Communications In Mathematical Physics - Volume 297

Commun. Math. Phys. 297, 1–43 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1031-x

Communications in

Mathematical Physics

Lyapunov Spectrum of Asymptotically Sub-additive Potentials De-Jun Feng1 , Wen Huang2 1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong.

E-mail: [email protected]

2 Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui,

P. R. China. E-mail: [email protected] Received: 23 April 2009 / Accepted: 19 January 2010 Published online: 19 March 2010 – © Springer-Verlag 2010

Abstract: For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map is not upper-semi continuous. 1. Introduction The present paper is devoted to the study of the multifractal behavior of Lyapunov exponents of asymptotically sub-additive potentials. This is mainly motivated by the recent works on the Lyapunov exponents of matrix products [26,23,24] and the Lyapunov exponents of differential maps on nonconformal repellers [6]. Before formulating our results, we first give some notation and backgrounds. We call (X, T ) a topological dynamical system (for short TDS) if X is a compact metric space and T is a continuous map from X to X . A sequence  = {log φn }∞ n=1 of functions on X is said to be a sub-additive potential if each φn is a continuous nonnegative-valued function on X such that 0 ≤ φn+m (x) ≤ φn (x)φm (T n x),

∀ x ∈ X, m, n ∈ N.

(1.1)

More generally,  = {log φn }∞ n=1 is said to be an asymptotically sub-additive potential if for any  > 0, there exists a sub-additive potential  = {log ψn }∞ n=1 on X such that lim sup n→∞

1 sup | log φn (x) − log ψn (x)| ≤ , n x∈X

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D. J. Feng, W. Huang

where we take the convention log 0 − log 0 = 0. Furthermore  is called an asymptotically additive potential if both  and − are asymptotically sub-additive, where − denotes {log(1/φn )}∞ n=1 . In particular,  is called additive if each φn is a continuous positive-valued function so that φn+m (x) = φn (x)φm (T n x) for all x ∈ X and m, n ∈ N; in this case, there is a continuous real function g on X such that φn (x) = n−1 exp( i=0 g(T i x)) for each n. Let  = {log φn }∞ n=1 be an asymptotically sub-additive potential on X . For any x ∈ X , we define log φn (x) (1.2) n and call it the Lyapunov exponent of  at x, provided that the limit exists. Otherwise we use λ (x) and λ (x) to denote the upper and lower limits respectively. It can be derived from Kingman’s sub-additive ergodic theorem (cf. [49], p. 231) that, for any μ ∈ E(X, T ), λ (x) = lim

n→∞

λ (x) = ∗ (μ)

for μ-a.e. x ∈ X,

where E(X, T ) denotes the space of ergodic T -invariant Borel probability measures on X and  log φn (x) dμ(x). (1.3) ∗ (μ) := lim n→∞ n This limit always exists and takes values in R∪{−∞}. (For details, see Proposition A.1.) In this paper we are mainly concerned with the distribution of the Lyapunov exponents of . More precisely, for any α ∈ R, define E  (α) = {x ∈ X : λ (x) = α},

(1.4)

which is called the α-level set of λ . We shall study the topological entropy h top (T, E  (α)) of E  (α) when α varies (here we are using the notion of topological entropy for arbitrary subsets of a compact space, introduced by Bowen in [12]; see Sect. 2.1). This is a general concept of multifractal analysis proposed by Barreira, Pesin and Schmeling [7], and it plays an important role in the dimension theory of dynamical systems [5,43]. For convenience we call h top (T, E  (α)), as a function of α, the Lyapunov spectrum of . A key ingredient in the above study is the topological pressure of . To introduce this concept, let X be endowed with the metric d. For any n ∈ N, define a new metric dn on X by     dn (x, y) = max d T k (x), T k (y) : k = 0, . . . , n − 1 . (1.5) For any  > 0, a set E ⊆ X is said to be a (n, )-separated subset of X if dn (x, y) >  for any two different points x, y ∈ E. For  = {log φn }∞ n=1 , we define 

Pn (T, , ) = sup φn (x) : E is a (n, )-separated subset of X . x∈E

It is clear that Pn (T, , ) is a decreasing function of . Define P(T, , ) = lim sup n→∞

1 log Pn (T, , ) n

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

3

and P(T, ) = lim→0 P(T, , ). We call P(T, ) the topological pressure of  with respect to T or, simply, the topological pressure of . If  is additive, P(T, ) recovers the classical (additive) topological pressure introduced by Ruelle and Walters (cf. [49, Chap. 9]). n−1 Let us return back to the study of the Lyapunov spectrum. When  = { i=0 f ◦ T i }∞ is an additive potential, the Lyapunov exponent λ is just equal to the Birkhoff  n=1 average of f . In this case, the topological entropy (or the Hausdorff dimension) of the level sets of Birkhoff averages has been extensively studied in the recent two decades (see, e.g., [1,2,7–9,13,16,20–23,27,29,34,41,42,47] and references therein). It is well known (see, e.g. [21,23,41]) that when (X, T ) is a transitive subshift of finite type and  is an additive potential, then E  (α) = ∅ ⇐⇒ α ∈ := {∗ (μ) : μ ∈ M(X, T )},

(1.6)

where M(X, T ) denotes the space of T -invariant Borel probability measures on X and h top (T, E  (α)) = sup{h μ (T ) : μ ∈ M(X, T ) with ∗ (μ) = α} = inf{P (q) − αq : q ∈ R}, ∀ α ∈ ,

(1.7)

where h μ (T ) denotes the measure-theoretic entropy of μ, P (q) := P(T, q) and q denotes the potential {q log φn }∞ n=1 . The first variational relation in (1.7) has been extended to any TDS satisfying the specification property [47]. Motivated by the study of the multifractal formalism associated to certain iterated function systems with overlaps, the Lyapunov spectrum of certain special sub-additive potentials  = {log φn }∞ n=1 on subshifts of finite type have been studied in [26,23,24], n−1 M(T i x), where M is a continuous function on X taking in which φn (x) =  i=0 values in the set of d × d matrices, and  ·  denotes the operator norm. It is known that in this general situation, (1.6) and (1.7) may both fail. The following is an example taken from [24]. Example 1.1. Let (X, T ) be the one-sided full shift over the alphabet {1, 2, 3, 4}. Let M(x) be a matrix function on X defined as M(x) = Mx1 for x = (x j )∞ j=1 , where Mi (1 ≤ i ≤ 4) are diagonal 4 × 4 matrices given by M1 = M2 = diag(1, 2, 0, 0), M3 = diag(1, 0, 3, 0), M4 = diag(1, 0, 0, 4). It is easily checked that P (q) =

q log 4, log 4,

if q ≥ 1 otherwise

and {α ∈ R : E  (α) = ∅} = {0, log 2, log 3, log 4}  [0, log 4] = {α ∈ R : ∗ (μ) = α for some μ ∈ M(X, T )}. Furthermore, E  (log 3) is a singleton and thus h top (T, E  (log 3)) = 0 < log 4 − log 3 = inf {−q log 3 + P (q)}. q∈R

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We remark that under some additional assumptions (e.g., positiveness or certain irreducibility) for the matrix function M, (1.6) and (1.7) still hold [23,24,26]. A natural question arises whether there exist some positive results without any additional assumptions. This is one of the original motivations of this paper. Indeed in this paper, we study the Lyapunov spectrum of general asymptotically subadditive potentials and asymptotically additive potentials on general TDS. Under this setting, the multifractal behavior may be quite irregular. For instance, we can construct a TDS (X, T ) and an additive potential  on X such that h top (T, E  (α)) < inf{P (q)−αq : q ∈ R}

∀ α ∈ := {∗ (μ) : μ ∈ M(X, T )}

(see Example 6.2). Nevertheless, we still have some positive results regarding the Lyapunov spectrum and its variational relations to measure-theoretic entropies and topological pressures. Some more properties are obtained when the corresponding TDS satisfies further assumptions (e.g., upper semi-continuity of the entropy map). To formulate our results, for an asymptotically sub-additive potential  = {log φn } on a general TDS (X, T ), we define β() = lim

n→∞

1 log sup φn (x). n x∈X

(1.8)

The limit exists and takes values in R ∪ {−∞} (see Lemma A.3). However if β() = −∞, it is easy to see that for all x ∈ X , λ (x) = −∞. To avoid trivialities we shall always assume that β() > −∞. For any q > 0, let q denote the sequence ∞ {q log φn }i=1 (which clearly is asymptotically sub-additive) and write P (q) = P (T, q) . The function P is called the pressure function of . When  is asymptotically additive on X , P can be defined over (−∞, ∞). Our main results are Theorems 1.1-1.4 formulated as follows: Theorem 1.1. Let (X, T ) be a TDS and  = {log φn }∞ n=1 an asymptotically sub-additive potential on X which satisfies β() > −∞. Then E  (β()) = ∅ and h top (T, E  (β())) = sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = β()} = sup{h μ (T ) : μ ∈ E(X, T ), ∗ (μ) = β()}. We emphasize that the above theorem only deals with the specific value α = β(), which is the largest possible value for λ (cf. Lemma A.3). Theorem 1.2. Let (X, T ) be a TDS such that the topological entropy h top (T ) is finite. Suppose that  = {log φn }∞ n=1 is an asymptotically sub-additive potential on X which satisfies β() > −∞. Then the pressure function P (q) is a continuous real convex function on (0, ∞) with P (∞) := limq→∞ P (q)/q = β(). Moreover, (i) For any t > 0, if α = P (t+) or α = P (t−), then ⎞ ⎛  E  (β)⎠ = inf {P (q) − αq} = P (t) − αt, lim h top ⎝T, →0

β∈(α−,α+)

q>0

where P (t−) and P (t+) denote the left and right derivatives of P at t, respectively. Moreover the first equality is also valid when α = P (∞).

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

5

(ii) For any t > 0 and any α ∈ [P (t−), P (t+)], inf {P (q) − αq} = lim sup{h μ (T ) : μ ∈ M(X, T ), |∗ (μ) − α| < }. →0

q>0

Furthermorethe above equality is validfor α = P (∞). (iii) For any α ∈ limt→0+ P (t−), P (∞) , inf {P (q) − αq} = sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α}.

q>0

By convexity, P may fail to be differentiable on a set which is at most countable; however the left and right derivatives of P exist everywhere. We remark that h top (T ) is finite for a lot of TDS’s such as expansive maps on compact metric space and Lipschitz continuous transformations on finite dimensional compact metric spaces (see e.g. [32, Sect. 3.2]), and asymptotically h-expansive TDS’s [39]. We need to mention that Theorem 1.2(i) only deals with the “fuzzy” level sets and it is not valid for the standard level sets E  (α). Indeed, there are examples such that h top (T, E  (α)) < inf {P (q) − αq} q>0

for any α = P (t+) or α = P (t−) with t > 0 (see e.g. Example 6.2). Nevertheless the results of Theorem 1.2 can be improved if we add an additional assumption that the entropy map μ → h μ (T ) is upper semi-continuous on M(X, T ). More precisely, we have Theorem 1.3. Under the condition of Theorem 1.2, we assume furthermore that the entropy map μ → h μ (T ) is upper semi-continuous on M(X, T ). Then (i) For any t > 0, if α = P (t+) or α = P (t−), then E  (α) = ∅ and h top (T, E  (α)) = inf {P (q) − αq} = P (t) − αt. q>0

(ii) For α ∈



 t>0 [P (t−),

P (∞)],

inf {P (q) − αq} = max{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α}.

q>0

(iii) If t > 0 such that t has a unique equilibrium state μt ∈ M(X, T ), then μt is ergodic, P (t) = ∗ (μt ), E  (P (t)) = ∅ and h top (T, E  (P (t))) = h μt (T ). A significant part of Theorem 1.3(i) is that we don’t need the differentiability assumption for P . To the best of our knowledge, this result is not known even in the additive case. It has a nice application in the multifractal analysis for certain probability measures on symbolic spaces (see Remark 4.9). We remark that the assumption of upper semi-continuity for the entropy map is quite essential for the results in Theorem 1.3. This assumption is satisfied by some natural TDS’s such as h-expansive TDS’s [11] and more generally, asymptotically h-expansive TDS’s [39] which include, for example, C ∞ transformations on Riemannian manifolds [14]. Without this assumption, the multifractal behavior may be very irregular and complicated. See Sect. 6 for some examples. We remark that the differentiability property of the pressure functions was studied in [37,38] for rational maps on the Riemann sphere for certain additive potentials. Meanwhile Theorems 1.1–1.3 are about asymptotically sub-additive potentials; our next theorem is concerned with asymptotically additive potentials. A TDS (X, T ) is

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D. J. Feng, W. Huang

called saturated if for any μ ∈ M(X, T ), we have G μ = ∅ and h top (T, G μ ) = h μ (T ), where G μ denotes the set of μ-generic points defined by ⎧ ⎫ n−1 ⎨ ⎬ 1 G μ := x ∈ X : δT j x → μ in the weak* topology as n → ∞ , ⎩ ⎭ n j=0

where δ y denotes the probability measure whose support is the single point y. It was shown independently in [22,44] that if a TDS (X, T ) satisfies the specification property, then (X, T ) is saturated. Theorem 1.4. Let (X, T ) be a TDS and let  be an asymptotically additive potential on X . Set = {∗ (μ) : μ ∈ M(X, T )}. Then is a bounded closed interval. Furthermore we have the following statements: (i) {α ∈ R : E  (α) = ∅} ⊆ . (ii) If h top (T ) < ∞, then P is a real convex function over R. Furthermore, α ∈ ⇐⇒ inf{P (q) − αq : q ∈ R} = −∞ ⇐⇒ inf{P (q) − αq : q ∈ R} ≥ 0. (iii) If h top (T ) < ∞ and the entropy map is upper semi-continuous, then for each α ∈ , sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α} = inf{P (q) − αq : q ∈ R}.  Furthermore, for α ∈ t∈R {P (t−), P (t+)} ∪ P (±∞), we have E  (α) = ∅ and h top (T, E  (α)) = inf{P (q) − αq : q ∈ R}, where P (+∞) := limq→+∞ P (q)/q and P (−∞) := limq→−∞ P (q)/q. (iv) Assume that (X, T ) is saturated. Then E  (α) = ∅ if and only if α ∈ . Furthermore, h top (T, E  (α)) = sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α} for any α ∈ . We remark that Theorem 1.4(iii)-(iv) extend previous results about the Lyapunov spectrum of continuous positive matrix-valued functions [23] and the Lyapunov spectrum of certain asymptotically additive potentials [35] on subshifts of finite type. In this paper, we also study the high dimensional Lyapunov spectrum. For a finite family of asymptotically sub-additive (resp. asymptotically additive) potentials i = k {log φn,i }∞ n=1 , i = 1, . . . , k, and a = (a1 , . . . , ak ) ∈ R , we define  log φn,i (x) E  (a) = x ∈ X : lim = ai for 1 ≤ i ≤ k . n→∞ n We indeed obtain the high dimensional versions of Theorems 1.2–1.4 regarding the properties about h top (T, E  (a)) and the corresponding variational relations (see Theorems 4.2, 4.8, 5.2). For instance, when (X, T ) is a saturated TDS such that the entropy map is upper-semi continuous, then for any asymptotically additive potentials i (i = 1, . . . , k), we have E  (a) = ∅ ⇐⇒ a ∈ A := {∗ (μ) : μ ∈ M(X, T )}

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

7

and h top (T, E  (a)) = sup{h μ (T ) : μ ∈ M(X, T ) with ∗ (μ) = a} = inf{P (q) − a · q : q ∈ Rk },

∀ a ∈ A, k where ∗ (μ) := ((1 )∗ (μ), · · · , (k )∗ (μ)), P (q) = P(T, i=1 qi i ) for q = (q1 , . . . , qk ), and a · q denotes the inner product of a and q (see Theorem 5.2(iii)-(iv)). As an application of the above result, we can improve a result of Barreira and Gelfert in [6] on Lyapunov exponents on nonconformal repellers. To see it, let be a repeller of a C 1 local diffeomorphism f : R2 → R2 , such that f satisfies a cone condition on (see [6] for the definition). Let i = {log φn,i }∞ n=1 (i = 1, 2) be two potentials given by φn,i (x) = σi (dx f n ), n ∈ N, i = 1, 2, where σi (dx f n )(i = 1, 2) denote the singular values of the differential of f n at x. Both 1 and 2 are asymptotically additive (see [6, Prop. 4]). Under the additional assumptions that f is C 1+δ and f has bounded distortion, Barreira and Gelfert showed that h top (T, E  (a)) = inf{P (q) − a · q : q ∈ Rk } for each gradient a of P (see [6, Theorem 1]). However according to our result, these two additional assumptions can be removed (although in this case P may be not differentiable) and the variational relation holds for each a ∈ A := {∗ (μ) : μ ∈ M(X, T )} (we remark that A contains the subdifferentials of P ; see Theorem 3.3). Below we give some further remarks. Remark 1.5. (i) In the definition of sub-additive potential  = {log φn }, we admit that φn (x) takes the value 0. As an advantage, we can cover the interesting case n−1 M(T i x), where M is an arbitrary continuous matrix-valued that φn (x) =  i=0 function. (ii) There are some natural examples of asymptotically sub-additive (resp. asymptotically additive) potentials which may not be sub-additive, such as the general potential  = {log μ(In (x))}∞ n=1 , where μ is a weak Gibbs measure on a full shift space over finite symbols and In (x) denotes the n th cylinder about x (cf. [28] and Prop. A.5(iv)). By the way, the quotient space of all asymptotically additive potentials on X under a certain equivalence relation is a separable Banach space endowed with some norm (cf. Remark A.6(ii)). These are two main reasons that we set up the theory for asymptotically sub-additive potentials rather than sub-additive potentials. (iii) For the proofs of Theorems 1.2–1.4, we first prove their higher dimensional versions by applying convex analysis and the thermodynamic formalism, then derive the one-dimensional versions. Although it looks a bit strange and there are relatively simple alternative approaches for the one-dimensional versions, however, the extension to higher dimensions along those approaches seems difficult. (iv) Let  = {log φn }∞ n=1 , where φn ’s are non-negative continuous functions on X satisfying φn+m (x) ≤ Cn φn (x)φm (T n x), ∀ n, m ∈ N, x ∈ X, where (Cn ) is a sequence of positive numbers with limn→∞ (1/n) log Cn = 0. We do not know whether  is asymptotically sub-additive. However, one can manage

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to prove Lemma A.2 and Theorem 3.1 for this  by an approach similar to [15]. Furthermore, Theorems 1.1–1.3 remain valid for this kind of potentials. The content of the paper is following. In Sect. 2, we give some preliminaries about topological entropy and topological pressures, and we also present and derive some results in convex analysis that are needed in the proof of our theorems. In Sect. 3, we introduce the asymptotically sub-additive thermodynamic formalism and we also set up a formula for the subdifferentials of pressure functions. The high dimensional versions of Theorem 1.2–1.3 are formulated and proved in Sect. 4. In particular, we give a class of sub-additive potentials on full shifts which satisfy part of (1.7) in Sect. 4. The high dimensional version of Theorem 1.4 is formulated and proved in Sect. 5. In Sect. 6, we give some examples about the irregular multifractal behaviors for additive potentials on TDS’s for which the entropy maps are not upper semi-continuous. In Appendix A, we give some properties about asymptotically sub-additive (resp. asymptotically additive potentials). In Appendix B, we summarize the main notation and conventions used in this paper. 2. Preliminaries In this section, we first give the definitions and some properties about the topological entropy of non-compact sets and the topological pressure of non-additive potentials, for which the reader is referred to [12,3,43,15] for more details. Then we present some notation and known facts in convex analysis and derive several results which are needed in the proofs of our main results. 2.1. Topological entropy. Let (X, d) be a compact metric space and T : X → X a continuous transformation. For any n ∈ N we define a new metric dn on X by     (2.1) dn (x, y) = max d T k (x), T k (y) : k = 0, . . . , n − 1 , and for every  > 0 we denote by Bn (x, ) the open ball of radius  in the metric dn around x, i.e., Bn (x, ) = {y ∈ X : dn (x, y) < }. Let Z ⊂ X and  > 0.  We say that an at most countable collection of balls  = {Bn i (xi , )}i covers Z if Z ⊂ i Bn i (xi , ). For  = {Bn i (xi , )}i , put n() = mini n i . Let s ≥ 0 and define exp(−sn i ), M(Z , s, N , ) = inf i

where the infinum is taken over all collections  = {Bn i (xi , )} covering Z , such that n() ≥ N . The quantity M(Z , s, N , ) does not decrease with N , hence the following limit exists: M(Z , s, ) = lim M(Z , s, N , ). N →∞

There exists a critical value of the parameter s, which we will denote by h top (T, Z , ), where M(Z , s, ) jumps from ∞ to 0, i.e. 0, s > h top (T, Z , ), M(Z , s, ) = ∞, s < h top (T, Z , ).

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

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It is clear to see that h top (T, Z , ) does not decrease with , and hence the following limit exists: h top (T, Z ) = lim h top (T, Z , ). →0

We call h top (T, Z ) the topological entropy of T restricted to Z or, simply, the topological entropy of Z , when there is no confusion about T . In particular we write h top (T ) for h top (T, X ). Here we recall some of the basic properties about the topological entropy. Proposition 2.1. ([12,43]) The topological entropy as defined above satisfies the following: (1) h top (T, Z 1 ) ≤ h top (T, Z 2 ) for any Z 1 ⊆ Z 2⊆ X . ∞ (2) h top (T, Z ) = supi h top (T, Z i ), where Z = i=1 Zi ⊆ X . (3) Suppose μ is an invariant measure and Z ⊆ X is such that μ(Z ) = 1, then h top (T, Z ) ≥ h μ (T ), where h μ (T ) is the measure-theoretic entropy. 2.2. Topological pressure. Let T : X → X be a continuous transformation of a compact metric space (X, d). For any n ∈ N, define the metric dn as in (2.1). For any  > 0, a set E ⊆ X is said to be a (n, )-separated subset of X if dn (x, y) >  for any two different points x, y ∈ E. Let  = {log φn (x)}∞ n=1 be a sequence of functions on X for which φn is non-negative for each n. We define 

Pn (T, , ) = sup φn (x) : E is a (n, )-separated subset of X . x∈E

It is clear that Pn (T, , ) is a decreasing function of . Define P(T, , ) = lim sup n→∞

1 log Pn (T, , ) n

and P(T, ) = lim→0 P(T, , ). We call P(T, ) the topological pressure of  with respect to T or, simply, the topological pressure of . 2.3. Subdifferentials of convex functions. We first give some notation and basic facts in convex analysis. For details, one is referred to [30,45]. k mBy a convex combination of points x1 , . . ., xm ∈ R we mean a linear combination k i=1 λi xi , where λ1 + · · · + λm = 1 and λ1 , . . . , λm ≥ 0. For any subset M of R , the convex hull conv(M) of M is the set of all convex combinations of points from M. Carathéodory’s Theorem says that for any subset M of Rk , the convex hull conv(M) is the set of all convex combinations of k + 1 points from M (cf. [45, Theorem 17.1]). Let C be a convex subset of Rk . A point x ∈ C is called an extreme point of C if x = py + (1 − p)z for some y, z ∈ C and 0 < p < 1, then x = y = z. The set of extreme points of C is denoted by ext(C). Minkowski’s Theorem says that for any nonempty compact convex subset C of Rk , C = conv(ext(C)) (cf. [30, Th. 2.3.4] or [45, Cor. 18.5.1]). Hence, according to Carathéodory’s Theorem and Minkowski’s Theorem, each point in a compact convex set C ⊂ Rk is a convex combination of k + 1 points from ext(C).

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A point x ∈ C is called an exposed point of C, if {x} is the intersection of C with some supporting hyperplane of C. The set of all exposed points of C will be denoted by expo(C). Straszewicz’ Theorem says for any compact convex set C in Rk , expo(C) is a dense subset of ext(C) (cf. [45, Theorem 18.6]). Let U be an open convex subset of Rk and f be a real continuous convex function on U . For x ∈ U , a ∈ Rk is called a subgradient of f at x, if for any y ∈ U one has f (y) − f (x) ≥ a · (y − x), where the dot denotes the dot product. The set of all subgradients at x is called the subdifferential of f at x and is denoted ∂ f (x). For x ∈ U , the subdifferential ∂ f (x) is always a nonempty convex compact set (cf. [45, Theorem 23.4]). Write ∂ e f (x) = ext(∂ f (x)). When ∂ e f (x) = {a}, we say that f is differentiable at x and write f  (x) = a. It is known that f is differentiable for almost every x ∈ U (cf. [30, Theorem 4.2.3]). In the case k = 1, ∂ f (x) = [ f  (x−), f  (x+)] and ∂ e f (x) = { f  (x−), f  (x+)}. Next we define   ∂ f (x) and ∂ e f (U ) = ∂ e f (x). (2.2) ∂ f (U ) = x∈U

x∈U

Proposition 2.2. Let U be an open convex subset of Rk and f be a real continuous convex function on U . Then for each x ∈ U and a ∈ ∂ e f (x), there exists a sequence (xn ) ⊂ U such that limn→∞ xn = x, f is differentiable at each point xn and a = limn→∞ f  (xn ). Proof. Let x ∈ U . Since expo(∂ f (x)) is dense in ∂ e f (x), we only need to show that the lemma holds when a ∈ expo(∂ f (x)). Fix such an a and write a = (a1 , . . . , ak ). Then there exists a non-zero vector t = (t1 , . . . , tk ) ∈ Rk such that t·b
for any b ∈ ∂ f (x)\{a}.

(2.3)

Since f is differentiable almost everywhere on U , there exists a sequence (xn ) ∈ U such that limn→∞ xn = x, f is differentiable at each xn and |xn − (x + t/n)| < n −2 for all n ∈ N.

(2.4)

Write an = f  (xn ). Note that the sequence (an ) is bounded because of the boundedness of (xn ). Hence by taking a subsequence if it is necessary, we can assume that limn→∞ an = a for some a ∈ Rk . In the following we show that a = a. Since an = f  (xn ), one has f (z) − f (xn ) ≥ an · (z − xn )

for any z ∈ U.

Letting n → ∞ yields f (z) − f (x) ≥ a · (z − x) for any z ∈ U , which implies a ∈ ∂ f (x). Meanwhile for each n ∈ N, f (x) − f (xn ) ≥ an · (x − xn )

and

f (xn ) − f (x) ≥ a · (xn − x).

Hence an · (xn − x) ≥ f (xn ) − f (x) ≥ a · (xn − x). That is, an · (t + nwn ) ≥ a · (t + nwn ),

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

11

where wn := xn − (x + t/n). Taking n → ∞ and noting that lim n|wn | = 0 (by (2.4)), n→∞ we have a · t ≥ a · t. Combining it with (2.3) and the fact a ∈ ∂ f (x), one has a = a. This finishes the proof of the proposition.   Proposition 2.3. Let Y be a compact convex subset of a topological vector space which satisfies the first axiom of countability (i.e., there is a countable base at each point) and U ⊆ Rk a non-empty open convex set. Suppose f : U × Y → R ∪ {−∞} is a map satisfying the following conditions: (i) (ii) (iii) (iv)

f (q, y) is convex in q; f (q, y) is affine in y; f is upper semi-continuous over U × Y ; g(q) := sup y∈Y f (q, y) > −∞ for any q ∈ U .

For each q ∈ U , denote I(q) := {y ∈ Y : f (q, y) = g(q)}. Then  ∂ f (q, y), ∂g(q) =

(2.5)

y∈I (q)

where ∂ f (q, y) denotes the subdifferential of f (·, y) at q. Proof. By (i)–(iv), g is a real convex function over U , and I(q) is a non-empty compact convex subset of Y for each q ∈ U . For convenience, denote by R(q) the righthand side of (2.5). A direct check shows that R(q) is a non-empty convex subset of Rk for each q ∈ U . We further show that for each q ∈ U , (c1) R(q) is compact; (c2) For each δ > 0, there exists γ > 0 such that R(t) ⊆ Bδ (R(q))

whenever t ∈ U, |t − q| < γ ,

where Bδ (R(q)) := {b ∈ Rk : d(b, R(q)) ≤ δ} is the closed δ-neighborhood of R(q) in Rk . To show (c1), let (an ) be a sequence in R(q). Take yn ∈ I(q) so that an ∈ ∂ f (q, yn ). Then f (t, yn ) − g(q) = f (t, yn ) − f (q, yn ) ≥ an · (t − q)

(2.6)

for each t ∈ U . Hence the sequence (an ) should be bounded (otherwise, there exists t ∈ U such that an · (t − q) is unbounded from above; however f (t, yn ) − f (q, yn ) = f (t, yn ) − g(q) ≤ g(t) − g(q)). Taking a subsequence if necessary, we assume that yn → y and an → a for some y ∈ I(q) and a ∈ Rk . Since f (t, ·) is upper semi-continuous, by (2.6) we have f (t, y) − f (q, y) = f (t, y) − g(q) ≥ a · (t − q) for each t ∈ U . This shows a ∈ R(q) and hence R(q) is compact. To show (c2), we use contradiction. Assume that (c2) does not hold. Then there exist δ > 0 and a sequence (tn ) in U with limn→∞ tn = q such that there exists an ∈ R(tn ) satisfying d(an , R(q)) > δ for each n. Take yn ∈ I(tn ) so that an ∈ ∂ f (tn , yn ). Then we have f (t, yn ) − g(tn ) = f (t, yn ) − f (tn , yn ) ≥ an · (t − tn )

(2.7)

12

D. J. Feng, W. Huang

for each t ∈ U . Similarly we can show that an is bounded. Hence by taking a subsequence if necessary, we can assume that yn → y and an → a for some y ∈ Y and a ∈ Rk . By the upper semi-continuity of f and the continuity of g, we have f (q, y) ≥ lim sup f (tn , yn ) = lim sup g(tn ) = g(q), n→∞

n→∞

which implies y ∈ I(q). Hence taking n → ∞ in (2.7) yields f (t, y) − f (q, y) = f (t, y) − g(q) ≥ lim sup( f (t, yn ) − g(tn )) ≥ a · (t − q) n→∞

for each t ∈ U . Hence a ∈ ∂ f (q, y). Thus a ∈ R(q), which contradicts the assumption that d(an , R(q)) > δ for all n. This finishes the proof of (c2). Now we are ready to show (2.5), i.e., ∂g(q) = R(q). For each a ∈ R(q), there exists y ∈ I(q) so that a ∈ ∂ f (q, y). Hence for each t ∈ U , g(t) − g(q) ≥ f (t, y) − g(q) = f (t, y) − f (q, y) ≥ a · (t − q).

(2.8)

This implies a ∈ ∂g(q) and thus ∂g(q) ⊇ R(q). In the end, we show that ∂g(q) ⊆ R(q) by contradiction. Assume that a ∈ ∂g(q) \ R(q). Since R(t) ⊆ ∂g(t), we have g(t) − g(q) ≤ b · (t − q),

∀ t ∈ U, b ∈ R(t).

(2.9)

Note that a ∈ ∂g(q). We have g(t) − g(q) ≥ a · (t − q) for all t ∈ U . This combining with (2.9) yields a · (t − q) ≤ b · (t − q),

∀ t ∈ U, b ∈ R(t).

(2.10)

Since a ∈ R(q) and R(q) is compact, there exists δ > 0 so that a ∈ Bδ (R(q)). Notice that Bδ (R(q)) is compact convex (since so is R(q)), there exists a vector e ∈ Rk such that |e| = 1 and a · e > b · e for any b ∈ Bδ (R(q)). By (c2), there exists γ > 0 such that R(t) ⊆ Bδ (R(q)) whenever |t − q| ≤ γ . Take a small 0 < γ˜ < γ such that t0 := q + (γ˜ /2)e ∈ U . Then a · (t0 − q) > b · (t0 − q) for any b ∈ R(t0 ), which contradicts (2.10). This proves ∂g(q) ⊆ R(q).   2.4. Conjugates of convex functions. Let f : Rk → R ∪ {+∞} be convex and not identically equal to +∞. Then the function f ∗ : Rk → R ∪ {+∞} defined by s → f ∗ (s) := sup{s · x − f (x) : x ∈ Rk } is called the conjugate function of f or Legendre transform of f . It is known that f ∗ is also convex and not identically equal to +∞ (cf. [30, p. 211]). Let f ∗∗ denote the conjugate of f ∗ . The following result is well known (cf. [45, Theorem 12.2]). Theorem 2.4. Let f : Rk → R ∪ {+∞} be convex and not identically equal to +∞. Let x ∈ Rk . Assume that f is lower semi-continuous at x, i.e., lim inf y→x f (y) ≥ f (x). Then f ∗∗ (x) = f (x).

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

13

As an application, we have Corollary 2.5. Let A be a non-empty convex set in Rk and g : A → R be a concave function. Set W (x) = sup{g(a) + a · x : a ∈ A},

x ∈ Rk ,

and G(a) = inf{W (x) − a · x : x ∈ Rk },

a ∈ Rk .

Then we have (i) G(a) = g(a) for a ∈ ri(A), where ri(A) denotes the relative interior of A. (ii) Assume in addition that A is closed. If g is upper semi-continuous at a ∈ A, then G(a) = g(a). Proof. Let f : Rk → R ∪ {+∞} be the function which agrees with −g on A but is +∞ everywhere else. Then f is convex and has A as its efficient domain, i.e., A = {x : f (x) < +∞}. By the definition of W and G, we have W = f ∗ and G = − f ∗∗ . However, f is lower semi-continuous on ri(A) (see, e.g., [45, Theorem 7.4]). Hence by Theorem 2.4, we have f ∗∗ (a) = f (a) for a ∈ ri(A), and thus G(a) = g(a) for a ∈ ri(A). This proves (i). To show (ii), assume that A is closed. Let a ∈ A so that g is upper semi-continuous at a. Then it is direct to check that f is lower semi-continuous at a. By Theorem 2.4, we have f ∗∗ (a) = f (a) and hence G(a) = g(a). This finishes the proof of (ii).   3. The Thermodynamic Formalism and Subdifferentials of Pressure Functions In this section, we first introduce a variational principle of topological pressures which plays a key role in the proofs of our main theorems. Then we set up a formula for the subdifferentials of pressure functions. Let (X, T ) be a TDS and let  = {log φn }∞ n=1 be an asymptotically sub-additive potential on a TDS (X, T ). Let λ , ∗ and β() be defined as in (1.2), (1.3) and (1.8). Some basic properties of λ , ∗ and β() are given in Appendix A. The following variational principle plays a key role in our analysis. Theorem 3.1. ([15]) The topological pressure P(T, ) of  satisfies the following variational principle: −∞, if ∗ (μ) = −∞ for all μ ∈ M(X, T ), P(T, ) = sup{h μ (T ) + ∗ (μ) : μ ∈ M(X, T ), ∗ (μ) = −∞}, otherwise. In particular if h top (T ) < ∞, then P(T, ) = sup{h μ (T ) + ∗ (μ) : μ ∈ M(X, T )}. The above theorem is only proved in [15, Theorem 1.1] for sub-additive potentials. However the proof given there works well for asymptotically sub-additive potentials, in which we only need to replace Lemma 2.3 in [15] by Lemma A.2 given in the Appendix. We remark that the variational principle for sub-additive potentials has been studied in [18,3,25,31,35,4,40] under additional assumptions on the corresponding sub-additive potential and TDS.

14

D. J. Feng, W. Huang

In the remaining part of this section, we present and prove a formula for the subdifferentials of pressure functions. We first give some notation. Let k ∈ N. For each i = 1, . . . , k, let i = {log φn,i }∞ n=1 be an asymptotically subadditive potential on (X, T ). Write Rk+ = {(x1 , x2 , . . . , xk ) : xi > 0, i = 1, 2, · · · , k} and  = (1 , 2 , . . . , k ). For μ ∈ M(X, T ), write ∗ (μ) = ((1 )∗ (μ), · · · , (k )∗ (μ)). k For q = (q1 , . . . , qk ) ∈ Rk+ , let q ·  = i=1 qi i denote the asymptotically k ∞ sub-additive potential { i=1 qi log φn,i }n=1 and write P (q) = P(T, q · ).

(3.1)

We call P the pressure function of . k Let β() = β( i=1 i ). Then by Theorem 3.1, if β() = −∞ then P (q) = −∞ for any q ∈ Rk+ . If β() > −∞, then β(1 ) > −∞, . . ., β(k ) > −∞. Proposition 3.2. Assume that h top (T ) < ∞ and β() > −∞. Then P is a real continuous convex function on Rk+ and ∂ P (Rk+ ) ⊆ (−∞, β(1 )] × (−∞, β(2 )] × · · · × (−∞, β(k )], where ∂ P (Rk+ ) is defined as in (2.2). Proof. By Lemma A.3(4), P (q) ∈ R for q ∈ Rk+ . The convexity of P over Rk+ just comes from Theorem 3.1, using the affine property of the maps μ → h μ (T ) and μ → (i )∗ (μ). Since P is also locally bounded on Rk+ , P is continuous on Rk+ . Fix q = (q1 , · · · , qk ) ∈ Rk+ . Define qλ = (q1 + λ, q2 , · · · , qk ) for λ > 0. Let a = (a1 , · · · , ak ) ∈ ∂ P . Then h top (T ) + λβ(1 ) +

k

qi β(i ) ≥ P (qλ ) ≥ P (q) + (qλ − q) · a = P (q) + λa1 .

i=1

Letting λ → ∞ one gets β(1 ) ≥ α1 . Similarly, we have αi ≤ β(i ) for i = 2, · · · , k.   For q ∈ Rk+ , let I(, q) denote the collection of invariant measures μ such that h μ (T ) + q · ∗ (μ) = P(T, q · ). If I(, q) = ∅, then each element I(, q) is called an equilibrium state for q · . In the following theorem, we set up a formula for the subdifferentials of P , which extends Ruelle’s derivative formula for the pressures of additive potentials (cf. [46, Exercise 5, p. 99], [41, Lemma 4] and [33, Theorem 4.3.5]). Theorem 3.3. Assume that h top (T ) < ∞, β() > −∞, and that the entropy map μ → h μ (T ) is upper semi-continuous. Then (i) For any q ∈ Rk+ , I(, q) is a non-empty compact convex subset of M(X, T ), and every extreme point of I(, q) is an ergodic measure (i.e., an extreme point of M(X, T )). Furthermore ∂ P (q) = {∗ (μ) : μ ∈ I(, q)}.

(3.2)

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

15

(ii) Assume in addition that i (i = 1, . . . , k) are all asymptotically additive. Then the above results hold for all q ∈ Rk . Proof. To prove (i), let q ∈ Rk+ . Then q· is an asymptotically sub-additive potential. We first show that I(, q) = ∅. By Theorem 3.1, there exists a sequence {μn } ⊂ M(X, T ) such that P (q) = limn→∞ h μn (T ) +q · ∗ (μn ). Let μ be a limit point of {μn }. Then by the upper semi-continuity of h (·) (T ) and (i )∗ (·), we have P (q) ≤ h μ (T ) + q · ∗ (μ). Applying Theorem 3.1 again we obtain P (q) = h μ (T ) + q · ∗ (μ), i.e., μ ∈ I(, q). Hence I(, q) = ∅. An identical argument shows that any limit point of I(, q) belongs to I(, q) itself. Therefore I(, q) is closed and thus compact. Now assume that μ is an extreme point of I(, q). We claim that μ is ergodic, i.e., μ is also an extreme point of M(X, T ). To see it, assume μ = pμ1 + (1 − p)μ2 for some p > 0 and μ1 , μ2 ∈ M(X, T ). Since h μ (T ) = ph μ1 (T ) + (1 − p)h μ2 (T ) and ∗ (μ) = p∗ (μ1 ) + (1 − p)∗ (μ2 ), we have P (q) = h μ (T ) + q · ∗ (μ)     = p h μ1 (T ) + q · ∗ (μ1 ) + (1 − p) h μ2 (T ) + q · ∗ (μ2 ) . By Theorem 3.1, we have μ1 , μ2 ∈ I(, q). Since μ is an extreme point of I(, q), we have μ1 = μ2 = μ. This shows that μ is also an extreme point of M(X, T ). Next we show (3.2). In Proposition 2.3, we take Y = M(X, T ), U = Rk+ . Define f : U × Y → R ∪ {−∞} by f (q, μ) = q · ∗ (μ) + h μ (T ). Then f satisfies the conditions (i)-(iv) in Proposition 2.3. The identity (3.2) just comes from (2.5). This finishes the proof of (i). Now we turn to the proof of (ii). Assume i (i = 1, . . . , k) are all asymptotically additive. Let q ∈ Rk . Then q ·  is also asymptotically additive. Clearly the above proof still works for this case (as a slightly different point, we should take U = Rk for the proof of (3.2)).   4. Multifractal Formalism for Asymptotically Sub-additive Potentials In this section, we establish the multifractal formalism for asymptotically sub-additive potentials. Let (X, T ) be a TDS. 4.1. Proof of Theorem 1.1. Let  = {log φn }∞ n=1 be an asymptotically sub-additive potential on (X, T ) with β() > −∞. For x ∈ X , we denote by V (x) the set of all  limit points in M(X ) of the sequence μx,n = (1/n) n−1 j=0 δT j x . This set is a non-empty compact subset of M(X, T ) for each x (cf. [12]). The following result of Bowen plays a key role in the proof of Theorem 1.1. Lemma 4.1 (Bowen [12]). For t ≥ 0, define R(t) = {x ∈ X : ∃ μ ∈ V (x) with h μ (T ) ≤ t}. Then h top (T, R(t)) ≤ t.

16

D. J. Feng, W. Huang

Proof of Theorem 1.1. Let α = β(). By Lemma A.3(2), there exists μ ∈ E(X, T ) so that ∗ (μ) = α. By Proposition A.1(1), μ(E  (α)) = 1. Thus E  (α) = ∅. Furthermore by Proposition 2.1(3), h top (T, E  (α)) ≥ h μ (T ). This indeed proves (4.1) h top (T, E  (α)) ≥ sup{h μ (T ) : μ ∈ E(X, T ), ∗ (μ) = α}.  Now assume ν ∈ M(X, T ) so that ∗ (ν) = α. Let ν = E (X,T ) θ dm(θ ) be the ergo dic decomposition of μ. By Proposition A.1(3), α = ∗ (ν) = E (X,T ) ∗ (θ ) dm(θ ). Since α ≥ ∗ (θ ) for each θ ∈ E(X, T ), we have α = ∗ (θ ) whenever θ ∈  , where  is a subset of E(X, T ) with m(  ) = 1. Hence   h ν (T ) + ∗ (ν) = (h θ (T ) + ∗ (θ )) dm(θ ) = (h θ (T ) + ∗ (θ )) dm(θ ) E (X,T )   ≤ sup{h μ (T ) : μ ∈ E(X, T ), ∗ (μ) = α} dm(θ ) + α 

= sup{h μ (T ) : μ ∈ E(X, T ), ∗ (μ) = α} + α. This proves sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α} ≤ sup{h μ (T ) : μ ∈ E(X, T ), ∗ (μ) = α}.

(4.2)

Next we prove that h top (T, E  (α)) ≤ sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α}.

(4.3)

Denote by t the right-hand side of the above inequality. We may assume that t < ∞, otherwise there is nothing remaining to prove. Let x ∈ E  (α) and μ ∈ V (x). Then there  i −1 is n i → ∞ such that μx,n i = n1i nj=0 δT j x → μ. By Lemma A.2, μ ∈ M(X, T ) log φn (x)

i ≤ ∗ (μ). Moreover α = ∗ (μ) by Lemma A.3(2). Hence and α = limi→∞ ni h μ (T ) ≤ t. It follows that

E  (α) ⊂ R(t) := {x ∈ X : ∃ μ ∈ V (x) with h μ (T ) ≤ t}. By Lemma 4.1, we have h top (T, E  (α)) ≤ h top (T, R(t)) ≤ t. This proves (4.3). Now Theorem 1.1 just follows from (4.1)–(4.3).   4.2. A high dimensional version of Theorem 1.2. In this subsection, we present and prove a high dimension version of Theorem 1.2. We first give some notation. Let k ∈ N. For each i = 1, . . . , k, let i = {log φn,i }∞ n=1 be an asymptotically sub-additive potential on (X, T ). For a = (a1 , . . . , ak ) ∈ Rk , let E  (a) = {x ∈ X : λi (x) = ai , i = 1, 2, . . . , k}. For any b = (b1 , . . . , bk ) ∈ Rk , define |b| := max{|bi | : i = 1, . . . , k}.

(4.4)

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

17

For x = (x1 , . . . , xk ) and y = (y1 , . . . , yk ) ∈ Rk , we write x ≥ y if xi ≥ yi for all 1 ≤ i ≤ k. For A ⊆ Rk , write cl+ (A) = {x ∈ Rk : ∃ (y j ) ⊂ A such that x ≥ y j and lim y j = x}. j→∞

(4.5)

For a real valued function f defined on a convex open set U ⊂ Rk , let ∂ f (x), ∂ e f (x), (x ∈ U ), ∂ f (U ) and ∂ e f (U ) be defined as in Sect. 2.3. The following result is a high dimensional version of Theorem 1.2. Theorem 4.2. Assume h top (T ) < ∞ and β() > −∞. Then P is a real continuous convex function on Rk+ . Moreover, (i) For any t ∈ Rk+ , if a ∈ ∂ e P (t), then ⎞ ⎛  lim h top ⎝T, E  (b)⎠ = inf {P (q) − a · q} = P (t) − a · t. →0

|b−a|<

q∈Rk+

Moreover the first equality is also valid when a ∈ cl+ (∂ e P (Rk+ )). (ii) For any t ∈ Rk+ , if a ∈ ∂ P (t), then   lim sup h μ (T ) : μ ∈ M(X, T ), |∗ (μ) − a| <  = inf {P (q) − a · q}. →0

q∈Rk+

Furthermore the above equality is valid for a ∈ cl+ (∂ P (Rk+ )). (iii) For any a ∈ ∂ P (Rk+ ) ∩ ri(A), inf {P (q) − a · q} = sup{h μ (T ) : ∗ (μ) = a},

q∈Rk+

where A := {a ∈ Rk : a = ∗ (μ) for some μ ∈ M(X, T )}, and ri denotes the relative interior (cf. [45]). Remark 4.3. When i (i = 1, . . . k) are all asymptotically additive, the results in Theorem 4.2 can be extended accordingly. Indeed, one can replace all the terms Rk+ in Theorem 4.2 by Rk , except the two terms in cl+ (∂ e P (Rk+ )) and cl+ (∂ P (Rk+ )). To prove Theorem 4.2, we need some preparations. For any a = (a1 , . . . , ak ) ∈ Rk and  > 0, define     1  G(a, n, ) := x ∈ X :  log φ,i (x) − ai  <  for all 1 ≤ i ≤ k and  ≥ n . (4.6)  We have the following Lemma 4.4. Assume that G(a, n, ) = ∅. Then (i) For any q = (q1 , . . . , qk ) ∈ Rk+ , h top (T, G(a, n, )) ≤ P (q) −

k i=1

(ai − )qi .

18

D. J. Feng, W. Huang

(ii) Assume furthermore that all i (i = 1, . . . , k) are asymptotically additive. Then for any q = (q1 , . . . , qk ) ∈ Rk , h top (T, G(a, n, )) ≤ P (q) − a · q + 

k

|qi |.

i=1

Proof. We first prove (i). Fix q = (q1 , . . . , qk ) ∈ Rk+ . It suffices to show that for any s < h top (T, G(a, n, )), P (q) ≥ s +

k (ai − )qi . i=1

Let s < h top (T, G(a, n, )) be given. By definition (cf. Sect. 2.1), there exists γ > 0 such that h top (T, G(a, n, ), γ ) > s. Therefore (cf. Sect. 2.1) ∞ = M(G(a, n, ), s, γ ) = lim M(G(a, n, ), s, N , γ ). N →∞

Hence there exists N0 such that M(G(a, n, ), s, N , γ ) ≥ 1, ∀ N ≥ N0 . Now take N ≥ max{n, N0 } and let F be a (N , γ )-separated subset of G(a, n, ) with the maximal cardinality. Then x∈F B N (x, γ ) ⊇ G(a, n, ). It follows #F · exp(−s N ) ≥ M(G(a, n, ), s, N , γ ) ≥ 1. Since

k

i=1 qi

log φ N ,i (x) ≥ N (

PN (T, q · , γ ) ≥



exp

x∈F

k

k

i=1 (ai

(4.7)

− )qi ) for each x ∈ G(a, n, ), we have !

qi log φ N ,i (x) ≥ #F · exp N

i=1

k

!! (ai − )qi

.

i=1

k (ai − )qi )). Combining this with (4.7) yields PN (T, q · , γ ) ≥ exp(N (s + i=1 k Taking N → ∞ we obtain P (T, q · , γ ) ≥ s + i=1 (ai − )qi . Hence we have P (q) = P (T, q · ) ≥ s +

k

(ai − )qi ,

i=1

which finishes the proof (i). The proof of (ii) is almost identical. The only difference is to use the inequality k

qi log φ N ,i (x) ≥ N

i=1

for each x ∈ G(a, n, ) and q ∈ Rk .

k i=1

 

! (ai qi − |qi |

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

19

As a corollary, we have Corollary 4.5. Let a = (a1 , . . . , ak ) ∈ Rk and  > 0. Let E  (a) be defined as in (4.4). Then ⎞ ⎛ k  h top ⎝T, E  (b)⎠ ≤ P (q) − (ai − )qi for any q = (q1 , . . . , qk ) ∈ Rk+ , |b−a|<

i=1

 whenever |b−a|< E  (b) = ∅. Furthermore if all i (i = 1, . . . , k) are asymptotically additive, then ⎛ h top ⎝T,

⎞



E  (b)⎠ ≤ P (q) − a · q + 

|b−a|<

whenever



|b−a|<

Proof. Observe that

k

|qi |, ∀ q = (q1 , . . . , qk ) ∈ Rk ,

i=1

E  (b) = ∅. 

|b−a|<

⎛ h top ⎝T,

E  (b) ⊆



∞

n=1

G(a, n, ). By Proposition 2.1, we obtain

⎞ E  (b)⎠ ≤ h top T,

|b−a|<

∞ 

! G(a, n, )

n=1

= sup h top (T, G(a, n, )). n≥1

By Lemma 4.4, we obtain the desired result.

 

Lemma 4.6. Assume h top (T ) < ∞ and β() > −∞. Let q ∈ Rk+ . Then for any  > 0, there exists ν ∈ E(X, T ) such that h ν (T ) + q · ∗ (ν) ≥ P (q) − . Proof. Let  > 0. By Theorem 3.1, there exists μ ∈ M(X, T ) such that h μ (T ) + q · ∗ (μ) ≥ P (q) − .  Let μ = E (X,T ) θ dm(θ ) be the ergodic decomposition of μ. Then by Proposition A.1(3), we have  (h θ (T ) + q · ∗ (θ )) dm(θ ) = h μ (T ) + q · ∗ (μ) ≥ P (q) − . E (X,T )

 Hence there exists at least one ν ∈ E(X, T ) such that h ν (T ) + q · ∗ (ν) ≥ P (q) − .  The following result is important in the proof of Theorem 4.2. Lemma 4.7. Assume h top (T ) < ∞ and β() > −∞. Let t ∈ Rk+ . Assume a ∈ ∂ e P (t). Then for any  > 0, there exists ν ∈ E(X, T ) such that |∗ (ν) − a| <  and |h ν (T ) − (P (t) − a · t)| < .

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D. J. Feng, W. Huang

Proof. By Proposition 3.2, P is a real continuous convex function on Rk+ . Let t = (t1 , . . . , tk ) ∈ Rk+ and  > 0. We first assume that P is differentiable at t. Let a = P (t)   and write a = (a1 , . . . , ak ). Set δ = min{ 3k , 3k|t| }. Choose γ0 > 0 such that |P (t + s) − P (t) − a · s| <δ |s|

for all s ∈ Rk with 0 < |s| ≤ γ0 .

(4.8)

Pick η such that 0 < η ≤ min{/3, δγ0 }. By Lemma 4.6, there exists ν ∈ E(X, T ) such that h ν (T ) + t · ∗ (ν) ≥ P (t) − η.

(4.9)

Meanwhile by Theorem 3.1, h ν (T ) + (t + s) · ∗ (ν) ≤ P (t + s)

for all s ∈ Rk with t + s ∈ Rk+ .

(4.10)

for all s ∈ Rk with t + s ∈ Rk+ .

(4.11)

Combining (4.10) and (4.9) yields P (t + s) − P (t) ≥ s · ∗ (ν) − η

Construct points si ∈ Rk (i = 1, . . . , k) by si = (si,1 , . . . , si,k ), where  0 if i = j, si, j = γ0 if i = j. Taking s = ±si in (4.11) yields P (t + si ) − P (t) η ≥ (i )∗ (ν) − γ0 γ0

P (t − si ) − P (t) η ≤ (i )∗ (ν) + . −γ0 γ0

and

Combining the above two inequalities with (4.8), we have |(i )∗ (ν) − ai | ≤ δ +

η ≤ 2δ < , i = 1, . . . , k, γ0

which combining with (4.9) and (4.10) yields h ν (T ) ≥ P (t)−η −

k

ti (ai + 2δ) ≥ P (t)−a · t − η − 2k|t|δ > P (t)−a · t−,

i=1

and h ν (T ) ≤ P (t) − t · ∗ (ν) ≤ P (t) −

k

ti (ai − 2δ) < P (t) − a · t + .

i=1

This proves the lemma in the case that P is differentiable at t. Now assume that P is not differentiable at t. Let a = (a1 , . . . , ak ) ∈ ∂ e P (t). Since P is a real continuous convex function on Rk+ , by Proposition 2.2, there exists a sequence (tn ) ⊂ Rk+ converging to t such that P (tn ) exists for each n and lim P (tn ) = a. n→∞ Choose a large integer n such that        P (tn ) − a <  and  P (tn ) − P  (tn ) · tn − (P (t) − a · t) <  . (4.12)   2 2

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

21

As proved in the last paragraph, we can choose ν ∈ E(X, T ) such that      ∗ (ν) − P  (tn ) <  and h ν (T ) − P (tn ) − P  (tn ) · tn  <  . (4.13)   2 2 Combining (4.12) with (4.13) yields |∗ (ν) − a| <  and |h ν (T ) − (P (t) − a · t)| < . This finishes the proof of the lemma.   Proof of Theorem 4.2. By Proposition 3.2, P is a real continuous convex function on Rk+ . We first prove part (i) of the theorem. Let t = (t1 , . . . , tk ) ∈ Rk+ and a = (a1 , . . . , ak ) ∈ ∂ e P (t). Let  > 0. Then by Lemma 4.7, there exists ν ∈ E(X, T ) such that |∗ (ν) − a| <  and |h ν (T ) − (P (t) − a · t)| < .

(4.14)

Since ν ∈ E(X, T ), by Proposition A.1(1), λi (x) = (i )∗ (ν) for ν-a.e. x ∈ X , i = 1, . . . , k. That is, ν(E  (∗ (ν))) = 1. By Proposition 2.1(3), h top (T, E  (∗ (ν))) ≥ h ν (T ). Since ∗ (ν) and h ν (T ) satisfy (4.14), we have ⎞ ⎛  E  (b)⎠ ≥ h top (T, E  (∗ (ν))) ≥ h ν (T ) ≥ P (t) − a · t − . h top ⎝T, |b−a|<

On the other hand by Corollary 4.5, we have ⎞ ⎛ k  h top ⎝T, E  (b)⎠ ≤ P (t) − (ai − )ti . |b−a|<

i=1

Combining the above two inequalities and letting  → 0, we obtain ⎞ ⎛  lim h top ⎝T, E  (b)⎠ = P (t) − a · t = inf {P (q) − a · q}. →0

q∈Rk+

|b−a|<

Now assume a ∈ cl+ (∂ e P (Rk )). Then there exist t j ∈ Rk+ and b j ∈ ∂ e P (t j ), j ∈ N such that a ≥ b j and lim j→∞ b j = a. Let  > 0. There exists a large j such that |a − b j | < 2 . Thus ⎞ ⎞ ⎛ ⎛   E  (b)⎠ ≥ h top ⎝T, E  (b)⎠ h top ⎝T, |b−b j |< 2

|b−a|<

≥ P (t j ) − b j · t j ≥ P (t j ) − a · t j ≥ inf {P (q) − a · q}, q∈Rk+

and hence

⎛ lim h top ⎝T,

→0

 |b−a|<

⎞ E  (b)⎠ ≥ inf {P (q) − a · q}. q∈Rk+

22

D. J. Feng, W. Huang

Meanwhile, the upper bound follows from Corollary 4.5. This finishes the proof of part (i). To show (ii), we first prove the following upper bound:   lim sup h μ (T ) : μ ∈ M(X, T ), |∗ (μ)−a| <  ≤ inf {P (q)−a · q} →0

q∈Rk+

(4.15) for any a = (a1 , . . . , ak ) ∈ Rk , where we take the convention sup ∅ = −∞. To see it, let q = (q1 , . . . , qk ) ∈ Rk+ and  > 0. Then by Theorem 3.1, for any μ ∈ M(X, T ) satisfying |∗ (μ) − a| < , h μ (T ) ≤ P (q) − q · ∗ (μ) ≤ P (q) −

k

(ai − )qi .

i=1

  k (ai − )qi . That is, sup h μ (T ) : μ ∈ M(X, T ), |∗ (μ) − a| <  ≤ P (q) − i=1 Letting  → 0 yields (4.15). Now we prove the lower bound. Assume a ∈ ∂ P (t) for some t ∈ Rk+ . Let  > 0. Then by Minkowski’s Theorem (cf. Sect. 2.3), there exist a j ∈ ∂ e P (t) and λ j ∈ [0, 1],  j = 1, . . . , k + 1, such that k+1 j=1 λ j = 1 and a=

k+1

λjaj.

(4.16)

j=1

By Lemma 4.7, there exist ν j ∈ E(X, T ), j = 1, . . . , k + 1 such that     ∗ (ν j ) − a j  < , h ν (T ) − (P (t) − a j · t) < . i Set ν =

k+1

j=1 λ j ν j .

(4.17)

Then ν ∈ M(X, T ) and

∗ (ν) =

k+1

λ j ∗ (ν j ), h ν (T ) =

j=1

k+1

λ j h ν j (T ).

j=1

Combining these with (4.16) and (4.17) yields |∗ (ν) − a| < , |h ν (T ) − (P (t) − a · t)| < .  Thus sup h μ (T ) : μ ∈ M(X, T ), |∗ (μ) − a| <  ≥ h ν (T ) ≥ P (t) − a · t − . Letting  → 0 yields the desired lower bound   lim sup h μ (T ) : μ ∈ M(X, T ), |∗ (μ) − a| <  

→0

≥ P (t) − a · t = inf {P (q) − a · q}. q∈Rk+

In the end, we assume a ∈ cl+ (∂ P ). Then there exist t j ∈ Rk+ and b j ∈ ∂ e P (t j ), j ∈ N such that a ≥ b j and lim j→∞ b j = a. Let  > 0. There exists a large j such

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

23

that |a − b j | < 2 . Thus   sup h μ (T ) : μ ∈ M(X, T ), |∗ (μ) − a| <      ≥ sup h μ (T ) : μ ∈ M(X, T ), ∗ (μ) − b j  < 2 ≥ P (t j ) − b j · t j ≥ P (t j ) − a · t j ≥ inf {P (q) − a · q}, q∈Rk+

and hence

  lim sup h μ (T ) : μ ∈ M(X, T ), |∗ (μ) − a| <  ≥ inf {P (q) − a · q}.

→0

q∈Rk+

This finishes the proof of part (ii). To show part (iii), let A = {a ∈ Rk : a = ∗ (μ) for some μ ∈ M(X, T )}. Clearly, A is non-empty and convex. Define g : A → R by g(a) = sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = a}. Then g is a real-valued concave function on A. Take W (x) = sup{g(a) + a · x : a ∈ A}, ∀ x ∈ Rk . Apply Corollary 2.5 to obtain inf{W (x) − a · x : x ∈ Rk } = g(a), ∀ a ∈ ri(A).

(4.18)

However, by Theorem 3.1, we have P (q) = W (q) for all q ∈ Rk+ . Now assume that a ∈ ∂ P (q) ∩ ri(A) for some q ∈ Rk+ . Then a ∈ ∂ W (q) ∩ ri(A). Hence g(a) = inf{W (x) − a · x : x ∈ Rk } = W (q) − a · q = P (q) − a · q = inf{P (x) − a · x : x ∈ Rk+ }. This finishes the proof of (iii) and hence the proof of Theorem 4.2.

 

Proof of Theorem 1.2. Here we show that Theorem 1.2 is just the one-dimensional version of Theorem 4.2. To see it, let (X, T ) be a TDS with h top (T ) < ∞ and let  = {log φn }∞ n=1 be an asymptotically sub-additive potential on X satisfying β() > −∞. Let t > 0. It is clear that ∂ e P (t) = {P (t−), P (t+)} and ∂ P (t) = [P (t−), P (t+)]. Thus   ∂ P (R+ ) = [P (t−), P (t+)] and ∂ e P (R+ ) = {P (t−), P (t+)}. t>0

t>0

(4.19) Moreover, cl+ (∂ P (R+ )) = ∂ P (R+ ) ∪ {P (∞)} and

cl+ (∂ e P (R+ )) = ∂ e P (R+ ) ∪ {P (∞)}.

(4.20)

Furthermore, by Lemma A.3(4), qβ() ≤ P (q) ≤ h top (T ) + qβ(), from which we obtain P (∞) := limq→∞ P (q)/q = β(). By the way, applying Theorem 4.2(ii), for

24

D. J. Feng, W. Huang

each a ∈ ∂ P (t) and any  > 0, there exists μ ∈ M(X, T ) such that |∗ (μ) − a| < . It implies that " # lim P (t−), P (∞) ⊆ int(A) = ri(A), (4.21) t→0+

where A := {a ∈ R : a = ∗ (μ) for some μ ∈ M(X, T )}. According to (4.19)– (4.21), Theorem 1.2 just follows from Theorem 4.2.   4.3. A high-dimensional version of Theorem 1.3. Let i = {log φn,i }∞ n=1 (i = 1, . . . , k) be asymptotically sub-additive potentials on a TDS (X, T ). Let  = (1 , . . . , k ). For δ > 0, we define  clδ+ (∂ P (Rk+ )) := cl+ ( ∂ P (t)), (4.22) {t∈Rk+ : ti ≥δ, i=1,2,··· ,k}

where cl+ (A) is defined as in (4.5). The following theorem is a high dimensional version of Theorem 1.3. Theorem 4.8. Assume h top (T ) < ∞, β() > −∞, and that the entropy map μ → h μ (T ) is upper semi-continuous on M(X, T ). Then (i) For any t ∈ Rk+ , if a ∈ ∂ e P (t), then E  (a) = ∅ and h top (T, E  (a)) = inf {P (q) − a · q} = P (t) − a · t. q∈Rk+

(ii) For a ∈



δ k δ>0 cl+ (∂ P (R+ )),

inf {P (q) − a · q} = max{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = a}.

q∈Rk+

(iii) If t ∈ Rk+ such that t ·  has a unique equilibrium state μt ∈ M(X, T ), then μt is ergodic, P (t) = ∗ (μt ), E  (P (t)) = ∅ and h top (T, E  (P (t))) = h μt (T ). Proof. We first prove (i). Fix t ∈ Rk+ . By Theorem 3.3, I(, t) is a non-empty compact convex subset of M(X, T ). We claim that for any b ∈ Rk the set Ib (, t) := {ν ∈ I(, t) : ∗ (ν) = b} is compact and convex. The convexity is clear. To show the compactness, assume that {νn } ⊂ Ib (, t) and νn converges to ν in M(X, T ). Then by the upper semi-continuity of h (·) (T ) and (i )∗ (·), i = 1, . . . , k, we have h ν (T ) + t · ∗ (ν) ≥ lim h νn (T ) + t · ∗ (νn ) = P (t). n→∞

By Theorem 3.1, ν ∈ I(, t) and furthermore, h ν (T ) = h νn (T ) = P (t) − t · b and ∗ (ν) = ∗ (νn ) = b. This is, ν ∈ Ib (, t). Hence Ib (, t) is compact. This finishes the proof of the claim. Now let a ∈ ∂ e P (t). By Theorem 3.3, the set Ia (, t) is non-empty. We are going to show further that Ia (, t) contains at least one ergodic measure. Since Ia (, t) is a non-empty compact convex subset of M(X, T ), by the Krein-Milman theorem (cf. [17,

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

25

p. 146]), it contains at least one extreme point, denoted by ν. Let ν = pν1 + (1 − p)ν2 for some 0 < p < 1 and ν1 , ν2 ∈ M(X, T ). Then P (t) = h ν (T ) + t · ∗ (ν) = p(h ν1 (T ) + t · ∗ (ν1 )) + (1 − p)(h ν2 (T ) + t · ∗ (ν2 )). By Theorem 3.1, ν1 , ν2 ∈ I(, t). By Theorem 3.3, ∗ (ν1 ), ∗ (ν2 ) ∈ ∂ P (t). Moreover, note that a = ∗ (ν) = p∗ (ν1 ) + (1 − p)∗ (ν2 ), we have ∗ (ν1 ) = ∗ (ν2 ) = a since a ∈ ∂ e P (t). That is, ν1 , ν2 ∈ Ia (, t). Since ν is an extreme point of Ia (, t), we have ν1 = ν2 = ν. It follows that ν is an extreme point of M(X, T ), i.e., ν is ergodic. By Proposition A.1(1), we have ν(E  (a)) = 1, and thus by Proposition 2.1(3), h top (T, E  (a)) ≥ h ν (T ) = P (t) − t · a = inf {P (q) − q · a}. q∈Rk+

However by Corollary 4.5, the upper bound h top (T, E  (a)) ≤ inf q∈Rk+ {P (q) − q · a} is generic. Thus we have the equality h top (T, E  (a)) = h ν (T ) = P (t) − t · a = inf {P (q) − q · a}. q∈Rk+

This finishes the proof of part (i). To show (ii), by Theorem 4.2(ii) we need only to show that for a ∈ clδ+ (∂ P ), inf {P (q) − a · q} ≤ max{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = a}.

q∈Rk+

To see it, we first assume that a ∈ ∂ P (t) for some t ∈ Rk+ . By Theorem 4.2(ii), there exists (ν j ) ⊂ M(X, T ) such that lim ∗ (ν j ) = a and lim sup h ν j (T ) ≥ inf {P (q) − a · q}.

j→∞

j→∞

q∈Rk+

Extract a subsequence if necessary so that lim j→∞ ν j = ν for some ν ∈ M(X, T ). Then h ν (T ) ≥ lim sup h ν j (T ) ≥ inf {P (q) − a · q} = P (t) − a · t j→∞

q∈Rk+

and ∗ (ν) ≥ lim sup j→∞ ∗ (ν j ) = a by the upper-semi continuity of h (·) (T ) and (i )∗ (·). Hence h ν (T ) ≥ P (t) − a · t ≥ P (t) − ∗ (ν) · t ≥ h ν (T ), which implies ∗ (ν) = a and h ν (T ) = P (t) − a · t. Next we assume that a ∈ clδ+ (∂ P (Rk+ )) for some δ > 0. Then there exists a sequence (t j ) ∈ Rk+ such that each entry of t j is greater than δ, and there exists a j ∈ ∂ e P (t j ) for each j such that a ≥ a j and lim j→∞ a j = a. By the above discussion, for each j ∈ N there exists μ j ∈ M(X, T ) such that ∗ (μ j ) = a j and h μ j (T ) = P (t j ) − a j · t j .

26

D. J. Feng, W. Huang

Extract a subsequence if necessary so that lim j→∞ μ j = μ for some μ ∈ M(X, T ). Thus ∗ (μ) ≥ lim j→∞ ∗ (μ j ) = a and h μ (T ) ≥ lim sup h μ j (T ) = lim sup(P (t j ) − a j · t j ) j→∞

j→∞

≥ lim sup(P (t j ) − a · t j ) j→∞

≥ lim sup(h μ (T ) + (∗ (μ) − a) · tj ) j→∞

≥ h μ (T ) +

k ((i )∗ (μ) − ai )δ. i=1

This implies that ∗ (μ) = a and h μ (T ) ≥ lim sup(P (t j ) − a · t j ) ≥ inf {P (q) − a · q}. q∈Rk+

j→∞

This finishes the proof of part (ii). Now we turn to prove (iii). We assume t ∈ Rk+ such that t · has a unique equilibrium state μt . By Theorem 3.3, ∂ P (t) = {∗ (μt )}. Now (iii) comes from parts (i) and (ii) of the theorem.   Proof of Theorem 1.3. It follows directly from Theorem 4.8, using the fact that in the one-dimensional case, ∂ e P (t) = {P (t+), P (t−)} for t > 0 and  δ>0

clδ+ (∂ P (R+ )) =



[P (t−), P (∞)].

t>0

  Remark 4.9. Theorem 1.3(i) has a nice application in the multifractal analysis of measures on symbolic spaces. Let μ be a fully supported Borel probability measure on the one-sided full shift space (, σ ) over a finite alphabet. Assume in addition that μ(In+m (x)) ≤ Cμ(In (x))μ(Im (σ n x)),

∀ x ∈ , n, m ∈ N,

(4.23)

where C > 0 is a constant and In (y) denotes the n th cylinder in  containing y. Let  be a potential on  given by  = {log μ(In (x))}∞ n=1 . By applying a general multifractal result of Ben Nasr [10], Testud [48] obtained that (formulated in our terminologies as) h top (σ, E  (α)) = inf{P (q) − αq : q > 0} whenever α = P  (t) for some t > 0, provided that P (t) exists at t. However by Proposition A.5(i),  is asymptotically sub-additive, hence by Theorem 1.3(i), the above variational relation actually holds for any α = P (t+) and α = P (t−) for each t > 0. Furthermore, the constant C in (4.23) can be replaced by Cn , where (Cn ) is a sequence of positive numbers satisfying limn→∞ (1/n) log Cn = 0 (cf. Remark A.6(iii)).

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

27

4.4. Lyapunov spectrum for certain sub-additive potentials on symbolic spaces. In this subsection, we assume that (X, T ) is the one-sided full shift over a finite set {1, 2, . . . , m}. That is, X = {1, . . . , m}N endowed with the standard metric d(x, y) = 2−n for x = ∞ and (y )∞ , where n is the largest integer so that x = y for 1 ≤ j ≤ n, and T (xi )i=1 i i=1 j j ∞  → (x ∞ is the shift map given by (xi )i=1 i+1 )i=1 . ∞ Let X ∗ denote the collection of finite words over {1, . . . , m}, i.e., X ∗ = i=1 {1, . . . , m}i . Assume that φ : X ∗ → [0, ∞) is a map (not identically equal to 0) satisfying the following two assumptions: (H1) φ(I J ) ≤ φ(I )φ(J ) for any I, J ∈ X ∗ ; (H2) There exist a sequence of positive integers (tn ) and a sequence of positive numbers (cn ) with limn→∞ tn /n = 0 = limn→∞ (1/n) log cn , such that for each I, J ∈ X ∗ with lengths |I | ≥ n, |J | ≥ n, there exists K ∈ X ∗ with |K | ≤ tn so that φ(I K J ) ≥ cn φ(I )φ(J ). ∞ . Let  = (log φn ) be a potential on X given by φn (x) = φ(x1 . . . xn ) for x = (xi )i=1 It is clear that  is sub-additive. Denote 1 φ(I )q , P(q) := lim sup log n→∞ n

where the sum is taken over the set of I ∈ {1, . . . , m}n with φ(I ) > 0. It is clear that P(q) = P(T, q) for q > 0. Although  is not necessary to be asymptotically additive, we still have the following rather complete result, as an analogue of our recent work [24] on the norm of matrix products. Theorem 4.10. Let  be given as above. Assume that P(q) ∈ R for each q < 0. Then {α ∈ R : E  (α) = ∅} = R ∩ [a, b], where a = limn→−∞ P(q)/q and b = limn→∞ P(q)/q. Furthermore, for α ∈ R ∩ [a, b], h top (T, E(α)) = inf{P(q) − aq : q ∈ R}. Proof of Theorem 4.10. Take a slight modification of the proof of Theorem 1.1 in [24].   We remark that under the condition of the above theorem, we always have b ∈ R. However it is possible that a = −∞. There are some natural maps φ : X ∗ → [0, ∞) which satisfy the assumptions m is a family of d × d real matrices so that there is no (H1)–(H2). For example, if {Mi }i=1 trivial proper linear subspace V ⊂ Rd with Mi V ⊆ V for all i, then the map φ defined by φ(x1 . . . xn ) = Mx1 . . . Mxn  satisfies (H1)-(H2) (see [24,19]). More generally, the m satisfies singular value functions for Mx1 . . . Mxn also satisfy (H1)-(H2) when {Mi }i=1 further mild irreducibility conditions (see [19]). 5. The Multifractal Formalism for Asymptotically Additive Potentials ∞ Let (X, T ) be a TDS. Let k ∈ N and let i = {log φn,i }∞ n=1 , i = {log ψn,i }n=1 (i = 1, 2, · · · , k) be asymptotically additive potentials on (X, T ). Furthermore assume

ψn,i (x) ≥ C(1 + δ)n

(i ∈ {1, 2, · · · , k}, n ∈ N, x ∈ X )

(5.1)

28

D. J. Feng, W. Huang

for some constants C, δ > 0. This assumption guarantees that (i )∗ (μ) = 0 (i = 1, 2, · · · , k) for each μ ∈ M(X, T ). For a = (a1 , . . . , ak ) ∈ Rk , denote  log φn,i (x) (5.2) E(a) := x ∈ X : lim = ai for 1 ≤ i ≤ k . n→∞ log ψn,i (x) In this section, we shall study the multifractal structure of E(a). For μ ∈ M(X, T ), the set G μ of μ-generic points is defined by ⎧ ⎫ n−1 ⎨ ⎬ 1 G μ := x ∈ X : δT j x → μ in the weak* topology as n → ∞ , ⎩ ⎭ n j=0

where δ y denotes the probability measure whose support is the single point y. Bowen [12] showed that h top (T, G μ ) ≤ h μ (T ) for any μ ∈ M(X, T ). Definition 5.1. A TDS (X, T ) is called saturated if for any μ ∈ M(X, T ), we have G μ = ∅ and h top (T, G μ ) = h μ (T ). It was shown independently in [22,44] that if a TDS satisfies the specification property (or a weaker form), then it is saturated. The main result in this section is the following. Theorem 5.2. Let (X, T ) be a TDS and let i , i (i = 1, 2, · · · , k) be asymptotically additive potentials on X satisfying the assumption (5.1). Let ⊂ Rk be the range of the following map from M(X, T ) to Rk : # " (k )∗ (μ) (1 )∗ (μ) (2 )∗ (μ) , ,··· , . μ→ (1 )∗ (μ) (2 )∗ (μ) (k )∗ (μ) For a ∈ , write H (a) = sup{h μ (T ) : μ ∈ M(X, T ), (i )∗ (μ) = ai (i )∗ (μ) for i = 1, 2, · · · , k}. (5.3) Then we have the following properties: (i) {a ∈ Rk : E(a) = ∅} ⊆ . (ii) If h top (T ) < ∞, then we have a ∈ ⇐⇒ inf{Pa (q) : q ∈ Rk } = −∞ ⇐⇒ inf{Pa (q) : q ∈ Rk } ≥ 0,    k where Pa (q) := P T, i=1 qi (i − ai i ) . (iii) Assume that h top (T ) < ∞ and the entropy map is upper semi-continuous. Then for any a ∈ , H (a) = inf Pa (q). q∈Rk

(iv) Assume that (X, T ) is saturated. Then E(a) = ∅ if and only if a ∈ . Furthermore h top (T, E(a)) = H (a),

∀ a ∈ .

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

29

We emphasize that in parts (i)–(iii) of the above theorem, we do not need to assume that (X, T ) is saturated. Proof. We first prove (i). Assume that E(a) = ∅ for some a = (a1 , . . . , ak ) ∈ Rk .  Take x ∈ E(a). Denote μx,n = (1/n) n−1 j=1 δT j x . Then there exists n j ↑ ∞ so that μx,n j → μ for some μ ∈ M(X, T ). Apply Lemma A.4(ii) (in which we take νn = δx ) to obtain log φn j ,i (x) (i )∗ (μ) = lim = ai , i = 1, . . . , k. j→∞ log ψn j ,i (x) (i )∗ (μ) Hence a ∈ . This proves (i). To show (ii), assume h top (T ) < ∞. For a = (a1 , . . . , ak ) ∈ Rk and μ ∈ M(X, T ), we denote τa (μ) = ((1 )∗ (μ) − a1 (1 )∗ (μ), . . . , (k )∗ (μ) − ak (k )∗ (μ)) . Clearly, τa (μ) ∈ Rk , and a ∈ ⇐⇒ τa (μ0 ) = 0 for some μ0 ∈ M(X, T ). Now assume a ∈ . Then there exists μ0 ∈ M(X, T ) such that τa (μ0 ) = 0. Apply Theorem 3.1 to obtain that ! k Pa (q) = P T, qi (i − ai i ) i=1

= sup{h μ (T ) + q · τa (μ) : μ ∈ M(X, T )} ≥ h μ0 (T ) ≥ 0 for each q = (q1 , . . . , qk ) ∈ Rk . Conversely, assume a ∈ . Write A = {τa (μ) : μ ∈ M(X, T )}.

(5.4)

By Lemma A.4(i), τa is a continuous affine function on M(X, T ), hence A is a compact convex set in Rk . a ∈ implies 0 ∈ A. Hence there exists a unit vector v ∈ Rk and c > 0 such that v · b < −c

for any b ∈ A.

By Theorem 3.1, we have for t > 0, Pa (tv) = sup{h μ (T ) + tv · τa (μ) : μ ∈ M(X, T )} ≤ sup{h μ (T ) − tc : μ ∈ M(X, T )} = h top (T ) − tc. Letting t → +∞, we obtain inf{Pa (q) : q ∈ Rk } = −∞. This finishes the proof of (ii). Next we prove (iii). Fix a = (a1 , . . . , ak ) ∈ . Define A as in (5.4). Since a ∈ , we have 0 ∈ A. Define g : A → R by g(t) = sup{h μ (T ) : μ ∈ M(X, T ), τa (μ) = t}.

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It is direct to check that g is concave and upper semi-continuous on A. By the definition of H (see (5.3)), we have H (a) = g(0). Define W (q) = sup{g(t) + q · t : t ∈ A},

∀ q ∈ Rk .

Then by Corollary 2.5(ii), we have g(t) = inf{W (q) − q · t : q ∈ Rk } for all t ∈ A. In particular, H (a) = g(0) = inf{W (q) : q ∈ Rk }.

(5.5)

However, by Theorem 3.1 and the definition of W , we have ! k W (q) = P T, qi (i − ai i ) =: Pa (q). i=1

Hence (5.5) implies H (a) = inf q∈Rk Pa (q). This finishes the proof (iii). In the end we prove (iv). We divide this proof into four steps. Step 1. For a ∈ , we have E(a) ⊇ G μ = ∅ for each μ ∈ M(X, T ) with (i )∗ (μ) = ai (i )∗ (μ) (i = 1, 2, · · · , k). To see this, let x ∈ G μ . By Lemma A.4(ii) (in which we take νn = δx ), we have limn→∞ (1/n) log φn,i (x) = (i )∗ (μ) and limn→∞ (1/n) log ψn,i (x) = (i )∗ (μ). It follows that x ∈ E(a). Hence E(a) ⊇ G μ . Step 2. Let a ∈ Rk so that E(a) = ∅. Then for each x ∈ E(a) and μ ∈ V (x)  (here V (x) denotes the set of limit points of μx,n = (1/n) n−1 j=0 δT j x ), we have (i )∗ (μ)/(i )∗ (μ) = ai for i = 1, 2, · · · , k. To show this, take such x and μ. Then there exists a subsequence sequence n  of natural numbers such that lim→∞ μn  ,x = μ. By Lemma A.4(ii) again (in which we take νn = δx ), we have 1 1 log φn  ,i (x) = (i )∗ (μ) and lim log ψn  ,i (x) = (i )∗ (μ) →∞ n  →∞ n  (i = 1, 2, · · · , k). lim

Since x ∈ E(a), we have log φn,i (x) = ai (i = 1, 2, · · · , k). n→∞ log ψn,i (x) lim

It follows that (i )∗ (μ)/(i )∗ (μ) = ai for i = 1, 2, · · · , k. Step 3. For a ∈ , we have h top (T, E(a)) ≥ H (a). To see it, let μ ∈ M(X, T ) so that (i )∗ (μ) = ai (i ) ∗ (μ) (i = 1, 2, · · · , k). By Step 1, E(a) ⊇ G μ , and hence h top (T, E(a)) ≥ h top (T, G μ ) = h μ (T ). This proves the inequality h top (T, E(a)) ≥ H (a). Step 4. For a ∈ , we have h top (T, E(a)) ≤ H (a). By Step 2, for each x ∈ E(a) and μ ∈ V (x), we have (i )∗ (μ)/(i )∗ (μ) = ai for i = 1, 2, . . . , k, and hence h μ (T ) ≤ H (a). It follows that E(a) ⊆ {x ∈ X : ∃ μ ∈ V (x) with h μ (T ) ≤ H (a)}. By Lemma 4.1, we have h top (T, E(a)) ≤ H (a). This finishes the proof of (iv).

 

We remark that (iii) of Theorem 5.2 can be proved alternatively by applying Proposition 3.15 in [29].

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

31

Proof of Theorem 1.4. Except the second part in (iii), all the statements listed in Theorem 1.4 follow from Theorem 5.2 (in which we take k = 1 and ψn (x) ≡ 1). In the following, we prove the second part in Theorem 1.4(iii): under the assumptions that h top (T ) < ∞ and the entropy map is upper semi-continuous, for any α ∈  :=     t∈R {P (t−), P (t+)} ∪ {P (±∞)}, we have E  (α)  = ∅ and h top (T, E  (α)) = inf{P (q) − αq : q ∈ R}. According to Corollary 4.5, it suffices to show that if α ∈ , then E  (α) = ∅ and h top (T, E  (α)) ≥ inf{P (q) − αq : q ∈ R}. For this purpose, we will show the following claim: Claim. For each α ∈ , there exists an ergodic measure ν such that ∗ (ν) = α and h ν (T ) ≥ inf{P (q) − αq : q ∈ R}. The claim will imply that ν(E  (α)) = 1 (by Kingman’s sub-additive ergodic theorem), and by Proposition 2.1(3), h top (T, E  (α)) ≥ h ν (T ) ≥ inf{P (q) − αq : q ∈ R}. In the following we prove the claim in a way similar to the proof of Theorem 4.8. First we consider the case α ∈ {P (t−), P (t+)} for some t ∈ R. Fix t and denote I(, t) = {μ ∈ M(X, T ) : P (t) = h μ (T ) + t∗ (μ)}. By Theorem 3.3(ii), I(, t) is a non-empty compact convex subset of M(X, T ). Furthermore, for any b ∈ R the set Ib (, t) := {ν ∈ I(, t) : ∗ (ν) = b} is compact and convex (may be empty). The convexity of Ib (, t) is clear. To show the compactness, assume that {νn } ⊂ Ib (, t) and νn converges to ν in M(X, T ). Then by the upper semi-continuity of h (·) (T ) and the continuity of ∗ (·), we have h ν (T ) + t∗ (ν) ≥ lim h νn (T ) + t∗ (νn ) = P (t). n→∞

By Theorem 3.1, ν ∈ I(, t) and furthermore, h ν (T ) = h νn (T ) = P (t) − tb and ∗ (ν) = ∗ (νn ) = b. This is, ν ∈ Ib (, t). Hence Ib (, t) is compact. Since α ∈ {P (t−), P (t+)} ⊂ ∂ P (t), by Theorem 3.3, Iα (, t) is non-empty. We are going to show further that Iα (, t) contains at least one ergodic measure. Since Iα (, t) is a non-empty compact convex subset of M(X, T ), by the Krein-Milman theorem, it contains at least one extreme point, denoted by ν. Let ν = pν1 + (1 − p)ν2 for some 0 < p < 1 and ν1 , ν2 ∈ M(X, T ). We will show that ν1 = ν2 , which implies that ν is ergodic. To see that ν1 = ν2 , note that P (t) = h ν (T ) + t∗ (ν) = p(h ν1 (T ) + t∗ (ν1 )) + (1 − p)(h ν2 (T ) + t∗ (ν2 )). By Theorem 3.1, ν1 , ν2 ∈ I(, t). By Theorem 3.3, ∗ (ν1 ), ∗ (ν2 ) ∈ ∂ P (t). Moreover, note that α = ∗ (ν) = p∗ (ν1 ) + (1 − p)∗ (ν2 ), we have ∗ (ν1 ) = ∗ (ν2 ) = α since α is an extreme point of ∂ P (t) (noting that ∂ P (t) = [P (t−), P (t+)]). That is, ν1 , ν2 ∈ Iα (, t). Since ν is an extreme point of Iα (, t), we have ν1 = ν2 = ν. Therefore, ν is ergodic. Since ν ∈ Iα (, t), we have ∗ (ν) = α, and h ν (T ) = P (t) − αt ≥ inf{P (q) − αq : q ∈ R}. This proves the claim in the case that α ∈ {P (t−), P (t+)} ⊂ ∂ P (t) for some t ∈ R.

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D. J. Feng, W. Huang

Next, we consider the case α ∈ {P (±∞)}. First assume that α = P (+∞). By Theorems 1.1 and 1.2, α = β() = max{∗ (μ) : μ ∈ M(X, T )}. By the convexity of P (·), there exists a sequence (t j ) ↑ +∞, such that P (t j ) := α j exists and α j ↑ α when j → ∞. As proved in the last paragraph, for each j ∈ N, there exists μ j ∈ M(X, T ) such that ∗ (μ j ) = α j and h μ j (T ) = P (t j )−α j t j . Extract a subsequence if necessary so that lim j→∞ μ j = μ for some μ ∈ M(X, T ). Then ∗ (μ) = lim j→∞ ∗ (μ j ) = α and h μ (T ) ≥ lim sup h μ j (T ) = lim sup(P (t j ) − α j t j ) j→∞

j→∞

≥ lim sup(P (t j ) − αt j ) ≥ inf {P (q) − αq}, q∈R

j→∞

where we use the facts α j ≤ α and t j > 0 for the second inequality. Now let μ =  θ dm(θ ) be the ergodic decomposition of μ ∈ M(X, T ). By Proposition A.1(3), ∗ (θ ) dm(θ ) = ∗ (μ) = α. By the way, we also have h θ (T ) dm(θ ) = h μ (T ) (cf. [49]). Note that ∗ (θ ) ≤ α for any θ . Hence there exists an ergodic measure ν such that ∗ (ν) = α and h ν (T ) ≥ h μ (T ) ≥ inf q∈R {P (q) − αq}, as desired. In the end, assume  (+∞). Since − is also asymptotically additive, there α = P (−∞). Then −α = P− exists an ergodic measure η such that (−)∗ (η) = −α, i.e., ∗ (η) = α, and h η (T ) ≥ inf {P− (q) − (−α)q} = inf {P (−q) + αq} = inf {P (q) − αq}. q∈R

q∈R

q∈R

This finishes the proof of the claim, and also the proof of Theorem 1.4.

 

6. Examples In this section, we give some examples regarding Lyapunov spectra on TDS’s on which the entropy map is not upper semi-continuous. The multifractal behaviors in this case are rather irregular and complicated. These examples also show that the conditions and results in Theorems 1.1, 1.2 and 4.2 are optimal. Example 6.1. There exist a TDS (X, T ) with h top (T ) < ∞ and an additive potential     = {log φn }∞ n=1 on X such that lim t→0+ P (t−), P (∞) = ∅ and for α = P (∞), h top (T, E  (α)) = sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α} < inf {P (q) − αq}. q>0

Construction. According to Krieger [36], for each i ∈ N, we can construct a Cantor set X i ⊆ [0, 1i ] × { 1i } and a continuous transformation Ti : X i → X i such that (X i , Ti ) is uniquely ∞ergodic (i.e., M(X i , Ti ) consists of a singleton) and h top (Ti ) = 1. Then we let X = i=1 X i ∪ {(0, 0)} and define T : X → X by  Ti (x) if x ∈ X i , T (x) = x if x = (0, 0). It is easy to check that (X, T ) is a TDS. Define a function g : X → R by  1 − 1/i if x ∈ X i , g(x) = 1 if x = (0, 0).

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

33

  n−1 ∞ j Let φn (x) = exp j=0 g(T x) for n ∈ N and x ∈ X . Then  = {log φn }n=1 is an additive potential on X . For i ∈ N, let μi denote the unique element in M(X i , Ti ). Let μ0 be the Dirac measure δ(0,0) at the point (0, 0). Then E(X, T ) = {μi : i = 0, 1, · · · } and thus  M(X, T ) =

∞

λi μi : λi ≥ 0 and

i=0

∞

λi = 1 .

i=0

By Theorem 3.1, we have P (q) = sup{h μ (T ) + q∗ (μ) : μ ∈ M(X, T )} 

∞ ∞ ∞ = sup h i=0 λi ∗ (μi ) : λi ≥ 0 and λi = 1 λi μi (T ) + q  = sup 

i=0 ∞



i=0



λi h μi (T ) + q∗ (μi ) : λi ≥ 0 and

i=0

∞



λi = 1

i=0

= sup qλ0 +

∞

λi (1 + q(1 − 1/i))) : λi ≥ 0 and

i=1

∞

λi = 1

i=0

 = max q, sup{1 + q(1 − 1/i)} = q + 1 for q > 0. i∈N

  Hence P (q) = 1 for q > 0 and P (∞) = 1. Thus limt→0+ P (t−), P (∞) = ∅. For α = P (∞) = 1, one has E  (α) = {(0, 0)}. Hence 0 = h top (T, E  (α)) = sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = α} < inf {P (q) − αq} = 1, q>0

as desired.

 

Example 6.2. There exist a TDS (X, T ) with h top (T ) < ∞, an additive potential  = {log φn }∞ n=1 on X such that for each α ∈ [β(), β()], h top (T, E  (α)) < inf {P (q) − αq}, q∈R

where β() := limn→∞

1 n

inf x∈X log φn (x).

Construction. Similar to the construction in Example 6.1, we construct Cantor sets 1 i ] × { |i|+1 } (i ∈ Z) and continuous transformations Ti : X i → X i such that X i ⊆ [0, |i|+1  |i| . Then let X = i∈Z X i ∪ {(0, 1)} ∪ (X i , Ti ) is uniquely ergodic and h top (Ti ) = |i|+1 {(0, −1)} and define T : X → X by  Ti (x) if x ∈ X i , T (x) = x if x = (0, 1) or (0, −1).

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D. J. Feng, W. Huang

It is clear that (X, T ) is a TDS. Define a continuous function h on X by ⎧ i ⎪ ⎨ |i|+1 if x ∈ X i , g(x) = 1 if x = (0, 1), ⎪ ⎩−1 if x = (0, −1).   n−1 j Let φn (x) = exp j=0 g(T x) for n ∈ N and x ∈ X . Then  = {log φn } is an additive potential on X with [β(), β()] = [−1, 1]. Similarly, it is not hard to verify that  |i| i P (q) = max q, −q, sup + q = 1 + |q|. |i| + 1 i∈Z |i| + 1 Hence P (∞) = P (0+) = 1, P (0−) = P (−∞) = −1 and  1 if q > 0,  P (q) = −1 if q < 0. i : i ∈ Z}∪{1, −1}. Furthermore It is easy to see that E  (α) = ∅ if and only if α ∈ { |i|+1 ⎧ i if α = |i|+1 for some i ∈ Z, ⎨ Xi E  (α) = {(0, 1)} if α = 1, ⎩ {(0, −1)} if α = −1.

Hence for α ∈ [β(), β()] = [−1, 1], h top (T, E  (α)) < 1 = inf {P (q) − αq}, q∈R

as desired. Keep in mind that {∗ (μ) : μ ∈ M(X, T )} = [−1, 1] by Lemma A.3.

 

Example 6.3. There exist a TDS (X, T ) with h top (T ) < ∞ and two additive potential 2 i = {log φn,i }∞ n=1 (i = 1, 2) on X such that ∂ P (R+ ) is one-dimensional set and for 2 any a ∈ ∂ P (R+ ), where  = (1 , 2 ), sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = a} < inf {P (q) − a · q}. q∈R2+

Construction. Similar to the previous two examples, we construct a Cantor set X i ⊆ 1 i [0, |i|+1 ] × { |i|+1 } and a continuous transformation Ti : X i → X i such that (X i , Ti ) is  uniquely ergodic and h top (Ti ) = 1. Then let X = i∈Z X i ∪ {(0, 1)} ∪ {(0, −1)} and define T : X → X by  Ti (x) if x ∈ X i , T (x) = x if x = (0, 1) or (0, −1). It is clear that (X, T ) is a TDS. Define two continuous functions g1 , g2 on X by ⎧ i ⎧ 0 if x ∈ X i , i ≥ 0, if x ∈ X i , i ≥ 0, ⎪ ⎪ ⎪ ⎪ i+1 ⎪ ⎪ ⎨ ⎨ 1 if x = (0, 1), 0 if x = (0, 1), and g2 (x) = −|i| g1 (x) = 2|i| ⎪ ⎪ ⎪ |i|+1 if x ∈ X i , i < 0, ⎪ |i|+1 if x ∈ X i , i < 0, ⎪ ⎪ ⎩ ⎩ 2 if x = (0, −1). −1 if x = (0, −1).

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

Set φn,i (x) = exp

 n−1

j=0 gi (T

j x)



35

for i = 1, 2, n ∈ N and x ∈ X . Then i =

{log φn,i }∞ n=1 , i = 1, 2, are two additive potentials on X with β(1 ) = 2, β(2 ) = 0. For i ∈ Z, let μi denote the unique element in M(X i , Ti ). Let μ∞ be the Dirac measure δ(0,1) at the point (0, 1). Let μ−∞ be the Dirac measure δ(0,−1) at the point (0, −1). To simplify, write Z = Z ∪ {±∞}. Then E(X, T ) = {μi : i ∈ Z}. A direct calculation by applying Theorem 3.1 yields that for q = (q1 , q2 ) ∈ R2+ , P (q) = max{1 + q1 , 1 + 2q1 − q2 }. Hence

 P (q)

=

(1, 0) (2, −1)

if q ∈ R2+ with q1 < q2 if q ∈ R2+ with q1 > q2

and ∂ P ((q, q)) = conv({(1, 0), (2, −1)}) for q > 0. Thus ∂ P (R2+ ) = conv({(1, 0), (2, −1)}) is one dimensional. Recall that A := {∗ (μ) : μ ∈ M(X, T )}. Clearly, A = conv({(0, 0), (1, 0), (2, −1)}) is a two-dimensional set and ∂ P (R2+ ) is just one edge in the convex set A. For a ∈ ∂ P (R2+ ), there exists unique t ∈ [0, 1] with a = t (1, 0)+(1−t)(2, −1). It is not hard to see that for μ ∈ M(X, T ), ∗ (μ) = a if and only if μ = tμ∞ + (1 − t)μ−∞ . Hence sup{h μ (T ) : μ ∈ M(X, T ), ∗ (μ) = a} = h tμ∞ +(1−t)μ−∞ (T ) = 0 < 1 = inf {max{1 + q1 , 1 + 2q1 − q2 } q∈R2+

− (tq1 + (1 − t)(2q1 − q2 ))} = inf {P (q) − a · q}, q∈R2+

as desired.

 

Acknowledgements. The first author was partially supported by the direct grant and RGC grants in the Hong Kong Special Administrative Region, China (projects CUHK400706, CUHK401008). The second author was partially supported by NSFC, 973 project (2006CB805903) and FANEDD (Grant 200520). The authors thank Yongluo Cao and Katrin Gelfert for their valuable comments. They also thank the anonymous referee for his helpful comments and suggestions that improved the manuscript.

Appendix A. Properties and Examples of Asymptotical Sub-additive Potentials In this Appendix, we give some properties and examples of asymptotically sub-additive (resp. asymptotically additive) potentials. Let (X, T ) be a TDS and let  = {log φn }∞ n=1 be an asymptotically sub-additive potential on a TDS (X, T ). Let λ and ∗ be defined as in (1.2)-(1.3). Proposition A.1. Let μ ∈ M(X, T ). Then we have the following properties:  (1) The limit ∗ (μ) = limn→∞ n1 log φn (x) dμ(x) exists  (which may take value −∞). Furthermore λ (x) exists for μ-a.e. x ∈ X , and λ (x) dμ(x) = ∗ (μ). In particular, when μ ∈ E(X, T ), λ (x) = ∗ (μ) for μ-a.e. x ∈ X . (2) The map ∗ : M(X, T ) → R ∪ {−∞} is upper semi-continuous and there is C ∈ R such that for all μ ∈ M(X, T ), λ (x) ≤ Cμ-a.e and ∗ (μ) ≤ C.

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D. J. Feng, W. Huang

 (3) Let  μ = θ dm(θ ) be the ergodic decomposition of μ ∈ M(X, T ). Then ∗ (μ) = ∗ (θ ) dm(θ ). Proof. In the case that  is sub-additive, statement (1) comes exactly from Kingman’s sub-additive ergodic theorem (cf. [49], p. 231). We shall show that it remains valid when  is asymptotically sub-additive. Fix such a . For  > 0, by definition, there exist a subadditive potential  = {log ψn }∞ n=1 and an integer n 0 such that | log φn (x)−log ψn (x)| ≤ n for any n ≥ n 0 and x ∈ X . Hence   1 1 lim sup log φn (x) dμ(x) ≤ lim log ψn (x) dμ(x) +  n→∞ n n→∞ n  1 ≤ lim inf log φn (x) dμ(x) + 2. n→∞ n Since the above inequalities hold for any  > 0, the limit for defining ∗ (μ) exists. Similarly, we have the inequalities 1 1 1 lim sup log φn (x) ≤ lim log ψn (x) +  ≤ lim inf log φn (x) + 2 n→∞ n n→∞ n n→∞ n  for μ-a.e. x, from which we derive that λ (x) exists μ-a.e and λ (x) dμ(x) = ∗ (μ). Furthermore, λ (x) = ∗ (μ)μ-a.e. when μ is ergodic. This proves (1). To see that ∗ is upper semi-continuous, let  > 0 and  be given as in the above paragraph. Suppose that {μi } is a sequence in M(X, T ) which converges to μ in the weak∗ topology. Then for any n ≥ n 0 and R ∈ R,  1 lim sup ∗ (μi ) ≤ lim sup ∗ (μi ) +  ≤ lim sup log ψn (x) dμi (x) +  i→∞ i→∞ i→∞ n  1 ≤ lim sup max {log ψn (x), R} dμi (x) +  i→∞ n  1 = max {log ψn (x), R} dμ(x) + . n Taking R → −∞ to obtain   1 1 lim sup ∗ (μi ) ≤ log ψn (x) dμ(x) +  ≤ log φn (x) dμ(x) + 2. n n i→∞ Letting n → ∞, we have lim supi→∞ ∗ (μi ) ≤ ∗ (μ). This proves the upper semicontinuity of ∗ . To give an upper bound for λ and ∗ , let D = max x∈X ψn 0 (x). Then log ψkn 0 (x) ≤ k log D by the subadditivity. Hence for μ-a.e x, λ (x) ≤  + lim sup k→∞

1 log ψkn 0 (x) ≤  + (log D)/n 0 . kn 0

Take integration with respect to μ to get ∗ (μ) ≤  + (log D)/n 0 .  To prove (3), we first assume that  is sub-additive. Let μ = θ dm(θ ) be the ergodic decomposition of μ ∈ M(X, T ). Let C1 = max x∈X | log φ1 (x)|. Then  1 (A.1) log φn (x) dθ (x) ≤ C1 for all θ ∈ , n ∈ N. n  Define h k (θ ) = 21k log φ2k (x) dθ (x) for θ ∈ M(X, T ) and k ∈ N. Since  is subadditive and θ is invariant, we have C1 ≥ h 1 (θ ) ≥ h 2 (θ ) ≥ · · · and h k (θ )  ∗ (θ ). By (A.1), we have

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

37

   1 1 log φ2k (x) dθ (x)dm(θ ) log φn (x) dμ(x) = lim n→∞ n k→∞ 2k    = lim lim h k (θ ) dm(θ ) = ∗ (θ ) dm(θ ), h k (θ ) dm(θ ) =

∗ (μ) = lim

k→∞

k→∞



where we use the monotone convergence theorem for the fourth equality. Hence we prove (3) in the case that  is sub-additive. Now assume that  is asymptotically sub-additive. For  > 0, let  be given as in the first paragraph of our proof. Then |∗ (θ )− ∗ (θ )| ≤  for any θ ∈ M(X, T ). It together with ∗ (μ) = ∗ (θ ) dm(θ )  yields ∗ (μ) − ∗ (θ ) dm(θ ) ≤ 2. Letting  → 0, we obtain the desired identity for . This finishes the proof.   Let M(X ) denote the space of Borel probability measures on X endowed with the weak-star topology. Then we have Lemma A.2. Suppose {νn }∞ n=1 is a sequence in M(X ). We form the new sequence 1 n−1 −i by μ = {μn }∞ n i=0 νn ◦ T . Assume that μn i converges to μ in M(X ) for n=1 n some subsequence {n i } of natural numbers. Then μ ∈ M(X, T ) and  1 log φn i (x) dνn i (x) ≤ ∗ (μ). lim sup i→∞ n i Proof. The lemma was proved in [15, Lemma 2.3] for the case that  is subadditive. Here we shall show that it can be extended to the case that  is asymptotically sub-additive. Let  be an asymptotically sub-additive potential on X and  > 0. Then there exist a sub-additive potential  = {log ψn }∞ n=1 on X and n 0 such that | log φn (x)−log ψn (x)| ≤ n for any n ≥ n 0 and x ∈ X . Hence   1 1 lim sup log φn i (x) dνn i (x) ≤ lim sup log ψn i (x) dνn i (x) +  ≤ ∗ (μ) +  i→∞ n i i→∞ n i ≤ ∗ (μ) + 2. Letting  → 0, we obtain the desired inequality for . Lemma A.3. Define β() = lim supn→∞ supx∈X

 

log φn (x) . n

Then

(1) β() ∈ R ∪ {−∞} and β() = lim inf n→∞ supx∈X log φnn (x) . (2) β() = sup{∗ (μ) : μ ∈ M(X, T )} and there exists an ergodic measure ν ∈ M(X, T ) such that β() = ∗ (ν). (3) The following conditions are equivalent: (a) β() = −∞; (b) λ (x) = −∞ for all x ∈ X ; (c) ∗ (μ) = −∞ for all μ ∈ M(X, T ); (d) P(T, ) = −∞. (4) If β() > −∞, then h top (T ) + β() ≥ P(T, ) ≥ β() > −∞. Moreover if we assume in addition that h top (T ) < ∞, then P(T, ) ∈ R. Proof. Let  > 0. Take a sub-additive potential  = {log ψn }∞ n=1 on (X, T ) such that | log φn (x) − log ψn (x)| < n for all n ≥ n 0 and x ∈ X . Let C = max x∈X |ψ1 (x)|. Then ψn (x) ≤ C n . Thus for n ≥ n 0 we have log φn (x) ≤ n(log C + ) and hence

38

D. J. Feng, W. Huang

supx∈X log φnn (x) ≤ log C + . This implies β() ∈ R ∪ {−∞}. Denote bn = supx∈X log ψn (x). Then by the sub-additivity of , bn+m ≤ bn + bm . It follows that lim inf n→∞ bn /n = lim supn→∞ bn /n and thus lim inf n→∞ supx∈X log φn (x)/n ≥ lim supn→∞ supx∈X log φn (x)/n − 2. Letting  → 0, we obtain lim inf sup log φn (x)/n = lim sup sup log φn (x)/n. n→∞ x∈X

n→∞ x∈X

This proves (1). For any μ ∈ M(X, T ), by Proposition A.1(1),  ∗ (μ) = lim

n→∞ X

log φn (x) log φn (x) dμ(x) ≤ lim sup sup = β(). n n n→∞ x∈X

Hence sup{∗ (μ) : μ ∈ M(X, T )} ≤ β(). Conversely, choose n i → ∞ and xi ∈ X  i −1 log φni (xi ) = lim supn→∞ supx∈X log φnn (x) . Let μn i = n1i nj=0 δT j xi such that limi→∞ ni for i ∈ N. Since M(X ) is compact, we may assume that μn i → μ for some μ ∈ M(X ). By Lemma A.2, μ ∈ M(X, T ) and limi→∞ n1i log φn i (x) dδxi ≤ ∗ (μ), i.e., log φn (xi )

i β() = limi→∞ ≤ ∗ (μ). Moreover, by Proposition A.1(3), there exists an ni ergodic measure ν ∈ M(X, T ) such that β() ≤ ∗ (ν). Clearly, β() = ∗ (ν). This proves (2). To show (3), note that the implications (a) ⇒ (b), (c) are direct. By (2), there exists an ergodic measure ν ∈ M(X, T ) such that β() = ∗ (ν). By Proposition A.1(1), λ (x) = β() for ν-a.e. x ∈ X . Hence β() = −∞ when (b) or (c) occurs. This shows that (b) or (c) implies (a). The equivalence of (c) and (d) comes from Theorem 3.1. This proves (3). Part (4) follows directly from (2) and Theorem 3.1.  

Now we give some properties of asymptotically additive potentials, which just follow from Proposition A.1(2) and Lemma A.2. Lemma A.4. Assume that  = {log φn }∞ n=1 is an asymptotically additive potential on (X, T ). Then (i) The map μ → ∗ (μ) is continuous on M(X, T ). ∞ (ii) Suppose {νn }∞ n=1 is a sequence in M(X ). We form the new sequence {μn }n=1  n−1 by μn = n1 i=0 νn ◦ T −i . Assume that μn i converges to μ in M(X ) for some subsequence {n i } of natural numbers. Then μ ∈ M(X, T ), and moreover 1 lim i→∞ n i

 log φn i (x) dνn i (x) = ∗ (μ).

(iii) := {∗ (μ) : μ ∈ M(X, T )} is an interval which equals [β(), β()], where β() := limn→∞ (1/n) inf x∈X log φn (x). In the end of this section, we give the following proposition. Proposition A.5. Let  = {log φn } be a potential on X (i.e., each φn is a non-negative continuous function on X ). We have the following statements.

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

39

(i) If there exists C ≥ 1 such that φn+m (x) ≤ Cφn (x)φm (T n x) for all x ∈ X and n, m ∈ N, then  is asymptotically sub-additive. (ii) If there exists C ≥ 1 such that 0 < C −1 φn (x)φm (T n x) ≤ φn+m (x) ≤ Cφn (x)φm (T n x) for all x ∈ X and n, m ∈ N, then  is asymptotically additive. (iii) If φn (x) > 0 for all n ∈ N, x ∈ X and there exists a continuous function g on X such that log φn+1 (x) − log φn (T x) → g(x) uniformly on X as n → ∞, then  is asymptotically additive. (iv)  is asymptotically additive if and only if for any  > 0, there exists an additive potential  = {log ψn }∞ n=1 on X such that lim sup n→∞

1 sup | log φn (x) − log ψn (x)| ≤ . n x∈X

(A.2)

Proof. To see (i), define  = {log ψn }∞ n=1 by ψn (x) = Cφn (x). Then ψn+m (x) = Cφn+m (x) ≤ C 2 φn (x)φm (x) = ψn (X )ψm (T n x). Hence  is sub-additive. Clearly, (log ψn (x) − log φn (x))/n = (log C)/n → 0 as n → ∞. Hence  is asymptotically sub-additive. This proves (i). Part (ii) follows directly from (i). To show (iii), define rn = supx∈X | log φn (x) − log φn−1 (T x) − g(x)|, with the conn−1 g ◦ T i . Then vention log φ0 (x) ≡ 0. It is clear that limn→∞ rn = 0. Let gn = i=0 ∞ G = {gn }n=1 is additive. Note that  n       log φi (T n−i x) − log φi−1 (T n−i+1 x) − g(T n−i x)  | log φn (x) − gn (x)| =    i=1

n n ≤ | log φi (T n−i x) − log φi−1 (T n−i+1 x) − g(T n−i x)| ≤ ri . i=1

i=1

n n (x) Hence lim supn→∞ supx∈X | log φn (x)−g | ≤ lim supn→∞ n1 i=1 ri = 0 as limn→+∞ n rn = 0. Hence  is asymptotically additive. The “if” part in (iv) is direct, we only need to show the “only if” part. Assume that  is asymptotically additive, that is, φn is positive continuous on X for each n and both −1 ∞ {log φn }∞ n=1 and {log(φn ) }n=1 are asymptotically sub-additive. We claim that for any  > 0, there exists K > 0 such that for each k ≥ K , there exists C,k > 0 so that

40

D. J. Feng, W. Huang

    n−1   1 j  log φn (x) − log φk (T x) ≤ n + C,k ,  k   j=0

∀ n ≥ 2k, x ∈ X.

Clearly the above inequality implies the “only if” part in (iv). Without loss of generality, we show that 1 log φk (T j x) + n + C,k , k n−1

log φn (x) ≤

∀ n ≥ 2k, x ∈ X,

(A.3)

j=0

for certain C,k > 0. Fix  > 0. Since  is asymptotically sub-additive, there exists a sub-additive potential  = {log ψn }∞ n=1 on X such that there is K > 0 and | log φn (x) − log ψn (x)| ≤

n , 2

∀ n ≥ K , x ∈ X.

(A.4)

Set C = max{1, supx∈X ψ1 (x)}. By [15, Lemma 2.2 ], 1 log ψk (T i x), ∀ x ∈ X, n ≥ 2k. k n−k

log ψn (x) ≤ 2k log C +

i=0

Combining the above inequality with (A.4), we have for k ≥ K , 1 log φk (T i x) k n−k

log φn (x) ≤ (2n − k)/2 + 2k log C +

i=0

1 log φk (T i x) k n−1

≤ (2n − k)/2 + 2k log C + Mk +

i=0

for all x ∈ X and n ≥ 2k, where Mk := max{1, supx∈X | log φk (x)|}. This proves (A.3), with C,k = 2k log C + Mk . We finish the proof of the proposition.   Remark A.6. (i) The potentials satisfying the assumption in Proposition A.5(iii) was considered by Barreira [3] in the study of the Hausdorff dimension of planar limit sets. (ii) Let Caa (X, T ) denote the collection of asymptotically additive potentials on X . Define an equivalence relation ∼ on Caa (X, T ) by  ∼  if  − lim = 0, where  − lim := lim sup n→∞

1 sup | log φn (x) − log ψn (x)| n x∈X

∞ for  = {log φn }∞ n=1 ,  = {log ψn }n=1 . Then it is not hard to see that the quotient space Caa (X, T )/ ∼ endowed with the norm  · lim is a separable Banach space.

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

41

Table 1. Main notation and conventions (X, T )  = {log φn }∞ n=1 β() λ (x), ∗ (μ) E  (α) M(X ) M(X, T ), E(X, T ) h μ (T ) P(T, ) P (q)  (±∞) P I(, q) h top (T, Z ), h top (T ) conv(M) ri(A) f∗ ext(C), expo(C) ∂ f (x), ∂ e f (x) ∂ f (U ), ∂ e f (U ) V (x) R+ cl+ (A) clδ+ (∂ P (Rk+ ))  = (1 , . . . , k ) β() ∗ (μ) P (q) I(, q) E  (a) Gμ

A topological dynamical system (Sect. 1) (Asymptotically sub-additive) potential (Sect. 1) β() = limn→∞ (1/n) log supx∈X φn (x) Lyapunov exponent of  at x (resp. with respect to μ) (Sect. 1) α-level set of λ (Sect. 1) Set of all Borel probability measures on X Set of T -invariant (resp. ergodic) Borel probability measures on X measure-theoretic entropy of T with respect to μ Topological pressure of  (Sect. 2.2) P(T, q) limq→∞ P (q)/q, limq→−∞ P (q)/q Set of equilibrium states of q Topological entropy of T with respect to Z (resp. Z = X ) (Sect. 2.1) Convex hull of M (Sect. 2.3) Relative interior of a convex set A Conjugate function of f (Sect. 2.4) Set of extreme points (resp. exposed points) of C (Sect. 2.3) Subdifferential of f at x (Sect. 2.3) ext(∂ f (x))   e x∈U ∂ f (x), x∈U ∂ f (x)  Set of limit points of the sequence μx,n = (1/n) n−1 j=0 δT j x in M(X ) (0, ∞) (cf. (4.5)) (cf. (4.22)) A family of asymptotically sub-additive potentials k β( i=1 i ) ((1 )∗ (μ), . . . , (k )∗ (μ)) P(T, q · ) Set of equilibrium states of q ·  (cf. (4.4)) Set of μ-generic points (see Sect. 5)

Appendix B. Main Notation and Conventions For the reader’s convenience, we summarize in Table 1 the main notation and typographical conventions used in this paper. References 1. Barral, J., Mensi, M.: Gibbs measures on self-affine Sierpinski carpets and their singularity spectrum. Ergod. Th. & Dynam. Sys. 27, 1419–1443 (2007) 2. Barral, J., Mensi, M.: Multifractal analysis of Birkhoff averages on ‘self-affine’ symbolic spaces. Nonlinearity 21, 2409–2425 (2008) 3. Barreira, L.: A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergod. Th. & Dyn. Sys. 16(5), 871–927 (1996) 4. Barreira, L.: Nonadditive thermodynamic formalism: equilibrium and Gibbs measures. Disc. Cont. Dyn. Sys. 16, 279–305 (2006) 5. Barreira, L.: Dimension and Recurrence in Hyperbolic Dynamics. Progress in Mathematics 272. Basel: Birkhauser ¨ Verlag, 2008

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6. Barreira, L., Gelfert, K.: Multifractal analysis for Lyapunov exponents on nonconformal repellers. Commun. Math. Phys. 267, 393–418 (2006) 7. Barreira, L., Pesin, Ya., Schmeling, J.: On a general concept of multifractality: multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal Rigidity. Chaos 7, 27–38 (1997) 8. Barreira, L., Saussol, B., Schmeling, J.: Higher-dimensional multifractal analysis. J. Math. Pures Appl. 81, 67–91 (2002) 9. Barreira, L., Schmeling, J.: Sets of “non-typical” points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116, 29–70 (2000) 10. Ben Nasr, F.: Analyse multifractale de mesures. C. R. Acad. Sci. Paris Sér. I Math. 319, 807–810 (1994) 11. Bowen, R.: Entropy-expansive maps. Trans. Amer. Math. Soc. 164, 323–331 (1972) 12. Bowen, R.: Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184, 125–136 (1973) 13. Brown, G., Michon, G., Peyrière, J.: On the multifractal analysis of measures. J. Stat. Phys. 66, 775– 790 (1992) 14. Buzzi, J.: Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100, 125–161 (1997) 15. Cao, Y.L., Feng, D.J., Huang, W.: The thermodynamical formalism for submultiplicative potentials. Disc. Cont. Dyn. Sys. 20, 639–657 (2008) 16. Chen, E.C., Küpper, T., Shu, L.: Topological entropy for divergence points. Ergod. Th. & Dyn. Sys. 25, 1173–1208 (2005) 17. Conway, J.B.: A Course in Functional Analysis. New York: Springer-Verlag, 1985 18. Falconer, K.J.: A subadditive thermodynamic formalism for mixing repellers. J. Phys. A 21(14), L737– L742 (1988) 19. Falconer, K.J., Sloan, A.: Continuity of subadditive pressure for self-affine sets. Real Anal. Exch. 34(2), 413–428 (2008) 20. Fan, A.H., Feng, D.J.: On the distribution of long-term time averages on symbolic space. J. Stat. Phys. 99, 813–856 (2000) 21. Fan, A.H., Feng, D.J., Wu, J.: Recurrence, dimension and entropy. J. Lond. Math. Soc. 64, 229–244 (2001) 22. Fan, A.H., Liao, L.M., Peyrière, J.: Generic points in systems of specification and Banach valued Birkhoff ergodic average. Disc. Cont. Dyn. Sys. 21, 1103–1128 (2008) 23. Feng, D.J.: Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices. Israel J. Math. 138, 353–376 (2003) 24. Feng, D.J.: Lyapunov exponents for products of matrices and multifractal analysis. Part II. General matrices. Israel J. Math. 170, 355–394 (2009) 25. Feng, D.J.: The variational principle for products of non-negative matrices. Nonlinearity 17, 447–457 (2004) 26. Feng, D.J., Lau, K.S.: The pressure function for products of non-negative matrices. Math. Res. Lett. 9, 363– 378 (2002) 27. Feng, D.J., Lau, K.S., Wu, J.: Ergodic limits on the conformal repeller. Adv. Math. 169, 58–91 (2002) 28. Feng, D.J., Olivier, E.: Multifractal analysis of the weak Gibbs measures and phase transition–application to some Bernoulli convolutions. Ergod. Th. & Dyn. Sys. 23, 1751–1784 (2003) 29. Feng, D.J., Shu, L.: Multifractal analysis for disintegrations of Gibbs measures and conditional Birkhoff averages. Ergod. Th. & Dyn. Sys. 29, 885–918 (2009) 30. Hiriart-Urruty, J.P., Lemaréchal, C.: Fundamentals of Convex Analysis. Berlin: Springer-Verlag, 2001 31. Käenmäki, A.: On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29, 419–458 (2004) 32. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1995 33. Keller, G.: Equilibrium States in Ergodic Theory. Cambridge: Cambridge University Press, 1998 34. Kesseböhmer, M.: Large deviation for weak Gibbs measures and multifractal spectra. Nonlinearity 14, 395–409 (2001) 35. Kesseböhmer, M., Stratmann, B.O.: A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. Ergod. Th. & Dyn. Sys. 24, 141–170 (2004) 36. Krieger, W.: On unique ergodicity. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II, Probability theory, Berkeley, CA: Univ. California Press, 1972, pp. 327–346 37. Makarov, N., Smirnov, S.: On “thermodynamics” of rational maps. I. Negative spectrum. Commun. Math. Phys. 211, 705–743 (2000) 38. Makarov, N., Smirnov, S.: On thermodynamics of rational maps. II. Non-recurrent maps. J. London Math. Soc. 67, 417–432 (2003) 39. Misiurewicz, M.: Topological conditional entropy. Studia Math. 55, 175–200 (1976) 40. Mummert, A.: The thermodynamic formalism for almost-additive sequences. Disc. Cont. Dyn. Sys. 16, 435–454 (2006)

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41. Olivier, E.: Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures. Nonlinearity 12, 1571–1585 (1999) 42. Olsen, L.: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. 82, 1591–1649 (2003) 43. Pesin, Y.B.: Dimension theory in dynamical systems. In: Contemporary Views and Applications. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press, 1997 44. Pfister, C.-E., Sullivan, W.G.: On the topological entropy of saturated sets. Ergod. Th. & Dyn. Sys. 27, 929–956 (2007) 45. Rockafellar, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press, 1970 46. Ruelle, D.: Thermodynamic formalism. In: The Mathematical Structures of Classical Equilibrium Statistical Mechanics. Encyclopedia of Mathematics and its Applications, 5, Reading, MA: Addison-Wesley Publishing Co, 1978 47. Takens, F., Verbitskiy, E.: On the variational principle for the topological entropy of certain non-compact sets. Ergod. Th. & Dyn. Sys. 23, 317–348 (2003) 48. Testud, B.: Phase transitions for the multifractal analysis of self-similar measures. Nonlinearity 19, 1201– 1217 (2006) 49. Walters, P.: An Introduction to Ergodic Theory. Berlin-Heidelberg-New York: Springer-Verlag, 1982 Communicated by S. Smirnov

Commun. Math. Phys. 297, 45–93 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1036-5

Communications in

Mathematical Physics

Holomorphic Factorization for a Quantum Tetrahedron Laurent Freidel1 , Kirill Krasnov2 , Etera R. Livine3 1 Perimeter Institute for Theoretical Physics, Waterloo N2L 2Y5, Ontario, Canada 2 School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK.

E-mail: [email protected]

3 Laboratoire de Physique, ENS Lyon, CNRS-UMR 5672, 46 Allée d’Italie, Lyon 69007, France

Received: 2 June 2009 / Accepted: 24 December 2009 Published online: 30 March 2010 – © Springer-Verlag 2010

Abstract: We provide a holomorphic description of the Hilbert space H j1 ,..., jn of SU(2)invariant tensors (intertwiners) and establish a holomorphically factorized formula for the decomposition of identity in H j1 ,..., jn . Interestingly, the integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. Our results provide a new interpretation for this quantity as being, in the limit of large conformal dimensions, the exponential of the Kähler potential of the symplectic manifold whose quantization gives H j1 ,..., jn . For the case n = 4, the symplectic manifold in question has the interpretation of the space of “shapes” of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coherent states. We describe how the holomorphic intertwiners are related to the usual real ones by computing their overlap. The semi-classical analysis of these overlap coefficients in the case of large spins allows us to obtain an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron. Our results are of direct relevance for the subjects of loop quantum gravity and spin foams, but also add an interesting new twist to the story of the bulk/boundary correspondence.

1. Introduction The main object of interest in the present paper is the space  SU(2) Hj = V j1 ⊗ · · · ⊗ V jn

(1)

of SU(2)-invariant tensors (intertwiners) in the tensor product of n irreducible SU(2) representations V j , dim(V j ) ≡ d j = 2 j + 1. Being a vector space with an inner product endowed from that in representation spaces V j this space is naturally a Hilbert space. It is finite-dimensional, with the dimension given by the classical formula:

46

L. Freidel, K. Krasnov, E. R. Livine

dim(Hj ) =

2 π



π

dθ sin2 (θ/2) χ j1 (θ ) . . . χ jn (θ ),

(2)

0

where χ j (θ ) = sin (( j + 1/2)θ )/ sin (θ/2) are the SU(2) characters. The intertwiners (1) figure prominently in many areas of mathematical physics. They are key players in the theory of angular momentum that has numerous applications in nuclear and particle physics, atomic and molecular spectroscopy, plasma physics, quantum chemistry and other disciplines, see e.g. [1]. A “q-deformed” analog of the space Hj plays the central role in quantum Chern-Simons (CS) theory and Wess-ZuminoWitten (WZW) conformal field theory, being nothing else but the Hilbert space of states of CS theory on a sphere with n marked points, or, equivalently, the Hilbert space of WZW conformal blocks, see e.g. [2]. A q-deformed version of the formula (2) is the famous Verlinde formula [3]. Finally, and this is the main motivation for our interest in (1), intertwiners play the central role in the quantum geometry or spin foam approach to quantum gravity. Indeed, in the quantum geometry (loop quantum gravity) approach the space of states of geometry is spanned by the so-called spin network states based on a graph . This space is obtained by tensoring together the Hilbert spaces L 2 (G) of square integrable functions on the group G = SU(2) – one for every edge e of the underlying graph – while contracting them at vertices v with invariant tensors to form a gauge-invariant state. Using the Plancherel decomposition the spin network Hilbert space can therefore be written as: L 2 (G  ) = ⊕ je H ( je ), where H ( je ) = ⊗v Hj v is the product of intertwiner spaces one for each vertex v of the graph . The Hilbert spaces Hj for n = 1, 2, 3 are either zero or one-dimensional. For n = 1 the space is zero dimensional, for n = 2 there is a unique (up to rescaling) invariant tensor only when j1 = j2 , and for n = 3 there is again a unique (up to normalization) invariant tensor when the “triangle inequalities” j1 + j2 ≥ j3 , j1 + j3 ≥ j2 , j2 + j3 ≥ j1 are satisfied. Thus, the first non-trivial case that gives a non-trivial dimension of the Hilbert space of intertwiners is n = 4. A beautiful geometric interpretation of states from H j1 ,..., j4 has been proposed in [4], where it was shown that the Hilbert space in this case can be obtained via the process of quantisation of the space of shapes of a geometric tetrahedron in R3 whose face areas are fixed to be equal to j1 , . . . , j4 . This space of shapes, to be defined in more details below, is naturally a phase space (of real dimension two) of finite symplectic volume, and its (geometric) quantization gives rise to a finite-dimensional Hilbert space H j1 ,..., j4 . In a recent work co-authored by one of us [5] the line of thought originating in [4] has been further developed. Thus, it was shown that the space of shapes of a tetrahedron is in fact a Kähler manifold of complex dimension one that is conveniently parametrized by Z ∈ C\{0, 1, ∞}. As we shall explain in more details below, this complex parameter is just the cross ratio of the four stereographic coordinates z i labelling the direction of the normals to faces of the tetrahedron. It was also shown in [5] by a direct argument that the two possible viewpoints on H j1 ,..., j4 – namely that of SU(2) invariant tensors and that of quantization of the space of shapes of a geometric tetrahedron – are equivalent, in line with the general principle of Guillemin and Sternberg [6] saying that (geometric) quantization commutes with symplectic reduction. Moreover, in [5] an explicit formula for the decomposition of the identity in H j1 ,..., j4 in terms of certain coherent states was given. The main aim of this paper is to further develop the holomorphic viewpoint on Hj introduced in [5], both for n = 4 and in more generality. Thus, we give several explicit proofs of the fact that the Hilbert space Hj of intertwiners can be obtained by quantization of a certain finite volume symplectic manifold Sj , where S stands for

Holomorphic Factorization for a Quantum Tetrahedron

47

shapes. The phase space Sj turns out to be a Kähler manifold, with convenient holomorphic coordinates given by a string of n −3 (suitably chosen) cross-ratios {Z 1 , . . . , Z n−3 } with Z i ∈ C\{0, 1, ∞}, and it is natural to use the methods of geometric quantization to get to Hj . Up to the “metaplectic correction” occurring in geometric quantization of Kähler manifolds (see more on this in the main text), the Hilbert space is constructed, see e.g. [7–9], as the space of holomorphic functions (z) integrable with the measure ¯ exp(−(z, z¯ )) k , where (z, z¯ ) is the Kähler potential, = (1/i)∂ ∂(z, z¯ ) is the symplectic form, and k is the (complex) dimension of the manifold. Alternatively, in the context of Kähler geometric quantization, one can introduce [10] the coherent states |z such that (z) = z| . Then the inner product formula can be rewritten as a formula for the decomposition of the identity operator in terms of the coherent states:  1 = k e−(z,¯z ) |z z|. (3) Our first main result in this paper is a version of formula (3) for the identity operator in the Hilbert space Hj . The corresponding coherent states shall be denoted by |j , Z ∈ Hj , where Z is a collective notation for the string Z 1 , . . . , Z n−3 of cross-ratio coordinates. We prove that: 1j = 8π 2

 n  d ji d2 Z Kˆ j (Z , Z¯ ) |j , Z j , Z |. 2π C

(4)

i=1

The integration kernel Kˆ j (Z , Z¯ ) here turns out to be just the n-point function of the AdS/CFT duality [11], given by an integral over the 3-dimensional hyperbolic space of a product of n so-called bulk-to-boundary propagators, see the main text for the details. A comparison between (4) and (3) shows that, in the semi-classical limit of all spins becoming large, the n-point function Kˆ j (Z , Z¯ ) must admit an interpretation of an exponential of the Kähler potential on Sj , and we demonstrate by an explicit computation in what sense this interpretation holds. Thus, as a corollary to our main result (4) we obtain a new and rather non-trivial interpretation of the bulk/boundary duality n-point functions. The formula (4) takes a particularly simple form of an integral over a single crossratio coordinate in the case n = 4 of relevance for the quantum tetrahedron. The coherent states |j , Z are holomorphic functions of Z that we shall refer to as holomorphic intertwiners. The resulting holomorphic description of the Hilbert space of the quantum tetrahedron justifies the title of this paper. In the second part of the paper we characterize the n = 4 holomorphic intertwiners |j , Z by projecting them onto a more familiar real basis in H j1 ,..., j4 . The real basis |j , k i j can be obtained by considering the eigenstates of the operators J(i) · J( j) representing the scalar product of the area vectors between the faces i and j. We compute these operators as second-order differential operators acting on functions of the crossratio Z , and use these results to characterise the pairing (or overlap) between the usual normalised intertwiners in the channel (i j) : |j , k i j , and our holomorphic intertwiner |j , Z . Denoting this pairing by i j Cjk ≡ i j j , k|j , Z , we find it to be given by the (α,β)

“shifted” Jacobi polynomials Pn 12

: ( j − j12 , j34 + j12 )

Cjk (Z ) = Njk Pk−34j34

(1 − 2Z ),

(5)

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L. Freidel, K. Krasnov, E. R. Livine

where ji j ≡ ji − j j , and Njk is a normalisation constant to be described below. This result can be used to express the H j1 ,..., j4 norm of the holomorphic intertwiner |j , Z in a holomorphically factorised form (for any choice of the channel i j):  dk |i j Cjk (Z )|2 , (6)

j , Z |j , Z = k

which gives another justification for our title. As a by-product of our analysis we also deduce several non-trivial facts about the holomorphic factorisation of the bulk/boundary 4-point function K j . The last step of our analysis is to discuss the asymptotic properties of the (normalized) overlap coefficients i j Cjk (Z ) for large spins and the related geometrical interpretation. This asymptotic analysis allows us to explicitly obtain an extremely non-trivial relation between the real and holomorphic description of the phase space of shapes of a geometric tetrahedron, and demonstrates the power of the methods developed in this paper. As a final and non-trivial consistency check of our analysis we show that the normalized overlap coefficient is sharply picked both in k and in Z around a value k(Z ) determined by the classical geometry of a tetrahedron. Our discussion has so far been quite mathematical, so we would now like to switch to a more heuristic description and explain the significance of our results for the field of quantum gravity. As we have already mentioned, the n = 4 intertwiner that we have characterized in this paper in most details plays a very important role in both the loop quantum gravity and the spin foam approaches. These intertwiners have so far been characterized using the real basis |j , k i j . In particular, the main building blocks of the spin foam models – the (15 j)-symbols and their analogs – arise as simple pairings of 5 of such intertwiners (for some choice of the channels i j). The main result of this paper is a holomorphic description of the space of intertwiners, and, in particular, an explicit basis in H j1 ,..., j4 given by the holomorphic intertwiners |j , Z . While the basis |j , k i j , being discrete, may be convenient for some purposes, the underlying geometric interpretation in it is quite hidden. Indeed, recalling the interpretation of the intertwiners from H j1 ,..., j4 as giving the states of a quantum tetrahedron, the states |j , k i j describe a tetrahedron whose shape is maximally uncertain. In contrast, the intertwiners |j , Z , being holomorphic, are coherent states in that they manage to contain the complete information about the shape of the tetrahedron coded into the real and imaginary parts of the cross-ratio coordinate Z . We give an explicit description of this in the main text. Thus, with the holomorphic intertwiners |j , Z at our disposal, we can now characterize the “quantum geometry” much more completely than it was possible before. Indeed, we can now build the spin networks – states of quantum geometry – using the holomorphic intertwiners. The nodes of these spin networks then receive a well-defined geometric interpretation as corresponding to particular tetrahedral shapes. Similarly, the spin foam model simplex amplitudes can now be built using the coherent intertwiners, and then the basic object becomes not the (15 j)-symbol of previous studies, but the (10 j)-(5Z )-symbol with a well-defined geometrical interpretation. Where this will lead the subjects of loop quantum gravity and spin foams remains to be seen, but the very availability of this new technology opens the way to many new developments and, we hope, will give a new impetus to the field that is already very active after the introduction of the new spin foam models in [12–16]. The organization of this paper is as follows. In Sect. 2 we describe how the phase space that we would like to quantize arises as a result of the symplectic reduction of a simpler phase space. Then, starting from an identity decomposition formula for the

Holomorphic Factorization for a Quantum Tetrahedron

49

unconstrained Hilbert space, we perform the integration along the directions orthogonal to the constraint surface by a certain change of variables. The key idea used in this section is that holomorphic states invariant under the action of SU(2) are also SL(2, C)invariant. In Sect. 3 we provide an alternative derivation of the identity decomposition formula starting from a formula established in [5], and also utilizing a change of variables argument. The two derivations that we give emphasize different geometric aspects of the problem, and are complementary. Then, in Sect. 4 we analyze the identity decomposition formula in the semi-classical limit of large spins and show that it takes the form precisely as is expected from the point of view of geometric quantization. This establishes that the n-point function of the bulk/boundary dualities is the (exponential of the) Kähler potential on the space of shapes. Sect. 5 specializes to the case n = 4 relevant for the quantum tetrahedron and characterizes the holomorphic intertwiners |j , Z by computing their overlap with the standard real intertwiners. This is done by considering the SU(2)-invariant operators given by the product of two face normals. Then in Sect. 6 we use the requirement of hermiticity of these geometric operators to put some constraints on the inner product kernel Kˆ j (Z , Z¯ ) given by the bulk/boundary 4-point function. We show that, remarkably, the hermiticity of these operators suggests that Kˆ j (Z , Z¯ ) holomorphically factorizes precisely in the form required for it to have a CFT 4-point function interpretation. In sect. 7 we study the asymptotic properties of the holomorphic intertwiners |j , Z projected onto the usual real intertwiners |j , k . We find that the overlap coefficients are peaked in both k and Z labels, and characterize the relation between the real and holomorphic descriptions of the space of shapes. We finish with a discussion. 2. The Space of Shapes and its Holomorphic Quantization 2.1. The space of shapes Sj as a symplectic quotient. In this subsection we recall the classical geometry behind the quantization problem we study. We start with a symplectic manifold obtained as the Cartesian product of n copies of the sphere with its standard SU(2)-invariant symplectic structure. Radii of the n spheres are fixed to be ji ∈ Z/2, i = 1, . . . , n and we denote by j = ( j1 , · · · , jn ) the n-tuple of spins. Thus, the space Pj we consider is parametrized by n vectors ji Ni , where Ni ∈ R3 are unit vectors Ni · Ni = 1. The space Pj is a 2n-dimensional symplectic manifold with symplectic form given by the sum of the sphere symplectic forms. There is a natural (diagonal) action of the group G = SU(2) on Pj , which is generated by the following Hamiltonian:  Hj ≡ ji Ni . (7) i

The space of shapes Sj that we are interested in is obtained by the symplectic reduction, that is by imposing the constraint Hj = 0 and then by considering the space of G = SU(2)-orbits on the constraint surface. This space can be thought of as that of n-faced polygons, with Ni being the unit face normals and ji being the face areas. The following notation for this symplectic reduction is standard: Sj = Pj //SU(2).

(8)

The usual theory of symplectic reductions tells us that the space of shapes is also a symplectic manifold. The fact of crucial importance for us is that the space Sj is also

50

L. Freidel, K. Krasnov, E. R. Livine

a Kähler manifold, i.e. is a complex manifold with a Hermitian metric (satisfying an integrability condition) such that the metric and the symplectic form arise as the real and imaginary parts of this Hermitian metric. Indeed, each of the n unit spheres is a Kähler manifold. The complex structure on the sphere is made explicit by the stereographic projection:   z + z¯ 1 z − z¯ 1 − |z|2 . (9) (N1 , N2 , N3 )(z) = , , 1 + |z|2 i 1 + |z|2 1 + |z|2 The complex structure on the sphere is then the usual complex structure on the complex z-plane, and the symplectic structure is: ω j (z) =

2 j dz ∧ d¯z ¯ j (z, z¯ ) dz ∧ d¯z , = ∂ ∂ 2 2 i (1 + |z| ) i

(10)

where  j (z, z¯ ) = 2 j log(1 + |z|2 ) is the Kähler potential. The complex structure on Pj is then just the product one, and, crucially, it turns out to commute with the action of SU(2) on Pj . Thus, the space of shapes Sj inherits from Pj a symplectic as well as a complex structure compatible with the symplectic structure, moreover the induced metric is positive and is hence a Kähler manifold. This is most easily seen via the Guillemin-Sternberg isomorphism [6] that expresses the symplectic quotient (8) as an unconstrained but complex quotient of the subset of Pj consisting of non-coincident points: Sj = {(z 1 , · · · , z n ) | z i = z j }/SL(2, C).

(11)

2.2. Holomorphic quantization of the unit sphere. In this subsection, as a preliminary step to the holomorphic quantization of the quotient (8) we remind the reader how the sphere can be quantized. According to the general spirit of the geometric quantization, and in the spirit of (3), the holomorphic quantization of the sphere of radius j is achieved via the SU(2) coherent states | j, z that satisfy the completeness relation:  dj d2 z 1j = | j, z j, z|, (12) 2π (1 + |z|2 )2( j+1) where d j = 2 j + 1 is the dimension of the representation V j and d2 z ≡ |dz ∧ d z¯ |. Note that with this convention d2 z is twice the canonical area form on the plane. The coherent states appearing in (12) are holomorphic, i.e. depend only on z and not on z¯ . They are normalized so that j, z| j, z = (1+|z|2 )2 j , which, when used in (12) immediately gives the correct relation Tr(1 j ) = d j . 2.3. Kinematical and physical Hilbert spaces. Let us start with a description of the Hilbert space obtained by quantizing the unconstrained phase space Pj . By the coadjoint orbits method, this is just the direct product on n irreducible representation spaces V ji of SU(2). Thus, our “kinematical” Hilbert space is: Hjkin  = V j1 ⊗ · · · ⊗ V jn .

(13)

A holomorphic description of each of these spaces has been given above. However, for our purposes it is more convenient to use the coherent states description. Each space V j

Holomorphic Factorization for a Quantum Tetrahedron

51

can then be described as spanned by holomorphic polynomials ψ(z) of degree less than 2 j, where z is the usual coordinate on the complex plane. A state in the tensor product depends on the n variables (z 1 , . . . , z n ) and the inner product is given by   n d ji d2 z i

ψ1 , ψ2 = ψ1 (z i )ψ2 (z i ), 2π (1 + |z i |2 )2( ji +1)

(14)

i=1

which, after the identification (z) ≡ ¯z | , is just n copies of the formula (12) above. As before, d j = 2 j + 1 and the convention for the measure is d 2 z = |dz ∧ d z¯ |. The action of SU(2) group elements in this description is given by: (Tˆ j (k t )ψ)(z) = (−β ∗ z + α ∗ )2 j ψ(z k ), where t denotes transposition and the action of SU(2) on the complex plane is   αz + β α β k= ∈ SU(2), zk = . −β ∗ α ∗ −β ∗ z + α ∗

(15)

(16)

Since, as is easily checked, (1 + |z k |2 ) = (−β ∗ z + α ∗ )−2 (1 + |z|2 ), and d 2 z/(1 + |z|2 )2 is an SU(2)-invariant measure, the inner product (14) is SU(2)-invariant. We are interested in computing the inner product of physical, i.e. SU(2)-invariant states: (Tˆ j1 (k) ⊗ . . . ⊗ Tˆ jn (k)ψ)(z 1 , . . . , z n ) = ψ(z 1 , . . . , z n ).

(17)

However, being holomorphic, such states are then automatically invariant under SUC (2) = SL(2, C). From this we immediately get:   n  a b g −2 ji ψ(z i ) = (cz i + d) ψ(z i ), where g = ∈ SL(2, C), (18) c d i=1

and the action of SL(2, C) on the complex plane by rational transformation is given in (38). Thus, the physical states are completely determined by their values on the moduli space {z 1 , . . . , z n }/SL(2, C). As we have already mentioned, and as is described at length in [5], this moduli space space is isomorphic to the space of shapes Sj . zg

2.4. Moduli space and cross-ratios. The integral in (14) is that over n copies of the complex plane. However, as we have seen above, on physical states the integrand has very simple transformation properties under SL(2, C). This suggests that the integral can be computed by a change of variables where one parametrizes z 1 , . . . , z n by an element of PSL(2, C) together with certain cross-ratios Z i , i = 4, . . . , n. Indeed, given the first three complex coordinates z 1 , z 2 , z 3 , there exists a unique PSL(2, C) transformation that maps these points to 0, 1, ∞ (and maps the points z i , i > 3 to Z i ). Let us use the inverse of this transformation to parametrize the unconstrained phase space by an element of SL(2, C) together with Z i . Explicitly, given an SL(2, C) element g and n − 3 cross-ratios Z i we can construct the n points {0, 1, ∞, Z i }g on the complex plane. Explicitly: z1 =

b , d

z2 =

a+b , c+d

z3 =

a , c

zi =

a Zi + b i ≥ 4. cZ i + d

(19)

52

L. Freidel, K. Krasnov, E. R. Livine

This gives us a map SL(2, C) × {Z 4 , . . . , Z n } → {z 1 , . . . , z n },

g × Z j → z i (g, Z j ),

(20)

 g which is such that (z 1 (g, Z j ), · · · z n (g, Z j )) = 0g , 1g , ∞g , Z i . This map is 2 : 1 since −g and g give the same image. The cross-ratios Z i , together with g (or a, b, c, d satisfying the relation ad − bc = 1) can be used as (holomorphic) coordinates on our space {z 1 , . . . , z n }. This change of variables is performed in details in Appendix B where we find the following relation between the integration measures: 

n 

Cn

i=1

= 8π 2

d2 z i F(z i , z i ) 

n 

Cn−3 i=4

 d2 Z i

dnorm g

SL(2,C)

F(z i (g, Z j ), z i (g, Z j )) n . + d|4 |c|4 i=4 |cZ i + d|4

|d|4 |c

(21)

Here dnorm g is the Haar measure on SL(2, C), normalized so that its compact SU(2) part measure is just the normalized measure on the unit three-sphere (see Appendix B). As before, the convention is that d2 z = |dz ∧ d¯z |. 2.5. The physical inner product. Given the transformation property (18) we can describe the functions ψ(z i ) by their values on the moduli space parametrized by Z i . Explicitly: n   g (cZ i + d)−2 ji (Z i ), (22) ψ(z i ) = ψ 0g , 1g , ∞g , Z i = d −2 j1 (c + d)−2 j2 c−2 j3 i=4

where we have defined a wave functional depending only on the cross ratios as given by the limit (Z i ) ≡ lim X −2 j3 ψ(0, 1, X, Z i ). X →∞

(23)

Now, starting from the expression (14) for the kinematical inner product, performing the change of variables from z 1 , . . . , z n to SL(2, C) × {Z 4 , . . . , Z n }, and substituting the expression (22) for the wave functional, we can reduce the inner product of two physical states to a simple integral over the cross-ratios only. We get

1 , 2 = 8π 2

  n n  dj d2 Z i Kˆ j (Z i , Z¯ i )1 (Z i )2 (Z i ), 2π i=1

(24)

i=4

where Kˆ j is given by a group integral Kˆ j (Z i , Z¯i ) =



dnorm g (|b|2 + |d|2 )−2( j1 +1) (|a + b|2 + |c + d|2 )−2( j2 +1)

SL(2,C)

(|a|2 + |c|2 )−2( j3 +1)

n

−2( ji +1)  |a Z i + b|2 + |cZ i + d|2 . i=4

(25)

Holomorphic Factorization for a Quantum Tetrahedron

53

It is not hard to see that this expression can be obtained from a kernel depending on n coordinates  n

−2( ji +1)  norm |cz i + d|2 + |az i + b|2 d g (26) K j (z i , z¯i ) := SL(2,C)

i=1

by taking the limit Kˆ j (Z i , Z i ) = lim |X |23 K j (0, 1, X, Z 4 , . . . , Z n ). X →∞

(27)

The formula (24) for the physical inner product is valid for all n and admits an illuminating reformulation in terms of the coherent states. Thus, under the identification (Z i ) ≡ j , Z¯ i | , we get: 1j = 8π 2

  n n  dj d2 Z i Kˆ j (Z i , Z¯ i ) |j , Z i j , Z i | , 2π i=1

(28)

i=4

which is the formula given in the Introduction. 2.6. Kernel as the n-point function of the bulk-boundary correspondence. In this subsection we explicitly relate the kernel K j (z i , z i ) that we encountered above, see (26), to an object familiar from the bulk-boundary duality of string theory. Indeed, observe that the integrand in (26) is SU(2)-invariant, so it is enough to integrate only over the quotient space H3 = SL(2, C)/SU(2) which is the 3 dimensional Hyperbolic space or Euclidean AdS space. This can be achieved by using the Iwasawa decomposition which states that any matrix of SL(2, C) can be decomposed as the product of a unitary matrix k ∈ SU(2), a diagonal Hermitian matrix and an upper triangular matrix. That is, any element g ∈ SL(2, C) can be uniquely written as 1

  1 −y ρ− 2 0 , k ∈ SU(2), ρ ∈ R+ , y ∈ C. g=k (29) 1 0 1 0 ρ2 The Haar measure on SL(2, C) written in terms of the coordinates k, ρ, y reads (see Appendix B): dnorm g = dk

dρ dyd y¯ , ρ3

(30)

where dk is the normalized SU(2) Haar measure, and the rest is just the standard measure on the hyperbolic space H3 whose metric is given by: ds 2 =

dρ 2 + dyd y¯ . ρ2

(31)

Now, expressing (26) in terms of the Iwasawa coordinates one explicitly sees that the integrand is independent of k, so one only has to perform the integration over the hyperbolic space H3 coordinatized by ρ, y. The first remark is that in these coordinates one immediately recognizes that the kernel is related to the n-point function of the bulkboundary correspondence of string theory [11]. Indeed, the integrand can be easily seen

54

L. Freidel, K. Krasnov, E. R. Livine

to be a product of the bulk-to-boundary propagators heavily used in AdS/CFT correspon−2( j+1)  dence of string theory. Thus, let us evaluate the quantity |cz + d|2 + |az + d|2 on an SL(2, C) group element appearing in the Iwasawa decomposition (29). Due to the SU(2)-invariance, it depends only on the hyperbolic part of the group element g, thus on 1



 −1 ρ 2 −yρ −1/2 ρ− 2 0 1 −y = h= . 1 1 0 1 0 ρ2 0 ρ2 Then setting explicitly a = ρ −1/2 , b = −yρ −1/2 , c = 0, d = ρ 1/2 and using the same convention  = 2( j + 1) as before, we see that the integrand is given by the product of the following quantities: K  (h, z) ≡

(ρ 2

ρ . + |z − y|2 )

(32)

This is just the usual expression for the bulk-boundary propagator, see [11], where h labels a point in the interior of H3 while z i label points in its asymptotic boundary. Thus, we have shown that the kernel (26) can be very compactly written as an integral over the bulk of H3 of a product of n bulk to boundary propagators:  K j (z i , z i ) =

d 3h H3

n 

K i (h, z i ).

(33)

i=1

This is the standard definition of the bulk-boundary duality n-point function, see [11]. 3. An Alternative Derivation of the Identity Decomposition Formula In the previous section we have given a simple derivation of the decomposition of the identity formula, with the starting point being the identity formula on the unconstrained Hilbert space. The key idea of the analysis above was to use the “analytic continuation” that implied that SU(2)-invariant holomorphic states are also SL(2, C)-invariant. This then allowed us to reduce the integral over z i to that over cross-ratios. However, some geometrical aspects remain hidden in this analysis. Thus, our general derivation made no reference to the symplectic potential on the constraint surface or metric on the orbits orthogonal to that surface. The aim of this section is to provide an alternative derivation of the decomposition formula that makes such geometrical aspects more manifest. 3.1. Decomposition of the identity in Hj . The basis of (holomorphic) coherent states | j, z in V j naturally extends to a basis in the Hilbert space of intertwiners SU(2)  , Hj = V j1 ⊗ · · · ⊗ V jn by performing the group averaging over G = SU(2) on a product of coherent states. This leads to the notion of “coherent intertwiner” [12] defined as  dg T j1 (g)| j1 , z 1 ⊗ · · · ⊗ T jn (g)| jn , z n . (34) ||j , z i ≡ SU(2)

Holomorphic Factorization for a Quantum Tetrahedron

55

These states are SU(2)-invariant by construction, and thus are vectors in Hj . Therefore, the operator quantizing the Hamiltonian constraint vanishes on them.  However, the labels of these coherent states do not satisfy the constraint Hj (z i ) ≡ i ji N (z i ) = 0 (although it can be argued that they are peaked on H = 0 in the large spin limit) and thus do not have the interpretation as states in the Hilbert space obtained by quantizing the space of shapes Sj . To relate them to some states obtained by quantizing Sj one can follow the GuilleminSternberg prescription [6] and integrate these states along the orbits orthogonal to the constraint surface. This was done in [5] where the following result for the projector onto Hj was obtained1  dj     i 1j = d 2 z i δ (3) (Hj (z i ))det G j (z i ) K j (z i , z i ) ||j , z i j , z i ||. (35) 2π i

i

There are two new ingredients in this formula. The first one is the determinant of the 3 by 3 matrix G j , which is the metric along the SL(2, C) orbits orthogonal to the constraint surface. Explicitly G ab j (z i ) =

n 

 ji δ ab − N a (z i )N b (z i ) .

(36)

i=1

The second ingredient is the n-point function K j (z i , z¯ i ) entering the measure of integration. It can be defined as the following integral over SL(2, C):  K j (z i , z i ) =

SL(2,C)

dnorm g

n

−2( ji +1) 

z i |g † g|z i .

(37)

i=1

Here |z is the SU(2) coherent state associated with the fundamental spin 1/2 representation. The normalization of the measure over SL(2, C) is given in Appendix B. It is not hard to see that the quantity (37) coincides with (26) and so is just the n-point function of the bulk-to-boundary dualities [11], as we have reviewed in the previous section. The integral  in (35) is taken over n copies of the complex plane subject to the closure constraint i ji N (z i ) = 0. Moreover, both the integrand and the measure are, in fact, SU(2)-invariant. Thus, this is an integral over our phase space of shapes Sj . By the Guillemin-Sternberg isomorphism (11) this space is isomorphic to the unconstrained space Pjs of n copies of the complex plane modulo SL(2, C) transformations. This means that a convenient set of coordinates on Sj is given by the SL(2, C)-invariant cross-ratios. So, the idea is now to express the integral in (35) in terms of the crossratios. 3.2. Cross-ratios and the holomorphic intertwiner. The conformal group SL(2, C) acts on the complex plane by fractional linear (or Möbius) transformations   az + b a b g ∈ SL(2,C). (38) z→z ≡ where g = c d cz + d 1 As compared to the formula (102) in [5], in order to emphasize the holomorphic structure of this formula, we have stripped out all the dependence on the factors (1 + |z|2 )2 . It is easy to see that these factors appearing in the states, the measure and the prefactors all cancel out to give the formula presented here.

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L. Freidel, K. Krasnov, E. R. Livine

The action on the coherent states is also easy to describe: g|z = (cz + d)|z g ,

(39)

where |z ≡ |1/2, z is the coherent state in the fundamental representation and one obtains a general coherent state simply by taking the 2 j’s power of the |z fundamental one. The n-point function K j (z i , z i ) is defined as an average over SL(2, C) and thus transforms covariantly under conformal transformation. Indeed, it is not hard to show that it satisfies the standard transformation property of a CFT n-point function: g

g

K j (z i , z i ) =

n 

|cz i + d|2i K j (z i , z i ), where i = 2( ji + 1).

(40)

i=1

Given n complex numbers z i we can parametrize the SL(2, C)-invariant set Sj by n − 3 cross-ratios: z i1 z 23 Zi ≡ , i = 4, .., n with z i j ≡ z i − z j . (41) z i3 z 21 In other words, there exists a SL(2, C) transformation g which maps (z 1 , z 2 , z 3 ) to (0, 1, ∞) and the remaining z i , i > 3 to the cross-ratios Z i . Explicitly, this SL(2, C) transformation is given by:     1 (z − z 1 )z 23 a b z 23 −z 1 z 23 =√ , or z g = . (42) c d z −z z 21 3 21 z 23 z 21 z 13 (z − z 3 )z 21 Thus, we can either use a group element and the cross-ratios (g, Z i ) or (z 1 , z 2 , z 3 , · · · , z n ) as a set of complex coordinates on the unconstrained phase space Pj . Moreover we can trade the SL(2, C) group element g for the elements (z 1 , z 2 , z 3 ), see, however, below for a subtlety related to the fact that this parametrization is not one-to-one. Then, using the transformation property (40) we can write the n-point function K j (z i , z i ) as K j (z i , z i ) = |z 12 |23 − |z 23 |−22 −23 |z 13 |−21 −23

n 

|z i3 |−2i Kˆ j (Z i , Z i ),

i=4

(43)

n

where  ≡ i=1 i . Here K j (Z i , Z i ) is defined via (27) and is a function of the cross-ratios only. As such it is SL(2, C) invariant. Now, the function to be integrated in the decomposition of the identity formula (35) is K j (z i , z i ) ||j , z i j , z i ||. Thus, let us also find an expression for the coherent intertwiners as functions of the cross-ratios Z i as well as z 1 , z 2 , z 3 . It is not hard to see from (39), and was shown explicitly in [5], that since ||j , z i is a state invariant under SU(2) it transforms covariantly under SL(2, C):  ¯ g ¯ i = 2 − i = −2 ji (cz i + d)i ||j , z i , where  (44) ||j , z i = i

is the dual conformal dimension. Thus, via a procedure similar to that employed for the n-point function K j (z i , z i ), we can express the covariant intertwiner state in terms of a state depending only on the cross-ratios: ¯ ¯ j /2− ¯j ¯ j −/2 ¯ ¯ j − ¯ ¯ j −  /2−   2 3 1 3 z 23 z 13

||j , z i = z 12 3

n  i=4

¯j − i

z i3

|j , Z i ,

(45)

Holomorphic Factorization for a Quantum Tetrahedron

57

n ¯ = i=1 ¯ ji . We shall refer to the state |j , Z that depends (holomorphically) where   only on the cross-ratios as the “holomorphic intertwiner”. In terms of the coherent intertwiner it is given by |j , Z i ≡ lim (−X ) j3 ||j , 0, 1, X, Z 4 , · · · , Z n . X →∞

(46)

Since the transformation properties of the two factors in our integrand are “inverse” of each other, their product has a very simple description in terms of the cross-ratio coordinates: K j (z i , z i ) ||j , z i j , z i || =

Kˆ j (Z i , Z i ) |j , Z i j , Z i | n , |z 12 |2(n−2) |z 23 |2(4−n) |z 13 |2(4−n) i=4 |z 3i |4

(47)

which is the main result of this subsection. Of particular interest to us are the two cases n = 3, 4. In the case n = 3 there is no cross-ratio coordinate and both Kˆ j1 , j2 , j3 ≡ Kˆ j (Z i , Z i )|n=3 and | j1 , j2 , j3 ≡ |j , Z i |n=3 are constants (depending only on the representation labels j ). Thus, in this case, the holomorphic intertwiner is (up to a normalization denoted by N j1 , j2 , j3 and computed below) just the projector onto the unique normalised SU(2) invariant state |0 . Thus for n = 3, the previous formula (47) takes the following form:   K j (z i , z i ) ||j , z i j , z i ||

n=3

=

N 2j1 , j2 , j3 |z 12 z 13 z 23 |2

Kˆ j1 , j2 , j3 |0 0|.

(48)

In the case n = 4, which is of main interest to us due to its relation to a quantum tetrahedron, there is a single cross-ratio parameter Z that possesses a nice geometrical interpretation (it can be expressed in terms of certain area and angle parameters of the tetrahedron, see [5]). In terms of this cross-ratio the formula (47) for n = 4 reads:   K j (z i , z i ) ||j , z i j , z i ||

n=4

=

Kˆ j (Z , Z ) z 41 z 23 |j , Z j , Z | , where Z ≡ . |z 12 z 34 |4 z 43 z 21 (49)

3.3. Measure and determinant. The integral in (35) is over the constraint surface, with the integration measure being

n   dμ(n) (z i ) ≡ d ji d2 N (z i ) δ (3) ji N (z i ) , (50) i=1

i

where d2 N (z) =

d2 z 1 2π (1 + |z|2 )2

(51)

is the normalized measure on the unit 2-sphere parametrised by z. The factors of (1+|z|2 )2 in the denominator are introduced for later convenience and are compensated in a formula below. Now that we have written the integrand in terms of the coordinates z 1 , z 2 , z 3 and Z i , i > 3, we need to obtain a similar representation for the measure dμ(n) (z i ). As a first step towards this goal, we notice that this is an SU(2) invariant measure on the constraint

58

L. Freidel, K. Krasnov, E. R. Livine (0)

surface that we denote by Pj := Hj−1 (0). Thus, it is given by a product of the Haar mea-

sure on SU(2) times the symplectic measure on the quotient manifold Sj = Pj(0) /SU(2). Apart from some numerical factors (see below), the non-trivial part of this decomposition, i.e. the symplectic measure is defined as follows. The symplectic structure on Pj is given by i ω ji (z i ), where ω j is the sphere symplectic structure (10). If we denote (0) by i : Pj → Pj the inclusion map and by π : P (0) → Sj the projection map, the  induced symplectic structure j on Sj is defined so that i ∗ ( i ω ji (z i )) = π ∗ ( j ). We now have the following formula for the integration measure: (n)

dμ

n

 1 dj n−3 j i , (z i ) = 4π dk ∧ 2π 2 ji (n − 3)! 2

(52)

i=1

where dk is the (normalized) Haar measure on SU(2). Proof. In this formula the j dependent prefactor just comes from the discrepancy between the normalisation of the symplectic measure 2 j and the unity decomposition measure d j = 2 j + 1. The numerical prefactor 1/(2π )n comes from the relative normalisation of the measure on the sphere relative to the symplectic measure d j d2 N = (1/2π )(d j /2 j)ω j . The additional factor of 4π 2 is the volume of the SU(2) group with respect to the unnormalized measure on SU(2) (see Appendix B). The fact that apart from this numerical factor the measure splits as a product is due to the fact that SU(2) vector fields are orthogonal with respect to the symplectic measure to vectors tangent to the constraint surface. Indeed, if Xˆ is a vector field denoting the action of SU(2), such that i Xˆ = −d H X , where is the symplectic form on Pj and ξ is a vector tangent to the constraint surface, then (ξ, Xˆ ) = −ξ i Xˆ = ξ(H X ) = 0. This implies that the determinant of the symplectic measure factorises as a product of determinant for each factor and therefore the measure factorizes.   To obtain a decomposition of the identity formula in terms of the cross-ratios we now only need to express the symplectic potential j in terms of the cross-ratio coordinates Z i , as well as the coordinates z 1 , z 2 , z 3 . We have explicitly computed this symplectic potential in the most interesting case for us, that is n = 4, with the result being:   j 

4 n=4

=2

i=1 (2 ji )

det(G j )

4

|z 43 z 21 |4

i=1 (1 + |z i

| 2 )2

dZ ∧ dZ . i

(53)

Proof. Instead of computing directly the induced symplectic structure it is equivalent but easier to compute the Poisson bracket of Z with Z . This is given by {Z , Z } =

 i (1 + |z i |2 )2 |∂zi Z |2 , 2 ji

(54)

i

which follows directly from the expression (10) for the symplectic 2-form on the sphere. Now using the definition of the cross-ratio (49), one can easily see that ∂z i Z = −

z jk z kl zl j , 2 z2 z 43 21

(55)

Holomorphic Factorization for a Quantum Tetrahedron

59

where (i, j, k, l) stands for an arbitrary (even) permutation of (1, 2, 3, 4). From this we get the Poisson bracket 4 2 2  |z i j |2 |z jk |2 |z ki |2 i i=1 (1 + |z i | ) 16 {Z , Z }= ji j j jk . 4 2 2|z 43 z 21 | (1+|z i | )2 (1+|z j |2 )2 (1+|z k |2 )2 i (2 ji ) i< j
(56) To finish the computation we need to recognize in this expression the determinant of the metric G j . This determinant is computed  explicitly in Appendix A, where we show that when the closure condition is satisfied i ji N (z i ) = 0, we have (for any n)    ji j j jk det G j (z i ) = 16 i< j
|z i j |2 |z jk |2 |z ki |2 . (1+|z i |2 )2 (1+|z j |2 )2 (1+|z k |2 )2

Using this we can rewrite the above formula for the Poisson brackets of Z , Z as 4 2 2 i i=1 (1 + |z i | ) det(G j ). {Z , Z } = 4 2|z 43 z 21 | i (2 ji ) Inverting this Poisson bracket gives the symplectic structure (53).

(57)

(58)

 

3.4. Holomorphic form of the decomposition of the identity. We now have all the ingredients to write the identity decomposition in a holomorphically factorized form. Taking into account the expression (52) for the measure, as well as the representation (47), we can now rewrite (35) as the following integral:

  n  n−3 n 2 2  d ji j i=1 (1+|z i | ) det G j (z i ) 2 n 1j = 8π 2π Sj (n−3)! 2 i (2 ji ) |z 12 |2(n−2) |z 23 |2(4−n) |z 13 |2(4−n) i=4 |z 3i |4 i=1

× Kˆ j (Z i , Z i ) |j , Z i j , Z i |,

(59)

where the integral is now over the quotient Sj . Note that we have dropped the integral over SU(2) since the integrand is SU(2)-invariant. To transform the result further, one just has to substitute here an expression for the measure n−3 j . Above we have found such an expression for the most interesting cases n = 3, 4, so let us analyze these cases. In the case n = 3 the integral drops out since the quotient space consists of a single point. The formula (57) for the determinant implies that the term in parenthesis is equal to unity. The identity decomposition (59) in this case thus becomes 1 d j d j d j Kˆ j , j , j N 2 , (60) π 1 2 3 1 2 3 j1 j2 j3 is the normalization coefficient of the coherent intertwiner introduced 1=

where N j1 , j2 , j3 in (48). In the case n = 4 the factors depending on z 1 , z 2 , z 3 in the symplectic 2-form (53) exactly cancel those in the parenthesis and the integral (59) simply becomes  4 1  d d2 Z Kˆ j (Z , Z ) |j , Z j , Z | , (61) 1j = ji 2π 2 C i=1

which is our previous result (28) specialized to the case n = 4.

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L. Freidel, K. Krasnov, E. R. Livine

4. The Kähler Potential on Sj and Semi-Classical Limit The purpose of this section is to study a geometrical interpretation of our main formula (28) in the case of large spins. From the perspective of geometric quantization of a Kähler manifold Sj we could expect that the integration measure in the physical inner product formula (24) is given by the exponential of the Kähler potential on Sj . Below we shall see that this is indeed the case in the limit of large spins.

4.1. A useful representation for the kernel. Let us start by remarking that, comparing the expression for the measure on Sj given by the formulae (28) and (59), we see that for any n we must have   n n n−3 det G j (z i ) |z 3i |4  2 |z 12 |2(n−2) |z 23 |2(4−n) |z 13 |2(4−n) i=4 j n d Zi . = 2 2 (n − 3)! 2 i (2 ji ) i=1 (1 + |z i | ) i=4

(62) The prefactor on the right-hand-side in this formula, i.e. e−2• (zi ,zi ) ≡

|z 12 |2(n−2) |z 23 |2(4−n) |z 13 |2(4−n) n 2 2 i=1 (1 + |z i | )

n

i=4 |z 3i |

4

,

(63)

turns out to play an important role in the geometrical description of the constraint surface. Let us give an expression for the function • (z i , z i ) in coordinates g, Z i . The change of coordinates is easily computed using the explicit map (19) between the z i ’s and the Z i ’s, and we get: 





e• (g,Z i ,Z i ) = |b|2 + |d|2 |a + b|2 + |c + d|2 |a|2 + |c|2 n

  |a Z i + b|2 + |cZ i + d|2 .

(64)

i=4

Considering the expression (25) for Kˆ j (Z , Z¯ ) in terms of an integral over g ∈ SL(2, C), we can now give the following useful representation for the kernel:  Kˆ j (Z i , Z¯i ) = dnorm g e−j (g,Z i ,Z i )−2• (g,Z i ,Z i ) , (65) SL(2,C)

where ej (g,Z i ,Z i ) ≡ (|b|2 + |d|2 )2 j1 (|a + b|2 + |c + d|2 )2 j2 (|a|2 + |c|2 )2 j3 n

2 ji  |cZ i + d|2 + |a Z i + b|2 ,

(66)

i=4

and the function • (g, Z i , Z i ) is given by (64) above. Note that • (g, Z i , Z i ) = j =1/2 (g, Z i , Z i ), where all the spins are taken equal to 21 .

Holomorphic Factorization for a Quantum Tetrahedron

61

4.2. The Kähler potential on Sj . The purpose of this subsection is to note that the function j (g, Z i , Z i ) that appears in the exponent of (65) is essentially the Kähler potential on the original unconstrained phase space Pj but written in coordinates g, Z i , and corrected by adding to it a holomorphic and an anti-holomorphic function of the coordinates on Pj . Indeed, we have:  j (g, Z i , Z i ) = 2 ji log(1 + |z i |2 ) + h j (g, Z i ) + h j (g, Z i ), (67) i

where h j (g, Z i ) = 2 j1 log(d) + 2 j2 log(c + d) + 2 j3 log(c) +

n 

2 ji log(cZ i + d) (68)

i=4

is a holomorphic function of g and the cross-ratio coordinates Z i (and thus the original coordinates z i ). We recognize the standard Kähler potential on n copies of the sphere plus a holomorphic and an anti-holomorphic function of the coordinates z i . Thus, the function j (g, Z i , Z i ) can also be used as the Kähler potential on the unconstrained phase space Pj , for an addition of a holomorphic and an anti-holomorphic function does not change the symplectic form and thus produces an equivalent potential. The Kähler potential on the constraint surface Sj is then simply obtained by evaluat ing i 2 ji log(1 + |z i |2 ), or equivalently j (g, Z i , Z i ), on this surface. For a general n and generic values of spins this Kähler potential on Sj is not easy to characterize in any explicit fashion. However, the described characterization is sufficient for seeing that the Kähler potential gets reproduced in the semi-classical limit of large spins by the kernel (25). For n = 4 and all spins being equal, we evaluate explicitly the Kähler potential in Sect. 7. 4.3. Kernel in the semi-classical limit. We first note that the formula (62) essentially computes for us the Pfaffian of the matrix of the symplectic two-form j . Recall that the Pfaffian of a 2n × 2n antisymmetric matrix is given by Pf(ω) =

1 2n n!

 a1 ···a2n ωa1 a2 · · · ωa2n−1 a2n

(69)

and that det(ω) = Pf(ω)2 . Thus, from the definition (63) of the function • (z i , z i ), we have Pf( j ) det(G j ) n = e−2• (zi ,zi ) , (70) 2 i=1 (2 ji ) with • (z i , z i ) given in terms of the g, Z i coordinates on the unconstrained phase space by Eq. (64). We can now turn to the main task of this section which is to study the holomorphic factorization formula (28) in the semi-classical limit where all spins are rescaled homogeneously ji → λji with λ → ∞. For simplicity, we restrict our attention to the case of main interest, which is n = 4, but all arguments remain essentially unchanged in the general n case. We now use the representation (65) for the kernel. When all the spins are rescaled we have:  Kˆ λj (Z , Z¯ ) = dg e−λj (g,Z )−2• (g,Z ) . (71) SL(2,C)

62

L. Freidel, K. Krasnov, E. R. Livine

In the limit where λ → ∞ the integral is dominated by the points where the action j (g, Z ) reaches its minimum. The analysis performed in [5] established that: (i) the minimum points of the action are the ones satisfying the closure condition Hj (z i (g, Z )) = 0; (ii) given any Z ∈ / {0, 1, ∞} there is a unique (up to g → −g) Hermitian g such that the closure condition is satisfied, we denote this solution by g(Z ) and z i (Z ) ≡ z i (g(Z ), Z ); (iii) the Hessian of the action is the metric G j . In brief, these results stem from the fact that any orbit on the space {z 1 , · · · , z 4 } generated by the transformations in the Hermitian direction of SL(2, C) crosses the constraint surface Hj = 0 only once. Moreover the metric along these orbits is given by G j . The fact that the second derivative of j along the Hermitian direction in SL(2, C) is given by a positive metric G j implies that the function is convex and possesses a unique minimum that lies on the constraint surface. Therefore, the asymptotic behaviour of the kernel (71) for large spins ji is given by Kˆ j (Z , Z¯ ) ∼ (2π ) 2

3

e−j (Z ,Z ) ,  e2• (Z ,Z ) det(G j (Z ))

(72)

where j (Z , Z ) ≡ j (g(Z ), Z , Z ) and the definition of • (Z , Z ) is similar, and G j (Z ) ≡ G j (z i (Z )). We have dropped the rescaling factor λ in order not to clutter the formulae. It is assumed here that all spins are uniformly large2 . We can now remark that the term in the denominator is familiar to us. Indeed using (70) we can rewrite it in terms of the symplectic potential:  (2π )3  Kˆ j (Z , Z¯ ) ∼ Pf( j ) e−j (Z ,Z )−• (Z ,Z ) . (73) 2 i (2 ji ) Let us note that det(G j (Z )) scales as j 3 and that Pf( j ) grows as j n−3 , i.e. as j for the case n=4. Using this asymptotic expression for the kernel we can write an asymptotic decomposition of the identity    i (2 ji ) 1j ∼ d2 Z Pf( j ) e−j (Z ,Z )−• (Z ,Z ) |j , Z j , Z |. (74) π C This is precisely what is expected from the perspective of geometrical quantization of Kähler manifolds. Indeed, as we have seen in the previous subsection, the quantity j (Z , Z ) is just the Kähler potential of the reduced phase space Sj . To give an interpretation to other terms appearing in this formula let us recall some standard facts about geometric quantization. In the “naive” geometrical quantization scheme which we used in this paper so far the states ψ0 are defined to be (after a choice of trivialization of the quantization bundle3 ) holomorphic functions (i-e holomorphic 0-form) on the phase space and the scalar product is given by  ||ψ0 ||2 = Pf(ω)e ˆ − |ψ0 |2 , (75) P 2 In other words, we take the limit j → ∞ while j/j = O(1/j 2 ), where j is the difference between i n any two spins and j = 1/n i=1 ji . 3 The quantization bundle is a Hermitian line bundle over the phase space P with curvature given by iω,

where ω is the symplectic two-form. If P is simply connected this quantization bundle is unique. It exists only  if ωˆ ≡ ω/2π is an integral two form, i.e. such that S ωˆ ∈ N for any closed surface S.

Holomorphic Factorization for a Quantum Tetrahedron

63

where ω is the symplectic form on P, ωˆ ≡ ω/2π is the integral two form and  is the ¯ Kähler potential ∂∂ = iω. This form of quantization is often called that of BargmannSegal in the physics literature, and geometric quantization in the mathematics literature, canonical references are [7–9]. It is well-known however that a more accurate geometrical quantization includes the so-called “metaplectic” or more appropriately half-form correction. In the geometrical quantization with half-form correction the states ψ1/2 are holomorphic half-forms on the phase space and the scalar product is given by ||ψ1/2 ||2 =



  Pf ωˆ (0)

 

  Pf ωˆ e− |ψ1/2 |2 ,

(76)

P

where 0 denotes the point at which  reaches its minimum. As an example illustrating the above discussion we mention that the quantization of the sphere given by (12) can be written (up to a normalization coefficient) in the “naive” Kähler form if one takes as the Kähler potential  j (z) ≡ 2 j ln(1 + |z|2 ) and as the sym¯ j (z) = 2 j/(1 + |z|2 )2 . With this choice the decomposition plectic structure ω j = ∂ ∂ of the identity (or the scalar product formula) reads dj 1j = 2j



d2 z Pf(ω j )e− j (z) | j, z j, z|, 2π

(77)

which coincides with the naive geometrical quantization (75) up to a prefactor d j /2 j that goes to 1 in the large spin limit. However, the correct prescription (76) gives (12) without the need for any prefactors. Indeed, in this case one chooses the Kähler potential to be  j (z) ≡ (2 j + 1) ln(1 + |z|2 ) (note the shift 2 j → 2 j + 1) and then (76) gives precisely (12). Comparing (76) and (74) we see that the Kähler potential j (Z , Z ) on the space of shapes Sj is correctly reproduced. Moreover, our result (74) reproduces not just the “naive” Kähler potential j , but even the metaplectic corrected one given in this case by j + • , which also amounts to the shift 2 ji → 2 ji + 1. It is worth emphasizing that the integration kernel in (24) is only equal to (minus) the exponential of the Kähler potential on Sj in the limit of large spins. For generic spins the two quantities differ by quantum corrections. Thus, for generic values of spins, the quantization of Sj provided by the identity decomposition formula (28), even though equivalent (in the sense that there is an isomorphism of the resulting Hilbert spaces) to the geometric quantization (76), is in details different from it. The isomorphism between the two Hilbert spaces is quite non-trivial, involves quantum corrections and is only unitary asymptotically (for large spins), see [17] for more details on this point. In this section we have only discussed the case n = 4, but it is easy to see that it generalizes without any difficulty to the arbitrary n case. The only novelty in the general case is a different numerical prefactor (i.e., the right-hand-side should be multiplied by 1/(2π )n−4 ) and the replacement of Z by Z i ’s. We would like to finish this section by pointing out that our results imply that the n-point function of the bulk-boundary correspondence of string theory has the interpretation of the (exponential of the) Kähler potential on the space of shapes S j . This is surprising, at least to the present authors, and appears to be a new result. It would be of interest to provide some more direct argument for why this is the case, but we leave this interesting question to further work.

64

L. Freidel, K. Krasnov, E. R. Livine

5. Invariant Operators on the Intertwiners Space In this section we will focus exclusively on the case n = 4 relevant for the quantum tetrahedron, even though some of our results can easily be extended to the general case. Now that we have defined the holomorphic intertwiners |j , Z and shown that they provide (28) an over-complete basis in the Hilbert space of intertwiners Hj = SU(2)  , we would like to describe and study the action of SU(2)-invariant V j1 ⊗ · · · ⊗ V j4 operators on this basis. It is well known that a generating set of such operators is given by 2 Ji j ≡ (J(i) + J( j) )2 = J(i) + J(2j) + 2 J(i) · J( j) ,

(78)

a , a = 1..3, act on the i th vector space V , with the action where the su(2) generators J(i) ji 2 of J(i) on V ji (and thus Hj ) being diagonal with eigenvalue ji ( ji + 1). The operators Ji j are then positive Hermitian operators acting on the intertwiners space Hj . Diagonalising these operators one gets an orthonormal basis in Hj denoted |j , k i j , where the subscript i j labels which “channel” one works with , and k is the “intermediate” spin in the corresponding channel, running from | ji − j j | to ( ji + j j ). These intertwiners are defined by the conditions

Ji j |j , k i j = k(k + 1)|j , k i j ,

ij

j , k  |j , k i j =

δk,k  . dk

(79)

Our goal in this section is to express the operators Ji j as second-order differential operators in the complex variable Z and then describe the eigenstates as holomorphic functions of Z . From this we will deduce the overlap function between the real and holomorphic intertwiners: ij

Cjk (Z ) ≡ i j j , k|j , Z .

(80)

We find that these eigenvectors are essentially hypergeometric polynomials. As a by-product of our results we obtain a check of the phased Gaussian ansatz for coherent states on the quantum tetrahedron [18,12], as well as an integral formula for the {6 j}-symbol in terms of Z . 5.1. Scalar product operators and their eigenfunctions. In this subsection we obtain and study Ji j as second-order differential operators in the complex variable Z . The action of Ji j on coherent states was computed in [5]. One finds that it acts as a holomorphic differential operator in the variables z i , z j on which the coherent intertwiner ||j , z i depends. Explicitly, Ji j = −z i2j ∂i ∂ j + 2z i j ( ji ∂ j − j j ∂i ) + ( ji + j j )( ji + j j + 1).

(81)

It is easy to check that this second-order differential operator has the correct spectrum by computing its action on polynomials in z i j = z i − z j . Thus, we have:  l    l Ji j z i j = ( ji + j j − l)( ji + j j − l + 1) z i j . (82) The identification k = ( ji + j j − l) ≥ 0 shows that such polynomials are indeed eigenvectors of eigenvalue k(k + 1). However, the state z li j is not an SU(2)-invariant vector.

Holomorphic Factorization for a Quantum Tetrahedron

65

In order to find eigenvectors in the space of invariant vectors (intertwiners), we need to express this operator as acting on functions of Z . This is an exercise in a change of variables. Indeed, recall that the holomorphic intertwiners |j , Z are related to the coherent intertwiners ||j , z i by a non trivial pre-factor (45): ||j , z 1 , . . . , z 4 = Pj (z 1 , . . . , z 4 ) |j , Z , j + j2 − j34 j34 − j12 j34 + j12 2 j4 z 23 z 31 z 43 ,

Pj (z 1 , . . . , z 4 ) = z 121

(83)

where we have denoted ji j ≡ ji − j j . One can now commute the differential operator Ji j through the pre-factor Pj (z 1 , . . . , z 4 ), and then translate the partial derivatives ∂i , ∂ j with respect to the complex labels z i and z j into the derivative ∂ Z with respect to the cross-ratio Z . We denote the resulting operator by i j : Pj (z 1 , . . . , z 4 ) i j |j , Z ≡ Ji j Pj (z 1 , . . . , z 4 ) |j , Z .

(84)

For definiteness, we now focus on the operator 12 . As we shall see, this operator happens to be the simplest one in the sense that it is precisely of the hypergeometric form, while to relate the other operators to the hypergeometric-form ones one needs to do some extra work, see below. In order to convert the ∂1 , ∂2 derivatives we use the following identities valid for an arbitrary function φ(Z ): z 42 z 31 Z ∂ Z φ, ∂2 φ = Z ∂ Z φ, z 21 z 41 z 23 z 21  1  ∂2 ∂1 φ = 2 Z (1 − Z )∂ Z2 φ + (1 − 2Z )∂ Z φ . z 21 ∂1 φ =

(85)

After straightforward but somewhat lengthy algebra one gets 12 = Z (Z − 1)∂ Z2 + (2( j34 + 1)Z − (1 + j34 − j12 )) ∂ Z + j34 ( j34 + 1).

(86)

One recognizes a second order differential operator of the hypergeometric type. It is then well-known that the hypergeometric function F(a, b; c|Z ) gives one of the two linearlyindependent solutions of (a,b;c) φ = 0, where (a,b;c) ≡ Z (Z − 1)∂ Z2 + ((a + b + 1)Z − c) ∂ Z + ab.

(87)

The SU(2)-invariant operator 12 − k(k + 1) is then a hypergeometric operator (a,b;c) with parameters a = −k + j34 , b = k + j34 + 1, c = j34 − j12 + 1.

(88)

However, since the operator (a,b;c) is second-order, there are two linearly-independent solutions. The question is then which linear combinations of them corresponds to the eigenvectors of 12 that we are after. The answer to this is as follows. When k respects the bounds max(| j12 |, | j34 |) ≤ k ≤ min( j1 + j2 , j3 + j4 ) one of the solutions of the arising hypergeometric equation is polynomial. This is the eigenfunction we are looking for, and it is given by (k − j12 )! (k − j34 )!( j34 − j12 )! ×F(−k + j34 , k + j34 + 1; j34 − j12 + 1; Z ).

(j −j , j +j ) Pˆk−34j34 12 34 12 (Z ) =

(89)

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L. Freidel, K. Krasnov, E. R. Livine

(a,b) (a,b) (a,b) Here Pn denotes the Jacobi polynomial and Pˆn (Z ) ≡ Pn (1−2Z ) is the shifted Jacobi polynomial (see Appendix E for some useful facts about the shifted polynomials). This expression is valid if j34 ≥ | j12 |, and we can assume that this inequality is satisfied without any loss of generality. Indeed, we can always assume that j12 ≥ 0 since otherwise we can exchange the role of 1 and 2 as is implied by the equality (j −j , j +j ) (j +j , j −j ) Pˆk−34j34 12 34 12 (Z ) = (−1)k− j34 Pˆk−34j34 12 34 12 (1 − Z ).

(90)

We can also assume that j34 ≥ j12 since otherwise we can exchange (12) with (34) using the exchange identity (k − j12 )!(k + j12 )! (j −j , j +j ) (j −j , j +j ) (−Z ) j12 − j34 Pˆk−12j12 34 12 34 (Z ), (91) Pˆk−34j34 12 34 12 (Z ) = (k − j34 )!(k + j34 )! which is valid if j12 ≥ j34 . A special case where all formulae simplify considerably is when all representations are equal, j1 = j2 = j3 = j4 = j, which correspond to a tetrahedron with all faces having the same area. Then our eigenfunctions reduce to the shifted Legendre polynomial: (0,0) (Z ) = Pi (1 − 2Z ) = Pˆi

i  2  i (−Z )i−l (1 − Z )l . l

(92)

l=0

Note that in this case the eigenvectors actually do not depend on the spin j. The dependence on j will nevertheless reappear in the normalization of these states. (j −j , j +j ) By construction the polynomials Pˆk−34j34 12 34 12 are eigenstates of the operator 12 . These eigenvectors give, up to normalization, matrix elements of the change of basis between the holomorphic intertwiner |j , Z and the usual orthonormal intertwiners |j , k 12 that diagonalize 12 . More precisely, if we define the overlap4 Cjk (Z ) ≡ j , k|j , Z ,

(93)

the above discussion shows that it is proportional to the Jacobi polynomial: (j −j , j +j ) Cjk (Z ) = Njk Pˆk−34j34 12 34 12 (Z ),

(94)

where the non-trivial normalization coefficient is given by the integral

Njk

−2

 4  2 1   ˆ ( j34 − j12 , j34 + j12 )  2 ˆ P = d Z K (Z , Z ) (Z ) d   . j j  i k− j34 2 2π i=1

Our task is now to determine these normalization coefficients. 4 From now on when we work in the channel 12 drop the superscript 12 to avoid notation cluttering.

(95)

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67

5.2. 3-point function and normalization of the 4-point intertwiner. Instead of computing the integral (95) directly, which is quite non-trivial, we will take an alternative route and express the normalized intertwiner |j , k as a combination of certain Clebsch-Gordan coefficients. To this end, we start by reminding the reader of some information about the coherent intertwiner for n=3. In this trivalent case the space of intertwiners is one dimensional and we have denoted in Sect. 2 the unique normalized 0|0 = 1 intertwiner by |0 . As the analysis of that section shows, the coherent intertwiner ||j , z i is proportional to |0 , with the proportionality coefficient being the normalization factor N j1 , j2 , j3 times a z i -dependent pre-factor. Explicitly: ||j , z 1 , z 2 , z 3 = C j1 , j2 , j3 (z 1 , z 2 , z 3 )|0 j + j2 − j3 − j1 + j2 + j3 j1 − j2 + j3 z 23 z 13 |0 .

= N j1 , j2 , j3 z 121

(96)

In (60) we have related the normalization coefficient N j1 , j2 , j3 to the 3-point function Kˆ j1 , j2 , j3 . The 3-point function can be computed explicitly, see (D9). One gets the result : N 2j1 , j2 , j3 =

[2 j1 ]![2 j2 ]![2 j3 ]! . (97) [ j1 + j2 + j3 + 1]![− j1 + j2 + j3 ]![ j1 − j2 + j3 ]![ j1 + j2 − j3 ]!

Now given the trivalent Clebsch-Gordan map we can construct the normalized 4-valent intertwiner – an eigenstate of J12 – by gluing two 3-valent intertwiners. Indeed, as described in [5], there is a gluing map g : Vk ⊗ Vk → C that can be represented in terms of coherent states as  d2 N (z) 2k g = dk z¯ | − 1/z ⊗ |z . (98) (1 + |z|2 )2k In terms of this gluing map the normalized intertwiner is given by  C j1 , j2 ,k (z 1 , z 2 , −1/z)Ck, j3 , j4 (z, z 3 , z 4 ) 2k 2 k z¯ d N (z). Cj (z i ) = dk (1 + |z|2 )2k

(99)

At first sight this integral seems quite cumbersome. However, we know that the integral being holomorphic and SU(2)-invariant is entirely determined by its values at the special points (0, 1, ∞, Z ). Thus, the overlap between the holomorphic |j , Z and the real |j , k intertwiners can be extracted as the limit Cjk (Z ) = lim X →∞ (−X )−2 j3 Cjk (0, 1, X, Z ). Using the explicit expression (96) for the SU(2)-invariant 3-point function we get the following integral representation for the overlap: Cjk (Z ) = (−1)s−2k di N j1 , j2 ,k Nk, j3 , j4 Ijk (Z ), where s = j1 + j2 + j3 + j4 and  (¯z + 1)k− j12 (z − Z )k− j34 k . Ij (Z ) ≡ d2 N (z) (1 + |z|2 )2k

(100)

In order to compute the integral we perform the following change of variables: √ √ iφ z 1 dφ = d2 N (z), ue =  , 1−u =  , du 2 2 2π 1 + |z| 1 + |z|

(101)

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then expand all the terms and perform the integration over φ. We are left with Ijk (Z ) =



(k − j12 )!(k − j34 )! (k − j12 − n)!(k − j34 − n)!(n!)2 n  1  n 2k−n × u (1 − u) du (−Z )k− j34 −n .

(102)

0

Now making use of the standard formula 

1

u n (1 − u)m =

0

n!m! , (n + m + 1)!

(103)

we recognized that the integral in question is proportional to the hypergeometric function Ijk (Z ) =

(k − j12 )!(k + j34 )! F(−k + j34 , k + j34 + 1; j34 − j12 + 1; Z ). (104) (2k + 1)!( j34 − j12 )!

Thus we get the final expression in terms of the Jacobi polynomial (94) with the normalization coefficient given by Njk = (−1)s−2k  ×

(2 j1 )!(2 j2 )!(2 j3 )!(2 j4 )!(k + j34 )!(k − j34 )! . ( j1 + j2 + k + 1)!( j3 + j4 + k + 1)!( j1 + j2 − k)!( j3 + j4 − k)!(k + j12 )!(k − j12 )!

By writing the factorials in this formula as -functions we can extend the definition of the normalization coefficient Njk beyond its initial domain of validity j34 ≤ k ≤ min( j1 + j2 , j3 + j4 ) (recall that we are under the assumption j34 ≥ | j12 |). Since 1/ (0) = 0 one sees however that Njk = 0 if k = j1 + j2 + 1 or k = j3 + j4 + 1. This implies that the Jacobi polynomial corresponding to this value is not normalisable with respect to our norm d 2 Z K j , and so this particular Jacobi polynomial is not part of the Hilbert space. To explore the other boundary k = j34 − 1 one first needs to rewrite the overlap in terms of the hypergeometric function Cjk = N˜ jk F and notice again that the normalization coefficient N˜ jk vanishes at the boundary k = j34 − 1, as long as j34 > j12 . In the case all ji ’s are equal to a given spin j, the expression (94) simplifies to Cjk (Z ) =

(2 j)!(2 j)! (−1)2k Pˆk (Z ). 2 j + k + 1 (2 j + k)!(2 j − k)!

(105)

We will need this expression in Sect. 7.

5.3. Other channels. In the previous two subsections we have studied the channel 12 and the associated operator J12 whose eigenstates |j , k 12 provided a real basis in the 4valent intertwiners Hilbert space. This choice of the channel is somewhat distinguished by the fact that, with our choice (49) for the cross-ratio coordinate Z , the second-order holomorphic operator 12 turned out to be precisely of the hypergeometric type (87) so that the eigenstates – the real intertwiners – are just the Jacobi polynomials (89).

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It is also interesting and important to compute the other channel operators and their eigenstates. Indeed, consider for example the channel 23. There is similarly an operator J23 and the basis in H j1 , j2 , j3 , j4 given by its eigenstates |j , k 23 . The overlap 23 j , k|j , l 12 is the 6 j-symbol, and this is why the other basis in the Hilbert space is of interest. We can similarly find the holomorphic representation of J23 by commuting it with the prefactor (83). With our choice (49) of the cross-ratio, however, the resulting holomorphic operator is not exactly of the type (87). Indeed, the computation is completely similar to the one performed in the 12 channel. We use: ∂3 φ =

z 24 z 31 Z ∂ Z φ, ∂2 φ = Z ∂ Z φ, z 23 z 43 z 23 z 21  1  ∂3 ∂2 φ(Z ) = 2 Z 2 (Z − 1)∂ Z2 φ + Z 2 ∂ Z φ . z 23

(106)

to obtain 23 = Z 2 (1 − Z )∂ Z2 + [( j1 + j2 − j34 + 2 j4 − 1)Z − 2( j1 + j4 )] Z ∂ Z −2 j4 ( j1 + j2 − j34 )Z + ( j1 + j4 )( j1 + j4 + 1).

(107)

Thus, this operator is not exactly of the hypergeometric type. However, as we shall explain below, its eigenfunctions are also related to Jacobi polynomials via a simple transformation of the cross-ratio coordinate. Let us also mention that one can similarly compute 13 with the result being: 13 = Z (Z − 1)2 ∂ Z2 + [( j1 + j2 − j34 + 2 j4 − 1)Z + j34 − j12 + 1] (1 − Z )∂ Z +2 j4 ( j1 + j2 − j34 )(Z − 1) + ( j2 + j4 )( j2 + j4 + 1), (108) which is also not of the hypergeometric type. It is then easy to check that: 12 + 13 + 23 = j1 ( j1 + 1) + j2 ( j2 + 1) + j3 ( j3 + 1) + j4 ( j4 + 1),

(109)

which is an expected relation that follows from the definition of Ji j and the condition J(1) + J(2) + J(3) + J(4) = 0 that is just the requirement of SU(2)-invariance of the intertwiner space. Thus, in view of (109) it is sufficient to compute only two scalar product operators, say 12 and 13 , in order to get the expression of all scalar product operators. Indeed these operators are symmetric, 12 = 21 , and operators with opposite labels are equal, 12 = 34 . Moreover, as we shall now explain, it is in fact sufficient to compute only one of these operators, for the two other inequivalent operators, as well as their eigenstates can be obtained by considering the action of the group of permutations acting on z 1 , . . . , z 4 . Thus, we denote by σi j the permutation that exchanges the variables ( ji , z i ) and ( j j , z j ) in the functional P(z 1 , . . . , z 4 ) defined in (83) and in the state ||j , z 1 , . . . , z 4 . The action of these permutations can then be extended to the holomorphic intertwiners by   (110) σˆ i j |j , Z ≡ Pj (z 1 , . . . , z 4 )−1 σi j Pj (z 1 , . . . , z 4 )|j , Z . Since Pj (z 1 , . . . , z 4 ) and σi j Pj (z 1 , . . . , z 4 ) have the same transformation properties g under conformal transformations, i.e., Pj (z i ) = i (cz i + d)−2 ji Pj (z i ), it follows that σi j is well defined as an operator acting purely on the cross-ratio Z . The action of all 24

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different permutations on the coherent intertwiner |j , Z is given in Appendix C. For instance, we have σˆ 12 |j , Z = (−1)s−2 j3 |j 12 , 1 − Z ,    Z 2 j2 2 j4  , σˆ 23 |j , Z = (−1) (1 − Z ) j 23 , Z −1    1 , σˆ 13 |j , Z = (−1)s Z 2 j4 j 13 , Z

(111)

where s = j1 + j2 + j3 + j4 and j i j ≡ σˆ i j (j ). Using (111) we can now understand why the computed above operators 13 , 23 turned out to be not of the hypergeometric type. Indeed, these operators can be obtained from 12 by conjugation with σˆ i j . We have: σˆ 23 12 σˆ 23 = 13 , σˆ 13 12 σˆ 13 = 23 . Let ˆ 23 the hypergeometus consider the channel 23 in more details. Thus, if we denote by  ric operator obtained from 12 (86) by exchanging j1 with j3 , then the operator 23 is related to this hypergeometric operator by a change of variables Z → Z −1 followed by the conjugation with Z 2 j4 , see (111). In other words, we have: 

˜ 23 Z 2 j4 F(Z −1 ). (23 F)(Z −1 ) = Z −2 j4  (112) Let us see that this is indeed the procedure that gives (107). To this end, let us first write 23 in terms of the variable X ≡ Z −1 . One gets: ˜ 23 = X (X − 1)∂ X2 + [2( j1 + j4 + 1)X − ( j1 + j2 − j34 + 2 j4 + 1)] ∂ X  2 j4 ( j1 + j2 − j34 ) + ( j1 + j4 )( j1 + j4 + 1). − X Now conjugating this operator with X −2 j4 one gets ˜ 23 X −2 j4 = X (X − 1)∂ X2 + [2( j1 − j4 + 1)X − ( j1 − j4 + j2 − j3 + 1)] ∂ X X 2 j4  + ( j1 − j4 )( j1 − j4 + 1), which is the hypergeometric operator obtained from 12 by exchanging j1 and j3 as expected. An interesting application of the above discussion is as follows. Let us consider the overlap 23 Cjk (Z ) of the holomorphic intertwiner |j , Z with the basis |j , k 23 diagonalizing J23 (and thus 23 ). We can then express these overlap coefficients in terms of the ones in the 12 channel given by the Jacobi polynomials: 23

Cjk (Z ) = (−1)s Z 2 j4 12 Cjk (Z −1 ).

(113)

This formula allows to get an interesting expression for the usual 6 j-symbol of SU(2). Indeed, the 6 j-symbol is given by the overlap between basis states in two different channels. For example, we can consider   j1 j2 k 23 12 . (114)

j , l|j , k = j3 j4 l

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Now inserting into this formula the holomorphic identity decomposition (61), and using (113) we get: 

j1 j2 k j3 j4 l

 =

 4 1  d d 2 Z Kˆ j (Z , Z¯ ) (−1)s Z¯ 2 j4 12 Cjl (1/ Z¯ ) 12 Cjk (Z ). i 2π 2 i=1

(115) Here the coefficients 12 Cjk (Z ) are essentially the Jacobi polynomials, see (94). 6. On the Bulk/Boundary 4-Point Function In this section we derive some non-trivial properties of the 4-point function Kˆ j (Z , Z¯ ) and compare them with what is known in the literature about this object. Here we shall use some of the facts about the action of the permutation group derived in the previous section. 6.1. Hermiticity and measure. The operator 12 studied in the previous section is a positive Hermitian operator with respect to the inner product on H j1 , j2 , j3 , j4 defined by the kernel Kˆ j (Z , Z¯ ). Indeed, it descends from (J(1) + J(2) )2 , which is obviously Hermitian. However, since it is given by a hypergeometric-type operator 12 = a,b,c , where a,b,c = Z (Z − 1)∂ Z2 + ((a + b + 1)Z − c)∂ Z − ab

(116)

with the hypergeometric parameters a = j34 , b = j34 + 1, c = j34 − j12 + 1, the hermiticity of 12 implies the equality 

φ|12 |φ = d 2 Z K j (Z , Z¯ ) φ(Z ) φ(Z )  = d 2 Z K j (Z , Z¯ ) φ(Z ) φ(Z )

(117)

(118)

for all holomorphic functions φ. In turn, integrating by parts, this leads to a constraint on the kernel Kˆ j (Z , Z¯ ). Thus, one finds that Kˆ satisfies a “balanced” hypergeometric equation in both Z and Z¯ :   a,b,c Kˆ j (Z , Z¯ ) = a,b,c Kˆ j (Z , Z¯ ),

(119)

 where  is the transpose of a,b,c , which is an operator of the same type, i.e. a,b,c = a  ,b ,c with:

a  = 1 − a = 1 − j34 , b = 1 − b = − j34 , c = 2 − c = 1 − j34 + j12 . (120) This equation strongly suggests that Kˆ (Z , Z¯ ) must be given by an expansion in terms of joint equal eigenvalue eigenstates of  and  . Moreover, because of the positivity of 12 these common eigenvalue should be positive. The space of positive

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 eigenvalue λ(λ + 1) eigenstates of a,b,c is (real) two-dimensional. Let us now discuss the eigenfunctions. When j12 ≥ j34 a convenient set of linearly independent solutions  of a,b,c φ = λ(λ − 1)φ, which is the same as the set of linearly independent solutions of − j34 +λ,1− j34 −λ,1− j34 + j12 φ = 0 is given by5

F(− j34 + λ, 1 − j34 − λ; 1 − j34 + j12 ; Z ), and Z j34 − j12 (Z − 1) j12 −λ F(− j12 + λ, j34 + λ; 2λ; 1/(1 − Z )).

(121)

The problem is to find which linear combination of these solutions arises in the holomorphic decomposition of K j . To see this, we note that, as reviewed in Appendix D, the kernel Kˆ j (Z , Z¯ ) admits a representation as a double series expansion around Z = 0. Thus, in particular it is regular at Z = 0. However, it must also be regular at Z = ∞. Indeed, the 4-point function K j (z i ) transforms covariantly under the permutations σi j of ( ji , z i ). This translates into non-trivial identities for the kernel Kˆ j (Z , Z¯ ), similar to those derived in the previous section for the holomorphic intertwiner. For instance, we have: Kˆ j1 , j2 , j3 , j4 (Z , Z¯ ) = Kˆ j2 , j1 , j3 , j4 (1 − Z , 1 − Z¯ )   1 1 −24 ˆ = |1 − Z | , , K j2 , j3 , j1 , j4 1 − Z 1 − Z¯

(122)

where i = 2( ji + 1). These should be compared with (111). The last equality plus our regularity condition at Z = 0 shows that Kˆ j (Z , Z¯ ) vanishes in the limit Z → ∞ as |Z |−24 , and so is regular. This immediately implies that the expansion of Kˆ j (Z , Z¯ ) should be in terms of functions regular at infinity. Only the second set of solutions in (121) is regular at infinity, so we must expect an expansion  Kˆ j (Z , Z¯ )= aλ |Z j34 − j12 (Z −1) j12 −λ F(− j12 +λ, j34 +λ; 2λ; 1/(1− Z ))|2 , (123) λ

where the sum is taken over λ ≥ j12 so that the whole expansion is regular at infinity. Thus, the requirement of hermiticity of the Ji j operators (of which we have considered only one, but the others lead to the same conclusion) strongly suggests that the kernel Kˆ j (Z , Z¯ ) holomorphically factorizes as indicated in (123). However, this holomorphic factorization that might seem surprising from the point of view taken in this article, is not at all surprising and in fact very much desired from the point of view of bulk/boundary dualities, that one would like to interpret Kˆ j (Z , Z¯ ) (or various closely related objects, see e.g. [19]) as the 4-point function of some CFT. If this interpretation is valid, then (123) is not surprising, and follows directly from the defining properties of the CFT as we now review. 5 Because for integral λ all coefficients entering the hypergeometric function are integers we cannot take a basis of two hypergeometric functions with the same argument as would be usual for generic hypergeometric functions. This is why we use here a mixed basis with arguments Z and 1/(1 − Z ). The existence of such a basis follows from Kummer’s relations. For the equation a,b,c φ(z) = 0 we take the set F(a, b; c; z) and z 1−c (z − 1)c−a−1 F(a − c + 1, 1 − b; a − b + 1; 1/(1 − z)).

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6.2. Holomorphic factorization in CFT. In this section we would like to understand to what extent the quantity Kˆ j (Z , Z¯ ) can be interpreted as a CFT 4-point function of operators φi of conformal dimensions 1 , . . . , 4 inserted at points 0, 1, ∞, Z . We base our discussion here on [20]. The first consideration of interest for us is to look at the limit of the 4-point function when the points z 1 → z 2 . The cross-ratio Z that we have been working with is not a convenient coordinate to study this limit, as Z ∼ 1/z 21 → ∞. Thus, let us introduce a different cross-ratio: U=

z 12 z 34 , z 13 z 24

U=

1 , 1− Z

(124)

which goes to zero in the limit z 1 → z 2 . Let us now recall some standard facts about 4-point functions in conformal field theory. A very important role in CFT is played by the so-called operator product expansion. This interprets the 4-point function as a sum over contributions of (primary and their descendants) operators of conformal dimension  and spin l to the operator product expansion of φ1 (z 1 ) and φ2 (z 2 ), where φ1,2 are the so-called primary operators of given conformal dimension 1,2 . To make this more precise, let us recall that, as we have witnessed in this paper already on many occasions, the CFT 4-point function transforms covariantly under conformal transformations and thus can in general be expressed as:   |z 24 | 12 1

φ1 (z 1 )φ2 (z 2 )φ3 (z 3 )φ4 (z 4 ) = |z 12 |1 +2 |z 34 |3 +4 |z 14 |   |z 14 | 34 × F(|U |2 , |V |2 ), (125) |z 13 | where i j = i −  j , the cross-ratio U is as introduced above (124), V = 1 − U , and F(|U |2 , |V |2 ) is the 4-point function as a function of the conformal invariants. The formula (125) is as given in [20]. With appropriate modifications it is valid in any dimensions, but only in 2 dimensions the two cross-ratios U, V are simply related as V = 1 − U . For later use we note that the 4-point function (125) is given as a function of the cross-ratio Z by: lim |X |23 φ1 (0)φ2 (1)φ3 (X )φ4 (Z ) = |1− Z |12 |Z |34 −12 F(|U |2 , |V |2 ),

X →∞

(126)

where U = 1/1 − Z , V = Z /Z − 1 are understood to be functions of Z . For any CFT, the function F(|U |2 , |V |2 ) can be decomposed into a sum of contributions of primary operators with given , l, where  is a conformal dimension and l is the angular momenta, so we have:  (l) F(|U |2 , |V |2 ) = a,l G  (|U |2 , |V |2 ), (127) ,l

where the sum is taken over the range of , l which constitutes the spectrum of the CFT. CFT’s for which this range is discrete and finite are called rational. But in general the sum here is an integral. This expansion corresponds to the operator product expansion O1 (z 1 , z¯ 1 )O2 (z 2 , z¯ 2 ) ∼ ,l a,l O,l (z, z¯ ). The requirement of unitarity states that all the coefficients a,l are positive. The representation (127) is also known as the partial wave decomposition, see [20] for more details on this, in particular for analogous formulae in higher dimensions.

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L. Freidel, K. Krasnov, E. R. Livine (l)

The partial waves G  (|U |2 , |V |2 ) can (in any dimension) be characterized completely. The situation is especially simple in two dimensions. These are functions satisfying the following eigenvalue differential equation: (l) 2 2 2 2 L 2 G (l)  (|U | , |V | ) = −C ,l G  (|U | , |V | ),

C,l = ( − 2) + l 2 , (128)

or its appropriate generalization to higher dimensions. Here L 2 = (1/2)Tr(L 2 ) is the “quadratic Casimir” for the operator L = L 1 + L 2 given by the sum of the generators of the conformal group acting on φ1 , φ2 . In two dimensions the operator L 2 factorizes into a sum of a holomorphic and an anti-holomorphic one with respect to U, U¯ , with each of these being (related to those) of the hypergeometric type. The solution that behaves as: 2 2 λ1 ¯ λ2 G (l)  (|U | , |V | ) ∼ U U ,

U → 0,

(129)

where λ1 =

1 ( + l), 2

λ2 =

1 ( − l) 2

(130)

is given by:   1 1 2 2 λ1 ¯ λ2   G (l) (|U | , |V | ) = U F λ − , λ + , 2λ ; U U 1 12 1 34 1  2 2   1 1 ×F λ2 − 12 , λ2 + 34 , 2λ2 ; U¯ 2 2 ¯ + U ↔ U.

(131)

Here F(a, b, c; z) is the usual hypergeometric function. See e.g. [20] for a demonstration of all these facts, as well as for a generalization to higher dimensions. We can now compare these standard CFT facts with the formula (123) we have been led to by the requirement of hermiticity in the previous subsection. Recalling (126) and rewriting everything in terms of the Z cross-ratio, we see that the 4-point function factorization formula takes precisely the form (123), where in (123) the sum is restricted to intermediate states with zero angular momentum l = 0 and thus λ1 = λ2 = λ. Thus, we learn that what the requirement of hermiticity suggests for the kernel Kˆ j (Z , Z¯ ) is in fact the standard CFT 4-point function partial wave decomposition formula. To summarize, we see that the kernel function Kˆ j (Z , Z¯ ) has all the properties of a CFT 4-point function. It behaves covariantly under the conformal transformations, of which at the level of the cross-ratio coordinate Z only the permutations (122) remain. It must moreover admit the holomorphic factorization of the standard CFT form. Quite remarkably as we have seen this holomorphic factorisation follows from the requirement of hermiticity of invariant operators. However, the underlying CFT is unknown. Moreover, it is even not obvious that there is an underlying unitary CFT, i.e., the one where the partial wave decomposition (123) gives rise to positive coefficients aλ . One could argue that this can be determined by explicitly computing the function K (Z , Z¯ ) for a given set of conformal dimensions 1 , . . . , 4 and then expanding the result into partial waves. However, the outcome of this certainly depends on the conformal dimensions chosen. It may be expected that for those integral dimensions of interest to us here  = 2( j +1) the situation must be simpler. However, the spectrum that should be expected in (123) does not seem to be known in the literature. What has been worked out in the applications to

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75

AdS/CFT correspondence of string theory is the first few terms of the expansion (123) (in the case of four dimensions) for some simple integral conformal dimensions, and these have been shown to match the boundary CFT predictions. However, no general result seems to be known even in the simplest case of two dimensions. We finish this section by pointing out that from the perspective taken in this paper, the positivity of aλ (and thus the existence of some underlying unitary CFT) is highly plausible since the kernel Kˆ j (Z , Z¯ ) is a strictly positive functional with the interpretation of an exponent of the Kähler potential on an appropriate moduli space, see Sect. 4. It is also tempting to think that the factorizability of a CFT n-point function in any dimension could be interpretable as following from the hermiticity of certain invariant operators, but we leave an attempt to prove all this to future work. 7. Asymptotic Properties of Intertwiners The purpose of this section is to describe the large spins asymptotic properties of the overlap coefficients Cjk (Z ) = j , k|j , Z characterizing the holomorphic intertwiners in the usual real basis. Recalling that this quantity consists of the normalization coefficient times the shifted Jacobi polynomial, we need to develop the asymptotic understanding of both of these pieces. As a warm-up, let us analyze the case n = 3, where there is only the normalization coefficient to consider. 7.1. A geometric interpretation of the n = 3 intertwiner. Recall that for n = 3 the coherent intertwiner ||j , z i , up to a simple z-dependent prefactor, see (96), is the normalization coefficient N j1 , j2 , j3 times the unique normalized intertwiner |0 . In this subsection we would like to develop a geometric interpretation for the normalization coefficient. This can be achieved by considering the limit of large spins. The normalization coefficient N j1 , j2 , j3 is given by (97) as a ratio of factorials. In order to evaluate it in the limit of large spins we use the Stirling formula √ n! = 2π n n n e−n φ(n), (132) where φ(n) = 1 + 1/12n + O(1/n 2 ). Up to terms of order O(1/j 2 ) one finds

2J3

2J2 √ ( j1 + j2 )2 − j32 ( j3 + j1 )2 − j22 +1 −2 N j1 j2 j3 ≈ 4π A( ji ) √ 4 j1 j2 4 j3 j1 j1 j2 j3

2J1 ( j2 + j3 )2 − j12 × , (133) 4 j2 j3 where  = j1 + j2 + j3 and 2Ji =  − 2 ji . In this formula  A( ji ) = [ j1 + j2 + j3 ][− j1 + j2 + j3 ][ j1 − j2 + j3 ][ j1 + j2 − j3 ]/4 denotes the area of the triangle of edge length ji . As such this formula is not particularly illuminating, but a nice geometric interpretation can be proposed by noting that the quantities in brackets are invariant under a simultaneous rescaling of all the spins. So, they can be rewritten in terms of unit length vectors. Thus, let us introduce three vectors j1 N1 , j2 N2 , j3 N3 , where Ni2 = 1 are unit

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vectors, with the condition that ji Ni satisfy the closure constraint j1 N1 + j2 N2 + j3 N3 = 0. Then these are the three edge vectors of a triangle with edge length ji . One can now check that ( j1 + j2 − j3 )( j1 + j2 + j3 ) |N1 − N2 |2 = , (134) j1 j2 which are essentially the factors appearing in the asymptotics (133) of the normalization coefficient. We can moreover see that the prefactor entering (133) has a nice geometrical interpretation as those arising from the determinant of the metric on orbits orthogonal to the constraint surface. Indeed, in case n = 3 this determinant, see (A5), is given simply by det G j =

j1 j2 j3 4 A2 ( ji ) 2 |N1 − N2 |2 |N2 − N3 |2 |N3 − N1 |2 = . 4 j1 j2 j3

(135)

Thus, approximating  + 1 ∼  at leading order in (133), the asymptotic evaluation of the normalization coefficient reads        |N1 − N2 | 4J3 |N3 − N1 | 4J2 |N2 − N3 | 4J1 −2 N j1 j2 j3 ≈ π det(G j ) . (136) 2 2 2 7.2. Asymptotics of the Jacobi polynomial. Let us now switch to the case of real interest – that of n = 4. In this subsection we study the asymptotics of the shifted Jacobi polynomials. Thus, we are interested in the limit ji → λji , λ → ∞. In order to avoid unnecessary cluttering of notation we do not make the parameter λ explicit, with the understanding that the evaluation is performed in the limit of uniformly large spins. As shown in Appendix E, a convenient integral representation for the shifted Jacobi polynomials is given by a contour integral: (j −j ,j +j ) Pˆk−34j34 12 34 12 (Z ) = Z j12 − j34 (1 − Z )− j12 − j34 I (Z ),  1 dω −S Z (ω) , I (Z ) ≡ e 2iπ ω

(137)

where the contour is around the origin and should avoid the other singularities at ω = −Z and ω = 1 − Z . The Z dependent action is given by S Z (ω) = (k − j34 ) ln ω − (k − j12 ) ln(Z + ω) − (k + j12 ) ln(1 − Z − ω). (138) Note that we have included in the action only the terms that are proportional to the large parameter λ, and this is the reason for leaving 1/ω outside of the exponent in (137). In the uniformly large spin limit this integral is evaluated by the steepest descent method by deforming the integration contour so that it passes through the stationary point.6 The stationary phase equation ∂ω S Z (ω) = 0 leads to a quadratic equation (k + j34 )ω2 + ω (( j34 + j12 )(Z − 1) + ( j34 − j12 )Z ) + ( j34 − k)Z (Z − 1) = 0 6 The steepest descent method used here may be not completely standard for some readers, so we briefly  review the basic idea. Consider the problem of computing the integral dzg(z) exp −λS(z) along some contour in the complex z-plane. Here S(z), g(z) are some holomorphic functions of z. Let us, for simplicity, assume that the stationary point z 0 : ∂z S(z)|z=z 0 = 0 is unique and does not coincide with any of the singularities of the integrand so that the contour can be deformed to pass through z 0 . In the limit λ → ∞ the integral can be evaluated as follows. Around the stationary point the “action” S(z) admits an expansion S(z) = S(z 0 ) + (1/2)S (2) (z 0 )(z − z 0 )2 + . . .. The second derivative S (2) (z 0 ) at the stationary point is a

Holomorphic Factorization for a Quantum Tetrahedron

77

√ whose solutions are ω± (Z ) ≡ L(Z ) ± Q(Z ), where L , Q are a linear and quadratic function of Z given by the following expressions: ( j34 + j12 )(Z − 1) + ( j34 − j12 )Z , (139) 2(k + j34 ) 2 − j 2 )Z (Z −1) ( j34 + j12 )2 (Z −1)2 +( j34 − j12 )2 Z 2 +2(2k 2 − j34 12 Q(Z ) ≡ . (140) 4(k + j34 )2

L(Z ) ≡ −

Recall now that for non-degenerate tetrahedra Z is complex. This means that in general ω± are complex numbers and the corresponding on-shell actions S± (Z ) ≡ S Z (ω± (Z )) are also complex. In the semi-classical limit of uniformly large spin, only the root possessing the smallest real value of S Z dominates, the other one being exponentially suppressed. Without loss of generality we can assume that this root is ω+ . Then we get 1 e−S+ (Z ) I (Z ) ∼  , ω (Z ) (2) i 2π S+ (Z ) +

(141)

(2)

where S± (Z ) ≡ ∂ω2 S Z (ω± (Z )). 7.3. The equi-area case: Peakedness with respect to k. In the “equi-area” case where all four representations are equal, ji = j, ∀i = 1, 2, 3, 4, all equations simplify considerably. Thus, the action (138) reduces to: S(ω) = k ln

ω . (Z + ω)(1 − Z − ω)

(142)

√ The two roots are given by ω± = ± Z (Z − 1). In order to compute the on-shell action it is convenient to introduce a complex angle (Z ) such that Z = cosh2 (Z ). Then the two roots are given by: 1 ω± = ± sinh  cosh  = ± sinh 2. 2

(143)

Changing  → − (which does not affect Z ) simply exchanges the two roots ω+ ↔ ω− . Then it is easy to check that (Z + ω± ) = cosh  e± , (1 − Z − ω± ) = ∓ sinh  e± ,

(144)

and so the on-shell action is given by: S± (Z ) = ∓2k(Z ) + ikπ(2l + 1),

l ∈ Z.

(145)

Footnote 6 continued complex number, which we parametrize as Aeiφ . Let us also introduce polar coordinates on the z-plane via z = z 0 + r eiθ . Then the action near the stationary point behaves as S(r, θ ) = S(z 0 ) + (1/2)Ar 2 ei(φ+2θ) . Therefore, the paths of steepest descent to the stationary point is θ = −φ/2. Along this path the integration measure is dz = e−iφ/2 dr and the resulting real integral can be evaluated using the usual steepest descent −iφ/2 g(z )e−λS(z 0 ) √2π/(λA). However, this can be written very compactly method with the result 0  being e as g(z 0 )e−λS(z 0 ) 2π/(λS (2) (z 0 )).

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L. Freidel, K. Krasnov, E. R. Livine 1

0.16

0.99

0.14 0.98

0.12

0.97

0.1

0.96

0.08 0.06

0.95

0.04

0.94

0.02

0.93 10

20

30

40

5

10

15

20

25

30

35

Fig. 1. On the left, we plot (as a function of k) the modulus of the (normalized) difference between the (shifted) Legendre polynomial Pˆk (Z ) and its approximation (147) for the value of the cross-ratio Z = exp(iπ/3) corresponding to the equilateral tetrahedron. We see that the asymptotic formula is already good at 2% from k = 8 and at 1% from k = 15. On the right, we’ve plotted (also as a function of k) the ratio between the binomial coefficient (2 j)!2 /(2 j − k)!(2 j + k)! and its approximation (148) for j = 20. We see that the approximation is excellent as long as k doesn’t get too close to its maximal value 2 j

The Hessian of S Z at the stationary points can also be computed and we find (2)

S± (Z ) = ∓

4k . sinh 2e±2

(146)

Let us now assume, for definiteness, that the real part of  is positive. Then the stationary point that dominates is ω+ and we have:  1 e(2k+1)(Z ) . Pˆk (Z ) ≈ (−1)k (147) 2π k sinh 2(Z ) One should keep in mind that both the cross-ratio Z and the angle parameter (Z ) are generically complex, since a real cross-ratio would correspond to a degenerate flat tetrahedron. Thus, there is both an exponential and oscillating behaviour of the integral. We have compared the asymptotic formula (147) for the Jacobi polynomial to the exact quantity numerically in Fig. 1. Let us now discuss the numerical prefactor coming from the normalization coefficient. As we have seen earlier, in the case ji = j, the 4-valent overlap coefficient is given by (105). Using the Stirling formula, it is easy to give the leading order behaviour of the pre-factor: e−4 j(x) (2 j)!2 ∼ √ , (x) ≡ (1/2)(1 − x) ln(1 − x) (2 j − k)!(2 j + k)! 1 − x2 +(1/2)(1 + x) ln(1 + x),

(148)

where x = k/2 j. Note that (x) ≈ x 2 for small x, so we have a Gaussian peaked near x = 0. The ratio between the approximation (148) and the exact quantity is plotted in Fig. 1. Putting everything together we get the semi-classical estimate:  k e(Z ) 1 (−1) C kj (Z ) ∼ e−4 j ((x)−x(Z )) . (149) (2 j)3/2 (1 + x) 2π x(1 − x 2 ) sinh 2(Z )

Holomorphic Factorization for a Quantum Tetrahedron

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We see that the exponent in (149) describes a Gaussian peaked at xc such that: d(x)  = Re((Z )),  d x x=xc

(150)

times an oscillating exponent exp [4i j xc Im((Z ))]. It is not hard to find xc . We have for the first derivative (1) (x) = (1/2) log[(1 + x)/(1 − x)] and thus xc (Z ) = tanh Re((Z )). In other words kc (Z ) = 2 j tanh Re((Z )).

(151)

It is natural to expect that the corresponding value kc should be the classical value associated with the tetrahedron in question, that is kc2 = j12 + j22 + 2 j1 j2 cos θ12 ,

(152)

where θ12 is the dihedral angle between the faces 1 and 2. It is also natural to expect that the phase factor 2Im((Z )) should be the conjugate variable to k, which is the angle ϕc between the edges (12) and (34) of the classical tetrahedra determined by Z . Putting the real and imaginary parts together, we should therefore expect the following relation between the cross-ratio parameter (Z ) and the geometrical variables e2 (Z ) =

2 j + kc iϕc e . 2 j − kc

(153)

This formula relates the two very different descriptions of the phase space of shapes of a classical tetrahedron – the real one in terms of the k, φ parameters and the complex one in terms of the cross-ratio coordinate Z . As is clear from this formula, the relation between the two descriptions is very non-trivial. Here we have only identified the simplest case of this relation when all areas are equal, leaving the general case to future studies. Below we shall check this geometrical interpretation in the case of the equilateral tetrahedron. Let us now explicitly write the exponent of (149) as a Gaussian peaked at x = xc times some prefactor. The imaginary part of the quantity in the exponent is just 2kIm((Z )) = kφc according to our real parametrization (153). For the real part we have (x) − xRe((Z )) = (xc ) − (1) (xc )xc + (1/2)(2) (xc )(x − xc )2 + · · ·. The second derivative is (2) = 1/(1 − x 2 ), while  − (1) x = (1/2) log(1 − x 2 ). Thus, going back to the parameter k, we see that the most essential part of the asymptotics (149) written in terms of the coordinates kc , φc , see (153), is given by the following Gaussian:   1 (k − kc )2 C kj (Z ) ∝

(154) + ik exp −2 j ϕ c c .  (4 j 2 − kc2 ) k2 2 j 1 − 4 jc2 An important feature of this state is the fact that its width σ = (4 j 2 − kc2 )/2 j =

2j cosh2 Re((Z ))

(155)

depends not only on j but also on the classical value kc (Z ). This is in qualitative agreement with the analysis performed in [18]. In this work a Gaussian ansatz for the semi-classical states was postulated and the width was calculated by asking that it is independent of the channel used. This led to an expression of the width in terms of

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L. Freidel, K. Krasnov, E. R. Livine

20000

15000

10000

5000

0

10

20

30

40

Fig. 2. We plot the modulus of the equi-area case state Cjk (Z ) (for j = 20) as a function of the spin label k,

for the value of the cross-ratio Z = exp(iπ/3) that corresponds to the equilateral tetrahedron.√It is obvious that the distribution looks Gaussian. We also see that the maximum is reached for kc = 2 j/ 3 ∼ 23.09, which agrees with our asymptotic analysis 0.035

70000

1.2e+12

0.03

60000

1e+12

0.025

50000

0.02

40000

0.015

30000

0.01

20000

0.005

10000

0

10

20

30

40

0

8e+11 6e+11 4e+11 2e+11 0 10

20

30

40

10

20

30

40

Fig. 3. We have plotted the modulus of the j = 20 equi-area state Cjk (Z ) for increasing cross-ratios Z = 0.1i, 0.8i, 1.7i. We see the Gaussian distribution progressively moving its peak from 0 to 2 j. This illustrates how changing the value of Z affects the semi-classical geometry of the tetrahedron

matrix elements of the Hessian of the Regge action. It would be interesting to check that this is indeed the case to provide an additional justification for the hypothesis made in [18] as well as to relate our explicit parametrization to the Regge action. The simplest example in which we can check everything is the regular equilateral tetrahedron, which corresponds to the value Z = exp(iπ/3). In this case, the complex angle is easily computed: √ 1+ 3 π  = ln +i . √ 4 2 This gives the position of the peak, the angle and the deviation: 2j kc = 2 j tanh Re() = √ , 3

ϕc =

π , σ = 2 j/ cosh2 Re() = 4 j/3. 2

This fits perfectly the expected classical values for an equilateral tetrahedron and the deviation σ agrees with that proposed in [18]. A plot of the modulus of the quantity Cjk (Z ) (for j = 20 and Z = exp(iπ/3)) as a function of the spin label k is given in Fig. 2 and confirms the above semi-classical analysis. Similar plots for other values of Z are given in Fig. 3 and show how the position of the peak depends on the cross-ratio coordinate Z .

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7.4. The equi-area case: Peakedness with respect to Z . In the previous subsection we have considered the overlap function Cjk (Z ) = j , k|j , Z and saw that, in the semiclassical limit of large spins, it is essentially a Gaussian in the k-variable peaked around some classical value kc parametrized by Z via (153). In this subsection we are going to continue our study of the equi-area case, and consider the peakedness properties of the same overlap function but now viewed as a function of the cross-ratio coordinate. To do this, it is important to obtain an expression for the Kähler potential j (Z , Z¯ ) on the constraint surface parametrized by Z , for, as we shall see below, it is the wave-function ¯ e−(1/2)j (Z , Z ) Cjk (Z ) that is peaked in Z . Thus, let us first obtain an explicit expression for the equi-area integration kernel Kˆ (Z , Z¯ ) that, as we know from the analysis of Sect. 4 is essentially the (exponential of the) Kähler potential on the constraint surface. A representation for Kˆ (Z , Z¯ ) as an integral over the orbits orthogonal to the constraint surface was given earlier in (71), and the following discussion established that in the semi-classical limit of large spins the Kähler potential on the constraint surface is essentially given by the function j (z i , z i ), given by e.g. (66), evaluated on the constraint surface. In the equi-area case this function can be computed explicitly. Thus, let us start with the function (63), which in the case n = 4 is given by:       N 1 − N 2 2  N 3 − N 4 2 |z 12 z 34 |2     , =  e−• (zi ,z i )  = 4 (156)    2 n=4 2 2 i=1 (1 + |z i | ) where we have used (A6) to write the second equality. Using |N1 − N2 |2 = 2(1−cos θ12 ), where θ12 is the dihedral angle between the faces 1 and 2, as well as the fact that in the equi-area case on the constraint surface we have cos θ12 = cos θ34 , and recalling the relation (152) between the parameter kc and θ12 , we can write the above formula as:  ¯  ≡ e−• (Z , Z ) = (1 − xc2 )2 , (157) e−• (zi ,z i ) 4 i=1 ji Ni =0

where, as before xc = kc /2 j. Now, the semi-classical Kähler potential j (Z , Z¯ ) in the equi-area case is equal to 2 j• (Z , Z¯ ), see (66) and (64). Let us write an expression for this Kähler potential in terms of the parameter (Z ). The quantity (1 − xc2 ) was computed in (155) and we get the following simple expression:  j (Z , Z¯ ) = 8 j ln [cosh Re()] .

(158)

Now, given the Kähler potential, we can compute the corresponding symplectic form ¯ 1 2 j d ∧ d ∂ Z ∂ Z¯  j dZ ∧ d Z¯ = − i i cosh2 Re() dRe() ∧ dIm() = 4j = dkc (Z ) ∧ dϕc (Z ), cosh2 Re()

j ≡

(159)

where in the last equality we have used the differential of (151) and the definition of ϕc = 2Im(). This demonstrates that kc and ϕc are canonically conjugate variables, as anticipated in the previous subsection. We would now like to compute the inner product j , k|j , l of two real intertwiners as an integral over the cross-ratio coordinate Z . In the semi-classical approximation

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L. Freidel, K. Krasnov, E. R. Livine

of large spins the integration kernel is found explicitly in (74), so we are interested in computing (in the equi-area case):   (2 j)2 ¯ (160) d2 Z Pf( j )e−(2 j+1)• (Z , Z ) C kj (Z )C lj (Z ).

j , k|j , l ∼ √ π To analyze this integral it is very convenient to switch to  coordinate instead of Z . The change of variables is easy to work out. Indeed, we have dZ = sinh(2)d, and the Pfaffian of the symplectic form is available from (159). Note that we get a factor of | sinh 2|2 from the change of integration measure, as well as a factor of | sinh 2|−1 from the Pfaffian. We also need to discuss the range of integration of . Rewriting the definition Z = cosh2 () in terms of the real R ≡ Re() and imaginary I ≡ Im() parts of  we get: Z=

1 1 i + cosh(2R) cos(2I) + sinh(2R) sin(2I). 2 2 2

(161)

It is then clear that the range R ∈ (−∞, ∞), 2I ∈ [0, 2π ] covers the whole Z -plane twice (note that in this parametrization there are two cuts in the Z -plane along the real axes starting each at ±1 and going to infinity). After this change of variables we get:   d2  | sinh 2| −(2 j+1)• (,) k 2 4j

j , k|j , l ∼ (2 j) e C j ()C lj (), (162) π 2 cosh Re() where, as we have computed above, • (, ) = 4 ln [cosh Re()], the integration over  is carried over the specified above domain, and an additional factor of 1/2 was introduced because the original domain is now covered twice. We can now substitute into this expression the semi-classical expression (149) to get:   j

j , k|j , l ∼ (163) d2  Cˆ kj ()Cˆ lj (), π where we have introduced the new states:  sinh 2 ¯ e−( j+1/2)• (,) Cjk () Cˆ kj () ≡ 2 j cosh Re() N (x, ) −4 j S (x,) ∼ (−1)k √ , e 2π k

(164)

where we have introduced, still with x = k/2 j: 1

e− 2 • (,)+R+iI 1  N (x, ) ≡  √cosh R , (1 + x) 1 − x 2 + ln cosh R,

S(x, ) ≡ (x) − x(R + iI) (165)

and wrote everything in terms of the real R ≡ Re() and imaginary parts I ≡ Im(). Taking into account the fact that with our conventions d2  = 2dRdI, we can easily perform the integral over I:  π 2dI e2i I (l−k) = 2π δk,l . 0

Holomorphic Factorization for a Quantum Tetrahedron

83

Thus the integral over I just imposes k = l. The integral over the real part R is more interesting. It can again be computed using the steepest descent method. The value Rc that dominates the integral is given by tanh Rc = x.

(166)

Remarkably, this is the same relation, see (151) that we have obtained in the previous subsection by extremizing the integrand with respect to x. One also sees that the value of Re(S) at the minimum is simply equal to 0 which reflects the fact that the states are normalized. To compute the integral over R in (163) it remains to compute the Hessian at the critical point, which is given by:  1  ∂ R2 Re(S) = . (167) Rc cosh2 Rc At the critical point the normalization coefficient N (x, ) also simplifies considerably since 1

e− 2 (Rc )+Rc 1   = 1, hence |N (x, Rc )| = √cosh R . c (1 + x) 1 − x 2

(168)

Then putting all the pieces together, the steepest descent evaluation of the remaining integral over R gives:   |N (x, Rc )|2 2π cosh2 Rc j δk,l

j , k|j , l = (2π δk,l ) = . (169) π 2π k 8j 2k This allows us to recover explicitly the expected orthonormality of the states |j , k at leading order in k (the exact normalization factor would be 1/(2k + 1) instead of 1/2k). This provides a highly non-trivial consistency check of all our asymptotic evaluations, i.e., of the asymptotic formulae for the kernel Kˆ j and the states Cjk (Z ). 8. Discussion In this paper we have introduced and studied in detail a holomorphic basis for the Hilbert space H j1 ,..., jn of SU(2) intertwiners. We have considered the general n case, but gave more details for the 4-valent intertwiners that can be interpreted as quantum states of a “quantum tetrahedron”. Our main result is the formula (24) for the inner product in H j1 ,..., jn in terms of a holomorphic integral over the space of “shapes” parametrized by the cross-ratio coordinates Z i . In the “tetrahedral” n = 4 case there is a single crossratio Z . Somewhat unexpectedly, we have found that the integration kernel Kˆ (Z i , Z¯ i ) is given by the n-point function of the bulk/boundary dualities of string theory, and this fact allowed us to give to Kˆ (Z i , Z¯ i ) an interpretation relating it to the Kähler potential on the space of “shapes”. The new viewpoint on the n-point functions Kˆ (Z i , Z¯ i ) as being a kernel in the inner product formula for H j1 ,..., jn also led us to the expectation that the n-point functions should satisfy the holomorphic factorization formula (123) of the type expected of an n-point function of a 2-dimensional conformal field theory. It would be of interest to develop this line of thought further by proving (123), computing the coefficients aλ and thus finding the spectrum of this CFT, as well as possibly even identifying

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L. Freidel, K. Krasnov, E. R. Livine

the underlying conformal field theory, if there is one. It would also be of considerable interest to see if a group-theoretic interpretation similar to the one developed here for the 2-dimensional n-point function also exists for d-dimensional objects, with the case of most interest for applications in string theory being of course d = 4. In spite of n-point functions of bulk/boundary correspondence of string theory showing a somewhat unexpected appearance, our results are most relevant to the subjects of loop quantum gravity and spin foam models. It is here that we believe the new techniques developed in this paper can lead to new advances, impossible without the new coherent states we introduced. Indeed, we have shown that the n=4 holomorphic intertwiners |j , Z parametrized by a single cross-ratio variable Z are true coherent states in that they form an over-complete basis of the Hilbert space of intertwiners and are semi-classical states peaked on the geometry of a classical tetrahedron. We have also shown how the new holomorphic intertwiners are related to the standard “spin basis” |j , k of intertwiners that are usually used in loop quantum gravity and spin foam models, and found that the change of basis coefficients are given by Jacobi polynomials. With the mathematics of the new holomorphic intertwiners hopefully being clarified by this work, the next step is to apply the techniques developed here to loop quantum gravity and spin foam models. In the canonical framework of loop quantum gravity, spin network states of quantum geometry are labeled by a graph as well as by SU(2) representations on the graph’s edges e and intertwiners on its vertices v. We can now put holomorphic intertwiners at the vertices of the graph, which introduce the new spin networks labeled by representations je and cross-ratios Z v . Since each holomorphic intertwiner can be associated to a classical tetrahedron, we could truly interpret these new spin network states as discrete geometries. In particular, geometrical observables such as the volume can be expected to be peaked on their classical values. This fact should be of great help when looking at the dynamics of the spin network states and when studying how they are coarse-grained and refined. We leave these interesting developments to future research. The holomorphic intertwiners are also of direct relevance to spin foam models. Indeed, spin foam amplitudes encode the dynamics of spin network states. The coherent intertwiners, introduced in [12], already led to significant improvement on the understanding of the semi-classical behavior of spin foams [14,15,21–23]. It is clear that the new holomorphic intertwiners developed in this paper (building upon [5]) should lead to further progress in the same direction. Indeed, the basic spin foam building blocks are the {15 j}-symbols, which is an SU(2) invariant labeled by 10 representations and 5 intertwiners. With holomorphic intertwiners at hand we can now define a {10 j, 5Z }-symbol. It is of great interest to understand the large spin behaviour of this new invariant, for it can be expected that this asymptotics contains a great deal of information about the classical geometry of a 4-simplex. It is also important to consider the case of the twice larger gauge group Spin(4) = SU(2) L × SU(2) R and construct the corresponding {10 jL , 5Z L }{10 j R , 5Z R }-symbols. These, with appropriate constraints between the labels jL , j R , Z L , Z R as in [13–16], should correspond to semi-classical 4-simplices in 4d gravity. Another very important task is to express the (area-)Regge action in terms of the spins and the cross-ratios Z . We leave all these exciting problems to future research and hope that the analysis performed in the present paper will provide a solid first step in the direction outlined. Acknowledgements. The authors would like to thank Florian Conrady for comments on the manuscript. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by

Holomorphic Factorization for a Quantum Tetrahedron

85

the Province of Ontario through the Ministry of Research & Innovation. KK was supported by an EPSRC Advanced Fellowship. EL was partly supported by a ANR “Programme Blanc” grant.

Appendix A: G Determinant Here we give a proof of the formula for the determinant of the metric G j on orbits orthogonal to the constraint surface. Let us recall that G j (z i ) is a 3 by 3 metric G ab j (z i ) =

n 

ji (δ ab − N a (z i )N b (z i )).

(A1)

i=1

We derive the following expression for the determinant of this matrix:    det G j (z i ) = 16 ji j j jk i< j
|z i j |2 |z jk |2 |z ki |2 (1 + |z i |2 )2 (1 + |z j |2 )2 (1 + |z k |2 )2

+ G j (Hj , Hj ),

(A2)

 where Hj = i ji N (z i ) is the closure vector and the sum is over all ordered triples belonging to {1, · · · , n}. When the closure condition is satisfied the last term vanishes and we recover the evaluation of the determinant that we use in the main text. Proof. G is a 3×3 matrix and so we can compute its determinant in terms of traces of its powers: det G =

 1 (Tr G)3 − 3(Tr G)(Tr G 2 ) + 2Tr G 3 . 6

(A3)

We compute: Tr G = 2

 i

Tr G 2 =

 i,k

Tr G = 3



ji ,

  ji jk 1 + (Ni · Nk )2 ,   ji jk jl 3(Ni · Nk )2 − (Ni · Nk )(Nk · Nl )(Nl · Ni ) ,

i,k,l

where we have introduced the convention Ni ≡ N (z i ). We can simplify the formula and get: 3 det G =



ji jk jl [1 − (Ni · Nk )(Nk · Nl )(Nl · Ni )] .

i,k,l

Using the trivial formula for the scalar product of two unit vectors: Ni · N k = 1 −

1 |Ni − Nk |2 , 2

(A4)

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L. Freidel, K. Krasnov, E. R. Livine

we expand the previous formula and obtain: 3 det G j =

1 ji jk jl |Ni − Nk |2 |Nk − Nl |2 |Nl − Ni |2 8 i,k,l

3 3 + ji jk jl |Ni − Nk |2 − ji jk jl |Ni − Nk |2 |Nk − Nl |2 . 2 4 i,k,l

i,k,l

The first term here is the one we would like to keep. It doesn’t vanish if and only if the indices i, k, l are all different. We can simplify the two remaining terms by noticing the symmetry under the exchange of the indices i, k, l, and using the scalar product formula the other way round, |Ni − Nk |2 = 2(1 − Ni · Nk ). We then obtain a much simpler formula (notice that this formula is valid for any number of vectors Ni ): det G j =

1  ji jk jl |Ni − Nk |2 |Nk − Nl |2 |Nl − Ni |2 24 i,k,l  + jl (Hj · Hj − (Hj · Nl )2 ) l

1  ji jk jl |Ni − Nk |2 |Nk − Nl |2 |Nl − Ni |2 + G j (Hj , Hj ), (A5) = 4 i
where we have introduced the closure vector Hj ≡ lowing identity:



i ji Ni .

Finally, we use the fol-

1 |z 12 |2 |N1 − N2 |2 = , 4 (1 + |z 1 |2 )(1 + |z 2 |2 )

(A6)

where z 12 = z 1 − z 2 , and derive the desired result.

 

Appendix B: Measures on SL(2, C) Here we obtain two expressions for the group-invariant measure on SL(2, C) that are used in the main text. We start from the following expression for the Haar measure on SL(2, C): dg = da ∧ db ∧ dc ∧ dd ∧ da¯ ∧ db¯ ∧ dc¯ ∧ dd¯ δ (2) (ad − bc − 1).

(B1)

Resolving the delta function we can express this measure as the product of the holomorphic and anti-holomorphic pieces, dg = dhol g ∧ dhol g with the holomorphic piece given by, e.g.: dhol g = −

da ∧ db ∧ dc . a

(B2)

Let us now introduce the following parametrization: z1 =

b a+b a , z2 = , z3 = . d c+d c

(B3)

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It is easy to compute dz 1 ∧ dz 2 ∧ dz 3 =

(ad − bc) (−adb ∧ dc ∧ dd + bda ∧ dc ∧ dd (c(c + d)d)2 − cda ∧ db ∧ dc + dda ∧ db ∧ dc) .

If one imposes the determinant condition ad − bc = 1 one gets dz 1 ∧ dz 2 ∧ dz 3 =

2dhol g . (c(c + d)d)2

(B4)

Note that since z 21 z 13 z 23 = (c(c + d)d)−2 we can equivalently rewrite this relation as dhol g =

dz 1 ∧ dz 2 ∧ dz 3 . 2z 21 z 23 z 13

(B5)

n dz in terms of the new coordinates a, b, c, d, We can now compute the measure ∧i=1 i Z i . The idea is to first decompose it in terms of z 1 , z 2 , z 3 and Z i and then use the previous relations to express it in terms of a, b, c, d and Z i . We have: n 

dz i = dz 1 ∧ dz 2 ∧ dz 3

i=1

 i

=2

dZ i (cZ i + d)2

n dhol g ∧i=4 dZ i . 2 2 2 2 d (c + d) c i (cZ i + d)

(B6) (B7)

Thus, the total measure is: n  i=1

n dg i=4 d2 Z i d zi = 4 4 , n |d| |c + d|4 |c|4 i=4 |cZ i + d|4 2

(B8)

where our convention is that d2 z = |dz ∧ d¯z |. Below we show that the relationship between the Haar measure on the group dg and the normalized one for which the volume of the SU(2) subgroup is normalized is given by dg = (2π )2 dnorm g, and thus the total measure is n n  dnorm g i=4 d2 Z i 2 2 d z i = (4π ) . (B9) n |d|4 |c + d|4 |c|4 i=4 |cZ i + d|4 i=1

The last subtlety comes from the fact that the map (g, Z i ) → (z i (g, Z i )) is 2 : 1 thus we have to insert an extra factor 1/2 when integrating and this leads us to the relation we were looking for: 

n 

Cn i=1

 d2 z i F(z i , z i ) = 8π 2



×

n 

Cn−3 i=4 g

dnorm g

SL(2,C)

d2 Z i

g

F(Z i , Z i ) n . 4 4 |d| |c + d| |c|4 i=4 |cZ i + d|4

(B10)

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We now compute the Haar measure in terms of the Iwasawa decomposition SL(2, C) = K AN . From (29) we have the following explicit parametrization: b = − cos(θ )eiφ ρ −1/2 n + sin(θ )eiψ ρ 1/2 , (B11) a = cos(θ )eiφ ρ −1/2 , −iψ −1/2 −iψ −1/2 −iφ 1/2 c = − sin(θ )e ρ , d = sin(θ )e ρ n + cos(θ )e ρ , where the range of the angular coordinates on the SU(2) part is θ ∈ [0, π/2] and φ, ψ ∈ [0, 2π ]. A simple explicit computation gives: da ∧ da¯ ∧ dc ∧ dc¯ =

dρ ∧ 2 sin(θ ) cos(θ )dθ ∧ dφ ∧ dψ. ρ3

(B12)

¯ which is just sin2 (θ )ρ −1 dn ∧ Thus, we only need to compute the dn ∧dn¯ part of dd ∧dd, 2 dn. ¯ Multiplying it all together and dividing by |c| we get: dg =

dρ ∧ dn ∧ dn¯ ∧ 2 sin(θ ) cos(θ )dθ ∧ dφ ∧ dψ. ρ3

(B13)

Now, the SU(2) measure that appears here is not a normalized one - the corresponding volume of SU(2) that it gives is (2π )2 . Thus, the normalized measure is given by: dnorm g =

dρ 1 ∧ dn ∧ dn¯ ∧ sin(2θ )dθ ∧ dφ ∧ dψ. 3 ρ 4π 2

(B14)

Appendix C: Permutations and Cross-Ratios In order to define the cross-ratio we need to choose an order among the z i . A permutation of z 1 , z 2 , z 3 , z 4 changes this order and naturally acts on the cross-ratios. The other cross ratios one obtains this way are given by Z=

z 41 z 23 z 42 z 31 , 1− Z = = σˆ 12 (Z ), z 43 z 21 z 43 z 21

z 41 z 32 Z = = σˆ 23 (Z ), Z −1 z 42 z 31

(C1)

as well as their inverses 1 = σˆ 13 (Z ), Z

1 = σˆ 12 σˆ 23 (Z ), 1− Z

Z −1 = σˆ 23 σˆ 12 (Z ). Z

(C2)

The reason why there are only 6 different cross-ratios while the number of permutations is 24, is that the initial cross-ratio Z is fixed by 4 permutations. These are generated by the identity and the three following permutations exchanging a pair of indices: σˆ 23 σˆ 14 , σˆ 13 σˆ 24 , σˆ 12 σˆ 34 .

(C3)

The whole permutation group (even the reflections (C3)) acts non-trivially on the prefactors (83) and thus on the holomorphic intertwiners |j , Z . The action of the non-trivial

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permutations is σˆ 12 |j , Z = (−1)s−2 j3 |j 12 , 1 − Z ,    Z 2 j2 2 j4  σˆ 23 |j , Z = (−1) (1 − Z ) j 23 , , Z −1    1 , σˆ 13 |j , Z = (−1)s Z 2 j4 j 13 , Z    1 2s−2 j3 2 j4  , σˆ 23 σˆ 12 |j , Z = (−1) (1 − Z ) j 231 , 1− Z    Z −1 , σˆ 12 σˆ 23 |j , Z = (−1)s+2 j2 (−Z )2 j4 j 312 , Z

(C4) (C5) (C6) (C7) (C8)

where s = j1 + j2 + j3 + j4 , j abcd = ( ja , jb , jc , jd ) and j abc = j abc4 etc. Appendix D: Bulk-Boundary Dualities n-Point Function In this section we are interested in reviewing some properties of the n-point function (33) 

 n  dρ ρ i 2 K j (z i , z i ) = d n , 3 (ρ 2 + |z i − n|2 )i R+ ρ C i=1

(D1)

where we have introduced the conformal dimensions: i ≡ 2( ji + 1). Because this function is covariant under SL(2, C), it can be expressed in terms of a function Kˆ j (Z i , Z i ) = lim X →∞ |X |23 K j (0, 1, X, Z i ) that only depends on the cross-ratios Z i . The quantities (D1) can be computed using the Feynman parameter technique proposed in this context already by Symanzik [24], see also [19,25]. The technique is based on the following formuli:  ∞ 1 1 = dt t λ−1 e−t z , (D2) zλ (λ) 0  ∞ dρ  i −  ti ρ 2 1 1− i /2  i i i S ρ e = ( i /2 − 1), (D3) ρ3 2 0 i   2π − 1 i, j ti t j |zi −z j |2 2 e S d2 n e− i ti |zi −n| = , (D4) S C  ¯ where S = i ti . In the last integral our measure convention is that d2 n = |dn ∧ dn|. The formuli given are an adaptation of the general ones reviewed in e.g. [25] to d = 2. Using these we find   n 1  2  ( − 1) K j (z i , z i ) = π n dti tii −1 S − e− S i, j ti t j |zi j | , n ( ) i R+ i=1 i=1

(D5)

 [24] is that the where z i j ≡ (z i − z j ) and  = i ( ji + 1). The crucial observation of  quantity S in (D5) can be modified without changing the integral to S = i ai ti , where

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again ai ≥ 0 are arbitrary positive coefficients not all zero. Indeed, supposing that S is of this form, we make the following change of variables:

 1 ak u k , u i ≡ √ ti , or ti = u i (D6) S k √  where we have used U ≡ i ai u i = S. The determinant of the arising n × n Jacobian matrix is easy to compute:   ∂t j ∂t j = 2U n . = δi j U + ai u j , det (D7) ∂u i ∂u i This leads to the simple expression in terms of the norms of the complex numbers z i j ≡ (z i − z j ):  ( − 1) K j (z i , z i ) = 2π n i=1 (i )

  n

du i u ii −1 e−

 i< j

u i u j |z i j |2

.

(D8)

i=1

The result is independent of ai , and so we can choose these numbers arbitrarily already at the level of (D5). Let us now specialize to the most interesting for us case n = 4. We are interested in computing the limit (27). We first note that we can reabsorb the multiplicative factor |X |23 by a rescaling of the Feynman parameter t3 → t3 /|X |2 . We then evaluate (D5) with z 1 = 0, z 2 = 1, z 3 = X → ∞ and the choice ai = δ3i that corresponds to S = t3 . This gives: π  ( − 1) Kˆ j (Z , Z )= n i=1 (i )

  n dti 1 2 3 − 4 − t1t t2 −(t1 +t2 +t4 )− t1t t4 |Z |2 − t2t t4 |1−Z |2 3 3 t t t t4 e 3 . ti 1 2 3 i=1

We now perform a change of variables t˜3 = t1 t2 /t3 , and write the result omitting the tilde on the new variable t3 . We get: π  ( − 1) Kˆ j (Z , Z ) = n i=1 (i ) ×e

−(t1 +t2 +t3 +t4 )−

  n dti i=1

ti

t3 t4 2 t3 t4 2 t2 |Z | − t1 |1−Z |

t11 +3 − t22 +3 − t3−3 t44

.

Now the case n = 3 can be recovered by putting 4 = 0. In this case the integral over t4 is trivial and we are left with  ( − 1) (1 +2 −3 )(1 −2 +3 )(−1 +2 + 3 ) , Kˆ j1 , j2 , j3 = π (1 )(2 )(3 ) (D9) where  = 1 + 2 + 3 . Returning to the case n = 4 we can, following [19], use the Barnes-Mellin expansion of the exponential  1 e−λ = (D10) ds (−s)λs , 2iπ

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with the integration contour running along the imaginary axes. Using this integral representation for the two exponentials containing the cross-ratio, and performing all Feynman parameter integrations, we get the Barnes-Mellin representation for the 4-point function  dtds π  ( − 1) Kˆ j (Z , Z ) = n [(−s)(−t)(1 + 3 −  − t) (2πi)2 i=1 (i ) ×(2 + 3 −  − s)  × ( − 3 + s + t)(4 + s + t)|Z |2s |1 − Z |2t . (D11) The double integral here is of Barnes-Mellin type with the integration contour running to the left of the imaginary axes. However, one must be careful when evaluating the integrals in terms of the pole contributions as i are integers and so there are double poles. Unfortunately, the integral expression (D11) is not very suitable for producing a series expansion in Z as it contains powers of |1 − Z |2 . It can be converted to a more suitable integral representation by assuming 1 + 3 −  ≥ 0 and using the Barnes lemma: 

dt (a + t)(b + t)(c − t)(−t)X t 2iπ (a + c)(b + c)(a)(b) F(a, b; a + b + c|1 − X ) = (a + b + c)  (a + n)(b + n) (1 − X )n = (a + c)(b + c) (a + b + c + n)n! n

(D12)

which is valid as long as a, b, a + c, b + c are not negative integers. Using this we can perform the t-integration and get a converging (for |Z | < 1) double power series expansion: ∞

π  ( − 1)  (1 − |1 − Z |2 )n K j (Z ) = n n! i=1 (i )



ds [(−s)(2 + 3 −  − s) 2πi n=0  ( − 3 + s + n)(4 + s + n) 2s . |Z | × (1 + s)(1 + 3 + 4 −  + s) (1 + 4 + 2s + n) (D13) The remaining Barnes-Mellin integral receives contributions from double poles, and an explicit formula for collecting these, as well as explicit expressions for the arising double power series for particular values of spins are given in [19].

Appendix E: The Shifted Jacobi Polynomial (a,b)

Jacobi Polynomials Pn Pn(a,b) (x) =

(x) can be defined in terms of the Rodrigues formula

 (−1)n −a −b n a+n b+n (1 − x) (1 − x) . (1 + x) ∂ (1 + x) x 2n n!

(E1)

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(a,b) Here we will give some useful formuli for the shifted Jacobi polynomial Pˆn (Z ) ≡ (a,b) Pn (1 − 2Z ). We define 1 − x = 2Z , 1 + x = 2(1 − Z ), and then −2∂x = ∂ Z . The “shifted” Rodriguez formula now reads

 1 (E2) Pˆn(a,b) (Z ) = Z −a (1 − Z )−b ∂ Zn Z a+n (1 − Z )b+n . n!

We can therefore express the shifted Jacobi polynomial as the following contour integral  dw (Z + w)n+a (1 − Z − w)n+b 1 (a,b) ˆ Pn (Z ) = , (E3) 2iπ Z a (1 − Z )b ωn+1 where the contour of integration is around ω = 0 in the positive direction and the contour should avoid the other poles at ω = −Z and ω = 1 − Z . The shifted Jacobi polynomials satisfy a recurrence relation given by (1 − 2Z ) Pˆn(a,b) =

2(n + 1)(n + a + b + 1) Pˆ (a,b) (1 + a + b + 2n)(2 + a + b + 2n) n+1 (b2 − a 2 ) + Pˆ (a,b) (2n + a + b)(2n + a + b + 2) n 2(n + a)(n + b) (a,b) + , Pˆ (2n + a + b)(2n + a + b + 1) n−1

(E4)

where all polynomials are evaluated at (1−2Z ). Now, using the fact that the action of the hypergeometric operator On ≡ Z (1− Z )∂ Z2 +[(a +1)−(a +b +2)Z ]∂ Z +n(n +a +b +1) on the Jacobi polynomial is diagonal: (a,b)

(a,b)

On Pn(a,b) = 0, On Pn−1 = [2n + a + b] Pn−1 , (a,b) (a,b) On Pn+1 = −[2n + a + b + 2] Pn+1 ,

(E5)

we can apply the hypergeometric operator to Eq. (E4), and obtain a first order differential recursion relation: Z (1 − Z )∂ Z Pˆn(a,b) =

n(n + 1)(n + a + b + 1) (a,b) Pˆ (2n + a + b + 1)(2n + a + b + 2) n+1 n(b − a)(n + a + b + 1) ˆ (a,b) + P (2n + a + b)(2n + a + b + 2) n (n + a)(n + b)(n + a + b + 1) ˆ (a,b) − . P (2n + a + b)(2n + a + b + 1) n−1

(E6)

Since these operators act at fixed (a, b) labels, it looks like we could define the action of Z and Z (1 − Z )∂ Z on the space of holomorphic intertwiners at fixed representation labels. Thus, taking into account the normalization coefficients of the normalised states we get an action of, say, Z of the type k−1 k ZCjk = αj (k)Cjk+1  + βj (k)C j + γj (k)C j .

(E7)

A closer look at these coefficients shows however that αj ( j1 + j2 ) (or αj ( j3 + j4 )) is infinite! This means that the operator of multiplication by Z is not defined on the entire Hilbert space. It is an operator with a “domain of definition” restricted to the states Cjk with k < max( j1 + j2 , j3 + j4 ). The same conclusion applies to the operator Z (1 − Z )∂ Z .

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References 1. Varshalovich, D.A., Moskalev, A.N., Khersonsky, V.K.: “Quantum Theory of Angular momentum: Irreducible Tensors, Spherical Harmonics, Vector Couplings, 3NJ symbols”, Singapore: World Scientific, 1988 2. Witten, E.: On Holomorphic Factorization Of WZW And Coset Models. Commun. Math. Phys. 144, 189 (1992) 3. Verlinde, E.P.: Fusion Rules And Modular Transformations In 2d Conformal Field Theory. Nucl. Phys. B 300, 360 (1988) 4. Barbieri, A.: Quantum tetrahedra and simplicial spin networks. Nucl. Phys. B 518, 714 (1998) 5. Conrady, F., Freidel, L.: Quantum geometry from phase space reduction. J. Math. Phys. 50, 123510 (2009) 6. Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67, 515–538 (1982) 7. Woodhouse, N.M.J.: Geometric quantization. Oxford mathematical monographs, Second edition, Oxford: Oxford Univ. Press, 1992 8. Tuynman, G.M.: Generalized Bergman kernels and geometric quantization. J. Math. Phys. 28, 573 (1987) 9. Tuynman, G.M.: Quantization: towards a comparison between methods. J. Math. Phys. 28, 2829 (1987) 10. Kirwin, D.: “Coherent States in Geometric Quantization”. http://arxiv.org/abs/math/0502026v2 [math.SG], 2005 11. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998) 12. Livine, E.R., Speziale, S.: A new spinfoam vertex for quantum gravity. Phys. Rev. D 76, 084028 (2007) 13. Engle, J., Pereira, R., Rovelli, C.: The loop-quantum-gravity vertex-amplitude. Phys. Rev. Lett. 99, 161301 (2007) 14. Freidel, L., Krasnov, K.: A New Spin Foam Model for 4d Gravity. Class. Quant. Grav. 25, 125018 (2008) 15. Livine, E.R., Speziale, S.: Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity. Europhys. Lett. 81, 50004 (2008) 16. Engle, J., Livine, E., Pereira, R., Rovelli, C.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B799, 136–149 (2008) 17. Hall, B.C., Kirwin, W.D.: Unitarity in “quantization commutes with reduction”. Commun. Math. Phys. 275(2), 401–422 (2007) 18. Rovelli, C., Speziale, S.: A semiclassical tetrahedron. Class. Quant. Grav. 23, 5861 (2006) 19. Arutyunov, G., Frolov, S., Petkou, A.C.: Operator product expansion of the lowest weight CPOs in N = 4 SYM(4) at strong coupling. Nucl. Phys. B 586, 547 (2000) [Erratum-ibid. B 609, 539 (2001)] 20. Dolan, F.A., Osborn, H.: Conformal partial waves and the operator product expansion. Nucl. Phys. B 678, 491 (2004) 21. Conrady, F., Freidel, L.: Path integral representation of spin foam models of 4d gravity. Class. Quant. Grav. 25, 245010 (2008) 22. Conrady, F., Freidel, L.: On the semiclassical limit of 4d spin foam models. Phys. Rev. D78, 104023 (2008) 23. Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Gomes, H., Hellmann, F.: Asymptotic analysis of the EPRL four-simplex amplitude. J. Math. Phys. 50, 112504 (2009) 24. Symanzik, K.: On Calculations in conformal invariant field theories. Lett. Nuovo Cim. 3, 734 (1972) 25. Krasnov, K., Louko, J.: SO(1,d+1) Racah coefficients: Type I Representations. J. Math. Phys. 47, 033513 (2006) Communicated by M.B. Ruskai

Commun. Math. Phys. 297, 95–127 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1033-8

Communications in

Mathematical Physics

Yukawa Couplings in Heterotic Compactification Lara B. Anderson1,5 , James Gray2 , Dan Grayson3 , Yang-Hui He2,4 , André Lukas2 1 Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street,

Philadelphia, PA 19104-6396, USA. E-mail: [email protected]; [email protected]

2 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road,

Oxford OX1 3NP, UK. E-mail: [email protected]; [email protected]; [email protected]

3 Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street,

Urbana, IL 61801, USA. E-mail: [email protected]

4 Merton College, Oxford OX1 4JD, UK 5 Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ 08540, USA

Received: 4 June 2009 / Accepted: 17 December 2009 Published online: 19 March 2010 – © Springer-Verlag 2010

Abstract: We present a practical, algebraic method for efficiently calculating the Yukawa couplings of a large class of heterotic compactifications on Calabi-Yau three-folds with non-standard embeddings. Our methodology covers all of, though is not restricted to, the recently classified positive monads over favourable complete intersection Calabi-Yau three-folds. Since the algorithm is based on manipulating polynomials it can be easily implemented on a computer. This makes the automated investigation of Yukawa couplings for large classes of smooth heterotic compactifications a viable possibility.

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yukawa Couplings in Heterotic Compactification . . . . . . . . . . . . . 2.1 Rephrasing in terms of cohomologies . . . . . . . . . . . . . . . . . 2.2 The arena: positive monad bundles over CICYs . . . . . . . . . . . Calculating Yukawa Couplings: General Procedure . . . . . . . . . . . . 3.1 Polynomial representation of line bundle cohomology . . . . . . . . 3.2 SU (3) vector bundles and E 6 GUTS . . . . . . . . . . . . . . . . . 3.2.1 Polynomial representatives for families in H 1 (X, V ). . . . . . . 3.2.2 Polynomial representatives for H 3 (X, ∧3 V ). . . . . . . . . . . 3.2.3 Computing Yukawa couplings. . . . . . . . . . . . . . . . . . . 3.2.4 A simple E 6 example. . . . . . . . . . . . . . . . . . . . . . . 3.3 SU (4) vector bundles and S O(10) GUTs . . . . . . . . . . . . . . 3.3.1 Polynomial representatives for Higgs multiplets in H 1 (X, ∧2 V ). 3.3.2 Computing Yukawa couplings. . . . . . . . . . . . . . . . . . . 3.4 SU (5) vector bundles and SU (5) GUTs . . . . . . . . . . . . . . . 3.4.1 Polynomial representatives for H 1 (X, ∧2 V ). . . . . . . . . . . 3.4.2 Polynomial representatives for H 2 (X, ∧4 V ). . . . . . . . . . .

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96 98 99 101 103 103 104 105 105 106 107 108 109 110 110 110 111

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3.4.3 Polynomial representatives for H 2 (X, ∧2 V ). . . 3.4.4 Computing Yukawa couplings. . . . . . . . . . . 4. An Example: One Higgs Multiplet and One Heavy Family 4.1 The model . . . . . . . . . . . . . . . . . . . . . . . 4.2 Engineering Higgs multiplets . . . . . . . . . . . . . 4.3 The mass matrix . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . A. Koszul Complex and Polynomial Representatives . . . . . B. Proof of Equivalence of Formulations . . . . . . . . . . . B.1 Chain complexes and bicomplexes . . . . . . . . . . B.2 Cup products in hypercohomology of sheaves . . . . B.3 Symmetric and exterior powers of complexes . . . .

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112 113 113 113 114 115 116 117 120 120 122 123

1. Introduction Some of the most important pieces of data defining a phenomenological theory of particle physics are the Yukawa couplings. Since these parameters determine particle masses and interactions, no theory’s phenomenology can be understood in even a rudimentary manner without knowledge of their values. In string phenomenology, Yukawa couplings are usually some of the first things one attempts to calculate once the low energy particle spectrum of a model is known [1–3]. However, despite their importance, in many cases it is not known how to carry out the calculations in practice. For compactifications of heterotic string theory and M-theory on Calabi-Yau threefolds, Yukawa couplings have been be calculated in only a relatively small number of cases. Examples include orbifold compactifications and heterotic models with “standard embedding”, that is, models where the gauge bundle is chosen to be the tangent bundle of the Calabi-Yau manifold. Aside from these, only a few other, isolated, examples appear [4–7], with some of these being closely related to the standard embedding. In this paper, we considerably improve this situation by providing a simple, easy to implement, algorithm for calculating the Yukawa couplings in a large class of heterotic compactifications on smooth Calabi-Yau spaces with “non-standard embedding”. In such compactifications the gauge fields are defined in terms of a general, poly-stable holomorphic vector bundle [8]. Our approach is in the spirit of the recent papers [9,10], where a systematic analysis of such general heterotic compactifications by means of computational algebraic geometry has been pursued. The manifolds we will consider are the favourable complete intersection CalabiYau (CICY) manifolds [11], which comprise a set of 4515 three-folds. In addition, we consider vector bundles defined over these manifolds that are built using the monad construction [12]. For simplicity, we will present our method for the case where certain bundle cohomology groups vanish, as summarised in Table 2, but it is likely that the basic ideas can be extended to all stable bundles on CICY manifolds. These conditions are automatically satisfied for positive monad bundles and a large number of “not too negative” monad bundles. The class of positive monad bundles has been recently studied in references [10,13–15], and the methods described in the present paper represent a further step towards a systematic analysis of their phenomenological properties. Our method calculates the Yukawa couplings that appear in the superpotential of the four-dimensional theory. They are related to the physical Yukawa couplings by a field rotation that brings the matter field kinetic terms into canonical form. Unfortunately, the matter field kinetic terms and, hence, the required field redefinitions, are not explicitly

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known, so the physical Yukawa couplings cannot be computed directly. Numerical calculations [16] may be the only way to overcome this common limitation, and we have nothing more to say about it in the present paper. In practice, this means that only certain invariants of the Yukawa couplings, which are unchanged under redefinition of the matter fields, can be regarded as physical. For example, in cases where one can talk about a Yukawa matrix, such as in a model with the S O(10) gauge group and a single Higgs representation in 10, the rank of this matrix is physically meaningful. Of the many different methods of constructing vector bundles, the monad construction is one that has been of consistent interest in the physics literature over the years (see, for example, references [10,13,17–19]). These constructions, which will be reviewed in Sect. 2.2, lend themselves nicely to methods of computational algebraic geometry. This feature, which allows us to systematically study large classes of such bundles at a time, is one of main motivations for focusing on monad bundles in this paper. We shall consider the cases of SU (n) bundles, where n = 3, 4, 5, corresponding to the GUT visible sector gauge groups E 6 , S O(10) and SU (5), respectively. We will give examples throughout our discussion, but in particular, in Sect. 4, we give a detailed presentation of an S O(10) model. We show how to engineer models with a single 10 multiplet of S O(10). In addition, we show that the rank of the Yukawa matrix for the 16 multiplets of a model engineered in this way is one. As a result, these cases correspond to compactifications with one heavy family. Before we delve into technicalities let us briefly outline the basic method for computing Yukawa couplings, which is quite simple in principle. For a heterotic compactification on a Calabi-Yau three-fold X , the families can be identified as elements of the cohomology group H 1 (X, V ) of the gauge bundle V . Anti-families correspond to H 2 (X, V )  H 1 (X, V ∗ ), but our focus will be on models without anti-families, a property which is automatic for positive monad bundles, as we will review. To be specific, let us discuss the case of an SU (3) bundle which leads to the low-energy gauge group E 6 and families in 27 representations. In this case, we are interested in the 273 Yukawa couplings, that is, we need to understand the map H 1 (X, V ) × H 1 (X, V ) × H 1 (X, V ) → H 3 (X, ∧3 V )  C (the last equivalence holds because ∧3 V  O X for an SU (3) bundle V and h 3 (X, O X ) = 1). It turns out, for monad bundles, that the “family cohomology group” H 1 (X, V ) can be represented by a quotient of polynomial spaces, containing polynomials of certain, well-defined, degrees. Likewise, we can represent the “Yukawa cohomology group” H 3 (X, ∧3 V ) by a quotient of polynomial spaces which, of course, must be one-dimensional. Let Q be a representative of the single class in this quotient and PI , where I, J, K , . . . = 1, . . . , h 1 (X, V ), a polynomial basis for the families. Then the Yukawa couplings λ I J K are obtained by multiplying three family representatives. The result represents an element in the one-dimensional Yukawa quotient space and must, hence, be proportional to Q. The constant of proportionality is precisely the desired Yukawa coupling, so [PI PJ PK ] = λ I J K [Q], where [·] denotes the class in the quotient space. Hence, calculating Yukawa couplings is reduced to a simple procedure of multiplying polynomials and projecting the result onto the class representative Q. For the cases of bundles with structure groups SU (4) and SU (5) the procedure is analogous although slightly more complicated. The plan of this paper is as follows. In the next section, we introduce the general methodology available for computing Yukawa couplings in heterotic compactifications and the arena in which we shall be working: positive monad bundles over the complete intersection Calabi-Yau manifolds. In Sect. 3, we proceed to outline the procedure for calculating Yukawa couplings in such compactifications. We split the discussion

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Table 1. A vector bundle V with structure group G can break the E 8 gauge group of the heterotic string into a GUT group H . The low-energy representations are found from the branching of the 248 adjoint of E 8 under G × H and the low-energy spectrum is obtained by computing the indicated bundle cohomology groups G×H

Breaking Pattern: 248 →

SU(3) × E6

(1, 78) ⊕ (3, 27) ⊕ (3, 27) ⊕ (8, 1)

SU(4) × SO(10)

(1, 45) ⊕ (4, 16) ⊕ (4, 16) ⊕ (6, 10) ⊕ (15, 1)

SU(5) × SU(5)

(1, 24) ⊕ (5, 10) ⊕ (5, 10) ⊕ (10, 5) ⊕ (10, 5) ⊕ (24, 1)

Particle Spectrum n 27 n 27 n1 n 16 n 16 n 10 n1 n 10 n 10 n5 n5 n1

= = = = = = = = = = = =

h 1 (V ) h 1 (V ∗ ) = h 2 (V ) h 1 (V ⊗ V ∗ ) h 1 (V ) h 1 (V ∗ ) = h 2 (V ) h 1 (∧2 V ) h 1 (V ⊗ V ∗ ) h 1 (V ) h 1 (V ∗ ) = h 2 (V ) h 1 (∧2 V ∗ ) h 1 (∧2 V ) h 1 (V ⊗ V ∗ )

into several subsections - one for each of the possible visible sector gauge groups of interest (E 6 , S O(10), and SU (5)). Section 4 contains a detailed discussion of a one-Higgs S O(10) model. In Sect. 5 we end with conclusions and prospects. A technical result required in the bulk of the text, as well as the proof that the polynomial-based procedure, outlined in Sect. 3, indeed reproduces the physical Yukawa couplings are presented in the Appendix. 2. Yukawa Couplings in Heterotic Compactification After a brief review of heterotic compactifications and Yukawa couplings, in Sect. 2.1 we will describe how the problem of calculating Yukawa couplings can be rephrased in terms of bundle cohomology groups. We will also hint at our method for calculating these interactions based thereon, leaving the technical details of the actual procedure to the following section. In addition, in Sect. 2.2, we shall describe the basic geometrical setup of our class of Calabi-Yau manifolds and bundles. Let us start by considering how Yukawa couplings are usually described in heterotic compactifications. The matter fields in Calabi-Yau compactifications of heterotic string theory and M-theory descend from the internal parts of the gauge fields and their superpartners. In the case where we have a visible sector gauge bundle V over the Calabi-Yau threefold X taking values in a subgroup G of E 8 , the low energy observable gauge group, H , is given by the commutant of G in E 8 . The arise in the  matter fields I ) with R I and decomposition of the adjoint of E 8 under G × H : 248 = I (RGI , R H G I being representations of the groups G and H respectively, indexed by I . See Table 1 RH for a complete list of the decompositions of the 248 of E 8 and associated cohomologies for standard heterotic theories. The reduction ansatz for the holomorphic part of the gauge field A in 10-dimensions is, to lowest order [20,21],  A= C iI u aI Tai + ABG . (2.1) I

Here ABG is the background gauge field vacuum expectation value satisfying the hermitian Yang-Mills equations. The first term in (2.1) gives rise to the four-dimensional

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matter fields C iI , where I is an index running over the terms in the decomposition of 248 I . The u a are bundle-valabove, and i runs over the dimension of each representation R H I ued harmonic 1-forms on X , taking values in the associated representation RGI of the bundle structure group V . Finally, the Tia are the relevant generators of the broken part of the original E 8 gauge group, that is, those broken generators that are not part of the bundle group G. The objects of interest in this paper are the trilinear couplings between the low energy matter fields C iI . A simple expression for the superpotential Yukawa couplings has been well known for a some time [8]:  ¯ f abc . λI J K ∝ u aI ∧ u bJ ∧ u cK ∧  (2.2) X

We have used a “proportional to” sign here to emphasise the fact that, without knowledge of the Kähler potential, we can not meaningfully make statements about the overall normalization. In Eq. (2.2), the holomorphic (3, 0) form has been denoted by  and the f abc are constants descending from the structure constants of E 8 , designed to make the above expression invariant under the bundle group G. This is the form for these couplings in the low energy theory as given to us by direct dimensional reduction. Naively, the evaluation of (2.2) is computationally awkward. On a given Calabi-Yau manifold, one would have to find explicit expressions for all of the forms involved and then integrate over the manifold. For (2, 1) matter fields in standard embedding models this has been explicitly carried out in references [2,22]. To repeat such an explicit calculation for non-standard embedding models would be technically very challenging and we will instead pursue a different, more algebraic approach. 2.1. Rephrasing in terms of cohomologies. The formula (2.2) has many appealing properties [8]. In particular, it is quasi-topological.1 It depends only on the cohomology class of the 1-forms u aI and not upon the actual representative form within that chosen class. Indeed, taking u aI → u aI + D aI , for example, one sees that the change to (2.2),  ¯ f abc , D aI ∧ u bJ ∧ u cK ∧  (2.3) X

vanishes upon integration by parts since both the 1-forms u b and the holomorphic ¯ are D closed. Given this observation, one can regard the matter fields as 3-form  being represented in the formula (2.2) by cohomology classes, and not just their harmonic representatives. This suggests that a simple description of Yukawa couplings in terms of topological quantities exists. To pursue this idea, we begin by rewriting the formula for the Yukawa couplings in the case where the bundle structure group G is SU (3). We can then calculate four dimensional couplings between three 27 multiplets of E 6 . The relevant structure constants in this case are f abc = abc and, hence, the combination u aI ∧ u bJ ∧ u cK abc is an SU (3) invariant harmonic 3-form. Up to an overall constant multiple there is, of course, only one such form on a Calabi-Yau 3-fold, namely the (3, 0) form . Thus we have that, u aI ∧ u bJ ∧ u cK abc = K I J K ,

(2.4)

1 Indeed, for standard embedding models, the Yukawa couplings for (1, 1) matter fields are topological and are given by the triple intersection numbers of the Calabi-Yau manifold.

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where K I J K are complex numbers. From   ¯ abc = K I J K ¯ u aI ∧ u bJ ∧ u cK ∧   ∧ , λI J K ∝ X

(2.5)

X

we see these numbers are proportional to the desired Yukawa couplings. Referring to Table 1 once more, we see that the families in the 27 representation of E 6 can be identified with the cohomology group H 1 (X, V ). Therefore, Eq. (2.4) defines a map that takes three of our bundle-valued 1-forms to a harmonic 3-form valued in the trivial bundle. Now, for an SU (n) bundle V we have that ∧n V ∼ = O X , where O X is the trivial line-bundle on X . Thus, (2.4) defines a map of the form H 1 (X, V ) × H 1 (X, V ) × H 1 (X, V ) → H 3 (X, ∧3 V ) ∼ = H 3 (X, O X ) ∼ = C, (2.6) where the last equivalence follows from the fact that h 3 (X, O X ) = 1. The main point of this paper is that, for a large class of compactifications, when the above cohomologies are represented by certain polynomial equivalence classes, there is a mathematically natural proposal for what the map implicit in Eq. (2.6) is. It is essentially the unique possibility and simply involves polynomial multiplication of cohomology representatives. In the next section, we present this proposal in detail and show that the results to which it gives rise have all of the properties one would expect. The rigorous proof that our method for calculating Yukawa couplings does indeed reproduce the physical formula (2.2) is somewhat technical and is thus presented in Appendix B. A similar procedure can be applied to the case of structure group G = SU (4) and a visible gauge group S O(10). For such models, we are interested in Yukawa couplings of the type 10 16 16, between two families in 16 representations and a Higgs multiplet in a 10 representation of S O(10). Note that the absence of anti-families in our models means there are no 16 representations. From Table 1, it is clear that families are still identified with the cohomology group H 1 (X, V ) while Higgs multiplets correspond to H 1 (X, ∧2 V ). The analogue of Eq. (2.6) is then H 1 (X, V ) × H 1 (X, V ) × H 1 (X, ∧2 V ) → H 3 (X, ∧4 V ) ∼ = H 3 (X, O X ) ∼ =C. (2.7) The appearance of the fourth wedge power, ∧4 V , means that one has to deal with polynomials of quite high degree in practical calculations. For this reason, it is useful to slightly reformulate the above mapping to H 1 (X, V ) × H 1 (X, V ) → (H 1 (X, ∧2 V ))∗ ∼ = H 2 (X, ∧2 V ), H p (X, W )

(2.8)

where the final equivalence follows from Serre duality [23] ,  H 3− p ∗ ∗ 2 2 ∗ (X, W ) , and the fact that ∧ V ∼ = ∧ V for SU (4) bundles. Hence, instead of mapping two families and a Higgs multiplet into a one-dimensional space of high degree we combine two families to represent an element in the Higgs cohomology group. The relevant Yukawa couplings are then given by expressing the result in terms of a basis of Higgs multiplets. In this case, we only need to deal with second wedge powers of V which, as we will see, implies lower polynomial degrees. Finally, the case where G = SU (5) can be dealt with in either of the two ways we have discussed so far. It turns out to be computationally more efficient to follow the second approach. From Table 1 we have three relevant multiplets, namely 10 multiplets associated to H 1 (X, V ), 5 multiplets associated to H 1 (X, ∧2 V ∗ ) and 5 multiplets associated to H 1 (X, ∧2 V ) (and since we are considering models without anti-families there

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are no 10 representations present). This gives rise to two types of Yukawa couplings that are schematically of the form 10 10 5 and 5 5 10. The corresponding maps in cohomology are H 1 (X, V ) × H 1 (X, V ) → (H 1 (X, ∧2 V ∗ ))∗ ∼ = H 2 (X, ∧2 V ), H 1 (X, ∧2 V ) × H 1 (X, ∧2 V ) → (H 1 (X, V ))∗ ∼ = H 2 (X, ∧4 V ),

(2.9) (2.10)

where once again the last equality follows from Serre duality and the fact that V ∗

= ∧4 V for an SU (5) bundle. We now need to discuss how the maps implied in (2.6), (2.8), (2.9) and (2.10) can actually be carried out explicitly. As we will see, within our class of models provided by CICY manifolds and monad bundles, the various cohomology groups can be represented by quotient spaces of polynomials and the maps amount to polynomial multiplication. To explore this in detail we now briefly describe the technical arena we will be working in - that of positive monad bundles over CICY manifolds - before we return to the problem of calculating Yukawa couplings in Sect. 3. 2.2. The arena: positive monad bundles over CICYs. In this paper, we will focus on heterotic compactifications involving vector bundles built via the monad construction [12]. In particular, we consider the class of positive monads2 defined over favourable CICY manifolds [11]. A systematic analysis of the stability and spectrum of this class has recently been completed in [10,13–15]. To begin, we recall that complete intersection CICY manifolds are defined by the zero loci of K polynomials { p j } j=1,...,K in an ambient space A = Pn 1 × · · · × Pn m given by a product of m projective spaces with dimensions nr . We denote the projective coordinates (r ) (r ) (r ) of each factor Pnr by (x0 , x1 , . . . , xnr ), its Kähler form by Jr , and the k th power of the hyperplane bundle by OPnr (k). The manifold X is called a complete intersection if the dimension of X equals the dimension of A minus the number of polynomials. To obtain three-folds X in this way we then need rm=1 nr − K = 3. Each of the defining homogeneous polynomials p j can be characterised by its multidegree q j = (q 1j , . . . , q mj ), where q rj specifies the degree of p j in the coordinates x(r ) of the factor Pnr in A. These polynomial degrees are conveniently encoded in a configuration matrix ⎤ ⎡ n1 1 P q1 q21 . . . q K1 ⎢ Pn 2 q 2 q 2 . . . q 2 ⎥ ⎢ 1 2 K ⎥ ⎥ ⎢ . (2.11) ⎢ . . .. . . .. ⎥ ⎢ .. .. . . . ⎥ ⎦ ⎣ Pn m q1m q2m . . . q Km m×K

Note that the j th column of this matrix contains the multi-degree of the polynomial p j .  The Calabi-Yau condition, c1 (T X ) = 0, is equivalent to the conditions Kj=1 q rj = nr + 1. In terms of this data, the normal bundle N of the CICY manifold X in A can be written as N =

K 

OA (q j ).

j=1 2 For reviews of this construction and some of its applications, see references [10,12,17].

(2.12)

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Here and in the following we employ the short-hand notation OA (k) = OPn1 (k 1 ) ⊗ · · · ⊗ OPnr (k r ) for line bundles on the ambient space A. In the notation given above, the famous quintic hypersurface in P4 is denoted as “[4|5]” and its normal bundle is N = OP4 (5). CICY threefolds have been completely classified [11] and of the 7890 manifolds, 4515 are favourable, that is, all of their Kähler forms, J , descend from those of the ambient projective space. This means that favourable CICY manifolds defined in an ambient space with m projective factors are characterized by h 1,1 (T X ) = m. We will focus on these favourable CICY manifolds in the following. A monad bundle, V , is defined by the short exact sequence f

0 → V → B −→ C → 0 , where rC rB   O X (bi ) , C = O X (c j ) B= i=1

(2.13)

j=1

are sums of line bundles with ranks r B and rC , respectively.3 From the exactness of (2.13), it follows that the bundle V is given by V = ker( f ).

(2.14)

From the above sequence, the rank, n, of V is n = rk(V ) = r B − rC .

(2.15)

For the structure group to be SU (n) rather than U (n) we need the first Chern class of V to vanish, hence c1r (V ) =

rB  i=1

bir −

rC 

car = 0.

(2.16)

a=1

The existence of sufficiently general maps f is guaranteed by demanding that crj ≥ bis ∀i, j, r, s. Viewing the sequence, (2.13), as one consisting of locally free modules, rather than vector bundles, we can think of f as a matrix f ai of polynomials with multi-degree ca − bi (and we will frequently make use of this notation in the following sections). Furthermore, from Eq. (2.14), the bundle moduli of V can be identified as the coefficients parameterizing the possible maps f (see [13] for a discussion). The term “positive” refers to monad bundles satisfying bir > 0 and crj > 0 ∀r, i, j. For the technical details of monad bundles, including the spectrum, moduli and such properties as slope-stability, we refer the reader to [10,13–15]. Here we will review one feature of positive monad bundles that will be of use to us in the following sections: Positive monads do not give rise to anti-generations, that is, H 2 (X, V ) = H 1 (X, V ∗ ) = 0. To see this, we consider the dual of the monad sequence (2.13) 0 → C ∗ → B ∗ → V ∗ → 0,

(2.17)

m1 3 More generally, a monad bundle is defined as the middle homology of a sequence of the form 0 → A −→

B → C → 0. This sequence is exact at A and C, and Im(m 1 ) is a subbundle of B [12]. In this paper we restrict ourselves, as is often done in the physics literature, to the case where Im(m 1 ) vanishes. We thus recover the description (2.13).

