 −r/ l 1 (z1 − z0 )1/k = Resz0 z2−1 Y (Y (u1 , z0 )v 1 , z2 ). δ 1/k k z2 z
(7.2)
2
As in Sect. 3 this shows that for u ∈ V the component operators of Y (u1 , z) on W form a Lie algebra. Let (M, Y ) be a weak γ k -twisted V -module. Set Y¯ (u, z) = Y (k (z)u, z1/k ). Lemma 7.1. The assertion of Lemma 3.2 holds in the present setting: for u ∈ V , d ¯ Y¯ (L(−1)u, z) = Y (u, z). dz Proof. The steps of the proof for Lemma 3.2 still hold in the present setting for (M, Y ) a γ k -twisted V -module rather than just a V -module. The analogue of Lemma 3.3 is Lemma 7.2. For u, v ∈ V such that γ u = ηlr u, we have 1 [Y¯ (u, z1 ), Y¯ (v, z2 )] = Resz0 z2−1 k
z1 − z 0 z2
−r/ l (z1 − z0 )1/k ¯ Y (Y (u, z0 )v, z2 ). δ 1/k z2 (7.3)
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Proof. From the twisted Jacobi identity, we have [Y (u, z1 ), Y (v, z2 )] =
Resx z2−1
z1 − x z2
−r/ l z1 − x Y (Y (u, x)v, z2 ) δ z2
(7.4)
for u ∈ V such that γ u = ηlr u. Following the steps of the proof of Lemma 3.3 using the commutator formula (7.4) instead of (3.10) gives (7.3). For u ∈ V such that γ u = ηlr define operators Yγ g on V ⊗k by Yγ g (u1 , z) = Y¯ (u, z) = Y (k (z)u, z1/k ), rj ηl Yγ g (uj +1 , z)
=
lim
−j
z1/n →ηn z1/n
Y (u , z) 1
(7.5) (7.6)
for j = 0, . . . , k − 1. Then as before, the operators Yγ g (uj , z) for u ∈ V and j = 1, . . . , k are mutually local and generate a local system A in the sense of [Li2]. Let σ be a map from A to A such that σ Yγ g (uj , z) = ηl−r Yγ g (uj +1 , z) for u ∈ V satisfying γ u = ηlr u, and for j = 1, . . . , k. Again A has the structure of a vertex algebra (A, YA ) and σ extends to an automorphism of A of order n such that M is naturally a weak σ -twisted A-module in the sense that Y (α(z), z1 ) = α(z1 ) for α(z) ∈ A. Lemma 7.3. The assertions of Lemmas 3.6, 3.7 and 3.8 hold for Yγ g and the corresponding local system. Proof. The steps of the proofs of Lemmas 3.6, 3.7 and 3.8 remain valid in the present setting. Let (M, Y ) be a weak γ k -twisted V -module. Define Tγkg (M, Y ) = (Tγkg (M), Yγ g ) = (M, Yγ g ). That is Tγkg (M, Y ) is M as the underlying vector space and the vertex operator Yγ g is given by (7.5) and (7.6) . Now repeating the proof of Theorem 3.9 under the present circumstances, we have the following result which is a generalization of Theorem 3.9. Theorem 7.4. (Tγkg (M), Yγ g ) is a weak γ g-twisted V ⊗k -module such that Tγkg (M) = M and Yγ g , defined by (7.5) and (7.6), is the linear map from V ⊗n to (End Tγkg (M))[[z1/n , z−1/n ]] defining the twisted module structure where n = o(γ g). Moreover, (1) (M, Y ) is an irreducible weak γ k -twisted V -module if and only if (Tγkg (M), Yγ g ) is an irreducible weak γ g-twisted V ⊗k -module. (2) M is an admissible γ k -twisted V -module if and only if Tγkg (M) is an admissible γ g-twisted V ⊗k -module. (3) M is an ordinary γ k -twisted V -module if and only if Tγkg (M) is an ordinary γ gtwisted V ⊗k -module.
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Let k
γ γg (V ) −→ Cw (V ⊗k ) Tγkg : Cw
(M, Y ) → (Tγkg (M), Yγ g ) = (M, Yγ g ) f → Tγkg (f ) = f γk
for (M, Y ) an object and f a morphism in Cw (V ). We have the following corollary to Theorem 7.4. γk
γg
Corollary 7.5. Tγkg is a functor from the category Cw (V ) to the category Cw (V ⊗k ) such γk
that: (1) Tγkg preserves irreducible objects; (2) The restrictions of Tγkg to Ca (V ) and γk
k
γg
k
C γ (V ) are functors from Ca (V ) and C γ (V ) to Ca (V ⊗k ) and C γ g (V ⊗k ), respectively. Note that if γ is the identity, Theorem 7.4 and Corollary 7.5 reduce to Theorem 3.9 and Corollary 3.10, respectively. Now let M = (M, Yγ g ) be a weak γ g-twisted V ⊗k -module. In analogy to Sect. 4, we set Uγkg (M) = M and define YU (u, z) = Yγ g ((k (zk )−1 u)1 , zk ) for u ∈ V to be the unique power series obtained by letting (zk )1/k = z. Lemma 4.1 remains valid in this case and Lemma 4.2 is modified as follows: Lemma 7.6. Let u, v ∈ V such that γ u = ηlr . Then on Uγkg (M), [YU (u, z1 ), YU (v, z2 )] = Resz0 z2−1
z1 − z 0 z2
−rk/ l z1 − z0 YU (Y (u, z0 )v, z2 ). δ z2 (7.7)
Proof. The γ g-twisted vertex operators satisfy [Yγ g (u1 , z1 ), Yγ g (v 1 , z2 )] 1 −1 z1 − z0 −r/ l (z1 − z0 )1/k = Resz0 z2 Yγ g (Y (u1 , z0 )v 1 , z2 ). δ 1/k k z2 z
(7.8)
2
Following the steps for the proof of Lemma 4.2 using (7.8) instead of (4.2) gives (7.7). Repeating the proof of Theorem 4.3 in the present setting gives: Theorem 7.7. With the notations as above, (Uγkg (M), YU ) is a weak γ k -twisted V module. Thus we have the following analogue of Theorem 4.4. γg
Theorem 7.8. Uγkg is a functor from the category Cw (V ⊗k ) of weak γ g-twisted V ⊗k γk
modules to the category Cw (V ) of weak γ k -twisted V -modules such that Tγkg ◦ Uγkg = idCwγ g (V ⊗k ) and Uγkg ◦Tγkg = id are isomorphic. Moreover,
γk
Cw (V )
γg
γk
. In particular, the categories Cw (V ⊗k ) and Cw (V )
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K. Barron, C. Dong, G. Mason γk
(1) The restrictions of Tγkg and Uγkg to the category of admissible V -modules Ca (V ) γg and to the category of admissible γ g-twisted V ⊗k -modules Ca (V ⊗k ), respectively, k give category isomorphisms. In particular, V is γ -rational if and only if V ⊗k is γ g-rational. k (2) The restrictions of Tγkg and Uγkg to the category of ordinary V -modules C γ (V ) and to the category of ordinary γ g-twisted V ⊗k -modules C γ g (V ⊗k ), respectively, give category isomorphisms. Recalling the conjugation functor h from Sect. 5, and again noting that the diagonal action of γ on V ⊗k commutes with that of the symmetric group so that γ hgh−1 = hγ gh−1 , we have the following corollary to Theorem 7.8. Corollary 7.9. Let g ∈ Sk be a k-cycle, let g = (12 · · · k) and let h1 be the unique element of Sk that fixes 1 and satisfies g = h1 gh−1 1 . Then we have the following isomorphism of categories: k
γ γg Tγkg = h1 ◦ Tγkg : Cw (V ) −→ Cw (V ⊗k )
(M, Y ) → (Tγkg (M), Yγ g ) = (M, Yh1 γ gh−1 ), 1
where Yh1 γ gh−1 (v, z) = Yγ g (h1 v, z) 1
and Yγ g is uniquely determined by (7.5) and (7.6). Moreover, Tγkg preserves irreducible, admissible and ordinary objects. Finally we deal with the case γ g when g is an arbitrary permutation in Sk . As in Sect. 6 we write g as a product of disjoint cycles g = hg1 · · · gp h−1 where the order of gi is ki i−1 such that i ki = k, and where gi permutes the numbers ( i−1 j =1 kj ) + 1, ( j =1 kj ) + 2, ..., ij =1 kj . The following theorem generalizes Theorem 6.4. Theorem 7.10. Let Wi be a weak γ ki -twisted V -module for i = 1, . . . , p. Given g ∈ Sk k and a decomposition g = hg1 · · · gp h−1 as above, h ◦ (Tγkg1 1 (W1 ) ⊗ · · · ⊗ Tγ gpp (Wp )) is a weak γ g-twisted V ⊗k -module. Moreover, k
(1) If each Wi is irreducible, then h ◦ (Tγkg1 1 (W1 ) ⊗ · · · ⊗ Tγ gpp (Wp )) is irreducible. (2) If each Wi is an admissible (resp., ordinary) V -module, then h ◦ (Tγkg1 1 (W1 ) ⊗ · · · ⊗ k
Tγ gpp (Wp )) is an admissible (resp., ordinary) γ g-twisted V ⊗k -module. (3) If V is a γ ki -rational vertex operator algebra for i = 1, . . . , p, then V ⊗k is γ grational and all irreducible γ g-twisted modules are given by h ◦ (Tγkg1 1 (W1 ) ⊗ · · · ⊗ k
Tγ gpp (Wp )), where Wi are irreducible γ ki -twisted V -modules. Of course the analogous symmetries discussed in Remark 5.2 and Remark 6.5 hold for Corollary 7.9 and Theorem 7.10, respectively. Acknowledgement. The authors thank James Lepowsky for pointing out some mistakes in earlier versions of this paper and for giving helpful comments on the paper’s exposition. We would also like to thank Hirotaka Tamanoi for valuable discussions.
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References [Ba]
Bantay, P.: Characters and modular properties of permutation orbifolds. Phys. Lett. B 419, 175– 178 (1998) [Bo] Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986) [BHS] Borisov, L., Halpern, M.B. and Schweigert, C.: Systematic approach to cyclic orbifolds. Internat. J. Modern Phys. A 13, 125–168 (1998) [DHVW1] Dixon, L.J., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B 261, 678–686 (1985) [DHVW2] Dixon, L.J., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds, II. Nucl. Phys. B 274, 285–314 (1986) [DVVV] Dijkgraaf, R. Vafa, C., Verlinde, E., Verlinde, H.: The operator algebra of orbifold models. Commun. Math. Phys. 123, 485–526 (1989) [DMVV] Dijkgraaf, R., Moore, G., Verlinde, E., Verlinde, H.: Elliptic genera of symmetric products and second quantized strings. Commun. Math. Phys. 185, 197–209 (1997) [D] Dong, C.: Twisted modules for vertex algebras associated with even lattices. J. Algebra 165, 91–112 (1994) [DL1] Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Math. Vol. 112, Boston: Birkhäuser, 1993 [DL2] Dong, C., Lepowsky, J.: The algebraic structure of relative twisted vertex operators. J. Pure Appl. Algebra 110, no. 3, 259–295 (1996) [DLM1] Dong, C., Li, H., Mason, G.: Regularity of rational vertex operator algebras. Adv. in Math. 132, 148–166 (1997) [DLM2] Dong, C., Li, H., Mason, G.: Twisted representations of vertex operator algebras. Math. Ann. 310, 571–600 (1998) [DLM3] Dong, C., Li, H., Mason, G.: Modular invariance of trace functions in orbifold theoryand generalized Moonshine. Commun. Math. Phys. 214, 1–56 (2000) [DM] Dong, C., Mason, G.: Nonabelian orbifolds and boson-fermion correspondence. Commun. Math. Phys. 163, 523–559 (1994) [FFR] Feingold, A.J., Frenkel, I.B. and Ries, J.F.X.: Spinor Construction of Vertex Operator Algebras, (1) Triality and E8 . Contemp. Math. 121. Providence, RI: Am. Math. Soc., 1991 [FHL] Frenkel, I.B., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs American Math. Soc. 104, 1993 [FLM1] Frenkel, I.B., Lepowsky, J. and Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc. Natl. Acad. Sci. USA 81, 3256–3260 (1984) [FLM2] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator calculus. In: Mathematical Aspects of String Theory, Proc. 1986 Conference, San Diego. ed. by S.-T. Yau, Singapore: World Scientific 1987, pp. 150–188 [FLM3] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Math. Vol. 134, Academic Press, 1988 [FKS] Fuchs, J., Klemm, A., Schmidt, M.G.: Orbifolds by cyclic permutations in Gepner type superstrings and in the corresponding Calabi–Yau manifolds. Ann. Phys. 214, 221–257 (1992) [GKO] Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro Algebra and superVirasoro algebras Commun. Math. Phys. 103, 105–119 (1986) [H] Huang, Y.: Two Dimensional Conformal Geometry and Vertex Operator Algebras. Progress in Math. Vol. 148, Boston: Birkhäuser, 1997 [KS] Klemm, A., Schmidt, M.G.: Orbifolds by cyclic permutations of tensor product conformal field theories. Phys. Lett. B245, 53–58 (1990) [Le1] Lepowsky, J.: Calculus of twisted vertex operators. Proc. Nat. Acad. Sci. USA 82, no. 24, 8295– 8299 (1985) [Le2] Lepowsky, J.: Perspectives on vertex operators and the Monster. In: Proc. 1987 Symposium on the Mathematical Heritage of Hermann Weyl, Duke Univ. Proc. Symp. Pure Math., American Math. Soc., 48, 1988, pp. 181–197 [Li1] Li, H.: An approach to tensor product theory for representations of a vertex operator algebra, Ph.D. thesis, Rutgers University, 1994
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[Li2]
[Z]
K. Barron, C. Dong, G. Mason
Li, H.. Local systems of twisted vertex operators, vertex superalgebras and twisted modules. In: Moonshine, the Monster, and related topics, South Hadley, MA, 1994 Contemporary Math. 193, 203–236 (1996) Zhu,Y.-C.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)
Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 227, 385 – 419 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Three-Point Functions for M N /SN Orbifolds with N = 4 Supersymmetry Oleg Lunin, Samir D. Mathur Department of Physics, The Ohio State University, Columbus, OH 43210, USA Received: 29 March 2001 / Accepted: 20 January 2002
Abstract: The D1–D5 system is believed to have an “orbifold point” in its moduli space where its low energy theory is a N = 4 supersymmetric sigma model with target space M N /SN , where M is T 4 or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N ; these expressions are “universal” in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space.
1. Introduction The D1–D5 system has been of great interest in recent developments in string theory. The system is described by a collection of n5 D5 branes which are wrapped on a 4manifold M (which can be a T 4 or a K 3 ) and n1 D1 branes parallel to the noncompact direction of the D5 branes and bound to them. This system has been very important for issues related to black holes, since it yields, upon addition of momentum excitations, a supersymmetric configuration which has a classical (i.e. not Planck size) horizon. In particular, the Bekenstein entropy computed from the classical horizon area agrees with the count of microstates for the extremal and near extremal black holes [1]. Further, the low energy Hawking radiation from the hole can be understood in terms of a unitary microscopic process, not only qualitatively but also quantitatively, since one finds an agreement of spin dependence and radiation rates between the semiclassically computed radiation and the microscopic calculation [2]. The AdS/CFT correspondence conjecture gives a duality between string theory on a spacetime and a certain conformal field theory (CFT) on the boundary of this spacetime [3]. The D1–D5 system gives a CFT which is dual to the spacetime AdS3 × S 3 × M.
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While it is possible to use simple models for the low energy dynamics of the D1– D5 system when one is computing the coupling to massless modes of the supergravity theory (as was done for example in the computations of Hawking radiation from the D1–D5 microstate), it is believed that the exact description of this CFT must be in terms of a sigma model with target space being a deformation of the orbifold M N /SN , which is the symmetric orbifold of N copies of M. (Here N = n1 n5 , and we must take the low energy limit of the sigma model to obtain the desired CFT.) In particular we may consider the “orbifold point” where the target space is exactly the orbifold M N /SN with no deformation. It was suggested in [4, 5] that this CFT does correspond to a certain point in the moduli space of the D1–D5 system. Thus this orbifold theory would be dual to string theory on AdS3 × S 3 × M 4 , but at this orbifold point the string theory is expected to be in a strongly coupled domain where it cannot be approximated by tree level supergravity on a smooth background. Recall that the Yang–Mills theory arising from D3 branes is dual to string theory on AdS5 × S 5 . The orbifold point of the D1–D5 system can be considered the analogue of free N = 4 supersymmetricYang–Mills theory, since it is the closest we get to a simple theory on the CFT side. Interestingly, it was found [6] that three point functions at large N in weakly coupled 4-d Yang–Mills theory arising from D3 branes were equal to the three point functions arising from the dual supergravity theory, even though the supergravity limit of string theory corresponded to strongly coupled Yang–Mills. It would be interesting to ask if there is any similar “protection” of three point function in the D1–D5 system. In this paper we find three point functions of a basic class of chiral operators in the orbifold theory. The orbifold group that we have is SN , the permutation group of N elements. This group is nonabelian, in contrast to the cyclic group ZN which has been studied more extensively in the past for computation of correlation functions in orbifold theories [7]. Though there are some results in the literature for general orbifolds [8], the study of nonabelian orbifolds is much less developed than for abelian orbifolds. It turns out however that the case of the SN orbifolds has its own set of simplifications which make it possible to develop a technique for computation of correlation functions for these theories. In [9] a method was developed to compute the correlation functions for twist operators in the bosonic CFT that emerges from sigma models with such orbifolds. The essential point in that computation was that for the permutation group the following simplification emerges. As in any orbifold theory, one can “undo” the effect of the twist operators by passing from the space where the theory is defined to a covering Riemann surface where fields are single valued. For orbifolds of the group SN , the path integral with twist insertions becomes an unconstrained path integral with no twist insertions (for one copy of the manifold M) on this covering surface. (Such a simplification would not happen if we pass to the covering surface for a general orbifold group; one would remove the twists by going to the cover, but the values of the fields in the path integral at one point on this cover could be constrained to be related to their values at other points on the cover.) The path integral on the covering space was then computed by using the conformal anomaly of the CFT. In this paper we extend the calculations of [9] to the case of theories with N = 4 supersymmetry. Thus we would obtain in particular results for the above mentioned “orbifold point” of the D1–D5 system. In order to be able to apply the formalism for any T 4 or K3 manifold that can appear in the description of the D1–D5 system, we construct a basic class of chiral operators in the theory in an abstract way, using only the definition of a twist operator of the permutation group, and the form of the superconformal algebra of the N = 4 CFT. We then compute explicitly the 3-point function of chiral operators,
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in the limit of large N , where the surviving contribution comes from the case where the covering surface is a sphere. We observe that the result is significantly simpler than the corresponding result in the bosonic theory. There are several earlier works that relate to the problem we are studying, in particular [10–16]. The plan of this paper is as follows. In Sect. 2 we construct the chiral operators corresponding to twist insertions. In Sect. 3 we review briefly the method of [9] for computing correlators of twist insertions, and explain its extension to the supersymmetric case. Section 4 discusses correlation functions of currents that are needed to extend the bosonic computation to the supersymmetric case, and also computes the 2-point functions. In Sect. 5 we compute a simple case of the 3-point functions of these chiral operators, and in Sect. 6 we do the general case. Section 7 is a discussion. 2. Constructing the Chiral Operators σn± 2.1. Twist operators in the bosonic theory. Let us start with the bosonic theory, and look at the definition of twist operators of the theory. We follow the notation used in [9], where the construction is discussed in more detail. The CFT is defined over the z plane. Over each point z the configuration in the target space is specified by an N-tuple of coordinates (X1 , . . . , XN ), where Xi is a collective symbol for the coordinates of a point in the i th copy of the manifold M. The fact that the target space is the orbifold of M N by the symmetric group SN means that the point (X1 , . . . , XN ) of M N is to be identified with the points obtained by any permutation of the Xi . First consider the path integral defining the theory in the absence of any twist operators. We can integrate over the Xi independently, without imposing the above identifications, to obtain a partition function that will be N ! times the partition function Z that would be obtained if we did take into account the identifications D[Xi ]e−S(Xi ) ≡ Z0N = N !Z, (2.1) i
where Z0 is the partition function when the target space is just one copy of M. In the above relation we have assumed that the contribution of the points where two or more of the coordinate sets Xi become equal is of measure zero in the path integral. (Thus note that (2.1) would not be true if the manifold M was replaced by a target space that had a finite number of points; the ignored configurations would then not have measure zero.1 ) To define twist operators consider an element of the permutation group in the form of a “single cycle” (1, 2, . . . , n). (Operators based on a product of two cycles can be regarded as two single cycle operators placed at the same point.) To insert a twist operator for this element of the permutation group at the point z = 0 we cut a hole |z| < around this point. As we go around this hole counterclockwise, we let the first copy of M go over into the second copy of M, and so on, returning to the first copy after n revolutions. Thus the twist operator changes the boundary conditions around z = 0. The correlation function of twist operators σi located at points zi is then defined to be the ratio of the path 1 Alternatively we can adopt (2.1) as part of the definition of the theory, taking two copies of the point (X1 , . . . , XN ) if X1 = X2 etc., so that (2.1) is true by construction. When the target space is a manifold the configurations affected are of measure zero.
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integral performed with these twisted boundary conditions to the path integral (2.1): σ1 (z1 ) . . . σk (zk )
≡
twisted
i
D[Xi ]e−S(X1 ...XN ) Z0N
.
(2.2)
Note that we have not yet specified the state at the edge of the hole |z| = . The choice of this state determines which operator we actually insert from the chosen twist sector. The choice of state that gives the lowest dimension operator from the given twist sector will be the one we will call σn ; other choices of state will give excited states in the same twist sector. To specify the state at this edge consider again the operator σn inserted at z = 0. We pass to local covering space (which we call ) by the map which locally looks like z ≈ at n ,
(2.3)
where a is a constant. The n copies of M involved in the twist give rise to a single copy of M on the cover . The hole |z| < gives a hole in the t space. We glue in a flat disc into this hole, and extend the path integral to perform on to the interior of this disc. This procedure effectively inserts the identity operator at the edge of the hole in the t space, and this gives the lowest dimension operator σn in the given twist sector. Note that this procedure inserts the identity operator with a given normalization. The twist operator thus constructed will be called σn (0); here is an essential regularization, and will cancel out in final expressions when we compute 3-point functions normalized by the 2-point functions of the operators.
2.2. Fermionic variables. In a supersymmetric theory each copy of M has in addition to the bosonic coordinates Xi a set of fermionic coordinates ψi . Upon insertion of a twist operator these fermionic variables from different copies of M are permuted around the twist insertion just like the bosonic variables, and the correlator of twist operators is defined in a manner analogous to (2.2). But we have to take some care in defining the state at the edge of the hole |z| < . When we pass to the covering space by the map (2.3) the fermionic variables transform as dz 1/2
ψt (t) = ψz (z)
dt
= ψz (z)a 1/2 n1/2 t
n−1 2
.
(2.4)
In the z plane we want to have the boundary condition that as we circle the insertion point z = 0 counterclockwise we get ψ1 → ψ2 → · · · → ψn → ψ1 . While for the spin zero variables Xi this meant that we just have X → X under transport around t = 0 in the t space, we now see that for the spin 1/2 fermionic variables ψt (t) → (−1)n−1 ψt (t)
(2.5)
under a counterclockwise rotation in the t space around t = 0. We now find a difference between the cases of n odd and n even. The n odd case is simpler, so we discuss it first.
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2.3. Twist operators in the supersymmetric theory for odd n. For odd n we see from (2.5) that ψt (t) is periodic around t = 0, so we can define the state at the edge of the hole just as in the bosonic case, by gluing in a flat disc to close the hole and continuing the fields x, ψ to the interior of the disc. This defines the twist operator σn (0), just as in the bosonic case. This operator however has no charge under the R-symmetry group of the N = 4 supersymmetric theory. Thus it is not a chiral operator of the theory, since a chiral operator has h = j, h¯ = j¯, where h, h¯ are the left and right dimensions of the operator and j, j¯ are the spins under the left and right SU(2) R-symmetry groups. We therefore seek a natural definition of a chiral operator that is based on the twist operators σn . Each copy of the manifold M gives rise to operators that yield an N = 4 supersymmetry algebra, for both the homomorphic and the antiholomorphic variables. Let Jzk,a (z) be the left SU(2) current of the CFT arising from the k th copy of M. The index a takes values 1,2,3. Let Jz+ = Jz1 + iJz2 , Jz− = Jz1 − iJz2 .
(2.6)
The operator Jza =
N i=1
Jzi,a (z)
(2.7)
is the diagonal element from the set of N SU(2) currents, and gives the left SU(2) current of the orbifold CFT. Now we note that in the presence of the twist operator σn (0) we can define the operators +(z)
J−m/n ≡
n dz k,+ Jz (z) e−2πim(k−1)/n z−m/n . 2π i
(2.8)
k=1
The integral over z is performed over the usual counterclockwise loop around the origin. The integrand is periodic around this loop, since Jzk,+ → Jzk+1,+ due to the cyclic permutation of copies of M around the twist insertion. We have called these operators +(z) J−m/n since they raise the dimension of the twist insertion by m/n. We have included a superscript (z) in these operators to denote the fact that they are operators on the z space; this will distinguish them from modes of the current operators on the t space which we will consider below. These operators raise the SU(2) charge under the diagonal SU(2) by one unit, as can be seen from (2.7) and Jz3 (z1 )Jzk,+ (z2 ) ∼
1 J k,+ (z2 ). z1 − z 2 z
(2.9)
In the t space the operation (2.8) becomes +(z) J−m/n
=
n dz k,+ Jz (z) e−2πim(k−1)/n z−m/n 2πi k=1 dt + + → . J (t) a −m/n t −m ≡ a −m/n J−m 2π i t
(2.10)
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In the t space the contour runs around a complete counterclockwise circle around t = 0. The n different currents Jzk,+ in the z plane give rise to the single current Jt+ on the covering space , which is the current for the single copy of M that describes the CFT +(z) on . To summarize, we find that the operator J−m/n in the z space gives the operator + J−m in the t space. Let us return to the construction of the chiral twist operators. We had observed that the twist operators σn which simply permuted copies of M had no charge, and thus had h > j and were not chiral operators. We now ask if we can stay in the same twist sector, but add other operators to σn such that we increase the charge of the resulting operator. Since σn had the minimum dimension in its own twist sector, the dimension of the operator will also go up in this process. But if we could achieve hfinal = qfinal then we would have constructed a chiral operator. +(z) The operators J−m/n allow us to make such a construction. We will compute the dimensions of our final chiral operators from the 2-point function in Sect. 4, but it is helpful to use alternative arguments to compute the dimension contributed by various components in the construction of the chiral operator, and we will do that below. Thus start with σn , which is just a twist operator that permutes the copies of M around its insertion point. The dimension of σn is ¯n = #n = #
c 1 6 1 1 1 n− = n− = n− . 24 n 24 n 4 n
(2.11)
This dimension can be deduced by looking at the CFT on a cylinder and noting that the twist operator changes a theory based on a set of n separate copies of M to a theory with one single copy of M but on a spatial section that is n times as long. Thus the vacuum energy of the ground state changes from −nc/24 to −(1/n)c/24, and the change gives the dimension of the twist operator. If M gives an N = 4 CFT based on a sigma model with 4 bosons and 4 fermions then we have c = 6, and (2.11) follows. To raise the charge of the operator with minimum increase in dimension consider the + application of J−1/n . The charge goes up by one unit, while the dimension increases by only 1/n. Note that this low cost in dimension is directly related to the existence of the twist which allows the fractional dimension charge operators (2.8); if we did not have a + twist then we could only apply J−1 which would increase q and h by the same amount, and so not bring an operator with h > q towards an operator with h = q. + to the identity operator at t = 0; thus we In the t space we have thus applied J−1 + + just get the state J−1 |0 NS = Jt (0), where |0 NS is the Neveu–Schwarz vacuum in the t space. + (in the t space), We might try to repeat this process with another application of J−1 + + but we find that J−1 J−1 |0 NS = 0. (This fact and other similar relations used below can be checked by using the commutation relations of the current algebra to find the norm of the state, or more simply by using a bosonic representation of the N = 4 algebra and observing that for the manipulations concerned any way of representing the algebra + + will yield the same results.) We also find that J−2 J−1 |0 NS = 0. Thus the next step is to + + construct J−3 J−1 |0 NS . We keep proceeding in this way, arriving at the operator (in the t space) + + + σn− ≡ J−(n−2) . . . J−3 J−1 |0 NS .
(2.12)
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The dimension of this operator (as seen from the z plane) is h ≡ #− n = #n +
3 n−2 n−1 1 + + ··· + = . n n n 2
(2.13)
The charge is q=
n−1 = #− n 2
(2.14)
and thus the operator is chiral. + The next current operator that we can apply to (2.12) is (in the t space) J−n . This raises the charge by one unit, but the dimension in the z plane also rises by one unit, so the resulting operator is another chiral operator: + + + + σn+ ≡ J−n J−(n−2) . . . J−3 J−1 |0 NS .
(2.15)
This operator has h ≡ #+ n =q =
n+1 . 2
(2.16)
+ If we apply the next allowed operator J−(n+2) then the charge goes up by one unit but n+2 the dimension goes up by n ; thus we would get h > q and an operator that is not chiral. To complete the construction we apply the same steps to the right moving sector using the current J¯+ . We thus obtain four chiral operators
σn−− , σn+− , σn−+ , σn++ .
(2.17)
It would appear that we could make other operators in this manner, for example by + + + + ˜ 2,−3/2 J + |0 NS . But as replacing . . . J−3 J−1 |0 NS by . . . J−2 J−2 |0 NS or . . . G1−3/2 G −1 we will show in the next section, the operators obtained by the latter constructions are proportional to the operators that we have made above, and so no new operators are obtained this way. There do exist other chiral operators in the theory, which use the details of the structure of the manifold M. For example there are 20 (1, 1) forms on K3 which give rise to chiral operators but only 4 (1, 1) forms on T 4 . While it should be possible to make and use these additional chiral operators to compute correlation functions with our general methods, we have not done so in this paper. Thus we will consider only the basic operators (2.12), (2.15) (and their counterparts for n even). 2.4. Twist operators in the supersymmetric theory for even n. For n even the construction of the chiral primaries from σn is slightly different. The operator σn just permutes the copies of M, so the fermionic variables ψ1 from the first copy cycle around and return to themselves after n rotations around the twist insertion in the z plane. But on the covering surface we see from (2.5) that ψt returns to itself after one rotation around t = 0 but with a change of sign. This means that we should not close the hole around t = 0 by just gluing in a disc and getting the state |0 NS at t = 0. Rather we need to insert an operator that creates a Ramond vacuum |0 R at t = 0, so that the fermion ψt will indeed be antiperiodic around t = 0. We will see that this operator must be a spin field S α , α = ±. We can extract all the relevant properties of this state |0 R without making any reference to the details of the manifold M, just using the fact that M gives rise to an
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N = 4 supersymmetric theory. The Ramond vacuum can be obtained by a spectral flow starting from the NS vacuum. We list in Appendix A the relevant formulae for spectral flow. The parameter giving the amount of spectral flow is η = 1. In the t space we have one copy of M so the CFT has c = 6. The NS vacuum has h = q = 0. Then we find that the R vacuum has h = 6/24 = 1/4, q = −1/2. We thus denote it as |0− R . We can act on this state by J0+ to obtain another degenerate R vacuum with h = 1/4, q = 1/2, J0+ |0− R = |0+ R . S±
(2.18)
|0±
The spin fields create the states R from |0 NS . Starting with |0− R let us make a state with the maximal charge to dimension ratio, just as we did in the case of n odd above. The first operator we apply is J0+ . The next + lowest dimension operator that we can apply is J−2 , and so on. We then find the chiral states + + + − σn− ≡ J−(n−2) . . . J−2 J0 |0 R .
(2.19)
The dimension and charge of this operator (as seen from the z plane) are h ≡ #− n = #n +
1 2 n−2 n−1 n−1 + + ··· + = , q= = #− n. 4n n n 2 2
(2.20)
1 Here the contribution #n arises just as in the case of n odd, and 4n is the contribution to the dimension in the z plane coming from the insertion of the spin field S − in the t plane (this field takes |0 NS to |0− R ). The fact that the dimension 1/4 in the t plane becomes 1/4n in the z plane can be seen from the form of the covering space map z = at n . We can apply one more current operator to obtain another chiral operator: + + + + − σn+ ≡ J−n J−(n−2) . . . J−2 J0 |0 R .
(2.21)
This operator has n+1 n+1 , q= . (2.22) 2 2 Applying current operators from the right moving sector in an analogous manner we again obtain four chiral operators of the form (2.17). The charges and dimensions of the operators have the same form in the case n even as in the case n odd. Notation. The chiral operators constructed from σn are denoted σn±± , and their dimenc 1 1 1 ¯± sions are denoted by #± n , #n . We will also use the notation #n = 24 (n− n ) = 4 (n− n ). #n is the contribution to the dimension of the chiral operator from the conformal anomaly. h ≡ #+ n =
2.5. Other members of the chiral operator representation. We have constructed states that have h = q and are thus chiral operators of the CFT. By charge conservation, any correlator of chiral operators will vanish. To find nonvanishing correlators we must look at the SU(2) representation of which the above chiral operator is the highest weight state |j, m = |j, j . These other states have the form k dz − (J0− )k |j, j = Jz (z) |j, j , (2.23) 2π i where Jz− is an element of the diagonal SU(2) (2.7). Thus the operators we study will be given by applications of J0+ and J0− operators to the twist operators σn .
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2.6. Universality in the construction of the chiral primaries. We note that the construction of the above chiral operators made no reference to the detailed structure of the manifold M. We used only the fact that the CFT based on M had N = 4 supersymmetry. Thus M could be a K3 space at a generic point in its moduli space, which is not simply an orbifold of a torus. Thus we are not working with a CFT which can be reduced to a free field theory. In the computations below we will use a bosonic representation of current operators to simplify the calculations; this is allowed because we will be working on a sphere. But it should be noted that these computations could all have been performed by using only the general properties of the N = 4 algebra. 3. The Method of Computing Correlation Functions In [9] a method was developed to compute the correlation functions for bosonic orbifolds M N /SN . We will find that the bosonic result can be extended to obtain the result in the supersymmetric case, after we take into account the current operators that were added to the twist operator σn to get the operators σn−− , etc. We review here briefly the method used in [9] and then indicate the way it will be extended to the supersymmetric case. 3.1. The method for bosonic orbifolds. Let us assume that we have only bosonic variables describing the sigma model with target space M. Consider the definition (2.2) of the correlation function of twist operators σn . The path integral performed with twisted boundary conditions becomes a path integral on the covering space . The holes at the location of the twist operators are closed by inserting a disc, and thus is a closed surface. In general may have several disconnected components, but we assume here that there is just one component, since all different components can be handled in the same way. The copies of M that are not involved in any of the twists give a contribution to the partition function that cancels out between the numerator and denominator of (2.2). The genus of the surface depends on the orders of the twist operators. At the insertion of the operator σnj (zj ) the covering surface has a branch point of order nj , which means that nj sheets of meet at zj . One says that the ramification order at zj is rj = nj − 1. Suppose further that over a generic point z here are s sheets of the covering surface . Then the genus g of is given by g=
1 rj − s + 1. 2
(3.1)
j
The path integral over the z plane with twist insertions becomes a path integral for a CFT based on only one copy of M, on the surface with no twist insertions or any other operator insertions. But the metric to be used on to compute the path integral is the metric induced from the z plane, and it is the dependence of the path integral on this metric that encodes the dependence of the correlation function on the location of twist operators in the z plane. We can compute the path integral for some fiducial metric g˜ on , provided we take into account the correction due to the metric change by using the conformal anomaly. If ds 2 = eφ d s˜ 2 , then the partition function Z (s) computed with the metric ds 2 is related to the partition function Z (˜s ) computed with d s˜ 2 through Z (s) = eSL Z (˜s ) ,
(3.2)
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where
c (3.3) d 2 t −g (˜s ) [∂µ φ∂ν φg (˜s )µν + 2R (˜s ) φ] 96π is the Liouville action [17]. Here c is the central charge of the CFT for one copy of M. We can choose the fiducial metric g˜ to be the flat metric dtd t¯ everywhere except for a set of isolated points. Then the metric induced from the z space is written as SL =
ds 2 = dzd z¯ = eφ dtd t¯.
(3.4)
Let us call the part of that excludes these isolated curvature points and the punctures arising from the twist insertions as the “regular region” of . It was shown in [9] that for the cases that we are interested in the contribution to the path integral from these (1) excluded points is zero, and thus we only have the contribution SL from the regular region. Since the metric is flat by construction in this region we get no contribution from the term R (˜s ) φ in the action (3.3). Thus we have 1 (1) S L = SL = (3.5) d 2 t[∂µ φ∂ µ φ]. 96π We rewrite (3.5) as 1 SL = − 96π
1 d t[φ∂µ ∂ φ] + 96π µ
2
∂
φ∂n φ.
Here ∂n is the normal derivative at the boundaries of . From (3.4) we find that dz d z¯ φ = log + log dt d t¯ so that ∂µ ∂ µ φ = ∂t ∂t¯φ = 0 and we get SL =
1 96π
(3.6)
(3.7)
(3.8)
∂
φ∂n φ.
(3.9)
Thus we get the desired correlation function of twist insertions on the z plane as a sum of contributions from local expressions (3.9) from a finite set of points on . An essential (and generally nontrivial) step in the calculation is finding the map t (z) that gives the branched cover of the z plane with the given ramifications at the insertions of the twist operators. In this way any correlation function of twist operators on the z plane for the theory M N /SN can be deduced from a knowledge of the partition functions Zg for the theory for one copy of M on Riemann surfaces of different genera g. The final primary fields of the CFT are not the twists σn but operators On which are made from σn = σ(i1 ,...,in ) by symmetrizing over all different ways in which the n indices ik involved in σn can be chosen from the set of indices i = 1, . . . , N which denote all the available copies of M. The correlation functions of the On are therefore just given by combinatorial factors multiplying correlators of the σn . Note that when we look at different choices of indices making up the On then we can get a finite number of different covering surfaces . But we find from the combinatorics that if N is large then for a given choice of orders of twist operators the leading contribution comes from the case when is a sphere [9]. The contribution from the cases where is higher genus is suppressed by a relative factor 1/N g .
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3.2. The supersymmetric case. The above result on combinatorics applies as much to the supersymmetric case as to the bosonic case. Thus we concentrate on the case where is a sphere in the present paper; this will give us the leading order result at large N . Let us consider the 3-point function of twist operators in the supersymmetric theory. As we saw in the previous section, the added complication in the supersymmetric case is that on we should not just close the puncture at the location of the twist operator by inserting the identity state |0 NS . Rather, we have to insert current operators at these locations, and for even n we also need an operator – the “spin field” – that takes the state |0 NS to |0− R . Thus is found in the same way as in the case of the bosonic orbifold, but instead of computing the path integral on we need to compute a correlation function on . As in the bosonic case (2.1), let Z0 be the partition function for the supersymmetric theory when the target space is one copy of M, Z0 = D[X, ψ]e−S[X,ψ] . (3.10) g
The partition function is computed in some chosen metric g on the z space. (In [9] the z space was chosen to be a large flat disc which was closed to a sphere by gluing an identical disc at its boundary.) Choosing the operators to be σn−− for concreteness, we define the correlation function analogous to (2.2), but with insertions of currents J + , J − required in the construction of the chiral operator: σ1−− (z1 ) · · · σk−− (zk ) ≡
s 1 D[Xm , ψm ]e−S(X1 ···XN ,ψ1 ···ψN ) Z0s twisted m=1 dqij × J ± (qij )(qij −zj )−nij /nj 2π i i,j
≡
Q . Z0s
(3.11)
We have assumed that s copies of M are joined by the twisted boundary conditions, so that the path integral over the remaining N − s copies of M cancels out between the numerator and denominator in (3.11). The qij are integrated over contours around the zj , and the integers nij are given by the form of the chiral operators discussed in the last section. Passing to the covering space , which we are assuming to be a sphere, we get the path integral for the theory with one copy of M but with a metric induced from the z space dqij −S[X,ψ] Q= Jt± (qij ) (qij − tj )−nij D[X, ψ]e S − (ti ) 2π i ginduced i,j i dq ij = eSL D[X, ψ]e−S[X,ψ] Jt± (qij ) S − (ti ) (qij − tj )−nij 2π i g i,j i
dq ij = eSL ( D[X, ψ]e−S[X,ψ] ) Jt± (qij ) S − (ti ) (qij − tj )−nij 2π i g i,j
i
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= eSL Z0
Jt± (qij )
i,j
dqij S − (ti ) . (qij − tj )−nij 2π i
(3.12)
i
Here S − are the spin field insertions for even n operators. SL is the Liouville action arising from the conformal anomaly when we rewrite the path integral for the metric ginduced (which is induced from the z space onto the t space) in terms of the fiducial metric g on the sphere that was used to define Z0 in (3.10). SL depends upon the cutoffs used in defining the twist operators. The 3-point function normalized by the 2-point functions is σ1−− (z1 )σ2−− (z2 )σ3−−† (z3 )
≡
σ1−− (z1 )σ2−− (z2 )σ3−−† (z3 )
σ1−− (0)σ1−−† (1) 1/2 σ2−− (0)σ2−−† (1) 1/2 σ3−− (0)σ3−−† (1) 1/2 (3.13)
The power of Z0 cancels out in (3.13), after we note the relation (3.1) and the fact that a correlator of the form σn σn† is proportional to Z0−n . Let us denote the Liouville action for the 3-point function by SL [σ1 σ2 σ3 ] and for the 2-point functions by SL [σ1 σ1† ], etc. Let us also write the correlator of currents and spin fields in (3.12) as J, S σ1 ...σk . Then we have σ1−− (z1 )σ2−− (z2 )σ3−−† (z3 )
†
†
†
1
1
1
J, S σ
†
= eSL [σ1 σ2 σ3 ]− 2 SL [σ1 σ1 ]− 2 SL [σ2 σ2 ]− 2 SL [σ3 σ3 ]
J, S
1 2
† 1 σ2 σ3 1 2 σ2 σ2†
† J, S
σ1 σ1
1
J, S 2
. (3.14)
σ3 σ3†
The contribution of the Liouville terms in the above equation was shown in [9] to be the three point function of twist operators in the bosonic orbifold theory, where the central charge was set to c = 6. (This is discussed in more detail below.) The computation of 3-point functions in the supersymmetric theory then just reduces, by (3.14), to a computation of correlation functions of currents and spin fields on the covering surface , for the 3-point function and for the 2-point functions in the denominator. The correlation functions of currents can of course be computed in any metric on , since they are independent of the metric. If all the chiral operators have odd n twist operators, then the only correlation functions that we need are correlators of current operators on . The other possible nonzero three point function is where two of the operators have even n twists. In both these cases the correlation functions (for a sphere) can be computed purely in terms of the properties of the chiral algebra of the N = 4 supersymmetric theory. (We can “undo” a contour of J + from around one point on and replace it by contours surrounding the other points in the correlator. The contour moves freely through any other operators J + on , while it picks up a contribution from locations of J − operators determined purely by the N = 4 algebra.) Given this fact we can use any convenient representation of this algebra without losing generality in the choice of M. We use a representation of current algebra in terms of free bosonic fields; the current operators are represented by exponentials and polynomials in these new bosonic variables. The spin fields are represented by exponentials in these bosons as well.
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397
We will normalize the current and spin field insertions in the next section such that J, S σ σ † = 1.
(3.15)
i i
It will also be convenient to use the notation σn−− (z) ≡
σn−− (z)
σn−− (0)σn−−† (1) 1/2
.
(3.16)
It is convenient to compute the fusion coefficients (and thus the 3-point functions) by computing σ1−− (0)σ2−− (a)σ3−−† (∞)
σ3−− (0)σ3−−† (∞)
−
−
−
−−− 2 ≡ |C1,2,3 | |a|−2(#1 +#2 −#3 ) .
(3.17)
From (3.14) and (B.4) we see that the LHS of (3.17) is σ1−− (0)σ2−− (a)σ3−−† (∞)
σ3−− (0)σ3−−† (∞)
= |C1,2,3 |12 |a|−2(#1 +#2 −#3 )
J, S σ
† 1 (0)σ2 (a)σ3 (∞)
J, S σ
† 3 (0)σ3 (∞)
,
(3.18) where we have used (3.15). Here C1,2,3 is the fusion coefficient (computed in [9]) of twist fields for a bosonic theory with c = 1. To summarize, the three point functions of the supersymmetric theory will be computed as the product of two contributions. The first part is the contribution of the conformal anomaly, which arises in the map of the z sphere to the t sphere. This part of the calculation is identical to that for the bosonic case, after we note that the central charge for the theory based on one copy of M is c = 6 for the field content of a N = 4 supersymmetric theory. The second part is the correlator of current operators and spin fields on ; these fields will all be represented by exponentials and polynomials of bosons on . 4. The Contribution of the Current Insertions 4.1. Representing the current algebra by free fields. As mentioned above when is a sphere we can use the following simplification. We need only the OPEs of the chiral algebra generators to get the correlation function, and so we can represent these generators in any manner that reproduces these OPEs. Let us therefore take a specific example of a system with N = 4 superconformal symmetry, and use it to construct the chiral algebra generators and their correlation functions on the sphere. We take two complex bosons (X1 and X2 ) and two complex fermions (31 and 32 ). The elements of the superconformal algebra for such a system are given by: 1 † a 1 1 T (z) = ∂Xi† ∂Xi + 3i† ∂3i − ∂3i† 3i , (4.1) 31 σ 31 + 32† σ a 32 , 2 2 √ 2 √ √ √ (4.2) G1 (z) = 232† ∂X1 − 231 ∂X2 , G2 (z) = 231† ∂X1 + 232 ∂X2 , √ √ † † √ √ † † † † ˜ 2 (z) = 231 ∂X + 23 ∂X . ˜ 1 (z) = 232 ∂X − 23 ∂X , G (4.3) G 1 1 2 1 2 2 J a (z) =
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O. Lunin, S. D. Mathur
Let us introduce the real components of the bosonic fields: X1 =
φ1 + iφ2 , √ 2
X2 =
φ3 + iφ4 √ 2
(4.4)
and bosonize the fermions: 31 = eiφ5 ,
32 = eiφ6 .
(4.5)
One can now rewrite the superconformal generators in terms of six real bosonic fields φi : J 3 (z) =
i (∂φ5 − ∂φ6 ) , 2
J + (z) = exp(iφ5 − iφ6 ),
J − (z) = exp(−iφ5 + iφ6 ),
1 † ∂φi ∂φi , 2
(4.6)
6
T (z) =
(4.7)
i=1
G1 (z) = exp(−iφ6 )(∂φ1 + i∂φ2 ) − exp(iφ5 )(∂φ3 + i∂φ4 ),
(4.8)
G2 (z) = exp(−iφ5 )(∂φ1 + i∂φ2 ) + exp(iφ6 )(∂φ3 + i∂φ4 )
(4.9)
˜ can be obtained from Ga by taking a complex conjugate. The components of G For later use it will be convenient to adopt a notation where all 6 bosons (4 original bosons and 2 from bosonizing the fermions) are grouped together into a vector φa . Then we write i ea ∂z φja (z), 2 j J + (z) = exp iea φja (z) , J 3 (z) =
j
G1 (z) =
J − (z) =
j
j
G2 (z) =
(4.10) exp −iea φja (z) ,
exp −ica φja (z) Ab ∂φjb (z) − exp ida φja (z) Bb ∂φjb (z) , (4.12)
j
(4.11)
exp ica φja (z) Bb ∂φjb (z) + exp −ida φja (z) Ab ∂φjb (z) .
(4.13)
Here we have A = (1, i, 0, 0, 0, 0),
B = (0, 0, 1, i, 0, 0),
c = (0, 0, 0, 0, 0, 1),
d = (0, 0, 0, 0, 1, 0).
(4.14)
Starting from the general form (4.10)–(4.13) and requiring that the elements of the chiral algebra satisfy their OPEs, we get the following constraints on real vectors c, d, e and complex vectors A, B: e · e = 2,
c · e = −1,
d · e = 1,
A · A∗ = B · B∗ = 2,
(4.15)
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399
c·d=e·A=e·B=c·A=c·B=d·A = d · B = A · B = A · A = B · B = A∗ · B = 0. e = d − c,
d · d = c · c = 1.
(4.16)
These are the only properties that we will explicitly use. It will be convenient not to use vectors d and c, but use instead e and f =c+d:
f · f = 2,
f · e = 0.
(4.17)
In this representation through free bosons it is easy to establish the claim made in the previous section about the possible chiral operators in the CFT. For odd n, the operators + + + σn− had the form Jn−2 . . . J−3 J−1 |0 NS . The same dimension and charge can be obtained + + + + ˜ 2,−3/2 J + . But as we argue now, these by replacing J−3 J−1 by J−2 J−2 or by G1−3/2 G −1 other operators will all be proportional to the original operator, and so there will be no further chiral operators with this charge and dimension in the twist sector of σn . n−1 3 This operator has charge n−1 2 under J and a dimension 2 . If we take any combiα α a ˜ J from (4.10)–(4.13) then we will get expressions nation of the generators T , G , G which are of the form “exponential in the φa ” times “polynomial in the φa ”. But the required dimension and charge are obtained only by the expression |p = b−p
2 /n
: exp ipea 8a (0) :,
p=
n−1 . 2
(4.18)
Here z ≈ bt n is the map taking the operator define in the z plane to its image in the t space. We will deduce the power of b in (4.18) below, but first let us look at the charge and dimension of the operator. The charge determines the exponential, and adding any polynomial or any exponential in a direction orthogonal to the vector ea increases the dimension without increasing the charge. This establishes the above claim that there is a unique operator with the desired quantum numbers. Note that using the free boson representation does not limit us to working with a free CFT; the same result could be proven by computing the determinant of the matrix of dot products between the above mentioned states, and finding that the matrix has rank 1 after use of the commutation relations of the chiral algebra. Let us now obtain (4.18) directly from the definition of the chiral operator in the z plane, and thus obtain the required power of b. Let us take n odd. We have from (2.10), dt + +(z) J−1/n σn (0) = b−1/n Jt (t)t −1 = b−1/n Jt+ (0) = b−1/n exp iea 8a (0) . 2π i (4.19) (Here and in what follows we will identify the operator in the z plane with its representation in the t space, letting it be clear from the notation which representation we are referring to.) If we apply another current operator then we get dt −k +(z) +(z) J−k/n J−1/n σn (0) = b−k/n b−1/n t exp iea 8a (t) exp iea 8a (0) . (4.20) 2π i
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From the OPE exp (iea 8a (t)) exp (iea 8a (0)) ∼ t 2 we see the claim made earlier that the lowest allowed value of k is 3. For k = 3 we get +(z) +(z) J−3/n J−1/n σn (0) = b−4/n exp 2iea 8a (0)
(4.21)
for the operator in the t space. Proceeding in this manner we find that +(z)
+(z)
+(z)
J−(n−2)/n . . . J−3/n J−1/n σn (0) = b−p
2 /n
exp ipea 8a (0) ,
p=
n−1 . 2
(4.22)
In a similar manner we can compute other sets of operators that would yield chiral states. Let z ≈ bt n (1 + ξ t + O(t 2 ))
(4.23)
be the map from t plane to the z plane near point t = 0. Then 2ξ +(z) +(z) +(z) J−2/n J−2/n σn (0) = J−2/n b−2/n : (ieb ∂8b (0) − ) exp iea 8a (0) : n = 2b−4/n : exp 2iea 8a (0) :, (4.24) +(z)
+(z)
so we get the same state as we would get from J−3/n J−1/n σn (0). (We had argued above that such had to be the case from a consideration of the charges and dimensions of the chiral operators.) Similarly we find that ˜ 2,−3/2 G1−3/2 J + σn (0) ∼ b−4/n : exp 2iea 8a (0) : . G −1
(4.25)
Let us now consider the case of even n. Now we need to insert a spin field in the t space to obtain the vacuum |0− R from |0 NS . We can reproduce the charge, dimension and OPEs of the spin field by using the operator exp ± 21 iea 8a (t) to represent the spin fields S ± . Again using the fact that the chiral operators have dimension equal to charge, we find that +(z)
+(z)
+(z)
J−(n−2)/n . . . J−2/n J0
σn (0) = b−p
2 /n
n−1 exp ipea 8a (0) , p = 2
(4.26)
is the unique state giving the chiral operator with h = q = n−1 2 in the twist sector of σn . Similarly, σn+ is given by taking in (4.26) the value p = n+1 2 . Note that we will be using the correlators of at most two spin fields on a sphere, and for such correlators the properties of the spin field encoded in this bosonic representation entails no loss of generality. In particular this representation is not equivalent to assuming that we are dealing with a theory that can be reduced to free bosons.
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401
4.2. Two point functions. We now wish to compute the 2-point functions of twist operators, for two reasons. First, we extract the dimensions of the operators from the 2-point functions. Second, to define the 3-point function we have to know the normalization of the operators that go into the computation, and this is done by choosing the normalizations of the 2-point functions of the operators with their own conjugates. We have seen that in the representation through free bosons the chiral operators take, when lifted to the t sphere, the form (4.18) |p = Ab−p
2 /n
: exp ipea 8a (0) :,
(4.27)
± where p = n±1 2 for σn . While the structure of the map z(t) fixes the b dependence in the last equation, the overall normalization constant A has not been determined so far. To do so, one needs to evaluate the two point function of such exponentials, and we will do this below. Let us consider the operator σn−− . We place this operator at z = 0 and its conjugate −−† at z = a. Note that if σn− corresponds to twist (1, . . . , n), and has charge n−1 σn 2 , −† then σn corresponds to the twist (n, . . . , 1) and has a charge −(n − 1)/2. Following the method outlined in the previous section, we go from the z sphere to the t sphere by the map [9]:
z=
at n . t n − (t − 1)n
(4.28)
The path integral in the presence of the chiral fields then becomes, on the t sphere, a path integral with the insertion of exponentials, but with no twists. The metric on the t sphere is to be induced from the z sphere, but we use a fiducial sphere metric for t while taking into account the change of metric by the conformal anomaly. Using the method outlined in Sect. 3 (Eq. (3.11), (3.12)) we get
σn−− (0)σn−−† (a) = Z01−n eSL
n − 1 2
, z = a|
n−1 ,z = 0 , 2
(4.29)
n−1 2 where n−1 2 , z = a| 2 , z = 0 is a correlation function of two exponentials (4.27):
n − 1
n−1 , z = 0 = |A|2 × 2 2 n−1 n−1 2 −(n−1)2 /2n |b0 | : exp i ea 8a (0) : |b1 |−(n−1) /2n : exp −i ea 8a (1) : 2 2 , z = a|
= |A|2 |b0 |−(n−1)
2 /2n
|b1 |−(n−1)
2 /2n
.
(4.30)
The values of b0 and b1 in (4.30) are determined by the asymptotic behavior of (4.28) near t = 0 and t = 1: z = a(−1)n−1 t n + O(t n+1 ), z = a + a(t − 1)n + O((t − 1)n+1 ),
b0 = a; b1 = a.
(4.31) (4.32)
2 To make this expression more compact, we use 8 to represent the total bosonic field, while in the rest of this paper 8 represents only the holomorphic part of such a field.
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O. Lunin, S. D. Mathur
Taking this into account, we get an expression for the correlator of two exponents: n − 1
n − 1 2 (4.33) , z = a , z = 0 = |A|2 |a|−(n−1) /n . 2 2 Note that in (3.14) the following shorthand notation has been used for the same two point function: n − 1
n − 1 J, S σ −− σ −−† ≡ , z = 1 ,z = 0 . (4.34) n n 2 2 Let us now choose A = 1,
|p = b−p
2 /n
: exp ipea 8a (0) : .
(4.35)
In this case the expression (4.34) equals unity, and we also get unity for the denominator in (3.14). Let us now go back to the complete two point function (4.29). The contribution of conformal anomaly was evaluated in [9] and it has the following form:
Z01−n eSL = B |a|−4#n .
(4.36)
Here B is a coefficient which depends on regularization parameters, but not on a. Then for the two point function we finally get: σn−− (0)σn−−† (a) = B |a|−4#n |a|−(n−1)
2 /n
= B |a|−2(n−1) .
(4.37)
n−1 ¯− Thus the dimension of the operator σn−− is #− n = #n = 2 . The dimensions of the other operators σn+− , σn−+ , σn++ can be computed in a similar manner.
± † 5. Three Point Function of the Form σn± σm± σm+n−1 To start with we will look at a special class of 3-point functions: ± † ± (z) . σn (0)σm± (a) σm+n−1
(5.1)
The fact that we have written only one superscript ± on each σn indicates that we are looking at only the holomorphic part of the 3-point function. The † operation changes a permutation (1, 2, . . . , n) to the inverse element of the permutation group (n, n − 1, . . . , 1), and also changes the sign of the J3 charge of the operator. If the permutations of order n and order m combine into a permutation with a single nontrivial cycle, then the maximal order of this cycle is n + m − 1, and this is the order of permutation chosen for the third operator in the above expression. The value of this subclass of correlators was computed by a recursion argument in [13], but we reproduce the result by our method as a warmup towards the general case which we study in the next section. n+1 + Note that σn− has charge n−1 2 while σn has charge 2 . Thus by charge conservation + † (z) = 0, σn+ (0)σm+ (a) σm+n−1 − † + − σn (0)σm (a) σm+n−1 (z) = 0, † + (z) = 0. σn− (0)σm− (a) σm+n−1
Three-Point Functions for Orbifolds with N = 4 Supersymmetry
403
The nonvanishing fusion coefficients will be denoted as −
−
−
−−− − σn− (0)σm− (a) = a #m+n−1 −#n −#m Cn,m,n+m−1 σn+m−1 (0),
σn+ (0)σm− (a) = a
+ − #+ m+n−1 −#n −#m
+−+ + Cn,m,n+m−1 σn+m−1 (0),
(5.2) (5.3)
To compute the fusion coefficients we use (3.18). In the expression (3.18) we have the factors coming from the conformal anomaly |C1,2,3 |12 |a|−2(#1 +#2 −#3 ) = a − 4 (1− n − m + n+m−1 ) (Cn,m,n+m−1 )6 , 1
1
1
1
1 1 1 1 n+ log n − m+ log m 12 n 12 m 1 1 1 + q+ + − 1 log q 12 n m 1 1 1 1 (q−1)! − 1+ − − log . 12 q n m (m−1)!(n−1)!
(5.4)
log |Cn,m,m+n−1 |2 = −
(5.5)
Here q = m + n − 1. The power of a in (5.4) arises from the dimensions of the bosonic c twists σn which are #n = 24 (n − n1 ). The remainder of the contribution to the OPE comes from the insertions of J + (and spin fields for even n) at the images the twist operators on . We thus have to compute a correlation function of these elements of the chiral algebra, and since we are on the sphere, we lose no generality by using a representation of the currents in terms of free bosons. In the previous section we have shown (Eq. (4.35) ) that the twist operator σn± (0) gives rise to the following insertion on the t sphere: −p2 /n |p = b0 : exp ipea 8a (0) :, (5.6) − where p = n−1 2 for σn and p = behavior of the map near t = 0:
n+1 2
for σn+ . The value of b0 is given by the leading
z = b0 t n + O(t n+1 ). Consider now a second insertion at the point z = a (which maps to t = 1): −q 2 /m |q = b1 : exp iqea 8a (1) :
(5.7)
(5.8)
m±1 2 ,
with q = depending on the charge of σm± (a). To evaluate b0 , b1 we recall the map used in [9] to go from the z sphere to the t sphere. This map was constructed in terms of the Jacobi polynomials: (n,−m)
z = at n Pm−1
(1 − 2t).
(5.9)
In particular we will need following asymptotic properties of this map: (m + n − 1)! n t near z = 0, n!(m − 1)! (m + n − 1)! z = a+a (t − 1)m near z = a, m!(n − 1)! (m + n − 2)! m+n−1 t z=a near z = ∞. (m − 1)!(n − 1)!
z=a
(5.10) (5.11) (5.12)
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The values of b0 and b1 can be seen from (5.10) and (5.11) to be: b0 = a
(m + n − 1)! n!(m − 1)!
b1 = a
(m + n − 1)! . m!(n − 1)!
(5.13)
Then the charge insertions at t = 0 and t = 1 give −q 2 /m : exp ipea 8a (0) : b1 : exp iqea 8a (1) : (5.14) −p2 /n−q 2 /m a(m + n − 1)! 2 2 = np /n mq /m : exp ipeb 8b (0) + iqeb 8b (1) : . (n − 1)!(m − 1)! (5.15)
−p2 /n
b0
To compute the contribution of (5.14) to the fusion coefficient we recall the way we had computed the fusion coefficients of the bosonic twists in [9]. The z plane was cut off at a large radius |z| = 1/δ, and a second disc was glued in at this boundary to make the z plane into a sphere with an explicitly chosen metric. The map z(t) taking multivalued functions on the z sphere to single valued functions on the covering space thus mapped a closed surface (a sphere) to a closed surface (which in the present case is also a sphere). It was important to explicitly close all surfaces in order to use the argument of the conformal anomaly to compute the correlation function. In the 3-point function we place operators at z = 0, z = a and z = ∞. We normalized the insertions at z = 0 and z = a. We could also normalize the insertion at z = ∞ and directly compute the 3-point function, but because the operator at infinity is in a different coordinate patch it is easier to proceed in a slightly different way. We place an operator σq∞ at infinity, without regard to its normalization, and compute the ratio σn (0)σm (a)σq∞ (∞)
σq (0)σq∞ (∞)
= Cmnq a −(#n +#m −#q )
(5.16)
(where σq is correctly normalized) to compute the OPE coefficient. To extend the same method to the present case we can get identical insertions at infinity in the case of σn (0)σm (a) and the case of σq (0) if we choose the map to the covering space for σq (0) to agree at large z with the map to the covering space for the insertion σn (0)σm (a). From (5.12) we see that the map for σm+n−1 (0) should be chosen to be z=a
(m + n − 2)! m+n−1 ≡ b∞ t m+n−1 t (m − 1)!(n − 1)!
near
z = ∞.
(5.17)
± Writing r = p + q for the charge of σn+m−1 (0) we get for the normalized charge insertion −r 2 /n
|r = b∞
: exp irea 8a (0) :,
(5.18)
where b∞ = a
(m + n − 2)! . (m − 1)!(n − 1)!
(5.19)
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405
Then we get from the charge insertions the contribution to (3.18), J, S σ1 (0)σ2 (a)σ3 (∞) J, S σ3 (0)σ3 (∞) −p2 /n
= =
b0
a (0)
: eipea 8
−q 2 /m
: b1
b (1)
: eipeb 8
−r 2 /(m+n−1) c b∞ : ei(p+q)ec 8 (0) −p2 /n −q 2 /m r 2 /(m+n−1) b0 b1 b∞ .
:
b (∞)
: e−i(p+q)eb 8
:
: e−i(p+q)ed
8d (∞)
:
:
(5.20)
−−− Let us now evaluate Cn,m,n+m−1 . In this case:
p=
n−1 , 2
q=
m−1 , 2
r=
n+m−2 . 2
(5.21)
Combining the contribution (5.20) with the contribution (5.4) we get −−− |Cn,m,n+m−1 |2
= |Cn,m,n+m−1 |12 |a|− 2 (1− n − m + n+m−1 ) |b0 |−2p 1
1
1
1
2 /n
|b1 |−2q
2 /m
|b∞ |2r
2 /(m+n−1)
, (5.22)
where we now write the combined holomorphic and antiholomorphic sector contributions. Substituting the value of the bosonic fusion coefficient (5.4) and making algebraic simplifications, we get −−− −− σn−− (0)σm−− (a) ∼ |Cn,m,n+m−1 |2 σn+m−1 (0)
(5.23)
with −−− |2 = |Cn,m,n+m−1
m+n−1 . mn
(5.24)
Note that the power of a cancels out in (5.23), reflecting the fact that in the supersym− − metric theory #− m+n−1 = #n + #m . Similarly one finds +−+ |Cn,m,n+m−1 |2 =
n . m(m + n − 1)
(5.25)
6. General Three Point Function for the Sphere Let us now consider the general 3-point function σn± σm± σq± . The twist operators carry representations of the su(2) × su(2) symmetry group. Thus representations of the first two twist operators have to combine to give the (conjugate) to the representation of the third twist operator. Since there is at most one way to combine two representations of su(2) to a third representation, we see that if we can compute the fusion coefficient for any chosen members of the three representations, then the general fusion coefficient can be deduced from this in terms of the Clebsch–Gordon coefficients. In the simple case studied in the last section all operators had charge vectors of the form |j, m = |Q, Q or |j, m = |Q, −Q , and so the charge could be represented as a pure exponential. The main additional complication in the general case is that all charge vectors cannot be taken to have this simple form. We will let the operator at z = 0 have
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the form |j, m = |Q, Q − d , while the operator at z = a has the form |j, m = |P , P
and the operator at z = ∞ has the form |j, m = |Q − d + P , −(Q − d + P ) . Thus the operator at z = 0 needs to be modified by the action of lowering operators acting on an exponential, and we carry out this modification below. The other steps in the calculation parallel those of the previous section. 6.1. Constructing the state |Q, Q − d . We will construct a family of twist operators by acting on the highest weight state (4.27) by the charge operator corresponding to J − . Using the fact that Jz− has holomorphic dimension unity we can rewrite the corresponding charge on the covering space: dt dz − dz Q− = exp −iea φja (z) = J (z) = exp −iea 8a (t) . 2πi 2πi 2π i j
(6.1) Acting by this charge on the exponent (4.27), we get: dz − dq 2 J (z)|Q = b−Q /n (q − t)−2Q : exp iQea 8a (t) − iea 8a (q) : . 2πi t 2π i (6.2) For the BPS operators constructed in Sect. 4, the product 2Q is an integer, so one can evaluate the integral in (6.2): 1 − a 2Q−1 a a Q : exp iQea 8 (t) := : exp iQea 8 (t) − iea 8 (q) : . ∂ (2Q − 1)! q q=t (6.3) Note that the action of Q− does not change the conformal dimension of the operator, since we get a polynomial of order 2Q − 1 multiplying the exponential, and Q2 = (Q − 1)2 + (2Q − 1). For the double action of the operator Q− on the BPS state one then gets: (Q− )2 : exp iQea 8a (t) : dq2 = : exp −iea 8a (q2 ) : 2πi 1 × lim ∂ 2Q−1 : exp iQea 8a (t)−iea 8a (q1 ) : q1 →t (2Q−1)! q1 1 = lim ∂ 2Q−1 q1 →t (2Q−1)! q1 dq2 (q2 −q1 )2 a a a × : exp iQe 8 (t)−ie 8 (q )−ie 8 (q ) : a a 1 a 2 2πi (q2 −t)2Q 1 1 2Q−1 = lim lim ∂q2Q−1 ∂ q1 →t q2 →t (2Q−1)! 1 (2Q−1)! q2 ×(q2 −q1 )2 : exp iQea 8a (t)−iea 8a (q1 )−iea 8a (q2 ) : . (6.4)
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407
One can generalize the above formula using induction: d (Q− )d : exp iQea 8a (t) : = lim . . . lim q1 →t
×
d
qd →t
(qj − qi ) : exp iQea 8a (t) − 2
i<j
d
l=1
1 ∂ 2Q−1 (2Q−1)! ql
(6.5)
iea 8a (ql ) : .
l=1
Let us evaluate the norm of the state (6.5). To do this we only need to use the structure of the SU(2) algebra: [Q+ , Q− ] = 2Q3 ,
[Q3 , Q± ] = ±Q± .
(6.6)
One can write the two point function of twist operators in terms of the norms of appropriate representations of SU(2):
n ± 1 n ± 1 2 d d m Q− σn± (0) Q+ σn±† (w) = Q− , . 2 2
(6.7)
Then standard manipulations give: d−1 d − m d−1 − d−1 Q, Q| Q+ Q Q |Q, Q = 2 |Q, Q
(Q − j ) Q, Q| Q+
− d−1
j =0
|Q, Q 2 , = d(2Q + 1 − d) Q 1/2 d d!(2Q)! |Q, Q . Q− |Q, Q = (2Q − d)!
(6.8)
Thus we can define the normalized state: |Q, Q − d =
(2Q − d)! d!(2Q)!
1/2
dp1 − J (p1 ) · · · 2π i
dpd − J (pd )|Q, Q . 2π i
(6.9)
In the bosonised formulation we get:
d (2Q − d)! 1/2 −Q2 /n 1 2Q−1 |Q, Q − d = b lim · · · lim ∂ q1 →t qd →t d!(2Q)! (2Q−1)! ql l=1 d d × (qj − qi )2 : exp iQea 8a (t) − iea 8a (ql ) : . (6.10) i<j
l=1
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6.2. The contribution of the current insertions. Let us work out the contribution from the current insertions on the t sphere. This contribution corresponds to the factor in (3.18), J, S σ1 (0)σ2 (a)σ3 (∞) J, S σ3 (0)σ3 (∞) .
(6.11)
We are looking for the fusion coefficient (Q, Q − d2 ) × (P , P ) → (Q + P − d, Q + P − d).
(6.12)
One should consider the fusion rule for the following CFT operators: |Q, Q − d 0 |P , P a ≈ A|Q + P − d, Q + P − d 0 . We first note that (Q− )d : exp iQea 8a (0) : : exp iP ea 8a (1) : d d 1 2Q−1 2 ≈ lim (qj − qi ) (1 − qk )−2P ∂ qi →0 (2Q − 1)! ql l=1 i<j k=1 × : exp i(P + Q − d)ea 8a (0) [1 + · · · ] :,
(6.13)
(6.14)
d
(6.15)
where the · · · in the last term indicate polynomials in ∂8a (0), ∂ 2 8a (0), etc. We are again using a computation analogous to (5.16), and will thus be computing the correlator of the above expression with a pure exponential operator at infinity. In this situation the terms represented by · · · in the above expression will give a contribution that is vanishing compared to the first term, and can thus be ignored. Taking into account the normalization (6.10), we finally get: A=
−Q2 /n −P 2 /n S 2 /q b0 b1 b∞
× lim
qi →0
d l=1
1 (2Q−1)!
(2Q − d)! 1/2 d!(2Q)! d d 2Q−1 ∂ql (qj − qi )2 (1 − qk )−2P . i<j
(6.16)
k=1
± The value of S in the above expression is S = q±1 2 for σq . Notation. It will be helpful in what follows to introduce the following notation. We will characterize the twist operator σn± by two numbers: n and 1n , where 1n = 1 for σn+ and 1n = −1 for σn− . We also introduce the notation
σn1n (d)
(6.17)
for the twist operator corresponding to the state (6.10). Here the value of Q is given by − n Q = n+1 2 and d is the number of lowering operators Q that have been applied to the state |Q, Q . To evaluate the expression A we need to use the map which takes the z sphere to the t sphere. In [9] we have shown that such a map is unique up to a SL(2, C) transformation
Three-Point Functions for Orbifolds with N = 4 Supersymmetry
409
and its explicit form is given in Appendix B. Here we will need only some properties of this map, namely its behavior near the ramification points: d1 !d2 !(n − d2 − 1)! n t , n!(d1 − n)!(n − 1)! d1 !d2 !(m − d2 − 1)! z≈a+a (t − 1)m , m!(d1 − m)!(m − 1)! d2 !(d1 − d2 − 1)!(d − 1 − d2 )! d1 −d2 z≈a . t d1 !(d1 − n)!(d1 − m)!
t →0:
z≈a
t →1: t →∞:
(6.18) (6.19) (6.20)
Using the above information, we get: (q + 1q )2 (m + 1m )2 (n + 1n )2 − − log d1 ! log A = − 4n 4m 4q (n+1n )2 (m+1m )2 (q+1q )2 + − − log(d1 −n)! + (n ↔ m) + (n ↔ q) 4n 4m 4q 2 (n + 1n ) + log (n!(n − 1)!) + (n ↔ m) + (n ↔ q) 4n d d d 1 2 −m−1m n −1 + log lim (q − q ) (1−q ) ∂qn+1 j i k qi →0 (n + 1n − 1)! l l=1 i<j k=1 2 1 P2 Q2 d!(n + 1n )! S − log + − − log a, (6.21) 2 (n + 1n − d)! q n m where 1 d = d2 + (1n + 1m − 1q + 1). 2
(6.22)
−−− . Collecting the contributions from the conformal 6.3. The fusion coefficient Cn,m,q anomaly and charge insertions, one finds the following OPE 1q
σn1n (d) (0)σm1m (a) ≈ a #q 1 (d),1m ,1q
n Cn,m,q
−#1nn −#1mm
1 (d),1m ,1q
n Cn,m,q
= (Cn,m,q )6 Aa −S
1
σq q (0),
2 /q+P 2 /n+Q2 /m
(6.23) (6.24)
Here we are writing only the holomorphic part of the OPE; the complete OPE will have representations of the two su(2) factors and will be constructed at the end. Taking into account the expression for the fusion coefficient of the bosonic twists found in [9] (see Appendix B), we finally get an expression for the holomorphic part of
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O. Lunin, S. D. Mathur
the fusion coefficient 1 (d),1 ,1
n m q log Cn,m,q = (−d2 + 2) log d1 ! + d2 (log(d1 − n)! + log(d1 − m)!) 1 +(d2 − 2) log d2 ! − 2 log(q − 1)! − log(mnq 3 ) + d2 (d2 − 1) log 2 2 d!(n + 1n )! 1 d + log D + log Ln+1n ,m+1m − log 2 (n + 1n − d)! 1 − (3 + 1n + 1m + 1q ) log d1 2 1n −1m −1q −1 + log(d1 − n)! 2 1+1n + log (n!(n − 1)!) + (n ↔ m) + (n ↔ q) . (6.25) 2
Here we introduced the notation: d d d 1 2 −m Ldn,m = lim . (q − q ) (1 − q ) ∂qn−1 j i k qi →0 (n − 1)! l l=1
i<j
(6.26)
k=1
To get a contribution form antiholomorphic sector, one should replace 1n , 1m and 1q in (6.25) by the corresponding value for the antiholomorphic part (1¯ n , 1¯ m or 1¯ q ). The expression (6.25) looks complicated for two reasons. Firstly it contains the discriminant D which is known only as a finite product (B.6). Secondly there is the term Ldn,m defined through derivatives and limits. First we note that (6.25) simplifies enormously for the low values of d2 . In particular, −−− for d = 0 and d = 1: one can easily evaluate Cn,m,q 2 2 −,−,− Cn,m,m+n−1
=
−(1),−,−
Cn,m,m+n−3 =
m+n−1 mn
1/2 ,
(6.27)
(m + n − 2)2 (n − 2)! mn(m + n − 3) (n − 1)!
1/2 .
(6.28)
The first of these two expressions is the result (5.25) derived for a special case in the previous section. Investigating a few further cases, and using the symmetry properties of C lead us to guess that the complicated expression (6.25) for the case of three operators σ − is equal to the following simple expression: −(d2 ),−,− Cn,m,q
=
d12 (d2 !)2 (n − d2 − 1)! mnq d2 !(n − 1)!
1/2 .
(6.29)
While we could not prove the agreement of (6.25) and (6.29) analytically, we have verified it for d2 ≤ 5 and arbitrary values of n and m using a symbolic manipulations
Three-Point Functions for Orbifolds with N = 4 Supersymmetry
411
program (Mathematica). Assuming the equality of these two forms of the result, one arrives at the following expression for the Ldn,m : Ldn,m = (d!)3 ×
$
d
(n + m + 1 − d)! d!(n − d)!(m − d)!
%d $
(n + m + 1 − 2d)! (n + m + 1 − d)!
%2
n+m+1−d (6.30) n + m + 1 − 2d
j 2d−2−j (j − n − 1)1−j (j − m − 1)1−j (j − n − m − 2 + d)j −d (6.31)
j =1
6.4. The fusion coefficients for general elements of the representations. From now on we will assume that (6.31) is correct, and use this to write compact expressions for general fusion coefficients. We can rewrite (6.29) in the more symmetric form: −(d2 ),−,− Cn,m,q
=
d12 [(n + m − q − 1)/2]![(n − m + q − 1)/2]! mnq (n − 1)!
1/2 .
(6.32)
Since we are considering the fusion of the different representations of SU(2), we anticipate the appearance of the appropriate 3j symbols in the final answer. In terms of the standard notation |j, m for the representations of SU(2), the operator σn− (0) can be written as |
n−1 n−1 , − d2 , 2 2
(6.33)
so the case computed in the above subsection corresponds to the following 3j symbol:
n−1 m−1 q−1 2 2 2 q−m m−1 1−q 2 2 2
.