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Table 2. List of vanishing conditions on the sums of line bundles B and C, defining the monad bundle, required for our calculation. All conditions are automatically satisfied for positive monads, due to the Kodaira vanishing theorem Case

Cohomologies required to vanish

E6

H 1 (X, B), H 3 (X, ∧3 B), H 2 (X, ∧3 B) H 1 (X, ∧2 B ⊗ C), H 2 (X, ∧2 B ⊗ C), H 1 (X, B ⊗ S 2 C) H 1 (X, B), H 1 (X, ∧2 B), H 2 (X, ∧2 B), H 1 (B ⊗ C) H 1 (X, B), H 1 (X, ∧2 B), H 1 (X, ∧4 B) H 2 (X, ∧4 B), H 1 (X, ∧3 B ⊗ C) H 2 (X, ∧2 B), H 1 (X, B ⊗ C)

S O(10) SU (5)

which gives rise to a long exact sequence · · · → H 1 (X, B ∗ ) → H 1 (X, V ∗ ) → H 2 (X, C ∗ ) → · · · .

(2.18)

Now, since B and C are sums of positive line bundles both H 1 (X, B ∗ ) and H 2 (X, C ∗ ) are zero from Kodaira’s vanishing theorem (see, for example, references [22,23]) so that H 1 (X, V ∗ ) = 0 follows immediately. Hence, there are no anti-families. With these preliminary definitions in hand we turn now to the calculation of Yukawa couplings. 3. Calculating Yukawa Couplings: General Procedure We shall consider in turn the three types of theories with E 6 , S O(10) and SU (5) lowenergy groups, corresponding respectively to the choices of an SU (n) bundle structure group with n = 3, 4, 5. A concrete SU (3) example will be presented in this section but, in the interest of brevity, we postpone doing the same for the more complicated S O(10) case until the next section. We do not give a detailed SU (5) example in this paper because the techniques are lengthy, while qualitatively the same as in the SU (3) and SU (4) cases. While the idea of computing Yukawa couplings using polynomial methods is based on the sheaf-module correspondence and should be quite general and widely applicable, the specific realisation discussed in this paper relies on a number of vanishing properties which we summarise in Table 2. These conditions are all automatically satisfied for positive monad bundles V , that is, when the sums of line bundles B and C that enter the monad sequence (2.13) consist of positive line bundles only. Given these conditions, we would like to derive polynomial representations for certain bundle cohomology groups and maps between them. It is useful to first discuss this problem for the main building blocks of the monad construction, line bundles.

3.1. Polynomial representation of line bundle cohomology. We begin with the simple case of a single projective space Pn with projective coordinates x = (x0 , . . . , xn ) and an associated graded ring R = C[x]. It is well-known that the sections, H 0 (Pn , OPn (k)) of the line bundle OPn (k) can be identified with the degree k polynomials in R. We denote the degree k part of R by Rk and write H 0 (Pn , OPn (k)) ∼ = Rk . The generalization to products of projective spaces, A = Pn 1 × · · · × Pn m , is straightforward. We denote the projective coordinates of the r th projective space by x(r ) and the

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associated multi-graded ring by R = C[x(1) , . . . , x(m) ].

(3.1)

Then the sections of the line bundle OA (k) can be identified with the multi-degree k polynomials in R, so H 0 (A, OA (k)) ∼ = Rk .

(3.2)

In our actual applications, we are of course interested in line bundles O X (k) on the CICY manifold X ⊂ A. They can be related to their ambient space cousins via a Koszul resolution and this leads to a method of calculating their cohomology and, in particular, their sections. The details of this argument are given in Appendix A but the final result is rather simple. Consider the polynomial ring (3.1), associated to our ambient space A, and the ideal p1 , . . . , p K  ⊂ R generated by the defining polynomials p j of the CICY manifold X . Then we can form the coordinate ring A=

R

p1 , . . . , p K 

(3.3)

of the CICY manifold X , which one can think of as the space of polynomials on X . In terms of the coordinate ring, the sections of the line bundle O X (k) are given by H 0 (X, O X (k)) ∼ = Ak ,

(3.4)

where the Ak denotes the multi-degree k part of A. This relation requires certain vanishing conditions, as detailed in Appendix A, which are all automatically satisfied for positive line bundles. The result (3.4) is in close analogy to its ambient space counterpart (3.2), so all that is required when dealing with line bundles on the CICY manifold X is passing from the full polynomial ring to the coordinate ring of X .

3.2. SU (3) vector bundles and E 6 GUTS. We start by considering the case of SU (3) bundles. From Table 1, the symmetry breaking pattern and decomposition of the matter field representations is E 8 ⊃ SU (3) × E 6 , 248 = (8, 1) ⊕ (1, 78) ⊕ (3, 27) ⊕ (3, 27).

(3.5) (3.6)

The (8, 1) term in this decomposition is associated with the cohomology group H 1 (X, V ⊗ V ∗ ) that counts the dimension of the bundle moduli space. Furthermore, when we are dealing with positive monads, as discussed above, anti-families in 27, corresponding to H 1 (X, V ∗ ), are absent. Hence, we are left with families in 27 multiplets, associated with the cohomology group H 1 (X, V ). The only type of Yukawa coupling is, therefore, of the form 27 27 27 and it can be calculated from the map (2.6). To do this we require polynomial representatives for the two cohomology groups involved, namely for H 1 (X, V ) and H 3 (X, ∧3 V ).

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3.2.1. Polynomial representatives for families in H 1 (X, V ). Looking at the long exact sequence in cohomology associated to the short exact monad sequence (2.13), we find that g

0 → H 0 (X, V ) → H 0 (X, B) −→ H 0 (X, C) → H 1 (X, V ) → H 1 (X, B) → · · · ,

(3.7)

where g is the induced map on sections associated to the monad map, f , in (2.13). For stable SU (n) bundles we know that H 0 (X, V ) = 0, and hence for the bundles of interest to us, we must have g injective. In addition, if we assume that H 1 (X, B) = 0, a condition which is always satisfied for positive monads as a consequence of Kodaira vanishing, it follows that H 1 (X, V ) ∼ =

H 0 (X, C)

. g H 0 (X, B)

(3.8)

From Eq. (3.4), both cohomology groups on the RHS can be represented in terms of the coordinate ring A of X , so we finally have rC Aca 1 ∼

. H (X, V ) = a=1 (3.9) rB g i=1 Abi The map g in this quotient is identical to the monad map in (2.13), viewed as a map on the associated modules. If we represent the monad map by a matrix f ai of polynomials B with multi-degrees ca − bi then its action on a vector of polynomials (qi ) ∈ ri=1 Abi is given by r  rC B   f ai qi ∈ Aca . (3.10) g((qi )) = i=1

a=1

This is the action of a polynomial matrix and it allows us to explicitly compute the polynomial quotient (3.9) once the monad map g ∼ ( f ai ) is specified. We note that the degrees of the various polynomials involved are given by the integer vectors bi and ca that define the monad bundle (2.13). We have obtained explicit polynomial representatives for the families and now turn to the “Yukawa cohomology group” H 3 (X, ∧3 V ). 3.2.2. Polynomial representatives for H 3 (X, ∧3 V ). Taking the exterior power sequence associated to our monad, as described in Appendix B of reference [13], and splitting it into short exact sequences we obtain 0 → ∧3 V → ∧3 B → K 1 → 0, 0 → K 1 → ∧2 B ⊗ C → K 2 → 0, 0 → K 2 → B ⊗ S 2 C → S 3 C → 0.

(3.11)

Here we have introduced the (co)-kernels K 1 and K 2 . The following pieces may be extracted from the associated long-exact sequences in cohomology.

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· · · → H 2 (X, ∧3 B) → H 2 (X, K 1 ) → H 3 (X, ∧3 V ) → H 3 (X, ∧3 B) → 0, (3.12) 1 2 1 2 2 2 · · · → H (X, ∧ B ⊗ C) → H (X, K 2 ) → H (X, K 1 ) → H (X, ∧ B ⊗ C) → · · · , (3.13) 0 2 0 3 1 1 2 . . . → H (X, B ⊗ S C) → H (X, S C) → H (X, K 2 ) → H (X, B ⊗ S C) → · · · . (3.14) We now assume that the following vanishing conditions: H 3 (X, ∧3 B) = 0, H 2 (X, ∧3 B) = 0, H 1 (X, ∧2 B ⊗ C) = 0, H 2 (X, ∧2 B ⊗ C) = 0, H 1 (X, B ⊗ S 2 C) = 0,

(3.15)

are satisfied. This is automatically the case for positive monad bundles as a consequence of the Kodaira vanishing theorem. Then one can combine the sequences (3.12), (3.13) and (3.14) to obtain, F

· · · → H 0 (X, B ⊗ S 2 C) −→ H 0 (X, S 3 C) → H 3 (X, ∧3 V ) → 0.

(3.16)

We therefore conclude that H 3 (X, ∧3 V ) ∼ =

H 0 (X, S 3 C)

. F H 0 (X, B ⊗ S 2 C)

Expressing this in terms of the coordinate ring via Eq. (3.4) as before, leads to  a≥b≥c Aca +cb +cc 3 3 ∼

. H (X, ∧ V ) =  F i,a≥b Abi +ca +cb

(3.17)

(3.18)

The map F is induced by the monad map f ∼ ( f ai ) and, acting on a tensor of polyno mials (qiab ) ∈ i,a≥b Abi +ca +cb , it can be written as r  B   F((qiab )) = qi(ab f c)i ∈ Aca +cb +cc , (3.19) i=1

a≥b≥c

where the brackets around the indices denote symmetrization. Since h 3 (X, ∧3 V ) = h 3 (X, O X ) = 1 we know that this polynomial quotient must be one-dimensional, although this is by no means obvious from the RHS of Eq. (3.18). For the example below we will explicitly verify that this is indeed the case. 3.2.3. Computing Yukawa couplings. From Eq. (3.9) we know that families are represented by a vector of polynomials (Pa )a=1,...,rC with multi-degrees ca , subject, of course, to the identifications implied by having to work in the coordinate ring of X and the quotient in Eq. (3.9). Let us pick a basis (PaI ), in family space, where I, J, K , . . . = 1, . . . , h 1 (X, V ) are the family indices. We can then form all possible symmetrized products, P(aI PbJ Pc)K , of these polynomials which are of degree ca + cb + cc . For each choice, (I, J, K ), of three families, these products form a three-index symmetric tensor (P(aI PbJ Pc)K ) which defines an element of a≥b≥c Aca +cb +cc and, hence, from Eq. (3.18), an element of the Yukawa cohomology group H 3 (X, ∧3 V ). That the polynomial degrees

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match in this way is non-trivial and, of course, necessary for our method to work. We can now pick a representative, (Q abc ), consisting of polynomials with multi-degree ca + cb + cc , whose class [(Q abc )] spans the quotient (3.18). Since we are dealing with a one-dimensional quotient, the class, [(P(aI PbJ Pc)K )], defined by the product of three families, must be proportional to [(Q abc )], so that we can write   (P(aI PbJ Pc)K ) = λ I J K [(Q abc )] . (3.20) The complex numbers λ I J K are of course the desired Yukawa couplings. Since the “comparison class” [(Q abc )] was chosen arbitrarily this only defines the Yukawa couplings up to an overall normalization and, of course, relative to the chosen basis in family space, as expected. 3.2.4. A simple E 6 example. Let us illustrate this procedure by a simple example on the quintic in P4 . The coordinate ring of the quintic is given by A=

C[x0 , . . . , x4 ] ,

p

(3.21)

where (x0 , . . . , x4 ) are projective coordinates on P4 and p is the defining quintic polynomial. We would like to consider the SU (3) monad bundle defined by f

0 → V → O X (1)⊕4 −→ O X (4) → 0

(3.22)

which is perhaps the simplest positive monad on the quintic. Note that, in this case, the monad map f can be represented by a vector f = ( f 1 , . . . , f 4 ) of four cubics in A. To make contact with the previous general notation, this means that the vectors bi and ca are, in fact, one-dimensional and explicitly given by b1 = b3 = b3 = b4 = (1) and c1 = (4). From Eq. (3.9) it follows that the families are represented by the quotient H 1 (X, V ) ∼ =



A4

g A⊕4 1



(3.23)

of quartic polynomials by the image of four linear polynomials. On a vector (q1 , . . . , q4 ) ∈ A⊕4 1 consisting of four linear polynomials, the map g acts as g((q1 , . . . , q4 )) =

4 

f i qi .

(3.24)

i=1

It is easy to count the dimension of this quotient. In general, the number of degree k polynomials in n + 1 variables is   n+k . (3.25) dim(C[x0 , . . . , xn ]k ) = n Hence, dim A4 = 70 and dim A⊕4 1 = 20. (In general, one has to correct for the fact that one is working with the coordinate ring, rather than the ring of all polynomials. In the present case we are dividing by an ideal generated by a quintic polynomial so that

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Table 3. The array of the 273 Yukawa couplings for the E 6 GUT associated to the SU (3) monad given in (3.22) on the quintic. There are 50 families of 27 multiplets, represented by H 1 (X, V ); we select three of these for illustrative purposes here, as indexed by I = 1, 2, 3. These are represented by the monomials x44 , x22 x32 and x02 x12 respectively (that is, these are the normal forms of the equivalence class of polynomials representing these families). The normal form of the comparison class in this calculation was x02 x12 x22 x32 x44 . The monad map is given by f = (x03 , x13 , x23 , x33 ) Y11I Y12I Y13I

I =1 0 0 0

I =2 0 0 1

I =3 0 1 0

Y21I Y22I Y23I

i =1 0 0 1

i =2 0 0 0

i =3 1 0 0

Y31I Y32I Y33I

i =1 0 1 0

i =2 1 0 0

i =3 0 0 0

degrees Ak , where k < 5 are not affected.) For sufficiently generic choices of polynomials f i , the map g is injective (as required by stability of V ) and we conclude that the quotient (3.23) has dimension 70 − 20 = 50. So we are dealing with a model with 50 families. For the Yukawa cohomology group (3.18) we have in the present case H 3 (X, ∧3 V ) ∼ =

A12 ,  F A⊕4 9

(3.26)

where F acts on a vector (r1 , . . . , r4 ) ∈ A⊕4 9 as F((r1 , . . . , r4 )) =

4 

f i ri .

(3.27)

i=1

As the degrees involved exceed 5, counting polynomials to determine the dimension of this quotient is not so simple any more. However, it is relatively straightforward to extract this information from the relevant Hilbert series which can be computed with computer algebra packages such as Macaulay and Singular [24,25]. It turns out that this dimension is indeed 1, as it must be from our general arguments. It should be noted that the computer algebra package Singular [25] is fast enough on a standard desktop machine to perform the calculation of the Yukawa couplings between all 50 families in a matter of minutes. A useful interface for Singular, designed for use by physicists, may be found here [26]. A sample of the result, for a given choice of family representatives and monad map, is given in Table 3. 3.3. SU (4) vector bundles and S O(10) GUTs. Having introduced our general method of computing Yukawa couplings for the case of SU (3) bundles, let us move on to consider the case of SU (4) bundles. From Table 1 we have the following symmetry breaking pattern and decomposition of the matter field representations: E 8 ⊃ SU (4) × S O(10), 248 = (15, 1) ⊕ (1, 45) ⊕ (4, 16) ⊕ (4, 16) ⊕ (6, 10).

(3.28) (3.29)

The (15, 1) term corresponds to bundle moduli that are counted by the cohomology group H 1 (X, V ⊗ V ∗ ). As in the E 6 case anti-families in (4, 16) multiplets are absent for positive monads. Therefore, the relevant Yukawa couplings are of the form 10 16 16

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and couple two families in 16 multiplets, associated to the cohomology group H 1 (X, V ), to a Higgs multiplet in 10, associated to the cohomology group H 1 (X, ∧2 V ). The associated Yukawa coupling can be computed by considering the map (2.8), so we need polynomial representations for H 1 (X, V ) and H 2 (X, ∧2 V ). Polynomial representatives for the families in H 1 (X, V ) can be worked out in exactly the same way as for the E 6 case and the result is given by Eqs. (3.9) and (3.10). 3.3.1. Polynomial representatives for Higgs multiplets in H 1 (X, ∧2 V ). For the 10 multiplets, corresponding to H 1 (X, ∧2 V )  H 2 (X, ∧2 V )∗ , we can choose to compute the cohomology that provides us with the lowest degree polynomial representatives and hence is the simplest and most efficient for commutative algebra. In this case, H 1 (X, ∧2 V ) provides the computationally easiest choice (see also the explicit example of Sect. 4). As a result, we introduce an exterior power sequence associated to the defining sequence of the monad, (2.13). Splitting the sequence up using the (co)-kernel K 3 we obtain the following: 0 → ∧2 V → ∧2 B → K 3 → 0, 0 → K 3 → B ⊗ C → S 2 C → 0.

(3.30)

These induce the following long exact sequences in cohomology: · · · → H 1 (X, ∧2 B) → H 1 (X, K 3 ) → H 2 (X, ∧2 V ) → H 2 (X, ∧2 B) → · · · , (3.31) F

· · · → H 0 (X, B ⊗ C) −→ H 0 (X, S 2 C) → H 1 (X, K 3 ) → H 1 (X, B ⊗ C) → · · · . (3.32) Thus, if H 1 (X, ∧2 B) = H 2 (X, ∧2 B) = 0, which are two of our vanishing conditions in Table 2 satisfied for all positive monads, we have that H 1 (X, K 3 ) ∼ = H 2 (X, ∧2 V ). 1 Together with the vanishing condition H (X, B ⊗C) ∼ = 0, again satisfied for all positive monads, this can be used in (3.32) to obtain H 1 (X, ∧2 V ) ∼ =

H 0 (X, S 2 C)

. F H 0 (X, B ⊗ C)

(3.33)

From Eq. (3.4) this translates to H (X, ∧ V ) ∼ = 1



2

F

a≥b 

Aca +cb

i,a

Abi +ca

.

(3.34)

The mapF is induced by the monad map f and, acting on a tensor of polynomials (qia ) ∈ i,a Abi +ca , it can be written as F((qia )) =

r C  i=1

 qi(a f b)i

∈

 a≥b

Aca +cb .

(3.35)

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3.3.2. Computing Yukawa couplings. We would now like to compute Yukawa couplings by mapping in the way indicated in (2.8). We note that, from Eq. (3.9), a basis in family space takes the form (PaI ), where I, J, K , . . . = 1, . . . , h 1 (X, V ) are family indices, and the polynomials are of multi-degree ca . A basis for the Higgs space (3.34) A ), where A = can be expressed in terms of multi-degree ca + cb polynomials (Hab 1, . . . , h 1 (X, ∧2 V ) numbers the Higgs multiplets and (ab) is a symmetrized index pair. Hence, the product of two polynomials representing families is precisely of the right multi-degree to be interpreted as an element of the Higgs polynomial space. We can, therefore, write      A (P(aI Pb)J ) = λ AI J (Hab ) (3.36) A

with λ AI J being the desired Yukawa couplings. An explicit example with just one Higgs multiplet will be discussed in the next section. 3.4. SU (5) vector bundles and SU (5) GUTs. The final case we shall consider is that of SU (5) bundles. For this case we have the following symmetry breaking pattern and decomposition of the matter field representations (to avoid confusion we have marked the GUT SU (5) group with a subscript GUT): E 8 ⊃ SU (5) × SU (5)GUT , 248 = (24, 1) ⊕ (1, 24) ⊕ (5, 10) ⊕ (5, 10) ⊕ (10, 5) ⊕ (10, 5).

(3.37) (3.38)

The absence of anti-generations for positive monad bundles implies that the (5, 10) states in the decomposition above are not present in the low energy spectrum as H 1 (X, V ∗ ) = 0. The relevant Yukawa couplings are then of the two types 5 10 10 and 10 5 5 and they have to be computed from the maps (2.9) and (2.10). This means we must have polynomial representations for H 1 (X, V ), H 1 (X, ∧2 V ), H 2 (X, ∧4 V ) and H 2 (X, ∧2 V ). We start, as in the other cases, by obtaining representatives for the cohomologies associated to the families residing in 10 multiplets. They correspond to the cohomology group H 1 (X, V ) and can be dealt with in exactly the same way as the 16 multiplets in the S O(10) case and the 27 multiplets for E 6 . Hence, their polynomial representatives are given by Eqs. (3.9) and (3.10) and we will not reproduce these expressions again here. 3.4.1. Polynomial representatives for H 1 (X, ∧2 V ). Polynomial representatives for the 5 multiplets in H 1 (X, ∧2 V ) may be obtained by similar techniques to those used to find the 10 multiplets in the S O(10) case (see Sect. 3.3.1). However, in the present case these particular representatives are not suitable for a calculation of Yukawa couplings following Eq. (2.10) since they do not square to the polynomial representatives for H 2 (X, ∧4 V ), as determined in the following section. We, therefore, have to follow a slightly more computationally complicated approach for the rank 5 bundles in this section. As usual, we use an exterior power sequence associated to the defining sequence of the monad. Splitting this sequence up, using the (co)-kernel K 4 , we obtain the two short exact sequences 0 → ∧2 V → ∧2 B → K 4 → 0, 0 → K 4 → B ⊗ C → S 2 C → 0.

(3.39)

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The corresponding long exact sequences in cohomology contain the parts f1

· · · → H 0 (X, ∧2 B) −→ H 0 (X, K 4 ) → H 1 (X, ∧2 V ) → H 1 (X, ∧2 B) → · · · , f2

(3.40) 0 → H 0 (X, K 4 ) → H 0 (X, B ⊗ C) −→ H 0 (X, S 2 C) → · · · . Given that H 0 (X, K 4 ) ∼ = Ker( f 2 ) it follows that f 2 ◦ f 1 = 0 and with the polynomial representatives  Abi +b j , (3.41) H 0 (X, ∧2 B) ∼ = i> j

H 0 (X, B ⊗ C) ∼ =



Abi +ca ,

(3.42)

Aca +cb ,

(3.43)

i,a

H 2 (X, S 2 C) ∼ =

 a≥b

we have the complex 

f1

Abi +b j −→

i> j



f2

Abi +ca −→

i,a



Aca +cb .

(3.44)

a≥b

On polynomial tensors (Q i j ) and (qia ) the two maps above act as ⎛ ⎞     f 1 ((Q i j )) = ⎝ f ai Q i j ⎠ , f 2 ((qia )) = qi(a f b)i , j

(3.45)

i

which confirms explicitly that f 2 ◦ f 1 = 0. The desired bundle cohomology H 1 (X, ∧2 V ) is now given, if H 1 (X, ∧2 B) = 0, by the cohomology of the above complex, that is, H 1 (X, ∧2 V ) 

Ker( f 2 ) . Im( f 1 )

(3.46)

3.4.2. Polynomial representatives for H 2 (X, ∧4 V ). Let us now obtain an appropriate polynomial description for the 10 multiplets in H 2 (X, ∧4 V ) as required for calculating the 10 5 5 Yukawa couplings from Eq. (2.10). Consider the exterior power sequence of the monad exact sequence, split by introducing (co)-kernels K 5 ,K 6 and K 7 , 0 → ∧4 V → ∧4 B → K 5 → 0, 0 → K 5 → ∧3 B ⊗ C → K 6 → 0, 0 → K 6 → ∧2 B ⊗ S 2 C → K 7 → 0, 0 → K 7 → B ⊗ S 3 C → S 4 C → 0.

(3.47) (3.48) (3.49) (3.50)

For our argument we require the following parts of the associated long exact sequences: · · · → H 1 (X, ∧4 B) → H 1 (X, K 5 ) → H 2 (X, ∧4 V ) → H 2 (X, ∧4 B) → · · · , f3

(3.51)

· · · → H (X, ∧ B ⊗ C) −→ H (X, K 6 ) → H (X, K 5 ) → H (X, ∧ B ⊗ C) → · · · , (3.52) 0

3

0

1

f4

1

0 → H 0 (X, K 6 ) → H 0 (X, ∧2 B ⊗ S 2 C) −→ H 0 (X, K 7 ) → · · · , 0 → H 0 (X, K 7 ) → H 0 (X, B ⊗ S 3 C) → H 0 (X, S 4 C) → · · · .

3

(3.53) (3.54)

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From our vanishing assumptions, which we remind the reader are automatically satisfied by the positive monads, H 1 (X, ∧4 B) = H 2 (X, ∧4 B) = 0 and, hence, the first of these sequences implies that H 2 (X, ∧4 V ) ∼ = H 1 (X, K 5 ). The last two sequences 0 0 2 tell us that H (X, K 6 ) injects into H (X, ∧ B ⊗ S 2 C) and H 0 (X, K 7 ) injects into H 0 (X, B ⊗ S 3 C). Introducing the polynomial representatives,  Abi +b j +bk +ca , (3.55) H 0 (X, ∧3 B ⊗ C) ∼ = i> j>k,a

H (X, ∧ B ⊗ S C) ∼ = 0

2

2



Abi +b j +ca +cb ,

(3.56)

Abi +ca +cb +cc ,

(3.57)

i> j,a≥b

H 0 (X, B ⊗ S 3 C) ∼ =



i,a≥b≥c

we can therefore combine (3.52)–(3.54) to form the complex    f3 f4 Abi +b j +bk +ca −→ Abi +b j +ca +cb −→ Abi +ca +cb +cc . i> j>k,a

i> j,a≥b

(3.58)

i,a≥b≥c

On polynomial tensors (Q i jka ) and (qi jab ) the above maps f 3 and f 4 act as   f 3 ((Q i jka )) = Q i jk(a f b)k , f 4 ((qi jab )) = qi j (ab f c) j . k

(3.59)

j

As before, the desired cohomology H 2 (X, ∧4 V ) is given, if H 1 (X, ∧3 B ⊗ C) = 0, by the cohomology of this complex, that is, Ker( f 4 ) H 2 (X, ∧4 V ) ∼ . = Im( f 3 )

(3.60)

3.4.3. Polynomial representatives for H 2 (X, ∧2 V ). Finally, we require polynomials to represent the 5 multiplets in H 2 (X, ∧2 V ), to calculate the 5 10 10 Yukawa couplings from Eq. (2.9). We once again consider the long exact sequence in cohomology induced by (3.39). This contains the following pieces: . . . → H 1 (X, ∧2 B) → H 1 (X, K 4 ) → H 2 (X, ∧2 V ) → H 2 (X, ∧2 B) → . . . , (3.61) f5

· · · → H 0 (X, B ⊗ C) −→ H 0 (X, S 2 C) → H 1 (X, K 4 ) → H 1 (X, B ⊗ C) → · · · . (3.62) Given the vanishing assumptions H 1 (X, ∧2 B) = H 2 (X, ∧2 B) = H 1 (X, B ⊗ C) = 0, which are automatically satisfied for positive monads, the first of these sequences implies that H 2 (X, ∧2 V ) ∼ = H 1 (X, K 4 ). Using this in the second sequence leads to H 2 (X, ∧2 V ) ∼ =

H 0 (X, S 2 C)

. f 5 H 0 (X, B ⊗ C)

Written in terms of polynomial representatives this means  a≥b Aca +cb 2 2 ∼ 

. H (X, ∧ V ) = f5 i,a Abi +ca

(3.63)

(3.64)

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113

On polynomial tensors (qia ) the map f 5 acts as    f 5 ((qia )) = qi(a f b)i .

(3.65)

i

3.4.4. Computing Yukawa couplings. We begin by summarising the polynomial representations for the various multiplets. For the families in 10 multiplets we have a basis of polynomials (PaI ) with multi-degrees ca , where I, J · · · = 1, . . . , h 1 (X, V ), as before. From Eq. (3.64), 5 multiplets are represented by multi-degree ca + cb polyA ), where A, B, · · · = 1, . . . , h 1 (X, ∧3 V ∗ ) and (ab) is a symmetric index nomials (Hab ¯ pair. Equation (3.46) shows that 5 multiplets can be represented by polynomials ( H¯ iaA ) ¯ B, ¯ · · · = 1, . . . , h 1 (X, ∧2 V ). Finally, from Eq. (3.60) of multi-degree bi + ca , where A, we have an alternative polynomial representation for the families in 10 by multi-degree bi + b j + ca + cb polynomials ( P˜iIjab ), where (i j) is an anti-symmetric and (ab) a symmetric index pair. Given these polynomial representatives, the 5 10 10 Yukawa couplings λ AI J and the 10 5 5 Yukawa couplings λ I A¯ B¯ can be computed from      A (P(aI Pb)J ) = λ AI J (Hab ) , (3.66) A

     ¯ A¯ ( H¯ [i|(a λ I A¯ B¯ ( P˜iIjab ) . H¯ |Bj]b) ) =

(3.67)

I

This concludes our general discussion. We now move on to give a comprehensively worked example of some physical interest in the S O(10) case. 4. An Example: One Higgs Multiplet and One Heavy Family 4.1. The model. As in our previous example, in Sect. 3.2.4, we consider the quintic in P4 . The coordinate ring is given by A=

C[x0 , . . . , x4 ] ,

p

(4.1)

where (x0 , . . . , x4 ) are the projective coordinates on P4 and p is the defining quintic polynomial. In this section, we will consider the following monad on the quintic: f

0 → V → O X (1)⊕7 −→ O X (2)⊕2 ⊕ O X (3) → 0.

(4.2)

This short exact sequence defines an SU (4) bundle and thus we are discussing an S O(10) GUT theory as in Sect. 3.3. The Yukawa couplings we shall calculate for this model are thus of the form 10 16 16. The monad map f can be written as f = ( f 1i , f 2i , f 3i ), where i = 1, . . . , 7 runs over the seven O X (1) line bundles and f 1i , f 2i are degree one polynomials in A while f 3i are degree two polynomials. From the general discussion in Sect. 3.3 it follows that the families in H 1 (X, V ) can be represented by polynomials as H 1 (X, V ) ∼ =

A⊕2 3 2 ⊕ A  , ⊕7 g A1

(4.3)

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where g = ( f 1i , f 2i , f 3i ). Given that dim A2 = 15, dim A3 = 35 and dim A1 = 5, and choosing the map g sufficiently general so it is injective, it follows that this model has 30 families. From Eq. (3.34), the Higgs multiplets in H 1 (X, ∧2 V ) can be represented by H 1 (X, ∧2 V ) ∼ =

⊕2 A⊕3 4 ⊕ A5 ⊕ A6 ,  ⊕7 F A⊕14 ⊕ A 3 4

(4.4)

where the map F has been defined in Eq. (3.35). If we write polynomials in the denominator of this quotient as (q(3)i , q˜(3)i , q(4)i )T , where i = 1, . . . , 7 and the first index indicates the polynomial degree, and polynomials in the numerator as (Q (4)1 , Q (4)2 , Q (5)1 , Q (4)3 , Q (6) , Q (5)2 )T , with the first index again indicating the polynomial degree, then this map can be explicitly written as ⎞ ⎛ ⎞ ⎛ f 0 0 1i Q (4)1 ⎜1 ⎟ 0 ⎟⎛ ⎜ 2 f 2i 21 f 1i Q (4)2 ⎟ ⎞ ⎜ ⎞ ⎛ ⎜ ⎟ ⎜ ⎟ q(3)i q(3)i ⎜ ⎟ ⎜1 ⎟ 1 0 ⎟ ⎜ Q (5)1 ⎟ ⎟ ⎜ 2 f 3i 2 f 1i ⎟ ⎜ q˜ ⎟=⎜ (4.5) F ⎝ q˜(3)i ⎠ = ⎜ ⎟ ⎝ (3)i ⎠ . ⎜Q ⎟ ⎜ ⎜ ⎟ f 2i 0 ⎟ ⎜ (4)3 ⎟ ⎜ 0 q(4)i q(4)i ⎜ ⎟ ⎝ Q (6) ⎠ ⎜ 0 f 3i ⎟ ⎝ 0 ⎠ 1 1 Q (5)2 0 2 f 3i 2 f 2i We note that this is a 6 × 21 matrix of polynomials. We can use this explicit map to compute the dimension of the quotient (4.4). For generic choices of the monad map f it turns out that this dimension is zero, so there are no Higgs multiplets. This confirms the general result, found in references [10,13], that h 1 (X, ∧2 V ) = 0 at a generic point in bundle moduli space. This generic case is of course of no interest in our context since Yukawa couplings of the form 10 16 16 are not present.

4.2. Engineering Higgs multiplets. To arrive at physically more interesting cases we have to understand how to engineer models with one (or possibly more than one) Higgs multiplet. This is typically not easy from a technical point of view and it was a particular challenge in the effort to find the exact MSSM spectrum from heterotic compactifications based on elliptically-fibered Calabi-Yau manifolds [27–29]. In the present framework, it is at least straightforward to state what needs to be done in principle. We need to make special choices for the polynomials defining the monad map f in such a way that the induced map F in Eq. (4.5) leads to the dimension-one quotient (4.4). At the same time, f still has to be sufficiently general so that V , as defined by the monad short exact sequence, is indeed a bundle rather than merely a sheaf.4 To examine this in detail we can consider F as a map between modules F : A(−3)⊕14 ⊕ A(−4)⊕7 −→ A(−4)⊕3 ⊕ A(−5)⊕2 ⊕ A(−6) and then examine the Hilbert function of Coker(F) at degree zero. As stated above, for generic choices of f , that is, at generic 4 Furthermore, one must be careful when choosing non-generic maps, f , that the map in question still defines a slope-stable bundle. For example, a necessary condition is that the induced map on sections, g : H 0 (X, B) → H 0 (X, C), is still injective, so that H 0 (X, V ) = 0 as required for stability. In the example described above, we can in fact go further since there is a necessary condition for stability of bundles on cyclic manifolds [10], namely that H 0 (X, ∧k V ) = 0 for k = 1 . . . rk(V ) − 1. This has been verified for the non-generic map in the example above, and the associated bundle is stable.

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115

points in bundle moduli space, the Hilbert function at degree zero vanishes. Another way of stating the same fact is that the Hilbert functions of the ideals

f 1i , f 2i , f 1i , f 3i , f 1i , f 2i , f 3i , f 3i , f 2i 

(4.6)

of the Calabi-Yau coordinate ring, that correspond to the images of the matrix rows in (4.5), are each individually zero at the appropriate degrees, that is at degrees (4, 4, 5, 4, 6). This suggests a simple way of engineering one Higgs multiplet. Rather than dealing with the full complication of the map (4.5) and its associated Hilbert function, we can focus on one row and produce a dimension one entry at the appropriate degree of the associated ideal, while keeping the dimensions zero for all other ideals. In particular, we can specialise the polynomials f 1i so that the ideal f 1i  has dimension one at degree 4. Since all of the other ideals depend on polynomials other than f 1i , one finds, upon doing this, that the Hilbert function for the remaining ideals can be kept zero at the appropriate degrees. As a result, the dimension of the quotient (4.4) is one and we have engineered an example with one Higgs multiplet. We still have to check that there exists a choice for f along the above lines that defines a bundle rather than just a sheaf. To do this we consider the explicit example f 1i = (40x3 + 94x4 , 117x3 + 119x4 , 449x3 + 464x4 + 266x0 + 195x1 + 173x2 , 306x2 , 273x3 , 259x3 + 291x4 , 76x3 + 98x2 ), (4.7) with the remaining polynomials in f being left generic. This choice has been engineered in the way described above and it can be verified that it indeed leads to precisely one Higgs multiplet. In addition, one can check that the locus in P4 where the polynomial matrix f degenerates (that is, where its rank is not maximal) does not intersect a sufficiently general quintic and, hence, leads to a bundle on the quintic (although not to a bundle on P4 ). 4.3. The mass matrix. We now wish to calculate the Yukawa couplings in this class of examples with one Higgs multiplet. To do so we first pick 30 family representatives P I = (P1I , P2I , P3I ) ∈ A⊕2 2 ⊕ A3 whose associated classes form a basis of (4.3). Fur⊕2 ther we choose a Higgs representative H = (H1 , . . . , H6 ) ∈ A⊕3 4 ⊕ A5 ⊕ A6 whose class spans the one-dimensional space (4.4). The Yukawa couplings then follow from Eq. (3.36) and form a symmetric matrix λIJ . Given that we do not know the matter field kinetic terms, the only physically significant property of this matrix is its rank. It turns out, with the map (4.7) this rank is precisely one. The monad in (4.2) gives rise to one massive family in four dimensions at the point specified by (4.7) in its bundle moduli space. In fact, this structure is somewhat more generic. Let us consider, more generally, an S O(10) model with a basis {PA } of F ≡ a Aca , such that {PI } ⊂ {PA } is a set of family representatives and a single Higgs multiplet represented by H ∈ H ≡ a≥b Aca +cb . Note that H = S 2 F, so the symmetric tensor products {PA ⊗ S PB } form a basis of H and we can introduce a hermitian scalar product, · , · , on H such that this basis is orthonormal. From Eq. (3.36) it then follows that the Yukawa matrix is given by the scalar product λ I J ∼ PI ⊗ S PJ , H . The  Higgs representative H can, of course, always be written as a linear combination H = A,B H AB PA ⊗ S PB . Inserting this into the above scalar product expression for the Yukawa couplings one finds that λ I J ∼ HI J .

(4.8)

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This means, in a case where the Higgs representative can be expressed in terms  of the family representatives, so that H = H P ⊗ S PJ , the Yukawa matrix I J I I,J and the matrix representing the Higgs are proportional. In particular, their rank has to be the same. The method of engineering models with one Higgs multiplet described above typically leads to a Higgs representative which can be written as the square of a vector v I , that is H = I,J v I v J PI ⊗ S PJ .5 This can be seen as follows. Let us choose a Higgs representative by taking  a so called “normal form” of a sufficiently generic linear combination of terms, A,B c A,B PA ⊗ S PB , where c A,B are some randomly generated coefficients. We take this normal form by performing the Buchberger Algorithm [30,31] on the linear combination relative to the module generated by the map polynomials defining the one dimensional class (4.4). Given our method of engineering a single Higgs, as discussed in the previous sub-section, the

T resulting Higgs representative will be of the form Q (4)1 , 0, 0, 0, 0, 0 . An inspection of the Buchberger algorithm [30,31] reveals that Q (4)1 in this expression will be a single monomial. It is in fact the “lagging monomial” of degree four that is not in the ideal

f 1i . That is, it is the degree four monomial that is lowest according to the monomial ordering used in the Buchberger algorithm, which does not appear as an element of

f 1i  ⊂ A. If this lagging monomial is a square, then clearly our Higgs representative is the square of a family representative. This is always the case for the type of example considered in Sects. 4.1 and 4.2. The f 1i are all linear polynomials for the monad given in (4.2). Given this, the lagging monomial is some variable to the fourth power and H = c (xi2 , 0, 0)T ⊗ S (xi2 , 0, 0)T for some i and some constant c. As a result, the matrix associated to the Higgs representative and, hence, the Yukawa matrix has rank one. We see that there is a close relation between our method for engineering one-Higgs models and obtaining precisely one heavy family. In conclusion we can state the following: A model in which one Higgs is engineered in the manner described in Sect. 4.2 will generically have one heavy family. 5. Conclusions In this paper we have introduced a simple algorithm for calculating Yukawa couplings in a wide class of heterotic models. The compactifications we have considered are on smooth Calabi-Yau spaces and are not restricted to the standard embedding. The methods can be used to calculate Yukawa couplings for a large class of bundles on complete-intersection Calabi-Yau manifolds. Such a systematic procedure for non-standard embedding models has not been presented in the literature before and, we believe, constitutes substantial advance. The key to our methodology is to obtain polynomial representatives for family and other relevant multiplets whose degrees are compatible with one another. In practice, this requires finding polynomial representatives for various cohomology groups whose degrees are such that our procedure of polynomial multiplication and reduction to normal form may be carried out. Because of the simple, algebraic nature of the resulting algorithm, the calculations can be carried out on a computer and we have done this in the text for a number of concrete examples. We should stress again that we have calculated the superpotential contributions to the Yukawa couplings. The Kähler potential for the matter fields remains an unknown 5 We would like to thank Tony Pantev for very helpful comments on this point.

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quantity in these non-standard embedding models. Nevertheless we have shown how some physically relevant information can be extracted from our results by focusing on quantities, such as the rank in the case of a Yukawa matrix, which are unaffected by the choice of basis in family space. The final example, presented in Sect. 4, demonstrates the power of these methods. This is an example of a smooth Calabi-Yau compactification leading to an S O(10) GUT. We have shown how one may isolate loci in bundle moduli space where the model has precisely one Higgs multiplet, residing in the 10 representation. Our approach based on polynomial representatives makes this conceptually rather simple and merely requires making specific choices for the polynomials defining the bundle. In practice, it is not always straightforward to find these but we have described a simple method to engineer viable cases. We have then shown that the structure of this one-Higgs model leads to a Yukawa matrix of rank one, and so to precisely one massive family. Moreover, the relation between our method of engineering one Higgs multiplet and obtaining one massive family seems to be more general, an observation which might be important for building heterotic models with a phenomenologically viable pattern of fermion masses. While our method of computing Yukawa couplings by multiplying polynomial representatives has been presented in the context of a particular class of models, the underlying mathematical structure – the sheaf-module correspondence – is quite general and we expect related methods to work for other Calabi-Yau and bundle constructions. This work should be of considerable utility in checking conclusions about the vanishing of Yukawa couplings resulting from the research presented in references [32,33]. Eventually, one would like to calculate Yukawa couplings in the context of more realistic models, where the GUT symmetry is broken due to Wilson lines. We expect that the methods described in this paper can be readily applied to such models, basically by projecting onto the various equivariant sub-spaces of the cohomology groups involved. Acknowledgements. We gratefully acknowledge enlightening discussions with Philip Candelas and Tony Pantev. D. G. is partially supported by NSF grants DMS 08-10948 and DMS 03-11378, L. A. is supported by the DOE under contract No. DE-AC02-76-ER-03071, J. G., by STFC, UK, Y.-H. H., by an Advanced Fellowship from the STFC, UK, as well as the FitzJames Fellowship of Merton College, Oxford, and A. L., by the EC 6th Framework Programme MRTN-CT-2004-503369. L. A., J. G. and A. L. would like to thank the organisers of the 2008 Vienna ESI workshop “Mathematical Challenges in String Phenomenology” where part of this work was completed. D. G. would like to thank Y.-H. H. for hospitality at the University of Oxford and Merton College, Oxford.

Appendices A. Koszul Complex and Polynomial Representatives In this appendix we justify the relation (3.4) between sections of line bundles on a CICY manifold and its coordinate ring. First let us recall the general set-up and the notation. We work in an ambient space A = Pn 1 × · · · × Pn m with projective coordinates (x(1) , . . . , x(m) ). Line bundles on A are denoted by OA (k) = OPn1 (k1 ) ⊗ · · · ⊗ OPnm (km ), where k = (k1 , . . . , km ). The associated ring R = C[x(1) , . . . , x(m) ]

(A.1)

is multi-graded by an m-dimensional grade vector k = (k1 , . . . , km ), where kr specifies the degree in the projective coordinates x(r ) of Pnr . The multi-degree k part of R is

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Table 4. The cohomology conditions that must be satisfied in order for the conclusions of Appendix A to hold ∀ 2 ≤ q ≤ K − 1, H q−2 (A, ∧q N ∗ ⊗ L) = 0 ∀ 3 ≤ q ≤ K − 1, H q−1 (A, ∧q N ∗ ⊗ L) = 0 q q ∗ ∀ 2 ≤ q ≤ K − 2, H q−1 (A, ∧q N ∗ ⊗ L) = 0 ∀ 3 ≤ q ≤ K − 2, H (A, ∧ N ⊗ L) = 0 H K −2 (A, ∧ K N ∗ ⊗ L) = 0, H K −1 (A, ∧ K N ∗ ⊗ L) = 0, H q (A, ∧q N ∗ ⊗ L) = 0 ∀ 1 ≤ q ≤ K − 1, ∀ 1 ≤ q ≤ K − 2, H q+1 (A, ∧q N ∗ ⊗ L) = 0 H K (A, ∧ K N ∗ ⊗ L) = 0.

denoted by Rk . Sections H 0 (X, OA (k)) of the line bundle OA (k) can be represented by polynomials of multi-degree k in R, so we write H 0 (X, OA (k)) ∼ = Rk .

(A.2)

A co-dimension K CICY manifold X ⊂ A is defined as the zero locus of homogeneous polynomials p1 , . . . , p K , and we denote the normal bundle of X in A by N . We define line bundles on X by restricting ambient space line bundles, that is O X (k) ≡ OA (k)| X . Moreover, we assume that the CICY manifold is “favourable”, that is, all line bundles on X are obtained in this way. The coordinate ring of X is given by A=

R ,

p1 , . . . , p K 

(A.3)

and it inherits the multi-grading from R. We denote by Ak the multi-degree k part of A. For the purpose of this appendix, we focus on line bundles L = OA (k) and their counterparts L = O X (k) on X which satisfy the vanishing conditions6 H q (∧κ N ∗ ⊗ L) = 0

(A.4)

for q > 0 and κ = 0, . . . , K . We note that, as a consequence of Kodaira’s vanishing theorem applied to line bundles L on the ambient space A, all positive line bundles fall into this class. Provided the above vanishing conditions are satisfied we want to show that H 0 (X, L) ∼ = Ak .

(A.5)

It is instructive to first do this for co-dimension one CICY manifolds, that is, for K = 1, before embarking on the general case. For K = 1 the Koszul resolution of L is given by the short exact sequence ·p

0 → N ∗ ⊗ L −→ L → L → 0,

(A.6)

where · p denotes multiplication by the defining polynomial p of X . This leads to the long exact sequence p

0 → H 0 (A, N ∗ ⊗ L) −→ H 0 (A, L) → H 0 (X, L) → H 1 (A, N ∗ ⊗ L) −→ · · · .

(A.7)

6 This can be slightly weakened without changing the result of this appendix. In fact we require the vanishing conditions stated in Table 4.

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Since H 1 (A, N ∗ ⊗ L) = 0 from the above vanishing assumptions one concludes that H 0 (X, L) ∼ =

p



H 0 (A, L)

. ⊗ L)

(A.8)

H 0 (A, N ∗

Combining this with Eq. (A.2) the desired statement (A.5) follows immediately. We now proceed to the case of general co-dimension K . While the basic structure of the argument remains unchanged from the K = 1 case a technical complication arises because the Koszul sequence 0 → ∧ K N ∗ ⊗ L → ∧ K −1 N ∗ ⊗ L → · · · → ∧κ N ∗ ⊗ L → · · · → N∗ ⊗ L → L → L → 0

(A.9)

is no longer short. A simple way of dealing with this is to break the sequence up into short exact sequences 0 → ∧ K N ∗ ⊗ L → ∧ K −1 N ∗ ⊗ L → C K −1 → 0, .. .

0 → Cκ+1 0 → C1

→ ∧κ N ∗ ⊗ L .. . →L

→ Cκ

→ 0,

→L

→ 0,

(A.10)

introducing co-kernels C1 , . . . , C K −1 . Here, we have κ = 1, . . . , K − 2 in the middle sequence. From our vanishing condition, the first of these sequences implies that H q (A, C K −1 ) = 0 for all q > 0. Further, from the long exact sequence associated to the middle sequence above and the vanishing conditions it follows that H q−1 (A, Cκ ) ∼ = H q (A, Cκ+1 ) for κ = 1, . . . , K − 2 and q = 2, . . . K + 3. Together, this means that H 1 (A, Cκ ) = 0 for κ = 1, . . . , K − 1 and, hence, the long exact sequences associated to (A.10) all break after three terms. This leads to the recursion relations H 0 (A, L) H 0 (X, L) ∼ , = 0 H (A, C1 ) H 0 (∧κ N ∗ ⊗ L) , H 0 (A, Cκ ) ∼ = H 0 (A, Cκ+1 ) H 0 (A, ∧ K N ∗ ⊗ L) H 0 (A, C K −1 ) ∼ , = 0 H (A ∧ K −1 N ∗ ⊗ L)

(A.11) (A.12) (A.13)

where κ = 1, . . . , K −2, which allow one to express H 0 (X, L) as a “chain of quotients”. However, since 0 → H 0 (A, ∧ K N ∗ ⊗ L) → H 0 (A, ∧ K −1 N ∗ ⊗ L) → · · · → H 0 (A, N ∗ ⊗ L) → H 0 (A, L) → H 0 (X, L) → 0 (A.14) is a complex, it is sufficient to keep the first quotient in this chain. Hence, we have H 0 (X, L) ∼ =

H 0 (A, L)

, p H 0 (A, N ∗ ⊗ L)

(A.15)

where p is the map induced by the defining polynomials p1 , . . . , p K of the CICY manifold. Using the polynomial representatives (A.2) for sections of line bundles in the ambient space this implies the desired Eq. (A.5).

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B. Proof of Equivalence of Formulations In this Appendix, as promised in the text, we give a formal mathematical proof of why calculating the Yukawa couplings using (2.2) is equivalent to the maps in cohomology as given in Sect. 2.1. The crucial point is, of course, that one may calculate a sheaf cohomology group with any acyclic resolution of that object. The different descriptions of the same cohomologies may be related, and operations, such as taking the image of maps, in one language may be translated to another. In what follows, we will go through in detail a full proof of the above claim. This requires a few more pieces of mathematical technology than utilized elsewhere in the paper, which we introduce as needed. B.1. Chain complexes and bicomplexes. The standard way to convert a bicomplex7 C = C·· (with horizontal differential d  : C p,q → C p−1,q and vertical differential d  : C p,q → C p,q−1 that commute) to a chain complex is to define the total complex Tot C  by setting (Tot C)n = C p+q=n p,q and setting the differential d : Tot C → Tot C of degree −1 to be x → d  (x) + (−1) p d  (x) for x ∈ C p,q . This is in accordance with the principle of signs [35] if C is the tensor product of two chain complexes and we identify the symbols d, d  , and d  . Given m ∈ Z and a chain complex B define the shifted chain complex B[m] by setting B[m] p = Bm+ p ; use the same differential, with no change in sign.8 Similarly, given m, n ∈ Z and a bicomplex C define the shifted bicomplex C[m, n] by setting C[m, n] p,q = Cm+ p,n+q . The formula for the differential in Tot C involves p but not q, so there is a simple isomorphism Tot(C[0, n]) ∼ = (Tot C)[n] involving just direct sums of identity maps, with no minus signs involved. Thus, if we think of a bicomplex C as being assembled from its rows C·,q for q ∈ Z, reindexing the rows results in shifting the total complex. If the bicomplex is zero outside the range 0 ≤ q ≤ N we will use the pictorial notation C = [C·,0 ← C·,1 ← · · · ← C·,N ] to indicate its assembly from its rows. No minus signs are to be used when assembling a bicomplex from chain complexes and maps between them in this way. ∼ =

In general, an isomorphism of chain complexes γ : Tot(C[m, n]) − → (Tot(C))[m+n] can be defined as (−1)mq times the identity map on the component (C[m, n]) p,q = C p+m,q+n . We omit the computation that γ is a chain map. A careful eye can discern something of degree m moving past something of degree q, in accordance with the principle of signs [35]. Given a map f : B → C of chain complexes we define the mapping cone9 by setting Cone f = Cone(C ← B) = Tot[C ← B]. There are isomorphisms Cone(C ← 0) ∼ = C and Cone(0 ← B) ∼ = B[−1], and the exact sequence 0 → [C ← 0] → [C ← B] → [0 ← B] → 0 of bicomplexes leads to an exact sequence 0 → C → Cone f → B[−1] → 0 of chain complexes. Given c ∈ C p and b ∈ B p−1 the element (c, b) ∈ (Cone f ) p satisfies d(c, b) = (dc + (−1) p−1 f b, db). 7 Warning: the standard definition of double complex in [37, p. 60] (see also [36, p. 174, Exer. 11] and [41, p. 8]) uses a sign convention different from the one we use here, namely d  d  = −d  d  . 8 Warning: this sign convention differs from the standard one implied by [37, p. 72, Ex. 1] and explicitly presented in [41, p. 9]. There the differential on B[m] is equal to (−1)m times the differential on B. Better notation for that concept, compatible with the principle of signs [35], would be [m]B. An isomorphism ∼ =

[m]B − → B[m], also compatible with the principle of signs, can be defined by x  → (−1)mp x for x ∈ B p . 9 Our definition of the mapping cone is not the usual one, see [41, p. 18, 20].

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A fundamental lemma in homological algebra states that a map C → D of first quadrant bicomplexes that is a quasi-isomorphism in each row induces a quasi-isomorphism on total complexes. The same statement applies to a map of third quadrant bicomplexes, or when rows are replaced by columns. A slightly stronger version, for filtered complexes, is proved in [34, Lemma 3.2]. This result is presented as the acyclic assembly lemma in [41]. f g Given a short exact sequence E : 0 → A − → B − → C → 0 of chain complexes, the corresponding map [0 ← A] → [C ← B] of bicomplexes is a quasi-isomorphism in each column, hence, according to the lemma, A[−1] → Cone(C ← B) is a quasi-isomorphism. Its inverse in the derived category composed with the map C → Cone(C ← B) gives a map ρ = ρ E : C → A[−1] in the derived category. We would like to compare the induced map ρ : H p C → H p−1 A with the connecting homomorphism ∂ = ∂ E : H p C → H p−1 A that appears in the long exact homology sequence. Given cycles c ∈ C p and a ∈ A p−1 , ∂[c] = [a] means that there is an element b ∈ B p such that gb = c and f a = db. The element (0, b) ∈ Cone(C ← B) p+1 satisfies d(0, b) = ((−1) p gb, db) = ((−1) p c, f a), so ((−1) p−1 c, 0) and (0, f a) are homologous elements of Cone(C ← B) p , which tells us that ρ[c] = (−1) p−1 [a], and thus ρ = (−1) p−1 ∂ on H p C. Now we consider longer extensions in the sense of Yoneda. Suppose we have an exact sequence E : 0 → A → Bn → · · · → B2 → B1 → C → 0 of chain complexes. The commutative diagram ··· o

0o

0o

0o

··· o

0o

Ao

0o

···

··· o

 0o

 Co

 B1 o

··· o

 Bn−1 o

 Bn o

 0o

···

of chain complexes can be regarded as a map of chain complexes of chain complexes. The corresponding map [0 ← · · · ← A] → [C ← B1 ← · · · ← Bn ] of bicomplexes is a quasi-isomorphism in each column (by exactness of E), hence the map A[−n] → Tot[C ← B1 ← · · · ← Bn ] is a quasi-isomorphism. Its inverse composed with the map C → Tot[C ← B1 ← · · · ← Bn ] gives a map ρ = ρ E : C → A[−n] in the derived category. Suppose we have another exact sequence F : 0 → C → Pm → · · · → P2 → P1 → Q → 0 of chain complexes, and consider the associated map ρ F : Q → C[−m]. Let E ∗ F : 0 → A → Bn → · · · → B2 → B1 → Pm → · · · → P2 → P1 → Q → 0 be the exact sequence obtained by splicing E to F along C; the differential in the middle is the composite map B1 → C → Pm . The following commutative diagram of bicomplexes, in which quasi-isomorphisms are indicated by ∼, shows that ρ E∗F = ρ E [−m] ◦ ρ F ; the simplicity of this formula and the absence of signs in it is a consequence of our choices above. A[−m − n] ∼

C[−m] ∼

Q

 / [Q ← P1 ← . . . ← Pm ]

 / [C ← B1 ← . . . ← Bn ][−m] 

∼

/ [Q ← P1 ← . . . ← Pm ← B1 ← . . . ← Bn ]

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Now let’s decompose our original sequence E by writing it as E = E 1 ∗ · · · ∗ E n , where E i : 0 → Di → Bi → Di−1 → 0, and Dn = A, D0 = C, and Di = im(Bi+1 → Bi ) for 0 < i < n. Then ρ E = ρ E 1 [−(n − 1)] ◦ · · · ◦ ρ E n−1 [−1] ◦ ρ E n . The resulting map H p (ρ E ) : H p C → H p−n A is thus a composite of connecting homomorphisms, up to sign. More precisely, H p (ρ E ) = H p (ρ E 1 [−(n − 1)] ◦ · · · ◦ ρ E n−1 [−1] ◦ ρ E n ) = H p (ρ E 1 [−(n − 1)]) ◦ · · · ◦ H p (ρ E n−1 [−1]) ◦ H p (ρ E n ) = H p−n+1 (ρ E 1 ) ◦ · · · ◦ H p−1 (ρ E n−1 ) ◦ H p (ρ E n ) = ((−1) p−n ∂ E 1 ) ◦ · · · ◦ ((−1) p−2 ∂ E n−1 ) ◦ ((−1) p−1 ∂ E n ) = (−1)( p−n)+···+( p−2)+( p−1) ∂ E 1 ◦ · · · ◦ ∂ E n−1 ◦ ∂ E n = (−1) pn+n(n+1)/2 ∂ E 1 ◦ · · · ◦ ∂ E n−1 ◦ ∂ E n . (This result was proved in [37, Chap. V, Prop. 7.1, p. 92]. See also the application in [37, Chap. V, Exer. 8, p. 105].) In particular, H0 (ρ E ) = (−1)n(n+1)/2 ∂ E 1 ◦ · · · ◦ ∂ E n−1 ◦ ∂ E n . Now suppose our chain complexes are bounded above and have their components drawn from an abelian category C with enough injectives, and suppose we are studying the right derived functors R p F of a left exact additive functor F : C → V, where V is an abelian category. A chain complex B with B p injective for each p is called injective. If E : · · · → C2 → C1 → C0 → . . . is a chain complex of such complexes (each bounded above), then it maps (injectively) to a chain complex E  : · · · → C2 → C1 → C0 → . . . of injective chain complexes (each bound above), so that for each p the map C p → C p is a quasi-isomorphism; moreover, if E is exact, then E  may be chosen to be exact; also, E  may be chosen so that C p = 0 for all p with C p = 0.10 In particular, a chain complex C maps via an injective quasi-isomorphism to an injective chain complex C  (an injective resolution). We set R F(C) = F(C  ) and R p F(C) = H p (F(C  )), thereby extending the usual definition of R F for objects (cohomology) of our category to chain complexes (hypercohomology). The usual arguments that show this definition is independent of the choice of injective resolution can be extended to cover this case. See [37, Chap. XVII] for a detailed discussion of hyperhomology and hypercohomology. See also [36, p. 183, Ex. 17] for a discussion of resolutions of complexes. When working with chain complexes of sheaves of abelian groups on a space X , we will use the same notation for sheaf cohomology and for sheaf hypercohomology, writing H p (X, C) whether C is a sheaf or a complex of sheaves (bounded above). In this context, one may use the flasque resolution C → G(C) constructed by Godement in [38, Chap. II, Sect. 4.3, p. 167]. The construction gives an exact functor from sheaves to flasque resolutions of them, hence, by applying it to each sheaf in a chain complex and then taking the total complex, it gives an exact functor from chain complexes to injective resolutions of them. Whether we use injective resolutions or flasque resolutions, the formula above for H0 (ρ E ) leads to an analogous formula on sheaf cohomology for H 0 (X, ρ E ) : H 0 (X, C) → H n (X, A) as a composite of connecting homomorphisms with a sign.

B.2. Cup products in hypercohomology of sheaves. In this section, the tensor product B ⊗ C of sheaves B and C on X may denote either: tensor product of sheaves of abelian groups; tensor product of sheaves of R-modules, where R is a commutative ring; or 10 We only sketch the proof. As in the construction of a Cartan-Eilenberg resolution of a chain complex, one writes the chain complex in terms of short exact sequences with maps from the tail end of one to the start of the next. Then one modifies the proof that the modules in a short exact sequence have injective resolutions that fit into a short exact sequence by replacing cokernels by pushouts at a certain point. In any case, for our intended application to sheaves on a topological space, we don’t really need this abstract formulation, because of the canonical Godement flasque resolution.

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tensor product of sheaves of O-Modules, where O is a sheaf of rings on X . It is an additive functor in each variable. Godement’s canonical flasque resolution G(C) of  a sheaf C in [38, Chap. II, Sect. 4.3, p. 167] begins with the map η : C → G 0 (C) = x∈X (i x )∗ C x , where i x : {x} → X is the inclusion map. The stalk of η at any point y ∈ X can be split by projecting onto the factor corresponding to y in the product, and that shows η is injective. Moreover, if B is another sheaf, then B ⊗ η is an injective map, for the same reason. The second step in Godement’s construction is G 1 (C) = G 0 (coker η), and the pattern continues. Because tensor product is always right exact, it follows that B ⊗ C → B ⊗ G(C) is a quasi-isomorphism, and that is also true when C is a chain complex of sheaves. This fact, which we may call (universal exactness), makes cup product operations possible, as we shall now see. (Our approach differs slightly from Godement’s original approach to cup products in [38, Chap. II, Sect. 6.1, p. 238], in that he emphasized the role of external tensor product sheaves on X × X . For a thorough and modern approach, see [40].) If B and C are chain complexes, the bicomplex B ⊗ C is defined by setting (B ⊗ C) p,q = B p ⊗ Cq . The vertical and horizontal differentials come from those of B and C, with no signs introduced, contrary to the standard convention [37, Chap. 4, Sect. 5]. By universal exactness of the Godement resolution, the map Tot(B ⊗ C) → Tot (G(B) ⊗ G(C)) is a quasi-isomorphism, hence the identity map on Tot(B ⊗ C) can be lifted to a map from Tot(G(B) ⊗ G(C)) to an injective resolution of Tot(B ⊗ C), unique up to homotopy. The resulting pairing H p (X, B)⊗ H q (X, C) → H p+q (X, Tot(B ⊗C)) is the cup product in hypercohomology. If a map Tot(B ⊗ C) → D is given, the resulting composite pairing H p (X, B) ⊗ H q (X, C) → H p+q (X, D) may also be called a cup product pairing. We may also assemble these maps into a single map H ∗ (X, B) ⊗ H ∗ (X, C) → H ∗ (X, D) of graded groups. Let’s examine compatibility of cup products with shifting. Composition with the map γ introduced above gives a map Tot(B[m]⊗C[n]) → D[m+n] that leads to a cup product pairing H p (X, B[m])⊗ H q (X, C[n]) → H p+q (X, D[m+n]). Identity maps can be used to compare this with the original cup product pairing H p−m (X, B) ⊗ H q−n (X, C) → H p+q−m−n (X, D), and the resulting discrepancy is the factor (−1)mq appearing in the definition of γ .

B.3. Symmetric and exterior powers of complexes. Suppose k ≥ 0 and C is a chain complex. Let C ⊗k denote the tensor product C ⊗ · · · ⊗ C of k copies of C. The symmetric group k acts on C ⊗k by permuting the factors, but a sign must be inserted to get an action on Tot C ⊗k , in accordance with the principle of signs [35]. Transposing adjacent factors involves a minus sign exactly when the two factors are both of odd degree, and the total sign can be determined by writing an arbitrary permutation as a product of adjacent transpositions. Another way of saying it that one excises the factors of even degree, collapsing to a possibly shorter tensor product, and then takes the sign of the residual permutation on the factors of odd degree. To see that that works, it suffices to consider the case k = 2. Let τ : C p ⊗ Cq → Cq ⊗ C p denote the (signed) transposition map defined by x ⊗ y → (−1) pq y ⊗ x. If x ∈ C p and y ∈ Cq , then τ (d(x ⊗ y)) = τ (d x ⊗ y + (−1) p x ⊗ dy) = (−1)( p+1)q y ⊗ d x + (−1) p+ p(q+1) dy ⊗ x = (−1) pq dy ⊗ x + (−1) pq+q y ⊗ d x = d((−1) pq y ⊗ x) = d(τ (x ⊗ y)).

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Assume for the rest of the section that we are working with coherent sheaves on a variety X over a field of characteristic 0, and that the tensor product B ⊗ C of sheaves denotes tensor product of sheaves of O X -Modules. Assume that B is a locally free finitely generated O-Module (vector bundle). Then B ⊗ C is an exact functor of the sheaf C. Hence, if C is an acyclic chain complex, so is B ⊗ C. If C → D is a quasi-isomorphism, then so is B ⊗ C → B ⊗ D (because being a quasi-isomorphism is determined by whether the mapping cone is acyclic, and formation of the mapping cone commutes with tensor product by B). Alternatively, assume that B is a chain complex of locally free sheaves, and that all our chain complexes are bounded above. Then if C → D is a quasi-isomorphism, then so is Tot(B ⊗C) → Tot(B ⊗ D). Finally, if C → D is a quasiisomorphism of chain complexes of locally free sheaves, then Tot C ⊗k → Tot D ⊗k is a quasi-isomorphism. Now let C be a chain complex, let S k C denote the part of Tot C ⊗k upon which k ⊗k acts trivially, and let ∧k C denote the part of Tot C  upon which k acts  by the sign of permutations. The projection operators (1/k!) σ ∈ k σ and (1/k!) σ ∈ k (−1)σ σ show that S k C and ∧k C appear functorially as direct summands of Tot C ⊗k . Moreover, if C → D is a quasi-isomorphism, then so are the induced maps S k C → S k D and ∧k C → ∧k D. Let’s compute symmetric and exterior powers of complexes of length 0 and length 1 in terms of symmetric and exterior powers of sheaves. Suppose C is a sheaf. When suggested by the notation, we convert C to a chain complex of length 0 by putting it in degree 0 and putting zeroes in the other positions; thus C[m] will denote the chain complex of length 0 with C in position −m and zeroes in the other positions. With this notation, we see that S k (C[m]) = (S k C)[km] if m is even and S k (C[m]) = (∧k C)[km] if m is odd, and ∧k (C[m]) = (∧k C)[km] if m is even and ∧k (C[m]) = (S k C)[km] if m d

is odd. Suppose now that C = [A ← − B] is a complex of length 1; recall that this notation puts A in degree 0 and B in degree 1. We wish to compute (S k C)q and (∧k C)q for q ∈ Z; for this purpose C∼ = = A ⊕ B[−1]. Then S k C ∼  we may passume qd = 0, so that  k ∼ ∼ S (A ⊕ B[−1]) = S A ⊗ S (B[−1]) = S p A ⊗ (∧q B)[−q] ∼ = p+q=k p+q=k  p q k k−q A ⊗ ∧q B, p+q=k (S A ⊗ ∧ B)[−q]. The general conclusion is that (S C)q = S and a similar argument shows that (∧k C)q = ∧k−q A ⊗ S q B. Suppose E : 0 → V → B → C → 0 is a short exact sequence of vector bundles. Then the map V → [C ← B][1] is a quasi-isomorphism, and hence so are the maps S k V → S k ([C ← B][1]) and ∧k V → ∧k ([C ← B][1]). In other words, we have long exact sequences S k E : 0 → S k V → S k B → S k−1 B ⊗ C → · · · → S k− p B ⊗ ∧ p C → · · · → ∧k C → 0, and ∧k E : 0 → ∧k V → ∧k B → ∧k−1 B ⊗ C → · · · → ∧k− p B ⊗ S p C → · · · → S k C → 0. (Suppose alternatively that 0 → A → B → V → 0 is an exact sequence of vector bundles. Then we have long exact sequences 0 → S k A → S k−1 A ⊗ B → · · · → S k− p A ⊗ ∧ p B → · · · → ∧k B → ∧k V → 0 and 0 → ∧k A → ∧k−1 A ⊗ B → · · · → ∧k− p A ⊗ S p B → · · · → S k B → S k V → 0).

Yukawa Couplings in Heterotic Compactification

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For an alternative presentation of the portion of these results that hold in any characteristic, see [39, Sect. 2]. For the case where V = 0 see the exposition in [36, p. 151, Ex. 1]. Our goal now is to relate ρ E to ρ S k E . We have a commutative diagram (H 1 (X, V ))⊗k ∼ =

/ H k (X, S k V ) ∼ =

 ⊗k (H 1 (X, [C ← O B][1]))

 / H k (X, S k ([C ← B][1])) O

(H 1 (X, C[1]))⊗k

/ H k (X, S k (C[1])),

where the horizontal maps are obtained by iterating the cup product pairings. Let E k = S k if k is even, and E k = ∧k if k is odd. Shifting the bottom row of the diagram above yields the following diagram. The vertical maps arise from identity maps, and the diagram commutes up to sign of (−1)((k−1)+(k−2)+···+1) = (−1)k(k−1)/2 , using our earlier computation: ⊗k (H 1 (X, C[1])) O ∼ =

/ H k (X, S k (C[1])) O ∼ =

(H 0 (X, C))⊗k

/ H 0 (X, E k C).

Splicing the two diagrams together and retaining only the top and bottom rows yields the following diagram, which commutes up to a sign of (−1)k(k−1)/2 : (H 1 (X,O V ))⊗k (ρ E )⊗k

(H 0 (X, C))⊗k

/ H k (X, S k V ) O ρSk E

/ H 0 (X, E k C).

The vertical maps can also be expressed in terms of connecting homomorphisms, as we have seen before. References 1. Greene, B.R., Kirklin, K.H., Miron, P.J., Ross, G.G.: 27**3 Yukawa Couplings For A Three Generation Superstring Model. Phys. Lett. B 192, 111 (1987) 2. Candelas, P.: Yukawa Couplings Between (2,1) Forms. Nucl. Phys. B 298, 458 (1988) 3. Candelas, P., Kalara, S.: Yukawa couplings for a three generation superstring compactification. Nucl. Phys. B 298, 357 (1988) 4. McOrist, J., Melnikov, I.V.: Summing the Instantons in Half-Twisted Linear Sigma Models. JHEP 0902, 026 (2009) 5. Donagi, R., Reinbacher, R., Yau, S.T.: Yukawa couplings on quintic threefolds. http://arxiv.org/abs/hepth/0605203v1, 2006 6. Donagi, R., He, Y.H., Ovrut, B.A., Reinbacher, R.: The particle spectrum of heterotic compactifications. JHEP 0412, 054 (2004) 7. Berglund, P., Parkes, L., Hubsch, T.: The Complete Matter Sector In A Three Generation Compactification. Commun. Math. Phys. 148, 57 (1992)

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8. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology. Cambridge: Cambridge Univ. Pr., 1987 9. Gabella, M., He, Y.H., Lukas, A.: An Abundance of Heterotic Vacua. JHEP 0812, 027 (2008) 10. Anderson, L.B., He, Y.H., Lukas, A.: Heterotic compactification, an algorithmic approach. JHEP 0707, 049 (2007) 11. Candelas, P., Dale, A.M., Lutken, C.A., Schimmrigk, R.: Complete Intersection Calabi-Yau Manifolds. Nucl. Phys. B 298, 493 (1988) 12. Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces. Basel: Birkhäuser Verlag, 1988 13. Anderson, L.B., He, Y.H., Lukas, A.: Monad Bundles in Heterotic String Compactifications. JHEP 0807, 104 (2008) 14. Anderson, L.B.: Heterotic and M-theory Compactifications for String Phenomenology. Oxford University DPhil Thesis, 2008, http://arxiv.org/abs/0808.3621v1[hep-th], 2008 15. Anderson, L.B., He, Y.H., Lukas, A.: Vector bundle stability in heterotic monad models. In preparation 16. Donaldson, S.K.: Some numerical results in complex differential geometry. http://arxiv.org/abs/math/ 0512625v1[math.DG], 2005. Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic. JHEP 0712, 083 (2007); Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical Calabi-Yau metrics. J. Math. Phys. 49, 032302 (2008). Braun, V., Brelidze, T., Douglas, M.R., Ovrut, B.A.: Calabi-Yau Metrics for Quotients and Complete Intersections. JHEP 0805, 080 (2008) 17. Blumenhagen, R., Moster, S., Weigand, T.: Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds. Nucl. Phys. B 751, 186 (2006) 18. Blumenhagen, R., Honecker, G., Weigand, T.: Loop-corrected compactifications of the heterotic string with line bundles. JHEP 0506, 020 (2005) 19. Distler, J., Greene, B.R.: Aspects of (2,0) String Compactifications. Nucl. Phys. B 304, 1 (1988) 20. Lukas, A., Ovrut, B.A., Waldram, D.: On the four-dimensional effective action of strongly coupled heterotic string theory. Nucl. Phys. B 532, 43 (1998) 21. Lukas, A., Ovrut, B.A., Stelle, K.S., Waldram, D.: The universe as a domain wall. Phys. Rev. D 59, 086001 (1999) 22. Hubsch, T.: Calabi-Yau manifolds: A Bestiary for physicists. Singapore: World Scientific, 1992 23. Hartshorne, R.: Algebraic Geometry, Springer. GTM 52, Springer-Verlag, 1977; Griffith, P., Harris, J., Principles of algebraic geometry. New York: Wiley-Interscience, 1978 24. Grayson, D., Stillman, M.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/ 25. Greuel, G.-M., Pfister, G., Schönemann, H.: Singular: a computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern (2001). Available at http://www. singular.uni-kl.de/ 26. Gray, J., He, Y.H., Ilderton, A., Lukas, A.: “STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology.” Comput. Phys. Commun. 180, 107–119 (2009); arXiv:0801.1508 [hep-th]. Gray, J., He, Y.H., Ilderton, A., Lukas, A.: “A new method for finding vacua in string phenomenology,” JHEP 0707 (2007) 023; Gray, J., He, Y.H., Lukas, A.: “Algorithmic algebraic geometry and flux vacua.” JHEP 0609 (2006) 031; The Stringvacua Mathematica package is available at: http://www-thphys.physics.ox.ac.uk/projects/Stringvacua/ 27. Braun, V., He, Y.H., Ovrut, B.A., Pantev, T.: “A heterotic standard model.” Phys. Lett. B 618, 252 (2005); “The exact MSSM spectrum from string theory.” JHEP 0605, 043 (2006) 28. Donagi, R., He, Y.H., Ovrut, B.A., Reinbacher, R.: Moduli dependent spectra of heterotic compactifications. Phys. Lett. B 598, 279 (2004) 29. Bouchard, V., Donagi, R.: An SU(5) heterotic standard model. Phys. Lett. B 633, 783 (2006) 30. Buchberger, B.: “An Algorithm for Finding the Bases Elements of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal” (German), Phd thesis, Univ. of Innsbruck (Austria), 1965; B. Buchberger, “An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations” (German), Aequationes Mathematicae 4(3), 374–383,1970; English translation can be found in: Buchberger, B., Winkler, F., eds.: “Gröbner Bases and Applications.” Volume 251 of the L.M.S. series, Cambridge: Cambridge University Press, 1998; Proc. of the International Conference “33 Years of Gröbner bases”; See B. Buchberger, “Gröbner Bases: A Short Introduction for Systems Theorists.” p1-19 Lecture Notes in Computer Science, Computer Aided Systems Theory - EUROCAST 2001, Berlin-Heidelberg: Springer, 2001, pp. 1–19 31. Gray, J.: A Simple Introduction to Grobner Basis Methods in String Phenomenology. http://arxiv.org/abs/ 0901.1662v1[hep-th], 2009 32. Anderson, L.B., Gray, J., Lukas, A., Ovrut, B.: The Edge Of Supersymmetry: Stability Walls in Heterotic Theory. Phys. Lett B 677, 190–194 (2009)

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33. Anderson, L.B., Gray, J., Lukas, A., Ovrut, B.: Stability Walls in Heterotic Theories. JHEP 0909, 026 (2009) 34. Avramov, L.L., Grayson, D.R.: Resolutions and cohomology over complete intersections, In: Computations in algebraic geometry with Macaulay 2, Algorithms Comput. Math., Vol. 8, Berlin: Springer, 2002, pp. 131–178 35. Boardman, J.M.: The principle of signs. Enseignement Math. (2) 12, 191–194 (1966) 36. Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitre 10. Algèbre homologique, Berlin: SpringerVerlag, 2007, (Reprint of the 1980 original [Paris: Masson]) 37. Cartan, H., Eilenberg, S.: Homological algebra. Princeton, N. J.: Princeton University Press, 1956 38. Godement, R.: Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Paris: Hermann, 1964 39. Grayson, D.R.: Adams operations on higher K -theory. K -Theory 6(2), 97–111 (1992) 40. Swan, R.G.: Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions. J. Pure Appl. Algebra 144(2), 169–211 (1999) 41. Weibel, C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge: Cambridge University Press, 1994 Communicated by N.A. Nekrasov

Commun. Math. Phys. 297, 129–168 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1030-y

Communications in

Mathematical Physics

Multifractal Analysis of Complex Random Cascades Julien Barral1 , Xiong Jin2 1 LAGA (UMR 7539), Département de Mathématiques, Institut Galilée, Université Paris 13, 99 avenue

Jean-Baptiste Clément, 93430 Villetaneuse, France. E-mail: [email protected]

2 INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. E-mail: [email protected]

Received: 5 June 2009 / Accepted: 3 December 2009 Published online: 19 March 2010 – © Springer-Verlag 2010

Abstract: We achieve the multifractal analysis of a class of complex valued statistically self-similar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new phenomena in multifractal analysis of continuous functions. In particular, we find examples of statistically self-similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval [0, ∞].

1. Introduction This paper deals with the multifractal formalism for functions and the multifractal analysis of a new class of statistically self-similar functions introduced in [7]. This class is the natural extension to continuous functions of the random measures introduced in [39] and considered as a fundamental example of multifractal signals model since the notion of multifractality has been explicitly formulated [21–23] (see also [5,16,24,35,45] for the multifractal analysis and thermodynamical interpretation of these measures). While the measures contructed in [39] provide a model for the energy dissipation in a turbulent fluid, the functions we consider may be used to model the temporal fluctuations of the speed measured at a given point of the fluid. Also, they provide an alternative to models of multifractal signals which use multifractal measures, either to make a multifractal time change in Fractional Brownian motions [4,42], or to build wavelet series [2,9]. We exhibit statistically self-similar continuous functions possessing the remarkable property to obey the multifractal formalism, and simultaneously to be nowhere locally Hölder continuous. Specifically, the support of their multifractal spectra does contain the exponent 0, and the set of points at which the pointwise Hölder exponent is 0 is dense in the support of the function. Moreover, these spectra can also be left-sided with singularity spectra supported by the whole interval [0, ∞] (see Fig. 3). These properties are new phenomena in multifractal analysis of continuous self-similar functions. Let us

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explain this in detail, by starting with some recalls and remarks on multifractal analysis of functions. Multifractal analysis is a natural framework to describe geometrically the heterogeneity in the distribution at small scales of the Hölder singularities of a given locally bounded function or signal f : U ⊂ Rn → R p (n, p ≥ 1). In this paper, we will work in dimension 1 with continuous functions f : I → R (or C), where I is a compact interval. The most natural notion of Hölder singularity is the pointwise Hölder exponent, which finely describes the best polynomial approximation of f at any point t0 ∈ I and is defined by h f (t0 ) = sup{h ≥ 0 : ∃P ∈ C[t], | f (t) − f (t0 ) − P(t − t0 )| = O(|t − t0 |h ), t → t0 }. Then, the multifractal analysis of f consists in computing the Hausdorff dimension of the Hölder singularities level sets, also called iso-Hölder sets E f (h) = {t ∈ I : h f (t) = h}, h ≥ 0. The mapping h ≥ 0 → dim H E f (h) is called the singularity spectrum of f (dim H stands for the Hausdorff dimension, whose definition is recalled at the end of this section); the support of this spectrum is the set of those h such that E f (h) = ∅. The function is called multifractal when at least two iso-Hölder sets are non-empty. Otherwise, it is called monofractal. When the function f is globally Hölder continuous, it has been proved in [25,31] that the exponent h f (t) can always be obtained through the asymptotic behavior of the wavelet coefficients of f located in a neighborhood of t, when the wavelet is smooth enough. Then, wavelet expansions have been used successfully to characterize the isoHölder sets of wide classes of functions [3,9,11,17,26,27,29,30], sometimes directly constructed as wavelet series (expansions in Schauder’s basis have also been used [32]). For most of these functions, the singularity spectrum can be obtained as the Legendre transform of a free energy function computed on the wavelet coefficients. This is the so-called multifractal formalism for functions, studied and developed rigorously in [27,28,30,31] after being introduced by physicists [23,21,22,46]. It is worth noting that for those functions mentioned above which satisfy the multifractal formalism, most of the time (see [27,32,29,30,9]) the wavelet expansion reveals that it is possible to closely relate the wavelet coefficients to the distribution of some positive Borel measure μ (sometimes discrete, as it can be shown for the saturation functions in Besov spaces [30]) satisfying the multifractal formalism for measures [13,47,48], for which the pointwise Hölder exponent is usually defined by h μ (t) = lim inf + r →0

log(μ(B(t, r )) . log(r )

(1)

In practice, it may happen to be difficult to extract a good enough characterization of the sets E f (h) from the function f expansion in wavelet series. This leads to seeking for other methods of h f (t) estimation, or exponents that are close to h f (t) and easier to estimate. The most natural alternative is the first order oscillation exponent of f defined as h (1) f (t) = lim inf + r →0

log(supt,s∈B(t,r ) | f (s) − f (t)|) log(r )

.

Multifractal Analysis of Complex Random Cascades

131

(1)

If not an integer, h f (t) is equal to h f (t). When the function f can be written as g ◦ θ , where g is a monofractal function of (single) Hölder exponent γ and μ = θ (the derivative of θ in the distributions sense) is a positive Borel measure satisfying the multifractal formalism for measures, we have a convenient way to obtain the singularity (1) spectrum of f associated with the exponent h f from that of μ (one exploits the equality

(1) h (1) f (t) = γ h θ (t) = γ h μ (t) at good points t). Such a representation f = g ◦ θ has been shown to exist for certain classes of deterministic multifractal functions mentioned above [33,43,53]. It turns out that for the functions considered in this paper, in general wavelet basis expansions are not enough tractable to yield accurate information on the iso-Hölder sets. Also, this class of functions is versatile enough to contain elements which can be naturally represented under the same form g ◦ θ as above, as well as elements for which such a natural decomposition does not exist. For these functions, inspired by the work achieved in [28,31], we are going to compute the singularity spectrum by using the m th order oscillation pointwise exponents (m ≥ 1) and consider associated multifractal formalisms. To our best knowledge, this approach has not been used to treat a non-trivial example before. We denote by ( f (m) )m≥1 the sequence of f derivatives in the distribution sense. If J is a non trivial compact subinterval of I , for m ≥ 1, let (m)

Osc f (J ) =

sup

[t,t+mh]⊂J

|m h f (t)|,

m−1 where 1h f (t) = f (t + h) − f (t) and for m ≥ 2, m f (t + h) − m−1 f (t) h h f (t) = h (1) (notice that Osc f (J ) = sups,t∈J | f (s) − f (t)|). Then, the pointwise oscillation exponent of order m ≥ 1 of f at t ∈ Supp( f (m) ) is defined as (m) h f (t)

(m)

= lim inf + r →0

log Osc f (B(t, r )) log r

.

We only consider points in Supp( f (m) ), because from the pointwise regularity point of view, Supp( f (m) ) is the only set over which we can learn non-trivial information thanks (m) to h f . Indeed, outside this closed set, the function f is locally equal to a polynomial of degree at most m − 1, so f is C ∞ . The pointwise Hölder exponent h f carries non-trivial information at points at which  f is not locally equal to a polynomial, that is points in m≥1 Supp( f (m) ).  (m) If t ∈ m≥1 Supp( f (m) ), it is clear that the sequence (h f (t))m≥1 is non decreasing. (m)

In fact, supm≥1 h f (t) = h f (t). This is a consequence of Whitney’s theorem on local approximation of functions by polynomial functions [54,52] (the result is in fact proved for bounded functions): For every m ≥ 1, there exists a constant Cm (independent of f ) such that for any subinterval J of I , there exists a polynomial function P of degree at most m − 1 such that (m)

| f (x) − P(x)| ≤ Cm Osc f (J ). This, together with the definition of h f yields the following statement, which is also established in [31] by using the wavelet expansion when f is uniformly Hölder.

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Proposition 1. If f : I → C is continuous, then for t ∈



m≥1 Supp( f

(m)

(m) ),

(m)

h f (t)

converges to h f (t). Moreover, if h f (t) < ∞, then h f (t) = h f (t) for all m > h f (t). Now, the multifractal analysis of f consists in computing singularity spectra like (m)

h ≥ 0 → dim H E f (h),

(2)

where for h ≥ 0 and m ∈ N+ ,

  (m) (m) E (m) (h) = t ∈ Supp( f ) : h (t) = h , f f

and for h ≥ 0, (∞)

Ef

(h) =

⎧ ⎨ ⎩

t∈



(∞)

Supp( f (n) ) : h f

(t) = h

n≥1

⎫ ⎬ ⎭

,

(∞) where h f (t) = h f (t) .

Proposition 1 yields (∞)

Ef

(m)

(h) = E f (h) (∀h ≥ 0, ∀ m > h).

Inspired by the multifractal formalisms for measures on the line [50,13,47,48,36] as well as multifractal formalism for functions in [28,31], it is natural to consider for each m ≥ 1 the L q -spectrum of f associated with the oscillations of order m, namely   q log sup Osc(m) (B ) i i f (m) , τ f (q) = lim inf r →0 log(r ) where the supremum is taken over all the families of disjoint closed intervals Bi of radius r with centers in Supp( f (m) ). For all h ≥ 0 and m ≥ 1, we have (Proposition 2) (m)

(m)

(m)

dim H E f (h) ≤ (τ f )∗ (h) = inf hq − τ f (q), q∈R

and due to Proposition 1, (∞)

dim H E f

(∞) ∗

(h) ≤ (τ f

(m)

) (h) := inf (τ f )∗ (h), m>h

(3)

(m)

a negative dimension meaning that E f (h) is empty. We will say that the multifrac-

tal formalism holds for f and m ∈ N+ ∪ {∞} at h ≥ 0 if E (m) f (h) is not empty and (m)

(m)

dim H E f (h) = (τ f )∗ (h).

(m)

When m = ∞, the exponent h f is naturally stable by addition of a C ∞ function, and so is the validity of the associated multifractal formalism. This is not the case when m < ∞ (see Corollary 1 for an illustration). As we said, our approach for the multifractal formalism is inspired by the “oscillation method" introduced in [28,31] for uniformly Hölder functions. There, quantities like τ (m) are computed by using balls centered at points of finer and finer regular grids, f (m)

and only for q ≥ 0. So our definition of τ f is more intrinsic, though equivalent. The choice q ≥ 0 in [31] corresponds to the introduction of some functions spaces related

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(m)

with the functions τ f that provide a natural link between wavelets and oscillations approach to the multifractal formalism when q ≥ 0 and f is uniformly Hölder. It is worth noting that thanks to this link, for any q ≥ 0, if we define n q as the smallest (n) integer n such that nq − 1 ≥ τ (n) f (q), then for all n ≥ n q , the function τ f coincides on the interval [q, ∞] with the scaling function τ W f associated with the so-called wavelet leaders in [28,31]. This implies that for h ≥ 0 such that the multifractal formalism holds (n) (∞) (∞) at h for m = ∞, even though E f (h) = E f (h) for all n ≥ [h] + 1, dim H E f (h) (n)

+ may be equal to (τ f )∗ (h) only for n  [h] + 1 as h tends to (τ W f ) (0 ). We now introduce the functions whose multifractal analysis will be achieved in this paper. = {0, . . . , b − 1}n (by We fix an integer b ≥ 2. For every n ≥ 0 we define A n  0 ∗ convention A contains the empty word denoted ∅), A = n≥0 A n , and A N+ = {0, . . . , b − 1}N+ . If n ≥ 1, and w = w1 · · · wn ∈ A n , then for every 1 ≤ k ≤ n, the word w1 . . . wk is denoted w|k , and if k = 0 then w|0 stands for ∅. Also, if t ∈ A N+ and n ≥ 1, t|n denotes the word t1 · · · tn and t|0 the empty word. We denote by π the natural projection of A N+ onto [0, 1]: If t ∈ A N+ , π(t) = ∞ −k k=1 tk b . When t ∈ [0, 1] is not a b-adic point, we identify it with the element of A N+ which represents its b-adic expansion, namely the element of π −1 ({t}). We consider a sequence of independent copies (W (w))w∈A ∗ of a random vector

W = (W0 , . . . , Wb−1 ) b−1 whose components are complex, integrable, and satisfy E( i=0 Wi ) = 1. Then, we define the sequence of functions  FW,n (t) =

t

b 0

n

n 

Wu k (u|k−1 ) du.

(4)

k=1

For q ∈ R let  b−1   q ϕW (q) = − logb E 1{Wi =0} |Wi | .

(5)

i=0

b−1 The assumption E( i=0 Wi ) = 1 implies that ϕW (1) ≤ 0 with equality if and only if W ≥ 0, i.e., the components of W are non-negative almost surely. In this case only, all the functions FW,n are non-decreasing almost surely. The following results are established in [7]. b−1 Theorem A [7] (Non-conservative case). Suppose that P( i=0 Wi = 1) > 0 and there exists p > 1 such that ϕW ( p) > 0. Suppose, moreover, either p ∈ (1, 2] or ϕW (2) > 0. 1. (Fn )n≥1 converges uniformly, almost surely and in the L p norm, as n tends to ∞, to a function F = FW , which is non-decreasing if W ≥ 0. Moreover, the function F is γ -Hölder continuous for all γ in (0, maxq∈(1, p] ϕW (q)/q).

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2. F satisfies the statistical scaling invariance property: F=

b−1 

1[i/b,(i+1)/b] · F(i/b) + Wi Fi ◦ Si−1 ,

(6)

i=0

where Si (t) = (t + i)/b, the random objects W , F0 , . . . , Fb−1 are independent, and the Fi are distributed like F and the equality holds almost surely. b−1 Theorem B. [7](Conservative case). Suppose that P( i=0 Wi = 1) = 1. 1. If there exists p > 1 such that ϕW ( p) > 0, then the same conclusions as in Theorem A hold. 2. (Critical case) Suppose that lim p→∞ ϕW ( p) = 0 (in particular ϕW is increasing and ϕW ( p) < 0 for all p > 1). This is equivalent to the fact that P(∀ 0 ≤ i ≤ b−1 b − 1, |Wi | ≤ 1) = 1 and i=0 P(|Wi | = 1) = 1. Suppose also that P(#{i : |Wi | = 1} = 1) < 1, and there exists γ ∈ (0, 1) such that, with probability 1, one of the two following properties holds for each 0 ≤ i ≤ b − 1 :  either |Wi | ≤ γ ,

i−1 i . (7) or |Wi | = 1 and k=0 Wi ∈ {(0, 1), (1, 0)} k=0 Wi , Then, with probability 1, (Fn )n≥1 converges almost surely uniformly to a limit F = FW which is nowhere locally uniformly Hölder and satisfies part 2 of Theorem A. When the components of W are non-negative (resp. positive), the function FW is is the measure considered in non-decreasing (resp. increasing) and the measure FW [39,35]. In the rest of the paper, we will work with the natural and more general model of function constructed as follows. Instead of considering only one multiplicative cascade, we consider a couple (W, L) of random vectors taking values in Cb × R∗+ b . We assume that both W and L satisfy the same property as W in the previous paragraph: b−1 b−1 E( i=0 Wi ) = 1 = E( i=0 L i ). We consider a sequence of independent copies (W (w), L(w))w∈A ∗ of (W, L), and we also assume that both W and L satisfy the assumptions of Theorem A or B. This yields almost surely two continuous, functions FW and FL , the former being increasing. The function we consider over [0, FL (1)] is F = FW ◦ FL−1 . When FW is non-decreasing, the measure F has been considered in [5], and also in b−1 L i = 1 almost surely. [1] under the assumption that i=0 b−1 Wi = If the components of W and L are deterministic real numbers and i=0 b−1 1 = i=0 L i , we recover the self-affine functions constructed in [10]. The multifractal analysis of these functions has been achieved in [27] by using their wavelet expansion (however, the endpoints of the spectrum are not investigated). It is also possible to use the alternative approach consisting in showing that FW can be represented as a monofractal (1) function in multifractal time [43,53], and then consider the exponent h F rather than h F . It turns out that such a time change also exists in the random case under restrictive assumptions on W , which include the deterministic case (see [7]). This is useful because,

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as we said, our calculations showed that in general in the random case it seems difficult to exploit the wavelet transform of F to compute its singularity spectrum. However, this approach could not cover all the cases since for the functions built in Theorem B(2), there is no natural time change (see [7]). Also, these functions are nowhere locally uniformly Hölder and do not belong to any critical Besov space (specifically, their singularity spectra have an infinite slope at 0), so that there is little expectation to characterize their pointwise Hölder exponents through their wavelet transforms. Using the m th order oscillation pointwise exponents provides an efficient alternative tool. We obtain the following results (for simplicity, we postpone to Sect. 2.6 the discussion of an extension under weaker assumptions). We discard the obvious case where W = L, for which F = Id[0,FL (1)] almost surely. Also, we assume that ϕ L > −∞ over R and 0 < L i < 1 almost surely. The first result concerns functions F with bell-shaped singularity spectra. We find that for some of these functions, the left endpoint of the support of their singularity spectra is equal to 0, and the slope of these spectra at 0 is ∞. This is a new phenomenon in the multifractal analysis of statistically self-similar continuous functions. b−1 Theorem 1 (Bell shaped spectra). Suppose that P( i=0 1{Wi =0} ≥ 2) = 1 and ϕW > −∞ over R. For q ∈ R, let τ (q) be the unique solution of the equation b−1  E( 1{Wi =0} |Wi |q L i−t ) = 1. i=0

The function τ is concave and analytic. With probability 1, 1. Supp(F (m) ) = Supp(F ) for all m ∈ N+ and dim H Supp(F ) = −τ (0). (m) ∗ (1) ∗ 2. For all h ≥ 0 and m ∈ N+ ∪ {∞}, dim H E (m) F (h) = (τ F ) (h) = (τ F ) (h), (m) (m) a negative dimension meaning that E F (h) is empty. Moreover, E F (h) = ∅ if (1) (τ F )∗ (h) = 0. In other words, for all m ∈ N+ ∪ {∞}, F obeys the multifractal formalism at every h ≥ 0 such that (τ F(m) )∗ (h) ≥ 0. In addition, if FW is built as in Theorem B(2) (critical case), the left endpoint of these singularity spectra is the exponent 0, and the corresponding level set is dense, with Hausdorff dimension 0. (m) 3. For all m ∈ N+ , τ F = τ on the interval J = {q ∈ R : τ (q)q − τ (q) ≥ 0}, and if q = sup(J ) < ∞ (resp. q := inf(J ) > −∞) then τ F(m) (q) = τ (q)q (resp. τ (q)q) over [q, ∞) (resp. (−∞, q]). Moreover, if there does not exist H ∈ (0, 1) such that for all 0 ≤ i ≤ b − 1 we have |Wi | ∈ {0, L iH } then τ is strictly concave over J ; otherwise, τ (q) = q H + τ (0) and F is monofractal with a Hölder exponent equal to H . Notice that −τ (0) < 1 if and only if at least one component of W vanishes with positive probability, and in this case the support of F is a Cantor set. In the next result, we get functions F obeying the multifractal formalism and for which the singularity spectra are left-sided, i.e., increasing, and with a support equal to the whole interval [0, ∞]. This is another new phenomenon in multifractal analysis of continuous staitistically self-similar functions.

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Fig. 1. Bell shaped spectrum in the case where the left endpoint is not 0

Fig. 2. Bell shaped spectrum in the critical case where the left endpoint is 0

b−1 Theorem 2 (Left-sided spectra). Suppose that P( i=0 1{Wi =0} ≥ 2) = 1 and ϕW (q) > −∞ over R+ . For q ∈ R+ , let τ (q) be defined as in Theorem 1. The function τ is concave, b−1 1{Wi =0} L i log(|Wi |)) = −∞, i.e. and analytic over (0, ∞). Suppose also that E( i=0  τ (0) = ∞. Finally, suppose that E (max0≤i≤b−1 |Wi |)−ε < ∞ for some ε > 0. Then, the same conclusions as in Theorem 1 hold. Moreover, the singularity spectra are left-sided, and h (m) F = ∞ for all m ∈ N+ ∪ {∞} on a set of full dimension in Supp(F ). In addition, if FW is built as in Theorem B(2) (critical case), the support of the spectra is [0, ∞]. Remark 1. Examples of left sided spectra do exist for some other (increasing) continuous functions over [0, 1] possessing self-similarity properties [40,51,41], but their spectra do not contain the left endpoint 0. It is also worth mentioning that in some Besov spaces of continuous functions, the generic singularity spectrum is left sided, supported by a compact interval, and linear; moreover, the left-end point of this spectrum is equal to 0 for critical Besov spaces [30,34]. In the critical case considered in this paper (Theorem B(2)), the slope of the singularity spectra at 0 is equal to ∞ because of the duality between h = τ (q) and q = (τ ∗ ) (h), and h → 0 corresponds to q → ∞.

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Fig. 3. Concave left-sided spectra with support [0, ∞] in the critical case

Remark 2. In the non-decreasing case (the components of W are nonnegative), results on the multifractal analysis of the measure μ = F have been obtained in several papers (which also deal with measures on Rd ). For the one dimensional case we are dealing with, the previous statements are substantial improvements of these results for the following reasons. At first, all these works only consider the first order oscillation exponent, which is sometimes computed only on the “distorded" grid associated with the increments of FL as described above [24,45,5], and not in the more intrinsic way (1). Moreover, in the papers which deal with the intrinsic exponent h μ , the assumptions on W and L are very strong: Their components must be bounded away from 0 and 1 by positive constants, and their sum must be equal to 1 almost surely [1,19]; moreover the result holds only for all h ≥ 0 such that τ ∗ (h) > 0 almost surely, and not almost surely for all h ≥ 0 such that τ ∗ (h) > 0. Also, the case of left sided spectra is not treated in these papers. Another important improvement concerns the computation of the endpoints of the singularity spectrum, which is a delicate issue; indeed it is already non-trivial to prove that the corresponding iso-Hölder sets are not empty. Our result includes the description of these endpoints, i.e. the endpoints of τ F∗ −1 (R+ ), without restriction on the behavior of τ . This is a progress with respect to the work achieved in [5] where the case when q = ∞ (resp. q = −∞ and limq→∞ (resp. limq→−∞ )τ (q)q − τ (q) = 0 was not worked out (in the present paper this is particularly important in the critical case of Theorem B(2)), and where the Hölder exponents are computed only on the grid naturally defined by FL . Also, the new method we introduce to study the endpoints could be used to deal with the same question for the general class of random measures considered in [8]. Remark 3. In the previous results, all the formalisms yield the same information. In particular our discussion on the link between the oscillations and wavelets methods developed in [31] shows that when F is uniformly Hölder, the multifractal formalism using wavelets also holds for such a function in the increasing part of the spectrum, without it be necessary to compute any wavelet transform.

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(

(1) * ) (h) G

* (h) * ( ’(q ), ( ’(q ))) 1

1

(1,1)

(1)

(1)

Fig. 4. Top: τG (q) = min{q − 1, τ (q)} for q ≥ 0. Bottom: (τG )∗ (h) = τ ∗ (h) for h ∈ [0, τ (q1 )], (1) (1) (τG )∗ (h) = τ ∗ (τ (q1 )) + q1 (h − τ (q1 )) for h ∈ (τ (q1 ), 1] and (τG )∗ (h) = 1 elsewhere

The next result illustrates the unstability of the exponents and spectra associated with the m th order oscillations by addition of a C ∞ function. Corollary 1. Let f be a complex valued C ∞ function over R+ such that for all m ∈ N+ the function f (m) does not vanish. Let F be as in Theorem 2 and let G = F + f . The functions F and G have the same multifractal behavior from the pointwise Hölder exponent point of view. For m ∈ N+ , let qm be the unique real number such that τ (qm ) = qm m − 1. (m) (m) With probability 1, for all m ∈ N+ , we have τG = τ F = τ over [qm , ∞), and (m) τG (q) = qm −1 for 0 ≤ q < qm . Moreover, for all m ∈ N+ , the multifractal formalism (m) ∗

holds at every h ∈ [0, τ (qm )] such that (τG ) (h) ≥ 0 as well as at h = m, and for all (m) (m) h ∈ (τ (qm ), m) we have dim H E G (h) = τ ∗ (h) < (τG )∗ (h). We end this section with additional definitions.

Definitions. The coding space. The word obtained by concatenation of u ∈ A ∗ and v ∈ A ∗ ∪ A N+ is denoted u · v and sometimes uv. For every w ∈ A ∗ , the cylinder with root w, i.e. {w · t : t ∈ A N+ } is denoted [w]. The σ -algebra generated in A N+ by the cylinders, namely σ ([w] : w ∈ A ∗ ) is denoted S . The set A N+ is endowed with the standard metric distance d(t, s) = inf{b−n : n ≥ 0, ∃ w ∈ A n , t, s ∈ [w]}. Then the Borel σ -algebra is equal to S . For every n ≥ 0, the length of an element of A n is by definition equal to n and we denote it |w|. For w ∈ A ∗ , we define Iw = [tw , tw +b−|w| ) and IwL = FL (Iw ). We denote by w − or w −1 (resp. w + or w +1 ) the unique element of A |w| such that tw − tw− = b−|w| (resp. tw+ − tw = b−|w| ) whenever tw = 0 (resp. tw = 1 − b−|w| ). We also denote w by w0 .

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Independent copies of FW and FL , and associated quantities. If w ∈ A ∗ , n ≥ 1 and [w] U ∈ {W, L}, we denote by FU,n the function constructed as FU,n , but with the weights [∅] = FU,n , and (U (w · v))v∈A ∗ . By construction, FU,n [w] FU,n (t)

 =

t

bn

0

n 

Uu k (w · u||w|+k−1 ) du.

k=1

[w] )n≥1 . We also define We denote by FU[w] the almost sure uniform limit of (FU,n

Q U (w) =

n 

Uwk (w|k−1 ).

k=1 (m)

(m)

(m) ([0, 1]) FU[w]

For m ≥ 1 we denote Osc FU ([0, 1]) by Z U and more generally Osc (m) Z U (w).

Also, we denote

(m) Osc FU (Iw )

by

(m) OU (w).

By construction, we have

(m) (m) (m) L Osc(m) F (Iw ) = Osc FW (Iw ) = OW (w) = |Q W (w)|Z W (w),

|IwL |

=

(1) Osc FL (Iw )

For (q, t) ∈ R2 let  b−1  q −t 1{Wi =0} |Wi | L i

(q, t) = E

=

(1) O L (w)

by

=

(1) Q L (w)Z L (w).

(8) (9)

  (q, t) = E Osc FW ([0, 1])q FL (1)−t .

and

i=0

(10) Hausdorff dimension. If (X, d) is a locally compact metric space, for D ∈ R, δ > 0, and E ⊂ X , let  HδD (E) = inf{ |Ui | D }, i∈I

 where the infimum is taken over the set of all the at most countable coverings i∈I Ui of E such that 0 ≤ |Ui | ≤ δ, where |Ui | stands for the diameter of Ui and by convention 0 D = 0. Then define H D (E) = lim HδD (E). δ0

(HδD (E) is by construction a non-increasing function of δ.) If D ≥ 0, H D (E) is called the D-dimensional Hausdorff measure of E. The Hausdorff dimension of E is the number dim H E = inf{D : H D (E) < ∞}. It is clear that we have dim H E < 0 if and only if dim H E = −∞ and E is the emptyset (see [20,44] for more details). We denote by ( , B, P) the probability space on which the random variables considered in this paper are defined. Finally, if f is a bounded C-valued function over an interval I , then  f ∞ stands for supt∈I | f (t)|.

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2. Proofs of Theorem 1, Theorem 2, and Corollary 1 The next three sections provide intermediate results yielding Theorem 1. Detailed proofs of these results are given in Sect. 3. The proof of Theorem 2 is almost the same as that of Theorem 1 and we outline it in Sect. 2.4. Corollary 1 is given in Sect. 2.5, and Sect. 2.6 provides weaker assumptions under which these results still hold, or partially hold. In the next three sections we work under the assumptions of Theorem 1. 2.1. Upper bound for the singularity spectra. Let f be a measurable bounded function from [0, 1] to R. Proposition 2. Let m ≥ 1. If Supp( f (m) ) = ∅ then for every h ≥ 0 we have (m) (m) dim H E f (h) ≤ (τ f )∗ (h), a negative dimension meaning that E f (h) is empty. Also, (m)

dim H Supp( f (m) ) ≤ dim B Supp( f (m) ) = −τ f (0), where dim B stands for the upper box dimension (see [20] for the definition). (1)

Remark 4. When f is non-decreasing and m = 1, the L q -spectrum τ f is nothing but the L q -spectrum of the measure f , and the inequality provided by Proposition 2 is familiar from the multifractal formalism for measures. Though the proof of the inequality is similar for m ≥ 2, for the reader’s convenience we will give a proof of Proposition 2 in Sect. A (see also [31] for similar bounds). We first need the following propositions. Proposition 3. With probability 1, Supp(F ) = ∅, and the function F is nowhere locally equal to a polynomial over the support of F . Consequently, Supp(F (m) ) = Supp(F ) for all m ≥ 1. Now for n ≥ 1, and (q, t) ∈ R2 define

 (m) (m) (m) (m) θ F,n (q, t) = Osc FW (Iw )q |IwL |−t and  θ F,n (q, t) = E θ F,n (q, t) , w∈A n

with the convention 0q = 0. Then define (m) (m) θ F,n (q, t) and  θ F(m) (q, t) = lim sup  (q, t), θ F(m) (q, t) = lim sup θ F,n n→∞

n→∞

as well as (m) (m) τ F,b (q) = sup{t ∈ R : θ F(m) (q, t) = 0} and  τ F,b (q) = sup{t ∈ R :  θ F(m) (q, t) = 0}.

Proposition 4. Let m ≥ 1. With probability 1, for all q ∈ R+ we have τ F(m) (q) ≥ (1) (1) (m) (m) (m) τ F,b (q) ≥  τ F,b (q), and for all q ∈ R∗− we have τ F (q) ≥ τ F,b (q) ≥  τ F,b (q).

(m) Moreover,  τ F,b (q) = τ (q) for all q <  q , where  q = max{ p : τ ( p) > 0} (by convention max(∅) = ∞).

Proof of the upper bound for the singularity spectra. Let m ≥ 1. Recall that J = {q ∈ R : τ (q)q −τ (q) ≥ 0}. Since τ is concave, we have J ⊂ (−∞,  q ]. Consequently, since (τ F(m) )∗ is concave, due to Proposition 4, with probability 1, for all h ≥ 0 we may have (m) (m) (m) (τ F )∗ (h) ≥ 0 only if τ ∗ (h) ≥ 0. In this case, we have dim H E F (h) ≤ (τ F )∗ (h) ≤ ∗ τ (h) by Proposition 2. Also, since 0 belongs to J , we have dim H Supp(F ) ≤ −τ (0).

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2.2. Lower bound for the singularity spectra. Let I = {τ (q) : q ∈ J }. We are going to distinguish the case h ∈ Int(I ) and the case h ∈ ∂ I . For q ∈ J and w ∈ A ∗ we set

Wq (w) = 1{Wi (w) =0} |Wi (w)|q (w)L i (w)−τ (q)

0≤i≤b−1

.

(11)

2.2.1. The case h ∈ Int(I ). At first we introduce some auxiliary measures. If q ∈ Int(J ), w ∈ A ∗ , n ≥ 1 and v ∈ A n let Q q[w] (v) =

n 

Wq,vk (w · v||w|+k−1 ),

(12)

k=1

and simply denote Q q[∅] (v) by Q q (v) = 1{Q W (v) =0} |Q W (v)|q Q L (v)−τ (q) . Then let Yq,n (w) =



Q q[w] (v).

v∈A n

Proposition 5. 1. With probability 1, for all q ∈ Int(J ) and w ∈ A ∗ , the sequence Yq,n (w) converge to a positive limit Yq (w). Moreover, for every n ≥ 1, σ ({Q U (w) : w ∈ A n−1 , U ∈ {W, L}}) and σ ({Yq (w) : w ∈ A n }) are independent, and the random variables Yq (w), w ∈ A n , are independent copies of Yq (∅), that we denote by Yq . 2. For every compact subinterval K of Int(J ), there exists p K > 1 such that p

E(supq∈K Yq K ) < ∞. 3. With probability 1, for all q ∈ Int(J ), the function μq ([w]) = Q q (w)Yq (w), w ∈ A ∗

(13)

defines a Borel measure on A N+ . Recall the definitions given at the end of Sect. 1. For m ≥ 1, t ∈ A N+ , U ∈ {W, L} and γ ∈ {−1, 0, +1} let (m),γ

αU

(m),γ

(t) (resp. αU

(m)

(t)) = lim inf (resp. lim sup) − n→∞

n→∞

logb Osc FU ((t|n )γ ) n

(recall that if w ∈ A ∗ , w − = w −1 , w = w 0 and w + = w +1 are defined in Sect. 1). The next proposition follows directly from the definition of the m th oscillation.

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Proposition 6. Let t ∈ A N+ and  t = FL (π(t)). 1. Let r ∈ (0, 1) and suppose that ∃ nr , nr ∈ N, It|Lnr ⊂ B( t, r ) ⊂ It|L− ∪ It|L ∪ It|L+ . nr

(14)

nr

nr

Then



(m) (m) (1) t, r )) ≤ 2m−1 O F (B( t, r )) ≤ 2m−1 OW (t|nr ) ≤ O F (B(

(1)

OW (w).

w∈{t|− ,t|n ,t|+ } nr

r

nr

(15) 2. Suppose that (14) holds for all r > 0 small enough and limr →0+ nr /nr = 1. Then, (1),γ

min{α W

(t) : γ = −1, 0, +1} (1),0 α L (t)

≤

(m) h F ( t)

(m),0

≤

αW

(1),γ min{α L (t)

(t)

: γ = −1, 0, +1}

. (16)

Recall that for (q, t) ∈ R2 we have defined b−1  

(q, t) = E 1{Wi =0} |Wi |q L i−t i=0

and τ (q) is the unique solution of (q, τ (q)) = 1. By construction, we have 

b−1 q L −τ (q) log(|W |) E 1 |W | {W = 0} i i i i=0 i (∂ /∂q)(q, τ (q)) 

. (17) = τ (q) = − −τ (q) b−1 (∂ /∂t)(q, τ (q)) E 1 |W |q L log(L ) i=0

{Wi =0}

i

i

i

Proposition 7. With probability 1, for all q ∈ Int(J ), for μq -almost every t ∈ Supp(μq ), ∂

log |Q W (t|n )| =− (q, τ (q)); −n ∂q ∂

log |Q W ((t|n )γ )| lim ∈ {− (q, τ (q)), +∞}, for γ ∈ {−1, 1}; n→∞ −n ∂q ∂

log Q L (t|n ) log Q L ((t|n )γ ) 2. lim = lim = (q, τ (q)), for γ ∈ {−1, 1}; n→∞ n→∞ −n −n ∂t (m) (m) logb Z U (t|n ) log Z U ((t|n )γ ) = lim = 0, for all m ≥ 1, U ∈ {W, L} and 3. lim n→∞ n→∞ n n γ ∈ {−1, 1}. log Yq (t|n ) ≥ 0. 4. lim inf n→∞ −n

1. lim

n→∞

Proof of the lower bound. Due to (13) and Proposition 7 (1), (2) and (4), with probability 1, for all q ∈ Int(J ), we have lim inf n→∞

log(μq ([t|n ])) ∂

∂

≥ −q (q, τ (q)) − τ (q) (q, τ (q)) −n ∂q ∂t  ∂

 (q, τ (q)) > 0, μq -a.e. = qτ (q) − τ (q) · ∂t

Multifractal Analysis of Complex Random Cascades

143

( ∂

∂t (q, τ (q)) > 0 due to our choice L i ∈ (0, 1)). Consequently, μq is atomless, and defining νq = μq ◦ π −1 ◦ FL−1 , we have νq (IwL ) = μq ([w]) for all w ∈ A ∗ . Thus, lim inf n→∞

log νq (InL (t)) ∂

≥ (qτ (q) − τ (q)) · (q, τ (q)), νq -almost everywhere, −n ∂t

where InL (t) is the unique interval IwL of generation n containing t. Now, Proposition 7 (2) and (3) as well as (9) also yield ∂

log |InL (t)| = (q, τ (q)) > 0, νq -almost everywhere, n→∞ −n ∂t lim

hence lim inf n→∞

log νq (InL (t)) ≥ qτ (q) − τ (q), νq -almost everywhere. log |InL (t)|

Consequently, we can apply the mass distribution principle ([49], Lemma 4.3.2) and we obtain dim H (νq ) ≥ qτ (q) − τ (q) = τ ∗ (τ (q)). We can also deduce from Proposition 7 that for μq -almost every t, for all m ≥ 1,  (1),γ (m) min{α W (t) : γ = −1, 0, +1} = α W (t) = − ∂

∂q (q, τ (q))/ log(b), (1),γ

min{α L

(t) : γ = −1, 0, +1} = α (1) L (t) =

∂

∂t (q, τ (q))/ log(b).

These properties imply that at νq -amost every  t, for r ∈ (0, 1) small enough, we can find integers nr and nr such that (15) holds with limr →0+ nr /nr = 1, and we have (m) (∞) for all m ≥ 1 h F (t) = τ (q). Due to Proposition 1, we also have h F (t) = τ (q). ∗ Since dim H (νq ) ≥ τ (τ (q)) we have the desired lower bound for the dimensions (m) of the sets E F (τ (q)), m ∈ N ∪ {∞}. The case q = 0 yields dim H Supp(F ) ≥ (1) dim H E F (τ (0)) ≥ −τ (0). Combining this with Proposition 4 we obtain that, with probability 1, for all m ∈ N+ , (m) (m) we have (τ (m) )∗F = τ ∗ over Int(I ). Since we also have τ F ≥ τ F,b ≥ τ over J , this (m)

yields τ F

(m)

= τ F,b = τ over J .

2.2.2. The case h ∈ ∂ I . Recall that q = inf J and q = sup J . Let h = lim τ (q), h¯ = lim τ (q), d¯ = lim τ (q)q − τ (q), d = lim τ (q)q − τ (q). q→q

q→q

q→q

q→q

¯ d¯ = τ ∗ (h) and d = τ ∗ (h). ¯ Moreover, with probability 1, d¯ = Then ∂ I = {h, h}, (m) ∗ (m) ∗ ∗ ∗ ¯ = (τ ) (h) ¯ for any m ≥ 1. τ (h) = (τ F ) (h) and d = τ (h) F (m) ¯ comes from the fact that there The difficulty in the study of E F (h) when h ∈ {h, h} (m) is no simple choice of a measure carried by E F (h) and whose Hausdorff dimension is larger than or equal to (τ F(m) )∗ (h). Even, it is not obvious to construct a point belonging (m) to E F (h). Neverthless such a measure can be constructed. (m) A measure μq partly carried by E F (h), for (q, h) ∈ {(q, h), (q, h)}.

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1. The case q ∈ {−∞, ∞}. Recall (11). We have τ ∗ (τ (q)) = ϕW (1) = −E( q

b−1 

Wq,i logb Wq,i ) = 0.

i=0

Moreover, ϕWq ( p) > −∞ in a neighborhood of 1+ . Consequently, it follows from Theorem 2.5 of [37] that, with probability 1, for all w ∈ A ∗ , the martingale  Q q[w] (v) log Q q[w] (v) Yq,n (w) = − v∈A n

converges to a limit Yq (w) (Yq (∅) = Yq ) as n → ∞, where the products Q q[w] (v) are defined in (12). Moreover, by construction, the branching property Yq (w) = b−1 ∗ i=0 Wq,i (w)Yq (wi) holds, the random variables Yq (w), w ∈ A , are identically γ distributed, and for γ > 0 we have E(Yq ) < ∞ if and only if γ < 1. We deduce from the branching property and our assumption on the probability that the components event {Yq = 0} is measurable with respect to the of W vanish that the  tail σ -algebra N ≥1 σ (W (w) : w ∈ n≥N A n ). Consequently, P(Yq > 0) = 1 since E(Yq ) > 0, and with probability 1, the branching property makes it possible to define on A N+ a measure μq by the formula μq ([w]) = Q q (w)Yq (w).

(18)

¯ h} and q ∈ {−∞, ∞} such that h = τ (q). With probability Proposition 8. Let h ∈ {h, 1, there exists a Borel set E h ⊂ A N+ of positive μq -measure such that for all t ∈ E h the same conclusions as in Proposition 7 (1) (2) (3) hold. (m) Then, the same arguments as in Sect. 2.2.1 yield νq (E (m) F (h)) > 0, hence E F (h) is not (m) empty and we get the desired lower bound dim H E F (h) ≥ 0 since τ ∗ (h) = 0. 2. The case q ∈ {−∞, ∞}. Let (qk )k≥0 be an increasing (resp. decreasing) sequence converging to q if q = ∞ (resp. q = −∞). For every k ≥ 0 and w ∈ A k , recall that by (11),

Wqk (w) = 1{Wi (w) =0} · |Wi (w)|qk L i (w)−τ (qk ) . 0≤i≤b−1

Then, for w ∈ A

∗,

n ≥ 1 and v ∈ A Q q[w] (v) =

n 

n,

instead of (12) we define

Wq|w|+k−1 ,vk (w · v||w|+k−1 ),

k=1

and simply denote Q q[∅] (v) by Q q (v). Then let  Yq,n (w) = Q q[w] (v), v∈A n

and simply denote Yq,n (∅) by Yq,n . The sequence (Yq,n (w))n≥1 is a non-negative martingale of expectation 1 which converges almost surely to a limit that we denote by Yq (w) (Yq if w = ∅). Since the set A ∗ is countable, all these random variables are defined  simultaneously. Moreover, the branching property Yq (w) = b−1 k=0 Q q,i (w)Yq (wi) also holds. Notice that by construction, given k ≥ 1, the random variables Yq (w), w ∈ A k , are independent and identically distributed.

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Proposition 9. The sequence (qk )k≥0 can be chosen so that there exists a > 0 such that for all w ∈ A ∗ the sequence (Yq,n (w))n≥1 converges in L 2 norm to a limit Yq and Yq (w)2 = O ba|w|/ log(|w|) . Fix a sequence (qk )k≥0 as in the previous proposition. For the same reason as in the case q ∈ {−∞, ∞}, we have P(Yq > 0) = 1 and with probability 1, the branching property makes it possible to define on A N+ a measure μq by the formula (18). ¯ h} and q ∈ {−∞, ∞} such that h = lim J q →q τ (q). Let Proposition 10. Let h ∈ {h, νq = μq ◦π −1 ◦ FL−1 . With probability 1, for every m ≥ 1, we have h (m) F (t) = h νq -almost everywhere and dim H (νq ) ≥ τ ∗ (h). Remark 5. In the case q ∈ {−∞, ∞}, it is possible to construct μq as in the case q ∈ {−∞, ∞} by using a sequence (J  qk )k≥0 converging to q. This avoids requiring Theorem 2.5 of [37] which is a strong result. Nevertheless, we are able to use this alternative only if ϕW (q) > −∞ for some q < −1. This is the case under the assumptions of Theorem 1, but this does not always hold under the weaker assumptions provided by Sect. 2.6. 2.3. The L q -spectra of F. We have seen at the end of Sect. 2.2.1 that, with probability (m) (m) 1, for all m ∈ N, τ F (q) = τ F,b (q) = τ (q) over J = [q, q]. It remains to show that (m)

τ F is differentiable at q (resp. q) and linear over [q, ∞) (resp. (−∞, q]) if q < ∞ (resp. q > −∞). We treat the case q < ∞ and leave the case q > −∞ to the reader.

(m) (m) (m) At first we notice that the equality τ F = τ F,b = τ over J implies that (τ F ) (q − ) = (m) (m) (m) (m) τ F,b (q)/q = τ (q)/q = h. Also, by concavity of τ F , we have τ F (q) ≤ τ F (q) + (τ F(m) ) (q − )(q − q) = τ (q) + τ (q)(q − q) = hq. To get the other inequality, and so (m) the differentiability of τ F at q, we use a simple idea inspired by the work achieved in (1) [45] which focuses on τ F,b in the case when the components of W are non-negative and

L = (1/b, . . . , 1/b). If q ≥ q and t ∈ R, we have ⎡ ⎤q/q   (m) (m) Osc FW (Iw )q |IwL |−t ≤ ⎣ Osc FW (Iw )q |IwL |−qt/q ⎦ , w∈A n

w∈A n

(m)

(m)

because q/q ≥ 1. Consequently, by definition we have τ F,b (q) ≥ (q/q)·τ F,b (q) = qh. τ F(m) (q)

(m) τ F,b (q)

This, together with Proposition 4, yields ≥ ≥ hq for q ≥ q. It remains to discuss the strict concavity of τ over J . Suppose τ is affine over a non-trivial sub-interval J of J . The analyticity of τ implies that it is affine over J (in fact over R under our assumptions), which is equivalent to saying that for all q, q ∈ J and λ ∈ [0, 1] we have   (19)

λq + (1 − λ)q , λτ (q) + (1 − λ)τ (q ) = 0, where is defined in (10). Let λ ∈ (0, 1) and q = q ∈ J . Applying the Hölder inequalb−1 −(1−λ)τ (q ) −λτ (q) 1{Wi =0} |Wi |λq L i |Wi |(1−λ)q L i shows that, in order to have ity to i=0 (19) it is necessary and sufficient that there exists C such that −τ (q)

1{Wi =0} |Wi |q L i



−τ (q )

= C1{Wi =0} |Wi |q L i

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almost surely. Thus, there exists H > 0, the slope of τ , such that |Wi | = L iH for all i, conditionally on Wi = 0. If the components of W are non-negative almost surely, by b−1 H L i ) = 1, hence H = 1 and W = L, the situation we construction this implies E( i=0 b−1 H have discarded. Otherwise, we have E( i=0 L i ) > 1, hence H ∈ (0, 1). 2.4. Proof of Theorem 2. We only have to deal with the exponent h = τ (0) = ∞. The rest of the study is similar to that achieved in the previous sections.   (w) = 1{Wi (w) =0} L i (w)−τ (0) For w ∈ A ∗ let W . By construction the com0≤i≤b−1  are non negative, we have ϕW ponents of W  (1) = 0, and ϕW  (1) > 0. Consequently, the

Mandelbrot measure on A N+ defined as μ0 = FW  (with the notations of Theorems A and B) is positive with probability 1. Moreover, it follows from the study achieved in [5] that dim H ν0 = −τ (0), where ν0 = μ0 ◦ π −1 ◦ FL−1 . Now, for a ∈ (0, 1), we define W (a) = (|Wi |∧a)

0≤i≤b−1 . We have h = lima→0 h(a), b−1 (a) where h(a) = −E i=0 1{Wi =0} L i log(Wi ) . By using the same techniques as in Sect. 3 we can prove that, with probability 1, for μ0 -almost every t, we have

log |Q W (a) ((t|n )γ )| = h(a), γ ∈ {−1, 0, 1}. n→∞ −n lim

(1) log Z W ((t|n )γ ) = n (1),γ 0, for all γ ∈ {−1, 0, 1}. Consequently, for μ0 -almost every t, min(α W (t) : γ ∈ {−1, 0, 1}) ≥ h(a). Since this holds for every a ∈ (0, 1), letting a tend to 0 yields (1),γ min(α W (t) : γ ∈ {−1, 0, 1}) = ∞ for μ0 -almost every t. Since there exists a > 0 (1),0 such that α L (t) ≤ a for all t (see Lemma 1) we conclude thanks to Proposition 6 that (1) for ν0 -almost every t we have h F (t) = ∞.

Also, due to our assumptions and Proposition 14, we have limn→∞

2.5. Proof of Corollary 1. Fix 1 ≤ m ∈ N. Recall that qm is the unique real number such that τ (qm ) = qm m − 1. (m) Let C > 0 such that Osc f (B) ≤ C|B|m for all subintervals B of [0, FL (1)]. For r > 0 let Br be a family of disjoint closed intervals B of [0, FL (1)] of radius r with centers in Supp(F (m) ). For any q ∈ R+ we have 

q −t Osc(m) ≤ 2q F+ f (B) · r

B∈Br

 (m) q q Osc(m) r −t F (B) + Osc f (B) B∈Br

⎛

≤ (2C)q · ⎝



q −t Osc(m) + F (B) r

B∈Br (m)

(m)



⎞ r qm−t ⎠ .

B∈Br (m)

(m)

By the definition of τG (q) this yields τG (q) ≥ min(τ F (q), qm −1) so (τG )∗ (h) ≤ (m) (m) τ ∗ (h) for h ∈ [0, τ (qm )] (we have used the equality τ F = τ ) and (τG )∗ (h) = 1 for h > m.

Multifractal Analysis of Complex Random Cascades

147

On the other hand, since we assumed that f (m) does not vanish, we deduce from (m) (m) (m) Theorem 2 that for any t ∈ [0, FL (1)] we have h G (t) = h F (t) if h F (t) < m and (m) (m) h G (t) = m if h F (t) > m. Thus ! ∗

τ (h), if h ∈ [0, m); (m) ∗ (m) τG (h) ≥ dim H E G (h) = 1, if h = m. (m)

This implies that (τG )∗ is equal to τ ∗ over [0, τ (qm )] and equal to h → τ ∗ (τ (qm )) + qm (h − τ (qm )) over [τ (qm ), m]. Taking the inverse Legendre transform implies that τG(m) (q) = min(τ F(m) (q), qm − 1) for all q ≥ 0. 2.6. Weaker assumptions. Theorem 1. If we only assume that ϕW > −∞ in a neighborhood J of [0, 1], then the multifractal formalisms holds for F at each h = τ (q) for all q ∈ J ∩ J . Also, the (m) functions τ F and τ coincide over J ∩ J . If, moreover, there exists q0 ∈ J such that τ ∗ (τ (q0 )) = 0, then either q0 > 0 and τ F(m) (q) = τ (q0 )q/q0 over [q0 , ∞) or q0 < 0 (m) and τ F (q) = τ (q0 )q/q0 over (−∞, q0 ]. Theorem 2. The same discussion as for Theorem 1 holds, except that J is a neighbor of [0, 1] in R+ . 3. Proofs of the Intermediate Results of Section 2 3.1. Proofs of the results of Section 2.1. Proof of Proposition 2. This is a consequence of Proposition 12. Proof of Proposition 3. The result could be obtained after achieving the multifractal analysis using the first order oscillation exponent. Nevertheless we find it valuable to have a proof only based on the the functional equation satisfied by the process F. b−1 We assumed that P( i=0 1{Wi =0} ≥ 2) = 1. Consequently, it follows from the def(1) that the event {Z inition of F W W = 0} is measurable with respect to the tail σ -algebra  p n≥0 σ ({W (w) : w ∈ ∪ p≥n A }) which contains only sets of probability 0 or 1. Since (1)

E(FW (1)) = 1, we have Z W > 0 with positive probability, hence almost surely. So Supp(F ) = ∅ almost surely. Now we prove that F is nowhere locally equal to a polynomial function over the support Supp(F ). At first, suppose that there exists 0 ≤ i ≤ b − 1 such that P(Wi = 0) > 0. Then, with probability 1, the interior of Supp(F ) is empty, since for every w ∈ A ∗ the probability that there exists v ∈ A ∗ such that Wi (w · v) = 0 is equal to 1. Thus F is nowhere locally equal to a polynomial function over Supp(F ). Now suppose that the components of W do not vanish and that there is a positive probability that there exists an interval IwL over which F is equal to a polynomial. [w] ◦ (FL[w] )−1 is a polynomial function. Due to the statisEquivalently, F [w] = FW tical self-similarity of the construction, the probability that F be itself a polynomial function is positive. Moreover, F is almost surely the uniform limit of the sequence

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(Fn = FW,n ◦ FL−1 ,n )n≥1 . The functions Fn are piecewise linear, and because we assumed W = L and the vectors (W (w), L(w)), w ∈ A ∗ , are independent, with probability 1, for every w ∈ A ∗ , there are infinitely many n such that the restriction of Fn to IwL is not linear, thus non differentiable. Consequently, the event {F is a polynomial} is mea surable with respect to the tail σ -algebra n≥0 σ ({W (w), L(w) : w ∈ ∪ p≥n A p }), so it has a probability equal to 1. For

0 ≤ i ≤ b − 2, let xi = FL (i/b). By construction, [i] − + ) = (W [i] [i+1] ) (0). ) = F (xi+1 we have (Wi /L i )(F ) FL (1) = F (xi+1 i+1 /L i+1 )(F Due to the independence between (W, L), F [i] and F [i+1] , we see that

all the terms in the [i] [i] FL (1) = (F [i+1] ) (0) = 0 previous equality must be deterministic, except if F

almost surely. In this later case, by statistical self-similarity we also have F (FL (1)) ≡ F (0) ≡ 0, and by induction over n ≥ 0 we see that F vanishes at all the endpoints of the intervals IwL , w ∈ A n . Thus F ≡ 0 and F is constant. This is in contradiction with F(0) = 0 and E (F(FL (1))) = E (FW (1)) = 1. Consequently, (W, L) must be deterb−1 b−1 Wi = 1 = i=0 Li ministic. Since we supposed that W = L, the assumption i=0 implies that |Wi | > L i for some 0 ≤ i ≤ b − 1. Let us write |Wi | = L iH with H < 1 (recall that L i < 1). Then, denoting by i ·n the word consisting in n letters i, we have Osc(1) (F, IiL·n ) ≥ |Wi |n = |IiL·n | H so F is not C 1 . This is a new contradiction, hence F is nowhere locally equal to a polynomial function. (m)

Proof of Proposition 4. We first establish the inequalities τ F (m)

(m)

(1)

≥ τ F,b over R+ and

τ F ≥ τ F,b over R∗− . By applying Theorem 2.3 in [7] to L we immediately have the following lemma:

Lemma 1. There exist a, a > 0 such that, with probability 1, there exists n 0 ∈ N such that for n ≥ n 0 , b−na ≤ inf w∈A n |IwL | ≤ supw∈A n |IwL | ≤ b−na . Moreover, with probability 1, for every ε > 0, there exists n ε such that L | |Iwi ≤ 1. L| w∈A n 0≤i≤b−1 |Iw

∀ n ≥ n ε , b−nε ≤ inf

inf

(20)

For P-almost every ω ∈ , we fix ε > 0, n 0 and n ε as in Lemma 1. Let n ε = max(n 0 , n ε ). Fix 0 < r ≤ minw∈A N+ |IwL |. n ε +1

Let Br be a family of disjoint closed intervals B of radius r with centers in Supp(F ). If B ∈ Br , by construction we can find three disjoint intervals IwLk , k = 1, 2, 3, with |wk | ≥ n ε + 1 such that B ⊂ IwL1 ∪ IwL2 ∪ IwL3 and r ≤ |IwLk | ≤ r b|wk |ε . Also, |IwLk | ≤ b−|wk |a so b|wk |ε ≤ r −ε/a . Thus r ≤ |IwLk | ≤ r 1−ε/a . 3 (1) L m−1 Osc(1) (B) ≤ 2m−1 We have Osc(m) k=1 Osc F (Iwk ), so for q ≥ 0 and F (B) ≤ 2 F t ∈ R we have (m)

Osc F (B)q r −t f (t, r ) ≤ 2(m−1)q 3q ·

3 

(1)

Osc F (IwLk )q |IwLk |−t ,

k=1

otherwise. Moreover, each such selected with f (t, r ) = 1 if t < 0 and f (t, r ) = interval IwLk meets at most 1 + r −ε/a elements of Br . Consequently,   (1) (m) Osc F (B)q r −t f (t, r ) ≤ 2(m−1)q 3q (1 + r −ε/a ) θ F,n (q, t). (21) r tε/a

B∈Br

n≥n ε +1

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149

(1)

Suppose that τ F,b (q) > −∞; otherwise there is nothing to prove. Due to the existence  (1) (1) (1) of a, by definition of τ F,b (q), if t < τ F,b (q) then we have n≥n ε +1 θ F,n (q, t) < ∞. (m)

(m)

Then, it follows from (21) and the definition of τ F (q) that τ F (q) ≥ t − (1 + |t|)ε/a. (m) (1) Since ε is arbitrary, we get τ F (q) ≥ τ F,b (q). On the other hand, for each B ∈ Br there exists IwL of maximal length included in B. We have 2r b−|w|ε ≤ |IwL | ≤ 2r . This yields 2r ≤ b|w|(ε−a) so (2r )ε/(a−ε) ≤ b−|w|ε whenever a > ε. Consequently, for ε small enough, we have (2r )1+ε/(a−ε) ≤ |IwL | ≤ 2r . Thus, if q < 0 we have (m)

(m)

Osc F (B)q (2r )−t f (t, r ) ≤ Osc F (IwL )q |IwL |−t , where f (t, r ) = 1 if t ≥ 0 and f (t, r ) = (2r )−tε/(a−ε) otherwise. Since the elements of Br are pairwise disjoint, this implies   (m) (m) Osc F (B)q (2r )−t f (t, r ) ≤ θ F,n (q, t), (22) n≥n ε +1

B∈Br (m)

(m)

and the same arguments as when q ≥ 0 yield τ F (q) ≥ τ F,b (q). (m)

(m)

(m)

(m)

To see that, with probability 1, τ F,b ≥  τ F,b , due to the concavity of τ F,b and  τ F,b , it (m)

(m)

is enough to show that given q ∈ R, we have τ F,b (q) ≥  τ F,b (q). 2 Let (q, t) ∈ R , and suppose that q < max{ p : ϕW ( p) > 0}. Due to Proposi(m) tion 14 we have ψ(q, t) < ∞. By using (8) we get  θ F,n (q, t) = (q, t)n (q, t) (m)

for all n ≥ 1. This yields  τ (q) = τ (q). Also, if t < τ (q) then (q, t) < 1 and   F,b (m) (m) (m)  n≥1 θ F,n (q, t) < ∞ so n≥1 θ F,n (q, t) < ∞ almost surely. This yields t < τ F,b (q). (m)

(m)

Since t is arbitrary we get τ F,b (q) ≥  τ F,b (q). To finish the proof, we notice that by construction, we have τ ( p) = 0 if and only if ϕW ( p) = 0. 3.2. Proofs of the results of Section 2.2.1. Proof of Proposition 5. This proof could be deduced from those of Lemma 4 and Corollary 5 of [5]. For reader’s convenience, we provide it. • Proof of (1) and (2). Recall in (11) that for q ∈ Int(J ) and w ∈ A ∗ ,

Wq (w) = 1{Wi (w) =0} |Wi (w)|q L i (w)−τ (q) . 0≤i≤b−1

The function can be extended to an analytic function in a complex neighborhood of J × C by b−1   z −t

(z, t) = E 1{Wi =0} |Wi | L i . i=0



(q)) b−1 For each q ∈ Int(J ) we have ∂ (q,τ = −E W log(L ) > 0 and (q, q,i i i=0 ∂t τ (q)) = 1, so there exists a neighborhood Vq of q in C such that for each z ∈ Vq there

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exists a unique τ (z) such that (z, τ (z)) = 1. Moreover, the mapping z → τ (z) is analytic. We define

Wz (w) = 1{Wi (w) =0} |Wi (w)|z L i (w)−τ (z) 0≤i≤b−1

as well as the mapping (z, p) ∈ Vq × [1, ∞) → M(z, p) =

b−1 

E(|Wz,i | p ).

i=0

∂M (q, 1+ ) < 0 , so there exists pq > ∂p 1 and an open neighborhood Vq of q in J such that supq ∈Vq M(q , p) < 1 for all p ∈ (1, pq ] (because p → M(q , p) is convex and M(q , 1) = 1). Now, we fix K a non-trivial compactsubinterval of Int(J ). It is covered by a finite number of such Vq i so that if VK = i Vq i we have supq∈V M(q, p K ) < 1, where p K = inf i pqi . K By a comparable procedure we can now find a complex neighborhood VK of VK such supz∈VK M(z, p K ) < 1. To prove the almost sure simultaneous convergence of the martingales (Yq,n (w))n≥1 , q ∈ K , we are going to use the argument developed to get Theorem 2 in [12]. For z ∈ VK and w ∈ A ∗ let The property τ ∗ (τ (q)) > 0 is equivalent to

Yz,n (w) =

n  

Wz,vk (w · v||w|+k−1 ),

v∈A n k=1

and denote Yz,n (∅) by Yz,n . Applying Proposition 13 to {V (w) = Wz (w)}w∈A ∗ yields for n ≥ 1, E(|Yz,n − Yz,n−1 | p K ) ≤ C p K M(z, p K )n ≤ C p K ( sup M(z, p K ))n , z∈VK

where Yz.0 = 1. Since, with probability 1, the functions z ∈ V → Yz,n , n ≥ 0, are analytic, if we fix a closed disc D(z"0 , 2ρ) included in V , the Cauchy formula yields supz∈D(z 0 ,ρ) |Yz,n − Yz,n−1 | ≤ ρ −1 ∂ D(z 0 ,2ρ) |Yu,n − Yu,n−1 | du/2π , so by using Jensen’s inequality and then Fubini’s Theorem we get    2π dt pK E sup |Yz,n − Yz,n−1 | E(|Yz 0 +2ρeit ,n − Yz 0 +2ρeit ,n−1 | p K ) ≤ 2 pK 2π 0 z∈D(z 0 ,ρ) ≤ 2 p K C p K (sup (z, p K ))n . z∈V

This implies that, with probability 1, z → Yz,n converges uniformly over the compact D(z 0 , ρ) to a limit Yz . This also implies that  supz∈D(z 0 ,ρ) Yz  p K < ∞. Since K can be covered by finitely many such discs, we get both the simultaneous convergence of (Yq,n )n≥1 to Yq for all q ∈ K and (2). Moreover, since Int(J ) can be covered by a countable increasing union of compact subintervals, we get the simultaneous convergence for all q ∈ Int(J ). The same holds simultaneously for all the functions q ∈ Int(J ) → Yq,n (w), w ∈ A ∗ , because A ∗ is countable.

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To finish the proof of (1) we need to establish that, with probability 1, q ∈ K → Yq does not vanish. Up to an affine transform, we can suppose that K = [0, 1]. If I is a closed dyadic subinterval of [0, 1], we denote by E I the event {∃ q ∈ I : Yq = 0}, and by I0 and I1 its two sons. At first, we note that since for each fixed q ∈ K a component of Wq vanishes if and only the same component of W vanishes too, each E I is a tail event. Consequently, if I is a closed dyadic subinterval of [0, 1] and P(E I ) = 1, then P(E I j ) = 1 for some j ∈ {0, 1}. Suppose that P(E [0,1] ) = 1. The previous remark yields a decreasing sequence (I (n))n≥0 of nested closed dyadic intervals such that P(E I (n) ) = 1. Let q0 be the unique element of I (n). Since q → Yq is continuous, we have P(Yq0 = 0) = 1. This contradicts the fact that the martingale (Yq0 ,n )n≥1 converges to Yq0 in L p K norm. • Proof of (3). This is a simple consequence of the fact that by construction we have for all n ≥ 1 and w ∈ A ∗ the branching property Yq,n+1 (w) =

b−1 

Wq,i (w) Yq,n (w · i).

i=0

Proof of Proposition 7. (1) We simply denote Q W by Q and we define  b−1  ∂

q −τ (q) 1{Wi =0} |Wi | L i log(|Wi |) . ξ(q) = − (q, τ (q)) = −E ∂q i=0

If ε > 0, n ≥ 1 and γ ∈ {−1, 0, 1} we define 1 E q,n,ε (γ ) = {t ∈ A N+ : Q((t|n )γ ) = 0 and en(ξ(q)−ε) |Q((t|n )γ )| ≥ 1}, −1 (γ ) = {t ∈ A N+ : Q((t|n )γ ) = 0 and en(ξ(q)+ε) |Q((t|n )γ )| ≤ 1}. E q,n,ε

Our goal is to prove that for any compact subinterval K of Int(J ) and ε > 0, ⎞ ⎛  λ μq (E q,n,ε (γ )⎠ < ∞ E ⎝ sup

(23)

q∈K n≥1

for all λ ∈ {−1, 1} and γ ∈ {−1, 0, 1}.  Then, with probability 1, for all q ∈ K , λ λ ∈ {−1, 1} and γ ∈ {−1, 0, 1}, the series n≥1 μq (E q,n,ε (γ )) is finite. Since Int(J ) can be written as a countable union of compact subintervals, this holds in fact for all q ∈ Int(J ). Consequently, from the Borel-Cantelli lemma applied to μq /μq  we deduce that, with probability 1, for all q ∈ Int(J ), for μq -almost every t ∈ A N+ , there exists N ≥ 1 such that for all n ≥ N and γ ∈ {−1, 0, 1}, either Q((t|n )γ ) = 0 or |Q((t|n )γ )| ∈ [e−n(ξ(q)+ε) , e−n(ξ(q)−ε) ]. Notice that when t ∈ Supp(μq ), we have Q((t|n )0 ) = Q(t|n ) = 0. Consequently, with probability 1, for all q ∈ K , for μq -almost every t, log |Q(t|n )| ∈ [ξ(q) − ε, ξ(q) + ε], −n log |Q((t|n )γ )| ∈ {+∞} ∪ [ξ(q) − ε, ξ(q) + ε] for γ ∈ {−1, 1}. lim n→∞ −n lim

n→∞

Since this holds for a sequence of positive ε tending to 0, we have the desired result.

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Now we prove (23). Fix K , a non-trivial compact subinterval of Int(J ). For η ≥ 0, q ∈ K and γ ∈ {−1, 0, 1}, by using a Markov inequality we get

η  1 (γ )) ≤ μq ([w])1{Q(wγ =0} en(ξ(q)−ε) · |Q(w γ )| , μq (E q,n,ε w∈A n −1 μq (E q,n,ε (γ )) ≤





−η μq ([w])1{Q(wγ ) =0} en(ξ(q)+ε) · |Q(w γ )| .

w∈A n

get

Since μq ([w]) = 1{Q ( w) =0} |Q q (w)|Yq (w), for λ ∈ {−1, 1} and γ ∈ {−1, 0, 1} we λ μq (E q,n,ε )≤



en(ληξ(q)−εη) 1{Q(w),Q(wγ ) =0} Q q (w)|Q(w γ )|λη Yq (w).

w∈A n

Now define Hnη,λ (q, γ ) =



en(ληξ(q)−εη) 1{Q(w),Q(wγ ) =0} Q q (w)|Q(w γ )|λη .

(24)

w∈A n

  η,λ Write K = [q0 , q1 ]. It follows from the independence between Hn (q, γ )

q∈K

and {Yq (w)}w∈A n ,q∈K that for n ≥ 1,  E(sup q∈K

λ μq (E q,n,ε (γ ))

≤ E(sup Yq )E sup q∈K

q∈K

 Hnη,λ (q, γ )

#    ≤ E(sup Yq ) E Hnη,λ (q0 , γ ) + q∈K

q1

q0

$% % #$ $ d η,λ $ $ E $ Hn (q, γ )$$ dq . dq

Lemma 2. Let λ ∈ {−1, 1} and γ ∈ {−1, 0, 1}. There exist constants C, δ > 0 and η∗ > 0 such that for any q ∈ K , η ∈ (0, η∗ ), λ ∈ {−1, 1} and n ≥ 1, $%& ! #$ $ d η,λ $  η,λ  $ ≤ Cne−nδ . max E Hn (q, γ ) , E $ Hn (q, γ )$$ dq Then (23) comes from the fact that E(supq∈K Yq ) < ∞ (see Proposition 5(2)). Proof of Lemma 2. Recall that ξ(q) = − ∂

∂q (q, −τ (q)). Since is twice continuously differentiable, we can choose η0 > 0 such that for η ∈ (0, η0 ), δη = inf εη − ληξ(q) − log ( (q + λη, −τ (q))) > 0.

(25)

q∈K

We now distinguish the cases γ = 0 and γ ∈ {−1, 1}. •

η,λ

The case γ = 0. Straightforward computations using the definition of Hn (q, 0) and taking into account the independence in the b-adic cascade construction yield a constant C K such that for all q ∈ K and n ≥ 1,   E Hnη,λ (q, 0) = (q + λη, −τ (q))n en(ληξ(q)−εη) ≤ e−nδη , $% #$ $ d η,λ $ $ E $ Hn (q, 0)$$ ≤ C K n (q + λη, −τ (q))n en(ληξ(q)−εη) ≤ C K ne−nδη . dq

Multifractal Analysis of Complex Random Cascades

•

153

The case γ = −1. For n ≥ 1 we have '

−

(w , w) =

w∈A n

n−1 '

' b−2 '

(u · i · gn−1−m , u · (i + 1) · dn−1−m ),

(26)

m=0 u∈Am i=0

where gn (resp. dn ) is the word consisting of n times the letter b − 1 (resp. 0). If w = u ·(i +1)·dn−1−m and w − = u ·i ·gn−1−m with u ∈ AmN+ and Q(w)Q(w − ) = 0 then Q q (w)|Q(w− )|λη = Q q (u)Q(u)λη Wq,i (u)|Wi+1 (u)|λη

n−1 

Wq,0 (w|k )|Wb−1 (w− |k )|λη .

k=m+1

Again, simple computations yield C K > 0 such that for all q ∈ K , n ≥ 1 and η ∈ (0, η0 ) we have $%% # #$ $ d $ max E(Hnη,λ (q, γ ), E $$ Hnη,λ (q, γ )$$ dq

n ≤ C K n (q + λη, τ (q))eληξ(q)−εη Sn (q, η), where Sn (q, η) =

)m n−1 (  E(Wq,0 )E(|Wb−1 |λη ) m=0

(q + λη, τ (q))

.

Due to (25), it is now enough to show that Sn (q, η) is uniformly bounded with respect to n, q ∈ K and η if η0 is small enough. This is due to the fact that the mapping (q, r ) → E(Wq,0 )E(|Wb−1 |r )/ (q + r, τ (q)) is continuous in a neighborhood of J × {0}, and by definition of Wq and it takes values less than 1 at points of the form (q, 0). • The case γ = 1. It uses the same ideas as the case γ = −1. (2) The proof is similar to the proof of (1). The only difference is that the components log Q L ((t|n )γ ) of L are positive so the limit of cannot be infinite. −n (m) (3) We denote Z U by Z . Fix K a non-trivial compact subinterval of Int(J ), λ ∈ {1, −1} and γ ∈ {−1, 0, 1}. For a > 1 and n ≥ 0 let λ (γ ) = {t ∈ A N+ : (Z (t|n )γ ))λ > a n }. E n,a

It is enough that we show that E(sup



q∈K n≥0

λ μq (E n,a )(γ )) < ∞.

(27)

Indeed, this implies that, with probability 1, for all q ∈ K , for μq -almost every t, if n is large enough then − log a ≤ lim inf n→∞

log Z ((t|n )γ ) log Z ((t|n )γ ) ≤ lim sup ≤ log a. n n n→∞

Since this holds for a sequence of numbers a tending to 1, we have the conclusion.

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We have λ (γ )) = sup sup μq (E n,a

q∈K

q∈K

= sup q∈K



1{Z (wγ )λ >a n } · μq ([w])

w∈A n



w∈A

Q q,n (w) · 1{Z (wγ )λ >a n } · Yq (w). n

By using the independence between σ (Q(w) : w ∈ A n ) and σ (Z (w γ ), Yq (w) : w ∈ A n , q ∈ K ), as well as the equidistribution of the random variables 1{Z (wγ )λ >a n } ·Yq (w), we get       λ E sup μq (E n,a (γ )) ≤ E 1{Z (wγ )λ >a n } · sup Yq (w0 ) E sup Hn0,0 (q, γ ) , 0

q∈K

q∈K

q∈K

γ

where Hn0,0 (q, γ ) is defined as in (24) and w0 is any element of A n such that w0 is defined. We learn from our computations in proving (1) that there exists a positive number C K such that E(supq∈K Hn0,0 (q, γ )) ≤ C(1+|K |)n. Moreover, the Hölder inequality yields   E 1{Z (wγ )λ >a n } · sup Yq (w0 ) ≤  sup Yq  p K P(Z λε > a εn )1−1/ p K 0

q∈K

q∈K

≤  sup Yq  p K [E(Z λε )]1−1/ p K a −nε(1−1/ p K ) , q∈K

λε ) < ∞ (this is possible thanks to Proposition 14). where ε >0 is chosen such that E(Z  λ Finally, E supq∈K μq (E n,a (γ )) = O(na −nε(1−1/ p K ) ) (with a > 1), hence (27) holds. (4) Fix K a non-trivial compact subinterval of Int(J ). For a > 1, n ≥ 0 and q ∈ K let λ Fn,a (q) = {t ∈ A N+ : Yq (t|n ) > a n }.

For η > 0, we have 

sup μq (Fn,a (q)) = sup q∈K

q∈K

1{Yq (w)>a n } · μq ([w])

w∈A n

≤ sup Q q,n (w)|q · a −nη Yq (w)1+η . q∈K

Consequently, taking η = p K − 1 and using the same kind of estimations as in the proof of (3) we obtain     E sup μq (Fn,a (q)) ≤ a −n( p K −1) E sup Hn0,0 (q, γ ) E(sup Yq K ) = O(na −n( p K −1) ), p

q∈K

q∈K

q∈K

hence the result. 3.3. Proofs of results of Section 2.2.2. We only deal with the case h = h, the case h = h¯ being similar. Then q = q > 0.

Multifractal Analysis of Complex Random Cascades

155

3.3.1. The case q < ∞ Proof of Proposition 8. At first, we specify a subset E h of E F (h) of positive μq -measure. For N ≥ 1, let E h (N ) = {t ∈ A N+ : ∀n ≥ N , μq ([t|n ]) ≤ 1}. With probability 1, there exists N ≥ 1 such that μq (E h (N )) > 0, otherwise, μq is concentrated on a finite number of singletons with a positive probability, which is impossible since by construction Supp(μq ) = Supp(F ) and dim Supp(F ) = −ϕ(0) > 0 almost surely. Thus, on a measurable set of probability 1, we can define the measurable function N (ω) = inf{N : μq (E h (N )) > 0}. Then we set E h (ω) = E h (N (ω)). (1) We denote Q W by Q and −∂ /∂q(q, τ (q)) by ξ(q). For ε > 0, γ ∈ {−1, 0, 1} and n ≥ 1 we define 1 E n,ε (γ ) = {t ∈ E h : Q(t|n ) = 0 and en(ξ(q)−ε) |Q((t|n )γ )| ≥ 1}, −1 E n,ε (γ ) = {t ∈ E h : Q(t|n ) = 0 and en(ξ(q)+ε) |Q((t|n )γ )|) ≤ 1}}.

The result will follow if we show that for any ε > 0 and λ ∈ {−1, 0, 1}, with probability 1,  λ μq (E n,ε (γ )) < ∞. (28) n≥1

We deal with the case γ = 0. Let θ and η be two numbers in (0, 1], that will be specified later. By using a Markov inequality and the definition of μq we can get λ (0)) ≤ μq (E n,ε



λη μq ([w])1{Q(w) =0} en(ξ(q)−λε) · Q(w)

 w∈A n μq ([w])≤1

≤





λη λ μq ([w])θ 1{Q(w) =0} en(ξ(q)−λε) · Q(w) = Sn,ε (θ, η),

w∈A n μq ([w])≤1

where λ (θ, η) = Sn,ε



en(ληξ(q)−εη) 1{Q(w) =0} Q q (w)θ Q(w)λη Yqθ .

w∈A n

Consequently, (28) will follow if we show that  λ E(Sn,ε (θ, η)) < ∞. n≥1

We have λ E(Sn,ε (θ, η)) = E(Yqθ )en(ληξ(q)−εη) (θq + λη, θ τ (q))n .

Let  ξ (q) =

∂

∂t (q, τ (q)).

By definition of ξ(q),  ξ (q) and −τ (q) we have

ληξ(q) + log (θq + λη, θ τ (q)) = −ξ(q)q(θ − 1) +  ξ (q)(θ − 1)τ (q) + O([q(θ − 1) + λη]2 )

(29)

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J. Barral, X. Jin

as (θ, η) → (1− , 0). Moreover, we have −ξ(q)q +  ξ (q)τ (q) = − ξ (q)(τ (q)q − τ (q)) = 0. It follows that if we fix η small enough and θ close enough to 1− we have en(ληξ(q)−εη) (θq + λη, θ τ (q))n ≤ e−nεη/2 . Since E(Yqθ ) < ∞, we get (29). In the case γ = −1, we leave the reader check that like in the proof of Proposition 7(1) we can find a constant C > 0 such that for θ, η ∈ (0, 1] we have λ (−1)) μq (E n,ε

≤C

· E(Yqθ )en(ληξ(q)−εη) (θq

+ λη, θ τ (q))

n

* +m n−1 E(W θ )E(|W λη  b−1 | ) q,0 m=0

(θq + λη, θ τ (q))

.

(2) The proof is similar to that of (1). (m) (3) We denote Z U by Z . Let θ, η ∈ (0, 1]. For n ≥ 1, γ ∈ {−1, 0, 1}, λ ∈ {−1, 1} λ (γ ) = {t ∈ E : Z ((t| )γ )λ > enε }. We have and ε > 0 let E n,ε h n  λ  (γ ))} ≤ μq E n,ε



μq ([w])θ e−nεη Z ((t|n )γ )λη .

w∈A n μq ([w])≤1

Thus,    λ (γ ) ≤ e−nεη (θq, θ τ (q))n E(Yq (w)θ Z (w γ )λη ). E μq E n,ε If we choose θ = 1 − η2/3 , then by using Hölder’s inequality we get,

η3/4 /(1+η3/4 ) 3/4 1/4 < ∞, E(Yq (w)θ Z (w)λη ) ≤ Yqθ 1+η3/4 E(Z λ(1+η )η ) since E(Yqθ ) < ∞ for θ ∈ (0, 1) and E(Z λβ ) < ∞ if |β| is small enough (see Proposition 14(2)). Moreover, by definition of τ (q), since  the λL i are smaller than 1, we have (θq, θ τ (q)) < 1. Consequently, n≥1 E μ(E n,ε (γ )) < ∞. We conclude as in the proof of Proposition 7(3). 3.3.2. The case q = ∞ Proof of Proposition 9. First we have the following lemma. n−1  Lemma 3. If

(2qk , 2τ (qk ))1/2 < ∞, then for every w ∈ A ∗ , Yq,n (w) n≥1

k=0

converges to Yq (w) in L 1 norm as n → ∞; in particular E(Yq (w)) = 1. Proof. An application of Proposition 13 to Yq,n − Yq,n−1 and p = 2 yields  n≥1

Yq,n − Yq,n−1 2 ≤ C

  n−1 n≥1 k=0

(2qk , 2τ (qk ))1/2 ,

(30)

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157

where we set Yq,0 = 1 and C is the supremum of the constants C p invoked in Proposition 13. Then, since Yq.n − Yq,n−1 1 ≤ Yq,n − Yq,n−1 2 we have the conclusion for w = ∅. Now, if m ≥ 1 we have  Yq = Q q (w) Yq (w), (31) w∈Am

where the random variables Yq (w), w ∈ A m , are identically distributed, as well as the discrete processes (Yq,n (w))n≥1 converging to them. Consequently, if Yq (w) is not the limit of (Yq,n (w))n≥1 in L 1 for some w ∈ A m , then E(Yq (w)) < 1 and the same holds for all w ∈ A m . In particular, (31) yields E(Yq ) < 1, which is in contradiction with the convergence in L 1 norm of (Yq,n )n≥1 . Now we specify the sequence (qk )k≥0 . We discard the obvious case where τ is affine and assume that τ is strictly concave. The graph of the function τ has the asymptote line l(q) = hq − τ ∗ (h) with τ ∗ (h) ∈ [0, 1). For δ ≥ 0, let lδ (q) = l(q) − δ. We deduce from the strict concavity of τ that for any δ ∈ (0, −τ (0) − τ ∗ (h)] there is a unique q(δ) > 0 such that lδ (q(δ)) = τ (q(δ)). Moreover, δ → q(δ) is continuous and strictly decreasing, and q(δ) → ∞ as δ → 0. Fix k0 such that log1k0 ∈ (0, −τ (0) − τ ∗ (h)), and for k ≥ 0, let δk = 1/ log(k0 + k). Then choose qk = q(δk ) for k ≥ 0. By using the definition of lδ and δ(·) as well as the concavity of τ , we obtain for all k ≥ 0, εk = τ (2qk ) − 2τ (qk ) = τ (2qk ) − 2lδk (qk ) ≥ lδk (2qk ) − 2lδk (qk ) = τ ∗ (h) + δk . (32) We also have for any conjugate pair (α, α ) such that 1/α + 1/α = 1,

(2qk , 2τ (qk )) = E

b−1 

 W2qk ,i L iεk

i=0

 b−1 1/α  b−1 1/α   εα α k ≤ E W2qk ,i Li . E i=0

i=0

Our

that τ as an asymptote at ∞ implies that τ (q)/q is increasing, so assumption b−1 α E i=0 Wq ,i = (α q , α τ (q )) ≤ (α q , τ (α q )) = 1 for all q > 0 and α > 1.

that ϕ L (q ) ∼ aq ¯ + c at ∞ with a¯ > 0, Also, the fact that the L i belong to (0, 1) implies  b−1 εk α ¯ k α/2 . = b−ϕ L (εk α) ≤ b−aε so by choosing α large enough we ensure E i=0 L i ¯ k /2 . Consequently, for all n ≥ 1, we have Thus (2qk , 2τ (qk )) ≤ b−aε n−1 

(2qk , 2τ (qk ))1/2 ≤ b−a¯ Sn /4 ,

k=0

 ∗ ∗ ∗ where Sn = n−1 k=0 εk . We have either τ (h) > 0 and Sn ≥ nτ (h)/2 or τ (h) = 0 and there exists n 0 ≥ 1 such that Sn ≥ n/(4 log(n)) for n ≥ n 0 . In both cases the conclusion of Lemma 3 holds.

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The same arguments as those used in the proof of Lemma 3 show that for w ∈ A ∗ and n ≥ 1 we have (with Y0 (w) = 1) ¯ |w|+n −S|w| )/4 . Yq,n (w) − Yq,n−1 (w)2 ≤ C b−n a(S

(33) ∗

¯ (h)/4 , an If τ ∗ (h) > 0, by using (32) we get Yq,n (w) − Yq,n−1 (w)2 ≤ Cb−n aτ upper bound which does not depend on w, and finally supw∈A ∗ Yq (w)2 < ∞. If τ ∗ (h) = 0, we have Yq,n (w) − Yq,n−1 (w)2 ≤ C ba¯ S|w| /4 · b−a¯ S|w|+n /4 . Thus,

Yq (w)2 ≤ 1 +



Yq,n (w) − Yq,n−1 (w)2

n≥1

≤ 1 + C · ba¯ S|w| /4



b−a¯ S|w|+n /4 ≤ 1 + C · ba¯ S|w| /4 L ,

n≥1



where L = n≥1 b−a¯ Sn /4 is finite because Sn ≥ n/4 log(n) for n large enough. Now, we notice that we also have by concavity of τ, τ (2qk ) − 2τ (qk ) = τ (2qk ) − 2lδk (qk ) ≤ l(2qk ) − 2lδk (qk ) = 2δk . Due to our choice for δk , this implies that for |w| large enough we have S|w| ≤

(34) 2|w| log |w| . a|w| ¯

Finally, there exists C > 0 such that for all w ∈ A ∗ we have Yq (w)2 ≤ C b 2 log |w| . Proof of Proposition 10. We first prove the following proposition ¯ h} and q ∈ {−∞, ∞} such that h = lim J q →q τ (q). Proposition 11. Let h ∈ {h, 1. Let m ≥ 1, γ1 , γ2 ∈ {−1, 0, 1} and δ1 , δ2 ∈ {0, 1}. With probability 1, for μq -almost (m) γ log |Q W (t|n )| · Z W (t|n1 )δ1 every t ∈ Supp(μq ), lim = h. n→∞ log Q (t| ) · Z (1) (t|γ2 )δ2 L n n L 2. For t ∈ A N+ , i ∈ {0, b − 1} and n ≥ 1 we define Nn (t) = max{0 ≤ j ≤ n : ∀ 0 ≤ k ≤ j, tn−k = i}. With probabilility 1, for μq -almost every t we have Nn (t) = o(n). 3. Let ε ∈ (0, 1), i ∈ {0, b − 1} and r ∈ {0, . . . , b − 1}. There exists α(ε) ∈ (0, ε) such that, with probability 1, for μq -almost every t, , for n large enough and n(1 − α(ε)) ≤ p ≤ n − 1 such that Wn, p (t|n ) := Wr (t| p−1 ) nk= p+1 Wi (t|k−1 ) = 0, we have log |Wn, p (t|n )|/ log O L (t|n ) ≥ −ε. Proof. (1) For γ1 , γ2 ∈ {−1, 0, 1}, δ1 , δ2 ∈ {0, 1}, m ≥ 1 and w ∈ A ∗ we simply denote ⎧ (m) γ1 δ1 ⎪ ⎨ OW (w) = Q W (w) · Z W (w ) , γ2 δ2 O L (w) = Q L (w) · Z (1) L (w ) , ⎪ ⎩ (1) (m) γ δ 2 2 Z (w) = Z L (w ) /Z W (w γ1 )δ1 . For ε > 0, n ≥ 1, and λ ∈ {−1, 1} we define λ = {t ∈ A N+ : OW (t|n ) = 0 and OW (t|n )λ < O L (t|n )λh+ε }. E n,ε

Multifractal Analysis of Complex Random Cascades

159

For any ηn > 0 we have 

λ μq (E n,ε )≤

μq (w)|Q W (w)|−ληn Q L (w)ληn h+εηn Z (w)ληn

w∈A n



=

Yq (w)Z (w)ληn

w∈A n

n 

Wwk (w|k−1 )qk−1 −ληn L wk (w|k−1 )ληn h+εηn −τ (qk−1 ) .

k=1

This yields λ )) ≤ E(Yq · Z ληn ) E(μq (E n,ε

n 

(qk−1 − ληn , τ (qk−1 ) − λhηn − εηn ). (35)

k=1

Let us make the following observation. For any qk , ηn > 0 we can write log (qk − ληn , τ (qk ) − λhηn − εηn ) 1 = ληn ξ(qk ) − (λhηn + εηn ) ξ (qk ) + (ζk )ηn2 2 1 ξ (qk ) + ληn ξ (qk )(τ (qk ) − h) + (ζk )ηn2 , = −εηn 2

(36)

where ζk = (ζ1 , ζ2 ) = s(qk , τ (qk )) + (1 − s)(qk − ληn , τ (qk ) − λhηn − εηn ) for some s ∈ [0, 1], and (ζk ) = λ2

2 ∂2 ∂2 2 ∂

(ζk ).

(ζ ) + (λh + ε)

(ζ ) + 2λ(λh + ε) k k ∂q 2 ∂t 2 ∂q∂t

Also, we have log (qk − ληn , τ (qk ) − λhηn − εηn ) = log (qk − ληn , τ (qk − ληn ) + λ(τ (qk ) − h)ηn + τ ( qk )ηn2 /2 − εηn ), where  qk ∈ [qk − ληn , qk ]. Since τ is concave and τ (q) tends to h at ∞, if ηn is small enough, then for k large enough we have λ(τ (qk ) − h)ηn + τ ( qk )ηn2 /2 − εηn ≤ −εηn /2 < 0, hence log (qk − ληn , τ (qk ) − λhηn − εηn ) < log (qk − ληn , τ (qk − ληn )) = 0. Hence, due to (36), 1 0 ≥ −εηn ξ (qk ) + ληn ξ (qk )(τ (qk ) − h) + (ζk ) · ηn2 . 2

(37)

Moreover, under our assumptions, the multifractal analysis of the Mandelbrot mea sure μ L = FL achieved in [5]) implies that for any random probability vector W b−1 b−1 b−1  i ) = 1 and E(   with E( i=0 W i=0 Wi log Wi ) < 0, −E( i=0 Wi logb L i ) is a Hölder exponent for μ L , so it must belong to [a, a], where 0 < a = lim τ FL (q)/q ≤ lim τ FL (q)/q = a < ∞. q→+∞

q→−∞

 = Wqk yields  Applying this with W ξ (qk )/ log(b) ∈ [a, a] for all k ≥ 0. Also, since qk  q¯ = +∞ we have τ (qk ) = ξ(qk )/ ξ (qk )  h < ∞, so supk≥0 ξ(qk ) < ∞. These properties together with (37) yield c = supk (ζk ) < ∞.

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For simplicity we define a := log(b) a and a := log(b) a. By using again the fact that τ (qk ) − h ≥ 0 as well as (35) and (36) we get n



λ )) ≤ e−εanηn +a( k=1 τ (qk )−h)ηn +cηn /2 · E(Yq (w) · Z ληn (w)). (38) E(μq (E n,ε . We take ηn = 1/ log(n m + n) for all n ≥ 1, where n m is an integer large 2

4λ (m) √log(n m +1)

enough so that for any λ ∈ {−1, 1} we have E((Z W )

) < ∞, as well

√ log(n m +1) E((Z (1) ) L ) 4λ

as < ∞ (the existence of n m comes from Proposition 14). Then due to Proposition 9, for n ≥ 1 and w ∈ A n we have (1)

(m)

E(Yq (w) · Z ληn ) ≤ Yq (w)2 · (Z L )λδ2 ηn 4 · (Z W )−λδ1 ηn 4 n

= O(b2n/ log(n) ). τ (q

(39) τ (q

Notice that k=1 k ) − h = o(n), since k ) − h  0 when k → ∞. Thus, due to our choice for ηn , for n large enough we have − εanηn + a(

n 

τ (qk ) − h)ηn + cηn2 = −εanηn + o(nηn ).

(40)

k=1

√ λ )) = O(b−εan/2 log(n) ). Then (39) and (40) together yield E(μq (E n,ε (2) Let us recall that E(μq ) = E(Yq ) = 1 and introduce on × A N+ the “Peyrière” probability measure Qq defined by # % Qq (A) = E 1 A (ω, t)μqω (dt) P(d(ω)) , A ∈ B ⊗ S . Notice that “Qq -almost surely” means “with probability 1, μq -almost everywhere”. Without loss of generality, we can assume that for i ∈ {0, b − 1} the sequence

−τ (q ) E(1{Wi =0} |Wi |qk L i k ) k≥0

has a limit f i as k → ∞, since this sequence takes values in the bounded interval [0, 1]. Now for n ≥ 1, i ∈ {0, b − 1} and (ω, t) ∈ × A N+ set f i,n (ω, t) = 1{i} (tn ). It is not difficult to show that the random variables f i,n , n ≥ 1 are Qq independent. Moreover, we have −τ (q ) 2  f i,n := EQq ( f i,n ) = EQq ( f i,n ) = E(Wqn−1 ,i ) = E(|Wi |qn−1 L i n−1 ).

Indeed, EQq ( f i,n ) =

 w∈A

=

n,

E(μq ([w]))

wn =i



E(Q q (w|n−1 ))E(Wqn−1 ,i (w|n−1 ))E(Yq (w)).

w∈A n , wn =i

Consequently,  f converges to f i as n → ∞, and on ( × A N+ , B ⊗ S, Qq ), the n i,n  martingale k=1 ( f i,k − EQq ( f i,k ))/k is bounded in L 2 norm by k≥1  f i,k (1 −

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161

 f i,k )/k 2 . It follows that the series k≥1 ( f i,k −EQh ( f i,k ))/k converges Qq -almost  surely, and the Kronecker lemma implies that nk=1 f i,n /n converges to f i Qq almost surely. This implies (1). (3) Let α ∈ (0, ε). Fix η ∈ (0, 1) and for n large enough let p ∈ [(1 − α)n, n − 1] be an integer. For (ω, t) ∈ × A N+ , let X n, p (ω, t) = log |Wn, p (t|n )|/ log O L (t|n ). We have  Qq (X n, p < −ε) = E(1{log |Wn, p (t|n )|/ log O L (t|n )<−ε} μq ([w])) w∈A n

≤



w∈A

E(Yq (w)Z L (w)ηε )E(Q q (w| p−1 )Q L (w| p−1 )ηε ) n

· E(Wq p−1 ,w p (w| p−1 )L w p (w| p−1 )ηε |Wr (w| p−1 )|η ) ·

n 

E(Wqk−1 ,wk (w|k−1 )L wk (w|k−1 )ηε |Wi (w|k−1 )|η )

k= p+1

for any η > 0. Applying the Cauchy-Schwarz inequality in the right hand side of the above inequality yields Qq (X n, p < −ε) ≤

p−1  k=1

(qk−1 , τ (qk−1 ) − ηε)

n 

(2qk−1 , 2τ (qk−1 ) − 2ηε)

k= p ηε ·Yq (w)2 · Z L 2 · |Wr |η 2

· (|Wi |η 2 )n− p .

Also, by using the same arguments as in the proof of (1) we can get ⎧ ⎪ ξ (qk−1 )ηε + O(ηε2 ) ⎨log (qk−1 , τ (qk−1 ) − ηε) = − log (2qk−1 , 2τ (qk−1 ) − 2ηε) ≤ log (2qk−1 , τ (2qk−1 ) − 2ηε) ⎪ ⎩ = − ξ (2qk−1 )ηε + O(ηε2 ). It follows that Qq (X n, p < −ε) ≤ C · en[−aηε+O(η

2 )]

· (|Wi |η 2 )αn .

Since |Wi |η 2 → 1 when η → 0, then we can find η small enough and α small enough such that for n large enough: Qq (X n, p < −ε) ≤ e−aηεn/2 , ∀ (1 − α)n ≤ p ≤ n.  Consequently, n≥1 Qq (∃ (1 − α)n ≤ m ≤ n : X n, p < −ε) < ∞, and the conclusion follows from the Borel-Cantelli lemma. Proof of Proposition 10. Due to Proposition 11(1), with probability 1, for μq almost every t ∈ A N+ , (notice that 1/ h can be infinite since h can be equal to 0), log |Q W (t|n )| log |Q W (t|n )| = lim = h, n→∞ log Q L (t|n ) n→∞ log Osc(1) (t| ) n lim

FL

lim

log Q L (t|n )

n→∞ log Osc(m) (t| ) n FW

=

1 , h

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and for γ ∈ {−1, 1}, (1)

log Z W ((t|n )γ )

lim

n→∞

(1)

log Osc FL (t|n )

= lim

n→∞

log Z L ((t|n )γ ) (m)

log Osc FW (t|n )

= 0.

Also, due to Lemma 1 and the fact that all the moments of Z L are finite, there exist ε > 0 such that for μq -almost every t ∈ A N+ , there exists n t,ε such that for all n ≥ n t,ε , we ¯ have Q L (t|n ) ∈ [b−n(a+ε) , b−n(a−ε) ]. In particular, for n large enough we have ( ) a − ε Nn (t) a¯ + ε Nn (t) log(Q L (t|n )/Q L (t|n−Nn (t) )) ∈ , . log Q L (t|n ) a¯ + ε n a−ε n Consequently, since Nn (t) = o(n) for μq -almost every t ∈ A N+ (Proposition 11(2)), we have lim

n→∞

log Q L (t|n−Nn (t) ) = 1, log Q L (t|n )

and if h = 0, we have 1 log Q W (t|n−Nn (t) ) log Q L (t|n−Nn (t) ) log Q L (t|n ) = lim · lim · lim n→∞ log Q L (t|n−Nn (t) ) n→∞ n→∞ log Q W (t|n ) h log Q L (t|n ) log Q W (t|n−Nn (t) ) = lim . n→∞ log Q W (t|n )

1 = h·1·

Moreover, let i ∈ {0, b − 1} and r ∈ {0, . . . , b − 1}, since L i ≤ 1, for any p ≤ n − 1 we have , log(L r (t| p−1 ) n−1 k= p+1 L i (t|k )) ≥ 0. lim inf (m) n→∞ log Osc FW (t|n ) Then, due to Proposition 11 (1) and (3), for μq -almost every t ∈ A N+ , for γ ∈ {−1, 1}, (1)

either lim inf n→∞

log Osc FW ((t|n )γ ) (1)

log Osc FL (t|n )

(1)

= ∞ or lim inf

log Osc FW ((t|n )γ )

n→∞

(1)

log Osc FL (t|n )

≥ h;

where the inequality is automatically true in the case h = 0, and (1)

lim inf n→∞

log Osc FL ((t|n )γ ) (m) log Osc FW (t|n )

≥

1 . h

We conclude from the fact that due to (15), for  t = FL (π(t)) we have (m)

log Osc F (B( t, r )) ≥ lim inf r →0 log r lim inf r →0

log r (m) log Osc F (B( t, r ))

min

lim inf

min

lim inf

γ =−1,0,1 n→∞

γ log Osc(1) FW ((t|n ) ) (1)

log Osc FL (t|n )

;

(1)

≥

γ =−1,0,1 n→∞

log Osc FL ((t|n )γ ) (m)

log Osc FW (t|n )

,

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where in the last inequality we have used Lemma 1. Consequently, (m) log Osc F (B( t, r )) = h, for μq ◦ π −1 ◦ FL−1 -almost every  t. lim r →0 log r

Almost surely dim H (νq ) ≥ τ ∗ (h). We only need to deal with the case where τ ∗ (h) > 0. We are going to prove that, with probability 1, for μq -almost every t ∈ A N+ , lim inf n→∞

log μq ([t|n ]) ≥ τ ∗ (h). log O L (t|n )

Then, due to the last claim of Lemma 1, the mass distribution principle (see [49], Lemma 4.3.2 or Sect. 4.1 in [20]) yields the conclusion. We set d = τ ∗ (h). Fix ε > 0, and for n ≥ 1 define E n,ε = {t ∈ Supp(μq ) : O L (t|n )−d+ε · μq ([t|n ]) ≥ 1}.    −d+ε · μ ([w]) ηn . For n ≥ 1 let ηn > 0 and set Sn,ε = n μ ([w]) O L (t|n ) q  w∈A q −(d−ε)η n Q (w)1+ηn Y (w)1+ηn For n ≥ 1 we have μq (E n,ε ) ≤ Sn,ε = w∈A n Q L (w) q q , 1+ηn Z L (w)ηn and E(Sn,ε ) = n−1

(q + q η , τ (q ) + (τ (q ) + d − ε)η ) · E(Y k k n k k n q (w) k=0   η n Z L (w) ). If we show that n≥1 E(Sn,ε ) < ∞, then the series n≥1 μq (E n,ε ) converges almost surely and the conclusion follows from the Borel Cantelli lemma. By an argument similar to those used in the proof of Proposition 8, we have log (qk + qk ηn , τ (qk ) + (τ (qk ) + d − ε)ηn ) = −ξ(qk )qk ηn +  ξ (qk )(τ (qk ) + d − ε)ηn + O(ηn2 ) = − ξ (qk )(τ ∗ (τ (qk )) − d + ε)ηn + O(ηn2 ) ≤ −aεηn + O(ηn2 ). Thus n−1 

(qk + qk ηn , τ (qk ) + (τ (qk ) + d − ε)ηn ) ≤ e−aεηn n+O(nηn ) . 2

k=0

Now since E(Yq (w)1+ηn Z L (w)ηn ) ≤ Yq (w)2

1+ηn

by taking ηn =

√1 n 0 +n

ηn

1−ηn

· Z L (w) 1−ηn 2

for n 0 large enough we will get E(Sn,ε ) = O(b

,

aεn − √ 2

log(n)

).

A. Appendix Proposition 12. Let M be a non-negative bounded and non-decreasing function defined over the subsets of Rd . Let Supp(M) = {t : ∀r > 0, M(B(t, r )) > 0} be the closed support of M. Suppose that Supp(M) is a non-empty compact set and define the L q -spectrum associated with M as the mapping namely 0 / q log sup i M(Bi ) , τ M (q) = lim inf r →0 log(r )

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where the supremum is taken over all the families of disjoint closed balls Bi of radius r with centers in Supp(M). We have dim B (Supp(M)) = −τ M (0), and for all h ≥ 0, & ! log(M(B(t, r )) ∗ = h ≤ τM dim E M (h) := t ∈ Supp(M) : lim inf (h), r →0+ log(r ) a negative dimension meaning that E M (h) is empty. Proof. The equality dim B (Supp(M)) = −τ M (0) is just the definition of the upper box dimension. Let h ≥ 0. Fix ε > 0. For every t ∈ E M (h), let (rt,k )k≥0 be a decreasing sequence h+ε ≤ M(B(t, r )) ≤ r h−ε . tending to 0 such that rt,k t,k t,k Fix δ > 0, and for each t ∈ E M (h) let kt be such that rt,kt ≤ δ. Now, for every n ≥ 0, let An = {t ∈ E M (h) : 2−(n+1) < rt,kt ≤ 2−n }. By the Besicovich covering theorem (see Theorem 2.7 in [44]) there exists an integer N such that for every n ≥ 0 we can find N disjoint subsets An,1 , . . . An,N of An such that each set An, j is at most countable, the balls of the form B(t, rt,kt ), t ∈ An, j , are pairwise disjoint, and  N  j=1 n≥0 t∈An, j B(t, rt,kt ) is a δ-covering of E M (h). (0+ )]. We have τ ∗ (h) = inf Suppose that h ∈ [0, τ M q∈R+ hq − τ M (q). Fix q ≥ 0 M such that τ M (q) > −∞ and then define Dε = (h + ε)q − τ M (q) + ε. We have HδDε (E M (h))

≤

N  

(2rt,kt )

Dε

≤2

n≥0 j=1 t∈An, j

≤ 2 Dε

N 



Dε

N  

(h+ε)q−τ M (q)+ε

rt,kt

n≥0 j=1 t∈An, j −τ (q)+ε

M(B(t, rt,k ))q rt,ktM

n≥0 j=1 t∈An, j

≤ 2 Dε 2|τ M (q)|

N  

M(B(t, 2−n ))q 2n(τ M (q)−ε) .

n≥0 j=1 t∈An, j

For each 1 ≤ j ≤ N , the family {B(t, 2−n )}t∈An, j can be divided into two disjoint 2−n -packing of Supp(M). Consequently, by definition of τ M (q), for n large enough,  M(B(t, 2−n ))q ≤ 2 · 2−n(τ M (q)−ε/2) t∈An, j

  −nε/2 < ∞. This yields dim E (h) ≤ D for all and HδDε (E M (h)) = O M ε n≥0 2 ε > 0, hence dim E M (h) ≤ hq − τ M (q). (0+ ). We have τ ∗ (h) = inf Now suppose that h > τ M q∈R− hq − τ M (q). Fix q ≤ 0 M such that τ M (q) > −∞ and then Dε = (h − ε)q − τ M (q) + ε. This time we have HδDε (E M (h)) ≤ 2 Dε 2|τ M (q)|

N  

M(B(t, 2−(n+1) ))q 2n(τ M (q)−ε) ,

n≥0 j=1 t∈An, j

and for each 1 ≤ j ≤ N , the family {B(t, 2−(n+1) )}t∈An, j is a 2−(n+1) -packing of Supp(M). We conclude as in the previous case.

Multifractal Analysis of Complex Random Cascades

165



(n) (n) Proposition 13. Let V (n) = (V0 , . . . , Vb−1 ) , be a sequence of random vectors

n≥1  b−1 (n) taking values in Cb , and such that E = 1. Let {V (w)}w∈A ∗ be a sequence i=0 Vi of independent vectors such that V (w) is distributed as V (|w|) for each w ∈ A ∗ . Define Z 0 = 1 and for n ≥ 1, Zn =

n  

Vwk (w|k−1 ).

w∈A n k=1

Let p ∈ (1, 2]. There exists a constant C p ≤ 2 p depending on p only such that for all n ≥ 1, b−1  n   (k) p p E |Vi | . E(|Z n − Z n−1 | ) ≤ C p k=1

i=0

See the proof of Theorem 1 in [6]. Proposition 14. We work under the assumptions of Theorem A or B. Let m ≥ 1 and U ∈ {W, L}. (m)

(1) If q > 1 and ϕU (q) > 0 then E((Z U )q ) < ∞. Moreover, if W satisfies the (m) assumptions of Theorem B(2) then ess sup Osc FW ([0, 1]) < ∞. (m)

(m)

(2) Define ψU (t) = E(e−t Z U ) for t ≥ 0. Let AU = max0≤i≤b−1 |Ui |. −q (m) If q > 0 and E(AU ) < ∞ then ψU (t) = O(t − p ) for all p ∈ (0, q). Conse(m) quently, E((Z U )− p ) < ∞ for all p ∈ (0, q). (m)

Proof of Proposition 14. (1) Since Osc FU ([0, 1]) ≤ 2m−1 Osc(1) (FU , [0, 1]) ≤ 2m

(m)

FU ∞ , this is a direct consequence of Theorems A and B (that ess sup Osc FW ([0, 1]) < ∞ when W satisfies the assumptions of Theorem B(2) is not stated in [7] but established in the proof of this theorem). (m) (m) (2) Since Osc FU ([0, 1]) ≥ Osc FU (Ii ) for all 0 ≤ i ≤ b − 1, by using (8) we get (m)

Z U ≥ b−1

b−1 

(m)

|U (i)| · Z U (i),

(41)

i=0 (m)

where the Z U (i) are independent copies of Z and they are independent of W . (m) Moreover, thanks to Proposition 3 applied to FU , we know that Z U > 0 almost surely for all m ∈ N+ . Also, with probability 1, we can define i 0 = max{0 ≤ i ≤ b − 1 : |Ui | = max0≤k≤b−1 |Uk | and i 1 = inf{0 ≤ i ≤ b − 1 : i = i 0 , Ui = 0}, A0 = |Ui0 | and A1 = |Ui1 |. −q Suppose that E(A0 ) < ∞. This clearly holds if ϕU (−q) > −∞ or if there exists a > 0 such that max0≤k≤b−1 |Uk | ≥ a almost surely (for instance a = 1/b is convenient when U is conservative). (m) Set ψ = ψU . By definition of ψ, we deduce from (41) and the fact that Z U is almost surely positive that ψ(t) ≤ E (ψ(A0 t)ψ(A1 t)) and limt→∞ ψ(t) = 0. Suppose that we have shown that ψ(t) = O(t − p ) at +∞, for all p ∈ (0, q).

166

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Then, for x > 0 we have P(Z (m) ≤ x) ≤ et x ψ(t) and choosing t = p/x yields P(Z ≤ x) = O(x p ) at 0+ . Hence E(Z − p ) < ∞ if h ∈ (0, q). Now we essentially use the elegant approach of [38] for the finiteness of the moments of negative orders of FU (1), when the components of W are non-negative (see also the references in [38] for this question). Let r > 1 and φ = ψ r . Due to the bounded convergence theorem we have limt→∞ E(ψ(A1 t)r/(r −1) ) = 0, so the Hölder inequality yields φ(t) = o(E(φ(t A0 )) at ∞. Let γ ∈ (0, 1) small enough −p to have γ E(A0 ) < 1, and let t0 > 0 such that φ(t) ≤ γ E(φ(t A0 )), t ≥ t0 .

(42)

i )i≥1 be a sequence of independent copies of A0 Since φ ≤ 1, for t ≥ t0 we Let ( A can prove by induction using (42) the following inequalities valid for all n ≥ 2:   φ(t) ≤ γ P(A0 t < t0 ) + γ E 1{A0 t≥t0 } φ(A0 t)   −p 1 t) ≤ γ E(A0 )(t0 /t) p + γ 2 E 1{A0 t≥t0 } φ(A0 A   −p 1 t) ≤ γ E(A0 )(t0 /t) p + γ 2 E φ(A0 A

−p −p 1 t) ≤ γ E(A0 )(t0 /t) p + γ 2 (E(A0 ))2 (t0 /t) p + γ 2 E 1{A0 A1 t≥t0 } φ(A0 A n

 −p n−1 t) . 1 · · · A (γ E(A0 ))k + γ n E 1{A0 A1 ··· An−1 t≥t0 } φ(A0 A ≤ (t0 /t) p

k=1 −p

Since ψ ≤ 1, and both γ and γ E(A0 ) belong to (0, 1), letting n tend to ∞ yields φ(t) = ψ(t)r = O(t − p ). Since r and p are arbitrary respectively in (1, ∞) and (0, q), we have the desired result. References 1. Arbeiter, M., Patszchke, N.: Random self-similar multifractals. Math. Nachr. 181, 5–42 (1996) 2. Arneodo, A., Bacry, E., Muzy, J.-F.: Random cascades on wavelet dyadic trees. J. Math. Phys. 39, 4142– 4164 (1998) 3. Aubry, J.M., Jaffard, S.: Random wavelet series. Commun. Math. Phys. 227, 483–514 (2002) 4. Bacry, E., Muzy, J.-F.: Log-infinitely divisible multifractal processes. Commun. Math. Phys. 236, 449– 475 (2003) 5. Barral, J.: Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theo. Probab. 13, 1027–1060 (2000) 6. Barral, J.: Generalized vector multiplicative cascades. Adv. Appl. Prob. 33, 874–895 (2001) 7. Barral, J., Jin, X., Mandelbrot, B.B.: Convergence of complex multiplicative cascades. Ann. Appl. Proba. (to appear) 8. Barral, J., Mandelbrot, B.B.: Random Multiplicative Multifractal Measures I, II, III. In: Lapidus, M., Frankenhuijsen, M.V. ed. Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Proc. Symp. Pure Math. 72(2), Providence, RI: Amer. Math. Soc., 2004, pp. 3–90 9. Barral, J., Seuret, S.: From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11, 589–614 (2005) 10. Bedford, T.: Hölder exponents and box dimension for self-affine fractal functions. Fractal Approximation. Constr. Approx. 5, 33–48 (1989) 11. Ben Slimane, M.: Multifractal formalism and anisotropic selfsimilar functions. Math. Proc. Camb. Phil. Soc. 124, 329–363 (1998) 12. Biggins, J.D.: Uniform Convergence of Martingales in the Branching Random Walk. Ann. Prob. 20, 137– 151 (1992) 13. Brown, G., Michon, G., Peyrière, J.: On the multifractal analysis of measures. J. Stat. Phys. 66, 775– 790 (1992) 14. Billingsley, P.: Ergodic Theory and information. New York: Wiley, 1965

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15. Chainais, P., Abry, P., Riedi, R.: On non scale invariant infinitely divisible cascades. IEEE Trans. Info. Th. 51, 1063–1083 (2005) 16. Collet, P., Koukiou, F.: Large deviations for multiplicative chaos. Commun. Math. Phys. 147, 329–342 (1992) 17. Durand, A.: Random wavelet series based on a tree-indexed Markov chain. Commun. Math. Phys. 283, 451–477 (2008) 18. Durrett, R., Liggett, R.: Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64, 275–301 (1983) 19. Falconer, K.J.: The multifractal spectrum of statistically self-similar measures. J. Theor. Prob. 7, 681– 702 (1994) 20. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications, 2nd Edition. New York: Wiley, 2003 21. Frisch, U., Parisi, G.: Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dymnamics, International School of Physics “Enrico Fermi”, Course 88, edited by M. Ghil, Amsterdam: North Holland, 1985, p. 84 22. Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Shraiman, B.I.: Fractal measures and their singularities: the characterisation of strange sets. Phys. Rev. A 33, 1141–1151 (1986) 23. Hentschel, H.G., Procaccia, I.: The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8, 435–444 (1983) 24. Holley, R., Waymire, E.C.: Multifractal dimensions and scaling exponents for strongly bounded random fractals. Ann. Appl. Probab. 2, 819–845 (1992) 25. Jaffard, S.: Exposants de Hölder en des points donnés et coefficients d’ondelettes. C. R. Acad. Sci. Paris, 308 Série I, 79–81 (1989) 26. Jaffard, S.: The spectrum of singularities of Riemann’s function. Rev. Math. Ibero-Amer. 12, 441– 460 (1996) 27. Jaffard, S.: Multifractal formalism for functions. I. Results valid for all functions. II Self-similar functions, SIAM J. Math. Anal. 28, 944–970 & 971–998 (1997) 28. Jaffard, S.: Oscillations spaces: Properties and applications to fractal and multifractal functions. J. Math. Phys. 39(8), 4129–4141 (1998) 29. Jaffard, S.: On lacunary wavelet series. Ann. Appl. Prob. 10(1), 313–329 (2000) 30. Jaffard, S.: On the Frisch-Parisi Conjecture. J. Math. Pures Appl. 79(6), 525–552 (2000) 31. Jaffard, S.: Wavelets techniques in multifractal analysis. In: Lapidus, M., Frankenhuijsen, M.V., ed. Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Proc. Symp. Pure Math. 72(2), Providencce, RI: Amer. Math. Soc., 2004, pp. 91–151 32. Jaffard, S., Mandelbrot, B.B.: Local regularity of nonsmooth wavelet expansions and application to the Polya function. Adv. Math. 120, 265–282 (1996) 33. Jaffard, S., Mandelbrot, B.B.: Peano-Polya motions, when time is intrinsic or binomial (uniform or multifractal). Math. Intelligencer 19, 21–26 (1997) 34. Jaffard, S., Meyer, Y.: On the pointwise regularity of functions in critical Besov spaces. J. Funct. Anal. 175, 415–434 (2000) 35. Kahane, J.P., Peyrière, J.: Sur certaines martingales de B. Mandelbrot. Adv. Math. 22, 131–145 (1976) 36. Lau, K.S., Ngai, S.M.: Multifractal measures and a weak separation condition. Adv. Math. 141, 45–96 (1999) 37. Liu, Q.: On generalized multiplicative cascades. Stoch. Proc. Appl. 86, 263–286 (2000) 38. Liu, Q.: Asymptotic Properties and Absolute Continuity of Laws Stable by Random Weighted Mean. Stoch. Proc. Appl. 95, 83–107 (2001) 39. Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades: divergence of hight moments and dimension of the carrier. J. Fluid. Mech. 62, 331–358 (1974) 40. Mandelbrot, B.B.: New “anomalous” multiplicative multifractals: left sided f (α) and the modelling of DLA. Phys. A 168(1), 95–111 (1990) 41. Mandelbrot, B.B., Evertsz, C.J.G., Hayakawa, Y.: Exactly self-similar left-sided multifractal measures. Phys. Rew. A 42, 4528–4536 (1990) 42. Mandelbrot, B.B.: Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, Berlin-Heidelberg-New York: Springer, 1997 43. Mandelbrot, B.B.: Gaussian Self-Affinity and Fractals. Berlin-Heidelberg-New York: Springer, 2002 44. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. In: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics 44, Cambridge: Cambridge University Press, 1995 45. Molchan, G.M.: Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179, 681–702 (1996) 46. Muzy, J.F., Bacry, E., Arneodo, A.: Wavelets and multifractal formalism for singular signals: application to turbulence data. Phys. Rev. Lett. 67, 3515–3518 (1991) 47. Olsen, L.: A multifractal formalism. Adv. Math. 116, 92–195 (1995)

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48. Pesin, Y.: Dimension theory in dynamical systems: Contemporary views and applications. Chicago Lectures in Mathematics, Chicago, IL: The University of Chicago Press, 1997 49. Peyrière, J.: A Singular Random Measure Generated by Spliting [0, 1]. Z. Wahrsch. Verw. Gebiete 47, 289–297 (1979) 50. Rényi, A.: Probability Theory, Amsterdam: North-Holland, 1970 51. Riedi, R., Mandelbrot, B.B.: Multifractal formalism for infinite multinomial measures. Adv. Appl. Math. 16(2), 132–150 (1995) 52. Sendov, Bl.: On the theorem and constants of H. Whitney. Constr. Approx. 3, 1–11 (1987) 53. Seuret, S.: On multifractality and time subordination for continuous functions. Adv. Math. 220, 936– 963 (2009) 54. Whitney, H.: On functions with bounded nth differences. J. Math. Pures Appl. 36, 67–95 (1957) Communicated by S. Smirnov

Commun. Math. Phys. 297, 169–187 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1029-4

Communications in

Mathematical Physics

Geometry of Maurer-Cartan Elements on Complex Manifolds, Zhuo Chen1 , Mathieu Stiénon2 , Ping Xu3 1 Department of Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China.

E-mail: [email protected]

2 Institut de Mathématiques de Jussieu, Université Paris Diderot, Batiment Chevaleret, case 7012,

75205 Paris cedex 13, France. E-mail: [email protected]

3 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.

E-mail: [email protected] Received: 12 June 2009 / Accepted: 6 November 2009 Published online: 31 March 2010 – © Springer-Verlag 2010

Abstract: The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite dimensional; we describe how homology and cohomology are related through the Evens-Lu-Weinstein duality module; and we describe a duality on Koszul-Brylinski homology, which generalizes the Serre duality of Dolbeault cohomology. Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lie bialgebroids . . . . . . . . . . . . . . . . . . . . . . 2.2 Hamiltonian operators . . . . . . . . . . . . . . . . . . . Maurer-Cartan Elements . . . . . . . . . . . . . . . . . . . . 3.1 The Lie bialgebroid stemming from a complex manifold 3.2 Extended Poisson structures . . . . . . . . . . . . . . . 3.3 Elliptic Lie algebroids . . . . . . . . . . . . . . . . . . . 3.4 Poisson cohomology . . . . . . . . . . . . . . . . . . . 3.5 Coisotropic submanifolds . . . . . . . . . . . . . . . . . 3.6 Poisson relations . . . . . . . . . . . . . . . . . . . . .

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 Research partially supported by NSFC grants 10871007 and 10911120391/A0109.  Research partially supported by NSF grants DMS-0605725 and DMS-0801129.

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Koszul-Brylinski Poisson Homology . 4.1 Koszul-Brylinski cochain complex 4.2 Evens-Lu-Weinstein duality . . . . 4.3 Proof of Theorem 4.3 . . . . . . . 4.4 Modular classes . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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1. Introduction Due to their close connection to mirror symmetry, noncommutative deformations of complex manifolds have recently generated increasing interest [5,20]. The KashiwaraKontsevich’s stacks of algebroids are one way of substantiating the abstract concept of quantum complex manifolds (or noncommutative deformations of complex manifolds) [6,7,16–18,20,35]. The quantization of the sheaf of holomorphic functions O X of a complex manifold X may no longer produce a sheaf of algebras but, instead, lead to a nonabelian gerbe over the complex manifold X [6,34] or, in Kontsevich’s terminology, a stack of algebroids. Roughly speaking, an algebroid à la Kontsevich consists of an open cover {Ui }i∈I of the complex manifold X, a sheaf of associative unital algebras Ai on each Ui , an isomorphism of algebras gi j : A j |Ui j → Ai |Ui j for each nonempty intersection Ui j , and an invertible element ai jk ∈ (Ui jk , Ai× ) for each triple intersection Ui jk . The isomorphisms gi j do not satisfy the usual cocycle condition. Instead, the equations gi j ◦ g jk ◦ gki = Ada −1 are satisfied as well as other compatibility conditions i jk

(among which a “tetrahedron equation”). In the terminology of [25], an algebroid à la ˇ Kontsevich would be described as an extension of a Cech groupoid by algebras. A stack of algebroids can be thought of as a Morita equivalence class (see [25]) of algebroids. A canonical abelian category of coherent sheaves can be defined on a quantum complex manifold using its stack of algebroids description [16–18,20]. It is well known that the semi-classical data attached to quantum real manifolds (i.e. star-algebras) are Poisson structures [1,2]. The cotangent bundle of a real Poisson manifold (M, π ) is endowed with a canonical Lie algebroid structure denoted by (T ∗ M)π . This Lie algebroid structure plays a central role in Poisson geometry. For instance, the Lichnerowicz-Poisson cohomology is simply the Lie algebroid cohomology of (T ∗ M)π with trivial coefficients. Evens-Lu-Weinstein discovered a procedure for constructing a canonical module over a given Lie algebroid. With the canonical module of (T ∗ M)π at hand, they interpreted Koszul-Brylinski homology as a Lie algebroid cohomology. According to Kontsevich’s formality theorem and Tsygan’s chain formality theorem, the Hochschild cohomology and Hochschild homology of a star algebra are isomorphic to the Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology of the underlying Poisson manifold. In the context of complex geometry, the semiclassical data associated to quantum complex manifolds are solutions of the Maurer-Cartan equation in the derived global sections R(X, ∧• T X[1]) of the sheaf of graded Lie algebras ∧• T X[1] of polyvector fields on X, which, according to Kontsevich’s formality theorem, classify the deformations of stacks of algebroids up to gauge transformations [6,20,35]. More precisely, a Maurer-Cartan element is an H = π + θ + ω ∈ 0,0 (∧2 T 1,0 X) ⊕ 0,1 (∧1 T 1,0 X) ⊕ 0,2 (∧0 T 1,0 X)

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(where 0, p (∧q T 1,0 X) denotes the space of ∧q T 1,0 X-valued (0, p)-forms on X) satisfying the following equations: ¯ + [ω, θ ] = 0, ∂ω ¯ + [ω, π] + 1 [θ, θ ] = 0, ∂θ 2

¯ + [θ, π ] = 0, ∂π [π, π ] = 0.

Holomorphic Poisson bivector fields are special cases of such Maurer-Cartan elements, as are holomorphic (0, 2)-forms. For this reason, complex manifolds endowed with such a Maurer-Cartan element H will be called extended Poisson manifolds. In a recent paper [30], one of the authors studied the Koszul-Brylinski homology of holomorphic Poisson manifolds, and established a duality on it using the general theory developed by Evens-Lu-Weinstein [12]. In this paper, in order to study the geometry of extended Poisson manifolds, we apply the Evens-Lu-Weinstein theory to complex Lie algebroids. Indeed, considering Maurer-Cartan elements as Hamiltonian operators (in the sense of [26]) deforming a Lie bialgebroid [27], we define a complex Lie algebroid, which mimics the role played by the cotangent Lie algebroid in real Poisson geometry. It is not surprising that, for a holomorphic Poisson structure, this complex Lie algebroid is the derived Lie algebroid of the (1,0) holomorphic cotangent Lie algebroid (T ∗ X)π , i.e. the matched pair T 0,1 X  (T ∗ X)π studied in [24,30]. Using this complex Lie algebroid, we introduce a Lichnerowicz-Poisson cohomology and a Koszul-Brylinski homology for extended Poisson manifolds, and study the relation between them. We extend the notion of coisotropic submanifolds of holomorphic Poisson manifolds to the “extended” setting. We give a criterion on the ellipticity of the complex Lie algebroid (in the sense of Block [4]) induced by a MaurerCartan element. And in the elliptic case, we obtain a duality, which we call Evens-LuWeinstein duality, on the Koszul-Brylinski homology groups. As was pointed out in [30] for the holomorphic Poisson case, this duality generalizes the Serre duality on Dolbeault cohomology. Note that, modulo gauge equivalences, our extended Poisson structures and Yekutieli’s Poisson deformations (see [35]) are equivalent. It would be interesting to explore the connection between our results on Poisson homology and Berest-EtingofGinzburg’s [3]. It would also be interesting to investigate if one can extend the method in this paper to study the Bruhat-Poisson structures of Evens-Lu on flag varieties [11] and the toric Poisson structures of Caine [8].

2. Preliminaries 2.1. Lie bialgebroids. A complex Lie algebroid [32] consists of a complex vector bundle A → M, a bundle map a : A → TC M called anchor, and a Lie algebra bracket [·, ·] on the space of sections (A) such that a induces a Lie algebra homomorphism from (A) to XC (M) and the Leibniz rule [u, f v] = (a(u) f ) v + f [u, v] is satisfied for all f ∈ C ∞ (M, C) and u, v ∈ (A). It is well-known that a Lie algebroid (A, [·, ·], a) is equivalent to a Gerstenhaber algebra ((∧• A), ∧, [·, ·]) [33]. On the other hand, for a Lie algebroid structure on a

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vector bundle A, there is also a degree 1 derivation d of the graded commutative algebra ((∧• A∗ ), ∧) such that d 2 = 0. The differential d is given by (dα)(u 0 , u 1 , . . . , u n ) =

n  (−1)i a(u i )α(u 0 , . . . , ui , . . . , u n ) i=0

+



(−1)i+ j α([u i , u j ], u 0 , . . . , ui , . . . , uj , . . . , u n ).

i< j

Indeed, a Lie algebroid structure on A is also equivalent to a differential graded algebra ((∧• A∗ ), ∧, d). Let A → M be a complex vector bundle. Assume that A and its dual A∗ both carry Lie algebroid structures with anchor maps a : A → TC M and a∗ : A∗ → TC M, brackets on sections (A) ⊗C (A) → (A) : u ⊗ v → [u, v] and (A∗ ) ⊗C (A∗ ) → (A∗ ) : α ⊗ β → [α, β]∗ , and differentials d : (∧• A∗ ) → (∧•+1 A∗ ) and d∗ : (∧• A) → (∧•+1 A). This pair of Lie algebroids (A, A∗ ) is a Lie bialgebroid [22,28,27] if d∗ is a derivation of the Gerstenhaber algebra ((∧• A), ∧, [·, ·]) or, equivalently, if d is a derivation of the Gerstenhaber algebra ((∧• A∗ ), ∧, [·, ·]∗ ). Since the bracket [·, ·]∗ (resp. [·, ·]) can be recovered from the derivation d∗ (resp. d), one is led to the following alternative definition. Proposition 2.1 ([33]). A Lie bialgebroid (A, A∗ ) is equivalent to a differential Gerstenhaber algebra structure on ((∧• A), ∧, [·, ·], d∗ ) (or, equivalently, on ((∧• A∗ ), ∧, [·, ·]∗ , d)). 2.2. Hamiltonian operators. Let (A, A∗ ) be a complex Lie bialgebroid, and H ∈ (∧2 A). We now replace the differential d∗ : (∧• A) → (∧•+1 A) by a twist by H: d∗H : (∧• A) → (∧•+1 A),

d∗H u = d∗ u + [H, u].

(1)

It follows from a simple verification that if H satisfies the Maurer-Cartan equation: 1 d∗ H + [H, H ] = 0, 2

(2)

then (d∗H )2 = 0 and ((∧• A), ∧, [·, ·], d∗H ) is again a differential Gerstenhaber algebra. Thus one obtains a Lie bialgebroid (A, A∗H ). A solution H ∈ (∧2 A) to Eq. (2) is called a Hamiltonian operator [26]. The Lie algebroid structure on A∗H can be described explicitly: the anchor and the Lie bracket are given, respectively, by a∗H = a∗ + a ◦ H  and [α, β]∗H = [α, β]∗ + [α, β] H . Here [α, β] H = L H  (α) β − L H  (β) α − d∗ H  (α)|β, for all α, β ∈ (A∗ ). We shall use A∗H to denote such a Lie algebroid and call it the H -twisted Lie algebroid of A∗ . Thus we obtain the following theorem, which was first proved in [26] by a different method.

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Theorem 2.2. If (A, A∗ ) constitutes a Lie bialgebroid, and H ∈ (∧2 A) is a Hamiltonian operator, then (A, A∗H ) is a Lie bialgebroid. 3. Maurer-Cartan Elements 3.1. The Lie bialgebroid stemming from a complex manifold. We fix a complex manifold X of complex dimension n with almost complex structure J . We regard the tangent bundle T X as a real vector bundle over X. The complexification of T X is denoted TC X, namely: TC X = T X ⊗ C. Similarly, TC∗ X = T ∗ X ⊗ C. Let J : TC X → TC X be the C-linear extension of the almost complex structure J , and T 1,0 X and T 0,1 X its +i and −i eigenbundles, respectively. We adopt the following notations: T p,q X = ∧ p T 1,0 X ⊗ ∧q T 0,1 X, (T p,q X)∗ = ∧ p (T 1,0 X)∗ ⊗ ∧q (T 0,1 X)∗ . Consider the following two vector bundles which are obviously mutually dual: A = T 1,0 X ⊕ (T 0,1 X)∗ ,

A∗ = T 0,1 X ⊕ (T 1,0 X)∗ .

(3)

We can endow A with a complex Lie algebroid structure. The anchor is the projection onto the first component:   ∂ ∂ a = i a(dz j ) = 0. i ∂z ∂z The bracket of two sections of T 1,0 X is their bracket as vector fields; the bracket of any pair of sections of (T 0,1 X)∗ is zero; and the bracket of a holomorphic vector field (i.e. a holomorphic section of the holomorphic vector bundle T 1,0 X) and an anti-holomorphic 1-form (i.e. an anti-holomorphic section of the holomorphic vector bundle (T 0,1 X)∗ ) is also zero. Thus     ∂ ∂ ∂ = 0, [dz = 0. , , dz ] = 0, and , dz i j j ∂z i ∂z j ∂z i Together with the Leibniz rule, the above three rules completely determine the bracket of any two arbitrary sections of A. Similarly, one endows A∗ with a complex Lie algebroid structure as well. It is simple to see that (A, A∗ ) constitutes a Lie bialgebroid. Indeed A and A∗ are transversal Dirac structures of the Courant algebroid TC X ⊕ TC∗ X, for they are the eigenbundles of the generalized complex structure on X induced by its complex manifold structure [15,13]. In the sequel we will use the symbols T 1,0 X  (T 0,1 X)∗ and T 0,1 X  (T 1,0 X)∗

(4)

A∗

to refer to A and when seen as Lie algebroids [24]. Moreover, one has  T i,0 X ⊗ (T 0, j X)∗ , ∧k A ∼ = i+ j=k

∧ A ∼ = k

∗



T 0,i X ⊗ (T j,0 X)∗ .

i+ j=k

The Lie algebroid differentials associated to the Lie algebroid structures on A∗ and A ¯ and ∂-operators, respectively: are the usual ∂-

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d∗ = ∂¯ : 0, j (T i,0 X) → 0, j+1 (T i,0 X), d = ∂ :  j,0 (T 0,i X) →  j+1,0 (T 0,i X). 3.2. Extended Poisson structures. Definition 3.1. An extended Poisson manifold (X, H ) is a complex manifold X equipped with an H ∈ (∧2 A) which is an Hamiltonian operator with respect to (A, A∗ ), i.e. 1 ∂¯ H + [H, H ] = 0. 2 In this case, H is called an extended Poisson structure.

(5)

Any H ∈ (∧2 A) decomposes as H = π + θ + ω, where π ∈ (T 2,0 X), θ ∈ (T 1,0 X ⊗ (T 0,1 X)∗ ) and ω ∈ ((T 0,2 X)∗ ). We will use the following notations to denote the bundle maps induced by natural contraction: θ : T 0,1 X → T 1,0 X, θ  : (T 1,0 X)∗ → (T 0,1 X)∗ , π  : (T 1,0 X)∗ → T 1,0 X, ω : T 0,1 X → (T 0,1 X)∗ . Note that θ  = −(θ )∗ . The following lemma is immediate. Lemma 3.2. An element H = π + θ + ω is an extended Poisson structure if and only if the following equations are satisfied: ¯ + [ω, θ ] = 0, ∂ω (6) 1 ¯ + [ω, π] + [θ, θ ] = 0, (7) ∂θ 2 ¯ + [θ, π ] = 0, ∂π (8) [π, π ] = 0. (9) Remark 3.3. When only one of the three terms of H is not zero, we are left with one of the following three special cases: (a) H = π is an extended Poisson if and only if π is a holomorphic Poisson bivector field. ¯ + 1 [θ, θ ] = 0. Moreover, if (b) H = θ is an extended Poisson if and only if ∂θ 2

θ ◦ θ − id is invertible, θ is equivalent to a deformed complex structure [19]. ¯ = 0. (c) H = ω is an extended Poisson if and only if ∂ω In fact, if [ω, π ] = 0, Eq. (7) implies that θ defines a deformed complex structure (under the assumption that θ ◦ θ − id is invertible). Then, according to Lemma 3.15 below, Eq. (6) is equivalent to ∂¯θ ω = 0, where ∂¯θ = ∂¯ + [θ, ·], and Eqs. (8)–(9) mean that π is a holomorphic Poisson tensor with respect to the deformed complex structure. Corollary 3.4. If H = π + θ + ω is an extended Poisson structure, then so is λπ + θ + λ−1 ω,

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for any λ ∈ C× . In particular, H ∨ = −π + θ − ω is an extended Poisson structure. Note that Maurer-Cartan elements as deformations of Lie bialgebroids or differential Gerstenhaber algebras were already considered by Cleyton-Poon [10] in their study of nilpotent complex structures on real six-dimensional nilpotent algebras. A natural question is: when will (A, A∗H ) arise from a generalized complex structure in the sense of Hitchin [15,13]? Let us recall the following: Lemma 3.5. (Lemma 6.1 in [29]). The graph {H  ξ + ξ ∈ A ⊕ A∗ } of H , which is clearly isomorphic to A∗H as a vector bundle, is the +i- (or −i-) eigenbundle of a generalized 

complex structure on X if and only if H ◦ H  − id A∗ is invertible. Here the map   H : A → A∗ is defined by H (u) = H  (u), ∀u ∈ A.

Again we let H = π + θ + ω be an extended Poisson structure on X. Relative to the  direct sum decompositions of A and A∗ , the endomorphisms H  and H are represented by the block matrices

     θ π θ π and H = H =  . ω θ  ω θ In turn, we have θ ◦ θ + π  ◦ ω H H =  ω ◦ θ + θ ◦ ω 



θ ◦ π + π ◦ θ .  ω ◦ π  + θ ◦ θ 

(10)

Proposition 3.6. Given an extended Poisson manifold (X, H ), let A = T 1,0 X  (T 0,1 X)∗ . Then A∗H is the (±i)-eigenbundle of a generalized complex structure if and 

only if H H  − id A∗ is invertible.

Example 3.7. If H = π (i.e. H is a holomorphic Poisson bivector field) or H = ω, it is  clear that H H  is zero. Hence, in these two situations, the extended Poisson structure on X is actually a generalized complex structure. Here is a simple example of extended Poisson structure, which does not arise from a generalized complex structure. Example 3.8. Consider the torus T = C/(Z + iZ) with its standard complex structure. Let z be the standard coordinate on T. Obviously, any θ = f (z, z¯ )

d ∧ d z¯ , dz

(11)

where f is a smooth C-valued function, is an extended Poisson structure. In this case,  H H  = | f |2 id. Hence A∗θ does not stem from a generalized complex structure provided that | f | = 1.

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3.3. Elliptic Lie algebroids. As in [4], we say that a complex Lie algebroid B is elliptic if Re ◦a B : B → T X is surjective. Here a B : B → TC X is the anchor map of B and Re : TC X → T X is the projection onto the real part. Theorem 3.9 ([4]). If B is an elliptic Lie algebroid over a compact complex manifold X, and E a finite rank complex vector bundle with a B-action as in [12], then all cohomology groups H• (B, E) are finite dimensional. It is therefore natural to ask when A∗H is elliptic. An easy calculation shows the following: Proposition 3.10. Let a∗H denote the anchor of A∗H and C : T 0,1 X → T 1,0 X the complex conjugation. The bundle maps Re ◦a∗H and F = (C + θ ) ⊕ π  : T 0,1 X ⊕ (T 1,0 X)∗ → T 1,0 X,

(12)

and the isomorphism of real vector bundles Re : T 1,0 X → T X fit into the commutative diagram ∗ T 0,1 X ⊕ (T 1,0 OOX) o H O o OORe Foooo ∗ OO◦a o O o O o OOO w oo o ' Re / T X. 1,0 T X

(13)

As a consequence, A∗H is an elliptic Lie algebroid if and only if F is surjective. Example 3.11. When H = π , or ω, it is clear that A∗H is elliptic. On the other hand, if we consider the torus T endowed with the bivector field θ of Example 3.8, the Lie algebroid A∗H is elliptic if and only if f is not identically 1. 3.4. Poisson cohomology. Definition 3.12. Given an extended Poisson manifold (X, H ), the cohomology of the Lie algebroid A∗H is called the Poisson cohomology of the extended Poisson structure, and denoted H• (X, H ). In other words, it is the cohomology of the cochain complex: ∂¯ H

∂¯ H

∂¯ H

· · · −→ (∧k A) −→ (∧k+1 A) −→ . . . , where

(∧k A)

= ⊕i+ j=k

0, j (T i,0 X)

and

∂¯ H

(14)

= ∂¯ + [H, ·].

Poisson cohomology is also called tangent cohomology by Kontsevich [21]. As an immediate consequence of Theorem 3.9 and Proposition 3.10, we have Corollary 3.13. If H is an extended Poisson structure on a compact complex manifold X and the map F (given by Eq. (12)) is surjective, then all Poisson cohomology groups are finite dimensional. Remark 3.14. When H is a holomorphic Poisson bivector field π , the cochain complex (14) is the total complex of the double complex as discussed in Corollary 4.26 in [24]. On the other hand, if H = θ ∈ 0,1 (T 1,0 X) is a Maurer-Cartan element such that θ ◦ θ − id is invertible, then θ defines a new complex structure on X according to Kodaira [19].

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The following lemma can be verified directly.

Lemma 3.15. Let H = θ ∈ 0,1 (T 1,0 X) be a Maurer-Cartan element such that θ ◦ θ − id is invertible. Then the Lie algebroid A∗H is isomorphic to Tθ1,0 X  (Tθ0,1 X)∗ , where Tθ1,0 X and Tθ0,1 X are, respectively, the +i and −i eigenbundles of the deformed almost complex structure Jθ : T X → T X. As a consequence, the differential operator ¯ d∗H in Eq. (1) is equal to ∂¯θ , the new ∂-operator of the deformed complex structure. Thus we have Proposition 3.16. If H = θ ∈ 0,1 (T 1,0 X) is a Maurer-Cartan element such that θ ◦ θ − id is invertible, then Hk (X, H ) ∼ = ⊕i+ j=k Hi (X, ∧ j Tθ X), where Tθ X denotes the holomorphic tangent bundle of the deformed complex manifold X. 3.5. Coisotropic submanifolds. Suppose that Y ⊆ X is a complex submanifold [19]. Set  N 1,0 Y = ξ ∈ (T 1,0 X|Y )∗ s.t. ξ |Y  = 0, ∀Y ∈ T 1,0 Y , and consider the subbundle K = T 0,1 Y ⊕ N 1,0 Y of A∗ . Definition 3.17. A complex submanifold Y of X is called coisotropic if H (u, v) = 0, for all u, v ∈ K . Example 3.18. If H = π is a holomorphic Poisson bivector field, then Y is coisotropic if and only if it is coisotropic in the usual sense, i.e. π (ξ1 , ξ2 ) = 0, ∀ξ1 , ξ2 ∈ N 1,0 Y , or π  (N 1,0 Y ) ⊆ T 1,0 Y . Example 3.19. If H = ω, then Y is coisotropic if and only if ι∗ ω = 0, where ι : Y → X is the embedding map. Example 3.20. If H = θ , then Y is coisotropic if and only if θ (T 0,1 Y ) ⊆ T 1,0 Y . It is well known that given a coisotropic submanifold C of a real Poisson manifold

 (P, π ), the conormal bundle N C = ξ ∈ Tc∗ P s.t. c ∈ C; ξ |X  = 0, ∀X ∈ Tc C is a Lie subalgebroid of the cotangent Lie algebroid (T ∗ P)π [31]. The following proposition can be considered as an analogue of this fact in the extended Poisson setting. Proposition 3.21. Let Y be a coisotropic submanifold of the extended Poisson manifold (X, H ). Then the vector subbundle K = T 0,1 Y ⊕ N 1,0 Y is a Lie subalgebroid of A∗H . That is, a∗H maps K into TC Y and for any smooth extensions  u , v ∈ (A∗H ) to X of any H two sections u, v ∈ (K ), the restriction to Y of [ u , v ]∗ is a section of K which does not depend on the choice of extensions.

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3.6. Poisson relations. Following Weinstein [31], we introduce the following Definition 3.22. Let (X 1 , H1 ) and (X 2 , H2 ) be extended Poisson manifolds. A Poisson relation from (X 2 , H2 ) to (X 1 , H1 ) is a coisotropic submanifold of the product manifold ∨ X1 × X∨ 2 (i.e. X 1 × X 2 endowed with the extended Poisson structure (H1 , H2 ), see Corollary 3.4). We call a holomorphic map f : X 2 → X 1 between extended Poisson manifolds (X 1 , H1 ) and (X 2 , H2 ) an extended Poisson map if its graph G f = {( f (x), x) s.t. x ∈ X 2 } ⊂ X 1 × X ∨ 2 is a Poisson relation. Proposition 3.23. Let (X 1 , H1 ) and (X 2 , H2 ) be extended Poisson manifolds, where the extended Poisson structures decompose as Hi = πi + θi + ωi (i = 1, 2). Then a holomorphic map f : X 2 → X 1 is an extended Poisson map if and only if f ∗ π2 = π1 ; f ∗ ω1 = ω2 ; and f ∗ ◦ θ2 = θ1 ◦ f ∗ . The proof is a direct verification and is left to the reader. As a consequence, we have Corollary 3.24. The composition of two extended Poisson maps is again an extended Poisson map. 4. Koszul-Brylinski Poisson Homology In this section we will introduce homology groups for extended Poisson manifolds based on the Evens-Lu-Weinstein module of a Lie algebroid. 4.1. Koszul-Brylinski cochain complex. First we recall the notion of Clifford algebras and spin representation. Let V be a vector space of dimension n endowed with a nondegenerate symmetric bilinear form (·, ·). Its Clifford algebra C(V ) is defined as the quotient of the tensor algebra ⊕nk=0 V ⊗k by the relations x ⊗ y + y ⊗ x = 2(x, y), with x, y ∈ V . It is naturally an associative Z2 -graded algebra. Up to isomorphisms, there exists a unique irreducible module S of C(V ) called spin representation [9]. The vectors of S are called spinors. An operator O on S is called even (or of degree 0) if O(S i ) ⊂ S i and odd (or of degree 1) if O(S i ) ⊂ S i+1 . Here i ∈ Z2 . If O1 and O2 are operators of degree d1 and d2 respectively, then their commutator is the operator O1 , O2  = O1 ◦ O2 − (−1)d1 d2 O2 ◦ O1 . Example 4.1. Let W be a vector space of dimension r . We can endow V = W ⊕ W ∗ with the non-degenerate pairing 1 (ξ1 (u 2 ) + ξ2 (u 1 )) , 2 where u 1 , u 2 ∈ W and ξ1 , ξ2 ∈ W ∗ . The representation of C(V ) on S = ⊕rk=0 ∧k W defined by u · w = u ∧ w and ξ · w = ιξ w, where u ∈ W , ξ ∈ W ∗ and w ∈ S, is the spin representation. Note that S is Z- and thus also Z2 -graded. (u 1 + ξ1 , u 2 + ξ2 ) =

Recall that E = TC X ⊕ TC∗ X admits the standard pseudo-metric 1 (X 1 + ξ1 , X 2 + ξ2 ) = ( ξ1 |X 2  + ξ2 |X 1 ) , 2

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where X i ∈ TC X and ξi ∈ TC∗ X. The corresponding Clifford bundle C(E) can be identified with the vector bundle (∧• TC X) ⊗ (∧• TC∗ X), under which the Clifford action of C(E) on the spinor bundle  (T p,q X)∗ ∧• TC∗ X = p,q

is given by (W ⊗ ξ ) · λ = (−1)

w(w−1) 2

ιW (ξ ∧ λ).

Here W ∈ ∧w TC X, ξ, λ ∈ ∧• TC∗ X, and the symbol ιW denotes the standard contraction

ιW ξ |X  = ξ |W ∧ X , for ξ ∈ ∧ p TC∗ X and X ∈ ∧ p−w TC X with p ≥ w. Let (X, H ) be an extended Poisson manifold of complex dimension n. Then A∗H is a Lie algebroid and the Evens-Lu-Weinstein module [12] of A∗H is the complex line bundle Q A∗H = ∧2n A∗H ⊗ ∧2n TC∗ X. The representation of A∗H on Q A∗H is given by ∇αH (α1 ∧ · · · ∧ α2n ⊗ μ) =

2n    α1 ∧ · · · ∧ [α, αi ]∗H ∧ · · · ∧ α2n ⊗ μ i=1

+α1 ∧ · · · ∧ α2n ⊗ L a∗H (α) μ, where α, α1 , . . . , α2n ∈ (A∗H ), μ ∈ (∧2n TC∗ X). A simple computation yields that Q A∗H ∼ = ∧n (T 1,0 X)∗ ⊗ ∧n (T 1,0 X)∗ . Accordingly, 1

L = Q A2 ∗ ∼ = ∧n (T 1,0 X)∗ = (T n,0 X)∗ H

A∗H -module

is also an and we use ∇ H again to denote the representation. Equivalently, we have an operator D H : (L ) → (A ⊗ L ),

(15)

such that ια D H s = ∇αH s, ∀α ∈ (A∗ ), s ∈ (L ), which allows us to define a differential operator d˘∗H : (∧k A ⊗ L ) → (∧k+1 A ⊗ L ) by d˘∗H (u ⊗ s) = (∂¯ H u) ⊗ s + (−1)k u ∧ D H s, for all u ∈ (∧k A) and s ∈ (L ). The following lemma is needed later.

(16)

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Lemma 4.2. The relation τ (X ⊗ s) = X · s, where in the r.h.s. X ∈ ∧k A is regarded as an element of the Clifford algebra C(E) and s ∈ L is regarded as an element in ∧• TC∗ X, defines an isomorphism of vector bundles  (T i, j X)∗ . τ : ∧k A ⊗ L → i− j=n−k

Equivalently, τ ((W ∧ ξ ) ⊗ s) = (−1) T w,0 X,

w(w−1) 2

ιW (ξ ∧ s) = (−1)

w(w−1) +n(k−w) 2

(ιW s) ∧ ξ,

(T 0,k−w X)∗

for W ∈ ξ∈ and s ∈ L . We define the inner product of H ∈ (∧2 A) with λ ∈ (∧• TC∗ X) as ι H λ = −H · λ. This coincides with the usual inner product of bivector fields with differential forms. Introduce ∂, ι H  = ∂ ◦ ι H − ι H ◦ ∂ : (∧• TC∗ X) → (∧• TC∗ X). Let us denote i, j (X) = ((T i, j X)∗ ). The following theorem is the main result in this section. Theorem 4.3. The diagram (∧k A ⊗ L ) τ





i− j=n−k

i, j (X)

d˘∗H

¯ ∂+∂,ι H

/

/ (∧k+1 A ⊗ L ) 



(17)

τ

i, j i− j=n−k−1  (X)

commutes.

 Definition 4.4. The cohomology of the cochain complex ( i− j=n−k i, j (X), ∂¯ + ∂, ι H ) is called the Koszul-Brylinski Poisson homology of the extended Poisson manifold (X, H ), and denoted H• (X, H ). Remark 4.5. (a) If H = π is a holomorphic Poisson bivector field, the cochain complex ( i− j=n−k i, j (X), ∂¯ + ∂, ι H ) is the total complex of a double complex. Its cohomology is the usual Koszul-Brylinski Poisson homology of a holomorphic Poisson manifold, as studied in detail by one of the authors [30]. ¯ = 0, the complex ( i− j=n−k i, j (X), ∂¯ + ∂, ι H ) (b) If H = ω ∈ 0,2 (X) with ∂ω  becomes ( i− j=n−k i, j (X), ∂¯ +(∂ω)∧). Its cohomology is the twisted Dolbeault cohomology. (c) If H = θ ∈ 0,1 (T 1,0 X) is a Maurer-Cartan element such that θ ◦θ −id is invertible, then θ defines a new complex structure on X. According to Lemma 3.15, the cochain complex ( i− j=n−k i, j (X), ∂¯ + ∂, ι H ) is isomorphic to ( i− j=n−k i, j ¯ θ (X), ∂¯θ ), where ∂¯θ is the ∂-Dolbeault operator of the deformed complex struci, j i, j ture. As a consequence, we have Hk (X, θ ) ∼ = ⊕ j−i=n−k Hθ (X), where Hθ (X) is the Dolbeault cohomology of the deformed complex structure.

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4.2. Evens-Lu-Weinstein duality. Consider a compact complex (and therefore orientable) manifold X with dimC X = n, a complex Lie algebroid B over X with rk C B = r . According to [12], the complex line bundle Q B = ∧r B ⊗ ∧2n TC∗ X is a module over 1

1

the complex Lie algebroid B. If Q B2 exists as a complex vector bundle, Q B2 becomes a B-module as well. There is a natural map 1

1

φ : (∧k B ∗ ⊗ Q B2 ) ⊗ (∧r −k B ∗ ⊗ Q B2 ) → (∧r B ∗ ⊗ Q B ) ∼ = (∧2n TC∗ X). Integrating, we get the pairing 1



1

(∧k B ∗ ⊗ Q B2 ) ⊗ (∧r −k B ∗ ⊗ Q B2 ) → C,

ξ ⊗ η →

X

φ(ξ ⊗ η).

(18)

The following result is essentially due to Evens-Lu-Weinstein [12] for the pairing, and to Block [4] for the non-degeneracy (see also [30]). Theorem 4.6. For a complex Lie algebroid B, with rk C B = r , over a compact manifold X, the pairing (18) induces a pairing 1

1

Hk (B, Q B2 ) ⊗ Hr −k (B, Q B2 ) → C. Moreover, if B is an elliptic Lie algebroid, this pairing is non-degenerate. Let (X, H ) be a compact extended Poisson manifold of complex dimension n. Consider the Lie algebroid B = (T 0,1 X  (T 1,0 X)∗ ) H . Applying Theorem 4.6 and Proposition 3.10, we obtain Theorem 4.7. Let (X, H ) be a compact extended Poisson manifold of complex dimension n, with H = π + θ + ω. Then the map  i, j (X) ⊗ k,l (X) → C : ζ ⊗ η → (ζ ∧ η)top X

induces a pairing on the Koszul-Brylinski Poisson homology: Hk (X, H ) ⊗ H2n−k (X, H ) → C.

(19)

Moreover, if the bundle map F = (C +θ ) ⊕ π  maps T 0,1 X ⊕ (T 1,0 X)∗ surjectively onto T 1,0 X, then all homology groups H• (X, H ) are finite dimensional vector spaces and the pairing (19) is non-degenerate. 4.3. Proof of Theorem 4.3. The following lemmas are needed. Lemma 4.8. For any u ∈ (∧ p A), λ ∈ •,• (X), one has ¯ ¯ · λ) = (∂u) ¯ · λ + (−1) p u · ∂λ. ∂(u

(20)

Lemma 4.9. For any u ∈ (∧ p A), v ∈ (∧q A), the Schouten bracket [u, v] is determined by [u, v] · λ = (−1)q+1 u, v, ∂λ, ∀λ ∈ •,• (X). Both lemmas can be proved by induction; this is left to the reader.

(21)

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Lemma 4.10. For any u ∈ (∧i A) and λ ∈ •,• (X), one has ∂, ι H (u · λ) = [H, u] · λ + (−1)i u · (∂, ι H λ).

(22)

In particular, for any smooth function f ∈ C ∞ (X, C), one has ∂, ι H ( f λ) = [H, f ] · λ + f ∂, ι H λ.

(23)

Proof. According to Eq. (21), we have [H, u] · λ =(−1)i+1 H, u, ∂λ =(−1)i (u · ∂(H · λ) − H · u · (∂λ)) + (H · (∂(u · λ)) − ∂(u · H · λ)) =(−1)i (u · ∂(H · λ) − u · H · (∂λ)) + (H · (∂(u · λ)) − ∂(H · u · λ)) = − (−1)i u · (∂, ι H λ) + ∂, ι H (u · λ).   A straightforward (though lengthy) computation shows the following: Lemma 4.11. Suppose that (z 1 , . . . , z n ) is a local holomorphic chart and H = π +θ +ω is given by H = π i, j

∂ ∂ p ∂ ∧ j + θq p ∧ d z¯ q + ωk,l d z¯ k ∧ d z¯ l , ∂z i ∂z ∂z

(24)

p

where π i, j , θq , and ωk,l are complex valued smooth functions on X. Then the H -twisted Lie algebroid structure on A∗H ∼ = T 0,1 X ⊕ (T 1,0 X)∗ can be expressed by:     ∂ ∂ ∂ p ∂ = i − θi , a∗H dz i = 2π i,q q , (25) a∗H i p ∂ z¯ ∂ z¯ ∂z ∂z   H  H  ∂ ∂ H j i j i, j j ∂ , = 2∂ωi, j , dz , dz = 2∂π , dz , i = ∂θi . (26) ∗ ∂ z¯ i ∂ z¯ j ∗ ∂ z¯ ∗ Lemma 4.12. Making the same assumptions as in Lemma 4.11, consider the local section s = dz 1 ∧ · · · ∧ dz n

(27)

1 2

of L = Q A∗ . The representation of A∗H on L is given by H

p

∇ H∂ s = − ∂ z¯ i

∂θi s, ∂z p

H ∇dz is = 2

∂π i, p s. ∂z p

(28)

Proof. Using Eq. (25), we compute LaH ( ∗

∂ ∂ z¯ i

) dz

j

j

= −dθi ,

L a∗H (dz i ) dz j = 2dπ i, j ,

LaH ( ∗

∂ ∂ z¯ i

) d z¯

j

= 0,

L a∗H (dz i ) d z¯ j = 0.

(29)

Write s2 = (

∂ ∂ ∧ · · · ∧ n ∧ dz 1 ∧ · · · ∧ dz n ) ⊗ (dz 1 ∧ · · · ∧ dz n ∧ d z¯ 1 ∧ · · · ∧ d z¯ n ). ∂ z¯ 1 ∂ z¯

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183

Then, using Eqs. (26) and (29), one obtains p

∇ H∂ s 2 = −2 ∂ z¯ i

∂θi 2 s , ∂z p

The conclusion thus follows immediately.

H 2 ∇dz is = 4

∂π i, p 2 s . ∂z p

 

Corollary 4.13. Locally, the operator D H in Eq. (15) is given by

p i, p ∂ ∂θ ∂π i − p d z¯ i ⊗ s, DH s = 2 ∂z p ∂z i ∂z

(30)

where s is defined in Eq. (27). We are now ready to prove Theorem 4.3. Proof of Theorem 4.3. We adopt an inductive approach. First we prove the commutativity of Diagram (17) for k = 0. Note that for any f ∈ C ∞ (X, C), u ∈ (∧k A) and s ∈ (L ), one has   by Eq. (16) τ d˘∗H ( f u ⊗ s) = τ f d˘∗H (u ⊗ s) + ((∂¯ f + [H, f ]) ∧ u) ⊗ s = f τ d˘∗H (u ⊗ s) + (∂¯ f + [H, f ]) · τ (u ⊗ s). On the other hand, if we write λ = τ (u ⊗ s), one has (∂¯ + ∂, ι H )τ ( f u ⊗ s) = (∂¯ + ∂, ι H )( f λ) ¯ + [H, f ] · λ + f ∂, ι H λ by Eq. (23) = ∂¯ f ∧ λ + f ∂λ = f (∂¯ + ∂, ι H )τ (u ⊗ s) + (∂¯ f + [H, f ]) · τ (u ⊗ s). It thus follows that the map τ ◦ d˘∗H − (∂¯ + ∂, ι H ) ◦ τ is C ∞ (X)-linear. Take a local holomorphic chart (z 1 , . . . , z n ) and write H locally as in Eq. (24) in Lemma 4.11. Again take s as in Eq. (27). For k = 0, we have d˘∗H s = D H s, which is given locally by Eq. (30). Then, we compute   τ d˘∗H s =



=2

p

∂θ ∂π i, p ∂ 2 − ip d z¯ i ∂z p ∂z i ∂z

· (dz 1 ∧ · · · ∧ dz n )

p n  ∂π i, p 1 i ∧ · · · ∧ dz n − ∂θi d z¯ i ∧ dz 1 ∧ · · · ∧ dz n . (−1)i+1 dz ∧ · · · ∧ dz ∂z p ∂z p i=1

Thus we have (∂¯ + ∂, ι H )s = ∂ι H (dz 1 ∧ · · · ∧ dz n ) ⎛  i ∧ · · · ∧ dz j ∧ · · · ∧ dz n (−1)i+ j−1 π i, j dz 1 ∧ · · · ∧ dz = ∂ ⎝2 i< j

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Z. Chen, M. Stiénon, P. Xu

+

n 

p (−1) p+1 θi d z¯ i ∧ dz 1 ∧ · · · ∧  dz p ∧ · · · ∧ dz n

p=1

⎞

−ωk,l d z¯ k ∧ d z¯ l ∧ dz 1 ∧ · · · · · · ∧ dz n ⎠ = τ (d˘∗H s). It thus follows that Diagram (17) indeed commutes when k = 0. Now assume that we have proved the commutativity of Diagram (17) when k ≤ m (where 0 ≤ m ≤ 2n−1). To prove the k = m +1 case, we consider a section (u ∧w)⊗s ∈ (∧m+1 A ⊗ L ), where u ∈ (A), w ∈ (∧m A) and s ∈ (L ). Then (∂¯ + ∂, ι H )τ ((u ∧ w) ⊗ s) = (∂¯ + ∂, ι H )(u · λ)

where λ = w · s

¯ · λ − u · ∂λ ¯ + [H, u] · λ − u · (∂, ι H λ) = ∂u H = ∂¯ u · λ − u · (∂¯ + ∂, ι H )λ   = τ (∂¯ H u ∧ w) ⊗ s − u · τ d˘∗H (w ⊗ s)

by Eqs. (20) and (22)

by assumption

= τ d˘∗H ((u ∧ w) ⊗ s). This concludes the proof.

 

4.4. Modular classes. The modular class of a Lie algebroid was introduced by Evens-Lu-Weinstein [12]. The following version for complex Lie algebroids appeared in the preprint version of [12] but not in the published paper. It is also implied in [14]. The presentation which we give below was communicated to us by Camille LaurentGengoux [23]. Let B be a complex Lie algebroid over a real manifold M, with rk C B = r and dim M = m. Its Evens-Lu-Weinstein module is Q B = ∧r B ⊗ ∧m TC∗ M. Consider the complex of sheaves ˜

dB dB dB S0 −→ S 1 −→ S 2 · · · −→ S r ,

(31)

where S0 is the sheaf of nowhere vanishing smooth complex valued functions on M; S • is the sheaf of sections of ∧• B ∗ ; d B is the usual Lie algebroid cohomology differential; and  d B f = d B log f = d Bf f , for all f ∈ C ∞ (U, C× ), where U is an arbitrary open subset of M. We denote its hypercohomology by  H• (B, C). Note that in Eq. (31), if we replace S0 by S 0 , the sheaf of smooth complex valued functions on M, and d˜B by the usual Lie algebroid differential d B , the hypercohomology of the resulting complex of sheaves dB

dB

dB

S 0 −→ S 1 −→ S 2 · · · −→ S r , H• (B, C)

(32)

of the complex Lie is isomorphic to the usual Lie algebroid cohomology algebroid B with trivial coefficients C since each S • is a soft sheaf. The exponential sequence 0 → Z → S → S → 0,

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where S (resp. S) stands for the complex of sheaves (32) (resp. (31)) and the locally constant sheaf Z is regarded as a complex of sheaves concentrated in degree 0, induces the long exact sequence · · · → Hi (M, Z) → Hi (B, C) →  Hi (B, C) → Hi+1 (M, Z) → · · · . ˇ double comNote that  H• (B, C) can be computed as the total cohomology of the Cech plex · ·O ·

· ·O ·

δ

Cˇ 2 (U;O S0 )

δ  dB

δ

Cˇ 1 (U;O S0 )

/ Cˇ 2 (U; S 1 ) O

/ Cˇ 2 (U; S 2 ) O

dB

/ ···

dB

/ ···

dB

/ ··· ,

δ

/ Cˇ 1 (U; S 1 ) O

dB

/ Cˇ 1 (U; S 2 ) O

δ  dB

(33)

δ dB

δ  dB

δ

Cˇ 0 (U; S0 )

· ·O ·

δ

/ Cˇ 0 (U; S 1 )

dB

/ Cˇ 0 (U; S 2 )

ˇ coboundary where U = {Ui }i∈I is a good open cover of M and δ is the usual Cech operator. Let (Ui )i∈I be a good open cover of M, and ωi a nowhere vanishing section of Q B over Ui . For all i, j ∈ I , there exists a unique nowhere vanishing function fi j ∈ C ∞ (Ui j , C× ) such that ωi = f i j ω j . It is clear from the construction that f i j f jk f ki = 1. Let ξi ∈ (B ∗ |Ui ) be the modular 1-form on Ui corresponding to ωi . That is, we have ∇ X ωi = ξi |X ωi for all X ∈ (B|Ui ), where ∇ denotes the canonical representation of B on Q B of [12]. It thus follows that ξi = ξ j +

d B fi j = ξ j + d˜B f i j . fi j

As a consequence, (ξi , f i j ) is a 1-cocycle of the double complex (33), and therefore defines a class in  H1 (B, C). Definition 4.14. The class in  H1 (B, C) defined by [(ξi , f i j )] is called the modular class of the complex Lie algebroid B, and denoted mod(B). Lemma 4.15. Consider the long exact sequence τ · · · → H1 (B, C) →  H1 (B, C) − → H2 (M, Z) → · · · .

The image of the modular class mod(B) under τ is the first Chern class c1 (Q B ) of Q B . When c1 (Q B ) = 0, the modular class mod(B) is the image of a class in H1 (B, C), which is defined exactly in the same way using a global nowhere vanishing section, as the usual modular class in [12]. A complex Lie algebroid B is said to be unimodular if its modular class vanishes. The following result follows immediately from Lemma 4.15.

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Corollary 4.16. A complex Lie algebroid B is unimodular if and only if c1 (Q B ) = 0 and for any fixed nowhere vanishing section ω ∈ (Q B ), the modular section ξ ∈ (B ∗ ) defined by ∇ X ω = ξ |X ω

(∀X ∈ (B))

is a coboundary, i.e. ξ = d B f for some f ∈ C ∞ (M, C). As a consequence, a complex Lie algebroid B is unimodular if and only if Q B is isomorphic to the trivial module C. Proposition 4.17. When B = T 0,1 X  A1,0 is the derived complex Lie algebroid [24,30] of a holomorphic Lie algebroid A over X, B is a unimodular complex Lie algebroid if and only if A is a unimodular holomorphic Lie algebroid, i.e. Q A is trivial as a holomorphic line bundle and there exists a holomorphic global section ω of Q A such that ∇ X ω = 0 for all X ∈ A. Definition 4.18. An extended Poisson manifold (X, H ) is unimodular if its corresponding complex Lie algebroid A∗H is unimodular. According to Theorem 4.3, we have Proposition 4.19. An extended Poisson manifold (X, H ) is unimodular if and only if there exists a nowhere vanishing (n, 0)-form ω ∈ n,0 (X) such that ¯ + ∂ι H ω = 0. ¯ + ∂, ι H ω = ∂ω ∂ω Remark 4.20. It is clear that, when H = 0, (X, H ) is unimodular means that X is Calabi-Yau. Thus one can consider a unimodular extended Poisson manifold (X, H ) as a generalized Calabi-Yau manifold. As an immediate consequence of the discussion above, we have Corollary 4.21. For any unimodular extended Poisson manifold (X, H ) of complex dimension n, we have Hk (X, H ) ∼ = H2n−k (X, H ). Acknowledgements This work benefited from a US–China Collaboration in Mathematical Research grant awarded by the National Science Foundations of China and the US. We would like to thank Penn State University (Chen), ETH Zurich (Xu) and Peking University (Stiénon and Xu) for their hospitality while work on this project was being done. We also wish to thank many people for useful discussions and comments, including Camille Laurent-Gengoux, Giovanni Felder, Jiang-Hua Lu, Pierre Schapira and Alan Weinstein.

References 1. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111(1), 61–110 (1978) 2. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. II. Physical applications. Ann. Phys. 111(1), 111–151 (1978) 3. Berest, Y., Etingof, P., Ginzburg, V.: Morita equivalence of Cherednik algebras. J. Reine Angew. Math. 568(2004), 81–98 (2004) 4. Block, J.: Duality and equivalence of module categories in noncommutative geometry I. http://arxiv.org/ abs/math/0509284v2[math.QA], 2009

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5. Bondal, A.: Non-commutative deformations and Poisson brackets on projective spaces. Max-PlanckInstitut Preprint 67 (1993) 6. Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Deformation quantization of gerbes. Adv. Math. 214(1), 230–266 (2007) 7. Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Deformations of gerbes on smooth manifolds. K theory and noncommutative geometry, pp. 349–392. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2008) 8. Caine, A.: Toric Poisson structures. http://arxiv.org/abs/0910.0229v1[math.SG], 2009 9. Chevalley, C.: The algebraic theory of spinors and Clifford algebras. In: Collected works. Vol. 2; Edited and with a foreword by Pierre Cartier and Catherine Chevalley; With a postface by J.-P. Bourguignon, Berlin: Springer-Verlag, 1997 10. Cleyton, R., Poon, Y.-S.: Differential Gerstenhaber algebras associated to nilpotent algebras. Asian J. Math., 12(2), 225–249 (2008) 11. Evens, S., Lu, J.-H.: Poisson harmonic forms, Kostant harmonic forms, and the S 1 -equivariant cohomology of K /T . Adv. Math., 142(2), 171–220 (1999) 12. Evens, S., Lu, J.-H., Weinstein, A.: Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50(200), 417–436 (1999); see also http://arxiv.org/ abs/dg-ga/961008v1, 1996 13. Gualtieri, M.: Generalized complex geometry, Ph.D. Thesis, Oxford University, 2003. http://arxiv.org/ abs/math/0401221v1[math.DG], 2004 14. Gualtieri, M.: Generalized complex geometry. http://arxiv.org/abs/math/0703298v2[math.DG], 2007 15. Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math., 54(3), 281–308 (2003) 16. Kashiwara, M.: Quantization of contact manifolds. Publ. Res. Inst. Math. Sci. 32(1), 1–7 (1996) 17. Kashiwara, M., Schapira, P.: Deformation quantization modules I: Finiteness and duality. http://arxiv. org/abs/0802.1245v3[math.QA], 2008 18. Kashiwara, M., Schapira, P.: Deformation quantization modules II: Hochschild class. http://arxiv.org/ abs/0809.4309v1[math.AG], 2008 19. Kodaira, K.: Complex manifolds and deformation of complex structures. Reprint of the 1986 English edition, In: Classics in Mathematics, Translated from the 1981 Japanese original by Kazuo Akao, Berlin: Springer-Verlag, 2005 20. Kontsevich, M.: Deformation quantization of algebraic varieties. In: EuroConférence Moshé Flato 2000, Part III (Dijon) Lett. Math. Phys. 56, no. 3, 271–294 21. Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2006) 22. Kosmann-Schwarzbach, Y.: Exact Gerstenhaber algebras and Lie bialgebroids. Acta Appl. Math. 41(1–3), 153–165 (1995) 23. Laurent-Gengoux, C.: Private communication 24. Laurent-Gengoux, C., Stiénon, M., Xu, P.: Holomorphic Poisson manifolds and holomorphic Lie algebroids. Int. Math. Res. Not. IMRN 2008:Art. ID rnn 088, 46 (2008) 25. Laurent-Gengoux, C., Stiénon, M., Xu, P.: Non-abelian differential gerbes. Adv. Math. 220(5), 1357–1427 (2009) 26. Liu, Z.-J., Weinstein, A., Xu, P.: Manin triples for Lie bialgebroids. J. Diff. Geom. 45(3), 547–574 (1997) 27. Mackenzie, K.C.H., Xu, P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73(2), 415–452 (1994) 28. Mackenzie, K.C.H., Xu, P.: Integration of Lie bialgebroids. Topology 39(3), 445–467 (2000) 29. Stiénon, M.: Generalized Moser Lemma. To appear in Trans. Amer. Math. Soc., available at http://arxiv. org/abs/math/0702718v1[math.DG], 2007 30. Stiénon, M.: Holomorphic Koszul-Brylinski Homology. http://arxiv.org/abs/0903.5065v1[math.DG], 2009 31. Weinstein, A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Japan 40(4), 705–727 (1988) 32. Weinstein, A.: The Integration Problem for Complex Lie Algebroids. From geometry to quantum mechanics. Progr. Math., vol. 252, pp. 93–109. Birkhäuser Boston, Boston (2007) 33. Xu, P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200(3), 545–560 (1999) 34. Yekutieli, A.: Central extensions of gerbes. http://arxiv.org/abs/0801.0083v2[math.AG], 2008 35. Yekutieli, A.: Lecture notes: twisted deformation quantization of algebraic varieties. http://arxiv.org/abs/ 0801.3233v2[math.AG], 2008 Communicated by S. Zelditch

Commun. Math. Phys. 297, 189–227 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1035-6

Communications in

Mathematical Physics

Unitary Representations of Nilpotent Super Lie Groups Hadi Salmasian Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa, ON K1N 6N5, Canada. E-mail: [email protected] Received: 13 June 2009 / Accepted: 10 December 2009 Published online: 30 March 2010 – © Springer-Verlag 2010

Abstract: We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for HeisenbergClifford super Lie groups.

1. Introduction 1.1. Background. One of the most elegant results in the theory of unitary representations is the Stone-von Neumann theorem, which yields a classification of irreducible unitary representations of the Heisenberg group. It is the starting point in the study of unitary representations of nilpotent Lie groups, in which it plays an essential role as well. Kirillov’s seminal work on unitary representations of nilpotent Lie groups showed that unitary representations can be obtained in a simple fashion, namely as induced representations from one-dimensional representations of certain subgroups which are called polarizing subgroups. From this, Kirillov deduced a well-behaved correspondence between irreducible unitary representations and coadjoint orbits. Physicists are interested in unitary representations of Lie superalgebras and super Lie groups1 and their applications, e.g. in the classification of free relativistic super particles in SUSY quantum mechanics (see [SaSt] and [FSZ]). Extensions of the Stone-von  This research is partially supported by an NSERC Discovery Grant. 1 We follow [DeMo] and [CCTV] in using the terms super Lie group and sub super Lie group. Nevertheless,

instead of Deligne and Morgan’s super Lie algebra we use the term Lie superalgebra merely because the latter is used in the literature more frequently.

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Neumann theorem to the Heisenberg-Clifford super Lie group, and the oscillator representation to the orthosymplectic case, have been studied widely by physicists as well as mathematicians (see [Ni,Lo, and BaGu]). Nevertheless, much of the work done on infinite-dimensional unitary representations of super Lie groups treats representations algebraically, without addressing the analytic aspects. When the even part of the Lie superalgebra is a reductive Lie algebra (e.g., for classical simple Lie superalgebras) the space of the representation can be identified with the space of K -finite analytic vectors of a unitary representation of the even part on a Hilbert space. This approach has been pursued in [FuNi]. However, this method is not applicable to more general super Lie groups, e.g., the nilpotent ones. Motivated by establishing a rigorous formalism for Mackey-Wigner’s little group method in the super setting, the authors of [CCTV] establish analytic foundations of the theory of unitary representations of super Lie groups. The key observation is that for infinite-dimensional representations, the action of the odd part of the Lie superalgebra is by unbounded operators, and thus one should consider densely defined operators. As shown in [CCTV], it turns out that the correct space to realize the action of the odd part is the (dense) subspace of smooth vectors (in the sense of [Kn, p. 52]) for the even part. 1.2. Our main results. The main goal of this work is to show that irreducible unitary representations (in the sense of [CCTV]) of nilpotent super Lie groups can be described in a way which is very similar to the classical work of Kirillov. More specifically, our results are as follows. (a) We generalize the notion of polarizing subalgebras of nilpotent Lie algebras to what we call polarizing systems in nilpotent super Lie groups. Let (N0 , n) be a nilpotent super Lie group. A polarizing system of (N0 , n) is a 6-tuple (M0 , m, , C0 , c, λ), where (M0 , m) is a sub super Lie group of (N0 , n),  : (M0 , m) → (C0 , c) is a homomorphism onto a super Lie group of Clifford type, and λ ∈ n∗0 . (There are extra conditions which are stated in Definition 6.1.1.) We show that every irreducible unitary representation of a nilpotent super Lie group is induced from a special Clifford module associated to a polarizing system (see part (a) of Theorem 6.1.1). The latter module is said to be consistent with the polarizing system. Conversely, we prove that induction from a consistent representation of a polarizing system always results in an irreducible unitary representation (see Theorem 6.2.1). In other words, we show that induction yields the following surjective map: ⎧ ⎫ ⎧ ⎫ Irreducible unitary ⎨ Ordered pairs of polarizing ⎬ ⎨ ⎬ systems of (N0 , n) and their −→ representations of (N0 , n) . ⎩ consistent representations ⎭ ⎩ up to unitary equivalence ⎭ (b) Given a λ ∈ n∗0 , we obtain a simple necessary and sufficient condition for the existence of a polarizing system (M0 , m, , C0 , c, λ) with a consistent representation. For every λ ∈ n∗0 , consider the symmetric bilinear form Bλ : n1 × n1 → R defined by Bλ (X, Y ) = λ([X, Y ]). In Sect. 6.4 we prove that such a polarizing system with a consistent representation exists if and only if Bλ is nonnegative definite.

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(c) We obtain a concrete geometric parametrization of irreducible unitary representations of (N0 , n). Let n+0 = { λ ∈ n∗0 | Bλ is nonnegative definite }. It is easily checked that n+0 is a union of coadjoint orbits. In Theorem 6.6.1 we prove that the inducing construction outlined above yields the following bijective correspondence: ⎧ ⎫ ⎨ Irreducible unitary representations of ⎬  (N0 , n) up to unitary equivalence N0 -orbits in n+0 ←→ . ⎩ ⎭ and parity change (d) As a simple application of our results, we obtain a proof of an analytic formulation of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups (see Sect. 5.2). We believe that this analytic formulation is new. We would like to mention that in [Ro], the author studies a generalization of the Stone-von Neumann theorem to the super case. Our approach has the advantage that it yields a concrete statement based on a rigorous and more general notion of unitary representation for super Lie groups, and avoids the assumption that the odd part have even dimension. (e) A consequence of part (b) of Theorem 6.1.1 is a numerical invariant of irreducible unitary representations of nilpotent super Lie groups. The value of the invariant is a positive integer, and is equal to one if and only if up to parity change the representation is purely even, i.e., in its Z2 -grading the odd summand is trivial. In conclusion, this work is yet another justification for fruitfulness of the approach pursued in [CCTV] to define and study unitary representations of super Lie groups rigorously.

1.3. Organization of the paper. This paper is organized as follows. Section 2 is devoted to recalling some basic definitions and facts about super Lie groups and their unitary representations. In Sect. 3 we recall the notion of induction of unitary representations from special sub super Lie groups which was introduced in [CCTV], and prove that it can be done in stages (see Proposition 3.2.1). Section 4 is devoted to studying the structure of nilpotent super Lie groups, proving a version of Kirillov’s Lemma, and classification of representations of super Lie groups of Clifford type. Section 5.1 contains a technical but important result. In this section we prove that under certain conditions a unitary representation is induced from a sub super Lie group of codimension one. Although this result is analogous to one of Kirillov’s original results, there are several delicate issues involving unbounded operators which need to be dealt with. Using the main result of Sect. 5.1, in Sect. 5.2 we obtain a proof of an analytic formulation of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups. In Sect. 6.1 we define polarizing systems, prove the surjectivity of the map from induced representations to irreducible representations, and show that if two polarizing systems yield the same representation then they correspond to the same coadjoint orbit. In Sect. 6.3 we prove the existence of a special kind of polarizing Lie subalgebra of n0 . This section is fairly technical and contains several lemmas, but the main point is to prove Lemma 6.3.4. In Sect. 6.4 we state and prove our main result on parametrization of representations by coadjoint orbits (see Theorem 6.6.1).

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2. Preliminaries 2.1. Notation and basic definitions. Recall that a densely defined operator T on a Hilbert space H is called symmetric if for every v, w ∈ D(T ) we have T v, w = v, T w, where D(T ) denotes the domain of T . By a Z2 -graded Hilbert space we mean a Hilbert space H with an orthogonal decomposition H = H0 ⊕ H1 . A densely defined linear operator T on H is called even (respectively, odd) if its domain D(T ) is Z2 -graded, i.e., D(T ) = D(T )0 ⊕ D(T )1 , where for every i ∈ {0, 1} we have D(T )i = D(T ) ∩ Hi , and for every v ∈ D(T )i we have T v ∈ Hi (respectively, T v ∈ H1−i ). For basic definitions and facts about Lie superalgebras, we refer the reader to [DeMo] and [Ka]. Unless explicitly stated otherwise, in this paper all Lie algebras and Lie superalgebras are over R. If g is a Lie superalgebra, its centre and universal enveloping algebra are denoted by Z(g) and U(g). Similarly, the centre of a Lie group G is denoted by Z(G). If a Lie group G acts on a vector space V, then the action of an element g ∈ G on a vector v ∈ V is denoted by g · v. Following [DeMo], our definition of a super Lie group is based on the notion of a Harish-Chandra pair. One can define a super Lie group concretely as follows. Definition 2.1.1. A super Lie group is a pair (G 0 , g) with the following properties: (a) g = g0 ⊕ g1 is a Lie superalgebra over R. (b) G 0 is a connected real Lie group with Lie algebra g0 which acts on g smoothly via R-linear automorphisms. (c) The action of G 0 on g0 is the adjoint action. The adjoint action of g0 on g is the differential of the action of G 0 on g. A super Lie group (H0 , h) is called a sub super Lie group of a super Lie group (G 0 , g) if H0 is a Lie subgroup of G 0 and h = h0 ⊕ h1 is a subalgebra2 of g such that h0 is the Lie subagebra of g0 corresponding to H0 and the action of H0 on h is the restriction of the action of G 0 on g. Let (G 0 , g) and (G 0 , g ) be arbitrary super Lie groups. A homomorphism  : (G 0 , g) → (G 0 , g ) consists of a Lie group homomorphism from G 0 to G 0 and a homomorphism of Lie superalgebras from g to g which are compatible with each other. We say  is surjective if both of these homomorphisms are surjective in the usual sense. If (π, H) is a unitary representation of a Lie group G on a Hilbert space H, then the subspace of smooth vectors of H for the action of G is denoted by H∞ . The infinitesimal action of the Lie algebra of G on H∞ is denoted by π ∞ . The definition of unitary representations of super Lie groups, which is given below, is originally introduced in [CCTV]. 2 In this paper, instead of the term sub super Lie algebra of [DeMo] we use the abbreviation subalgebra.

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Definition 2.1.2. A unitary representation of (G 0 , g) is a triple (π, ρ π , H) such that H is a Z2 -graded Hilbert space endowed with a unitary representation π of G 0 , and ρ π : g1 → EndC (H∞ ) is an R-linear map with the following properties. (a) For every g ∈ G 0 , π(g) is an even operator on H. (b) For every X ∈ g1 , ρ π (X ) is an odd linear operator. Moreover, ρ π (X ) is symmetric, i.e., for every v, w ∈ H∞ , we have ρ π (X )v, w = v, ρ π (X )w. (c) For every X, Y ∈ g1 and v ∈ H∞ , we have

√ ρ π (X )ρ π (Y )v + ρ π (Y )ρ π (X )v = − −1π ∞ ([X, Y ])v.

(d) For every g ∈ G 0 and X ∈ g1 , we have ρ π (g · X ) = π(g)ρ π (X )π(g −1 ). Remark. 1. One can combine ρ π and π ∞ to obtain a representation of g in H∞ , where π√ −1 ∞ π an element X 0 + X 1 ∈ g0 ⊕ g1 acts by π (X 0 ) + e 4 ρ (X 1 ). Consequently, from [CCTV, Prop. 1] it follows that for every X ∈ g0 , Y ∈ g1 , and v ∈ H∞ we have ρ π ([X, Y ])v = π ∞ (X )ρ π (Y )v − ρ π (Y )π ∞ (X )v. 2. From the closed graph theorem for Fréchet spaces, it follows that for every X ∈ g1 , ρ π (X ) is a continuous operator on H∞ .

Given two unitary representations (π, ρ π , H) and (π , ρ π , H ) of (G 0 , g), by an

intertwining operator from (π, ρ π , H) to (π , ρ π , H ) we mean an even bounded linear transformation T : H → H such that for every g ∈ G 0 and X ∈ g1 we have

T π(g) = π (g)T and Tρ π (X ) = ρ π (X )T . (Note that if T π(g) = π (g)T for every g ∈ G 0 , then T H∞ ⊆ H ∞ .)

Two unitary representations (π, ρ π , H) and (π , ρ π , H ) of (G 0 , g) are said to be unitarily equivalent if there exists an isometry T : H → H which is also an intertwining operator. Note that it follows that T H∞ = H ∞ .

From now on, to indicate that two unitary representations (π, ρ π , H) and (π , ρ π , H ) are unitarily equivalent, we write

(π, ρ π , H)  (π , ρ π , H ). A unitary representation (π, ρ π , H) of (G 0 , g) is called irreducible if H does not have any proper (G 0 , g)-invariant closed Z2 -graded subspaces. By [CCTV, Lemma 5], a representation (π, ρ π , H) is irreducible if and only if every intertwining operator from (π, ρ π , H) to itself is scalar. From every unitary representation (π, ρ π , H) we can obtain a new unitary representation (π, ρ π , H), where  is the parity change operator. The operator  can be considered as a special case of the tensor product, namely tensoring (π, ρ π , H) with a trivial (0|1)-dimensional representation. The unitary representations (π, ρ π , H) and (π, ρ π , H) are said to be the same up to parity change. Note that they are not necessarily unitarily equivalent.

From now on, to indicate that two unitary representations (π, ρ π , H) and (π , ρ π , H ) are identical up to unitary equivalence and parity change, we write

(π, ρ π , H)  (π , ρ π , H ).

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2.2. Stability of unitary representations. A remarkable feature of unitary representations as defined in Definition 2.1.2 is their stability, i.e. that one can replace the space H∞ with a variety of dense and invariant subspaces. Stability is needed even for justifying that the restriction of a unitary representation of a super Lie group (G 0 , g) to a sub super Lie group (H0 , h) is well defined, i.e., that the restriction determines a unique unitary representation up to unitary equivalence. The result that justifies the latter statement is [CCTV, Prop. 2]. For the reader’s convenience, and for future reference in this article, we would like to record a slightly simplified formulation of the statement of this proposition. Proposition 2.2.1. [CCTV, Prop. 2]. Let (G 0 , g) be a super Lie group and (π, H) be a unitary representation of G 0 . Suppose B is a dense, Z2 -graded, and G 0 -invariant subspace of H, and {ρ(X )} X ∈g1 is a family of densely defined linear operators on H with the following properties: (a) (b) (c) (d) (e) (f) (g) (h)

B ⊆ H∞ . If X ∈ g1 , then B ⊆ D(ρ(X )). ρ(X ) is symmetric for every X ∈ g1 . For every X ∈ g1 and i ∈ {0, 1} we have ρ(X )Bi ⊆ H1−i . If X, Y ∈ g1 and a, b ∈ R then ρ(a X + bY ) = aρ(X ) + bρ(Y ). π(g)ρ(X )π(g −1 ) = ρ(g · X ) for all g ∈ G 0 and X ∈ g1 . For every X, Y ∈ g1 we have ρ(X )B ⊆ D(ρ(Y )). For every X, Y ∈ g1 and v ∈ B we have √ ρ(X )ρ(Y )v + ρ(Y )ρ(X )v = − −1π ∞ ([X, Y ])v.

Then the following statements hold: (i) For every X ∈ g1 , the operator ρ(X ) is essentially self adjoint, and the closure ρ(X ) of ρ(X ) satisfies H∞ ⊆ D(ρ(X )). (ii) Suppose that for every X ∈ g1 and v ∈ H∞ , we set ρ π (X )v = ρ(X )v. Then for every X ∈ g1 we have ρ π (X ) ∈ EndC (H∞ ). Moreover, (π, ρ π , H) is a unitary representation of (G 0 , g).

(iii) Let (π , ρ π , H) be a unitary representation of (G 0 , g) in the same Z2 -graded Hilbert space H. Suppose that for every g ∈ G 0 we have π (g) = π(g), and for every



X ∈ g1 and v ∈ B we have ρ π (X )v = ρ π (X )v. Then (π , ρ π , H)  (π, ρ π , H), and the intertwining isometry T : H → H yielding this unitary equivalence is the identity map. We conclude this section with a simple but useful lemma about polarizing subalgebras. Let g be a Lie algebra and fix λ ∈ g∗ . Recall that a Lie subalgebra m of g is called a polarizing subalgebra corresponding to λ if m is a maximal isotropic subspace of g for the skew-symmetric bilinear form ωλ : g × g → R defined by ωλ (X, Y ) = λ([X, Y ]). Lemma 2.2.1. Let g be a nilpotent Lie algebra, λ ∈ g∗ , and m ⊆ g be a Lie subalgebra. If λ = 0 and m is a polarizing subalgebra of g corresponding to λ, then there exists an X ∈ m such that λ(X ) = 0. Proof. Suppose, on the contrary, that m ⊆ ker λ. Then from λ = 0 it follows that g  m. If Ng(m) = {Y ∈ g | [Y, m] ⊆ m},

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then Ng(m) is a Lie subalgebra of g and Ng(m)  m. Choose an X ∈ Ng(m) such that X ∈ / m and set m = m ⊕ RX . It is easy to check that [m , m ] ⊆ m and thus

λ([m , m ]) = {0} which contradicts maximality of dimension of m.  

3. Special Induction 3.1. Realization of the induced representation. Let (G 0 , g) be a super Lie group and (H0 , h) be a sub super Lie group of (G 0 , g0 ), i.e., H0 ⊆ G 0 and h ⊆ g. As in [CCTV, §3.2], we assume that (H0 , h) is special, i.e., that h1 = g1 . For every unitary representation (σ, ρ σ , K) of (H0 , h), the representation of (G 0 , g) induced from (σ, ρ σ , K) is defined in [CCTV, §3]. We recall the definition of special induction only in a case which we need in this paper, i.e., when the Lie groups G 0 and H0 are unimodular. In this case, to define the representation (π, ρ π , H) of (G 0 , g) induced from (σ, ρ σ , K), one fixes a G 0 -invariant measure μ on H0 \G 0 and defines H as the space of measurable functions f : G 0 → K such that (a) For

any g ∈ G 0 and h ∈ H0 , we have f (hg) = σ (h) f (g). (b) H0 \G 0 || f (g)||2 dμ < ∞. The action of every g ∈ G 0 on every f ∈ H is the usual right regular representation, i.e., if g, g ∈ G 0 then (π(g) f ) (g ) = f (g g). Recall that H∞ is the space of smooth vectors of (π, H). It is well-known that H∞ ⊆ C ∞ (G 0 , K), where C ∞ (G 0 , K) denotes the space of smooth functions f : G 0 → K (see [Po, Th. 5.1] or [CG, Th. A.1.4]). Moreover, one can check that for every f ∈ H∞ , we have f (G 0 ) ⊆ K∞ . Let H∞,c be the space consisting of functions f : G 0 → K such that f ∈ H ∩ C ∞ (G 0 , K) and Supp( f ) is compact modulo H0 . It is shown in [CCTV, Prop. 4] that H∞,c ⊆ H∞ , the subspace H∞,c is dense in H, and for every X ∈ g0 we have π ∞ (X )H∞,c ⊆ H∞,c . The action of g1 is initially defined on H∞,c . For every X ∈ g1 and f ∈ H∞,c , one defines 

ρ π (X ) f (g) = ρ σ (g · X ) ( f (g)).

(3.1)

From Proposition 2.2.1 it follows that the domain of the closure of ρ π (X ) contains H∞ , and consequently the induced representation (π, ρ π , H) is well-defined. We will denote the latter representation by (G ,g)

Ind(H00 ,h) (σ, ρ σ , K).

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3.2. Special induction in stages. A basic but important property of special induction is that it can be done in stages. The proof of this property is not difficult, but it is not mentioned in [CCTV] explicitly. For the reader’s convenience, we would like to sketch it below. Proposition 3.2.1. Suppose that (G 0 , g) is a super Lie group, (K 0 , k) is a special sub super Lie group of (G 0 , g), and (H0 , h) is a special sub super Lie group of (H0 , h). Assume that G 0 , K 0 , and H0 are unimodular, and let (σ, ρ σ , K) be a unitary representation of (H0 , h). Then (G ,g)

(G ,g)

(K ,k)

Ind(H00 ,h) (σ, ρ σ , K)  Ind(K 00 ,k) Ind(H00 ,h) (σ, ρ σ , K).

(3.2)

Proof. Set (G ,g)

(K ,k)

(π, ρ π , H) = Ind(H00 ,h) (σ, ρ σ , K) , (η, ρ η , L) = Ind(H00 ,h) (σ, ρ σ , K), and (G ,g)

(ν, ρ ν , V) = Ind(K 00 ,k) (η, ρ η , L). Thus H, L, and V are function spaces introduced in Sect. 3.1 which realize the corresponding induced representations. Let H∞,c be the subspace of H defined in Sect. 3.1. We define L∞,c and V∞,c similarly. The intertwining map T : (π, H) → (ν, V) is given in [Ma, §4]. We recall the definition of T . Given a function f : G 0 → K such that f ∈ H∞,c , the function T f : G 0 → L is obtained as follows. For every g ∈ G 0 and k ∈ K 0 , (T f (g)) (k) = f (kg). One can normalize the involved measures such that for every f ∈ H∞,c , we have ||T f || = || f ||. 0 Fix an f ∈ H∞,c and a V ∈ g1 . Since f is a smooth vector for π = Ind G H0 σ and T is an interwining isometry, T f is a smooth vector for H0 0 ν = Ind G H0 Ind K 0 σ.

By Proposition 2.2.1, to prove Proposition 3.2.1 it suffices to show that for every f ∈ H∞,c and V ∈ g1 , Tρ π (V ) f = ρ ν (V )T f.

(3.3)

Since Supp( f ) is compact modulo H0 , it follows readily that Supp(T f ) is compact modulo K 0 , and for every g ∈ G 0 the support of T f (g) : K 0 → K is compact modulo H0 . From [CCTV, Prop. 4] it follows that T f ∈ V∞,c and for every g ∈ G 0 we have T f (g) ∈ L∞,c . By the definition of special induction given in Sect. 3.1,

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the action of ρ ν (V ) on T f is given by (3.1). For the same reason, the action of ρ η (V ) on T f (g) is given by (3.1). Thus for every g ∈ G 0 , and k ∈ K 0 ,  ν  ρ (V )T f (g) (k) = ρ η (g · V ) (T f (g)) (k) = ρ σ (kg · V ) ( (T f (g)) (k) ) = ρ σ (kg · V ) ( f (kg)). To finish the proof of (3.3) note that  π  Tρ (V ) f (g) (k) = ρ π (V ) f (kg) = ρ σ (kg · V ) ( f (kg)) =

 ν ρ (V )T f (g) (k).  

4. Reduced Forms and Super Lie Groups of Clifford Type 4.1. The reduced form of a super Lie group. Contrary to the case of locally compact groups, super Lie groups do not necessarily have faithful representations. The next lemma presents a simple but key example of elements of the Lie superalgebra which lie in the kernel of every unitary representation. Lemma 4.1.1. Let (G 0 , g) be a super Lie group and (π, ρ π , H) be a unitary representation of (G 0 , g). If X 1 , ..., X m ∈ g1 satisfy m

[X i , X i ] = 0

i=1

then ρ π (X i ) = 0 for every 1 ≤ i ≤ m. Proof. We have m

i=1

√

√ m m −1 ∞ −1 ∞ π ( [X i , X i ]) = 0. ρ (X i ) = − π ([X i , X i ]) = − 2 2 π

2

i=1

i=1

Since every ρ π (X i ) is symmetric, for every v ∈ H∞ we have m m

ρ π (X i )v, ρ π (X i )v = v, ρ π (X i )2 v = 0. i=1

Therefore for every 1 ≤ i ≤ m we have ρ π (X i )v = 0.  

i=1

ρ π (X

i )v, ρ

π (X

i )v

= 0, which implies that

The proof of the following proposition is easy by induction. Proposition 4.1.1. Let (G 0 , g) be a super Lie group. Let a(1) be the ideal of g generated by all X ∈ g1 such that [X, X ] = 0. For every m > 1, let a(m) be the ideal of g generated by elements X ∈ g1 such that [X, X ] ∈ a(m−1) for every unitary  . Then (m) acts trivially on representation (π, ρ π , H) of (G 0 , g), the Z2 -graded ideal ∞ a m=1  ∞ (m) , and π ∞ (X ) = 0 if X ∈ g ∩ (m) . H, i.e., ρ π (X ) = 0 if X ∈ g1 ∩ ∞ 0 m=1 a m=1 a

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 (m) is a Z -graded ideal of g. Note that a(1) ⊆ a(2) ⊆ · · · and the set a[g] = ∞ 2 m=1 a Therefore we have a[g] = a[g]0 ⊕ a[g]1 . Let A0 be the normal subgroup of G 0 with Lie algebra a[g]0 . Then (A0 , a[g]) is a sub super Lie group of (G 0 , g). Moreover, setting G 0 = G 0 /A0 and g = g/a[g] we obtain a nilpotent super Lie group (G 0 , g). From now on, the super Lie group (G 0 , g) will be called the reduced form of (G 0 , g). Obviously, the categories of unitary representations of (G 0 , g) and (G 0 , g) are equivalent. If (G 0 , g) is a super Lie group with the property that a[g] = {0}, then the super Lie group (G 0 , g) and the Lie superalgebra g are called reduced. 4.2. Heisenberg-Clifford super Lie groups. In this section we introduce an important example of nilpotent super Lie groups which will be used in the rest of this paper. Let (w, ω) be a super symplectic vector space, i.e., a Z2 -graded vector space w = w0 ⊕ w1 endowed with a nondegenerate bilinear form ω :w×w→R with the following properties: (a) ω(w0 , w1 ) = ω(w1 , w0 ) = {0}. (b) Restriction of ω to w0 is a symplectic form. (c) Restriction of ω to w1 is a symmetric form. Consider the Z2 -graded vector space n = w ⊕ R, where n0 = w0 ⊕ R and n1 = w1 . We define a (super)bracket on n as follows. For every P, Q ∈ w and a, b ∈ R, we set [(P, a), (Q, b)] = (0 , ω(P, Q)). One can easily check that with this bracket n becomes a Lie superalgebra. The latter Lie superalgebra is called a Heisenberg-Clifford Lie superalgebra. If N0 denotes the simply connected nilpotent Lie group with Lie algebra n0 , then the super Lie group (N0 , n) is called a Heisenberg-Clifford super Lie group. It may sometimes be more convenient to work with an explicit basis of the HeisenbergClifford Lie superalgebra. One can always find a basis {Z , X 1 , . . . , X m , Y1 , . . . , Ym , V1 , . . . , Vn }

(4.1)

of n such that (a) n0 = SpanR {Z , X 1 , . . . , X m , Y1 , . . . , Ym } and n1 = SpanR {V1 , . . . , Vn }. (b) For every 1 ≤ i ≤ m we have [X i , Yi ] = Z . (c) For every 1 ≤ j ≤ n we have [V j , V j ] = c j Z , where c j ∈ {1, −1}. (d) Z ∈ Z(n).

(4.2)

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4.3. Nilpotent supergroups. Recall that a Lie superalgebra g is called nilpotent if g appears in its own upper central series. (Equivalently, g is called nilpotent if its lower central series has only finitely many nonzero terms.) In this paper, a super Lie group (N0 , n) is called nilpotent if it has the following properties: (a) n is a nilpotent Lie superalgebra. (b) N0 is a connected, simply connected, nilpotent Lie group. It follows that the exponentional map exp : n0 → N0 is an analytic diffeomorphism which results in a bijective correspondence between Lie subgroups and Lie subalgebras. 4.4. Structure of reduced forms. Our next task is to state and prove a generalization of Kirillov’s lemma [CG, Lem. 1.1.12]. The proof of this generalization is a slight modification of that of the original result. Recall that Z(n) denotes the centre of n. Definition 4.4.1. A nilpotent Lie superalgebra n is said to be of Clifford type if one of the following properties hold: (a) n = {0}. (b) n is a Heisenberg-Clifford Lie superalgebra such that dim n0 = 1 and the restriction of ω to n1 is positive definite. In other words, n is of Clifford type if either n = {0} or n satisfies both of the following properties: (a) dim n0 = 1 and Z(n) = n0 . (b) There exists a basis {Z , V1 , . . . , Vl }

(4.3)

of n such that Z ∈ Z(n0 ), V1 , . . . , Vl ∈ n1 , and for every 1 ≤ i ≤ j ≤ l we have [Vi , V j ] = δi, j Z . A nilpotent super Lie group (N0 , n) is said to be of Clifford type whenever n is of Clifford type. Note that the zero-dimensional Lie superalgebra and the (unique) Lie superalgebra n which satisfies dim n = dim n0 = 1 are also considered to be of Clifford type. Up to parity change, irreducible unitary representations of their corresponding super Lie groups are one-dimensional and purely even. Up to parity change, trivial representation is the only such representation of the first case. For the second case, these representations are unitary characters of the even part. Proposition 4.4.1. Let n be a reduced nilpotent Lie superalgebra which satisfies dim n > 1 and dim Z(n) = 1. Then exactly one of the following two statements is true: (a) There exist three nonzero elements X, Y, Z ∈ n0 such that n = RX ⊕ RY ⊕ RZ ⊕ w, where w = w0 ⊕ w1 is a Z2 -graded subspace of n, [X, Y ] = Z , and Z ∈ Z(n). Moreover, the vector space n = RY ⊕ RZ ⊕ w is a subalgebra of n, and Y ∈ Z(n ). (b) n is a Lie superalgebra of Clifford type.

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Proof. Obviously Z(n) ⊆ n0 , since if X = 0 and X ∈ n1 ∩ Z(n), then [X, X ] = 0 which contradicts the assumption that n is reduced. Fix an arbitrary nonzero Z ∈ Z(n). Since n is nilpotent, we have Z(n/Z(n)) = {0}. Let Z1 (n) denote the Z2 -graded ideal of n which corresponds to Z(n/Z(n)) via the quotient map q : n → n/Z(n). Obviously Z1 (n)  Z(n). First assume that Z1 (n) ∩ n0  Z(n). We show that statement (a) of the proposition holds. Choose an arbitrary Y ∈ Z1 (n) ∩ n0 such that Y ∈ / Z(n), and consider the map adY : n → Z(n). Since Y ∈ n0 and Y ∈ / Z(n), there exists an element X ∈ n0 such that [X, Y ] = 0. After an appropriate rescaling, we can assume that [X, Y ] = Z . We can now take w to be a Z2 -graded complement of RY ⊕ RZ in n , where n = {V ∈ n | [V, Y ] = 0}. Next assume that Z1 (n) ∩ n0 ⊆ Z(n). We use induction on dim n to show that n is of Clifford type. Choose an arbitrary nonzero V1 ∈ Z1 (n) ∩ n1 . Since n is reduced, after an appropriate rescaling we can assume that [V1 , V1 ] = ±Z . If dim n = 2, then the proof is complete. Next assume dim n > 2. Set n = ker(ad V1 ). Obviously n is a subalgebra of n and n = n ⊕ RV1 as vector spaces. It is easy to see that n is reduced, Z(n ) = Z(n), and Z1 (n ) = Z1 (n) ∩ n . Therefore dim Z(n ) = 1 and Z1 (n ) ∩ n 0 ⊆ Z(n ). By induction hypothesis, there exists a basis {Z , V2 , . . . , Vl } for n such that Z ∈ Z(n ), Vl ∈ n 1 for every l > 1, and [Vi , V j ] = δi, j Z for every 1 < i ≤ j ≤ l. After rescaling V1 appropriately, we have [V1 , V1 ] = ±Z . If [V1 , V1 ] = Z , then the proof is complete. If [V1 , V1 ] = −Z , then it follows that [V1 + V2 , V1 + V2 ] = 0, contradicting the assumption that n is reduced.  

4.5. Representations of super Lie groups of Clifford type. Throughout this section, we assume that (C0 , c) is a super Lie group of Clifford type such that c = {0}. Let {Z , V1 , . . . , Vl } be the basis of c given in (4.3), and (π, ρ π , H) be an irreducible unitary representation of (C0 , c). By [CCTV, Lem. 5], the action of ρ π (Z ) is via multiplication by a scalar. It follows that H∞ = H, i.e., for every 1 ≤ i ≤ l we have D(ρ π (Vi )) = H.

(4.4)

Fix 1 ≤ i ≤ l. Since ρ π (Vi ) is symmetric, it is closable [Co, p. 316]. Thus (4.4) implies that ρ π (Vi ) is a closed operator. Consequently, by the closed graph theorem, ρ π (Vi ) is a bounded, self adjoint operator. √ Since ρ π (V1 ) is a self adjoint operator and π ∞ (Z ) = 2 −1ρ π (V1 )2 , it follows that for every v ∈ H we have √ π ∞ (Z )v = a −1v, where a ≥ 0.