(6.34)
One can easily evaluate this particular type of the 3j symbol (see, for example [18]):
n−1 m−1 q−1 2 2 2 q−m m−1 1−q 2 2 2
=
(m − 1)!(q − 1)! d1 ![(m + q − n − 1)/2]!
1/2 .
(6.35)
We are to take the positive sign of the square root here and in all similar expressions in what follows. The fact that the coefficients (6.35) are positive allow us to take such square roots and thus write separate factors for the holomorphic and aniholomorphic fusion coefficients, without introducing any extra phase factors. Using the above expression, we can rewrite (6.32) in the final form:
−(d2 ),−,− Cn,m,q
n+m−q−1 ]! [ n+q−m−1 ]! [ m+q−n−1 ]! d12 d1 ! [ 2 2 2 = mnq (n − 1)! (m − 1)! (q − 1)! n−1 m−1 q−1 2 2 2 . × q−m m−1 1−q 2
2
2
1/2
(6.36)
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O. Lunin, S. D. Mathur
In particular, it is convenient to define the reduced fusion coefficient, which describes a fusion of different representations of SU (2) and does not specify the “‘orientation of the spins”, −−− = Cˆ n,m,q
n+m−q−1 ]! [ n+q−m−1 ]! [ m+q−n−1 ]! d12 d1 ! [ 2 2 2 mnq (n − 1)! (m − 1)! (q − 1)!
1/2 .
(6.37)
Then the fusion rule for the specific example of twist operators σ −− reads: σn−−(s1 ,¯s1 ) (0)σm−−(s2 ,¯s2 ) (a) ∼ |a|−(2#n +2#m −2#q ) σq−−(s3 ,¯s3 ) (0) n−1 m−1 q−1 n−1 m−1 q−1 −−− 2 2 2 2 2 2 2 × |Cˆ n,m,q | s1 s2 s 3 s¯1 s¯2 s¯3 1 ,1 ,1q
n m 6.5. The fusion coefficients Cm,n,q
1n ,1m ,1q Cm,n,q . We
(6.38)
. We have also analyzed the general fusion coeffi-
+−+ , in cients present the computation of another case, the coefficient Cm,n,q Appendix C. The final result for the reduced fusion coefficients is: 1n ,1m ,1q Cˆ n,m,q
1/2 2 1n n + 1m m + 1q q + 1 ! αn !αm !αq ! = , 4mnq (n + 1n )! (m + 1m )! (q + 1q )! (6.39)
where 1 n + 1n + m + 1m + q + 1q + 1, 2 αn = − n − 1n − 1. =
(6.40) (6.41)
Note that the parameters 1n , 1m , 1q can be chosen independently for the holomorphic and antihlomorphic parts of the twist operator. The full fusion coefficient is then a product of (6.39) from the left and right sides, together with the Clebsch-Gordon coefficients from the left and right su(2) representations. 6.6. Combinatoric factors and large N limit. The twist operators we have considered so far do not represent proper fields in the conformal field theory. In the orbifold CFT there is one twist field for each conjugacy class of the permutation group, not for each element of the group [8]. The true CFT operators that represent the twist fields can be constructed by summing over the group orbit, for example: ¯
On1n 1n =
λn 1n 1¯ n σh(1,... ,n)h−1 . N!
(6.42)
h∈G
Here G is the permutation group SN and the normalization constant λn can be determined from the normalization condition. Namely if one starts from normalized σ operators: ¯
¯
σn1n 1n (0)σn1n 1n † (1) = 1,
(6.43)
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413
then for O we get: ¯
¯
On1n 1n (0)On1n 1n † (1) = λ2n n ¯
(N − n)! 1n 1¯ n ¯ σn (0)σn1n 1n † (1) . N!
(6.44)
¯
Requiring the normalization On1n 1n (0)On1n 1n † (1) = 1, we find the value of λn : $ % n(N − n)! −1/2 λn = . (6.45) N! Let us now look at the three point function. For the case when the covering surface is a sphere, the three point function is [9]: √ mnq(N − n)!(N − m)!(N − q)! 1q 1¯ q † 1n 1¯ n 1m 1¯ m (z) = On (0)Om (1)Oq √ (N − s)! N ! ¯
1 1¯ q †
¯
× σn1n 1n (0)σm1m 1m (1)σq q
(z) ,
with s = 21 (n + m + q − 1). Now we analyze the behavior of the combinatoric factors for arbitrary genus g but in the limit where N is taken to be large while the orders of twist operators (m, n and q) as well as the parameter g are kept fixed. There are s different fields X i involved in the 3-point function, and these fields can be selected in ∼ N s ways. Similarly the 2-point function of σn will go as N n since n different fields are to be selected. Thus the 3-point function of normalized twist operators will behave as N s−
n+m+q 2
= N −(g+ 2 ) . 1
(6.46)
Thus in the large N limit the contributions from surfaces with high genus will be suppressed, and in the leading order the answer can be obtained by considering only contributions from the sphere (g = 0). This is presicely the case that we have analyzed in detail, and knowing the amplitude for operators σ one can easily extract the leading order of the CFT correlation function:
1 1¯ ¯ 1m 1¯ m On1n 1n (0)Om (1)Oq q q (z) & 1√ 1 1 1¯ ¯ ¯ = . mnq σn1n 1n (0)σm1m 1m (1)σq q q (z) sphere + O N N 3/2
(6.47)
7. Discussion In this paper we have computed the contribution to the 3-point function for the case when the covering surface is a sphere. The construction of the chiral operators and the method of computing correlators used only the fact that the CFT based on the manifold M had N = 4 supersymmetry; thus the computation is not restricted to orbifolds that can be obtained from free fields. The result we obtain is independent of the details of M, and thus exhibits a “universal” property of CFTs arising from orbifolds M N /SN . A nontrivial aspect of the computation is that we have an orbifold group that is nonabelian – this fact required us to develop a new way to compute correlators [9]. The method we use is particular to the symmetric group SN however, and it is not immediately clear if it could be extended to other nonabelian groups.
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The sphere contribution is the dominant contribution at large N , but we can in principle calculate the contribution of surfaces with g > 0 to get an exact result for finite N. Note that for a given choice of orders of the chiral operators, the number of different surfaces that can contribute is finite, though it grows with the orders of the operators. In the case of the bosonic orbifold we had shown in [9] that the correlator for twist operators on the z plane for the orbifold M N /SN could be written in terms of the partition functions Zg for the theory with one copy of M but on a worldsheet of genus g. For the supersymmetric case we get the following extension of this result: the correlator of the chiral operators considered here can be written in terms of the Zg and a finite number of derivatives of Zg with respect to moduli, where the moduli arise from both the shape of the Riemann surface and from the “current algebra moduli” that couple to the su(2) currents. This result follows from the fact that we map the correlator of twist operators to the covering surface as in the bosonic case, but we get some correlators of currents that need to be computed on . The current insertions can be removed by using the current algebra Ward identity, but in the process we generate derivatives with respect to the moduli. The supersymmetric orbifold that we have studied is expected to be a point in the D1– D5 system moduli space, and the latter system is dual to string theory on AdS3 ×S 3 ×M. In this context we recall some observations that were made in [9] relating orbifold correlation functions to the dual string theory. First, it was found that to get a leading order 3-point function of twist operators (which requires that the genus of be zero) we must satisfy some restrictions on the orders of the twists. These restrictions turn out to be identical to the fusion rules of the su(2) WZW model, which restrict the 3-point functions of string states at tree level in the dual string theory. Secondly, we note that the orbifold theory has expressed the correlators of twist operators as a sum of contributions from different genus Riemann surfaces, with the contribution of higher genera surfaces being suppressed by 1/N g . This is reminiscent of the genus expansion in the dual √ string theory, where higher genus amplitudes are suppressed by a similar factor (1/ N is the coupling constant of the string modes). These facts are exciting, since they indicate that we may be able to see aspects of the dual supergravity geometry by analyzing correlators in the CFT. It would be interesting to compare the 3-point functions that we have computed to the 3-point functions of the supergravity field in the dual theory, to see if there is any analogue of the nonrenormalization that was found in the case of D-3 branes and the dual theory on AdS5 × S 5 . Some 3-point supergravity amplitudes have been computed in [21], but these correspond to scalar fields which are expected to be supersymmetry descendents of the chiral fields that we have worked with. Another supergravity 3-point correlation function was computed in [14], and there is a significant agreement in overall form between this result and the correlators that we get. However the calculation of [14] used a step where a symmetrization was performed over the three fields involved in the correlator, and to reproduce the result obtained from the orbifold computation we would have to choose a somewhat different way to symmetrize (instead of summing the squares of the three momenta we would have to sum the momenta and then square the result). We hope to return to this comparison at a later point. The D1–D5 system for the black holes studied in [1][2] arises by wrapping the space direction of the D1–D5 CFT on a circle, with periodic boundary conditions for the fermions on this circle. Thus we need to study the CFT in the Ramond sector, rather than in the NS sector. We can compute correlation functions in the Ramond sector if we can compute correlators with insertions of spin fields, since these spin fields map
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the vacuum on the plane to the Ramond vacuum. (These insertions of spin fields are different from the spin fields that we have encountered in the present paper; the latter arose only after the map to the t space, while the former would be inserted to change boundary conditions in the original z space.) We have computed examples of correlators in the Ramond sector, which correspond to the amplitude for a simple “black hole state” to absorb and re–emit a quantum; this calculation will be presented elsewhere [22]. In the dual theory on AdS3 × S 3 × M the NS and Ramond sector are connected by a spectral flow as well. It was suggested in [23] that a family of singular conical defect spacetimes would represent this spectral flow, but it was argued recently in [24] that the generic spacetime in the flow would in fact be smooth, and the spacetime deformation corresponding to spectral flow away from the NS vacuum was computed to first order. Acknowledgements. We are grateful to A. Jevicki, M. Mihailescu, S. Ramgoolan, and S. Frolov for patiently explaining their results to us. We also benefited greatly from discussions with S. Das, E. D’Hoker, C. Imbimbo, D. Kastor, F. Larsen, E. Martinec, S. Mukhi, Z. Qiu, L. Rastelli and S.-T. Yau.
A. N = 4 Superconformal Algebra ∂T (w) c 2T (w) + , + 2 z−w (z − w) 2(z − w)4 iεij k J k (w) c J i (z)J j (w) = , + z−w 12(z − w)2 ∂J i (w) J i (w) , T (z)J i (w) = + z−w (z − w)2 2T (w)δba 2cδba 2(σ i )a b ∂J i (w) 4(σ i )a b J i (w) ˜ b (w) = Ga (z)G − + , (A.1) − z−w 3(z − w)3 z−w (z − w)2 ˜ ˜ ∂Ga (w) 3Ga (w) ˜ a (w) = ∂ Ga (w) + 3Ga (w) , T (z)Ga (w) = , T (z)G + 2 z−w 2(z − w) z−w 2(z − w)2 i b ˜ (σ i )a b Gb (w) ˜ a (w) = Gb (w)(σ ) a . J i (z)Ga (w) = − , J i (z)G 2(z − w) 2(z − w) ˜ b are related by complex conjugation: The elements of Ga and G ˜ a (z) = Ga (z) † . (A.2) G T (z)T (w) =
In terms of modes we have:
c [Lm , Ln ] = (m − n)Lm+n + m(m2 − 1)δm+n,0 , 12 c j k i Jmi , Jn = iεij k Jm+n + mδm+n,0 , Lm , Jni = −nJm+n , 12 c(4r 2 − 1) a i ˜ b,s = 2δ a Lr+s − 2(r − s)(σ i )a b Jr+s Gar , G + δb δr+s,0 , b 12 ˜ a,r , G ˜ b,s = 0, Gar , Gbs = 0, G ' ( m ˜ a,r = m − r G ˜ a,m+r , Lm , Gar = Lm , G − r Gam+r , 2 2 1 ˜ b,m+r (σ i )b a . ˜ a,r = 1 G Jmi , Gar = − (σ i )a b Gbm+r , Jmi , G 2 2
(A.3)
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As was pointed out by Schwimmer and Seiberg [19], there is a family of equivalent representations of the superconformal algebra, which differ only by the boundary conditions: ˜ 2 (z) = −eiπη G ˜ 2 (e2iπ z). J ± (z) = e∓2iπη J ± (e2iπ z), G1 (z) = −eiπη G1 (e2iπ z), G (A.4) The equivalence can be established by the action of the spectral flow 3 : η 3 cη2 , J (z) + z 24z2 cη Jη3 (z) = J 3 (z) − , Jη± (z) = z∓η J ± (z), 12z Tη (z) = T (z) −
η 2
G1η (z) = z G1 (z), ˜ 1η (z) = z G
− η2
G2η (z) = z
˜ 1 (e2iπ z), G
− η2
(A.5) (A.6)
G2 (z),
(A.7)
η 2
˜ 2 (e2iπ z). ˜ 2η (z) = z G G
(A.8)
In particular, if we started with “vacuum state” |χ :
dz p+1 z T (z)|χ = 2πi
dz p 3 z J (z)|χ = 0, 2π i
p ≥ 0.
(A.9)
in η = 0 sector, then the charges in η sector are given by:
dz cη2 zTη (z)|χ = , 2πi 24
cη dz 3 Jη (z)|χ = − , 2π i 12
c = 6.
(A.10)
To have a well defined expression for J (z) one has to choose an odd integer η, but for odd η the fermionic operators become integer–moded, i.e. we obtain a Ramond sector of the theory. We will pick the simplest possible value: η = 1. Thus, the starting point of our flow (i.e. NS sector) corresponds to η = 0, while at η = 1 we get a Ramond sector. We will also need relations between modes in NS and R sectors (primed modes are from the R sector): c c ± (J 3 ) n = Jn3 − δn,0 , (J ± ) n = Jn∓1 , (A.11) δn,0 , 24 12 ˜ 1,n = G ˜ ˜ 2,n = G ˜ (G1 ) n = G1n+ 1 , (G2 ) n = G2n− 1 , G G . 1,n− 1 , 2,n+ 1
Tn = Tn − Jn3 + 2
2
2
2
In the NS sector there was a bound on a maximal possible charge: |j | ≤ h, which saturated only for (anti)chiral fields. In particular, chiral fields had h = j . In the Ramond c and it is saturated only by images of chiral fields. sector this bound becomes: h ≥ 24 3 Since the R symmetry group for N = 4 theory is SU (2), there is a three parametric family of spectral flows. We will perform such a flow only along the J 3 direction
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B. Three Point Function for the Bosonic Orbifold In this appendix we will summarize the results of [9], where the three point function on the bosonic SN orbifold was calculated. In particular, in [9] we evaluated the contribution to the three point function for the case where the covering space is a sphere. In this case the map corresponding to the three point function σn (0)σm (a)σq (∞)
(B.1)
−1 (−n,−d1 −d2 +n−1) (1 − 2t) Pd2 (1 − 2t) .
(B.2)
is given by: (n,−d1 −d2 +n−1)
z = at n Pd1 −n
(α,β)
This map involves Jacobi polynomials Pn as d1 =
1 (n + m + q − 1), 2
(x) and the values of d1 and d2 are defined d2 =
1 (n + m − q − 1). 2
(B.3)
One can easily show that the asymptotic behavior of this map is given by (6.18), (6.19), (6.20). We recall that the three point function (B.1) was evaluated in [9] by considering the conformal anomaly for the map (B.2). More precisely, this three point function can be written in terms of the Liouville action describing the map (B.2) and the Liouville actions corresponding to two point functions σn (0)σn (z) : σn (0)σm (a)σq (∞)
= exp SL (σn (0), σm (a), σq (∞)) − SL (σq (0)σq (∞)) σq (0)σq (∞)
1 1 1 × exp − SL (σn (0), σn (1)) − SL (σm (0), σm (1)) + SL (σq (0), σq (1)) . 2 2 2 c
≡ |Cn,m,q |2c |a|− 12 (n+m−q−1/n−1/m+1/q)
(B.4)
Cn,m,q is the fusion coefficient of σn and σm to σq . The evaluation of the three point function leads to the following result [9]: q n−1 1 m−1 q −1 log |Cn,m,q |2 = log − log n − log m + log(q) 6 mn 12 12 12 n−1 d1 !d2 ! (d1 − m)! − log 12n n!(n − 1)! (d1 − n)! m−1 d1 !d2 ! (d1 − n)! − log 12m m!(m − 1)! (d1 − m)! q −1 (q − 1)!d2 ! (d1 − d2 )!) + log (B.5) 12q (d1 − n)!(d1 − m)! d1 ! 1 d2 1 3d2 − 4 + d2 (d2 − 1) log 2 − log n + log D + log d2 ! 3 6 3 6 $ % d2 d1 ! (n − 1)! n + d2 − 1 − log log − 6 n!(d1 − n)! 6 (n − d2 − 1)! d1 − d2 + 3 (d1 − d2 )! d1 + d2 − n (d1 + d2 − n)! − log − log . 6 d1 ! 6 (d1 − n)!
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This expression involves the discriminant of the Jacobi polynomials D, which is known as the following finite product [20]: (−n,−d1 −d2 +n−1)
D ≡ Dd2 ×
d2
= 2−d2 (d2 −1)
(B.6)
j j +2−2d2 (j − n)j −1 (j − d1 − d2 + n − 1)j −1 (j − d1 − 1)d2 −j .
j =1
Note that the combination of Liouville actions entering (B.4) is the combination needed for the SUSY orbifold (3.13). Thus we can take the result for |Cn,m,q | as the contribution of the conformal anomaly even for the supersymmetric case. +,−,+ C. Calculation of Cˆ n,m,q −,−,− In Sect. 6 we have calculated Cˆ n,m,q and presented the result for the general reduced 1n ,1m ,1q ˆ fusion coefficient Cn,m,q . In this section we will assume the relation (6.26) and, using +,−,+ it, we will evaluate another fusion coefficient Cˆ n,m,q which will give another case of the result (6.39). Using the general expression (6.25), one can find a ratio: +(d ),−,+
2 Cn,m,q
=
−(d ),−,−
2 Cn,m,q
n!(n − 1)!q!(q − 1)! (d1 !(d1 − m)!)2
(n − d2 )(n − d2 + 1) n(n + 1)
1/2
2 Ldn+1,m−1 2 Ldn−1,m−1
. (C.1)
To evaluate a ratio of limits one can use the fact that for an integer l > 1: % d $ j − l − 2 1−j (l − d − 1)! 2 (l − d)d+1 (l − d + 1)d = . j −l (l − 1)! l
(C.2)
j =1
Then one gets: 2 Ldn+1,m−1 2 Ldn−1,m−1
$
(n − d2 − 1)!(m + n − d2 − 1)! = (m + n − 2d2 − 1)!(n − 1)!
%2
n − d m + n − 2d2 − 1 . n m + n − d2 − 1
(C.3)
To define the reduced fusion coefficient we also need the relation between appropriate 3j symbols:
m−1 n+1 2 2 q−m+2 m−1 2 2
q+1 2 − 1+q 2
$ =
q(q + 1) (d1 + 1)(d1 + 2)
%1/2
n−1 m−1 q−1 2 2 2 q−m m−1 1−q 2 2 2
,
(C.4)
which can be deduced from the general expression:
a b c c − b b −c
=
[2b]![2c]! [a + b + c + 1]![b + c − a]!
1/2 .
(C.5)
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Collecting all this information together, one finds the reduced fusion coefficient: %1/2 $ d1 − m + 1 +,−,+ −,−,− (d1 + 1)(d1 + 2) (d1 − m + 1)(d1 − m + 2) Cˆ n,m,q = Cˆ n,m,q (q + 1)(n + 1) nq d1 1/2 (d1 − m + 1)2 (d1 + 2)! (d1 − q)!(d1 − n)!(d1 − m + 2)! = , (C.6) mnq (n + 1)! (m − 1)! (q + 1)! which agrees with (6.39). Proceeding in a similar manner for other values of the indices 1n , 1m , 1q we arrive at (6.39). References 1. Strominger, A. and Vafa, C.: Phys. Lett. B379, 99 (1996), hep–th/9601029 2. Callan, C. and Maldacena, J.: Nucl. Phys. B472, 591 (1996), hep–th/9602043; Dhar, A., Mandal, G. and Wadia, S.R.: Phys. Lett. B 388, 51 (1996), hep-th/9605234; Das, S.R. and Mathur, S.D.: Nucl. Phys. B478, 561 (1996), hep–th/9606185; Das, S.R. and Mathur, S.D.: Nucl. Phys. B 482, 153 (1996), hep-th/9607149; Maldacena, J. and Strominger, A.: Phys. Rev. D55, 861 (1997), hep–th/9609026 3. Maldacena, J.: Adv. Theor. Math. Phys. 2, 231 (1998), hep–th/9711200; Gubser, S., Klebanov, I. and Polyakov, A.: Phys. Lett. B428, 105 (1998), hep–th/9802109; Witten, E.: Adv. Theor. Math. Phys. 2, 253 (1998), hep–th/9802150; Aharony, O., Gubser, S.S., Maldacena, J., Ooguri, H. and Oz, Y.: Phys. Rept. 323, 183 (2000), hep– th/9905111 4. Seiberg, N. and Witten, E.: JHEP 9904, 017 (1999), hep–th/9903224 5. Larsen, F. and Martinec, E.: JHEP 9906, 019 (1999), hep–th/9905064; de Boer, J.: Nucl. Phys. B548, 139 (1999), hep–th/9806104; Dijkgraaf, R.: Nucl. Phys. B543, 545 (1999), hep–th/9810210 6. Freedman, D., Mathur, S.D., Rastelli, L. and Matusis, A.: Nucl. Phys. B546, 96 (1999), hep–th/9804058; Lee, S., Minwalla, S., Rangamani, M. andSeiberg, N.: Adv. Theor. Math. Phys. 2, 697 (1999), hep– th/9806074 7. Dixon, L., Friedan, D., Martinec, E., and Shenker, S.: Nucl. Phys. B282, 13 (1987) 8. Dijkgraaf, R., Vafa, C., Verlinde, E. and Verlinde, H.: Commun. Math. Phys. 123, 485 (1989) 9. Lunin, O. and Mathur, S.D.: hep-th/0006196, to appear in Commun. Math. Phys 10. Hamidi, S. and Vafa, C.: Nucl. Phys. B279, 465 (1987) 11. Arutyunov, G.E. and Frolov, S.A.: Theor. Math. Phys. 114, 43 (1998), hep–th/9708129; Nucl. Phys. B524, 159 (1998), hep–th/9712061 12. Bantay, P.: Phys. Lett. B419, 175 (1998), hep–th/9708120 13. Jevicki, A. , Mihailescu, M. and Ramgoolam, S.: Nucl. Phys. B577, 47 (2000), hep–th/9907144 14. Mihailescu, M.: JHEP 0002, 007 (2000), hep–th/9910111. 15. Evslin, J., Halpern, M.B. and Wang, J.E.: Int. J. Mod. Phys. A14, 4985 (1999), hep-th/9904105; de Boer, J., Evslin, J., Halpern, M.B. and Wang, J.E.: Int. J. Mod. Phys. A15, 1297 (2000), hep-th/9908187; Halpern, M.B. and Wang, J.E.: hep-th/0005187 16. Argurio, R., Giveon, A. and Shomer, A.: JHEP 0012, 003 (2000), hep-th/0009242 17. Friedan, D.: Introduction To Polyakov’s String Theory. In: Recent Advances in Field Theory and Statistical Mechanics, ed. by J. B. Zuber and R. Stora. Amsterdam: North–Holland, 1984 18. Edmonds, A.R.: Angular momentum in quantum mechanics. Princeton, NJ: Princeton University Press, 1974 19. Schwimmer, A. and Seiberg, N.: Phys. Lett. B184, 191 (1987) 20. Szego, G.: Orthogonal Polynomials. Providence, RI: American Mathematical Society, 1959 21. Arutyunov, G., Pankiewicz, A. and Theisen, S.: Phys. Rev. D 63, 044024 (2001), hep-th/0007061 22. Lunin, O. and Mathur, S.D.: In preparation 23. David, J.R., Mandal, G., Vaidya, S. and Wadia, S.R.: Nucl. Phys. B 564, 128 (2000), hep-th/9906112 24. Mathur, S.D.: hep-th/0101118 Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 227, 421 – 460 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Hyperbolic Low-Dimensional Invariant Tori and Summations of Divergent Series G. Gallavotti1 , G. Gentile2 1 INFN, Dipartimento di Fisica, Università di Roma 1, 00185 Roma, Italy 2 Dipartimento di Matematica, Università di Roma 3, 00146 Roma, Italy
Received: 9 July 2001 / Accepted: 26 October 2001
Abstract: We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coefficients that grow at most as a power of a factorial and a remainder that to any order N is bounded by the (N + 1)-st power of the argument times a power of N !. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory. 1. Introduction 1.1. The model. Consider the Hamiltonian 1 1 H = ω · A + A · A + B · B + εf (α, β), (1.1) 2 2 where (α, A) ∈ Tr × Rr and (β, B) ∈ Ts × Rs are conjugated variables, · denotes the inner product both in Rr and in Rs , and ω is a vector in Rr satisfying the Diophantine condition |ω · ν| > C0 |ν|−τ
∀ν ∈ Zr \ {0},
(1.2)
with C0 > 0 and τ ≥ r − 1; we shall define by Dτ (C0 ) the set of rotation vectors in Rr satisfying (1.2). We also write eiν·α fν (β), (1.3) f (α, β) = ν∈Zr
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and set d = r + s. We shall suppose that f is analytic in a strip of width κ > 0 around the real axis of the variables α, β, so that there exists a constant F such that |fν (β)| ≤ F e−κ|ν| for all ν ∈ Zr and all β ∈ Ts . 1.2. Low-dimensional tori. The equations of motion for the system (1.1) are α˙ = ω + A, β˙ = B, ˙ = −ε∂α f (α, β), A B ˙ = −ε∂β f (α, β).
(1.4)
For ε = 0 the system of Eqs. (1.4), with initial data (α 0 , β 0 , 0, 0), admits the solution α(t) = α 0 + ωt, β(t) = β , 0 (1.5) A(t) = 0, B(t) = 0, which corresponds to a r-dimensional torus (KAM torus): the first r angles rotate with angular velocity ω1 , . . . , ωr , while the remaining s remain fixed to their initial values. Note that (1.4) can be written as α¨ = −ε∂α f (α, β), (1.6) β¨ = −ε∂β f (α, β), so that we obtain closed equations for the angle variables: once a solution has been found for them, it can be used to find the action components by a simple integration. We look for solutions of (1.6), for ε = 0, conjugated to (1.5), i.e. we look for solutions of the form α(t) = ψ + a(ψ, β 0 ; ε), (1.7) β(t) = β 0 + b(ψ, β 0 ; ε), for some functions a and b, real analytic and 2π -periodic in ψ ∈ Tr , such that the motion in the variable ψ is ψ˙ = ω. We shall prove the following result. Theorem 1.1. Consider the equations of motion (1.6) for ω ∈ Dτ (C0 ), and suppose β 0 to be such that ∂β f0 (β 0 ) = 0, ∂β2 f0 (β 0 ) is negative definite.
(1.8)
There exist a constant ε0 > 0 and, for all ε ∈ (0, ε0 ), two functions a(ψ, β 0 ; ε) and b(ψ, β 0 ; ε), real analytic and 2π-periodic in ψ ∈ Tr , such that (1.7) is a solution of (1.6).
Hyperbolic Low-Dimensional Invariant Tori
423 complex ε−plane
Fig. 1. Analyticity domain D0 for the hyperbolic invariant torus. The cusp at the origin is a second order cusp. The figure corresponds to the case in (1.8) of the Theorem 1.1
Remarks. (1) As it is well known, and as it will appear from the proof, the solutions whose existence is stated by the theorem cannot be expected to be analytic in ε at ε = 0. Furthermore, if the second condition in (1.8) is replaced with ∂β2 f0 (β 0 ) is positive definite,
(1.9)
then the same conclusions hold for ε ∈ (−ε0 , 0). (2) The proof will yield more detailed information on the regularity of the considered tori, as we shall point out. In particular the analyticity domain is much larger, see the heart-like domain D0 in Fig. 1 below (and the discussion in the forthcoming Sect. 5.3). In fact we think that our technique can lead to prove existence of many elliptic invariant tori, i.e. for a large set of negative ε’s, and to understand some of their analyticity properties; see Sect. 6 for further remarks and results.
1.3. Contents of the paper. The paper is organized as follows. In Sect. 2 we introduce the main graph techniques which will be used, and we prove through them the formal solubility of the equations of motions (known from [JLZ]): this is enough if one wants to prove existence and analyticity of periodic solutions, (see Remark 2.3 below), but it requires new arguments to obtain existence of quasi-periodic solutions. Such arguments are developed in the following sections: in Sect. 3 we introduce the concept of self-energy graph, which will play a crucial rôle, and we describe the basic cancellation mechanisms which will be used in Sect. 4 to perform a suitable resummation of the series. In Sect. 5 we shall use such results in order to prove the convergence of the resummed series and its analyticity properties. Finally in Sect. 6 we make some conclusive remarks, and briefly discuss possible generalizations and extensions of the results. The main technical aspects of the proof will be relegated to the Appendices.
1.4. Comparison with other papers. The problem considered here is a priori stable in the sense of [CG]: the low-dimensional invariant tori are degenerate in absence of perturbations. Hamiltonians of the form (1.1) were explicitly studied in [T], in a more general formulation (see Sect. 6 below), and in [JLZ], in a more particular case. The problem usually considered in the literature essentially corresponds to a Hamiltonian of the form 1 2 1 1 A + ω · A + B2 − β 2 + f (A, B, α, β), 2 2 2
f0 (A, B, β) = O(A3 + B3 + β 3 ), (1.10)
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where is a, a priori fixed, nondegenerate matrix (so that before the perturbation is switched on the invariant torus at A = 0 has a priori a well defined stability property, i.e. its elliptic or hyperbolic stability is already well defined); this case is called a priori unstable in [CG]. The system (1.10) has been widely studied in the hyperbolic case > 0, in the elliptic case < 0, and in the mixed case. The general hyperbolic problem has been studied in [Mo]; in [Gr] the stable and unstable manifolds of the tori are also determined. The elliptic and mixed cases have been considered in great detail in several papers starting with [Me]; the reader will find, besides original results, a complete description of the subsequent results and the relevant references in the recent paper [R], with some very recent further results, on a subject that remains under intense study, in [XY, BKS,Y]. Our case is of the form of (1.10) with replaced by ε; in our case the perturbation is small because it is proportional to ε, while in (1.10) one makes also use of the possibility of taking A, B, β small to obtain a small perturbation. By classical perturbation analysis our case can be reduced to the theory of (1.10). We consider the novelty of this paper to be the technical analysis of the analyticity, in ε, of the resonant (i.e. of dimension lower than maximal) hyperbolic invariant tori of (1.1) in a region as large as Figure 1 based on the Lindstedt series method; the same analyticity domain can be obtained by a careful analysis and a nontrivial extension of the methods of [Mo]. √ This is partially done in [T], where C ∞ dependence on ε was proved at ε = 0. And it is done in a more special case in the paper [JLZ], where a scenario very similar to the one provided by our conjecture (see below) emerges. Closer to our approach is the analysis in [CF]: however the model studied there differs from ours (see (2.24) of [CF]), and existence of hyperbolic low-dimensional tori can be obtained for it without the need of performing the resummations which are on the contrary essential in our case. The technique of [CF] can be extended to cover also our case (which coincides with Eq. (2.22) of [CF]), but it would still make reference to the coordinate changes which are characteristic of the methods of [Mo] (called “classical transformation theory” in [CF]). In fact one is also interested in asking whether the analyticity region in ε can be extended further to reach some points on the negative real axis and whether the analytic continuation to ε < 0 of the parametric equations of the hyperbolic tori can be interpreted as the parametric equations of elliptic tori. We do not address the latter question: the analysis performed in the present paper at first suggests to us that by the same methods it should be possible to prove the following. Conjecture 1.1. Consider the equations of motion (1.6) for ω ∈ Dτ (C0 ), and suppose β 0 to be such that ∂β f0 (β 0 ) = 0, ∂β2 f0 (β 0 ) is negative definite.
(1.11)
Is it possible that there exist constants ε0 > 0, ξ > 0 and a subset Iε0 ⊂ [−ε0 , 0] ξ with length ≥ (1 − ε0 ) ε0 such that the functions of Theorem 1.1 above are analytically continuable outside the domain D0 along vertical lines which start at points interior to D0 and end on Iε0 , where their boundary value is real and gives the parametric equations of an invariant torus for all ε ∈ Iε0 on which the motion according to (1.6) is ψ˙ = ω?
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The extended domain shape, near the origin, suggested in the above conjecture is illustrated in the following Fig. 1 . complex ε−plane
Fig. 1 . The domain D0 of Fig. 1 can be further extended? The conjecture above asks whether the extended analyticity domain could possibly be represented (close to the origin) as here: the domain reaches the real axis at cusp points which are in Iε0 and correspond, in the complex ε-plane, to the elliptic tori which are the analytic continuations of the hyperbolic tori. The analytic continuation would be continuous through the real axis at the points of Iε0 . The cusps would be at least quadratic
2. Formal Solubility of the Equations of Motion 2.1. Formal expansion and recursive equations. We look for a formal expansion a(ψ; ε) = b(ψ; ε) =
∞
ε k a(k) (ψ) =
k=1
ν∈Zr
∞
k (k)
ε b (ψ) =
k=1
eiν·ψ aν (ε) = e
iν·ψ
bν (ε) =
ν∈Zr
∞ k=1 ∞
εk
ν∈Zr
ε
k
ν∈Zr
k=1
eiν·ψ aν(k) , (2.1) e
iν·ψ
b(k) ν ,
where we have not explicitly written the dependence on β 0 . Then to order k the equations of motion (1.6) become , (ω · ν)2 aν(k) = [∂α f ](k−1) ν (k−1) , (ω · ν)2 b(k) ν = ∂β f ν
(2.2)
where, given any function F admitting a formal expansion F (ψ; ε) =
∞ k=1
εk
ν∈Zr
eiν·ψ Fν(k) ,
(2.3)
(k)
we denote by [F ]ν the coefficient with Taylor label k and Fourier label ν. We can write [∂α f ](k−1) ν
p
p+q
(kj ) 1 1 ∗ (kj ) p+1 q = ∂β fν 0 (β 0 ) aν j bν j , (iν 0 ) p! q! p≥0 q≥0
j =1
j =p+1
(2.4)
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where 0 < kj < k for all j = 1, . . . , p + q and the ∗ denotes that the sum has to be performed with the constraints 1+
p+q
kj = k,
ν0 +
j =1
p+q
ν j = ν,
(2.5)
j =1
and, analogously, p
p+q
(kj ) (k−1) 1 1 ∗ (k ) q+1 = aν jj bν j , ∂β f ν (iν 0 )p ∂β fν 0 (β 0 ) p! q! p≥0 q≥0
j =1
j =p+1
(2.6) with the same meaning of the symbols. 2.2. Tree formalism. By iterating (2.2), (2.4) and (2.6), one finds that one can represent (k) (k) graphically aν and bν in terms of trees. The definition and usage of graphical tools based on tree graphs in the context of KAM theory has been advocated recently in the literature as an interpretation of the work [E]; see for instance [G1, GG, BGGM] and [BaG]. A tree θ (see Fig. 2 below) is defined as a partially ordered set of points, connected by lines. The lines are oriented toward the root, which is the leftmost point; the line entering the root is called the root line. If a line ! connects two points v1 and v2 and is oriented from v2 to v1 we say that v2 ≺ v1 and we shall write v1 = v2 and ! = !v2 ; we shall say also that ! exits from v2 and enters v1 . There will be two kinds of points: the nodes and the leaves. The leaves can only be endpoints, i.e. they have no lines entering them, but an endpoint can be either a node or a leaf. The lines exiting from the leaves play a very different rôle with respect to the lines exiting from the nodes, as we shall see below. We shall denote by v0 the last (i.e. leftmost) node of the tree, and by !0 the root line; for future convenience we shall write v0 = r but r will not be considered a node.
v5 ν v1 v1 !0 root
ν=ν !0
ν v0
v6 v3 v7
v0 v2
v4
v8 v9 v10 v11
Fig. 2. A tree θ with 12 nodes; one has pv0 =2,pv1 =2,pv2 =3,pv3 =2,pv4 =2. The length of the lines should be the same but it is drawn of arbitrary size
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We shall denote by V (θ ) the set of nodes, by L(θ) the set of leaves and by (θ) the set of lines. For any !v ∈ θ fixed, we shall say that the subset of θ containing !v as well as all nodes w v and all lines connecting them is a subtree of θ with root v : of course a subtree is a tree. Given a tree, with each node v we associate a mode label ν v ∈ Zr , and to each leaf v a leaf label κv ∈ N. The quantity k = |V (θ)| +
κv
(2.7)
v∈L(θ)
is called the order of the tree θ. With any line ! exiting from a node v we associate two labels γ! , γ! assuming the symbolic values α, β and a momentum label ν ! ∈ Zr , which is defined as ν ! ≡ ν !v =
νw,
(2.8)
w∈V (θ ) wv
while with any line ! exiting from a leaf v we associate only the labels γ! = γ! = β. We can associate with each node also some labels depending on the entering lines and on the exiting one: the branching labels pv and qv , denoting how many lines ! having the label γ! = α and, respectively, γ! = β enter v, and the label δv , defined as 1, if γ!v = β, δv = (2.9) 0, if γ!v = α. Then with each node v we associate a node factor Fv =
1 1 pv +(1−δv ) qv +δv ∂β fν v (β 0 ), iν v p v ! qv !
(2.10)
which is a tensor of rank pv + qv + 1, while with each leaf v we associate a leaf factor (to be defined recursively, see below) (κ )
L v = b0 v ,
(2.11)
which is a tensor of rank 1 (i.e. a vector); to each line ! exiting from a node v we associate a propagator G! ≡ δγ! ,γ!
11 , (ω · ν ! )2
(2.12)
which is a (diagonal) d × d matrix, while no small divisor is associated with the lines exiting from the leaves. For consistency we can define G! ≡ δγ! ,γ! δγ! ,β 11,
(2.13)
for lines exiting from leaves, so that a propagator G! is in fact associated with each line.
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Remark 2.1. Note that we can write (2.12) in the form G!,αα G!,αβ , G! = G!,βα G!,ββ
(2.14)
where G!,αα , G!,αβ , G!,βα and G!,ββ are r × r, r × s, s × r and s × s matrices. By construction one has G!,αβ = GT!,βα = 0, and GT (−x) = G† (x) = G(x);
G! = G(ω · ν ! ),
(2.15)
here and henceforth T and † denote, respectively, the transposed and the adjoint of a matrix. 2.3. Tree values and reduced tree values. Call -k,ν,γ the set of all trees of order k with ν !0 = ν and γ!0 = γ , if !0 is the root line. Set r, for γ = α, dγ = (2.16) s, for γ = β; we can define an application Val : -k,ν,γ → Rdγ , defined as
Val(θ ) = Fv Lv G! , v∈V (θ)
v∈L(θ)
which is called the value of the tree θ . We can define also
Val (θ ) = Fv Lv v∈V (θ)
v∈L(θ)
(2.17)
!∈(θ)
G! ,
(2.18)
!∈(θ)\!0
where, as usual, !0 denotes the root line; Val (θ) is called the reduced value of the tree θ . The following cancellation is proved in Appendix A1. Lemma 2.1. Suppose that for all trees θ ∈ -k,ν,γ the set (θ) \ !0 does not contain any lines ! with momentum ν ! = 0. Then Val (θ) is well defined and Val (θ) = 0. (2.19) θ ∈-k,0,α
2.4. Existence of formal solutions. The following result states the existence of formal solutions to (1.6) which are conjugated to the unperturbed motion (1.5), provided the value β 0 is suitably fixed. Lemma 2.2. One can write, formally, for all ν ∈ Zr \ {0}, Val(θ), aν(k) = θ∈-k,ν,α
b(k) ν
=
θ∈-k,ν,β
(2.20) Val(θ),
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429
(k)
while a0 ≡ 0 and
−1 (k) b0 = − ∂β2 f0 (β 0 )
Val (θ),
(2.21)
θ∈-∗k+1,0,β
where the quantities Val(θ ) and Val (θ) are defined by (2.17) and (2.18), respectively, (k) and ∗ imposes the constraint that the tree whose reduced value is given by ∂β2 f0 (β 0 ) b0 has to be discarded from the set -k+1,0,β . If one has ∂β f0 (β 0 ) = 0,
(2.22)
det ∂β2 f0 (β 0 ) = 0, (k)
(k)
(k)
then there exists a unique way to fix b0 for all k ∈ N such that aν and bν are finite for all ν ∈ Zp \ {0} to all perturbative orders k. About the proof. The proof of (2.20) is by induction. In order to show that it is possible (k) to fix uniquely b0 so that the existence of a formal solution follows, the key is to realize (k) that no division by zero occurs in the recursive solution of (2.2): the coefficients b0 are determined precisely by imposing the validity of this property for the lines ! with γ! = γ! = β. In fact the condition to avoid dividing by zero takes, to all orders k in ε, (k) (k) the form ∂β2 f0 (β 0 ) b0 = some vector determined recursively, so that b0 is defined by exploiting the assumption (2.22). A further key point is to realize that the lines ! with γ! = γ! = α and carrying ν ! = 0 never appear, and the previous lemma is enough to imply this. Details of the proof are given in Appendix A2. Remark 2.2. By (2.2) and by Lemma 2.2 one has
∂α f
∂β f
(k) ν
(k) ν
=
Val (θ),
θ∈-k,ν,α
=
(2.23)
Val (θ),
θ∈-k,ν,β
as one realizes by comparing (2.17) with (2.18). Remark 2.3. As it will follow from the analysis performed in the next sections, the tools described above are sufficient to prove the convergence (hence the analyticity) of the perturbative expansions (2.1), for ε small enough, in the case of periodic solutions (i.e. r = 1): in fact we shall see that the main technical difficulties shall arise from the problem of bounding the propagators, while, in the case of periodic solutions, we can simply bound G! by the inverse of the rotation vector ω (which is a number in such a case).
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3. Self-Energy Graphs 3.1. Trimmed trees. With respect to the papers [GG, BGGM] and [BaG], the trees here (k) carry also “leaves”: each leaf can be decomposed in terms of trees, because b0 is given by
−1 (k) (k+1) b0 = − ∂β2 f0 (β 0 ) G0 ,
(3.1)
(k+1)
expressed as the sum of reduced values of trees of order k + 1 (see Apwith G0 pendix A2); more precisely (k+1) (k) G0 = Val (θ) − ∂β2 f0 (β 0 )b0 ≡ Val (θ), (3.2) θ∈-∗k+1,0,β
θ∈-k+1,0,β
where ∗ has been defined after (2.21): it recalls that the tree whose reduced value is (k) given by ∂β2 f0 (β 0 ) b0 has to be discarded from the set -k+1,0,β . Of course each leaf can contain other leaves and so on. If each time a leaf is encountered, it can be decomposed into trees, at the end we have that the value of a tree θ can be expressed as product of factors which are values of trees without leaves, that we can call, as in [Ge], trimmed trees. The sum of the orders of all the so obtained trimmed trees is equal to k, if the tree θ belonged to -k,ν,γ ; moreover for all trimmed trees the order equals exactly the number of nodes, as it follows from (2.7) by using that a trimmed tree has no leaves. 3.2. Multi-scale decomposition and clusters. Given a vector ω ∈ Dτ (C0 ), define ω0 = 2τ C0−1 ω. Then there exists a sequence {γn }n∈Z+ , with γn ∈ [2n−1 , 2n ], such that |ω0 · ν| − γp ≥ 2n+1 if 0 < |ν| ≤ 2−(n+3)/τ , (3.3) for all n ≤ 0 and for all p ≥ n, and |ω0 · ν| = γn for all ν ∈ Z and for all n ≤ 0; the existence of such a sequence (depending on ω) is proved by proposition in Sect. 3 of [GG]. Given a line ! with momentum ν ! we say that ! has scale label n! = 1 if |ω0 · ν ! | ≥ γ0 ,
(3.4)
γn−1 ≤ |ω0 · ν ! | < γn .
(3.5)
and scale label n! = n ∈ Z \ Z+ if
Once the scale labels have been assigned to the lines one has a natural decomposition of the tree into clusters. A cluster T on scale n is a maximal set of nodes and lines connecting them such that all the lines have scales n ≥ n and there is at least one line with scale n; if a cluster T is contained inside a cluster T we shall say that T is a subcluster of T . The mT ≥ 0 lines entering the cluster T and the possible exiting line (unique if existing at all) are called the external lines of the cluster T ; given a cluster T on scale n, we shall denote by nT = n the scale of the cluster. We call T (θ) the set of all clusters in a tree θ .
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Given a cluster T ∈ T (θ ), call V (T ), L(T ) and (T ) the set of nodes, the set of leaves and the set of lines of T , respectively. Let us define also νT = νv , (3.6) v∈V (T )
and denote by T0 (θ ) the set of all clusters T with ν T = 0. Given a cluster T call T0 the subset of T obtained from T by eliminating all the nodes and lines of the subclusters T ⊂ T such that ν T = 0, and denote by V (T0 ) and (T0 ) the set of nodes and lines, respectively, in T0 . 3.3. Self-energy graphs. We call self-energy graphs of a tree θ the clusters T ∈ T (θ) such that (1) T has only one entering line !2T and one exiting line !1T , (2) T ∈ T0 (θ ), i.e. νT ≡ ν v = 0,
(3.7)
v∈V (T )
(3) one has
|ν v | ≤ 2−(n+3)/τ ,
(3.8)
v∈V (T0 )
where n!2 = n is the scale of the line !2T . T
We say that the line !1T exiting a self-energy graph T is a self-energy line; we call a normal line any line of the tree which is not a self-energy line. Given a self-energy graph T ∈ T (θ) we say that a self-energy graph T ∈ T (θ) contained in T is maximal if there are no other self-energy graphs internal to T and containing T . We say that a self-energy graph T has height DT = 0 if it does not contain any other self-energy graphs, and that it has height DT = D ∈ Z+ , recursively, if it contains maximal self-energy graphs with height D − 1. Given a line ! ∈ (T0 ) with momentum ν ! , its reduced momentum ν 0! is defined as ν 0! = νw, ! ≡ !v , (3.9) w∈V (T0 ) wv
and it can be given a scale n0! such that
γn0 −1 ≤ ω0 · ν 0! < γn0 ; !
!
(3.10)
we call n0! the reduced scale of the line !. Remark 3.1. (1) Given a self-energy graph T , for all lines ! ∈ (T ), one can write, by setting ! = !v , ν ! = ν 0! + σ! ν,
(3.11)
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v6 T
v3 !1T
v4 v2
T
v5
v1
!2T
v7 T
Fig. 3. An example of three clusters symbolically delimited by circles, as visual aids, inside a tree (whose remaining lines and clusters are not drawn and are indicated by the bullets); not all labels are explicitly shown. The scales (not marked) of the lines increase as one crosses inward the circles boundaries: recall, however, that the scale labels are ≤ 0. If the mode labels of (v4 , v5 ) add up to 0 the cluster T is a self-energy graph. If the mode labels of (v4 , v5 , v2 , v6 ) add up to 0 the cluster T is a self-energy graph and such is T if the mode labels of (v1 , v2 , v7 , v4 , v5 , v2 , v6 ) add up to 0. The graph T is maximal in T . If the three clusters T , T , T are self-energy graphs then their heights are respectively 2, 1, 0
where ν ≡ ν !2 is the momentum flowing through the line !2T entering T , while σ! is T
defined as follows: writing ! ≡ !v then σ! = 1 if !2T enters a node w v and σ! = 0 otherwise. (2) Note that the entering line !2T must have, by the condition (3.7), the same momentum as the exiting line !1T , hence, by construction, the same scale n!2 = n!1 . T T (3) The notion of self-energy graphs has been introduced by Eliasson who named them “resonances”, [E]. We change the name here not only to avoid confusion with the notion of mechanical resonance (which is related to a rational relation between frequencies of a quasi-periodic motion) but also because the “tree expansions” that we use here (also basically due to Eliasson) can be interpreted, [GGM], as Feynman graphs of a suitable field theory. As such they correspond to classes of self-energy graphs: we use here the correspondence to perform resummation operations typical of renormalization theory.
3.4. Value of a self-energy graph. Given a self-energy graph T and denoting by |V (T )| the number of nodes in T , define the self-energy value as
VT (ω · ν) = ε |V (T )|
v∈V (T )
Fv
v∈L(T )
Lv
G! ,
|V (T )| ≥ 1, (3.12)
!∈(T )
seen as a function of ω · ν, if ν ≡ ν !2 = ν !1 is the momentum flowing through the T T external lines of the self-energy graph T . Recall that we are considering trimmed trees, so that no leaves can appear; see Sect. 3.1.
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We can have four types of self-energy graphs depending on the types α or β of the labels γ!1 and γ!2 : T
T
γ!1
γ!2
α α β β
α β α β
T
1. 2. 3. 4.
T
(3.13)
Given a tree θ , define Nn (θ) = {! ∈ (θ) : n! = n} ,
(3.14)
and M(θ) =
|ν v | .
(3.15)
v∈V (θ)
Call Nn∗ (θ ) the number of normal lines on scale n and call Rn (θ) the number of selfenergy lines on scale n. Of course Nn (θ) = Nn∗ (θ) + Rn (θ).
(3.16)
Then the following result holds; it is a version of the key estimate of Siegel’s theory in the interpretation of Bryuno [B] and Pöschel [P]. This is proved as in [G1] or [BaG], for instance; however, for completeness, a proof is also given in Appendix A3. Lemma 3.1. For any tree θ ∈ -κ,ν,γ one has Nn∗ (θ) ≤ c M(θ) 2n/τ ,
(3.17)
for some constant c.
3.5. Localization operators. For any self-energy graph T we define LVT (ω · ν) ≡ VT (0) + (ω · ν) ∂VT (0),
(3.18)
where ∂VT denotes the first derivative of VT with respect to its argument; the quantity VT (0) is obtained from VT (ω · ν) by replacing ν ! with ν 0! in the argument of each propagator G! , while ∂VT (0) is obtained from VT (ω ·ν) by differentiating it with respect to x = ω · ν, and thence replacing ν ! with ν 0! in the argument of each propagator G! . We shall call L the localization operator and LVT (ω · ν) the localized part of the self-energy value.
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v2
!1T
T
V1
v1
v4
!
v5 v3
!2T
V2
v6
Fig. 4. The sets V1 and V2 in a self-energy graph T ; note that, even if they are drawn like circles, the sets V1 and V2 are not clusters. One has ν 0! =ν v3 +ν v4 +ν v5 +ν v6 and ν=ν !2 ; of course ν !1 =ν !2 and ν 0! =−(ν v1 +ν v2 ) by T
T
T
definition of self-energy graph. The black balls represent the remaining parts of the trees. The labels are not explicitly shown.
3.6. Families of self-energy graphs. Given a tree θ containing a self-energy graph T , we can consider all trees obtained by changing the location of the nodes in T0 (note that T0 is defined after (3.6)) which the external lines of T are attached to: we denote by FT0 (θ ) the set of trees so obtained, and call it the self-energy family associated with the self-energy graph T . And we shall refer to the operation of detaching and reattaching the external lines, by saying that we are shifting such lines. Of course shifting the external lines of a self-energy graph produces a change of the propagators of the trees. In particular since all arrows have to point toward the root, some lines can revert their arrows. Moreover the momentum can change, as a reversal of the arrow implies a change of the partial ordering of the nodes inside the self-energy graph and a shifting of the entering line can add or subtract the contribution of the momentum flowing through it. More precisely, if the external lines of a self-energy graph T are detached then reattached to some other nodes in V (T ), the momentum flowing through the line ! ∈ (T ) can be changed into ±ν 0! + σ ν, with σ ∈ {0, 1}: if we call V1 and V2 the two disjoint sets into which ! divides T (see Fig. 2), such that the arrow superposed on ! is directed from V2 to V1 (before detaching the external lines), then the sign is + if the exiting line is reattached to a node inside V1 and it is − otherwise, while σ = 1 if the entering line is reattached to a node inside V2 when the sign is + and to a node inside V1 when the sign is −, and σ = 0 otherwise. Referring to (3.13) for the notion of type of self-energy graph one shows the existence of the following cancellations (the proof is in Appendix A4).
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Lemma 3.2. Given a tree θ , for any self-energy graph T ∈ T (θ) one has
θ ∈FT0 (θ)
LVT (ω · ν) =
0, (ω · ν)B FT
0
if T is of type 1, (θ) , if T is of type 2,
(ω · ν)BF , if T is of type 3, T0 (θ) A if T is of type 4, FT0 (θ) ,
(3.19)
where ν = ν !2 , the sum is over the self-energy family associated with T , and AFT0 (θ) , T and B BF FT0 (θ) are matrices s × s, r × s and s × r, respectively, depending only T0 (θ) on the self-energy graph T ; in particular they are independent of the quantity ω · ν. 3.7. First step toward the resummation of self-energy graphs. Let θ be a tree θ ∈ -ν,k,γ with a self-energy graph T . Define θ0 = θ \ T as the set of nodes and lines of θ outside T (of course θ0 is not a tree). Consider simultaneously all trees such that the structure θ0 outside of the self-energy graph is the same, while the self-energy graph itself can be arbitrary, i.e. T can be replaced by any other self-energy graph T with kT ≥ 1. This allows us to define as a formal power series the matrix
M(ω · ν; ε) =
VT (ω · ν),
(3.20)
θ=θ0 ∪T
where the sum is over all trees θ such that θ \ T is fixed to be θ0 and the mode labels of the nodes v ∈ V (T ) have to satisfy the conditions (1)–(3) in Sect. 3.3 defining the self-energy graphs. The following property holds (the proof is in Appendix A5) as an algebraic identity between formal power series. Lemma 3.3. The following two properties hold: (1) (M(x; ε))T = M(−x; ε), and (2) (M(x; ε))† = M(x; ε); the latter means that the matrix M(x; ε) is self-adjoint. Remark 3.2. (1) The function M(ω · ν; ε) depends on ε but, by construction, it is independent of θ0 : hence we can rewrite (3.20) as M(ω · ν; ε) =
VT (ω · ν),
(3.21)
T
where the sum is over all self-energy graphs of order k ≥ 1 with external lines with momentum ν. (2) In (3.20) or (3.21), if γn−1 ≤ |ω0 · ν| < γn , the sum has to be restricted to the self-energy graphs T on scale nT ≥ n + 3. Writing, for any line ! ∈ T0 , ν ! as in (3.11) one has −τ −τ |ν v | ≥ 2τ 2n+3 , (3.22) ω0 · ν 0! > 2τ C0−1 C0 ν 0! ≥ 2τ v∈V (T0 )
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while |ω0 · ν| < 2n , so that, by using again (3.11), one obtains |ω0 · ν ! | > 2τ 2n+3 − 2n > 2n+2 ,
(3.23)
which implies n! ≥ n + 3. (3) The matrix M(ω · ν; ε) can be written as M(ω · ν; ε) =
Mαα (ω · ν; ε) Mαβ (ω · ν; ε) , Mβα (ω · ν; ε) Mββ (ω · ν; ε)
(3.24)
where Mαα (ω · ν; ε), Mαβ (ω · ν; ε), Mβα (ω · ν; ε) and Mββ (ω · ν; ε) are r × r, r × s, s × r and s × s matrices. It is easy to realize that (up to convergence problems to be discussed in Sect. 5) Mαα (ω · ν; ε) = O(ε 2 (ω · ν)2 )), Mαβ (ω · ν; ε) = O(ε 2 (ω · ν)),
(3.25)
Mββ (ω · ν; ε) = O(ε) + O(ε (ω · ν) ). 2
2
The proportionality of Mαα (ω · ν; ε) to (ω · ν)2 and of Mαβ (ω · ν; ε) to ω · ν is a consequence of Lemma 3.2. First order computations already give, for instance, Mββ (ω · ν; ε) = ε∂β2 f0 (β 0 ) + O(ε 2 ) + O(ε 2 (ω · ν)2 ), 1 Mαβ (ω · ν; ε) = −2ε 2 i (ω · ν) (ω · ν 2 )3 ν +ν =0 1 2 |ν 1 |+|ν 2 |<2−(n+3)/τ
,
ν 1 ∂β fν 1 (β 0 )∂β2 fν 2 (β 0 )−ν 21 fν 1 (β 0 )ν 2 ∂β fν 2 (β 0 ) +O(ε 3 (ω · ν))
(3.26) where γn−1 ≤ |ω0 · ν| < γn . Therefore Mββ (ω · ν; ε) = 0 by hypothesis (see (1.8)), and Mαβ (ω · ν; ε) is generically nonvanishing. (4) Lemma 3.3 implies that, by defining the matrices BF and BF as in Lemma 3.2, T0 (θ) T0 (θ) one has BF T
0 (θ)
= − BF T
T 0 (θ)
.
(3.27)
3.8. Changing scales. When shifting the lines external to the self-energy graphs, the momenta of the internal lines can change. As a consequence in principle also the scale labels could change; however this does not happen, as the following result shows; for the proof see Appendix A6. Lemma 3.4. For all lines ! ∈ (θ) one has n! = n0! ; in particular this implies that, when shifting the lines external to the self-energy graphs of a tree θ , the scale labels n! of all line ! ∈ (θ ) do not change.
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437
4. Resummations of Self-Energy Graphs: Renormalized Propagators 4.1. Renormalized trees. So far we considered formal power expansions in ε. By introducing the function h = (hα , hβ ) = (a, b), we can write the function h(ψ, β 0 ; ε) ≡ h(ψ; ε) as h(ψ; ε) = eiν·ψ hν (ε), (4.1) ν∈Zr
because we are looking for a solution periodic in ψ ∈ Tr . In terms of the formal power expansion envisaged in Sect. 3, we can define a solution “approximated to order k” as hν(≤k) (ε) =
k k =1
ε k hν(k ) ,
hγ(kν) =
Val(θ),
(4.2)
θ∈-k ,ν,γ
where -k ,ν,γ is defined in 2.3, and Val(θ) is given by (2.17).
[k]
However we can define a different sequence of approximating functions h (ψ; ε), formally converging to the formal solution (as we shall see in Proposition 5.4 below), by defining it iteratively as follows. Denote by -R k,ν,γ the set of all trees of order k without self-energy graphs and with labels ν !0 = ν and γ!0 = γ associated with the root line; we shall call -R k,ν,γ the set of renormalized trees of order k (and with labels ν and γ associated with the root line). Given a tree θ ∈ -R k,ν,γ and a cluster T ∈ T (θ), by extension we shall say that T is a renormalized cluster. We can also consider a self-energy graph which does not contain any other self-energy graph: we shall say that such a self-energy graph is a renormalized self-energy graph; of course no one of such clusters can appear in any tree in -R k,ν,γ . For a renormalized tree θ of arbitrary order k , define
[k] [k−1] Val (θ ) = Fv Lv G! , (4.3) v∈V (θ)
v∈L(θ)
!∈(θ)
with the dressed propagators given by [0] G! = (ω · ν ! )−2 11δγ! ,γ! , −1 [k] G! = (ω · ν ! )2 11 − M [k] (ω · ν ! ; ε) ,
(4.4)
for k ≥ 1,
where the sequence {M [k] (ω · ν; ε)}k∈N is iteratively defined as the sum of the values of all renormalized self-energy graphs which can be obtained by using the propagators [k−1] G! , i.e. as M [k] (ω · ν; ε) = VT[k] (ω · ν), renormalized T
VT[k] (ω
· ν) = ε |V (T )|
v∈V (T )
Fv
v∈L(T )
Lv
!∈(T )
[k−1]
G!
,
(4.5)
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where |V (T )| is the number of nodes in T ; we can also define M [0] (ω · ν; ε) ≡ 0. The leaf factors Lv in (4.3) are recursively defined as
−1 Lv = b0[k,κv ] = − ∂β2 f0 (β 0 )
Val
[k]
(θ),
(4.6)
θ∈-R∗ κv ,0,β
where Val
[k]
(θ ) =
Fv
v∈V (θ)
Lv
v∈L(θ)
[k−1]
!∈(θ)\!0
G!
,
(4.7)
and ∗ has the same meaning as after (3.2). To avoid confusing the value of a renormalized tree with the tree value introduced in (2.13), we shall call (4.3) the renormalized value of the (renormalized) tree. Then we shall write [k] [k] h (ψ; ε) = eiν·ψ hν (ε), [k]
hν (ε) =
ν∈Zr ∞ k =1
[k,k ]
ε k hν
(ε),
[k,k ]
hγ ν (ε) =
[k]
Val (θ),
(4.8)
θ∈-R k ,ν,γ
where the last formula holds for ν = 0, because for ν = 0, one has (4.6) for γ! = β, [k,k ]
while hγ 0
≡ 0.
Remark 4.1. Note that if we expand the quantity M [k] (ω · ν; ε) in powers of ε, by [k−1] expanding the propagators G! , we reconstruct the sum of the values of all self-energy graphs containing only self-energy graphs with height D ≤ k. Therefore if we expand M [k+1] (ω · ν; ε) in powers of ε we obtain the same terms as if expanding M [k] (ω · ν; ε), plus the sum of the values of all the self-energy graphs containing also self-energies graphs with height k + 1, which are absent in the self-energy graphs contributing to M [k] (ω · ν; ε). Such a result will be used in Appendix A7 in order to prove the following result. [k]
Lemma 4.1. The power series defining the functions hν (ε), truncated at order k ≤ k, (≤k ) coincide with the functions hν (ε) given by (4.2). 5. Convergence of the Renormalized Perturbative Expansion 5.1. Domains of analyticity and norms. Consider in the complex ε-plane the domain Dε0 (ϕ) in Fig. 5 below: if ϕ denotes the half-opening of the sector Dε0 (ϕ), then the radius of the circle delimiting Dε0 (ϕ) will be of the form ε(ϕ) = (π − ϕ)ε0 (see below). We shall define # · # an algebraic matrix norm (i.e. a norm which verifies #AB# ≤ #A# #B# for all matrices A and B); for instance # · # can be the uniform norm. [k] The propagators G! in (4.4) satisfy interesting k-independent bounds described and proved below.
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439
ε-plane Dε0 (ϕ) −x 2
Fig. 5. The domain Dε0 (ϕ) in the complex ε-plane: the half-opening angle of the sector is ϕ<π but, otherwise, arbitrary, and the radius of the circle delimiting Dε0 (ϕ) is given by ε(ϕ) = (π − ϕ)ε0
Proposition 5.1. Let Dε0 (ϕ) be obtained from the disk of diameter ε0 > 0 in the complex ε-plane by taking out a sector of half-opening π −ϕ around the negative real axis. Assume [k] [k] that the propagators G! ≡ G (ω · ν ! ; ε) satisfy [k] T [k] G (x; ε) = G (−x; ε),
[k] G (x; ε) <
2 1 π − ϕ x2
(5.1)
for all |ε| < (π − ϕ)ε0 , if ε0 is small enough. Then there is a constant Bf such that, [k]
summing over all renormalized trees θ with |V (θ)| = V nodes the values |Val (θ)|, one has [k,V ]
hγ ν
≤
θ∈-R V ,ν,γ
|ε|B V f [k] , Val (θ) ≤ π −ϕ
[k] |ε|Bf V e−κs/8 , Val (θ) ≤ π − ϕ R
(5.2)
θ ∈-V ,ν,γ M(θ )=s
B V +1 f [k,V ] 2 −1 |ε|V , b0 ≤ ∂β f0 (β 0 ) π −ϕ
for all s > 0. Remark 5.1. Note that, although the propagators are no longer diagonal, they still satisfy the same property as (2.15), which is the crucial one which is used in order to prove both the Lemmata 2.1 and 2.2 about the formal solubility of the equations of motion and the Lemmata 3.2 and 3.3 about the formal cancellations between tree values. Proof of Proposition 5.1. We can consider first trees without leaves, so that the tree values are given by (4.3) with L(θ ) = ∅. [k] The hypothesis (5.1) implies that for all propagators G! one has [k] G! ≤ C1 2−2n! ,
C1 =
2 π −ϕ
2τ +2 C0
2 .
(5.3)
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Therefore the contribution from a single tree (see Sect. 2.3) is bounded for all n0 ≤ 0 by V |ε|V C1 2−2(n0 −1) v∈V (θ)
0 −1 n
1 n/τ q +1 C1 2−2c|ν v |n2 , |ν v |pv +1 ∂βv fν v (β 0 ) pv !qv ! n=−∞
(5.4) where V = |V (θ )|, having used that, for all trees θ ∈ -R k,ν,γ , the number Nn (θ) of lines with scale n in θ satisfy the bound
Nn (θ ) ≤ c M(θ) 2n/τ = c 2n/τ
|ν v |,
(5.5)
v∈V (θ)
for some constant c: an estimate which follows from the proof of Lemma 3.1 (see Sect. A3.3). The bound (5.5) is used in deriving (5.3) for all lines ! ∈ (θ) with scale n! < n0 , while for the lines ! with scale n! ≥ n0 we have used simply that the propagators are bounded by C1 2−2(n0 −1) . If we use (recall that we are supposing that there are no leaves) p+1 1 p+1 8 ≤ (p + 1) eκ|ν|/8 , |ν| p! κ 1 q+1 q ∂β fν (β 0 ) ≤ C2 F e−κ|ν| , q! (pv + qv ) = k − 1,
(5.6)
v∈V (θ)
for some constant C2 , and if we choose n0 so that κ + 2c 8
∞
n2−n/τ ≤
n=|n0 |+1
κ , 4
(5.7)
(e.g. we can choose n0 = min{0, −2τ log 2 log((1 − 2−1/τ )κ/(16c log 2)), then we obtain the first bound in (5.2), where we can take Bf = Df , with Df = D0 C02 2−(n0 −1) F
e−κ|ν|/4 ,
(5.8)
ν∈Zr
for some positive constant D0 . This follows after summing over all renormalized trees with V nodes and without leaves: this can be easily done. The sum over the mode labels can be performed by using the decay factors e−κ|ν v |/8 , while the sum over all the possible tree shapes gives a constant to the power k. Furthermore the value κ/8 is so small that with our choices of the constants an extra factor exp[−κM(θ )/8] has been bounded by 1 so that if, instead, the value of M(θ) is fixed we obtain the second bound of (5.2). So far we considered only trees without leaves. If we want to consider also tree with leaves, we can proceed in the following way.
Hyperbolic Low-Dimensional Invariant Tori
441 [k]
Given a tree θ (with leaves) of order k , we can write its renormalized value Val (θ) as the product of the value of a trimmed tree θ times the factors of its leaves: simply look at (4.3)), and interpret the renormalized value of the trimmed tree θ as
[k] [k−1] , (5.9) Val (θ ) = Fv G! v∈V (θ)
!∈(θ)
while the product
Lv
(5.10)
v∈L(θ)
represents the product of the leaf factors (4.6) associated to the |L(θ)| leaves of θ ; note that in (5.9) we can completely neglect the propagators associated to the lines exiting from the leaves, as (2.13) trivially implies. [k] The only effect of the leaves on Val (θ) is through the presence of some extra derivatives ∂β acting on the node factors corresponding to some nodes v ∈ V (θ); in particular the momenta of the lines ! ∈ (θ) are completely independent of the leaves (which contribute 0 to such momenta). Each leaf whose factor contributes to (5.10) can be written as a sum of values of renormalized trees θ1 , . . . , θ|L(θ)| , according to (4.6); for each such tree, say θj , we can [k]
write Val (θj ) as a product of the renormalized value of the trimmed tree θ j times the product of the factors of its |L(θj )| leaves. And so on: we iterate until only trimmed trees are left. The sum of the orders of all trimmed trees equals the order k of the tree θ. Then we can see how the analysis performed above in the case of trees without leaves can be modified when trees with leaves are also taken into account. First of all note that if, when considering the trees whose renormalized values contribute to the leaf factor (4.6), we retain only the trees without leaves, we can repeat the analysis leading to (5.8), with the only difference that (as it can be read from (4.6)) one
−1 [k] acting on the reduced value Val (θ) and the tree θ has has a matrix ∂β2 f0 (β 0 ) order κv + 1 (hence V + 1, if V is the number of nodes of θ, as we are supposing that θ has no leaves), so that the first bound in (5.2) has to be replaced with
−1 [k] [k,V ] Val (θ) b0 ≤ − ∂β2 f0 (β 0 ) θ∈-R∗ V ,ν,γ
D V +1 f |ε|V , ≤ ∂β2 f0 (β 0 )−1 π −ϕ
(5.11)
so that we have an extra factor
D f 2 −1 C3 = ∂β f0 (β 0 ) π −ϕ
(5.12)
with respect to the bound one obtains for (5.2): this yields the third bound in (5.2) for leaves arising from trees which do not contain other leaves. Now we consider any tree of order k , and we decompose it in a collection of trimmed trees (as described above) θ 0 , θ 1 , θ 2 , . . . , such that the root of θ 0 is the root r of θ, while the root ri of each other trimmed tree θ i , i ≥ 1, coincides with a node of some other
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trimmed tree. Moreover the propagators of the root lines of the trimmed trees θ j , j ≥ 0, can be neglected by the definition (2.13). Then the value of the tree θ becomes the product of (factorising) values of trimmed trees. Then we can define the clusters as done in Sect. 3, with the further constraint that all lines internal to a cluster have to belong to the same trimmed tree. Then for each trimmed tree the cancellation mechanisms described in the previous sections apply, and for each of them the same bound as before is obtained. Therefore for θ 0 we can repeat the same analysis as for trees without leaves with the only difference that the third of (5.6) does not hold anymore, and it has to be replaced with (5.13) (pv + qv ) = k − 1 + |L(θ)|; v∈V (θ)
as noted before the presence of the leaves implies that, for each of them, there is a derivative ∂β acting on the node factor of some node v ∈ V (θ), so that, with respect to |L(θ)| (one for leaf). the bound (5.2), we obtain an extra factor C2 Now we can consider the trimmed trees θ 1 , . . . , θ |L(θ)| , and proceed in the same way. With respect to the previous case, for each trimmed tree θ j we obtain an extra factor C2 for each leaf attached to some node of θ j . Furthermore, as all the trimmed trees except θ 0 contribute to leaves, there is also an extra factor C3 for each of them. At the end, instead of the first bound in (5.2) with Df given by (5.8), we obtain
|ε|Df π −ϕ
k
(C2 C3 )L ;
(5.14)
as the total number of leaves is less than the total number of lines with vanishing momentum (hence less than k ), we obtain the first bound in (5.2), provided that one replaces the previous value (5.8) for Bf with B f = D f C 2 C3 .
(5.15)
The sum over the trees can be performed exactly as in the previous case. In the same way one discusses the second and the third bound in (5.2), which follow with the constant Bf given by (5.15)). This completes the proof. Proposition 5.2. Let Dε0 (ϕ) be as in Proposition 5.1; then the matrices M [k] (ω · ν; ε) satisfy for ε ∈ Dε0 (ϕ) the relation
T M [k] (x; ε) = M [k] (−x; ε).
(5.16)
Let also [k] [k] (ω · ν; ε) Mαβ (ω · ν; ε) Mαα ; (ω · ν; ε) = [k] [k] Mβα (ω · ν; ε) Mββ (ω · ν; ε)
M
[k]
(5.17)
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443
then, if γq−1 ≤ |ω0 · ν| < γq and ε0 is small enough, the submatrices Mγ[k]γ (ω · ν; ε) can be analytically continued in the full disk |ω0 · ν| ≤ γq and satisfy the bounds [k] Mαα (x; ε) ≤ (|ε|/(π − ϕ))2 C x 2 , (k) Mαβ (x; ε) ≤ (|ε|/(π − ϕ))2 C |x| , [k] Mββ (x; ε) − ε ∂β2 f0 (β 0 ) ≤ (|ε|/(π − ϕ))2 C x 2 ,
(5.18)
[k]
for all k ∈ N and for a suitable constant C. As a consequence G! verify (5.1) for all k ≥ 1, and therefore (5.2) holds for all k ≥ 1. Proof of Proposition 5.2. We consider the matrices M [k] defined in (4.5) and suppose inductively that M [k] verifies (5.18) and the analyticity property preceding it for 0 ≤ k ≤ p − 1; note that the assumption holds trivially for k = 0. Note also that (5.18) imply [k] that the propagators G! verify (5.1) for ε0 small enough and ε ∈ Dε0 (ϕ). To define M [k] we must consider the renormalized self-energy graphs T and evaluate [p−1] , according to (4.5). their values by using the propagators G! [p−1]
Given x = ω · ν such that γq−1 ≤ |x| < γq for some q ≤ 0, the propagators G! have an analytic extension to the disk |x| < γq+2 and, under the hypotheses (5.16) and (5.18), verify the symmetry property and the bound in (5.1), as is shown in Appendix A8. We have (see (4.5)) 0
h=q+3
renormalized T nT =h
M [p] (x; ε) =
[p]
VT ,h (x),
(5.19)
[p]
where by appending the label h to VT (x) we distinguish the contributions to M [p] (x; ε) coming from self-energy graphs T on scale h (which is constrained to be ≥ q + 3; see Remark 3.2, (2)). [p] The value VT (x) is analytic in x for |x| ≤ γh+2 and the sum over all T ’s with V nodes is bounded by [p] (|ε|Bf )V −κ2−h/τ /8 e , VT ,h (x) ≤ 1 − e−κ/8 T
(5.20)
|V (T )|=V
because the mode labels ν v of the nodes v ∈ V (T ) must satisfy v∈V (T ) |ν v | > 2−h/τ (recall that we are dealing with renormalized trees, so that for all clusters T ∈ T (θ) one has T0 = T , and use (3.8) and use Remark 3.2, (2)). Since the symmetry property expressed by (5.1) for k = p is implied by (5.16) and this is the only property of the propagators that one needs in order to check the algebraic Lemmata 3.2 and 3.3 (see Remark 5.1), we can conclude that the same cancellation [p] mechanisms extend to the renormalized self-energy values VT (ω·ν) (see Remarks A4.6 [p] and A5.3). Therefore we see that VT ,h,γ γ (x) will vanish at x = 0 to order σγ γ , if we
444
G. Gallavotti, G. Gentile
set
σγ γ
[p]
2, for γ = α and γ 1, for γ = α and γ = 1, for γ = β and γ 0, for γ = β and γ
= α, = β, = α, = β;
(5.21)
[p]
moreover VT ,h,ββ (x) − VT ,h,ββ (0) vanishes to order 2 at x = 0. By the analyticity in x for |x| ≤ γh+2 and by the maximum principle (Schwarz’s lemma) we deduce from (5.10) that one has σγ γ [p] (|ε|Bf )V −κ2−h/τ /8 x e , VT ,h,γ γ (x) ≤ 1 − e−κ/8 γh+2 T |V (T )|=V
2 [p] (|ε|Bf )V −κ2−h/τ /8 x [p] e . VT ,h,ββ (x) − VT ,h,ββ (0) ≤ 1 − e−κ/8 γh+2 T
(5.22)
|V (T )|=V
−h/τ −2h 2 < B1 < ∞ and that V ≥ 2 for Therefore we can use that 0h=q+3 e−κ2 (γ , γ ) ∈ {(α, α), (α, β), (β, α)}, while V ≥ 1 for (γ , γ ) = (β, β ), and the proof is complete.
5.2. Convergence of the sequence {M [k] (ω · ν; ε)}k∈N . It also follows that there exists the limit lim M [k] (x; ε) = M [∞] (x; ε),
(5.23)
k→∞
with M [∞] (x; ε) analytic in ε in Dε0 (ϕ): in fact the following result holds (the proof is in Appendix 6.3). Lemma 5.1. For all k ≥ 1 one has [k+1] (x; ε) − M [k] (x; ε) ≤ Bˆ 1 Bˆ 2k ε02k , M
(5.24)
for some constants Bˆ 1 and Bˆ 2 . 5.3. Fully renormalized expansion. We can now define the “fully renormalized” expansion of the parametric equations of the invariant torus as the sum of the values of the [k−1] renormalized trees evaluated according to (4.3) with G! replaced by G
[∞]
−1 (x; ε) = x 2 11 − M [∞] (x; ε) ,
x = ω · ν!.
(5.25)
The above discussion shows that the series converges for all ε ∈ Dε0 and that it coincides [k] with the limit for k → ∞ of h (ψ; ε), which therefore exists.
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445
The radius of the domain Dε0 (ϕ) is (π − ϕ) ε0 , if ϕ is the half-opening of the sector [∞] Dε0 (ϕ), because the norms of the propagators G (x; ε) are bounded by 2/(x 2 (π −ϕ)) (see A8.3). [k] Therefore the functions h (ψ; ε) converge in a heart-like domain Dε0 (ϕ) = D0 , (5.26) −π≤ϕ<π
whose boundary, for negative ε close to 0, is such that Im (ε) is proportional to (Re (ε))2 . Proposition 5.3. There exist positive constants ε0 , B, B˜ 1 and B˜ 2 , such that if (k)
hRγ ν (ε) = Val
[∞]
(θ ) =
Val
[∞]
-R k,ν,γ
Fv
(θ),
v∈V (θ)
Fv
v∈(θ)
!∈L(θ)
[∞]
G!
(5.27) ,
the renormalized series h
[∞]
(ψ; ε) =
∞ k=1
εk
ν∈Zr
(k)
eiν·ψ hRν (ε)
(5.28)
converges in the heart-shaped domain (5.26) and its coefficients are bounded by (k) (k) N !−1 ∂εN hRν (ε) < N !2τ +1 B N B˜ 1 B˜ 2k , for N ≥ 0, hRν (ε) ≤ B˜ 1 B˜ 2k , (5.29) uniformly in ε ∈ D0 . 5.4. Comments about (5.29). We leave out, for simplicity, the proof that the N !2τ +1 is the appropriate power of N! that follows from our analysis. Although it is quite clear that one has obtained a remainder bound proportional to a power of N !, derived already in [∞] [JLZ], we evaluated it explicitly in the hope that the power series expansion of h (ψ; ε) (hence of h(ψ; ε), see Proposition 5.4 below) at ε = 0 could be shown to be summable in the sense of the Borel transforms or of its extensions. Since we have analyticity of h in the domain D0 of Fig. 1 in Sect. 1 we would need that the remainder in (5.29) behaves at most as N!2 , see [CGM]. Since τ ≥ r − 1 and r ≥ 2 (in order to have quasi-periodic solutions) we see that (5.29) is not compatible with the general theory. Therefore one needs more information than just (5.29) in order to be able to reconstruct from the power series at the origin the full equation of the invariant torus. [∞]
5.5. Conclusions. In Appendix A10 we show that the function h (ψ; ε), i.e. the limit [k] for k → ∞ of the approximated functions h (ψ; ε), solves the equations of motion (1.6), so proving the following proposition: this concludes the proof of Theorem 1.1.
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Proposition 5.4. One has, formally (i.e. order by order in the expansion in ε around ε = 0) h
[∞]
[k]
(ψ; ε) ≡ lim h (ψ; ε) = h(ψ; ε), k→∞
(5.30)
where h(ψ; ε) is the formal power series which solves Eq. (1.6). 6. Concluding Remarks 6.1. Some extensions. The case of more general Hamitonians of the form H = h0 (A) + εf (α, A),
(6.1)
with (α, A) ∈ Td × A, where A is an open domain in Rd , should be easily studied as the case treated here to show existence and regularity of invariant tori associated with rotation vectors ω ∈ Rd among whose components there are s rational relations, while the independent ones verify a Diophantine condition. 6.2. Periodic orbits. The fully resonant case r = 1 corresponds to periodic orbits is of course a special case of our theory, but it is well known. Note that in such a case the series expansion envisaged in Sect. 2 is sufficient to prove existence (and analyticity) of the periodic solutions, and no resummation is needed; see Remark 2.3.
6.3. (Lack of) Borel summability. As pointed out in the concluding sentence of Sect. 5 the results that we have are not sufficient to imply (extended) Borel summability of the formal power series at the origin of the parametric equations of the torus, i.e. of h(ψ; ε). The resummations that lead to the construction of h(ψ; ε) are therefore of a different type from the well known ones associated with the Borel transforms.
Appendix A1. Proof of Lemma 2.1 A1.1. Proof of the lemma. Part I: Neglecting the factorials. Consider all contributions arising from the trees θ ∈ -0,k,α : we group together all trees obtained from each other by shifting the root line, i.e. by changing the node which the root line exits and orienting the arrows in such a way that they still point toward the root. We call F(θ) such a class of trees (here θ is any element inside the class). The reduced values Val (θ ) of such trees θ ∈ F(θ) differ because (1) there is a factor iν v depending on the node v to which the root line is attached (see the definition (2.9) of Fv ), and (2) some arrows change their directions; more precisely, when the root line is detached from the node v0 and reattached to the node v, if P(v0 , v) = {w ∈ V (θ) : v0 ' w ' v} denotes the path joining the node v0 to the node v, all the momenta flowing through the lines ! along the path P(v0 , v) change their signs, the factorials of the node factors corresponding to the nodes joined by them can change, and the propagators G! are replaced transposed.
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The change of the signs of the momenta simply follows from the fact that
ν v = 0,
(A1.1)
v∈V (θ)
as θ ∈ -k,0,α : by the property (2.15), the propagator does not change. The change of the factorials contributing to the node factors is due to the fact that for the nodes along the path P(v0 , v), an entering line can become an exiting line and vice versa, so that the labels pv and qv can be transformed into pv ± 1 and qv ± 1, q +δ respectively: this does not modify the factor (iν v )pv +(1−δv ) ∂βv v fν v (β 0 ) in (2.9) – up to the factor iν v (if the root line is attached to v), which has been already taken into account –, as one immediately checks, but it can produce a change of the factorials. If we neglect the change of the factorials, i.e. if we assume that all combinatorial factors are the same, by summing the reduced values of all possible trees inside the class F(θ ) we obtain a common value times i times (A1.1), and the sum gives zero.
A1.2. Proof of the lemma. Part II: Taking into account the factorials. One can easily show that a correct counting of the trees implies that all factorials are in fact equal: to do this it is convenient to use topological trees instead of the usual semitopological used so far (we follow the discussion in [BeG]). We briefly outline the differences between the two kinds of trees, deferring to [G2] and [GM] for a more detailed discussion of the differences between what finally amounts to a different way to count trees. Define a group of transformations acting on trees generated by the following operations: fix any node v ∈ V (θ) and permute the subtrees entering such a node. We shall call semitopological trees the trees which are superposable up to a continuous deformation of the lines, and topological trees the trees for which the same happens modulo the action of the just defined group of transformations. We define equivalent two trees which are equal as topological trees. Then we can still write (2.15) restricting the sum over the set of all nonequivalent topological trees of order k with labels ν !0 = ν and γ!0 = γ (we can denote it by top -k,ν,γ ), provided that to each node v ∈ V (θ) we associate a combinatorial factor which is not the (pv !qv !)−1 appearing in (2.9). In fact for topological trees the combinatorial factor associated to each node is different, because we have to look now to how the subtrees emerging from each node differ. For semitopological trees we have a factor (pv !qv !)−1 for each node v, where pv and qv are the numbers of lines ! with γ! = a and γ! = β, respectively, entering v: except for the labels γ! , we are disregarding the kinds of the subtrees entering v, so that in this way we are counting as different many trees otherwise identical. On the contrary, in the case of topological trees, we consider one and the same tree those trees that are different as semitopological trees, but have the same value because they just differ in the order in which identical subtrees enter each node v: therefore, if sv,1 , . . . , sv,jv are the number of entering lines to which are attached subtrees of a given shape and with the same labels (so that sv,1 + · · · + sv,jv = pv + qv , 1 ≤ jv ≤ pv + qv ), the combinatorial factor, for each node, becomes 1 pv !qv ! 1 · = ; pv !qv ! sv,1 ! . . . sv,jv ! sv,1 ! . . . sv,jv !
(A1.2)
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note in the second factor in the above formula the multinomial coefficient corresponding to the number of different semitopological trees corresponding to the same topological tree, for each node. (k) (k) So in terms of topological trees aν and bν can be expressed as a sum of tree values top Val (θ ), where
top Fv Lv G! , (A1.3) Valtop (θ ) = v∈V (θ)
v∈L(θ)
!∈(θ)
where top
Fv
=
pv +(1−δv ) 1 q +δ iν v ∂βv v fν v (β 0 ). sv,1 ! . . . sv,jv !
(A1.4)
Still, when computing the combinatorial factors inside each family F(θ), they do differ. But this is actually an apparent, not a real discrepancy. In fact, due to symmetries in the tree (that is, to the fact that the subtrees emerging from some node are sometimes equal, i.e. that some sv,i are greater than 1), the actual number of topological trees in a given family F(θ ) is less than the total number of trees obtained by the action of the group of transformations: in other words some trees obtained by the action of the group are equivalent as topological trees. When moving the root line from a node v0 to another node v1 , so transforming a tree θ into a tree θ1 ∈ F(θ), for some nodes w along the path P (v0 , v1 ) the factor 1/sw,i ! can turn into 1/(sw,i − 1)!, but then this means that the same topological tree could be formed by the action of sw,i different transformations of the group: each of the sw,i equivalent subtrees entering w contains a node such that, by attaching to it the root line, the same topological tree is obtained. Therefore, by counting all trees obtained by the action of the group, the corresponding topological tree value is in fact counted sw,i times, so to avoid overcounting one needs a factor 1/sw,i : this gives back the same combinatorial factor 1/sw,i !. Analogously one discusses the case of a factor 1/sw,i ! turning into 1/(sw,i + 1)!, simply by noting that the same argument as above can be followed also in this case by changing the rôles of the two nodes v0 and v1 . A1.3. Remark. The proof of the lemma relies only on the property (2.15) of the propa[k] gators, so that also the function h (ψ; ε) is well defined for all k ∈ N. Appendix A2. Proof of Lemma 2.2 (k)
A2.1. Proof. In order to prove the lemma we shall show by induction that a0 can (k) be arbitrarily fixed and b0 can be uniquely fixed in order to make formally solvable Eqs. (2.2). (1) (1) For k = 1 it is straightforward to realize that aν and bν are well defined for all r ν ∈ Z \ {0}, by using the first condition in (1.8). (k ) (k ) Then, for k > 1, assume that all a0 and b0 , with k < k − 1, have been fixed, (k )
(k )
and that, as a consequence, all aν and bν are well defined for k < k and for all ν ∈ Zr \ {0}. (k−1) By (2.16) and by Lemma 2.1, in (2.2) one has [∂α f ]ν = 0, so that the equation (k) for a(k) is formally soluble, and a0 can be arbitrarily fixed, for instance equal to 0.
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In the second equation in (2.2) one can write (k) + Gν(k) , ∂β f ν = ∂β2 f0 (β 0 )b(k−1) ν
(A2.1)
(k)
where the function Gν takes into account all contributions except the one explicitly (k) written, and, by construction, all terms appearing in Gν can depend only on factors (k ) b0 of orders k ≤ k − 2. We can choose (k−1)
b0
−1 (k) = − ∂β2 f0 (β 0 ) G0 ,
(A2.2)
where the second condition in (1.8) has been used, so that also the equation for b(k) (k) becomes formally soluble. Of course also b0 is left undetermined: it will have to be fixed in the next iterative step. To complete the proof of the lemma one has still to show that the sums over the Fourier labels can be performed, but this is a trivial fact for ω ∈ Dτ (C0 ). A2.2. Remark. The same proof applies to the renormalized trees introduced in Sect. 4.1. Appendix A3. Proof of Lemma 3.1 A3.1. Inductive bounds. We prove inductively on the number of nodes of the trees the bounds Nn∗ (θ ) ≤ max{0, 2 M(θ) 2(n+3)/τ − 1},
(A3.1)
where M(θ ) is defined in (3.15). First of all note that if M(θ) < 2−(n+3)/τ then Nn (θ) = 0 as in such a case for any line ! ∈ (θ ) one has |ω0 · ν ! | > 2τ |ν ! |−τ > 2τ M(θ)−τ > 2τ 2n+3 ,
(A3.2)
by the Diophantine hypothesis (1.2) and by the definition of ω0 given in Sect. 3.2. A3.2. Bound on Nn∗ (θ ). If θ has only one node the bound is trivially satisfied because, if v is the only node in V (θ ), one must have M(θ) = |ν v | ≥ 2−n/τ in order that the line exiting from v is on scale ≤ n: then 2 M(θ) 2(n+3)/τ ≥ 4. If θ is a tree with V > 1 nodes, we assume that the bound holds for all trees having V < V nodes. Define En = (2 2(n+3)/τ )−1 : so we have to prove that Nn∗ (θ) ≤ max{0, M(θ ) En−1 − 1}. If the root line ! of θ is either on scale = n or a self-energy line with scale n, call θ1 , . . . , θm the m ≥ 1 subtrees entering the last node v0 of θ . Then Nn∗ (θ) =
m i=1
Nn∗ (θi ),
hence the bound follows by the inductive hypothesis.
(A3.3)
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If the root line ! is normal (i.e. it is not a self-energy line) and it has scale n, call !1 , . . . , !m the m ≥ 0 lines on scale ≤ n which are the nearest to ! (this means that no other line along the paths connecting the lines !1 , . . . , !m to the root line is on scale ≤ n). Note that in such a case !1 , . . . , !m are the entering line of a cluster T on scale > n. If θi is the subtree with !i as root line, one has Nn∗ (θ) = 1 +
m i=1
Nn∗ (θi ),
(A3.4)
so that the bound becomes trivial if either m = 0 or m ≥ 2. If m = 1 then one has T = θ \ θ1 , and the lines ! and !1 are both with scales ≤ n; as !1 is not entering a self-energy graph, then |ω0 · ν ! | ≤ 2n ,
|ω0 · ν !1 | ≤ 2n ,
and either ν ! = ν !1 and one must have (recall that T0 is defined after (3.6)) |ν v | ≥ |ν v | > 2−(n+3)/τ = 2En > En , v∈V (T )
(A3.5)
(A3.6)
v∈V (T0 )
or ν ! = ν !1 , otherwise T would be a self-energy graph (see (3.7) and (3.8)). If ν ! = ν !1 , then, by (A3.5) one has |ω0 · (ν ! − ν !1 )| ≤ 2n+1 , which, by the Diophantine condition (1.2), implies |ν ! − ν !1 | > 2 2−(n+1)/τ , so that again |ν v | ≥ ν ! − ν !1 > 2 2−(n+2)/τ > 21/τ +2 En > En , (A3.7) v∈V (T )
as in (A3.6). Therefore in both cases we get M(θ ) − M(θ1 ) =
|ν v | > En ,
(A3.8)
v∈T
which, inserted into (A3.4) with m = 1, gives, by using the inductive hypothesis, Nn∗ (θ ) = 1 + Nn∗ (θ1 ) ≤ 1 + M(θ1 ) En−1 − 1 ≤ 1 + M(θ) − En En−1 − 1 ≤ M(θ) En−1 − 1,
(A3.9)
hence the bound is proved also if the root line is normal and on scale n. A3.3 Remark. The same argument proves the bound (5.4) for renormalized trees, by using the observation that there are no self-energy lines in the renormalized trees. Appendix A4. Proof of Lemma 3.2 A4.1. Factorials. As for the proof of the Lemma 2.2 we ignore the factorials: to take them into account one can reason as said in Appendix A1.
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A4.2. Self-energy graphs of type 1. First we prove that T VT (0) = 0. Given a tree θ consider all trees which can be obtained by shifting the entering line !2T . Note that the trees so obtained are contained in the self-energy graph family FT0 (θ). Corresponding to such an operation VT (0) changes by a factor iν v if v is the node which the entering line is attached to, as all node factors and propagators do not change. By (3.6) the sum of all such values is zero. Then consider ∂VT (0). By construction
(n ) (n ) (A4.1) Fv ∂G! ! G! ! , ∂VT (0) = !∈(T )
v∈V (T )
! ∈(T )\!
where all propagators have to be computed for ω · ν = 0, and d (n! ) (n ) x = ω · ν. G! (ω · ν 0! + σ! x) , ∂G! ! = x=0 dx
(A4.2)
The line ! divides V (T ) into two disjoint set of nodes V1 and V2 , such that !1T exits from a node inside V1 and !2T enters a node inside V2 : if ! = !v one has V2 = {w ∈ (T ) : w v} and V1 = V (T ) \ V2 . By (3.4), if νv , ν2 = νv , (A4.3) ν1 = v∈V1
v∈V2
one has ν 1 + ν 2 = 0. Then consider the families F1 (θ) and F2 (θ) of trees obtained as follows: F1 (θ ) is obtained from θ by detaching !1T then reattaching to all the nodes w ∈ V1 and by detaching !2T then reattaching to all the nodes w ∈ V2 , while F2 (θ) is obtained from θ by reattaching the line !1T to all the nodes w ∈ V2 and by reattaching the line !2T to all the nodes w ∈ V2 ; note that F1 (θ) ∪ F2 (θ) ⊂ FT0 (θ). As a consequence of such an operation the arrows of some lines ! ∈ (T ) change their directions: this means that for some line ! the momentum ν ! is replaced with −ν ! and the propagators G! are replaced with their transposed GT! . As the propagators satisfy (2.15) no overall change is produced by such factors, except for the differentiated propagator which can change sign: one has a different sign for the trees in F1 (θ) with respect to the trees in F2 (θ ). Then by summing over all the possible trees in F1 (θ) we obtain a value i 2 ν 1 ν 2 times a common factor, while by summing over all the possible trees in F2 (θ ) we obtain −i 2 ν 1 ν 2 times the same common factor, so that the sum of two sums gives zero. A4.3. Self-energy graphs of type 2. Given a tree θ with a self-energy graph T consider all trees obtained by detaching the exiting line, then reattaching to all the nodes v ∈ V (T ); note again that the trees so obtained are contained in the self-energy graph family FT0 (θ). In such a case again some momenta can change sign, but the corresponding propagator does not change (reason as above for self-energy graphs of type 1). So at the end we obtain a common factor timesiν v , where v is the node which the exiting line is attached to. By (3.6) again we obtain T VT (0) = 0. A4.4. Self-energy graphs of type 3. To prove that T VT (0) = 0 simply reason as for 2 T VT (0) in the case (1), by using that the entering line !T has γ!2 = α. T
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A4.5. Self-energy graphs of type 4. Given a tree θ with a self-energy graph T consider the contribution to ∂VT (0) in which a line ! is differentiated (see (A4.1)). The line ! divides V (T ) into two disjoint set of nodes V1 and V2 , such that !1T exits from a node v1 inside V1 and !2T enters a node v2 inside V2 : if ! = !v one has V2 = {w ∈ V (T ) : w v} and V1 = V (T ) \ V2 . By (3.6), with the notations (A4.3), one has ν 1 + ν 2 = 0. Then consider the tree obtained by detaching !1T from v1 , then reattaching to the node v2 and, simultaneously, by detaching !2T from v2 , then reattaching to the node v1 ; note that the tree so obtained is inside the class FT0 (θ). As a consequence of such an operation the arrows along the path P connecting v1 to v2 change their directions: this means that for such lines ! the momentum ν ! is replaced with −ν ! , but the propagators are even in the momentum, so that no overall change is produced by such factors, if not because of the differentiated propagator (which is along the path by construction) which changes sign. For all the other lines (i.e. the lines not belonging to P) the propagator is left unchanged. Since a derivative with respect to β acts on both the nodes v1 and v2 , the shift of the external lines does not produce any change on the node factors (except for the factorials, that we are not explicitly considering, as said at the beginning of this subsection). Then by summing over the two considered trees we obtain zero because of the change of sign of the differentiated propagator.
A4.6. Remark. To prove the Lemma 3.2 we only use that the propagators satisfy (2.15), so that the same proof applies also to the renormalized self-energy graphs (see the Proposition 5.2), where there are no self-energy lines and the propagators are given by (4.4). Appendix A5. Proof of Lemma 3.3 A5.1. Proof of the property (1). Given a self-energy graph T with momentum ν flowing through the entering line !2T , call P the path connecting the exiting line !1T to the entering line !2T . Then consider also the self-energy graph T obtained by taking !1T as the entering line and !2T as the exiting line and by taking −ν as the momentum flowing through the (new) entering line !1T : in this way the arrows of all the lines along the path P are reverted, while all the subtrees (internal to T ) having the root in P are left unchanged. This implies that the momenta of the lines belonging to P change signs, while all the other momenta do not change. Since all propagators G! are transformed into GT! the property (2.15) implies that the entry ij of the matrix M(ω · ν; ε) corresponding to the self-energy graph T is equal to the entry j i of the matrix M(−ω ·ν; ε); then the assertion follows. A5.2. Proof of the property (2). Given a self-energy graph T , consider also the selfenergy graph T obtained by reverting the sign of the mode labels of the nodes v ∈ V (T ), and by swapping the entering line with the exiting one. In this way the arrows of all the lines along the path P joining the two external lines are reverted, while all the subtrees (internal to T ) having the root in P are left unchanged. It is then easy to realize that the complex conjugate of VT (ω · ν) equals VT (ω · ν), by using the form of the node factors (2.10), and the fact that one has fν∗ (β) = f−ν (β) and G† (ω · ν) = G(ω · ν) (see (2.15)).
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A5.3 Remark. The lemma has been proved without making use of the exact form of the propagator, but only exploiting the fact that it satisfies the property (2.15): therefore, once more, the proof applies also to the renormalized trees as a consequence of the first relation in (5.1), and it gives (5.6).
Appendix A6. Proof of Lemma 3.4 A6.1. Set-up. Given a tree θ ∈ -k,ν,γ , consider a self-energy graph T with height D. Call T (2) ⊂ T (3) ⊂ . . . ⊂ T (D) the resonances containing T , and set T = T (1) ; denote by n = n1 > n2 > n3 > . . . > nD the scales of the lines entering such resonances. For any ! ∈ (T0 ), one can write ν ! = ν 0! + σ! ν,
(eA6.1)
where ν is the momentum of the line !2T (with scale n) entering T (see (3.9)). A6.2. Proof. By shifting the lines entering all the resonances containing !, the momentum ν ! can change into a new value ν˜ ! (as it can be seen by applying iteratively (3.10)) in such a way that |ω0 · ν˜ ! | differs from ω0 · ν 0! by a quantity bounded by γn + γn2 + + γnD ≤ 2n + 2n2 + . . . + 2nD < 2n+1 . . . . −(n+3)/τ 0 , by definition of the self-energy graph, we can apply (3.2) and As ν ! < 2 conclude that |ω0 · ν 0! | has to be contained inside an interval [γp−1 , γp ], with p = n0! ≥ n + 3 (see Remark 3.2, (2)), at a distance at least 2n+1 from the extremes: therefore the quantity |ω0 · ν˜ ! | still falls inside the same interval [γp−1 , γp ]. In particular this implies the identity n! = n˜ ! = n0! ,
(A6.2)
if n˜ ! is defined as the integer such that γn˜ ! −1 ≤ |ω0 · ν˜ ! | < γn˜ ! . Appendix A7. Proof of Lemma 4.1 A7.1. Set-up. We have to prove that, for all k ≤ k, one has [k]
) (k ) [h (ψ; ε)](k ν = hν , [k]
(A7.1)
which yields that the functions h and h admit the same power series expansions up to order k. [k] (k ) (k ) Of course both hν and [h (ψ; ε)]ν are given by sums of several contributions; in the same way [M [k] (ω · ν; ε)](k ) can be expressed as the sum of several terms. We shall (k ) prove that, given any tree value Val(θ) contributing to hν and any self-energy value VT (x) corresponding to a self-energy graph T of order k , one can find the same terms [k] (k ) contributing, respectively, to [h (ψ; ε)]ν and to [M [k] (x; ε)](k ) , and vice versa. The proof will be by induction on k = 1, . . . , k.
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(k )
A7.2. Remarks. (1) Note that [h (ψ; ε)]ν [k]
(k ) (ψ; ε)]ν
is the coefficient to order k
[k,k ]
is not the same as hν
(ψ; ε): the quantity (k)
[h that one obtains by developing h (ψ; ε) in powers of ε. [k] (k ) (2) The contributions to [h (ψ; ε)]ν and to [M [k] (x; ε)](k ) can arise only from trees [k]
(k )
[k ]
(k )
of order k ≤ k . Furthermore [h (ψ; ε)]ν = [h (ψ; ε)]ν and [M [k] (x; ε)](k ) = [M [k ] (x; ε)](k ) : this simply follows from Remark 4.1 and the trivial observation that, for [k] k > k , the self-energy graphs [trees] whose values contribute to M [k] (x; ε) [h (ψ; ε)]
[k ]
but not to M [k ] (x; ε) [h (ψ; ε)] are those containing also self-energy graphs with height D > k : such contributions are of order at least D in ε, so that they can not [k] contribute to [M [k] (x; ε)](k ) [[h (ψ; ε)](k ) ]. A7.3. Starting from h. The case k = 1 is trivial. Suppose that, given k ≤ k, the assertion is true for all k < k : then we show that it is true also for k . Consider a tree θ ∈ -k,ν,γ and let Val(θ) be its value. Denote by T1 , . . . , TN the maximal self-energy graphs in θ, and by k1 , . . . , kN the number of nodes that they contain, respectively; the number of nodes external to the self-energy graphs will be k0 = k − k1 − . . . − kN . Call θ0 the tree obtained from θ by replacing each chain of self-energy graphs together with their external lines with a new line carrying the same momentum of the external lines. The tree θ0 will have k0 + 1 lines: by construction each line !i of θ0 corresponds to a chain of pi self-energy graphs, with pi ≥ 0, of orders Ki1 , . . . , Kipi , such that pi k0
Kij = k − k0 .
(A7.2)
i=1 j =1
R Then consider θ0 as a tree θ R ∈ -R k0 ,ν,γ ; let !i be the line in θ which corresponds to the line with the same name in θ0 . For each line !i ∈ (θ R ), by setting xi = ω · ν !i , the propagator is of the form
−1 [k−1] [k−1] [k−1] G!i =G (xi ; ε), G (xi ; ε) = xi2 11 − M [k−1] (xi ; ε) , (A7.3)
and it can be expanded in powers of M [k−1] (xi ; ε) as 1 1 1 1 [k−1] G (xi ; ε) = 2 11 + M [k−1] (xi ; ε) 2 + M [k−1] (xi ; ε) 2 M [k−1] (xi ; ε) 2 + . . . . xi xi xi xi (A7.4) We can consider the contribution 1 1 1 1 [M [k−1] (xi ; ε)](Ki1 ) 2 . . . 2 [M [k−1] (xi ; ε)](Kipi ) 2 2 x xi xi xi
(A7.5)
to (A7.4). Note that Kij < k − 1 for all i, j , by construction, so that by the Remark A7.2, (2), we can write [M [k−1] (xi ; ε)](Kipi ) = [M [k] (xi ; ε)](Kipi ) = [M [Kipi ] (xi ; ε)](Kipi ) ,
(A7.6)
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for all j = 1, . . . , pi and for all i = 1, . . . , k0 . Hence by the inductive hypothesis, we can deduce that, for all i, j , there is a contribution to [M [k−1] (xi ; ε)](Kij ) = [M [k] (xi ; ε)](Kij ) which corresponds to the considered resonance in θ . As a consequence [k] (k ) we can also conclude that there is a term contributing to [h (ψ; ε)]ν which is the same as the considered tree value Val(θ). Of course if instead of a tree value we had considered a self-energy value, the same argument should have applied, so that the assertion follows. [k]
A7.4. Starting from h . The construction described in Sect. A7.3 can be used in the opposite direction, in order to prove that each term of order k in ε which is obtained by [k] truncating h to order k corresponds to a term contributing to h(k ) . Proof of the bound (5.1) from (5.8) A8.1. Set-up. Consider the matrix −1 = x 2 11 − M [k] (x; ε) = + ε 2 x=1 + ε 2 x 2 =2 , A(x; ε) = G[k] (x; ε) with
2 x 11 0 αα 0 = , 0 ββ 0 x 2 11 − ε∂β2 f0 (β 0 ) 0 B(ε) , =1 = −B(ε) 0 [k] [k] (x; ε) −Mαβ −Mαα (x; ε) − ε 2 xB(ε) =2 = , [k] [k] (x; ε) − ε 2 xB(ε) −Mββ (x; ε) + ε∂β2 f0 (β 0 ) −Mβα
(A8.1)
=
(A8.2)
where −1 B(ε) = ε2 x LVT (ω · ν),
(A8.3)
T
with the sum running over all self-energy graphs of type 2, so that one has #=1 # ≤ C,
#=2 # ≤ C,
(A8.4)
for some positive constant C. The matrix is a block matrix which induces a natural decomposition Rd = Rr ⊕Rs ; the eigenvalues of the block αα ≡ |Rr are all equal to x 2 , while the eigenvalues of the block ββ = |Rs are of the form λj = x 2 + aj ε, with aj > 0. Set 1 = + η=1 , with η = ε2 x. Define B(x; ε) = eηX A(x; ε)e−ηX = eηX 1 e−ηX + ε 2 x 2 eηX =2 e−ηX ≡ B0 (x; ε) + ε 2 x 2 eηX =2 e−ηX ; of course B(x; ε) has the same eigenvalues as A(x; ε).
(A8.5)
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A8.2. Block-diagonalization. Consider B0 (x; ε): we shall fix the matrix X in such a way that B0 (x; ε) is block-diagonal up to order η2 . So we look for X such that 11 + ηX + O(η2 ) ( + η=1 ) 11 − ηX + O(η2 ) = + ηJ1 + O(η2 ), (A8.6) with
J J J1 = 1,αα 1,αβ J1,βα J1,ββ
J1,αα 0 = . 0 J1,ββ
(A8.7)
By expanding to first order (A8.6) we obtain [X, ] + =1 = J1 ,
(A8.8)
−1 , Xαβ = −B(ε) ββ − x 2 11 −1 B(ε), Xβα = − ββ − x 2 11
(A8.9)
−1 −1 1 2 C ββ − x 2 11 = ∂ f0 (β 0 ) ≤ , ε β ε
(A8.10)
while imposing (A8.7) gives
where
for some constant C. Furthermore, by choosing Xαα = 0 and Xββ = 0, one obtains J1 ≡ 0. Then it follows that one has B(x; ε) = + O(η2 X 2 ) + O(ε 2 x 2 ) = + O(ε 2 x 2 ),
(A8.11)
B −1 (x; ε) = −1 + O(ε 2 x 2 ),
(A8.12)
so that
where the eigenvalues of −1 are of the form either 1/x 2 or 1/(x 2 + aj ε). Therefore one has 2 [k] G (x; ε) ≤ eηX B −1 e−ηX ≤ 2 , x
(A8.13)
which proves the bound in (5.1) for real ε. A8.3. Bounds in the complex plane. The above analysis applies also for complex values of ε. Consider the domain D0 represented in Fig. 5, with half-opening angle ϕ < π . For [k] ε ∈ D0 one has that the norms of G (x; ε))−1 are bounded from below by x2 (π − ϕ) , 2 when ϕ is close to π .
(A8.14)
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Appendix A9. Proof of Lemma 5.1 A9.1. Set-up. Both M [k] (ω · ν; ε) and M [k+1] (ω · ν; ε) can be expressed by (4.5): the [k] only difference is that one has to use the propagators G! for M [k+1] (ω · ν; ε). This means that there is a correspondence 1-to-1 between the graphs contributing to M [k] (ω · ν; ε) and those contributing to M [k+1] (ω · ν; ε), so that we can write M [k] (ω · ν; ε) − M [k+1] (ω · ν; ε) = VT[k,k+1] (ω · ν), VT[k,k+1] (ω
· ν) = ε
|V (T )|
renormalized T
Fv
v∈V (T )
[k−1]
!∈(T )
G!
−
[k]
!∈(T )
G!
.
(A9.1)
For each renormalized self-energy T we can write VT[k,k+1] (ω · ν) as sum of V = |V (T )| terms corresponding to trees whose lines have all the propagators of the form [k−1] [k] either G! or G! , up to one which has a new propagator given by the difference [k−1] [k] G! − G! . A9.2. Remark. Note that the scales of all lines are uniquely fixed by the momenta, so [k−1] [k] that both propagators G! and G! admit the same bounds (see (5.6)). A9.3. Bounds. We can order the lines in (T ) and construct a set of V subsets 1 (T ), . . . , V (T ) of (T ), with |j (T )| = j , in the following way. Set 1 (T ) = ∅, 2 (T ) = !1 , if !1 is the root line of θ and, inductively for 2 ≤ j ≤ V − 1, j +1 (T ) = j (T ) ∪ !j , where the line !j ∈ (T ) \ j (T ) is connected to j (T ); of course V (T ) = (T ). Then
VT[k,k+1] (ω · ν) = ε |V (T )| Fv v∈V (T )
V j =1
!∈j (T )
[k−1]
G!
[k−1] [k] G!j − G!j
[k]
!∈(T )\j (T )
G!
(A9.2)
,
where, by construction, the sets j (θ) are connected (while of course the sets (θ) \ j (θ ) in general are not). We can write
[k] [k−1] [k] [k−1] G!j − G!j = G!j (A9.3) M [k] (ω · ν !j ; ε) − M [k−1] (ω · ν !j ; ε) G!j , so that in (A9.2) we can bound [k−1] [k−1] [k] G! G!j G!j !∈j (T )
≤
2
2+2τ
C0−1
1 2k
n=−∞
!∈(T )\j (T )
2
2
−2nNn (θ)
[k] G! (A9.4)
,
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where the power 2 (with respect to (5.9)) is due to the fact that in the product two propagators correspond to the line !j . This means that VT[k,k+1] (ω · ν) admits the same bound as the square of VT[k] (ω · ν) times the supremum (over ν) of the norms [k] (A9.5) M (ω · ν; ε) − M [k−1] (ω · ν; ε) . If we perform the sum over all self-energy graphs in (A9.1) and we use that the first non-trivial terms correspond to graphs with V = 2 nodes, we obtain, for k ≥ 1, [k+1] [k] (ω · ν; ε) − M [k] (ω · ν; ε) ≤ Cε 2 M M (ω · ν; ε) − M [k−1] (ω · ν; ε) , ν∈Zr
(A9.6) for some constant C. Therefore the lemma follows. Appendix 10. Proof of Proposition 5.4 A10.1. Set-up. Define Val
[∞]
(θ ) =
Fv
v∈V (θ) [∞]
Lv
v∈L(θ)
!∈(θ)
[∞]
G!
,
(A10.1)
[∞]
where the propagators G! = G (ω · ν ! ; ε) are defined in (5.19); we shall denote by [∞] [∞] G the operator with kernel G (ω · ν; ε) in Fourier space. Then one has [∞]
hγ (ψ; ε) =
∞
ε k eiν·ψ
k=1 ν∈Zr
Val
[∞]
(θ),
(A10.2)
θ∈-R k,ν,γ
which we can represent, in a more compact notation, as [∞] R h (ψ; ε) = Val (θ ; ψ; ε),
(A10.3)
θ∈-R
R where -R is the set of all renormalized trees, and, for θ ∈ -R k,ν,γ ⊂ - , we have defined R
Val (θ ; ψ; ε) = εk eiν·ψ Val
[∞]
(θ).
(A10.4)
The function h(ψ; ε) solving the equations of motion (1.6) is formally defined as the solution of the functional equation h(ψ; ε) = G∂ψ f (ψ + h(ψ; ε)) , [0]
where G = (iω · ∂)−2 = G is the operator with kernel G(x) = x 2 . We have the following result.
(A10.5)
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459
A10.2. Lemma. One has G(x) M [∞] (x; ε) + (G[∞] (x; ε))−1 = 11. [∞]
A10.3. Proof. By definition one has G (x; ε) = (G−1 (x) − M [∞] (x; ε))−1 , so that [∞] G−1 (x) = (G (x; ε))−1 + M [∞] (x; ε); then the assertion follows. [∞]
A10.4. Conclusions. The following result shows that the function h (ψ; ε) formally solves the equation of motions (1.6); as the analysis of the previous sections shows that [∞] the function h (ψ; ε) is well defined and it is, order by order, equal to the formal solution envisged in Sect. 3, we have proved the proposition.
A10.5. Lemma. The function h
[∞]
(ψ; ε) defined by (A10.3) formally solves (A10.4).
A10.6. Proof. We shall show that (A10.3) solves (A10.5). One has ∞ [∞] p 1 p+1 [∞] G∂ψ f ψ + h (ψ; ε) = G ∂ψ f (ψ) h (ψ; ε) p! p=0
=G
∞ p=0
1 p+1 R R ∂ψ f (ψ) Val (θ1 ; ψ; ε) . . . Val (θp ; ψ; ε) p! R R θ1 ∈-
θp ∈-
[∞] −1 R =G G Val (θ ; ψ; ε), θ∈-∗R
(A10.7) where -∗R differs from -R as it contains also trees which can have only one selfenergy graph with exiting line !0 , if, as usual, !0 denotes the root line of θ ; the operator [∞] G(G )−1 takes into account the fact that, by construction, to the root line !0 an operator R [∞] G is associated, while in Val (θ ; ψ; ε), by definition, a propagator G is associated. Then we can write (A10.5), by explicitly separating the trees containing such a selfenergy graph from the others, [∞] G∂ψ f ψ + h (ψ; ε) [∞] −1 [∞] R R G M [∞] Val (θ ; ψ; ε) + Val (θ; ψ; ε) =G G
[∞] [∞]
θ∈-R [∞] −1 [∞]
θ∈-R
= G M h (ψ; ε) + (G ) h (ψ; ε) [∞] [∞] [∞] = G M [∞] + (G )−1 h (ψ; ε) = h (ψ; ε), (A10.8) where Lemma A10.2 has been used in the last line. Note that at each step only absolutely converging series have been dealt with; then the assertion is proved.
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Acknowledgements. We are indebted to V. Mastropietro for his criticism in the early stages of this work. We thank also H. Eliasson for useful discussions.
References [BaG]
Bartuccelli, M.V., Gentile, G.: Lindstedt series for perturbations of isochronous systems. A review of the general theory. Rev. Math. Phys. 14, 121–171 (2002) [BeG] Berretti, A., Gentile, G.: Scaling properties for the radius of convergence of Lindstedt series: generalized standard maps. J. Math. Pures Appl. 79, 691–713 (2000) [BGGM] Bonetto, F., Gallavotti, G., Gentile, G.: Lindstedt series, ultraviolet divergences and Moser’s theorem. Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4) 26, 545–593 (1998) [B] Bryuno, A.D.: Analytic form of differential equations. I, II. (Russian). Trudy Moskov. Mat. Obšˇc 25, 119–262 (1971); ibid. 26, 199–239 (1972) [BKS] Bricmont, J., Kupiainen, A., Schenkel, A.: Renormalization group for the Melnikov problem for PDE’s. Commun. Math. Phys. 221, 101–140 (2001) [CGM] Caliceti, E., Grecchi, V., Maioli, M.: The distributional Borel summability and the large coupling B4 lattice fields, Commun. Math. Phys. 104, 163–174 (1986); Erratum, Commun. Math. Phys. 113, 173–176 (1987) [CG] Chierchia, L., Gallavotti, G.: Drift and diffusion in phase space. Ann. Inst. H. Poincarè Phys. Théor. 60, 1–144 (1994); Erratum, Ann. Inst. H. Poincaré Phys. Théor. 68, 135 (1998) [CF] Chierchia, L., Falcolini, C.: Compensations in small divisor problems. Commun. Math. Phys. 175, 135–160 (1996) [E] Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions. Math. Phys. Electron. J. 2 (1996), Paper 4, http:// mpej.unige.ch [G1] Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164, 145–156 (1994) [G2] Gallavotti, G.: Twistless KAM tori, quasiflat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review. Rev. Math. Phys. 6, 343–411 (1994) [GG] Gallavotti, G., Gentile, G.: Majorant series convergence for twistless KAM tori. Ergodic Theory Dynam. Systems 15, 857–869 (1995) [Ge] Gentile, G.: Whiskered tori with prefixed frequencies and Lyapunov spectrum. Dynam. Stability of Systems 10, 269–308 (1995) [GGM] Gentile, G., Gallavotti, G., Mastropietro, V.: Field theory and KAM tori. Math. Phys. Electron. J. 1 (1995), Paper 1, http:// mpej.unige.ch [GM] Gentile, G., Mastropietro, V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in Classical mechanics. A review with some applications. Rev. Math. Phys. 8, 393–444 (1996) [Gr] Graff, S.M.: On the conservation for hyperbolic invariant tori for Hamiltonian systems. J. Differential Equations 15, 1–69 (1974) [JLZ] Jorba, A., Llave, R., Zou, M.: Lindstedt series for lower dimensional tori. In Hamiltonian systems with more than two degrees of freedom (S’Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 533, Ed. C. Simó, Dortrecht: Kluwer Academic Publishers, 1999, 151–167 [Me] Mel’nikov, V.K.: On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function. Soviet Math. Dokl. 6, 1592–1596 (1965); A family of conditionally periodic solutions of a Hamiltonian systems. Soviet Math. Dokl. 9, 882–886 (1968) [Mo] Moser, J.: Convergent series expansions for quasi periodic motions. Math. Ann. 169, 136–176 (1967) [P] Pöschel, J.: Invariant manifolds of complex analytic mappings near fixed points, in Critical Phenomena, Random Systems, Gauge Theories, Les Houches, Session XLIII (1984), Vol. II, , Ed. K. Osterwalder & R. Stora, Amsterdam: North Holland, 1986, pp. 949–964 [R] H. Rüssmann: Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dynam. 6, 119–204 (2001) [T] Treshchëv, D.V.: The mechanism of destruction of resonant tori of Hamiltonian systems. Math. Sb. 180, 1325–1439 (1989); translation in Math. USSR-Sb. 68, 181–203 (1991); see also the MathSciNet review by M.B. Sevryuk. http://www.ams.org [XY] Xu, J., You, J.: Persistence of Lower Dimensional Tori Under the First Melnikov’s Non-resonance Condition. Nanjing University Preprint, 1–21, 2001, J. Math. Pures Appl. 80, 1045–1067 (2001) [Y] Yuan, X.: Construction of quasi-periodic breathers via KAM techniques. Commun. Math. Phys. 226, 61–100 (2002) Communicated by A. Kupiainen
Commun. Math. Phys. 227, 461 – 481 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Ergodic Theory of Infinite Dimensional Systems with Applications to Dissipative Parabolic PDEs Nader Masmoudi, Lai-Sang Young Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA. E-mail:
[email protected];
[email protected] Received: 15 May 2001 / Accepted: 30 November 2001
Abstract: We consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general (nonlinear) dissipative parabolic PDEs. Results on invariant measures and their exponential mixing properties are proved, and applications to 2D Navier–Stokes systems are included. 1. Introduction This paper concerns the ergodic theory of a class of nonlinear dissipative PDEs of parabolic type. Leaving precise statements for later, we first give an indication of the nature of our results. We view the equation in question as a semi-group or dynamical system St on a suitable function space H , and assume the existence of a compact attracting set (as in Temam [15], Chapter 1). To this deterministic system, we add a random force in the form of a “kick” at periodic time intervals, defining a Markov chain X with state space H . We assume that the combined effect of the semi-group and our kicks sends balls to compact sets. Under these conditions, the existence of invariant measures for X is straightforward. The goal of this paper is a better understanding of the set of invariant measures and their ergodic properties. In a state space as large as ours, particularly when the noise is bounded and degenerate, the set of invariant measures can, in principle, be very large. In this paper, we discuss two different types of conditions that reduce the complexity of the situation. The first uses the fact that for the type of equations in question, high modes tend to be contracted. By actively driving as many of the low modes as needed, we show that the dynamics resemble those of Markov chains on RN with smooth transition probabilities. In particular, the set of ergodic invariant measures is finite, and every aperiodic ergodic measure is exponentially mixing. The second type of conditions we consider is when all of the Lyapunov exponents of X are negative. As in finite dimensions, we show under This research is partially supported by grants from the NSF
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these conditions that nearby orbits cluster together in a phenomenon known as “random sinks”. The conditions in the last paragraph give a general understanding of the structure of invariant measures; they alone do not guarantee uniqueness. (Indeed, it is not the case that for the equations in question, invariant measures are always unique; see Theorem 3.) For uniqueness, one needs to guarantee that there are places for distinct ergodic components to meet. To this end, we have identified some conditions expressed in terms of existence of special sequences of controls. These conditions are quite special; however, they are easily verified for the equations of interest. Assuming these conditions, the uniqueness of the invariant measure follows readily. In the case of negative Lyapunov exponents, there is, in fact, a stronger form of uniqueness or stability, namely that all solutions independent of initial conditions become asymptotically close to one other as time goes to infinity. This work is inspired by a number of recent papers on the uniqueness of invariant measure for the Navier–Stokes equations ([5, 1, 3, 7, 8]), and by [4], which proves uniqueness of invariant measure for a different equation. With the exception of [7] and [8], all of the other authors worked with unbounded noise. Naturally there is overlap among these papers and with the first part of ours. More detailed references will be given as the theorems are stated. Instead of working directly with specific PDEs, we have elected to prove our ergodic theory results for general randomly perturbed dynamical systems on infinite dimensional Hilbert spaces satisfying conditions compatible with the PDEs of interest. This allows us to make more transparent the relations between the various dynamical properties and the mechanisms responsible for them. Once our “abstract” results are in place, to apply them to specific equations, it suffices to verify that the conditions in the theorems are met. (In this regard, we are influenced by [7], which takes a similar approach.) This paper is organized as follows. Before proceeding to a discussion of our “abstract results”, we first give a sample of their applications. This is done in Sect. 2. Sections 3 and 4 treat the two types of conditions that lead to simpler structures for invariant measures. In each case, we begin with a general discussion and finish with proofs of concrete results for PDEs which we now state.
2. Statement of Results for PDEs This section contains precise formulations of results on PDEs that can be deduced from our “abstract theory”. The theorems below are proved in Sects. 3.3 and 4.3.
2.1. The Navier–Stokes system. The first application of our general results is to the 2-D incompressible Navier–Stokes equations in the 2-torus T2 = (R/2π Z)2 . We consider the randomly forced system ∂t u − ν u + u · ∇u = −∇p + ∞ k=1 δ(t − k)ηk (x),
div(u) = 0,
u(t = 0) = u0 ,
(1)
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where u0 (x) ∈ L2 (T2 ), div(u0 ) = 0, u0 = 0, ν > 0 is the viscosity, and where the ηk ’s are i.i.d. random fields which can be expanded as ηk (x) =
∞
bj ξj k ej (x) .
(2)
j =1
Here the Hilbert space in question is
H = {u, u ∈ L2 (T2 ), div(u) = 0, and
u = 0} ,
and {ej , j ≥ 1} is the orthonormal basis consisting of the eigenfunctions of the Stokes operator − ej + ∇pj = λj ej , div(ej ) = 0, with λ1 ≤ λ2 ≤ · · · . We assume that ξj k , j, k ∈ N, are independent random variables where ξj k is distributed according to a law which has a positive Lipschitz density ρj with respect to the Lebesgue measure on 2 2 [−1, 1]. Finally, the bj are required to satisfy ∞ j =1 bj = a < ∞ for some a > 0. From (1), we define a Markov chain uk with values in H given by uk = u(k + 0, ·). That is to say, if St is the semi-group generated by the unforced Navier–Stokes equation, i.e. Eq. (1) without the term ∞ δ(t − k)ηk (x), and S = S1 , then k=1 uk+1 = S(uk ) + ηk . Theorem 1 (Uniqueness of invariant measure and exponential mixing). For the system above, there exists N ≥ 1 depending only on the viscosity ν and on a such that if bj = 0 for all 1 ≤ j ≤ N , then the Markov chain uk has a unique invariant measure µ in H . Moreover, for all u0 ∈ H , the distribution k of uk converges exponentially fast to µ in the sense that for every test function f : H → R of class C 0,σ , σ > 0, there exists C = C(f, u0 ) such that for all k ≥ 1, fd k − < Cτ k f dµ for some τ < 1 depending only on the Hölder exponent σ . Papers [7] and [8] together contain a proof of the uniqueness of invariant measure part of Theorem 1; these papers rely on ideas different from ours. While this manuscript was being written, we received electronic preprints [9] and [10] which together prove the results in Theorem 1 using methods similar to ours. Theorem 1’. The result in Theorem 1 holds if we replace L2 by H s , any s ∈ N, and ∞ impose the restriction j =1 λsj bj2 = a 2 < ∞ on the noise. Remark. In the theorems above, we can also treat noises that are bounded but not compact provided that we consider the Markov chain uk = u(k − 0, ·) or, equivalently, uk+1 = S(uk + ηk ). An example of bounded, noncompact noise satisfying the conditions of Theorems 1 and 1’ is the following: Let VN be the span of {e1 , e2 , ..., eN }, and consider ηk =
N j =1
bj ξj k ej + ηk ,
(3)
where bj = 0 for all j, 1 ≤ j ≤ N , and ηk are i.i.d. random variables with a law supported on a bounded set in VN⊥ .
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Our next result gives a stronger form of uniqueness than the previous one. It guarantees, under the assumption of negative Lyapunov exponents, that independent of initial condition, all the solutions eventually come together and evolve as one, their time evolution depending only on the realization of the noise. Lyapunov exponents are defined in Sect. 4.1. Having only negative Lyapunov exponents means, roughly speaking, that infinitesimally the semi-group is contractive on average along typical orbits. More regularity is required for the next result; thus we work in H 2 . Let A(0) ⊂ H 2 denote the closure of the set of points accessible under the Markov chain uk with u0 = 0. Theorem 2 (Asymptotic uniqueness of solutions independent of initial conditions). Consider the system defined by (1) with H = H 2 and where the ηk are i.i.d. with a law which has bounded support. Assume there is an invariant measure µ supported on A(0) such that all of its Lyapunov exponents are strictly negative. Then (a) µ is the unique invariant measure the Markov chain uk has in H ; (b) there exists λ < 0 such that for almost every sequence of ηk and every pair of initial conditions u0 , u0 ∈ H , there exists C = C(u0 , u0 ) such that if uk+1 = S(uk ) + ηk and uk+1 = S(uk ) + ηk for all k ≥ 0, then uk − uk ≤ Ceλk
∀k≥0.
Observe that for this result very little is required of the structure of the noise. Remark. We will explain in Sect. 4.4 (see Remark) that for fixed positive viscosity, S is a uniform contraction near 0, and so it continues to be a contraction for sufficiently small bounded noise. Very small but unbounded noise is treated in [12]. As noise level increases, it is likely that there is a range where S is no longer a contraction but all of its Lyapunov exponents remain negative. Indeed, for any ergodic invariant measure µ of the Navier–Stokes system, the largest Lyapunov exponent λ1 is either < 0, = 0, or > 0: λ1 > 0 can be interpreted as “temporal chaos”; λ < 0 implies the asymptotic uniqueness of solutions as we have shown; the case λ1 = 0 is sometimes regarded as less significant because it can often be perturbed away. Of these three possibilities, the only one that has been proved to occur is λ1 < 0. Remark. Theorems 1 and 2 apply to other nonlinear parabolic equations for which all solutions of the unforced equation relax to their unique stable stationary solutions.
2.2. The real Ginzburg–Landau equation. Our second application is to the following equation, which, following [4], we refer to as the real Ginzburg–Landau equation. We consider a periodic domain in one space dimension, i.e. T = (R/2π Z), and consider the system ∂t u − ν u − u + u3 = ∞ k=1 δ(t − k)ηk (x), (4) u(t = 0) = u0 . Here H = L2 (T), {ej , j ≥ 0} is the orthonormal basis defined by − ej = λj ej , λ1 ≤ λ2 ≤ · · · , ν > 0 is a positive constant, and the ηk ’s are i.i.d. random fields which
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465
can be expanded as ηk (x) =
∞
bj ξj k ej (x).
(5)
j =0
We assume the same conditions on ξj k and bj as in the first paragraph of Sect. 2.1. The unforced equation in (4) is somewhat more unstable than the (unforced) Navier– Stokes equation. It has at least three stationary solutions: two stable ones, namely u = 1 and u = −1, and an unstable one, namely u = 0. Our next result shows that the number of invariant measures vary depending on how localized the forcing is, particularly in the zeroth mode. Theorem 3 (Number of ergodic measures). Consider the Markov chain uk defined by the system in (4). 2 2 2 (a) There exists α > 0 such that if ∞ j =0 |bj | = a ≤ α , then there are at least two different invariant measures. (b) There exists N depending only on ν and on a such that if bj = 0 for all 0 ≤ j ≤ N , then the number of ergodic invariant measures is finite. (c) If bj = 0 for all 0 ≤ j ≤ N and b0 > 1, then the invariant measure is unique, and for every initial condition u0 ∈ H , the distribution of uk converges to it exponentially fast in the sense of Theorem 1. In contrast to part (a), we observe that to obtain uniqueness of the invariant measure, we may take bj , 1 ≤ j ≤ N , to be arbitrarily small as long as they are > 0, and the forcing in the zeroth mode, i.e. b0 ξ0k , can be arbitrarily weak as long as its law has a tail which extends beyond [−1, 1]. As will be explained in Sect. 3.4, the condition b0 > 1 above can, in fact, be replaced by b0 > κ for a smaller κ. Theorem 3 complements [4], which drives high rather than low modes, and proves uniqueness for unbounded noise using techniques very different from ours.
3. Invariant Measures and Their Ergodic Properties 3.1. Formulation of abstract results. Setting and notation. Let S : H → H be a transformation of a separable Hilbert space H , and let ν be a probability measure on H . We consider the Markov chain X = {un , n = 0, 1, 2, · · · } on H defined by either (I) un+1 = S(un ) + ηn
or
(II)
un+1 = S(un + ηn )
where η0 , η1 , · · · are i.i.d. with law ν. The following notation is used throughout this paper: BH (R) or simply B(R) denotes the ball of radius R in H , i.e. B(R) = {u ∈ H, u ≤ R}; K denotes the support of ν; and given an initial distribution 0 of u0 , the distribution of un under X is denoted by n . If T : H → H is a mapping and µ is a measure on H , then T∗ µ is the measure defined by (T∗ µ)(E) = µ(T −1 (E)).
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Standing Hypotheses. (P1) (a) S(B(R)) is compact ∀R > 0; (b) ∀R > 0, ∃MR > 0 such that ∀u, v ∈ B(R), Su − Sv ≤ MR u − v. (P2) ∀a > 0, ∃R0 = R0 (a) such that if K ⊂ B(a), then ∀R > 0, ∃N0 = N0 (R) ∈ Z+ such that for u0 ∈ B(R), un ∈ B(R0 ) ∀n ≥ N0 . (P3) ∃γ < 1 such that given R > 0, there is a finite dimensional subspace V ⊂ H such that if PV and PV ⊥ denote orthogonal projections from H onto V and V ⊥ respectively, then ∀u, v ∈ B(R), PV ⊥ S(u) − PV ⊥ S(v) ≤ γ u − v. (P4) (a) K is compact if X is defined by (I), bounded if X is defined by (II). (b) Let V be given by (P3) with R = R0 . Then ν = (PV )∗ ν × (PV ⊥ )∗ ν where (PV )∗ ν has a density ρ with respect to the Lebesgue measure on V , 5 := {ρ > 0} has piecewise smooth boundary and ρ|5 is Lipschitz. We remark that (P1)–(P3) are selected to reflect the properties of general (nonlinear) parabolic PDEs. Definition 3.1. A probability measure µ on H is called an invariant measure for X if 0 = µ implies n = µ for all n > 0. Lemma 3.1. Assume (P1), (P2) and (P4)(a). Then (i) X has an invariant measure; (ii) there exists a compact set A ⊂ B(R0 ) on which all invariant measures of X are supported. Proof. Let A0 = B(R0 ). For n > 0, let An = S(An−1 ) + K in the case of (I) and An = S(An−1 + K) in the case of (II). Then each An is compact, and by (P2), An ⊂ A0 for all n ≥ some N0 . Let ∞ 0 −1 A = ∪N i=0 (∩k=0 AkN0 +i ) .
Then A is compact, contained in B(R0 ), and satisfies S(A) + K = A. To construct an invariant measure for X , pick an arbitrary u0 ∈ A, and let 0 = δu0 , the Dirac measure at u0 . Then any accumulation point of the sequence { n1 i
0, there exists τ = τ (σ ) < 1 such that the following holds for every f : H → R of class C 0,σ : for µ-a.e. u0 , there exists C = C(f, u0 ) such that < Cτ n for all n ≥ 1. fd n − f dµ Let X n denote the n-step Markov chain associated with X .
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Theorem A (Structure of invariant measures). Assume (P1)–(P4). Then (1) X has at most a finite number of ergodic invariant measures. (2) If (X n , µ) is ergodic for all n ≥ 1, then (X , µ) is exponentially mixing for Hölder continuous observables. The reasons behind these results are that under (P1)–(P4), X resembles a Markov chain on RN whose transition probabilities have densities. One expects, therefore, the same type of decomposition into ergodic and mixing components. We now give a condition that guarantees the uniqueness of invariant measures and other convergence properties. This condition is expressed in terms of the existence of special sequences of controls; it is quite strong, but is easily verified for the PDEs under consideration. (C) Given ε0 > 0 and R > 0, there is a finite sequence of controls ηˆ 0 , · · · ηˆ n such that for all u0 , u0 ∈ B(R), if uk+1 = S(uk ) + ηˆ k and uk+1 = S(uk ) + ηˆ k for k < n, then un − un < ε0 . Theorem B (Sufficient condition for uniqueness and mixing). Assume (P1)–(P4) and (C). Then (1) X has a unique invariant measure µ, and (X , µ) is exponentially mixing; (2) ∃τ = τ (σ ) < 1 such that ∀f ∈ C 0,σ and for every u0 ∈ H , there exists C s.t. f d n − f dµ < Cτ n for all n ≥ 1. Recalling that the invariant measure µ is supported on a (relatively small) compact subset of H , we remark that the assertion in (2) above is considerably stronger than the usual notion of exponential mixing: it tells us about initial conditions far away from the support of µ. This property is reminiscent of the idea of Sinai-Ruelle-Bowen measures for attractors in finite dimensional dynamical systems.
3.2. Proofs of abstract results (Theorems A and B). We will prove Theorems A and B for the case where X is defined by (I); the proofs for (II) are very similar. Also, to avoid the obstruction of main ideas by technical details, we will assume (PV )∗ ν is the normalized Lebesgue measure on 5 := {u ∈ V , u ≤ r} for some r > 0; the general case is messier but conceptually not different. Let M = MR0 , where R0 is given by (P2) and MR0 is as defined in (P1). The following notation is used heavily: Given u0 and η = (η0 , η1 , η2 , . . . ) ∈ K N , we define ui (η) inductively by letting u0 (η) = u0 and ui (η) = S(ui−1 (η)) + ηi−1 for i > 0. Notation such as ui (η0 , . . . , ηn−1 ) for a finite sequence (η0 , · · · , ηn−1 ) with i ≤ n has the obvious meaning, as does ui (η) for given u0 . Lemma 3.2 (Matching Lemma). Let δ = r(2M)−1 . There is a set 9 ⊂ K N with ν N (9) > 0 such that ∀u0 , u0 ∈ B(R0 ) with u0 − u0 < δ, there is a measurepreserving map : : 9 → K N with the property that ∀η ∈ 9, un (η) − un (:(η)) ≤ u0 − u0 γ n
∀n ≥ 0 .
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By virtue of (P4)(b), K = 5 × E where 5 ⊂ V is as above and E ⊂ V ⊥ . We write η0 = (η01 , η02 ) with η01 ∈ 5, η02 ∈ E. Since all of our operations take place in V , it is convenient to introduce the notation Kε := {u ∈ 5, u ≤ ε} × E, so that in particular Kr = K. Proof. Suppose u0 − u0 < δ. We define :(1) : K 2r = Kr−Mδ → H by (η0 , η0 ) = :(1) (η0 ) := (η01 + PV S(u0 ) − PV S(u0 ), η02 ). 1
2
Observe that (ii) η0 1 < (r − Mδ) + M · u0 − u0 < r, so that :(1) (K 2r ) ⊂ K; (ii) :(1) preserves ν-measure; and (iii) for η0 ∈ K 2r , if u1 = u1 (η0 ) and u1 = u1 (:(1) (η0 )), then PV u1 = PV u1
PV ⊥ u1 − PV ⊥ u1 < γ u0 − u0 .
and
We may, therefore, repeat the argument above with (u1 , u1 ) in the place of (u0 , u0 ), defining for each u1 = u1 (η0 ), η0 ∈ K 2r , a map from Kr−Mδγ = Kr(1− 1 γ ) to K. Put 2
together, this defines an injective map :(2) : K 2r ×Kr(1− 1 γ ) → K 2 which carries the ν 2 2
measure to ν 2 -measure such that for each (η0 , η1 ) ∈ K 2r × Kr(1− 1 γ ) , if u2 = u2 (η0 , η1 ) 2
and u2 = u2 (:(2) (η0 , η1 )), then PV u2 = PV u2 and PV ⊥ u2 −PV ⊥ u2 < γ 2 u0 −u0 . Continued ad infinitum, this process defines a map : : 9 := K 2r × Kr(1− 1 γ ) × Kr(1− 1 γ 2 ) × · · · → K N 2
2
with the desired properties. Clearly, ν(9) = ;i≥0 (1 − 21 γ i )D > 0, where D =dimV . Proof of Theorem A(1). Recall that if µ is an ergodic invariant measure for X , then by the Birkhoff Ergodic Theorem, n1 n−1 δui (η) → µ for µ-a.e. u0 and ν N -a.e. η = 0 (η0 , η1 , · · · ). This together with Lemma 3.2 implies that if µ, and µ are ergodic measures and there exist u0 ∈ supp(µ) and u0 ∈ supp(µ ) with ||u0 − u0 < δ, then µ = µ . Since all invariant measures of X are supported on the compact set A (Lemma 3.1), it follows that there cannot be more than a finite number of them. Proof of the uniqueness of invariant measure part of Theorem B. From the last paragraph, we know that all the ergodic components of µ are pairwise ≥ δ apart in distance. Thus condition (C) with ε0 = δ and R = R0 guarantees that there is at most one ergodic component. We remark that the uniqueness of invariant measure results in Theorem 1 and Theorem 3(c) follow immediately from the preceding discussion once the abstract hypotheses (P1)–(P4) and (C) are checked for these equations. The next lemma is used only to prove the general result in Theorem A(2); it is not needed for the applications in Theorems 1–3. (Both the Navier–Stokes and Ginzberg– Landau equations satisfy much stronger conditions, making this argument unnecessary.) Let B(u, ε) denote the ball of radius ε centered at u, and let P n (·|u) denote the n-step transition probability given u. In the language introduced earlier, if 0 = δu , then P n (·|u) = n (·).
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Lemma 3.3. Let µ be an invariant measure with the property that (X n , µ) is ergodic for all n ≥ 1. We fix B = B(u, ˜ ε˜ ), where u˜ ∈ supp µ and ε˜ > 0. Then there exist N0 ∈ Z+ N 0 and α0 > 0 such that P (B|u) ≥ α0 for every u ∈ supp µ. Proof. Pick arbitrary u0 ∈ suppµ. Until nearly the end of the proof, the discussion pertains to this one point. Consider the “restricted distribution" ˆ n defined by ˆ n (G) = ν n {(η0 , · · · , ηn−1 ) : ηi ∈ Kr−Mδγ i ∀i < n and un (η0 , · · · , ηn−1 ) ∈ G}, where δ is as in Lemma 3.2, and let Wn denote the support of ˆ n . Claim 1. d(u0 , ∪n>0 Wn ) = 0. −1 ˆ converges weakly to a ˆ Proof. By compactness, a subsequence of n1 n−1 i i=0 ( i (A)) probability measure µ˜ on A (where A is as in Lemma 3.1). Since the restrictions on ηi become milder and milder as i → ∞, µ˜ is an invariant measure for X . By construction, all the ˆ i are supported on supp µ, so we must have µ˜ = µ, for we know from Theorem A(1) that all the other ergodic invariant measures have their supports bounded away from supp µ. Let N = N (u0 ) be such that d(u0 , WN ) < ε, where ε < δ is a small positive number to be determined. Claim 2. For all k ≥ 0 and u ∈ WkN , ∃u ∈ W(k+1)N such that u − u < γ kN ε. Proof. The claim is true for k = 0 by choice of N . We prove it for k = 1: Let u0 ∈ WN be such that u0 − u0 < ε, and fix an arbitrary u ∈ WN . By definition, there exist ηi ∈ Kr−Mδγ i such that u = uN (η0 , · · · , ηN−1 ). We wish to use the proximity of u0 to u0 and the Matching Lemma to produce (η0 , · · · , ηN −1 ) with the property that ) ∈ W2N and uN − uN < εγ N . To obtain the first property, it uN (η0 , · · · , ηN−1 is necessary to have ηi ∈ Kr−Mδγ i+N for all i < N. We proceed as follows: since u0 − u0 < ε and η0 ∈ Kr−Mδ , ∃η0 ∈ Kr−Mδ+Mε such that u1 (η0 ) − u1 (η0 ) < εγ ; similarly ∃η1 ∈ Kr−Mδγ +Mεγ such that u2 (η0 , η1 ) − u1 (η0 , η1 ) < εγ 2 , and so on. (See the proof of Lemma 3.2.) Thus ηi ∈ Kr−Mγ i (δ−ε) , and assuming ε is sufficiently small that δγ N < (δ − ε), we have ηi ∈ Kr−Mδγ i+N . To prove the assertion for k = 2, we pick an arbitrary u ∈ W2N , which, by definition, is equal to vN from some v0 ∈ WN . Since we have shown that there exists v0 ∈ W2N with v0 − v0 < γ N ε, it suffices to ∈W 2N ε. repeat the argument above to obtain vN 3N with vN − vN < γ Claim 3. There exists k1 = k1 (u0 ) s.t. for k ≥ k1 , P kN (B|u) ≥ ˆ kN (B(u, ˜ 2ε˜ )) > 0 for all u ∈ H with u − u0 < δ. Proof. Let N (W, ε) denote the ε-neighborhood of W ⊂ H . If follows from Claim 2 that if NkN := N (WkN , 2ε ki=0 γ iN ), then NkN ⊂ N(k+1)N for all k. Moreover, the ergodicity of (X N , µ) together with an observation similar to that in Claim 1 shows that ˜ 4ε˜ ) = ∅ for large enough k. If the closure of ∪k NkN contains suppµ. Thus NkN ∩ B(u, ∞ iN ˜ 2ε˜ )) > 0. Now for u with u − u0 < δ, the entire 2ε i=1 γ < 4ε˜ , then ˆ kN (B(u, restricted distribution ˆ n starting from u0 can be coupled to a part of the (unrestricted) ˜ 2ε˜ )). distribution starting from u. Thus for sufficiently large n, P n (B|u) ≥ ˆ n (B(u,
470
N. Masmoudi, L.-S. Young (1)
(n)
To finish, we cover supp µ with a finite number of δ-balls centered at u0 , . . . , u0 , (i) (i) and choose N0 = kˆ1 Nˆ , where kˆ1 = maxi k1 (u0 ) and Nˆ = ;i N (u0 ). The lemma is ε ˜ ˜ 2 )), where ˆ N0 is the restricted distribution starting proved with α0 = mini ˆ N0 (B(u, (i) from u0 . From Lemma 3.2, we see that associated with each pair of points (u0 , u0 ) with u0 − u0 < δ, there is a cascade of matchings between un and un , leading to the definition of a measure-preserving map : : 9 := K 2r × Kr(1− 1 γ ) × Kr(1− 1 γ 2 ) × · · · → K N 2
2
with the property that for η ∈ 9, ui (η) − ui (:(η)) ≤ γ i u0 − u0
for all i ≤ n.
The main goal in the next proof is, in a sense, to extend : to all of K N by attempting repeatedly to match the orbits that have not yet been matched. Proof of Theorem A(2). We consider for simplicity the case N0 = 1. Let u0 , u0 ∈ supp µ, and let n and n denote the distributions of un and un respectively. We seek to define a measure-preserving map : : K N → K N and to estimate the difference between n and n by In := f d n − f d n ≤ |f (un (η)) − f (un (:(η)))| dν N (η) . Let B be a ball of diameter δ centered at some point in suppµ. By Lemma 3.3, P (B|u0 ) ≥ α0 , and P (B|u0 ) ≥ α0 . Matching u1 ∈ B to u1 ∈ B, we define a measurepreserving map :(1) : 9˜ 1 → K for some 9˜ 1 ⊂ K with |9˜ 1 | = α0 . This extends, by the Matching Lemma, to a measure-preserving map : : 91 = 9˜ 1 × 9 → K N . The map :|91 represents the cascade of future couplings initiated by :(1) . Suppose now that : has been defined on ∪k≤n 9k , where 9k is the set of η matched at step k. More precisely, 91 , 92 , · · · , 9n are disjoint subsets of K N , and each 9k is of the form 9k = 9˜ k × 9 for some 9˜ k ⊂ K k ; the matching of uk and uk in B that takes place at step k defines a map :(k) : 9˜ k → K k , while the cascade of future matchings initiated by :(k) results in the definition of : : 9˜ k × 9 → K N . We now explain how to ˜ n = K n \ ∪k≤n 9 (n) , where 9 (n) = 9˜ k × 9 (n−k−1) is the first n-factor define 9n+1 . Let G k k ˜ n ; the in 9k . Consider the restricted distribution ˜ n+1 defined by (η0 , · · · , ηn−1 ) ∈ G corresponding distribution ˜ n+1 is defined similarly. By Lemma 3.3, an α0 -fraction of these two distributions can be matched, defining an immediate matching :(n+1) : ˜ n × K and |9˜ n+1 | = α0 |G ˜ n |. Future couplings that result 9˜ n+1 → K n+1 with 9˜ n+1 ⊂ G (n+1) N ˜ define : : 9n+1 → K with 9n+1 = 9n+1 × 9. from : ˜ n ) decreases exponentially. This requires a little argument, for even We claim that ν n (G though at each step a fraction of α0 of what is left is matched, our matchings are “leaky”, (n) meaning not every orbit defined by a sequence in 9k can be matched to something ˜ n ), we write K N \∪k≤n 9k as the disjoint reasonable at the (n+1st ) step. To estimate ν n (G N ˜ union Gn ∪ Hn , where Gn = Gn × K . The dynamics of (Gn , Hn ) → (Gn+1 , Hn+1 ) are as follows: An α0 -fraction of Gn leaves Gn at the next step; of this part, a fraction of
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;i≥0 (1 − 21 γ i )D (recall that D is the dimension of V ) goes into 9n+1 (see Lemma 3.2) while the rest goes into Hn+1 . At the same time, a fraction of Hn returns to Gn+1 . We claim that this fraction is bounded away from zero for all n. To see this, consider one 9k (n) (n) (n+1) at a time, and observe (from the definition of 9k ) that |(9k × K) \ 9k | ∼ const n−k ˜ |9k |γ . Combinatorial Lemma. Let a0 , b0 > 0, and suppose that an and bn satisfy recursively an+1 ≥ (1 − α0 )an + α1 bn
and
bn+1 ≤ (1 − α1 )bn + α0 an
for some 0 < α0 , α1 < 1. Then there exists c > 0 such that
an bn
> c for all n.
The proof of this purely combinatorial lemma is left as an exercise. We deduce from ˜ n ) ≤ Cβ n for some C > 0 and β < 1. it that inf n |Gn |/|Hn | > 0, which implies ν n (G n This in turn implies that |9n+1 | ≤ Cβ . Proceeding to the final count, we let f : supp µ → R be such that |f | < C1 and |f (u) − f (v)| < C1 u − vσ . Then In ≤
˜n G
|f (un (η0 , . . . , ηn−1 ))|dν n + +
k≤n
(n)
9k
(n) K n −:(n) (∪k≤n 9k )
|f (un (η0 , . . . , ηn−1 ))|dν n
|f (u( η0 , . . . , ηn−1 )) − f (un (:(n) (η0 , . . . , ηn−1 )))|dν n
≤ 2C1 · Cβ n +
(6)
Cβ k−1 · C1 (δγ n−k )σ
k≤n
≤ const n · [max(β, γ σ )]n ≤ const · τ n . Since these estimates are uniform for all pairs u0 , u0 , we obtain by integrating over u0 that fd n − f dµ ≤ const · τ n . Proof of Theorem B. We will prove, in the next paragraph, that assertion (2) in Theorem B holds for any invariant measure µ of X . From this (1) follows immediately: since (X , µ) is exponentially mixing, it is ergodic; and since µ is chosen arbitrarily, it must be the unique invariant measure. To prove the claim above, we pick arbitrary u0 ∈ H , u0 ∈ A, and compare their distributions n and n as we did in the proof of Theorem A(2). First, by waiting a suitable period, we may assume that n is supported in B(R0 ) (where R0 is as in (P2)). By condition (C) with ε0 = δ, where δ is as in Lemma 3.2, there is a set of controls of length N0 and having ν N0 -measure α0 for some α0 > 0 that steer the entire ball B(R0 ) into a set of diameter < δ. The estimate for | f d n − f d ˆ n | now proceeds as in Theorem A(2), with the use of these special controls taking the place of Lemma 3.3 to guarantee that an α0 -fraction of what is left is matched every N0 steps. Averaging u0 with respect to µ, we obtain the desired result.
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3.3. Applications to PDEs: Proofs of Theorems 1 and 3. In this subsection, we prove the theorems related to PDEs stated in Sect. 2.1. Proof of Theorem 1. We will prove that the abstract hypotheses (P1)–(P4) and (C) hold for the incompressible Navier–Stokes equation in L2 for the type of noise specified. Let S(u0 ) = u(t = 1) where u is the solution of the Navier–Stokes equation with initial data u0 , and let uk = S(uk−1 ) + ηk . Most of the computations below are classically known (see for instance [2, 14]); we include them for completeness. We start by recalling a few properties of the Navier–Stokes equation in the 2-D torus. First, the following energy estimate holds for all t > 0: 1 ||u(t)||2L2 + ν 2 Since
t 0
||∇u||2L2 =
1 ||u0 ||2L2 . 2
(7)
u = 0, we have the Poincaré inequality ||∇u||L2 ≥ ||u||L2 .
(8)
||S(u)||L2 ≤ e−ν ||u||L2 ;
(9)
From (7) and (8), it follows that
thus (P2) is satisfied by taking R0 (a) > 1−e1 −ν a. On the other hand, for any two solutions u and v with initial conditions u0 and v0 , we have 1 2 2 ∂t ||u − v||L2 + ν||∇(u − v)||L2 ≤ | (u − v).∇v(u − v)| 2 ≤ C||∇v||L2 ||u − v||L2 ||u − v||H 1 (10) ν C ≤ ||u − v||2H 1 + ||∇v||2L2 ||u − v||2L2 . 2 ν (Hölder and Sobolev inequalities are used to get the second line, and the Cauchy– Schwartz inequality is used to get the third.) Then, applying a Gronwall lemma, we get ||S(u0 ) − S(v0 )||2L2 + ν
1 0
||(u − v)(s)||2H 1 ds ≤ CR ||u0 − v0 ||2L2 .
(11)
Here and below, CR denotes a generic constant depending only on R, an upper bound on the L2 norm of u0 , and on the viscosity ν. (P1)(b) follows from (11). To prove that (P3) holds, we use (11), (7) and a Chebychev inequality to deduce the existence of a time s, 0 < s < 1, such that ν||(u − v)(s)||2H 1 ≤ 4CR ||u0 − v0 ||2L2 , ν||u(s)||2H 1 < 2R 2 and ν||v(s)||2H 1 < 2R 2 . Combining these estimates with energy estimates in H 1 for t > s, namely, t 1 1 ||∇u(t)||2L2 + ν || u||2L2 = ||∇u(s)||2L2 , 2 2 s t 1 1 2 ||∇v(t)||L2 + ν || v||2L2 = ||∇v(s)||2L2 , 2 2 s
(12) (13)
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1 ∂t ||u − v||2H 1 + ν||u − v||2H 2 ≤ ||u − v||H 2 ||u − v||H 1 ||u||H 2 + ||v||H 2 (14) 2
1 ν ≤ ||u − v||2H 2 + ||u||2H 2 + ||v||2H 2 ||u − v||2H 1 , 4 ν integrating (14) between s and 1 and using again a Gronwall lemma, we deduce easily that ||S(u0 ) − S(v0 )||H 1 ≤ CR ||u0 − v0 ||L2 .