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201

If a = 0, then for every V ∈ c1 the symmetric operator ρ π (V ) satisfies ρ π (V )2 = 0, and it follows immediately that ρ π (V ) = 0. Therefore (π, ρ π , H) is a trivial representation. Next suppose a > 0. Let Z − a denote the two-sided ideal of U(c) generated by Z − a. We can set ρ(V ) = ρ π (V ) for every V ∈ c1 , and then extend ρ to a homomorphism ρ : U(c)/Z − a → EndC (H) of associative (super)algebras. In this fashion, from a representation of (C0 , c) we obtain a representation of U(c)/Z − a on a complex Z2 -graded vector space. Fix a nonzero vector v ∈ H0 and consider the subspace W ⊆ H defined by W = SpanC {ρ(W )v | W ∈ U(c)}. Since U(c) is finite dimensional, W is finite dimensional as well, and hence it is a closed subspace of H. It is easily seen that W is a Z2 -graded, c-invariant (and hence (C0 , c)-invariant) subspace of H. Since (π, ρ π , H) is irreducible, it follows that W = H. Therefore we have proved that every irreducible unitary representation of (C0 , c) is finite dimensional. Next observe that U(c)/Z − a is isomorphic (as a Z2 -graded algebra) to a complex Clifford algebra. It is a well-known result in the theory of Clifford modules that up to parity change, a complex Clifford algebra has a unique nontrivial finite dimensional irreducible Z2 -graded representation. (See [LaMi, Chap. 5] or [CCTV, Lem. 11].) If dim c1 is odd, then the choice of Z2 -grading does not matter, whereas if this dimension is even, then parity change yields two non-isomorphic modules. Conversely, fixing an a > 0 and an irreducible Z2 -graded module K for the complex Clifford algebra U(c)/Z − a, one can obtain an irreducible unitary3 representation (σμ , ρ σμ , Kμ ) of (C0 , c), where Kμ = K as a vector space and μ : c0 → R is an R-linear functional such that μ(Z ) = a and √ for every W ∈ c0 and v ∈ Kμ , σμ∞ (W )v = μ(W ) −1 v. (4.5) Note that the condition a > 0 implies that μ([V, V ]) > 0 for every V ∈ c1 . In conclusion, if (σ0 , ρ σ0 , K0 ) denotes the (1|0)-dimensional trivial representation of (C0 , c), then we have proved the following statement.4 Proposition 4.5.1. Let (C0 , c) be a super Lie group of Clifford type. Suppose that (π, ρ π , H) is an irreducible unitary representation of (C0 , c). Then there exists a unique R-linear functional μ : c0 → R satisfying μ([V, V ]) ≥ 0 for every V ∈ c1 such that (π, ρ π , H)  (σμ , ρ σμ , Kμ ). The representation (π, ρ π , H) is trivial if and only if μ = 0. 3 Unitarity of this module follows from standard constructions of Clifford modules. See [LaMi] or [CCTV, Sect. 4.2]. 4 Strictly speaking, there is ambiguity in the choice of (σ , ρ σμ , K ) up to parity change. However, for μ μ our purposes the choice of Z2 -grading does not really matter, since special induction commutes with parity change.

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5. Realization as Induced Representations 5.1. Codimension-one induction. Throughout this section (N0 , n) will be a reduced nilpotent super Lie group such that dim n > 1 and dim Z(n) = 1. We assume that (N0 , n) is not of Clifford type. Hence part (a) of Proposition 4.4.1 holds for n. Let n = n 0 ⊕ n 1 , X , Y , and Z be as in part (a) of Proposition 4.4.1. Let (N0 , n ) be the sub super Lie group of (N0 , n) corresponding to n . Let (π, ρ π , H) be an irreducible unitary representation of (N0 , n). By [CCTV, Lem. 5], there exists a real number b ∈ R such that for every t ∈ R and v ∈ H, we have π(exp(t Z ))v = etb

√ −1

v.

The main goal of this section is to prove the following. Proposition 5.1.1. Suppose that the restriction of π ∞ to Z(n) is nontrivial, i.e., b = 0. Then there exists an irreducible unitary representation (σ, ρ σ , K) of (N0 , n ) such that (N ,n)

π  Ind(N0 ,n ) (σ, ρ σ , K). 0

The rest of this section is devoted to the proof of Proposition 5.1.1. The proof is inspired by that of [CG, Prop. 2.3.4], but there are several crucial technical points that our proof deviates from the argument given in [CG]. Set h = SpanR {X, Y, Z }. Clearly, h is a Heisenberg Lie subalgebra of n0 , corresponding to a Heisenberg Lie subgroup H of N0 . Fix an i ∈ {0, 1}. The space Hi is an N0 -invariant subspace of H, and we will denote this representation of N0 by (πi , Hi ). Let Hi∞ be the space of smooth vectors of (πi , Hi ). 0 σ , where σi is a From the proof of [CG, Prop. 2.3.4] it follows that πi  Ind N N i 0

unitary representation of N0 . For the reader’s convenience, we give an outline of the argument. By the Stone-von Neumann theorem in the form stated in [CG, 2.2.9], there exist a Hilbert space Ki and a linear isometry Si : Hi → L 2 (R, Ki )

(5.1)

such that Si intertwines the action of H , where the action of H on L 2 (R, Ki ) is given as follows : for every s, t ∈ R and f ∈ L 2 (R, Ki ), (πi (exp(t X )) f ) (s) = f (s + t), (πi (exp(tY )) f ) (s) = ebts (πi (exp(t Z )) f ) (s) = e

√

√

−1

tb −1

f (s),

f (s).

Lemma 2.3.2 of [CG] is still valid, and Lemma 2.3.1 of [CG] implies that for every g ∈ N0 , there exists a family {Tg,t }t∈R of unitary operators from Ki to Ki such that for every f ∈ Hi∞ , g ∈ N0 , and t ∈ R, we have (πi (g) f ) (t) = Tg,t ( f (t)). The rest of the argument, i.e., showing that the choice of σi (g) = Tg,0

(5.2)

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203

0 defines a unitary representation of N0 on Ki , and that πi  Ind N σ , follows the proof N0 i of [CG, Prop. 2.3.4] mutatis mutandis. 0 Since πi  Ind N σ , the Hilbert space Hi can be realized as a space of functions N0 i from N0 to Ki (see Sect. 3.1). If we pick this realization of Hi , then the isometry Si of (5.1) is given by a simple formula which we now describe. Let

L 0 = {exp(t X ) | t ∈ R}

(5.3)

be the one-parameter subgroup of N0 corresponding to X . Then after normalizing the inner products, the map Si : Hi → L 2 (R, Ki )

(5.4)

is given by Si f (t) = f (exp(t X )). Let Hi∞,c denote the subspace of Hi∞ consisting of functions with compact support modulo N0 . Let C πi (R, Ki ) and Ccπi (R, Ki ) be the subspaces of L 2 (R, Ki ) defined by C πi (R, Ki ) = Si Hi∞ and Ccπi (R, Ki ) = Si Hi∞,c . Let K = K0 ⊕ K1 be the orthogonal direct sum of K0 and K1 and the isometry S : H∞,c → L 2 (R, K) be defined as S = S0 ⊕ S1 . We also set Ccπ (R, K) = S H∞,c and C π (R, K) = SH∞ . As usual, let C ∞ (R) denote the space of complex valued smooth functions on R, and Cc∞ (R) denote the subspace of C ∞ (R) consisting of functions with compact support. Lemma 5.1.1. Assume the above notation. (a) If φ ∈ C ∞ (R) and f ∈ Ccπi (R, Ki ) then φ f ∈ Ccπi (R, Ki ). (b) If φ ∈ Cc∞ (R) and f ∈ C πi (R, Ki ) then φ f ∈ Ccπi (R, Ki ). Proof. Let L 0 be defined as in (5.3). Every element n ∈ N0 can be written uniquely as a product n = n · l of an element n ∈ N0 and an element l ∈ L 0 . Consider the function ψ : N0 → C defined by ψ(n) = φ(l). It is easily seen that ψ is smooth. To prove part (a) it suffices to show that if h ∈ Hi∞,c , then ψ h ∈ Hi∞,c , and the latter inclusion follows from the description of smooth vectors for induced representations in [Po, Th. 5.1] or [CG, Th. A.1.4]. The proof of part (b) is similar.   Lemma 5.1.2. Let V ∈ n1 . If φ ∈ Cc∞ (R) and f ∈ C π (R, K) then ρ π (V )(φ f ) = φρ π (V ) f.

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Proof. Let Mχa : L 2 (R, K) → L 2 (R, K) be the operator of multiplication by χa , i.e., for every h ∈ L 2 (R, K) and t ∈ R, (Mχa h) (t) = χa (t)h(t), where χa (t) = eat

√ −1 .

By Lemma 5.1.1, for every i ∈ {0, 1} we have Mχa (Ccπi (R, Ki )) ⊆ Ccπi (R, Ki ).

From [Y, V ] = 0 it follows that for every a ∈ R, ρ π (V )Mχa = Mχa ρ π (V ).

(5.5)

Choose a sequence {φn }∞ n=1 of elements of SpanC { χa | a ∈ R } such that lim φn f = φ f and

n→∞

lim φn ρ π (V ) f = φρ π (V ) f,

(5.6)

n→∞

where the convergences are in L 2 (R, K). The sequence {φn }∞ n=1 can be found as follows. By the Stone-Weierstrass theorem, for every positive integer n one can choose a function φn ∈ SpanC { χa | a ∈ R } which is periodic with period 2n, and max { |φ(t) − φn (t)| } ≤

−n≤t≤n

1 . n

Since elements of C π (R, K) are smooth vectors for the action of the Heisenberg group H , they are Schwartz functions from R to K. Suppose n is large enough such that Supp(φ) ⊂ {x ∈ R | − n ≤ x ≤ n}. Now we have



||(φ − φn ) f ||2 ≤

n −n

 ||(φn (t) − φ(t)) f (t)||2 dt +

1 ≤ 2 × 2n × max{|| f (t)||2 } + |t|≤n n



|t|>n

||(φn (t) − φ(t)) f (t)||2 dt

1 + max{|φ(t)|} n t∈R

2  |t|>n

|| f (t)||2 dt.

Since f ∈ C π (R, K), when n grows to infinity the last line above converges to zero. Since ρ π (V ) f ∈ C π (R, K), the same reasoning applies to ρ π (V ) f instead of f as well. Since φn ∈ SpanC { χa | a ∈ R }, it follows from (5.5) that ρ π (V )φn f = φn ρ π (V ) f. The operator ρ π (V ) is symmetric, hence it is closable (see [Co, p. 316]). In particular, from (5.6) and the fact that φ f ∈ D(ρ π (V )) it follows that ρ π (V )(φ f ) = φ ρ π (V ) f.  

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Consider the map  : N0 × R → N0 defined as (n , s) = n · exp(s X ).

(5.7)

The map  is bijective and the Campbell-Baker-Hausdorff formula implies that it is smooth. Moreover, for every x = (n , s) ∈ N0 × R, if by means of the left action of N0 we identify the tangent spaces at x and (x) with n 0 ⊕ R and n0 , then the derivative D(x) : n 0 ⊕ R → n0 of  at x is given by the following formula: For every (Q, t) ∈ n 0 ⊕ R, D(x)(Q, t) = (exp(−s X ) · Q, t). Note that exp(−s X ) · Q ∈ n 0 because N0 is a normal subgroup of N0 (see [CG, Lem. 1.1.8]). From the formula for D it follows immediately that D(x) is invertible at every x ∈ N0 × R. Hence the inverse mapping theorem implies that  −1 is smooth. Consequently, a function f : N0 → Ki is smooth if and only if f ◦  : N0 × R → Ki is smooth. Lemma 5.1.3. Let f : R → Ki be a smooth function such that f (0) = 0. Set  f (t) if t = 0, t g(t) = f (0) otherwise. Then g is a smooth function as well, and g (n) (0) =

f (n+1) (0) n+1 .

Proof. For t = 0, the lemma is trivial. We prove the lemma for t = 0 by induction on n, (n+1) in each step proving that g (n) (0) = f n+1(0) . For n = 0 the latter statement is obvious. We will assume that the statement is true for some n, and prove it for n + 1. If t = 0, then g

(n)

(t) =

n  

n k=0

Therefore

lim

t→0

g (n) (t) − g (n) (0) = lim t→0 t

k

(−1)n−k (n − k)!t −n+k−1 f (k) (t).

 n

n  k=0

k

 (−1)

n−k

(n − k)! t f k

(k)

(t) −

t n+1 f (n+1) (0) n+1

t n+2

which is a limit of the form lim

t→0

h 1 (t) , h 2 (t)

where h 1 and h 2 are continuously differentiable functions and h 1 (0) = h 2 (0) = 0. It follows that the above limit is equal to h 1 (t) t→0 h 2 (t) lim

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in case the latter limit exists. But we have n

tk (−1)n−k f (k+1) (t) k! k=0  n 

t k−1 n−k (k) (−1) f (t) + n! − t n f (n+1) (0) (k − 1)!

h 1 (t) = n!

=t f n

k=1 (n+1)

(t) − t n f (n+1) (0),

while h 2 (t) = (n + 2)t n+1 . In conclusion, we have h 1 (t) (t n f (n+1) (t) − t n f (n+1) (0)) f (n+2) (0) = lim , =

t→0 h 2 (t) t→0 (n + 2)t n+1 n+2 lim

which implies that g (n+1) (0) =

f (n+2) (0) n+2 .

 

Lemma 5.1.4. Let q be a positive integer and f : Rq × R → Ki be a smooth function such that for every x ∈ Rq , we have f (x, 0) = 0. Define ⎧ ⎨ f (x,t) if t = 0, t g(x, t) = ⎩ ∂f ∂t (x, 0) otherwise. Then g(x, t) is smooth, and indeed ∂ n+1 f (x, 0) ∂n g ∂t n+1 (x, 0) = . (5.8) ∂t n n+1 Proof. That g(x, t) is smooth when t = 0 is trivial. From Lemma 5.1.3 it follows that n for every integer n ≥ 0 and every x ∈ Rq , ∂∂t ng (x, 0) exists and equality (5.8) holds. Every differential operator in x1 , ..., xq , t is a linear combination of operators D of the form D=

∂ a 1 ∂ b1 ∂ a 2 ∂ b2 ∂ a k ∂ bk · · · , ∂ xia11 ∂t b1 ∂ xia22 ∂t b2 ∂ xiakk ∂t bk

where i 1 , ..., i k ∈ {1, ..., q} and a1 , ..., ak , b1 , ..., bk ∈ {0, 1}. (For example, if k = 3, a1 = a3 = 1, a2 = 0, b1 = b2 = 1, b3 = 0, i 1 = 3, i 3 = 2, and 1 ≤ i 2 ≤ q then 2 D = ∂∂x3 ∂t∂ 2 ∂∂x2 .) In order to complete the proof of the lemma, it suffices to show that for every such D and every x ∈ Rq , the partial derivative Dg(x, 0) exists. For every 1 ≤ i ≤ q and n ≥ 0 we have  ⎧ n  ∂ 1 ∂f ⎨ n if t = 0, n ∂ ∂ g ∂t t ∂ xi (x, t)   (x, t) = n+1 ∂f ∂ ⎩ 1 ∂ xi ∂t n (x, 0) otherwise. n+1

∂t n+1 ∂ xi

Thus for every (x, t) ∈ Rq × R the partial derivative ∂∂xi ∂∂t ng (x, t) exists and if we set ⎧ ∂f ⎨ ∂ xi (x,t) if t = 0, t g1 (x, t) = ⎩ ∂ ∂ f (x, 0) otherwise, n

∂t ∂ xi

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then we have ∂ ∂n g ∂ n g1 (x, t) = (x, t). ∂ xi ∂t n ∂t n Note that Lemma 5.1.3 implies that ∂∂tgn1 (x, t) exists for every n ≥ 0. By repeating the above argument one can show that Dg(x, t) exists and is equal to n

∂ b1 +···bk g2 (x, t), ∂t b1 +···+bk where

⎧   ⎪ a1 +···+ak ⎪ 1 ∂ ⎪ if t = 0, ⎪ ⎨ t ∂ x a1 ∂ x a2 · · · ∂ x ak f (x, t) i1 i2 ik   g2 (x, t) = ⎪ ⎪ ∂ ∂ a1 +···+ak ⎪ f (x, 0) otherwise. ⎪ ∂t a1 ⎩ ∂ xi1 ∂ xia22 · · · ∂ xiakk

The existence of

∂ b1 +···bk ∂t b1 +···+bk

g2 (x, t) follows from Lemma 5.1.3.

 

Lemma 5.1.5. Let f ∈ Ccπi (R, Ki ) satisfy f (0) = 0. Then there exists a function g ∈ Ccπi (R, Ki ) such that for every t ∈ R we have f (t) = tg(t). Proof. Let f = Si h where Si is the operator defined in (5.4) and h ∈ Hi∞,c . Set h 1 = h ◦ , where  : N0 × R → N0 is the map defined in (5.7). For every (n , t) ∈ N0 × R we have h 1 (n , t) = t h 2 (n , t), where h 2 : N0 × R → Ki is defined as follows. 

h 2 (n , t) =

h 1 (n ,t) t ∂h 1

∂t (n , 0)

if t = 0, otherwise.

Lemma 5.1.4 implies that h 2 is smooth. From the description of smooth vectors for induced representations given in [Po, Th. 5.1] or [CG, Th. A.1.4] it follows that the function h 3 = h 2 ◦  −1 belongs to Hi∞,c . To complete the proof, we set g = Si h 3 .   Lemma 5.1.6. Let V ∈ n1 . Suppose that f ∈ Ccπi (R, Ki ) satisfies f (0) = 0. Then  π ρ (V ) f (0) = 0. Proof. By Lemma 5.1.5 we have f (t) = t g(t), where g ∈ Ccπi (R, Ki ). Since g has compact support, it is not hard to see that there exists a function ψ ∈ Cc∞ (R) such that ψ(t) = 1 for t ∈ Supp(g) ∪ {0}. It follows that f = ψ f . Set ψ1 (t) = t ψ(t). By Lemma 5.1.2 we have T f = T ψ f = T (ψ1 g) = ψ1 T g, hence (T f )(0) = ψ1 (0) ((T g)(0)) = 0, which completes the proof.

 

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Remark. 1. Let V ∈ n1 and suppose that f ∈ Ccπi (R, Ki ) satisfies f (t◦ ) = 0 for some t◦ ∈ R. Then (πi (exp(t◦ X )) f ) (0) = 0, hence by Lemma 5.1.6,   π ρ i (V ) f (t◦ ) = πi (exp(t◦ X ))ρ πi (V ) f (0)  (5.9) = ρ πi (exp(t◦ X ) · V ) πi (exp(t◦ X )) f (0) = 0. 2. An immediate consequence of (5.9) is that ρ π (V )Ccπ (R, K) ⊆ Ccπ (R, K). For every V ∈ n1 we define a family of linear operators ∞ ∞ ∞ TV,t : K∞ 1 ⊕ K2 → K1 ⊕ K2

as follows. For every i ∈ {0, 1}, t◦ ∈ R, and v ∈ Ki∞ , choose an f ∈ Ccπi (R, Ki ) such that f (t◦ ) = v. For instance, one can fix ϕ ∈ Cc∞ (R) such that ϕ(t0 ) = 1, and take f = Si (h ◦  −1 ) where h : N0 × R → Ki is given by h(n , t) = ϕ(t)σi (n )v. Now set TV,t v = (ρ π (V ) f )(t). Lemma 5.1.6 and the remark after this lemma imply that the operators TV,t are well defined. Since ρ π (V ) is odd, it follows that the TV,t ’s are odd operators. We now set σ = σ0 ⊕ σ1 , and for any W ∈ n1 define ρ σ (W ) : K∞ → K∞ by ρ σ (W )v = TW,0 v. Our next task is to verify that the triple (σ, ρ σ , K) satisfies the conditions of Definition 2.1.2. Linearity of ρ σ and condition (a) of Definition 2.1.2 are obvious. Next we prove that for every W ∈ n1 the operator ρ σ (W ) is symmetric. Suppose, on the contrary, that ρ σ (W ) is not symmetric, and let v, w ∈ K∞ such that ρ σ (W )v, w = v, ρ σ (W )w. Choose ϕ ∈ Cc∞ (R) such that φ(0) = 1, and consider two functions f v , f w : N0 × R → K defined by f v (n , t) = ϕ(t)σ (n )v and f w (n , t) = ϕ(t)σ (n )w. The functions f v ◦  −1 and f w ◦  −1 belong to H∞,c . Let gv , gw ∈ Ccπ (R, K) be defined by gv = S( f v ◦  −1 ) and gw = S( f w ◦  −1 ). It is readily seen that  π ρ (W )gv (t) = Texp(t X )·W,0 (gv (t)). If {W, W1 , ..., Wr } is a basis containing W for n1 , then exp(t X ) · W = γ0 (t)W +

r

i=1

γi (t)Wi ,

(5.10)

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209

where for every 0 ≤ i ≤ r the function γi (t) : R → R is smooth. Moreover, lim γ0 (t) = 1 and for every 1 ≤ i ≤ r, lim γi (t) = 0.

t→0

t→0

Now  π ρ (W )gv (t) = Texp(t X )·W,0 (gv (t)) = γ0 (t)TW,0 (gv (t)) +

r

 = ϕ(t) γ0 (t)TW,0 v +

γi (t)TWi ,0 (gv (t))

i=1 r

 γi (t)TWi ,0 v .

i=1

But

 lim γ0 (t)TW,0 v +

t→0

r

 = TW,0 v = ρ σ (W )v

γi (t)TWi ,0 v

(5.11)

i=1

and



∞

  ρ π (W )gv (t), gw (t)  dt −∞    ∞ r

2 = ϕ(t)  γ0 (t)TW,0 v + γi (t)TWi ,0 v , w  dt. (5.12)

π

ρ (W )gv , gw  =

−∞

Similarly we have  lim γ0 (t)TW,0 w +

t→0

i=1

r

 γi (t)TWi ,0 w

= TW,0 w = ρ σ (W )w

(5.13)

i=1

and



∞

  gv (t), ρ π (W )gw (t)  dt −∞    ∞ r

2 = ϕ(t)  v, γ0 (t)TW,0 w + γi (t)TWi ,0 w  dt. (5.14)

gv , ρ π (W )gw  =

−∞

i=1

From (5.11), (5.12), (5.13) and (5.14) it follows that if Supp(ϕ) is small enough, then ρ π (W )gv , gw  = gv , ρ π (W )gw , which contradicts the fact that ρ π (W ) is symmetric. Condition (c) of Definition 2.1.2 can be verified as follows. Let V, W ∈ g1 and v ∈ K∞ . Choose an f ∈ Ccπ (R, K) such that f (0) = v. We have      σ ρ (V )ρ σ (W )+ρ σ (V )ρ σ (W ) v = ρ π (V ) ρ π (W ) f (0)+ ρ π (V ) ρ π (W ) f (0) √  = − −1 π ∞ ([V, W ]) f (0).

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Since [V, W ] ∈ n 0 , from (5.2) it follows that  ∞ π ([V, W ]) f (0) = σ ∞ ([V, W ]) ( f (0)) = σ ∞ ([V, W ])v which completes the proof of condition (c). Finally, we prove condition (d) of Definition 2.1.2. Let V ∈ n1 , g ∈ N0 , and w ∈ K∞ . Let f ∈ Ccπ (R, K) be such that f (0) = w. Using (5.2) we have  ρ σ (g · V )w = ρ π (g · V ) f (0)   = π(g)ρ π (V )π(g)−1 f (0)    = σ (g) ρ π (V )π(g −1 ) f (0)     = σ (g) ρ σ (V ) π(g −1 ) f (0)    = σ (g) ρ σ (V ) σ (g −1 ) ( f (0)) = σ (g)ρ σ (V )σ (g −1 )w. To finish the proof of Proposition 5.1.1, note that the unitary representations (π, ρ π , H) and (N ,n)

Ind(N0 ,n ) (σ, ρ σ , K) 0

H∞,c .

are identical on This follows from the fact that for every V ∈ n1 , t ∈ R, and f ∈ Ccπ (R, K) we have (ρ π (V ) f )(t) = ρ σ (exp(t X ) · V ) ( f (t)). Consequently, Proposition 2.2.1 implies that these representations are unitarily equivalent. Since (π, ρ π , H) is assumed to be irreducible, it follows that (σ, ρ σ , K) is irreducible as well. 5.2. Stone-von Neumann theorem for Heisenberg-Clifford supergroups. In this section we show how to use Proposition 5.1.1 to prove a generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups. Let (N0 , n) be a Heisenberg-Clifford super Lie group (see Sect. 4.2). For every P, Q ∈ n1 , the value of the bracket [P, Q] lies in n0 = R, and hence can be thought of as a real number. Consider the symmetric bilinear form B : n1 × n1 → R defined by B(P, Q) = [P, Q]. Let (π, ρ π , H) be an irreducible unitary representation of (N0 , n). By [CCTV, Lem. 5] the action of Z(N0 ) is via multiplication by a unitary character χ : Z(N0 ) → C× . When B is a definite form, we say that the character χ agrees with B if there exists a positive real number c such that for every P ∈ n1 , χ ([P, P]) = ecB(P,P)

√

−1

.

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If χ is the trivial character, then it is easily seen that the sub super Lie group (Z(N0 ), [n1 , n1 ] ⊕ n1 ) of (N0 , n) belongs to the kernel of (π, ρ π , H). Consequently, (π, ρ π , H) yields an irreducible representation of the abelian Lie group N0 /Z(N0 ). It follows that (π, ρ π , H) is a one-dimensional representation obtained from a unitary character of N0 /Z(N0 ). If χ is not trivial, then we have the following result. Theorem 5.2.1. Suppose that the unitary character χ : Z(N0 ) → C× is nontrivial. (a) If B is an indefinite form, then there are no irreducible unitary representations of (N0 , n) with central character χ . (b) Suppose that B is a definite form. If χ agrees with B, then up to unitary equivalence and parity change there exists a unique irreducible unitary representation of (N0 , n) with central character χ . If χ does not agree with B, then such a unitary representation does not exist. Proof. Part (a) follows from the fact that a[n] = [n1 , n1 ] ⊕ n1 . Part (b) is proved by induction on the dimension of N0 as follows. Let {Z , X 1 , X 2 , . . . , X m , Y1 , . . . , Ym , V1 , . . . , Vn } be the basis of n given in (4.1), and (π, ρ π , H) be an irreducible unitary representation of (N0 , n) with central character χ . By Proposition 5.1.1 we have (N ,n)

(π, ρ π , H) = Ind(N0 ,n ) (σ, ρ σ , K), 0

where n = SpanR {Z , X 2 , ..., X m , Y1 , ..., Ym , V1 , .., Vn }. Moreover, from the proof of Proposition 5.1.1 it follows that σ ∞ (Y1 ) = 0. Therefore (σ, ρ σ , K) factors through a representation of a Heisenberg-Clifford super Lie group (N0

, n

), where n

= n /r and r = SpanR {Y1 }. Since dim N0

< dim N0 , the proof is completed by induction on dim N0 . Details are left to the reader.   Remark. Suppose that χ is nontrivial, B is definite, and χ agrees with B. Then part (b) of Theorem 5.2.1 can be refined slightly as follows. When dim n1 is even, there exist two irreducible unitary representations which are not unitarily equivalent. However, when dim n1 is odd, we obtain a unique such representation up to unitary equivalence. Indeed the restriction to (Z(N0 ), [n1 , n1 ] ⊕ n1 ) of such a representation is a countable direct sum of modules for a complex Clifford algebra, and when dim n1 is even there are two nonisomorphic such modules [LaMi, Chap. 5]. The details are left to the reader. 6. Polarizing Systems and Main Theorems 6.1. Polarizing systems. Throughout this section (N0 , n) is a (not necessarily reduced) nilpotent super Lie group. Definition 6.1.1. A polarizing system of (N0 , n) is a 6-tuple (M0 , m, , C0 , c, λ) where (a) (M0 , m) is a special sub super Lie group of (N0 , n).

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(b) λ : n0 → R is an R-linear functional and m0 is a polarizing subalgebra of n0 corresponding to λ. (c) (C0 , c) is a super Lie group of Clifford type and  is a surjective homomorphism  : (M0 , m) → (C0 , c). (d) m0 ∩ ker  = m0 ∩ ker λ. Let (M0 , m, , C0 , c, λ) be a polarizing system of (N0 , n) and (σμ , ρ σμ , Kμ ) be the irreducible unitary representation of (C0 , c) associated to a linear functional μ : c0 → R (see Sect. 4.5). One can compose (σμ , ρ σμ , Kμ ) with the map  : (M0 , m) → (C0 , c) and obtain an irreducible unitary representation (σμ ◦ , ρ σμ ◦ , Kμ ) of (M0 , m). The representation (σμ , ρ σμ , Kμ ) is said to be consistent with the polarizing system if for every W ∈ m0 , λ(W ) = μ ◦ (W ).

(6.1)

We will see below that consistent representations play a special role in the classification of irreducible unitary representations. Theorem 6.1.1. Let (π, ρ π , H) be an irreducible unitary representation of a nilpotent super Lie group (N0 , n). (a) There exists a polarizing system (M0 , m, , C0 , c, λ) and an irreducible unitary representation (σμ , ρ σμ , Kμ ) of (C0 , c) which is consistent with (M0 , m, , C0 , c, λ) such that (N ,n)

(π, ρ π , H) = Ind(M00 ,m) (σμ ◦ , ρ σμ ◦ , Kμ ).

(6.2)

(b) Suppose that (M0 , m ,  , C0 , c , λ ) is another polarizing system and (σμ , ρ σμ , Kμ ) is a representation of (C0 , c ) consistent with (M0 , m ,  , C0 , c , λ ) such that (N0 ,n) σμ ◦ (π, ρ π , H) = Ind(M , Kμ ).

,m ) (σμ ◦ , ρ 0

Then there exists an n ∈ N0 such that λ = Ad∗ (n)(λ). Moreover, the super Lie groups (C0 , c) and (C0 , c ) are isomorphic.

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213

Proof. Part (a) is proved by induction on dim n. There are three cases to consider: Case I. (N0 , n) is not reduced. In this case (π, ρ π , H) factors through the reduced form (N 0 , n) of (N0 , n), and dim n < dim n. Let us denote this representation of (N 0 , n) by (π , ρ π , H). By induction hypothesis, there exists a polarizing system (M 0 , m, , C 0 , c, λ) of (N 0 , n) and a representation (σμ , ρ σμ , Kμ ) of (C 0 , c) which is consistent with (M 0 , m, , C 0 , c, λ) such that (π , ρ π , H) = Ind(N 0 ,n) (σμ ◦ , ρ σμ ◦ , Kμ ). (M 0 ,m)

Let q : (N0 , n) → (N 0 , n) be the quotient map and set (M0 , m, , C0 , c, λ) = (q−1 (M 0 ), q−1 (m),  ◦ q, C 0 , c, λ ◦ q). It is easily checked that (M0 , m, , C0 , c, λ) is a polarizing system of (N0 , n), and (σμ , ρ σμ , Kμ ) is consistent with (M0 , m, , C0 , c, λ). Case II. (N0 , n) is reduced and dim Z(n) > 1. Since the action of Z(n) is via scalar multiplication [CCTV, Lem. 5], it is easily seen that (π, ρ π , H) factors through a representation of a quotient (N0 , n ) of (N0 , n), where the kernel of the quotient corresponds to a subalgebra of codimension one in Z(n). Again dim n < dim n, and an argument similar to Case I above applies. Case III. (N0 , n) is reduced and dim Z(n) = 1. In this case one of the statements of Proposition 4.4.1 must hold. If statement (b) of Proposition 4.4.1 holds, then by Proposition 4.5.1 there is nothing left to prove. Next suppose that statement (a) of Proposition 4.4.1 holds. If the restriction of π ∞ to Z(n) is trivial, then an argument similar to Case II above applies. If the restriction of π ∞ to Z(n) is not trivial, then from Proposition 5.1.1 it follows that there exists an irreducible unitary representation (σ, ρ σ , K) of (N0 , n ) such that (N0 ,n) σ (π, ρ π , H) = Ind(N

,n ) (σ, ρ , K). 0

(N0 , n )

is the super Lie group identified by statement (a) of Proposition 4.4.1. Here By induction hypothesis, there exists a polarizing system (M0 , m ,  , C0 , c , λ ) of (N0 , n ) and an irreducible unitary representation (σμ , ρ σμ , Kμ ) of (C0 , c ) which is consistent with (M0 , m ,  , C0 , c , λ ) such that (N ,n )



(σ, ρ σ , K) = Ind(M0 ,m ) (σμ ◦  , ρ σμ ◦ , Kμ ). 0

By Proposition 3.2.1 we have (N ,n)



(π, ρ π , H) = Ind(M0 ,m ) (σμ ◦  , ρ σμ ◦ , Kμ ).

(6.3)

0

Let λ˜ be an arbitrary R-linear extension of λ to n0 , and set ˜ (M0 , m, , C0 , c, λ) = (M0 , m ,  , C0 , c , λ). To show that (M0 , m, , C0 , c, λ) is a polarizing system of (N0 , n), it suffices to check that m0 is a polarizing subalgebra of n0 corresponding to λ. Let X, Y, Z ∈ n0 be chosen as in part (a) of Proposition 4.4.1. From Z ∈ Z(n 0 ) and the fact that m0 is a polarizing

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subalgebra of n 0 corresponding to λ , it follows that Z ∈ m0 . Hence for every t ∈ R and v ∈ Kμ , √  σμ ◦  (exp(t Z )) v = etλ(Z ) −1 v. Using (6.3) and the realization of induced representations given in Sect. 3.1 it is easy to check that for every t ∈ R and v ∈ H, π(exp(t Z ))v = etλ(Z )

√ −1

v.

Since the restriction of π ∞ to Z(n) is assumed to be nontrivial, it follows that λ(Z ) = 0. Consider the skew-symmetric bilinear form ωλ : n0 × n0 → R defined by ωλ (V, W ) = λ([V, W ]), and let ωλ be the restriction of ωλ to n 0 × n 0 . Since m0 is a maximal isotropic subspace of n 0 , we have dim m0 =

1 (dim n 0 + dim s λ ), 2

where s λ is the radical of ωλ . To show that m0 is a maximal isotropic subspace of ωλ , it suffices to prove that dim sλ = dim s λ − 1, where sλ is the radical of ωλ . Let V ∈ sλ , and write V = a X + W , where a ∈ R and W ∈ n 0 . From [Y, n 0 ] = {0} it follows that ωλ (V, Y ) = λ([V, Y ]) = aλ(Z ) which implies that a = 0, i.e. V ∈ n 0 . Consequently, sλ ⊆ s λ . Moreover, [Y, n 0 ] = {0} implies that Y ∈ s λ , but λ([X, Y ]) = 0 implies that Y ∈ / sλ . Thus dim sλ < dim s λ , from

which it readily follows that dim sλ = dim sλ − 1. Finally, verifying that (σμ , ρ σμ , Kμ ) is consistent with (M0 , m, , C0 , c, λ) is trivial. Next we prove part (b) of Theorem 6.1.1. Suppose that χ : C0 → C× (respectively,

χ : C0 → C× ) is the central character of (σμ , ρ σμ , Kμ ) (respectively, (σμ , ρ σμ , Kμ )). Since m0 is a polarizing subalgebra of n0 corresponding to λ and (σμ , ρ σμ , Kμ ) is conN0 χ ◦  is irreducible. Since sistent with (M0 , m, , C0 , c, λ), the representation Ind M 0 N0 π = Ind M σ ◦ , it follows that the unitary representation (π, H) of the nilpotent Lie 0 μ

N0 group N0 is a direct sum of dim Kμ copies of Ind M χ ◦ . With a similar argument, 0 one can see that (π, H) is a direct sum of dim Kμ copies of the irreducible unitary N0

representation Ind M

χ ◦  . Consequently, dim Kμ = dim Kμ , which immediately 0

implies that (C0 , c) and (C0 , c ) are isomorphic. Moreover, we have N0 N0

χ ◦   Ind M Ind M

χ ◦ 0 0

and Kirillov theory for nilpotent Lie groups (e.g., [CG, Th. 2.2.4]) implies that λ = Ad∗ (n)(λ) for some n ∈ N0 .

 

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215

Corollary 6.1.1. Let (π, ρ π , H) be an irreducible unitary representation of a nilpotent Lie supergroup (N0 , n), and (π, H) be the unitary representation of N0 obtained as a restriction of (π, ρ π , H) to the even part. Then there exists an irreducible unitary representation (σ, K) of N0 such that (π, H) is a direct sum of 2l copies of (σ, K), where l is a nonnegative integer. Proof. Since special induction commutes with restriction to the even part, this follows immediately from part (a) of Theorem 6.1.1 and the fact that dim Kμ = 2l for some l ≥ 0.   Remark. Suppose that an irreducible unitary representation (π, ρ π , H) is given by (6.2). If we set κ(π, ρ π , H) = dim c, then by part (b) of Theorem 6.1.1 the positive integer κ(π, ρ π , H) does not depend on the choice of the polarizing system and hence is an invariant of (π, ρ π , H). In fact using Corollary 6.1.1 one can see that κ(π, ρ π , H) can be obtained as follows. Consider the representation (π, H) of the Lie group N0 obtained by restriction of (π, ρ π , H) to the even part of (N0 , n). The representation (π, H) is always a direct sum of 2r copies of an irreducible unitary representation (π , H ) of N0 , where r is a nonnegative integer. In the latter case, we have  2r if (π, ρ π , H)  (π, ρ π , H), π κ(π, ρ , H) = 2r + 1 otherwise. In particular, when r = 0 the representation (π, ρ π , H) is purely even and therefore κ(π, ρ π , H) = 1. 6.2. Irreducibility of codimension-one induction. In this section we prove that induction from a polarizing system always yields an irreducible unitary representation. Theorem 6.2.1. Let (M0 , m, , C0 , c, λ) be a polarizing system of (N0 , n). Suppose that (σμ , ρ σμ , Kμ ) is the representation of (C0 , c) consistent with this polarizing system. Then the unitary representation (N ,n)

(π, ρ π , H) = Ind(M00 ,m) (σμ , ρ σμ , Kμ ) is irreducible. Proof. We prove the theorem by induction on dim n. If λ = 0, then m = n and ker  ⊇ n0 , which implies that c = {0} and therefore (π, ρ π , H) is the trivial representation. Without loss of generality, from now on we assume that λ = 0. There are three cases to consider. Case I. (N0 , n) is not reduced. Recall that a[n] is a Z2 -graded ideal of n. Since a[n] = {0}, we have Z(n) ∩ a[n] = {0}. Indeed let e(0) = a[n] and for any positive integer j, set e( j+1) = [n, e( j) ]. Let j0 = min{ j | e( j) = {0} }. Then e( j0 −1) ⊆ a[n] ∩ Z(n). Let W ∈ Z(n) ∩ a[n]. Since Z(n) ∩ a[n] is Z2 -graded, we can choose W suitably such that W ∈ n0 or W ∈ n1 . If W ∈ n1 then obviously W ∈ m1 , and if W ∈ n0 , then W ∈ m0 because otherwise m 0 = m0 + RW is a Lie subalgebra of n0 with the property that λ([m 0 , m 0 ]) = {0}, and the latter implies that (M0 , m, , C0 , c, λ) does not satisfy part (b) of Definition 6.1.1. Therefore we have shown that Z(n) ∩ a[n] ⊆ m. Our next task is to show that (W ) = 0 for every W ∈ Z(n) ∩ a[n]. Without loss of generality, we can assume W ∈ n0 or W ∈ n1 . If W ∈ n1 , then we have [W, W ] = 0

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which implies that [(W ), (W )] = 0. Since (C0 , c) is reduced, we have (W ) = 0. If W ∈ n0 , then for every v ∈ Kμ we have √  ∞ σμ ◦ (W ) v = μ ◦ (W ) −1 v. Since λ = 0, by Lemma 2.2.1 and surjectivity of  it follows that μ = 0. From the realization of induced representations given in Sect. 3.1 and the fact that W ∈ Z(n) it is easily seen that for every v ∈ H, √ √ π ∞ (W )v = λ(W ) −1 v = μ ◦ (W ) −1 v. If (W ) = 0, then μ ◦ (W ) = 0 from which it follows that π ∞ (W ) = 0, which contradicts the fact that by Proposition 4.1.1 we have π ∞ (W ) = 0. Set s = Z(n) ∩ a[n] and consider the super Lie group (N0 , n ), where n = n/s. (Thus N0 = N0 /S0 , where S0 = { exp(t V ) | V ∈ s0 }.) Obviously (π, ρ π , H) factors through (N0 , n ). We denote this representation of (N0 , n ) by (π , ρ π , H). Moreover, Z(n) ∩ a[n] ⊆ Z(n) ⊆ m. Since ker  ∩ m0 = ker λ ∩ m0 we have Z(n) ∩ a[n] ∩ n0 ⊆ ker λ. Therefore the polarizing system (M0 , m, , C0 , c, λ) corresponds via the quotient map q : n → n

to a polarizing system (M0 , m ,  , C0 , c, λ ) of (N0 , n ). Moreover, (σμ , ρ σμ , Kμ ) is consistent with (M0 , m ,  , C0 , c, λ ). We can express (π , ρ π , H) as (N ,n )

(π, ρ π , H) = Ind(M0 ,m ) (σμ ◦  , ρ σμ ◦  , Kμ ). 0

Since dim n < dim n, by the induction hypothesis it follows that (π , ρ π , H) (and hence (π, ρ π , H)) is irreducible. Case II. (N0 , n) is reduced and Z(n) ∩ ker λ = {0}. In this case Z(n) ∩ ker λ is an ideal of n, and the fact that m0 is a polarizing subalgebra of n0 corresponding to λ implies that Z(n) ⊆ m0 . The representation (π, ρ π , H) factors through (N0 , n ), where n = n/Z(n) ∩ ker λ, and the polarizing system (M0 , m, , C0 , c, λ) corresponds via the quotient map q : n → n to a polarizing system (M0 , m ,  , C0 , c, λ ) of (N0 , n ). The rest of the argument is similar to Case I. Case III. (N0 , n) is reduced and Z(n) ∩ ker λ = {0}. It follows that dim Z(n) = 1, hence one of the statements of Proposition 4.4.1 should hold. If statement (b) of Proposition 4.4.1 holds, then there is essentially nothing left to prove. From now on we assume that statement (a) of Proposition 4.4.1 holds. Let X, Y, Z , n , and w be as in part (a) of Proposition 4.4.1. Our first task is to show that without loss of generality, we can assume that λ(Y ) = 0 and λ(Z ) = 0. Indeed one can modify the choice of the polarizing system as follows. Since m0 is a polarizing Lie subalgebra of n0 corresponding to λ, we should have Z ∈ m0 , and since Z(n) ∩ ker λ = {0}, we should have λ(Z ) = 0. For every n ∈ N0 , we have a polarizing system ( n M0 n −1 , Ad(n)(m),  ◦ Ad(n −1 ), C0 , c, Ad∗ (n)(λ) )

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217

in (N0 , n) and one can see that (N ,n) (σμ −1 0 n ,Ad(n)(m))

0 (π, ρ π , H)  Ind(n M

◦  ◦ Ad(n −1 ), ρ σμ ◦◦Ad(n

In particular, if we set n = exp(t◦ X ), where t◦ =

λ(Y ) λ(Z ) ,

−1 )

, Kμ ).

then

 ∗ λ(Y ) Ad (n)(λ) (Y ) = λ(Y − Z ) = 0. λ(Z ) The condition Z(n) ∩ ker (Ad∗ (n)(λ)) = {0} is easy to check as well. From now on we assume that λ(Y ) = 0 and λ(Z ) = 0. Our next task is to prove that without loss of generality we can also assume that m ⊆ n . Suppose, on the contrary, that m  n . In this case we show that (π, ρ π , H) is unitarily equivalent to a representation

(N ,n)



(π , ρ π , H ) = Ind(M0 ,m ) (σμ ◦  , ρ σμ ◦ , Kμ ), 0

where (σμ , ρ σμ , Kμ ) is consistent with a polarizing system (M0 , m ,  , C0 , c, λ) which satisfies m ⊆ n . To this end, first note that in part (a) of Proposition 4.4.1, we can choose X such that λ(X ) = 0 and m = RX ⊕ RZ ⊕ w 0 ⊕ n1 , where w 0 is a subspace of n 0 such that λ(w 0 ) = 0. Indeed since m  n , we can choose X such that X ∈ m0 . If λ(X ) = 0, then since Z ∈ m and λ(Z ) = 0 we can substitute X by )



X − λ(X λ(Z ) Z . In a similar fashion we can choose a complement w0 to RZ in m ∩ n which is included in ker λ. Next note that Y ∈ / m0 because otherwise λ([X, Y ]) = λ(Z ) = 0 which contradicts the fact that m0 is a polarizing subalgebra of n0 corresponding to λ. Consider the subalgebra m of n defined by m = RY ⊕ RZ ⊕ w 0 ⊕ n1 . To show that m is a subalgebra of n , note that [m 0 , m 0 ] ⊆ [w 0 , w 0 ] ⊆ m0 ∩ n 0  m 0

(6.4)

and [n1 , n1 ] ⊆ n 0 ∩ m0  m 0 . Let M0 be the Lie subgroup of N0 corresponding to m 0 . We define  : (M0 , m ) → (C0 , c) as follows. For every W ∈ RZ ⊕ w 0 ⊕ n1 and W ∈ RY we set  (W + W ) = (W ). We now prove that (M0 , m ,  , C0 , c, λ) is a polarizing system and (σμ , ρ σμ , Kμ ) is consistent with it. From a calculation similar to (6.4) it follows that λ([m 0 , m 0 ]) ⊆ λ([m0 , m0 ]) = {0}.

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Moreover, Y ∈ / m0 because otherwise we have Z ∈ [m0 , m0 ] and λ(Z ) = 0 which contradicts the fact that m0 is a polarizing subalgebra of n0 corresponding to λ. Therefore we have dim m 0 = dim m0 , which implies that m 0 is a polarizing subalgebra of n0 . Using [Y, n ] = {0} it is easy to check that part (c) of Definition 6.1.1 holds. Part (d) of Definition 6.1.1 follows from λ(X ) = λ(Y ) = 0 and λ(Z ) = 0. Finally, one can check that (σμ , ρ σμ , Kμ ) is consistent with (M0 , m ,  , C0 , c, λ).

To prove that (π, ρ π , H)  (π , ρ π , H ), it suffices to show that (M

,m

)

(M

,m

)



Ind(M00 ,m) (σμ ◦ , ρ σμ ◦ , Kμ )  Ind(M0 ,m ) (σμ ◦  , ρ σμ ◦ , Kμ ),

(6.5)

0

where M0

= M0 M0 and m

= m + m , i.e., m

= RX ⊕ RY ⊕ RZ ⊕ w 0 ⊕ n1 . Since m

is Z2 -graded, we can express it as m

= m

0 ⊕ m

1 . Observe that the vector space w 0 is in fact an ideal of m

0 . To prove the latter statement, note that since m0 and m 0 are polarizing subalgebras of n0 corresponding to λ, we should have [m0 , m0 ] ⊆ RX ⊕ w 0 and [m 0 , m 0 ] ⊆ RY ⊕ w 0 , which imply that RX ⊕ w 0 and RY ⊕ w 0 are Lie subalgebras of n0 . Since w 0 = (RX ⊕ w 0 ) ∩ (RY ⊕ w 0 ), the vector space w 0 is in fact a Lie subalgebra of both of RX ⊕ w 0 and RY ⊕ w 0 . But in a nilpotent Lie algebra, any Lie subalgebra of codimension one is an ideal. Therefore w 0 is an ideal in both RX ⊕ w 0 and RY ⊕ w 0 . It follows that w 0 is an ideal in the Lie algebra generated by RX ⊕ w 0 and RY ⊕ w 0 , i.e., in m

0 = RX ⊕ RY ⊕ RZ ⊕ w 0 . Next we obtain the unitary equivalence of (6.5). Let E 0 = { exp(t Z ) | t ∈ R } and χ : E 0 → C× be the unitary character given by χ (exp(t Z )) = etλ(Z )

√

−1

.

If (π L , ρ π L , H L ) denotes the representation on the left hand side of (6.5), then we can realize H L as L 2 (R, Kμ ) such that the action of (M0

, m

) is given as follows. For every y, t ∈ R, W ∈ w 0 , and f ∈ L 2 (R, Kμ ), we have (π L (exp(t X )) f ) (y) (π L (exp(tY )) f ) (y) (π L (exp(t Z )) f ) (y)  π L (exp(W ) f (y)

= χ (t y) f (y), = f (y + t), = χ (t) f (y), = f (y).

Moreover, if f ∈ L 2 (R, Kμ ) is in the Schwartz space then from [Y, n1 ] = {0} it follows that for every W ∈ n1 and y ∈ R we have (ρ π L (W ))(y) =  (exp(yY ) · W ) ( f (y)) = (W ) ( f (y)).

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Similarly, if (π R , ρ π R , H R ) denotes the representation on the right-hand side of (6.5), then (π R , ρ π R , H R ) can also be realized on L 2 (R, Kμ ) as follows. For every x, t ∈ R, W ∈ w 0 , and f ∈ L 2 (R, Kμ ), we have (π R (exp(t X )) f ) (x) (π R (exp(tY )) f ) (x) (π R (exp(t Z )) f ) (x)  π R (exp(W ) f (x)

= = = =

f (x + t), χ (−t x) f (x), χ (t) f (x), f (x).

Moreover, if f ∈ L 2 (R, Kμ ) is indeed in the Schwartz space, then for every W ∈ n1 and x ∈ R we have (ρ π R (W ))(x) =  (exp(x X ) · W ) ( f (x)) = (W ) ( f (x)), where the last equality follows from the fact that (X ) = 0 and thus 1  (exp(x X ) · W ) =  (W + [X, W ] + [X, [X, W ]] + · · · ) 2 1 = (W + [X, W ] + [X, [X, W ]] + · · · ) 2 1 = (W ) + [(X ), (W )] + [(X ), [(X ), (W )]] + · · · 2 = (W ). It is now easy to check that the isometry T : H L → H R which intertwines (π L , ρ π L , H L ) and (π R , ρ π R , H R ) is given by the Fourier transform, i.e.,  ∞ T f (x) = χ (x y) f (y)dy. −∞

We now complete the proof of Case III. The proof closely follows an argument that is given in [CG, p. 63]. Recall that as shown above, we can assume that m ⊆ n . It follows that (M0 , m, , C0 , c, λ) is a polarizing system in (N0 , n ). Since dim n < dim n, by induction hypothesis the representation (N ,n )



(π

, ρ π , H

) = Ind(M00 ,m) (σμ ◦ , ρ σμ ◦ , Kμ )

(6.6)

is irreducible. Since Z ∈ Z(n ), by [CCTV, Lem. 5] there exists a real number b ∈ R such that for every t ∈ R and v ∈ H

we have π

(exp(t Z ))v = etb

√ −1

v.

Recall that λ(Z ) = 0 and λ(Y ) = 0. Since Z ∈ Z(n) ∩ n 0 ⊆ m0 , from (6.6) and the realization of the induced representation (see Sect. 3.1) it follows that π

(exp(t Z ))v = etλ(Z )

√ −1

v,

and therefore b = 0. Next observe that (N ,n)



(π, ρ π , H) = Ind(N0 ,n ) (π

, ρ π , H

), 0

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and by Sect. 3.1 we can assume H = L 2 (R, H

), where for every f ∈ L 2 (R, H

), s ∈ R, t ∈ R, and n ∈ N0 we have (π(exp(t X )) f ) (s) = f (s + t)

(6.7)

and (π(n) f )(s) = π

(exp(s X )n exp(−s X )) ( f (s)). In particular, since SpanR {X, Y, Z } is a Heisenberg Lie algebra, we have (π(exp(tY )) f ) (s) = estb

√

−1

f (s).

Moreover, if f ∈ L 2 (R, H

) is a smooth vector for the action of π and has compact support, then for every W ∈ n1 and s ∈ R we have  π

ρ (W ) f (s) = ρ π (exp(s X ) · W ) ( f (s)). Let T : L 2 (R, H

) → L 2 (R, H

) be a bounded even linear operator which intertwines (π, ρ π , H) with itself. To complete the proof of Case III, it suffices to show that T is a scalar multiple of the identity. From [CG, Lem. 2.3.3], [CG, Lem. 2.3.2] and [CG, Lem. 2.3.1] it follows that there exists a family {Tt }t∈R of even linear operators Tt : H

→ H

such that ||Tt || ≤ ||T || for every t ∈ R, and for every f ∈ L 2 (R, H

) we have T f (t) = Tt ( f (t)). One can check that Tt intertwines the action of the repre

sentation (πt

, ρ πt , H

) of (N0 , n ) which is defined by





πt

(n) = π

(exp(t X )n exp(−t X )) and ρ πt (W ) = ρ π (exp(t X ) · W ).



But (πt

, ρ πt , H

) is irreducible, and from [CG, Lem. 5] it follows that for every t ∈ R, the operator Tt is multiplication by a scalar γ (t). From (6.7) it follows that γ (t) does not depend on t, i.e., T is a scalar multiple of identity.   6.3. Existence of suitable polarizing subalgebras. In this section n = n0 ⊕ n1 will be a nilpotent Lie superalgebra. In this section we prove the existence of a special kind of polarizing subalgebras in n0 . The main goal of this section is to prove Lemma 6.3.4. For every λ ∈ n∗0 we consider the symmetric bilinear form Bλ : n1 × n1 → R defined by Bλ (X, Y ) = λ([X, Y ]). We denote the radical of Bλ by rλ . Lemma 6.3.1. Suppose λ ∈ n∗0 and Bλ is nonnegative definite. If X ∈ n1 is an isotropic vector, i.e., it satisfies Bλ (X, X ) = 0, then X ∈ rλ . Proof. Suppose, on the contrary, that there exists an element Y ∈ n1 such that Bλ (X, Y ) = 0. Then for every s ∈ R we have Bλ (X + sY, X + sY ) = Bλ (X, X ) + 2sBλ (X, Y ) + s 2 Bλ (Y, Y ). Since Bλ (X, X ) = 0, one can find an s ∈ R such that Bλ (X + sY, X + sY ) < 0, which contradicts the fact that Bλ is nonnegative definite.  

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221

Lemma 6.3.2. For every B, C, D ∈ n1 , we have [B, [C, D]] ∈ rλ . Proof. Let n(1) ⊇ n(2) ⊇ · · · be the lower central series of n, which is defined by n(1) = n and n(i+1) = [n, n(i) ] for every i ≥ 1. Note that n(i) = {0} for i  0. We use a backward induction, assuming that the statement of the lemma holds for every B, C, D such that [B, [C, D]] ∈ n(r +1) , and proving it for every B, C, D such that [B, [C, D]] ∈ n(r ) . Fix B, C, D ∈ n1 such that [B, [C, D]] ∈ n(r ) and set X = [B, [C, D]]. By Lemma 6.3.1, to prove the lemma it suffices to show that λ([X, X ]) = 0. By the Jacobi identity we have λ([X, X ]) − λ([B, [[C, D], X ]]) + λ([[C, D], [X, B]]) = 0. To complete the proof of the lemma, we show that λ([B, [[C, D], X ]]) = 0 and λ([[C, D], [X, B]]) = 0. The equality λ([B, [[C, D], X ]]) = 0 follows from the induction hypothesis, since X ∈ n1 and [X, [C, D]] ∈ [n(r ) , n] = n(r +1) . To show that λ([[C, D], [X, B]]) = 0, note that by the Jacobi identity λ([[X, B], [C, D]]) + λ([C, [D, [X, B]]]) − λ([D, [[X, B], C]]) = 0, and the induction hypothesis applies to [D, [X, B]] and [C, [X, B]].

 

Lemma 6.3.3. We have λ([[n1 , n1 ], [n1 , n1 ]]) = {0}. Proof. Let A, B, C, D ∈ n1 be arbitrarily chosen. By the Jacobi identity, λ([[A, B], [C, D]]) + λ([C, [D, [A, B]]]) − λ([D, [[A, B], C]]) = 0. The equality λ([[A, B], [C, D]]) = 0 follows from the fact that by Lemma 6.3.2 we have [D, [A, B]] ∈ rλ and [C, [A, B]] ∈ rλ .   Lemma 6.3.4. There exists a polarizing subalgebra m0 of n0 corresponding to λ such that m0 ⊇ [n1 , n1 ]. Proof. Since [n1 , n1 ] is an ideal of n0 , we can find a sequence of ideals of n0 such as n0 = i(1) ⊃ i(2) ⊃ i(3) ⊃ · · · ⊃ i(r −1) ⊃ i(r ) = {0}, such that for every 1 < j ≤ r we have dim i( j−1) = dim i( j) + 1 and moreover for some 1 ≤ s ≤ r we have [n1 , n1 ] = i(s) . For every 1 ≤ j ≤ r , let ( j)

ωλ : i( j) × i( j) → R ( j)

be the skew-symmetric bilinear form defined by ωλ (X, Y ) = λ([X, Y ]) and let q( j) be ( j) the radical of ωλ . By a result of M. Vergne (see [CG, Th. 1.3.5]) the vector space q(1) + · · · + q(r ) is indeed a polarizing Lie subalgebra of n0 corresponding to λ. Lemma 6.3.3 implies that ωλ(s) is zero, hence q(s) = [n1 , n1 ].  

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6.4. Existence of polarizing systems. Throughout this section (N0 , m) will be a nilpotent super Lie group. Let (M0 , m, , C0 , c, λ) be a polarizing system in (N0 , n) and (σμ , ρμσ , Kμ ) be a representation of (C0 , c) which is consistent with this polarizing system. Let Bλ denote the bilinear form on n1 defined in Sect. 6.3. Obviously, for every X ∈ n1 we have Bλ (X, X ) = λ([X, X ]) = μ ◦ ([X, X ]) = μ([(X ), (X )]) ≥ 0. Consequently, Bλ is nonnegative definite. Conversely, let λ ∈ n∗0 be such that Bλ is nonnegative definite. From Lemma 6.3.4 it follows that there exists a sub super Lie group (M0 , m) of (N0 , n) such that m1 = n1 and m0 is a polarizing Lie subalgebra of n0 corresponding to λ. Let kλ = { X ∈ m0 | λ(X ) = 0 } and set j = kλ ⊕ rλ . Lemma 6.4.1. The vector space j is an ideal in m. Proof. Since λ([m0 , m0 ]) = 0, we have [m0 , m0 ] ⊆ kλ and therefore [m0 , kλ ] ⊆ kλ . Next we prove that [kλ , m1 ] ⊆ rλ . To this end, first note that by the Jacobi identity for every A ∈ kλ , B ∈ rλ , and C ∈ n1 we have −[[A, B], C] − [[B, C], A] + [[C, A], B] = 0, and therefore −λ([[A, B], C]) − λ([[B, C], A]) + λ([[C, A], B]) = 0. But λ([[B, C], A]) = 0 because [B, C] ∈ [n1 , n1 ] ⊆ m0 , and λ([[C, A], B]) = 0 because B ∈ rλ . Consequently, for every A ∈ kλ and B ∈ rλ we have ad A B ∈ rλ . It follows that ad A descends to a linear transformation ad A : n1 /rλ → n1 /rλ . The bilinear form Bλ induces a positive definite bilinear form Bλ : n1 /rλ × n1 /rλ → R. Next observe that for every A ∈ kλ and every V, W ∈ n1 we have −λ([W, [A, V ]]) + λ([A, [V, W ]]) + λ([V, [W, A]]) = 0. Moreover, λ([A, [V, W ]]) = 0 since [A, [V, W ]] ∈ [m0 , m0 ] ⊆ kλ . Therefore for every v, w ∈ n1 /rλ we have Bλ (ad A v, w) = −Bλ (v, ad A w). In other words, ad A is skew-symmetric. Since ad A is also nilpotent, it follows that ad A = 0. Therefore [kλ , n1 ] ⊆ rλ . Next we prove that [m0 , rλ ] ⊆ rλ . To this end, first note that by the Jacobi identity, for every A ∈ m0 , B ∈ rλ , and C ∈ n1 we have −λ([[A, B], C]) − λ([[B, C], A]) + λ([[C, A], B]) = 0.

Unitary Representations of Nilpotent Super Lie Groups

223

But λ([[B, C], A]) = 0 because [[B, C], A] ∈ [m0 , m0 ] ⊆ kλ , and λ([[C, A], B]) = 0 because B ∈ rλ . It follows that λ([[A, B], C]) = 0, and consequently, as C ∈ n1 is arbitrary, we have [A, B] ∈ rλ . Finally, the inclusion [m1 , rλ ] ⊆ kλ follows from the definition of kλ .   Lemma 6.4.2. The quotient Lie superalgebra m/j is reduced. Proof. It suffices to prove that for every X ∈ n1 such that [X, X ] ∈ kλ , we have X ∈ rλ . But this follows immediately from Lemma 6.3.1.   Proposition 6.4.1. Let (M0 , m) be a sub super Lie group of (N0 , n) such that m0 is a polarizing subalgebra of n0 corresponding to λ. Then there exists a polarizing system (M0 , m, , C0 , c, λ) and a representation (σμ , ρ σμ , Kμ ) of (C0 , c) which is consistent with this polarizing system. Moreover, up to unitary equivalence and parity change the representation (σμ ◦ , ρ σμ ◦ , Kμ ) of (M0 , m) is unique. Proof. By Lemma 2.2.1 the quotient Lie superalgebra m/j is either zero or has a one dimensional even part. Moreover, m/j is zero if and only if λ = 0. Since the case λ = 0 can be easily dealt with, from now on we assume that λ = 0, and consequently m/j is nonzero. Since m/j is reduced and nilpotent, we have Z(m/j) = m0 /kλ and hence dim Z(m/j) = 1. Therefore from Proposition 4.4.1 it follows that m/j is of Clifford type. (Note that one may have dim m/j = 1.) Let K λ be a closed subgroup of M0 with Lie algebra kλ and  : (M0 , m) → (M0 /K λ , m/j) be the natural quotient map. Then (M0 , m, , M0 /K λ , m/kλ , λ) is a polarizing system. Moreover, up to unitary equivalence and parity change, there exists a unique irreducible unitary representation (σμ , ρ σμ , Kμ ) of (M0 /K λ , m/j) which is consistent with this polarizing system. Next we prove the uniqueness claim of Proposition 6.4.1. Without loss of generality, we can assume λ = 0. Consider another polarizing system (M0 , m, , C0 , c, λ) and a consistent irreducible unitary representation (σμ , ρ σμ , Kμ ) of (C0 , c). Observe that j ⊆ ker .

(6.8)

Indeed for every X ∈ kλ we have μ ◦ (X ) = λ(X ) = 0 which implies that (X ) = 0. Similarly, for every X ∈ rλ we have μ ◦ ([X, X ]) = λ([X, X ]) = 0, which implies that [(X ), (X )] = ([X, X ]) = 0. But since c is reduced, it follows that (X ) = 0. This completes the proof of (6.8). From (6.8) it follows that there exists an epimorphism  : (M0 /K λ , m/j) → (C0 , c), which satisfies  ◦  = . However, any epimorphism between super Lie groups of Clifford type is indeed an isomorphism. From Proposition 4.5.1 it follows that (σμ ◦ , ρ σμ ◦ , Kμ )  (σμ ◦  ◦ , ρ σμ ◦◦ , Kμ )  (σμ ◦ , ρ σμ ◦ , Kμ ), which completes the proof.

 

224

H. Salmasian

Let (M0 , m, , C0 , c, λ) be a polarizing system with a consistent representation (σμ , ρ σμ , Kμ ). From now on, (π, ρ π , H) will denote the induced unitary representation (N ,n)

(π, ρ π , H) = Ind(M00 ,m) (σμ ◦ , ρ σμ ◦ , Kμ ).

(6.9)

Recall the definition of a[n] from Sect. 4.1. Since m is an ideal of n, we have m ⊇ a[n]

(6.10)

because m contains all of the generators of a[n]. Lemma 6.4.3. a[n]0 ∩ Z(n) ⊆ ker λ. Proof. The proof of this lemma is essentially given throughout the proof of Case I of Theorem 6.2.1, and therefore here we only give a sketch of the proof. Let (π, ρ π , H) be as in (6.9), and assume X ∈ a[n]0 ∩ Z(n) and λ(X ) = 0. Using the definition of the induced representation, it is not difficult to see that π ∞ (X ) = 0, which contradicts Proposition 4.1.1.   6.5. Relation between (π, ρ π , H) and λ. Let λ ∈ n∗0 be such that Bλ is nonnegative definite. By Proposition 6.4.1 there exists a polarizing system (M0 , m, , C0 , c, λ), and by Theorem 6.2.1 the representation obtained by induction from a consistent representation of the polarizing system is irreducible. Our next task is to show that if we choose different polarizing systems, we always obtain the same representation. Proposition 6.5.1. Up to unitary equivalence and parity change, the representation (π, ρ π , H) is uniquely determined by λ. Proof. We prove the proposition by induction on dim n. The argument is similar to the proof of Theorem 6.2.1. Let (M0 , m, , C0 , c, λ) (respectively, (M0 , m ,  , C0 , c , λ)) be a polarizing system with a consistent representation (σμ , ρ σμ , Kμ ) (respectively, (σμ , ρ σμ , Kμ )). (Note that the two polarizing systems are associated to the same λ.) Suppose that (π, ρ π , H)

(respectively, (π , ρ π , H )) is the induced representation defined as in (6.9). Our main goal is to prove that

(π, ρ π , H)  (π , ρ π , H ).

(6.11)

There are three cases to consider. Case I. (N0 , n) is not reduced. As in the proof of Case I in Theorem 6.2.1, we can show that a[n] ∩ Z(n) = {0}. Moreover, a[n] ∩ Z(n) ⊆ m ∩ m , and using Lemma 6.4.3 we can see that for every W ∈ a[n] ∩ Z(n) we have (W ) = 0 and  (W ) = 0. Set s = a[n] ∩ Z(n) and consider the corresponding sub super Lie group (S0 , s) of (N0 , n). Since s is an ideal of n, we have a quotient homomorphism q : (N0 , n) → (N0 /S0 , n/s).

Unitary Representations of Nilpotent Super Lie Groups

225

The polarizing system (M0 , m, , C0 , c, λ) corresponds via q to a polarizing system (M0 /S0 , m/s, q , C0 , c, λq ) in (N0 /S0 , n/s), where λq ∈ (n/s)∗ satisfies λq ◦ q = λ. If we set (N /S ,n/s)

(πq , ρ πq , Hq ) = Ind(M00 /S00 ,m/s) (σμ ◦ q , ρ σμ ◦q , Kμ ), then (π, ρ π , H)  (πq ◦q, ρ πq ◦q , Hq ). From the other polarizing system and its consis

tent representation one can obtain another representation (πq , ρ πq , Hq ) of (N0 /S0 , n/s) which is defined in a similar way. Since dim n/s < dim n, induction hypothesis implies that

(πq , ρ πq , Hq )  (πq , ρ πq , Hq ) from which (6.11) follows immediately. Case II. (N0 , n) is reduced and Z(n)∩ker λ = {0}. In this case Z(n)∩ker λ is an ideal of n and Z(n) ∩ ker λ ⊆ m ∩ m . Set s = Z(n) ∩ ker λ and let (S0 , s) be the corresponding sub super Lie group of (N0 , n). As in Case I above, using the quotient map q : (N0 , n) → (N0 /S0 , n/s) we can obtain new polarizing systems and consistent representations for (N0 /S0 , n/s). The rest of the argument is similar to that of Case I above. Case III. (N0 , n) is reduced and Z(n) ∩ ker λ = {0}. In this case the proof is very similar to that of Case III in Theorem 6.2.1. Without loss of generality we can assume that n is not of Clifford type. From Z(n) ∩ ker λ = {0} it follows that dim Z(n) = 1. Let X, Y, Z , and n be as in part (a) of Proposition 4.4.1. As shown in the proof of Case III in Theorem 6.2.1, we can choose X, Y, Z suitably such that there exist polarizing systems



(M 0 , m, , C0 , c, λ) and (M 0 , m ,  , C0 , c , λ)

(6.12)

in (N0 , n) with the following properties: (a) λ = Ad∗ (n)(λ) for some n ∈ N0 . (b) m ⊆ n and m ⊆ n . (c) (σμ , ρ σμ , Kμ ) is consistent with (M 0 , m, , C0 , c, λ).



(d) (σμ , ρ σμ , Kμ ) is consistent with (M 0 , m ,  , C0 , c , λ). (N ,n) (e) If (π , ρ π , H) = Ind(M0 ,m) (σμ ◦ , ρ σμ ◦ , Kμ ) then 0

(π , ρ π , H)  (π, ρ π , H).





(f) If (π , ρ π , H ) = Ind(N0 ,n) (σμ ◦  , ρ σμ ◦ , Kμ ) then

(M 0 ,m )







(π , ρ π , H )  (π , ρ π , H ). Let (N0 , n ) be the sub super Lie group of (N0 , n) corresponding to n . Since dim n < dim n, by induction hypothesis we have Ind

(N0 ,n ) (σ (M 0 ,m) μ

◦ , ρ σμ ◦ , Kμ )  Ind

and (6.11) follows by Proposition 3.2.1.

 

(N0 ,n )

(M 0 ,m )





(σμ ◦  , ρ σμ ◦ , Kμ )

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6.6. Geometric parametrization of representations. Let (π, ρ π , H) be an irreducible unitary representation of a nilpotent super Lie group (N0 , n). One can associate a coadjoint orbit O ⊆ n∗0 to (π, ρ π , H) as follows. Let (π, H) denote the restriction of (π, ρ π , H) to N0 . From Corollary 6.1.1 it follows that (π, H) is a direct sum of finitely many copies of an irreducible unitary representation (σ, K) of N0 . By classical Kirillov theory [CG], the representation (σ, K) is associated to a coadjoint orbit O ⊆ n∗0 . Theorem 6.1.1 shows that (π, ρ π , H) is induced from a consistent representation of a polarizing system (M0 , m, , C0 , c, λ), where λ ∈ O. Our last theorem puts together the results in this paper to obtain a geometric parametrization of irreducible unitary representations of (N0 , n) by coadjoint orbits. Recall that n+0 = { λ ∈ n∗0 | Bλ is nonnegative definite }. Theorem 6.6.1. For a nilpotent super Lie group (N0 , n), the process of associating a coadjoint orbit O ⊆ n∗0 to an irreducible unitary representation (π, ρ π , H) of (N0 , n) yields a bijection between equivalence classes of irreducible unitary representations (up to unitary equivalence and parity change) and N0 -orbits in n+0 . Proof. By part (a) of Theorem 6.1.1, any irreducible unitary representation (π, ρ π , H) of (N0 , n) is induced from a consistent representation of a polarizing system (M0 , m, , C0 , c, λ), and as shown in Sect. 6.4, it follows that λ ∈ n+0 . By part (b) of Theorem 6.1.1, if (π, ρ π , H) is induced from a consistent representation of another polarizing system, (M0 , m ,  , C0 , c , λ ), then λ and λ are in the same N0 -orbit. Moreover, once we fix a λ ∈ n+0 , by Proposition 6.4.1 there always exists an associated polarizing system and a consistent representation, and by Proposition 6.5.1, up to unitary equivalence and parity change all such polarizing systems yield the same irreducible unitary representation of (N0 , n).   Remark. One can actually prove that for every irreducible unitary representation (π, ρ π , H) of (N0 , n), the space [n1 , [n1 , n1 ]] acts trivially, i.e., ρ π (X ) = 0 for every X ∈ [n1 , [n1 , n1 ]].

(6.13)

Indeed if λ ∈ n+0 then from Lemma 6.3.3, Lemma 6.3.4, and Lemma 6.4.1 it follows that [n1 , [n1 , n1 ]] ⊆ rλ . Consequently, when (π, ρ π , H) is induced from a consistent representation of a polarizing system (M0 , m, , C0 , c, λ), we have ([n1 , [n1 , n1 ]]) = 0. Statement (6.13) now follows from the realization of the induced representation given in Sect. 3.1 and the fact that [n1 , [n1 , n1 ]] is N0 -invariant. Another more direct way to prove (6.13) is to use the method of proof of Proposition 6.5.1. Statement (6.13) can be used to obtain slightly different proofs for the main results of this paper. We thank the referee for suggesting this statement and the second method of proof. Acknowledgements. After the first draft of this article was written, we realized that M. Duflo had previously worked on the same problem and obtained similar results which were not published. We would like to thank him for extremely illuminating conversations, and his encouragement to write this article. One of the referees provided numerous helpful comments, some of which substantially simplified the proofs in Sect. 6.3. We thank the referee for those comments and for reading the paper carefully.

Unitary Representations of Nilpotent Super Lie Groups

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References [BaGu] Bars, I., Günaydin, M.: Unitary representations of non-compact supergroups. Commun. Math. Phys. 91, 31–51 (1983) [CCTV] Carmeli, C., Cassinelli, G., Toigo, A., Varadarajan, V.S.: Unitary representations of super lie groups and applications to the classification and multiplet structure of super particles. Commun. Math. Phys. 263, 217–258 (2006) [Co] Conway, J.B.: A Course in Functional Analysis. Graduate Texts in Mathematics, 96, New York: Springer-Verlag, 1985 [CG] Corwin, L.J., Greenleaf, F.P.: Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples. Cambridge Studies in Advanced Mathematics, 18. Cambridge: Cambridge University Press, 1990 [DeMo] Deligne, P., Morgan, J. W.: Notes on supersymmetry (following Joseph Bernstein). In: Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Providence, RI: Amer. Math. Soc., pp 41–97, 1999 [FSZ] Ferrara, S., Savoy, C.A., Zumino, B.: General massive multiplets in extended supersymmetry. Phys. Lett. B 100(5), 393–398 (1981) [FuNi] Furutsu, H., Nishiyama, K.: Classification of irreducible super unitary representations of su( p, q/n). Commun. Math. Phys. 141, 475–502 (1991) [Ka] Kac, V.G.: Lie superalgebras. Adv. in Math. 26(1), 8–96 (1977) [Kn] Knapp, A. W.: Representation theory of semisimple groups. An overview based on examples. (Reprint of the 1986 original.) Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 2001 [LaMi] Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, 38. Princeton, NJ: Princeton University Press, 1989 [Lo] Lo, W.T.: Super theta functions and the weil representations. J. Phys. A: Math. Gen. 27, 2739– 2748 (1994) [Ma] Mackey, G.: Induced representations of locally compact groups I. Ann. Math. (Second Series) 55(1), 101–139 (1952) [Ni] Nishiyama, K.: Super dual pairs and highest weight modules of orthosymplectic lie algebras. Adv. Math. 104, 66–89 (1994) [Po] Poulsen, N.S.: On c∞ vectors and intertwining bilinear forms for representations of lie groups. J. Funct. Anal. 9, 87–120 (1972) [Ro] Rosenberg, J. M.: A Selective History of the Stone-von Neumann Theorem. In: Operator algebras, quantization, and noncommutative geometry, Contemp. Math. 365, Providence, RI: Amer. Math. Soc., 2004, pp. 123–158 [SaSt] Salam, A., Strathdee, J.: Unitary representations of super-gauge symmetries. Nucl Phys. B 80, 499– 505 (1974) [Va] Varadarajan, V. S.: Introduction to Harmonic Analysis on Semisimple Lie Groups. Cambridge Studies in Advanced Mathematics, 16. Cambridge: Cambridge University Press, 1989 Communicated by Y. Kawahigashi

Commun. Math. Phys. 297, 229–264 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1041-8

Communications in

Mathematical Physics

Phase Transition and Correlation Decay in Coupled Map Lattices A. de Maere UCL, FYMA, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium. E-mail: [email protected] Received: 17 June 2009 / Accepted: 23 January 2010 Published online: 30 March 2010 – © Springer-Verlag 2010

Abstract: For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya’s probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling. This implies the exponential decay, in space and in time, of the correlation functions of the invariant measures. 1. Introduction It is now well-known that infinite dimensional systems are radically different from their finite dimensional counterparts, and perhaps the most striking difference is the phenomenon of phase transition. In general, finite dimensional systems tend to have only one natural measure, also called phase. For infinite dimensional systems, the picture is quite different: weakly coupled systems tend to have only one natural measure and strongly coupled systems may have several. This picture also holds for Coupled Map Lattices (CML). CML are discrete time dynamical systems generated by the iterations of a map on a countable product of compact spaces. The map is the composition of a local dynamic with strong chaotic properties and a coupling which introduce some interaction between the sites of the lattice. CML were introduced by Kaneko [1,2], and they can be seen as an infinite dimensional generalization of interval maps. Their natural measures are the SRB measures and in this case, the definition of SRB measure is a measure invariant under the dynamic with finite dimensional marginals of bounded variation. The unicity of the SRB measure for weakly Coupled Map Lattices has been thoroughly studied in various publications [3–16]. Despite many numerical results on the existence of phase transition for strongly coupled map lattices (see for instance [17–20]), there are still few analytical results on the  Partially supported by the Belgian IAP program P6/02.

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subject. The first rigorous proof of the existence of a phase transition was performed by Gielis and MacKay [21], who constructed a bijection between some Coupled Map Lattices and Probabilistic Cellular Automata (PCA) and relied on the existence of a phase transition for the PCA to prove the existence of a phase transition for the CML. But their result requires the assumption that the coupling does not destroy the Markov partition of the single site dynamics, and this hypothesis is clearly not true for general Coupled Map Lattices. Other publications are following this approach by considering specific coupling that preserve the Markov partition [22,23]. Later, Bardet and Keller [24] proved the existence of a phase transition for a more natural coupled map lattice emulating Toom’s probabilistic cellular automata, using a standard Peierls argument. The purpose of this article is to extend these results for a Coupled Map Lattice with a very general local dynamic and a coupling behaving like Stavskaya’s PCA. 2. Description of the Model and Main Results 2.1. General setup. Let I = [−1, 1], and X = I Z . The Coupled Map Lattice is given by a map T : X → X , where T = Φ ◦ τ Z with τ : I → I the local dynamic and Φ : X → X the coupling. The evolution of initial signed Borel measures under the dynamic is given by the transfer operator T , also called the Perron-Frobenius operator, which is defined by T μ(ϕ) = μ(ϕ ◦ T ). Let m Z be the Lebesgue measure on X . Let C(X ) be the set of continuous real-valued functions on X , and | · |∞ be the sup norm on this space. For every finite Λ ⊂ Z, let |Λ| be the cardinality of Λ, πΛ : X → I Λ be the canonical projector from X to I Λ , m Λ the Lebesgue measure on I Λ , and πΛ μ the restriction of μ to I Λ . Then, for every signed Borel measure μ, the total variation norm is defined by    |μ| = sup μ(ϕ)  ϕ ∈ C(X ) and |ϕ|∞ ≤ 1 . (1) Consider L 1 (X ), the space of signed Borel measures such that |μ| < ∞ and πΛ μ is absolutely continuous with respect to m Λ for every finite Λ ⊂ Z. We immediately see that if the map T is piecewise continuous, Proposition 7 from the Appendix implies that |T μ| ≤ |μ| .

(2)

Note that if μ is a probability measure, its total variation norm is always equal to 1. It is well-known that the total variation norm is not sufficient to study the spectral properties of Coupled Map Lattices [12], and that the bounded variation norm also plays an important role. Let · be the bounded variation norm, defined by    μ = sup μ(∂ p ϕ)  p ∈ Z , ϕ ∈ C 1 (X ) and |ϕ|∞ ≤ 1 . (3)    It can be seen that the space B(X ) = μ ∈ L 1 (X )  μ < ∞ , endowed with the norm · is a Banach  x space. If we use the fact that for any continuous function ϕ, we have ϕ(x) = ∂ p 0 p ϕ(ξ p , x = p ) dξ p , we can also prove that |μ| ≤ μ .

(4)

Phase Transition and Correlation Decay in Coupled Map Lattices

Following an original idea of Vitali [25], we also consider     1 Var Λ μ = sup μ (∂Λ ϕ )  ϕ ∈ CΛ (X ) and |ϕ|∞ ≤ 1

231

(5)

for any finite Λ ⊂ Z, where ∂Λ denotes the derivative with respect to all the coordinates 1 (X ) is the set of continuous functions ϕ such that ∂ ϕ is also continuous. in Λ and CΛ Λ We already note that  Var ∅μ = |μ| . sup p∈Z Var { p} μ = μ In general, we do not expect the variation Var Λ μ to be bounded uniformly in Λ. In fact, even for a totally decoupled measure of bounded variation μ, it is straightforward to check that Var Λ μ will grow exponentially with |Λ|. Consequently, it is natural to consider the following θ -norm, for some θ > 1: |||μ|||θ = sup θ −|Λ| Var Λ μ. Λ⊂Z

(6)

For any K > 0, α > 0 and θ ≥ 1, let B(K , α, θ ) be the set of measures in B(X ) such that for all finite Λ ⊂ Z, we have   1(0,1]Λ μ ≤ K α |Λ| , (7) θ where (0, 1]Λ ⊆ X is the set of configurations x such that x p ∈ (0, 1] for every p ∈ Λ and for any A ⊆ X, 1 A is used as an operator acting on measures through 1 A μ(ϕ) = μ (1 A ϕ) . Let us just give an example of some measure in B(K , α, θ ). If |·| L 1 (I ) and · BV are respectively the total variation norm and bounded variation norm on functions, if h (−) and h (+) are two probability densities of bounded variation on [−1, 0] and (0, 1] respectively and if μ = p∈Z h(x p ) dx p with h = αh (+) + (1 − α)h (−) for some α ∈ [0, 1], we can check that

|Ω∩Λ| |Ω\Λ| h BV . Var Ω 1(0,1]Λ μ ≤ α |Λ| h (+) BV



  Hence, as long as θ > max{ h (+) BV , h BV }, we have 1(0,1]Λ μθ ≤ α |Λ| and so μ belongs to B(1, α, θ ). 2.2. Assumptions on the dynamic. We will assume the following properties of the dynamic. The coupling Φ : X → X depends on some parameter  ∈ [0, 1] and is explicitly given by ⎧ if x p > 0 and x p+1 > 0 ⎨ xp Φ (x) p = x p − 1 +  if x p > 0 and x p+1 ≤ 0 . ⎩x + if x p ≤ 0 p The coupling Φ has a behavior similar to Stavskaya’s probabilistic cellular automata (see [26,27] for more details on Stavskaya’s PCA). Indeed, if both x p and x p+1 are strictly positive, x p will be sent to the interval (0, 1], and if x p or x p+1 are negative,

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x p will be sent on [−1, 0], except if x p is in the small subset (−, 0] ∪ (1 − , 1]. If  is close to 0, the system is strongly coupled, and if  is close to 1, the system is weakly coupled. On the other hand, we will assume that the single site dynamics τ is a piecewise expanding map τ : I → I such that: – ∃ ζ1 , . . . , ζ N +1 ∈ [−1, 1], where −1 = ζ1 < · · · < ζ N +1 = 1 and Ji = (ζi , ζi+1 ), such that the and uniformly C 2 (I ).  restriction of τ to the interval Ji is monotone  2  – κ = inf τ  > 2 and there is some D0 > 0 such that κ min|Ji | + | τ 2 |∞ ≤ D0 . (τ ) – The map τ has two non-trivial invariant subsets [0, 1] and [−1, 0], and the dynamic restricted to these subsets is mixing. For the sake of simplicity, we will assume the map on [−1, 0] to be the translation of the map on (0, 1]: for every x ∈ (0, 1], τ (x − 1) = τ (x) − 1. If Pτ is the Perron-Frobenius operator associated to τ and λ0 = κ2 , these assumptions imply the Lasota-Yorke inequality [28]:

m

P h ≤ λm h BV + D0 |h| L 1 (I ) , (8) τ 0 1−λ0 BV and this inequality puts strong constraints on the spectrum of Pτ as an operator acting on functions of bounded variation. Indeed, the Ionescu Tulcea-Marinescu theorem [29,30] shows us that the spectrum of Pτ in the space of functions of bounded variation consists of the doubly degenerate eigenvalue 1 with the rest of the spectrum contained inside a circle of radius λ0 . Since [−1, 0] and (0, 1] are invariant subsets, we know that we can choose the two invariant densities associated to the eigenvalue 1 to be respectively concentrated on (+) [−1, 0] and (0, 1]. Let h (−) inv and h inv be these two eigenvectors. The Gelfand formula implies then that we can always choose ς ∈ (0, 1) with ς > λ0 and c > 0 such that, for any function h on [−1, 0] of bounded variation and any m ∈ N, we have      m  (−)  P h − ≤ c ς m h BV . (9) h(x) dx h inv   τ L 1 (I )

A similar result also holds for h (−) inv and any function h on (0, 1] of bounded variation. Let T (D0 , c, ς ) be the set of maps τ satisfying the above assumptions for given values of D0 > 0, c > 0 and ς ∈ (0, 1) and arbitrary values of λ0 ∈ (0, ς ). We can see for instance that the Bernoulli shift or the maps introduced in [31,32] extended on the interval [−1, 1] using the symmetry assumptions all belong to one of the T (D0 , c, ς ). 2 It can be seen that, if some map τ belongs to T (D   0 , c, ς ), then τ = τ ◦ τ also belongs to T (D0 , c, ς ). And since inf (τ 2 )  ≥ (inf τ  )2 , this implies that if T (D0 , c, ς ) is not empty, it contains maps with arbitrary large values of κ. This point will become important later. Let E  = τ −1 (−, 0] ∪ τ −1 (1 − , 1]. The assumptions  on τ imply that the Lebesgue measure of E  is of order at most . Indeed, since τ   is bounded from below and since τ preserves the intervals [0, 1] and [−1, 0], we know that the preimage of (−, 0] ∪ (1 − , 1] under τ consists of intervals of length at most κ , and there are at most N such intervals. One could be worried about the fact that N seems to be unbounded in the assumptions on τ , but this is not the case, because N ≤ min2i |Ji | and so, N ≤ κ D0 . Therefore, we have |E  | ≤

N  ≤ D0 . κ

(10)

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For the commodity, we also introduce the following constants: λ1 =

4 D0 |E  | + , κ 2

D1 =

4 . κ mini |Ji |

(11)

2.3. Main results. We immediately see that for any value of , the measure μ(+) inv defined as  (+) μ(+) h inv (x p ) dx p inv = p∈Z

is always invariant under T . If  is close to 1, we can consider the system as a small perturbation of the case  = 1, and use a simple modification of the decoupling technique introduced by Keller and Liverani [16] to prove that μ(+) inv is indeed the unique SRB meais totally decoupled, it trivially has the property of exponential decay sure. Since μ(+) inv of correlation in space. Furthermore, as a direct consequence of the decay of correlation in time for the single-site dynamics (which comes from (9), see [33] for more details.), we also have the decay of correlations in time for μ(+) inv . We will prove in Sect. 3 that if we decrease the strength of the coupling, other SRB measures may appear, and the system therefore undergoes a phase transition. For this, let us first define α0 as    D0 |E  | 1 . (12) α0 = max λ1 + λ21 + 2D1 |E  |, 2 1 − λ0 Since |E  | ≤ D0 , we have



lim α0 = lim λ1 = lim

→0

→0

→0

4 D0 |E  | + κ 2

 =

4 . κ

(13)

Then, the existence of a phase transition is a consequence of this theorem: Theorem 1 (Existence of a phase transition). Assume that τ belongs to T (D0 , c, ς ) and that κ > 108. Then, there is some 0 > 0 such that, if  ∈ [0, 0 ): (+) • the dynamic T admits another SRB measure μ(−) inv = μinv , • μ(−) inv belongs to B(K 0 , 3α0 , θ0 ) with α0 < θ0 =

2α0 |E  |

K0 =

1 27

defined in (12), and K 0 and θ0 given by

1 . 2(1 − 27α0 )(1 − 3α0 )

(14)

The strategy used in the proof of this result is similar to the one used by Bardet and Keller in [24] in the sense that is also used for a Peierls argument, but the contour estimates are done in a different way, giving us a stronger result which allows us to prove in Sect. 4 that a wide class of initial measures converges exponentially fast towards μ(−) inv . Theorem 2 (Exponential convergence to equilibrium). Assume that τ belongs to T (D0 , c, ς ) and that κ is larger than some κ1 that depends on D0 , c and ς . Then, there is some 1 ∈ (0, 1) such that, if  ∈ [0, 1 ], there is some σ < 1 such that for any K > 0 there is some constant C > 0 such that    μ( ϕ ◦ T t ) − μ(−) ( ϕ )  ≤ C |Λ| σ t |ϕ|∞ inv

for any probability measure μ in B(K , 3α0 , θ0 ) and for any continuous function ϕ depending only on the variables in Λ ⊂ Z.

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Eventually, we will show in Sect. 5 that Theorem 2 implies the exponential decay of correlations for the invariant measure μ(−) inv both in space and time, therefore showing is not only a SRB measure, but an extremal one. that μ(−) inv Proposition 1 (Exponential decay of correlations in space). Under the assumptions of Theorem 2, there is some positive constant C such that for any bounded continuous functions ϕ and ψ depending on finitely many variables, respectively in Λ and Ω, we have  (−)   μ (ϕ ψ) − μ(−) (ϕ) μ(−) (ψ)  ≤ C |Λ ∪ Ω| σ d(Λ,Ω) |ϕ|∞ |ψ|∞ , inv inv inv where d(Λ, Ω) = inf{ |q − p| | q ∈ Λ, p ∈ Ω } is the distance between Λ and Ω. Proposition 2 (Exponential decay of correlations in time). Under the assumptions of Theorem 2, for any functions ϕ and ψ depending only on the variables in some finite 1 (X ), there is some constant C Λ ⊂ Z, with ϕ ∈ C(X ) and ψ ∈ CΛ ϕ,ψ > 0 such that   (−)  μ (ϕ ◦ T t ψ) − μ(−) (ϕ) μ(−) (ψ)  ≤ Cϕ,ψ |Λ| σ t . inv inv inv 3. Existence of a Phase Transition 3.1. Cluster expansion. For n ∈ N fixed and for any finite Λ ⊂ Z, let E(Λ) ⊆ X be the set of configurations x such that T n x p > 0 for every p ∈ Λ, where T n x p is a notation for (T n x) p . Then, to any x in E(Λ), we can associate a cluster Γ ⊆ Z × {0, . . . , n} using the following rules: 1. At time n, we add every ( p, n) with p ∈ Λ to Γ . 2. For every t ∈ {0, . . . , n − 1} and starting from t = n − 1, if ( p, t + 1) already belongs to Γ , T t x p > 0 and T t x p+1 > 0, we add ( p, t) and ( p + 1, t) to Γ . An example of such a cluster can be found in Fig. 1. Let g be the application mapping x onto Γ , and G(Λ) be the image of E(Λ) under g. Then  E(Λ) = g −1 Γ, (15) Γ ∈G (Λ)

Fig. 1. Example of a cluster. White points are negative sites and black points positive sites

Phase Transition and Correlation Decay in Coupled Map Lattices

235

or equivalently, in term of characteristic functions:  1g−1 Γ (x). 1(0,1]Λ (T n x) =

(16)

If we define ∂Γ by     / Γ or ( p + 1, t − 1) ∈ /Γ , ∂Γ = ( p, t) ∈ Γ  t = 0 or ( p, t − 1) ∈

(17)

Γ ∈G (Λ)

and if we define Γt = { q ∈ Z | (q, t) ∈ Γ } and ∂Γt = { q ∈ Z | (q, t) ∈ ∂Γ } the restrictions of respectively Γ and ∂Γ to time t, we can see that the characteristic function of g −1 Γ can be rewritten as 1g−1 Γ ( x ) ⎤ ⎡ n      ⎣ = 1 − 1(0,1] (T t x p ) 1(0,1] (T t x p+1 ) ⎦ 1(0,1] (T t x p ) t=0

=

n 

p∈Γt

p∈∂Γt+1

1 E(Γ,t) (T t x),

(18)

t=0

where E(Γ, t) ⊆ X is defined by 1 E(Γ,t) (x)     1 − 1(0,1] (x p ) 1(0,1] (x p+1 ) . 1(0,1] (x p ) = p∈Γt

(19)

p∈∂Γt+1

Let us pick some arbitrary Γ ∈ G(Λ). The cluster Γ can be split in connected parts, respectively Γ (k) for k = 1, . . . , c with c the number of connected parts. For any connected part of Γ , say Γ (k) , we define Λ(k) = { p | ( p, n) ∈ Γ (k) }. The outer boundary of Γ (k) is now a closed loop, and we can always choose the orientation of the loop to be clockwise. The outer path of Γ (k) is now defined as the part of the closed loop that goes from (sup Λ(k) , n) to (inf Λ(k) , n). The outer paths associated to the cluster of Fig. 1 have been drawn at Fig. 2. One can see that the cluster Γ (k) is unequivocally defined by its outer path and that the outer path only makes jumps along the edges (+1, −1), (−1, 0) and (0, +1). Let n (k) d ,

Fig. 2. Outer paths associated to the cluster of Fig. 1

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A. de Maere

(k) (k) n (k) = v and n h be the number of jumps in these directions respectively, and let ∂Γ (k) (k) (k) ∂Γ ∩ Γ . Then, since the outer path starts in (sup Λ , n) and ends in (inf Λ , n), and since there is always an horizontal edge between two sites of the outer path belonging to ∂Γ (k) , we have ⎧  (k)  (k) (k) ⎪ ⎨n h ≥ n d + Λ  − 1 (k) n (k) d − nv = 0 ⎪ ⎩∂Γ (k)  ≥ n (k) + 1. h

We can nowgo back to the  cluster Γ by summing the c connected parts and cover (k) c (k) defining n d = ck=1 n (k) k=1 n v and n h = k=1 n h yields: d , nv = ⎧ ⎪ ⎨n h ≥ n d + |Λ| − c nd = nv ⎪ ⎩|∂Γ | ≥ n + c. h

(20)

If we want to estimate the probability with respect to some initial signed measure μ that at time n all the sites in Λ are positive, we can now use (16) and (18): 1(0,1]Λ T μ = n

 Γ ∈G (Λ)

T 1g−1 Γ μ = n

 Γ ∈G (Λ)

1(0,1]Λ

n−1 

T 1 E(Γ,t)



μ.

(21)

t=0

  Of course, we assumed that the product of operators n−1 t=0 T 1 E(Γ,t) is time-ordered. The expansion of Eq. (21) can be the starting point of what is called in Statistical Mechanics a Peierls argument: indeed, if we can prove that for any fixed cluster, the weight of the cluster decays exponentially with its size in some sense that we still have to clarify, and if we can prove that the number of clusters of fixed size grows at most exponentially with the size of the cluster, we can find an upper bound on the probability that all sites in some Λ ⊂ Z are positive at some time n ∈ N with a simple geometric series. But before giving all the details of the Peierls argument, let us review some of the properties of Var Λ and |||·|||θ . 3.2. Generalized Lasota-Yorke inequalities. An important result for Interval Maps and Coupled Map Lattices is the Lasota-Yorke inequality [28] which controls the growth of · under the iterations of T . In this section, we will see that we can also control the growth of Var Λ through a simple generalization of the usual Lasota-Yorke inequality. Proposition 3 (Generalized Lasota-Yorke inequalities). For every finite Λ in Z and every μ ∈ L 1 (X ) such that Var Λ μ < ∞, we have Var Λ (T μ) ≤

 Ω⊆Λ

λ0 |Ω| D0 |Λ\Ω| Var Ω μ.

1 (X ). Then, if x is restricted to one of the Proof. Let ϕ be some arbitrary function in CΛ p intervals Ji , we see that ϕ ◦ T is differentiable with respect to x p and since we assumed

Phase Transition and Correlation Decay in Coupled Map Lattices

237

  that inf τ   > 0, we get ∂ p (ϕ ◦ T ) τ  (x p )     ϕ◦T 1 − 1 Ji (x p ) (x p ) ϕ ◦ T . = 1 Ji (x p ) ∂ p  τ (x p ) τ

1 Ji (x p ) (∂ p ϕ) ◦ T = 1 Ji (x p )

(22)

Now, for every p ∈ Z and i ∈ {1, . . . , N }, we introduce the operators Δi, p and Ri, p : ψ(ζi+1 , x = p ) − ψ(ζi , x = p ) Δi, p ψ(x) = , ζi+1 − ζi  xp   Ri, p ψ(x) = ∂ p ψ(ξ p , x = p ) − Δi, p ψ(ξ p , x = p ) dξ p

(23)

ζi

  ζi+1 − x p  x p − ζi  ψ(x) − ψ(ζi , x = p ) − ψ(ζi+1 , x = p ) − ψ(x) . = ζi+1 − ζi ζi+1 − ζi (24) One might remark that if ψ is a piecewise continuously differentiable function with  its discontinuities located at the boundaries of the intervals Ji , the function i 1 Ji (x p ) Ri, p ψ(x) vanishes at the boundaries of the Ji and is therefore not only piecewise continuously differentiable but also continuous with respect to x p . Moreover, the definition of Ri, p implies that 1 Ji (x p )∂ p ψ = 1 Ji (x p )∂ p (Ri, p ψ) + 1 Ji (x p )Δi, p ψ.

(25)

Therefore, using (25) in (22), we find 1 Ji (x p ) (∂ p ϕ) ◦ T        1  ϕ◦T ϕ◦T = 1 Ji (x p ) ∂ p Ri, p + Δi, p − τ  (x p ) ϕ ◦ T τ  (x p ) τ  (x p )   = 1 Ji (x p ) ∂ p Ki, p + Di, p (ϕ ◦ T ),

(26)

where the operators Ki, p and Di, p are defined by 

Ki, p Di, p

 ψ , : ψ → Ri, p τ  (x p )     1 ψ − : ψ → Δi, p (x p ) ψ.  τ (x p ) τ

(27)

For the proof of the usual Lasota-Yorke inequality, we just have to perform this construction for some fixed p in Z. But since we have multiple derivatives, we will iterate this for every p in Λ. For any iΛ = {i p } p∈Λ , we define the set J (iΛ ) = {x | ∀ p ∈ Λ :

238

A. de Maere

x p ∈ Ji p }. Since the operators ∂ p , Ki,q and Di,s commute as long as q, p and s are different, we have ⎛ ⎞  μ ( ( ∂Λ ϕ) ◦ T ) = μ ⎝ 1 J (iΛ ) ( ∂Λ ϕ) ◦ T ⎠ ⎛ = μ⎝

iΛ

 iΛ

=



⎞   ∂ p Ki p , p + Di p , p (ϕ ◦ T ) ⎠

1 J (iΛ )

⎛

μ⎝

p∈Λ



Ω⊆Λ

⎡

1 J (iΛ ) ∂Ω ⎣

⎤⎡



Ki p , p ⎦ ⎣

p∈Ω

iΛ



⎤

⎞

Di p , p ⎦ (ϕ ◦ T ) ⎠ .

(28)

p∈Λ\Ω

If for every Ω ⊆ Λ, we define the function: ⎡ ⎤⎡ ⎤    1 J (iΛ ) ⎣ Ki p , p ⎦ ⎣ Di p , p ⎦ (ϕ ◦ T ), ψΩ = p∈Ω

iΛ

p∈Λ\Ω

we can see that, by definition of the operators Ki p , p , ψΩ vanishes when x p = ζi , for each p ∈ Ω. Therefore, as long as p is in Ω, we have   xp ψΩ (x) = 1 Ji (ξ p )∂ p ψΩ (ξ p , x = p ) dξ p . (29) i

−1

Iterating this for every p ∈ Ω and taking the derivative with respect to all these variables yields: ⎡ ⎤⎡ ⎤    ∂Ω ψΩ = 1 J (iΛ ) ∂Ω ⎣ Ki p , p ⎦ ⎣ Di p , p ⎦ (ϕ ◦ T ). (30) p∈Ω

iΛ

p∈Λ\Ω

Therefore, (28) becomes μ ( ( ∂Λ ϕ) ◦ T ) =



μ ( ∂Ω ψΩ ),

Ω⊆Λ

but since ψΩ is piecewise continuous with respect to x =Ω , continuous and piecewise continuously differentiable with respect to xΩ , we can apply Proposition 8 from the Appendix, and we get ⎡ ⎤⎡ ⎤        μ ( ( ∂Λ ϕ) ◦ T ) ≤ sup  ⎣ Ki p , p ⎦ ⎣ Di p , p ⎦ (ϕ ◦ T ) Var Ω ( μ ). Ω⊆Λ

iΛ

p∈Ω

p∈Λ\Ω

∞

(31) We can now check that for any continuous function ψ, by the definition of Ki p , p and Di p , p from (27), Ri p , p and Δi p , p from (24), and λ0 and D0 from the assumptions on τ ,

Phase Transition and Correlation Decay in Coupled Map Lattices

239

we have

 Di

p,p

 ψ ∞

    Ki , p ψ  ≤ 1  Ri , p ψ  ≤ 2 |ψ|∞ ≤ λ0 |ψ|∞ , p p ∞ ∞ κ κ      1  1 ≤ Δi p , p ψ ∞ +    |ψ|∞ , κ τ    ∞   τ  2  |ψ|∞ ≤ D0 |ψ|∞ . + ≤ κ mini |Ji |  (τ  )2 ∞

(32)

(33)

Consequently, from (31), we get the expected result: μ ( ( ∂Λ ϕ) ◦ T ) ≤ |ϕ|∞

 Ω⊆Λ

λ0 |Ω| D0 |Λ\Ω| Var Ω μ.

 

A first consequence of Proposition 3 is the Lasota-Yorke inequality. Indeed, if we take Λ to be a singleton, and recall that sup p∈Z Var { p} μ = μ , we have T μ = sup Var { p} T μ ≤ λ0 μ + D0 |μ| . p∈Z

(34)

This implies that the operator T t is uniformly bounded in B(X ), because t−1 t−1    

t

t k  k 

T μ ≤ λt μ + D0 μ T ≤ λ + D λ μ λ0 k |μ|   0 0 0 0

%

≤ λ0 t +

k=0

D0 1−λ0

&

k=0

μ .

(35)

Therefore, if we take as initial measure m (−) , the Lebesgue measure concentrated

on  t m (−) is uniformly bounded in B(X ) because m (−) = 2, T [−1, 0], the sequence n1 n−1 t=0 and we can choose a subsequence which converges weakly to an invariant measure in B(X ). Let μ(−) inv be such an invariant measure. Another important consequence of Proposition 3 is the fact that the transfer operator T is bounded in the θ -norm, for θ large enough. Corollary 1. For any θ ≥

D0 1−λ0

and any μ ∈ L 1 (X ) with bounded θ -norm, we have |||T μ|||θ ≤ |||μ|||θ .

Proof. We use Proposition 3 and the fact that θ −|Ω| Var Ω μ ≤ |||μ|||θ : θ −|Λ| Var Λ (T μ) ≤ θ −|Λ| % ≤ λ0 +



(λ0 θ )|Ω| D0 |Λ|−|Ω| |||μ|||θ

Ω⊆Λ &|Λ| D0 θ

% The lower bound on θ then implies λ0 +

D0 θ

&

|||μ|||θ .

≤ 1 and so |||T μ|||θ ≤ |||μ|||θ .  

240

A. de Maere

3.3. The Peierls argument. The bottom line of the Peierls argument is to show that the number of clusters of a fixed size grows at most exponentially with the size, and that the probability of having a large cluster decays exponentially with the size of the cluster. The first estimate, sometimes called the entropic estimate, is quite standard. However, the second estimate, also called the energetic estimate, will become problematic in the case of CML. Indeed, for any finite Λ ⊂ Z, if E  Λ ⊆ X is the set of configurations x such that x p ∈ E  for any p ∈ Λ, we know that the Lebesgue measure of E  Λ is smaller than |E  |Λ , but we do not expect this to be true for an arbitrary signed measure, even if this measure is of bounded variation. For a measure of bounded variation, the best estimate one can find is |1 E Λ μ| ≤ |E  | μ . Therefore, we need to introduce extra regularity conditions on the initial measures. For instance, one could follow Bardet and Keller [24] and consider only totally decoupled initial measures. But in order to prove the exponential convergence to equilibrium, we will need to apply the Peierls argument to an invariant measure which is not totally decoupled as long as  = 0. We will solve this problem in a new approach that relies on Var Λ and the θ -norm. First, let us see how Var Λ allows us to control |1 E Λ μ|. If we define the operator E p by  xp E p ψ(x) = 1 E  (ξ p ) ψ(ξ p , x = p ) dξ p , (36) 0

the symmetry assumption on τ implies that |E  ∩ [0, 1]| = |E  ∩ [−1, 0]| =

|E  | 2

and so   E p ψ 

∞

≤ |ψ|∞

sup

x p ∈[−1,1]

   

xp

0

  1 E  (ξ p ) dξ p  ≤

|E  | 2

|ψ|∞ .

(37)

We can check that 1 E  (x p )ψ(x) = ∂ p E p ψ(x). Hence, if Ω and Λ are two disjoint subsets of Z, we have ⎛ ⎡ ⎤ ⎞  1 E  Λ μ(∂Ω ϕ) = μ( 1 E  Λ ∂Ω ϕ ) = μ ⎝ ∂Λ∪Ω ⎣ Ep⎦ ϕ ⎠ ≤

%

|E  | 2

&|Λ|

p∈Λ

Var Ω∪Λ μ |ϕ|∞ .

This finally implies an estimate on 1 E  Λ μ with the appropriate exponential decay: Var Ω 1 E  Λ μ ≤

%

|E  | 2

&|Λ|

Var Ω∪Λ μ.

(38)

However, if the assumption Λ ∩ Ω = ∅ is not fulfilled, we can not use such a simple method without having to consider second derivatives with respect to some variables, which we do not expect to behave nicely. But the dynamic can help us, and with the generalized Lasota-Yorke inequalities, we have:

Phase Transition and Correlation Decay in Coupled Map Lattices

241

Lemma 1. For any measure μ, any cluster Γ , any finite Ω ⊆ Z and any Λ ⊆ Ω, we have: % &   Var Ω T 1 E(Γ,t) 1 E  Λ μ ≤ λ1 |V1 | D1 |Λ\V1 | λ0 |V0 | D0 |Ω\(Λ∪V0 )| Var V1 ∪V0 μ, V1 ⊆Λ V0 ⊆Ω\Λ

where λ1 and D1 were defined in (11) and E(Γ, t) in (19). Proof. We start by applying the development of (28) to the measure 1 E(Γ,t) 1 E  Λ μ: T 1 E(Γ,t) 1 E  Λ μ ( ∂Ω ϕ ) ⎛ ⎤ ⎞ ⎡   = μ ⎝ 1 E(Γ,t) 1 J (iΩ ) 1 E  Λ ⎣ (∂ p Ki p , p + Di p , p )⎦ (ϕ ◦ T ) ⎠ .

(39)

p∈Ω

iΩ

We first consider the characteristic functions of E(Γ, t) and J (iΩ ). Since the partition Ji is finer than the intervals (0, 1] or [−1, 0], we know that if the configuration x is fixed outside Ω, 1 E(Γ,t) 1 J (iΩ ) is either  identically 0 or identically 1 as a function of xΩ restricted to J (iΩ ). Therefore, iΩ 1 E(Γ,t) 1 J (iΩ ) can always be rewritten as  iΩ ciΩ 1 J (iΩ ) , where the ciΩ are some discontinuous functions depending only on the variables outside Ω and taking only values 0 and 1. So (39) can be rewritten as T 1 E(Γ,t) 1 E  Λ μ ( ∂Ω ϕ ) ⎛ ⎤ ⎞ ⎡   = μ ⎝ ciΩ 1 J (iΩ ) 1 E  Λ ⎣ (∂ p Ki p , p + Di p , p )⎦ (ϕ ◦ T ) ⎠ .

(40)

p∈Ω

iΩ

We now focus on the characteristic function of E  Λ . Since Λ ⊆ Ω, we have ⎤ ⎡   ciΩ 1 J (iΩ ) 1 E  Λ ⎣ (∂ p Ki p , p + Di p , p )⎦ (ϕ ◦ T ) iΩ

=



⎡ ciΩ 1 J (iΩ ) ⎣

×⎣

 p∈Λ

iΩ

⎡

p∈Ω



⎤

(1 E  (x p )∂ p Ki p , p + 1 E  (x p )Di p , p )⎦ ⎤

(∂ p Ki p , p + Di p , p )⎦ (ϕ ◦ T ).

(41)

p∈Ω\Λ

But 1 E  (x p )ψ is equal to ∂ p E p ψ, with the operator E p introduced in (36), and therefore we have 1 E  (x p )Di p , p ψ = ∂ p E p Di p , p ψ. And, since τ restricted to Ji is monotone, we know that E  ∩ Ji is always an interval, let us say [ai , bi ]. Therefore, for any interval Ji and any coordinate p ∈ Z, we can define the operator:  xp Si, p ψ = ψ(ai , x = p ) + 1 E  ∂ p ψ, (42) ai

242

A. de Maere

and see that, as long as x p belongs to Ji , ∂ p Si, p ψ = 1 E  ∂ p ψ and that   we immediately 1 J (x p )Si, p ψ  ≤ |ψ|∞ . Hence, if x p ∈ Ji : p i ∞ 1 E  (x p )∂ p Ki p , p ψ = ∂ p Si, p Ki p , p ψ. Eventually, Eq. (41) can be rewritten as 

⎡ ciΩ 1 J (iΩ ) 1 E  Λ ⎣

iΩ

=



⎡ ciΩ 1 J (iΩ ) ⎣



⎤ (∂ p Ki p , p + Di p , p )⎦ (ϕ ◦ T )

p∈Ω



⎤⎡

(∂ p Si p , p Ki p , p + ∂ p E p Di p , p )⎦ ⎣

p∈Λ

iΩ



⎤ (∂ p Ki p , p + Di p , p )⎦ (ϕ ◦ T ).

p∈Ω\Λ

Here, we would like to apply directly Proposition 8, but this is impossible because the operator Si p , p destroys the regularization introduced by Ki p , p . Indeed, if ψ is continuously differentiable on the intervals Ji , the function i p 1 Ji (x p )Ki p , p ψ(x) is not only continuously differentiable on the intervals Ji but also continuous with respect to x p , because Ki p , p ψ(x)  vanishes at the boundaries of the intervals Ji . However, this is no longer true for i p 1 Ji (x p )Si p , p Ki p , p ψ(x), and we need to apply once again the operator Ri p , p from (24) in order to regularize the discontinuities of the function. From (25), we have 1 Ji p ∂ p Si p , p Ki p , p ψ = 1 Ji p ∂ p Ri p , p Si p , p Ki p , p ψ + 1 Ji p Δi p , p Si p , p Ki p , p ψ, and this implies that ⎡



ciΩ 1 J (iΩ ) 1 E  Λ ⎣

iΩ

=



⎡ ciΩ 1 J (iΩ ) ⎣



⎤ (∂ p Ki p , p + Di p , p )⎦ (ϕ ◦ T )

p∈Ω



⎤

(∂ p (Ri p , p Si p , p Ki p , p + E p Di p , p ) + Δi p , p Si p , p Ki p , p )⎦

p∈Λ

iΩ

⎡

⎤    ×⎣ ∂ p Ki p , p + Di p , p ⎦ (ϕ ◦ T ) p∈Ω\Λ

=







⎡ ciΩ 1 J (iΩ ) ⎣

V1 ⊆Λ V0 ⊆Ω\Λ iΩ

⎡



×⎣ 

⎤⎡

Δi p , p Si p , p Ki p , p ⎦ ⎣



V1 ⊆Λ V0 ⊆Ω\Λ

⎤ ∂ p (Ri p , p Si p , p Ki p , p + E p Di p , p )⎦

p∈V1

p∈Λ\V1

=





⎤⎡ ∂ p Ki p , p ⎦ ⎣

p∈V0

∂V

0 ∪V1

ϕ˜ V

0 ∪V1

,

where, for V1 and V0 fixed, ϕ˜ V0 ,V1 is defined as



⎤ Di p , p ⎦ (ϕ ◦ T )

p∈Ω\(Λ∪V0 )

(43)

Phase Transition and Correlation Decay in Coupled Map Lattices

ϕ˜ V0 ,V1 =

 iΩ

⎡



ciΩ 1 J (iΩ ) ⎣

⎡

×⎣

⎤ (Ri p , p Si p , p Ki p , p + E p Di p , p )⎦

p∈V1



243

⎤⎡

Δi p , p Si p , p Ki p , p ⎦ ⎣

p∈Λ\V1



⎤⎡



Ki p , p ⎦ ⎣

⎤ Di p , p ⎦ (ϕ ◦ T ).

p∈Ω\(Λ∪V0 )

p∈V0

(44) We can now conclude: if we insert (43) into (40), we find     μ ∂V0 ∪V1 ϕ˜ V0 ,V1 . T 1 E(Γ,t) 1 E  Λ μ ( ∂Ω ϕ ) = V1 ⊆Λ V0 ⊆Ω\Λ

But since ϕ˜ V0 ,V1 is continuous and piecewise continuously differentiable with respect to x V1 ∪V0 , and piecewise continuous with respect to the other variables, we can apply Proposition 8 and we get     Var V1 ∪V0 μ ϕ˜ V0 ,V1 ∞ . (45) T 1 E(Γ,t) 1 E  Λ μ ( ∂Ω ϕ ) ≤ V1 ⊆Λ V0 ⊆Ω\Λ

    Using  Ri p , p ψ ∞ ≤ 2 |ψ|∞ ,  Si p , p ψ ∞ ≤ |ψ|∞ , the bounds on Ki p , p from (32), on Di p , p from (33) and on E p from (37), altogether with the definition of λ1 and D1 from (11) yields     (Ri , p Si , p Ki , p + E p Di , p )ψ  ≤ 2 2 + |E  | D0 |ψ|∞ ≤ λ1 |ψ|∞ , p p p p ∞ κ 2   2 2 Δi , p Ki , p ψ  ≤ ≤ D1 |ψ|∞ . p p ∞ mini |Ji | κ   Therefore, ϕ˜ V0 ,V1 ∞ is bounded by  ϕ˜ V

0 ,V1

 

∞

|V |

|Λ\V1 |

≤ λ1 1 D 1

|V0 |

λ0

|Ω\(Λ∪V0 )|

D0

|ϕ|∞ .

And so, (45) becomes T 1 E(Γ,t) 1 E  Λ μ ( ∂Ω ϕ )   |V | |Λ\V | |V | |Ω\(Λ∪V0 )| ≤ |ϕ|∞ λ1 1 D 1 1 λ0 0 D 0 Var V1 ∪V0 μ.

 

V1 ⊆Λ V0 ⊆Ω\Λ

We have now all the tools to complete the Peierls argument. Theorem 3. Assume that τ belongs to T (D0 , c, ς ) and that κ > 108. Then, there is some 0 > 0 such that, if  < 0 , for any measure μ in B(K , α, θ0 ) with K < ∞, 2α0 1 and θ0 = |E , we have α < 27 |    1(0,1]Λ T n μ  with α  = 3 max{α0 , α} <

1 9

θ0

≤

K |Λ| α , 2(1 − 9α  )(1 − α  )

and α0 defined in (12).

244

A. de Maere

Proof. We start with the contour expansion of (21):    1(0,1]Λ T n μ 

θ0



≤

Γ ∈G (Λ)

  n−1       T 1 E(Γ,t) μ  .  1(0,1]Λ  

(46)

θ0

t=0

However, for some fixed cluster Γ , we know by the definition of E(Γ, t) in (19) that if x belongs to E(Γ, t) and T x belongs to E(Γ, t + 1), x has to be in E ∂Γt+1 with ∂Γt+1 = { q | (q, t + 1) ∈ ∂Γ }. So 1 E(Γ,t+1) (T x) 1 E(Γ,t) (x) = 1 E(Γ,t+1) (T x) 1 E(Γ,t) (x) 1 E  ∂Γt+1 (x). Therefore, we can insert the characteristic functions of E  ∂Γt+1 at every time t in each term of the sum in (46), and we get:    1(0,1]Λ T n μ 

θ0



≤

Γ ∈G (Λ)

  n−1   & %   T 1 E ∂Γt+1 1 E(Γ,t) μ  .  1(0,1]Λ   

(47)

θ0

t=0

But, for any measure ν and any finite subset Ω, if we first apply Lemma 1 and then use inequality (38), we have % & Var Ω T 1 E ∂Γt 1 E(Γ,t−1) ν    |V | |(Ω∩∂Γt )\V1 | |V0 | |Ω\(∂Γt ∪V0 )| −|Ω| λ1 1 D 1 λ0 D 0 ≤ θ0

−|Ω|

θ0

V1 ⊆Ω∩∂Γt V0 ⊆Ω\∂Γt

× Var V1 ∪V0 (1 E ∂Γt \Ω ν)    −|Ω| ≤ θ0  ×

|V1 |

λ1

|(Ω∩∂Γt )\V1 |

D1

|V0 |

λ0

|Ω\(∂Γt ∪V0 )|

D0

V1 ⊆Ω∩∂Γt V0 ⊆Ω\∂Γt

|E  | 2

|∂Γt \Ω|

−|Ω|

≤ θ0



Var V1 ∪V0 ∪(∂Γt \Ω) (ν)  |(Ω∩∂Γt )\V1 | |Ω\(∂Γt ∪V0 )| (λ1 θ0 )|V1 | D1 (λ0 θ0 )|V0 | D0

V1 ⊆Ω∩∂Γt V0 ⊆Ω\∂Γt

 θ0 |E  | |∂Γt \Ω| |||ν|||θ0 × 2       D1 |Ω∩∂Γt | D0 |Ω\∂Γt | θ0 |E  | |∂Γt \Ω| |||ν|||θ0 . λ0 + ≤ λ1 + θ0 θ0 2 

By definition of θ0 ,

θ0 |E  | 2

≤ α0 , and we can check that

   D1 1 2 λ1 + λ1 + 2D1 |E  | , λ1 + ≤ α0 ⇐⇒ α0 ≥ θ0 2   D0 1 D0 |E  | . λ0 + ≤ 1 ⇐⇒ α0 ≥ θ0 2 1 − λ0

(48)

Phase Transition and Correlation Decay in Coupled Map Lattices

245

So, the definition of α0 in (12) implies that, if we take the supremum over all finite Ω in (48), we get     (49) T 1 E Λt 1 E(Γ,t−1) ν  ≤ α0 |∂Γt | |||ν|||θ0 . θ0



We now go back to Eq. (47). We apply Corollary 2, inequality (49), use the assumption that μ belongs to B(K , α, θ0 ), define α  = 3 max{α0 , α}, recall the definition of ∂Γ , and we find   n−1            1(0,1]Λ T n μ  ≤ 1 T 1 μ Λ     E(Γ,t) (0,1] θ0   t=0 Γ ∈G (Λ) θ0  n−1        % &      T 1 E ∂Γt+1 1 E(Γ,t) 1(0,1]Λ0 μ  ≤      t=0 Γ ∈G (Λ) θ0   n   |∂Γt |  t=1 ≤ α0 (50) 1(0,1]∂Γ0 μ  θ0

Γ ∈G (Λ)

≤



K α0

Γ ∈G (Λ)

≤



 K

Γ ∈G (Λ)

n

α 3

t=1 |∂Γt |

|∂Γ |

α |∂Γ0 |

.

(51)

We can now count the number of clusters. A cluster Γ is unequivocally determined by its outer path and there are at most 3n d +n v +n h outer paths with n d , n v and n h edges in the diagonal, vertical and horizontal directions respectively. We have seen in (20) that n d = n v and that there is some k ≥ 0 such that n h = n d + |Λ| − c + k and |∂Γ | ≥ n h + c = n d + |Λ| + k. Therefore, (51) becomes:   n d +|Λ|+k ∞  ∞  ∞    3n d +|Λ|−c+k α 1(0,1]Λ T n μ ≤ K 3 . (52) θ0 3 n d =0 k=0 c=1

Now, we have seen in (46) that lim→0 α0 = κ4 . If we assume that κ > 108, there is 1 some 0 such that  < 0 implies α0 < 27 . We assume then that κ > 108 and  < 0 . 1 Since we also assumed that α < 27 , we have α  < 19 and the geometric series in (52) converge. This yields:   1(0,1]Λ T n μ ≤ θ 0

K |Λ| α . 2(1 − 9α  )(1 − α  )

 

Now, Theorem 1 is a straightforward consequence of Theorem 3. Proof (Theorem 1). Let us start by proving that μ(−) inv belongs to B(K 0 , 3α0 , θ0 ). Since Var Λ m (−) ≤ 2|Λ| , we know that the measure m (−) belongs to B(1, 0, 2). But, by definition of θ0 , we have θ0 =

2α0 D0 ≥ 2. ≥ |E  | 1 − λ0

(53)

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Therefore, m (−) also belongs to B(1, 0, θ0 ). We can now apply Theorem 3. If κ > 108, there is some 0 such that, if  < 0 , T t m (−) belongs to B(K 0 , 3α0 , θ0 ) for any t and (+) 3α0 < 19 . Therefore, μ(−) inv too belongs to B(K 0 , 3α0 , θ0 ). Since μinv does not belong to (+) (−) any B(K , α, θ ) as long as α < 1, this also proves that μinv = μinv .   4. Exponential Convergence to Equilibrium In the previous section, we defined the measure μ(−) inv as a converging subsequence of 1 n−1 t (−) t=0 T m . However, it was actually unnecessary to take the limit in the sense of n Cesaro and to restrict ourselves to some subsequence, because, as we will see in this section, m (−) and many other initial probability measures converge exponentially fast to μ(−) inv . Let us start by choosing some arbitrary positive integer γ and considering the wellordering ≺, defined by 0 ≺ −1 ≺ −2 ≺ · · · ≺ −γ ≺ +1 ≺ −γ − 1 ≺ +2 ≺ −γ − 2 . . . . With this ordering, we see that all the sites influenced by 0 after γ iterations of the dynamic are the γ + 1 first sites. Let ≺q be the translation of this well-ordering at any site q of Z. Then, for any q and p in Z, we define the operator Πq p : ⎛ ⎞    (−)  (−) Πq p μ(ϕ) ≡ μ(Πq p ϕ) = μ ⎝ ϕ h inv (xs ) dxs − ϕ h inv (xs ) dxs ⎠ , (54) s≺q p

sq p

where h (−) inv was defined as the invariant measure of the local map τ concentrated on [−1, 0]. Note that as long as ϕ does not depend on x p , Πq p ϕ is identically zero. Furthermore, for any ϕ depending only on the variables in Λ, an arbitrary finite subset of Z, and for any signed measure of zero mass μ, we have, for any q ∈ Z: ⎛ ⎞      (−)  (−) Πq p μ(ϕ) = μ⎝ ϕ h inv (xs ) dxs − ϕ h inv (xs ) dxs ⎠ p∈Z

p∈Z

= μ(ϕ) − μ(1)

s≺q p



ϕ

sq p



h (−) inv (x s ) dx s = μ(ϕ).

s∈Λ

Since this is true for any continuous function, this implies that any signed measure of zero mass μ can be decomposed as  Πq p μ. (55) μ= p∈Z

We can also see that the operator Πq p is bounded both in total variation norm, bounded variation norm and θ -norm, as stated in the next lemma: Lemma 2. For any measure μ in B(X ), any q, p ∈ Z and any θ ≥  ⎧    ⎪ ⎨ Πq p μ ≤ 2 |μ| % & D0

Πq p μ ≤ 2 1−λ0 μ ⎪  ⎩  Πq p μ ≤ 2 |||μ|||θ . θ

D0 1−λ0 ,

we have

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247

Proof. For the two first inequalities, it is sufficient to prove that, for any Λ ⊂ Z, we have &|Λ| % D0 Var Λ Πq p μ ≤ 2 1−λ Var Λ (μ). 0 1 (X ) with |ϕ| ≤ 1, we consider So, for any function ϕ in CΛ ∞ ⎛ ⎞ ⎛ ⎞    (−)  (−) Πq p μ(∂Λ ϕ) = μ ⎝ ∂Λ ϕ h inv (xs ) dxs ⎠ − μ ⎝ ∂Λ ϕ h inv (xs ) dxs ⎠ . s≺q p

sq p

(56) If we define Ω = { k ∈ Λ | q q k ≺q p } , we see that the derivatives with respect to xΛ\Ω commute with the first integral of (56). And the same can be done for the second integral of (56) with Ω  = { k ∈ Λ | q q k q p }. And so, by definition of Var Ω or Var Ω  , we get ⎛ ⎞   (−) h (xs ) dxs ⎠ Πq p μ(∂Λ ϕ) = μ ⎝∂Λ\Ω ∂Ω ϕ inv

⎛ −μ ⎝∂Λ\Ω 

s≺q p

 ∂Ω  ϕ



⎞ ⎠ h (−) inv (x s ) dx s

sq p

     (−)   ≤ Var Λ\Ω (μ)  ∂Ω ϕ h inv (xs ) dxs    s≺q p ∞      (−)   +Var Λ\Ω  (μ)  ∂Ω  ϕ h inv (xs ) dxs    sq p ∞

(−) |Ω|

|Ω  |

≤ Var Λ\Ω (μ) h inv BV |ϕ|∞ + Var Λ\Ω  (μ) h (−) (57) inv BV |ϕ|∞ .

(−)

We can find an upper bound on h inv BV with the Lasota-Yorke inequality of τ from 8. Indeed, if h (−) leb is the Lebesgue measure concentrated on [−1, 0], we have n (−) τ n h (−) leb BV ≤ λ0 h leb BV +





and it implies that h (−) inv BV ≤

D0 1−λ0

D0 (−) |h | 1 , 1 − λ0 leb L (I )

(−) because τ n h (−) leb converges to h inv . Hence, since

Var Λ\Ω μ ≤ Var Λ μ, Ω ⊆ Λ, and since we already saw that

D0 1−λ0

≥ 1, (57) becomes

&|Ω| & Ω % % D0 D0 | | + Var Var Λ Πq p μ ≤ Var Λ\Ω (μ) 1−λ  (μ) Λ\Ω 1−λ 0 0 &|Ω| &|Ω  | % % D0 D0 + Var Λ (μ) 1−λ ≤ Var Λ (μ) 1−λ 0 0 &|Λ| % D0 Var Λ (μ), ≤ 2 1−λ 0 and this proves the two first inequalities of the lemma.

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The bound in the θ -norm is a consequence of (57). Indeed, if we multiply each side D0

of the inequality by θ −|Λ| , and use θ ≥ 1−λ ≥ h (−) inv , we get 0   Πq p μ ≤ |||μ|||θ + |||μ|||θ . θ

 

Let us now consider some signed measure of zero mass μ and some continuous function ϕ on X depending only on the variables xΛ for some finite Λ in Z, and carry out the decomposition of (55) after every γ iteration of the dynamic. If we assume that t = m γ for some m ∈ N, we get  T t μ(ϕ) = Π pm−1 pm T γ Π pm−2 pm−1 . . . T γ Π0 p0 μ(ϕ). p∈Zm+1

If F(Λ) is the set of p such that pm ∈ Λ and every pk belongs to { pk−1 − γ , . . . , pk−1 }, we notice that every configuration of p not belonging to F(Λ) does not contribute to the sum. Indeed, if pm does not belong to Λ, Π pm−1 pm ϕ is identically zero because ϕ does not depend on x pm , so Π pm−1 pm ν(ϕ) = ν(Π pm−1 pm ϕ) = 0 for any measure ν. And if p does not belong to {q − γ , . . . , q}, we see that, from the definition of ≺, Πq p ϕ does  not depend on any variable xs with s ∈ {q − γ , . . . , q}, and so Πq p ϕ ◦ T γ does not depend on xq . Therefore, Πq p T γ Πsq ν = 0 for any s ∈ Z and any measure ν. Hence, if we define T˜ q p = T γ Πq p ,

(58)

and apply Lemma 2, the expansion becomes    t   γ T μ(ϕ) ≤  Π p p T γ Π p p m−1 m m−2 m−1 . . . T Π0 p0 μ(ϕ) p∈F (Λ)

≤2

    T˜ pm−2 pm−1 . . . T˜ 0 p0 μ |ϕ|∞ .

(59)

p∈F (Λ)

In the next subsection, we will prove with a decoupling argument that the dynamic restricted to a pure phase, namely the operator T˜ q p 1[−1,0] (x p )1[−1,0] (x p+1 ), is a contraction in B(X ). 4.1. Decoupling in the pure phases. The idea behind the decoupling in the pure phase is to reproduce the decoupling argument of Keller and Liverani [14,16], but instead of considering the coupling as a perturbation of the identity, we will consider the coupling as a perturbation of a strongly coupled dynamic for which we can prove the exponential convergence to equilibrium in the pure phases. The decoupled dynamic at site p is given ( p) ( p) ( p) by T0 = Φ0 ◦ τ Z , where the coupling Φ0 is explicitly given by  Φ,q (x) if q = p ( p) Φ0,q (x) = . (60) Φ0, p (x = p+1 , −1) if q = p The next proposition shows that this slight modification of the coupling does not change too much the dynamic when applied to a measure 1[−1,0] (x p+1 )μ.

Phase Transition and Correlation Decay in Coupled Map Lattices

249

Proposition 4. Let μ ∈ B(X ) and p ∈ Z. Then     ( p)  Φ 1[−1,0] (x p+1 ) μ − Φ0 1[−1,0] (x p+1 ) μ  ≤ 2  μ . Proof. For the demonstration of this proposition, we will basically follow the lines of ( p) the proof of Proposition 5 in [14]. If Ft = t Φ + (1 − t) Φ0 , we can state that & %   ( p) μ 1[−1,0] (x p+1 ) ϕ ◦ Φ − μ 1[−1,0] (x p+1 ) ϕ ◦ Φ0    1 = μ 1[−1,0] (x p+1 ) ∂t (ϕ ◦ Ft ) dt 0 ' (  1    μ 1[−1,0] (x p+1 ) ∂q ϕ ◦ Ft ∂t Ft,q dt. (61) = 0

q ( p)

But we can check that 1[−1,0] (x p+1 ) ∂t Ft,q = 1[−1,0] (x p+1 ) (Φ,q (x) − Φ0,q (x)) is equal to 0 if q = p, and to 1[−1,0] (x p+1 )  if q = p. So, the sum over q reduces to the term q = p and becomes & %   (q) μ 1[−1,0] (x p+1 ) ϕ ◦ Φ − μ 1[−1,0] (x p+1 ) ϕ ◦ Φ0  1   = (62) μ 1[−1,0] (x p+1 ) ∂ p ϕ ◦ Ft dt. 0

But, if we define the function ψ:  ψ(x) = 0

xp

(∂ p ϕ ◦ Ft )(ξ p , x = p ) dξ p ,

we can check that ψ is continuous with respect to x p , piecewise continuously differentiable, that ∂ p ψ = ∂ p ϕ ◦ Ft , and as long as x p+1 belongs to [−1, 0], we have ∂ p Ft, p = 1 and  x p    sup ψ(x) = sup  (∂ p ϕ ◦ Ft )(ξ p , x = p ) dξ p  x:x p ≤0 x:x p ≤0 0  x p    ∂ p (ϕ ◦ Ft )(ξ p , x = p ) = sup  dξ p  ∂ p Ft, p x:x p ≤0 0  x p     = sup  ∂ p (ϕ ◦ Ft )(ξ p , x = p ) dξ p  x:x p ≤0

0

≤ 2 |ϕ ◦ Ft |∞ ≤ 2 |ϕ|∞ . With Proposition 8 and Corollary 2, the definition of ψ implies that     μ 1[−1,0] (x p+1 ) ∂ p ϕ ◦ Ft = μ 1[−1,0] (x p+1 ) ∂ p ψ ≤ 2 |ϕ|∞ μ , and we conclude by inserting this inequality into Eq. (62): & %   ( p) μ 1[−1,0] (x p+1 ) ϕ ◦ Φ − μ 1[−1,0] (x p+1 ) ϕ ◦ Φ0 ≤ 2  |ϕ|∞ μ .

 

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A. de Maere

This estimate allows us to control the difference between the original dynamic T and ( p) = Φ0 ◦ τ Z when the site p + 1 stays negative:

( p) T0

Proposition 5. For any measure μ ∈ B(X ), p ∈ Z and m ∈ N, we have  % &m  m   ( p) D0  T 1[−1,0] (x p+1 ) μ − T0 1[−1,0] (x p+1 ) μ ≤ 1+m 1−λ0 2  μ . Proof. Once again, we follow the proof of Theorem 6 in [14]. We define T¯ = T 1[−1,0] ( p) ( p) (x p+1 ), and T¯0 = T0 1[−1,0] (x p+1 ). Then, with the help of a simple telescopic sum, we have ( p) m T¯0 μ = T¯ m μ +

m−1 

% & ( p) m−k−1 ¯ ( p) T0 − T¯ T¯ k μ. T¯0

(63)

k=0

Then, taking the total variation norm of this expansion and applying Proposition 4 to ( p) control the difference between Φ0 and Φ , we get m      % &  ¯m  ¯ ( p) m−k−1 ¯ ( p)  ( p) m  ¯ T0 − T¯ T¯ k−1 μ μ ≤ T μ − T0 T0 k=1

m % m %    & &   ¯ ( p)  ( p)   ≤  T0 − T¯ T¯ k−1 μ ≤  Φ0 − Φ τ Z 1[−1,0] (x p+1 )T¯ k−1 μ k=1

k=1

m %  &   ( p)  ≤  Φ0 − Φ 1[−1,0] (x p+1 )τ Z T¯ k−1 μ k=1

m %  &   ( p)  ≤  Φ0 1[−1,0] (x p+1 ) − Φ 1[−1,0] (x p+1 ) τ Z T¯ k−1 μ k=1

≤

m 





2  τ Z T¯ k−1 μ .

(64)

k=1

But T¯ satisfies a Lasota-Yorke inequality, because of Corollary 2:







T¯ μ ≤ λ0 1[−1,0] (x p+1 )μ + D0 1[−1,0] (x p+1 )μ ≤ λ0 μ + D0 |μ| ,

(65)

and τ Z also satisfies the same inequality, as a consequence of (8). Therefore

%

&

Z ¯ k−1

D0 μ , μ ≤ λk0 + 1−λ

τ T 0 and inequality (64) becomes  m     ¯m ( p) m  ¯ T ≤ μ − T μ 2  λk0 +   0 k=1

D0 1 − λ0

 μ ≤

1 + m D0 2  μ . 1 − λ0

 

We are now ready to prove that if μ is a signed measure of bounded variation concentrated on x p ∈ [−1, 0] and x p+1 ∈ [−1, 0], the operator T˜ q p acting on μ is a contraction:

Phase Transition and Correlation Decay in Coupled Map Lattices

251

Theorem 4. For any measure μ in B(X ), and any q and p in Z, we have



˜ qp

T 1[−1,0] (x p )1[−1,0] (x p+1 )μ ≤ σ1 μ , where σ1 is given by ) % & % &2 γ −n γ D0 D0 n λ + + λ + cς σ1 = 2 1−λ 0 0 1−λ0 0

* 1+n D0 1−λ0

(4  + |E  |) .

Proof. Remember that T˜ q p was defined in (58) as T γ Πq p . If we apply the Lasota-Yorke inequality (35) to T γ −n , for some strictly positive n < γ , we have

γ

T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ

γ −n

≤ λ0 T n Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ

  D0  n (66) T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ . + 1−λ 0 Applying once again the Lasota-Yorke to the first term of this inequality, using Lemma 2 and Corollary 2, we get

n

T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ



 ≤ λn Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ + D0 Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ 0

≤2 ≤2

D0 1−λ0 D0 1−λ0

1−λ0

D0 1−λ0

μ + 2 |μ|   n 1 + λ0 μ .

λn0

And so, inequality (66) becomes



˜ qp

T 1[−1,0] (x p )1[−1,0] (x p+1 )μ

% & γ −n γ D0 ≤ 2 1−λ λ + λ 0 0 μ + 0

D0 1−λ0

  n T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ .

(67)

For the second term in (67), we first note that, by the definition of Πq p in (54) and the fact that h (−) inv is concentrated on [−1, 0], we have, for any s ∈ Z and any measure ν: Πq p 1[−1,0] (xs )ν = 1[−1,0] (xs )Πq p 1[−1,0] (xs )ν.

(68)

We can therefore introduce an operator 1[−1,0] (x p+1 ) in front of Πq p in the second term of (67):  n  T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ   = T n 1[−1,0] (x p+1 )Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ . Since x p+1 is initially negative, either it stays negative up to time n or there is a sign flip at some intermediate time k:  n  T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ   n ≤  T 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ +

n−1     k  n−k  1(0,1] (x p+1 ) T 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ . T k=1

(69)

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A. de Maere

We can then apply Proposition 5 to the first term of (69), and replace the initial dynamic ( p) by T0 up to an error that grows at most linearly with time. This, together with Lemma 2 and Corollary 2, leads to     T 1[−1,0] (x p+1 ) n Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ %  &n  ( p)  ≤  T0 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ

1 + n D0

+2  Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ

1 − λ0 %  &n  ( p)  ≤  T0 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ +4

D0 1 + n D0  μ . 1 − λ0 1 − λ0

(70)

Now that we are left with the decoupled dynamic at site p, we can take advantage of the mixing properties of the local dynamic τ as in [16]. Indeed, for any measure ν, we see that % &n ( p) T0 1[−1,0] (x p+1 ) 1[−1,0] (x p )ν(ϕ) ( ' n  ( p) n ( p) t 1[−1,0] (T0 x p+1 ) . = ν 1[−1,0] (x p ) ϕ(T0 x) t=0 ( p) t 1[−1,0] (T0 x p+1 )

Here, does not depend on x p , but only on the variables x> p . More( p) n over, the sign of x p is initially fixed to be negative, therefore, T0 x p = τ n x p and ( p) n T0 xq for q = p depends only on the sign of x p which is fixed and negative. So, the dynamic is actually the product of two dynamics, τ n acting on x p , and T n acting on x = p with fixed negative boundary conditions in x p . If we define T = p xq = T (x = p , −1)q , and remember that τ preserves the signs, we see that % &n ( p) T0 1[−1,0] (x p+1 ) 1[−1,0] (x p )ν(ϕ) ( ' n  n n n t 1[−1,0] (T x p+1 ) = ν 1[−1,0] (τ x p ) ϕ(τ x p , T = p x = p )    ≤  τ n × 1 = p ν  |ϕ|∞ .

t=0

If we apply this inequality to the first term of (70), together with (68) to create a 1[−1,0] (x p ) in front of Πq p , we find  % &n   ( p)  T0 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ %  &n  ( p)  =  T0 1[−1,0] (x p+1 ) 1[−1,0] (x p )Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ    ≤  τ n × 1 = p Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ . But, if ϕ is some continuous function, with |ϕ|∞ ≤ 1, and if we define ψ:   (−) h inv (ξs ) dξs , ψ(x) = 1[−1,0] (x p+1 ) ϕ(ξ≺q p , xq p ) s≺q p

Phase Transition and Correlation Decay in Coupled Map Lattices

253

we see that 

  τ n × 1 = p Πqp 1[−1,0] (x p )1[−1,0] (x p+1 )μ (ϕ)     (ξ ) dξ = μ 1[−1,0] (x p ) ψ(τ n x p , x = p ) − ψ(τ n ξ p , x = p )h (−) p inv p   x p   = μ 1[−1,0] (x p )∂ p . ψ(τ n ξ p , x = p )dξ p − (x p + 1) ψ(ξ p , x = p )h (−) (ξ ) dξ p p inv −1

And now, by definition of the bounded variation norm, inequality (9), and the fact that |ψ|∞ ≤ |ϕ|∞ ≤ 1, we have   n  τ × 1 = p Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ (ϕ)  1  ψ(τ n ξ p , x = p ) 1[−1,x p ] (ξ p ) dξ p − (x p + 1) ≤ μ sup  x p ∈[−1,0] −1   



 (ξ ) dξ sup c ς n 1[−1,x p ] BV ≤ 2 c ς n μ . × ψ(ξ p , x = p )h (−) p  ≤ μ inv p ∞

x p ∈[−1,0]

We can now go back to (70). Indeed, we just proved that %  &n  ( p)   T0 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ ≤ 2 c ς n μ . If we insert this bound into (70), we have     T 1[−1,0] (x p+1 ) n Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ   D0 1 + n D0 n ≤ 2cς +4  μ . 1 − λ0 1 − λ0

(71)

And consequently (69) can be rewritten as  n  T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ   D0 1 + n D0 n ≤ 2cς +4  μ 1 − λ0 1 − λ0 n−1     k  n−k  1(0,1] (x p+1 ) T 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ + T k=1

.

(72)

We are then left with the cases where a sign flip happens at some time k. Since we know that at time k − 1, x p+1 belongs to [−1, 0], and at time k, x p+1 belongs to (0, 1], x p+1 at time k − 1 has to belong to the small set E  . Therefore, applying (38) with Ω = ∅ and Λ = { p + 1}, we get    k   n−k 1(0,1] (x p+1 ) T 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ T    k−1   ≤ 1 E  (x p+1 ) T 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ

k−1 |E  |



(73) ≤ Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ .

T 1[−1,0] (x p+1 ) 2

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A. de Maere

 k−1 D0 But from (65), we can check that T 1[−1,0] (x p+1 ) ν ≤ (λk−1 + 1−λ ) ν . This 0 0 inequality and Lemma 2 applied to (73) implies:    k   n−k 1(0,1] (x p+1 ) T 1[−1,0] (x p+1 ) Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ T  



|E  | D0

Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ

λk−1 ≤ + 0 2 1 − λ0   D0 D0 k−1 μ . ≤ |E  | λ0 + 1 − λ0 1 − λ0 We insert this inequality into (72), take the geometric series as an upper bound on the sum, and we get:  n  T Πq p 1[−1,0] (x p )1[−1,0] (x p+1 )μ % D0 ≤ 2 c ς n + 4 1−λ 0 % ≤ 2 c ςn +

1+n D0 1−λ0

n−1 % &  |E  | λk−1  μ + + 0

D0 1−λ0

&

D0 1−λ0

μ

k=1

D0 1+n D0 1−λ0 1−λ0

& (4  + |E  |) μ .

And finally, we insert this inequality in (67):





˜ qp

T 1[−1,0] (x p )1[−1,0] (x p+1 )μ

% & % &2 γ −n γ 1+n D0 D0 D0 n μ ≤ 2 1−λ λ + + λ + cς 0 0 1−λ0 1−λ0 (4  + |E  |) μ 0 * ) % & % &2 γ −n γ 1+n D0 D0 D0 n |) μ , |E λ + + λ + cς ≤ 2 1−λ  + (4  0 0 1−λ0 1−λ0 0 and by definition of σ1 , we therefore proved that





˜ qp

T 1[−1,0] (x p )1[−1,0] (x p+1 )μ ≤ σ1 μ .

 

4.2. Polymer expansion. We are now at a turning point of our reasoning. Indeed, the Peierls argument from Sect. 3 tells us that the probability of having positive sites is small with respect to some class of initial measures, and Theorem 4 allows us to control the dynamic restricted to the negative phase. Combining these two arguments, a contour estimate and a decoupling estimate, is usually called a polymer expansion in Statistical Physics, and we will see that it implies the exponential convergence to equilibrium for a wide class of initial measures. Theorem 5. Assume that τ belongs to T (D0 , c, ς ) for D0 , c > 0 and ς ∈ (0, 1) given and that κ is larger than some κ1 that depends on D0 , c and ς . Then, there is some 1 ∈ (0, 1) such that, if  ∈ [0, 1 ], there is some σ < 1 such that for any K > 0 there is some constant C > 0 such that   t T μ(ϕ) ≤ C |Λ| σ t μ |ϕ|∞ for any signed measure of zero mass μ in B(K , 3α0 , θ0 ) with α0 defined in (12) and θ0 defined in (14), and for any continuous function ϕ depending only on the variables in Λ ⊂ Z.

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Proof. Assume for the beginning that t = mγ for some m ∈ N, and let tk = kγ . The expansion of (59) gives us      t  T μ(ϕ) ≤ 2 (74) T˜ pm−2 pm−1 . . . T˜ 0 p0 μ |ϕ|∞ . p∈F (Λ)

We define and, for any p ∈ F(Λ): G(p) = {( p0 , t0 ), ( p0 + 1, t0 ), . . . , ( pm−1 , tm−1 ), ( pm−1 + 1, tm−1 )}. If P(G(p)) is the set of subsets of G(p), then, for any Ω ∈ P(G(p)) and for any t, we can define Ωt = {q | (q, t) ∈ Ω} , the projection of Ω on time t. Then, we introduce the set F(Ω, t), defined by  1 F(Ω,t) = 1(0,1]Ωt 1[−1,0] (xq ). (75) q : (q,t)∈G(p)\Ω

If we sum over all possibilities, (74) can be rewritten as       t  T μ(ϕ) ≤ 2 T˜ pm−2 pm−1 1 F(Ω,tm−1 ) . . . T˜ 0 p0 1 F(Ω,0) μ |ϕ|∞ . p∈F (Λ) Ω∈P (G(p))

(76) We note that the number of terms in the sums grows at most exponentially with t. Indeed, |F(Λ)| ≤ |Λ| γ m and |P(G(p))| ≤ 22m , so t

|F(Λ)| |P(G(p))| ≤ |Λ| (4γ ) γ .

(77)

Consider now P  , the set of the Ω ∈ P(G(p)) such that at least of the m sets Ωtk are empty. If Ω belongs to P  , we know that at least m2 of the m operators T˜ pk pk+1 1 F(Ω,tk ) are bounded in bounded variation norm by Theorem 4:





Ωtk = ∅ ⇒ T˜ pm−2 pm−1 1 F(Ω,tm−1 ) ν ≤ σ1 ν , m 2

and the other operators T˜ pk pk+1 1 F(Ω,tk ) are bounded by (35), Lemma 2 and Corollary 2:



% & % &



D0 D0 ν . Ωtk = ∅ ⇒ T˜ pm−2 pm−1 1 F(Ω,tm−1 ) ν ≤ 2 1−λ λ + 0 1−λ 0 0 Therefore, for any Ω ∈ P  , using the fact that μ ∈ B(K , 3α0 , θ0 ), we have     ˜ pm−2 pm−1 1 F(Ω,tm−1 ) . . . T˜ 0 p0 1 F(Ω,0) μ T % % & % && m m 2 D0 D0 ≤ 2 1−λ λ + σ12 μ 0 1−λ0 0 +% % & % && , m 2 D0 D0 ≤ 2 1−λ λ σ1 + K θ0 . 0 1−λ 0 0

(78)

If Ω does not belong to P  , we know that we have at least m2 characteristics functions of (0, 1], and we will use the Peierls argument to show that this only happens with small probability. We start by defining the sequence of measures μk by  μ0 = 1 F(Ω,0) μ . (79) μk = 1 F(Ω,tk ) T˜ pk−2 pk−1 1 F(Ω,0) μk−1

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Then, using Lemma 2, we can see that         ˜ pm−2 pm−1 1 F(Ω,tm−1 ) . . . T˜ 0 p0 1 F(Ω,0) μ = T˜ pm−2 pm−1 μm−1  ≤ 2 |||μm−1 |||θ0 . T

(80)

    Assume that we already picked up some Λk ⊂ Z and consider 1(0,1]Λk μk  . By θ0

definition of μk , and since F(Ω, tk ) ⊂ (0, 1]Ωtk , we get         1(0,1]Λk μk  ≤ 1(0,1]Λk 1 F(Ω,tk ) T˜ pk−2 pk−1 μk−1  θ0 θ0       p p ≤ 1(0,1]Λk ∪Ωtk T˜ k−2 k−1 μk−1  . θ0

We can now apply inequality (50) to the measure μk−1 :  (k)   (k)       ∂Γ −∂Γ   tk−1   α0 1(0,1]Λk μk  ≤ 1 θ0

Γ (k) ∈G (Λk ∪Ωtk )

(k) ∂Γ (0,1] tk−1

  Π pk−1 pk μk−1  . θ0

If we define Λk−1 = {q | (q, tk−1 ) ∈ Γ (k) }, we then find      (k)         α0 ∂Γ −|Λk−1 | 1(0,1]Λk−1 Π pk−1 pk μk−1  . 1(0,1]Λk μk  ≤ θ0

θ0

Γ (k) ∈G (Λk ∪Ωtk )

Now, remember that the operator Πq p only integrates over [−1, 0]. Therefore 1(0,1]Λk−1 Π pk−1 pk = 1(0,1]Λk−1 Π pk−1 pk 1(0,1]Λk−1 , and using Corollary 2 and Lemma 2, we get      (k)         α0 ∂Γ −|Λk−1 | Π pk−1 pk 1(0,1]Λk−1 μk−1  1(0,1]Λk μk  ≤ θ0

Γ (k) ∈G (Λk ∪Ωtk )



≤2

Γ (k) ∈G (Λ

α0

 (k)  ∂Γ −|Λk−1 |

    1(0,1]Λk−1 μk−1  . θ0

k ∪Ωtk )

θ0

(81)

We can iterate this inequality to find an upper bound on |||μm−1 |||θ0 . Starting from Λm−1 = ∅:    (m−1)   −|Λm−2 |    |||μm−1 |||θ0 ≤ 2 α0 ∂Γ 1(0,1]Λm−2 μm−2  θ0

Γ (m−1) ∈G (Ωtm−1 )



≤ 2m−1

...

Γ (m−1) ∈G (Ωtm−1 )

    × 1(0,1]Λ0 μ0  .



α0

 (m−1)  ∂Γ −|Λm−2 |

. . . α0

Γ (1) ∈G (Ωt1 ∪Λ1 )

(82)

θ0

But we can also see that     1(0,1]Λ0 μ0 

θ0

 (1)  ∂Γ −|Λ0 |

    = 1(0,1]Λ0 1 F(Ω,0) μ

θ0

    ≤ 1(0,1]Ω0 ∪Λ0 μ . θ0

Phase Transition and Correlation Decay in Coupled Map Lattices

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-m−1 (k)  For convenience, let us define Γ (0) = Ω0 ∪ Λ0 × {0} and Γ = k=0 Γ , and let Γ denote the sum over all Γ (k) from the previous inequality. We can check that tk 

  |∂Γt | = ∂Γ (k)  − |Λk−1 | .

t=tk−1 +1

And so m−1 

tm−1    |∂Γt | = |∂Γ | − |∂Γ0 | . (∂Γ (k)  − |Λk−1 |) =

k=1

t=1

Since Ω0 ∪ Λ0 = ∂Γ0 , (82) then becomes      |||μm−1 |||θ0 ≤ 2m−1 α0 |∂Γ |−|∂Γ0 | 1(0,1]∂Γ0 μ .

(83)

θ0

Γ

We can now use the assumption that μ belongs to B(K , 3α0 , θ0 ), and for any Ω ∈ P  , we get       ˜ pm−2 pm−1    1 F(Ω,tm−1 ) . . . T˜ 0 p0 1 F(Ω,0) μ ≤ 2m α0 |∂Γ |−|∂Γ0 | 1(0,1]∂Γ0 μ T Γ

≤ K 2m



θ0

α0 |∂Γ |−|∂Γ0 | (3α0 )|∂Γ0 | .

Γ

Consider now the outer paths associated to the cluster Γ . We can check that the number of diagonal, vertical and horizontal edges have to be equal and that if Γ has c connected parts, n d = n v = n h ≥ m2 − c. Since |∂Γ | ≥ n h + c from (20) still holds, we can see 1 , we have that, as long as α0 < 81 ∞      ˜ pm−2 pm−1 0 p0 m ˜ ≤ K 2 T 1 . . . T 1 μ  F(Ω,tm−1 ) F(Ω,0) 

∞ 

33n h (3α0 )n h +c

c=1 n h = m2 −c m K 2m (81α0 ) 2 26(1 − 81α0 ) √ K (18 α0 )m . ≤ 26(1 − 81α0 )

≤

(84)

We can now move to the conclusion of this proof. We need to choose σ ∈ (0, 1), γ and n and the parameters of the model (namely κ and ) such that ⎧ 1 ⎪ ⎪ α0 < , ⎪ ⎪ 81 ⎪ ⎨ √ σγ , 18 α0 < (85) 4γ ⎪ ⎪ 1 +% % & % && , γ ⎪ ⎪ σ 2 D0 D0 ⎪ ⎩ 2 1−λ . λ0 + 1−λ σ1 < 0 0 4γ We start by choosing σ ∈ (0, 1), γ and n independently of κ such that +% % & % &&  , 21 σγ D0 D0 D0  γ −n 2 1−ς . ς + 1−ς 2 1−ς ς + ς γ + cς n ≤ 4γ

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A. de Maere

We have seen that we can always choose ς such that λ0 ≤ ς , and by definition of σ1 , it implies that +% % & % && , 1 σγ 2 D0 D0 . λ σ1 ≤ + lim 2 1−λ 0 1−λ 0 0 →0 4γ Using (13), we now choose κ1 > 0 such that, if κ ≥ κ1 , the following inequalities are satisfied: ⎧ 1 ⎪ ⎨ lim α0 < , →0 81 (86) √ σγ ⎪ ⎩ lim 18 α0 < . →0 4γ Under these assumptions, there is some 1 ∈ (0, 1) such that, if  < 1 , inequalities (78) and (84) can be rewritten as    ˜ pm−2 pm−1  1 F(Ω,tm−1 ) . . . T˜ 0 p0 1 F(Ω,0) μ T +% % & % && , m σt 2 D0 D0 ≤ K θ0 2 1−λ λ σ + ≤ K θ , 0 1 0 1−λ0 0 (4γ )m    ˜ pm−2 pm−1  1 F(Ω,tm−1 ) . . . T˜ 0 p0 1 F(Ω,0) μ T ≤

√ K σt K (18 α0 )m ≤ . 26(1 − 81α0 ) 26(1 − 81α0 ) (4γ )m

Therefore, for some constant C, we have, for any Ω:    ˜ pm−1 pm  1 F(Ω,tm−1 ) . . . T˜ p0 p1 1 F(Ω,0) Π0 p0 μ ≤ C (4γ )−m σ t . T With this estimate, we control all the terms of in the sum of (76), and since the number of terms in the sum is controlled by (77), we get   t t t T μ(ϕ) ≤ |Λ| (4γ ) γ C K (4γ )− γ σ t ≤ C |Λ| σ t |ϕ|∞ . (87) Finally, if t is not a multiple of γ , we know that t can be rewritten as t = mγ + t  ,  for some m ∈ N and t  < γ . Then, we can apply (87) to μ and ϕ ◦ T t , which depends  only on the variables in a set of size at most t |Λ|, and we find  t  T μ(ϕ) ≤ C t  |Λ| σ mγ |ϕ|∞ ≤ γ C |Λ| σ t |ϕ|∞ , (88) σγ which is exactly the promised result, up to a redefinition of the constant C.   Theorem 2 is now a trivial consequence of Theorem 5. Proof (Theorem 2). We already know from Theorem 1 that μ(−) inv belongs to B(K 0 , 3α0 , θ0 ). And since μ belongs to B(K , 3α0 , θ0 ),     1(0,1]Λ (μ − μ(−) ) ≤ K (3α0 )|Λ| |μ| + K 0 (3α0 )|Λ| μ(−) ) ≤ (K + K 0 ) (3α0 )|Λ| . inv θ inv 0

K

 So, for = K + K 0 , (μ − μ(−) inv ) is a signed measure of zero mass in B(K , 3α0 , θ0 ), and Theorem 5 implies that   μ(ϕ ◦ T ) − μ(−) (ϕ) ≤ C |Λ| σ t |ϕ|∞ .   inv

Phase Transition and Correlation Decay in Coupled Map Lattices

259

5. Exponential Decay of Correlations Theorem 2 implies that the spatial correlations of μ(−) inv decay exponentially: Proof (Proposition 1). Since m (−) belongs to B(1, 3α0 , θ0 ), we know that for any continuous function ϕ depending only on the variables in Σ:   (−) m (ϕ ◦ T t ) − μ(−) (ϕ) ≤ C  |Σ| σ t |ϕ|∞ . (89) inv If we choose t = d(Λ, Ω) − 1, ϕ ◦ T t and ψ ◦ T t depends on different variables, and since m (−) is a totally decoupled, we have m (−) (ϕ ◦ T t ψ ◦ T t ) = m (−) (ϕ ◦ T t ) m (−) (ψ ◦ T t ). Therefore  (−)   μ (ϕψ) − μ(−) (ϕ) μ(−) (ψ)  inv inv inv    t (−)  (−) t (−) t (−)    ≤  μ(−) inv (ϕψ) − T m (ϕψ) + T m (ϕψ) − μinv (ϕ) T m (ψ)  (−)  (−) (−) +  μinv (ϕ) T t m (−) (ψ) − μinv (ϕ) μinv (ψ)     t (−)   t (−)  (−) t (−)     T m (ψ)  ≤  μ(−) inv (ϕψ) − T m (ϕψ) + T m (ϕ) − μinv (ϕ)   (−) (−) t (−)  +  μ(−) inv (ϕ) T m (ψ) − μinv (ϕ) μinv (ψ) . This inequality, together with (89), yields  (−)   μ (ϕψ) − μ(−) (ϕ) μ(−) (ψ)  inv inv inv

≤ C  |Λ ∪ Ω| σ t |ϕ|∞ |ψ|∞ + C  |Λ| σ t |ϕ|∞ |ψ|∞ + C  |Ω| σ t |ϕ|∞ |ψ|∞ ≤ 2 C  |Λ ∪ Ω| σ t |ϕ|∞ |ψ|∞ .

If we take C = 2C  , we have the exponential decay of correlation in space for the invariant measure μ(−)  inv .  The proof of the exponential decay in time also follows from standard arguments, but we have to put some additional regularity assumptions on ψ because ψμ(−) inv has to belong to B(K , 3α0 , θ0 ). Proof (Proposition 2). Without loss of generality, assume that |ϕ|∞ ≤ 1 and |ψ|∞ ≤ 1. Consider then the measure μψ defined by (−) (−) μψ (ϕ) ≡ μ(−) inv (ϕ ψ) − μinv (ϕ) μinv (ψ).

In order to prove the exponential decay in time, we just have to prove that μψ satisfies the assumptions of Theorem 5. It is easy to check that μψ is a signed measure of zero (−) mass. And since μ(−) inv (ϕ) μinv already belongs to B(K 0 , 3α0 , θ0 ), we just have to prove (−) that ψμinv also belongs to B(K , 3α0 , θ0 ) for some K > 0. Since ψ depends only on the variables inside some finite set Λ ⊂ Z and belongs to 1 (X ), we have CΛ ψ∂Λ˜ ϕ˜ =

 V ⊆Λ˜

(−1)

  ˜  Λ\V 

% & ˜ Λ\V ∂V ϕ∂ ψ , ˜

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A. de Maere

for any Λ˜ ⊆ Z and any ϕ˜ ∈ C 1˜ (X ). Therefore Λ   ϕ˜ ψ ∂Λ∩Λ ψ ∂Λ˜ ϕ˜ = ∂Λ\Λ ˜ ˜   % &  ˜   (Λ∩Λ)\V  ϕ˜ ∂(Λ\Λ)\V (−1) ∂V ∪(Λ\Λ) ψ . = ˜ ˜ ˜ V ⊆Λ∩Λ

This implies that   Var Λ˜ 1(0,1]Ω ψμ(−) inv ≤

 ˜ V ⊆Λ∩Λ

     (−) ∂ 1 Var V ∪(Λ\Λ) μ ψ Ω   ˜ ˜ (0,1] inv (Λ∩Λ)\V

∞



≤ K 0 (3α0 )|Ω|

θ0

    ˜ V ∪(Λ\Λ) 

    ψ ∂(Λ∩Λ)\V  . ˜ ∞

˜ V ⊆Λ∩Λ

1 (X ) and since I Λ is a compact set, there is some constant C  such Since ψ belongs to CΛ   ψ  ≤ C  for any set V . Therefore that ∂(Λ∩Λ)\V ˜ ∞

  |Ω|  Var Λ˜ 1(0,1]Ω ψμ(−) inv ≤ K 0 C (3α0 ) ≤ K 0 C  (3α0 )|Ω|



θ0

    ˜ V ∪(Λ\Λ) 

˜ V ⊆Λ∩Λ



θ0

    |V |+Λ˜ −|Λ|

.

V ⊆Λ

    −Λ˜ 

and taking the supremum over all finite Multiplying each side of the inequality by θ0 Λ˜ ⊂ Z yields:    1(0,1]Ω ψμ(−)  ≤ K 0 C  (3α0 )|Ω| θ0 |V |−|Λ| inv θ 0

≤ K0C 



V ⊆Λ

1 + θ0 θ0

|Λ|

(3α0 )|Ω| .

So, ψμ(−) inv belongs to B(K , 3α0 , θ0 ) for some K > 0. Therefore, μψ also belongs to B(K , 3α0 , θ0 ) for another K > 0, and we can apply Theorem 5 to the signed measure of zero mass μψ . This yields:  t    T μψ (ϕ) = μ(−) (ϕ ◦ T t ψ) − μ(−) (ϕ) μ(−) (ψ) ≤ Cϕ,ψ |Λ| σ t .   inv inv inv A. Regularization Estimates We assigned to this Appendix all the regularization estimates needed in this article. Let Λ be some arbitrary finite subset of Z. Let η : I Λ → R be a non-negative, real-valued function in C0∞ (RΛ ), the space of continuous functions on RΛ with compact support. Then, if ϕ : I Λ → I Λ is integrable, one can define the regularization of ϕ for every  > 0,     x−w −|Λ| ϕ(w) dw, (90) η(w) ϕ(x + w) dw =  η (η  ϕ) (x) =  IΛ IΛ

Phase Transition and Correlation Decay in Coupled Map Lattices

261

where we assumed ϕ to be identically zero outside I Λ . This regularization of ϕ has the following properties: Proposition 6. For all ϕ ∈ C 0 (I Λ ), we have 1. |η  ϕ|∞ ≤ |ϕ|∞ . 2. lim |(η  ϕ) − ϕ|∞ = 0. →0

3. Moreover, if ϕ ∈ C 1 (I Λ ), then ∂ p (η  ϕ) = (η  ∂ p ϕ). The proof of these results can be found in textbooks (see for instance [34] or [35]). However, the functions we are interested in are usually not continuous but only piecewise continuous in the following sense: we say that ϕ-is piecewise continuous on I Λ if there is a finite family of open intervals Ii such that i I i = I and ϕ is continuous on Ii1 × · · · × Ii|Λ| for every {i p } p∈Λ . The following proposition shows that the piecewise continuity of the function ϕ will not play an important role if ϕ is integrated with respect to a measure in L 1 (X ): Proposition 7. Let μ ∈ L 1 (X ) and ϕ be a piecewise continuous on I Λ . Then: lim μ( η  ϕ ) = μ( ϕ ).

→0

Proof. Take some arbitrary α > 0. From (90), we know that   |μ( η  ϕ − ϕ )| ≤ μ(dx) dw |ϕ(x + w) − ϕ(x)| η(w). IΛ

(91)

Let B be a strip of size  around the discontinuities of ϕ. If x does not belong to B , x + w and x are in the same Ii1 × · · · × Ii|Λ| , and the piecewise continuity of ϕ implies that, if  is small enough, |ϕ(x + w) − ϕ(x)| ≤ α. If x belongs to B , we have the following trivial bound: |ϕ(x + w) − ϕ(x)| ≤ 2 |ϕ|∞ . So, (91) becomes |μ( η  ϕ − ϕ )| ≤ μ(Bc ) α + 2μ(B ) |ϕ|∞ ≤ μ(1) α + 2μ(B ) |ϕ|∞ . When  goes to 0, both terms of the sum vanishes. This is trivial for the first one, and for the second one, we just have to notice that when  goes to zero, B shrinks to a set of zero Lebesgue measure in I Λ and that μ on I Λ is absolutely continuous with respect to the Lebesgue measure.   However, we are not only interested in the L 1 (X ) norm but also in the variations seminorms, such as Var Λ . The following lemma shows that in some sense, the regularizations and the derivatives do commute: Lemma 3. Let Λ be some finite subset of Z, and ϕ : I Λ → R such that: / Λ , – ϕ is piecewise continuous with respect to x p for p ∈  – ϕ is continuous with respect to x p for p ∈ Λ , – ϕ is piecewise continuously differentiable with respect to x p for p ∈ Λ . Then, for every μ such that Var Λ μ < ∞, we have lim μ (η  ∂Λ ϕ) = lim μ (∂Λ (η  ϕ)) .

→0

→0

262

A. de Maere

Proof. The proof of the result is a simple recurrence on the size of Λ . If Λ = { p}, the continuity of ϕ with respect to x p implies that  xp ∂ p ϕ(ξ p , x = p ) dξ p . ϕ(x) = ϕ(0, x = p ) + 0

Let us then define ϕ :



ϕ (x) = η  ϕ(0, x = p ) +

xp 0

η  ∂ p ϕ(ξ p , x = p ) dξ p .

Then, ∂ p ϕ (x) = η  ∂ p ϕ, and     lim μ η  ∂ p ϕ − ∂ p (η  ϕ) = lim μ ∂ p ϕ − ∂ p (η  ϕ) →0

→0

≤ μ lim |ϕ − η  ϕ|∞ . →0

But now, since both ϕ and η  ϕ tends towards ϕ in the sup norm when  goes to zero, this completes the proof for Λ = { p}. Take now some arbitrary Λ , and assume that for some q ∈ Λ , the property is true in Λ \ {q}. Therefore    lim μ (η  ∂Λ ϕ − ∂Λ (η  ϕ)) = lim μ ∂Λ \{q} η  ∂q ϕ − ∂q (η  ϕ) . →0

→0

Let’s then define



ϕ (x) = η  ϕ(0, x =q ) +

xq 0

η  ∂q ϕ(ξq , x =q ) dξq .

Then, once again, ∂q ϕ = η  ∂q ϕ. So    lim μ ∂Λ \{q} η  ∂q ϕ − ∂q (η  ϕ) ≤ Var Λ μ lim |ϕ − η  ϕ|∞ = 0. →0

→0

 

Proposition 8. For every finite Λ ⊂ Z, and ϕ : X → R continuous and piecewise continuously differentiable with respect to every x p with p ∈ Λ, and piecewise continuous with respect to the other variables x =Λ . Then μ (∂Λ ϕ) ≤ Var Λ (μ) |ϕ|∞ . Proof. Since ∂Λ ϕ is piecewise continuous, if η is some positive symmetric mollifier for ϕ, we have μ (∂Λ ϕ) = lim μ (η  ∂Λ ϕ) .

(92)

→0

But then, using Lemma 3, Eq. (92) becomes   μ ∂Λ ∂ p ϕ = lim μ (∂Λ η  ϕ) →0

≤ lim Var Λ (μ) |η  ϕ|∞ ≤ Var Λ (μ) |ϕ|∞ . →0

And, as a direct corollary, we have

 

Phase Transition and Correlation Decay in Coupled Map Lattices

263

Corollary 2. For any interval [a, b] ⊆ [−1, 1], every p and every Λ, we have: Var Λ (1[a,b] (x p )μ) ≤ Var Λ (μ). Proof. If p ∈ / Λ, the result is a direct consequence of Proposition 8. For any function ϕ 1 (X ), p ∈ in CΛ / Λ implies that 1[a,b] (x p )∂Λ ϕ = ∂Λ 1[a,b] (x p )ϕ, and since 1[a,b] (x p )ϕ is continuously differentiable with respect to the variables in Λ, and piecewise continuous with respect to the other variables, we have μ(1[a,b] (x p )∂Λ ϕ) = μ(∂Λ 1[a,b] (x p )ϕ) ≤ Var Λ (μ) |ϕ|∞ . 1 (X ), we can define If p ∈ Λ, for any function ϕ ∈ CΛ  xp ψ(x) = ϕ(a, x = p ) + 1[a,b] (ξ p ) ∂ p ϕ(ξ p , x = p ) dξ p . a

We immediately see that ψ is continuous, piecewise continuously differentiable, and ∂ p ψ = 1[a,b] (ξ p )∂ p ϕ. Therefore μ(1[a,b] (ξ p )∂Λ ϕ) = μ(∂ p ψ) ≤ Var Λ (μ) |ψ|∞ . But the definition of ψ implies that |ψ|∞ ≤ |ϕ|∞ , and we therefore have μ(1[a,b] (ξ p )∂Λ ϕ) ≤ Var Λ (μ) |ϕ|∞ .

 

Acknowledgements. The author would like to thank Jean Bricmont, Carlangelo Liverani and Christian Maes for useful comments and discussions.

References 1. Kaneko, K.: Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Prog. Theor. Phys. 72(3), 480–486 (1984) 2. Kaneko, K.: Theory and applications of Coupled Map Lattices. New York: Wiley & Sons, 1993 3. Bunimovich, L., Sinai, Y.: Spacetime chaos in coupled map lattices. Nonlinearity 1(4), 491–516 (1988) 4. Pesin, Y.B., Sinai, Y.: Space-time chaos in chains of weakly interacting hyperbolic mappings. Adv. in Sov. Math. 3, 165–198 (1991) 5. Bricmont, J., Kupiainen, A.: Coupled analytic maps. Nonlinearity 8(3), 379–396 (1995) 6. Bricmont, J., Kupiainen, A.: High temperature expansions and dynamical systems. Commun. Math. Phys. 178(3), 703–732 (1996) 7. Bricmont, J., Kupiainen, A.: Infinite-dimensional SRB measures. Physica D 103, 18–33 (1997) 8. Maes, C., Van Moffaert, A.: Stochastic stability of weakly coupled lattice maps. Nonlinearity 10(3), 715–730 (1997) 9. Baladi, V., Degli Esposti, M., Isola, S., Järvenpää, E., Kupiainen, A.: The spectrum of weakly coupled map lattices. J. Math. Pures Et Appl. 77(6), 539–584 (1998) 10. Jiang, M., Pesin, Y.B.: Equilibrium Measures for Coupled Map Lattices: Existence, Uniqueness and Finite-Dimensional Approximations. Commun. Math. Phys. 193(3), 675–711 (1998) 11. Keller, G.: An ergodic theoretic approach to mean field coupled maps, fractal geometry and stochastics. Progr. Probab. 46, 183–208 (1998) 12. Järvenpää, E., Järvenpää, M.: On the definition of SRB-measures for coupled map lattices. Commun. Math. Phys. 220(1), 1–12 (2001) 13. Keller, G., Liverani, C.: Coupled map lattices without cluster expansion. Disc. Cont. Dyn. Syst. 11(2–3), 325–335 (2004) 14. Keller, G., Liverani, C.: A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps. Volume 671 Lecture Notes in Physics, Berlin-Heidelberg-New York: Springer, 2005, pp. 115–151 15. Järvenpää, E.: SRB-measures for coupled map lattices. Volume 671 of Lecture notes in physics, Berlin: Springer, 2005, pp. 95–114

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16. Keller, G., Liverani, C.: Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Commun. Math. Phys. 262(1), 33–50 (2006) 17. Miller, J., Huse, D.: Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice. Phys. Rev. E 48(4), 2528–2535 (1993) 18. Boldrighini, C., Bunimovich, L., Cosimi, G., Frigio, S., Pellegrinotti, A.: Ising-type transitions in coupled map lattices. J. Stat. Phys. 80(5-6), 1185–1205 (1995) 19. Boldrighini, C., Bunimovich, L., Cosimi, G., Frigio, S., Pellegrinotti, A.: Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry. J. Stat. Phys. 102(5), 1271–1283 (2001) 20. Schmüser, F., Just, W.: Non-equilibrium behaviour in unidirectionally coupled map lattices. J. Stat. Phys. 105(3–4), 525–559 (2001) 21. Gielis, G., MacKay, R.: Coupled map lattices with phase transition. Nonlinearity 13(3), 867–888 (2000) 22. Just, W.: Equilibrium phase transitions in coupled map lattices: A pedestrian approach. J. Stat. Phys. 105(1–2), 133–142 (2001) 23. Blank, M., Bunimovich, L.A.: Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity 16(1), 387–401 (2003) 24. Bardet, J.-B., Keller, G.: Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling. Nonlinearity 19, 2193 (2006) 25. Vitali, G.: Sui gruppi di punti e sulle funzioni di variabili reali. Atti Accad. Sci. Torino 43, 229–246 (1908) 26. Stavskaya, O.: Invariant gibbs measures for markov chains on finite lattices with local interaction. Sbornik: Mathematics 21(3), 395–411 (1973) 27. Dobrushin, R., Kryukov, V., Toom, A., editors. Stochastic cellular systems: ergodicity, memory, morphogenesis. Manchester: Manchester University Press, 1990 28. Lasota, A., Yorke, J.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186, 481–488 (1973) 29. Ionescu Tulcea, C., Marinescu, G.: Théorie ergodique pour des classes d’opérations non completement continues. Ann. Math. 52(1), 140–147 (1950) 30. Hennion, H.: Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Am. Math. Soc. 118(2), 627–634 (1993) 31. Liverani, C.: Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78(3–4), 1111–1129 (1995) 32. Keller, G.: Interval maps with strictly contracting Perron-Frobenius operators. Int. J. Bifurcation Chaos 9(9), 1777–1783 (1999) 33. Baladi, V.: Positive Transfer Operators and Decay of Correlations. Volume 16 of Advanced Series in Nonlinear Dynamics. Singapore: World Scientific Publishing, 2000 34. Giusti, E.: Minimal surfaces and functions of bounded variations. Boston: Birkhäuser, 1984 35. Ziemer, W.: Weakly Differentiable Functions. New York: Springer-Verlag, 1989 Communicated by A. Kupiainen

Commun. Math. Phys. 297, 265–281 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-0997-8

Communications in

Mathematical Physics

On the Zero-Temperature Limit of Gibbs States Jean-René Chazottes1 , Michael Hochman2 1 Centre de Physique Théorique, CNRS-École Polytechnique, 91128 Palaiseau Cedex,

France. E-mail: [email protected]

2 Department of Mathematics, Fine Hall, Princeton University, Washington Rd.,

Princeton, NJ 08540, USA. E-mail: [email protected] Received: 1 July 2009 / Accepted: 15 October 2009 Published online: 12 March 2010 – © Springer-Verlag 2010

Abstract: We exhibit Lipschitz (and hence Hölder) potentials on the full shift {0, 1}N such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are “exponentially decaying” interactions on the configuration space {0, 1}Z for which the zero-temperature limit of the associated Gibbs measures does not d exist. In higher dimension, namely on the configuration space {0, 1}Z , d ≥ 3, we show that this non-convergence behavior can occur for the equilibrium states of finite-range interactions, that is, for locally constant potentials. 1. Introduction 1.1. Background. The central problem in equilibrium statistical mechanics or thermodynamic formalism is the description of families of Gibbs states for a given interaction. Their members are parametrized by inverse temperature, magnetic field, chemical potential, etc. The ultimate goal is then to describe the set of Gibbs states as a function of these parameters. The zero temperature limit is especially interesting since it is connected to “ground states”, that is, probability measures supported on configurations with minimal energy [14, Appendix B.2]. The purpose of this article is to shed some light on the zero-temperature limit in the case of classical lattice systems, that is, systems with a configuration space of the d form F Z , where F is a finite set. We consider shift invariant, summable interactions  = ( B ) B⊆Zd ,|B|<∞ . For every β > 0, we denote by G(β) the (nonempty) set of Gibbs states of  at inverse temperature β. It contains at least one shift-invariant Gibbs state [6]. The question we are interested in is: What is the limiting behavior of G(β) as β → +∞ ? When G(β) is a singletom, we denote by μβ its unique (and necessarily shiftinvariant) element. Then the previous question becomes: Does the limit of μβ exist, as β → +∞?

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(Limits of measures should be understood in the weak-* sense.) For the class of interactions we consider, shift-invariant Gibbs states are also equilibrium states. To define them, we need to introduce the function  1 ϕ(x) :=  B (x). |B| B0

This function can be physically interpreted as the contribution of the lattice site 0 to the energy in the configuration x.1 Equilibrium states at inverse temperature β > 0 are then shift-invariant measures which maximize the quantity  Pβ (ν) := βϕdν + h(ν) (1.1) over all shift-invariant probability measures ν on F Z . Here h(ν) is the KolmogorovSinai entropy of ν, and the supremum is called the (topological) pressure. When the lattice is one dimensional and the potential ϕ is Hölder, there is a unique Gibbs measure which is also the unique equilibrium measure. For d > 1 the notions generally are not equivalent and there may be multiple Gibbs states, even for finite-range interactions, but any shift-invariant Gibbs measure is an equilibrium state. As in [14], we define zero-temperature  equilibrium states as those shift-invariant probability measures which maximize2 ϕdν among all shift-invariant measures ν. It can be proven that the weak-∗ accumulation points of equilibrium states for a given interaction as β → +∞ are necessarily zero-temperature equilibrium states for that interaction. Zero-temperature equilibrium states are related to ground states (see [14] for details). d

1.2. The one-dimensional case. Let us make a few remarks about the ergodic perspective. Fix the usual metric d(x, y) = 2− min{k : xi =x j

for |i|
on F Z . For a number of reasons, the usual class of “potentials” ϕ : F Z → R which are studied are Hölder continuous ones. First, for these potentials the Gibbs measure μβϕ is unique for each β > 0 (no phase transition). Second, this class of potentials arises naturally in the theory of differentiable dynamical systems (e.g. Axiom A diffeomorphisms): By choosing a suitable Markov partition of the phase space one can code such a diffeomorphism to a subshift of finite type in F Z [1], and under this coding smooth potentials lift to Hölder ones. And third, Hölder potentials correspond to the natural objects in statistical mechanics, namely “exponentially decaying” interactions ( B ) [13, Chap. 5]. We also note that the case when ϕ is locally constant corresponds to interactions of finite range; see below. There is a trick, due to Sinai, which allows one to reduce the study to a “one-sided” subshift of finite type of F N and a potential ϕ which depends only on “future” coordinates [1]. Thus it suffices to study the one-sided full shift, and our question can be formulated as follows: For Hölder continuous ϕ on F N , when does lim μβϕ exist? β→+∞

1 Since the interaction is shift-invariant, we can take any lattice site. Other definitions are possible [13, Sect. 3.2], but all lead to the same expected value under a given shift-invariant measure. 2 Or minimizes, depending on the sign convention for ϕ.

On the Zero-Temperature Limit of Gibbs States

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The existence of the zero-temperature limit has been verified in a number of situations, but, surprisingly, a systematic study of this question began only recently. When d = 1 and ϕ is locally constant (i.e. the interaction is finite range), the zero-temperature limit was proved to exist in [3] and was described explicitly in [4,10]. In this case, the zero-temperature limit is supported on the union of finitely many transitive subshifts of finite type and is a convex combination of the entropy-maximizing measure on them. The case d = 1 with F a countable set was studied in [9,11]. Another class of examples where convergence may be verified arises as follows. Let X ⊆ F N be a subshift (a closed non-empty shift-invariant set) and define ϕ = ϕ X by ϕ(y) = −d(y, X ) = − inf{d(y, x) : x ∈ X }. This is a Lipschitz function on F N with ϕ| X = 0 and ϕ ≤ 0. The ground states of ϕ X are then precisely the measures supported on X , and it follows that all accumulation points of {μβϕ , β > 0} are invariant measures supported on X . In particular, when X has only one invariant measure μ (i.e. is uniquely ergodic), all accumulation points coincide, and we have μβϕ → μ as β → +∞. The only example of non-convergence of which we are aware is by van Enter and Ruszel [15]. The example is of a nearest-neighbor potential model, but is defined over a continuous state space F (the circle). This state of affairs has led to the belief that over finite state spaces convergence should generally hold. Our first result is a counterexample, showing that this is not the case: Theorem 1.1. There exist subshifts X ⊆ {0, 1}N so that, for the Lipschitz potential ϕ X (y) = −d(y, X ), the sequence μβϕ does not converge (weak-*) as β → +∞. This theorem holds more generally for one-sided or two-sided mixing shifts of finite type. Our construction gives reasonable control over the dynamics of X and of the dynamics, number and geometry of the limit measures. An interesting consequence of the construction is that the set of limit measures need not be convex. We discuss these issues in Sect. 4.

1.3. The multi-dimensional case. Our second result concerns higher dimension: nonconvergence can also arise when d ≥ 3, even for finite-range interactions. As we noted, in dimension d = 1 the zero temperature limit is known to exist in this case [3,4,10]. While the methods used in the one-dimensional case are fairly classical and quite well-known in the dynamics community, the study of zero-temperature limits and ground states in higher dimensions turns out to be closely connected to symbolic dynamics, and our results rely heavily on recent progress in understanding of multidimensional subshifts of finite type, where computation theory plays a prominent role. Recall that a shift 3 of finite type X ⊆ {0, 1}Z is a subshift defined by a finite set L of patterns and the condition that x ∈ X if and only if no pattern from L appears in x. Given L ⊆ {0, 1} E one can define the finite-range interaction ( B ) B⊆Zd ,|B|<∞ by   E (x) =

0 x| E ∈ L , −1/|E| otherwise

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and  B = 0 for B = E; the associated potential on {0, 1}Z is   1 0 x| E ∈ L ϕ L (x) :=  B (x) = −1 otherwise. |B| d

B0

 d Clearly an invariant measure μ on {0, 1}Z satisfies ϕ L dμ = 0 if and only if μ is supported on X ; thus the shift-invariant ground states are precisely the shift invariant measures on X . In this sense ϕ L is similar to ϕ X (although there are some delicate differences, as we shall see in Sect. 5). The main result of [7] provides a general method for transferring one-dimensional constructions to higher-dimensional SFTs with corresponding directional dynamics. Using this we are able to adapt the construction from Theorem 1.1 to the multidimensional case with a finite range potential. Recall that an equilibrium state for a potential ϕ at inverse temperature β is a shift-invariant measure which maximizes the functional Pβ of Eq. (1.1) Theorem 1.2. For d ≥ 3 there exist locally constant (i.e. finite-range) potentials ϕ on d {0, 1}Z such that for any sequence (μβϕ )β>0 in which μβϕ is an equilibrium state (i.e. shift-invariant Gibbs measure), the limit limβ→+∞ μβϕ does not exist. This implies, in particular, that any family (μβϕ )β>0 of shift-invariant Gibbs measures fails to converge as β → +∞. The point is that this holds for every such family. In the presence of a phase transition one can often find non-convergent families of measures as above (e.g. in the Ising model below the critical temperature one can choose Gibbs measures which alternate between the + and - phase), but in the known cases there certainly do exist convergent families. A more precise way to state the previous theorem is to say that the set-valued sequence (G (β))β>0 does not converge in Hausdorff distance. Let us say a few words about the limitations of this result. First, it seems likely that our examples support non-shift-invariant Gibbs measures, i.e. Gibbs measures which are not equilibrium states, and, furthermore, we do not know it the statement extends to them. Hence the requirement of shift-invariance. As for the restriction d ≥ 3, the method used in our construction, which produces a potential of the form ϕ L above, relies on the results from [7] which at present are not available in d = 2; but probably they hold in that case as well. Problem. For d ≥ 2, do there exist finite-range potentials on the d-lattice such that every family of Gibbs states (μβϕ )β>0 fails to converge as β → +∞? In the next section we construct the subshift X of Theorem 1.1. Section 3 contains the analysis and proof of Theorem 1.1. Section 4 contains some remarks and problems. Section 5 discusses the multidimensional case. 2. Construction of X For each k ≥ 0 we define by induction integers k , and finite sets of blocks Ak , Bk ⊆ {0, 1}k . The construction uses an auxiliary sequence of integers N1 , N2 , . . ., with N1 . . . Nk determining Ai , Bi , i for i ≤ k. Here we treat the Nk as given, but in fact at each stage we are free to choose Nk+1 based on the construction so far, and during the analysis in the next section we impose conditions on the relation between k and Nk+1 .

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269

Begin with 0 = 5, and let A0 = {00000, 01000}, B0 = {11111, 10111}. Next, given Ak−1 , Bk−1 and k−1 and the parameter Nk , let ck be a block containing k−1 every block in (Ak−1 ∪ Bk−1 )2 +1 , e.g. enumerate all these blocks and concatenate them. We proceed in one of two ways, depending on whether k is odd or even. We denote by ab the concatenation of blocks a, b of symbols, and by a k the k-fold concatenation of a block a. • If k is odd, let Ak = {ck a Nk : a ∈ Ak−1 }, Bk = {ck b1 b2 . . . b Nk : bi ∈ Bk−1 }. • If k is even, set Ak = {ck a1 a2 . . . a Nk : ai ∈ Ak−1 }, Bk = {ck b Nk : b ∈ Bk−1 }. Thus Ak , Bk consist of blocks of the same length, which we denote k . Note that k can be made arbitrarily large by increasing Nk . If we assume that Nk is large enough then one can identify the occurrences of ck in any long enough subword of length 2k of a concatenation of blocks from Ak ∪ Bk . This is shown by induction: first one shows that one can identify the Ak−1 ∪ Bk−1 -blocks, and then ck is identifiable because it contains blocks from both Ak−1 and Bk−1 . For a set  let  ∗ denote the set of all concatenations of elements from a set . Given a finite set L ⊆ {0, 1}∗ let L =



T n (L N )

n

denote the subshift of all shifts of concatenations of blocks from L. Note that   consisting if L  ⊆ L ∗ , then L  ⊆ L . Let L k = Ak ∪ Bk , so that L k+1 ⊆ L ∗k , and define X=

∞

L k .

k=1

Alternatively, X is the set of points x ∈ {0, 1}N such that every finite block in x appears as a sub-block in a block from some L k .

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3. Analysis of the Zero-Temperature Limit We make some preliminary observations. For u ∈ L k let f i (u) = frequency of i in u. Then the following is clear from the construction: Lemma 3.1. If Nk /k−1 increases rapidly enough, then f 0 (u) > f 0 (u) < 13 for u ∈ Bk .

2 3

for u ∈ Ak and

In fact it can be shown that X supports two ergodic measures, respectively giving mass > 23 and < 13 to the cylinder [0]. The construction is designed so that the ratio |Ak |/|Bk | fluctuates between very large and very small. More precisely, one may verify the following: Lemma 3.2. If Nk /k−1 is sufficiently large, then • If k is odd then |Bk | > |Ak |100 . • If k is even then |Ak | > |Bk |100 . The next two lemmas show that for certain values of β the measure μβϕ concentrates mostly on blocks from L k . Let Yk = {x ∈ {0, 1}N : x|[i,i+k −1] ∈ L k for some i ∈ [0, k − 1]}. Yk ⊆ {0, 1}N is an open and closed set. Lemma 3.3. For β = 23k , μβϕ (Yk ) > 1 − 2−k . Proof. If x ∈ / Yk , then we certainly have d(x, X ) > 2−2k . Therefore,   βϕdμβϕ = −βd(y, X )dμβϕ (y) < −23k · 2−2k μβϕ ({0, 1}N \ Yk ) = 2k (μβϕ (Yk ) − 1). Since h(μβϕ ) ≤ 1 we have Pβ (μβϕ ) ≤



βϕdμβϕ + 1 ≤ 2k μβϕ (Yk ) − (2k − 1).

Finally, choosing ν to be an invariant measure supported on X we have Pβ (ν) = h(ν) ≥ 0, hence Pβ (μβϕ ) ≥ Pβ (ν) ≥ 0. Combining these we have the desired inequality.   Lemma 3.4. For β = 23k , for all large enough n at least half of the mass of μβϕ is concentrated on sequences u ∈ {0, 1}n which can be decomposed as u = v1  . . . v2  . . . vm ,

(3.1)

where vi ∈ L k , the symbol  represents blocks of 0, 1’s (which may vary from place to place), and at least a (1 − 2−k )-fraction of indices j ∈ [0, n) lie in one of the vi .

On the Zero-Temperature Limit of Gibbs States

271

Proof. Let Yk = {x ∈ {0, 1}N : x|[0,k −1] ∈ L k }. Since Yk = that

k −1 i=0

T −i Yk , from the previous lemma and shift-invariance of μβϕ , we see μβϕ (Yk ) >

1 (1 − 2−k ). k

Since μβϕ is ergodic (being the unique Gibbs measure), by the ergodic theorem, for n large enough at least half the mass of μβϕ is concentrated on points x ∈ X such that 1 1 #{i ∈ [0, n − 1] : T i x ∈ Yk } > (1 − 2−k ). n k Since the beginning of an L k -block is uniquely determined (because the ck blocks can be identified uniquely) we also have that if y ∈ Yk , then T i y ∈ / Yk for all 1 ≤ i < k . Thus if u is the initial n-segment of a point x as above, then there is a representation of u of the desired form.   Next, we obtain a lower bound on Pβ (μβϕ ): Lemma 3.5. If k is odd and β = 2−3k , then Pβ (μβϕ ) >

log |Bk | k − 23k 2−k 2 . k

A similar statement holds for even k and Ak . Proof. Let ν be the entropy-maximizing measure on Bk . Since  Pβ (μβϕ ) ≥ Pβ (ν) = h(ν) − βϕdν and h(ν) =

log |Bk | k ,

it suffices to show that  k βϕ(y)dν(y) > −23k 2−k 2 .

(3.2)

Indeed, if y ∈ Bk then y = ab1 b2 . . ., where bi ∈ Bk and a is the tail segment of a block in Bk . Since, by construction, every concatenation of 2k + 1 blocks from Bk appears in X , it follows that the initial segment of y of length k 2k appears in X , and   therefore d(y, X ) < 2−k 2 k , and (3.2) follows.  The last component of the proof is to show that, for β = 23k , the measures μβϕ concentrate alternately Bk and Ak . This is essentially due to the fact that by the lemmas above, μβϕ is mostly supported on the blocks of L k , and because of the appearance of entropy in the variational formula, it tends to give approximately equal mass to these blocks. Since |Bk |/|L k | → 1 along the odd integers and |Ak |/|L k | → 1 along the even ones, this implies that μβϕ will alternately be supported mostly on Bk and Ak . Here are the details. Denote by [u] the cylinder set defined by a block u ∈ {0, 1}∗ .

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Proposition 3.6. If Nk increases sufficiently rapidly, then for all δ > 0 and all sufficiently large k, if we set βk = 2−3k , then: if k is odd,   μβk ϕ [u] ≥ 1 − δ, (3.3) u∈Bk

and if k even, μβk ϕ

 

[u] ≥ 1 − δ.

u∈Ak

Proof. We assume that Nk increases rapidly enough for the previous lemmas to hold and furthermore that, writing H (t) = −t log t − (1 − t) log(1 − t), H (2−k ) →0 log |Bk |/k and 

23k 2−k 2 k →0 log |Bk |/k as k → ∞ along the odd integers, and similarly, with Ak in place of Bk , as k → ∞ along the even integers. This condition is easily satisfied by choosing Nk large enough at each stage, since for fixed k, as we increase Nk the numerator decays to 0 but the denominator does not. Under these hypotheses we establish the proposition for odd k, the case of even k being similar. Thus, we assume that |Bk | > |Ak |100 . Fix δ > 0 and suppose that (3.3) fails for some k. For all large enough n Lemma 3.4 implies that at least half the mass of μβk ϕ is concentrated on points whose initial n-segment is of the form (3.1), and, by the ergodic theorem and the assumed failure of (3.3), if n is large, then with μβk ϕ -probability approaching 1 the fraction of vi ’s that belong to Ak in the decomposition (3.1) is at least δ. For such an n we now perform a standard estimate to bound the entropy of μβk ϕ . Applying e.g. Stirling’s formula, the number of different ways the ’s can appear in u is  n  −k ≤ 2 H (2 )n . ≤ r − r <2

k ·n

The positions of ’s determines the positions of the vi , and given this, the number of ways to fill in the vi so that at least a δ-fraction of them come from Ak is bounded from above by n/ k r =δn/k

|Ak |r |Bk |n/k −r ≤

n · |Ak |δn/k |Bk |(1−δ)n/k . k

Using the bound |Ak | ≤ |Bk |1/100 and setting δ = δ ·

99 , 100

On the Zero-Temperature Limit of Gibbs States

273

we get ≤

n  · |Bk |(1−δ )n/k . k

Thus, for arbitrarily large n, half the mass of μβk ϕ is concentrated on a set E k ⊆ {0, 1}n of cardinality |E k | ≤ 2n H (2

−k )+log n−log 

k



· 2(1−δ )n log |Bk |/k .

It follows from this and the Shannon-McMillan theorem that h(μβk ϕ ) ≤ (1 − δ  )

log |Bk | + H (2−k ), k

hence, since ϕ ≤ 0, we have Pβk (μβk ϕ ) ≤ h(μβk ϕ ) ≤ (1 − δ  )

log |Bk | + H (2−k ). k

Substituting the lower bound from Lemma 3.5, we have log |Bk | log |Bk | k − 23k 2−k−1 2 < (1 − δ  ) + H (2−k ). k k By our assumptions about the growth of Nk the inequality above is possible only for finitely many k. This completes the proof.   1 We can now prove Theorem 1.1. For δ = 100 choose the sequence Nk so that the conclusion of the last proposition holds. Since the density of 0’s in the blocks a ∈ Ak is > 23 and the density in the blocks b ∈ Bk is < 13 , it follows that for k large enough and βk = 2−3k ,

1 −δ 3 2 μβk ([0]) > + δ 3 μβk ([0]) <

if k is odd, if k is even.

Hence (μβϕ )β≥0 does not weak-* converge. 4. Remarks 4.1. Topological dynamics of X . In our example X is minimal. Indeed, any block a ∈ L k appears in ck+1 and hence in every block in L k+1 , so a appears in X with bounded gaps. Note that there are also minimal (non-uniquely ergodic) systems X for which the zerotemperature limit exists. One can easily modify the construction to endow X with other dynamical properties, e.g. one can make X topologically mixing (our example is not, in fact it has a periodic factor of order 5). It is also simple to obtain positive entropy of X (and the limiting measures): form the product of the given example with a full shift.

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4.2. Measurable dynamics of the zero-temperature limits. In our example, (μβϕ )β≥0 has two ergodic accumulation points, and one can show that the convex combinations of these two are also accumulation points. In general, the set of accumulation points need not contain ergodic measures, even when the zero-temperature limit exists. This is true even of locally constant potentials [4,10], and one can also construct examples which are simpler to analyze. For example, if X ⊆ {0, 1}N is a subshift invariant under involution 0 ↔ 1 of {0, 1}N , and if X has precisely two invariant measures μ , μ which are exchanged by this involution, then for the potential ϕ X (y) = −d(y, X ) we will have limβ→+∞ μβϕ = 21 μ + 21 μ . The set of accumulation points also need not be convex. Using the same scheme as above one can construct a subshift X ⊆ {1, 2, 3}Z with three invariant measures μ(i) , i = 1, 2, 3, by maintaining three sets of blocks Ak , Bk , Ck at each stage (rather than two). At each step of the construction we choose the smallest of the sets and concatenate its blocks freely, but concatenate the blocks of the others in a constrained way, so that at the next stage the sizes of the selected set is much larger than the other two, which have not changed much in relative size. For each n there are always two sets (the two which are not growing very much at that stage) for which the number of n-blocks in one is much greater than in the other. Thus the Gibbs measures at the appropriate scale will have very small contributions from the smaller of these sets, and the accumulation points of μβϕ will lie near the boundary of the simplex spanned by the μ(i) (in our example there were only two sets and at each step one grew at the expense of the other; thus the relative number of n- blocks achieved all intermediate ratios). Regarding the ergodic nature of the accumulation points, the same periodicity of order five that obstructs topological mixing causes the ergodic invariant measures on X (i.e. the ergodic zero-temperature limits) to have e−2πi/5 in their spectrum, but this can be avoided by introducing spacers into the construction. In this way one can make the limiting ergodic measures weak or strong mixing, and possibly K . Finally, we have the following variant of Radin’s argument from [12]. Let μ be an ergodic probability measure for some measurable transformation of a Borel space, and h(μ) < ∞. By the Jewett-Krieger theorem [5] there is a subshift X on at most h(μ) + 1 symbols whose unique shift-invariant measure ν is isomorphic to μ in the ergodic theory sense. For the potential ϕ X , all accumulation points of μβϕ are invariant measures on X , so they all equal ν; thus μβϕ → ν as β → +∞. This shows that the zero-temperature limit of Gibbs measures can have arbitrary isomorphism type, subject to the finite entropy constraint, and raises the analogous question for divergent potentials: Problem. Given arbitrary ergodic measures μ , μ of the same finite entropy, can one construct a Hölder potential ϕ whose Gibbs measures μβϕ have two ergodic accumulation points as β → +∞, isomorphic respectively to μ , μ ? 4.3. Maximization of marginal entropy. Letϕ be a Hölder potential and M the set of invariant probability measures μ for which ϕdμ is maximal. It is known that if μ is an accumulation point of (μβϕ )β>0 , then μ ∈ M, and furthermore μ maximizes h(μ) subject to this condition. In the example constructed above the potential ϕ had two ϕ-maximizing ergodic measures μ , μ , and the key property that we utilized was that their marginals at certain scales had sufficiently different entropies. In fact, the measure maximizing the marginal entropy on {0, 1}n for certain n was alternately very close to μ and to μ .

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It is an interesting question if such a connection between zero-temperature convergence and marginal entropy exists in general. Let ϕ be a Hölder potential, and for each n let M∗n denote the set of marginal distributions produced by restricting μ ∈ M to {0, 1}n . The entropy function H (·) is strictly concave on M∗n , and therefore there is a unique μ∗n ∈ M∗n maximizing the entropy function. Let Mn = {μ ∈ M : μ|{0,1}n = μ∗n }. This is the set of ϕ-maximizing measures which maximize entropy on n-blocks. Note that the diameter of Mn tends to 0 as n → ∞ in any weak-* compatible metric. Hence we can interpret Mn → μ in the obvious way. Problem. Is the existence of a zero-temperature limit for ϕ equivalent to existence of lim Mn ? More generally, do (μβϕ )β≥0 and (Mn )n≥0 have the same accumulation points? 5. The Multidimensional Case In this section we apply the main theorem of [7] to obtain a locally constant potential (i.e. a finite-range interaction) in dimension d ≥ 3 such that any associated family of equilibrium measures does not converge as β → +∞; contrast this with the positive result for locally constant potentials in dimension one [3,4,10]. Note that in dimension d ≥ 2, the possibility of failure of the 0-temperature limit to exist for finite-range potentials is known over continuous state spaces [15]; it is the fact that the alphabet is finite and the potential is locally constant that is of interest here. Our methods do not work in d = 2, because the results of [7] are not known in that case, but probably a more direct construction is possible. 5.1. SFTs and their subdynamics . The metric on {0, 1}Z is defined by3 d

d(x, y) = 2− min{u : x(u) = y(u)} , where · is the sup-norm. We denote by T the shift action on {0, 1}Z and write T1 , . . . , Td for its generators. Let d

E n = {−n, . . . , 0, , . . . , n}d denote the discrete d-dimensional cube of side 2n + 1. A subshift X is a shift of finite type (SFT) if there is an n and finite set of patterns L ⊆ {0, 1} E n such that X = {x ∈ {0, 1}Z : no pattern from L appears in x}. d

(Note: here L determines the forbidden patterns, which is the opposite of its role in L .) A pattern a is said to be locally admissible if it does not contain any patterns from L; it is globally admissible if it appears in X , i.e. it can be extended to a configuration on all of Zd which does not contain patterns from L. These two notions are distinct, and it is 3 The dimension of the ambient space is also denoted d but no confusion should arise.

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formally impossible to decide in general, given L, whether a locally admissible word is globally admissible. If we write  ϕ L (y) =

−1 y| E n ∈ L , 0 otherwise

(5.1)

 k then every invariant measure μ on {0, 1}Z satisfies ϕ L dμ ≤ 0 with equality if and only if μ is supported on X . Thus for any SFT X there is a locally constant potential whose maximizing measures are precisely the invariant measures on X . d Given a subshift X ⊆ {0, 1}Z , we may consider the restricted one-parameter action of T1 on X . We shall say that the topological dynamical system (X, T1 ) is a (one-dimend sional) subaction of (X, T ). To each partition C = {C1 , . . . , Cm } of {0, 1}Z into closed and open sets we associate to each x ∈ X its itinerary x C given by the action of T1 and the partition C, i.e. x → x C ∈ {1, . . . , m}Z is defined by x C (i) = j if and only if T1i x ∈ C j . The subshift X C = {x C : x ∈ X } ⊆ {1, . . . , m}Z is a factor, in the sense of topological dynamics, of the subaction (X, T1 ). For a subshift Y ⊆ {0, 1}Z write L k (Y ) ⊆ {0, 1}k for the set of k-blocks appearing in Y ; note that for any sequence k(i) → ∞ the sets L k(i) , i = 1, 2, . . ., determine Y . The main result of [7] says that the subaction of SFTs can be made to look like an arbitrary subshift, as long as the subshift is constructive in a certain formal sense. The version we need is the following: Theorem. Let A be an algorithm that for each i computes4 an integer n(i) and a set L i ⊆ {0, 1, . . . , r }n(i) such that L i ⊇ L i+1 . Then there is an alphabet , an SFT 3 3 X ⊆  Z of entropy 0 and a closed and open partition C = {C0 , C1 , . . . , Cr } of  Z such that L n(i) (X C ) = L i , and consequently X C = ∩ L i . Furthermore, the partition elements Ci can be made invariant under the shifts T2 and T3 . To apply this one usually begins with a subshift Y which has been constructed in some explicit manner, and a computable sequence n(i) (e.g. n(i) = i), and derives an algorithm which from i computes L n(i) (Y ); one then gets an SFT X and partition C so that X C = Y . This means that for nearly all practical purposes (e.g. the construction of counterexamples) one can realize arbitrary dynamics as the subdynamics of an SFT.5 From the result for dimension d = 3 the same is easily seen to hold for d ≥ 3, but it is not known whether this holds in dimension d = 2. 4 A stronger statement can be made in which the computability is replaced with semi-computability of an appropriate family of blocks, and then one obtains (nearly) a characterization; but we do not need this here. 5 Nevertheless, one should bear in mind that the family of SFTs (and the set of algorithms) is countable.

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5.2. A modified one-dimensional example. For notational convenience, for the rest of the paper we concentrate on the case d = 3, the general case being similar. Realizing a specific subshift (such as the one from Sect. 2) as the subaction of an SFT X does not in itself give good control over the equilibrium measures of ϕ X or ϕ L . Indeed, the size of L n (X C ) is exponential in n, which implies similar growth of the corresponding set L n (X ), but does not guarantee exponential growth in n 3 , which is the appropriate scale for 3-dimensional subshifts. Thus for example we can have h(X C ) > 0 but h(X ) = 0. In order to use subactions to control entropy of the full Z3 action we rely on a trick by which the frequency of symbols in X C can be used to control pattern counts in a certain extension of X . This approach was used in [2,8]. We begin by modifying the main example of this paper so as to control frequencies rather than block counts. We define a sequence of integers k and sets of blocks Ak , Bk ⊆ {0, 1, 2}k by induction, using an auxiliary sequence N1 , N2 . . . of integers. Start with 0 = 2 and A0 = {00, 01}, B0 = {00, 02}. Next, given k define Ak = {a 1+2

k−1

: a ∈ Ak−1 },

+Nk

k−1

2 Nk k−1 : b ∈ Bk−1 },

k−1

1 Nk k−1 : a ∈ Ak−1 },

Bk = {b1+2 and for k even define Ak = {a 1+2 Bk = {b1+2

k−1

+Nk

: b ∈ Bk }.

Let k be the common length of blocks in the sets above, i.e. k = k−1 (2k−1 +1 + Nk ). Note that 1k ∈ Ak and 2k ∈ Bk . As k → ∞ the frequency of 0’s in the blocks of Ak , Bk tends to 0, and the frequency of 1’s and 2 s tends, respectively, to 1, and we can control the relative speed at which k (·) such that given N1 , . . . , Nk−1 and they do so. More precisely, there is a function N k (N1 , . . . , Nk−1 ) we have Nk ≥ N f 0 (a) > 100 f 0 (b) f 0 (b) > 100 f 0 (a)

for k odd, a ∈ Ak , b ∈ Bk , for k even, a ∈ Ak , b ∈ Bk .

(Recall that f 0 (x) is the frequency of the symbol 0 in x.) Define Y =

∞

Ak ∪ Bk .

k=1

Similarly define Y1 =



Ak

and Y2 =



Bk

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(notice that these are decreasing intersections). Note that the only invariant measures on Y are the point masses at the fixed points 1∞ ∈ Y1 and 2∞ ∈ Y2 . We denote  k (N1 . . . Nk−1 ) (5.2) k = k−1 (|A K | + |Bk |) Mk + N k (N1 , . . . , Nk−1 ), the set L  (Y ) is in fact (so k ≥ k ) and note that as long as Nk ≥ N k k can be comindependent of Nk and depends only on N1 , . . . , Nk−1 . We also note that N k (N1 , . . . , Nk−1 ) puted explicitly, and in particular the function (k, N1 , . . . , Nk−1 ) → N is a formally computable function.

5.3. Controlling pattern counts in a 3-dimensional SFT. We now incorporate the subshift Y constructed above into a 3-dimensional SFT and use the control over the frequency of symbols in Y to gain control of the pattern counts of an associated SFT. 3 First, some notation: for a subshift X ⊆  Z write L n (X ) = {x| E n : x ∈ X } ⊆  E n , where E n = {−n, . . . , n}3 . This is the same notation we used for one-dimensional subshifts, but the meaning will be clear from the context. We remark that if the (topological) entropy of X is 0 then |L n (X )| = o(|E n |). Apply Theorem (5.1) to Y (or, rather, to an algorithm that computes a sequence L n(k) (Y ); we shall be more precise later about the algorithm used). We obtain a zero3 entropy SFT X ⊆  Z and C = {C0 , C1 , C2 } a T2 , T3 -invariant partition so that C X = Y. Next, for x ∈ X and u = (u 1 , u 2 , u 3 ) ∈ Z3 , if x C (u 1 ) = 0 (i.e. if T1u 1 x ∈ C0 ) we “color” the site with one of the two colors 0 , 0 . Otherwise we leave it “blank”. Collect 3 all such colorings into a new subshift  X . Formally,  X ⊆ X × {0 , 0 , blank}Z is defined by  X = {(x, y) ∈ X × {0 , 0 , blank} : y(u) = blank if x C (u 1 ) = 0}. For x = (x1 , x2 ) ∈  X we also write x C instead of x1C . One may verify that  X is an SFT.  =  × {0 , 0 , blank} for the alphabet of  We write  X and write  L for the finite set of  is locally patterns whose exclusion defines  X . We may assume that if a pattern over   admissible for L then the pattern induced from its first component is locally admissible for L. Notice that, since C0 , C1 , C2 are invariant under T2 , T3 , the pattern of symbols 0 , 0 in a point x ∈  X is the union of affine planes whose direction is spanned by (0, 1, 0), (0, 0, 1). The sequence of coordinates at which these planes intersect the x-axis corresponds to the location of 0-s in x C , and on each plane the symbols 0 , 0 are distributed as randomly as possible, i.e. given the arrangement of affine planes there is no restriction on the combinations of 0 , 0 that may appear in them. It follows that if a ∈ {0, 1, 2}{−n,...,n} is a block in Y then #{(x, y)| E n : (x, y) ∈  X and x C |{−n,...,n} = a} = 2 f0 (a)|E n |+o(|E n |) (the term o(|E n |) comes from the pattern growth of X , which has entropy 0).

On the Zero-Temperature Limit of Gibbs States

279

Write  X 1 = {x ∈  X : x C ∈ Y1 },  X : x C ∈ Y2 }. X 2 = {x ∈  Then, for k large enough, the frequency gap between blocks in Ak and Bk translates into |L k (  X 1 )| > |L k (  X 2 )|1/10 X 2 )| > |L k (  X 1 )|1/10 |L k ( 

k odd, k even.

Compare this with Lemma 3.2.  5.4. Local versus global admissibility. For ϕ = ϕ  X , i.e. ϕ(y) = −d(y, X ), one can adapt the analysis in Sect. 3 and show that μβϕ does not have a limit as β → +∞. Let us review this argument. Fix β = 2−3k , and set p = 1, 2 according to whether k is odd or even, and write q = 2 − p for the other index. First, as in Lemma 3.5, we prove a lower bound on Pβ (μβϕ ) by constructing a measure νk whose blocks (i.e. square patterns) are overwhelmingly drawn from L k (X p ), making it nearly ϕ  X -maximizing, and with entropy close to |E1 | |L k (  X p )|. This forces the entropy of μβϕ to be similar. Second, k we use the fact that most of the mass of μβϕ concentrates on blocks from L k (  X ) and the fact that L k (  X p )  L k (  X q ) to deduce that in order for μβϕ to have entropy near 1   |E k | |L k ( X p )|, it must be mostly concentrated on X p . This argument is similar to that in Proposition 3.6. We are now interested in proving the same thing for the potential ϕ L (given in (5.1)) instead of ϕ  X . The first part of the analysis above carries over with only minor modifications.  However, the second part runs into difficulties. Notice that ϕ  X dμ ≈ 0 implies that  nearly all the μβϕ -mass is concentrated on patterns in L k ( X ), but ϕ L dμ ≈ 0 tells us L; they do not have to be only that μβϕ -most blocks on E k are locally admissible for  globally admissible, giving us little control of their structure. To pull things through, we will make use of the following observation: it is not necessary for us to know that most of the mass of μβϕ concentrates on L k (  X ). Instead, it suffices that it concentrates on L k (  X ), where  is as in Eq. (5.2). This is because L k (  X p ) is already much larger than L k (  X q ), so we can argue as in the first part of the proof of Proposition 3.6. Thus, to complete the construction we want to ensure that if a block a ∈  E k is locally admissible then a| E  is globally admissible, i.e. belongs to L k (  X ). k A simple compactness argument establishes the following general fact: For any SFT and m ∈ N there is an R so that if b ∈  E R is locally admissible then b| E m is globally admissible. In general, however, R depends in a very complicated way on both the SFT and m, and in fact is not formally computable given these parameters. For our purposes we require finer control than this. Luckily, an inspection of the proof in [7] gives the following: Theorem 5.1. Let A be an algorithm that from i computes n(i) ∈ N and L i ⊆ {0, 1, . . . , r }n(i) such that L i ⊇ L i+1 . Denote by τi the number of timesteps required for the computation on input i. Then the SFT X from (Theorem 5.1) can

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be chosen so that, for Ri = Ri (|A|, τ1 , . . . , τi ) , if a ∈  E Ri is locally admissible then a| E n(i) is globally admissible, and furthermore the function Ri (. . .) is computable. Here τi and A are taken with respect to some fixed universal Turing machine. 5.5. Completing the construction: The fine print. We now specify an algorithm A which, given i, computes sequences n(i) ∈ N and L i ⊆ {0, 1, 2}n(i) so that L i ⊇ L i+1 . The even elements n(2k) are the lengths k associated to a sequence Nk in the construction in Sect. 5.2, i.e. Nk =

n(2k) − n(2k − 2) · 21+n(2k−2) · n(2k − 2)

The odd elements of the sequence are k (N1 , . . . , Nk−1 ). n(2k − 1) = k = k−1 N Note that, having determined n(i), the blocks in Y of length n(i) depend only on N1 , . . . , N[n(i)/2] and not on any future choices of parameters of the construction. Hence L i = L n(i) (Y ) is well defined given n(1), . . . , n(i) and may be computed from this data. Thus at the i th stage of the computation we will write L n(i) (Y ) even though strictly speaking Y is not yet defined. On input i the algorithm is as follows. Case 0 i = 1. Output n(1) = 1, L 1 = {0, 1, 2}. Case 1 i = 2k − 1. Recursively compute N1 , . . . , Nk−1 , and output k (N1 , . . . , Nk−1 ), = k−1 N n(i) = k L i = L n(i) (Y ). Case 2 i = 2k. Recursively compute Nm ,m < k and the time τ1 , . . . , τi−1 spent by the algorithm when run on each of the inputs j = 1, . . . , i − 1. Let Nk

=

(max{n(i − 1), R(|A|, τ1 , . . . , τi−1 )})2

and output n(i) Li

= =

k = Nk k−1 , L n(i) (Y ).

Realizing such an algorithm (which can simulate itself) is a non-trivial but standard exercise in computation theory. We can now sketch the remainder of the proof of Theorem 1.2. Using A as input to 3 Theorem 5.1 we obtain an SFT X ⊆  Z and associated partition C = {C0 , C1 , C2 } of 3 X as explained {0, 1, 2}Z , invariant under T2 , T3 , such that X C = Y . Next, form the SFT  above, defined by a set  L of excluded patterns. For β = 23k let μβϕ be an equilibrium measure associated to the potential ϕ L. −3k , where dμ > −c2 By the definition of equilibrium measures we have ϕ βϕ L

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281

 | is the maximal entropy achieved by an invariant measure on the full shift c = log |  Z3 ; in Sect. 3 this constant was 1. Thus in a μβϕ -typical configuration the density  √ of patterns from  L is < c2−3k . Hence for r = k and large enough k, with μβϕ probability > 1 − 2−2k , a configuration x satisfies that x| Er is globally admissible. By our choice of k = n(2k) we have r ≥ R(|A|, N1 , . . . , Nk ), so x| E n(2k−1) is globally admissible. But since n(2k − 1) ≥ k , we are in the situation described at the end of the previous subsection, and this is enough to conclude that μβϕ is mostly concentrated on  X 2 , depending on k mod 2; so μβϕ diverges along β = 2−3k . X 1 or  Acknowledgements. We are grateful to María Isabel Cortez for pointing out a gap in an early version of this paper. We are also grateful to A. C. D. van Enter for useful comments.

References 1. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Volume 470 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, revised edition, 2008. With a preface by David Ruelle, edited by Jean-René Chazottes 2. Boyle, M., Schraudner, M.: Zd shifts of finite type without equal entropy full shift factors. J. Diff. Eq. Appl. 15(1), 47–52 (2009) 3. Brémont, J.: Gibbs measures at temperature zero. Nonlinearity 16(2), 419–426 (2003) 4. Chazottes, J.-R., Gambaudo, J.-M., Ugalde, E.: Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials. Preprint, 2009, available at http://arxiv.org/ abs/0903.1212v1[math.DS], to appear in Ergod. Th. Dynam. Sys. (2010) 5. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, Vol. 527. Berlin: Springer-Verlag, 1976 6. Georgii, H.-O.: Gibbs measures and phase transitions. In: de Gruyter (ed.), Studies in Mathematics, vol. 9, pp xiv+525. Walter de Gruyter & Co., Berlin (1988) 7. Hochman, M.: On the dynamics and recursive properties of multidimensional symbolic systems. Invent. Math. 176(1), 131–167 (2009) 8. Hochman, M., Meyerovitch, T.: A characterization of the entropies of multidimensional shifts of finite type. Anals of Mathematics, 171(3) (May 2010), available at http://pjm.math.berkeley.edu/annals/ta/ 080814-Hochman/080814-Hochman-v1.pdf 9. Jenkinson, O., Mauldin, R.D., Urba´nski, M.: Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type. J. Stat. Phys. 119(3–4), 765–776 (2005) 10. Leplaideur, R.: A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity 18(6), 2847–2880 (2005) 11. Morris, I.D.: Entropy for zero-temperature limits of Gibbs-equilibrium states for countable-alphabet subshifts of finite type. J. Stat. Phys. 126(2), 315–324 (2007) 12. Radin, C.: Disordered ground states of classical lattice models. Rev. Math. Phys. 3(2), 125–135 (1991) 13. Ruelle, D.: Thermodynamic Formalism. 2nd ed. Cambridge Mathematical Library. Cambridge: Cambridge University Press, 2004 14. van Enter, A.C.D., Fernández, R., Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72(5–6), 879–1167 (1993) 15. van Enter, A.C.D., Ruszel, W.M.: Chaotic temperature dependence at zero temperature. J. Stat. Phys. 127(3), 567–573 (2007) Communicated by G. Gallavotti

Commun. Math. Phys. 297, 283–297 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1013-z

Communications in

Mathematical Physics

Bounds on the Minimal Energy of Translation Invariant N -Polaron Systems Marcel Griesemer1 , Jacob Schach Møller2 1 Fachbereich Mathematik, Universität Stuttgart, 70550 Stuttgart,

Germany. E-mail: [email protected]

2 Department of Mathematical Sciences, Aarhus University,

8000 Århus C, Denmark. E-mail: [email protected] Received: 4 July 2009 / Accepted: 4 December 2009 Published online: 25 February 2010 – © Springer-Verlag 2010

Abstract: For systems of N charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U > 0 and α > 0 measuring the electron-electron √ and the electron-phonon coupling strengths. They are constrained by the condition 2α < U , which follows from the dependence of U and α on electrical properties of the crystal. large N asymptotic behavior of √ We show that the √ at 2α = U and that 2α ≤ U is necessary for therthe minimal energy E N changes √ modynamic stability: for 2α > U the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and E N behaves like −N 7/3 . 1. Introduction We study a system of N electrons in a polar (ionic) crystal, modelled by a Hamiltonian derived by H. Fröhlich [12]. The model takes into account the electron-electron Coulomb repulsion, and a linear interaction of the electrons with the longitudinal optical phonons. The model is called the ’large polaron’ model, since it assumes that a polaron (dressed electron) extends over a region which is large compared to the ion-ion spacing. In particular the underlying discrete (and infinite) crystal is replaced by a continuum. See [7,11,19]. As is well-known, linear electron-phonon couplings induce an effective pair attraction between electrons. This attraction competes with the electron-electron repulsion and may cause a phase-transition as the electron-phonon interaction strength increases. This mechanism is behind the production of Cooper pairs in the BCS model of low temperature superconductivity, and in high-Tc superconductivity the role of many-polaron systems is being investigated [1,8,17]. The Fröhlich Hamiltonian depends on two non-negative dimensionless quantities, U and α. The constant U is the electron-electron repulsion strength, and α is the Fröhlich

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electron-phonon coupling constant. Physically relevant models must satisfy the constraint, cf. [4,28], √ 2α < U. In this paper we prove upper and lower bounds on the minimal energy E N of the N -electron Fröhlich√Hamiltonian for all N and all non-negative values of U , α. In the unphysical regime 2α ≥ U , our results imply that E N ∼ −N 7/3 . In the physical regime we find that E N √ ≥ −C N 2 , thus establishing a sharp transition in the large is due to the mediated attraction N -asymptotics of E N at 2α = U . This transition √ 2α = U in the limit of large N . In fact, between electrons overcoming the repulsion at √ the quantity U − 2α appears in our analysis as an effective Coulomb coupling strength. We also demonstrate that E N ≤ −α N and E N +M ≤ E N + E M in the physical regime. We do not know whether or not E N is an extensive quantity, but if it is not extensive, then this must be due to electron-phonon correlations, cf. Proposition A.3. We pause this discussion to introduce the mathematical model. The Fröhlich Hamiltonian describing N electrons in a polar crystal reads N   1  √ − 2 x + α(x ) + Hph + U VC ,

(1.1)

=1

where the number operator

 Hph =

R3

a ∗ (k)a(k)dk,

accounts for the kinetic energy of the phonons while the field operator   1  ik·x e a(k) + e−ik·x a ∗ (k) dk, (x) = R3 c0 |k| is responsible for the electron-phonon interaction. Here c0 := 23/4 π . Finally the electron-electron interaction is given by the sum of two-body Coulomb potentials VC (x1 , . . . , x N ) =

 1≤i< j≤N

1 . |xi − x j |

We work in units where the frequency of the longitudinal optical phonons, ωLO , Planck’s constant h¯ , and the electron band mass are equal to one. Let F denote the symmetric Fock space over L 2 (R3 ). The Hamiltonian (1.1) defines a symmetric quadratic form on H = ∧ N L 2 (R3 ) ⊗ F, but, a priori, it is not well defined as a self-adjoint operator. For that one must first impose an ultraviolet cutoff on the electron-phonon interaction: Let  > 0, and define the cutoff Hamiltonian as H N , =

N   1  √ − 2 x + α (x ) + Hph + U VC , =1

where

  (x) =

|k|≤

 1  ik·x e a(k) + e−ik·x a ∗ (k) dk. c0 |k|

N -Polaron Systems

285

N The operators H N , are self-adjoint on D(Hph ) ∩ D( =1 x ), by the Kato-Rellich theorem, and it is well known, cf. [2,6,13,14,27], that H N , converges, as  → ∞, in the norm-resolvent sense to a semi-bounded, self-adjoint operator, which we denote by H N . This implies that E N = lim E N ,

(1.2)

→∞

if E N , := inf σ (H N , ) and E N := inf σ (H N ). The main goal of this paper is to investigate the large N behavior of the minimal energy E N as a function of α and U . Our first result is an upper bound in the regime √ 2α > U . √ Theorem 1.1. There is a constant C such that for all N and for 2α ≥ U ≥ 0,

√ 7 1 E N ≤ ( 2α − U )2 N 3 E PTF + C N − 17 . Here E PTF < 0 is given by (1.3) below. Theorem 1.1 is proved variationally by using Pekar’s ansatz in terms of a product state, which is known to give the correct ground state energy for N = 1, 2 in the large α limit [9,25,26]. Taking the expectation value in a state f ⊗ η ∈ ∧ N L 2 (R3 ) ⊗ F and explicitly minimizing with respect to η we arrive at a Hartree-Fock type energy which is then estimated by a Thomas-Fermi energy. This allows us to scale out all parameters and we are left with the bound in Theorem 1.1, where E PTF = EPTF (ρ) :=

EPTF (ρ),  inf ρ≥0, ρ(x)d x=1 2 23 3 10 (6π )



5

R3

ρ(x) 3 d x −

(1.3)  1 2

R6

ρ(x)ρ(y) d xd y. |x − y|

(1.4)

We note that in the error term in Theorem 1.1 the exponent 1/17 can be replaced by any number less than 2/33 at the expense of a larger and divergent constant C. To show that the variational upper bound from Theorem 1.1 has the right asymptotics in N and α, we provide the following lower bound: √ Theorem 1.2. There exists C > 0 such that for all N and 2α ≥ U ≥ 0, √ 7 7 1 1 (1.5) E N ≥ −CG ( 2α − U )2 N 3 − Cα 2 N 3 − 9 − 3N 9 , where the constant C G is defined by (1.6). This lower bound is obtained, essentially, by completing the  with respect √ square to creation and annihilation operators in the expression Hph + α nj=1 (x j ). The √ computation brings out an effective Coulomb interaction with coupling strength − 2α. Unfortunately, it also yields an infinite self-energy, which must be dealt with before completing the square. For that we use a commutator argument from [25], which is responsible for the error term in Theorem 1.2. The resulting effective Hamiltonian with an attractive Coulomb potential is bounded below by the ‘gravitational collapse’ bound N  j=1

− 21  j −

 1≤ j<≤N

7 1 ≥ −CG N 3 |x j − x |

(1.6)

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due to Lévy-Leblond [20, Theorem 2]. Hence the presence of the constant CG in Theorem 1.2. √ We now turn to the physical regime 2α < U . Here our lower bound holds for fermionic and bosonic particles alike and hence it will not be optimal in the fermionic case. Together with Theorem √ 1.1 it demonstrates, however, that the model undergoes a sharp transition at α = U/ 2. √ Theorem 1.3. For 0 < 2α < U , EN ≥ −



16 2 2 3π α N

+3

U √ . U − 2α

The proof of Theorem 1.3 is based on our estimate of the ultraviolet part of the electron-phonon interaction in the proof of Theorem 1.2. We do not know yet how to incorporate the Pauli-Principle into that estimate. In view of Proposition A.3 we expect a lower bound linear in N in the fermionic case. Last but not least there are the following universal variational upper bounds for E N and E N +M . Theorem 1.4. For all N ,M, α and U we have E N ≤ −α N , E N +M ≤ E N + E M . The bound E 1 ≤ −α is well known from [10,19] and it agrees with the result of a formal computation of E 1 by second order perturbation theory [11]. Also, it is consistent with Haga’s computation of E 1 including α 2 -terms1 [11,16]. The bound E N ≤ −α N follows from the estimates E 1 ≤ −α and E N ≤ N E 1 , the latter of which is a consequence of the second result of Theorem 1.4. We remark that E N +M ≤ E N + E M holds quite generally for translation invariant N -particle systems with interactions that go to zero with increasing particle separation. In particular it holds for fermions and for distinguishable particles alike. Numerically computed upper bounds on E(N )/N , for N = 2 through N = 32 can be found in the literature [5], but in the case of fermions they are not refined enough to be consistent with the bound E N +M ≤ E N + E M . In this paper we have omitted spin, but the Fermi statistics is taken into account. There are only few small modifications necessary for treating fermions with q spin states, such as the factor of q −2/3 in front of the Thomas-Fermi kinetic energy, which alters the upper bound in Theorem 1.1 by a factor of q 2/3 . The many-polaron model has also been studied with a confining potential of the form N =1 W (x  ), W (0) = 0 and W ≥ 0 included in the Hamiltonian √ [18]. We could include such a potential in our work as well, but, at least in the regime 2α > U this would not affect the leading large N behaviour of E N . 2. Upper Bounds on E N In this section we prove Theorem 1.1 and Theorem 1.4. Since E N = lim→∞ E N , we only need to deal with the self-adjoint operator H N , . Let f ∈ D N = ∧ N L 2 (R3 ) ∩ 1 There is a sign error in Feynman’s quote of Haga’s result.

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H 1 (R3N ) be normalized and recall that the one-particle density matrix γ and the density function ρ associated with f are defined by  f (x, x2 , . . . , x N ) f (x , x2 , . . . , x N )d x2 · · · d x N , (2.1) γ (x, x ) := N R3(N −1)  ρ(x) := γ (x, x) = N | f (x, x2 , . . . , x N )|2 d x2 · · · d x N . (2.2) R3(N −1)

In this paper the Fourier transform ρˆ of the density function ρ, or of any other function, is defined by:  ρ(k) ˆ = e−ik·x ρ(x)d x, R3

that is, without a factor of (2π )−3/2 . √ Proposition 2.1. Suppose 2α ≥ U . Then for every one-particle density matrix γ on L 2 (R3 ) with 0 ≤ γ ≤ 1, Tr[γ ] = N , Tr[−γ ] < ∞, and for ρ(x) := γ (x, x), EN

√ ≤ ( 2α − U )2



 1 2

Tr[−γ ] −

 √ −U ( 2α − U ) 21

1 2

ρ(x)ρ(y) d xd y R6 |x − y|

|γ (x, y)|2 d xd y. R6 |x − y|

Proof. This proof is based on the estimate E N , ≤  f ⊗ η, H N , f ⊗ η for suitable normalized f ∈ D N and η ∈ F. We begin by observing that the expectation value of the interaction operator in a state f ⊗ η may be represented in the following two ways: if f and η are normalized, then  f ⊗ η,  =

N 

  (x ) f ⊗ η

=1

R3N

| f (x1 , . . . , x N )|2

N 

V,η (x ) d x1 . . . d x N

(2.3)

=1

= η,  (ρ)η,

(2.4)

where V,η (x) := η,  (x)η, ρ is the density associated with f , and   (ρ) :=

R

= η

Hence if we define H N , :=

3

ρ(x) (x) d x

|k|≤

N

 1  ∗ ρ(k)a(k) ˆ + ρ(k)a ˆ (k) dk. c0 |k|

1 =1 [− 2 

+

√ αV,η (x )] + U VC , then η

 f ⊗ η, H N , f ⊗ η =  f, H N , f  + η, Hph η.

(2.5)

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M. Griesemer, J. S. Møller η

The ground state energy of the N -body Hamiltonian H N , is bounded above by its ground state energy in the Hartree-Fock approximation. By Lieb’s variational principle, [23] and [3, Coro. 1], this Hartree-Fock ground state energy is bounded above by    U  1 √ ρ(x)ρ(y) − |γ (x, y)|2 E HN , γ + (γ , η) := Tr −  + αV d xd y ,η F 2 2 R6 |x − y| (2.6) for any one-particle density matrix γ with Tr[γ ] = N and ρ(x) = γ (x, x). Hence, in view of (2.5), we conclude that E N , ≤ E HN , F (γ , η) + η, Hph η

(2.7)

for all normalized η ∈ F. In order to minimize the right hand side with respect to η, we use that (2.3) equals (2.4). It follows, by Lemma A.2, that √  α inf α Tr(V,η γ ) + η, Hph η = − 2 η∈F ,η=1 c0

 |k|≤

2 |ρ(k)| ˆ dk. |k|2

(2.8)

By combining (2.6), (2.7), and (2.8) and then letting  → ∞ we arrive at EN ≤

1 2

Tr[−γ ] + (U −

 √ 2α) 21

 U ρ(x)ρ(y) |γ (x, y)|2 d xd y − d xd y 2 R6 |x − y| R6 |x − y| (2.9)

for any one-particle density matrix √ γ with Tr(γ ) = N and ρ(x) = γ (x, x). Here (A.3) and (1.2) were used also. In the case 2α = U it is clear from (2.9) √ or from (2.5) with η being the vacuum vector, that E N ≤ 0. In the case where β := 2α − U > 0, we choose the density matrix γ on the form γ = Uβ  γ Uβ∗ with Uβ defined by (Uβ ϕ)(x) := β 3/2 ϕ(βx). γ (βx, βy) The proposition then follows from Uβ∗ Uβ = β 2  and from γ (x, y) = β 3  by a simple change of variables in the integrals of (2.9).   The second ingredient for proving Theorem 1.1 is the following lemma. 2 3 1 3 Lemma  2.2. Let g ∈ H (R ) with g = 1. Then for every ρ ∈ L (R ) with ρ2 ≥ 0 and R3 ρ(x)d x = N there exists a density matrix γ such that γ (x, x) = (ρ ∗ |g| )(x) and  2 5 ρ(x) 3 d x + N ∇g2 . Tr[−γ ] = 35 (6π 2 ) 3

R3

Proof. For the reader’s convenience, we recall the proof from [22, p. 621]. Let M : R6 → R be defined by M( p, q) = 1 if | p| ≤ (6π )2/3 ρ(q)1/3 and M( p, q) = 0 otherwise. Then   −3 M( p, q)dpdq = ρ(q)dq = N , (2π ) 6 R3  R  2 5 (2π )−3 p 2 M( p, q)dpdq = 35 (6π ) 3 ρ(q) 3 dq. (2.10) R6

R3

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289

We define γ by γ = (2π )−3

 R6

M( p, q) pq dpdq,

where  pq is the rank one projection given by  g pq (x)ϕ(x)d x,  pq ϕ = g pq R3

g pq (x) = ei px g(x − q).

 It follows that γ (x, x) = R3 |g(x − q)|2 ρ(q) dq, and from Tr[− pq ] = ∇g pq 2 = p 2 + ∇g2 + 2 p · g, −i∇g, and (2.10) we find the asserted expression for Tr[−γ ].

 

Proposition 2.1 and Lemma 2.2 suggest the definition of a Polaron Thomas-Fermi functional by   2 5 ρ(x)ρ(y) 3 EPTF (ρ) := 10 d xd y, (2.11) (6π 2 ) 3 ρ(x) 3 d x − 21 R3 R6 |x − y| where ρ ∈ L 1 (R3 ) ∩ L 5/3 (R3 ) and ρ ≥ 0. If ρ N (x) := N 2 ρ(N 1/3 x), then ρ N 1 = N ρ1 and 7

EPTF (ρ N ) = N 3 EPTF (ρ).  Hence it suffices to consider densities ρ with ρ(x)d x = 1. Let      E PTF := inf EPTF (ρ)ρ ≥ 0, ρ(x)d x = 1 R3

which is finite by Lemma A.1. Lemma 2.3. E PTF < 0. Proof. Given ρ ∈ L1 (R3 ) ∩ L 5/3 (R3 ) with ρ ≥ 0 and R −3 ρ(R −1 x). Then R3 ρ R (x)d x = 1 for all R > 0 and   5 −2 3 2 23 −1 1 3 EPTF (ρ R ) = R 10 (6π ) ρ(x) d x − R 2 R3

This is negative for R large enough.



R6

ρ d x = 1, let ρ R (x) = ρ(x)ρ(y) d xd y. |x − y|

 

Proof of Theorem 1.1. Let g ∈ L 2 (R3 ) be given by g(x) = √ (2π )−3/4 e−x /4 and set −3/2 g(x/ε), so that gε  = 1 for all ε > 0. Let β = 2α − U ≥ 0. If β = 0 gε (x) = ε then E N ≤ 0 by Proposition 2.1. Hence it remains to consider the case β > 0. Every density function ρ N ∈ L 1 (R3 ) with ρ N 1 = N is of the form ρ N (x) = N 2 ρ(N 1/3 x) with ρ1 = 1. From Proposition 2.1 and Lemma 2.2 combined it follows that   2 5 ρ N ,ε (x)ρ N ,ε (y) 3 d xd y + N ∇gε 2 , (6π 2 ) 3 ρ N (x) 3 d x − 21 β −2 E N ≤ 10 3 6 |x − y| R R (2.12) 2

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M. Griesemer, J. S. Møller

2 = e−k 2 /2 . Then where ρ N ,ε = ρ N ∗ |gε |2 . Suppose 1 < μ < 6/5 and let f (k) := |g| 2 ρ   N ,ε (k) = ρ N (k)|gε | (k) = ρ N (k) f (εk) and 1 − | f (k)|2 ≤ 1. |k|μ−1 k=0

sup

(2.13)

By definition of f , by (2.13), and by Lemma A.1,   ρ N (x)ρ N (y) ρ N ,ε (x)ρ N ,ε (y) d xd y − d xd y 6 6 |x − y| |x − y| R R  ρ N (k)|2 1 2 | (1 − | f (εk)| ) dk = 2π 2 R3 |k|2  1 − | f (εk)|2 | ρ N (k)|2 1 μ−1 = ε dk 2π 2 |εk|μ−1 |k|3−μ R3  | ρ N (k)|2 1 μ−1 ≤ ε dk 3−μ 2π 2 R3 |k|  ρ(x)ρ(y) 2+ μ3 μ−1 μ−2 cμ =N ε 2(2π ) d xd y. c3−μ R6 |x − y|μ Combining this estimate with (2.12), we see that 7

β −2 E N ≤ N 3 EPTF (ρ) + N ε−2 ∇g2 +N

2+ μ3 μ−1

ε

(2π )

μ−2



ρ(x)ρ(y) d xd y c3−μ R6 |x − y|μ cμ

for all ρ ∈ L 1 (R3 ) with ρ1 = 1. If {ρn } ⊂ L 1 (Rn ) is a minimizing sequence, EPTF (ρn ) → E PTF as n → ∞, then ρn 5/3 is bounded uniformlyin n, by definition of EPTF and by (A.1) with μ = 1. It follows, again by (A.1), that ρn (x)ρn (y)/|x − y|μ d xd y is bounded uniformly in n for μ < 6/5. Therefore, in the limit n → ∞, we obtain μ

7

β −2 E N ≤ N 3 E PTF + 41 N ε−2 + N 2+ 3 εμ−1 Cμ , where the constant Cμ is finite for μ < 6/5 and where ∇g2 = 1/4 was used. Upon optimizing with respect to ε we arrive at 7

9+5μ

β −2 E N ≤ N 3 E PTF + N 3+3μ Dμ with a new constant Dμ . This bound with the choice μ = 37/31 < 6/5 proves Theorem 1.1. Proof of Theorem 1.4. We only need to prove that E 1 ≤ −α. The bound E N ≤ −α N will then follow from E N +M ≤ E N + E M as pointed out in the Introduction. Following Nelson [27] we introduce √    α 1 eik·x a(k) − e−ik·x a ∗ (k) dk. B := − 2 c0 |k|≤ i(1 + k )|k| 2

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291

Then eiB H1, e−iB =

1 2



p 2 + 2a ∗ · p + 2 p · a + a 2 + (a ∗ )2 + 2a ∗ a + Hph − αe , (2.14)

where

√  α k eikx a(k) dk, c0 |k|≤ (1 + k 2 )|k| 2  1 1 dk. e := 2 c0 |k|≤ |k|2 (1 + k 2 ) a :=

2

From (2.14) we see that, for all normalized f ∈ L 2 (R3 ),    f ⊗ , eiB H1, e−iB f ⊗  = f, (− 21 ) f − αe ,

(2.15)

where  ∈ F denotes the vacuum vector. Since inf σ (−) = 0 it follows from (2.15) that E 1, ≤ −αe , where  1 1 lim e = 2 dk = 1. 2 →∞ 3 c0 R |k| (1 + k 2 ) 2 This concludes the proof of the first bound in Theorem 1.4. A result similar to E N +M ≤ E N + E M is expressed by Theorem 6 in [15]. A copy of the proof of that theorem, with small modifications due to the differences of the Hamiltonians, also proves the desired bound here. In fact, the main part of the proof of [15, Theorem 6] is Eq. (19) and the equation thereafter, which show that the interaction between electrons mediated by bosons decreases with increasing particle separation. This part remains valid for the coupling function χ|k|≤ /(c0 |k|) of the Hamiltonian H N , . Other parts of the proof are simplified due to the fact the phonon dispersion relation ω L O is constant and hence a local operator with respect to the boson position as measured by i∇k .   3. Lower Bounds on E N In this section we prove Theorems 1.2 and 1.3. The first step is to make sure that phonons with large momenta contribute to lower order in N . To this end, for given K , , δ, κ > 0, we define the operator H N ,,K :=

− 21 (1 − κ) N √  + α

N 

 + (1 − δ)Hph + U VC

=1



=1 |k|≤

|k|2

−  e 4K 2  ik·x e a(k) + e−ik·x a ∗ (k) dk. c0 |k|

Of course, later on, δ, κ ∈ (0, 1) and K <  → ∞. The following result, in the case N = 1, is essentially due to Lieb and Thomas [25]. While a sharp cutoff |k| ≤ K is used in [25], we work with a Gaussian cutoff since we need the Fourier transform of the cutoff to be positive in the proof of Lemma 3.2.

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M. Griesemer, J. S. Møller

Lemma 3.1. Suppose K , , α and U are positive, 0 < δ < 1 and let κ := √ √ where I∞ := ( 2 − 1)/ π . Then 3 . 2δ Proof. For each  ∈ {1, . . . , N }, we introduce three high momenta modes by  () () T j (k)a(k)dk, j ∈ {1, 2, 3}, Z j := H N , ≥ H N ,,K −

8α N 3K δ I∞ ,

(3.1)

R3

() T j (k)

:=

√

− |k|

2

1 − e 4K 2 αχ (k) k j e−ik·x , c0 |k|3

k j ∈ R being the j th component of k ∈ R3 and χ the characteristic function of the set {|k| ≤ }. For later use we compute the inner product of two functions T j() . By straightforward computations,  α () () I, T j (k)T j (k) dk = δ j j (3.2) 3K K R3 where

√  R s2 2 (1 − e− 4 )2 I R := ds. π 0 s2

√ √ √ Note that 4π/c02 = 2/π and that I∞ = lim R→∞ I R = ( 2 − 1)/ π as defined in the statement of the lemma. By definition of H N ,,K , H N , = H N ,,K +

N

 κ () −  + I K , + δ Hph , 2

(3.3)

=1

() I K ,

:=

√

 α

|k|2

|k|≤

−  1 − e 4K 2  ik·x e a(k) + h.c. dk, c0 |k| ()

where we introduced the operators I K , associated with the ultraviolet part of the electron-phonon interaction. The key ingredient of this proof is that () I K ,

=

3  

()

()∗

p, j , Z j − Z j



,

(3.4)

j=1

where p, j := −i∂/∂x, j . This identity implies that 3      () ()∗  p, j η(Z () η, I K , η ≤ 2 j − Z j )η j=1

≤

3 κ 2 ()∗ 2 η, − η + η, −(Z () j − Z j ) η 2 κ j=1

≤

3 κ 4 () ()∗ η, − η + η, (Z ()∗ Z () j j + Z j Z j )η, 2 κ j=1

(3.5)

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293

where κ > 0 is to be selected, and the estimate ()∗ () () ()∗ 2 1 + Z ()∗ Z () |η, (Z () j ) η| ≤ Z j ηZ j η ≤ 2 η, (Z j Z j j j )η

was used. From (3.2) and I/K ≤ I∞ it is clear that 3 3     ()∗ () () ()∗ ()∗ () () ()∗ (Z j Z j + Z j Z j ) = 2Z j Z j + Z j , Z j j=1

j=1

≤

α 2α I∞ Hph + I∞ . 3K K

(3.6)

Combining (3.5) and (3.6) we arrive at ±

N  =1

()

I K , ≤

N κ 8α N I∞ 4α N I∞ Hph + , (− ) + 2 3κ K κK =1

which, by (3.3) and the choice κ = 8α N I∞ /(3K δ), proves the lemma.

 

Lemma 3.2. Suppose K , , α, U and κ are positive, and 0 < δ ≤ 1/2. Then  √ N  2α 2α N K − U VC − √ .  − H N ,,K ≥ − 21 (1 − κ) 1−δ π

(3.7)

=1

Proof. By completing the square in annihilation and creation operators, that is, by using Lemma A.2, we see that 2 √  N  − |k|  α e 4K 2  ik·x e (1 − δ)Hph + a(k) + e−ik·x a ∗ (k) dk c0 |k|≤ |k|

=1

2 N  − |k|  e 2K 2 ik·(x j −x ) α e dk ≥− (1 − δ)c02 j,=1 R3 |k|2 |k|2   e− 2K 2 2α αN K =− eik·(x j −x ) dk − √ . (1 − δ) π (1 − δ)c02 j< R3 |k|2

(3.8)

The integral in (3.8) represents the electrostatic energy of two spherically symmetric, non-negative charge distributions centered at x j and x , respectively, each distribution having total charge one, see (A.3). Hence Newton’s theorem, [24, Theorem 9.7], implies that 

2

e

− |k| 2 2K

eik·(x j −x ) dk ≤

2π 2 . |x j − x |

2 R3 |k| √ Since c02 = 2π 2 2, it follows that (3.8) is bounded below by √ 2α αN K VC − − √ , 1−δ (1 − δ) π

which proves the lemma.

 

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M. Griesemer, J. S. Møller

Proof of Theorem 1.2. We shall combine Lemmas 3.1 and 3.2 with suitable choices for √ δ and K . First, suppose that 0 < δ ≤ 1/2 and that κ ∈ (0, 1). Since 2α − U ≥ 0, by assumption of Theorem 1.2, the constant multiplying the potential VC in Lemma 3.2 is positive, and hence, after the scaling transformation −1 √ 2α −U (1 − κ)x x→ 1−δ we may apply (1.6) and find that √

2  N  2α  1 1−δ − U 2α N K inf σ −  − VC − √ H N ,,K ≥ − 1−κ 2 π =1 √

2 2α 7 1−δ − U 2α N K N3 − √ , ≥ −CG 1−κ π where CG is chosen such that (1.6) holds true. We now make the choices 1

2

δ = 21 N − 9 and K = 13 32I∞ α N 1+ 9 , which imply that κ, as defined in Lemma 3.1, obeys κ = 21 N −1/9 = δ. Using that √ √ √ (1 − t)−1 ≤ 1 + 2t, for 0 ≤ t ≤ 1/2, that U ≤ 2α, and I∞ / π = ( 2 − 1)/π ≤ 1/(2π ), we find that √ 2 7 32 2 2+ 2 α N 9 2α(1 + 2δ) − U (1 + 2κ)N 3 − H N ,,K ≥ −CG 3π √  7 32 2 7 − 1 α N3 9 ≥ −CG ( 2α − U )2 + 16α 2 δ (1 + 2κ)N 3 − 3π √  32 7 7 1 7 1 α2 N 3 − 9 ≥ −CG ( 2α − U )2 N 3 + 18α 2 N 3 − 9 − 3π

√ 7 7 1 32 α2 N 3 − 9 . = −CG ( 2α − U )2 N 3 − 18CG + 3π √

 

Proof of Theorem 1.3.√Finally we consider the case, where U − 2α > 0. In Lemma 3.2 we choose δ = (U − 2α)/(2U ) and K = 8α N I∞ /(3δ), so that κ = 1 in Lemma 3.1, and √ √ U − 2α 2α = > 0. U− 1−δ 2(1 − δ) From Lemma 3.1 and Lemma 3.2 it hence follows that 16I∞ 2U 2α N K 3 3U EN ≥ − √ = − √ α2 N 2 − √ − √ , 2δ π 3 π U − 2α U − 2α √ √  where I∞ / π = ( 2 − 1)/π ≤ 1/(2π ). 

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295

A. Auxiliary Results Lemma A.1. Suppose that ρ ∈ L 1 (R3 ) ∩ L 5/3 (R3 ), ρ ≥ 0, 0 < μ < 6/5, and let ρ N (x) = N 2 ρ(N 1/3 x). Then 

5μ ρ(x)ρ(y) 2− 5μ d xd y ≤ aμ ρ1 6 ρ 56 , μ 3 R6 |x − y|   2 c3−μ ρ(x)ρ(y) |ρ(k)| ˆ d xd y = (2π )−μ dk, μ 3−μ cμ R3 |k| R6 |x − y|   2 1 ρ(x)ρ(y) |ρ(k)| ˆ d xd y = dk, 2π 2 R3 |k|2 R6 |x − y|   ρ N (x)ρ N (y) ρ(x)ρ(y) 2+ μ3 d xd y = N d xd y, μ μ 6 6 |x − y| R R |x − y|

(A.1) (A.2) (A.3) (A.4)

where aμ :=

4π 3

μ 3

6 5μ

1+ μ 3

−1+ μ 2 6 , −1 5μ

μ μ cμ := π − 2 ( ) 2

in (A.1) and (A.2), respectively. Inequality (A.1), in the special case μ = 1, implies that EPTF (ρ) is bounded below, and moreover, that ρ5/3 is bounded uniformly for densities ρ with ρ1 = 1 and EPTF (ρ) ≤ E PTF + 1. Proof of Lemma A.1. For each R > 0, by Hölder’s inequality, 

ρ(y) dy = μ R3 |x − y|

 |x−y|≤R

≤

8π 6 − 5μ

ρ(y) dy + |x − y|μ

2

5

 |x−y|≥R

ρ(y) dy |x − y|μ

6

R 5 −μ ρ 5 + R −μ ρ1 . 3

By optimizing this bound w.r.to R > 0 we obtain (A.1). Equation (A.2) follows from [24, Coro. 5.10]. The factor (2π )−μ stems from the differences in the definition of the Fourier transform. Equation (A.3) is the important special case μ = 1 from (A.2), and (A.4) is straightforward to verify by a change of variables. Lemma A.2. Suppose f ∈ L 2 (R3 ). Then, for every δ > 0, 

  1 δa ∗ (k)a(k) + f (k)a(k) + f (k)a ∗ (k) dk ≥ −  f 2 , 3 δ R

and the lower bound is attained by the expectation value in the coherent state η ∈ F, η = 1, defined by a(k)η = −δ −1 f (k)η.

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M. Griesemer, J. S. Møller

Proof. By completing the square in creation and annihilation operators 

  δa ∗ (k)a(k) + f (k)a(k) + f (k)a ∗ (k) dk R3  

   f (k) | f (k)|2 f (k) ∗ a(k) + − = δ a (k) + dk δ δ δ R3 1 ≥ −  f 2 . δ  

Proposition A.3. Suppose that η ∈ F with  f  = η = 1,

√ 2α ≤ U . Then for all N ,  > 0, and all f ∈∧ N L 2 (R3 ),

 f ⊗ η, H N , f ⊗ η ≥ −cL2

5 6



2 3π

2

3

U2N,

where cL = 1.68 or any other constant for which the Lieb-Oxford inequality holds. Proof. As in the proof of Theorem 1.1,  f ⊗ η, H N , f ⊗ η

√ =  f, (− 21  + U VC ) f  + η, (Hph + α (ρ))η  √ ρ(x)ρ(y) d xd y. ≥  f, (− 21  + U VC ) f  − 2α 21 R6 |x − y|

Using the Lieb-Thirring [24, Theorem 2.15] and the Lieb-Oxford inequalities [22] we find that  f ⊗ η, H N , ( f ⊗ η)   √ 5 ≥ cLT ρ(x) 3 d x + (U − 2α) 21 R3

 4 ρ(x)ρ(y) d xd y − U cL ρ(x) 3 d x, R6 |x − y| R3 (A.5)

3 3π 2/3 where cLT = 10 (2) and cL = 1.68 or any other constant for which the Lieb-Oxford inequality is satisfied. From ρ(x)4/3 = ρ(x)5/6 ρ(x)1/2 and the CauchySchwarz inequality, for every ε > 0,

 R3

4 3



ρ(x) d x ≤

5 3

1  2

1 2

ρ(x) d x ρ(x)d x R3 R3 

 5 1 ≤ 21 ε ρ(x) 3 d x + ρ(x)d x . ε R3 R3

The estimates (A.5) and (A.6) with ε = 2cLT /U cL prove the proposition.

(A.6)  

N -Polaron Systems

297

References 1. Alexandrov, A.S., Mott, N.: Polarons and Bipolarons. Singapore: World Scientific, 1996 2. Ammari, Z.: Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: The Nelson model. Math. Phys. Anal. Geom. 3, 217–285 (2000) 3. Bach, V.: Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147, 527–548 (1992) 4. Brosens, F., Klimin, S.N., Devreese, J.T.: Variational path-integral treatment of a translation invariant many-polaron system. Phys. Rev. B 71, 214301 (2005) 5. Brosens, F., Klimin, S.N., Devreese, J.T.: Path-integral approach to the ground-state energy of a homogeneous polaron gas. Phys. Rev. B 77, 085308 (2008) 6. Cannon, J.T.: Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators. J. Funct. Anal. 8, 101–152 (1971) 7. Devreese, J.T.: Polarons. In: Encyclopedia of Applied Physics, G. L. Trigg, E. H. Immergut, eds., Vol. 14, Weinhein: Wiley-VCH, 1996, pp. 383–409 8. Devreese, J.T., Tempere, J.: Large-polaron effects in the infrared spectrum of high-Tc cuprate superconductors. Solid State Commun. 106, 309–313 (1998) 9. Donsker, M.D., Varadhan, S.R.: Asymptotics for the polaron. Comm. Pure Appl. Math. 36, 505– 528 (1983) 10. Feynman, R.P.: Slow electrons in a polar crystal. Phys. Rev. 97, 660–665 (1955) 11. Feynman, R.P.: Statistical Mechanics. A Set of Lectures, Frontiers in Physics, Reading, MA: W. A. Benjamin, Inc., 1972 12. Fröhlich, H.: Electrons in lattice fields. Adv. in Phys. 3, 325–362 (1954) 13. Fröhlich, J.: Existence of dressed one-electron states in a class of persistent models. Fortschr. Phys. 22, 159–198 (1974) 14. Gerlach, B., Löwen, H.: Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?. Rev. Mod. Phys. 63, 63–90 (1991) 15. Griesemer, M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210, 321–340 (2004) 16. Haga, E.: Note on the slow electrons in a polar crystal. Prog. Theoretical Phys. 11, 449–460 (1954) 17. Hartinger, Ch., Mayr, F., Deisenhofer, J., Loidl, A., Kopp, T.: Large and small polaron excitations in La2/3(Sr/Ca)1/3Mn O3 films. Phys. Rev. B 69, 100403 (2004) 18. Klimin, S.N., Fomin, V.M., Brosens, F., Devreese, J.T.: Characterization of shell-filling of interacting polarons in a quantum dot through their optical absorption. Physica E 22, 494–497 (2004) 19. Lee, T.D., Low, F.E., Pines, D.: The motion of slow electrons in a polar crystal. Phys. Rev. 90, 297– 302 (1953) 20. Lévy-Leblond, J.-M.: Nonsaturation of gravitational forces. J. Math. Phys. 10, 806–812 (1968) 21. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 70A, 444–446 (1977) 22. Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53, 603–604 (1981), Erratum 54, 311 (1981) 23. Lieb, E.H.: Variational principle for many-fermion systems. Phys. Rev. Lett. 46, 457–459 (1981), Erratum 47, 69 (1981) 24. Lieb, E.H., Loss, M.: Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, Providence, RI: Amer. Math. Soc., 2001 25. Lieb, E.H., Thomas, L.E.: Exact ground state energy of the strong-coupling polaron. Commun. Math. Phys. 183, 511–519 (1997), Erratum 188, 499–500 (1997) 26. Miyao, T., Spohn, H.: The bipolaron in the strong coupling limit. Ann. Henri Poincaré 8, 1333– 1370 (2007) 27. Nelson, E.: Interaction of non-relativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964) 28. Verbist, G., Peeters, F.M., Devreese, J.T.: Large bipolarons in two and three dimensions. Phys. Rev. B 43, 2712–2720 (1991) Communicated by H. Spohn

Commun. Math. Phys. 297, 299–344 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1038-3

Communications in

Mathematical Physics

Kinetically Constrained Lattice Gases N. Cancrini1 , F. Martinelli2 , C. Roberto3 , C. Toninelli4 1 Dip. Matematica, Univ. L’Aquila, I-67010 L’Aquila, Italy.

E-mail: [email protected]

2 Dip. Matematica, Univ. Roma Tre, Largo S.L.Murialdo, 00146, Roma, Italy.

E-mail: [email protected]

3 L.A.M.A., Univ. Marne-la-Vallée, 5 bd Descartes, 77454 Marne-la-Vallée, France.

E-mail: [email protected]

4 L.P.M.A. and CNRS-UMR 7599, Univ. Paris VI-VII, 4, Pl. Jussieu, 75252 Paris, France.

E-mail: [email protected] Received: 22 June 2009 / Accepted: 16 December 2009 Published online: 8 April 2010 – © Springer-Verlag 2010

Abstract: Kinetically constrained lattice gases (KCLG) are interacting particle systems which show some of the key features of the liquid/glass transition and, more generally, of glassy dynamics. Their distintictive signature is the following: i) reversibility w.r.t. product i.i.d. Bernoulli measure at any particle density and ii) vanishing of the exchange rate across any edge unless the particle configuration around the edge satisfies a proper constraint besides hard core. Because of degeneracy of the exchange rates the models can show anomalous time decay in the relaxation process w.r.t. the usual high temperature lattice gas models particularly in the so-called cooperative case, when the vacancies have to collectively cooperate in order for the particles to move through the systems. Here we focus on the Kob-Andersen (KA) model, a cooperative example widely analyzed in the physics literature, both in a finite box with particle reservoirs at the boundary and on the infinite lattice. In two dimensions (but our techniques extend to any dimension) we prove a diffusive scaling O(L 2 ) (apart from logarithmic corrections) of the relaxation time in a finite box of linear size L. We then use the above result to prove a diffusive decay 1/t (again apart from logarithmic corrections) of the density-density time autocorrelation function at any particle density, a result that has been sometimes questioned on the basis of numerical simulations. The techniques that we devise, based on a novel combination of renormalization and comparison with a long-range Glauber type constrained model, are robust enough to easily cover other choices of the kinetic constraints. Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Kinetically Constrained Lattice Gases (KCLG) . . . . . . . . . . . . . . . . 302 2.1 The Markov process and the spectral gap . . . . . . . . . . . . . . . . . 306

 This work was partially supported by the GRDE GREFI-MEFI, by the French Ministry of Education through the ANR BLAN07-2184264 grant and by the European Research Council through the “Advanced Grant” PTRELSS 228032.

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2.2 0-1 KCLG: ergodicity and exchangeability thresholds 3. Kob-Andersen (KA) Model . . . . . . . . . . . . . . . . 4. Main Ideas and Results . . . . . . . . . . . . . . . . . . 5. Renormalization and Long Range Constraints . . . . . . 6. Spectral Gap of KA Model: Proof of Theorem 4.1 . . . . 7. Spectral Gap of AGL: Proof of Theorem 5.5 . . . . . . . 8. Polynomial Decay to Equilibrium: Proof of Theorem 4.2 . 9. Appendix: Properties of KA Model . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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308 310 312 313 318 322 333 340 343

1. Introduction Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice Zd with the usual hard core exclusion. A configuration is therefore defined by assigning to each vertex x ∈ Zd its occupation variable, η(x) ∈ {0, 1}, which represents an empty or occupied site respectively. The evolution is given by a continuous time Markov process of Kawasaki type, namely with rate cx,y (η) the occupation variables at the end points of an unoriented bond e = (x, y) of Zd are exchanged. The exchange rate is equal to one if the current configuration satisfies an apriori specified local constraint and zero otherwise. In the former case we say that the exchange is legal. A key feature of the constraint is that it does not depend on the occupation variables η(x), η(y) so d that any Bernoulli product measure μ p on {0, 1}Z , where p is the particle density, is automatically an invariant reversible measure for the process. However, at variance with the simple symmetric exclusion process (SSEP) which corresponds to the unconstrained choice ce (η) ≡ 1 for any bond (x, y), KCLG have several other invariant measures. This is related to the fact that there exist blocked configurations, namely configurations for which all exchange rates are equal to zero. KCLG have been introduced in the physics literature (see [24] for a review) to model liquid/glass transition and more generally the glassy dynamics which occurs in different systems, e.g. granular materials. In particular they were devised to mimic the fact that the motion of a molecule in a dense liquid can be inhibited by the geometrical constraints created by the surrounding molecules. The exchange rates are devised to encode this local caging mechanism and thus they typically require a minimal number of empty sites in a proper neighborhood of e = (x, y) in order for the exchange at e to be legal, i.e. to have ce = 1. KCLG are usually classified into cooperative and non-cooperative models. Definition 1.1. A model is said to be non-cooperative if its rates are such that it is possible to construct a proper finite group of vacancies, the so-called mobile cluster, with the following two properties: (i) for any configuration it is possible to move the mobile cluster to any other position in the lattice by a sequence of legal exchanges; (ii) any exchange is legal if the mobile cluster is in a proper position in its vicinity. All models which are not non-cooperative are said to be cooperative. From the point of view of the modelisation of the liquid/glass transition, cooperative models are the most relevant ones. Indeed, very roughly speaking, non-cooperative models are expected to behave like a re-scaled SSEP with the mobile cluster playing

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the role of a single vacancy. Therefore they are not very suitable to describe the rich behavior of glassy dynamics. Let us start by recalling some fundamental problems which require for KCLG new ideas and techniques from those used to study SSEP or other high temperature lattice gas models. A first basic question is whether the infinite volume process is ergodic, namely whether zero is a simple eigenvalue for the generator of the Markov process in L2 (μ p ). This would in turn imply relaxation to μ p in the L2 (μ p ) sense. The constraints which are chosen in the physics literature in order to model the caging mechanism render the dynamics increasingly slow as p is increased. Therefore it is possible that the process undergoes a transition from an ergodic to a non-ergodic regime when the particle density p crosses a critical value pc ∈ (0, 1). This has indeed been conjectured for some cooperative models [15,17,26]. The same issue for non-cooperative models is trivially solved for any p < 1 because of the μ p -almost sure existence of the mobile cluster. When ergodicity holds, the next natural issue is to establish the large time behaviour of the infinite volume process started from the reversible equilibrium measure at time zero. A typical quantity to be considered is the density-density time autocorrelation function. This problem in turn is related to the scaling with the system size of the relaxation time (i.e. inverse spectral gap) in a finite box. Recall that for SSEP such a scaling is diffusive, i.e. O(L 2 ) if L denotes the linear size of the system, and that the induced time decay of the density-density time autocorrelation function is proportional to t −d/2 . Numerical simulations for the Kob-Andersen model suggest the possibility of an anomalous slowing down at high density [15,21] which could correspond to an anomalous scaling of the relaxation time in finite volume. Finally one would like to investigate the large time behaviour of a tagged particle and the evolution of macroscopic density profiles, namely the hydrodynamic limit of the process. For some cooperative models it has been conjectured that a diffusive/non-diffusive transition would occur at a finite critical density: both the self-diffusion coefficent of the tagged particle and the macroscopic diffusion coefficient of the hydrodynamic equation would be strictly positive below this critical density and zero above [15,17]. To our knowledge, the existing rigorous answers to the above questions are the following: Non-cooperative models. In this case much more is known because the existence of mobile finite clusters greatly simplifies the analysis and allows the application of standard familiar techniques (e.g. paths arguments) already developed for lattice gases and exclusion models. In [6] it is proven in certain cases that both the spectral gap and the log Sobolev constant in finite volume of linear size L with boundary sources scale as O(L 2 ). Furthermore for the same models it is established that the self-diffusion coefficient of the tagged particle is strictly positive. Moreover the hydrodynamic limit has been succesfully analyzed for a special class of gradient type in [13]. Cooperative models. In [29,30] for a large class of models it has been proven that pc = 1, namely ergodicity always holds (see instead [28] for a choice of the constraints which certainly leads to pc < 1). The self-diffusion coefficient is instead analyzed in [27] where positivity is proved only modulo a conjecture on the behavior of random walks on a random environment. Finally, we recall that the Glauber version of KCLG, the so-called Kinetically Constrained Spin Models (KCSM), have also been very much studied in physics literature [4,10–12,24] and that some features of their long time behaviors have been

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rigorously analyzed in [1,8,9,16]. In particular in [9] an important correction to the “exact solution” of the East model based on non-rigorous methods was obtained. In conclusion, apart from the ergodicity problem studied in [30], cooperative KCLG remain mathematically largely unexplored. The main contribution of this paper is to introduce for the first time suitable new ideas and techniques to partially cover this gap. We mainly focus on the cooperative model which has been most studied in the physics literature, the so-called Kob Andersen (KA) model, which was introduced in [15] and subsequently studied in several works [3,17,18,21,25,26,29,30]. KA actually denotes a class of models on Zd characterized by a integer parameter j ∈ [2, d] and defined by the following nearest neighbour exchange rates: cx,y = 1 iff at least j − 1 neighbours of x different from y are empty and at least j − 1 neighbours of y different from x are empty too. It is immediate to verify that KA is always a cooperative model. For example if j = d = 2, a fully occupied double stripe which spans the lattice can never be destroyed. Thus any finite cluster of vacancies cannot be mobile since it cannot overcome the double stripe. Nevertheless in [29,30] it has been proven that the infinite volume process is always ergodic at any finite density, namely pc = 1. This contradicts previous claims [15,17,26] on the existence of a finite critical density. Our two main results concern the KA model in two dimensions with j = 2 (which is the only possible choice when d = 2) but actually both extend to higher dimensions (with much more effort and more cumbersome reasoning). This choice was made in order to present the overall strategy stripped from unnecessary complications. In Theorem 4.1 we consider the model in a box of linear size L with sources, i.e. Glauber moves, at the boundary sites. We establish upper and lower bounds of order 1/L 2 (apart from logarithmic corrections) for the spectral gap at any density. Thus the scaling is the same as for the unconstrained case, in contrast with previous conjectures of an anomalous scaling at high density suggested by numerical evidences of a strong slowing down of the dynamics [15,17]. However, contrary to what happens in high temperature Ising type lattice gases, there is no uniformity in the particle density. In Theorem 4.2 instead, we establish a diffusive 1/t decay (apart from logarithmic corrections) for the infinite volume time auto-correlation of local functions as for SSEP. A rough sketch of the main new ideas which are needed to overcome the problems posed by the cooperative constraints is presented in Sect. 4. Although we have devised our techniques for the KA model, they can be easily extended to analyze other cooperative KCLG (and all non-cooperative models) via a proper modification of the choice of the constraints for the auxiliary constrained Kawasaki and Glauber dynamics discussed in Sect. 5. 2. Kinetically Constrained Lattice Gases (KCLG) In this section we define a general setting for the class of models that will be analyzed later on and provide the main characterization of their ergodicity threshold (see Proposition 2.16). Setting and notation. Lattices, distances and neighbourhoods. The models considered here are defined on the integer lattice Zd with sites x = (x1 , . . . , xd ) and basis vectors e1 = (1, . . . , 0),

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Fig. 1. The various neighborhoods of a vertex x in two dimensions

e2 = (0, 1, . . . , 0), . . ., ed = (0, . . . , 1). On Zd we will consider the Euclidean norm x, the 1 (or graph theoretic) norm x1 and the sup-norm x∞ . The associated distances will be denoted by d(·, ·), d1 (·, ·) and d∞ (·, ·) respectively. A bond is a couple of sites (x, y) with d1 (x, y) = 1 (couples are meant to be non ordered so that (x, y) ≡ (y, x)).  For any set , E  will denote  the set of all bonds with both sites in , namely E  = (x, y) ∈ 2 : d1 (x, y) = 1 . For any set A ⊂ Zd and site x ∈ Zd , we denote by A + x the set translated of x, namely A + x := {y : ∃z ∈ A s.t. y = z + x}. For any vertex x we define its neighborhoods (see Fig. 1) Nx = {y ∈ Zd : d1 (x, y) = 1}, Nx∗ = {y ∈ Zd : d∞ (x, y) = 1}, Kx = {y ∈ Nx : y = x +

d 

αi ei , αi  0},

i=1

K∗x = {y ∈ Nx∗ : y = x +

d 

αi ei , αi  0}.

i=1

The exterior neighborhood (∂+ ) and *-neighborhood (∂+∗ ), the interior neighbor∗ ) neighborhood are defined as hood (∂− ) and *-neighborhood (∂− ∂+  := {x ∂+∗  := {x ∂−  := {x ∗ ∂−  := {x

∈ /  : d1 (x, )  1}, ∈ /  : d∞ (x, )  1}, ∈  : d1 (x, c )  1}, ∈  : d∞ (x, c )  1}.

Furthermore it is useful to introduce the following additional oriented sets: i

∂ +  := {x ∈ / ; x − ei ∈ } , i

d ∂ +  := ∪i=1 ∂ + ,   ∗ ∂ +  := ∪x∈ K∗x \, i

∂ −  := {x ∈ ; x + ei ∈ / } , i

d ∂ −  := ∪i=1 ∂ − .

In order to remember the notation we may observe that: * means using the d∞ distance, ∂ means taking only an oriented part of the boundary and +/− means exterior/interior.

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Geometric sets and paths. The following notions of rectangles, cubes, cylinders, geometric paths, double-paths and crossings will be used throughout the work. Definition 2.1 (Rectangles, cubes and cylinders). A rectangle R is a set of sites of the form R := [a1 , b1 ] × · · · × [ad , bd ] with ai , bi ∈ Z. Given a length  ∈ Z+ , Q  is the cube of side , Q  := [0,  − 1] × . . . [0,  − 1]. Finally, for any x ∈ , N ∈ Z+ and i ∈ {1, . . . , d}, we define the cylinder of radius N around x in the i th direction as   N Tx,i = y ∈ Zd : y j ∈ [x j − N , x j + N ] for all j = i

(2.1)

N () :=  ∩ T N . and for any  ⊂ Zd , we let Tx,i x,i

The following property of cylinders can be immediately verified N ( ) ∩ , then z ∈ T N (); if Claim 2.2. Choose x ∈ Zd and ,  ⊂ Zd . If z ∈ Tx,i x,i N () ⊂ T N ( ).  ⊂  , then Tx,i x,i

Definition 2.3 (Geometric paths, double-paths and crossings). Given x, y ∈ Zd , a sequence (x (1) , . . . , x (n) ) is a geometric path from x to y and we denote it by γx y

if: x (1) = x, x (n) = y, x ( j) = x ( j ) for any j = j and d1 (x (k) , x (k−1) ) = 1 for any k = 2, . . . , n. For any γx y we also define the corresponding geometric double-path as ∗ γ x y := γx,y ∪ ∂ + γx,y . Given  ⊂ Zd we say that γx,y is inside  and write γx,y ⊂  if x (i) ∈  for all i ∈ {1, . . . , n}. Given z ∈ Zd (e ∈ E Zd ) we say that z is inside γx,y and write z ∈ γx,y (e belongs to γx,y or e ∈ γx,y ) if there exists i ∈ {1, . . . , n} such that z = x (i) (if there exists i ∈ {1, . . . , n − 1} such that (x (i) , x (i+1) ) = e). Given a rectangle R = [a1 , b1 ] × · · · × [ad , bd ] we say that a path γx,y is crossing R in direction i if: γ x,y ⊂ R and xi = ai and yi = bi − 1. Finally, if d = 2 and i = 1 (i = 2) we say that the path is left-right (top-bottom) crossing. The following two-dimensional results can be easily verified Claim 2.4. Given a rectangle R = [a1 , b1 ] × [a2 , b2 ]: (i) if γx,y is left-right crossing R and γx, ˜ y˜ is top-bottom crossing R they have (at least) one common point z ∈ γx,y ∩ γx, ˜ y˜ . (ii) given R = [a1 , b1 ]×[a2 , b2 ] (R = [a1 , b1 ]×[a2 , b2 ]) with a1  a1 and b1  b1 (a2  a2 and b2  b2 ), if γx,y is left-right (top-bottom) crossing R, then it is also left-right (top-bottom) crossing R .

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The probability space and the good event. Consider a finite probability space (W, ν) with ν(w) > 0 for any w ∈ W . We will denote by G ⊂ W a distinguished event in W which will be referred to as the “good event” and by ρ ≡ ν(G) its probability. Given d consider the configuration space  = W Z equipped with the product (W, ν) we will measure μ := x∈Zd νx , where νx ≡ ν. If W = {0, 1} then ν is completely determined by the parameter p := ν(1) and μ is a Bernoulli product measure, μ ≡ μ p . Similarly we define  and μ for any subset  ⊂ Zd . Given ω ∈  for each x ∈ Zd we denote by ω(x) the value of ω at site x and we say that x is good if ω(x) ∈ G. Elements of  ( ) will be denoted by Greek letters ω, η (ω , η ) etc. and the variance w.r.t. μ (μ ) by Var μ (Var μ ). We drop the measure from the variance by adopting the simpler notation Var (Var  ) when confusion does not arise. We will use the shorthand notation μ( f ) (μ ( f )) to denote the expected value of any f ∈ L 1 (μ). Given a configuration ω ∈  and a set  ⊂ Zd , we call ω the restriction of ω to . Given two configurations ω, τ ∈  we call ω · τ the configuration that equals ω in  and equals τ in Zd \. Furthermore, for any 1 , 2 ,  ⊂ Zd with 1 ∪ 2 =  and 1 ∩ 2 = ∅ and any two configurations ω, τ ∈  we set ω1 · τ2 for the configuration that equals ω inside 1 and τ inside 2 . A function f :  → R that depends on finitely many variables {ω(x)}x∈Zd will be called local. Given a configuration ω ∈  for any bond e = (x, y) ∈ E Zd we denote by ωe (or sometimes by ω x y ) the configuration ω with the occupation variables at x and y exchanged, ⎧ ⎨ ω(z) ωe (z) = ω x y (z) := ω(x) ⎩ ω(y)

if z ∈ / {x, y} if z = y if z = x.

We define Te :  →  to be the operator acting as Te (ω) = ωe and we use the symbol ∇e to denote ∇e f (ω) := f (ωe ) − f (ω). When W = {0, 1} we also denote by ω x the configuration flipped at x, namely  ω (z) := x

ω(z) 1 − ω(z)

if z = x if z = x,

and we define as before Tx :  →  as Tx (ω) = ω x and ∇x as ∇x f (ω) := f (ω x ) − f (ω). Finally, we introduce the notions of G-equivalence, good paths and good crossings (recall Definition 2.3). Definition 2.5 (G-equivalence). We say that ω and ω are G-equivalent in  and write G,

ω ⇐⇒ ω if for all x ∈  the following holds: ω(x) ∈ G iff ω (x) ∈ G. Definition 2.6 (Good paths and good crossings). Given a configuration ω we say that a path γx,y is good for ω if ω(z) ∈ G for any z ∈ γ x y \x. Given a configuration ω, a rectangle R and a path γx,y we say that γx,y is a good crossing in R in direction i if: γx,y is crossing in R in direction i, γx,y is good and ω(x) ∈ G. In d = 2 if i = 1 (i = 2) we use the notation good left-right (top-bottom) crossing. It is immediate to verify that if γx,y ⊂  is good for ω, then it is also good for any ω which is G-equivalent to ω in \x (where here and in the sequel we let \x := \{x}).

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2.1. The Markov process and the spectral gap. The interacting particle models that we study here are Kawasaki type Markov processes in  which are reversible w.r.t. the product measure μ. When considered in  they will instead be a mixture of Glauber (on a proper inner boundary) and Kawasaki (inside ) dynamics. Each model is characterized by a collection of influence classes {Ce }e∈E Zd . For any bond e, Ce is a collection of subsets of Zd which satisfies the following basic hypothesis: / A; (a) independence of e: for all e ∈ E Zd and all A ∈ Ce , e ∈ (b) translation invariance: Ce + x = Cx+e for all e; (c) finite range interaction: there exists r < ∞ such that for any bond e = (u, v), any element of Ce is contained in ∪rj=1 {y : d1 ({u, v}, y) = j}. Definition 2.7 Given a bond e ∈ E Zd we will say that the constraint at e is satisfied by the configuration ω if ce (ω) equals one, where 1 if there exists a set A ∈ Ce such that ω(y) ∈ G for all y ∈ A ce (ω) = 0 otherwise. On the whole lattice Zd the process of interest for us can be informally described as follows. Each bond e = (x, y) waits an independent mean one exponential time and then, provided that the current configuration ω satisfies the constraint at e, the values ω(x) and ω(y) are exchanged. Standard methods (see e.g. [19]) show that the Markov semigroup Pt associated to this process is self-adjoint on L2 (μ) for any choice of ν (i.e. for any product measure μ) and the corresponding infinitesimal generator L (i.e. the operator such that Pt := et L ) is a non-positive self-adjoint operator which acts on local functions as  

 L f (ω) = (2.2) ce (ω) f ωe − f (ω) . e∈E Zd

The corresponding Dirichlet form on L2 (μ) is Dμ ( f ) := −μ ( f · L f ) which can be rewritten as    f ∈ L2 (μ). μ ce (∇e f )2 Dμ ( f ) = e∈E Zd

In the whole work, when confusion does not arise, we will omit the index μ from the Dirichlet form. It is important to notice that due to the fact that the rates are not bounded away from zero, the reversible measure μ is not in general the only invariant measure for the process. In particular there exist initial configurations that are blocked forever (all exchange rates are zero) and any measure concentrated on them is invariant too. An interesting question is therefore whether μ is ergodic or mixing for the Markov process generated by L. To this purpose it is useful to recall the following well known result (see e.g. Theorem 4.13 in [19]). Denote by Lμ the generator of the semigroup Pt extended by continuity to L2 (μ). Theorem 2.8. The following are equivalent: (a) limt→∞ Pt f = μ( f ) in L2 (μ) for all f ∈ L2 (μ). (b) 0 is a simple eigenvalue for Lμ .

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Clearly (a) above implies that limt→∞ μ ( f Pt g) = μ( f )μ(g) for any f, g ∈ L2 (μ), i.e. μ is mixing. Up to now we have considered the infinite volume version of the models. If instead we restrict the generator (2.2) to a finite set  ⊂ Zd with edge set E  , the corresponding continuous-time Markov chain is in general not ergodic on  due to the presence of the constraints and to the conservative character of the dynamics. A natural possibility to restore ergodicity and to make μ an invariant measure for the chain is to freeze the external configuration to a proper reference configuration τ (the boundary condition) and to add a collection of sources (i.e. Glauber moves) on a proper set S ⊂  (the source set). Let M ⊂ Zd \ be s.t. τ (x) ∈ G (τ (x) ∈ G) for x ∈ M (for x ∈ M). The finite volume generator will depend on the boundary condition τ only via M (the boundary set). More precisely Definition 2.9. The finite volume generator with source set S, boundary set M, source constraints {cx, }x∈S and on-site distribution ν is given by G,ν K L = L, M + LS ,

where, for any f :  → R, K L, M f (ω) =





 M ce, (ω) f ωe − f (ω) ,

(2.3)

(2.4)

e∈E 

LG,ν S f =



cx, (ω)(νx ( f ) − f ),

(2.5)

x∈S

M are defined through the infinite volume conand the finite volume exchange rates ce, straints by M ce, (ω) := ce (ω · τ ),

(2.6)

where τ is any configuration satisfying τ (z) ∈ G for all z ∈ M and τ (z) ∈ G otherwise. The source rates cx, (ω) are either one or zero according to whether the particle configuration ω satisfies or not a proper constraint which does not depend on ω(x). In the sequel we will always drop the sub/superscripts M, S, ν from the notation whenever confusion does not arise. Informally, the above definition means that in addition to the Kawasaki dynamics, each vertex of the source set waits an independent mean one exponential time and then, provided the corresponding source constraint is satisfied, the value ω(x) is refreshed with a new value sampled in W with ν and the whole procedure starts again. The generator L is a non-positive self-adjoint operator on L2 ( , μ ), where μ is now fixed by the choice of the on-site probability measure ν in the Glauber term (2.5). The corresponding Dirichlet form D ( f ) is given by K G D ( f ) = D, M( f ) + DS ( f )    

 M μ ce, (∇e f )2 + μ cx, Var x ( f ) . = e∈E 

(2.7) (2.8)

x∈S



 2 Here Var x ( f ) ≡ dν(ω(x)) f 2 (ω) − dν(ω(x)) f (ω) denotes the local variance with respect to the variable ω(x) computed while the other variables are held fixed.

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To the generator L we associate the Markov semigroup Pt := et L with reversible invariant measure μ and the spectral gap gap(L ) :=

inf

f  =const f ∈L2 (μ)

D ( f ) . Var  ( f )

(2.9)

Remark 2.10. The chain generated by L is not ergodic for all choices of the boundary sets M, S. The interesting choices for us will be those for which the resulting chain is irreducible. We conclude this paragraph with the notion of domination. Definition 2.11 (Domination). In the above setting let {Ce }e∈E Zd be another choice of influence classes. Denote by ce (ω) and L the corresponding rates and generator. If for all ω ∈  and all e ∈ E Zd it holds ce (ω) ≤ ce (ω), then we say that L is dominated by L (or, equivalently, that the rates ce are dominated by ce ). The term domination here means that the KCLG associated to L is more constrained than the one associated to L. If L is dominated by L , for any  and any choice of the K K boundary sets M, D , M ( f )  D,M ( f ) holds. In particular (2.9) yields Lemma 2.12. Fix  ⊂ Zd . If L is dominated by L and we define L  and L with the same choice for M, S and with cx, ≡ c x, , then gap(L  ) ≤ gap(L ). 2.2. 0-1 KCLG: ergodicity and exchangeability thresholds. In the physics literature, the on-site configuration space W is always the two-state space W = {0, 1} which represent the empty and occupied configuration, respectively. We call such models 0-1 KCLG. The on-site distribution ν is now completely defined by specifying the parameter p := ν(1) d which can be varied in [0, 1]. The probability μ over  = {0, 1}Z is thus a product Bernoulli(p) measure, μ ≡ μ p . The good set G is conventionally chosen as the empty state {0} and we denote by q := 1 − p its probability (thus q corresponds to ρ for a generic KCLG). Note that the Simple Symmetric Exclusion Process (SSEP) is a 0-1 KCLG with the trivial choice ce ≡ 1. Recall that on a finite volume  ⊂ Zd the generator (2.3) explicitly depends on the choice of ν which is here completely defined by the parameter q. We denote by L (q) the corresponding generator. From Definition 2.11 and Lemma 2.12 it follows immediately that the spectral gap for a 0-1 KCLG in finite volume is upper bounded by the spectral gap of SSEP in the same region. For 0-1 KCLG it is natural to define the critical value qc = inf{q ∈ [0, 1] : 0 is a simple eigenvalue of Lq }, where, with a slight abuse of notation, we let Lq := Lμ1−q be the infinite volume generator extended by continuity to L2 (μ1−q ). We now relate qc (sometimes called ergodicity threshold) to another threshold of the dynamics. For this purpose we need to define the notion of allowed paths and exchangeable configurations (with respect to a given choice of the constraints), which are valid also for generic (i.e. non-0-1) KCLG.

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Definition 2.13 (Allowed paths). Given η, σ ∈  , a sequence of configurations Pη,σ = (η(1) , η(2) , . . . , η(n) ) starting at η(1) = η and ending at η(n) = σ is an allowed configuration path (or simply allowed path) from η to σ inside  if for any i = 1, . . . , n − 1 there exists either a bond ei ∈ E  with η(i+1) = (η(i) )ei and ceM (η(i) ) = 1 or a site xi ∈ S with η(i+1) = (η(i) )xi i , and such that cxi (η(i) ) = 1. We also say that n is the length of the path and write |Pη,σ | = n. Furthermore, given η ∈  and a geometric path γx,y = (x (1) , . . . x (n) ), we (i) ) = 1 for any i = 1, . . . n − 1, say that γ is an allowed geometric path for η if ceM i , (η where we let η(1) = η and η(i+1) = (η(i) )e with ei := x (i+1) , x (i) . Note that the above definitions depend on the choice of the source and boundary set S, M. When  = Zd we will mean S, M = ∅. i

Definition 2.14 (-Connected configurations). Given η, σ ∈  , we say that they are -connected if there exists (at least) one allowed configuration path Pη,σ inside . Definition 2.15 ((e, )-Exchangeable configurations). A configuration η ∈  is (e, )-exchangeable if η and ηe are -connected. We denote by Ee (Ee ) the set of (e, Zd )-exchangeable ((e, )-exchangeable) configurations. With the above notation we can define the exchangeability threshold as   qex := inf q ∈ [0, 1] : μ1−q (∩e∈E Zd Ee ) = 1 . Using the simple fact that the rates ce (ω) are increasing functions w.r.t. the partial order in  for which ω  ω iff ω (x) ∈ G whenever ω(x) ∈ G, it is easy to check that μ1−q (∩e∈E Zd Ee ) = 1 if q > qex . We shall now prove that qex coincides with qc by using a strategy analogous to the one of [6], Prop. 5.1. Proposition 2.16. qc = qex . Proof. Assume that q < qex . Let f be the indicator function of the set of all configurations that are (e, Zd )-exchangeable for each e. By construction Var( f ) = 0. Furthermore, since f is left invariant by the dynamics, Lq f = 0 almost surely w.r.t. μ = μ1−q . Hence 0 is not a simple eigenvalue of Lq and q  qc . Assume now that q > qex . Consider a function f ∈ L2 (μ) with Lq f = 0, which implies Dμ ( f ) = 0. We will now show that in turn this implies that f is constant μ-a.s. For this purpose we will show that Dμ ( f ) = 0 implies    (2.10) μ |∇e f |2 = 0. e∈Zd

Then the fact that f is constant μ-a.s. immediately follows by using the well known fact that the simple symmetric exclusion process which has the unconstrained Dirichlet form in (2.10) is ergodic at any density (see e.g. [19]). We are thus left with proving (2.10). Suppose

that (2.10) does not hold, then there exists at least one bond e such that μ |∇e f |2 > 0. We will now show that this leads to a contradiction. For any η ∈ Ee we can fix once and for all an allowed configuration path Pη→ηe and let

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e e Aen = {η ∈ Ee : |P  η→η | = n} and A1 = ∅ by convention. Since q > qex , μ(Ee ) = 1. ∞ e Thus μ ∪n=2 An = 1 and



μ |∇e f |

2



=

∞   Aen

n=2

dμ(η)| f (ηe ) − f (η)|2 .

(2.11)

Writing a telescopic sum, and using Cauchy-Schwartz inequality, for any η ∈ Aen we get by the very definition of the path, | f (η ) − f (η)| = e

2

n−1 

2 f (η

(i+1)

(i)

) − f (η )

i=1

 (n − 1) = (n − 1)

n−1   i=1 n−1 

f (η(i+1) ) − f (η(i) )

2

 2 cei (η(i) ) f (η(i+1) ) − f (η(i) ) ,

i=1

where in the last step we could insert the constrained rates because the path is allowed (see in Definition 2.13). It follows that     dμ(η)| f (ηe ) − f (η)|2  C(n) μ cb |∇b f |2 , (2.12) Aen

b∈E Zd :d1 (e,b)  n+1

where the constant C(n) = max ω

max

b∈E Zd :d1 (e,b)  n+1

{η : Pη→ηe  (ω, ωb )}

takes into account the number of possible choices of configuration η such that the path d Pη→ηe crosses a given couple (ω, ωb ). One can choose C(n)  ecn for some  constant c = c(q, d). Since by assumption Dμ ( f ) = 0, it follows that μ cb |∇b f |2 = 0 for all

 b ∈ E Zd . Thus (2.11) and (2.12) lead immediately to μ |∇e f |2 = 0 and the proof of (2.10) is complete.   3. Kob-Andersen (KA) Model In this section we define the Kob-Andersen (KA) model [15] and recall some of its properties. KA is a 0-1 KCLG on Zd with influence classes Ce = {A ∪ B : (A, B) ⊂ Nx \{y} × N y \{x} with |A|, |B| ≥ j − 1}, where j is a parameter satisfying 1 < j  d. Recalling Definition 2.7 for the rates and the fact that the good event is the empty state for 0-1 KCLG, this means the following: for any two neighbouring sites x and y at least j − 1 neighbors of x belonging to Nx \{y} and j −1 neighbors of y belonging to N y \{x} should be empty in order for the exchange between x and y to be allowed. See Fig. 2 left (right) for an example of an allowed (not allowed) exchange when d = j = 2. Another way to formulate this rule is to say that when ω(x) = 1 and ω(y) = 0 (when ω(x) = 0 and ω(y) = 1), the jump of the particle

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311

Fig. 2. The bond e = (x, y) with Nx \{y} (inside the continuous line) and N y \{x} (inside the dashed line). In the left (right) figure the exchange of the value at x and y is (is not) allowed for KA model with d = 2, j = 2

from x to y (from y to x) occurs iff the particle before and after the move has at least j empty neighbors. The choice j = 1 is not considered among the KA models because it corresponds to the simple symmetric exclusion process (SSEP). The choices j > d are instead excluded because at any finite density zero is not a unique eigenvalue of their generator. In other words the model on infinite volume is uninteresting because it is never ergodic [29,30], namely qc = 1. This can be readily verified by noticing that for any j > d it is possible to construct a finite set of particles that can never be moved under the dynamics. For example for the choice d = 2, j = 3 a two by two fully occupied square can never be destroyed. Let us now discuss boundary and source choices for the model on a finite volume,  ⊂ Zd . For simplicity we discuss only the case of a rectangular region . For all 1 < j  d a choice which renders the generator ergodic with invariant measure μ corresponds to imposing fully occupied boundary conditions and unconstrained particle source on ∂− . This is the choice which is usually considered in the physics literature and which in our notation corresponds to the choice M = ∅, S = ∂−  and cx, (ω) = 1. We will make here the more constrained choice M = ∅, S = ∂ −  and cx, (ω) = 1 which is also ergodic. We will now recall some properties of KA obtained in [30]. We start by introducing the notion of framed and frameable configurations. Definition 3.1 (Framed and frameable configurations). Fix a set  ⊂ Zd and a configuration ω ∈ . We say that ω is -framed if ω(x) = 0 for any x ∈ ∂− . Let ω() be the configuration equal to ω inside  and equal to 1 outside . We say that ω is -frameable if there exist a -framed configuration σ () with at least one allowed configuration path Pω() →σ () inside . (By definition any framed configuration is also frameable). The following results, which are valid in any dimension d and for any 1 < j  d, have been derived in [30] and will play a key role in the proof of Theorem 4.1. For sake of completeness we present their proof in Appendix 9. Recall Definitions 2.14 and 2.15 of connectedness and exchangeability, then Lemma 3.2. Consider a rectangle R ⊂ Zd and a configuration ω which is R-framed. Then, for any bond e = (x, y) ∈ R, ω is (e, R)-exchangeable, namely ω ∈ EeR . Corollary 3.3. Consider a rectangle R ⊂ Zd and a couple of configurations  ω, σ which are both R-framed and have the same number of particles inside R, x∈R ω(x) =  x∈R σ (x). Then σ and η are R-connected.

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Lemma 3.4. For any , let F := {ω ∈  : ω is Q  − frameable}. Then for any q ∈ (0, 1], for any j ∈ (1, d], any ε > 0, there exists 0 = 0 (ε, q, j) such that if  ≥ 0 , then μ1−q (F ) > 1 − ε. As a consequence of the equivalence between the ergodicity and exchangeability thresholds (see Proposition 2.16) and of the above lemmas we get that for any 1 < j ≤ d the ergodicity threshold qc is zero (see [30]). We formalize the result into a theorem whose proof is also postponed to Appendix 9. Theorem 3.5. For any d ≥ 1 and any 1 < j ≤ d the ergodicity threshold qc of the KA model on Zd with parameter j verifies qc = 0. 4. Main Ideas and Results Consider the KA model in d = 2 with j = 2. We call L K A the generator on the infinite volume Z2 and L KQ LA (q) the generator on the finite cube Q L ⊂ Z2 with boundary-source choice (M, S) = (∅, ∂ − Q L ) and parameter q. Theorem 4.1. For any q > 0 there exists a constant C = C(q) > 0 and a constant c > 0 independent on q such that for any L, −2  c 1 − (1 − q)3 L 2  gap(L KQ LA (q))−1  C(q)L 2 (log L)4 . Theorem 4.2. Let Pt be the semigroup associated to L K A (q). For any q > 0 there exists a constant C = C(q) > 0 such that for any local function f , Var μ (Pt f )  C

(log t)5  f 2∞ t

∀t > 0,

where μ = μ1−q . Remark 4.3. From the variational characterization of the spectral gap it follows immediately that the result of Theorem 4.1 holds also for the less constrained choice S = ∂− Q L which is usually considered in the physics literature. Actually, the proof of Theorem 4.1 will lead to the following stronger result. For any f :  Q L → R, K ( f ) + C L(log L)2 D∂G Var Q L ( f )  C L 2 (log L)4 D Q L

− QL

K ( f ) and D G where D Q L

∂− QL L KQ LA (q).

( f ),

( f ) are respectively the Kawasaki and Glauber term of the

Dirichlet form of This implies that the result of Theorem 4.1 holds also if K A (q) is the sources are slowed down of (log L)2 /L, namely if the Glauber term of L 2 multiplied by a factor (log L) /L. Remark 4.4. With much more effort, the proof of Theorems 4.1 and 4.2 can be generalized in any dimension d ≥ 3 for all j, 1 < j  d. The bounds for the spectral gap remain of order L 2 (but with higher correction in the log terms). For the decay to equilibrium we α get Var μ (Pt f )  C (logt t)  f 2∞ for some constant α(d). The latter result is probably not optimal: we expect the decay to be of order C/t d/2 . Remark 4.5. Any 0-1 KCLG which is dominated by KA-2f verifies the same bounds as in Theorem 4.1. The upper (lower) bound follows from Corollary 2.12 and comparison with KA-2f (with SSEP).

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313

Sketch of the main ideas. In order to explain our approach let us quickly review a simple route to prove Theorem 4.1 in the context of the SSEP. One first establishes a Poincaré inequality w.r.t. an auxiliary Dirichlet form with pure Glauber moves and reversible w.r.t. μ (a trivial fact since μ is a product measure) and then one transfers the Glauber moves from the bulk to the boundary using the exchange moves of the SSEP along apriori chosen geometrical paths (the so-called “path argument” see e.g. [20,23]). That gives almost immediately the diffusive scaling of the spectral gap. More or less the same technique can be applied to “non-cooperative” models. A completely different scenario is presented when considering “cooperative models” like the Kob-Andersen model. Indeed the above Poincaré inequality is now completely useless because we are not guaranteed that from a site x ∈  where a Glauber move is performed we can reach the boundary by a sequence of allowed exchange moves. In other words nobody guarantees that in the current configuration the holes are “cooperating” in such a way that the new particle created at x can be moved to the boundary. It is precisely this loss of uniformity that requires new ideas that, to the best of our knowledge, were completely absent before our work. The way out and a major novelty of our approach is to prove a modified Glaubertype Poincaré inequality in which the creation/annihilation move in the bulk occurs only if the holes in the current configuration are cooperating in a way to be able to move to the boundary the extra hole created at x. This forces us to consider an auxiliary Glauber process with very long range (essentially from the inner bulk to the boundary) constraints and now the existence of the corresponding Poincaré inequality is highly non trivial. One could naively think that the long range constraint should be of the form “there exists a path of holes from x to the boundary along which a particle can move with legal exchanges”. However, when the density of particles is high (and therefore the density of holes is small) the probability of such an event is exponentially small in the distance from the boundary, and most of the times the constraint will not be satisfied. That brings up the second set of new ideas, namely to consider a “renormalized Kob-Andersen model” with a much richer structure than just particle/hole and for which the “effective holes” have a high density (see model AKG below). For the new model we carry out the program just illustrated above and then finally we go back to the original Kob-Andersen model via standard comparison techniques. 5. Renormalization and Long Range Constraints In this section we define two auxiliary models: one with purely Glauber dynamics and long range constraints (AGL) and another one with Kawasaki dynamics plus Glauber sources (AKG). Thus AGL belongs to the class of kinetically constrained spin models (KCSM) while AKG is a kinetically constrained lattice gas (KCLG). Both models are defined with an arbitrary on-site probability space (W, ν) and good event G ⊂ W , at variance with the specific choice W = (0, 1), ν(1) = p and G = 0 of the KA model. We will establish the positivity of the spectral gap for AGL (Theorem 5.5). This will be a key ingredient to prove both the lower bound on the spectral gap (Theorem 4.1) and the polynomial decay to equilibrium (Theorem 4.2) for KA. By combining Theorem 5.5 with proper path arguments we will deduce a 1/L 2 lower bound for the spectral gap of AKG (Theorem 5.6). It is by using the latter result and a suitable renormalization procedure that we will cast KA into AKG and deduce the desired 1/L 2 lower bound for the spectral gap of the KA model (Theorem 4.1). The peculiar choice of the constraints for both the auxiliary models is motivated by this final renormalization procedure and

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x

y

x

x

y

x

y

x

y

y

x

x

y

y

x

y

Fig. 3. The sets A1 , . . . , A8 which belong to the influence class Ce for AKG model. We depict the case e = x, x + e1

should be properly modified when one ultimately wishes to study KCLG which are different from KA. Let us start by defining the influence classes which characterize the Kawasaki dynamics of AKG. We set Ce := {A1 , A2 , A3 , A4 , A5 , A6 , A7 , A8 },

(5.1)

where Ai are defined as follows (see Fig. 3): A1 A3 A5 A7

= = = =

(x (x (x (x

+ e2 , x + 2e1 , x + e1 + e2 ), A2 = (x − e2 , x + 2e1 , x + e1 + e2 ), + e2 , x + 2e1 , x + e1 − e2 ), A4 = (x − e2 , x + 2e1 , x + e1 − e2 ), − e1 , x + e2 , x + e1 + e2 ), A6 = (x − e1 , x + e2 , x + e1 − e2 ), − e1 , x − e2 , x + e1 + e2 ), A8 = (x − e1 , x − e2 , x + e1 − e2 ),

if e = x, x + e1 and A1 A3 A5 A7

= = = =

(x (x (x (x

+ e1 , x + 2e2 , x + e1 + e2 ), A2 = (x − e1 , x + 2e2 , x + e1 + e2 ), + e1 , x + 2e2 , x + e2 − e1 ), A4 = (x − e1 , x + 2e2 , x + e2 − e1 ), − e2 , x + e1 , x + e1 + e2 ), A6 = (x − e2 , x + e1 , x + e2 − e1 ), − e2 , x − e1 , x + e1 + e2 ), A8 = (x − e2 , x − e1 , x + e2 − e1 ),

if e = x, x + e2 . The following results follow immediately from the above definitions: Remark 5.1. The influence classes of AKG dominate (see Definition 2.11) those of the KA model with d = j = 2. Indeed for all i there exists A ⊂ Nx \y and B ⊂ N y \x s.t. Ai = A ∪ B and either |A| = 1 and |B| = 2 or |A| = 2 and |B| = 1. Thus Ai also belongs to the influence class of KA and the inequality among the rates of KA and AKG required to have domination immediately follows from Definition 2.7. Fix a configuration ω, then a geometric path which is good (see Definition 2.6) is also allowed for ω with the choice of AKG constraints (see Definition 2.13).

Kinetically Constrained Lattice Gases

315 akg

For any  ⊂ Z2 the finite volume generator of AKG, L , is then defined as in (2.3). M (ω) The corresponding Kawasaki term is defined as in (2.4) with exchange rates ce, ∗ defined by (2.6) with boundary set M = ∂ +  and ce (ω) as in Definition 2.7 with influence classes (5.1). The Glauber term is defined by (2.5) with source set S = ∂ −  and constraints  1 if (ω · τ ) (z) ∈ G for any z ∈ K∗x cx, (ω) = , (5.2) 0 otherwise ∗

∗

where τ ∈  is any configuration such that τ (z) ∈ G (τ (z) ∈ G) if z ∈ ∂ +  (z ∈ ∂ + ). The finite volume generator of AGL is instead of purely Glauber type and is defined by the following action on local functions:  agl N L,N f (ω) = cx, (ω) (νx ( f ) − f (ω)) , x∈ N are defined as where the constraints cx, ⎧ 1 if |Gx,N , (ω)|  1 ⎪ ⎪ ⎪ ⎪ ⎪ 0 otherwise ⎨ N cx, (ω) = ⎪ ⎪ ⎪ 1 if (ω · τ ) (z) ∈ G for any z ∈ K∗x ⎪ ⎪ ⎩ 0 otherwise

if x ∈ / ∂ − , (5.3) if x ∈ ∂ − , ∗

∗

where τ ∈  is any configuration such that τ (z) ∈ G (τ (z) ∈ G) if z ∈ ∂ +  (z ∈ ∂ + ) and Gx,N , is the set of all geometric paths which are allowed for ω with the AKG con1 2 straints (see Definition 2.13) and go from x to the East (∂ − ) or North (∂ − ) interior boundary of  never leaving a tube of width N centered at x. In formulas Gx,N , (ω) :=

2  

N {γx,y ⊂ G : γx,y is allowed for AKG; γx,y ⊂ Tx,i ()},

i=1 y∈∂ i  −

(5.4) where G is the overall set of geometric paths. Recalling Definition 2.5, the following property for the set of geometric paths can be immediately verified: G,\x

Remark 5.2. Let ω, ω ∈  be such that ω ⇐⇒ ω . Then the following holds for any x ∈  and N > 0: Gx,,N (ω) = Gx,,N (ω ). agl

The Dirichlet form associated to L,N is given by    N D,N ( f ) = μ cx, Var x ( f ) ,

f :  → R.

x∈

Recall from Sect. 2.1 that μ is the product measure μ := results hold:



x∈ νx .

(5.5) The following

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Claim 5.3. For any  ⊂ Z2 and any choice of W, ν and G, the generator L,N is reversible w.r.t. μ and it is ergodic. N (ω) does not depend on Proof. Reversibility follows immediately from the fact that cx, the value of ω(x). Ergodicity follows by noticing that the constraint is verified on all ˜ := {x ∈  : K∗x ∩  = ∅} whose existence can be proved by induction because sites   is finite. We can thus make the configuration of these sites good. Then we can render ˜ : K∗x ∩ (\) ˜ = ∅} good since they have the constraint verified. all sites {x ∈ (\)   akg

Claim 5.4. For any  ⊂ Z2 and any choice of W, ν and G, the generator L reversible w.r.t. μ and ergodic on  .

is

Proof. Reversibility follows, as for any other KCLG, from the independence of cx on ωx and from the independence of cx,y on ωx and ω y . In order to prove ergodicity it is sufficient to show that for any η ∈  there exists an allowed path which connects η to σ , where σ is a completely good configuration, σ (x) ∈ G for all x ∈ . In order ∗ to construct the path we start by using the source terms to make good all sites in ∂ − . ∗ ˜ := \∂ −  is finite, there should exist at least one site x ∈  ˜ such that: Then, since  ∗ ∗ ∗ ∗ ˜ = ∅, thus Kx ⊂ (∂ +  ∪ ∂ − ). Thus we can exchange the occupation variable Kx ∩  ∗ in x with the (good) occupation variable of a site in ∂ − . The latter site can then be restored to good by using the sources. The procedure may then be iterated until making the whole configuration good.   Recall from Sect. 2.1 that ρ is the probability of the good event G ⊂ W, ρ := ν(G). Our main results concerning the auxiliary models are the following: Theorem 5.5. There exist ρ1 ∈ (0, 1) and A > 0 independent of W and ν such that if ρ > ρ1 , then for any rectangle  = [0, L 1 ] × [0, L 2 ], agl

gap(L,N ) ≥

1 , 2

provided that N ≥ A(log(max(L 1 , L 2 ))2 . Theorem 5.6. There exists ρ0 ∈ (0, 1) independent of W and ν and a constant C = C(|W |, ν) such that if ρ ≥ ρ0 , then for any cube Q L , gap(L Q L )−1  C L 2 (log L)4 . akg

In [9] we have devised a technique which allows to prove the positivity of the spectral gap for a large class of KCSM. However AGL does not belong to this class because its constraints are not local (while the proof in [9] relies on the hypothesis that the influence classes of the KCSM have finite range). Some additional efforts and a proper extension of the technique in [9] is thus required to establish Theorem 5.5. We postpone this rather technical proof to Sect. 7 and proceed with the proof of the result for AKG. Proof of Theorem 5.6. We prove the theorem for ρ0 = ρ1 , with ρ1 defined by Theorem 5.5. Recalling the definition of spectral gap (2.9), the expression (5.5) for

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the Dirichlet form of AGL and using Theorem 5.5, for any ρ > ρ1 and f :  Q L → R, the following holds:    N Var Q L ( f )  2 μ Q L cx,Q Var ( f ) , (5.6) x L x∈Q L N where cx,Q have been defined in (5.3), N := A(log L)2 and A and ρ1 are those defined L in Theorem 5.5. Our aim is to bound the r.h.s. with the Dirichlet form of AKG model. This will be achieved via proper path arguments. We start by rewriting Var x ( f ) as

Var x ( f )(ω) =

1 2



μ(w)μ(w )( f (ω Q L \x · wx ) − f (ω Q L \x · wx ))2 .

(5.7)

w,w ∈W

N (ω) = 1 we choose once and for all a path in Then for all x and ω such that cx,Q L Gx,Q L ,N (5.4) which we will call γ (x, ω). We make the latter choice in order that G,L\x

γ (x, ω) = γ (x, ω ) for all ω, ω such that ω ⇐⇒ ω , which is possible thanks to Remark 5.2. If we set γ (x, ω) = (x (1) , . . . , x (n) ) from the above definition it follows N (Q )∪T N (Q )) immediately that x (1) = x, x (n) ∈ ∂ − Q L ∩TxN (Q L ) := ∂ − Q L ∩(Tx,1 L L x,2 2

and n(x, ω) < L(2 A(log L) + 1). For any ω ∈  Q L and w, w ∈ W , we are now ready

x (ω) from ω Q L \x · w x to ω Q L \x · w x as follows: to define a configuration path Pw→w (1) x := (ω Pw→w , . . . , ω(2n) )

(5.8)

with ω(1) = ω Q L \x · wx ; ω(i+1) = Tei ω(i) , with ei := (x˜ (i) , x˜ (i+1) ) := (x (i) , x (i+1) ) for all i ∈ {1, . . . , n − 1}; ω(n) = ω(n−1) · wx (n) ; ω(i+1) = Tei ω(i) , with ei := Q \x (n) L

(x˜ (i) , x˜ (i+1) ) := (x (2n+1−i) , x (2n+2−i) ) for all i ∈ {n, . . . , 2n − 1}. It is immediate to verify that ω(2n) = ω Q L \x · wx . Note that, even if we do not write it for simplicity of notation, ω(i) depends on w, w , x, ω. Recall Definition 2.13. The following properties N can be immediately verified by using the definition (5.3) for cx,Q (ω). L N (ω) = 1 and g ∈ G: Claim 5.7. For any x, ω such that cx,Q L x (5.8) is allowed for AKG; (i) the path Pw→w

G,Q L \x

(ii) for i ∈ {1, . . . , 2n}\n + 1, ω(i) Q L \{x˜i ,x˜i+1 } · gx˜ (i) · gx˜ (i+1) ⇐⇒ ω; (n+1) (n) L \x

(iii) ω Q

G,Q L \x

· gx (n) ⇐⇒ ω.

x (ω), via a teleThus, recalling Definition 2.13 for the meaning of (σ, σ e ) ∈ Pw→w scopic sum and Cauchy-Schwartz inequality we get for any ω,

 2 N (ω) f (ω Q L \x · wx ) − f (ω Q L \x · wx ) cx,Q L  2L(2 A(log L)2 + 1)

2n 

 2 cei ,Q L (ω(i) ) ∇ei f (ω(i) )

i=1

i=n (n)

+ 2cx (n) ,Q L (ω(n) )( f (ω Q L \y · w y ) − f (ω(n) ))2 ,

(5.9)

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where ce,Q L and cx,Q L are the Kawasaki and source Glauber rates for the AKG model. Then, by plugging (5.7) and (5.9) into (5.6) we upper bound Var Q L ( f ) by ⎡ |W |2 max

w,w ∈W

  x∈Q L ω

⎢ (n) (n)

(n) 2 μ Q L (ω) ⎢ ⎣2cx (n) ,Q L (ω )( f (ω Q L \y · w y ) − f (ω )) ⎤

+

2n 

 2 ⎥ 2L(2 A(log L)2 + 1)cei ,Q L (ω(i) ) ∇ei f (ω(i) ) ⎥ ⎦.

(5.10)

i=1

i=n (i) satisfies x any ω By construction of the path Pw→w

μ Q L (ω) μ(w) .  C := max w,w ∈W μ(w ) μ Q L (ω(i) ) Hence, inverting the summations, (5.10) is bounded above by   e K x DQ ( f ) |W |2 C L 2 (2 A(log L)2 + 1)2 max  ω : (σ, σ ) ∈ P L w→w w,w ,x,σ,e   (n) (1) (n) + 2|W |2 L(2 A(log L)2 + 1) max  ω : ω = σ ; x = x; x = z D∂G

w,w ,x,σ,z

 8C|W |3 A2 L 2 (log L)4 D Q L ( f ),

− QL

(f)

(5.11)

K ( f ), D G where D Q L ( f ) is the Dirichlet form for AKG and D Q ( f ) are its Kawasaki L ∂− QL

and Glauber parts. In order to derive the last inequality we have bounded the number x . To perform this bound of configurations ω such that a chosen σ, σ e belongs to Pw→w we used as a key ingredient the fact that from the knowledge of (σ, σ e ) we can reconstruct ω modulo the configuration in x (or completely if (σ, σ e ) = (ω(i) , ω(i+1) ) with i  n − 1) and from the knowledge of ω(n) and x (n) we can reconstruct ω. This in turn is true thanks to properties (ii) and (iii) of Claim 5.7. The proof is then completed by combining the variational characterization of the spectral gap with the upper bound (5.11) for Var Q L ( f ).   6. Spectral Gap of KA Model: Proof of Theorem 4.1 Since KA dominates the Symmetric Simple Exclusion Process (SSEP), a lower bound on the inverse of the spectral gap as L 2 uniform on q follows from Claim 2.12 and from the standard results for SSEP, see e.g. [6]. In order to obtain the stronger lower bound of Theorem 4.1 which guarantees that, at variance with SSEP, the spectral gap on scale L −2 is not bounded away from zero at all density and instead it vanishes as q → 0 we πx πx consider the test function x∈Q L cos( L−1 ) cos( L−1 )ηx . The term (1−(1−q)3 )2 comes from the presence of the kinetic constraints: in order for the exchange of the occupation variables at x and y to be allowed there should be at least one empty site among the three nearest neighbours of x different from y and at least one empty site among the three neighbors of y different from x. We will now prove the upper bound by using the 1/L 2 bound for the spectral gap of AKG (Theorem 5.6) combined with a renormalization technique similar to the one we used in [9].

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Proof of Theorem 4.1. Let ρ0 be the threshold density defined in Theorem 5.6. Thanks to Lemma 3.4 we can choose an integer length scale  such that μ(F ) > ρ0 , where F is the set of configurations which are Q  -frameable. For any z ∈ Z2 we define Q z ⊂ Z2 as Q z := Q  + z. Then we define the renormalized lattice Z2 () :=  Z2 . Given L˜ s.t. ˜ is integer we also define the renormalized cube associated to Q ˜ ⊂ Z2 as L := L/ L Q˜ L := Z2 () ∩ Q L˜ . Note that Q˜ L contains L × L sites and that ∪x∈ Q˜ L Q x = Q L˜ . Consider the probability space W = {0, 1} Q  equipped with ν = μ Q  . The two prob 2 2 ability spaces ({0, 1}Z , μ) and (W Z () , x∈Z2 () νx ) coincide. Furthermore we have  ˜ μ Q L˜ = ν Q˜ L , where ν A = x∈A∩Z2 () νx . Thus if we consider AKG on Q L with W = {0, 1} Q  , ν = μ Q  and good event F , by Theorem 5.6 there exists a constant C = C(, q) such that Var Q L˜ ( f )  C L 2 (log L)4



  μ Q L˜ c˜e ( f ◦ Te − f )2

e∈E Q L

+ C L 2 (log L)4



 μ Q L˜ c˜x Var Q x ( f ) ,

(6.1) (6.2)

x∈∂ − Q L

where for any bond e = (x, y) ∈ E Q L , we let Te = T(x,y) :  →  be the operator that exchanges the configuration inside Q x and Q y , namely ⎧ ⎪ ⎨ ω(z) Qx Q y Te ω(z) = ω (z) := ω(z + y − x) ⎪ ⎩ ω(z + x − y)

if z ∈ / Qx ∪ Q y if z ∈ Q x if z ∈ Q y

and c˜e (ω) and c˜x (ω) are defined as follows. c˜e is the indicator function of the event that there exists A ∈ Ce with Ce defined in (5.1) s.t. Q z ∈ F for any z ∈ A ∩ Q L . Instead c˜x is the indicator function of the event that Q z ∈ F for any z ∈ K∗x ∩ Q L . The proof of Theorem 4.1 is then completed by the following key Lemma 6.1 and explicit counting (left to the reader).   

For any x ∈ ∂ − Q L let E(x) := E Q L˜ ∩ E Q x ∪z∈K∗x E Q z and for any bond e = 

(x, y), E(e) := E Q L˜ ∩ E Q x ∪ E Q y ∪z∈∂+ (x,y) E Q z . In order to avoid confusion we K A the rates of KA. call here ce, Lemma 6.1. There exists a constant C = C (, q) > 0 such that: (i) for any bond e = (x, y) ∈ E Q L ,      A 2

f) μ Q L˜ " ce ( f ◦ Te − f )2  C μ Q L˜ ceK ,Q (∇ e ˜ e ∈E(e)

+ C

L



z∈(Q x ∪Q y )∩∂

(with possibly (Q x ∪ Q y ) ∩ ∂ − Q L˜ = ∅);

μ Q L˜ (Var z ( f )) − Q L˜

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(ii) for any x ∈ ∂ − Q L ,   

 A 2

f) μ Q L˜ " cx Var Q x ( f )  C μ Q L˜ ceK ,Q (∇ e ˜ L

e ∈E(x)



+ C

μ Q L˜ (Var z ( f )) .

z∈Q x ∩∂ − Q L˜

Proof. The proof is based on path techniques. In all the proof C will denote a positive constant that depends on , q but never on L, and that might change from line to line. Finally, unless explicitly stated, an allowed path means allowed w.r.t. KA rates. (i) Fix a bond e = (x, y) ∈ E Q L and a configuration ω such that " ce (ω) = 1 (otherwise the result trivially holds). Since " ce (ω) = 1 there exists A ⊂ Ce s.t. for any z ∈ A ∩ Q L , ω Q z ∈ F holds. We analyze separately the case A ⊂ Q L (a) and A ⊂ Q L (b). (a) We let y = x + e1 and A = A1 = x + e2 , x + e1 + e2 , x +2e1 (the other cases can be treated analogously). By subsequently applying Claim 9.2 we can construct an allowed path ω, . . . ω(M) inside Q x ∪ Q y ∪z∈A Q z such that ω(M) = Te (ω) and M(ω)  C() uniformly in ω. Using a telescopic sum and the Cauchy-Schwartz inequality , we get  M−1 2     2 (m+1) (m) ce ( f ◦ Te − f ) = μ Q L˜ (ω)" ce (ω) f (ω ) − f (ω ) μ Q L˜ " ω

C

 ω

C

 ω

C

μ Q L˜ (ω)

M−1 

m=1



cem ,Q L˜ (ω(m) ) f (ω(m+1) ) − f (ω(m) )

2

m=1

μ Q L˜ (ω)



e ∈E(e)

 σ,e

ce ,Q L˜ (σ ) (∇e f )2 (σ )1{(σ,e ):(σ,σ e )∈Pω,T

e (ω) }

  μ Q L˜ ce ,Q L˜ (∇e f )2 ,

where as usual, (σ, σ e ) ∈ Pω,Te (ω) means that there exists m such that σ = ω(m)

and σ e = ω(m+1) . In the last line we inverted the summations, used the fact that any σ ∈ Pω,Te (ω) satisfies μ Q L˜ (σ ) = μ Q L˜ (ω) and differs from ω on at most 92 sites. (b) Assume that y = x + e2 , A = A1 = x + e1 , x + e1 + e2 , x +2e2 and that x + e1 , y + e1 ∈ Q L (i.e. Q x+e1 , Q y+e1 ∈ Q L˜ ) and x + 2e2 ∈ Q L (i.e. Q x+e2 ∈ Q L˜ ) as in Fig. 4 (the other cases can be treated analogously). Thanks to the presence of sources on ∂ − Q L˜ 1

we can create zeros on ∂ − (Q x ∪ Q y ). Indeed, if z (1) , . . . , z (m 1 ) denote the sites inside 1

∂ − (Q x ∪ Q y ) for which ω(z (i) ) = 1, i = 1, . . . , m 1 , enumerated from top to bottom, the z (i)

path (ω(1) , . . . , ω(m 1 ) ) with ω(1) = ω and ω(m i+1 ) = ω(m i ) is allowed. Furthermore 1

1

ω(m 1 ) (u) = 0 for all u ∈ ∂ − (Q x ∪Q y ) and ω(m 1 ) (u) = ω(u) for u ∈ ∂ − (Q x ∪Q y ). Since (m 1 ) (m 1 ) , . . . , ω(m 2 ) ) ω Q x+2e2 ∈ F (and thus ω x+2 e2 ∈ F ), there exists an allowed path (ω Q

inside Q x+2e2 such that ω(m 2 ) is Q x+2e2 -framed (see Fig. 4). Claim b guarantees the existence of a sequence of allowed exchanges (ω(m 2 ) , . . . , ω(m 3 ) ) inside Q x+2e2 which bring the empty upper line of this frame adjacent to the bottom empty line. Then it is easy to verify that we can rigidly shift the double empty line downwards thanks to the presence of 1 the empty sites on ∂ − (Q x ∪Q y ). Therefore, for any chosen bond e = x, x +e1 ∈ Q x ∪Q y

Kinetically Constrained Lattice Gases

321

Qy

Qy

Qx

Qx

Fig. 4. Proof of Lemma 6.1(i), case (b). The configurations ω(m 1 ) on the left and ω(m 2 ) on the right. The dashed line corresponds to the rectangle R

we can shift the double empty line till the position x ·e2 +1 and then perform the exchange. The case e = x, x + e2 can be dealt by first creating a second empty column adjacent to 1 the one on ∂ − (Q x ∪ Q y ) (via exchanges plus source terms) and then shifting horizontally this double empty column till the position x · e1 + 1. In conclusion, since we can perform any internal exchange in Q x ∪ Q y , we can construct a path (ω(m 3 ) , . . . , ω(m 4 ) ) with ω(m 4 ) = Te (ω(m 3 ) ). As before, we can now reconstruct the initial configuration ω 1 outside Q x ∪ Q y and then also on ∂ − (Q x ∪ Q y ) by using the sources again. Thus we have constructed an allowed path Pω,Te ω = (ω(1) , . . . , ω(M) ) inside Q x ∪ Q y ∪ Q x+2e2 with M(ω)  C() uniformly in ω. Using that μ Q L (ω)  Cμ Q L (ω(m) ) for any m, the same kind of computation as before (telescopic sum, Cauchy-Schwartz inequality, inverting the summations, explicit counting...) leads to  M−1 2     2 (m+1) (m) μ Q L˜ " ce ( f ◦ Te − f ) = μ Q L˜ (ω)" ce (ω) f (ω ) − f (ω ) C

 e ∈E(e)

ω

μ Q L˜

  ce ,Q L˜ (∇e f )2 + C

m=1



μ Q L˜ (Var z ( f )) .

1 z∈∂ − (Q x ∪Q y )∩∂ − Q L

(ii) Let x ∈ ∂ − Q L and ω be such that " cx (ω) = 1. Assume that x + e1 ∈ / Q L and z = x + e2 ∈ Q L as in Fig. 5 (the other cases are similar). Using the Poincaré inequality for the unconstrained Glauber dynamics inside Q x (or the tensorisation property, see e.g. [2]) leads to    Var Q x ( f )  pq μ Q x (∇ y f )2 . y∈Q x

Thus we have to estimate terms of the form f (ω y ) − f (ω). This will be done by constructing a proper allowed path which is depicted in Fig. 5. Assume that ω(y) = 1 (the case ω(y) = 0 can be treated analogously). Since " cx (ω) = 1, Q z is frameable.

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Fig. 5. From left to right, the configurations ω(m 1 ) , ω(m 2 ) , ω(m 3 ) , ω(m 3 +1) and ω(m 3 +2) . The dotted line corresponds to the rectangle R

Hence, there exists an allowed path (ω(1) , . . . , ω(m 1 ) ) inside Q z s.t. ω(1) = ω and ω(m 1 ) 1 1 is Q z -framed. Thanks to the sources on ∂ − Q x , we can create zeros on ∂ − Q x . Indeed, 1

if z (m 1 +1) , . . . , z (m 2 ) denote the sites inside ∂ − Q x for which ω(z (m) ) = 1, enumerated z (i)

from top to bottom, the path (ω(m 1 ) , . . . , ω(m 2 ) ) with ω(m 1 +i) = ω(m 1 ) is allowed. 1

1

Furthermore ω(m 2 ) (z) = 0 for all z ∈ ∂ − Q x and ω(m 2 ) (z) = ω(z) for z ∈ ∂ − (Q x ∪ Q y ). 2 Qx ∪ ∂+ Qx

and let t be the unique site such that Kt∗ ⊂ ∂ − R and t1 = t + e1 . Set R := Suppose ω(t) = 0 (the case ω(t) = 1 can be treated analogously). Then we can shift the top empty line of the frame of Q z near the bottom one and, by using this double empty 1 line plus the empty column on ∂ − Q x we can perform any exchange inside Q x analogously to what we did in point (i). In particular we can construct a path ω(m 2 ) , . . . ω(m 3 ) which moves the particle from y to t and then perform the exchange on t, t1 and use the source to force on t1 an empty site. In other words the path (ω(m 3 ) , ω(m 3 +1) , ω(m 3 +2) ) t,t

(m +1) with ω(m 3 +1) = ω(m 3 ) 1 and ω(m 3 +2) = ω Q ˜3\t1 · 0t1 is allowed (see Fig. 5). Thus we L

1

have reached a configuration which corresponds to ω y on all site in Q x \∂ − Q x . Then we can reconstruct the configuration ω y following the inverse of the path (ω, ω(m 2 ) ). Thus we have shown the existence of an allowed path Pω,ω y = (ω(1) , . . . , ω(M) ) with M(ω)  C() uniformly in ω. Using the same routine arguments (Cauchy-Schwartz inequality, telescopic sum,…) as in point (i) leads to the expected result.  

7. Spectral Gap of AGL: Proof of Theorem 5.5 In this section we prove Theorem 5.5 which establishes the positivity of the spectral gap for the auxiliary model AGL. This result has in turn been used as a key ingredient for the proof of the 1/L 2 lower bound for the spectral gap of KA and will be also used in Sect. 8 to prove the polynomial decay to equilibrium. The main tool here is an extension of the bisection-constrained method we introduced in [9] which we have here properly modified to account for the long range constraints of AGL. As a result the proof is quite technical and lengthy due to easy but cumbersome geometric results. These are

Kinetically Constrained Lattice Gases

323

necessary to establish the existence of the paths which guarantee that the long range constraints of AGL are satisfied. We start by a monotonicity remark ˜ ⊂ Z2 and N , N > 0 with N  N and  ⊂ . ˜ For any x ∈  Remark 7.1. Fix ,  and for all ω: N (ω)  c N (ω); (i) cx, x, ˜ then c N (ω)  c N (ω). (ii) if ∂ −  ⊂ ∂ − , x, ˜ x,

Proof of Theorem 5.5. In what follows we will drop the superscript agl from the generator. Thanks to the monotonicity of the spectral gap established by Lemma 2.12 and to the property of the rates in Remark 7.1 (i), it is enough to prove the result when N = A(log(max(L 1 , L 2 ))2 . We start by recalling a simple geometric result of [5] which we will use. Let lk := (3/2)k/2 , and let Fk be the set of all rectangles  ⊂ Z2 which, modulo translations and permutations of the coordinates, are contained in [0, lk+1 ]×[0, lk+2 ]. The main property of Fk is that each rectangle in Fk \Fk−1 can be obtained as a “slightly overlapping union” of two rectangles in Fk−1 . More precisely we have: Lemma 7.2 ([5], Prop. 3.2). For all k ∈ Z+ , for all  ∈ Fk \Fk−1 there exists a finite 1/3 (i) sk 1√ sequence {(i) 1 , 2 }i=1 in Fk−1 , where sk := lk , such that, letting δk := 8 lk − 2, (i)

(i)

(i)  = 1 ∪ 2 , (i) (ii)  d(\(i) 1 , \  2 ) ≥ δk ,  ( j) ( j) (i) ∩  = ∅, if i = j. ∩  ∩  (iii) (i) 1 2 1 2 Let k¯ be such that  ∈ Fk¯ \Fk−1 = lk¯ , thus N = ¯ . Then max(L 1 , L 2 )  lk−1+1 ¯ 2 2 A(log(max(L 1 , L 2 )))  Nk¯ , where Nk := A(log lk ) . By using again the monotonicity properties of Lemma 2.12 and Remark 7.1 (i) we immediately get gap(L,N )−1  gap(L,Nk¯ )−1  γk¯  sup γk , k

where we define

 γk :=

sup

k =1,...,k

 sup

∈Fk \Fk −1

gap(L,Nk )−1 .

(7.1)

Therefore, to prove the theorem, it is enough to show that there exist ρ1 ∈ (0, 1) and A > 0 independent of W and ν such that for any ρ > ρ1 , sup γk  2.

(7.2)

k

The strategy to prove (7.2) will be to establish a proper iterative inequality between γk and γk−1 . Let us fix k,  ∈ Fk \Fk−1 , and let  = 1 ∪ 2 with 1 , 2 ∈ Fk−1 satisfying the properties described in Lemma 7.2 above. Without loss of generality we can assume that the faces of 1 and of 2 parallel to e1 lay on the faces of  and that, along that direction, 1 comes before 2 (see Fig. 6). Set " I ≡ 1 ∩ 2 and write, for I in the first concreteness, " I = [a˜ 1 , b˜1 ] × [a˜ 2 , b˜2 ]. Lemma 7.2 implies that the width of " ˜ direction, b1 − a˜ 1 , is at least δk . Set B1 = \2 , B2 = 2 and I = [a1 , b1 ] × [a2 , b2 ] with a1 = a˜ 1 , b1 = a˜ 1 + A2 log(3/2) log(lk ), a2 = a˜ 2 and b2 = b˜2 . Notice that our choice implies b1 − a1 ≤ Nk − Nk−1 . We also assume that k is sufficiently large so that δk  b1 − a1 . Then the following geometric properties can be immediately verified.

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N Fig. 6. We depict the regions 1 , 2 = B2 ,  = 1 ∪ 2 , I˜ = 1 ∩ 2 , I ⊂ I˜, and the cylinder Tx,1k () m R (here m = 4). (inside the dotted-dashed line). The set 2 is divided into rectangles of width c j , 2 = ∪i=1 j The dashed lines inside the R j ’s stand for the paths which are good left-right crossings and thus guarantee m that ω ∈ ∩i=1 Ri . The bold dashed line inside I is a good top-bottom crossing which guarantees ω ∈ I

Claim 7.3. (i) I ⊂ " I; Nk (ii) If x ∈ I , then I ⊂ Tx,2 ; N

N

Nk . (iii) If x is such that I ∩ Tx,2k−1 = ∅, then I ∪ Tx,2k−1 ⊂ Tx,2

We will now define a constrained block dynamics on  with blocks B1 and B2 and prove that it has a positive spectral gap. Then from its Dirichlet form we will reconstruct the Dirichlet form of AGL (5.5) and establish the desired recursive inequality between γk and γk−1 . To this purpose, in analogy with the strategy adopted in [9], we have to define a proper good event on the block B2 which should occur in order to allow refreshing of the configuration on B1 . Recall Definition 2.6 and define the event I := {ω : ω has a good top-bottom crossing in I }. In analogy with what is done in [14, see Proof of Lemma (11.73)], we define the following natural partial order on the set of top-bottom crossing paths in I . We say that γ is to the right of γ if it lies inside the connected (with respect to d1 distance) region of I which stays to the right of γ . For any ω ∈ I we can then define the geometric set ω which is its right-most good top-bottom crossing and let ω be the corresponding double-path (see Fig. 8). Finally we set C := { ⊂ B2 : ∃ ω s.t. ω = }. By geometrical considerations we have the following. Claim 7.4. For all ω ∈ I, (i) there exists a unique right-most good top-bottom crossing ω of ω in I ; (ii) for any  ∈ C the event {ω : ω = } does not depend on the values of ω to the left of .

Kinetically Constrained Lattice Gases

325

Fig. 7. Left: An example of top bottom crossing with k-bounded oscillation inside the stripe I. Right: The bold dashed line is a good top-bottom crossing γ which does not have k-bounded oscillations: at h there is an oscillation which goes beyond h − A/2 log(3/2) log lk . The dotted-dashed line is a good left-right bottom crossing of the rectangle I j . Note that γ cannot be the rightmost good top-bottom crossing, γ  = ω . Indeed a good top-bottom crossing to the right of γ can be constructed by using γ and the part of the good left-right crossing from site A to B

Next we need to introduce the notion of paths which “do not oscillate too much” in the vertical direction, namely Definition 7.5 (Path with k-bounded oscillation). Consider a geometric path γx y = (x (1) = x, . . . , x (n) = y). Let (1)

(n)

(1)

(n)

x 1 = min{x1 , . . . , x1 }, x 1 = max{x1 , . . . , x1 }, x 2 = min{x2(1) , . . . , x2(n) }, x 2 = max{x2(1) , . . . , x2(n) }. We say that γx y has k-bounded oscillations if for all h ∈ [x 2 , x 2 ] the following h-condition holds. Let 1 ≤ i 1 < i 2 < · · · < i m ≤ n be the indexes of the points of γx,y with height h, namely {x (i1 ) , x (i2 ) , . . . , x (im ) } := γx y ∩ ([x 1 , x 1 ] × {h}). Then the h-condition (depicted in Fig. 7) requires that for all i = i 1 , i 1 + 1, i 1 + 2, . . . , i m , x (i) ∈ [x 1 , x 1 ] × [h − A/2 log(3/2) log lk , h + A/2 log(3/2) log lk ]. With this notation we can define the event J := {ω ∈ I : ω has k-bounded oscillations} and the geometric set C" ⊂ C as C" := { ⊂ B2 : ∃ ω s.t. ω =  and ω has k-bounded oscillations}. Lemma 7.2 guarantees that B2 = 2 ∈ Fk−1 . Thus there exist integers a, b which are bounded from above by lk+1 such that 2 is a translated copy of [0, a] × [0, b]. If b  Nk−1 /2 we decompose 2 into m disjoint rectangles 2 = ∪mj=1 R j (see Fig. 6) with each R j being a translation of [0, a]×[0, c j ], where c j satisfies Nk−1  2c j  2Nk−1 −1 for any j = 1, . . . , m and m verifies m  (2lk+1 /Nk−1 ) (the bounds Nk−1 /2 < b  lk+1 guarantee that we can perform such a procedure). Thanks to the bounds on c j the following property can then be immediately verified:

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Claim 7.6. For any x ∈  there exists j ∈ 1, . . . , m such that R j ⊂ Tx,1k−1 (2 ). We are now ready to define the good event on the block B2 which will enter in the definition of the block dynamics. Definition 7.7 (B2 -good configurations). Call b the size of B2 in direction 2. If b < Nk−1 /2 we say that ω is B2 -good iff ω ∈ I. If b  Nk−1 /2 we say that ω is B2 -good iff ω ∈ ∩mj=1 R j ∩ J , where R j := {ω : ω has a good left-right crossing in R j } (see Fig. 6). Z The  block dynamics, which is again defined on  = W and reversible w.r.t. μ = x∈Z2 νx , is then defined as follows. The block B2 waits a mean one exponential random time and then its current configuration is refreshed with a new one sampled from μ B2 . The block B1 does the same but now the configuration is refreshed only if the current configuration ω is B2 -good. Thus the generator of this auxiliary chain acts on local functions as

 Lblock f (ω) = c1 (ω) μ B1 ( f ) − f (ω) + μ B2 ( f ) − f (ω), (7.3) 2

where c1 is the characteristic function of the event that ω is B2 -good, namely   1IJ (ω) mj=1 1IR j (ω) if b  Nk−1 /2 c1 (ω) := , 1II if b < Nk−1 /2

(7.4)

where we recall that b is the vertical size of 2 (and of ). The Dirichlet form associated to (7.3) is

 Dblock ( f ) = μ c1 Var B1 ( f ) + Var B2 ( f ) . (7.5) Denote by γblock () the inverse spectral gap of Lblock . The following bound, whose proof relies on the fact that c1 (ω) depends only on ω B2 , can be proven as in [9, Prop. 4.4]. Proposition 7.8. Let εk ≡ max I P(ω is not B2 -good), where the max I is taken over the sk possible choices of the pair (1 , 2 ). Then γblock () ≤

1 √ . 1 − εk

Thus, by using the standard Poincaré inequality for the block auxiliary chain and Proposition 7.8 as well as (7.5), we get that for any f :  → R, $ #

 1 μ c1 Var B1 ( f ) + Var B2 ( f ) . Var  ( f ) ≤ (7.6) √ 1 − εk We will now reconstruct the Dirichlet form of AGL (5.5) from the two terms on the right hand side of (7.6). Let us start with the second term. By construction, there exists k ∈ {1, . . . , k − 1} such that B2 ∈ Fk \Fk −1 . Thus, using the definition (7.1) for γk−1 , we have    

Nk μ Var B2 ( f ) ≤ γk−1 μ cx,B Var ( f ) . x 2 x∈B2

By using both monotonicity properties stated in Remark 7.1, it is immediate to verify the following property

Kinetically Constrained Lattice Gases

327

Fig. 8. The volume B1 , the stripe I and the rightmost good top bottom crossing ω (empty sites within the dashed line). The whole set of empty sites is instead ω . The set Bω is the region which lies inside the shaded region N

Nk k Claim 7.9. For all x ∈ B2 , all ω and all k  k − 1, cx,B (ω)  cx, (ω). 2

Therefore we have

  

 Nk μ Var B2 ( f ) ≤ γk−1 μ cx, Var x ( f ) .

(7.7)

x∈B2

The r.h.s. of the latter is nothing but the contribution carried by the set B2 to the full Dirichlet form (5.5) when N = Nk .

 Let us now examine the more complicated term μ c1 Var B1 ( f ) . For any B2 -good configuration ω recall that ω is the right-most good top-bottom crossing of ω in I . Then divide B1 ∪ I into two connected components (with respect to the distance d1 ): the sites on the right of ω , and those on the left. We shall call Bω the sites of B1 ∪ I \ω on the left of ω (see Fig. 8). Notice that if ω and ω are B2 -good and ω = ω , then Bω = Bω . In other words Bω is unequivocally defined by ω . Thus, with a slight abuse of notation, for any  ∈ C we let B be the Bω which corresponds to all ω with ω = . If we observe that Var B1 ( f ) and c1 (ω) depend only on ω B2 , we use the independence of 1I{ω =} from ω I (Claim 7.4 (ii)) and let I := B ∩ I we can write   

 μ c1 Var B1 ( f ) = μ 1I{ω =} 1I∩mj=1 R j Var B1 ( f ) =

 

∈C"

μ(ω B2 \I )

∈C" ω B2 \I

=

 

∈C" ω B2 \I

 ωI

μ(ω B2 \I )

μ(ω I )1I{ω =} 1I∩mj=1 R j Var B1 ( f )

 ω I \I

μ(ω I \I )1I{ω =} 1I∩mj=1 R j

 ω I

(7.8)

μ(ω I ) Var B1 ( f )

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when b  Nk−1 /2 and    

μ 1I{ω =} Var B1 ( f ) μ c1 Var B1 ( f ) = ∈C

=

 

μ(ω B2 \I )

∈C ω B2 \I

=

 

 ωI

μ(ω B2 \I )

∈C ω B2 \I

μ(ω I )1I{ω =} Var B1 ( f )

 ω I \I

μ(ω I \I )1I{ω =}

 ω I

(7.9)

μ(ω I ) Var B1 ( f )

when b < Nk−1 /2. Then we can in both cases upper bound the last term by using the convexity of the variance which implies  ω I

ν(ω I ) Var B1 ( f ) ≤ Var B ( f ).

(7.10)

Now for any x ∈ B and any ωB ∈ B , let c x (ωB ) be the indicator function of the N event that there exists a direction i ∈ {1, 2} and a geometric path γx,y inside Tx,ik−1 (B ) from x to some y ∈ ∂ − B ∪(∂+  ∩B ) which is allowed for the AKG model if τ (t) ∈ G ∗ for any t ∈ ∂ + B (see Fig. 9). It is then possible to define the generator LB obtained Nk from L,Nk by substituting the rates cx, with c x . The generator LB with Dirichlet form DB is ergodic and reversible with respect to μB . Denote by gap(LB ) the associated spectral gap. Applying the Poincaré inequality leads to Var B ( f ) ≤ gap(LB )−1



μB (c x Var x ( f )) .

(7.11)

x∈B

Claim 7.10. For any ω and any x ∈ B , N

k−1 (ω)  c x (ω). cx,B 1 ∪I

Proof. The result follows immediately from the definition of c¯ (see Fig. 9).

 

Claim 7.11. gap(LB )−1 ≤ gap(L B1 ∪I,Nk−1 )−1 ≤ γk−1 . Proof. For any f ∈ L2 (B , μB ) we have D BN1 ∪I ( f ) ≤ DBN ( f ) and Var B ( f ) = Var B1 ∪I ( f ). The first property follows by using Claim 7.10. The second property follows from the product structure of the measure μ B1 ∪I . The first inequality of the claim then follows at once from the variational characterization of the spectral gap. Furthermore, since B1 ∪ I ∈ Fk−1 there should exist k ∈ {1, . . . , k − 1} such that B1 ∪ I ∈ Fk \Fk −1 . Thus since Nk  Nk−1 from the monotonicity Remark 7.1(i) we get gap(L B1 ∪I,Nk−1 )−1  gap(L B1 ∪I,Nk )−1 . We can now use the definition (7.1) of γk−1 to get the second inequality.  

Kinetically Constrained Lattice Gases

329

N

k−1 Fig. 9. Inside the dotted-dashed line we depict the cylinder Tx,B . The bold dashed line represents . The 1 ∪I light dashed line is instead a path belonging to Gx,Nk−1 ,B1 ∪I . It is immediate to verify that this implies the existence of an AKG allowed path inside B from x to a site y ∈ ∂+  ∩ B , thus c¯x = 1. From the drawings it ∗ B , hence the necessity to introduce is also clear that it does not necessarily imply a path which ends at y ∈ ∂− 

N

the additional rates c¯x in B instead of using cx,k−1 B

By Putting together (7.11) with Claim 7.11 yields  Var B ( f ) ≤ γk−1 μB (c x Var x ( f )) x∈B

which, together with (7.8), (7.9) and (7.10) gives ⎞ ⎛    

 1I∩mj=1 R j 1I{ω =} c x Var x ( f ) ⎠ (7.12) μ c1 Var B1 ( f )  γk−1 μ ⎝ ∈C"

x∈B

when b  Nk−1 /2 and

⎞ ⎛    

 1I{ω =} c x Var x ( f ) ⎠ μ ⎝ μ c1 Var B1 ( f )  γk−1 ∈C

(7.13)

x∈B

when b < Nk−1 /2. We now wish to upper-bound the terms which appear in front of Nk , in order Var x in the right-hand sides of (7.12) and (7.13) with the long range rates cx, to upper-bound the right-hand side of (7.6) with the full Dirichlet form (5.5) by using (7.7) and (7.12) or (7.7) and (7.13) according to the value of the height b of the rectangle . Once this is achieved we will divide the left and right side by Var  ( f ) and take the sup on f in order to gain an inequality between γk and γk−1 which, properly iterated, will lead to the desired bound (7.2). Claim 7.12. For any ω,  ∈ C" and x ∈ B , Nk 1I∩mj=1 R j 1I{ω =} c x (ω)  1I{ω =} cx, (ω).

(7.14)

For any ω,  ∈ C and x ∈ B , Nk 1I{ω =} c x (ω)  1I{ω =} cx, (ω).

(7.15)

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N. Cancrini, F. Martinelli, C. Roberto, C. Toninelli

Proof. It is sufficient to show that when c¯x (ω) = 1 the right hand side also equals one. We recall that c¯x (ω) = 1 guarantees that there exists (at least) one direction i and a N geometric path γx,y inside Tx,ik−1 (B ) from x to some y ∈ ∂ − B ∪ (∂+  ∩ B ) which is AKG allowed. Let γx,y be one of such paths and distinguish three cases: (a) y ∈ ∂ − ; (b) y ∈ ∂ −  and i = 1; (c) y ∈ ∂ −  and i = 2. Nk =1 (a) From the definition of the rates in formula (5.3) it follows immediately that cx, N

Nk () ⊃ Tx,ik−1 (B ), thus γx,y ∈ Gx,Nk , . since Tx,i (b) In this case y ∈ ∂+  ∩B , thus there exists y with d1 (y, y ) = 1 and y ∈  = ω . This implies that ω(y ) is good and either y ∈ ω or there exists y

∈ ω with d1 (y , y

) ∈ (1, 2). By using this together with the existence of the AKG allowed path γx,y allows to conclude that there always exists a path γ y,y

AKG allowed. We should now distinguish the case (i) b < Nk−1 /2 and (ii) b  Nk−1 /2. Case (i) Nk () can be handled very simply by noticing that in this case ∀x ∈ B , I ⊂ Tx,1 Nk Nk holds. Therefore cx, = 1 thanks to the path γx,z := γx,y · γ y,y

· γ y

,z ⊂ Tx,1 (), 2 where z ∈ ∂−  and γ y

,z is a subset of the good top-bottom crossing (ω). Case (ii) requires a bit more work and the use of the left-rightmost crossings in the R j rectanm R (ω) = 1 guarantee that there exists a good gles. Claim 7.6 and the fact that 1I∩i=1 j

∗  ∩ T k−1 (). path γw,z inside Tx,1k−1 () with w ∈ ω ∩ Tx,1k−1 () and z ∈ ∂− x,1 " (ω) has k-bounded oscillations. Thus by recalling Definition 7.5 Since  ∈ C, and noticing that Nk−1 + A/2 log(3/2) log(lk ) ≤ Nk , there exists a good path Nk . γ y

,w ⊂ ω inside Tx,1 (c) Define as in the previous case y

and γ y,y

. By construction (since y

∈ ω ) there exists a good path γ y

,z ⊂ I (which is a subset of the top bottom crossing 2 . Furthermore Claim 7.3(iii) guarantees that in this case of I ) for some z ∈ ∂− Nk Nk (ω) = 1 is guaranteed if we consider the overall path I ⊂ Tx,2 (). Thus cx, γx,z := γx,y · γ y,y

· γ y

,z .   N

N

N

If we finally plug (7.14) in the r.h.s. of (7.12) or (7.15) in the r.h.s. of (7.13) we obtain in both cases ⎞ ⎛  N

 k μ c1 Var B1 ( f ) ≤ γk−1 μ ⎝ cx, Var x ( f )⎠ ⎛ ≤ γk−1 μ ⎝

x∈B



⎞ Nk cx, Var x ( f )⎠ .

(7.16)

x∈B1 ∪I

Thus, by using (7.6), (7.7), (7.16) and (5.5) we get   $ #    1 Nk γk−1 D,Nk ( f ) + μ cx, Var x ( f ) . (7.17) Var  ( f ) ≤ √ 1 − εk x∈I

1/3

Recalling Lemma 7.2 we can now averaging over the sk = lk possible choices  of (k) (k) (k) (k) (k ) (k ) k k =∅ the sets 1 , 2 which verify I ⊂ 1 ∩ 2 and 1 ∩ 2 ∩ 1 ∩ 2

Kinetically Constrained Lattice Gases

331

for k = k . Thus we get # Var  ( f ) ≤

1 √ 1 − εk

$

# $ 1 γk−1 1 + D,Nk ( f ) sk

(7.18)

which in turn, dividing by Var  and taking the sup over f , implies # γk ≤

1 √ 1 − εk

$#

1 1+ sk

$ γk−1 ≤ γk0

k )



j=k0

1 √ 1 − εj

#

1 1+ sj

$ ,

(7.19)

where k0 is the smallest integer such that δk0 > 1 and εk has been defined in Proposition 7.8. By plugging the results of Claim 7.13 and 7.14 below into (7.19) the proof of Theorem 5.5 is completed with the choice ρ1 = max(ρ˜1 , ρ¯1 ).   Claim 7.13. There exists ρ˜1 ∈ (0, 1) and A0 > 0 such that for ρ > ρ˜1 and A > A0 , εk 

2 . lk

(7.20)

Claim 7.14. For all  ∈ (0, 1) there exists ρ¯1 ∈ (0, 1) such that for ρ > ρ¯1 , γk0  2 − .

(7.21)

Proof of Claim 7.13. Recalling the definition of εk given in Proposition 7.8 and Definition 7.7 for B2 -good configurations we should distinguish two cases: (i) b  Nk−1 /2 and (ii) b < Nk−1 /2, where b is the length of 2 in direction 2. (i) By using the FKG inequality we get m   ) ν({ω is B2 -good}) = μ ∩mj=1 R j ∩ J  μ(J ) μ(R j ).

(7.22)

j=1

Next we can decompose I into m smaller disjoint rectangles, I = ∪mj=1 I j , with each I j being a translation of [0, a] × [0, h j ], where a = A/2 log(3/2) log(lk ), m < 2lk+1 /(A/2 log(3/2) log lk ) and h j verifies A log(3/2)/4 log lk < 2h j < A/2 log(3/2) log lk (this procedure is possible thanks to the bounds on b, Nk−1 /2  b  lk+1 ). Define the events I j := {ω : ω has a left-right good crossing in I j }. It is then easy to prove by recalling Definition 7.5 and by an inspection of the right Fig. 7 that

J ⊂ ∩mj=1 I j ∩ I,

(7.23)

namely the occurrence of a good top bottom crossing in I plus good left right crossing in each I j guarantee that the rightmost top bottom crossing of I has k-bounded oscillations.

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By using standard percolation results, there exists ρ1 < 1 and α > 0 such that if ρ > ρ1 and k is sufficiently large, then μ(R j )  1 − lk+1 exp(−αc j ) ∀ j = 1, . . . , m, μ(I j )  1 − (b1 − a1 ) exp(−αh j ) ∀ j = 1, . . . , m , μ(I)  1 − lk+1 exp(−α[b1 − a1 ]),

(7.24) (7.25) (7.26)

where we recall that c j is the height of the rectangle R j which verifies c j  Nk /2, b1 − a1 = A/2 log(3/2) log lk is the width of the strip I , m verifies m  (2lk+1 /Nk−1 ), and Nk = A(log(lk ))2 . Provided that A is large enough the desired inequality (7.20) immediately follows from (7.22), (7.23), (7.24), (7.25) and (7.26). (ii) In this case ν({ω is B2 -good}) = μ(I)

(7.27)

and again, provided that A is large enough, the desired inequality (7.20) immediately follows from (7.26).   Proof of Claim 7.14. Choose k  k0 and a rectangle R ∈ Fk \Fk−1 . Let 1 (2 ) be the length of R in the e1 (e2 ) direction. We label the 1 2 sites of R from the bottom left one from left to right and bottom to top as x1 , . . . , x1 2 . We also let B0 = R, B1 = x1 and B2 = R\x1 and we consider the following block dynamics. The block B2 waits a mean one exponential random time and then the current configuration inside it is refreshed with a new one sampled from μ B2 . The block B1 does the same but now the configuration is refreshed only if the current configuration ω in B2 is such that the path γx1 ,x1 = x1 , . . . , x1 which goes straight towards the right of x1 up to the border of R is good. By using Poincaré inequality together with the same strategy as in [9, Prop. 4.4] to evaluate the spectral gap of this auxiliary dynamics we get # $ 1 * Var R ( f )  μ R 1I{γx ,x is good} Var B1 ( f ) + Var B2 ( f ) 1 1 1 − 1 − q 21   1 * (7.28) D R,Nk ( f ) gap(L B2 ,Nk )−1 ,  1 − 1 − q 21 where to get the last inequality we use the fact that ∂ − B2 ⊂ ∂ − R and 1I{γx Nk cx,R .

1 ,x1

is good} 

The variational characterization of the spectral gap together with (7.28) leads to

gap(L R,Nk )

−1



1−

*

1 1 − q 21

−1 gap(L R\x1 ,Nk ) .

(7.29)

We can then let B˜ 0 := R\x1 and divide it into B˜ 1 := x2 and B˜ 2 := R\(x1 ∪ x2 ) and proceed analogously to get inequality (7.29) with R\x1 in the left-hand side and R\(x1 ∪ x2 ) on the right-hand. By proceeding iteratively we finally get γk 0 

1 * , (1 − 1 − q 2k0 )k0

which concludes the proof of the claim.

 

(7.30)

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333

8. Polynomial Decay to Equilibrium: Proof of Theorem 4.2 In order to establish polynomial decay to equilibrium in infinite volume we start by reducing as usual the dynamics to a finite volume thanks to the finite speed of propagation. Then we follow a soft spectral theoretic argument introduced in [7] which requires a bound of the variance with the Dirichlet form of the process. Establishing this bound is the difficult step here due to the presence of the kinetic constraints. In order to obtain this result we use the positivity of the spectral gap of AGL (Theorem 5.5) combined with path and renormalization arguments. Proof of Theorem 4.2. Let f be a local function of zero mean value and fix a large time t. Consider L˜ = at with the constant a defined by Lemma 8.1 below. By translation invariance of the system we can assume that f has support  f at the center of the cube Q L˜ . Then, we have Q L˜

Var μ (Pt f )  2Pt f − Pt Q

Q L˜

f 2∞ + 2 Var μ Q ˜ (Pt L

f ),

(8.1)

tL

where Pt L˜ = e Q L˜ and L Q L˜ is defined with boundary-source choice (M, S) = (∅, ∂ − Q L˜ ). A standard property known as finite speed of propagation (see [22]) asserts that for a proper C, if a is chosen large enough for all t, Q L˜

f 2∞  Ce−t/C  f 2∞

Pt f − Pt

(8.2)

holds. Putting together (8.1) and (8.2) with Lemma 8.1 concludes the proof.

 

Lemma 8.1. There exists a > 0 such that for all t > 0 if we let L˜ := at, Q L˜

Var μ Q ˜ (Pt L

f)  C

(log t)5  f 2∞ t

holds. Q

Q L˜

Proof of Lemma 8.1. Let g = P2t L˜ f . By reversibility Var μ Q ˜ (Pt L Thus if 

μ Q L˜ ( f, g)

2

  f 2∞

(log t)5 Q Var μ Q ˜ (Pt L˜ f ), L t

f ) = μ Q L˜ ( f, g).

(8.3)

the desired result follows immediately. We are therefore left with proving (8.3). Note Q Q Q that by definition ∂t∂ Var μ Q ˜ (Pt L˜ f ) = −2D Q L˜ (Pt L˜ f )  0. Hence, Var μ Q ˜ (Pt L˜ f ) L L is a decreasing function in t and Q

Q L˜

Var ν Q˜ (g) = Var μ Q ˜ (P2t L˜ f )  Var μ Q ˜ (Pt L

L

L

f ).

(8.4)

˜ = at/ is integer, f has support  f ⊂ Fix t and a and choose  > 0 s.t.: L := L/ x Q := Q  + x for a proper x and μ(F ) > max(ρ0 , ρ˜0 ), where ρ0 and ρ˜0 are the thresholds defined in Theorem 5.6 and Claim 8.2 respectively (this is possible thanks to Lemma 3.4). Then define the renormalized lattice and the renormalized cube as in Sect. 6, namely Z2 () := Z2 and Q˜ L := Z2 ()∩ Q L˜ . As already noticed in Sect. 6, if we

334

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consider the probability space W = {0, 1} Q  equipped with ν = μ Q  the two probabil 2 2 ity spaces ({0, 1}Z , μ) and (W Z () , x∈Z2 () νx ) coincide. Furthermore μ Q L˜ = ν Q˜ L ,  where ν A = x∈A∩Z2 () νx . Choose B > 0 such that M := B log L divides L and L/M is odd and consider the following rectangles on Z2 (): Hk := [0, L − 1] × [(k − 1)M, k M − 1], Vk := [(k − 1)M, k M − 1] × [0, L − 1],

L , M L k = 1, . . . , . M

k = 1, . . . ,

It is immediate to verify that the renormalized cube Q˜ L can be written as the disjoint union of both sets of rectangles, Q˜ L = ∪k Vk = ∪k Hk (see Fig. 10). Define on W = {0, 1} Q  the good event G = F . With this choice and recalling Definition 2.6 we let Vk := {ω : ω has a good top-bottom crossing in Vk }, Hk := {ω : ω has a good left-right crossing in Hk }, L/M

 L ,B := {ω : ω ∈ ∩k=1 (Vk ∩ Hk )}. Then, by using the Cauchy-Schwartz inequality and the fact that ν Q L˜ ( f ) = 0 we get   2  2 2 μ Q L˜ ( f, g) = ν Q˜ L ( f, g)  2ν Q˜ L (1 f g)2 + 2ν Q˜ L (1 − 1 ) f ( g − ν Q˜ L (g)) , (8.5) where from now on we drop the indexes L and B from . Let us deal first with the second term. By using again the Cauchy-Schwartz inequality we get  2 ν Q˜ L (1 − 1 ) f (g − ν Q˜ L (g))  ν Q˜ L (1 − 1 ) Var ν Q˜ (g) f 2∞ , L

C C Q  Var ν Q˜ (g) f 2∞  Var μ Q ˜ (Pt L˜ f ) f 2∞ , L L L at

(8.6)

where the second inequality relies on Claim 8.2 below and we used (8.4) in order to derive the third inequality. Let us now consider the first term of (8.5). Without loss of generality we can assume that f has support  f ⊂ (∪x∈Hk0 Q x ) ∩ (∪x∈Vk0 Q x ) with k0 = L/2M (see Fig. 10). As explained in Sect. 7 (see before Claim 7.4), there is a natural partial order on the set of top-bottom crossing paths in Vk that allows to the define the right-most one. Thus, for any ω ∈ Vk , we define ωV,k to be its right-most good top-bottom crossing. Analogously for each ω ∈ Hk we define ωH,k to be its up-most good left-right crossing. As usual, V,k H,k we let ω and ω be the corresponding double paths. For any ω ∈  we can then L/M let ω := {ωH,k , ωV,k }k=1 and define the geometric set C := { : ∃ ω ∈  s.t. ω = }. By geometrical considerations one can verify that points (i) and (ii) of Claim 7.4 are valid for all ω ∈ Vk , and analogous statements are valid for all ω ∈ Hk . Let Rk0 := (Hk0 ∪ Hk0 +1 ) ∩ (Vk0 ∪ Vk0 +1 ). For any chosen ω ∈  we divide the region Rk0 into two connected (with respect to the distance d1 ) components : the sites on the right of

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335

Fig. 10. The box Q˜ L ⊂ Z2 () divided into the rectangles Hk and Vk with k = 1, . . . L/M (here L/M = 6). The shaded region represents the support of f ,  f ⊂ Hk0 ∩ Vk0 . We depict a configuration ω ∈ . The continuous non straight lines which form a grid represent the rightmost good top-bottom crossing of each Vk and the up-most good left-right crossing of each Hk . The region delimited by the bold black line is Bω . We choose a site x ∈ Bω , assume that c¯x (ω) = 1 and we associate to x the corresponding site u ∈ Qˆ L/M . Then we fix v ∈ ∂− Qˆ L/M and we choose a geodesic path γu,v (dashed line). The green path is the geometric path γx,x ∗ which is allowed for the AKG model and is composed of two parts. The first path is from x to ∂− B ∪ (∂+  ∩ B ) and is guaranteed by c¯x = 1. The second path is guaranteed by ω ∈  and uses the grid  of good crossing following the sites which belong to the geodesic path γu,v V,k +1

ω 0 and those on the left. We call BωV the latter set. Analogously we consider the two H,k +1 connected components which correspond to the sites above ω 0 and those below. We call BωH the latter set. Then we define Bω := BωV ∩ BωH and, with a slight abuse of notation, for any  ∈ C we let B be the Bω which corresponds to all ω with ω = . With this notation we have ν Q˜ L (1 f g)2 ≤  f 2∞

 ∈C

 ν Q˜ L Var B (g)χ ,

where we used the fact that χ (σ ) does not depend on the value of σ inside B and the hypothesis νB ( f ) = 0. Then, by using Lemma 8.3 below and Lemma 6.1 and some explicit counting we get 

ν Q˜ L (1 f g)

2

 K ≤ C(log L)5  f 2∞ D Q (g) + D∂G ˜ L

− Q L˜

 (g) .

(8.7)

By the spectral decomposition of −L Q L˜ in L2 (μ Q L˜ ) and the bound 2tλe−2tλ  1/e, we have    ∞ K G DQ g(−L (g) + D (g) = μ )g = λe−2tλ d E λ (g) Q L˜ Q L˜ ∂ − Q L˜ L˜ 0  ∞ 1 1 1 Q Var μ Q ˜ (g)  Var μ Q ˜ (Pt L˜ f ), d E λ (g) = (8.8)  L L 2et 0 2et 2et

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where the last inequality comes from (8.4). Therefore the desired inequality (8.3) follows from (8.5), (8.6), (8.7), (8.8) and L˜ = at and the proof is concluded.   Claim 8.2. There exists ρ˜0 < 1 s.t. for μ(F )  ρ˜0 we have C . L Proof of the claim. By translation invariance, the probability that there is a good leftright crossing in Hk or that there is a good top bottom crossing in Vk do not depend on k. Hence, let α(L) = ν Q˜ L (Vk ) = ν Q˜ L (Hk ). Using the FKG inequality, we get that ν Q˜ L (1 − 1 ) ≤

ν Q˜ L (1 − 1 ) ≤ 1 − α 2L/M . Standard percolation results [14] guarantee the existence of a constant c > 0 such that, provided the probability for a site to be good is above a certain percolation threshold ρ˜0 , then α ≥ 1 − Le−cM = 1 − Le−cB log L . Hence, provided B is large enough, ν Q˜ L

+ C 2L −cB log L log(1 − Le ) ≤ . (1 − 1 ) ≤ 1 − exp M L

This achieves the proof of the claim.



 

Let Te , " ce and " cx be defined as in Sect. 6, then Lemma 8.3. For any f there exists a positive constant C such that    

 ce (Te f − f )2 ν Q˜ L Var B ( f )χ  C(log L)5 ν Q˜ L " ∈C

e∈E Q˜

+ C(log L)5

L



x∈∂ − Q˜ L

ν Q˜ L (" cx Var x ( f )) .

Proof of the lemma. The proof of this result makes use of the positivity of the spectral gap of the AGL model (Theorem 5.5) and involves a renormalization technique in the same spirit as the one used to prove the lower bound for the spectral gap of KA (Theorem 4.1). Fix  ∈ C . Let N := A(log(2M))2 with A defined as in Theorem 5.5. For any x ∈ B and any ω ∈ B , we let c x (ω) be the indicator function of the N (B ) or inside T N (B ) with event that there exists a geometric path γx,y inside Tx,1  x,2  y ∈ ∂ − B ∪ (∂+  ∩ B ) and such that this path is allowed with the choice of the AKG ∗ constraints for the configuration (ωB · τ )(z) with τ (t) ∈ G for any t ∈ ∂ + B . It is then possible to define the correspondent Glauber generator in the volume B as  LB f = c x (νx ( f ) − f ). x∈B

Applying the Poincaré inequality leads to Var B (g) ≤ gap(LB )−1



νB (c x Var x (g)) .

(8.9)

x∈B

Analogously to Claim 7.10 and 7.11, one can prove the following bounds with respect to the rates and the spectral gap of the AGL model

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337

Claim 8.4. For any ω and x ∈ B , N (ω)  c¯x (ω) cx,R k 0

and gap(LB )−1  gap(L Rk

agl

0 ,N

)−1 .

If we choose B (the constant which enters in the definition of M, the short size of the renormalized rectangles) in order that there exists k ∈ N s.t. 2M = 2B log L = lk with lk = (3/2)k/2 , then N = A(log lk )2 = Nk , where Nk is the one used in Sect. 7. Furthermore since Rk0 is a cube of linear size 2M = lk it belongs to Fk−1 \Fk−2 , where Fk are the set of rectangles defined in Sect. 7. Therefore by recalling the definition of γk (Eq. 7.1) and the result of Theorem 5.5 we get agl

gap(L Rk

0 ,N

)−1  2.

(8.10)

This, together with (8.9) yields 

 ν Q L (c x χ Var x (g)) . ν Q˜ L Var B (g)χ  2

(8.11)

x∈B

As usual we can rewrite the variance as  2 1  Var x (g)(ω) = ν(w)ν(w ) g(ω Q˜ L \x · w ) − g(ω Q˜ L \x · w) . (8.12) 2

w,w ∈W

Our aim is now to reconstruct the move from ω Q˜ L \x · w to ω Q˜ L \x · w via proper paths by using the properties which are guaranteed if c x χ = 1. We start by noticing that the renormalized cube Q˜ L can also be seen as the union of squares whose side has length M, i.e. as a subset Qˆ L/M of Z2 (M). Then, for any (u, v) ∈ Qˆ L/M × ∂ − Qˆ L/M we choose once and for all a path γu,v with γuv = (u = t (1) , t (2) , . . . , t (m−1) , v = t (m) ) among the geodesic paths inside Qˆ L/M from u to v such that, for any t ∈ √ γuv , the Euclidean distance between t and the straight line segment [u, v] is at most 2/2 (see Fig. 10). Fix x ∈ B and suppose c¯x = 1. Then there exists at least one geometric path (x1 = x, . . . , x m ) which is allowed for ω with AKG constraints and with x m ∈ ∂ − B ∪ (∂+  ∩ B ). Then let u ∈ Qˆ L/M be the square which contains x m , i.e. x m ∈ Q u (see Fig. 10). If χ = 1, by using the grid  of vertical and horizontal good crossings, we can now construct in a unique way a geometric path x m , . . . , x n = x ∗ (1) inside Q˜ L which is allowed for AKG constraints and such that x m , . . . , x m 1 ∈ Q t , (2) x (m 1 +1) , . . . , x (m 2 ) ∈ Q t and so on (see Fig. 10). In conclusion we have constructed an AKG allowed path γx x ∗ = (x 1 , . . . , x n = x ∗ ) with x ∗ ∈ ∂ − Q˜ L ∩ Q v (overall green path in the figure). We construct such path for any couple ω, x with c¯x (ω) = 1 and ω ∈  and we perform this choice in order that the path is the same for any two configurations which are G-equivalent inside Q˜ L \x. For i = 1, . . . , n−1, let ei = (x (i) , x (i+1) ) ∈ E Q˜ L . Then, for any w, w ∈ W , we can define the path (1) x = P x (ω, v, ) = (ω , . . . , ω(2n) ) Pw→w w→w

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from ω(1) = ω Q˜ L \x · w to ω(2n) = ω Q˜ L \x · w by ω(i+1) = Tei ω(i) for i = 1, . . . , n − 1, (n−1) Q L \x ∗

ω(n) = ω ˜

· w and ω(i+1) = Te2n−i ω(i) for i = n + 1, . . . , 2n − 1. It is then easy

˜ L , more precisely for x is an allowed path for the AKG model on Q to verify that Pw→w (i) (n) cx ∗ (ω ) = 1. any i = n, " cei (ω ) = 1, and " For any e = (z, z ) ∈ E Q˜ L , define t (e) := (t : z ∈ Q t ) and the weight function ψ by ψ(e) := j + 1, where j := d1 (t (e), u). We will denote by

x |ψ := 2 |Pw→w

n−1  i=1

1 ψ(ei )

x . Using a telescopic sum, and the Cauchythe weighted length of the path Pw→w Schwartz inequality with weight ψ, we get that for any w, w ∈ W ,

c¯x (ω)χ (ω)( f (ω Q L \x · w ) − f (ω Q L \x · w))2 2n−1 2  = c¯x (ω)χ (ω) f (ω(i+1) ) − f (ω(i) ) i=1 x |ψ  χ (ω)2|Pw→w



ψ(e)" ce (σ )(∇e f )2 (σ )1{(σ,σ e )∈P

} x w→w

σ,e (n) Q L \x ∗

+ χ (ω)2" cx ∗ (ω(n) )( f (ω ˜

(n) Q L \x ∗

· w ) − f (ω ˜

· w))2 .

(8.13)

By construction, uniformly in x, ω and w, w ∈ W , we have

2 x |ψ  C M |Pw→w

2L/M  j=1

1  C M 2 log(L/M)  C(log L)3 . j

(8.14)

We get from (8.11), (8.12), (8.13) and (8.14) that  

 ν Q˜ L Var B ( f )χ ≤ C(log L)3 ν Q˜ L (ω)c x (ω)χ (ω) 

×

x∈B ω

ν(w)ν(w )

w,w ∈W

+2

 

x∈B ω



ψ(e)" ce (σ )(∇e f )2 (σ )1{(σ,σ e )∈P

} x w→w

σ,e

ν Q˜ L (ω)χ (ω)" cx ∗ (ω(n) ) Var x ∗ ( f )(ω(n) ).

x satisfies Note that by construction, any σ ∈ Pw→w

ν Q˜ (ω) L ν Q˜ (σ ) L

 C. Hence, using the trivial

bound c x ≤ 1, taking the average with respect to the 2L/M possible v ∈ ∂ − Qˆ L/M , and inverting the summations, we get

Kinetically Constrained Lattice Gases

 ∈C

≤

339

 ν Q˜ L Var B ( f )χ

 CM (log L)3 ν Q˜ L (σ )" ce (σ )(∇e f )2 (σ ) L σ,e ⎧ ⎫ ⎪ ⎪ ⎨ ⎬     e × max χ (ω)1 ψ(e)  {(σ,σ )∈P } x w→w ⎪ w,w ∈W ⎪ ⎩ ⎭ ∈C x∈B ω ˆ v∈∂ − Q L/M

+

CM (log L)3 L

×

⎧ ⎪ ⎨ ⎪ ⎩







σ

∗ x ∗ ∈∂ − Q L

μ Q L (σ )" cx ∗ (σ ) Var x ∗ ( f )(σ )

  

χ (ω)1{O(ω,x)=(σ,x ∗ )}

v∈∂ − Qˆ L/M ∈C x∈B ω

⎫ ⎪ ⎬ ⎪ ⎭

.

(8.15)

Note that ω can be reconstructed from x, σ and e, at the exception of one site where the value of the configuration might be unknown. This, analogously to what occurred in the proof of 5.6, is true thanks to the fact that we have chosen the geometric path from x to the border in such a way that paths are equal for any two configurations which are G-equivalent inside Q˜ L \x. Hence for any σ, e,     max χ (ω)1{(σ,σ e )∈P x }

w,w ∈W

w→w

v∈∂ − Qˆ L/M ∈C x∈B ω

  ,  |W |M 2 max max  v : t (e) ∈ γ u,v

w,w ∈W ,ω,x

(8.16)

where v is running over ∂ − Qˆ L/M , x ∈ Q k0 ,  ∈ C and t (e) ∈ Qˆ L/M is such that if e = (z, z ) then z ∈ Q t (e) . Recalling that γuv is a geodesic path, by the Theorem of Thales (see Fig. 11) given , u and e, one has  {v : t (e) ∈ γuv } ≤ C

L . Mψ(e)

Fig. 11. The set of admissible v such that t (e) ∈ γuv

(8.17)

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It follows that ⎧ ⎪ ⎨  ψ(e) max max σ,e w,w ∈W ⎪ ⎩ ˆ

v∈∂ − Q L/M

   ∈ x∈B ω

χ (ω)1{(σ,σ e )∈P

} x w→w

⎫ ⎪ ⎬ ⎪ ⎭

 C L M. (8.18)

Then, a similar reasoning gives     max∗ χ (ω)1{O(ω,x)=(σ,x ∗ )}  C M 2 . σ,x

(8.19)

v∈∂ − Qˆ L/M ∈C x∈B ω

Plugging (8.16), (8.17), (8.18) and (8.19) into (8.15) the proof of the lemma is concluded.   9. Appendix: Properties of KA Model In this section we prove Lemma 3.2 and 3.4 in the case d = j = 2, since they have been used in the proof of Theorem 4.1. We also prove that pc = 1 for the two-dimensional KA model, Theorem 3.5. The proofs follow the arguments sketched in [30]. We start with the trivial observation that if a region is framed then its empty borders can be rigidly shifted in the interior of the region by proper allowed paths. More precisely we have Claim 9.1. Consider a rectangle R = [0, n] × [0, m] and fix a configuration ω which is R-framed. Then (a) Let σ be such that σ (z) = ω(z − e2 ) if z = i e1 + m e2 with i ∈ {1, . . . , n − 1}, σ (z) = ω(z) otherwise. There exists an allowed path Pω,σ inside R. (b) Let σ be such that σ (z) = ω(z − e1 ) if z = ne1 + i e2 with i ∈ {0 . . . , m − 1}, σ (z) = ω(z) otherwise. There exists an allowed path Pω,σ inside R. Proof. (a) Consider the geometric path x1 , . . . , xn−2 with x1 = (n − 1)e1 + (m − 1)e2 , xi+1 = xi − e1 . It is immediate to verify that ω(1) , . . . , ω(n+1) with ω(1) = ω, ω(i+1) = (ω(i) )xi ,xi −e2 is an allowed path from ω to σ . (b) The proof follows along the same lines as (a).   Proof of Lemma 3.2. Let R = [0, n] × [0, m]. In order to prove the result it is clearly sufficient to show that for any framed ω and any e = (x, y) ∈ E R such that ω(x) = ω(y) there exists an allowed path Pω,ωe . Suppose that y = x + e1 (the other cases can be treated analogously) and let x = x1 e1 + x2 e2 . By repeatedly using the path constructed in the proof of Claim b (a) we can construct an allowed path from ω to ω˜ with ω(z) ˜ = 0 if z · e2 = x2 + 1, ω(z) ˜ = ω(z) if z · e2  x2 and ω(z) ˜ = ω(z − e2 ) if z · e2 > x2 + 1. Then ce (ω) ˜ = 1 and we can perform the exchange on x, y. Finally, using the reverse of the path to go from ω to ω, ˜ we reconstruct the initial configuration on all sites z = x, y.   Claim 9.2. Choose e = (x, y) ∈ E Z2 and  ∈ N and let Q x := Q  + x, Q y := Q  + y. If there exists A ∈ Ce with Ce defined in (5.1) s.t. ω Q z ∈ F for all z ∈ A, then there

exists an allowed path inside Q x ∪ Q y ∪z∈A Q z from ω to ωe for all e ∈ E Q x ∪Q y .

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341

Fig. 12. The path to frame a 6 × 6 configuration which has at least two empty sites on each of the four independent sides of the central framed 4 × 4 square

Proof. The result can be proved analogously to Lemma 3.2. We give a rough sketch of the procedure since some additional efforts are required in the construction of the allowed path. Let e = x, x + e1 , e ∈ Q x and A = x − e1 , x + e2 , x + e1 − e2 . We first construct the frames inside the Q z with z ∈ A. Then we construct a double empty line adjacent to Q x ∪ Q y by properly shifting the part of the frames in Q z which are far away (thanks to Claim b). Finally we can rigidly shift the double empty line of Q x+e1 inside Q x ∪ Q y and bring it as before near the desired bound to perform the exchange. The reason why we construct and shift the double empty lines is a technical trick which is necessary since, at variance with the situation of Lemma 3.2, we do not have here complete frames but frames which cover only two adjacent sides of Q x .   Consider a rectangle R = [a1 , b1 ] × [a2 , b2 ] and let " ∂+i R := {x ∈  : x + ei ∈ }. 1 Divide ∂+∗ R into four non-intersecting sets R1 := ∂ + R ∪ ((b1 + 1)e1 + (a2 − 1)e2 ), 2 R2 := ∂ + R ∪ ((b1 + 1)e1 + (b2 + 1)e2 ), R3 := " ∂+1 R ∪ ((a1 − 1)e1 + (b2 + 1)e2 ), and 2 " R4 := ∂+ R ∪ ((a1 − 1)e1 + (a2 − 1)e2 ). We call the Ri ’s the independent sides of R. The following property can be easily verified: Claim 9.3. If η is R-framed and there exists at least two empty sites inside each of the ∗ four independent sides of R, then η is R ∪ ∂ + R frameable (see for example Fig. 12).

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Fig. 13. The bottom left site is x − 2(e1 + e2 ) and the whole region is Q(3, x − 2(e1 + e2 )). The sites touched by the dashed lines correspond (from inside to outside) to Q(1, x), ∂+∗ Q(1, x) and ∂+∗ Q(2, x − (e1 + e2 )). The continuous lines delimit the four independent sides of ∂+∗ Q(1, x) and ∂+∗ Q(2, x − (e1 + e2 )). The depicted configuration belongs to F 0 (x, 2)

Proof of Lemma 3.4. For any integer n, any x ∈ Z2 , let Q(n, x) = Q 2n + x. Note that Q(n + 1, x − e1 − e2 ) = Q(n, x) ∪ ∂+∗ Q(n, x). Let F 0 (x, n) be the set of configurations such that Q(1, x) is empty and for each i ∈ (1, n − 1) there are at least two empty sites on each independent side of Q(i, x − (i − 1)(e1 + e2 )) (see Fig. 13). A direct calculation gives μ(F 0 (n, x)) = q 4

n  4 ) 1 − (1 − q)2k−1 − (2k − 1)q(1 − q)2k−2 . k=2

Thus for any q > 0, μ(F 0 (n, x)) converges to a non-zero limit (independent from x) when n tends to infinity. Moreover, the limit n → ∞ and q → 0 is computed in [30]:  1 log(1 − y + y log y) lim lim q log μ(F 0 (n, x)) = 4α := 2 dy. n→∞ q→0 y 0 Fix q ∈ (0, 1). From the limit above, there exists n 0 = n 0 (q) such that for any n ≥ n 0 , μ(F 0 (n, x))  e−α/q .

(9.1)

Now for any n, any x ∈ Z2 , we define F 1 (n, x) as the set of all configurations ω ∈  such that every horizontal and vertical row (of length n) inside Q(n, x) has at least two empty sites. All the sets F 1 (n, x) with x ∈ Z2 have the same probability. By FKG inequality one gets 2n  μ(F 1 (n, x)) ≥ 1 − (1 − q)n − nq(1 − q)n−1 . (9.2) √ Finally, we of smaller boxes  into a collection √ 2 √ of size . Assum√ divide the box 2Q√ ing that  ∈ N we let Z ( ) := Z and Q˜ √ := Z( ) ∩ Q  . Then Q  = √ ∪x∈ Q˜ √ Q( , x) (see Fig. 14). Then we set Q √ := (x ∈ Q˜ √ : x = 2i e1 + 2 j e2 with  i, j ∈ N) and define √ √ A := {ω : ∃x ∈ Q √ s.t. ω ∈ F 0 ( , x )}

Kinetically Constrained Lattice Gases

343

√ Fig. 14. The large box is Q  divided into small boxes of size . The bottom left site of each small box is a point in Q˜ √ . The bottom left sites of the white boxes are the points in Q √ . On the right is an example of √ √ "√ configuration ω ∈ F 1 ( , y ) for some y ∈ Q √ \ Q 

and



√ √ B := {ω : ω ∈ F 1 ( , x) ∀x s.t. x = z  with z ∈ Q˜ √ \Q √ }.

By using Claim 9.3, it follows by construction that A ∩ B ⊂ F . Furthermore, thanks to (9.1) and (9.2) and using the fact that the events which define A and B are independent, we get #  /2 $  /2 √ √ μ(F 1 ( , 0)) μ(F ) ≥ μ(A )μ(B ) ≥ 1 − 1 − μ(F 0 ( , 0)) √   −α/q 2  e−c (1−q) ≥ 1 − e−ce for a proper constant c = c(q). Thus for any  there exists 0 (, q) such that for  > 0 we get μ(F ) > 1 − .   Proof of Theorem 3.5. Let  be a cube of size a. Then lim→∞ μ(∩e∈E  Ee ) = μ(∩e∈E Z2 Ee ). For a given bond e = (x, y) we let Fe be the set of configuration inside F which remain Q  -frameable even if we fill both x and y, namely Fe := {ω ∈  : ω Q  \(x,y) · 1x · 1 y is Q  − frameable }. By proceeding along the same lines as the proof of the above Lemma 3.4, it is easy to verify that there exists c1 , c2 > 0 Q s.t. μ(Fe ) > 1 − c1 exp(−c2 ). Furthermore Fe ⊂ Ee  ⊂ Ee . Indeed if η ∈ Fe then e both η and η are frameable, thus there exists an allowed path from η to ηe which goes through the corresponding framed configurations which in turn are connected thanks to Corollary 3.3. Therefore μ(∩e∈E  Ee )  1−|E  |(1−μ(Fe ))  1−22 a 2 c1 exp(−c2 ), which goes to one when  → ∞ and the proof is concluded.  

References 1. Aldous, D., Diaconis, P.: The asymmetric one-dimensional constrained Ising model: rigorous results. J. Stat. Phys. 107(5-6), 945–975 (2002) 2. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques, Paris: Société Mathématique de France, 2000, with a preface by Dominique Bakry and Michel Ledoux

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3. Barrat, A., Kurchan, J., Loreto, V., Sellitto, M.: Edwards measures for powders and glasses. Phys. Rev. Lett. 85, 5034–5038 (2000) 4. Berthier, L., Garrahan, J.P., Whitelam, S.: Dynamic criticality in glass forming liquids. Phys. Rev. Lett. 92, 185705–185709 (2004) 5. Bertini, L., Cancrini, N., Cesi, F.: The spectral gap for a Glauber-type dynamics in a continuous gas. Ann. Inst. H. Poincaré Prob. Stat. 38(1), 91–108 (2002) 6. Bertini, L., Toninelli, C.: Exclusion processes with degenerate rates: Convergence to equilibrium and tagged particle. J. Stat. Phys. 117, 549–580 (2004) 7. Cancrini, N., Martinelli, F.: On the spectral gap of Kawasaki dynamics under a mixing condition revisited. J. Math. Phys. 41(3), 1391–1423 (2000) 8. Cancrini, N., Martinelli, F., Roberto, C., Toninelli, C.: Facilitated spin models: recent and new results. Proceedings of Prague summer school 2006 on Mathematical Statistical Mechanics, available at http://arxiv.org/abs/0712.1934v1[math.PR], 2007 9. Cancrini, N., Martinelli, F., Roberto, C., Toninelli, C.: Kinetically constrained spin models. Probab. Th. and Rel. Fields 140, 459–504 (2008) 10. Eisinger, S., Jackle, J.: A hierarchically constrained kinetic ising model. Z. Phys. B 84, 115–124 (1991) 11. Evans, M.R., Sollich, P.: Glassy time-scale divergence and anomalous coarsening in a kinetically constrained spin chain. Phys. Rev. Lett 83, 3238–3241 (1999) 12. Fredrickson, G.H., Andersen, H.C.: Kinetic ising model of the glass transition. Phys. Rev. Lett. 53, 1244– 1247 (1984) 13. Goncalves, P., Landim, C., Toninelli, C.: Hydrodinamic limit for a particle system with degenerate rates. Annals of IHP Prob. Stat. 45, 887–909 (2009) 14. Grimmett, G.: Percolation. Second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 321, Berlin: Springer-Verlag, 1999 15. Kob, W., Andersen, H.C.: Kinetic lattice-gas model of cage effects in high-density liquids and a test of mode-coupling theory of the ideal-glass transition. Phys. Rev. E 48, 4359–4363 (1993) 16. Kordzakhia, G., Lalley, S.: Ergodicity and mixing properties of the northeast models. J. Appl. Probab. 43(3), 782–792 (2006) 17. Kurchan, J., Peliti, L., Sellitto, M.: Aging in lattice-gas models with constrained dynamics. Europhys. Lett 39(4), 365–370 (1997) 18. Lawlor, A., De Gregorio, P., Bradley, P., Sellitto, M., Dawson, K.A.: Geometry of dynamically available empty space is the key to near-arrest dynamics. Phys. Rev. E 72, 021401 (2005) 19. Liggett, T.M.: Interacting particle systems. New York: Springer-Verlag, 1985 20. Lu, S.L., Yau, H.-T.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys. 156(2), 399–433 (1993) 21. Marinari, E., Pitard, E.: Spatial correlations in the relaxation of the kob-andersen model. Europhys. Lett. 69(2), 35–241 (2005) 22. Martinelli, F.: Lectures on Glauber dynamics for discrete spin models, In: Lectures on probability theory and statistics (Saint-Flour, 1997), Berlin: Springer, 1999, pp. 93–191 23. Quastel, J.: Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45(6), 623–679 (1992) 24. Ritort, F., Sollich, P.: Glassy dynamics of kinetically constrained models. Adv. in Phys. 52(4), 219–342 (2003) 25. Sellitto, M., Arenzon, J.J.: Free-volume kinetic models of granular matter. Phys. Rev. E 62, 7793 (2000) 26. Franz, S., Mulet, R., Parisi, G.: Kob-andersen model: A nonstandard mechanism for the glassy transition. Phys. Rev. E 65(2), 021506 (1987) 27. Toninelli, C., Biroli, G.: Dynamical arrest, tracer diffusion and kinetically constrained lattice gases. J. Stat. Phys. 117, 7–54 (2004) 28. Toninelli, C., Biroli, G.: A new class of cellular automata with a discontinuous glass transition. J. Stat. Phys. 130, 83–112 (2008) 29. Toninelli, C., Biroli, G., Fisher, D.S.: Spatial structures and dynamics of kinetically constrained models for glasses. Phys. Rev. Lett. 92(1-2), 185504 (2004) 30. Toninelli, C., Biroli, G., Fisher, D.S.: Kinetically constrained lattice gases for glassy systems. J. Stat. Phys. 120(1-2), 167–238 (2005) Communicated by H. Spohn

Commun. Math. Phys. 297, 345–370 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1012-0

Communications in

Mathematical Physics

Random Quantum Channels I: Graphical Calculus and the Bell State Phenomenon Benoît Collins1,2 , Ion Nechita2 1 Département de Mathématique et Statistique, Université d’Ottawa, 585 King Edward,

Ottawa, ON K1N6N5, Canada. E-mail: [email protected]

2 Université de Lyon, Institut Camille Jordan, 43 Blvd du 11 Novembre 1918,

F-69622 Villeurbanne Cedex, France. E-mail: [email protected] Received: 26 June 2009 / Accepted: 7 December 2009 Published online: 21 February 2010 – © Springer-Verlag 2010

Abstract: This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel. As an application, we study variations of random matrix models introduced by Hayden [7], and show that their eigenvalues converge almost surely. In particular we obtain, for some models, sharp improvements on the value of the largest eigenvalue, and this is shown in further work to have new applications to minimal output entropy inequalities.

1. Introduction, Motivation & Plan The theory of random matrices is a field of its own, but whenever it comes to applications, the driving idea is almost always that although it is very difficult to exhibit matrices having specified properties, a suitably chosen random matrix will have very similar properties as the original matrix with a high probability. This idea has been used successfully for example in operator algebra with the theory of free probability. In 2007 for the first time, a similar leitmotiv was used with success by Hayden in [7] and Hayden-Winter in [9] to disprove the Rényi entropy additivity conjecture for a wide class of parameters p. A proof for the most important case p = 1 was even announced by Hastings in [6] with probabilistic arguments of a different nature. This is arguably the most important conjecture in quantum information theory, and the random matrix models introduced by Hayden and their modifications due to Hastings seemed very new from our random matrix points of view. This paper is therefore an attempt to understand these matrix models with random matrix techniques. For this purpose, we introduce a formalism that is very close to that of planar algebras of Jones [10], and we suggest that the language of quantum gates and planar algebras should be considered as very closely related to each other.

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As an application of our formalism, we obtain Theorems 5.2 and 6.3, which we summarize for comparison purposes in one single theorem below in the case of fixed dimension for the ancilla space: Theorem 1.1. Let Z be the output under a random bi-channel 1 ⊗ 2 of a (maximally entangled) Bell state. We assume that the size of the ancilla space k is fixed, and the size of the input and output size tend to infinity (see Sects. 5 and 6 for the explicit definitions). Then, • If 1 and 2 are chosen independently, almost surely, the asymptotic eigenvalues of Z are 1/k 2 with multiplicity k 2 and 0 with multiplicity n 2 − k 2 . • If 1 is chosen at random and 2 = 1 , then the eigenvalues of the matrix Z converge almost surely towards: k1 + k12 − k13 , with multiplicity one, k12 − k13 , with multiplicity k 2 − 1 and 0, with multiplicity n 2 − k 2 . Our paper is organized as follows. In Sect. 2.1 we recall known facts about integration over unitary groups and their large dimensional asymptotics. This is nowadays known as Weingarten calculus. In Sect. 3, we introduce a graphical model to represent (random) matrices arising in random quantum calculus. Section 4 gives a theoretical method for computing expectations with our graphical model and in the last two sections we give explicit applications of these techniques to random quantum channels. More precisely, in Sect. 5 we investigate tensor products of two independent quantum channels and in Sect. 6 we look at a product of a random channel U with the channel U defined by the conjugate unitary U . Limit theorems presented in this paper are just a sample of what can be accomplished with the calculus developed in Sect. 4. New results will be given in the forthcoming papers [3,4].

2. Background on Weingarten Calculus and Quantum Channels 2.1. Weingarten calculus. This section contains some basic material on unitary integration and Weingarten calculus. A more complete exposition of these matters can be found in [2,5]. We start with the definition of the Weingarten function. Definition 2.1. The unitary Weingarten function Wg(n, σ ) is a function of a dimension parameter n and of a permutation σ in the symmetric group S p . It is the inverse of the function σ → n #σ under the convolution for the symmetric group (#σ denotes the number of cycles of the permutation σ ). Notice that the function σ → n #σ is invertible when n is large, as it behaves like e as n → ∞. If n < p the function is not invertible any more, but we can keep this definition by taking the pseudo-inverse and the theorems below will still hold true (we refer to [5] for historical references and further details). We shall use the shorthand notation Wg(σ ) = Wg(n, σ ) when the dimension parameter n is clear from context. The function Wg is used to compute integrals with respect to the Haar measure on the unitary group (we shall denote by U(n) the unitary group acting on an n-dimensional Hilbert space). The first theorem is as follows: n pδ

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Theorem 2.2. Let n be a positive integer and i = (i 1 , . . . , i p ), i = (i 1 , . . . , i p ), j = ( j1 , . . . , j p ), j = ( j1 , . . . , j p ) be p-tuples of positive integers from {1, 2, . . . , n}. Then  Ui1 j1 · · · Ui p j p Ui1 j1 · · · Ui p j p dU = U (n)



δi1 i 

σ,τ ∈S p

If p = p  then

σ (1)

. . . δi p i 

σ ( p)

δ j1 j 

τ (1)

. . . δ jp j

τ ( p)

Wg(n, τ σ −1 ).

(1)

 U (n)

Ui1 j1 · · · Ui p j p Ui1 j1 · · · Ui   j   dU = 0. p

p

(2)

Since we shall perform integration over large unitary groups, we are interested in the values of the Weingarten function in the limit n → ∞. The following result encloses all the data we need for our computations about the asymptotics of the Wg function; see [2] for a proof. Theorem 2.3. For a permutation σ ∈ S p , let Cycles(σ ) denote the set of cycles of σ . Then  Wg(n, σ ) = Wg(n, c)(1 + O(n −2 )) (3) c∈Cycles(σ )

and Wg(n, (1, . . . , d)) = (−1)d−1 cd−1



(n − j)−1 ,

(4)

−d+1 j d−1

where ci =

(2i)! (i+1)! i!

is the i th Catalan number.

As a shorthand for the quantities in Theorem 2.3, we introduce the function Mob on the symmetric group. Mob is invariant under conjugation and multiplicative over the cycles; further, it satisfies for any permutation σ ∈ S p : Wg(n, σ ) = n −( p+|σ |) (Mob(σ ) + O(n −2 )),

(5)

where |σ | = p − #σ is the length of σ , i.e. the minimal number of transpositions that multiply to σ . We refer to [5] for details about the function Mob. We finish this section by a well known lemma which we will use several times towards the end of the paper. This result is contained in [13]. Lemma 2.4. The function d(σ, τ ) = |σ −1 τ | is an integer valued distance on S p . Besides, it has the following properties: • the diameter of S p is p − 1; • d(·, ·) is left and right translation invariant; • for three permutations σ1 , σ2 , τ ∈ S p , the quantity d(τ, σ1 ) + d(τ, σ2 ) has the same parity as d(σ1 , σ2 ); • the set of geodesic points between the identity permutation id and some permutation σ ∈ S p is in bijection with the set of non-crossing partitions smaller than π , where the partition π encodes the cycle structure of σ . Moreover, the preceding bijection preserves the lattice structure.

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We end this section by the following definition which generalizes the trace function. For some matrices A1 , A2 , . . . , A p ∈ Mn (C) and some permutation σ ∈ S p , we define    Tr Ai1 Ai2 · · · Aik . Tr σ (A1 , . . . , A p ) = c∈Cycles(σ ) c=(i 1 i 2 ··· i k )

We also put Tr σ (A) = Tr σ (A, A, . . . , A). 2.2. Quantum channels. In Quantum Information Theory, a quantum channel is the most general transformation of a quantum system. Quantum channels generalize the unitary evolution of isolated quantum systems to open quantum systems. Mathematically, we recall that a quantum channel is a linear completely positive trace preserving map  from Mn (C) to itself. The trace preservation condition is necessary since quantum channels should map density matrices to density matrices. The complete positivity condition can be stated as ∀d  1,  ⊗ Id : Mnd (C) → Mnd (C) is a positive map. The following two characterizations of quantum channels turn out to be very useful. Proposition 2.5. A linear map  : Mn (C) → Mn (C) is a quantum channel if and only if one of the following two equivalent conditions holds: (1) Stinespring dilation. There exists a finite dimensional Hilbert space K = Cd , a density matrix Y ∈ Md (C) and an unitary operator U ∈ U(nd) such that   (X ) = U,Y (X ) = Tr K U (X ⊗ Y )U ∗ , ∀X ∈ Mn (C). (6) (2) Kraus decomposition. There exists an integer k and matrices L 1 , . . . , L k ∈ Mn (C) such that (X ) =

k 

L i X L i∗ , ∀X ∈ Mn (C),

i=1

and k 

L i∗ L i = In .

i=1

It can be shown that the dimension of the ancilla space K in the Stinespring dilation theorem can be chosen d = dim K = n 2 and that the state Y can always be considered to be a rank one projector. A similar result holds for the number of Kraus operators: one can always find a decomposition with k = n 2 operators. In the final two sections of the paper we study a model of random quantum channels originating from the Stinespring dilation formula (6). We shall be interested in the spectral properties of the elements in the image of such random channels. Quantum channels will be the main field of application of the graphical calculus we develop in the next two sections, our aim being the treatment of some additivity problems in quantum information theory.

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3. Graphical Model In this section, we lay out the foundation for the graphical calculus we shall develop later. We introduce a graphical formalism for representing tensors and tensor contractions that is adapted to quantum information theory. We start at an abstract level, with a purely diagrammatic axiomatization and then we study the Hilbert representations, where graph-theoretic objects shall be associated with concrete elements of Hilbert spaces.

3.1. Diagrams and tensors. Diagrams, boxes, decorations and wires. Our starting point is a set S˜ endowed with an involution without fixed point ∗. The set S˜ splits as S  S ∗ according to the involution. Elements of S˜ are called decorations. A diagram is a collection of decorated boxes and possibly wires (or strings) connecting the boxes along their decorations according to rules which we shall specify. In terms of graph theory, a diagram is an unoriented (multi-)graph whose vertices are boxes, and whose edges are strings. Each vertex comes with a (possibly empty) n-tuple of indices (or decorations or labels) in S˜ n . The number n of decorations may depend on the vertex. We say that two diagrams are isomorphic if they are isomorphic as multi-graphs with labeled vertices. A box is an elementary diagram from which we can construct more elaborate diagrams by putting boxes together and possibly wiring them together. Each box B of a diagram has attached to it a collection of n(B) decorations in S˜ n(B) . The union of the decorations attached to a box B is denoted by S(B)  S ∗ (B). Graphically, boxes are represented by rectangles with symbols corresponding to the decorations attached to them (see Fig. 1). We take the convention that decorations in S ∗ are represented by empty (or white) symbols and decorations in S by full (or black) symbols. Each decoration is thought of as having potentially up to two attachment points. An inner one (which is attached to the box it belongs to) and an outer one, which we shall allow to be attached to a string later on.

Constructing new diagrams out of old ones. Given a family of existing diagrams (e.g. boxes) there exist several ways of creating new diagrams. (1) One can put diagrams together, i.e. take their disjoint union (when it comes to taking representations in Hilbert spaces, this operation will amount to tensoring). One diagram can be viewed as a box. This amounts to specifying an order between the boxes. (2) Given a diagram A and a complex number x, one can create a new diagram A = xA.

Fig. 1. A box M

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(3) Given two boxes A, B having the same n-tuples of decorations, one can define A+ B. This axiom and the previous one (together with evident relations such as A+ A = 2 A which we don’t enumerate in detail) endow the set of identically decorated diagrams with a structure of a complex vector space. (4) One can add wires to an existing diagram (or between two diagrams that have been put together). A wire is allowed between the outer attachment of two decorations only if the decorations have the same shape and different shadings. Such a wire can be created if and only if the two candidate decorations have their outer attachments unoccupied. (5) There exists an anti-linear involution on the diagrams, denoted by ∗. This operation does nothing on the wires. On the boxes, it reverts the shading of the decorations. The involution ∗ is conjugate linear. Hilbert structure. We shall now consider a concrete representation of the diagrams introduced above as tensors in Hilbert spaces. We start by assuming that the set S of full (or black) decorations corresponds to a collection of finite dimensional Hilbert spaces S = {V1 , V2 , · · · }. An important fact that will be useful later is that each Hilbert space Vi comes equipped with an orthonormal basis {e1 , e2 , . . . , edim Vi }. Our aim is to define a ∗-linear map T between the diagrams and tensors in products of Hilbert spaces in the above class and their duals. By duality, white decorations correspond to dual spaces S ∗ = {V1∗ , V2∗ , · · · }. With these conventions, boxes can be seen as tensors whose legs belong to the vector spaces corresponding to its decorations. In a diagram, symbols of the same shape denote isomorphic spaces, but the converse may be false. A particular space Vi (or Vi∗ ) can appear several times in a box. The reader acquainted with quantum mechanics might think of white shapes as corresponding to ‘bras’ and black shapes corresponding to ‘kets’, but we shall get back to quantum mechanical notions later. To a box B we therefore associate a tensor ⎡ ⎤ ⎡ ⎤   TB ∈ ⎣ Vi ⎦ ⊗ ⎣ V j∗ ⎦ . (7) i∈S(B)

j∈S ∗ (B)

Using the canonical duality between tensors and multilinear maps, TB can also be seen as a function   TB : Vj → Vi . j∈S ∗ (B)

i∈S(B)

We use freely partial duality results, and for example, an element of V ⊗ W ∗ can as well be seen as an element of L(W, V ) or L(V ∗ , W ∗ ). Equation (7) defines the map T from the collection of boxes to the collection of vectors in Hilbert spaces obtained by tensoring finitely many copies of Vi , i ∈ S(B)∪S ∗ (B). This map is denoted by T : B  → TB and we now explain how we can extend it to all diagrams. A wire connecting two decorations of the same shape (corresponding to some Hilbert space V ) is associated with the identity map (or tensor) I : V → V . Together with our duality axiom, it also corresponds to a canonical tensor contraction (or trace) C : V ∗ ⊗ V → C.

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We denote the set of wires in a diagram D by C(D). With this notation, a diagram D is associated with the tensor T obtained by applying all the contractions (“wires”) to the product of tensors represented by the boxes. One is left with a tensor ⎡ ⎤    ⎣ ⎦ TD = C TB . C∈C (D )

B box of D

This is well defined (provided that one specifies one total order on the boxes): the order of the factors in the product does not matter, since wires act on different spaces. For a box B, we denote by F S(B) ⊂ S(B) the subset of black decorations which have no wires attached (we call such a decoration free). F S ∗ (B) is defined in the same manner for white decorations (dual spaces). With this notation, the tensor TD associated to a diagram D can be seen in two ways: as an element of a Hilbert space ⎡ ⎤ ⎡ ⎤   TD ∈ ⎣ V j∗ ⎦ ⊗ ⎣ Vi ⎦,  j∈ B F S ∗ (B)

 i∈ B F S(B)

or, equivalently, as a linear map TD :

  j∈ B F S ∗ (B)

Vj →

  i∈ B F S(B)

Vi .

We need two further axioms to ensure that we are indeed dealing with acceptable Hilbert representations. (1) A diagram such that all outer attachments of its decorations are occupied by wires corresponds canonically to an element in C. In addition, a trivial box with a given decoration of type i closed on itself by a wire into a loop takes a value in N. This value is called the dimension of Vi . (2) Given a diagram D, if it is canonically paired to its dual D∗ by strings, the result lies in R+ .

Special diagrams. To make our calculus useful, we need to introduce a few special diagrams (equivalently, boxes) satisfying some specific axioms. (1) The trivial box. A wire connecting two identically shaped decorations of different shading corresponds to the identity map I : V → V . We shall call this box the trivial or the identity box (see Fig. 2). It satisfies the identity axiom represented in Fig. 3.

Fig. 2. Trivial box

Fig. 3. Trivial axiom

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(2) Bras and kets. The simplest boxes one can consider are vectors and linear forms. Following the quantum mechanics ‘bra’ and ‘ket’ vocabulary, vectors, or (1, 0)tensors have no white decorations and only one black decoration, whereas linear forms (or (0,1)-tensors) have one white label and no black labels. Since our Hilbert spaces come equipped with fixed basis, we introduce some special notation for the ket |e1  = e1 and the bra e1 | = e1 , · corresponding to the first vector, as in Fig. 4 (the choice of the first element of the basis being of course arbitrary). (3) The Bell state. Since each space V ∈ S comes equipped with a particular fixed basis dim V , we can define the bra Bell state as the tensor (it is in fact a linear form) {ei }i=1 Bell∗V =

dim V

ei∗ ⊗ ei∗ ,

i=1

and its ket counterpart (which is a vector in V ⊗ V ) Bell V =

dim V

ei ⊗ ei .

i=1

This notation is needed in the sense that Bell states are not canonical and are not well defined from the sole data of V . They rely on some additional real structure of the vector space V which can be encoded by the data of an explicit basis. Bell states are represented in Fig. 5(a). They satisfy the graphical axiom in Fig. 5(b). Bell states play a central role in our formalism; we shall see later that they allow us to define the transposition of a box and even to consider wires connecting identical decorations. (4) Unitary boxes. Boxes associated to unitary matrices U satisfy the graphical axiom depicted in Fig. 6 which corresponds to the identities UU ∗ = U ∗ U = I. 3.2. Examples. Let us now look at some simple diagrams which illustrate this formalism. Suppose that each diagram in Figure 7 comes equipped with two vector spaces V1 and V2 which we shall represent respectively by circle and square shaped symbols.

Fig. 4. Fixed ket and bra

(a)

(b)

Fig. 5. Bell states and axiom

Fig. 6. Unitary axioms

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In the first diagram, M is a tensor (or a matrix, depending on which point of view we adopt) M ∈ V1∗ ⊗ V1 , and the wire applies the contraction V1∗ ⊗ V1 → C to M. The result of the diagram Da is thus TDa = Tr(M) ∈ C. In the second diagram, again there are no free decorations, hence the result is the complex number TDb = y, Mx. Finally, in the third example, N is a (2, 2) tensor or a linear map N ∈ L(V1 ⊗ V2 , V1 ⊗ V2 ). When one applies to the tensor N the contraction of the couple (V1 , V1∗ ), the result is the partial trace of N over the space V1 : TDc = Tr V1 (N ) ∈ L(V2 , V2 ). Bell states allow us to introduce the transposition operation for a tensor (or a box) as follows. We define, as usual, transposition for a matrix M (or a tensor M ∈ V ∗ ⊗ V ) and we extend it in a trivial way to more general situations. Graphically, the box corresponding to the transposed tensor M t is defined in Fig. 8(a); it consists in connecting an appropriate Bell state to each decoration of the box. Note however that this operation is different from the involution ∗ applied to the same box. Moreover, Bell states allow for wires connecting identical shaped symbols of the same color, as in Fig. 8(b). Such noncanonical tensor contractions (V ⊗ V → C or V ∗ ⊗ V ∗ → C) are shorthand graphical notations for the corresponding diagram containing a Bell state, and we shall use them quite often in what follows. Also, for reasons which shall be clear later, we shall sometimes make substitution M = (M ∗ )t . Finally, by grouping two Bell states together, one obtains the (non-canonical) tensor E (Fig. 9), called “the maximal entangled state”. It corresponds to the tensor E=

dim V dim V i=1

ei∗ ⊗ ei∗ ⊗ e j ⊗ e j ∈ V ∗ ⊗ V ∗ ⊗ V ⊗ V.

j=1

The reader with background in quantum information will notice that the maximally entangled state we just defined is not normalized in order to be a density matrix. The reader with background in planar algebra theory will recognize a multiple of the Jones projection. The diagram in Fig. 10 is of crucial importance in what follows. It corresponds to a quantum channel in its Stinespring representation (see Eq. (6)). Round shaped inputs and outputs correspond to the Hilbert space H and squares correspond to the “environment” K. We shall study such diagrams with random interaction unitaries U in Sects. 5 and 6.

(a)

(b)

(c)

Fig. 7. Some simple diagrams

(a) Fig. 8. Bell states and transposition

(b)

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Fig. 9. The maximal entangled state E

Fig. 10. Diagram for a quantum channel in its Stinespring form

3.3. Comments on other existing graphical calculi. The above formalism is the one that seemed the most compatible with Weingarten calculus. Here, we comment about already existing graphical formalisms, in the hope that this section will serve as a dictionary for the reader acquainted with one of the calculi below. Our calculus is mainly inspired by Bob Coecke’s Kindergarten Quantum Mechanics [1]. However we choose not to orient the strings; rather, we separate with color (black/ white) the vector spaces and their duals, therefore there is only one possible pairing. A common feature of the two calculi is the central place occupied in the formalisms by Bell states. V.F.R. Jones’s theory of planar algebras [10] is also connected to our graphical calculus. One of our diagrammatic axioms is the existence of a Bell state. This is very closely related to the axioms of Temperley Lieb algebras and the diagrammatic for a Jones projection. Most of our calculus could take place in Jones’ bipartite graph planar algebra. However it is not clear whether planarity plays an important role in our calculus. More generally, one could view our calculus as fitting in the frame of traced monoidal (or tensor) categories. Here, our objects are our elementary family of Hilbert spaces, their duals and all their finite Hilbert tensor products. The monoidal structure corresponds graphically, consists in copying two diagrams side to side, and amounts to taking the tensor product of the Hilbert spaces. The trace corresponds to the conditional expectations obtained by our wiring procedure. We refer for example to [11]. 4. Planar Expansion In this section, we consider diagrams that may contain random matrices. This is where probability theory comes into play; we focus on the case where the random elements appearing in the diagrams are random unitary Haar-distributed on some finite dimensional unitary group. Our task is to compute expectation values E(D) of diagrams D containing boxes associated with random unitary operations. We shall write E(D) as a weighted sum of some diagrams obtained from D that do not contain anymore random tensors. 4.1. Expectation of a diagram containing random independent unitary matrices. Suppose we have a diagram D that has boxes of two types: either boxes of type U, U ∗ , U or U t , where U is a unitary random variable in a fixed space of type End(⊗i∈I Vi ), distributed according to the Haar measure on the unitary group U(n) on this space,

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or other boxes which are independent (as classical random variables) from U (this includes deterministic boxes or tensors). We shall now present an algorithm for computing the expectation of such a diagram with respect to the probability law of U . Before describing the algorithm, let us note that if a diagram contains several independent Haar unitary matrices, one can recursively apply the algorithm to compute the expectation over all the random unitary matrices appearing in the diagram. The first step in our algorithm is to ensure that D contains only boxes of type U and U . This can be done by using the transposition transformation via Bell states, and replacing U t boxes by U boxes with the opposite shading of the decorations and U ∗ boxes by U boxes. Next, we introduce a concept of removal of boxes U and U . A removal r is a procedure which transforms a diagram D into a new diagram Dr which does not contain either U or U boxes. In other words, r is a way to remove random unitaries U from a diagram D. The set of all admissible removal procedures for a diagram D will be denoted by Rem(D). We now move on to describe removal procedures and how they operate on diagrams. First of all, a removal is not possible if the number of boxes U in D is different from the number of boxes U . In such a case, the set Rem(D) will be defined to be empty. This rule is the diagrammatic equivalent of Eq. (2) from Theorem 2.2. Assuming that the number of U boxes and U boxes is the same, a removal r is a way to pair decorations of the U and U boxes appearing in a diagram. More precisely, r is the data of a pairing α of the white decorations of U boxes with the white decorations of U boxes, together with a pairing β between the black decorations of U boxes and the black decorations of U boxes. Assuming that D contains p boxes of type U and that the boxes U (resp. U ) are labeled from 1 to p, then r = (α, β), where α, β are permutations of S p . Given a removal r ∈ Rem(D), we construct a new diagram Dr associated to r , which has the important property that it no longer contains boxes of type U or U . We proceed in the following way: one starts by erasing the boxes U and U but keeps the decorations attached to them. Assuming that one has labeled the erased boxes U and U with integers from {1, . . . , p}, one connects all the (inner parts of the) white decorations of the i th erased U box with the corresponding (inner parts of the) white decorations of the α(i)th erased U box. In a similar manner, one uses the permutation β to connect black decorations. A diagrammatic explanation of the above algorithm is described in Fig. 11. In the Appendix, the above algorithm is illustrated on an example. We are now ready to state the main result of this section.

Fig. 11. Removal: elimination of boxes and pairing of decorations

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Theorem 4.1. The following holds true:  EU (D) =

Dr Wg(n, αβ −1 ).

r =(α,β)∈Rem(D )

Proof. This is just Weingarten calculus of Theorem 2.2 applied to our graphical conventions.   When more than one independent unitary matrices U, V, . . . are present in a diagram, we proceed by induction over each independent matrix: we successively remove U , V , etc. One can check directly that the order of the induction does not change the final result. This is compatible with the probabilistic property of the expectation, EU,V (D) = EV (EU (D)). Theorem 4.1 might just look like a reformulation of known results, but without this graphical method, obtaining the main results of this paper is extremely cumbersome and very counterintuitive. Let us now comment on the first step of our removal algorithm, replacing U ∗ boxes with U boxes. The purpose of such a substitution is purely practical: later in the removal procedure, we pair decorations of the same color. If we should have decided to work with U and U ∗ boxes, one should always pair decorations of different colors, and this can turn out to be rather cumbersome when doing combinatorics. On the other hand, each time we replace a U ∗ box by a U box, we introduce two more Bell states into our diagram (see Fig. 8); although we decided not to display such states and rather to allow wires connecting decorations of the same color, this operation increases in some sense the “complexity” of the diagram. In the next sub-section we present a warm-up toy example of Theorem 4.1. Further applications of the above theorem will be considered in a forthcoming paper [4], where problems from free probability theory will be treated using similar techniques. 4.2. First example: partial tracing a randomly rotated matrix. As a first application of the graphical formalism, we consider the following problem. Let X ∈ Mnk (C) be a deterministic matrix. In a manner similar to random quantum channels, we define, for a fixed integer parameter k  1, the random matrix Y = Tr k [U XU ∗ ] ∈ Mn (C), where U ∈ U(nk) is a Haar distributed random unitary matrix. Notice that we are considering here non-normalized traces. Using our graphical formalism, we shall compute the moments of Y , E[Tr(Y p )] for all p  1. After replacing U ∗ boxes with U , one gets the diagram in Fig. 12, where round decorations correspond to Cn and square ones to Ck . Note that in Fig. 12 there are p groups “U XU ” wired together. By Theorem 4.1, the expectation (with respect to the Haar measure of the unitary group U(nk)) of this diagram is a weighted sum (with Weingarten weights) of diagrams

Fig. 12. The diagram for Tr(Y p )

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Dr obtained after the removal of U and U boxes. Such diagrams Dr contain only X blocks and loops of different types. Let us compute the value of a diagram Dr , where the removal is given by r = (α, β) ∈ S 2p . Suppose we number the boxes from 1 to p and the permutations α connects the white decorations and β the black decorations. The point here is that the permutation α will be responsible for the loops appearing in Dr , whereas β will connect X boxes. We start by counting the number of loops in Dr . Loops are of two types, the ones containing the top, square decorations (a loop of this type has a value of k) and the ones that come from the bottom, round decorations, each with a value of n. Since the top decorations are already connected (in the original diagram D) by the identity permutation, the number of loops of the first type is given by the number of cycles of α, #α; they give a total contribution of k #α . Loops corresponding to round decorations are initially connected by the cycle γ = ( p p − 1 · · · 3 2 1) ∈ S p .

(8)

Hence, the number of such loops is #(γ −1 α) and they give a total contribution of −1 −1 n #(γ α) . In conclusion, the total contribution of loops is k #α n #(γ α) . The contribution of the X boxes is straightforward to compute, since these boxes are connected only by β. After the removal we get #β connected components of powers of X , and the total contribution is Tr β (X ). Putting all these factors together, we obtain the following proposition: Proposition 4.2. The mean p th moment of the random matrix Y = Tr k [U XU ∗ ] is given by  −1 E[Tr(Y p )] = k #α n #( α) Tr β (X ) Wg(αβ −1 ). (9) α,β∈S p

In the particular case where X is a rank one projector, one has Tr β (X ) = 1 for all permutations β and since ⎛ ⎞−1 p−1   Wg(nk, σ ) = ⎝ (nk + j)⎠ , σ ∈S p

j=0

one obtains the following simplification: ⎛ ⎞−1 p−1   −1 E[Tr(Y p )] = ⎝ (nk + j)⎠ k #α n #(γ α) . j=0

(10)

α∈S p

We will discuss at length the simplifications that occur when dealing with rank one projectors in the forthcoming paper [4]. 5. Tensor Products of Independent Random Quantum Channels In the present section and in the next one, we consider two different models of tensor products of random quantum channels. In both cases, we first fix an interesting input state X 12 (the Bell state) and investigate the random matrix [1 ⊗ 2 ](X 12 ).

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Our two models correspond to the choice of two different random pairs (1 , 2 ). In both models, the channels 1 , 2 are defined by random unitary matrices U1 , U2 via the Stinespring representation introduced in Eq. (6). The difference between the two cases lies in the correlation between the random matrices U1 and U2 . In the first model, interaction unitaries U1,2 are independent Haar distributed random matrices. The second model, which is more involved, deals with the case where U1 is distributed according to the Haar measure and U2 = U1 . This choice introduces an interesting symmetry into the problem and such a construction has become classical in quantum information theory [6,7]. Asymptotic results in this case shed light on the interesting phenomenon that the output of the product channel has one “large” eigenvalue. We call this phenomenon the Bell state phenomenon. In order to simplify the notation, we shall assume that Y1 and Y2 are rank-one projectors and that n 1 = n 2 = n, k1 = k2 = k. Before looking in detail at the two models of interest, let us make one brief comment on the choice of the input state of the channels. It is clear that if one chooses an input state which factorizes X 12 = X 1 ⊗ X 2 , then [1 ⊗ 2 ](X 12 ) = 1 (X 1 ) ⊗ 2 (X 2 ), and there is no correlation (classical or quantum) between the channels. In order to avoid such trivial situations, one has to choose an input state which is entangled. An obvious choice (given that n 1 = n 2 = n) is to take X 12 = E n , the n-dimensional Bell state (or the maximally entangled state), and we shall use this state in what follows. The first model, although new, does not bring strikingly new results from the random matrix point of view. We treat it here as an illustration of what our calculus can allow to compute, and as a point of comparison with the second model.

Independent interaction unitaries. In the remainder of this section, we consider two independent realizations U1 = U and U2 = V of Haar-distributed unitary random matrices on U(nk). For both channels the state of the environment is a rank-one projector and we are interested in the n 2 × n 2 random matrix Z = [U ⊗ V ](E n ), where E n is the maximal entangled Bell state (notice the 1/n normalization) En =

n 1  |ei e j | ⊗ |ei e j |. n i, j=1

The diagram associated with the (2,2) tensor Z is drawn in Fig. 13; we chose to represent by squares decorations corresponding to Ck and by circles decorations corresponding to Cn . As usual, we are interested in computing the moments E[Tr(Z p )] for all p  1 using the graphical method. We start by replacing U ∗ (resp. V ∗ ) blocks by U (resp. V ) blocks. An important point here is that there are two types of blocks corresponding to the independent random unitary matrices U and V (when computing the p th moment of Z , there are p blocks of each type). This has two important consequences: when expanding the diagram in order to compute the expectation of the trace, one can only pair U blocks with U blocks and V blocks with V blocks; “cross-pairings” between U blocks and V

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359

Fig. 13. Z = U ⊗ V (E n )

blocks are not allowed by the expansion algorithm. This algorithm proceeds iteratively, first by removing, say, the U blocks (the V blocks being treated as constants) and then by removing the V blocks. Hence “cross-pairings” cannot occur. The second consequence of the presence of two independent Haar unitary matrices is that in the final expression for the expectation of the diagram, there will be two Weingarten weights, one for each independent unitary integration. Lemma 5.1. The following holds true (γ is the cycle permutation defined in Eq. (8)): E[Tr(Z p )]  =

k #αU +#αV n #(γ

−1 α )+#(γ −1 α )+#(β −1 β )− p U V U V

Wg(αU βU−1 ) Wg(αV βV−1 ).

αU ,βU ,αV ,βV ∈S p

(11) Proof. As has already been stated, the expectation with respect to both unitary matrices U and V can be seen as the result of two removal procedures, and hence the Weingarten sum shall be indexed by a pair of removals (rU , r V ). In other words, the Weingarten sum shall be indexed by 2 pairs of permutations, one for each type of block; we shall denote them by αU , βU , αV , βV ∈ S p . The four permutations are responsible for pairing blocks in the following way (1  i  p): (1) the white decorations of the i th U -block are paired with the white decorations of the αU (i)th U block; (2) the black decorations of the i th U -block are paired with the black decorations of the βU (i)th U block; (3) the white decorations of the i th V -block are paired with the white decorations of the αV (i)th V block; (4) the black decorations of the i th V -block are paired with the black decorations of the βV (i)th V block. The diagram associated with Tr(Z p ) contains, aside from the random unitary blocks, deterministic bras and kets . However, these boxes have a trivial contribution of 1 to the final result. Hence, the result of the graph expansion is a (sum over a) collection of loops, multiplied by some scalar factor. The different contributions of a quadruple (αU , βU , αV , βV ) ∈ S 4p are given by (recall that circles correspond to n-dimensional spaces and squares correspond to k-dimensional spaces): 1. “ U ”-loops: k #αU ; 2. “ V ”-loops: k #αV ; −1 3. “ U ”-loops: n #(γ αU ) ;

360

4. 5. 6. 7. 8. 9. 10.

B. Collins, I. Nechita −1

“ V ”-loops: n #(γ αV ) ; “ U ”-loops: none; “ V ”-loops: none; −1 “ U, V ”-loops: n #(βU βV ) ; normalization factors 1/n from the Bell matrices E n : n − p ; Weingarten weights for the U -matrices: Wg(αU βU−1 ); Weingarten weights for the V -matrices: Wg(αV βV−1 ).

Adding all these contributions, we obtain the announced exact closed-form expression.  

Asymptotics. The preceding expression is intractable at fixed n and k, so we study two asymptotic regimes: (I) n fixed, k → ∞; (II) k fixed, n → ∞. At this stage, before looking into each particular asymptotic regime, we can make an important observation. Notice that in the preceding expression, aside from the factor −1 n #(βU βV ) , the general term of the sum factorizes into a “(αU , βU )” part and a “(αV , βV )” part. This was to be expected, since the coupling between the two channels is realized −1 by the input state E n which has a contribution of n #(βU βV )− p . Let us also note that a third interesting asymptotic regime, n, k → ∞, k/n → c > 0 will be studied using the same methods in a forthcoming paper. Theorem 5.2. In the first regime, n fixed, k → ∞, the output of the tensor product of the channels, for large values of k, is close to the chaotic state ρ∗ =

In 2 . n2

In the second regime, k fixed, n → ∞, the asymptotic eigenvalues of Z are 1/k 2 with multiplicity k 2 and 0 with multiplicity n 2 − k 2 . Proof. Using the standard asymptotics for the Weingarten functions, k→∞

−1

k→∞

−1

Wg(αU βU−1 ) ∼ (nk)− p−|αU βU | Mob(αU βU−1 ) and Wg(αV βV−1 ) ∼ (nk)− p−|αV βV | Mob(αV βV−1 ), the power of k appearing in a general (αU , βU , αV , βV ) term is −1

k p−|αU |+ p−|αV |− p−|αU βU

|− p−|αV βV−1 |

−1

= k −(|αU |+|αV |+|αU βU

|+|αV βV−1 |)

.

It is obvious that all the terms converge to zero, except the one with αU = βU = αV = βV = id. Using Mob(id) = 1, we conclude that lim E[Tr(Z p )] = n 2−2 p .

k→∞

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361

One can restate this in terms of the empirical eigenvalue distribution of the n 2 × n 2 matrix Z : 2

μZ =

n 1  δλi (Z ) −→ δ1/n 2 . k→∞ n2 i=1

In other words, the output of the tensor product of the channels, for large values of k, is close to the chaotic state ρ∗ =

In 2 . n2

As for the second regime, using similar considerations, we obtain lim E[Tr(Z p )] = k 2−2 p .

n→∞

  Note that both regimes presented here are trivial to some extent. We could prove at only a small additional cost that the convergence of the eigenvalues is actually almost sure. See the next section for the technique of proof, a direct adaptation of the BorelCantelli lemma. Finally, we want to emphasize that the asymptotic behavior of the output in the second regime changes drastically when considering the conjugate case, and that this fact will have very important consequences in the theory of quantum information. 6. Tensor Products of Conjugate Random Quantum Channels We have seen that tensor products of independent random channels have an eigenvalue behavior close to the single channel case (see [12] for the treatment of the single channel case) - despite the fact that the input state is maximally entangled. In this section, we consider the case where U1 = U , U2 = U and U ∈ U(nk) is a Haar uniform random unitary matrix. Tensor products of channels of this type are now classical in the literature (see [6,7]). One of the reasons why channels of this particular form receive such attention is that one can show that the product channel has a “trivial large eigenvalue” of order 1/k. We shall provide a graphical proof of this fact later, in Lemma 6.6. Again, we are interested in the moments of the random matrix Z = U ⊗ U (E n ), depicted in the Fig. 14. This time calculations are more complicated, because only one unitary matrix appears in the product channel. This means that in the removal algorithm, one can pair boxes

Fig. 14. Z = U ⊗ U (E n )

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from U with boxes from U , thus obtaining more complicated patterns. Another consequence of the fact that we use only one random unitary matrix is that the Weingarten sums are indexed by only one pair of permutations (α, β) ∈ S22p . In order to count the loops obtained after the graph expansion, we label the U and the U boxes in the following manner: 1T , 2T , . . . , p T for the U boxes of the first channel (T as “top”) and 1 B , 2 B , . . . , p B for the U boxes of the second channel (B as “bottom”). We shall also order the labels as {1T , 2T , . . . , p T , 1 B , 2 B , . . . , p B }  {1, . . . , 2 p}. A removal r = (α, β) ∈ S22p of the random (U and U ) boxes connects the decorations in the following way: (1) the white decorations of the i th U -block are paired with the white decorations of the α(i)th U block; (2) the black decorations of the i th U -block are paired with the black decorations of the β(i)th U block. Some explicit diagrams obtained by the graph expansion algorithm are illustrated in the Appendix. Next, we introduce two fixed permutations γ , δ ∈ S2 p which will be useful in counting the loops. The permutation γ represents the initial wiring of the decorations (before the graph expansion) and δ accounts for the wires between the decorations (which come from E n ). More precisely, for all i, γ (i T ) = (i − 1)T , γ (i B ) = (i + 1) B , and δ(i T ) = i B , δ(i B ) = i T .

(12)

We are now ready to compute the average moments of the random matrix Z . Lemma 6.1. The following holds true  −1 E[Tr(Z p )] = k #α n #(αγ )+#(βδ)− p Wg(αβ −1 ).

(13)

α,β∈S2 p

Proof. With the notations introduced above, we can now count the contributions for each individual pairing (α, β): (1) (2) (3) (4) (5) (6)

“ ”-loops: k #α ; −1 “ ”-loops: n #(αγ ) ; “ ”-loops: none; −1 “ ”-loops: n #(βδ ) = n #(βδ) (notice that δ is an involution); normalization factors 1/n from the E n matrices: n − p ; Weingarten weights for the U -matrices: Wg(αβ −1 ). Adding up all the above contributions, we obtain the claimed formula.

 

In the rest of the section, we shall focus on the asymptotic regime n fixed, k → ∞, and in the next section we shall look into the more interesting case k fixed, n → ∞. Before stating the asymptotic result, let us make two preliminary remarks. In the case of the conjugate product channel, since only one unitary matrix appears in the diagrams, there is a notable difference concerning the asymptotics of the Weingarten function: Wg(αβ −1 ) ∼ (nk)−2 p−|αβ

−1 |

Mob(αβ −1 ).

One can easily compute the following quantities involving the permutations γ and δ which will be useful later, when doing asymptotics: |γ | = 2 p − 2, |δ| = p, |γ δ| = p.

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Proposition 6.2. In the asymptotic regime where n is fixed and k → ∞, the random matrix Z converges to the chaotic state ρ∗ =

In 2 . n2

Proof. Computing the asymptotic trace for large k gives  −1 −1 −1 E[Tr(Z p )] ∼ k −(|α|+|αβ |) n p−(|αγ |+|βδ|+|αβ |) Mob(αβ −1 ). α,β∈S2 p

Minimizing the power of k above gives |α|+|αβ −1 |  0, with equality iff α = β = id, hence lim E[Tr(Z p )] = n 2−2 p ,

k→∞

and the conclusion is the same as in the case of two independent quantum channels: the output Z is asymptotically close to the chaotic state ρ∗ .   6.1. Conjugate channels, the Bell phenomenon. We are left with studying the most interesting regime, k fixed and n → ∞. Our main result is as follows: Theorem 6.3. In the regime of k fixed, n → ∞, the eigenvalues of the matrix Z converge almost surely towards: • k1 + k12 − k13 , with multiplicity one; • k12 − k13 , with multiplicity k 2 − 1; • 0, with multiplicity n 2 − k 2 . Note that it follows from the Stinespring theorem that Z has at most k 2 non-zero eigenvalues. Therefore a moment approach is possible for the proof. We start with a technical lemma about the structure of geodesics between the specific permutations γ and δ introduced in Eq. (12).   Lemma 6.4. For 1  i  p, let τi be the transposition i T , (i − 1) B . Then the permutations  α on the geodesic γ → α → δ are indexed by subsets A ⊆ {1, . . . , p}: α = γ i∈A τi . Moreover, for such a permutation, we have  2p − 2 if A = ∅, |α| = 2 p − |A| if A = ∅. Proof. If A = ∅, then |α| = |γ | = 2 p − 2. Otherwise, after computing the action of α,  iB if i ∈ A, T α(i ) = T (i − 1) if i ∈ / A;  iT if (i + 1) ∈ A, α(i B ) = / A; (i + 1) B if (i + 1) ∈ it is easy to see that each element i of A spans a cycle of α and thus |α| = 2 p − #α = 2 p − |A|.  

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We split this proof of Theorem 6.3 in two steps: first we prove the convergence in expectation, and then we prove the almost sure convergence. Proof of the convergence in expectation. Using the same asymptotic formula as in the previous section (this time for large n), the quantity one wants to minimize in this case is |αγ −1 | + |βδ| + |αβ −1 | = |γ −1 α| + |α −1 β| + |β −1 δ|  |γ −1 δ| = p. Equality is attained when γ → α → β → δ is a geodesic in S2 p . Using this observation, we obtain  −1 E[Tr(Z p )] ∼ k −(|α|+|αβ |) Mob(αβ −1 ). γ →α→β→δ

It turns out that we can compute exactly the last sum as follows. First, notice that the −1 β → γ −1 δ. geodesic condition γ → α → β → δ can be restated as id → γ −1 α → γ p −1 −1 But γ δ is a product of p transpositions with disjoint support: γ δ = i=1 τi , where  T B τi is the transposition i , (i − 1) for all 1  i  p. As in Lemma 6.4, permutations on a geodesic between id and γ −1 δ are parameterized by subsets of {1, . . . , p} as follows. Permutations γ −1 α and γ −1 β lie on a geodesic between id and γ −1 δ (i.e. id → γ −1 α → γ −1 β → γ −1 δ) if and only if there exist two subsets ∅ ⊆ A ⊆ B ⊆ {1, 2, . . . , p} such that  τi , γ −1 α = i∈A

γ

−1

β=



τi .

i∈B

For two such permutations, it is obvious that |α −1 β| = |(γ −1 α)−1 γ −1 β| = |B \ A|. In order to compute |α|, we rely on Lemma 6.4. Since α −1 β is a product of |B \ A| transpositions of disjoint support, it follows by [5] that Mob(α −1 β) = (−1)|B\A| and we are left with the following expression:   E[Tr(Z p )] ∼ k −(2 p−2+|B|) (−1)|B| + k −(2 p−|A|+|B\A|) (−1)|B\A| . ∅= A⊆B

B

Using the multinomial identities 

x|A| = (1 + x) p and

∅⊆A⊆{1,..., p}



x|A| y |B\A| = (1 + x + y) p ,

∅⊆A⊆B⊆{1,..., p}

we obtain the asymptotic traces of the output matrix Z :     1 1 1 p 1 1 p p 2 + − 3 + (k − 1) 2 − 3 . E[Tr(Z )] ∼ k k2 k k k We conclude that the matrix Z has, asymptotically, the following eigenvalues:

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• k1 + k12 − k13 , with multiplicity one; • k12 − k13 , with multiplicity k 2 − 1;  • 0, with multiplicity n 2 − k 2 .  Next we move on to the proof of almost sure convergence. We would like to mention about the proof below that we believe that it should be possible to prove that  2  = O(n −2 ) E Tr(Z p ) − E Tr(Z p ) simply by observing that the function U → Tr(Z p ) is Lipschitz on the unitary group and by applying a Gromov-Milman type concentration measure argument. We refer to [8] for an exposition of such techniques. The authors acknowledge that this approach might be slightly less cumbersome in the specific case of this proof. However, we chose to keep our proof of a combinatorial nature for the sake of coherence. Proof of the almost sure convergence. It is a standard technique in probability theory that in order to show the almost-sure convergence of the eigenvalues of Z to their respective limits, it suffices for the covariance series to converge, for all values of p: ∞   2  < ∞. E Tr(Z p ) − E Tr(Z p ) n=1

Indeed, this inequality together with the Borel-Cantelli lemma imply that almost surely as n → ∞, Tr(Z p ) → E Tr(Z p ). The two ingredients which make the proof work are the following. The first one is the fact that the error one makes when approximating the Weingarten function with its dominating term is of the order −2: Wg(α) = (nk)−( p+|α|) (Mob(α) + O((nk)−2 )). This follows from Theorem 2.3 and the definition of Mob below. The second ingredient is contained in the geodesic inequality |γ −1 α| + |α −1 β| + |β −1 δ|  |γ −1 δ| = p. Earlier, we have completely described the set of couples (α, β) which saturate the equality. It turns out that one can say more on the values of the function (α, β) → E(α, β) = |γ −1 α| + |α −1 β| + |β −1 δ| − p, as follows. Applying Lemma 2.4 two times, it is clear that the values taken by E(α, β) are all even, E(α, β) = 1, and thus  −1 E[Tr(Z p )] = k #α n #(αγ )+#(βδ)− p Wg(αβ −1 ) α,β∈S2 p

=

 γ →α→β→δ

k

−(|α|+|αβ −1 |)

Mob(αβ

−1

 )+O

1 n2

 .

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We start by computing the easiest term of the covariance, namely E[Tr(Z )] = p

2



k

−(|α1 |+|α1 β1−1 |+|α2 |+|α2 β2−1 |)

Mob(α1 β1−1 ) Mob(α2 β2−1 ) +

 O

γ →α1 →β1 →δ γ →α2 →β2 →δ

 1 . n2

The second term E[Tr(Z p )2 ] is more difficult to estimate and one needs to introduce the permutations γ¯ , δ¯ ∈ S4 p : γ¯ = (1T 2T · · · p T )(( p + 1)T ( p + 2)T · · · (2 p)T )(1 B 2 B · · · p B ) × (( p + 1) B ( p + 2) B · · · (2 p) B ); δ¯ = (1T 1 B )(2T 2 B ) · · · ( p T p B )(( p + 1)T ( p + 1) B ) · · · ((2 p)T (2 p) B ). With this notation, we have  −1 ¯ E[Tr(Z p )2 ] = k #α n #(α γ¯ )+#(β δ)− p Wg(αβ −1 ) α,β∈S4 p

=



k −(|α|+|αβ

−1 |)

n 2 p−(|γ¯

−1 α|+|α −1 β|+|β −1 δ|) ¯

Mob(αβ −1 ) + O



α,β∈S4 p

1 n2

 .

¯  |γ¯ −1 δ| ¯ = 2 p. Since both γ¯ and δ¯ One can easily show that |γ¯ −1 α|+|α −1 β|+|β −1 δ| T,B T,B T,B leave invariant the sets {1 , 2 , . . . , p } and {( p + 1)T,B , ( p + 2)T,B , . . . , 2 p T,B }, geodesic couples (α, β) are obtained as direct sums α = α1 ⊕ α2 , β = β1 ⊕ β2 , where α1 , β1 ∈ S({1T,B , 2T,B , . . . , p T,B }), α2 β2 ∈ S({( p + 1)T,B , ( p + 2)T,B , . . . , 2 p T,B }) are such that γ1 → α1 → β1 → δ1 and γ2 → α2 → β2 → δ2 are geodesics (the permutations γ1,2 and δ1,2 are defined in an obvious way). One has also that |α| = |α1 | + |α2 |, |β| = |β1 | + |β2 | and that Mob(αβ −1 ) = Mob(α1 β1−1 ) Mob(α2 β2−1 ). Putting all this together, one gets the final expression which ends the proof E[Tr(Z p )2 ] =

 γ →α1 →β1 →δ γ →α2 →β2 →δ

−1 −1 k −(|α1 |+|α1 β1 |+|α2 |+|α2 β2 |) Mob(α1 β1−1 ) Mob(α2 β2−1 ) + O



 1 . n2

  6.2. Generalization of Theorem 6.3 and discussion. We finish this paper by studying a generalization of the model investigated in the previous section. We consider the case where k is a fixed integer, and t ∈ (0, 1) is a fixed number (possibly a function of k). For each n, we consider a random unitary matrix U ∈ Mnk (C), and a projection qn of Mnk (C) of rank pn such that pn /(nk) ∼ t as n → ∞. Our model of a random quantum channel is  : M pn (C) → Mn (C)

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Fig. 15. Z = U ⊗ U (E tnk )

Fig. 16. Re-writing Tr(Z n E n )

given by (X ) = Tr k (U XU ∗ ), where the density matrix X satisfies X  qn (in other words we consider the isomorphism qn Mnk (C)qn  M pn (C)). Graphically, our model amounts to Fig. 15. As usual, we are interested in the process given by the eigenvalues of Z as n → ∞ (in our setup, k is fixed). Here, with almost the same techniques as in Theorem 6.3, we obtain the following result. Theorem 6.5. Almost surely, as n → ∞, the random matrix  ⊗ (E tnk ) ∈ Mn 2 (C) has non-zero eigenvalues converging towards ⎞ ⎛ ⎜ 1−t 1−t 1−t⎟ ⎟ γ (t) = ⎜ ⎝t + k 2 , k 2 , . . . , k 2 ⎠.    k 2 −1 times

Proof. Theorem 6.3 is a particular case of this theorem with t = (1 − 1/k) and it has been worked out in great detail. Therefore we leave it to the reader to work out the appropriate modifications to this case.   The striking fact here is that the largest eigenvalue behaves almost surely like t + (1 − t)/k 2 . The existence of a large eigenvalue was already anticipated by Hayden in [7], Lemma II.2. The lemma below is a slight generalization of Hayden’s lemma, following his idea. Lemma 6.6. In the above model, the largest eigenvalue is at least t. For the sake of being self contained, we give a proof of this fact. Moreover, the proof is graphical, using our diagrammatic calculus developed in Sect. 3.

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Proof. Following [7], it is enough to prove that   1 1 Tr(Z n E n ) =  √ Belln , U ⊗ U (E tkn ) √ Belln   t. n n In order to accomplish this, we use the diagram invariance to stretch outside the inner parts of the diagram and then we notice that the resulting diagram is of the form X, X  2 for some X ∈ Ck , see Fig. 16. Introducing the orthogonal projector A = Ik 2 −E k ∈ Mk 2 (C), it is obvious that X, AX   0 and thus the inequality in Fig. 17 holds. Note that we have replaced the identity operator Ik 2 connecting X and X ∗ by the maximally entangled state on Ck ⊗ Ck . Now we use the unitary axioms on each of the two connected diagrams above and we obtain the result in Fig. 18. Putting all the factors together, we get Tr(Z n E n ) 

1 (tnk)2 = t, tn 2 k 2

and then, since E n is an orthogonal projector, we conclude that the largest eigenvalue of Z n is at least t.   To conclude, let us compare Theorem 6.5 and Lemma 6.6. Independently of the choice of t and k, the value that we obtain almost surely for the largest eigenvalue in Theorem 6.5 improves strictly the lower bound for the largest eigenvalue obtained in Lemma 6.6, as t + (1 − t)/k 2 > t. For fixed t, the relative improvement (t + (1 − t)/k 2 )/t becomes small as k becomes big. On the other hand, if t  k −2 , Lemma 6.6 does not bring new information, as the largest eigenvalue is always at least k −2 , whereas Theorem 6.5 brings new information. So, the relative improvement is optimal for t ∼ k −2 . This heuristic study leads us to think that it is possible to improve known counterexamples about the various entropy additivity conjectures, and this is the object of study of our second paper [3].

Fig. 17. Replacing Ik 2 by E k

Fig. 18. Application of the unitary axioms

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Acknowledgements. One author (B.C.) would like to thank P. Hayden for an enlightening talk and conversations in Guadalajara during the summer 2007. This is where he discovered random quantum channels and additivity problems. He is also grateful to the audiences of two preliminary lectures about these results in ´ Sendai and Yokohama. He also thanks T. Hayashi, R. Burstein and P. Sniady for discussions at various stages about related topics. Both authors would like to thank the referee for a careful reading of the manuscript and for useful suggestions of improvement. This research was conducted partly in Lyon 1 and Ottawa. The authors are grateful to these institutions for hosting their research. One author (I. N.) benefited from funding by the conference “Random matrices, related topics and applications” at C.R.M. and the mini-workshop “Introduction to infinite-dimensional topological groups”, in Ottawa organized by M. Neufang. He thanks the organizers of these two events for making a first visit to eastern Canada possible. B.C.’s research was partly funded by ANR GranMa and ANR Galoisint. The research of both authors was supported in part by NSERC grant RGPIN/341303-2007.

Appendix In this appendix we present an explicit example of graphical computation of expectation values using Theorem 4.1. We look at the second moment of the matrix Z introduced in Sect. 6, Fig. 14, Z = U  ⊗ U (E n ). Before the graph expansion procedure, the diagram associated to the random variable Tr(Z 2 ) is depicted in Fig. 19. Note that all the decorations are attached to wires, so the diagram corresponds to a (random) scalar value. To compute the second moment of Z , we use Theorem 4.1. Since there are 4 U and 4 U boxes in the diagram, the Weingarten sum will be indexed by a pair (α, β) of permutations from S4 . We choose to order the U boxes in clockwise order, starting from the top-left corner. For each permutation pair, we construct the corresponding “removed” diagram, where the unitary boxes have been removed, and extra-wires have been added between the labels, according to the two permutations. In Fig. 20, we present four such removed diagrams, corresponding to the pairs (id, id), (id, δ), (δ, id) and (δ, δ) respectively, where δ is the “vertical” permutation δ = (14) (23). Since the diagrams contain only loops, traces of powers of rank-one projectors (which are equal to one) and the n −2 normalization scalar, the contributions of each diagram are easily calculated: 1 4 4 n k , n2 1 = 2 n6k 4, n 1 = 2 n4k 2, n

D(a) = Did,id = D(b) = Did,δ D(c) = Dδ,id

Fig. 19. Diagram for Tr(Z 2 )

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(a)

(b)

(c)

(d)

Fig. 20. Four terms in the graph expansion of E Tr(Z 2 )

D(d) = Dδ,δ =

1 6 2 n k . n2 −1

When taking into account the Weingarten weights Wg(nk, αβ −1 ) ∼ (nk)−4−|αβ | , one can see that the last diagram above D(d) · Wg(nk, id) has the largest total contribution. Of course, one needs to consider all (4!)2 permutation couples and then find the dominating terms. The proofs of Theorems 5.2 and 6.3 proceed in this way, by a detailed analysis of the cycle structure of the different permutations. References 1. Coecke, B.: Kindergarten quantum mechanics — lecture notes. In: Quantum Theory: Reconsideration of Foundations—3, AIP Conf. Proc. 810, Melville, NY: Amer. Inst. Phys., 2006, pp. 81–98 2. Collins, B.: Moments and Cumulants of Polynomial random variables on unitary groups, the ItzyksonZuber integral and free probability. Int. Math. Res. Not. 17, 953–982 (2003) 3. Collins, B., Nechita, I.: Random quantum channels II: Entanglement of random subspaces, Renyi entropy estimates and additivity problems. http://arxiv.org/abs/0906.1877v2[math.PR], 2009 4. Collins, B., Nechita, I.: Gaussianization and eigenvalue statistics for Random quantum channels (III) http://arxiv.org/abs/0910.1768v2[quant-ph], 2009 ´ 5. Collins, B., Sniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006) 6. Hastings, M.B.: Superadditivity of communication capacity using entangled inputs. Nature Physics 5, 255–257 (2009) 7. Hayden, P.: The maximal p-norm multiplicativity conjecture is false. http://arxiv.org/abs/0707. 3291v1[quant-ph], 2007 8. Hayden, P., Leung, D., Winter, A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95– 117 (2006) 9. Hayden, P., Winter, A.: Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1. Commun. Math. Phys. 284(1), 263–280 (2008) 10. Jones, V.F.R.: Planar Algebras. http://arxiv.org/abs/math/9909027v1[math.QA], 1999 11. Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119(3), 447–468 (1996) 12. Nechita, I.: Asymptotics of random density matrices. Ann. Henri Poincaré 8(8), 1521–1538 (2007) 13. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. Volume 335 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 2006 Communicated by M.B. Ruskai

Commun. Math. Phys. 297, 371–400 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-0992-0

Communications in

Mathematical Physics

Incompressible Limit of the Compressible Magnetohydrodynamic Equations with Periodic Boundary Conditions Song Jiang1 , Qiangchang Ju2 , Fucai Li3 1 LCP, Institute of Applied Physics and Computational Mathematics,

P.O. Box 8009, Beijing 100088, P.R. China. E-mail: [email protected]

2 Institute of Applied Physics and Computational Mathematics,

P.O. Box 8009-28, Beijing 100088, P.R. China. E-mail: [email protected]

3 Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China.

E-mail: [email protected] Received: 1 July 2009 / Accepted: 26 October 2009 Published online: 6 February 2010 – © Springer-Verlag 2010

Dedicated to Professor Ling Hsiao on the occasion of her 70th birthday. Abstract: This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the case of the well-prepared initial data.

1. Introduction Magnetohydrodynamics (MHD) studies the dynamics of compressible quasineutrally ionized fluids under the influence of electromagnetic fields. The applications of magnetohydrodynamics cover a very wide range of physical objects, from liquid metals to cosmic plasmas. The compressible viscous MHD equations in the isentropic case take the form (see, e.g., [14,15,21]) ˜ = 0, ∂t ρ˜ + div(ρ˜ u)

(1.1)

˜ ×H ˜ + μ ˜ + div(ρ˜ u˜ ⊗ u) ˜ + ∇ P˜ = (curl H) ˜ ˜ u˜ + (μ˜ + λ˜ )∇(divu), (1.2) ∂t (ρ˜ u) ˜ − curl (u˜ × H) ˜ = −curl (˜ν curl H), ˜ ˜ = 0. divH ∂t H

(1.3)

Here x ∈ Td , a torus in Rd , d = 2 or 3, t > 0, the unknowns ρ˜ denote the density, ˜ = ( H˜ 1 , . . . , H˜ d ) ∈ Rd the magnetic field, u˜ = (u˜ 1 , . . . , u˜ d ) ∈ Rd the velocity, and H respectively. The constants μ˜ and λ˜ are the shear and bulk viscosity coefficients of the flow, respectively, satisfying μ˜ > 0 and 2μ˜ + d λ˜ > 0; the constant ν˜ > 0 is the magnetic ˜ ρ) diffusivity acting as a magnetic diffusion coefficient of the magnetic field. P( ˜ is the pressure-density function and here we consider the case

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˜ ρ) P( ˜ = a ρ˜ γ ,

(1.4)

where a > 0 and γ > 1 are constants. The well-posedness of the Cauchy problem and initial boundary value problems for (1.1)-(1.3) have been investigated recently. The global existence of weak solutions to the compressible MHD equations with general initial data was obtained by Hu and Wang [9,10] (also see [6] on “variational solutions”). From the physical point of view, one can formally derive the incompressible models from the compressible ones when the Mach number goes to zero and the density becomes almost constant. Based on this observation, Hu and Wang [11] proved the convergence of the weak solutions of the compressible MHD equations (1.1)-(1.3) to a weak solution of the viscous incompressible MHD equations. Jiang, Ju and Li [12] obtained the convergence towards the strong solution of the ideal incompressible MHD equations in the whole space by using the dispersion property of the wave equation if both the shear viscosity and the magnetic diffusion coefficients go to zero. In this paper, we shall extend the results on the Cauchy problem in [12] to the periodic case. First, we consider the well-prepared initial data for which the oscillations will never appear. We will rigorously show the weak solutions of the compressible MHD equations converge to the strong solution of the ideal incompressible MHD equations in the periodic domain if both the shear viscosity and the magnetic diffusion coefficients go to zero, as well as to the strong solution of the viscous incompressible MHD equations. Furthermore, we shall also give the rates of convergence which are not obtained in [11,12]. Secondly, we consider the case of general initial data. For this case the oscillations (acoustic waves) will appear. Comparing with [12] where the Cauchy problem was dealt with, the acoustic waves in the current situation will lose the dispersion property and will interact with each other. Thus, here we have to impose more regular conditions than L 2 on the initial data to control the oscillating parts. In addition, we have to assume that the Sobolev norm of the oscillating parts is comparable to the magnetic diffusion coefficient in order to deal with the general initial data. We will rigorously prove the convergence of the weak solutions of the compressible MHD equations to the strong solution of the incompressible MHD equations, as well as to the strong solution of the partial viscous incompressible MHD equations. To begin our argument, we first give some formal analysis. Formally, by utilizing the identity ˜ 2 ) = 2(H ˜ · ∇)H ˜ + 2H ˜ × curl H, ˜ ∇(|H| we can rewrite the momentum equation (1.2) as ˜ 2 ) + μ ˜ · ∇)H ˜ − 1 ∇(|H| ˜ ˜ + div(ρ˜ u˜ ⊗ u) ˜ + ∇ P˜ = (H ˜ ∂t (ρ˜ u) ˜ u˜ + (μ˜ + λ)∇(div u). 2 (1.5) By the identities ˜ = ∇ divH ˜ − H ˜ curl curl H and ˜ = u(div ˜ − H(div ˜ ˜ · ∇)u˜ − (u˜ · ∇)H, ˜ ˜ ˜ + (H curl (u˜ × H) H) u)

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˜ = 0, the magnetic field equation (1.3) can be expressed together with the constraint divH as ˜ + (divu) ˜ + (u˜ · ∇)H ˜ − (H ˜ · ∇)u˜ = ν˜ H. ˜ ˜ H ∂t H

(1.6)

We introduce the scaling ˜ ˜ ρ(x, ˜ t) = ρ  (x, t), u(x, t) = u (x, t), H(x, t) = H (x, t), and assume that the viscosity coefficients μ, ˜ ξ˜ , and ν˜ are small constants and scaled like μ˜ = μ , λ˜ = λ , ν˜ = ν  ,

(1.7)

where  ∈ (0, 1) is a small parameter and the normalized coefficients μ , λ , and ν  satisfy μ > 0, 2μ + dλ > 0, and ν  > 0. With the preceding scalings and the pressure function (1.4), the compressible MHD equations (1.1), (1.5), and (1.6) take the form ∂t ρ  + div(ρ  u ) = 0, ∂t (ρ  u ) + div(ρ  u ⊗ u ) +

(1.8) a∇(ρ  )γ 2

1 = (H · ∇)H − ∇(|H |2 ) + μ u + (μ + λ )∇(divu ), 2

(1.9)

∂t H + (divu )H + (u · ∇)H − (H · ∇)u = ν  H , divH = 0. (1.10) √ Moreover, by replacing  by aγ , we can always assume a = 1/γ . Now, we investigate the incompressible limit of the compressible MHD equations (1.8)-(1.10). Formally let  → 0 in Eqs. (1.8)-(1.10), then we obtain from the momentum equation (1.9) that ρ  converges to some function ρ(t) ¯ ≥ 0. If we further assume that the initial datum ρ0 is of order 1 + O() (this can be guaranteed by the initial energy bound (2.4) below), then we can expect that ρ¯ = 1. Thus, the continuity equation (1.8) gives div u = 0. Furthermore, using the assumption μ → 0, ν  → 0 as  → 0,

(1.11)

we obtain the following ideal incompressible MHD equations (suppose that the limits u → u and H → H exist) 1 ∂t u + (u · ∇)u − (H · ∇)H + ∇ p + ∇(|H|2 ) = 0, 2 ∂t H + (u · ∇)H − (H · ∇)u = 0, divu = 0, divH = 0.

(1.12) (1.13) (1.14)

In Sect. 3 we shall rigorously prove that the weak solutions of the compressible MHD equations (1.8)-(1.10) converge to, as  → 0, the strong solution of the ideal incompressible MHD equations (1.12)-(1.14) for the well-prepared initial data in the time interval where the strong solution of (1.12)-(1.14) exists. Furthermore, the convergence rates are obtained. To show these results, since the viscosity coefficients go to zero, we lose the spatial compactness property of the velocity and the magnetic field, and the arguments in [11] do not work here. To overcome such difficulty, we shall carefully exploit the energy arguments.

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Next, if we assume that the shear and the bulk viscosity coefficients and the magnetic diffusivity coefficient satisfy μ → μ > 0, λ → λ, ν  → ν > 0 as  → 0,

(1.15)

then the compressible MHD equations (1.12)-(1.14) formally converge to the incompressible MHD equations (suppose that the limits u → u and H → H exist) 1 ∂t u + (u · ∇)u − μu + ∇ p − (H · ∇)H + ∇(|H|2 ) = 0, 2 ∂t H + (u · ∇)H − (H · ∇)u − νH = 0, div u = 0, divH = 0.

(1.16) (1.17) (1.18)

In Sects. 3 and 4 we shall prove the convergence to the strong solution of the incompressible viscous MHD equations (1.16)-(1.18) for both the well-prepared and the general initial data. Furthermore, the convergence rates are also obtained for the well-prepared initial data. For the general initial data, we shall also show the convergence to the strong solution of the partial viscous incompressible MHD equations (that is, μ = 0 and ν > 0 in (1.16)-(1.18)). There are a lot of studies on the compressible MHD equations in the literature. Besides the aforementioned results, the interested reader can see [13,17] on the global smooth solutions with small initial data and see [5,24] on the local strong solution with general initial data. We also mention the work [25] where a MHD model describing the screw pinch problem in plasma physics was discussed and the global existence of weak solutions with symmetry was obtained. Before ending the Introduction, we give the notation used throughout the current q paper. We denote the space L 2 (Td ) by q

L 2 (Td ) = { f ∈ L loc (Td ) : f 1{| f |≥1/2} ∈ L q , f 1{| f |≤1/2} ∈ L 2 }. We use the letters C and C T to denote various positive constants independent of , but C T may depend on T . For convenience, we denote by H r ≡ H r (Td ) (r ∈ R) the standard Sobolev space. For any vector field v, we denote by Pv and Qv the divergence-free part and the gradient part of v, respectively. Namely, Qv = ∇−1 (divv) and Pv = v − Qv. We state our main results in Sect. 2 and present the proofs for the well-prepared case in Sect. 3 and the ill-prepared case in Sect. 4, respectively. 2. Main Results We first recall the local existence of strong solutions to the ideal incompressible MHD equations (1.12)-(1.14) in the torus Td . The proof can be found in [4,23]. Proposition 2.1 ([4,23]). Assume that the initial data (u, H)|t=0 = (u0 , H0 ) satisfy u0 , H0 ∈ H s (s > d/2 + 1), and div u0 = 0, divH0 = 0. Then, there exist a T ∗ ∈ (0, ∞) and a unique solution (u, H) ∈ L ∞ ([0, T ∗ ), H s ) to the ideal incompressible MHD equations (1.12)-(1.14) satisfying, for any 0 < T < T ∗ , div u = 0, divH = 0, and   (2.1) sup ||(u, H)(t)|| H s + ||(∂t u, ∂t H)(t)|| H s−1 + ||∇ p(t)|| H s−1 ≤ C T . 0≤t≤T

Remark 2.1. The local existence of strong solutions to the incompressible viscous MHD equations (1.16)-(1.18) was also established in [4,23].

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We prescribe the initial conditions to the compressible MHD equations (1.8)-(1.10) as ρ  |t=0 = ρ0 (x), ρ  u |t=0 = ρ0 (x)u0 (x) ≡ m0 (x), H |t=0 = H0 (x),

(2.2)

and assume that ρ0 ≥ 0, ρ0 ∈ L γ , ρ0 |u0 |2 ∈ L 1 , H0 ∈ L 2 , divH0 = 0, m0 = 0 for a.e. ρ0 = 0. (2.3) Moreover, we assume that the initial data also satisfy the following uniform bound      γ  1  2 1 2 a ρ0 |u0 | + |H0 | + 2 (ρ0 ) − 1 − γ (ρ0 − 1) d x ≤ C. (2.4) 2  (γ − 1) Td 2 The initial energy inequality (2.4) implies that ρ0 is of order 1 + O(). Under the above assumptions, it was proved in [9] that the compressible MHD equations (1.8)-(1.10) with initial data (2.2)-(2.4) has a global weak solution. More precisely, we have Proposition 2.2 ([9]). Let γ > d/2. Suppose that the initial data (ρ0 , u0 , H0 ) satisfy the assumptions (2.3) and (2.4). Then the compressible MHD equations (1.8)-(1.10) with the initial data (2.2) enjoy at least one global weak solution (ρ  , u , H ) satisfying (1) ρ  ∈ L ∞ (0, ∞; L γ )∩C([0, ∞), L r ) for all 1 ≤ r < γ , ρ  |u |2 ∈ L ∞ (0, ∞; L 1 ), 2γ

γ +1 H ∈ L ∞ (0, ∞; L 2 ), and u ∈ L 2 (0, T ; H 1 ), ρ  u ∈ C([0, T ], L weak ), 2γ

γ +1 H ∈ L 2 (0, T ; H 1 ) ∩ C([0, T ], L weak ) for all T ∈ (0, ∞); (2) the energy inequality  t E  (t) + D (s)ds ≤ E  (0)

(2.5)

0

holds with the finite total energy      γ  1  2 1 2 a   ρ |u | + |H | + 2 (ρ ) −1−γ (ρ − 1) (t) E (t) ≡ 2  (γ − 1) Td 2 and the dissipation energy 

D (t) ≡ μ |∇u |2 + (μ + λ )|divu |2 + ν  |∇H |2 (t); Td

(2.6)

(2.7)

(3) the continuity equation is satisfied in the sense of renormalized solutions, i.e.,   ∂t b(ρ  ) + div(b(ρ  )u ) + b (ρ  )ρ  − b(ρ  ) divu = 0 (2.8) for any b ∈ C 1 (T) such that b (z) is constant for z large enough; (4) Eqs. (1.8)-(1.10) hold in D (Td × (0, ∞)). The main results of this paper can be stated as follows.

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Theorem 2.3. Let s > d/2+2 and μ +λ > 0. Suppose that the initial data (ρ0 , u0 , H0 ) satisfy the conditions presented in Proposition 2.2. Assume further that   |ρ0 − 1|2 1(|ρ0 −1|≤δ) d x + |ρ0 − 1|γ 1(|ρ0 −1|>δ) d x ≤ C 2 , (2.9) Td Td  (2.10) || ρ0 u0 − u0 ||2L 2 (T3 ) ≤ C, ||H0 − H0 ||2L 2 (T3 ) ≤ C for any δ ∈ (0, 1), where u0 and H0 are defined in Proposition 2.1. We assume that the shear viscosity μ and the magnetic diffusion coefficient ν  satisfy μ =  α , ν  =  β

(2.11)

for some constants α, β > 0 satisfying 0 < α + β < 2. Let (u, H) be the smooth solution to the ideal incompressible MHD equations (1.12)-(1.14) defined on [0, T ∗ ) with (u, H)|t=0 = (u0 , H0 ). Then, for any 0 < T < T ∗ , the global weak solution (ρ  , u , H ) of the compressible MHD equations (1.8)-(1.10) established in Proposition 2.2 satisfies    2 |ρ − 1| 1(|ρ  −1|≤δ) d x + |ρ  − 1|γ 1(|ρ  −1|>δ) d x ≤ C T  2 , (2.12) Td Td || ρ  u − u||2L 2 (T3 ) ≤ C T  σ , ||H − H||2L 2 (T3 ) ≤ C T  σ (2.13) for any t ∈ [0, T ], where σ = min{α, β, 1 − (α + β)/2}. The proof of Theorem 2.3 is based on the combination of the modulated energy method, motivated by Brenier [1], the weak convergence method and the refined energy analysis. Masmoudi [19] made use of such an idea to study the incompressible, inviscid limit of the compressible Navier-Stokes equations in both the whole space and the torus. Comparing with the proof in [19], here we have to overcome the difficulties caused by the strong coupling of the hydrodynamic motion and the magnetic field. Furthermore, we can use an idea similar to that described above to obtain the convergence of the compressible MHD equations (1.8)-(1.10) to the incompressible viscous MHD equations (1.16)-(1.18). In fact, we have the following result. Theorem 2.4. Let s > d/2+2 and μ +λ > 0. Suppose that the initial data (ρ0 , u0 , H0 ) satisfy the conditions presented in Proposition 2.2. Assume further that   |ρ0 − 1|2 1(|ρ0 −1|≤δ) d x + |ρ0 − 1|γ 1(|ρ0 −1|>δ) d x ≤ C 2 , (2.14) Td Td  (2.15) || ρ0 u0 − u0 ||2L 2 (T3 ) ≤ C, ||H0 − H0 ||2L 2 (T3 ) ≤ C for any δ ∈ (0, 1) and for some u0 , H0 ∈ H s (T3 ), satisfying divu0 = 0, divH0 = 0. We also assume that the shear viscosity μ and the magnetic diffusion coefficient ν  satisfy (1.15). Let (u, H) be the smooth solution to the incompressible MHD equations (1.16)-(1.18) with (u, H)|t=0 = (u0 , H0 ). Then, for any 0 < T < T ∗∗ ( T ∗∗ is the maximal time of existence for (1.16)-(1.18)), the global weak solution (ρ  , u , H ) of the compressible MHD equations (1.8)-(1.10) established in Proposition 2.2 satisfies

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that ∇u and ∇H converge strongly to ∇u and ∇H in L 2 (0, T ; L 2 (Td )), respectively. Moreover, for any t ∈ [0, T ], we have   |ρ  − 1|2 1(|ρ  −1|≤δ) d x + |ρ  − 1|γ 1(|ρ  −1|>δ) d x ≤ C T  2 , (2.16) Td

Td

  || ρ  u − u||2L 2 (T3 ) ≤ C T √ , ||H − H||2L 2 (T3 ) ≤ C T √ . μν μν

(2.17)

To show Theorem 2.4, besides the techniques mentioned above, we have to employ a new technique, that is, to modulate both the total energy and the partial dissipative energy simultaneously. Moreover, the dissipative effect of the viscous terms is also carefully exploited to obtain the desired results. Remark 2.2. Comparing with Theorem 2.3, we have gotten the better convergence rates (2.17) than (2.13) when the shear viscosity and the magnetic diffusion coefficient tend to some positive constants. Some results in Theorem 2.4 can be extended to the case of general initial data. More precisely, we shall obtain the convergence of the compressible MHD equations (1.8)(1.10) to the incompressible MHD equations (1.16)-(1.18) for the general initial data under the conditions that the oscillating parts of the initial data have higher regularity and the Sobolev norm of the oscillation parts is comparable to the magnetic diffusion coefficient. This implies that the influence of oscillations on the magnetic field can be balanced by the diffusive effect of the magnetic field, which is one of the new ingredients in our paper. To describe the result, we write ρ  = 1 + ϕ  and denote

2a 1   (x, t) = ((ρ  )γ − 1 − γ (ρ  − 1)).  γ −1 We will use the above approximation  (x, t) instead of ϕ  , since we can not obtain any bound for ϕ  in L ∞ (0, T ; L 2 ) directly if γ < 2. Theorem 2.5. Let s > 2+d/2 and 2μ+dλ > 0. Suppose that the initial data (ρ0 , u 0 , H0 ) satisfy the conditions presented in Proposition 2.2. Moreover, we assume that ρ0 u0 converges strongly in L 2 to some u˜ 0 satisfying Q u˜ 0 ∈ H s−1 , H0 converges strongly in L 2 to some H0 with Td H0 (x)d x = 0,  |t=0 = 0 converges strongly in L 2 to some ϕ0 ∈ H s−1 , and ϕ0  H 2 + ||Q u˜ 0 || H 2 ≤ c0 ν

(2.18)

for some constant c0 > 0. Let (u, H) be the smooth solution to the incompressible MHD equations (1.16)-(1.18) with (u, H)|t=0 = (u0 , H0 ) ∈ H s (T3 ) satisfying u0 = P u˜ 0 and divH0 = 0. Then, for any 0 < T < T ∗∗ ( T ∗∗ is the maximal time of existence for (1.16)-(1.18)), the global weak solution (ρ  , u , H ) of the compressible MHD equations (1.8)-(1.10) established in Proposition 2.2 satisfies γ

(1) ρ  converges strongly to 1 in C([0, T ], L 2 (Td )); (2) ∇H converges strongly to ∇H in L 2 (0, T ; L 2 (Td )); (3) H converges strongly to H in L ∞ (0, T ; L 2 (Td ));

378

S. Jiang, Q. Ju, F. Li

√ (4) √ P( ρ  u ) converges strongly to u in L ∞ (0, T ; L 2 (Td )); (5) ρ  u converges weakly to u in H −1 (0, T ; L 2 (Td )). By slightly modifying the proof of Theorem 2.5, we can obtain the convergence of compressible MHD equations to the partial viscous incompressible MHD equations when the shear viscosity goes to zero and the magnetic diffusion coefficient goes to a positive constant. The partial viscous incompressible MHD equations correspond to the case of turbulent flow with very high Reynolds number (where the viscosity of flow can be ignored, see [16]). Theorem 2.6. Let s > 2 + d/2. Suppose that the conditions in Theorem 2.5 hold. Moreover, we assume that ν  → ν > 0, 2μ + λ → 2θ > 0 as  → 0, and μ =  α for some constant 0 < α < 1. Let (u, H) be the smooth solution to the following partially viscous incompressible MHD equations: 1 ∂t u + (u · ∇)u + ∇ p − (H · ∇)H + ∇(|H|2 ) = 0, 2 ∂t H + (u · ∇)H − (H · ∇)u − νH = 0, divu = 0, divH = 0, with (u, H)|t=0 = (u0 , H0 ) ∈ H s (T3 ) satisfying u0 = P u˜ 0 and divH0 = 0. Then, for any 0 < T < T ∗∗ (T ∗∗ is the maximal time of existence for (1.16)-(1.18)), the global weak solution (ρ  , u , H ) of the compressible MHD equations (1.8)-(1.10) established in Proposition 2.2 satisfies (1) (2) (3) (4) (5)

γ

ρ  converges strongly to 1 in C([0, T ], L 2 (Td )); ∇H converges strongly to ∇H in L 2 (0, T ; L 2 (Td )); H √ converges strongly to H in L ∞ (0, T ; L 2 (Td ));   P( in L ∞ (0, T ; L 2 (Td )); √  ρ u ) converges strongly to u −1 ρ u converges weakly to u in H (0, T ; L 2 (Td )).

Remark 2.3. The assumption that 0 converges strongly in L 2 to some ϕ0 in fact implies γ that ϕ0 converges strongly to ϕ0 in L 2 . Remark 2.4. When taking H ≡ 0 in (1.1)-(1.3), the MHD equations reduce to the classical compressible Navier-Stokes equations. The low Mach number limit problem of the compressible Navier-Stokes equations has been investigated extensively, for instance, see [2,3,7,8,18]. The interested reader can refer to the survey article [20] for more related results. Remark 2.5. We point out that our arguments in the present paper can be applied to the case of H ≡ 0. In this case, we obtain the convergence of the compressible NavierStokes equations to the incompressible Euler or Navier-Stokes equations with general initial data, extending thus the results in [18,19].

Incompressible Limit of MHD Equations

379

3. Proof of Theorems 2.3 and 2.4 In this section, we shall prove our convergence results for the case of well-prepared initial data by combining the modulated energy method, the weak convergence method, and the refined energy analysis. Proof of Theorem 2.3. We divide the proof into several steps. Step 1: Basic energy estimates and compact arguments. By the assumptions on the initial data we obtain, from the energy inequality (2.5), that the total energy E  (t) has a uniform upper bound for a.e. t ∈ [0, T ], T > 0. This uniform bound implies that ρ  |u |2 and ((ρ  )γ − 1 − γ (ρ  − 1)) / 2 are bounded in L ∞ (0, T ; L 1 ) and H is bounded in L ∞ (0, T ; L 2 ). Using the analysis in [18], we obtain   1  1  2 |ρ − 1| 1{|ρ  −1|≤ 1 } + |ρ − 1|γ 1{|ρ  −1|≥ 1 } ≤ C, (3.1) 2 2 2 2 Td  Td  which implies (2.12) and γ

ρ  → 1 strongly in C([0, T ], L 2 (Td )).

(3.2)

As in [18], we know that u is bounded in L 2 (0, T ; L 2 ). Furthermore, the fact that ρ  |u |2 and |H |2 are bounded in L ∞ (0, T ; L 1 ) implies the following convergence (up to the extraction of a subsequence n ): ρ  u converges weakly-∗ to some J in L ∞ (0, T ; L 2 (Td )), H converges weakly-∗ to some K in L ∞ (0, T ; L 2 (Td )). Thus, to finish our proof, we need to show that J = u and K = H in some sense and the inequalities (2.13) hold, where (u, H) is the strong solution to the ideal incompressible MHD equations (1.12)-(1.14). Step 2: The modulated energy functional and the uniform estimates. We first recall the energy inequality of the compressible MHD equations (1.8)-(1.10), i.e., for almost all t, there holds  t 

1   2  2  2  ρ (t)|u | (t) + |H | (t) + ( (t)) + μ |∇u |2 2 Td 0 Td  t  t    2  +(μ + λ ) |divu | + ν |∇H |2 d d 0 T  0 T

1   2  2  2 ρ0 |u0 | + |H0 | + (0 ) . (3.3) ≤ 2 Td The conservation of energy for the ideal incompressible MHD equations (1.12)-(1.14) reads 

1 1 2 2 |u| (t) + |H| (t) = |u0 |2 + |H0 |2 . (3.4) 2 Td 2 Td Using u to test the momentum equation (1.9), we obtain    t  1 (ρ  u · u)(t) + ρ  u · (u · ∇)u − (H · ∇)H + ∇ p + ∇(|H|2 ) 2 Td 0 Td   t     (ρ u ⊗ u ) · ∇u + (H · ∇)H · u − μ ∇u · ∇u = ρ0 u0 · u0 . − 0

Td

Td

(3.5)

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S. Jiang, Q. Ju, F. Li

Similarly, using H to test the magnetic field equation (1.10), one gets  t  t  (H · H)(t) + H · [(u · ∇)H − (H · ∇)u] + ν  ∇H · ∇H d d d T 0 T 0 T  t          + H0 · H0 . (3.6) (divu )H + (u · ∇)H − (H · ∇)u · H = 0

Td

Td

Summing up (3.3) and (3.4), and inserting (3.5) and (3.6) into the resulting inequality, we can deduce the following inequality by a straightforward computation    1 | ρ  u − u|2 (t) + |H − H|2 (t) + ( )2 (t) 2 Td  t  t  t |∇u |2 + (μ + λ ) |divu |2 + ν  |∇H |2 +μ d d d 0 T 0 T 0 T  t  t  t     ≤μ ∇u · ∇u + ν ∇H · ∇H − ρ  u · [(H · ∇)H)] 0 Td 0 Td 0 Td  t  t − (H · ∇)H · u + H · [(u · ∇)H − (H · ∇)u] 0 Td 0 Td  t     1 t + (divu )H + (u · ∇)H − (H · ∇)u · H + ρ  u · ∇(|H|2 ) 2 0 Td 0 Td  t  t + ρ  u · [(u · ∇)u + ∇ p] − (ρ  u ⊗ u ) · ∇u 0 Td 0 Td  



   + ( ρ − 1) ρ u · u (t) − ( ρ  − 1) ρ  u · u (0) Td Td    1 + | ρ  u − u|2 (0) + |H − H|2 (0) + (0 )2 . (3.7) 2 Td √ We first deal with the right-hand side of the inequality (3.7). Denoting w = ρ  u − u and Z = H − H, integrating by parts, and using the fact that div H = 0, div u = 0 and div H = 0, we find that  t  t   − ρ u · [(H · ∇)H)] − (H · ∇)H · u 0 Td 0 Td  t   1 t + H · [(u · ∇)H − (H · ∇)u] + ρ  u · ∇(|H|2 ) d d 2 0 T 0 T  t        + (divu )H + (u · ∇)H − (H · ∇)u · H 0 Td  t  t =− ρ  u · [(H · ∇)H)] + (H · ∇)u · H d d 0 T 0 T  t  t  + (u · ∇)H · H − (H · ∇)u · H d d 0 T 0 T  t  t   1 t   − (u · ∇)H · H + (H · ∇)H · u + ρ  u · ∇(|H|2 ) d d d 2 0 T 0 T 0 T  t  t    = (1 − ρ )u · [(H · ∇)H)] + [(H − H) · ∇]u · (H − H) 0

Td

0

Td

Incompressible Limit of MHD Equations

 t +

Td

0

381

[(H − H) · ∇]H · (u − u) −

 t 0

Td

[(u − u) · ∇]H · (H − H)

  1 t + (ρ  − 1)u ∇(|H|2 ) 2 0 Td  t  t ≤ (1 − ρ  )u · [(H · ∇)H)] + ||Z (s)||2L 2 ||∇u(s)|| L ∞ ds 0 Td 0  t

+ ||w (s)||2L 2 + ||Z (s)||2L 2 ||∇H(s)|| L ∞ ds 0  t + (Z · ∇)H · [(1 − ρ  )u ] 0 Td  t     1 t − [(1 − ρ  )u ] · ∇ H · Z + (ρ  − 1)u ∇(|H|2 ) 2 0 Td 0 Td and

(3.8)

 t

 t     ρ u · ((u · ∇)u + ∇ p) − (ρ  u ⊗ u ) · ∇u 0 Td 0 Td  t  t   =− (w ⊗ w ) · ∇u + (ρ  − ρ  )u · ((u · ∇)u) d d 0 T 0 T  t  t    + [( ρ u − u) · ∇]u · w + ρ  u · ∇ p 0 Td 0 Td  2  t |u|   . − ( ρ u − u) · ∇ d 2 0 T

(3.9)

Substituting (3.8) and (3.9) into the inequality (3.7), we conclude that  t ||w (t)||2L 2 + ||Z (t)||2L 2 + || (t)||2L 2 + 2μ |∇u |2 0 Td  t  t +2(μ + λ ) |divu |2 + 2ν  |∇H |2 0 Td 0 Td  t ≤ 2C (||w (τ )||2L 2 + ||Z (τ )||2L 2 )(||∇u(τ )|| L ∞ + ||∇H(τ )|| L ∞ )dτ 0

+||w (0)||2L 2 + ||Z (0)||2L 2 + ||0 ||2L 2 + 2

6 

Ri (t),

(3.10)

i=1

where R1 (t)



 t

Td



 t

=μ ∇u · ∇u + ν ∇H · ∇H, 0 Td 0 Td  



R2 (t) = ( ρ  − 1) ρ  u · u (t) − ( ρ  − 1) ρ  u · u (0) Td Td  t + (ρ  − ρ  )u · ((u · ∇)u), d 0 T  t  t    R3 (t) = [(1 − ρ  )u ] · ∇ H · Z , (Z · ∇)H · [(1 − ρ  )u ] − 0



0

Td

382

S. Jiang, Q. Ju, F. Li

R4 (t) = R5 (t)

=

 t 0

 t 0

R6 (t) = −

Td

0

 t Td

0

(ρ  − 1)u · ∇(|H|2 ),

 

ρ u · ∇ p,

Td t



1 2

(1 − ρ  )u · [(H · ∇)H)] +

Td

( ρ  u − u) · ∇



|u|2 2

 .

Step 3: Convergence of the modulated energy functional. To show the convergence of the modulated energy functional (3.10) and to finish our proof, we have to estimate the reminders Ri (t), i = 1, . . . , 6. First, in view of (3.1) and the following two elementary inequalities: √ | x − 1|2 ≤ M|x − 1|γ , |x − 1| ≥ δ, γ ≥ 1, √ | x − 1|2 ≤ M|x − 1|2 , x ≥ 0,

(3.11) (3.12)

for some positive constants M and 0 < δ < 1, we obtain  Td

 | ρ  − 1|2 =

|ρ  −1|≤ 21

 | ρ  − 1|2 +

|ρ  −1|≥ 21



≤M





|ρ  −1|≤ 21

| ρ  − 1|2

|ρ − 1| + M 2

|ρ  −1|≥ 21

|ρ  − 1|γ

≤ M 2 .

(3.13)

Now, we begin to estimate the terms Ri (t), i = 1, . . . , 6. For the term R1 (t), by Young’s inequality and the regularity of u and H, we have |R1 (t)| ≤

μ 2

 t 0

Td

|∇u |2 +

ν 2

 t 0

Td

|∇H |2 + C T μ + C T ν  .

(3.14)

For the term R2 (t), by Hölder’s inequality, the estimate (3.13), the assumption on √ the initial data, the estimate on ρ  u , and the regularity of u, we infer that |R2 (t)| ≤ C + ||u(t)|| L ∞

 Td

+||[(u · ∇)u](t)|| L ∞

| ρ  − 1|2

 t  0

≤ C T .

Td

 1  2

Td

| ρ  − 1|2

ρ  |u |2

1 2

 21  t  0

Td

ρ  |u |2

 21 (3.15)

For the term R3 (t), making use of the inequality (3.13), the basic inequality (2.5), the estimates on u and H , the regularity of H, the assumption (2.11), and Sobolev’s imbedding theorem, we get

Incompressible Limit of MHD Equations

383

|R3 (t)| ≤ (||[(H · ∇)H](t)|| L ∞ + ||∇H(t)|| L ∞ · ||H(t)|| L ∞ ) ×

 t  Td

0

|



ρ

− 1|

2

 21  t 

+||∇H(t)|| L ∞

Td

0

Td

0

 t 

 2

|u |

| ρ  − 1|2

≤ C T  + || ρ  − 1|| L ∞ (0,T ;L 2 )



t

0

1 2

 21 

||u (τ )|| L 6 ||H (τ )|| L 3 dτ

||u (τ )||2H 1 dτ



 21 

t 0

||H (τ )||2H 1 dτ

 21



≤ C T  + C T (T ||u || L ∞ (0,T ;L 2 ) + ||∇u || L 2 (0,T ;L 2 ) ) ×(T ||H || L ∞ (0,T ;L 2 ) + ||∇H || L 2 (0,T ;L 2 ) )



1 1 ≤ C T  + C T  T + (μ )− 2 · T + (ν  )− 2 ≤ CT  + CT σ ≤ CT σ ,

(3.16)

where σ = 1 − (α + β)/2. For the term R4 (t), one can utilize the inequality (3.13), the estimates on u and √   √ √ ρ u , the regularity of H, and ρ  − 1 = ρ  − ρ  + ρ  − 1 to deduce |R4 (t)|

≤ (||[(H · ∇)H](t)|| ×

  t 0

Td

+ ||∇(|H| )|| 2

L∞

|u |2

 21

 t  +

Td

0

L∞

)

 t  Td

0

ρ  |u |2

| ρ  − 1|2

 21

 21 

≤ C T .

(3.17)

Using (2.1), (3.1) and (3.2), the term R5 (t) can be bounded as follows: |R5 (t)|

 t        = ρ u · ∇ p  0 Td    t         = ((ρ − 1) p)(t) − ((ρ − 1) p)(0) − (ρ − 1)∂t p  Td

≤

|ρ  −1|≤ 21

|ρ  − 1|2

 +

|ρ  −1|≥ 21

0

 1  2

1 | p(t)|2

T3

 1 ⎡ γ ⎣ |ρ  − 1|γ

 t  +

Td

0



|ρ  −1|≤ 21

T3



|ρ − 1|

| p(t)|

+

γ γ −1

T3 γ

1 Td

| p(0)|2

 γ −1

 1  2 2

1 



2

|∂t p(t)|2

2

2

 +

T3

| p(0)|

γ γ −1

 γ −1 γ

⎤ ⎦

384

S. Jiang, Q. Ju, F. Li

 t  +



|ρ − 1|

|ρ  −1|≥ 21

0

γ

 1  γ Td

|∂t p(t)|

γ γ −1

 γ −1 γ

≤ C T ( +  2/κ ) ≤ C T ,

(3.18)

where κ = min{2, γ } and we have used the conditions s > 2 + d/2 and γ > 1. Finally, to estimate the term R6 (t), we rewrite it as  2  t |u|    R6 (t) = − ( ρ u − u) · ∇ 2 0 Td  2  t   2  t |u| |u| − ρ  ( ρ  − 1)u · ∇ ρ  u · ∇ = 2 2 0 Td 0 Td   (t) + R62 (t), (3.19) = R61 where  R61 (t) =  R62 (t)

=

 t 0

 t 0

Td

ρ  ( ρ  − 1)u · ∇



Td



(ρ − 1)∂t

|u|2 2





|u|2 2

 ,

  2   2     |u| |u|   − (ρ − 1) (t) − (ρ − 1) (0) . 2 2 Td  (t) and R  (t), we arrive at the following Applying arguments similar to those used for R61 5 boundedness:   |R6 (t)| ≤ |R61 (t)| + |R62 (t)| ≤ C T .

(3.20)

Inserting the estimates (3.14)-(3.20) into (3.10) and applying Gronwall’s inequality, we conclude ||w (t)||2L 2 + ||Z (t)||2L 2 + || (t)||2L 2

≤ C¯ ||w (0)||2L 2 + ||Z (0)||2L 2 + ||0 ||2L 2 + C T  σ , for a.e. t ∈ [0, T ], where

  ¯ C = exp C

T

[||∇u(τ )|| L ∞ + ||∇H(τ )|| L ∞ ] dτ

< +∞.

(3.21)

(3.22)

0

Now, letting  go to 0, we obtain K = H in L ∞ (0, T ; L 2 ) and J = u in L ∞ (0, T ; L 2 ). The inequality (2.13) follows from (2.10) and (3.21) directly. Thus, we complete the proof.   Proof of Theorem 2.4. For simplicity we assume here that μ ≡ μ, λ ≡ λ, and ν  ≡ ν are constants, independent of , satisfying μ > 0, μ + λ > 0, and ν > 0. The case (1.7) can be treated similarly. The proof of Theorem 2.4 is similar to that of Theorem 2.3. Since the viscosity is involved here, we have to modulate the part of the dissipation energy in the energy inequality (2.5). We state the main different points in the proof here.

Incompressible Limit of MHD Equations

385

From the basic energy inequality (2.5), we obtain that, for a.e. t ∈ [0, T ], ρ  |u |2 and ((ρ  )γ − 1 − γ (ρ  − 1)) / 2 are bounded in L ∞ (0, T ; L 1 ), H is bounded in L ∞ (0, T ; L 2 ), ∇u is bounded in L 2 (0, T ; L 2 ), and ∇H is bounded in L 2 (0, T ; L 2 ). Therefore, we have γ

ρ  → 1 strongly in C([0, T ], L 2 (Td )), and u is bounded in L 2 (0, T ; L 2 ). The boundedness of ρ  |u |2 and |H |2 in L ∞ (0, T ; L 1 ) implies the following convergence (up to the extraction of a subsequence n ): ρ  u converges weakly-∗ to some J¯ in L ∞ (0, T ; L 2 (Td )), ¯ in L ∞ (0, T ; L 2 (Td )). H converges weakly-∗ to some K ¯ = H in some sense, where Our main task in this section is to show that J¯ = u and K (u, H) is the strong solution to the viscous incompressible MHD equations (1.16)-(1.18). Next, we shall also modulate the energy inequality (2.5). The conservation of energy for the viscous incompressible MHD equations (1.16)-(1.18) reads  t 

1

1 2 2 2 2 |u| + |H| (t) + μ|∇u| + ν|∇H| = |u0 |2 + |H0 |2 . 2 Td 2 Td 0 Td (3.23) Similarly to Step 2, we use u to test the momentum equation (1.9) to deduce  Td

  1 ρ  u · (u · ∇)u − (H · ∇)H − μu + ∇ p + ∇(|H|2 ) 2 0 Td      (ρ u ⊗ u ) · ∇u + (H · ∇)H · u − μ∇u · ∇u = ρ0 u0 · u0 .

(ρ  u · u)(t) +

−

 t 0

Td

 t

Td

(3.24) Then, we test (1.10) by H to infer that  Td

(H · H)(t) +

0

 t

+ 0

 t

Td

Td

H · [(u · ∇)H − (H · ∇)u − νH] + ν

  (divu )H + (u · ∇)H − (H · ∇)u · H =

 Td

 t 0

Td

H0 · H0 .

∇H · ∇H (3.25)

Summing up (3.3) and (3.23), and inserting (3.5), (3.6) with μ ≡ μ and λ ≡ λ, (3.24), and (3.25) into the resulting inequality, we deduce the following inequality by a straightforward calculation:

386

S. Jiang, Q. Ju, F. Li

   1 | ρ  u − u|2 (t) + |H − H|2 (t) + ( )2 (t) 2 Td  t  t   ν t +μ |∇u − ∇u|2 + |∇H − ∇H|2 + (μ + λ) |divu |2 2 0 Td 0 Td 0 Td     ν t ν t + |∇H |2 + |∇H|2 2 0 Td 2 0 Td  t  t   ≤− ρ u · [(H · ∇)H)] − (H · ∇)H · u Td

0

 t

Td

0

  1 t + H · [(u · ∇)H − (H · ∇)H] + ρ  u · ∇(|H|2 ) 2 0 Td 0 Td  t  t         + (divu )H + (u · ∇)H − (H · ∇)u · H + μ (1 − ρ  )u u 0

Td

0

Td

 t +  +



Td

ρ  u · [(u · ∇)u + ∇ p] −



( ρ  − 1) ρ  u · u (t) −

0

Td



0

 t Td

Td

(ρ  u ⊗ u ) · ∇u



( ρ  − 1) ρ  u · u (0)

  | ρ  u − u|2 (0) + |H − H|2 (0) + (0 )2 .

1 (3.26) 2 Td By virtue of (3.8) and (3.9), we can rewrite the inequality (3.26) as follows:    1 | ρ  u − u|2 (t) + |H − H|2 (t) + ( )2 (t) 2 Td  t   ν t  2 +μ |∇u − ∇u| + |∇H − ∇H|2 2 0 Td 0 Td  t     ν t ν t  2  2 +(μ + λ) |divu | + |∇H | + |∇H|2 2 0 Td 2 0 Td 0 Td    1 | ρ  u − u|2 (0) + |H − H|2 (0) + (0 )2 ≤ 2 Td +

+R2 (t) + R4 (t) + R5 (t) + R6 (t) + R7 (t) + R8 (t),

(3.27)

where R2 (t) and Ri (t), i = 4, 5, 6, are the same as before, and  t  R7 (t) = μ (1 − ρ  )u u, Td

0

R8 (t)

=

 t 0

 t   (Z · ∇)H · [(1 − ρ )u ] − 

Td

Form the previous arguments on

R2 (t)

|R2 (t)| +

6  i=4

and



[(1 −

0 Td  Ri (t), i =

|Ri (t)| ≤ C T .



 ρ  )u ] · ∇ H · Z .

4, 5, 6, we get (3.28)

Incompressible Limit of MHD Equations

387

Now, we estimate R7 (t) and R8 (t). Using the inequality (3.13), Hölder’s inequality, √ √ √ the estimates on u and ρ  u , the regularity of u, and ρ  − 1 = ρ  − ρ  + ρ  − 1, we obtain |R7 (t)|

≤ μ||u(t)|| L ∞ ×

Td

0

  t

 2

Td

0

 t 

|u |

 21

|



ρ

− 1|

 t  +



Td

0

 21

2

 2

ρ |u |

 21 

≤ C T .

(3.29)

For the term R8 (t), we can make use of (3.13), (2.5), and the estimates on u and H , the regularity of H, the assumption (2.11), and Sobolev’s imbedding theorem to deduce |R8 (t)| ≤ (||[(H · ∇)H](t)|| L ∞ + ||∇H(t)|| L ∞ · ||H(t)|| L ∞ ) ×

 t  0

Td

| ρ  − 1|2

 21  t 

 t 

+||∇H(t)|| L ∞

Td

0

Td

0

|u |2

| ρ  − 1|2

≤ C T  + || ρ  − 1|| L ∞ (0,T ;L 2 )





0

t

 21 

1 2





||u (τ )|| L 6 ||H (τ )|| L 3 dτ

||u (τ )||2H 1 dτ

 21 

t 0

||H ||2H 1 (τ )dτ

 21



≤ C T  + C T (T ||u || L ∞ (0,T ;L 2 ) + ||∇u || L 2 (0,T ;L 2 ) ) ×(T ||H || L ∞ (0,T ;L 2 ) + ||∇H || L 2 (0,T ;L 2 ) )



1 1 ≤ C T  + C T  T + μ− 2 · T + ν − 2 √ √ ≤ C T  + C T / μν ≤ C T / μν.

(3.30)

Now, substituting (3.28)-(3.30) into (3.27) and applying Gronwall’s inequality, we conclude ||w (t)||2L 2 + ||Z (t)||2L 2 + || (t)||2L 2 √

≤ C¯ ||w (0)||2L 2 + ||Z (0)||2L 2 + ||0 ||2L 2 + C T / μν , for a.e. t ∈ [0, T ], (3.31) where C¯ is defined by (3.22). Combining (2.15) with (3.31) we obtain (2.17). Substituting (3.31) into (3.27), we conclude that ∇u converges to ∇u strongly in L 2 (0, T ; L 2 (Td )) and ∇H to ∇H strongly in L 2 (0, T ; L 2 (Td )). This completes the proof of Theorem 2.4.  

388

S. Jiang, Q. Ju, F. Li

4. Proof of Theorem 2.5 In this section we shall study the incompressible limit of the compressible MHD equations (1.8)-(1.10) with general initial data. Compared with the case of the well-prepared initial data, the main difficulty here is to control the oscillations caused by the initial data. For simplicity, we assume here that μ ≡ μ, λ ≡ λ, and ν  ≡ ν are constants, independent of , satisfying μ > 0, 2μ + dλ > 0, and ν > 0. Proof of Theorem 2.5. As stated in the proof of Theorem 2.4, we obtain from the basic energy inequality (2.5) that, for a.e. t ∈ [0, T ], ρ  |u |2 and ((ρ  )γ − 1 − γ (ρ  − 1)) / 2 are bounded in L ∞ (0, T ; L 1 ), H is bounded in L ∞ (0, T ; L 2 ), ∇u is bounded in L 2 (0, T ; L 2 ), and ∇H is bounded in L 2 (0, T ; L 2 ). Therefore, we have γ

ρ  → 1 strongly in C([0, T ], L 2 (Td )),

(4.1)

and u is bounded in L 2 (0, T ; L 2 ). The fact that ρ  |u |2 and |H |2 are bounded in L ∞ (0, T ; L 1 ) gives the following convergence (up to the extraction of a subsequence n ): ρ  u converges weakly-∗ to some J¯ in L ∞ (0, T ; L 2 (Td )), ¯ in L ∞ (0, T ; L 2 (Td )). H converges weakly-∗ to some K ¯ = H in some sense, where Our main task in this section is to show that J¯ = u and K (u, H) is the strong solution to the incompressible viscous MHD equations (1.16)-(1.18). The key point is to control the oscillations caused by the initial data. This can be done as follows. Step 1: Description and cancelation of the oscillations. In order to describe the oscillations caused by the initial data, we employ the “filtering” method which has been used previously by several authors, see [2,7,18,19]. We project the momentum equation (1.9) on the “gradient vector-fields” to find 1 ∂t Q(ρ  u ) + Q[div(ρ  u ⊗ u )] − (2μ + λ)∇divu + ∇(|H |2 ) 2  1 a   γ    −Q[(H · ∇)H ] + 2 ∇ (ρ ) − 1 − γ (ρ − 1) + 2 ∇(ρ  − 1) = 0.  

(4.2)

Noticing ρ  = 1 + ϕ  , we can write (1.8) and (4.2) as

where

F

∂t ϕ  + divQ(ρ  u ) = 0,

(4.3)

∂t Q(ρ  u ) + ∇ϕ  = F ,

(4.4)

is given by 1 F = −Q[div(ρ  u ⊗ u )] + (2μ + λ)∇divu − ∇(|H |2 ) 2  a   γ    +Q[(H · ∇)H ] − 2 ∇ (ρ ) − 1 − γ (ρ − 1) . 

(4.5)

τL Therefore, we introduce the following group defined by L(τ  ) = e , τ ∈ R, where L is the operator defined on D0 × (D )d with D0 = {φ ∈ D , Td φ(x)d x = 0}, by     −div v φ . = L −∇φ v

Incompressible Limit of MHD Equations

389

Then, it is easy to check that eτ L is an isometry on each H r × (H r )d for all r ∈ R and for all τ ∈ R. Denoting     ¯ ) φ φ(τ = eτ L , v v¯ (τ ) we have ∂ φ¯ = −div¯v, ∂τ Thus,

∂ 2 φ¯ ∂τ 2

∂ v¯ ¯ = −∇ φ. ∂τ

− φ¯ = 0.

In the sequel, we shall denote      t ϕ ϕ   U = , V =L − Q(ρ  u ) Q(ρ  u )  and use the following approximations:        ¯ = ¯ = L −t √   , V √   , U Q( ρ u ) Q( ρ u )  which satisfy ¯  || ||U − U

2γ

L ∞ (0,T ;L γ +1 (Td ))

→ 0 as  → 0.

(4.6)

With this notation, we can rewrite Eqs. (4.3)-(4.4) as ∂t U =

1 LU + F! , 

or equivalently   t ! ∂t V = L − F , 

(4.7)

where (and in what follows) ! v denotes (0, v)T .  It is easy to check that F , given by (4.5), is bounded in L 2 (0, T ; H −s0 (Td )) for some s0 (s0 ∈ R). Hence, V is compact in time. Moreover, by virtue of the energy inequal2γ

ity (2.5) and the boundedness of the linear projector P, V ∈ L ∞ (0, T ; L γ +1 (Td )) uniformly in . Thus, ¯ in L r (0, T ; H −s (Td )) V converges strongly to some V

(4.8)

for all s > s0 and 1 < r < ∞. Denote θ ≡ 2μ + λ, L1 (τ ) the first component of L(τ ), and L2 (τ ) the last d components of L(τ ). If we had sufficient compactness in space, then we could pass the limit in (4.7) and obtain the following limit system for the oscillating parts: ¯ + Q1 (u, V) ¯ + Q2 (V, ¯ V) ¯ − ∂t V

θ ¯ V = 0, 2

(4.9)

390

S. Jiang, Q. Ju, F. Li

where u is the strong solution of the viscous incompressible MHD equations (1.16)(1.18), Q1 is a linear form of V defined by 

1 Q1 (v, V) = lim τ →∞ τ

τ 0

 0 ds, L(−s) div(v ⊗ L2 (s)V + L2 (s)V ⊗ v) 

(4.10)

and Q2 is a bilinear form of V defined by 1 Q2 (V, V) = lim τ →∞ τ



τ 0



0 L(−s) div(L2 (s)V ⊗ L2 (s)V) +

 γ −1 2 2 ∇(L1 (s)V)

ds (4.11)

for any divergence-free vector field v ∈ L 2 (Td )d and any V = (φ, ∇q)T ∈ L 2 (Td )d+1 . Actually, the convergence in (4.10) and (4.11) can be guaranteed by the following proposition: Proposition 4.1 ([19]). For all v ∈ L r1 (0, T ; L 2 ) and V ∈ L r2 (0, T ; L 2 ), we have the following weak convergences (r1 and r2 are such that the products are well defined):   t 0 = Q1 (v, V), (4.12) w − lim L − div(v ⊗ L2 ( t )V + L2 ( t )V ⊗ v) →0     t 0 = Q2 (V, V). w − lim L − t t t 2 div(L ( )V ⊗ L ( )V) + γ −1 →0  2  2  2 ∇(L1 (  )V) (4.13) 

The viscosity term in the oscillation equations (4.9) is obtained by the following proposition: Proposition 4.2 ([19]). Suppose that the same hypothesis as in Proposition 4.1 on V holds. Then, we have θ 1 V = lim τ →∞ τ 2



τ 0

 0 ds. L(−s) θ L2 (s)V 

(4.14)

The following propositions, the proof of which can be found in [19], play an important role in our subsequent analysis. Proposition 4.3 ([19]). For all v, V, V1 and V2 (regular enough to define all the products), we have 

 T

d

d T

Td

Q1 (v, V)V = 0,

Td

Q2 (V, V)V = 0,

(4.15)

[Q1 (v, V1 )V2 + Q1 (v, V2 )V1 ] = 0,

(4.16)

[Q2 (V1 , V1 )V2 + 2Q2 (V1 , V2 )V1 ] = 0.

(4.17)

Incompressible Limit of MHD Equations

391

Proposition 4.4 ([19]). Using the symmetry of Q2 , we can extend the equality (4.13) in Proposition 4.1 to the case:     t 0   w − lim L − 1 t t t t →0  2 div L2 (  )V1 ⊗ L2 (  )V2 + L2 (  )V2 ⊗ L2 (  )V1   0 = Q2 (V1 , V2 ). + γ −1 (4.18) t t 2 ∇(L1 (  )V1 ⊗ L1 (  )V2 ) Moreover, the above identity holds for V1 ∈ L q (0, T ; H r ) and V2 ∈ L p (0, T ; H −r ) with r ∈ R and 1/ p + 1/q = 1. Also, (4.18) can be extended to the case where we replace V2 in the left-hand side by a sequence V2 such that V2 converges strongly to V2 in L p (0, T ; H −r ). Step 2: The modulated energy functional and uniform estimates. Let V0 be the solution of the following system: ∂t V0 + Q1 (u, V0 ) + Q2 (V0 , V0 ) −

θ V0 = 0 2

(4.19)

with initial data V0 |t=0 = (ϕ0 , Q u˜ 0 )T ,

(4.20)

where u is the strong solution of the viscous incompressible MHD equations (1.16)-(1.18) with initial velocity u0 . From [19], we know that the Cauchy problem (4.19)-(4.20) has a unique global strong solution. In order to prove the convergence results in Theorem 2.5, we have to bound the term " "   "2   "2 " " " "  t  2 " ρ  u − u − L2 t V0 " " V0 " + H − H L 2 (Td ) + " − L1 " " " 2 d .   L 2 (Td ) L (T ) To this end, we first recall the following energy inequality of the compressible MHD equations (1.8)-(1.10):  t 

1   2  2  2 ρ (t)|u | (t) + |H | (t) + ( (t)) + μ |∇u |2 2 Td 0 Td  t  t  2 +(μ + λ) |divu | + ν |∇H |2 0 Td 0 Td 

1 ρ0 |u0 |2 + |H0 |2 + (0 )2 , for a.e. t ∈ [0, T ]. (4.21) ≤ 2 Td On the other hand, the conservation of energy for the incompressible viscous MHD equations (1.16)-(1.18) reads 

1

 t 1 |u|2 (t) + |H|2 (t) + μ|∇u|2 + ν|∇H|2 = |u0 |2 + |H0 |2 . 2 Td 2 Td 0 Td (4.22) For the system (4.19), Proposition 4.3 implies that   Q1 (u, V0 )V0 = 0, Q2 (V0 , V0 )V0 = 0, Td

Td

392

S. Jiang, Q. Ju, F. Li

from which the following equality follows:    1 θ 1 0 2 0 2 |V | + |∇V | = |V0 (t = 0)|2 . 2 Td 2 Td 2 Td

(4.23)

Using L1 ( t )V0 as a test function and noticing ρ  = 1 + ϕ  , we obtain the following weak formulation of the continuity equation (1.8):      # # $ $ #τ $

t τ 1 t 0  V ϕ (t) + V0 ϕ  + div(ρ  u )L1 V0 div L2 L1   0 Td   Td  t  #τ $ 0  − ∂t V ϕ = L1 ϕ0 ϕ0 . (4.24)  Td 0 Td We use u and L2 ( t )V0 to test the momentum equation (1.9) respectively, to deduce  Td

−

 t Td

0

and

  1 ρ  u · (u · ∇)u − (H · ∇)H − μu + ∇ p + ∇(|H|2 ) 2 0 Td      (ρ u ⊗ u ) · ∇u + (H · ∇)H · u−μ∇u · ∇u = ρ0 u0 · u0 (4.25)

(ρ  u · u)(t) +

 t

Td

      # #τ $ $ t 1 t V0 ρ  u · ∇ L1 ρ  u · L2 V0 (t) +   0 Td  Td  t  t #τ $ # #τ $ $ ∂t V0 · (ρ  u ) − V0 − L2 (ρ  u ⊗ u ) · ∇ L2   0 Td 0 Td  t # #τ $ $ # # τ $ $

+ V0 + (μ + λ)divu div L2 V0 μ∇u · ∇ L2   0 Td  t  t #τ $ 1  2 # # τ $ 0$ − V0 − |H | div L2 V (H · ∇)H · L2   0 Td 0 Td 2   t  # #τ $ $  1  γ −1  2 − ϕ + ( ) div L2 V0 = ρ0 u0 · Q u˜ 0 . (4.26) 2  Td 0 Td 



Similarly, we test (1.10) by H to get   t  t   (H · H)(t) + H · [(u · ∇)H − (H · ∇)u − νH] + ν ∇H · ∇H Td 0 Td 0 Td  t          + (divu )H + (u · ∇)H − (H · ∇)u · H = H0 · H0 . (4.27) 0

Td

Td

Summing up (4.21), (4.22) and (4.23), inserting (4.24)-(4.27) into the resulting inequality, and using the fact  t # # $  t $ τ ∂t V0 · U = L ∂t V 0 · V  ,  0 Td 0 Td

Incompressible Limit of MHD Equations

393

we deduce, after a straightforward calculation, the following inequality:    2   2 &  %      t t 1  0  2  ρ  u − u − L2 V  (t) + |H − H| (t) +  − L1 V0  (t)  2 Td    t  # # τ $ $2 ν  t    V0  + +μ |∇H − ∇H|2 ∇ u − u − L2 d d  2 0 T 0 T  t  #   # $ # τ $ $2 μ t   V0  |∇H |2 + |∇H|2 + (μ + λ) + div u − u − L2 2 0 Td  0 Td   # τ $ 2 1     V0  (0) + |H − H|2 (0) ≤  ρ u − u − L2 2 Td  8  # τ $ 2    +  − L1 V0  (0) + Ai (t), 

(4.28)

i=1

where

         t t  0     ( ρ −1) ρ u · u + L2 ( −ϕ )L1 V (t)− V0 (t) = d d   T T       t    V0 (0) ( ρ − 1) ρ u · u + L2 − d  T      t   ( − ϕ )L1 (4.29) V0 (0), + d  T  t  t    A2 (t) = ρ u ·∇p −μ (ρ  − 1)u u, (4.30) A1 (t)



Td

0

 t

0

Td

 t

# #τ $ $ V0 ∇u · ∇ L2  0 Td 0 Td  t # #τ $ $ −(μ + λ) V0 , divu · div L2 d  0 T  t   t # τ $ 2 θ   A4 (t) = − V0  |∇V0 |2 + μ ∇L2 d 2 0 Td  0 T  t  # τ $ 2   V0  , +(μ + λ) divL2 d  0 T  t  t A5 (t) = − ρ  u · [(H · ∇)H)] − (H · ∇)H · u A3 (t) = −

θ 2

0

 t + 0

Td

Td

 t + 0

Td

V0 · V − μ

0

Td

H · [(u · ∇)H − (H · ∇)H] +

1 2

 t 0

Td

(4.31)

(4.32)

ρ  u · ∇(|H|2 )

  (divu )H + (u · ∇)H − (H · ∇)u · H,

(4.33)

394

S. Jiang, Q. Ju, F. Li

 t

 t

#τ $ $ # V0 (ρ  u ⊗ u ) · ∇u + L2 d d  0 T 0 T  t #τ $ $ # γ −1 − V0 , (4.34) ( )2 div L2 d 2  0 T  t

A7 (t) = Q1 (u, V0 ) + Q2 (V0 , V0 ) · V , (4.35)

A6 (t) =

0

A8 (t) = −

ρ  u · [(u · ∇)u] −

Td

 t 0

1  2 # # τ $ 0$ |H | div L2 V −  Td 2

 t 0

Td

(H · ∇)H · L2

#τ $ 

V0 . (4.36)

Step 3: Convergence of the modulated energy functional. To show the convergence of the modulated energy functional (4.28), we need to estimate the remainders Ai (t), i = 1, . . . , 8. In the sequel, we will denote by ω (t) any sequence of time-dependent functions which converges to 0 uniformly in t. For convenience, we also denote √ w ≡ ρ  u − u − L2 ( t )V0 , Z ≡ H − H, and   ≡  − L1 ( t )V0 . For the term A1 (t), we employ (2.5), (3.13), the regularity of u and (4.6), and follow a procedure similar to that for R2 (t), to obtain |A1 (t)| ≤ C T  + ω (t).

(4.37)

On the other hand, the term A2 (t) has the same bound as R5 (t) + R7 (t), thus |A2 (t)| ≤ C T .

(4.38)

To bound the term A3 (t), we integrate by parts and use the fact L2 ( t )V0 = ∇ q˜  for some function q˜  and Proposition 4.2 to infer  t #τ $ # #τ $ $ V0 = μ V0 ∇u · ∇ L2 u · L2 d d   0 T 0 T  t # τ $ 0  · V0 =μ L −  d  u 0 T  t μ ¯ · V0 + ω (t), = V 2 0 Td  t  t #τ $ # #τ $ $  0 V = (μ + λ) V0 −(μ + λ) divu · div L2 u · L2   0 Td 0 Td   μ+λ t ¯ · V0 + ω (t), = V 2 0 Td −μ

 t

and −

θ 2

 t 0

Td

V0 · V = −

θ 2

 t 0

Td

¯ · V0 + ω (t). V

Incompressible Limit of MHD Equations

395

Thus, recalling θ = 2μ + λ, one has A3 (t) = ω (t).

(4.39)

Similarly, it follows from Proposition 4.2 that A4 (t) = ω (t).

(4.40)

Recalling (3.8) and using Hölder’s inequality, the inequalities (2.5) and (3.13), the regularity of H, and Sobolev’s imbedding theorem, we conclude  t  t    A5 (t) ≤ (1 − ρ )u · [(H · ∇)H)] + ||Z (τ )||2L 2 ||∇u(τ )|| L ∞ dτ Td

0

0

 t

||w (τ )||2L 2 + ||Z (τ )||2L 2 ||∇H(τ )|| L ∞ dτ + 0

 t

#τ $

V0 (Z · ∇)H · (1 − ρ  )u + L2 d  0 T  t   #τ $  − V0 · ∇ H · Z (1 − ρ  )u + L2 d  0 T   1 t + (ρ  − 1)u ∇(|H|2 ) 2 0 Td  t

||w (τ )||2L 2 + ||Z (τ )||2L 2 · [||∇u(τ )|| L ∞ + ||∇H(τ )|| L ∞ ] dτ ≤ +

0

 t

+ 0

#τ $

# #τ $ $ V0 − L2 V0 · ∇ H · H dτ + C T . (H · ∇)H · L2   Td (4.41)

The term A6 (t) can be rewritten as  t # #τ $ $ V0 A6 (t) = − (w ⊗ w ) · ∇ u + L  0 Td −

γ −1 2  t

 t 0

# #τ $ $ V0 |  |2 div L2  Td

#τ $ $ # V0 ρ  u ⊗ u + L2 d  0 T #τ $ $  # #τ $ $ # V0 ⊗ ρ  u · ∇ u + L V0 + u + L2    t # #τ $ $ # #τ $ $ # #τ $ $ V0 ⊗ u + L2 V0 · ∇ u + L V0 u + L2 +    0 Td

−

 t + 0



# #τ $ $ γ − 1  # τ $ 0 2 V  div L2 V0 L1 2   Td

396

S. Jiang, Q. Ju, F. Li

# #τ $ $ V0  div L2 V0    2  t  t |u| . (ρ  − ρ  )u · ((u · ∇)u) − ( ρ  u − u) · ∇ + d d 2 0 T 0 T (4.42)

−(γ − 1)L1

#τ $

We have to bound all the terms on the right-hand side of (4.42). Keeping in mind that div v = 0 and applying Proposition 4.1, one obtains  t

# #τ $ $ # #τ $ $ # #τ $ $ V0 ⊗ L2 V0 · ∇ u + L2 V0 L2    0 Td    # $  # #τ $ $ τ γ −1 t  2 V0  div L2 V0 + L1 d 2   0 T  t  # # $ $ # # $ $ τ τ V0 ⊗ L2 V0 div L2 =− d   0 T $ γ − 1  # τ $ 0 2 # 0 ∇ L1 V  · V +! + u 2   t # τ$ L − =−  0 Td   $ # 0 0 u   τ  0    τ  0  γ −1   τ  0 2 · V + ! div L2  V ⊗ L2  V + 2 ∇ L1  V   t # $ =− Q2 (V0 , V0 ) · V0 + ! u + ω (t) = ω (t). (4.43) 0

Td

From Proposition 4.4 we get  t

 #τ $ #τ $  # #τ $ $ V0 ⊗ ρ  u + ρ  u ⊗ L2 V0 · ∇ v + L2 V0 L2    0 Td  t #τ $ # #τ $ $ V0  div L2 V0 (γ − 1)L1 − d   0 T  t  # # $ #τ $ $ τ V0 ⊗ ρ  u + ρ  u ⊗ L2 V0 div L2 =   0 Td $ # #τ $ $ # #τ $ V0  · u + L2 V0 +(γ − 1)∇ L1    t # τ$ = L −  0 Td

−

Incompressible Limit of MHD Equations

397





0

          √ √ div L2 τ V0 ⊗ ρ  u + ρ  u ⊗ L2 τ V0 +(γ −1)∇ L1 τ V0  $ # u · V0 + ! ⎛ ⎞ 0 # τ$      ⎟ √ √ ⎜ L − = ⎝ div L2 τ V0 ⊗ Q( ρ  u ) + Q( ρ  u ) ⊗ L2 τ V0 ⎠ d      0 T + (γ − 1)∇ L1 τ V0  $ # u · V0 + !  t

 t

# τ$ L − +  0 Td $ # u · V0 + ! =

 t

 Td

0





0

      √ √ div L2 τ V0 ⊗ P( ρ  u ) + P( ρ  u ) ⊗ L2 τ V0

# $ # $ ¯ · V0 + ! 2Q2 (V0 , V) u + Q1 (u, V0 ) · V0 + ! u + ω (t).

(4.44)

Similarly, we have  t

  # #τ $ $ V0 ρ  u ⊗ u + u ⊗ ρ  u · ∇ u + L  0 Td  t ¯ 0 + ω (t), = Q1 (u, V)V

−

Td

0

(4.45)

and  t

#τ $ #τ $ $  # #τ $ $ V0 + L2 V0 ⊗ u · ∇ u + L V0 u ⊗ L2 d    0 T  t =− Q1 (u, V0 )V0 + ω (t). 

0

Td

(4.46)

On the other hand, using the basic energy equality (2.5) and the regularity of u, following the same process as for R6 (t), we obtain  t   2   t  |u|    )u · ((u · ∇)u) −  u − u) · ∇  ≤ C T . (ρ − ρ ( ρ  d d 2  0

T

0

T

(4.47) The term A7 (t) can be rewritten as A7 (t) =

 t 0

Td



¯ + ω (t). Q1 (u, V0 ) + Q2 (V0 , V0 ) · V

(4.48)

398

S. Jiang, Q. Ju, F. Li

Substituting (4.43)-(4.48) into (4.42), we conclude that |A6 (t)| + |A7 (t)|

 ≤C

t

(||w ||2 + ||  ||2 )

0

 " #τ $ "  " " dτ + C T  + ω (t). V0 " × ∇u L ∞ + "∇L2  L∞

(4.49)

Thus, we insert the estimates on Ai (t) (i = 1, · · · , 7) into (4.28) to obtain  t  #    # τ $ $2 1    2  2  2 V0  |w | + |Z | + | | (t) + μ ∇ u − u − L2 d 2 Td  0 T  t # $ ν |∇Z |2 + |∇H |2 + |∇H|2 + 2 0 Td  t  # # τ $ $2   +(μ + λ) V0  div u − u − L2 d  0 T  t

||w (τ )||2L 2 + ||Z (τ )||2L 2 ≤ 0

#τ $

V0  L ∞ ) dτ · ||∇u(τ )|| L ∞ + ||∇H(τ )|| L ∞ + ∇L2     1 |w |2 + |Z |2 + |  |2 (0) + C T  + ω (t) + A8 (t) + A9 (t), (4.50) + 2 Td where  t

#τ $

# #τ $ $ V0 − L2 V0 · ∇ H · H . (4.51) (H · ∇)H · L2   0 Td  ¯  = 1d Now, to deal with A8 (t) and A9 (t), we denote H H (x)d x to deduce 0 |T | Td 0 from the magnetic field equation (1.10) that A9 (t)

=

 Td



H (x, t)d x =

Td

¯ 0 |Td |. H0 (x)d x = H

(4.52)

The assumption that H0 converges strongly in L 2 to some H0 implies    

Td

 

H0 (x) − H0 (x)d x 

 ≤

Td

|H0 (x) − H0 (x)|d x 1

≤ |Td | 2

 Td

|H0 (x) − H0 (x)|2 d x

1 2

→ 0 as  → 0,

whence ¯ 0 as  → 0. ¯ 0 → H H

(4.53)

Incompressible Limit of MHD Equations

399

Using Hölder’s inequality, Sobolev’s imbedding theorem, Poincáre’s inequality, the isometry property of L, and (4.52) and (4.53), we find that  ν t |A8 (t)| + |A9 (t)| ≤ [||∇H ||2 + ||∇H ||2 ](τ )dτ + ω(t) 4 0  t 1 + (ϕ0 2H 2 + ||Qu0 ||2H 2 ) [||∇H ||2 + ||∇H ||2 ](τ )dτ. ν 0 (4.54) Thus, substituting (4.54) into (4.50) and using (2.18), we deduce by Gronwall’s inequality that, for almost all t ∈ [0, T ], ||w (t)||2L 2 + ||Z (t)||2L 2 + ||  (t)||2L 2 % &  2  2  2  ≤ C˜ ||w (0)|| L 2 + ||Z (0)|| L 2 + ||0 − ϕ0 || L 2 + C T  + sup ω (s) ,

(4.55)

0≤s≤t

where

  ˜ C = exp C

T



||∇u(τ )|| L ∞ + ||∇H(τ )|| L ∞ + ∇L2

0

#τ $ 

V0  L ∞ ) dτ

< +∞.

Step 4: End of the proof of Theorem 2.5. Letting  go to zero in (4.55), we see that H ¯ = H. Combining (4.50) with converges strongly to H in L ∞ (0, T ; L 2 (Td )). Hence, K (4.55), we can easily prove that ∇H converges strongly to ∇H in L 2 (0, T ; L 2 (Td )). Next, it suffices to prove (4) and (5) in Theorem 2.5. Noting that P(L2 ( t )V0 ) = 0 and the fact that the projection operator P is a bounded linear mapping from L 2 to L 2 , we obtain, with the help of (4.55), that "    " " " t  u − u − L P V0 " ρ sup P( ρ  u ) − u L 2 = sup " 2 " 2 "  0≤t≤T 0≤t≤T L "   " " " t 0"   ≤ sup " " ρ u − u − L2  V " 2 0≤t≤T L → 0 as  → 0.

(4.56)

Therefore, (4) is proved. Utilizing (4.1), we deduce from (1.8) that div u converges weakly to 0 in H −1 ((0, T ) × Td ). Thus we obtain easily that Qu converges weakly 2 d to 0 √ in H −1 (0, T ; L 2 (Td )). In view of the fact that u is bounded in L 2 (0, √T ; L (T )) 2 d  and ρ converges strongly to 1 in C([0, T ], L (T )), we see that Q( ρ u ) con−1 (0, T ; L 2 (Td )). Obviously, the fact √ρ  u = P(√ρ  u ) + verges weakly to 0 in H √ √ Q( ρ  u ) implies the weak convergence of ρ  u to u in H −1 (0, T ; L 2 (Td )). The proof of Theorem 2.5 is finished.   Theorem 2.6 can be shown by slightly modifying the proof of Theorem 2.5, and therefore, we omit its proof here. Acknowledgement. This work was done while Fucai Li was visiting the Institute of Applied Physics and Computational Mathematics, he thanks the institute for its hospitality. Jiang was supported by the National Basic Research Program (Grant No. 2005CB321700) and NSFC (Grant No. 40890154). Ju was supported by NSFC (Grant No. 10701011). Li was supported by NSFC (Grant No. 10501047, 10971094).

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References 1. Brenier, Y.: Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. PDE 25, 737–754 (2000) 2. Danchin, R.: Zero Mach number limit for compressible flows with periodic boundary conditions. Amer. J. Math. 124, 1153–1219 (2002) 3. Desjardins, B., Grenier, E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455, 2271–2279 (1999) 4. Duvaut, G., Lions, J.L.: Inéquations en thermoélasticité et magnéto-hydrodynamique. Arch. Rat. Mech. Anal. 46, 241–279 (1972) 5. Fan, J., Yu, W.: Strong solution to the compressible MHD equations with vacuum. Nonlinear Anal. Real World Appl. 10, 392–409 (2009) 6. Fan, J., Yu, W.: Global variational solutions to the compressible magnetohydrodynamic equations. Nonlinear Anal. 69, 3637–3660 (2008) 7. Gallagher, I.: A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations. J. Math. Kyoto Univ. 40, 525–540 (2001) 8. Hoff, D.: The zero-Mach limit of compressible flows. Commun. Math. Phys. 192, 543–554 (1998) 9. Hu, X., Wang, D.: Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Rat. Mech. Anal., to appear, http://arXiv.org/ abs/0904.3587v1[math.AP], 2009 10. Hu, X., Wang, D.: Global solutions to the three-dimensional full compressible magnetohydrodynamic flows. Commun. Math. Phys. 283, 255–284 (2008) 11. Hu, X., Wang, D.: Low Mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J. Math. Anal. 41, 1272–1294 (2009) 12. Jiang, S., Ju, Q.C., Li, F.C.: Combined incompressible and inviscid limit of the compressible magnetohydrodynamic equations in the whole space. http://arXiv.org/abs/0905.3937v1[math.AP], 2009 13. Kawashima, S.: Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics. Japan J. Appl. Math. 1, 207–222 (1984) 14. Kulikovskiy, A.G., Lyubimov, G.A.: Magnetohydrodynamics. Reading, MA: Addison-Wesley, 1965 15. Laudau, L.D., Lifshitz, E.M.: Electrodynamics of continuous media. 2nd ed., New York: Pergamon, 1984 16. Lei, Z., Zhou, Y.: BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin. Dyn. Syst. 25, 575–583 (2009) 17. Li, F.C., Yu, H.J.: Optimal Decay Rate of Classical Solution to the Compressible Magnetohydrodynamic Equations. Preprint, 2009 18. Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998) 19. Masmoudi, N.: Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 199–224 (2001) 20. Masmoudi, N.: Examples of singular limits in hydrodynamics, In: Evolutionary Equations, Vol. III, Handbook of Differential Equations. Amsterdam: North-Holland, 2007, pp. 275–375 21. Polovin, R.V., Demutskii, V.P.: Fundamentals of Magnetohydrodynamics. New York: Consultants, Bureau, 1990 22. Schochet, S.: Fast singular limits of hyperbolic PDEs. J. Diff. Eqns. 114, 476–512 (1994) 23. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635–664 (1983) 24. Vol’pert, A.I., Khudiaev, S.I.: On the Cauchy problem for composite systems of nonlinear equations. Mat. Sbornik 87, 504–528 (1972) 25. Zhang, J.W., Jiang, S., Xie, F.: Global weak solutions of an initial boundary value problem for screw pinches in plasma physics. Math. Models Methods Appl. Sci. 19, 833–875 (2009) Communicated by P. Constantin

Commun. Math. Phys. 297, 401–426 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1037-4

Communications in

Mathematical Physics

Proof of the Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics with Degree of Mobility Greater than Two Volodymyr Kiosak1, , Vladimir S. Matveev2, 1 Odessa, Ukraina. E-mail: [email protected] 2 Institute of Mathematics, FSU Jena, 07737 Jena, Germany. E-mail: [email protected]

Received: 7 July 2009 / Accepted: 27 November 2009 Published online: 20 April 2010 – © Springer-Verlag 2010

Abstract: Degree of mobility of a (pseudo-Riemannian) metric is the dimension of the space of metrics geodesically equivalent to it. We prove that complete metrics on (n ≥ 3)−dimensional manifolds with degree of mobility ≥ 3 do not admit complete metrics that are geodesically equivalent to them, but not affinely equivalent to them. As the main application we prove an important special case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the standard round sphere (up to a finite cover and multiplication of the metric by a constant). 1. Introduction 1.1. Definitions and result. Let M be a connected manifold of dimension n ≥ 3, let g be a (Riemannian or pseudo-Riemannian) metric on it. We say that a metric g¯ on the same manifold M is geodesically equivalent to g, if every g-geodesic is a reparametrized g-geodesic. ¯ We say that they are affine equivalent, if their Levi-Civita connections coincide. As we recall in Sect. 2.1, the set of metrics geodesically equivalent to a given one (say, g) is in one-to-one correspondence with the nondegenerate solutions of Eq. (9). Since Eq. (9) is linear, the space of its solutions is a linear vector space. Its dimension is called the degree of mobility of g. Locally, the degree of mobility of g coincides with the dimension of the set (equipped with its natural topology) of metrics geodesically equivalent to g. The degree of mobility is at least one (since const·g is always geodesically equivalent to g) and is at most (n + 1)(n + 2)/2, which is the degree of mobility of simply-connected spaces of constant sectional curvature.  Partially supported by DFG (SPP 1154).  Partially supported by DFG (SPP 1154 and GK 1523).

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Our main result is: Theorem 1. Let g be a complete Riemannian or pseudo-Riemannian metric on a connected M n of dimension n ≥ 3. Assume that for every constant c = 0 the metric c · g is not the Riemannian metric of constant curvature +1. If the degree of mobility of the metric is ≥ 3, then every complete metric g¯ geodesically equivalent to g is affine equivalent to g. The assumption that the metrics are complete is important: the examples constructed by Solodovnikov [70,71] show the existence of complete metrics with a big degree of mobility (all metrics geodesically equivalent to such metrics are not complete). Theorem 2. Let g be a complete Riemannian or pseudo-Riemannian metric on a closed (= compact, without boundary) connected manifold M n of dimension n ≥ 3. Assume that for every constant c = 0 the metric c · g is not the Riemannian metric of constant curvature +1. Then, at least one the following possibilities holds: • the degree of mobility of g is at most two, or • every metric g geodesically equivalent to g¯ is affine equivalent to g. Remark 1. In the Riemannian case, Theorem 1 was proved in [57, Th. 16] and in [56]. The proof uses observations which are wrong in the pseudo-Riemannian situation; we comment on them in Sect. 1.2. Our proof for the pseudo-Riemannian case is also not applicable in the Riemannian case, since it uses lightlike geodesics in an essential way. In Sect. 2.5, we give a new, shorter (modulo results of our paper) proof of Theorem 1 for the Riemannian metrics as well. Remark 2. In the Riemannian case, Theorem 2 follows from Theorem 1, since every Riemannian metric on a closed manifold is complete. In the pseudo-Riemannian case, Theorem 2 is a separate statement. Remark 3. Moreover, the assumptions that the metric is complete and the dimension is ≥ 3 could be removed from Theorem 2: by [60, Cor. 5.2] and [61, Cor. 1], if the degree of mobility of g on a closed (n ≥ 2)−dimensional manifold is at least three, then for a certain constant c = 0 the metric c · g is the Riemannian metric of curvature 1, or every metric geodesically equivalent to g is affine equivalent to g. The proofs in [60] and [61] are nontrivial; the proof of [60, Cor. 5.2] is in particular based on the results of Sect. 2.3.5 of the present paper. 1.2. Motivation I: Projective Lichnerowicz conjecture. Recall that a projective transformation of the manifold (M, g) is a diffeomorphism of the manifold that takes (unparametrized) geodesics to geodesics. The infinitesimal generators of the group of projective transformations are complete projective vector fields, i.e., complete vector fields whose flows take (unparameterized) geodesics to geodesics. Theorem 1 allows us to prove an important special case of the following conjecture, which folklore attributes (see [57] for discussion) to Lichnerowicz and Obata (the latter assumed in addition that the manifold is closed, see, for example, [26,63,77]): Projective Lichnerowicz Conjecture. Let a connected Lie group G act on a complete connected pseudo-Riemannian manifold (M n , g) of dimension n ≥ 2 by projective transformations. Then, it acts by affine transformations, or for a certain c ∈ R \ {0} the metric c · g is the Riemannian metric of constant positive sectional curvature +1.

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We see that Theorem 1 implies Corollary 1. The projective Lichnerowicz Conjecture is true under the additional assumption that the dimension n ≥ 3 and that the degree of mobility of the metric g is ≥ 3. Indeed, the pullback of the (complete) metric g under the projective transformation is a complete metric geodesically equivalent to g. Then, by Theorem 1, it is affine equivalent to g, i.e., the projective transformation is actually an affine transformation, as it is stated in Corollary 1. Corollary 1 is thought to be the most complicated part of the solution of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics. We do not know yet whether the Lichnerowicz conjecture is true (for pseudo-Riemannian metrics), but we expect that its solution (= proof or counterexample) will require no new additional ideas beyond those from the Riemannian case. To support this optimistic expectation, let us recall that the projective Lichnerowicz conjecture was recently proved for Riemannian metrics [51,57]. The proof contained three parts: (i) proof for the metrics with the degree of mobility 2 ([57, Th. 15], [51, Chap. 4]), (ii) proof under the assumption dim(M) ≥ 3 for the metrics with the degree of mobility ≥ 3 ([57, Th. 16]), (iii) proof under the assumption dim(M) = 2 for the metrics with the degree of mobility ≥ 3, [51, Cor. 5 and Th. 7]. The most complicated (= lengthy; it is spread over [57, §§3.2–3.5, 4.2]) part was the proof under the additional assumptions (ii). The proof was based on the Levi-Civita description of geodesically equivalent metrics, on the calculation of the curvature tensor for Levi-Civita metrics with degree of mobility ≥ 3 due to Solodovnikov [70,71], and on global ordering of eigenvalues of j ai := ai p g pj , where ai j is a solution of (9), due to [6,54,74]. This proof can not be generalized to the pseudo-Riemannian metrics. More precisely, a pseudo-Riemannian analog of the Levi-Civita theorem is much more complicated, calculations of Solodovnikov essentially use positive-definiteness of the metric, and, as examples show, the global j ordering of eigenvalues of ai is simply wrong for pseudo-Riemannian metrics. Thus, Theorem 1 and Corollary 1 close the a priori most difficult part of the solution of the Lichnerowicz conjecture for the pseudo-Riemannian metrics. Let us now comment on (i), (iii), from the viewpoint of the possible generalization of the Riemannian proof to the pseudo-Riemannian case. We expect that this is possible. More precisely, the proof of (i) is based on a trick invented by Fubini [18] and Solodovnikov [70], see also [48,50,51]. The trick uses the assumption that the degree of mobility is two to double the number of PDEs (for a vector field v to be projective for the metric g), and to lower the order of this equation (the initial equations have order 2, the equations that we get after applying the trick have order 1). This of course makes everything much easier; moreover, in the Riemannian case, one can explicitly solve this system [18,64,70]. After doing this, one has to analyze whether the metrics and the projective field are complete; in the Riemannian case it was possible to do. The trick survives in the pseudo-Riemannian setting. The obtained system of PDE was solved for the simplest situations (for small dimensions [11,58], or under the additional assumption that the minimal polynomial of a ij coincides with the characteristic polynomial). We expect that the other part of the program could be realized for pseudoRiemannian metrics as well, though of course it will require a lot of work.

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Now let us comment on the proof under the assumptions (iii): dim(M) = 2, degree of mobility is ≥ 3. The initial proof of [51] uses the description of quadratic integrals of geodesic flows of complete Riemannian metrics due to [28]. This description has no analog for pseudo-Riemannian metrics. Fortunately, one actually does not need this description anymore: in [11,58] a complete list of 2-dimensional pseudo-Riemannian metrics admitting a projective vector field was constructed; the degree of mobility for all these metrics has been calculated. The metrics that are interesting for the setting (iii) are the metrics (2a, 2b, 2c) of [11, Th. 1] and (3d) of [58, Th. 1], because all other metrics admitting projective vector fields have constant curvature or degree of mobility equal to 2. All these metrics are given by relatively simple formulas using only elementary functions. In order to prove the projective Lichnerowicz conjecture in the setting (iii), one needs to understand which metrics from this list could be extended to a bigger domain; it does not seem to be too complicated. For the metrics (2a, 2b,2c) of [11, Theorem 1] it was already done in [38]. Under the additional assumption that M 2 is closed, two dimensional pseudoRiemannian version of the Licherowicz conjecture was proved in [61, Theorem 6]. As a consequence of Theorem 1, we obtain the following simpler version of the Lichnerowicz conjecture. Corollary 2. Let Projo (respectively, Affo ) be the connected component of the Lie group of projective transformations (respectively, affine transformations) of a complete connected pseudo-Riemannian manifold (M n , g) of dimension n ≥ 3. Assume that for no constant c ∈ R\{0} the metric c · g is the Riemannian metric of constant positive curvature +1. Then, the codimension of Affo in Projo is at most one. Indeed, it is well known (see, for example [57], or more classical sources acknowelged therein) that a vector field is projective if the (0, 2)−tensor a := L v g −

1 trace(g −1 L v g) · g n+1

(1)

is a solution of (9), where L v is the Lie derivative with respect to v. Moreover, the projective vector field is affine, iff the function (10) constructed by ai j given by (1) is constant. Now, let us take two infinitesimal generators of the Lie group Projo , i.e., two complete projective vector fields v and v. ¯ In order to show that the codimension of Affo in Projo is at most one, it is sufficient to show that a linear combination of these vector fields is 1 an affine vector field. We consider the solutions a := L v g − n+1 trace(g −1 L v g) · g and 1 a¯ := L v¯ g − n+1 trace(g −1 L v¯ g) · g of (9). If a, a, ¯ and g are linearly independent, the degree of mobility of the metric is ≥ 3. Then, Corollary 1 implies Projo = Affo . Thus, a, a, ¯ g are linearly dependent. Since the function λ := 21 g pq g pq , i.e., the function (10) constructed by a = g, is evidently constant, there exists a nontrivial linear combination aˆ of a, a¯ such that the corresponding λˆ given by (10) is constant. Since the mapping v → a := L v g −

1 trace(g −1 L v g) · g n+1

is linear, the linear combination of v, v¯ with the same coefficients is an affine vector field.  

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1.3. Motivation II: New methods for investigation of global behavior of geodesically equivalent metrics. The theory of geodesically equivalent metrics has a long and fascinating history. First non-trivial examples were discovered by Lagrange [35]. Geodesically equivalent metrics were studied by Beltrami [5], Levi-Civita [36], Painlevé [65] and other classics. One can find more historical details in the surveys [3,62] and in the introduction to the papers [42,43,46,47,53,56,57,74]. The success of general relativity made it necessary to study geodesically equivalent pseudo-Riemannian metrics. The textbooks [15,23,66,67] on pseudo-Riemannian metrics have chapters on geodesically equivalent metrics. In the popular paper [76], Weyl stated a few interesting open problems on geodesic equivalence of pseudo-Riemannian metrics. Recent references (on the connection between geodesically equivalent metrics and general relativity) include Ehlers et al [16,17], Hall and Lonie [20,24,25], Hall [21,22]. In many cases, local statements about Riemannian metrics could be generalised for the pseudo-Riemannian setting, though sometimes this generalisation is difficult. As a rule, it is very difficult to generalize global statements about Riemannian metrics to the pseudo-Riemannian setting. The theory of geodesically equivalent metrics is not an exception: most local results could be generalized. For example, the most classical results: the Dini/Levi-Civita description of geodesically equivalent metrics [12,36] and the Fubini Theorem [18] were generalised in [2,7–10]. Up to now, no global (if the manifold is closed or complete) methods for investigation of geodesically equivalent metrics were generalized for the pseudo-Riemannian setting. More precisely, virtually every global result on geodesically equivalent Riemannian metrics was obtained by combining the following methods. • “Bochner technique”. This is a group of methods combining local differential geometry and the Stokes theorem. In the theory of geodesically equivalent metrics, the most standard (de-facto, the only) way to use the Bochner technique was to use tensor calculus to canonically obtain a nonconstant function f such as g f = const · f , where const ≥ 0, which of course can not exist on closed Riemannian manifolds. An example could be derived from our paper: from Eq. (55) it follows that (g λ),k = 2(n + 1)Bλ,k . Thus, for a certain const ∈ R we have (g (λ + const)) = 2(n + 1)B(λ + const). If B is positive, g is Riemannian, and M is closed, this implies that the function λ is constant, which is equivalent to the statement that the metrics are affine equivalent. The first application of this technique in the theory of geodesically equivalent metrics is due to the Japan geometry school of Yano, Tanno, and Obata, see for example [27]. Also, the mathematical schools of Odessa and Kazan were extremely strong in this group of methods, see the review papers [3,62], and the references inside these papers. Of course, since for pseudo-Riemannian metrics the equation g f = const · f could have solutions for const ≥ 0, this technique completely fails in the pseudoRiemannian case. • “Volume and curvature estimations”. For geodesically equivalent metrics g and g, ¯ the repametrisation of geodesics is controlled by a function φ given by (5). This function also controls the difference between Ricci curvatures of g and g. ¯ Playing with this, one can obtain obstructions for the existence of positively definite geodesically equivalent metrics with negatively definite Ricci-curvature (assuming the manifold is closed, or complete with finite volume). Recent references include [29,68]. This method essentially uses the positive definiteness of the metrics. • “Global ordering of eigenvalues of a ij ”. The existence of a metric g¯ geodesically equivalent to g implies the existence of integrals of special form (we recall one of

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the integrals in Lemma 1) for the geodesic flow of the metric g [39,42,43]. In the Riemannian case, analysis of the integrals implies global ordering of the eigenvalues of  1  g) ¯ n+1 i p g¯ g pj , where g¯ i p is the tensor dual to g¯ i j , see [6,54,74]. the tensor a ij := det( det(g) Combining it with the Levi-Civita description of geodesically equivalent metrics, one could describe topology of closed manifolds admitting geodesically equivalent Riemannian metrics [33,40,41,44–47,49,52]. Though the integrability survives in the pseudo-Riemannian setting [6,73], the global ordering of the eigenvalues is not valid anymore (there exist counterexamples), so this method also is not applicable to the pseudo-Riemannian metrics. Our proofs (we explain the scheme of the proofs in the beginning of Sect. 2) use essentially new methods. We would like to emphasize here once more that the last step of the proof, which uses the local results to obtain global statements, is based on the existence of lightlike geodesics, and, therefore, is essentially pseudo-Riemannian. A similar idea was used in [30], where it was proved that complete Einstein metrics are geodesically rigid: every complete metric geodesically equivalent to a complete Einstein metric is affine equivalent to it. We expect further application of these new methods in the theory of geodesically equivalent metrics. 1.4. Additional motivation: Superintegrable metrics. Recall that a metric is called superintegrable, if the number of independent integrals of special form is greater than the dimension of the manifold. Superintegrable systems are nowadays a hot topic in mathematical physics, probably because almost all exactly solvable systems are superintegrable. There are different possibilities for the special form of integrals; de facto the most standard special form of the integrals is the so-called Benenti integrals, which are essentially the same as geodesically equivalent metrics, see [4,6,34]. Theorem 2 of our paper shows that complete Benenti-superintegrable metrics of nonconstant curvature cannot exist on closed manifolds, which was a folklore conjecture. 2. Proof of Theorems 1, 2 In Sect. 2.1, we recall standard facts about geodesically equivalent metrics and fix the notation. In Sect. 2.2, we will prove Lemma 2, which is a purely linear algebraic statement. Given two solutions of Eq. (11), it gives us Eq. (27). The coefficients in the equation are a priori functions. We will work with this equation for a while: In Sect. 2.3.1, we prove (Lemma 5) that (under the assumptions of Theorem 1) one of the coefficients of (27) is actually a constant. Later, we will show (Lemma 8) that the metric g determines the constant uniquely. Equation (27) will be used in Sect. 2.3.6. The main result of this section is Corollary 8. This corollary gives us (under assumptions of Theorem 1) an ODE that must be fulfilled along every lightlike geodesic, and that controls the reparameterization that produces g-geodesics from g-geodesics. ¯ The ODE is relatively simple and could be solved explicitly (Sect. 2.4). Analyzing the solutions, we will see that the geodesic is complete with respect to both metrics iff the function controlling the reparametrization of the geodesics is a constant, which implies that the metrics are affine equivalent. This proves Theorem 1 provided lightlike geodesics exist. As we mentioned in the Introduction, Theorem 1 was already proved [45,57] for Riemannian metrics. Nevertheless, for self-containedness, in

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Sect. 2.5 we give a new proof for Riemannian metrics as well, which is much shorter than the original proof from [45,57]. The proof of Theorem 2 will be done in Sect. 2.6. The idea is similar: we analyze a certain ODE along lightlike geodesics (this ODE will easily follow from Eq. (55), which is an easy corollary of Eq. (27)), and show that the assumption that the manifold is closed implies that the solution of the ODE coming from the metric g¯ is constant, which implies that g and g¯ are geodesically equivalent. 2.1. Standard formulas we will use. We work in tensor notation with the background metric g. That means, we sum with respect to repeating indexes, use g for raising and lowering indexes (unless we explicitly say otherwise), and use the Levi-Civita connection of g for covariant differentiation. As it was known already for Levi-Civita [36], two connections  =  ijk and ¯ = ¯ ijk have the same unparameterized geodesics, if and only if their difference is a pure trace: there exists a (0, 1)-tensor φ such that ¯ ijk =  ijk + δki φ j + δ ij φk .

(2)

The reparametrizations of the geodesics for  and ¯ connected by (2) are done accord¯ the curve γ (τ (t)) is ing to the following rule: for a parametrized geodesic γ (τ ) of , a parametrized geodesic of , if and only if the parameter transformation τ (t) satisfies the following ODE:     dτ  1 d p (3) φ p γ˙ = log   . 2 dt dt (We denote by γ˙ the velocity vector of γ with respect to the parameter t, and assume summation with respect to repeating index p.) If  and ¯ related by (2) are Levi-Civita connections of metrics g and g, ¯ then one can find explicitly (following Levi-Civita [36]) a function φ on the manifold such that its differential φ,i coincides with the covector φi : indeed, contracting (2) with respect p p to i and j, we obtain ¯ pi =  pi + (n + 1)φi . On the other hand, for the Levi-Civita p

connection  of a metric g we have  pk = φi =

1 ∂ log(|det (g)|) . 2 ∂ xk

Thus,

   det(g) ¯  ∂ 1 = φ,i log  2(n + 1) ∂ xi det(g) 

(4)

for the function φ : M → R given by φ :=

   det(g) ¯  1 . log  2(n + 1) det(g) 

(5)

In particular, the derivative of φi is symmetric, i.e., φi, j = φ j,i. The formula (2) implies that two metrics g and g¯ are geodesically equivalent if and only if for a certain φi (which is, as we explained above, the differential of φ given by (5)) we have g¯i j,k − 2g¯ i j φk − g¯ik φ j − g¯ jk φi = 0,

(6)

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where “comma” denotes the covariant derivative with respect to the connection . Indeed, the left-hand side of this equation is the covariant derivative with respect to ¯ and vanishes if and only if ¯ is the Levi-Civita connection for g. , ¯ Equations (6) can be linearized by a clever substitution: consider ai j and λi given by ai j = e2φ g¯ pq g pi gq j ,

(7)

λi = −e φ p g¯

(8)

2φ

pq

gqi ,

where g¯ pq is the tensor dual to g¯ pq : g¯ pi g¯ pj = δ ij . It is an easy exercise to show that the following linear equations for the symmetric (0, 2)-tensor ai j and (0, 1)-tensor λi are equivalent to (6). ai j,k = λi g jk + λ j gik .

(9)

Remark 4. For dimension 2, the substitution (7,8) was already known to R. Liouville [37] and Dini [12], see [11, Sect. 2.4] for details and a conceptual explanation. For arbitrary dimension, the substitution (7,8) and Eq. (9) are due to Sinjukov [69]. The underlying geometry is explained in [13,14]. Note that it is possible to find a function λ whose differential is precisely the (0, 1)tensor λi : indeed, multiplying (9) by g i j and summing with respect to repeating indexes i, j we obtain (g i j ai j ),k = 2λk . Thus, λi is the differential of the function λ :=

1 pq g a pq . 2

(10)

In particular, the covariant derivative of λi is symmetric: λi, j = λ j,i . We see that Eqs. (9) are linear. Thus the space of the solutions is a linear vector space. Its dimension is called the degree of mobility of the metric g. We will also need integrability conditions for Eq. (9) (one obtains them substituting p p the derivatives of ai j given by (9) in the formula ai j,lk − ai j,kl = ai p R jkl + a pj Rikl , which is true for every (0, 2)−tensor ai j ) p

p

ai p R jkl + a pj Rikl = λl,i g jk + λl, j gik − λk,i g jl − λk, j gil .

(11)

The integrability condition in this form was obtained by Sinjukov [69]; in equivalent form, it was known to Solodovnikov [70]. As a consequence of these integrability conditions, we obtain that every solution ai j of (9) must commute with the Ricci tensor Ri j : p

p

ai R pj = a j Ri p .

(12)

To show this, we “cycle” Eq. (11) with respect to i, k, l, i.e., we sum it with itself after renaming the indexes according to (i → k → l → i) and with itself after renaming the indexes according to (i → l → k → i). The first term at the left-hand side of p p p the equation will disappear because of the Bianchi equality Rikl + Rkli + Rlik = 0, the right-hand side vanishes completely, and we obtain p

p

p

a pi R jkl + a pk R jli + a pl R jik = 0.

(13)

Multiplying with g jk , using symmetries of the curvature tensor, and summing over the p p repeating indexes we obtain a pi Rl − a pl Ri = 0, i.e., (12).

Proof of Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics

409

Remark 5. For further use, let us recall that Eqs. (9) are of finite type (they close after two differentiations [14,62,69]). Since they are linear, and since in view of (10) they could be viewed as equations on ai j only, linear independence of the solutions on the whole connected manifold implies linear independence of the restriction of the solutions to every neighborhood. Thus, the assumption that the degree of mobility of g (on a connected M) is ≥ 3 implies that the degree of mobility of the restriction of g to every neighborhood is also ≥ 3. We will also need the following statement from [39,74]. We denote by co(a)ij the classical comatrix (adjugate matrix) of the (1, 1)-tensor a ij viewed as an n × n-matrix. co(a)ij is also a (1, 1)-tensor. Lemma 1 ([39,74]). If the (0, 2)-tensor ai j satisfies (9), then the function I : T M → R, (

x , ξ ) → g pq co(a)γp ξ γ ξ q 

∈M

(14)

∈Tx M

is an integral of the geodesic flow of g. Recall that a function is an integral of the geodesic flow of g, if it is constant along the orbits of the geodesic flow of g, i.e., if for every parametrized geodesic γ (t) the function I (γ (t), γ˙ (t)) does not depend on t. Remark 6. If the tensor ai j comes from a geodesically equivalent metric g¯ by formula (7), the integral (14) is    det(g) 2/(n+1)  I (x, ξ ) =  g(ξ, ¯ ξ ). det(g) ¯  In this form, Lemma 1 was already known to Painlevé [65]. 2.2. An algebraic lemma. Lemma 2. Assume symmetric (0, 2) tensors ai j , Ai j , λi j and i j satisfy p

p

ai p R jkl + a pj Rikl = λli g jk + λl j gik − λki g jl − λk j gil , p

p

Ai p R jkl + A pj Rikl = li g jk + l j gik − ki g jl − k j gil ,

(15)

where gi j is a metric and R ijkl is its curvature tensor. Assume ai j , Ai j , and gi j are linearly independent at the point p. Then, at the point, λi j is a linear combination of ai j and gi j . Remark 7. We would like to emphasize here that, though the lemma is formulated in the tensor notation, it is a purely algebraic statement (in the proof we will not use differentiation, and, as we see, no differential condition on a, A is required). Moreover, we can replace R ijkl by any (1,3)-tensor having the same algebraic symmetries (with respect to g) as the curvature tensor, so that for example the fact that the first equation of (15) coincides with (11) will not be used in the proof (but of course this will be used in the applications of Lemma 2). The underlying algebraic structure of the lemma is explained in the last section of [9].

410

V. Kiosak, V. S. Matveev

Proof. First observe that Eqs. (15) are unaffected by replacing ai j → ai j + a · gi j , λi j → λi j + λ · gi j , Ai j → Ai j + A · gi j , i j → i j + · gi j for arbitrary a, λ, A, ∈ R. Therefore we may suppose, without loss of generality, that ai j , λi j , Ai j , i j are trace-free, i.e., ai j g i j = λi j g i j = Ai j g i j = i j g i j = 0.

(16)

Our assumptions become that ai j and Ai j are linearly independent and our aim is to show that λi j = const · ai j . We multiply the first equation of (15) by All and sum over l. After renaming l → l, we obtain p

q

p

q

p

p

ai p R jkq Al + a pj Rikq Al = λ pi Al g jk + λ pj Al gik − λki A jl − λk j Ail . p

q

p

(17) q

We use symmetries of the Riemann tensor to obtain ai R pjkq Al = ai Rqk j p Al = p q ai Aql Rk j p . After substituting this in (17), we get p

q

p

q

p

p

ai Aql Rk j p + a j Aql Rki p = λ pi Al g jk + λ pj Al gik − λki A jl − λk j Ail .

(18)

Let us now symmetrize (18) by l, k,     p q q p q q ai Aql Rk j p + Aqk Rl j p + a j Aqk Rli p + Aql Rki p p

p

p

p

= λ pi Al g jk +λ pj Al gik − λki A jl − λk j Ail + λ pi Ak g jl + λ pj Ak gil −λli A jk − λl j Aik .

(19) We see that the components in brackets are the left-hand side of the second equation of (15) with other indexes. Substituting (15) in (19), we obtain p

p

p

p

ai pl g jk + ai pk g jl − jl aik − jk ail + a j pl gik + a j pk gil − il a jk − ik a jl p

p

p

p

= λ pi Al g jk + λ pj Al gik − λki A jl − λk j Ail + λ pi Ak g jl + λ pj Ak gil − λli A jk − λl j Aik .

(20) Collecting the terms by g, we see that (20) is can be written as  p  p p p ai pl − λ pi Al g jk + ai pk − λ pi Ak g jl     p p p p + a j pl − λ pj Al gik + a j pk − λ pj Ak gil = jl aik + jk ail + il a jk + ik a jl − λki A jl − λk j Ail − λli A jk − λl j Aik . (21) After denoting p

p

τil := ai pl − Al λ pi ,

(22)

Eq. (21) can be written as τil g jk + τik g jl + τ jl gik + τ jk gil = jl aik + jk ail + il a jk + ik a jl − λki A jl − λk j Ail − λli A jk − λl j Aik . (23)

Proof of Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics

411

Multiplying (23) by g jk , contracting with respect to j, k, and using (16), we obtain   p p p p (n + 2)τil + τ jk g jk gil = pl ai + i p al − λ pi Al − λlp Ai (22)

= τil + τli .

(24)

We see that the right-hand side is symmetric with respect to i, l. Then, be  so should the left-hand-side implying τil = τli . Then, Eq. (24) implies nτil + τ jk g jk gil = 0 implying τil = 0. Then, Eq. (23) reads 0 = jl aik + jk ail + il a jk + ik a jl − λki A jl − λk j Ail − λli A jk − λl j Aik . (25) We alternate (25) with respect to j, k to obtain 0 = jl aik + ik a jl − λki A jl − λl j Aik − kl ai j − i j akl + λ ji Akl + λlk Ai j . (26) Let us now rename i ↔ k in (26) and add the result with (25). We obtain jl aik + ik a jl − λki A jl − λl j Aik = 0. In other words, α aβ + β aα = λβ Aα + λα Aβ , where α and β stand for the symmetric indices jl and ik, respectively. But it is easy to check that a non-zero simple symmetric tensor X αβ = Pα Q β + Pβ Q α determines its factors Pα and Q β up to scale and order (it is sufficient to check, for example, by taking Pα and Q β to be basis vectors). Since ai j and Ai j are supposed to be linearly independent, it follows that λi j = const · ai j , as required.   2.3. Local results. Within this section, we assume that (M, g) is a connected Riemannian or pseudo-Riemannian manifold of dimension n ≥ 3. Recall that the degree of mobility of a metric g is the dimension of the space of the solutions of (9). Lemma 3. Suppose that the degree of mobility of g is ≥ 3. Then for every solution ai j of (9), where λi is the differential of the function λ given by (10), there exists an open dense subset N of M each of whose points admits an open neighborhood U , a constant B, and a function μ on U , such that the hessian of λ satisfies on U the equation λ,i j = μgi j + Bai j .

(27)

Proof. If a = const·g, then λ is constant and the lemma holds with N = M, μ ≡ B = 0. Otherwise there exists a solution A of (9) such that a, A, g are linearly independent. We denote by i the (0, 1)-tensor from Eq. (8) corresponding to A, i.e., i = ,i for := 21 A pq g pq . Then the integrability conditions (11) for the solutions a and A are given by (15) (with λi j = λ,i j and i j = ,i j ). Let N be the set of all x ∈ M which admit a neighborhood on which a, A, g are either pointwise linearly independent or pointwise linearly dependent. Being a union of open sets, N is open. N is also dense in M: every nonempty open set U ⊂ M either consists only of points where a, A, g are linearly dependent, then U ⊂ N ; or it contains a point where a, A, g are linearly independent and which is therefore contained in N . By definition every point in N has an open connected neighborhood U on which one of two possibilities holds:

412

V. Kiosak, V. S. Matveev

(a) a, A, g are pointwise linearly independent. Then, by Lemma 2, λ,i j = μgi j + Bai j , where μ and B are functions; they are unique and smooth because of linear independence. Our goal is to show that B is actually a constant, this will be done in Sect. 2.3.3. (b) a, A, g are pointwise linearly dependent. Then there exist a nonempty open con1 2

nected subset U of U and (smooth) functions c, c on U such that on U , we have 1

2

1

2

1 2

a+ c A+ c g ≡ 0 or A+ c a+ c g ≡ 0. (To see that c, c can be chosen to be smooth, distinguish three cases: the span of a, A, g has on U pointwise dimension 1; or A, g are linearly independent somewhere; or a, g are linearly independent somewhere.) 1 2

We will prove in Sect. 2.3.1 that c, c are actually constants. (Lemma 5 can be applied here because if a or A had the form const · g on U , then also on M, in contradiction to linear independence.) Thus a, A, g are linearly dependent on U and therefore on M. This contradiction rules out case (b). 2.3.1. Linear dependence of three solutions over functions implies their linear dependence over numbers. We will use the following statement (essentially due to Weyl [75]); its proof can be found for example in [74], see also [9, Lemma 1 in Sect. 2.4]. Lemma 4. Suppose ai j and Ai j are solutions of (9). Assume a = f · A, where f is a function. Then f is actually a constant. Our main goal is the following lemma, which settles the case (b) of the proof of Lemma 3. 1 2

Lemma 5. Suppose for certain functions c, c the solutions a, A (of (9) on a connected manifold (M n≥3 , g)) satisfy 1

2

ai j = c gi j + c Ai j .

(28) 1 2

We assume in addition that A is not const · g. Then the functions c, c are constants. Remark 8. Though we will use that the dimension of the manifold is at least three, the statement is true in dimension two as well provided the curvature of g is not constant, see [33]. 1

2

Proof of Lemma 5. We assume that c,k or c,k is not zero everywhere, and find a contradiction. Differentiating (28) and substituting (9) and its analog for the solution A, we obtain 1

2

2

2

λi g jk + λ j gik = c,k gi j + c i g jk + c j gik + c,k Ai j ,

(29)

which is evidently equivalent to 1

2

τi g jk + τ j gik = c,k gi j + c,k Ai j ,

(30)

2

where τi = λi − c i . We see that for every fixed k the left-hand side is a sym1

2

metric matrix of the form τi v j + τ j vi . If c,k is not proportional to c,k at some point

Proof of Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics

413

x ∈ M, this will imply that gi j also is of the form τi v j + τ j vi at x, which contradicts the nondegeneracy of g. Thus there exists a function f with 1

2

c,k = f · c,k .

(31) 2

At each point x there exists a nonzero vector ξ = (ξ k ) ∈ Tx M such that ξ k c,k = 0. Multiplying (30) with ξ k and summing with respect to k, we see that the right-hand side vanishes, and obtain the equation τi v j + τ j vi = 0, where vi := ξ k gik . Since vi = 0, we 2

2

obtain τi = 0 at x; hence Eq. (30) reads f · c,k gi j = − c,k Ai j everywhere on M. Since 2

the covector field c,k is pointwise nonzero on some nonempty connected open subset U of M, this equation implies f · gi j = −Ai j on U . By Lemma 4, f is constant on U . By Remark 4, it is constant globally, which contradicts the assumptions.   2.3.2. In dimension 3, only metrics of constant curvature can have the degree of mobility ≥ 3 Lemma 6. Assume that the conformal Weyl tensor Cihjk of the metric g on (a connected) M n≥3 vanishes. If the curvature of the metric is not constant, the degree of mobility of g is at most two. Since the conformal Weyl tensor Cihjk of any metric on a 3-dimensional manifold vanishes, a special case of Lemma 6 is Corollary 3. The degree of mobility of each metric g of nonconstant curvature on M 3 is at most two. Proof of Lemma 6. It is well-known that the curvature tensor of spaces with Cihjk = 0 has the form Rihjk = Pkh gi j − P jh gik + δkh Pi j − δ hj Pik , (32)   1 R Ri j − 2(n−1) gi j (and therefore Pkh = Ppk g ph ). We denote by P where Pi j := n−2 R the trace of Pkh ; easy calculations give us P = 2(n−1) . Substituting Eqs. (32) in the integrability conditions (11), we obtain p

p

p

p

a pi Pl g jk − a pi Pk g jl + ali P jk − aki P jl + a pj Pl gik − a pj Pk gil + al j Pik − ak j Pil = λl,i g jk + λl, j gik − λk,i g jl − λk, j gil . (33) Multiplying (33) with g jk and summing with respect to repeating indexes, and using the symmetry of Pi j due to (12), we obtain p

a pi Pl = λl,i −

P Pˆ 2λ ali + gli + Pil , n n n

(34)

p p where Pˆ = g qγ a pq Pγ − λ , p . Substituting (34) in (33), we obtain

0=

2λ 2λ 2λ 2λ Pil g jk − Pik g jl + P jl gik − P jk gil n n n n P P P P + ali P jk − aki P jl + al j Pik − ak j Pil − ail g jk + aik g jl − a jl gik + a jk gil . n n n n (35)

414

V. Kiosak, V. S. Matveev

Alternating Eq. (35) with respect to j, k, renaming i ←→ k, and adding the result to (35), we obtain 2λ 2λ P P Pil g jk − P jk gil + ali P jk − ak j Pil − ail g jk + a jk gil = 0, n n n n

(36)

which is evidently equivalent to 2λ 2λ Pil g jk − P jk gil + ali n n



   P P P jk − g jk − ak j Pil − gil = 0. n n

(37)

Hence (in view of P jk − Pn g jk = 0 because by assumption the curvature of g is not constant) there exists a nonempty open set U such that every solution ai j of (9) is on U a smooth linear combination of gi j and Pi j . Thus every three solutions g, a, aˆ of (9) are on U linearly dependent over functions. By Lemma 5, they are on U , and therefore everywhere, linearly dependent over numbers.   2.3.3. Case (a) of Lemma 3: Proof that B = const. We consider a neighborhood U ⊆ M n≥3 such that a, A, g are linearly independent at every point of the neighborhood; by Lemma 5, almost every point has such neighborhood. Remark 9. Within the whole paper we understand “almost everywhere” and “almost every” in the topological sense: a condition is fulfilled almost everywhere (or in almost every point) if and only if it the set of the points where it is fulfilled is everywhere dense. In the beginning of the proof of Lemma 3, we explained that at every point of the neighborhood Eq. (27) holds for certain smooth functions μ and B. Our goal is to show that B is actually a constant (on U ). Because of Corollary 3, we can assume n = dim(M) ≥ 4. Indeed, otherwise by Corollary 3 the curvature of the metrics is constant, and the metric is Einstein. Then, by [30, Cor. 1], Eq. (27) holds. Within the proof, we will use the following equations, the first one is (9), the second follows from Lemma 3:

ai j,k = λi g jk + λ j gik (38) λ,i j = μgi j + Bai j . Our goal will be to show that B is constant. We assume that it is not the case and show that for a certain covector field u i and functions α, β on the manifold we have ai j = αgi j + βu i u j . Later we will show that this gives a contradiction with the assumption that the degree of mobility is three. We consider the equation λi, j = μgi j + Bai j . Taking the covariant derivative ∇k , we obtain (9)

λi, jk = μ,k gi j + B,k ai j + Bai j,k = μ,k gi j + B,k ai j + Bλi g jk + Bλ j gik .

(39)

p

By definition of the Riemannian curvature, we have λi, jk − λi,k j = λ p Ri jk . Substituting (39) in this equation, we obtain p

λ p Ri jk = μ,k gi j + B,k ai j − μ, j gik − B, j aik + Bλ j gik − Bλk gi j .

(40)

Proof of Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics

415

Now, substituting the second equation of (38) in (11), we obtain  p p a pi R jkl + a pj Rikl = B ali g jk + al j gik − aki g jl − ak j gil .

(41)

p

We multiply this equation by λl and sum over l. Using that a pi R jkq λq is evidently equal p q to ai Rk j p λq , we obtain  p q p q (42) ai Rk j p λq + a j Rki p λq = B aiq λq g jk + a jq λq gik − aki λ j − ak j λi . q

q

Substituting the expressions for Rk j p λq and Rki p λq , we obtain 1

1

2

2

τ i a jk + τ j aik + τ i g jk + τ j gki − B, j aip a pk − B,i a jp a pk = 0, 1

(43)

2

p p p where τ i := ai B, p − μ,i + 2Bλi and τ i := ai μ, p − 2Bλ p ai . Now let us work with (43): we alternate the equation with respect to i, k to obtain: 1

2

1

2

τ i a jk + τ i g jk − B,i a jp a pk − τ k a ji − τ k g ji + B,k a jp a pi = 0.

(44)

We rename j ↔ k and add the result to (43): we obtain 1

2

τ i a jk + τ i g jk = B,i a jp a pk . 1

(45)

2

Remark 10. If B = const on U , then τ i a jk + τ i g jk = 0. Since by Lemma 4 a jk is not 1

proportional to g jk , we have τ i = 0, which implies that μ,i = 2Bλi . 1

2

The condition (45) implies that under the assumption B = const the covectors τ i , τ i 1

2

and B,i are collinear: Moreover, for certain functions c, c, 1

2

1

1

2

2

c B,i = τ i , c B,i = τ i , c a jk + c g jk = a jp a pk .

(46)

Taking the ∇k derivative of the last formula of (46), we obtain p

1

p

2

1

1

λ p a j gik + λi a jk + λ p ai g jk + λ j aik = c,k ai j + c,k gi j + c λi g jk + c λ j gik . Alternating the last formula with respect to i and k, we obtain: 3

3

4

4

τ i a jk − τ k ai j + τ i g jk − τ k gi j = 0, 3

1

1

4

(47)

2

where τ i = λi + c,i , τ i = λ p a p i − c λi + c,i . Let us explain that this equation implies 3

4

either ai j = αgi j + βu i u j (which was our goal), or τ = τ = 0. 3

4

We fix a point x ∈ U and assume that τ i = 0 at the point. Then, τ i = 0 as well. For every vector ξ ∈ Tx M we multiply (47) by ξ j and sum with respect to j. Denoting A(ξ )k := a jk ξ j and G(ξ )k := g jk ξ j , we obtain 3

3

4

4

τ i A(ξ )k − τ k A(ξ )i + τ i G(ξ )i − τ k G(ξ )i = 0.

(48)

416

V. Kiosak, V. S. Matveev 3

Then, the (at most two-dimensional) subspaces of Tx∗ M generated by {τ i , A(ξ )i } and 4

by {τ i , G(ξ )i } coincide. Since the metric g is nondegenerate, varying ξ we obtain all 4

possible elements of Tx∗ M as G(ξ )i , so the subspaces generated by {τ i , G(ξ )i } are all 4

possible at most two-dimensional subspaces containing τ i , and the subspace generated 4

by {τ i } is the intersection of all such subspaces. Similarly, the subspace generated by 3

3

3

4

{τ i } is the intersection of subspaces generated by {τ i , A(ξ )i }. Thus, τ i = −α τ i for a certain constant α, and Eq. (47) looks like 3

3

τ i (a jk − αg jk )− τ k (ai j − αg jk ) = 0.

(49)

3

We take η ∈ Tx M such that ηk τ k = 0, multiply (49) by ηk and sum over k. We obtain that 3 3

A(η) = αG(η) for all such η. Thus, for a certain const β we have ai j = αgi j + β τ i τ j as we claimed. 3 4 In the case where τ and τ vanish identically on U , using (46), (9) and the defini3

4

tion of τ and τ , we obtain λα aiα =

1

(n+2)c−2λ λi , n+4

j

i.e., that λα is an eigenvector of ai . 3

Differentiating this equation and substituting (38), (46), (9), and τ = 0, we obtain ⎞ ⎞ ⎛ ⎛ 1 1 c −2λ c −2λ (n + 2) (n + 2) 1 2 ⎝μ+ c B − B ⎠ ai j = ⎝ μ − λ p λ p − c B ⎠ gi j − 2λi λ j . n+4 n+4 Assume that the coefficient of ai j vanishes identically on U . Since gi j has rank ≥ 4 and λi λ j has rank ≤ 1, the coefficient of gi j vanishes identically on U , and thus the covector field λi vanishes identically on U . Differentiating λi = 0, and using λi j = μgi j + Bai j and Lemma 5, we see that either a = const · g on U and therefore everywhere, in contradiction to our linear independence assumption; or B ≡ 0 on U , in contradiction 3

4

to the choice of U . This shows that also in the case τ =τ ≡ 0 there exist a nonempty open subset U of U and functions α, β on U and a covector field u on U with ai j = αgi j + βu i u j . Let us now explain that if ai j is not proportional to g and ai j = α(x)gi j + β(x)u i u j for every point x of some neighborhood, then α is a smooth function, and β (resp. u i ) can be chosen to be smooth function (resp. smooth covector field), probably in a smaller j neighborhood. Indeed, under these assumptions α is the eigenvalue of ai of (algebraic and geometric) multiplicity precisely n − 1. Then, it is a smooth function. Then, βu i u j is a smooth (0, 2)-tensor field. Since ai j and gi j are not proportional, βu i u j is not zero and we can choose β = ±1. Then, we have precisely two choices for the covector u i (x) at every point x and in a small neighborhood we can choose u i (x) smoothly. Thus, under the assumptions of this section, for every solution ai j of (9), we have (for certain functions α1 , α2 and a covector field u i ) ai j = α1 gi j + α2 u i u j .

(50)

For the solution Ai j an analog of Eq. (50) holds so (in a possible smaller neighborhood) we also have (for certain functions β1 , β2 and a covector field vi ) Ai j = β1 gi j + β2 vi v j .

(51)

Proof of Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics

417

Without loss of generality, we can assume that ai j + Ai j (which is certainly a solution of (9)) is also not proportional to gi j , otherwise we replace Ai j by 21 Ai j . Then, ai j + Ai j = γ1 gi j + γ2 wi w j .

(52)

Subtracting (52) from the sum of (50) and (51), we obtain (γ1 − α1 − β1 )gi j = α2 u i u j + β2 vi v j − γ2 wi w j .

(53)

Since the tensor gi j is nondegenerate, its rank coincides with the dimension of M that is at least 4. The rank of the tensor α2 u i u j + β2 vi v j − γ2 wi w j is at most three. Thus the coefficient (γ1 − α1 − β1 ) must vanish, which implies that α2 u i u j + β2 vi v j = γ2 wi w j .