(15)
For any γ > 0 and R > 0, we may take N large enough that if VN :=span{e1 , e2 , ..., eN }, then CR ||u||L2 ≤ γ ||u||H 1 ∀u ∈ VN⊥ . This together with (15) proves (P3). Finally, property (C) is satisfied by taking ηi = 0 for 1 ≤ i ≤ n0 , where n0 is large enough that Re−νn0 ≤ ε0 (see (9)). The product structure of the noise ν 1 in property (P4)(b) holds because ξj k in (2) are independent; the assumption on PV ∗ ν holds because bj = 0 for 1 ≤ j ≤ N , where N is as in (P3) and the law for ξj k has density ρj . Proof of Theorem 1’. We now prove (P1)–(P4) and (C) in H s . To prove (P1)(b), we use the energy estimates 1 ∂t ||u||2H s + ν||u||2H s+1 ≤ C||u||H s ||u||H s+1 ||u||H 1 2 ν C ≤ ||u||2H s+1 + ||u||2H 1 ||u||2H s , 2 ν 1 ∂t ||u − v||2H s + ν||u − v||2H s+1 2 ≤ C||u − v||H s ||u − v||H s+1 (||u||H s+1 + ||v||H s+1 ) ν C ≤ ||u − v||2H s+1 + (||u||2H s+1 + ||v||2H s+1 )||u − v||2H s , 2 ν
(16)
(17)
and Gronwall’s lemma between times 0 and 1. To prove (P3), we proceed as in the case of L2 , showing the existence of a time τ , 0 < τ < 1, such that ||(u−v)(τ )||H s+1 ≤ 4CR ||u0 −v0 ||H s and ||u(τ )||H s+1 , ||v(τ )||H s+1 ≤ 4CR , where ||u0 ||H s , ||v0 ||H s < R. Then using (16) and (17) with s replaced by s + 1 and integrating between τ and 1, we deduce that ||S(u0 ) − S(v0 )||H s+1 ≤ CR ||u0 − v0 ||H s ,
(18)
from which we obtain (P3). To prove (P2), we make use of the regularizing effect of the Navier–Stokes equation in 2-D ||S(u0 )||H s ≤ Cs (||u||L2 ),
(19)
where Cs is a function depending only on s (see [14]). Since BH s (a) ⊂ BL2 (a), we know from (P3) for L2 that if u0 ∈ BL2 (R), we have un ∈ BL2 (R0 ) ∀n ≥ some N0 . Taking Rs = Cs (R0 ) + a, we get that un ∈ BH s (Rs ) ∀n ≥ N0 . To prove (C), we argue as in L2 , taking ηi = 0, 1 ≤ i ≤ n0 , for large enough n0 and appealing to the fact that Cs (r) → 0 as r → 0. 1 We hope our dual use of the symbol ν as viscosity and as noise does not lead to confusion.
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We remark that (P2) and (C) above can be proved directly without going through L2 . Next we move on to the real Ginzburg–Landau equation. Proof of Theorem 3. For simplicity, we take ν = 1. (a) We need to prove that there exist two disjoint stable sets A1 and A−1 , stable in the sense that ∀u ∈ A±1 , S(u) + η ∈ A±1 ∀η ∈ K. Let A1 = {u ∈ H, ||u − 1||L2 ≤ β},
(20)
where β is a constant to be determined. We recall for each φ ∈ R the energy estimate 1 (21) ∂t ||u − φ||2L2 + ||∇(u − φ)||2L2 + u(u − 1)(u + 1)(u − φ) dx = 0. 2 T Substituting φ = 1 in (21), we get 1 ∂t ||u − 1||2L2 + ||∇(u − 1)||2L2 ≤ − 2
T
u(u + 1)(u − 1)2 dx.
(22)
Now for any φ with 0 < φ < 1, we have u(u + 1)(u − 1)2 ≥ φ(φ + 1)(u − 1)2 u(u + 1)(u − 1)2 ≥ −1
if
u≥φ ∀u.
or
u ≤ −1 − φ,
(23)
Hence u(u + 1)(u − 1)2 dx ≥ (1{u≥φ} + 1{u≤−1−φ} )φ(φ + 1)(u − 1)2 − meas{u ≤ φ}. T
T
Since the first term on the right side is ≥ φ(φ + 1)||u − 1||2L2 −
T
1{−1−φ
we see for φ ≤ 1/4 that φ(φ + 1)(φ + 2)2 ≤ 3, so that u(u + 1)(u − 1)2 dx ≥ φ(φ + 1)||u − 1||2L2 − 4 meas{u ≤ φ} . T
(24)
Assuming β < 1/4 so that A1 ∩ {u > 3/4} = ∅, the Poincaré inequality yields for ψ < 3/4 that ||(u − ψ)1{u<ψ} ||L2 ≤ C meas {u ≤ ψ}||∇(u1{u<ψ} )||L2 ≤ C meas {u ≤ ψ}||∇u||L2 ,
(25)
the factor meas{u ≤ ψ} coming from the scale invariance. On the other hand, for φ < ψ, we have meas {u ≤ φ} ≤
1 ||(u − ψ)1{u<ψ} ||2L2 . (φ − ψ)2
(26)
β2 . (1 − ψ)2
(27)
For u ∈ A1 , we also have meas {u ≤ ψ} ≤
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Putting together (25), (27) and (26), and choosing for instance φ = 1/4 and ψ = 1/2, we have 1 ||(u − ψ)1{u<ψ} ||2L2 (1 − ψ)4 β4 1 ≥ 4 meas {u ≤ φ}(φ − ψ)2 (1 − ψ)4 . β
||∇u||2L2 ≥
(28) (29)
Taking β so that β 4 ≤ 18 (φ − ψ)2 (1 − ψ)4 (e.g. β ≤ 1/8), (22) and (24) yield 1 1 ∂t ||u − 1||2L2 + ||∇u||2L2 ≤ −φ(φ + 1)||u − 1||2L2 2 2
(30)
as long as u ∈ A1 . Hence ||S(u) − 1||L2 ≤ e−φ(φ+1) ||u − 1||L2 .
(31)
Finally, taking a small enough, namely a ≤ β(1 − e−φ(φ+1) ),
(32)
we see that A1 is stable under the Markov chain. Applying Lemma 3.1 ((P1) and (P4)(a) are easily satisfied and (P2) is replaced by the stability of A1 ), we deduce that there is at least one invariant measure supported in A1 . A symmetric argument produces an invariant measure in A−1 . Clearly these two measures are distinct. (b) We need to verify (P3) and (P4)(b); the arguments are similar to those in the proof of Theorem 1. The assertion then follows from Theorem A(1). (c) We explain how to verify condition (C). First, we use the regularizing effect of the Laplacian to deduce that for u0 with u0 L2 < R, ||u1 ||L∞ ≤ ||u1 ||H 1 ≤ CR . Then, using the maximum principle for parabolic equations, we get ∂t g ≤ g − g 3
where
g(t) = max |u| . x∈T
(33)
Choosing n0 large enough that −b0 < g(n0 ) < b0 and taking η0 = η1 = · · · = ηn0 = 0, we obtain un0 +1 ∞ < b0 . Let ηn0 +1 = ηn0 +2 = b0 e0 . Then un0 +1 = S(un0 )+ηn0 +1 > 0, and so S(un0 +1 ) > 0. Thus 1 < un0 +2 < C = 3b0 . Taking ηn0 +3 = ηn0 +4 = · · · ηn0 +n1 = 0 for large enough n1 , we can arrange to have ||un0 +n1 − 1||L2 as small as we wish. Notice that in the argument above, we took b0 > 1 to make sure that after arranging for un0 +1 ∞ to be ≈ 1, we obtain un0 +2 > 1 after two kicks in a suitable direction. It is clear that with more kicks the condition b0 > 1 can be relaxed.
4. Dynamics with Negative Lyapunov Exponents 4.1. Formulation of abstract results. We consider a semi-group St on H and a Markov chain X defined by (I) or (II) in the beginning of Sect. 3.1. In order for Lyapunov exponents to make sense, we need to impose differentiability assumptions.
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(P1’) (a) S(B(R)) is compact ∀R > 0; (b) S is C 1+Lip , meaning for every u ∈ H , there exists a bounded linear operator Lu : H → H with the property for all h ∈ H , 1 {S(u + εh) − S(u) − Lu (εh)} = 0 ε→0 ε lim
(34)
and ∀R > 0, ∃MR such that ∀u, v ∈ B(R), Lu − Lv ≤ MR u − v. Since Lemma 3.1 clearly holds with (P1) replaced by (P1’), we let A be as in Sect. 3. Proposition 4.1. Assume (P1’), (P2) and (P4)(a), and let µ be an invariant measure for X . Then there is a measurable function λ1 on H with −∞ ≤ λ1 < ∞ such that for µ-a.e. u0 and ν N -a.e. η = (η0 , η1 , η2 , . . . ), lim
n→∞
1 log Lun−1 ◦ · · · ◦ Lu1 ◦ Lu0 = λ1 (u0 ) . n
Moreover, λ1 is constant µ-a.e. if (X , µ) is ergodic. This proposition follows from a direct application of the Subadditive Ergodic Theorem [6] together with the boundedness of Lu on A (see also Lemma 4.1 below). We will refer to the function or, in the ergodic case, number λ1 as the top Lyapunov exponent of (X , µ). This section is concerned with the dynamics of X when λ1 < 0. We begin by stating a result, namely Theorem C, which gives a general description of the dynamics when λ1 < 0. This result, however, is not needed for our application to PDEs. The proof of Theorem 2 uses only Theorem D, which is independent of Theorem C. Let µ be an invariant measure of X . Theorem C concerns the conditional measures of µ given the past. That is to say, we view X as starting from time −∞, i.e. consider . . . , u−2 , u−1 , u0 , u1 , u2 , . . . defined by un+1 = Sun + ηn ∀n ∈ Z where . . . , η−2 , η−1 , η0 , η1 , η2 , . . . are ν-i.i.d. Then for ν Z -a.e. η = (. . . , η−1 , η0 , η1 , . . . ), the conditional probability of µ given η− := (. . . , η−2 , η−1 ) is well defined. We denote it by µη . Theorem C (Random sinks). Assume (P1’), (P2) and (P4)(a), and let µ be an ergodic invariant measure with λ1 < 0. Then there exists k0 ∈ Z+ such that for ν Z -a.e. η ∈ K Z , µη is supported on exactly k0 points of equal mass. This result is well known for stochastic flows in finite dimensions (see [11]). In the next theorem we impose a condition slightly stronger than (C) in Sect. 3.1 to obtain the type of uniqueness result needed for Theorem 2. (C’) There exists uˆ 0 ∈ H such that for all ε0 > 0 and R > 0, there is a finite sequence of controls ηˆ 0 , · · · ηˆ n such that for all u0 ∈ B(R), if uk+1 = Suk + ηˆ k and uˆ k+1 = S uˆ k + ηˆ k for all k < n, then un − uˆ n < ε0 . For u ∈ H , we define the accessibility set A(u) as follows: let A0 (u) = {u}, An (u) = S(An−1 (u)) + K for n > 0, and A(u) = ∪n≥0 An (u).
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Theorem D (Asymptotic uniqueness of solutions independent of initial condition). Assume (P1’), (P2), (P4)(a) and (C’). Suppose there is an ergodic invariant measure µ supported on A(uˆ 0 ) for which λ1 < 0. Then µ is the only invariant measure X has, and the following holds for ν N -a.e. η = (η0 , η1 , · · · ): ∀u0 , u0 ∈ H,
un (η) − un (η) ≤ Ceλn
where λ is any number > λ1 and C =
∀n > 0,
C(u0 , u0 , λ).
Roughly speaking, Theorem D allows us to conclude that all the orbits are eventually “the same” once we know that the linearized flows along some orbits are contractive. This passage from a local to a global phenomenon is made possible by condition (C’), which in the abstract is quite special but is satisfied by a number of standard parabolic PDEs. 4.2. Proofs of abstract results (Theorems C and D). Let A be the compact set in Lemma 3.1, and let K denote the support of ν as before. We consider the dynamical system F : K N × A → K N × A defined by F (η, u) = (σ η, S(u) + η0 ), where η = (η0 , η1 , η2 , . . . ) and σ is the shift σ (η0 , η1 , η2 , . . . ) = (η1 , η2 , . . . ). The following is straightforward.
operator,
i.e.
Lemma 4.1. Let µ be an invariant measure of X in the sense of Definition 3.1. Then F preserves ν N × µ, and (F, ν N × µ) is ergodic if and only if (X , µ) is ergodic in the sense of Definition 3.2. Our next lemma relates the top Lyapunov exponent of a system, which describes the average infinitesimal behavior along its typical orbits, to the local behavior in neighborhoods of these orbits. A version applicable to our setting is contained in [13]. Let B(u, α) = {v ∈ H, v − u < α}. Proposition 4.2. [13] Let µ be an invariant measure, and assume that λ1 < 0 µ-a.e. Then given ε > 0, there exist measurable functions α, γ : K N × A → (0, ∞) and a measurable set G ⊂ K N × A with (ν N × µ)(G) = 1 such that for all (η, u0 ) ∈ G and v0 ∈ B(u0 , α(η, u0 )), vn (η) − un (η) < γ (η, u0 ) e(λ1 +ε)n ∀n ≥ 0. We first prove Theorem D, from which Theorem 2 is derived. Proof of Theorem D. From (P2), it follows that we need only to consider initial conditions in B(R0 ). Fix ε > 0 and let α and G be as in Proposition 4.2 for the dynamical system (F, ν N × µ). We make the following choices: 99 (1) Let α0 > 0 be a number small enough that (ν N × µ){α > 2α0 } > 100 . Covering 1 the compact set A(uˆ 0 ) with a finite number of 2 α0 -balls, we see that there exists u˜ 0 ∈ A(uˆ 0 ) such that
91 := {η ∈ K N : B(u˜ 0 , α0 ) ⊂ B(u, α(η, u)) for some u with (η, u) ∈ G} has positive ν N -measure.
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(2) Since u˜ 0 ∈ A(uˆ 0 ), there is a sequence of controls (η˜ 0 , . . . , η˜ k−1 ) that puts uˆ 0 in B(u˜ 0 , 21 α0 ). Choose δ > 0 and 92 ⊂ K k with ν k (92 ) > 0 such that if u0 ∈ B(uˆ 0 , δ) and (η0 , . . . , ηk−1 ) ∈ 92 , then uk (η0 , . . . , ηk−1 ) ∈ B(u˜ 0 , α0 ). (3) Condition (C’) guarantees that there exists a sequence of controls (ηˆ 0 , . . . , ηˆ j −1 ) that puts the entire ball B(R0 ) inside B(uˆ 0 , 21 δ). Choose 93 ⊂ K j with ν j (93 ) > 0 such that every sequence (η0 , . . . , ηj −1 ) ∈ 93 puts B(R0 ) inside B(uˆ 0 , δ). Let 9 ⊂ K N be the set defined by {(η0 , . . . , ηj −1 ) ∈ 93 ; (ηj , . . . , ηj +k−1 ) ∈ 92 ; (ηj +k , ηj +k+1 , . . . ) ∈ 91 } . Clearly, ν N (9) > 0. The following holds for ν N -a.e. η : Fix η, and let Bn denote the nth image of B(R0 ) for this sequence of kicks. By the ergodicity of (σ, ν N ), there exists N such that σ N η ∈ 9. Choosing N ≥ N0 (R0 ), we have, by (P2), that BN ⊂ B(R0 ). The choice in (3) then guarantees that BN+j ⊂ B(uˆ 0 , δ), and the choice in (2) guarantees that BN+j +k ⊂ B(u˜ 0 , α0 ). By (1), BN+j +k ⊂ B(u, α(u, σ N+j +k η)) for some u with (σ N+j +k η, u) ∈ G. Proposition 4.2 then says that when subjected to the sequence of kicks defined by σ N+j +k η, all orbits with initial conditions in BN+j +k converge exponentially to each other as n → ∞. Hence this property holds for all orbits starting from B(R0 ) when subjected to η. Theorem D is proved. Proceeding to Theorem C, the measures µη defined in Sect. 4.1 are called the sample or empirical measures of µ. They have the interpretation of describing what one sees at time 0 given that the system has experienced the sequence of kicks η− = (· · · , η−2 , η−1 ). The characterization of µη in the next lemma is useful. We introduce the following notation: Let Sη0 : H → H be the map defined by Sη0 (u) = Su + η0 ; for a measure µ on H , Sη0 ∗ µ is the measure defined by (Sη0 ∗ µ)(E) = µ(Sη−1 E). 0 Lemma 4.2. Let µ be an invariant measure for X . Then for ν Z -a.e. η (. . . , η−2 , η−1 , η0 , . . . ), (Sη−1 Sη−2 · · · Sη−n )∗ µ converges weakly to µη . Proof. Fix a continuous function ϕ : A → R, and define ϕ (n) : K Z → R by ϕ (n) (η) = ϕ d((Sη−1 Sη−2 · · · Sη−n )∗ µ) = ϕ(Sη−1 Sη−2 · · · Sη−n (u))dµ(u).
=
(35)
−n −n -measurable, where B−1 is the σ -algebra on K Z generated by coordiThen ϕ (n) is B−1 −n+1 ) = ϕ (n−1) . nates η−1 , · · · , η−n . Since Sη−n ∗ µ dν(η−n ) = µ, we have E(ϕ (n) |B−1 (n) The martingale convergence theorem then tells us that ϕ convergence ν Z -a.e. to a −∞ function measurable on B−1 . It suffices to carry out the argument above for a countable dense set of continuous functions ϕ.
Lemma 4.3. Given δ > 0, ∃N = N (δ) ∈ Z+ such that for ν Z -a.e. η, there is a set Eη consisting of ≤ N points such that µη (Eη ) > (1 − δ). Proof. Let α and γ be the functions in Proposition 4.2 for the dynamical system (F, ν N × µ). Given δ > 0, we let α0 , γ0 > 0 be constants with the property that if G = {(η, u) : α(η, u) ≥ α0 , γ (η, u) ≤ γ0 }
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and
479
9 = {η ∈ K N : µ{u : (η, u) ∈ G} > 1 − δ},
then ν N (9) > 1 − δ. Consider η ∈ K Z such that (i) µη = lim(Sη−1 Sη−2 · · · Sη−n )∗ µ and (ii) (η−n , η−n+1 , . . . ) ∈ 9 for infinitely many n > 0. By Lemma 4.2 and the ergodicity of (σ, ν Z ), we deduce that the set of η satisfying (i) and (ii) has full measure. We will show that the property in the statement of the lemma holds for these η. Fix a cover {B1 , . . . , BN } of A by α20 -balls, and let η be as above. We consider n arbitrarily large with (η−n , η−n+1 , . . . ) ∈ 9. For each i, 1 ≤ i ≤ N , such that Bi ∩ {u ∈ H : ((η−n , η−n+1 , . . . ), u) ∈ G} = ∅, pick an arbitrary point u(i) in this set. Our choices of G and 9 ensure that µ(∪i B(u(i) , α0 )) > 1 − δ, and that the diameter of (Sη−1 Sη−2 · · · Sη−n )B(u(i) , α0 ) is ≤ γ0 α0 e(λ+ε)n . We have thus shown that a set of µη -measure > 1 − δ is contained in ≤ N balls each with diameter ≤ γ0 α0 e(λ+ε)n . The result follows by letting n → ∞. To prove Theorem C, we need to work with a version of (F, ν N × µ) that has a past. Let F˜ : K Z × A → K Z × A be such that F˜ : (η, u) ' → (σ η, Sη0 u), and let ν Z ∗ µ be the measure which projects onto ν Z in the first factor and has conditional probabilities µη on η-fibers. That ν Z ∗ µ is F˜ -invariant follows immediately from Lemma 4.2. It is also easy to see that (F˜ , ν Z ∗ µ) is ergodic if and only if (F, ν N × µ) is. Proof of Theorem C. It follows from Lemma 4.3 that for ν Z -a.e. η, µη is atomic, with possibly a countable number of atoms. We now argue that there exists k0 ∈ Z+ such that for a.e. η, µη has exactly k0 atoms of equal mass. Let h(η) = supu∈H µη {u}. To see that h is a measurable function on K Z , let P (n) , n = 1, 2, · · · , be an increasing sequence of finite measurable partitions of A such that diamP (n) → 0 as n → ∞. Then for each P ∈ P (n) , η ' → µη (P ) is a measurable function, as are hn := maxP ∈P (n) µη (P ) and h := limn hn . Observe that h(σ η) ≥ h(η), with > being possible in principle since Sη0 is not necessarily one-to-one. However, the measurability of h together with the ergodicity of (σ, ν Z ) implies that h is constant a.e. Let us call this value h0 . From the last lemma we know that h0 > 0. To finish, we let X = {(η, u) ∈ K Z × A : µη {u} = h0 }. Then X is a measurable set, (ν Z ∗ µ)(X) > 0 and F˜ −1 X ⊃ X. This together with the ergodicity of (F˜ , ν Z ∗ µ) implies that (ν Z ∗ µ)(X) = 1, which is what we want.
4.3. Application to PDEs: Proof of Theorem 2. Let St be the semi-group generated by the (unforced) Navier–Stokes system, and let S = S1 . Lemma 4.4. S is C 1+Lip in H 2 (R2 ). Proof. It is easy to see that Lu is defined by Lu w = ψ(1), where ψ is the solution of the linear problem ∂t ψ + U.∇ψ + ψ.∇U − ν ψ = −∇p, (36) ψ(t = 0) = w , divψ = 0,
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where U denotes the solution of the Navier–Stokes system with initial data u. That Lu is linear, continuous and goes from H 2 to H 2 is obvious. To prove that (34) holds, let U and V be the solutions of the Navier–Stokes system with initial data u and u + Lw respectively. Then y = V − U − Lψ satisfies ∂t y + (U + Lψ).∇y + y.∇V + L 2 ψ.∇ψ − y = −∇p (37) y(t = 0) = 0 , div(y) = 0. By a simple computation, we get that ||y(t = 1)||H 2 ≤ C(1 + ||w||2H 2 )L 2 , where here and below C denotes a constant depending only on the H 2 norm of u. To prove that Lu is Lipschitz, i.e., ||(Lu − Lv )w||H 2 ≤ C||u − v||H 2 ||w||H 2 ,
(38)
we define Lv w = φ(1), where φ solves an equation analogous to (36) with V in the place of U , V being the solution with initial condition v. The desired estimate ||(ψ − φ)(t = 1)||H 2 is obtained by subtracting this equation from (36). Remark 4.1. We observe here that the top Lyapunov exponent is negative if the noise is sufficiently small. We will show, in fact, that given any positive viscosity ν, if a (see Sect. 2.1 for definition) is small enough, then S : H 2 → H 2 is a contraction on the ball ν of radius 2C . Rewriting Eqs. (16) and (17) with s = 2, we have ∂t ||u||2H 2 + ν||u||2H 3 ≤
∂t ||u − v||2H 2 + ν||u − v||2H 3 ≤
C2 ||u||4H s , ν
(39)
C2 (||u||2H 3 + ||v||2H 3 )||u − v||2H 2 . ν
(40)
ν ν and a noise with a ≤ 2C (1 − e−ν/4 ), it follows from (39) For u0 with ||u0 ||H 2 ≤ 2C and a Gronwall lemma that ν 2 ||S(u0 )||2H 2 ≤ e−ν/2 , 2C
from which we obtain ||u1 ||H 2 ≤
ν 2C .
1
ν 0
Moreover, from (39), we have that ||u||2H 3 ≤
ν3 , 16C 2
so that if v is another solution of the Navier–Stokes system with ||v0 ||H 2 ≤ (40) gives
ν 2C ,
then
ν
||u − v||2H 2 ≤ ||u0 − v0 ||2H 2 e−ν+ 8 . Proof of Theorem 3. It suffices to check the hypotheses of Theorem D: (P1’) is proved in Lemma 4.4, and we explained in the proof of Theorem 1’ why (C’) holds with uˆ 0 = 0.
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References 1. Bricmont, J., Kupiainen, A., Lefevere, R.: Exponential mixing for the 2D Navier–Stokes dynamics, 2000 preprint 2. Constantin, P., Foias, C.: Navier–Stokes equations. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press, 1988, x+190 pp. 3. E, W., Mattingly, J., Sinai, Ya.G.: Gibbsian Dynamics and ergodicity for the stochastically forced 2D Navier–Stokes equation. 2000 preprint. 4. Eckmann, J.-P., Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. 2000 preprint 5. Flandoli, F., Maslowski, B.: Ergodicity of the 2D Navier–Stokes equations. NoDEA 1, 403–426 (1994) 6. Kingman, J.F.C.: Subadditive processes. Ecole d’été des Probabilités de Saint-Flour. Lecture Notes in Math. 539. Berlin–Heidelberg–New York: Springer, 1976 7. Kuksin, S., Shirikyan, A.: Stochastic dissipative PDEs and Gibbs measures. Commun. Math. Phys. 213 , no. 2, 291–330 (2000) 8. Kuksin, S., Shirikyan, A.: On dissipative systems perturbed by bounded random kick-forces. 2000 preprint 9. Kuksin, S., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDE’s. 2001 preprint 10. Kuksin, S., Piatnitski, A.L., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDEs. II. 2001 preprint 11. Le Jan, Y.: Equilibre statistique pour les produits de diffeomorphismes aleatoires independants. Ann. Inst. Henri Poincare (Probabilites et Statistiques) 23, 111–120 (1987) 12. Mattingly, J.: Ergodicity of 2D Navier–Stokes Equations with random forcing and large viscosity. Commun. Math. Phys. 206, 273–288 (1999) 13. Ruelle, D.: Characteristic exponents and invariant manifolds in Hilbert space. Ann. Math. 115, 243–290 (1982) 14. Temam, R.: Navier–Stokes equations and nonlinear functional analysis. Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics 66. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1995, xiv+141 pp. 15. Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Second edition.Applied Mathematical Sciences 68. New York: Springer-Verlag, 1997, xxii+648 pp. Communicated by P. Constantin
Commun. Math. Phys. 227, 483 – 514 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Random Wavelet Series Jean-Marie Aubry , Stéphane Jaffard Département de Mathématiques, Faculté des Sciences et Technologies, Université Paris XII, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail: [email protected]; [email protected] Received: 15 December 2000 / Accepted: 20 December 2001
Abstract: This paper concerns the study of functions which are known through the statistics of their wavelet coefficients. We first obtain sharp bounds on spectra of singularities and spectra of oscillating singularities, which are deduced from the sole knowledge of the wavelet histograms. Then we study a mathematical model which has been considered both in the contexts of turbulence and signal processing: random wavelet series, obtained by picking independently wavelet coefficients at each scale, following a given sequence of probability laws. The sample paths of the processes thus constructed are almost surely multifractal functions, and their spectrum of singularities and their spectrum of oscillating singularities are determined. The bounds obtained in the first part are optimal, since they become equalities in the case of random wavelet series. This allows to derive a new multifractal formalism which has a wider range of validity than those that were previously proposed in the context of fully developed turbulence. 1. Introduction The statistical study of fully developed turbulence started in 1941 with Kolmogorov [19]. Based on a dimensional analysis, he predicted that the scaling function τ (p) of the velocity field v, defined by |v(x + δx) − v(x)|p dx ∼ |δx|τ (p) , should be linear: τ (p) = p/3. Subsequent experiences in wind tunnels showed that τ (p) is actually a strictly concave function, which is believed to be universal and central to This work was performed while the author was at the Department of Mathematics, University of California at Davis. Also at the Institut Universitaire de France.
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the understanding of turbulence (see for instance [10]). This problem was addressed by Kolmogorov and Obukhov [20] and Mandelbrot [23], among others. Frisch and Parisi [9] proposed an explanation which has received the name multifractal: they interpret the nonlinearity of τ (p) as the signature of the presence of several kinds of Hölder singularities. Suppose that the Hölder exponent of v has value h on a set of (Hausdorff) dimension d(h). This function, called spectrum of singularities, is linked to τ (p) by the multifractal formalism, here stated in dimension 1 (although turbulence is a 3D phenomenon, its data are accessed only through the one-dimensional trace of the velocity field, measured in wind tunnels by the hot wire technique): d(h) = 1 + inf hp − τ (p). p
According to this formula, a linear τ (p) would indeed lead to a spectrum of singularities supported by only one value h0 . It should be noted that this relation is only based on a heuristic argument, and is not necessarily true for arbitrary functions, see [11, 12, 15] and our discussion in Sect. 3.3. A more precise information than τ (p) would be given by the distribution of the velocity increments at all scales δx. Then, the knowledge of these distributions could allow to test the validity of more sophisticated models of turbulence. This way was explored by Castaing et al. [5], who proposed continuous cascade models for the p.d.f. of the increments. To our knowledge, however, one has not yet constructed a process satisfying Castaing’s continuous equation. One problem met is that the family of p.d.f. of the increments of any given function at all scales are interdependent, so that it is by no means obvious to determine if a given (continuous) family of p.d.f. are indeed p.d.f. of increments. One way to eliminate this problem has been proposed by Arneodo et al. [2]: instead of modeling p.d.f. of increments at all scales, they propose to model p.d.f. of wavelet coefficients on an orthonormal wavelet basis. The wavelet coefficients Cj,k = f (t)ψ(2j t − k)dt can be interpreted as smoothed increments of f at the scale 2−j (because ψ is well localized and has a vanishing integral); therefore, it is reasonable to assume that, for a given j , the collections of Cj,k should follow Castaing’s statistics when δx = 2−j . The random cascades of Arneodo et al. have some drawbacks: their correlations between scales have the rigid structure of the dyadic tree; nevertheless they allow to explore basic hypotheses on turbulence statistics. Note also that, if it is possible to estimate sequences of histograms of wavelet coefficients at all dyadic scales, it is impossible to determine the correlations between all wavelet coefficients, see [25], especially since the statistics are not Gaussian. Therefore, any model that incorporates specific correlations is in practice impossible to validate on real-life data. We will adopt a different point of view: dropping all correlations between wavelet coefficients, but making no assumption on their distributions (except for a general regularity hypothesis). We then have a fairly general model that can be fitted to any statistics (note that correlations with limited range can be added, but would not change our qualitative conclusions). The advantage is that, under this hypothesis, we can completely derive the spectrum of singularities d(h), as well as other quantities related to the local oscillations of the functions considered. Of course, this general model can be fitted to Castaing’s model (this will be discussed in [4]). Our results also give new information on models that are already used as Bayesian a priori for signal and image processing [27, 29].
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Random wavelet series were first introduced and studied in [14] in a very simplified case, where the non-zero coefficients at a given scale take only one value. However, even in this extremely simple model, it turned out that the sample paths are multifractal. Our goal in this paper is to compute the properties that can be deduced from the knowledge of the wavelet coefficients p.d.f. at each scale. We will obtain upper bounds for d(h) based on histograms of the wavelet coefficients, which are sharper than the bounds obtained by the usual Fenchel–Legendre transform technique, and we will show that these upper bounds become equalities in the case of random wavelet series. We will actually go beyond this analysis; for, when the computation of this spectrum is involved, it often happens that the Hölder exponent is not precise enough, in the sense that it doesn’t take into account the local oscillations of the function. Indeed, a given Hölder exponent h at x0 allows for many different behaviors near x0 : for instance cusp-like singularities, such as |x − x0 |h or very oscillatory behaviors, such as
1 gh,β (x) := |x − x0 | sin |x − x0 |β h
(1)
for β > 0. The functions gh,β are the most simple examples of chirps at x0 . In signal analysis, this notion is expected to give a model for functions whose “instantaneous frequency” increases fast at some time (see [16]). The oscillating singularity exponent β measures how fast the instantaneous frequency of (1) diverges at x0 . (We will give a precise definition of β for an arbitrary function in Sect. 2.2). Furthermore, such local oscillations can make the multifractal formalism wrong, see [1]. The introduction at each point of both exponents h and β thus has two advantages: first, it gives much more complete information on the pointwise behavior; second, it leads to extensions of the multifractal formalism which have a wider range of validity, as shown in [13]. In Theorem 2, we will also determine the spectrum of oscillating singularities d(h, β) of random wavelet series (i.e. the Hausdorff dimension of the set E(h, β) of points where a given sample path has Hölder exponent h and oscillating singularity exponent β). It should be noticed at this point that different functions with the same histograms of wavelet coefficients at each scale can have completely different spectra of singularities, see [11]; in other words, it is not only the histograms of wavelet coefficients which are important in multifractal analysis, but also the positions of these coefficients. Therefore no formula that gives the spectrum from these histograms can be valid in all generality. But we may expect that some formulas are “more valid than others”; indeed, we will show that, if the values of the coefficients are i.i.d. random variables, there exists an almost sure spectrum (which is a deterministic function); we will explicitly compute this spectrum and show that it differs from the spectra proposed up to now. This study will lead us to consider countable intersections and differences of some random fractals with interesting properties (Sect. 5); they are related to the sets with large intersection previously introduced and studied by Falconer [8].
2. Results Valid for All Functions Since we are interested in local properties of wavelet expansions, it is more convenient to work with periodic wavelets which are obtained by a periodization of a usual wavelet basis, see [6, 22], and are thus defined on the unit torus T := R/Z. Extensions to R and higher dimension are straightforward. We use a wavelet ψ in the Schwartz class such
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that the periodized wavelets j
j
2 2 ψj,k (x) := 2 2
ψ(2j (x − l) − k),
l∈Z
j ∈ N, 0 ≤ k < 2j , form (together with the function ϕ(x) := 1) an orthonormal basis of L2 (T). We suppose that ψ has “enough” vanishing moments, in a sense to be made precise later (Proposition 4.1). Thus any one-periodic function f can be written Cj,k ψj,k (x), f (x) = j,k
where the wavelet coefficients of f are given by 1 Cj,k := 2j ψj,k (t)f (t) dt. 0
2.1. Histograms of coefficients. Let us now define some quantities that will be pertinent in our study. For each j , let Nj (α) := # k, |Cj k | ≥ 2−αj . We note for α ≥ 0, λ(α) := lim sup j →+∞
ρ(α, ε) := lim sup j →∞
log2 (Nj (α)) , j log2 (Nj (α + ε) − Nj (α − ε)) , j
and ρ(α) := inf ρ(α, ε). ε>0
(The reader should keep in mind the following heuristic interpretation: there are about 2λ(α)j coefficients larger than 2−αj , and about 2ρ(α)j coefficients of size of order 2−αj .) 2.2. Oscillating singularity exponents. We will study the sets of points where f has a given Hölder exponent, and local oscillations as in (1). We first have to define a pointwise oscillation exponent. Two definitions have been used previously, and we will briefly discuss them in order to motivate our choice. If f ∈ L∞ (R), denote by f (−n) an iterated primitive of f of order n. A consequence (−n) of the oscillations of (1) near x0 is that gh,β is C α+n(β+1) (x0 ) (the increase of regularity at x0 is not 1 at each integration, as would be expected for an arbitrary function, but β + 1). This remark motivated the following definition introduced by Y. Meyer, see [16]. Definition. Let h ≥ 0 and β > 0. A function f ∈ L∞ (R) is a chirp of type (h, β) at (n) x0 if, for every n ≥ 0, f can be written as f = gn , where gn ∈ C h+n(1+β) (x0 ).
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Such functions, under an additional regularity assumption (Definition 2), can be characterized by size estimates of their wavelet transform, see [16]. One immediately meets some difficulties when using this definition for experimental data. Indeed it is not stable when one adds to f a function g which is arbitrarily smooth, but not C ∞ : for 1 1 instance |x| 3 sin x1 is a chirp of type 13 , 1 while |x| 3 sin x1 + |x|h (if h > 1 and h∈ / 2Z) is a chirp of type 13 , 0 , even if h is chosen arbitrarily large. Nonetheless the 1 strongest singularity at 0 is the chirp x 3 sin x1 , and one actually observes its oscillatory behavior after magnifying enough the graph near the origin, see [1]. Thus the local oscillations are not reflected in the chirp exponents since this function is a chirp of type ( 13 , 0) at the origin, and the oscillation exponent β, if defined as above, is a very unstable quantity. This drawback can be avoided by introducing a slightly different definition of oscillating singularities which agrees with the definition of a chirp for functions such as (1), and has the required stability properties with respect to the addition of “smooth noise”. Consider 1 f0 (x) := |x − x0 |h g + O(|x − x0 |h ), (2) |x − x0 |β where h > h; the first term describes the local behavior of f0 near x0 , so that, if x x → 0 g(t)dt ∈ L∞ (R), the oscillating singularity exponent at x0 should be (h, β). Let ht (x0 ) denote the Hölder exponent of the fractional primitive of order t at x0 of a function f . More precisely, if f is locally bounded, we denote by ht (x0 ) the Hölder exponent at x0 of the function t
ft := (Id −")− 2 (φf ), where φ is a C ∞ compactly supported function satisfying φ(x0 ) = 1. In the case of the function f0 defined by (2), for t small enough, ht (x0 ) = h + (1 + β)t: the increase of pointwise Hölder regularity at x0 after a fractional integration of very small order t is (1 + β)t. This remark motivated the following definition of [1]. Definition 1. Let f : Rd → R be a locally bounded function and x0 such that ht (x0 ) < +∞. The oscillating singularity exponents of f at x0 are defined by
∂
−1 . (h, β) := h(x0 ), ht (x0 )
∂t t=0 This definition makes sense because, for a given x0 , the function t → ht (x0 ) is differentiable (with a possible derivative of +∞, so that β can be infinite), see [1]. The following proposition of [3] shows that this definition does adequately recapture the oscillatory behavior of (2). We first need to make a minimal regularity hypothesis. Definition 2. We say that a function f is uniform Hölder if f ∈ C r (T).
(3)
r>0
This condition will also be necessary for our main theorems; see Appendix A for a more complete discussion.
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Proposition 2.1. If f is uniform Hölder, then for any h < h(x0 ) and any β < β(x0 ), it can be written 1 h f (x) = |x − x0 | g + r(x), |x − x0 |β where r(x) ∈ C α (x0 ) for an α > h(x0 ), and g is indefinitely oscillating, i.e., g has bounded primitives of all orders on R+ and on R− . Note that, in the first example of random wavelet series studied in [14], chirp exponents were considered; however it is not difficult to see that, in this very particular case, the chirp exponent is everywhere identical to the oscillating singularity exponent. In the general setting of the present paper, they usually differ; and, for the reasons discussed above, it makes more sense to determine oscillating singularity exponents. We will obtain two kinds of results concerning either spectra of singularities or spectra of oscillating singularities. The first kind of results will be a priori upper bounds deduced from ρ(α). These bounds hold for any function which is uniform Hölder. The other type of results will be specific to random wavelet series, and we will show that, in this case, the upper bounds become equalities. 2.3. General upper bounds for spectra. As soon as we talk about pointwise Hölder regularity, f has to be locally bounded. This condition cannot be characterized by a condition on the modulus of wavelet coefficients (see [26]), so if we want to measure Hölder and oscillating singularity exponent with wavelet coefficients, we need to make a stronger assumption such as (3). This uniform Hölder regularity is not the weakest known condition on the modulus of wavelet coefficients that ensures local boundedness. In Appendix A, we present a weak uniform Hölder regularity which is sufficient for Proposition 2.1 to hold, as well as many of the results in the following sections. However, we shall also see that Theorems 1 and 2 cannot hold with only the weak hypothesis. This is why, in the following, (strong) uniform Hölder regularity is assumed, even though it is sometimes less than optimal. The following result will be proved in Sect. 4.2. Theorem 1. If f is uniform Hölder, its spectrum of oscillating singularities satisfies for all h ≥ 0, 0 ≤ β < +∞, h d(h, β) ≤ (1 + β)ρ . (4) 1+β Its spectrum of singularities satisfies, for h ≥ 0, ρ(α) α∈(0,h] α
d(h) ≤ h sup
(5)
(with the convention that sup(∅) = −∞). Remark. If f is a given function, using two different wavelet bases might lead to two different functions ρ and hence to two different upper bounds for d(h), and similarly for d(h, β). More generally, if ρ depends on the wavelet basis, it would be important to determine the quantities that can be deduced from ρ and are “wavelet-invariant” (for instance, it is the case for the concave hull of ρ, because of its relation with η). We intend to consider the general problem in a forthcoming paper.
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In order to relate this result to previously published ones, we recall the definition of the scaling function, which was first introduced by physicists in the context of fully developed turbulence, see [10]. Definition 3. The scaling function η(p) is defined for p > 0 by any of the formulas s p ,∞ η(p) := sup s : f ∈ Bp,loc j −1 2 1 := lim inf − log2 2−j |Cj,k |p j →+∞ j k=0 1 := lim inf − log2 2−j 2−αpj Nj (α) dα . j →+∞ j
The first two definitions coincide because of the wavelet characterization of Besov spaces; the second and the third coincide because j −1 2
|Cj,k |p =
k=0
2−αpj dNj (α)
= pj
2−αpj Nj (α) dα.
The following proposition, proved in Sect. 4.2, relates η(p) to ρ(α). Note that absolutely no regularity assumption is made here (not even that f is a function). Proposition 2.2. For any periodic tempered distribution f , we have η(p) = inf (αp − ρ(α) + 1) . α≥0
(6)
Let us now deduce some implications of Theorem 1. If f is uniform Hölder, it is shown in [15] that there exists a unique critical exponent pc such that η(pc ) = 1. We will see, after the proof of Theorem 1, that in this case (5) implies the classical bound d(h) ≤ inf (ph − η(p) + 1), p≥pc
(7)
proved in [11] (see also [15]). Nonetheless, (5) clearly yields a sharper estimate if ρ(α) is not concave. This remark is important since it shows a situation where strictly more information can be deduced from the histogram of the wavelet coefficients than from the scaling function, or from the set of Besov spaces to which the function considered belongs. As a consequence, a multifractal formalism obtained by claiming equality in (5) has a domain of validity which is strictly larger than for the classical multifractal formalism based on equality in (7) (this will be developed in Sect. 3.3). 3. Random Wavelet Series In this section, we study the random processes obtained by first choosing an (almost) arbitrary sequence of histograms of wavelet coefficients at each scale, and then drawing at random each wavelet coefficients at each scale inside the corresponding histogram, independently. We will see that, for this general class of random processes, the sample paths are multifractal, and the spectra almost surely satisfy equality in (5) and (4).
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3.1. Distribution of the coefficients. We shall use the following notation: bold symbols (ρj , ρ, λ, . . . ) denote deterministic quantities that are derived from the probability laws. Thin symbols (Nj , ρ, λ, . . . ) denote empirical quantities, that are measured on one sample path. Let (*, F, P) be a probability space. We suppose that, at each scale j , the wavelet coefficients of the process are drawn independently with a given law; ρj will denote the common probability measure of the 2j random variables Xj,k := − log2 (|Cj,k |)/j (the signs, or arguments, of the wavelet coefficients have no consequence for Hölder regularity, therefore, we do not need to make any assumption on them). The measure ρj thus satisfies
P Cj,k ≥ 2−αj = ρj ((−∞, α]) and E(Nj (α)) = 2j ρj ([0, α]). We note for α ≥ 0,
log2 2j ρj ([0, α]) , j j →+∞ log2 2j ρj ([α − ε, α + ε]) ρ(α, ε) := lim sup j j →+∞ λ(α) := lim sup
(8)
ρ(α) := inf ρ(α, ε).
(9)
and ε>0
We call ρ(α) the upper logarithmic density of the sequence ρj . Definition 4. We say that j
f :=
−1 2
Cj k ψj k
j ∈N k=0
is a Random Wavelet Series (R.W.S.) if there exists γ > 0 such that α < γ implies ρ(α) < 0. We assume from now on that this requirement is satisfied. The following propositions give the relationships between the quantities we defined, and show what are the “admissible” functions λ and ρ that can be obtained by (8) and (9). They will be proved in Sect. 4.3. Proposition 3.1. The function λ is nondecreasing and for all α, λ(α) ≤ 1; for any α < γ , λ(α) < 0. Conversely, for any function λ verifying these properties, there exists a R.W.S. such that (8) holds. Proposition 3.2. The function ρ is upper semi-continuous and for all α, ρ(α) ≤ 1; for any α < γ , ρ(α) < 0. Conversely, for any function ρ verifying these conditions, there exists a R.W.S. such that (9) holds.
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491
We note λ¯ the upper closure of λ: the hypograph of λ¯ is the closure of the hypograph of λ. λ¯ is the only càdlàg function which coincides with λ almost everywhere. Or, equivalently since λ is increasing, ¯ λ(α) := lim λ(α ). α →α +
Proposition 3.3. For all α ≥ 0, ¯ λ(α) = sup ρ(α ).
(10)
α ≤α
Conversely, if (10) holds, as well as the conditions specified in Propositions 3.1 and 3.2, there exists a R.W.S. such that both (8) and (9) hold. Remark 3.1. Since the proof of Proposition 3.3 uses only the definitions (8) and (9), it ¯ is easy to see that (10) also holds for the corresponding empirical quantities λ(α) and ρ(α) of any function. Proposition 3.4. Let
2j ρj ([α − ε, α + ε]) = +∞ . W = α∀ε > 0, j ∈N
With probability one, for all α ≥ 0,
ρ(α) −∞
ρ(α) = and
λ(α) =
λ(α) −∞
if α ∈ W, else
(11)
if α ≥ inf(W ), else.
(12)
Corollary 3.5. A R.W.S. is almost surely uniform Hölder. Proof. From (11) and Definition 4, almost surely, ρ(α)
= −∞ when α < γ . This implies that, with a possible finite number of exceptions, Cj k ≤ 2−γj ; this is equivalent to f ∈ C γ (T). Remark. It is easy to check that ρ(α) > 0 ⇒ α ∈ W and that ρ(α) < 0 ⇒ α ∈ W . In cases where ρ(α) = 0 ⇒ α ∈ W , (11) boils down to: ρ(α) if ≥ 0, ρ(α) = −∞ else. Remark. Let α0 be defined by
α0 := inf γ , sup ρ(α) = sup ρ(α) [0,γ ]
R+
(possibly α0 = +∞). Since η(p) is computed for positive values of p, the infimum in (6) can be computed on [0, α0 ]. We can thus replace ρ(α) by ρ(α) in (6); in this case, because of (6), η(p) depends only on the concave hull of ρ(α) on [0, α0 ]; but, because of (10), the concave hulls of ρ(α) and of λ(α) coincide on [0, α0 ]; it follows that, almost surely, η(p) = inf (αp − λ(α) + 1). α≥0
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3.2. Almost everywhere regularity and exact spectra. From now on, we suppose that ρ(α) takes a positive value for at least one value of α. Let W be defined as in Proposition 3.4, hmin := inf(W ) and
hmax
ρ(α) := sup α>0 α
−1
.
The supremum of ρ(α) α on an interval (0, h] cannot be achieved at a point in the neighborhood of which λ(α) is constant, so ¯ ρ(α) λ(α) = sup , α∈(0,h] α α∈(0,h] α sup
and, since λ¯ is càdlàg, this supremum is clearly achieved. In particular, we will denote by α˜ the largest value of α for which ¯ λ(α) α∈(0,hmax ] α sup
is achieved (or α˜ = 0 if hmax = 0). Theorem 2. Let f be a random wavelet series. With probability one, f has the following properties: • For almost every x, hf (x) = hmax
(13)
and βf (x) =
1 − 1. ˜ λ(α)
(14)
• The spectrum of oscillating singularities of f is defined for (h, β) in the rectangle h [hmin , hmax ] × 0, −1 , hmin where
d(h, β) = (1 + β)ρ
h 1+β
.
(15)
• The spectrum d(h) is defined for h ∈ [hmin , hmax ], where ¯ λ(α) ρ(α) = h sup . α∈(0,h] α α∈(0,h] α
d(h) = h sup
(16)
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493
Remark. The assertions (16) and (15) are stronger than stating that, for each h and β, d(h) or d(h, β) has almost surely a given value, which would not be sufficient to determine the spectrum of singularities and the spectrum of oscillating singularities of almost every sample path. By inspecting (16), it is clear that d(h) need not be concave, which shows another possible occurrence of the failure of the classical multifractal formalism (this will be developed in Sect. 3.3 below). The function h → supα∈(0,h] h ρ(α) α is increasing on (0, hmax ] and takes the value d(h) = 1 for h = hmax , which is compatible with (13). Similarly, using Formula (15), we see that 1 1 ˜ d hmax , ρ(hmax λ(α)) −1 = ¯ α) ˜ λ(α) ˜ λ( 1 ˜ = ρ(α) ¯ α) ˜ λ( = 1, which is compatible with (14). Comparing Theorem 1 and Theorem 2, we see that both spectra d(h) and d(h, β) of random wavelet series take the largest possible values compatible with the bounds (5) and (4), which shows first that these bounds are optimal, and second that random wavelet series strive to have their Hölder singularities on sets of dimension as large as possible. Note finally that a random wavelet series satisfies the very natural property d(h) = sup d(h, β). β≥0
(17)
3.3. Multifractal formalisms. A multifractal formalism is a formula which allows in many cases to compute numerically the spectrum of singularities d(h) of a function, usually based on its wavelet coefficients. As an illustration, let us assume that we are given a function f with an upper logarithmic density of histograms ρ(α) as in Fig. 1 (and α < αmin ⇒ ρ(α) = −∞). Two main methods are used to derive a multifractal formalism. The first one consists in deducing formulas directly from “box-counting” arguments. This leads to the original Fenchel–Legendre transform formula of Frisch and Parisi (Fig. 2), which asserts that d(h) = d1 (h), where d1 (h) := inf (hp − η(p) + 1). p≥0
An alternative formula, based on a large deviation-type argument, simply asserts that d(h) = ρ(h); let us sketch its justification. If the Hölder exponent at x0 is h and is determined by the wavelet coefficients such that the support of the corresponding wavelet includes x0 (this is called a “cusp-type” singularity at x0 ), then (18) implies that these wavelet coefficients are of the order of magnitude of 2−hj . If the corresponding dimension is d(h), we expect to find about 2d(h)j such coefficients, but we know that there are about 2ρ(h)j of them, hence the formula.
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1
0
α
αmin Fig. 1. Example of ρ(α)
1
h
0 Fig. 2. The Fenchel–Legendre transform spectrum d1 (h)
A less intuitive method for deriving a multifractal formalism consists first in obtaining sharp upper bounds for d(h), based either on “Besov-type information” (summed up in the scaling function η(p)) or based on wavelet histograms (in which case the information is summed up in the function ρ(α)). In both cases, the multifractal formalism asserts that, for a “large” class of functions, these upper bounds must be saturated; “large” meaning either in the sense of Baire categories if we only have a function space setting, or almost surely if we have a precise probabilistic model (which is the case in the present paper). This leads us to two additional possible multifractal formalisms. The first one states that d(h) = d2 (h), where d2 (h) is the quasi-sure spectrum (Fig. 3) d2 (h) := inf (hp − η(p) + 1), p≥pc
where pc is the only value of p for which η(p) = 1, see [15]).
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1
0
hc
h
Fig. 3. The quasi-sure spectrum d2 (h)
The second one states that d(h) = d3 (h), where d3 (h) is the almost-sure spectrum studied in this paper (Fig. 4), ρ(α) . α∈(0,h] α
d3 (h) := h sup
1
0
hc
h
Fig. 4. The almost-sure spectrum d3 (h)
Let us now compare these different formulas. We saw that, for any Hölder function, d(h) ≤ d3 (h) ≤ d2 (h).
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Of course, d1 (h) ≤ d2 (h) and ρ(h) ≤ d3 (h). Finally, since η(p) is the Fenchel–Legendre transform of ρ(h), d1 (h) is the increasing concave hull of ρ(h), so that ρ(h) ≤ d1 (h). These are the only inequalities between spectra that hold in all generality; indeed d(h) can be larger than d1 (h), see [15], ρ(h) can be smaller than d(h), as shown in the present paper, and it can be larger than d(h), as shown in [11]. Figure 5 recapitulates these inequalities (−→ means ≤ for any uniform Hölder function). d(h) −−−−−→ d3 (h) −−−−−→ d2 (h) ρ(h) −−−−−→ d1 (h) Fig. 5. Comparison between multifractal formalisms
Note that the inequality d3 (h) ≤ d2 (h) fits to the classical “rule of thumb” which holds for Fourier series: quasi-all Fourier series display the worst possible regularity, whereas random Fourier series display more regularity, see [18] and [21]. Since quasi-sure results are expected to display the worst possible case (in terms of regularity) and almost-sure results, the best possible case, it is interesting to determine when they coincide. It is clearly the case if and only if ρ(α) is concave for α ≤ hc , where hc = η (pc ) is the critical Hölder exponent introduced in [15] (indeed, for h ≥ hc , both spectra always coincide). Thus, when this additional condition holds, quasi-sure and almost-sure spectra coincide, which is a very strong mathematical argument in favor of the corresponding multifractal formalism. 4. First Proofs In this section the propositions and theorems stated in Sects. 2 and 3 are proved. The proof of Theorem 2 will be finished in Sect. 5. 4.1. Regularity and wavelet coefficients. We first recall the characterizations of Hölder and oscillating singularity exponents using wavelet coefficients, see [1, 16]. Proposition 4.1. If f is uniform Hölder, the Hölder exponent of f at x0 is hf (x0 ) = lim inf inf j →∞
k
log(|Cj,k |) , log(2−j + |k2−j − x0 |)
(18)
provided that ψ has at least hf (x0 ) +1 vanishing moments and continuous derivatives, each of them having fast decay. We call any sequence (jn , kn ) such that log(|Cjn ,kn |) → hf (x0 ) + |kn 2−jn − x0 |)
log(2−jn
a minimizing sequence for the wavelet coefficients of f at x0 . For β ≥ 0, let Zx0 (β) ⊂ N×Z be the set of indices (j, k) such that |k2−j − x0 |1+β ≤ 2−j . Note that Zx0 (β) is increasing with β. The following proposition is proved in [1].
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Proposition 4.2. If f is uniform Hölder, its oscillating singularity exponent βf (x0 ) at x0 is the infimum of the β such that there exists a minimizing sequence in Zx0 (β). Propositions 4.1 and 4.2 will now be reformulated in a way easier to use in our proofs. If α ≥ 0, let F j (α) := {k : |Cj,k | ≥ 2−αj } and if d ≤ 1, let
k2−j − 2−dj , k2−j + 2−dj , E j (α, d) := k∈F j (α)
and
E(α, d) := lim sup E j (α, d) := j →∞
E m (α, d).
(19)
j ∈N m≥j
Since E(α, d) is increasing in α and decreasing in d, for all x ∈ T, γx0 (α) := sup {d, x0 ∈ E(α, d)} defines an increasing function α → γx0 (α), which is bounded by 1. Let γ¯x0 be the upper closure of γx0 . Note that supα set of α for which its maximum.
γ¯ (α) supα x0α
γx0 (α) α
= supα
γ¯x0 (α) α ,
and that the
is attained is a non-empty compact; we denote by α(x0 )
Proposition 4.3. If f is uniform Hölder, for all x0 , γx (α) −1 α(x0 ) hf (x0 ) = sup 0 = α γ¯x0 (α(x0 )) α>0
(20)
and βf (x0 ) =
1 − 1. γ¯x0 (α(x0 ))
(21)
We can also define H (α, d) := {x ∈ T, α(x) = α, γ¯x (α(x)) = d} ; it has the property that x0 ∈ H (α, d) ⇐⇒ hf (x0 ) =
α d
and βf (x0 ) =
Proof of Proposition 4.3. Let x0 be fixed and let h(x0 ) := {0, . . . , 2jn
(22)
inf α γx α(α) . 0
1 d
− 1.
If x0 ∈ E(α, d),
− 1} such that there exist sequences jn → +∞ and kn ∈
Cj ,k ≥ 2−αjn and |kn 2−jn − x0 | ≤ 2−djn . n n
Because of (18), it follows that hf (x0 ) ≤ αd , hence that hf (x0 ) ≤ γx α(α) ; since this holds 0 for any α > 0, we obtain that hf (x0 ) ≤ h(x0 ). Conversely, let h > hf (x0 ). Then there exist sequences jn → +∞ and kn ∈ {0, . . . , 2jn − 1} such that |Cjn ,kn | ≥ (2−jn + |kn 2−jn − x0 |)h .
(23)
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Let d be any accumulation point of the sequence ε > 0, there exist infinitely many n such that
log2 (|kn 2−jn −x0 |) . −jn
2−(d+ε)jn ≤ 2−jn + |kn 2−jn − x0 | ≤ 2−(d−ε)jn .
If d ≤ 1, for any
(24)
Because of (23), it follows that |Cjn ,kn | ≥ 2−jn h(d+ε) , so that x0 ∈ E(h(d +ε), d −ε). Thus, for all ε > 0, h(d + ε) α ≥ inf ; α γx0 (α) d −ε it follows that h ≥ h(x0 ). If d > 1, E(α, d) is not defined but then for infinitely many n, |Cjn ,kn | ≥ 2−hjn , so x0 ∈ E(h, 1) and γx0 (h) ≥ 1, which implies h ≥ h(x0 ). In any case, h > hf (x0 ) implies h ≥ h(x0 ), hence hf (x0 ) ≥ h(x0 ). 1 Now let β(x0 ) := γ¯x (α(x − 1 and let β ∈ (0, β(x0 )). In order to show that 0 )) 0 βf (x0 ) ≥ β(x0 ), we need to show that there exists no minimizing sequence in Zx0 (β) (see Proposition 4.2). Suppose that (jn , kn ) is such a sequence, and let d = lim inf n→∞
log2 (2−jn + |kn 2−jn − x0 |) ; −jn
since (jn , kn ) belongs to Zx0 (β), d≥
1 1 > . 1+β 1 + β(x0 )
For any ε > 0, and infinitely many n, we have both log2 (2−jn + |kn 2−jn − x0 |) ≥ −(d + ε)jn
(25)
log2 (|Cjn kn |) ≤ h(x0 ) + ε, + |kn 2−jn − x0 |)
(26)
and
log2
(2−jn
hence log2 (|Cjn kn |) ≥ −(h(x0 ) + ε)(d + ε)jn . In other words, as soon as α > h(x0 )d, 1 we have x0 ∈ E(α, d), which is a contradiction with the fact that d > 1+β(x . 0) 1 1 . This implies that x0 ∈ E(α(x0 ) + n1 , 1+β ) for If β > β(x0 ), γ¯x0 (α(x0 )) > 1+β all n > 1, which means that there exists infinitely many j and k such that |Cj k | ≥ −
j
2−j (α(x0 )+ n ) and |k2−j − x0 | ≤ 2 1+β . By taking jn ≥ n among those, and the corresponding kn , we constructed a minimizing sequence in Zx0 (β). With Proposition 4.2, this proves that βf (x0 ) ≤ β(x0 ). 1
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499
4.2. Properties of histograms. Proof of Proposition 2.2. Let α be fixed and ρ0 < ρ(α). For any ε > 0, there exists jn → +∞ such that Njn (α + ε) − Njn (α − ε) ≥ 2ρ0 jn , and therefore 2−jn
|Cjn ,k |p ≥ 2−jn 2ρ0 jn 2−(α+ε)jn p ,
k
so that η(p) ≤ (α + ε)p − ρ0 + 1. Since this holds for any ε > 0, any ρ0 < ρ(α) and any α ≥ 0, it follows that, for all p > 0, η(p) ≤ inf (αp − ρ(α) + 1) . α≥0
Conversely, let A > 0, B be larger than the order of the distribution f , and η > 0. For all α ∈ [−B, A], there exists ε > 0 such that for all ε ≤ ε, |ρ(α, ε ) − ρ(α)| ≤ η. Thus, there exist α1 , . . . , αN and ε1 , . . . , εN ≤ ε such that the intervals [αi − εi , αi + εi ] cover [−B, A], and ∀i, |ρ(αi , εi ) − ρ(αi )| ≤ η. Thus, for each (αi , εi ) there exists Ji such that ∀j ≥ Ji , Nj (αi + εi ) − Nj (αi − εi ) ≤ 2j 2(ρ(αi )+η)j .
(27)
Taking for J the maximum of the Ji , it follows that ∀j ≥ J , 2−j
|Cj,k |p ≤ 2−j
k
N
2−(αi −εi )jp 2(ρ(αi )+η)j + 2j 2−Ajp
i=1
(the last term corresponds to wavelet coefficients smaller than 2−Aj ). Thus ∀j ≥ J, 2−j |Cj,k |p ≤ N 2(εp+η)j 2supα (−αp+ρ(α)−1)j + 2j 2−Ajp . k
Since ε and η can be chosen arbitrarily small and A arbitrarily large, it follows that, for any p > 0, η(p) ≥ − sup (−αp + ρ(α) − 1) = inf (αp − ρ(α) + 1) , α≥0
α≥0
and Proposition 2.2 is proved.
We now obtain the upper bounds for spectra. To fix the notations, we recall that for s > 0 and ε > 0 (possibly ε = +∞), Hεs (A) := inf |I |s , (28) r∈R(A,ε)
I ∈r
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where R(A, ε) is the set of all countable coverings r of A with intervals I of lengths |I | ≤ ε. The s-Hausdorff (outer) measure is defined by Hs (A) := lim Hεs (A) ε→0
and the Hausdorff dimension by dimH (A) := inf s, Hs (A) = 0 . Proof of Theorem 1. Let h > 0 be given. If f is uniform Hölder, the set of points x0 , where hf (x0 ) = h can be expressed as α H α, . h 0<α≤h
αs−hρ(α) Let ν > 0 be fixed. We take α ∈ [ν, h], µ > 0, s = h ρ(α) α + µ, and ε ∈ (0, h+s ). By definition of ρ(α), there exist η ∈ (0, ε) and j0 ∈ N such that for all j ≥ j0 ,
Nj (α + η) − Nj (α − η) ≤ 2j (ρ(α)+ε) . Let us define the auxiliary set Eη (α, d) := lim sup Eηj (α, d), j →∞
where Eηj (α, d) := E j (α + η, d)\E j (α − η, d). Clearly, for all γ ∈ (α − η2 , α + η2 ),
γ α−η ⊂ Eη α, . H γ, h h
j At scale j ≥ j0 , Eη α, α−η is covered by Nj (α + η) − Nj (α − η) segments of size h
2−j
α−η h
, and (Nj (α + η) − Nj (α − η))2−j
α−η h s
≤ 2j (ρ(α)+ε−s
α−η h )
sα
s
≤ 2j (ρ(α)− h +ε(1+ h )) , whose series converges. This proves that γ ρ(α) H γ, ≤s=h + µ. dimH h α η η γ ∈(α− 2 ,α+ 2 )
Taking a finite sub-covering of [ν, h] by the (α − η2 , α + η2 ), we get α ρ(α) H α, ≤ h sup + µ. dimH h ν≤α≤h α ν≤α≤h
(29)
Random Wavelet Series
501
To conclude, we make µ → 0 and remark that this result stays valid when we take the union over ν in a sequence converging to 0. So finally, dimH
0<α≤h
α ρ(α) H α, ≤ h sup , h 0<α≤h α
which proves (5). Equation (29) also implies that for all d ≤ 1, dimH (H (α, d)) ≤
ρ(α) d , hence (4).
Let us now show that, if f ∈ C r (T), (5) implies (7). From the definition of pc , it follows that sup (−αpc + ρ(α)) ≤ 0,
α≥0
and (5) implies d(h) ≤ hpc .
(30)
Let now λc (α) be the concave hull of λ(α) (λc (α) is the smallest concave function such that ∀α ≥ 0, λc (α) ≥ λ(α)). Note that λc (α) is also the concave hull of ρ(α). Let hc = η (pc ). The line y = λ c (hc ) is tangent to the curve y = λc (α) at α = hc , see [15]. Therefore, if h ≤ hc , the function α → λc (α) − α dh is increasing on [0, h], and sup[0,h] αh ρ(α) ≤ λc (h), so (5) implies in this case that for all h ≤ hc , d(h) ≤ λc (h).
(31)
The scaling function η(p) is a concave increasing function, see [11]. From Proposition 2.2, it follows that 1 − η(p) = sup (−αp + ρ(α)) . α≥0
Thus 1−η(p) and λc (α) are convex conjugates, so that (30) and (31) together are nothing but the classical bound (7).
4.3. Properties of λ and ρ. Proof of Proposition 3.1. It is clear that λ is nondecreasing and bounded by 1, and the R.W.S. hypothesis (Definition 4) implies λ(α) < 0 as soon as α < γ because of Proposition 3.3 proved hereafter. Let us prove the converse result. The natural choice ρj ([0, α]) := 2j (λ(α)−1) would not necessarily be right-continuous, so we modify it slightly as follows. Remark that since λ is monotonous, its discontinuities are at most countable. For j ≥ 1, let D(j ) be the set of the first discontinuities of λ such that, for 1 ≤ k ≤ j 2 , there is at most one of √ , √k . them in k−1 j j
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J.-M. Aubry, S. Jaffard
We pose √ ) 2j λ( √kj ) − 2j λ( k−1 j ω(j, k) := + − 2j λ (α) − 2j λ (α)
k−1
, √kj = ∅ , √ , √k if α ∈ D(j ) ∩ k−1 j j if D(j ) ∩
√
j
where λ− (α) and λ+ (α) are respectively the left and right limits of λ at α. 1 3 Let ρ˜j be the measure supported on [j − 2 , j 2 ] and defined on this interval by j 1 −j ρ˜j := 2 + 2 ω(j, k)δ √k . j j 2
k=1
The total mass of ρ˜j is bounded by 1 + √1j − j 21√j ; ρj is obtained as follows: If the total mass of ρ˜j is larger than 1, we divide ρ˜j by this total mass; if it is smaller than 1, we add cj δj √j , where cj is chosen so that the total mass becomes 1. The sequence ρj thus constructed clearly has the properties announced. Proof of Proposition 3.2. For any γ > 0, there exists ε > 0 such that ρ(α, 2ε) ≤ ρ(α) + γ . On the other hand, as soon as |α − α | < ε, ρj ([α − 2ε, α + 2ε]) ≥ ρj ([α − ε, α + ε]), hence ρ(α, 2ε) ≥ ρ(α , ε), so finally ρ(α ) ≤ ρ(α) + γ , so that ρ is upper semicontinuous. It is obvious that ρ(α) ≤ 1, and ρ(α) < 0 if α < γ is precisely the hypothesis made in Definition 4. We now prove the converse part. Let ω(j, k) :=
2j ρ(α)
sup √ , k+1/2 √ ] α∈[ k−1/2 j j
and ω(j, ˜ k) be defined by ω(j, ˜ k) :=
if ω(j, k) ≤ j 3 . else
0 ω(j, k)
Then let ρj be the measure supported on [j − 2 , j 2 ] and defined by 1
3
j 2−j ρj = 2 ω(j, k)δ √k + cj δj √j , j j 2
k=1
where cj is the positive constant such that the total mass of ρj is 1. In this case also, one easily checks that the sequence ρj has the properties announced. ¯ Proof of Proposition 3.3. The upper semi-continuity of λ¯ means that for all α, λ(α) ≥ ¯ ). But for α > α, λ(α ¯ ) ≥ ρ(α), hence λ(α) ¯ lim supα →α λ(α ≥ ρ(α). Since λ¯ is ¯ increasing, this implies that λ(α) ≥ supα ≤α ρ(α ).
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For the other way, take ε > 0. By definition of ρ, for any α ≤ α there exists r(α ) > 0 and J (α) such that ∀j ≥ J (α),
ρj ([α − r(α ), α + r(α )]) ≤ 2j (ρ(α )+ε) .
(32)
From the covering of the compact set [0, α] by the open intervals (α − r(α ), α + r(α )) we extract a finite sub-covering B(αi , r(αi )), 1 ≤ i ≤ n . Then, for j large enough,
ρj ([0, α]) ≤
n
2j (ρ(αi )+ε) ≤ n2j (supα ≤α ρ(α )+ε)
i=1
hence ¯ λ(α) ≤ sup ρ(α ) + ε
(33)
α ≤α
and Proposition 3.3 follows. We now treat the converse part. If we took the same ρj as in the proof of Proposition 3.2, then (8) would yield λ¯ instead of λ. We need to take care of the discontinuities of λ. Let D(j ) be defined as in the proof of Proposition 3.1, and ω(j, k) :=
supα∈[ k−1 √ , k+1 √ ] ρ(α) − 1 j
j
2j λ+ (α) − 2j λ− (α)
√ , √k ] = ∅, if D(j ) ∩ ( k−1 j j √ , √k ] if α ∈ D(j ) ∩ ( k−1 j j
.
The rest of the construction is the same as in the proof of Proposition 3.2, and one checks that the sequence ρj has the properties announced. Before proving Proposition 3.4, let us state a technical lemma. Lemma 4.4. Let α < β be fixed. With probability 1, there are infinitely many wavelet coefficients satisfying
2−βj ≤ Cj k ≤ 2−αj if and only if
j
(34)
2j ρj ([a, b]) = +∞.
Proof. Suppose that
j
2j ρj ([a, b]) < +∞. Then for j fixed, the probability that at j
least one wavelet coefficient satisfies (34) is 1 − (1 − ρj ([a, b]))2 , which is of order 2j ρj ([a, b]) for j large enough. The Cj k being independent, by the Borel-Cantelli lemma, with probability 1, there can be only a finite number of wavelet coefficients satisfying (34). Conversely, suppose that the series diverges. If lim supj 2j ρj ([a, b]) > 0 then the result is trivial; otherwise 2j ρj ([a, b]) → 0 and we can use the same equivalence as above, to conclude again with the Borel–Cantelli lemma.
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J.-M. Aubry, S. Jaffard
We write Nj (α, β) the number of wavelet coefficients satisfying (34). Proof of Proposition 3.4. Note that the set W is closed. R\W being open, it is a countable n union of open intervals (αn , βn ). Let m ∈ N be such that m1 < βn −α and let α ∈ 2 1 1 [αn + m , βn − m ]. Since α ∈ W , by Lemma 4.4, ∃>(α) and J (α) such that j ≥ J (α) ⇒ Nj (α − >(α), α + >(α)) = 0. From the covering of [αn + m1 , βn − m1 ] with intervals (α − >(α), α + >(α)) we extract a finite sub-covering centered on α1 , . . . , αk ; we note J (n, m) := sup1≤i≤k J (αi ). Thus for j ≥ J (n, m), Nj (αn + m1 , βn − m1 ) = 0 is an event of probability 1. Taking the intersection over n and m of all those events, we get that with probability 1, for all n, m, there exists j such that Nj (αn + m1 , βn − m1 ) = 0; this is equivalent to saying that for all α ∈ W , ρ(α) = −∞. Let us now consider the α such that ρ(α) > 0. We denote by Gj (a) = ρj ((−∞, a]) the common distribution of the 2j random variables Xj,k := − log2 (|Cj,k |)/j . The standard way to study properties of the distribution of a large number of independent draws of a random variable is to reduce it to the case where this random variable is uniformly distributed on the interval [0, 1]; this is done by writing the random variables Xj,k under the form Xj,k = G−1 j (ξ ), where ξ is equidistributed on [0, 1]. Thus we now j suppose that we have n = 2 independent draws ξ 1 , . . . , ξ n of the random variable ξ . Let Fn (t) :=
1 #{ξ i : ξ i ≤ t}; n
Fn (t) is the empirical distribution function of the (ξ i )i=1,...,n . The uniform empirical process is defined by 1
αn (t) := n 2 (Fn (t) − t). We will use independent copies of the empirical process for each n = 2j . The increments of the empirical process can be estimated using the following result which is a particular case of Lemma 2.4 of Stute [30]. Lemma. There exist √ two positive constants C1 and C2 such that, if 0 < l < 1/8, nl ≥ 1 and 8 ≤ A ≤ C1 nl, √ C2 − A2 P sup |αn (t) − αn (s)| > A l ≤ e 64 . l |t−s|≤l √ (log2 (n))2 For a fixed j and n = 2j , we pick l = and A = C1 nl = C1 log2 (n). n Therefore (C log (n))2 (log2 (n))2 C2 n − 1 642 P sup |αn (t) − αn (s)| > C1 e . ≤ √ (log2 (n))2 n But, if |αn (t) − αn (s)| ≤ C1 satisfies
(log2 (n))2 √ , n
the number N (t, s) of ξ k in the interval (s, t]
1 (log2 (n))2 , √ |N (t, s) − n(t − s)| ≤ C1 √ n n
Random Wavelet Series
505
so that N (t, s) = n(t − s) + O(log(n)2 ), and there are between n(t − s)/2 and 2n(t − s) of the ξ k in any interval of length (log2 (n))2 . n
Coming back to the random variables Xj,k , it follows that, in any interval [a, b)
satisfying ρj ([a, b)) ≥
j2 , 2j
there are between 2j −1 ρj ([a, b)) and 2j +1 ρj ([a, b)) of the
C12 j 2 −j 64 ). Let Ij k := [k2 , (k 2 2 C1 j k = 0, . . . , 2j − 1: with probability at least 1 − C2 22j j −2 exp(− 64 ), for 2 j j (at most 2 ) intervals Ij k which satisfies ρj (Ij,k ) ≥ 2j ,
Xjk with probability at least 1 − C2 2j j −2 exp(−
+ 1)2−j ), any of the
2j −1 ρj (Ij,k ) ≤ #{Xjk ∈ Ij,k } ≤ 2j +1 ρj (Ij,k ). C12 j 2 Since the series 22j j −2 exp(− 64 ) is convergent, by the Borel–Cantelli lemma, with probability 1 the above event happens for all j with only a finite number of exceptions. But if α ∈ [0, 1) and ρ(α) > 0, this α will be in infinitely many Ij k such that ρj (Ij,k ) ≥ j2 , 2j
hence ρ(α) = ρ(α). The same proof is valid ∀n for α ∈ [n, n + 1), so finally, with probability 1, ρ(α) > 0 ⇒ ρ(α) = ρ(α). The last case we have to consider is when α ∈ W0 := {α ∈ W, ρ(α) = 0}. The previous reasoning provides only an upper bound: ρ(α) ≤ 0. For > > 0, let {(αi − >, αi + >), i ∈ N} be a countable covering of W0 such that for all i, αi ∈ W0 . Applying Lemma 4.4 to each of these intervals, we see that with probability
1, for all i, there are infinitely many wavelet coefficients such that 2−αi −> ≤ Cj k ≤ 2−αi +> . It follows that
for
all α ∈ W0 , there are infinitely many wavelet coefficients such that 2−α−2> ≤ Cj k ≤ 2−α+2> , hence ρ(α) ≥ 0.
4.4. Almost everywhere regularity. Proposition 4.5. Let f be a random wavelet series. The following events hold with probability one: • For every x ∈ T, h(x) ≤ hmax .
(35)
• For almost every x ∈ T, for all α ≥ 0, γx (α) = λ(α).
Proof. By definition of hmax , for all ε > 0, there exists an α1 such that ρ(α1 ) −
(36)
α1
hmax ≤
ε. There exists also a sequence jn → +∞ such that, with probability at least 1 − jn−2 , there are at least 2(ρ(α1 )−ε)jn coefficients Cjn ,k satisfying 2−(α1 +ε)jn ≤ |Cjn ,k | ≤ 2−(α1 −ε)jn .
(37)
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J.-M. Aubry, S. Jaffard
If we condition by this event, for all jn ≥ J (and J chosen large enough), the locations k of the coefficients satisfying (37) are picked at random among the 2jn possible locations. We can apply Lemma 1 of [14], which is a consequence of classical results (see [28]) concerning random coverings of the circle and which implies that, with probability 1, every x belongs to lim sup n→∞
k −(ρ(α1 )−2ε)jn k −(ρ(α1 )−2ε)jn , − 2 , + 2 2jn 2 jn k
where the union is taken on the k verifying (37). It follows that, with probability one, for any ε > 0 and any x ∈ T, hf (x) ≤
α1 + ε , ρ(α1 ) − 2ε
therefore (35) holds. Let us now prove the second part of Proposition 4.5. In the following, |E| denotes the Lebesgue measure of a measurable set E. Let α be a given positive number. Clearly, if λ(α) = −∞, then for all x ∈ T, γx (α) = −∞. If λ(α) = 0, then for almost every x, γx (α) = 0. If λ(α) > 0, let ε ∈ Q, ε > 0. By the Borel–Cantelli lemma, |E(α, λ(α) + ε)| = 0, and applying Lemma 1 of [14] as above, almost surely |E(α, λ(α) − ε)| = 1. Therefore, if E(α) :=
E(α, λ(α) − ε)\E(α, λ(α) + ε),
ε∈Q,ε>0
almost surely, |E(α)| = 1. We can take α in a countable set Q, which is defined as the set of α such that λ(α) > 0 and α is rational or λ has a discontinuity at α. Then, if E :=
E(α),
α∈Q
almost surely, |E| = 1, and by construction, ∀x ∈ E, ∀α ∈ Q, γx (α) = λ(α). When α ∈ Q, λ is continuous at α and by density of Q we get the same result.
Combined with Proposition 4.3, this result yields the almost everywhere Hölder and oscillating singularity exponents of f . We thus obtained one point of the spectra of f , namely
α˜ d ˜ λ(α)
α˜ 1 =d , − 1 = 1. ˜ λ(α) ˜ λ(α)
For the other values of the spectra of f , we want to calculate the dimension of the sets H (α, d), defined by (22), which form a partition of T. These sets are random, because they depend on the repartition of the wavelet coefficients.