(54)

We see that the rank of α2 u i u j + β2 vi v j is at most one, which implies that u i is proportional to vi (the coefficient of the proportionality is a function). Thus (54) implies that wi is proportional to u i as well. Thus ai j , Ai j , and gi j are linearly dependent over functions, which implies by Lemma 5 that they are linearly dependent over numbers. This is a contradiction to the assumptions, which proves the remaining part of Lemma 3. 2.3.4. The constant B is universal. Let (M n≥3 , g) be a connected pseudo-Riemannian manifold. Assume the degree of mobility of g is ≥ 3, let (ai j , λi ) be a solution of Eqs. (9) such that ai j = const · gi j for every const ∈ R. Then, in a neighborhood of almost every point there exist a constant B and a function μ such that Eqs. (38) hold. Note that the constant B determines the function μ: indeed, multiplying (27) by g i j and summing with respect to i, j we obtain λi ,i = nμ − 2Bλ. Our goal is to prove the statement announced in the title of the section: we would like to show that the constant B is the same in all such neighborhoods (which in particular implies that Eqs. (38) hold at all points with one universal constant B and one universal function μ). We will need the following Corollary 4. Let ai j , λi satisfy Eqs. (38) in a neighborhood U ⊆ (M, g) with a certain constant B and a smooth function μ. Then the function λ given by (10) satisfies the equation  λ,i jk − B 2λ,k gi j + λ, j gik + λ,i g jk = 0. (55) Remark 11. This equation is a famous one; it naturally appeared in different parts of differential geometry. Obata and Tanno used this equation trying to understand the connection between the eigenvalues of the laplacian g and the geometry and topology of the manifold. They observed [64,72] that the eigenfunctions corresponding to the second eigenvalue of the Laplacian of the metrics of constant positive curvature −B on the sphere satisfy Eq. (55). Tanno [72] and Hiramatu [27] related the equations to projective vector fields. Tanno has shown that for every solution λ of this equation the vector field λ,i is a projective vector field (assuming B = 0), Hiramatu proved the reciprocal statement under certain additional assumptions. As it was shown by Gallot [19], see also [1,59,60], decomposability of the holonomy group of the cone over a manifold implies the existence of a nonconstant solution of Eq. (55) on the manifold.

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Proof of Corollary 4. Covariantly differentiating (27) and replacing the covariant derivative of ai j by (9) we obtain (55) from Remark 10 if a = const · g. If a = const · g, we have λ,i = 0, thus (55) holds as well.   Corollary 5. Let the degree of mobility of a metric g on a connected (n > 3)dimensional M be ≥ 3. Assume (ai j , λi ) is a solution of (9). Then, if λi = 0 at a point, then the set of the points such that λi = 0 is everywhere dense. Remark 12. The assumption that the degree of mobility of g is ≥ 3 is important: LeviCivita’s description of geodesically equivalent metrics [36] immediately gives counterexamples. Proof of Corollary 5. Combining Lemma 3, Remark 10, and Corollary 4, we obtain that in a neighborhood of almost every point λ given by (8) satisfies (55). By [72, Prop. 2.1], the vector field λi is a projective vector field (almost everywhere, and, therefore, everywhere) on (M, g). As it was shown for example in [23, Th. 21.1(ii)], if it is not zero at a point, then it is not zero at almost every point.   Corollary 6. Let ai j , λi satisfy Eqs. (38) in a neighborhood U with a certain constant B and a smooth function μ. Let λ be the function constructed by (10). Then for every geodesic γ (t) the following equation holds (at every t ∈ γ −1 (U )): d3 d λ(γ (t)) = 4Bg(γ˙ (t), γ˙ (t)) · λ(γ (t)), 3 dt dt

(56)

where γ˙ denotes the velocity vector of the geodesic γ , and g(γ˙ (t), γ˙ (t)) := gi j γ˙ i γ˙ j . Proof. Multiplying (55) by γ˙ i γ˙ j γ˙ k and summing with respect to i, j, k we obtain (56).   Lemma 7. Let (M n≥3 , g) be a connected manifold and (ai j , λi ) be a solution of (9). Assume almost every point has a neighborhood such that in this neighborhood there exists a constant B and a smooth function μ such that Eq. (27) is fulfilled. Then the constant B is the same in all such neighborhoods. Proof. It is sufficient to prove this statement locally, in a sufficiently small neighborhood of arbitrary point. We take a small neighborhood U , two points p0 , p1 ∈ U , and two neighborhoods U ( p0 ) ⊂ U, U ( p1 ) ⊂ U of these points. We assume that our neighborhoods are small enough and that we can connect every point of U ( p0 ) with every point of U ( p1 ) by a unique geodesic lying in U . We assume that Eq. (27) holds in U ( pi ) with the constant B := Bi ; our goal is to show that B0 = B1 . Suppose it is not the case. We consider all geodesics γ p, p0 lying in U connecting all points p ∈ U ( p1 ) with p0 , see Fig. 1. We will think that γ (0) = p0 and γ (1) ∈ U ( p1 ). For every such geodesic γ p, p0 (t) there exists a point q p, p0 := γ p, p0 (t p, p0 ) on this geodesic such that for all t ∈ [0, t p, p0 ) the following conditions are fulfilled: 1. Equations (38) are fulfilled with B = B0 in a small neighborhood of γ (t), and 2. for no neighborhood of γ p, p0 (t p, p0 ) Eqs. (38) are fulfilled with B = B0 . Then, at every such point γ p, p0 (t p, p0 ) we have that ai j = n2 λgi j . Indeed, the trace-free version of (27) is   1 2 (57) λ,i j − λ,kk = B ai j − λgi j , n n

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419

q p1 p0 Fig. 1. The geodesics γ p, p0 , their velocity vectors at p0 , and the point q p, p0 = γ p, p0 (t p, p0 ) on one of these geodesics

implying that B is the coefficient of proportionality of two smooth tensors. If ai j = n2 λgi j at γ p, p0 (t p, p0 ), we have ai j − n2 λgi j = 0, and B can be prolonged to a smooth function in a small neighborhood of γ p, p0 (t p, p0 ). Since it is locally-constant, it is (the same) constant at all points of the neighborhood of γ p, p0 (t p, p0 ) contradicting the conditions 1, 2. Moreover, at every such point γ p, p0 (t p, p0 ) we have λi = 0. Indeed, otherwise we multiply (55) by g i j and sum with respect to i, j. We obtain λi ,ik = 2(n + 1)Bλk . We again have that B is the coefficient of proportionality of two smooth tensors. Arguing as above we obtain that λi = 0 at every point γ p, p0 (t p, p0 ). d Since at every point γ p, p0 (t p, p0 ) we have λi = 0, we have that dt λ(γ p, p0 (t))|t=t p, p0 = 0. Then, the set of all such γ p, p0 (t p, p0 ) contains a smooth (connected) hypersurface (because the set of zeros of the derivatives of the solutions of Eq. (56) depends smoothly on the initial data and on g(γ˙ , γ˙ )). We denote this hypersurface by H . Since λi = 0 at every point of H , the function λ is constant (we denote it by λ˜ ∈ R) on H . Now let us return to the geodesics γ p, p0 connecting points p ∈ U ( p1 ) with p0 . We consider the integral I given by (14). Direct calculations show that at every point q where ai j = c · gi j the integral is given by I (ξ ) = cn−1 g(ξ, ξ )

(58)

(for every tangent vector ξ ∈ Tq M). As we explained above, every such geodesic passing through a point of H has a point such that ai j = c · gi j , where c = n2 λ˜ is  a constant. Since the integral is constant on the orbits, we have that I γ˙ p, p0 (0) =  cn−1 · g γ˙ p, p0 (0), γ˙ p, p0 (0) . Then, the measure of the subset {ξ ∈ T p0 M | I (ξ ) = cn−1 · g(ξ, ξ )} ⊆ T p0 M is not zero. Since this set is given by an algebraic equation, it must coincide with the whole T p0 M. Then, ai j = c · gi j at the point p0 . Since we can replace p0 by every point of its neighborhood U ( p0 ), we obtain that ai j = cn−1 · gi j at every point of U ( p0 ). By Remark 5, a = cn−1 · g on the whole manifold.   2.3.5. The metric g uniquely determines B. By Lemma 3, under the assumption that the degree of mobility is ≥ 3, for every solution a of (9) there exists a constant B such that Eq. (27) holds on a suitable open set. In this chapter we show that the constant B is the same for all (nontrivial) solutions ai j , i.e., the metric determines it uniquely.

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Lemma 8. Suppose two nonconstant functions f, F : M n → R on a connected manifold (M n , g) of dimension n > 1 satisfy  f ,i jk − b 2 f ,k gi j + f , j gik + f ,i g jk = 0, (59)  F,i jk − B 2F,k gi j + F, j gik + F,i g jk = 0, where b and B are constants. Assume that there exists a point where the derivative of f is nonzero and a point where the derivative of F is nonzero. Then, b = B. Proof. By definition of the curvature, for every function f , we have f ,i jk − f ,ik j = p f p Ri jk ; replacing f ,i jk by the right-hand side of the first equation of (59) we obtain.  p f , p Ri jk = b f ,k gi j − f , j gik . (60) The same is true for the second equation of (59):  p F, p Ri jk = B F,k gi j − F, j gik .

(61)

Multiplying (60) by F, k , summing with respect to repeating indexes and using (61) we obtain   B F, p f , p gi j − F, j f ,i = b F, p f , p gi j − F,i f , j . (62) Multiplying by g i j and summing with respect to repeating indexes, we obtain B(n − 1)F, p f , p = b(n − 1)F, p f , p . If F, p f , p = 0 we are done: B = b. Assume F, p f , p = 0. Then, (62) reads B F, j f ,i = bF,i f , j . Since by Corollary 5 there exists a point where F, j and f ,i are both nonzero, we obtain again B = b. Then, f ,i is proportional to F, j . Hence, B = b.   2.3.6. An ODE along geodesics. Lemma 9. Let g be a metric on a connected M n≥3 of degree of mobility ≥ 3. For a metric g¯ geodesically equivalent to g, let us consider ai j , λi , and φ given by (7, 8, 5). Then, there exist constants B, B¯ such that the following formula holds: φi, j − φi φ j = −Bgi j + B¯ g¯ i j .

(63)

Proof. We covariantly differentiate (8) (the index of differentiation is “j”); then we substitute the expression (6) for g¯i j,k to obtain λi, j = −2e2φ φ j φ p g¯ pq gqi − e2φ φ p, j g¯ pq gqi + e2φ φ p g¯ ps g¯ sl, j g¯ lq gqi = −e2φ φ p, j g¯ pq gqi + e2φ φ p φs g¯ ps gi j + e2φ φ j φl g¯ lq gqi ,

(64)

where g¯ pq is the tensor dual to g¯ pq , i.e., g¯ pi g¯ pj = δ ij . We now substitute λi, j from (27), use that ai j is given by (7), and divide by e2φ for cosmetic reasons to obtain e−2φ μgi j + B g¯ pq g pj gqi = −φ p, j g¯ pq gqi + φ p φs g¯ ps g¯i j + φ j φl g¯ lq gqi .

(65)

Multiplying with g iξ g¯ ξ k , we obtain φk, j − φk φ j = (φ p φq g¯ pq − e−2φ μ) g¯ k j − Bgk j .  

b¯

(66)

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The same holds with the roles of g and g¯ exchanged (the function (5) constructed by the interchanged pair g, ¯ g is evidently equal to −φ). We obtain − φk; j − φk φ j = (φ p φq g pq − e2φ μ) ¯ gk j − B¯ g¯ k j ,  

(67)

b

where φi; j denotes the covariant derivative of φi with respect to the Levi-Civita connection of the metric g. ¯ Since the Levi-Civita connections of g and of g¯ are related by the formula (2), we have −φk; j − φk φ j = −φk, j + 2φk φ j −φk φ j = −(φk, j − φk φ j ).  

−φk; j

We see that the left hand side of (66) is equal to minus the left hand side of (67). Thus, b · gi j − B¯ · g¯i j = B · gi j − b¯ · g¯i j holds on U . Since the metrics g and g¯ are not ¯ and the formula (66) coincides with (63).  proportional on U by assumption, b¯ = B,  Corollary 7. Let g, g¯ be geodesically equivalent metrics on a connected M n≥3 such that the degree of mobility of g is ≥ 3. ... We consider a (parametrized) geodesic γ (t) of the ˙ φ¨ and φ the first, second and third derivatives of the function metric g, and denote by φ, φ given by (5) along the geodesic. Then, there exists a constant B such that for every geodesic γ the following ordinary differential equation holds: ... ˙ 3, φ = 4Bg(γ˙ , γ˙ )φ˙ + 6φ˙ φ¨ − 4(φ) (68) where g(γ˙ , γ˙ ) := gi j γ˙ i γ˙ j . Since lightlike geodesics have g(γ˙ , γ˙ ) = 0 at every point, a partial case of Corollary 7 is Corollary 8. Let g, g¯ be geodesically equivalent metrics on a connected M n≥3 such that the degree of mobility of g is ≥ 3. Consider a (parametrized) lightlike geodesic ... ˙ φ¨ and φ the first, second and third derivatives of γ (t) of the metric g, and denote by φ, the function φ given by (5) along the geodesic. Then, along the geodesic, the following ordinary differential equation holds: ... ˙ 3. φ = 6φ˙ φ¨ − 4(φ) (69) Proof of Corollary 7. If φ ≡ 0 in a neighborhood U , the equation is automatically fulfilled. Then, it is sufficient to prove Corollary 7 assuming φi is not constant. The formula (63) is evidently equivalent to φi, j = B¯ g¯i j − Bgi j + φi φ j .

(70)

Taking the covariant derivative of (70), we obtain φi, jk = B¯ g¯i j,k + φi,k φ j + φ j,k φi .

(71)

Substituting the expression for g¯i j,k from (6), and substituting B¯ g¯i j given by (63), we obtain ¯ g¯ i j φk + g¯ik φ j + g¯ jk φi ) + φi,k φ j + φ j,k φi φi, jk = B(2 = B(2gi j φk + gik φ j + g jk φi ) + 2(φk φi, j + φi φ j,k + φ j φk,i ) − 4φi φ j φk .

(72)

Contracting with γ˙ i γ˙ j γ˙ k and using that φi is the differential of the function (5) we obtain the desired ODE (68).  

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2.4. Proof of Theorem 1 for pseudo-Riemannian metrics. Let g be a metric on a connected M n≥3 . Assume that for no constant c = 0 the metric c · g is Riemannian, which in particular implies the existence of lightlike geodesics. Let g¯ be geodesically equivalent to g. Assume both metrics are complete. Our goal is to show that φ given by (5) is constant, because in view of (2) this implies that the metrics are affine equivalent. Consider a parameterized lightlike geodesic γ (t) of g. Since the metrics are geodesically equivalent, for a certain function τ : R → R the curve γ (τ ) is a geodesic of g. ¯ Since the metrics are complete, the reparameterization τ (t) is a diffeomorphism d τ : R → R. Without loss of generality we can think that τ˙ := dt τ is positive, otherwise we replace t by −t. Then, Eq. (3) along the geodesic reads φ(t) =

1 log(τ˙ (t)) + const0 . 2

(73)

Now let us consider Eq. (69). Substituting 1 φ(t) = − log( p(t)) + const0 2 in it (since τ˙ > 0, the substitution is global), we obtain ... p = 0.

(74)

(75)

The solution of (75) is p(t) = C2 t 2 + C1 t + C0 . Combining (74) with (73), we see that τ˙ = C t 2 +C1 t+C . Then 2

1

0

 τ (t) =

t t0

dξ + const. C2 ξ 2 + C1 ξ + C0

(76)

We see that if the polynomial C2 t 2 + C1 t + C0 has real roots (which is always the case if C2 = 0, C1 = 0), then the integral explodes in finite time. If the polynomial has no real roots, but C2 = 0, the function τ is bounded. Thus, the only possibility for τ to be a diffeomorphism is C2 = C1 = 0 implying τ (t) = C10 t + const1 , implying τ˙ = C10 , implying φ is constant along the geodesic. Since every two points of a connected pseudo-Riemannian manifold such that for no constant c the metric c · g is Riemannian can be connected by a sequence of lightlike geodesics, φ is a constant, so that φi ≡ 0, and the metrics are affine equivalent by (2).   2.5. Proof of Theorem 1 for Riemannian metrics. As we already mentioned in the Introduction and at the beginning of Sect. 2, Theorem 1 was proved for Riemannian metrics in [45,57]. We present an alternative proof, which is much shorter (modulo the results of the previous sections and a nontrivial result of Tanno [72]). We assume that g is a complete Riemannian metric on a connected manifold such that its degree of mobility is ≥ 3. Then, by Corollary 4, the function λ is a solution of (55). If the metrics are not affine equivalent, λ is not identically constant. Let us first assume that the constant B in Eq. (55) is negative. Under this assumption, Eq. (55) was studied by Obata [64], Tanno [72], and Gallot [19]. Tanno [72] and Gallot [19] proved that a complete Riemannian g such that there exists a nonconstant function

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λ satisfying (55) must have a constant positive sectional curvature. Applying this result in our situation, we obtain the claim. Now, let us suppose B ≥ 0. Then, one can slightly modify the proof from Sect. 2.4 to obtain the claim. More precisely, substituting (74) in (68), we obtain the following analog of Eq. (75): ... p = 4Bg(γ˙ , γ˙ ) p. ˙ (77) If B = 0, the equation coincides with (75). Arguing as in Sect. 2.4, we obtain that φ is constant along the geodesic. If B > 0, the general solution of Eq. (77) is C + C + e2

√

Bg(γ˙ ,γ˙ )·t

+ C− e−2

Then, the function τ satisfies the ODE τ˙ =  τ (t) =

t

t0

C+C+ e2

√

√

C + C+

+ C− e−2

.

1

Bg(γ˙ ,γ˙ )·t +C

dξ

√ e2 Bg(γ˙ ,γ˙ )·ξ

Bg(γ˙ ,γ˙ )·t

√

−e

Bg(γ˙ ,γ˙ )·ξ

(78) √ −2 Bg(γ˙ ,γ˙ )·t

+ const.

implying (79)

If one of the constants C+ , C− is not zero, the integral (79) is bounded from one side, or explodes in finite time. In both cases, τ is not a diffeomorphism of R on itself, i.e., one of the metrics is not complete. The only possibility for τ to be a diffeomorphism of R on itself is C+ = C− = 0. Finally, φ is a constant along the geodesic γ . Since every two points of a connected complete Riemannian manifold can be connected by a geodesic, φ is a constant, so that φi ≡ 0, and the metrics are affine equivalent by (2).   Remark 13. A similar idea (contracting the equation with lightlike geodesic and investigating the obtained ODE along the geodesic) was recently used in [31,59]

2.6. Proof of Theorem 2. Let g be a complete pseudo-Riemannian metric on a connected closed manifold M n such that for no const = 0 the metric const · g is Riemannian (if g is Riemannian, Theorem 2 follows from Theorem 1). We assume that the degree of mobility of g is ≥ 3. Our goal is to show that every metric g¯ geodesically equivalent to g is actually affine equivalent to g. We consider the function λ constructed by (10) for the solution ai j of (9) given by (7). We consider a lightlike geodesic γ (t) of the metric g, and the function λ(γ (t)). By d3 Corollary 6, the function λ(γ (t)) satisfies the ODE dt 3 λ(γ (t)) = 0. Hence λ(γ (t)) = C2 t 2 + C1 t + C0 . If C2 = 0, or C1 = 0, then the function λ is not bounded; that contradicts the compactness of the manifold. Thus λ(γ (t)) is constant along every lightlike geodesic. Since every two points can be connected by a sequence of lightlike geodesics, λ is constant. Thus λi = 0, implying in view of (8) that φi = 0, implying in view of (6) that the metrics are affine equivalent.   Acknowledgements. We thank Deutsche Forschungsgemeinschaft (Priority Program 1154 — Global Differential Geometry and research training group 1523 — Quantum and Gravitational Fields) and FSU Jena for partial financial support, and Alexei Bolsinov and Mike Eastwood for useful discussions. We also thank Abdelghani Zeghib for finding a misprint in the main theorem in the preliminary version of the paper, and Graham Hall for his grammatical and stylistic suggestions.

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31. Kiosak, V., Matveev, V.S.: There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics. C. R. Acad. Sci. Paris, Ser. I 347, 1067–1069 (2009) 32. Koenigs, G.: Sur les géodesiques a intégrales quadratiques. Note II from “Lecons sur la théorie générale des surfaces,” Vol. 4, New York: Chelsea Publishing, 1896 33. Kruglikov, B.S., Matveev, V.S.: Strictly non-proportional geodesically equivalent metrics have h top (g) = 0. Erg. Th. Dyn. Syst. 26(1), 247–266 (2006) 34. Kruglikov, B.S., Matveev, V.S.: Vanishing of the entropy pseudonorm for certain integrable systems. Electron. Res. Announc. Amer. Math. Soc. 12, 19–28 (2006) 35. Lagrange, J.-L.: Sur la construction des cartes géographiques. Novéaux Mémoires de l’Académie des Sciences et Bell-Lettres de Berlin, 1779 36. Levi-Civita, T.: Sulle trasformazioni delle equazioni dinamiche. Ann. di Mat., Serie 2a 24, 255– 300 (1896) 37. Liouville, R.: Sur les invariants de certaines équations différentielles et sur leurs applications. Journal de l’École Polytechnique 59, 7–76 (1889) 38. Manno, G., Matveev, V.S.: 2-dim metrics admitting two projective vector fields near the points where the vector fields are linearly dependent. In preparation 39. Matveev, V.S., Topalov, P.J.: Trajectory equivalence and corresponding integrals. Reg. and Chaotic Dyn. 3(2), 30–45 (1998) 40. Matveev, V.S., Topalov, P.J.: Geodesic equivalence of metrics on surfaces, and their integrability. Dokl. Math. 60(1), 112–114 (1999) 41. Matveev, V.S., Topalov, P.J.: Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. ERA-AMS 6, 98–104 (2000) 42. Matveev, V.S., Topalov, P.J.: Quantum integrability for the Beltrami-Laplace operator as geodesic equivalence. Math. Z. 238, 833–866 (2001) 43. Matveev, V.S., Topalov, P.J.: Integrability in theory of geodesically equivalent metrics. J. Phys. A. 34, 2415–2433 (2001) 44. Matveev, V.S.: Geschlossene hyperbolische 3-Mannigfaltigkeiten sind geodätisch starr. Manuscripta Math. 105(3), 343–352 (2001) 45. Matveev, V.S.: Low-dimensional manifolds admitting metrics with the same geodesics. In: Contemporary Mathematics, Providence, RI: Amer. Math. Soc., 308 2002, pp. 229–243 46. Matveev, V.S.: Three-dimensional manifolds having metrics with the same geodesics. Topology 42(6), 1371–1395 (2003) 47. Matveev, V.S.: Hyperbolic manifolds are geodesically rigid. Invent. Math. 151, 579–609 (2003) 48. Matveev, V.S.: Die Vermutung von Obata für Dimension 2. Arch. Math. 82, 273–281 (2004) 49. Matveev, V.S.: Projectively equivalent metrics on the torus. Diff. Geom. Appl. 20, 251–265 (2004) 50. Matveev, V.S.: Solodovnikov’s theorem in dimension two. Dokl. Math. 69(3), 338–341 (2004) 51. Matveev, V.S.: Lichnerowicz-Obata conjecture in dimension two. Commun. Math. Helv. 81(3), 541–570 (2005) 52. Matveev, V.S.: Closed manifolds admitting metrics with the same geodesics. Proceedings of SPT2004 (Cala Gonone). River Edge, NJ: World Scientific, 2005, pp. 198–209 53. Matveev, V.S.: Beltrami problem, Lichnerowicz-Obata conjecture and applications of integrable systems in differential geometry. Tr. Semin. Vektorn. Tenzorn. Anal 26, 214–238 (2005) 54. Matveev, V.S.: The eigenvalues of Sinjukov’s operator are globally ordered. Mathematical Notes 77(3–4), 380–390 (2005) 55. Matveev, V.S.: Geometric explanation of Beltrami theorem. Int. J. Geom. Methods Mod. Phys. 3(3), 623–629 (2006) 56. Matveev, V.S.: On degree of mobility of complete metrics. Adv. Stud. Pure Math. 43, 221–250 (2006) 57. Matveev, V.S.: Proof of projective Lichnerowicz-Obata conjecture. J. Diff. Geom. 75, 459–502 (2007) 58. Matveev, V.S.: Two-dimensional metrics admitting precisely one projective vector field. Math. Ann., accepted, http://arxiv.org/abs/0802.2344v2 [math.DG], 2010 59. Matveev, V.S.: Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist. J. Diff. Geom. and Its Appl. 28(2), 236–240 (2010) 60. Matveev, V.S., Mounoud, P.: Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications. Ann. Glob. Anal. Geom. doi:10.1007/s10455-010-9211-7 (2010) 61. Matveev, V.S.: Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudoRiemannian metrics, http://arxiv.org/abs/1002.3934 [math.DG], 2010 62. Mikes, J.: Geodesic mappings of affine-connected and Riemannian spaces. Geometry, 2. J. Math. Sci. 78(3), 311–333 (1996) 63. Nagano, T., Ochiai, T.: On compact Riemannian manifolds admitting essential projective transformations. J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 33, 233–246 (1986)

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64. Obata, M.: Riemannian manifolds admitting a solution of a certain system of differential equations. Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965), Tokyo: Nippon Hyoronsha, 1966, pp. 101–114 65. Painlevé, P.: Sur les intégrale quadratiques des équations de la Dynamique. Compt.Rend. 124, 221–224 (1897) 66. Petrov, A.Z.: Einstein spaces. London: Pergamon Press. XIII, 1969 67. Petrov, A.Z.: New methods in the general theory of relativity. (in Russian). Moscow: Izdat. “Nauka”, 1966 68. Shen, Z.: On projectively related Einstein metrics in Riemann-Finsler geometry. Math. Ann. 320(4), 625–647 (2001) 69. Sinjukov, N.S.: Geodesic mappings of Riemannian spaces (in Russian). Moscow: “Nauka”, 1979 70. Solodovnikov, A.S.: Projective transformations of Riemannian spaces. Uspehi Mat. Nauk (N.S.) 11, no. 4(70), 45–116 (1956) 71. Solodovnikov, A.S.: Geometric description of all possible representations of a Riemannian metric in Levi-Civita form. Trudy Sem. Vektor. Tenzor. Anal. 12, 131–173 (1963) 72. Tanno, S.: Some differential equations on Riemannian manifolds. J. Math. Soc. Japan 30(3), 509–531 (1978) 73. Topalov, P.: Geodesic hierarchies and involutivity. J. Math. Phys. 42(8), 3898–3914 (2001) 74. Topalov, P.J., Matveev, V.S.: Geodesic equivalence via integrability. Geometriae Dedicata 96, 91–115 (2003) 75. Weyl, H.: Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung. Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1921; “Selecta Hermann Weyl”, Basel Stuttgart: Birkhäuser Verlag, 1956 76. Weyl, H.: Geometrie und Physik. Die Naturwissenschaftler 19, 49–58 (1931); “Hermann Weyl Gesammelte Abhandlungen”, Band 3, Berlin-Heidelberg: Springer-Verlag, 1968 77. Yamauchi, K.: On infinitesimal projective transformations. Hokkaido Math. J. 3, 262–270 (1974) Communicated by P.T. Chru´sciel

Commun. Math. Phys. 297, 427–445 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1039-2

Communications in

Mathematical Physics

Relaxation Times for Hamiltonian Systems Alberto Mario Maiocchi, Andrea Carati Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy. E-mail: [email protected]; [email protected] Received: 10 July 2009 / Accepted: 1 December 2009 Published online: 6 April 2010 – © Springer-Verlag 2010

Abstract: Usually, the relaxation times of a gas are estimated in the frame of the Boltzmann equation. In this paper, instead, we deal with the relaxation problem in the frame of the dynamical theory of Hamiltonian systems, in which the definition itself of a relaxation time is an open question. We introduce a lower bound for the relaxation time, and give a general theorem for estimating it. Then we give an application to a concrete model of an interacting gas, in which the lower bound turns out to be of the order of magnitude of the relaxation times observed in dilute gases.

1. Introduction The definition and the estimate of relaxation times are problems of central interest when one attempts describing macroscopic systems through microscopic Hamiltonian models. In the case of gases, these problem are tackled, and solved, in the frame of the Boltzmann equation (see [1]). In such a frame the existence of a relaxation time is somehow obvious, due to the irreversible character of the equation, and the estimate is obtained in terms of the eigenvalues of the linearized equation, about the equilibrium solution. On the other end, Boltzmann equation refers to a reduced description, while we want to tackle the problem considering the complete system. This would require to estimate the time needed for an initial measure in phase space to relax to an asymptotic one. This approach was followed, for example, in [2], for a reversible dissipative model, which should mimic a coupling of the system of interest with two (or more) heat reservoirs. In the present work, instead, we tackle the problem from the point of view of the dynamical theory of Hamiltonian systems, for systems which are isolated. In this perspective, a partial answer to the problem is given by Kubo’s linear response theory [3]. Indeed, such a theory enables one, at least in principle, to compute in microscopic terms the macroscopic transport coefficients, and then, via macroscopic equations, the

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relaxation time. From our point of view, however, this answer is not completely satisfactory, because it appeals to macroscopic irreversible equations, which should preliminarily be deduced from the microscopic ones. A related but different approach is followed here, whose main scheme can be sketched as follows. From linear response theory we take the starting point, namely, the idea of following the time evolution of the probability distribution function in phase space (and not in the reduced μ–space, as in the Boltzmann equation), when a perturbation −h A( p, q) to the original Hamiltonian H0 ( p, q) is introduced. Still following Kubo, we then choose to concentrate our attention on a particular observable, namely the one conjugated to the perturbing field, in the familiar sense in which pressure is conjugated to volume and magnetization to the magnetic field. As an example, later on in this paper we will deal with the simple case in which the perturbing field is gravity, and the conjugate observable is the height of the center of mass. Then, our attention is addressed at defining and estimating the relaxation time. In the spirit of the Kubo approach it is natural to say that equilibrium is attained when the time derivative of the distribution function is negligible. The aim of the present paper is indeed to give a lower bound tin f to the relaxation time, looking at the evolution not of the distribution function itself, but of the time–derivative of the variable conjugated to the perturbation A in the Hamiltonian, which is strictly related to the distribution function (see formula (11) below). It is easily seen that the time–derivative of the conjugate def

variable is the function B = [H0 , A], namely, the time–derivative of A with respect to the flow generated by the full Hamiltonian H (here, [·, ·] denotes Poisson bracket), so that this is the quantity on which we will concentrate in this paper. Having chosen the relevant function, namely B, we make use of the easily established properties (see later) that its expectation vanishes at equilibrium, and that its time–derivative is positive at the initial time. Thus a lower bound tin f to the relaxation time is provided by the time before which the time–derivative of the expectation of B is proven to be positive. The problem is then that one should make use of suitable a priori estimates on the dynamics, since an explicit integration of the equations of motion is lacking. This can actually be implemented following the main idea introduced in paper [4], which was concerned with Hamiltonian perturbation theory in the thermodynamic limit. In such a paper, a procedure is given which, for any L2 function f of phase space with respect to the Gibbs measure, allows one to provide an upper bound to Ut f − f 2 , by knowing an upper bound to [ f, H ]2 . Here, H is the Hamiltonian of the system, and Ut the corresponing unitary evolution group. The estimate of the lower bound tin f is provided by formula (8) of Theorem 1, which is stated and proved in Sect. 2. Such a proof is given for an ample class of Hamiltonian systems, which are the ones considered in most rigorous works in Statistical Mechanics (see [5]). In Sect. 3 the general theorem is applied to the case of a gas of interacting point– particles enclosed in a cubic box, to which the gravity force is added as a perturbation. To this aim, we give an interesting estimate of the s–point correlation function for a gas interacting through a stable and tempered two–body potential, which is here obtained by extending some old results of Bogolyubov et al. [6] and of O. Penrose [7]. The lower bound to the relaxation time thus found turns out to be comparable with the typical relaxation times observed in dilute gases. Some further comments are given in Sect. 4.

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2. General Theorem about Relaxation Times We consider an isolated Hamiltonian system, with phase space M, and with an invariant measure with respect to the unperturbed Hamiltonian H0 . One could think that, in principle, one has to take the microcanonical measure, but, in view of the ensemble equivalence for large N (see, for instance, [8]), we will take, instead, the Gibbs measure at inverse temperature β, i.e., the measure with density ρ0 given by ρ0 =

1 exp (−β H0 ), Z0

Z 0 being the partition function. Suppose at time 0 a perturbation −h A( p, q) is introduced, where A is a given function on phase space and the parameter h > 0 controls the size of the perturbation. So, at positive times the Hamiltonian is H1 = H0 − h A. The corresponding Gibbs density (at the same β) will be denoted by ρ1 . Our aim is to find a sensible lower bound for the relaxation time to the final equilibrium with respect to the full Hamiltonian H1 . To this end, along the scheme sketched in the Introduction, following Kubo we consider the observable B defined by B = [A, H0 ], i.e., the time derivative of the perturbation A with respect to the flow generated by the full Hamiltonian H1 . We then consider the probability density ρ, solution of Liouville’s equation relative to the total Hamiltonian H1 with initial condition ρ(0) = ρ0 , and look at the evolution of the expectation of B, i.e., we look at the quantity  B(t) = B ρ(t) d p dq. The quantity of interest actually will be its increment def

B(t) = B(t) − B(0). Writing ρ in the form def

ρ(t) = ρ0 + ρ(t), one has

(1)

 B(t) =

B ρ(t) d p dq.

(2)

We will show that under the familiar conditions which entail reversibility (namely, that both H0 and A are even in the momenta), the quantity B vanishes not only (as it is obvious) at time zero, but also at equilibrium with respect to the full Hamiltonian H1 . This is due to the fact that the expectations of B with respect to the Gibbs densities ρ0 and ρ1 corresponding to the Hamiltonians H0 and H1 , both vanish by symmetry, because ρ0 and ρ1 are even in the momenta, whereas B is odd. On the other hand, it turns out that B is initially an increasing function of time, since its time–derivative is positive at time 0, as it will be shown later. Thus, the time–derivative of B has to become negative at some time if equilibrium with respect to the full Hamiltonian has to be attained, and

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consequently a lower bound to the relaxation time is provided by the time tin f up to which the time derivative of B is guaranteed to be positive. We thus define the lower bound tin f by tin f = sup t ∗ ,

(3)

d B(t) > 0 for all 0 < t < t ∗ , dt

(4)

def

where t ∗ is such that

or tin f = +∞ if d B(t) > 0 ∀t > 0. dt Notice that our definition makes sense also for h = 0, in which case one has tin f = 0, as can be seen by formula (13) below. The problem is then to estimate the rate of growth of B. Now, on the one hand, following Kubo we know that B(t) is strictly related to the time–autocorrelation of B (see (13) below). On the other hand, we can make use of the main result obtained in paper [4], in which it was shown how to estimate the time–autocorrelation of B in terms of the Hamiltonian. Indeed, from the main result of that paper one easily obtains the following property: an a priori estimate of the type [B, H0 ]0 ≤ η B0 , (with the norm defined below) implies that the time–evolution of B is slow if η is small, or, more precisely, that the lower bound tin f to the relaxation time defined by (3), (4) is inversely proportional to η. Here,  · 0 is the norm on L20 (M), the Hilbert space of square integrable complex functions on M, with respect to ρ0 . We will also have to consider the Hilbert space L21 (M) of the square integrable complex functions with respect to ρ1 . The corresponding L2 –norm will be denoted by  · 1 . Under the rather natural condition (5) given below, which ensures the smallness of the “change” of the Gibbs measure induced by the perturbation −h A, for a large class of observables it can be proven that the two norms just introduced are asymptotically equivalent as h → 0. Indeed one has the following lemma, whose proof is deferred to Appendix A. Lemma 1. Assume there exist δ > 0 such that   d p dq eδ A ρ0 < +∞ and M

M

d p dq e−δ A ρ0 < +∞.

Then, for all real functions f on M satisfying at least one of the conditions      2  2  f  < +∞,  f  < +∞, 0

1

one has  f 21 −  f 20 = o(1), as h → 0.

(5)

(6)

Relaxation Times for Hamiltonian Systems

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We are now able to give an estimate for tin f in terms of η, which is provided by the following Theorem 1. It will be seen that some technical hypotheses, namely those given in (7) below, are required just in order that at least one of the conditions (6) of Lemma 1 is satisfied. Theorem 1. Let the unperturbed Hamiltonian H0 ( p, q) be even in the momenta and bounded from below, and consider a perturbation −h A( p, q), with h > 0 and A even in the momenta. Suppose A and H0 are such that hypothesis (5) of Lemma 1 is satified. With B = [A, H0 ], suppose furthermore that the following technical conditions are satisfied:        4     (7) B  < +∞; [B, H0 ]2  < +∞; [B, A]2  < +∞. 0

0

0

Then a lower bound to the relaxation time defined by (3), (4) is given by √ 2 + o(1), as h → 0, tin f ≥ η

(8)

where η is such that [B, H0 ]0 < η B0 .

(9)

Remark. It may appear that some conditions are too restrictive if the theorem has to be used in the thermodynamic limit, but it turns out that such a difficulty can be overcome. For example, H0 was required to have a finite lower bound, call it D; however, the result is found not to depend on the value of D. So, D can grow with the number N of degrees of freedom, without affecting the validity of the theorem, provided D is finite for any finite N . A similar argument also applies to conditions (5) and (7), so the theorem holds for any system, however large it may be. Then, in order to pass to the thermodynamic limit, it suffices to have that η tends uniformly to a finite limit as N increases. Proof of Theorem 1. First of all, we notice that the time evolution of the perturbation ρ satisfies, by Liouville’s equation, the differential equation ∂ρ = [H0 − h A, ρ] − h [A, ρ0 ] , ∂t

(10)

with ρ(0) = 0 as initial condition. Such an equation admits a unique solution in the Hilbert space L21 (M). Indeed, Eq. (10) is a linear inhomogeneous first–order differential equation in L21 (M) of the form ˆ + f, x˙ = Ox def where the operator Oˆ = [H1 , ·] generates a semigroup of unitary evolution transformations (see for example [9]). Thus, since the second term h[A, ρ0 ] at the r.h.s. belongs to L21 (M), as will be shown below, the solution is known to exist and be unique (see Theorem 3.3, p. 104, of [10]). Such a solution is given by a simple adaptation of the variation of constants formula, namely by  t ρ(x, t) = βh ds B(s x)ρ0 (s x), (11) 0

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where x = ( p, q) denotes a point of phase space M, and t the flow generated by H1 . Notice that, as the initial datum vanishes, one obviously has Oˆ ρ(0) ∈ L21 (M). We show now that [A, ρ0 ] ∈ L21 (M), too. To this end, we first notice that [A, ρ0 ]1 = Bρ0 1 ≤

eβ D B1 , Z0

(12)

def

where D = inf p,q H0 . On the other hand, iterating the Schwarz inequality gives    1    1   2   4 B0 ≤  B 2  ≤  B4 < +∞, 0

0

in which the first hypothesis of (7) was used.1 By virtue of such an hypothesis, we can also apply Lemma 1 to B and observe that, for h small enough, B1 is finite. Thus, by (12) it is proved that [A, ρ0 ] belongs to L21 (M). We now look at the expectation B(t) and at its increment B(t). By using (11) for ρ in (2), one finds for B(t) the expression   t B(t) = βh dx ds B(s x) B(x)ρ0 (s x). (13) M

0

Using the shorthand f (xt ) = f (−t x), one has then:  d B(t) = βh dx B(x−t ) B(x) ρ0 (x−t ), dt M or equivalently (due to preservation of Lebesgue measure),  d B(t) = βh dx B(xt ) B(x) ρ0 (x). dt M

(14)

At this point we remark that the integral in (14) could be evaluated in a quite simple way, if there appeared ρ1 in place of ρ0 . Indeed, due to the unitarity of the flow, for any f in L21 (M) one would have  1 dx f (xt ) f (x) ρ1 (x) =  f 21 −  f (xt ) − f (x)21 , (15) 2 M and thus, on account of hypothesis (9) of the theorem, the thesis would follow by using Theorem 1 of [4] (see below). The rest of the proof is devoted to show that the error made by taking ρ0 in place of ρ1 is negligible in the limit h → 0. To this end, we suitably rewrite (14) in the form    d B(t) = βh dx B(xt ) B(x) ρ1 (x) − dx B(xt )B(x) (ρ1 (x) − ρ0 (x)) dt M  M   ≥ βh dx B(xt ) B(x) ρ1 (x) −  dx B(xt )B(x) (ρ1 (x) − ρ0 (x)) . M

M

(16) 1 According to the same reasoning, the square of the norm of a function will be bounded from above by the norm of the squared function.

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First, we show that the second term at the r.h.s. vanishes as h → 0. Indeed, by Schwarz’s inequality we have    1   2   2 2  dx B(xt )B(x) (ρ1 (x) − ρ0 (x)) ≤ dx B (xt )B (x) ρ1 (x) γ˜ (h), (17)   M

M

where we have defined def

γ˜ (h) =



 M

dx

2 ρ0 (x) − 1 ρ1 (x). ρ1 (x)

This function coincides with the one defined by (50) in Appendix A. As there shown, one has γ˜ (h) → 0 as h → 0. We then make use of (15), by replacing B 2 for f and neglecting the negative term, in order to find an upper bound to the r.h.s. of (17): one has, in fact, 1      2   2 2 dx B (xt )B (x) ρ1 (x) ≤ B 2  . M

1

In order to show that B 2 1 is finite, we use Lemma 1, whose hypotheses are satisfied owing to the first inequality of (7). Thus, as γ˜ (h) → 0 for h → 0, one gets      (18) dx B(xt )B(x) (ρ1 (x) − ρ0 (x)) = o(1) as h → 0.  M

We then come to the first term at the r.h.s. of (16), which, using (15) again, can be estimated as  1 dx B(xt ) B(x) ρ1 (x) = B21 − B(xt ) − B(x)21 . (19) 2 M We now make use of Theorem 1 of [4], which ensures that, if [B, H1 ]1 < η˜ B1 is satisfied, then one has B(xt ) − B(x)1 < ηt ˜ B1 .

(20)

Now, to give an estimate for η, ˜ we notice that the following inequalities hold as h → 0: [B, H1 ]1 ≤ [B, H0 ]1 + h [B, A]1  1 2 ≤ [B, H0 ]20 + o(1)  1 2 + h [B, A]20 + o(1) . Here, in the first line the triangle inequality was used, while the second line is a consequence of Lemma 1, the hypotheses of which are satisfied by virtue of the second and the third inequalities in (7). Hence, by hypothesis (9) we obtain [B, H1 ]1 ≤ [B, H0 ]0 + o(1) B1 < η B0 + o(1) B1 ≤ (η + o(1)) B1 , so that η˜ = η + o(1).

(21)

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Eventually, by replacing in (16) the estimates (18) and (19), one has 

d η2 t 2 2 B(t) ≥ βh 1 − − o(1) − o(1) · t B21 . dt 2

(22)

Therefore, in the limit as h → 0, the time derivative of B(t) remains positive for √ 2 t< + o(1), η 

and this completes the proof.

3. A Gas in a Gravitational Field We study now a gas of N interacting particles enclosed in a tridimensional box of side L and total volume V = L 3 . Our aim is to show that Theorem 1 holds if we take as a simple example of perturbation the force of gravity, in which case the conjugated variable will be the displacement on the vertical axis of the center of mass of the system. For what concerns the interaction of the particles with the walls, due to the form of the conjugated variable it turns out that only the interaction with the horizontal walls will matter. Thus we limit ourselves to choose a particular form for the interaction potential with the horizontal walls. The unperturbed Hamiltonian H0 is then def

H0 =

N p2 j

2

j=1

+ U N (q1 , . . . , q N ),

(23)

where p αj ∈ (−∞, +∞), q αj ∈ (−L/2, L/2), α = 1, 2, 3, q 3j = z j , and U N denotes the potential energy of the system, which we take of the form def

UN =



(qi − q j ) +

1≤i< j≤N

N

f (z i ),

(24)

i=1

where  represents the mutual interaction potential between the particles and f the interaction with the horizontal walls. The choice of the possible forms of  and f is restricted by some technical conditions, which guarantee the existence of a suitable upper bound to the configuration integrals. In fact, the main difficulty which is encountered in applying Theorem 1 to the present case is the estimate of the distribution function for s particles, often called the s–point correlation function. We thus define   def Fs(N ) (q1 , . . . , qs ) = V s d3 qs+1 . . . d3 q N D N (q1 , . . . , q N ), (25) V

V

where def

D N (q1 , . . . , q N ) = and def





QN =

d3 q1 . . . V

V

 1 exp −βU N (q1 , . . . , q N ) , QN

(26)

 d3 q N exp −βU N (q1 , . . . , q N ) .

(27)

Relaxation Times for Hamiltonian Systems

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For sufficiently low densities ρ = N /V , one can prove a useful lemma which relates Fs(N ) to

 def n s (q1 , . . . , qs ) = exp −βUs (q1 , . . . , qs ) , (28) under suitable conditions for the potentials  and f . We consider  to be a stable and tempered potential in the familiar sense (see, for example, [5]). Then, in Appendix B we prove the following Lemma 2. Let β be fixed. Let  be a stable potential, i.e. suppose that b > 0 exists such that β (qi − q j ) ≥ −sb; (29) 1≤i< j≤s

furthermore, assume that  is tempered and that V is so large that one has    e−2 def  −β(r)  V. I = − 1 d3 r < e 2(e + 1) R3

(30)

Let f be continuous on the open interval (−L/2, L/2), nonnegative and such that  L 2 L def L˜ = (31) dz e−β f (z) > . L 2 −2 Then, for all densities ρ satisfying     1 L˜ 1 1 ρ < min − , 2b+1 , I L 2 4e I the following inequality holds. Fs(N ) (q1 , . . . , qs )

√

 2s e s Is n s (q1 , . . . , qs ), <√ exp 2I e−1

where I1 = I and, for s ≥ 2,      def max 1 − e−β(r) , e2sb 1 − eβ(r) d3 r. Is = R3

(32)

(33)

(34)

In order to use such a lemma and have, at the same time, a physically relevant model without complicating too much the computations, we take  and f equal to the repulsive part of the Lennard–Jones potential, namely, given by ε , x − y12 ⎡ 12  1 1 def ⎣ f (z) = δ + z + L2 z− def

(x − y) =

L 2

12 ⎤ ⎦,

(35)

where ε and δ are positive parameters, chosen in a convenient way (see hypothesis (37) of Theorem 2). According to the general scheme previously discussed, we add at time t = 0 a perturbation −h A, and for the observable A we make the choice

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def

A =

N

z j.

(36)

j=1

We then have Theorem 2. Let H0 and A be given by (23) and (36) respectively. Let the parameters ε, δ and L in (35) be such that 1

1

(βε) 12 < L/3 and (βδ) 12 < L/5,

(37)

where β is the inverse temperature. Then, for all densities ρ which satisfy ρ<

1.5 · 10−2 1

(βε) 4

,

one has the estimate tin f ≥ t0 + o(1), as h → 0, with 1 1 t0 = (βδ) 24 Lβ c and

1  +∞ 2 1 13e def u 12 e−u du ≈ 14.5. c = 2 √ e−1 0

(38)

(39)

(40)

Remark 1. The theorem is stated for a dimensionless Hamiltonian. By inserting the 1 1 proper dimensions, condition (βδ) 12 < L/5 becomes (βδ) 12 σ < L/5, where σ is the characteristic parameter for the range of the Lennard-Jones potential. This just expresses the requirement that the interactions with the walls have a range which is negligible with respect to L. Notice that we inserted the factor 1/5 at the r.h.s. just in order to fix a numerical value for c, but any other reasonable choice would not affect the result. A similar reasoning holds for the other requirement in (37). Notice, however, that the value of t0 does not depend on the interaction potential2 , which affects only the density up to which the result is valid, through formula (38). Remark 2. The time t0 , once the correct dimensional constants have been introduced, becomes 1 1 t0 = (βδ) 24 βm Lσ , c where m is the mass (of molecular order) of each particle. Therefore, for macroscopic systems in which L is of the order of magnitude of 1 m, while σ ≈ 10−10 m, at room temperature one gets t0 ≈ 10−8 s, a value which is of the order of magnitude of the typical relaxation times measured in gases. In the same way, condition (38) is proved to hold for ordinary densities, namely, of the order of magnitude of 1024 m −3 . Further comments will be given in the next section. Proof. The proof consists in showing that the hypotheses of Theorem 1 hold, with √ 2c η= . 1 √ (βδ) 24 Lβ 2 This is true for every potential  for which Lemma 2 can be applied (see the discussion concerning (42) and (43) in the proof).

Relaxation Times for Hamiltonian Systems

437

In the first place, (5) is satisfied for any δ, because it involves integrals of continuous functions over a compact, and the integral over the p coordinates is equal to 1. As far as the other hypotheses are concerned, we first notice that in the present case one has B=

N

p zj .

j=1

The integral over the momenta of any power of B can be easily turned into a combination of terms of the form 

 +∞ −βp 2 , d p p n exp 2 −∞ which are finite for any n, thus proving the first of (7). In particular, one has  2N B0 = . β

(41)

Clearly, one also has [B, A] = N , and hence the third of (7) holds. We then compute [B, H0 ]0 . One has [B, H0 ] = −

N

f  (z j ),

(42)

j=1

with f  (z j ) = −12δ

N

⎡ ⎣

j=1

zj +



13

1 L 2

+

13 ⎤

1 zj −

L 2

⎦,

since the contribution due to the pair interaction is the z–component of the sum of all the internal forces, and consequently it vanishes. Function (42), and its powers too, are actually singular at some point in phase space, but they diverge there as a power, while the density ρ0 vanishes as an exponential, making the norm finite. So the second of (7) is proved, as well. There finally remains the task of providing an estimate for the quantity η, which is an upper bound for the ratio [B, H0 0 /B0 . To this end, from (42) we get [B, H0 ]2 =

N

f  (z j ) f  (zl ),

j,l=1

and we have to estimate its ρ0 norm. It is convenient to integrate by parts: we use the equality 1 d d −βU N 1 d  z −βU N  Fj e − Flz f  (z j )e−βU N e − f  (z j ) f  (zl )e−βU N = 2 β dzl dz j β dzl δ jl  f (z j )e−βU N , + (43) β

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A. M. Maiocchi, A. Carati

where δ jl is the Krönecker delta and by F jz we denote the z component of the force exerted on the j th particle by the other particles, i. e. d def (qi − qk ). F jz = − dz j 1≤i
We observe that the first term at the r.h.s. of (43) has a vanishing inegral, being the derivative of a function which vanishes at the boundaries. Moreover, for what concerns  the second term, we point out that the quantity j F jz vanishes, being the z component of the sum of all the internal forces. The same remark is in order for the sum over l of the terms in the third place. The only term left is thus the fourth one, which we write in the form δ jl times the function ⎡ 14  14 ⎤ 1 1  1 156 δ ⎣ ⎦ e−βU N . + (44) f (z j )e−βU N = β β z j + L2 z j − L2 Therefore, we have to compute N identical integrals of the latter quantity, depending only on one coordinate. We remark here that conditions (37), (38) are sufficient to ensure that Lemma 2 can be used. In fact, our  is certainly stable, tempered and nonnegative, while f is continuous on (−L/2, L/2); furthermore, it is easy to verify the remaining hypotheses, on account of a simple integration for (30) and of numerical computations of the integrals appearing in (31) and (32). Making use of Lemma 2 for F1(N ) one finds that the modulus of the integral of (44), which is independent of j, is smaller than ⎡ 14  14 ⎤  L 2 1 1 312 e δ ⎦ e−β f (z) . dz ⎣ + (45) √ ( e − 1)β L − L2 z + L2 z − L2 The two terms in the integral (45) are identical, due to symmetry. Each of them is bounded from above on account of the inequality  14 

 L  L 2 1 1 βδ −β f (z) dz e ≤ dz 14 exp − 12 z z z + L2 − L2 0  +∞ 1 1 u 12 e−u du. ≤ 13 12(βδ) 12 0 In conclusion, we can infer that [B, H0 ]20 <

N c2 25

1

β 12 δ 12

,

(46)

and a quick comparison with (41) shows that one has √ 2c B0 . [B, H0 ]0 < 1 √ (βδ) 24 Lβ This relation, on account of Theorem 1, leads to formula (39).

(47) 

Relaxation Times for Hamiltonian Systems

439

4. Conclusions We have provided by Theorem 1 a lower bound to the relaxation times in Hamiltonian systems, and shown that in the case of an interacting gas enclosed in a box such an estimate is of the order of magnitude of the typical relaxation times measured in gases. This fact seems to indicate that the interactions with the walls, which we have considered in the present work, might have sensible effects even when one is interested in investigating relaxations of observables related to internal interactions. We now add some comments concerning possible further developments. The first point concerns the hypothesis made in Theorem 1, that H0 and A are even in the momenta, which actually is not at all essential. Indeed, if such an hypothesis is not satisfied, it suffices to define tin f in a different way, namely, as the time up to which the time–derivative of B remains larger than, for example, 21 dtd B(0). This way, by √ inequality (22) one could prove Theorem 1, except for setting 1/η in place of 2/η in (8). We decided to deal with the case of reversible Hamiltonians just because it is a very important one; furthermore, in such a case the relaxation time can be defined with no reference to arbitrary features, as the factor 1/2 introduced above. As a more interesting fact, we are confident that our line of reasoning may be extended to the case of perturbations of a finite size h, because this would just entail to consider the norms in L21 (M) rather than in L20 (M). Indeed, if we substitute [B, H1 ] for [B, H0 ] in hypothesis (9) and use there the norm  · 1 instead of  · 0 , we can directly set η˜ = η in (20) and there is no need of deducing (21), so that the second and third conditions in (7) are no more required. Moreover, the first condition in (7) can be replaced by the condition that B 2 1 is finite, which makes trivial the proof that [A, H0 ] is in L21 (M). The really open problem that remains in order to implement an extension to the case of finite h, at least for macroscopic systems, is the estimate of the difference of the two norms in (17). The estimate which appears in such a formula has the serious flaw of increasing exponentially with the number N of particles. This occurs because the upper bound provided there, which is an immediate consequence of the Schwarz inequality, is valid for all functions in L2 . A way to improve such an estimate would be to restrict oneself to perturbations having some suitable characteristic features. In particular, the work [11] of Lanford seems to suggest a good starting point. There it is pointed out that the only observables of interest in describing a macroscopic system are the ones he calls finite range observables,3 namely, observables which are sums of terms depending only on the position of a finite number of particles. The difference of the two norms in question can then be evaluated for each term and this should lead to an estimate which doesn’t increase too much with N . We think that, if one limits oneself to considering a smaller class of functions, there is a good chance that the problem of the number of degrees of freedom is overcome, and that some results are obtained also for the case of perturbations of finite size. These interesting investigations are left for possible future works. Acknowledgements. We thank very much Professor C. Cercignani and L. Galgani for useful comments and discussions.

3 As a matter of fact, he explains that this definition is chosen to make things simpler and is too restrictive. He gives also a reference to Ruelle’s book [5] in which it is shown how to deal with a broader class of observables, which represents the class of real interest.

440

A. M. Maiocchi, A. Carati

A. Proof of Lemma 1 We take as starting point the obvious equality    d p dq f 2 ρ1 = dp dq f 2 ρ0 + M

M

M

d p dq f 2 (ρ1 − ρ0 )

which, by the Schwarz inequality, gives    

   d p dq f (ρ1 − ρ0 ) ≤

1

2

M

d p dq f ρ0 4

M

2



γ (h),

(48)

with def

γ (h) =



 M

d p dq

2 ρ1 − 1 ρ0 . ρ0

One can also write 

d p dq e2hβ A ρ0  − 1, hβ A ρ 2 d p dq e 0 M

γ (h) = M

(49)

as is seen by expanding the square and using the fact that ρ0 and ρ1 are the densities of the Gibbs measures corresponding to H0 and H1 , respectively. It is also of interest to provide an upper bound to the l.h.s. of (48) in terms of ρ1 rather than of ρ0 . Indeed, one has    

M

   d p dq f (ρ1 − ρ0 ) ≤ 2

M

d p dq f ρ1 4

1 2

γ˜ (h),

where, in a way similar to (49), one gets 2 ρ0 − 1 ρ1 ρ1 M   

 = d p dq ehβ A ρ0 d p dq e−hβ A ρ0 − 1.

def





γ˜ (h) =

d p dq

M

M

Now, we observe that the functions γ (h) and γ˜ (h) can take arbitrarily small values as h goes to 0, if (5) is satisfied. Indeed, by their definitions, they are always nonnegative quantities. Thus, in order to show, for example, that γ (h) < ε for any fixed positive ε, it will suffice that one has  2hβ A ρ 0 M d p dq e  2 < 1 + ε. hβ A ρ0 M d p dq e To this end, let us note that  1  ≤ d p dq e−hβ A , hβ A ρ 0 M M d p dq e

Relaxation Times for Hamiltonian Systems

441

according to the Schwarz inequality. We combine this estimate with the Hölder inequalδ ity, on whose account, if h < cβ and condition (5) holds, one has  M

d p dq e

±chβ A

 ρ0 ≤

M

≤K

chβ δ

d p dq e

±δ A

 chβ  ρ0

δ

M

d p dq ρ0

(1− chβ ) δ

,

(50)

for some K > 0, and we eventually obtain    2

 2hβ A ρ 0 2hβ A −hβ A M d p dq e d p dq e ρ0 d p dq e ρ0   ≤ hβ A ρ 2 M M 0 M d p dq e ≤K

4hβ δ

→ 1 as h → 0.

  δ δ log(1+ε) The immediate consequence is that, if we take h < min 2β , 4 β log K , then γ (h) is less than ε. An analogous argument, still based on inequality (50), ensures that γ˜ (h) takes arbitrarily small values as h goes to 0, too. Hence, the difference between  f 20 and  f 21 vanishes with h, provided f 2 belongs to L20 (M) or L21 (M).  B. Proof of Lemma 2 The lemma is proved by using the results that were obtained in [6] in deducing the Mayer–Montroll equation. A relevant difference from the classical works in this field is that here we have to deal with an external field, too. In order to connect the computations with the ones used in the absence of such a field, we change the coordinates from q j to q˜ j , according to  zj q˜ 1j = q 1j , q˜ 2j = q 2j , z˜ j = dx e−β f (x) . (51) −L/2

This change of coordinates is well defined, because f is continuous and the following inequality holds: 

d˜z j L L . >0 ∀z j ∈ − , dz j 2 2 The volume differential, thus, changes in accordance with e−β f (z j ) d3 q j = d3 q˜ j , making the external field disappear in the integration; we denote by V˜ the domain of integration which takes the place of V . We must also notice that the pair potential (qi −q j ) is replaced by ˜ q˜ i , q˜ j ) def ( = (q˜i1 − q˜ 1j , q˜i2 − q˜ 2j , z i (˜z i ) − z j (z˜j )), which is a function of the coordinates of both particles, and not only of their difference, as was the case for , but it is still symmetric under the exchange of q˜ i and q˜ j . Furthermore, we point out here that we must consider a different distribution function in the modified phase space. Such a function, which we call D˜ N , is chosen by asking that it preserves the volume element under the change of coordinates from qi to q˜ i , i. e. that it fulfills the requirement

442

A. M. Maiocchi, A. Carati

D˜ N (q˜ 1 , . . . , q˜ N ) d3 q˜ 1 . . . d3 q˜ N = D N (q1 , . . . , q N ) d3 q1 . . . d3 q N , if q˜ i = q˜ i (qi ) for i = 1, . . . , N . It is apparent that one has ⎡ ⎤ 1 ˜ q˜ i , q˜ j )⎦ . ( exp ⎣−β D˜ N (q˜ 1 , . . . , q˜ N ) = QN 1≤i< j≤N

(N )

Accordingly, we will study the modified s–point correlation F˜s , defined in a way sim˜ The relation with the function Fs(N ) , ilar to (25), replacing D N with D˜ N and q with q. which is the one we want to bound from above, would be   s (N ) (N ) f (z i ) . (52) Fs (q1 , . . . , qs ) = F˜s (q˜ 1 (q1 ), . . . , q˜ s (qs )) exp −β i=1

We can now repeat the deduction of the Mayer–Montroll equations in these new coordinates, by writing   V s Q N −s (N ) 3 ∗ ˜ d q˜ 1 . . . d3 q˜ ∗N −s D˜ N −s (q˜ 1∗ , . . . , q˜ ∗N −s ) Fs (q˜ 1 , . . . , q˜ s ) = QN V˜ V˜  N −s    ∗ ˜ f s (q˜ 1 , . . . , q˜ s ; q˜ i ) + 1 n˜ s (q˜ 1 , . . . , q˜ s ), (53) × i=1

with def f˜s (q˜ 1 , . . . , q˜ s ; y˜ ) =

s 

 ˜ q˜ i , y˜ ) − 1, exp −β (

i=1

⎡

n˜ s (q˜ 1 , . . . , q˜ s ) = exp ⎣−β def



⎤ ˜ q˜ i , q˜ j )⎦ . (

1≤i< j≤s

We come then to the problem of finding an upper bound for the fraction Q N −s /Q N and for the term in square brackets in (53). As far as the fraction is concerned, it is shown4 in [6] that ⎛ ⎞   QM 1 ˜ q˜ i , q˜ j )⎠ ≥ d3 q˜ 1 . . . d3 q˜ M−1 exp ⎝−β ( Q M−1 Q M−1 V˜ V˜ 1≤i< j≤M−1    M−1   ˜  ˜ q˜ l ) × d3 q˜ 1 − − 1 . e−β (q, V˜

l=1

˜ we obtain that the term in square brackets at the r.h.s. of this inequality Integrating over q, is bounded from below by    L˜ L˜ ˜ ˜ q˜ l )   V − (M − 1) sup d3 q˜ e−β (q, − 1 ≥ V − (M − 1)I, L L q˜ ∈V˜ V˜ l

4 See formulas (3.20) and (3.21) in [6].

Relaxation Times for Hamiltonian Systems

443

with L˜ defined by (31) in the statement of Lemma 2. Therefore, if hypothesis (32) holds, one has V Q M−1 1 ≤ 2. ≤ ˜ QM L/L − ρI Thus, we can write  V Q N −i V s Q N −s = ≤ 2s , QN Q N −i+1 s

(54)

i=1

and get the required upper bound. As regards the term in square brackets in (53), instead, we expand the product and we get that such a term is equal to 1+

N −s

k=1

   k  N −s 1 (N −s) ∗ 3 ∗ 3 ∗ ˜ ˜ q q d . . . d (q˜ 1 , . . . , q˜ k∗ ). f˜s (q˜ 1 , . . . , q˜ s ; q˜ i∗ ) F˜k 1 k k k V V˜ V˜ i=1

(N −s) We know an uniform upper bound for F˜k , namely,

sup q˜ 1 ∈V˜ ,...,q˜ k ∈V˜

   ˜ (N −s)   Fk ≤

(2e2b+1 )k  , 1 − exp 2ρe2b+1 I − 1

(55)

which holds in the hypotheses V+I ˜ ˜ 2e2b+1 ( L/L)(V ( L/L) − I)

< 1 and

V ˜ ˜ 4e2b+2 ( L/L)(V ( L/L) − I)

< 1.

These conditions are certainly satisfied if hypothesis (30) holds. Inequality (55) has been proved5 by Bogolyubov et al. in work [6], which was dealing with pair potentials depending only on the distance between two particles. On the other hand, the hypothesis on the dependence on distance is not crucial, and a proof can be produced in the weaker hypothesis of potentials symmetric under the exchange of i and j. Indeed, the only difference from the proof given in [6] would be in the construction of the functions νi , but we show in Lemma 3 that it is possible to construct such functions in the present case, too. The νi are introduced in connection with the symmetrization operator πl , which acts on the function f (q1 , . . . , qs ) through the formula πl f (q1 , q2 , . . . , ql−1 , ql , ql+1 , . . . , qs ) = f (ql , q2 , . . . , ql−1 , q1 , ql+1 , . . . , qs ). One has the following lemma, whose proof can be found in Appendix C. Lemma 3. Suppose there exists a positive constant b such that, for all s and (q1 , . . . , qs ), the potential ϕ(qi , q j ) = ϕ(q j , qi ) satisfies

β ϕ(qi , q j ) ≥ −sb.

1≤i< j≤s 5 See the deduction of formula (4.10) in [6].

(56)

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A. M. Maiocchi, A. Carati

Then, for all s, there exist measurable functions νi (q1 , . . . , qs ), having values in the interval [0, 1], and such that s

νi (q1 , . . . , qs ) = 1, νk (q1 , . . . , qs ) = πk ν1 (q1 , . . . , qs ),

(57)

i=1

with the inequality β



ϕ(qi , q j ) > −2b

(58)

j=i

holding if νi (q1 , . . . , qs ) = 0. The only difference with respect to the functions used in [6] is that here the functions νi are not invariant under the rotation group. This, however, does not affect the proof, because it turns out that the proof given in [6] can be repeated word by word. This gives the upper bound (55). For what concerns f˜s , we adapt to the present situation the reasoning followed by O. Penrose in work [7], which deals with hard–core potentials, in order to provide an upper bound in this case. Indeed, we notice that, by condition (29), one has 1 + f˜s (q˜ 1 , . . . , q˜ s ; y) =

s 

 ˜ q˜ i , y˜ ) ≤ e2sb . exp −β (

(59)

i=1

Defining

 def ˜ q˜ i , y˜ ) − 1 gi = exp −β (

and gi± = max(0, ±gi ) ≥ 0, we can use the upper bound in (59) and the fact that gi ≥ −1, in order to prove by induction on s that 1−

s i=1

gi− ≤

s s  (1 + gi ) ≤ 1 + e2sb i=1

i=1

gi+ . 1 + gi+

We can use this relation in (59) to obtain 

s   e2sb gi+  ˜ . max gi− ,  f s (q˜ 1 , . . . , q˜ s ; y) ≤ 1 + gi+ i=1

Then, the integral over y of this quantity, for every choice of (q˜ 1 , . . . , q˜ s ), is smaller than s Is , with Is defined by (34) if s ≥ 2, while the case of s = 1 is trivial. If one, eventually, recalls that

 N −s Nk , < k! k one can then bound (53) from above by F˜s(N ) (q˜ 1 , . . . , q˜ s ) ≤ 2s

  exp 2ρe2b+1 s Is   n˜ s (q˜ 1 , . . . , q˜ s ). 1 − exp 2ρe2b+1 I − 1

Observing that, for √ lower than 1/4eI , the denominator of the fraction in the √ densities r.h.s. is larger than ( e − 1)/ e, the thesis is finally proved by going back to the initial coordinates and using relation (52). 

Relaxation Times for Hamiltonian Systems

445

C. Proof of Lemma 3 We show how to construct the functions νi having the properties required in the lemma. First, we fix a value for s and consider the subsets Ai of the configuration space X , which are defined, for all i ≤ s, by ⎧ ⎫ ⎨ ⎬ def Ai = x : β ϕ(qi , q j ) > −2b , ⎩ ⎭ j=i

def

where x = (q1 , . . . , qs ). We observe now that, due to condition (56), one has + Ai = X. i

We can thus choose the function ν1 (x) as ν1 (x) = N (x)χ A1 (x), where χ A is the characteristic function of the set A and the function N (x), which is introduced to normalize the sum, takes the value 1/n if k belongs to n sets Ai but not to n + 1 of them. Obviously, N (x) takes values in the set {1, 1/2, . . . , 1/s}. Then, if we construct νk by νk = πk ν1 , it is clear that the functions νi satisfy condition (58), because πk χ A1 = χ Ak and because of the definition of Ai . On the other hand, condition (57) holds, too, because πk N (x) = N (x). Indeed, the belonging of x to sets different from A1 and Ak is not affected by the action of πk , while πk x belongs to A1 if and only if x belongs to Ak and πk x belongs to Ak if and only if x belongs to A1 . Thus, the number of sets to which x belongs does not change under the action of πk , and this completes the proof.  References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Cercignani, C.: The Boltzmann equation and its applications. New York: Springer-Verlag, 1988 Gallavotti, G.: Statistical Mechanics. Berlin: Springer-Verlag, 2000 Kubo, R.: J. Phys. Soc. Japan 12, 570 (1957) Carati, A.: J. Stat. Phys. 128, 1057 (2007) Ruelle, D.: Statistical Mechanics, Rigorous Results. New York: Benjamin, 1969 Bogolyubov, N.N., Khatset, B.I., Petrina, D.Ya.: Ukrainian J. Phys. 53, Special Issue, 168 (2008), available at http://www.ujp.bitp.kiev.ua/files/file/papers/53/special_issue/53SI34p.pdf; Russian original in Teoret. i Mate. Fiz. 1:2, 251 (1969) Penrose, O.: J. Math. Phys. 6, 1312 (1963) Minlos, R.A.: Introduction to Mathematical Statistical Physics. Providence RI: Amer. Math. Soc., 2000 Koopman, B.O.: Proceedings of the National Academy of Sciences 17, 315 (1931) Showalter, R.E.: Hilbert Space Methods for Partial Differential Equations. London: Pitman, 1977 Lanford, O.E.: In: Statistical Mechanics and Mathematical Problems. Berlin: Springer-Verlag, 1973, p. 1

Communicated by G. Gallavotti

Commun. Math. Phys. 297, 447–474 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1040-9

Communications in

Mathematical Physics

Conformal Mappings and Dispersionless Toda Hierarchy II: General String Equations Lee-Peng Teo Department of Applied Mathematics, Faculty of Engineering, University of Nottingham Malaysia Campus, Jalan Broga, 43500, Semenyih, Selangor Darul Ehsan, Malysia. E-mail: [email protected] Received: 13 July 2009 / Accepted: 8 December 2009 Published online: 30 March 2010 – © Springer-Verlag 2010

Abstract: In this article, we classify the solutions of the dispersionless Toda hierarchy into degenerate and non-degenerate cases. We show that every non-degenerate solution is determined by a function H(z 1 , z 2 ) of two variables. We interpret these non-degenerate solutions as defining evolutions on the space D of pairs of conformal mappings (g, f ), where g is a univalent function on the exterior of the unit disc, f is a univalent function on the unit disc, normalized such that g(∞) = ∞, f (0) = 0 and f  (0)g  (∞) = 1. For each solution, we show how to define the natural time variables tn , n ∈ Z, as complex coordinates on the space D. We also find explicit formulas for the tau function of the dispersionless Toda hierarchy in terms of H(z 1 , z 2 ). Imposing some conditions on the function H(z 1 , z 2 ), we show that the dispersionless Toda flows can be naturally restricted to the subspace  of D defined by f (w) = 1/g(1/w). ¯ This recovers the result of Zabrodin (Theor Math Phy 12:1511–1525, 2001).

1. Introduction This paper is a continuation of our previous work [23], where we considered evolutions of conformal mappings described by an infinite hierarchy of dispersionless Toda flows [19,20] which satisfies the string equation. In this paper, we are going to consider the general evolutions of conformal mappings that can be described by dispersionless Toda hierarchies and derive the corresponding string equations. Dispersionless Toda hierarchy was introduced in [19,20] as a dispersionless limit of the Toda lattice hierarchy [25]. It describes the evolutions of two formal power series

L(w) = r (t)w +

∞  n=0

−1 ˜ u n+1 (t)w −n , L(w) = r (t)w −1 +

∞  n=0

u˜ n+1 (t)w n ,

(1.1)

448

L.-P. Teo

with respect to an infinite set of time variables tn , n ∈ Z, denoted collectively by t, by the following Lax equations: ∂L = {Bn , L} , ∂tn

  ∂ L˜ = Bn , L˜ . ∂tn

(1.2)

Here Bn is defined by    1 L(w)n 0 , n ≥ 1, B0 (w) = log w, Bn (w) = L(w)n >0 + 2    1˜ n ˜ Bn (w) = L(w) L(w)n , n ≤ −1, + <0 0 2 and the Poisson bracket is defined by {F1 (w), F2 (w)} = w

∂ F1 (w) ∂ F2 (w) ∂ F2 (w) ∂ F1 (w) −w . ∂w ∂t0 ∂w ∂t0

As mentioned in our previous paper [23], the integrable structures of conformal mappings have attracted considerable interest since the work of Wiegmann and Zabrodin [26]. Given a simple analytic curve C that separates 0 and ∞ on the extended ˆ let + be the interior domain that contains the origin and − the extecomplex plane C, rior domain. Denote by g(w) the unique conformal map mapping the exterior of the unit disc D∗ to the exterior domain which satisfies the normalization conditions g(∞) = ∞ and g  (∞) > 0. Wiegmann and Zabrodin [26] defined the time variables tn , n ≥ 1, in terms of the harmonic moments of the exterior domain − :

1 1 z −n d 2 z = z −n z¯ dz, tn = − − πn 2πin  C and defined t0 in terms of the area of the interior domain:

1 1 t0 = d2z = z¯ dz. π 2πi C + They showed that by taking t−n = −t¯n for n ≥ 1, the evolutions of (L(w) = ˜ g(w), L(w) = 1/g(1/w)) ¯ with respect to tn , n ∈ Z, satisfy the dispersionless Toda hierarchy. This solution of the dispersionless Toda hierarchy satisfies the string equation   −1 ˜ = 1. (1.3) L(w), L(w) A tau function τ (t) was also constructed and shown to generate the harmonic moments of the interior domain + . More precisely, it was proved that for n ≥ 1,

∂ log τ 1 1 = zn d 2 z = z n z¯ dz. ∂tn π 2πi + C This work of Wiegmann and Zabrodin has later been reinterpreted and elaborated from different perspectives such as the tau function and inverse potential problem [12], interface dynamics, Laplacian growth or the Hele-Shaw flow problem [1,3,5,8,10,16,17,33, 32], Dirichlet boundary value problem [7,15], quantum field theory of free bosons [21], large N -limit of normal matrix ensemble [27,29,30] and string equation and string theory

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[2,4,11]. The extension of these to multiply connected domains have been considered in [13,14,24,31]. As an integrable hierarchy, the dispersionless Toda hierarchy was proposed in the real domain where all the time variables and the coefficients r, u n , u˜ n , n ≥ 1, of L and L˜ are real-valued. Moreover, tn , n ∈ Z, are independent variables. However, as one can see from the discussion above, for the solution provided by Wiegmann and Zabrodin, the time variables tn , n ∈ Z, are in general complex-valued, so are the coefficients u n , u˜ n , n ≥ 1. Moreover, the time variables are independent over R, but are not independent over C, since for n ≥ 1, t−n and tn are related by t−n = −t¯n . This leads ˜ to the relation L˜ −1 (w) = L(1/w) ¯ of the two power series L(w) and L(w). Therefore, for this solution, only half of the flows are independent (over C). In other words, it can be considered as a solution of a reduction of the dispersionless Toda hierarchy. In our previous work [23], we extended the work of Wiegmann and Zabrodin by considering pairs of conformal mappings (g, f ), where g is conformal on the exterior of the unit disc D∗ normalized such that g(∞) = ∞, and f is conformal on the unit disc normalized such that f (0) = 0. f and g are only required to be related by f  (0)g  (∞) = 1. We showed that by suitably defining tn , n ∈ Z, which are in general complex-valued and independent over C, the dynamics of the pair of conformal maps (g, f ) is described by the dispersionless Toda flows, but with tn , n ∈ Z, and r, u n , u˜ n , n ∈ N, complex-valued, and r, u n , u˜ n , n ∈ N, are holomorphic in tn , n ∈ Z. We also constructed a tau function for the hierarchy which is real-valued. By restricting our flows to the subspace t−n + t¯n = 0 for n ≥ 1, and t0 = t¯0 , we recovered the flows considered by Wiegmann and Zabrodin. On the other hand, by restricting our flows to the subspace where tn = t¯n , n ∈ Z, or equivalently where f and g have real coefficients, we obtained the usual solution of the dispersionless Toda hierarchy where all the variables are real. The tau function we constructed can then be interpreted as the free energy of a two Hermitian matrix model [6]. From a different perspective, we have considered a particular solution of the complexified version of the dispersionless Toda hierarchy which satisfies the string equation (1.3) and interpreted it as describing evolutions of pairs of conformal mappings. Since the works on evolutions of conformal mappings have found applications in lots of different areas, it is natural to ask what are the general solutions of dispersionless Toda hierarchies and whether they can be interpreted as evolutions of conformal mappings. In fact, soon after the work [26], Zabrodin [28] has shown that one can obtain   ˜ L(w) = g(w), L(w) = 1/g(1/w) ¯ as more general solutions of the dispersionless Toda hierarchy which satisfy the generalized string equation   1 −1 ˜ = , (1.4) L(w), L(w) ˜ −1 )−1 ) ∂z ∂z¯ U(L(w), L(w where U(z, z¯ ) is a real-valued function of z and z¯ , by defining the time variables tn , n ≥ 0, as



1 1 z −n ∂z U(z, z¯ )dz, n ≥ 1, t0 = ∂z U(z, z¯ )dz, tn = 2πin C 2πi C and let t−n = −t¯n for n ≥ 1. A tau function for the problem was also derived by using the electrostatic variational principle. The particular case considered in [26] corresponds to choosing U(z, z¯ ) = z z¯ . As in [26], the solutions provided by Zabrodin [28] should be considered as solutions to reductions of the complexified dispersionless Toda hierarchy characterized by t−n + t¯n = 0 for n ≥ 1 and t0 = t¯0 .

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In this paper, we consider general solutions of the complexified dispersionless Toda hierarchy where r, u n , u˜ n , n ∈ N, depend holomorphically on tm , m ∈ Z. We show that these solutions can be interpreted as describing evolutions of pairs of conformal mappings (g, f ) when one defines the time variables tn , n ∈ Z, appropriately. We also construct a real-valued tau function for each solution of the dispersionless Toda hierarchy. Under certain reality conditions, we show that the solutions of Zabrodin [28] can be considered as the restriction of our solution to the subspace defined by t−n + t¯n = 0, n ≥ 1, and t0 = t¯0 . From the perspective of the dispersionless Toda hierarchy, this work answers the question: What is the general solution of dispersionless Toda hierarchy and what is the corresponding tau function? From the perspective of conformal mappings, this work characterizes all different complex coordinates on the space of pairs of conformal mappings which can give rise to dispersionless Toda flows.

2. Generalized Grunsky Coefficients and Faber Polynomials To make this paper self-contained, we review again the concepts of generalized Grunsky coefficients and Faber polynomials here. Let F(z) = α1 z + α2 z 2 + . . . be a function univalent in a neighborhood of the origin and G(z) = βz + β0 + β1 z −1 + . . . be a function univalent in a neighborhood of ∞ such that α1 β = 1. We define the generalized Grunsky coefficients bm,n , m, n ∈ Z, and Faber polynomials Pn (z), n ∈ Z, of the pair (G, F) by the following formal power series expansion: ∞

log

log

log

∞

 G(z) − G(ζ ) = b0,0 − bmn z −m ζ −n , z−ζ G(z) − F(ζ ) = b0,0 − z F(z) − F(ζ ) =− z−ζ log

G(z) − w =− βz

m=1 n=1 ∞  ∞ 

bm,−n z −m ζ n ,

m=1 n=0

∞ ∞  

b−m,−n z m ζ n ,

m=0 n=0 ∞  Pn (w) −n n=1

n

z

(2.1)

,

∞

log

F(z)  P−n (w) n w − F(z) = log − z . w α1 z n n=1

For m ≥ 0, n ≥ 1, b−m,n := bn,−m , and by convention, P0 (w) := log w (which is not a polynomial). By definition, the Grunsky coefficients are symmetric, i.e., bm,n = bn,m for all m, n ∈ Z. The coefficient b0,0 is given explicitly by − log α1 = log β, where α1 = F  (0) and β = G  (∞). For n ≥ 1, Pn (w) is a polynomial of degree n in w and P−n (w) is a polynomial of degree n in 1/w, which can be defined alternatively by   Pn (w) = (G −1 (w)n )≥0 , P−n (w) = F −1 (w)−n . ≤0

 n Here when S is a subset of integers and A(w)  = nnAn w is a (formal) power series, we denote by (A(w)) S the truncated sum n∈S An w .

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From (2.1), we can deduce the following: ∞

log

∞

 G(z) = b0,0 − b0,m z −m , z

 F(z) =− b0,−m z m , z

log

m=1

(2.2)

m=0

and for n ≥ 1, Pn (G(z)) = z n + n

∞ 

bnm z −m ,

Pn (F(z)) = nbn,0 +n

m=1

∞ 

bn,−m z m ,

m=1

(2.3) P−n (G(z)) = −nb−n,0 + n

∞ 

b−n,m z −m ,

P−n (F(z)) = z −n + n

m=1

∞ 

b−n,−m z m .

m=1

It follows that P0 (G(z))G  (z) =

∞

1  mb0,m z −m−1 , + z m=1

P0 (F(z))F  (z)

∞

1  = − mb0,−m z m−1 , z m=1

Pn (G(z))G  (z) = nz n−1 − n

∞ 

mbnm z −m−1 ,

m=1

Pn (F(z))F  (z)

=n

∞ 

mbn,−m z

(2.4) m−1

,

m=1  P−n (G(z))G  (z) = −n

∞ 

mb−n,m z −m−1 ,

m=1  P−n (F(z))F  (z)

= −nz

−n−1

+n

∞ 

mb−n,−m z m−1 .

m=1

3. Dispersionless Toda Hierarchy and its General Solutions As discussed in the Introduction, the dispersionless Toda hierarchy is a hierarchy of equations which can be put into the Lax form (1.2). In this section, we first review some basic facts we need later. We then classify and characterize the solutions of the dispersionless Toda hierarchy. ˜ are power series of the form 3.1. Orlov-Schulman functions. First, recall that if (L, L) (1.1) that satisfy the dispersionless Toda hierarchy (1.2), their Orlov-Schulman functions ˜ are functions M(w) and M(w) of the form

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M(w) =

∞ 

ntn L(w)n + t0 +

n=1 ∞ 

˜ M(w) =−

∞ 

vn L(w)−n ,

n=1 −n ˜ nt−n L(w) + t0 −

n=1

∞ 

(3.1) n ˜ v−n L(w)

n=1

that satisfy the Lax equations ∂M = {Bn , M} , ∂tn

  ˜ ∂M ˜ , = Bn , M ∂tn

and the canonical Poisson relations {L, M} = L,

n ∈ Z,

  ˜ M ˜ = L. ˜ L,

(3.2)

(3.3)

˜ of the dispersionless 3.2. Phi function φ and tau function τ . Given a solution (L, L) Toda hierarchy (1.2), there exists a phi function φ(t) and a tau function τ (t) such that ∂φ ∂vn ∂ log τ = and = vn for all n ∈ Z. ∂tn ∂t0 ∂tn ˜ respecHere v0 := φ. If we let G(z) and F(z) be formally the inverses of L(w) and L(w) ˜ tively, i.e., L(G(z)) = z and L(F(z)) = z, and let bm,n , m, n ∈ Z, be the generalized Grunsky coefficients of the pair (G, F), then ⎧ ⎪−|mn|bm,n (t), if mn = 0, ∂ 2 log τ (t) ⎨ = |m|bm,0 (t), (3.4) if m = 0, n = 0, ⎪ ∂tm ∂tn ⎩−2b (t), if m = n = 0. 0,0 Conversely, if bm,n , m, n ∈ Z, are the generalized Grunsky coefficients of the pair ˜ = F −1 (w)) (G, F), and τ (t) is a function satisfying (3.4), then (L(w) = G −1 (w), L(w) is a solution of the dispersionless Toda hierarchy. ˜ of the dis3.3. Riemann-Hilbert data. The Riemann-Hilbert data of a solution (L, L) persionless Toda hierarchy is a pair of functions U (w, t0 ) and V (w, t0 ) of the variables w and t0 that satisfy ˜ = V (L, M), L˜ = U (L, M), M

(3.5)

and the canonical Poisson relation {U, V } = w

∂U ∂ V ∂ V ∂U −w = U. ∂w ∂t0 ∂w ∂t0

(3.6)

It was shown in [20] that there always exists a Riemann-Hilbert data for any solutions of the dispersionless Toda hierarchy. Conversely, it was also proved that if U (w, t0 ) and V (w, t0 ) are functions satisfying the canonical Poisson relation (3.6), and L, L˜ are func˜ are functions of the form (3.1), and they satisfy (3.5), then tions of the form (1.1), M, M ˜ is a solution of the dispersionless Toda hierarchy with Orlov-Schulman functions (L, L) ˜ M(w) and M(w).

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3.4. General solutions of dispersionless Toda hierarchy. Now we come to the classification and characterization of the solutions of the dispersionless Toda hierarchy. Suppose ˜ is a solution of the dispersionless Toda hierarchy with Orlov-Schulman functions (L, L) ˜ M(w), M(w) and Riemann-Hilbert data U (w, t0 ) and V (w, t0 ). We have two cases. First, if U is independent of t0 , i.e., ∂U/∂t0 = 0, then the canonical Poisson relation (3.6) implies that U  (w) = 0 and U (w) t0 + U1 (w), wU  (w)

V (w, t0 ) =

for an arbitrary function U1 (w). In this case, we find that ˜ = U (L) M + U1 (L), L˜ = U (L), M LU  (L) and 

 L, L˜ = 0.

This should be considered as a degenerate solution of the dispersionless Toda hierarchy since the relation L˜ = U (L) implies that for all n ∈ Z, the set of equations   ∂ L˜ = Bn , L˜ ∂tn follows immediately from the set of equations ∂L = {Bn , L} . ∂tn In the second case, ∂U/∂t0 does not vanish. Then the inverse function theorem implies that we can solve t0 as a function of w and w˜ from w˜ = U (w, t0 ). More precisely, there exists a function A(w, w) ˜ such that w˜ = U (w, A(w, w)) ˜ . Moreover, ∂U ˜ ∂A ∂w (w, A(w, w)) (w, w) ˜ = − ∂U , ∂w ˜ ∂t0 (w, A(w, w))

∂A (w, w) ˜ = ∂ w˜

1 ∂U ∂t0

˜ (w, A(w, w))

.

˜ Obviously, ∂A/∂ w˜ = 0. Define the function A(w, w) ˜ by ˜ A(w, w) ˜ = V (w, A(w, w)) ˜ . Then from the canonical Poisson relation (3.6), we have ∂ A˜ ∂ V ∂ V ∂A 1 = + = ∂U ∂w ∂w ∂t0 ∂w ∂t

0



∂ V ∂U ∂U ∂ V − ∂w ∂t0 ∂w ∂t0

 =−

w˜ ∂A (w, w). ˜ w ∂ w˜

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This gives ∂ ∂w



A˜ − w˜

 =

∂ ∂ w˜



 A . w

Consequently, there exists a function H (w, w) ˜ such that A (w, w) ˜ = w∂w H (w, w) ˜ , A˜ (w, w) ˜ = −w∂ ˜ w˜ H (w, w) ˜ .

(3.7)

The condition ∂A/∂ w˜ = 0 is then equivalent to ∂ 2H = 0. ∂w∂ w˜

(3.8)

    ˜ = −L∂ ˜ ˜ H L, L˜ . M = L∂L H L, L˜ , M L

(3.9)

The relations (3.5) become

Moreover, 

   ∂U  1 L, L˜ = L L, A L, L˜ = , ˜ ∂t0 ∂L ∂L˜ H(L, L)

which is the string equation for this solution. It is interesting to note that the function ˜ can be interpreted as a generating function for the canonical transformation H(L, L) (3.5) performed by U and V . Conversely, suppose H(w, w) ˜ is a function satisfying (3.8), and L, L˜ are functions ˜ of the form (1.1), M, M are functions of the form (3.1), so that (3.9) holds. Define the ˜ functions A(w, w) ˜ and A(w, w) ˜ by the relations (3.7). Then ∂A/∂ w˜ = 0. Therefore, we can solve w˜ as a function of w and t0 from the equation t0 = A(w, w), ˜ which we denote by U (w, t0 ), so that t0 = A(w, U (w, t0 )). Moreover, ∂A ∂U = − ∂∂w , A ∂w ∂ w˜

∂U 1 = ∂A . ∂t0 ∂ w˜

Define the function V (w, t0 ) by ˜ V (w, t0 ) = A(w, U (w, t0 )). The relations (3.9) are then equal to (3.5). On the other hand, A ∂V ∂ A˜ ∂ A˜ ∂U ∂ A˜ ∂ A˜ ∂∂w = + = − , ∂w ∂w ∂ w˜ ∂w ∂w ∂ w˜ ∂ A ∂ A˜ ∂V ∂ A˜ ∂U w˜ = = ∂∂A . ∂t0 ∂ w˜ ∂t0 ∂ w˜

∂ w˜

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Therefore, ˜

{U, V } = w

∂A ∂U ∂ V ∂U ∂ V = −w ∂∂w −w = w˜ = U (w, t0 ). A ∂w ∂t0 ∂t0 ∂w ∂ w˜

˜ is a solution of the dispersionless Toda hierarchy with OrlovIn other words, (L, L) ˜ Schulman functions M(w) and M(w). Summarizing, we have shown the following. ˜ is a solution of the dispersionless Toda hierarchy with Proposition 3.1. Suppose (L, L) ˜ Orlov-Schulman functions M(w) and M(w), then one of the following two cases holds: ˜ = 0. In this case, there exists two functions U and U1 of z such that Case I {L, L} U  (z) = 0 and ˜ = U (L) M + U1 (L). L˜ = U (L), M LU  (L) ˜ = 0. In this case, there exists a function H(z 1 , z 2 ) such that Case II {L, L} ∂z 1 ∂z 2 H(z 1 , z 2 ) = 0, and     ˜ = −L∂ ˜ z 2 H L, L˜ . M = L∂z 1 H L, L˜ , M

(3.10)

In this case, the string equation is 

 L, L˜ =

1 ˜ ∂L ∂L˜ H(L, L)

.

Conversely, we have ˜ are functions of the form (1.1), M, M ˜ are functions of the Proposition 3.2. If (L, L) form (3.1), and if U and U1 are two functions of z so that U  (z) = 0 and ˜ = U (L) M + U1 (L), L˜ = U (L), M LU  (L) ˜ is a solution of the dispersionless Toda hierarchy with Orlov-Schulman then (L, L) ˜ functions M(w) and M(w). ˜ are functions of the form (1.1), M, M ˜ are functions of the Proposition 3.3. If (L, L) form (3.1), and if H(z 1 , z 2 ) is a function of z 1 and z 2 such that ∂z 1 ∂z 2 H(z 1 , z 2 ) = 0, and     ˜ = −L∂ ˜ z 2 H L, L˜ , M = L∂z 1 H L, L˜ , M ˜ is a solution of the dispersionless Toda hierarchy with Orlov-Schulman then (L, L) ˜ functions M(w) and M(w).

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From the results above, we see that we can use the string equation to classify the ˜ is a dispersionless Toda hierarchy into degenerate and non-degenerate cases. (L, L) degenerate solution if and only if   L, L˜ = 0. However, the string equation does not determine the solution uniquely. In the degenerate case, we see that the solutions are determined by two functions U and U1 . In the non-degenerate case, i.e., when ˜ = {L, L}

1 ˜ ∂L ∂L˜ H(L, L)

= 0,

the solution is also only determined up to two arbitrary functions. More precisely, for any two functions H1 (z 1 ) and H2 (z 2 ) of z 1 and z 2 respectively, the functions H(z 1 , z 2 ) and H(z 1 , z 2 ) + H1 (z 1 ) + H2 (z 2 ) give rise to the same string equation. In the following section, we are mainly going to discuss the non-degenerate solutions of the dispersionless Toda hierarchy. We are going to show that the roles of the two auxiliary functions H1 (z 1 ) and H2 (z 2 ) are just shifting the origin of the time coordinates tn , n ∈ Z. 4. Dispersionless Toda Flows on Space of Pairs of Conformal Mappings Let D and D∗ be respectively the unit disc and its exterior. As in [23], we introduce the following spaces of conformal mappings:   S∞ = g : D∗ → C univalent  g(w) = bw + b0 + b1 w −1 + . . . ; b = 0;  ˆ , 0∈ / g(D∗ ); g is extendable to a C 1 homeomorphism of C.   S0 = f : D → C univalent  f (w) = a1 w + a2 w 2 + . . . ; a1 = 0;  ˆ , ∞∈ / f (D); f is extendable to a C 1 homeomorphism of C.    D = (g, f )  g ∈ S∞ , f ∈ S0 ; f  (0)g  (∞) = a1 b = 1. . − + + ∗ Let − 1 = g(D ) and 1 be its exterior, 2 = f (D) and 2 be its exterior. C1 and C2 denotes the C 1 curves C1 = g(S 1 ) and C2 = f (S 1 ) respectively.1 The main objective of this paper is to interpret the dispersionless Toda flows as an integrable structure on the space D of pairs of conformal mappings, by identifying L(w) ˜ with g(w) and L(w) with f (w). As discussed in the previous section, the solutions of the dispersionless Toda hierarchy can be classified into degenerate and non-degenerate solutions. For the degenerate solutions, L˜ can be expressed as a nontrivial function U of L, which is independent of the time variables. This implies that the dispersionless Toda flows are restricted to the subspace of D defined by f (w) = U (g(w)), and the time variables tn , n ∈ Z, will not be independent. Therefore, we will not study the degenerate flows in this paper. We focus on the non-degenerate flows where the time variables are locally coordinates of the space D. 1 The notations used here is a little bit different from those used in [23].

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Although the results in the previous section shows that any function H(z 1 , z 2 ) with ∂z 1 ∂z 2 H(z 1 , z 2 ) = 0 gives rise to a solution of the dispersionless Toda hierarchy, this is not the end of the story. In this section, we are going to show that given H(z 1 , z 2 ) with ∂z 1 ∂z 2 H(z 1 , z 2 ) = 0, how the time variables tn , n ∈ Z, are defined in terms of g(w) and f (w). We will prove that these time coordinates can play the role of local coordinates on the space D. We are also going to define the phi function φ and the tau function τ in terms of H(z 1 , z 2 ). To be more concrete, one can assume that μ

H(z 1 , z 2 ) = z 1 z 2−ν , μ, ν ∈ Z\{0}, or a linear combination of these functions. However, the results of this section does not depend on the specific form of the function H(z 1 , z 2 ). Nevertheless, we assume throughout that H(z 1 , z 2 ) is an analytic function of z 1 and z 2 on C\{0}. The case considered in [23] is the special case where H(z 1 , z 2 ) = z 1 z 2−1 . We begin with the definitions of tn , n ∈ Z, and vn , n ∈ Z, in terms of H(z 1 , z 2 ), and g(w) and f (w). From (3.1) and (3.10), we find that tn , n ∈ Z, and vn , n ∈ Z, should be defined as: For n ≥ 1,

1 tn = ∂z H(g(w), f (w))g(w)−n dg(w) 2πin S 1 1

1 = ∂z H(z, f ◦ g −1 (z))z −n dz, 2πin C1 1

1 ∂z H(g(w), f (w)) f (w)n d f (w) t−n = 2πin S 1 2

1 ∂z H(g ◦ f −1 (z), z)z n dz, = 2πin C2 2

(4.1) 1 vn = ∂z 1 H(g(w), f (w))g(w)n dg(w) 2πi S 1

1 ∂z H(z, f ◦ g −1 (z))z n dz, = 2πi C1 1

1 ∂z H(g(w), f (w)) f (w)−n d f (w) v−n = 2πi S 1 2

1 ∂z H(g ◦ f −1 (z), z)z −n dz. = 2πi C2 2 The function t0 is defined as



1 1 ∂z H(g(w), f (w))dg(w) = ∂z H(z, f ◦ g −1 (z))dz t0 = 2πi S 1 1 2πi C1 1



1 1 ∂z H(g(w), f (w))d f (w) = − ∂z H(g ◦ f −1 (z), z)dz. (4.2) =− 2πi S 1 2 2πi C2 2 The function v0 := φ will be defined later. Since tn , n ∈ Z, only depends explicitly on g(w) and f (w), but not their complex conjugates, they are holomorphic functions on D. In the following, we are going to show that the set of variables tn , n ∈ Z, are local coordinates on D, by showing that there are independent vector fields ∂n , n ∈ Z, on D such that ∂n tm = δn,m . For this purpose, we define the functions S+ (z), S− (z), S˜+ (z) and S˜− (z) by

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∂z 1 H(ζ, f ◦ g −1 (ζ )) 1 S± (z) = dζ, z ∈ ± 1, 2πi C1 ζ −z

∂z 2 H(g ◦ f −1 (ζ ), ζ ) 1 dζ, z ∈ ± S˜± (z) = − 2. 2πi C2 ζ −z

(4.3)

It is easy to see that in a neighbourhood of z = 0, we have S+ (z) =

∞ 

ntn z n−1 , S˜+ (z) = −

n=1

∞ 

v−n z n−1 .

(4.4)

n=1

In a neighbourhood of z = ∞, ∞

S− (z) = −

∞

t0  t0  vn z −n−1 , S˜− (z) = − + nt−n z −n−1 . − z z n=1

(4.5)

n=1

Let G and F be the inverse functions of g and f respectively. Denote by bm,n , m, n ∈ Z, and Pn (w), n ∈ Z, the generalized Grunsky coefficients and Faber polynomials of (G, F). Proposition 4.1. There are independent vector fields ∂n , n ∈ Z, on D such that ∂n tm = δn,m . Proof. We begin by constructing the vector fields ∂n , n ∈ Z. First, define the functions un (w), w ∈ S 1 , n ∈ Z, by un (w) = −

∞  Pn (w) = un;m w m+1 . f  (w)g  (w)∂z 1 ∂z 2 H(g(w), f (w)) m=−∞

(4.6)

Given a vector field ∂ on D, notice that ∂g(w) = (∂ log b)w + lower power terms in w, g  (w) ∂ f (w) = (∂ log a1 )w + higher power terms in w. f  (w) Since a1 b = 1, this implies that ∂ log a1 = −∂ log b. Therefore, if we define vector fields ∂n on D by   ∞  1  −m+1 ∂n g(w) = g (w) un;0 w + un;−m w , 2 m=1   (4.7) ∞  1  m+1 un;m w ∂n f (w) = f (w) − un;0 w − , 2 m=1

then Pn (w) ∂n g(w) ∂n f (w) (w) = − − = u . n g  (w) f  (w) f  (w)g  (w)∂z 1 ∂z 2 H(g(w), f (w))

(4.8)

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Since the functions Pn (w), n ∈ Z, are independent, the functions un (w), n ∈ Z, are independent. Therefore the vector fields ∂n , n ∈ Z, are also independent. In fact, we can say more. Since P0 (w) = 1/w, and for n ≥ 1, Pn (w) ∼ w n−1 +  (w) ∼ w −n−1 + lower negative power terms in w, lower positive power terms in w, P−n we can conclude that the vector fields ∂n span the tangent space of D. Now for z ∈ C1 ,       f ∂n ∂z 1 H z, f ◦ g −1 (z) = ∂z 1 ∂z 2 H z, f ◦ g −1 (z) ∂n f −  ∂n g ◦ g −1 (z) g = Pn (g −1 (z))(g −1 ) (z).

(4.9)

Using (4.3), (4.4), (4.5) together with (4.9) and (2.4), we find that in a neighbourhood of z = 0, 

∞  Pn (g −1 (ζ ))(g −1 ) (ζ ) 1 nz n−1 , if n ≥ 1, m−1 dζ = m(∂n tm )z = 0, if n ≤ 0. 2πi C1 ζ −z m=1

(4.10) In a neighbourhood of z = ∞,

∞ Pn (g −1 (ζ ))(g −1 ) (ζ ) ∂n t0  −1 (∂n vm )z −m−1 = + dζ z 2πi C1 ζ −z m=1 ⎧ ∞ −1 + −m−1 , ⎪ if n = 0, ⎨z  m=1 mb0,m z ∞ = −n m=1 mbnm , if n ≥ 1, (4.11) ⎪ ⎩−n ∞ mb −m−1 , if n ≤ −1. −n,m z m=1 Similarly, for z ∈ C2 ,       g −1 −1 − ∂n ∂z 2 H g ◦ f (z), z = −∂z 1 ∂z 2 H g ◦ f (z), z ∂n g −  ∂n f f ◦ f −1 (z) = Pn ( f −1 (z))( f −1 ) (z).

(4.12)

This, together with (4.3), (4.4), (4.5) and (2.4) imply that in a neighbourhood of z = 0, ⎧ ∞ ⎪ mb0,−m z m−1 , if n = 0, ∞ ⎨ m=1   ∞ m−1 m−1 (∂n v−m )z = −n m=1 mbn,−m z (4.13) , if n ≥ 1, ⎪ ⎩−n ∞ mb m−1 , if n ≤ −1. m=1 z −n,−m m=1 In a neighbourhood of z = ∞, −

∂n t0 + z

∞ 

m(∂n t−m )z −m−1

m=1

⎧ −1 ⎪ if n = 0, ⎨−z , = 0, if n ≥ 1, ⎪ ⎩nz −n−1 , if n ≤ −1.

(4.14)

Compare the coefficients on both sides of (4.10), (4.11) and (4.14), we find that ∂n tm = δm,n for all n, m ∈ Z, which is the assertion of the proposition.



(4.15)

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If {en }n∈Z is a set of local coordinates on D, then (4.15) implies that  ∂tm k∈Z

In other words, the Z × Z matrix 

∂ek

∂tm ∂en

∂n ek = δm,n .

 −∞<m,n<∞

is invertible with inverse (∂n em )−∞<m,n<∞ . This implies that {en }n∈Z can be locally solved as functions of {tn }n∈Z , and   ∂em = (∂n em )−∞<m,n<∞ , ∂tn −∞<m,n<∞ i.e., the actions of the vector field ∂n on the local coordinates {em }m∈Z are the same as the action of the partial derivatives ∂/∂tn . This shows that {tn }n∈Z can be taken to be local coordinates on D and the vector fields ∂n , n ∈ Z, constructed above coincides with the partial derivatives ∂/∂tn . From (4.11) and (4.13), we also obtain  ∂vm −|mn|bm,n , if n = 0, = m = 0. (4.16) |m|bm,0 , if n = 0, ∂tn From (4.6) and (4.7), we observe that the partial derivatives ∂g(w) , ∂tn

∂ f (w) ∂tn

(4.17)

depend on H(z 1 , z 2 ) only through ∂z 1 ∂z 2 H(z 1 , z 2 ). Therefore, these partial derivatives are not changed if we replace H(z 1 , z 2 ) by H(z 1 , z 2 ) + H1 (z 1 ) + H2 (z 2 ). Since the partial derivatives (4.17) determine g(w) and f (w) up to the initial condition, we can conclude that the roles of the two auxiliary functions H1 (z 1 ) and H2 (z 2 ) are to change the origin of the coordinates tn , n ∈ Z. In fact, this can also be observed in a different way. From the definitions (4.1) and (4.2), we find that if we replace H(z 1 , z 2 ) by H (z 1 , z 2 ) = H(z 1 , z 2 )+H1 (z 1 )+H2 (z 2 ), then the time variables tn , n ∈ Z, are changed to tn , n ∈ Z, where ⎧  1 −n ⎪ if n ≥ 1, ⎨ 2πin ∞ ∂z H1 (z)z dz,  tn = tn + cn , cn = 0, if n = 0, ⎪ ⎩ 1  ∂ H (z)z n dz, if n ≤ −1. 2πin 0 z 2 In fact, the functions vn , n ∈ Z\{0} are also changed to vn , where  1  n if n ≥ 1, 2πi ∞ ∂z H1 (z)z dz,  vn = vn + dn , dn =  1 −n if n ≤ −1. 2πi 0 ∂z H2 (z)z dz,

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Since dn , n ∈ Z\{0} are independent of t, we find that replacing H(z 1 , z 2 ) by H (z 1 , z 2 ) = H(z 1 , z 2 ) + H1 (z 1 ) + H2 (z 2 ) does not change the relations (4.16) which characterize the functions vn up to constants. We have used the forms of the Orlov-Schulman functions and the Riemann-Hilbert data to help us define the variables vn , n = 0, in terms of f (w) and g(w). Unfortunately, this does not lead to a definition of v0 , or equivalently, the phi function φ, which should be defined so that  ∂v0 ∂vn |n|b0,n , if n = 0, = = (4.18) −2b0,0 , if n = 0. ∂tn ∂t0 In other words, v0 = φ generates the coefficients of bn,0 , n ∈ N, of log g −1 (z) and the coefficients b−n,0 , n ∈ N, of log f −1 (z). In the following, we define the function v0 and show that it satisfies (4.18). Proposition 4.2. The function v0 defined by  g(w) ∂z 1 H(g(w), f (w))g  (w) log 1 w S  H(g(w), f (w)) f (w)  − dw + ∂z 2 H(g(w), f (w)) f (w) log w w

1 v0 = 2πi



(4.19)

satisfies  ∂v0 |n|b0,n , = −2b0,0 , ∂tn

if n = 0, if n = 0.

(4.20)

Proof. A straightforward computation gives  ∂ g(w) f (w) ∂z 1 H(g(w), f (w))g  (w) log + ∂z 2 H(g(w), f (w)) f  (w) log ∂tn w w  H(g(w), f (w)) − w   ∂ ∂g(w) g(w) ∂ f (w) f (w) ∂z 1 H(g(w), f (w)) +∂z 2 H(g(w), f (w)) = log log ∂w ∂tn w ∂tn w   2 ∂ H ∂n f (w) ∂n g(w) + f  (w)g  (w) −  (g(w), f (w)) ∂z 1 ∂z 2 f  (w) g (w)   f (w) g(w) − log . × log w w This together with the definition (4.19) of v0 and (4.8), give ∂v0 1 = ∂tn 2πi

S1

  f (w) g(w) − log dw. Pn (w) log w w

(4.21)

462

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For n ≥ 1, (4.21) and (2.4) give



∂v0 1 g(w) g −1 (z) −1  = Pn (w) log Pn (g −1 (z))(g −1 ) (z) log dw = dz ∂tn 2πi S 1 w 2πi C1 z

−n g −1 (z) n−1 z dz = nb0,n , = log 2πi C1 z

−1 f (w) ∂v0  dw = P−n (w) log ∂t−n 2πi S 1 w

1 f −1 (z)  = dz P−n ( f −1 (z))( f −1 ) (z) log 2πi C2 z

−n f −1 (z) −n−1 = z log dz = nb0,−n . 2πi C2 z For n = 0, since P0 (w) = 1/w, we have 

 g(w) f (w) 1 ∂v0 1 log − log dw = log b − log a1 = 2 log b = −2b0,0 . = ∂t0 2πi S 1 w w w

We would like to remark that if we replace H(z 1 , z 2 ) by H (z 1 , z 2 ) = H(z 1 , z 2 ) + H1 (z 1 ) + H2 (z 2 ), then v0 is changed to v0 , where



1 1  ∂z H1 (z) log zdz + ∂2 H2 (z) log zdz v0 − v0 = 2πi ∞ 2πi 0 is independent of t. From (4.16), (4.20) and the symmetry of the Grunsky coefficients, we find that ∂vn ∂vm = for all n, m ∈ Z. ∂tm ∂tn This gives the integrability condition for the tau function of the dispersionless Toda hierarchy, which is a function τ defined so that ∂ log τ = vn . ∂tn

(4.22)

In the context of matrix models, the function log τ is known as the free energy. Since in this paper, the variables tn , n ∈ Z, are complex-valued, (4.22) does not determine log τ uniquely. However, since the coefficients of g(w) and f (w) and the functions vn , n ∈ Z, depend holomorphically on tn , it is natural to require log τ to be a real-valued function so that ∂ log τ = v¯n . ∂ t¯n + Define the functions (z), z ∈ − 1 and (z), z ∈ 2 so that in a neighbourhood of z = ∞,

(z) =

∞  vn n=1

n

z −n .

(4.23)

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In a neighbourhood of z = 0, (z) =

∞  v−n

n

n=1

zn .

(4.24)

Notice that  (z) = S− (z) + t0 /z and   (z) = − S˜+ (z). Let J1 (z 1 , z 2 ) and J2 (z 1 , z 2 ) be two functions defined so that −

∂ J1 ∂ J2 ∂ 2H (z 1 , z 2 ) = (z 1 , z 2 ) = H(z 1 , z 2 ) (z 1 , z 2 ). ∂z 2 ∂z 1 ∂z 1 ∂z 2

(4.25)

We claim that the tau function of the dispersionless Toda hierarchy is given by τ = |T|2 ,

(4.26)

where T is a holomorphic function of t defined by

 1 t0 v0 + ∂z 1 H(g(w), f (w))g  (w) (g(w)) log T = 2 4πi S 1  + ∂z 2 H(g(w), f (w)) f  (w)( f (w)) dw

  1 J1 (g(w), f (w))g  (w) + J2 (g(w), f (w)) f  (w) dw. + 1 8πi S

(4.27)

Before proceeding to the proof, we would like to comment that Eq. (4.25) only defines the function J1 (z 1 , z 2 ) up to a function of z 1 and the function J2 (z 1 , z 2 ) up to a function of z 2 . If we replace J1 (z 1 , z 2 ) and J2 (z 1 , z 2 ) by J1 (z 1 , z 2 ) = J1 (z 1 , z 2 ) + J˜( z 1 ) and J2 (z 1 , z 2 ) = J2 (z 1 , z 2 ) + J˜2 (z 2 ) respectively, then log T = log T +

1 8πi

∞

1 J˜1 (z)dz + 8πi



J˜2 (z)dz.

0

Namely, log τ  and log τ only differ by a constant independent of t. On the other hand, (4.25) implies that there exists a function J0 (z 1 , z 2 ) so that ∂ J0 (z 1 , z 2 ) = −J1 (z 1 , z 2 ), ∂z 1

∂ J0 (z 1 , z 2 ) = J2 (z 1 , z 2 ). ∂z 2

Consequently,



 J1 (g(w), f (w))g (w)dw = − ∂z 1 J0 (g(w), f (w))g  (w)dw S1 S1

= ∂z 2 J0 (g(w), f (w)) f  (w)dw S1

J2 (g(w), f (w)) f  (w)dw. = S1

Now we prove (4.22).

464

L.-P. Teo

Proposition 4.3. The function τ defined by (4.27) satisfies ∂ log τ = vn , ∂tn

∂ log τ = v¯n , for all n ∈ Z. ∂ t¯n

(4.28)

Therefore it is the tau function of the dispersionless Toda hierarchy. Proof. Since log τ = log T + log T and log T is a holomorphic function of t, it suffices to show that ∂ log T = vn for all n ∈ Z. ∂tn We write the function log T defined by (4.27) as the sum of three terms Z1 , Z2 and Z3 , where Z1 = t0 v0 /2,

 1 ∂z 1 H(g(w), f (w))g  (w) (g(w)) Z2 = 1 4πi S  + ∂z 2 H(g(w), f (w)) f  (w)( f (w)) dw, and Z3 =

1 8πi



 S1

 J1 (g(w), f (w))g  (w) + J2 (g(w), f (w)) f  (w) dw.

It follows immediately from (4.20) that  v0 ∂Z1 2 − t0 b0,0 , = |n| ∂tn t0 b0,n , 2

if n = 0, if n = 0.

(4.29)

For Z2 , a straightforward computation gives  ∂  ∂z 1 H(g(w), f (w))g  (w) (g(w)) + ∂z 2 H(g(w), f (w)) f  (w)( f (w)) ∂tn   ∂g(w) ∂ f (w) ∂

(g(w))+∂z 2 H(g(w), f (w)) ( f (w)) = ∂z 1 H(g(w), f (w)) ∂w ∂tn ∂tn   ∂n f (w) ∂n g(w) + g  (w) f  (w)∂z 1 ∂z 2 H(g(w), f (w)) −  ( (g(w))−( f (w))) f  (w) g (w) + ∂z 1 H(g(w), f (w))g  (w)∂n (g(w)) + ∂z 2 H(g(w), f (w)) f  (w)∂n ( f (w)). Together with (4.8), this implies that

∂Z2 1 = P  (w) ( (g(w)) − ( f (w))) dw ∂tn 4πi S 1 n

 1 ∂z H(g(w), f (w))g  (w)∂n (g(w)) + 4πi S 1 1  + ∂z 2 H(g(w), f (w)) f  (w)∂n ( f (w)) dw. Now the definitions (4.23), (4.24) imply that 

1 0, if n = 0,  P (w) ( (g(w)) − ( f (w))) dw = vn /2, if n = 0. 4πi S 1 n

(4.30)

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On the other hand, the definitions (4.23), (4.24) and the Eqs. (4.16), (2.2) and (2.3) imply that ⎧ −1 ⎪− log g (z) + b0,0 , if n = 0, z ∂ (z) ⎨ = −Pn (g −1 (z)) + z n , if n ≥ 1, ⎪ ∂tn ⎩ −Pn (g −1 (z)) + nbn,0 , if n ≤ −1, ⎧ −1 ⎪ − log f z (z) − b0,0 , if n = 0, ⎨ ∂(z) = −Pn ( f −1 (z)) + nbn,0 , if n ≥ 1, ⎪ ∂tn ⎩ −Pn ( f −1 (z)) + z n , if n ≤ −1. Therefore,

 1 ∂z H(g(w), f (w))g  (w)∂n (g(w)) 4πi S 1 1  + ∂z 2 H(g(w), f (w)) f  (w)∂n ( f (w)) dw   1 v0 /2 + t0 b0,0 + 4πi f (w))P0 (w)dw, if n = 0, S 1 H(g(w),  = 1 vn /2 − |n|t0 bn,0 /2 + 4πi S 1 H(g(w), f (w))Pn (w)dw, if n = 0. Together with (4.29) and (4.30), we find that

∂Z1 ∂Z2 1 + = vn + H(g(w), f (w))Pn (w)dw. ∂tn ∂tn 4πi S 1

(4.31)

Now we consider Z3 . Using (4.25) and (4.8), we have  ∂  J1 (g(w), f (w))g  (w) + J2 (g(w), f (w)) f  (w) ∂tn   ∂g(w) ∂ f (w) ∂ J1 (g(w), f (w)) + J2 (g(w), f (w)) = ∂w ∂tn ∂tn   ∂n g(w) ∂n f (w)   −  + 2H(g(w), f (w)) f (w)g (w)∂z 1 ∂z 2 H(g(w), f (w)) g  (w) f (w)   ∂ ∂g(w) ∂ f (w) J1 (g(w), f (w)) = + J2 (g(w), f (w)) ∂w ∂tn ∂tn  − 2H(g(w), f (w))Pn (w). Therefore, ∂Z3 1 =− ∂tn 4πi

S1

H(g(w), f (w))Pn (w)dw.

Together with (4.31), we conclude that ∂ log τ ∂ log T = = vn , ∂tn ∂tn which is the assertion of the proposition.



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Although it is not obvious from the definition, one can show that if the function H(z 1 , z 2 ) is replaced by the function H(z 1 , z 2 ) + H1 (z 1 ) + H2 (z 2 ), then the tau function τ is changed to τ  , where log τ  and log τ differ by log τ  − log τ =

∞ 

∞ 

dn tn +

n=−∞

d¯n t¯n + C,

n=−∞

which is a linear function of t and ¯t . On the other hand, using the definitions (4.23) and (4.24), one can show that the function Z2 is given by   ∞  1  tn vn + t−n v−n . Z2 = 2 n=1

n=1

Consequently, the holomorphic part of the tau function τ is 

∞  1  1 J1 (g(w), f (w))g  (w) tn vn + T = exp 2 n=−∞ 8πi S 1   + J2 (g(w), f (w)) f  (w) dw . Equations (4.16), (4.20) and (4.28) imply that the generalized Grunsky coefficients bm,n of the pair of univalent functions (g −1 , f −1 ) can be generated by the tau function τ : 1 ∂ 2 log τ , if mn = 0, |mn| ∂tm ∂tn 1 ∂ 2 log τ = , if m = 0, n = 0, |m| ∂tm ∂t0 1 ∂ 2 log τ = , if m = 0, n = 0, |n| ∂t0 ∂tn 1 ∂ 2 log τ =− , if m = n = 0. 2 ∂t02

bm,n = − bm,n bm,n bm,n

(4.32)

Therefore, (2.2) can be rewritten as ∞

 1 ∂ 2 log τ 1 ∂ 2 log τ g −1 (z) =− log − z −m , z 2 ∂t02 m ∂t0 ∂tm m=1 ∞

 1 ∂ 2 log τ 1 ∂ 2 log τ f −1 (z) = log − zm . 2 z 2 ∂t0 m ∂t0 ∂t−m

(4.33)

m=1

This shows that the coefficients of the conformal mappings can be expressed as second partial derivatives of the tau function. For n ≥ 1, define 1 n Bn (w) = (g(w)n )>0 + (g(w)n )0 = Pn (w) − bn,0 , 2 2 1 n B−n (w) = ( f (w)−n )<0 + ( f (w)−n )0 = P−n (w) + b−n,0 . 2 2

Conformal Mappings and Dispersionless Toda Hierarchy II

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Then (2.3) and (4.32) imply that Bn (g −1 (z)) = z n −

∞

 1 ∂ 2 log τ 1 ∂ 2 log τ − z −m , 2 ∂tn ∂t0 m ∂tn ∂tm m=1

Bn ( f −1 (z)) = B−n (g

−1

∂ 2 log τ

1 2 ∂tn ∂t0

∞  1 ∂ 2 log τ m − z , m ∂tn ∂t−m

∂ 2 log τ

m=1 ∞ 

1 (z)) = − − 2 ∂t−n ∂t0

B−n ( f −1 (z)) = z −n

m=1

1 ∂ 2 log τ −m z , m ∂t−n ∂tm

(4.34)

∞

 1 ∂ 2 log τ 1 ∂ 2 log τ + − zm . 2 ∂t−n ∂t0 m ∂t−n ∂t−m m=1

From (4.33) and (4.34), we find that for all n ∈ Z,   ∞  ∂g −1 1 ∂ 2 log τ −m 1 ∂ 3 log τ ∂Bn ◦ g −1 −1 (z) = g (z) − − z (z), = g −1 (z) 2 ∂tn 2 ∂tn ∂t0 m ∂tn ∂t0 ∂tm ∂t0 m=1   ∞  1 ∂ 2 log τ m ∂ f −1 1 ∂ 3 log τ ∂Bn ◦ f −1 −1 −1 (z) = f (z) − z (z) (z). = f ∂tn 2 ∂tn ∂t02 m ∂tn ∂t0 ∂t−m ∂t0 m=1 Therefore, by chain rule, we find that for all n ∈ Z, ∂g(w) ∂g −1 ∂g(w) ∂Bn ◦ g −1 ∂g(w) =− ◦ g(w) = −w ◦ g(w) ∂tn ∂w ∂tn ∂w ∂t0   ∂g −1 ∂g(w) ∂Bn ∂Bn ◦ g(w) ◦ g −1 = −w ◦ g −1 + ∂w ∂t0 ∂w ∂t0 ∂Bn (w) ∂g(w) ∂Bn (w) ∂g(w) =w = {Bn (w), g(w)} . −w ∂w ∂t0 ∂t0 ∂w Similarly, ∂ f (w) = {Bn (w), f (w)} . ∂tn These are precisely the Lax equations of the dispersionless Toda hierarchy (1.2). By setting n = 0 in (4.8) and using the fact that P0 (w) = 1/w, we find that {g(w), f (w)} = wg  (w)

∂ f (w) ∂g(w) 1 , −w f  (w) = ∂t0 ∂t0 ∂z 1 ∂z 2 H(g(w), f (w))

which is the string equation.

(4.35)

468

L.-P. Teo

5. Reduction to Subspaces of D In this section, we consider the reduction of the dispersionless Toda hierarchy to some subspaces of D. First we consider the subspace R of D consists of (g, f ) satisfying g(w) ¯ = g(w),

f (w) ¯ = f (w),

or equivalently, g and f has real coefficients. Assume that H(z 1 , z 2 ) is real-valued when z 1 and z 2 are real. It is easy to check from (4.1), (4.2) and (4.19) that restricted to R, all the variables tn , vn , n ∈ Z, are real-valued. Therefore, the subspace R can be defined by the condition t¯n = tn for all n ∈ Z. In other words, if H(z 1 , z 2 ) is real-valued when z 1 and z 2 are real, the Toda flows ∂/∂tn , n ∈ Z, on D naturally restrict to the subspace R and give rise to solutions of the real-valued dispersionless Toda hierarchy. The tau function is given by T defined in (4.27). Next we consider the subspace  of D consists of (g, f ), where f (w) =

1 g(1/w) ¯

.

(5.1)

In this case, if the function H(z 1 , z 2 ) is defined such that the function   U(z, z¯ ) = H z, z¯ −1 is real-valued, then (4.1), (4.2) and (4.19) show that restricted to , if n ≥ 1,



  1 1 ∂z 1 H g(w), g(w) −1 g(w)−n dg(w) = ∂z U(z, z¯ )z −n dz, tn = 2πin S 1 2πin C1



  1 1 ∂z 2 H g(w), g(w) −1 g(w) −n dg(w) −1 = ∂z¯ U(z, z¯ )¯z −n d z¯ , t−n = 2πin S 1 2πin C1



1 1 ∂z U(z, z¯ )z n dz, v−n = ∂z¯ U(z, z¯ )¯z n d z¯ , vn = 2πi C1 2πi C1 and if n = 0,



1 −1 ∂z U(z, z¯ )dz = ∂z¯ U(z, z¯ )d z¯ , 2πi C1 2πi C1

1 v0 = (∂z U(z, z¯ ) log zdz − ∂z¯ U(z, z¯ ) log z¯ d z¯ ) . 2πi C1 t0 =

(5.2)

It is easy to see that on the space , t−n = −t¯n ,

v−n = −v¯n for n = 0, and t0 = t¯0 , v0 = v¯0 .

Therefore the subspace  is characterized by t−n = −t¯n for all n = 0 and t0 = t¯0 . In particular, t0 is real-valued. Let ∂ ∂ ∂ = − , n ≥ 1, ∂tn ∂tn ∂ t¯−n

∂ ∂ ∂ = + . ∂t0 ∂t0 ∂ t¯0

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These are well-defined vector fields on the subspace . Moreover, since T and vn , n ∈ Z, are holomorphic functions of tn , n ∈ Z, on D, we find that restricted to , ⎧ ⎪ if mn = 0, ⎨−|mn|bm,n , ∂ log T ∂vm ∂vm ∂ log T = = vn , = = |m|bm,0 , if m = 0, n = 0, (5.3) ⎪ ∂tn ∂tn ∂tn ∂tn ⎩−2b , if m = n = 0. 0,0 Moreover, the restriction of the tau function T (4.27) to  is given by

  1 t0 v0 + ∂z U (z, z¯ ) (z)dz − ∂z¯ U (z, z¯ ) (¯z )d z¯ log T = 2 4πi C1

1 + (V1 (z, z¯ )dz + V2 (z, z¯ )d z¯ ) , 8πi C1

(5.4)

where V1 (z, z¯ ) and V2 (z, z¯ ) are defined so that −∂z¯ V1 (z, z¯ ) = ∂z V2 (z, z¯ ) = U(z, z¯ )∂z ∂z¯ U(z, z¯ ). Since U(z, z¯ ) is real-valued, we can choose V1 (z, z¯ ) and V2 (z, z¯ ) such that they satisfy V2 (z, z¯ ) = −V1 (z, z¯ ). Equation (5.4) then shows that log T is real-valued. Equation (5.3) implies that ∂ 2 log T ∂ 2 log T ∂ 2 log T = −mnbm,n , = −mnb−m,−n , = mnbm,−n , ∂tm ∂tn ∂ t¯m ∂ t¯n ∂tm ∂ t¯n (5.5) ∂ 2 log T ∂ 2 log T ∂ 2 log T = nbn,0 , = −nb−n,0 , = 2b0,0 . ∂tn ∂t0 ∂ t¯n ∂t0 ∂t02 We can then show as at the end of Sect. 4 that restricted to , ∂g(w) = {Bn (w), g(w)} , ∂tn

  ∂g(w) = − Bn (w), ¯ g(w) , ∂ ¯tn

for all n ≥ 1. If we further assume that U(z, z¯ ) is regular at z = 0, this is precisely the general conformal mapping problem considered by Zabrodin in [28]. In this case, the functions t0 and v0 (5.2) can be rewritten as 1 1 t0 = ∂z ∂z¯ U(z, z¯ )dzd z¯ , v0 = ∂z ∂z¯ U(z, z¯ ) log |z|2 dzd z¯ . + + π π 1 1 Moreover, (5.4) can be further simplified to   1 t0 v0 + ∂z ∂z¯ U(z, z¯ ) (z) + (¯z ) d 2 z log T = 2 2π +1 1 − U(z, z¯ )∂z ∂z¯ U(z, z¯ )d 2 z 2π +1   ∞ ∞   1 1 tn vn + U(z, z¯ )∂z ∂z¯ U(z, z¯ )d 2 z. = t¯n v¯n − t0 v0 + 2 2π +1 n=1

n=1

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This agrees with the tau function derived in [28] using the electrostatic variational principle. The string equation (4.35) is equivalent to   1  . g(w), g(1/w) ¯ = ∂z ∂z¯ U g(w), g(1/w) ¯ The interest on the dispersionless Toda flows on the space  is partly motivated by its intimate relations with the Dirichlet boundary value problem. This is first observed in [21] and later discussed in detail in [15], for the special solution where U(z, z¯ ) = z z¯ . For general U(z, z¯ ), such a relation also exists. Recall that for a simply connected domain  that contains the ∞, the Dirichlet boundary value problem seeks a harmonic function u(z) on  with prescribed boundary value u 0 (z) on the boundary C = ∂ of . This problem has a unique solution which can be written in terms of the Dirichlet Green’s function G  (z 1 , z 2 ):

1 u(z) = − u 0 (ζ )∂n G  (z, ζ )|dζ |, 2π C where ∂n is the normal derivative on C, and the Dirichlet Green’s function G  (z 1 , z 2 ) can be written in terms of the Riemann mapping G = g −1 :  → D∗ by    G(z 1 ) − G(z 2 )  .  G  (z 1 , z 2 ) = log  G(z 1 )G(z 2 ) − 1  Notice that the relation (5.1) gives F(z) =

1 G (1/¯z )

,

where F = f −1 . Setting F(z) = 1/G(1/¯z ) in (2.1), we find that ∞

log

 G(z) = log β + b0,−n z¯ −n , z¯ n=1

log

G(z)G(z 2 ) − 1 = 2b0,0 + z 1 z¯ 2

∞ 

b−n,0 z¯ 2−n −

n=1

∞ 

bn,0 z 1−n −

n=1

∞  ∞ 

bm,−n z 1−m z¯ 2−n .

m=1 n=1

Together with the other identities in (2.1) and (5.5), we find that        G(z 1 ) − G(z 2 )   = log  1 − 1  + 1 D(z 1 )D(z 2 ) log T, G  (z 1 , z 2 ) = log    z1 z2  2 G(z 1 )G(z 2 ) − 1

(5.6)

where D(z) is the operator D(z) =

∞

∞

n=1

n=1

 z −n ∂  z¯ −n ∂ ∂ + + . ∂t0 n ∂tn n ∂ t¯n

In other words, the tau function T can be used to generate the coefficients of the Dirichlet Green’s function. Notice that (5.6) is independent of the function U(z, z¯ ) which specifies the solution of the dispersionless Toda hierarchy. In other words, it is an universal relation that has to be satisfied for any dispersionless Toda flows that is restricted to the space . It can be regarded as the compact form of the dispersionless Hirota equations [20,22] for the dispersionless Toda hierarchies on .

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6. Special Cases μ

In this section, we consider the special case where H(z 1 , z 2 ) = z 1 z 2−ν , μ, ν are nonzero integers. The relation (3.10) is then equal to ˜ = νLμ L˜ −ν , M = μLμ L˜ −ν , M

(6.1)

or equivalently, Lμ =

˜ L˜ ν M ML−μ , L˜ −ν = . ν μ

This Riemann-Hilbert problem was proposed by Takasaki in [18], and studied by Alonso and Medina in [2]. ˜ (3.1), and setting L(w) = g(w) and L(w) ˜ Using the definition of M and M = f (w), (6.1) implies that μg(w)μ f (w)−ν = μ

νg(w) f (w)

−ν

∞ 

ntn g(w)n + t0 +

n=1 ∞ 

=−

∞ 

vn g(w)−n ,

n=1

nt−n f (w)

−n

+ t0 −

n=1

∞ 

(6.2) v−n f (w) . n

n=1

Multiplying the first equation by g  (w)/g(w) and the second equation by f  (w)/ f (w), subtracting the two resulting equations and integrating with respect to w, we find that g(w)μ f (w)ν =

∞ 

∞

tn g(w)n + t0 log

n=1 ∞ 

−

n=1

g(w)  vn − g(w)−n w n n=1

t−n f (w)

−n

∞

f (w)  v−n + f (w)n . − t0 log w n

(6.3)

n=1

The functions tn , n ∈ Z, are given by



μ ν g(w)μ−1 f (w)−ν dg(w) = g(w)μ f (w)−ν−1 d f (w), t0 = 2πi S 1 2πi S 1

μ g(w)μ−n−1 f (w)−ν dg(w), n ≥ 1, (6.4) tn = 2πin S 1

−ν g(w)μ f (w)−ν+n−1 d f (w), n ≥ 1, t−n = 2πin S 1 and the functions vn , n ∈ Z, are given by

μ g(w)μ+n−1 f (w)−ν dg(w), n ≥ 1, vn = 2πi S 1

−ν g(w)μ f (w)−ν−n−1 d f (w), n ≥ 1, (6.5) v−n = 2πi S 1   

g (w) g(w) f  (w) f (w) 1 1 log −ν log − dw. g(w)μ f (w)−ν μ v0 = 2πi S 1 g(w) w f (w) w w

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The functions J1 (z 1 , z 2 ) and J2 (z 1 , z 2 ) in (4.25) can be chosen to be μ 2μ−1 −2ν J1 (z 1 , z 2 ) = − z 1 z2 , 2

ν 2μ J2 (z 1 , z 2 ) = − z 1 z 2−2ν−1 . 2

Consequently the holomorphic part of log τ is

 ∞ 1  1 μg(w)2μ−1 f (w)−2ν g  (w) log T = tn vn − 2 n=−∞ 16πi S 1  + νg(w)2μ f (w)−2ν−1 f  (w) dw. Using (6.2), (6.4) and (6.5), we find that

1 μg(w)2μ−1 f (w)−2ν g  (w)dw 2πi S 1 

 ∞ ∞  1 n −n = ntn g(w) + t0 + vn g(w) g(w)μ−1 f (w)−ν dg(w) 2πi S 1 n=1 n=1   ∞  1 = ntn vn + t02 , 2 μ n=1

1 νg(w)2μ f (w)−2ν−1 f  (w)dw 2πi S 1 

  ∞ ∞  1 = nt−n f (w)−n + t0 − v−n f (w)n g(w)μ f (w)−ν−1 d f (w) − 2πi S 1 n=1 n=1   ∞  1 = nt−n v−n + t02 . 2 ν n=1

Therefore, log T = −

1 8



  ∞  ∞ n 1 1 2 1 n  1 1  1− + t0 + t0 v0 + tn vn + t−n v−n . 1− μ ν 2 2 2μ 2 2ν n=1

n=1

On the other hand, one can show by integration by parts that



1 1 2μ−1 −2ν  μg(w) f (w) g (w)dw = νg(w)2μ f (w)−2ν−1 f  (w)dw. 2πi S 1 2πi S 1 This gives the nontrivial identity 2ν

∞ 

ntn vn + νt02 = 2μ

n=1

∞ 

nt−n v−n + μt02 .

n=1

When μ = ν ∈ N, the function U(z, z¯ ) = H(z, z¯ −1 ) = |z|2μ

Conformal Mappings and Dispersionless Toda Hierarchy II

473

is real-valued and regular at z = 0. By setting f (w) =

1 g(1/w) ¯

in (6.4) and (6.5), we find that restricted to the subspace ,



μ μ2 −μ t0 = z μ−1 z¯ μ dz = |z|2μ−2 d 2 z = z μ z¯ μ−1 d z¯ , 2πi C1 π 2πi C1 +1

μ μ2 μ−n−1 μ z z¯ dz = − z −n |z|2μ−2 d 2 z tn = 2πin C1 πn − 1

μ z μ z¯ μ−n−1 d z¯ = −t¯−n , =− 2πin C1

μ μ2 vn = z μ+n−1 z¯ μ dz = z n |z|2μ−2 d 2 z 2πi C1 π +1

μ z μ z¯ μ+n−1 d z¯ = −v¯−n , =− 2πi C1

  μ2 μ v0 = z μ−1 z¯ μ log zdz − z μ z¯ μ−1 log z¯ d z¯ = |z|2μ−2 log |z|2 d 2 z. + 2πi C1 π 1 Moreover, the restriction of the tau function T to  is equal to  ∞   t02 t0 v0 1  n  1− + + tn vn + t¯n v¯n , log T = − 4μ 2 2 2μ n=1

agreeing with the result in [28]. Acknowledgements. We would like to thank the anonymous referee for the useful comments.

References 1. Alonso, L.M.: Genus-zero Whitham hierarchies in conformal-map dynamics. Phys. Lett. B. 641, 466–473 (2006) 2. Alonso, L.M., Medina, E.: Solutions of the dispersionless Toda hierarchy constrained by string equations. J. Phys. A 37, 12005–12017 (2004) 3. Alonso, L.M., Medina, E.: Exact solutions of integrable 2D contour dynamics. Phys. Lett. B 610, 277–282 (2005) 4. Alonso, L.M., Medina, E., Manas, M.: String equations in Whitham hierarchies: tau-functions and Virasoro constraints. J. Math. Phys. 47, 083512 (2006) 5. Bauer, M., Bernard, D.: 2D growth processes: SLE and Loewner chains. Phys. Rep. 432, 115–221 (2006) 6. Bertola, M.: Free energy of the two-matrix model/dToda tau-function. Nucl.Phys. B 669, 435–461 (2003) 7. Crowdy, D.: The Benney hierarchy and the Dirichlet boundary problem in two dimensions. Phys. Lett. A 343, 319–329 (2005) 8. Agam, O., Bettelheim, E., Wiegmann, P., Zabrodin, A.: Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime. Phys. Rev. Lett. 88, 236801 (2002) 9. Alonso, L.M., Medina, E.: Exact solutions of integrable 2D, contour dynamics. Phys. Lett. B 610, 277– 282 (2005) 10. Abanov, Ar., Mineev-Weinstein, M., Zabrodin, A.: Multi-cuts solutions of Laplacian growth. http://arxiv. org/abs0812.2622v2[nlin.SI], 2009

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11. Kostov, I.K.: String equation for string theory on a circle. Nucl. Phys. B 624, 146–162 (2002) 12. Kostov, I.K., Krichever, I.M., Mineev-Weinstein, M., Zabrodin, A., Wiegmann, P.B.: The τ -function for analytic curves. In: “Random Matrix Models and Their Applications”, Math. Sci. Res. Inst. Publ., Vol. 40, Cambridge: Cambridge Univ. Press, 2001, pp. 285–299 13. Krichever, I., Mineev-Weinstein, M., Wiegmann, P., Zabrodin, A.: Laplacian Growth and Whitham Equations of Soliton Theory. Physica D 198, 1–28 (2004) 14. Krichever, I., Marshakov, A., Zabrodin, A.: Integrable structure of the dirichlet boundary problem in multiply-connected domains. Commun. Math. Phys. 259, 1–44 (2005) 15. Marshakov, A., Wiegmann, P., Zabrodin, A.: Integrable structure of the Dirichlet boundary problem in two dimensions. Commun. Math. Phys. 227(1), 131–153 (2002) 16. Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: Integrable structure of interface dynamics. Phys. Rev. Lett. 84, 5106–5109 (2000) 17. Mineev-Weinstein, M., Zabrodin, A.: Whitham-Toda hierarchy in the Laplacian growth problem. J. Nonlin. Math. Phys. 8, 212–218 (2001) 18. Takasaki, K.: Dispersionless Toda hierarchy and two-dimensional string theory. Commun. Math. Phys. 170(1), 101–116 (1995) 19. Takasaki, K., Takebe, T.: SDiff(2) Toda equation–hierarchy, tau function, and symmetries. Lett. Math. Phys. 23, 205–214 (1991) 20. Takasaki, K., Takebe, T.: Integrable hierarchies and dispersionless limit. Rev. Math. Phys. 7, 743–808 (1995) 21. Takhtajan, L.A.: Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves. Current Correlation Functions. Lett. Math. Phys. 56, 181–228 (2001) 22. Teo, L.P.: Analytic functions and integrable hierarchies—characterization of tau functions. Lett. Math. Phys. 64(1), 75–92 (2003) 23. Teo, L.P.: Conformal mappings and dispersionless Toda hierarchy. Commun. Math. Phys. 292(2), 391– 415 (2009) 24. Teodorescu, R., Bettelheim, E., Agam, O., Zabrodin, A., Wiegmann, P.: Normal random matrix ensemble as a growth problem. Nucl. Phys. B 704, 407–444 (2005) 25. Ueno, K., Takasaki, K.: Toda Lattice hierarchy. In: “Group Representations and Systems of Differential Equations”. Adv. Stud. Pure Math., Vol. 4, Amsterdam: North Holland, 1984, pp. 1–95 26. Wiegmann, P.B., Zabrodin, A.: Conformal maps and integrable hierarchies. Commun. Math. Phys. 213, 523–538 (2000) 27. Wiegmann, P.B., Zabrodin, A.: Large scale correlations in normal and general non-Hermitian matrix ensembles. J. Phys. A 36, 3411–3424 (2003) 28. Zabrodin, A.: Dispersionless limit of Hirota equations in some problems of complex analysis. Theor. Math. Phys. 12, 1511–1525 (2001) 29. Zabrodin, A.: New applications of non-Hermitian random matrices. Ann. Henri Poincar´e 4(2), S851–S861 (2003) 30. Wiegmann, P.B., Zabrodin, A.: Large N Expansion for Normal and Complex Matrix Ensembles. In: “Frontiers in Number Theory, Physics, and Geometry I”. Berlin-Heidelberg: Springer, 2006, 213–229 31. Zabrodin, A.: Whitham hierarchy in growth problems. Theor. Math. Phys. 142, 166–182 (2005) 32. Zabrodin, A.: Matrix models and growth processes: From viscous flows to the quantum Hall effect. In: “Applications of Random Matrices in Physics”, Nato Science Series, Series II: Mathematics, Physics and Chemistry, Vol. 221, Dordrecht: Springer, 2006, pp. 261–318 33. Zabrodin, A.: Growth processes related to the dispersionless Lax equations. Physica D 235, 101–108 (2007) Communicated by M. Aizenman

Commun. Math. Phys. 297, 475–514 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0943-9

Communications in

Mathematical Physics

Quantum Variance of Maass-Hecke Cusp Forms Peng Zhao Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA. E-mail: [email protected] Received: 15 July 2009 / Accepted: 13 August 2009 Published online: 28 October 2009 – © Springer-Verlag 2009

Abstract: In this paper we study quantum variance for the modular surface X = \H, where  = S L 2 (Z) is the full modular group. We evaluate asymptotically the quantum variance, which is introduced by S. Zelditch and describes the fluctuations of a quantum observable. It is shown that the quantum variance is equal to the classical variance of the geodesic flow on S ∗ X , the unit cotangent bundle of X , but twisted by the central value of the Maass-Hecke L-functions. 1. Introduction Let X = \H be the arithmetic modular surface, where  = S L(2, Z) and H the upper y half plane. On X we have the normalized invariant hyperbolic measure dμ = π3 d xd , y2 and the Beltrami-Laplace operator  2  ∂ ∂2  = y2 . + ∂ x 2 ∂ y2 In addition to the Laplace operator, we have the commuting family of Hecke operators Tn , n ≥ 1; and it is well known that all Tn ’s commute with . For the Maass-Hecke eigenforms, i.e., the Maass form φ satisfies φ + λφ = 0, Tn φ = λφ (n)φ, 1 4

+ t 2 and has a Fourier expansion of the type √  φ(z) = y ρφ (n)K it (2π |n|y)e(nx),

with Laplacian eigenvalue

n=0

where K it is the K -Bessel function and ρφ (n) is proportional to the n th Hecke eigenvalue λφ (n), i.e. ρφ (n) = λφ (n)ρφ (1).

476

P. Zhao

We are mainly interested in the distribution of the probability measures on X , dμ j := |φ j (z)|2 dμ, as j → ∞. The quantum unique ergodicity conjecture (QUE) for X is a deep problem concerning the mass equidistribution of automorphic forms on the modular surface X . Its goal is to understand the limiting behavior of Laplacian eigenfunctions. This is an important problem in mathematical physics and number theory and has its origin from Quantum Chaos, which attempts to understand the relation between classical physics and quantum physics. In the number theory setting, we consider the case of Maass cusp forms associated with large Laplace eigenvalues for X . Schnirelman [30], Colin de Verdiere [4] and Zelditch [36] have shown in general that if the geodesic flow on the unit cotangent bundle of a manifold Y is ergodic, (by a similar definition for X , we form dμ j and dμ on Y ) there exists a full density subsequence {φ jk } of {φ j } which becomes equidistributed as jk → ∞, i.e.   lim ψdμ jk = ψdμ. jk →∞ Y

Y

C0∞ (Y ).

for any Schwartz function ψ ∈ The geodesic flow being ergodic means almost all orbits of the flow become equidistributed. The above result can be viewed as the quantum analogue of the geodesic flow being geodesic. Works of Hejhal-Rackner [12] and Rudnick-Sarnak [27] suggest that there are no exceptional subsequences such that dμ j → dμ as j → ∞, that is called quantum unique ergodicity. More precisely, the quantum unique ergodicity conjecture for X , recently established by the works of Lindenstrauss and Soundararajan, states that: Conjecture 1. For any Jordan measurable compact region A ⊂ X , we have   lim dμ j = dμ. j→∞

This is equivalent to

A



 lim

j→∞ X

A

ψdμ j =

ψdμ X

for any Schwartz function ψ ∈ C0∞ (X ). From Watson’s explicit triple product formula [34], let f be a Maass-Hecke eigenform,   

21 , f ⊗ sym2 φ j ( 21 , f ) 2 | f (z)dμ j | = ,

(1, sym2 f ) (1, sym2 φ j )2 X where is the completed L-function with the infinite part included. The special values L(1, sym2 ·) have not much effect since we have the following effective bounds due to Iwaniec and Hoffstein-Lockhart, for any  > 0: 2  λ− j  L(1, sym φ j )  λ j .

Thus, the subconvexity bound of the degree 6 triple product L-function   1  1 , f ⊗ sym2 ϕ j = o λ 2j L 2

Quantum Variance of Maass-Hecke Cusp Forms

477

would imply the above conjecture. The General Riemann hypothesis would imply the Lindelof Hypothesis   1 L , f ⊗ sym2 ϕ j = O, f (λj ), 2 hence Conjecture 1 holds under GRH. Moreover, it predicts the rate of convergence for the equidistribution  1   − + . f (z)dμ j = O, f λ j 4 X

In [23], Luo and Sarnak proved the quantum unique ergodicity with the above rate on the average, i.e. ∞ (X ), then Theorem 1. Let f ∈ C0,0

  | λ j ≤λ

1

f dμ j |2  λ 2 + X

where the implied constant depends on  and f . For CM forms, the above conjecture also holds by the work of Sarnak [29]. Lindenstrauss [19] has established quantum unique ergodicity for compact arithmetic quotients of H by ergodic methods. In the case of X = \H, he showed the quantum unique ergodicity conjecture is true up to a constant c, where 0 ≤ c ≤ 1. Holowinsky and Soundararajan recently proved the holomorphic version of QUE for X [11]. Combined with the work of Lindenstrauss, Soundararajan [31] has proven the QUE conjecture for X . We will study this equidistribution problem by calculating the quantum variance. This variance sum  Sψ (λ) = |μ j (ψ)|2 λ j ≤λ

is first introduced by Zelditch [36]. It describes the fluctuations of a quantum observable ∞ (X ). For the classical observable ψ ∈ C ∞ (Y ),

O p(ψ)μ j , μ j  = μ j (ψ) for ψ ∈ C0,0 0,0 where Y =  \ S L 2 (R), as studied by Ratner [26], the fluctuations along a generic orbit of the geodesic flow obey the central limit theorem, i.e. the distribution of 1 √ T



T

ψ(Gt (g))dt

0

become Gaussian with mean 0 and the variance given by the non-negative Hermitian ∞ (Y ): form on C0,0  VC (φ, ψ) =

∞



−∞ \S L(2,R)

  φ g

t

e2 0 t 0 e− 2

ψ(g)dgdt.

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P. Zhao

∞ (X ), it can be diagonalized by the Maass cusp forms φ with Restricting VC to C0,0 Laplacian eigenvalue λ = 41 + t 2 and the eigenvalue of VC on φ equals [24]

VC (φ, φ) =

|

1

2π |

4

1 2



|4  . − it |2

−

it 2

It is shown that the means of the classical observable and the quantum observable are the same by the work of Zelditch [36]. There should be a deep relation between the quantum variance and the classical variance. In [7 and 8], the physicists conjectured that for a generic quantum chaotic system, one expects a central limit theorem for the statistical fluctuations of the quantum observable. For the quantum variance, it should correspond to the classical variance VC (ψ, ψ). In [24], Luo and Sarnak examined the analogous quantum variance for holomorphic Hecke eigenforms. The automorphic forms they considered are the holomorphic cusp ∞ (X ), forms in Sk () of even integral weight k for . They proved that for φ, ψ ∈ C0,0 as K → ∞,  

L(1, Sym2 f )μ f (φ)μ f (ψ) ∼ B(φ, ψ)K ,

k≤K f ∈Hk ∞ (X ). It turns out that B(φ, ψ) where B(φ, ψ) is a non-negative Hermitian form on C0,0 has some important symmetric properties. Thus B is diagonalized by the orthonormal basis of Maass-Hecke cusp forms and the eigenvalues of B at ψ j is π2 L( 21 , ψ j ). We will turn to these similar properties for our case in Sect. 3. In this paper, we generalize the result in [24] to Maass cusp forms. In contrast to the case of the holomorphic cusp forms, the real analytic case is more challenging and complicated. New difficulties arise when one applies the Kuznetsov formula instead of the Petersson formula, in view of the occurrence of the continuous spectrum contribution, the subtle analysis of the hypergeometric functions and the sum of Kloosterman sum weighted by the more involved Hankel transform. It turns out that the quantum variance is equal to the classical variance of the geodesic flow on S ∗ X , the unit cotangent bundle of X , but twisted by the central value of the Maass-Hecke L-functions. More precisely, we obtain the following:

Theorem 2. Let ϕ j (z) be the j th Maass-Hecke eigenform, with the Laplacian eigenvalues λ j = 41 + t 2j . For φ, ψ ∈ C(X ), fix  > 0, we have 

 e

t −T 2 − j1− T

+e

−

t j +T T 1−

2

L(1, sym2 ϕ j )μ j (φ)μ j (ψ)

tj 1

= T 1− V (φ, ψ) + O(T 2 + ), V is diagonalized by the orthonormal basis {φ j } of Maass-Hecke cusp forms and the eigenvalue of V at a Maass-Hecke cusp form φ is  L

 1 , φ VC (φ, φ). 2

Quantum Variance of Maass-Hecke Cusp Forms

479

In fact, the weights in the theorem are mildly varying and have no effect on this asymptotic. To prove this theorem, we first establish a preliminary theorem for the incomplete Poincare series with the test functions satisfying some analytic conditions in view of the Kuznetsov formula. We then remove the analyticity condition on the test functions by using the Fourier transform to obtain the general asymptotic formula for the quantum variance sum. For the incomplete Poincaré series, we can take advantage of unfolding the integral and represent the integral as expressions involving the Fourier coefficients in quadratic polynomials, which allow us to apply the Kuznetsov formula and transfer the spectral sum into the sum of Kloosterman sums. The treatment of the non-diagonal terms contribution is very subtle. For the continuous spectrum part, it turns out that it is not negligible and contributes to the main term. The preliminary theorem is as follows. Let h(t) be a smooth function with compact support. Define the Poincaré series (m = 0):  Ph,m (z) = h(y(γ z))e(mx(γ z)). γ ∈∞ \

For the continuous spectrum part, we consider the Eisenstein series E(z, s), and define  d xd y 1 μt (ψ) = ψ(z)|E(z, + it)|2 2 . 2 y X Theorem 3. Let ϕ j denote the eigenfunctions of the Laplacian, with the corresponding eigenvalues λ j = 41 +t 2j . For m 1 , m 2 = 0, h 1 , h 2 smooth functions with compact support for i = 1, 2, with fixed  > 0, we have 

t +T 2  − t j −T 2 − j1− 1− +e T e T L(1, sym2 ϕ j )μ j (Ph 1 ,m 1 )μ j (Ph 2 ,m 2 ) tj



∞

e

+ 0

=T



1−

−

t−T T 1−

2

+e

−

t+T T 1−

2

|ζ (1 + 2it)|2 μt (Ph 1 ,m 1 )μt (Ph 2 ,m 2 )dt

1  B(Ph 1 ,m 1 , Ph 2 ,m 2 ) + O T 2 + .

The Laplacian  and the Hecke operators Tn are self-adjoint with respect to B. Remark 1. (i) The weight e sum to the range

t −T 2 − j1− T

+e

−

t j +T T 1−

2

is in effect localizing the spectrum

|t j − T |  T 1− . (ii) In order to apply the Kuznetsov trace formula, we consider the weighted version sums of the quantum variance in (1.2). The weights L(1, sym2 ϕ j ) are mildly varying. For them, we have the following bounds due to Iwaniec and HoffsteinLockhart: 2  t − j  L(1, sym ϕ j )  t j .

Thus, we can remove these weights using the methods in [16].

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P. Zhao

(iii) The quantum variance V (ψ) is the same as the variance for the fluctuations of the arithmetic measures on the modular surface obtained by collecting all closed geodesics. See the work of Luo, Rudnick and Sarnak [20]. We end the Introduction with an outline of this paper. In Sect. 2, we first establish a preliminary version of Theorem 3 for the incomplete Poincaré series Pm,h with h satisfing some special analytic conditions in order to apply ∞ (X ) the Kutznesov formula. Now these Poincaré series form a dense subspace of C0,0 and the unfolding method allows us to analyze μ j (Pm,h ) in terms of sums over Fourier coefficients. We average over the orthonormal basis for the cuspidal subspace by the Kutznesov formula, and deal with the diagonal, non-diagonal terms separately. Here the subtle test functions arise due to the complicated hypergeometric functions. For non-diagonal terms, we apply the Fourier transform in p. 59 [2], 

J2it (x) − J−2it (x) (y) = −i cos(x cosh(π y)). sinh(π t)

The continuous spectrum parts is not negligible and contributes to the main term. We then remove the analyticity condition on h by using the Fourier transform to obtain the general asymptotic formula for the quantum variance sum Sψ (λ). Sect. 3 and Sect. 4 are devoted to analyze the structure of the quadratic form B. In Sect. 3, we obtain the self-adjointness of B with respect to the Laplacian  and the Hecke operators using the series expression as obtained in Theorem 3. To be precise, we analyze the bilinear form B and using the series expression as obtained in Sect. 2, we prove it is self-adjoint with respect to the Laplacian  and the Hecke operators Tn , n ≥ 1, i.e., for Poincaré series ψ1 , ψ2 , B satisfies the symmetries B(ψ1 , ψ2 ) = B(ψ1 , ψ2 ),

(1)

B(Tn ψ1 , ψ2 ) = B(ψ1 , Tn ψ2 ).

(2)

and for n ≥ 1,

This part is more complicated than the holomorphic case in [24] due to the subtle series ∞ (X ). expression of B. In Sect. 4, we then diagonalize and extend B to C0,0 In Sect. 5, we establish the second part of Theorem 2. We compute the value of B on φ(z), an even Maass-Hecke cuspidal eigenform. After applying Watson’s formula, the quantum variance sum over φ j boils down to averaging L(1/2, φ ⊗ sym2 (φ j )) with respect to the spectrum. By Rankin-Selberg theory, we can express this sum in a suitable series to which the Kuznetsov formula is applied. For the continuous spectrum, it can be shown that it is small enough by Jutila’s subconvexity bound [17]. The resulting asymptotics is used to determine explicitly the quantum variance. Therefore, we complete the proof of Theorem 2.

2. Poincaré Series Let ϕ j denote the j th Hecke-Maass eigenform with the corresponding Laplacian eigenvalue λ j = 41 + t 2j , Hecke eigenvalues λ j (n) and we normalize ϕ j 2 = 1. Let h(t) be

Quantum Variance of Maass-Hecke Cusp Forms

481

an even real analytic function on R satisfying h (n) (t)  (1 + |t|)−N for any n > 0 and any large N , and h(t)  t 10 when t → 0. Define the Poincaré series (m ∈ Z, m = 0):  Ph,m (z) = h(y(γ z))e(mx(γ z)). γ ∈∞ \

We normalize ρ j (n) = cosh(π t j )1/2 a j (n), we have [22], |a j (1)|2 =

2 , a j (n) = a j (1)λ j (n). L(1, sym2 ϕ j )

Thus, we have

⎛

 < Ph,m , |ϕ j |2 > =

\H

|ϕ j (z)|2 ⎝



⎞ h(y(γ z))e(mx(γ z))⎠ dμ(z)

γ ∈∞ \

  1 G(s) −3 2π λ (k)λ (k + m) 2 ds = j j 2 s L(1, sym ϕ j ) (σ ) 2πi (π k) k=0,−m      2s + it j  2s − it j × f (k, t j , s), (3) ( 21 + it j )( 21 − it j ) where 

1

f (k, t j , s) = (1 + m/k)

it j 0

τ

s/2−1

(1 − τ )

s/2−1

 −s/2−it j 2τ m 2 1+ + τ (m/k) dτ. k

Here we also applied the formulas (6.576) and (9.111) in [9]  ∞

m  r dr r s−1 K it j (r )K it j 1 + k 0  s  1

s −3−s it j =2 + it j  − it j (1 + m/k)  τ s/2−1 (1 − τ )s/2−1 2 2 0 −s/2−it j  2τ m + τ (m/k)2 × 1+ dτ. k We can show f (k, t j , s) is even in t j by the integral identities of p.959, 9.131 in [9] since  



m 2 m it j s s + it j , , s, 1 − 1 + 1+ F k 2 2 k  



m it j (s)(−it j ) s m 2 s   F  + it j , , it j + 1, 1 + = 1+ k 2 2 k  2s − it j  2s   



−it j (s)(it j ) m s m 2 s  s  F s . − it j , , −it j + 1, 1 + + 1+ k 2 2 k  2 + it j  2 We also applied the unfolding integral, and Mellin transform:  ∞ dy G(s) = h(y)y −s . y 0

482

P. Zhao

For the Gamma functions in (3) we can get an asymptotic formula so that we can use the Mellin inversion:  σ +i∞ 1 h(y) = G(s)y s ds 2πi σ −i∞ for any σ > 0. By the Stirling Formula, for any vertical strip 0 < a ≤ (s) ≤ b, as t j → ∞, we have     s−1

 2  2s + it j 1 + it j 1 + Os (t −1 = 1 j ) . 2  2 + it j Similarly, we have 

s

2  21 Thus, 

− it j − it j





=

1 − it j 2

 s−1

 2 1 + Os (t −1 j ) .

s

   + it j  2s − it j    + it j  21 − it j  s−1 

  1 2 2

s−1 + tj 1 + Os (t −1 1 + Os (t −1 = j ) = (t j ) j ) , 4

2  21

(4)

as t j → ∞. 2.1. The Kuznetsov formula. The strategy to obtain the asymptotic for the variance is to take a smoothed version of the above sum and apply the Kuznetsov trace formula. From (3), (4) and Mellin inversion, we obtain  π < Ph,m , |μ j |2 >= λ j (k)λ j (k + m)H (t j , k, m) + O(t −1− ), j 4t j L(1, sym2 ϕ j ) k=0,−m

where 

1

H (t j , k, m) = 0

 1+

1+

m k

2τ m k

+

it j τ m2 k2

⎛ ⎞ √ t j τ (1 − τ ) 1 ⎠ dτ. h⎝  2 τ (1 − τ ) π k 1 + 2τ m + τ m k

k2

Thus, by the multiplicativity of Hecke eigenvalues, we have < Ph 1 ,m 1 , |μ j |2 > < Ph 2 ,m 2 , |μ j |2 >         m1 m2 π2 λ j k2 k2 + λ j k1 k1 + = d1 d2 16t 2j L2 (1, sym2 ϕ j ) d |m ,d |m k ,k 1

1

2

2 1

2

× H1 (t j , d1 k1 , m 1 )H2 (t j , d2 k2 , m 2 ) + O(t −2− ) j         m1 m2 π2 k k k k a + a + = j 1 1 j 2 2 d1 d2 32t 2j L(1, sym2 ϕ j ) d1 |m 1 ,d2 |m 2 k1 ,k2

× H1 (t j , d1 k1 , m 1 )H2 (t j , d2 k2 , m 2 ) + O(t −2− ). j

Quantum Variance of Maass-Hecke Cusp Forms

483

Consider  

1−ε 2 1−ε 2 e−((t j −T )/T ) + e−((t j +T )/T ) L(1, sym2 ϕ j ) j≥1

× < Ph 1 ,m 1 , |μ j |2 > < Ph 2 ,m 2 , |μ j |2 >         π2 m1 m2  = a j k1 k1 + a j k2 k2 + ), h(t j )+ O(t −1− j 32 d d 1 2 t d1 |m 1 ,d2 |m 2 k1 ,k2

j

where 1 1−ε 2 1−ε 2  h(t j ) = 2 H1 (t j , d1 k1 , m 1 )H2 (t j , d2 k2 , m 2 )(e−((t j −T )/T ) + e−((t j +T )/T ) ). tj Applying Kuznetsov’s formula [18] to the inner sum, we obtain       m1 m2  a j k1 k1 + a j k2 k2 + h(t j ) d1 d2

 tj



 m m k1 k1 + d 1 ,k2 k2 + d 2

δ =

1

2

π2



∞ −∞

t tanh(π t) h(t)dt −

2 π



∞

0

 h(t)dit (k12 + k1 m 1 /d1 ) |ζ (1 + 2it)|2

2i  −1 × dit (k22 + k2 m 2 /d2 )dt + c S(k12 + k1 m 1 /d1 , k22 + k2 m 2 /d2 ; c) π c ⎛  ⎞  ∞ 4π (k12 + k1 m 1 /d1 )(k22 + k2 m 2 /d2 )  ⎠ t h(t) dt. × J2it ⎝ c cosh(π t) −∞

Here

S(m, n; c) =

 ad≡1( mod c)

 e

dm + an c

is the Kloosterman sum and   d1 it dit (n) = . d2 d1 d2 =n



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P. Zhao

Thus, we have 2 2     − (t j −T )/T 1−ε − (t j +T )/T 1−ε e L(1, sym2 ϕ j ) +e tj

< Ph 1 ,m 1 , |μ j |2 > < Ph 2 ,m 2 , |μ j |2 > ⎛



 δ  ∞ m m π 2  ⎝ k1 k1 + d11 ,k2 k2 + d22 t tanh(π t) h(t)dt = 32 π2 −∞ d1 ,d2 ,k1 ,k2   2 ∞ h(t) − dit (k12 + k1 m 1 /d1 )dit (k22 + k2 m 2 /d2 )dt π 0 |ζ (1 + 2it)|2 2i  −1 c S(k12 + k1 m 1 /d1 , k22 + k2 m 2 /d2 ; c) + π c ⎛  ⎞ ⎞  ∞ 4π (k12 + k1 m 1 /d1 )(k22 + k2 m 2 /d2 )  h(t) ⎠t × J2it ⎝ dt ⎠ , c cosh(π t) −∞

(5)

(6)

Next, we will estimate these three terms respectively.

2.2. The Diagonal Term. Since k1 (k1 + md11 ) = k2 (k2 + md22 ) has at most finitely many solutions if m 1 /d1 = m 2 /d2 , the integer solutions to k1 (k1 + md11 ) = k2 (k2 + md22 ) are only k1 = k2 if md11 = md22 . Thus, the diagonal terms are π2



32 ×

∞

−∞

   1−ε 2 1−ε 2 e− (t j −T )/T + e−((t j +T )/T ) 



t H1 (t, d1 k, m 1 )H 2 (t, d2 k, m 2 )dt,

m 1 /d1 =m 2 /d2 k≥1

where H1 (t, d1 k, m 1 )H 2 (t, d2 k, m 2 )      1 1 m1 m2 1 cos t (2τ − 1) cos t (2η − 1) = d1 k d2 k 0 0 τ η(1 − τ )(1 − η) ⎛ ⎞ ⎛ ⎞ √ √ ⎜ ⎟ ⎜ ⎟ t τ (1 − τ ) t η(1 − η) ⎟ h2 ⎜ ⎟ dτ dη. × h1 ⎜   ⎝ ⎠ ⎝ ⎠ τ m 21 ηm 22 2ηm 2 2τ m 1 π d1 k 1 + d1 k + d 2 k 2 π d2 k 1 + d2 k + d 2 k 2 1

2

(n)

For i = 1, 2, restricting h i on R and h i satisfy h i (t)  (1 + |t|)−N for any n > 0 sufficiently large N and h i (t)  t 10 when t → 0. Thus, h i are continuous uniformly on R. For the sum over k, we estimate it as

Quantum Variance of Maass-Hecke Cusp Forms



485

H1 (t, d1 k, m 1 )H 2 (t, d2 k, m 2 )

k≥1

=

⎞ √ ∞ ⎜ ⎟ t τ (1−τ ) m2 m1 ⎟ t (2τ −1) cos t (2η−1) h 1 ⎜ cos  ⎝ 2⎠ d1 k d2 k 0 2τ m 1 τ m 1 π d1 k 1 + d1 k + d 2 k 2

0

0

⎛ ⎜ × h2 ⎜ ⎝





 1 1

√ t η(1 − η)  2 π d2 k 1 + 2ηm d2 k +





⎛

1

⎞ ηm 22 d22 k 2

⎟ 1 ⎟ dk ⎠ τ η(1 − τ )(1 − η) dτ dη + O(1)

     √ t τ (1 − τ ) m1 m2 = cos t (2τ − 1) cos t (2η − 1) h 1 d1 k d2 k π d1 k 0 0 0  √  t η(1 − η) 1 dk × h2 dτ dη + O(1) π d2 k τ η(1 − τ )(1 − η)



  √   1  1  ∞ cos π m 1 ξ(2τ − 1) cos π m 2 ξ(2η − 1) d1 d2 ξ τ (1 − τ ) t = h1 π 0 0 0 τ η(1 − τ )(1 − η) d1  √  ξ η(1 − η) dξ dτ dη + O(1). × h2 d2 ξ2 

1 1 ∞



Therefore, we obtain the main term of the diagonal term:

π 3/2 64

T (1−)

  m1 m2 d1 = d2

1 1 ∞

0

0

cos

π m1 d1 ξ(2τ

 − 1) cos( πdm2 2 ξ(2η − 1))

τ η(1 − τ )(1 − η)

0

 √   √  ξ τ (1 − τ ) ξ η(1 − η) dξ h2 × h1 dτ dη, d1 d2 ξ2

(7)

i.e.

π 3/2 (1−) T 64

m 1 /d1 =m 2 /d2 0





1

× 0

cos





π m2 d2 ξ(2η

∞ 1

cos

π m1 d1 ξ(2τ

0

  √ − 1) h 2 ξ η(1−η) d2

η(1 − η)

  √ ) − 1) h 1 ξ τd(1−τ 1

τ (1 − τ )

dη

dξ . ξ2

dτ

(8)

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P. Zhao

2.3. The Non-diagonal term. The non-diagonal term is: 

 



m m    S k1 k1 + d11 , k2 k2 + d22 ; c c

d1 |m 1 k1 ,k2 c≥1 d2 |m 2



⎛



m1  4π k k2 + k k + 1 2 1 d1 ⎜ × J2it ⎜ ⎝ c R

m2 d2

⎞ ˜ ⎟ h(t)t ⎟ ⎠ cosh(π t) dt,

where ˜ = 1 H1 (t, d1 k1 , m 1 )H2 (t, d2 k2 , m 2 )(e−((t−T )/T 1−ε )2 + e−((t+T )/T 1−ε )2 ), h(t) t2 ⎛ ⎞ it √  1 1 + mk t τ (1 − τ ) 1 ⎠ dτ ; hj ⎝  H j (t, k, m) = 2 2τ m τ m 2 τ (1 − τ ) 0 1 + k + k2 π k 1 + 2τ m + τ m2 k

k

for j = 1, 2. Let x =

 m m 4π k1 k2 (k1 + d 1 )(k2 + d 2 ) 1

2

c

; the inner integral in the non-diagonal terms is

 J2it (x) − J−2it (x) ˜ 1 IT (x) = h(t) tanh π tdt. 2 R sinh(π t) Since tanh(π t) =sgn(t) + O(e−π |t| ) for large |t|, we can remove tanh(π t) by getting a negligible term O(T −N ) for any N > 0. Next we apply the Parseval identity and the Fourier transform in [2],  J2it (x) − J−2it (x) (y) = −i cos(x cosh(π y)). sinh(π t) By evaluation of the Fresnel integrals, we have −i IT (x) = 2



∞

 e

√  2 xy 1− − 2 −T /T

0

 ×

1 1

cos

m1 d1 k



+e

xy 2 (2τ −1)

√  2  xy 1− − 2 +T /T



cos

m2 d2 k



xy 2 (2η−1)



2 xy

τ η(1−τ )(1−η) ⎞ ⎛ ⎞  √ √ xy xy τ (1 − τ ) η(1 − η) ⎜ ⎟ ⎜ ⎟ 2 2 ⎟ h2 ⎜ ⎟ dτ dη × h1 ⎜   ⎝ ⎠ ⎝ ⎠ 2 2 τ m1 ηm 2 2ηm 2 2τ m 1 π d1 k 1 + d1 k + d 2 k 2 π d2 k 1 + d2 k + d 2 k 2 0

⎛



0



π  dy × cos x − y + √ . 4 πy

1

2

Quantum Variance of Maass-Hecke Cusp Forms

487

Thus, the non-diagonal terms are equal to 

 



m1 m2  −i    S k1 k1 + d1 , k2 k2 + d2 ; c 2

c

d1 |m 1 k1 ,k2 c≥1 d2 |m 2

+e

√  2  xy 1− − +T /T 2

⎛ × h1 ⎝



xy √ τ (1 − τ ) 2

⎠ h2 ⎝

π d1 k1

 e

√  2 xy 1− − 2 −T /T

0

1 1

2 xy 0 ⎞ ⎛

∞

cos

m1 d1 k



0 xy √ η(1 − η) 2

π d2 k2

xy 2 (2τ

⎞



  − 1) cos dm22k x2y (2η − 1)

τ η(1 − τ )(1 − η)

⎠ dτ dη cos(x − y + π ) √dy . 4 πy

(n)

Since both h 1 (t) and h 2 (t) satisfy h i  (1 + |t|)−N for any n > 0 and sufficiently large N , and h i (t)  t 10 when t → 0, the above sum is concentrated on    xy − T    2     T 1−   T 2   1

x yτ (1 − τ )  T k12 x yη(1 − η)   T . k22

T − 10  1

T − 10

Thus we can get the following range: 

xy ∼ T. 2

Note that here x ∼ k1 k2 c−1 , the ranges for k1 , k2 , c are as follows:  21 τ (1 − τ )  k1  T 20 + τ (1 − τ ),   21 T 1− 2 η(1 − η)  k2  T 20 + η(1 − η), 

T 1− 2



1

c  yT 10 . (In the holomorphic case [24], one could get k1 ∼ k2 ∼ T and y  c, so by partial integration, the terms with c  T  contribute O(1).) Here by the above relations and partial integration sufficiently many times, we will get a sufficiently large power of y, k1 and k2 occurring in the denominator, so we get 1 the terms with c  T 10 + contribute O(1). Denote the above sum as    S(k1 (k1 + d1 |m 1 k1 ,k2 c≥1 d2 |m 2

m1 d1 ), k2 (k2

c

+

m2 d2 ); c)

Jk1 ,k2 ,c + O(1).

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P. Zhao

√ xy Making the change of variable t = T 2 , we get Jk1 ,k2 ,c is

2   3  ∞  (t−1) 2 1 22 2(t T )2 π − − − (t+1) − +e T e T sin −x + − √ t x 4 πx 0

 

 1  1 cos m 1 t T (2τ − 1) cos m 2 t T (2η − 1) d1 k d2 k × τ (1 − τ ) η(1 − η) 0 0  √   √  t T τ (1 − τ ) t T η(1 − η) h2 dτ dηdt. ×h 1 π d1 k1 π d2 k2 By Taylor expansion,

 4πi xi = c =

So we can write



f c (k1 , k2 )=ec

×e

k1 k2

m1 k1 + d1



m2 k2 + d2



  m 2 k1 m 1 k2 2k1 k2 + + +··· . d2 d1



Jk1 ,k2 ,c where

2πi c



    m 2 k1 m 1 k2 =  ec − 2k1 k2 + + f c (k1 , k2 ) , d2 d1

3  

2  ∞ (t+1) 2 22 m 1 m 2 m 21 k2 m 22 k1 1 − (t−1) − − − +· · · √ e T − + e T − 2d1 d2 4d12 k1 4d22 k2 t πx 0



   2  1  1 cos m 1 t T (2τ − 1) cos m 2 t T (2η − 1) i 2(t xT ) − π4 d1 k d2 k 

× h1

√

0

0



t T τ (1 − τ ) h2 π d1 k1



τ η(1 − τ )(1 − η) √  t T η(1 − η) dτ dηdt, π d2 k2

2πi z

and we use the notation ec (z) = e c . Reducing the summation over k1 , k2 into congruence classes mod c, we have,           m1 m2 m 2 k1 m 1 k2 , k2 k2 + ; c ec − 2k1 k2 + f c (k1 , k2 ) S k1 k1 + + d1 d2 d2 d1 k1 ,k2 ≥1           m2 m2a m1b m1 ,b b + ; c ec − 2ab + S a a+ + = d1 d2 d2 d1 a,b mod c  × f c (k1 , k2 ) k1 ≡a,k2 ≡b

mod c

      m2 m1 ,b b + ;c S a a+ d1 d2 u,v mod c a,b mod c ⎞ ⎛         m2 m1 ⎝ × ec − 2ab + +u a + +v b f c (k1 , k2 )ec (−uk1 − vk2 )⎠. d2 d1

1 = 2 c





k1 ,k2

Quantum Variance of Maass-Hecke Cusp Forms

489

Apply the Poisson summation for the sum in k1 , k2 and obtain,

   u k1 f c (k1 , k2 )ec (−uk1 − vk2 ) = f c (k1 , k2 )e l1 − c R2 k1 ,k2 l1 ,l2

v  k2 dk1 dk2 . + l2 − c We can assume |u| ≤ 2c , |v| ≤ 2c ; by partial integration sufficiently many times, we get  

u  v  f c (k1 , k2 )ec (−uk1 − vk2 )= f c (k1 , k2 )e − k1 − k2 dk1 dk2 + O(T −A ) c c R2 k1 ,k2

for any A > 1. For (u, v) = (0, 0), by partial integration sufficiently many times, we also obtain  

u v  f c (k1 , k2 )e − k1 − k2 dk1 dk2  T −A , c c R2 1

for any A > 0. Thus only (u, v) = (0, 0) contributes. We can allow c  T 10 + in the 2 c-summation; notice that we have the term kT1 kc2 in f c (k1 , k2 ), so by partial integration sufficiently many times,   f c (k1 , k2 )dk1 dk2  c−A T 2 , R2

for any A > 0. For fixed di , m i (i = 1, 2), denote           m2 m2a m1b m1 ,b b + ; c ec − 2ab + . Sc = S a a+ + d1 d2 d2 d1 a,b

mod c

Thus, the non-diagonal contribution is     Sc    2 f c (k1 , k2 )dk1 dk2 + O(1) c R2 d1 |m 1 c≥1 d2 |m 2

=

 

 

d1 |m 1 c≥1 d2 |m 2



∞

×



e

Sc c2

2 − (t−1) − T

0



1 1

×

cos



  R2

+e

ec

m 2 k2 m 2 k1 m1m2 − 12 − 22 2d1 d2 4d1 k1 4d2 k2

2 − (t+1) − T

m1 d1 k1 t T (2τ



1 i e t

2(t T )2 π x −4

0

√

3

22 √ πx





 − 1) cos dm2 k22 t T (2η − 1)

τ η(1 − τ )(1 − η)    t T η(1 − η) dτ dηdtdk1 dk2 + O(1) × h2 π d2 k2 0



 h1

√  t T τ (1 − τ ) π d1 k1

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P. Zhao

=T

 

 

Sc ζ8 3



∞

×



e

2 − (t−1) − T

0



1 1

×

R2

c2

d1 |m 1 c≥1 d2 |m 2

cos



 

+e

ec

2 − (t+1) − T

m1ξ d1 (2τ −1)



m2ξ m 1 m 2 m 21 φ − 2 − 22 2d1 d2 4d1 ξ 4d2 φ



3

22

√ 3 π (ξ φ) 2

e(ξ φc)

cos

m2φ d2 (2η−1)



 √  ξ τ (1−τ ) h1 π d1

τ η(1−τ )(1−η) 0  √   φ η(1 − η) dτ dηdtdξ dφ + O(1) × h2 π d2   3   2φ 2ξ   m m S 22 m ζ m c 8 1 2 1− 1 2 =T  ec − 2 − 2 3 2d1 d2 4d1 ξ 4d2 φ (ξ φ) 23 R2 c2 d |m c≥1 0

1

1

d2 |m 2



1 1

cos

m1ξ d1 (2τ



 − 1) cos md22φ (2η − 1)

× e(ξ φc) τ η(1 − τ )(1 − η) 0 0  √   φ η(1 − η) dτ dηdξ dφ + O(1). × h2 π d2

 √  ξ τ (1 − τ ) h1 π d1

2.4. The continuous spectrum part. Next, we study the continuous spectrum integrals: 

 

d1 |m 1 k1 ,k2 ≥1 0 d2 |m 2

∞

˜ h(t) dit (k12 + k1 m 1 /d1 )dit (k22 + k2 m 2 /d2 )dt. |ζ (1 + 2it)|2

Recall that |ζ (1 + 2it)| ≥ c log−2/3 (2 + |t|) for some c > 0 [32],   d1 it = O(n  ), dit (n) = d2 d1 d2 =n

and



2 2 t+T 1 − t−T − ˜ = H1 (t, d1 k1 , m 1 )H2 (t, d2 k2 , m 2 ) e T 1− + e T 1− h(t) , t2 ⎛ ⎞ it √  1 1 + mk t 1 τ (1 − τ ) ⎠ dτ hj ⎝  H j (t, k, m) = 2 τ (1 − τ ) 2τ m τ m2 0 1 + 2τ m + τ m2 πk 1 + + k

k

k

k2

for j = 1, 2. The Fourier expansion of E(z, 21 + it) is     1 ∞ 1 1 1 2y 2  1 +it y 2 −it + dit (n)K it (2π ny) cos(2π nx), E z, + it =y 2 +it +ϕ 2 2 ξ(1+2it) n=1

Quantum Variance of Maass-Hecke Cusp Forms

where

 ϕ

491



ξ(2it) , ξ(1 + 2it)

s  s ζ (s). ξ(s) = π − 2  2

1 + it 2

=

Thus, we have ⎞ ⎛   1 < Ph,m , |μt |2 > = |E z, + it |2 ⎝ h(y(γ z))e(mx(γ z))⎠ dμ(z) 2 \H γ ∈∞ \  ∞

 m    r  dr    . = dit (k)dit (k + m) K it j (r )K it j 1+  r h k 2π |k| r 0 



k=0,−m

Then, by the similar calculation in Sect. 2.1, we obtain the continuous spectrum integral is

2 2  

∞

e

−

t−T T 1−

+e

−

t+T T 1−

|ζ (1 + 2it)|2 < Ph 1 ,m 1 , |μt |2 > < Ph 2 ,m 2 , |μt |2 >dt.

0

2.5. The asymptotic formula for the variance. From the above three subsection’s results, we obtain the following: Proposition 1. Let ϕ j denote the j th Maass-Hecke eigenform with Laplacian eigeny . For m 1 , m 2 = 0, values λ j = 41 + t 2j . Denote |μt |2 as the measure |E(z, 21 + it)|2 d xd y2 (n)

let h 1 , h 2 be even analytic functions satisfying h i (t)  (1 + |t|)−N for any n > 0 and sufficiently large N , h i (t)  t 10 when t → 0, i = 1, 2, fix  > 0; we have 

t +T 2

 j  − t j −T 2 − e T 1− + e T 1− L 1, sym2 ϕ j < Ph 1 ,m 1 , |μ j |2 > < Ph 2 ,m 2 , |μ j |2 > tj

 +

∞

 e

−

t−T T 1−

2

+e

−

t+T T 1−

2

|ζ (1 + 2it)|2 < Ph 1 ,m 1 , |μt |2 > < Ph 2 ,m 2 , |μt |2 >dt

0

1  = T 1− B(Ph 1 ,m 1 , Ph 2 ,m 2 ) + O T 2 + , where B(Ph 1 ,m 1 , Ph 2 ,m 2 ) π 3/2 = 64 

1

× 0





∞ 1

cos

π m1 d1 ξ(2τ

  √ ) − 1) h 1 ξ τd(1−τ 1

τ (1 − τ ) 

 √ cos πdm2 2 ξ(2η − 1) h 2 ξ η(1−η) d2 dξ dη 2 η(1 − η) ξ

m 1 /d1 =m 2 /d2 0

dτ

0

(9)

492

P. Zhao

+

 

 

3

R2

c2

d1 |m 1 c≥1 d2 |m 2

2   × e d1 d2 ξ φc



 

Sc ζ8 

ec

m2ξ m2φ m1m2 − 12 − 22 2d1 d2 4d1 φ 4d2 ξ



3

22

3

(ξ φ) 2

1 1

cos (π m 1 d2 ξ(2τ − 1)) cos (π m 2 d1 φ(2η − 1)) τ η(1 − τ )(1 − η) 0 0    × h 1 (ξ d2 τ (1 − τ ))h 2 (φd1 η(1 − η))dτ dηdξ dφ . (10)

Next, we remove the restriction on h 1 and h 2 by using Fourier transformation so that we can obtain a more general asymptotic formula for the quantum variance sum Sψ (λ). For any ϕ(x) smooth function with compact support on R+ , write ϕ(x) = 2 e−x x A ψ(x), where A is a sufficiently large even integer; thus, by Fourier inversion,  ∞  ∞ 2 −x 2 A ˆ ˆ ϕ(x) = e x ψ(y)e(x y)dy = ψ(y)(e(x y)e−x x A )dy, −∞

−∞

where ˆ ψ(y) =



∞ −∞

ψ(x)e(−x y)d x,

ˆ is analytic. h t (x) = e(xt)e−x x A satisfy the condition since ψ(x) ∈ C0∞ (0, ∞), ψ(y) of Proposition 2.1 for each t, so we have 2

t +T  −( t j −T )2 −( j )2 (e T 1− + e T 1− )L(1, sym2 ϕ j ) Ph,m 1 , |μ j |2  Ph,m 2 , |μ j |2 

tj



∞

+

(e

−(

t−T 2 ) T 1−

+e

−(

t+T 2 ) T 1−

0

)|ζ (1 + 2it)|2 Ph 1 ,m 1 , |μt |2  Ph 2 ,m 2 , |μt |2 dt 1

= T 1− B(Ph,m 1 , Ph,m 2 ) + O(T 2 + ). We also have 

Pϕ,m (z) =

ϕ(y(γ z))e(mx(γ z))

γ ∈∞ \

 

=

γ ∈∞ \ ∞

 =

−∞

 =

∞

−∞

ˆ ψ(t)

∞

−∞

 −(y(γ z))2 A ˆ ψ(t)e(t y(γ z))e (y(γ z)) dt e(mx(γ z))

 

2 e(t y(γ z))e−(y(γ z)) (y(γ z)) A e(mx(γ z))dt γ ∈∞ \

ˆ ψ(t)P dt. 2 e(xt)e−x x A ,m

(11)

Thus, we have 

Pϕ,m , |μ j |2  =

∞ −∞

ˆ , |μ j |2 dt. ψ(t) P 2 e(xt)e−x x A ,m

(12)

Quantum Variance of Maass-Hecke Cusp Forms

493

So, for any ϕ1 , ϕ2 smooth with compact support, t +T  −( t j −T )2 −( j )2 (e T 1− + e T 1− )L(1, sym2 ϕ j ) Pϕ1 ,m 1 , |μ j |2  Pϕ2 ,m 2 , |μ j |2 

tj



=

∞



∞

−∞ −∞

ψˆ 1 (t1 )ψˆ2 (t2 )



(e

−(

t j −T 2 ) T 1−

+e

−(

t j +T 2 ) T 1−

)L(1, sym2 ϕ j )

tj

× Pe(xt )e−x 2 x A ,m , |μ j |  Pe(xt )e−x 2 x A ,m , |μ j |2 dt1 dt2 1 2 2  ∞  1∞ 1− = B(Pe(xt )e−x 2 x A ,m , Pe(xt )e−x 2 x A ,m )dt1 dt2 ψˆ 1 (t1 )ψˆ2 (t2 )T 2

1

−∞ −∞

+O(T =T

1−

1 2 −

1

2

2

) 1

B(Pϕ1 ,m 1 , Pϕ2 ,m 2 ) + O(T 2 + ).

For the last equality, we applied the expression of B in Proposition 1. Therefore, we obtain the preliminary version for first part of Theorem 3: Theorem 4. Let ϕ j denote the eigenfunctions of the Laplacian, with the corresponding eigenvalues λ j = 41 +t 2j . For m 1 , m 2 = 0, h 1 , h 2 smooth functions with compact support for i = 1, 2, and fixed  > 0, we have t +T  −( t j −T )2 −( j )2 (e T 1− + e T 1− )L(1, sym2 ϕ j ) Ph 1 ,m 1 , |μ j |2  Ph 2 ,m 2 , |μ j |2 

tj



∞

+ =T

0 1−

(e

−(

t−T 2 ) T 1−

+e

−(

t+T 2 ) T 1−

)|ζ (1 + 2it)|2 < Ph 1 ,m 1 , |μt |2 > < Ph 2 ,m 2 , |μt |2 >dt 1

B(Ph 1 ,m 1 , Ph 2 ,m 2 ) + O(T 2 + ),

where B satisfies the expression in Proposition 2.1. Remark 2. If any incomplete Poincaré series in the above theorem is replaced by incomplete Eisenstein series, i.e. m i = 0 with zero mean, the theorem is still valid. For the case m 1 = m 2 = 0, there is a slight change for B:

√ 

√  ξ τ (1−τ )  1 h 2 ξ η(1−η)   ∞  1 h1 d1 d2 dξ dτ dη 2 τ (1 − τ ) η(1 − η) ξ 0 0 d1 ,d2 ≥1 0



 √ √ ξ η(1−η) ξ τ (1−τ )  1  ∞ 1  h h 1 2 d1 ≥1 d2 ≥1 d1 d2 dξ dτ dη 2 . = τ (1 − τ ) η(1 − η) ξ 0 0 0 By the Euler-MacLaurin summation formula, we have √   √  ∞   ξ τ (1 − τ )  ξ τ (1 − τ ) dα =− h1 b2 (α)H1 , d1 α α2 0 d1 ≥1

where b2 (α) is the Bernoulli polynomial of degree 2, H1 (x) = (h 1 (x)x 2 ) . For the sum over d2 , we have a similar expression.

494

P. Zhao

3. Symmetry Properties of B In this section we analyze the bilinear form B and using the series expression as obtained in the last chapter, we prove it is self-adjoint with respect to the Laplacian  and the Hecke operators Tn , n ≥ 1, i.e., for Poincaré series ψ1 , ψ2 , B satisfies the symmetries B(ψ1 , ψ2 ) = B(ψ1 , ψ2 ),

(13)

B(Tn ψ1 , ψ2 ) = B(ψ1 , Tn ψ2 ).

(14)

and for n ≥ 1,

x be the differential operator on 3.1. Self-adjointness of Laplacian for B. Let L m = L m ∞ C0 (0, ∞) given by

L m h(x) = (x 2

d2 − 4π 2 m 2 x 2 )h(x). dx2

If we define the inner product on C0∞ (0, ∞) by  (h 1 , h 2 ) =

∞

h 1 (x)h 2 (x)

0

dx , x2

then L m is symmetric with respect to (, ), i.e. (L m h 1 , h 2 ) = (h 1 , L m h 2 ). It is easy to check (h(y)e(mx)) = (L m h)(y)e(mx), hence Ph,m = PL m h,m .

(15)

To prove (13), we first deal with the case of m 1 m 2 = 0. We want to show B(PL m 1 h 1 ,m 1 , Ph 2 ,m 2 ) = B(Ph 1 ,m 1 , PL m 2 h 2 ,m 2 ). For the diagonal part, it suffices to prove 

∞ 1

0

0



1

×

cos( πdm1 1 ξ(2τ − 1))L m 1 h 1



 √ ξ τ (1−τ ) d1

τ (1 − τ )

√  cos( πdm2 2 ξ(2η − 1))h 2 ξ η(1−η) d2

0

is symmetric with respect to h 1 , h 2 .

η(1 − η)

dη

dτ

dξ ξ2

Quantum Variance of Maass-Hecke Cusp Forms

495

By integration by parts, ∞ 1

 0



cos( πdm1 1 ξ(2τ − 1))h 1

0





1

×

cos



 √ ξ τ (1−τ ) d1

d12 π m2 d2 ξ(2η

  √ − 1) h 2 ξ η(1−η) d2

η(1 − η)

0

dτ

dηdξ

 √  ) cos( πdm1 1 ξ(2τ − 1))h 1 ξ τd(1−τ 1 =− √ d1 τ (1 − τ ) 0 0 0

√   π m2 cos( d2 ξ(2η − 1))h 2 ξ η(1−η) d2 × dτ dηdξ √ d2 η(1 − η)



 √ √  ∞  1  1 h  ξ τ (1−τ ) h 2 ξ η(1−η) 1 d1 d2 + √ η(1 − η) d τ (1 − τ ) 0 0 0      π m1 π m2 × sin ξ(2τ − 1) cos ξ(2η − 1) d1 d2      π m1 π m2 π m2 π m1 (2τ − 1)+cos ξ(2τ−1) sin ξ(2η−1) (2η−1) dτ dηdξ. × d1 d1 d2 d2 

∞ 1 1



The first integral in the above sum is symmetric with respect to h 1 and h 2 . For the second integral, by integration by parts in terms of τ , it is easy to check the first term including sin( πdm1 1 ξ(2τ − 1)) cos( πdm2 2 ξ(2η − 1)) can be cancelled with the other term in the Laplacian operator L m 1 ,

√   ∞  1 m 2 cos( π m 1 ξ(2τ − 1))h 1 ξ τ (1−τ ) 1 d d 1 1 4π 2 dτ d12 0 0

√   1 cos( π m 2 ξ(2η − 1))h 2 ξ η(1−η) d2 d2 dηdξ. × η(1 − η) 0 For the second term including cos( πdm1 1 ξ(2τ − 1)) sin( πdm2 2 ξ(2η − 1)), by integration by parts in terms of ξ and η, it equals

√  ) π m1 2  ∞  1 cos( ξ(2τ − 1))h 1 ξ τd(1−τ m d 1 1 −π 2 22 dτ τ (1 − τ ) d2 0 0

√   1 cos( π m 2 ξ(2η − 1))h 2 ξ η(1−η) d2 d2 dηdξ. × η(1 − η) 0 It is symmetric since md11 = md22 . For the non-diagonal part, by change of variables ξ→

m2 ξ, d2

φ→

m1 φ, d1

496

P. Zhao

we symmetrize the integral kernel to sin A cos B cos C 3

τ η(1 − τ )(1 − η)(ξ φ) 2

,

where π m1m2ξ π m1m2φ π − − + 2π d1 d2 m 1 m 2 cξ φ, 4 2cd1 d2 φ 2cd1 d2 ξ B = π m 1 m 2 ξ(2τ − 1) and C = π m 1 m 2 φ(2η − 1). A=−

We can verify the following symmetry:     d2 d2 ξ 2 2 −4π 2 m 21 m 22 ξ 2 τ (1−τ ) K (ξ, φ)= φ 2 2 −4π 2 m 21 m 22 φ 2 η(1−η) K (ξ, φ), dξ dφ where K (ξ, φ) =



ξ φ sin A cos B cos C,

and A, B, C as above. To show this, we check the symmetry in the following terms: d2 − 4π 2 m 21 m 22 ξ 2 τ (1 − τ ))K (ξ, η) dξ 2  π m1m2 π m1m2η + + 2π m 1 m 2 d1 d2 ηc)2 = −ξ 2 sin A cos B cos C ξ η(− 2cd1 d2 η 2cd1 d2 ξ 2  π m1m2 π m1m2η + −2ξ 2 cos A sin B cos C ξ η(− + 2π m 1 m 2 d1 d2 ηc) 2cd1 d2 η 2cd1 d2 ξ 2 × π m 1 m 2 (2τ − 1)  π m1m2 π m1m2η η +ξ 2 cos A cos B cos C (− + + 2π m 1 m 2 d1 d2 ηc) ξ 2cd1 d2 η 2cd1 d2 ξ 2  π m1m2η ) +ξ 2 cos A cos B cos C ηξ (− cd1 d2 ξ 3  −ξ 2 sin A cos B cos C ηξ (2π m 1 m 2 (2τ − 1))2  η 2 (2π m 1 m 2 (2τ − 1)) −ξ sin A sin B cos C ξ  −4π 2 m 21 m 22 ξ 2 sin A cos B cos C ηξ τ (1 − τ )  − sin A cos B cos C ξ η.

(ξ 2

By some careful calculation, it can be seen the above terms are symmetric with respect to ξ and η, for example, the sum of the first one and the last two can be canceled and

Quantum Variance of Maass-Hecke Cusp Forms

497

only some symmetric terms left. More precisely, these three parts are   −2π 2 m 21 m 22 ξ 2 sin A cos B cos C ηξ − ξ 2 sin A cos B cos C ξ η  2 π m1m2 π m1m2η × − + 2π m 1 m 2 d1 d2 ηc + 2cd1 d2 η 2cd1 d2 ξ 2   = −2π 2 m 21 m 22 ξ 2 sin A cos B cos C ηξ − ξ 2 sin A cos B cos C ξ η     π 2 m 21 m 22 π m1m2 2 π m1m2η 2 2 + + (2π m m d d ηc) − × 1 2 1 2 2cd1 d2 η 2cd1 d2 ξ 2 2c2 d12 d22 ξ 2 −2π 2 m 21 m 22 +

2π 2 m 21 m 22 η2 ξ2

  = −2π 2 m 21 m 22 (ξ 2 + η2 ) sin A cos B cos C ηξ − π 2 m 21 m 22 ξ 2 sin A cos B cos C ξ η  2 2  1 η 1 2 + + (2d1 d2 ηc) − 2 2 2 2 × 2cd1 d2 η 2cd1 d2 ξ 2 2c d1 d2 ξ  = −2π 2 m 21 m 22 (ξ 2 + η2 ) sin A cos B cos C ηξ − π 2 m 21 m 22 sin A cos B cos C  2 5/2  2 3/2 1 ξ η 1 1 2 5/2 + + (2d1 d2 c) (ξ η) − 2 2 2 . × 2cd1 d2 η3/2 2cd1 d2 ξ 5/2 2c d1 d2 Thus, combining (3.3) and the expression of B in Theorem 2.2, we obtain (3.1) when m 1 m 2 = 0. If m 1 m 2 = 0, it can be shown by using the continuity argument for the desymmetrized integral kernel.

3.2. Self-adjointness of Hecke operators for B. In order to prove (14), we can check it for each Hecke operator T p , where p is a prime, i.e. B(T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B(Ph 1 ,m 1 , T p Ph 2 ,m 2 ). Following the strategy in [24], we can use the fact (see the proof of Theorem 6.9 in [14]) 

Tn Ph,m (z) =

(

d|(m,n)

d2 1 ) 2 Ph( ny ), mn (z), d2 d2 n

(16)

and the explicit evaluation of Sc, m 1 , m 2 (γ ) to verify it. m2

Write

d2

B(Ph 1 ,m 1 , Ph 2 ,m 2 ) = B∞ (Ph 1 ,m 1 , Ph 2 ,m 2 ) + B f (Ph 1 ,m 1 , Ph 2 ,m 2 ), where B∞ (Ph 1 ,m 1 , Ph 2 ,m 2 ) =





d1 |m 1 ,d2 |m 2 0 m 1 /d1 =m 2 /d2

∞ 1 0

h1

 √ ξ τ (1−τ ) d1

τ (1−τ )

 dτ 0

1

h2

 √ ξ η(1−η) d2

η(1−η)

dη

dξ ξ2

498

P. Zhao

and B f (Ph 1 ,m 1 , Ph 2 ,m 2 ) =

 

(

d1 |m 1 c≥1 d2 |m 2



1 1

× 0

0

Sc ζ8



3

c2

2iφηm 2

3

m2φ m2ξ m1m2 22 ec ( − 12 − 22 ) e(ξ φc) 2d1 d2 4d1 ξ 4d2 φ (ξ φ) 23 R2

−

2iξ τ m 1

d1 e d2 h1 τ η(1 − τ )(1 − η)

 √   √  ξ τ (1 − τ ) φ η(1 − η) h2 dτ dηdξ dφ). 2π d1 2π d2

We consider the following 4 cases: (1) (2) (3) (4)

If If If If

p  m 1 m 2 , B∞ (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B∞ (Ph 1 ,m 1 , T p Ph 2 ,m 2 ); p  m 1 m 2 , B f (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B f (Ph 1 ,m 1 , T p Ph 2 ,m 2 ); pa  (m 1 , m 2 ), B∞ (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B∞ (Ph 1 ,m 1 , T p Ph 2 ,m 2 ); pa  (m 1 , m 2 ), B f (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B f (Ph 1 ,m 1 , T p Ph 2 ,m 2 ).

To prove (1), we use the fact 1

T p Ph,m (z) = p − 2 Ph( p·), pm (z) from (16). Also, from the conditions d1 | pm 1 , d2 |m 2 and making the change of variable d1 → pd1 , we have

pm 1 d1

=



h2

m2 d2

we have p|d1 . Thus,

B∞ (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) 1

= p − 2 B∞ (Ph 1 ( p·), pm 1 , Ph 2 ,m 2 ) =p



− 21



∞ 1

d1 |m 1 ,d2 |m 2 0 m 1 /d1 =m 2 /d2

h1



 √ ξ τ (1−τ ) d1

τ (1 − τ )

0

1

dτ 0

 √ ξ η(1−η) d2

η(1 − η)

dη

dξ ξ2

1

= p − 2 B∞ (Ph 1 ,m 1 , Ph 2 ( p·), pm 2 ) = B∞ (Ph 1 ,m 1 , T p Ph 2 ,m 2 ). For (2), we have 1

B f (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = p − 2 B f (Ph 1 ( p·), pm 1 , Ph 2 ,m 2 )  3     Sc ζ8 pm 1 m 2 p 2 m 21 φ m 22 ξ 22 − 12 =p  ec ( − e(ξ φc − ) 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 | pm 1 c≥1 d2 |m 2

2iφηm 2 2iξ τ pm 1  √   √  − d 1 pξ τ (1 − τ ) φ η(1 − η) e d2 h2 dτ dηdξ dφ) × h1 2π d1 2π d2 0 0 τ η(1−τ )(1 − η)   3     Sc ζ8 22 pm 1 m 2 p 2 m 21 φ m 22 ξ − 21 =p  ec − e(ξ φc) − 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 |m 1 c≥1



1 1

d2 |m 2

 × 0

1 1 0

2iφηm 2

−

2iξ τ pm 1

d1 e d2 h1 τ η(1−τ )(1−η)



⎞ √   √  pξ τ (1−τ ) φ η(1 − η) h2 dτ dηdξ dφ ⎠ 2π d1 2π d2

Quantum Variance of Maass-Hecke Cusp Forms

+p

 

− 21

 

Sc ζ8



1 1

× 0

0

2iφηm 2

 

3

c2

d1 |m 1 c≥1 d2 |m 2 −

499 3

ec (

R2

m2ξ m 1 m 2 m 21 φ 22 − 2 −d 22 ) e(ξ φc) 2d1 d2 4d1 ξ 4d2 φ (ξ φ) 23

⎞  √   √  ξ τ (1−τ ) φ η(1−η) h2 dτ dηdξ dφ ⎠. 2π d1 2π d2

2iξ τ m 1

d1 e d2 h1 τ η(1−τ )(1−η)

The above two sums correspond to the conditions p  d1 , and p|d1 respectively. Similarly, we have B f (Ph 1 ,m 1 , T p Ph 2 ,m 2 )   3   2 φ p2 m 2 ξ m 1   S 22 pm ζ m c 8 1 2 −2 1 2 =p  ec − e(ξ φc) − 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 |m 1 c≥1 d2 |m 2

⎞  √   √  ξ τ (1 − τ ) pφ η(1 − η) e h2 dτ dηdξ dφ ⎠ h1 τ η(1 − τ )(1 − η) 2π d 2π d2 1 0 0   3   2φ 2ξ m m 1   S 22 m ζ m c 8 1 2 −2 1 2 +p  ec − e(ξ φc) − 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2

 1 ×

1

2iφηpm 2 2iξ τ m 1 − d d2 1

d1 |m 1 c≥1 d2 |m 2

 1 × 0

1 0

⎞  √   √  ξ τ (1 − τ ) φ η(1 − η) e h1 h2 dτ dηdξ dφ ⎠. τ η(1 − τ )(1 − η) 2π d1 2π d2 2iφηm 2 2iξ τ m 1 d2 − d1

Making the change of variables ξ → ξp , φ → pφ, we can see B f (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B f (Ph 1 ,m 1 , T p Ph 2 ,m 2 ) , i.e., we have proved (2). In the cases (3) and (4), we use the fact 1

1

T p Ph,m (z) = p − 2 Ph( p·), pm (z) + p 2 P

h

where if p  m, we understand that P

h

 · p

, mp

 (z), · m p ,p

(z) = 0.

Thus, for the case (3), we have B∞ (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) 1

1

= p − 2 B∞ (Ph 1 ( p·), pm 1 , ph 2 ,m 2 ) + p 2 B∞ (Ph 1 ( · ), m 1 , Ph 2 ,m 2 ) p

p

= A + B. Similarly, B∞ (Ph 1 ,m 1 , T p Ph 2 ,m 2 )

 1 1 = p − 2 B∞ (Ph 1 ,m 1 , ph 2 ( p·), pm 2 ) + p 2 B∞ Ph 1 , m 1 , P p

= A1 + B1

h2



m2 · p , p



500

P. Zhao

Next, we check that A( p|d1 ) A( p  d1 ) B( p  d2 ) B( p|d2 )

= = = =

A1 ( p|d2 ), B1 ( p  d1 ), A1 ( p  d2 ), B1 ( p|d1 ).

Hence, we get (3). The proof of (4) is the most tedious one and we will use the induction to prove that. We have B f (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) 1

1

= p − 2 B f (Ph 1 ( p·), pm 1 , Ph 2 ,m 2 ) + p 2 B f



 m , h 1 p· , p1

P

 Ph 2 ,m 2 .

From the expression of (10), it equals   3     Sc ζ8 pm 1 m 2 p 2 m 21 φ m 22 ξ 22 − 21 − p  e − e(ξ φc) c 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 | pm 1 c≥1 d2 |m 2

⎞ 2iφηm 2 2iξ τ pm  √   √  − d 1 1 pξ τ (1 − τ ) φ η(1 − η) e d2 × h1 h2 dτ dηdξ dφ ⎠ 2πd1 2πd2 0 0 τ η(1 − τ )(1 − η)   3   1   Sc ζ8 pm 1 m 2 p 2 m 21 φ m 22 ξ 22 − +p 2  e − e(ξ φc) c 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 | pm 1 c≥1 1 1



d2 |m 2

⎞ 2iφηm 2 2iξ τ pm  √   √  − d 1 1 e d2 pξ τ (1 − τ ) φ η(1 − η) × h2 dτ dηdξ dφ ⎠ h1 2πd1 2πd2 0 0 τ η(1 − τ )(1 − η)   3   2m2φ m2ξ p 1   S 22 pm ζ m c 8 1 2 −2 1 2 =p  ec − e(ξ φc) − 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 |m 1 c≥1 

1 1

d2 |m 2

⎞ 2iφηm 2 2iξ τ pm  √   √  − d 1 1 pξ τ (1 − τ ) φ η(1 − η) e d2 h1 h2 dτ dηdξ dφ ⎠ × 2πd1 2πd2 0 0 τ η(1 − τ )(1 − η)   3   1   Sc ζ8 m 1 m 2 m 21 φ m 22 ξ 22 2 − +p  ec − e(ξ φc) 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 |m 1 c≥1 1 1



d2 |m 2



1 1

× 0

0

2iφηm 2

−

2iξ τ m 1

d1 e d2 h1 τ η(1 − τ )(1 − η)

⎞  √   √  ξ τ (1 − τ ) φ η(1 − η) h2 dτ dηdξ dφ ⎠ . 2πd1 2πd2

We denote the above sum as I1 + I2 . Similarly, B f (Ph 1 ,m 1 , T p Ph 2 ,m 2 ) =p

− 21

1 2

B f (Ph 1 ,m 1 , Ph 2 ( p·), pm 2 ) + p B f

 Ph 1 ,m 1 ,

P ·  m2 h2 p , p

 .

Quantum Variance of Maass-Hecke Cusp Forms

501

From the expression of (10), it equals   3     22 pm 1 m 2 p 2 m 21 φ m 22 ξ Sc ζ8 − 21 p  e − e(ξ φc) − c 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 c≥1 d1 | pm 1 d2 |m 2

⎞ 2iφηm 2 2iξ τ pm  √   √  − d 1 1 pξ τ (1 − τ ) φ η(1 − η) e d2 × h1 h2 dτ dηdξ dφ ⎠ 2πd1 2πd2 0 0 τ η(1 − τ )(1 − η)   3   1   Sc ζ8 pm 1 m 2 p 2 m 21 φ m 22 ξ 22 − + p2  e − e(ξ φc) c 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 | pm 1 c≥1 

1 1

d2 |m 2

⎞ 2iφηm 2 2iξ τ pm  √   √  − d 1 1 pξ τ (1 − τ ) φ η(1 − η) e d2 h1 h2 dτ dηdξ dφ ⎠ 2πd1 2πd2 0 τ η(1 − τ )(1 − η)   3     Sc ζ8 pm 1 m 2 p 2 m 21 φ m 22 ξ 22 −  e − e(ξ φc) c 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 |m 1 c≥1

1 1

 × 0

1

= p− 2

d2 |m 2

⎞ 2iφηm 2 2iξ τ pm  √   √  − d 1 1 pξ τ (1 − τ ) φ η(1 − η) e d2 × h1 h2 dτ dηdξ dφ ⎠ 2πd1 2πd2 0 0 τ η(1 − τ )(1 − η)   3   1   Sc ζ8 22 m 1 m 2 m 21 φ m 22 ξ 2  ec − e(ξ φc) − +p 3 2d1 d2 4d12 ξ 4d22 φ (ξ φ) 23 R2 c2 d1 |m 1 c≥1 

1 1

d2 |m 2

1 1

 × 0

0

2iφηm 2

−

2iξ τ m 1

d1 e d2 h1 τ η(1 − τ )(1 − η)

⎞  √   √  ξ τ (1 − τ ) φ η(1 − η) h2 dτ dηdξ dφ ⎠ 2πd1 2πd2

= I I1 + I I2 .

According to whether or not p|(c, ∗, ∗) in Sc,∗,∗ , we can decompose the above sums I1 , I2 , I I1 , I I2 into the following 8 sums: I1 = I11 + I12 ,

I2 = I21 + I22 ,

Note if p|(c, ∗, ∗), Sc,∗,∗ = 0 unless Sc, |m 1 p| , |m 2 | d1

d2

I I1 = I I11 + I I12 ,

p 2 |c.

I I2 = I I21 + I I22 .

Let c = we have   δ( p, c1 ) 2 , = Sc , |m 1 | , |m 2 | p 1 − 1 d p pd2 1 p 2 c1 ;

 − I  where δ( p, c1 ) = 0 if p|c1 ; δ( p, c1 ) = 1 if p  c1 . Hence we can write I11 = I11 11 correspondingly. Similarly we have   δ( p, c1 ) 2 , Sc, |m 1 | , |m 2 | = Sc , |m 1 | , |m 2 | p 1 − 1 2 p pd1 d2 p d1 pd2  − I  , and write I21 = I21 21

Sc, |m 1 | , |m 2 p| = Sc d1

d2

|m 1 | |m 2 | 1 , pd , d 1 2

  δ( p, c1 ) , p2 1 − p

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P. Zhao

 − I I  , and write I I11 = I I11 11

Sc, |m 1 | , |m 2 | = Sc d1

pd2

|m 1 | |m 2 | 1 , pd , 2 1 p d2

  δ( p, c1 ) , p 1− p 2

 − I I  corresponding to p|c or not. and write I I21 = I I21 1 21  + I = I I + I I . By the induction hypothesis on ( mp1 , mp2 ), we have I11 21 11 21 2 We have Scp,a,b = p Sc,a,b and St p2 ,ap,b = 0 if p  bc. Using this and the evaluation of Sc,a,b we can verify that

I12 ( p|d1 ) = I I12 ( p|d2 ), where I12 ( p|d1 ) means the partial sum of I12 in which p|d1 ). Similarly, we have I12 ( p  d1 , p  d2 , p  c) = I I12 ( p  d2 , p  d1 , p  c), I12 ( p  d1 , p  d2 , p  c) = I I12 ( p  d2 , p  d1 , p  c),  ( p  d1 , p 2 |m 2 /d1 ), I12 ( p  d1 , p 2 |d2 , p  c) = I11  I12 ( p  d1 , p 2 |d2 , p  c) = I11 ( p  d1 , p  m 2 /d2 ),  I I11 ( p  d2 , p 2 |m 1 /d1 ) =  I I11 ( p  d2 , p  m 1 /d1 ) =  I11 ( p|d1 ) = I22 ( p|d2 ) = I22 ( p  d2 , p  d1 , p  c) = I22 ( p  d2 , p  d1 , p  c) =

I I12 ( p  d2 , p 2 |d1 , p  c), I I12 ( p  d2 , p  c),  I I11 ( p|d2 ), I I22 ( p|d1 ), I I22 ( p  d1 , p  d2 , p  c), I I22 ( p  d1 , p  d2 , p  c),

 ( p  d2 , p 3 |m 1 /d1 ), I22 ( p  d2 , p 2 |d1 , p  c) = I21  I22 ( p  d2 , p|c) = I21 ( p  d2 , p 2  m 1 /d1 ),   I I21 ( p|d1 ) = I21 ( p|d2 ),  I I22 ( p  d1 , p 2 |d2 , p  c) = I I21 ( p  d1 , p 3 |m 2 /d2 ),  I I22 ( p  d1 , p|c) = I I21 ( p  d1 , p 2  m 2 /d2 ).

Hence we deduce from the above identities that B f (T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B f (Ph 1 ,m 1 , T p Ph 2 ,m 2 ). This completes the proof of B(T p Ph 1 ,m 1 , Ph 2 ,m 2 ) = B(Ph 1 ,m 1 , T p Ph 2 ,m 2 ); for each T p , p is a prime. Thus, the bilinear form B(·, ·) defined on the space spanned by Ph,m ’s is self-adjoint with respect to the Hecke operators Tn , n ≥ 1.

Quantum Variance of Maass-Hecke Cusp Forms

503

4. Extension and Diagonalization of B ∞ (X ), where X =  \ H. In this section, we apply the method in [24] to extend B to C0,0 If we restrict B to the Maass-Hecke cusp forms, it can be shown that B is diagonalized by the orthonormal basis of Hecke-Maass cusp forms. The idea of the extension is to ∞ (X ) can choose a partition of unity subordinate to a covering of X , then any ψ ∈ C0,0 be written as a linear combination of Ph,m , where h is smooth with compact support functions. From the last chapter’s symmetry properties of B, it is easy to see that B can be diagonalized by the orthonormal basis of the subspace of Hecke-Maass cusp forms. Let P be the projection map from H to X , and construct the following open sets with compact closures in H whose projections to X form a locally finite open covering of X : D00 is a neighborhood of i and the restriction of P on D00 is two to one. D01 is a 2πi neighborhood of e 3 and the restriction of P on D01 is three to one:   1 D02 = z|I(z) < 2, |R(z)| < , |z| > 1 , 2   1 D03 = z|I(z) < 2, − ≤ |R(z)| < 0, |z| > 1 2   1 ∪ z|I(z) < 2, −1 < |R(z)| ≤ − , |z + 1| > 1 . 2

For k ≥ 1, let

 k  3 Dk1 = z| < I(z) < 3k+1 , −1 < |R(z)| < 0 , 2   k 1 3 1 k+1 . Dk2 = z| < I(z) < 3 , − < |R(z)| < 2 2 2

Let { f k j } be the partition of unity subordinate to the above covering of X , then each f k j can be viewed as an automorphic function with respect to . We can extend f k, j to a smooth ∞ periodic function f˜k, j on H. There exists y0 > 0 such that f˜k, j are all supported in the half-plane y ≥ y0 , where f˜k, j (z) = f k, j (z), except when k = 0 and j = 2, 3. ∞ (X ); we have Let ψ ∈ C0,0  ψ(z) = f k, j (z)ψ(z), k, j

and f k, j (z)ψ(z) =

1 n k, j



f˜k, j (γ z)ψ(γ z),

γ ∈∞ \

where n k, j = 2 if k = 0, and j = 0; n k, j = 3 if k = 0, and j = 1; n k, j = 1, if otherwise. Expanding f˜k, j (z)ψ(z) into the Fourier series in x, we have  h k, j,m (y)e(mx), f˜k, j (z)ψ(z) = m∈Z

504

P. Zhao

where h k, j,m (y) are smooth with compact support and satisfy the rapid decay condition h k, j,m (y)  A y −A |m|−A for any A > 0. Thus, we have the following decomposition: ψ(z) =

 1 Ph ,m (z) n k, j k, j,m k, j m∈Z

 1  1 2 = Ph k, j,m ,m (z) + Ph k, j,0 ,0 (z) − Ph 0,0,0 ,0 (z) − Ph 0,1,0 ,0 (z) n k, j 2 3 k, j m=0

k, j

 1 1 2 = Ph k, j,m ,m (z) + PH,0 (z) − Ph 0,0,0 ,0 (z) − Ph 0,1,0 ,0 (z), n k, j 2 3 k, j m=0

where H (y) ∈ Cc∞ (0, ∞) is defined as H (y) =



h k, j,0 (y).

k, j

 This follows from the fact that k, j f k, j (z) = 1, and f˜k, j (z) are all supported in the half-plane y ≥ y0 , where we have  1     h k, j,0 (y) = f˜k, j (z) ψ(z)d x 0

k, j

 =

k, j 1 

0

 =

 f k, j (z) + ( f˜0,2 (z) − f 0,2 (z)) + ( f˜0,3 (z) − f 0,3 (z)) ψ(z)d x

k, j 1

(( f˜0,2 (z) − f 0,2 (z)) + ( f˜0,3 (z) − f 0,3 (z)))ψ(z)d x,

0

since

1 0

ψ(z)d x = 0. Moreover we have  1 2 (PH,0 (z) − Ph 0,0,0 ,0 (z) − Ph 0,1,0 ,0 (z))dμ = 0. 2 3 X

Let 1 2 Ph ,0 (z) − Ph 0,1,0 ,0 (z) = Ph,0 (z), 2 0,0,0 3

PH,0 (z) − with

1 2 h = H − h 0,0,0 − h 0,1,0 . 2 3 We have ψ(z) =

 1 Ph ,m (z) + Ph,0 (z), n k, j k, j,m k, j m=0

Quantum Variance of Maass-Hecke Cusp Forms

with

505

 Ph,0 (z)dμ = 0. X

∞ (X ), we have Thus, it follows from Theorem 4 that for ψ and φ in C0,0



e

tj

−(

 +

=T

t j −T 2 ) T 1−

∞

e

0 1−

−(

+e

−(

t j +T 2 ) T 1−

t−T 2 ) T 1−

+e



−(

L(1, sym2 ϕ j ) ψ, |μ j |2  φ, |μ j |2 

t+T 2 ) T 1−

 |ζ (1 + 2it)|2 < ψ, |μt |2 > < φ, |μt |2 >dt

1

B(ψ, φ) + O(T 2 + ),

where 

B(ψ, φ) =

k1 , j1 ,m 1 =0;k2 , j2 ,m 2 =0

+



k1 , j1 ,m 1 =0

+

1 B(Ph k1 , j1 ,m 1 ,m 1 , Ph k2 , j2 ,m 2 ,m 2 ) n k1 , j1 n k2 , j2

1 B(Ph k1 , j1 ,m 1 ,m 1 , Ph φ ,0 ) n k1 , j1



1

k2 , j2 ,m 2 =0

n k2 , j2

B(Ph ψ ,0 , Ph k2 , j2 ,m 2 ,m 2 )

+B(Ph ψ ,0 , Ph φ ,0 ).

(17)

Since for φ and ψ we have the rapid decay condition h k, j,m (y)  A y −A |m|−A for any A > 0, we can see the above series converges abosolutely. From the last section’s results it follows that the bilinear form B(ψ, φ) is defined on ∞ (X ) × C ∞ (X ) and satisfies C0,0 0,0 B(ψ, φ) = B(ψ, φ), and for n ≥ 1, B(Tn ψ, φ) = B(ψ, Tn φ). If we restrict B to the Maass-Hecke cusp forms, we can show the continuous spec1 trum integral contributes O(T 2 − ), see Sect. 5.2.3. Hence, we obtain the first part of Theorem 2. Now restrict B to the Maass-Hecke cusp forms, it is easy to see B is diagonalized by the orthonormal basis of this subspace from the symmetry properties of B. If ψ, φ are two distinct Maass-Hecke cusp forms, then for n ≥ 1, B(Tn ψ, φ) = B(ψ, Tn φ). We have λn (ψ)B(ψ, φ) = λn (φ)B(ψ, φ).

506

P. Zhao

Since there must be an n such that λn (ψ) = λn (φ) for ψ and φ, we obtain B(ψ, φ) = 0 for distinct Maass-Hecke cusp forms. Thus, we have shown that B is diagonalized by the orthonormal basis of the subspace of the Maass-Hecke cusp forms. We are going to calculate the eigenvalue of B on such a Maass-Hecke cusp form in the next section. 5. Eigenvalue of B Let φ(z) be an even Maass-Hecke cuspidal eigenform for the modular group , with the Laplacian eigenvalue λφ = 41 + tφ2 . We consider the original asymptotic 

 e

t −T 2 − j1− T

+e

−

t j +T T 1−

2

L(1, sym2 φ j )| μφ j , φ|2 .

j≥1

After applying Watson’s formula on | μφ j , φ|2 , the quantum variance sum over φ j boils down to averaging L(1/2, φ ⊗ sym2 (φ j )). By Rankin-Selberg theory, we can express this sum in a suitable series to which the Kuznetsov formula is applied. 5.1. Watson’s formula. Let L(s, φ) be the associated standard L-function, which admits analytic continuation to the whole complex plane and satisfies the functional equation:     s − itφ s + itφ −s

φ (s) := π   L(s, φ) = φ (1 − s). 2 2 Moreover, we have  s 

s  s  + itφ  − itφ L(s, sym2 (φ)),

sym2 (φ) (s) = π −3s/2  2   2     2 s + itψ s + itψ s + itψ −3s + itφ  − itφ 

ψ⊗sym2 (φ) (s) = π  2 2 2       s −itψ s −itψ s −itψ + itφ  − itφ  L(s, ψ ⊗ sym2 (φ));  2 2 2 here ψ(z) is also a Maass-Hecke cuspidal eigenform. By Watson’s formula [34], we have | μφ , ψ|2 =

ψ⊗sym2 (φ) ( 21 ) ψ ( 21 )

sym2 (φ) (1)2 sym2 (ψ) (1) it

L( 21 , ψ)L( 21 , ψ ⊗ sym2 (φ)) cosh(π tψ )|( 41+ 2ψ )|4 |aφ (1)|2 = (1+O(tφ−1 )) tφ L(1, sym2 (φ))L(1, sym2 (ψ)) it

L( 21 , ψ ⊗ φ ⊗ φ)|( 41 + 2ψ )|4 |aφ (1)|2 (1 + O(tφ−1 )). = tφ L(1, sym2 (φ))

Quantum Variance of Maass-Hecke Cusp Forms

507

5.2. The Kuznetsov formula. Next, we compute 

 e

t −T 2 − j1− T

+e

−

t j +T T 1−

2

L(1, sym2 φ j )| μφ j , ψ|2 .

j≥1

Let  be the cuspidal automorphic form on G L(3) which is the Gelbart-Jacquet lift of the cusp form φ, with the Fourier coefficients a (m 1 , m 2 ) [3], where 

a (m 1 , m 2 ) =

λ

 m2 , 1 λ ( , 1)μ(d), d d

m

d|(m 1 ,m 2 )

1

and 

λ (r, 1) =

λφ (t 2 ).

s 2 t=r

The Rankin-Selberg convolution L(s, ψ ⊗ sym2 (φ)) is represented by the Dirichlet series, 

L(s, ψ ⊗ sym2 (φ)) =

λψ (m 1 )a (m 1 , m 2 )(m 1 m 22 )−s ,

m 1 ,m 2 ≥1

where λψ (r ) is the r th Hecke eigenvalue of ψ. Since 1

ψ⊗sym2 (φ) (1/2) = πi

 (2)

ψ⊗sym2 (φ) (s + 1/2)

ds , s

we have the following approximate functional equation: 



L(1/2, ψ ⊗ sym (φ j )) = 2 2

λψ (m 1 )a j (m 1 , m 2 )(m 1 m 22 )−1/2 V

m 1 ,m 2 ≥1

m 1 m 22



t 2j

where 

γ (1/2 + s, ψ ⊗ sym2 (φ j )) ds , γ (1/2, ψ ⊗ sym2 (φ j )) s (2)       s + itψ s + itψ s + itψ + it j  − it j  γ (s, ψ ⊗ sym2 (φ j )) = π −3s  2 2 2       s − itψ s − itψ s − itψ + it j  − it j  .  2 2 2 V (y) =

1 2πi

y −s

,

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P. Zhao

Thus, we have 

t +T 2  − t j −T 2 − j1− 1− +e T e T L(1, sym2 φ j )| μφ j , ψ|2 j≥1

 =L

t

    − e 1 1 1 , ψ | + it |4 2 4 2

j −T T 1−

2

+e tj

t j ≥1

×(1/2, ψ ⊗ sym (φ j ))

−

t j +T T 1−

2

|a j (1)|2 L

2

 =L

t

    − 1 1 e 1 , ψ | + it |4 2 4 2

j −T T 1−

+e tj 

t j ≥1



×

−

2

λψ (m 1 )a j (m 1 , m 2 )(m 1 m 22 )−1/2 V

t j +T T 1−

2

|a j (1)|2

m 1 m 22



t 2j

m 1 ,m 2 ≥1

     3n n2   4 d 1 1 μ(d) 1 1 2 + it  λψ (dn 1 )V = L , ψ  (n 1 n 22 )−1/2 3 2 2 4 2 t d2 j 

t j ≥1 d≥1

t −T 2 − j1−

−

n 1 ,n 2 ≥1

2

t j +T

+ e T 1− × |a j (1)|2 λ j (n 1 , 1)λ j (n 2 , 1) tj        4 1 1 μ(d) 1  ,ψ  + it  =L λψ (ds12 t1 )V 3 2 4 2 d2 e

T

 ×

d 3 s12 t1 s24 t22

 =L

×



t j ≥1 d≥1 s1 ,s2 ,n 1 ,n 2 ≥1

t −T 2

t +T 2 j − 1− − j1− T e +e T |a j (1)|2 λ j (t12 )λ j (t22 ) (s12 t1 s24 t22 )−1/2 tj

t 2j     1 1 1 μ(d) , ψ | + it |4 3 2 4 2 d2



 V

d 3 s12 t1 s24 t22



e

d≥1

t −T 2 − j1− T

t 2j

t j ≥1

+e tj



λψ (ds12 t1 )(s12 t1 s24 t22 )−1/2

s1 ,s2 ,n 1 ,n 2 ≥1

t +T 2 − j1− T

|a j (1)|2 λ j (t12 )λ j (t22 ).

For the inner sum, by the Kuznetsov formula, we have 

 V

d 3 s12 t1 s24 t22 t 2j

t j ≥1

δ(t1 , t2 ) = π2 −



2 π





−∞

 V

0



∞

∞

e

V

t −T 2 − j1− T

−

+e tj 

d 3 s12 s24 t1 t22 t2

d 3 s12 t1 s24 t22 t2



e

−

t j +T T 1−

e t−T T 1−

− 2

2

|a j (1)|2 λ j (t12 )λ j (t22 )

t−T T 1−

2

−

+e t+T T 1−

+e t|ζ (1 + 2it)|2

−

t+T T 1−

2

tanh(π t)dt

2

dit (t12 )dit (t22 )dt

Quantum Variance of Maass-Hecke Cusp Forms

509

   3 2 4 2  S(t 2 , t 2 ; c)  ∞ d s 1 s 2 t1 t2 4π t1 t2 1 2 V + J2it c c t2 −∞ c≥1 

2 2 dt − t−T − t+T 1− 1− T T × e +e . cosh(π t) Next, we will estimate the above three sums respectively. 5.2.1. The diagonal term The diagonal term is

2   ∞  t−T 2     1 1 1 − 1− − t+T 4 1− , ψ | + itψ | +e T s2−2 e T L 2 4 2 −∞ s2 ≥1  2 4  s1 s2 × s1−1 λψ (s12 )V tanh(π t)dt. T2 s1 ≥1

For the sum over s1 , we have   −s   2s4 2)  s s24 λ (s ds 1 ψ −1 1 2 1 , s1 λψ (s12 )V U (s) = t 2s+1 2 T2 2πi (2) T s s s ≥1 s ≥1 1 1

1

where Ut (s) = (1 + Pt (s))





s+1/2−itψ 2



1/2−itψ 2

 

 



s+1/2+itψ 2 1/2+itψ 2



 ,

and Pt (s) =

 N +1   pr +1 (s) |s| + O tr tN

1≤r ≤N

is an analytic function in Rs ≥ −2. pr +1 (s) is a polynomial of degree at most r + 1. Also, we have  λψ (s 2 ) 1 1 L(s, sym2 ψ). = s s1 ζ (2s)

s1 ≥1

Thus, moving the line of integration in the sum over s1 to R(s) = −1/4 + , we get   s12 s24 1 −1 2 L(1, sym2 ψ) + O(T −1/2+ ). s1 λψ (s1 )V = T2 ζ (2) s1 ≥1

Therefore, we get the diagonal terms contribute as follows:     4  1 1 π 1− 1 2  T ,ψ  + itψ  + O(T 1/2+ ). L(1, sym ψ)L 4 2 4 2

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P. Zhao

5.2.2. The Non-diagonal term Next, we estimate the non-diagonal term:  μ(d) d≥1



d

1 2



∞

−∞

 

J2it

4π t1 t2 c ,

λψ (ds12 t1 )(s12 t1 s24 t22 )−1/2

s1 ,s2 ,t1 ,t2 ≥1

∞ −∞

Let x =

3 2

4π t1 t2 c



 V

d 3 s12 t1 s24 t22 t2

 e

 S(t 2 , t 2 ; c) 1 2 c c≥1

−

t−T T 1−

2

+e

−

t+T T 1−

2

dt . cosh(π t)

and the inner integral in the non-diagonal terms is

J2it (x) − J−2it (x) V sinh π t



d 3 s12 t1 s24 t22 t2

 e

−

t−T T 1−

2

+e

−

t+T T 1−

2

tanh(π t)dt.

Since tanh(π t) = sgn(t) + O(e−π |t| ) for large |t|, we can remove tanh(π t) by getting a negligible term O(T −N ) for any N > 0. Applying the Parseval identity, the Fourier transform in [2], 

J2it (x) − J−2it (x) (y) = −i cos(x cosh(π y)), sinh(π t)

and the evaluation of the Fresnel integrals, the integral is   

2 2 ∧    d 3 s12 t1 s24 t22 1 ∞ J2it (x) − J−2it (x) ∧ − t−T − t+T T 1− T 1− (y) V + e (y)dy e 2 −∞ sinh π t t2   

2 2 ∧  d 3 s12 t1 s24 t22 −i ∞ − t−T − t+T 1− 1− T T (cos(x cosh(π y))) V +e (y)dy = e 2 −∞ t2

2 ∧     3 2 4 2  t−T 2  d s 1 t 1 s 2 t2 1 2 2 −i ∞ − 1− − t+T 1− T T cos(x + π x y ) +e (y)dy = e V 2 −∞ 2 t2 

 

2 2  d 3 s12 t1 s24 t22 π  −i ∞

− t−T − t+T 1− 1− T T cos(x − y + ) V = +e e 2 0 4 t2   dy xy × √ 2 πy   x y 2 ⎞⎞ ⎛ ⎛   x y 2  2 −T 2 +T  ∞

− − 2 4 2 3  ⎟⎟ dy 4d s1 t1 s2 t2 ⎜ π ⎜ −i T 1− T 1− ⎜ ⎜ ⎟⎟ √ V cos x − y + +e = ⎝e ⎠⎠ π y 2 0 4 ⎝ xy

Quantum Variance of Maass-Hecke Cusp Forms

511

⎛

⎛ ⎜  ⎜  4d 3 s12 t1 s24 t22 −i ∞

π ⎜ −1 cos(4π t1 t2 c − y + ) ⎜ = V ⎜ 2 0 4 ⎜ 4π t1 t2 c−1 y ⎝ ⎛ ⎜ −⎝

4π t1 t2 c−1 y +T 2 T 1−

+ e

⎛

⎜ ⎜ ⎜ −⎝

⎜ ⎜e ⎜ ⎜ ⎝

4π t1 t2 c−1 y −T 2 T 1−

⎞2 ⎟ ⎠

⎞2 ⎞⎞ ⎟ ⎟⎟ ⎠ ⎟⎟

⎟⎟ dy ⎟⎟ √ . ⎟⎟ π y ⎟⎟ ⎠⎠

Here the entire equation is up to an error of O(T −N ). Thus, the non-diagonal terms are concentrated on T 2 − T 2−  t1 t2 c−1 y  T 2 . So, we can assume d 3 s12 t1 s24 t22  T 2+ since V (ξ ) has exponential decay as ξ → ∞. By partial integration, the terms with c  T  and t1 t2  T 2−4 contribute O(1). So we 2−4 , therefore we have t  T 5 , also the sum over can assume c  T  and t1 t2  2 √T 2π t1 t2 c−1 y s1 and s2 converges. Let t = ; the inner integral is T √  ∞

 T c − 2 − 2 e−((t−1)/T ) + e−((t+1)/T ) 2π t1 t2 0 

4d 3 s12 t1 s24 t22 π  −1 2 cos(4π t1 t2 c − (t T ) c/(2π t1 t2 ) + ) V dt. 4 t2T 2 From Hecke’s bound



λψ (r )r −1/2  R  ,

r ≤R

where α ∈ R and the Hecke relation λψ (r1r2 ) =



μ(d)λψ (r1 /d)(r2 /d)

d|(r1 ,r2 )

and partial summation, we get the non-diagonal terms contribute O(T 5 ). 5.2.3. The continuous spectrum part To evaluate the continuous part, we need to rewrite  μ(d) d≥1

d 

×

3 2

 s1 ,s2 ,t1 ,t2 ≥1



∞

V 0

λψ (ds12 t1 )(s12 t1 s24 t22 )−1/2

d 3 s12 t1 s24 t22 t2



e

−

t−T T 1−

2

−

t+T T 1−

+e t|ζ (1 + 2it)|2

2

dit (t12 )dit (t22 )dt

with respect to the L-function, then we can use some subconvexity bounds of the L-function to show the continuous part is also small.

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The Eisenstein series E(z, s) has Fourier expansion  φ(n, s)Ws (nz), E(z, s) = y s + φ(s)y 1−s + n=0

where Ws (z) is the Whittaker function given by 1

Ws (z) = 2|y| 2 K s− 1 (2π |y|)e(x) 2

and K s (y) is the K -Bessel function,

  √  s − 21 ζ (2s − 1) φ(s) = π (s)ζ (2s)

with ζ (s) the Riemann zeta function and 1

φ(n, s) = π s (s)−1 ζ (2s)−1 |n|− 2 η(n, s) with η(n, s) =

 a s− 12 . d

ad=n

Corresponding to the continuous spectrum part, i.e., for the Eisenstein series, we can take advantage of unfolding integral,   1 d xd y ϕ j (z)E z, + it E(z, s) 2 2 y X    ∞ 1 1 d xd y = ϕ j (z)E z, + it y s 2 2 y 0 0     ∞ 1  ∞ 1 1 1 1 2 = λ j (n)K it j (2πny) cos(2πnx) y 2 +it + ϕ y + it y 2 −it 2 0 0 n=1 1 ∞  2y 2 d xd y + n it σ−2it (n)K it (2πny) cos(2πnx)y s ξ(1 + 2it) y2 n=1  ∞   d xd y  ∞ 1 = λ j (n)K it j (2πny)n it σ−2it (n)K it (2πny) y s ξ(1 + 2it) 0 y n=1  ∞ it   λ j (n)n σ−2it (n)  ∞ 1 d xd y = K it j (2π y)K it (2π y)y s ξ(1 + 2it) ns y 0 n=1 







s+it j +it s+it j −it s−it j +it s−it j −it    L(ϕ j , s − it)L(ϕ j , s + it) 2π −s  2 2 2 2 = . ζ (2s) ξ(1 + 2it) (s) 

I j (s) =

Thus,

ϕ j , μt  = 2π

−2it

|



1 4

+

it j 2



|2 

|ζ (1 + 2it)|2  1 L ϕ j , − 2it . 2 



1 4

−

it j 2



− it  41 +   | 21 + it |2

it j 2

− it





1 L ϕj, 2



Quantum Variance of Maass-Hecke Cusp Forms

513

Therefore, the continuous part contributes

2 2  

∞

e 0

|



1 4

−

−

t−T T 1−

itψ 2

+e

−

t+T T 1−



− it  41 +   | 21 + it |4

itψ 2

1 |L |ζ (1 + 2it)|2  − it |2 dt.



 1 + it, ψ |2 2

By the Stirling formula and Jutila’s bound, the subconvex bound,   L 21 + it, ψ j  (κ j + t)1/3+ , 1

we obtain the continuous part contributes O(T 2 + ). 6. Conclusion Combining last section’s results, we conclude that 

t +T 2 j  − t j −T 2 − e T 1− + e T 1− L(1, sym2 φ j )| μφ j , ψ|2 j≥1

=

π 1− T L(1, sym2 ψ)L 4



   4  1 1 1 , ψ  + itψ  + O(T 1/2+ ). 2 4 2

Since L(1, sym2 ψ) = 2 < ψ, ψ > cosh(π tψ ), we obtain the eigenvalue of B at ψ is  L

1 ,ψ 2

 |



itψ 2



|4  .  2π | 21 − itψ |2 1 4

−

Therefore, we complete the proof of Theorem 2. Acknowledgements. This is a part of the author’s Ph.D thesis and he would like to thank his advisor, Professor Wenzhi Luo, for his guidance and encouragement. The author would like to thank Professor Peter Sarnak, Professor Steve Zelditch and Professor Zeev Rudnick for their interest in this work.

References 1. Artin, E.: Ein mechanisches System mit quasi-ergodischen Bahnen. Abh. Math. Sem. d. Hamburgischen Universitat, pp. 170–175 (1924) 2. Bateman, H.: Tables of Integral Transforms. McGraw-Hill, New York, 1954 3. Bump, D.: Automorphic Forms on G L(3, R). Lecture Notes in Mathematics, Vol. 1083, Berlin-Heidelberg-NewYork: Springer, 2008 4. Colin de Verdiere, Y.: Ergodicite et fonctions propres du laplacien. Commun. Math. Phy. 102, 497–502 (1985) 5. Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982)

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6. Erdelyi, A., et al.: Higher Transcendental Functions. Vol. 2 New York: McGraw-Hill Book Company, 1953 7. Eckhardt, B., Fishman, et al.: Approach to ergodicity in quantum wave functions. Phys. Rev. E 52, 5893 (1995) 8. Feingold, M., Peres, A.: Distributin of matrix elements of chaotic systems. Phy. Rev. A 34, 591 (1986) 9. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 4th ed., New York: Academic Press, 1965 10. Hoffstein, J., Lockhart, P.: Coefficients of Maass forms and the Siegel zero. Ann. of Math. (2) 140(1), 161– 181 (1994) 11. Holowinsky, R., Soundararajan, K.: Mass equidistribution of Hecke eigenforms. http://arXiv.org/abs/ 0809.1636v1[math.NT], 2008 12. Hejhal and Rackner: On the topography of Maass wave forms. Exper. Math. 1(4), 275–305 (1992) 13. Hopf, E.: Statistik der geodeischen Linien in Mannigfaltigkeiten negativer Krmmung. Leipzig Ber. Verhandl. Shs. Akad. Wiss. 91, 261–304 (1939) 14. Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, Vol. 17, Providence, RI: Amer. Math. Soc., 1997 15. Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications 53, Providence, RI: Amer. Math. Soc., 2004 16. Iwaniec, H., Luo, W., Sarnak, P.: Low lying zeros of families of L-functions. Inst. Hautes Etudes Sci. Publ. Math. No. 91, 55–131 (2000) 17. Jutila, M.: The spectral mean square of Hecke L-functions on the critical line. Publ. Inst. Math. (Beograd) (N.S.) 76(90), 41–55 (2004) 18. Kuznetsov, N.V.: Petersson’s Conjecture for Cusp Forms of Weight Zero and Linnik’s Conjecture. Math. USSR Sbornik 29, 299–342 (1981) 19. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2) 163(1), 165–219 (2006) 20. Luo, W., Rudnick, Z., Sarnak, P.: The variance of arithmetic measures associated to closed geodesics on the modular surface. http://arXiv.org/abs/0810.3331v2[math.NT], 2009 21. Luo, W.: Zeros of Hecke L-functions associated with cusp forms. Acta Arith. 71(2), 139–158 (1995) 22. Luo, W.: Values of symmetric square L-functions at 1. J. Reine Angew. Math. 506, 215–235 (1999) 23. Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on P S L 2 (Z ) \ H . Inst. Hautes Etudes Sci. Publ. Math. No. 81, 207–237 (1995) 24. Luo, W., Sarnak, P.: Quatum variance for Hecke eigenforms. 2004 Ann. Sci. Cole Norm. Sup. (4) 37(5), 769–799 (2004) 25. Ratner, M.: The rate of mixing for geodesic and horocycle flows. Erg. Theory Dynam. Sys. 7, 267–288 (1987) 26. Ratner, M.: The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16, 181–197 (1973) 27. Rudnick, Z., Sarnak, P.: The behavior of eigenstates of hyperbolic manifolds. Commun. Math. Phys. 161, 195–213 (1994) 28. Sarnak, P.: Arithmetic Quantum Chaos. The Schur Lectures, 2003 29. Sarnak, P.: Estimates for Rankin-Selberg L-functions and quantum unique ergodicity. J. Funct. Anal. 184(2), 419–453 (2001) 30. Schnirelman, A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk. 29, 181–182 (1974) 31. Soundararajan, K.: Quantum unique ergodicity for S L 2 (Z ) \ H. http://arxiv.org/abs/0901.4060v1[math. NT], 2009 32. Titchmarsh, E.C.: The Theory of the Riemann Zeta-function. 2nd ed., Edited and with a preface by D. R. Heath-Brown. New York: The Clarendon Press / Oxford University Press, 1986 33. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge / New York: Cambridge University Press / The Macmillan Company, 1944 34. Watson, T.: Central Value of Rankin Triple L-function for Unramified Maass Cusp Forms. Princeton thesis, 2004 35. Weyl, H.: Zur Abschatzung von ξ(1 + it). Math. Z. 10, 88–101 (1921) 36. Zelditch, S.: On the rate of quantum ergodicity. Commun. Math. Phys. 160, 81–92 (1994) 37. Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) Communicated by S. Zelditch

Commun. Math. Phys. 297, 515–528 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0944-8

Communications in

Mathematical Physics

Suspending Lefschetz Fibrations, with an Application to Local Mirror Symmetry Paul Seidel Department of Mathematics, M.I.T., 77 Massachusetts Avenue, Cambridge, MA 02139, USA. E-mail: [email protected] Received: 16 July 2009 / Accepted: 29 July 2009 Published online: 25 October 2009 – © Springer-Verlag 2009

Abstract: We describe the behaviour of Fukaya categories under “suspension”, which means passing from the fibre of a Lefschetz fibration to the double cover of the total space branched along that fibre. As an application, we consider the mirrors of canonical bundles of toric Fano surfaces. 1. Introduction Let Y be a smooth toric del Pezzo surface, and K Y the total space of its canonical bundle. Let D b (Coh(K Y )) be the bounded derived category of coherent sheaves on K Y , and DYb (Coh(K Y )) the full subcategory consisting of complexes whose cohomology is supported on the zero-section Y ⊂ K Y . The mirror of K Y (see [10, Section 8] or [8]) is the hypersurface H = {(x, y) ∈ (C∗ )2 × C2 : y1 y2 + p(x) = z},

(1.1)

Here, p : (C∗ )2 → C is the superpotential mirror to Y (following [7 or 9]), and z is any regular value of p. H is an affine threefold with trivial canonical bundle. Hence, it has a Fukaya category Fuk(H ), whose objects are compact exact Lagrangian submanifolds equipped with gradings and Spin structures. This is an A∞ -category over C. Consider the associated derived category D(Fuk(H )). Theorem 1.1. There is a full embedding of triangulated categories, DYb (Coh(K Y )) → D(Fuk(H )).

(1.2)

To explain the strategy of the proof, we need to return to Y itself. Homological mirror symmetry for such varieties, first considered by Kontsevich [11], is now a theorem (see [15] for Y = CP 2 , [2] for several more cases, and [1,19] for all remaining ones). On the other hand, Segal [14] and Ballard [3] independently proved a result which

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describes, on the level of the underlying A∞ -algebras, the relation between D b (Coh(Y )) and DYb (Coh(K Y )). The missing piece, which we provide here, is a corresponding statement about the relation between the Fukaya category of the Lefschetz fibration p and that of the hypersurface H . We can prove this relation only partially, which is why we get an embedding (1.2) instead of the ultimately desired equivalence. The proof of Theorem 1.1 is in fact an application of a more general result about suspending Lefschetz fibrations (Theorem 6.4), which means passing from p(x) to p(x) − y12 − · · · − ym2 for any number of variables. Independently of the present work, Futaki and Ueda [6] have applied similar methods to p(x) − y k for k ≥ 2. However, their interest is directed towards mirror symmetry for Landau-Ginzburg models, hence their computations concern only directed Fukaya categories. 2. Algebraic Suspension Fix a ground field K, which will be used throughout. We will describe a suspension construction, which starts from a pair (A, B) consisting of an A∞ -algebra B and an A∞ -subalgebra A ⊂ B, and produces another pair (Aσ , B σ ) of the same kind. In fact Aσ is isomorphic to A in a straightforward way, but B σ is usually not even quasi-isomorphic to B. We will see that in general, the construction leads to a loss of information, hence cannot be reversed in a meaningful (that is, quasi-isomorphism invariant) way. Here is the most elementary description. As a graded vector space B σ = A+ ⊕ A− ⊕ B[−1],

(2.1)

where both A+ and A− are copies of A. The shift [−1] means that elements of B σ of degree r are triples (a+ , a− , b), where a± ∈ A have degree r , and b ∈ B has degree r − 1. The differential is μ1Bσ (a+ , a− , b) = (μ1A (a+ ), μ1A (a− ), −μ1B (b) − a+ + a− ).

(2.2)

The higher order A∞ -structure maps, for d ≥ 2, are μdBσ ((ad,+ , ad,− , bd ), . . . , (a1,+ , a1,− , b1 ))  = μdA (ad,+ , . . . , a1,+ ), μdA (ad,− , . . . , a1,− ), d   (−1) a1,− +···+ ai−1,− +1 μdB (ad,+ , . . . , ai+1,+ , bi , ai−1,− , . . . , a1,− ) .

(2.3)

i=1

The notational convention is that a = deg(a) − 1 is the reduced degree. The subspace Aσ ⊂ B σ just consists of triples of the form (a, a, 0). Example 2.1. It maybe makes sense to write down the construction in the simpler special case of dga’s, taking into account the differences in sign conventions. Namely, if A ⊂ B are dga’s, the dga structure on the suspension B σ is given by d(a+ , a− , b) = (da+ , da− , db − (−1)deg(a+ ) a+ + (−1)deg(a− ) a− ),

(a2,+ , a2,− , b2 ) · (a1,+ , a1,− , b1 ) = (a2,+ a1,+ , a2,− a1,− , a2,+ b1 + (−1)deg(a1,− ) b2 a1,− ). (2.4)

Lefschetz Fibrations

517

The fact that (2.2), (2.3) satisfy the A∞ -associativity equations can be checked by hand, but there is also an equivalent description which makes these equations obvious. Start with the following contractible complex of vector spaces: 1

C = {· · · 0 → K − → K → 0 · · · },

(2.5)

where the nontrivial generators are in degrees −1 and 0. Let homK (C, C) be the differential graded algebra of endomorphisms of C. Consider the tensor product B ⊗ homK (C, C), which is again an A∞ -algebra. As a graded vector space, B ⊗ homK (C, C) = B[1] ⊕ B+ ⊕ B− ⊕ B[−1],

(2.6)

where B+ is B tensored with the endomorphisms of the degree 0 part of C, and B− similarly for the degree −1 part of C. Define B σ as an A∞ -subalgebra of B ⊗homK (C, C) in the obvious way. This leads to the formulae described above. For instance, the additional terms in the differential (2.2) are precisely the ones inherited from the differential on homK (C, C). Remark 2.2. One can recast the previous description in slightly different language as follows. Assume that B is strictly unital, with identity element e. Consider it as an A∞ -category with a single object V , and let Tw(B) be the associated A∞ -category of twisted complexes. This contains a contractible object, which is the mapping cone S = Cone(e : V → V ). Then homTw(B) (S, S) = B ⊗ homK (C, C)

(2.7)

Bσ

as an A∞ -subalgebra of this (our sign as an A∞ -algebra, and one can again define conventions differ from those of [17, Section 3k], essentially because we think of twisted complexes as formal tensor products with graded vector spaces on the right side, rather than the left one). There is a special class of examples where the effect of suspension is quite simple. To explain this, we need the notion of an A∞ -bimodule over A, which is a graded vector space P together with operations s|1|r

μP

: A⊗s ⊗ P ⊗ A⊗r −→ P[1 − r − s]

(2.8)

for all r, s ≥ 0, satisfying appropriate equations. A straightforward example is the diagonal bimodule, which is A itself with d−i|1|i−1

μA

(ad , . . . , a1 ) = (−1) a1 +···+ ai−1 +1 μdA (ad , . . . , a1 ).

(2.9)

We will also need the shift operation, which takes P to the shifted vector space P[−1] with operations s|1|r

μP [−1] (ar +s , . . . , ar +1 , p, ar , . . . , a1 ) = (−1) a1 +···+ ar +1 μP (ar +s , . . . , ar +1 , p, ar , . . . , a1 ). s|1|r

s|1|r

(2.10)

Hence, the shifted diagonal bimodule A[−1] has μA[−1] = μs+1+r . For a more thorough A exposition of the theory of A∞ -bimodules and their morphisms, see [16,18] (our sign conventions are related to those in [16] by reversing the ordering of the entries in (2.8), which brings things in line with [17]).

518

P. Seidel

For any given A∞ -algebra A and A-bimodule P, one can define the trivial extension A∞ -algebra A ⊕ P, which is the direct sum of vector spaces equipped with the structure maps  μdA⊕P ((ad , pd ), . . . , (a1 , p1 )) = μdA (ad , . . . , a1 ), d   d−i|1|i−1 (−1) a1 +···+ ai−1 +1 μP (ad , . . . , ai+1 , pi , ai−1 , . . . , a1 ) .

(2.11)

i=1

This obviously contains A as an A∞ -subalgebra. The suspension of B = A ⊕ P is B σ = A+ ⊕ A− ⊕ A[−1] ⊕ P[−1], with 0|1|0

μ1Bσ (a+ , a− , a, p) = (μ1A (a+ ), μ1A (a− ), −μ1A (a) − a+ + a− , μP

( p))

(2.12)

and, for d ≥ 2, μdBσ ((ad,+ , ad,− , ad , pd ), . . . , (a1,+ , a1,− , a1 , p1 ))  = μdA (ad,+ , . . . , a1,+ ), μdA (ad,− , . . . , a1,− ), d  (−1) a1,− +···+ ai−1,− +1 μdA (ad,+ , . . . , ai+1,+ , ai , ai−1,− , . . . , a1,− ), i=1 d 

d−i|1|i−1

μP

 (ad,+ , . . . , ai+1,+ , pi , ai−1,− , . . . , a1,− ) .

(2.13)

i=1

This contains the A∞ -subalgebra of elements of the form (a, a, 0, p), which is isomorphic to the trivial extension A ⊕ P[−1]. Moreover, the induced differential on the quotient B σ /(A ⊕ P[−1]) ∼ = A ⊕ A[−1] is acyclic. We have therefore shown: Lemma 2.3. If B is a trivial extension A⊕P, then its suspension B σ is quasi-isomorphic to the trivial extension A ⊕ P[−1].

3. A Topological Viewpoint This section is a digression from our main argument, and its only purpose is to build some topological intuition. Let U be a topological manifold with boundary W = ∂U (in fact, more general topological spaces and subspaces are also possible). Let A = C ∗ (U ) be the dga of singular cochains with coefficients in K. Set B = C ∗ (U ) ⊕ C ∗ (U, W )[1], with the differential and multiplication given, in terms of the standard differential δ and cup-product on cochains, by d(b, c) = (δb + (−1)deg(c) c, δc), (b2 , c2 ) · (b1 , c1 ) = (b2 b1 , b2 c1 + (−1)deg(b1 ) c2 b1 ).

(3.1)

By definition, the map B → C ∗ (W ), (b, c) → b|W , is a quasi-isomorphism of dga’s. The point of replacing C ∗ (W ) by its quasi-isomorphic version B is that it allows us to view the restriction map C ∗ (U ) → C ∗ (W ) as the inclusion A → B.

Lefschetz Fibrations

519

Consider the suspension of B, which following Example 2.1 is the dga B σ = C ∗ (U )⊕ with

C ∗ (U ) ⊕ C ∗ (U )[−1] ⊕ C ∗ (U, W )

d(a+ , a− , b, c) = (δa+ , δa− , δb − (−1)deg(a+ ) a+ + (−1)deg(a− ) a− + (−1)deg(c) c, δc), (a2,+ , a2,− , b2 , c2 ) · (a1,+ , a1,− , b1 , c1 ) = (a2,+ a1,+ , a2,− a1,− , a2,+ b1 + (−1)deg(a1,− ) b2 a1,− , a2,+ c1 + c2 a2,− ).

(3.2)

Let W σ = U+ ∪W U− be the manifold obtained by gluing two copies of U along their common boundary. This comes with a dga homomorphism C ∗ (W σ ) −→ B σ , a −→ (a|U+ , a|U− , 0, a|U+ − a|U− ).

(3.3)

This implicitly uses the fact that the restrictions a|U± agree along their boundaries, so that their difference is a relative cochain. As a map of chain complexes, (3.3) sits in a commutative diagram with short exact rows, 0

/ C ∗ (W σ , U− )

/ C ∗ (W σ )

/ C ∗ (U− )

/0

0

 / C ∗ (U, W )

 / Bσ

 / C ∗ (U ) ⊕ C ∗ (U ) ⊕ C ∗ (U )[−1]

/ 0. (3.4)

The lower row consists of the maps c → (c, 0, 0, c) and (a+ , a− , b, c) → (a+ −c, a− , b). The left hand vertical arrow is restriction from W σ to U+ = U , and the right hand one is a → (a, a, 0) with respect to the identification U− = U . The two last-mentioned maps are quasi-isomorphisms, hence so is (3.3). Next, W σ is naturally the boundary of a manifold U σ which is homotopy equivalent to U , namely U σ = U × [−1, 1] / (x, t) ∼ (x, s) for x ∈ W .

(3.5)

The maps induced by projection U σ → U and inclusion W σ → U fit into a commutative diagram of dga’s, C ∗ (U )



/ C ∗ (U σ )

/ C ∗ (W σ )





 Aσ

(3.6)



/ Bσ .

Hence, in this context suspension is the algebraic counterpart of the topological process of passing from (U, W ) to (U σ , W σ ). 4. General Properties We return to general pairs (A, B). The inclusion A ⊂ B makes B into an A-bimodule, in the same way as in (2.9). It contains the diagonal bimodule A, so the quotient B/A is an A-bimodule as well, and we have a short exact sequence ι

π

→B− → B/A → 0. 0→A−

(4.1)

520

P. Seidel

˜ together with an Lemma 4.1. Suppose that we have two pairs (A, B) and (A, B), A-bimodule quasi-isomorphism φ : B˜ → B which restricts to the identity on A. Then there is an induced quasi-isomorphism of A∞ -algebras φ σ : B˜ σ → B σ . Proof. Our given φ has components φ s|1|r : A⊗s ⊗ B˜ ⊗ A⊗r → B[−r − s]. Define φ σ,1 (a+ , a− , b) = (a+ , a− , φ 0|1|0 (b)),

(4.2)

which is obviously a quasi-isomorphism, and for d ≥ 2, φ σ,d ((ad,+ , ad,− , bd ), . . . , (a1,+ , a1,− , b1 ))   d  φ d−i|1|i−1 (ad,+ , . . . , ai+1,+ , bi , ai−1,− , . . . , a1,− ) . = 0, 0,

(4.3)

i=1

It is straightforward to check the A∞ -homomorphism equations.



Lemma 4.2. Suppose that there is an A∞ -bimodule map ξ : B/A → B whose composition with the projection π : B → B/A is a quasi-isomorphism from B/A to itself. Then B σ is quasi-isomorphic to the trivial extension A∞ -algebra A ⊕ (B/A)[−1]. Proof. Consider the A-bimodule B˜ = A ⊕ (B/A). The inclusion ι and the map ξ combine to yield an A-bimodule map ι ⊕ ξ : B˜ → B. Just seen as a map of chain complexes, this sits in a commutative diagram 0

/A

0

 /A

id

/ B˜ ι⊕ξ

 /B

/ B/A

/0

(4.4)

(π ◦ξ )0|1|0

 / B/A

/ 0,

hence is itself a quasi-isomorphism. Now consider B˜ as an A∞ -algebra, namely the trivial extension of A by B/A. By Lemma 4.1, our bimodule map induces a quasiisomorphism between the suspensions B˜ σ and B σ . But Lemma 2.3 says that the first of these is quasi-isomorphic to A ⊕ (B/A)[−1], which implies the desired result.

Lemma 4.3. Take any (A, B). Then there is a map ξ σ : B σ /Aσ → B σ of bimodules over Aσ = A whose composition with the projection B σ → B σ /Aσ is a quasi-isomorphism from B σ /Aσ to itself. Proof. Consider the subspace of B σ consisting of elements of the form (a+ , 0, b). This is a sub-bimodule over Aσ . Projection from that bimodule to B σ /Aσ is an isomorphism, hence admits a unique strict inverse.

The principal consequence of the discussion above is the following: Proposition 4.4. For any (A, B), the double suspension B σ σ is an A∞ -algebra quasiisomorphic to the trivial extension A ⊕ (B/A)[−2]. Proof. Apply Lemma 4.3 to (A, B), and then Lemma 4.2 to (Aσ , B σ ). The consequence is that B σ σ is quasi-isomorphic to the trivial extension of Aσ ∼ = A by the bimodule (B σ /Aσ )[−1]. Projection B σ /Aσ = A ⊕ B[−1] → (B/A)[−1] is a quasiisomorphism of A-bimodules, and induces a quasi-isomorphism of the associated trivial extension A∞ -algebras.

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5. Fukaya Categories Let (M, ω) be a 2n-dimensional symplectic manifold, and (V1 , . . . , Vm ) an ordered collection of Lagrangian spheres in M. To apply Floer-Fukaya theory, in its simplest form, we want to impose a number of restrictions: • ω should be exact, in fact we want to choose a particular primitive ω = dθ . To deal with the resulting inevitable non-compactness of M, we assume that it can be exhausted by a sequence of relatively compact open subsets U1 ⊂ U2 ⊂ · · · with smooth boundaries, such that the Liouville vector field dual to θ points strictly outwards along each ∂Ui . • c1 (M) should be trivial, in fact we want to choose a trivialization of the canonical bundle. • The Vi should be exact, and we equip them with gradings as well as Spin structures. Actually, exactness (θ |Vi = d Fi for some function) is nontrivial only in the lowest dimension n = 1. The existence of gradings requires zero Maslov index, which is again nontrivial only for n = 1. Finally, the Spin structure is unique up to isomorphism unless n = 1, in which case we always choose the nontrivial Spin structure on the circle. In this framework, we have well-defined Floer cohomology groups HF ∗ (Vi , V j ), which are graded vector spaces over our fixed coefficient field K. Following Fukaya (see for instance [17] for an exposition) one constructs an A∞ -category B, with objects (V1 , . . . , Vm ), whose morphism spaces are the Floer cochain groups CF ∗ (Vi , V j ). We sometimes find it convenient to adopt an equivalent point of view. Namely, consider the semisimple ring R = Ke1 ⊕ · · · ⊕ Kem , where ei2 = ei , ei e j = 0 for all i = j.

(5.1)

A∞ -categories with m numbered objects can also be viewed as A∞ -algebras over R. Concretely, consider the direct sum B= CF ∗ (Vi , V j ), (5.2) i, j

with its natural R-bimodule structure (left multiplication with ek projects to the direct summand j = k, and right multiplication to the direct summand i = k). This carries A∞ -multiplications, obtained in the straightforward way from those of the previously mentioned A∞ -category, which are compatible with the R-bimodule structure. We will mostly use the point of view of (5.2), but switch back freely to A∞ -categorical language whenever that simplifies the discussion. As explained in [13, Theorem 3.2.1.1] or [17, Lemma 2.1], we can modify the A∞ structure on B to a quasi-isomorphic one which is strictly unital. Assume from now on that this has been done, without changing notation. Concretely, strict unitality means that there is a unique inclusion R → B, compatible with the R-bimodule structure, such that μ1B (ei ) = 0, μ2B (a, ei ) = aei ,

μ2B (ei , a) = (−1) a −1 ei a, μdB (. . . , ei , . . . ) = 0 for all d ≥ 3.

(5.3)

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In particular, we can then define the directed A∞ -subalgebra of B to be the subspace A= R⊕ CF ∗ (Vi , V j ) ⊂ B, (5.4) i< j

which is automatically closed under all the A∞ -multiplications. We will need one general property of the pair (A, B). Proposition 5.1. The quotient B/A, with its natural A-bimodule structure, is quasiisomorphic to A∨ [−n], which is the dual of the diagonal bimodule shifted upwards by n. Proof. We first explain the analogous but much simpler argument on the level of cohomology. There are Poincaré duality isomorphisms HF ∗ (Vi , V j ) ∼ = HF n−∗ (V j , Vi )∨

(5.5)

(generally, the sign of these depends on orientations of the Lagrangian spheres, but in our case there are preferred orientations determined by the gradings). The direct sum of these yields an isomorphism H (B) ∼ = H (B)∨ [−n], which is compatible with the structure of both as bimodules over H (A). By definition of the directed subalgebra, the composition H (A) → H (B) ∼ = H (B)∨ [−n]  H (A)∨ [−n]

(5.6)

is zero, which means that we get an induced map H (B)/H (A) = H (B/A) → H (A)∨ [−n]. It is straightforward to verify that this is an isomorphism. On the cochain level, one proceeds as follows: Step 1. There is a quasi-isomorphism of A-bimodules, η : B → B ∨ [−n], which on cohomology induces the maps (5.5). This is a weak version of the Calabi-Yau property of Fukaya categories. For coefficient fields K containing R, Fukaya [5] has proved the strong form of that property (cyclicity) for a single Lagrangian submanifold. An extension of his arguments to finitely many submanifolds should be unproblematic, and would be enough for the applications to mirror symmetry (taking K = C). There is actually another argument leading to the weaker form, but which works for arbitrary K; we’ll mention it briefly, omitting all details. The general algebraic framework for homomorphisms B → B ∨ [−n] was set up in [18, Sect. 5]. A look at the pictures there makes it clear how such a map can be constructed in the Fukaya-theoretic context by looking at moduli spaces of infinite strips with added marked boundary points (strictly speaking, the resulting bimodule map applies to the B obtained directly from geometry, but of course it carries over to any quasi-isomorphic A∞ -structure). Step 2. The composition of A-bimodule maps η

A → B −→ B ∨ [−n]  A∨ [−n]

(5.7)

is nullhomotopic. Recall that A is an A∞ -algebra over R. All our bimodules and maps between them are compatible with that structure (as in [16], for instance). Secondly, A is strictly unital, and therefore every homomorphism between strictly unital A-bimodules is chain

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homotopic to a strictly unital one. We apply this to η and, without changing notation, assume that it is strictly unital. Concretely, this means that ηs|1|r : A⊗ R s ⊗ R B ⊗ R A⊗ R r → B ∨ [−n]

(5.8)

vanishes whenever any of the A entries is equal to some ei . Because of this, the R-bimodule structure, and directedness, the only possible nonzero component of (5.7) is the linear part ei Aei −→ ( ei Aei )∨ = R ∨ [−n] (5.9) R= i

i

which vanishes for degree reasons. Step 3. There is an A-bimodule map η˜ : B/A → A∨ [−n] which fits into a diagram, commutative up to homotopy, B/A UUU OO UUUU UUUUη˜ UUUU UUUU * η ∨ / / / A∨ [−n] B [−n] B

(5.10)

This is a consequence of the previous step, and the fact that a short exact sequence of A∞ -bimodules induces a long exact sequence of morphism spaces. Namely, denoting by C the differential graded category of bimodules, we have a long exact sequence · · · → H (homC(B/A, A∨ )) −→ H (homC(B, A∨ )) −→ H (homC(A, A∨ )) −→ · · · .

(5.11)

By construction, the cohomology level map induced by η˜ is obtained by taking some splitting of the projection H (B) → H (B)/H (A) and composing it with H (B) ∼ = H (B)∨ [−n] → H (A)∨ [−n]. The result is independent of the choice of splitting, and is an isomorphism, hence η˜ is a quasi-isomorphism.

6. Lefschetz Fibrations Picard-Lefschetz theory can be formulated in various frameworks and degrees of generality. Even though the results here are of a purely symplectic nature, we prefer to keep close to the original algebro-geometric context for expository reasons. Hence, by a Lefschetz fibration we will mean a map p : X −→ C

(6.1)

of the following kind. X is a smooth affine algebraic variety, equipped with a Kähler form ω which is exact and complete (in the sense of Riemannian geometry). Additionally, we assume that X carries a holomorphic volume form η. The function p is regular (which means a polynomial), has only nondegenerate critical points, and additionally satisfies the following Palais-Smale type property: Any sequence (xk ) of points in X such that | p(xk )| is bounded and (∇ p)xk → 0 has a convergent subsequence.

(6.2)

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This condition, similar to that of “tameness” in singularity theory [4], implies the welldefinedness of symplectic parallel transport maps away from the critical points (because the norm of the parallel transport vector field is essentially the inverse of that of ∇ p). In fact, the entire package of symplectic Picard-Lefschetz theory, see for instance [17, Chapter 3], applies.

Example 6.1. Take X = (C∗ )n+1 , with the “logarithmic” Kähler form ω = k dre(log ±1 ] be a Laurent polynomial satisfying the (z k )) ∧ dim(log(z k )). Let p ∈ C[z 1±1 , . . . , z n+1 following two conditions: its Newton polytope contains the origin in its interior; and it is nondegenerate in the sense of [12]. Then (6.2) holds (compare [4, Prop. 3.4], which is an analogue for ordinary polynomials). In fact, these conditions imply the stronger property that ∇ p is a proper function on X . The suspension of a Lefschetz fibration is defined as p σ (x, y) = p(x) − y 2 : X σ = X × C −→ C.

(6.3)

To be precise, X σ carries the standard product Kähler form ωσ = ω + 2i dy ∧ d y¯ , and the holomorphic volume form ησ = η ∧ dy. If p satisfies (6.2), then so does its suspension. We will now analyze the geometry of the suspension (this material is also covered in [17, Sect. 18], in a closely related setup and with more details). Denote by X z and X zσ the fibres of p and p σ , respectively, over z ∈ C. Suppose, just for simplicity of notation, that each fibre of p contains only one critical point, and also that there are no critical values on the positive real half-axis R+ ⊂ C. Projection to the y-variable is itself a Lefschetz fibration p˜ : X 0σ −→ C.

(6.4)

Its fibres are p˜ −1 (z) = X z 2 , and the symplectic connection is the pullback of that on p by the double covering of the base. Suppose that we have chosen a distinguished basis (γ1 , . . . , γm ) of vanishing paths for p, all starting at z = 0 and otherwise avoiding R+ . Let (V1 , . . . , Vm ) be the associated collection of vanishing cycles, which are Lagrangian spheres in X 0 . Then √ √ √ √ (6.5) (γ˜1 , . . . , γ˜2m ) = ( γ1 , . . . , γm , − γ1 , . . . , − γm ) is a distinguished basis of vanishing paths for p, ˜ and by our previous observation concerning the symplectic connection, the associated basis of vanishing cycles consists of two copies of that for p: (V˜1 , . . . , V˜2m ) = (V1 , . . . , Vm , V1 , . . . , Vm ).

(6.6)

Since there are two instances of each vanishing cycle Vi in that collection, one can join the respective critical points by a matching path μi [17, Sect. 16g], which is simply the composition of γ˜i and γ˜i+m , or equivalently the preimage of γi under the double branched cover. The matching cycles for these paths form a collection (V1σ , . . . , Vmσ ) of Lagrangian spheres in the total space of p, ˜ which of course also happens to be the fibre X 0σ . These spheres have two interpretations. On one hand, each Vi bounds a Lefschetz thimble, which is a Lagrangian ball in X , fibered over γi . The preimage of this ball under the branched covering X 0σ → X is Viσ (this follows directly from the definition of matching cycle). Alternatively, we can use γi as a vanishing path for p σ itself, and then Viσ is the vanishing cycle associated to that path (this is a standard braid monodromy argument, compare [17, Lemma 18.2]).

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Passing to Floer-Fukaya theory, we now have the following algebraic structures: • B is the A∞ -algebra over R = Km associated to V1 , . . . , Vm ⊂ X 0 . Here, we assume that the vanishing cycles have been equipped with gradings and Spin structures. As in Sect. 5, we also assume that B has been made strictly unital, so that we have a directed A∞ -subalgebra A. • Likewise, let B˜ be the A∞ -algebra over R˜ = K2m associated to V˜1 , . . . , V˜2m ⊂ X 0 , where the gradings are borrowed from those of the Vi , and A˜ ⊂ B˜ its directed subalgebra. The algebraic relation between these two is straightforward, in view of (6.6). In categorical terms, B˜ is obtained from B by introducing two isomorphic copies of each existing object. Equivalently, as an algebra it is given by B˜ = B ⊗K mat 2 (K),

(6.7)

where mat 2 (K) is the associative algebra of matrices of size two. If one thinks in terms of such matrices, then the directed A∞ -subalgebras are related by   A 0 ˜ ˜ ⊂ B. (6.8) A= BA Let’s give another and even more explicit description. The morphism spaces in A˜ are ⎧ ⎪ if i ≤ j ≤ m or m < i ≤ j, ⎨e j Aei ˜ e j Aei = e j−m Bei (6.9) if i ≤ m < j, ⎪ ⎩0 if i > j; and the A∞ -structure is induced from that of B (more precisely, only the A∞ -structure of A and the structure of B as an A∞ -bimodule over A enter into the construction of ˜ We introduce two more A∞ -structures: A). ˜ be the • Think of A˜ as an A∞ -category with objects (V˜1 , . . . , V˜2m ), and let Tw(A) associated category of twisted complexes. Consider the particular twisted complexes Si = Cone(ei : V˜i → V˜i+m ),

(6.10)

where the arrow is ei ∈ homA˜ (V˜i , V˜i+m ) = homB (Vi , Vi ) = ei Bei , in other words is σ to be the full A -subcatderived from the identity elements in B. We then define Balg ∞ ˜ with objects (S1 , . . . , Sm ). This can also be seen as an A∞ -algebra egory of Tw(A) over R = Km , as usual, and we denote by Aσalg its directed subalgebra. σ • Bgeom is the A∞ -algebra associated to V1σ , . . . , Vmσ ⊂ X 0σ , and (after modifying that to make it strictly unital, as usual) Aσgeom is its directed A∞ -subalgebra. σ ) is the suspension (Aσ , B σ ) of the pair (A, B). Lemma 6.2. (Aσalg , Balg

Here, we are implicitly generalizing the material of Sect. 2–4 to the context of A∞ -algebras over R, which is entirely unproblematic. With this in mind, the lemma is merely a reformulation of the description given in Remark 2.2. In fact, if we consid˜ ∼ ered the objects Si as lying in Tw(B) = Tw(B), the associated endomorphism algebra would be exactly B ⊗ homK (C, C) as previously considered in Sect. 2. Restricting morphisms to A˜ ⊂ B˜ yields the subspace B σ ⊂ B ⊗ homK (C, C). Of course, the directed subalgebras are then also the same.

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Fig. 1.

Lemma 6.3. Provided that char(K) = 2, and that the gradings of the Viσ have been σ is quasi-isomorphic to B σ chosen appropriately, Balg geom . This is a direct application of results from [17, Sect. 10] to the Lefschetz fibration (6.4) (technical tricks used in proving those results are responsible for the restriction on the ground field; there is no reason to believe that this restriction is fundamentally necessary). Figure 1 shows the basis of vanishing paths for that fibration, as well as the matching paths (in the interest of clarity, the position of the critical points has been distorted somewhat). In such a situation, [17, Prop. 18.21] says that the full A∞ -subcategory of the Fukaya category Fuk(X 0σ ) consisting of the associated matching cycles is ˜ More precisely, the object corresponding quasi-isomorphically embedded into Tw(A). σ th to the i matching cycle Vi (with suitably chosen grading) is the mapping cone of the lowest degree nontrivial morphism V˜i → V˜i+m . But in our case, H (homA˜ (V˜i , V˜i+m )) = H F ∗ (Vi , Vi ) = H ∗ (S n ; K), so the lowest degree nontrivial morphism is represented by the identity ei , leading to (6.10). On general grounds [13, Theorem 3.2.2.1], the σ ∼ Bσ quasi-isomorphism Balg = geom can be made strictly unital, and then it restricts to a quasi-isomorphism of the associated directed A∞ -subalgebras. Theorem 6.4. Assume that char(K) = 2. Let (A, B) be the pair of A∞ -algebras associated to p, for some choice of vanishing paths. Then the corresponding pair for p σ is quasi-isomorphic to the algebraic suspension (Aσ , B σ ). This follows directly from Lemmas 6.2 and 6.3. Moreover, applying Propositions 4.4 and 5.1, we obtain the following consequence: Corollary 6.5. Let p σ σ be the double suspension of p. Take any basis of vanishing cycles, and let (Aσ σ , B σ σ ) be the associated pair of A∞ -algebras. Suppose that char(K) = 2. Then B σ σ is the trivial extension of Aσ σ by the bimodule (Aσ σ )∨ [−n − 2] (n + 2 is the complex dimension of the fibre of p σ σ ). Remark 6.6. p σ σ : X σ σ = C2 × X → C carries a fibrewise circle action. It seems plausible to think that an appropriate S 1 -equivariant version of B σ σ would recover the information lost during the suspension process (the corresponding statement is true in the elementary topological counterpart of this process, discussed in Section 3, as one can see by applying the localization theorem in equivariant cohomology). 7. Local Mirror Symmetry Let Y be any smooth complex projective variety, and ι : Y → K Y its embedding into the canonical bundle as the zero-section. Take any collection of objects E 1 , . . . , E m ∈

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D b (Coh(Y )). Let C be the A∞ -algebra over R = Cm , defined using suitable resolutions (Dolbeault, Cech, or injective), which underlies H (C) =

m

Hom∗Y (E i , E j ),

(7.1)

i, j=1

where the Homs are morphisms of any degree in the derived category. There is a similar algebra D for the objects ι∗ E i ∈ DYb (Coh(K Y )). Theorem 7.1 (Segal [14, Theorem 4.2], Ballard [3, Prop. 4.14]). D is quasi-isomorphic to the trivial extension C ⊕ C ∨ [−dimC (K Y )]. Now suppose that the E i generate D b (Coh(Y )), which implies that the ι∗ E i generate DYb (Coh(K Y )). Consider D as an A∞ -category with objects (ι∗ E 1 , . . . , ι∗ E m ), and let D(D) be its derived category, defined using twisted complexes as H 0 (Tw(D)). In that case DYb (Coh(K Y )) ∼ = D(D).

(7.2)

Proof of Theorem 1.1. Take a toric del Pezzo surface Y and its mirror p : X → C. The mirror is a Laurent polynomial which, for generic choice of coefficients, has nondegenerate critical points and satisfies the conditions from Example 6.1 (the specific choice of coefficients is irrelevant, since the space of allowed choices is connected). Fix a basis of vanishing paths for p. Let (V1 , . . . , Vm ) be the resulting vanishing cycles in some regular fibre X z , and A the associated directed A∞ -algebra. As mentioned in the Introduction, we know from [1,2,19] that there is a full exceptional collection (E 1 , . . . , E m ) in D b (Coh(Y )) such that A is quasi-isomorphic to the associated A∞ -algebra C (in fact, in all these cases one can choose the vanishing cycles so that the E i form a strong exceptional collection, which simplifies computations somewhat; but that is not strictly relevant for our purpose). On the other hand, let L i = Viσ σ ⊂ H = X zσ σ be the corresponding vanishing cycles for the double suspension. By Corollary 6.5, the full subcategory of Fuk(H ) with objects (L 1 , . . . , L m ) is quasi-isomorphic to the trivial extension A ⊕ A∨ [−3]. On the other hand, Theorem 7.1 says that D is quasi-isomorphic to C ⊕ C ∨ [−3]. Using (7.2), we get a full embedding DYb (Coh(K Y )) ∼ = D(D) ∼ = D(A ⊕ A∨ [−3]) → D(Fuk(H )).

(7.3)

Acknowledgements. This problem was brought to my attention by Matthew Ballard. I would like to thank him, Mohammed Abouzaid, Mark Gross, and Ivan Smith for useful conversations. NSF grant DMS-0652620 provided partial support.