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507
5. Random Fractals 5.1. Upper limit of random segments. Though the sets E(α, d), defined by (19), are not those that appear in the definition of the spectra, as an intermediate step, we need to determine their dimension and to show that they fall in the category of the “sets with large intersection”. Proposition 5.1. For any α > 0, d ≥ λ(α), dimH (E(α, d)) ≤ Proof. Let s >
λ(α) d .
λ(α) . d
Then
2−sdj < ∞,
j ∈N k∈F j (α)
! and since j >j0 E j (α, d) is a covering of E(α, d) for any j0 ∈ N, this implies that Hs (E(α, d)) = 0, and dimH (E(α, d)) ≤ s. Note that this bound depends only on the histogram of the wavelet coefficients, and not on the random process considered here. The lower bound will be specific to the model we consider, and uses the following result (Theorem 2 of [14]). Proposition 5.2. Let Snt := (xn − snt , xn + snt ) for a sequence of xn ∈ T and sn > 0. We note B t = lim sup Snt . n→∞
If |B 1 | = 1, then for all t ≥ 1, dimH (B t ) ≥ 1/t. Corollary 5.3. Almost surely, for all α > 0, d ≥ λ(α), dimH (E(α, d)) ≥
λ(α) . d
(38)
Proof. Thanks to Proposition 5.2, we only have to prove that almost surely, for all α, |E(α, λ(α))| = 1. By the Borel–Cantelli lemma, for each α this is true almost surely, following the fact that lim supj →∞ |E j (α, λ(α))| = 1, which implies that the sum of the lengths of the segments whose lim sup constitutes E(α, λ(α)) diverges. As in the proof of Proposition 4.5, we take α in the countable set Q which is the set of α such that λ(α) > 0 and α is rational or λ has a discontinuity at α. So (38) is true almost surely for all α ∈ Q and any d. If α ∈ Q, λ is continuous at α; since E(α, d) is increasing in α and decreasing in d, (38) still holds by density of Q.
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J.-M. Aubry, S. Jaffard
5.2. Sets with large intersection. Actually, we have something more. We recall that the class of sets with large intersection G s (T), defined in [8], is"the class of Gσ -sets (countable intersections of open sets) E ⊂ T such that dimH i∈N fi (E) ≥ s for all sequences of similarity transformations (fi,i∈N ). It is also the maximal class of Gσ sets of Hausdorff dimension at least s, that is closed under countable intersections and similarities (the original setting was in R; we make the obvious modifications for working on T). This property is somewhat counter-intuitive, because one usually expects for “generic” sets E and F that dimH (E ∩ F ) = dimH (E) + dimH (F ) − 1 (1 corresponds here to the dimension of T) (for precisions on what is meant by “generic”, and a sufficient condition for this relation to hold, see [24, 31]). On the contrary, if E and F belong to G s (T), their intersection still has dimension s. With this notion, and without any extra hypothesis, the conclusion of Proposition 5.2 can be improved, as follows. 1
Proposition 5.4. Under the hypotheses of Proposition 5.2, B t ∈ G t (T). Proof. According to Theorem B of [8], it suffices to prove that for any sequence of similarity transformations (fk ), we have 1 fk (B t ) ≥ . (39) dimH t k≥1
The original proof of Proposition 5.2, which relies on the construction of a generalized Cantor set included in B t , may be adapted to that purpose. We will prove the following lemma later.
Lemma 5.5. If lim supn Sn = 1, then for any interval I ⊂ T, for any c < 1, there exists a finite set of indices E ⊂ N such that for all n, n ∈ E, n = n , Sn ∩ Sn = ∅, Sn ⊂ I , and |Sn | ≥ c|I |. n∈E
We apply this lemma to T with c = 21 , obtaining a set of indices E0 . The segments n ∈ E0 , form the first stage of our generalized Cantor set. For each n0 ∈ E0 , we apply Lemma 5.5 to Snt 0 with c = √1 , obtaining a new set 2 of indices E1 (n0 ). But notice that the segments f1 (Sn ), n ∈ N can also be used in
Lemma 5.5, because if f1 is a similarity, lim supn f1 (Sn ) = 1. We do this on each Sn , n ∈ E1 (n0 ), again with c = √1 . At this point, we obtained a family of disjoint segments Snt ,
2
Sn ∩ f1 (Sn ) filling (in Lebesgue measure) at least half of each of the Snt 0 , n0 ∈ E0 . The (Sn ∩ f1 (Sn ))t form the second stage of the Cantor set. The construction of the next stages follows the same principle. At each step, we introduce a new similarity, so that at step k + 1 we obtain a family of disjoint segments Sn ∩ f1 (Sn ) ∩ · · · ∩ fk (Snk ) filling at least half of the segments from the previous stage. The (Sn ∩ f1 (Sn ) ∩ · · · ∩ fk (Sn(k) ))t form the stage number k + 1. In " the end, K is the intersection of all the stages. It is straightforward to check that K ⊂ k≥1 fk (B t ), and, exactly as in [14], that the natural measure µK supported by K 1
has the scaling property µK (I ) ≤ C|I | t log(|I |)2 for any interval I ⊂ T, which proves that dimH (K) ≥ 1t .
Random Wavelet Series
509
Proof of Lemma 5.5. We can suppose that the |Sn | are taken in decreasing order. Let n1 be the first index such that Sn1 ⊂ I , n2 the first index such that Sn2 ⊂ I \Sn1 , and so on. Suppose that 1. x ∈ I ∩ lim sup Sn ; 2. x is never covered by any of the 3Snj , j ∈ N. Then 1 implies that there exists m such that x ∈ Sm , and by 2 we know that around x there is enough space not covered by any of the Snj , nj ≤ m to fit Sm (because their sizes are decreasing). So m would be eventually selected as one of the nj , which is clearly in contradiction with 2.
!
As a consequence, j 3Snj → |I |, so there exists a finite set of indices E(I ) such
!
that n∈E(I ) 3Sn ≥ 21 |I |, hence n∈E(I ) |Sn | ≥ 16 |I |. ! But we can apply the same argument to each of the segments composing I \ n∈E(I )Sn , obtaining a new family of disjoint segments. The total covering is now larger than ( 16 + 16 (1 − 16 ))|I |. This construction yields, after n steps, a disjoint covering larger than un |I |, where u0 = 0 and un+1 = un + 16 (1 − un ). Since un → 1 when n → ∞, Lemma 5.5 is proved. Corollary 5.6. Almost surely, for all α > 0, d ≥ λ(α), E(α, d) ∈ G
λ(α) d
(T).
The proof is similar to the proof of Corollary 5.3. In particular, Corollary 5.6 shows that the dimension of the difference of two such sets (which is needed below) cannot be deduced from the dimension of their intersection, because the latter is as big as the dimension of the smallest set. Nevertheless, Proposition 5.7. Let E ⊂ T be a fixed set and E = lim sup Si where the Si are open segments with centers uniformly and independently drawn on T, and i |Si | < ∞. Then almost surely dimH (E\E ) = dimH (E). Proof. If dimH (E) = 0, E has at least one point, which is almost surely not in E (the latter being of measure zero, by the Borel–Cantelli lemma). If dimH (E) > 0, for all 0 < s < dimH (E), Hs (E) = +∞, thus according to the Frostman lemma, there exists a non-zero measure µ with support in E, such that for all x ∈ T for all r > 0, µ(B(x, r)) ≤ r s . It is well known (see for instance [7]) that the existence of such a measure implies conversely that dimH (E) ≥ s. If this measure was with support in E\E , the lemma would thus be proved. Unfortunately, since F := T\E is not closed in T, the support of µF (the measure restricted to F ) is not necessarily included in E ∩ F , so we cannot use this measure directly. " However Fk := i≥k T\Si is compact, so µFk has its support in E ∩ Fk . We just need to prove that µ(Fk ) > 0. For i ≥ K, K large enough,
E µ(T\Si ) = µ(E)(1 − |Si |), n n # E µ = µ(E) T\Si (1 − |Si |), i=K
E(µ(FK )) = µ(E)
i=K ∞ # i=K
(1 − |Si |) > 0,
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hence with non-zero probability, µ(FK ) > 0. The measure µFK allows thus to prove that dimH (E\E ) = dimH (E ∩ F ) ≥ s with a non-zero probability. But dimH (E\E ) is a tail random variable (it doesn’t depend on any finite number of the Si ), obeying Kolmogorov zero-one law, so almost surely dimH (E\E ) ≥ s. To conclude, we take s in a sequence si $ dimH (E). Remark. Here the set E is fixed and E is random. It follows that the same result holds with E random if its building process is independent from E . Proposition 5.7 can naturally be extended to the case where a finite number of Ei , i ∈ E, is subtracted from E. This is not true in general if E is countable; but in a more precise case, we can combine the ideas of Propositions 5.4 and 5.7. Proposition 5.8. Let B t be fixed as in Proposition 5.2, and E a random set as in Propo1 sition 5.7. Then almost surely B t \E ∈ G t (T). 1
Proof. Theorem B in [8] gives another characterization for B t ∈ G t (T): it is that for 1/t any dyadic segment I of size 2−j , we have H∞ (I ∩ B t ) = 2−j/t . Now we want to 1/t calculate H∞ (I ∩ B t \E ). With the notations of Proof 5.7, for all k ∈ N, ∞
# 1/t E H∞ (I ∩ B t ∩ Fk ) = 2−j/t (1 − |Si |), i=k
hence E(H∞ (I ∩ B t \E )) = 2−j/t ; since it is a tail variable, this means that with 1/t probability one, H∞ (I ∩ B t \E ) = 2−j/t . There are only countably many dyadic segments, so almost surely, this is true for all dyadic I . 1/t
1
This time, because the class of sets with large intersection G t (T) is stable under ! 1 countable intersection, B t \ i∈N Ei is still in G t (T). We shall need this in the proof of Proposition 5.9 below. 5.3. Lower bound for the dimension of H (α, d). Proposition 5.9. Almost surely, for all α > 0, d ≥ ρ(α), dimH (H (α, d)) ≥
ρ(α) . d
Before entering the proof, let us make a few remarks. As usual when we want to lower bound the dimension of a set with a rather non-geometric definition, we look for a geometrically simpler set that is small enough to fit in H (α, d), but large enough to obtain a lower bound for the Hausdorff dimension. One of the difficulties here is that we want an almost sure property for all α and d. To ensure this, we will make this “almost sure”, that is, the exceptional event of probability zero, depend only on parameters belonging to a countable set. The large intersection property, which forced us to make the detour by Propositions 5.7 and 5.8, actually saves us there, and here is why. We want to prove that a certain property (a lower bound for the dimension) is almost surely true for a whole family of sets depending on some real parameters.
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We will express these sets as a countable intersection of some other sets that depend only on rational parameters, and we prove for each of these sets, almost surely, the slightly stronger property that it belongs to a G s (T). This property is then almost sure for all the rational parameters, and we know that it is stable by countable intersection: it will thus be true, almost surely, for the real parameters. Proof of Proposition 5.9. Let ε1 , ε2 > 0 and Eεj1 ,ε2 (α, d) := E j (α + ε2 , d)\E j (α − ε1 , d), Eε1 ,ε2 (α, d) := lim sup Eεj1 ,ε2 (α, d) j →∞
(this is just a complicated version of Eε (α, d), but now we can take ε1 and ε2 independently). Using Proposition 5.4, we see that [almost surely, for any α > 0, d > ρ(α), ε1 ∈ Q+α and ε2 ∈ Q − α], we have Eε1 ,ε2 (α, d) ∈ G s (T), where s ≥ ρ(α) d is the dimension of Eε1 ,ε2 (α, d). Let d0 ∈ (ρ(α), d) ∩ Q and
Gε1 ,ε2 (α, d) := Eε1 ,ε2 (α, d)\ G(α, d) :=
Eε1 ,ε2 (α, d ),
(40)
d >d
Gε1 ,ε2 (α, d),
ε1 ,ε2 >0
ˆ G(α, d) := G(α, d)\
(41)
E(α , d0 ),
α <α
˜ ˆ G(α, d) := G(α, d)\
α<α <α
(42)
E α −α 2
α + α ,d . 2
(43)
Naturally, we take d , α , α rational so that the intersections and unions are countable. For any d > d, the dimension of Eε1 ,ε2 (α, d ) is [always] strictly less than s, so Eε1 ,ε2 (α, d)\Eε1 ,ε2 (α, d ) is [almost surely, for any α > 0, d > ρ(α), ε1 ∈ Q + α and ε2 ∈ Q − α] in G s (T) as well. By stability of G s (T) under countable intersection, this implies in (40) that [almost surely, for any α > 0, d > ρ(α), ε1 ∈ Q + α and ε2 ∈ Q − α], Gε1 ,ε2 (α, d) ∈ G s (T) ρ(α)
and then in (41), that [almost surely, for any α > 0, d > ρ(α)], G(α, d) ∈ G d (T). Now if we take a close look at (42) and (43), each of the sets that are subtracted is independent from G(α, d), so by Proposition 5.8, [almost surely, for any α > 0, ρ(α) ˜ d > ρ(α)], G(α, d) is still in G d (T), and in particular its dimension is larger than ρ(α) d . ˜ ˜ To conclude, we only have to show that G(α, d) ⊂ H (α, d). Let x ∈ G(α, d) and ε > 0. 1. For ε1 , ε2 < ε, E(α + ε, d − ε) contains Eε1 ,ε2 (α, d), hence x ∈ E(α + ε, d − ε). 2. When ε and ε are small enough, x cannot belong to E(α − ε, d − ε αd + ε ) because in (42) we subtracted all E(α − ε, d0 ). 3. Suppose x ∈ E(α + ε, d + ε αd ). Then either (cases are non exclusive)
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(a) there exist α < α and infinitely many (j, k) such that |k2−j − x| < 2−(d+ε α ) j and 2 2 |Cj k | ≥ 2−(α )j ; (b) there exist α ∈ (α, α + ε) and infinitely many (j, k) such that |k2−j − x| < j d 2−(d+ε α ) and 2−(α+ε)j ≤ 2 2 |Cj k | ≤ 2−(α )j ; (c) for all α < α < α , there exist infinitely many (j, k) such that |k2−j − xl < j d 2−(d+ε α ) and 2−(α )j ≤ 2 2 |Cj k | ≤ 2−(α )j . But 3(a) is ruled out by (42), 3(b) is ruled out by (43), and 3(c) by (40) and (41). So finally x ∈ E(α + ε, d + ε αd ). These three properties 1, 2, and 3 are sufficient (and indeed necessary) for x ∈ H (α, d): the proof is now complete. Proof of Theorem 2. First, (13) and (14) were already seen as a consequence of Propositions 4.1 and 4.5. Together with the upper bound given by Theorem 1, Proposition 5.9 proves directly (15). To prove (16), it suffices to remark that for all β, d(h) ≥ d(h, β), hence d(h) ≥ supβ d(h, β). So, if the upper bound in Theorem 1 is attained for d(h, β), the same is true for d(h).
A. Weak Hölder Uniform Regularity As we mentioned in Sect. 2.3, the uniform Hölder regularity condition (3) is not the weakest hypothesis that ensures local boundedness, and thus that the Hölder and oscillating singularity exponents can be recovered from the modulus of the wavelet coefficients. In Propositions 2.1, 4.1, 4.2, 4.3, as well as for (4) in Theorem 1, it can be replaced by the following condition [16]: Definition 5. We say that f is a weak uniform Hölder function if for all n > 0, there exist Cn such that for all j > 0, |Cj,k | ≤
Cn . jn
(44)
This is equivalent to the following uniform Hölder condition: for all n ∈ N, there exists C(n) such that for all x, y, |f (x) − f (y)| ≤
C(n) . (1 + | log(|x − y|)|)n
If the R.W.S. hypothesis (Definition 4) is replaced by $ % n log(j ) (H) There exists nj → ∞ such that ρj is supported in j j , +∞ , then almost every sample path of the random wavelet series is weak uniform Hölder. Also note that Propositions 3.1, 3.2 and 3.3 still hold if Definition 4 is replaced by (H) (the proofs were designed for this case). Unfortunately, we cannot make weak uniform Hölder regularity our basic hypothesis for this paper. Even for the deterministic upper bound in Theorem 1, (strong) uniform Hölder regularity is required for (5). The following example shows what can happen if f is only weak uniform Hölder.
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Example 1. At each scale j , take 2log(j ) equidistributed (up to round-off errors) wavelet 2 coefficients of size 2− log(j ) (the other coefficients being set equal to 0). Then ρ(α) = −∞ if α = 0, and ρ(0) = 0. But it is not difficult to see that f has everywhere a Hölder exponent equal to 0, so d(0) = 1. 3
However, (4) holds even if f is not uniform Hölder. Example 1 thus shows a situation where d(0) = 1, but for all 0 ≤ β < ∞, d(0, β) ≤ 0 (and d(0, +∞) = 1). Here is its random counterpart, showing that (strong) uniform Hölder regularity is also needed for Theorem 2. Example 2. Take
ρj := 2−j log(j )3 δ log(j )2 + 1 − 2−j log(j )3 δj log(j ) .
(45)
j
With probability one, the spectrum of singularities of f is reduced to d(0) = 1, whereas d(0, β) = 0 for any β ≥ 0 (and d(0, +∞) = 1). In this case, naturally, hmin = hmax = 0 so (16) doesn’t make sense. But the problem is more serious than that: if we add a uniform Hölder random wavelet series to (45), then hmax > 0 but still for almost every x, hf (x) = 0 (the big wavelet coefficients in (45) “mask” everything else), so (13) also fails in this case. However, uniform Hölder regularity may not be the weakest condition necessary for Theorems 1 and 2; this question is still open. References 1. Arneodo, A., Bacry, E., Jaffard, S., Muzy, J.-F.: Singularity spectrum of multifractal functions involving oscillating singularities. J. Fourier Anal. Appl. 4 (2), 159–174 (1998) 2. Arneodo, A., Bacry, Muzy, J.-F.: Random cascades on wavelet dyadic trees. J. Math. Phys. 39 (8), 4142– 4164 (1998) 3. Aubry, J.-M.: Representation of the singularities of a function. Appl. Comput. Harmonic Anal. 6, 282–286 (1999) 4. Aubry, J.-M., Jaffard, S.: Random wavelet series: Theory and applications. To be presented at Fractal 2002, Granada 5. Chillà, F., Peinke, J., Castaing, B.: Multiplicative process in turbulent velocity statistics: A simplified analysis. J. Phys. II France 6 (4), 455–460 (1996) 6. Daubechies, I.: Ten Lectures on Wavelets, Volume 61 of CBMS-NSF regional conference series in applied mathematics. SIAM, 1992 7. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. Chichester: John Wiley & Sons, 1990 8. Falconer, K.J.: Sets with large intersection properties. J. Lond. Math. Soc. (2) 49, 267–280 (1994) 9. Frisch, U.: Fully developed turbulence and intermittency. In M. Ghil, ed., Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, Volume 88, International School of Physics Enrico Fermi, Amsterdam: North-Holland, June 1983, pp. 71–88 10. Frisch, U.: Turbulence : The legacy of A.N. Kolmogorov. Cambridge: Cambridge University Press, 1995 11. Jaffard, S.: Multifractal formalism for functions Part I: Results valid for all functions. SIAM J. Math. Anal. 28(4), 944–970 (1997) 12. Jaffard, S.: Multifractal formalism for functions Part II: Selfsimilar functions. SIAM J. Math. Anal. 28 (4), 971–998 (1997) 13. Jaffard, S.: Oscillation spaces: Properties and applications to fractal and multifracal functions. J. Math. Phys. 39 (8), 4129–4141 (1998) 14. Jaffard, S.: Lacunary wavelet series. Ann. Appl. Probab. 10 (1), 313–329 (2000) 15. Jaffard, S.: On the Frish–Parisi conjecture. J. Math. Pures Appl. 79 (6), 525–552 (2000) 16. Jaffard, S., Meyer, Y.: Wavelet methods for pointwise regularity and local oscillations of functions. Mem. Amer. Math. Soc. 123, 587 (Sept. 1996)
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17. Jaffard, S., Meyer, Y.: On the pointwise regularity of functions in critical Besov spaces. J. Funct. Anal. 175, 415–434 (2000) 18. Kahane, J.-P.: Some random series of functions. Cambridge: Cambridge University Press, 1985 19. Kolmogorov, A. N.: C. R. Acad. Sci. URSS 30, 301–305 (1941) 20. Kolmogorov, A. N.: J. Fluid Mech. 13, 82–85 (1962) 21. Körner, T.: Kahane’s Helson curve. J. Fourier Anal. Appl., pp. 325–346 (1995). Special Issue: Proceedings of the conference in honor of J.-P. Kahane 22. Lemarié, P.-G., Meyer, Y.: Ondelettes et bases hilbertiennes. Rev. Mat. Iber. 2 (1/2), 1–18 (1987) 23. Mandelbrot, B. B.: Intermittent turbulence in self similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974) 24. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Volume 44 of Cambridge studies in advanced mathematics. Cambridge: Cambridge University Press, 1995 25. Meneveau, C., Sreenivasan, K.: Measurement of f (α) from scaling of histograms and applications to dynamical systems and fully developed turbulence. Phys. Letters A 137, 103–112 (1989) 26. Meyer, Y.: Ondelettes et Opérateurs I : Ondelettes. Actualités Mathématiques. Paris: Hermann, 1990 27. Müller, P., Vidakovic, B. (editors): Bayesian Inference in Wavelet Based Models, Volume 141 of Lect. Notes Stat., Berlin–Heidelberg–New York: Springer-Verlag, 1999 28. Shepp, L.A.: Covering the circle with random arcs. Israel J. Math. 11, 328–345 (1972) 29. Simoncelli, E.: Bayesian denoising of visual images in the wavelet domain. Lect. Notes Stat. 141, 291–308 (1999) 30. Stute, W.: The oscillation behavior of empirical processes. Ann. Probab. 10, 86–107 (1982) 31. Tricot, C.: Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91, 57–74 (1982) Communicated by J. L. Lebowitz
Commun. Math. Phys. 227, 515 – 539 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Phase-Averaged Transport for Quasi-Periodic Hamiltonians Jean Bellissard1,2 , Italo Guarneri3,4,5 , Hermann Schulz-Baldes6 1 2 3 4 5 6
Université Paul-Sabatier, 118 route de Narbonne, 31062 Toulouse, France Institut Universitaire de France Università dell’Insubria a Como, via Valleggio 11, 22100 Como, Italy Istituto Nazionale per la Fisica della Materia, via Celoria 16, 20133 Milano, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy University of California at Irvine, CA 92697, USA
Received: 30 May 2001 / Accepted: 2 January 2002
Abstract: For a class of discrete quasi-periodic Schrödinger operators defined by covariant representations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl– Heisenberg–Gabor lattices of coherent states is given. 1. Introduction This work is devoted to proving a lower bound on the diffusion exponents of a class of quasiperiodic Hamiltonians in terms of the multifractal dimensions of their density of states (DOS). The class of models involved describes the motion of a charged particle in a perfect two-dimensional crystal with 3-fold, 4-fold or 6-fold symmetry, submitted to a uniform irrational magnetic field. Irrationality means that the magnetic flux through each lattice cell is equal to an irrational number θ in units of the flux quantum. As shown by Harper [Har] in the specific case of a square lattice with nearest neighbor hopping, the Landau gauge allows to reduce such models to a family of Hamiltonians each describing the motion of a particle on a 1D chain with quasiperiodic potential. The latter representation gives a strongly continuous family H = (Hω )ω∈T of selfadjoint bounded operators on the Hilbert space 2 (Z) of the chain indexed by a phase ω ∈ T = R/2πZ. This family satisfies the covariance relation THω T−1 = Hω+2πθ (here T represents the operator of translation by one site along the chain). The phase-averaged diffusion exponents β(q), q > 0, of H are defined by: T dt qβ(q) q e−ıHω t |φ T∼ φ|eıHω t |X| dω ↑∞ T , 2T T −T
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denotes the position operator on the chain. The DOS of the family H is the where X Borel measure N defined by phase-averaging the spectral measure with respect to any site. Its generalized multifractal dimensions DN (q) for q = 1 are formally defined by
R
dN (E)
E+ε
E−ε
dN (E )
q−1
∼
ε↓0
ε (q−1)DN (q) .
A somewhat imprecise statement of the main result of this work is: whenever θ/2π is a Roth number [Her] (namely, for any > 0, there is c > 0 such that |θ − p/q| ≥ c/q 2+ for all p/q ∈ Q), and for the class of models mentioned above, the following inequality holds for all 0 < q < 1: β(q) ≥ DN (1 − q).
(1)
This result can be reformulated in terms of two-dimensional magnetic operators on the lattice and then gives an improvement of the general Guarneri–Combes–Last lower bound [Gua, Com, Las] by a factor 2. More precise definitions and statements will be given in Sect. 2. The inequality (1) has been motivated by work by Piéchon [Pie], who gave heuristic arguments and numerical support for β(q) = DN (1 − q) for q > 0, valid for the Harper model and the Fibonacci chain (for the latter case, a perturbative argument was also given). It was theoretically and numerically demonstrated by Mantica [Man] that the same exact relation between spectral and transport exponents is also valid for the Jacobi matrices associated with a Julia set. This result was rigorously proven in [GSB1, BSB]. For the latter operators, the DOS and the local density of states (LDOS) coincide. Numerous works [Gua, Com, Las, GSB2, GSB3, BGT] yield lower bounds on the quantum diffusion of a given wave packet in terms of the fractal properties of the corresponding LDOS. These rigorous lower bounds are typically not optimal as shown by numerical simulations [GM, KKKG]. Better lower bounds are obtained if the behaviour of generalized eigenfunctions is taken into account [KKKG]. Kiselev and Last have proven general rigorous bounds in terms of upper bounds for the algebraic decay of the eigenfunctions [KL]. However, in most models used in solid state physics, the Hamiltonian is a covariant strongly continuous family of self-adjoint operators [Bel] indexed by a variable which represents the phase or the configuration of disorder. The measure class of the singular part of the LDOS may sensitively depend on the phase [DS]. In addition, the multifractal dimensions are not even measure class invariants [SBB] (unlike the Hausdorff and packing dimensions). This raises concerns about the practical relevance of bounds based on multifractal dimensions of the LDOS in this context. The bound (1) has a threefold advantage: (i) it involves the DOS, which is phase-averaged; (ii) it does not require information about eigenfunctions; (iii) the exponent of phase-averaged transport is the one that determines the low temperature behaviour of the conductivity [SBB]. The present formulation uses the C∗ -algebraic framework introduced by one of the authors for the study of homogeneous models of solid state physics. While referring to [Bel, SBB] for motivations and details, in the opening Sect. 2 we briefly recall some of the basic notions. A precise statement of our main results is also given in Sect. 2, along with an outline of the logical structure of their proofs. In the subsequent sections we present more results and proofs.
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2. Notations and Results A number α ∈ R is of Roth type if and only if, for any > 0, there is a constant c > 0 such that for all rational numbers p/q the following inequality holds: α − p ≥ c . (2) q q 2+ Most properties of numbers of Roth type can be found in [Her]. They form a set of full Lebesgue measure containing all algebraic numbers (Roth’s theorem). θ > 0 will be called a Roth angle if θ/2π is a number of Roth type. The rotation algebra Aθ [Rie] is the smallest C ∗ -algebra generated by two unitaries U and V , such that U V = eıθ V U . It is convenient to set Wθ (m) = e−ıθm1 m2 /2 U m1 V m2 , whenever m = (m1 , m2 ) ∈ Z2 . The Wθ (m)’s are unitary operators satisfying Wθ (l)Wθ (m) = eı(θ/2)l∧m Wθ (l + m), where l ∧ m = l1 m2 − l2 m1 . The unique trace on Aθ (θ/2π irrational) is defined by Tθ (Wθ (m)) = δm,0 . A strongly continuous action of the torus T2 on Aθ is given by ((k1 , k2 ), Wθ (m)) ∈ T2 × Aθ → eı(m1 k1 +m2 k2 ) Wθ (m). The associated ∗-derivations are denoted by δ1 , δ2 . For n ∈ N, one says A ∈ C n (Aθ ) if δ1m1 δ2m2 A ∈ Aθ for all positive integers m1 , m2 satisfying m1 + m2 ≤ n. Aθ admits three classes of representations that will be considered in this work. The 1D-covariant representations is a faithful family (πω )ω∈R of representations on 2 (Z) are the shift and the defined by πω (U ) = T and πω (V ) = eı(ω−θ X) , where T and X position operator respectively, namely Tu(n) = u(n − 1),
Xu(n) = nu(n),
∀u ∈ 2 (Z).
It follows that πω+2π = πω (periodicity) and that Tπω (·)T−1 = πω+θ (·) (covariance). Moreover ω → πω (·) is norm continuous. In the sequel, it will be useful to denote by |n = un (n ∈ Z) the canonical basis of 2 (Z) defined by un (n ) = δn,n . The 2Drepresentation (or the GNS-representation of Tθ ) is given by the magnetic translations on 2 (Z2 ) (in symmetric gauge): π2D (Wθ (m))ψ(l) = eıθm∧l/2 ψ(l − m),
ψ ∈ 2 (Z2 ).
The position operators on 2 (Z2 ) are denoted by (X1 , X2 ). The Weyl representation πW acts on L2 (R). Let Q and P denote the position and momentum operators defined by Qφ(x) = xφ(x) and P φ = −ıdφ/dx whenever φ belongs to the Schwartz space S(R). It is known that Q and P are essentially selfadjoint and satisfy the canonical commutation rule [Q, P ] = ı1. Then πW is defined by πW (U ) = eı
√ θP
,
πW (V ) = eı
√ θQ
.
For every θ > 0, πW and π2D are unitarily equivalent and faithful. More results about Aθ are reviewed in Sect. 3.2. The group SL(2, Z) acts on Aθ through the automorphisms ηS (Wθ (m)) = Wθ (Sm), S ∈ SL(2, Z). S is called a symmetry if S = ±1 and supn∈N S n < ∞. Of special interest are the 3-fold, 4-fold and 6-fold symmetries 0 −1 0 −1 1 −1 , S4 = , S6 = , S3 = 1 −1 1 0 1 0
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respectively generating the symmetry groups of the hexagonal (or honeycomb), square and triangular lattices in dimension 2. In this work, the Hamiltonian H = H ∗ is an element of Aθ . Of particular interest are Hamiltonians invariant under some symmetry S ∈ SL(2, Z), that is ηS (H ) = H . The most prominent among such operators is the (critical) Harper Hamiltonian on a square lattice H4 = U + U −1 + V + V −1 . For the sake of concreteness, let us write out its covariant representations u ∈ 2 (Z). πω (H4 )u(n) = u(n + 1) + u(n − 1) + 2 cos(nθ + ω)u(n), √ √ Its Weyl representation is πW (H4 ) = 2 cos( θQ) + 2 cos( θP ). Further examples are the magnetic operator on a triangular lattice H6 = U + U −1 + V + V −1 + e−ıθ/2 U V + e−ıθ/2 U −1 V −1 as well as on a hexagonal lattice (which reduces to two triangular ones [Ram]). For H = H ∗ ∈ Aθ let us introduce the notations Hω = πω (H ) and H2D = π2D (H ). Its density of states (DOS) is the measure N defined by (see, e.g., [Bel]) R
dN (E)f (E) = Tθ (f (H )) = 0|f (H2D )|0 = lim
0→∞
1 Tr0 (Hω ), 0
f ∈ C0 (R). (3)
Here |0 denotes the normalized state localized at the origin of Z2 , Tr0 (A) = 0 n=1 n|A|n and the last equality in (3) holds uniformly in ω. For a Borel set 1 ⊂ R and a Borel measure µ, the family of generalized multifractal dimensions is defined by
q−1 log 1 dµ(E) 1 dµ(E ) exp(−(E − E )2 T 2 ) 1 Dµ± (1; q) = lim ± , 1 − q T →∞ log(T ) (4) where lim+ and lim− denote lim sup or lim inf respectively. The gaussian exp(−(E − E )2 T 2 ) may be replaced by the indicator function on [E − T1 , E + T1 ] without changing the values of the generalized dimensions [GSB3, BGT]. Let now H ∈ C 2 (Aθ ). The diffusion exponents of H2D are defined by ± β2D (H, 1; q) = lim
T →∞
± log(M2D (H, 1; q, ·)T )
q log(T )
,
q ∈ (0, 2],
(5)
where M2D (H, 1; q, t) = 0|χ1 (H2D )eıH2D t (|X1 |q + |X2 |q )e−ıH2D t χ1 (H2D )|0, (6) +T and f (·)T denotes the average −T dtf (t)/2T of a measurable function t ∈ R → f (t) ∈ R. The phase-averaged diffusion exponents of the covariant family (Hω )ω∈R are defined as in (5) as growth exponents of 2π dω ˆ q e−ıHω t χ1 (Hω )|0. M1D (H, 1; q, t) = (7) 0|χ1 (Hω )eıHω t |X| 2π 0 Because H ∈ C 2 (Aθ ) and q ∈ (0, 2], M2D (H, 1; q, t) and M1D (H, 1; q, t) are finite. ± (H, 1; q) and β ± (H, 1; q) take values in the interval [0, 1] as long as Moreover, β2D 1D the boundary of 1 lies in gaps of H [SBB].
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Main Theorem. Let θ be a Roth angle and H = H ∗ ∈ C 2 (Aθ ). (i) For any Borel subset 1 ⊂ R and q ∈ (0, 1), ± ± (H, 1; q) ≥ DN (1; 1 − q). β2D
(8)
(ii) Let H be invariant under some symmetry S ∈ SL(2, Z). Then, for any Borel subset 1 ⊂ R and q ∈ (0, 1), ± ± (H, 1; q) ≥ DN (1; 1 − q). β1D
(9)
Remark 1. Existing lower bounds (inequalities proved in [GSB3, BGT]) yield ± (H, 1; q) ≥ 1 D ± (1; 1/(1 + q)), where the factor 1 stems from the dimension β2D 2 N 2 ± ± (1; 1 − q) ≥ DN (1; 1/(1 + q)), so inequality (8) of physical space. In addition, DN substantially improves such bounds. The same is true of the inequality in Theorem 1 below which is actually the key to the bounds (8) and (9). This crucial improvement follows from an almost-sure estimate on the growth of the generalized eigenfunctions in the Weyl representation (cf. Proposition 4 below) which in turn follows from number-theoretic estimates. As in [KL], a control on the asymptotics of the generalized eigenfunctions then leads to an improved lower bound on the diffusion coefficients (here by a factor 2 at q = 0). Remark 2. The bound (8) is of practical interest especially if H is invariant under some symmetry. Non-symmetric Hamiltonians may lead to ballistic motion and absolutely continuous spectral measures (as it is generically the case for the non-critical Harper Hamiltonian, see [Jit] and references therein). In this situation, the bound becomes trivial because both sides in (9) are equal to 1. Remark 3. Numerical results [TK, RP] as well as the Thouless property [RP] support that DN (−1) = 21 in the case of the critical Harper Hamiltonian H4 for Diophantine θ/(2π). According to (9), one thus expects β1D (H4 , R; 2) ≥ 21 . Remark 4. Numerical simulations by Piéchon [Pie] for the Harper model with some strongly incommensurate θ/(2π ) indicate that (9) may actually be an exact estimate. Piechon also gave a perturbative argument supporting the equality β1D (H ; q) = DN (1− q) in the case of the Fibonacci Hamiltonian, and verified it numerically. The techniques of the present article do not apply to the Fibonacci model which has no phase-space symmetry. + − (1; q) = DN (1; q) Remark 5. Our proof forces q ∈ (0, 1) (see Lemma 3). If DN for all q = 1, the large deviation technique of [GSB3] leads to (8) for all q > 0 (if H ∈ C ∞ (Aθ )) and (9) for all q ∈ (0, 2]. Numerical results [TK, RP] suggest that the upper and lower fractal dimensions indeed coincide for Diophantine θ/(2π ). This is unlikely for θ/(2π) Liouville (compare [Las]).
Remark 6. Two-sided time averages are used for technical convenience. Important steps forwards of the proof are summarized below. Associated with the ηS with symmetry S there is a harmonic oscillator Hamiltonian HS invariant under ground state φS ∈ S(R), see Sect. 3.3. In the case of S4 (relevant to the critical Harper model) this is the conventional harmonic oscillator Hamiltonian HS 4 = (P 2 + Q2 )/2, and φS is the gaussian state. Let ρS be the spectral measure of HW = πW (H ) with respect to φS .
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Proposition 1. Let θ > 2π. There are two positive constants c± such that for any Borel subset 1 ⊂ R, c− N (1) ≤ ρS (1) = φS |χ1 (HW )|φS ≤ c+ N (1). In particular, N and ρS have the same multifractal exponents. The Hamiltonian HS will be used to study transport in phase space. Similarly to Eqs. (5) and (6), moments of the phase space distance and growth exponents thereof can be defined in the Weyl representation as follows: q/2
MW (H, 1; q, t) = φS |χ1 (HW )eıtHW HS e−ıtHW χ1 (HW )|φS , log(MW (H, 1; q, ·)T ) ± (H, 1; q) = lim ± . βW T →∞ q log(T ) Proposition 2. Let θ > 2π and H = H ∗ ∈ C 2 (Aθ ). For q ∈ (0, 2], ± ± βW (H, 1; q) = β2D (H, 1; q).
ηS for some Proposition 3. Let θ > 2π and H = H ∗ ∈ C 2 (Aθ ) be invariant under symmetry S ∈ SL(2, Z). Then ± ± βW (H, 1; q) ≤ β1D (H, 1; q),
q ∈ (0, 2].
Thanks to Propositions 1, 2 and 3 and since θ may be replaced by θ + 2π without changing the 1D and 2D-representations, the Main Theorem is a direct consequence of the following: Theorem 1. Let H = H ∗ ∈ C 2 (Aθ ) and θ > 2π be a Roth angle. Then, for any Borel subset 1 ⊂ R, ± βW (H, 1; q) ≥ Dρ±S (1; 1 − q),
∀q ∈ (0, 1).
The proof of Theorem 1 will require two technical steps that are worth being mentioned here. The first one requires some notations. Given a symmetry S, let 6S be the projection onto the HW -cyclic subspace HS ⊂ H of φS . Using the spectral the(n) orem, there is an isomorphism between HS and L2 (R, dρS ). If (φS )n∈N denotes the (n) orthonormal basis of eigenstates of HS in H, let 7n,S (E) be the representative of 6S φS 2 in L (R, dρS ). Then: Proposition 4. Let H = H ∗ ∈ C 2 (Aθ ) and let θ be a Roth angle. Then for any > 0 there is c > 0 such that ∞
|7n,S (E)|2 e−δ(n+1/2) ≤ c δ −(1/2+) ,
∀0 < δ < 1,
ρS -a.e. E ∈ R.
n=0
Remark 7. This result is uniform (ρS -almost surely) with respect to the spectral parameter E and to δ. In particular, integrating over E with respect to ρS shows that N−1 (n) 2 1/2+ ). This is possible because of the following complen=0 6S φS = O(N mentary result proved in the Appendix:
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Proposition 5. Let H = H ∗ ∈ Aθ . Then HW has infinite multiplicity and no cyclic vector. The second technical result concerns the so-called Mehler kernel of the Hamiltonian HS , notably the integral kernel of the operator e−tHS in the Q-representation: MS (t; x, y) = x|e−tHS |y.
(10)
Proposition 6. Let θ be a Roth angle. Then, for all > 0, as t ↓ 0 sup |MS (t; x + 2π m1 θ −1/2 , y + θ 1/2 m2 )| = O(t −1/2− ). 0≤x≤2πθ −1/2 ,0≤y≤θ 1/2 m∈Z2
3. Weyl’s Calculus This chapter begins with a review of basic facts about Weyl operators, the rotation algebra and implementation of symmetries therein. The formulas are well-known (e.g. [Per, Bel94] and mainly given in order to fix notations, but for the convenience of the reader their proofs are nevertheless given in the Appendix. The chapter also contains a new and compact solution of the frame problem for coherent states (Sect. 3.4). 3.1. Weyl operators. Let H denote the Hilbert space L2 (R). Given a vector a = (a1 , a2 ) ∈ R2 , the associated Weyl operator is defined by: W(a) = eı(a1 P +a2 Q) ⇔ W(a)ψ(x) = eıa1 a2 /2 eıa2 x ψ(x + a1 ),
∀ψ ∈ H. (11)
The Weyl operators are unitaries, strongly continuous with respect to a and satisfy W(a)W(b) = eıa∧b/2 W(a + b),
a ∧ b = a1 b2 − a2 b1 .
(12)
The following weak-integral identities are verified in the Appendix: ψ|W(a)−1 |ψW(a) = W(b)|ψψ|W(b)−1 =
R2
R2
d 2 b ıa∧b W(b)|ψψ|W(b)−1 , e 2π d 2 a ıb∧a ψ|W(a)−1 |ψW(a). e 2π
(13) (14)
Applying (13) to φ and setting a = 0 leads to φ=
R2
d 2b ψ|W(b)−1 |φW(b)ψ, 2π
φ, ψ ∈ H,
ψ = 1.
(15)
In particular, any non-zero vector in H is cyclic for the Weyl algebra {W(a)|a ∈ R2 }. If ψ ∈ H, the map a ∈ R2 → ψ|W(a)|ψ ∈ C is continuous, tends to zero at infinity and belongs to L2 (R2 ), whereas ψ ∈ S(R) if and only if this map belongs to S(R2 ).
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3.2. The rotation algebra. The rotation algebra Aθ , its representations (πω )ω∈R , π2D and πW as well as the tracial state Tθ and ∗-derivations δ1 , δ2 were defined in Sect. 2. Here we give some complements, further definitions and the short proof of Proposition 5. The trace is faithful and satisfies the Fourier formula: A= al Wθ (l), al = Tθ (Wθ (l)−1 A). (16) l∈Z2
In addition, Tθ (A) =
2π 0
dω m|πω (A)|m = l|π2D (A)|l, 2π
∀A ∈ Aθ , ∀m ∈ Z, ∀l ∈ Z2 . (17)
The ∗-derivations satisfy δj Wθ (m) = ımj Wθ (m), j = 1, 2. It follows from (16) that A ∈ C ∞ (Aθ ) if and only if the sequence of its Fourier coefficients is fast decreasing. If A ∈ C ∞ (Aθ ) and A is invertible in Aθ , then A−1 ∈ C ∞ (Aθ ). The position operator (X1 , X2 ) defined on the space s(Z2 ) of Schwartz sequences in 2 (Z2 ) forms a connection [Con] in the following sense: Xj (π2D (A)φ) = π2D (δj A)φ + π2D (A)Xj φ
∀A ∈ C ∞ (Aθ ),
φ ∈ s(Z2 ). (18) √ √ Similarly, if (∇1 , ∇2 ) is defined on S(R) by ∇1 = −ıQ/ θ, ∇2 = ıP / θ , then ∇j (πW (A)ψ) = πW (δj A)ψ + πW (A)∇j ψ
∀A ∈ C ∞ (Aθ ),
ψ ∈ S(R). (19)
Then S(R) is exactly the set of C ∞ -elements of H with respect to ∇. In particular, if ψ ∈ S(R) and A ∈ C ∞ (Aθ ), then πW (A)ψ ∈ S(R). For the Weyl representation, let us use the notations √ πW (Wθ (m)) = Wθ (m) := W( θm), ∀m ∈ Z2 . (20) It can be seen as a direct integral of 1D-representations by introducing the family (Gω )ω∈R of transformations from H into 2 (Z), ω − nθ (Gω φ)(n) = θ −1/4 φ , ∀φ ∈ H. (21) √ θ Then a direct computation (given in the Appendix) shows that: θ dωGω φ|πω (A)|Gω ψ, A ∈ Aθ , φ, ψ ∈ H. φ|πW (A)|φ =
(22)
0
θ In particular, φ2 = 0 dωGω φ22 . The link between πW and π2D will be established in Sect. 4.2. It follows from a theorem by Rieffel [Rie] that the commutant of πW (Aθ ) is the von Neumann algebra generated by πW (Aθ ), where θ /2π = 2π/θ and πW (Wθ (l)) = Wθ (l). The following result is proven in the Appendix:
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Proposition 7 (The generalized Poisson summation formula). Tψθ :=
Wθ (l)|ψψ|Wθ (l)−1 =
l∈Z2
θ ψ|Wθ (m)−1 |ψWθ (m). 2π 2
(23)
m∈Z
By Eq. (23), ψ ∈ S(R) implies Tψθ ∈ C ∞ (Aθ ). It follows immediately from Eq. (23) that, given ψ ∈ S(R), there is a positive element in Aθ , denoted Fψθ , such that Tψθ = (θ/2π)πW Fψθ . Moreover ψ|πW (A)|ψ = Tθ AFψθ ,
∀A ∈ Aθ .
(24)
3.3. Symmetries. It is well-known that S ∈ SL(2, R) can be uniquely decomposed in a torsion, a dilation and a rotation as follows: S=
ab cd
=
10 κ1
λ 0 0 λ−1
cos s − sin s sin s cos s
,
with κ = (ac+db)/(a 2 +b2 ), λ = (a 2 +b2 )1/2 , eıs = (a −ıb)(a 2 +b2 )−1/2 . Moreover, if S ∈ SL(2, R), then there is a unitary transformation FS acting on H such that W(Sa) = FS W(a)FS−1 ,
a ∈ R2 ,
(25)
as shows the above decomposition as well as the following result, the proof of which is deferred to the Appendix: Proposition 8. For any κ, λ, s ∈ R, λ = 0, up to a phase F 1
0 κ 1
F λ
0 0 λ−1
F cos s
− sin s sin s cos s
= e−ıκQ
2 /2
= e−ı ln(λ)(QP +P Q)/2 ,
= e−ıs(Q
,
2 +P 2 −1)/2
(26)
.
Note in particular that FS FS = zFSS for z ∈ C, |z| = 1. Furthermore, if 0 < s < π,
dy 2 2 F cos s − sin s φ(x) = (27) eı cos s(x +y )−2xy /2 sin s φ(y). √ sin s cos s 2π sin s R In the special case s = π/2, namely for the matrix S4 (see Sect. 2), this gives the usual Fourier transform dy FS4 φ(x) = (28) √ e−ıxy φ(y). 2π R
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For the case of the 3-fold and 6-fold symmetries S3 and S6 , acting on a hexagonal or a triangular lattice (see Sect. 2), Eqs. (26) and (27) give dy ıπ/12 FS3 φ(x) = e √ e−ıx(x+2y)/2 φ(y), 2π R (29) dy −ıπ/12 FS6 φ(x) = e √ e−ıy(2x−y)/2 φ(y). 2π R Now suppose that S ∈ SL(2, R) satisfies S r = 1 for some r ∈ N, r ≥ 2 and S n = 1 for n < r. It will be convenient to introduce the following operator acting on H: r−1 1 n 2 −n 1 FS Q FS = K|MS |K, HS = 2r 2 n=0
r−1
1 n MS = S |e2 e2 |(S t )n , r n=0
R2 .
Note that HS4 = (P 2 + where K = (P , Q) and {e1 , e2 } is the canonical basis of 2 n Q )/2. There is 0 ≤ n ≤ r − 1 such that S e2 ∧ e2 = 0, so MS is positive definite and can be diagonalized by a rotation: −1 + cos γ − sin γ cos γ − sin γ µS 0 . MS = sin γ cos γ sin γ cos γ 0 µ− S 2 Hence HS is unitarily equivalent to the harmonic oscillator Hamiltonian (µ+ SP + − 2 µS Q )/2. Therefore, 1/4
∞ µ+ 1 (n) (n) + − 1/2 S n+ |φS φS |, µ = (µS µS ) , λ= , (30) HS = µ 2 µ− S n=0 (n)
(0)
where the φS are the eigenstates. The ground state is denoted φS ≡ φS . Proposition 9. Up to a phase, the ground state is given by + µ− %e(σS ) 1/4 −σS x 2 /2 S cos γ + ı µS sin γ φS (x) = e , σs = , π − µ+ cos γ + ı µ sin γ S S
(31)
and the Mehler kernel (10) by − (x−y)
2 tanh (tµ)−1 +(x+y)2 tanh (tµ) 4(λ2 cos γ 2 +λ−2 sin γ 2 )
2
−2
) e ı(x 2 −y 2 ) 2sin (2γ 2)(λ −λ 4(λ cos γ +λ−2 sin γ 2 ) . e MS (t; x, y) = λ 2π sinh (tµ)(λ2 cos γ 2 + λ−2 sin γ 2 ) (32)
By construction, FS HS FS∗ = HS , so that FS φS = eıδS φS for some phase δS . Thus, it is possible to choose the phase of FS such that FS φS = φS . Such is the case for FSi in Eqs. (28) and (29). Recall from Sect. 2 that ±1 = S ∈ SL(2, Z) is called a symmetry of Aθ if supn∈Z S n < ∞. Since the set of M ∈ SL(2, Z) with M ≤ c is finite (for any 0 < c < ∞), and since S = ±1, there is an integer r ∈ N∗ such that S r = 1 and S n = 1 for 0 < n < r. So the two eigenvalues are {e±ıϕs }, with rϕs = 0 (mod 2π) and ϕs = 0, π. In particular Tr(S) = 2 cos ϕs ∈ Z, implying r ∈ {3, 4, 6} and ϕs ∈ {±π/3, ±π/2, ±2π/3}. Any S ∈ SL(2, Z) defines a ∗-automorphism ηS of Aθ through ηS (Wθ (m)) = Wθ (Sm). According to the above, πW ( ηS (Wθ (m))) = FS πW (Wθ (m))FS−1 .
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3.4. θ -traces and θ -frames. Definition 1. A vector ψ ∈ H will be called θ-tracial if ψ|Wθ (l)|ψ = Tθ (Wθ (l)) = δl,0 for all l ∈ Z2 . Equivalently, the family (Wθ (l)ψ)l∈Z2 is orthonormal. Using the commutation rules (12), it is possible to check that ψ is θ -tracial if and only if W(a)ψ is θ -tracial for any a ∈ R2 . It also follows from Eq. (23) that ψ is θ -tracial if and only if Tψθ = (θ/2π)1. Such θ -tracial states exist under the following condition: Theorem 2. There is a θ -tracial vector ψ ∈ H if and only if θ ≥ 2π. If θ > 2π there is a θ-tracial vector in S(R). For θ ≥ 2π , denote by 6ψ the projection on the orthocomplement of the ψ-cyclic subspace πW (Aθ )ψ ⊂ H. There is a projection Pψ ∈ Aθ satisfying πW (Pψ ) = 6ψ and Tθ (Pψ ) = 1 − 2π/θ . In particular, ψ is also Aθ -cyclic for θ = 2π . Proof. If ψ is θ -tracial, then (θ/2π ) = ψ|Tψθ |ψ = l∈Z2 |Wθ (l)ψ|ψ|2 ≥ ψ2 = 1. If θ > 2π, for 0 < ε < min (2π, θ − 2π ), let φ be a C ∞ function on R such that 0 ≤ φ ≤ 1, with support in [0, 2π + ε], such that φ = 1 on [ε, 2π ], and φ(x)2 + φ(x + 2π)2 = 1 whenever 0 ≤ x ≤ ε. Using (22), φ is θ-tracial (after normalization), and belongs to S(R). If θ = 2π, the same argument holds with ε = 0. Then φ ∈ H, but it is not smooth anymore. Let ψ be θ-tracial. Exchanging the rôles of θ and θ , the Poisson summation formula implies 2π Wθ (m)|ψψ|Wθ (m)−1 = ψ|Wθ (l)−1 |ψWθ (l). Tψθ = θ 2 2 m∈Z
l∈Z
Tψθ
is the desired orthonormal projection which, due to the r.h.s., is Hence 6ψ = 1 − the Weyl representative of an element Pψ ∈ Aθ . Its trace is Tθ (Pψ ) = 1 − 2π/θ . If ' θ = 2π, since the trace is faithful, Tψθ = 1, so that ψ is cyclic. & Definition 2. A vector ψ ∈ H is called a θ -frame, if there are constants 0 < c < C < ∞ such that c1 ≤ Tψθ ≤ C1. This definition is in accordance with the literature ([Sei] and references therein) where the complete set (Wθ (l)ψ)l∈Z2 is called a frame. The principal interest of frames −1 is due to the following: any vector φ ∈ H can be decomposed as φ = Tψθ (Tψθ ) φ = θ −1 ∗ l∈Z2 cl Wθ (l)ψ, where cl = ψ|Wθ (l) (Tψ ) |φ. If ψ ∈ S(R) and φ ∈ S(R), then −1/2 ψ (cl )l∈Z2 ∈ s(Z2 ). Further note that, if ψ is a θ -frame, then ψˆ = (θ/2π )1/2 (T θ ) ψ
is θ-tracial. In addition, if ψ ∈ S(R) then ψˆ ∈ S(R). The next result shows that so-called Weyl–Heisenberg or Gabor lattices constructed with a gaussian mother state are frames if only the volume of the chosen phase-space cell is sufficiently small. This was proved in [Sei], but the present proof is new and covers more general cases. Suppose S ∈ SL(2, R) satisfies S r = 1 for some r. Using the results of Sect. 3.3 and Eq. (11), it is possible to compute φS |W(a)|φS = e−|a|S /4 , 2
|a|2S =
− 2 2 µ+ S a1 + µ S a2 . µ
(33)
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Theorem 3. For θ > 2π, φS is a θ -frame in S(R). Proof. The proof below is given for φ0 ≡ φS4 , but the same strategy works for any φS . 2 Thanks to Poisson’s formula (23) and Eq. (33), Tφθ0 ≤ (θ/2π ) m e−θ|m| /4 . It is therefore enough to find a positive lower bound. Since πW is faithful, it is enough to −θ |m|2 /4 Wθ (m) is itself bounded from below in Aθ . Writing show that T0 = me θ = 2π + δ with δ > 0, there is a ∗-isomorphism between Aθ and the closed subalgebra of A2π ⊗ Aδ generated by (W2π (m) ⊗ Wδ (m))m∈Z2 . It is enough to show −θ |m|2 /4 that Tˆ0 = W2π (m) ⊗ Wδ (m) is bounded from below in A2π ⊗ Aδ . me A2π is abelian and ∗-isomorphic to C(T2 ), provided W2π (m) is identified with the map κ = (κ1 , κ2 ) ∈ T2 → (−1)m1 m2 eıκ·m ∈ C. Hence it is enough to show that 2 Tˆ0 (κ) = m (−1)m1 m2 e−θ |m| /4+ıκ·m Wδ (m) is bounded from below in Aδ uniformly in κ. Since the Weyl representation √ is faithful, Wδ (m) can be replaced by Wδ (m). Using Eq. (13) with ψ = φ0 and a = δm, it is thus enough to show that T˜0 (κ) =
R2
√ √ d 2b I(κ1 + δb2 , κ2 − δb1 )W(b)|φ0 φ0 |W(b)−1 , 2π
where I(κ) =
(−1)m1 m2 e−π|m|
2 /2+ı(κ·m)
,
(34)
m∈Z2
is bounded from below. Clearly the function I is 2π -periodic in both of its arguments. Hence, decomposing the integral into a sum of integrals over the shifted unit cell C = √ [0, 2π) × [0, 2π) and using Wδ (a) = W(2π a/ δ) gives T˜0 (κ) =
d 2a a + κˆ a + κˆ −1 I(a)Wδ l + |φ0 φ0 |Wδ l + , 2πδ 2π 2π 2 C
l∈Z
where κˆ = (κ2 , −κ1 ). The Poisson summation formula applied to the summation over m1 in (34) gives a sum over an index n1 . Changing summation indexes n2 = m2 − n1 shows √ 2 that I(κ) = 2e−κ1 /2π |f (κ1 + ıκ2 )|2 , where f is the holomorphic entire function 2 given by f (z) = n∈Z e−πn −nz . It can be checked that f (z + 2ıπ ) = f (z) and that f (z + 2π) = ez+π f (z). Moreover, using the Poisson summation formula, f does not vanish on γ , the boundary of C oriented clockwise. As I has no poles, the number of zeros of f within C counted with their multiplicity is given by γ df/2ıπf . Using the periodicity properties of f , this integral equals 1. Moreover, a direct calculation shows that the unique zero with multiplicity 1 of f lies at the center π(1 + ı) of C. Hence there is a constant c1 > 0 such that |f (π + ıπ + reıϕ )| ≥ c1 r 2 for all ϕ ∈ [0, 2π ). Let Br denote the ball of size r around π(1 + ı). Replacing this shows
−1 2 2a d a + κ ˆ a + κ ˆ c r 1 . 1− Wδ Tφδ0 Wδ T˜0 (κ) ≥ δ 2π 2π Br 2π As Tφδ0 ≤ c2 1, T˜0 (κ) ≥ 1c1 r 2 (1−c2 r 2 /2)/δ. Choosing r small enough, T˜0 (κ) is bounded from below by a positive constant uniformly in κ. & '
Phase-Averaged Transport for Quasi-Periodic Hamiltonians
527
4. Comparison Theorems 4.1. Proof of Proposition 1 . For normalized φ ∈ H, ρφ denotes the spectral measure of HW relative to φ. Proposition 1 is a corollary of the following result: Theorem 4. For θ ≥ 2π, for any normalized θ -frame φ ∈ H and any Borel subset 1 of R, 2π θ 2π −1 (Tφθ ) −1 N (1) ≤ ρφ (1) ≤ Tφ N (1). θ θ
(35)
Proof. Equation (24) leads to ρφ (1) = Tθ χ1 (H )Fφθ ≤ Fφθ N (1), and to
−1 −1 N (1) = Tθ χ1 (H )Fφθ (Fφθ ) ≤ ρφ (1)(Fφθ ) .
Since Tφθ = θ/2ππW (Fφθ ), the theorem follows.
' &
4.2. Proof of Proposition 2 . Let θ > 2π . The ground state φS of HS is a θ -frame −1/2 according to Theorem 3. Let ψS = (θ/2π )1/2 (TφθS ) φS be the associated θ -tracial vector. Further set HS = πW (Aθ )ψS . In this section, πW denotes the restriction of the Weyl representation to HS . A unitary transformation U : HS → 2 (Z2 ) is defined by (Uφ)(l) = ψS |Wθ (l)−1 |φ,
φ ∈ HS , l ∈ Z2 .
Then UπW (A)U ∗ = π2D (A) for all A ∈ Aθ . Moreover U : S(R) ∩ HS → s(Z2 ). As UψS = |0, MW (H, 1; q, t) = 0|χ1 (H2D )eıH2D t (UHS U ∗ )q/2 e−ıH2D t χ1 (H2D )|0. Recall that HS is a polynomial of degree two in Q and P . From (19) follows UQU ∗ = −θ 1/2 X1 + A1 ,
UP U ∗ = −θ −1/2 X2 + A2 ,
where l|A1 |m = ψS |Wθ (l − m)|QψS and l|A2 |m = ψS |Wθ (l − m)|P ψS . Because ψS , QψS and P ψS are in S(R), A1 and A2 are bounded operators. Using the standard operator inequalities |AB| ≤ A|B| and |A + B| ≤ 2(|A| + |B|) and the commutation relation [X1 , X2 ] = 0, it is now possible to deduce MW (H, 1; q, t) ≤ c1 M2D (H, 1; q, t) + c2 for two positive constants c1 and c2 . An inequality M2D (H, 1; q, t) ≤ c1 MW (H, 1; q, t) + c2 is obtained similarly. This implies Proposition 2.
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4.3. Proof of Proposition 3. Lemma 1. Let Y1 , . . . , YN be selfadjoint operators on H with common domain which satisfy [Ym , Yn ] = ıcm,n 1. Then, if c = maxm,n (|cm,n |) > 0 and if 0 ≤ α ≤ 1, α
N N N 1 2α 2 Yn ≤ Yn ≤ Yn2α + 2N (N − 1)cα . N n=1
n=1
(36)
n=1
Proof. For α = 0, 1 both inequalities are trivial. For 0 < α < 1 the following identity holds: sin (π α) A = π α
∞ 0
dv A , 1−α v v+A
(37)
2 for a positive operator A. If A = N n=1 Yn , then the left-hand inequality in (36) follows 2 from Yn ≤ A and from the operator monotonicity of A/(v + A) = 1 − v/(v + A). On the other hand N 1 1 A Yn = Y n + Y n Yn , . v+A v+A v+A n=1
The first term of each summand is bounded by Yn2 /(v+Yn2 ). Noting Yn Yn , (v + A)−1 = Yn (v +A)−1 [A, Yn ] (v +A)−1 , and using the commutation rules for the Yn ’s, the second term in the r.h.s. is estimated by 1 1 1 cm,n Yn |cm,n |, Ym ≤2 −2ı v + A v + A v + c0 m,n m,n where c0 is the infimum of the spectrum of A. In the latter inequality Yn2 ≤ A has been used. By definition, there are m, n such that cm,n = c > 0 so that Yn2 + Ym2 = (Ym − ıYn )(Ym + ıYn ) + c1 ≥ c1. Hence c0 ≥ c. Integrating over v, using Eq. (37), and remarking that m,n |cm,n | ≤ N (N − 1)c gives the result. & ' If S ∈ SL(2, Z) is a symmetry such that S r = 1, the operators Yn = FSn QFS−n satisfy the hypothesis of Lemma 1, because calculating the derivative of (25) at a = 0 shows that each Yn is linear in P and Q. Clearly HS = 1/(2r) rn=1 Yn2 . If H ∈ Aθ is S-invariant, then HS (t) = 1/(2r) rn=1 FSn Q2 (t)FS−n , where A(t) = eıtHW Ae−ıtHW whenever A is an operator on H. Therefore, if 0 ≤ q ≤ 2, the inequality (36) leads to (with χ1 = χ1 (HW )) φS |χ1 HS (t)q/2 χ1 |φS ≤ r(2r)−q/2 φS |χ1 |Q(t)|q χ1 |φS + 2r(r − 1)
c q/2 , 2r
where Fs φS = φS has been used. Proposition 3 is then a direct consequence of the ± (H, 1; q), β ± (H, 1; q) and of the following lemma: definitions of the exponents β1D W
Phase-Averaged Transport for Quasi-Periodic Hamiltonians
529
Lemma 2. Let φ ∈ S(R), θ ≥ 2π and q ≥ 0. Then, there are two positive constants c0 , c1 such that, for any element B ∈ Aθ , 2π dω ∗ q q Bω |0 + c1 , φ|BW |Q| BW |φ ≤ c0 0|Bω∗ |X| 2π 0 where BW = πW (B) and Bω = πω (B). Proof. Definition (21) and identity (22) of Sect. 3.2 lead to θ ω − nθ ω − n θ ∗ |Q|q BW |φ = θ (q−1)/2 dω φ φ|BW φ n|Kω |n , √ √ θ θ 0 n,n ∈Z q Bω . Since Kω is a positive operator, the Schwarz inequality with Kω = Bω∗ |(ω/θ ) − X| gives |n|Kω |n | ≤ (n|Kω |n + n |Kω |n )/2. Both terms can be bounded similarly. The covariance property of πω (see Sect. 3.2) gives n|Kω |n = 0|Kω−nθ |0. Since φ ∈ S(R), summing up over n first, then over n, there are constants C, c1 such that ∗ |Q|q BW |φ ≤ C dx|φ(x)|0|Kx √θ |0 φ|BW R q B √ |0 + c1 , ≤C dx|φ(x)|0|Bx∗√θ |X| x θ R
q ), valid for q ≥ 0 and some suitable ≤ Cq (|x|q + |X| where the inequality |x − constant Cq , has been used. Thanks to the periodicity of πω , the r.h.s. of the latter estimate can be written as 2π ω − 2π n dω φ + c1 , q Bω |0 sup r.h.s. ≤ √ 0|Bω∗ |X| √ θ θ 0<ω<2π n 0 q X|
completing to the proof of the lemma.
' &
5. Bounds on Phase-Space Transport Section 5.1 is devoted to the proof of Theorem 1 assuming Propositions 4 and 6 which in turn are proven in the subsequent sections. 5.1. Proof of Theorem 1. The proof goes along the lines of [GSB3] and is reproduced here for the sake of completeness. As shown in [GSB3], the time average f (·)T of a non-negative function can be replaced by the gaussian average dt 2 2 g f (·)T = √ e−t /4T f (t), R 2T π without changing the values of the growth exponents, provided f has at most powerlaw increase. Let 1 ⊂ R be a Borel set and ψ1 (t) = e−ıtHW χ1 (HW )φS . Since x α ≥ (1 − e−x ) whenever 0 ≤ α ≤ 1 and x ≥ 0, for any δ > 0 one has g g Mq (H, 1; T ) T ≥ δ −q/2 ψ1 2 − ψ1 (t)|e−δHS |ψ1 (t) . T
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J. Bellissard, I. Guarneri, H. Schulz-Baldes
For 11 ⊂ 1, 1c1 will denote the complement 1 \ 11 . The decomposition of ψ1 into ψ11 + ψ1c1 gives rise to the following lower bound: g Mq (H, 1; T ) T ≥ δ −q/2 ψ11 2 − A11 ,11 (T , δ) − 2%eA11 ,1c1 (T , δ) ,
g where A11 ,12 (T , δ) := ψ11 (t)|e−δHS |ψ12 (t) T . Using the spectral decomposition of HS (see Eq. (30) in Sect. 3.3), it is easy to get 2 2 A11 ,12 (T , δ) = dρS (E) dρS (E )e−(E−E ) T 11 ∞
×
12
7n,S (E)7n,S (E )e−δµ(n+1/2) .
n=0
The Schwarz inequality 2|ψ1 |ψ2 | ≤ ψ1 2 + ψ2 2 applied to the sum on the r.h.s., together with Proposition 4, lead to 2 2 |A11 ,12 (T , δ)| ≤ c δ −(1/2+) dρS (E) dρS (E )e−(E−E ) T , 11
12
for a suitable constant c . For α > 0, let 11 = 1(α, T ) be chosen as −α−1/ log(T ) −(E−E )2 T 2 −α 1(α, T ) = E ∈ 1 T . ≤ dρS (E )e ≤T 1
By definition of ρS it follows then that
g Mq (H, 1; T ) T ≥ δ −q/2 ρS (1(α, T )) 1 − c δ −(1/2+) T −α
≥ cT qα/(1+2) ρS (1(α, T )), for suitable c , c, and the choice δ = (2cT −α )2/(1+2) . The final step uses Lemma 3 below, which is a variation of a result in [BGT]. Choosing p = 1 − q/(1 + 2) therein, the definition of the multifractal dimensions completes the proof of Theorem 1. & ' Lemma 3. Let ρ be a positive measure on R with compact support I and define for T > 0, 2 2 g Iα (T ) = E ∈ I T −α−1/ log(T ) ≤ dρ(E )e−(E−E ) T = ρ(BT (E)) ≤ T −α . I
Then, for all p ∈ [0, 1], there is α = α(p, T ) and a constant c such that
p−1 cT (p−1)α g ρ(Iα (T )) ≥ dρ(E) ρ(BT (E)) . log(T ) I g Proof. Let κ > 0 and set O0 = E ∈ supp(ρ)|ρ(BT (E)) ≤ T −κ . In addition, for j = 1, . . . , κ log(T ) let g Oj = E ∈ supp(ρ)T −κ+(j −1)/ log(T ) ≤ ρ(BT (E)) ≤ T −κ+j/ log(T ) .
Phase-Averaged Transport for Quasi-Periodic Hamiltonians
Then
g
dρ(E)ρ(BT (E))p−1 ≤
531
g
dρ(E)ρ(BT (E))p−1 + κ log(T ) max O0
g
j =1...κ log(T ) Oj
dρ(E)ρ(BT (E))p−1 .
(38)
Let j = j (T , p) be the index where the maximum is taken, and then set α = α(T , p) = κ − j log(T ). It only remains to show that the O0 -term is subdominant if only κ is chosen sufficiently big. To do so, the support of ρ is covered with intervals (Ak )k=1...K of length 1/T . Then K ≤ T |supp(ρ)| (where |A| denotes the diameter of A). If g ak = inf{ρ(BT (E))|E ∈ Ak ∩ O0 }, then ak ≤ T −κ by definition of O0 . More 2 2 g over ρ(BT (E)) ≥ Ak ∩O0 dρ(E )e−(E−E ) T . In particular, if E ∈ Ak ∩ O0 , then g |E − E |T ≤ 1 implying ρ(BT (E)) ≥ e−1 ρ(Ak ∩ O0 ) and thus, ρ(Ak ∩ O0 ) ≤ eak . Hence (p − 1 ≤ 0): p g p−1 dρ(E)ρ(BT (E))p−1 ≤ ρ(Ak ∩ O0 )ak ≤ e ak ≤ eT 1−κp |supp(ρ)|. O0
k≤K
k≤K
Hence choosing κ = 2/p, for example, provides a subdominant contribution in (38) such that (38) fulfills the desired bound. & ' 5.2. Proof of Proposition 4. This section √is devoted to the proof of Proposition 4 as = e2ıπQ/ θ = Wθ (0, 1) commutes with πW (Aθ ), it suming Proposition 6. Since U ) has a joint spectrum commutes, in particular, with HW . Therefore the pair (HW , U contained in R × T. Let mS denote the spectral measure of the pair relative to φS defined by )|φS , dmS (E, η)F (E, eıη ) = φS |F (HW , U ∀F ∈ C0 (R × T). R×T
The marginal probabilities associated with mS are respectively dρS (E), the spectral . measure of HW , and dηGθη/2π φS 22 θ/(2π ) for η ∈ T, the spectral measure of U Thanks to the Radon–Nikodym theorem, mS can be written either as 2π θ dmS (E, η)F (E, eıη ) = dη dµ(θη/2π) (E)F (E, eıη ), (39) 2π 0 R×T R (where µω is the spectral measure of Hω relative to Gω φS ), or as 2π dmS (E, η)F (E, eıη ) = dρS (E) dνE (η)F (E, eıη ), R×T
R
(40)
0
for some probabilty measure νE depending ρS -measurably upon E. Due to the spectral theorem, for every n ∈ Z, there is a function gn (ω, ·) ∈ L2 (R, µω ) such that dµω (E)f (E)gn (ω, E). (41) Gω φS |f (Hω )|n = R
In the following lemma, g˜ n (η, E) stands for θ −1/4 gn (θ η/2π, E):
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J. Bellissard, I. Guarneri, H. Schulz-Baldes
Lemma 4. Let ψ ∈ S(R). Then the representative in L2 (R, ρS ) of the projection of ψ on the HW -cyclic component of φS is given by 2π ˜ ψ(E) = dνE (η) g˜ n (η, E)ψ (η − 2π n)θ 1/2 /2π . 0
n∈Z
Proof. ψ˜ is defined by φS |f (HW )|ψ = On the other hand, thanks to Eq. (22), φS |f (Hω )|ψ =
θ 0
˜
R dρS (E)f (E)ψ(E)
dωGω φS |f (Hω )|Gω ψ =
n∈Z 0
θ
for every f ∈ C0 (R).
dωGω φS |f (Hω )|n(Gω ψ)(n).
Then, using the definition (41) of gn together with Eqs. (39) and (40), and changing from ω to η, gives the result. & ' Proof of Prop. 4. Let 1 ⊂ R be a Borel set and, for δ > 0, let Q(1, δ) be defined by ∞ Q(1, δ) = dρS (E) e−δ(n+1/2) |7n,S (E)|2 . 1
n=0 (n)
Thanks to Lemma 4 applied to the eigenstates φS of HS (see Eq. (30)), it can be written as Q(1, δ) = 1 dρS dνE (η)dνE (η ) m,m g˜ m (η, E)g˜ m (η, E) · · · ···
∞
n=0 e
−δ(n+1/2) φ (n) ((η S
(n)
− 2π m)θ 1/2 /2π )φS ((η − 2π m )θ 1/2 /2π ).
The last sum on the r.h.s. of this identity reconstructs the Mehler kernel of Eq. (32) with t = δ/µ. It will be convenient to define (42) Gδ (E; x) = dνE (η ) MS (δ/µ; x, (η − 2π m )θ 1/2 /2π ) . m
Since the Mehler kernel decays fast, this sum converges. Using the Schwarz inequality together with the symmetry (m, η) ↔ (m , η ), Q(1, δ) can be bounded from above by Q(1, δ) ≤ dρS dνE (η) |g˜ m (η, E)|2 Gδ E; (η − 2π m)θ 1/2 /2π . m
1
Thanks to Eqs. (39) and (40), and changing again from η to ω, this bound can be written as θ dω 2 1/2 |g Q(1, δ) ≤ E; (ω − mθ )/θ . dµ (E) (ω, E)| G ω m δ θ 1/2 1 m 0 If now Pω is the projection on the Hω -cyclic component of Gω φS in 2 (Z), the definition (41) of gm and the covariance lead to the following inequality: dµω (E) |gm (ω, E)|2 f (E) = m|Pω f (Hω )Pω |m ≤ 0|f (Hω−mθ )|0,
Phase-Averaged Transport for Quasi-Periodic Hamiltonians
533
valid for f ∈ C0 (R), f ≥ 0, because Hω commutes with Pω and the latter is a projection. (0) Let then µω be the spectral measure of Hω relative to the vector |0. The previous estimate implies θ (0) Q(1, δ) ≤ θ −1/2 dω dµω−mθ (E)Gδ E; (ω − mθ )/θ 1/2 m
≤ θ −1/2
0
∞ ∞
dω
1
1
1/2 dµ(0) (E)G E; ω/θ . δ ω
(0)
Since µω is 2π-periodic with respect to ω, the latter integral can be decomposed into a sum over intervals of length 2π leading to the following estimate: 2π 1/2 Q(1, δ) ≤ θ −1/2 E; (ω + 2π k)/θ . dω dµ(0) (E) G δ ω 0
1
k∈Z
Definitions (17) of the trace on Aθ , (3) of the DOS and (42) of Gδ give 2π dN (E)dνE (η) Q(1, δ) ≤ 1/2 θ 1×[0,2π] 1/2 MS δ/µ; ω + 2π k , (η − 2π m)θ . × θ 1/2 2π (k,m)∈Z2
The result of Proposition 6 can now be used. Remarking that νE is a probability, and using the equivalence between ρS and the DOS (Theorems 3 and 4 combined), the last estimate implies Q(1, δ) ≤ c ρS (1)δ −(1/2+) , for some suitable constant c . Since this inequality holds for all Borel subsets 1 of R, the Proposition 4 is proven. & '
5.3. Proof of Proposition 6. If α = θ/2π ∈ [0, 1] is an irrational number, a rational approximant is a rational number p/q, with p, q prime to each other, such that |α − p/q| < q −2 . The continued fraction expansion [a1 , · · · , an , · · · ] of α [Her], provides an infinite sequence pn /qn of such approximants, the principal convergents, recursively defined by p−1 = 1, q−1 = 0, p0 = 0, q0 = 1 and sn+1 = an+1 sn + sn−1 if s = p, q. It can be proved (see [Her] that α is a number of Roth type (see Eq. (2) in Prop. 7.8.3) < ∞ for all > 0. Sect. 2) if and only if ∞ a /q n+1 n n=1 The proof of Proposition 6 relies upon the so-called Denjoy–Koksma inequality [Her]. Let ϕ be a periodic function on R with period 1, of bounded total variation Var(ϕ) over a period interval. Then (see [Her], Theorem 3.1) Theorem (Denjoy–Koksma inequality). Let α ∈ [0, 1] be irrational and let ϕ be a real valued function on R of period one. Then, if p/q is a rational approximant of α, q ϕ(x + j α) − q j =1
1 0
dyϕ(y) ≤ Var(ϕ).