References 1. Auroux, D., Katzarkov, L., Orlov, D.: Mirror symmetry for Del Pezzo surfaces: vanishing cycles and coherent sheaves. Invent. Math. 166, 537–582 (2006) 2. Auroux, D., Katzarkov, L., Orlov, D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. of Math. (2) 167, 867–943 (2008) 3. Ballard, M.: Sheaves on local Calabi-Yau varieties. Preprint, http://arxiv.org/abs/0801.3499v1[math. AG], 2008

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4. Broughton, S.: Milnor number and the topology of polynomial hypersurfaces. Invent. Math. 92, 217–241 (1988) 5. Fukaya, K.: Cyclic symmetry and adic convergence in Lagrangian Floer theory. Preprint, http://arxiv. org/abs/0907.4219v1[math.DG], 2009 6. Futaki, M., Ueda, K.: Homological mirror symmetry for Brieskorn-Pham singularities. Expository article written for the Japan Geometry Symposium, 2009 7. Givental, A.: Homological geometry and mirror symmetry. In: Proceedings of the International Congress of Mathematics, Zürich, Volume 1, Basel: Birkhäuser, 1994, pp. 472–480 8. Gross, M.: Examples of special Lagrangian fibrations. In: Symplectic Geometry and Mirror Symmetry (Seoul), River Edge, NJ: World Sci. Publishing, 2001, pp. 81–109 9. Hori, K., Vafa, C.: Mirror symmetry. Preprint, http://arxiv.org/abs/hep-th/0002222v3, 2000 10. Vafa, C., Hori, K., Iqbal, A.: D-branes and mirror symmetry. Preprint, http://arxiv.org/abs/hep-th/ 0005247v2, 2000 11. Kontsevich, M.: Lectures at ENS Paris, Spring 1998. Set of notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet, H. Randriambololona 12. Kouchnirenko, A.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976) 13. Lefevre, K.: Sur les A∞ -catégories. PhD thesis, Université Paris 7, 2002 14. Segal, E.: The A∞ deformation theory of a point and the derived categories of local Calabi-Yaus. J. Algebra 320(8), 3232–3268 (2008) 15. Seidel, P.: More about vanishing cycles and mutation. In: K. Fukaya, Y.-G. Oh, K. Ono, G. Tian, eds., Symplectic Geometry and Mirror Symmetry (Proceedings of the 4th KIAS Annual International Conference), River Edge, NJ: World Scientific, 2001, pp. 429–465 16. Seidel, P.: A∞ -subalgebras and natural transformations. Homology, Homotopy Appl. 10, 83–114 (2008) 17. Seidel, P.: Fukaya categories and Picard-Lefschetz theory. Zürich: European Math. Soc., 2008 18. Tradler, T.: Infinity-inner-products on A-infinity-algebras. J. Homotopy Relat. Struct. 3, 245–271 (2008) 19. Ueda, K.: Homological mirror symmetry for toric del Pezzo surfaces. Commun. Math. Phys. 264, 71–85 (2006) Communicated by A. Kapustin

Commun. Math. Phys. 297, 529–551 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0969-z

Communications in

Mathematical Physics

Z-Actions on AH Algebras and Z2 -Actions on AF Algebras Hiroki Matui Graduate School of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan. E-mail: [email protected] Received: 17 July 2009 / Accepted: 21 September 2009 Published online: 2 December 2009 – © Springer-Verlag 2009

Abstract: We consider Z-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two Z-actions with the Rohlin property on such a C ∗ -algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer Z2 -actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants. 1. Introduction Classification of group actions is one of the most fundamental subjects in the theory of operator algebras. For AFD factors, a complete classification is known for actions of countable amenable groups. However, classification of automorphisms or group actions on C ∗ -algebras is still a far less developed subject, partly because of K -theoretical difficulties. For AF and AT algebras, A. Kishimoto [9–12] showed the Rohlin property for a certain class of automorphisms and obtained a cocycle conjugacy result. Following the strategy developed by Kishimoto, H. Nakamura [24] showed that aperiodic automorphisms on unital Kirchberg algebras are completely classified by their K K -classes up to K K -trivial cocycle conjugacy. As for Z N -actions, Nakamura [23] introduced the notion of the Rohlin property and classified product type actions of Z2 on UHF algebras. T. Katsura and the author [8] gave a complete classification of uniformly outer Z2 -actions on UHF algebras by using the Rohlin property. For Kirchberg algebras, M. Izumi and the author [7] classified a large class of Z2 -actions and also showed the uniqueness of Z N -actions on the Cuntz algebras O2 and O∞ . The present article is a continuation of these works. In the first half of this paper, we study single automorphisms (i.e. Z-actions) on a unital simple classifiable AH algebra. More precisely, we prove the Rohlin type theorem

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(Theorem 4.8) for an automorphism α of a unital simple AH algebra A with real rank zero and slow dimension growth under the assumption that αr is approximately inner for some r ∈ N and A has finitely many extremal tracial states. Furthermore, we also prove that if two automorphisms α and β of a unital simple AH algebra with real rank zero and slow dimension growth have the Rohlin property and α ◦ β −1 is asymptotically inner, then the two Z-actions generated by α and β are cocycle conjugate (Theorem 4.9). These results are generalizations of Kishimoto’s work for AF and AT algebras ([9–12]). For the proofs, we need to improve some of the arguments in [11,12] concerning projections and unitaries in central sequence algebras. As a byproduct, it will be also shown that for A as above, the central sequence algebra Aω satisfies Blackadar’s second fundamental comparability question (Proposition 3.8). In the latter half of the paper, we study Z2 -actions on a unital simple AF algebra A with a unique trace. We first show a Z-equivariant version of the Rohlin type theorem for single automorphisms (Theorem 5.5), and as its corollary we obtain the Rohlin type theorem for a Z2 -action ϕ : Z2  A such that ϕ(r,0) and ϕ(0,s) are approximately inner for some r, s ∈ N (Corollary 5.6). This is a generalization of [23, Theorem 3]. Next, by using a Z-equivariant version of the Evans-Kishimoto intertwining argument [4], we classify uniformly outer locally K K -trivial Z2 -actions on A up to K K -trivial cocycle conjugacy (Theorem 6.6). This is a generalization of [8, Theorem 6.5]. We remark that K K -triviality of α ∈ Aut(A) is equivalent to α being approximately inner, because A is assumed to be AF. For classification invariants, we employ the OrderExt group introduced in [13]. The crossed product of A by the first generator ϕ(1,0) is known to be a unital simple AT algebra with real rank zero. The second generator ϕ(0,1) naturally extends to an automorphism of the crossed product and its OrderExt class gives the invariant of the Z2 -action ϕ. However we do not know the precise range of the invariant in general. 2. Preliminaries We collect notations and terminologies relevant to this paper. For a Lipschitz continuous map f between metric spaces, Lip( f ) denotes the Lipschitz constant of f . Let A be a unital C ∗ -algebra. For a, b ∈ A, we mean by [a, b] the commutator ab − ba. For a unitary u ∈ A, the inner automorphism induced by u is written by Ad u. An automorphism α ∈ Aut(A) is called outer, when it is not inner. A single automorphism α is often identified with the Z-action induced by α. An automorphism α of A is said to be asymptotically inner, if there exists a continuous family of unitaries {u t }t∈[0,∞) in A such that α(a) = lim Ad u t (a) t→∞

for all a ∈ A. When there exists a sequence of unitaries {u n }n∈N in A such that α(a) = lim Ad u n (a) n→∞

for all a ∈ A, α is said to be approximately inner. The set of approximately inner automorphisms of A is denoted by Inn(A). Two automorphisms α and β are said to be asymptotically (resp. approximately) unitarily equivalent if α ◦ β −1 is asymptotically (resp. approximately) inner. The set of tracial states on A is denoted by T (A). We mean by Aff(T (A)) the space of R-valued affine continuous functions on T (A). The dimension map D A : K 0 (A) → Aff(T (A)) is defined by D A ([ p])(τ ) = τ ( p). For a

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homomorphism ρ between C ∗ -algebras, K 0 (ρ) and K 1 (ρ) mean the induced homomorphisms on K -groups. As for group actions on C ∗ -algebras, we freely use the terminology and notation described in [7, Def. 2.1]. Let A be a separable C ∗ -algebra and let ω ∈ βN \ N be a free ultrafilter. We set c0 (A) = {(an ) ∈ ∞ (N, A) | lim an  = 0},

A∞ = ∞ (N, A)/c0 (A),

cω (A) = {(an ) ∈ ∞ (N, A) | lim an  = 0},

Aω = ∞ (N, A)/cω (A).

n→∞ n→ω

We identify A with the C ∗ -subalgebra of A∞ (resp. Aω ) consisting of equivalence classes of constant sequences. We let A∞ = A∞ ∩ A ,

Aω = Aω ∩ A

and call them the central sequence algebras of A. A sequence (xn )n ∈ ∞ (N, A) is called a central sequence if [a, xn ] → 0 as n → ∞ for all a ∈ A. A central sequence is a representative of an element in A∞ . An ω-central sequence is defined in a similar way. When α is an automorphism on A or an action of a discrete group on A, we can consider its natural extension on A∞ , Aω , A∞ and Aω . We denote it by the same symbol α. Next, we would like to recall the definition of uniform outerness introduced by Kishimoto and the definition of the Rohlin property of Z N -actions introduced by Nakamura. Definition 2.1 ([9, Def. 1.2]). An automorphism α of a unital C ∗ -algebra A is said to be uniformly outer if for any a ∈ A, any non-zero projection  p ∈ A and any ε > 0, there exist projections p1 , p2 , . . . , pn in A such that p = pi and  pi aα( pi ) < ε for all i = 1, 2, . . . , n. We say that an action α of a discrete group on A is uniformly outer if αg is uniformly outer for every element g of the group other than the identity element. Let N be a natural number. Let ξ1 , ξ2 , . . . , ξ N be the canonical basis of Z N , that is, ξi = (0, 0, . . . , 1, . . . , 0, 0), where 1 is in the i th component. For m = (m 1 , m 2 , . . . , m N ) and n = (n 1 , n 2 , . . . , n N ) in Z N , m ≤ n means m i ≤ n i for all i = 1, 2, . . . , N . For m = (m 1 , m 2 , . . . , m N ) ∈ N N , we let mZ N = {(m 1 n 1 , m 2 n 2 , . . . , m N n N ) ∈ Z N | (n 1 , n 2 , . . . , n N ) ∈ Z N }. For simplicity, we denote Z N /mZ N by Zm . Moreover, we may identify Zm = Z N /mZ N with {(n 1 , n 2 , . . . , n N ) ∈ Z N | 0 ≤ n i ≤ m i −1 ∀i = 1, 2, . . . , N }. Definition 2.2 ([23, Def. 1]). Let ϕ be an action of Z N on a unital C ∗ -algebra A. Then ϕ is said to have the Rohlin property, if for any m ∈ N there exist R ∈ N and m (1) , m (2) , . . . , m (R) ∈ N N with m (1) , . . . , m (R) ≥ (m, m, . . . , m) satisfying the following: For any finite subset F of A and ε > 0, there exists a family of projections eg(r )

(r = 1, 2, . . . , R, g ∈ Zm (r ) )

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in A such that R   r =1 g∈Zm (r )

(r )

eg(r ) = 1, [a, eg(r ) ] < ε, ϕξi (eg(r ) ) − eg+ξi  < ε

for any a ∈ F, r = 1, 2, . . . , R, i = 1, 2, . . . , N and g ∈ Zm (r ) , where g + ξi is understood modulo m (r ) Z N . It is clear that if ϕ : Z N  A has the Rohlin property, then ϕ is uniformly outer. We recall the definition of tracial rank zero introduced by H. Lin. Definition 2.3 ([14,15]). A unital simple C ∗ -algebra A is said to have tracial rank zero if for any finite subset F ⊂ A, any ε > 0 and any non-zero positive element x ∈ A there exists a finite dimensional subalgebra B ⊂ A with p = 1 B satisfying the following: (1) [a, p] < ε for all a ∈ F. (2) The distance from pap to B is less than ε for all a ∈ F. (3) 1 − p is Murray-von Neumann equivalent to a projection in x Ax. In [16], H. Lin gave a classification theorem for unital separable simple nuclear C ∗ -algebras with tracial rank zero which satisfy the UCT ([16, Theorem 5.2]). Indeed, the class of such C ∗ -algebras agrees with the class of all unital simple AH algebras with real rank zero and slow dimension growth. 3. Central Sequences Lemma 3.1. Let A be a unital separable approximately divisible C ∗ -algebra. Then for any m ∈ N, there exists a unital embedding of Mm ⊕ Mm+1 into A∞ . Proof. Let l = m 2 − 1. For any finite subset F ⊂ A and ε > 0, there exists a unital finite dimensional subalgebra B ⊂ A such that every central summand of B is at least l × l and [a, b] < ε for any a ∈ F and b ∈ B with b ≤ 1. It is easy to find a unital subalgebra C of B such that C ∼  = Mm ⊕ Mm+1 , which completes the proof.  The following is Lemma 3.6 of [11]. Lemma 3.2. Let A be a unital simple AT algebra with real rank zero. For any finite subset F ⊂ A and ε > 0, there exist a finite subset G ⊂ A, δ > 0 and k ∈ N such that the following holds. If p, q ∈ A are projections satisfying k[ p] ≤ [q], [a, p] < δ and [a, q] < δ for all a ∈ G, then there exists a partial isometry v ∈ A such that v ∗ v = p, vv ∗ ≤ q and [a, v] < ε for all a ∈ F. We generalize the lemma above to AH algebras. Lemma 3.3. The above lemma also holds for any unital simple AH algebra with slow dimension growth and real rank zero. Proof. Let A be a unital simple AH algebra with slow dimension growth and real rank zero and let Q be the UHF algebra such that K 0 (Q) ∼ = Q. By the classification theorem ([1,2,5]), A ⊗ Q is a unital simple AT algebra with real rank zero. Let F ⊂ A be a finite subset and ε > 0. Applying the lemma above to {a ⊗ 1 | a ∈ F} ⊂ A ⊗ Q and

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ε/2 > 0, we obtain a finite subset G ⊂ A ⊗ Q, δ > 0 and k ∈ N. We may assume G = {a ⊗ 1 | a ∈ G 0 } ∪ {1 ⊗ b | b ∈ G 1 }, where G 0 and G 1 are finite subsets of A and Q, respectively. We may further assume that G 0 contains F and δ is less than ε/2. We will prove that G 0 , δ and 4k meet the requirement. Suppose that p, q ∈ A are non-zero projections satisfying 4k[ p] ≤ [q], [a, p] < δ and [a, q] < δ for all a ∈ G 0 . By Lemma 3.1, M2 ⊕ M3 embeds into A∞ . Hence there exist a projection r and a partial isometry s such that r ≤ q, s ∗ s = r, ss ∗ ≤ q − r, 4[r ] > [q] and [a, s] < δ, [a, r ] < δ ∀a ∈ G 0 . From 4k[ p] ≤ [q] < 4[r ], we get k[ p] < [r ]. It follows that there exists a partial isometry u ∈ A ⊗ Q such that u ∗ u = p ⊗ 1, uu ∗ ≤ r ⊗ 1 and [a ⊗ 1, u] < ε/2 for all a ∈ F. We may assume that u belongs to some A ⊗ Mm ⊂ A ⊗ Q. Put u = (u i, j )1≤i, j≤m . Define w = (wi, j )1≤i, j≤m+1 ∈ A ⊗ Mm+1 by ⎧ ⎪ if 1 ≤ i, j ≤ m ⎨u i, j wi, j = su i,1 if i = m + 1 and j = m + 1 ⎪ ⎩0 if i = m + 1. It is not so hard to see that w∗ w = p ⊗ 1 ∈ A ⊗ Mm+1 and ww ∗ ≤ q ⊗ 1 ∈ A ⊗ Mm+1 . Moreover, one has [a ⊗ 1, w] < ε for all a ∈ F. Let v = u ⊕ w ∈ A ⊗ (Mm ⊕ Mm+1 ). Then v ∗ v = p ⊗ 1, vv ∗ ≤ q ⊗ 1 and [a ⊗ 1, v] < ε for all a ∈ F. By [3] (see also [1,5]), A is approximately divisible. By Lemma 3.1, there exists a unital homomorphism from Mm ⊕ Mm+1 to A∞ , and so there exists a unital homomorphism π from A ⊗ (Mm ⊕ Mm+1 ) to A∞ such that π(a ⊗ 1) = a for a ∈ A. It follows that π(v)∗ π(v) = p, π(v)π(v)∗ ≤ q and [a, π(v)] < ε for all a ∈ F.   Remark 3.4. By using the lemma above and [27, Theorem 4.5], one can show the following. Let A be a unital simple AH algebra with real rank zero and slow dimension growth. Then for any α ∈ Aut(A), there exists α˜ ∈ Aut(A) such that α˜ has the Rohlin property in the sense of [12, Def. 4.1] and α˜ is asymptotically unitarily equivalent to α. The following is a well-known fact. We have been unable to find a suitable reference in the literature, so we include a proof for completeness. Lemma 3.5. Let A be a unital separable C ∗ -algebra and let ( pn )n be a central sequence of projections. For any extremal trace τ ∈ T (A), one has lim |τ (apn ) − τ (a)τ ( pn )| = 0

n→∞

for all a ∈ A. Proof. First, we deal with the case that there exists ε > 0 such that τ ( pn ) ≥ ε for all n ∈ N. Consider a sequence of states ϕn (a) =

τ (apn ) τ ( pn )

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on A. Let ψ be an accumulation point of {ϕn }n . Since ( pn )n is a central sequence and τ ( pn ) is bounded from below, one can see that ψ is a trace. For any positive element a ∈ A, it is easy to see ϕn (a) ≤ ε−1 τ (a), and so ψ(a) ≤ ε−1 τ (a). Hence, ψ is equal to τ , because τ is extremal. We have shown that any accumulation point of {ϕn } is τ , which implies ϕn converges to τ . Therefore, |τ (apn ) − τ (a)τ ( pn )| goes to zero. Next, we consider the general case. Fix a ∈ A. Take ε > 0 arbitrarily. We would like to show that |τ (apn ) − τ (a)τ ( pn )| is less than ε for sufficiently large n. We may assume a ≤ 1. Put C = {n ∈ N | τ ( pn ) ≥ ε/2}. If n ∈ / C, then evidently |τ (apn ) − τ (a)τ ( pn )| is less than ε. By applying the first part of the proof to ( pn )n∈C , we have |τ (apn ) − τ (a)τ ( pn )| < ε for sufficiently large n ∈ C, thereby completing the proof.   Lemma 3.6. Let A be a unital simple separable C ∗ -algebra with tracial rank zero. Suppose that A has finitely many extremal traces. For any finite subset F ⊂ A and ε > 0, there exist a finite subset G ⊂ A and δ > 0 such that the following hold. If p, q ∈ A are projections satisfying [a, p] < δ, [a, q] < δ ∀a ∈ G and τ ( p) + ε < τ (q) for all τ ∈ T (A), then there exists a partial isometry v ∈ A such that v ∗ v ≤ p, vv ∗ ≤ q, [a, v] < ε ∀a ∈ F and τ ( p

− v ∗ v)

< ε for all τ ∈ T (A).

Proof. The proof is by contradiction. Suppose that the assertion does not hold for a finite subset F ⊂ A and ε > 0. We would have central sequences of projections ( pn )n and (qn )n such that τ ( pn ) + ε < τ (qn ) ∀τ ∈ T (A), n ∈ N and any partial isometry does not meet the requirement. Use tracial rank zero to find a projection e ∈ A and a finite dimensional unital subalgebra E ⊂ e Ae such that the following are satisfied. • For any a ∈ F, [a, e] < ε/4. • For any a ∈ F there exists b ∈ E such that eae − b < ε/4. • τ (1 − e) < ε for all τ ∈ T (A). Since ( pn )n and (qn )n are central sequences of projections, we can find projections pn and qn in A ∩ E such that  pn − pn  → 0 and qn − qn  → 0 as n → ∞. We would like to show that, for sufficiently large n, there exists a partial isometry vn ∈ e Ae ∩ E such that vn∗ vn = epn and vn vn∗ ≤ eqn . Let {e1 , e2 , . . . , em } be a family of minimal central projections in e Ae ∩ E such that e1 + e2 + · · · + em = e. Clearly ei (e Ae ∩ E ) is a unital simple C ∗ -algebra with tracial rank zero and the space of tracial states on ei (e Ae ∩ E ) is naturally identified with T (A). By Lemma 3.5, for sufficiently large n, one has τ (ei pn ) < τ (ei qn ) for all τ ∈ T (A) and i = 1, 2, . . . , m, because A has finitely many extremal traces. Hence [ei pn ] ≤ [ei qn ] in K 0 (ei (e Ae ∩ E )). It follows that there exists a partial isometry vn ∈ e Ae ∩ E such that vn∗ vn = epn and vn vn∗ ≤ eqn . Besides, τ ( pn − vn∗ vn ) = τ ( pn (1 − e)) < ε and [a, vn ] < ε for all a ∈ F. This is a contradiction.   By using Lemma 3.6 and 3.3, we can show the following.

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Lemma 3.7. Let A be a unital simple AH algebra with slow dimension growth and real rank zero. Suppose that A has finitely many extremal traces. For any finite subset F ⊂ A and ε > 0, there exist a finite subset G ⊂ A and δ > 0 such that the following hold. If p, q ∈ A are projections satisfying [a, p] < δ, [a, q] < δ ∀a ∈ G and τ ( p) + ε < τ (q) for all τ ∈ T (A), then there exists a partial isometry u ∈ A such that u ∗ u = p, uu ∗ ≤ q and [a, u] < ε for all a ∈ F. Proof. Notice that A has tracial rank zero by [15, Prop. 2.6]. Suppose that a finite subset F ⊂ A and ε > 0 are given. By applying Lemma 3.3 to F and ε/2, we obtain a finite subset F1 ⊂ A, ε1 > 0 and k ∈ N. By applying Lemma 3.6 to F ∪ F1 ∪ F1∗ and min{ε1 /4, ε/k, ε/2}, we obtain a finite subset G ⊂ A, δ > 0. We would like to show that G ∪ F1 and min{δ, ε1 /2} meet the requirement. Suppose that p, q ∈ A are projections satisfying [a, p] < min{δ, ε1 /2}, [a, q] < min{δ, ε1 /2} ∀a ∈ G ∪ F1 and τ ( p) + ε < τ (q) for all τ ∈ T (A). By Lemma 3.6, there exists a partial isometry v ∈ A such that v ∗ v ≤ p, vv ∗ ≤ q, [a, v] < min{ε1 /4, ε/2} ∀a ∈ F ∪ F1 ∪ F1∗ and τ ( p − v ∗ v) < ε/k for all τ ∈ T (A). Let p = p − v ∗ v and q = q − vv ∗ . One has τ (q ) = τ (q − vv ∗ ) = τ (q) − τ ( p) + τ ( p − v ∗ v) > ε, and so k[ p ] ≤ [q ]. One also has [a, p ] < ε1 and [a, q ] < ε1 for all a ∈ F1 . By Lemma 3.3, we obtain a partial isometry w ∈ A such that w∗ w = p , ww ∗ ≤ q and [a, w] < ε/2 for all a ∈ F. Put u = v + w. It is easy to see u ∗ u = p, uu ∗ ≤ q and [a, u] < ε for all a ∈ F.   Any tracial state τ ∈ T (A) naturally extends to a tracial state on Aω and we write it by τω . Proposition 3.8. Let A be a unital simple AH algebra with slow dimension growth and real rank zero. Suppose that A has finitely many extremal traces. If p, q ∈ Aω are projections satisfying τω ( p) < τω (q) for all τ ∈ T (A), then there exists v ∈ Aω such that v ∗ v = p and vv ∗ ≤ q. In particular, Aω satisfies Blackadar’s second fundamental comparability question. Proof. Let ( pn )n and (qn )n be ω-central sequences of projections such that lim τ ( pn ) < lim τ (qn )

n→ω

n→ω

for all τ ∈ T (A). Since A has finitely many extremal traces, there exists ε > 0 such that C = {n ∈ N | τ ( pn ) + ε < τ (qn ) for all τ ∈ T (A)} ∈ ω. We choose an increasing sequence {Fm }∞ m=1 of finite subsets of A whose union is dense in A. By applying Lemma 3.7 to Fm and ε/m, we obtain a finite subset G m ⊂ A and δm > 0. We may assume that {G m }m is increasing and {δm }m is decreasing. Put Cm = {n ∈ C | [a, pn ] < δm and [a, qn ] < δm for all a ∈ G m } ∈ ω.

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For n ∈ Cm \Cm+1 , by Lemma 3.7, there exists a partial isometry u n such that u ∗n u n = pn , u n u ∗n ≤ qn and [a, u n ] < ε/m for all a ∈ Fm . For n ∈ N \ C1 , we let u n = 0. Then (u n )n is a desired ω-central sequence of partial isometries.   The following is Lemma 4.4 of [12]. Lemma 3.9. Let A be a unital simple AT algebra with real rank zero. For any finite subset F ⊂ A and ε > 0, there exist a finite subset G ⊂ A and δ > 0 such that the following holds. If u : [0, 1] → A is a path of unitaries satisfying [a, u(t)] < δ for all a ∈ G and t ∈ [0, 1], then there exists a path of unitaries v : [0, 1] → A satisfying v(0) = u(0), v(1) = u(1), [a, v(t)] < ε ∀a ∈ F, t ∈ [0, 1] and Lip(v) < 5π + 1. We generalize the lemma above to AH algebras. Lemma 3.10. The above lemma also holds for any unital simple AH algebra with slow dimension growth and real rank zero, the Lipschitz constant being bounded by 11π . Proof. Let A be a unital simple AH algebra with slow dimension growth and real rank zero and let Q be the UHF algebra such that K 0 (Q) ∼ = Q. By the classification theorem ([1,2,5]), A ⊗ Q is a unital simple AT algebra with real rank zero. Let F ⊂ A be a finite subset and ε > 0. We may assume that F is contained in the unit ball of A. Applying the lemma above to {a ⊗ 1 | a ∈ F} ⊂ A ⊗ Q and ε/2 > 0, we obtain a finite subset G ⊂ A ⊗ Q and δ > 0. We may assume G = {a ⊗ 1 | a ∈ G 0 } ∪ {1 ⊗ b | b ∈ G 1 }, where G 0 and G 1 are finite subsets of A and Q, respectively. We will prove that G 0 and δ meet the requirement. Suppose that u : [0, 1] → A is a path of unitaries satisfying [a, u(t)] < δ for all a ∈ G 0 and t ∈ [0, 1]. Choose N ∈ N so that u( Nk ) − u( k+1 N ) < ε/2 for every k = 0, 1, . . . , N −1. Put u k = u(k/N ). By the lemma above, we can find continuous paths x : [0, 1] → A ⊗ Q, yk : [0, 1] → A ⊗ Q and z k : [0, 1] → A ⊗ Q for k = 1, 2, . . . , N −1 such that x(0) = u 0 ⊗ 1, x(1) = u N ⊗ 1, [a ⊗ 1, x(t)] < ε/2 ∀a ∈ F, t ∈ [0, 1], yk (0) = u 0 ⊗ 1,

yk (1) = u k ⊗ 1, [a ⊗ 1, yk (t)] < ε/2 ∀a ∈ F, t ∈ [0, 1],

z k (0) = u k ⊗ 1, z k (1) = u N ⊗ 1, [a ⊗ 1, z k (t)] < ε/2 ∀a ∈ F, t ∈ [0, 1], and Lip(x), Lip(yk ), Lip(z k ) are less than 5π + 1. We may assume that the ranges of x, yk , z k are contained in A ⊗ Mn for some Mn ⊂ Q. Put m = n(N − 1). We would like to construct a path of unitaries v : [0, 1] → A ⊗ (Mm ⊕ Mm+1 ) such that Lip(v) < 11π , v(0) = u 0 ⊗ 1, v(1) = u N ⊗ 1 and [a ⊗ 1, v(t)] < ε for all a ∈ F and t ∈ [0, 1]. First, let x˜ : [0, 1] → A ⊗ Mm be the direct sum of N −1 copies of x : [0, 1] → A ⊗ Mn . Next, by using y1 , y2 , · · · , y N −1 , we can find a path y˜ : [0, 1] → A ⊗ Mm+1 such that

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y˜ (0) = u 0 ⊗ 1, y˜ (1) = diag(u 0 , u 1 , · · · , u 1 , u 2 , · · · , u 2 , · · · , u N −1 , · · · , u N −1 )     

 n

n

n

[a ⊗ 1, y˜ (t)] < ε/2 ∀a ∈ F, t ∈ [0, 1] and Lip( y˜ ) < 5π + 1. Likewise, by using z 1 , z 2 , . . . , z N −1 , we can find a path z˜ : [0, 1] → A ⊗ Mm+1 such that z˜ (0) = diag(u 1 , · · · , u 1 , u 2 , · · · , u 2 , · · · , u N −1 , · · · , u N −1 , u N )     

 n

n

n

z˜ (1) = u N ⊗ 1, [a ⊗ 1, z˜ (t)] < ε/2 ∀a ∈ F, t ∈ [0, 1] and Lip(˜z ) < 5π + 1. Since  y˜ (1) − z˜ (0) < ε/2, if ε is sufficiently small, there exists a path w : [0, 1] → Mm+1 such that w(0) = u 0 ⊗ 1, w(1) = u N ⊗ 1, [a ⊗ 1, w(t)] < ε ∀a ∈ F, t ∈ [0, 1], and Lip(w) < 11π . Then v = x˜ ⊕ w is the desired path. By [3] (see also [1,5]), A is approximately divisible. By Lemma 3.1, there exists a unital homomorphism from Mm ⊕ Mm+1 to A∞ , and so there exists a unital homomorphism π from A ⊗ (Mm ⊕ Mm+1 ) to A∞ such that π(a ⊗ 1) = a for a ∈ A. It follows that the path v˜ : [0, 1]  t → π(v(t)) ∈ A∞ satisfies ˜ = u N , [a, v(t)] ˜ < ε ∀a ∈ F, t ∈ [0, 1] v(0) ˜ = u 0 , v(1) and Lip(v) ˜ < 11π , which completes the proof.   4. Automorphisms of AH Algebras In this section, we discuss the Rohlin property of automorphisms of AH algebras. For a ∈ A, we define a2 = sup τ (a ∗ a)1/2 . τ ∈T (A)

If A is simple and T (A) is non-empty, then ·2 is a norm. Proposition 4.1. Let A be a unital simple separable C ∗ -algebra with tracial rank zero and let  ⊂ Aut(A) be a finite subset containing the identity. Suppose that there exists a sequence of projections (en )n in A satisfying the following property: (1) γ (en )γ (en )2 → 0 for any γ , γ ∈  such that γ = γ . (2) 1 − γ ∈ γ (en )2 → 0. (3) For every a ∈ A, we have [a, en ]2 → 0. Then there exists a sequence of projections ( f n )n in A satisfying the following. (1) γ ( f n )γ ( f n ) → 0 for any γ , γ ∈  such that γ = γ . (2) en − f n 2 → 0. (3) For every a ∈ A, we have [a, f n ] → 0.

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Proof. This is almost the same as [19, Proposition 5.4]. In [19, Proposition 5.4], the finite set  is assumed to be an orbit of a single automorphism γ of finite order. The proof, however, does not need this.   The following is a variant of [9, Lemma 3.1] and [26, Theorem 2.17]. See [26, Definition 2.1] for the definition of the tracial Rohlin property. Theorem 4.2. Let A be a unital simple separable C ∗ -algebra with tracial rank zero. Suppose that A has finitely many extremal traces. Let α be an automorphism of A such that α m is uniformly outer for any m ∈ N. Then α has the tracial Rohlin property. Proof. Let {τ1 , . . . , τd } be the set of extremal tracial states of A and let (πi , Hi ) be the GNS representation associated with τi . It is well-known that πi (A) is a hyperfinite d πi . Note that, for a bounded sequence II1 -factor (see [26, Lemma 2.16]). Let ρ = i=1 (an )n in A, ρ(an ) converges to zero in the strong operator topology if and only if an 2 d converges to zero. We regard A as a subalgebra of N = ρ(A) ∼ = i=1 πi (A) and denote the extension of the automorphism α to N by α. ¯ Let k be the minimum positive integer such that τi ◦ α k = τi for all i = 1, 2, . . . , d. In the same way as [9, Lemma ( j) ( j) 3.1], for any l ∈ N, one can find a sequence { f 0 , . . . , f kl−1 } of orthogonal families of kl−1 ( j) projections in N such that i=0 f i = 1, ( j)

[a, f i ] → 0 ∀a ∈ A, ( j)

( j)

α( ¯ f i ) − f i+1 → 0 ∀i = 0, 1, . . . , kl−1 ( j)

( j)

in the strong operator topology as j → ∞, where f kl = f 0 . By [26, Lemma 2.15], ( j) we may replace the projections f i with projections of A. From the proposition above, we can conclude that α has the tracial Rohlin property.   The following is a well-known fact, but we include the proof for the reader’s convenience. Lemma 4.3. Let A be a unital separable C ∗ -algebra and let α ∈ Inn(A). For any separable subset C ⊂ A∞ , there exists a unitary u ∈ A∞ such that uxu ∗ = α(x) for all x ∈ C. Proof. We choose an increasing sequence {Fn }n∈N of finite subsets of A whose union is dense in A. We can find a sequence of unitaries (vn )n in A such that vn avn∗ − α −1 (a) < n −1 for all a ∈ Fn , because α is approximately inner. We may assume that C is countable. Let C = {x1 , x2 , . . . } and let (xi, j ) j be a representative of xi . There exists an increasing sequence (m(n))n of natural numbers such that [vn , xi, j ] < n −1 ∀ j ≥ m(n) for any i = 1, 2, . . . , n, because (xi, j ) j is a central sequence. Since α is in Inn(A), one can find a sequence of unitaries (wn )n in A such that wn awn∗ − α(a) < n −1

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for all a in α −1 (Fn ) ∪ {xi, j | i = 1, . . . , n, m(n) ≤ j < m(n+1)}. For j ∈ N, find n ∈ N so that m(n) ≤ j < m(n+1) and define a unitary u j by u j = wn vn . It is easy to see [u j , a] < 2/n ∀a ∈ Fn and u j xi, j u ∗j − α(xi, j ) < 2/n ∀i = 1, . . . , n, and so the proof is completed.   We quote the following theorem by Lin and Osaka from [20]. See [20, Def. 2.4] for the definition of the tracial cyclic Rohlin property. Since we need to discuss an ‘equivariant version’ of this theorem later, we would like to include the proof briefly. Theorem 4.4 ([20, Theorem 3.4]). Let A be a unital simple separable C ∗ -algebra and suppose that the order on projections in A is determined by traces. Suppose that α ∈ Aut(A) has the tracial Rohlin property. If αr is in Inn(A) for some r ∈ N, then α has the tracial cyclic Rohlin property. Proof. Take m ∈ N and ε > 0 arbitrarily. Let l = r (mr + 1). Since α has the tracial Rohlin property and the order on projections is determined by traces, there exists a central sequence of projections (en )n such that lim en α i (en ) = 0 ∀i = 1, 2, . . . , l−1

n→∞

and lim τ (1 − (en + α(en ) + · · · + αl−1 (en )) = 0 ∀τ ∈ T (A).

n→∞

Let e ∈ A∞ be the image of (en )n and define e˜ by e˜ =

r −1 

α i(mr +1) (e).

i=0

It follows from the lemma above that there exists a partial isometry v ∈ A∞ such that v ∗ v = e˜ and vv ∗ = α(e). ˜ The C ∗ -algebra C generated by v, α(v), . . . , α mr −1 (v) is isomorphic to Mmr +1 and its unit is e˜ + α(e) ˜ + · · · + α mr (e). ˜ The rest of the proof is exactly the same as that of [9, Lemma 4.3] and we omit it.   Remark 4.5. The following was shown by Lin in [17, Theorem 3.4]. Let A be a unital simple separable C ∗ -algebra with tracial rank zero. Suppose that α ∈ Aut(A) has the tracial cyclic Rohlin property and that there exists r ∈ N such that K 0 (αr )|G = id G for some subgroup G ⊂ K 0 (A) for which D A (G) is dense in D A (K 0 (A)). Then A α Z has tracial rank zero. By using Lemma 3.3, we can show the following.

540

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Lemma 4.6. Let A be a unital simple AH algebra with slow dimension growth and real rank zero. Suppose that α ∈ Aut(A) has the tracial cyclic Rohlin property and that there exists r ∈ N such that τ ◦ αr = τ for any τ ∈ T (A). Then, for any m ∈ N, there exist projections e, f ∈ A∞ and a partial isometry v ∈ A∞ such that v ∗ v = f, vv ∗ ≤ e,

f +

mr 

α i (e) = 1

i=0

and

α mr +1 (e)

= e.

Proof. Suppose that we are given m ∈ N. Let l = r (mr + 1). Since α has the tracial cyclic Rohlin property, we can find central sequences of projections (en )n and ( f n )n such that fn +

l−1 

α i (en ) → 1, en − αl (en ) → 0,

i=0

sup τ ( f n ) → 0

τ ∈T (A)

as n → ∞. There exists a central sequence of projections (e˜n )n such that lim e˜n −

n→∞

r −1 

α i(mr +1) (en ) = 0.

i=0

Then fn +

mr 

α i (e˜n ) → 1, e˜n − α mr +1 (e˜n ) → 0

i=0

as n → ∞. It is also easy to see τ (α(e˜n )) = τ (e˜n ) for all τ ∈ T (A), and so τ (e˜n ) goes to (mr + 1)−1 for all τ ∈ T (A). Therefore, by Lemma 3.3, one can find a central sequence of partial isometries (vn )n such that vn∗ vn = f n and vn vn∗ ≤ e˜n for sufficiently large n, which completes the proof.   By using the lemma above, we can show the following theorem. Theorem 4.7. Let A be a unital simple AH algebra with slow dimension growth and real rank zero. Suppose that α ∈ Aut(A) has the tracial cyclic Rohlin property. If there exists r ∈ N such that τ ◦ αr = τ for any τ ∈ T (A), then α has the Rohlin property. Proof. Suppose that we are given M ∈ N. Choose a natural number m ∈ N so that m ≥ M and m ≡ 1 (mod r ). Let k, l be sufficiently large natural numbers satisfying k ≡ l ≡ 1 (mod r ). By the lemma above, we can find projections e, f ∈ A∞ and a partial isometry v ∈ A∞ such that v ∗ v = f, vv ∗ ≤ e,

f +

klm−1 

α i (e) = 1

i=0

˜ w ∈ A∞ by and α klm (e) = e. Define e, e˜ =

k−1  i=0

α ilm (e)

and

k−1 1  ilm w=√ α (v). k i=0

Z-Actions on AH Algebras and Z2 -Actions on AF Algebras

541

Then e˜ is a projection and w is a partial isometry satisfying f +

lm−1 

α i (e) ˜ = 1, αlm (e) ˜ = e˜

i=0

and 2 w ∗ w = f, ww ∗ ≤ e, ˜ αlm (w) − w ≤ √ . k Let D be the C ∗ -algebra generated by w, α(w), . . . , αlm−1 (w). Then D is isomorphic to Mlm+1 and the unit 1 D of D is equal to f +ww∗ +· · ·+αlm−1 (ww ∗ ). From the spectral property of α restricted to D, if k and l are sufficiently large, we can obtain projections p0 , . . . , pm−1 , q0 , . . . , qm of D such that m−1  i=1

pi +

m 

qi = 1 D , α( pi ) ≈ pi+1 , α(qi ) ≈ qi+1 ,

i=1

where pm = p0 and qm+1 = q0 . We define projections pi in A∞ by pi = pi +

l−1 

α i+ jm (e˜ − ww ∗ ).

j=0 , q0 , . . . , qm meet the requirement. See [9,10] for Then the projections p0 , . . . , pm−1 details.  

Combining the theorems above, we obtain the following theorem which is a generalization of [11, Theorem 2.1]. Theorem 4.8. Let A be a unital simple AH algebra with slow dimension growth and real rank zero and let α ∈ Aut(A). Suppose that A has finitely many extremal traces and that αr is approximately inner for some r ∈ N. Then the following are equivalent. (1) α has the Rohlin property. (2) α m is uniformly outer for any m ∈ N. We can also generalize [12, Theorem 5.1] by using Lemma 3.10 instead of [12, Lemma 4.4]. Theorem 4.9. Let A be a unital simple AH algebra with slow dimension growth and real rank zero. If α, β ∈ Aut(A) have the Rohlin property and α is asymptotically unitarily equivalent to β, then there exist μ ∈ Inn(A) and a unitary u ∈ A such that Ad u ◦ α = μ ◦ β ◦ μ−1 . The proof is similar to that of [12, Theorem 5.1] and we omit it. As an application of the theorems above, we can show the following, which will be used in Sect. 6.

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H. Matui

Lemma 4.10. Let A be a unital simple AF algebra with finitely many extremal traces and let α be an approximately inner automorphism of A such that α m is uniformly outer for all m ∈ N. For any finite subset F ⊂ A and ε > 0, there exist a finite subset G ⊂ A and δ > 0 satisfying the following: If u : [0, 1] → A is a path of unitaries such that [a, u(t)] < δ and u(t) − α(u(t)) < δ ∀a ∈ G, t ∈ [0, 1], then there exists a path of unitaries v : [0, 1] → A such that v(0) = u(0), v(1) = u(1), [a, v(t)] < ε, v(t) − α(v(t)) < ε, ∀a ∈ F, t ∈ [0, 1] and Lip(v) < 2π . of Mn (C) such that Sp(xn ) = {ωk | k = 0, 1, . . . , n−1}, Proof. Let xn be a unitary √ where ω = exp(2π −1/n). One can find an increasing sequence {An }∞ n=1 of unital finite dimensional subalgebras of A such that n An is dense in A and there exists a unital embedding πn : Mn ⊕ Mn+1 → An+1 ∩ A n . Let yn = πn (xn ⊕ xn+1 ). Define an automorphism σ of A by σ = limn→∞ Ad(y1 y2 . . . yn ). Then σ is approximately inner and σ m is uniformly outer for all m ∈ N. We would like to show that the assertion holds for σ . Suppose that we are given F ⊂ A and ε > 0. Without loss of generality, we may assume that there exists n ∈ N such that F is contained in the unit ball of An . Applying [8, Lemma 4.2] to ε/2, we obtain a positive real number δ1 > 0. We may assume δ1 is less than min{2, ε}. Choose a finite subset G ⊂ A and δ2 > 0 so that if z : [0, 1] → A is a path of unitaries such that [a, z(t)] < δ2 for all a ∈ G and t ∈ [0, 1], then there exists a path of unitaries z˜ : [0, 1] → A ∩ A n such that z(t) − z˜ (t) < δ1 /6. Let δ = min{δ1 /6, δ2 }. Suppose that u : [0, 1] → A is a path of unitaries such that [a, u(t)] < δ and u(t) − σ (u(t)) < δ ∀a ∈ G, t ∈ [0, 1]. ˜ < δ1 /6. By the choice of δ, we can find u˜ : [0, 1] → A ∩ A n such that u(t) − u(t) We may assume that there exists m > n such that the range of u˜ is contained in Am . Put y = yn yn+1 . . . ym−1 ∈ Am ∩ A n . Then [y, u(t)] ˜ = u(t) ˜ − σ (u(t)) ˜ < u(t) − σ (u(t)) + δ1 /3 < δ + δ1 /3 ≤ δ1 /2 for every t ∈ [0, 1]. Hence [y, u(t) ˜ u(0) ˜ ∗ ] is less than δ1 . It follows from [8, Lemma 4.2] that one can find a path of unitaries w : [0, 1] → Am ∩ A n such that w(0) = 1, w(1) = u(1) ˜ u(0) ˜ ∗ , Lip(w) ≤ π + ε and [y, w(t)] < ε/2 for every t ∈ [0, 1]. Note that yw(t)y ∗ is equal to σ (w(t)). By perturbing w(t)u(0) a little bit, the required v : [0, 1] → A is obtained. Suppose that α is an approximately inner automorphism of A such that α m is uniformly outer for all m ∈ N. By Theorem 4.8 and Theorem 4.9, there exist μ ∈ Aut(A) and a unitary u ∈ A such that Ad u ◦ α = μ ◦ σ ◦ μ−1 . Moreover, one can choose u arbitrarily close to 1, because A is AF (see [6,12]). Therefore the assertion also holds for α.   Remark 4.11. In the proof of the lemma above, it is easily seen that (the Z-action generated by) σ is asymptotically representable ([7, Def. 2.2]). Hence the automorphism α stated in the lemma above is also asymptotically representable. Besides, it is not so hard to see that the crossed product C ∗ -algebra A σ Z is a unital simple AT algebra with real rank zero. Therefore, A α Z is a unital simple AT algebra with real rank zero, too.

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5. The Rohlin Property of Z2 -Actions on AF Algebras In this section, we would like to show that certain Z2 -actions on an AF algebra have the Rohlin property. This is a generalization of Nakamura’s theorem [23, Theorem 3]. Throughout this section, we keep the following setting. Let A be a unital simple separable C ∗ -algebra with tracial rank zero and suppose that A has a unique tracial state τ . Suppose that automorphisms α, β ∈ Aut(A) and a unitary w ∈ A satisfy β ◦ α = Ad w ◦ α ◦ β αm

and induce

◦ β n is uniformly outer a Z2 -action on A∞ .

for all (m, n) ∈ Z2 \ {(0, 0)}. We remark that α and β

Lemma 5.1. For any m 1 , m 2 ∈ N, there exists a central sequence of projections (en )n in A such that lim τ (en ) =

n→∞

1 m1m2

and lim β j (α i (en ))β l (α k (en )) = 0

n→∞

for all (i, j) = (k, l) in {(i, j) | 0 ≤ i ≤ m 1 −1, 0 ≤ j ≤ m 2 −1}. Proof. Set I = {(i, j) | 0 ≤ i ≤ m 1 −1, 0 ≤ j ≤ m 2 −1}. Let (πτ , Hτ ) be the GNS representation associated with τ . It is well-known that πτ (A) is a hyperfinite II1 -factor (see [26, Lemma 2.16]). For (i, j) ∈ Z2 , we put ϕ(i, j) = α i ◦β j . Then ϕ : Z2 → Aut(A) ¯ Since α m ◦ β n is is a cocycle action of Z2 . We denote its extension to πτ (A) by ϕ. 2 uniformly outer for all (m, n) ∈ Z \ {(0, 0)}, ϕ¯ is an outer cocycle action of Z2 on πτ (A) . It follows from [25] that there exists a sequence of projections (en )n in πτ (A) such that  β j (α i (en )) → 1, [x, en ] → 0 ∀x ∈ πτ (A) (i, j)∈I

and β j (α i (en ))β l (α k (en )) → 0 ∀(i, j), (k, l) ∈ I with (i, j) = (k, l) in the strong operator topology as n → ∞. By [26, Lemma 2.15], we may replace en with projections in A. Applying Proposition 4.1 to  = {β j ◦ α i | (i, j) ∈ I }, we obtain the conclusion.   Let (en )n be the projections as in the lemma above. If αr is in Inn(A) for some r ∈ N and m 1 is large enough, then we can construct a central sequence of projections (en )n in A such that lim en (en + α(en ) + · · · + α m 1 −1 (en )) − en  = 0,

n→∞

α(en ) ≈ en and τ (en ) ≈ m 1 τ (en ), by using the arguments in [9, Lemma 3.1] (see also [23, Lemma 6]). Consequently, we get the following:

544

H. Matui

Lemma 5.2. If αr is in Inn(A) for some r ∈ N, then for any m ∈ N, there exists a central sequence of projections (en )n in A such that lim τ (en ) =

n→∞

1 , m

lim en − α(en ) = 0

n→∞

and lim en β j (en ) = 0

n→∞

for all j = 1, 2, . . . , m−1. Our next task is to achieve the cyclicity condition β m (en ) ≈ en . Lemma 5.3. Suppose that A is AF. Suppose that a projection e ∈ A∞ and a partial isometry u ∈ A∞ satisfy e = α(e) and e = u ∗ u = uu ∗ . Then there exists a partial isometry w ∈ A∞ such that w ∗ w = ww ∗ = e and u = w ∗ α(w). Proof. Theorem 4.8 tells us that α possesses the Rohlin property. We can modify the standard argument deducing stability from the Rohlin property (see [4,6]) and apply it to the unitary u + (1 − e). We leave the details to the readers.   Lemma 5.4. Suppose that either of the following holds: (1) A is AF, αr is approximately inner for some r ∈ N and β s is approximately inner for some s ∈ N. (2) αr is approximately inner for some r ∈ N and there exist a natural number s ∈ N and a sequence of unitaries (u n )n in A such that lim u n − α(u n ) = 0 and

n→∞

lim u n au ∗n − β s (a) = 0 ∀a ∈ A.

n→∞

Then for any m ∈ N, there exists a central sequence of projections (en )n such that 1 , lim en − α(en ) = 0, m n→∞ for all j = 1, 2, . . . , m−1 and lim τ (en ) =

n→∞

lim en β j (en ) = 0

n→∞

lim en − β m (en ) = 0.

n→∞

Proof. Choose a large natural number l such that l ≡ 1 (mod s). By using Lemma 5.2 and the assumption that β s is in Inn(A) for some s ∈ N, one can find a projection e ∈ A∞ and a partial isometry v ∈ A∞ such that e = α(e), v ∗ v = e, vv ∗ = β(e)

and

eβ j (e) = 0 ∀ j = 1, 2, . . . , l−1

in the same way as the proof of Theorem 4.4. Moreover, we have lim τ (en ) = l −1 ,

n→∞

where (en )n is a representative sequence of e consisting of projections. Note that β j (e) is fixed by α, because e is a central sequence. In the case (2), clearly we may further assume v = α(v). In the case (1), the lemma above applies to v ∗ α(v) and yields w ∈ A∞ satisfying w ∗ w = e, ww ∗ = e and v ∗ α(v) = w ∗ α(w). By replacing v with vw∗ , we get v = α(v), too. Then the conclusion follows from exactly the same argument as [9, Lemma 4.3].  

Z-Actions on AH Algebras and Z2 -Actions on AF Algebras

545

Theorem 5.5. Suppose that the conclusion of Lemma 5.4 holds. Then for any m ∈ N, there exist projections e and f in A∞ such that α(e) = e, α( f ) = f, β m (e) = e, β m+1 ( f ) = f and m−1 

β i (e) +

m 

i=0

β j ( f ) = 1.

j=0

Proof. Let (en )n be the central sequence of projections obtained in Lemma 5.4. Define fn = 1 −

m−1 

β j (en ).

j=0

There exists a sequence of unitaries (u n )n in A such that u n → 1 as n → ∞ and u n α(en ) u ∗n = en for sufficiently large n. The Z-action on en Aen generated by Ad u n ◦ α is uniformly outer, and so it has the tracial Rohlin property by Theorem 4.2 (or [26, Theorem 2.17]). It follows that, for any k ∈ N, there exists a central sequence of projections (e˜n )n such that e˜n ≤ en ,

lim τ (e˜n ) = 1/mk, and

n→∞

lim e˜n α i (e˜n ) = 0 ∀i = 1, 2, . . . , k−1.

n→∞

Let e, f, e˜ ∈ A∞ be the images of (en )n , ( f n )n , (e˜n )n , respectively. By Lemma 3.3, there exists a partial isometry v such that v ∗ v = f and vv ∗ ≤ e. ˜ We define a partial isometry v˜ ∈ A∞ by k−1 1  i α (v). v˜ = √ k i=0

Then one has √ ˜ < 2/ k. v˜ ∗ v˜ = f, v˜ v˜ ∗ ≤ e and v˜ − α(v) By a standard trick on central sequences, we may assume α(v) ˜ = v. ˜ Thus, we have obtained the α-invariant version of the conclusion of Lemma 4.6. We can complete the proof by the same argument as in Theorem 4.7.   The following is a generalization of [23, Theorem 3]. Corollary 5.6. Let ϕ : Z2  A be a Z2 -action on a unital simple AF algebra A with unique trace. When ϕ(r,0) and ϕ(0,s) are approximately inner for some r, s ∈ N, the following are equivalent. (1) ϕ has the Rohlin property. (2) ϕ is uniformly outer. Proof. This immediately follows from Theorem 5.5 and [23, Remark 2] (see also [22, Remark 2.2]).  

546

H. Matui

The next corollary also follows from Theorem 5.5 immediately, because condition (2) of Lemma 5.4 is satisfied in this case. See [7, Def. 2.2] for the definition of approximate representability. Corollary 5.7. Let ϕ : Z2  A be an approximately representable Z2 -action on a unital simple AH algebra A with real rank zero and slow dimension growth. Suppose that A has a unique trace. Then the following are equivalent. (1) ϕ has the Rohlin property. (2) ϕ is uniformly outer. 6. Classification of Certain Z2 -Actions on AF Algebras In this section, we will show a classification result of a certain class of Z2 -actions on unital simple AF algebras. We freely use the terminology and notation introduced in [7, Def. 2.1]. For an automorphism α of a C ∗ -algebra A, we write the crossed product C ∗ -algebra A α Z by C ∗ (A, α) and the implementing unitary by λα . The mapping torus M(A, α) is defined by M(A, α) = { f ∈ C([0, 1], A) | α( f (0)) = f (1)}. When A is an AF algebra, ‘K K -triviality’ of α ∈ Aut(A) is equivalent to K 0 (α) = id, and also equivalent to α being approximately inner. The following theorem is a Z-equivariant version of Theorem 4.9. Let A be a unital simple AF algebra with unique trace and let α ∈ Inn(A). Let Aut T (C ∗ (A, α)) denote the set of all automorphisms of C ∗ (A, α) commuting with the dual action α. ˆ For i = 1, 2, we suppose that an automorphism βi ∈ Aut(A) and a unitary wi ∈ A are given and satisfy βi ◦ α = Ad wi ◦ α ◦ βi . Then βi extends to β˜i ∈ Aut T (C ∗ (A, α)) by setting β˜i (λα ) = wi λα . Suppose further that α m ◦βin is uniformly outer for all (m, n) ∈ Z2 \{(0, 0)} and that βisi is approximately inner for some si ∈ N. Theorem 6.1. In the setting above, if β˜1 and β˜2 are asymptotically unitarily equivalent, then there exist an approximately inner automorphism μ ∈ AutT (C ∗ (A, α)) and a unitary v ∈ A such that μ|A is also approximately inner and μ ◦ β˜1 ◦ μ−1 = Ad v ◦ β˜2 . Proof. We can apply the argument of [24, Theorem 5] to β˜1 and β˜2 in a similar fashion to [7, Theorem 4.11]. By Remark 4.11, (the Z-action generated by) α is asymptotically representable. Then [7, Theorem 4.8] implies that β˜1 and β˜2 are T-asymptotically unitarily equivalent. Moreover, by Theorem 5.5, we can find Rohlin projections for β˜i in the fixed point algebra (A∞ )α . Hence, by using Lemma 4.10 instead of [24, Theorem 7], the usual intertwining argument shows the statement.   Let us recall the OrderExt invariant introduced in [13]. Let G 0 , G 1 , F be abelian groups and let D : G 0 → F be a homomorphism. When ξ:

ι

q

0 −−−−→ G 0 −−−−→ E ξ −−−−→ G 1 −−−−→ 0

Z-Actions on AH Algebras and Z2 -Actions on AF Algebras

547

is exact, R is in Hom(E ξ , F) and R ◦ ι = D, the pair (ξ, R) is called an order-extension. Two order-extensions (ξ, R) and (ξ , R ) are equivalent if there exists an isomorphism θ : E ξ → E ξ such that R = R ◦ θ and ξ:

0 −−−−→ G 0 −−−−→   

ξ :

0 −−−−→ G 0 −−−−→ E ξ −−−−→ G 1 −−−−→ 0

E ξ −−−−→ ⏐ ⏐ θ

G 1 −−−−→ 0   

is commutative. Then OrderExt(G 1 , G 0 , D) consists of equivalence classes of all orderextensions. As shown in [13], OrderExt(G 1 , G 0 , D) is equipped with an abelian group structure. The map sending (ξ, R) to ξ induces a homomorphism from OrderExt (G 1 , G 0 , D) onto Ext(G 1 , G 0 ). Let B be a unital C ∗ -algebra with T (B) non-empty. We denote by Aut 0 (B) the set of all automorphisms γ of B such that K 0 (γ ) = K 1 (γ ) = id and τ ◦γ = τ for all τ ∈ T (B). When B is a unital simple AT algebra with real rank zero, Aut0 (B) equals Inn(B). Let D B : K 0 (B) → Aff(T (B)) denote the dimension map defined by D B ([ p])(τ ) = τ ( p). As described in [13], there exist natural homomorphisms η˜ 0 : Aut0 (B) → OrderExt(K 1 (B), K 0 (B), D B ) and η1 : Aut 0 (B) → Ext(K 0 (B), K 1 (B)). The following is the main result of [13]. See [18,21] for further developments. Theorem 6.2 ([13, Theorem 4.4]). Suppose that B is a unital simple AT algebra with real rank zero. Then the homomorphism η˜ 0 ⊕ η1 : Inn(B) → OrderExt(K 1 (B), K 0 (B), D B ) ⊕ Ext(K 0 (B), K 1 (B)) is surjective and its kernel equals the set of all asymptotically inner automorphisms of B. By using this OrderExt invariant, we introduce an invariant of certain Z2 -actions as follows. Let A be a unital simple AF algebra and let ϕ : Z2  A be an action of Z2 on A. Suppose that ϕ is uniformly outer and locally K K -trivial (i.e. locally approximately inner). We write B = C ∗ (A, ϕ(1,0) ). Then ϕ(0,1) extends to ϕ˜(0,1) ∈ Aut(B) by setting ϕ˜(0,1) (λϕ(1,0) ) = λϕ(1,0) . Let ι : A → B = C ∗ (A, ϕ(1,0) ) be the canonical inclusion. One can check the following immediately: • K 0 (ι) is an isomorphism from K 0 (A) to K 0 (B). • The connecting map ∂ : K 1 (B) → K 0 (A) in the Pimsner-Voiculescu exact sequence is an isomorphism and ∂ −1 ([ p]) = [λϕ(1,0) ι( p) + ι(v(1 − p))] for any projection p ∈ A, where v is a unitary of A satisfying vpv ∗ = ϕ(1,0) ( p). • The map ι∗ : T (B) → T (A) sending τ to τ ◦ ι is an isomorphism and satisfies D B (K 0 (ι)(x))(τ ) = D A (x)(ι∗ (τ )) for x ∈ K 0 (A) and τ ∈ T (B). From these properties, we can obtain a natural isomorphism ζϕ(1,0) : OrderExt(K 1 (B), K 0 (B), D B ) → OrderExt(K 0 (A), K 0 (A), D A ). In addition, it is easy to see K 0 (ϕ˜(0,1) ) = K 1 (ϕ˜ (0,1) ) = id and τ ◦ ϕ˜(0,1) = τ for all τ ∈ T (B), that is, ϕ˜(0,1) belongs to Aut 0 (B).

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Lemma 6.3. In the setting above, η1 (ϕ˜(0,1) ) ∈ Ext(K 0 (B), K 1 (B)) is zero. Proof. There exists a natural commutative diagram 0 −−−−→ C0 ((0, 1), B) −−−−→ M(B, ϕ˜(0,1) ) −−−−→   ⏐ ⏐ ⏐ ⏐

B −−−−→ 0  ⏐ι ⏐

0 −−−−→ C0 ((0, 1), A) −−−−→ M(A, ϕ(0,1) ) −−−−→ A −−−−→ 0, where the horizontal sequences are exact. From the naturality of the six-term exact sequence, we obtain the commutative diagram 0 −−−−→ K 1 (B) −−−−→ K 0 (M(B, ϕ˜(0,1) )) −−−−→ K 0 (B) −−−−→ 0    ⏐ K (ι) ⏐ ⏐ ⏐ 0 ⏐ ⏐ 0 −−−−→ K 1 (A) −−−−→ K 0 (M(A, ϕ(0,1) )) −−−−→ K 0 (A) −−−−→ 0, where the horizontal sequences are exact. Since K 1 (A) is zero and K 0 (ι) is an isomorphism, we can conclude η1 (ϕ˜ (0,1) ) = 0.   Definition 6.4. In the setting above, we define our invariant [ϕ] by [ϕ] = ζϕ(1,0) (η˜ 0 (ϕ˜ (0,1) )) ∈ OrderExt(K 0 (A), K 0 (A), D A ). Proposition 6.5. Let ϕ, ψ : Z2  A be uniformly outer, locally K K -trivial Z2 -actions on a unital simple AF algebra A. If ϕ and ψ are K K -trivially cocycle conjugate, then [ϕ] = [ψ]. Proof. For μ ∈ Inn(A), it is straightforward to see that the Z2 -action μ ◦ ϕ ◦ μ−1 has the same invariant as ϕ. Hence, it suffices to show [ϕ] = [ϕ u ] for any ϕ-cocycle {u n }n∈Z2 . u Define an isomorphism π from C ∗ (A, ϕ(1,0) ) to C ∗ (A, ϕ(1,0) ) by u π(λϕ(1,0) ) = u ∗(1,0) λϕ(1,0) and π(a) = a ∀a ∈ A,

where A is identified with subalgebras of the crossed products. For γ ∈ Aut(C ∗ (A, u ϕ(1,0) )), one can check u (η˜ 0 (γ )) ∈ OrderExt(K 0 (A), K 0 (A), D A ), ζϕ(1,0) (η˜ 0 (π −1 ◦ γ ◦ π )) = ζϕ(1,0)

where η˜ 0 in the left-hand side is defined for C ∗ (A, ϕ(1,0) ) and η˜ 0 in the right-hand side u is defined for C ∗ (A, ϕ(1,0) ). We also have u u u ◦ π )(a) = π −1 (ϕ˜ (0,1) (a)) = ϕ˜(0,1) (a) = (Ad u (0,1) ◦ ϕ˜(0,1) )(a) ∀a ∈ A (π −1 ◦ ϕ˜(0,1)

and u u u (π −1 ◦ ϕ˜(0,1) ◦ π )(λϕ(1,0) ) = (π −1 ◦ ϕ˜(0,1) )(u ∗(1,0) λϕ(1,0) ) u u = π −1 (ϕ˜ (0,1) (u ∗(1,0) )λϕ(1,0) )

u = ϕ(0,1) (u ∗(1,0) )u (1,0) λϕ(1,0)

= u (0,1) ϕ(0,1) (u ∗(1,0) )u ∗(0,1) u (1,0) λϕ(1,0) = u (0,1) ϕ(1,0) (u ∗(0,1) )u ∗(1,0) u (1,0) λϕ(1,0) = u (0,1) ϕ(1,0) (u ∗(0,1) )λϕ(1,0) = u (0,1) λϕ(1,0) u ∗(0,1)

= (Ad u (0,1) ◦ ϕ˜(0,1) )(λϕ(1,0) ).

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u Thus π −1 ◦ ϕ˜ (0,1) ◦ π = Ad u (0,1) ◦ ϕ˜(0,1) . Since inner automorphisms are contained in the kernel of η˜ 0 , we obtain

ζϕ(1,0) (η˜ 0 (ϕ˜(0,1) )) = ζϕ(1,0) (η˜ 0 (Ad u (0,1) ◦ ϕ˜(0,1) )) u ◦ π )) = ζϕ(1,0) (η˜ 0 (π −1 ◦ ϕ˜(0,1) u u (η˜ 0 (ϕ˜(0,1) )), = ζϕ(1,0)

which completes the proof.   Theorem 6.6. Let ϕ, ψ : Z2  A be uniformly outer, locally K K -trivial Z2 -actions on a unital simple AF algebra A with unique trace. The following are equivalent: (1) [ϕ] = [ψ]. (2) ϕ and ψ are K K -trivially cocycle conjugate. Proof. (2)⇒(1) was shown in the proposition above without assuming that A has a unique trace. Let us consider the other implication (1)⇒(2). By Theorem 4.8 and Theorem 4.9, we may assume that there exists a unitary u ∈ A such that ψ(1,0) = Ad u ◦ϕ(1,0) . By Theorem 4.8 and Remark 4.11 (or Remark 4.5), the crossed product C ∗ -algebra C ∗ (A, ϕ(1,0) ) is a unital simple AT algebra with real rank zero. Clearly ϕ(0,1) extends to ϕ˜(0,1) ∈ Aut(C ∗ (A, ϕ(1,0) )) by ϕ˜(0,1) (a) = a ∀a ∈ A and ϕ˜(0,1) (λϕ(1,0) ) = λϕ(1,0) . Since ψ(0,1) ◦ ϕ(1,0) = Ad(ψ(0,1) (u ∗ )u) ◦ ϕ(1,0) ◦ ψ(0,1) , we can extend ψ(0,1) to ω ∈ Aut(C ∗ (A, ϕ(1,0) )) by ω(a) = a ∀a ∈ A and ω(λϕ(1,0) ) = ψ(0,1) (u ∗ )uλϕ(1,0) . In order to apply Theorem 6.1 to ϕ˜(0,1) and ω, we would like to check that these automorphisms are asymptotically unitarily equivalent. There exists an isomorphism π : C ∗ (A, ϕ(1,0) ) → C ∗ (A, ψ(1,0) ) defined by π(a) = a ∀a ∈ A and π(λϕ(1,0) ) = u ∗ λψ(1,0) . As mentioned in the proof of Proposition 6.5, for any γ ∈ Aut(C ∗ (A, ϕ(1,0) )), one has ζϕ(1,0) (η˜ 0 (γ )) = ζψ(1,0) (η˜ 0 (π ◦ γ ◦ π −1 )). Moreover it is easy to see that π ◦ ω ◦ π −1 is equal to ψ˜ (0,1) , which is defined by ψ˜ (0,1) (a) = a ∀a ∈ A and ψ˜ (0,1) (λψ(1,0) ) = λψ(1,0) . It follows that ζϕ(1,0) (η˜ 0 (ω)) = ζψ(1,0) (η˜ 0 (π ◦ ω ◦ π −1 )) = ζψ(1,0) (η˜ 0 (ψ˜ (0,1) )) = [ψ] = [ϕ] = ζϕ(1,0) (η˜ 0 (ϕ˜ (0,1) )), and so η˜ 0 (ω) = η˜ 0 (ϕ˜(0,1) ). By Lemma 6.3, η1 (ω) = η1 (ϕ˜(0,1) ) = 0. Therefore, by Theorem 6.2, ϕ˜(0,1) and ω are asymptotically unitarily equivalent.

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Then, Theorem 6.1 applies and yields an approximately inner automorphism μ ∈ Aut T (C ∗ (A, ϕ(1,0) )) and a unitary v ∈ A such that μ|A is in Inn(A) and μ ◦ ω ◦ μ−1 = Ad v ◦ ϕ˜(0,1) .

(6.1)

By restricting this equality to A, we get (μ|A) ◦ ψ(0,1) ◦ (μ|A)−1 = Ad v ◦ ϕ(0,1) .

(6.2)

Let z ∈ A be the unitary satisfying μ(λϕ(1,0) ) = zλϕ(1,0) . Then (μ|A) ◦ ψ(1,0) ◦ (μ|A)−1 = (μ|A) ◦ Ad u ◦ ϕ(1,0) ◦ (μ|A)−1 = Ad μ(u)z ◦ ϕ(1,0) . (6.3) From (6.1), one can see that (μ ◦ ω ◦ μ−1 )(λϕ(1,0) ) = (μ ◦ ω)(μ−1 (z ∗ )λϕ(1,0) ) = μ(ψ(0,1) (μ−1 (z ∗ ))ψ(0,1) (u ∗ )uλϕ(1,0) ) = (Ad v ◦ ϕ(0,1) )(z ∗ μ(u ∗ ))μ(u)zλϕ(1,0) = vϕ(0,1) (z ∗ μ(u ∗ ))v ∗ μ(u)zλϕ(1,0) is equal to (Ad v ◦ ϕ˜ (0,1) )(λϕ(1,0) ) = vλϕ(1,0) v ∗

= vϕ(1,0) (v ∗ )λϕ(1,0) .

Hence one obtains vϕ(0,1) (μ(u)z) = μ(u)zϕ(1,0) (v).

(6.4)

It follows from (6.2), (6.3), (6.4) that ψ and ϕ are K K -trivially cocycle conjugate.   Remark 6.7. We do not know the precise range of our invariant which takes its values in OrderExt. At least, the following observation shows that the range does not exhaust OrderExt. Let ϕ : Z2  A be a locally K K -trivial and uniformly outer Z2 -action on a unital simple AF algebra. Suppose that (ξ, R) is a representative of [ϕ] ∈ OrderExt(K 0 (A), K 0 (A), D A ). Since ξ:

ι

q

0 −−−−→ K 0 (A) −−−−→ E ξ −−−−→ K 0 (A) −−−−→ 0

is exact and R : E ξ → Aff(T (A)) satisfies R ◦ ι = D A , there exists a homomorphism R0 : K 0 (A) → Aff(K 0 (A))/ Im D A such that R0 (q(x)) = R(x) + D A (K 0 (A)) for any x ∈ E ξ . It is easy to see R0 ([1 A ]) = 0, because the implementing unitary λϕ(1,0) is fixed by ϕ˜(0,1) . Thus, [ϕ] belongs to the subgroup {[(ξ, R)] ∈ OrderExt(K 0 (A), K 0 (A), D A ) | R0 ([1 A ]) = 0}. When A is a UHF algebra, one can see that this subgroup coincides with the range of the invariant introduced in [8]. Therefore, Theorem 6.6 yields a new proof of [8, Theorem 6.5].

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Commun. Math. Phys. 297, 553–596 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-0998-7

Communications in

Mathematical Physics

Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras Chengming Bai1 , Li Guo2 , Xiang Ni1 1 Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071,

P.R. China. E-mail: [email protected]; [email protected]

2 Department of Mathematics and Computer Science, Rutgers University,

Newark, NJ 07102, USA. E-mail: [email protected] Received: 23 July 2009 / Accepted: 22 October 2009 Published online: 11 February 2010 – © Springer-Verlag 2010

Abstract: We generalize the classical study of (generalized) Lax pairs, the related O-operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, extended O-operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized r -matrix ansatz and the related double Lie algebra structures. Relationship between extended O-operators and the extended classical Yang-Baxter equation is established, especially for self-dual Lie algebras. This relationship allows us to obtain an explicit description of the Manin triples for a new class of Lie bialgebras. Furthermore, we show that a natural structure of PostLie algebra is behind O-operators and fits in a setup of the triple Lie algebra that produces self-dual nonabelian generalized Lax pairs. Contents 1. 2.

3.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonabelian Generalized Lax Pairs and Extended O-Operators . . 2.1 Nonabelian generalized Lax pairs . . . . . . . . . . . . . . . 2.2 Extended O-operators and double Lie brackets . . . . . . . . 2.3 Adjoint representations and Baxter Lie algebras . . . . . . . Extended O-Operators, the Extended CYBE and Type II Quasitriangular Lie Bialgebras . . . . . . . . . . . . . . . . . . 3.1 Lie bialgebras and the extended CYBE . . . . . . . . . . . . 3.2 Extended O-operators and the ECYBE . . . . . . . . . . . . 3.3 Extended O-operators (of mass 1) and type II CYBE . . . . 3.4 Type II quasitriangular Lie bialgebras . . . . . . . . . . . . Self-Dual Lie Algebras and Factorizable (Type II) Quasitriangular Lie Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Extended O-operators and the ECYBE on self-dual Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 Factorizable quasitriangular Lie bialgebras . . . . . . . . . . 4.3 Factorizable type II quasitriangular Lie bialgebras . . . . . . 5. O-Operators, PostLie Algebras and Dendriform Trialgebras . . . 5.1 O-operators and PostLie algebras . . . . . . . . . . . . . . . 5.2 Dendriform trialgebras and PostLie algebras: a commutative diagram . . . . . . . . . . . . . . . . . . . . 6. Triple Lie Algebras and Examples of Nonabelian Generalized Lax Pairs . . . . . . . . . . . . . . . . . . . . . . . 6.1 Triple Lie algebra and a typical example of nonabelian generalized Lax pairs . . . . . . . . . . . . . . . . . . . . . 6.2 The case of PostLie algebras . . . . . . . . . . . . . . . . . Appendix: Extended O-Operators and Affine Geometry on Lie Groups References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction This paper is devoted to a systematic study of the integrable Hamiltonian systems and the related (generalized) classical Yang-Baxter equation (CYBE) in a broad context that generalizes or extends the studies of Bordemann [10], Hodge and Yakimov [25], Kosmann-Schwarzbach and Magri [29], and Semenov-Tian-Shansky [44]. Since their introduction by Lax in 1968, Lax pairs have become important in giving conservation laws in an integrable system. In connection with r -matrices satisfying the classical Yang-Baxter equation (CYBE), Poisson commuting conservation laws could be constructed. Main contributors in this direction include Adler [1], Babelon and Viallet [4,5], Belavin and Drinfeld [8,9,17], Faddeev [21], Kostant [30], Reyman and Semenov-Tian-Shansky [41,44], Sklyanin [46,47] and Symes [49,50]. In [10] Bordemann introduced the notions of generalized Lax pairs and generalized r -matrix ansatz. He achieved this through replacing the well-known Lax equation [32] dL = [L , M] dt by dL = −ρ(M)L , dt

(1)

where ρ is any representation of a Lie algebra g in a representation space V , M is a g-valued function on the phase space and L is a V -valued function on the phase space, reducing to the Lax equation when V is taken to be g and ρ is taken to be the adjoint representation. In this generality, the correct framework to extend the classical r -matrices is through their operator forms, later called O-operators by Kupershmidt [31]. The classical Yang-Baxter equation, through its operator form and tensor form, plays a central role in relating several areas in mathematics. For the most part, the operator form is more convenient in application to integrable systems. For example, the modified classical Yang-Baxter equation is solely defined in the operator form. Nevertheless, the tensor form of the CYBE is the classical limit of the quantum Yang-Baxter equation, and its solutions give rise to important concepts such as (coboundary) Lie bialgebras. Thus it is desirable to work with both forms of the CYBE. In the present paper, we keep both forms of the CYBE in mind while we generalize the previous works. For the operator form, we further generalize the work of Bordemann and

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Kupershmidt by introducing the concepts of an O-operator of weight λ (for a constant λ) and an extended O-operator. This is motivated by our attempt to extend generalized Lax pairs of Bordemann to nonabelian generalized Lax pairs, by still considering Eq. (1) but replacing the representation space V by any Lie algebra a and the representation ρ by any Lie algebra homomorphism from g to Der(a) consisting of derivations of a. The setting of Bordemann is recovered when a is taken to be an abelian Lie algebra. We extend the generalized r -matrix ansatz of Bordemann to the nonabelian context and show that extended O-operators ensure the consistency of a Lie structure on a∗ defined by the r -matrices. For the tensor form, we introduce the concept of the extended classical Yang-Baxter equation and establish their relationship with extended O-operators as in the case of (the tensor form and operator form) of the CYBE. We further extend the well-known work of Drinfeld on quasitriangular Lie bialgebras from the CYBE to what we dubbed type II quasitriangular Lie bialgebras from a case of the extended classical Yang-Baxter equation, called the type II CYBE. The corresponding Drinfeld’s doubles and Manin triples are studied carefully as in the classical case by Hodge and Yakimov [25], for their importance in the classification of the Poisson homogeneous spaces and symplectic leaves of the corresponding Poisson-Lie groups [18,25,45,53]. As it turns out, an O-operator of weight λ is related to the concept of a PostLie algebra that has recently arisen from the quite different context of operads [52]. More precisely, an O-operator, paired with a g-Lie algebra, gives a PostLie algebra. In particular, Baxter Lie algebras and quasitriangular Lie bialgebras give rise to PostLie algebras. Furthermore the well-known relation [12] between pre-Lie algebras and dendriform dialgebras, in connection with the classical relation between associative algebras and Lie algebras, can be extended to that between PostLie algebras and dendriform trialgebras. Quite unexpectedly, this “digression” of O-operators to PostLie algebra is tied up with our primary application of O-operators in studying nonabelian generalized Lax pairs: We introduce the concept of a triple Lie algebra to construct self-dual nonabelian generalized Lax pairs and show that a natural example of a triple Lie algebra is provided by the PostLie algebra from a Rota-Baxter operator action on a complex simple Lie algebra. We next give a summary of this paper. We begin our study by introducing the concept of a nonabelian generalized Lax pair. We write down a “nonabelian generalized r -matrix ansatz” to produce Poisson commuting conservation laws. The idea is to use the Lie-Poisson structure on the representation space (equipped with a Lie bracket) to twist the “generalized r -matrix ansatz” of Bordemann [10]. In geometry, this construction might be understood as “twisting” a Hamiltonian system (Poisson bracket) by the Hamiltonian system (Lie-Poisson bracket) on the dual space of a Lie algebra. The notions O-operator of weight λ and extended O-operator of weight λ with extension β of mass (ν, κ, μ) (for constants (ν, κ, μ)) appear naturally when we investigate sufficient conditions for the double Lie algebra structures needed for the existence of the ansatz. To generalize the well-known relationship between the operator form and tensor form of the CYBE, we introduce in Sect. 3 the concept of an extended CYBE and relate it to extended O-operators. Applications to Lie bialgebras are given. In particular, we study in detail the structure of the Manin triple of a type II quasitriangular Lie bialgebra. In Sect. 4, we study the case of self-dual Lie algebras. The idea is to use a nondegenerate symmetric and invariant bilinear form of a self-dual Lie algebra to identify the adjoint representation and coadjoint representation [44]. Some new aspects on Lie bial-

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gebras are given along this approach, for example, new examples of (type II) factorizable quasitriangular Lie bialgebras are provided. We show in Sect. 5 that there naturally exists an algebraic structure behind an Ooperator of weight λ, namely, the PostLie algebra which Vallette discovered in a study of operads [52]. We also reveal a relation between PostLie algebras and dendriform trialgebras of Loday and Ronco [36] by a commutative diagram. In Sect. 6, we provide a framework of triple Lie algebras to construct a class of nonabelian generalized Lax pairs for which the corresponding r -matrix ansatz can be written down explicitly [13]. We show that PostLie algebras provide natural instances of such triple Lie algebras. Finally in an Appendix, we give a geometric explanation of extended O-operators. Conventions: In this paper, the base field is taken to be R of real numbers unless otherwise specified. This is the field from which we take all the constants and over which we take all the associative and Lie algebras, vector spaces, linear maps and tensor products, etc. All Lie algebras, vector spaces and manifolds are assumed to be finite-dimensional, although many results still hold in the infinite-dimensional case. 2. Nonabelian Generalized Lax Pairs and Extended O-Operators We begin with generalizing the generalized Lax pairs of Bordemann [10] further to nonabelian generalized Lax pairs. By studying generalized r -matrix ansatz and double Lie algebra structures in this context, we are motivated to introduce the concept of an extended O-operator, generalizing the work of Bordemann and Kupershmidt [31] in several directions. The case of adjoint representations is studied separately. 2.1. Nonabelian generalized Lax pairs. We first introduce a suitable replacement of Lie algebra representations in order to extend generalized Lax pairs to the nonabelian context. Definition 2.1. (i) Let (g, [ , ]g), or simply g, denote a Lie algebra g with Lie bracket [ , ]g. (ii) For a Lie algebra b, let Der R b denote the Lie algebra of derivations of b. (iii) Let a be a Lie algebra. An a-Lie algebra is a triple (b, [ , ]b, π ) consisting of a Lie algebra (b, [ , ]b) and a Lie algebra homomorphism π : a → Der R b. To simplify the notation, we also let (b, π ) or simply b denote (b, [ , ]b, π ). (iv) Let a be a Lie algebra and let (g, π ) be an a-Lie algebra. Let a · b denote π(a)b for a ∈ a and b ∈ g. According to [26], if (b, π ) is an a-Lie algebra, then there exists a unique Lie algebra structure on the direct sum g = a ⊕ b of the underlying vector spaces a and b such that a and b are subalgebras and [x, y] = π(x)y for x ∈ a and y ∈ b. Further, a is a subalgebra and b is an ideal of the Lie algebra g.  Let (P, w) be a Poisson manifold with the Poisson bivector w ∈ 2 T (M) which induces a Poisson bracket {, } on C ∞ (P). A smooth function f on P, which is called an observable, determines a Hamiltonian vector field X f by X f g ≡ { f, g}, g ∈ C ∞ (P). If a Hamiltonian system is modeled by a Poisson manifold (P, w) (the phase space of the system) and a Hamiltonian H ∈ C ∞ (P), its time-evolution is given by the following

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integral curves of the Hamiltonian vector field X H on P corresponding to H: X H( f ) ≡ {H, f }, ∀ f ∈ C ∞ (P). It follows that df = {H, f }, dt in the sense that (d/dt)( f (m(t))) = {H, f }(m(t)) for an integral curve m(t) of X H. As usual, an observable f is called a conservation law or conserved if {H, f } = 0. Two conservation laws f 1 , f 2 on a Poisson manifold are in involution or Poisson commuting if { f 1 , f 2 } = 0. Moreover, a Hamiltonian system (P, w, H) is called completely integrable if it has the maximum number of conserved observables in involution [13]. An important procedure to obtain Poisson commuting observables and completely integrable Hamiltonian systems is through the concept of Lax pairs [32] which was generalized by Bordemann [10] to generalized Lax pairs. We now generalize this further to the following concept. Definition 2.2. (i) A nonabelian generalized Lax pair for a Hamiltonian system (P, w, H) is a quintuple (g, ρ, a, L , M) satisfying the following conditions: (a) g is a (finite-dimensional) Lie algebra; (b) (a, ρ) is a (finite-dimensional) g-Lie algebra with the Lie algebra homomorphism ρ : g → DerR (a); (c) L : P → a is a smooth map, (d) M : P → g is a smooth map such that d L( p)X H( p) = −ρ(M( p))L( p), ∀ p ∈ P.

(2)

(ii) A nonabelian generalized Lax pair (g, ρ, a, L , M) is said to be self-dual if a is equipped with a nondegenerate symmetric bilinear form B : a ⊗ a → R such that B([x, y]a, z) = B(x, [y, z]a), ∀x, y, z ∈ a,

(3)

B(ρ(ξ )x, y) + B(x, ρ(ξ )y) = 0, ∀ξ ∈ g, x, y ∈ a.

(4)

Note that a bilinear form on a Lie algebra satisfying Eq. (3) is called invariant and a Lie algebra endowed with a nondegenerate symmetric invariant bilinear form is called a self-dual Lie algebra [22]. By the chain rule, Eq. (2) is equivalent to dL = −ρ(M)L . dt

(5)

Remark 2.3. (i) When the Lie bracket on a happens to be trivial, the g-Lie algebra (a, ρ) becomes a representation of g and the nonabelian generalized Lax pair becomes the generalized Lax pair in the sense of Bordemann [10]. (ii) For a = g and ρ = ad, Eq. (5) is the usual Lax equation. Moreover, the Lax pair can be realized as a nonabelian generalized Lax pair in two different ways, by either taking ρ to be ad and a to be the Lie algebra g, or taking ρ to be ad and a to be the underlying vector space of g equipped with the trivial Lie bracket.

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Let G be a connected Lie group whose Lie algebra is g such that ρ exponentiates to a representation of G in V which we shall also call ρ. We first show that, as in the case of Lax pairs and generalized Lax pairs [10], nonabelian generalized Lax pairs also give conservation laws. Proposition 2.4. Let (g, ρ, a, L , M) be a nonabelian generalized Lax pair for a Hamiltonian system (P, w, H). If f : a → R is a G-invariant smooth function, i.e., f (ρ(g)x) = f (x) for all g ∈ G and x ∈ a, then f ◦ L is a conservation law, i.e., d( f ◦ L) = { f ◦ L , H} = 0. dt Proof. Since G-invariant functions are always constant on each G-orbit, we have d f (x)ρ(ξ )x = 0, ∀ξ ∈ g, x ∈ a. So d d ( f ◦ L) = d f (L) L = −d f (L)ρ(M)L = 0. dt dt  Let {ei }1≤i≤dima be a basis of a and {T A }1≤A≤dimg be a basis of g. For any  i  A  A j i x = x ei ∈ a and ξ = ξ T A ∈ g, we set (ρ(ξ )x)i = ξ x ρ A j . On the i A A, j  other hand, suppose that the Lie algebra structure on a is given by [ei , e j ]a = cikj ek . k

The Poisson bracket { f ◦ L , h ◦ L} for arbitrary smooth functions f, h : a → R is  ∂f ∂h { f ◦ L , h ◦ L} = ◦ L j ◦ L{L i , L j }. (6) i ∂x ∂x i, j

Now consider smooth maps which we shall call classical r -matrices (following [10]) r+ , r− : a × P → a ⊗ g and make the following nonabelian generalized r -matrix ansatz:  j {L i , L j }( p) = − r+i A (L( p), p)ρ Ak L k ( p) A,k

+

 A,k

jA

r− (L( p), p)ρ iAk L k ( p) −



j

θi ( p)cik L k ( p),

(7)

k

where θ : P → a is a smooth function and θi = x i ◦ θ : P → R, 1 ≤ i ≤ dima. When θ = 0, the third term on the right hand side vanishes and the ansatz is reduced to Bordemann’s generalized r -matrix ansatz [10]. Generalizing the work of Bordemann, we next show that the nonabelian generalized r -matrix ansatz gives Poisson commuting conservation laws. Proposition 2.5. Let (g, ρ, a, L , M) be a nonabelian generalized Lax pair for a Hamiltonian system (P, w, H) allowing for classical r -matrices that obey Eq. (7). Then for two real-valued G-invariant and Ada-invariant functions f and h on a, we have { f ◦ L , h ◦ L} = 0.

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Proof. Using Eq. (6), we get { f ◦ L , h ◦ L} = −

 ∂f  ∂f ∂h ∂h j jA ◦ Lr+i A j ◦ Lρ Ak L k + ◦ Lρ iAk L k j ◦ Lr− i ∂x ∂x ∂ xi ∂x

i,A, j,k

i,A, j,k

 ∂h ∂f j − ◦ Lcik L k i ◦ Lθi . ∂x j ∂x i, j,k

The underlined terms are zero because of infinitesimal G-invariance and Ada-invariance of f and h.  As pointed out by Bordemann in [10], for a = g (with the trivial bracket) and ρ = ad, the classical r -matrices take values in g ⊗ g, and the above conclusion becomes the classical fact [4] that arbitrary trace polynomials of L Poisson commute among themselves. The Lie bracket conditions on the left hand side of Eq. (7) impose consistency restraints on the classical r -matrices on the right hand side. In the case of constant r -matrices (namely L-independent) that we will consider below, as observed by Bordemann, the space spanned by the component functions L i behaves like a finite-dimensional Lie subalgebra of the Poisson algebra of functions on the phase space (P, w) since the right-hand side of Eq. (7) is linear in L. Suppose one wants to collectively investigate all nonabelian generalized Lax pairs that are defined on a given Hamiltonian system (P, w, H), that have given g, ρ and a, and that satisfy Eq. (7) with given classical r -matrices r+ and r− . Then one is led to the following stronger condition than the above mentioned consistency restraints imposed on Eq. (7): Condition 2.6. The quantities ij

fk ≡ −

 A

j

r+i A ρ Ak +



jA

j

r− ρ iAk − θi cik

A

should be the structure constants of a Lie structure on a∗ . To obtain an index-free form of Condition 2.6, we first give the following lemma. Lemma 2.7. Let g be a Lie algebra and (a, ρ) be a g-Lie algebra. Let B : a ⊗ a → R be a nondegenerate bilinear form on a which can be identified as an invertible linear map ϕ : a → a∗ through B(x, y) = ϕ(x), y, ∀x, y ∈ a.

(8)

Let (a∗ , ρϕ ) be the g-Lie algebra through ϕ by transporting the g-Lie algebra structure on a. More precisely, define the Lie bracket on a∗ by [a ∗ , b∗ ]a∗ = ϕ([ϕ −1 (a ∗ ), ϕ −1 (b∗ )]a), ∀a ∗ , b∗ ∈ a∗

(9)

and define a linear map ρϕ : g → EndR (a∗ ), ρϕ (ξ )a ∗ ≡ ϕρ(ξ )ϕ −1 (a ∗ ), ∀a ∗ ∈ a∗ , ξ ∈ g.

(10)

If B satisfies Eq. (4), then ρϕ is just the dual representation ρ ∗ of ρ which is defined by ρ ∗ (ξ )a ∗ , x = −a ∗ , ρ(ξ )x, ∀ξ ∈ g, x ∈ a, a ∗ ∈ a∗ . In this case, (a∗ , ρ ∗ ) is a g-Lie algebra with the Lie bracket defined by Eq. (9).

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Proof. If B satisfies Eq. (4), then for any ξ ∈ g, x, y ∈ a, ϕ(ρ(ξ )x), y = −ϕ(x), ρ(ξ )y ⇒ ϕρ(ξ ) = ρ ∗ (ξ )ϕ, ∀ξ ∈ g. Hence ρϕ = ρ ∗ . So the conclusion holds.



Assume that a is equipped with a nondegenerate symmetric bilinear form B : a ⊗ a → R for which the nonabelian generalized Lax pair (g, ρ, a, L , M) is self-dual. Let a∗ be equipped with the Lie bracket defined by Eq. (9). By Lemma 2.7, (a∗ , [ , ]a∗ , ρ ∗ ) is a g-Lie algebra. Since B is nondegenerate and symmetric, we can choose a basis {ei }1≤i≤dima of a such that bi j ≡ B(ei , e j ) = ϕ(ei ), e j  = 0, if i = j; bii ≡ B(ei , ei ) = ϕ(ei ), ei  = 0. Thus, ϕ(ei ) = bii ei∗ , where {ei∗ }1≤i≤dima is the dual basis of {ei }1≤i≤dima. Since j B([ei , e j ]a, ek ) + B(e j , [ei , ek ]a) = 0, we have cikj bkk + cik b j j = 0. Therefore,  ei e j [ei∗ , e∗j ]a∗ = ϕ[ϕ −1 (ei∗ ), ϕ −1 (e∗j )]a = ϕ([ , ]a) = bii b j j

k

−

cikj bkk ek bii b j j

=

 k

j

cik ek

bii

.

Now we set θi ≡ bλii for λ ∈ R. On the other hand, since a ⊗ g  Hom(a∗ , g), r+ and  iA r− can be considered as linear maps a∗ → g : x = xi ei∗ → r± (x) ≡ xi r± T A . Set i,A

k ≡ a∗ , π ≡ ρ ∗ , ξ · x ≡ π(ξ )x, x ∈ k, ξ ∈ g. Then Condition 2.6 can be reformulated as follows: Condition 2.8. (Double Lie algebra structure) The product [x, y] R ≡ r+ (x) · y − r− (y) · x + λ[x, y]k, ∀x, y ∈ k, defines a Lie bracket on k. Define r ≡ (r+ + r− )/2, β ≡ (r+ − r− )/2.

(11)

Then r± = r ± β. Moreover, we have the following result: Proposition 2.9. Condition 2.8 holds if and only if for any x, y, z ∈ k, (i) [x, y] R = r (x) · y − r (y) · x + λ[x, y]k ⇔ β(x) · y + β(y) · x = 0, (ii) ([r (x), r (y)]g − r ([x, y] R )) · z + ([r (y), r (z)]g − r ([y, z] R )) · x + ([r (z), r (x)]g − r ([z, x] R )) · y = 0. To simplify notations, for an expression η(x, y, z) in x, y and z, we denote η(x, y, z) + cycl. = η(x, y, z) + η(y, z, x) + η(z, x, y).

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Proof. Obviously, condition (i) is equivalent to the fact that [, ] R is skew-symmetric. Now we prove that condition (ii) is equivalent to the fact that [, ] R satisfies the Jacobi identity. In fact, for all x, y, z ∈ k, [[x, y] R , z] R = r ([x, y] R ) · z − r (z) · (r (x) · y) + r (z) · (r (y) · x) − λr (z) · [x, y]k + λ[r (x) · y, z]k − λ[r (y) · x, z]k + λ2 [[x, y]k, z]k, [[z, x] R , y] R = r ([z, x] R ) · y − r (y) · (r (z) · x) + r (y) · (r (x) · z) − λr (y) · [z, x]k + λ[r (z) · x, y]k − λ[r (x) · z, y]k + λ2 [[z, x]k, y]k, [[y, z] R , x] R = r ([y, z] R ) · x − r (x) · (r (y) · z) + r (x) · (r (z) · y) − λr (x) · [y, z]k + λ[r (y) · z, x]k − λ[r (z) · y, x]k + λ2 [[y, z]k, x]k. Then the conclusion follows from the fact that (k, π ) = (a∗ , ρ ∗ ) is a g-Lie algebra.



2.2. Extended O-operators and double Lie brackets. We will next study the conditions in Proposition 2.9 in order to understand double Lie algebra structures and nonabelian generalized Lax pairs. For this purpose, we introduce the following concepts. Definition 2.10. Let (g, [ , ]g) be a Lie algebra and let (k, [ , ]k, π ) be a g-Lie algebra. Let ν, κ, μ and λ be constants (in R). (i) A linear map β : k → g is called antisymmetric (of mass ν) if νβ(x) · y + νβ(y) · x = 0 for any x, y ∈ k. (ii) A linear map β : k → g is called g-invariant (of mass κ) if κβ(ξ ·x) = κ[ξ, β(x)]g, for any ξ ∈ g, x ∈ k. (iii) A linear map β : k → g is called equivalent (of mass μ) if μβ([x, y]k) · z = μ[β(x) · y, z]k, for any x, y, z ∈ k. (iv) Let β : k → g be antisymmetric of mass ν, g-invariant of mass κ and equivalent of mass μ. Let r : k → g be a linear map. The pair (r, β) or simply r is called an extended O-operator of weight λ with extension β of mass (ν, κ, μ) if [r (x), r (y)]g − r (r (x) · v − r (y) · x + λ[x, y]k) = κ[β(x), β(y)]g + μβ([x, y]k), ∀x, y ∈ k.

(12)

(v) A linear map r : k → g is called an O-operator of weight λ if [r (x), r (y)]g = r (r (x) · y − r (y) · x + λ[x, y]k), ∀x, y ∈ k.

(13)

(vi) Let (k, [ , ]k, π ) be the g-Lie algebra (g, [ , ]g, ad). Then an O-operator r : g → g becomes what is known as a Rota-Baxter operator of weight λ satisfying [r (x), r (y)]g = r ([r (x), y]g + [x, r (y)]g + λ[x, y]g), ∀x, y ∈ g.

(14)

A Lie algebra equipped with a Rota-Baxter operator is called a Rota-Baxter Lie algebra. Remark 2.11. (i) We include the parameters ν, κ, μ, λ in the definition in order to uniformly treat the different cases when the parameters vary.

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(ii) Rota-Baxter operators on associative algebras were introduced by the mathematician Glenn Baxter [7] in 1960 and have recently found many applications especially in the algebraic approach of Connes and Kreimer to renormalization of quantum field theory [14,15]. For further details, see the survey articles [20,23,43]. See also [6] for the relationship between Rota-Baxter operators on associative algebras and the associative CYBE motivated by the study of this paper. (iii) If λ = 0, then r is an O-operator of weight λ if and only if r/λ is an O-operator of weight 1. (iv) When λ = 1, the difference of the two sides of Eq. (13) has appeared in the work of Kosmann-Schwarzbach and Magri under the name Schouten curvature, which is the algebraic version of the contravariant analogue of the Cartan curvature of the Lie-algebra valued one-form on a Lie group (see [29] for details). When k in Definition 2.10 is taken to be a vector space regarded as an abelian Lie algebra, then (k, π ) is a g-Lie algebra means that π : g → gl(k) is a linear representation of g. Thus the above definition has the following variation (with ν = κ). Definition 2.12. Let g be a Lie algebra and V be a vector space. Let ρ : g → gl(V ) be a linear representation of g. Suppose that β : V → g is an antisymmetric of mass κ, g-invariant of mass κ linear map. Let r : V → g be a linear map. The pair (r, β) or simply r is called an extended O-operator with extension β of mass κ if [r (u), r (v)] − r (r (u) · v − r (v) · u) = κ[β(u), β(v)], ∀u, v ∈ V.

(15)

When κ = 0, we obtain the O-operator defined by Kupershmidt [31] and (the operator form of) the classical Yang-Baxter equation (CYBE) of Bordemann [10]. When κ = −1, Eq. (15) is called the modified classical Yang-Baxter equation (MCYBE) in [10,28,44]. The following theorem displays the close connection between extended O-operators and the double Lie algebra structures on k in Condition 2.8. Theorem 2.13. Let g be a Lie algebra and (k, π ) be a g-Lie algebra. Let r± : k → g be two linear maps, λ, ν, κ, μ ∈ R and r and β be defined by Eq. (11). (i) Suppose r is an extended O-operator of weight λ with extension β of mass (ν, κ, μ) for ν = 0. Then Condition 2.8 holds. (ii) Suppose β satisfies β(ξ · x) = [ξ, β(x)]g, for all ξ ∈ g, x ∈ k, that is, β is g-invariant of mass 1 (or equivalently, a g-module homomorphism). Then r satisfies Eq. (12) for κ = −1, μ = ±λ if and only if the following equation holds: [r± (x), r± (y)]g − r± ([x, y] R ) = 0, ∀x, y ∈ k.

(16)

Proof. (i ) In order to prove that Eq. (12) implies the Jacobi identity for the bracket [, ] R on k, it is enough to prove that (k[β(x), β(y)]g + μβ([x, y]k)) · z + cycl. = 0. In fact, we will prove that k[β(x), β(y)]g · z + cycl. = 0

(17)

μβ([x, y]k) · z + cycl. = 0.

(18)

and

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563

Equation (17) has already been proved by Bordemann [10]. In order to be self-contained, we give the details. For any x, y, z ∈ k, k[β(x), β(y)]g · z = kβ(x)(β(y) · z) − kβ(y) · (β(x) · z) = −kβ(β(y) · z) · x − kβ(β(z) · x) · y (by antisymmetry) = −k[β(y), β(z)]g · x − k[β(z), β(x)]g · y (by g − invariance). So Eq. (17) follows immediately. Moreover, = = = = =

μβ([x, y]k) · z = −μβ(z) · [x, y]k (by antisymmetry) −μ[β(z) · x, y]k − μ[x, β(z) · y]k μ[β(x) · z, y]k + μ[x, β(y) · z]k (by antisymmetry) μβ(x) · [z, y]k − μ[z, β(x) · y]k + μβ(y) · [x, z]k − μ[β(y) · x, z]k −μβ([z, y]k) · x − μβ([x, z]k) · y + 2μ[β(x) · y, z]k (by antisymmetry) μβ([y, z]k) · x + μβ([z, x]k) · y + 2μβ([x, y]k) · z (by equivalence).

Therefore, Eq. (18) holds. So by Proposition 2.9, Condition 2.8 holds. (ii) A direct computation gives [(r ± β)(x), (r ± β)(y)]g − (r ± β)(r (x) · y − r (y) · x + λ[x, y]k) = [r (x), r (y)]g − r (r (x) · y − r (y) · x +λ[x, y]k) + [β(x), β(y)]g ∓ λβ([x, y]k) ± ([r (x), β(y)]g −β(r (x) · y) + [β(x), r (y)]g + β(r (y) · x)) = [r (x), r (y)]g − r (r (x) · y − r (y) · x + λ[x, y]k) + [β(x), β(y)]g ∓ λβ([x, y]k), where the last equality follows from g-invariance of mass 1. So (ii) holds.



Remark 2.14. When the bracket [, ]k on k is trivial and κ = −1, Proposition 2.9 and Theorem 2.13 give Theorem 2.18 in [10]. The following results give the relations of O-operators with Eq. (16) and extended O-operators. Theorem 2.15. Let g be a Lie algebra and (k, π ) be a g-Lie algebra. Let r± : k → g be two linear maps and let λ ∈ R and r and β be defined by Eq. (11). Suppose that β is antisymmetric of mass ν = 0, g-invariant of mass κ = 0 and equivalent of mass λ. (i) (k± , [ , ]± , π ) are g-Lie algebras, where (k± , [ , ]± ) are the new Lie algebra structures on k defined by [x, y]± ≡ λ[x, y]k ± 2β(x) · y, ∀x, y ∈ k.

(19)

(ii) r is an extended O-operator of weight λ with extension β of mass (ν, −1, ±λ) for ν = 0 if and only if r± : k∓ → g is an O-operator of weight 1, where k∓ is equipped with the Lie bracket [, ]∓ defined by Eq. (19). Proof. (i) Since β is antisymmetric, [, ]± is antisymmetric. Moreover, for any x, y, z ∈ k, we have [[x, y]± , z]± + cycl. = [λ[x, y]k ± 2β(x) · y, z]± + cycl. = (λ2 [[x, y]k, z]k ± 2λ[β(x) · y, z]k ± 2λβ([x, y]k) · z + 4β(β(x) · y) · z) + cycl. = (λ2 [[x, y]k, z]k ± 4λβ([x, y]k) · z + 4[β(x), β(y)]g · z) + cycl.,

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where the last equality follows from the g-invariance of mass κ = 0 and equivalence of mass λ. So by Theorem 2.13 the Jacobi identity for the bracket [, ]± on k holds. Moreover, for any ξ ∈ g, we have ξ · [x, y]± = λξ · [x, y]k ± 2ξ · (β(x) · y) = λ[ξ · x, y]k + λ[x, ξ · y]k ± 2β(ξ · x) · y ± 2β(x) · (ξ · y) (by g − invariance) = [ξ · x, y]± + [x, ξ · y]± . So (k± , π ) equipped with the bracket [, ]± on k is a g-Lie algebra. (ii) The last conclusion follows from Theorem 2.13, Item (i) and the following computations: [r± (x), r± (y)]g − r± ([x, y] R ) = [r± (x), r± (y)]g − r± (r± (x) · y − r± (y) · x + λ[x, y]k ∓ β(x) · y ± β(y) · x) = [r± (x), r± (y)]g − r± (r± (x) · y − r± (y) · x + [x, y]∓ ) (by antisymmetry).  When k in Theorem 2.15 is taken to be a vector space regarded as an abelian Lie algebra, we obtain the following conclusions. Corollary 2.16. Let g be a Lie algebra and V be a vector space. Let ρ : g → gl(V ) be a linear representation of g. Suppose that β : V → g is antisymmetric of mass κ = 0 and g-invariant of mass κ = 0. (i) (V± , [ , ]± , ρ) are g-Lie algebras, where (V± , [ , ]± ) are the Lie algebra structures on V defined by [u, v]± ≡ ±2β(u) · v, ∀u, v ∈ V.

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(ii) Let r : V → g be a linear map. Then r is an extended O-operator with extension β of mass −1 if and only if r ± β : V∓ → g are O-operators of weight 1, where V∓ are equipped with the Lie brackets [ , ]∓ defined by Eq. (20). 2.3. Adjoint representations and Baxter Lie algebras. We now consider the case of adjoint representations. If k = g with the trivial Lie bracket and π = ad, then by Proposition 2.9, Theorem 2.13 and Theorem 2.15 we have the following conclusion. Proposition 2.17. Let g be a Lie algebra and R, β : g → g be two linear maps. Let β be antisymmetric of mass κ and g-invariant of mass κ, i.e., the following equation holds: κβ([x, y]) = κ[β(x), y] = κ[x, β(y)], ∀x, y ∈ g.

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Suppose that R is an extended O-operator with extension β of mass κ, i.e., the following equation holds: [R(x), R(y)] − R([R(x), y] + [x, R(y)]) = κ[β(x), β(y)], ∀x, y ∈ g. Then the product [x, y] R = [R(x), y] + [x, R(y)], ∀x, y ∈ g,

(22)

Nonabelian Generalized Lax Pairs

565

defines a Lie bracket on g. On the other hand, if β satisfies Eq. (21) for κ = 0, then (g± , [ , ]± , ad) are g-Lie algebras, where (g± , [ , ]± ) are the new Lie algebra structures defined by [x, y]± ≡ ±2[β(x), y], ∀x, y ∈ g.

(23)

Moreover, R is an extended O-operator with extension β of mass −1, i.e., Eq. (22) holds for κ = −1, if and only if R ± β : g∓ → g are O-operators of weight 1, where g∓ are equipped with the Lie brackets [ , ]∓ defined by Eq. (23). Remark 2.18. Let g be a Lie algebra. A linear endomorphism β of g satisfying Eq. (21) for κ = 0 is called an intertwining operator in [40], where it is used to construct compatible Poisson brackets. If β : g → g is an intertwining operator on g, then it is also an averaging operator [2,43] in the Lie algebraic context, namely, [β(x), β(y)] = β([x, β(y)]) = β([β(x), y]), ∀x, y ∈ g, and is a Nijenhuis tensor, namely, [β(x), β(y)] + β 2 ([x, y]) = β([β(x), y] + [x, β(y)]), ∀x, y ∈ g.

(24)

Let the g-Lie algebra (k, π ) be (g, ad). It is obvious that β = id : g → g satisfies the conditions of Proposition 2.9, Theorem 2.13 and Theorem 2.15 and in this case, Eq. (12) takes the following form (set r = R): [R(x), R(y)] − R([R(x), y] + [x, R(y)] + λˆ [x, y]) = κ[x, ˆ y], ∀x, y ∈ g, (25) ˆ by Theorem 2.15, R satisfies Eq. (25) if for λˆ = λ and κˆ = κ + μ. When κˆ = −1 ± λ, and only if R ± id is a Rota-Baxter operator of weight λˆ ∓ 2. Note that when λˆ = 0, Eq. (25) takes the following form: [R(x), R(y)] − R([R(x), y] + [x, R(y)]) = κ[x, y], ∀x, y ∈ g,

(26)

for κ = κ. ˆ When κ = −1, Eq. (26) becomes [R(x), R(y)] − R([R(x), y] + [x, R(y)]) = −[x, y], ∀x, y ∈ g.

(27)

A Lie algebra equipped with a linear endomorphism satisfying Eq. (27) is called a Baxter Lie algebra in [10]. We note the difference between a Baxter Lie algebra and a Rota-Baxter Lie algebra defined in Definition 2.10. Moreover, the equivalence of the facts that R satisfies Eq. (27) and R ± id is a Rota-Baxter operator of weight ∓2 was pointed out in [19,44]. 3. Extended O-Operators, the Extended CYBE and Type II Quasitriangular Lie Bialgebras In this section, we define the extended CYBE and apply the study in Sect. 2 to investigate the relationship between extended O-operators and the extended CYBE. We also introduce the concept of type II quasitriangular Lie bialgebras from type II CYBE as a parallel concept of quasitriangular Lie bialgebras from CYBE. We then explicitly describe the Drinfeld’s doubles and Manin triples of type II quasitriangular Lie bialgebras.

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3.1. Lie bialgebras and the extended CYBE. We recall the following concepts [13]. Definition 3.1. Let g be a Lie algebra. (i) A Lie bialgebra structure on g is a skew-symmetric R-linear map δ : g → g ⊗ g, called co-commutator, such that (g, δ) is a Lie coalgebra and δ is a 1-cocycle of g with coefficients in g ⊗ g, that is, it satisfies the following equation: δ([x, y]) = (ad(x) ⊗ id + id ⊗ ad(x))δ(y) −(ad(y) ⊗ id + id ⊗ ad(y))δ(x), ∀x, y ∈ g. (ii) A Lie bialgebra (g, δ) is called coboundary if δ is a 1-coboundary, that is, there exists an r ∈ g ⊗ g such that δ(x) = (ad(x) ⊗ id + id ⊗ ad(x))r, ∀x ∈ g.

(28)

We usually denote the coboundary Lie bialgebra by (g, r ) or simply g. (iii) A Manin triple is a triple (a, a+ , a− ) of Lie algebras together with a nondegenerate symmetric invariant bilinear form B( , ) on a, such that (a) a+ and a− are Lie subalgebras of a; (b) a = a+ ⊕ a− as vector spaces; (c) a+ and a− are isotropic for B( , ). We recall the following basic results on Lie bialgebras and Manin triples. Proposition 3.2. ([17]) Let (g, δ) be a Lie bialgebra. Let D(g) ≡ g ⊕ g∗ . Then (D(g), g, g∗ ) is a Manin triple with respect to the bilinear form B p ((x, a ∗ ), (y, b∗ )) = a ∗ , y + x, b∗ , ∀x, y ∈ g, a ∗ , b∗ ∈ g∗ ,

(29)

on D(g). Explicitly, the Lie algebra structure on D(g) is given by [(x, a ∗ ), (y, b∗ )]D(g) = ([x, y] + ad∗ (a ∗ )y − ad∗ (b∗ )x, [a ∗ , b∗ ]δ + ad∗ (x)b∗ −ad∗ (y)a ∗ ), ∀x, y ∈ g, a ∗ , b∗ ∈ g∗ , (30) where the Lie algebra structure [ , ]δ on g∗ is defined by [a ∗ , b∗ ]δ , x = a ∗ ⊗ b∗ , δ(x), ∀x ∈ g, a ∗ , b∗ ∈ g∗ .

(31)

D(g) is called the Drinfeld’s double for the Lie bialgebra (g, r ). Proposition 3.3. ([13]) Let g be a Lie algebra and r ∈ g⊗g. The linear map δ defined by Eq. (28) is the commutator of a Lie bialgebra structure on g if and only if the following conditions are satisfied for all x ∈ g: (i) (ad(x) ⊗ id + id ⊗ ad(x))(r + σ (r )) = 0, that is, the symmetric part of r is invariant. (ii) (ad(x)⊗id⊗id+id⊗ad(x)⊗id+id⊗id⊗ad(x))([r12 , r13 ]+[r12 , r23 ]+[r13 , r23 ]) = 0. Here σ : g⊗2 → g⊗2 is the twisting operator defined by σ (x ⊗ y) = y ⊗ x, ∀x, y ∈ g.

Nonabelian Generalized Lax Pairs

In the following we call r =

567



ai ⊗ bi ∈ g⊗2 skew-symmetric (resp. symmetric) if

i

r = −σ (r ) (resp. r = σ (r )). Moreover, we use the notations (in the universal enveloping algebra U (g)): r12 =



ai ⊗ bi ⊗ 1, r13 =

i



ai ⊗ 1 ⊗ bi , r23 =



i

1 ⊗ ai ⊗ bi ,

i

and [r12 , r13 ] = =



 [ai , a j ] ⊗ bi ⊗ b j , [r13 , r23 ] i, j

ai ⊗ a j ⊗ [bi , b j ], [r23 , r12 ] =

i, j



a j ⊗ [ai , b j ] ⊗ bi .

i, j

The equation C(r ) ≡ [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = 0

(32)

is called the (tensor form of) the classical Yang-Baxter equation (CYBE). One should not confuse it with the (operator form of) CYBE of Bordemann [10], though under certain conditions the former is equivalent to a particular case of the latter that we will elaborate next. A coboundary Lie bialgebra (g, r ) arising from a solution of CYBE is said to be quasitriangular, whereas a coboundary Lie bialgebra (g, r ) arising from a skew-symmetric solution of CYBE is said to be triangular [9,13]. Note that for any coboundary Lie bialgebra (g, r ), the condition (i) in Proposition 3.3 holds automatically. For any r = ai ⊗ bi ∈ g ⊗ g, we set i

r21 =



bi ⊗ ai ⊗ 1, r32 =

i



1 ⊗ bi ⊗ ai , r31 =



i

bi ⊗ 1 ⊗ ai .

i

Moreover, we set [(a1 ⊗ a2 ⊗ a3 ), (b1 ⊗ b2 ⊗ b3 )] = [a1 , b1 ] ⊗ [a2 , b2 ] ⊗ [a3 , b3 ], ∀ ai , bi ∈ g, i = 1, 2, 3. Definition 3.4. Let g be a Lie algebra. Fix  ∈ R. The equation [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = [(r13 + r31 ), (r23 + r32 )]

(33)

is called the extended classical Yang-Baxter equation of mass  (or ECYBE of mass  in short). Remark 3.5. (i) When  = 0 or r is skew-symmetric, then the ECYBE of mass  is the same as the CYBE in Eq. (32).

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(ii) If the symmetric part β of r is invariant, by the proof of Theorem 3.9 below, for any a ∗ , b∗ , c∗ ∈ g∗ , we have [r13 + r31 , r23 + r32 ], a ∗ ⊗ b∗ ⊗ c∗  = = = ∗ ∗ ∗ [r13 + r31 , r23 + r32 ], a ⊗ b ⊗ c  = = =

4[β(a ∗ ), β(b∗ )], c∗  4β(ad∗ (β(a ∗ ))b∗ ), c∗  [r23 + r32 , r12 + r21 ], a ∗ ⊗ b∗ ⊗ c∗ , 4[β(a ∗ ), β(b∗ )], c∗  −4β(ad∗ (β(b∗ ))a ∗ ), c∗  [r12 + r21 , r13 + r31 ], a ∗ ⊗ b∗ ⊗ c∗ .

So in this case, the ECYBE of mass  is equivalent to either one of the following two equations: [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = [r23 + r32 , r12 + r21 ], [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = [r12 + r21 , r13 + r31 ]. 3.2. Extended O-operators and the ECYBE. We now study the relationship between extended O-operators and solutions of the ECYBE, generalizing the well-known relationship between the operator form and tensor form of the CYBE [29]. Let g be a Lie algebra and r ∈ g ⊗ g. Since g is assumed to be finite-dimensional, we will be able to identify r with the linear map r : g∗ → g through r (a ∗ ), b∗  = a ∗ ⊗ b∗ , r , ∀a ∗ , b∗ ∈ g∗ .

(34)

We will do this throughout the rest of the paper. Moreover, r t : g∗ → g is defined as a ∗ , r t (b∗ ) = a ∗ ⊗ b∗ , r , ∀a ∗ , b∗ ∈ g∗ . Note that r t is just the linear map (from g∗ to g) induced by σ (r ). We also use the following notations: α = (r − σ (r ))/2 = (r − r t )/2, β = (r + σ (r ))/2 = (r + r t )/2,

(35)

that is, α and β are the skew-symmetric part and symmetric part of r respectively, and in this case r = α + β and r t = −α + β. Lemma 3.6. Let g be a Lie algebra and β ∈ g ⊗ g be symmetric. Then the following conditions are equivalent: (i) β ∈ g ⊗ g is invariant, that is, (ad(x) ⊗ id + id ⊗ ad(x))β = 0, for any x ∈ g; (ii) β : g∗ → g is antisymmetry, that is, ad∗ (β(a ∗ ))b∗ + ad∗ (β(b∗ ))a ∗ = 0, for any a ∗ , b ∗ ∈ g∗ ; (iii) β : g∗ → g is g-invariant, that is, β(ad∗ (x)a ∗ ) = [x, β(a ∗ )], for any x ∈ g, a ∗ ∈ g∗ . Proof. Bordemann in [10] pointed out the equivalence of (ii) and (iii). For completeness, we shall prove (i)⇔(ii) and (i)⇔(iii). In fact, for any x ∈ g, a ∗ , b∗ ∈ g∗ , (ad(x) ⊗ id + id ⊗ ad(x))β, a ∗ ⊗ b∗  = β, −(ad∗ (x)a ∗ ) ⊗ b∗  + β, −a ∗ ⊗ (ad∗ (x)b∗ ) = a ∗ , [x, β(b∗ )] + [x, β(a ∗ )], b∗  (by symmetry) = ad∗ (β(b∗ ))a ∗ + ad∗ (β(a ∗ ))b∗ , x.

Nonabelian Generalized Lax Pairs

569

So (i)⇔(ii). Moreover, (ad(x) ⊗ id + id ⊗ ad(x))β, a ∗ ⊗ b∗  = β, −(ad∗ (x)a ∗ ) ⊗ b∗  + β, −a ∗ ⊗ (ad∗ (x)b∗ ) = −β(ad∗ (x)a ∗ ) + [x, β(a ∗ )], b∗ . So (i)⇔(iii).



Note that the condition (i) in Lemma 3.6 is exactly the condition (i) of Proposition 3.3. Lemma 3.7. ([29]). Let g be a Lie algebra and r ∈ g ⊗ g. Let α, β : g∗ → g be the two linear maps given by Eq. (35). Then the bracket [, ]δ defined by Eq. (31) satisfies [a ∗ , b∗ ]δ = ad∗ (r (a ∗ ))b∗ + ad∗ (r t (b∗ ))a ∗ , ∀a ∗ , b∗ ∈ g∗ .

(36)

Moreover, if the symmetric part β of r is invariant, then [a ∗ , b∗ ]δ = ad∗ (α(a ∗ ))b∗ − ad∗ (α(b∗ ))a ∗ , ∀a ∗ , b∗ ∈ g∗ .

(37)

We supply a proof to be self-contained. Proof. Let {ei }1≤i≤dimg be a basis of g and {ei∗ }1≤i≤dimg be its dual basis. Then the first conclusion holds due to the following equations:   [ek∗ , el∗ ]δ = ek∗ ⊗ el∗ , δ(es )es∗ = ek∗ ⊗ el∗ , (ad(es ) ⊗ id + id ⊗ ad(es ))r es∗ s

s

 k l = (atl cst + akt cst )es∗ = ad∗ (r (ek∗ ))el∗ + ad∗ (r t (el∗ ))ek∗ . s,t

The last conclusion follows from Lemma 3.6.



The above lemma motivates us to apply the study in Sect. 2. More precisely, we have the following results. Proposition 3.8. Let g be a Lie algebra and r ∈ g ⊗ g. Let α, β : g∗ → g be two linear maps given by Eq. (35). Suppose that β, regarded as an element of g ⊗ g, is invariant. (i) (g, r ) becomes a (coboundary) Lie bialgebra if α is an extended O-operator with extension β of mass κ ∈ R, namely the following equation holds: [α(a ∗ ), α(b∗ )] − α(ad∗ (α(a ∗ ))b∗ − ad∗ (α(b∗ ))a ∗ ) = κ[β(a ∗ ), β(b∗ )], ∀a ∗ , b∗ ∈ g∗ .

(38)

(ii) ([29]) The following conditions are equivalent: (a) α is an extended O-operator with extension β of mass −1, i.e., Eq. (38) with κ = −1) holds; (b) r (resp. −r t ) satisfies the following equation: [r (a ∗ ), r (b∗ )] = r ([a ∗ , b∗ ]δ ), ∀a ∗ , b∗ ∈ g∗

(39)

(resp. [(−r t )(a ∗ ), (−r t )(b∗ )] = (−r t )([a ∗ , b∗ ]δ ), ∀a ∗ , b∗ ∈ g∗ ); (40)

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(c) r (resp. −r t ) is an O-operator of weight 1, that is, r (resp. −r t ) satisfies the following equation: [r (a ∗ ), r (b∗ )] = r (ad∗ (r (a ∗ ))b∗ − ad∗ (r (b∗ ))a ∗ + [a ∗ , b∗ ]− ), ∀a ∗ , b∗ ∈ g∗ ,

(41)

(resp. [(−r t )(a ∗ ), (−r t )(b∗ )] = (−r t )(ad∗ ((−r t )(a ∗ ))b∗ − ad∗ ((−r t )(b∗ ))a ∗ + [a ∗ , b∗ ]+ ), ∀a ∗ , b∗ ∈ g∗ )

(42)

where the brackets [, ]± on g∗ are defined by [a ∗ , b∗ ]± ≡ ±2ad∗ (β(a ∗ ))b∗ , ∀a ∗ , b∗ ∈ g∗ ,

(43)

and (g∗ , ad∗ ) equipped with the brackets [, ]± on g∗ are g-Lie algebras. Proof. (i) By Lemma 3.7, we see that (g, r ) becomes a (coboundary) Lie bialgebra if the bracket [ , ]δ defined by Eq. (36) is a Lie structure on g∗ . Further by Lemma 3.6, β is antisymmetric of mass ν = 0 and g-invariant of mass κ = 0. Then the conclusion follows from Theorem 2.13.(i) by setting (k, π ) = (g∗ , ad∗ ) with trivial Lie bracket, r+ = r and r− = −r t . (ii) It follows from Theorem 2.13 and Theorem 2.15 by setting (k, π ) = (g∗ , ad∗ ) with trivial Lie bracket, r+ = r and r− = −r t .  The following theorem establishes a close relationship between extended O-operators on a Lie algebra g and solutions of the ECYBE in g. Theorem 3.9. Let g be a Lie algebra and let r ∈ g ⊗ g which is identified as a linear map from g∗ to g. Define α and β by Eq. (35). Suppose that the symmetric part β of r is invariant. Then r is a solution of ECYBE of mass κ+1 4 : [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] =

κ +1 [(r13 + r31 ), (r23 + r32 )] 4

if and only if α is an extended O-operator with extension β of mass κ, i.e., Eq. (38) holds. Proof. Let r =



u i ⊗ vi ∈ g ⊗ g for u i , vi ∈ g, then

i, j

 [r12 , r13 ], a ∗ ⊗ b∗ ⊗c∗  = [u i , u j ], a ∗ vi , b∗ v j , c∗  = −r (ad∗ (r t (b∗ ))a ∗ ), c∗ , i, j

 [r12 , r23 ], a ⊗b ⊗ c  = u i , a ∗ [vi , u j ], b∗ v j , c∗  = −r (ad∗ (r (a ∗ ))b∗ ), c∗ , ∗

∗

∗

i, j

[r13 , r23 ], a ∗ ⊗ b∗ ⊗ c∗  =

 u i , a ∗ u j , b∗ [vi , v j ], c∗  = [r (a ∗ ), r (b∗ )], c∗ . i, j

Therefore, r is a solution of CYBE in g if and only if Eq. (39) holds, i.e., [r (a ∗ ), r (b∗ )] = r (ad∗ (r (a ∗ ))b∗ + ad∗ (r t (b∗ ))a ∗ ), ∀a ∗ , b∗ ∈ g∗ .

Nonabelian Generalized Lax Pairs

571

Therefore, by Proposition 3.8, for any a ∗ , b∗ , c∗ ∈ g∗ , we have that [α(a ∗ ), α(b∗ )] − α(ad∗ (α(a ∗ ))b∗ − ad∗ (α(b∗ ))a ∗ ) − κ[β(a ∗ ), β(b∗ )], c∗  = [α(a ∗ ), α(b∗ )] − α(ad∗ (α(a ∗ ))b∗ − ad∗ (α(b∗ ))a ∗ ) + [β(a ∗ ), β(b∗ )] −(κ + 1)[β(a ∗ ), β(b∗ )], c∗  = [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ], a ∗ ⊗ b∗ ⊗ c∗ −(κ + 1)[β13 , β23 ], a ∗ ⊗ b∗ ⊗ c∗  r13 + r31 r23 + r32 , ], a ∗ ⊗ b∗ ⊗ c∗ . = [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] − (κ + 1)[ 2 2 So r is a solution of the ECYBE of mass (κ + 1)/4 if and only if α is an extended O-operator with extension β of mass κ.  Therefore by Proposition 3.8 and Theorem 3.9 (for κ = −1), we have the following conclusion: Corollary 3.10. ([29]). Let g be a Lie algebra and r ∈ g ⊗ g. Let α, β : g∗ → g be two linear maps given by Eq. (35). Suppose that β, regarded as an element of g ⊗ g, is invariant. Then the following conditions are equivalent: (i) r is a solution of the CYBE; (ii) (g, r ) is a quasitriangular Lie bialgebra; (iii) r (resp. −r t ) is an O-operator of weight 1, that is, r (resp. −r t ) satisfies Eq. (41) (resp. Eq. (42) with g∗ equipped with the bracket [, ]− (resp. [, ]+ ) defined by Eq. (43). (iv) α is an extended O-operator with extension β of mass −1, i.e., α and β satisfy Eq. (38) with k = −1; (v) r (resp. −r t ) satisfies Eq. (39) (resp. Eq. (40). 3.3. Extended O-operators (of mass 1) and type II CYBE. Proposition 3.8 and Theorem 3.9 reveal close connections of extended O-operators α : g∗ → g (defined by Eq. (38)) with coboundary Lie bialgebras and ECYBE. Thus we would like to study these operators in more detail. Note that, for κ = η2 κ  with κ, κ  ∈ R and η ∈ R× , α is an extended O-operator with extension β of mass κ if and only if α is an extended O-operator with extension ηβ of mass κ  . Thus we only need to consider the cases when κ = 0, 1, −1. The case of κ = −1 is considered in Corollary 3.10. The case of κ = 0 has been considered by Kupershmidt [31] as remarked before. So we will next focus on the case when κ = 1: [α(a ∗ ), α(b∗ )] − α(ad∗ (α(a ∗ ))b∗ − ad∗ (α(b∗ ))a ∗ ) = [β(a ∗ ), β(b∗ )], ∀a ∗ , b∗ ∈ g∗ . (44) Note here β regarded as an element of A ⊗ A is invariant (Lemma 3.6). Definition 3.11. Let g be a Lie algebra and r ∈ g ⊗ g. Then [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] =

1 [r13 + r31 , r23 + r32 ] 2

is called the type II Classical Yang-Baxter Equation (type II CYBE).

(45)

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C. Bai, L. Guo, X. Ni

The following conclusion follows directly from Theorem 3.9 for κ = 1. Proposition 3.12. Let g be a Lie algebra and r ∈ g ⊗ g. Let α, β : g∗ → g be two linear maps given by Eq. (35). Suppose that β, regarded as an element of g ⊗ g, is invariant. Then r is a solution of type II CYBE if and only if α is an extended O-operator with extension β of mass 1, i.e., Eq. (44) holds. In this case, (g, r ) becomes a coboundary Lie bialgebra. Corollary 3.13. Let g be a Lie algebra and r ∈ g ⊗ g. Let α, β : g∗ → g be the two linear maps given by Eq. (35). Suppose that β, regarded as an element of g ⊗ g, is √ invariant. Define gˆ = g ⊗ C = g ⊕ ig, where i = −1, and regard gˆ as a real Lie algebra. The following conditions are equivalent: (i) r is a solution of the type II CYBE. (ii) α is an extended O-operator with extension β of mass 1. (iii) Regarding α and iβ as linear maps from gˆ ∗ = g∗ ⊕ ig∗ to gˆ , α is an extended O-operator with extension iβ of mass −1. (iv) α ± iβ are solutions of the CYBE in gˆ . (v) α ± iβ, regarded as linear maps from gˆ ∗ = g∗ ⊕ ig∗ to gˆ , satisfy (α ± iβ)([a ∗ , b∗ ]δ ) = [(α ± iβ)(a ∗ ), (α ± iβ)(b∗ )], ∀a ∗ , b∗ ∈ g∗ ⊂ gˆ ∗ = g∗ ⊕ ig∗ , where the Lie algebra structure [, ]δ on

g∗

(46)

is given by Eq. (37).

Proof. By Proposition 3.12, we have (i)⇔(ii). It follows from the definition of extended O-operators that (ii)⇔(iii). Moreover, applying Proposition 3.8 to gˆ , we have (iii)⇔(iv). To prove (iv)⇔(v), we note that Proposition 3.8 also gives the equivalence of (iv) with the equation (α ± iβ)([u, v]δ ) = [(α ± iβ)(u), (α ± iβ)(v)], ∀u, v ∈ gˆ ∗ = g∗ ⊕ ig∗ ,

(47)

where [u, v]δ = ad∗ (α(u))v − ad∗ (α(v))u, ∀u, v ∈ gˆ ∗ = g∗ ⊕ ig∗ . Then (iv)⇔(v) follows since Eq. (47) ⇔ Eq. (46) by the definition of extended O-operators.  3.4. Type II quasitriangular Lie bialgebras. Considering the important role played by the Manin triple and Drinfeld’s double from a Lie bialgebra in the classification of the Poisson homogeneous spaces and symplectic leaves of the corresponding Poisson-Lie groups [18,25,45,53], it is important to investigate such Manin triple, as in [25,33,48]. However, explicit structures for Manin triples have been obtained only in special cases, such as for quasitriangular Lie bialgebras in [25]. Making use of the relationship between type II CYBE and extended O-operators as displayed in Proposition 3.12, we consider the following class of Lie bialgebras and obtain similar explicit constructions of their Manin triples. Definition 3.14. A coboundary Lie bialgebra (g, r ) is said to be type II quasitriangular if it arises from a solution r of type II CYBE given by Eq. (45). Our strategy is to express the Drinfeld’s double D(g) as an extension of a Lie algebra by an abelian Lie algebra, both derived from the extended O-operator associated to the solution r of the type II CYBE. We then obtain the structure of the Manin triple explicitly in terms of this extension.

Nonabelian Generalized Lax Pairs

573

3.4.1. An Lie algebra extension associated to a type II quasitriangular Lie bialgebra We obtain the Lie algebra extension from a type II quasitriangular Lie bialgebra by an exact sequence. Let g be a Lie algebra and r ∈ g ⊗ g. Define the symmetric and skew-symmetric parts α and β by Eq. (35). Lemma 3.15. With the same conditions as above, suppose that (g, r ) is a Lie bialgebra and β is invariant. (i) For any x ∈ g, a ∗ ∈ g∗ , we have ad∗ (a ∗ )x = −[x, α(a ∗ )] + α(ad∗ (x)a ∗ ). (ii) If r is a solution of type II CYBE, then [(−α(a ∗ ), a ∗ ), (−α(b∗ ), b∗ )]D(g) = (−[β(a ∗ ), β(b∗ )], 0), ∀a ∗ , b∗ ∈ g∗ . Proof. (i) By Lemma 3.7, for any x ∈ g, a ∗ , b∗ ∈ g∗ , we have ad∗ (a ∗ )x, b∗  = x, [b∗ , a ∗ ]δ  = x, −ad∗ (α(a ∗ ))b∗ + ad∗ (α(b∗ ))a ∗  = −[x, α(a ∗ )] + α(ad∗ (x)a ∗ ), b∗ , where the last equality follows from the fact that α is skew-symmetric. (ii) Since r is a solution of type II CYBE and β is invariant, by Proposition 3.12, α and β satisfy Eq. (44). So by Lemma 3.7 and Item (i), for any a ∗ , b∗ ∈ g∗ we have [(−α(a ∗ ), a ∗ ), (−α(b∗ ), b∗ )]D(g) = ([α(a ∗ ), α(b∗ )] − ad∗ (a ∗ )α(b∗ ) + ad∗ (b∗ )α(a ∗ ), [a ∗ , b∗ ]δ −ad∗ (α(a ∗ ))b∗ + ad∗ (α(b∗ ))a ∗ ) = ([α(a ∗ ), α(b∗ )] + [α(b∗ ), α(a ∗ )] − α(ad∗ (α(b∗ ))a ∗ ) −[α(a ∗ ), α(b∗ )] + α(ad∗ (α(a ∗ ))b∗ ), 0) = (−[α(a ∗ ), α(b∗ )] + α(ad∗ (α(a ∗ ))b∗ − ad∗ (α(b∗ ))a ∗ ), 0) = (−[β(a ∗ ), β(b∗ )], 0).  Now let (g, r ) be a type II quasitriangular Lie bialgebra. By Proposition 3.3, β ∈ g⊗g is invariant. Regarding β as a linear map from g∗ to g, we define f = Imβ, f⊥ = Kerβ. Then by Lemma 3.6, f is an ideal of g. On the other hand, define gˆ = g ⊗ C = g ⊕ ig, √ where i = −1, and regard gˆ as a real Lie algebra. Let D(g) ≡ g ⊕ g∗ be the Drinfeld’s double defined in Proposition 3.2. Proposition 3.16. With the notations explained above, define two linear maps ± : D(g) → gˆ by ± (x, a ∗ ) = x + α(a ∗ ) ± iβ(a ∗ ), ∀x ∈ g, a ∗ ∈ g∗ .

(48)

Then ± are homomorphisms of Lie algebras. Moreover, Ker+ = Ker− is an abelian Lie subalgebra of D(g).

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Proof. First, it is obvious that for any x, y ∈ g, ± ([x, y]D(g) ) = [± (x), ± (y)]gˆ . On the other hand, by Corollary 3.13.(v), Eq. (46) holds, that is, for any a ∗ , b∗ ∈ g∗ , we have ± ([a ∗ , b∗ ]D(g) ) = [± (a ∗ ), ± (b∗ )]gˆ . Furthermore, by Lemma 3.6 and Lemma 3.15.(i), we have ± ([x, a ∗ ]D(g) ) = ± (ad∗ (x)a ∗ − ad∗ (a ∗ )x) = α(ad∗ (x)a ∗ ) − ad∗ (a ∗ )x ± iβ(ad∗ (x)a ∗ ) = [x, α(a ∗ )] ± i[x, β(a ∗ )] = [x, (α ± iβ)(a ∗ )]gˆ = [± (x), ± (a ∗ )]gˆ .

So ± are homomorphisms of Lie algebras. Moreover, it is easy to show that Ker+ = Ker− = {(−α(a ∗ ), a ∗ )|a ∗ ∈ f⊥ = Kerβ}. By Lemma 3.15.(ii), for any a ∗ , b∗ ∈ f⊥ = Kerβ, we have [(−α(a ∗ ), a ∗ ), (−α(b∗ ), b∗ )]D(g) = (−[β(a ∗ ), β(b∗ )], 0) = (0, 0). So Ker+ = Ker− is an abelian Lie subalgebra of D(g).  Equip the space f⊥ = Kerβ with the structure of an abelian Lie algebra. Define a linear map ι : f⊥ → D(g) by ι(a ∗ ) = (−α(a ∗ ), a ∗ ), ∀a ∗ ∈ f⊥ . Then ι is in fact an embedding of Lie algebras whose image coincides with Ker+ = Ker− . On the other hand, the images of ± in gˆ = g ⊕ ig are g ⊕ iImβ = g ⊕ if, which is a Lie subalgebra of gˆ . Thus we have Proposition 3.17. The sequences 0

/ f⊥

ι

/ D(g)

±

/ g ⊕ if

/0

(49)

are exact. As a special case, we have Corollary 3.18. ([34]). Let g be a Lie algebra and r ∈ g ⊗ g. Define α and β by Eq. (35). Suppose that β is invariant and invertible (regarded as a linear map from g∗ to g). If (g, r ) is a type II quasitriangular Lie bialgebra, then ± : D(g) → g ⊕ ig are isomorphisms of Lie algebras. Proof. In this case, Ker+ = Ker− = 0 and Im+ = Im− = g ⊕ ig.



Nonabelian Generalized Lax Pairs

575

3.4.2. Description of the extension According to Proposition 3.17, D(g) is an extension of g ⊕ if by the abelian Lie algebra f⊥ . So there is an induced representation of g ⊕ if on f⊥ and the extension is uniquely defined by an element of H 2 (g ⊕ if, f⊥ ). To describe these structures explicitly, we need to fix two splittings S± : g ⊕ if → D(g) of Eq. (49) in the category of vector spaces, that is, ± ◦ S± = idg⊕i f such that S(0) = 0. In fact, suppose that s : f → g∗ is a right inverse of β : g∗ → f ⊂ g, that is, β ◦ s = idf, then the desired splittings S± : g ⊕ if → D(g) are defined by S± (x + i y) = x ∓ αs(y) ± s(y), ∀x ∈ g, y ∈ f. Recall that the construction of a Lie algebra h by a h-module V associated to a cohomology class [τ ] ∈ H 2 (h, V ) is the vector space h ⊕ V equipped with the bracket [(x, u), (y, v)] = ([x, y], x · v − y · u + τ (x, y)), ∀x, y ∈ h, u, v ∈ V. We denote such extension by h τ V . Returning to D(g), we shall write down the actions of g⊕if on f⊥ and the cohomology classes τ± explicitly. Lemma 3.19. The actions of g⊕if on f⊥ induced from the extensions defined by Eq. (49) are given by (x + i y) ·± a ∗ = ad∗ (x)a ∗ , for any x ∈ g, y ∈ f, a ∗ ∈ f⊥ . Proof. According to Lemma 3.15, for any x ∈ g, y ∈ f, a ∗ ∈ f⊥ , we have [S± (x + i y), ι(a ∗ )] = [x ∓ α(s(y)) ± s(y), −α(a ∗ ) + a ∗ ] = [x, −α(a ∗ ) + a ∗ ] ± [β(s(y)), β(a ∗ )] = −[x, α(a ∗ )] − ad∗ (a ∗ )x + ad(x)a ∗ = ι(ad(x)a ∗ ). So the actions are given by (x + i y) ·± a ∗ = ι−1 ([S(x + i y), ι(a ∗ )]) = ad∗ (x)a ∗ .



Theorem 3.20. Define two forms τ± : (g ⊕ if) ⊗ (g ⊕ if) → f⊥ by τ± (x1 + i y1 , x2 + i y2 ) = ±(ad∗ (x1 )s(y2 ) − ad∗ (x2 )s(y1 ) − s([x1 , y2 ]) + s([x2 , y1 ])), for any x1 , x2 ∈ g, y1 , y2 ∈ f⊥ . Then the forms τ± are 2-cocycles and D(g) ∼ = (g ⊕ if) τ± f⊥ .

(50)

Proof. The cohomology classes associated to the extensions defined by Eq. (49) are the classes of the 2-cocycles (x1 , x2 ∈ g, y1 , y2 ∈ f⊥ ) ι−1 ([S± (x1 + i y1 ), S± (x2 + i y2 )] − S± ([x1 + i y1 , x2 + i y2 ])) = ι−1 ([x1 ∓ α(s(y1 )) ± s(y1 ), x2 ∓ α(s(y2 )) ± s(y2 )] − S± ([x1 , x2 ] − [y1 , y2 ] + i([x1 , y2 ] + [y1 , x2 ])) = ι−1 ([x1 , x2 ] + [x1 , ±(−α(s(y2 )) + s(y2 ))] + [±(−α(s(y1 )) + s(y1 )), x2 ] + [−α(s(y1 )) + s(y1 ), −α(s(y2 )) + s(y2 )] − [x1 , x2 ] + [y1 , y2 ] ± α(s([x1 , y2 ] + [y1 , x2 ])) ∓ s([x1 , y2 ] + [y1 , x2 ])) = ι−1 (±ι(ad∗ (x1 )(s(y2 ))) ∓ ι(ad∗ (x2 )(s(y1 ))) − [β(s(y1 )), β(s(y2 ))] + [y1 , y2 ] ∓ι(s([x1 , y2 ] + [y1 , x2 ]))) = ±(ad∗ (x1 )s(y2 ) − ad∗ (x2 )s(y1 ) − s([x1 , y2 ]) + s([x2 , y1 ])),

where the third equality follows from Lemma 3.15.



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3.4.3. The embeddings of g and g∗ in D(g) and the description of the Manin triple We now apply the isomorphisms in Eq. (50) to describe the structure of the Manin triple (D(g), g, g∗ ) explicitly in terms of (g ⊕ if) τ± f⊥ . It is clear that from the identifications defined by Eq. (50), g is embedded in D(g) by x → (x, 0) τ± 0 ∈ (g ⊕ if) τ± f⊥ , ∀x ∈ g.

(51)

Moreover, for any a ∗ ∈ g∗ , we have a ∗ − s(β(a ∗ )) ∈ f⊥ and a ∗ = S± (α(a ∗ ) ±iβ(a ∗ )) + ι(a ∗ − s(β(a ∗ ))). So the embeddings of g∗ in (g ⊕ if) τ± f⊥ ∼ = D(g) are given by a ∗ → (α(a ∗ ) ± iβ(a ∗ )) τ± (a ∗ − s(β(a ∗ ))).

(52)

To describe the embeddings of g∗ more explicitly, we first recall some results in [25] about classification of subalgebras of extensions of the form h τ V , where h is a Lie algebra, V is an h-module and τ ∈ H 2 (h, V ). Let p : h τ V → h and q : h τ V → V be the projections p(h, u) = h and q(h, u) = u for any h ∈ h, u ∈ V . Theorem 3.21. ([25]). Let b be a Lie subalgebra of h and W be a b-submodule of V . Let φ : b → V /W be a 1-cochain whose coboundary is − ◦ τ |b, where  denotes the projection V → V /W . Define φ

bW = {(x, u)|x ∈ b, u + W = φ(x)}. φ

Then bW is a Lie subalgebra of h τ V . Conversely, if k is a Lie subalgebra of h τ V , φ then k is of the form bW , where b = p(k), W = k ∩ V and φ : b → V /W is given by φ(x) = q( p −1 (x)) + W , for any x ∈ b. We now identify g∗ with its embedded images inside (g ⊕ if) τ± f⊥ . It follows from Eq. (52) that W = Kerα ∩ Kerβ and b± = ± (g∗ ) = {α(a ∗ ) ± iβ(a ∗ )|a ∗ ∈ g∗ }, where ± are defined by Eq. (48). Furthermore the projections p± |g∗ : g∗ → ± (g∗ ) factor through the isomorphisms p¯ ± : g∗ /W → b± given by p¯ ± (a ∗ + W ) = α(a ∗ ) ± iβ(a ∗ ), ∀a ∗ ∈ g∗ , respectively. Hence the 1-cochains φ± : b± → ¯f⊥ = f⊥ /W of Theorem 3.21 in this situation are given by −1 −1 φ± (x + i y) = p¯ ± (x + i y) − sβ p¯ ± (x + i y) −1 (x + i y) ∓ s(y). = p¯ ±

(53)

Thus we have Theorem 3.22. The images of g∗ inside D(g) under the isomorphisms D(g) ∼ = (g ⊕ φ if) τ± f⊥ coincide with the subalgebras b± W± respectively, where b± = ± (g∗ ), W = Kerα ∩ Kerβ and φ± : b± → ¯f⊥ are described by Eq. (53). Remark 3.23. One can define a type II quasitriangular Poisson-Lie group as a simply connected Poisson-Lie group whose tangent Lie bialgebra is a type II quasitriangular Lie bialgebra. Moreover, one can investigate the above descriptions of the structure of D(g) and the embeddings of g and g∗ in D(g) in the context of (type II quasitriangular) Poisson-Lie groups. For the corresponding discussion of quasitriangular Lie bialgebras and quasitriangular Poisson-Lie groups, see the study in [25].

Nonabelian Generalized Lax Pairs

577

We end our explicit description of the Manin triple (D(g), g, g∗ ) in terms of the isomorphisms in Eq. (50) by expressing the bilinear form B p in Eq. (29). For any d = x + i y τ± η ∈ (g ⊕ if) τ± f⊥ , x ∈ g, y ∈ f, η ∈ f⊥ , define ± (d) ≡ x − α(η) ∓ α(s(y)) ∈ g, ± (d) ≡ η ± s(y) ∈ g∗ . Using Eq. (51) and Eq. (52), it is obvious that the compositions of the isomorphisms (g ⊕ if) τ± f⊥ ∼ = D(g) ∼ = g ⊕ g∗ are given by d → (± (d), ± (d)) respectively. Therefore, the bilinear forms given by Eq. (29) on (g ⊕ if) τ± f⊥ ∼ = D(g) satisfy B± (d1 , d2 ) ≡ ± (d1 ), ± (d2 ) + ± (d2 ), ± (d1 ). 4. Self-Dual Lie Algebras and Factorizable (Type II) Quasitriangular Lie Bialgebras We will focus on extended O-operators on self-dual Lie algebras and the related (type II) factorizable quasitriangular Lie bialgebras in this section. We first obtain finer properties of the various extended O-operators (in Eq. (22) and Eq. (38)) and the ECYBE in this context. We then apply these properties to provide new examples of (type II) factorizable quasitriangular Lie bialgebras. 4.1. Extended O-operators and the ECYBE on self-dual Lie algebras. Definition 4.1. Let g be a Lie algebra and B : g ⊗ g → R be a bilinear form. Suppose that R : g → g is a linear endomorphism of g. Then R is called self-adjoint (resp. skew-adjoint) with respect to B if B(R(x), y) = B(x, R(y)) (resp. B(R(x), y) = −B(x, R(y))) for any x, y ∈ g. Lemma 4.2. Let g be a Lie algebra and B : g ⊗ g → R be a nondegenerate symmetric invariant bilinear form. Let ϕ : g → g∗ be defined from B by Eq. (8). Suppose that β : g → g is an endomorphism that is self-adjoint with respect to B. Then for a given κ ∈ R, β is antisymmetric of mass κ and g-invariant of mass κ, i.e., it satisfies Eq. (21), if and only if β˜ = βϕ −1 : g∗ → g is antisymmetric of mass κ and g-invariant of mass κ, i.e., ˜ ∗ )], ∀x ∈ g, a ∗ ∈ g∗ , ˜ ∗ (x)a ∗ ) = κ[x, β(a κ β(ad

(54)

˜ ∗ ))b∗ + κad∗ (β(b ˜ ∗ ))a ∗ = 0, ∀a ∗ , b∗ ∈ g∗ . κad∗ (β(a

(55)

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C. Bai, L. Guo, X. Ni

Proof. When κ = 0, the conclusion is obvious. Now we assume κ = 0. Since B is symmetric and β is self-adjoint with respect to B, for any a ∗ , b∗ ∈ g∗ and x = ϕ −1 (a ∗ ), ˜ ∗ ), b∗  = y = ϕ −1 (b∗ ) ∈ g, we have β(x), ϕ(y) = ϕ(x), β(y). Hence β(a ∗ ∗ ˜ ˜ a , β(b ), that is, β as an element of g ⊗ g is symmetric. So by Lemma 3.6, Eq. (54) and Eq. (55) are equivalent. On the other hand, since B is symmetric and invariant and β is self-adjoint with respect to B, for any z ∈ g, we have ˜ ∗ ))b∗ , z = b∗ , [z, β(x)] = B(y, [z, β(x)]) ad∗ (β(a = B([y, z], β(x)) = B(β([y, z]), x), ˜ ∗ ))a ∗ , z = B(x, [z, β(y)]). ad∗ (β(b ˜ ∗ ))b∗ + ad∗ (β(b ˜ ∗ ))a ∗ = 0 if and only if β([y, z]) = Since B is nondegenerate, ad∗ (β(a [β(y), z], which is equivalent to the fact that β satisfies Eq. (21) for k = 0. So the conclusion follows.  Proposition 4.3. Let g be a Lie algebra and B : g ⊗ g → R be a nondegenerate symmetric invariant bilinear form. Let ϕ : g → g∗ be defined from B by Eq. (8). Suppose that R and β are two linear endomorphisms of g and β is self-adjoint with respect to B. Let κ ∈ R be given. (i) R is an extended O-operator with extension β of mass κ, i.e., β satisfies Eq. (21) and R and β satisfy Eq. (22), if and only if R˜ = Rϕ −1 : g∗ → g is an extended O-operator with extension β˜ = βϕ −1 : g∗ → g of mass κ, i.e., β˜ satisfies Eq. (54) ˜ where the linear and Eq. (55) and R˜ and β˜ satisfy Eq. (38) for α = R˜ and β = β, map ϕ : g → g∗ is defined by Eq. (8). (ii) Suppose in addition that R is skew-adjoint with respect to B. Then r± = R˜ ± β˜ regarded as an element of g ⊗ g is a solution of ECYBE of mass κ+1 4 if and only if R is an extended O-operator with extension β of mass κ. Proof. (i) First, by Lemma 4.2 we know that β is antisymmetric of mass κ and g-invariant of mass κ if and only if β˜ = βϕ −1 is antisymmetric of mass κ and g-invariant of mass κ. On the other hand, since B is symmetric and invariant, for any x, y, z ∈ g, we have B([x, y], z) = B(x, [y, z]) ⇔ ϕ([x, y]), z = ϕ(x), [y, z] ⇔ ϕ(ad(y)x) = ad∗ (y)ϕ(x). For any x, y ∈ g, put written as

a∗

=

ϕ(x), b∗

(56)

= ϕ(y). Since ϕ is invertible, Eq. (22) can be

˜ ∗ ), R(b ˜ ∗ )] − R(ϕ([ ˜ ˜ ∗ ), ϕ −1 (b∗ )] + [ϕ −1 (a ∗ ), R(b ˜ ∗ )])) = k[β(a ˜ ∗ ), β(b ˜ ∗ )]. [ R(a R(a By Eq. (56), the above equation is equivalent to ˜ ∗ )] − R(ad ˜ ∗ ( R(a ˜ ∗ ))b∗ − ad∗ ( R(b ˜ ∗ ))a ∗ ) = κ[β(a ˜ ∗ ), R(b ˜ ∗ ), β(b ˜ ∗ )]. [ R(a So R is an extended O-operator with extension β of mass κ if and only if R˜ = Rϕ −1 : g∗ → g is an extended O-operator with extension β˜ of mass κ. (ii) Furthermore, if R is skew-adjoint with respect to B, then R(x), ϕ(y) + ϕ(x), ˜ ∗ ) = 0, that is, R˜ regarded as an element ˜ ∗ ), b∗  + a ∗ , R(b R(y) = 0. Hence  R(a of g ⊗ g is skew-symmetric. Therefore, the conclusion (ii) follows from Item (i) and Theorem 3.9. 

Nonabelian Generalized Lax Pairs

579

As special cases of Proposition 4.3.(ii), we have Corollary 4.4. Under the same assumptions as in Proposition 4.3.(ii), we have (i) If κ = −1, then r± = R˜ ± β˜ as an element of g ⊗ g is a solution of the CYBE (Eq. (32)), namely (g, r± ) is a quasitriangular Lie bialgebra, if and only if R is an extended O-operator with extension β of mass −1, that is, β satisfies Eq. (21) for κ = 0 and R and β satisfy Eq. (22) for κ = −1. (ii) If κ = 1, then r± = R˜ ± β˜ as an element of g ⊗ g is a solution of type II CYBE (Eq. (45)), namely (g, r± ) is a type II quasitriangular Lie bialgebra, if and only if R is an extended O-operator with extension β of mass 1, that is, β satisfies Eq. (21) for κ = 0 and R and β satisfy Eq. (22) for κ = 1. Remark 4.5. Conclusion (i) in the above corollary in the special case when β = idg can also be found in [28]. 4.2. Factorizable quasitriangular Lie bialgebras. Recall that a quasitriangular Lie bialgebra (g, r ) is said to be factorizable if the symmetric part of r regarded as a linear map from g∗ to g is invertible. Factorizable quasitriangular Lie bialgebras are related to the factorization problem in integrable systems [39]. Next we will provide some new examples of factorizable quasitriangular Lie bialgebras. Lemma 4.6. Let G be a simply connected Lie group whose Lie algebra is g. Let N be a linear transformation of g which induces a left invariant (1, 1) tensor field on G. If there exists a left invariant torsion-free connection ∇ on G such that N is parallel with respect to ∇, then N is a Nijenhuis tensor, that is, it satisfies Eq. (24). Proof. Since N is parallel with respect to ∇, for any x, y ∈ g, we have that N (∇xˆ yˆ (e)) = ∇xˆ N (y)∧ (e), where x, ˆ yˆ are the left invariant vector fields generated by x, y ∈ g respectively and e is the identity element of G. Moreover, since ∇ is torsion-free, for any x, y ∈ g, we show that [N (x), N (y)] + N 2 ([x, y]) = ∇ N (x)∧ N (y)∧ (e) − ∇ N (y)∧ N (x)∧ (e) +N 2 (∇xˆ yˆ (e)) − N 2 (∇ yˆ x(e)) ˆ

= N (∇ N (x)∧ yˆ (e)) − N (∇ yˆ N (x)∧ (e))

ˆ +N (∇xˆ N (y)∧ (e)) − N (∇ N (y)∧ x(e)) = N ([N (x), y] + [x, N (y)]). 

Lemma 4.7. Let (g, r ) be a triangular Lie bialgebra, that is, r is a skew-symmetric solution of CYBE. Suppose that r regarded as a linear map from g∗ to g is invertible. Define a family of linear maps Nλ1 ,λ2 ,λ3 ,λ4 : D(g) = g ⊕ g∗ → D(g) = g ⊕ g∗ by Nλ1 ,λ2 ,λ3 ,λ4 (x, a ∗ ) = (λ1r (a ∗ ) + λ2 x, λ3 r −1 (x) + λ4 a ∗ ), ∀x ∈ g, a ∗ ∈ g∗ , λi ∈ R, i = 1, 2, 3, 4.

(57)

Then Nλ1 ,λ2 ,λ3 ,λ4 is skew-adjoint with respect to the bilinear form B p defined by Eq. (29) if and only if λ2 + λ4 = 0.

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The lemma is interesting on its own right since the simply connected Lie group corresponding to the Lie algebra in the lemma is a symplectic Lie group ([13,16,17]). Proof. In fact, for any x, y ∈ g, a ∗ , b∗ ∈ g, we have Bp (Nλ1 ,λ2 ,λ3 ,λ4 (x, a ∗ ), (y, b∗ )) + Bp ((x, a ∗ ), Nλ1 ,λ2 ,λ3 ,λ4 (y, b∗ )) = Bp ((λ1 r (a ∗ ) + λ2 x, λ3 r −1 (x) + λ4 a ∗ ), (y, b∗ )) + Bp ((x, a ∗ ), (λ1 r (b∗ ) + λ2 y, λ3 r −1 (y) + λ4 b∗ )) = λ1 r (a ∗ ), b∗  + λ2 x, b∗  + λ3 r −1 (x), y + λ4 a ∗ , y + λ3 x, r −1 (y) + λ4 x, b∗  + λ1 a ∗ , r (b∗ ) +λ2 a ∗ , y = (λ2 + λ4 )(x, b∗  + a ∗ , y),

where the last equality follows from r being skew-symmetric. So the conclusion follows.  Lemma 4.8. With the conditions and notations in Lemma 4.7, the linear operator Nλ1 ,λ2 ,λ3 ,λ4 defined by Eq. (57) is a Nijenhuis tensor on D(g), that is, it satisfies Eq. (24) on D(g). Proof. Let D(G) be the corresponding simply connected double Lie group of the Drinfeld’s double D(g), where G denotes the simply connected Poisson-Lie group of the Lie bialgebra (g, r ). Then it is easy to see that the following equation defines a left invariant torsion-free connection (in fact, according to [16], it is also flat) on D(G): ∇(x,a ∗ )∧ (y, b∗ )∧ (e) = (r (ad∗ (x)r −1 (y)) + ad∗ (a ∗ )y, ad∗ (r (a ∗ ))b∗ +ad∗ (x)b∗ ), ∀x, y ∈ g, a ∗ , b∗ ∈ g∗ , where (x, a ∗ )∧ , (y, b∗ )∧ are the left invariant vector fields generated by (x, a ∗ ), (y, b∗ ) ∈ D(g) respectively and e is the identity element of D(G). We only need to prove that the tensor Nλ1 ,λ2 ,λ3 ,λ4 defined by Eq. (57) is parallel with respect to the above connection, since then Nλ1 ,λ2 ,λ3 ,λ4 satisfies Eq. (24) on D(g) by Lemma 4.6. Now by Lemma 3.7, Corollary 3.10 and Lemma 3.15.(i), for any a ∗ , b∗ ∈ g∗ , ad∗ (a ∗ )r (b∗ ) = −[r (b∗ ), r (a ∗ )] + r (ad∗ (r (b∗ ))a ∗ ) = −r ([b∗ , a ∗ ]δ ) + r (ad∗ (r (b∗ ))a ∗ ) = r (ad∗ (r (a ∗ ))b∗ ). Moreover, for any x, y ∈ g, ∇(x,a ∗ )∧ Nλ1 ,λ2 ,λ3 ,λ4 (y, b∗ )∧ (e) = ∇(x,a ∗ )∧ (λ1r (b∗ ) + λ2 y, λ3 r −1 (y) + λ4 b∗ )∧ (e) = (λ1r (ad∗ (x)b∗ ) + λ2 r (ad∗ (x)r −1 (y)) + λ1 ad∗ (a ∗ )r (b∗ ) + λ2 ad∗ (a ∗ )y, λ3 ad∗ (r (a ∗ ))r −1 (y) + λ4 ad∗ (r (a ∗ ))b∗ + λ3 ad∗ (x)r −1 (y) + λ4 ad∗ (x)b∗ ), Nλ1 ,λ2 ,λ3 ,λ4 (∇(x,a ∗ )∧ (y, b∗ )∧ (e)) = Nλ1 ,λ2 ,λ3 ,λ4 (r (ad∗ (x)r −1 (y)) + ad∗ (a ∗ )y, ad∗ (r (a ∗ ))b∗ + ad∗ (x)b∗ ) = (λ1r (ad∗ (r (a ∗ ))b∗ ) + λ1r (ad∗ (x)b∗ ) + λ2 r (ad∗ (x)r −1 (y)) + λ2 ad∗ (a ∗ )y, λ3 ad∗ (x)r −1 (y) + λ3r −1 (ad∗ (a ∗ )y) + λ4 ad∗ (r (a ∗ ))b∗ + λ4 ad∗ (x)b∗ ). Therefore by Eq. (58), we get ∇(x,a ∗ )∧ Nλ1 ,λ2 ,λ3 ,λ4 (y, b∗ )∧ (e) = Nλ1 ,λ2 ,λ3 ,λ4 (∇(x,a ∗ )∧ (y, b∗ )∧ (e)). Thus, Nλ1 ,λ2 ,λ3 ,λ4 is parallel with respect to ∇, as needed.



(58)

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Proposition 4.9. Let (g, r ) be a triangular Lie bialgebra. Let B p be the bilinear form on D(g) = g ⊕ g∗ given by Eq. (29) and let ϕ : D(g) = g ⊕ g∗ → D(g)∗ = g ⊕ g∗ be the linear map induced by B p through Eq. (8) for B = B p . Define a family of linear endomorphisms of D(g) by Rμ (x, a ∗ ) ≡ (μr (a ∗ ) + x, −a ∗ ), ∀x ∈ g, a ∗ ∈ g∗ , μ ∈ R. Define r˜±,μ ≡ Rμ ϕ −1 ± ϕ −1 and regard r˜±,μ as elements of D(g) ⊗ D(g). Then (D(g), r˜±,μ ) are factorizable quasitriangular Lie bialgebras. Proof. First we prove that, for any μ ∈ R, Rμ is an extended O-operator with extension id : D(g) → D(g) of mass −1, that is, it satisfies Eq. (27) on D(g). Recall the Lie algebra structure of D(g) is given by Eq. (30). Then, for any x, y ∈ g, a ∗ , b∗ ∈ g∗ , we have [Rμ (x, a ∗ ), Rμ (y, b∗ )]D(g) = [(μr (a ∗ ) + x, −a ∗ ), (μr (b∗ ) + y, −b∗ )]D(g) = ([μr (a ∗ ) + x, μr (b∗ ) + y] + ad∗ (−a ∗ )(μr (b∗ ) + y) − ad∗ (−b∗ )(μr (a ∗ ) + x), [a ∗ , b∗ ]δ − ad∗ (μr (a ∗ ) + x)b∗ + ad∗ (μr (b∗ ) + y)a ∗ ). On the other hand, [(x, a ∗ ), (y, b∗ )]D(g) = ([x, y] + ad∗ (a ∗ )y − ad∗ (b∗ )x, [a ∗ , b∗ ]δ + ad∗ (x)b∗ − ad∗ (y)a ∗ ), ∗ ∗ Rμ ([Rμ (x, a ), (y, b )]D(g) ) = (−μr ([a ∗ , b∗ ]δ ) + μ2 r (ad∗ (r (a ∗ ))b∗ ) + μr (ad∗ (x)b∗ ) + μr (ad∗ (y)a ∗ ) +μ[r (a ∗ ), y] + [x, y] − ad∗ (a ∗ )y − μad∗ (b∗ )r (a ∗ ) − ad∗ (b∗ )x, [a ∗ , b∗ ]δ − μad∗ (r (a ∗ ))b∗ − ad∗ (x)b∗ − ad∗ (y)a ∗ ), Rμ ([(x, a ∗ ), Rμ (y, b∗ )]D(g) ) = (−μr ([a ∗ , b∗ ]δ ) − μr (ad∗ (x)b∗ ) − μ2 r (ad∗ (r (b∗ ))a ∗ ) − μr (ad∗ (y)a ∗ ) + μ[x, r (b∗ )] + [x, y] + μad∗ (a ∗ )(r (b∗ )) + ad∗ (a ∗ )y + ad∗ (b∗ )x, [a ∗ , b∗ ]δ + ad∗ (x)b∗ + μad∗ (r (b∗ ))a ∗ + ad∗ (y)a ∗ ). Therefore, by the fact that r is a homomorphism of Lie algebras (see Corollary 3.10), we get [Rμ (x, a ∗ ), Rμ (y, b∗ )]D(g) + [(x, a ∗ ), (y, b∗ )]D(g) = Rμ ([Rμ (x, a ∗ ), (y, b∗ )]D(g) ) +Rμ ([(x, a ∗ ), Rμ (y, b∗ )]D(g) ). On the other hand, from the proof of Lemma 4.7, we know that Rμ is skew-adjoint with respect to the nondegenerate symmetric invariant bilinear form B p . So the conclusion follows from Corollary 4.4.(i) by setting g = D(g), R = Rμ , β = idD(g) and B = B p .  Note that when μ = 0, then Proposition 4.9 gives a special case of the famous “Drinfeld’s double construction” [28] (in the original construction there is no restriction that g is triangular, or even coboundary).

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Proposition 4.10. Let (g, r ) be a triangular Lie bialgebra such that r regarded as a linear map from g∗ to g is invertible. Define two families of linear endomorphisms on D(g) by Nμ (x, a ∗ ) = (x, μr −1 (x) − a ∗ ), μ ∈ R; Nκ1 ,κ2 (x, a ∗ ) = (κ1r (a ∗ ) + κ2 x,

1 − κ22 −1 r (x)−κ2 a ∗ ), κ1 , κ2 ∈ R, κ22 = 1, κ1 = 0, κ1

for any x ∈ g, a ∗ ∈ g∗ . Let ϕ : D(g) = g ⊕ g∗ → D(g)∗ = g ⊕ g∗ be the linear map induced by the bilinear form B p given by Eq. (29) through Eq. (8) for B = B p . Define N˜ ±,μ ≡ Nμ ϕ −1 ± ϕ −1 , N˜ ±,κ1 ,κ2 ≡ Nκ1 ,κ2 ϕ −1 ± ϕ −1 and regard N˜ ±,μ and N˜ ±,κ1 ,κ2 as elements of D(g) ⊗ D(g). Then (D(g), N˜ ±,μ ) and (D(g), N˜ ±,κ1 ,κ2 ) are factorizable quasitriangular Lie bialgebras. Proof. In fact, according to Lemma 4.8, Nμ and Nκ1 ,κ2 satisfy Eq. (24) on D(g). Moreover, it is straightforward to check that Nμ2 = id and Nκ21 ,κ2 = id. So both of them satisfy Eq. (27) on D(g). On the other hand, by Lemma 4.7, they are skew-adjoint with respect to the nondegenerate symmetric invariant bilinear form B p . So the conclusion follows from Corollary 4.4.(i) by setting g = D(g), R = Nμ or Nκ1 ,κ2 , β = idD(g) and B = Bp.  4.3. Factorizable type II quasitriangular Lie bialgebras. We now consider the “factorizable” case of type II quasitriangular Lie bialgebras. Definition 4.11. A type II quasitriangular Lie bialgebra (g, r ) is called factorizable if the symmetric part β of r regarded as a linear map from g∗ to g is invertible. The following conclusion is the type II analogue of the “factorizable” property of quasitriangular Lie bialgebras [39]. Proposition 4.12. Let (g, r ) be a factorizable type II quasitriangular Lie bialgebra. Put r˜ = α + iβ : g ⊕ ig → g ⊕ ig, where α and β are defined by Eq. (35). Then any element x ∈ g admits a unique decomposition: x = x+ + x− , with (x+ , x− ) ∈ Im(˜r ⊕ r˜ t ) ⊂ g ⊕ ig, where r˜ and r˜ t are restricted to linear maps from ig∗ ⊂ g ⊕ ig to g ⊕ ig. Proof. Since r˜ + r˜ t = 2iβ and β : g∗ → g is invertible, we have x = r˜ (

β −1 (x) β −1 (x) ) + r˜ t ( ) ∈ Im(˜r ⊕ r˜ t ) ⊂ g ⊕ ig, ∀x ∈ g. 2i 2i

On the other hand, if there exist a ∗ , b∗ ∈ g∗ such that x = r˜ (ia ∗ ) + r˜ t (ia ∗ ) = r˜ (ib∗ ) + r˜ t (ib∗ ). Then 0 = r˜ (ia ∗ − ib∗ ) + r˜ t (ia ∗ − ib∗ ) = −2β(a ∗ − b∗ ). Since β : g∗ → g is invertible, we obtain a ∗ = b∗ . So the conclusion follows.  The following result provides a class of factorizable type II quasitriangular Lie bialgebras (hence a new class of (coboundary) Lie bialgebras).

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Proposition 4.13. Let (g, r ) be a triangular Lie bialgebra such that r regarded as a linear map from g∗ to g is invertible. Let B p be the bilinear form on D(g) = g ⊕ g∗ given by Eq. (29) and let ϕ : D(g) = g ⊕ g∗ → D(g)∗ = g ⊕ g∗ be the linear map induced by B p through Eq. (8) for B = B p . Define a family of linear endomorphisms on D(g) by Jλ,μ (x, a ∗ ) = (λr (a ∗ ) + μx,

−1 − μ2 −1 r (x) − μa ∗ ), λ, μ ∈ R, λ = 0. λ

Set r˜±,λ,μ ≡ Jλ,μ ϕ −1 ± ϕ −1 and regard r˜±,λ,μ as elements of D(g) ⊗ D(g). Then (D(g), r˜±,λ,μ ) are factorizable type II quasitriangular Lie bialgebras. Proof. In fact, according to Lemma 4.8, for any λ, μ ∈ R, Jλ,μ satisfies Eq. (24) on 2 = −id. So J D(g). Moreover, it is straightforward to check that Jλ,μ λ,μ satisfies Eq. (26) for κ = 1 on D(g). On the other hand, by Lemma 4.7, Jλ,μ is skew-adjoint with respect to the nondegenerate symmetric invariant bilinear form B p . So the conclusion follows from Corollary 4.4.(ii) by setting g = D(g), R = Jλ,μ , β = idD(g) and B = B p .  Remark 4.14. (i) A linear transformation on a Lie algebra g satisfying Eq. (24) and J 2 = −id is called a complex structure on g. Suppose a Lie algebra is self-dual with respect to a nondegenerate symmetric invariant bilinear form. According to Corollary 4.4.(ii), a complex structure on this Lie algebra that is self adjoint with respect to the bilinear form gives rise to a coboundary Lie bialgebra structure on this Lie algebra. This idea was pursued further in [34] in the study of Poisson-Lie groups. (ii) The complex structure J−1,0 has already been found in [16]. 5. O-Operators, PostLie Algebras and Dendriform Trialgebras In this section, we reveal a PostLie algebra structure underneath the O-operators. We then show that there is a close relationship between PostLie algebras and dendriform trialgebras of Loday and Ronco [36] in parallel to the relationship [42] between Pre-Lie algebras and dendriform bialgebras. 5.1. O-operators and PostLie algebras. We begin with recalling the concept of a PostLie algebra from an operad study [52]. Definition 5.1. ([52]). A (left) PostLie algebra is a R-vector space L with two bilinear operations ◦ and [, ] which satisfy the relations: [x, y] = −[y, x],

(59)

[[x, y], z] + [[z, x], y] + [[y, z], x] = 0,

(60)

z ◦ (y ◦ x) − y ◦ (z ◦ x) + (y ◦ z) ◦ x − (z ◦ y) ◦ x + [y, z] ◦ x = 0,

(61)

z ◦ [x, y] − [z ◦ x, y] − [x, z ◦ y] = 0,

(62)

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for all x, y ∈ L. Equation (59) and Eq. (60) mean that L is a Lie algebra for the bracket [, ], and we denote it by (G(L), [, ]). Moreover, we say that (L , [, ], ◦) is a PostLie algebra structure on (G(L), [, ]). On the other hand, it is straightforward to check that L is also a Lie algebra for the operation: {x, y} ≡ x ◦ y − y ◦ x + [x, y], ∀x, y ∈ L .

(63)

We shall denote it by (G(L), {, }) and say that (G(L), {, }) has a compatible PostLie algebra structure given by (L , [, ], ◦). A homomorphism between two PostLie algebras is defined as a linear map between the two PostLie algebras that preserves the corresponding operations. Remark 5.2. (i) The notion of PostLie algebra was introduced in [52] (in its “right version”), where it is pointed out that PostLie, the operad of PostLie algebras, is the Koszul dual of ComTrias, the operad of commutative trialgebras. (ii) If the bracket [, ] in the definition of PostLie algebra happens to be trivial, then a PostLie algebra is a pre-Lie algebra [11]. Lemma 5.3. Let (L , [, ], ◦) be a PostLie algebra. Define ρ : L → gl(L) by ρ(x)y = x ◦ y for any x, y ∈ L. Then (G(L), [ , ], ρ) is a (G(L), { , })-Lie algebra. Proof. By Eq. (61), ρ is a representation of (G(L), { , }). Then by Eq. (62), ρ is a Lie algebra homomorphism from (G(L), {, }) to Der R (G(L)).  Theorem 5.4. Let g be a Lie algebra and (k, π ) be a g-Lie algebra. Let r : k → g be an O-operator of weight λ. (i) The following operations define a PostLie algebra structure on the underlying vector space of k: [x, y] ≡ λ[x, y]k, x ◦ y ≡ r (x) · y, x, y ∈ k,

(64)

where [, ]k is the original Lie bracket of k. (ii) r is a Lie algebra homomorphism from G(k) to g, where k is taken as a PostLie algebra with the operations ([, ], ◦) defined in Eq. (64). (iii) If Ker(r ) is an ideal of (k, [, ]k), then there exists an induced PostLie algebra structure on r (k) given by [r (x), r (y)]r ≡ λr ([x, y]k), r (x) ◦r r (y) ≡ r (r (x) · y), ∀x, y ∈ k.

(65)

Further, r is a homomorphism of PostLie algebras. Proof. (i) Since k is a Lie algebra, Eq. (59) and Eq. (60) obviously hold. Furthermore, for any x, y, z ∈ k, we have z ◦ (y ◦ x) − y ◦ (z ◦ x) + (y ◦ z) ◦ x − (z ◦ y) ◦ x + [y, z] ◦ x = r (z) · (r (y) · x) − r (y) · (r (z) · x) + r (r (y) · z) · x − r (r (z) · y) · x + λr ([y, z]k) · x = ([r (z), r (y)]g − r (r (z) · y − r (y) · z + λ[z, y]k)) · x = 0. So Eq. (61) holds. Similarly, Eq. (62) holds, too. (ii) By Definition 5.1, for any x, y ∈ k we have r ({x, y}) =r (x ◦ y − y ◦ x +[x, y]) = r (r (x) · y −r (y) · x + λ[x, y]k) = [r (x), r (y)]g.

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(iii) We first prove that the multiplications given by Eq. (65) are well-defined. In fact, let x1 , y1 , x2 , y2 ∈ k be such that r (x1 ) = r (x2 ) and r (y1 ) = r (y2 ). Since x1 − x2 , y1 − y2 ∈ Ker(r ) and Ker(r ) is an ideal of (k, [, ]k), we have r (x1 ) ◦r r (y1 ) = r (r (x1 ) · y1 ) = r (r (x2 + (x1 − x2 )) · (y2 + (y1 − y2 ))) = r (r (x2 ) · y2 + r (x2 ) · (y1 − y2 )) = r (r (x2 ) · y2 ) + [r (x2 ), r (y1 − y2 )]g +r (r (y1 − y2 ) · x2 ) − λr ([x2 , y1 − y2 ]k) = r (r (x2 ) · y2 ) = r (x2 ) ◦r r (y2 ). Also, [r (x1 ), r (y1 )]r = [r (x2 ) + r (x1 − x2 ), r (y1 ) + r (y1 − y2 )]r = [r (x2 ), r (y2 )]r . Furthermore, we have r ([x, y]) = [r (x), r (y)]r and r (x ◦ y) = r (x) ◦r r (y) for any x, y ∈ k. Thus, (r (k), [, ]r , ◦r ) is a PostLie algebra since applying r to the PostLie algebra axioms of (k, [, ], ◦) gives the PostLie algebra axioms of (r (k), [, ]r , ◦r ). Finally, the last statement in Item (iii) is clear.  Corollary 5.5. Let g be a Lie algebra. Then there is a compatible PostLie algebra structure on g if and only if there exists a g-Lie algebra (k, π ) and an invertible O-operator r : k → g of weight 1. Proof. Suppose that g has a compatible PostLie algebra structure given by (L , [, ], ◦), that is, G(L) = g. By Lemma 5.3, (G(L), ρ, [, ]) is a g-Lie algebra, where ρ : L → gl(L) is defined as ρ(x)y = x ◦ y for any x, y ∈ L. Moreover, the equation {x, y} = x ◦ y − y ◦ x + [x, y] means that id : G(L) → G(L) = g is an O-operator of weight 1. Furthermore, id is obviously invertible. Conversely, suppose that (k, π ) is a g-Lie algebra and r : k → g is an invertible O-operator weight 1. Since Ker(r ) = {0}, by Theorem 5.4, there is a PostLie algebra structure on r (k) = g given by Eq. (65) for λ = 1. Moreover, it is obvious that (r (k) = g, [, ]r , ◦r ) (for λ = 1) is a compatible PostLie algebra structure on (g, [, ]g).  Corollary 5.6. Let g be a Lie algebra and R : g → g be a Rota-Baxter operator of weight λ ∈ R, that is, it satisfies Eq. (14). Then there is a PostLie algebra structure on g given by [x, y] ≡ λ[x, y]g, x ◦ y ≡ [R(x), y]g, ∀x, y ∈ g.

(66)

If in addition, R is invertible, then there is a compatible PostLie algebra structure on g given by [x, y] ≡ λR([R −1 (x), R −1 (y)]g), x ◦ y ≡ R([x, R −1 (y)]g), ∀x, y ∈ g.

Proof. The conclusion follows from Theorem 5.4.



We next give examples of PostLie algebras by applying Corollary 5.6. Example 5.7. Let g be a complex simple Lie algebra, h be its Cartan subalgebra,  be its root system and + ⊂  be the set of positive roots (with respect to some fixed order). For α ∈ , let gα ⊂ g be the corresponding root space. Put n± = ⊕α∈+ g±α , b± = h + n± . Then we have g = b+ + n− as decomposition of two subalgebras. Let

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Pb+ : g → b+ → g and Pn− : g → n− → g be the projections onto the subalgebras b+ and n− respectively. Then −Pb+ and −Pn− are Rota-Baxter operators of weight 1. Define new operations on g as follows: [x, y] ≡ [x, y]g, x ◦b+ y ≡ −[Pb+ (x), y]g, ∀x, y ∈ g.

(67)

By Corollary 5.6, ([, ], ◦b+ ) defines a PostLie algebra structure on g. If {Hi }i=1,...,n ∪ {X α }α∈+ ∪ {X −α }α∈+ is a basis of g, then the PostLie operations defined by Eq. (67) can be computed as follows: [x, y] = [x, y]g, X −α ◦b+ y = 0, Hi ◦b+ H j = 0, Hi ◦b+ X β = −β, αi X β , X α ◦b+ Hi = α, αi X α , X α ◦b+ X β = −Nα,β X α+β , ∀x, y ∈ g, α ∈ + , β ∈ . Similarly, with the same bracket [ , ] and with x ◦n− y ≡ −[Pn− (x), y]g, we obtain another PostLie algebra structure ([, ], ◦n− ) on g. The following result is interesting considering the importance of the Baxter Lie algebra in integrable systems [10,44]. Corollary 5.8. Let (g, R) be a Baxter Lie algebra, that is, R : g → g satisfies Eq. (27). Define the following operations on the underlying vector space of g by    R±1 (x), y , ∀x, y ∈ g. [x, y] ≡ [x, y]g, x ◦± y ≡ ∓2 g Then (g, [, ], ◦± ) are PostLie algebras. Proof. From the discussion at the end of Sect. 2.3, we show that (R ± 1)/(∓2) both are Rota-Baxter operators of weight 1. So the conclusion follows from Corollary 5.6.  By Corollary 3.10 and Theorem 5.4, we also obtain the following close relation between quasitriangular Lie bialgebras and PostLie algebras. Corollary 5.9. Let (g, r ) be a quasitriangular Lie bialgebra. Define β ∈ g ⊗ g by Eq. (35). Then [a ∗ , b∗ ] ≡ −2ad∗ (β(a ∗ ))b∗ , a ∗ ◦ b∗ ≡ ad∗ (r (a ∗ ))b∗ , ∀a ∗ , b∗ ∈ g∗ , defines a PostLie algebra structure on g∗ . If in addition, r regarded as a linear map from g∗ to g is invertible, then the following operations define a compatible PostLie algebra structure on g: [x, y] ≡ −2r (ad∗ (β(r −1 (x)))r −1 (y)), x ◦ y ≡ r (ad∗ (x)r −1 (y)), ∀x, y ∈ g. It is obvious that for any Lie algebra (g, [, ]), (g, [, ], −[, ]) is a PostLie algebra. Moreover, we have the following conclusion: Theorem 5.10. Let (g, [, ]) be a semisimple Lie algebra. Then any PostLie algebra structure (g, [, ], ◦) (on g, [, ])) is given by x ◦ y = [ f (x), y], ∀x, y ∈ g, where f : g → g is a Rota-Baxter operator of weight 1.

Nonabelian Generalized Lax Pairs

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Proof. Let L ◦ be the left multiplication operator with respect to ◦, that is, L ◦ (x)y = x ◦ y for any x, y ∈ g. Then by Eq. (62), L ◦ is a derivation of the Lie algebra g. Since g is semisimple, every derivation of g is inner. Therefore, there exists a linear map f : g → g such that x ◦ y = L ◦ (x)y = ad f (x)y = [ f (x), y], ∀x, y ∈ g. Moreover, by Eq. (61), we see that [[ f (y), f (z)], x] = [ f ([ f (y), z] + [y, f (z)] + [y, z]), x], ∀x, y, z ∈ g. Since the center of g is zero, f is a Rota-Baxter operator of weight 1.



Remark 5.11. In fact, the above conclusion can be extended to a Lie algebra g satisfying that the center of g is zero and every derivation of g is inner (such a Lie algebra is called complete [37]). On the other hand, note that f is a Rota-Baxter operator of weight 1 if and only if R = 2 f + 1 is an extended O-operator with extension id : g → g of mass −1, i.e., R satisfies Eq. (27). In particular, the classification of the linear maps satisfying Eq. (27) for every complex semisimple Lie algebra was given in [44]. 5.2. Dendriform trialgebras and PostLie algebras: a commutative diagram. Dendriform dialgebras [35] and trialgebras [36] are introduced with motivation from algebraic K -theory and topology. Dendriform dialgebras are known to give pre-Lie algebras. We will show that a more general correspondence holds between dendriform trialgebras and PostLie algebras. Definition 5.12. ([36]) A dendriform trialgebra (A, ≺, ", ·) is a vector space A equipped with three bilinear operations {≺, ", ·} satisfying the following equations: (x ≺ y) ≺ z = x ≺ (y  z), (x " y) ≺ z = x " (y ≺ z), (x  y) " z = x " (y " z), (x " y) · z = x " (y · z), (x ≺ y) · z = x · (y " z), (x · y) ≺ z = x · (y ≺ z), (x · y) · z = x · (y · z), for x, y, z ∈ A. Here  ≡≺ + " +·. According to [36], the product given by x  y = x ≺ y + x " y + x · y defines an associative product on A. Moreover, if the operation · is trivial, then a dendriform trialgebra is a dendriform dialgebra [35]. Proposition 5.13. Let (A, ≺, ", ·) be a dendriform trialgebra. Then the products [x, y] ≡ x · y − y · x, x ◦ y ≡ x " y − y ≺ x, ∀x, y ∈ A

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make (A, [, ], ◦) into a PostLie algebra. Proof. We will only prove Axiom (62). The other axioms are similarly proved. For any x, y, z ∈ A, we have z ◦ [x, y] − [z ◦ x, y] − [x, z ◦ y] = z " (x · y − y · x) − (x · y − y · x) ≺ z − (z " x − x ≺ z) · y +y · (z " x − x ≺ z) − x · (z " y − y ≺ z) + (z " y − y ≺ z) · x = z " (x · y) − (z " x) · y − z " (y · x) + (z " y) · x − (x · y) ≺ z + x · (y ≺ z) + (y · x) ≺ z − y · (x ≺ z)+(x ≺ z) · y −x · (z " y) + y · (z " x) − (y ≺ z) · x = 0. 

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It is easy to see that Eq. (63) and Eq. (68) fit into the commutative diagram: Dendriform trialgebra

x≺y+x"y+x·y

/ Associative algebra

[x,y]=x·y−y·x x◦y=x"y−y≺x

 PostLie algebra

xy−yx

x◦y−y◦x+[x,y]

 / Lie algebra

When the operation · of the dendriform trialgebra and the bracket [, ] of the PostLie algebra are trivial, we obtain the following commutative diagram introduced in [12] (see also [2,3,42]): Dendriform dialgebra

x≺y+x"y

x"y−y≺x

 Pre-Lie algebra

/ Associative algebra

xy−yx

x◦y−y◦x

 / Lie algebra

6. Triple Lie Algebras and Examples of Nonabelian Generalized Lax Pairs Our primary goal in this section is to apply our study of PostLie algebras in Sect. 5 to study integrable systems. To construct nonabelian generalized Lax pairs, we formulate the setup of a triple Lie algebra that is consistent with the classical r -matrix approach to integrable systems [13,28,44]. We then show that new situations where this setup applies are provided by PostLie algebras from Rota-Baxter operators on complex simple Lie algebras. 6.1. Triple Lie algebra and a typical example of nonabelian generalized Lax pairs. We introduce the following concept to obtain self-dual nonabelian generalized Lax pairs. Definition 6.1. A triple Lie algebra consists of the following data (g, [ , ]0 , ρ, [ , ], B, r, λ) where (i) (g, [ , ]0 ) is a Lie algebra; (ii) [ , ] is another Lie bracket on the underlying vector space of g and ρ : g → gl(g) is a representation of (g, [ , ]0 ) such that (g, [ , ], ρ) is a (g, [, ]0 )-Lie algebra. Denote x · y ≡ ρ(x)y, for any x, y ∈ g; (iii) B : g ⊗ g → R is a nondegenerate symmetric bilinear form such that Eq. (3) and Eq. (4) hold for (a, [, ]a) = (g, [, ]). (iv) r is in g ⊗ g such that the corresponding linear map r : g∗ → g through Eq. (34) has the property that the following bilinear operation defines a Lie bracket on g: [x, y]r ≡ r˜ (x) · y − r˜ (y) · x + λ[x, y], ∀x, y ∈ g, for certain λ ∈ R and for r˜ ≡ r ϕ : g → g where ϕ is defined by Eq. (8).

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A triple Lie algebra is so named because of the three Lie algebra structures [ , ]0 , [ , ] and [ , ]r on the same underlying vector space g. It often happens that the invariant condition in Eq. (3) implies Eq. (4), so Eq. (3) is enough in a triple Lie algebra. This is the case in the following classical example. This is also the case of PostLie algebras considered in Sect. 6.2. Example 6.2. An example of triple Lie algebra is the following well-known setting considered by Semenov-Tian-Shansky [13,28,44] in integrable systems. Let (g, [ , ]0 ) be a semisimple Lie algebra. Let ρ = ad be the adjoint representation. Let (g, [ , ]) be (g, [ , ]0 ) and let B( , ) be its Killing form. Let r be a skew-symmetric solution of the generalized classical Yang-Baxter equation (GCYBE): (ad(x) ⊗ id ⊗ id + id ⊗ ad(x) ⊗ id + id ⊗ id ⊗ ad(x))([r12 , r13 ] +[r12 , r23 ] + [r13 , r23 ]) = 0, ∀x ∈ g. Then Eq. (69) with λ = 0 defines a Lie bracket on the underlying vector space of g. Remark 6.3. (i) Let G be a simply connected Lie group whose Lie algebra is g. Then any representation ρ : g → gl(g) is determined by a left invariant flat connection ∇ on G through ρ(x)y ≡ ∇xˆ yˆ (e), ∀x, y ∈ g. Here x, ˆ yˆ are the left invariant vector fields generated by x, y ∈ g and e is the identity element of G. Moreover, a bilinear form B satisfying Eq. (4) for (a, [, ]a) = (g, [, ]) corresponds to a left invariant pseudo-Riemannian metric which is compatible with the connection ∇ [38]. (ii) By the study in Sect. 2, an obvious ansatz satisfying condition (iv) in Definition 6.1 is that r˜ is an extended O-operator of weight λ with extension β of mass (ν, κ, μ) for ν = 0. For a triple Lie algebra, there exists a Lie-Poisson structure [51] on g∗ , defined by { f, g}r (a ∗ ) ≡ [d f (a ∗ ), dg(a ∗ )]r , a ∗ , ∀ f, g ∈ C ∞ (g∗ ), a ∗ ∈ g∗ .

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Proposition 6.4. Given a triple Lie algebra (g, [ , ]0 , ρ, [ , ], B, r, λ) in Definition 6.1, any two smooth functions on g∗ that are invariant under the dual representation of ρ and the coadjoint representation of (g, [ , ]) are in involution with respect to the Lie-Poisson structure. Proof. If f and g are two smooth functions on g∗ that are invariant under the dual representation of ρ and the coadjoint representation of g, then { f, g}r (a ∗ ) = ρ(˜r (d f (a ∗ )))dg(a ∗ ), a ∗  − ρ(˜r (dg(a ∗ )))d f (a ∗ ), a ∗  + λ[d f (a ∗ ), dg(a ∗ )], a ∗  = −dg(a ∗ ), ρ ∗ (˜r (d f (a ∗ )))a ∗  + d f (a ∗ ), ρ ∗ (˜r (dg(a ∗ )))a ∗  + λd f (a ∗ ), ad∗ (dg(a ∗ ))a ∗  = 0, as needed.



The above proposition motivates us to consider Hamiltonian systems on g∗ with the Lie-Poisson structure {, }r .

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Theorem 6.5. Let a triple Lie algebra (g, [ , ]0 , ρ, [ , ], B, r, λ) be given. Let H (the Hamiltonian) be a smooth function on g∗ which is invariant under the dual representation of ρ and the coadjoint representation of (g, [ , ]). Let {ei }1≤i≤dimg be a basis of g with dual basis {ei }1≤i≤dimg with respect to B. Let ≡



ei ⊗ ei ∈ g ⊗ g

(71)

i

be the Casimir element. Let L , M : g∗ → g be smooth maps defined by L(a ∗ ) = (a ∗ ⊗ 1)() and M(a ∗ ) = r˜ (dH(a ∗ )), a ∗ ∈ g∗ . Then (g, ρ, g, L , M) is a self-dual nonabelian generalized Lax pair for the Hamiltonian system (g∗ , { , }r , H) in the sense of Definition 2.2. Proof. For any f ∈ C ∞ (g∗ ), we have d f (a ∗ ) = {H, f }r dt = ρ(˜r (dH(a ∗ )))d f (a ∗ ), a ∗  − ρ(˜r (d f (a ∗ )))dH(a ∗ ), a ∗  + λ[dH(a ∗ ), d f (a ∗ )], a ∗  = −d f (a ∗ ), ρ ∗ (˜r (dH(a ∗ )))a ∗ , ∀a ∗ ∈ g∗ . Since B satisfies Eq. (4) for (a, [, ]a) = (g, [, ]), it is easy to show that (cf. Lemma 4.2) (ρ(x) ⊗ id + id ⊗ ρ(x)) = 0, ∀x, y ∈ g.

(72)

Then d L(a ∗ ) = −((ρ ∗ (˜r (dH(a ∗ )))a ∗ ) ⊗ id)() = (a ∗ ⊗ 1)((ρ(M(a ∗ )) ⊗ id)) dt (73) = −(a ∗ ⊗ 1)((id ⊗ ρ(M(a ∗ )))). Hence d L(a ∗ ) = −ρ(M(a ∗ ))((a ∗ ⊗ 1)()) = −ρ(M(a ∗ ))L(a ∗ ). dt Therefore (g, ρ, g, L , M) is a self-dual nonabelian generalized Lax pair.



The invariant condition under the dual representation of ρ holds automatically in some interesting cases, such as in Example 6.2 and Sect. 6.2. This is also true for Corollary 6.8. Remark 6.6. Consider the triple Lie algebra in Example 6.2 and take H to be a smooth function on g∗ which is invariant under the coadjoint representation of (g, [ , ]). Applying Theorem 6.5, we have d L(a ∗ ) = [L(a ∗ ), M(a ∗ )], ∀a ∗ ∈ g∗ , dt that is, (L , M) is a Lax pair in the ordinary sense [13].

Nonabelian Generalized Lax Pairs

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We next show that (g, ρ, g, L , M) admits a certain “nonabelian generalized r -matrix ansatz”. First, the Poisson bracket of smooth functions on g∗ defined by Eq. (70) can be extended to g-valued functions in an obvious way: with the notations as above, let E and F be two g-valued smooth functions on g such that   E= E s es , F = Fs es , s

where E s , Fs ∈ C ∞ (g∗ ), then {E, F}r =

s

 {E s , Ft }r es ⊗ et . s,t

Suppose that r is skew-symmetric (resp. symmetric) and     r= ast es ⊗ et = − ats es ⊗ et (resp. r = ast es ⊗ et = ats es ⊗ et ). s,t

s,t

s,t

s,t

  Then r˜ (es ) = r (ϕ(es )) = − t ats et (resp. r˜ (es ) = r (ϕ(es )) = t ats et ). Set  k   t et . Since L(a ∗ ) = t] = k ek and e · es = ˜ , e ] = d e , [e , e c d [e s t k s l k st k st t ls  ∗ s ∗ ∗ s L s (a )e , where L s (a ) = es , a , we have   {L s , L t }r (a ∗ )es ⊗ et = [d L s (a ∗ ), d L t (a ∗ )]r , a ∗ es ⊗ et {L , L}r (a ∗ ) = s,t

s,t

  = [es , et ]r , a ∗ es ⊗ et = ˜r (es ) · et − r˜ (et ) · es +λ[es , et ], a ∗ es ⊗et s,t

=



s,t

−als el · et + alt el · es , a ∗ es ⊗ et + λ

s,t,l

(resp. {L , L}r (a ∗ ) =



dstk ek , a ∗ es ⊗ et .

s,t,k

  als el · et − alt el · es , a ∗ es ⊗ et + λ dstk ek , a ∗ es ⊗ et ). s,t,l

s,t,k

However, by Eq. (72) we have   el · es ⊗ es = − es ⊗ el · es . s

Letting

a∗

s

⊗ 1 act on both sides of the above equation, we see that   a ∗ , el · es es = − a ∗ , es el · es . s

s

Therefore −als el · et , a ∗ es ⊗ et = als et , a ∗ es ⊗ el · et , alt el · es , a ∗ es ⊗ et = −alt es , a ∗ el · es ⊗ et . t . In concluFurthermore, since B([es , et ], ek ) = −B(et , [es , ek ]), we have dstk = −d˜sk sion, we obtain the “nonabelian generalized r -matrix ansatz” that we are looking for (Eq. (7)).

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Theorem 6.7. When r is skew-symmetric (resp. symmetric), the self-dual nonabelian generalized Lax pair in Theorem 6.5 satisfies  t s t {L , L}r = {als clk ek , a ∗  − alt clk ek , a ∗  − λd˜sk ek , a ∗ }es ⊗ et , s,t,l,k

(resp. {L , L}r =



t s t {−als clk ek , a ∗  + alt clk ek , a ∗  − λd˜sk ek , a ∗ }es ⊗ et ).

s,t,l,k

Thus by Proposition 2.5, we have Corollary 6.8. With the conditions in Theorem 6.7, for any two smooth functions f and g on g that are invariant under the representation ρ and the adjoint representation of (g, [ , ]), we have { f ◦ L , g ◦ L}r = 0. 6.2. The case of PostLie algebras. We now apply Rota-Baxter operators and PostLie algebras to give an example of a triple Lie algebra. Theorem 6.9. Let (g, [, ]g) be a complex simple Lie algebra. Let R : g → g be a Rota-Baxter operator of weight 1. Let ([ , ], ◦) denote the PostLie algebra structure on g given by Eq. (66) for λ = 1. Let (g, ρ, [ , ]) denote the (g, {, })-Lie algebra given by Lemma 5.3. Let B denote the Killing form on g. Suppose there exists an r ∈ g ⊗ g such that [x, y]r ≡ ρ(˜r (x))y − ρ(˜r (y))x + λ˜ [x, y] = [R(˜r (x)), y]g +[x, R(˜r (y))]g + λ˜ [x, y]g, ∀x, y ∈ g,

(74)

defines a Lie bracket on the underlying vector space of g, where λ˜ ∈ R and r˜ ≡ r ϕ : g → g and ϕ is defined by Eq. (8). Then ˜ is a triple Lie algebra. (i) (g, { , }, ρ, [ , ], B, r, λ) (ii) Let H (the Hamiltonian) be a smooth function on g∗ which is invariant under the coadjoint representation of (g, [ , ]). Let  be the Casimir element in Eq. (71). Let L , M : g∗ → g be smooth maps defined by L(a ∗ ) = (a ∗ ⊗ 1)() and M(a ∗ ) = r˜ (dH(a ∗ )), a ∗ ∈ g∗ . Then (g, ρ, g, L , M) is a self-dual nonabelian generalized Lax pair for the Hamiltonian system (g∗ , { , }r , H), where { , }r is the Lie-Poisson structure defined in Eq. (70). (iii) If r is symmetric or skew-symmetric, then for any two smooth functions f and g on g that are invariant under the adjoint representation of (g, [ , ]), we have { f ◦ L , g ◦ L}r = 0. Proof. (i) Since B is the Killing form, it satisfies Eq. (3) for (a, [, ]a) = (g, [, ]). Moreover, we have B([R(x), y], z) + B(y, [R(x), z]) = 0 ⇔ B(ρ(x)y, z) +B(y, ρ(x)z) = 0, ∀x, y, z ∈ g, that is, B also satisfies Eq. (4) for (a, [, ]a) = (g, [, ]).

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(ii) If H is a smooth function which is invariant under the coadjoint action of G, then H is also invariant under the dual representation of ρ since for any x ∈ g, a ∗ ∈ g∗ , dH(a ∗ ), ρ ∗ (x)(a ∗ ) = −[R(x), dH(a ∗ )], a ∗  = dH(a ∗ ), ad∗ (R(x))a ∗  = 0. By Theorem 6.5, (g, ρ, g, L , M) is a self-dual nonabelian generalized Lax pair. (iii) In this case, f and g are also invariant under the representation ρ since by definition ρ(x)y = [R(x), y], for any x, y ∈ g. Then the conclusion follows from Corollary 6.8. 

Appendix: Extended O-Operators and Affine Geometry on Lie Groups In this Appendix, motivated by [10], we provide a geometric explanation of the extended O-operators. Let K be a simply connected Lie group whose Lie algebra is k. Let ∇ be a left invariant connection on K , which, according to [27], is specified by a linear map r˜ : k → gl(k) through r˜ (x) · y ≡ ∇xˆ yˆ (e), ∀x, y ∈ k, where x, ˆ yˆ are the left invariant vector fields generated by x, y ∈ k respectively and e is the identity element of K . Define a linear map r : k → gl(k) by r (x) · y ≡ ∇xˆ yˆ (e) −

λ λ [x, y]k = r˜ (x) · y − [x, y]k, ∀x, y ∈ k. 2 2

Let g be the Lie subalgebra of gl(k) generated by all r (x). Then r is a linear map from k to g. Furthermore, for any x, y ∈ k, we have [x, y] R ≡ r (x) · y − r (y) · x + λ[x, y]k λ λ = r˜ (x) · y − [x, y]k − r˜ (y) · x + [y, x]k + λ[x, y]k 2 2 = r˜ (x) · y − r˜ (y) · x = ∇xˆ yˆ (e) − ∇ yˆ x(e). ˆ So if [, ] R defines a Lie bracket on the underlying vector space of k and K R denotes the corresponding simply connected Lie group, then the left invariant connection determined by ∇xˆ yˆ (e) = r (x) · y +

λ [x, y]k 2

is torsion-free, where x, y ∈ k and e is the identity element of K R . Now we assume that k is a g-Lie algebra, that is, the image of r belongs to Der R (k), the Lie subalgebra consisting of the derivations of k. This is equivalent to ∇xˆ ([y, z]k)∧ (e) = [∇xˆ yˆ (e), z]k + [y, ∇xˆ zˆ (e)]k, ∀x, y, z ∈ k.

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Next we compute the curvature tensor R( , ) of ∇: R(x, ˆ yˆ )ˆz (e) = (∇xˆ ∇ yˆ − ∇ yˆ ∇xˆ − ∇[x,y]∧R )ˆz (e) = r (x) · (r (y) · z) +

λ λ [x, r (y) · z]k + r (x) · [y, z]k 2 2

λ2 [x, [y, z]k]k − r (y) · (r (x) · z) 4 λ λ λ2 − r (y) · [x, z]k − [y, r (x) · z]k − [y, [x, z]k]k 2 2 4 λ − r ([x, y] R ) · z − [r (x) · y, z]k 2 λ λ2 + [r (y) · x, z]k − [[x, y]k, z]k 2 2 λ2 = ([r (x), r (y)]g − r ([x, y] R )) · z − [[x, y]k, z]k, 4 +

where the Lie bracket [, ]g on g is the commutator bracket of linear transformations. Since [, ]k satisfies the Jacobi identity, we can re-interpret the “Jacobi identity condition” in Proposition 2.9.(ii) as the first Bianchi’s identity for the curvature tensor of a torsion-free connection. Theorem. With the same notations as above, suppose that k is a g-Lie algebra and [, ] R defines a Lie bracket on the underlying vector space of k. Denote K R for the corresponding simply connected Lie group. Let β : k → g be a linear map such that β is g-invariant of mass κ and also of mass μ, i.e., the following equations hold: κβ(ξ · x) = κ[ξ, β(x)]g, μβ(ξ · x) = μ[ξ, β(x)]g, ∀ξ ∈ g, x ∈ k. Let r and β satisfy Eq. (12). Then the corresponding curvature tensor (of the left invariant torsion-free connection ∇) Re (x, y)z ≡ κ[β(x), β(y)]g · z + μβ([x, y]k) · z −

λ2 [[x, y]k, z]k, ∀x, y, z ∈ k, 4

is g-invariant, that is, ξ · Re (x, y)z − Re (x, y)ξ · z − Re (ξ · x, y)z − Re (x, ξ · y)z = 0, ∀x, y, z ∈ k, ξ ∈ g. In particular, setting ξ = r (w), w ∈ k, then the curvature tensor is covariantly constant which in turn is equivalent to the Lie group K R being an affine locally symmetric space. Proof. The first statement depends on a direct computation. Moreover, combining with the fact that ∇ is torsion-free, we see that K R is affine locally symmetric (cf. [27]).  Remark. The above conclusion is a generalization of Theorem 3.7 in [10]. Acknowledgements. C. Bai was supported in part by the National Natural Science Foundation of China (10621101, 10920161), NKBRPC (2006CB805905) and SRFDP (200800550015). L. Guo was supported by NSF grant DMS 0505445 and thanks the Chern Institute of Mathematics at Nankai University for hospitality.

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32. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968) 33. Levendorskii, S.L., Soibelman, Y.S.: Algebras of functions on compact quantum groups, Schubert cells, and quantum tori. Commun. Math. Phys. 139, 141–170 (1991) 34. Liu, Z.-J., Qian, M.: Generalized Yang-Baxter equations, Koszul operators and Poisson-Lie groups. J. Diff. Geom. 35, 399–414 (1992) 35. Loday, J.-L.: Dialgebras. In: Dialgebras and Related Operads, Lecture Notes in Math. 1763, Berlin: Springer, 2001, pp. 7–66 36. Loday, J.-L., Ronco, M.: Trialgebras and families of polytopes. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory. Comtemp. Math. 346, 369–398 (2004) 37. Meng, D.J.: Some results on complete Lie algebras. Comm. Algebra 22, 5457–5507 (1994) 38. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976) 39. Reshetikhin, N., Semenov-Tian-Shansky, M.: Quantum R-matrices and factorization problems. J. Geom. Phys. 5, 533–550 (1998) 40. Reyman, A., Semenov-Tian-Shansky, M.: Compatible Poisson structures for Lax equations: an r-matrix approach. Phys. Lett. A 130, 456–460 (1988) 41. Reyman, A., Semenov-Tian-Shansky, M.: Group-theoretical methods in the theory of finite-dimensional integrable systems. In: Integrable Systems II. Dynamical Systems VII, Encyclopaedia of Math. Sciences, Vol. 16. Berlin: Springer-Verlag, 1994, pp. 116–220 42. Ronco, M.: Primitive elements in a free dendriform algebra. In: New Trends in Hopf Algebra Theory (La Falda, 1999). Contemp. Math. 267, 245–263 (2000) 43. Rota, G.-C.: Baxter operators, an introduction. In: Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, Joseph P.S. Kung, Ed., Boston: Birkhäuser, 1995, pp. 504–512 44. Semenov-Tian-Shansky, M.: What is a classical R-matrix? Funct. Anal. Appl. 17, 259–272 (1983) 45. Semenov-Tian-Shansky, M.: Dressing transformations and Poisson group actions. Publ. RIMS, Kyoto Univ. 21, 1237–1260 (1985) 46. Sklyanin, E.K.: On complete integrability of the Landau-Lifschitz equation. Preprint Leningr. Otd. Mat. Inst. E-3-79 (1979) 47. Sklyanin, E.K.: The quantum inverse scattering method. Zap. Nauch. Sem. LOMI 95, 55–128 (1980) 48. Stolin, A.: Some remarks on Lie bialgebra structures on simple complex Lie algebras. Comm. Algebra 27, 4289–4302 (1999) 49. Symes, W.: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math. 59, 13–51 (1980) 50. Symes, W.: Hamiltonian group actions and integrable systems. Physica D 1, 339–374 (1980) 51. Vaisman, I.: Lecture on the Geometry of Poisson Manifolds. Progress in Mathematics 118, Basel: Birkhäuser Verlag, 1994 52. Vallette, B.: Homology of generalized partition posets. J. Pure Appl. Algebra 208, 699–725 (2007) 53. Yakimov, M.: Symplectic leaves of complex reductive Poisson-Lie groups. Duke Math. J. 112, 453–509 (2002) Communicated by A. Connes

Commun. Math. Phys. 297, 597–619 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1052-5

Communications in

Mathematical Physics

Noncommutative Resolutions of ADE Fibered Calabi-Yau Threefolds Alexander Quintero Vélez1,2 , Alex Boer2 1 Mathematics Department, University of Glasgow, University Gardens,

Glasgow G12 8QW, UK. E-mail: [email protected]

2 Mathematisch Instituut, Universiteit Utrecht, P. O. Box 80010, 3508 TA,

Utrecht, The Netherlands. E-mail: [email protected]; [email protected] Received: 4 July 2008 / Accepted: 12 March 2010 Published online: 17 May 2010 – © Springer-Verlag 2010

Abstract: In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by Cachazo et al. (Geometric transitions and N = 1 quiver theories. http://arxiv.org/abs/hep-th/0108120v2, 2001). The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities. The construction is in terms of a noncommutative algebra introduced by Ginzburg (Calabi-Yau algebras. http://arxiv.org/abs/math/0612139v2, 2006) which we call the “N = 1 ADE quiver algebra”. 0. Introduction In recent years, there has been a great deal of interest in noncommutative algebra in connection with algebraic geometry, particularly in the study of singularities and their resolutions. The underlying idea in this context is that the resolutions of a singularity are closely linked to the structure of a noncommutative algebra. The case of Kleinian singularities X = C2 /G, for G a finite subgroup of SL(2, C), was the first non-trivial example of this phenomenon, studied in [12]. It was shown that the minimal resolution of X is a moduli space of representations of the preprojective algebra associated to the action of G. This preprojective algebra is known to be Morita equivalent to the skew group algebra C[x, y] # G, so we could alternatively use this algebra to construct the minimal resolution of X . Later, M. Kapranov and E. Vasserot [22] showed that there is a derived equivalence between C[x, y] # G and the minimal resolution of X . A similar statement was established by T. Bridgeland, A. King and M. Reid [8] for crepant resolutions of quotient singularities X = C3 /G arising from a finite subgroup G ⊆ SL(3, C). In this case the crepant resolution of X is realized as a moduli space of representations of the McKay quiver associated to the action of G, subject to a certain natural commutation relations (see [14] and Sect. 4.4 of [20]). Various steps in the direction mentioned above have been taken in a series of papers [21,30,31,37,42], where several concrete examples have been discussed. More abstract

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approaches have also been put forward in [6,29]. The lesson to be drawn from these works is that for some singularities it is possible to find a noncommutative algebra A such that the representation theory of this algebra dictates in every way the process of resolving these singularities. More precisely, it is shown that: • • • •

the centre of A is the coordinate ring of the singularity; the algebra A is finitely generated as a module over its centre; resolutions of the singularity are realized as moduli spaces of representations of A; the category of finitely generated modules over A is derived equivalent to the category of coherent sheaves on an appropriate resolution.

Following the terminology of M. Van den Bergh (cf. [40,41]) we may think of A as a “noncommutative resolution”. This phenomenon also appears naturally in string theory in the context of “geometric engineering”. There the singularity X should be a Calabi-Yau threefold and one studies Type IIB string theory compactified on X . It turns out that a collection of D-branes located at the singularity gives rise to a noncommutative algebra A, which can be described as the path algebra of a quiver with relations. For a fixed quiver Q, this construction only depends on a “noncommutative function” called the superpotential. As a consequence, the aforementioned derived equivalence establishes a correspondence between two different ways of describing a D-brane: as an object of the derived category of coherent sheaves on a crepant resolution of X and as a representation of the quiver Q. Now, let us explain the situation on which we will focus. We shall study a kind of singular Calabi-Yau threefolds obtained by fibering the total space of the semi-universal deformation of a Kleinian singularity over a complex plane, subsequently termed ADE fibered Calabi-Yau threefolds. They have been defined and studied in the work of F. Cachazo, S. Katz and C. Vafa [11] from the point of view of N = 1 quiver gauge theories. The quiver diagrams of interest here are the extended Dynkin quivers of type A, D or E. Following [20], for such a quiver Q, we associate a noncommutative algebra Aτ (Q) which we call the “N = 1 ADE quiver algebra”. The choice of τ is encoded in the fibration data. The goal of this paper is to show that the N = 1 ADE quiver algebra realizes a noncommutative resolution of the ADE fibered Calabi-Yau threefold associated with Q and τ . The proof of this result depends on two ingredients. On the one hand, we use the results in [12], on the construction of deformations of Kleinian singularities and their simultaneous resolutions in terms of Q. On the other hand, we use the results in [17] to construct a Morita equivalence between Aτ (Q) and a noncommutative crepant resolution Aτ in the sense of Van den Bergh; this allows us to use the techniques developed in [40] to show that the derived category of finitely generated modules over Aτ (Q) is equivalent to the derived category of coherent sheaves on any small resolution of the ADE fibered Calabi-Yau threefold. Some related results using different methods were obtained by B. Szendr˝oi in [38]. He considers threefolds X fibered over a general curve C by ADE singularities and shows that D-branes on a small resolution of X are classified by representations with relations of a Kronheimer-Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. The correspondence is given by a derived equivalence between a small resolution of X and a sheaf of noncommutative algebras on C. In particular, there is a substantial overlap between Sect. 3 of this paper and the results of Ref. [38]. The structure of the paper is as follows. Section 1 will be devoted to setting up the physical and mathematical context of our work. Even though many of the statements outlined in this section do not constitute rigorous mathematics, they provide motivation and background for what follows. In Sect. 2 we define the N = 1 ADE quiver algebra

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Aτ (Q), describe some of its basic structure and prove that small resolutions of ADE fibered Calabi-Yau threefolds are realized as moduli spaces of representations of Aτ (Q). We conclude with a discussion of the derived equivalence between Aτ (Q) and a small resolution of an ADE fibered Calabi-Yau threefold in Sect. 3.

1. Physical and Mathematical Context This section is a digression providing a general context and motivation for what we are doing. The setup for our discussion is the reverse geometric engineering of singularities. Let us start with some background about D-branes. Recall that type II superstrings are described by maps from a Riemann surface , the “worldsheet” as it is called, to a ten-dimensional “spacetime” manifold M. In the simplest instance, a D-brane is a submanifold of M on which open strings can end. This means that if a D-brane is present, then one needs to consider maps from a Riemann surface with boundaries to M such that the boundaries are mapped to a given submanifold W ⊆ M. In this case one says that there is a D-brane wrapped on W . If W is connected and has dimension p + 1 we will refer to the brane as a D p-brane. W itself is referred to as the “worldvolume” of the D-brane. In what follows, we assume that the underlying ten-dimensional space M is decomposed as M = R1,3 × X , where R1,3 denotes the four-dimensional Minkowski space and X is a six-dimensional space given by a Calabi-Yau threefold. We will specialize to D-branes in M whose worldvolume is of the form W = R1,3 × S with S ⊆ X . Forgetting about the manifold R1,3 for the moment, we will speak of D-branes wrapping S. D-branes are generally more complicated objects than just submanifolds in an ambient spacetime, because in string theory they are realized as boundary conditions for a certain auxiliary quantum field theory on the Riemann surface . More concretely, the data specifying the boundary conditions for the auxiliary theory on  include a choice of a rank r vector bundle E on S and a connection on it. From a physical viewpoint, such bundle should be thought of as r coincident D-branes wrapped on the same submanifold S. When these facts are properly taken into account, it turns out that the dynamics on the worldvolume S is an N = 1 supersymmetric gauge theory with gauge group U(r ). For the present discussion we will only consider D-branes in the open string topological B-model, in which case the submanifolds are complex submanifolds and the vector bundles are holomorphic. The above point of view can be generalised if one takes into account that a holomorphic vector bundle defined on a complex submanifold S defines a coherent sheaf i ∗ E (with i : S → X being the inclusion map). We are thus lead to consider coherent sheaves with support on a submanifold of X . However, more generally one would like to describe collections of D-branes and anti-D-branes.1 As explained in [1], among many other references, this forces us to consider not only coherent sheaves but complexes of coherent sheaves. Furthermore, maps between complexes are represented by tachyons and localization on quasi-isomorphisms is expected to be realized by renormalization group flow. Assembling this information, the picture that emerges is that D-branes do correspond to objects in the derived category of coherent sheaves on X . Now let us briefly discuss the geometric engineering of gauge theories. The idea behind geometric engineering is to look at the gauge theories that arise on D-branes at singularities. To be more concrete, we take spacetime to be M = R1,3 × X , 1 An anti-D-brane has all the same physical properties as an arbitrary D-brane, modulo the fact that they try to annihilate each other.

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where X is a Calabi-Yau threefold with an isolated singularity at P ∈ X and consider a D3-brane wrapped on R1,3 × {P}. In terms of the derived category this D3-brane is represented by the skyscraper sheaf O P on X . We want to determine the gauge theory on the worldvolume of such D3-brane. The crucial thing to note here is that O P is marginally stable to decay into a collection of so-called “fractional” branes E i .2 Each fractional brane may appear with multiplicity3 αi and so is associated to a factor of U(αi ) in the worldvolume gauge theory. Most notably, these branes are bound together by open string excitations corresponding to classes of morphisms φi j ∈ Hom(E i , E j ). As a result, the effective dynamics on the worldvolume R1,3 of the D3-brane is an N = 1 supersymmetric gauge theory whose matter content can be conveniently encoded in a quiver with relations coming from a “superpotential”. This will therefore be what is known as an N = 1 quiver gauge theory. We can be somewhat more precise about this. Let Q be a quiver with vertex set I and denote by CQ the corresponding path algebra. A superpotential is a formal sum of oriented cycles on the quiver, i.e. an element of the vector space CQ/[CQ, CQ]. On this space we can define for every arrow a a “derivation” ∂a that takes any occurrences of the arrow in an oriented cycle and removes them leading to a path from the head of a to its tail. An N = 1 quiver gauge theory consists of a quiver Q together with a choice of a superpotential W and a dimension vector α ∈ N I . Given an N = 1 quiver gauge theory, one can construct the N = 1 quiver algebra A = CQ/ (∂a W | a ∈ Q) . We now explain how this data gives rise to an N = 1 supersymmetric gauge theory on R1,3 . Suppose we are given an N = 1 quiver gauge theory (Q, W, α). Representations of the quiver Q of dimension vector α are given by elements of the vector space  Rep(Q, α) = HomC (Cαt (a) , Cαh(a) ), a∈Q

where h(a) and t (a) denote the head and tail vertices of  an arrow a. The isomorphism classes correspond to orbits of the group GL(α) = i∈I GL(αi , C) acting by conjugation.  We define a Hermitian inner product on Rep(Q, α) via the trace form (x, y) = a∈Q tr(xa ya∗ ), where ∗ denotes the adjoint map. Let U(α) denote the prod uct of unitary groups i∈I U(αi ). This is a maximal compact subgroup of GL(α) and acts on Rep(Q, α), preserving the Hermitian structure. The corresponding moment map  μα : Rep(Q, α) → i∈I u(αi ) is given by ⎛ ⎞   √ μα (x)i = −1 ⎝ xa xa∗ − xa∗ xa ⎠ . h(a)=i

t (a)=i

Now consider the superpotential W . Recall that it is required to be a sum of oriented  cycles in Q. If W = i1 ,...,ir ai1 · · · air , then the function Wα : Rep(Q, α) → C given by  Wα (x) = tr(xai1 · · · xair ) i 1 ,...,ir

2 The existence of this decay is argued for in [2]. In modern language, this construction amounts to intro-

ducing a notion of Bridgeland stability for D-branes; see, for example, [5]. 3 The multiplicities α are uniquely determined via the condition  α ch(E ) = ch(O ) (as asserted in i i P i i Eq. (2.1) of [43]).

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is invariant under the action of GL(α), and thus also U(α). The upshot of all this is that an N = 1 quiver gauge theory gives rise to a quadruple (U(α), Rep(Q, α), μα , Wα ). According to [18, Supersolutions, Sect. 6.2], the latter is the data one needs to specify an N = 1 supersymmetric gauge theory on R1,3 . The gauge group is given by U(α). The field content of the theory associated with the quiver is encoded as follows. We associate to each vertex i an N = 1 vector multiplet Ai = (Ai , λi , Di ) and to each arrow a an N = 1 chiral multiplet a = (φa , ψa , Fa ). In the first (vector) multiplet, Ai is a connection on some principal U(αi ) bundle over R1,3 , λi is spinor with values in the adjoint bundle and Di is an auxiliary field with values in the adjoint bundle. Letting Ri be the fundamental αi -dimensional representation of U(αi ), there is an associated vector bundle E i . If Ri∗ is the dual representation of Ri –sometimes called “antifundamental representation”– then the associated vector bundle is the dual bundle to E i . In the chiral multiplet, φa is a section of E t∗(a) ⊗ E h(a) , ψa is a spinor with values in E t∗(a) ⊗ E h(a) , and Fa is an auxiliary field with values in E t∗(a) ⊗ E h(a) . The chiral multiplets are therefore often called “bifundamental fields”. The explicit Lagrangian, together with the relevant supersymmetry transformations, is given in [18, Th. 6.33]. Now we give a description of the moduli space of classical vacua of an N = 1 quiver gauge theory. Stealing a look at Theorem 6.33 of [18], the moduli space of classical vacua of such a quiver gauge theory is M0 = μ−1 α (0) ∩ Crit(Wα )/ U(α). Here ‘Crit’ denotes the set of critical points. In general, Wα drops to a regular function W α on the symplectic quotient μ−1 α (0)/ U(α) and the moduli space of vacua is the set of critical points of W α on this quotient. There is another mathematical interpretation of this process, as a quotient in the sense of GIT: we complexify the group U(α) to GL(α), and consider the action of GL(α) on Rep(Q, α). It turns out that we can identify M0 with the set of critical points of Wα on the affine quotient variety Rep(Q, α)  GL(α). Let us restate the above in terms of the N = 1 quiver algebra A. Let Rep(A, α) denote the closed subspace of Rep(Q, α) corresponding to representations for A. The group GL(α) acts naturally on this variety, and the orbits correspond to isomorphism classes of representations. It is pointed out in [33, Prop. 3.8] that the set of critical points of Wα is precisely Rep(A, α). Altogether this implies that the moduli space of vacua M0 admits an alternate presentation as an affine quotient Rep(A, α)GL(α). At least in passing, we should mention that this construction has a noncommutative-geometric interpretation in which A is viewed as the noncommutative coordinate ring of the “critical locus” of W ; see, for instance, [16]. It is also useful to make the following remark. The moduli space of classical vacua M0 is well known to have several irreducible components, typically referred to as “branches”. This feature was considered in [19] to be reflected by the fact that Rep(A, α) is in general a reducible variety. In this reference it was argued that there exists a unique top-dimensional irreducible component of Rep(A, α), which we denote by V A (α). The “Higgs branch” of the moduli space of classical vacua is given by the affine quotient V A (α)  GL(α). The other components of M0 are commonly referred to as “Coulomb branches”. Now we come to the central point. One of the main insights in [26], further explored in [3], was that for the N = 1 quiver gauge theory associated to a D3-brane on an isolated Calabi-Yau singularity X , the Higgs branch of the moduli space of classical vacua recovers the geometry of X . A bit more precisely, we are asserting that ∼ V A (α)  GL(α) ⊆ M0 . X=

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Here the dimension vector α is fixed in terms of the multiplicities αi of the fractional branes. From this perspective, the worldvolume gauge theory of the D-brane is the primary concept, whereas the spacetime itself is a secondary, derived concept. The above discussion suggests that it would be possible to reconstruct singular Calabi-Yau threefolds from N = 1 quiver gauge theories. Following the terminology in [4], we call this process “reverse geometric engineering of singularities”. The basic idea of this construction may be summarised as follows. One is given an N = 1 quiver gauge theory so that the corresponding N = 1 quiver algebra A is finitely generated as a module over its centre Z (A). Then Z (A) is itself the coordinate ring of a three-dimensional variety, to be identified as the singularity X . We must show, then, that the Higgs branch of the moduli space of classical vacua coincides with the variety Spec Z (A). (This ties in with a general principle of noncommutative algebraic geometry espoused in [28] and further developed in [30].) A more physical version of this statement is to say that Spec Z (A) will correspond to the “spacetime” in which closed strings propagate, while A is associated to a noncommutative algebraic geometry that D-branes see. In a similar vein, one can expect that crepant resolutions of the Calabi-Yau singularity X = Spec Z (A) are realized as moduli spaces of the D-brane theory, in the presence of Fayet-Iliopoulos parameters. Let us spell out more clearly what we mean by this. If θ : Z I → Z satisfies θ (α) = 0, then there are notions of θ -stable and θ -semistable elements of V A (α), there is a GIT quotient V A (α) θ GL(α), and a natural map πθ : V A (α) θ GL(α) → V A (α)  GL(α), which is a projective morphism; see Sect. 2.3 below for a more precise description. Moreover, if α is such that the general element of V A (α) is a simple representation of A, then πθ is a birational map of irreducible varieties. The quotient V A (α)θ GL(α) is a quasiprojective variety whose points are in bijection with S-equivalence classes of θ -semistable elements of V A (α). This is usually called the “mesonic moduli space with Fayet-Iliopoulos parameters θ ”. It turns out that, in many cases, if θ is chosen so that θ -semistables are θ -stable, then the corresponding mesonic moduli space V A (α) θ GL(α) is a smooth Calabi-Yau threefold and so πθ is a crepant resolution of the Calabi-Yau singularity X = Spec Z (A). We actually can go somewhat further along these lines. Let Y be any crepant resolution of the singularities of X = Spec Z (A). A point that we mentioned earlier but did not elaborate upon, is that D-branes on Y should be properly regarded as objects in Db (Coh(Y )), the bounded derived category of coherent sheaves on Y . The foregoing discussion makes it highly plausible that the N = 1 quiver gauge theory would give a different description of these D-branes in terms of representations of the N = 1 quiver algebra A. This statement can be made more precise by saying that there is an equivalence of triangulated categories Db (Coh(Y )) ∼ = Db (mod– A), where Db (mod–A) is the bounded derived category of finitely generated right modules over A. In M. Van den Bergh’s terminology, A is a “noncommutative crepant resolution” of X = Spec Z (A). The situation that we actually wish to apply this to is the case of ADE fibered CalabiYau threefolds and their small resolutions; see Sect. 2.1 below for details. The relevant N = 1 quiver gauge theory was written down in [11] (see also [10,23,44]). We will explicitly carry out the previous construction in the subsequent sections.

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2. ADE Fibered Calabi-Yau Threefolds and Their Small Resolutions Revisited This section studies the reverse geometric engineering of ADE fibered Calabi-Yau threefolds and their small resolutions along the lines indicated in the previous section. Before we do so, we describe our general setup and fix notation. 2.1. General setup. Let G be a finite subgroup of SL(2, C), let C2 /G be the corresponding Kleinian singularity and let π0 : Y0 → C2 /G be its minimal resolution. The exceptional divisor C of π0 is known to be a union of projective lines intersecting transversally, and the graph whose vertices correspond to the irreducible components of C, with two vertices joined if and only if the components intersect, is a Dynkin diagram of type A, D or E. Now let Z (CG) be the centre of the group algebra CG, and let h be the codimension one hyperplane in Z (CG) formed by all central elements which have trace zero in CG. According to the McKay correspondence, the dual space h∗ carries a root system associated to the Dynkin diagram . Write W for the Weyl group of this root system. It is then a fairly standard result (see, for example, [34] and references therein) that the Kleinian singularity C2 /G has a semi-universal deformation, a flat family ϕ : X → h/W whose fiber over 0 is C2 /G. Furthermore, Brieskorn and Tyurina [9,39] showed that this family admits a simultaneous resolution after making the base change h → h/W. More precisely, the family X ×h/W h may be resolved explicitly and one obtains a simultaneous resolution Y → X ×h/W h of ϕ inducing the minimal resolution Y0 → C2 /G. The situation can be conveniently summarised by the diagram Y II / X ×h/W h II II II II  I$ h

/X  / h/W

Using these observations we can define a broader class of Calabi-Yau threefolds as follows. We want to obtain a Gorenstein Calabi-Yau threefold X by fibering the total space of the semiuniversal deformation of a Kleinian singularity C2 /G over a complex plane. To make things more concrete, let t : C → h be a polynomial map. Via the defining equation for the family X ×h/W h, we can view X as the total space of a one parameter family defined by t. Similarly, the simultaneous resolution Y → X ×h/W h can be used to construct a Calabi-Yau threefold Y . That is, we get a cartesian diagram

π

Y

/Y

 X

 / X ×h/W h

 C

t

 /h

where Y is the pullback of Y by t and X is the pullback of X ×h/W h by t. One can show that if t is sufficiently general, then Y is smooth, X is Gorenstein with an isolated singular point, and π : Y → X is a small resolution. The genericity condition is that t

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is transverse to the hyperplanes ρ ⊆ h orthogonal to each positive root ρ of h. This class of Calabi-Yau threefolds we call ADE fibered Calabi-Yau threefolds. Note that if t = 0, then the singular threefold X is isomorphic to a direct product of the form C2 /G × C. In particular X has a line of Kleinian singularities. The resolution Y is isomorphic to the direct product Y0 × C. This of course is not a small resolution, as it has an exceptional divisor over a curve. The main point for us here is that ADE fibered Calabi-Yau threefolds are related to C2 /G × C by a complex structure deformation. For a full discussion of these matters consult [36,38].

2.2. ADE fibered Calabi-Yau threefolds revisited. In this subsection we show how to construct ADE fibered Calabi-Yau threefolds in terms of a noncommutative algebra, which we call the “N = 1 ADE quiver algebra”. This confirms what was suggested earlier more informally. We start by summarising some of the neccessary definitions. Let Q be an extended Dynkin quiver with vertex set I , and let h(a) and t (a) denote the head and tail vertices of an arrow a ∈ Q. The double Q of Q is the quiver obtained by adding a reverse arrow a ∗ : j → i for each arrow a : i → j in Q. We denote by  the quiver obtained from Q by attaching an additional edge-loop u i for each vertex Q   i ∈ I . We write CQ  and C Q for the path algebras of Q and Q. Now let B = i∈I Cei be the semisimple commutative subalgebra of CQ spanned by the trivial paths and consider the algebra B[u] of polynomials in an indeterminate  u with coefficients in B. For an element τ ∈ B[u], we will write τ (u) = i τi (u)ei , where τi ∈ C[u]. Let CQ ∗ B B[u] denote the free product4 of CQ with B[u] over B. We have an isomorphism ∼

=  : u −→ CQ ∗ B B[u] −→ C Q

 i

ui .

This isomorphism sends the element ei uei to u i , the additional edge-loop at the vertex i. We also have an isomorphism B[u] =

 i∈I

∼ =  Cei ⊗ C[u] −→ i∈I C[u i ] : ei ⊗ u −→ u i .

Therefore, choosing an element τ ∈ B[u] amounts to choosing a collection of polynomials {τi ∈ C[u i ] | i ∈ I }. If τ ∈ B[u] then the N = 1 ADE quiver algebra determined by Q is defined by ⎞ ⎛   [a, a ∗ ] − τi (u)ei  ⎝ a∈Q ⎠. Aτ (Q) = C Q i∈I u is a central element Compare this with the definition of [20, Sect. 4.3]. We point out that the defining relations for Aτ (Q) are generated by the superpotential   W =u [a, a ∗ ] − ηi (u)ei , a∈Q

i∈I

4 “Free product” is a misnomer in this context: if R is a commutative ring and A and B are R-algebras then A ∗ R B is the coproduct in the category of R-algebras.

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where each ηi ∈ C[u] satisfies ηi (u) = τi (u). We also note that if τ (u) is identified with   by the relations the element i τi (u i ) then Aτ (Q) is the same as the quotient of C Q  h(a)=i

aa ∗ −



a ∗ a − τi (u i ) = 0, au i = u j a,

t (a)=i

for each vertex i, and for each arrow a : i → j in Q. This is helpful when considering  which representations of Aτ (Q), as they can be identified with representations V of Q satisfy   Va Va ∗ − Va ∗ Va − τi (Vu i ) = 0, Va Vu i = Vu j Va , (∗) h(a)=i

t (a)=i

for each vertex i, and for each arrow a : i → j in Q. The following is immediate from what we have just seen. Lemma 2.1. Let V be a simple representation of Aτ (Q). Then there exists a λ such that Vu i v = λv for all i ∈ I and all v ∈ Vi . Proof. By virtue of (∗), we may regard the collection of C-linear maps {Vu i : Vi → Vi | i ∈ I } as an endomorphism φ : V → V . Because V is simple, it follows from Schur’s lemma that φ must act as a scalar multiple λ · id V of the identity. The latter obviously implies the assertion.

 of dimension vector α are given by elements of If α ∈ N I , then representations of Q the variety ⎛ ⎞     αt (a) αh(a) αh(a) αt (a) ⎠ αi  ⎝ Rep( Q, α) = HomC (C ,C )⊕HomC (C ,C ) ⊕ EndC (C ) . a∈Q

i∈I

 α) corresponding to repWe denote by Rep(Aτ (Q), α) the closed subspace of Rep( Q, τ resentations for A (Q). The group GL(α) acts on both these spaces, and the orbits correspond to isomorphism classes. We have the following easily verified result.  Lemma 2.2. If x ∈ Rep(Aτ (Q), α) then i tr τi (xu i ) = 0. Proof. Given a ∈ Q, we have tr(xa xa ∗ ) = tr(x a xa ∗ ). Taking traces to relations (∗) and summing over all vertices i ∈ I , one obtains i tr τi (xu i ) = 0, as required.

We define V Q (τ, α) to be the subset of Rep(Aτ (Q), α) consisting of the representations x for which there exists a λ (depending on x) with the property that each xu i acts diagonally on Cαi by multiplication by λ, i.e., for any i ∈ I and v ∈ Vi , we will have xu i v = λv. It is clear that this is a locally closed subset of Rep(Aτ (Q), α), so a variety. In view of Lemma 2.1, we immediately deduce that V Q (τ, α) contains the open subset S Q (τ, α) consisting of simple representations of Aτ (Q). The next result is an easy consequence of Lemma 2.2.  Corollary 2.3. If x ∈ V Q (τ, α) then i αi τi (λ) = 0.

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We are now ready to start our study of ADE fibered Calabi-Yau threefolds. Let G be a finite group of SL(2, C) and let C2 /G be the corresponding Kleinian singularity. Let ρ0 , . . . , ρn be the irreducible representations of G with ρ0 trivial, and let V be the natural 2-dimensional representation of G. The McKay graph of G is the graph with vertex set I = {0, 1, . . . , n} and with the number of edges between i and j being the multiplicity of ρi in V ⊗ ρ j . According to the McKay correspondence, this graph is an extended Dynkin diagram of type A, D or E. Let Q be the quiver obtained from the McKay graph by choosing any orientation of the edges, and let δ ∈ N I be the vector with δi = dim ρi . In [11], Sect. 5, it is pointed out that the moduli space of classical vacua M0 = Rep(Aτ (Q), δ)  GL(δ) contains a branch (the Higgs branch) which is an ADE fibered Calabi-Yau threefold. With the above formulation, we wish to present a clear proof of this claim. First, however, it will be convenient to provide the following piece of information. The space of representations of Q of dimension vector δ can be identified with a cotangent bundle Rep(Q, δ) ∼ = Rep(Q, δ) × Rep(Q, δ)∗ ∼ = T ∗ Rep(Q, δ). This has a natural symplectic structure, and associated to the action of GL(δ) there is a  moment map μδ : Rep(Q, δ) → i∈I gl(δi , C) given by   μδ (x)i = xa xa ∗ − xa ∗ xa . t (a)=i

h(a)=i



Identifying τ ∈ C I with the element of i∈I gl(δi , C) whose i th component is τi times the identity matrix of size δi , one can consider the fiber μ−1 δ (τ ) and the affine quo−1 (τ )  GL(δ). The elements of μ (τ ) correspond to representations of tient variety μ−1 δ δ dimension vector δ of a certain algebra, the deformed preprojective algebra τ (Q) of [17]. More directly relevant for us is that, letting h = {τ ∈ C I | δ · τ = 0}, there is a 2 flat family ϕ : μ−1 δ (h)  GL(δ) → h whose fiber over 0 is C /G. By the discussion at the end of Sect. 8 of [17], this family is obtained from the semi-universal deformation of C2 /G by lifting through the Weyl group. Now  let x be any element of V Q (τ, δ). By Corollary 2.3, the dimension vector δ satisfies i δi τi (λ) = 0. Furthermore, for any vertex i we have μδ (x)i = τi (λ). Therefore one can identify V Q (τ, δ) with the fiber product V Q (τ, δ)

/ μ−1 (h)

 C

 /h

δ

 τ

μδ

where  τ : C → h is the map corresponding to τ . Observe that GL(δ) acts naturally on V Q (τ, δ) in such a way that all maps in the fiber product are equivariant (where the action on C is trivial). Now V Q (τ, δ) → C is flat since it is the pullback of μδ , which is flat by [17, Lem. 8.3]. From this it follows that the map V Q (τ, δ)  GL(δ) → C is also flat and surjective. Incidentally, if τ ∈ h, then μ−1 δ (τ ) is irreducible by [15, Lem. 6.3], which implies that every fiber of the map V Q (τ, δ) → C is irreducible. It then follows from [15, Lem. 6.1] that V Q (τ, δ) is irreducible. On the other hand, we may apply Theorem 6.7 of [15] to

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infer that Aτ (Q) has a simple representation of dimension vector δ. This allows us to conclude that the set S Q (τ, δ) of simple representations must be dense. We have now accumulated all the information necessary to prove the following result. Theorem 2.4. Assume that  τ is sufficiently general. Then the affine quotient variety V Q (τ, δ)  GL(δ) is isomorphic to the ADE fibered Calabi-Yau threefold associated with Q and  τ. −1 Proof. Let C[μ−1 δ (h)] and C[h] be the coordinate rings of μδ (h) and h, respectively. Then the coordinate ring of V Q (τ, δ) is given by

C[V Q (τ, δ)] = C[μ−1 δ (h)] ⊗C[h] C[u]. Since GL(δ) is linear reductive and acts trivially on C[u], we see that GL(δ) C[V Q (τ, δ)]GL(δ) ∼ ⊗C[h] C[u]. = C[μ−1 δ (h)]

Accordingly, we have V Q (τ, δ)  GL(δ) ∼ = μ−1 δ (h)  GL(δ) ×h C. Hence we obtain the affine quotient V Q (τ, δ)  GL(δ) as the fiber product V Q (τ, δ)  GL(δ)

/ μ−1 (h)  GL(δ)

 C

 /h

δ

 τ

ϕ

Since, by hypothesis,  τ is sufficiently general, the desired assertion follows.

We illustrate with the following concrete example. n , so that δi = 1 for all vertices i. The arrows Example 2.5. Suppose that Q is of type A  has shape ai in Q connect vertices i and i + 1 (identifying n + 1 with zero). Thus Q u i−1

u1

5 j

... ...

*  U  ai−1

a0

∗ ai−1

a0∗ u0

5  |

" U u

ui

ai∗

an∗

ai

an

 Nj un

... ...

* P u u i+1

As above let τ ∈ B[u]; recall that it is specified by a set of polynomials {τi ∈ C[u i ] | 0 ≤  of dimension δ involves placing a i ≤ n}. Because δ = (1, . . . , 1), a representation of Q one-dimensional vector space at each vertex i and assigning a complex number to each

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 δ) with the space Cn+1 × Cn+1 × Cn+1 arrow ai , ai∗ , u i . Hence, we may identify Rep( Q, so that Rep(Aτ (Q), δ) ∼ = {(xi , yi , λi ) | −xi yi + xi+1 yi+1 = τi (λi ), 0 ≤ i ≤ n} ⊆ Cn+1 × Cn+1 × Cn+1 . Also one can identify V Q (τ, δ) ∼ = {(xi , yi , λ) | −xi yi + xi+1 yi+1 = τi (λ), 0 ≤ i ≤ n} ⊆ Cn+1 × Cn+1 × C.

n The relations for V Q (τ, δ) lead to the condition that i=0 τi (λ) = 0. From this it follows n+1 that the map  τ = (τ0 , . . . , τn ) : C → C corresponding to τ has its image in h. Without loss of generality it is possible to suppose that τi = ti − ti+1 for some polynomial map t = (t0 , . . . , tn ) : C → h. Now the action of GL(δ) = (C× )n+1 on V Q (τ, δ) is by −1 (xi , yi , λ) −→ (gi+1 gi−1 xi , gi gi+1 yi , λ)

for (gi ) ∈ GL(δ) and (xi , yi , λ) ∈ V Q (τ, δ). It is easily seen that the ring of invariants C[V Q (τ, δ)]GL(δ) is generated by x = x0 · · · xn , y = y0 · · · yn , z i = xi yi , 0 ≤ i ≤ n. These invariants satisfy the relation x y = z0 · · · zn . On the other hand, the relations for V Q (τ, δ) imply that zi = zn −

i 

τ j (λ), 0 ≤ i ≤ n.

j=0

Bearing in mind that

i

j=0 τ j (λ) n 

= t0 (λ) − ti+1 (λ), we derive

z i = (n + 1)(z n − t0 (λ)).

i=0

Setting z =

1 n+1

n

i=0 z i ,

we therefore deduce that z i = z + ti+1 (λ), 0 ≤ i ≤ n.

The conclusion is that the affine quotient variety V Q (τ, δ)  GL(δ) is given as the hypersurface    n   (z + ti+1 (λ)) ⊆ C4 , (x, y, z, λ)  x y = i=0

which is the total space of the family describing the An fibration over the λ-plane.

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2.3. Small resolutions of ADE fibered Calabi-Yau threefolds revisited. We continue with the same hypothesis and notation as in the previous subsection. Our main aim in this paragraph is to show how small resolutions of ADE fibered Calabi-Yau threefolds can be obtained as moduli spaces of representations of the N = 1 ADE quiver algebra Aτ (Q). This is a further confirmation of the picture we have developed. We begin with some generalities. Let A be a commutative ring, let S be a graded ring with S0 = A, and let φ : A → R be a commutative ring homomorphism. Then we may construct the tensor product T = S ⊗ A R of S and R over A by means of φ. We consider T as a graded ring by Td = Sd ⊗ A R. Clearly T0 = R. We then get the following simple observation. Lemma 2.6. With the notation above, we have Proj T ∼ = Proj S ×Spec A Spec R. Proof. We note first that φ : A → R extends to a homomorphism ψ : S → T of graded  : U → Proj S be rings (preserving degrees). Let U = {p ∈ Proj T | p  ψ(S+ )} and ψ the morphism determined by ψ. Since ψ(S+ ) generates T+ , it is clear that U = Proj T . Now fix a homogeneous f ∈ S, and set g = ψ( f ). Then we get an isomorphism from S( f ) ⊗ A R to T(g) by assigning s/ f n ⊗ r to (s ⊗ r )/g n . This implies that Spec T(g) ∼ =

Spec S( f ) ×Spec A Spec R, whence the result follows. Let us now proceed with the construction of small resolutions of ADE fibered Calabi-Yau threefolds. For convenience of reference, we first record some background information. Recall that the flat family ϕ : μ−1 δ (h)  GL(δ) → h realizes the semi2 universal deformation of C /G, or rather its lift through a Weyl group action. Let θ : Z I → Z satisfy θ (δ) = 0. Then θ determines a character χθ of GL(δ) mapping (gi )  GL(δ) to i det(gi )θi . We denote by C[μ−1 the set of θ -semi-invariants of weight n, δ (h)]nθ i.e. those functions on which GL(δ) acts by the character χθn . We say that x ∈ μ−1 δ (h) GL(δ) −1 is θ -semistable if there is a θ -semi-invariant f ∈ C[μδ (h)]nθ with n ≥ 1, such that f (x) = 0. We say that x is θ -stable if, in addition, the stablizer of x in GL(δ) is (h) | f (x) = 0} is closed. finite and the action of GL(δ) on {x ∈ μ−1 δ   Following [25],  GL(δ) −1 −1 is the categorical the GIT quotient μδ (h) θ GL(δ) = Proj n≥0 C[μδ (h)]nθ

−1 SS SS quotient μ−1 δ (h)θ / GL(δ), where μδ (h)θ is the open subset parametrizing θ -semi−1 stable elements of μδ (h). A parameter θ is generic if every θ -semistable element is −1 S θ -stable. In this case, μ−1 δ (h) θ GL(δ) is the geometric quotient μδ (h)θ  GL(δ), −1 −1 −1 S where μδ (h)θ parametrizes θ -stable elements of μδ (h). Since C[μδ (h)]GL(δ) is a subalgebra of the graded ring defining μ−1 δ (h) θ GL(δ), the Proj construction induces −1 a projective morphism ψθ from μδ (h) θ GL(δ) to μ−1 δ (h)  GL(δ). We write ϕθ to denote the composition of ψθ and ϕ. Using results of Cassens and Slodowy [12, Sect. 7] one can show that, for generic θ , the family ϕθ : μ−1 δ (h) θ GL(δ) → h induces a simultaneous resolution of ϕ. Now consider V Q (τ, δ) ⊆ Rep(Aτ (Q), δ). Just as for μ−1 δ (h), one can consider θ -semistable and θ -stable points of V Q (τ, δ). Again one can form a GIT quotient V Q (τ, δ) θ GL(δ) and we get a projective morphism

πθ : V Q (τ, δ) θ GL(δ) −→ V Q (τ, δ)  GL(δ). We have seen earlier that V Q (τ, δ) is irreducible, and the general element is a simple representation of Aτ (Q), hence θ -stable. Thus the morphism πθ is a birational map of irreducible varieties.

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We are at last in a position to attain our main objective, which is to prove the following result. Theorem 2.7. Assume that  τ is sufficiently general. If θ is generic, then πθ is a small resolution of the ADE fibered Calabi-Yau threefold associated with Q and  τ. Proof. We keep the notation employed in the proof of Theorem 2.4. To begin with, we observe that there is an isomorphism GL(δ)

C[V Q (τ, δ)]nθ

GL(δ) ∼ ⊗C[h] C[u]. = C[μ−1 δ (h)]nθ

Invoking Lemma 2.6, it follows that ⎛ ⎞ ⎛ ⎞   GL(δ) GL(δ) ⎠ ×h C, Proj ⎝ C[V Q (τ, δ)]nθ ⎠ ∼ C[μ−1 = Proj ⎝ δ (h)]nθ n≥0

n≥0

∼ which entails V Q (τ, δ) θ GL(δ) = θ GL(δ) ×h C. Therefore we obtain the GIT quotient V Q (τ, δ) θ GL(δ) as the fiber product μ−1 δ (h)

V Q (τ, δ) θ GL(δ)

/ μ−1 (h) θ GL(δ) δ

 C

 /h

 τ

ϕθ

The required result now follows from our hypothesis on  τ.

We end this section with the following illustration of Theorem 2.7. n . We use the notations introExample 2.8. Assume that Q is a quiver of Dynkin type A duced in Example 2.5. Consider the generic stability parameter θ = (−n, 1, . . . , 1). n β By definition, the ring of θ -semi-invariants is spanned by the monomials i=0 xiαi yi i satisfying −α0 + αn + β0 − βn = −n and −αi + αi−1 + βi − βi−1 = 1 for 1 ≤ i ≤ n. Given j = 0, . . . , n − 1, put u j = x0 · · · x j , v j = y j+1 · · · yn . Then we have the following relations xv j = u j z j+1 · · · z n , 0 ≤ j ≤ n − 1, yu j = v j z 0 · · · z j , 0 ≤ j ≤ n − 1, u j vk = u k v j z k+1 · · · z j , 0 ≤ k < j ≤ n − 1, or, using the fact that z i = z + ti+1 (λ) for 0 ≤ i ≤ n, xv j = u j

n 

(z + ti+1 (λ)), 0 ≤ j ≤ n − 1,

i= j+1

yu j = v j

j 

(z + ti+1 (λ)), 0 ≤ j ≤ n − 1,

i=0

u j vk = u k v j

j  i=k+1

(z + ti+1 (λ)), 0 ≤ k < j ≤ n − 1.

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Analyzing possibilities for αi , βi (0 ≤ i ≤ n) it is easily seen that the ring of θ -semi-invariants is generated as a polynomial ring by u I v I  = u i1 . . . u i p vi1 . . . viq , where I = {i 1 , . . . , i p } is a multi-index of {0, . . . , n − 1} and I  = {i 1 , . . . , i q } denotes the complementary index. It is also not difficult to see that this space is the module over C[V Q (τ, δ)]GL(δ) generated by f 0 = v0 · · · vn−2 vn−1 , f 1 = v0 · · · vn−2 u n−1 , ··· f n = u 0 · · · u n−2 u n−1 . We observe next that C[u I v I  | I = {i 1 , . . . , i p }, I  = {i 1 , . . . , i q }] = C[u 0 , v0 ] ∗ · · · ∗ C[u n−1 , vn−1 ], where ∗ denotes the Segre product of polynomial rings. Thus we have C[V Q (τ, δ)]GL(δ) [ f 0 , . . . , f n ] ∼ C[V Q (τ, δ)]GL(δ) [u 0 , v0 ] ∗ · · · ∗ C[V Q (τ, δ)]GL(δ) [u n−1 , vn−1 ]. = Therefore the Proj quotient V Q (τ, δ) θ GL(δ) can be identified with a closed subvariety of C4 × (P1 )n with (u j : v j ) the homogeneous coordinates on the j th P1 . Now let U0 , U1 , . . . , Un be the open subsets of C4 × (P1 )n defined by U0 = {v0 = 0}, Uk = {u k−1 = 0, vk = 0}, 1 ≤ k ≤ n − 1 Un = {u n−1 = 0}, and on Uk , let ξk = vk−1 /u k−1 ,

ηk = u k /vk .

Direct computations show that V Q (τ, δ) θ GL(δ) ∩ Uk is defined by equations ⎛ ⎞ k−1  (u j : v j ) = ⎝1 : ξk (z + ti+1 (λ))⎠ , for j < k − 1, i= j+1

⎛ (u j : v j ) = ⎝ηk

j 

⎞

(z + ti+1 (λ)) : 1⎠ , for j > k,

i=k+1

and x = ηk

n 

(z + ti+1 (λ)),

i=k+1

y = ξk

k−1 

(z + ti+1 (λ)),

i=0

z = ξk ηk − tk+1 (λ).

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From the explicit analysis in [24, Sect. 4], one sees that the GIT quotient V Q (τ, δ) θ GL(δ) is isomorphic to a small resolution of the threefold for a An fibration over the λ-plane. 3. Derived Equivalence In this section, it is shown how to describe the derived category of a small resolution of an ADE fibered Calabi-Yau threefold in terms of the associated N = 1 ADE quiver algebra, in the spirit of noncommutative crepant resolutions of M. Van den Bergh. Assertions of this sort have already been considered in [38]. Our work is mostly based on the ideas and constructions of [17] and [40]. 3.1. The algebra Aτ . As explained in the previous section, the N = 1 ADE quiver algebra contains enough information to reconstruct ADE fibered Calabi-Yau threefolds and their small resolutions. Here we introduce yet another noncommutative algebra whose center is related to the coordinate ring of an ADE fibered Calabi-Yau threefold via Morita equivalence. The discussion borrows largely from [17]. We begin by setting up notation. Let G be a finite subgroup of SL(2, C). For our convenience, we denote by R = C[u] the ring of polynomials in a dummy variable u. The group G acts naturally on the ring Rx, y of noncommuting polynomials, with the action of G on R being trivial, so one can form the skew group algebra Rx, y # G. We use Z (RG) to denote the centre of the group algebra RG. For τ ∈ Z (RG) we define the algebra Aτ as the quotient Aτ = (Rx, y # G)/(x y − yx − τ ). This algebra was introduced and studied by W. Crawley-Boevey and M. Holland in [17]. (In the notation of [17], Aτ corresponds to S R,τ .) Observe that if τ = 0 then we recover the skew group algebra R[x, y] # G. In other words, Aτ is a flat deformation of R[x, y] # G for every choice of τ . The algebra Aτ carries a natural filtration, given by deg x = deg y = 1, deg u = 0 and deg g = 0 for any g ∈ G. Let gr Aτ denote the associated graded algebra. It was explicitly demostrated in [17, Lemma 1] that gr Aτ ∼ = R[x, y] # G. As a consequence, the arguments in [17, Sect. 1] go through and show that Aτ is a prime noetherian maximal order which is Auslander-Gorenstein and Cohen-Macaulay of GK dimension 3. The reader may want to consult  [35] for some background. Now let e = |G|−1 g∈G g be the averaging idempotent, viewed as an element in Aτ . Define a subalgebra C τ of Aτ to be e Aτ e. The increasing filtration on Aτ induces a filtration on C τ . It is well known that C 0 ∼ = R[x, y]G . Further e lies in the degree τ zero part of the filtration of A and therefore gr C τ ∼ = R[x, y]G . This allows = e gr Aτ e ∼ G τ us to lift properties from R[x, y] to C ; we refer again to [35] for details. In particular, C τ is a noetherian integral domain of GK dimension 3. Note finally that C τ is a flat deformation of the coordinate ring of the associated singular Calabi-Yau threefold C2 /G × C. We anticipate that if the trace of τ on the group algebra RG is zero, then C τ is a commutative ring, and it occurs as the coordinate ring of the fiber of the semi-universal deformation of C2 /G × C. On the other hand, if τ has nonzero trace on RG, then C τ is a noncommutative ring. In order to make further progress we need to bring in the notion of noncommutative crepant resolution introduced by M. Van den Bergh [40,41]. Let R be an integrally

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closed Gorenstein domain. If A is an R-algebra that is finite as an R-module, then A is said to be homologically homogeneous if A is a maximal Cohen-Macaulay R-module and gldim Ap = dim Rp for all p ∈ Spec R. A noncommutative crepant resolution of R is a homologically homogeneous R-algebra of the form A = End R (M), where M is a finitely generated reflexive R-module. We remind the reader that an R-module M is said to be reflexive if the natural morphism M → Hom R (Hom R (M, R), R) is an isomorphism. Our aim now is to show that Aτ is a noncommutative crepant resolution of C τ . The following preliminary result will clear our path. Lemma 3.1. Aτ e is a finitely generated reflexive C τ -module. In addition, Aτ ∼ = EndC τ (Aτ e). Proof. One can adapt the techniques of [17, Lem. 1.4] to the present Note first situation. m τ e Aτ is a finitely generated ideal of Aτ . We write Aτ e Aτ = τ and x = that A x A i i i=1  some ri j , si j ∈ Aτ . For each a ∈ Aτ , we have ae ∈ Aτ e = (Aτ e Aτ )e, j ri j esi j for 

 and so ae = i x i ai e = i, j ri j esi j ai e. This proves that the elements ri j generate Aτ e as a C τ -module. For the condition on the endomorphism ring, there are natural inclusions Aτ ⊆ EndC τ (Aτ e) ⊆ End Aτ (Aτ e Aτ ). Let Q denote the simple artinian quotient ring of Aτ . The fact that Qe ∼ = Q ⊗ Aτ Aτ e τ τ implies that EndC τ (A e) ⊆ EndC τ (Qe). But C is a maximal order in eQe, so EndC τ (Qe) = EndeQe (Qe). Because Q is simple artinian, we also have Q ∼ = EndeQe (Qe). Thus the endomorphism ring EndC τ (Aτ e) can be identified with a subring of Q. From this it follows that EndC τ (Aτ e) ∼ = {q ∈ Q | q Aτ e ⊆ Aτ e}. Similarly, it can be shown that End Aτ (Aτ e Aτ ) ∼ = {q ∈ Q | q Aτ e Aτ ⊆ Aτ e Aτ } = Aτ , the latter equality being an immediate consequence of the definition of a maximal order. The conclusion is that Aτ ∼ = EndC τ (Aτ e), as asserted. It remains to check that Aτ e is reflexive. A similar argument to the one above can be applied to show that Aτ ∼ = EndC τ (e Aτ ). Hence Aτ e ∼ = HomC τ (e Aτ , e Aτ )e ∼ = HomC τ (e Aτ , C τ ), and e Aτ ∼ = HomC τ (Aτ e, C τ ), = e HomC τ (Aτ e, Aτ e) ∼ proving that Aτ e ∼ = HomC τ (HomC τ (Aτ e, C τ ), C τ ). This completes the proof of the lemma.

We are now ready to prove our promised result. Proposition 3.2. The algebra Aτ is a noncommutative crepant resolution of C τ .

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Proof. By Lemma 3.1, it suffices to show that Aτ is homologically homogeneous. We already know that Aτ is Cohen-Macaulay. Further, by Lemma 3.1 and [32, Cor. 6.18], Aτ has finite global dimension. The desired assertion now follows by appealing to [40, Lem 4.2].

We now study the relationship between the algebra Aτ and the N = 1 ADE quiver algebra. Keeping our earlier notation, the irreducible representations of G are ρ0 , . . . , ρn , with ρ0 trivial, and I = {0, 1, . . . , n}. Let Q be the quiver with vertex set I obtained by choosing any orientation of the McKay graph, and let δ ∈ N I be the vector with ∼ δi = dim ρi . Fix an isomorphism CG = i∈I Mat δi (C) and for every ordered pair ( p, q), 1 ≤ p, q ≤ δi , take ei pq to be the matrix with p, q entry 1 and zero elsewhere. Given i ∈ I , put f i = ei11 . Then { f 0 , . . . , f n } is a set of nonzero orthogonal idempotents with the property CG f i ∼ = ρi for all i ∈ I . Hence we get that f = f 0 + · · · + f n is idempotent. Furthermore f = e, so e = e f = f e. Observe also that the map R I → Z (RG) 0  given by τ → i∈I (τi /δi ) f i is a bijection, and we use this to identify R I and Z (RG). Before going on to give the connection between the algebra Aτ and the N = 1 ADE quiver algebra Aτ (Q), it is convenient to point out the following description of Aτ (Q). We keep the notation of Sect. 2.2. Following Crawley-Boevey and Holland [17], given an element τ ∈ R I we define  R,τ (Q) to be ⎞ ⎛    RQ ⎝ [a, a ∗ ] − τi ei ⎠ . a∈Q

i∈I

Because u is central, we must have R Q = CQ ⊗ R ∼ = CQ ∗ B B[u]. Hence it follows that Aτ (Q) ∼ =  R,τ (Q). With this understood, we get the following. Proposition 3.3. Aτ is Morita equivalent to Aτ (Q) and C τ ∼ = e0 Aτ (Q)e0 . Proof. The first part of the proposition follows from the the fact that f Aτ f ∼ = Aτ (Q) established in [17, Th. 3.4]. Under this isomorphism, e corresponds to the trivial path e0 . Using that e = e f = f e, we have C τ = e Aτ e = e f Aτ f e ∼ = e0 Aτ (Q)e0 , as desired.

For simplicity of notation we fix an isomorphism between C τ and e0 Aτ (Q)e0 and henceforth identify C τ = e0 Aτ (Q)e0 . Recall from Sect. 2.2 that one can identify V Q (τ, δ) with the fiber product μ−1 δ (h)×hSpec R. Now, the coordinate ring of Rep(Q, δ) is the polynomial ring C[sapq | a ∈ Q, 1 ≤ p ≤ δh(a) , 1 ≤ q ≤ δt (a) ], where the indeterminate sapq picks out the p, q entry of the matrix xa , corresponding to x ∈ Rep(Q, δ). It is fairly straightforward to see that V Q (τ, δ) has coordinate ring R[sapq ]/Jτ , where Jτ is generated by the elements δt (a)   h(a)=i r =1

sapr sa ∗ rq −

h(a)  δ

t (a)=i r =1

sa ∗ pr sarq − δ pq τi

 for each vertex i and for 1 ≤ p, q ≤ δi . Letting  = i δi there is a natural ring homomorphism R Q → Mat  (R[sapq ]) sending an arrow a to the matrix whose entries are the

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relevant sapq . By our previous remark this homomorphism descends to a map Aτ (Q) → Mat  (C[V Q (τ, δ)]). Since δ0 = 1, this restricts to a homomorphism e0 Aτ (Q)e0 → C[V Q (τ, δ)]. One easily checks that the elements in the image of this map are invariant under the action of GL(δ). In this way we get a map φτ : C τ → C[V Q (τ, δ)]GL(δ) . It  follows from [17, Cor. 8.12] that if τ ∈ R I satisfies i δi τi = 0, then the map φτ is an isomorphism. Thus, we arrive at the following result.  Proposition 3.4. If i δi τi = 0, then C τ ∼ = C[V Q (τ, δ)]GL(δ) . One immediate consequence of this is that C[V Q (τ, δ)]GL(δ) is an integrally closed domain, so the quotient scheme V Q (τ, δ)  GL(δ) is normal. Another application of Proposition 3.4 is given by the following.  Corollary 3.5. If i δi τi = 0, then the rings Aτ and C τ have Krull dimension 3. Proof. Using Lemma 3.1 and [32, Cor. 13.4.9] one sees immediately that the rings Aτ and C τ are PI rings, and so their Krull dimension coincides with their GK dimension. The assertion follows.

We finish this subsection with an observation which will be central to our main result. Here we denote the centres of Aτ and C τ by Z (Aτ ) and Z (C τ ) respectively. Proposition 3.6. The map φ : Aτ → C τ given by φ(a) = eae for all a in Aτ restricts to an algebra isomorphism from Z (Aτ ) to Z (C τ ). Proof. It is a straightforward calculation to show that φ| Z (Aτ ) is an algebra homomorphism with image in Z (C τ ). To see that it is an algebra isomorphism we construct the inverse map. First we note that an element ξ in Z (C τ ) implements a C τ -endomorphism of Aτ e via right multiplication by ξ . Thanks to Lemma 3.1, this endomorphism can be regarded as an element aξ of Aτ . Then the algebra homomorphism ψ : Z (C τ ) → Aτ given by ψ(ξ ) = aξ for all ξ in Z (C τ ) has its image in Z (Aτ ) because the right multiplication by ξ on Aτ e commutes with left multiplication by Aτ . It is readily verified that this homomorphism is inverse to φ| Z (Aτ ) . This completes the proof of the proposition.

τ This result the centre of shows a second vital feature of C : its structure determines Now, if i δi τi = 0, then we know from Proposition 3.4 that C τ is commutative. According to Proposition 3.6, in this case C τ ∼ = Z (Aτ ).

Aτ .

3.2. Brief account of Van den Bergh’s construction. In this subsection we describe some of Van den Bergh’s results concerning noncommutative crepant resolutions. We first consider the following more general situation. Let R be a commutative noetherian algebra over C and let A be an R-algebra which is finitely generated as an R-module. Let {e0 , . . . , en } be a complete set of primitive orthogonal idempotents in A and set I = {0, 1, . . . , n}. We wish to construct a moduli space of A-modules. To do this we introduce a stability condition. Let us fix a field K and a ring homomorphism R → K . If M is a finite dimensional A ⊗ R K -module, its dimension vector dim M is the element of N I whose i th component is dim K (ei M). Let θ be a homomorphism Z I → Z. A finite dimensional A ⊗ R K -module M is said to be θ -stable (or θ -semistable) if θ (dim M) = 0, but θ (dim M  ) > 0 (or θ (dim M  ) ≥ 0) for every proper submodule M  ⊆ M. As usual, we say that θ is generic

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for α if every θ -semistable A ⊗ R K -module of dimension α is θ -stable. Note that such a θ exists if and only if α is indivisible, meaning that the αi have no common divisors. As a matter of fact, the condition θ (β) = 0 for all 0 < β < α ensures θ is generic. Next we recall the notion of family from [21]. Fix a dimension vector α ∈ N I . A family of A-modules of dimension α over an R-scheme S is a locally free sheaf F over S together with an R-algebra homomorphism A → End S (F ) such that ei F has constant rank αi for all i ∈ I . Two such families F and F  are equivalent if there is a line bundle L on S and an isomorphism F ∼ = F  ⊗O S L . Finally we say that a family F is θ -stable (or θ -semistable) if for every field K and every morphism φ : Spec K → S we have that φ ∗ F is θ -stable (or θ -semistable) as A ⊗ R K -module. We have the following result, see [40, Prop. 6.2.1]. Proposition 3.7. If θ is generic, then the functor which assigns to a scheme S the set of equivalence classes of families of θ -stable A-modules of dimension α over S is representable by a projective scheme Mθ (A, α) over X = Spec R. We now illustrate how to use this result to construct a crepant resolution starting from a noncommutative one. Let R be an integrally closed Gorenstein domain admitting a noncommutative crepant resolution A = End R (M)  and set X = Spec R. We assume, for simplicity, that X is irreducible. Let M = i∈I Mi be any decomposition of M corresponding to idempotents e0 , . . . , en ∈ A = End R (M), and let α ∈ N I be the vector with αi = rank Mi . By Proposition 3.7 we know that, for generic θ , there is a fine moduli space Mθ (A, α) of θ -stable A-modules of dimension α. Let us denote by φ : Mθ (A, α) → X the structure morphism. If we let U ⊆ X be the open subset over which M is locally free then it follows from [40, Lem. 6.2.3] that φ −1 (U ) → U is an isomorphism. Each point y ∈ φ −1 (U ) is a θ -stable A-module of dimension α so there is an embedding U → Mθ (A, α). Let W ⊆ Mθ (A, α) be the irreducible component of Mθ (A, α) containing the image of this morphism. Then W is fine, in that W is projective and there is a universal sheaf U on W × X . We denote by P the restriction of U to W . Notice that P is a sheaf of (left) A-modules on W . Now let Db (Coh(W )) denote the bounded derived category of coherent sheaves on W and Db (mod–A) the bounded derived category of finitely generated right modules over A. The method of Bridgeland, King and Reid generalises to prove the following result, see [40, Th. 6.3.1]. Theorem 3.8. Let the setting be as above. If dim(W × X W ) ≤ dim X + 1, then φ : W → . L X is a crepant resolution and the functors R (−⊗L OW P) and −⊗ A RH om W (P, OW ) define inverse equivalences between Db (Coh(W )) and Db (mod– A). 3.3. Application to our situation. We now return to the concrete situation of Sect. 3.1. Our main aim is to show how the ideas developed in the previous subsection can be used to prove that any small resolution of an ADE fibered Calabi-Yau threefold is derived equivalent to the corresponding N = 1 ADE quiver algebra. We start with some preliminary observations. We have seen in Proposition 3.2 that the algebra Aτ is a noncommutative crepant resolution of C τ . Hereafter we assume that τ ∈ R I satisfies i δi τi = 0. As we pointed out earlier, this implies that C τ ∼ = C[V Q (τ, δ)]GL(δ) . Setting X = Spec C τ , it = Z (Aτ ) ∼ follows, from Theorem 2.4, that X is isomorphic to an ADE fibered Calabi-Yau threefold. We shall again let ρ0 , . . . , ρn denote the irreducible representations of G with ρ0 trivial, and set I = {0, . . . , n}. For each i ∈ I , let f i be the idempotent in CG with

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CG f i ∼ elements of Aτ . Then = ρi . As previously emphasized, we may regard the f i ’s as τ τ τ τ f i A e is a submodule of A e for all i ∈ I and A e = i∈I f i A e. Bearing in mind that f i Aτ e ∼ = HomCG (ρi , Aτ e), we have δi = dim ρi = rank( f i Aτ e). Let Mθ (Aτ , δ) be the moduli space, as constructed in the previous subsection, of θ -stable Aτ -modules of dimension δ (equivalently, isomorphic to CG), and let W be the irreducible component of Mθ (Aτ , δ) that maps birationally to X . With the aid of Theorem 3.8, we easily derive the following. Proposition 3.9. With the notation above, W is a crepant resolution of X and there is an equivalence of categories between Db (Coh(W )) and Db (mod– Aτ ). Proof. Define  to be the diagonal of W × W . As in the previous subsection, we write φ : W → X for the structure morphism. This is a birational projective mapping, so it is closed. Let us take non-empty open subsets V ⊆ W and U ⊆ X , such that φ restricts to an isomorphism φ : V → U . Denote by Z the complement of V . We may assume without loss of generality that φ(Z ) ∩U = ∅. It therefore follows that W × X W ⊆  ∩ (Z × Z ). Since dim X = 3 we have dim Z ≤ 2, which ensures that dim(W × X W ) ≤ 4. Now we are in the situation of Theorem 3.8 and the assertion follows.

We now apply Proposition 3.9 to prove the result promised in the beginning of this subsection. Theorem 3.10. Let the context be as above. If π : Y → X is a small resolution of the ADE fibered Calabi-Yau threefold X , then there is an equivalence of categories Db (Coh(Y )) ∼ = Db (mod–Aτ (Q)), where Aτ (Q) is the associated N = 1 ADE quiver algebra. Proof. It is well-known (see, e.g. [13, Prop. 16.4]) that π is a crepant resolution. Owing to Proposition 3.9, there exists another crepant resolution φ : W → X associated to Aτ . Let f : Y → W be the birational map over X such that f is isomorphic in codimension 1. Then, by [27, Th. 6.38], f is a composition of finitely many flops. A result of Bridgeland [7, Th. 1.1] provides an equivalence of categories Db (Coh(Y )) ∼ = Db (Coh(W )). Invoking Propositions 3.9 and 3.3, we therefore deduce that Db (Coh(Y )) ∼ = Db (mod–Aτ ) ∼ = Db (mod–Aτ (Q)), as we wished to show.

Acknowledgements. We would like to thank Balázs Szendr˝oi for helpful remarks and e-mail correspondence. A.Q.V. is grateful to Jan Stienstra for his constant guidance and also for some essential conversations. A.Q.V. is also grateful to Tom Bridgeland, Michel Van den Bergh and Marjory Jane Macleod for valuable discussions related to this work. A.B. thanks A.Q.V. for introducing him to this problem.

References 1. Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. In: Progress in string theory, Hackensack, NJ: World Sci. Publ., 2005, pp. 1–152 2. Aspinwall, P.S.: D-branes, -stability and θ -stability. In: Snowbird lectures on string geometry. Volume 401 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2006, pp. 1–13 3. Beasley, C., Greene, B.R., Lazaroiu, C.I., Plesser, M.R.: D3-branes on partial resolutions of abelian quotient singularities of Calabi-Yau threefolds. Nucl. Phys. B. 566(3), 599–641 (2000)

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4. Berenstein, D.: Reverse geometric engineering of singularities. J. High Energy Phys. (4), No. 52, 18, (2002) 5. Bergman, A.: Stability conditions and branes at singularities. J. High Energy Phys. (10), 073, 19, (2008) 6. Bocklandt, R., Le Bruyn, L., Symens, S.: Isolated singularities, smooth orders, and Auslander regularity. Comm. Algebra 31(12), 6019–6036 (2003) 7. Bridgeland, T.: Flops and derived categories. Invent. Math. 147(3), 613–632 (2002) 8. Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc. 14(3), 535–554 (electronic) (2001) 9. Brieskorn, E.: Die Auflösung der rationalen Singularitäten holomorpher Abbildungen. Math. Ann. 178, 255–270 (1968) 10. Cachazo, F., Fiol, B., Intriligator, K., Katz, S., Vafa, C.: A geometric unification of dualities. Nucl. Phys. B 628(1–2), 3–78 (2002) 11. Cachazo, F., Katz, S., Vafa, C.: Geometric transitions and N = 1 quiver theories. http://arxiv.org/abs/ hep-th/0108120v2, 2001 12. Cassens, H., Slodowy, P.: On Kleinian singularities and quivers. In: Singularities (Oberwolfach, 1996), Volume 162 of Progr. Math., Basel: Birkhäuser, 1998, pp. 263–288 13. Clemens, H., Kollár, J., Mori, S.: Higher-dimensional complex geometry. Astérisque 166, 144 pp. (1989) 14. Craw, A., Ishii, A.: Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient. Duke Math. J. 124(2), 259–307 (2004) 15. Crawley-Boevey, W.: Geometry of the moment map for representations of quivers. Compositio Math. 126(3), 257–293 (2001) 16. Crawley-Boevey, W., Etingof, P., Ginzburg, V.: Noncommutative geometry and quiver algebras. Adv. Math. 209(1), 274–336 (2007) 17. Crawley-Boevey, W., Holland, M.P.: Noncommutative deformations of Kleinian singularities. Duke Math. J. 92(3), 605–635 (1998) 18. Deligne, P., Freed, D.S.: Supersolutions. In: Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Providence, RI: Amer. Math. Soc., 1999, pp. 227–355. 19. Forcella, D., Hanany, A., He, Y.-H., Zaffaroni, A.: The master space of N = 1 gauge theories. J. High Energy Phys. (8), 012, 72 (2008) 20. Ginzburg, V.: Calabi-Yau algebras. http://arxiv.org/abs/math/0612139v2, 2006 21. Gordon, I., Smith, S.P.: Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities. Math. Ann. 330(1), 185–200 (2004) 22. Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(3), 565–576 (2000) 23. Katz, S.: ADE geometry and dualities. Workshop on Algebraic Geometry and Physics, Lisbon, 2004 24. Katz, S., Morrison, D.R.: Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups. J. Alg. Geom. 1(3), 449–530 (1992) 25. King, A.D.: Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45(180), 515–530 (1994) 26. Klebanov, I.R., Witten, E.: Superconformal field theory on threebranes at a Calabi-Yau singularity. Nucl. Phys. B 536(1–2), 199–218 (1999) 27. Kollár, J., Mori, S.: Birational geometry of algebraic varieties. Volume 134 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 1998, with the collaboration of C. H. Clemens, A. Corti, translated from the 1998 Japanese original 28. Kontsevich, M.: Formal (non)commutative symplectic geometry. In: The Gel’ fand Mathematical Seminars, 1990–1992, Boston, MA: Birkhäuser Boston, 1993, pp. 173–187 29. Le Bruyn, L.: Noncommutative compact manifolds constructed from quivers. AMA Algebra Montp. Announc., Paper 1, 5 pp. (electronic) (1999) 30. Le Bruyn, L.: Non-commutative algebraic geometry and commutative desingularizations. In: Noncommutative algebra and geometry, Volume 243 of Lect. Notes Pure Appl. Math., Boca Raton, FL: Chapman & Hall/CRC, 2006, pp. 203–252 31. Le Bruyn, L., Symens, S.: Partial desingularizations arising from non-commutative algebras. http://arxiv. org/abs/math/0507494v1[math.RA], 2005 32. McConnell, J.C., Robson, J. C.: Noncommutative Noetherian rings. Pure and Applied Mathematics (New York), with the cooperation of L. W. Small, A Wiley-Interscience Publication, Chichester: John Wiley & Sons, 1987 33. Segal, E.: The A∞ deformation theory of a point and the derived categories of local Calabi-Yaus. J. Algebra 320(8), 3232–3268 (2008) 34. Slodowy, P.: Simple singularities and simple algebraic groups. Volume 815 of Lecture Notes in Mathematics. Berlin: Springer, 1980 35. Stafford, J.T., Zhang, J.J.: Homological properties of (graded) Noetherian PI rings. J. Algebra 168(3), 988– 1026 (1994)

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36. Szendr˝oi, B.: Artin group actions on derived categories of threefolds. J. Reine Angew. Math. 572, 139– 166 (2004) 37. Szendr˝oi, B.: Non-commutative Donaldson-Thomas invariants and the conifold. Geom. Topol. 12(2), 1171–1202 (2008) 38. Szendr˝oi, B.: Sheaves on fibered threefolds and quiver sheaves. Commum. Math. Phys. 278(3), 627–641 (2008) 39. Tjurina, G.N.: Resolution of singularities of flat deformations of double rational points. Funkcional. Anal. i Priložen. 4(1), 77–83 (1970) 40. van den Bergh, M.: Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel, Berlin: Springer, 2004, pp. 749–770 41. Vanden Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004) 42. Wemyss, M.: Reconstruction algebras of type A. http://arxiv.org/abs/0704.3693v3[math.AG], 2008 43. Wijnholt, M.: Parameter space of quiver gauge theories. Adv. Theor. Math. Phys. 12(4), 711–755 (2008) 44. Zhu, X.: Representations of N = 1 AD E quivers via reflection functors. Michigan Math. J. 54(3), 671–686 (2006) Communicated by A. Connes

Commun. Math. Phys. 297, 621–651 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1054-3

Communications in

Mathematical Physics

Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds Spyridon Alexakis1, , Rafe Mazzeo2, 1 Department of Mathematics, University of Toronto, Bahen Centre, 40 st. George Street,

Toronto, Ontario M5S2E4, Canada. E-mail: [email protected]

2 Department of Mathematics, Stanford University, Stanford, CA 94305, USA.

E-mail: [email protected] Received: 9 September 2008 / Accepted: 14 March 2010 Published online: 6 May 2010 – © Springer-Verlag 2010

Abstract: We study the renormalized area functional A in the AdS/CFT correspondence, specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3-manifolds (and somewhat more broadly, Poincaré-Einstein spaces). Our main results include an explicit formula for the renormalized area of such a minimal surface Y as an integral of local geometric quantities, as well as formulæ for the first and second variations of A which are given by integrals of global quantities over the asymptotic boundary loop γ of Y . All of these formulæ are also obtained for a broader class of nonminimal surfaces. The proper setting for the study of this functional (when the ambient space is hyperbolic) requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary. We show that this moduli space is a smooth Banach manifold and develop a Z-valued degree theory for the natural map taking a minimal surface to its boundary curve. We characterize the nondegenerate critical points of A for minimal surfaces in H3 , and finally, discuss the relationship of A to the Willmore functional. 1. Introduction There is an interesting nonlinear asymptotic boundary problem in which one seeks a minimal submanifold in hyperbolic space with prescribed asymptotic boundary a submanifold in the sphere at infinity. This was treated conclusively by Anderson [2,3] in the early 1980’s using techniques from geometric measure theory; the solutions he obtained are absolutely volume minimizing with respect to compact variations. One may also pose this problem when the ambient space is a convex cocompact hyperbolic manifold, or even more generally a conformally compact manifold (M n+1 , g) (all definitions are reviewed in Sect. 2), and it is not hard to extend the existence theory to these settings.  This research was partially conducted during the period the author served as a Clay Research Fellow.

 Supported by the NSF under Grant DMS-0505709.

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Here, however, we focus mostly on the special case of properly embedded minimal surfaces Y 2 in M = H3 / , where  is a convex cocompact subgroup (a particular case is M = H3 itself), with boundary curve ∂Y = γ an embedded closed curve γ ⊂ ∂ M. Beyond Anderson’s aforementioned work, in this particular setting there is also a rich existence theory of minimal (not necessarily minimizing) surfaces of arbitrary genus by de Oliveira and Soret [30], see also Coskunzer [12,13]. It turns out that there is a well-defined Hadamard regularization of the area of such a minimal surface, and this renormalized area is our central concern. Roughly speaking, our goal is to obtain a local formula for this renormalized area, i.e. one involving integrals of local geometric quantities, and then to use this to study the variational theory of the renormalized area functional A. In order to do this properly, we must study the moduli space of all properly embedded minimal surfaces with embedded asymptotic boundary, as this is the natural domain of A. This leads to a subsidiary investigation of the structure of these moduli spaces and the degree theory for the natural map taking a minimal surface to its boundary curve. We next calculate the first and second variations of A; interestingly, these are expressed as integrals of global quantities over the boundary curve. While we do not touch on all aspects of this variational theory, we are able to characterize the nondegenerate critical points amongst surfaces in hyperbolic space, and give some estimates for the numerical range of A. Finally, we show the relationship of A to the much-studied Willmore functional W, which suggests that in the hyperbolic setting A is the correct conformally natural generalization of W to surfaces with boundary. There are strong motivations from the AdS/CFT correspondence in string theory for studying the renormalized area, and we shall explain some of these below, after describing the mathematical context more carefully. Degree theory. Our results about minimal surfaces parallel a number of known results about Poincaré-Einstein (PE) spaces, so we describe these together. Let (M, g) be a PE space; this means that M is a manifold with boundary, g = ρ −2 g where, ρ is a boundary defining function for ∂ M and g is smooth and nondegenerate up to the boundary, and g is Einstein. There is a well-defined conformal class c(g) on ∂ M, called the conformal infinity of g, which should be regarded as the asymptotic boundary value of g. The space of all PE metrics (with some fixed regularity) on the interior of a given manifold with boundary M is a Banach manifold, and the conformal infinity map from this to the space of conformal structures on ∂ M (which also has the structure of a Banach manifold) is Fredholm of degree 0. These facts were proved by Anderson [5], see also Biquard [8] and Lee [25]. Most existence results for PE metrics are perturbative in nature, but Anderson established a scheme to obtain a much broader existence theory when dim M = 4 using degree theory [6]. One key ingredient is the properness of this conformal infinity boundary value map over the preimage of scalar positive conformal classes on ∂ M. There are substantial technicalities in making all of this work; recent work of Chang and Yang [10] clarifies some of this. We first prove an analogous result for properly embedded minimal submanifolds: Theorem 1.1. Suppose that M = H3 /  is a convex cocompact hyperbolic manifold, and let Mk (M) be the space of properly embedded minimal surfaces in M of genus k with asymptotic boundary curve a C 3,α embedded closed (but possibly disconnected) curve in ∂ M. Let E denote the space of all C 3,α closed embedded curves in ∂ M. Then both Mk (M) and E are Banach manifolds, and the natural map  : Mk (M) −→ E is a smooth proper Fredholm map of index 0.

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These properties of  imply the existence of a Z-valued degree for it, which yields many refinements of the existence theory for these minimal surfaces. Some consequences will be described in Sect. 4. The proof of most of this uses various well-known tools, hence this can be regarded as a good toy model for the corresponding result about four-dimensional Poincaré-Einstein spaces. Note that the use of degree theory for the boundary map of minimal surfaces goes back to work of Tromba [38] in the 1970’s and White [42] in the 1980’s, and indeed those papers provided some of the inspiration for Anderson’s proposal to use degree theory in the Einstein setting. A special case of this degree theory, for genus zero surfaces, was developed in [12]. Renormalized volume and area, and their variations: Now return to the PE setting. Assuming the conformal infinity of the PE metric g is sufficiently regular, then g itself has an expansion up to some order at the boundary. When dim M = 4, this has the form g=

d x 2 + h(x) , x2

h(x) ∼ h 0 + x 2 h 2 + x 3 h 3 + · · · ;

(1.1)

here each h j is a symmetric 2-tensor on ∂ M; in particular, h 0 is a metric representing c(g) and h 3 is trace- and divergence-free with respect to h 0 . All other h j are determined in terms of these two tensors. Furthermore, x is a special boundary defining function naturally associated to the choice of h 0 . The volume form d Vg has a corresponding expansion d Vg ∼

A0 A2 + + A4 + · · · ; x4 x2

the x −3 and x −1 terms are absent due to the absence of the h 1 term and the vanishing trace of h 3 . The volume of {x ≥ } is obviously finite for each > 0 and has an expansion as  0 of the form Vol ({x ≥ }) ∼

α0 α1 + V(M, g) + · · · . + 3

The constant term in this expansion is by definition the renormalized volume of (M, g). The key fact, first proved by the physicists Henningson and Skenderis [22], cf. [18] for a careful mathematical treatment, is that this is well-defined independent of the choice of metric h 0 ∈ c(g). The definition of renormalized volume extends to arbitrary dimensions, and they show that it is well defined when dim M is even; when dim M is odd, however, it is not well-defined and has a simple transformation law under change of representative h 0 . For simplicity here we focus on the four-dimensional case. Using the Einstein condition in the Gauss-Bonnet formula, Anderson [5] noted that  1 4π 2 χ (M) − |W |2 d Vg ; (1.2) V(M, g) = 3 6 M here W is the Weyl tensor, and the integral is convergent since |W |2 is pointwise conformally invariant of weight −4. Anderson also computed a formula giving the infinitesimal variation of the renormalized volume in the direction of an infinitesimal Einstein deformation κ:  1 DV|g (κ) = − κ0 , h n d Vh 0 , (1.3) 4 ∂M

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in terms of the leading term in the expansion κ ∼ κ0 + xκ1 +· · ·. A much easier derivation of this formula is given in [1]. Again, this extends immediately to all even dimensions. It follows from this that (when n is even), Hn , and indeed any convex cocompact hyperbolic quotient Hn / , is a critical point of V. The variational problem for renormalized volume remains unstudied. When dim M = 4, V is closely related to the σ2 functional of the underlying incomplete metric on M, and there are some interesting rigidity results using it, see [9]. There are also several nicely geometric results about renormalized volume in 3 dimensions [24,40] (recall that it depends on some choices here, so one has not simply a number but rather a functional on a given conformal class of the boundary surface). Shortly after [22], and motivated by the same string-theoretic concerns, Graham and Witten [20] proved the existence of a well-defined renormalized area A for properly embedded minimal submanifolds Y in a PE space where the boundary of Y is also embedded in ∂ M. Two dimensions is critical for minimal surfaces in roughly the same way that four dimensions is critical for Einstein metrics, so it is reasonable that the results above about renormalized volume of four-dimensional PE metrics have analogues for properly embedded minimal surfaces, and this is indeed true. Our second main result is an explicit formula for A (in the general Poincaré-Einstein setting) and its first and second variations (in the hyperbolic setting): Theorem 1.2. Let Y ∈ M(M) have a C 3,α embedded boundary curve γ . Then  1 A(Y ) = −2π χ (Y ) − | k|2 d A, 2 Y where  k is the trace-free second fundamental form of Y ; the integral is convergent since | k|2 d A is invariant under conformal changes of the ambient metric. Furthermore, if 0 is not in the spectrum of the Jacobi operator L Y (in which case we say that Y is nondegenerate), and φ˙ is a Jacobi field on Y (i.e. L Y φ˙ = 0), which thus corresponds to a one-parameter family of minimal surfaces around Y , then (relative to a normalization which will be explained later), φ˙ ∼ φ˙ 0 + x φ˙ 1 + · · · and  ˙ 0 u 3 ds; DA|Y (φ) = −3 (φ) γ

x3

in the expansion for the function u which gives a graph here u 3 is the coefficient of parametrization of Y over the vertical cylinder γ × [0, x0 ). Furthermore,   1 2  φ˙ 0 φ˙ 3 ds; D A (φ, φ) = − Y 2 γ as we explain later, this shows that the Hessian of A is represented by the Dirichlet-to-Neumann operator for the Jacobi operator L Y . Finally, if M = H3 , the unique nondegenerate critical points of A are the totally geodesic copies of H2 (so γ is a round circle). Physical motivation. We now turn to the physical precursors of all of this. Maldacena’s pioneering work [27] proposes that in the large ’t Hooft coupling regime, the expectation value of the Wilson loop operator corresponding to some closed loop γ ⊂ ∂ M should be given by the area of the minimal surface Y ⊂ M n+1 with asymptotic boundary γ . The papers [27] and [14] already point out that one must introduce an area renormalization, which motivated [20]. Quite recently, it has also been suggested ([34,35]) that this renormalized area be used to measure the entanglement entropy of a particular region in the CFT. More specifically,

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[34] (see also [23]) proposes an ‘area law’: for the model (Hn+1 , Sn ), the information of a domain  ⊂ Sn should correspond to the region in Hn+1 enclosed by a minimal submanifold with asymptotic boundary ∂ (which need not be well-defined, of course, since the minimal submanifold is not unique), and in particular, the entanglement entropy of a domain  ⊂ S2 should correspond to the renormalized area of the minimal surface Y ⊂ H3 with boundary γ = ∂ (see formula (1.5) in [34]). This assertion is checked in the lowest dimensional case n = 1 in [34,35], and special examples are also presented in [23] for n = 2 – but the validity of the assertion in higher dimensions is disputed in [36]. Motivated by these proposals, substantial effort has been devoted in several recent physics papers to understanding the geometric features and renormalized area of various simple cases of minimal surfaces in H3 . For example, in [14] the authors compute the renormalized area of totally geodesic planes; in [23] Hirata and Takayanagi study the existence of minimal surfaces with two disconnected circles as asymptotic boundaries and also estimate the renormalized area of those surfaces; Maldacena [27] studies the case of a rectangle where the length T of one side approaches infinity. Furthermore, Drukker-Gross-Ooguri, [14] and Polyakov-Rychkov, [33] have sought to check the proposed formula in [27] relating the expectation value of the Wilson loop in CFT with the renormalized area of a minimal surface in AdS. This verification involves calculating the first and second variations of the renormalized area functional with respect to deformations of the loop γ . Since those authors did not have a usable explicit formula for the renormalized area, their calculations required justification for dropping certain divergent terms; in contrast, our local formulæ allow for straightforward calculations. Since there seems to be active and continuing interest in these proposals relating renormalized area with the expectation values of Wilson loop operators, the loop equation and to entanglement entropy, we hope that our results will facilitate further investigations in this area. Our paper is structured as follows: in Sect. 2 we present some background material needed for this work, on Poincaré-Einstein metrics, uniformly degenerate elliptic operators and embedded minimal surfaces in convex co-compact hyperbolic 3-manifolds. The local formula for A is proved in Sect. 3, and certain global aspects of this functional are studied in Sect. 5. The intervening Sect. 4 develops the moduli space theory of properly embedded minimal surfaces. This provides the correct setting in which to derive the first and second variation formulæ, which appears in Sect. 6. Section 7 characterizes the nondegenerate critical points of A when the ambient space is hyperbolic 3-space, H3 . Finally, in Sect. 8 we discuss the relationship of A and the Willmore functional. As a general disclaimer, we have written this paper to focus primarily on the case of surfaces in three-dimensional hyperbolic manifolds. Most of the results generalize in various ways: for example, it is usually fairly straightforward to reprove these results for minimal surfaces in smoothly conformally compact asymptotically hyperbolic threemanifolds. Some of the results also extend to minimal surfaces of higher codimension in higher dimensional Poincaré-Einstein (or even again just smoothly conformally compact asymptotically hyperbolic) spaces. Except in a few instances we have not stated nor emphasized these generalizations. 2. Geometric and Analytic Preliminaries We now give precise definitions of the spaces and submanifolds we shall be working with and explain some of their properties. We also discuss some basic results about elliptic operators on these spaces.