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J. Bellissard, I. Guarneri, H. Schulz-Baldes
Proposition 6 is a direct consequence of the definition of the Mehler kernel (see Eq. (32)) and of the following result: Lemma 5. If δ > 0, let Fδ be the function on R2 defined by Fδ (x, y) = δ(x + y)2 + δ −1 (x − y)2 . If α is a number of Roth type, then for any a > 0, > 0, there is c > 0 such that sup e−aFδ (x+k,y+mα) ≤ c δ − , ∀δ ∈ (0, 1). x,y∈R
(k,m)∈Z2
2 2 2 Proof. Let (x0 , y 0 ) ∈ R be fixed and set L = {(x0 + k, y0 + mα) ∈ R |(k, m) ∈ Z }. −aF (x +k,y +mα) δ 0 0 If S(x0 , y0 ) = , then S is periodic of period 1 in x0 and of k,m e period α in y0 . Therefore, it is enough to assume 0 ≤ x0 < 1 and 0 ≤ y0 < 1 (since 0 < α < 1). For 0 < σ < 1 and for j ∈ N, let Lj be the set of points (x, y) ∈ L for which j 2 δ −σ ≤ Fδ (x, y) < (j + 1)2 δ −σ . Thus
S(x0 , y0 ) ≤
∞
e−aj
2 δ −σ
|Lj |,
(43)
j =0
where |A| denotes the number of points in A. Lj is contained in an elliptic crown with axis along the two diagonals x = ±y. In particular, (x, y) ∈ Lj ⇒ max{|x|, |y|} ≤ (j + 1)δ −(1+σ )/2 and |x − y| ≤ (j + 1)δ (1−σ )/2 . (44) If j ≥ 1, the number of points contained in Lj can be estimated by counting the number of rectangular cells of sizes (1, α) centered at points of L and meeting the elliptic crown. Since this crown is included inside the square max{|x|, |y|} ≤ (j + 1)δ −(1+σ )/2 it is enough to count such cells meeting this square. Such cells are all included inside the square C = {(x, y) ∈ R2 | max{|x|, |y|} ≤ (j + 2)δ −(1+σ )/2 } (since δ ≤ 1). Hence the number of such cells is certainly dominated by the ratio of the area of C to the area of each cell, namely (j + 2) 2 −(1+σ ) |Lj | ≤ δ . α Therefore, the part of the sum in (43) coming from j ≥ 1 converges to zero as δ ↓ 0. In particular, it is bounded by a constant c1 that is independent of (x0 , y0 ). Thus, it is sufficient to consider the term j = 0 only. Let ϕ be the function on R defined by ϕ(x) = k∈Z χI (x + y0 − x0 + k), where I is the interval I = [−δ (1−σ )/2 , δ (1−σ )/2 ] ⊂ R. It is a periodic function of period 1 with Var(ϕ) = 2. Moreover, using (44) it can be checked easily that S(x0 , y0 ) ≤ c1 +
|m|<M
ϕ(mα) ≤ c1 +
M−1
(ϕ(mα) + ϕ(−mα)) ,
m=0
provided M ≥ 3δ −(1+σ )/2 /α. For indeed, (x, y) ∈ L0 only if |y0 + mα| ≤ δ −(1+σ )/2 for some m ∈ Z. Let then n ∈ N be such that qn ≤ M < qn+1 , where the pn /qn ’s are the principal convergents of α. Replacing M by qn+1 in the r.h.s. gives an upper
Phase-Averaged Transport for Quasi-Periodic Hamiltonians
535
bound. By the Denjoy-Koksma inequality, the r.h.s. is therefore bounded from above by c1 + 4qn+1 δ (1+σ )/2 . Since α is a number of Roth type, qn+1 ≤ (an+1 + 1)qn ≤ c2 · qn1+σ , thanks to Prop. 7.8.3 in [Her] (see above). It is important to notice that c2 only depends upon α and the choice of the exponent σ . Collecting all inequalities, gives S(x0 , y0 ) ≤ c1 +
12 · c2 −2σ . δ α
Choosing σ = /2 and remarking that none of the constants on the r.h.s. depends on ' (x0 , y0 ) leads to the result. & Appendix: Proofs of Various Results on Weyl Operators Proof of Eqs. (13) and (14). Due to the polarization principle, (13) is equivalent to d 2 b ıa∧b |ψ|W(b)|ψ|2 . φ|W(a)|φψ|W(a)|ψ = e (45) R2 2π By inverse Fourier transform, (45) is equivalent to d 2 a ıb∧a |φ|W(b)|ψ|2 = e φ|W(a)|φψ|W(a)|ψ, R2 2π
(46)
which is equivalent to (14), so that it is sufficient to prove (45). Using (11), db1 db2 dx r.h.s. of (45) = 2π R2 R × dyφ(x)φ(y)ψ(x + b1 )ψ(y + b1 )eı(b2 (x−y+a1 )−a2 b1 ) . R
The integral over b2 can be immediately evaluated by R db2 eıb2 (x−y+a1 ) = 2π δ(y −x − a1 ). Thus the integration over y is elementary. Changing variable from b1 to x = x + b1 therefore gives a1 a2 a1 a2 dx dx φ(x)φ(x + a1 )eıa2 x+ı 2 ψ(x )ψ(x + a1 )e−ıa2 x −ı 2 , r.h.s. of (45) = R
R
which is precisely the l.h.s. of (45).
' &
Proof of Eq. (22). It is sufficient to verify (22) for the generators A = Wθ (m), m ∈ Z2 , of Aθ . For such A, θ dω ω − nθ ω − lθ r.h.s. of (22) = φ n|πω (Wθ (m)|lψ . √ √ √ θ n,l∈Z θ θ 0 As n|πω (Wθ (m)|l = eıθm1 m2 /2 eı(ω−lθ)m2 δn,l+m1 , the sum over n can be immediately computed, and the one over l can be combined with the integral over ω in order to give dx x − m1 θ ıθ m1 m2 ıxm2 x ψ √ r.h.s. of (22) = e 2 e . √ φ √ θ θ θ R
536
J. Bellissard, I. Guarneri, H. Schulz-Baldes
√ √ Changing variable y = (x − m1 θ )/ θ and identifying W( θm) shows √ r.h.s. of (22) = dyφ(y) W( θ m)ψ (y), R
' &
namely the l.h.s. of (22).
Proof of Proposition 7. For f ∈ S(R2 ), let f˜ be its symplectic Fourier transform defined by (l, m ∈ R2 ): d 2 m ıl∧m d 2 l ım∧l ˜ ˜ f (l) = f (m). f (m), ⇔ f (m) = e e R2 2π R2 2π Then the classical Poisson summation formula reads f (m) = 2π f˜(2π l). m∈Z2
l∈Z2
√ √ Setting f (m) = φ|W( θ m)|φψ|W( θ m)|ψ, Eq. (46) leads to 2 1 2π ˜ f (l) = ψ|W √ l |φ . θ θ Inserting this into the Poisson summation formula and recalling the notation (20) gives (23). & ' Proof of Eq. (24). By (16) and (20), πW (A) = Thus ψ|πW (A)|ψ =
l∈Z2
al ψ|Wθ (l)|ψ = Tθ
al Wθ (l) with al = Tθ (Wθ (l)−1 A).
ψ|Wθ (l)|ψWθ (l)−1 A .
l∈Z2
l∈Z2
Comparing with the Poisson summation formula (23) shows (24).
' &
Proof of Proposition 8. Because of the freedom of phase and relation (12), it is sufficient to verify all implementation formulas (25) for the Weyl operators eıQ and eıP or equivalently (on the domain of) their generators Q and P . Concerning the first formula in (26), it thus follows from the identities e−ıκQ
2 /2
QeıκQ
2 /2
= Q,
e−ıκQ
2 /2
P eıκQ
2 /2
= κQ + P . √ Next let us consider the dilations on L2 (R) defined by (D(a)φ)(x) = ea φ(ea x). It generators are computed by d ı (D(a)φ)(x) = (QP + P Q)φ(x), da 2 a=0 so that for a = −ln(λ),
# e
−ı ln λ(QP +P Q)/2
φ(x) =
1 x φ . λ λ
Phase-Averaged Transport for Quasi-Periodic Hamiltonians
537
This immediately allows us to verify 1 Q, λ = λP ,
e−ı ln λ(QP +P Q)/2 Qeı ln λ(QP +P Q)/2 = e−ı ln λ(QP +P Q)/2 P eı ln λ(QP +P Q)/2
which proves the second formula √ in (26). To prove the last√one, we use the annihiliationcreation operators a = (Q−ıP )/ 2 and a ∗ = (Q+ıP )/ 2.As (P 2 +Q2 −1)/2 = a ∗ a ∗ ∗ ∗ and e−ısa a aeısa a = eıs a, the formula follows after decomposing W(a) into a and a . ∗a −ısa , notably (Kφ) = dyk(x, y)φ(y). Finally we search the integral kernel for K = e (n) (n) (n) ısn If φS4 are the Hermite functions, then KφS4 = e φS4 . Equivalently, k has to satisfy ay k = eıs ax∗ k and Kφ (0) = φ (0) (here the index on the a’s indicate with respect to which 2 2 variable the operator acts). An Ansatz k(x, y) = e−b(x +y )+cxy+d leads to the integral kernel in (27). & ' Proof of Proposition 9. Let us set cos γ − sin γ R= , sin γ cos γ
D=
λ0 0 λ1
.
Then, using the notations and formulas in Subsect. 3.3, µ −1 −1 HS = (RD)t K|(RD)t K = µFR FD HS4 FD FR , φS = FR FD φS4 . (47) 2 Now φS4 is known to be the normalized gaussian. Using the implementation formulas of Proposition 8, it is straightforward to calculate the gaussian integrals giving (31). The Mehler kernel MS4 (t; x, y) for HS4 = (P 2 + Q2 )/2 is well-known (and can be read of (27) at imaginary time). Using (47) and the definition (10), −1 −1 MS (t; x, y) = dx dy x|FR FD |x MS4 (t; x , y )y |FD FR |y. R
R
The gaussian integrals herein give rise to (32). & ' Let us conclude with the proof of the complementary result given in Sect. 2. Proof of Proposition 5. The commutant B of the abelian C∗ -algebra generated by HW contains the commutant of πW (Aθ ), that is the von Neumann algebra πW (Aθ ) generated by πW (Aθ ). As πW (Aθ ) is of type II1 [Sak], there exist ∗-endomorphisms ηq : Matq×q → B for every q ∈ N (here Matq×q denotes the complex q × q matrices). According to the spectral theorem, H decomposes according to the multiplicity of πW (H ): H = ⊕n≥1 L2 (Xn , µn ) ⊗ Cn ⊕ L2 (X∞ , µ∞ ) ⊗ 2 (N), where the µn ’s are positive measures with pairwise disjoint supports Xn ⊂ R. In this representation, πW (H ) = ⊕n≥1 Mult(E)⊗1n ⊕Mult(E)⊗1∞ (here Mult(E) denotes the multiplication by the identity on R) and B = ⊕n≥1 L∞ (Xn , µn )⊗Matn×n ⊕L∞ (X∞ , µ∞ )⊗B(2 (N)). Let Pn be the projection on L2 (Xn , µn ) ⊗ Cn . Then Pn BPn = L∞ (Xn , µn ) ⊗ Matn×n . Moreover φn,x (B) = Pn BPn (x) defines a ∗-endomorphism from B to Matn×n for µn -almost all x ∈ Xn . Combining with ηq , one gets ∗-endomorphisms φn,x ◦ ηq : Matq×q → Matn×n for any q satisfying φn,x ◦ ηq (1q ) = 1n . This is impossible for any q > n so that Xn = ∅ for all n ≥ 1. If HW had a cyclic vector, its spectrum would be simple. & '
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Acknowledgements. We would like to thank B. Simon, R. Seiler and S. Jitormiskaya for very useful comments. The work of H. S.-B. was supported by NSF Grant DMS-0070755 and the DFG Grant SCHU 1358/1-1. J.B. wants to thank the Institut Universitaire de France and the MSRI at Berkeley for providing support while this work was in progress.
Note added in proof. After this work was completed we learned that S. Tcheremchantsev in “Mixed lower bounds for quantum transport” extended Lemma 3 to arbitrary values of q. This implies that the Main Theorem (i) holds for all q > 0 and (ii) for q ∈ (0, 2]. References [BSB]
Barbaroux, J.-M., Schulz-Baldes, H.: Anomalous transport in presence of self-similar spectra. Annales I.H.P. Phys. Théo. 71, 539–559 (1999) [BGT] Barbaroux, J.M., Germinet, F., Tcheremchantsev, S.: Nonlinear variation of diffusion exponents in quantum dynamics. C.R. Acad. Sci. Paris 330, série I, 409–414 (2000); Fractal Dimensions and the Phenomenon of Intermittency in Quantum Dynamics. Duke Math. J. 110, 161–193 (2001) [Bel] Bellissard, J.: K-theory of C∗ -algebras in solid state physics. In: Statistical Mechanics and Field Theory: Mathematical Aspects, Lecture Notes in Physics 257, edited by T. Dorlas, M. Hugenholtz, M. Winnink, Berlin: Springer-Verlag, 1986, pp. 99–156 ; Gap labelling theorems for Schrödinger operators. In: From Number Theory to Physics, Berlin: Springer, 1992, pp. 538–630 [Bel94] Bellissard, J.: Lipshitz continuity of gap boundaries for Hofstadter-like spectra. Commun. Math. Phys. 160, 599–613 (1994) [Com] Combes, J.-M.: In:Differential Equations with Applications to Mathematical Physics. Ames, W.F., Harrell, E.M., Herod J.V., eds, Boston: Academic Press, 1993 [Con] Connes, A.: Noncommutative Geometry. London: Academic Press, 1994 [DS] Deift, P., Simon, B.: Almost periodic Schr'ödinger operators. III. The absolutely continuous spectrum in one dimension. Commun. Math. Phys. 90, 389–411 (1983) [Gua] Guarneri, I.: Spectral properties of quantum diffusion on discrete lattices. Europhys. Lett. 10, 95– 100 (1989); On an estimate concerning quantum diffusion in the presence of a fractal spectrum. Europhys. Lett. 21, 729–733 (1993) [GM] Guarneri, I., Mantica, G.: Multifractal Energy Spectra and Their Dynamical Implications. Phys. Rev. Lett. 73, 3379–3383 (1994) [GSB1] Guarneri, I., Schulz-Baldes, H.: Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar spectra. Rev. Math. Phys. 11, 1249–1268 (1999) [GSB2] Guarneri, I., Schulz-Baldes, H.: Lower bounds on wave packet propagation by packing dimensions of spectral measures. Elect. J. Math. Phys. 5 (1999) [GSB3] Guarneri, I., Schulz-Baldes, H.: Intermittent lower bound on quantum diffusion. Lett. Math. Phys. 49, 317–324 (1999) [Har] Harper, P. G.: Single Band Motion of Conduction Electrons in a Uniform Magnetic Field. Proc. Phys. Soc. Lond. A 68, 874–878 (1955) [Her] Herman, M. R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publications I.H.E.S. 49, 5–233 (1979) [Jit] Jitomirskaya, S.: Metal-Insulator Transition for the Almost Mathieu Operator. Ann. of Math. 150, 1159–1175 (1999) [Las] Last, Y.: Quantum Dynamics and decomposition of singular continuous spectra. J. Funct. Anal. 142, 402–445 (1996) [KKKG] Ketzmerick, R., Kruse, K., Kraut, S. and Geisel, T.: What determines the spreading of a wave packet?. Phys. Rev. Lett. 79, 1959–1962 (1997) [KL] Kiselev, A., Last, Y.: Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains. Duke Math. J. 102, 125–150 (2000) [Man] Mantica, G.: Quantum intermittency in almost periodic systems derived from their spectral properties. Physica D 103 , 576–589 (1997); Wave Propagation in Almost-Periodic Structures. Physica D 109, 113–127 (1997) [Per] Perelomov, A.: Generalized Coherent States and Their Applications. Berlin: Springer, 1986 [Pie] Piéchon, F.: Anomalous Diffusion Properties of Wave Packets on Quasiperiodic Chains. Phys. Rev. Lett. 76, 4372–4375 (1996) [Ram] Rammal, R.: Landau level spectrum of Bloch electron in a honeycomb lattice. J. Phys. France 46, 1345–1354 (1985)
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[Rie] [RP] [Sak] [SBB] [Sei] [TK]
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Rieffel, M. A.: C∗ -algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981) Rüdinger, A., Piéchon, F.: Hofstadter rules and generalized dimensions of the spectrum of Harper’s equation. J. Phys. A 30, 117–128 (1997) Sakai, S.: C∗ -algebras and W∗ -algebras. Berlin: Springer, 1971 Schulz-Baldes, H., Bellissard, J. Anomalous transport: A mathematical framework. Rev. Math. Phys. 10, 1–46 (1998) Seip, K.: Density theorems for sampling and interpolation in the Bargmann-Fock space I. J. Reine Angew. Math. 429, 91–106 (1992) Tang, C., Kohmoto, M.: Global scaling properties of the spectrum for a quasiperiodic Schrödinger equation. Phys. Rev. B 34, 2041–2044 (1986)
Communicated by M. Aizenman
Commun. Math. Phys. 227, 541 – 550 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Invariance Properties of Induced Fock Measures for U(1) Holonomies J. M. Velhinho Departamento de Física, Universidade da Beira Interior, R. Marquês D’Ávila e Bolama, 6201-001 Covilhã, Portugal. E-mail: [email protected] Received: 19 July 2001 / Accepted: 7 January 2002
Abstract: We study invariance properties of the measures in the space of generalized U(1) connections associated to Varadarajan’s r-Fock representations. 1. Introduction Holonomies are the starting point for a rigorous approach to quantum gravity – often called “loop quantum gravity” – carried throughout the last decade. It is based on Ashtekar’s formulation of general relativity as a gauge theory [As], loop variables [GT, RoSm], C ∗ -algebra techniques [AI2, Ba1] and integral and functional calculus in spaces of generalized connections [AL1,AL3, MM, Ba2] (an excellent review of both the fundamentals and the most recent developments in this field can be found in [T]). Since the early days of this approach, free Maxwell theory has been a preferred testing ground for new ideas, especially in what concerns the relation between background independent representations of holonomy algebras and the standard Fock representation for smeared fields [ARS,AI1,AR]. Recently, Varadarajan revisited this subject, and proposed a family of representations for a kinematical Poisson algebra of U (1) holonomies and certain functions of the electric fields [Va1,Va2]. Varadarajan’s work allowed the emergence of Fock states within the framework of generalized connections and is therefore a promising starting point to close the gap between non-perturbative loop quantum gravity states and low energy states [AL4] (see also [T] for a general discussion of the issue of semiclassical analysis in loop quantum gravity). In the present work we study (quasi-)invariance and mutual singularity properties of the measures associated to Varadarajan’s representations. These are measures on the space A/G of generalized U (1) connections that can be obtained, by push-forward, from the standard Maxwell–Fock measure (see [AI2,AR,ARS] for previous work along these lines and also [AL4] for a projective construction of the measures). We will show that the measures are singular with respect to each other and are singular with respect to the measure µ0 of Ashtekar and Lewandowski. This implies, in particular, that the Fock
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states are not in L2 (A/G, µ0 ) (but rather in an extension thereof [Va2]). It also follows that the measures are not quasi-invariant with respect to the natural action of A/G on itself, which is an obstruction to the quantization of the usual smeared electric fields. On the other hand, the measures on A/G inherit quasi-invariance properties directly related to the electric operators considered by Varadarajan. This work is organized as follows. In Sect. 2 we review the Fock representation, whereas the loop approach is reviewed in Sect. 3. Varadarajan’s measures are presented in Sect. 4 and studied in Sect. 5, which contains our main results. We conclude with a brief discussion, in Sect. 6. 2. Smeared Fields and Fock Representation This section briefly reviews some aspects of the Schrödinger representation of the usual Fock space for the Maxwell field, following [GV, ReSi, GJ, BSZ]. We use spatial coordinates (x a ), a = 1, 2, 3, and units such that c = h¯ = 1. The Euclidean metric δab in R3 is used to raise and lower indices whenever necessary. As is well known, connections A and electric fields E do not give rise to well defined quantum operators. In the Fock framework, they are replaced by smeared versions A() = Aa a d 3 x and E(λ) = λa E a d 3 x, where belongs to the (nuclear) space E∞ of smooth and fast decaying transverse vector fields and λ to the (nuclear) space (A/G)∞ of smooth and fast decaying transverse connections. The Poisson bracket between these basic observables is (1) A(), E(λ) = λa a d 3 x , to which correspond the Weyl relations: V(λ) U() = ei
λa a d 3 x
U()V(λ).
(2)
∗ , µ , where The usual Fock representation can be realized in the Hilbert space L2 E∞ ∗ is the space of tempered distributional 1-forms (the topological dual of E ) and µ E∞ ∞ is the Gaussian measure defined by: 1 iφ() a −1/2 b 3 e dµ (φ) := exp − d x , (3) δab (−) ∗ 4 E∞ where is the Laplacian in R3 . It is well known that µ is quasi-invariant with respect ∗ : to the action of (A/G)∞ on E∞ ∗ φ → φ + λ , λ ∈ (A/G)∞ , E∞ ∗ is defined by where φ + λ ∈ E∞
(φ + λ)() := φ() +
λa a d 3 x , ∀ ∈ E∞ .
(4)
(5)
(Recall that a measure µ is quasi-invariant with respect to a group of transformations T if the push-forward measure T∗ µ has the same zero measure sets as µ, ∀T , i.e. (T∗ µ)(B) =
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0 if and only if µ(B) = 0.) One therefore has an unitary representation V of the abelian group (A/G)∞ as translations: ∗ dµ,λ (φ) V(λ)ψ (φ) = ψ(φ − λ) , ψ ∈ L2 E∞ (6) , µ , dµ (φ) where µ,λ is the push-forward of the measure µ by the map (4) and dµ,λ /dµ is the Radon-Nikodym derivative. (The existence of both the Radon-Nikodym derivative and of its inverse is equivalent to quasi-invariance.) A representation of the Weyl relations (2) is achieved with the following representation U of E∞ : (7) U()ψ (φ) = e−iφ() ψ(φ) . ˆ ˆ Since both representations U and V are continuous, the quantized fields A() and E(λ) can be identified with the generators of the one-parameter groups U(t) and V(tλ), respectively. 3. Holonomies and Haar Measure This section briefly reviews the loop approach to the Maxwell field, following in essence the general framework for gauge theories with a compact (not necessarily abelian) group (see e.g. [T] and references therein). Notice, however, that the presentation of the uniform measure µ0 [AL1] and the quantization of electric fields [AL3] are considerably simpler in the U (1) case. In the loop approach the configuration variables are (traces of) holonomies rather then smeared connections. Let us then consider U (1) holonomies Tα (A) := ei
α
Aa dx a
(8)
associated with piecewise analytic loops on R3 . It is convenient to eliminate redundant loops, i.e. one identifies two loops α and β such that Tα (A) = Tβ (A) ∀A. Such classes of loops are called hoops. The set HG of all U (1) hoops is an abelian group under the natural composition of loops. The set of holonomy functions Tα , α ∈ HG, is an abelian ∗-algebra. The C ∗ completion in the supremum norm is called the U (1) holonomy algebra HA [AI2,AL1]. It turns out that HA is isomorphic to the algebra of continuous functions on the space A/G of generalized connections, where A/G is the set of all group morphisms from the hoop group HG to U (1). In order to describe the isomorphism, let us consider the functions α : A/G → U (1): ¯ := A(α) ¯ , A¯ → α (A)
(9)
¯ where α ∈ HG and A(α) denotes evaluation. The space A/G is compact in the weakest topology such that all functions α are continuous. It is a key result that A/G is homeomorphic to the spectrum of HA [MM,AL1,AL2], with the functions α corresponding to Tα . (Cyclic) representations of HA are in 1-1 correspondence with positive linear functionals on HA. By the above isomorphism, those are in turn in 1-1 correspondence with
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Borel measures in A/G. Given a measure µ, one thus has a representation the Hilbert space L2 A/G, µ : ¯ = α (A)ψ( ¯ ¯ ∀ψ ∈ L2 A/G, µ . A), (Tα )ψ (A)
of HA in
(10)
The associated positive linear functional ϕ is defined by: ϕ(Tα ) = 1,
(Tα )1 =
A/G
α dµ .
(11)
In the U (1) case, A/G is a topological group [AL1, Ma] with multiplication ¯ A¯ , A¯ ∈ A/G, α ∈ HG, A¯ A¯ (α) = A¯ (α)A(α),
(12)
¯ −1 ). Let us consider the Haar measure µ0 and the associand inverse A¯ −1 (α) = A(α ated representation 0 of HA [AL1]. Since µ0 is invariant, we also have an unitary representation V0 of the group A/G in L2 A/G, µ0 : ¯ ∀ψ ∈ L2 A/G, µ0 . ¯ = ψ(A¯ A), V0 (A¯ )ψ (A)
(13)
The representation V0 leads to smeared electric operators, as follows. For λ ∈ (A/G)∞ , let A¯ λ denote the element of A/G defined by holonomies, i.e. A¯ λ (α) := Tα (λ), ∀α ∈ HG. Restricting V0 to elements A¯ λ and the representation 0 to the functions Tα , one obtains the commutation relations: V0 (λ)
0 (Tα )
= ei
α
λa dx a
0 (Tα )V0 (λ) ,
(14)
where V0 (λ) := V0 (A¯ λ ). The action of V0 (λ) is particularly simple for the dense space of finite linear combinations of functions α : V0 (λ) α = ei
α
λa dx a
α .
(15)
Let us consider the one-parameter unitary group V0 (tλ), t ∈ R, and let dV0 (λ) be its self-adjoint generator. From (14) one finds the commutator:
dV0 (λ),
0 (Tα ) =
α
λa dx a
0 (Tα ) ,
(16)
showing that the operators 0 (Tα ) and dV0 (λ) give a quantization ofthe Poisson algebra of holonomies Tα and smeared electric fields E(λ), in L2 A/G, µ0 . In this representation, the states α describe one-dimensional excitations of the electric field along loops, or electric flux "quanta", and are therefore called loop states [GT, RoSm]. These type of excitations are, of course, absent in Fock space. On the other hand, neither are the familiar Fock n-particle states or coherent states obviously related to loop states.
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4. r-Fock Measures In this section we present Varadarajan’s r-Fock representations of the U (1) holonomy algebra HA from the measure theoretic point of view. Let us start with hoop form factors [ARS,AR,AI2]. Given a hoop α, the form factor Xα is the transverse distributional vector field such that
(17) Xαa (x)Aa (x)d 3 x = Aa dx a , ∀A. α
Consider the one-parameter family of functions on R3 : fr (x) =
1 2π 3/2 r 3
e−x
2 /2r 2
,
(18)
where r > 0. The smeared form factors are smooth and fast decaying transverse vector fields, i.e. elements of E∞ , defined by: a (x) Xα,r
:=
fr (y − x)Xαa (y)d 3 y .
(19)
One thus has, for each r, a map α → Xα,r from hoops to E∞ . Notice that the composition of hoops is preserved, i.e. Xαβ,r = Xα,r + Xβ,r [AR]. Smeared form factors can be used to define measurable maps from the space of ∗ to A/G. Consider then the family of maps * : E ∗ → distributional connections E∞ r ∞ ¯ A/G given by φ → Aφ,r , where A¯ φ,r (α) := eiφ(Xα,r ) ∀α ∈ HG.
(20)
Since the σ -algebra of measurable sets in A/G is the smallest one such that all functions α (9) are measurable, one sees that *r is measurable if and only if the maps α ◦ *r : ∗ → U (1) are measurable for all α ∈ HG, which is clearly true, since they can be E∞ obtained as a composition of measurable maps: φ → φ(Xα,r ) → eiφ(Xα,r ) . One can now use the maps *r to push-forward the Fock measure µ , thus obtaining a family of measures µr := (*r )∗ µ on A/G. By definition µr (B) = µ (*−1 r B) ∀ measurable set B ⊂ A/G.
(21)
Each of the measures µr provides us with a Hilbert space L2 A/G, µr and a representation r of HA. The associated positive linear functional ϕr is: ϕr (Tα ) =
A/G
¯ dµr (A) ¯ = α (A)
∗ E∞
eiφ(Xα,r ) dµ (φ) .
(22)
Expression (22) shows that the representation r is the r-Fock representation considered by Varadarajan in [Va1], µr being the r-Fock measure in A/G whose existence was proved in [Va2].
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5. Properties of the r-Fock Measures The present section contains our main results. We show that the r-Fock measures µr are all mutually singular, and are singular with respect to the Haar measure µ0 . We study also (quasi-)invariance properties of the r-Fock measures µr and their relation to the quantization of certain twice smeared electric fields introduced in [Va1]. Let Diff be the group of (analytic) diffeomorphisms of R3 . The natural action of Diff on the (piecewise analytic) curves of R3 induces an action on the hoop group HG: HG × Diff (α, ϕ) → ϕα , and therefore one has an action of Diff in A/G, given by ∗ ¯ ϕ ∈ Diff, A¯ ∈ A/G, α ∈ HG . ϕ A¯ (α) = A(ϕα),
(23)
(24)
It can be seen that the maps ϕ ∗ : A/G → A/G are continuous [AL1,AL2, Ba1]. The Haar measure µ0 is invariant under the action of Diff, since no background geometric structure is used in its definition [AL1]. The induced measures µr , on the other hand, are not invariant, due to the appearance of the Euclidean metric δab in the construction of the Fock measure µ . From now on we will restrict our attention to the Euclidean group, i.e., the subgroup of Diff of transformations that preserve the Euclidean metric. It is clear that the measures µr are invariant under these transformations, given the well known Euclidean invariance of the Fock measure. Besides being invariant, the Fock measure is moreover ergodic with respect to the action of the Euclidean group (see e.g. [BSZ,Ve2]), which means that the only invariant ∗ functions in L2 E∞ , µ are the constant functions. This ergodic property is shared by the measures µr , since if an invariant and non-constant function ψ were to exist in L2 A/G, µr , then the pull-back ψ ◦ *r would define an invariant and non-constant ∗ function in L2 E∞ , µ . An important fact is that the Haar measure µ0 on A/G is also ergodic under the action of the Euclidean group, as follows from more general results proven in [MTV]. Thus, all measures µr , r ∈ R+ , and µ0 are invariant and ergodic under the action of the same group. From well known results in measure theory (see e.g. [Ya]), this is only possible if all these measures are mutually singular, meaning that each measure of the set {µr , r ∈ R+ }∪µ0 is supported on a subset of A/G which has zero measure with respect to all the other measures (recall that a subset X of a space M is said to be a support for the measure µ on M if any measurable subset Y on the complement, Y ⊂ X c , has measure zero). It is thus proven that Theorem 1. The measures in the set {µr , r ∈ R+ } ∪ µ0 are all singular with respect to each other. Theorem 1 leads to the conclusion that none of the measures µr is quasi-invariant under the action of A/G on itself. This follows from the fact that A/G is a compact group, which implies that any quasi-invariant measure is in the equivalence class of the Haar measure, meaning that it must have the same zero measure sets (see e.g. [Ki, 9.1]). Thus Corollary 1. The measures µr , r ∈ R+ , are not quasi-invariant. We saw in Sect. 3 how the quantization of smeared electric fields canbe obtained from an unitary representation of the group A/G in the Hilbert space L2 A/G, µ0 . From
Invariance Properties of Induced Fock Measures for U(1) Holonomies
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the corollary we conclude that such an unitary representation of A/G is not available in the Hilbert spaces L2 A/G, µr . One should thus look for the quantization of different functions of the electric fields. Varadarajan showed in [Va1] that certain “Gaussian-smeared smeared” electric fields can be consistently quantized in the r-Fock representations. In the remaining we will relate the quantization of these functions to quasi-invariance properties of the r-Fock measures µr . We will start by establishing the quasi-invariance properties, which, as expected, follow from the quasi-invariance of the Fock measure under the action (4). Let us consider the restriction of the maps *r (20) to (A/G)∞ , i.e., we consider the maps
such that
(A/G)∞ λ → A¯ λ,r ∈ A/G
(25)
a 3 ¯ Aλ,r (α) = exp i λa (x)Xα,r (x)d x , ∀α ∈ HG .
(26)
It is clear that A¯ λ+λ ,r = A¯ λ,r A¯ λ ,r , and therefore the group (A/G)∞ acts on the space A/G as a subgroup of the full group A/G. Let us denote this action by .r : ¯ → A¯ λ,r A¯ . (A/G)∞ × A/G (λ, A)
(27)
For any given λ ∈ (A/G)∞ , let µλ,r denote the push-forward of the measure µr by ¯ The measure µλ,r is completely determined by the integrals of the map A¯ → A¯ λ,r A. ¯ continuous functions F (A): ¯ ¯ ¯ . F (A)dµλ,r (A) = F A¯ λ,r A¯ dµr (A) (28) A/G
A/G
We need only to consider the functions α (9), and therefore the measure µλ,r is determined by the following map from HG to C: ¯ . A¯ λ,r A¯ (α)dµr (A) (29) α → A/G
One gets from (12), (22) and (26): a ¯ = exp i λa (x)Xα,r (x)d 3 x A¯ λ,r A¯ (α)dµr (A) A/G
∗ E∞
eiφ(Xα,r ) dµ (φ).
(30)
∗ , one gets further: Recalling the action (4) of (A/G)∞ on E∞ ¯ = A¯ λ,r A¯ (α)dµr (A) ei(φ+λ)(Xα,r ) dµ (φ) ∗ E∞
A/G
=
∗ E∞
eiφ(Xα,r ) dµ,λ (φ) ,
(31)
where the measure µ,λ is the push-forward of µ by the map φ → φ +λ (4). Recalling also the arguments of Sect. 4, one sees easily that the measure µλ,r coincides with (*r )∗ µ,λ , the push-forward of µ,λ by the map *r (20). Since the Fock measure µ is quasi-invariant under the action of (A/G)∞ , this is sufficient to prove that, for any
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r ∈ R+ , the measure µr is quasi-invariant with respect to the action .r (27). For if B ⊂ A/G is such that µr (B) = 0, we then have µ (*−1 r B) = 0, by definition of the push-forward measure. The quasi-invariance of µ then shows that µ,λ (*−1 r B) = 0, ∀λ ∈ (A/G)∞ , which in turn is equivalent to µλ,r (B) = 0. Thus Theorem 2. The measure µr is quasi-invariant with respect to the action .r , for any given r. Using we define, for any r, a natural unitary representation Vr of (A/G)∞ in this result, L2 A/G, µr : ¯ = Vr (λ)ψ (A)
dµλ,r ¯ ¯ ψ Aλ,r A , λ ∈ (A/G)∞ , ψ ∈ L2 A/G, µr , dµr
(32)
where dµλ,r /dµr is the Radon-Nikodym derivative. One can easily work out the commutation relations between Vr (λ) and the r-Fock representation r (Tα ) of holonomies: Vr (λ)
a 3 (T ) = exp i λ (x)X (x)d x r α a α,r
r (Tα )Vr (λ) .
(33)
Let us consider the self-adjoint generator dVr (λ) of the one-parameter unitary group Vr (tλ), t ∈ R. Notice that the existence of dVr (λ), or the continuity of the one-parameter ∗ group Vr (tλ), follows from the continuity of the representation V (6) in L2 E∞ , µ . From (33) one obtains the following commutator:
dVr (λ),
r (Tα )
=
a λa (x)Xα,r (x)d 3 x
r (Tα ) .
(34)
This commutator is indeed the quantization of a given classical Poisson bracket, as realized by Varadarajan [Va1]. Consider then, for each r, the following functions of the electric field, parametrized by elements of (A/G)∞ : a
E (x) → Er (λ) :=
λa (x)
a
fr (x − y)E (y)d y d 3 x , 3
(35)
where fr is given by (18). The functions Er (λ) are referred to as “Gaussian-smeared smeared electric fields” in [Va1]. The Poisson bracket between these functions and the holonomies is Tα , Er (λ) = i
a λa (x)Xα,r (x)d 3 x Tα ,
(36)
showing that dVr (λ) can be seen as the quantization in L2 A/G, µr of the classical function Er (λ).
Invariance Properties of Induced Fock Measures for U(1) Holonomies
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6. Discussion Varadarajan’s use of smeared form factors allowed an embedding of distributional connections into A/G, overcoming the fact that the natural embedding of connections is not extensible to distributions. This step is very welcome, since A/G can be seen as a common measurable space from which both Fock states and loop states can be defined. In particular, the r-Fock measures µr give natural images L2 A/G, µr of the Fock space [Va1]. The Fock states, however, cannot be regarded as elements of L2 A/G, µ0 , the kinematical Hilbert space for loop states, as a consequence of the mutual singularity between the Haar measure and the r-Fock measures, which we have proven. Nevertheless, Fock states can be exhibited within the framework of loop states. In fact, Varadarajan [Va2] showed that the Fock states can be realized as elements of a natural extension of L2 A/G, µ0 , e.g. the dual Cyl∗ of the space of cylinder functions in A/G. Notice that such an extension from L2 A/G, µ0 to Cyl∗ is already required in loop quantum gravity, in order to solve the diffeomorphism constraint [ALMMT]. A suitable generalization of Varadarajan’s work to quantum gravity is, therefore, expected to produce an embedding of Minkowskian Fock-like states (describing “gravitons” of a semiclassical or low energy effective theory) into the space Cyl∗ of non-perturbative physical loop states. These issues are currently under investigation (see [AL4] and also [T] for a more general approach to semiclassical analysis). In these efforts, measure theory in A/G plays a relevant role, e.g. in the definition of quantum operators and in the analysis of the physical contents of the states. A good understanding of the r-Fock measures and their relation to the Haar measure may therefore be important to further developments. In order to complement our measure theoretical results, we would like to conclude with a brief comment regarding topological aspects. Although the r-Fock measures are supported in irrelevant sets with respect to the Haar measure, it can be shown on the topological side that every conceivable support of a r-Fock measure is dense in A/G 1 . The r-Fock measures µr are therefore faithful2 , just like the Haar measure µ0 [AL1]. (Recall that a Borel measure is said to be faithful if every non-empty open set has non-zero measure, which is readily seen to be equivalent to the denseness of every conceivable support. Turning to representations, a measure in A/G is faithful if and only if the corresponding representation of the holonomy algebra HA is faithful.) The fact that the Haar measure and the r-Fock measures are all faithful and mutually singular means that one can find a family of mutually disjoint dense sets, each of which supports a different measure. Notice finally that dense sets in A/G that do not contribute to the Haar measure were already known, e.g. the set of smooth connections [MM] or a considerable extension of it given in [MTV]. In the present case, however, one has new measures, living on µ0 -irrelevant sets. Acknowledgements. I thank José Mourão, Jerzy Lewandowski, Madhavan Varadarajan and Roger Picken. This work was supported in part by PRAXIS/2/2.1/FIS/286/94 and CERN/P/FIS/40108/2000.
1 The crucial result, pointed out by M. Varadarajan, is that * (E ∗ ) is dense. The denseness of smaller r ∞ ∗ and the continuity of * with respect supports follows from the faithfulness of the Fock measure µ in E∞ r ∗ to an appropriate topology in E∞ . 2 An independent proof using projective arguments is given in [AL4].
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References [As]
Ashtekar, A.: Lectures on non-Perturbative Canonical Quantum Gravity. Singapore: World Scientific, 1991 [ARS] Ashtekar, A., Rovelli, C., Smolin, L. S.: Phys. Rev. D44, 1740 (1991) [AR] Ashtekar, A., Rovelli, C.: Class. Quantum Grav. 9, 1121 (1992) [AI1] Ashtekar, A., Isham, C. J.: Phys. Lett. B274, 393 (1992) [AI2] Ashtekar, A., Isham, C. J.: Class. Quant. Grav. 9, 1433 (1992) [AL1] Ashtekar, A., Lewandowski, J.: Representation Theory of Analytic Holonomy C 0 Algebras. In: Knots and Quantum Gravity, ed. J. Baez, Oxford: Oxford University Press, 1994 [AL2] Ashtekar, A., Lewandowski, J.: J. Math. Phys. 36, 2170 (1995) [AL3] Ashtekar, A., Lewandowski, J.: J. Geom. Phys. 17, 191 (1995) [AL4] Ashtekar, A., Lewandowski, J.: Class. Quant. Grav. 18, L117 (2001) [ALMMT] Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., Thiemann, T.: J. Math. Phys. 36, 6456 (1995) [BSZ] Baez, J., Segal, I., Zhou, Z.: Introduction to Algebraic and Constructive Quantum Field Theory, Princeton: Princeton University Press, 1992 [Ba1] Baez, J.: Lett. Math. Phys. 31, 213 (1994) [Ba2] Baez, J.: Diffeomorphism Invariant Generalized Measures on the Space of Connections Modulo Gauge Transformations. In: Proceedings of the Conference on Quantum Topology, ed. D. Yetter Singapore: World Scientific, 1994 [GV] Gelfand, I.M., Vilenkin, N.: Generalized Functions. Vol. IV, New York: Academic Press, 1964 [GT] Gambini, R., Trias, A.: Phys. Rev. D23, 553 (1981) [GJ] Glimm, J., Jaffe, A.: Quantum Physics. New York: Springer Verlag, 1987 [Ki] Kirillov, A.A.: Elements of the Theory of Representations. Berlin: Springer Verlag, 1975 [MM] Marolf, D., Mourão, J.M.: Commun. Math. Phys. 170, 583 (1995) [MTV] Mourão, J.M., Thiemann, T., Velhinho, J.M.: J. Math. Phys. 40, 2337 (1999) [Ma] Martins, J.: Mecânica Quântica em Espaços de Conexões. Instituto Superior Técnico, 2000. Unpublished [ReSi] Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, SelfAdjointeness. New York: Academic Press, 1975 [RoSm] Rovelli, C., Smolin, L.: Nucl. Phys. B331, 80 (1990) [T] Thiemann, T.: Introduction to Modern Canonical Quantum General Relativity. To appear in “Living Reviews” [Va1] Varadarajan, M.: Phys. Rev. D61, 104001 (2000) [Va2] Varadarajan, M.: Phys. Rev. D64, 104003 (2001) [Ve2] Velhinho, J.M.: Métodos Matemáticos em Quantização Canónica de Espaços de Fase não Triviais. Ph.D. Dissertation (Universidade Técnica de Lisboa, Instituto Superior Técnico, 2001) [Ya] Yamasaki, Y.: Measures on Infinite Dimensional Spaces. Singapore: World Scientific, 1985 Communicated by H. Nicolai
Commun. Math. Phys. 227, 551 – 585 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Gauge Theoretical Equivariant Gromov–Witten Invariants and the Full Seiberg–Witten Invariants of Ruled Surfaces Ch. Okonek1 , A. Teleman2,3 1 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland.
E-mail: [email protected]
2 LATP, CMI, Université de Provence, 39 Rue F. J. Curie, 13453 Marseille Cedex 13, France.
E-mail: [email protected]
3 Faculty of Mathematics, University of Bucharest, Bucharest, Romania
Received: 22 February 2001 / Accepted: 16 January 2002
Abstract: Let F be a differentiable manifold endowed with an almost Kähler structure (J, ω), α a J -holomorphic action of a compact Lie group Kˆ on F , and K a closed normal subgroup of Kˆ which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F, α, K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface . Our main results concern the special case of the triple (Hom(Cr , Cr0 ), αcan , U (r)), where αcan denotes the canonical action of Kˆ = U (r) × U (r0 ) on Hom(Cr , Cr0 ). We give a complex geometric interpretation of the corresponding moduli spaces of solutions in terms of gauge theoretical quot spaces, and compute the invariants explicitly in the case r = 1. Proving a comparison theorem for virtual fundamental classes, we show that the full Seiberg–Witten invariants of ruled surfaces, as defined in [OT2], can be identified with certain gauge theoretical Gromov–Witten invariants of the triple (Hom(C, Cr0 ), αcan , U (1)). We find the following formula for the full Seiberg–Witten invariant of a ruled surface over a Riemann surface of genus g: −signc,[F ]
SWX,(O1 ,H0 ) (c) = 0, signc,[F ] SWX,(O1 ,H0 ) (c)(l)
= signc, [F ]
g
i≥max(0,g− w2c )
ic ∧ l, lO1 , i!
Partially supported by: EAGER – European Algebraic Geometry Research Training Network, contract No HPRN-CT-2000-00099 (BBW 99.0030), and by SNF, nr. 2000-055290.98/1.
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where [F ] denotes the class of a fibre. The computation of the invariants in the general case r > 1 should lead to a generalized Vafa-Intriligator formula for “twisted” Gromov– Witten invariants associated with sections in Grassmann bundles. 1. Introduction 1.1. The general set up. Let F be a differentiable manifold, ω a symplectic form on F , and J a compatible almost complex structure. Let α be a J -holomorphic action of a compact Lie group Kˆ on F , and let K be a closed normal subgroup of Kˆ which leaves ω invariant. ˆ Put K0 := K/K, and let π be the projection of Kˆ onto this quotient. We fix an invariant inner product on the Lie algebra kˆ of Kˆ and denote by pr k : kˆ → k the orthogonal projection onto the Lie algebra of K. The topological data of our moduli problem are a K0 -bundle P0 on a compact oriented ˆ consisting of a differentiable 2-manifold , and an equivalence class c of pairs (λ, h) λ ˆ ˆ π -morphism P − → P0 and a homotopy class h of sections in the associated bundle ˆ (λ , hˆ ) are equivalent if there exists an isomorphism E := Pˆ ×Kˆ F . Two pairs (λ, h), Pˆ → Pˆ over P0 which maps hˆ onto hˆ . The pair (P0 , c) should be regarded as the discrete parameter on which our moduli problem depends. It plays the same role as the data of a SU(2)- or a PU(2)-bundle in Donaldson theory, or the data of an equivalence class of Spinc -structures in Seiberg– Witten theory. λ ˆ of c we denote by λ (c) ⊂ ( , E) the union For every representant (Pˆ − → P0 , h) ˆ of all homotopy classes h of sections in E for which (λ, hˆ ) ∈ c. This set λ (c) is the union of the homotopy classes in the orbit of hˆ with respect to the action of the group π0 (Aut P0 (Pˆ )) on the set π0 ( ( , E)). In other words, λ (c) is the saturation of hˆ with respect to the AutP0 (Pˆ )-action on the space of sections. λ ˆ of c. Let µ be a K-equivariant ˆ Now fix a representant (Pˆ − moment map → P0 , h) for the restricted K-action α|K on F , let g be a metric on , and let A0 be a connection on P0 . The triple p := (µ, g, A0 ) is the continuous parameter on which our moduli problem depends. It plays the role of the Riemannian metric on the base manifold in Donaldson theory [DK], or the role of the pair (Riemannian metric, self-dual form) in Seiberg– Witten theory [W2, OT1, OT2]. The orthogonal projection pr k induces a bundle projection which we denote by the same symbol pr k : Pˆ ×ad kˆ −→ Pˆ ×ad k. Since Kˆ acts J -holomorphically, a connection Aˆ in Pˆ defines an almost holomorphic structure JAˆ in the associated bundle E; JAˆ agrees with J on the vertical tangent bundle TE/ of E and it agrees with the holomorphic structure Jg defined by g on on the ˆ A-horizontal distribution of E. Our gauge group is G = Aut P0 (Pˆ ) ( , Pˆ ×Ad K), and it acts from the right on our configuration space A := AA0 (Pˆ , λ) × λ (c).
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Here AA0 (Pˆ , λ) is the affine space of connections Aˆ in Pˆ which project onto A0 via λ. ˆ ϕ) ∈ A we consider the equations For a pair (A, ϕ is JAˆ holomorphic (Vp ) 0. pr k #FAˆ + µ(ϕ) = These vortex type equations are obviously gauge invariant. The first condition of (Vp ) can be rewritten as ∂¯Aˆ ϕ = 0, where ∂¯Aˆ ϕ ∈ ( , #0,1 (ϕ ∗ (TE/ ))) is the (0, 1)-component of the derivative dAˆ ϕ ∈ ( , #1 (ϕ ∗ (TE/ ))). ˆ these equations were independently found and In the particular case where K = K, studied in [Mu1, CGS, G]. ˆ the moduli space of solutions of the equations (Vp ) Denote by M = Mp (λ, h) modulo gauge equivalence. Let A∗ be the open subspace of A consisting of irreducible pairs, i.e. of pairs with trivial stabilizer, and denote by M∗ the moduli space of irreducible solutions; M∗ can be ∗ regarded as a subspace of the infinite dimensional quotient B ∗ := A G of irreducible pairs. The space B ∗ becomes a Banach manifold after suitable Sobolev completions. The parameters p for which M∗ = M are called bad parameters, and the set of bad parameters is called the bad locus or the wall. Our purpose is to define invariants for triples (F, α, K) by evaluating certain tautological cohomology classes on the virtual fundamental class of moduli spaces M corresponding to good parameters, provided these spaces are compact (or have a canonical compactification) and possess a canonical virtual fundamental class. The invariants will depend on the choice of the discrete parameter (P0 , c), and a chamber C, i.e. a component of the complement of the bad locus in the space of continuous parameters. Some general ideas for the construction of Gromov–Witten type invariants associated with moduli spaces of solutions of vortex-type equations have been outlined in [CGS]; in [Mu2] such invariants are rigorously defined in the special case where F is compact Kähler and K = Kˆ = S 1 . Note that our program is fundamentally different: Our main construction begins with an important new idea, the parameter symmetry group ˆ K0 := K/K. This group, whose introduction was motivated by our previous work on Seiberg–Witten theory, leads to an essentially new set up and plays a crucial role in the following. Without it none of our main results could even be formulated. Our first aim is to construct tautological cohomology classes on the infinite dimensional quotient B ∗ . ˆ Note first that any section ϕ ∈ ( , E) can be regarded as a K-equivariant map Pˆ → F . ˆ Therefore one gets a K-equivariant evaluation map ev : A∗ × Pˆ → F, ˆ over which is obviously G-invariant. Let Pˆ := A∗ ×G Pˆ be the universal K-bundle ∗ ˆ B × . The evaluation map descends to a K-equivariant map ( : A∗ ×G Pˆ → F
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which can be regarded as the universal section in the associated universal F -bundle A∗ ×G E. Let (∗ : HK∗ˆ (F, Z) → H ∗ (B ∗ × , Z) ˆ be the map induced by ( in K-equivariant cohomology. Using the same idea as in Donaldson theory, we define for every c ∈ H ∗ˆ (F, Z) and β ∈ H∗ ( ) the element K δ c (β) ∈ H ∗ (B ∗ , Z) by δ c (β) := (∗ (c)/β. Recall that one has natural morphisms η∗
B(π)∗
ˆ Z) −−→ H ∗ (F, Z) H ∗ (BK0 , Z) −−−−→ H ∗ (B K, Kˆ which are induced by the natural maps η
B(π)
E Kˆ ×Kˆ F − → B Kˆ −−−→ BK0 . Let κˆ : → B Kˆ be a classifying map for the bundle Pˆ , and let κ0 := B(π ) ◦ κˆ be the corresponding classifying map for P0 . ˆ Denote by hˆ ∗ the morphism H ∗ˆ (F, Z) → H ∗ ( , Z) defined by h. K
Proposition 1.1. The assignment (c, β) → δ c (β) has the following properties: 1. It is linear in both arguments. 2. For any homogeneous elements c ∈ H ∗ˆ (F, Z), β ∈ H∗ ( , Z) of the same degree, K one has c ∗ δ (β) = hˆ (c), β · 1H 0 (B∗ ,Z) . 3. For any homogeneous elements c, c ∈ H ∗ˆ (F, Z), one has K
δ
c∪c
([∗]) = δ c ([∗]) ∪ δ c ([∗]).
4. For any homogeneous elements c, c ∈ H ∗ˆ (F, Z), β ∈ H1 ( , Z) one has K
δ
c∪c
(β) = (−1)
degc
δ c (β) ∪ δ c ([∗]) + δ c ([∗]) ∪ δ c (β).
5. Let (βi )1≤i≤2g( ) be a basis of H1 ( , Z). Then for any homogeneous elements c, c ∈ H ∗ˆ (F, Z) one has K
δ c∪c ([ ]) = δ c ([ ]) ∪ δ c ([∗]) + δ c ([∗]) ∪ δ c ([ ]) − (−1)degc
2g( )
δ c (βi ) ∪ δ c (βj )(βi · βj ).
i,j =1
6. For every c0 ∈ H ∗ (BK0 , Z) one has δ c∪(η
∗ B(π)∗ c ) 0
(β) = δ c (κ0∗ (c0 ) ∩ β).
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The properties 1–5 follow from general properties of the slant product, whereas the last property follows from the natural identification Pˆ ×Kˆ K0 B ∗ × P0 . To every pair of homogeneous elements c ∈ H ∗ˆ (F, Z), β ∈ H∗ ( , Z) satisfying K c degc ≥ degβ we associate the symbol , considered as an element of degree β degc − degβ. Let A = A(F, α, K, c)be the graded-commutative graded Z-algebra which is genc erated by the symbols , subject to the relations which correspond to the properties β 1–6 in the proposition above. This algebra depends only on the homotopy type of our topological data. c The assignment → δ c (β) defines a morphism of graded-commutative Zβ algebras δ : A → H ∗ (B ∗ , Z). λ ˆ of c Now fix a discrete parameter (P0 , c) and choose a representant (Pˆ − → P0 , h) as above. Choose a continuous parameter p not on the wall. When the moduli space ˆ ∗ is compact and possesses a virtual fundamental class [Mp (λ, h) ˆ ∗ ]vir , then Mp (λ, h) this class defines an invariant (P0 ,c)
GGWp
(F, α, K) : A(F, α, K, c) −→ Z,
given by (P0 ,c)
GGWp
ˆ ∗ ]vir . (F, α, K)(a) := δ(a), [Mp (λ, h)
The 6 properties listed in the proposition above show that: Remark 1.2. Let G be a set of homogeneous generators of H ∗ˆ (F, Z), regarded as a K graded H∗ (BK 0 , Z)-algebra. Then A is generated as a graded Z-algebra by elements of c the form with c ∈ G, β ∈ H∗ ( , Z), and degc > degβ. β Suppose for example that we are in the simple situation where Kˆ splits as Kˆ = U (r) × K0 and F is contractible. In this case the graded algebra ˆ Z) = H ∗ (BU (r), Z) ⊗ H ∗ (BK0 , Z) HK∗ˆ (F, Z) = H ∗ (B K, is generated as a H ∗ (BK0 , Z)-algebra by the universal Chern classes ci ∈ H ∗ (BU (r), Z), 1 ≤ i ≤ r, and one has a natural identification A Z[u1 , . . . , ur , v2 , . . . , vr ] ⊗ #
∗
r i=1
H1 (X, Z)i .
(I )
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Here ui =
ci ci , vi = have degree 2i and 2i − 2 respectively, whereas [∗] [ ] ci β ∈ H1 ( , Z) H1 ( , Z)i := β
is a copy of H1 ( , Z) whose elements are homogenous of degree 2i − 1. Note also that in the case Kˆ = K × K0 , Pˆ splits as the fibre product of a K-bundle P and P0 , and the gauge group G can be identified with Aut(P ). ˆ Similarly, the universal K-bundle Pˆ over B ∗ × splits as the fibre product of the ∗ universal K-bundle P := A ×G P with the K0 -bundle pr ∗ (P0 ). If K = U (r), one can use this bundle to give a geometric interpretation of the images ci via δ of the classes ui , vi , ∈ H1i ( , Z) defined above: β c δ(ui ) = ci (P)/[∗], δ(vi ) = ci (P)/[ ], δ i = ci (P)/β. β In the special case r = 1, one just gets A Z[u] ⊗ #∗ (H1 ( , Z)). This shows that when Kˆ = S 1 × K0 and F is contractible, the gauge theoretical Gromov–Witten invariants can be described by an inhomogeneous form (P0 ,c)
GGWp
(F, α, S 1 ) ∈ #∗ (H 1 ( , Z))
setting
(P ,c) GGWp 0 (F, α, S 1 )(l)
:= δ(
ˆ ∗ ]vir uj ∪ l), [Mp (λ, h)
j ≥0
for any l ∈
#∗ (H
ˆ represents c and p is a good continuous parameter. 1 ( , Z)). Here (λ, h)
1.2. Special cases. Twisted Gromov–Witten invariants. This is the special case K = {1}. Here the gauge group G is trivial, the moduli space M is the space of JA0 -holomorphic sections of the bundle E, and giving c is equivalent to fixing a homotopy class h0 of sections in P0 ×K0 F . The resulting invariants, when defined, should be regarded as twisted Gromov–Witten invariants, because we have replaced the space of F -valued maps on in the definition of the standard Gromov–Witten invariants ([Gr, LiT, R]) by the space of sections in a F -bundle P0 ×K0 F . These invariants are associated with the almost Kähler manifold F , the K0 -action, and they depend on the discrete parameter (P0 , h0 ) and the continuous parameter A0 . The invariants are defined on a graded algebra A(F, α, h0 ) obtained by applying the construction above in this special case. Note that even in the particular case when the bundle P0 is trivial, varying the parameter connection A0 provides interesting deformations of the usual Gromov–Witten moduli spaces. In some situations one can prove a transversality result with respect to the parameter A0 and then compute the standard Gromov–Witten invariants using a general parameter.
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Equivariant symplectic quotients. This is the special case where the K-action on µ−1 (0) is free and µ is a submersion around µ−1 (0). In this case our data define a symplec −1 tic factorization problem, and one has a symplectic quotient Fµ := µ (0) K with an induced compatible almost complex structure and an induced almost holomorphic K0 -action αµ . When K0 = {1}, the system (F, α, K, µ) should be called symplectic factorization problem with additional symmetry, since the symplectic manifold F was endowed with a larger symmetry than the Hamiltonian symmetry used in performing the symplectic factorization. For any homotopy class h0 of sections in P0 ×K0 Fµ one can consider the twisted Gromov–Witten invariants of the pair (Fµ , αµ ) corresponding to the parameters (P0 , h0 ) ˆ as follows. We choose a section and A0 . One can associate to h0 a class c(h0 ) = [λ, h] ϕ0 ∈ h0 regarded as a K0 -equivariant map P0 → Fµ , put Pˆ := P0 ×Fµ µ−1 (0) endowed λ ˆ action and the obvious morphism Pˆ − with the natural K→ P0 , and let hˆ be the class ˆ of the section defined by the K -equivariant map (p0 , f ) → f . It is then an interesting and natural problem to compare the twisted Gromov–Witten invariants of the pair (Fµ , αµ ) with the gauge theoretical Gromov–Witten invariants of the initial triple (F, α, K) via the natural morphism A(F, α, K, c(h0 )) → A(Fµ , αµ , h0 ). In the non-twisted case K0 = {1}, this problem was treated in [G, CGS]. 1.3. Main results. In Sect. 2 we study the moduli spaces Mt (E, E0 , A0 ) of solutions of the vortex type equations over Riemann surfaces ( , g), associated with the triple (Hom(Cr , Cr0 ), αcan , U (r)) and the moment map µt (f ) = 2i f ∗ ◦ f − itid, t ∈ R. In Sect. 2.1 we introduce the gauge theoretical quot space GQuot E E0 of a holomorphic bundle E0 on a general compact complex manifold X. The space GQuot E E0 parametrizes ∞ the quotients of E0 with locally free kernels of fixed C -type E, and can be identified with the corresponding analytical quot space when X is a curve. We prove a transversality result (Proposition 2.4) which states that, when X is a curve, GQuot E E0 is smooth and has the expected dimension for an open dense set of holomorphic structures E0 in a fixed C ∞ -bundle E0 . In Sect. 2.2 we use the Kobayashi–Hitchin correspondence for the vortex equation [B] over a Riemann surface ( , g), to identify the irreducible part M∗t (E, E0 , A0 ) of Volg ( ) Mt (E, E0 , A0 ) with the gauge theoretical moduli space of 2π t - stable pairs. The latter can be identified with a gauge theoretical quot space when t is sufficiently large (Corollary 2.8, Proposition 2.9). In Sect. 2.3 we prove transversality and compactness results for the moduli spaces Mt (E, E0 , A0 ). In Sect. 3 we introduce formally our gauge theoretical Gromov–Witten invariants for the triple (Hom(Cr , Cr0 ), αcan , U (r)) and prove an explicit formula in the abelian case r = 1. We define the invariants using Brussee’s formalism of virtual fundamental classes associated with Fredholm sections [Br] applied to the sections cutting out the moduli spaces M∗t (E, E0 , A0 ). The comparison Theorem 3.2 states that one can alternatively
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use the virtual fundamental class of the corresponding moduli space of stable pairs. This provides a complex geometric interpretation of our invariants. The results of Sect. 2, and a complex geometric description of the abelian quot spaces as complete intersections in projective bundles, enables us to explicitly compute the full invariant in the abelian case r = 1: Theorem. Put v = χ (Hom(L, E0 )) − (1 − g( )). The Gromov–Witten invariant (E ,c ) GGWp 0 d (Hom(C, Cr0 ), αcan , U (1)) ∈ #∗ (H 1 ( , Z)) is given by the formula g( ) (r0 )i (E0 ,cd ) r0 GGWp (Hom(C, C ), αcan , U (1))(l) = ∧ l, lO1 , i! i≥max(0,g( )−v)
for any l ∈ #∗ (H1 ( , Z)). Here is the class in #2 (H1 ( , Z)) given by the intersection form on , and lO1 is the generator of #2g( ) (H 1 ( , Z)) defined by the complex orientation O1 of H 1 ( , R). As an application we give in Sect. 3.4 an explicit formula for the number of points in certain abelian quot spaces of expected dimension 0. This answers a classical problem in Algebraic Geometry. A generalisation of this result to the case r > 1 requires a wall-crossing formula for the non-abelian invariants. The main result of Sect. 4 is a natural identification of the full Seiberg–Witten invariants of ruled surfaces with certain abelian gauge theoretical Gromov–Witten invariants. This result is a direct consequence of two important comparison theorems: The stanπ dard description of the effective divisors on a ruled surface X := P(V0 ) −→ over a curve, identifies the Hilbert schemes of effective divisors on X with certain quot schemes associated with symmetric powers of the 2-bundle V0 over . In Sect. 4.1 we show that, if one replaces the Hilbert schemes and the quot schemes by their gauge theoretical analoga GDou, GQuot, one has Theorem. For every C ∞ - line bundle L on , there is a canonical isomorphism of complex spaces ∨ GDou(π ∗ (L) ⊗ OP(V0 ) (n)) GQuot L S n (V0 ) which maps the virtual fundamental class [GDou(π ∗ (L) ⊗ OP(V0 ) (n))]vir to the virtual vir
∨ . fundamental class GQuotL S n (V0 ) On the other hand, the gauge theoretical Douady space on the left can be identified with the moduli space of monopoles on X which corresponds to the π ∗ (L) ⊗ OP(V0 ) (n)twisted canonical Spinc -structure. In Sect. 4.2, we show that this identification respects virtual fundamental classes too. Combining all these results we see that the full Seiberg–Witten invariant of the ruled surface X as defined in [OT2] can be identified with a corresponding gauge theoretical Gromov–Witten invariant for the triple (Hom(C, Cn+1 ), αcan , S 1 ). Using the explicit formula proven in Sect. 3, one gets an independent check of the universal wall-crossing formula for the full Seiberg–Witten invariant in the case b+ = 1. 2. Moduli Spaces Associated with the Triple (Hom(Cr , Cr0 ), αcan , U (r)) Because of the very technical compactification problem, we will not introduce our gauge theoretical invariants formally in the general framework described in Sect. 1.1. Instead
Gromov–Witten Invariants and Seiberg–Witten Invariants of Ruled Surfaces
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we specialize to the case K = U (r), Kˆ = U (r) × U (r0 ), and F = Hom(Cr , Cr0 ) ˆ endowed with the natural left K-action. The K-action on F has the following family of moment maps, µt (f ) =
i ∗ f ◦ f − it id, t ∈ R, 2
ˆ which are all K-equivariant. Since F is contractible, one has only one homotopy class of sections in any fixed F -bundle. Hence in this case our topological data reduce to the data of a differentiable Hermitian bundle E0 of rank r0 and a class of differentiable Hermitian bundles E of rank r. ˆ as Therefore, when we fix the bundle E0 , the set of equivalence classes of pairs (λ, h) above can be identified with Z via the map E → deg(E). We will denote the class corresponding to an integer d by cd . Our moduli problem becomes now: Let A0 be a fixed Hermitian connection in E0 and let E0 be the associated holomorphic bundle. Classify all pairs (A, ϕ) consisting of a Hermitian connection A in E and a (A, A0 )-holomorphic morphism ϕ : E → E0 such that the following vortex type equation is satisfied: 1 i#FA − ϕ ∗ ◦ ϕ = −t idE . 2 Our first purpose is to show that, in a suitable chamber, the moduli space of solutions of this equation can be identified with a certain space of quotients of the holomorphic bundle E0 . This remark will allow us later to describe the invariants explicitly in the abelian case r = 1.
2.1. Gauge theoretical quot spaces. Let E0 be a holomorphic bundle of rank r0 on a compact connected complex manifold X of dimension n, and fix a differentiable vector bundle E of rank r on X. There is a simple gauge theoretical way to construct a complex space GQuot E E0 parametrizing equivalence classes of pairs (E, ϕ) consisting of a holomorphic bundle E of C ∞ -type E and a sheaf monomorphism ϕ : E <→ E0 ; in other words, GQuotE E0 parametrizes the quotients of E0 with locally free kernel of fixed ∞ C -type E. Denote by E0 the underlying differentiable bundle of E0 and by ∂¯0 the corresponding ¯ Dolbeault operator. Let A(E) be the space of semiconnections in E and put ¯ A¯ := A(E) × A0 Hom(E, E0 ). Let G C := (X, GL(E)) be the complex gauge group of the bundle E. A pair (δ, ϕ) ∈ A¯ will be called: – simple if its stabilizer with respect the natural action of G C is trivial, – integrable if δ 2 = 0 and ∂¯δ,∂¯0 ϕ = 0. We denote by A¯ simple the open subspace of simple pairs, and by B¯ simple its G C quotient; B¯ simple becomes a possibly non-Hausdorff Banach manifold after suitable Sobolev completions. Using similar methods as in [LO] one can construct a finite dimensional – but possibly non-Hausdorff – complex subspace Msimple (E, E0 ) of B¯ simple parametrizing the G C orbits of simple integrable pairs. This construction has been carried out in [Su].
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It is easy to see – using Aronszajin’s theorem [A] – that any pair (δ, ϕ), such that ϕx : Ex → E0,x is injective in at least one point x ∈ X, is simple. Put A¯ inj := {(δ, ϕ) ∈ A¯ simple | ∃ x ∈ X with ϕx injective}, ¯ inj and B¯ inj := A . GC Proposition 2.1. After sufficiently high Sobolev completions, the open subspace B¯ inj of B¯ simple becomes an open Hausdorff submanifold of B¯ simple . Proof. Use the subscript ( )k to denote Sobolev L2k -completions. The Sobolev index is chosen sufficiently large, such that L2k becomes an L2l module for any l ≥ k. Let (δ1 , ϕ1 ), inj (δ2 , ϕ2 ) ∈ A¯ k be two pairs whose orbits [δ1 , ϕ1 ], [δ2 , ϕ2 ] cannot be separated. Then inj there exists a sequence of pairs (δ1n , ϕ1n ) ∈ A¯ k and a sequence of gauge transformations C gn ∈ Gk+1 such that (δ1n , ϕ1n ) → (δ1 , ϕ1 ), (δ1n , ϕ1n ) · gn → (δ2 , ϕ2 ). Put (δ2n , ϕ2n ) := (δ1n , ϕ1n ) · gn . With this notation, one has gn ◦ δ2n = δ1n ◦ gn , ϕ2n = ϕ1n ◦ gn .
(1)
The first relation can be rewritten as n δ12 (gn ) = 0,
(2)
n is the semiconnection δ n ⊗ (δ n )∨ induced by δ n , δ n in End(E). Put where δ12 1 2 1 2
fn :=
1 gn . # gn #k
Since the sequence (fn ) is bounded in L2k , we may suppose, passing to a subsequence if necessary, that (fn ) converges weakly in L2k to an element f12 ∈ A0 (End(E))k . Now n converges to δ := δ ⊗ (δ )∨ in L2 to see that (δ (f )) use (2) and the fact that δ12 12 1 2 12 n k converges strongly to 0 in L2k . This implies, by standard elliptic estimates, that (fn ) is bounded in L2k+1 . Therefore, passing again to a subsequence if necessary, we may suppose that the convergence of (fn ) to f12 is strong in L2k ; the limit must fulfill δ12 (f12 ) = 0, # f12 #k = 1.
(3)
The second relation in (1) implies # ϕ1n ◦ fn #k 1 = . # gn #k # ϕ2n #k The right-hand term converges to c12 :=
#ϕ1 ◦f12 #k #ϕ2 #k .
Taking n → ∞ in (1), we get
ϕ1 ◦ f12 = c12 ϕ2 .
(4)
Similarly we get a Sobolev endomorphism f21 ∈ A0 (End(E))k and a constant c21 ∈ R satisfying δ21 (f21 ) = 0, # f21 #k = 1, ϕ2 ◦ f21 = c21 ϕ1 . (5)
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Put f1 := f12 ◦ f21 , f2 := f21 ◦ f12 . Using (3) and (5) we find δ11 (f1 ) = δ22 (f2 ) = 0,
ϕ1 ◦ f1 = c12 c21 ϕ1 ,
ϕ2 ◦ f2 = c12 c21 ϕ2 .
(6)
Suppose that c12 = 0 or c21 = 0. Then by (4) or (5), f12 (respectively f21 ) would vanish on the non-empty open set where ϕ1 (respectively ϕ2 ) is injective. But f12 (respectively ∗ δ (u) = 0 (respectively δ ∗ δ (u) = 0), f21 ) is a solution of the Laplace equation δ12 12 21 21 where the Laplace operator on the left has the same symbol as the usual Dolbeault Laplace operator. By Aronszajin’s identity theorem, this would imply f12 = 0 (or f21 = 0), which contradicts (3) or (5). Therefore, we must have c12 = 0 and c21 = 0. Now formula (6) shows that f1 = c12 c21 idE on the open set where ϕ1 is injective hence, by Aronszajin’s theorem again, f1 = c12 c21 idE everywhere. This implies that the endomorphism g12 := c112 f12 is a bundle isomorphism. Moreover, by (3) and (4), g12 satisfies δ1 ◦ g12 − g12 ◦ δ2 = 0, ϕ1 ◦ g12 = ϕ2 , so that the pairs (δi , ϕi ) are gauge equivalent and [δ1 , ϕ1 ] = [δ2 , ϕ2 ].
% $
Note that an integrable pair (δ, ϕ) is in A¯ inj if and only if ϕ defines an injective sheaf homomorphism. Definition 2.2. The gauge theoretical quot space GQuot E E0 of quotients of E0 with locally free kernels of C ∞ -type E is defined as the open subspace simple GQuot E (E, E0 ) ∩ B¯ inj E0 := M
of Msimple (E, E0 ). Note that GQuot E E0 is a Hausdorff complex space of finite dimension. For a holomorphic bundle E0 on a compact complex manifold, denote by Quot E E0 the complex analytic quot space parametrizing coherent quotients of E0 with locally free kernel of C ∞ -type E [Dou]. When E0 is a bundle on an algebraic complex manifold endowed with an ample line bundle, denote by Quot PE0 the Grothendieck quot scheme over C, parametrizing algebraic coherent quotients of E0 with Hilbert polynomial P . With these notations, one has: Remark 2.3. 1. Our gauge theoretical quot space GQuot E E0 can be identified with the . complex analytic quot space QuotE E0 This identification is an isomorphism of complex spaces, but a rigorous proof of this fact is very difficult [LL]. 2. If E0 is a bundle on an algebraic complex manifold endowed with an ample line bundle, then Quot E E0 can be identified with the underlying complex space of the open PE −PE
consisting of coherent quotients of E0 with locally free subscheme of Quot E0 0 ∞ kernel of C -type E [S]. 3. When X is a complex curve endowed with an ample line bundle of degree 1, the C ∞ type of a vector bundle on X is determined by its Hilbert polynomial. Furthermore, since torsion free sheaves on curves are locally free, GQuotE E0 parametrizes in this case all quotients of E0 with Hilbert polynomial PE0 − PE . In other words, on curves one has natural identifications of complex spaces PE −PE
E 0 GQuotE E0 Quot E0 Quot E0
.
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Note that, on curves, the first part of the integrability condition is automatically satisfied. Put d0 := deg(E0 ), d := deg(E). A simple transversality argument shows that Proposition 2.4. Let X be a curve, and put v(r0 , r, d, d0 ) := χ (E ∨ ⊗E0 )−χ (E ∨ ⊗E). The gauge theoretical quot space GQuotE E0 is smooth and has the expected dimension v(r0 , r, d, d0 ) for a dense open set of holomorphic structures E0 in E0 . Proof. We identify the space of holomorphic structures in a bundle over a curve with the affine space of semiconnections in the usual way. ¯ inj ¯ Let GQuotE E0 ⊂ B × A(E0 ) be the parametrized gauge theoretical moduli space, i.e. the moduli space of solutions ([δ, ϕ], ∂¯0 ) of the equation ∂¯δ,∂¯0 ϕ = 0. E ¯ The space GQuotE E0 is the fibre of the projection GQuot E0 → A(E0 ) over the semicon¯ 0 ) corresponding to E0 . nection ∂¯0 ∈ A(E ¯ 0 ) → A0,1 Hom(E, E0 ) the map (δ, ϕ, ∂¯0 ) → ∂¯ ¯ ϕ, We denote by f : A¯ inj × A(E δ,∂0 and by f¯ the induced section in the bundle
¯ 0 )] ×G C A0,1 Hom(E, E0 ) [A¯ inj × A(E ¯ 0 ). We will show that (after suitable Sobolev completions) f¯ is regular over B¯ inj × A(E in every point of its vanishing locus, hence GQuot E E0 becomes a smooth submanifold of ¯ 0 ). Equivalently, we will show that f is a submersion in every point of its B¯ inj × A(E vanishing locus Z(f ): Let ξ = (δ, ϕ, ∂¯0 ) ∈ Z(f ) and let β ∈ A0,1 Hom(E, E0 ) be L2 -orthogonal to imdξ (f ). One has ∂ f (δ, ϕ, ∂¯0 )(α) = α ◦ ϕ, α ∈ A0,1 End(E0 ). ∂(∂¯0 ) Note that the map End(Cr0 ) → Hom(Cr , Cr0 ) given by A → A ◦ ( is surjective when ( is injective. Therefore the image of ∂(∂∂¯ ) f (δ, ϕ, ∂¯0 ) contains the space 0
0 (U, #0,1 Hom(E, E0 )) of (0, 1)-forms with compact support contained in U , for every open set U ⊂ on which ϕ is a bundle monomorphism. This shows that β vanishes on U as a distribution, hence as a Sobolev section as well. ∂ , which is the first differential But β must also be orthogonal to the image of ∂(δ,ϕ) of the elliptic complex associated with the solution (δ, ϕ) and parameter ∂¯0 . This means that β is a solution of an elliptic system with scalar symbol, so that another application of Aronszajin’s identity theorem gives β = 0. ¯ Since the projection GQuotE E0 → A(E0 ) is proper, the set of regular values is open. ¯ By Sard’s theorem – which applies since the projection GQuot E E0 → A(E0 ) is a smooth Fredholm map defined on a Hausdorff manifold with countable basis [Sm] – the set of regular values is also dense. $ % A stronger form of this result refers to the embedding of quot spaces E GQuotE F0 <→ GQuot E0
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induced by a sheaf monomorphism ψ : F0 → E0 with torsion quotient. The map ψ defines a bundle isomorphism over the complement of the finite set S = supp(E0 F0 ). Let U be a small neighbourhood of S. Any holomorphic structure E0 in E0 which coincides with E0 on U defines a holomorphic structure ψ ∗ E0 on F0 which coincides with F0 on U . We denote by A¯ U,E0 (E0 ) the space of holomorphic structures in E0 which coincide with E0 on U . Using Aronszajin and Sard theorems again, one can prove the following simultaneous regularity result Proposition 2.5. Let X be a curve. The gauge theoretical quot spaces GQuot E E0 and E GQuotψ ∗ E0 are smooth and have the expected dimensions for a dense open set of holomorphic structures E0 ∈ A¯ U,E0 (E0 ). 2.2. Moduli spaces of vortices and stable holomorphic pairs of type (E, E0 ) . Consider again the case K = U (r), Kˆ = U (r) × U (r0 ), F = Hom(Cr , Cr0 ), and let (X, g) be a compact n-dimensional Kähler manifold. Let E0 be a fixed Hermitian bundle of rank r0 on X endowed with a fixed integrable Hermitian connection A0 , and denote by E0 the corresponding holomorphic bundle. We also fix a Hermitian bundle E of rank r and denote by d its degree deg(E) = c1 (E) ∪ [ωgn−1 ], [X]. Our original gauge theoretical problem can be extended to this more general setting: For a given real number t, classify all pairs (A, ϕ) consisting of an integrable Hermitian connection in E and a (A, A0 )-holomorphic morphism ϕ ∈ A0 Hom(E, E0 ) such that the following vortex type equation is satisfied: 1 i#FA − ϕ ∗ ◦ ϕ = −t idE . 2
(VtA0 )
We denote by Mt (E, E0 , A0 ) the moduli space cut out by the equation (VtA0 ) and the integrability equations FA02 = 0, ∂¯A,A0 ϕ = 0. The space Mt (E, E0 , A0 ) is a subspace of the quotient 0 B(E, E0 ) = A(E) × A Hom(E, E0 ) Aut(E) . The irreducible part M∗t (E, E0 , A0 ) ⊂ Mt (E, E0 , A0 ) is the open subspace consisting of orbits with trivial stabilizer; this is a real analytic finite dimensional subspace of the free quotient 0 ∗ B ∗ (E, E0 ) = [A(E) × A Hom(E, E0 )] Aut(E) , which becomes a Banach manifold after suitable Sobolev completions. The (slope) stability concept which corresponds to this gauge theoretical problem is well-known [B, HL]: Let τ be a real constant with deg(E)/rk(E) > −τ . A pair (E, ϕ) consisting of ϕ a holomorphic bundle of C ∞ -type E and a holomorphic sheaf morphism E − → E0 is 1 τ -(semi)stable if for every nontrival subsheaf F ⊂ E one has µ(E/F) (≥) −τ if rk(F) < r, µ(F) (≤) −τ if F ⊂ ker(ϕ). 1 The τ -stability introduced here corresponds to the slope δ -stability of [HL] for δ = deg(E) + τ rk(E). 1 1
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Definition 2.6. The gauge theoretical moduli space Mst τ (E, E0 ) of τ -stable pairs of type (E, E0 ) is the open subspace of Msimple (E, E0 ) consisting of τ -stable pairs. Note that it is not at all obvious that – even when (X, g) is a projective curve – the moduli space Mst τ (E, E0 ) can be identified with the underlying complex space of the corresponding quasi-projective moduli space of stable pairs of [HL]. For a proof of this fact we refer to [LL]. Theorem 2.7 (Kobayashi–Hitchin correspondence for the equations (VtA0 )). The moduli space M∗t (E, E0 , A0 ) of irreducible solutions of the equation (VtA0 ) can be identified with the gauge theoretical moduli space Mst τ (E, E0 ) of τ -stable pairs where τ=
(n − 1)!Volg (X) t. 2π
Proof. The set theoretical identification follows from the work of Bradlow [B] and Lin [Lin]. The identification as real analytic spaces follows as in [LT] and [OT1]. Corollary 2.8. Suppose r = 1. Then one has Mt (E, E0 , A0 ) = M∗t (E, E0 , A0 ) for deg(E) 2π t = − (n−1)!Vol and g (X) rk(E) Mt (E, E0 , A0 ) =
∅
deg(E) 2π if t < − (n−1)!Vol g (X) rk(E)
deg(E) GQuot E if t > − 2π (n−1)!Volg (X) rk(E) . E0
Proof. Indeed, integrating the equation (VtA0 ) over X one finds t >−
deg(E) 2π (n − 1)!Volg (X) rk(E)
when (VtA0 ) has solutions with non-vanishing ϕ-component. Conversely, if t > deg(E) 2π − (n−1)!Vol , any solution (A, ϕ) must have ϕ = 0, so it must be irreducible. g (X) rk(E) Using the theorem we get an isomorphism Mt (E, E0 , A0 ) Mst τ (E, E0 ), where τ > − deg(E) rk(E) . Since any non-trivial morphism defined on a holomorphic line
deg(E) E bundle is generically injective, we see that Mst τ (E, E0 ) = GQuot E0 if τ > − rk(E) and r = 1. $ %
In the non-abelian case, one has the following generalization of Corollary 2.8: Proposition 2.9. There exists a constant c(E0 , E) such that, for all τ ≥ c(E0 , E) the following holds: (i) For every τ -semistable pair (E, ϕ), ϕ is injective. (ii) Every pair (E, ϕ) with ϕ injective is τ -stable. E (iii) There is a natural isomorphism Mst τ (E, E0 ) = GQuot E0 .
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For all sufficiently large t ∈ R, one has Mt (E, E0 , A0 ) = M∗t (E, E0 , A0 ) and a natural identification Mt (E, E0 , A0 ) = GQuotE E0 . Proof. (i) Note first that, if ker(ϕ) = 0, the second inequality of the stability condition for F = ker(ϕ) implies deg(im(ϕ)) ≥ d + τ rk(ker(ϕ)). But im(ϕ) is a non-trivial subsheaf of the fixed bundle E0 , so one has an estimate of the form deg(im(ϕ)) ≤ C(E0 ), where C(E0 ) = sup deg(G) [Ko]. Therefore, as soon as G ⊂E0
τ > c(E0 , E) := max
1≤i≤r−1
C(E0 ) − d , i
any τ -semistable pair (E, ϕ) has an injective ϕ. (ii) Suppose now that ϕ is injective. The second part of the stability condition becomes empty, hence we only have to show that deg(F) < d + τ (r − rk(F)) for all subsheaves F of E with 0 < rk(F) < r. But if τ is larger than c(E0 , E), it follows that d + τ (r − s) > C(E0 ) for all 0 < s < r. The inequality above is now automatically satisfied, since F can be regarded as a subsheaf of E0 via ϕ. (iii) This follows directly from (i), (ii) and Definition 2.6. The last statement follows from (iii) and the fact that any solution with generically injective ϕ-component is irreducible. $ % Corollary 2.8 shows that in the abelian case the moduli space M∗t (E, E0 , A0 ) is either empty or can be identified with a quot space. In the non-abelian case, the space of parameters (t, g, A0 ) has a chamber structure which can be very complicated. The wall in this parameter space consists of those points (t, g, A0 ) for which reducible solutions (A, ϕ) appear in the moduli space Mt (E, E0 , A0 ). Note that a solution (A, ϕ) is reducible if and only if either ϕ = 0, or A is reducible and ϕ vanishes on an A-parallel summand of E. When the parameter (t, g, A0 ) crosses the wall, the corresponding moduli space changes by a “generalized flip” [Th1, Th2, OST]. Let E := A∗ ×G E be the universal complex bundle over B ∗ × associated with E. This bundle is the dual of the vector bundle P ×U (r) Cr , where P is the universal K-bundle introduced in Sect. 1.1. In order to compute the gauge theoretical Gromov– Witten invariants we will need an explicit description of the restriction of this bundle to M∗t (E, E0 , A0 ) × . The following proposition provides a complex geometric interpretation of this bundle via the isomorphism given by Corollary 2.8, Proposition 2.9. Proposition 2.10. Suppose that t is large enough so that the Kobayashi–Hitchin correspondence defines an isomorphism M∗t (E, E0 , A0 ) GQuot E E0 . Via this isomorphism the restriction of the universal bundle E to M∗t (E, E0 , A0 ) × can be identified with ∗ (E ) → Q over GQuot E × . the kernel of the universal quotient p 0 E0
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2.3. Transversality and compactness for moduli spaces of vortices. We first prove a simple regularity result for moduli spaces of vortices over curves. Proposition 2.11. Let X be a curve. (i) The moduli space M∗t (E, E0 , A0 ) is smooth of expected dimension in every point [A, ϕ] with ϕ generically surjective. (ii) There is a dense second category set C ⊂ A(E0 ) such that, for every A0 ∈ C and every t ∈ R, the open part Mt (E, E0 , A0 )inj ⊂ M∗t (E, E0 , A0 ), consisting of classes of pairs with generically injective ϕ-component, is smooth of expected dimension. Proof. (i) Since the Kobayashi–Hitchin correspondence is an isomorphism of real analytic spaces, it suffices to study the regularity of the moduli space Mst τ (E, E0 ) in (E, E ) which corresponds to [A, ϕ]. The first differential the point [∂¯A , ϕ] ∈ Mst 0 τ D∂1¯
A ,ϕ
: A1 End(E) × A0 Hom(E, E0 ) → A0,1 Hom(E, E0 )
in the elliptic complex associated with the τ -stable pair (∂¯A , ϕ) is given by D∂1¯
A ,ϕ
(α, φ) = ∂¯A,A0 φ − ϕ ◦ α.
It suffices to see that, after suitable Sobolev completions, the first order differential operator D∂1¯ ,ϕ is surjective. Let β ∈ A0,1 Hom(E, E0 ) be L2 -orthogonal to A
im(D∂1¯ ,ϕ ). Note that the linear map End(Cr ) → Hom(Cr , Cr0 ) given by A → A ( ◦ A is surjective when ( is surjective. Therefore, as in the proof of Proposition 2.4, we find that β vanishes as distribution, hence as a Sobolev section as well, on the open set where ϕ is surjective. But since β solves an elliptic second order system with scalar symbol, it follows that β = 0. (ii) Note that Mt (E, E0 , A0 )inj can be identified via the Kobayashi–Hitchin correpondence with an open subspace of GQuot E E0 . Therefore the statement follows from Proposition 2.4. $ % Theorem 2.12. Let (X, g) be a compact Kähler manifold of dimension n, E and E0 Hermitian bundles on X of ranks r and r0 respectively. Suppose that either n = 1 or r = 1. Then the moduli spaces Mt (E, E0 , A0 ) are compact for every t ∈ R and for every integrable Hermitian connection A0 ∈ A(E0 ). In particular, the moduli space GQuotE E0 is compact if X is a curve or rk(E) = 1. Proof. The Hermite–Einstein type equation 1 i#FA − ϕ ∗ ◦ ϕ = −tidE 2 implies µ(E) −
(n − 1)!V olg (X) (n − 1)! t. # ϕ #2 = − 2π 4πr
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The Weitzenböck formula for holomorphic sections in the holomorphic Hermitian bundle E ∨ ⊗ E0 with Chern connection B := A∨ ⊗ A0 yields ¯ i#∂∂(ϕ, ϕ) = (i#FB (ϕ), ϕ) − |∂B ϕ|2 ≤ ((i#FA0 ) ◦ ϕ − ϕ ◦ (i#FA ), ϕ) = (i#FA0 (ϕ), ϕ) − (i#FA , ϕ ∗ ◦ ϕ) 1 = (i#FA0 (ϕ), ϕ) + t|ϕ|2 − |ϕ ∗ ◦ ϕ|2 . 2 Notice that |ϕ ∗ ◦ ϕ|2 ≥ 1r |ϕ|4 . Let x0 be a point where the supremum of the function 0 |ϕ|2 is attained, and let λA M be the supremum of the highest eigenvalues of the Hermitian bundle endomorphism i#FA0 . By the maximum principle we get 1 2 2 0 ¯ 0 ≤ [i#∂∂|ϕ| ]x0 ≤ (λA |ϕ(x0 )|4 . M + t)|ϕ(x0 )| − 2r Therefore we have the following a priori C 0 -bound for the second component of a solution of (VtA0 ): 0 sup |ϕ|2 ≤ max(0, 2r(λA M + t)).
X
Now, if r = 1, one can bring A in Coulomb gauge with respect to a fixed connection A0 in E by a gauge transformation gA . Moreover, one can choose gA so that the projection of gA (A) − A0 on the kernel of the operator d + + d ∗ : iA1 (X) −→ i[(A0,2 (X) + A2,0 (X) + A0,0 (X)ωg ) ∩ A2 (X)] (which coincides with the harmonic space iH1 (X) in the Kählerian case) belongs to a fixed fundamental domain D of the lattice iH 1 (X, Z). Now standard bootstrapping arguments apply as in the case of the abelian monopole equations [KM]. If X is a curve, then the contraction operator # is an isomorphism, so one gets an a priori L∞ -bound for the curvature of the connection component. The result follows now from Uhlenbeck’s compactness theorems for connections with Lp -bound on the curvature [U]. $ % Corollary 2.13. Let X be a projective manifold endowed with an ample line bundle H , and let PL be the Hilbert polynomial of a locally free sheaf L of rank 1 with respect to PE −PL H . Then the analytic quot space Quot E0 0 is compact. Proof. Indeed, by Remark 2.3, the gauge theoretical quot space GQuot L E0 is an open PE0 −PL . But any torsion free sheaf on subspace of the underlying analytic space of Quot E0 ∞ X with Hilbert polynomial PL is a line bundle of C -type L, so that the open embedding PE −PL
0 GQuotL E0 <→ Quot E0
is surjective.
% $
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3. The Definition of the Invariants and an Explicit Formula in the Abelian Case 3.1. Virtual fundamental classes for Fredholm sections and the definition of the invariants. We explain briefly – following [Br] – the definition and the basic properties of virtual fundamental classes of vanishing loci of Fredholm sections. For simplicity we discuss only the compact case. Let E be a Banach bundle over the Banach manifold B, and let σ be a Fredholm section of index d in E with compact vanishing locus Z(σ ). Fix a trivialization θ of the real line bundle det(Index(Dσ )) in a neighbourhood of Z(σ ). One can associate with ˇ ˇ these data a Cech homology class [Z(σ )]vir θ ∈ Hd (Z(σ ), Z) in the following way: Notice first that one can choose a finite rank subbundle F ⊂ E|U of the restriction of E to a sufficiently small neighbourhood U of Z(σ ) in B such that Dx σ + Fx = Ex for every x ∈ Z(σ ). This shows that the induced section σ¯ in the quotient bundle E|U /F is regular in the points of Z(σ ), hence it is also regular on a neighbourhood V ⊂ U of Z(σ ) in B. Put M := Z(σ¯ |V ). Then M is a smooth closed submanifold of V of dimension m := d + rkF . Denote by or(M) the orientation sheaf of M, and let [M] ∈ Hˇ mcl (M, or(M)) be the ˇ fundamental class of M in Cech homology with closed supports. Notice that the restriction σ |M takes values in the subbundle F |M of E|M , and that the real line bundle #max (TM )∨ ⊗#max (F |M ) can be identified with det(Index(Dσ ))|M ; therefore it comes with a natural trivialization induced by θ|M . Let e(F |M , σ |M ) ∈ Hˇ rkF (M, M \ Z(σ |M ), or (F |M )) be the localized Euler class of (F |M , σ |M ). Using the trivialization of #max (TM )∨ ⊗ #max (F |M ), the virtual fundamental class is defined as ˇ [Z(σ )]vir θ := e(F |M , σ |M ) ∩ [M] ∈ Hd (Z(σ ), Z). ˇ ˇ In this definition the cap product pairs Cech cohomology and Cech homology with closed supports ([Br], Lemma 13). The homology class obtained in this way is well-defined, i.e. it does not depend on the choice of the subbundle F and the submanifold M used in the definition. Note also, that when Z(σ ) is locally contractible, e.g. when Z(σ ) is locally homeomorphic to a ˇ real analytic set, then its Cech homology can be identified with its singular homology. In this case one gets a well defined virtual fundamental class [Z(σ )]vir θ ∈ Hd (Z(σ ), Z).
Remark 3.1. When the section σ is regular in every point of its vanishing locus, then Z(σ ) is either empty or a smooth manifold of dimension d which comes with a natural orientation induced by θ . In this case, [Z(σ )]vir θ coincides with the usual fundamental class [Z(σ )]θ of this oriented manifold. We will omit the index θ when there is a natural choice of a trivialization, e.g. when B is a complex Banach manifold, E is a holomorphic Banach bundle, and σ is holomorphic. The virtual fundamental class has the following two fundamental properties which will play an important role in this section ([Br], Proposition 14):
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Associativity Property. Let E be a Banach vector bundle on a Banach manifold B, and let σ be a Fredholm section in E with compact vanishing locus Z(σ ). Let 0 → E → E → E → 0
(e)
be an exact sequence of bundles and suppose that the section σ ∈ (B, E ) induced by σ is regular2 in the points of its vanishing locus B := Z(σ ). Let σ ∈ (B , E |B ) be the section in E |B defined by σ . The inclusion i : B ⊂ B induces: 1. An homeomorphism Z(σ ) Z(σ ) which is an isomorphism of real (complex) analytic spaces if B, E, E , E , σ and the morphisms in the exact sequence (e) are real (complex) analytic. 2. An identification of virtual fundamental classes [Z(σ )]vir [Z(σ )]vir . Note that one has a well defined map Hˇ ∗ (Z(σ )) → H∗ (B) induced by the composition Z(σ ) <→ M → B and the identification Hˇ ∗ (M) = H∗ (M). With this remark, we can state Homotopy Invariance. Let (σt )t∈[0,1] be a smooth 1-parameter family of sections in E such that the vanishing locus of the induced section in the bundle pr ∗B (E) over B × [0, 1] is compact. Then the images of [Z(σ0 )]vir and [Z(σ1 )]vir in Hd (B) coincide. Now we can introduce our gauge theoretical Gromov–Witten invariants for the triple (Hom(Cr , Cr0 ), αcan , U (r)). Let be a compact oriented 2-manifold, and let E, E0 be Hermitian bundles on of ranks r, r0 and degrees d, d0 respectively. Choose a continuous parameter p = (t, g, A0 ) as in Sect. 2. The moduli space M∗t (E, E0 , A0 ) can be regarded as the vanishing locus of a Fredholm section vtA0 in the vector bundle A∗ ×G [A0,1 Hom(E, E0 ) ⊕ A0 Herm(E)] over B ∗ . The section vtA0 is defined by the G-equivariant map 1 (A, ϕ) → (∂¯A,A0 ϕ, i#FA − ϕ ∗ ◦ ϕ + tidE ). 2 Moreover, this moduli space is compact for good parameters (t, g, A0 ) by Theorem 2.12. We trivialize the determinant line bundle det(IndexDvtA0 ) in the following way: The kernel (cokernel) of the intrinsic derivative of DvtA0 in a solution [A, ϕ] can be identified with the harmonic space H1 (H2 ) of the elliptic deformation complex associated with this solution. But the Kobayashi–Hitchin correspondence identifies these harmonic spaces with the corresponding harmonic spaces of the elliptic complex associated with the simple holomorphic pair (∂¯A , ϕ). We orient the kernel and the cokernel of the intrinsic derivative using the complex orientations of the latter harmonic spaces. Following the general formalism described in Sect. 1.1, we put (E0 ,cd )
GGWp
(Hom(Cr , Cr0 ), αcan , U (r))(a) := δ(a), [M∗t (E, E0 , A0 )]vir
2 A section in a Banach bundle with fibre # is regular in a point x of its vanishing locus if the associated #-valued map with respect to a trivialization around x is a submersion in x. In the non-Fredholm case, this condition is stronger than the surjectivity of the intrinsic derivative, but it is equivalent to this condition if the base manifold is a Hilbert manifold.
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for any good continuous parameter p = (t, g, A0 ) and any element r
a ∈ A(F, α, K, c) = Z[u1 , . . . , ur , v2 , . . . , vr ] ⊗ #∗ H1 ( , Z)i . i=1
We have seen that, on curves, the moduli space Msimple (E, E0 ) of simple pairs of type (E, E0 ) can be regarded as the vanishing locus of a Fredholm section v¯ E0 of complex index χ (Hom(E, E0 )) − χ (End(E)) in the Banach bundle A¯ simple ×G C A0,1 Hom(E, E0 ) over the non-Hausdorff Banach manifold B¯ simple . However, since Msimple (E, E0 ) is in general non-Hausdorff, one cannot endow it with a virtual fundamental class. On the other hand, by Theorem 2.7, the open subspace Mstable (E, E0 ) of τ -stable pairs of τ type (E, E0 ) is always Hausdorff, and it is also compact if the corresponding parameter p = ( Vol2π τ, g, A0 ) is good. The following proposition shows that for good parameters g ( )
p, Mstable (E, E0 ) can be endowed with a virtual fundamental class, and that the isomorτ phism given by Theorem 2.7 maps [M∗t (E, E0 , A0 )]vir onto [Mstable (E, E0 )]vir . Thereτ fore one can use the complex geometric virtual fundamental classes [Mstable (E, E0 )]vir τ to compute the gauge theoretical Gromov–Witten invariants. This proposition is a particular case of a more general principle which states that the Kobayashi–Hitchin type correspondence associated to a complex geometric moduli problem of “Fredholm type” respects virtual fundamental classes. The proof below can be adapted to the general case. Theorem 3.2. Let p = (t, g, A0 ) be a good parameter, let E0 be the holomorphic bundle Volg ( ) defined by ∂¯A0 in E0 , and put τ := 2π t. Then (i) Mstable (E, E0 ) is compact and has a Hausdorff neighbourhood in B¯ simple ; it comes τ with a virtual fundamental class induced by the restriction of v¯ E0 to such a neighbourhood. (ii) The isomorphism given by the Kobayashi–Hitchin correspondence maps the virtual vir fundamental class [M∗t (E, E0 , A0 )]vir onto [GQuot E E0 ] . Proof. We apply the Associativity Property of the virtual fundamental classes to the following exact sequence of Banach bundles over B ∗ : 0 → A∗ ×G A0,1 Hom(E, E0 ) → A∗ ×G [A0,1 Hom(E, E0 ) ⊕ A0 Herm(E)] → → A∗ ×G A0 Herm(E) → 0. The section (vtA0 ) in A∗ ×G A0 Herm(E) induced by vtA0 is given by the G-equivariant map 1 (A, ϕ) → i#FA − ϕ ∗ ◦ ϕ + tidE . 2 Using the fact that this map comes from a formal moment map, one can prove ([LT] ch 4, [OT1]) that: 1. (vtA0 ) is regular around Z(vtA0 ),
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2. The natural map ρ : Z((vtA0 ) ) → B¯ simple given by [A, ϕ] → [∂¯A , ϕ] induces a bijection Z(vtA0 ) = M∗t (E, E0 , A0 ) → Mstable (E, E0 ), τ and is étale around Z(vtA0 ). Since, by Theorem 2.12, M∗t (E, E0 , A0 ) is compact for a good parameter p = (t, g, A0 ), it follows that Mstable (E, E0 ) is compact, and that ρ maps a sufficiently small neighτ ∗ bourhood V of Mt (E, E0 , A0 ) in Z((vtA0 ) ) isomorphically onto a neighborhood U of Mstable (E, E0 ) in B¯ simple . This neighbourhood must be Hausdorff, because B ∗ is τ Hausdorff. Via the natural identification ρ ∗ (A¯ simple ×G C A0,1 Hom(E, E0 )) = [A∗ ×G A0,1 Hom(E, E0 )] Z((v A0 ) ) , the section
(vtA0 )
to the section
induced by
v¯ E0 . This
vtA0
in
[A∗
×G
A0,1 Hom(E, E
0 )]
t
A
Z((vt 0 ) )
corresponds
shows that:
1. The vanishing locus of v¯ E0 |U is exactly Mstable (E, E0 ), so we can define τ vir as the virtual fundamental class defined by v¯ E0 | . (E, E )] [Mstable 0 U τ 2. The map ρ maps [Z((vtA0 ) |V )]vir onto [Mstable (E, E0 )]vir . τ The statement follows now directly from the Associativity Property.
% $
In the non-abelian case one has a very complicated chamber structure and the invariants jump when the continuous parameter p crosses the wall. Proving a wall crossing formula for these jumps is an important but very difficult problem. In the abelian case the situation is much simpler: Recall that in this case the invariants are determined by an inhomogenous form (E0 ,cd )
GGWp
(Hom(C, Cr0 ), αcan , S 1 ) ∈ #∗ (H 1 ( , Z)).
Corollary 2.8 yields the following: Proposition 3.3. For any fixed topological data (E0 , d), there are exactly two chambers in the space of parameters. The “interesting chamber” C + – in which the moduli space can be non-empty – is defined by the inequality t >−
2π deg(E). Volg (X)
For any parameter p = (t, g, A0 ) in this chamber, the corresponding gauge theoretical Gromov–Witten moduli space coincides with the gauge theoretical quot space GQuot L E0 , where L is a line bundle of degree d on and E0 is the holomorphic structure in E0 associated with the connection A0 . The wall is defined by the equation t = − Vol2π deg(E), which does not involve the g (X) third parameter A0 , hence one cannot cross the wall by varying only A0 . In order to compute the invariants in the “interesting chamber” C + , we need an explicit description of the abelian quot spaces. We will see that these quot spaces can be described as subspaces of a projective bundle over a component of Pic( ), defined as the intersection of finitely many divisors representing the Chern class of the relative hyperplane line bundle.
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3.2. Quot spaces in the abelian case. We begin with the following known results [Gh]: Let F0 be a vector bundle on a curve , and let m be a sufficiently negative integer such that H 1 (M∨ ⊗ F0 ) = 0 for all M ∈ Picm ( ). line bundle on Picm ( ) × , by V the locally free sheaf
Denote by P a Poincaré ∨ ∗ [pr Picm ( ) ]∗ Hom(P, pr (F0 )) on Picm ( ), and by P(V) its projectivization in Grothendieck’s sense. Applying the projection formula to the projective morphism p : P(V) × → Picm ( ) × , we get p∗ (Hom(p ∗ (P)(−1), pr ∗ (F0 )))
∨
= Hom(P, pr ∗ (F0 )) ⊗ [pr Picm ( ) ]∗ [pr Picm ( ) ]∗ [Hom(P, pr ∗ (F0 )] ,
hence on P(V) × there is a canonical monomorphism ν pr ∗Picm ( )× (P)(−1) − → pr ∗ (F0 ).
Let M be a differentiable line bundle of degree m. Proposition 3.4. Choose m sufficiently negative such that H 1 (M∨ ⊗ F0 ) = 0 for all ∗ M ∈ Picm ( ). Then the quotient pr (F0 ) im(ν) is flat over P(V), and the associated morphism P(V) → Quot M F0 is an isomorphism. α → Ox , where x ∈ is a simple point, induces an epimorAn epimorphism F0 − α˜ ∗ → OP(V )×{x} on P(V) × . The composition α˜ ◦ ν can be regarded, phism pr (F0 ) − by adjunction, as a morphism
pr ∗Picm ( )×{x} (P|Picm ( )×{x} )(−1) → OP(V )×{x} , hence as a section σα in the line bundle pr ∗Picm ( ) (Px )∨ (1) over the projective bundle P(V) P(V) × {x}. Here Px is the line bundle on Picm ( ) corresponding to P|Picm ( )×{x} . Proposition 3.5. Let Z0 be a finite set of simple points in , and consider for each x ∈ Z0 an epimorphism αx : F0 → Ox . Put α = αx : F0 → Ox , Z := Z(σαx ), x∈Z0
x∈Z0
x∈Z0
and let p be the projection Z × → . Then , for all m sufficiently negative, the ∗ quotient p (ker α) im(ν|Z× ) is flat over Z, and the induced morphism Z → Quot M ker α is an isomorphism. Let now L be a differentiable line bundle of degree d and E0 a holomorphic bundle of rank r0 and degree d0 on . Let H be an ample line bundle on and n ∈ N sufficiently large such that E0∨ ⊗ H ⊗n is globally generated. Then the cokernel of a generic morphism ⊕r0 O → E0∨ ⊗ H ⊗n
has the form
k i=1
Oxi with k := −d0 + nr0 deg(H ) distinct simple points xi ∈ .
Dualizing, one gets an exact sequence ρ
⊕r0 0 → E0 ⊗ H ⊗−n → O − →
k i=1
Oxi → 0.
(∗)
Gromov–Witten Invariants and Seiberg–Witten Invariants of Ruled Surfaces
573
⊕r0 The i th component ρi : O → Oxi of ρ is defined by a non-trivial linear form i r 0 ρ : C → C. Note also that one has a natural isomorphism ⊗−n
L⊗H Quot L E0 Quot E ⊗H ⊗−n 0
which identifies the corresponding virtual fundamental classes. Thus we can replace L by L := L ⊗ H ⊗−n and E0 by E0 := E0 ⊗ H ⊗−n . The exact sequence (∗) shows now that, at least as a set, QuotL can be identified E 0
with the subspace of QuotL ⊕r0 consisting of quotients O
⊕r0
⊕r0 0 → L − → O → O ϕ
ϕ(L ) → 0
⊕r0 ⊕r0 with ρ i (ϕ(xi )) = 0. By Proposition 3.5 applied to F0 = O of the free sheaf O and M = L , this identification is also an isomorphism of complex spaces, and we have
Corollary 3.6. Let P be a Poincaré line bundle over Picd ( )× , with d := deg(L ) = d − ndeg(H ). If n is sufficiently large, then QuotL can be identified with the analytic E0 subspace Z of the projective bundle P := P([(pr Picd ( ) )∗ (P∨ )⊕r0 ]∨ )
over Picd ( ) which is cut out by the sections σρi ∈ (P , pr ∗ d (Pxi )∨ (1)). The Pic ( ) kernel of the universal quotient over Z × is the restriction of the line bundle pr ∗
Picd ( )×
(P)(−1) to Z × .
L Consider the embedding j : Quot L E0 <→ P given by the identification Quot E0 = and Corollary 3.5. Denote by π the projection of the projective bundle P onto QuotL E 0
its basis Picd ( ), and let ι : Picd ( ) → Pic−d ( ) be the natural identification given by M → M−1 .
Lemma 3.7. Via the isomorphism M∗t (E, E0 , A0 ) Quot L E0 defined by the Kobayashi– Hitchin correspondence one has c δ(u)|M∗t (E,E0 ,A0 ) = j ∗ [c1 (π ∗ (Px∨ )(1))], δ 1 = j ∗ [(ι ◦ π )∗ (β)]. β M∗ (E,E ,A ) t
0
0
In the second formula we used the natural identification
H1 ( , Z) = H 1 (Pic−d ( ), Z). Proof. By Proposition 2.12 and Corollary 3.6 we find that the universal bundle E over M∗t (E, E0 , A0 ) × QuotL E0 × is given by E j ∗ [pr ∗
Picd ( )×
(P)(−1)].
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Therefore, the restriction to M∗t (E, E0 , A0 ) × of the line bundle associated with the universal U (1)-bundle P (see Sect. 1.1) is j ∗ [pr ∗ d (P∨ )(1)]. The formulae given Pic ( )× in Sect. 1.1 yield (P∨ )(1))/[x] = j ∗ [c1 (π ∗ (Px∨ )(1))], δ(u) = j ∗ c1 (pr ∗ d Pic ( )× c1 (P∨ )(1))/β = j ∗ c1 (pr ∗ d (P∨ ))/β = j ∗ c1 (pr ∗ d δ Pic ( )× Pic ( )× β = j ∗ (π ∗ (c1 (P∨ )/β)). To get the second equality we used the fact that c1 (OP (1)) has type (2, 0) with respect to the Künneth decomposition of H ∗ (P × , Z). Note now that the line bundle P1 := (ι × id )∗ (P∨ ) is a Poincaré line bundle on Pic−d ( ) × . We get c δ 1 = j ∗ ((ι ◦ π )∗ (c1 (P1 )/β)). β But the assignment β → c1 (P1 )/β gives the standard identification H1 ( , Z) H 1 (Pic−d ( ), Z). $ % 3.3. The explicit formula. We can now prove the following Theorem 3.8. Put g := g( ), v = v(r0 , 1, d, d0 ) := χ (Hom(L, E0 ))−(1−g). Let lO1 be the generator of #2g (H 1 ( , Z)) defined by the complex orientation O1 of H 1 ( , R). The Gromov–Witten invariant (E0 ,cd )
GGWp
(Hom(C, Cr0 ), αcan , S 1 ) ∈ #∗ (H 1 ( , Z))
is given by the formula (E ,c ) GGWp 0 d (Hom(C, Cr0 ), αcan , S 1 )(l)
g
=
i≥max(0,g−v)
(r0 )i ∧ l, lO1 i!
for any p in the interesting chamber C + . Proof. In the following computation we make the identifications
L L GQuot L E0 Quot E0 Quot E
0
using the notations from above. By the homotopy invariance of the virtual class, we can use a general holomorphic structure E0 . For such a structure E0 , the quot space Quot L E0 is smooth and has the expected dimension v by Proposition 2.4. L Since the codimension of QuotL E0 = Quot E in P is k and this subspace is smooth, 0
it follows from Corollary 3.6 that the section σ :=
k
i=1
σρi is regular along its vanishing
locus. We can normalize the Poincaré line bundle P such that the line bundles Px , x ∈ are topologically trivial. Then Corollary 3.6 shows that the fundamental class [QuotL ] ∈ Hv (P , Z) is Poincaré dual to c1 (OP (1))k . E 0
Gromov–Witten Invariants and Seiberg–Witten Invariants of Ruled Surfaces
575
By Lemma 3.7 we see that our problem reduces to the computation of the direct image of the homology classes P D(c1 (OP (1))i |QuotL ) via the push-forward morphism E0
(ι◦π◦j )
∗ −d H∗ (Quot L ( ), Z). E0 , Z) −−−−−→ H∗ (Pic
Using the same arguments as in [OT2], the direct image of P D(c1 (OP (1))i |QuotL ) E0
in H∗ (Pic−d ( ), Z) can be identified with the Segre class sk+i of the vector bundle pr ∗ −d (P1 )⊕r0 over Pic−d ( ). The Chern classes of this bundle can be determined Pic ( ) by applying the Grothendieck–Riemann–Roch theorem to the map
Picd ( ) × → Picd ( ).
% $
3.4. Application: Counting quotients. We give a purely complex geometric application of our computation. Suppose that we have chosen the integers r, r0 , d, d0 such that the expected dimension v(r0 , r, d, d0 ) = χ (Hom(E, E0 )) − χ (End(E)) of the corresponding quot spaces is 0. Suppose also that for a particular bundle E0 of rank r0 and degree d0 the quot space QuotE E0 has dimension 0. We do not require that it is smooth. The problem is to estimate the number of points of such a 0-dimensional quot space. Using the results above one can easily prove the following result. Proposition 3.9. Suppose v(r0 , r, d, d0 ) = dim(QuotE E0 ) = 0. Then (i) The length of the 0-dimensional complex space QuotE E0 is an invariant which does not depend on E0 , but only on the integers r, r0 , d, d0 . g g (ii) When r = 1, this invariant is r0 , and the set Quot E E0 has at most r0 elements. Proof. The virtual fundamental class of a complex space Z which is cut out by a holomorphic section of index 0 with finite vanishing locus is just dim(OZ,z )[z] ∈ H0 (Z, Z). z∈Z
This follows easily from the definition of the virtual fundamental class. It suffices now to use the Homotopy Invariance of virtual fundamental classes. The second statement follows directly from Theorem 3.7. $ % We close this subsection with the following remarks: 1. The quot spaces QuotE E0 cannot be regarded as fibres of a flat family as the holomorphic structure E0 in E0 varies; the first statement is therefore not a consequence of the invariance of the length of zero dimensional spaces under deformations. 2. The main ingredient used in our proof is the fact that the quot spaces can be defined as vanishing loci of Fredholm sections and that the virtual fundamental class of the vanishing locus of such a section is invariant under continuous deformations. 3. It is more difficult to get the result above with purely complex geometric methods. In the particular case r0 = 2 the inequality in (ii) was obtained with such methods by Lange [L]. In the smooth case, the equality of (ii) was proven by Oxbury [Oxb].
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4. It is an interesting problem to compute the invariant introduced in (i) also in the non-abelian case, i.e. for r > 1. This reduces to the computation of the gauge theoretical Gromov–Witten invariant for (Hom(Cr , Cr0 ), αcan , U (r)) in the chamber which corresponds to t + 0. The main difficulty is that, in the non-abelian case, there are many chambers, not just two. 5. More generally, consider spaces of holomorphic sections in a Grassmann bundle Gr (E0 ) over . The quot spaces GQuot E E0 are natural compactifications of these spaces to which the tautological cohomology classes extend naturally. These quot spaces map surjectively onto the Uhlenbeck compactifications of the spaces of sections in Gr (E0 ). It should therefore be possible to compare our non-abelian invariants with the twisted Gromov– Witten invariants (Sect. 1.2) associated with sections in Grassmann bundles, and this should lead to an interesting generalization of the Vafa-Intriligator formula [BDW, W1]. 4. Gauge Theoretical Gromov–Witten Invariants and the Full Seiberg–Witten Invariants of Ruled Surfaces 4.1. Douady spaces of ruled surfaces and quot spaces on curves. Let V0 be a holomorphic bundle of rank 2 on a curve and let X = P(V0 )3 be the corresponding ruled surface; we denote by π : X → the projection map. Let M a line bundle of Chern class m := df + ns on X, where f is the Poincaré dual of a fibre and s = c1 (OP(V0 ) (1)). An elementary computation shows that for every holomorphic structure M in M one has π∗ (M) S n (V0 ) ⊗ L, where L is a holomorphic line bundle of degree d on the base curve . Moreover, the assignment L → M := π ∗ (L) ⊗ OP(V0 ) (n) defines an isomorphism Picd ( ) → Picdf +ns (X). Let Hilb(m) stand for the Hilbert scheme of effective divisors on X representing the homology class P D(m) Poincaré dual to m. The family of identifications H 0 (X, M) = H 0 ( , π∗ (M)) for M ∈ Picm (X) gives rise to an isomorphism of schemes over C Hilb(m) QuotPSn (V0 ) ,
(I)
where P is the Hilbert polynomial P = PS n (V0 ) − PL∨ [Ha]. Notice that both moduli spaces in (I) have complex analytic as well as gauge theoretical versions. In complex analytic geometry one defines Douady spaces Dou(m) of effective divisors representing P D(m) and, more generally, complex analytic quot spaces. These analytic objects are isomorphic to the underlying complex spaces of the corresponding algebraic geometric objects, as explained in Sect. 2.1. The gauge theoretical quot spaces have been introduced in Sect. 2. The gauge theoretical Douady space is defined as follows: Let M be a differentiable line bundle on X. The gauge theoretical Douady space GDou(M) is the space of equivalence classes of simple pairs (d, f), consisting of a holomorphic structure d in M and a non-trivial d-holomorphic section in M. By the results of [LL] one has a natural identification of complex spaces GDou(M) = Dou(c1 (M)). 3 We use here the Grothendieck convention for the projectivization of a bundle.
Gromov–Witten Invariants and Seiberg–Witten Invariants of Ruled Surfaces
577
As explained in Sect. 2, the complex analytic quot space Quot PSn (V0 ) can be identified ∨
with the gauge theoretical quot space GQuot L S n (V0 ) , where L is a fixed smooth bundle of degree d on . We will need a gauge theoretical version of the isomorphism (I) which allows us to compare the virtual fundamental classes of the corresponding gauge theoretical complex spaces. We begin by defining the virtual fundamental class of GDou(M): inj ¯ Let A¯ X := A(M) × A0 (M), let A¯ X be the open subset of pairs whose section component does not vanish identically, and put ¯ inj inj B¯X = AX C . GX inj Over the Hausdorff Banach manifold B¯X consider the bundles inj 0,i−1 E i := A¯ X ×G C [A0,i (M)], X ⊕A X
and the bundle morphisms D i : E i → E i+1 given by i ¯ −uf − dv). D(d,f) (u, v) = (∂u,
Let E be the bundle
E := ker(D 2 ) ⊂ E 2 ,
whose fibre in a point [d, f] ∈ B¯X is inj
0,1 E(d,f) = {(u, v) ∈ A0,2 X × A (M)| dv + uf = 0}.
The fact that ker(D 2 ) is a subbundle of E 2 is crucial for our construction. It follows from the following Remark 4.1. After suitable Sobolev completions, the morphism D 2 : E 2 → E 3 is a bundle epimorphism on B¯ inj . The proof uses a similar argument as the proof of Proposition 2.4. The moduli space GDou(M) is the vanishing locus of the Fredholm section s in E, C -equivariant map given by the GX (d, f) → (−Fd02 , d(f)). The index of this section is w(m) = χ (M) − χ (OX ). Note that the section in E 2 defined by the same formula as s is not Fredholm. Definition 4.2. The virtual fundmental class of GDou(M) is defined as the virtual fundamental class [GDou(M)]vir ∈ Hw(m) (GDou(M), Z) associated with the pair (E, s). Now let V0 be the underlying differentiable bundle of V0 , H the underlying differentiable bundle of OP(V0 ) (1), and let h be the corresponding semiconnection in H . Fix a smooth line bundle L of degree d on .
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Ch. Okonek, A. Teleman
inj ¯ Similarly as above let A¯ := A(L) × A0 (L ⊗ S n (V0 )), let A¯ be the open subset of pairs whose section component is nondegenerate on a non-empty open set, and put inj
¯ inj B¯ = A
C G
as in Sect. 2. inj Over the Hausdorff Banach manifold B¯ consider the bundles F i defined by inj 0,i−1 F i := A¯ ×G C [A0,i (L ⊗ S n (V0 ))]. ⊕A
∨
Put F := F 2 . With this notation, the moduli space GQuot L S n (V0 ) , which was introduced in Sect. 2, is the vanishing locus of the Fredholm section s in F given by the equivariant map (δ, ϕ) → ∂¯δ,δ0 ϕ. Here ∂¯δ,δ0 stands for the semiconnection in L ⊗ S n (V0 ) associated with δ and the semiconnection δ0 in V0 corresponding to the holomorphic structure V0 . The virtual funda∨ vir was defined as the virtual fundamental class associated mental class [GQuotL S n (V0 ) ] with (F, s). We can now state the main result of this section: Theorem 4.3. Let L be a differentiable line bundle on , and put M := π ∗ (L) ⊗ H ⊗n . There is a canonical isomorphism of complex spaces ∨
GDou(M) GQuotL S n (V0 ) which maps the virtual fundamental class [GDou(M)]vir onto the virtual fundamental ∨ vir class [GQuotL S n (V0 ) ] . Remark 4.4. One can prove a similar identification of moduli spaces for much more general fibrations; in general, however, this identification will not respect the virtual fundamental classes. Proof of Theorem 4.3. To every pair (δ, ϕ) consisting of a semiconnection in L and a section ϕ ∈ A0 (L ⊗ S n (V0 )) = A0 Hom(L∨ , S n (V0 )) we associate the pair ¯ ˜ ϕ) (δ, ˜ ∈ A(M) × A0 (M) defined by δ˜ := π ∗ (δ) ⊗ hn , ϕ([e]) ˜ := ϕ(e) for e ∈ V0∨ \ {0 − section}. We will prove – using the Associativity Property of virtual fundamental classes inj inj ˜ ϕ) – that the assignment (δ, ϕ) → (δ, ˜ induces an embedding B¯ <→ B¯X which vir maps the virtual fundamental class [GDou(M)] onto the virtual fundamental class ∨ vir [GQuotL S n (V0 ) ] :
Gromov–Witten Invariants and Seiberg–Witten Invariants of Ruled Surfaces
579
Consider the vertical subbundle TX/ of TX , and define E to be the linear subspace of E whose fibre in a point [d, f] is E(d,f) := {(u, v) ∈ E(d,f) | u = 0, v|TX/ = 0} 0,1 = {(u, v) ∈ A0,2 X × A (M)| u = 0, v|TX/ = 0, dv = 0}.
The last two conditions mean that v defines a section in the line bundle M ⊗ π ∗ (#0,1 ) which is d-holomorphic on the fibres. Claim. After suitable Sobolev completions, E is a subbundle of E. Proof. We will omit Sobolev indices to save on notations. The proof uses the fact that the fibres of π are projective lines in an essential way. Let d be an arbitrary semiconnection in M. Since in the line bundle H ⊗n over P1 all semiconnections are gauge equivalent, C , unique modulo G C , such that g · d and there exists a gauge transformation gd ∈ GX d ⊗n h coincide on the fibres. Moreover, one can choose gd to depend smoothly on d. The with the fixed space gauge transformation gd identifies the space E(d,f) n E0 = {α ∈ A0 (M ⊗ π ∗ (#0,1 ))| α|fibres is h⊗n − holomorphic} A0,1 ( , L ⊗ S (V0 )).
The union
E(d,f)
(d,f)∈A¯ X
inj
inj 0,1 becomes therefore a trivial subbundle of A¯ X × [A0,2 X × A (M)]. Since it is gauge invariant, this subbundle descends to a subbundle E of E, as required. $ %
Now consider the exact sequence 0 → E → E → E → 0 and let s be the section in E induced by s. Claim. The section s is regular in every point of its vanishing locus. Proof. We have to show that 1 E(d,f) = im(D(d,f) ) + E(d,f)
˜ ϕ). for every pair (d, f) of the form (d, f) = (δ, ˜ In other words, we must prove that, for such pairs (d, f), the natural map 2 E(d,f) −→ E(d,f) = H(d,f) 1 im(D(d,f) ) 2 is surjective. Here H(d,f) denotes the second cohomology group associated with the elliptic deformation complex of the holomorphic pair (d, f). But it is easy to see that the 2 2 . In image of this map coincides with the image of the pull-back map H(δ,ϕ) → H(d,f)
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order to check the surjectivity of this map, put M := (M, d), L := (L, δ) and consider the following morphism of long exact cohomology sequences: 2 → H(d,f) → H 2 (X, OX ) → H 2 (X, M) H 1 (X, M) ↑ ↑ ↑ ↑ 2 H 1 ( , L ⊗ S n (V0 )) → H(δ,ϕ) → 0 → 0
One has H 2 (X, OX ) = 0, and the first vertical map is an epimorphism for all n ≥ 0, since H 0 ( , R 1 π∗ (M)) = H 0 ( , L ⊗ R 1 π∗ (OP(V0 ) (n))) = 0. The case n < 0 is not inj inj interesting, since in this case B¯X = B¯ = ∅. This shows that the natural morphism of elliptic complexes F ∗ → E ∗ induces an 2 2 epimorphism H(δ,ϕ) → H(d,f) as desired. Claim. One has natural identifications inj Z(s ) = B¯ , F = E |Z(s ) , s = s .
Proof. Indeed, when (d, f) ∈ Z(s ), then d is integrable and df vanishes on the vertical tangent space. Applying the gauge transformation gd if necessary, we may assume that d coincides with h on the fibres. We fix a semiconnection δ 0 in L. Since the difference ¯ d − π ∗ (δ 0 ) ⊗ h⊗n ∈ A0,1 X vanishes on the vertical tangent space and is ∂-closed, it must ∗ be the pull-back of a (0, 1)-form α on . But this implies that d = π (δ 0 + α) ⊗ h⊗n = δ˜α , where δα := δ 0 + α. Similarly, the condition df|TX/ = 0 implies that f is h⊗n holomorphic on the fibres, hence it has the form ϕ, ˜ where ϕ is a section of L ⊗ S n (V0 ). Theorem 4.3 follows now directly from the Associativity Property of virtual fundamental classes. $ %
4.2. Comparison of virtual fundamental classes of Seiberg–Witten moduli spaces and Douady spaces. Let (X, g) be a Kähler surface and let KX be the differentiable line bundle underlying the canonical bundle of X . Every Hermitian line bundle M on X defines a Spinc -structure γM : #1X → RSU (#0 (M) ⊕ #0,2 (M), #0,1 (M)), obtained by tensoring the canonical Spinc -structure with M. The determinant bundle of this Spinc -structure is M ⊗2 ⊗ KX−1 . The assignment [M] → [γM ] induces a bijective correspondence between the group of isomorphism classes of Hermitian line bundles, which can be identified with H 2 (X, Z), and the set of equivalence classes of Spinc structures on X. Let β ∈ A1,1 R be a closed form. The Kobayashi–Hitchin correspondence for the γM Seiberg–Witten monopole equations [OT1,OT2] states that the moduli space WX,β of 0 0,2 solutions (A, A) ∈ A(L) × [A (M) ⊕ A (M)] of the twisted monopole equations γM A A D γ =0 (SWβ M ) ¯ 0, γM FA+ + 2π iβ + = 2(A A) can be identified with the gauge theoretical Douady space GDou(M) (respectively GDou(KX ⊗ M −1 )) when (2c1 (M) − c1 (KX ) − [β]) ∪ [ωg ], [X] < 0 (respectively > 0).
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The fact that this identification is an isomorphism of real analytic spaces was proved in [Lu]. One has the following stronger result: Theorem 4.5. The Kobayashi–Hitchin correspondence for the Seiberg–Witten equaγM vir tions induces an isomorphism which maps the virtual fundamental class [WX,β ] , computed with respect to the complex orientation data, onto the virtual fundamental class [GDou(M)]vir (respectively onto (−1)χ(M) [GDou(KX ⊗ M −1 )]vir ) when (2c1 (M) − c1 (KX ) − [β]) ∪ [ωg ], [X] < 0 (respectively > 0). Proof. Let C0 be the standard connection induced by the Levi-Civita connection in the line bundle KX−1 . Using the substitutions A := C0 ⊗ B ⊗2 with B ∈ A(M) and A =: ϕ + α ∈ A0 (M) ⊕ A0,2 (M), the configuration space of unknowns becomes A = A(M) × [A0 (M) ⊕ A0,2 (M)], and a pair (B, ϕ + α) solves the twisted monopole γ equation (SWβ M ) if −FA02 + α ⊗ ϕ¯ = 0,
∂¯B (ϕ) − i#∂B (α) = 0, ¯ = 0. i#g (FA + 2π iβ) + (ϕ ϕ¯ − ∗(α ∧ α)) We denote by A∗ the open subspace of A with non-trivial spinor component, and by B ∗ ∗ its quotient A G by the gauge group G = C ∞ (X, S 1 ). Let ei (B, ϕ, α), i = 1, . . . , 3 stand for the map of A defined by the left hand term of the i th equation above. This map induces a section ε i in a certain bundle H i over B ∗ which is associated with the principal G-bundle A∗ → B ∗ . γ The Seiberg–Witten moduli space Wβ M is the analytic subspace of B ∗ cut out by the Fredholm section ε = (ε1 , ε2 , ε3 ) in the bundle H := ⊕H i , and the virtual fundamental γ class [Wβ M ]vir is by definition the virtual fundamental class in the sense of Brussee [Br], associated with this section and the complex orientation data. We define a bundle morphism q : H → H 2 := A∗ ×G A0,2 (M) by 1 q(B,ϕ,α) (x 1 , x 2 , x 3 ) = ∂¯B x 2 + x 1 ϕ. 2 One easily checks that 1 q ◦ ε(B, ϕ, α) = ( |ϕ|2 + ∂¯B ∂¯B∗ )α. 2 Suppose now that (2c1 (M) − c1 (KX ) − [β]) ∪ [ωg ], [X] < 0. Integrating the third equation over X, one sees that any solution of the equations has a nontrivial ϕcomponent. The space ASW of solutions is therefore contained in the open subspace A◦ consisting of triples (B, ϕ, α) with ϕ = 0. But the operator ( 21 |ϕ|2 + ∂¯B ∂¯B∗ ) is invertible for ϕ = 0. It follows that on B ◦ := A◦ /G the section ε := q ◦ ε is regular around its vanishing locus Z(ε ), and that the submanifold Z(ε ) ⊂ B ◦ is just the submanifold cut out by the equation α = 0. One checks that q is a bundle epimorphism on B ◦ . Set H := ker q. The Associativity γ Property shows now that the virtual fundamental class [Wβ M ]vir can be identified with the virtual fundamental class associated with the Fredholm section ε := ε|Z(ε ) ∈ (Z(ε ), H |Z(ε ) ).
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In other words, the virtual fundamental class of the Seiberg–Witten moduli space can be identified with the virtual fundamental class of the moduli space V( 2s −π#g β) (M) of ( 2s − π#g β)-vortices in M [OT1]. Recall that V( 2s −π#g β) (M) is defined as the space of equivalence classes of pairs (B, ϕ) ∈ A(M) × [A0 (M) \ {0}] satisfying the equations (−FA02 , ∂¯B ϕ) = 0, s 1 i#g FB + ϕ ϕ¯ + ( − π #g β) = 0. 2 2 Here the first equation is considered as taking values in the subspace 0,1 ¯ G1B,ϕ := {(u, v) ∈ A0,2 X ⊕ A (M)| ∂B v + uϕ = 0}.
0 More precisely, let C ∗ be the quotient C ∗ := A(M) × [A (M) \ {0}] G , and let G1 be the subbundle of the associated bundle
0,1 A(M) × [A0 (M) \ {0}] ×G [A0,2 X ⊕ A (M)] over C ∗ , whose fibre in [B, ϕ] is G1B,ϕ . Let G2 be the trivial bundle C ∗ × A0 (X) and G := G1 ⊕ G2 . The left-hand terms of the equations above define sections g i in the bundles Gi , and the section g = (g 1 , g 2 ) is Fredholm. So far we have shown that the virtual fundamental class of the Seiberg–Witten moduli space can be identified with the virtual fundamental class of the moduli space V( 2s −π#g β) (M) associated with the Fredholm section g and the complex orientations. To complete the proof, we have to identify the virtual fundamental class [V( 2s −π#g β) (M)]vir with the virtual fundamental class [GDou(M)]vir of the corresponding gauge theoretical Douady space. This is again an application of the general principle which states the Kobayashi–Hitchin-type correspondence between moduli spaces associated with Fredholm problems respects virtual fundamental classes. We proceed as in the proof of Theorem 3.2: Consider the exact sequence π 0 −→ G1 −→ G −→ G2 −→ 0
of bundles over C ∗ . The section g 2 = π ◦ g comes from a formal moment map, so one can show: 1. g 2 is regular around Z(g), 2. the natural map ρ : Z(g 2 ) → B¯ inj induces a bijection
Z(g) = V( 2s −π#g β) (M) → GDou(M), and is étale around Z(g). Using the notations of Sect. 4.1, one obtains a natural identification ρ ∗ (E) = G1 |Z(g 2 ) , and g 1 |Z(g 2 ) corresponds via this identification to the section s which defines the virtual fundamental class [GDou(M)]vir . The result follows now by applying again the Associativity Property of virtual fundamental classes. $ %
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Recall from [OT2] that with any compact oriented 4-manifold X with b+ = 1 one ± can associate a full Seiberg–Witten invariant SWX,( (c) ∈ #∗ H 1 (X, Z) which O 1 ,H0 ) depends on an equivalence class c of Spinc -structures, an orientation O1 of H 1 (X, R), and a component H0 of the hyperquadric H of H 2 (X, R) defined by the equation x·x = 1. By the homotopy invariance of the virtual fundamental classes, one has Remark 4.6. The Seiberg–Witten invariants defined in [OT2] using the generic regularity of Seiberg–Witten moduli spaces with respect to Witten’s perturbation [W2], coincide with the Seiberg–Witten invariants defined in [Br] using the virtual fundamental classes of these moduli spaces. Combining Theorems 2.8, 3.2, 3.8, 4.3, 4.5 we obtain Corollary 4.7. Consider a ruled surface X = P(V0 ) over the Riemann surface of genus g, and a class c of Spinc -strucures on X. Let c be the Chern class of the determinant line bundle of c and let wc := 41 (c2 − 3σ (X) − 2e(X)) be the index of c . Denote by [F ] the class of a fibre of X over , by c ∈ #2 (H1 (X, Z)) = #2 (H 1 (X, Z))∨ the element defined by 1 c (a, b) := c ∪ a ∪ b, [X], 2 and let lO1 be the generator of #2g (H 1 (X, Z)) corresponding to O1 . The full Seiberg–Witten invariant of X corresponding to c, the complex orientation O1 of the cohomology space H 1 (X, R), and the component H0 of H which contains the Kähler cone, is given by ± SWX,( (c) = 0 O 1 ,H0 ) if c, [F ] = 0; when c, [F ] = 0, it is given by −signc,[F ]
SWX,(O1 ,H0 ) (c) = 0, signc,[F ] SWX,(O1 ,H0 ) (c)(l)
= signc, [F ]
g
i≥max(0,g− w2c )
ic ∧ l, lO1 . i!
Remarks 1. This result cannot be obtained directly using the Kobayashi–Hitchin correspondence for the Seiberg–Witten equations, because the Douady spaces of divisors on ruled surfaces are in general oversized, non-reduced, and they can contain components of different dimensions. Moreover, it is not clear at all whether one can achieve regularity by varying the holomorphic structure V0 in V0 . This shows that the quot ∨ L∨ spaces of the form Quot L S n (V0 ) are very special within the class of quot spaces Quot E0 with E0 C ∞ - equivalent to S n (V0 ). The theory of gauge theoretical Gromov–Witten invariants and the comparison Theorem 4.3 show that one can however compute ∨ the full Seiberg–Witten invariant of X using a quot space Quot L E0 with E0 a general holomorphic bundle C ∞ -equivalent to S n (V0 ), although such a quot space cannot be identified with a space of divisors of X.
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2. The result provides an independent check of the universal wall-crossing formula for the full Seiberg–Witten invariant, proven in [OT2]. Note however that, in the formula given in [OT2], the sign in front of uc , which corresponds to c above, is wrong. The error was pointed out to us by Markus Dürr, who also checked the corrected formula for a large class of elliptic surfaces [Dü]. References [A]
Aronszajin, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of the second order. J. de Math. Pures et Appl. 9, 36, 235–249 (1957) [BDW] Bertram, A., Daskalopoulos, G., Wentworth, R.: Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians. J. A. M. S. 9, No. 2, 529–571 (1996) [B] Bradlow, S.B.: Special metrics and stability for holomorphic bundles with global sections. J. Diff. Geom. 33, 169–214 (1991) [Br] Brussee, R.: The canonical class and the C ∞ properties of Kähler surfaces. New York J. Math. 2, 103–146 (1996) [CGS] Cieliebak, K., Gaio, A.R., Salamon, D.: J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions. Preprint math/9909122 [DK] Donaldson, S., Kronheimer, P.: The Geometry of Four-Manifolds. Oxford: Oxford Univ. Press, 1990 [Dou] Douady, A.: Le problème des modules pour les variétés analytiques complexes. Sem. Bourbaki nr. 277, 17-eme annee 1964/65 [Dü] Dürr, M.: Virtual fundamental classes and Poincaré invariants. Zürich, in preparation [G] Gaio,A.R.P.: J-holomorphic curves and moment maps. Ph. D. Thesis, University of Warwick, November 1999 [Gh] Ghione, F.: Quot schemes over a smooth curve. Indian Math. Soc. 48, 45–79 (1984) [Gr] Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985) [Ha] Hartshorne, R.: Algebraic geometry. Berlin–Heidelberg–New York: Springer-Verlag, 1977 [HL] Huybrechts, D., Lehn, M.: Stable pairs on curves and surfaces. J. Alg. Geom. 4, 67–104 (1995) [K] Kobayashi, S.: Differential geometry of complex vector bundles. Princeton, NJ: Princeton Univ. Press, 1987 [KM] Kronheimer, P., Mrowka, T.: The genus of embedded surfaces in the projective plane. Math. Res. Letters 1, 797–808 (1994) [L] Lange, H.: Höhere Sekantenvarietäten und Vektorbündel auf Kurven. Manuscripta Math. 52, 63–80 (1985) [Lin] Lin, T.R.: Hermitian–Yang–Mills–Higgs Metrics and stability for holomorphic vector bundles with Higgs Fields. Preprint, Rutgers University, New Brunswick, NJ [LiT] Li, J., Tian, G.: Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds. In: Topics in symplectic 4-manifolds (Irvine, CA, 1996), First Int. Press Lect. Ser., I, Cambridge, MA: Internat. Press, 1998, pp. 47–83 [LL] Lübke, M., Lupa¸scu, P.: Isomorphy of the gauge theoretical and the deformation theoretical moduli space of simple holomorphic pairs. Preprint (2001) [Lu] Lupa¸scu, P.: Seiberg–Witten equations and complex surfaces. Ph. D. Thesis, Zürich University, 1998 [LO] Lübke, M., Okonek, Ch.: Moduli spaces of simple bundles and Hermitian-Einstein connections. Math. Ann. 276, 663–674 (1987) [LT] Lübke, M., Teleman, A.: The Kobayashi–Hitchin correspondence. Singapore: World Scientific Publishing Co., 1995 [Mu1] Mundet i Riera, I.: A Hitchin–Kobayashi correspondence for Kaehler fibrations. J. Reine Angew. Math. 528, 41–80 (2000) [Mu2] Mundet i Riera, I.: Hamiltonian Gromov–Witten invariants. prep. math. SG/0002121 [OST] Okonek, Ch., Schmitt, A., Teleman, A.: Master spaces for stable pairs. Topology 38, No 1, 117–139 (1998) [OT1] Okonek, Ch., Teleman, A.: The Coupled Seiberg–Witten Equations, Vortices, and Moduli Spaces of Stable Pairs. Int. J. Math. 6, No. 6, 893–910 (1995) [OT2] Okonek, Ch., Teleman, A.: Seiberg–Witten invariants for manifolds with b+ = 1, and the universal wall crossing formula. Int. J. Math. 7, No. 6, 811–832 (1996) [Oxb] Oxbury, W. M.: Varieties of maximal line subbundles. Math. Proc. Cambridge Phil. Soc. 129, no. 1, 9–18 (2000) [R] Ruan, Y.: Topological sigma model and Donaldson-type invariants in Gromov theory. Duke Math. J. 83, no. 2, 461–500 (1996) [S] Serre, J. P.: Géometrie algébrique et Géometrie analytique. Ann. Inst. Fourier 6, 1–42 (1956)
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[Sm] [Su] [Th1] [Th2] [U] [W1] [W2]
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Smale, S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861–866 (1965) Suyama, Y.: The analytic moduli space of simple framed holomorphic pairs. Kyushu J. Math. 50, 65–68 (1996) Thaddeus, M.: Stable pairs, linear systems and the Verlinde formula. Invent. Math. 117, 181–205 (1994) Thaddeus, M.: Geometric invariant theory and flips. JAMS 9, 691–725 (1996) Uhlenbeck, K.: Connections with Lp bounds on curvature. Commun. Math. Phys. 83, 31–42 (1982) Witten, E.: The Verlinde algebra and the cohomology of the Grassmannian. In: Geometry, Topology and Physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Cambridge, MA: Internat. Press, 1995, pp. 357–422 Witten, E.: Monopoles and four-manifolds. Math. Res. Letters 1, 769–796 (1994)
Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 227, 587 – 603 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
Simulation of Topological Field Theories by Quantum Computers Michael H. Freedman1 , Alexei Kitaev1, , Zhenghan Wang2 1 Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA 2 Indiana University, Dept. of Math., Bloomington, IN 47405, USA
Received: 4 May 2001 / Accepted: 16 January 2002
Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.
1. Introduction A topological quantum field theory (TQFT) is a mathematical abstraction, which codifies topological themes in conformal field theory and Chern–Simons theory. The strictly 2-dimensional part of a TQFT is called a topological modular functor (TMF). It (essentially) assigns a finite dimensional complex Hilbert space V () to each surface and to any (self)-diffeomorphism h of a surface a linear (auto)morphism V (h) : V () → V ( ). We restrict attention to unitary topological modular functors (UTMF) and show that a quantum computer can efficiently simulate transformations of any UTMF as a transformation on its computational state space. We should emphasize that both sides of our discussion are at present theoretical: the quantum computer which performs our simulation is also a mathematical abstraction – the quantum circuit model (QCM) [D,Y]. On leave from Landau Institute for Theoretical Physics, Moscow.
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Very serious proposals exist for realizing this model, perhaps in silicon, e.g. [Ka], but we will not treat this aspect. There is a marked analogy between the development of the QCM from 1982 Feynman [Fey] to the present, and the development of recursive function theory in the 1930’s and 1940’s. At the close of the earlier period, “Church’s thesis” proclaimed the uniqueness of all models of (classical) calculation: recursive function theory, Turing machine, λcalculus, etc.... This result was refined in the 1960s, by showing that most “natural” models are polynomially equivalent to the Turing machine. The present paper can be viewed as supporting a similar status for QCM as the inherently quantum mechanical model of calculation. The modern reconsideration of computation is founded on the distinction between polynomial time and slower algorithms. Of course, all functions computed in the QCM can be computed classically, but probably not in comparable time. Assigning to an integer its factors, while polynomial time in QCM [Sh] is nearly exponential time, exp(O(n1/3 poly(log n))) (an emphiric bound, the proved one is even worse) according to the most refined classical algorithms. The origin of this paper is in thought [Fr] that since ordinary quantum mechanics appears to confer a substantial speed up over classical calculations, that some principle borrowed from the early, string, universe might go still further. Each TQFT is an instance of this question since their discrete topological nature lends itself to translation into computer science. We answer here in the negative by showing that for a unitary TQFT, the transformations V (h) have a hidden poly-local structure. Mathematically, V (h) can be realized as the restriction to an invariant subspace of a transformation gi on the state space of a quantum computer where each gi is a gate and the length of the composition is linear in the length of h as a word in the standard generators, “Dehn twists” of the mapping class group = diffeomorphisms ()/identity component. Thus, we add evidence to the unicity of the QCM. Several variants and antecedents of QCM, including quantum Turing machines, have previously been shown equivalent (with and without environmental errors)[Y]. From a physical standpoint, the QCM derives from Schrödinger’s equation as described by Feynman [Fey] and Lloyd [Ll]. Let us introduce the model. Given a decision problem, the first or classical phase of the QCM is a classical program, which designs a quantum circuit to “solve” instances of the decision problem of length n. A quantum circuit is a composition Un of operators or gates gi ∈ U(2) or U(4) taken from some fixed list of rapidly computable matrices1 , e.g. having algebraic entries. The following short list suffices to efficiently approximate any other choice of gates [Ki]: 0 1
1 , 0
1 0
0 , i
1 0 and 0 0
0 0 1 0 0 √1 0
2 √1 2
1 √ . 2 −1 √ 0 0
2
The gates are applied on some tensor power space (C2 )⊗k(n) of “k qubits” and models a local transformation on a system of k spin 21 particles. The gate g acts as the identity on all but one or two tensor factors where it acts as a matrix as above. This is the middle or quantum phase of the algorithm. The final phase is to perform a local von Neumann measurement on a final state ψfinal = Un (ψinitial ) (or a commuting family of the same) to extract a probabilistic answer to the decision problem. (The initial states’ ψinitial must also be locally constructed.) In this phase, we could declare that observing a certain 1 The i th digit of each entry should be computable in poly(i) time.
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eigenvalue with probability ≥ 23 means “yes”. We are interested only in the case where the classical phase of circuit design and the length of the designed circuit are both smaller than some polynomial in n. Decision problems which can be solved in this way are said to be in the computational class BQP: bounded-error quantum polynomial. The use of C2 , the “qubit”, is merely a convenience, any decomposition into factors of bounded dimension gives an equivalent theory. We say U is a quantum circuit over Cp if all tensor factors have dimension = p. Following Lloyd [Ll], note that if a finite dimensional quantum system, say (C2 )⊗k , evolves by a Hamiltonian H , it is physically reasonable to assert that H is poly-local, L ∼ ∼
H = H , where the sum has ≤ poly(k) terms and each H = H ⊗ id, where H i=1
acts nontrivially only on a bounded number (often just two) qubits and as the identity on the remaining tensor factors. Now setting Plank’s constant h = 1, the time evolution is given by Schrödinger’s equation: Ut = e2πitH whereas gates can rapidly approximate [Ki] any local transformation of the form e2πitH . Only the nonabelian nature of the L
unitary group prevents us from approximating Ut directly from e2πiH . However, by the Trotter formula:
i=1
A/n+B/n n 1 A+B e =e +O , n
where the error O is measured in the operator norm. Thus, there is a good approximation to Ut as a product of gates:
2πi t H 1 2πi nt HL n 2 1 n Ut = e ...e +L ·O . n Because of the rapid approximation result of [Ki], in what follows, we will not discuss quantum circuits restricted to any small generating set as in the example above, rather we will permit a 2 × 2 or 4 × 4 unitary matrix with algebraic number entries to appear as a gate. In contrast to the systems considered by Lloyd, the Hamiltonian in a topological theory vanishes identically, H = 0, a different argument - the substance of this paper - is needed to construct a simulation. The reader may wonder how a theory with vanishing H can exhibit nontrivial unitary transformations. The answer lies in the Feynman pathintegral approach to QFT. When the theory is constructed from a Lagrangian (functional on the classical fields of the theory), which only involves first derivatives in time, the Legendre transform is identically zero [At], but may nevertheless have nontrivial global features as in the Aharonov-Bohm effect. Before defining the mathematical notions, we would make two comments. First, the converse to the theorem is also true. It has been shown recently [FLW] that a particular UTMF allows efficient simulation of the universal quantum computer. Second, we would like to suggest that the theorem may be viewed as a positive result for computation. Modular functors, because of their rich mathematical structure, may serve as higher order language for constructing a new quantum algorithm. In [Fr], it is observed that the transformations of UTMF’s can readily produce state vectors whose coordinates are computationally difficult evaluations of the Jones and Tutte polynomials. The same is now known for the state vector of a quantum computer, but the question of whether any useful part of this information can be made to survive the measurement phase of quantum computation is open.
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2. Simulating Modular Functors We adopt the axiomatization of [Wa] or [T] to which we refer for details. Also see, Atiyah [At], Segal [Se], and Witten [Wi]. A surface is a compact oriented 2-manifold with parameterized boundaries. Each boundary component has a label from a finite set L = {1, a, b, c, . . . } with involution , 1 = 1. In examples, labels might be representations of a quantum group up to a given level or positive energy representations of a loop group, or some other algebraic construct. Technically, to avoid projective ambiguities each surface is provided with a Lagrangian subspace L ⊂ H1 (; Q) and each diffeomorphism f : → is provided with an integer “framing/signature” so the dynamics of the theory is actually given by a central extension of the mapping class group. Since these extended structures are irrelevant to our development, we suppress them from the notation. We use the letter below to indicate a label set for all boundary components, or in some cases, those boundary components without a specified letter as label. Definition 1. A unitary topological modular functor (UTMF) is a functor V from the category of (labeled surfaces with fixed boundary parameterizations, label preserving diffeomorphisms which commute with boundary parameterizations) to (finite dimensional complex Hilbert spaces, unitary transformations) which satisfies: 1. Disjoint union axiom: V (Y1 Y2 , 1 2 ) = V (Y1 , 1 ) ⊗ V (Y2 , 2 ). 2. Gluing axiom: let Yg arise from Y by gluing together a pair of boundary circles with dual labels, x glues to x , then V (Yg , ) =
V (Y, (, x, x )).
x" L
3. Duality axiom: reversing the orientation of Y and applyingto labels corresponds to replacing V by V ∗ . Evaluation must obey certain naturality conditions with respect to gluing and the action of the various mapping class groups. 4. Empty surface axiom: V (φ) ∼ =C. ∼ C, if a = 1 . 5. Disk axiom: Va = V (D, a) = 0, if a = 1 C, if a = b ∼ 6. Annulus axiom: Va,b = V A, (a, b) = 0, if a = b 7. Algebraic axiom: The basic data, the mapping class group actions and the maps F and S explained in the proof (from which V may be reconstructed if the Moore and Seiberg conditions are satisfied, see [MS] or [Wa] 6.4, 1–14) is algebraic over Q for some bases in Va , Va,a , and Vabc , where Vabc denotes V P , (a, b, c) for a (compact) 3-punctured sphere P . 3-punctured spheres are also called pants. Comments. (1) From the gluing axiom, V may be extended via dissection from simple pieces D, A, and P to general surfaces . But V () must be canonically defined: this looks quite difficult to arrange and it is remarkable that any nontrivial examples of UTMFs exist.
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(2) The algebraic axiom is usually omitted, but holds for all known examples. We include it to avoid trivialities such as a UTMF where action by, say, a boundary twist is multiplication by a real number whose binary expansion encodes a difficult or even uncomputable function: e.g. the i th bit is 0 iff the i th Turing machine halts. If there are nontrivial parameter families of UTMF’s, such nonsensical examples must arise – although they could not be algebraically specified. In the context of bounded accuracy for the operation of diffeomorphisms V (h), Axiom 7 may be dropped (and simulation by bounded accuracy quantum circuits still obtained), but we prefer to work in the exact context since in a purely topological theory exactness is not implausible. (3) Axiom 2 will be particularly important in the context of a pants decomposition of a surface . This is a division of into a collection of compact surfaces P having the topology of 3-punctured spheres and meeting only in their boundary components which we call “cuffs”. Definition 2. A quantum circuit U : (Cp )⊗k → (Cp )⊗k =: W is said to simulate on W (exactly) a unitary transformation τ : S → S if there is a C-linear imbedding i : S ⊂ (Cp )⊗k invariant under U so that U ◦ i = i ◦ τ . The imbedding is said to intertwine τ and U. We also require that i be computable on a basis in poly(k) time. Since we prove efficient simulation of the topological dynamics for UTMFs V , it is redundant to dwell on “measurement” within V, but to complete the computational model, we can posit von Neumann type measurement with respect to any efficiently computable frame F in Vabc . The space Cp above, later denoted X = Cp , is defined by X := ⊕ Vabc and the computational space W := X ⊗k . We have set S := V () (a,b,c)∈L3
and assumed is divided into k “pants”, i.e. Euler class () = −k. Any frame F extends to a frame for V () via the gluing axiom once a pants decomposition of is specified. Thus, measurement in V becomes a restriction of measurement in W . It may be physically more natural to restrict the allowable measurements on V () to cutting along a simple closed curve γ and measuring the label which appears. Mathematically, this amounts to transforming to a pants decomposition with γ as one of its decomposition or “cuff” curves and then positing a Hermitian operator with eigenspaces equal to the summands of V () corresponding under the gluing axiom to labels x on γ − and x on γ + , x"L. A labeled surface (, ) determines a mapping class group M = M(, ) = “isotopy classes of orientation preserving diffeomorphisms of preserving labels and commuting with boundary parameterization”. For example, in the case of an n-punctured sphere with all labels equal (distinct), M = SFB(n), the spherical framed braid group M = PSFB(n), the pure spherical framed braid group . To prove the theorem below, we will need to describe a generating set S for the various M’s and within S chains of elementary moves which will allow us to prepare to apply any s2 ∈ S subsequent to having applied s1 ∈ S. Each M is generated by Dehn-twists and braid-moves (See [B]). A Dehn-twist Dγ is specified by drawing a simple closed curve (s.c.c.) γ on , cutting along γ , twisting 2π to the right along γ and then regluing. A braid-move Bδ will occur only when a s.c.c. δ cobounds a pair of pants with two boundary components of : If the labels of the boundary components are equal then Bδ braids them by a right π -twist. In the case that all labels are equal, there is a rather short list of D and B generators indicated in Fig. 1 below. Also sketched in Fig. 1 is a pants decomposition of diameter = O log b1 () ,
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c
D γ ’s
c
c
c
c c
c
c
c
c
γ
γ'
A
Bδ ’s Fig. 1.
meaning the graph dual to the pants decomposition has diameter order log the first Betti number of . The s.c.c. γ ( δ) label Dehn (braid) generators Dγ ( and Bδ ). Figure 1 contains a punctured annulus A; note that the composition of oppositely oriented Dehn twists along the two “long” components of ∂A, γ and γ yield a diffeomorphism which moves the punctures about the loop γ . The figure implicitly contains such an A for each (γ , p), where p is a preferred puncture. The γ curves come in three types: (1) The loops at the top of the handles which are curves (“cuffs”) of the pants decomposition, (2) loops dual to type 1, and (3) loops running under adjacent pairs of handles (which cut through up to O log(b1 ) many cuffs). (See Fig. 1, where cuffs are marked by a “c”.) Each punctured annulus A is determined as a neighborhood (of a s.c.c. γ union an arc η from γ to p). To achieve general motions of p around , we require these arcs to be “standard” so that for each p, π1 (, p) is generated by {η · γ · η−1 }, where = with punctures filled by disks, and the disk corresponding to p serving as a base point. This list of generators is only linear in the first Betti number of . In the presence of distinct labels, many of the Bδ are illegal (they permute unequal labels). In this case, quadratically many generators are required. Figure 2 displays the replacements for the B’s, and additional A’s and D’s. Figure 2 shows a collection of B’s sufficient to effect arbitrary braiding within each commonly-labeled subset of punctures, a quadratically large collection of new Dehn curves {"} allowing a full twist between any pair of distinctly labeled punctures. (If the punctures are arranged along a convex arc of the Euclidean cell in , then each " will be the boundary of a narrow neighborhood of the straight line segment joining pairs of dissimilarly labeled punctures.) Finally a collection of punctured annuli, which enable one puncture pi from each label – constant subset to be carried around each free homotopy class from {γ }(respecting the previous generation condition for π1 (, pi ). Thus for distinct labels the generating sets are built from curves of type γ , γ , " and δ by Dehn twists around γ , γ , and ", braid moves around δ. Denote by ω, any such curve: ω ∈ 1 = {γ } ∪ {γ } ∪ {"} ∪ {δ} .
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Bδ Dε γ'
A
A
Fig. 2.