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2.1. Conformally compact and Poincaré-Einstein spaces and convex cocompact hyperbolic 3-manifolds. A Riemannian manifold (M, g) is called conformally compact if M is the interior of a smooth compact manifold with boundary and g = ρ −2 g, where ρ is a defining function for ∂ M and g is a metric smooth and nondegenerate up to ∂ M. Any such metric is complete and has sectional curvatures tending to −|dρ|2g (q) upon approach to any point q ∈ ∂ M. In particular, if |dρ|2g is constant along ∂ M, we say that (M, g) is asymptotically hyperbolic (AH). To any conformally compact metric g one may associate a conformal class on ∂ M:    c(g) = ρ 2 g 

 T∂M

,

which is obviously independent of the choice of the defining function ρ. This conformal equivalence class is called the conformal infinity of g. Any AH metric has a normal form, due to Graham and Lee [19]. Let (M, g) be an AH space and fix any metric h 0 representing the conformal class c(g). Then there is a unique defining function x for ∂ M, defined in some neighborhood U of the boundary, which satisfies the two conditions |d log x|2g ≡ 1,

g|T ∂ M = h 0 , where g = x 2 g.

The flow lines for the gradient ∇ g x give a product decomposition U ∼ = [0, x0 ) × ∂ M, in terms of which the pullback of the metric g takes the form g=

d x 2 + h(x) , x2

h(x) ∼ h 0 + xh 1 + x 2 h 2 + · · · .

(2.4)

The defining function x associated to the boundary metric h 0 will be called a special boundary defining function (bdf). A case of particular interest is when (M, g) is Poincaré-Einstein (PE), which means simply that it is both conformally compact and Einstein. These metrics were introduced by Fefferman and Graham [16] as a way of canonically associating a Riemannian metric on an ambient (n + 1)-manifold to a conformal class on an n-manifold, with the goal of finding new conformal invariants on the boundary via Riemannian invariants of the ambient manifold. If the conformal infinity of such a g is smooth, then the family of tensors h(x) in (2.4) has a complete expansion in powers of x (and also powers of x n−1 log x when n = dim X is odd, n ≥ 5). The coefficients h 0 and h n−1 are formally undetermined, but all other h j can be expressed as local differential operators applied to these two coefficients; it is thus natural to think of the pair (h 0 , h n−1 ) as the Cauchy data of g. In this paper we shall be primarily concerned with the three-dimensional case. If (M 3 , g) is PE, then M is isometric to a convex cocompact quotient H3 / . (Convex cocompact means that  is geometrically finite and has no parabolic elements; equivalently, the quotient by  of the convex hull (in H3 ) of the limit set () is compact in M.) The Fefferman-Graham expansion for g simplifies then, and has a special form where only h 0 , h 2 and h 4 are nonzero, see [15] and Epstein’s Appendix in [31]. These coefficients can be calculated in terms of the metric and second fundamental form of any one of the level sets {x = const. }, and the special bdf x then has the property that − log x is the distance function to this level set (up to an additive constant).

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2.2. Uniformly degenerate operators. We shall be using results about the mapping and regularity properties for elliptic operators which are uniformly degenerate. The theory here is drawn from [28], but see also [25]. Let X be a manifold with boundary, and suppose that (x, y) is a local chart near some boundary point, where x is a boundary defining function and y restricts to coordinates along the boundary. A differential operator L is called uniformly degenerate if in any such chart it takes the form  L= a j,α (x, y)(x∂x ) j (x∂ y )α . j+|α|≤m

We assume that the coefficients are smooth, or at least C 2,α up to ∂ X . There is a welldefined uniformly degenerate symbol  0 σm (L)(x, y, ξ, η) = a j,α (x, y)ξ j ηα j+|α|=m

and L is elliptic in this category of objects if this symbol is invertible for all (x, y) and (ξ, η) = 0. Unlike in the standard interior case, there is a further model which must be studied, called the normal operator, which is defined by  N (L) = a j,α (0, y)(t∂t ) j (t∂v )α , j+|α|≤m

where (t, v) are linear coordinates on the half-space R+t × Rv ,  + 1 = dim X . Finally, for any such operator, we define its set of indicial roots to be the values of μ for which L x μ = O(x μ+1 ). (This definition must be modified  slightly when L is a system.) These values are the roots of the indicial polynomial j≤m a j0 (0, y)μ j , so (in the scalar case) there are exactly m such values. For simplicity, we now restrict to the case where the degree of L is 2, and list the indicial roots as μ1 and μ2 . We shall let these operators act on weighted Sobolev and Hölder spaces of functions. By definition H0k (X ) consists of functions which lie in L 2 along with all derivatives up to order k with respect to the vector fields x∂x and x∂ y . Similarly, k,α 0 denotes the Hölder space where the derivatives and difference quotients are measured with respect to these same vector fields. If E is any function space, then x μ E denotes the set of functions x μ v where v ∈ E. The basic result we need is the following: Proposition 2.1. Let L be a uniformly degenerate operator of degree 2 on the compact manifold with boundary X , and suppose that L is uniformly degenerate elliptic. If N (L) : t μ−1/2 H02 (dtdv) −→ t μ−1/2 L 2 (dtdv) is an isomorphism for one value of μ ∈ (μ1 , μ2 ), then it is an isomorphism for every μ ∈ (μ1 , μ2 ), and for all such μ, (X ) −→ x μ k,α L : x μ k+2,α 0 0 (X ) is Fredholm, with nullspace contained in x μ2 ,α 0 (X ) for every . If N (L) is only surjective as a map t μ−1/2 H02 (dtdv) → t μ−1/2 L 2 (dtdv) but its nullspace is nontrivial, then L itself still has closed range of finite codimension, but an infinite dimensional kernel. The proof is contained in [28].

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2.3. Properly embedded minimal surfaces with embedded asymptotic boundary. As explained in the Introduction, there is a rich existence theory for properly embedded minimal or area-minimizing surfaces in convex cocompact hyperbolic 3-manifolds. Something not treated in Anderson’s original investigations is the boundary regularity. One expects that a properly embedded minimal surface Y is as regular as its asymptotic boundary curve γ . This problem and its generalization to higher dimensional minimal codimension one submanifolds was investigated by Lin [26], Hardt and Lin [21] and Tonegawa [39]. The higher codimension case has apparently not been treated at all, but can be handled in a fairly straightforward way using the theory of uniformly degenerate elliptic operators; we shall come back to this in a later paper. The general result, which we state only, but do not prove here, is as follows: in general dimensions and codimensions, if γ is smooth then any corresponding minimal Y with ∂Y = γ is polyhomogeneous at the boundary, i.e. has an expansion in powers of any defining function for M restricted to Y ; when dim Y is even, only positive integer powers appear, while if dim Y is odd, then powers of x k log x also appear. For the case of importance here, the proof is fully contained in the various papers cited above. Proposition 2.2. Let γ be a C k,α embedded curve in ∂ M, where M = H3 /  is convex cocompact, k ∈ N, 0 < α < 1. Then Y is C ∞ in the interior of M and C k,α up to γ = Y ∩ ∂ M. One mild generalization is that the same regularity result holds if Y ⊂ M, where (M, g) is three-dimensional smoothly conformally compact asymptotically hyperbolic. The proof below can then be adapted directly, though the equations all involve higher order correction terms, hence the arguments are messier. We discuss some features of the proof in order to bring out some consequences. This result is local in γ , so we may as well suppose that M = H3 and focus on the behaviour of Y near some fixed point p ∈ γ . Using the upper half-space model with coordinates y ∈ R2 , x > 0, place p at the origin and choose a local arc-length parametrization γ (s) for γ (with respect to the standard Euclidean metric on R2 ). Let  denote the vertical cylinder over γ , i.e.  = {(y, x) ∈ R2 × R+ : y ∈ γ }; thus near the origin,  = {(γ (s), x)}. Choose two smooth families of minimal hemispheres, i.e. totally geodesic copies of H2 , which lie completely inside and outside of γ , respectively, and which are tangent to γ , and let  ± be the envelopes of these families. These are smooth mean-convex surfaces tangent to  along γ , and it is straightforward to use them as barriers to deduce that Y must lie in the open set between  − and  + . It follows that Y is vertical along γ , or equivalently, that its unit normal with respect to the Euclidean metric on the upper halfspace is tangent to R2 = {x = 0} along γ . We now write Y as a horizontal graph over . More specifically, if N = N (s) is the unit normal (again with respect to the Euclidean metric) at a point of , then there is a scalar function u(s, x) and a neighbourhood U of the origin so that Y ∩ U = {F(s, x) := (γ (s) + u(s, x)N (s), x) : |s| < , x < }. The argument above implies that u(s, 0) = ∂s u(s, 0) = 0. The regularity of Y along γ is equivalent to that of this function u, and the key point is that u is a solution of a uniformly degenerate elliptic partial differential equation F(u) = 0 corresponding to the minimality of Y , which we derive now. The function F induces a coordinate chart on Y ; let the indices 1 and 2 refer to the s and x coordinates,

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respectively. Letting T = γ  (s), then Fs = (1 − κu)T + u s N ,

Fx = u x N + ∂x ,

where κ is the curvature of γ . For convenience below, write w = 1 − κu. The inward pointing g unit normal is equal to Fx × Fs = J −1 (−u s T + wN − u x w∂x ), ν= J := u 2s + w 2 (1 + u 2x ). (2.5) |Fx × Fs | The coefficients of the first fundamental form and its inverse are 



2 1 1 + u 2x −u x u s w + u 2s u x u s ij (g i j ) = , and (g . ) = u x us 1 + u 2x J 2 −u x u s w 2 + u 2s Next, we compute that Fss = (ws − κu s )T + (u ss + κw)N , Fsx = −κu x T + u xs N , 

1 w(u ss + κw) − u s (ws − κu s ) wu xs + κu x u s . (k i j ) = − wu xs + κu x u s wu x x J

Fx x = u x x N ,

Finally, use the general formula ki j = eφ (k i j + ∂ν φ g i j ) relating the second fundamental forms of Y of two conformally related metrics g = e2φ g. Here φ = − log x and ν is as in (2.5), so the matrix (ki j ) is equal to 

1 w(u ss + κw)−u s (ws −κu s )−x −1 u x w(w 2 + u 2s ) wu xs + κu x u s − x −1 u 2x u s w . − wu xs + κu x u s − x −1 u 2x u s w wu x x − x −1 u x w(1 + u 2x ) Jx The equation of minimality, i.e. that g i j ki j = H = 0, is then given by the expression F(u) := (1 + u 2x ) [w(u ss + κw) − u s (ws − κu s )] − 2u x u s (wu xs + κu x u s )

wu x 2 + w(w 2 + u 2s )u x x − 2 u s + w 2 (1 + u 2x ) = 0. (2.6) x The coefficient 1/x in this last term makes this a degenerate elliptic equation. Assume that γ is at least C 3 ; we compute the first few coefficients in the expansion of u(s, x) as x  0. Set u ∼ u 2 (s)x 2 + u 3 (s)x 3 + · · · (since we already know that u vanishes to second order). Inserting this into F(u) = 0 yields that u 2 (s) = 21 κ(s), but u 3 (s) is formally undetermined by the equation. In other words, this coefficient must depend globally on Y . Just as in the Fefferman-Graham expansion for PE metrics, all higher terms in the expansion for u are determined by γ and u 3 and their derivatives, so we regard (γ , u 3 ) as the Cauchy data for the minimal surface Y . Using the unique continuation theorem from [29], it is straightforward to show that if Y1 and Y2 are two minimal surfaces with the same Cauchy data (γ , u 3 ) (even locally), then Y1 ≡ Y2 . This global coefficient u 3 plays a central role in our work. As a side remark for the moment, consider C 3 surfaces with boundary Y ⊂ M with ∂Y ⊂ ∂ M, which intersect ∂ M orthogonally (this makes sense since M has a conformal structure). Any such Y can still be represented near the boundary as a normal graph over the vertical cylinder  over its boundary curve γ , and the graph function still vanishes to second order. It is no longer necessarily true that u 2 = 21 κ. The second fundamental form now satisfies 

1 (2u 2 − κ) + 3u 3 + O(x) −2u 2 + O(x 2 ) (ki j ) = x ; −2u 2 + O(x 2 ) −3u 3 + O(x) note that we now have only |k|g = O(x) unless 2u 2 = κ in which case |k|g = O(x 2 ).

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Finally, since the Jacobian term J = |Fs × Fx | = 1 + O(x 2 ), we see that in these coordinates, the area form equals 1 + O(x 2 ) dsd x. x2 Writing Y = Y ∩ {x ≥ }, then by definition, the renormalized area of Y is the constant term in the expansion  length(γ ) + A(Y ) + O( ). (2.7) dA = Y dA =

In order for this to be interesting, we must show that A(Y ) is well-defined, independent of the choice of special bdf x. This was done by Graham and Witten [20]; their key observation, which is particularly simple in this low-dimensional setting, is that if h 0 and  h 0 = e2χ0 h 0 are two representatives of the conformal class c(g), corresponding to special bdf’s x and  x , respectively, then  x = eχ x, where χ (x, y) = O(x 2 ). This means 2 that  x = x + O(x ), and hence in the new coordinate system ( s,  x ) on Y , one still has dA =  x −2 (1 + O( x 2 )) d sd x . From this, the claim about well-definedness of A(Y ) is immediate. The only property about Y needed for this argument to work is that it is at least C 2 and meets ∂ M orthogonally. Thus even in this broader setting there is still a welldefined notion of renormalized area of Y . To maintain the distinction, we shall denote this extended renormalized area functional by R rather than A when the surface Y is not minimal. 3. A Formula for Renormalized Area We now express the renormalized area of a properly embedded minimal surface Y in M in terms of its Euler characteristic and an integral of local invariants. In fact, since it is not much more complicated to do so, we find an expression for the renormalized area when Y lies in an arbitrary Poincaré-Einstein space of any dimension and is not necessarily minimal, but still meets ∂ M orthogonally. Proposition 3.1. Let (M n+1 , g) be a PE space and γ ⊂ ∂ M a C 3,α embedded curve, and suppose that Y 2 ⊂ M is a properly embedded minimal surface with asymptotic boundary γ , an embedded closed curve in ∂ M. Then the renormalized area A of Y is equal to   1 2  A(Y ) = −2π χ (Y ) − |k| d A + W1212 d A, (3.8) 2 Y Y where  k is the trace-free second fundamental form of Y and W1212 is the Weyl curvature of g evaluated on any orthonormal basis for T Y . In particular, the integrals on the right are convergent. More generally, if Y is any properly embedded surface (not necessarily minimal) which extends to be a C 2 surface with boundary in M intersecting the boundary orthogonally, then (with the convention that H = (tr k)/2), the renormalized area is equal to  

1 2 2  2|H | − |k| d A + A(Y ) = −2π χ (Y ) + W1212 d A. 2 Y Y Proof. We begin with some preliminary observations and calculations.

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First, denote by Ri jk and (RY )i jk the components of the curvature tensor of g and of the induced metric on Y , respectively. The Ricci curvature of g satisfies Ri j = −ngi j , and from the standard decomposition of the curvature tensor of an Einstein metric, the components of the Weyl tensor for g are given by Wi jk = Ri jk + gik g j − gi g jk .

(3.9)

Fix a point p ∈ Y and choose an oriented orthonormal basis {e1 , . . . , en+1 } for T p M such that e1 and e2 are an oriented basis for T p Y . Now, denoting by kisj , i, j = 1, 2, s = 3, . . . , n + 1, the components of the second fundamental form of Y at p, the GaussCodazzi equations become R1212 = (RY )1212 −

n+1  s=3

1 2 s s s s k| . (k11 k22 − k12 k12 ) = (RY )1212 − |H |2 + | 2

s +k s = 2H s , so k s =  kiis + H s To check this last equality, simply note that for each s, k11 22 ii and kisj =  kisj for i = j, and hence n+1 n+1

  s s   s s s s s 2 ( k11 k11 k22 − k12 k12 = + H s )( k22 + H s ) − ( k12 ) s=3

s=3 n+1  1 s 2 s 2 s 2 = (H s )2 − (( k12 ) ). k ) + (k22 ) + 2( 2 11 s=3

Combined with (3.9), this gives 1 ˆ2 − |H |2 − W1212 = −1. (RY )1212 + |k| 2

(3.10)

This equation holds at each point. The first term on the left is simply the Gauss curvature K of Y ; for simplicity, we continue to write W1212 for the third term on the left, noting that it is independent of orthonormal frame. Now integrate over Y = Y ∩ {x ≥ } to obtain    

1 2 2 ˆ 2|H | − |k| d A − K dA − W1212 d A = − d A. 2 Y Y Y Y By the Gauss-Bonnet theorem, since χ (Y ) = χ (Y ) for small enough,   K d A = 2π χ (Y ) − κ ds, Y

γ

where κ is the geodesic curvature of the boundary γ := ∂Y in Y and ds is the length element with respect to the metric induced by g. Altogether we get    

1 2|H |2 − | d A = −2π χ (Y ) + κ ds + k|2 d A + W1212 d A. 2 Y Y γ Y To proceed further, we use the formula that as  0,  length (γ ) + O( ). κ ds = γ

(3.11)

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Deferring the proof of this for a moment, using the basic definition of renormalized area via Hadamard regularization in (2.7), we find that    

1 A(Y ) = −2π χ (Y ) + lim 2|H |2 − | k|2 d A + W1212 d A . (3.12) →0 2 Y Y In order to show that the second and third terms on the right have limits as  0, recall the transformation law  ki j (g), ki j (e2φ g) = eφ for the trace-free second fundamental form  k(g) under the conformal change of ambient metric from g to e2φ g (this is true no matter the dimension or codimension of the submanifold Y ). When dim Y = 2, | k(e2φ g)|2e2φ g d Ae2φ g = | k(g)|2g d A g . Similarly, the components of the Weyl tensor transform as Wi jk (e2φ g) = e−2φ Wi jk (g) ⇒ W1212 (e2φ g) d Ae2φ g = W1212 (g) d A g . Thus these two potentially worrisome terms do have a limit. Similarly, even when Y is not minimal, by the calculations in Sect. 2, the mean curvature H is O(x), so its integral has a limit too. It remains to prove (3.11). Denote by κ the geodesic curvature of γ with respect to the metric g = x 2 g and n the interior g-unit normal to ∂Y in Y . Since u ∼ u 2 x 2 +u 3 x 3 + O(x 3+α ), it follows that n = (1 + O(x 2 ))∂x + V , where g(V, ∂x ) = 0. Now, geodesic curvature also transforms nicely under conformal re-scalings: κ = (κ + ∂n log x). Since g(Fss , ∂x ) = 0 at x = 0, we deduce κ = O( ), hence κ = 1 + O( 2 ); recalling too that ds = −1 ds, we obtain finally  length (γ ) + O( ), κ ds = γ as claimed.

 

 The expression 21 Y (2H 2 − | k|2 ) d A is finite only when Y intersects ∂ M orthogonally. Indeed, using the notation and formulæ from Sect. 2.3 again, suppose that Y is written as a normal graph over the vertical cylinder  over the boundary curve γ , but do not assume  that u x (s, 0)≡ 0. Now H = 2u x (s, 0) + O(x), as follows from the formulæ H = x H + 2∂ν (log x) and ∂ν x = g(ν, ∂x ) = u x (s, 0) + O(x). Recalling again that | k|2 d A is conformally invariant, we see that 1 2



(2H 2 − | k|2 ) d A = Y

4|u x (s, 0)|2 + O(log )

does not have a limit as  0 unless u x (s, 0) ≡ 0. This is consistent with the fact that the definition of renormalized area A(Y ) via Hadamard regularization is independent of choice of special bdf x only when this same condition is satisfied.

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4. The Moduli Spaces M k (M) Fix the convex cocompact hyperbolic 3-manifold (M, g) and an integer k ≥ 0. We define k (M) to be the space of all properly embedded surfaces of genus k which extend to M M as C 3,α submanifolds with boundary and which intersect ∂ M orthogonally, and Mk (M) the subspace of all such surfaces which are minimal. In this section we study the structure of these moduli spaces, which are the natural domains for the renormalized area functional, as well as some properties of the natural map  which assigns to any such Y its asymptotic boundary ∂Y = γ , which is a C 3,α closed (but possibly disconnected) embedded curve in ∂ M. It is a standard fact that the space of all C 3,α surfaces Y ⊂ M with boundary γ k (M) defined above is clearly a closed lying in ∂ M is a Banach manifold. The space M submanifold (of infinite codimension). Similarly, the space E of all C 3,α closed embedded (but not necessarily connected) curves γ ⊂ ∂ M is also a Banach manifold. The corresponding structure for the smaller space of minimal surfaces is also true. Proposition 4.1. For each k, Mk (M) is a Banach manifold. Proof. Fix any Y ∈ Mk (M) and assume for the moment that ∂Y = γ is actually a C ∞ embedded curve in ∂ M. We construct a coordinate neighbourhood around Y in Mk which in the generic (nondegenerate) setting is modelled on a small ball around 0 in the space of Jacobi fields for the minimal surface operator on Y which are C 3,α up to ∂Y ; this ball in turn is identified with a small ball in the space of C 3,α normal vector fields along γ . We make this nondegeneracy condition explicit below. To set this up, let ν be the unit normal (with some fixed choice of orientation) along Y . If φ is any scalar function on Y which is small in C 3,α , we can define a new surface Y0,φ = {exp p (φ( p)ν( p)) : p ∈ Y }, which we call a normal graph over Y . The mean curvature of Y0,φ is computed by a nonlinear elliptic second order operator F(φ). The precise expression of this operator is rather complicated, but its linearization has the familiar form DF|φ=0 := L Y = Y + |AY |2 − 2; here AY is the second fundamental form of Y and Y is its Laplacian with respect to the induced metric. This Jacobi operator, L Y , is an elliptic uniformly degenerate operator of order 2. Its normal operator is N (L Y ) = t 2 ∂t2 + t 2 ∂v2 − 2 since the second fundamental form AY vanishes at ∂Y ; the leading (second order) term is just the Laplacian on the hyperbolic plane, so N (L Y ) = H2 − 2. The indicial roots are μ1 = −1, μ2 = 2, hence solutions of L Y u = 0 satisfy u ∼ a(y)x −1 + · · · or u ∼ a(y)x 2 + · · ·. Note that since the g- and g-unit normals are related by ν = xν, in the case where u blows up as x  0, the product uν = (xu)ν behaves like (a(y) + O(x))ν, or in other words, the solutions growing at this rate are the ones which are bounded (but not blowing up) at x = 0 with respect to g, and hence correspond to moving the boundary curve γ nontrivially. In this g normalization, the decaying Jacobi fields vanish like

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x 3 , which should be no surprise. In any case, it follows directly from self-adjointness and integration by parts that N (L Y ) : t μ H02 (H2 ; t −2 dtdv) = t 1+μ H02 (H2 ; dtdu) −→ t 1+μ L 2 (H2 ; dtdu) is invertible when μ = 0. By Proposition 2.1, this is true for any −1 < μ < 2 and for any μ in this range, μ 0,α L Y : x μ 2,α 0 −→ x 0

(4.13)

is Fredholm of index zero. We call the minimal surface Y nondegenerate if the nullspace K μ of this mapping contains only 0 for any μ ∈ (−1, 2); in this case, (4.13) is surjective. In general, its cokernel is canonically identified with K μ in the following sense. First note that by Proposition 2.1 again, K μ ⊂ x 2 2,α 0 (Y ) (indeed, if γ is smooth, any u ∈ K μ is polyhomogeneous, i.e. has full tangential regularity), so we may as well μ 0,α drop the subscript  μ. Next, if f ∈ x 0 (Y ) lies in the range of (4.13), f = L Y w, then obviously Y u f d AY = Y u L Y w d AY = 0 for all u ∈ K . (Note that this integral makes sense since μ > −1.) However, this gives precisely the correct number of linear conditions, so this necessary condition is also sufficient. To study Mk (M), we must consider a broader class of deformations of Y where the boundary curve γ also varies. Let ν = x −1 ν be the unit normal to Y with respect to the conformally compactified metric g = x 2 g. This vector field extends smoothly to Y , and its restriction to γ = ∂Y is the unit normal N to this curve in ∂ M with respect to h 0 .1 Any nearby curve can be written as a normal graph γψ = {exp p (ψ( p)N ( p)) : p ∈ γ } (where now exp is with respect to h 0 ). We now define an extension operator E which assigns to any small ψ a surface Yψ,0 which is ‘approximately minimal’ and which has ∂Yψ,0 = γψ . To do this, let u be the graph function for Y over the cylinder . We define a new graph function u ψ in some neighbourhood {x < } of the boundary such that j u ψ (s, 0) = ψ(s), and ∂x u ψ (s, 0), j = 1, 2 is determined by the formal expansion of 3 solutions for F; ∂x u ψ (s, 0) could be chosen freely, but we set it equal to u 3 (s). Now let Uψ = χ u ψ + (1 − χ )u where χ is a cutoff function which equals 1 near x = 0. It is not hard to check that F(Uψ ) ∈ x μ 1,α for some 0 < μ < 2. The extension E can be chosen to depend smoothly on ψ. We then have that ˆ =w DE|0 (ψ) is a function on Y which satisfies w ∼ x −1 ψˆ as x  0 and L Y w = O(x μ ) for some μ ∈ (0, 2). Finally, perturb Yψ,0 to a normal graph over it using the unit normal for Yψ,0 and as a graph function any small φ ∈ x μ 2,α 0 (Y ). The resulting surface will be denoted Yψ,φ , and we write its mean curvature as F(ψ, φ). Thus if B is a small neighbourhood of the origin in C 3,α (γ ) × x μ 3,α 0 (Y ), then F : B −→ x μ 1,α 0 (Y ) is a smooth mapping. 1 See (2.4) for a definition of h . 0

(4.14)

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A neighbourhood of Y in Mk (M) is identified with the space of solutions to F(ψ, φ) = 0, and so may be studied by the implicit function theorem. Note that ˆ φ) ˆ = L Y (DE(ψ) ˆ + φ). ˆ DF|(0,0) (ψ, When Y is nondegenerate, D2 F|(0,0) = L Y on x μ 2,α 0 (Y ) is already surjective; this yields the existence of a smooth map G defined in a neighbourhood of 0 in C 3,α (γ ) to x μ 2,α 0 (Y ) such that F(ψ, G(ψ)) ≡ 0, and so that all elements of the nullspace of F near (0, 0) are of this form. In the degenerate case, we must show that by allowing ψˆ to vary over some suitable finite dimensional subspace of infinitesimal deformations of γ , we can still obtain a surjective map. If this were to fail, then there would exist a nontrivial u ∈ K such that ˆ L Y (DE(ψ) ˆ + φ) ˆ ⊥ u. Write η = DE(ψ). ˆ Then for all ψˆ and φ,    ˆ = ˆ − (η + φ)n ˆ · ∇u = −2 ψu ˆ 0, L Y (η + φ)u n · ∇(η + φ)u 0= Y

γ

γ

which implies that u 0 (the leading coefficient of x 2 in the expansion of u) is orthogonal ˆ This is impossible, since if it were true, then all coefficients in the entire to every ψ. expansion would vanish, hence u 0 would vanish to infinite order; this violates [29].This proves that F is always surjective as a function of both (ψ, φ), and hence finally that Mk (M) is a smooth Banach manifold in a neighbourhood of Y .   Proposition 4.2. The natural map  : Mk (M) −→ E(∂ M) given by (Y ) = ∂Y is Fredholm with index 0. (Recall that this means that if Y ∈ Mk (M), then DY is a Fredholm map from TY Mk (M) to T(Y )  with the dimensions of its kernel and cokernel equal to one another.) ˆ + φ, ˆ Proof. Let K denote the nullspace of L Y acting on functions of the form DE|0 (ψ) 2,α 3,α μ ˆ ˆ ψ ∈ C (γ ), ψ ∈ x 0 (Y ). If Y is nondegenerate, then K contains no elements of ˆ so DY is an isomorphism. If Y is degenerate, however, then the proof the form (0, φ), above shows that if  = dim(K ∩ x μ 2,α 0 (Y )), so that dim ker DY = , then we can make L Y surjective by supplementing x μ 2,α 0 (Y ) with an -dimensional space H of ˆ (and we may even assume that each ψˆ is C ∞ ). Let H  be functions of the form DE0 (ψ) any choice of complement of H in C 3,α (γ ). The implicit function theorem shows that ˆ ˆ there exists a smooth map G from H  to H ⊕ x μ 3,α 0 (Y ) such that F(ψ, G(ψ)) ≡ 0, so the codimension of the range of DY is  too. Hence ind (DY ) = 0, as claimed.   The final general result about these moduli spaces is contained in Proposition 4.3.  is a proper mapping. Proof. We must show that if γ j is a sequence of elements in E such that γ j → γ in C 3,α , and if Y j ∈ Mk (M) has ∂Y j = γ j , then (possibly after passing to a subsequence) Y j converges to a properly embedded minimal surface Y with genus k and ∂Y = γ .

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Let  be the vertical cylinder over γ , and let u j be the horizontal graph function corresponding to the surface Y j . A priori, the function u j is only defined on some vertical strip where x < j . The first step is to show that j can be chosen independent of j. The only thing which prevents these graphs from existing on a uniform strip would be if the u j did not have a uniform gradient bound, or in other words, that there exists a sequence (s j , x j ) with s j in the parameter interval for γ and x j  0, and such that |∇u j (s j , x j )| = 1, say (any positive number would do), and |∇u j (s j , x j )| < 1 for all s and for x < x j . The gradient and norm here are with respect to the Euclidean metric. Perform a rescaling by a factor x1j , centered at the point (γ (s j ), 0), and then a translation and rotation to move (s j , 0, 0) to the origin in R2 and to make the rescaled curve γ j j in the upper half-space, defined tangent to the y 1 -axis. The result is a minimal surface Y in a ball of expanding radius tending to infinity, which passes through (0, 0, 0) in the boundary, and which can be expressed as a horizontal graph y 2 = F j (y 1 , x) over some u (s ,x ) large ball in the vertical (y 1 , x)-plane. By construction, F j (0, 1) = j xjj j ; by Rolle’s theorem, this is bounded by |∂x u j (s j , x j )| for some x j < x j , hence by construction |F j (0, 1)| ≤ 1 and |∇ F j (0, 1)| = 1. Passing to a subsequence, as j → ∞ this minimal surface converges to a com ⊂ H3 whose boundary is the limit of rescalings of γ , i.e. a plete minimal surface Y straight line, and which can be expressed as a horizontal graph y 2 = F(y 1 , x) over all  is not of R y 1 × R+x with |F(0, 1)| ≤ 1. However, by construction, the tangent space of Y vertical at the point (0, 1, F(0, 1)), which contradicts the fact that the unique minimal surface in hyperbolic space with boundary a straight line is a totally geodesic plane. This argument proves that the graph functions u j are defined on a uniform interval [0, ], and moreover that the boundary curves at height x = are also converging in C 3,α (in fact, in C ∞ by interior elliptic estimates). Notice that this already proves that no handles can slide off to infinity, provided the boundary curves remain uniformly smooth enough. Let Y j, = Y j ∩ {x ≥ }. This is now a sequence of compact minimal surfaces with boundary in the convex set {x ≥ } ⊂ M. The proof will be finished if we can prove that these surfaces have a convergent subsequence. This in turn follows from the results of Anderson [4] and White [41]. In order to apply their results, it suffices to show that the genera of the Y j, remain bounded, which is obvious by definition, and that the areas of these surfaces are also bounded. This follows from the Gauss-Bonnet theorem: since each Y j is minimal, its Gauss curvature satisfies K ≤ −1, which implies that   (−K ) d A = −2π χ (Y j ) + κ ds. Area (Y j, ) ≤ ∂Y j,

Y j,

Now, the first term is fixed, so we must show that the second term is bounded. But this is immediate from standard elliptic estimates applied to the graph function u j for Y j in the annulus /2 < x < 2 since is now fixed.   As explained in [40], it is important to work with a slightly different regularity condition: we shall replace C k,α by the closure in this space of C ∞ . This smaller subspace is separable, whereas C k,α is not, so with this new regularity restriction (which we shall k (M) and E(∂ M) are separable Banach manifolds. not comment on further) both M Using all of these facts, we may now define the degree of  by  deg() = (−1)n(Y ) , Y ∈−1 (γ )

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where γ is a regular value of , so each Y ∈ −1 (γ ) is nondegenerate, and where n(Y ) denotes the number of negative eigenvalues of −L Y . This degree is a well-defined invariant on each component of E(∂ M) (once we have fixed the integer k and the component of Mk (M) mapping to that isotopy class of boundary curves). For example, when M = H3 , and γ is any convex curve, then by the maximum principle, there is exactly one properly embedded minimal surface Y with ∂Y = γ , and necessarily, its genus is 0. This proves that when M is the entire hyperbolic space, then deg (0 ) = 1 while deg (k ) = 0 for k > 0, on the component of E containing connected curves. This has some interesting consequences. For example, Anderson [3] displayed a connected curve which bounds a minimal surface of genus k > 0; by genericity, we can assume that this curve is regular for , and since the degree equals zero, we obtain the existence of yet another element in Mk (H3 ) with boundary equal to this same curve. On the other hand, de Oliveira and Soret [30] construct stable properly embedded minimal surfaces in H3 with arbitrary genus, where the boundary curve has any prescribed number of components. Here too, for any given boundary curve, we conclude the existence of at least one other element of Mk with that boundary curve which is unstable. It would be interesting to compute the degree of  precisely in some of these other cases. 5. Area Minimization and Renormalized Area We now investigate the role of locally area minimizing surfaces in the study of renormalized area for minimal and nonminimal properly embedded surfaces. 5.1. Renormalized area of absolute minimizers. Proposition 5.1. Let γ be a C 3,α embedded curve in ∂ M which bounds in M. Suppose that Y1 and Y2 are two properly embedded minimal surfaces with ∂Y1 = ∂Y2 = γ . If Y1 is area minimizing in M, then A(Y1 ) ≤ A(Y2 ), and equality holds if and only if Y2 is also an area minimizer. Proof. Fix a special boundary defining function x and set Y j, = Y j ∩ {x ≥ } and γ j, = ∂Y j, . The functions u 1 and u 2 for these two surfaces agree up to order three, so in terms of any local coordinate s on γ , |u 1 (s, x) − u 2 (s, x)| ≤ C x 3 , with corresponding estimates for the first 3 derivatives. If S denotes the region between γ1, and γ2, in {x = }, this gives Area (S ) = O( ). Recalling that Area (Y ) = −1 length (∂Y ) + A(Y ) + O( ), we obtain Area (Y2, ) − Area (Y1, ) = A(Y2 ) − A(Y1 ) + O( ), since ∂Y1 = ∂Y2 .  =Y Thus if A(Y1 ) > A(Y2 ), then the new (nonsmooth) surface Y1, 2, ∪ S would eventually have area smaller than Y1, . Indeed, by the inequalities above,  Area (Y1, ) ≤ Area (Y2, ) + C1 ≤ Area (Y1, ) + A(Y2 ) − A(Y1 ) + C2 < Area (Y1, )

for small enough. This contradicts the fact that Y1 is area-minimizing.

 

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5.2. The renormalized area spectrum. The result in this last subsection suggests the consideration of the set-valued function Sk

E  γ −→ {A(Y ) : Y ∈ Mk (M) ∂Y = γ }, which we call the renormalized area spectrum (of degree k). Properness of the boundary map ensures that Sk (γ ) is always a compact set. Proposition 5.1 implies that if Y0 is any absolutely area minimizing surface with ∂Y0 = γ , then A(γ ) := A(Y0 ) is a lower bound for Sk (γ ) for every k. On the other hand, trivially by (3.1), this set is bounded above by −2π χ (Y ) = 2π(2k +  − 2), where  is the number of components of γ . In other words, Sk (γ ) ⊂ [A(γ ), 2π(2k +  − 2)].

(5.15)

Note furthermore that the upper limit is never attained unless there exists a totally geodesic minimal surface Y with ∂Y = γ , which never happens unless γ is a round circle. The lower bound is attained if and only if there exists a genus k absolute area minimizer with boundary γ . 5.3. Minimizers of the extended renormalized area functional. In Sect. 4.2 of [23], Hirata and Takayanagi assert that minimizers of the extended renormalized area functional amongst all surfaces with a given boundary are necessarily area minimizing surfaces with this same asymptotic boundary. Furthermore, Polyakov has communicated to us his suggestion (based on his work [32]) that the area-minimizers amongst all surfaces with a given boundary and a given genus must also minimize a functional that depends on a quadratic expression of the extrinsic curvature. Both these assertions follow easily from our techniques. Recall that we are denoting the extension of A to this setting by R. Proposition 5.2. Let γ be a C 3,α closed curve in ∂ M which bounds in M. Then the infimum of R(Y ), where Y ranges over the set of all C 3,α surfaces with ∂Y = γ which intersect ∂ M orthogonally is attained only by absolutely area-minimizing surfaces. Proof. Let Y be any C 3,α surface which intersects ∂ M at ∂Y = γ . We must show that if R(Y ) ≤ R(Y  ) for all other such surfaces Y  , then Y is absolutely area-minimizing. Fix a decreasing sequence j  0 and let Y j = Y ∩ {x ≥ j } and γ j = ∂Y j . Let Y j denote an area minimizing surface with ∂Y j = γ j . Following the original existence proof by Anderson, possibly after passing to a subsequence, we may assume that Y j → Y  ,

(5.16)

where Y  is properly embedded and area-minimizing with ∂Y  = γ . Standard results imply that the convergence is in C ∞ in the interior, and also C 3,α up to the boundary. This may be proved by an argument very similar to that used in establishing properness in Proposition 4.3. Indeed, if the convergence were not C 1 , we would be able to take a suitable rescaling and obtain a limiting surface which has boundary a straight line in R2 = ∂H3 but which is not a totally geodesic plane, which would be a contradiction. Knowing this, if the convergence were not C 3,α , we could again rescale and extract a limiting surface which converges to a totally geodesic plane, but not smoothly, which also contradicts standard convergence results for minimal surfaces.

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Denote the genus of Y  by k, so Y  minimizes A in ∪s Ms (M) amongst all minimal surfaces with the same boundary and of arbitrary genus. We now claim that Area (Y j ) ≥ Area (Y j ) =

1 length(γ ) + A(Y  ) + o(1). j

(5.17)

The first inequality is by definition. As for the second equality, we proceed in steps. Since  := Y  ∩ {x ≤ ρ} Y j → Y  in C 3,α , there exists ρ > 0 so that each of the annuli Y j \Y j,ρ j    and Y \Yρ := Y ∩ {x ≤ ρ} is a normal graph over some portion of the vertical cylinder  is the graph of a function u ,  of height ρ above γ . In particular, suppose that Y j \Y j,ρ j   defined on some band  j = { j ≤ x ≤ ρ} and Y \Yρ is the graph of a function u defined on the band {0 ≤ x ≤ ρ}. Clearly the areas of the portions where x ≥ ρ converge, i.e.  ) → Area (Y  ). Thus if we denote by Y  j, ρ, the portion of Y  that lies Area (Y j,ρ j ρ below the hyperplane x = ρ and above the hyperplane x = j it suffices to show that lim

j→∞



   Area (Y j,ρ, − Area (Y \Y ) = 0. j j,ρ j

This follows by direct computation. First write u j (x, s) = x 2 u˜ j (x, s), u(x, s) = x 2 u(x, ˜ s), so that u˜ j → u˜ in C 1 . Denote by Ju , Ju j the Jacobians for parametrizations of these surfaces in the (s, x) coordinate charts as in Sect. 2.3. Using the formulæ from that section we compute that, with constants independent of j,    

  ρ  Ju − Ju     j    = − Area (Y \Y ) dsd x   Area (Y j,ρ,   j j,ρ j 2  j γ  x  ρ  

 1  2  ≤C u s − u 2j,s − (1 − κu)2 (1 + u 2x ) − (1 − κu 2j )(1 + u 2j,x )  dsd x 2 x γ j  ρ ≤ C |u˜ j − u| ˜ + |u˜ j,s − u˜ s | + |u˜ j,x − u˜ x | dsd x → 0 (5.18) j

γ

as j → ∞. This proves (5.17). Subtracting −1 j length (γ ) from (5.17) and passing to the limit gives R(Y ) ≥ R(Y  ) = A(Y  ). The second step of the proof is to show that if R(Y ) = R(Y  ) then Y must be an area-minimizer. By Proposition 5.1, it suffices to show that Y is minimal. If it is not, then its mean curvature H is nonvanishing in some open set. We can then perturb Y locally, in some small ball Y ∩ Br ( p), to a new surface Y  which is also smooth, but has smaller area in this ball. Clearly the areas of the truncations to x ≥ satisfy Area (Y  ) < Area (Y  ) for small, which in turn implies that R(Y ) < R(Y  ), contrary to what we proved above.  

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6. First and Second Variations of Renormalized Area We now begin the variational analysis of the renormalized area functional A on each of the moduli spaces Mk (M), as well as for its extension to the unconstrained spaces k (M). (Recall the definitions of Mk (M), M k (M) from the first paragraph in Sect. 4; M in particular M is a convex, co-compact hyperbolic 3-manifold.) The first variation formula for A on Mk is formally analogous to the corresponding first variation formula for the renormalized volume of Poincaré-Einstein metrics in even dimensions; this formula appears in a paper by Anderson [5], see also Albin [1] for a simpler approach. A formula for the second variation of renormalized volume does not seem to have been computed in the Poincaré-Einstein setting. 6.1. The first variation. We shall compute the first variation of A at any Y ∈ Mk (M); slightly more generally, we compute the first variation of the extended functional R k (M). Actually, we compute DR only applied to compactly supported at any Y ∈ M k (M). perturbations, and show this is not well-defined for arbitrary variations in M As before, fix a special boundary defining function x on M, and for any Y ∈ Mk (M) let u 3 denote the free third order term in the expansion of the graph function u of Y with respect to x over its vertical cylinder. Theorem 6.1. Fix any Y ∈ Mk (M) with ∂Y = γ . Let Yt denote a smooth one-paramk (M), ∂Yt = γt , with Y0 = Y (so the surface Y0 is minimal but the eter curve in M surfaces Yt , t = 0, need not be). Write Yt as a normal graph Yφ over Y0 = Y via Yt = {Ft ( p) = exp p (φt ( p)ν( p)) : p ∈ Y }, where φ0 = 0. Setting φ˙ =



d  dt φt t=0 ,

then φ˙ ∼ x −1 φ˙ 0 + · · ·, and we have   d  φ˙ 0 u 3 ds. A(Y ) = −3 t dt t=0 γ

On the other hand, if Y ∈  Mk (M) and φt = 0 outside some compact set K ⊂ Y , then   d  ˙ A. A(Y ) = 2 H φd t dt  Y

t=0

Proof. By virtue of (3.8), we have     d  1 d  2 2 A(Y ) = (tr k ) − |k | d A t t t t , dt t=0 2 dt t=0 Yt

(6.19)

where kt and d At denote the second fundamental form and area form on Yt (tr kt denotes the trace of the second fundamental form of Yt ). In order to compute this, introduce the following notation. Fix any point p ∈ Y and choose an orthonormal moving frame {e1 , e2 } on Y near p, as well as a unit normal vector field ν, so that e1 and e2 are the principal directions and ∇ei e j = −κi δi j ν at p (the κi are the principal curvatures of Y ). Define ηi (t) = (Ft )∗ (ei ), so ηi (0) = ei but {η1 (t), η2 (t)} is no longer orthonormal when t = 0. We define gi j (t) = ηi (t), η j (t)

and

ki j (t) = −∇ηi (t) η j (t), ν(t) ,

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where ν(t) is the unit vector orthogonal to both η1 (t) and η2 (t) with ν(0) = ν = e3 . ˙ when t = 0. Finally, write T = F∗ ∂t , so T = φν In the following, all terms are to be computed eventually at p. We first compute that     ˙ = φκ ˙ i ei + φ˙ i ν, ∇T ηi |t=0 = ∇T ∇ηi F t=0 = ∇ei ∇T F t=0 = ∇ei φν which yields immediately  ˙ i ei + φ˙ i ν, e j + ei , φκ ˙ j e j + φ˙ j ν = 2φk ˙ ij, T gi j (t)t=0 = φκ ˙ i j . It is also since ki j = κi δi j in this frame. This also implies that T g i j (t)|t=0 = −2φk standard that  d  ˙ ki j ) d A 0 . F ∗ (d At ) = φ(tr dt  t=0

Finally,

   T ki j (t)t=0 = −T ∇ηi η j , ν = − ∇T ∇ηi η j , ν + ∇ηi η j , ∇T ν t=0 .

By assumption, ∇ei e j is orthogonal to Y , while ∇ν ν is tangential, so the second term on the right vanishes. By definition of the curvature tensor, the other term equals ˙ j e j + φ˙ j ν), ν − φR(ν, ˙ −∇ηi ∇T η j + ∇[T,ηi ] η j + R(T, ηi )η j , ν = −∇ei (φκ ei )e j , ν . Observe that ˙ = φκ ˙ i ei + φ˙ i ν − φ˙ i ν − φκ ˙ i ei = 0, [T, ηi ] = ∇T ei − ∇ei (φν) so expanding out yields that  ˙ i j + φ(k ˙ 2j δi j − φ˙ i j + R3i3 j = −(∇ 2 φ) ˙ ◦ k)i j + R3i3 j . T ki j t=0 = φκ Putting these formulæ together, we compute that   

d  1 2 2 2˙ φ −∇ k i j + φ˙ tr k d A (tr k ) − |k | d A = − t t t i j dt t=0 2 Yt Y   ˙ 3i3 i tr k − R3i3 j k i j ) d A. + (tr k)3 − 3|k|2 (tr k) + 2tr (k ◦ k ◦ k) d A + 2 φ(R Y

Y

(6.20) (Here tr (k ◦ k ◦ k) = g i p g jq g r kir k pj kq .) A simple calculation using the principal curvature decomposition shows that (tr k)3 − 3|k|2 (tr k) + 2tr (k ◦ k ◦ k) = 0, while since M is hyperbolic, R3i3 i tr k − R3i3 j k i j = −tr k. This proves     1 d  2 2 2˙ ij ˙ tr k d A + 2 φ˙ tr k d A. φ (tr k ) − |k | d A = − −∇ k +  φ t t t i j dt t=0 2 Yt Y Y (6.21)

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   The next step is to evaluate Y −∇ 2 φ˙ i j k i j +  φ˙ tr k d A. To do this, we integrate by parts on Y = Y ∩ {x ≥ } to get 

˙ k − φ˙ tr k d A ∇ 2 φ, − Y     j ˙ n tr k − k(∇ φ, ˙ n) ds, (6.22) = ∇ φ, φ˙ i (∇ j ki j − ∇ i k j ) d A + γ

Y

where γ = ∂Y and n is the g-unit normal in Y to γ . The contracted Codazzi equation states that ∇ j ki j = ∇i tr k + Ricνi , but in fact Ricνi = −2gνi = 0 since the i index refers to a vector tangent to Y . Hence k (M) then since φ˙ is compactly supported, only the boundary terms remain. If Y ∈ M these boundary terms vanish. If Y ∈ Mk (M), however, then tr k = 0 so the first part of the integrand in the boundary integral vanishes and we only have to evaluate the second one. To do this, revert to the (s, x) coordinates introduced in Sect. 2. In terms of these, the expression becomes  φ˙ i g i j k j n  ds. γ

To calculate this, we first note that the unit normal n = n 1 ∂s + n 2 ∂x has coefficients which satisfy n 1 = O(x 3 ), n 2 = 1 + O(x 2 ). Thus we may as well set  = 2 and n 2 = 1. Note also that g i j = x 2 g i j and ds = x −1 ds, where s is arclength on γ with respect to g. (This differs slightly from our earlier convention.) The terms in the integrand are thus

φ˙ 1 x 2 (g 11 k12 + g 12 k22 ) + φ˙ 2 x 2 (g 21 k12 + g 22 k22 ) ds. Finally, recall that φ˙ ∼ x −1 φ˙ 0 and use the expansions for the ki j and gi j to deduce that this reduces to  φ˙ 0 u 3 ds + O( ). −3 γ

This completes the proof.

 

From this formula we deduce Corollary 6.1. Suppose that Y ∈ Mk (M) is a critical point for A. If Y is a nondegenerate element in Mk (i.e. it has no decaying Jacobi fields), then the coefficient u 3 in its boundary expansion vanishes identically. In general, if Y is degenerate, then we may only conclude that u 3 is orthogonal to the finite dimensional space spanned by leading coefficients ψ0 of Jacobi fields ψ ∼ ψ0 x −1 + ψ1 x 0 + · · · along Y . This is equivalent to the statement that u 3 lies in the (finite dimensional) span of all leading coefficients ψ3 of decaying Jacobi fields ψ ∼ ψ3 x 2 + · · · on Y . k (M) is critical for R, then Y is necessarily minimal, and u 3 = 0 regardless If Y ∈ M of whether or not Y is degenerate in Mk (M). We remark that it would be quite interesting to know if there exist degenerate critical points of A.

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6.2. The second variation. We now derive the second variation formula for A at a general element Y ∈ Mk . The calculation is more involved than for the first variation formula, but is actually fairly elementary since it relies primarily on some detailed trigonometric calculations. Before stating the result, we set up the basic notation. We use the same language as in Theorem 6.1. Let Yt be any 1-parameter family of surfaces in Mk (M), with corresponding family of boundary curves γt in ∂ M. The calculations are local, so we assume that M = H3 and we work in standard upper half-space coordinates. Let N t denote the inward-pointing unit normal to γt with respect to the Euclidean metric on R2 . Let (s, v) be Fermi coordinates around γ0 , i.e. corresponding to the map (s, v) → γ0 (s) + v N 0 (s). Then γt can be written as a normal graph, v = ψt (s), where ψ0 ≡ 0 and ψ˙ N 0 is the normal variation vector field. Similarly, introduce Fermi coordinates for the Euclidean metric around Y0 , using the inward unit normal ν; the cylindrical coordinates (s, x) parametrize Y0 , and we write Yt as a normal graph v = φt (s, x) using the g-unit normal ˙ so φ˙ is a solution of the Jacobi equation L Y0 φ˙ = 0. ν. The variation vector field is φν, The regularity theory for this equation shows that it has an expansion ˙ x) ∼ φ˙ 0 (s) + φ˙ 2 (s)x 2 + φ˙ 3 (s)x 3 + · · · , φ(s, ˙ If Y0 is nondegenerate, the coefficient of x 3 is determined by a where φ˙ 0 (s) = ψ(s). global Dirichlet-to-Neumann map in terms of the leading coefficient, which we write as 1  φ˙ 3 = ∂x3  φ˙ = DY0 φ˙ 0 . x=0 6 Theorem 6.2. The second variation of A along the family Yt is given by   2  ˙ ˙ φ˙ 0 φ˙ 3 ds, D A (φ, φ) = −3 Y0

γ0

or equivalently, when Y0 is nondegenerate,  φ˙ 0 DY0 φ˙ 0 ds. −3 γ0

Proof. Let us begin by setting up some notation. Write each Yt as a horizontal graph over the vertical cylinder t on the curve γt , i.e. via (γt (s), x) + u t (s, x)N t . Through the remainder of this proof, subscripts refer only to t-dependence but not derivatives, and we often write w˙ for ∂t w. Set 1 κt (s)x 2 + u 3,t (s)x 3 + O(x 3+α ). 2 The unit tangent and normal vectors of γt are u t (s, x) =

1 Tt =  2 (∂s φt ) + (1 − κ0 φt )2 1 Nt =  2 (∂s φt ) + (1 − κ0 φt )2

  (1 − κ0 φt )T 0 + ∂s φt N 0 ,   −∂s φt T 0 + (1 − κ0 φt )N 0 ,

so the component along N t of the variation vector field φ˙ N 0 is

−1/2 1 − κ0 φt N t , φ˙ N 0 = φ˙ t  = φ˙ t 1 + (∂s φt /(1 − κ0 φt )2 . (∂s φt )2 + (1 − κ0 φt )2

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This is equal to φ˙ t (1 + O(t 2 )). Hence the first variation of A at Yt is given by  d A(Yt ) = −3 u 3,t φ˙ t (1 + O(t 2 )) d s¯ . dt γt Differentiating once more at t = 0 yields      d2  u˙ 3,0 φ˙ 0 + u 3,0 φ¨ 0 d s¯ . A(Yt ) = −3 2 dt γ0 t=0 The second term in the integrand is an artifact of the parametrization. Indeed, recall that for the composition of any two functions, A(s) and s = Y (t), (A(Y (t)) = A (Y0 )(Y  )2 + A (Y0 )Y  , but the Hessian of A corresponds only to the first term on the right. In other words,   2  ˙ ˙ D A (φ, φ) = −3 u˙ 3,0 φ˙ 0 d s¯ . (6.23) Y0

γ0

The key point is to relate φ˙ 3 to u˙ 3,0 . The long calculation that follows involves expressing φt in terms of u t and then calculating ∂t ∂x3 at (x, t) = (0, 0). The relationship between these two terms arises as follows. Fix any P ∈ Y0 with x coordinate x(P) = x0 sufficiently small. Let λ˜ be the line emanating from P ∈ Y0 in the direction ν(P). Denote by P˜ and R˜ its points of intersection with t and Yt , respectively. Note ˜ one encounters that assuming γ0 is convex near P, then moving from P along the line λ, ˜ we orient coordinates so that all quantities are positive in this situation. first P˜ then R; By definition, φt (P) is the distance from Y0 to Yt along λ˜ , and it can be decomposed as ˜ which we call ˜ and χ, the sum of two signed distances, from P to P˜ and from P˜ to R, ˜ respectively: ˜ = |P P| ˜ + | P˜ R| ˜ = ˜ + χ˜ . φt = |P R| We show how to relate each of the two quantities ˜ and χ˜ to u t in turn. ˜ Let x0 be the x coordinate of the point P and α the angle between First consider . λ˜ and the plane x = x0 . From (2.5) the orthogonal projection of ν(P) onto this plane is spanned by −∂s u T 0 + (1 − κ0 u 0 )N 0 , and ∂x u (1 − κ0 u 0 ) sin α =  2 (∂s u) + (1 − κ0 u 0 )2 (1 + (∂x u 0 )2 )

⇒

α = κ0 x + 3u 3,0 x 2 + O(x 3 ).

Now let λ1 denote the line in the plane x = x0 from P in the direction of this projection of ν, and P 1 = λ1 ∩ t . Setting 1 = |P P 1 | (interpreted as a signed distance), since P P 1 P˜ is a right triangle, we have ˜ =

1 1 1 = 1 (1 + α 2 + · · · ) = 1 (1 + κ02 x 2 + 3κ0 u 3,0 x 3 + · · · ). cos α 2 2

We shall prove below that     = ψt , ∂x 1  1  x=0

x=0

= 0, and

  ∂x3 1 

x=0

= 6u 3,0 + O(t 2 ),

(6.24)

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and hence, after some calculation,

  ∂t ∂x3 ˜

x=t=0

= 18κ0 u 3,0 φ˙ 0 .

(6.25)

To prove (6.24), we examine 1 more closely. Consider the (very thin) triangle T in the plane x = x0 with sides the segments of the line λ1 , the line λ¯ emanating from P with direction N 0 , and the line tangent to γt at the point of intersection λ¯ ∩ γt . Let 2 , ¯ and ˆ denote the lengths of these sides, respectively, and let β and ω denote the angles between λ1 ∩ λ¯ and λ¯ ∩ γt . Note that 1 −∂s uT 0 + (1 − κ0 u 0 )N 0 , N 0 cos β =  2 (∂s u 0 ) + (1 − κ0 u 0 )2  2

−1/2 1 1 ∂s κ0 x 2 + ∂s u 3,0 x 3 + · · · + · · · =1− = 1 + (∂s u 0 /(1 − κ0 u 0 ))2 2 2 1 ⇒ β = ∂s κ0 x 2 + ∂s u 3,0 x 3 + · · · . (6.26) 2 In addition, since π/2 − ω is the angle between N 0 and N t ,

π 1 π − ω = 1 − (∂s φt + O(t 2 ))2 + · · · ⇒ ω = − ∂s φt + O(t 2 ). cos 2 2 2 The law of sines for this triangle is the set of equalities ˆ 2 ¯ = = . sin β sin ω sin(π − β − ω) Since ω is bounded away from 0, the first equality and (6.26) imply that ˆ = O(x 2 ), which in turn shows that 2 = 1 + O(x 4 ). Thus for computing third derivatives, we may as well work with 2 . The second equality yields

¯ sin ω = 1 − tan β cot ω + O(x 4 ) . 2 = ¯ sin(π − β − ω) cos β Recalling that ¯ = u 0 + ψt , and using the expressions for β and ω above, we obtain         1 2 4 2 2 ∂s κ0 x + · · · ∂s φt + O(t )  = (ψt + u 0 ) 1 + O(x ) 1 − 2 1 = ψt + κ0 x 2 + u 3,0 x 3 + O(t 2 ) + O(x 4 ), 2 which yields (6.24). We turn to the computation of χ˜ . The idea is much the same. Let (s0 , x0 ) denote the cylindrical coordinates of P, i.e. as a horizontal graph over the cylinder 0 , and (s0 , x0 ) the cylindrical coordinates of the point R˜ = λ˜ ∩ Yt , when written as a horizontal graph over t . Since R˜ = P + φt (s, x)ν and ν = N 0 − κ0 x∂x + O(x 2 ), we obtain s0 = s0 + O(x 2 ),

x0 = x0 − κ0 xφt + O(x 2 ). u t (s0 , x0 )

(6.27)

We shall compute the horizontal displacement of Yt from t in the plane x = x0 . For this, let γ˜t = t ∩ {x = x0 }, which is just the vertical translate of γt into this plane. Abusing notation slightly, write P˜ for the point on γ˜t which is directly below the intersection of λ˜ ∩ t (which had been called P˜ above). Finally, let Q˜ denote the ˜ i.e. so that R˜ = Q˜ + cN t for some constant c. point on γ˜t which is nearest to R,

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˜ The sides P˜ R˜ and R˜ Q˜ have lengths χ˜ cos α and Consider the thin triangle P˜ Q˜ R. ˜ We find that u t (s0 , x0 ), respectively. Let η and ζ be the angles at the vertices P˜ and Q. η=

π + O(t) + O(x 2 ), 2

ζ =

π + O(x 2 ). 2

By the law of sines again, u t (s0 , x0 )/ sin η = χ˜ / sin ζ , so χ˜ cos α = u t (s0 , x0 )(1 + O(x 4 ))(1 + O((t + x 2 )2 )) = u t (s0 , x0 ) + O(t 2 ) + O(x 4 ). Expanding in a Taylor series, using (6.27) and recalling that u t = O(x 2 ), we find that u t (s0 , x0 ) = u t (s0 , x0 ) + ∂s u t (s0 , x0 )O(x 2 ) + ∂x u t (s0 , x0 )(−κ0 xφt + O(x 2 )) + O(t 2 + x 4 ) = ut (s0 , x0 ) − κ0 κt φt x 2 − 3κ0 u 3,t φt x 3 + O(t 2 + x 4 ) 1 κt − κ0 κt φt x 2 + (u 3,t − 3κ0 u 3,t φt )x 3 + · · · ; = 2 also, as before, 1/ cos α = 1 + O(x 2 ), so altogether we obtain   ˙ = 6u˙ 3,0 − 18κ0 u 3,0 ψ. ∂t ∂x3 χ˜  t=x=0

We have now proved that     ∂t ∂x3 φt  = ∂t ∂x3 (˜ + χ˜ ) x=t=0

x=t=0

˙ = 6DY0 (φ˙ 0 ) = 6u˙ 3,0 + 18κ0 u 3,0 ψ.

Inserting this into (6.23) yields the formulæ in the statement of the theorem.

 

7. Critical Points of A in H3 In this section we prove that the only nondegenerate critical points of renormalized area for proper minimal surfaces in all of H3 are the totally geodesic planes, the boundary curves of which are circles. The proof requires a preliminary geometric lemma about osculating circles of plane curves, which is perhaps of independent interest, and then proceeds via a refined version of the asymptotic maximum principle. Recall that the osculating circle C at a point p ∈ γ is a circle which makes second order contact with γ at that point; its curvature, the inverse of its radius, is therefore the same as that for γ at this point of intersection. By inscribed we simply mean that C remains entirely within the closure of one of the two components of S 2 \γ . Proposition 7.1. Let γ be a C 2 embedded loop in S 2 = ∂H3 and  one component of S 2 \γ . Then there exists a point p ∈ γ so that the osculating curve of γ at p lies in . Remark 7.1. As the proof makes clear, there may be many such points. Of course, we could equally well have replaced  by the other component of S 2 \γ . Proof. We begin with a few elementary observations. First, the entire question is invariant under Möbius transformations, hence we may freely apply such transformations to reduce the problem to one that is easier to visualize. Second, if C  is any circle inscribed in  which is locally maximal (in the sense that for any continuous family of inscribed circles C  ( ) with C  (0) = C  , the family of radii r  ( ) reaches a local maximum at

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= 0) then necessarily either C  is tangent to γ at two or more distinct points, or else C  is tangent to γ at a single point and is the osculating circle there. The reason is that if there is only one point of contact, p = C  ∩ γ , then the curvature of the circle 1/r  is greater than or equal to κ( p), the curvature of γ at p. If this inequality is strict and there are no other points of contact, then we could increase the radius of C  slightly while keeping it inside . Thirdly, and slightly more complicated, we claim that if C  arises as a limit of inscribed circles C j such that each C j ∩ γ contains at least two points P j = Q j and dist (P j , Q j ) → 0 as j → ∞, then the limit C  is necessarily an inscribed osculating circle. To see this, choose for each j a Möbius transformation F j which carries C j into a fixed straight line in R2 , say the y1 -axis. We also suppose that dist (F j (P j ), F j (Q j )) = dist (P j , Q j ), so that F j does not diverge, that F j carries  to the lower half-plane, and that lim F j (P j ) = lim F j (Q j ) is the origin. Each curve F j (γ ) lies in the upper half-plane and is tangent to the y1 -axis at two points which are converging to the origin. Clearly, the curve must be a graph over the axis between these two points for j large enough, say of some function f j . By the intermediate value theorem, there is a sequence of points t j → 0 such that f j (t j ) = 0. Taking a limit, we see that the limiting curve is flat to second order at the origin and lies entirely in the closed upper half-plane, which proves the claim. We can now proceed with the proof. Let K denote the set of pairs (P, Q) ∈ γ × γ , P = Q such that there is an inscribed circle which is tangent to γ at these two points (and possibly other points as well). By the second remark above, if this set were empty, then there would have to be an inscribed osculating circle already. So assume K = ∅. We claim that the closure of K must intersect the diagonal, which by the third remark above would produce an inscribed osculating circle: If it did not, then K would be compact in γ × γ , and hence there would be a point (P  , Q  ) = (γ (t1 ), γ (t2 )) such that the difference in parameter values |t2 − t1 | is minimal (for some fixed parametrization of the curve). Let C  be the corresponding circle. Conformally transform so that C  is the y1 -axis. Then the images of P  and Q  lie on this axis and the transformed curve lies entirely on or above the axis. If the image of P  is the left-most point of tangency of the curve with the y1 -axis, fix another point P  just to the right for which the horizontal component of the downward pointing normal is positive. There is a maximal radius for which a circle tangent to the curve at P  remains in the component of the lower half-plane, and this circle is obviously tangent to the curve at another point Q  to the left of the image of Q  . This shows the existence of another pair (P  , Q  ) strictly between the pair (P  , Q  ) for which there is an inscribed circle tangent at these two points, which is a contradiction. This finishes the proof.   Now we turn to the main result of this section. Theorem 7.1. Let Y ∈ Mk (H3 ) be nondegenerate and suppose that ∂Y = γ is C 3,α and connected, and Y is a critical point for A. Then ∂Y is a round circle and Y is a totally geodesic disk. Remark 7.2. Using Corollary 6.1, the proof below actually shows that if Y is critical for the extended renormalized area functional R, then the same conclusion holds. Proof. First, by the results in the previous section, we know that Y must be a minimal surface and also that the formally undetermined term u 3 in its expansion must vanish. Using the result about osculating circles, and applying a conformal transformation, we reduce to the case where γ is a closed curve in R2 which lies entirely in the closed

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upper half-plane, and which is tangent to second order to the y1 -axis at the origin. We now write some neighbourhood of 0 ∈ Y as a horizontal graph over the (y1 , x)-plane, i.e. Y = {y2 = u(y1 , x)} for |y1 | < δ, x < δ. Set s = y1 for simplicity. This function satisfies the minimal surface equation, which in this coordinate system takes the form F(u) = (1 + u 2x )u ss − 2u s u x u sx + (1 + u 2s )u x x −

2(1 + u 2x + u 2s ) u x = 0. (7.28) x

(Note that this is simply the specialization of (2.6) when the base of the cylinder  is flat, so κ = 0 and w = 1.) Furthermore, since Y lies to one side of the (s, x) plane, u ≥ 0 everywhere, and it has the asymptotic expansion u(s, x) ∼ a(s)x 2 + O(x 3+α ).

(7.29)

The absence of the x 3 term is because Y is critical for A; furthermore, a(0) is the one half the curvature of γ at 0, and hence because it osculates the line there, a(0) = 0. Now choose any β ∈ (0, α) and define u c = u − cx 3+β for c > 0. By the various properties above, if we fix δ then choose c sufficiently small, the function u c ≥ 0 on all four sides of the rectangle |s| ≤ δ, 0 ≤ x ≤ δ. However, using (7.29) and the fact that β < α, we also have that u c (0, x) < 0 for x sufficiently small. This means that the minimum of u c is achieved somewhere strictly inside this rectangle, and of course at that point (sc , xc ), we have F(u c )(sc , xc ) ≥ 0. On the other hand, we compute that F(u − cx 3+β ) = F(u) − L u (cx 3+β ) + O(cx 6+2β ); the first term on the right vanishes, while the operator in the second term is the linearization of F at u, L u v = (1 + u 2x )vss − 2u s u x vsx + (1 + u 2s )vx x + 2(u x u ss − u s u sx )vx + vs + 2(u s u x x − u x u sx )vs −

2(1 + u 2x + u 2s ) 4u 2 4u s u x vx − x vx − vs . x x x

By (7.29) and scale-invariant Schauder estimates, x −2 |u|+ x −1 |∇u|+|∇ 2 u| ≤ K . Hence altogether F(u c ) ≤ −cβ(3 + β)x 1+β + K cx 2+β , where K is independent of c and δ. Choosing δ sufficiently small, we can ensure that F(u c ) < 0 everywhere in this box, which contradicts that it is positive at (sc , xc ). This proves the theorem.   8. Connections with the Willmore Functional It is not particularly surprising that the functional A is connected with the Willmore functional, which by definition is the total integral of the square of mean curvature. In this final section we explore some of these relationships. Fixing a special bdf x, then the expansion of g = x 2 g has only even powers, so its natural extension to a Z2 -invariant metric on the double of M across its boundary k (M) can also be doubled to a closed C 2,1 is smooth; furthermore, any surface Y ∈ M surface. We denote these doubles by 2M and 2Y , respectively.

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Proposition 8.1. If Y ∈ Mk (M), then 1 A(Y ) = − W(2Y ). 2 Proof. We have already noted that | k|2 d A g = | k|2 d A g . Next, observe that    2

 κ 1 − κ 2 2   = 2 H 2 − κ 1κ 2 ,  k  = 2  2  where the κ j are the principal curvatures of Y with respect to g. Thus finally 

2 A(Y ) = −2π χ (Y ) − H − κ 1 κ 2 d Ag Y 

1 1 2 = −2π χ (Y ) − H − κ 1 κ 2 d A g = −2π χ (Y ) − W(2Y ) + 2π χ (Y ), 2 2Y 2 since χ (2Y ) = 2χ (Y ).

 

On the other hand, it does not seem to be the case that the extended functional R k (M) has any simple connection with the Willmore functional. Nonetheless, this on M result recasts the program of finding extremals of A in a different and more classical light. Let us conclude by explaining this a bit further. The conformal invariance of the Willmore functional makes its variational theory quite challenging. The existence of minimizers of the Willmore functional for closed surfaces in R3 with genus less than or equal to some fixed constant is proved in [37], and this result is sharpened in [7], where it is shown how to prevent a ‘drop of genus’ in these minimization arguments. We have just shown that the problem of finding extrema, and in particular, maxima, for A on some given moduli space Mk (M) is equivalent to finding constrained extrema (in particular, minima) of W on 2M with respect to the metric g, within the restricted class of surfaces which are invariant under the Z2 involution and which are minimal with respect to g. If it were possible to adapt the arguments from [37] to this ambiently curved setting, we could prove the existence of such extrema. This may well be subtle, and the strengthened result from [7] may not be available, even when M = H3 . Indeed, consider that we have proved that there are no nondegenerate critical points for A on Mk (H3 ) when k > 0. This indicates that any extremizing sequence Y j , e.g. one for which A(Y j ) tends to the supremum, probably does not converge to a surface of the same genus. Acknowledgements. The first author is very grateful to Chris Herzog, Juan Maldacena and A. M. Polyakov for useful conversations. The second author wishes to thank Joel Hass, Steve Kerckhoff and particularly Brian White for helpful conversations.

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Commun. Math. Phys. 297, 653–686 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1053-4

Communications in

Mathematical Physics

N -Vortex Equilibria for Ideal Fluids in Bounded Planar Domains and New Nodal Solutions of the sinh-Poisson and the Lane-Emden-Fowler Equations Thomas Bartsch1 , Angela Pistoia2, , Tobias Weth3 1 Mathematisches Institut, Justus-Liebig-Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany.

E-mail: [email protected]

2 Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16,

00161 Roma, Italy. E-mail: [email protected]

3 Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany.

E-mail: [email protected] Received: 26 September 2008 / Accepted: 9 February 2010 Published online: 18 May 2010 – © Springer-Verlag 2010

Abstract: We prove the existence of equilibria of the N -vortex Hamiltonian in a bounded domain  ⊂ R2 , which is not necessarily simply connected. On an arbitrary bounded domain we obtain new equilibria for N = 3 or N = 4. If  has an axial symmetry we obtain a symmetric equilibrium for each N ∈ N. We also obtain new stream functions solving the sinh-Poisson equation −ψ = ρ sinh ψ in  with Dirichlet boundary conditions for ρ > 0 small. The stream function ψρ induces a stationary velocity field vρ solving the Euler equation in . On an arbitrary bounded domain we obtain velocitiy fields having three or four counter-rotating vortices. If  has an axial symmetry we obtain for each N a velocity field vρ that has a chain of N counter-rotating vortices, analogous to the Mallier-Maslowe row of counter-rotating vortices in the plane. Our methods also yield new nodal solutions for other semilinear Dirichlet problems, in particular for the Lane-Emden-Fowler equation −u = |u| p−1 u in  with p large. Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . Some Properties of the Green’s Function The Symmetric Case . . . . . . . . . . . Avoiding the Domain’s Boundary . . . . Avoiding Collisions . . . . . . . . . . . Three Vortices . . . . . . . . . . . . . . Four Vortices . . . . . . . . . . . . . . . The sinh-Poisson Equation . . . . . . . . The Lane-Emden-Fowler Equation . . .

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654 659 663 666 672 678 680 681 683

 Supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

654

T. Bartsch, A. Pistoia, T. Weth

1. Introduction Point-vortex dynamics for ideal fluids in planar domains is a classical topic in mathematical fluid dynamics, beginning with the fundamental work of Helmholtz [26] and Kirchhoff [27]. It also has some practical interest because a two-dimensional model seems to be valid approximately in a number of important applications, e. g. in the study of large scale atmospheric or oceanographic turbulences or the study of flows on soap films. The theory of two-dimensional turbulence and point-vortex dynamics has therefore attracted great attention; we just refer to the surveys and monographs [2,28,31,36,41] and the references therein. The dynamics of N point-vortices x1 , . . . , x N ∈  on a bounded or unbounded domain  in the plane is governed by a Hamiltonian system ⎧ ∂H d xi1 ⎪ ⎪ ⎨ i dt = ∂ x (x1 , . . . , x N ); i,2 ⎪ ∂H d xi2 ⎪ ⎩ i =− (x1 , . . . , x N ); dt ∂ xi1

i = 1, . . . , N .

(1.1)

Here i , i = 1, . . . , N , is the strength of the i th vortex xi . If i > 0 (resp. i < 0) then the point vortex induces an anti-clockwise (resp. clockwise) particle motion. The Hamiltonian H for the dynamics on the entire plane  = R2 is given by H (x1 , . . . , x N ) = −

N 1  i  j log |xi − x j |. 2π i, j=1

(1.2)

i = j

There are many results for (1.1) in the plane and we just refer to the references mentioned above. Concerning the existence and the classification of (relative) equilibria (vortex crystals) of (1.1) in the plane, the reader may find recent results in [3,37]. Compared with the large number of papers on point-vortex dynamics in the plane there is much less work dealing with domains   R2 , where boundary effects play an important role. Somewhat scattered in the literature, a number of papers deal with point-vortex dynamics in special domains like a strip or a semidisk; we refer to [36, Chap. 3] and the references therein. Our goal in this paper is to contribute to this problem on a general bounded domain  ⊂ R2 with C 3 -boundary. In particular, the domain need neither be convex nor symmetric, although we also have a result if  is symmetric with respect to reflection at a line through it. Let G(x, y) = g(x, y) −

1 log |x − y| 2π

be the Green’s function of the Dirichlet Laplacian in . Here g :  ×  → R is its regular part and we write h :  → R, h(x) = g(x, x), for the Robin’s function. Then the Hamiltonian is given by the function H (x1 , . . . , x N ) =

N  i=1

i2 h(xi ) +

 i= j

i  j G(xi , x j ),

(1.3)

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

655

which is a generalization of the classical Kirchhoff-Routh path function (see [22, Sect. 7] for a rigorous derivation of this Hamiltonian). H is defined on the configuration space   F N  = (x1 , . . . , x N ) ∈  N : xi = x j for i = j . Even the existence of equilibria of (1.1), i. e. critical points of H , is highly nontrivial. This is due to the fact that F N  is not a complete manifold being a bounded open subset of R2N , and that H (x) may remain bounded as x → ∂(F N ). Therefore standard methods from critical point theory do not apply, in particular it is not sufficient to prove the Palais-Smale condition for H and to find a linking. The single vortex case N = 1 is exceptional. Here the Hamiltonian is just a multiple of the Robin function h, so that a single vortex moves along the level sets of h and equilibria are merely the critical points of h. We refer to [22,25] for a discussion of h in terms of single vortex motion and further properties of h. If N > 1, all i = 1 and  is not simply connected, it has been proved in [16] that H has a critical point. This case is somewhat easier because one may set H (x) = +∞ for those x = (x1 , . . . , xk ) satisfying xi = x j for some i = j. In [18] special contractible, dumbbell shaped domains are constructed where H with i = 1, i = 1, . . . , N , has multiple critical points. Concerning the case that some i are positive, others negative we are only aware of the paper [5] which deals mostly with the case N = 2, 1 = −1, 2 = 1. This case is considerably simpler since here H is bounded from above and H (x) → −∞ as x → ∂(F2 ()). In [5] the existence of at least two critical points of H has been obtained for any smooth bounded domain. More critical points exist if  has topology, depending on the Lusternik-Schnirelman category of the configuration space F2 /(x1 , x2 ) ∼ (x2 , x1 ) of unordered pairs in . Moreover, if  is symmetric with respect to the reflections at both coordinate axes and contains the origin, then H has a critical point x = (x1 , x2 ) ∈ F2  with x2 = −x1 lying on a coordinate axis. In addition, in this symmetric case [5] contains also a result for N = 3, yielding a symmetric critical point (x1 , 0, −x1 ) with x1 on a coordinate axis. In our first result we improve the result from [5] in two directions: We weaken the symmetry hypothesis and we obtain results for arbitrary N ≥ 2. Concerning the symmetry we only require that the domain is symmetric with respect to a line through it, which we assume to be R × {0}. Theorem 1.1. Suppose  is symmetric with respect to the reflection at R × {0} and  ∩ R × {0} = ∅. Then for every N ∈ N, H with i = (−1)i has a critical point (x1∗ , . . . , x N∗ ) with xi∗ = (ti , 0) ∈  ∩ R × {0}, t1 < t2 < · · · < t N . In Theorem 1.1 it is not required that  is simply connected nor that  ∩ R × {0} is connected. In fact, if  ∩ R × {0} consists of several intervals we obtain multiple critical points which are moreover stable under certain perturbations of H . The precise statement of this improved version of Theorem 1.1 is Theorem 3.3 below. It would be interesting to determine the positions of the equilibria for special domains where H is known explicitly. However, even in the simplest case where  is the unit ball in R2 , one has to solve an algebraic system which gets more and more complicated as N increases. Here we can determine the positions of equilibria in some cases: √ • if N = 2 then t2 = −t1 = 5 − 2 ∼ 0.4858, √ 4 • if N = 3 then t2 = 0 and t3 = −t1 = 1/ 5 ∼ 0.6687, • if N = 4 then t3 = −t2 ∼ 0.2696 and t4 = −t1 ∼ 0.8457,

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• if N = 5 then t3 = 0, t4 = −t2 ∼ 0.2558 and t5 = −t1 ∼ 0.7526. The values for N = 4 and N = 5 have been obtained with computer assistance. The problem becomes considerably more difficult on an arbitrary C 3 - bounded domain  ⊂ R2 when N ≥ 3 and some i are positive, others negative. Here we have the following result whose proof turns out to be much harder than the one of Theorem 1.1, though it is only concerned with the cases N = 3 and N = 4. Theorem 1.2. In any bounded domain  ⊂ R2 with C 3 -boundary, H has a critical point if N = 3 or N = 4 and i = (−1)i . Again we point out that it is not required that  is simply connected. The proof of Theorem 1.2 relies on two points: 1. a new linking phenomenon for H ; 2. a careful investigation of H near the boundary of F N () in R2N in order to find relatively compact gradient flow lines. Parts of the proof hold for general N and for general nonzero i , i = 1, . . . , N . In particular, in Sect. 4 we prove in full generality that if H is bounded from above along a flow line of the gradient flow of H , then each component of this trajectory stays away from the boundary of the domain. Other parts of the proof only work in the case where i = (−1)i . However, even in this case a major problem is to exclude possible collisions, i. e. gradient flow lines of H in F N () along which (at least) two components converge towards the same point in . In particular, if N = 4 and three of the i have the same sign, the fourth the opposite sign, then such collisions do occur; see [21]. The same holds for N ≥ 5 and arbitrary (non-constant) sign distributions of the i . We conjecture that Theorem 1.2 holds if the i satisfy  i  j = 0 for every subset J ⊂ {1, . . . , N } with |J | ≥ 3. (1.4) i, j∈J, i= j

Parts of our argument work if (1.4) holds. We plan to come back to this conjecture in future work. Remark 1.3. The proofs yield also information about the Morse index of the solution x ∗ from Theorem 1.1 or Theorem 1.2 as a critical point of H . For instance, in the case N = 3 of Theorem 1.2 the solution has Morse index 5, in the case N = 4 the Morse index is 6, assuming that x ∗ is nondegenerate. Thus, they are saddle points of H and, as equilibria of (1.1) they are unstable. Somewhat surprisingly, the function H appears also when one is interested in solutions of certain nonlinear elliptic problems. In fact, H is a singular limit functional for the sinh-Poisson equation

−ψ = ρ sinh ψ in ; (1.5) ψ =0 on ∂; as ρ → 0. The sinh-Poisson equation arises in several important applications, particularly as the vorticity equation in classical hydrodynamics [10,11,30,32,38]. In this context solutions of (1.5) can be thought of as streamfunctions yielding stationary solutions of the Euler flow in . Our theorems above do not directly apply to find solutions of (1.5) because we need to find critical points of functions F : F N  → R that are close to H on compact subsets of F N  so that we have no control on F or ∇ F near ∂F N . We first state an existence result for (1.5) in the nonsymmetric case.

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

657

Theorem 1.4. Fix N = 3 or N = 4. Then there exists ρ0 > 0 such that for 0 < ρ < ρ0 , the sinh-Poisson equation (1.5) has a solution ψρ with the following properties: For any sequence ρn → 0 there exists x ∗ ∈ F N  such that (along a subsequence) the vorticity field ρn sinh ψρn 8π

N 

(−1)i δxi∗

weakly in the sense of measures in .

i=1

The blow-up point x ∗ ∈ F N  is a critical point of H with i = (−1)i .  Passing from the streamfunction ψ to the velocity field v = J ∇ψ with J = 0 −1 we obtain a stationary solution of the Euler flow in . If N = 3 (resp. 1 0 N = 4) the solution vρ corresponding to the streamfunction ψρ from Theorem 1.4 has two vortices rotating clockwise and one vortex (resp. two vortices) rotating anti-clockwise. The proof of Theorem 1.4 will also give a precise description of the limit profile of the streamfunctions as ρ → 0. In the symmetric situation we have the following Theorem 1.5. Suppose  is symmetric with respect to the reflection at R × {0} and  ∩ R × {0} = ∅. Then fixing N ∈ N, there exists ρ0 = ρ0 (N ) > 0 such that for 0 < ρ < ρ0 , (1.5) has a solution ψρ with the following properties: For any sequence ρn → 0 there exists x ∗ ∈ F N  such that (along a subsequence) the vorticity field ρn sinhψρn 8π

N  (−1)i δxi∗

weakly in the sense of measures in .

i=1

The blow-up point x ∗ ∈ F N  is a critical point of H with i = (−1)i , and it satisfies x ∗ = (x1∗ , . . . , x N∗ ), xi∗ = (ti , 0), t1 < t2 < · · · < t N . As with Theorem 1.1 one can prove a multiplicity result if  ∩ R × {0} has several connected components. Thus on a domain  as in Theorem 1.5 we obtain for every N ∈ N an N -chain of counter-rotating vortices aligned on the symmetry line  ∩ R × {0}. These solutions seem to be analogues of the Mallier-Maslowe row of counter-rotating vortices in the plane [30]. Here the streamfunction is explicitly given by

√ √  √  cosh(εx2 / 2) − ε cos(x1 / 2) ε cos(x1 / 2) ϕε (x) = log = −2arctanh , √ √ √ cosh(εx2 / 2) + ε cos(x1 / 2) cosh(εx2 / 2) which solves − ψ =

(1 − ε2 ) sinh ψ 2

in R2 .

(1.6)

It would be interesting to understand whether this streamfunction can be obtained as a limit of streamfunctions ψ N from Theorem √ 1.5 on an expanding sequence of domains √  N , e. g. on the ball  N = B(0, N π/ 2) centred at the origin with radius N π/ 2. If ρ0 = ρ0 (N ,  N ) from Theorem 1.5 could be chosen to be uniform in N , then for on  N having N extrema in  exists a solution ψ N of (1.5)  0 <√ρ < ρ0√there √ −N π/ 2, N π/ 2 × {0}, just like ϕε with ε = 1 − 2ρ. The streamfunctions ψ N

658

T. Bartsch, A. Pistoia, T. Weth

then converge locally in H 1 towards a solution of (1.6), possibly ϕ√1−2ρ . We leave this to future research. We note that our existence results – being derived by a combination of topological and analytical techniques – hold for a very large class of domains. They also provide precise information on the location of the vortex centers in the limit ρ → 0, and the proofs in Sect. 8 give a precise description of the limit profile. However, our methods do not yield explicit formulae for the streamfunctions. In this respect, they complement the study of classes of exact vortices (i.e., exact solutions to the steady Euler equations) with complex analytic methods. Exemplarily we note that in [12–14] an approach based on the Schwarz function of suitably chosen boundary curves of simply connected domains is used to detect symmetric multipolar vortices. Moreover, in [15] analytical formulae for the Kirchhoff-Routh path function in multiply connected domains are derived. It would be extremely interesting to compare the different approaches and results for domains where the complex analytic methods apply. Generalizations of Theorems 1.1 and 1.2 can also be used to obtain new solutions for other singular limit equations. In fact, in [16] critical points of functions close to H yield solutions of the Liouville-type equation  −u = ε2 eu in ; (1.7) u=0 on ∂; for ε → 0. In [19,20] H appears as a singular limit energy function for the LaneEmden-Fowler equation  −u = |u| p−1 u in ; (1.8) u=0 on ∂; for p → ∞. We just state consequences of our results for (1.8). Theorem 1.6. Fix N = 3 or N = 4. Then there exists p0 > 0 such that for p > p0 , (1.8) has a solution u p with the following property: For any sequence pn → ∞ there exists x ∗ ∈ Fn  such that (along a subsequence) pn |u pn |

pn −1

u pn 8π e

N 

(−1)i δxi∗

weakly in the sense of measures in .

i=1

The blow-up point x ∗ is a critical point of H with i = (−1)i , i = 1, · · · , N . Theorem 1.7. Suppose  is symmetric with respect to the reflection at R × {0} and  ∩ R × {0} = ∅. Fix N ∈ N. Then there exists PN > 0 such that for p > p N , (1.8) has a solution u p with the following property: For any sequence pn → ∞ there exists x ∗ ∈ F N  such that (along a subsequence) pn |u pn | pn −1 u pn 8π e

N 

(−1)i δxi∗

weakly in the sense of measures in .

i=1

The blow-up point x ∗ is a critical point of H with i = (−1)i , i = 1, . . . , N , and it satisfies x ∗ = (x1∗ , . . . , x N∗ ), xi∗ = (ti , 0), t1 < t2 < . . . < t N .

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

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The paper is organized as follows. In Sect. 2 we collect a few properties of the Dirichlet Green’s function in . In Sect. 3 we prove a generalization of Theorem 1.1. Section 4 is devoted to an investigation of H (x) when at least one component xi approaches the domain’s boundary. These results hold for a general smooth bounded domain, general N , and general nonzero i . This is not the case in Sect. 5, where we study the behavior of H near collisions. We can prove the results here only under the conditions of Theorem 1.2. In Sect. 6 we prove Theorem 1.2 in the case N = 3, and in Sect. 7 we prove it in the case N = 4. As mentioned above, in order to prove our results about the Dirichlet problems (1.5) and (1.8) we shall in fact extend Theorems 1.1 and 1.2 to cover functions close to H on compact subsets of F N . Theorems 1.4 and 1.5 will then be proved in Sect. 8, Theorems 1.6 and 1.7 in Sect. 9. There the reader can also find a description of the limit profile of the solutions. 2. Some Properties of the Green’s Function Let  ⊂ R2 be a bounded domain with C 3 -boundary. As in the Introduction we write 1 G(x, y) = g(x, y) − 2π log|x − y| for the Green’s function of the Dirichlet Laplacian in ; g :  ×  → R is its regular part, and h(x) := g(x, x) the Robin’s function. Recall that G ≥ 0 and that G, g are symmetric. The following facts are well known. Lemma 2.1. g :  ×  → R and h :  → R satisfy: (i) (ii) (iii) (iv)

g is bounded from above (and thus also h); h(x) → −∞ as dist(x, ∂) → 0; h(x) + h(y) ≤ 2g(x, y) for all x, y ∈ ; For every ε > 0 there is a constant C1 = C1 (, ε) > 0 such that |h(x)| + |∇h(x)| ≤ C1

for every x ∈  with dist(x, ∂) ≥ ε

and |G(x, y)|+|∇x G(x, y)|+|∇ y G(x, y)| ≤ C1 for every x, y ∈  with |x − y| ≥ ε. A proof of 2.1 (iii) can be found in [4, p. 197]. We also need a result concerning the behavior of the regular part g near the boundary. We fix ε0 > 0 small so that the reflection at ∂ is well defined in 0 := {x ∈  : dist(x, ∂) < ε0 }; we denote it by 0 → R2 \, x → x. It is of class C 2 since ∂ is of class C 3 . There exists C > 0 depending only on  such that |x − y| ≤ C for all x, y ∈ 0 . |x − y|

(2.1)

This can be seen, for instance, by considering a tubular neighborhood U ⊂ R2 of ∂ and a C 1 -diffeomorphism T : ∂ × (−1, 1) → U such that T (∂ × [0, 1)) = U ∩  and T (∂ × (−1, 0]) = U \, and such that the reflection at the boundary corresponds to (x, t) → (x, −t). Lemma 2.2. There exists a constant C2 = C2 () > 0 such that ψ(x, y) := g(x, y) − 1 2π log |x − y| satisfies |ψ(x, y)| + |∇x ψ(x, y)| + |∇ y ψ(x, y)| ≤ C2

for every x, y ∈ 0 .

660

T. Bartsch, A. Pistoia, T. Weth

Proof. We need some notation and basic estimates; ci > 0 will denote various positive constants. Let p : 0 → ∂ be the projection onto the boundary and let d : 0 → R, dx = d(x) = | p(x) − x| be the distance to the boundary. The functions p and d are of class C 2 because ∂ is of class C 3 . Moreover, ∇d(x) = −νx , where νx denotes the outward normal at the point p(x) ∈ ∂. It is easy to see that there exist c1 , c2 > 0 such that |x¯ − y| ≥ c1 dx

for any x, y ∈ 0 ,

(2.2)

and |z − w, νz | ≤ c2 |z − w|2

for any z, w ∈ ∂.

(2.3)

This uses that ∂ is C 2 . Next observe that |x¯ − y|2 = | p(x) − p(y)|2 + dx2 + d y2 + 2dx d y νx , ν y  + 2 p(x) − p(y), dx νx + d y ν y  from which we deduce, using the analogous equation for |x − y¯ |2 and (2.3), that     (2.4) |x¯ − y|2 − |x − y¯ |2  ≤ c3 (dx + d y )| p(x) − p(y)|2 , and also that |x¯ − y|2 ≥ c4 | p(x) − p(y)|2

(2.5)

for some c3 , c4 > 0. Here we also used the fact that νx , ν y  > 0 if x and y are close enough. The bound |ψ(x, y)| ≤ c

for x, y ∈ 0

follows from the maximum principle because for x ∈  fixed, the function ψ(x, y) is 1 harmonic in y with boundary values 2π log |x−y| , and (2.4) and (2.5) imply |x−y| ¯ c5 ≤

|x¯ − y|2 ≤ c6 . |x − y¯ |2

In order to obtain the bound |∇x ψ(x, y)| ≤ c for x, y ∈ 0

(2.6)

|ψ(x, y)| ≤ c7 dx for x, y ∈ 0

(2.7)

we will show:

and |x ψ(x, y)| ≤

c7 for x, y ∈ 0 dx

(2.8)

for some c7 > 0. The estimate (2.6) follows from (2.7) and (2.8), taking into account the inequality (see [24], Theorem 3.9):     2  sup |d(x)∇x ψ(x, y)| ≤ C sup |ψ(x, y)| + sup d (x)x ψ(x, y) , x∈B

where B :=

x∈B

B(x  , d

x

  /2), x

x∈B

∈ 0 , and the constant C does not depend on x  and y.

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

661

Observe that (2.7) is again a consequence of the maximum principle because for 1 x ∈ 0 fixed, the function ψ(x, y) is harmonic in y with boundary values 2π log |x−y| . |x−y| ¯ The estimates (2.4) and (2.5) with y = y¯ yield   2  1 − |x − y|  ≤ c3 dx ,  |x¯ − y|2  c4 which implies    1 |x − y|  ≤ c8 dx , log max |ψ(x, y)| = max  y∈∂ y∈∂ 2π |x¯ − y|  for some c8 > 0. Let us prove (2.8). Since the function g(x, y) is harmonic in x, it suffices to prove that for fixed y ∈ 0 the function w(x) := 21 log |x¯ − y|2 satisfies |w(x| ≤

c9 , dx

(2.9)

with c9 > 0 independent of y. Setting ∂i := ∂xi we compute: ∂i w(x) =

2  1 (x¯ j − y j )∂i x¯ j |x¯ − y|2 j=1

and ⎛ ⎞2 2  1 ⎝ (x¯ j − y j )∂i x¯ j ⎠ ∂ii2 w(x) = −2 |x¯ − y|4 j=1

+

2    1 2 2 (∂ , x ¯ ) + ( x ¯ − y )∂ x ¯ i j j j j ii |x¯ − y|2 j=1

which implies ⎛ ⎞2 2 2   1 ⎝ (x¯ j − y j )∂i x¯ j ⎠ w = −2 |x¯ − y|4 i=1

+

1 |x¯ − y|2

2 

j=1

  (∂i x¯ j )2 + (x¯ j − y j )∂ii2 x¯ j .

(2.10)

i, j=1

Moreover, since x¯ = x + 2dx νx we have ∂i x¯ j = δi j − 2νx i νx j + 2dx ∂i νx j and ∂ii2 x¯ j = −2∂i νx i νx j − 4νx i ∂i νx j + 2dx ∂ii2 νx j .

(2.11)

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Plugging this into (2.10), and using the identity ⎞2 ⎛ 2 2 2    ⎝ (x¯ j − y j )(δi j − 2νx i νx j )⎠ = |x¯ − y|2 2 (δi j − 2νx i νx j )2 i=1

j=1

i, j=1

we deduce that w(x) = O(1/dx ). This concludes the proof of (2.9), hence of (2.8) and (2.6). It remains to prove the bound |∇ y ψ(x, y)| ≤ c for any x, y ∈ 0 .

(2.12)

Here it is enough to show that   2   ∇x 1 log |x¯ − y|  ≤ c for any x, y ∈ 0 .  2 |x − y¯ |2 

(2.13)

In fact, claim (2.12) will follow from (2.6) and (2.13), taking into account that ψ(x, y) = ψ(y, x) −

|x¯ − y| 1 log . 2π |x − y¯ |

From (2.11) we deduce  ∂i

1 |x¯ − y|2 log 2 |x − y¯ |2



xi − y¯i  x¯ j − y j + ∂i x¯ j |x − y¯ |2 |x¯ − y|2 2

=−

j=1

x¯ − y, νx  x¯ − y, ∂i νx  xi − y¯i x¯i − yi + − 2νx i + 2dx |x − y¯ |2 |x¯ − y|2 |x¯ − y|2 |x¯ − y|2   p(x) − p(y), νx  1 1 − 2νx i = (xi − y¯i ) − |x¯ − y|2 |x − y¯ |2 |x¯ − y|2 x¯ − y, ∂i νx  + 2dx |x¯ − y|2   1 2dx νx i + 2d y ν y i − 2νx i dx νx + d y ν y , νx  + 2 |x¯ − y| = O(1), =−

 using (2.2), (2.3), (2.4), (2.5) and the fact that |νx − ν y | = O(|x − y|).  Finally we prove a result on the behaviour of G on lines through the domain. Lemma 2.3. For any line L = {a + tv : t ∈ R} with a, v ∈ R2 , v = 0, and such that  ∩ L = ∅ there exists a constant C > 0 so that G(a + r v, a + sv) − G(a + r v, a + tv) ≥ −C

for all r < s < t.

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

663

Proof. Clearly we may assume that L = R × {0}. Suppose by contradiction that there are points xn = (rn , 0), yn = (sn , 0), z n = (tn , 0) ∈  ∩ L with rn < sn ≤ tn and G(xn , yn ) − G(xn , z n ) → −∞.

(2.14)

Since G ≥ 0 this implies G(xn , z n ) → ∞, and therefore |rn − tn | → 0 as n → ∞. Passing to a subsequence, we may assume that xn , yn , z n → w ∈  ∩ L. We first suppose that w ∈ . Then g(xn , yn ) − g(xn , z n ) ≥ C˜ > −∞ for all n, so that G(xn , yn ) − G(xn , z n ) ≥ C˜ +

|rn − tn | 1 1 log ≥ C˜ + log 1 = C˜ 2π |rn − sn | 2π

for all n, contradicting (2.14). Hence w ∈ ∂. We may assume that w = (0, 0) ∈ ∂ ∩ L, so that rn , sn , tn → 0 as n → ∞. Moreover we may assume that either 0 < rn < sn < tn for all n

or rn < sn < tn < 0 for all n.

(2.15)

Now Lemma 2.2 yields G(xn , yn ) =

|xn − yn | 1 1 1 log |xn − yn | + O(1) − log |xn − yn | = O(1) + log 2π 2π 2π |xn − yn |

and G(xn , z n ) = O(1) +

|xn − z n | 1 log , 2π |xn − z n |

so that G(xn , yn ) − G(xn , z n ) = O(1) + For n large, we have

|xn −z n | |xn −yn |

≥

|xn −z n | |xn −yn |

1 log 2π



|xn − yn | |xn − z n | · . |xn − z n | |xn − yn |

by (2.15), so that

G(xn , yn ) − G(xn , z n ) ≥ O(1) +

1 log 1 = O(1), 2π

contradicting (2.14).   3. The Symmetric Case In this section we prove a generalization of Theorem 1.1. Let  ⊂ R2 be symmetric  1 0 . We also require that the fixed point set with respect to the reflection S = 0 −1 S  = {x ∈  : Sx = x} =  ∩ R × {0} is nonempty. It is not necessary that  is simply connected nor that  S is an interval. We fix N ∈ N and – throughout this section – we consider the Hamiltonian H in the case where i = (−1)i for i = 1, . . . , N . Due to the symmetry of  we have H (Sx1 , . . . , Sx N ) = H (x1 , . . . , x N ) for x ∈ F N (),

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T. Bartsch, A. Pistoia, T. Weth

hence a critical point x ∈ F N ( S ) of H constrained to F N ( S ) is a critical point of H . We set R := {t = (t1 , . . . , t N ) ∈ R N : t1 < t2 < · · · < t N , ((t1 , 0), . . . , (t N , 0)) ∈ } ⊂ R N and E : R → R,

E(t) := H ((t1 , 0), . . . , (tn , 0)).

Observe that R is a bounded open subset of R N . Proposition 3.1. E(t) → −∞ as dist(t, ∂R) → 0. As a consequence, E is bounded from above and satisfies the Palais-Smale condition. In fact, this is equivalent to Proposition 3.1. Postponing the proof of Proposition 3.1 we deduce two consequences. Corollary 3.2. E achieves a local maximum in every connected component of R. The second consequence is our multiplicity result for symmetric domains generalizing Theorem 1.1. Theorem 3.3. Suppose  S is the disjoint union of k intervals I1 , . . . , Ik . Then for any choice of integers i 0 = 0 < 1 ≤ i 1 ≤ · · · ≤ i k−1 ≤ i k = N , H has a critical point x ∈ F N ( S ) such that xi j +1 , . . . , xi j+1 ∈ I j+1 for j = 0, . . . , k − 1. This critical point is stable in the following sense. If Jn : F N  → R is a sequence of S-invariant C 1 -functionals converging towards H uniformly on compact subsets of F N ( S ) then for n large Jn has a critical point x (n) ∈ F N ( S ) with Jn (x (n) ) → H (x), and (n) (n) xi j +1 , . . . , xi j+1 ∈ I j+1 for j = 0, . . . , k − 1. Proof. The set R0 := {(t1 , . . . , t N ) ∈ R : ti j +1 , . . . , ti j+1 ∈ I j+1 , j = 0, . . . , k − 1} is a connected component of R. By Corollary 3.2, E has a local maximum t in R0 , hence (t, 0) is a critical point of H . Moreover, the set K ε := {t ∈ R0 : E(t) ≥ E(t − ε} is compact as a consequence of Proposition 3.1. If Jn : F N  → R is an S-invariant C 1 -functional and Jn → H uniformly on K ε , then E n : R0 → R,

E n (t) = Jn ((t1 , 0), . . . , (tn , 0)),

achieves a local maximum in K ε provided Jn − H  L ∞ (K ε ) < ε/2. The corollary follows easily.   We now turn to the Proof of Proposition 3.1. For simplicity of notation we define for s, t ∈ R with (s, 0), (t, 0) ∈ : g0 (s, t) = g((s, 0), (t, 0)), h 0 (t) = g0 (t, t), G 0 (s, t) = g0 (s, t) −

1 log |s − t|, 2π

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

665

where g is the regular part of the Green’s function G as before. So we have for t ∈ R: E(t) =

N 

 (−1)i+ j G 0 (ti , t j ).

h 0 (ti ) +

i= j

i=1



Step 1. i= j (−1)i+ j G 0 (ti , t j ) is bounded from above. For i = 1, . . . , N − 1 and t = (t1 , . . . , t N ) ∈ R we set G i (t) :=

N 

(−1)i+ j G 0 (ti , t j ),

j=i+1

so that N −1   (−1)i+ j G 0 (ti , t j ) = 2 G i (t). i= j

i=1

If N − i is even, we find N −i

G i (t) =

2 

(−G 0 (ti , ti+2k−1 ) + G 0 (ti , ti+2k )) ≤

k=1

N −i C 2

by Lemma 2.3. If N − i is odd, we obtain N −i−1 2

G i (t) =



(−G 0 (ti , ti+2k−1 ) + G 0 (ti , ti+2k )) − G 0 (ti , t N ) ≤

k=1

k −i −1 C 2

also by Lemma 2.3. Step 1 follows immediately. Step 2. E(t) → −∞ as dist(t, ∂R) → 0. Consider a sequence t n = (t1n , . . . , t Nn ) ∈ R such that dist(t n , ∂R) → 0 as n → ∞. If dist((tin , 0), ∂) → 0 for at least one component then E(t) → −∞ by Step 1 and because the Robin’s function is bounded above and satisfies h(x) → −∞ as dist(x, ∂) → 0. Thus it remains to consider the case where lim inf dist((tin , 0), ∂) > 0 for every i = 1, . . . , N ,

(3.1)

n ti+1 − tin → 0 for some i ≤ N − 1.

(3.2)

n→∞

and

By (3.1) the first two sums in E(t n ) =

N  i=1

h 0 (tin ) +

 1  (−1)i+ j g0 (tin , t nj ) − (−1)i+ j log |tin − t nj | 2π i= j

i= j

remain bounded as n → ∞. We write  i= j

(−1)i+ j log |tin − t nj | = 2

N −1 

N 

i=1 j=i+1

(−1)i+ j log |tin − t nj | = 2

N −1  i=1

log ψi (t n ),

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T. Bartsch, A. Pistoia, T. Weth

where, if N − i is even, N −i

2 n  |ti+2k − tin | ψi (t ) = ≥ 1, n |ti+2k−1 − tin |

n

k=1

and, if N − i is odd, N −i−1

2 n  |ti+2k − tin | 1 1 ≥ n . ψi (t ) = n n n |t N − ti | |ti+2k−1 − tin | |t N − tin |

n

k=1

It remains to show that ψi (t n ) → ∞ as n → ∞ for some i = 1, . . . , N − 1. Let i be maximal satisfying (3.2). If i = N − 1 then |t Nn − t Nn −1 | → 0 as n → ∞, hence ψ N −1 (t n ) → ∞ as n → ∞. Otherwise we may assume that, after passing to a subsequence, 1 ≥ |t nj+1 − t nj | ≥ δ > 0 for some δ > 0 and all j > i, n ∈ N. δ If N − i is even, (3.2) and (3.3) imply

(3.3)

N −i

2 n n − tn| |t n − tin |  |ti+2k − tin | |ti+2 δ i ψi (t n ) = i+2 ≥ ≥ n →∞ n n n n n n |ti+1 − ti | |ti+2k−1 − ti | |ti+1 − ti | |ti+1 − tin |

k=2

as n → ∞. And if N − i ≥ 3 is odd, (3.2) and (3.3) yield ⎛ ⎞ N −i−1 2 n n n n  |t − ti | ⎜ |ti+2k − ti | ⎟ 1 δ2 ≥ ψi (t n ) = i+2 ⎝ ⎠ n − tn| n n − tn| → ∞ |ti+1 |ti+2k−1 − tin | |t Nn − tin | |ti+1 i i k=2

as n → ∞.   4. Avoiding the Domain’s Boundary As mentioned in the Introduction we need to control the behavior of H near ∂(F N ) in order to find relatively compact gradient flow lines. It is the goal of this section to show that along certain gradient flow lines of H each component stays away from ∂. Here  ⊂ R2 can be an arbitrary bounded planar domain with C 3 -boundary. Moreover, we consider the Hamiltonian H for general N ∈ N and general nonzero i , i = 1, . . . , N . The problem of collisions will be dealt with in the next section where we cannot work in this generality. The gradient flow of H solves the equation ⎧ ⎨ ∂ ϕ(x, t) = ∇ H (ϕ(x, t)),  (4.1) ∂t ⎩ ϕ(x, 0) = x, and is a map ϕ : G = {(x, t) : x ∈ F N , T − (x) < t < T + (x)} → F N , where T − (x) < 0 < T + (x) are the maximal existence times for the trajectory t → ϕ(x, t) in the negative and positive direction. Our main result in this section is

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

667

Proposition 4.1. If x ∈ F N  satisfies limt→T + (x) H (ϕ(x, t)) < ∞, then {ϕ(x, t) : 0 ≤ t < T + (x)} ∩ ∂( N ) = ∅. Here the closure and the boundary are taken in (R2 ) N . Thus as long as H is bounded above along a flow line, each component of this flow line stays away from the boundary of the domain. For the proofs of these propositions we need some notation. We fix ε0 > 0 and 0 = {x ∈  : dist(x, ∂) < ε0 } such that for every x ∈ 0 there exists a unique point p(x) ∈ ∂ with dist(x, ∂) = |x − p(x)|, hence x − p(x) ⊥ T p(x) ∂. Then p : 0 → ∂ is a C 1 -function. For x ∈ 0 we denote by x = 2 p(x) − x the reflection of x at ∂ as in Sect. 2, and we x− p(x) write q(x) = |x− p(x)| for the interior normal. Lemma 4.2. There is C3 = C3 () > 0, independent of N and i , i = 1, . . . , N , such that 

N N  k2 1  2 for every x ∈ F N 0 , − N C3 ˜ ∇xk H (x), q(xk ) ≥ π |xk − xk | k=1

k=1

where ˜ = maxi=1,...,N i2 . Proof. Let ψ be defined as in Lemma 2.2. For k = 1, . . . , N we consider the functions Ak : F N 0 → R,

Ak (x) = k2 ∇xk ψ(xk , xk ) + 2

N 

k  j ∇xk ψ(xk , x j ).

j=1

j=k

Lemma 2.2 implies |Ak (x)| ≤ 2N C2 ˜ for every x ∈ F N 0 and every k; here C2 = C2 () depends only on . For x ∈ F N 0 we have ∇xk H (x) = k2 ∇h(xk ) + 2

N  j=1

 k  j ∇xk g(xk , x j ) +

xk − x j 2π|xk − x j |2



j=k

=

k2 ∇x log |xk − xk | + Ak (x) 2π k  N xk − x j 1 1 + k  j ∇x (log |xk − x j | + log |xk − x j |) − π 2 k |xk − x j |2 j=1

j=k



xk − xk T xk − xk + (id − Dp(xk )) + Ak (x) |xk − xk |2 |xk − xk |2   N xk − x j xk − x j 1 1 T xk − x j k  j − − Dp(x )) − + (id l π 2 |xk − x j |2 |xk − x j |2 |xk − x j |2

2 = k 2π

j=1

j=k

668

T. Bartsch, A. Pistoia, T. Weth

⎛ =

1⎜ ⎜ 2 x k − x k + π ⎝ k |xk − xk |2

j=1

⎞   xk − x j xk − x j ⎟ x j − xk 1 ⎟ − l  j + 2 2 2 |xk − x j | |x j − xk | |xk − x j |2 ⎠

j=k

⎛ +

N 

⎞

1 ⎜ ⎜− 2 Dp(xk )T xk − xk + 2π ⎝ k |xk − xk |2

N 

k  j Dp(xk )T

j=1

xk − x j ⎟ ⎟ + Ak (x). |xk − x j |2 ⎠

j=k

Since Dp(x)q(x) = 0 for every x ∈ 0 , we deduce: N  ∇xk H (x), q(xk ) k=1

=

  N N k2 x j − xk 1 xk − x j 1 1 + k  j − , q(xk ) π |xk − xk | π 2 |xk − x j |2 |x j − xk |2 k=1

−

j=1

xk − x j , q(xk ) |xk − x j |2



j=k

+

N  Ak (x), q(xk ) k=1

  N N k2 x j − xk 1 xk − x j 1 1 + = k  j − , q(xk ) − q(x j ) π |xk − xk | π 2 |xk − x j |2 |x j − xk |2 k=1

j=1

j
xk − x j , q(xk ) − q(x j ) − |xk − x j |2

+

N  Ak (x), q(xk ). k=1

We note that there is a constant C = C() > 0 such that |q(x) − q(y)| ≤ C|x − y| for x, y ∈ 0 , x = y. Hence, for j = k,    xk − x j , q(xk ) − q(x j )   ≤ C,    |x j − xk |2 and, using (2.1),



xk − x j x j − xk − , q(xk ) − q(x j ) |xk − x j |2 |x j − xk |2  1 1 + |xk − x j | ≤ C  , ≤C |xk − x j | |x j − xk |

where C  = C  () > 0 also depends only on the domain. Consequently, N N  k2 N (N − 1) 1 − (C + C  )˜ − 2N 2 C2 ˜ ∇xk H (x), q(xk ) ≥ π |xk − xk | π k=1 k=1 

N k2 1  2  ˜ ≥ − N (2πC2 + C + C ) . π |xk − xk | k=1

The claim follows for C3 := 2πC2 + C + C  .  

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

669

For 0 < δ < ε0 we set M1δ := {x ∈ F N  : dist(xi , ∂) ≤ δ for some i}.

(4.2)

and define δ (x) := {i : dist(xi , ∂) ≤ δ} ⊂ {1, . . . , N }

for x ∈ M1δ .

Lemma 4.3. For every ε > 0 there exists 0 < δ < ε0 with the following property. If x ∈ M1δ satisfies dist(xi , ∂) ≥ ε for every i ∈ {1, . . . , N }\δ (x), then  ∇x j H (x), q(x j ) ≥ N + 1. j∈δ (x)

Proof. Let C3 = C3 () be given as in Lemma 4.2. Moreover, for given ε > 0, let C1 = C1 (, 2ε ) be given as in Lemma 2.1(iv). We choose δ such that !  mink=1,...,N k2 ε , . (4.3) 0 < δ < min ˜ + N 2 C3 ) ˜ 2 2(π(N + 1 + 2N 2 C1 ) Let x ∈ M1δ satisfy dist(x j , ∂) ≥ ε for every j ∈ {1, . . . , N }\δ (x). Since the following argument is independent of the order of the i , we may assume that δ (x) = {1, . . . , m}, so that dist(xi , ) < δ for i ≤ m

and

dist(xi , ) ≥ ε for m + 1 ≤ i ≤ N . (4.4)

Let us now write H (x) = H1 (x) + H2 (x), where H1 (x) =

m 

i2 h(xi ) +



i  j G(xi , x j )

i= j

i=1

and H2 (x) =

N 

i2 h(xi ) +



i  j G(xi , x j ),

(i, j)∈M

i=m+1

where M := {(i, j) ∈ {1, . . . , N } : i = j, i ≥ m + 1 or j ≥ m + 1}. By Lemma 4.2 with m instead of N we have 

m m  k2 1  2 ˜ (4.5) − m C3  . ∇xk H1 (x), q(xk ) ≥ π |xk − xk | k=1

k=1

Moreover,   m   m         i  j ∇xk G(xi , x j ), q(xk )  ∇xk H2 (x), q(xk ) =      k=1 (i, j)∈M

k=1

≤ ˜

m 



|∇xk G(xi , x j ), q(xk )| ≤ ˜

k=1 (i, j)∈M

= 2˜

m  N  i=1 j=m+1

m   k=1 (i, j)∈M

|∇xi G(xi , x j )|.

|∇xk G(xi , x j )|

670

T. Bartsch, A. Pistoia, T. Weth

Since |xi − x j | ≥ ε − δ ≥

ε 2

for 1 ≤ i ≤ m and m + 1 ≤ j ≤ N ,

and C1 = C1 (, 2ε ), we conclude by Lemma 2.1(iv) that  m      ˜  ∇xk H2 (x), q(xk ) ≤ 2m(N − m)C1 ˜ ≤ 2N 2 C1 .   k=1

Therefore m  1 ∇xk H (x), q(xk ) ≥ π k=1

m  k=1

 k2 2 ˜ − m C3 ˜ − 2N 2 C1 . |xk − xk |

Since for k ∈ {1, . . . , m} we have dist(xk , ∂) ≤ δ and therefore |xk − xk | ≤ 2δ, we conclude that m  1 ∇xk H (x), q(xk ) ≥ π k=1

 m 1  2 2 k − m C3 ˜ − 2N 2 C1 ˜ ≥ N + 1 2δ k=1

by (4.3), as claimed.   Lemma 4.4. There exists 0 < δ0 < ε0 such that |∇ H | > 1 on M1δ0 . Proof. Suppose by contradiction that there exists a sequence (x n )n in F N  with |∇ H (x n )| ≤ 1 for all n and min1≤i≤N dist(xin , ∂) → 0 as n → ∞. Passing to a N

subsequence, we may assume that x n → x¯ ∈ F N  =  . Let  ⊂ {1, . . . , N } be the set of indices i with x¯i ∈ ∂, and let ε > 0 be such that dist(x nj , ∂) ≥ ε

for all n ∈ N and j ∈ {1, . . . , N }\.

We now choose δ = δ(ε) according to Lemma 4.3. For n large enough we then have x n ∈ M1δ and δ (x n ) = , so Lemma 4.3 implies        |∇xk H (x n )| N + 1 ≤  ∇xk H (x n ), q(xkn ) ≤   k∈

≤

N 

k∈

|∇xk H (x n )| ≤ N |∇ H (x n )|,

k=1

and therefore |∇ H (x n )| ≥ assumed.  

N +1 N

> 1 for n large enough. This contradicts what we have

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

671

Now we come to the Proof of Proposition 4.1. For x ∈ F N  we set x t := ϕ(x, t) for T − (x) < t < T + (x). We first consider the case T + (x) < ∞. Then for 0 ≤ s < t < T + (x) we have " |x − x | ≤ t

s

s

=

t

√ |∇ H (x )|dτ ≤ t − s τ

" s

t

τ

1/2

|∇ H (x )| dτ 2

√ √ (t − s)(H (x t ) − H (x s )) ≤ c t − s N

with c := limt→T + (x) H (x t ) − H (x). Hence x t → x¯ ∈ F N  =  as t → T + (x). Suppose by contradiction that x¯ ∈ ∂( N ). Let  ⊂ {1, . . . , N } be the set of indices i with x¯i ∈ ∂, and let ε > 0 be such that dist(x tj , ∂) ≥ ε

for all t ∈ [0, T + (x)) and j ∈ {1, . . . , k}\.

We put m := || and consider the function d : m 0 → R,

d(x1 , . . . , xm ) =

m  i=1

dist(xi , ∂) =

m 

|xi − p(xi )|.

(4.6)

i=1

Then d is of class C 1 , and ∇d(x) = (q(x1 ), . . . , q(xm )) ∈ (R2 )m for every x ∈ m 0. We choose δ = δ(ε) according to Lemma 4.3. For t sufficiently close to T + (x) we have x t ∈ M1δ and δ (x t ) = , so Lemma 4.3 implies  ∂ d((xkt )k∈ ) = q(xkt ), ∇xk H (x t ) ≥ k + 1 > 0 ∂t k∈

for t sufficiently close to T + (x). This however contradicts the fact that d((xkt )k∈ ) → 0 as t → T + (x). It remains to consider the case T + (x) = ∞. We suppose by contradiction that there exist a sequence (tn )n in [0, ∞) such that tn → ∞ and x tn → x¯ ∈ (∂) N as n → ∞. Using the upper bound for H , it is easy to find a sequence (sn ) in [0, ∞) with |tn − sn | → 0 and ∇ H (x sn ) → 0 as n → ∞. In particular, x sn → x¯ as n → ∞, so x sn ∈ M1δ0 for n large enough. This however contradicts Lemma 4.4. Since in both cases we came to a contradiction, the proof is finished.   As an immediate consequence of the proof of Proposition 4.1 we also obtain – in some cases – that H satisfies the Palais-Smale condition: (PS) any sequence (x n ) in F N  with ∇ H (x n ) → 0 and H (x n ) bounded uniformly in n has a convergent subsequence, Proposition 4.5. H satisfies (PS) if one of the following conditions hold: (i) (ii)

1 , . . . ,  N have the same sign; N = 2 and sign 1 = −sign 2 .

672

T. Bartsch, A. Pistoia, T. Weth

Proof. Consider a (PS)c -sequence (x n ) in F N . Then ∇ H (x n ) → 0 implies by Lemma 4.4 that x n ∈ / M1δ0 , hence x n → x¯ ∈  N along a subsequence. It follows that 1  H (x n ) = − i  j log |xin − x nj | + O(1) as n → ∞. 2π i= j

Consequently, x¯ ∈ F N  because otherwise H (x n ) → +∞ if all i have the same sign, and H (x n ) → −∞ if N = 2 and sign 1 = −sign 2 .   The two cases in Proposition 4.5 are precisely those considered in [5,16,18]. As mentioned in the Introduction, the Palais-Smale condition is not very helpful for functionals on incomplete manifolds if the functional stays bounded near the boundary of the manifold. 5. Avoiding Collisions In this section we investigate H near the collision set {x ∈ F N  : xi = x j for some i = j}. We also study functionals that are close to H uniformly on compact subsets but cannot be controlled near the boundary of  nor near collisions. For this we need to restrict our attention to the case of at most four vortices and i = (−1)i . So we fix N ∈ {2, 3, 4}, and from now on H has the form H (x) =

N 

h(xi ) +

i=1



(−1)i+ j G(xi , x j ).

i= j

Collisions in general do occur so some restrictions on the i ’s are necessary. We can prove some, but not all results of this section under the condition (1.4). Lemma 5.1. Let 2 ≤ k ≤ N , σ : {1, . . . , N } → {1, . . . , N } be an arbitrary permutation, and consider the C 1 -function Jk,σ : F N R2 → R defined by ⎛ ⎞   σi +σ j ⎠. (−1)σi +σ j log |xσi − xσ j | = log ⎝ |xσi − xσ j |(−1) Jk,σ (x) = 1≤i< j≤k

1≤i< j≤k

Then we have

|∇ Jk,σ (x)| ≥ max

x0 ∈R2

k 

−1/2 |xσi − x0 |2

for all x ∈ F N R2 .

i=1

Proof. Define σ˜ : F N R2 → F N R2 , σ˜ x = (xσ1 , . . . , xσ N ) and ⎛ ⎞  σi +σ j ⎠. J˜(x) = log ⎝ |xi − x j |(−1) J˜ : F N R2 → R, 1≤i< j≤k

Then Jk,σ = J˜ ◦ σ˜ and |∇ J˜(σ˜ (x))| = |∇ Jk,σ (x)|

for all x ∈ F N R2 ,

(5.1)

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

673

since σ˜ is a linear isometry. Fix x0 ∈ R2 and x ∈ Fk R2 , put x¯0 = (x0 , . . . , x0 ) ∈ (R2 ) N and consider j : (0, ∞) → R defined by j (r ) = J˜(σ˜ [x¯0 + r (x − x¯0 )]) = Jk,σ (x¯0 + r (x − x¯0 )) ⎛ ⎞  σi +σ j ⎠. = log ⎝ |r (xσi − xσ j )|(−1) 1≤i< j≤k



Then j  (1) = 1≤i< j≤k (−1)σi +σ j . We claim that j  (1) = 0. This is obvious in the cases k = 2 and k = 3, since then the sum consists of an odd number of terms. If k = 4, then also N = 4 by assumption, and therefore j  (1) =

4 

(−1)σi +σ j =

4 

(−1)i+ j = −2.

i, j=1

i, j=1

i< j

i< j

In any case we infer

1 ≤ | j (1)| = |∇ J˜(σ˜ (x)), σ˜ x − x¯0 | ≤ |∇ J˜(σ˜ (x))| 

k 

1/2 |xσi − x0 |

2

.

i=1

Together with (5.1), the claim follows.

 

Next we set M2δ := {x ∈ F N  : dist(xi , x j ) ≤ δ for some i = j}

(5.2)

and Mδ := M1δ ∪ M2δ , where M1δ has been defined in (4.2). Lemma 5.2. There exists δ > 0 such that |∇ H | > 1 in Mδ . Proof. Suppose by contradiction that there exists δn → 0 and points x n ∈ Mδn with |∇ H (x n )| ≤ 1

for all n.

(5.3)

N

Then, after passing to a subsequence, x n → x¯ ∈  as n → ∞. Lemma 4.4 implies that x¯ ∈  N . Since 2 ≤ N ≤ 4 there is a permutation σ : {1, . . . , N } → {1, . . . , N } such that one of the following alternatives hold: Case 1. There is 2 ≤ k ≤ N such that x0 := x¯σ1 = x¯σ2 = · · · = x¯σk

and

xσi = xσ j = x0 for i, j > k, i = j.

(5.4)

Case 2. N = 4 and x¯σ1 = x¯σ2 = x¯σ3 = x¯σ4 .

(5.5)

We first consider Case 1, and we use the fact that H (x) = Jk,σ as in Lemma 5.1 and K (x) =

N  i=1

+2



h(xi ) + 2

g(xσi , xσ j ) + 2

1≤i< j≤k

 k+1≤i< j≤N

(−1)σi +σ j G(xσi , xσ j ).



− π1 Jk,σ (x)

+ K (x) with

(−1)σi +σ j G(xσi , xσ j )

i=1,...,k j=k+1,...,N

(5.6)

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Since ∇ K remains bounded in a neighborhood of x¯ by (5.4), Lemma 5.1 implies that there exists C > 0 with −1/2

k  1 |∇ H (x n )| ≥ |xσni − x0n |2 −C for all n ∈ N . π i=1

Consequently, since xσni → x0 for i = 1, . . . , k, |∇ H (x n )| → ∞ as n → ∞, contrary to (5.3). It remains to consider Case 2. In this case we use the fact that 1 H (x) = − L(x) + M(x), π where

  σ1 +σ2 σ3 +σ4 |xσ3 − xσ4 |(−1) L(x) = log |xσ1 − xσ2 |(−1)

and M(x) =

4 

h(xi ) + 2g(xσ1 , xσ2 ) + 2g(xσ3 , xσ4 ) + 2

i=1



(−1)σi +σ j G(xσi , xσ j ). (5.7)

i=1,2 j=3,4

It is easy to see that, as a consequence of (5.5), ∇ M is bounded in a neighborhood of x¯ and |∇ L(x n )| → ∞ as n → ∞. We therefore deduce that |∇ H (x n )| → ∞ as n → ∞, contrary to (5.3). The proof is finished.   An immediate consequence of Lemma 5.2 is Corollary 5.3. If N = 3 or N = 4, sign i = (−1)i , then H satisfies the Palais-Smale condition. When dealing with the nonlinear elliptic problems (1.5) and (1.8), H appears as a singular limit functional. Here we need to find critical points of functionals F : F N  → R which are close to H uniformly on compact sets, but not necessarily uniformly in all of F N . It is therefore unknown whether or not F satisfies the Palais-Smale condition. In order to deal with this problem we consider the gradient flows of H and F. Let ϕ : G → F N  be the gradient flow of H defined in (4.1). Lemma 5.4. If x ∈ F N  is such that limt→T + (x) H (ϕ(t, x)) < ∞, then T + (x) = ∞. Proof. As before we put x t := ϕ(x, t), and we assume by contradiction that T + (x) < ∞. N As in the proof of Proposition 4.1 we deduce that x t → x¯ = (x¯1 , . . . , x¯ N ) ∈ F N  =  N as t → T + (x). Clearly x¯ ∈  \F N , since otherwise the trajectory could be continued + beyond T (x). By Proposition 4.1, x¯ ∈ ∂( N ), hence x¯ ∈  N \F N . Since 2 ≤ N ≤ 4 as in the proof of Lemma 5.2 there is a permutation σ : {1, . . . , N } → {1, . . . , N } such that one of the following alternatives hold: Case 1. There is 2 ≤ k ≤ N such that x0 := x¯σ1 = x¯σ2 = · · · = x¯σk

and

xσi = xσ j = x0 for i, j > k, i = j.

(5.8)

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

675

Case 2. N = 4 and x¯σ1 = x¯σ2 = x¯σ3 = x¯σ4 .

(5.9)

We first consider Case 1, and use as in the proof of Lemma 5.2 the fact that H (x) = − π1 Jk,σ (x)+ K (x) with Jk,σ given as in Lemma 5.1 and K as in (5.6). Since ∇ K remains bounded in a neighborhood of x¯ by (5.8), Lemma 5.1 implies that there exists C > 0 with −1/2

k 1  t t t 2 |∇ H (x )| ≥ |xσi − x0 | −C for 0 ≤ t < T + (x). π i=1

Since xσt i → x0 for i = 1, . . . , k, there is t1 ∈ (0, T + (x)) such that 1 |∇ H (x )| ≥ 2π t

k 

−1/2 |xσt i

−

x0t |2

for t1 ≤ t < T + (x).

i=1

For t ≥ t1 we therefore have " t " t |∇ H (x s )| 1 t t1 s 2 H (x ) − H (x ) = |∇ H (x )| ds ≥  1/2 ds 2π t1 k t1 s − x |2 |x 0 i=1 σi  1/2 " t " t ∂ k |x s − x0 |2 s i=1 σi ∂s 1 |x˙ | 1 =  1/2 ds ≥ −  1/2 ds 2π t1 k 2π k t1 s − x |2 s −x |2 |x |x 0 0 i=1 σi i=1 σi  1/2 t1 k 2 i=1 |x σi − x 0 | 1 log  = 1/2 , k 2π t 2 i=1 |x σi − x 0 | so that H (x t1 ) − H (x t ) → ∞ as t → T + (x), contrary to the assumption. It remains to consider Case 2. In this case we use the fact that   1 σ1 +σ2 σ3 +σ4 + M(x), H (x) = − log |xσ1 − xσ2 |(−1) |xσ3 − xσ4 |(−1) π with M as in (5.7). Since M is bounded in a neighborhood of x¯ as a consequence of (5.9) and since (−1)σ1 +σ2 = (−1)σ3 +σ4 , we conclude that |H (x t )| → ∞ as t → T + (x). Because H is increasing along flow lines, this implies H (x t ) → ∞ as t → T + (x), contrary to our assumption.   Now we consider C 1 -functions F : F N  → R which are close to H in the following sense. We fix δ as in Lemma 5.2 and require: |∇ F − ∇ H | <

1 δ and |F − H | < 4 8

in F N \Mδ/4 .

(5.10)

We do not assume anything on F in Mδ/4 , in particular we do not know whether F satisfies the Palais-Smale condition. Set K := (∇ F)−1 (0) and let V : F N \K → R2N be a pseudo-gradient vector field for F such that the first inequality in (5.10) holds for

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V instead of ∇ F. More precisely, we require that V is locally Lipschitz continuous and satisfies ⎧ 1 ⎪ in F N \K , ⎨ |V | < 2|∇ F|, ∇ F, V  > |∇ F|2 2 (5.11) ⎪ ⎩ |V − ∇ H | < 1 in F  \(M )\K .  N δ/4 4 V can be constructed in a standard way using a partition of unity argument. We now define a vector field v : F N \K → R2N by setting v(x) =

dist(x, Mδ/2 ) V (x) dist(x, Mδ/2 ) + dist(x, F N \Mδ ) dist(x, F N \Mδ ) ∇ H (x). + dist(x, Mδ/2 ) + dist(x, F N \Mδ )

Clearly v is locally Lipschitz continuous and satisfies v ≡ ∇ H in Mδ/2

and

v ≡ V in F N  \(Mδ/4 )\K .

(5.12)

Moreover we have Lemma 5.5. If x ∈ Mδ \Mδ/4 , then v(x), ∇ F(x) ≥ 41 . Proof. Observe that |v(x) − ∇ H (x)| < 1/4 because |V (x) − ∇ H (x)| < 1/4 by (5.11). Using this, (5.10) and Lemma 5.2 we deduce v(x), ∇ F(x) = v(x) − ∇ H (x), ∇ F(x) − ∇ H (x) + v(x) − ∇ H (x), ∇ H (x) +∇ H (x), ∇ F(x) − ∇ H (x) + |∇ H (x)|2 1 1 1 ≥ |∇ H (x)|2 − |∇ H (x)| − ≥ . 2 4 4   We consider the flow ψ : Gψ → F N  induced by v: 

∂ ψ(x, t) = v(ψ(x, t)) ∂t ψ(0, x) = x.

(5.13)

Here Gψ = {(x, t) : x ∈ F N \K , Tψ− (x) < t < Tψ+ (x)}, and Tψ− (x) < 0 < Tψ+ (x) are the maximal existence times for the trajectory t → ψ(x, t) in the negative and positive direction. Fixing some # $ a > sup H (x) : x ∈ F N \Mδ/2 (5.14) we have the following: Lemma 5.6. If x ∈ F N  satisfies H (x) = a, then H (ψ(x, t)) > a for 0 < t < Tψ+ (x).

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

Proof. It suffices to show that ∂  0 <  H (ψ(x, t)) = ∇ H (x), v(x) ∂t t=0

677

for x ∈ F N  with H (x) = a.

This follows from the fact that H−1 (a) ⊂ Mδ/2 and v ≡ ∇ H in Mδ/2 , while |∇ H | > 1 in Mδ/2 by Proposition 5.2.   Proposition 5.7. If x ∈ F N  satisfies limt→Tψ+ (x) H (ψ(x, t)) < ∞, then there exists a sequence tn → Tψ+ (x) such that ψ(x, tn ) → x ∗ ∈ F N  and ∇ F(x ∗ ) = 0. Proof. We write again x t := ψ(x, t). We first exclude the case that there exists t0 < Tψ+ (x) such that x t ∈ Mδ/2 for t0 ≤ t < Tψ+ (x).

(5.15)

If (5.15) holds, then the trajectory coincides with a flow line of the negative gradient flow of H for t ≥ t0 . Therefore Lemma 5.4 implies that Tψ+ (x) = ∞, so there must exist a sequence sn → ∞ such that ∇ H (x sn ) → 0. Since x sn ∈ Mδ for every n, this contradicts the fact that |∇ H | > 1 in Mδ . Now we argue by contradiction and suppose that there exists ε > 0 such that xt ∈ / K ε := {x ∈ F N  : dist(x, K ) ≤ ε} for all 0 ≤ t < Tψ+ (x).

(5.16)

Since (5.15) is false and F N \K is bounded, the theory of ordinary differential equations implies Tψ+ (x) = ∞. We claim that there exists t1 > 0 such that x t ∈ F N \(K ε ∪ Mδ/4 ) for t ≥ t1 .

(5.17)

Suppose by contradiction that this is false. Since (5.15) is false as well and (5.16) holds, there must exist times 0 < s1 < t1 ≤ t1 < s2 ≤ s2 < t2 < . . . such that for all n we have 

x sn , x sn ∈ F N \Mδ/2 , and

⎧ ⎪ ⎨ Mδ/2 \Mδ/4 t x ∈ x t ∈ Mδ/2 ⎪ ⎩ t x ∈ F N \Mδ/4



x tn , x tn ∈ Mδ/4

 for sn < t < tn and tn < t < sn+1 ,

for tn ≤ t ≤ tn , for sn ≤ t ≤ sn .

Note that, by Lemma 5.2, " tn " tn δ ≤ |∇ H (x t )| dt ≤ |∇ H (x t )|2 dt = H (x tn ) − H (x sn ) 4 sn sn and, similarly, δ   ≤ H (x sn+1 ) − E(x tn ). 4 

Moreover, since v ≡ ∇ H in Mδ/2 , we find that H (x tn ) ≤ H (x tn ). Finally, assumption (5.10) and Lemma 5.5 imply H (x sn ) ≥ F(x sn ) −

δ δ δ   ≥ F(x sn ) − ≥ H (x sn ) − . 8 8 4

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T. Bartsch, A. Pistoia, T. Weth

Combining this, we find that 





H (x sn+1 ) − H (x sn ) = H (x sn+1 ) − H (x sn+1 ) + (H (x sn+1 ) − H (x tn )) 

+ (H (x tn ) − H (x tn )) + (H (x tn ) − H (x sn )) δ ≥ 4 → ∞ as n → ∞, contrary to our for all n, and therefore H (x sn ) ≥ H (x s1 ) + (n−1)δ 4 main assumption. We thus conclude that (5.17) holds. Now limt→∞ H (x t ) < ∞ implies limt→∞ F(x t ) < ∞ by (5.10), hence ∇ F(x tn ), v(x tn ) → 0 for some sequence tn → ∞. Lemma 5.5 and (5.17) imply that x tn ∈ / Mδ , hence v(x tn )) = V (x tn ) by (5.12) and ∇ F(x tn ) → 0 by (5.11). Since F N \Mδ/4 is compact, we deduce dist(x tn , K ) → 0, contradicting (5.16).   6. Three Vortices In this section we prove Theorem 1.2 in the case N = 3. As before, for simplicity we assume i = (−1)i . We begin with bounding H from above on the set L3  := {x ∈ F3  : x1 − x2 + r (x3 − x2 ) = 0 for some r > 0}. Lemma 6.1. supL3  H < ∞. Proof. For x = (x1 , x2 , x3 ) ∈ F3  we have H (x) =

3 

h(xi ) − 2G(x1 , x2 ) + 2G(x1 , x3 ) − 2G(x2 , x3 )

i=1

|x1 − x2 | 1 log π |x1 − x3 | + h(x3 ) + 2g(x1 , x3 ) − 2G(x2 , x3 ).

= (h(x1 ) + h(x2 ) − 2g(x1 , x2 )) +

Now 2g(x1 , x2 ) ≥ h(x1 ) + h(x2 ), G ≥ 0 and c := sup g < ∞, ×

according to Lemma 2.1 and sup h ≤ c, so we conclude that H (x) ≤

|x1 − x2 | 1 log + 3c. π |x1 − x3 |

For x ∈ L3  we have |x1 − x3 | > |x1 − x2 | and therefore H (x) ≤ 3c.   In the sequel we assume without loss of generality that 0 ∈  and fix ρ > 0 such that the closed ball B(0, 2ρ) ⊂ . Using complex notation for the elements of  ⊂ R2 = C, we set γ0 : S 1 → F3 , γ0 (ζ ) = xζ := (ρζ, 0, 2ρ),

(6.1)

and  := {γ : S 1 → F3  | γ is homotopic to γ0 }. Here S 1 = {ζ ∈ C : |ζ | = 1}, and γ being homotopic to γ0 means that there exists a continuous deformation H : S 1 ×[0, 1] → F3  with H (ζ, 0) = γ0 (ζ ) and H (ζ, 1) = γ (ζ ) for all ζ ∈ S 1 . We have the following intersection property.

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

679

Lemma 6.2. γ (S 1 ) ∩ L3  = ∅ for every γ ∈ . Proof. Let H : S 1 × [0, 1] → F3  be a deformation as above. We define two deformations h 1 , h 2 : S 1 × [0, 1] → C\{0} by setting h 1 (ζ, t) := H1 (ζ, t) − H2 (ζ, t) and h 2 (ζ, t) := H3 (ζ, t) − H2 (ζ, t)) where Hi (ζ, t) ∈  is the i th component of H (ζ, t) ∈ F3 . For fixed t the winding number of the curve ζ → h 1 (ζ, t) around the origin is 1 for t = 0, hence it is 1 for t = 1. Similarly, the winding number of the curve ζ → h 2 (ζ, t) around the origin is 0 for t = 0, hence it is 0 for t = 1. Therefore the homotopy S 1 × [0, 1] → C, (ζ, t) → th 1 (ζ, 1) + (1 − t)h 2 (ζ, 1), must have a zero, say at ζ ∈ S 1 , t ∈ (0, 1). It follows that h 1 (ζ, 1) + which immediately implies γ (ζ ) ∈ L3 .  

1−t t h 2 (ζ, 1)

=0

For our results about the sinh-Poisson and the Lane-Emden-Fowler equation we need to deal with C 2− -functions F : F3  → R which are close to H in the sense of Sect. 5. Thus (5.10) holds with δ as in Lemma 5.2. Since F is not required to satisfy the PalaisSmale condition, defining c as in (6.3) with H replaced by F does not work. Instead we shall apply Proposition 5.7 using the flow ψ from (5.13). Theorem 6.3. If F : F3  → R satisfies (5.10) with δ as in Lemma 5.2 then it has a critical point. Proof. According to Proposition 5.7 it is sufficient to find x ∈ F3  satisfying lim

t→T + (x)

H (ψ(x, t)) < ∞.

(6.2)

In fact, we shall find x ∈ S := {xζ : ζ ∈ S 1 } satisfying (6.2), where xζ is defined in (6.1). Arguing by contradiction, if (6.2) is false for every x ∈ S then#there exists T (x) ∈ R with$ H (ψ(x, T (x))) = max{a, 1+supL3  H }, where a > sup H (x) : x ∈ F3 \Mδ/2 is as in (5.14). As a consequence of Lemma 5.6 we have ψ(x, t) ∈ Mδ/2 for t > T (x). Moreover, the flow ψ coincides with the gradient flow ϕ of H in Mδ/2 by (5.12). Lemmas 5.2 and 5.6 now imply that T is continuous. So we obtain a homotopy H : S 1 × [0, 1] → F3 ,

H (ζ, t) := ψ(xζ , t T (xζ )),

with the property H (ζ, 0) = xζ and H (H (ζ, 1)) = max{a, 1 + supL3  H }. This implies H (ζ, 1) ∈ / L3  for every ζ ∈ [0, 2π ], contradicting the intersection lemma 6.2.   Proof of Theorem 1.2 for three vortices. This follows immediately from Theorem 6.3.   Lemmas 6.1 and 6.2 imply that c := sup min H (γ (ζ )) ≤ sup H . γ ∈ ζ ∈S 1

L3 

(6.3)

It is tempting to conjecture that c is a critical value of H but we are not able to prove this.

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T. Bartsch, A. Pistoia, T. Weth

7. Four Vortices In this section we prove Theorem 1.2 in the case N = 4. Here we begin with bounding H from above on the set L4  := {x ∈ F4  : (x1 , x2 , x3 ), (x2 , x3 , x4 ) ∈ L3 }. Lemma 7.1. supL4  H < ∞. Proof. For x = (x1 , x2 , x3 , x4 ) ∈ F4  we have with Lemma 2.1 and c = sup× g: H (x) = (h(x1 ) + h(x2 ) − 2g(x1 , x2 )) + (h(x3 ) + h(x4 ) − 2g(x3 , x4 )) 1 |x1 − x2 | 1 |x3 − x4 | + log + log π |x1 − x3 | π |x2 − x4 | +2g(x1 , x3 ) + 2g(x2 , x4 ) − 2G(x1 , x4 ) − 2G(x2 , x3 ) ≤ 4c.   Similarly to Sect. 6 we now assume without loss of generality that 0 ∈  ⊂ C and fix ρ > 0 such that B(0, 4ρ) ⊂ . Setting γ0 : S 1 × S 1 → F4 , γ0 (ζ1 , ζ2 ) := (ρζ1 , 0, 3ρ, 3ρ + ρζ2 ) and  := {γ : S 1 × S 1 → F4  | γ is homotopic to γ0 }, we have again an intersection property: Lemma 7.2. γ (S 1 × S 1 ) ∩ L4  = ∅ for every γ ∈ . Proof. Let H : S 1 × S 1 × [0, 1] → F4  be a deformation from γ0 to γ . For t ∈ [0, 1] we define f t : S 1 × S 1 × [0, 1] × [0, 1] → C × C by setting f t (ζ1 , ζ2 , s1 , s2 ) = (gt (ζ1 , ζ2 , s1 ), h t (ζ1 , ζ2 , s2 )) with gt (ζ1 , ζ2 , s1 ) = s1 (H1 (ζ1 , ζ2 , t) − H2 (ζ1 , ζ2 , t)) + (1 − s1 )(H3 (ζ1 , ζ2 , t) − H2 (ζ1 , ζ2 , t)) and h t (ζ1 , ζ2 , s2 ) = s2 (H2 (ζ1 , ζ2 , t) − H3 (ζ1 , ζ2 , t)) + (1 − s2 )(H4 (ζ1 , ζ2 , t) − H3 (ζ1 , ζ2 , t)), where again Hi denotes the i th component of the deformation H . Observe that γ (ζ ) = H (ζ, to f 1 (ζ, s) = 0 for some s ∈ (0, 1)2 . Clearly f t ∈  1) ∈ L4  is equivalent  C 0 (S 1 )2 × [0, 1]2 , C2 depends continuously on t ∈ [0, 1]. Moreover f t (ζ, s) = 0 for

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

681

every s ∈ ∂([0, 1]2 ) and every ζ ∈ (S 1 )2 . Thus f t induces a homomorphism in singular homology     ( f t )∗ : Z ∼ = H4 (S 1 )2 × [0, 1]2 , (S 1 )2 × ∂([0, 1]2 ) → H4 C2 , C2 \{0} ∼ = Z, and the degree of f t , deg( f t ) = ( f t )∗ (1) ∈ Z, is independent of t. For t = 0 we have f 0 (ζ, s) = (s1 ρζ1 + 3(1 − s1 )ρ, −3s2 ρ + (1 − s2 )ρζ2 ) which has nonzero degree. This can be checked easily upon observing that f 0 has a single nondegenerate zero, namely (−1, 1, 3/4, 1/4). It follows that deg( f 1 ) = deg( f 0 ) = 0, hence f 1 must have a zero and we are done.   The following theorem follows as in Sect. 6. Theorem 7.3. If F : F4  → R satisfies (5.10) with δ as in Lemma 5.2 then it has a critical point. Theorem 1.2 for four vortices is an immediate consequence. Using Lemmas 7.1 and 7.2 we have c := sup min H (γ (ζ )) ≤ sup H , γ ∈ ζ ∈(S 1 )2

L4 

(7.1)

so c ought to be a critical value of H . 8. The sinh-Poisson Equation In this section we recall how the Hamiltonian H (see (1.3)) appears as a limit functional for the sinh-Poisson problem (1.5) in the limit ρ → 0, and we prove Theorems 1.4 and 1.5. We require some facts which we state without proofs, referring to [5,16] for details. First we introduce the limit profile problem " U 2 − U = e in R , eU < +∞. (8.1) R2

In [7,29] it is shown that all the solutions of (8.1) take the form Uδ,x (y) = log

8δ 2 , (δ 2 + |y − x|2 )2

x, y ∈ R2 , δ > 0.

(8.2)

In the following, let P u denote the projection of u ∈ H1 () onto H01 (), which is defined (in the weak sense) by P u = u in ,

P u = 0 on ∂.

(8.3)

We fix N ∈ N and i ∈ {−1, +1}, i = 1, . . . , N . For x = (x1 , . . . , x N ) ∈ F N  we consider the function Vx ∈ H01 (),

Vx (y) :=

N  i=1

i P Uδi ,xi (y),

(8.4)

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where the concentration parameters δi satisfy the relation ⎛ ⎡ √ δi = di ρ,

1 ⎜ ⎢ di = di (x) = √ exp ⎝4π ⎣h(xi ) + 8

⎤⎞ N 

⎥⎟ i  j G(xi , x j )⎦⎠ .

j=1 j =i

We look for solutions to (1.5) of the form ψρ = Vx + φρ , where φρ is a higher order term in the expansion of ψρ which satisfies suitable orthogonality conditions. The next step is to reduce problem (1.5) to finite dimensions by the Ljapunov-Schmidt method. In order to do this we introduce the following nonlinear auxiliary problem: ⎧  ⎪ (Vx + φ) + ρ sinh (Vx + φ) = ci, j eUδi ,xi Z i j in , ⎪ ⎪ ⎪ i=1,...,N ⎪ ⎪ ⎪ j=1,2 ⎨ φ = 0 on ∂, (8.5) ⎪ " ⎪ ⎪ ⎪ ⎪ eUδi ,xi Z i j φ = 0 if i = 1, . . . , N , j = 1, 2, ⎪ ⎪ ⎩ 

for some coefficients ci, j ; here Z i j := [16, Lem. 4.1] and [5, Prop. 3.2]):

∂U ∂yj



y−xi δi

 . We have the following result (see

Proposition 8.1. For any ε > 0 there exists ρ0 > 0 such that for any ρ ∈ (0, ρ0 ) and x ∈ F N  with |xi − xj | ≥ ε and dist(xi , ∂) ≥ ε, problem (8.5) has a unique solution φρ (x), (ci, j (x))i, j ∈ H01 () × R N ×2 . Furthermore, the function F N  → H01 (), x → φρ (x), is of class C 1 . We note that Proposition 8.1 provides a solution Vx + φρ (x) to problem (1.5) if we find a point x ∈ F N  such that the coefficients ci, j (x) in (8.5) are all equal to zero. In order to achieve this we introduce the energy functional Iρ : H01 () → R given by " " 1 Iρ (u) := |∇u|2 d x − ρ cosh ud x, 2   whose critical points are solutions to (1.5), and we introduce the finite-dimensional restriction   + + Iρ (x) := Iρ Vx + φρ (x) . Iρ : F N  → R, According to [16, Lem. 5.1] and [5, Lem. 4.1] the following result holds. Lemma 8.2. If xρ is a critical point of + Iρ , then Vxρ + φρ (xρ ) is a critical point of Iρ , hence a solution to problem (1.5). Next, we expand + Iρ as ρ goes to 0 (see [16, Lem. 6.1] and [5, Lem. 4.2]). Lemma 8.3. The functional + Iρ satisfies + Iρ (x) = −8N π log ρ + 24N π log 2 − 16N π − 32π 2 H (x) + rρ (x)

for x ∈ F N ,

1 (F ) as ρ → 0. Here H : F  → R is where the term rρ (x) goes to zero in Cloc N  N the Hamiltonian defined in (1.3).

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

683

Proof of Theorem 1.4. We consider the special case where N = 3 or N = 4 and i = (−1)i , i = 1, . . . , N . Define, for ρ > 0, Fρ : F N  → R,

Fρ (x) = H (x) −

1 rρ (x) 32π 2

(8.6)

1 (F ) as ρ → 0, the conwith rρ (x) as in Lemma 8.3. Since rρ (x) goes to zero in Cloc N dition (5.10) is satisfied for F = Fρ and ρ small enough. Hence Theorems 6.3 and 7.3 yield the existence of critical points of Fρ . Clearly, these points are also critical for + Iρ , and by Lemma 8.2 they give rise to solutions of (1.5).  

Proof of Theorem 1.5. For general N , i = (−1)i , i = 1, . . . , N and ρ > 0, we consider the functional Fρ as in (8.6). By Theorem 3.3, Fρ has a critical point for ρ small enough, and again this yields a solution of (1.5).   9. The Lane-Emden-Fowler Equation In this section we recall how H (see (1.3)) appears as a limit functional for the LaneEmden-Fowler equation (1.8) as p → +∞, and we prove Theorems 1.6 and 1.7. Again we state some facts without proofs, referring to [19,20] for details. We also use some results obtained in [6]. Let U := U1,0 as given in (8.2), and let V be the unique radial solution of V + eU V =

1 2 U U e , 2

which satisfies 

V (y) = ν log |y| + O |y|

−1



" as |y| → +∞, ν =

+∞

t 0

t2 − 1 v(t)dt. t2 + 1

Moreover, let W be the unique radial solution of W + eU W = w in R2 , where

 w := e

U

1 2 1 3 1 4 1 2 VU − V − U − U + VU . 2 3 8 2

W satisfies the asymptotic estimate   W (y) = ω log |y| + O |y|−1

"

+∞

as |y| → +∞, ω =

t 0

Let also Vδ,x (y) := V



y−x δ



 , Wδ,x (y) := W

y−x δ

t2 − 1 w(t)dt. t2 + 1

,

x, y ∈ R2 , δ > 0,

and let P Vδ,x resp. P Wδ,x denote the projections of Vδ,x , Wδ,x , respectively onto H01 () (see (8.3)).

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T. Bartsch, A. Pistoia, T. Weth

For fixed N ∈ N and i ∈ {−1, +1}, i = 1, . . . , N , define the functions Ux ∈ H01 ()

Ux (y) :=

k  i=1



i 2 p−1

P Uδi ,xi (x) +

γ μi

1 1 P Vδi ,xi (x) + 2 P Wδi ,xi (y) p p

(9.1)

for x = (x1 , . . . , x N ) ∈ F N ; here the concentration parameters δi are defined as p

δi = μi e− 4 ,

i = 1, . . . , k,

with μi = μi ( p, x) uniquely determined by   log δi ω ν ω ν+ log(8μi4 ) = 8π H (xi , xi ) 1 − − 2 + 4p 4p p p 2   μ p−1 ω ν − 2 . i  j i 2 G(xi , x j ) 1 − + 8π 4p 4p j=i μ jp−1 We look for solutions to (1.8) of the form u p = Ux +  p , where  p is a higher order term in the expansion of u p satisfying suitable orthogonality conditions. For this we introduce the following nonlinear auxiliary problem: ⎧  p−1 ⎪ (U + ) + |U + | + ) = ci, j eUδi ,xi Z i, j in , (U x x x ⎪ ⎪ ⎪ ⎪ i=1,...,N ⎪ ⎨ j=1,2 (9.2)  = 0 on ∂, ⎪ ⎪ " ⎪ ⎪ ⎪ ⎪ ⎩ eUδi ,xi Z i, j  = 0 if i = 1, . . . , k, j = 1, 2, 

for some coefficients ci, j ; here Z i j = 2.3] we have the following result.

∂U ∂yj



y−xi δi

 is as before. According to [20, Prop.

Proposition 9.1. For any ε > 0 there exists p0 > 0 such that for any p > p0 and x ∈ F N with |xi − x j | ≥ dist(xi , ∂) ≥ ε, problem (9.2) has a unique  ε and 1 solution  p (x), (ci, j (x))i, j ∈ H0 () × R N ×2 . Furthermore, the function F N  → H01 (), x →  p (x), is of class C 1 . Proposition 9.1 provides a solution Vx +  p (x) to problem (1.8) if we find a point x ∈ F N  such that coefficients ci, j (x) in (8.5) are all equal to zero. As before we introduce the energy functional " " 1 1 1 2 J p : H0 () → R, J p (u) := |∇u| d x − |u| p+1 d x, 2  p+1  whose critical points are solutions to (1.8). The finite dimensional restriction of this functional is   J+p : F N  → R, J p (x) := J p Ux +  p (x) , and the following result holds (see [20, Lem. 2.4]).

Existence of Equilibria of N-Vortex Hamiltonian in a Bounded Domain

685

Lemma 9.2. If x p is a critical point of J+p , then Ux p +  p (x p ) is a critical point of J p , hence a solution to problem (1.8). Next we expand J+p as p goes to +∞ (see [20, Lem. 2.5]). Lemma 9.3. The functional J+p satisfies 4π e c 32π 2 e 1 8π e log p J+p (x) = N + N − H (x) + 2 r p (x), −N 2 2 2 p p p p p 1 (F ) as p → ∞. Here H is defined as in where the term r p (x) goes to zero in Cloc N  , e U (1.3) and c = 8π e + 2 R2 (e U − V )(z)dz.

Proof of Theorem 1.6. We consider the special case where N = 3 or N = 4 and i = (−1)i , i = 1, . . . , N . Define, for p > 0, F p : F N  → R,

F p (x) = H (x) −

1 r p (x), 32π 2 e

(9.3)

1 (F ) as p → ∞, the with r p (x) as in Lemma 9.3. Since r p (x) goes to zero in Cloc N condition (5.10) is satisfied for F = F p and p large enough. Hence Theorems 6.3 and 7.3 yield the existence of critical points of F p . Clearly, these points are also critical for J+p , and by Lemma 8.2 they give rise to solutions of (1.8).  

Proof of Theorem 1.5. For general N , i = (−1)i , i = 1, . . . , N and p > 0, we consider the functional F p as in (9.3). By Theorem 3.3, F p has a critical point for p large enough, and again this yields a solution of (1.8).   Acknowledgements. We would like to thank Eugenio Montefusco for his help in computer assistance. We also thank the referee for helpful remarks. T.B. and T.W. thank the Universita di Roma “La Sapienza” for the invitation and hospitality during several visits.

References 1. Adimurthi, Grossi, M.: Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity. Proc. Amer. Math. Soc. 132, 1013–1019 (2004) 2. Aref, H.: Point vortex dynamics: A classical mathematical playground. J. Math. Phys. 48, 1–22 (2007) 3. Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.: Vortex crystals. Adv. Appl. Mech. 39, 1–79 (2002) 4. Bandle, C., Flucher, M.: Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s n+2

5. 6. 7. 8. 9. 10.

equations U = eU and U = U n−2 . SIAM Reviews 38, 191–238 (1996) Bartolucci, D., Pistoia, A.: Existence and qualitative properties of concentrating solutions for the sinhPoisson equation. IMA J. Appl. Math. 72, 706–729 (2007) Chae, D., Imanuvilov, O.: The existence of non-topological multivortex solutions in the relativistic selfdual Chern-Simons theory. Commun. Math. Phys. 215, 119–142 (2000) Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615– 623 (1991) Chorin, A.J.: Vorticity and Turbulence. Applied Mathematical Sciences 103, New York: Springer-Verlag, 1994 Chorin, A.J., Marsden, E.J.: A Mathematical Introduction to Fluid Mechanics. Second Edition. Texts in Applied Mathematics 4, New York: Springer-Verlag, 1990 Chow, K.W., Ko, N.W.M., Leung, R.C.K., Tang, S.K.: Inviscid two dimensional vortex dynamics and a soliton expansion of the sinh-Poisson equation. Phys. Fluids 10, 1111–1119 (1998)

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11. Chow, K.W., Tsang, S.C., Mak, C.C.: Another exact solution for two dimensional, inviscid sinh Poisson vortex arrays. Phys. Fluids 15, 2437 (2003) 12. Crowdy, D.: A class of exact multipolar vortices. Phys. Fluids 11, 2556–2564 (1999) 13. Crowdy, D.: The construction of exact multipolar equilibria of the two-dimensional Euler equations. Phys. Fluids 14, 257–267 (2002) 14. Crowdy, D., Marshall, J.: Growing vortex patches. Phys. Fluids 16, 3122–3130 (2004) 15. Crowdy, D., Marshall, J.: Analytical formulae for the Kirchhoff-Routh path function in multiply connected domains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461, 2477–2501 (2005) 16. del Pino, M., Kowalczyk, M., Musso, M.: Singular limits in Liouville-type equations. Calc. Var. Part. Diff. Eq. 24, 47–81 (2005) 17. El Mehdi, K., Grossi, M.: Asymptotic estimates and qualitative properties of an elliptic problem in dimension two. Adv. Nonlinear Stud. 4, 15–36 (2004) 18. Esposito, P., Grossi, M., Pistoia, A.: On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. Poincaré, Anal. Non Lin. 22, 227–257 (2005) 19. Esposito, P., Musso, M., Pistoia, A.: Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent. J. Diff. Equ. 227(1), 29–68 (2006) 20. Esposito, P., Musso, M., Pistoia, A.: On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity. J. London Math. Soc. (3) 94, 497–519 (2007) 21. Esposito, P., Wei, J.: Nonsimple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation. Calc. Var. Part. Diff. Eq. 34, 341–375 (2009) 22. Flucher, M., Gustafsson, B.: Vortex Motion in Two-dimensional Hydromechanics. Preprint, TRITA-MAT1997-MA-02, 1997 23. Flucher, M., Wei, J.: Asymptotic shape and location of small cores in elliptic free-boundary problems. Math. Z. 228, 683–703 (1998) 24. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. 2nd edition. Berlin-New York: Springer-Verlag, 1983 25. Gustafsson, B.: On the convexity of a solution of Liouville’s equation. Duke Math. J. 60, 303–311 (1990) 26. Helmholtz, H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858) 27. Kirchhoff, G.: Vorlesungen über Mathematische Physik. Leipzig: Teubner, 1876 28. Kraichnan, R.H., Montgomery, D.: Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619 (1980) ∂ 2 log λ

29. Liouville, J.: Sur l’équation aux difference partielles ∂u∂v ± 2λa 2 = 0. J. de Math. 18, 71–72 (1853) 30. Mallier, R., Maslowe, S.A.: A row of counter rotating vortices. Phys. Fluids A 5, 1074–1075 (1993) 31. Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids. Applied Mathematical Sciences 96, New York: Springer-Verlag, 1994 32. Montgomery, D., Matthaeus, W.-H., Stribling, W.T., Matinez, D., Oughton, S.: Relaxation in two dimensions and the sinh-Poisson equation. Phys. Fluids A 4, 3–6 (1992) 33. Moseley, J.L.: Asymptotic solutions for a Dirichlet problem with an exponential nonlinearity. SIAM J. Math. Anal. 14, 719–735 (1983) 34. Moseley, J.L.: A two-dimensional Dirichlet problem with an exponential nonlinearity. SIAM J. Math. Anal. 14, 934–946 (1983) 35. Nagasaki, K., Suzuki, T.: Asymptotic analysis for a two dimensional elliptic eigenvalue problem with exponentially dominated nonlinearity. Asymptotic Analysis 3, 173–188 (1990) 36. Newton, P.K.: The N -vortex Problem. Berlin: Springer-Verlag, 2001 37. Newton, P.K.: N-vortex equilibrium theory. Discr. Cont. Dyn. Syst. 19, 411–418 (2007) 38. Pasmanter, R.A.: On long-lived vortices in 2-D viscous flows, most probable states of inviscid 2-D flows and soliton equation. Phys. Fluids 6, 1236–1241 (1994) 39. Ren, X., Wei, J.: On a two-dimensional elliptic problem with large exponent in nonlinearity. Trans. Amer. Math. Soc. 343, 749–763 (1994) 40. Ren, X., Wei, J.: Single point condensation and least energy solutions. Proc. Amer. Math. Soc. 124, 111– 120 (1996) 41. Saffman, P.G.: Vortex dynamics. Cambridge: Cambridge University Press, 1992 42. Spruck, J.: The elliptic sinh Gordon equation and the construction of toroidal soap bubbles. In: Calculus of variations and partial differential equations (Trento, 1986), Lecture Notes in Math. 1340, Berlin: Springer-Verlag, 1988, pp. 275–301 43. Weston, V.H.: On the asymptotic solution of a partial differential equation with exponential nonlinearity. SIAM J. Math. Anal. 9, 1030–1053 (1978) Communicated by P. Constantin

Commun. Math. Phys. 297, 687–732 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1055-2

Communications in

Mathematical Physics

Quantum Group as Semi-infinite Cohomology Igor B. Frenkel, Anton M. Zeitlin Department of Mathematics, Yale University, 442 Dunham Lab, 10 Hillhouse Avenue, New Haven, CT 06511, USA. E-mail: [email protected]; [email protected] Received: 18 December 2008 / Accepted: 19 February 2010 Published online: 7 May 2010 – © Springer-Verlag 2010

Abstract: We obtain the quantum group S L q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c + c¯ = 26. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of S L q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory. Contents 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uq (sl(2)), its Representations and Intertwining Operators . . . . . . . . Braided Vertex Operator Algebra on the Lattice of Fock Spaces . . . . . Uq (sl(2)) and the Homology of Local Systems . . . . . . . . . . . . . . The Construction of Intertwiners between Fock Spaces and Braided VOA  on Fκ = λ≥0 (V(λ),κ ⊗ Vλ ) . . . . . . . . . . . . . . . . . . . . . . . Identification of the Semi-infinite Cohomology for Fκ ⊗ F−κ . . . . . . Further Developments and Conjectures . . . . . . . . . . . . . . . . . .

. . . .

. . . .

687 691 698 703

. . 714 . . 720 . . 729

1. Introduction Soon after the original discovery of the theory of quantum groups and their representations by Drinfeld [7] and Jimbo [27], several mathematicians and physicists have realized their profound relation to the representation theory of affine Lie algebras and conformal  http://math.yale.edu/~az84; http://www.ipme.ru/zam.html.

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field theory [10,15,16,24,34,36]. Eventually, this relation has been accomplished in the precise form of equivalence of certain tensor categories of representations [17,25,28]. The nonstandard tensor product of the representations of affine Lie algebras of the same level, motivated by two dimensional conformal field theory, becomes natural in the context of vertex operator algebras (VOA) [21]. For any simple Lie algebra g let Cq be a category of type I finite-dimensional representations of the quantum group Uq (g) and let Ck be a category of standard modules of the affine Lie algebra g. The simple objects of both categories are indexed by positive highest weights, which in the case g = sl(2) can be identified with Z+ . The equivalence of braided tensor categories Cq and Ck , can be made transparent if one considers two other intermediate equivalent categories Cq ∼ = Cκ ∼ = Cc ∼ = Ck

(1.1)

based, respectively, on the homology of configuration systems and certain representations of the W-algebra, corresponding to g (see [38]). The first isomorphism has been intensively studied by Varchenko et al. (see [41,42] and references therein); the second isomorphism in the special case g = sl(2), when W-algebra is just the Virasoro algebra, is implicit in the work of Feigin and Fuks [12]; the third isomorphism is a version of the quantum Drinfeld-Sokolov reduction developed in [11]. The equivalence of representation categories points to a direct relation between the regular representations of the quantum group and affine Lie algebra, i.e. between Drinfeld’s deformed algebra of functions on the group and WZW conformal field theory. The latter does not have a vertex operator algebra structure that would place it in the context of tensor categories, but it admits a remarkable modification that does have a VOA structure ([23] and references therein). Besides, it turns out that the central charge of this modified regular VOA is precisely the one that yields a nonzero semi-infinite cohomology. A modified regular VOA can also be defined for W-algebras at the critical central charge; in the special case of Virasoro algebra, the central charge is equal to 26, pointing to a relation with string theory [23]. In fact, the relation between the theory of semi-infinite cohomology and string theory [20] has been realized at about the time of the discovery of quantum groups and played an important role in the development of both subjects. In particular, Lian and Zuckerman have shown in [33] that the zero semi-infinite cohomology of vertex operator algebras inherits a natural structure of associative and commutative algebra, which was realized as a “ground ring” in string theory [43,44]. In the case of modified regular VOA, the semi-infinite cohomology was identified as the center of the corresponding quantum group [23]. In the present paper, we obtain the full quantum group S L q (2) via the semi-infinite cohomology. Since the semi-infinite cohomology of any VOA is necessarily commutative we need to replace the modified regular VOA by a certain generalization, known as braided VOA [19,34] (based on previous work [15,35,40]). The latter is constructed from the tensor product of two braided VOA’s with the complementary charges of precisely the same values as in the modified regular VOA, which ensures nontriviality of the semi-infinite cohomology [20]. In our paper, we treat in detail only the case of the Virasoro algebra, since it is the most interesting for pure mathematical reasons and because it has important applications in physics. An extension of our construction to  and other types of W and affine Lie algebras is straightforward though technically sl(2) more difficult. The key to our realization of a quantum group is an algebra isomorphism H

∞ 2 +0

(Vir, Cc, Fc ⊗ Fc¯ ) ∼ = S L q (2),

(1.2)

Quantum Group as Semi-infinite Cohomology

689

where c + c¯ = 26, and q depends on c. The braided VOA Fc (and similarly Fc¯ ) can be realized on the space ⊕λ≥0 (V(λ),c ⊗ Vλ ),

(1.3)

where Vλ and V(λ),c are the corresponding simple modules in the equivalent braided tensor categories Cq and Cc of quantum algebra Uq (sl(2)) and the Virasoro algebra. The most transparent way to describe the braided VOA structure arises from the equivalence of both categories to the intermediate one Cκ in (1.1). As we mentioned before, the category Cκ is constructed from the homology of configuration spaces and it provides a geometric realization of the purely algebraic structure of Fc . However, it is more convenient to consider a generic version C˜κ of Cκ , which also exists for the other braided categories in (1.1), and we also obtain their equivalences ∼ C˜κ ∼ C˜q = = C˜c ∼ = C˜k .

(1.4)

In these categories we allow the weights to be arbitrary complex numbers, therefore the simple objects in each category are indexed not by Z+ as in (1.1) but by C. They are again simple highest weight modules with fixed central extensions as in (1.1) though for the generic weights in C\Z+ they coincide with the Verma modules and the contragradient Verma modules. It is also convenient to realize them as certain Fock spaces. At the generic highest weights we have semisimplicity in all four categories of (1.4), which allow us to define the structure coefficients of our generic categories in a complete parallel with the classical categories in (1.1). We verify that the structure coefficients of C˜q , C˜κ , C˜c at the generic weights are the same (the category C˜k is not considered in this paper but the equivalence with the other three categories is straightforward). Then, using the Fock space realization of the Verma modules we analytically continue certain structure coefficients that allow us to relate the structure coefficients in (1.4) with the ones in (1.1). Note that for g = sl(2) the relation between (1.1) and (1.4) can be restated in terms of analytic continuation of quantum 6 j-symbols and the corresponding identities. The full analysis of the categories in (1.4) and their equivalences at the integral points is an interesting problem, the solution of which, however, is not necessary for the main goal of the present paper. The structure of the homology of configuration spaces that determines C˜κ and the isomorphism with the representation category of the quantum group C˜q was extensively studied by Varchenko in [41,42] for an arbitrary type of Lie algebra. On the other hand, ˆ in the Fock space realization of the representations of Virasoro and sl(2) Lie algebras, the intertwining operators are given by integrals of vertex operators via certain cycles that precisely belong to H om’s of the category C˜κ . This yields the other two isomorphisms of (1.1) from their generic counterparts by a limiting procedure that we explain in our paper. In particular, this leads to a realization of the space (1.3) as a subspace of the Fock space S(a−1 , a−2 , . . . ) ⊗ S(β) ⊗ C[Z]

(1.5)

with the natural action of the Heisenberg algebra [am , an ] = 2κm δm+n,0

(1.6)

and the 1-dimensional q-deformed harmonic oscillator γβ − qβγ = q −N , [N , β] = β, [N , γ ] = −γ .

(1.7)

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The verification of the axioms of braided VOA for Fc is essentially identical to the proof of the equivalence of these tensor categories (see [38]). Though the latter equivalence has been studied in many sources (though often in disguised form [10,34]), to make the paper self contained and explicit, we provide all the necessary details that are needed for the verification of the braided VOA structure. We also explicitly identify the representatives of the semi-infinite cohomology classes that yield the generators A, B, C, D of the quantum group S L q (2) and verify the defining relations AB = q −1 B A, AC = q −1 C A, B D = q −1 D B, C D = q −1 DC, AD − D A = (q −1 − q)BC, BC = C B, AD − q −1 BC = 1.

(1.8)

Our results open new perspectives of the relation of semi-infinite cohomology and string theory. In fact, the coupling of Fc with 0  c  1 and Fc¯ with 25  c  26 can be interpreted as a coupling of a minimal model and the Liouville model with complementary central charges c + c¯ = 26, which arises in the so-called minimal string theory (see [37] and references therein). This indicates that our realization of the quantum group as the semi-infinite cohomology might admit a geometric interpretation in terms of string theory providing a new link between two subjects. Thus, our construction might be viewed as a step towards an invariant geometric description of the untamed noncommutative structure of quantum group. The paper is organized as follows. Section 2 is devoted to some basic facts about Uq (sl(2)). We recall results useful in the following: the statements about Verma and dual Verma modules, and the relations between associated intertwiners. We also derive the polynomial (q-oscillator) realization for the intertwiner between dual Verma modules. The third section is devoted to the data we will work with throughout this paper. Namely, we consider the lattice of Fock modules and the braided VOA on this space associated with Feigin-Fuks realization of Virasoro algebra. In this section we recall basic facts about irreducible Virasoro modules for the generic values of central charge: we partly use the tools which were introduced by Felder in the more complicated case of rational conformal field theory [14]. In Sect. 4, we study the geometry of local systems associated with the multivalued function corresponding to a certain correlator from the braided VOA constructed in Sect. 3. Our constructions are motivated by the heuristic constructions of Gomez and Sierra [24] and rigorous results of [22,41]. These geometric considerations allow us to construct in Sect. 5 the braided vertex algebra of intertwiners between Fock spaces and then braided VOA on the space Fc (see (1.3)). In Sect. 6, we consider a certain “double” of the braided VOA from Sect. 5. Namely, we examine the structure of the braided VOA on the space F = Fc ⊗ Fc¯ . It appears that ∞ there is a hereditary ring structure on the semi-infinite cohomology of H 2 +· (V ir, Cc, F). As we already mentioned above, one can explicitly calculate this semi-infinite cohomology on the zero level and show that it (as a space) coincides with S L q (2). Lian and Zuckerman introduced an associative product on the space of the semi-infinite cohomology of VOA. Applying the certain modification of the Lian-Zuckerman construction to our braided VOA, we reproduce the multiplicative structure of S L q (2) on the zero level of the semi-infinite cohomology space. In the last section, we outline possible extensions of the results in this paper.

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2. Uq (sl(2)), its Representations and Intertwining Operators 2.1. Basic facts and notations. Let Uq (sl(2)) be the Hopf algebra over C(q) with generators E, F, q ±H and commutation relations: q ±H E = q ±2 Eq ±H , q ±H F = q ∓2 Fq ±H , q H − q −H [E, F] = . q − q −1

(2.1)

The comultiplication is given by (q ±H ) = q ±H ⊗ q ±H , (E) = E ⊗ q H + 1 ⊗ E, (F) = F ⊗ 1 + q −H ⊗ F.

(2.2)

The universal R-matrix for Uq (sl(2)), which is an element of a certain completion of Uq (sl(2)) ⊗ Uq (sl(2)), is given by: H ⊗H

R = C, C =q 2 ,  (q − q −1 )k k E ⊗ Fk, q k(k−1)/2 = [k]!

(2.3)

k 0

q n −q −n

where [n] = q−q −1 and [n]! = [1][2] . . . [n]. For any given pair V, W of representations, the R-matrix gives the following commutativity isomorphism: Rˇ = P R : V ⊗ W → W ⊗ V , where P is a permutation: P(v ⊗ w) = w ⊗ v. We denote by Mλ the Verma module with highest weight λ ∈ C. We will say that the weight λ is generic, if λ ∈ / Z. In the case λ ∈ Z+ one obtains an irreducible finite dimensional representation Vλ (of dimension λ + 1) by means of the quotient: Vλ = Mλ / F λ+1 vλ ,

(2.4)

where vλ is the vector corresponding to the highest weight in Mλ . Let us define an algebra anti-automorphism τ : Uq (sl(2)) → Uq (sl(2)) by setting τ (E) = Fq H , τ (F) = Eq −H , τ (q H ) = q H , τ (ab) = τ (b)τ (a). Then τ is a coalgebra automorphism: (τ ⊗ τ )(x) = (τ (x)) and τ (R) = R 21 = P(R), where P(a ⊗ b) = b ⊗ a. For every module M, let the contragradient module M c be the restricted dual to M with the action of Uq (sl(2)) given by < gv ∗ , v > = < v ∗ , τ (g)v >, v ∈ M, v ∗ ∈ M c , g ∈ Uq (sl(2)).

(2.5)

Note that (M1 ⊗ M2 )c ∼ = M1c ⊗ M2c and for λ ∈ Z+ we have Vλc ∼ = Vλ . From (2.4) we have an embedding for λ ∈ Z+ : Vλ ⊂ Mλc ,

(2.6)

which leads to the following exact sequence: 0 → Vλ → Mλc → V−λ−2 → 0, and for λ ∈ Z−1 we have

Mλc

∼ = Vλ .

(2.7)

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2.2. Intertwining operators. In the following the intertwiners for Verma modules and their contragradient counterparts will play a crucial role. Namely, in this paper we will consider the elements of H om(Mμc ⊗ Mλc , Mνc ) and its dual H om(Mν , Mμ ⊗ Mλ ), such that λ, μ, ν ∈ C and μ + λ − ν ∈ 2Z. In the generic case if ν ≤ μ + λ then dim H om(Mν , Mμ ⊗ Mλ ) = 1 and otherwise dim H om(Mν , Mμ ⊗ Mλ ) = 0. We will use the following notation for the intertwining operators from H om(Mμc ⊗ Mλc , Mνc ): νμλ (· ⊗ ·) : Mμc ⊗ Mλc → Mνc ,

(2.8)

and the ones from H om(Mν , Mμ ⊗ Mλ ): μλ ν (·) : Mν → Mμ ⊗ Mλ .

(2.9)

We will also need intertwiners from H om(Vν , Vμ ⊗ Vλ ), where μ, ν, λ ∈ Z+ and their dual from H om(Vμ ⊗ Vλ , Vν ). They can be reconstructed from the intertwiners above by means of the following projections/embeddings: μλ

Pμ ⊗Pλ

ν

Mν −−→ Mμ ⊗ Mλ −−−−→ Vμ ⊗ Vλ , νμλ

i μ ⊗i λ

(2.10)

Vμ ⊗ Vλ −−−→ Mμc ⊗ Mλc −−→ Mνc , where Pξ is a standard projection on the irreducible module from the corresponding Verma module, and i ξ is an embedding of the finite dimensional irreducible module into a contragradient Verma module. It is clear that the first expression gives the element from H om(Vν , Vλ ⊗ Vμ ) and the second one corresponds to H om(Vμ ⊗ Vλ , Vν ). Similarly, one can construct intertwiners from H om(Mμ ⊗ Vλ , Mν ) and H om(Mμ ⊗ Vλ , Mν ). Let μλ ν us denote the elements of H om(Vν , Vμ ⊗ Vλ ) and H om(Vμ ⊗ Vλ , Vν ) as φν and φμλ correspondingly. It is known [8] that there exist identifications H om(Mν , Mμ ⊗ Mλ ) H om(Vν , Vμ ⊗ Vλ )

∼ = Singν (Mμ ⊗ Mλ ), ∼ = Singν (Vμ ⊗ Vλ ),

(2.11) (2.12)

where Singν denotes the space of singular vectors of the weight ν. The explicit form of the above isomorphism is given by the following map: μλ μλ ν → ν (vν ),

(2.13)

where vν is the highest weight vector in Mν . In the case of the generic values of the weights of appropriate modules, we have the following proposition, expressing the bilinear relations for the intertwining operators. Proposition 2.1. Let λi (i = 0, 1, 2, 3) be generic. Then there exists an invertible operator  BM

λ0 λ2

λ1 λ3



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such that the following diagram is commutative:



⊕ρ H om(Mρ , Mλ1 ⊗ Mλ2) ⊗H om(Mλ0 , Mρ ⊗ Mλ3 )



λ0 λ1  λ2 λ3 ⊕ξ H om(Mξ , Mλ ⊗ Mλ ) 1 3 / ⊗H om(Mλ0 , Mξ ⊗ Mλ2 )

BM

i

i



PR

H om(Mλ0 , Mλ1 ⊗ Mλ2 ⊗ Mλ3 )

 / H om(Mλ0 , Mλ1 ⊗ Mλ3 ⊗ Mλ2 ),

where ρ ∈ {λ1 + λ2 − 2k, k ∈ Z+ }, ξ ∈ {λ1 + λ3 − 2k, k ∈ Z+ }, and i is an isomorphism. The proof follows from the complete reducibility of the tensor product Verma modules in the case of generic highest weights. Using the notation we introduced above, one can write the statements of Proposition 2.1 as follows:    ξλ λ1 λ2 ρλ3 M λ0 λ1 λξ 1 λ3 λ0 2 , (1 ⊗ P R)ρ λ0 = Bρξ (2.14) λ2 λ3 ξ

 M where Bρξ

λ0 λ2

λ1 λ3

 are the matrix coefficients of the operator B M , which depend λμ

on the normalization of the intertwining operators ν . We will fix such normalization later. For the dual intertwiners λμν , we have a similar identity:    ρ ξ M λ0 λ1 λξ λ0 2 λ1 λ3 . λρλ0 3 λ1 λ2 (1 ⊗ P R) = Bξρ (2.15) λ2 λ3 ξ

In the case of integer weights, the following proposition gives the bilinear algebraic relations between the compositions of the intertwiners of finite-dimensional modules (see e.g. [8]). Proposition 2.2. Let λi ∈ Z+ (i = 0, 1, 2, 3). Then there exists an invertible operator   λ0 λ1 BV λ2 λ3 such that the following diagram is commutative:

 ⊕ρ H om(Vρ , Vλ1 ⊗ Vλ2) ⊗H om(Vλ0 , Vρ ⊗ Vλ3 )

BV

λ0 λ1 λ2 λ3

i



 / ⊕ξ H om(Vξ , Vλ1 ⊗ Vλ3) ⊗H om(Vλ0 , Vξ ⊗ Vλ2 ) i



H om(Vλ0 , Vλ1 ⊗ Vλ2 ⊗ Vλ3 )

PR

 / H om(Vλ0 , Vλ1 ⊗ Vλ3 ⊗ Vλ2 ),

where |λ1 + λ2 | ≥ ρ ≥ |λ1 − λ2 |, |λ3 + ρ| ≥ λ0 ≥ |λ3 − ρ|, |λ1 + λ3 | ≥ ξ ≥ |λ1 − λ3 |, |λ2 + ξ | ≥ λ0 ≥ |λ2 − ξ |, and i is an isomorphism.

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Using the notation we introduced above, one can tion 2.2 as follows, similarly to (2.14), (2.15):   λ0 ρλ V (1 ⊗ P R)φρλ1 λ2 φλ0 3 = Bρξ λ2 ξ   λ0 λ0 ρ V φ (1 ⊗ P R) = Bξρ φρλ λ λ 3 1 2 λ2 ξ

write the statements of Proposi λ1 ξλ φ λ1 λ3 φλ0 2 , λ3 ξ  λ1 ξ φ λ0 φ . λ3 ξ λ2 λ1 λ3

(2.16) (2.17)

The relations (2.16), (2.17) provide the main structure coefficients of the braided tensor category Cq of finite-dimensional representations of Uq (sl(2)). Similarly, the relations (2.14) and (2.15) allow us to define a generic version of Cq which we denote by C˜q . We define objects of C˜q to be infinite sums of modules from the usual category O of Uq (sl(2)), though we still may require that all the weight spaces are finite dimensional. This is sufficient to include the tensor products of Verma modules. Note that H om spaces between single Verma module and tensor products of Verma modules are finite-dimensional. Then the relations (2.14), (2.15) again determine the structure coefficients of the category C˜q at the generic weights. For the integral weights the dimensions of H om spaces might jump up (in particular, dim H om(Mν , Mμ ⊗ Mλ ) might be greater than 1, see e.g. [2]) and in order to extend the structure of the category C˜q to the integral weights, one needs a careful study of the analytic continuation of the structure coefficients. In the next subsection we give an explicit realization of the simple objects and H om’s in both categories Cq and C˜q that will allow us to relate directly their structure coefficients. 2.3. Polynomial realization for Mλc and the formula for intertwining operator. Let’s consider two variables β, ζ . We claim that the space Fλ = C[β]ζ λ carries a structure of q the Uq (sl(2)) module and one can identify it with Mλc . Let’s introduce γ = ∂β , where q ∂β is a Jackson’s q-derivative: q

∂β f (β) =

f (qβ) − f (q −1 β) . β(q − q −1 )

(2.18)

λ . These vectors span all Fλ . Moreover, the following We denote vm,λ ≡ β m ζ λ ∈ M statement holds. Proposition 2.3. Let ∂β , ∂ζ denote usual partial derivatives with respect to β, γ correspondingly. Then the following identification: E = q H γ , F = β[ζ ∂ζ − N ]q −H , H = ζ ∂ζ − 2N , −a

−q where N = β∂β is a number operator (for a = ζ ∂ζ − N , [a] = qq−q −1 ), gives a c structure of Uq (sl(2))-module on Fλ , such that Fλ is isomorphic to Mλ . a

Proof. One can find that the action of generators on basis vectors vm,λ is given by q −H Evm,λ = [m]vm−1,λ , Fq H vm,λ = [λ − m]vm+1,λ , q

±H

vm,λ = q

±(λ−2m)

vm,λ .

(2.19)

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λ is isomorhpic to M c . Moreover, one can easily get The resulting module over M λ that vm,λ corresponds to [m]!(F m vλ )∗ , where vλ is the highest weight vector in Mλ , and

vλ∗ , vλ  = 1.   Next we obtain an explicit form for the intertwining operator νμλ : Mμc ⊗ Vλ → Mνc in the realization of Proposition 2.3. Let us denote the coefficients of νμλ in such a way:

μ λ ν vn,ν , (2.20) νμλ (vm,μ ⊗ v,λ ) = m  n where we have (λ − 2) + (μ − 2m) = (ν − 2n). We also make the following notation s = λ+μ−ν 2 . At first, we consider the case  = 0. From the basic property of the intertwiner ˜ νμλ = q 2 νμλ (q −H ⊗ E˜ + E˜ ⊗ 1), where E˜ = q −H E, E we get a recurrent relation:



μ λ ν μ λ ν [n] = q 2m−μ+2 [] m  n m −1 n−1

μ λ ν . + q 2 [m] m−1  n−1 For  = 0 one obtains:

μ λ ν μ 2 [n] = q m 0 n m−1

λ 0

ν [m]. n−1

We normalize the intertwining operator by the condition:

μ λ ν = 1. s 0 0 Therefore, νμλ (vm,μ ⊗ v0,λ ) = q 2(m−n)



 m (ν) vn , n

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

and one can write the explicit formula for νμλ (· ⊗ v0,λ ) in the polynomial realization. Namely, νμλ (vm,μ ⊗ v0,λ ) =

ζ ν−μ q 2s q s (∂β ) vm,μ . [s]!

(2.26)

In order to obtain the general formula, we use again the basic property of the intertwiner, namely:  νμλ = q −2 νμλ (1 ⊗ F + F  ⊗ q H ), where F  = Fq H . F

(2.27)

Therefore,  · ⊗q λ−2 v,λ ) + F  νμλ (· ⊗ v,λ ). (2.28) q −2 [λ − ]νμλ (· ⊗ v+1,λ ) = −q −2 νμλ ( F

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Hence, νμλ (· ⊗ v,λ ) =

q 2 [λ − ( − 1)] . . . [λ] ·

 

−k q (λ−2)k ζ F

k=0

λ−2s

[s]!

k gk (, q), (∂β )s (−1)k F q

(2.29)

where 

gk (, q) =

q

−2(r1 +···+rk )

=q

−k(−1)

0r1 <···
   . k q

(2.30)

Therefore, the following statement holds. Proposition 2.4. Let λ ∈ Z+ . Then the polynomial realization for the operator νμλ (v,λ ) ≡ νμλ (· ⊗ v,λ ) : Mμc → Mνc ,

(2.31)

= s in the case of  = 0, is given where νμλ ∈ H om(Mμc ⊗ Vλ , Mνc ), such that μ+λ−ν 2 by (2.26) and, if  > 0, the explicit expression is: νμλ (v,λ ) =

q 2 [λ] . . . [λ − ( − 1)][s]!      −k λ−2s q s k (−1)k q k(λ−−1) (∂β ) F , · F ζ k q

(2.32)

k=0

where  d  F˜ = β ζ −N . dζ

(2.33)

In the case of generic λ, formula (2.32) gives the polynomial realization for the intertwiner from H om(Mμc ⊗ Mλc , Mνc ) Note that the operator we have constructed above, represents an element from H om(Mμc ⊗ Vλ , Mνc ). We note that in [39] the intertwiners H om(Mν , Mμ ⊗ Vλ ) were studied in the higher rank case. One can show that the intertwiner H om(Mμc ⊗ Vλ , Mνc ), which we have constructed in Proposition 2.4, can be extended in a unique way to another one, from H om(Mμc ⊗ Mλc , Mνc ). In fact, one can allow  to take values below λ and, therefore, one can write down the expression (2.22) in the case when  = λ + 1,



μ λ ν μ λ ν [n] = q 2m−μ+2 [λ + 1] m λ+1 n m λ n−1

μ λ ν . (2.34) + q 2 [m] m−1 λ+1 n−1

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The expression above gives a recurrent relation for the matrix elements

μ λ ν (2.35) m λ+1 n

μ λ ν which are the and allows us to express them by means of the elements m λ n coefficients we already know from the previous calculations for the intertwiner with Mλc reduced to Vλ . Once we know the expression for (2.35), we can calculate the matrix coefficients of νμλ (vλ+1,λ ). The coefficients for νμλ (vλ+k,λ ), where k > 1 can be deduced as before, by means of the action of F˜ operator (2.27). Hence we have the following statement. Corollary 2.1. There is a unique extension of νμλ ∈ H om(Mμc ⊗ Vλ , Mνc ) to the intertwining operator νμλ  ∈ H om(Mμc ⊗ Mλc , Mνc ), such that νμλ  (·, i λ ·) = νμλ (·, ·), where i λ is an inclusion i λ : Vλ → Mλc . The construction above shows that there exists a continuation of intertwiners from the generic values of weights to the integer values. Therefore, restricting the relation (2.15) to the subspaces, which in the case of the integer λi (i = 0, 1, 2, 3) corresponds to the embedding of the irreducible finite-dimensional modules, we find out that the relation between braiding matrices is the one provided by the proposition below. Proposition 2.5. Let λi ∈ Z+ (i = 0, 1, 2, 3). There exists a continuation of the elements M such that of the braiding matrix Bρ,ξ     λ0 λ1 M λ0 λ1 V Bρξ = Bρξ , (2.36) λ2 λ3 λ2 λ3 where ρ, ξ ∈ Z+ , such that λ1 + λ2 ≥ ρ ≥ |λ1 − λ2 |, λ3 + ρ ≥ λ0 ≥ |λ3 − ρ|, λ1 + λ3 ≥ ξ ≥ |λ1 − λ3 |, λ2 + ξ ≥ λ0 ≥ |λ2 − ξ |. Proof. In this section we gave the explicit construction of the intertwining operator from H om(Mμc ⊗ Mλc , Mνc ). It is important that the construction holds both in the case of generic points and in the case of integer weights. It makes sense to consider the c c c element ν˜ ˜ ∈ H om(Mμ+ ⊗ Mλ+ , Mν+ ), where 0 < 1,2 << 1, λ, μ ∈ Z+ 2 1 +2 1 μ˜ λ and the normalization of this intertwiner is the one from Proposition 2.4. Let us consider the identity (2.15) in the case of the -regularized intertwiners as above, such that λ˜ i > 0 (i = 0, 1, 2, 3). We will consider two limits with respect to regularization parameters. At first, we evaluate the limit λ˜ 2 → λ2 , λ˜ 3 → λ3 . Since the intertwining operators exist in this case (see Proposition  2.4. and  Corollary 2.1), the limit ˜ ˜ λ λ 0 1 at integer points λ , λ is M of the corresponding braiding matrix elements Bρξ 2 3 λ2 λ3 c finite. Now it is possible to restrict the intertwining operators to Vλ2 ⊂ Mλ2 , Vλ3 ⊂ Mλc3 . In a similar manner one can take the limit λ˜ 1 → λ1 and again, thanks to Proposition 2.4, the limit of the elements of the braiding matrix exists. Then it is possible to restrict ν . Hence, interwiners to Vλ1 ⊂ Mλc1 , i.e. we get braiding relations between operators φμλ one obtains that the analytic continuation of the elements of the braiding matrix to integer points exists and, when λi , ρ, ξ satisfy the conditions stated in the proposition, they coincide with the appropriate elements of the braiding matrix B V .  

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 λ0 λ1 of the category Cq can λ2 λ3 be expressed by quantum 6 j-symbols, which in its turn are given by the balanced basic hypergeometric functions 4 φ3 with the  integral values of parameters [30]. Similarly, the λ λ 0 1 M of the category C˜q are also given by the functions structure coefficients Bρξ λ2 λ3 4 φ3 with three arbitrary complex parameters corresponding to λ1 , λ2 , λ3 and three other parameters corresponding to λ0 , ρ, ξ , restricted by the integrality conditions of Proposition 2.1. This gives an analytic continuation of 6 j-symbols in 3 out of 6 parameters as well as relations between them. V It is well known that the structure coefficients Bρξ

3. Braided Vertex Operator Algebra on the Lattice of Fock Spaces 3.1. Virasoro algebra: basic facts. The Virasoro algebra [L n , L m ] = (n − m)L m+n +

c 3 (n − n)δn,−m 12

(3.1)

and its representations has been extensively studied for many years (see e.g. [18] and references therein). Here we need just basic facts. Let us denote by Mh,c and Vh,c the Verma module and irreducible module (with highest weight h), correspondingly. Throughout the paper we will consider only generic values of c. This means that if we parametrize the central charge c in such a way:

1 c = 13 − 6 κ + , (3.2) κ where the parameter κ ∈ R\Q. Then we have the following proposition (see e.g. [18]). Proposition 3.1. For a generic value of c, Verma module Mh,c has a unique singular vector in the case if h = h m,n , where h m,n =

1 2 1 1 (m − 1)κ + (n 2 − 1)κ −1 − (mn − 1). 4 4 2

(3.3)

This singular vector occurs on the level mn, i.e. the value of L 0 is h m,n + mn. In the following we will be interested in the modules with h = h 1,n = (λ), where λ = n − 1, (λ) = − λ2 + λ(λ+2) 4κ . Corollary 3.1. Let c be generic and λ ≥ 0, then V(λ),c = M(λ),c /M(λ)+λ+1,c , where V(λ),c is the irreducible Virasoro module with the highest weight (λ). For λ < 0 and generic values of c the irreducible module is isomorphic to the Verma one, namely, V(λ),c = M(λ),c . 3.2. Braided VOA on the lattice of Fock spaces. Let us consider the Heisenberg algebra [an , am ] = 2κmδn+m,0

(3.4)

and denote by Fλ,κ the Fock module associated to this algebra. Namely, Fλ,κ = S(a−1 , a−2 , . . . ) ⊗ 1λ , such that an 1λ = 0 if n > 0 and a0 1λ = λ1λ (λ ∈ C), where

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the elements 1λ form an additive group (i.e. 1λ · 1μ = 1λ+μ ), isomorphic to C. It is well known (see e.g. [18,21] and references therein) F0,κ gives rise to the vertex  that operator algebra, generated by the field a(z) = an z −n−1 , such that deg(a(z)) = 1 which has the following operator product expansion (OPE): 2κ . (z − w)2

a(z)a(w) ∼

(3.5)

We will denote this vertex algebra as F0,κ (a). Moreover, the following is true. Proposition 3.2. The vertex algebra F0,κ (a) has a vertex operator algebra structure, where the vertex operator, corresponding to the Virasoro element, is given by the following formula: L(z) =

1 κ−1  : a(z)2 : + a (z), 4κ 2κ

(3.6)

 such that L(z) = n L n z −n−2 and L n satisfy Virasoro algebra relations with the central 1 ). charge c = 13 − 6(κ + κ Now let us consider the following space: Fˆκ = ⊕λ∈Z⊕Zκ Fλ,κ .

(3.7)

Below we will show that Fˆκ carries a more general structure than VOA, namely the braided VOA. Let us consider the following operator: λa0

X(λ, z) = 1λ z 2κ e



λ 2κ

 n>0

a−n n n z



e

 λ  − 2κ n>0

an −n n z



,

(3.8)

where λ ∈ Z ⊕ Zκ. It is clear that X(λ, z)10 |z=0 = 1λ . Moreover, denoting Xn 1 ,...,n k (λ, z) ≡: a (n 1 −1) (z) . . . a (n k −1) (z)X(λ, z) :,

(3.9)

where symbol “: :” stands for the Fock space normal ordering and a (n) (z) =  the n 1 d a(z), one can see that n! dz Xn 1 ,...,n k (λ, z)10 |z=0 = a−n 1 . . . a−n k 1λ .

(3.10)

In such a way, we build the correspondence v → Y (v, z) =



v(n) z −n−1 ,

(3.11)

n∈Z

such that v ∈ Fˆκ and v(n) ∈ End( Fˆκ ). Let |z| > |w|, then λμ

X(λ, z)X(μ, w) = (z − w) 2κ (X(λ + μ, w) + · · · ),

(3.12)

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where dots stand for the terms regular in (z − w). Let us consider the paths  1 (z + w) + (w − z)eπit , 2  1 (z + w) + (z − w)eπit , z(t) = 2

w(t) =

(3.13)

where t ∈ [0, 1], and let Az,ω denote the monodromy around these paths, then Az,w (X(λ, z)X(μ, w)) = q

λμ 2

X(μ, w)X(λ, z),

(3.14)

πi

where q = e κ . The expression above should be understood in a weak sense, i.e. the analytic continuation is performed for the matrix elements of the corresponding operator products. Moreover, the matrix elements of operator product expansion (i.e. correlator, see below) X(λ, z)X(μ, w) exist in the domain |z| > |w| and the analytic continuation relates it to the matrix elements of operator product expansion X(μ, w)X(λ, z), which converge in the domain |w| > |z|. The same property (3.14) is true if we take Xn 1 ,...,n k (λ, z) instead of X(λ, z) operators. The following statement summarizes all the properties of the resulting algebraic object, which is a particular example of the braided vertex operator algebra, which we will consider in Sect. 5. Proposition 3.3. 1) There exists a linear correspondence v → Y (v, z) =



v(n) z −n−1

(3.15)

n∈Z

such that v ∈ Fˆκ and v(n) ∈ End Fˆκ . 2) Let |z| > |w| and vξ ∈ Fξ,κ , vη ∈ Fη,κ , where ξ, η ∈ C. Then   Az,w Y (vξ , z)Y (vη , w) = q ξ η/2 Y (vη , w)Y (vξ , z).

(3.16)

3) There is a vector 1 = 10 , which satisfies Y (1, z) = Id Fˆκ ,

Y (v, z)1|z=0 = v

(3.17)

for any v ∈ Fˆκ . 4) There exists an element D ∈ End( Fˆκ ) such that D1 = 0,

[D, Y (v, z)] =

d Y (v, z), dz

∀v ∈ Fˆκ .

(3.18)

5) There exists an element ω ∈ Fˆκ such that Y (ω, z) =



L n z −n−2

n∈Z

and L n satisfy the relations of Virasoro algebra with L −1 = D.

(3.19)

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We note here that similar objects were studied in [4], where they were called “abelian intertwining algebras”. In the following we will consider the subalgebra of the described above braided vertex κ = ⊕λ∈Z Fλ,κ . Let’s introduce two vertex operator algebra, related to the subspace F operators X+s (z) = X(−2, z) and X− (z) = X(2κ, z), which in the physics literature are s usually called “screening operators”. We have the following statement. Lemma 3.1. Operators X± s (z) have conformal dimension 1, i.e   n+1 ± w [L n , X± (w)] = ∂ X (w) . w s s

(3.20)

Proof. It is easy to see that the OPE of L(z) with X± s has the following form: L(z)X± s (w) ∼

X± ∂X± s (w) s (w) . + 2 (z − w) (z − w)

(3.21)

Then the statement of Lemma follows from the Cauchy theorem.

 

The expressions of the form

v1∗ , Y (u n , z n )...Y (u 1 , z 1 )v2 ,

(3.22)

where |z n | > · · · > |z 1 | > 0, v2 , u 1 , . . . , u n ∈ Fˆκ and v1∗ is the element of the dual space Fˆκ∗ , are usually called corr elator s. One can prove (via Wick theorem for the Fock space normal ordering) the following lemma (see e.g. [3]), which gives the explicit value for the correlator of operators (3.8). Lemma 3.2. Let z i ∈ R (i = 1, . . . , n), such that z n > · · · > z 1 > 0 and 1∗ν is the highest weight element in the dual Fock space such that 1∗ν , 1μ  = δν,μ . Then

1∗ν , X(μn , z n ) · · · X(μ1 , z 1 )1μ0  = δν,μn +···+μ1 +μ0



(z i − z j )

μi μ j 2κ

.

(3.23)

i> j

In the following we will be interested in the functions corresponding to the specific type of correlators, namely:

1∗ν , X(λn , z n ) . . . X+s (x ) . . . X+s (x1 ) . . . X(λ1 , z 1 )1λ0  = z (x1 , . . . , x )δν,λn +···+λ1 +λ0 −2 ,

(3.24)

where z stands for (z 1 , . . . , z n ) and z (x1 , . . . , x ) =

 i< j

(xi − x j )2/κ

 j, p

(xi − z p )−λ p /κ



(z p − z q )

λ p λq 2κ

. (3.25)

p
This will motivate our constructions in the next section, where we will consider the local system generated by the branches of this function. In the next subsection, we will study the relation between the Fock spaces and Virasoro modules.

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3.3. Irreducible highest weight Virasoro modules and the screening charge. In the previous subsection, we introduced Fock modules Fλ,κ and also screening operators X± s (z). The next proposition will explain how to construct the so-called “screening charge”, associated with X− s (z), which acts on Fλ,κ . Proposition 3.4. Let Q − be the operator Q − : Fλ,κ → Fλ+2κ,κ , where λ ∈ Z, which is defined by means of the following formula:  dz − Q−v = (3.26) Xs (z)v, v ∈ Fλ , C z 2 2πi then (i) (ii)

the action of Q − is well defined on Fλ,κ , i.e. the operator X− s (z 1 ) has only integer powers in the OPE with Y (v, z 2 ); the operator Q − commutes with the action of Virasoro operators: [Q − , L n ] = 0.

Proof.

(i) follows from the fact that λ X− s (z)X(λ, w) = (z − w) (X(λ + 2κ, w) + . . . ) ,

(3.27)

where dots stand for the terms with regular powers in (z − w). (ii) comes from the following calculation:      dw dzz n+1 n+1 − [L n , Q − ] = L(z)X− w (w) = dw∂ X (w) . w s s C 2πi Cw 2πi C (3.28)   The kernel of Q − gives an irreducible representation of the Virasoro algebra, namely, we have the following statement. Proposition 3.5. The kernel of the operator Q − acting on Fλ,κ , where λ ∈ Z, is given by the following expressions: ker Q − | Fλ,κ = V(λ),κ , λ  0, (λ) = − ker Q − | Fλ,κ = 0, λ < 0.

λ λ(λ + 2) + , 2 4κ

(3.29)

Proof. From the previous proposition we already know that Q − maps Fλ to Fλ , where λ(λ+2) λ = λ + 2κ. Therefore, (λ ) = λ+2 2 + 4κ . Let’s write the Virasoro characters for  Fλ and Fλ :  chFλ,κ = q (λ) (1 − q r )−1 , r

chFλ ,κ = q

(λ)+λ+1



(1 − q r )−1 .

(3.30)

r

Let λ  0, then chFλ,κ − chFλ ,κ = q (λ) (1 − q λ+1 )

 r

(1 − q r )−1 = chV(λ),κ .

(3.31)

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So, in order to prove the proposition we need to show that Im Q − = Fλ,κ . However it is easy to see that the vector, corresponding to the operator : a λ+1 (z)X(λ, z) :, is mapped by Q − to the highest weight state in Fλ . Therefore, we have proved the statement for λ  0. In the case of λ < 0 it is not hard to see that the highest weight from Fλ,κ is not mapped to 0 by Q − . Since Q − commutes with L n we find that ker Q − | Fλ,κ = 0, where λ < 0.   Corollary 3.2. The space Fλ,κ gives a realization for the dual Verma module of the Virasoro algebra, namely, we have the following exact sequence for λ ≥ 0 (cf. (2.7)): 0 → V(λ),κ → Fλ,κ → V(−λ−2),κ → 0,

(3.32)

and Fλ,κ ∼ = V(λ),κ for λ ∈ Z≤−1 . 4. Uq (sl(2)) and the Homology of Local Systems 4.1. Local systems on configuration space and homology. In this section, using the combination of approaches of [16,24,41], we consider geometric versions of the objects we introduced in Sect. 2, namely finite dimensional irreducible modules, Verma modules and intertwiners, or, in other words, some part of the braided tensor category associated to Uq (sl(2)). Here we give the sketch of results, considering only the results which we will need for our further constructions. For more details one should consult [16] and [41]. The key object for us will be the function z (x1 , . . . , x ) (3.25), which gives the value to the correlator with screening operators X+s . We consider the following data: λ1 , . . . , λn ∈ C, z 1 , . . . , z n ∈ C, z i = z j , z i = 0, πi κ ∈ R\Q. Let λ = λ1 + · · · + λn , q = e κ . Let us denote by H the set of hyperplanes xi = x j , i, j = 1, . . . , , xi = z k , i = 1, . . . , , k = 1, . . . , n in C = {(x1 , . . . , x )}. Now we consider the 1-dimensional local system S (see e.g. [8]) over C \H such that its flat sections are s(x) = α (univalent branch of z (x)) such that α ∈ C and (see 3.25)    λ p λq z (x1 , . . . , x ) = (xi − x j )2/κ (xi − z p )−λ p /κ (z p − z q ) 2κ , (4.1) i< j

j, p

p
where z stands for (z 1 , . . . , z n ). Now we give a prescription how one should choose a section of the corresponding local system. For any pair of indices i, j we define  ρ log(x −x ) i j   e Re xi > Re x j ρ , (4.2) Br (xi − x j ) = ρ log(x −x j i) e Re xi < Re x j where log(x) is the principal branch of the logarithm defined for Re x > 0 by the condition log x ∈ R+ for x > 0. Similarly, we define Br() =



Br(xi − x j )2/κ



Br(xi − z k )−λk /κ



Br(z i − z j )

λi λ j 2κ

.

(4.3)

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I. B. Frenkel, A. M. Zeitlin

r1

...

...

z1

rn ...

zn

Fig. 1. Basic chains

One can see that if z i − z j ∈ / iR and a region D ⊂ C is such that xi − x j ∈ / iR on D, then Br() is a section of S over D. One can define twisted -cells associated with this local system: it is a pair ( , s), where  ⊂ C \H is a singular -cell and s is the univalent branch of . In such a way one can define a boundary operator on the twisted -cell ∂ by the formula ∂ ( , s) = (∂ , s|∂ ). In this section our main objects will be homology groups H (C \H, S ) = ker ∂ /I m∂+1 modulo the permutation of the coordinates. Namely, there is a natural action of the permutation group  on a set of coordinates x1 . . . x such that this action preserves H and S . So, let us denote the antisymmetric parts of the corresponding homology groups as follows: H− (z 1 , . . . , z n , λ1 , . . . , λn ) = H− (C \H, S ).

(4.4)

4.2. Gomez-Sierra contours, Verma modules and irreducible modules. In this subsection we introduce an additional construction, which gives the geometric description of the tensor product of Verma modules. We consider the twisted chains of a specific kind, namely loops, over which xi are running, emanating from some fixed point a ∈ C around points z 1 , . . . , z n (see e.g. [16,41,42]). For calculations, it is useful to move the reference point a to infinity. In such a way, these loops will be transformed to the infinite chains with boundaries involving the point at infinity, see e.g. the chains G r1 r2 ...,rn (z 1 , . . . , z n , λ1 , . . . , λn ) on Fig. 1 (see [24]) together with the associated branch of z chosen in accordance with (4.3). At first they were introduced by Gomez and Sierra in [24], therefore in the following we will refer to these loops as Gomez-Sierra contours. These chains also appeared in [22] as dual to certain relative homology cycles. Note that the authors of [24] view these chains as integration contours (without caring about convergence) for some function f = z (x1 , . . . , x )A(z , x), where A(z , x) is a meromorphic function of (z k − zl ), (xi − x j ), (z k − x j ). Each of the contours on Fig. 1 goes counter-clockwise along the cut (provided by the branches of ) and around one of the points z i in such a way that r1 + · · · + rn = . We can treat them in a formal way as relative cycles with respect to the set of hyperplanes corresponding to the point at infinity (xi = ∞). Let us consider the space S (z 1 , . . . , z n ; λ1 , . . . , λn ), spanned by these cycles modulo the homotopy trivial ones. We will be interested in the antisymmetric part of it with respect to the action of the permutation group . In the following we denote such space as S− (z 1 , . . . , z n ; λ1 , . . . , λn ).

(4.5)

Let us ndenote by Mλ1 ⊗ · · · ⊗ Mλn [λ − 2] the subspace of weight λ − 2 (where λ = i=1 λi ) of the tensor product of the corresponding Verma modules of Uq (sl(2)).

Quantum Group as Semi-infinite Cohomology

k1

...

...

ki

705

k i+1 ...

kn ...

...

z1

ki

...

...

...

zi+1

zi

k1

^Δ i,i+1 (F) zn

ki+1 ...

...

zi+1

zi

z1

kn ...

...

zn

Fig. 2. The action of coproduct

There is a one-to-one a map: Mλ1 ⊗ · · · ⊗ Mλn [λ − 2]

ϕz

/ S − (z 1 , . . . , z n ; λ1 , . . . , λ ), 

(4.6)

such that for k1 + · · · + kn = , ϕz : F k1 vλ1 ⊗ · · · ⊗ F kn vλn −→ G k1 ,...,kn (z 1 , . . . , z n ; λ1 , . . . , λ ).

(4.7)

Therefore, there is one-to-one map between ⊕ S− (z 1 , . . . , z n ; λ1 , . . . , λn ) and Mλ1 ⊗ · · · ⊗ Mλn . Moreover, the geometric realization of the action of the F-generator is given by the statement below. Proposition 4.1. The following diagram is commutative: Mλ1 ⊗ · · · ⊗ Mλi ⊗ Mλi+1 ⊗ · · · ⊗ Mλn [λ − 2]

ϕz

/ S − (z 1 , . . . , z n ; λ1 , . . . , λn ) 

ˆ i,i+1 (F) 

i,i+1 (F)

Mλi+1

 Mλ1 ⊗ · · · ⊗ Mλi ⊗ ⊗ · · · ⊗ Mλn [λ − 2 − 2]

(4.8)

ϕz

 / S − (z 1 , . . . , z n ; λ1 , . . . , λn ). +1

ˆ i,i+1 (F) is represented by Fig. 2. where the map  Proof. In order to prove the proposition it is enough to do it in the case of two points z 1 , z 2 . Namely,

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I. B. Frenkel, A. M. Zeitlin

k2

k1 ...

k1+1 =

...

z1

k2 ...

z1

z2

+

−λ1 +2 k1

q

+

...

k1

z2

k2+1 ...

...

z1

z2

Fig. 3. The action of coproduct for two points

ˆ 1,2 (F)G k ,k (z 1 , z 2 ; λ1 , λ2 )  1 2 −λ1 −2k1 +1 G k1 ,k2 +1 (z 1 , z 2 ; λ1 , λ2 ) = G +1 k1 +1,k2 (z 1 , z 2 ; λ1 , λ2 ) + q   = ϕz (F ⊗ 1 + q −H ⊗ F)F k1 vλ1 ⊗ F k2 vλ2   = ϕz F(F k1 vλ1 ⊗ F k2 vλ2 ,

(4.9)

where the q-factor in the second line comes from the structure of branches of , see also Fig. 3. Therefore, the proposition is proven.   We can also give geometric meaning to the action of F on the whole tensor product of Verma modules. One just needs to consider the contour, which embraces not only two (as it was in Proposition 4.1), but all n families of Gomez-Sierra contours. The last statement in this subsection gives geometric meaning to the R-matrix. Proposition 4.2. Let Re z 1 < · · · < Re z i < Re z i+1 · · · < Re z n . Then the following diagram is commutative: Mλ1 ⊗ · · · ⊗ Mλn [λ − 2]

ϕz



Rˇ i,i+1



Mλ1 ⊗ · · · ⊗ Mλn [λ − 2]

/ S − (z , λ )

(4.10)

Ai,i+1

ϕsi (z )

   / S − si (z ), si (λ)  , 

 ) = (λ1 , . . . , λi+1 , λi , . . . , λn ). Here where si (z ) = (z 1 , . . . , z i+1 , z i , . . . , . . . , z n ), si (λ Ai,i+1 is the monodromy operator along the path:  1 (z i + z i+1 ) + ((z i − z i+1 )eπit , z i (t) = 2 (4.11)  1 πit (z i + z i+1 ) + ((z i+1 − z i )e , z i+1 (t) = 2

Quantum Group as Semi-infinite Cohomology

n1

707

n2

n1 ...

...

z1

n2 ...

...

12

z2

z2 z1

n1

n1 ...

n2 ...

...

z2

z1

Fig. 4. Action of the monodromy operator

z1 • · · · zi •



? • z i+1 · · · • z n .

(4.12)

Proof. The proof follows the same steps as in [36]. In order to prove this we note that is enough to prove it just for two points. In other words, we consider A12 G n 1 ,n 2 (z 1 , z 2 , λ1 , λ2 ) and reexpress it as follows (Fig. 4): A12 G n 1 ,n 2 (z 1 , z 2 , λ1 , λ2 ) n1  n 1 −k λ1 (λ2 −2n 2 )  ˆ 2 =q Cnλ11 (k) (F) G n 2 +k,0 (z 2 , z 1 , λ2 , λ1 ).

(4.13)

k=0

In order to determine the coefficients Cnλ11 (k), we use induction. Suppose, we had n + 1 contours over z 1 . Then the expression in this case can be decomposed as follows: A12 G n 1 +1,n 2 (z 1 , z 2 , λ1 , λ2 ) n1  n 1 −k+1 λ1 (λ2 −2n 2 )  ˆ 2 =q Cnλ11 (k) (F) G n 2 +k,0 (z 2 , z 1 , λ2 , λ1 ) −q

k=0 n1  −2λ1 +2n 1 )

 n 1 −k ˆ Cnλ11 (k) (F) G n 2 +k+1,0 (z 2 , z 1 , λ2 , λ1 ).

(4.14)

k=0

Therefore, the recurrent relation is: Cλn 11 +1 (k) = Cnλ11 (k) − q −2λ1 +2n 1 Cnλ11 (k − 1).

(4.15)

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I. B. Frenkel, A. M. Zeitlin

Let’s define the q-deformation of the binomial coefficient in the following way: (n)q ! n , (4.16) ≡ k q (k)q !(n − k)q ! −1 where (n)q = qq−1 and (n)q ! = (n)q (n − 1)q . . . 1. For the binomial coefficients we have the following relation:



n+1 n n 2n−2k+2 = +q . (4.17) k k − 1 q2 k q2 q2 n

Therefore, the expression for Cnλ (k) is the following one: n . Cnλ (k) = (−1)k q −2kλ+k(k−1) k q2

(4.18)

On the other hand, ((F))n−k = =

n−k  n−k r =0 n−k 

r

q2

(F n−k−r ⊗ F r )(q −r H ⊗ 1)

q −2r (n−k−r )

r =0



n−k (q −r H ⊗ 1)(F n−k−r ⊗ F r ). (4.19) r 2 q

Therefore, A12 G n 1 ,n 2 (z 1 , z 2 , λ1 , λ2 ) =q

λ1 (λ2 −2n 2 ) 2

×q

n 1 n 1 −k 

(−1)k q 2kλ1 +k(k−1)

k=0 r =0 −2r (n−k−r ) −r H

(q



n1 n1 − k k q2 r q2

⊗ 1)G n 1 +n 2 −r,r (z 2 , z 1 ; λ2 , λ1 ).

(4.20)

Using the fact that

n1 n1 − k n1 − r n1 = k q2 r q2 r k q2 q2

(4.21)

and changing the summation, we rewrite the expression above as q

λ1 (λ2 −2n 2 ) 2

×q

n 1 n 1 −r 

(−1) q

k=0 r =0 −2r (n−k−r ) −r H

(q

k 2kλ1 +k(k−1)



n1 n1 − r r q2 k q2

⊗ 1)G n 2 +n 1 −r,r (λ2 , λ1 ; z 2 , z 1 ).

(4.22)

Next, we use the formula (1 − z)(1 − q 2 z) . . . (1 − q 2n−2 z) =

n  n (−1)k q k(k−1) z k . k q2 k=0

(4.23)

Quantum Group as Semi-infinite Cohomology

709

Let us take n = n 1 − r and z = q −2λ1 +2r . Therefore,     1 − q −2λ1 +2r . . . 1 − q 2(n 1 −r )−2 q −2λ1 +2r

n  n1 − r (−1)k q k(k−1) q −2kλ1 +2r k . = k 2 q

(4.24)

k=0

We rewrite the expression above as follows: q

λ1 (λ2 −2n 2 ) 2

n 1 n 1 −r    −2λ+2n 1 −2 −2r (n−r ) n 1 1−q q r q2 r =0 =1

×(q −r H ⊗ 1)G n 2 +n 1 −r,r (λ2 , λ1 ; z 2 , z 1 ).

(4.25)

Therefore, the final result is: q

λ1 (λ2 −2n 2 ) 2

n1      −2λ1 +2n 1 −2s −2(−n 1 ) n 1 1−q q  q2 =0 s=1

×(q −(n 1 −)H ⊗ 1)G n 2 +,n 1 − (λ2 , λ1 ; z 2 , z 1 ) n1   λ1 (λ2 −2n 2 )    2 =q 1 − q −2λ+2n 1 −2s q (−n 1 −)(λ2 −2n 2 ) =0 s=1

n1 × G n +,n 1 − (z 2 , z 1 ; λ2 , λ1 ).  q2 2

(4.26)

Now we need another formula:  

E  F n 1 vλ1 =

[n 1 − s + 1][λ1 − n 1 + s]F n− vλ1 .

(4.27)

s=1

We see that         1 − q −2λ1 +2n 1 −2s = (q − q −1 ) q −λ1 +n 1 −s [λ1 − n 1 + s], s=1  

s=1

q −λ1 +n 1 −s = q (−λ1 +n 1 ) q

−(−1) 2

s=1

,

s=1

n 1 (n 1 +1) (n 1 )q 2 ! n1 q 2 [n]! = (+1) (n1 −)(n1 −+1) =  q2 ()q 2 !(n 1 − )q 2 ! []![n 1 − ]! 2 q 2 q

=

(4.28)

 1  2 [n 1 − s + 1]q − +n 1  . []! s=1

Now let’s collect all q-factors: q

λ1 (λ2 −2n 2 ) 2

=q

q (−λ1 +n 1 ) q

−(+1) 2

(λ1 −2n 1 +2)(λ2 −2n 2 −2) 2

q

q − q −(n 1 −)(λ2 −2n 2 )

(−1) 2

2

.

(4.29)

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I. B. Frenkel, A. M. Zeitlin

Hence, we have   A12 G n 1 ,n 2 (z 1 , z 2 ; λ1 , λ2 ) ⎛⎛ ⎞ ⎞  (−1) (q − q −1 ) q 2 = ϕz 1 ,z 2 ⎝⎝ F  ⊗ E  ⎠ F n 2 vλ2 ⊗ F n 1 vλ1 ⎠ []! 0   = ϕz 1 ,z 2 Rˇ · (F n 2 vλ2 ⊗ F n 1 vλ1 ) . This finishes the proof.

(4.30)

 

4.3. Singular vectors and intertwiners. In Proposition 4.1 we defined the action of the F-generator and the coproduct in a geometric way. However, the action of the E-generator is implicitly defined in the following statement. Proposition 4.3. Let Re z 1 < · · · < Re z n . There is a natural isomorphism between the spaces of singular vectors from Mλ1 ⊗ · · · ⊗ Mλn [λ − 2] and cycles Z − (z 1 , . . . , z n ; λ1 , . . . , λn ), i.e. chains from S− (z 1 , . . . , z n ; λ1 , . . . , λn ) annihilated by the boundary operator. Proof. At first, let us consider the case of one point z, i.e. we will apply the boundary operator to G  (z, λ). −1 (z, λ), where { pt} stands for the It is clear that ∂G  (z, λ) = { pt} × α (q)G −1 ∞-point and α (q) is some constant. Analyzing carefully the q-factors, we find that G  ), G  (z, λ) = { pt} × ( E

(4.31)

where G  = E

−1 

  −1 q 2(n−k−1) 1 − q 2λ+4k G −1 .

(4.32)

k=0

Each term in the sum above corresponds to the application of the boundary operator to each of the  loops, while q-factors appear from different values of ψz on the boundaries. From here one can deduce that  −1    2(n−k−1) −2λ+4k −1 G  = ϕz E q (1 − q )F vλ 

k=0

 = ϕz (q − q −1 )q H E F  vλ .

(4.33)

 and, therefore, of the boundary operator reproduces the action Hence, the action of E of E  = (q − q −1 )q H E. It is easy to see that (E  ) = q H ⊗ E  + E  ⊗ 1. Now let us apply the boundary operator to G k1 ,k2 (z 1 , z 2 ; λ1 , λ2 ). We have ∂G k1 ,k2 (z 1 , z 2 ; λ1 , λ2 )   = { pt} × G k−1 α (q) + q −λ1 −2k1 G k−1 α (q) 1 −1,k2 k1 1 ,k2 −1 k2   = { pt} × ϕz (E  )F k1 vλ1 ⊗ F k2 vλ2 .

(4.34)

Quantum Group as Semi-infinite Cohomology

z1

711

z2

z1

z2

Fig. 5. Pochhammer loop P 1 decomposed

One can continue it to the case of n points obtaining that the action of the boundary operator is equivalent to the action of (q − q −1 )q H E on the product of Verma modules. Therefore, we have a one-to-one map between singular vectors in the product of Verma  modules and cycles in ⊕ S− (z 1 , . . . , z n ; λ1 , . . . , λn ).  In [41], Varchenko found the homological description of singular vectors in the tensor product of Verma modules. Let us recall the notation (4.4) of Subsect. 4.1. Theorem 4.1. There is a natural isomorphism between H− (z 1 , z 2 ..., z n ; λ1 , λ2 ...λn ) and singular vectors on the level λ − 2 in the tensor product of n Verma modules, i.e. Sing(Mλ1 ⊗ · · · ⊗ Mλn )[λ − 2], where λ = λ1 + · · · + λn . According to the statement of Theorem 4.1 and Proposition 4.3 we can identify  , z ) with H − (λ  , z ). Then, restricting the result of Proposition 4.2 to the subspace Z − (λ  of singular vectors and its geometric counterpart, we rederive the result of Varchenko [41]. Theorem 4.2. The following diagram commutes: Singλ−2 (Mλ1 ⊗ · · · ⊗ Mλi ⊗ Mλi+1 ⊗ · · · ⊗ Mλn )

i

Rˇ i,i+1

 Singλ−2 (Mλ1 ⊗ · · · ⊗ Mλi +1 ⊗ Mλi+1 ⊗ · · · ⊗ Mλn )

/ H − (z 1 , . . . , z n ; λ1 , . . . , λn ) 

(4.35)

Ai,i+1 i

 / H − (z 1 , · · · , z n ; λ1 , . . . , λn ). 

Here i is an isomorphism and Ai,i+1 is the monodromy operator along the paths:  1 (z i + z i+1 ) + ((z i − z i+1 )eπit , 2  1 (z i + z i+1 ) + ((z i+1 − z i )eπit . z i+1 (t) = 2 z i (t) =

(4.36)

In the case of  = 1, it is easy to see that one of the singular vectors is given by the Pochhammer loop P 1 , which can be easily decomposed in terms of chains from S− (z 1 , z 2 ; λ1 , λ2 ) (see Fig. 5). We recall that in Sec. 2 we identified the space of Singν (Mλ1 ⊗ Mλ2 ) with H om(Mν , Mλ1 ⊗ Mλ2 ). Let’s consider the cycle from Hs− (z 1 , z 2 ; λ1 , λ2 ), corresponding to an element of Singλ1 +λ2 −2s (Mλ1 ⊗ Mλ2 ). We will denote it by Pλs1 λ2 and draw it schematically as the product of s Pochhammer cycles (in case it is nonzero), motivated

712

I. B. Frenkel, A. M. Zeitlin

z1 ...

...

zi

... zn

^ μν Φλ i

...

... S

z1...

...

...

zi

w

... zn

Fig. 6. Intertwiner’s action

by analogy with the s = 1 case. One can notice that there is exactly one Pλs1 λ2 , i.e. singular vector on the level λ1 + λ2 − 2s if λ1 , λ2 ≥ 0 and 0 ≤ s ≤ λ1 + λ2 − |λ1 − λ2 |. Now one can construct an intertwiner, using this geometric language and the equivalence between the H om-spaces and the spaces of singular vectors. In order to do this, ˆ μν (z, w), constructed as it is shown on Fig. 6. Namely, one should consider the map  λi − this is a map from S− (z 1 , . . . , z n ; λ1 , . . . , λn ) to S+s (z 1 , . . . , z i , w, z i+1 , . . . , z n ; λ1 , . . . , λi−1 , μ, ν, λi+1 , . . . , λn ). It is constructed by means of puncturing an additional point w inside the Gomez-Sierra contours surrounding z i such that Re z i < Re w < s winding around them. This operator is preRe z i+1 and the insertion of the cycle Pμν μν s corresponds to a singular cisely a geometric version of an intertwiner λi , since Pμν vector. Namely, the following proposition holds. Proposition 4.4. Let Re z 1 < · · · < Re z n . Then the following diagram is commutative: Mλ1 ⊗ · · · ⊗ Mλi ⊗ · · · ⊗ Mλn [λ − 2] 

i

μν i

(4.37)

ˆ μν (z i ,w)  λ

λ

Mλ1 ⊗ · · · ⊗ Mμ ⊗ Mν ⊗ · · · ⊗ Mλn [λ − 2 − 2s]

/ S − (z 1 , . . . , z n ; λ1 , . . . , λn )  i

i

 − / S+s (z 1 , . . . , z i , w, z i+1 , . . . , z n ; λ1 , . . . , λi−1 , μ, ν, λi+1 , . . . , λn ),

μν

where λi is the element of H om(Mλi , Mμ ⊗ Mν ) corresponding to the singular vector from Singλi (Mμ ⊗ Mν ) geometrically represented by Pλs1 λ2 , and Re z i < Re w < Re z i+1 . Therefore, for generic values of weights we have the following proposition which gives the bilinear relations between “geometric” intertwiners, defined above. Proposition 4.5. Let Re z 1 < Re z 2 < Re z 3 and λi (i = 0, 1, 2, 3) be generic. Then we have   ˆ ρλ3 (z 1 , z 3 ) ˆ λρ1 λ2 (z 1 , z 2 ) A2,3  λ0    M λ0 λ1 ˆ λ1 λ3 (z 1 , z 3 ) ˆ ξ λ2 (z 1 , z 2 ),  Bρξ (4.38) = ξ λ0 λ2 λ3 ξ

μ

ˆ in the expression above act on the space S − (z ; λ  ). where the intertwiners  νλ 

Quantum Group as Semi-infinite Cohomology

713

Proof. This statement is a consequence of Propositions 2.1 and 4.2. We already know that the action of the monodromy operatoris equivalent to the action of the R-matrix ˆ ρλ3 (z 1 , z 3 ) corresponds to ˆ λρ1 λ2 (z 1 , z 2 ) on the product of modules. Therefore, A2,3  λ0 ρλ

the expression (1 ⊗ P R)λρ1 λ2 λ0 3 . On the other hand, we know that this is equal (see    M λ0 λ1 ξλ λξ 1 λ3 λ0 2 . Thus the proposition is proven.  (2.14)) to ξ Bρξ  λ2 λ3

4.4. Irreducible representations of Uq (sl(2)) via local systems. In this section, we considered relations between tensor products of Verma modules and local systems on configuration space. There is also a possibility to include the irreducible modules of Uq (sl(2)) in such a correspondence. In order to do that one should understand the factorization map Mλ → Mλ /M−λ−2 (for λ ≥ 0) in the geometric context. This can be established by considering the relative cycles (w.r.t. the set of xi = z j ) instead of, usual ones. Namely, we denote by G˜ r1 ,r2 ...rn [z 1 , . . . , z n , λ1 , . . . , λn ] the same geometric object as before, but consider it as a relative cycle not only with respect to the hyperplanes corresponding to the reference point, but also with respect to the set xi = z p , where i = 1, . . . ,  and p = 1, . . . , n. Let us denote the antisymmetric (w.r.t. the action of the group ) part of the space of such cycles factorized by homologically trivial ones as S˜− [z 1 , . . . , z n ; λ1 , . . . , λn ]. One can give an explicit geometric picture of transfer from the representatives corresponding to absolute cycles (associated with basic contours) to relative cycles. It can be achieved by means of the shrinking of the appropriate contours in G r1 ,r2 ...rn (z 1 , . . . , z n , λ1 , . . . , λn ) and the moving all the resulting rays from z k to ∞ on the left-hand side of the cut, as it is shown in Fig. 7. Counting the q-powers at the points z 1 , . . . , z n , we obtain that the elements represented by G˜ r1 ,r2 ...rn [z 1 , . . . , z n ] ∈ S˜− [z 1 , . . . , z n ; λ1 , . . . , λn ], will be annihilated if ri ≥ λi and λi ∈ Z+ (i = 1, . . . , n). In the example of one point, as shown  on Fig. 7, the resulting factor, corresponding to the right hand side of the picture, is: k (1−q 2(λ−k) ) (each multiplier is a contribution from a different basic contour), which vanishes when k reaches λ. In such a way we have a natural map between ⊕ S˜− [z 1 , . . . , z n ; λ1 , . . . , λn ] and Vλ1 ⊗ · · · ⊗ Vλn . Namely, it is the map: Vλ1 ⊗ · · · ⊗ Vλn [λ − 2]

ϕ˜ z

/ S˜ − [z 1 , . . . , z n ; λ1 , . . . , λ ], 

(4.39)

such that for k1 + · · · + kn = , ϕ˜ z : F k1 vλ1 ⊗ · · · ⊗ F kn vλn −→ G˜ k1 ,...,kn [z 1 , . . . , z n ; λ1 , . . . , λ ].

(4.40)

This allows us to formulate a statement. Proposition 4.6. Let λi ∈ Z+ (i = 1, . . . , n). Then the following diagram is commutative: Mλ1 ⊗ · · · ⊗ Mλn [λ − 2]

ϕz



p1...n



Vλ1 ⊗ · · · ⊗ Vλn [λ − 2]

/ S − (z , λ)  f 1...n

ϕ˜ z

 / S˜ − [z ; λ],  

(4.41)

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...

...

z λ

z λ

Fig. 7. Shrinking of a cycle

where the map p1...n is just the composition of projections on the corresponding irreducible modules and the map f 1...n is given by the composition of shrinking maps as demonstrated on Fig. 7. Similar construction of the tensor product of finite-dimensional representations was used in [22] in the framework of [41]. Our last task will be to give the geometric meaning to the relations between intertwiners, corresponding to finite-dimensional representations. Proposition 4.7. Let Re z 1 < Re z 2 < Re z 3 and λi ∈ Z+ (i = 0, 1, 2, 3). Then we have   ˆ λρ1 λ2 (z 1 , z 2 ) ˆ ρλ3 (z 1 , z 3 ) A2,3  λ0    λ0 λ1 ˆ λ1 λ3 V ˆ ξ λ2 (z 1 , z 2 ), ξ (z 1 , z 3 ) Bρξ (4.42) = λ0 λ2 λ3 ξ

ˆ μ in the expression above act on the space where ρ, ξ ∈ Z+ and the intertwiners  νλ  ]. S˜− [z ; λ Propositions 4.5 and 4.7 are the geometric versions of Propositions 2.1 and 2.2. Formula (4.42) is very important for deriving the relation between the intertwiners of Fock space modules, which we will study in the next section. 5. The Construction of Intertwiners between Fock Spaces and Braided VOA  on Fκ = λ≥0 (V(λ),κ ⊗ Vλ ) 5.1. Intertwiners between Fock spaces. Let’s consider the multivalued function z (x1 , . . . , xs ) (3.25). As we already know, this function determines a local system on a configuration space. One can consider differential forms on Cs of the following kind: z 1 ,...,z n (x1 , . . . , xs )A(z 1 , . . . , z n ; x1 , . . . , xs )dx 1 ∧ · · · ∧ dx s

(5.1)

and integrate them over cycles in Hs− (Cs \H, S ), if A(z 1 , . . . , z n ; x1 , . . . , xs ) is symmetric in xi . Now we reduce the number of z-variables to two of them, namely, we choose them to be z, 0 and consider the following expression for A: ∗

Aλ,v,v (z, 0; x1 , . . . , xs ) = < v ∗ , : X(λ, z)X+s (x1 ) . . . X+s (xs ) : v >,

(5.2)

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715

∗ where v ∈ Fμ,κ and v ∗ ∈ Fλ+μ−2s, κ . If we integrate the corresponding differential form s − s over a cycle Pλμ ∈ Hs (C \H, S ), which we constructed in the previous section, one can consider the resulting expression as a matrix of some operator

ϕ(·, z) : Fμ,κ → Fλ+μ−2s,κ .

(5.3)

Moreover, the following statement holds [13]. Proposition 5.1. Let λ, μ, ν ∈ Z such that ν ≤ λ + μ. Then there exists an intertwining operator νλμ (z) : Fλ,κ ⊗ Fμ,κ → Fν,κ [[z, z −1 ]]z (ν)−(μ)−(λ) ,

(5.4)

i.e. the operator, such that L n · νλμ (z) = νλμ (z)z,0 (L n ),

(5.5)

  dξ n+1  −m−2 ξ (ξ − z) Lm ⊗ 1 + 1 ⊗ Ln. 2πi m

(5.6)

where  z,0 (L n ) = z

In the particular case when the first argument is the highest weight vector 1λ ∈ Fλ,κ , the explicit expression for the matrix elements of an intertwiner are given by the following formula: 

v ∗ , νλμ (z)(1λ ⊗ v) = 0,z (x1 , . . . , xs ) s Pλμ

∗

× v , :

X(λ, z)X+s (x1 ) . . . X+s (xs )

∗ where v ∈ Fμ,κ , and v ∗ ∈ Fλ+μ−2s, κ (s =

: vdx 1 ∧ · · · ∧ dx s ,

(5.7)

λ+μ−ν 2 ).

Proof. One can construct an intertwining operator by means of the following procedure. Let’s consider the correlator

v ∗ , Y (u, z)X+s (x1 ) . . . X+s (xs )v,

(5.8)

where u ∈ Fλ,κ and z, x1 , , , xs ∈ R such that z > x1 > · · · > xs . This expression can be rewritten in the form u 0,z (x1 , . . . , xs ) f v,v ∗ (z, x 1 , . . . , x s ),

(5.9)

u (z, x , . . . , x ) is a rational function of z, x , . . . , x . One can make the anawhere f v,v ∗ 1 s 1 s lytic continuation of this multivalued function to the complex domain, using the branches of 0,z (x1 , . . . , xs ). Therefore, one can define the matrix elements of the intertwiner by

v ∗ , νλμ (z)(u ⊗ v)  u 1 s 0,z (x1 , . . . , xs ) f v,v = ∗ (z, x 1 , . . . , x s )dx ∧ · · · ∧ dx . s Pλμ

(5.10)

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In order to check the property (5.5), one needs to consider the expression  dξ n+1 ∗

v , ξ L(ξ )Y (u, z)X+s (x1 ) . . . X+s (xs )v, C 2πi

(5.11)

where the contour C is a circle including point 0 and z. The formula above can be reexpressed in the following way:  dξ n+1

v ∗ , ξ (ξ − z)−n−2 Y (L n · u, z)X+s (x1 ) . . . X+s (xs )v z 2πi + Y (u, z)X+s (x1 ) . . . X+s (xs )L n · v s  Y (u, z)X+s (x1 ) . . . ∂xi (xin+1 X+s (xi )) . . . X+s (xs ) · v. (5.12) + i=1

Since in the formula (5.10) we are integrating over the closed cycle, terms with derivatives of screening operators disappear due to Stokes theorem and we arrive to the formula (5.5). One can obtain the coefficient z (ν)−(μ)−(λ) by a simple change of variables. Namely, replacing xi → xzi we get an expression for the nonsingle-valued multiplier from < v ∗ , νλμ (z)(u ⊗ v) >: ν

λμ s(s − 1) λ + μ + − . 2κ κ 2κ = (ν) − (λ) − (μ).  

(z 1 − z 2 )λμ , One can easily see that νλμ

νλμ = s +

(5.13)

As in the case of Uq (sl(2)) we will be interested in the composition of intertwinρ ing operators. The composition of intertwiners λλ03 ρ (z)λ2 λ1 (w) is understood as an operator, acting on Fλ3 ⊗ Fλ2 ⊗ Fλ1 . We recall that when we studied local systems, we have introduced the operators ˆ μν (z 1 , z 2 ) geometrically representing intertwining operators. They obey the braid λ ing relation, repeating the one for the finite-dimensional representations of Uq (sl(2)). Namely, for 0 < Re z 1 < Re z 2 ,   ˆ ρλ3 (0, z 2 ) ˆ λρ1 λ2 (0, z 1 ) Az 1 ,z 2  λ0    λ0 λ1 ˆ λ1 λ3 V ˆ ξ λ2 (0, z 1 ), ξ (0, z 2 ) Bρξ (5.14) = λ0 λ2 λ3 ξ

where the expression above acts on ⊕ S˜− [λ1 , λ2 , λ3 ; 0, z 1 , z 2 ] and λi ∈ Z+ , (i = 0, 1, 2, 3). If one integrates suitable expressions, like the integrand of (5.10) over the cycles from the expression above, one arrives to the following statement: Proposition 5.2. Let z 1 , z 2 ∈ R, such that 0 < z 1 < z 2 and λi ≥ 0 (i = 0, 1, 2, 3), κ > maxλi . Then the following relation holds:   ρ Az 1 ,z 2 λλ03 ρ (z 2 )λ2 λ1 (z 1 ) (P ⊗ 1)    λ0 λ1 ξ V λλ02 ξ (z 1 )λ3 λ1 (z 2 ), = Bρξ (5.15) λ2 λ3 ξ

where P is the interchange operator, namely P(v1 ⊗ v2 ) = v2 ⊗ v1 .

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717

Using the explicit values for the “screened” correlators in the Coulomb gas formalism studied in the physics literature (see e.g. [5,6]), one can show that the relation (5.15) can be analytically continued in κ to any value in R\Q.  5.2. Braided λ (V(λ),κ ⊗ Vλ ). Let’s consider the following space:  VOA on Fκ = Fκ = (V ⊗ V ). Below we will show that there exists a structure of a λ λ∈Z+ (λ),κ braided VOA on this space. First of all, we define the following map:  ν Y : v ⊗ a → Y (v ⊗ a, z) = νλμ (z)(v ⊗ ·) ⊗ φμλ (· ⊗ a). (5.16) ν,μ

Here v ∈ Fλ,κ and a ∈ Vλ for some λ ∈ Z. As a consequence of Proposition 3.4 we have the following statement. Lemma 5.1. [Q − , νμλ (z)(v ⊗ ·)] = 0 if v ∈ V(λ),κ ⊂ Fλ,κ . Hence, [Q − ⊗ 1, Y (v ⊗ a, z)] = 0 if v ∈ V(λ),κ . Therefore, Y acts as follows: Y : Fκ → End(Fκ ){z}.

(5.17)

Let 0 < z 1 < z 2 , z 1 , z 2 ∈ R, vi ∈ V(λi ),κ (i = 1, 2, 3), ai ∈ Vλi (i = 1, 2, 3). Then Az 2 ,z 1 (Y (v1 ⊗ a1 , z 2 )Y (v2 ⊗ a2 , z 1 )) (v3 ⊗ a3 ) ⎛ ⎞  ρ ρ ν ⎠· νλ1 ρ (z 2 )λ2 λ3 (z 1 ) ⊗ φρλ φ = Az 2 ,z 1 ⎝ 1 λ3 λ2 λ1 ,λ2 ,ν,ρ

(v1 ⊗ v2 ⊗ v3 ) ⊗ (a3 ⊗ a2 ⊗ a1 ) ⎛ ⎞    ν λ ξ ρ 3 V ν ⎠· ⊗ φρλ =⎝ νλ2 ξ (z 1 )λ1 λ3 (z 2 )Bρξ φ 1 λ3 λ2 λ2 λ1 λ1 ,λ2 ,ν,ρ,ξ

(v2 ⊗ v1 ⊗ v3 )(a3 ⊗ a2 ⊗ a1 ) ⎛ ⎞  ξ ξ =⎝ νλ2 ξ (z 1 )λ2 λ3 (z 2 ) ⊗ φξνλ2 φλ3 λ1 ⎠ · λ1 ,λ2 ,ν,ξ

(v2 ⊗ v1 ⊗ v3 ) ⊗ (a3 ⊗ =





(2)

(5.18)

(1)

r i a1 ⊗ r i a2 )

i (1) Y (v2 ⊗ ri a2 , z 1 )Y (v1

(2)

⊗ ri a1 , z 1 )(v3 ⊗ a3 ).

i

Hence, we have proved the following proposition: Proposition 5.3. Map Y satisfies the commutativity condition, namely let z, w ∈ R, such that 0 < z < w, then Az,w (Y (v1 ⊗ a1 , w)Y (v2 ⊗ a2 , z))  Y (v2 ⊗ ri(1) a2 , z)Y (v1 ⊗ ri(2) a1 , w), = where R =



(1) i ri

i (2)

⊗ ri

is the universal R-matrix for Uq (sl(2)).

(5.19)

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I. B. Frenkel, A. M. Zeitlin

The next proposition shows that the Virasoro action is compatible with correspondence Y . Proposition 5.4. (i) There is a natural action of the Virasoro algebra on F˜ κ = ⊕λ∈Z+ V(λ),κ ⊗ Vλc , namely L n · (v ⊗ a) = (L n · v) ⊗ a, where v ∈ Fλ,κ , a ∈ Mλc for some λ. Moreover, there exists an element ω˜ ∈ V(0),κ ⊗ V0 , namely ω˜ = ω ⊗ v0 (v0 is  ˜ z) = L(z) = n L n z −n−2 is the the only basis element in V0 ), such that Y (ω, Virasoro element. (ii) [L −1 , Y (v ⊗ a, z)] = ∂z Y (v ⊗ a, z). Proof. (i) is obvious. (ii) follows from the explicit definition of the intertwining operator νλμ (z).

 

Finally, we show that the associativity condition holds (see e.g [21]). First, we will prove the following lemma. Lemma 5.2. Map Y satisfies the creation property, namely: (i) Y (v ⊗ a, z)1 = e z L −1 (v ⊗ a), (ii) e z 2 L −1 Y (v ⊗ a, z 1 )e−z 2 L −1 = Y (v ⊗ a, z 1 + z 2 ). Proof. At first, we prove (ii). Let ad L −1 · = [L −1 , ·]. Then ad L −1 Y (v ⊗ a, z) = ∂wn Y v ⊗ a, z). Now ∞ n  z2 n ∂ Y (v ⊗ a, z 1 ) = Y (v ⊗ a, z 1 + z 2 ). n! z 1

(5.20)

e z 2 L −1 Y (v ⊗ a, z 1 )e−z 2 L −1 = Y (v ⊗ a, z 1 + z 2 ).

(5.21)

n=0

Therefore,

(i) follows as an easy consequence.

 

Proposition 5.5. Let t, w, z ∈ R, such that 0 < t < w < z. Then Y (v1 ⊗ a1 , z)Y (Y (v2 ⊗ a2 , w − t)v3 ⊗ a3 , t) 1 = Y (Y (v1 ⊗ a1 , z − w)v2 ⊗ a2 , w) Y (v3 ⊗ a3 , t)1.

(5.22)

Proof. At first, we recall the quasitriangular property of the universal R-matrix: (I ⊗ )R = R 13 R 12 ,

(5.23)

or, in components: (1)

ri

(2)

⊗ (ri ) =



(1) (2)

(2)

(2)

ri r j ⊗ r j ⊗ ri .

(5.24)

i, j

We combine it with Lemma 5.2 and Proposition 5.3 and derive (5.22) as follows. Let us denote by Az,w the inverse of our usual monodromy: | z• •< w . (5.25)

Quantum Group as Semi-infinite Cohomology

719

Then, Y (v1 ⊗ a1 , z)Y (v2 ⊗ a2 , w)Y (v3 ⊗ a3 , t)1 ⎞ ⎛  (1) (2) = At,w ⎝ Y (v1 ⊗ a1 , z)Y (v3 ⊗ r j a3 , t)Y (v2 ⊗ r j a3 , w)⎠ 1 j

⎛  (1) (2) (2) = At,z At,w ⎝ Y (v3 ⊗ r j r j a3 , t)Y (v1 ⊗ r j a1 , z)· j



Y (v2 ⊗ r (2) j a2 , w)1

⎛ ⎞  (1) (1) (2) (2) Y (v3 ⊗ ri r j a3 , t)Y (v1 ⊗ ri a1 , z)ewL −1 v2 ⊗ r j a2 ⎠ = At,z At,w ⎝ i, j

⎛ ⎞  (1) (1) (2) (2) = At,z At,w ⎝ Y (v3 ⊗ ri r j a3 , t)ewL −1 Y (v1 ⊗ ri a1 , z − w)v2 ⊗ r j a2 ⎠ 1 i, j

⎞ ⎛  (1) (1) (2) (2) = At,z At,w ⎝ Y (v3 ⊗ ri r j a3 , t)Y (Y (v1 ⊗ ri a1 , z − w)v2 ⊗ r j a2 , w)⎠ 1 i, j

= Y (Y (v1 ⊗ a1 , z − w)v2 ⊗ a2 , w) Y (v1 ⊗ a3 , t)1. In the last line, we used the quasitriangular property.

(5.26)

 

All these properties allow us to give a general definition of general braided VOA (cf. Proposition 3.3 and see also [38]). Definition. Let V = ⊕λ∈I Vλ be a direct sum of graded complex vector spaces, called sectors: Vλ = ⊕n∈Z+ Vλ [n], indexed by some set I . Let λ , λ ∈ I be complex numbers, which we will call conformal weights of the corresponding sectors. We say that V is a braided vertex operator algebra, if there are distinguished elements 0 ∈ I such that 0 = 0, 1 ∈ V0 [0], linear maps D : V → V, R : V ⊗ V → V ⊗ V and the linear correspondence  λλ Yλ1 2 (z), (5.27) Y(·, z)· : V ⊗ V → V{z}, Y = λ,λ1 ,λ2

where Yλλ1 λ2 (z) ∈ H om(Vλ1 ⊗ Vλ2 , Vλ ) ⊗ z λ −λ1 −λ2 C[[z, z −1 ]],

(5.28)

such that the following properties are satisfied: i) Vacuum property: Y(1, z)v = v, Y(v, z)1|z=0 = v. ii) Complex analyticity: for any vi ∈ Vλi , (i = 1, 2, 3, 4) the matrix elements

v4∗ , Y(v3 , z 2 )Y(v2 , z 1 )v1  regarded as a formal Laurent series in z 1 , z 2 , converge in the domain |z 2 | > |z 1 | to a complex analytic function r (z 1 , z 2 ) ∈ z 1h 1 z 2h 2 (z 1 − z 2 )h 3 C[z 1±1 , z 2±1 , (z 1 − z 2 )−1 ], where h 1 , h 2 , h 3 ∈ C. d Y(v, z). iii) Derivation property: Y(Dv, z)1 = dz

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I. B. Frenkel, A. M. Zeitlin

iv) Braided commutativity (understood in a weak sense):  Az 1 ,z 2 (Y(v, z 1 )Y(u, z 2 )) = Y(u i , z 2 )Y(vi , z 1 ),

(5.29)

i

 where R(u ⊗ v) = i u i ⊗ vi . v) There exists an element ω ∈ V0 , such that  Y (ω, z) = L n z −n−2

(5.30)

n∈Z

and L n satisfy the relations of Virasoro algebra with L −1 = D. vi) Associativity (understood in a weak sense): Y(Y(u, z 1 − z 2 )v, z 2 ) = Y(u, z 1 )Y(v, z 2 ).

(5.31)

In such a way, we have constructed the operators satisfying all necessary properties of the VOA algebra on Fκ . Therefore, we proved the following statement. Theorem 5.1. The correspondence Y : Fκ → End(Fκ ){z} defined by (5.16) gives a braided VOA structure on Fκ . 6. Identification of the Semi-infinite Cohomology for Fκ ⊗ F−κ 6.1. Semi-infinite cohomology: a reminder. In this subsection, we just recall basic facts about semi-infinite cohomology for Virasoro algebra and provide some basic statements which allow to compute explicit formulas for cycles. In the special case of the Virasoro algebra, semi-infinite forms can be realized by means of the following super Heisenberg algebra: {bn , cm } = δn+m,0 , n, m ∈ Z.

(6.1)

One can construct a Fock module  in such a way:  = {b−n 1 . . . b−n k c−m 1 . . . c−m  1; ck 1 = 0, k  2; bk 1 = 0, k  −1}.

(6.2)

This Fock module has a VOA structure on it, namely, one can define two quantum fields:   b(z) = bm z −m−2 , c(z) = cn z −n+1 , (6.3) m

n

which according to the commutation relations between modes have the following operator product: b(z)c(w) ∼

1 . z−w

(6.4)

The Virasoro element is given by the following expression: L  (z) = 2 : ∂b(z)c(z) : + : b(z)∂c(z) :, such that b(z) has conformal weight 2, and c(z) has conformal weight −1.

(6.5)

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721

The central charge of the corresponding Virasoro algebra is equal to -26. One can define the following operator: N g (z) =: c(z)b(z) :,

(6.6)

which is known as the ghost current. The reason for such name is the following  number dz N g (z) gives an integer grading to the Fock module , one. The operator N g = 2πi namely, N g 1 = 0, [N g , bn ] = −bn , [N g , cm ] = cm .

(6.7)

Next, we consider the semi-infinite cohomology (BRST) operator [29,20]. Let V be the space, equipped with the structure of the VOA, where the Virasoro algebra has central charge, equal to 26. Let’s consider the tensor product V ⊗ . This space has a structure of VOA, such that the central charge of the Virasoro algebra is equal to 0. Moreover, the space V ⊗  has integer grading with respect to the operator N g . The following proposition holds (see e.g. [20]). Proposition 6.1. The operator of ghost number 1,

 dz 1 3 J B (z), J B (z) =: L V (z) + L  (z) c(z) : + ∂ 2 c(z), Q= 2πi 2 2

(6.8)

is nilpotent: Q 2 = 0 on V ⊗ . The space V ⊗  is known as a semi-infinite cohomology complex, where the differential is the Q-operator known in physics literature as the BRST operator. The grading in the complex is given by ghost number operator N g . The k th cohomology group is ∞ usually denoted as H 2 +k (V ir, Cc, V ). The following operator product expansions will be helpful for us below (see e.g. [18]): 3 1 J B (z)b(w) ∼ + N g (w) (z − w)2 (z − w)2   1 L V (w) + L  (w) , + z−w 1 J B (z)c(w) ∼ c(w)∂c(w), (6.9) z−w a J B (z)Y (a, w) ∼ c(w)Y (a, w) (z − w)2 1 + (a ∂c(w)Y (a, w) + c(w)∂w Y (a, w)) . z−w As a consequence we obtain the following statement. Corollary 6.1. [Q, c(z)] = c(z)∂c(z), [Q, b(z)] = L  (z) + L V (z), [Q, Y (a, z)] = a ∂c(w)Y (a, w) + c(w)∂w Y (a, w).

(6.10)

In the formulas above “a” denotes the highest weight vector w.r.t. the {L nV } Virasoro algebra and Y (a, z) denotes the corresponding vertex operator. Finally, we recall the following crucial fact about semi-infinite cohomology.

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Lemma 6.1. Let  ∈ V , such that  is Q-closed and L 0  = . Then if  = 0,  = Q for some  ∈ V . In other words, semi-infinite cohomology is nontrivial only on the level L 0 = 0. Proof. We know that [Q, b0 ] = L 0 , therefore, [Q, b0 ] = Qb0  = . Therefore, for  = 0,  = −1 b0 .

(6.11)

 

6.2. Double of the braided vertex algebra and semi-infinite cohomology. We recall that q above we defined the braided VOA structure on the space Fκ = ⊕λ (V(λ),κ ⊗ Vλ ). We πi

q

also recall that Vλ is the irreducible representation for Uq (sl(2)), where q = e κ . Let’s consider the space F = Fκ ⊗ F−κ . Proposition 6.2. The space F = Fκ ⊗ F−κ possesses a structure of braided VOA such that the Virasoro algebra has central charge 26. Proof. It is clear that one can define a map Yˆ : Fκ ⊗ F−κ → End(Fκ ⊗ F−κ ){z}

(6.12)

Yˆ ((v ⊗ a) ⊗ (v¯ ⊗ a), ¯ z) = Y (v ⊗ a, z)Y (v¯ ⊗ a, ¯ z),

(6.13)

in the following way:

−1

q q where v ∈ Fλ,κ , a ∈ Vλ , v¯ ∈ Fμ,−κ , a¯ ∈ Vμ . It is easy to check that the map Yˆ satisfies all properties of the braided vertex algebra, where the commutativity is given by:

Az,w Yˆ ((v1 ⊗ a1 ) ⊗ (v¯1 ⊗ a¯ 1 ), z) Yˆ ((v2 ⊗ a2 ) ⊗ (v¯2 ⊗ a¯ 2 ), w)   (1) (1) = Yˆ (v2 ⊗ ri a2 ) ⊗ (v¯2 ⊗ r¯i a¯ 2 ), w   (2) (2) ×Yˆ (v1 ⊗ ri a1 ) ⊗ (v¯1 ⊗ r¯i a2 ), z ,

(6.14)

 (1) (2)  (1) (2) where R = i ri ⊗ri is the universal R-matrix for Uq (sl(2)) and R¯ = i r¯i ⊗ r¯i is the universal R-matrix for Uq −1 (sl(2)). The vertex operator, corresponding to the Virasoro element for F, is given by L(z) = L κ (z) + L −κ (z),

(6.15)

where L κ (z) and L −κ (z) are the vertex operators associated with Virasoro elements of Fκ and F−κ correspondingly. From Proposition 3.2 the central charge, corresponding to Virasoro algebra generated by L(z) is given by



1 1 + 13 + 6 κ + = 26. (6.16) c = 13 − 6 κ + κ κ  

Quantum Group as Semi-infinite Cohomology

723

In such a way, F ⊗  also possesses a braided VOA structure. It is clear that Q-closed terms form a braided VOA subalgebra in F ⊗ . It is important to note that since the nontrivial cohomology occurs only on the level L 0 = 0, then nontrivial cycles with respect to Q belong to the subspace Fr = ⊕λ∈Z+ Fκ (λ) ⊗ F−κ (λ), since this is the subspace containing all the zeros of L 0 in the generic case. Moreover, we have the following statement. Proposition 6.3. The cycles with respect to Q in F involve only integer powers of formal variables in the operator products up to the cohomologically trivial terms. Proof. Let’s look on the operator product Yˆ ((v1 ⊗ a1 ) ⊗ (v¯1 ⊗ a¯ 1 ), z) (v2 ⊗ a2 ) ⊗ (v¯2 ⊗ a¯ 2 ),

(6.17)

where vi ⊗ ai ∈ Fκ (λi ), v¯1 ⊗ a¯ 1 ∈ F−κ (λ¯ i ) for some λi , λ¯ i ∈ Z. This product belongs to the following space: 

! Fκ (λ3 ) ⊗ F−κ (λ¯ 3 )z 12 (λ3 ) · [z, z −1 ] ,

(6.18)

λ3 ,λ¯ 3

¯ λ¯ 3 ) − (λ1 ) − (λ2 ) − ( ¯ λ¯ 1 ) − ( ¯ λ¯ 2 ) and (λ) = where 12 (λ3 ) = (λ3 ) + ( λ¯ (λ¯ +2) λ λ(λ+2) ¯ ¯ λ¯ / Q this operator product contains − 2 + 4κ , (λ) = − 2 − 4κ . We see that for κ ∈ a “regular” part (i.e. just integer powers of z) only in the case if λ1 = λ¯ 1 , λ2 = λ¯ 2 . This regular part will correspond to the sector λ3 = λ¯ 3 . Hence, Yˆ r , the reduction of Yˆ to the regular part, is equivalent to the reduction of Yˆ to the subspace Fr = ⊕λ (Fκ (λ) ⊗ F−κ (λ)). But this is precisely, what we need to do if we want to cancel the Q-exact terms in case if v1 ⊗ v¯1 and v2 ⊗ v¯2 are Q-closed.   The last calculation we do in this section corresponds to the explicit form of the commutativity relation in the simplest nontrivial case, when λ = 1. There are only two vectors q

q −1

in each of V1 , V1 , i.e. the highest weight and the lowest weight vectors. We denote them as a+ , a− and a¯ + , a¯ − correspondingly. Let us make the following notation: ¯ , v ⊗ a− ⊗ v¯ ⊗ a¯ + = A(v,v)

¯ , v ⊗ a+ ⊗ v¯ ⊗ a¯ − = D (v,v)

¯ , v ⊗ a+ ⊗ v¯ ⊗ a¯ + = B (v,v) (v,v) ¯ v ⊗ a− ⊗ v¯ ⊗ a¯ − = C ,

(6.19)

for any v ∈ V(λ),κ , v¯ ∈ V(λ),− ¯ κ . Then the following proposition holds. Proposition 6.4. Let vi ⊗ v¯i ∈ V(1),κ ⊗ V(1),− ¯ κ (i = 1, 2) be Q-closed. The commutativity relation on F leads to the following relations, which hold in F up to Q-exact

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I. B. Frenkel, A. M. Zeitlin

terms: Az,w (B1 (z)A2 (w)) ≈ A2 (w)B1 (z)q −1 + (q − q −1 )q −1 B2 (w)A1 (z), Az,w (B1 (z)C2 (w)) ≈ C2 (w)B1 (z) + (q − q −1 )(D2 (w)A1 (z) −A2 (w)D1 (z)) − (q − q −1 )2 B2 (w)C1 (z), Az,w (B1 (z)D2 (w)) ≈ D2 (w)B1 (z)q − B2 (w)A(z)q(q − q −1 ), Az,w (A1 (z)C2 (w)) ≈ C2 (w)A1 (z)q − A2 (w)C1 (z)q(q − q −1 ), Az,w (D1 (z)A2 (w)) ≈ A2 (w)D1 (z) + (q − q −1 )B2 (w)C2 (z), Az,w (D1 (z)C2 (w)) ≈ C2 (w)D1 (z)q −1 + (q − q −1 )q −1 D2 (w)C1 (z), (6.20) Az,w (A1 (z)B2 (w)) ≈ B2 (w)A1 (z)q −1 , Az,w (C1 (z)B2 (w)) ≈ B2 (w)C1 (z), Az,w (D1 (z)B2 (w)) ≈ B2 (w)D1 (z)q, Az,w (C1 (z)A2 (w)) ≈ A2 (w)C1 (z)q, Az,w (C1 (z)D2 (w)) ≈ D2 (w)C1 (z)q −1 , Az,w (A1 (z)D2 (w)) ≈ D2 (w)A1 (z) + (q −1 − q)B2 (w)C1 (z), where Si (S = A, B, C, D, i = 1, 2) stands for S (vi ,v¯i ) (S = A, B, C, D, i = 1, 2). Proof. One can obtain all these relations by means of direct use of the R-matrix. Really, in the fundamental representation the R-matrix acts as follows:   H ⊗H R=q 2 1 + (q − q −1 )E ⊗ F , (6.21)   H ⊗H 1 − (q − q −1 )E ⊗ F , R¯ = q − 2 since the higher powers of E and F act as 0. Since we chose a− = Fa+ and a¯ − = F¯ a¯ + , the result can be obtained by direct computation. We will give here the explicit computation of the first line in (6.20) (all other relations can be derived in a similar way). Namely, Az,w (B1 (z)A2 (w)) = Az,w (a+ ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(z)(a− ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(w)) ¯ − ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(w)(a+ ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(z) ≈ R R(a = q −1 (a− ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(w)(a+ ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(z) +(q − q −1 )q −1 (a+ ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(w)(a− ⊗ v1 ⊗ a¯ + ⊗ v¯1 )(z) = A2 (w)B1 (z)q −1 + (q − q −1 )q −1 B2 (w)A1 (z).

(6.22)  

This proposition will be used in the identification of the ring structure of H F) that we will now define.

∞ 2 +·

(V ir, Cc,

Quantum Group as Semi-infinite Cohomology

725

6.3. Lian-Zuckerman associative algebra and S L q (2). Let’s consider the following ∞ algebraic operation on the H 2 +· (V ir, Cc, F):

U (z)V . (6.23) μ(U, V ) = Resz z ∞

Here U, V are representatives of H 2 +· (V ir, Cc, F) and U (z) is shorthand notation for the vertex operator corresponding to the vector U . Since [Q, A(z)]V = Q A(z)V and U (z)Q B = QU (z)B for any A, B ∈ F, one obtains that the operation μ is well defined, ∞ i.e. does not depend on the choice of representatives in H 2 +· (V ir, Cc, F). Then the following statement holds. Proposition 6.5. The operation μ being considered on H and satisfies the following commutativity relation:

where Rˆ =

∞ 2 +·

(V ir, Cc, F) is associative

μ(U, V ) = μ(ˆri(1) V, rˆi(2) U )(−1)|U ||V | ,



(1) i rˆi

(2)

⊗ rˆi

(6.24)

= R R¯ and | · | denotes the ghost number.

Proof. The proof follows the same steps as in the “abelian” case (nonbraided VOA) studied in [33]. Due to Proposition 6.3 we can limit ourselves to the consideration only of the regular terms in operator products (all other contributions correspond to the Q-exact terms). At first, let us prove the commutativity relation: (1)

(2)

μ(U, V ) − (−1)|U ||V | μ(ˆri V, rˆi U ) = Resw Resz−w =



(−1)i Resw Resz−w

i 0

=

 (−1)i i 0

= =

i +1

 (−1)i i>0



(U (z − w)V ) (w)1 (1 + (z − w)/w) w 2

i +1

(U (z − w)V ) (w)1 (z − w)−i wi+2

Resw Resz−w L −1

Resw Resz−w

QResw Resz−w b−1

i>0

(U (z − w)V ) (w)1 (z − w)−i wi+1

(Qb−1 + b−1 Q) (U (z − w)V ) (w)1 (z − w)−i wi+1

(U (z − w)V ) (w)1 (z − w)−i wi+1

(6.25) ∞

for any U, V which are the representatives of the cohomology classes H 2 +· (V ir, Cc, F). Therefore, the commutativity relation holds. Now, let us prove the associativity of μ; μ (μ(U, V ), W ) − μ (U, μ(V, W )) = Resw Resz−w −Resz Resw

(U (z − w)V ) (w)W (z − w)w

U (z)V (w)W U (z)V (w)W = Resz Resw zw (1 − wz )zw

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I. B. Frenkel, A. M. Zeitlin

+(−1)|U ||V |



(1)

Resw Resz

k

−Resz Resw

(2)

(ˆrk V )(w)(ˆrk U )(z)W (1 − wz )w 2

 U (z)V (w)W U (z)V (w)W Resz Resw i+1 −i+1 = zw z w i>0

+(−1)|U ||V |

 k

(1)

Resw Resz

i 0

(2)

(ˆrk V )(w)(ˆrk U )(z)W z −i wi+2

 1 (L −1 U )(z)V (w)W Resz Resw = i +1 z i+1 w −i i 0

+(−1)|U ||V |

(1) (2)  1 (L −1rˆk U )(w)ˆrk V (z)W Resw Resz i +1 z −i wi+1 k

i>0

= Qn(U, V, W ) + n(QU, V, W ) + (−1)|U | n(U, QV, W ) +(−1)|U |+|V | n(U, V, QW ),

(6.26)

where n is given by  1 (b−1 U )(z)V (w)W Resz Resw n(U, V, W ) = i +1 z i+1 w −i i 0

+(−1)|U |+|V |

(1)) (2)  1 (b−1 rˆk U )(z)(ˆrk V )(w)W Resw Resz . (6.27) i +1 z −i wi+1 i 0

Therefore, μ is associative on semi-infinite cohomology. Thus, the proposition is proven.   F.

The next proposition gives a possibility to compute the semi-infinite cohomology of

Proposition 6.6. Let F = ⊕λ∈Z+ (V(λ),κ ⊗ V(λ),− ¯ κ ). Then i) ii)

F has a VOA structure; ∞ H 2 +k (V ir, Cc, F(λ)) = Cδk,0 ⊕ Cδk,3 , where F(λ) = V(λ),κ ⊗ V(λ),− ¯ κ.

The proof is given in [23], following the results of [31,32]. It is not hard to calculate the ∞ ∞ explicit formulas for the representatives of H 2 +0 (V ir, Cc, F) and H 2 +3 (V ir, Cc, F). Proposition 6.7. (i) The operators corresponding to the representatives of H ∞ (V ir, Cc, F(1)) and H 2 +3 (V ir, Cc, F(1)) have the following explicit form: −κ −1 : bc : (z)(z), 0 (z) = L κ −1 (z) − L −1 (z) − κ

3 (z) = c∂c∂ 2 cL tot −1 (z),

∞ 2 +0

(6.28)

−κ where we denoted by L κ n , L n the Virasoro algebra generators in V(λ),κ , −κ V(λ),− correspondingly and Ln ≡ Lκ ¯ n + L n . The operator (z) corresponds κ to the tensor product of highest weight states in V(1),κ and V(1),− ¯ κ;

Quantum Group as Semi-infinite Cohomology

(ii)

727

0 , 1 (the vector corresponding to 0 (z) and the vacuum state) generate all ∞ H 2 +0 (V ir, Cc, F) by means of the bilinear operation μ.

Proof.

(i) First of all, we have the following relation, which follows from (6.1): [Q, ](z) = −∂c + c∂.

(6.29)

" dz κ Let L κ (z) be the Virasoro element in V(λ),κ . Then L κ −1 (w) = 2πi L (z)(w). Therefore,   dz κ dz κ L (z)[Q, ](w) = [Q, L κ (z)](w). (6.30) [Q, L −1 ](w) = 2πi 2πi It is easy to see that   dz κ dz [Q, L (z)] = 2∂ 2 cL κ (z) + c∂ L κ (z). 2πi 2πi

(6.31)

Hence, 

dz [Q, L κ (z)](w) = (1)∂ 2 c + ∂cL κ −1 , 2πi  dz κ −κ L (z)[Q, ](w) = −∂c L¯ κ −1  + cL −1 ∂, 2πi

(6.32)

which lead to −κ 2 [Q, L κ −1 (w)] = (1)∂ c + cL −1 ∂.

(6.33)

It is not hard to see that −κ [Q, (L κ −1 − L −1 )(w)] =

3 2 κ 2 κ 2 ∂ c + c((L − −1 ) − (L −1 ) ). 2κ

(6.34)

Another necessary fact which we will need is that there is a linear dependence −κ 2 between (L κ −1 )  and L −2 , namely, 2 (1 + 2(1)) L κ −2 , 3  −κ 2 κ 2 ¯ 1 + 2(1) L −2  (L − −1 )  = 3 2 (L κ −1 )  =

(6.35)

or 2 −1 κ (L κ −1 )  = κ L −2 ,

κ 2 −1 −κ (L − −1 )  = −κ L −2 .

(6.36)

Therefore, −κ −1 [Q, (L κ −1 − L −1 )(w)] = κ



3 2 ∂ c + cL −2  . 2

(6.37)

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I. B. Frenkel, A. M. Zeitlin

Let us consider another term from 0 , namely, : bc : (w)(w),  1 dz [Q, : bc :](z)(w) [Q, : bc : (w)(w)] = w 2πi (z − w)

 + : bc : (z) [Q, (w)] , 3 [Q, : bc : (w)] = j B (w) = c(w)L(w)+ : c∂cb : (w) + ∂ 2 c(w), (6.38) 2  1 dz j B (z)(w) = − ∂ 2 c + ∂c∂ + cL −2  z−w 2 w 2πi 3 + : bc∂c :  + ∂ 2 c. 2 On the other hand,  dz : bc : (z) (−∂c(w)(w) + c(w)∂(w)) z−w w 2πi 1 = ∂ 2 c(w)(w) − ∂c(w)∂(w)− : bc∂c : (w)(w). 2

(6.39)

Therefore, [Q, : bc : (w)(w)] =

3 2 ∂ c(w) + cL −2 (w), 2

(6.40)

and the operator −κ −1 (L κ : bc : (w)(w) −1 − L −1 )(w) − κ

(6.41)

corresponds to the closed state in F(1) ⊗ . At the same time, it is the only Q-closed state of ghost number 0 in F(1) ⊗ , which belongs to the kernel of L 0 ; therefore, due to Theorem 3.7 it is not exact. Hence, it represents the cohomology ∞ class from H 2 +0 (V ir, Cc, F). It is not hard to check that 3 = c∂c∂ 2 cL −1 (z) ∞ represents the cohomology class H 2 +3 (V ir, Cc, F). Part (ii) follows from the Theorem 3.7. of [23].   Our next aim is to use the above information to calculate H understand its underlying multiplicative structure.

∞ 2 +·

(V ir, Cc, F) and to

Proposition 6.8. The zeroth and third semi-infinite cohomology groups of F are H

∞ 2 +0

∞ (V ir, Cc, F) ∼ = H 2 +3 (V ir, Cc, F) ∼ =

 λ0

q

q −1

Vλ ⊗ Vλ

.

(6.42)

Proof. The proof evidently follows from the fact: H for λ ≥ 0.

 

∞ 2 +0

∞ (V ir, Cc, F(λ)) ∼ = H 2 +3 (V ir, Cc, F(λ)) ∼ =C

(6.43)

Quantum Group as Semi-infinite Cohomology

729

∞

Theorem 6.1. H 2 +0 (V ir, Cc, F(1)) generates all H multiplication μ, and the generating set A, B, C, D ∈ H

∞ 2 +0

∞ 2 +0

(V ir, Cc, F) by means of

q q (V ir, Cc, F(1)) ∼ = V1 ⊗ V1

−1

satisfies the following relations: AB = B Aq −1 , C B = BC, D B = B Dq, C A = ACq, AD − D A = (q −1 − q)BC, C D = DCq −1 , AD − q −1 BC = 1,

(6.44)

or, equivalently,  H

∞ 2 +0

Proof. The statement that H

 (V ir, Cc, F), μ ∼ = S L q (2).

∞ 2 +0

q

q −1

(V ir, Cc, F(1)) = V1 ⊗ V1

(6.45) forms a generating set

q q −1 (V ir, Cc, F) = ⊕λ∈Z≥0 Vλ ⊗ Vλ , is a consequence of the fact that by means q q multiplication of V1 with each other one can obtain any Vλ and the explicit

∞ 2 +0

for H of tensor form of vertex operators (since they act as intertwiners). One can choose generators A, B, C, D in the same way as we did in Proposition 6.4. Then all the relations except for the last one follow from the commutativity relation of μ. One can obtain the last relation by means of the explicit consideration of the action of vertex operators (more precisely, corresponding intertwiners) and normalizing A, B, C, D appropriately.   In this section, we found how the multiplication structure on S L q (2) emerges from the braided VOA structure on F via Lian-Zuckerman construction. However, S L q (2) is a Hopf algebra and the natural question is whether the comultiplication structure has the same origin. The answer is yes, and it is provided by means of the notion of a vertex operator coalgebra (VOCA) [26]. If one has a map Y giving a structure of a VOA algebra on some space V with a nondegenerate bilinear pairing ( , ), preserving the Virasoro action, the structure of VOCA is given by Y c : V → V ⊗ V [[z, z −1 ]], such that (Y c (z)u, v ⊗ w) = (u, Y (v, z)w), where u, v, w ∈ V [26]. One can extend this definition to the case of braided VOA, namely, define the structure of a braided VOCA algebra by the same formula. Therefore, the definition of the Lian-Zuckerman operation μ gives rise to the dual object μ : (μ (u), v ⊗ w) ≡ Resz z −1 (Y c (z)u, v ⊗ w) = Resz z −1 ((u, Y (v, z)w) = (u, μ(v, w)).

(6.46)

Since μ is dual to μ with respect to the canonical pairing, we obtain that μ in the specific case of braided VOA F gives the comultiplication structure on S L q (2). We leave the proof of the consistency of multiplication and comultiplication as an exercise. 7. Further Developments and Conjectures We already discussed in the Introduction that thanks to the equivalence of categories   of Virasoro and sl(2) Lie algebras one can replace the Virasoro modules by their sl(2)  Lie counterparts and obtain the quantum group as a semi-infinite cohomology of the sl(2)

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algebra. Also the extension to arbitrary affine and W-algebras is possible thanks to the general results of Varchenko [41,42] and Styrkas [38], though some technical difficulties might arise, related e.g. to the definition of semi-infinite cohomology of W-algebras, etc. (see e.g. [1]). However, we believe that the most interesting further developments will appear not from generalizations but from the deepening of our construction, and better understanding its relation with physics. The first possible extension of our construction comes from the existence of double lattice (3.3) of Verma modules with singular vectors and fixed central charge. The semi-infinite cohomology of the Virasoro algebra with values in the corresponding double lattice of the Fock spaces must yield the modular double of the quantum group introduced in [9]. The second development of our construction results from the special features of the rational level case that we conscientiously avoided in the present paper. It is well known that the quantum group at roots of unity has a remarkable finite-dimensional subquotient. Its realization via semi-infinite cohomology might require finding a corresponding subquotient braided VOA. The analogous modified regular VOA should give rise to the finite-dimensional Verlinde algebra as conjectured in [23]. The third possible development follows from the replacements of simple modules of Virasoro algebra and quantum group by the big projective modules as in [23]. In fact, the braided VOA Fκ has the appropriate size for such an extension but has a degenerate structure resulting from the absence of linking of Fκ (λ) and Fκ (−λ − 2). The way to achieve such linking is suggested by the construction of the heterogeneous VOA in [23], and will require an additional exponential term in the action of the Virasoro algebra. This exponential term and the free bosonic realization of the Virasoro algebra, known as Coulomb gas in the physics literature, reveal the relation of our construction to the quantum Liouville model and minimal string theory. In particular, the latter theory also uses the pairing of representations of the Virasoro algebra with complementary central charges c + c¯ = 26. We conjecture that our braided VOA and its projective counterpart discussed above admits a geometric (and path integral) interpretation, so that the semiinfinite Lie algebra cohomology acquires a natural meaning as cohomology in a certain infinite dimensional geometry as well. Then the quantum group will finally find its truly geometric setting. Acknowledgements. We are indebted to P.I. Etingof, Y.-Z. Huang, A.A. Kirillov Jr., M.M. Kapranov, J. Lepowsky and G.J. Zuckerman for fruitful discussions. I.B.F. thanks K. Styrkas for a joint research on braided vertex operator algebras 10 years ago. A.M.Z. also wants to thank the organizers of the Simons Workshop 2008, where this work was partly done. The research of I.B.F. was supported by NSF grant DMS-0457444.

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40. Tsuchiya, A., Kanie, Y.: Vertex operators in the conformal field theory on P1 and monodromy representations of braid group. Lett. Math. Phys. 13, 303–312 (1987) 41. Varchenko, A.: Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Adv. in Math. Phys. 21, Singapore-River Edge, NJ: World Scientific, 1995 42. Varchenko, A.: Special functions, KZ type equations and representation theory. http://arxiv.org/abs/math/ 0205313v2[math.QA], 2002 43. Witten, E.: Ground ring of two-dimensional string theory. Nucl. Phys. B373, 187–213 (1992) 44. Witten, E., Zwiebach, B.: Algebraic Structures and Differential Geometry in 2D String Theory. Nucl. Phys. B377, 55–112 (1992) Communicated by A. Connes

Commun. Math. Phys. 297, 733–758 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1056-1

Communications in

Mathematical Physics

Bubbling Solutions for Relativistic Abelian Chern-Simons Model on a Torus Chang-Shou Lin1 , Shusen Yan2 1 Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University,

Taipei 106, Taiwan. E-mail: [email protected]

2 Department of Mathematics, The University of New England, Armidale, NSW 2351, Australia.

E-mail: [email protected] Received: 31 December 2008 / Accepted: 14 March 2010 Published online: 11 May 2010 – © Springer-Verlag 2010

Abstract: We prove the existence of bubbling solutions for the following ChernSimons-Higgs equation:  1 u e (1 − eu ) = 4π δ p j , in , 2 ε N

u +

j=1

where  is a torus. We show that if N > 4 and p1 = p j , j = 2, . . . , N , then for small ε > 0, the above problem has a solution u ε , and as ε → 0, u ε blows up at the vertex point p1 , and satisfies 1 u e (1 − eu ) → 4π N δ p1 . ε2 This is the first result for the existence of a solution which blows up at a vertex point. 1. Introduction The Chern-Simons theories were developed to explain certain condensed matter phenomena, anyon physics, superconductivity, quantum mechanics and so on. In this paper, we will study the (2 + 1)-dimensional relativistic Abelian Chern-Simons-Higgs model on a torus . This theory was proposed in [9,10] in an attempt to explain superconductivity of type 2. A self-dual system for stationary solutions is derived and can be further reduced to an elliptic partial differential equation with exponential nonlinearity:  1 u u e (1 − e ) = 4π δ p j , in , ε2 N

u +

j=1

(1.1)

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C.-S. Lin, S. Yan

where δ p j (x) is the Dirac measure at p j ∈ , and ε is the Chern-Simons constant. See [9,10,17,21] and the references therein. Note that pi and p j may coincide for some i = j. For a torus , it is well-known (see for example [18]) that there exists a constant ε∗ > 0, such that if ε ≥ ε∗ , (1.1) has no solution, while if ε < ε∗ , there are at least two solutions for (1.1), one of which is the maximal solution, and the other one can be obtained through the mini-max variational method. As ε → 0, the maximal solution tends to 0 uniformly in any compact subset of \{ p1 , . . . , p N }. But the second solution has a different asymptotic behavior. In [4], it is proved that for any sequence of solution u n with εn > 0, one of the following holds true: (a) u n → 0 uniformly in any compact subset of \{ p1 , . . . , p N }; (b) u n + ln ε12 is bounded from above; n (c) there is a finite set S = {q1 , . . . , q L } ⊂  and x1,n , . . . , x L ,n ∈ , such that as n → +∞, x j,n → q j , u n (x j,n ) + ln

1 → +∞, ∀ j = 1, . . . , L , εn2

and u n (x) + ln

1 → −∞, uniformly on any compact subset of \S. εn2

Moreover,  1 un e (1 − eu n ) → M j δq j , 2 εn L

M j ≥ 8π,

j=1

in the sense of measure. In case (c), q j is called a blow up point for the solution u n . Let u 0 (x) = −4π

N 

G(x, p j ),

(1.2)

j=1

where G(x, p j ) is the Green function of − in  with singularity at p j . That is, G(x, p j ) satisfies 1 −G(x, p j ) = δ p j − , || where || is the measure of . For the second solution, Choe [3] proved the following theorem: Theorem A. Assume that N > 2 and let u ε be a sequence of solutions of (1.1), which is obtained from the mountain pass lemma in the variational theory. As ε → 0, u ε blows up at exactly one point q ∈ \{ p1 , . . . , p N }, satisfying u 0 (q) = max x∈ u 0 (x) and 1 uε e (1 − eu ε ) → 4π N δq , ε2 in the sense of measure.