Since various ω s intersect, it is not possible to realize all ω simultaneously as cuffs in a pants decomposition. However, we can start with the “base point” pants decomposition D indicated in Fig. 1 (note γ of type(1) are cuffs in D, but γ of types (2) or (3) are not) and for any ω find a short path of elementary moves: F and S (defined below) to a pants decomposition Dω containing ω as a cuff. Lemma 2.1. Assume = S 2 , disk, or annulus, and D the standard pants decomposition sketched in Fig. 1. Any ω as above, can be deformed through O log b1 () F and S moves to a pants decomposition Dω in which ω is a cuff. We postpone the proof of the lemma and the definition of its terms until we are partly into the proof of the theorem and have some experience passing between pants decompositions. Theorem 2.2. Suppose V is a UTMF and h : → is a diffeomorphism of length n in the standard generators for the mapping class group of described above (see Figs. 1 and 2). Then there are constants depending only on V , c = c(V ) and p = p(V ) such that V (h) : V () → V () is simulated (exactly) by a quantum circuit operating on “qupits” Cp of length ≤ c · n · log b1 (). The collection {cuffs} refers to the circles along which the pants decomposition decomposes; the “seams” are additional arcs, three per pant which cut the pant into two hexagons. Technically, we will need each pant in D to be parameterized by a fixed 3-punctured sphere so these seams are part of the data in D; for simplicity, we choose seams to minimize the number of intersections with {ω}. The theorem may be extended to cover a more general form of input. The original algorithm [L] which writes a Dα , α a s.c.c., as a word in standard generators Dγ is super-exponential. We define the combinatorial length of α, (α), to be the minimum number of intersections as we vary α by isotopy of α with {cuffs} ∪ {seams}. The best upper-bound (known to the authors) to the length L of Dα as a word in the mapping class group spanned by a fixed generating set is of the form L(Dα ) < super-exponential function f (). For this reason, we consider as input V (h), where h is a composition of k Dehn twists on α1 , . . . , αk and j braid moves along β1 , . . . , βj in any order.
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Then V (h) is costed as the sum of the combinatorial length of the simple closed curves needed to write h as Dehn twists and braid moves within the mapping class group, j
k
i=1
i=1
(h) := (βi ) + (αi ). We obtain the following extension of the theorem. The map h∗ : V () −→ V () is exactly simulated by a quantum circuit QC with length (QC) ≤ 11(h) composed of algebraic 1 and 2-qupit Cp gates. Extension.2
Pre-Proof. Some physical comments will motivate the proof. V () are quantized gauge fields on (with a boundary condition given by labels ) and can be regarded as a finite dimensional space of internal symmetries. This is most clear when genus () = 0 , is a punctured sphere, the labeled punctures are “anyons” [Wil] and the relevant mapping class group is the braid group which moves the punctures around the surface of the sphere. An internal state ψ"V () is transformed to U(b)ψ ∈ V () under the functorial representation of the braid group. For U(b) to be defined the braiding must be “complete” in the sense that the punctures (anyons) must return setwise to their initial position. Infinitesimally, the braiding defines a Hamiltonian H on V ()⊗E, where E is an infinite dimensional Hilbert space which encodes the position of the anyons. The projection of H into V () vanishes which is consistent with the general covariance of topological theories. Nevertheless, when the braid is complete, the evolution U of H will leave V () invariant and it is U|V () = U which we will simulate.Anyons inherently reflect nonlocal entanglement so it is not to be expected that V () has any (natural) tensor decomposition and none are observed in interesting examples. Thus, simulation of U as an invariant subspace of a tensor product (Cp )⊗k is the best result we can expect. The mathematical proof will loosely follow the physical intuition of evolution in a super-space by defining, in the braid case (identical labels and genus = 0), two distinct imbeddings “odd” and odd
−→
p ⊗k = W and constructing the local evolution by gates acting on even (C ) “even”, V () −→ the target space. The imbeddings are named for the fact that in the usual presentation of the braid group, the odd (even) numbered generators can be implemented by restricting an action on W to image odd V () evenV () .
Proof. The case genus () = 0 with all boundary components carrying identical labels (this contains the classical, uncolored Jones polynomial case [J, Wi]) is treated first. For any number q of punctures (q = 10 in the illustration) there are two systematic ways of 8 dividing into pants (3-punctured spheres) along curves α = {α1 , . . . , αq−3 } or along 8
β = {β1 , . . . , βq−3 } so that a sequence of q F moves (6j -moves in physics notation) 8
8
transforms α to β . Let X = (a,b,c)" L3 Vabc be the orthogonal sum of all sectors of the pants Hilbert ⊗(q−2) := W is the sum over all space. Distributing over , the tensor power X labelings of the Hilbert space for (q − 2) pants. Choosing parameterizations, W is identified with both the label sum space (cut8 ) and sum ( 8 ). Now is assembled α 8
8
cut β
from the disjoint union by gluing along α or β so the gluing axiom defines imbeddings 8
8
i( α ) and i( β ) of V (, ) as a direct summand of X ⊗(q−2) = W . 2 Lee Mosher has informed us that the existence of the linear bound f () (but without control of the constants) follows at least for closed and single punctured surfaces from his two papers [M1] and [M2].
Simulation of Topological Field Theories by Quantum Computers
α1
595
α2 α5
α3
α4
3 6j
α5 α3
4 6j
β3
β1
3 6j
β1
β2
β5
β4
β3
Fig. 3.
Consider the action of braid move about α. This acts algebraically as θ(αi ) on a single 8 X factor of W and as the identity on other factors. This action leaves i( α ) V (, ) invariant and can be thought of as a “qupit” gate: θ(αi ) = V (braidαi ) : X → X, 8
where dimension dim(X) = p. Similarly the action of V (braidβi ) leaves i( β ) invariant. 8
8
It is well known [B] that the union of loops α ∪ β determines a complete set of generators of the braid group. The general element ω, which we must simulate by an action on W is a word in braid moves on α’s and β’s. Part of the basic data – implied by the gluing axiom for a UTMF is a fixed identification between elementary gluings: Fabcd :
x" L
Vxab ⊗ V x cd −→
y" L
Vybc ⊗ V y da
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M. H. Freedman, A. Kitaev, Z. Wang
a
d
a 2
3
d 3
2
2
2
Fabcd 1
1 3
b
2
2
b
c
3
c
Fig. 4.
corresponding to the following two decompositions of the 4-punctured sphere into two pairs of pants (the dotted lines are pant “seams”, the uncircled number indicate boundary components, the letters label boundary components, and the circled numbers order the pairs of pants.): For each F , we choose an extension to a unitary map F : X X −→ X X. Then extend F to F by tensoring with identity on the q − 4 factors unaffected by F . The composition of q F ’s, extended to q F ’s, corresponding to the q moves illustrated in the case q = 10 by Fig. 3. (For q > 10 imagine the drawings in Fig. 3 extended 8
8
periodically.) These define a unitary transformation T : W → W with T ◦ i( α ) = i( β ). The word ω in the braid group can be simulated by τ on W , where τ is written as a composition of the unitary maps T , T −1 , θ(αi ), and θ(βj ). For example, β5 α1 β2−1 α1 α3 would be simulated as τ = T −1 ◦ θ (β5 ) ◦ T ◦ θ (α1 ) ◦ T −1 ◦ θ(β2−1 ) ◦ T ◦ θ(α1 ) ◦ θ(α3 ). As described τ has length ≤ 2q length ω. The dependence on q can be removed by dividing into q2 overlapping pieces i , each i a union of 6 consecutive pants. Every 8
8
loop of α ∪ β is contained well within some piece i so instead of moving between two fixed subspaces iα (V ) and iβ (V ) ⊂ W , when we encounter a βj , do constantly many F operations to find a new pants decomposition modified locally to contain βj . Then θ(βj ) may be applied and the F operations reversed to return to the α pants decomposition. The resulting simulation can be made to satisfy length τ ≤ 7 length ω. This completes the braid case with all bounding labels equal - an important case corresponding to the classical Jones polynomial [J]. Proof of Lemma. We have described the F -move on the 4-punctured sphere both geometrically and under the functor. The S-move is between two pants decompositions on the punctured torus T − . (Filling in the puncture, a variant of S may act between two distinct annular decomposition of T 2 . We suppress this case since, without topological parameter, there can be no computational complexity discussion over a single surface.) By [Li] or [HT] that one may move between any two pants decompositions via a finite sequence of moves of three types: F , S, and diffeomorphism M supported on the interior of a single pair of pants (see the Appendix [HT]). To pass from D, our “base point” decomposition, to Dω , F and S moves alone suffice and the logarithmic count is a consequence of the log depth nest of cuff loops of D on the planar surface obtained by
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597
3 2
3 2
Sa
a
1
1
a Fig. 5. V (S):
Vax xˆ −→
x∈L
Vay yˆ
y∈L
cutting along type (1) γ curves. Below we draw examples of short paths of F and S moves taking D to a particular Dω . The logarithmic count is based on the proposition. Proposition 2.3. Let K be a trivalent tree of diameter = d and f be a move, which locally replaces { } and with { }, then any two leaves of K can be made adjacent by ≤ d moves of type f . (Here we consider abstract trees rather than ones imbedded in the plane.) Passing from K to a punctured sphere obtained by imbedding (K, univalent vertices) into ( 21 R 3 , R 2 ), thickening and deleting the boundary R 2 , the f move induces the previously defined F move. Some example of paths of F , S moves (Fig. 6). Continuation of the proof of the theorem. For the general on numerous case, we compute imbeddings of V () into W (rather than on two: iα V () and iβ V () as in the braid case). Each imbedding is determined by a pants decomposition and the imbedding changes (in principle) via the lemma every time we come to a new literal of the word ω. Recall that ω ∈ M, the mapping class group, is now written as a word in the letters (and their inverses) of type Dγ , Dγ , D" , and Bδ . Pick as a home base a fixed pants decomposition D0 corresponding to i0 V () ⊂ W . If the first literal is a twist or braid along the s.c.c. ω, then apply the lemma to pass through a sequence of F and S moves from D0 to D1 containing ω as a “cuff” curve. As in the braid case, choose extensions F and S to unitary automorphisms of W and applying V to the composition gives a transformation T1 of W such that i1 = T1 ◦ i0 , i1 being the inclusion V () → W associated with D 1 . Now execute the first literal ω1 of ω as a transformation θ(ω1 ), which −1 leaves i1 V () invariant and satisfies: θ(ωi ) ◦ i1 = i1 ◦ V (ω1 ). Finally apply T1 to return to the base inclusion i0 V () . The previous three steps can now be repeated for the second literal of ω: follow T1−1 ◦ θ(ω1 ) ◦ T1 by T2−1 ◦ θ(ω2 ) ◦ T2 . Continuing in this way, we construct a composition τ which simulates ω on W : . τ = Tn−1 ◦ θ(ωn ) ◦ Tn−1 . . ◦ T1−1 ◦ θ(ω1 ) ◦ T1 .
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M. H. Freedman, A. Kitaev, Z. Wang γ'
ω S
ω
F
ω
ω
ω
F
ω
F
F
F
ω
Fig. 6.
From Lemma 2.1 the length of this simulation by one corresponding to S and θ(ωi ) and two (corresponding to F moves) qupit gates is proportional to n =length ω and log b1 (), where p = dim(X). Proof of Extension. What is at issue is the number of preparatory moves to change the base point decomposition D to Dγ containing γ = αi or βi as a cuff curve 1 ≤ i ≤ k or j . We have defined the F and S moves rigidly, i.e. with specified action on the seams. This was necessary to induce a well defined action on the functor V . Because of this rigid choice, we must add one more move – an M move – to have a complete set of moves capable of moving between any two pants decompositions of a surface (compare [HT]). The M move is simply a Dehn twist supported in a pair of pants of the current pants decomposition; it moves the seams (compare Chapter 5 [Wa]). Note that if M is a +1 Dehn twisit in a s.c.c. ω then, under the functor, V (M) is a restriction of θ(ω) in the notation above. As in [HT], the cuff curves of D may be regarded as level curves of a Morse function f : → R + , constant on boundary components which we assume to have minimum complexity (= total number of critial points) satisfying this constraint. Isotope α (we drop the index) on to have the smallest number of local maximums with respect to f and is disjoint from critical points of f on . Now generically deform f in a thin annular neighborhood of γ so that γ becomes a level curve. Consider the graphic G of the deformation ft , 0 ≤ t ≤ 1. For regular t the Morse function ft determines a pants decomposition: let the 1- complex K consist of / ∼ where x ∼ y if x and y belong to the same component of a level set of ft , and let
Simulation of Topological Field Theories by Quantum Computers
1) −ε cuff
+ε cuff
+ε cuff or
F-move
599
−ε cuff F-move
double critical level
double critical level 2)
moves: F 0 M 0 F (as shown below) double critical level
M
F
3)
F
−ε moves: M 0 M 0 F 0 S 0 F (as shown below)
+ε +ε
−ε
double critical level
F
S
M
2 0
S
2 M moves adjust seams to standard position Fig. 7.
L ⊂ K be the smallest complex to which K collapses relative to endpoints associated to boundary components. For example in Fig. 8, the top tree does not collapse at all while in the lower two trees the edge whose end is labeled, “local max” is collapsed away. The preimage of one point from each intrinsic 1-cell of L not containing a boundary point constitutes a {cuffs} determining a pants decomposition Dt . For singular t0 , let Dt0 −" and Dt0 +" may differ or may agree up to isotopy. The only change in D occurs when t is a crossing point for index= 1 handles where the two critical points are on the same connected component of a level set ft−1 (r). There are essentially only three possible “Cerf-transitions” and they are expressible as a product of 1, 2, or 3 F and S moves together with braid moves whose number we will later bound from above. The Cerf
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M. H. Freedman, A. Kitaev, Z. Wang
γ local max
γ local max
γ Fig. 8. Pulling γ down yields F ◦ F
γ'
pull γ' down, cancel local max yields F o M o F Fig. 9.
transitions on D are shown in Fig. 7, together with their representation as compositions of elementary moves. Critical points of f |γ become critical points of ft of the same index once the deformation as passed an initial "0 > 0, and before any saddle-crossings have occurred. Let P be a pant from the composition induced by f and δ ⊂ γ ∩ P an arc. Applying the connectivity criterion of the previous paragraph, we can see that flattening a local maxima can effect at most the two cuff circles which δ meets, and these by elementary Cerf transition shown in Fig. 8. If γ crosses the seam arcs then the transitions are of the Cerf type, precomposed with M-moves to remove these crossings as shown in Fig. 9. Dynamically seam crossings by γ produce saddle connections in the Cerf diagram. The total number of these twists is bounded by length (γ ). The number of flattening moves as above is less than or equal |γ ∩ cuffs| ≤ length(γ ). The factor of 11 in the statement allows up to 5 F , S, and M moves for expressing each Cerf singularity which arises in passing from D◦ to Dγ and the same factor of 5 to pass back from Dγ to D◦ again, while saving at least one step to implement the twist or build move along γ . This completes the proof of the extension.
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601
We should emphasize that although, we have adopted an “exact” model for the operation of the UTMF, faithful simulation as derived above does not depend on a perfectly accurate quantum circuit. Several authors have proved a threshold theorem [Ki, AB], and [KLZ]: If the rate of large errors acting on computational qubits (or qupits) is small enough, the size of ubiquitous error small enough, and both are uncorrelated, then such a computational space may be made to simulate with probability ≥ 23 an exact quantum circuit of length = L. The simulating circuit must exceed the exact circuit in both number of qubits and number of operations by a multiplicative factor ≤ poly (log L). 3. Simulating TQFT’s We conclude with a discussion about the three dimensional extension, the TQFT of a UTMF. In all known examples of TMF’s there is an extension to a TQFT meaning that b∗
it is possible to assign a linear map V () → V ( ) subject to several axioms [Wa] and [T] whenever and cobounds a bordism b (with some additional structure). The case of bordisms with a product structure is essentially the TMF part of the theory. Unitarity is extended to mean that if the orientation of the bordism b is reversed to b, we have b∗† = (b)∗ . It is known that a TMF has at most one extension to a TQFT and conjectured that this extension always exists. Non-product bordisms correspond to some loss of information of the state. This can be understood by factoring the bordism into pieces consisting of a product union 2-handle: × I ∪ h. The 2-handle h has the form (D 2 × I, ∂D 2 × I ) and is attached along the subspace ∂D 2 × I . The effect of attaching the handle will be to “pinch” off an essential loop ω on and so replace an annular neighborhood of ω by two disks turning into a simpler surface . It is an elementary consequence of the axioms that if b = × I ∪ h then b∗ is a projector as follows: Let D be a pants decomposition containing ω as a dissection curve. There are two cases: (1) ω appears as the first and second boundary components of a single pant called P0 or (2) ω appears as the first boundary component on two distinct pants called P1 and P2 . V () =
V \P Va ac , with label c on ∂ P = 0 3 0 , ˆ c" L
or =
a" L
labels a" L
Vabc
Vade ˆ
case (1),
V \ P1 ∪ P2 , appropriate labels ,
case (2).
In case (2), there may be a relation b = cˆ and/or d = eˆ depending on the topology of D. The map b∗ is obtained by extending linearly from the projections onto summands:
Va ac ˆ −→ V111
canonically
∼ =
V1 ,
(case 1)
a,c L
a,b,c,d,e " L
Vabc
or Vade −→ V1bbˆ ˆ
V1d dˆ
canonically
∼ =
Vbbˆ
Vd dˆ .
(case 2)
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If the orientation on b is reversed the unitarity condition implies that b determines an injection onto a summand with a formula dual to the above. Thus, any bordism’s morphism can be systematically calculated. In quantum computation, as shown in [Ki], a projector corresponds to an intermediate binary measurement within the quantum phase of the computation, one outcome of which leads to cessation of the other continuation of the quantum circuits operation. Call such a probabilistically abortive computation a partial computation on a partial quantum circuit. Formally, if we write the identity as a sum of two projectors: idV = 0 + 1 , and let U0 and U1 be unitary operators on an ancillary space A with U0 (|0 ) = |0 and U1 |0 = |1 . The unitary operator 0 ⊗ U0 + 1 ⊗ U1 on V ⊗ A when applied to |v ⊗ |0 is |0 v ⊗ |0 + |1 v ⊗ |1 so continuing the computation only if the indicator |0 ∈ A is observed simulates the projection 0 . It is clear that the proof of the theorem can be modified to simulate 2-handle attachments as well as Dehn twists and braid moves along s.c.c.’s ω to yield: Scholium 3.1. Suppose b is an oriented bordism from 0 to 1 , where i is endowed with a pants decomposition Di . Let complexity (b) be the total number of moves of four types: F , S, M, and attachment of a 2-handle to a dissection curve of a current pants decomposition that are necessary to reconstruct b from (0 , D0 ) to (1 , D1 ). Then there is a constant c (V ) depending on the choice of UTQFT and p(V ) as before (for the TQFTs underlining TMF) so that b∗ : V (0 ) → V (1 ) is simulated (up to a non-topological factor of the form ν n2 , where n2 is the number of 2-hanles attached) by a partial quantum circuit over Cp of length ≤ c complexity (b). In general, the difference between topological objects (such as b∗ or closed 3-manifold invariants) and quantum mechanical ones (the evolution and probability) is related to critical points of a Morse function. A similar phenomenon for links in R 3 has been mentioned in [FKLW]. This subject will be addressed in detail in a forthcoming paper by S. Bravyi andA. Kitaev, “Quantum invariants of 3-manifolds and quantum computation”. Acknowledgements. We would like to thank Greg Kupperberg and Kevin Walker for many stimulating discussions on the material presented here.
References [At] [AB]
Atiyah, M.: Topological quantum field theories. Publ. Math. IHES 68, 175–186 (1989) Aharonov, D. and Ben-Or, M.: Fault-tolerant quantum computation with constant error. LANL e-print quan-ph/9611025 [B] Birman, J.: Braids, links, and mapping class groups. Ann. Math. Studies, Vol. 82 [D] Deutsch, D.: Quantum computational networks. Proc. Roy. Soc. London, A425, 73–90 (1989) [Fey] Feynman, R.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982) [FKLW] Freedman, M.H., Kitaev, A., Larsen, M.J. and Wang, Z.: Topological Quantum Computation; LANL e-print quant-ph/0101025 [FLW] Freedman, M.H., Kitaev, A., Larsen, M.J. and Wang, Z.: A modular functor which is universal for quantum computation. LANL e-print quant-ph/0001108 [Fr] Freedman, M.H.: P/NP, and the quantum field computer. Proc. Natl. Acad. Sci., USA 95, 98–101 (1998) [HT] Hatcher, A. and Thurston, W.: A presentation for the mapping class group of a closed orientable surface. Topology 19, no. 3, 221–237 (1980) [J] Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomial. Ann. Math. 126, 335–388 (1987) [Ka] Kane, B.: A silicon-based nuclear spin quantum computer. Nature 393, 133–137 (1998) [Ki] Kitaev, A.: Quantum computations: algorithms and error correction. Russian Math. Survey 52:61, 1191–1249 (1997)
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[KLZ] [Li] [Ll] [M1] [M2] [MS] [Se] [Sh] [T] [Wa] [Wil] [Wi] [Y]
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Knill, E., Laflamme, R. and Zurek, W.: Threshold Accuracy for Quantum Computation. LANL e-print quant-ph/9610011, 20 pages, 10/15/96 Lickorish, W.: A representation of orientable, combinatorial 3-manifolds. Ann. Math. 76, 531–540 (1962) Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996) Mosher, L.: Mapping class groups are automatic. Ann. of Math. 142, 303–384 (1995) Mosher, L.: Hyperbolic extensions of groups. J. of Pure and Applied Alg. 110, 305–314 (1996) Moore, G. and Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989) Segal, G.: The definition of conformal field theory. Preprint (1999) Shor, P.W.: Algorithms for quantum computers: Discrete logarithms and factoring. Proc. 35th Annual Symposium on Foundations of Computer Science. Los Alamitos: IEEE Computer Society Press, CA: pp. 124–134 Turaev, V.G.: Quantum invariant of knots and 3-manifolds. de Gruyter Studies in Math., Vol. 18 Walker, K.: On Witten’s 3-manifold invariants. Preprint, 1991 Wilczek, F.: Fractional statistics and anyon superconductivity. Teaneack, NJ: World Scientific Publishing Co., Inc., 1990 Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989) Yao, A.: Quantum circuit complexity. Proc. 34th Annual Symposium on Foundations of Computer Science. Los Alamitos, CA: IEEE Computer Society Press, pp. 352–361
Communicated by P. Sarnak
Commun. Math. Phys. 227, 605 – 622 (2002)
Communications in
Mathematical Physics
© Springer-Verlag 2002
A Modular Functor Which is Universal for Quantum Computation Michael H. Freedman1 , Michael Larsen2 , Zhenghan Wang2 1 Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA 2 Indiana University, Dept. of Math., Bloomington, IN 47405, USA
Received: 4 May 2001 / Accepted: 18 February 2002
Abstract: We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor’s state space. A computational model based on Chern–Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation has topological implications which will be considered elsewhere.
1. Introduction
The idea that computing with quantum mechanical systems might offer extraordinary advantages over ordinary “classical” computation has its origins in independent writings of Benioff [B], Manin [M] and Feynman [Fey]. Feynman explained that local “quantum gates”, the basis of his model, can efficiently simulate the evolution of any finite dimensional quantum system evolving under a local Hamiltonian Ht and by extension any renormalizable system. The details of this argument are (much clarified) in [Ll]. Topological quantum field theories (TQFTs), although possessing a finite dimensional Hilbert space, lack a Hamiltonian – the derivative of time evolution on which the Feynman– Lloyd argument is based. In [FKW], we provide a different argument for the poly-local nature of TQFTs showing that quantum computers efficiently simulate these as well. Here we give a converse to this simulation result. The Feynman–Lloyd argument is reversible, so we may summarize the situation as:
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(1) finite dimensional local1 quantum systems. (2) quantum computers (meaning the quantum circuit model QCM [D, Y]), (3) certain topological modular functors (TMFs). Each can efficiently simulate the others. We wrote TMF above instead of TQFT as a matter of notation because we use only the conformal blocks and the action of the mapping class groups on these – not the general morphisms associated to 3-dimensional non-product bordisms. The two dimensional aspects of a (2 + 1)-dimensional TQFT are referred to as a TMF. 2. A Universal Quantum Computer The strictly 2-dimensional part of a TQFT is called a topological modular functor (TMF). The most interesting examples of TMFs are given by the SU(2) Witten–Chern–Simons theory at roots of unity [Wi]. These examples are mathematically constructed in [RT] using quantum groups (see also [T, Wa]). A modular functor assigns to a compact surface (with some additional structures detailed below) a complex vector space V () and to a diffeomorphism of the surface (preserving structures) a linear map of V (). In the cases considered here V () always has a positive definite Hermitian inner product , h and the induced linear maps preserve , h , i.e. are unitary. The usual additional structures are fixed parameterizations of each boundary component, a labeling of each boundary component by an element of a finite label set L with an involution ˆ : L → L, and a Lagrangian subspace L of H1 (, Q) ([T, Wa]). Since our quantum computer is built from quantum-SU(2)-invariants of braiding, and the intersection pairing of a planar surface is 0, L = H1 (; Q) and can be ignored. The parameterization of boundary components can also be dropped at the cost of losing the overall phase information in the system which in any case is not physical. Mathematically this means that all unitaries should be regarded as projective. In three dimensional terms, this parameterization becomes the framing of a “Wilson” loop and is essential to well definedness of the phase of the Jones–Witten invariants. In our context it may be neglected. The involution ˆ is simply the identity since the SU(2)-theory is self-dual. In fact, we can manage by only 2π i considering the SU(2)-Chern–Simons theory at q = e r , r = 5 and so our label set will be the symbols {0, 1, 2, 3} which are the quantum group analogs of the 0th , 1st , 2nd , and 3rd symmetric powers of the fundamental representation of SU(2) in C2 . Note that in our notation, 0 labels the trivial representation, not 1. Since we are suppressing boundary parameterizations, we may work in the disk with n marked points thought of us crushed boundary components. Because we only need the “uncolored theory” to make a universal model, each marked point is assigned the label 1, and the boundary of the disk is assigned the label 0. We consider the action of the braid group B(n) which consists of diffeomorphisms of the disk which leave the n marked points and the boundary set-wise invariant modulo those isotopic to the identity leaving all marked points fixed. The braid group has the well-known presentation: B(n) = {σ1 , . . . , σn−1 | σi σj σi−1 σj−1 = id if |i − j | > 1 σi σj σi = σj σi σj if |i − j | = 1}, where σi is the half right twist of the i th marked point about the i + 1st marked point. 1 Local refers to the ubiquitous physical assumption that the Hamiltonian contains only k-body terms for k ≤ some fixed n. Note that such Hamiltonians well approximate lattice models with interactions which decay exponentially.
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To describe a fault-tolerant computational model “Chern–Simons 5” CS5, we must deal with the usual errors arising from decoherence as well as a novel “qubit smearing error” resulting from imbedding the computational qubits within a modular functor super-space. To explain our approach we initially ignore all errors; in particular formula (1) below is a simplification valid only in the error-free context. In fact, it is within the bounds of physical realism to study “Exact Chern–Simons 5” ECS 5, a model in which it is assumed that no errors occur in the implementation of the Jones representation from the braid group to the modular functor V . This may seem strange given that the major focus of the field of quantum computation has, since 1995, been on fault tolerance. The point is that topology represents a potential alternative path toward computational stability. Topology can confer physical error correction where the traditional approach within qubit models is a kind of software error correction. By definition topological structures, such as braids, are usually discrete so small variations do not risk confusing one type with another. The idea that the discreteness in topology can be used to protect quantum information first appears in [Ki1], though not yet in the context of a computational model. In that paper Kitaev uses perturbation theory to calculate an exponential decay, proportional to e−const. L , L a length scale, in the probability of one important source of error (tunneling of virtual excitations). Thus “ECS 5 computation” might be implemented in practice by adjusting the length scale L (in this context the distance at which punctures − physically anyons − must be kept separated) by a factor polylogrithmic in computation length. Perhaps a more likely implementation would be a hybrid scheme in which topology is used to reach the rather demanding threshold [P] required for software error correction. In this case modular functors and the usual theory of fault tolerance must be fitted together. This is possible using the perspective in [AB] and an argument for this sketched within the proof of Thm. 2.2. However, a comprehensive discussion of the interaction of the environment with topological degrees of freedom, and how computational stability can be achieved in this context is beyond the scope of this article. In fact recent work [AHHH] suggests that earlier interaction models which assume an uncorrelated environment may be too naive. We expect that the best framework for this discussion has not yet been constructed. The state space Sk = (C2 )⊗k of our quantum computer consists of k qubits, that is the disjoint union of k spin= 21 systems which can be described mathematically as the tensor product of k copies of the state space C2 of the basic 2-level system, i
C2 = span(|0, |1). For each even integer k, we will choose an inclusion Sk → V (D 2 , 3k marked points) = V (D 2 , 3k) and show how to use the action of the braid group B(3k) on the modular functor V to (approximately) induce the action of any poly-local unitary operator U : Sk → Sk . That is we will give an (in principle) efficient procedure for constructing a braid b = b(U) so that i ◦ U = V (b) ◦ i.
(1)
To see that this allows us to simulate the QCM, we need to explain: (i) what we mean by the hypothesis “poly-local” on U, (ii) what “efficient” means, (iii) what the effect of the two types of errors are on line (1), and (iv) what measurement consists of within our model. We begin by explaining how to map Sk into V and how to perform 1 and 2 qubit gates. Let D be the unit 2-dimensional disk and 12 13 21 22 23 10k + 1 10k + 2 10k + 3 11 , , , , , ,... , , , 100k 100k 100k 100k 100k 100k 100k 100k 100k
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be a subset of 3k marked points on the x-axis. Without giving formulae the reader should picture k disjoint sub-disks Di , 1 ≤ i ≤ k, each containing one clump of 3 marked points in itsinterior (these will serve to support qubits in a manner explained k below) and further disks Di,j , 1 ≤ i < j ≤ k, containing Di and Dj , but with 2 Dij ∩ Dl = ∅, l = i or j (which will allow 2-qubit gates). Strictly speaking, among the larger subdisks, we only need to consider Di,i+1 , 1 ≤ i, i + 1 < k, and could choose a standard (linear) arrangement for these but there is no cost in the exposition to considering all Di,j above which will correspond in the model to letting any two qubits interact. Also, curiously, we will see that any of the numerous topologically distinct arrangements for the {Di,j } within D may be selected without prejudice. Restricting to q = e2πi/5 , define Vkl to be the SU(2) Hilbert space of k marked points in the interior with labels equal 1 and l label on ∂D. We need to understand the many ways in which Vm0 arises via the “gluing axiom” ([Wa]) from smaller pieces. The axiom provides an isomorphism: V (X ∪γ Y ) ∼ = ⊕all consistent labelings l V (X, l) ⊗ V (Y, l),
(2)
where the notation has suppressed all labels not on the 1-manifold γ along which X and Y are glued. The sum is over all labelings of the components of γ satisfying the conditions that matched components have equal labels. According to SU(2)-Chern– Simons theory [KL], for three-punctured spheres with boundary labels a, b, c, the Hilbert space Vabc ∼ = C if (i) a + b + c = even, (ii) a ≤ b + c, b ≤ a + b, c ≤ a + b (triangle inequalities), (iii) a + b + c ≤ 2(r − 2);
(3)
and Vabc ∼ = 0 otherwise. The gluing axiom together with the above information allows an inductive calculation of Vkl , where the superscript denotes the label on ∂D. We easily calculate that dimV31 = 2, dimV33 = 1, dimV60 = 5, dimV62 = 8.
(4)
Line (4) motivates taking V (Di , its 3 marked points and boundary all label 1) =: Vi ∼ = C2 as our fundamental unit of computation, the qubit. Note that when V has only a lower index, 1 ≤ i ≤ k, it denotes the qubit supported in the disk Di . We fix the choice of k
k
an arbitrary “complementary vector” v in the state space of D\ ∪ Di v ∈ V (D\ ∪ Di , i=1
i=1 Vcomplement (To
all boundary labels = 1 except the label on the boundary of D is 0) =: keep this space nontrivial, we have taken k even.) Using v, the gluing axiom defines an injection: k
⊗v iv : (C2 )⊗k ∼ = ⊗ Vi → i=1
k ⊗ Vi ⊗ Vcomplement
i=1
as summand
→
0 . V3k
(5)
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This composition iv determines the inclusion of the computational qubits within the 0 . Observe in the calculation of line (9) below that the complementary modular functor V3k vector v will evolve to different v but this will be irrelevant to the measurement which is made at the end of the computation. The reader familiar with [FKW] will notice that we use here a dual approach. In that paper, we imbedded the modular functor into a larger Hilbert space that is a tensor power; here we imbedded a tensor power into the modular functor. The action of B(3) on Di yields 1-qubit gates, whereas two qubit gates will be constructed using the action of the six strand braid group B(6) on Di,j . Supposing our quantum computer Sk is in state s, a given v as above determines a state iv (s) = s ⊗ v ∈ 0 . Now suppose we wish to evolve s by a 2-qubit gate g ∈ P U (4) acting unitarily on V3k 2 Ci ⊗ C2j and by id on C2l , l = i or j . Using the gluing axiom (2) and the inclusion (5), we may write s=
th ⊗ uh ,
(6)
h
where {th } is a basis or partial basis for Vi ⊗ Vj ∼ = C2i ⊗ C2j and uh ∈ ⊗l=i,j C2l ,
so s ⊗ v = h (th ⊗ uh ) ⊗ v. Decomposing along γ = ∂D i,j , we may write v = α0 ⊗ β0 + α2 ⊗ β2 , where α. ∈ V Di,j \(D i ∪ Dj ), . on γ , . = 0 or 2 and β. ∈ V D\(∪l=i,j Dl ∪ Dij ), . on γ , and 0 on ∂D . Thus s⊗v =
th ⊗ u h ⊗ α 0 ⊗ β 0 +
h
th ⊗ uh ⊗ α2 ⊗ β2 .
(7)
h
An element of B(6) applied to the 6 marked points in Di ∪ Dj ⊂ Dij acts via a representation ρ 0 ⊕ ρ 2 =: ρ on V 0 (Dij , 6 pts) ⊕ V 2 (Dij , 6 pts), where the superscript denotes the label appearing when the surface is cut along γ . In particular B(6) acts on each factor th ⊗ α0 and th ⊗ α2 in (7). Note th ⊗ α 0 belongs to the summand of V 0 (Dij , 6 pts) corresponding to boundary labels on ∂ Dij \(Di ∪ Dj ) = 0, 1, 1. There is an additional 1-dimensional summand corresponding to boundary labels 0,3,3with 0,1,3 and 0,3,1 excluded by the triangle inequality (ii) in (3) above. Similarly th ⊗ α2 belongs to the summand of V 2 (Dij , 6 pts) with boundary labels=2,1,1. There are additional summands corresponding to (2,1,3), and (2,3,1) of dimensions 2 each. Ideally we would find a braid b = b(g) ∈ B(6) so that ρ 0 (b)(th ⊗ α0 ) = gth ⊗ α0 and ρ 2 (b)(th ⊗ α2 ) = gth ⊗ α2 . Then referring to (7) we easily check that ρ(b)(s ⊗ v) =
(gth ) ⊗ uh ⊗ v,
(8)
h
i.e. ρ(b) implements the gate g on the state space Sk of our quantum computer. In practice there are two issues: (i) we cannot control the phase of the output of either ρ 0 or ρ 2 , and (ii) these outputs will be only approximations of the desired gate g. The phase issue (i) leads to a change of the complimentary vector v → v as follows as seen on line (9) below. This is harmless since ultimately we only measure the qubits.
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s⊗v =
th ⊗ uh ⊗ α0 ⊗ β0 +
h
th ⊗ u h ⊗ α 2 ⊗ β 2
h
⇓ gate ρ(b)(s ⊗ v) = ω0 =
gth ⊗ uh ⊗ α0 ⊗ β0 + ω2
h
ω0 gth ⊗ uh ⊗ α0 ⊗ β0 +
h
=
gth ⊗ uh ⊗ α2 ⊗ β2
h
ω2 gth ⊗ uh ⊗ α2 ⊗ β2
h
(gth ⊗ uh ) ⊗ (ω0 α0 ⊗ β0 + ω2 α2 ⊗ β2 ) h
(gth ⊗ uh ) ⊗ v . =:
(9)
h
The approximation issue is addressed by Theorem 2.1 below. Theorem 2.1. There is a constant C > 0 so that for any positive . and for all unitary g : C2i ⊗ C2j → C2i ⊗ C2j , there is a braid bl of length ≤ l in the generators σi and their inverses σi−1 , 1 ≤ i ≤ n − 1, so that:
ω0 ρ 0 (bl ) − g ⊕ id1 + ω2 ρ 2 (bl ) − g ⊕ id4 ≤ . for some unit complex numbers (phases) ωi , i = 0, 2 whenever . satisfies k l ≤ C · log(1/.) for k ≥ 2.
(10)
(11)
We use to denote the operator norms and the subscripts on id indicate the dimension of the orthogonal component in which we are trying not to act. Proof. The main work in proving Theorem 2.1 is to show that the closure of the image of the representation ρ : B(6) → U(5) × U(8) contains SU(5) × SU(8). Once this is accomplished the estimate (10) follows with some exponent ≥ 2 from what is called the Solovay–Kitaev theorem [So, Ki2, KSV]. This is a rapid effective approximation theorem originally established in SU(2) with an exponent > 2 but in the last reference proved in SU(n) for all n, with same exponent k ≥ 2. Also by [KSV] there is a log2 (1/.) time classical algorithm which can be used to construct the approximating braid bl as a word in {σi } and {σi−1 }. The action ρ(b) “approximately” executes the gate g on Sk but not in the usual sense of approximation since the image of the state space ρ(b) (iv (Sk )) is only approximately iv (Sk ). This impression in the location of the computational qubits within a larger Hilbert space can be called “smearing”. We convert this “smearing of qubits” to errors of the type usually considered in the fault tolerant literature. After each g is approximately executed k
by ρ(b) we measure the labels around U ∂Di to project the new state ρ(b)(s ⊗ v) i=1
into the form s ⊗ v , s ∈ Sk , with probability 1 − O(. 2 ), |s − s| ≤ O(.). With probability O(. 2 ) the label measurement around ∂Di does not yield one; in this case V 1 (Di ; 3 pts.) ∼ = V31 ∼ = C2 has collapsed to V33 ∼ = C and it is as if a qubit has been
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“traced out” of our state space. More specifically, if the label 3 is measured on ∂Di , we replace V 3 (Di , its 3 marked pts.) with a freshly cooled qubit V 1 (Di , 3 pts.) with (say) a completely random initial state which we have been saving for such an occasion. The reader may picture dragging Di off to the edge of the disk D and dragging the ancillae D in as its replacement (and then renaming D as Di .) The hypothesis that such ancillae are available is discussed below. The error model of [AB] is precisely suited to this situation; Aharanov and Ben-Or show in Chapter 8 that a calculation on the level of “logical” qubits can be kept precisely on track with a probability ≥ 23 provided the ubiquitous errors at the level of “physical” qubits are of norm ≤ O(.) (even if they are systematic and not random) and the large errors (in our case tracing a qubit) have probability also ≤ O(.) for some threshold constant . > 0. For this, and all other fault tolerant models, entropy must be kept at bay by ensuring a “cold” stream of ancillary |0’s. In the context of our model we must now explain both the role of measurement and ancilla. Given any essential simple closed curve γ on a surface , the gluing formula reads: V () = ⊕l∈L V (cutγ , l)
(12)
so “measuring a label” means that we posit for every γ a Hermitian operator Hγ with eigenvalues distinguishing the summands of the r.h.s. of (12) above. For a more comprehensive computational study, we would wish to posit that if γ has length = L, then Hγ can be computed in poly(L) time. For the present purpose we only need that Hγ , γ = ∂Di or ∂Di,j can be computed in constant time. Beyond measuring labels, we hypothesize that there is some way of probing the quantum state of the smallest 1 ∼ C2 , nontrivial building blocks in the theory. For us these are the k qubits = V3,i = i 1 ≤ i ≤ k, where the index i refers to the qubit supported in Di . Fix a basis {|0, |1} for V31 and posit for each Di , 1 ≤ i ≤ k, with label 1 on its boundary, an observable 1 0 1 → V 1 which acts as the Pauli matrix Hermitian operator σzi : V3,i in the fixed 3,i 0 −1 i basis {|0, |1} for that qubit. In concrete terms, this Pauli operator σz has eigen vectors |0 and |2, where 0 and 2 are the two possible labels which can appear on the simple closed curve αi ⊂ Di which separates exactly two of the three punctures from ∂Di . The Pauli matrix σzi might be implemented by first fusing a pair of the punctures in Di and then measuring the resulting particle type. This then is our repertoire of measurement: Hγ is used to “unsmear physical qubits” after each gate and the σz ’s to read out the final state (according to the usual “von Neumann” statistical postulate on measurement) after the computation is completed. In fault tolerant models of computation it is essential to have available a stream of “freshly cooled” ancillary qubits. If these are present from the start of the computation, even if untouched, they will decohere from errors in employing the identity operator. In the physical realization of a quantum computer, unless stored zeros were extremely stable there would have to be some device (inherently not unitary!) for resetting ancillae to |0, e.g. a polarizing magnetic field. As a theoretical matter, unbounded computation requires such resetting. As discussed near the beginning of this section, in a topological model such as V () it is not unreasonable to postulate that |0 ∈ V31 = V 1 (Di , 3 pts.) is stable if not involved in any gates. An alternative hypothesis is that there is some mechanism outside the system analogous to the polarizing magnetic field above which can “refrigerate” ancillae in the state |0 until they are to be used. We refer below to either of these as the “fresh ancilli” hypothesis. To correct the novel qubit smearing errors, we already encountered the need for ancilli which we took to be an easily maintained random
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1 2
0
. Other uses of ancilli within fault tolerant schemes require a known 0 21 pure state |0. Let us now return to line (1). Let U be the theoretical output of a quantum circuit C of (i.e. composition of) gates to be executed on the physical qubit level so as to faulttolerantly solve a problem instance of length n. We assume the problem is in BQP and that the above composition has length ≤ poly(n). Actually, due to error, C will output a completely positive trace preserving super-operator O, called a physical operator. Now simulate C in the modular functor V a gate at a time by a succession of braidings and Hγ -measurements. With regard to parallelism (necessary in all fault tolerant schemes), notice that disjoint 2 qubit gates can be performed simultaneously if Di,j ∩ Di ,j = ∅. For example this can always be arranged in the linear QCM for gates acting in Di,i+1 and Dj,j +1 provided i + 1 = j, j + 1 = i, and i = j , and even this model is known to be fault tolerant [AB]. From line (9), the complementary vector v ∈ Vcomplement evolves probabilistically as the simulation progresses. Different v’s will occur as a tensor factor in a growing number of probabilistically weighted terms. However, the various v − factors are in the end inconsequential; they simply label a computational state (to be observed with some probability) and are never read by the output measurements σzi . We fix terminology and state the main theorems. QCM denotes the exact quantum circuit model. It is known that a quantum circuit operating in the presence of certain kinds of error can still simulate an exact QCM with only polylogrithmic cost in space and time. The basic error model permits gate error of arbitrary super−operator norm (to include identity gates) at some low rate, e.g. . ≈ 10−6 per operation site, but demands independence. This error model is enlarged (while retaining efficient simultability) in two ways in [AB] which are important to use here. First (see line 2.6 [AB]), as long as the probability of these arbitrary errors, which include tracing a qubit, is dominated by the independent case along the “fault-path” correlations are permitted. Second small systematic errors are permitted everywhere in the model provided they are small enough, e.g. unitaries may have systematic error of, again, about one part in 10−6 . Let BQP denote the class of decision problems which can be solved with probability ≥ 43 by an exact quantum circuit designed by a classical algorithm in time poly(L), where L is the length of the problem instance M. This same class can be solved in poly-time by a (slightly) error-prone QC. Let CS5 denote the model of computation described in this section. It is based on the Chern–Simons theory of SU(2) at the fifth root of unity q = e2πi/5 . We review its structure here; a list of generating “braid gates” is given in Sect. 3. The functor 1 , it contains k-qubits, i : S → V 1 and can be assigned a is the Hilbert space V3k v k 3k standard initial state α ∈ iv (Sk ). The 3k-strand braid group B(3k) acts unitarily by ρ 1 and a classical poly-time algorithm converts a circuit C in the QCM to a word in on V3k B(3k). Note that the braid group can be implemented in parallel (most of it generators commute) in imitation of that essential feature of quantum circuits. The model has two kinds of measurements Hγ and σzi , but only the later is allowed in the exact version of the model ECS5. In CS5 we envision access to “fresh ancilli”, in ECS5 there is 1 no need for these. The action ρ(b) of the braid b produces an evolution of α ⊗ v.V3k i to a probabilistic mixture of states γl = αl ⊗ vl with probability pl . Performing σz measurements 1 ≤ i ≤ k, then samples γl and observes only the αl factor. Classical poly (L)-time post−processing of these k observations can be permitted in the model but equivalently this step can be folded back into the quantum circuit phase to make the observation of σz1 on the first qubit the one and only read-out.
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Without error-correction no model ECS5 included can compute for very long if subjected errors of any constant size or probability > 0. However we explicitly assume that CS5 faces the kinds of environmental error analyzed in [AB] in addition to its intrinsic “gate errors” (from the approximate output of the Solovay–Kitaev theorem) and qubit smearing errors inherent in the model. Specifically for some small δ > 0 permit (1) δ-small systematic errors in each operation σi± or identity and (2) a probability of large environmental errors, which is dominated by the probability of independent individual errors of probability < δ each. Theorem 2.2. Given a problem in BQP and an instance M of length L a classical poly-time algorithm can convert the quantum circuit C for M into a braid b.B(3k). 1 and measuring σ 1 will correctly solve M with probability Implementing ρ(b) on V3k z 3 ≥ 4 . The number of marked points to be braided space (= 3k) and the length of the braiding exceed the size of the original circuit C by at most a multiplicative poly(log(L)) factor. Taken in triples, the points support represent the “physical qubits” of the [AB] fault tolerant model. Thus CS5 provides a model which efficiently and fault tolerantly simulates the computations of QCM. We note that the use of label measurements Hγ introduces non-unitary steps in the middle of our simulation. As usual the probability 3 4 is independent w.r.t trials and so converges exponentially to 1 upon repetition of the entire procedure. Proof. The proof relies heavily on Chapter 8 [AB] to reduce the QCM to a linear quantum circuit (with state space Sk ) stable under a very liberal error model – one permitting small systematic errors plus
rare large but uncorrelated qubit errors or trace over a qubit. In the final state γ = pl γl , each γl admits a tensor decomposition according to the geometry: D = (∪i Di ) ∪ (complement), but along the k boundary components ∪i ∂Di all choices of labels 1 or 3 may appear. In writing βl = αl ⊗ vl we must remember that associated to l is an element [l] ∈ {1, 3}k which defines the subspace [l]-sector, of the modular functor in which γl lies. All occurrences of the label 3 correspond to a C tensor factor, C ∼ = V33 ∼ = V 3 (Di , 3 pts) ⊂ V (Di , 3 pts) whereas the label 1 corresponds to a 2 C factor. Thus in the [AB] framework each label 3 corresponds to a “lost” or according to our replacement procedure Di ←→ D , a traced qubit. (Losing an occasional qubit from the computational space Sk is the price we pay to “unsmear” Sk within the modular functor.) Theorem 2.1 implies that for a braid length = O( .12 ) a qubit will be traced with probability O(. 2 ) and if no qubit is lost the gate will be performed with error O(.) on pure states. Factoring a mixed state as a probabilistic combination of pure states and passing the error estimate across the probabilities we see that for δ > 0 sufficiently small, the O(.) error bound holds with high probability on the observed γ6 . Thus for . sufficiently small (estimated ≈ 10−6 [AB]), observing αl amounts to sampling from an error prone implementation of the quantum circuit C. The error model is not entirely random in that the approximation procedure used to construct b will have systematic biases. This implies that the O(.) errors introduced in the functioning of each gate are not random and must be treated as “malicious”. The error model explained in Chapter 8 [AB] permits such small errors to be arbitrary as long as the large error, e.g. qubit losses, occurs with a probability dominated by a small constant independent of the qubit and the computational history. This is consistent with the assumptions on the CS5 model. This completes the proof of Theorem 2.2 modulo the proof of the density Theorem 4.1. We now turn to the exact variant ECS5, in which we assume that all the braid groups act exactly (no error) on the modular functor V . The only difference in the algorithm
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for modeling the QCM in ECS5 is the simplification that Hγ measurements are not performed in the middle of the simulation, but only at the very end, prior to reading out the qubits Sk with σzk measurements. Theorem 2.3. There is an efficient and strictly unitary simulation of QCM by ECS5. Thus given a problem instance M of length L in BQP, there is a classical poly(L) time algorithm for constructing a braid b as a word of length poly(L) in the generators σi , 1 ≤ i ≤ polyL). Let k be another polynomial function of L. Applying b to a standard initial state, ψinitial ∈ V 0 (D, 3k), results in a state ψfinal ∈ V 0 (D, 3k), so that the results of Hγ on ∂Di followed by σzi measurements on ψfinal correctly solve the problem instance M with probability ≥ 43 . Proof. In the quantum circuit C for M (implied by the problem lying in BQP) count the number n of gates to be applied. Use line (11) to approximate each gate g by a braid b of length l so that the operator norm error ||ρ(b) − g|| of the approximating gate will be less than .n−1 , for some fixed . > 0. The composition of n braids which gate-wise simulate the quantum circuit introduces an error on operator norm < .. It follows that the approximation of the desired unitary by the braid results in a 8final so that the absolute angle | < (8final , 8final )| ≤ 2 arcsin 2. . The application of our two measurement steps will therefore return an answer nearly as reliable as the original quantum circuit C: The probability ρ that the sequential measurements Hγ and σz1 (which is defined if and only if Hγ projects to V 1 (D, 3pts.)) will give different results for 8final and 8final is ≤ sin 2 arcsin 2. < .. So with probability 1 − p > 1 − . the final measurement |0 or |1 will be the same in the quantum circuit C and the ECS5 model. Remark. Theorem 2.2 and 2.3 are complementary. One provided additional fault tolerance – fault tolerance beyond what might be inherent in a topological model – but at the cost of introducing intermediate non-unitary steps (i.e. measurements). The other eschews intermediate measurements and so gives a strictly unitary simulation, but cannot confer additional fault tolerance. It is an interesting open technical problem whether fault tolerance and strict unitarity can be combined in a universal model of computation based on topological modular functors. Looking ahead to a possible implementation, however, intermediate measurements as in the fault tolerant model do not seem undesirable. 3. Jones’ Representation of the Braid Groups A TMF gives a family of representations of the braid groups and mapping class groups. In this section, we identify the representations of the braid groups from the SU(2) modular functor at primitive roots of unity with the irreducible sectors of the representation discovered by Jones whose weighted trace gives the Jones polynomial of the closure link of the braid [J1, J2]. To prove universality of the modular functor for quantum computation, we only use this portion of the TMF. Therefore, we will focus on these representations. First let us describe the Jones representation of the braid groups explicitly following [We]. To do so, we need first to describe the representation of the Temperley-Lieb2π i Jones algebras Aβ,n . Fix some integer r ≥ 3 and q = e r . Let [k] be the quantum integer defined as [k] =
k
−k 2 1 −1 q 2 −q 2
q 2 −q
1
. Note that [−k] = −[k], and [2] = q 2 + q
−1 2
. Then
β := [2]2 = q + q¯ + 2 = 4cos 2 ( πr ). The algebras Aβ,n are the finite dimensional C ∗ -algebras generated by 1 and projectors e1 , · · · , en−1 such that
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1. ei2 = ei , and ei∗ = ei , 2. ei ei±1 ei = β −1 ei , 3. ei ej = ej ei if |i − j | ≥ 2, ∞
and there exists a positive trace tr : U Aβ,n → C such that tr(xen ) = β −1 tr(x) for all n=1
x ∈ Aβ,n . The Jones representation of Aβ,n is the representation corresponding to the G.N.S. construction with respect to the above trace. An important feature of the Jones representation is that it splits as a direct sum of irreducible representations indexed by some 2-row Young diagrams, which we will refer to as sectors. A Young diagram λ = [λ1 , . . . , λs ], λ1 ≥ λ2 ≥ · · · ≥ λs is called a (2, r) diagram if s ≤ 2 (at most (2,r) denote all (2, r) diagrams with n nodes. two rows) and λ1 − λ2 ≤ r − 2. Let ∧n (2,r) (2,r) be all standard tableaus {t} with shape λ satisfying the inGiven λ ∈ ∧n , let Tλ ductive condition which is the analogue of (iii) in (3): when n, n−1, . . . , 2, 1 are deleted from t one at a time, each tableau appeared is a tableau for some (2, r) Young diagram. (2,r) over all The representation of Aβ,n is a direct sum of irreducible representations πλ (2,r) for a fixed (2, r) Young diagram λ is (2, r) Young diagrams λ. The representation πλ (2,r) (2,r) be the complex vector space with basis {v t , t ∈ Tλ }. Given given as follows: let Vλ (2,r) a generator ei in the Temperley–Lieb–Jones algebra and a standard tableau t ∈ Vλ . Suppose i appears in t in row r1 and column c1 , i + 1 in row r2 and column c2 . Denote
[dt,i +1] by dt,i = c1 − c2 − (r1 − r2 ), αt,i = [2][d , and βt,i = αt,i (1 − αt,i ). They are both t,i ] 2 + β 2 . Then we define non-negative real numbers and satisfy the equation αt,i = αt,i t,i (2,r)
πλ
(ei )(v t ) = αt,i v t + βt,i v gi (t) ,
(13) (2,r)
where gi (t) is the tableau obtained from t by switching i and i + 1 if gi (t) is in Tλ . (2,r) If gi (t) is not in Tλ , then αt,i is 0 or 1 given by its defining formula. This can occur (2,r) in several cases. It follows that πλ with respect to the basis {v t } is a matrix consisting of only 2 × 2 and 1 × 1 blocks. Furthermore, the 1 × 1 blocks are either 0 or 1, and the 2 × 2 blocks are αt,i βt,i . (14) βt,i 1 − αt,i 2 + β 2 implies that (14) is a projector. So all eigenvalues of e The identity αt,i = αt,i i t,i are either 0 or 1. The Jones representation of the braid groups is defined by
ρβ,n (σi ) = q − (1 + q)ei .
(15)
Combining (15) with the above representation of the Temperley–Lieb–Jones algebra, we get Jones’ representation of the braid groups, denoted still by ρβ,n : ρβ,n : Bn → Aβ,n → U(Nβ,n ),
(2,r) where the dimension Nβ,n = λ∈∧(2,r) dimVλ grows asymptotically as β n . n When |q| = 1, as we have seen already, Jones’ representation ρβ,n is unitary. To verify that ρ(σi )ρ ∗ (σi ) = 1, note ρ ∗ (σi ) = q¯ − (1 + q)e ¯ i∗ . So we have ρ(σi )ρ ∗ (σi ) =
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q q¯ + (1 + q)(1 + q)e ¯ i ei∗ − (1 + q)ei − (1 + q)e ¯ i∗ = 1. We use the fact ei∗ = ei and ei2 = ei to cancel out the last 3 terms. From the definition, ρβ,n also splits as a direct sum of representations over (2, r)Young diagrams. A sector corresponding to a particularYoung diagram λ will be denoted by ρλ,β,n . Now we collect some properties about the Jones representation of the braid groups into the following: Theorem 3.1. (i) For each (2, r)-Young diagram λ, the representation ρλ,β,n is irreducible. (ii) The matrices ρλ,β,n (σi ) for i = 1, 2 generate an infinite subgroup of U(2) modulo center for r = 3, 4, 6, 10. (iii) Each matrix ρλ,β,n (σi ), 1 ≤ i ≤ n − 1, has exactly two distinct eigenvalues −1, q. (iv) For the (2,5)-Young diagram λ = [4, 2], n = 6, the two eigenvalues −1, q of every ρλ,β,6 (σi ) have multiplicity of 3 and 5 respectively. The proofs of (i) and (ii) are in [J2]. For (iii), first note that the matrix ρλ,β,n (σ1 ) is a diagonal matrix with respect to the basis {v t } with only two distinct eigenvalues −1, q. Now (iii) follows from the fact that all braid generators σi are conjugate to each other. For (iv), simply check the explicit matrix for ρλ,β,6 (σ1 ) at the end of this section. Now we identify the sectors of the Jones representation with the representations of the braid groups coming from the SU(2) Chern–Simons modular functor. The SU(2) Chern–Simons modular functor CSr of level r has been constructed several times in the literature (for example, [RT, T, Wa, G]). Our construction of the modular functor CSr is based on skein theory [KL]. The key ingredient is the substitute of Jones–Wenzl idempotents for the intertwiners of the irreducible representations of quantum groups [RT, T, Wa]. This is the same SU(2) modular functor as constructed using quantum groups in [RT] (see [T]) which is regarded as a mathematical realization of the Witten–Chern– Simons theory. All formulae we need for skein theory are summarized in Chapter 9 of √ 2π i [KL] with appropriate admissible conditions. Fix an integer r ≥ 3. Let A = −1·e− 4r , 2 4 and s = A , and q = A . (Note the confusion caused by notations. The q in [KL] is A2 which is our s here. But in Jones’ representation of the braid groups [J2], q is A4 . In all formulae in [KL], q should be interpreted as s in our notation.) The label set L of the modular functor CSr will be {0, 1, . . . , r − 2} and the involution is the identity. We are interested in a unitary modular functor and the one in [G] is not unitary. We claim that if we follow the same construction of [G] using our choice of A and endow all state spaces of the modular functor with the following Hermitian inner product, the resulting modular functor CSr is unitary. The relevant Hilbert space structure has also been constructed earlier by others (e.g. in [KS, KSVo]). Given a surface , a pants decomposition of determines a basis of V (): each basis element is a tensor product of the basis elements of the constituent pants. The desired inner products are determined by axiom (2.14) [Wa] if we specify an inner product on each space Vabc . Our choice of A makes all constants S(a) appearing in the axiom (2.14) [Wa] positive. Consequently, positive definite Hermitian inner products on all spaces Vabc determine a positive definite Hermitian inner product on V (). The vector space Vabc of the three punctured sphere Pabc is defined to be the skein space of the disk Dabc enclosed by the seams of the punctured sphere Pabc . The numbering of the three punctures induces a numbering of the three boundary “points” of the disk Dabc labeled by {a, b, c}. Suppose t is a tangle on Dabc in the skein space of Dabc , and let t¯ be the tangle on Dabc obtained by reflecting the disk Dabc through the first
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boundary point and the origin. Then the inner product , h : Vabc × Vabc → C is as follows: given two tangles s and t on Dabc , their product s, th is the Kauffman bracket evaluation of the resulting diagram on S 2 obtained by gluing the two disks with s and t¯ on them respectively, along their common boundaries with matching numberings. Extending , h on the skein space of Dabc linearly in the first coordinate and conjugate linearly in the second coordinate, we obtain a positive definite Hermitian inner product on Vabc . It is also true that the mapping class groupoid actions in the basic data respect this Hermitian product, and the fusion and scattering matrices F and S also preserve this product. So CSr is indeed a unitary modular functor. This modular functor CSr defines representations of the central extension of the mapping class groups of labeled extended surfaces, in particular for n-punctured disks Dnm with all interior punctures labeled 1 and boundary labeled m. If m = 1, then the mapping class group is the braid group Bn . If m = 1, then the mapping class group is the spherical braid group SBn+1 = M(0, n + 1). Recall that we suppress the issues of framing and central extension as they are inessential in our discussion. Also the representation of the mapping class groups coming from CSr will be denoted simply by ρr . Theorem 3.2. Let Dnm be as above. (1) If m + n is even, and m = 1, then ρr is equivalent to the irreducible sector of the m−n Jones representation ρλ,β,n for the Young diagram λ = [ m+n 2 , 2 ] up to phase. (2) If n is odd, and m = 1, then the composition of ρr with the natural map ι : Bn → SBn+1 is equivalent to the irreducible sector of the Jones representation ρλ,β,n for n−1 the Young diagram λ = [ n+1 2 , 2 ] up to phase. The equivalence of these two representations was first established in a non-unitary version [Fu]. A computational proof of this theorem can be obtained following [Fu]. So we will be content with giving some examples for r = 5. To get a universal set of gates using these matrices, all we need is to realize the Solovay-Kitaev theorem by an algorithm for any prescribed precision [KSV, NC]. For the (2, 5) Young diagram λ = [2, 1], n = 3 with an appropriate ordering of the basis: −1 0 ρ[2,1],β,3 (σ1 ) = , 0 q √ q2 [3] − qq+1 q+1 √ , where quantum [3] = q + q¯ + 1. ρ[2,1],β,3 (σ2 ) = [3] 1 − qq+1 − q+1 For the (2, 5) Young diagram λ = [3, 3], n = 6, the representation is 5-dimensional. With an appropriateordering of thebasis, we have: −1 q −1 ρ[3,3],β,6 (σ1 ) = , q q √ q2 q [3] − q+1 q+1 q √[3] 1 − q+1 − q+1 √ 2 q q [3] ρ[3,3],β,6 (σ2 ) = − q+1 . q+1 √ − q [3] − 1 q+1
q+1
q
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For the (2, 5) Young diagram λ = [4, 2], n = 6, the representation is 8-dimensional. Here the inductive condition on basis elements make one standard tableau illegal, so the representation is not 9-dimensional as it would be if r > 5. This is the restriction analogous to (iii) in (3) for the modular functor. With an appropriate ordering of the basis: −1 q −1 q ρ[4,2],β,6 (σ1 ) = . −1 q q q 4. A Density Theorem In this section, we prove the density theorem. Theorem 4.1. Let ρ := ρ[3,3] ⊕ ρ[4,2] : B6 → U(5) × U(8) be the Jones representation 2π i of B6 at the 5th root of unity q = e 5 . Then the closure of the image of ρ(B6 ) in U(5) × U(8) contains SU(5) × SU(8). By Theorem 3.2, this is the same representation ρ := ρ 0 ⊕ρ 2 : B6 → U(5)×U(8) in the SU(2) Chern–Simons modular functor at the 5th root of unity used in Sect. 2 to build a universal quantum computer. In the following, a key fact used is that the image matrix of each braid generator under the Jones representation has exactly two eigenvalues {−1, q} whose ratio is not ±1. This strong restriction allows us to identify both the closed image and its representation. Proof. First it suffices to show that the images of ρ[3,3] and ρ[4,2] contain SU(5) and SU(8), respectively. Supposing so, if K = ρ(B6 ) ∩ (SU(5) × SU(8)), then the two projections p1 : K → SU(5) and p2 : K → SU(8) are both surjective. Let N2 (respectively N1 ) be the kernel of p1 (respectively p2 ). Then N1 (respectively N2 ) can be identified as a normal subgroup of SU(5) (respectively SU(8)). By Goursat’s Lemma (p. 54, [La]), the image of K in SU(5)/N1 × SU(8)/N2 is the graph of some isomorphism SU(5)/N1 ∼ = SU(8)/N2 . As the only nontrivial normal subgroups of SU(n) are finite groups, this is possible only if N1 = SU(5) and N2 = SU(8). Therefore, K = SU(5) × SU(8). The proofs of the density for ρ[3,3] and ρ[4,2] are similar. So we prove both cases at the same time and give separate argument for the more complicated case ρ[4,2] when necessary. Let G be the closure of the image of ρ[3,3] (or ρ[4,2] ) in U(5) (or U(8)) which we will try to identify. By Theorem 3.1, G is a compact subgroup of U(m) (m = 5 or 8) of positive dimension. Denote by V the induced m-dimensional faithful, irreducible complex representation of G. The representation V is faithful since G is a subgroup of U(m). Let H be the identity component of G. What we actually show is that the derived group of H , Der(H ) = [H, H ], is actually SU(m). We will divide the proof into several steps. Claim 1. The restriction of V to H is an isotypic representation, i.e. a direct sum of several copies of a single irreducible representation of H .
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Proof. As G is compact, V = ⊕P VP , where P runs through some irreducible representations of H , and VP is the direct sum of all the copies of P contained in V . Since H is a normal subgroup, and the braid generators σi topologically generate G, the σi ’s permute transitively the isotypic components VP [CR, Sect. 49]. If there is more than 1 such component, then some σi acts nontrivially, so it must permute these blocks. Now we need a linear algebra lemma: Lemma 4.2. Suppose W is a vector space with a direct sum decomposition W = ⊕ni=1 Wi , and there is a linear automorphism T such that T : Wi → Wi+1 1 ≤ i ≤ n cyclically. Then the product of any eigenvalue of T with any nth root of unity is still an eigenvalue of T . Proof. Choose a basis of W consisting of bases of Wi , i = 1, 2, . . . , n. If k is not a multiple of n, then trT k = 0, as all diagonal entries are 0 with respect to the above basis. repeat.) Consider all values of trT m =
mLet {λi } be all eigenvalues of T . (They may λi (m = 1, 2, . . . ) which are sums of mth powers of all eigenvalues of T . These if we simultaneously multiply sums of mth powers of {λi } are invariant
mall the eigenvalues
m m m th root of unity ω: m {λi } by an λi which is equal to n (ωλ ) = ω λ = ω i i
m trT m = λi because when m is not a multiple of n, they are both 0, and when m is, ωm = 1. These values trT m uniquely determine the eigenvalues of T , and therefore the set of the eigenvalues of T is invariant under multiplication by any nth root of unity. Back to Claim 1, if there is more than one isotypic component, then some σi will have an orbit of length at least 2. It is impossible to have an orbit of length 3 or more by the above lemma as this will lead to at least 3 eigenvalues. If the orbit is of length 2 and as ρ(σi ) has only two eigenvalues {a, b}, by the lemma, {−a, −b} are also eigenvalues. It follows that a = −b which is impossible when q = −1. Claim 2. The restriction of V to H is an irreducible representation. Proof. By Claim 1, V |H has only one isotypic component. If V |H is reducible, then the isotypic component is a tensor product V1 ⊗V2 , where V1 is the irreducible representation of H in the isotypic component and V2 is a trivial representation of H with dimV2 ≥ 2. If V1 is 1-dimensional, then ρ(σi ), i = 1, 2 generate a finite subgroup of U(m) modulo center which is excluded by Theorem 3.1. So we have dimV1 ≥ 2. Now we recall a fact in representation theory: a representation of a group ρ : G → GL(V ) is irreducible if and only if the image ρ(G) of G generates the full matrix algebra End(V ). As V1 is an irreducible representation of H , the image ρ(H ) generates End(V1 ) ⊗ id2 , where the subscript of id indicate the tensor factor. As the elements σi normalize H , they also normalize the subalgebra End(V1 ) ⊗ id2 in End(V1 ⊗ V2 ). Consequently they act as automorphisms of the full matrix algebra End(V1 ). Any automorphism of a full matrix algebra is a conjugation by a matrix, so the braid generators σi act via conjugation (up to a scalar multiple) as invertible matrices in End(V1 ) ⊗ id2 modulo its centralizer. It is not hard to see the centralizer of End(V1 ) ⊗ id2 in End(V1 ⊗ V2 ) is id1 ⊗ End(V2 ). Therefore, the braid generators σi act via conjugation as invertible matrices in End(V1 )⊗ End(V2 ), i.e. they preserve the tensor decomposition. This is impossible by the following eigenvalue analysis. Consider a braid generator σi , its image ρ(σi ) is a tensor product of two matrices each of sizes at least 2. Since ρ(σi ) has only two eigenvalues, neither factor matrix can have 3 or more eigenvalues. If both factor matrices have two eigenvalues, the fact that ρ(σi ) has 2 eigenvalues in all implies that the ratio of these two eigenvalues is ±1 which is forbidden. If one factor matrix is trivial, then ρ(σi ) acts trivially on this
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factor. As all braid generators are conjugate to each other, so the whole group G will act trivially on this factor which implies that V is a reducible representation of G. This case cannot happen either, as V is an irreducible representation of G. Claim 3. The derived group, Der(H ) = [H, H ], of H is a semi-simple Lie group, and the further restriction of V to Der(H ) is still irreducible. Proof. By Claim 2, V |H is a faithful, irreducible representation, so H is a reductive Lie group [V, Theorem 3.16.3]. It follows that the derived group of H is semi-simple. It also follows that the derived group and the center of H generate H . By Schur’s lemma, the center act by scalars. So V |Der(H ) is still irreducible. Claim 4. Every outer automorphism of Der(H ) has order 1, 2, or 3. First we recall a simple fact in representation theory. If V is an irreducible representation of a product group G1 × G2 , then V splits as an outer tensor product of irreducible representations of Gi , i = 1, 2. The restriction of V to G1 has only one isotypic component, and the restriction of V to G2 lies in the centralizer of the image of G1 . So the representation splits. Proof. It suffices to prove the same statement for the universal covering Der uc (H ) of Der(H ), as the automorphism group of Der(H ) is a subgroup of the automorphism group of Der uc (H ). For the 5-dimensional case: as 5 is a prime, Der uc (H ) is a simple group. Any outer automorphism of a simple Lie group is of order 1, 2, or 3. This follows from the fact that any outer automorphism of a simple Lie group is an outer automorphism of its Dynkin diagram together with the A-G classification of Dynkin diagrams [V]. For the 8-dimensional case, if Der uc (H ) is a simple group, it can be handled as above, so we need only to consider the split cases. If Der uc (H ) splits into two simple factors, then one factor must be SU(2): of all simply connected simple Lie groups, only SU(2) has a 2-dimensional irreducible representation. So the outer automorphism group is either Z2 when both factors are SU(2), or the same as the outer automorphism group of the other simple factor. Our claim holds. If there are three simple factors, they must all be SU(2). The outer automorphism group is the permutation group on three letters S3 . Again our claim is true. Claim 5. For each braid generator σi , we can choose a corresponding element σ˜i lying in the derived group Der(H ) which also has exactly two eigenvalues, whose ratio is not ±1. The multiplicity of each eigenvalue of σ˜i is the same as that of σi . (The choice of σ˜i is not unique, but its two eigenvalues have ratio q.) Proof. Since Der(H ) is still a normal subgroup of G, and the braid generators σi normalize Der(H ), so they determine outer-automorphisms of Der(H ). By Claim 4, an outer-automorphism of Der(H ) is of order 1, 2, or 3. Hence σi6 acts as an inner automorphism of Der(H ). By Schur’s lemma, each σi6 is the product of an element in Der(H ) with a scalar, though the decomposition is not unique. Fix a choice for an element σ˜i in Der(H ). Then it has exactly two desired eigenvalues. To complete the proof of Theorem 4.1, we summarize our situation: we have a nontrivial semi-simple group Der uc (H ) with an irreducible unitary representation. Furthermore, it has a special element x whose image under the representation has exactly two distinct eigenvalues whose ratio is not ±1.
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For the 5-dimensional case, Der uc (H ) is a simple Lie group. Going through the list [MP] of pairs (G, D ), where G is a simply connected Lie group and D a dominant weight, the only possible 5-dimensional irreducible representations are as follows: rank=1, (SU(2), 4D1 ), rank=2, (Sp(4), D2 ) on p. 52 of [MP], and rank=4, (SU(5), Di ), i = 1, 4 on p. 30. By examining the possible eigenvalues, we can exclude the first two cases as follows: for the first case, suppose α, β are the two eigenvalues of the above element x in SU(2), then under the representation 4D1 the eigenvalues of the image of x are α i β j , i + j = 4, where i and j both are non-negative integers. The only possibility is two eigenvalues whose ratio is ±1. For the second case, since 5 is an odd number, any element in the image has a real eigenvalue. Other eigenvalues come in mutually reciprocal pairs. Again the only possibility is two eigenvalues whose ratio is ±1. Therefore, the only possible pair is the third case which gives Der uc (H ) = SU(5). As V is a faithful representation of Der(H ), the image of Der(H ) is the same as that of Der uc (H ) which is SU(5). The 8-dimensional case for ρ[4,2] is similar. By [MP], pairs we see the possible for simply connected simple groups are SU(2), 7D + D , SU(3), D on p. 26 of 1 1 2 [MP], Spin(7), D3 on p. 40, Sp(8), D1 on p. 56, Spin(8), Di , i = 1, 3, 4 on p. 66 and SU(8), Di , i = 1, 7 on p. 36, where Di is the fundamental weight. The same eigenvalue analysis will exclude all but the SU(8), Di case. The proof follows the same pattern as above with the following novelties. Case 2 is the adjoint representation of SU(3), if the special element x ∈ SU(3) has eigenvalues {α, β, γ }, the image matrix of x will have eigenvalue 1 with multiplicity 2 and all six pair-wise ratios of {α, β, γ }, so they are ±1. For Case 4, recall that if λ is an eigenvalue of a symplectic matrix, so is λ−1 with the same multiplicity, thus there are candidates for the special element x, but all such elements have the property that the multiplicity for both eigenvalues is 4. Notice by Theorem 3.1 (iv), the multiplicity of the two distinct eigenvalue in ρ(σ ˜ i ) is 3 and 5, respectively. Case 5 is done just as Case 4. This excludes all the unwanted simple groups. We have to consider also the product cases. For a product of two or three simple factors, the same analysis of eigenvalues as at the end of the proof of Claim 2 excludes them. Actually, there are only four cases here: SU(2) × SU(2), SU(2) × SU(4), SU(2) × Sp(4) and SU(2) × SU(2) × SU(2). This completes the proof of our density theorem. Acknowledgement. We would like to thank Alexei Kitaev for conversations on our approach.
References [AB]
Aharanov, D. and Ben-Or, M.: Fault tolerant quantum computation with constant error. quantph/9906129 [AHHH] Alicki, R., Horodecki, M., Horodecki, P., Horodecki, R.: Dynamical description of quantum computing: Generic nonlocality of quantum noise. quant-ph/0105115 [B] Benioff, P.: The computer as a physical system: A microscopic quantum mechanical Hamitonian model of computers as represented by Turing machines. J. Stat. Phys. 22(5), 563–591 (1980) [CR] Curtis, C. and Reiner, I.: Representation theory of finite groups and associate algebras. Pure and Applied Math. Vol XI, New York: Interscience Publisher, 1962 [D] Deutsch, D.: Quantum computational networks. Proc. Roy. Soc. London A425, 73–90 (1989) [Fey] Feynman, R.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982) [FKW] M. Freedman, A. Kitaev, and Z. Wang: Simulation of topological field theories by quantum computers, quant-ph/0001071 [Fu] Funar, L.: On the TQFT representations of the mapping class groups. Pac. J. Math. 188, 251–274 (1999) [G] Gelca, R.: Topological quantum field theory with corners based on the Kauffmann bracket. Comment. Math. Helv. 72, 210–243 (1997)
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M.H. Freedman, M. Larsen, Z. Wang
[J1]
Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomial. Ann. Math. 126, 335–388 (1987) Jones, V.F.R.: Braid groups, Hecke algebras and type I I1 factors. In: Geometric methods in operator algebras, Proc. of the US-Japan Seminar, Kyoto, July 1983 Kauffmann, L. and Lins, S.: Temperley-Lieb recoupling theory and invariants of 3-manifolds. Ann. Math. Studies, Vol 134, Princeton, NJ: Princeton Univ. Press, 1994 Kitaev, A.: Fault-tolerant quantum computation by anyons. quant-ph/9707021, July (1997) Kitaev, A.: Quantum computations: Algorithms and error correction. Russ. Math. Surv. 52 61, 1191–1249 (1997) Karowski, M. and Schrader, R.: A combinatorial approach to topological quantum field theory and invariants of graphs. Commun. Math. Phys. 151, 355–402 (1992) Karowski, M., Schrader, R. and Vogt, E.: Invariants of three manifolds, unitary representations of the mapping class groups and numerical calculations. Experiment. Math. 6, 312–352 (1997) Kitaev, A. Yu., Shen, A. and Vyalyi, M.: Classical and quantum computation. To be published by AMS, approx. 250 pages Lang, S.: Algebra, 2nd edition, Reading, MA: Addison–Wesley Publishing Company, 1984 Lloyd, S.: Universal quantum simulators. Science 273, 1073–1078 (1996) Manin, Y.: Computable and uncomputable. (in Russian). Moscow: Sovetskoye Radio, 1980 Mckay, W. and Patera, J.: Tables of dimensions, indices, and branching rules for representations of simple Lie algebras. Lecture Notes in Pure and Applied Math. Vol 69, New York: Marcel Dekker, 1981 Nielsen, M. and Chuang, I.: Quantum computation and Quantum information. Cambridge: Cambridge Univ. Press, 2000 Preskill, J.: Fault tolerant quantum computation, quant-ph/9712048 Reshetikhin, N. and Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, no. 3, 547–597 (1991) Turaev, V.: Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Math. Vol 18, 1994 Varadarajan, V.S.: Lie groups, Lie algebras and their representaions, Graduate Texts in Math. Vol. 102, Berlin–Heidelberg–New York: Springer-Verlag, 1984 Walker, K.: On Witten’s 3-manifold invariants. Preprint, 1991 Wenzl, H.: Hecke algebras of type An and subfactors. Invent. Math. 92, 349–383 (1988) Witten, E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121, 351–399 (1989) Yao, A.: Quantum circuit complexity, Proc. 34th Annual Symposium on Foundations of Computer Science, Los Alamitos, CA: IEEE Computer Society Press, pp. 352–361
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Communicated by M. Aizenman