Bubbling Solutions for Abelian Chern-Simons Model

735

When N = 1, 2, it was shown in [19] that the conclusion of Theorem A does not necessarily hold. Theorem A is the first result on the existence of bubbling solutions for the Chern-Simons-Higgs equation on a torus, but it only tells us that (1.1) has solutions which blow up at a regular point q of the function u 0 . A question raised in [17] is whether (1.1) has a solution which blows up at one of the vortex points p j . This kind of solution is physically meaningful, because the order parameter eu is always zero at any vortex. However, the existence of such solutions implies even in a tiny neighborhood of a vertex, the order parameter may have a positive lower bound, which makes superconductivity possible. In this paper, we will give a positive result to the problem raised by Tarantello. The main result of this paper is the following: Theorem 1.1. Let  ⊂ R 2 be a torus and N > 4. If p1 = p j , ∀ j = 1, there is an ε0 > 0, such that for any ε ∈ (0, ε0 ), (1.1) has a solution u ε , satisfying that as ε → 0,  1 uε  e 1 − eu ε → 4π N δ p1 . ε2 The solution u ε in Theorem 1.1 satisfies a stronger inequality: max u ε (x) ≥ c > −∞, x∈

although u ε + ln ε12 → −∞ uniformly in any compact subset of \{ p1 }. As a result, p1 is a blow-up point of u ε . As far as we know, Theorem 1.1 is the first result on the existence of solutions for (1.1), which blow up at a vortex point. It is also natural to ask whether there is a sequence of bubbling solutions with max u ε (x) → −∞, x∈

as ε → 0. For the case N = 2, the readers can refer to [15,19] for the existence of such solutions. For the general case, we will discuss it in a future work. It seems hard to find a min-max theorem as in [3,16,19] to prove the existence of a solution blowing up at a vortex point. In this paper, we will use the contraction mapping theorem to prove Theorem 1.1. The crucial step in this procedure is to construct a good approximation solution ϕε for (1.1), such that the linear operator L ε of (1.1) at ϕε is invertible, although L −1 ε has a large norm. Let us point out the question of the existence of a solution which blows up at several vortex points is still open. In this paper, we will also study the system of equations corresponding to the relativistic Abelian Chern-Simons model involving two Higgs scalar fields and two gauge fields on a torus : ⎧ N1 ⎪ 1 v u ⎪ u = e (e − 1) + 4π δpj ⎪ 2 ⎨ ε j=1 (1.3) N2 ⎪ ⎪ ⎪v = 12 eu (ev − 1) + 4π δ . q ⎩ j ε j=1

The readers can refer to [7,8,12,13] and references therein for the background physics. The mathematical analysis of the system (1.3) has been recently initiated in [13] where the problem (1.3) is studied on the plane. More recently, the paper [6] studied the uniqueness of topological solutions and existence of non-topological solutions for (1.3)

736

C.-S. Lin, S. Yan

in a plane. In [14], it is shown that there is a constant ε∗ > 0, such that if ε < ε∗ , (1.3) has at least two solutions, while if ε ≥ ε∗ , (1.3) has no solution. It is interesting to study this problem on a torus since the solutions of the Chern Simons equations depend on the topology of underlying space. For example, the relativistic Abelian Chern-Simons theory (scalar equation) ([1]), non-Abelian SU(3) ChernSimons theory ([20]), Maxwell-Chern-Simons theory ([2,17]) models on a torus are well studied. In spite of similarity with the scalar Abelian Chern Simons equation and Abelian Higgs vortex equations (see [11] for details), the system (1.3) is difficult not only since it involves two equations, but also due to the “mixed” nature of the nonlinear terms. Although it is difficult to obtain a blow-up solution for the system by using the mountain pass lemma as in [3,16,19], it turns out that the method we use to prove Theorem 1.1 applies to (1.3) in some special case. Since system (1.3) is symmetric in (u, v) except the terms involving the Dirac measure, as in [13], we make the following the change of variable u → u + v, v → u − v. Then the system (1.3) can be transformed into an equivalent problem ⎧ N1 N2 u−v ⎪ 2 u 1 u+v ⎪ 2 − 1 e 2 + 4π u = e − e δ + 4π δq j ⎪ p 2 2 2 j ⎨ ε ε ε j=1

⎪ ⎪ ⎪ ⎩v =

1 e ε2

u+v 2

−

1 e ε2

u−v 2

+ 4π

N1

j=1

δ p j − 4π

j=1

N2

(1.4)

δq j .

j=1

Define u˜ 0 (x) = −4π

N1 

G(x, p j ) − 4π

j=1

N2 

G(x, q j ),

j=1

and v˜0 (x) = −4π

N1 

G(x, p j ) + 4π

j=1

N2 

G(x, q j ).

j=1

Making a further change of variable u → u˜ 0 + u, v → v˜0 + v, we can reduce system (1.4) to ⎧ u˜ 0 −v˜0 u−v 4π(N1 +N2 ) ⎨u = 2 eu˜ 0 +u − 1 e u˜ 0 2+v˜0 + u+v 2 − 1 e 2 + 2 + , in ; || ε2 ε2 ε2 u˜ 0 −v˜0 u−v 4π(N −N ) 1 + ⎩v = 1 e u˜ 0 2+v˜0 + u+v 1 2 2 − 2 + e 2 , in . || ε2 ε2

(1.5)

Note that the second equation in (1.5) is uniquely solvable for any given u. Of course, v depends on u. The main observation for (1.5) is that if N1 = N2 , then the second equation of (1.5) becomes

 v+v˜0 v+v˜0 1 u˜ 0 +u v = 2 e 2 e 2 − e− 2 . (1.6) ε

Bubbling Solutions for Abelian Chern-Simons Model

737

If we further assume that p1 = q1 , then the function v˜0 is regular near p1 . As a result, the constant −v˜0 ( p1 ) can be used as an approximate solution of (1.6) near p1 for any given u. Note that v + v˜0 ≈ 0 near p1 , we find that near p1 , the first equation in (1.5) is a perturbation of the following equation:

 u˜ 0 +u 2 8π N1 u˜ 0 +u 2 u = 2 e + −e . ε || So, using the method as in the proof of Theorem 1.1, we can obtain Theorem 1.2. Let  ⊂ R 2 be a torus. Suppose that N1 = N2 = N > 4, p1 = q1 and p1 = p j , q j for j > 1. Then, there is an ε0 > 0, such that for any ε ∈ (0, ε0 ), (1.5) has a solution (u ε , vε ), satisfying vε → 4π

N 

G( p1 , p j ) − 4π

j=2

N 

G(q1 , q j ), uniformly in ;

j=2

and 1 u˜ 0 +u ε e 2 ε2

 u˜ 0 +u ε 2 1−e → 4π N δ p1 ,

as ε → 0. Theorem 1.2 roughly states that if the two terms in (1.3) involving the Dirac measures are equal at p1 and the total weights of the two equations are equal, then (1.3) has a solution (u ε , vε ), such that u ε and vε have the same bubbling behavior at p1 . That is, u ε − vε has no bubble at p1 . We need the condition N1 = N2 and p1 = q1 to show that (1.5) has a solution (u ε , vε ), such that vε is nearly a constant. Let us point out that if N1 = N2 , or p1 = qi , i = 1, . . . , N2 , it is still an open problem whether (1.3) has a solution concentrating at p1 . We will use that contraction mapping theorem to prove Theorem 1.1. The first crucial step in this paper is to construct a good approximation solution ϕε , which will be presented in Sect. 2. The most difficult part of this paper is the analysis of the linear operator L ε corresponding to ϕε , which has the form L ε u = u + ε12 f ε (x)u, where f ε (x) has compact support and f ε (x) may not have fixed sign. Here, we will encounter a typical difficulty for an elliptic problem in R 2 : the L 2 norm of the gradient of u can not control any L p norm of u. Thus, the operator L ε will not generate naturally a space in which we can carry out our analysis. Instead, we have to work on some weighted spaces which are not related to L ε in a natural way. The estimate of the norm of L ε involves lots of complicated analysis, so we put this part in the appendixes. In Sect. 3, we will prove Theorem 1.1. Since the proof of Theorem 1.2 is just a bit more technically complicated than that of Theorem 1.1, we sketch it in Sect. 4. 2. The Approximate Solutions Recall that u 0 (x) = −4π

N  j=1

G(x, p j ).

738

C.-S. Lin, S. Yan

We consider the following equivalent problem of (1.1): u +

 4π N 1 u 0 +u  , in . 1 − eu 0 +u = e 2 ε ||

(2.1)

In this section, we will construct an approximate solution for (2.1). Suppose that N > 4. Then by Theorem 2.1 of [5], we know that the following problem has a unique radial solution V (|x|): V + |x|2 e V (1 − |x|2 e V ) = 0, in R 2 ; (2.2) as t → +∞. −t V (t) → 2N , Let γ (x, p) be the regular part of G(x, p). That is, γ (x, p) = G(x, p) −

1 1 ln . 2π |x − p|

We define the approximate solution ϕε for (2.1) as follows: 



|x − p1 | + 4π N (γ (x, p1 ) − γ ( p1 , p1 )) 1 − ε2N −4−θ + M, ϕε (x) = V ε x ∈ Bdε ( p1 ), (2.3) while ϕε (x) = V



 

 dε 1 ln dε − γ ( p1 , p1 ) 1 − ε2N −4−θ + M, + 4π N G(x, p1 ) + ε 2π x ∈ \Bdε ( p1 ), (2.4)

where V (|x|) is the solution of (2.2), θ > 0 is a fixed small constant, dε is the constant to make ϕε ∈ C 1 , and  1 + 4π γ ( p1 , p1 ) + 4π G( p1 , p j ). ε N

M = 2 ln

(2.5)

j=2

We will prove that the constant dε has a following estimate: θ

dε = (a¯ + o(1)) ε 2N −4 , for some constant a¯ > 0.

(2.6)

In the rest of this section, we explain how to find this approximate solution ϕε . We can decompose (2.1) into two problems: 4π N , in , ||

(2.7)

 1 u 0 +u 2  u 0 +u 2 1 − e = 0, in . e ε2

(2.8)

u 1 = and u 2 + For any constant M,

u 1 = 4π N (γ (x, p1 ) − γ ( p1 , p1 )) + M. is a solution of (2.7).

(2.9)

Bubbling Solutions for Abelian Chern-Simons Model

739

On the other hand, if we blow up (2.8) at p1 , we then obtain the following limit problem: V + |x|2 e V (1 − |x|2 e V ) = 0, in R 2 . By Theorem 2.1 of [5], if β > 8, the following problem has a unique radial solution: V + |x|2 e V (1 − |x|2 e V ) = 0, in R 2 ; (2.10) −t V (t) → β, as t → +∞. Let V be the solution of (2.10), where the constant β is to be determined later. So, near

|x− p1 | p1 , u 2 (x) ≈ V . ε For a constant dε > 0 small with dε >> ε, we define 

|x − p1 | + u 1 (x), x ∈ Bdε ( p1 ), ϕ˜ε (x) = V ε where u 1 (x) is defined in (2.9). Then in Bdε ( p1 ),



 |x − p1 |2 V |x−εp1 | 4π N 1 |x − p1 |2 V |x−εp1| 1 − − −ϕ˜ε = 2 e e ε ε2 ε2 ||   2 |x− p1 |2 1 u 0 (x)+ϕ˜ε (x)−u 0 (x)−u 1 (x)+ln |x− p21 | u 0 (x)+ϕ˜ε (x)−u 0 (x)−u 1 (x)+ln 2 ε ε = 2e 1−e ε −

4π N . ||

Since |x − p1 |2 ε2 N  1 = 4π γ (x, p1 ) + 4π G(x, p j ) + ln 2 − 4π N (γ (x, p1 ) − γ ( p1 . p1 )) − M, ε

−u 0 (x) − u 1 (x) + ln

j=2

if we choose  1 + 4π γ ( p1 , p1 ) + 4π G( p1 , p j ), ε N

M = 2 ln

(2.11)

j=2

then, near p1 , −ϕ˜ε ≈ So, we find ϕ˜ε (x) = V



|x − p1 | ε

 4π N 1 u 0 (x)+ϕ˜ε (x) u 0 (x)+ϕ˜ε (x) . 1 − e − e ε2 ||  + 4π N (γ (x, p1 ) − γ ( p1 , p1 )) + M, x ∈ Bdε ( p1 ), (2.12)

where M is given by (2.11).

740

C.-S. Lin, S. Yan

 We only define ϕ˜ε near p1 . But we need to define ϕ˜ε in . Noting that V dεε → −∞, we expect that in \Bdε ( p1 ), φ˜ ε ≈ 0. So we can define

  dε 1 +4π N G(x, p1 )+ ϕ˜ε (x) = V ln dε −γ ( p1 , p1 ) + M, x ∈ \Bdε ( p1 ). ε 2π (2.13) For any V , which is the solution of (2.10), the function ϕ˜ ε is continuous. But as an approximate solution of a second order equation, we need to choose a suitable β in (2.10) to make ϕ˜ε ∈ C 1 . If ϕ˜ε ∈ C 1 (), then

 1 1 dε V = −2N holds. ε ε dε If we let tε =

dε ε

→ +∞, then − tε V (tε ) = 2N .

(2.14)

Thus, we see that β must satisfy β = 2N . Unfortunately, (2.14) has no solution. To solve this problem, we modify ϕ˜ε as follows: 



|x − p1 | +4π N (γ (x, p1 )−γ ( p1 , p1 )) 1−ε L + M, x ∈ Bdε ( p1 ), ϕε (x) = V ε (2.15) while



  dε 1 + 4π N G(x, p1 ) + ln dε − γ ( p1 , p1 ) 1 − ε L + M, ε 2π x ∈ \Bdε ( p1 ), (2.16)

ϕε (x) = V

where L > 0 is a constant to be determined, M is given by (2.11), and dε > 0 satisfies



 1 1 dε V = −2N 1 − ε L . ε ε dε Let tε =

dε ε

→ +∞, then 

− tε V (tε ) = 2N 1 − ε L .

It follows from Lemma 2.6 of [5] that as t → +∞, V (t) = −2N ln t + I −

a t 2N −4

+O

and V (t) = −



2N a(2N − 4) + +O t t 2N −3

(2.17)

1



t 2N −2



 , 2N −1 1

t

,

Bubbling Solutions for Abelian Chern-Simons Model

741

where a > 0 and I are some constant. Thus, tε satisfying 

a(2N − 4) 1 − 2N −4 + O 2N −2 = −2N ε L , tε tε which implies that there is a constant a¯ > 0, L

tε = (a¯ + o(1)) ε− 2N −4 .

(2.18)

So, we find L

dε = (a¯ + o(1)) ε1− 2N −4 .

(2.19)

In order to keep dε << 1, we need to choose L, such that 1 − 2NL−4 > 0. So, we take L = 2N − 4 − θ,

(2.20)

where θ > 0 is a fixed small constant. With such L, we see that (2.6) comes from (2.19). 3. Solutions Concentrating at a Vortex Point In this section, we will construct a solution for (2.1) with u ε = ϕε + ω, where the approximate solution ϕε is defined in (2.3) and (2.4). So, ω should solve L ε,1 ω = gε,1 (x, ω),

(3.1)

L ε,1 ω = ω + f ε,1 (x)ω,

(3.2)

where

  ⎧

  |x− p1 | |x− p | |x− p1 | |x− p | ⎪ V +2 ln ε 1 +2 ln ε 1 ⎨− 1 e V ε ε − 1 , x ∈ B2d ( p1 ); 2e ε2 f ε,1 (x) = ⎪ ⎩ 0, x ∈ \B2d ( p1 ),

(3.3) d > 0 is a fixed constant, and gε,1 (x, s) = f ε,1 (x)s +

 4N π 1 u 0 +ϕε +s  u 0 +ϕε +s − ϕε . e e −1 + 2 ε ||

(3.4)

By (2.3) and (2.4), we have 4N π −ϕε + || ⎧  |x− p | |x− p | |x− p | |x− p | 1 ⎨− 2 e V ( ε 1 )+2 ln ε 1 e V ( ε 1 )+2 ln ε 1 − 1 − ε = ⎩− 4π N ε2N −4−θ , ||

4π N 2N −4−θ , || ε

x ∈ Bdε ( p1 ); x ∈ \Bdε ( p1 ).

(3.5)

742

C.-S. Lin, S. Yan

Let us introduce two function spaces X α,ε and Yα,ε , such that for any ξ ∈ X α,ε , we not only control the L 2 norm of ξ and ξ , but also take into account the concentration at p1 . For this purpose, we fix a small α > 0 and define 1

α

ρ(x) = (1 + |x|)1+ 2 , ρ(x) ˆ =

α

(1 + |x|) (ln(2 + |x|))1+ 2

.

We say a function ξ is in X α,ε if

ξ 2X α,ε = ε4 (ξ˜ )ρ 2L 2 (B

2d/ε )

+ ξ 2L 2 (\B ( p )) d 1

+ ξ˜ ρ ˆ 2L 2 (B

2d/ε )

+ ξ 2L 2 (\L 2 (B

d ( p1 ))

< +∞,

(3.6)

where ξ˜ (y) = ξ( p1 + εy), Bt = Bt (0). On the other hand, we say ξ ∈ Yα,ε if

ξ 2Yα,ε = ε4 ξ˜ ρ 2L 2 (B

2d/ε )

+ ξ 2L 2 (\L 2 (B

d ( p1 ))

< +∞.

(3.7)

By using Theorem B.1 in Appendix B, we are now ready to prove the main result of this paper. Proof of Theorem 1.1. It follows from Theorem B.1 that L ε,1 is invertible from X α,ε to Yα,ε . Rewrite (3.1) as ω = Bε,1 ω =: L −1 ε,1 gε,1 (x, ω), ω ∈ X α,ε . Fix a small constant σ > 0. Define   Sε,1 = ω : ω ∈ X α,ε , ω L ∞ + ω X α,ε ≤ ε1−σ . We will prove that Bε,1 is a contraction map from Sε,1 to Sε,1 . So, by the contraction mapping theorem, there is an ωε ∈ Sε,1 , such that ωε = Bε,1 ωε . First, we prove that Bε,1 maps Sε,1 to Sε,1 . It follows from Theorem B.1 that 1

Bε,1 ω L ∞ () + Bε,1 ω X α,ε ≤ C ln gε,1 (x, ω) Yα,ε . ε To estimate gε,1 (x, ω) Yα,ε , by (3.4) and (3.5), we first note that for x ∈ Bdε ( p1 ),  1 u 0 +ϕε +ω  u 0 +ϕε +ω e −1 e 2 ε  

1 V ( |x− p1 | )+2 ln |x− p1 | V ( |x− p1 | )+2 ln |x− p1 | ε ε ε ε e − 2e − 1 + O ε2N −4−θ . ε (3.8)

gε,1 (x, ω) = f ε,1 (x)ω +

On other hand, by (2.5), for x ∈ Bdε ( p1 ),

 |x − p1 | u 0 + ϕε = −4π G(x, p j ) + V ε j=1 

+ 4π N (γ (x, p1 ) − γ ( p1 , p1 )) 1 − ε2N −4−θ N 



 1 + 4π γ ( p1 , p1 ) + 4π G( p1 , p j ) ε j=2

 |x − p1 | |x − p1 | =V + 2 ln + O(|x − p1 |), ε ε N

+ 2 ln

(3.9)

Bubbling Solutions for Abelian Chern-Simons Model

743

which, together with (3.8), gives

 

1 V ( |x− p1 | )+2 ln |x− p1 | 2 2N −4−θ ε ε gε,1 (x, ω) = O |ω| . e + |x − p | + O ε 1 ε2 (3.10) As a result, ε4 gε,1 (εx + p1 , ω)ρ(x) 2L 2 (B ) dε /ε



 V (x)+2 ln |x| |ω(εy + p1 )|2 + ε|x| ρ(x) 2L 2 (B ) + O ε4 ≤ C e dε /ε



 1 ≤ C ε4(1−σ ) + ε2 d x + O ε4 ≤ Cε2 , 2(2N −4)−α R 2 (1 + |x|) α

since |ω| ≤ ε1−σ , |x|ρ(x) ≤ C(1 + |x|)2+ 2 and 2(2N − 4) − α > 2 if α > 0 is small. On the other hand, in \Bdε ( p1 ),

 C V ( |x− p1 | )+2 ln |x− p1 | C ε 2N ε ε | f ε,1 (x)| ≤ 2 e ≤ 4 , (3.11) ε ε dε and

 

2N  1 u +ϕ  |x− p | d  e 0 ε  ≤ C e V ( εε )+2 ln ε 1 ≤ C ε .  ε2  ε2 ε4 dε

Combining (3.4), (3.5), (3.11) and (3.12), we obtain 

gε,1 (x, ω) = O ε2N −4−θ , x ∈ \Bdε ( p1 ). α

(3.12)

(3.13)

α

So, from ρ(x) = (1 + |x|)1+ 2 ≤ Cε−1− 2 in B2d/ε \Bdε /ε , ε4 gε,1 (εx + p1 , ω)ρ(x) 2L 2 (B

2d/ε \Bdε /ε )

≤ Cε2(2N −2−θ) ε−4−α ≤ Cε2 ,

and

gε,1 (x, ω) L 2 (\Bd ( p1 )) ≤ Cε2N −4−θ ≤ Cε2 . So, we have proved

gε,1 (x, ω) Yα,ε ≤ Cε,

(3.14)

which gives 1

Bε,1 ω L ∞ () + Bε,1 ω X α,ε ≤ C ln gε,1 (x, ω) Yα,ε ≤ ε1−σ . ε So, Bε,1 maps Sε,1 to Sε,1 . Next, we show that Bε,1 is a contraction map. For any ω1 , ω2 ∈ Sε,1 , we have

Bε,1 ω1 − Bε,1 ω2 L ∞ () + Bε,1 ω1 − Bε,1 ω2 X α,ε 1 ≤ C ln gε,1 (x, ω1 ) − gε,1 (x, ω2 ) Yα,ε . ε

(3.15)

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C.-S. Lin, S. Yan

But gε,1 (x, ω1 ) − gε,1 (x, ω2 ) = f ε,1 (x)(ω1 − ω2 ) +

   1 1 2(u 0 +ϕε ) 2ω1 e e − e2ω2 − 2 eu 0 +ϕε eω1 − eω2 . 2 ε ε (3.16)

Using (3.11) and (3.12), we deduce from (3.16),

gε,1 (x, ω1 ) − gε,1 (x, ω2 ) L 2 (\Bd ( p1 )) = O(ε) ω1 − ω2 L 2 (\Bd ( p1 )) , (3.17) and noting that ρ ρˆ −1 ≤ C(1 + |x|)2+2α ≤ Cε−2−2α in Bd/ε \Bdε /ε ,   ε2 gε,1 (εx + p1 , ω1 (εx + p1 )) − gε,1 (εx + p1 , ω2 (εx + p1 )) ρ L 2 (Bd/ε \Bdε /ε ) = ε2N −4−θ−2−2α (ω1 (εx + p1 ) − ω2 (εx + p1 )) ρ ˆ L 2 (Bd/ε \Bdε /ε ) .

(3.18)

On the other hand, by the mean value theorem, there is a t between ω1 and ω2 , such that    1 2(u 0 +ϕε ) 2ω1 1 e e − e2ω2 − 2 eu 0 +ϕε eω1 − eω2 2 ε ε  1 2(u 0 +ϕε +t) = 2 2e − eu 0 +ϕε +t (ω1 − ω2 ) ε

   1 1 2(u 0 +ϕε ) u 0 +ϕε e |t||ω = 2 2e2(u 0 +ϕε ) − eu 0 +ϕε (ω1 − ω2 ) + O + e − ω | 1 2 ε ε2

 1 = 2 2e2(u 0 +ϕε ) − eu 0 +ϕε (ω1 − ω2 ) ε

  1 2(u 0 +ϕε ) u 0 +ϕε e (|ω +O + e | + |ω |)|ω − ω | , 1 2 1 2 ε2 since |t| ≤ |ω1 | + |ω2 |). Using (3.16) and (3.9), we see that for x ∈ Bdε ( p1 ), |x− p1 | |x− p1 | |x− p1 | |x− p1 |  1 gε,1 (x, ω1 ) − gε,1 (x, ω2 ) = 2 e2(V ( ε )+2 ln ε ) + e V ( ε )+2 ln ε ε ×O (|x − p1 | + |ω1 | + |ω2 |))|ω1 − ω2 |. Therefore,   ε2 gε,1 (εx + p1 , ω1 (εx + p1 )) − gε,1 (εx + p1 , ω2 (εx + p1 )) ρ L 2 (Bdε /ε ) ≤ C (ε + ω1 L ∞ + ω2 L ∞ ) (ω1 (εx + p1 ) − ω2 (εx + p1 )) ρ ˆ L 2 (Bdε /ε ) . (3.19) Combining (3.17), (3.18) and (3.19), we obtain

gε,1 (x, ω1 ) − gε,1 (x, ω2 ) Yα,ε ≤ C (ε + ω1 L ∞ + ω2 L ∞ ) ω1 − ω2 X α,ε ≤ Cε1−σ ω1 − ω2 X α,ε , (3.20) which, together with (3.15), gives

Bε,1 ω1 − Bε,1 ω2 L ∞ () + Bε,1 ω1 − Bε,1 ω2 X α,ε ≤ So, we have proved that Bε,1 is a contraction map.  

1

ω1 − ω2 X α,ε . (3.21) 2

Bubbling Solutions for Abelian Chern-Simons Model

745

4. Bubbling Solutions for the System In this section, we will discuss the system under the assumption that p1 = q1 and N1 = N2 . Since the proof of Theorem 1.2 is similar to the proof of Theorem 1.1, we just sketch it. Let ϕε be the function defined in (2.3) and (2.4), where in the definition of the constant M in (2.5), we need to replace u 0 by u˜ 0 . Denote ∗

v = 4π

N1 

G( p1 , p j ) − 4π

N2 

j=2

G( p1 , q j ).

j=2

Our objective is to prove that (1.5) has a solution of the form u ε = 2ϕε + ωε , v = v ∗ + ηε , where (ωε , ηε ) is a perturbation term. For any ω, we consider the second equation of (1.5):

∗  v +v˜0 +η v ∗ +v˜0 +η 1 ϕε +u˜ 0 +ω η = 2 e 2 e 2 − e− 2 . ε

(4.1)

Rewrite (4.1) as follows: L ε,2 η = gε,2 (x, ω, η),

(4.2)

L ε,2 η = η + f ε,2 (x)η,

(4.3)

where



1 f ε,2 (x) = − ε2 e 0,

V

|x− p1 | ε

 +2 ln

|x− p1 | ε

gε,2 (x, s, t) is defined as gε,2 (x, s, t) = f ε,2 (x)t +

1 2ϕε +s+u˜ 0 e 2 ε2

, x ∈ B2d ( p1 ); x ∈ \B2d ( p1 ),

∗  v +v˜0 +t v ∗ +v˜0 +t e 2 − e− 2 .

(4.4)

(4.5)

It follows from Theorem B.1 that L ε,2 is an isomorphism from X α,ε to Yα,ε . Note that under the condition p1 = q1 , v˜0 has no singularity at p1 . Using the contraction mapping theorem as in Sect. 3, we can prove Proposition 4.1. Fix τ1 ∈ (0, 1). Then for any ω ∈ X α,ε with ω L ∞ () + ω X α,ε ≤ ετ1 , there is an ηε ∈ X α,ε , ηε = ηε (ω), satisfying (4.2). Moreover, 1

ηε L ∞ () + ηε X α,ε ≤ Cε ln , ε

(4.6)

and

ηε (ω1 ) − ηε (ω2 ) L ∞ + ηε (ω1 ) − ηε (ω2 ) X α,ε

 1 2 ≤ Cε ln

ω1 − ω2 X α,ε . ε (4.7)

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C.-S. Lin, S. Yan

Proof. The proof of the existence part and (4.6) is similar to the proof of Theorem 1.1. We thus omit it. To prove (4.7), we let η˜ ε =: η1 − η2 , ηi = ηε (ωi ), i = 1, 2. Then L ε,2 η˜ ε = gε,2 (x, ω1 , η1 ) − gε,2 (x, ω2 , η2 ). By Theorem B.1, we have 1

η˜ ε L ∞ + η˜ ε X α,ε ≤ C ln gε,2 (x, ω1 , η1 ) − gε,2 (x, ω2 , η2 ) Yα,ε ε 1 ≤ C ln gε,2 (x, ω1 , η1 ) − gε,2 (x, ω2 , η1 ) Yα,ε ε  + gε,2 (x, ω2 , η1 ) − gε,2 (x, ω2 , η2 ) Yα,ε  1 1 1 ≤ C ln Cε ln ω1 − ω2 Yα,ε + Cε ln η˜ ε X α,ε . (4.8) ε ε ε So we obtain (4.7).   Proof Theorem 1.2. For any ω ∈ X α,ε with ω L∞ + ω X α,ε ≤ ετ1 , let ηε (ω) be the map obtained in Proposition 4.1. We substitute vε = v ∗ + ηε (ω) into the first equation of (1.5). Then, ω satisfies L ε,1 ω = g˜ ε,1 (x, ω, ηε (ω)),

(4.9)

where L ε,1 is the linear operator defined in (3.2) and (3.3), and 2 u˜ 0 +ϕε +s 1 u˜ 0 +ϕε +s v˜0 +v∗ +t e − 2e 2 + 2 2 ε ε 1 u˜ 0 +ϕε +s − v0 +v∗ +t 8π N1 2 − ϕε . − 2e 2 + ε ||

g˜ ε,1 (x, s, t) = f ε,1 (x)s +

By (4.6), we find that g˜ ε,1 (x, ω, ηε (ω)) ≈ gε,1 (x, ω), where gε,1 (x, ω) is defined in (3.4). So we can prove 1

g˜ ε,1 (x, ω, ηε (ω)) Yα,ε ≤ Cε ln . ε Moreover, from (4.7), we can deduce

g˜ ε,1 (x, ω1 , ηε (ω1 )) − g˜ ε,1 (x, ω2 , ηε (ω2 )) Yα,ε



1 2 ≤ Cε ln

ω1 − ω2 X α,ε . ε  

Appendix A. The Invertibility of Linear Operators in R2 In this section, we consider the invertiblity of the following linear operator: ˜ = w + a(x)w, in R 2 , Lw

(A.1)

Bubbling Solutions for Abelian Chern-Simons Model

747

where a(x) is a continuous function satisfying C

, β¯ > 2. (1 + |x|)β¯    Here, we do not assume that a is negative. So R 2 |Du|2 + au 2 may not be a norm. We ˜ in a suitable weighted space. need to discuss the invertiblity of Lw Define |a(x)| ≤

1

α

ρ(x) = (1 + |x|)1+ 2 , ρ(x) ˆ =

α

(1 + |x|) (ln(2 + |x|))1+ 2

,

where α > 0 is a fixed small constant. Let X α be the closure of C0∞ (R 2 ) under the norm



ξ X α =

1

 R2

|ξ |2 ρ 2 +

R2

ξ 2 ρˆ 2

2

,

while Yα is the closure of C0∞ (R 2 ) under the norm 1



ξ Yα =

ξ ρ

2 2

R2

2

.

Then we have, ˜ = h in Theorem A.1. Suppose that L˜ has trivial kernel in L ∞ (R 2 ). If w satisfies Lw R 2 , and w ∈ X α and h ∈ Yα , then

w X α + w L ∞ (R 2 ) ≤ C h Yα , where C > 0 is a constant, independent of w and h. Proof. The proof of this theorem is exactly the same as that of Theorem 4.1 in [5].   By Theorem 2.1 in [5], we know that if N > 4, then u + eu (1 − eu ) = 4π δ0 , in R 2 ,

(A.2)

has a unique radial solution W (|x|), satisfying −t W (t) → 2N as t → +∞. In the following, we consider the invertibility of the linear operator: L 1 w = w + e W (r ) (1 − 2e W (r ) )w, in R 2 . For this purpose and the discussion of the system, we also introduce L 2 w = w − e W (r ) w, in R 2 . Lemma A.2. Suppose that w is a bounded solution of L i w = 0, i = 1, 2. Then w = 0.

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C.-S. Lin, S. Yan

Proof. The result for i = 1 follows from the first part of Corollary 2.8 in [5]. Suppose that w is bounded in R 2 satisfying L 2 w = 0. It is easy to check that for any φ ∈ C0∞ (R 2 ) with φ ≥ 0,    w+ φ ≥ φw = e W (|x|) w+ φ ≥ 0. w>0

R2

R2

Thus, w+ is a bounded sub-harmonic function in R 2 . So w+ must be a constant. Similarly, w− is a bounded super-harmonic function in R 2 . So w− must be a constant. As a result, w = 0.   One of the main results in this section is Theorem A.3. L i is an isomorphism from X α to Yα . Moreover, if w satisfies L i w = h in R 2 , and w ∈ X α and h ∈ Yα , then

w X α + w L ∞ (R 2 ) ≤ C h Yα , where C > 0 is a constant, independent of w and h. Proof. It follows from Theorem A.1, Lemma A.2 that we only need to prove that L i is a onto map from X α to Yα . We consider L 2 first. By the definition of Yα , we only need to show that for any h ∈ C0∞ (R 2 ), there is a w ∈ X α , such that L 2 w = h. Suppose that spt h ⊂ B R (0). Consider w − e W w = h, in Bn (0), w ∈ H01 (Bn (0)). By the Riesz representation theorem, (A.3) has a solution wn , satisfying  (|Dwn |2 + e W wn2 ) ≤ C.

(A.3)

(A.4)

R2

We claim that |wn | ≤ C. To prove this, for any n > R and x ∈ B R (0), sup |wn | ≤ C wn L 2 (B2 (x)) + C h L 2 (B2 (x))

B1 (x)

1

≤ C e 2 W wn L 2 (B2 (x)) + C h L 2 (B2 (x)) ≤ C. Moreover, we have wn − e W wn = 0, in Bn (0)\B R (0), from which we find |wn (y)| ≤ max |wn (x)| ≤ C, ∀ y ∈ Bn (0)\B R (0). |x|=R

So we have prove (A.5).

(A.5)

Bubbling Solutions for Abelian Chern-Simons Model

749

From (A.4) and (A.5), we know that there is w, satisfying  w − e W w = h, in R 2 , w ∈ L ∞ (R 2 ), (|Dw|2 + e W w 2 ) ≤ C.

(A.6)

R2

We claim that w ∈ X α . In fact, noting that ρ 2 e W ≤ C, we obtain    |w|2 ρ 2 ≤ h2ρ2 + e2W w 2 ρ 2 2 2 2 R R R  W 2 ≤C +C e w < ∞, R2

which, together with w ∈ L ∞ (R 2 ), implies w ∈ X α . Next, we consider L 1 . Write L 1 w = L 2 w + 2e W (1 − e W )w. Then L 1 w = h, h ∈ Yα is equivalent to −1 W W w = L −1 2 (2e (1 − e )w) + L 2 h, w ∈ X α .

(A.7)

It is easy to check that for any w ∈ X α , 2e W (1 − e W )w ∈ Yα by using the fact that −1 W W W ≤ C. Thus, L −1 2 (2e (1−e )w) is defined. On the other hand, L 2 (2e (1− W e )w) is a compact operator in X α . To check this, we just need to use

e2W ρ 2 ρˆ −2

(i) e2W ρ 2 ρˆ −2 → 0 as |x| → ∞; (ii) any bounded sequence in X α has a subsequence which is strongly convergent for any R > 0 in   1 Yα,R = ξ :

B R (0)

ξ 2ρ2

2

<∞ .

W W By the Fredholm alternative, (A.7) has solution if w = L −1 2 (2e (1 − e )w) just has zero solution. By Lemma A.2, we obtain the result.  

Appendix B. Invertibility of Some Linear Operators In this section, we will discuss the invertibility of the following linear operators L ε,1 w = w +

1 f ε,1 (x)w, ε2

L ε,2 w = w +

1 f ε,2 (x)w, ε2

and

where



  ⎧ |x− p | |x− p1 | W ⎨−e W ε 1 ε 2e − 1 , x ∈ B2d ( p1 ); f ε,1 (x) = ⎩ 0, x ∈ \B2d ( p1 ),

750

C.-S. Lin, S. Yan

f ε,2 (x) = −e 0,

W

|x− p1 | ε



, x ∈ B2d ( p1 ); x ∈ \B2d ( p1 ),

d > 0 is a fixed small constant, and W (|x|) is the solution of u + eu (1 − eu ) = 4π δ0 , in R 2 ,

(B.1)

satisfying −t W (t) → 2N as t → +∞. First, let us recall that ξ ∈ X α,ε if ξ 2X α,ε < ∞, where ξ 2X α,ε is defined in (3.6); while ξ ∈ Yα,ε if ξ 2Yα,ε < ∞, where ξ 2Yα,ε is defined in (3.7). The main result of this section is the following: Theorem B.1. L ε,i is an isomorphism from X α,ε to Yα,ε . Moreover, if wε ∈ X α,ε and h ε ∈ Yα,ε satisfy L ε,i wε = h ε , then there is a constant C > 0, independent of ε, such that 

1 ∞

h ε Yα,ε .

wε L () + wε X α,ε ≤ C ln ε When h ε has compact support in B2d ( p1 ), after blowing-up at p1 , the problem L ε,i w = h ε is a perturbation problem of L i w = h ε (εx + p1 ) discussed in Appendix A. So we just need to concentrate on the case that h ε = 0 in B2d ( p1 ). We will prove Lemma B.2. Assume h ε = 0 in B2d ( p). Suppose that wε ∈ X α,ε and h ε ∈ Yα,ε satisfy wε +

1 f ε,i (x)wε = h ε . ε2

Then there is a constant C > 0, independent of ε, such that 

1

h ε Yα,ε .

wε L ∞ () + wε X α,ε ≤ C ln ε The proof of Lemma B.2 is quite long. We leave it to the end of this section. Proof of Theorem B.1. Let |x− p1 | |x− p1 |  1 L˜ ε,1 w = w + 2 e W ( ε ) 1 − 2e W ( ε ) w, ε

and |x− p1 | 1 L˜ ε,2 w = w − 2 e W ( ε ) w. ε

Then, we may regard L ε,i as a cut-off of L˜ ε,i . Suppose that wε ∈ X α,ε and h ε ∈ Yα,ε satisfy L ε,i wε = h ε .

Bubbling Solutions for Abelian Chern-Simons Model

751

Let h ∗ε = h ε in B2d ( p), and h ∗ε = 0 in \B2d ( p). By Theorem A.3, there is wε∗ ∈ X α , such that L˜ ε,i wε∗ = h ∗ε , in R 2 , and

 R2



 21 |w˜ ∗ |2 ρ 2 + |w˜ ∗ |2 ρˆ 2 + w ∗ L ∞ ≤ C

R2

|h˜ ∗ |2 ρ 2

1 2

,

(B.2)

where w˜ ∗ (y) = w ∗ (εy + p1 ) and h˜ ∗ (y) = h ∗ (εy + p1 ). Let ξ ∈ C0∞ (B2d+δ ( p1 )), with ξ = 1 in B2d ( p1 ), 0 ≤ ξ ≤ 1, where δ > 0 is a small number. Then L ε,i (wε − ξ wε∗ ) = h¯ ε , where

 h¯ ε = h ε − ξ h ∗ε − 2Dξ Dwε∗ − ξ wε∗ + L˜ ε,i (ξ wε∗ ) − L ε,i (ξ wε∗ ) .

Note that h¯ ε = 0 in B2d ( p). By Lemma B.2, 1 1

wε − ξ wε∗ X α,ε + wε − ξ wε∗ L ∞ ≤ C ln h¯ ε Yα,ε ≤ C ln h ε Yα,ε . ε ε

(B.3)

In the last inequality, we have used (B.2). So, we obtain

wε L ∞ () + wε X α,ε ≤ wε − ξ wε∗ X α,ε + wε − ξ wε∗ L ∞ + ξ wε∗ X α,ε + ξ wε∗ L ∞ 

1

h ε Yα,ε . ≤ C ln ε

(B.4)

To show that L ε,i is an onto map, we can rewrite L ε,i w = h, as w = −( − I )

−1



 1 f ε,i (x)w + w + ( − I )−1 h, w ∈ X α,ε . ε2

Since  is bounded, it is easy to check that −( − I )−1 ( ε12 f ε,i (x)w + w) is compact. Thus the result follows from the Fredholm alternative.   We now go back to the proof of Lemma B.2. In Lemma B.2, we assume h ε = 0 in B2d ( p1 ). So we need to study the solution of the following problem: wε +

1 f ε,i (x)wε = 0, x ∈ B2d ( p1 ). ε2

(B.5)

Define f 1 (t) = e W (t) (1 − 2e W (t) ) and f 2 (t) = −e W (t) . It is easy to show that the following problem: ξ + f i (t)ξ = 0, ξ(0) = 1,

(B.6)

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C.-S. Lin, S. Yan

has a radial solution ξi , satisfying ξi (t) = ξ¯i ln t + Ii + O



1 t τi

 ,

(B.7)

for some constants ξ¯i = 0, Ii and τi > 0. The major step in the proof of Lemma B.2

 is to show that the solution of (B.5) is a perturbation of the radial solution ξi |x−εp1 | . The following decomposition lemma is crucial to the proof of Lemma B.2. Lemma B.3. Let wε,i be a solution of (B.5). Then

 |x − p1 | + φε , in B2d ( p1 ), wε,i (x) = wε,i ( p1 )ξi ε where ξi is the solution of (B.6), φε satisfies φε + ε12 f ε,i (x)φε = 0, x ∈ B2d ( p1 ), φε ( p1 ) = 0.

(B.8)

Moreover, the maximum point xε of |φε | in B2d ( p1 ) satisfies |xε − p1 | → 2d, as ε → 0, and (up to a subsequence) φε 2 → φ, in Cloc (B2d \{ p1 }), με where με = max x∈B2d ( p1 ) |φε (x)|, and φ is a harmonic function in B2d ( p1 ) with φ( p1 ) = 0. Proof. Step 1. Let xε ∈ B2d ( p1 ) be a maximum point of |φε | in B2d ( p1 ). We claim |xε − p1 | ≥ r0 > 0. We argue by contradiction. Suppose that xε → 0. First, we claim that |xε − p1 | → +∞. ε

(B.9)

p1 ) Otherwise, define φ¯ ε (y) = φεφ(εy+ . Then |φ¯ ε | ≤ 1 and φ¯ ε (yε ) = 1 with yε = xε −ε p1 → ε (x ε ) y0 ∈ R 2 . Using |φ¯ ε | ≤ 1 , we may assume that φ¯ ε → φ, and φ¯ is a bounded solution of

φ + f i (|x|)φ = 0. By Lemma A.2, φ¯ = 0, which contradicts φ¯ ε (yε ) = 1. So, we have proved the claim. Let rε = |xε − p1 |. Define φ˜ ε (y) =

φε (rε y + p1 ) . φε (xε )

Bubbling Solutions for Abelian Chern-Simons Model

753

Then, φ˜ ε + |φ˜ ε | ≤ 1, and φ˜ ε

x ε − p1 rε



1 f ε,i (rε y + p1 )φ˜ ε = 0, ηε2

= 1, where ηε =

ε rε .

Using (B.9), we see that φ˜ ε con-

verges uniformly to a bounded harmonic function φ in any compact subset of R 2 \{0}. By Liouville theorem, φ = 1 in R 2 . In particular, for any large constant C > 0, and C −1 ≤ |x−rεp1 | ≤ C, we have φε (x) 1 ≥ . φε (xε ) 2

(B.10)

Without loss of generality, we may assume φε (xε ) > 0. Define φε∗ (r )

 = 0

2π

φε (r, θ ) dθ, r = |x − p1 |.

Then, φε∗ satisfies

φε∗ + ε12 f ε,i (x)φε∗ = 0, in B2d (0) φε∗ (0) = 0.

(B.11)

Thus, by the uniqueness of the initial value problem for the ordinary differential equations, φε∗ = 0. On the other hand, by (B.10), φε∗ (r ) > 0, C −1rε ≤ r ≤ Crε . This is a contradiction. So, we have proved |xε − p1 | ≥ r0 > 0.

 Step 2. Since f i |x−εp1 | → 0 uniformly in any compact subset of B2d ( p1 )\{ p1 }, we

1 (B ( p )\{ p }). We claim may assume that μφεε converge to a harmonic function φ in Cloc 2d 1 1 φ( p1 ) = 0 if φ = 0. Suppose not. We may assume that φ( p1 ) > 0. Then, we can find r0 > 0, such that φε (x) > c με , |x − p1 | = r0 , for some constant c > 0. Define

φε∗ (r ) =

 0

2π

φε (r, θ ) dθ.

Then φε∗ (r0 ) > 0. So we can obtain a contradiction as in Step 2. Therefore, φ( p1 ) = 0. Finally, if φ = 0, the maximum point of |φ| must be on the boundary of B2d ( p1 ). Thus, |xε − p1 | → 2d. If φ = 0, then φε (x) = o(1)με in B2d−θ ( p1 ) for any θ > 0. So we also have |xε − p1 | → 2d.   Proof of Lemma B.2. We argue by contradiction. Suppose that there are εn → 0, wn ∈ X α,εn , h n ∈ Yα,εn , h n = 0 in B2d ( p1 ), satisfying wn +

1 f ε,i (x)wn = h n , in , ε2

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C.-S. Lin, S. Yan

such that



wn X α,εn + wn L ∞ () = 1, h n Yα,εn

1 = o(1) ln ε

−1

.

(B.12)

By Lemma B.3, wn has the following decomposition:

 |x − p1 | + φn , in B2d ( p1 ), wn = m n ξi ε with m n = wn ( p1 ). We may assume that m n ξ¯i ≥ 0. Thus,

 |x − p1 | ≥ 0, ∀|x − p1 | ≥ d. m n ξ0 ε Recall μn = max x∈B2d ( p1 ) |φn (x)|. Now we define μ˜ n =

max

x∈B 7 d ( p1 )

|φn (x)|.

(B.13)

4

The

crucial  step to obtain a contradiction is to show that φn is a perturbation term of m n ξi |x−εp1 | . We will prove lim

n→+∞

μn m n ln

1 εn

< +∞,

lim

n→+∞

μ˜ n m n ln

1 εn

= 0.

(B.14)

Assume (B.14) at the moment. We are now ready to obtain a contradiction. By (B.14), we find 



1 1 = O m n ln , in B2d ( p1 ), (B.15) wn = m n ξi + O m n ln εn εn which, together with the L p estimate, gives

wn L ∞ (\Bd ( p1 )) C

f ε,i (x)wn L 2 (\Bd ( p1 )) + C h n L 2 (\Bd ( p1 )) ε2 ≤ wn L ∞ (∂ Bd ( p1 )) + C h n L 2 (\Bd ( p1 )) + Cε2N −2 wn L 2 (\Bd ( p1 )) . ≤ wn L ∞ (∂ Bd ( p1 )) +

(B.16)

Thus, we obtain

wn L ∞ (\Bd ( p1 )) ≤ (1 + O(ε2N −2 )) wn L ∞ (∂ Bd ( p1 )) + C h n L 2 (\Bd ( p1 )) 

1 1 −1 ≤ C|m n | ln + o(1) ln . (B.17) εn εn Combining (B.15) and (B.17), we are led to

wn L ∞ () ≤ C|m n | ln



1 1 −1 + o(1) ln . εn εn

(B.18)

Bubbling Solutions for Abelian Chern-Simons Model

755

On the other hand, by definition, we have

wn 2X α,ε = ε4 w˜ n ρ 2L 2 (B

2d/ε )

+ w˜ n ρ ˆ 2L 2 (B ) 2d/ε

+ wn 2L 2 (\L 2 (B

+ wn 2L 2 (\B ( p )) d 1

 2 2 = − f ε,i (εy + p1 )wn (εy + p1 ) + εn h n (εy + p1 ) ρ L 2 (B ) + w˜ n ρ ˆ 2L 2 (B ) 2d/ε

+ h n 2L 2 (\L 2 (B ( p )) d 1

d ( p1 ))

2d/ε

≤ C|m n | ln



1 1 −1 + o(1) ln . εn εn

+ wn 2L 2 (\B ( p )) d 1 (B.19)

In the last inequality, we have used (B.15) and (B.17). But wn 2X α,ε + wn L ∞ () = 1. Thus, (B.18) and (B.19) gives |m n | ln

1 ≥ a0 > 0. εn

(B.20)

A direct consequence of (B.20) and (B.14) is wn (x) > 0, ∀ x ∈ B 7 d ( p1 )\Bd ( p1 ). 4

On the other hand, from (B.17), we obtain

wn L ∞ (\Bd ( p1 )) ≤ (1 + O(ε2N −2 )) wn L ∞ (∂ Bd ( p1 )) + C h n L 2 (\Bd ( p1 ))

 

d 1 −1 + max φn (x) + o(1) ln ≤ m n ξi . |x− p1 |=d εn εn

(B.21)

φn μ˜ n

converges uniformly in B 7 d ( p1 )\Bd ( p1 ) to φ˜ and φ˜ is a harmonic function 4 in B2d ( p1 ), we deduce from the maximum principle that if φ˜ = 0, the maximum point tn of φn in B 7 d ( p1 )\Bd ( p1 ) satisfies tn ≥ 23 d. If φ˜ = 0, then, maxd≤|y− p1 |≤ 3 d φn = 4 2 o(1)μ˜ n . So in this case, we also have that the maximum point tn of φn in B 7 d ( p1 )\ Since

Bd ( p1 ) satisfies tn ≥ 23 d. Thus, we obtain from (B.21) that

 |tn | + φn (tn ) = wn L ∞ (∂ B|tn | ( p1 )) ≤ wn L ∞ (\Bd ( p1 )) m n ξi εn

 

d 1 −1 + max φn (x) + o(1) ln ≤ m n ξi . |x− p1 |=d εn εn

4

(B.22)

But φn (tn ) ≥ max|x− p1 |=d φn (x). So (B.22) gives

m n ξi

|tn | εn



≤ m n ξi

d εn



 1 −1 + o(1) ln . εn

(B.23)

Using (B.7), we find



  

    1 −1 |tn | |d| m n ξ¯i ln ≤ m n ξ¯i ln + Ii + O εnσ + Ii + O εnσ + o(1) ln . εn εn εn

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C.-S. Lin, S. Yan

Thus, 3 1 |tn | 1 ξ¯i ln m n ln m n ln ≤ ξ¯i ln ≤ o(1). 2 εn d εn This is a contradiction to (B.20), since we assume m n ξ¯i > 0. To finish the proof of this theorem, it remains to prove (B.14). Let τn = max x∈B2d ( p1 ) |wn (x)|. From (B.17), we find

wn L ∞ (\Bd ( p1 )) ≤ (1 + O(ε2N −2 )) wn L ∞ (∂ Bd ( p1 )) + C h n L 2 (\Bd ( p1 )) ≤ (1 + O(ε2N −2 ))τn + C h n Yα,εn .

(B.24)

h n Yα,εn → 0. τn

(B.25)

We first claim that

Suppose that τn ≤ C h n Yα,εn . Then, (B.24) gives

wn L ∞ () ≤ C h n Yα,εn .

(B.26)

But from (B.19), 1 = wn X α,εn + wn L ∞ ()



≤ C wn L ∞ () + C h n Yα,εn ≤ C h n Yα,εn

1 = o(1) ln εn

−1

.

This is a contradiction. Thus, we prove (B.25). Next, we show μn ≤ C|m n | ln

1 . εn

(B.27)

Suppose we have |m n | ln

1 εn

μn

→ 0.

(B.28)

Then, from Lemma B.3 and (B.28), we find τn = Let w˜ n =

wn τn .

max (φεn (x) + o(με )) = (1 + o(1))μn .

x∈B2d ( p1 )

(B.29)

Then |w˜ n | ≤ 1 in B2d ( p1 ), and w˜ n satisfies w˜ n +

1 hn f εn ,i (x))w˜ n = . 2 εn τn

(B.30)

Using (B.25) and the L p estimate, we can deduce from (B.30) that

w˜ n L ∞ (\B2d ( p1 )) ≤ C +

C h n L 2 (\B2d ( p1 )) τn

≤ C.

Bubbling Solutions for Abelian Chern-Simons Model

757

Thus, w˜ n is a bounded function in . So, w˜ n converges in Cloc (\{ p1 }) to a bounded harmonic function ω1 . Noting that h n = 0 in B2d ( p1 ), similar to the proof of Lemma B.3, we can prove that the maximum point xn of w˜ n satisfies |xn − p1 | ≥ r0 > 0. Thus, ω1 = 1. On the other hand, from Lemma B.3, (B.28) and (B.29), wn μn → φ, in B2d ( p1 ). μn τn

w˜ n =

So, φ = ω1 = 1 in B2d ( p1 ). But φ( p1 ) = 0. This is a contradiction and (B.27) is proved. Using Lemma B.3 and (B.27), we obtain τn ≤ C|m n | ln Define w¯ n =

wn |m n | ln

1 εn

1 1 + μn ≤ C|m n | ln . εn εn

(B.31)

. Then w¯ n is bounded in B2d ( p1 ) and satisfies 1 hn f ε ,i (x)w¯ n = εn2 n |m n | ln

w¯ n +

1 εn

.

(B.32)

From (B.25) and (B.31), we find

h n Yα,εn |m n | ln

≤

1 εn

h n Yα,εn τn τn |m n | ln

1 εn

→ 0.

As a result, we can deduce as above that w¯ n → C in Cloc (\{ p1 }). But w¯ n =

m n ξi |m n | ln

+

1 εn

μn |m n | ln

1 εn

φn . μn

Taking a limit in the above relation, we obtain C = C1 + lim

n→+∞

μn |m n | ln

1 εn

φ, in B2d ( p1 )\{ p1 },

where C1 is a constant. If φ = 0, then φ is not a constant, since φ( p1 ) = 0. Thus lim

n→+∞

μn |m n | ln

1 εn

= 0.

So, we prove (B.14). If φ = 0, then μ˜ n = o(1)μn . So, in this case, we obtain the result from (B.27).  

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References 1. Caffarelli, L.A., Yang, Y.S.: Vortex condensation in the Chern-Simons Higgs model: an existence theorem. Commun. Math. Phys. 168(2), 321–336 (1995) 2. Chae, D., Imanuvilov, O.Y.: Non-topological multivortex solutions to the self-dual Maxwell-ChernSimons-Higgs systems. J. Funct. Anal. 196(1), 87–118 (2002) 3. Choe, K.: Asymptotic behavior of condensate solutions in the Chern-Simons-Higgs theory. J. Math. Phy. 48, 103501 (2007) 4. Choe, K., Kim, N.: Blow-up solutions of the self-dual Chern-Simons-Higgs vertex equation. Ann. I.H. Poincaré Anal., Non Linéaire 25, 313–338 (2008) 5. Chan, H., Fu, C.-C., Lin, C.-S.: Non-topological multi-vortex solutions to the self-dual Chern-SimonsHiggs equation. Commun. Math. Phys. 231(2), 189–221 (2002) 6. Chern, J.-L., Chen, Z.-Y., Lin, C.-S.: Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles. Preprint 7. Dunne, G. V.: Aspects of Chern-Simons theory. In: Aspects topologiques de la physique en basse dimension/Topological aspects of low dimensional systems (Les Houches, 1998), Les Ulis: EDP Sci., 1999, pp. 177–263 8. Dziarmaga, J.: Low energy dynamics of [U (1)] N Chern-Simons solitons and two dimensional nonlinear equations. Phys. Rev. D 49, 5469–5479 (1994) 9. Hong, J., Kim, Y., Pac, P.: Multivortex solutions of the abelian Chern-Simons-Higgs theory. Phys. Rev. Lett. 64, 2230–2233 (1990) 10. Jackiw, R., Weinberg, E.: Self-dual Chern-Simons vortex. Phy. Rev. Lett. 64, 2234–2237 (1990) 11. Jaffe A., Taubes, C.: Vortices and Monopoles: Progr. Phys. Vol. 2, Boston, MA: Birkhäuser Boston, 1990 12. Kim, C., Lee, C., Ko, P., Lee, B.-H: Schrödinger fields on the plane with [U (1)] N Chern-Simons interactions and generalized self-dual solitons. Phys. Rev. D (3) 48, 1821–1840 (1993) 13. Lin, C.-S., Ponce, A.C., Yang, Y.: A system of elliptic equations arising in Chern-Simons field theory. J. Funct. Anal. 247(2), 289–350 (2007) 14. Lin, C.-S., Prajapat, J.V.: Vortex condensates for relativistic Abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus. Commun. Math. Phys. 288, 311–347 (2009) 15. Lin, C.-S., Wang, C.-L.: Elliptic functions, Green functions and the mean field equation on tori. Ann. of Math., to appear, available at http://pjm.math.berkeley.edu/editorial/uploads/annals/accepted/080814Wang-VL.pdf 16. Tarantello, G.: Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys. 37(8), 3769–3796 (1996) 17. Tarantello, G.: Self-dual gauge field vortices: an analytical approach, Berlin-Heidelberg-NewYork, Springer, 2007 18. Nolasco, M., Tarantello, G.: On a sharp Sobolev-type inequality on two dimensional compact manifolds. Arch. Rat. Mech. Anal. 154, 161–195 (1998) 19. Nolasco, M., Tarantello, G.: Double vortex condensates in the Chern-Simons-Higgs theory. Calc. Var. and PDE 9, 31–94 (1999) 20. Nolasco, M., Tarantello, G.: Vortex condensates for the SU(3) Chern-Simons theory. Commun. Math. Phys. 213(3), 599–639 (2000) 21. Yang, Y.: Solitons in field theory and nonlinear analysis. Springer Monographs in Mathematics. New York: Springer-Verlag, 2001 Communicated by A. Kupiainen

Commun. Math. Phys. 297, 759–816 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1058-z

Communications in

Mathematical Physics

On the Fundamental Solution of a Linearized Homogeneous Coagulation Equation Miguel Escobedo1 , J. J. L. Velázquez2 1 Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, E–48080 Bilbao, Spain.

E-mail: [email protected]

2 ICMAT (CSIC-UAM-UC3M-UCM) Facultad de Matemáticas, Universidad Complutense, E–28040

Madrid, Spain. E-mail: [email protected] Received: 3 March 2009 / Accepted: 18 February 2010 Published online: 8 May 2010 – © Springer-Verlag 2010

Abstract: We describe the fundamental solution of the equation that is obtained by linearization of the coagulation equation with kernel K (x, y) = (x y)λ/2 , around the steady state f (x) = x −(3+λ)/2 with λ ∈ (1, 2). Detailed estimates on its asymptotics are obtained. Some consequences are deduced for the flux properties of the particles distributions described by such models. 1. Introduction Under rather general conditions on the kernel K (x, y), a symmetric homogeneous function in x and y, the Cauchy problem for the Smoluchowski coagulation equation:

Q[ f ] =

1 2

 0

x

∂f = Q[ f ], ∂t

(1.1) 

∞

K (x − y, y) f (x − y) f (y) dy − f (0, x) = f in (x),

K (x, y) f (x) f (y) dy, (1.2)

0

∞

(1.3)

has a global solution f (t, x) for all initial data f in (x) ≥ 0 such that 0 (1+x) f in (x)d x < ∞. Moreover, this solution satisfies the same estimate for all t > 0. Equations (1.1), (1.2) describe the aggregation process of particles of mass x and y with rate K (x, y), assuming that the distribution of particles are uncorrelated at all times. In this context the quantity:  ∞ x f (t, x) d x (1.4) 0

represents the total mass of particles in the system. On the other hand, it is known that, when the kernel is of the form K (x, y) = x α y β + x β y β with α ≥ 0, β ≥ 0 and α + β = λ > 1, the solutions to the Cauchy

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M. Escobedo, J. J. L. Velázquez

problem for the Smoluchowski equation undergo the so-called gelation phenomenon. This means that there exists a positive time Tg < ∞ such that, for all t < Tg , 

∞



∞

x f (t, x) d x =

0

x f in (x) d x

(1.5)

x f in (x) d x.

(1.6)

0

and for all t > Tg , 

∞ 0



∞

x f (t, x) d x < 0

It is also known that when λ ≤ 1 gelation does not occur and mass is conserved for all time (cf. [7,19]). Two interesting open questions are related with this phenomenon. One is to describe the solution f as t → Tg . We consider here the second one, which is to understand the behaviour of the solution f after the gelling time Tg . Although no general result is known, several partial results indicate that before the gelling time, at least for a large family of initial data, the solutions to the coagulation equation decay exponentially fast as x → +∞. The Smoluchowski equation (1.1) has a discrete counterpart for which an explicit exact gelling solution was constructed in [14] for K (k, j) = k j. Such a solution decays exponentially fast before the gelling time, and as a power law after that time. The exponential decay, before the gelling time, was later shown in [8] for the continuous equation (1.1), λ = 2 and several initial data. Moreover, it has also been formally shown in [4,8] that, for several initial data and λ ∈ (1, 2], the solution of (1.1) decays, after gelling, like x −(3+λ)/2 as x → +∞ (see [13] for more detailed references). On the other hand, it was proved in [9] that x −(3+λ)/2 is the only possible power law decay for the solutions of (1.1) after gelation. Our main purpose is to prove that for the coagulation kernel K (x, y) = (x y)λ/2 , λ ∈ (1, 2)

(1.7)

and any initial data f in , regular near the origin and such that: f in (x) ∼ x −(3+λ)/2 as x → +∞,

(1.8)

the problem (1.1)–(1.3) has a solution f satisfying f (t, x) ∼ a(t) x −(3+λ)/2 as x → +∞. Moreover, this solution satisfies  ∞ d x f (x, t)d x = −2πa 2 (t) for all t > 0, dt 0

(1.9)

(1.10)

which was formally shown in [6] for the discrete equation. By (1.10) the total mass of the solution f is decreasing. This loss of mass is a characteristic feature of the solutions of (1.1),(1.2) after the gelation time. The choice of exponents λ < 2 is natural, because λ ≤ 2 excludes instantaneous gelation or non existence of solutions ([3,6,18]) and λ = 2 is one of the “explicit” cases which has been treated using the Laplace transform (cf. [8]).

Linearized Homogeneous Coagulation Equation

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In order to prove the existence of classical solutions of (1.1)-(1.3) after gelation we will use the same approach as in [10,11]. The starting point of this approach is to linearize around an initial data f in satisfying f in ≈ x −(3+λ)/2 for x large and to derive detailed estimates on the solutions of the resulting linear equation. ∂g = L(g, f in ). ∂t

(1.11)

To this end we will need some rather delicate estimates on the asymptotics of the solutions as x tends to infinity. Moreover, even to prove solvability of the linearized problem (1.11) is nontrivial. We will obtain it treating this problem as a perturbation of the problem obtained replacing f in by its asymptotics as x tends to infinity: ∂g = L(g). ∂t

(1.12)

In order to carry on this program we need to derive detailed estimates about the solutions of (1.12). This will be the main goal of this paper. The linearized equation around the weak solution x −(3+λ)/2 may be introduced more directly as follows. Consider a solution f (t, x) of the coagulation equation with an initial data f in satisfying (1.8). If one is interested in the behaviour of f (t, x) for x large it is natural to scale the variables as follows: x = R x, y = R y, t = R −(λ−1)/2 t and f (t, x) = R −(3+λ)/2 FR (t, x). In these new variables, Eq. (1.1) reads (FR )t = Q[FR ] and the initial data Fin satisfies now: FR (0, x) = R (3+λ)/2 f in (R x) ∼ (x)−(3+λ)/2 as R → +∞. The limit of the function FR as R → +∞, if it exists, would then solve the same Eq. (1.1) with initial data x −(3+λ)/2 . Therefore the linear problem (1.12) appears naturally as the linearisation of the coagulation Eq. (1.1) in the region x >> 1. Notice, however that in the region where x is small the function FR is bounded and the approximation by means of the power law x −(3+λ)/2 cannot be valid. The analysis of that region would lead naturally to the study of a boundary layer whose description requires the analysis of the operator L. This will be made in a forthcoming work. On the other hand, the linearized equation (1.12) has some interest by itself. It is indeed a simple model to describe a set of particles at equilibrium, whose density distribution is given by x −(3+λ)/2 , and where a small set of particles is introduced, whose distribution ϕ(x) is considered as a small perturbation. The particles so introduced start to collide both between themselves and with the particles in the background. The equilibrium density distribution x −(3+λ)/2 is then perturbed. The distribution density function of the resulting set of particles may then be seen at any time t as the equilibrium distribution x −(3+λ)/2 and a remaining perturbation ϕ(t, x). The linear equation (1.12) only takes into account the collisions of the “particles in the perturbation” with the background and describes how the distribution of these particles evolves in time. It neglects the collisions between particles in the perturbation. This could be a reasonable approximation as long as the perturbation ϕ(t, x) remains small. Notice that the number of clusters in the background as well as the number of particles (the total mass) are infinite (since neither x −(3+λ)/2 nor x 1−(3+λ)/2 are integrable in (0, +∞)), but the number of clusters and particles in the initial perturbation are finite. Our results show the following: – There is instantaneously an infinite number of “perturbed clusters”, although their mass is finite. – As t → +∞, the number of perturbed particles (the mass in the perturbation) tends to zero, but the number of perturbed clusters remains infinite.

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– The total flux of particles is perturbed at t finite but tends to the flux corresponding to the original equilibrium distribution as t → +∞. Our results are obtained using classical Fourier analysis and the Wiener Hopf method, in a similar way as we did for the linearized Uehling Uhlenbeck operator in [10] although with an important difference. This is the regularising effect of the operator L, absent in the operator studied in [10], and coming from the fact that L is similar to the half derivative operator. The fundamental solution of (1.12) has then very different properties than that obtained in [10]. In Sect. 2 we shortly describe the conventions used for the names of the different constants in the paper. In Sect. 3 we state our main results and transform the integro differential equation (1.12) to a Carleman equation in the complex plane. In Sect. 4 we state the fundamental properties of the auxiliary function Φ appearing in the Carleman equation. This equation is solved in Sects. 5 and 6 using the classical Cauchy integral, which gives an explicit solution. We prove in Sect. 7 that the fundamental solution obtained in this paper is unique in a suitable functional class. In Sects. 8 and 9, the precise asymptotics of the solutions are obtained. Section 10 describes how to solve the initial value problem associated to the linearised coagulation equation. Some properties of the fluxes of particles described by the solutions are considered in Sect. 11. We have finally Sects. 12, 13 and 14 where some necessary technical results are collected.

2. A Guide about the Names of the Constants used in this Paper In this paper we will use different letters to denote the numerical constants used in the arguments. In order to make easier the reading we will describe the role that constants with different names have in the arguments. Unless specifically stated similar names will be used for these constants in independent arguments. We will denote as C a positive constant whose value can change from line to line that depends only on λ and in the variables mentioned in each specific lemma. We denote as ε a positive constant that can be made arbitrarily small. This constant will be always an exponent appearing in estimate containing also a multiplicative constant depending on ε. We will denote as δ a positive constant, perhaps small and depending only on λ. Occasionally we will denote also as a similar positive constants appearing in exponential functions. We will use ε0 , ε1 , . . . , δ1 , δ2 , . . . to denote small positive constants used in technical arguments. The names εk s will be reserved for variables which are sent to zero or infinity respectively at the end of the argument, and the names δk s for constants that must remain strictly positive until the end of the argument. We will denote as βk ∈ R the names used to characterize some horizontal lines in the complex plane used in contour integration arguments involving the variables η, y, ξ .  V and G in (5.1), (6.6), The values β0 , β1 , β2 are used to define the functions G, (9.1) respectively. They take values in the ranges defined by (5.2), (6.13), (9.2). Due to the fact that the three mentioned functions play a central role in the arguments, the corresponding constants β0 , β1 , β2 will be used repeatedly. On the other hand we will use the names β3 , β4 , . . . , β10 to denote horizontal lines in the variables y, ξ that will arise in contour deformation arguments. These constants will be used only once. Finally we will denote as γ1 , γ2 real numbers characterizing horizontal lines in the variable Y = y − ξ.

Linearized Homogeneous Coagulation Equation

763

Other letters used for the constants are B that characterizes the domains D (ξ, B), L that measures the width of some strips of the complex plane and M that denotes a large constant at two places. 3. The Linearized Equation We start this section writing the precise expression of the linearized equation (1.12). Proposition 3.1. The linearized equation of (1.1)-(1.2) with K (x, y) = (x y)λ/2 around the solution x −(3+λ)/2 is ∂g = L(g), (3.1) ∂t  x/2   L(g) = (x − y)−3/2 − x −3/2 y λ/2 g(y)dy 0



x/2 

 (x − y)λ/2 g(x − y) − x λ/2 g(x) y −3/2 dy 0  ∞ √ y λ/2 g(y)dy − 2 2x (λ−1)/2 g(x). −x −3/2

+

(3.2)

x/2

Proof. By the symmetry of the kernel K :   1 x/2 1 x K (y, x − y) f (y) f (x − y)dy = K (y, x − y) f (y) f (x − y)dy. 2 0 2 x/2 Then we may write Eq. (1.1)–(1.2) as follows:  x/2   ∂f = (x − y)λ/2 f (x − y) − x λ/2 f (x) y λ/2 f (y)dy ∂t 0  ∞ K (x, y) f (x) f (y)dy. −

(3.3)

(3.4)

x/2

If we linearize around the x −(3+λ)/2 , define f = x −(3+λ)/2 + g and neglect quadratic terms on g we obtain (3.1), (3.2). Remark 3.2. The second term in the right-hand side of (3.2) can be seen as some kind of half derivative operator applied to function x λ/2 g(x). This will appear again in the Fourier analysis that will be made later on the linearized equation. Remark 3.3. In order for the first integral in the right hand side of (3.2) to be defined we need y 1+λ/2 g(y) to be integrable at the origin. For the second integral we need some kind of regularity of g(x) with respect to x, for example y λ/2 g(y) γ -Hölder continuous with γ > 1/2. Finally, for the last one we need y λ/2 g(y) to be integrable as y → ∞. Assuming power like behaviours we then need bounds on g of the form: g(y) ≤ C y −λ/2−r as y → +∞, g(y) ≤ C y −λ/2−ρ as y → 0,

(3.5) (3.6)

for some r > 1 and ρ < 2. We will solve the problem in functional spaces where (3.5) and (3.6) are satisfied in some averaged sense because we solve (3.1), (3.2) with initial value a Dirac mass.

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Definition 3.4.



E = g ∈ M + (R+ ); ||g||E < +∞ ,     1 1 ||g||E = sup R 3/2 dg + sup R (3+λ)/2 dg , R [R/2,R] R [R/2,R] R≤1 R≥1

where M + (R+ ) is the set of nonnegative Radon measures in R+ . For all T > 0 we define:

HT = g ∈ L ∞ (0, T ); E , ||g||HT = sup ||g(t)||E . 0≤t≤T

We state now the main results of this paper. Theorem 3.5. For all x0 > 0, there exists a unique solution of (3.1), (3.2) g ∈ HT for any T > 0. satisfying  ∞ lim g(t, x, x0 ) ϕ(x) d x = ϕ(x0 ) (3.7) t→0 0

for all ϕ ∈ Cc (R+ ). Moreover g ∈ C ∞ (R+ × R+ ) and has the self similar form  λ−1 1 x g(t, x, x0 ) = g t x0 2 , , 1 . x0 x0

(3.8)

Remark 3.6. The self similar form (3.8) just follows from the rescaling properties of the problem (3.1), (3.2), (3.7). Notice that due to (3.8) it is enough to restrict our analysis to x0 = 1. Remark 3.7. It may be relevant to notice the strong regularising effect of Eq. (3.1) (3.2) compared to other kinetic models like for instance the Uehling-Uhlenbeck equation considered in [10]. Theorem 3.8. Let g be the solution of (3.1), (3.2) that has been obtained in Theorem 3.5. Then, there exists positive constants δ and 1 , only depending on λ, such that for any 0 < ε < ε1 the following statements hold: For all t ≥ 1: 2

g(t, x, 1) = t λ−1 ϕ1 (σ ) + ϕ2 (t, σ ),

(3.9)

where σ is the self similar variable: 2

σ = t λ−1 x, and the functions ϕ1 and ϕ2 satisfy the following estimates:  4−λ 

3 for 0 ≤ σ < 1 a1 σ − 2 + Oε σ − 2 +ε ϕ1 (σ ) =  −(1+λ−ε)  − 3+λ a2 σ 2 + Oε σ for σ > 1, where a1 and a2 are two explicit constants, ⎧   2 −δ − 23 +δ ⎨ b1 (t) σ − 23 + O t λ−1 for 0 ≤ σ < 1 σ   2 ϕ2 (t, σ ) = 3+λ 3+λ ⎩ b2 (t) σ − 2 + O t λ−1 −δ σ − 2 −δ for σ > 1,

(3.10)

(3.11)

(3.12)

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765 2

where b1 and b2 are two continuous functions such that |b1 (t)| + |b2 (t)| ≤ C t λ−1 −δ . For all 0 < t < 1: ⎧   − 32 − 32 − 32 +δ ⎪ t x for 0 ≤ x ≤ 21 , + b (t) x + O t x ⎪ 3 ⎪ ⎪   ⎨ − 3+λ − 3+λ − 3+λ −δ for x ≥ 23 , (3.13) 4 (t) x 2 + O t x 2 g(t, x, 1) = a3 tx 2 + b ⎪ ⎪ ⎪ t 1−2ε ⎪ for t 2 < |x − 1| < 21 , ⎩ Oε 3 −ε |x−1| 2

where a3 is an explicit numerical constant and b3 and b4 are continuous functions such that |b3 (t)| + |b4 (t)| ≤ Ct 1+δ . Finally: lim t 2 g(t, 1 + t 2 χ , 1) = Ψ (χ ) uniformly on compact subsets of R, (3.14)

t→0

where the function Ψ is given by:

Ψ (χ ) =

π χ 3/2

0

e

− πχ

, for all χ ≥ 0, for all χ < 0.

(3.15)

Remark 3.9. We use the standard notation f = O(g) to denote that | f | ≤ Cg in the region under consideration with C depending only on λ. The notation f = Oε (g) has a similar meaning but with C depending also on ε. Remark 3.10. For t ≥ 1, (3.11) (3.12) imply: ⎧   1−2δ 1 1 3 3 3 ⎪ ⎪ a1 t − λ−1 x − 2 + O t − λ−1 x − 2 +δ + t − λ−1 −δ x − 2 , ⎪ ⎪ ⎪ 2 ⎨ for 0 < x < t − λ−1 ,   g(t, x, 1) = λ+1 λ+1 − λ−1 − λ+1+2δ − λ−1 −δ − 3+λ − 3+λ − 3+λ −δ ⎪ λ−1 2 2 2 x +O t x +t x ⎪ ⎪ a2 t ⎪ ⎪ 2 ⎩ − λ−1 for x > t .

(3.16)

Our strategy to solve the problem (3.1), (3.2), (3.7) is to use Fourier analysis. The resulting problem is explicitly solvable by means of the Wiener Hopf method [2]. Using the representation formula for the solution, we then prove Theorem 3.5 and Theorem 3.8 by deriving suitable a priori estimates. Related arguments have been used in [10].

3.1. Fourier variables. We reformulate the original problem using Fourier variables. To this end we define x = e X , X ∈ R, as well as the Fourier transform  1  G(t, ξ ) = √ e−i X ξ G(t, X )d X, G(t, X ) = g(t, e X ). (3.17) 2π R Then, the problem (3.1), (3.2), (3.7) reads in terms of the new variables:    ∂G  t, ξ + λ − 1 i Φ ξ + λ − 1 i , (t, ξ ) = G ∂t 2 2 1  ξ) = √ , G(0, 2π

(3.18) (3.19)

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where the function Φ is given by: Φ(ξ ) = −

√ 2 π Γ (iξ + 1 + λ2 ) Γ (iξ +

λ+1 2 )

.

(3.20)

 t) = √1 M(g)(−i ξ ), where M(g) is Indeed, this is a consequence of the fact that G(ξ, 2π the Mellin transform defined in Sect. 12 (cf. (12.1). Notice that Φ(ξ ) = P(iξ + 1 + λ/2), where P is as in (12.5). The fact that the function g(t, ·, 1) ∈ E implies that:  ·) is analytic in S = {ξ ∈ C; I m(ξ ) ∈ (3/2, (3 + λ)/2)}. G(t,

(3.21)

Problems of the form (3.18), (3.19) and (3.21) are a particular case of so called Carleman’s equations (cf. [2]).

4. The Auxiliary Function We now summarise some properties of the function Φ in (3.20). Proposition 4.1. The function Φ is a meromorphic function in C with simple poles at:  λ ξ p (n) = i 1 + + n , n = 0, 1, · · · , 2

(4.1)

 1+λ ξz (n) = i n + , n = 0, 1, · · · . 2

(4.2)

and whose zeros are:

For all M > 0 fixed:  √ Φ(ξ ) = − 2π (1 + i Q) Q ξ +

√   1 2π (1 + i Q)i  1 λ + +O Qξ ξ 8 4 |ξ |3/2

as Re(ξ ) → ∞, uniformly on I m(ξ ) ∈ (−M, M), and where the function Q is defined as: Q ≡ Q(ξ ) = sgn (Re(ξ ))

(4.3)

with sgn(0) = 0. Proof. The properties of poles and zeros are a consequence of Proposition 12.2 (Fig. 1). The asymptotic behaviour of Φ(ξ ) as |Re(ξ )| → +∞ follows from Proposition 12.3 and Taylor’s expansion.

Linearized Homogeneous Coagulation Equation

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Fig. 1. Some relevant zeros and poles of the function Φ

5. Solving (3.18)-(3.19)  analytic in the strip S. Since we are Our goal is to solve the problem (3.18)-(3.19) with G interested in deriving a solution G (t, X ) in the sense of distributions, we want to obtain  as |Re (ξ )| → ∞. We will actually obtain, for the particular solution boundedness of G  ·) constructed here, exponential decay, something that means that G (t, ·) ∈ C ∞ G(t, for t > 0. The following transformation allows to reduce (3.18), (3.19) to an equation for a function depending only on one variable ξ : √   2 V (ξ ) 2i(ξ −y) 2i (ξ − y)  λ−1 t dy Γ − G (t, ξ ) = − √ λ−1 πi (λ − 1) I m(y)=β0 V (y)

(5.1)

for some β0 ∈ (3/2, 2).

(5.2)

5.1. A heuristic derivation of (5.1). A heuristic explanation for the formula (5.1) can be given using the Laplace transform. Suppose that we define the Laplace transform of  (t, ξ ) in t as: G  ∞  (t, ξ ) e−zt dt.  G G (z, ξ ) = 0

Then, (3.18), (3.19) becomes:   λ−1 1 λ−1   i Φ ξ+ i +√ . z G (z, ξ ) = G t, ξ + 2 2 2π

(5.3)

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The solution of this equation can be formally reduced to (5.11) by means of the transformation:  2i  log (−z) ξ V (ξ ) H (z, ξ ) . (5.4) G (z, ξ ) = exp − λ−1  (·, ξ ) can be expected to The reason for using log (−z) instead of log (z) is that G be analytic for Re (z) > a for some a ∈ R and in this form the function log (−z) can be expected to be analytic in this region assuming that arg (z) ∈ (−2 π, 0) . The transformation (5.4) brings (5.3) to 2i  λ−1 1 e λ−1 log(−z)ξ H (z, ξ ) − H z, ξ + i =√ . 2 2π zV (ξ )

(5.5)

Equation (5.5) can be transformed into a Riemann-Hilbert problem by means of the following conformal mapping: 4π

H (z, ξ ) = h (z, ζ ) , ζ = e λ−1 (ξ −β0 i)

(5.6)

where, for the sake of simplicity we will write, with some slight abuse of notation V (ξ ) = V (ζ ) . Then (5.5) becomes: 4π

h (z, ζ + i0) − h (z, ζ − i0) =

e λ−1 β0 α(z) ζ α(z) , ζ ∈ R+ √ 2π zV (ζ )

(5.7)

with h analytic in C\R+ and: α (z) =

1 arg (−z) . 2πi

It is well known that the solution of Riemann-Hilbert problems can be obtained using ζ α(z) Wiener Hopf methods (cf. [16,17]). However, in this particular case, assuming that V (ζ ) satisfies suitable boundedness estimates for small and large ζ , we can solve (5.7) just using Cauchy’s formula to obtain: 4π

1 e λ−1 β0 α(z) 1 h (z, ζ ) = √ 2πi 2π z



∞

0

s α(z) ds , V (s) (s − ζ )

and, using (5.6): H (z, ξ ) =

1 1 1 √ 2πi 2π z



∞ −∞

4π α(z)

dy e λ−1 y . 4π (ξ −y) V (y) 1 − e λ−1

(5.8)

It then follows from (5.4) that:  (z, ξ ) = 1 √1 1 V (ξ ) G 2πi 2π z



∞ −∞

e

4π α(z) λ−1 (y−ξ )

V (y)

dy 4π

1 − e λ−1 (ξ −y)

and inverting the Laplace transform we finally obtain (5.1).

,

(5.9)

Linearized Homogeneous Coagulation Equation

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5.2. Proving that (5.1) solves (3.18) (3.19). We have: Lemma 5.1. Suppose that V (η) is analytic in the strip S and that V satisfies     1  − π |y|   e (λ−1)  |y| + 1 |dy| < ∞   I m(y)=β0 V (y) for any β0 ∈

3

2, 2



(5.10)

, as well as:   λ−1 λ−1 i Φ η+ i V (η) = −V η + 2 2

(5.11)

   (t, ξ ) by means of (5.1) for I m (ξ ) > β0 . Then G  can be for I m (η) ∈ 23 , 2 . Define G 3  extended analytically to S and it solves (3.18), (3.19) for I m (ξ ) ∈ 2 , 2 . Remark 5.2. In formula (5.11) and all the remainder of the paper we denote as |dy| the total variation of the signed measure dy. Proof. It just follows by direct computation. Indeed, notice that Stirling’s formula, that is uniformly valid for Γ (z) , arg (z) ∈ (−π + ε0 , π − ε0 ) with ε0 > 0 (cf. [1]) implies: π |y|    − (λ−1)   Γ − 2i (ξ − y)  ≤ C R e√   λ−1 |y| + 1

for |ξ | ≤ R, I m (y) = β0 . Therefore, the integral  on the right-hand side of (5.1) con (t, ξ ) verges for any ξ ∈ S∩ ξ : I m (ξ ) ∈ β0 , 3+λ due to (5.10) and the function G 2 satisfies:     1  − π |y| |dy|    e (λ−1) √   < ∞, |ξ | ≤ R. G (t, ξ ) ≤ C R   |y| + 1 I m(y)=β0 V (y)  in S. Taking β0 arbitrarily close to 23 we obtain analyticity of G  in (5.1) can be computed by means Moreover, the derivative with respect to t of G of: √     −y)  ∂G 2 V (ξ ) 2i(ξ 2i (ξ − y) −1 t λ−1 +1 dy, Γ − (t, ξ ) = √ ∂t λ−1 πi (λ−1) I m(y)=β0 V (y) (5.12) where we have used zΓ (z) = Γ (z + 1). On the other hand, using (5.1) we obtain:   √   V ξ + (λ−1) 2i(ξ −y) 2 i − 1) 2 (λ  t, ξ + i = −√ t λ−1 −1 G 2 V (y) πi (λ − 1) I m(y)=β0  2i (ξ − y) + 1 dy, ×Γ − λ−1 and using (5.11) and (5.12), (3.18) follows.

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It only remains to check (3.19). To this end we use contour deformation and the residue theorem to transform (5.1) into:  (t, ξ ) = √1 − G r (t, ξ ) , G 2π √   2 V (ξ ) 2i(ξ −y) 2i (ξ − y) r (t, ξ ) = √ t λ−1 Γ − dy, G λ−1 πi (λ − 1) I m(y)=I m(ξ )+δ1 V (y) with δ1 > 0 small. Using (5.10) it follows that: 2δ1    G r (t, ξ ) ≤ C R t λ−1 , ξ ∈ S ∩ {|ξ | ≤ R} ,

r converges to zero as t → 0 uniformly in bounded sets of ξ, whence and therefore G (3.19) follows.

6. On the solutions of (5.11) Equation (5.11) admits infinitely many solutions. Indeed, given any solution V par t (ξ ) we can obtain any other one by means of: V (ξ ) = V par t (ξ ) p (ξ ) , where λ−1 i . p (ξ ) = p ξ + 2 

(6.1)

Notice that (6.1) has infinitely many solutions, some of them being e4π  ξ/(λ−1) ,  ∈ N, and linear combinations of them. Given such a non uniqueness a natural and essential question is then how to choose one of them. We may state several sufficient conditions  is the Fourier transform of a tempered distribution. First we that would ensure that G  want the function G to be defined. This is guaranteed by the condition (5.10) above.  ξ ) is globally bounded with However, this condition is not sufficient to prove that G(t, respect to ξ . The difficulty comes from the fact that, if the behaviours of V(ξ ) are too disparate as Re(ξ ) tends to plus or minus infinity, the quotient VV(ξ(ξ+Y) ) may be strongly increasing in some regions of the integral in (5.1). A sufficient condition to avoid this difficulty is to have: |V(ξ )| ≈ e B± |ξ | , |B± | ≤

π (λ − 1)

(6.2)

as Re(ξ ) → ±∞. The decay rate of the Gamma function in (5.1) may then control the possible growth of the quotient VV(ξ(ξ+Y) ) uniformly on ξ . Another requirement that we need for the function V comes from the requirement  must be analytic in the strip S. This is ensured by imposing also that V is also that G analytic in S. We will then construct a function V analytic in that strip, satisfying Eq. (5.11) for I m(ξ ) ∈ (3/2, 2), and conditions (5.10) and (6.2).

Linearized Homogeneous Coagulation Equation

771

A particular solution of (5.11) can be easily obtained using Cauchy’s formula. To this end we take the logarithm of both sides of (5.11) to obtain:     λ−1 λ−1 log (V (ξ )) = log V ξ + i + log −Φ ξ + i (6.3) 2 2 or equivalently   λ−1 log V ξ − i = log (V (ξ )) + log (−Φ (ξ )) . 2

(6.4)

Let us take any β1 such that Φ (ξ ) has no zeros nor poles along the line I m (ξ ) = β1 . We define: 4π

ψ (ζ ) = log (V (ξ )) , ζ = e λ−1 (ξ −β1 i) , Q (ζ ) = log (−Φ (ξ )) . Equation (6.4) then becomes ψ (ζ + i0) = ψ (ζ − i0) + Q (ζ − i0) , ζ ∈ R+

(6.5)

with ψ analytic in C\R+ . Taking into account that |Q (ζ )| ≤ C (1 + |log (ζ )|) we can obtain a particular solution of (6.5) as:    1 1 1 ψ (ζ ) = − Q (s) ds, 2πi R+ s−ζ s+1 where the term 1/(s + 1) has been added to the classical Cauchy integral in order to ensure the convergence of the integral. Then, returning to the variable ξ we obtain: V par t,β1 (ξ ) 

2 = exp (λ − 1) i



 log (−Φ (η)) I m(η)=β1

1 4π

1 − e λ−1 (ξ −η)

−



1 4π

1 + e− λ−1 η

 dη ,

(6.6)   where I m (ξ ) ∈ β1 − λ−1 2 , β1 . Formula (6.6) provides a particular solution of (5.11). On the other hand, we can obtain an infinite family of solutions of (6.1) given by: 4π

p (ξ ) = e λ−1 ξ ,  ∈ Z.

(6.7)

Let us define a family of solutions of (5.11): 4π 

V (ξ ) = e λ−1 ξ V par t,β1 (ξ ) .

(6.8)

Actually, using Fourier series, it can be seen that any solution of (6.1) can be written as an infinite linear combination of the functions V (ξ ). The formula (6.6) does not define uniquely the function V par t,β1 unless we prescribe the value of β1 and the argument of the function ln(−Φ(η)). The different possible choices of this argument just differ by a factor 2π  i and therefore the resulting functions V par t,β1 would differ by a multiplicative factor (6.7). Proposition 4.1 implies that arg(−Φ(η)) → π/4 + 2 π  i as Re(η) → +∞. In order to avoid exponential factors in

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M. Escobedo, J. J. L. Velázquez

some of the forthcoming formulas, we determine uniquely the function ln(−Φ(η)) by choosing: lim

Re(η)→+∞

arg(−Φ(η)) =

π . 4

(6.9)

Notice that in the formula (6.6) there exists an infinite possibility of choices of the constant β1 . These functions may be extended analytically moving ξ and simultaneously the contour of integration in such a way that the condition −(λ − 1)/2 < I m(ξ − η) < 0 always holds. The only true obstruction to extend analytically the functions V par t,β1 arises from crossing with the contour deformation the zeros or poles of the function Φ. 1 are such that: Suppose that ξsing is a zero or a pole of Φ and β1 , β ξsing −

1 1 1 < ξsing + . < β1 < ξsing < β 2 2

Then  4π −n V par t,β1 (ξ ) e λ−1 (ξsing −ξ ) − 1 =− , 4π V par t,β1 (ξ ) 1 + e λ−1 ξsing where

 n=

1 if ξsing is a zero −1 if ξsing is a pole.

(6.10)

(6.11)

Combining (6.10) with (6.6) we can then extend any function V par t,β1 to the whole complex plane as a meromorphic function. As it could be expected the different functions V par t,β1 can be related to each other by means of linear combinations of functions of the form given in (6.8). In order to obtain the function V(ξ ) with the properties requested above, it is sufficient to take  λ−1 (6.12) , β1 , V(ξ ) = V par t,β1 (ξ ), I m(ξ ) ∈ β1 − 2 with

 β1 ∈

2+λ 3+λ , . 2 2

(6.13)

Moving the contour of integration if needed, inside the strip I m(ξ ) ∈ (2+λ)/2, (3+λ)/2 we obtain that the function V(ξ ) has no zeros nor poles in the whole strip S. It only remains to check that this function satisfies the two conditions (5.10) and (6.2). It follows from Proposition 13.2 in Sect. 13 that: Cε e

− 12



π λ−1 +ε



|ξ |

1

π

≤ |V (ξ )| ≤ Cε e− 2 ( λ−1 −ε)|ξ | for ξ ∈ S

(6.14)

for δ > 0 arbitrarily small and Cδ > 0 a constant depending on δ. This behaviour implies both (5.10) and (6.2). Finally, we can extend V meromorphically using (5.11). The positions of the poles and zeros of V can then be obtained using (3.20) as well as the properties of the Gamma function.

Linearized Homogeneous Coagulation Equation

773

Summarizing, we have shown: Proposition 6.1. The function V(ξ ) defined by means of (6.6), (6.12) with β1 as in (6.13) can be extended analytically to the strip S and meromorphically to the whole complex plane. It satisfies Eq. (5.11) as well as the estimates (6.14). Morever, V(ξ ) = 0 in all the strip S and we have the following representation formula:   V(ξ ) 2 ln (−Φ(η)) = exp V(y) (λ − 1)i I m η=β1    1 1 × − dη (6.15) 4π 4π 1 − e λ−1 (ξ −η) 1 − e λ−1 (y−η) for ξ and y such that β1 − (λ − 1)/2 < I m(ξ ) < β1 and β1 − (λ − 1)/2 < I m(y) < β1 . The poles of the function V are:  1+λ λ−1 +n+k i, n = 1, 2, . . . , k = 0, 1, . . . , (6.16) 2 2  λ λ−1 1+ −k i, k = 1, 2, . . . (6.17) 2 2 The zeros of the function V are  λ λ−1 1+ +n+k i, n = 1, 2, . . . , k = 0, 1, . . . , 2 2  1+λ λ−1 −k i, k = 1, 2, . . . . 2 2

(6.18) (6.19)

 ξ ) defined by means of (5.1) with V defined in PropoCorollary 6.2. The function G(t, sition 6.1, solves (3.18), (3.19). 7. Uniqueness of Solutions In this section we prove the uniqueness of the solution g in HT stated in Theorem 3.5. Suppose that there exist two solutions g1 , g2 in the space HT of the problem (3.1), (3.2), (3.7). Then the difference g = g1 − g2 ∈ HT , solves (3.1), (3.2) and satisfies g(t, ·)  0  ξ ) as in (3.17). Notice that since g ∈ HT we have that as t → 0+ . We then define G(t,  ·) is analytic and bounded in S for 0 ≤ t ≤ T and it satisfies (3.18) in (0, T ) × S. G(t,  ξ ) = 0 uniformly in compact sets of ξ . Let Moreover due to (3.7) we have limt→0+ G(t, σ (t) be a C ∞ cut-off function satisfying σ (t) = 1 for 0 ≤ t ≤ T /2, σ (t) = 0 if t ≥ T .  ξ ) σ (t), then Define G(t, ξ ) = G(t,   λ−1 ∂G λ−1 (t, ξ ) = G t, ξ + i Φ ξ+ i + r (t, ξ ), (7.1) ∂t 2 2 where the function r is bounded in (0, T ) × S and r (t, ·) ≡ 0 for 0 ≤ t ≤ T /2. Taking the Laplace transform on t of (7.1) we obtain:    (z, ξ ) = G  z, ξ + λ − 1 i Φ ξ + λ − 1 i +  zG r (z, ξ ), (7.2) 2 2 Re(z) > 0, ξ ∈ S, (7.3)

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where, for some positive constant C: T

| r (z, ξ )| ≤ C e− 2

for all ξ ∈ S, Re(z) > 0.

Re(z)

(7.4)

 as G =G  par t + G hom , where G  par t Due to the linearity of (7.2), (7.3) we can split G may be obtained as in the heuristic argument in Subsect. 5.1 (cf. (5.9)): 

 par t (z, ξ ) = 1 V (ξ ) G 2πi z

∞

e

−∞

4π α(z) λ−1 (y−ξ )

V (y)

r˜ (z, ξ ) dy 4π

1 − e λ−1 (ξ −y)

,

(7.5)

hom solves and where G   λ−1 λ−1   i Φ ξ+ i , z G hom (z, ξ ) = G hom z, ξ + 2 2 Re(z) > 0, ξ ∈ S.

(7.6) (7.7)

We assume in (7.5) that V(ξ ) is defined as in (6.6), (6.12) (cf. also Proposition 6.1). Then due to our choice of log(−z) we have α(z) ∈ (−3/4, −1/4) for Re(z) > 0.  par t in (7.5) Therefore using also (6.14) and (7.4) it follows that the integral defining G  par t (·, ξ ) is analytic in Re(z) > 0 for any ξ ∈ S. is convergent for any ξ ∈ S and G Moreover,  par t (z, ξ )| ≤ C e− 2 |G T

Re(z)

, for all ξ ∈ S, Re(z) > 0.

(7.8)

hom we define (cf. (5.4)) To study the function G 2i

H (z, ξ ) =

hom (z, ξ ) e λ−1 log(−z) ξ G , V(ξ )

(7.9)

and we then define h(z, ξ ) by means of (5.6). Then h(z, ·) is analytic on C\R+ and h (z, ζ + i0) = h (z, ζ − i0) , ζ ∈ R+ ,

(7.10)

 (6.14), (7.8), (7.9) and (5.6), whence h(z, ·) is analytic in C\{0}. The boundedness of G, 1

|h(z, ζ )| ≤ Cε (z)|ζ | 2 −α(z) max{|ζ |−ε , |ζ |ε },

(7.11)

where ε can be chosen arbitrarily small. Liouville’s theorem then implies h(z, ζ ) = 0 hom = 0 and G =G  par t . Laplace’s inversion formula then yields whence G G(t, ξ ) =

1 2π i



b+i ∞

b−i ∞

 par t (z, ξ ) e z t dz G

(7.12)

 ξ ) = 0 for all 0 ≤ t < T /2 and for any b > 0. Therefore, (7.8) implies G(t, ξ ) = G(t, all ξ ∈ S, whence g(t, x) = 0 for 0 ≤ t ≤ T /2 and x ∈ R+ and the result follows.

Linearized Homogeneous Coagulation Equation

775

 ξ) 8. Decay Estimates for the Function G(t, For any L > 0 we define T L := S ∪ {ξ ∈ C; |I m(ξ )| ≤ L , |ξ | ≥ 2L}, where S is defined in (3.21). The main result of this section is the following. Proposition 8.1. For any L > 0 there exists positive constants C and a, depending only  defined in (5.1) satisfies: on L, such that the function G √

 ξ )| ≤ Ce−a |ξ |t , |G(t,         2 √   1/2  ∂  3/2  ∂   ≤ Ct e−a |ξ |t  1 + |ξ | + 1 + |ξ | G(t, ξ ) G(t, ξ )  ∂ξ  ∂ξ 2  

(8.1) (8.2)

for all t ∈ (0, 1), ξ ∈ T L . We will prove this proposition by means of a general lemma that provides estimates of a class of integrals using contour deformation and the Laplace method. This technical result is one of the key ingredients of this paper. In order to formulate it we need some definitions. 1 1 Θ(σ, Y ) := − , (8.3) 4π 4π − λ−1 σ (−σ +Y ) λ−1 1−e 1 − e  2 Ψ (ξ, Y, t) := ln (−Φ(η)) Θ(η − ξ, Y )dη (λ − 1) i I m η=β1   2iY 2iY 2iY 1 2iY − ln(t) − + − ln , (8.4) λ−1 λ−1 λ−1 2 λ−1 Γ (z) . (8.5) A(z) := √ 2π e−z z z−1/2 Stirling’s formula yields lim A(z) = 1 uniformly as |z| → ∞ and arg(z) ∈ (−π + ε, π − ε) for any ε > 0 small. For any ξ ∈ C, and B > 0 we set:     Q    D(ξ, B) := Z ∈ C; I m(Z ) < 0, |I m(Z )| ≤ B  Re(Z ) + |ξ |  , 8  sgn(Re(Z )) = sgn(Re(ξ )) (8.6) (cf. Fig. 6 in Sect. 14), with Q as in (4.3). We then have: √ Lemma 8.2. Suppose that L > 0, B > 2 π , γ1 > 0. Assume that for every ξ ∈ T L the function m is such that m(ξ, ·) is analytic in    λ−1 . D(ξ, B) ∪ Z ∈ C; I m(Z ) ∈ −γ1 , γ1 + 2 Let us consider the function: W (t, ξ ) =

 I m(Y )=−γ1

m(ξ, Y )eΨ (ξ,Y,t) dY.

(8.7)

Then there exists ξ0 > 0 sufficiently large and C > 0, both depending on L , B and γ1 , such that:

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M. Escobedo, J. J. L. Velázquez

– If |m(ξ, Y )| ≤ 1

(8.8)

then |W (t, ξ )| ≤ C e−a|ξ |

1/2 t

(8.9)

for all t ∈ [0, 1] and all ξ such that |Re(ξ )| ≥ ξ0 and ξ ∈ T L . – If |m(ξ, Y )| ≤ (1 + |Y |) and m(ξ, 0) = 0

(8.10)

then, |W (t, ξ )| ≤ C t |ξ |1/2 e−a|ξ |

1/2 t

(8.11)

for all t ∈ [0, 1] and all ξ such that |Re(ξ )| ≥ ξ0 and ξ ∈ T L . √ Proof of Lemma 8.2. We introduce the new variable Z as: Y = |ξ | Z . Then the function Ψ becomes:   2  ln (−Φ(η)) Θ(η − ξ, |ξ |Z )dη Ψ (ξ, Y, t) = Ψ (ξ, Z , t) = (λ − 1) i I m η=β0 + λ−1 −ε 2    2i Z 2i Z 1 2i Z 2i Z ln(t) + − − √ ln − |ξ | λ−1 λ−1 λ − 1 2 |ξ | λ−1  2i Z 1 (8.12) − − √ ln |ξ |1/2 . λ − 1 2 |ξ | (ξ, Z , t) can be extended analytically as a function of Z to the set The function Ψ D(ξ, B) ∪ B√|ξ |/8 (0) (cf. Lemma 14.1). Moreover we prove in Lemma 14.3 the exis(ξ, Z , t) with the asymptotics (14.11). tence of a critical point Z c of the function Ψ (ξ, Z , t) and m(ξ, Y ) we can obtain Using the analyticity properties of the functions Ψ the new representation formula for W using contour deformation: W (t, ξ ) =



 |ξ |

Ct

  eΨ (ξ,Z ,t) m(ξ, |ξ |Z )d Z ,

(8.13)

where Ct is defined as (see Fig. 2):



Ct = Z 1 + R− ∪ γ3,t (M) ∪ γ1,t (M) ∪ γ2,t (M) ∪ Z 2 + R+ , Z 1 = Re(Z c ) − M t + i γ1 t, Z 2 = Re(Z c ) + M t + i γ1 t with γ,t (M),  = 1, 2, 3 and M are as in (14.19)-(14.21). Let R > 0 be as in Lemmas (14.8) and (14.9). We now consider separately the two different cases |ξ |2 t ≥ R and |ξ |2 t ≤ R bounded.

Linearized Homogeneous Coagulation Equation

777

Fig. 2. The curve Ct

The estimate of W (t, ξ ) for |ξ |2 t ≥ R. We write the function W (t, ξ ) as follows: W (t, ξ ) = I1 + I2 ,     eΨ (ξ,Z ,t) m(ξ, |ξ |Z )d Z , I1 = |ξ | Ct ∩{Z ; |Z −Z c |≤ε t}     eΨ (ξ,Z ,t) m(ξ, |ξ |Z )d Z I2 = |ξ | Ct ∩{Z ; |Z −Z c |≥ε t}

(8.14) (8.15) (8.16)

for some positive constant ε to be fixed. We estimate the integral I1 using Lemmas 14.1–14.5 and Taylor’s expansion we obtain: (ξ, Z , t) = Ψ (ξ, Z c , t) − Ψ

√ |ξ | 1 √ (1 + δ(ξ, Z , t)) |Z − Z c |2 , 2 2π t (1 + i Q)

(8.17)

where |δ(ξ, Z , t)| can be made arbitrarily small if |ξ |t 2 ≥ R and ε sufficiently small. Therefore:    I1= |ξ |eΨ (ξ,Z c ,t)

Ct ∩{Z ; |Z −Z c |≤ε t}

e

− 21

√

√

|ξ | (1+δ(ξ,Z ,t))|Z −Z c |2 2π t (1+i Q)

 m(ξ, |ξ |Z )d Z

which gives, in the case (8.8): |I1 (t, ξ )| ≤ C e−a

√

|ξ | t

if |ξ | t 2 ≥ R, 0 < t < 1,

(8.18)

and in the case (8.10): √  |I1 (t, ξ )| ≤ C t |ξ | e−a |ξ | t

if |ξ | t 2 ≥ R, 0 < t < 1.

(8.19)

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M. Escobedo, J. J. L. Velázquez

We must now estimate the I2 given by (8.16). To this end we split the integral as follows: I2 = I2,1 + I2,2 + I2,3 ,     I2,1 = |ξ | eΨ (ξ,Z ,t) m(ξ, |ξ |Z )d Z , Ct ∩{Z ; |Z −Z c |≥ε t}∩γt (M)     I2,2 = |ξ | eΨ (ξ,Z ,t) m(ξ, |ξ |Z )d Z , √ Ct \γt (M)∩{Z ; |Z |≤ε1 |ξ |}     I2,3 = |ξ | eΨ (ξ,Z ,t) m(ξ, |ξ |Z )d Z , √ Ct \γt (M)∩{Z ; |Z |≥ε1 |ξ |}

where γt (M) is the portion of the curve Ct along which I m(Z ) < γ1 t and ε1 is as in Lemma 14.9. Using Lemma 14.8 it follows that, in the case (8.8): √      I2,1  ≤ CeΨ(ξ,Z c ,t) e−δ0 |ξ | t |ξ | |m(ξ, |ξ |Z )|d Z ≤ Ce

Ct ∩{Z ; |Z −Z c |≥ε t}∩γt (M)

√ (ξ,Z c ,t) −δ0 |ξ | t  Ψ



|ξ | t ≤ CeΨ (ξ,Z c ,t) e−

e

√ δ0 |ξ | 2

t

,

(8.20)

and in the case (8.10): √ √     δ |ξ | I2,1  ≤ CeΨ(ξ,Z c ,t) e−δ0 |ξ | t ( |ξ | t)2 ≤ C( |ξ | t) eΨ(ξ,Z c ,t) e− 0 2 t . (8.21)

The estimate of I2,2 follows using Lemma 14.9. In the case (8.8) we obtain: √ √     I2,2  ≤ C |ξ | e−a |ξ ||Z | d Z = Ce−a M |ξ |t , |Z |≥M t

and in case (8.10) we have:     I2,2  ≤ C |ξ |

|Z |≥M t

e−a

√

|ξ ||Z |



(8.22)

√  |ξ | Z d Z ≤ C( |ξ | t)e−a M |ξ |t , (8.23)

≥ R. since The third integral I2,3 is estimated using Proposition 13.2. To this end we use the variable Y and the identity:   iY Γ 2λ−1 2i Y V(ξ ) ,  eΨ (ξ,Z ,t) = t − λ−1 V(ξ + Y ) A 2 i Y |ξ |2 t

λ−1

where A is defined by formula (8.5). Then           m(ξ, Y )   V(ξ )  − 2iY  2iY       λ−1   dY. (8.24) |I2,3 | ≤ Γ   t λ−1  I m(Y )=γ1 ,|Y |≥ε1 |ξ |  A 2 i Y  V(Y + ξ ) λ−1 Using that γ1 > 0, 0 ≤ t ≤ 1 and Stirling’s formula, it follows that, in both cases (8.8) and (8.10):     V(ξ )  − π |Y |  e λ−1 dY.  |I2,3 | ≤ (1 + |Y |)  V(Y + ξ )  I m(Y )=γ1 ,|Y |≥ε1 |ξ |

Linearized Homogeneous Coagulation Equation

779

Proposition 13.2 gives the following bounds:  − π |Y | |I2,3 | ≤ C (1 + |Y |) eε π |ξ | e 2(λ−1) dY ≤ Ce

I m(Y )=γ1 ,|Y |≥ε1 |ξ | π − 2(λ−1) ε1 |ξ | ε π |ξ |

.

e

(8.25)

This last term can be estimated by the right hand side of (8.9) choosing ε sufficiently small. Combining this with (8.20) and (8.22) we obtain (8.9) for |ξ |t 2 ≥ R. In the case (8.10), we have for some δ1 > 0, δ1

|I2,3 | ≤ Ce−δ1 |ξ | ≤ Ce− 2 |ξ | e

−

δ1 2 t2

δ1 √

≤ Ce− 2

δ

1 |ξ | t − 2 t 2

e

which is estimated by the right hand side of (8.11). Combining this with (8.21) and (8.23) we obtain (8.11) for |ξ |t 2 ≥ R. This ends the proof of the estimate of I2 and then of Proposition 8.1 in the domain where |ξ | t 2 ≥ R, 0 < t < 1. The estimate of W (t, ξ ) for |ξ |2 t ≤ R.. We suppose first that condition (8.8) holds. We split the integral in (8.33) in two pieces: W (t, ξ ) = J1 + J2 ,  J1 (t, ξ ) =

I m Y =−γ1 t, |Y |≤ε0 |ξ |

m(ξ, Y ) V(ξ ) − 2iY   t λ−1 Γ i Y V(Y + ξ ) A 2λ−1



2iY dY, λ−1

(8.26)

√  2 m(ξ, Y ) V(ξ ) − 2iY   t λ−1 J2 (t, ξ ) = √ πi (λ − 1) I m Y =−γ1 t, |Y |≥ε0 |ξ | A 2 i Y V(Y + ξ ) λ−1  2iY dY. ×Γ λ−1 The integral J2 is estimated with the same argument used to bound the integral I2,3 in (8.24), |J2 | ≤ Ce−δ1 |ξ | . We rewrite J1 as: J1 (t, ξ ) = where



 |ξ | t

γ

I m(ζ )=− √|ξ1 | ,|ζ |≤ε0

√

|ξ | t

(8.27)

  eΨ (ξ,ζ t,t) m(ξ, |ξ | ζ t)dζ,

   √   2iζ 2iζ i Q π4  1 − ln + ln 2 π e Ψ (ξ, ζ t, t) = − |ξ | t λ−1 λ−1   1 2i ζ 1  + h(ξ, ζ t, t). − ln t |ξ |1/2 − ln 2 2 λ−1

Notice that Lemma 14.2 implies: |h(ξ, ζ t, t)| ≤ C

 |ζ |2 t 2 +

1 |ξ |



(8.28)

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M. Escobedo, J. J. L. Velázquez

√ (ξ, ζ t, t)) for t |ζ | ≤ ε0 |ξ |, ζ t ∈ D(ξ, B), |ξ | ≥ ξ0 . A detailed computation of Re(Ψ yields     1  π (ξ, ζ t, t)) ≤ |ξ | t C − Re(Ψ |ζ | − ln t |ξ |1/2 + Cε0 |ξ | t |ζ | + C 2(λ − 1) 2    1  π |ζ | − ln t |ξ |1/2 + C ≤ |ξ | t C − 4(λ − 1) 2 √

√ for ζ ∈ Bε0 √|ξ |/t (0) ∩ ζ ∈ C; I m(ξ ) = −γ1 / |ξ | , |ξ | > ξ0 , |ξ | t ≤ R and for √ some positive constant C independent of ξ and ζ . Using again that t |ξ | ≤ R it follows that, in the case (8.8):  1/2  π √ − λ−1 |ξ | t|ζ | |dζ | |J1 (t, ξ )| ≤ C |ξ | t e √ , √ γ1 |ξ | |ζ | I m(ζ )=− √ ,|ζ |≤ε0 t |ξ |

and computing the integral: |J1 (t, ξ )| ≤ C. Combining this estimate with (8.27), estimate (8.9) of the lemma follows. In the case (8.10) we deform the contour of integration in formula (8.7) using the analyticity properties of the function m(ξ, Y ) as well as the fact that m(ξ, 0) = 0 to obtain:  π (λ − 1) λ−1 V(ξ ) W (t, ξ ) = − m ξ, i  t A(−1) 2 V λ−1 2 i +ξ  + m(ξ, Y )eΨ (ξ,Y,t) dY I m(Y )=γ1 + λ−1 2

=

  π(λ − 1) λ−1 λ−1 m ξ, i Φ i +ξ t A(−1) 2 2  m(ξ, Y )eΨ (ξ,Y,t) dY = K 1 + K 2 , + I m(Y )=γ1 + λ−1 2

(8.29)

using (5.11). By (8.10), m(ξ, λ−1 2 i) is uniformly bounded. Using then Proposition 4.1 we obtain that |K 1 | ≤ C(1 + |ξ |1/2 )t.

(8.30)

We are then left with the integral K 2 which is formally very similar to (8.7) although the integral contour is different. We split this integral in two terms J1 and J2 like in (8.26). The term J2 is bounded as (8.27) by a similar argument as before with one small difference which is that the term m(ξ, Y ) is now bounded as (1 + |Y |). The term J1 can be written as:   m(ξ, Y ) V(ξ ) − 2iY 2iY   J1 (t, ξ ) = dY. t λ−1 Γ 2 i Y V(Y + ξ ) λ−1 I m Y =γ1 + λ−1 2 , |Y |≤ε0 |ξ | A λ−1

Linearized Homogeneous Coagulation Equation

781

    π +δ |Y |   Using Lemma 14.1, we may estimate  VV(Y(ξ+ξ) )  by Ce 2(λ−1) 0 in the domain of inte 2iY  2γ1  − λ−1  we may estimate gration. On the other hand, since I m Y = γ1 + λ−1  by t 1+ λ−1 . t 2 Therefore, using the decay of the Gamma function:    π 2γ1 π 1+ λ−1 2(λ−1) +δ0 |Y | − λ−1 |Y | |J1 (t, ξ )| ≤ Ct e e |d Y | (8.31) I m(Y )=γ1 + λ−1 2

≤ Ct

2γ1 1+ λ−1

.

(8.32)

Combining (8.30) and (8.31) we deduce the estimate (8.11) when |ξ | t 2 ≤ R. This concludes the proof of Lemma 8.2.

Proof of Proposition 8.1. Using the change of variables: y − ξ = Y in (5.1) we obtain √   2 V(ξ ) − 2iY 2iY  λ−1 G(t, ξ ) = − √ dY, (8.33) Γ t λ−1 πi (λ − 1) I m Y =−γ1 V(Y + ξ )  ξ ) as where γ1 is a positive constant sufficiently small. We rewrite the function G(t, follows. √   2 2iY  ξ) = √ dY, (8.34) G(t, eΨ (ξ,Y,t) A λ−1 πi (λ − 1) I m(Y )=−γ1 where the functions Ψ and θ are given by (8.4) and (8.3). In order to obtain estimates (8.1) and (8.2) for bounded values of ξ we use contour deformation. In particular, crossing the pole at Y = 0 in integral (8.34) and using the residue theorem we obtain:  ξ ) = √1 + G 1 (t, ξ ), G(t, 2π √   2 V(ξ ) − 2iY 2iY  λ−1 t dY. Γ G 1 (t, ξ ) = − √ λ−1 πi (λ − 1) I m Y =γ1 V(Y + ξ )

(8.35) (8.36)

Using Proposition 13.2 it follows that 2γ1

1 (t, ξ )| ≤ Ct λ−1 |G uniformly for ξ in bounded sets. This yields estimate (8.1) for ξ in bounded sets. If we differentiate in (8.36) with respect to ξ we obtain: √   2iY ∂  2 2iY − λ−1 dY, (8.37) m (ξ, Y )t Γ G(t, ξ ) = − √  λ−1 ∂ξ  πi (λ − 1) I m Y =γ1  V(ξ ) ∂ m  (ξ, Y ) =  . (8.38) ∂ξ V(Y + ξ ) Using the analyticity properties of the functions m  (ξ, Y ) we deform the integration contour in the integral (8.37). The first singularity that is met is the pole of function  2iY Γ λ−1 located at Y = (λ − 1)i/2. (This point is below the first zero of the function

782

M. Escobedo, J. J. L. Velázquez

V(ξ + Y ) which is located at Y = (2 + (λ/2) − ξ ) i and 2 + (λ/2) − ξ > (λ − 1)/2.) We then deduce √ λ−1 ∂  2 2π i) (8.39) t m  (ξ, G(t, ξ ) = √ 2 ∂ξ  πi (λ − 1) √   2iY 2iY 2 − λ−1 −√ m  (ξ, Y )t Γ dY, (8.40) λ−1 πi (λ − 1) I m Y =γ1 + λ−1 2 using the analyticity properties of the functions m  (ξ, Y ) we deduce estimate (8.2) for bounded values of ξ . In order to complete the proof of Proposition 8.1 we apply the method of the stationary phase. To this end we differentiate the expression in (8.34) with respect to ξ , and obtain: √      ∂  ∂ Ψ 2 2iY eΨ (ξ,Y,t) dY (ξ, Y, t) A G(t, ξ ) = √ λ−1 ∂ξ  πi (λ − 1) I m Y =−γ1 ∂ξ  (8.41) for  = 1, 2 and use Lemma 8.2 with three choices for m(t, ξ ): √

 2 2i Y , A λ−1 πi (λ − 1) √  2 ∂ ψ 2i Y m(ξ, Y ) = √ (ξ, Y, t) A (|ξ | + 1) ∂ξ  λ−1 πi (λ − 1) m(ξ, Y ) = √

(8.42) (8.43)

∂ ψ for  = 1, 2. (Notice that ∂ψ ∂ξ (ξ, Y, t) and ∂ξ 2 (ξ, Y, t) are independent of t.) The function m in (8.42) satisfies condition (8.8). On the other hand, using Lemma 14.10, it follows that for the choices (8.43),  = 1, 2: |m(ξ, Y )| ≤ C|Y |. Moreover, by the definition of the function Ψ (cf. (8.4)) 2

∂ψ ∂ (ξ, Y ) = ∂ξ ∂ξ

  V(ξ ) ln , V(ξ + Y )

and therefore m(ξ, 0) = 0 with the choice (8.43) and  = 1. This follows by a similar argument for the choice (8.43),  = 2. Applying then Lemma 8.2 the estimates (8.1) and (8.2) follow for |ξ | sufficiently large and this concludes the proof of Proposition 8.1.

We consider now the case t > 1.  defined in (5.1) satisfies that for any ε0 > 0 arbitrarily Lemma 8.3. The function G small, there exist two positive constants κ1 and a such that 1

 ξ )| ≤ κ1 t − λ−1 +ε0 e−a |G(t,

√

|ξ |

for all ξ such that I m(ξ ) ∈ (3/2, (3 + λ)/2) and all t > 1.

(8.44)

Linearized Homogeneous Coagulation Equation

783

Fig. 3. The curve D

Proof. We deform the integration contour in the expression (8.33) to the line: I m(Y ) = (λ−1)/2 −ε0 with ε0 arbitrarily small in order to avoid the √ zero of the function V(ξ +Y ) at ξ + Y = 1. We then use the change of variables Y = |ξ | Z , and deform the contour of integration in the Z variable to the curve D in Fig. 3 to obtain: √ √  √  2i √ 2 |ξ | 2i |ξ |Z  − |ξ | Z (ξ,Z ,1) Ψ  Z) = √ d Z. (8.45) t λ−1 e A G(t, λ−1 πi (λ − 1) D Then,

  √     Ψ(ξ,Z ,1) 1 2i |ξ |Z  − λ−1 +ε0   d Z. |G(t, Z )| ≤ C |ξ | t A e λ−1  D

We argue now in the same way as in the previous case with t = 1 splitting the integral in the same pieces to obtain (8.44).

9. Estimates on the Function G(t, X)  ξ ) to obtain the function: We may now take the inverse Fourier transform of G(t,  1  ξ )dξ G(t, X ) = √ ei X ξ G(t, (9.1) 2π I m(ξ )=β2 with

 β2 ∈

3 3+λ , . 2 2

(9.2)

Proposition 9.1. For all t > 0 the function G(t, ·) defined by (9.1) belongs to C ∞ (R) and for any fixed R > 0 and ε0 > 0, it satisfies:    ∂ G  C,R   (9.3)  ∂ X  (t, X ) ≤ t 2(1+) , for 0 < t ≤ 1, |X | ≤ R,  = 0, 1, 2, · · · ,    ∂ G    ≤ C,ε0 ,R , for t ≥ 1, |X | ≤ R,  = 0, 1, 2, · · · , (t, X ) (9.4) 1 ∂ X  t λ−1 −ε0 for suitable positive constants C,R and C,ε0 ,R .

784

M. Escobedo, J. J. L. Velázquez

Proof. This proposition is just a consequence of (9.1) as well as from (8.1) for t ∈ (0, 1) and (8.44) for t > 1.

The estimates (9.3) and (9.4) provide the regularity result for g in Theorem 3.5, but they do not give any detailed description of the function G in the different regions of X and t. We derive such results in the remainder of this section. 9.1. Behaviour as t → 0, 0 ≤ |X | ≤ 1: Heuristics. We begin by showing that the function G behaves like a mollifier of the Dirac measure when t → 0 and for X small. The asymptotic profile of the solution G(t, X ) can be heuristically guessed as follows. If we assume that the function g, solution of (3.1) with g(0, x) ∼ δ(x − 1) is small, then Eq. (3.1) may be approximated, for t and x − 1 small, as  1/2 ∂g g(x − y) − g(x) = dy, g(x, 0, 1) = δ(x − 1). (9.5) ∂t y 3/2 0 The equation in (9.5) describes the probability distribution for the size of a particle, initially equal to one, and increasing its size by an amount y at the rate 1/y 3/2 . If we write the function g as  1 g(t, x) = √  g (t, x) ei k x dk, 2π R applying the Fourier transform we obtain: e−i k ∂ g (t, k) = m(k)  g (t, k),  g (k, 0) = √ , ∂t 2π  1/2 −i k y e −1 dy. m(k) = 3/2 y 0 The multiplier m(k) can be computed explicitly but its exact formula is not needed to compute the asymptotics of the function g(t, x). The only relevant information that we really need is √ m(k) ≈ −2 π k i, for |k| → +∞. Then, √ e−i k  g (t, k) ≈ √ e−2 π k i t , for |k| → +∞, 2π

and inverting the Fourier transform we obtain  x −1 2 as t → 0, g(t, x) ≈ 2 Ψ t t2 with

Ψ (χ ) =

π χ 3/2

0

e

− πχ

for χ ≥ 0, for χ < 0.

(9.6)

Linearized Homogeneous Coagulation Equation

785

Finally, since x = e X , x − 1 = e X − 1 ∼ X when X ∼ 0 we obtain  2 X as t → 0. G(t, X ) ≈ 2 Ψ t t2

(9.7)

The fact that the support of G(t, X ) is contained in R+ is a consequence of the interpretation of (9.5) in terms of coagulation of particles given above. 9.1.1. The region X = O(t 2 ) We now prove that the fundamental solution G(t, X ) behaves in the self similar form (9.7) in the region where X = O(t 2 ) using the explicit representation formula given by (5.1), (6.15) and (9.1). We are interested in the limit limt→0+ t 2 G(t, t 2 χ ) = Ψ (χ ) in compact sets of χ . To this end we rewrite (9.1) as follows:    1  t, η dη. (9.8) G(t, χ ) = √ eiη χ G 2 t 2π t 2 I m(η)=β2 t 2 Formula (9.8) that in order to compute Ψ (χ ) we need to obtain   suggests η  limt→0+ G t, t 2 for η such that I m(η) = β2 t 2 . To this end we use the expression (8.33) that implies: √   2 V(η/t 2 ) − 2iY 2iY  η/t 2 ) = − √ λ−1 Γ t dY. (9.9) G(t, λ−1 πi (λ − 1) I m Y =−γ1 V(Y + η/t 2 ) Proposition 9.2.   lim+ t 2 G(t, t 2 χ ) = ψ(χ )

t→0

uniformly for χ in compact sets of R where ψ(χ ) is as in (9.6). The proof of Proposition 9.2 requires several lemmas. Lemma 9.3. For all ε0 > 0 and M > 0 there exists a function h ε0 ,M (t) such that lim h ε0 ,M (t) = 0

(9.10)

  √ √   V(η/t 2 ) − 2 i Y iY − 2λ−1 ln(2 π i η)   λ−1 −e  ≤ h ε0 ,M (t)  V(Y + η/t 2 ) t

(9.11)

t→0+

and

1 2 for all Y such that |I m(Y )| ≤ 1/4, |Re(Y )| ≤ t | ln t| , for I m(η/t ) in compact subsets of (3/2, (3 + λ)/2) and ε0 ≤ |η| ≤ M. Moreover there is δ0 > 0 small (depending on ε0 and M) such that:    V(η/t 2 ) − 2 i Y  |Y |   ≤ C e 3π4 λ−1 λ−1 (9.12)  V(Y + η/t 2 ) t 

for all Y such that |I m(Y )| ≤ 1/4, |Re(Y )| ≤ (3/2, (3 + λ)/2) and ε0 ≤ |η| ≤ M.

δ0 , t2

for I m(η/t 2 ) in compact subsets of

786

M. Escobedo, J. J. L. Velázquez

Remark 9.4. In Lemmas 9.3 until 9.6 and in Proposition 9.2 we choose the branch of the function square root as follows: √

θ

z = |z|1/2 ei 2 with θ ∈ (−π, π ].

Proof. Lemma 9.3 is a consequence of Lemma 14.1. By formula (6.15) we have:    V(η/t 2 ) 2 2 = exp ln (−Φ(ζ )) Θ(ζ − ζ /t , Y )dζ V(η/t 2 + Y ) (λ − 1) i I m ζ =β1     = exp F η/t 2 , t Y/ |η| , and using (14.3):    √   2     F η/t 2 , t Y/ |η| + 2 i Y ln 2 π i η  ≤ C t 2 |Y | .   λ−1 |η| Therefore, in the region where |Re(Y )| ≤ 1/(t| ln t|) we obtain (9.11). On the other hand, if |Re(Y )| ≤ δ0 /t 2 we deduce (9.12) using that:    √     π  Re 2 i Y ln 2 π i η  ≤   2(λ − 1) |Y | + C λ−1 as |η| → +∞ and for I m(Y ) = −γ1 with γ1 as in (9.9).



Lemma 9.5. For all positive constants M, ε0 such that M > ε0 : √    η √ √ iY 2 2iY − 2λ−1 ln(2 π i η)  dY, e lim G t, 2 = − √ Γ t→0+ t λ−1 πi (λ − 1) I m(Y )=−γ1 uniformly for I m(η/t 2 ) in compact subsets of (3/2, (3 + λ)/2) and ε0 ≤ |η| ≤ M. Proof. We split the integral in (9.9) as follows:    t, η G t2 √   − 2 V(η/t 2 ) − 2iY 2iY λ−1 t dY =√ Γ λ−1 πi (λ − 1) I m Y =−γ1 , |Re(Y )|≤ 2 1 V(Y + η/t 2 ) t | ln t| √   − 2 V(η/t 2 ) − 2iY 2iY λ−1 Γ =√ t dY λ−1 πi (λ − 1) I m Y =−γ1 , 2 1 ≤|Re(Y )|≤ δ20 V(Y + η/t 2 ) t | ln t| t √   2 V(η/t 2 ) − 2iY 2iY λ−1 Γ +√ t dY λ−1 πi (λ − 1) I m Y =−γ1 , |Re(Y )|≥ δ20 V(Y + η/t 2 ) t

= J1 + J2 + J3 . √ 2 J1 = − √ πi (λ − 1)  ×

I m Y =−γ1 , |Re(Y )|≤

 1 t 2 | ln t|

√ √ 2i Y V(η/t 2 ) − 2 i Y λ−1 − e − λ−1 ln(2 π i η) t V(Y + η/t 2 )



Linearized Homogeneous Coagulation Equation

787

2iY dY λ−1 √   √ √ iY 2iY 2 − 2λ−1 ln(2 π i η) +√ e Γ dY λ−1 πi (λ − 1) I m Y =−γ1 , √   √ √ 2i Y 2 2iY dY. (9.13) e− λ−1 ln(2 π i η) Γ −√ λ−1 πi (λ − 1) I m Y =−γ1 , |Re(Y )|≥ 2 1 

×Γ

t | ln t|

Using that     2i Y √   π π 2 i Y   − λ−1 ln( iη)  |Y | ≤ C e− (λ−1) |Y | , and Γ e  ≤ C e 2(λ−1)  λ−1 the last term in (9.13) is estimated as:  √    2 √  πi (λ − 1) I m Y =−γ1 , |Re(Y )|≥  ≤ Ce

−

1 t 2 | ln t|

e

iY − 2λ−1

√ √ ln(2 π i η)

 Γ

   2iY dY  λ−1 

a t 2 | ln t|

for some positive constant a. In the first term of (9.13), using (9.11) in Lemma 9.3 we  π are led to estimate: h ε0 ,M (t) I m Y =−γ1 e− (λ−1) |Y | |dY | which tends to zero as t → 0+ by (9.10). We now consider the integral J3 . This term is estimated using the estimates of the function V proved in Proposition 13.2 and Stirling’s formula: 2 γ1

|J3 | ≤ Cε t − λ−1 e

−

1 t2



π δ0 λ−1 −2εM



2 γ1

≤ Cδ t − λ−1 e

−

a t2

for some positive constant a choosing δ sufficiently small. The integral J2 is estimated using (9.12) in Lemma 9.3:  3π π − a e 4(λ−1) |Y | e− λ−1 |Y | |dY | ≤ Ce t 2 | ln t| |J2 | ≤ C δ I m Y =−γ1 ,

1 ≤|Re(Y )|≤ 20 t 2 | ln t| t

for some positive constant a.



Lemma 9.6. √   √ √ √ √ 2i Y 2 2iY dY = − 2 π e−2 π η i . e− λ−1 ln(2 π i η) Γ √ λ−1 πi (λ − 1) I m(Y )=−γ1 Proof. The change of variable 2iY/(λ − 1) = s yields √   √ √ iY 2 2iY − 2λ−1 ln(2 π i η) dY e Γ √ λ−1 πi (λ − 1) I m(Y )=−γ1  √ 1 = −√ e−s ln(2 π η i) Γ (s) ds. 2 π Re(s)=2γ1 /(λ−1) This integral can then be computed adding the residues of the integrand at the poles s = −n of the Gamma function.

788

M. Escobedo, J. J. L. Velázquez

Proof of Proposition 9.2. We split the integral in (9.8) t 2 G(t, χ ) = I1 + I2 + I3 , where 1 Ik = √ 2π

 {I m(η)=β2 t 2 }∩Dk

   t, η dη eiη χ G t2

with D1 = Bε0 (0), D2 = B M (0)\Bε0 (0), D3 = C\B M (0), where ε0 and M are for awhile arbitrary positive constants with ε0 < M. Using Proposition 8.1:   (9.14) |I1 | + |I3 | ≤ C ε0 + e−a M . The integral I2 is estimated using the previous lemmas:   √  √ 1  t, η − 2 π e−2 π η i )dη eiη χ (G I2 = √ 2 t 2π {I m(η)=β2 t 2 }∩D2  √ + eiη χ e−2 π η i dη = I2,1 + I2,2 . {I m(η)=β2 t 2 }∩D2

The first term is bounded as  |I2,1 | ≤ C

  η √  √   G t, 2 − 2 π e−2 π η i  |dη| 2 t {I m(η)=β2 t }∩D2

and tends to zero by Lemma 9.5 and Lemma 9.6. And,  √ lim+ I2,1 = eiη χ e−2 π η i dη. t→0

{I m(η)=0}∩D2

Arguing as in the derivation of (9.14) we obtain that      √ √   iη χ −2 π η i i η χ −2 π η i  ≤ C ε0 + e−a M .  e e dη − e e dη   {I m(η)=0}∩D2

I m(η)=0

We have then shown that, for t sufficiently small (depending on ε0 and M):    √ 2  i η χ −2 π η i t G(t, x) − e e dη ≤ 2C(ε0 + e−a M ),  I m(η)=0

which means exactly: lim+ t 2 G(t, χ ) =

t→0



eiη χ e−2

√

π ηi

dη.

(9.15)

I m(η)=0

The integral in the right hand side of (9.15) may be calculated explicitly using contour deformation. For χ < 0 the contour is sent to the region where I m(η) → −∞ and the integral gives zero. When χ > 0 the deformation is made to the upper half plane in such a way that it avoids the cut along the half line η ∈ i R+ and the integral is reduced to  ∞  √  π −π sin 2 π λ e−χ λ dλ = 3/2 e χ . 2 χ 0



Linearized Homogeneous Coagulation Equation

789

9.1.2. Estimates of G(t, X ) for t 2 ≤ X ≤ 1. We derive a self similar estimate for the function G in this region as t → 0. Proposition 9.7. For any ε ∈ (0, 1/2), there exists a positive constant Cε such that |G(t, X )| ≤ C

t 1−2 ε 3

|X | 2 −ε

for t 2 ≤ |X | ≤ 1.

Proof. We integrate by parts twice in formula (9.1) and obtain:    ∂2 1 1  ξ )dξ. G(t, X ) = √ ei X ξ − 1 G(t, (9.16) 2 ∂ξ 2 2π X I m(ξ )=β2   Using that ei X ξ − 1 ≤ Cε |X |1/2+ε |ξ |1/2+ε for ε ∈ [0, 1/2], as well as (8.2) we deduce:  √ t Cε |X |1/2+ε |ξ |1/2+ε e−t |ξ | |dξ | 2 3/2 X I m(ξ )=β2 (1 + |ξ | )  √ Cε t |dξ | ≤ e−t |ξ | 1−ε , 3 |ξ | |X | 2 −ε I m(ξ )=β2

|G(t, X | ≤

and the result follows.

9.2. Behaviour of G(t, X ) for t ≥ 1. The behaviour of the function G as t → +∞ has a self similar structure. This is seen by writing the function G(t, X ), given in (5.1) and (9.1), in terms of the variable 2 ln(t), (9.17) λ−1    i V(ξ ) − 2 i y 2i t λ−1 Γ − (ξ − y) G(t, X ) = dξ eiξ θ dy π(λ − 1) I m(ξ )=β2 λ−1 I m(y)=β0 V(y) (9.18) θ =X+

with 3/2 < β0 < β2 < (3 + λ)/2 (cf. Lemma 5.1). Moving the integration contour of y downward in the expression of G(t, X ), the first singularity to be met in the integrand of (9.18) is y = i which is a zero of V(y) (cf. Proposition 6.1). This gives 2

G(t, X ) = t λ−1 Ψ1 (θ ) + G 1 (t, X ), where   2i 2 iξ θ V(ξ ) Γ − (ξ − i) , (9.19) dξ e Ψ1 (θ ) = λ − 1 I m(ξ )=β2 V (i) λ−1    i V(ξ ) − 2 i y 2i iξ θ G 1 (t, X ) = dξ e dy t λ−1 Γ − (ξ − y) , π(λ − 1) I m(ξ )=β2 λ−1 I m(y)=β3 V(y) (9.20)

790

M. Escobedo, J. J. L. Velázquez

and now β3 ∈ ((3 − λ)/2, 1). Notice that V (i) = 0. Indeed, differentiating (5.11) we obtain:  λ+1 i , V (i) = 2πiV 2 where we use that Φ((λ + 1)/2) = 0 and Φ ((λ + 1)/2) = −2π i (cf. (3.20)). By Proposition 6.1, V( λ+1 2 i)  = 0. Then:   1 2i Ψ1 (θ ) = (i − ξ ) . (9.21) dξ eiξ θ V(ξ )Γ  λ+1  λ−1 π(λ − 1) i V 2 i I m(ξ )=β2 We study now the function Ψ1 (θ ) and give its behaviour as θ → ±∞. Proposition 9.8. There exists  > 0 such that the following estimates hold:   4−λ  − 2 +ε θ − 32 θ +O e Ψ1 (θ ) = a1 e as θ → −∞,   3+λ as θ → +∞, Ψ1 (θ ) = a2 e− 2 θ + O e−(λ+1−ε)θ

(9.22) (9.23)

where, (cf. (3.11)), a1 =

V((1 + λ2 )i)

2i (λ − 1) V( λ+1 2 i)

and a2 = −

Γ



λ+1 λ−1

2π i

 V(2 i) V( λ+1 2 i)

.

Proof of Proposition 9.8. We use again contour deformation. In order to obtain the behaviour as θ → −∞ we deform the contour integration in Ψ downward. The first singularity of the integrand that we meet is ξ = 3i/2 which is a pole of V(ξ ). Using (5.11) and (3.20) we obtain:   3i λ Res V, ξ = = −i V 1+ i , (9.24) 2 2  V(2i) (3 + λ) i =− . (9.25) Res V, ξ = 2 4π i Therefore V ((1 + λ/2)i) − 3 θ 2i e 2 (λ − 1) V ((λ + 1)/2)i)   1 2i iξ θ (ξ − i) , + dξ e V (ξ )Γ −   λ−1 π(λ − 1) i V λ+1 I m(ξ )=β4 2 i   λ+1 Γ λ−1 V (2 i) − 3+λ θ e 2 Ψ1 (θ ) = − 2π i V ( λ+1 2 i)   1 2i iξ θ (ξ − i) , + dξ e V (ξ )Γ − λ−1 π(λ − 1) i V ( λ+1 2 i) I m(ξ )=β5

Ψ1 (θ ) =

(9.26)

(9.27)

where β4 ∈ ((4 − λ)/2, 3/2), β5 ∈ ((3 + λ)/2, 1 + λ). We have derived (9.26), (9.27) deforming the contour of integration upward and downward respectively.

Linearized Homogeneous Coagulation Equation

791

  2i Proposition 13.2 ensures that the function V(ξ )Γ − λ−1 (ξ − i) is integrable and then, for Re θ ≤ 0:      2i iξ θ  (ξ − i)  ≤ Ce−β4 Re(θ) , dξ e V(ξ )Γ − (9.28)  λ − 1 I m(ξ )=β4 while for θ ≥ 0:    

I m(ξ )=β5

dξ eiξ θ V(ξ )Γ

   2i − (ξ − i)  ≤ Ce−β5 Re(θ) . λ−1

Proposition 9.8 follows from (9.26), (9.27), (9.28) and (9.29).

(9.29)



9.2.1. Estimate of G 1 (t, θ ) in formula (9.20). We have the following lemma. Lemma 9.9. There exist δ0 > 0 and C > 0 such that, for all t > 1:

|G 1 (t, θ )| e

2

3θ 2

≤ C t λ−1 −δ0

for all θ ≤ 0,

(9.30)

3+λ 2 θ

2 λ−1 −δ0

for all θ ≥ 0,

(9.31)

|G 1 (t, θ )| e

≤Ct

where G 1 is given by (9.20). Proof of Lemma 9.9. The function G 1 may be written as:  i G 1 (t, θ ) = dξ eiξ θ H1 (t, ξ ), π(λ − 1) I m(ξ )=β2   V(ξ ) − 2 i y 2i H1 (t, ξ ) = t λ−1 Γ − (ξ − y) . dy V(y) λ−1 I m(y)=β3

(9.32) (9.33)

 The function H1 is estimated in the same way as the function √ G(t, ξ ) in Lemma 8.3. To this end we first perform the change of variables: y = ξ + |ξ | Z and obtain:  √   |ξ |Z +ξ ) V(ξ ) 2i  − 2 i ( λ−1 H1 (t, ξ ) = |ξ | d Z Γ |ξ | Z . t √ β3 −β2 λ−1 V( |ξ |Z + ξ ) I m(Z )= √ |ξ |

(9.34) Then we deform the contour of integration in (9.34) to the new contour D1 (cf. Fig. 4). We then need to bound:    √     |ξ |Z +ξ ) V(ξ ) 2i  − 2 i ( λ−1  |ξ | Γ |ξ | Z  |dy|  V(√|ξ |Z + ξ ) t λ−1 D1 that may be estimated following the same arguments as in the proof of Lemma 8.3. The only difference with the argument used there is how to bound the contribution of the time dependent term. However arguing √ as in the proof of Lemma 8.3 and taking into account that along D1 we have I m( |ξ |Z + ξ ) ≤ β3 we finally obtain: 2

|H1 (t, ξ )| ≤ Ct λ−1 −δ e−a

√

|ξ |

.

(9.35)

792

M. Escobedo, J. J. L. Velázquez

Fig. 4. The curve D1

Using (9.1) it then follows 2

|G 1 (t, θ )| ≤ C R t λ−1 −δ for all |θ | ≤ R. To conclude the proof of the Lemma it only remains to obtain estimates as θ → ±∞. To this end we use contour deformation.To obtain the estimate (9.31) as θ → +∞ we 1 (t, ξ ) is located at ξ = (3 + λ) i/2 deform the contour upward. The first singularity of G (see Proposition 6.1). Therefore: 3+λ

G 1 (t, θ ) = b2 (t) e− 2 θ + Q 1 (t, θ ),  Q 1 (t, θ ) = dξ eiξ θ H2 (t, ξ ), I m(ξ )=β6

H2 (t, ξ ) =

i π(λ − 1)

 dy I m(y)=β3

(9.36) (9.37)

V(ξ ) − 2 i y t λ−1 Γ V(y)

 −

2i (ξ − y) , (9.38) λ−1

where β6 ∈ ((3 + λ)/2, (1 + λ)), and V(2 i) b2 (t) = 2π i(λ − 1)



2i y

t − λ−1 Γ dy V(y) I m(y)=β3

  2i 3+λ − i−y . λ−1 2

Since β3 ∈ ((3 − λ)/2, 1), we deduce: 2

|b2 (t)| ≤ Ct λ−1 −δ0 , for t > 1.

(9.39)

 ξ ) in Lemma 8.3. We The function H2 is estimated√ in the same way as the function G(t, change variables as y = ξ + |ξ | Z to obtain:  √   |ξ |Z +ξ ) V(ξ ) 2i  − 2 i ( λ−1 H2 (t, ξ ) = |ξ | t dy Γ |ξ | Z . √ β3 −β6 λ−1 V( |ξ |Z + ξ ) I m(Z )= √ |ξ |

(9.40)

Linearized Homogeneous Coagulation Equation

793

Fig. 5. The curve D2

Then we deform the integration contour in (9.40) to D2 (cf. Fig. 5). Since along this new √ 3 −β6 contour I m(Z ) = β√ , we have I m( |ξ |Z + ξ ) ≤ β3 and then |ξ | |t −2 i

√

|ξ |Z +ξ λ−1

2 β1

2

| ≤ Ct λ−1 = t λ−1 −δ ,

whence 2

|H2 (t, ξ )| ≤ Ct λ−1 −δ e−a

√

|ξ |

.

(9.41)

Using now (9.37) we deduce that 2

|Q 1 (t, θ )| ≤ Ct λ−1 −δ e−β6 θ .

(9.42)

Combining this with (9.39), estimate (9.31) follows. The estimate (9.30) for θ → −∞ is obtained in a very similar way. We deform downward the contour of the integral (9.33) and we continue the proof as for (9.31). This concludes the proof of Lemma 9.9.

9.3. Behaviour as 0 ≤ t ≤ 1, |X | → +∞. For small values of time the solution is described in the (t, X ) variables as follows. Lemma 9.10. There exists positive constants δ and δ1 such that, for 0 ≤ t ≤ 1 , the following estimates hold: ⎧  3  ⎪ ⎪ e− 32 X t + b3 (t)e− 23 X + O e− 2 −δ1 X t as X → −∞ ⎨   3+λ  G(t, X ) = (9.43) 3+λ 3+λ − +δ1 X ⎪ ⎪ t as X → +∞, ⎩ a3 e− 2 X t + b4 (t)e− 2 X + O e 2 where a3 =

V (2i)   1+ λ2 i ≤ Ct 1+δ .

4π V

|b3 (t)| + |b4 (t)|

(cf. (3.13)) and b3 and b4 are continuous functions such that

794

M. Escobedo, J. J. L. Velázquez

Proof of Lemma 9.10. Using (8.33) and (9.1) we obtain: i G(t, X ) = π(λ − 1)



 e

iξ X

I m(ξ )=β2

V(ξ ) − 2 i Y dξ dy t λ−1 Γ V(ξ + Y) I m(Y )=−γ1



2iY , λ−1

   (t, ·) decays where β0 ∈ 23 , 3+λ and γ1 > 0 small. Proposition 8.1 implies that G 2 exponentially in the ξ variable in the region ξ ∈ T L . This provides the convergence of all the integrals used in this proof if L is taken sufficiently large.  (t, ξ ) that We can now deform the contour of integration on ξ crossing the poles of G are due to the poles of V(ξ ). The closest poles are at ξ = 23 i, ξ = 3+λ i respectively. We 2 deform the contour upwards if X > 0 and downwards if X < 0. We then obtain using (9.24), (9.25): G(t, X )



2i    i λ 2iY t − λ−1 Y − 32 X = −2π V 1+ i e Γ dy 3 π(λ − 1) 2 λ−1 V( 2 i + Y ) I m(Y )=−γ1    i V(ξ ) − 2 i Y 2iY + t λ−1 Γ eiξ X dξ dy π(λ − 1) I m(ξ )=β7 V(ξ + Y ) λ −1 I m(Y )=−γ1 (9.44) ≡ J1 + J2 , 2i   − λ−1 Y 2iY t V(2i) i − 3+λ X Γ dy 3+λ G(t, X ) = − e 2 2π(λ − 1) λ−1 I m(Y )=−γ1 V( 2 i + Y )    2 i V(ξ ) − i Y 2iY + t λ−1 Γ eiξ X dξ dy π(λ − 1) I m(ξ )=β8 V(ξ + Y ) λ−1 I m(Y )=−γ1 ≡ J1 + J2 , (9.45)

where β7 = 23 − δ , β8 = 3+λ 2 + δ with δ > 0 small. The terms J2 in (9.44), (9.45) can be estimated easily. Indeed, they can be written in the form:  1  (t, ξ ) dξ,  = 7, 8. G(t, X ) = √ eiξ X G 2π I m(ξ )=β Integrating by parts we obtain: G(t, X ) = − √ We can now estimate

∂2 ∂ξ 2





1 2π X 2

 I m(ξ )=β

dξ eiξ X

 ∂2  G (t, ξ ) . 2 ∂ξ

  (t, ξ ) using Proposition 8.1. It then follows that: G

  √ t  iξ X  −a |ξ |t |dξ |  e e   1 + |ξ |3/2 I m(ξ )=β  √ Ceβ X t −a |ξ |t |dξ | .   ≤ e 3/2 X2 I m(ξ )=β 1 + |ξ |

C |J2 | ≤ 2 X

Linearized Homogeneous Coagulation Equation

795

In order to estimate the integral we split it as follows:   √ t −a |ξ |t |dξ |   = e [...] |dξ | 3/2 I m(ξ )=β 1 + |ξ | I m(ξ )=β , |ξ |≤ 12 t  + [...] |dξ | I m(ξ )=β , |ξ |≥

1 t2

≤ Ct + Ct 2 . Then |J2 | ≤ Cteβ X for |X | ≥ 1, where  = 7 for X < 0 and  = 8 for X > 0. In order to compute the terms J1 in (9.44), (9.45)  deform the contour on Y upwards. We cross the first pole of the function  we 2iY Γ λ−1 for Y = 0. However, this point is not a pole of the integrand, because the     functions V 23 i + Y , V 3+λ the first pole 2 i + Y have also a pole at Y = 0. Therefore 3  i. Notice that V i + Y does not of the integrand that is found is the one at Y = λ−1 2 2 λ 3 λ 3+λ λ−1 have a zero before, since 23 + λ−1 2 = 1 + 2 < 2 + 2 . On the other hand 2 + 2 = 1 + λ, λ λ while the first zero of V (η) is at η = 2 + 2 and 1 + λ < 2 + 2 . Then, after deforming the integral contour as indicated, the terms J1 can be written as: 3

J1 = e− 2 X t −

2i V λ−1

2i    3 λ 2iY t − λ−1 Y 1+ i e− 2 X Γ dy 3 2 λ−1 V( 2 i + Y ) I m(Y )=γ2

3

= e− 2 X (t + J3 ) for X < 0, and V(2i) t    4 π V 1 + λ2 i 2i   2iY i V(2i) − 3+λ X t − λ−1 Y e 2 Γ − dy 3+λ 2 π (λ − 1) λ−1 V( 2 i + Y ) I m(Y )=γ2   3+λ V(2i) t for X > 0, = e− 2 X   + J3  4 π V 1 + λ2 i

J1 = e−

3+λ 2 X

where γ2 > (λ − 1)/2 is such that γ2 − (λ − 1)/2 is small. In both cases there exist positive constants δ and Cδ such that for all t > 1: |J3 | ≤ Cδ t 1+δ , |J1 | ≤ Cteβ X for |X | ≥ 1, where  = 7 for X < 0 and  = 8 for X > 0. It then follows that: |J1 | + |J2 | ≤ Cteβ X for |X | ≥ 1, where, as before,  = 7 for X < 0 and  = 8 for X > 0, and Lemma 9.10 follows.



796

M. Escobedo, J. J. L. Velázquez

10. The Initial Value Problem Using the fundamental solution g obtained in Theorem 3.5 we can obtain a solution of the initial value problem ∂h = L [h] , ∂t h (0, x) = h 0 (x) ,

(10.1) (10.2)

with L [ · ] defined in (3.2). Assuming that there are not difficulties with the integrals written below, we would expect, due to the linearity of the problem (10.1), (10.2) the following representation formula for their solutions (cf. Theorem 3.5): 

∞

h (t, x) = 0

 λ−1 x dy . h 0 (y) g t y 2 , , 1 y y

(10.3)

We first precise sufficient conditions on h 0 that allow to define h (t, x) in (10.3).   Theorem 10.1. Suppose that the function h 0 ∈ C R+ satisfies 

1

|h 0 (y)| y λ dy +

0



∞

|h 0 (y)| dy < ∞.

(10.4)

1

Then the function h (t, x) defined for t ≥ 0, x > 0 by means of (10.3) solves the initial value problem (10.1), (10.2). The proof of this theorem reduces to a detailed analysis of the conditions on h 0 yielding integrability of the right-hand side of (10.3). Under more stringent assumptions on h 0 it is possible to use Theorem 3.5 to derive more detailed information on the asymptotics of the solutions of (10.1), (10.2) for x → 0 and x → ∞. The meaning of this asymptotics will be explained in the next section. Theorem 10.2. Suppose that 3

|h 0 (x)| ≤ C x − 2 +δ , 0 < x ≤, ε > 0, |h 0 (x)| ≤ C x −(1+δ) , x ≥ 1, ε > 0.

(10.5) (10.6)

Then the function h (t, x) given in (10.3) satisfies for any t > 0,   3 3  h (t, x) − A− (t) x − 2  ≤ B− (t) x − 2 +δ for 0 < x ≤ 1,   3+λ  3+λ  h (t, x) − A+ (t) x − 2  ≤ B+ (t) x − 2 −δ for x ≥ 1 for suitable functions A− (t) , A+ (t) , B− (t) , B+ (t) . Detailed proofs of these two results will be given in [12].

(10.7) (10.8)

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797

11. Particle Fluxes for Singular Solutions of the Coagulation Equation It is well known that (1.1), (1.2) may be written as ∂ ∂ (x f ) = − j ( f ), ∂t  ∂x x ∞ y K (y, z) f (y) f (z) dz dy j ( f )(t, x) = 0

(11.9) (11.10)

x−y

see [20], as well as [5] for similar formulas in the self similar regime and [15] for an application of the function j to handle with solutions which are singular near the origin. The number j ( f )(t, x) represents the flux of particles at x produced by the collisions. In particular:  R2 d x f (t, x) d x = j ( f )(t, R1 ) − j ( f )(t, R2 ). (11.11) dt R1 Using the formula (11.10) it follows that a solution f of (1.1), (1.2) with the asymptotics f (t, x) ∼ A(t) 3+λ as x → ∞ can be interpreted as a particle distribution yielding a flux x

2

of particles to infinity: lim j ( f )(t, R) = 2π A2 (t).

R→+∞

Notice in particular that f (t, x) = o(x −(3+λ)/2 ) as x → +∞ implies that the mass of the particle distribution is conserved. The function f s (x) = Ax −(3+λ)/2 can be thought as a singular steady state of (1.1), (1.2) because j ( f s ) = 2π A2 . We recall that (3.1), (3.2) has been obtained using f (x, t) = x −

3+λ 2

+ g (x, t)

(11.12)

in (1.1), (1.2) and keeping just linear terms on g. Then, we can derive formulas for the particle fluxes associated to (3.1), (3.2) linearizing (11.10), (11.14):  x ∞   y −1/2 z λ/2 g(z) + y 1+λ/2 z −3/2 g(y) dz dy (11.13) jlin (g)(t, x) = 0

and d dt



x−y



R2

= jlin (g)(t, R1 ) − jlin (g)(t, R2 ).

x g(t, x) d x

(11.14)

R1

Using g(t, x) ∼ a(t) x −3/2 as x → 0 it follows that for all t > 0, lim R→0+ jlin (g)(t, R) = 0. Therefore the perturbation g does not modify the incoming flux of mass that is the one of f s (x) = x −(3+λ)/2 . Moreover  R d xg (t, x) d x = − jlin (g)(t, R). (11.15) dt 0 Integrating (11.15) we obtain:   R x g(0, x)d x = 0

t 0

 jlin (g)(s, R)ds +

R

x g(t, x)d x. 0

(11.16)

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Since the solution g (t, x) obtained in Theorem 3.5 satisfies g(t, x) ∼ a(t) x −(3+λ)/2 as x → +∞, taking the limit R → ∞ in (11.16) it follows that: 

∞



0



t

x g(0, x)d x = 4π

∞

a(s) ds + 0

x g(t, x)d x.

The self-similar asymptotics (3.9)-(3.12) implies limt→∞ 

∞

 x g(0, x)d x = 4π

0

(11.17)

0

∞ 0

xg (t, x) d x = 0. Then:

∞

a(s) ds.

(11.18)

0

The left-hand side of (11.18) is the initial total mass of the perturbation. The right hand side of (11.18) is the total amount of particles contained in clusters of infinite size. Equation (11.18) means that all the excess of particles initially introduced in the system move as t → +∞ to an infinitely large cluster. 12. The Function Φ We take now the Mellin transform on both hands of the equation. We recall that the Mellin transform of a function g(y) is defined as:  ∞ M(g)(s) = y s−1 g(y)dy. (12.1) 0

Taking the Mellin transform of the right hand side of (3.1), (3.2) we obtain after straightforward calculations:   ∂ λ−1 3 M(g)(s) = M(g) s + P −s + , (12.2) ∂t 2 2  ∞   P(s) = θ 1/2−s (θ − 1)−3/2 − θ −3/2 dθ 2



  √ 2−s −2 2 (1 − θ )−3/2 θ s−1 − 1 dθ + s 1/2 = I1 (s) + I2 (s) + I3 (s) + I4 . 1

+

(12.3) (12.4)

Remark 12.1. If the function g satisfies the estimates (3.5), (3.6) for some r > 1 and ρ < 2 such that ρ < r we will have that its Mellin transform is well defined in the strip Res ∈ (λ/2 + ρ, λ/2 + r ). 12.1. The function P(s). We consider in this section the auxiliary function P obtained by taking the Mellin transform of Eq. (3.1), (3.2) and first rewrite it in terms of Gamma functions. Proposition 12.2. The function P(s) defined in (12.3) can be written as √ 2 π Γ (s) P(s) = − . Γ (s − 1/2)

(12.5)

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799

The function P(s) is meromorphic on the whole complex plane. It has simple zeros and poles at the points sz (n) =

1 − n, n = 0, 1, 2, . . . , 2

(12.6)

and s p (n) = −n, n = 0, 1, 2, . . . , respectively Proof. We can write the term I1 of (12.4) as

  ∞

I1 (s) =

θ

1 1− θ

1/2−s −3/2

θ

2

−3/2

 − 1 dθ.

−3/2  and integrating each term of the Using the Binomial Theorem to expand 1 − θ1 resulting series we obtain: I1 (s) + I3 (s) = 2−s

∞ n=0

 2−n −3/2 (−1)n . n s+n

(12.7)

Integrating by parts we obtain:  √ √ I2 (s) = 2 2 − 2 2(1/2)s−1 − 2(s − 1)

1

(1 − θ )−1/2 θ s−2 dθ.

(12.8)

1/2

In order to compute the last term in (12.8) we write: 

1

(1 − θ )

−1/2 s−2

θ

1/2

 1  1/2 −1/2 s−2 dθ = (1 − θ ) θ dθ − (1 − θ )−1/2 θ s−2 dθ. (12.9) 0

0

Expanding (1 − θ )−1/2 using the Binomial Theorem and integrating each term of the resulting series:

  1/2 ∞  2 −1/2 (−1)+1 − −1/2 s−2 −s + 2 (1 − θ ) θ dθ = 2 . (12.10) +1 s−1 +s 0 =0

Moreover, 2 + s−1

∞ n=0



2(1 − 1/2)−1/2 −1/2 (−1)n+1 −n 2 = n+1 n+s s−1 −

1 2(s − 1)

∞ n=0

 −3/2 (−1)n −n 2 , n+1 n+s (12.11)

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  −3/2 −1/2 1 as well as the Binomial Theorem. = − 2(n+1) n n+1 Combining (12.10) and (12.11) we deduce

√   1/2 ∞  1 −3/2 (−1)n 2−n −1/2 s−2 −s 2 2 (1−θ ) θ dθ = 2 − . n+1 s − 1 2(s − 1) n+s 0

where we have used that

n=0

(12.12) Therefore, using (12.3), (12.7), (12.8), (12.9) and (12.12), 

1

P(s) = −2(s − 1) 0

(1 − θ )−1/2 θ s−2 dθ = −

√ 2 2 Γ (s) . Γ (s − 1/2)

(12.13)

The properties of zeros and poles are consequence of the properties of the Gamma function.

12.2. Behaviour of Φ at infinity. Proposition 12.3. The following asymptotic formulas hold:   √ 1 3 as |I m (s)| → ∞ +O 2 P (s) = −2 π s 1 − 8s s

(12.14)

uniformly in sets where arg (s) ∈ (−π + ε0 , π − ε0 ) for any ε0 > 0. Proof. Formula (12.14) is a consequence of (12.5) as well as the asymptotic formula:   √ 1 1 1 Γ (z) ∼ 2π (z)z− 2 e−z 1 + as |z| → ∞ (12.15) +O 2 12z z that is uniformly valid in sets arg (z) ∈ (−π + ε0 , π − ε0 ) for any ε0 > 0. Then 1  √ 1   2 π (s)s− 2 s − 21 2 1 1 + O as |s| → ∞, P (s) = −   1 s2 1 s− 2 21 s−2 e uniformly in sets arg (s) ∈ (−π + ε0 , π − ε0 ) and arg (s − 1/2) ∈ (−π + ε0 , π − ε0 ) for any ε0 > 0. Notice that    1 √ 1 1 (s)s− 2 = e 1−  as |s| → ∞, +O 2 1 1   s− 2 s 8 s−2 s − 21 uniformly in the same sets as above, whence   √ 3 1 as |s| → ∞, P (s) = −2 π s 1 − +O 2 8s s uniformly in sets arg (s) ∈ (−π + ε0 , π − ε0 ) for any ε0 > 0. and (12.14) follows.



Linearized Homogeneous Coagulation Equation

801

13. On the Properties of the Function V We prove now some auxiliary results used to prove the results of the paper. Lemma 13.1. Suppose that f and h are two analytic functions in the cone ! " C(2ε0 ) = ζ ∈ C; ζ = |ζ |eiθ , θ ∈ (−2ε0 , 2ε0 ) for some ε0 > 0 and real valued in R+ . Let us also assume that 

∞

| f (r eiθ )| + |h(r eiθ )| dr < +∞, for any θ ∈ (−2ε0 , 2ε0 ), 1 + r2 0 lim f (ζ ) = θ1 and lim f (ζ ) = θ2 , |ζ |→0 |ζ |→∞  1 as |ζ | → 0, |ζ | → ∞, ζ ∈ C(2ε0 ). | f (ζ )| = o |ζ |

(13.1) (13.2) (13.3)

Then, the function F(ζ ) =

1 2πi

 0

∞

 (h(s) + i f (s))

1 1 − ds s−ζ s+1

is analytic in the domain: ! " D(ε0 ) = ζ ∈ R; ζ = |ζ |eiθ , |ζ | > 0, θ ∈ (−ε0 , 2π + ε0 ) ,

(13.4)

(13.5)

where R is the Riemann surface associated to the function ln ζ . Moreover, θ1 ln ζ + i H (ζ ) + o(ln |ζ |), as ζ → 0, ζ ∈ D(ε0 ), 2π θ2 ln ζ + i H (ζ ) + o(ln |ζ |), as |ζ | → +∞, ζ ∈ D(ε0 ), F(ζ ) = − 2π F(ζ ) = −

where the function H (ζ ) is a real valued function defined by   ∞ 1 1 1 H (ζ ) = − − ds. h(s) 2π 0 s−ζ s+1

(13.6) (13.7)

(13.8)

Proof of Lemma 13.1. Using Lemma C.2 of [10] with the function f we obtain (13.6) and (13.8). On the other hand, the condition (13.1) ensures that the function F is well defined.

Proposition 13.2. Let V(ξ ) defined by (6.6) and (6.12). Then, for any ε > 0 arbitrarily small and all M > 0 arbitrarily large, there exist two positive constants C1,ε,M , C2,ε,M such that 1

π

1

π

C1,ε,M e− 2 ( λ−1 +ε)|ξ | ≤ |V(ξ )| ≤ C2,ε,M e− 2 ( λ−1 −ε)|ξ |

(13.9)

uniformly for I m(ξ ) in compact sets of (3/2, (3 + λ)/2) as well as for all ξ such that |Re(ξ )| ≥ 1, |I m(ξ )| ≤ M.

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M. Escobedo, J. J. L. Velázquez

Proof of Proposition 13.2. Given ξ ∈ S we can represent the function V by (6.6) and (6.12) for β1 satisfying (6.13). In order to simplify some of the calculations we use the following change of variables. 4π

ζ = e λ−1 (ξ −β1 ) , ν(ζ ) = V(ξ ), ϕ(ζ ) = Φ(ξ ).

(13.10) (13.11) (13.12)

The function ln (−ϕ(s)) may be written as ln (−ϕ(s)) = ln (|ϕ(s)|) + i arg (−ϕ(s)) .

(13.13)

The functions ln (|ϕ(s)|) and arg (ϕ(s)) satisfy the hypothesis required to h and f respectively in Lemma 13.2. In particular, by Proposition 4.1 and the fact that Re(−Φ(y)) > 0 for all y such that I m(y) ∈ ((2 + λ)/2, (3 + λ)/2) we may normalize the argument of the function ln (−ϕ(s)) such that: π lim arg (−ϕ(ζ )) = − , ζ →0 4

lim arg (−ϕ(ζ )) =

ζ →∞

π . 4

(13.14)

Applying Lemma 13.1 it follows that   ∞ 1 1 1 1 − ds = ln(−ζ ) + i H (ζ ) + o(ln |ζ |) ln (−ϕ(s)) 2πi 0 s−ζ s+1 8 as ζ → 0, ζ ∈ D(ε0 ), (13.15)   ∞ 1 1 1 1 ds = − ln(−ζ ) + i H (ζ ) + o(ln |ζ |) ln (−ϕ(s)) − 2πi 0 s−ζ s+1 8 as ζ → ∞, ζ ∈ D(ε0 ). (13.16) The two estimates in (13.9) follow, for I m(ξ ) in compact sets of (3/2, (3 + λ)/2), by taking exponentials in both sides of (13.15) and (13.16) and inverting the change of variables (13.10)-(13.12). In order to prove the estimate for ξ in the region |Re(ξ )| ≥ 1 and |I m(ξ )| ≤ M, we extend analytically the function V(ξ ) to such regions using (5.11) as well as the fact that, by Proposition 4.1, we have for some positive constants C1 and C2 : C1 |ξ |1/2 ≤ |Φ(ξ )| ≤ C2 |ξ |1/2 for |Re(ξ )| ≥ 1 and |I m(ξ )| ≤ M.



14. Contour Deformation Estimates We must estimate in Sects. 8 and 9 several integral expressions of the form  eΨ (ξ,Y,t) m (ξ, Y ) dY I m Y =−γ1

for a given function Ψ but different functions m. This is done using contour deformation combined with Laplace’s method. We collect in this section some technical results about the function Ψ (ξ, Y, t) and its critical points.

Linearized Homogeneous Coagulation Equation

803

Fig. 6. D(ξ, B) when Re(ξ ) > 0 and then Q = 1

14.1. The critical point. We first compute the critical points of the function Ψ (ξ, Y, t) (ξ, Z , t) (cf. (8.12)). defined by (8.4) or equivalently, those of Ψ 14.1.1. The case 0 < t < 1 We start with the following: Lemma 14.1. Consider the function F(ξ, Z ) defined by means of   2 F(ξ, Z ) = ln (−Φ(η)) Θ(η − ξ, |ξ |Z )dη (λ − 1) i I m(η)=β1

(14.1)

with β1 as in (6.13), ξ ∈ C and (β1 − I m(ξ )) λ−1 1 < I m(Z ) ≤ . √ (β1 − I m(ξ )) − √ 2 |ξ | |ξ |

(14.2)

For any constant B > 0 there exists L > 0, and ξ0 , both depending on B, such that, for all ξ ∈ T L ∩ (Bξ0 (0))c , the function F(ξ, ·) can be extended analytically on the variable Z to the domain Z ∈ D(ξ, B) ∩ B√|ξ |/8 (0), where D(ξ, B) is as in (8.6). Moreover, there exists a positive constant C depending on B such that         F(ξ, Z ) + 2 i ln (−Φ(ξ )) |ξ | Z  ≤ C Z 2 + O 1 (14.3)   (λ − 1) |ξ | for Z ∈ D(ξ, B) ∩ B√|ξ |/8 (0) and , ξ ∈ T L ∩ (Bξ0 (0))c . Proof. The function F is well defined in (14.2) since in that domain the variable Z is in the region where the function Θ is analytic. The function F(ξ, ·) given by (14.1) is then )) analytic in the strip |I m(Z )| ≤ δ0 (I√m(ξ for some δ0 (I m(ξ )) sufficiently small. |ξ | We now claim that for any fixed constant B > 0, there exists L > 0 (large) depending on B such that for ξ ∈ T L the function F(ξ, ·) may be extended analytically to the region D(ξ, B) Fig. 6. To prove this we derive new representation formulas of the function F performing suitable contour deformations in the variable η in (14.1). Notice that the singularities of the integrand in (14.1) are contained in the set Re(η) = √ 0, η = ξ + (λ − 1)/2 and η = ξ + |ξ | Z + (λ − 1)/2 for  ∈ Z. Given Z 0 in the region above, let  Z 0 be such that: Re(  Z 0 ) = Re(Z 0 ), |I m(  Z 0 )| ≤

δ0 (I m(ξ )) √ |ξ |

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M. Escobedo, J. J. L. Velázquez

Fig. 7. The curve C2

√ and√the integral curve I m (η) = β1 lies between the two points ξ +  Z 0 |ξ | and ξ +  Z 0 |ξ | + (λ − 1)i/2. (Notice that this is possible since δ0 (I m(ξ )) may be made as small as we need and I m(η) > I m(ξ )). Consider the vertical segment of the complex plane connecting Z 0 and  Z0: Zθ = (1 − θ )  Z 0 + θ Z 0 , θ ∈ [0, 1]. We then obtain an analytic extension of F(ξ, ·) varying θ continuously from 0 to one and deforming continuously the contour I m η = β0 + θ−1 2 −ε to a new contour C2 (cf. Fig. 7), in such a way that: √ √ – it always passes between ξ +  Z θ |ξ | and ξ +  Z θ |ξ | + (λ − 1)i/2 – we do not change the original integration contour for √     |Re(Z 0 ) |ξ |  ≥ η − ξ − Z . |ξ |  Re 0 2 The first condition ensures that the integration contour never crosses any of the singu −1 √ 4π larities of the function 1 − e− λ−1 (η−ξ −Z |ξ |) . The second one ensures that it does −1  4π . not cross either of the singularities of 1 − e λ−1 (η−ξ ) Finally, since sgn(Re(Z 0 )) = sgn(Re(ξ )), the new integration contour never crosses the line Re(η) = 0, where the singularities of ln (−Φ(η)) are located. To estimate this integral we write   2 ln (−Φ(η)) Θ(η − ξ, |ξ |Z )dη (λ − 1) i C2   2 ln (−Φ(ξ )) Θ(η − ξ, |ξ |Z )dη = (λ − 1) i C2    2 Φ(η) Θ(η − ξ, |ξ |t Z )dη + ln (λ − 1) i C2 Φ(ξ ) = I1 + I2 .

(14.4)

Linearized Homogeneous Coagulation Equation

805

The first integral, I1 is computed explicitly :

I1 =

 2 ln (−Φ(ξ ))  2i ln (−Φ(ξ )) |ξ | Z . |ξ | Z = − (λ − 1) i (λ − 1)

(14.5)

In order to compute the second integral we have to distinguish the cases Re(ξ ) → −∞ and Re(ξ ) → +∞. Since both may be treated using similar arguments let us treat only the case Re(ξ ) → +∞. In that case we decompose I2 as follows

I2 =

  2 [· · · ] dη + [· · · ] dη (λ − 1) i C2 ,Re(η)>0,|η−ξ |≤ |ξ4| C2 ,Re(η)>0,|η−ξ |> |ξ4|  2 [· · · ] dη = I2,1 + I2,2 + I2,3 . (14.6) + (λ − 1) i C2 ,Re(η)<0

In I2,1 we use Proposition 4.1 and Taylor’s expansion to obtain: √  η 1− Φ(η) = √  Φ(ξ ) ξ 1−

(2λ+1)i 8η

+ O(|η|−2

(2λ+1)i 8ξ

+ O(|ξ |−2

 .

Therefore: 

Φ(η) ln Φ(ξ )





η−ξ =O ξ





1 +O ξ2

, as Re(ξ ) > 1, |η − ξ | ≤

|ξ | . 4

We now estimate the two following integrals for all Z ∈ D(ξ, B) and |Z | ≤ first one can be bounded as: 

 C2 ,Re(η)>0,|η−ξ |≤ |ξ4|

 |η − ξ | Θ(η − ξ, |ξ |Z )dη |ξ |      σ Θ(σ, |ξ |Z ) dσ

 1 |ξ | C2 ,Re(σ )>−Re(ξ ), |σ |≤ |ξ4|     1   ≤ σ Θ(σ, |ξ |Z ) dσ |ξ | C2 ,Re(σ )>−Re(ξ ), |σ |≤2|ξ |1/2 Z     1   + σ Θ(σ, |ξ |Z ) dσ |ξ | C2 ,Re(σ )>−Re(ξ ), |σ |≥2|ξ |1/2 Z =

≤ C Z2 + C

e−a|ξ | |ξ |

1/2 |Z |

for Re(ξ ) > ξ0 ,

√

|ξ | 8 .

The

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M. Escobedo, J. J. L. Velázquez

and the second one:  C2 ,Re(η)>0,|η−ξ |≤ |ξ4|

  1   − ξ, |ξ |Z )  dη Θ(η |ξ |2      |ξ |Z ) dσ Θ(σ, |ξ |

 1 |ξ |2 C2 ,Re(σ )>−Re(ξ ),|σ |≤ 4     1   ≤ Θ(σ, |ξ |Z )   dσ 2 1/2 |ξ | C2 ,Re(σ )>−Re(ξ ),|σ |≤2|ξ | Z     1   + 2 Θ(σ, |ξ |Z ) dσ 1/2 |ξ | C2 ,Re(σ )>−Re(ξ ),|σ |≥2|ξ | Z  1/2 1/2 |Z | e−a|ξ | |Z | e−a|ξ | |Z | 1 2 ≤ C 3/2 + C ≤ C |Z | + 3 + C for Re(ξ ) > ξ0 , |ξ | |ξ |2 |ξ | |ξ |2 =

where in both cases C2 = C2 − ξ , ξ0 is a positive constant sufficiently large, depending on B and a and C are positive constants which may depend on B but are independent on Re(ξ ) and Z ∈ D(ξ, B). We then deduce that, for all Re(ξ ) > ξ0 :  1/2 1 e−a|ξ | |Z | 2 |I2,1 | ≤ C |Z | + 3 + C . (14.7) |ξ | |ξ | In the integral I2,2 we use the fact that when |η − ξ | > |ξ4| and Re(ξ ) > ξ0 , the function Θ(η − ξ, Y ) has an exponential decay Ce−a|ξ | with C and a as above as well as the inequality | ln(−Φ(η))| ≤ C| ln(|η − ξ | + |ξ |)| ≤ C|(ln(|η − ξ |)| + | ln(|ξ |)|) for η large. For η of order one we use that ln(−Φ(η)) is of order one to derive a similar estimate. Then    |I2,2 | ≤ e−a|η−ξ | (| ln(|η − ξ |)| + | ln(|ξ |)|) dη = O e−a|ξ | . (14.8) |η−ξ |> |ξ4|

Finally, the estimate of I2,3 follows using the same cut-off properties of the function Θ since Re(η) < 0 and Re(ξ ) → +∞ implies that Re(η − ξ ) > C|ξ |. The final estimate of I2 , by (14.6), is then,  1/2 e−a|ξ | |Z | 1 2 . (14.9) |I2 | ≤ C |Z | + 3 + C |ξ | |ξ | Using Proposition 4.1, (8.12), (14.4), (14.5) and (14.9) the lemma follows.



Lemma 14.2. Given B > 0, let ξ0 , L be the ones given by Lemma 14.1. Then there exists a constant C > 0 depending on B such that the function h defined by means of:    √   2i Z 2i Z i Q π4  Ψ (ξ, Z , t) = − |ξ | 1 + ln t − ln + ln 2 π e λ−1 λ−1   1 2i Z 1  + h(ξ, Z , t) (14.10) − ln |ξ |1/2 − ln 2 2 λ−1 satisfies

 |h(ξ, Z , t)| ≤ C

 Z2 + O

for ξ ∈ T L ∩ (Bξ0 (0))c , Z ∈ D(ξ, B) ∩ B√|ξ |/8 (0).

1 |ξ |



Linearized Homogeneous Coagulation Equation

807

Proof. This lemma is a direct consequence of Lemma (14.1). The only difficulty in order  defined in (8.12) comes from the term V(ξ )/V(ξ + Y ) which to estimate the function Ψ corresponds to the integral term. This term, given by formula (6.15) may also be written as: V(ξ ) √ V(ξ + |ξ |Z )   2 = exp (λ − 1) i I m η=β1

Λ(ξ, Z ) =





ln (−Φ(η)) Θ(η − ξ, |ξ |Z )dη .



(ξ, ·, t), the precise result is the following. As for the critical point Z c of Ψ √ Lemma 14.3. Suppose √ that B > 2 π let ξ0 , L be the ones given by Lemma 14.1. Let us define δ0 = (λ−1) π /4. There exists R > 0 depending on B, such that for all 0 ≤ t ≤ 1 and ξ ∈ T L ∩ (Bξ0 (0))c , |ξ | t 2 ≥ R, there exists a unique point Z c ∈ D(ξ, B)\Bδ0 t (0) (ξ, Z c , t)/∂ Z = 0. Moreover the following asymptotics holds uniformly such that ∂ Ψ for 0 ≤ t ≤ 1:   √ 2 i Zc 1 = 2πt (1 + i Q)) 1 + O √ as |Re(ξ )| t 2 → ∞. (14.11) λ−1 |ξ | t  given by (8.12) gives for all Z ∈ Proof. Computing the derivative of the function Ψ D(ξ, B): √ 4π √   8π |ξ | i ∂Ψ e λ−1 ( |ξ |Z −η+ξ ) (ξ, Z , t) = − ln (−Φ(η))  2 dη 4π √ ∂Z (λ − 1)2 C2 ( |ξ |Z −η+ξ ) λ−1 1−e     2i(Z /t) 1 2i ln − √ + |ξ | + λ−1 λ−1 2 |ξ |Z 2i (14.12) ln |ξ |1/2 . + λ−1 We compute the leading term of the integral in the right hand side of (14.12) as |ξ | → ∞:  I (ξ, Z ) =

4π

e λ−1 (

C2

 = where C2 = C2 − ξ − Z ∈ D(ξ, B), |Z | ≤ B:

√ |ξ |Z −η+ξ )

ln (−Φ(η))  2 dη 4π √ 1 − e λ−1 ( |ξ |Z −η+ξ )

4π    e− λ−1 σ ln −Φ(σ + ξ + |ξ |Z )  2 dσ, 4π − λ−1 σ C2 1−e

√

|ξ | Z . Using Proposition 4.1 we have that, uniformly for

 2       σ σ +O ln −Φ(σ + ξ + |ξ |Z ) = ln −Φ(ξ + |ξ |Z ) + A(ξ ) |ξ | |ξ |2

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for |ξ | → +∞, where A(ξ ) is a bounded function of sgn(Re(ξ )). It then follows:   λ−1  I (ξ, Z ) = − ln −Φ(ξ + |ξ |Z ) 4π 4π 4π  2   e− λ−1 σ e− λ−1 σ σ A(ξ ) σ dσ + O + 2  2 dσ 4π 4π |ξ | C2 |ξ |2 C2 1 − e− λ−1 σ 1 − e− λ−1 σ   (14.13) +O e−a|ξ | as |ξ | → +∞ uniformly for Z ∈ D(ξ, B), |Z | ≤ B, where a > 0 is independent on ξ , Z , where we 4π σ − λ−1  have used C2  e 4π 2 dσ = − λ−1 4π . 1−e

− λ−1 σ

On the other hand, in order to estimate the second term in the right-hand side of (14.13) we deform the integration contour C2 to a horizontal line at a bounded distance of the real axis. The resulting integral can then be bounded by a positive constant independent of ξ , Z . The third term in the right hand side of (14.13) can be bounded using the specific form of C2 as:  2 |Z |3 |Z | + 1/2 . C |ξ | |ξ | We notice that by Proposition 4.1, we have:     |Z | ln −Φ(ξ + |ξ |Z ) = ln (−Φ(ξ )) + O √ , as |ξ | → +∞ |ξ | uniformly for Z ∈ D(ξ, B), |Z | ≤ B. Combining everything we deduce:  λ−1 |Z | I (ξ, Z ) = − , as |ξ | → +∞ ln (−Φ(ξ )) + O 4π |ξ |1/2

(14.14)

uniformly for Z ∈ D(ξ, B), |Z | ≤ B. Combining (14.12) and (14.14) it follows: √      ∂Ψ 2i |ξ | 2i(Z /t) (λ − 1)i (ξ, Z , t) = ln (−Φ(ξ )) + ln + √ + ln |ξ |1/2 ∂Z (λ − 1) λ−1 4 |ξ |Z  |Z | as |ξ | → +∞ +O |ξ |1/2 uniformly for Z ∈ D(ξ, B), |Z | ≤ B. Using Rouché’s Theorem it then follows that for |ξ |t 2 sufficiently large, ξ ∈ T L ∩ /∂ Z )(ξ, Z , t) = 0 in Z ∈ D(ξ, B)\Bδ0 t (0) (Bξ0 (0))c there exists a unique root of (∂ Ψ satisfying the asymptotics (14.3) and the lemma follows.

. We now derive estimates for higher order derivatives of Ψ Lemma 14.4. Under the same conditions as in Lemma 14.3 the following asymptotics holds: √    ∂ 2Ψ 1 2 i |ξ | 1 + O (ξ, Z , t) = as |ξ | t 2 → +∞, √ c ∂ Z2 (λ − 1) Z c |ξ |t uniformly in 0 ≤ t ≤ 1.

Linearized Homogeneous Coagulation Equation

809

 with respect to Z is: Proof. The second derivative of Ψ √ √ 4π   ∂ 2Ψ 1 e λ−1 (Z |ξ |−η+ξ ) 8π |ξ |i 2 |ξ | i + =− [ln (−Φ(η))]  2 dη + √ 2 2 4π ∂Z (λ − 1) C2 (λ − 1) Z 2Z 2 1 − e λ−1 (Z |ξ |−η+ξ ) √ √ 4π  e λ−1 (Z |ξ |−η+ξ ) 1 Φ (η) 8π |ξ |i 2 i |ξ | + =− dη + .  2 √ 2 4π (λ − 1)2 C2 Φ(η) (λ − 1) Z 2Z (Z |ξ |−η+ξ ) 1 − e λ−1

Taking Z = Z c and using Lemma 14.3, we deduce, uniformly in 0 ≤ t ≤ 1, √ √ 4π   ∂ 2Ψ e λ−1 (Z c |ξ |−η+ξ ) Φ (η) 2 i |ξ | 8π |ξ |i (ξ, Z c , t) = −  2 dη √ 4π ∂ Z2 (λ − 1) Z c (λ − 1)2 C2 Φ(η) 1 − e λ−1 (Z c |ξ |−η+ξ )  1 , (14.15) +O t

as |ξ |t 2 → +∞. Using Proposition 4.1: √ √ 4π 4π  e λ−1 (Z c |ξ |−η+ξ ) e λ−1 (Z c |ξ |−η+ξ ) Φ (η) 1 1 J (ξ ) =  2 dη =  2 dη √ √ 4π 4π 2 C2 η (Z c |ξ |−η+ξ ) C2 Φ(η) 1 − e λ−1 1 − e λ−1 (Z c |ξ |−η+ξ ) √ 4π     e λ−1 (Z c |ξ |−η+ξ ) 1 −a|ξ | + , as |ξ | → +∞, O dη + O e  2 √ 4π |ξ |2 C2 1 − e λ−1 (Z c |ξ |−η+ξ )



(14.16) where a is a positive constant independent on ξ . The first term in the right hand side is estimated as: √ √ 4π 4π  e λ−1 (Z c |ξ |−η+ξ ) 1 e λ−1 (Z c |ξ |−η+ξ ) 1  2 dη =  2 dη √ √ 4π 4π ξ C2 (Z c |ξ |−η+ξ ) C2 η 1 − e λ−1 1 − e λ−1 (Z c |ξ |−η+ξ ) √ 4π   e λ−1 (Z c |ξ |−η+ξ ) η−ξ + O  2 dη √ 4π |ξ |2 C2 1 − e λ−1 (Z c |ξ |−η+ξ )   +O e−a|ξ | as |ξ | → +∞.



The first term in the right hand side can be computed explicitly and gives −(λ − 1)/4π ξ . The second one is estimated using the form of the contour C2 and is bounded by O (|Z c |/|ξ |). Therefore, the first and second terms in the right hand side of (14.16) can be estimated respectively as O(1/|ξ |), O(1/|ξ |3/2 )as |ξ | → +∞. Then  1 as |ξ | → +∞. J (ξ ) = O |ξ |

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Using (14.15) we obtain

√   ∂ 2Ψ 2 i |ξ | 1 as |ξ |t 2 → +∞ (ξ, Z , t) = + O c ∂ Z2 (λ − 1) Z c t

whence Lemma 14.4 follows.



Lemma 14.5. Suppose that Z c , B and δ0 are as in Lemma 14.3. Then the following asymptotics holds:  3  √  ∂ Ψ  |ξ |  =O as |ξ | t 2 → +∞, (ξ, Z , t)  ∂ Z3  t2 uniformly in 0 ≤ t ≤ 1, Z ∈ D(ξ, B), |Z | ≤ B. Proof of Lemma 14.5. √ 4π    Φ (η) ∂ 3Ψ e λ−1 (Z |ξ |−η+ξ ) 8π |ξ |3/2 i (ξ, Z , t) = −  2 dη √ 4π ∂ Z3 (λ − 1)2 C2 Φ(η) 1 − e λ−1 (Z |ξ |−η+ξ ) √ 2 i |ξ | 1 − − 3. (14.17) 2 (λ − 1) Z Z √ The last two√terms of (14.17) are bounded as C( |ξ |/t 2 + 1/t 3 ). and this can be estimated as C |ξ |/t 2 for |ξ |t 2 >> 1. On the other hand we may bound the first term in the right hand side of (14.17) using    Φ (η)  C   ,  ≤  Φ(η)  1 + |η|2

the form of the contour C2 . The term under consideration is then bounded by a constant. Combining all these estimates for the terms in the right hand side of (14.17) the lemma follows.

Lemma 14.1 and Lemma 14.3 yield Corollary 14.6. For all 0 < t < 1:  1 2 i Z  √ 1  c (ξ, Z c , t) = − |ξ | 2π t (1 + i Q) − ln |ξ |1/2 − ln Ψ + O(1), 2 2 λ−1 as |ξ |t 2 → +∞. Proof. Using (14.11) and (14.10) we deduce    √   2i Z c 2i Z c i Q π4  1 + ln(t) − ln + ln 2 π e Ψ (ξ, Z c , t) = − |ξ | λ−1 λ−1  2 i Zc 1 + O(1) − ln 2 λ−1   √ 2 i Zc 1  1/2  1 + O(1) − ln = − |ξ | 2π t (1 + i Q) − ln |ξ | 2 2 λ−1 (14.18) as |ξ |t 2 → +∞.



Linearized Homogeneous Coagulation Equation

811

Remark 14.7. Notice that the sign of the real part of the main term in the asymptotic  as t 2 |ξ | → ∞ is essential in these arguments. expansion of Ψ (ξ, Z , t) is “well behaved” in a region In the next lemma we prove that the function Ψ |Z − Z c | ≥ εt and |Z | ≤ M t for M > 0 large. Lemma 14.8. For all ε > 0 and M > 0 large, there exists δ0 > 0 and R > 0 such that (ξ, Z , t) satifies: the function Ψ  (ξ, Z , t) ≤ Re Ψ (ξ, Z c , t) − δ0 |ξ | t Re Ψ when Z lies in the curve γt (M) = γ1,t (M) ∪ γ2,t (M) ∪ γ3,t (M)\ {Z ; |Z − Z c | ≤ ε t} , where γ1,t (M) = {Z ; Z = Z c + λ, λ ∈ R, |λ| ≤ M t} , $

# γ2,t (M) = Z ; Z = Z c + M t + λ i, λ ∈ 0, |I m(Z c )| + γ1 , $

# γ3,t (M) = Z ; Z = Z c − M t + λ i, λ ∈ 0, |I m(Z c )| + γ1

(14.19) (14.20) (14.21)

for all ξ and t such that |ξ | t 2 > R. Proof. Let us consider the auxiliary function: √  Θ(ξ, Ω, t) = −2 π |ξ | Ω t [1 − ln(Ω) + ln(Ω0 )] , 1 iZ Ω= √ , λ−1 πt π

Ω0 = ei Q 4 .

(14.22) (14.23) (14.24)

By Lemma 14.1: (ξ, Z , t) = Θ(ξ, Ω, t) − Ψ

1  1/2  ln |ξ | + h(ξ, Z , t). 2

Moreover, it is easily checked that Ω0 is the critical point of the function Θ(ξ, Ω, t). , conBy Lemma 14.3 we already know that Z c , the critical point of the function Ψ verges to Ω0 as |ξ | t 2 → +∞. We now study the behaviour of the function Θ along the curve obtained from γ (M) using the change of variable (14.23). Due to the conver and its critical point Z c when |ξ | t 2 → +∞ this will gence properties of the function Ψ be enough in order to prove the statement in Lemma 14.8. We first consider the curve corresponding to γ1,t (M). It is then enough to consider Re Θ(ξ, Ω, t) along the points: Ω = Ω0 + √λ i, λ ∈ R. A straightforward calculation yields: 2

√  Re Θ(ξ, Ω, t) = − 2 π |ξ | t ψ(σ ), σ = 1 + Qλ π   1 1  ψ(σ ) = 1 − ln 1 + σ 2 + ln 2 − σ − ar ctgσ . 2 2 4

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Since ψ (σ ) = − π4 + ar ctg(σ ) the point σ0 = 1 is a strict minimum for the function ψ. It is also easily checked that π as σ → +∞, 4 3π ψ(σ ) ∼ − as σ → −∞. 4 ψ(σ ) ∼

It follows that, for any ε > 0 there exists δ0 > 0 such that if |σ − 1| > ε, ψ(σ ) − ψ(1) ≥ δ0 . Arguing by continuity this yields the statement of the lemma when Z lies in the curve γ1,t (M). The evolution of the function θ along the two curves corresponding to γ2,t (M) and γ3,t (M) is studied with very similar arguments. We then only consider the case of the curve γ2,t (M). It is then enough to consider Re Θ(ξ, Ω, t) along the points: Ω = r ei ϕ ,   I m r ei ϕ − Ω0 = ±

M . π(λ − 1)

A straightforward calculation gives:    √  πQ −ϕ . Re Θ(ξ, Ω, t) = −2 π |ξ | Ω t r cos ϕ(1 − ln r ) − r sin ϕ 4 For Z ∈ γ2,t (M) and the constant M sufficiently large, we have that ϕ > π/4 + δ, r cos ϕ ≤ 2 and r > M/2. Similarly, if Z ∈ γ3,t (M) and the constant is M sufficiently large, we have that ϕ < −π/4 − δ, r cos ϕ ≤ 2 and r > M/2. Therefore, we have in both cases:  πQ − ϕ > δ0 > 0, r cos ϕ(1 − ln r ) − r sin ϕ 4 for M large enough.  and its critical point Z c Using again the convergence properties of the function Ψ 2 when |ξ | t → +∞ this yields the statement in Lemma 14.8 when Z lies in the curves γ2,t (M), γ3,t (M).

 In the following Lemma √ we extend the behaviour of Ψ (ξ, Z , t) to the region |Z − Z c | ≥ εt and |Z | ≤ δ1 |ξ | for some δ1 > 0 sufficiently small. Lemma 14.9. For all M > 0 large, there exists a > 0, ε1 > 0 and R > 0 such that the (ξ, Z , t) satifies: function Ψ (ξ, Z , t) ≤ −a Re Ψ

 |ξ | |Z |,

√ for all Z ∈ C1 such that M t ≤ |Z | ≤ ε1 |ξ | and all ξ and t such that |ξ | t 2 > R.

Linearized Homogeneous Coagulation Equation

813

Proof. We only need to check that the function Θ defined in the proof of the previous lemma behaves linearly when |Ω| → +∞ and Re(Ω) remains constant. This follows from Re(Θ(ξ, Ω, t)) ≤ −

π 3/2  |ξ | t |Ω| 2

uniformly for Ω = i |Ω| + O(1). Using (14.23) we deduce Re(Θ(ξ, Ω, t)) ≤ −

|Z | π |ξ | 2 λ−1

for all Z ∈ C1 such that |Z | ≥ M t assuming that M > 0 is sufficiently large. Using Lemma 14.1 we have:   1 (ξ, Z , t) ≤ Re(Θ(ξ, Ω, t)) + C Z 2 + O Re Ψ |ξ |   π |Z | 1 + C Z2 + O ≤− |ξ | 2 λ−1 |ξ |    π 1 − ε1 + O ≤ − |ξ | |Z | 2(λ − 1) |ξ |3/2 t √ for all Z ∈ C1 such that M t ≤ |Z | ≤ ε1 |ξ |. The result follows for ε1 small enough and |ξ |1/2 t → +∞.

Lemma 14.10. For all B > 0 there exists ξ0 and C > 0 such that    ∂ Ψ  |Y |    ∂ξ  (ξ, Y, t) ≤ C |ξ | ,  = 1, 2 √ for Y = Z |ξ |, |Y | ≤ |ξ8| , Z ∈ D(ξ, B) and |Re(ξ )| > ξ0 . Proof. Let Ψ be given: Ψ (ξ, Y, t) =

 2 ln (−Φ(η)) Θ(η − ξ, Y )dη (λ − 1) i I m η=β1   2iY 2iY 2iY 1 2iY − ln(t) − + − ln . λ−1 λ−1 λ−1 2 λ−1

Differentiating with respect to ξ we obtain:  2 ∂Ψ ∂Θ (ξ, Y, t) = (η − ξ, Y )dη ln (−Φ(η)) ∂ξ (λ − 1) i I m η=β1 ∂ξ  2 Φ (η) = Θ(η − ξ, Y )dη. (λ − 1) i I m η=β1 Φ(η) By Proposition 4.1 we have    Φ (η)  C    Φ(η)  ≤ 1 + |η|

(14.25)

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√ in the domain Y/ |ξ | ∈ D(ξ, B). We deform the contour of integration to C2 defined in Fig. 4. Then we split the integral in two pieces:  1 ∂Ψ | (ξ, Y, t)| ≤ C |Θ(η − ξ, Y )||dη| |ξ | 1 + |η| ∂ξ η∈C2 ,|η−ξ |≤ 4  1 |Θ(η − ξ, Y )||dη| = J1 + J2 . +C |ξ | 1 + |η| η∈C2 ,|η−ξ |≥ 4 By the exponential decay of the function Θ: J2 ≤ Ce−a|ξ | for some positive constant a. On the other hand, Θ(η − ξ, Y ) =

1 1−e

4π − λ−1 (η−ξ )

−

1 1−e

4π λ−1 (−(η−ξ )+Y )

.

The integral J1 is then divided in two parts. The first, J1,1 is the integral along the “vertical part of the curve” C2 . The second, J1,2 is along the horizontal part of that curve, where I m(η) = β1 . In the integral J1,1 , Re(η) is bounded and therefore |Θ(η − ξ, Y )| ≤ C. Since the total length of the integration curve of J1,1 is of order |Y | we deduce that J1,1 ≤ C|Y |/(1 + |ξ |). We split the integral J1,2 as follows:  C J1,2 ≤ |Θ(η − ξ, Y )|dη 1 + |ξ | I m(η)=β1 ,|η−ξ |≤2 |Y |  + |Θ(η − ξ, Y )|dη , I m(η)=β1 ,|η−ξ |≥2 |Y |  |Θ(η − ξ, Y )|dη I m(η)=β1 ,|η−ξ |≥2 |Y |

  4π 4π   e− λ−1 σ − e λ−1 (Y −σ )   ≤  dη.  4π 4π − σ (Y −σ )  I m(η)=β1 ,|σ |≥2 |Y | (1 − e λ−1 )(1 − e λ−1 ) 

We use now that, if Re(σ ) > 2 |Y |:   4π 4π   2π e− λ−1 σ − e λ−1 (Y −σ )     ≤ Ce− λ−1 |σ | , 4π 4π − σ (Y −σ )  (1 − e λ−1 )(1 − e λ−1 ) and if Re(σ ) < −2|Y |,     4π 4π − λ−1 σ (Y −σ ) λ−1   4π 6π e −e Y − λ−1 |σ |   ≤ Ce λ−1    e . 4π 4π   − σ (Y −σ )  1 − e λ−1  1 − e λ−1 The last remaining term is easily estimated by:   |Θ(η − ξ, Y )||dη| ≤ C I m(η)=β1 ,|η−ξ |≤2 |Y |

I m(η)=β1 ,|η−ξ |≤2 |Y |

|dη| ≤ C|Y |.

Linearized Homogeneous Coagulation Equation

815

It then follows that J1,2 ≤ Ce−a|Y | /(1 + |ξ |) for positive constant C and a. This ends the proof of (14.25) for  = 1. Similarly,   Φ (η) ∂ 2Ψ 2 (ξ, Y, t) = Θ(η − ξ, Y )dη ∂ξ 2 (λ − 1) i I m η=β1 Φ(η) with

   Φ (η)  C   ≤   Φ(η)  (1 + |η|)2

√ for Y/ |ξ | ∈ D(ξ, B), again by Proposition 4.1. The proof of (14.25) follows then from the same arguments as those of the proof of (14.25) for  = 2.

Acknowledgements. Both authors acknowledge the hospitality and support of the MPI for Mathematics in the Sciences (Leipzig) where this research was begun. JJLV is supported by Grant MTM2007-61755. He thanks Universidad Complutense for its hospitality and also thanks the Humboldt Foundation for support. M.E. is supported by Grants MTM2008-03541 and IT-305-07.

References 1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1972 2. Balk, A.M., Zakharov, V.E.: Stability of Weak-Turbulence Kolmogorov Spectra. In: Zakharov, V.E. (ed.), Nonlinear Waves and Weak Turbulence, A. M. S. Translations Series 2, Vol. 182, Providence, RI: Amer. Math. Soc., 1998, pp. 1–81 3. Carr, J., da Costa, F.P.: Instantaneous gelation in coagulation dynamics. Z. Angew. Math. Phys. 43, 974–983 (1992) 4. van Dongen, P.G.J., Ernst, M.H.: Cluster size distribution in irreversible aggregation at large times. J. Phys. A 18, 2779–2793 (1985) 5. van Dongen, P.G.J., Ernst, M.H.: Scaling solutions of Smoluchowski’s coagulation equation. J. Stat. Phys. 50, 295–329 (1988) 6. van Dongen, P.G.J.: On the possible occurrence of instantaneous gelation in Smoluchowski’s coagulation equation. J. Phys. A: Math. Gen. 20, 1889–1904 (1987) 7. Dubovski, P.B., Stewart, I.W.: Existence, Uniqueness and Mass Conservation for the CoagulationFragmentation Equation. Math. Meth. Appl. Sciences 19, 571–591 (1996) 8. Ernst, M.H., Ziff, R.M., Hendriks, E.M.: Coagulation processes with a phase transition. J. Colloid and Interface Sci. 97, 266–277 (1984) 9. Escobedo, M., Mischler, S., Perthame, B.: Gelation in coagulation and fragmentation models. Commun. Math. Phys. 231(1), 157–188 (2002) 10. Escobedo, M., Mischler, S., Velázquez, J.J.L.: On the fundamental solution of the linearized UehlingUhlenbeck equation. Arch. Rat. Mech. Anal. 186, 309–349 (2007) 11. Escobedo, M., Mischler, S., Velázquez, J.J.L.: Singular Solutions for the Uehling Uhlenbeck Equation. Proc. Roy. Soc. Edinburgh 138A, 67–107 (2008) 12. Escobedo, M., Velázquez, J.J.L.: On a linearized coagulation equation. In preparation 13. Leyvraz, F.: Scaling Theory and Exactly Solved Models in the Kinetics of Irreversible Aggregation. Phys. Repts. 383(2–3), 95–212 (2003) 14. Leyvraz, F., Tschudi, H.R.: Singularities in the kinetics of coagulation processes. J. Phys. A 14, 3389– 3405 (1981) 15. Makino, J., Fukushige, T., Funato, Y., Kokubo, E.: On the mass distribution of planetesimals in the early runaway stage. New Astronomy 3, 411–417 (1998) 16. Muskhelishvili, N.I.: Singular Integral Equations. Translated from second edition, Moscow (1946) by J.R.M. Radok. Groningen: Noordhof, 1953 17. Noble, B.: Methods based on the Wiener-Hopf Technique. Second edition, New York: Chelsea Publishing Company, 1988 18. Stewart, I.W.: On the coagulation-fragmentation equation. Z. Angew. Math. Phys. 41, 917–924 (1990) 19. Stewart, I.W.: Density conservation for a coagulation equation. Z. Angew. Math. Phys. 42, 746–756 (1991)

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20. Tanaka, H., Inaba, S., Nakaza, K.: Steady-state size distribution for the self-similar collision cascade. Icarus 123, 450–455 (1996) 21. Wagner, W.: Post-gelation behaviour of a spatial coagulation model. Electronic J. Prob. 11, 893–933 (2006) Communicated by A. Kupiainen

Commun. Math. Phys. 297, 817–839 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1068-x

Communications in

Mathematical Physics

Inverse Hyperbolic Problems and Optical Black Holes G. Eskin Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. E-mail: [email protected] Received: 3 March 2009 / Accepted: 16 March 2010 Published online: 23 May 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com

To the memory of Leonid Romanovich Volevich Abstract: In this paper we state a uniqueness theorem for the inverse hyperbolic problem in the case of a finite time interval. We apply this theorem to the inverse problem for the equation of the propagation of light in a moving medium (the Gordon equation). Then we study the existence of black and white holes for the general second order hyperbolic equation and for the Gordon equation and we discuss the impact of this phenomenon on the inverse problems. 1. The Introduction and the Statement of the Main Theorem We start with the formulation of the inverse problem. Let  be a smooth bounded domain in Rn . Consider a second order hyperbolic equation in a cylinder  × (−∞, +∞): n 

1 ∂ √ n g(x) ∂ x (−1) j j,k=0

 

(−1)n g(x)g jk (x)

∂u(x0 , x) ∂ xk

 = 0,

(1.1)

where x = (x1 , . . . , xn ) ∈ , x0 ∈ R is the time variable, g(x) = (det[g jk (x)]nj,k=0 )−1 . We assume that g jk (x) = g k j (x) are real-valued smooth functions in C ∞ () independent of x0 .  The hyperbolicity of (1.1) means that the quadratic form nj,k=0 g jk (x)ξ j ξk has one positive and n negative eigenvalues, ∀x ∈ . We assume in addition that g 00 (x) > 0, x ∈ ,

(1.2)

i.e. (1, 0, . . . , 0) is not a characteristic direction, and that n  j,k=1

g jk (x)ξ j ξk < 0

for ∀(ξ1 , . . . , ξn ) = (0, . . . , 0), ∀x ∈ ,

(1.3)

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G. Eskin

i.e. the quadratic form (1.3) is negative definite. Note that (1.3) is equivalent to the condition that any direction (0, ξ ) is not characteristic for ∀x ∈ .

(1.4)

We shall give later another equivalent characterization of the condition (1.3). We shall study the initial-boundary value problem for Eq. (1.1) in  × R, u(x0 , x) = 0 for x0 << 0, x ∈ , u|∂×R = f,

(1.5) (1.6)

where f (x0 , x  ) has compact support in ∂ × R. Consider the Dirichlet-to-Neumann (DN) operator ⎞− 1  2  ∂u  jk pr ⎝ ⎠ f = g (x) νk (x) g (x)ν p νr   ∂x j p,r =1 j,k=0  ⎛

n 

n 

,

(1.7)

∂×R

where ν0 = 0, (ν1 , . . . , νn ) is the unit outward normal to ∂ ⊂ Rn with respect to the Euclidean metric, u(x0 , x) is the solution of (1.1), (1.5), (1.6). Let 0 ⊂ ∂ be an open subset of ∂. We shall say that the DN operator  is given on 0 × (0, T0 ) if we know  f |0 ×(0,T0 ) for any smooth f with support in  0 × [0, T0 ]. Let X be a closed compact subset of ∂ × (−∞, +∞). For each distribution f on ∂ × (−∞, +∞), supp f ⊂ X , denote by D+ ( f ) the support in  × (−∞, +∞) of the solution of the initial-boundary value problem for (1.1) with u = 0 for x0 << 0, x ∈ , u|∂×(−∞,+∞) = f . We define the forward domain of influence of X as the closure of the set ∪supp f ⊂X D+ ( f ), where the union is taken over all f on ∂ × (−∞, +∞) with supports in X . Analogously, let D− ( f ) be the support in  × (−∞, ∞) of the solution of the initial boundary value problem for (1.1) with u = 0 for x0 >> 0, x ∈ , u|∂×(−∞,+∞) = f . Then D− (X ) is the closure of the union of all D− ( f ) with supp f ⊂ X . We shall give a geometrical description of the set D+ (X ). Let [g jk (x)]nj,k=0 = ([g jk ]nj,k=0 )−1 , i.e. [g jk (x)]nj,k=0 is the pseudo-Riemannian metric tensor. We say that γ (s) = (x0 (s), x1 (s), . . . , xn (s)), s ∈ [0, s0 ] is a forward time-like path (ray) if γ (s) is continuous and piece-wise smooth and each smooth segment γ (i) (s) of γ (s) satisfies n  j,k=0

g jk (x(s))

d x j d xk > 0, si ≤ s ≤ si+1 , ds ds

(1.8)

d x0 ds

> 0. Then D+ (X ) is the closure in  × (−∞, +∞) of all time-like rays starting on X . Note that the vertical ray x0 = s, x = x (0) , s ≥ 0, is time-like iff g00 (x (0) ) > 0. Proposition 1.1. Condition (1.3) holds iff g00 (x) > 0, i.e. the ray x0 = s, x ∈ , s ≥ 0, is time-like.

Inverse Hyperbolic Problems and Optical Black Holes

819

Proof. We have g00 (x) =

det[g jk ]nj,k=1 g −1 (x)

.

Note that ⎞2 ⎛ n 0 j  g (x) ⎠ g jk (x)ξ j ξk = ⎝ g 00 (x)ξ0 + ξj √ g00 (x) j,k=0 j=1 ⎛ ⎞2 n n 0 j (x)   g −⎝ g jk (x)ξ j ξk . ξj⎠ + √ g (x) 00 j=1 j,k=1 n 

Since the quadratic form quadratic form

n j,k=0

g jk ξ j ξk has the signature (+1, −1, . . . , −1) the

⎛

⎞2 n n j0 (x)   g −⎝ g jk (x)ξ j ξk ξj⎠ + √ g (x) 00 j=1 j,k=1  is negative definite. Therefore nj,k=1 g jk (x)ξ j ξk either has only negative eigenvalues, or it has one zero eigenvalue, or it has one positive eigenvalue. In the first case sgn(det[g jk ]nj,k=1 ) = (−1)n , in the second case det[g jk (x)]nj,k=1 = 0, and in the third case sgn(det[g jk ]nj,k=1 ) = (−1)n−1 . Since sgn g(x) = (−1)n we get that g00 (x) > 0 iff condition (1.3) holds.

Let T+ be the smallest number such that D+ ( 0 × {x0 = 0}) ⊃  × {x0 = T+ }. Analogously let T− be the smallest number such that D− ( 0 × {x0 = T− }) ⊃  × {x0 = 0}. We shall require that T0 > T+ + T− .

(1.9)

Consider the following change of variables in  × (−∞, +∞): y0 = x0 + a(x), y = ϕ(x),

(1.10)

˜ a(x) ∈ C ∞ (), where y = ϕ(x) is a diffeomorphism of  onto , ϕ(x) = I on  0 ⊂ ∂, a(x) = 0 on  0 .

(1.11)

Note that if u(y ˜ 0 , y) = u(x0 , x), where (y0 , y) and (x0 , x) are related by (1.10), (1.11), then u(y ˜ 0 , y) satisfies an equation of the form n  j,k=0

∂  n ∂yj (−1) g(y) ˜ 1

  ∂ u(y ˜ 0 , y) jk n =0 (−1) g(y) ˜ g˜ (y) ∂ yk

(1.12)

820

G. Eskin

˜ × (−∞, +∞) with initial and boundary conditions in  ˜ u(y ˜ 0 , y) = 0 for y0 << 0, y ∈ ,  u| ˜ ∂ ×R = f (y0 , y ), ˜

(1.13) (1.14)

where f (x0 , x  ) = f (y0 , y  ) on 0 × (−∞, +∞). ˜ be the DN operator on 0 ×(−∞, +∞) corresponding to (1.12), (1.13), (1.14). Let  The following theorem holds (cf. [E1], Theorem 2.3) Theorem 1.1. Let (1.1), (1.5), (1.6) and (1.12), (1.13), (1.14) be two hyperbolic ˜ × R, respectively, and ∂ ∩ ∂  ˜ ⊃ 0 . initial-boundary value problems in  × R and  Suppose conditions (1.2) and (1.3) are satisfied for (1.1) and (1.12). Suppose supp f ⊂ ˜ are equal on 0 × (0, T0 )  0 × [0, T0 ] in (1.6) and (1.14). If the DN operators  and  and if T0 > T+ + T− (cf. (1.9)) then there exists a change of variables (1.10), (1.11) such that [g˜ jk (y)]nj,k=0 = J (x)[g jk (x)]nj,k=0 J T (x), where y0 = x0 + a(x), y = ϕ(x), J (x) =

1 ax (x) 0 DDϕ(x) x

(1.15)

 (1.16)

is the Jacobi matrix of (1.10). Note that condition (1.9) is required for (1.1) only. The condition (1.15) is equivalent to [g˜ jk (y)]−1 = (J T (x))−1 [g jk (x)]−1 J −1 (x), or [g jk (x)]nj,k=0 = J T (x)[g˜ jk (y)]nj,k=0 J (x).

(1.17)

The equality (1.17) can be rewritten as the equality of differential forms n  j,k=0

g jk (x)d x j d xk =

n 

g˜ jk (y)dy j dyk .

(1.18)

j,k=1

Theorem 1.1 is a refinement of Theorem 2.3 in [E1] when the time-interval (−∞, +∞) is replaced by (0, T0 ), and it does not require any changes in the proof of Theorem 2.3 in [E1] (see also [E2]). Now we shall describe the content of the rest of the paper. In Sect. 2 we apply Theorem 1.1 to the inverse problem for the equation of the propagation of light in a moving medium (the Gordon equation). In Sects. 3 and 4 we find the conditions for the existence of the black or white holes for the equation of the form (1.1). Note that the black or white holes in § 4 are stable, i.e. they change only slightly when the metric is slightly changed. The phenomenon of black holes for the wave equation (1.1) attracted the attention of physicists (see, for example, the survey of M.Visser [V], and the book “Artificial black holes” by M.Novello, M.Visser and G.Volovik [NVV]). The physicists were interested in studying these black holes (they are called either artificial black holes, or acoustic

Inverse Hyperbolic Problems and Optical Black Holes

821

black holes, or optical black holes) as a “laboratory” for the black holes arising in the Einstein equations of general relativity. The relation between the inverse problems in §§ 1–2 and the black holes in §§ 3–4 is the following: Black or white holes may appear only when the condition (1.3) in the uniqueness Theorem 1.1 is violated. The existence of black or white holes leads to the non-uniqueness of the solution of the inverse problem. 2. The Inverse Problem for the Gordon Equation Consider the equation of the propagation of light in a moving medium. Let w(x) = (w1 (x), w2 (x), w3 (x)) be the velocity of flow and let (v (0) , v (1) , v (2) , v (3) ) be the corresponding four-velocity vector v

(0)

− 21 − 21   w j (x) |w|2 |w|2 ( j) = 1− 2 , v = 1− 2 , 1 ≤ j ≤ 3, c c c

where c is the speed of light in the vacuum (cf. [G,NVV,LP]). The equation for the propagation of light was found in 1923 by Gordon (cf. [G]) and it has the form (1.1) when g jk (x), 0 ≤ j, k ≤ n, n = 3, are g jk (x) = η jk + (n 2 (x) − 1)v j (x)v k (x),

(2.1)

[η jk ] = [η jk ]−1 is the Lorenz metric tensor: η jk = 0 when j = k, η00 = 1, η j j = −1, √ 1 ≤ j ≤ 3, n(x) = ε(x)μ(x) is the refraction index. Note that [g jk (x)] = [g jk ]−1 has the form (cf. [LP]): g jk (x) = η jk + (n −2 (x) − 1)v j vk , 0 ≤ j, k ≤ n,

(2.2)

where v0 = v 0 , v j = −v j , 1 ≤ j ≤ n. The metric (2.2) is called the Gordon metric and the corresponding equation of the form (1.1) is called the Gordon equation. We shall consider the Gordon equation in  × (−∞, +∞), where  has the form  = 0 \ ∪mj=1  j , 0 is diffeomorphic to a ball,  j are smooth domains,  j ∩ k = ∅ when j = k and ∪mj=1  j ⊂ . Domains  j are called obstacles. We shall study the following initial-boundary value problem for the Gordon equation: u(x0 , x) = 0 for x0 << 0, x ∈ , u(x0 , x)|∂ j ×R = 0, 1 ≤ j ≤ m, u(x0 , x)|∂0 ×R = f (x0 , x),

(2.3) (2.4) (2.5)

i.e. 0 = ∂0 , where 0 is the same as in Theorem 1.1. One can consider also the case when on some or all obstacles  j the zero Dirichlet boundary condition is replaced by the zero Neumann boundary condition. Note that the condition (1.2) always holds for the Gordon equation g 00 = 1 + (n 2 − 1)(v 0 )2 > 0.

(2.6)

The condition (1.3) holds iff |w(x)|2 <

c2 n 2 (x)

.

(2.7)

822

G. Eskin

To prove (2.7) note that the principal symbol of the Gordon equation is ⎛ ξ02 − |ξ |2 + (n 2 − 1) ⎝

n 

⎞2 v jξj⎠ .

j=0

Therefore the quadratic form ⎛ − |ξ |2 + (n 2 − 1) ⎝

n 

⎞2 v jξj⎠

(2.8)

j=1

 is negative definite iff (n 2 −1)|v|2 < 1. Since v 0 = 1 − wj c

|w|2 c2

− 1 2

 , vj = 1 −

|w|2 c2

− 1 2

, 1 ≤ j ≤ n, we get (2.7). We shall study the inverse problem for the Gordon equation. The case of slowly moving medium (cf. [LP]) was considered in [E1]. The Dirichlet-to-Neumann operator corresponding to the Gordon equation has the form (cf. (1.7)):  n n    jk 2 j k ∂u 2 r 2 − 21  f = [η + (n (x) − 1)v v ] νk | − 1 + (n − 1)( v νr ) |  ,  ∂x j r =1

j,k=0

∂0 ×R

where ν(x) = (ν1 , . . . , νn ) is the unit outward normal vector to ∂0 , ν0 = 0, u |∂0 ×R = f . We shall impose some restriction on the flow w(x) = (w1 (x), . . . , wn (x)). Let x = x(s), s ∈ [0, s0 ] be a trajectory of the flow, i.e. ddsx = w(x(s)), w(x(s)) = 0, s ∈ [0, s0 ]. Consider the trajectories that either start and end on ∂0 or are closed curves in . Our assumption is that The union of all such trajectories is a dense set in .

(2.9)

Theorem 2.1. Let [g jk (x)]nj,k=0 and [g˜ jk (y)]nj,k=0 be two Gordon metrics in domains ˜ respectively, and ∂0 = ∂  ˜ 0 . Consider two initial-boundary value problems  and , (1.1), (1.5), (1.6) and (1.12), (1.13), (1.14) corresponding to the metrics [g jk (x)]nj,k=0 and [g˜ jk (y)]nj,k=0 , respectively. Assume that the condition (2.7) holds for both metrics and assume that the refraction coefficients n 2 (x) and n˜ 2 (y) are constant. Assume also ˜ correspondthat the flow w(x) satisfies the condition (2.9). If the DN operators  and  ing to [g jk (x)]nj,k=0 and [g˜ jk (y)]nj,k=0 are equal on ∂0 × (0, T0 ), where T0 satisfies (1.9) for Eq. (1.1), then we have ˜ n 2 = n˜ 2 ,  = , and the flows w(x) and w(x) ˜ are equal. Proof. Analogously to Remark 2.2 in [E1] we canfind the symbol of the DN operator ˜ on ∂0 × (0, T0 ) in the “elliptic” region (cf. [E1]) and retrieve n 2 (x)∂ . Since  =  0   we get that n 2 (x)∂ = n˜ 2 (x)∂ . Therefore n 2 = n˜ 2 in 0 since we assume that n 2 0 0 and n˜ 2 do not depend on x. Applying Theorem 1.1 to the case of the Gordon equation we

Inverse Hyperbolic Problems and Optical Black Holes

823

get that there exists a change of variables (1.10) such that a(x) = 0 on ∂0 , ϕ(x) = I on ∂0 and (1.15), (1.18) hold where g jk (x) = η jk + (n 2 − 1)v j (x)v k (x), 0 ≤ j, k ≤ n, g˜ jk (y) = η jk + (n 2 − 1)v˜ j (y)v˜ k (y), 0 ≤ j, k ≤ n.

(2.10)

We have from (1.10) dy0 = d x0 +

n 

axk (x)d xk ,

k=1 n 

dy j =

(2.11)

ϕ j xk (x)d xk , 1 ≤ k ≤ n.

k=1

Substituting (2.11) into (1.18) and taking into account that d x0 , d x1 , . . . , d xn are arbitrary we get g00 (x) = g˜ 00 (y). Therefore (cf. (2.2)) 1 + (n −2 − 1)v02 (x) = 1 + (n˜ −2 − 1)v˜02 (y).

(2.12)

Note that (1.15) is equivalent to g˜ jk (y) =

n 

g pr (x)ϕ j x p ϕkxr ,

(2.13)

p,r =0

where ϕ0 = x0 + a(x). In particular, g˜ 00 (y) = g 00 (x) + 2

n 

g p0 (x)ax p (x) +

n 

g pr (x)ax p axr (x).

(2.14)

p,r =1

p=1

In the case of Gordon metrics we have 1 + (n˜ 2 − 1)(v˜ 0 (y))2 = 1 + (n 2 − 1)(v 0 (x))2 + 2

n 

(n 2 − 1)v 0 (x)v p (x)ax p (x)

p=1

−

n 

(ax x p (x))2 +

n 

(n 2 − 1)v p (x)vr (x)ax p (x)axr (x).

(2.15)

p,r =1

p=1

Denote |∇a(x)|2 =

n 

ax2p (x), a0 (x) =

p=1

n 1 w p (x)ax p (x). c p=1

Since n = n˜ and since (2.12) implies that v 0 = v˜ 0 we get a02 + 2a0 (x) −

|∇a|2 = 0. − 1)

(v 0 )2 (n 2

(2.16)

824

G. Eskin

It follows from (2.16) that either  a0 = −1 −

|∇a|2 , − 1)

(2.17)

|∇a|2 . − 1)

(2.18)

1+

(v 0 )2 (n 2

1+

(v 0 )2 (n 2

or  a0 = −1 +

Let x = x(s) be a trajectory of the flow, i.e. d xk = wk (x(s)), 1 ≤ k ≤ n, 0 ≤ s ≤ 1. ds Then da(x(s))  d xk = = ca0 (x(s)), axk (x(s)) ds ds n

k=1

i.e. ca0 (x(s)) is the derivative of a(x(s)) along the trajectory of the flow. Suppose x = x(s), 0 ≤ s ≤ 1, is a trajectory that starts and ends on ∂0 . Therefore a(x(0)) = a(x(1)) = 0 since a(x) = 0 on ∂0 . Then a0 (x(s)) can not satisfy (2.17) since da(x(s)) = ca0 (x(s)) < 0 on [0, 1]. Therefore da(x(s)) = ca0 (x(s)) satisfies ds ds da(x(s)) ≥ 0 on [0, 1]. Since a(x(0)) = a(x(1)) = 0 we must have Eq. (2.18), i.e. ds a(x(s)) = 0 and ∇a(x(s)) = 0 on [0, 1]. In the case when x = x(s), 0 ≤ s ≤ 1, is a <0 closed trajectory, i.e. x(0) = x(1), a(x(s)) can not again satisfy (2.17) since da(x(s)) ds on [0, 1] and a(x(s)) satisfies (2.18) only when ∇a(x(s)) = 0 and a(x(s)) = const on [0, 1]. Since the condition (2.9) holds we have ∇a(x) = 0 in , and since a|∂0 = 0, we get that a = 0 in . Applying (2.13) for k = 0, 1 ≤ j ≤ n, and for 1 ≤ j, k ≤ n and taking into account that a ≡ 0 we get (n˜ − 1)v˜ (y)v˜ (y) = (n − 1) 2

0

j

2

n 

v 0 (x)v p (x)ϕ j x p (x),

(2.19)

p=1

η

jk

+ (n˜ − 1)v˜ (y)v˜ (y) = − 2

j

k

+

n 

ϕ j x p ϕkx p

p=1 n 

(2.20)

(n 2 − 1)v p (x)vr (x)ϕ j x p ϕkxr , 1 ≤ j, k ≤ n.

p,r =1

Using again that n˜ = n and v 0 = v˜ 0 we get from (2.19) v˜ j (y) =

n  p=1

v p (x)ϕ j x p (x), 1 ≤ j ≤ n, v˜ 0 (y) = v 0 (x).

(2.21)

Inverse Hyperbolic Problems and Optical Black Holes

825

Substituting (2.21) into (2.20) we obtain η jk = −

n 

ϕ j x p (x)ϕkx p (x), 1 ≤ j, k ≤ n.

(2.22)

ϕ 2j x p (x) = 1, ϕ j |∂0 = x j , 1 ≤ j ≤ n.

(2.23)

p=1

In particular, we have n  p=1

The Cauchy problem (2.23) has a unique solution in  : ϕ j = x j , 1 ≤ j ≤ n. Therefore ˜ = . Also ϕ j (x) = x j implies (cf. (2.21)) that y = ϕ(x) = x in . This implies that  v˜ j (x) = v j (x), 1 ≤ j ≤ n. Therefore w˜ j (x) = w j (x), 1 ≤ j ≤ n.

3. Optical Black Holes In this section we explore the situation when the condition (1.3) is not satisfied. Denote by S the surface in  given by the equation (x) = 0, where (x) = det[g jk (x)]nj,k=1 .

(3.1)

We assume that S is a smooth and closed surface. Denote by ext the exterior of S and by int the interior of S. We assume that > 0 in ext ∩ , < 0 in int near S. It follows from Proposition 1.1 that the equation of S can be written in the form g00 (x) = 0 and g00 (x) > 0 in ext ∩ , g00 < 0 in int near g00 (x) = 0. We shall write often the equation of S in the form S(x) = 0 and we assume that ∂ S(x) ∂ x is an outward normal to the surface S(x) = 0. In the case of the Gordon metric the equation of S has the form (n 2 (x) − 1)

n  (v j (x))2 − 1 = 0, j=1

or, equivalently, n 

w 2j (x) −

j=1

c2 = 0, n 2 (x)

(3.2)

and the domain int is given by the inequality n  j=1

w 2j (x) >

c2 n 2 (x)

near S(x) = 0. Suppose that S(x) = 0 is a characteristic surface of Eq. (1.1), i.e. n  j,k=1

g jk (x)Sx j (x)Sxk (x) = 0 when S(x) = 0.

(3.3)

826

G. Eskin

Let x j = x j (s), ξ j = ξ j (s), 0 ≤ j ≤ n, s ≥ 0 be a null-bicharacteristic of (1.1), i.e. n  dx j g jk (x(s)ξ j (s), x j (0) = y j , 0 ≤ j ≤ n, =2 ds

(3.4)

dξ p =− ds

(3.5)

k=0 n 

jk

gx p (x(s))ξ j (s)ξk (s), ξ p (0) = η p , 0 ≤ p ≤ n,

j,k=0

where x(s) = (x1 (s), . . . , xn (s)), η0 = 0, η = 0. The null-bicharacteristic means that n 

g jk (x(s))ξ j (s)ξk (s) = 0.

(3.6)

j,k=0

Note that if n 

g jk (y)η j ηk = 0,

(3.7)

j,k=0

then (3.6) holds for all s ∈ R. The bicharacteristic (null-bicharacteristic) is a curve in T0 (Rn+1 ) = Rn+1 × (Rn+1 \ {0}) and its projection on Rn+1 is called a geodesic (null-geodesic). It is easy to show that the null-geodesic satisfies the equation: n 

g jk (x(s))

j,k=0

d x j d xk = 0. ds ds

(3.8)

Denote by G the matrix [g jk (x)]nj,k=0 . Then G −1 = [g jk (x)]nj,k=0 . It follows from (3.4) that ddsx = 2G ξ (s), where x(s) = (x0 (s), x(s)), ξ = (ξ0 , ξ ). Then ξ = 21 G −1 ddsx and (3.6) implies     1 1  −1 d x  d x −1 d x −1 d x   GG ,G = G , , (3.9) 0 = (G ξ , ξ ) = 4 ds ds 4 ds ds that is equivalent to (3.8). Fix an arbitrary point y on S(x) = 0. Denote by K + (y) the following half-cone in n+1 R : K + (y) = {ξ = (ξ0 , ξ ) ∈ Rn+1 : (G(y)ξ , ξ ) ≥ 0, ξ0 ≥ 0}.

(3.10)

Let K + (y) be the dual half-cone in Rn+1 : ˙ x) ˙ ≥ 0, x˙0 > 0}. K + (y) = {x˙ = (x˙0 , x) ˙ ∈ Rn+1 : (G −1 x,

(3.11)

˙ ξ ) ≥ 0 for any x˙ ∈ K + (y) and any ξ ∈ K + (y). (x,

(3.12)

Note that

Consider the null-bicharacteristic ( x (s), ξ (s)) (cf. (3.4), (3.5), (3.6)) with the following initial conditions: x0 (0) = y0 , x(0) = y, ξ0 (0) = 0, ξ(0) = Sx (y),

(3.13)

Inverse Hyperbolic Problems and Optical Black Holes

827

where S(y) = 0. We have (cf. (3.4)):  d x0 (0) =2 g 0 j (y)Sx j (y). ds n

(3.14)

j=1

Proposition 3.1. Let (0, b(y)) be a characteristic direction, i.e. n 

g jk (y)b j (y)bk (y) = 0.

j=1

Suppose n 

g j0 (y)b j (y) > 0.

j=1

Then (0, b(y)) ∈  K + (y) and the half-cone K + (y) is contained in the half-space {(α0 , α1 , . . . , αn ) : nj=1 α j b j (y) ≥ 0}. + 00 2 Proof. n Let εj0 > 0 be small. Then (ε, b1 (y), b2 (y), . . . , bn (y)) ∈ K (y) since g (y)ε + 2ε j=1 g (y)b j (y) > 0 (cf. (3.10)). For any (x˙0 , x) ˙ ∈ K + (y) we have (cf. (3.12)):

x˙0 ε +

n 

x˙ j b j (y) ≥ 0,

j=1

 i.e. K + (y) is contained in the half-space εα0 + nj=1 α j b j (y) ≥ 0. Taking the limit when ε → 0 we prove Proposition 3.1 It follows from the hyperbolicity of (1.1) and from (3.3) that the  right hand side of (3.14) is not zero. In fact, the hyperbolicity implies that the equation nj,k=0 g jk (y)ξ j ξk = (1)

(2)

0 has two distinct real roots ξ0 (ξ ), ξ0 (ξ ) for any ξ = 0. Taking ξ = Sx (y) we  get g 00 (y)ξ02 + 2 nj=1 g 0i (y)ξ0 Sx j (y) = 0, i.e. ξ0(1) = 0, ξ0(2) = −2(g 00 (y))−1 n 0j

j=1 g (y)Sx j (y)  = 0. We assumed that Sx (x) is the outward normal to the S(x) = 0. Since S(x) satisfies (3.3) when S(x) = 0 we have that either n 

g 0 j (y)Sx j (y) > 0, ∀y such that S(y) = 0,

(3.15)

j=1

or n 

g 0 j (y)Sx j (y) < 0, S(y) = 0.

(3.16)

j=1

If (3.15) holds then by Proposition 3.1 (0, Sx (y)) ∈ K + (y), ∀y, S(y) = 0, and K + (y) is contained in the half-space (0, Sx (y)) · (x˙0 , x) ˙ ≥ 0. Note that the forward domain of influence of the point y, S(y) = 0, consists of all forward time-like rays and null-geodesics that have the direction (x˙0 , x) ˙ ∈ K + (y) at the

828

G. Eskin

point y. All directions of K + (y) except (1, 0, . . . , 0) are pointed inside ext × R. Therefore the forward domain of influence of the surface {S(x) = 0} × R does not intersect int × R. In such case the surface {S(x) = 0} × R is called the boundary of a white hole. Note that the forward domain of influence of ext × R does intersect int × R. Consider now the case when (3.16) holds. Then (cf. Proposition 3.1) the dual halfcone K + (y) is contained in the half-space x˙ · (0, −Sx (y)) ≥ 0 and all x˙ ∈ K + (y) (except (1, 0, . . . , 0)) are pointed inside int ×R. Therefore the domain of influence of int ×R is contained in int × R. The surface {S(x) = 0} × R is called the boundary of a black hole in this case. We proved the following theorem: Theorem 3.1. Let S(x) = det[g jk ]nj,k=1 = 0 be a closed and smooth characteristic surface, g00 (x) > 0 in ext ∩ , g00 (x) < 0 in int near S(x) = 0. Let (0, Sx (x)) be the outward normal to the surface {S(x) = 0} × R. Then {S(x) = 0} × R forms a white hole if (3.15) holds, and a black hole if (3.16) holds. The application of Theorem 3.1 to the Gordon equation yields the following result (cf. [V]):  2 Theorem 3.2. Let S(x) = 0 be the surface nj=1 w 2j (x) = n 2c(x) and let {S(x) = 0}×R be a characteristic surface for the Gordon equation. Then {S(x) = 0} × R forms a white hole if w(x) = (w1 (x), . . . , wn (x)) is pointed inside ext when S(x) = 0 and {S(x) = 0} × R forms a black hole if w(x) is pointed inside int when S(x) = 0. Proof. Since {S(x) = 0} × R is a characteristic surface for the Gordon equation we have (cf. (2.8)): ⎞2 ⎛ n 2 − 1)  (n ⎝ |Sx (x)|2 =  w j (x)Sx j (x)⎠ . 2 c2 1 − |w| j=1 c2 Therefore (c2 − |w|2 )|Sx |2 = (n 2 − 1)(w(x) · Sx (x))2 .

(3.17)

2

Since |w|2 = n 2c(x) when S(x) = 0 we get |Sx |2 |w|2 = (w · Sx )2 . This last equality holds iff Sx (x) = α(x)w(x), where S(x) = 0, α(x) = 0. Since Sx (x) is an outward normal we have that α(x) > 0 if w(x) is pointed outwardly, and α(x) < 0 if w(x) is pointed 2  w j (x) (cf. (2.10)). We inwardly. In the case of the Gordon equation g 0 j (x) = (n −1) |w|2 have

n j=1

c 1−

g 0 j (x)S

x j (x) =

2 (n −1)2  α(x)|w|2 . c 1− |w|2

c2

Therefore (3.15) holds when w(x) is

c

pointed into ext and (3.16) holds when w(x) is pointed into int . In the first case we have a white hole and in the second case we have a black hole.

We shall study now the impact of the existence of the black or white hole on the uniqueness of the inverse problem. Consider the case of a white hole. Then the domain of the dependence (i.e. the backward domain of influence) of any point (y0 , y) ∈ int × (−∞, +∞) is contained in int × (−∞, +∞). If u(x0 , x) is the solution of the initial-boundary value problem

Inverse Hyperbolic Problems and Optical Black Holes

829

(1.1), (1.5), (1.6) we get that u = 0 in int × (−∞, +∞) by the uniqueness of the Cauchy problem. Therefore the change of the coefficients of (1.1) in int × (−∞, +∞) does not change the solution of the initial-boundary value problem (1.1), (1.5), (1.6) in ( ∩ ext ) × (−∞, +∞) and it does not change the Cauchy data on ∂0 × (−∞, +∞). Therefore the boundary data on ∂0 × (−∞, +∞) are not able to determine the coefficients of (1.1), (modulo (1.10), (1.11)) in int , i.e. we have a nonuniqueness. Consider now the case of a black hole. Then the domain of influence of each point (y0 , y) ∈ int × R is contained in int × R and the domain of dependence of each point (y0 , y) ∈ ext × R is contained in ext × R. Consider the initial-boundary value problem (1.5), (1.6) for two hyperbolic equations L (i) u (i) = 0 in  × (−∞, +∞) whose coefficients differ in int . Since the domain of dependence of any (y0 , y) ∈ ext × R is contained in ext × R the solutions u (i) , i = 1, 2, of the initial-boundary value problem (1.5), (1.6) are equal in ext × (−∞, +∞). Therefore the (1) and (2) are equal on ∂0 × (−∞, +∞), i.e. the boundary measurements are equal despite the fact that the coefficients of L (1) and L (2) differ in int . Therefore we again have a non-uniqueness of the solution of the inverse problem. Change the formulation of the initial-boundary value problem allowing nonzero initial condition in int × R or allowing the nonzero right hand of (1.1) with support in int × R. Then u(x0 , x) will be nonzero in int × (−∞, +∞) but this will not effect the solution in ext × R. 4. Black Holes Inside the Ergosphere Let S(x) = 0, int , ext be the same as in § 3. Borrowing the terminology from general relativity we shall call S(x) = 0 the ergosphere and e = int ∩ { (x) < 0} ∩  is an ergoregion (cf. [V]). We assume that (x) < 0 in e \ S. If S = {x : S(x) = 0} is a characteristic surface then (int ∩ ) × R is either a black or white hole (cf. Theorem 3.1). Now consider the case when S is not characteristic at any point y ∈ S, i.e. n 

g jk (y)ν j (y)νk (y) = 0, ∀y ∈ S,

(4.1)

j,k=1

 where ν(y) is the outward normal to S. Note that the quadratic form nj,k=1 g jk (x)ξ j ξk has the signature (1, −1, . . . , −1) when x ∈ e . The question is whether there exists a black or white hole inside e × R. Consider the case n = 2, i.e. the case of two space dimensions. In this case there are two families of characteristic curves S ± = const: 2 

g jk (x)Sx±j Sx±k = 0

(4.2)

j,k=1

in a neighborhood of any point of e . Equation (4.2) can be factored and we get in the 22 region (1) e = {x ∈ e : g (x)  = 0}:  ∂ S±  ∂ S ± (x)  12 g 22 (x) + g (x) ± − (x) = 0. ∂ x2 ∂ x1 Note that (x) = g 11 (x)g 22 (x) − (g 12 (x))2 < 0 in e .

830

G. Eskin

We shall derive an equation for the characteristic curves S ± (x) = const that holds in a neighborhood of any point in e . Let U1 be the set in e , where either g 22 (x) = 0 or g 12 (x) > 0. Denote by fU+1 (x) a nonzero vector field fU+1 (x) = (g 12 (x) + √ − (x), g 22 (x)), x ∈ U1 . Let U2 be the set where either g 11 (x) = 0 or√ g 12 (x) < 0 and + + 11 12 denote by fU2 (x) the nonzero vector field fU2 (x) = (g (x), g (x) − − (x)), x ∈ U2 . Let Uic = e \ Ui , i = 1, 2. Note that U1c ∩ U2c = {x : g 11 = g 22 = g 12 = 0} = ∅, since the rank of [g jk ]2j,k=1 is at least 1. Therefore U1 ∪ U2 = e . Let Vi , i = 1, 2, be a small neighborhood of Uic , i = 1, 2, such that V 1 ∩ V 2 = ∅. Denote U˜ i = Ui \ Vi , i = 1, 2. Then U˜ 1 ∪ U˜ 2 = e . Note that √ g 11 (x) g 12 − − = (4.3) = λ(x) = 0 in U˜ 1 ∩ U˜ 2 , √ g 22 (x) g 12 (x) + − √ √ since g 11 g 22 = (g 12 − − )(g 12 + − ). Extend λ(x) as a nonzero function from U˜ 1 ∩ U˜ 2 to U˜ 2 such that λ(x) is continuous in U˜ 2 and smooth in U˜ 2 \ S, and define f + (x) = fU+1 (x) in U˜ 1 ,

f + (x) = λ−1 (x) fU+2 (x) in U˜ 2 .

(4.4)

Then f + is a continuous is smooth in e \ S. Analogously √ nonzero vector field in e that√ let fU−3 (x) = (g 12 − − , g 22 ), fU−4 (x) = (g 11 , g 12 + − ), where U3 is the set where either g 22 = 0 or g 12 < 0 and U4 is the set where either g 11 = 0 or g 12 > 0. Then fU−3 (x) = λ1 (x) fU−4 (x) in U˜ 3 ∩ U˜ 4 and λ1 = 0 in U˜ 3 ∩ U˜ 4 . Here U˜ 3 ⊂ U3 , U˜ 4 ⊂ U4 are similar to U˜ i , i = 1, 2, in (4.4). Extending λ1 (x) from U˜ 3 ∩ U˜ 4 to U˜ 4 we get a vector field f − (x) in e that is continuous in e and smooth in e \ S. We have that (4.2) is equivalent to f 1± (x)Sx±1 (x) + f 2± (x)Sx±2 (x) = 0,

(4.5)

where f ± (x) = ( f 1± (x), f 2± (x)). Note that f ± (x) = (0, 0) for any x ∈ e and f + (x) = f − (x), ∀x ∈ e \ S. Since the rank of [g jk (y)]2j,k=1 is 1 on S there exists a smooth and nonzero b(y) = (b1 (y), b2 (y)) such that 2 

g jk (y)bk (y) = 0, j = 1, 2, y ∈ S.

(4.6)

k=1

Since 2 

g jk (y)b j (y)bk (y) = 0,

(4.7)

j,k=1

we get from (4.1) that b(y) is not co-linear with ν(y), ∀y ∈ S. It follows from (4.5) with Sx± (y) replaced by b(y) that f 1± (y)b1 (y) + f 2± (y)b2 (y) = 0, ∀y ∈ S.

(4.8)

Inverse Hyperbolic Problems and Optical Black Holes

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Note that f + (y) = f − (y) when y ∈ S. Therefore the condition (4.1) implies that f ± (y) is not tangential to S for any y ∈ S. Therefore changing f ± (x) to − f ± (y) if needed we will assume that f ± (y) is pointed inside e for any y ∈ S. Consider differential equations d xˆ ± (σ ) = f ± (x ± (σ )), xˆ ± (0) = y, y ∈ S, σ ≥ 0. dσ

(4.9)

d ± ± Let S ± (x) = const be a characteristic curve. Then (4.5), (4.9) imply that dσ S (xˆ (σ )) = ± ± ± ± 0, i.e. S (xˆ (σ )) = S (y), σ ≥ 0. Therefore x = xˆ (σ ) are the parametric equations  of the characteristic curves S ± (x) = S ± (y). It was shown in §3 that 2j=1 g 0 j (y)b j (y) = 0 if b(y) = (b1 , b2 ) satisfies (4.7). Changing b(y) to −b(y) if needed we assume that 2 

g 0 j (y)b j (y) > 0, ∀y ∈ S.

(4.10)

j=1

We shall denote by S ± × R the characteristic surfaces in e × R, where S ± (x) = const and x0 ∈ R. Let (y0 , y ± ) be any point of S ± ×R. Consider the null-bicharacteristic (3.4), (3.5) with initial conditions x0± (0) = y0 , x ± (0) = y ± , ξ ± (0) = Sx± (y ± ), ξ0± (0) = 0. Note that x = x ± (s), x0± = x0 (s), s ≥ 0 is the corresponding null-geodesic. We shall show that S ± (x ± (s)) = S ± (y), i.e. this null-geodesic remains on the characteristic surface S ± × R. We have 2 

± g jk (x ± (s))ξ ± j (s)ξk (s) = 0,

(4.11)

j,k=0

 since 2j,k=0 g jk η±j ηk± = 0, η0± = 0, η± = Sx± (y ± ) (cf. (3.6), (3.7)). Note that ξ0± (s) = ξ ± (0) = 0 since [g jk (x)]2j,k=0 is independent of x0 (cf. (3.5)). Since S ± (x)  is a solution of the eiconal equation 2j,k=1 g jk (x)Sx±j (x)Sx±k (x) = 0 in an open neighborhood of y ± in e , we have that (cf. [CH])  Note that

d x0± d x ± ds , ds

2 dx±  j j=1

ds



ξ ± (s) = Sx± (x ± (s)). = 2G(0, ξ ± (s)). Therefore

Sx±j (x ± (s)) =

2 dx±  j j=1

ds

ξ± j (s) = 2

2 

± g jk (x ± (s))ξ ± j (s)ξk (s) = 0,

j,k=0

d ± ± ds S (x (s))

= 0, ∀s. Therefore the projection of the null-bicharacteristic x0 = x0± (s), x = x ± (s), x ± (0) = ± y , ξ0 = 0, ξ = ξ ± (s) on the (x1 , x2 )-plane belongs to the curve S ± (x) = S ± (y ± ). Fix arbitrary y ∈ S. Denote by + (y) the intersection of the half-plane

i.e.

ξ1 b1 (y) + ξ2 b2 (y) > 0

(4.12)

with e . Analogously let − (y) be the intersection of the half-plane ξ1 b1 (y) + ξ2 b2 (y) < 0

(4.13)

832

G. Eskin

with e . Let x = xˆ ± (σ ) be the solution of (4.9), xˆ ± (0) = y, σ ≥ 0. Note that f + (y) = + − f − (y) ∈  (y) ∩  (y). However for σ > 0 and small, either xˆ + (σ ) belongs to + (y) − − and xˆ (σ ) to  (y), or vice versa. For definiteness let xˆ + (σ ) ∈ + (y) for 0 < σ < ε and xˆ − (σ ) belongs to − (y) for 0 < σ < ε. Condition (4.1) and the continuity in y imply xˆ + (σ ) ∈ + (y), xˆ − (σ ) ∈ − (y), 0 < σ < ε, for all y ∈ S. Each xˆ ± (σ ) is a parametric equation of the characteristic curve S ± (x) = S ± (y), i.e. S ± (xˆ ± (σ )) = S ± (y), ∀σ ≥ 0.

(4.14)

Note that the curve S ± (x) = S ± (y) is contained in ± (y) when x = y and close to y. Consider now two null-bicharacteristics x0 = x0+ (s), x = x + (s), ξ0 = 0, ξ = ξ + (s), s ≥ 0,

(4.15)

with initial conditions x0+ (0) = 0, x + (0) = y, ξ + (0) = b and x0 = x0− (s), x = x − (s), ξ0 = 0, ξ = ξ − (s), s ≥ 0,

(4.15 )

with initial conditions x0− (0) = 0, x − (0) = y, ξ − (0) = −b. Note that  d x0+ (s) =2 g j0 (x + (s))ξ +j (s) > 0, ds 2

(4.16)

j=1

since (4.10) holds. Also we have that d x0− (s) < 0, s ≥ 0, ds

(4.17)

since ξ − (0) = −b. Therefore x0 = x0+ (s), x = x + (s) is a forward null-geodesics since x0 is increasing when s is increasing and x = x0− (s), x = x − (s) is a backward null-geodesics. It follows from (3.5) and (4.6) that dx± j (0) ds

=2

2 

g jk (y)bk (y) = 0, j = 1, 2.

k=1

However d x ± (s) = 2G(x ± (s))(0, ξ ± (s)) = 0, s > 0, ds since (x) = 0 in e \ S. + Since (0, b(y)) ∈ K + (y) (cf. (3.10)) the dual half-cone K + (y) is contained in  (y). Therefore the projection x = x + (s), s ≥ 0, of the null-bicharacteristic (4.15) satisfies S + (x + (s)) = S + (y), s ≥ 0.

(4.18)

Analogously, the projection x = x − (s) of the null-bicharacteristic (4.15 ) on (x1 , x2 )plane satisfies S − (x − (s)) = S − (y), s ≥ 0.

(4.18 )

Inverse Hyperbolic Problems and Optical Black Holes

833

Comparing (4.14) and (4.18), (4.18 ) we get that x = x ± (s), x = xˆ ± (σ ) are a different parametrization of the same curve, i.e. x ± (s ± (σ )) = xˆ ± (σ ), σ ≥ 0, where s ± (0) = 0,

ds ± (σ ) dσ

> 0 for σ > 0.

(4.19)



Remark 4.1. Note that if x = x0 (s), x = x(s), ξ0 = 0, ξ = ξ(s), satisfies (3.4), (3.5), then x = x0 (−s), x = x(−s), ξ0 = 0, ξ = −ξ(−s) also satisfies (3.4), (3.5). Therefore changing s ≥ 0 to −s in (4.15 ) and combining (4.15) and (4.15 ) we get a forward nullbicharacteristic x = x0 (s), x = x(s), ξ0 = 0, ξ = ξ(s), defined on (−∞, +∞) with initial conditions x0 (0) = 0, x(0) = y, ξ(0) = b. The projection of this bicharacteristic on the (x1 , x2 )-plane has a singularity (caustic) at x = y.

Let e ⊂  ∩ int ∩ { < 0}. We assume that the boundary of e consists of S = ∂int and S1 . Assume that < 0 on S1 and e is diffeomorphic to an annulus domain in R2 . Let y be an arbitrary point on S1 . Consider the forward cone of influence K + (y). Let (0) K + (y) be the projection of this cone on the (x1 , x2 )-plane. We assume that (0)

N (y) · x˙ > 0, ∀y ∈ S1 , ∀x˙ ∈ K + (y),

(4.20)

where N (y) is the outward unit normal to S1 and x˙ = (x˙1 , x˙2 ) ∈ R2 is any vector (0) in K + (y). In particular, S1 is not a characteristic curve. Note that condition (4.20) is equivalent to the condition that d x ± (s1± ) > 0, x ± (s1± ) = y, ∀y ∈ S1 , (4.21) ds where x ± (s) is the projection on the (x1 , x2 )-plane of two forward null-bicharacteristics such that x = x ± (s) are parametric equations of two characteristics of the form (4.2) N (y) ·

d x ± (s ± )

passing through y ∈ S1 . Note that 0ds 1 > 0 when x ± (s1± ) ∈ S1 . We shall say that conditions (4.20 ) or (4.21 ) are satisfied if N (y) · x˙ < 0 in (4.20) ± or N (y) · d xds(s) < 0 in (4.21). Theorem 4.1. Suppose ∂e = S0 ∪ S1 , where (x) = 0 on S, (x) < 0 on e \ S. Suppose (4.1) holds on S. Suppose also that either (4.20) or (4.20 ) is satisfied on S1 . Then there exists a Jordan curve S0 (x) = 0 between S and S1 such that S0 × R is a characteristic surface, i.e. S0 × R is a boundary of either a black or a white hole. Proof. It was shown already that condition (4.1) implies the existence of two null-bicharacteristics x0 = x0± (s), x = x ± (s), ξ0± = 0, ξ = ξ ± (s), s ≥ 0, such that x ± (0) = ± y ∈ S and x ± (s) after reparametrization s = s ± (σ ), s ± (0) = 0, dsdσ(σ ) > 0, σ > 0, coincide with the solution x = xˆ ± (σ ) of the differential equation (4.9): x ± (s(σ )) = xˆ ± (σ ), σ ≥ 0. Moreover d x0+ (s) ds

d x0− (s) ds

< 0, i.e. x0 decreases when σ (or s) increases, and

> 0. Suppose condition (4.20) is satisfied. We shall show that there is no solution x = xˆ − (σ ) of d xˆ − (σ ) = f − (x(σ ˆ )), dσ x(0) ˆ = y ∈ S that reaches S1 , i.e. xˆ − (σ1 ) = y (1) where y (1) ∈ S1 , σ1 > 0.

(4.22)

834

G. Eskin

Suppose such x = xˆ − (σ ) exists. When σ > σ1 , x = xˆ − (σ ) leaves e , since S1 is not characteristic. Note that for the null-bicharacteristic whose projection is x = x − (s(σ )) = xˆ − (σ ), the time variable x0 = x0− (s) is decreasing when σ is increasing. Therefore x = xˆ − (σ ) leaves e when x0 is decreasing. From the other side the condition (4.20) implies that the projection of all null bi-characteristics passing through y (1) ∈ S1 leave e when x0 increases. This contradiction proves that the limit set of the trajectory x = xˆ − (σ ) is contained inside e . By the Poincaré-Bendixson theorem (cf. [H]) there exists a limit cycle, i.e. a closed periodic solution x = z − (σ ) of (4.22) which has no points of self-intersection. Let S0 (x) = 0 be the equation of this orbit. Then S0 × R is a characteristic surface where S0 = {x : S0 (x) = 0}. Other solutions of (4.22) are spiraled around S0 when x0 → −∞. Now we shall assume that the condition (4.20 ) (or (4.21 )) is satisfied. These conditions mean that the projection on the (x1 , x2 )-plane of any null-bicharacteristic passing through S1 enters e when x0 increases. From the other side, consider the solution of d xˆ + (σ ) = f + (xˆ + (σ )), σ ≥ 0, dσ

(4.23)

that starts on S : x(0) ˆ = y ∈ S. We claim that this solution can not reach S1 . In fact if d x0+ (0) + > 0 for the nullx = xˆ (σ ) reaches S1 it will leave S1 when σ > σ1 . We have ds bicharacteristics whose projection on the (x1 , x2 )-plane is x = xˆ + (σ ) after the change of the parameter s = s + (σ ). Therefore x = x + (s) leaves e when x0 increases. This contradiction shows that the limit set of x = x + (σ ) is contained inside e . Again by the Poincaré-Bendixson theorem there exists a closed characteristic curve S0 (x) = 0 without points of self-intersection. All other solutions of (4.22) spiral around S0 when x0 → +∞. Remark 4.2. Without additional assumptions it is impossible to specify whether S0 × R is a boundary of a white or a black hole. For example, assume that the condition (4.20) is satisfied. Then S0 is a limit cycle for − the equation d xˆdσ(σ ) = f − (xˆ − (σ )). Since f − (x) = f + (x) for all x ∈ e we have that f + (x), x ∈ S0 is pointed either into the interior of S0 for all x ∈ S0 or into the exterior of S0 . In the first case S0 ×R forms a black hole and in the second case S0 ×R forms a white hole. In the second case we can apply the Poincaré-Bendixson theorem to Eq. (4.23) in the annulus domain between S and S0 . Then there exists a limit cycle S2 for the equation d xˆ + (σ ) = f + (x(σ ˆ )) between S and S1 and S2 × R that is the boundary of a white hole. dσ Also there exists a white or black hole formed by S3 × R, where S3 (x) = 0 is a limit cycle for (4.23) between S0 and S1 . Analogously if condition (4.20 ) is satisfied then the limit cycle S0 for Eq. (4.23) forms a white hole if f − (x) restricted to S0 is pointed in the exterior of S0 and there are two additional black or white holes if f − (x)| S0 is pointed into the interior of S0 . Remark 4.3. The black or white holes described in Theorem 4.1 are stable in the sense that if we perturb slightly [g jk (x)]2j,k=0 then assumptions of Theorem 4.1 remain valid and therefore the limit cycle solution will still exist. This is in contrast with black or white holes obtained by Theorem 3.1: when we perturb [g jk ]3j,k=0 then the surface S = { (x) = 0} may cease to be characteristic and the black or white hole at S × R disappears.

Inverse Hyperbolic Problems and Optical Black Holes

835

Now we shall formulate the application of Theorem 4.1 to the case of the Gordon equation. Theorem 4.2. Let ∂e = S∪ S1 . Assume that |w(x)|2 =

c2 n 2 (x)

on S and |w(x)|2 >

c2 n 2 (x)

in e \ S. Suppose w(y) is not co-linear with ν(y) on S, where ν(y) is the outward unit normal to S. Suppose that either 1

(n 2 (x) − 1) 2 (v(x) · N ) > 1 on S1 ,

(4.24)

or 1

(4.24 )

(n 2 (x) − 1) 2 (v(x) · N ) < −1 on S1 ,

 1  2 − 2 w(x) where v(x) = 1 − |w(x)| 2 c , N (x) is the outward unit normal to S1 . c Then there exists a Jordan curve S0 (x) = 0 between S and S1 such that S0 × R is a characteristic surface, i.e. S0 × R is the boundary of either black or white hole. Proof. Note that (y) = 0 is equivalent to |w|2 = Gordon equation and (x) < 0 implies |w|2 >

c2 n 2 (y)

c2 , n 2 (y)

y ∈ S in the case of the

in e \ S.

Since (y) = 0 there exists a smooth b(y) = 0, y ∈ S, such that 0, j = 1, 2 (cf. (4.6)). For the Gordon equation we have

2 k=1

g jk (y)bk (y) =

−b j (y) + (n 2 − 1)(v · b)v j = 0, j = 1, 2, i.e. b(y) is co-linear with w(y) on S. Therefore b(y) is not co-linear with ν(y) since w(y) is not co-linear, ∀y ∈ S. Note that condition (4.10) has the form (n 2 − 1)(v · b)v 0 > 0, i.e. b and w has the same direction. We take b = w. Therefore f + (y) = f − (y) is orthogonal to w(y) and is not tangential to S, ∀y ∈ S. Let y (1) be any point on S1 . Suppose (4.24) holds. Consider any null-bicharacteristic whose projection is passing through y (1) . The equation 2 

g jk (y (1) )ξ j (y (1) )ξk (y (1) ) = 0

j,k=1

has the following form for the Gordon equation: − |ξ(y (1) )|2 + (n 2 (y (1) ) − 1)(v(y (1) ) · ξ(y (1 )))2 = 0.

(4.25)

0 (s) = 2(n 2 − 1)v 0 (v · ξ ) > 0 implies that (v(x(s)) · ξ(s)) > 0. It follows Note that d xds from (4.25) that

v·ξ =

|ξ | (n 2

1

− 1) 2

.

(4.26)

We have 2  1 dx j =2 g jk ξk = −2ξ j + 2(n 2 − 1)(v · ξ )v j = −2ξ j + 2(n 2 − 1) 2 |ξ |v j , ds k=1

j = 1, 2.

(4.27)

836

G. Eskin

Take the inner product of (4.27) with N , where N is outward unit normal to S1 . We get 1 dx · N = −2ξ · N + 2|ξ |(n 2 − 1) 2 (v · N ). ds 0 (s) Since |ξ · N | ≤ |ξ | we get, using (4.24), that ddsx · N > 0 on S1 . Note that d xds > 0. Therefore any null-bicharacteristic escapes e when s > s0 (or when x0 > x0 (s0 )) where x(s0 ) ∈ S1 . 1 Therefore by Theorem 4.1 we have a characteristic surface S0 × R. If (n 2 − 1) 2 (v · N ) < −1 then all null-bicharacteristics are entering e when x0 increases. Therefore again we can apply Theorem 4.1. As in Remark 4.3 the black and white holes described in Theorem 4.2 are stable. Now we shall refine Theorem 4.2 and specify when the characteristic surface S0 × R is the boundary of a white hole and when it is the boundary of a black hole. We shall say that the flow w(x) = (w, (x), w2 (x)) is incoming if for any closed simple curve  ⊂ e containing S1 there exists at least one point y ∈  such that w(y) · ν(y) > 0, where ν(y) is a normal to  pointed inside . Analogously, the flow w(x) is called outgoing in e if for any  there exists y ∈  such that w(y) · ν(y) < 0.



Theorem 4.3. Consider the Gordon equation in e , where e is the same as in Theorems 4.1 and 4.2. Suppose w(y) is not colinear with the normal to S, ∀y ∈ S, and (4.24) holds on S1 . Assume in addition that the flow w(x) is incoming. Then there exists a black hole bounded by S0 × R, where S0 is a Jordan curve between S and S1 . If (4.24 ) holds and the flow w(x) is outgoing then there exists a white hole with the boundary S0 × R. Proof. Suppose (4.24) holds. It was proven in Theorem 4.2 that there exists a characteristic surface S0 × R. Now using that the flow w(x) is incoming we shall prove that S0 × R is the boundary of a black hole. Take any y ∈ S0 . Let b1 (y) be a normal to S0 . Choose the direction of b1 (y) such  that 2j=1 g 0 j (y)b j1 (y) = 2(n 2 (y) − 1)v 0 (y)(v(y) · b1 (y)) > 0, i.e. b1 (y) · w(y) > 0). It follows from Proposition 3.1 that K + (y) (cf. (3.11)) is contained in the half-space (α0 , α1 , α2 ) · (0, b11 (y), b21 (y)) ≥ 0. Since the flow w(x) is incoming there exists y0 ∈ S0 such that w(y0 ) · ν(y0 ) > 0, where ν(y0 ) is pointed inside S0 . Since b1 (y0 ) · w(y0 ) > 0 and b1 (y0 ) is co-linear with ν(y0 ) we get that b1 (y0 ) is also pointed inside S0 . Since b1 (y) is continuous in y we have that b1 (y) is pointed inside S0 for all y ∈ S0 . Therefore (x˙0 , x) ˙ ∈ K + (y) are pointed inside S0 × R, i.e. S0 × R is the boundary of a black hole. Suppose that (4.24 ) is satisfied and the flow w(x) is outgoing. Let b1 (y) be the same as above, i.e. b1 (y) is a normal to S0 and b1 (y) · w(y) > 0. Since the flow w(x) is outgoing there exists y0 ∈ S0 such that w(y0 ) · ν(y0 ) < 0, where ν(y0 ) is pointed inside S0 . Then b1 (y0 ) is a normal to S0 that is pointed outside of S0 . Therefore K + (y0 ) is pointed outside of S0 × R. The same is true for any y ∈ S0 by the continuity. Therefore S0 × R is the boundary of a white hole.

Example 4.1. (Acoustic black hole (cf. [V])) Consider a fluid flow in a vortex with the velocity field v = (v 1 , v 2 ) =

B A ˆ rˆ + θ, r r

(4.28)

Inverse Hyperbolic Problems and Optical Black Holes

837

    x1 x2 x2 x1 where r = |x|, rˆ = |x| , |x2 , θ = − |x| , |x| , A and B are constants. The inverse of the metric tensor in this case has the form 1 1 j , g 0 j = g j0 = v , 1 ≤ j ≤ 2, ρc ρc 1 (−c2 δi j + v i v j ), 1 ≤ j, k ≤ 2, = ρc

g 00 = g

jk

(4.29)

where c is the sound speed, ρ is the density. It is convenient to use polar coordinates in (4.2). Assuming c = 1, ρ = 1 we have  2   2  2   2 A ∂S B ∂S 2 AB ∂ S ∂ S 1 + − 1 + − = 0, (4.30) r2 ∂r r 3 ∂r ∂θ r4 r2 ∂θ or (cf. (4.5)) 

⎛ ⎞   ± ± 2 + B2 AB ∂S A2 A 1 ⎝ ⎠ ∂S + − 1 ± − r2 ∂r r3 r4 r2 ∂θ

(4.31)

√ for r < A2 + B 2 . The ergosphere, i.e. the curve√where = 0 has the form A2 + B 2 = r 2 . We shall consider the domain 0 < r < A2 + B 2 or any compact domain √ e = {r1 ≤ r ≤ r0 } where r0 = A2 + B 2 , r1 < |A|. The condition (4.1) is satisfied when B = 0. The condition (4.20) is satisfied when A > r1 and the condition (4.20’) holds when A < −r1 . Consider the case A > 0, B > 0. The system (4.9) has the following form in polar coordinates: dr + AB  2 dθ + = A2 − r 2 , = + A + B2 − r 2, (4.32) ds ds r 2 1 − Br 2 dθ − (s) dr − (s) = −1, = AB √ , (4.33) ds ds A2 + B 2 − r 2 r + y dθ ˆ , |y| = r0 , s ≥ 0. Note that ddsx = dr r ± (0) = r0 , θ ± (0) = |y| ds rˆ + r ds θ. We wanted to ensure is not zero. Since √ that the vector field in the right hand sides of (4.32), (4.33) AB 2 + B 2 − r 2 = 0 when A = r, B > 0 we divided by r 2 − A2 in (4.33) and − A r   2 √ (r 2 −A2 ) 1− B2 AB r 2 2 2 used that r − A + B − r = AB √ 2 2 2 . It is clear that r 2 = A2 is the limit r

+ A +B −r

cycle for Eq. (4.32). In the notations of Theorem 4.2 (cf. (4.10), (4.29)) we have b(y) = v(y) = rA0 rˆ + rB0 θˆ , √ where |y| = r0 = A2 + B 2 . We have, (cf. (4.12), (4.13)):     dx+ dr + A d θˆ (s) B · b(y) = rˆ + r0 θˆ · rˆ + θˆ ds ds ds r0 r0 AB 2 AB 2 + + B r02 − r 2 (s) + O(r0 − r (s)) > 0 for 0 ≤ s ≤ ε. =− r0 r0

838

G. Eskin −

Analogously d xds(s) · b(y) < 0 for 0 < s < ε, ε is small. Note that r = r + (s), θ = θ + (s) is the projection on the (x1 , x2 )-plane (after a reparametrization) of the null-bicharacteristic with initial conditions x + (0) = y, |y| = r0 , x0+ (0) = 0, ξ0+ (0) = 0, ξ(0) = d x + (s)

0 b(y) and ds > 0, ∀s ≥ 0. When x0 → +∞, r = r + (s), θ = θ + (s) spirals toward the limit cycle r = A. Also r = r − (s), θ = θ − (s) is the projection (after a reparametrization) of the nullbicharacteristic with initial conditions x0− (0) = 0, x − (0) = y, ξ0− = 0, ξ(0) = −b(y)

d x − (s)

0 < 0. Note that r = r − (s), θ = θ − (s) reaches r = r1 when x0 → −∞. and ds Therefore r = A is the boundary of a white hole. When A < −r1 , B > 0 we have, analogously to (4.32), (4.33):

dr + = −1, ds

dr − (s) = A2 − r 2 , ds

2

1 − Br 2 , √ AB A2 + B 2 − r 2 r − AB  2 dθ − (s) = − A + B2 − r 2, ds r

dθ + = ds

(4.34) (4.35)

√ y 2 2 2 , |y| = r0 , and we modified (4.34) since AB r ± (0) = r0 , θ ± (0) = |y| r + A + B −r − − = 0 when r = |A|. Now r = r (s), θ = θ (s) spirals towards r = |A| when x0 → −∞ and r = r + (s), θ = θ + (s) reaches r = r1 when x0 → +∞. Therefore r = |A| is the boundary of a black hole. A similar result holds when B < 0. The difference is that the spiraling of the solutions towards r = |A| changes from the clockwise to the counter-clockwise or vice versa. Example 4.2. Consider a generalization of Example 4.1 when ˆ v(x) = A(r )ˆr + B(r )θ, where r1 ≤ r ≤ r0 , A(r ), B(r ) are smooth, A2 (r0 )+ B 2 (r0 ) = 1, A2 (r )+ B 2 (r ) > 1 on [r1 , r0 ), B(r ) > 0 on [r1 , r0 ], A(r )+1 has simple zeros α1 , . . . , αm 1 on (r1 , r0 ), A(r )−1 has simple zeros β1 , . . . , βm 2 on (r1 , r0 ), βk = α j , ∀ j, k, |A(r1 )| > 1. Equation (4.31) has the form  (A2 (r ) − 1)Sr± + (A(r )B(r ) ± A2 + B 2 − 1)Sθ± = 0. (4.36) Let

√ dr + (s) dθ + A(r )B(r ) + A2 + B 2 − 1 = A(r ) − 1, = , ds ds A(r ) + 1 √ dθ − A(r )B(r ) − A2 + B 2 − 1 dr − (s) = A(r ) + 1, = . ds ds A(r ) − 1

(4.37) (4.38)

Note that the right hand sides in (4.37), (4.38) are smooth since the singularities at A(r ) + 1 = 0 and A(r ) − 1 = 0 are removable. It follows from (4.37), (4.38) that {|x| = α j } × R, j = 1, . . . , m 1 , {|x| = βk } × R, k = 1, . . . , m 2 , are boundaries of black or white holes. In particular if α1 = min j,k (α j , βk ) and A(r1 ) < −1 then r = α1 is the boundaries of a black hole. If β1 = min j,k (α j , βk ) and A(r1 ) > 1 then r = β1 is the boundary of a white hole.

Inverse Hyperbolic Problems and Optical Black Holes

839

Remark 4.4. (Axially symmetric metrics) Consider Eq. (1.1) in  × R, where  is a three-dimensional domain. Let (r, θ, ϕ) be the spherical coordinates of x = (x1 , x2 , x3 ), i.e. x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ . Suppose that g jk are independent of ϕ : g jk = g jk (r, θ ). Consider the characteristic surface S independent of x0 and ϕ: 3  j,k=1

g jk (r, θ )

∂S ∂S = 0, ∂ x j ∂ xk

(4.39)

where ∂S ∂ S ∂r ∂ S ∂θ = + , k = 1, 2, 3. ∂ xk ∂r ∂ xk ∂θ ∂ xk Substituting (4.40) into (4.39) we get an equation:  2  2 ∂S ∂S ∂S ∂S 11 12 22 + a (r, θ ) + 2a (r, θ ) = 0. a (r, θ ) ∂r ∂r ∂θ ∂θ

(4.40)

(4.41)

We assume that a jk (r, θ ) are also independent of ϕ. We consider (4.41) in the twodimensional domain ω such that δ1 ≤ r ≤ δ2 , 0 < δ3 < θ < π − δ4 when (r, θ ) ∈ ω. Imposing on ω and a i j (r, θ ), 1 ≤ i, j ≤ 2, the conditions of Theorems 3.1 and 4.1, we shall prove the existence of black or white holes in  × R with the boundary of the form S0 × S 1 × R, where ϕ ∈ S 1 , x0 ∈ R and S0 is a closed Jordan curve in the (r, θ ) plane. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References [B]

Belishev, M.: Recent progress in the boundary control method. Inverse Problems 23(5), R1– R67 (2007) [CH] Courant, R., Hilbert, D.: Methods of Mathematical Physics. Vol. II, New York, London: WileyInterscience, 1962 [E1] Eskin, G.: Optical aharonov-bohm effect: inverse hyperbolic problem approach. Commun. Math. Phys. 284, 317–343 (2008) [E2] Eskin, G.: A new approach to the hyperbolic inverse problems II: global step. Inverse Problems 23, 2343–2356 (2007) [G] Gordon, W.: Ann. Phys. (Leipzig) 72, 421 (1923) [H] Hartman, F.: Ordinary differential equations. New York: J.Wiley & Son, 1964 [KKL] Katchalov, A., Kurylev, Y., Lassas, M.: Inverse boundary spectral problems. Boca Baton: Chapman&Hall, 2001 [LP] Leonhardt, V., Piwnicki, P.: Phys. Rev. A60, 4301 (1999) [NVV] Novello, M., Visser, M., Volovik, G.: (editors): Artificial black holes. Singapore: World Scientific, 2002 [U] Unruh, W.: Phys. Rv. Lett. 46, 1351 (1981) [V] Visser, M.: Acoustic black holes, horizons, ergospheres and hawking radiation. Class. Quant. Grav. 15(6), 1767–1791 (1998) Communicated by P. Constantin

Commun. Math. Phys. 297, 841–893 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1057-0

Communications in

Mathematical Physics

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles Francesco D’Andrea1 , Giovanni Landi2 1 Département de Mathématique, Université Catholique de Louvain, Chemin du Cyclotron 2,

B-1348, Louvain-La-Neuve, Belgium. E-mail: [email protected]

2 Dipartimento di Matematica e Informatica, Università di Trieste, Via A. Valerio 12/1,

I-34127 Trieste, Italy, and INFN, Sezione di Trieste, Trieste, Italy. E-mail: [email protected] Received: 5 April 2009 / Accepted: 7 January 2010 Published online: 19 May 2010 – © Springer-Verlag 2010

Abstract: We present several results on the geometry of the quantum projective plane. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding ‘classical’ characteristic classes (via Fredholm modules); complete diagonalization of gauged Laplacians on these line bundles; ‘quantum’ characteristic classes via equivariant K-theory and q-indices. Contents 1. 2.

3. 4.

5.

6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . The Quantum Projective Plane and its Symmetries . . . . . . . 2.1 The algebra of ‘infinitesimal’ symmetries . . . . . . . . . 2.2 The quantum SU(3) group . . . . . . . . . . . . . . . . . 2.3 The quantum 5-sphere and the quantum projective plane . Hermitian Vector Bundles . . . . . . . . . . . . . . . . . . . . 3.1 Hermitian bundles of any rank . . . . . . . . . . . . . . . 3.2 Line bundles . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . 4.1 Fredholm modules and their characters . . . . . . . . . . . 4.2 The rank and the 1st Chern number of a projective module 4.3 The 2nd Chern number of a projective module . . . . . . . 4.4 Chern numbers of line bundles . . . . . . . . . . . . . . . The Differential Calculus . . . . . . . . . . . . . . . . . . . . 5.1 Holomorphic and antiholomorphic forms . . . . . . . . . . 5.2 The wedge product . . . . . . . . . . . . . . . . . . . . . 5.3 Hodge star and a closed integral . . . . . . . . . . . . . . 5.4 The exterior derivatives . . . . . . . . . . . . . . . . . . . Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.1 The anti-selfdual connections . . . . . . . . . 6.2 Gauged Laplacians . . . . . . . . . . . . . . 7. Quantum Characteristic Classes . . . . . . . . . . 7.1 Equivariant K-theory and K-homology . . . . 7.2 The example of CPq2 . . . . . . . . . . . . . 8. Concluding Remarks . . . . . . . . . . . . . . . . Appendix A. Proof of Proposition 5.1 . . . . . . . . . Appendix B. Some General Facts on Calculi . . . . . . Appendix C. An Alternative Proof of Theorem 7.2 for n Appendix D. Irreducible ∗-Representations of CPq2 . . References . . . . . . . . . . . . . . . . . . . . . . . .

F. D’Andrea, G. Landi

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1. Introduction Quantum spaces are revealing rich geometrical structures and are the subject of intense research activities. In this paper we present a construction of monopoles on the quantum projective plane CPq2 . By a monopole we mean a line bundle over CPq2 , that is to say a ‘rank 1’ (in a sense to be made precise) finitely generated projective module over the coordinate algebra A(CPq2 ), endowed with a connection having anti-selfdual curvature; these are described in Sect. 6. Necessary for the anti-selfduality was a differential calculus and a Hodge star operator on forms. In Sect. 5 we give a full differential ∗-calculus on CPq2 with an Hodge star operator which satisfies all the required properties. Both the K-theory and the K-homology groups of the quantum projective plane are known: K 0 (A(CPq2 ))  Z3  K 0 (A(CPq2 )), while K 1 (A(CPq2 )) = 0 = K 1 (A(CPq2 )). Thus, a finitely generated projective module over A(CPq2 ) is uniquely identified by three integers. Hermitian vector bundles over a homogeneous space – actually the corresponding modules of sections – can be equivalently described as vector valued functions over the total space which are equivariant for a suitable action of the structure group: that is to say, one thinks of sections as ’equivariant maps’. This construction still makes sense for quantum homogeneous spaces. In Sect. 3 we describe general Hermitian ‘vector bundles’ on CPq2 as modules of equivariant elements; we then specialize to line bundles, for which we construct explicit projections (they are all finitely generated projective modules). The generators of the K-homology are obtained in Sect. 4 where we also compute their pairings with the line bundle projections; the resulting numbers are interpreted as the rank (always 1 for line bundles) and 1st and 2nd Chern numbers of the underlying ‘vector bundles’. In particular, we identify three natural generators for the K-theory of CPq2 and three dual generators for its K-homology. Classically, topological invariants are computed by integrating powers of the curvature of a connection, and the result depends only on the class of the vector bundle, and not on the chosen connection. In order to integrate the curvature of a connection on the quantum projective space CPq2 one needs ‘twisted integrals’; moreover, one does not get integers any longer but rather q-analogues of integers. In Sect. 7 we give some general result on equivariant K-theory and K-homology and corresponding Chern-Connes characters, and then focus on CPq2 with equivariance under the action of the symmetry algebra Uq (su(3)). In particular, we construct twisted cocycles that, when paired with the line bundle projections, give q-analogues of monopole and instanton numbers of the line bundle. As a corollary, we obtain that when the deformation parameter q is

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

843

transcendental the equivariant K 0 -group has (at least) a countable number of generators, U (su(3)) (A(CPq2 )) ⊃ Z∞ . K0 q After this introduction, we start in Sect. 2 with some basic results on the geometry of the quantum projective plane CPq2 and its symmetry algebra Uq (su(3)). One of our motivations for the present work was to study quantum field theories on noncommutative spaces. The construction of monopoles on CPq2 that we present here (instanton configurations on CPq2 will be reported in [10]) is a step in this direction. In this spirit, in Sect. 6.2 we study gauged Laplacian operators acting on modules of sections of monopole bundles. Their complete diagonalization is made possible by the presence of many symmetries. From the point of view of physics, such an operator would describe ‘excitations moving on the quantum projective space’ in the field of a magnetic monopole. In the limit q → 1 it provides a model of quantum Hall effect on the projective plane. Notations. Throughout this paper the real deformation parameter will be taken to be 0 < q < 1. The symbol [x] denotes the q-analogue of x ∈ C, [x] :=

q x − q −x . q − q −1

For n a positive integer the q-factorial is [n]! := [n][n − 1] . . . [2][1], with [0]! := 1 and the q-binomial is given by   n m

:=

[n]! . [m]![n − m]!

For convenience we define the q-trinomial coefficient as [ j, k, l]! = q −( jk+kl+l j)

[ j + k + l]! . [ j]![k]![l]!

(1.1)

We shall use Sweedler notation for coproducts (h) = h (1) ⊗ h (2) with a sum understood. Finally, all algebras will be unital ∗-algebras over C, and their representations will be implicitly assumed to be unital ∗-representations. 2. The Quantum Projective Plane and its Symmetries We recall from [7] some results on the geometry of the quantum projective plane. 2.1. The algebra of ‘infinitesimal’ symmetries. Let Uq (su(3)) denote the Hopf ∗-algebra generated by elements {K i , K i−1 , E i , Fi }i=1,2 , with ∗-algebra structure K i = K i∗ and Fi = E i∗ , and relations [K i , K j ] = 0, K i E i K i−1 = q E i ,

[E i , Fi ] =

K i2 − K i−2 , q − q −1

[E i , F j ] = 0 if i = j,

K i E j K i−1 = q −1/2 E j if i = j,

and (Serre relations) E i2 E j − (q + q −1 )E i E j E i + E j E i2 = 0

∀i = j.

(2.1)

844

F. D’Andrea, G. Landi

Additional relations for the Fi ’s are obtained from the above ones by taking their ∗-conjugated. Our Uq (su(3)) is the ‘compact’ real form of the Hopf algebra denoted U˘ q (sl(3)) in Sect. 6.1.2 of [13]. With the q-commutator defined as [a, b]q := ab − q −1 ba, relations (2.1) can be rewritten in the form [E i , [E j , E i ]q ]q = 0 or [[E i , E j ]q , E i ]q = 0. Counit, antipode and coproduct are given by (K i ) = 1, (E i ) = (Fi ) = 0, −1 S(K i ) = K i , S(E i ) = −q E i ,

S(Fi ) = −q −1 Fi ,

(K i ) = K i ⊗ K i , (E i ) = E i ⊗ K i + K i−1 ⊗ E i , (Fi ) = Fi ⊗ K i + K i−1 ⊗ Fi , for i = 1, 2. An easy check on generators gives for the square of the antipode: S 2 (h) = (K 1 K 2 )4 h (K 1 K 2 )−4 ,

for all h ∈ Uq (su(3)) .

(2.2)

For obvious reasons we denote Uq (su(2)) the Hopf ∗-subalgebra of Uq (su(3)) generated by the elements {K 1 , K 1−1 , E 1 , F1 }, while Uq (u(2)) denotes the Hopf ∗-subalgebra generated by Uq (su(2)) and L = K 1 K 22 and L −1 = (K 1 K 22 )−1 . We shall also use the notation Uq (u(1)) for the Hopf ∗-subalgebra generated by L and L −1 . We need finite-dimensional highest weight irreducible representations for which the K i ’s are positive operators; these are all the finite-dimensional irreducible representations that are well defined for q → 1. For Uq (su(3)) these representations are labelled by a pair of non-negative integers (n 1 , n 2 ) ∈ N2 . Basis vectors of the representation (n 1 , n 2 ) carry a multi-index j = ( j1 , j2 , m) satisfying the constraints ji = 0, 1, 2, . . . , n i , i = 1, 2,

and

1 2 ( j1

+ j2 ) − |m| ∈ N.

(2.3)

The representation (n 1 , n 2 ) is conveniently described as follows. We let |↑ denote its highest weight vector, i.e. E i |↑ = 0 and K i |↑ = q n i /2 |↑ for i = 1, 2, and we let X in 1 ,n 2 ∈ Uq (su(3)) be the element n 1 − j1

X nj11,,nj22,m := N nj11,,nj22,m

k=0





q −k( j1+j2+k+1) n 1 − j1 21 ( j1 + j2 )−m+k n − j −k j +k F1 [F2 , F1 ]q 1 1 F2 2 , [ j1 + j2 + k + 1]! k (2.4)

with N nj11,,nj22,m

  j1 + j2 [  + m]! [n 2 − j2 ]![ j1 ]! [n 1 + j2 + 1]![n 2 + j1 + 1]! . := [ j1 + j2 + 1] j +2 j 1 2 [ 2 − m]! [n 1 − j1 ]![ j2 ]! [n 1 ]![n 2 ]![n 1 + n 2 + 1]!

the vector space Vn 1 ,n 2 underlying the representation is spanned by the vectors Then,

i := X n 1 ,n 2 |↑ . Using the commutation rules of Uq (su(3)) one proves that in this basis i

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

845

the representation is given by (cf. [7, Sect. 2]) K 1 | j1 , j2 , m := q m | j1 , j2 , m , 3

1

K 2 | j1 , j2 , m := q 4 ( j1 − j2 )+ 2 (n 2 −n 1 −m) | j1 , j2 , m ,  E 1 | j1 , j2 , m := [ 21 ( j1 + j2 ) − m][ 21 ( j1 + j2 ) + m + 1] | j1 , j2 , m + 1 ,  E 2 | j1 , j2 , m := [ 21 ( j1 + j2 ) − m + 1] A j1 , j2 | j1 + 1, j2 , m − 21  + [ 21 ( j1 + j2 ) + m] B j1 , j2 | j1 , j2 − 1, m − 21 , with Fi the transpose of E i , and with coefficients given by [n 1 − j1 ][n 2 + j1 + 2][ j1 + 1] , A j1 , j2 := [ j1 + j2 + 1][ j1 + j2 + 2] ⎧ ⎨ [n 1 + j2 + 1][n 2 − j2 + 1][ j2 ] if j1 + j2 = 0, B j1 , j2 := [ j1 + j2 ][ j1 + j2 + 1] ⎩ 1 if j1 + j2 = 0. 

It is a ∗-representation for the inner product i|i  = δi,i  . With our notation, the high

est weight vector is |↑ = n 1 , 0, 21 n 1 . The representation ρ n 1 ,n 2 : Uq (su(3)) →   End(Vn 1 ,n 2 ), has matrix elements ρi,n 1j,n 2 (h) := i|h| j . Since L = K 1 K 22 commutes with all elements of Uq (su(2)), any irreducible representation σ,N of Uq (u(2)) is the product of a representation of Uq (su(2)) (these are labelled by the ‘spin’  with  ∈ 21 N), and a representation of charge N of the Hopf ∗-subalgebra Uq (u(1)) generated by L, that is σ,N (L) = q N . Restricting the representation ρ n 1 ,n 2 of Uq (su(3)) to Uq (u(2)) we get: n 1 +n 2 ρ n 1 ,n 2 U (u(2))  q 2=0

min{,n 1 −}

σ,N ,

N +n 1 −n 2 =max{−,−n 2 } 3

and thus the representations σ,N of Uq (u(2)) appearing as components in at least one representation of Uq (su(3)) are only those for which  + 13 N ∈ 13 Z, or equivalently  + N ∈ Z (since 2 ∈ Z). For later use (in Sect. 6.2) we need the Casimir operator; it is the operator given (in a slightly enlarged algebra, cf. [7]) by     Cq = (q − q −1 )−2 (H + H −1 ) (q K 1 K 2 )2 + (q K 1 K 2 )−2 + H 2 + H −2 − 6     + q H K 22 + q −1 H −1 K 2−2 F1 E 1 + q H −1 K 12 + q −1 H K 1−2 F2 E 2 +q H [F2 , F1 ]q [E 1 , E 2 ]q + q H −1 [F1 , F2 ]q [E 2 , E 1 ]q , with H := (K 1 K 2−1 )2/3 . In the representation ρ n 1 ,n 2 it has spectrum Cq V = [ 13 (n 1 − n 2 )]2 + [ 13 (2n 1 + n 2 ) + 1]2 + [ 13 (n 1 + 2n 2 ) + 1]2 . n 1 ,n 2

(2.5)

(2.6)

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2.2. The quantum SU(3) group. The deformation A(SUq (3)) of the Hopf ∗-algebra of representative functions of SU(3) is given in [18] (see also [13], Sect. 9.2). As a ∗-algebra it is generated by elements u ij , with i, j = 1, 2, 3, having commutation relations j

j

u ik u k = qu k u ik , u ik u kj = qu kj u ik , ∀i < j, [u li , u k ] = 0, [u ik , u l ] = (q − q −1 )u li u k , ∀i < j, k < l. j

j

There is also a cubic relation  π ∈S3

j

(−q)||π || u 1π(1) u 2π(2) u 3π(3) = 1,

with the sum over all permutations π of the three elements {1, 2, 3} and ||π || is the length of π . The ∗-structure is given by (u ij )∗ = (−q) j−i (u lk11 u lk22 − qu lk21 u lk12 ), with {k1 , k2 } = {1, 2, 3}  {i} and {l1 , l2 } = {1, 2, 3}  { j}, as ordered sets. Thus for example (u 11 )∗ = u 22 u 33 − qu 23 u 32 . Coproduct, counit and antipode are the standard ones:  j u ik ⊗ u kj , (u ij ) = δ ij , S(u ij ) = (u i )∗ . (u ij ) = k

There is a non-degenerate dual pairing (cf. [13], Sect. 9.4) , : Uq (su(3)) × A(SUq (3)) → C, which allows one to define left  and right  canonical actions of Uq (su(3)) on A(SUq (3)), 



and a  h = h, a(1) a(2) . h  a = a(1) h, a(2) , By using the counit on these equations one gets that h, a = (h  a) = (a  h),

(2.7)

for all h ∈ Uq (su(3)) and a ∈ A(SUq (3)). Also, it is known that the left (resp. right) canonical action is dual to the right (resp. left) regular action. For the case at hand this is the statement that 





x h y, a = x, a(1) h, a(2) y, a(3) = h, y  a  x , (2.8) for all x, y, h ∈ Uq (su(3)) and a ∈ A(SUq (3)). On generators the actions are given by 1

j

1

j

K i  u k = q 2 (δi+1,k −δi,k ) u k , j

u k  K i = q 2 (δi+1, j −δi, j ) u k , j

j

j

E i  u k = δi,k u i+1 , j

u k  E i = δi+1, j u ik ,

j

j

Fi  u k = δi+1,k u i , j

u k  Fi = δi, j u i+1 k .

(2.9)

By the Peter-Weyl theorem [13] a linear basis {ti,n 1j,n 2 } of A(SUq (3)) is defined implicitly by h, ti,n 1j,n 2 = ρi,n 1j,n 2 (h),

∀h ∈ Uq (su(3)).

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

From the definition it follows that (ti,n 1j,n 2 ) =



847

n 1 ,n 2 k ti,k

n 1 ,n 2 ⊗ tk, and (ti,n 1j,n 2 )∗ = j

S(t nj,i1 ,n 2 ) (that is, t n 1 ,n 2 is a unitary matrix). Also, this matrix transforms according to the representation ρ n 1 ,n 2 under the left/right canonical action:  n ,n n ,n  n ,n n 1 ,n 2 h  ti,n 1j,n 2 = ti,k1 2 ρk,1j 2 (h), ti,n 1j,n 2  h = ρi,k1 2 (h)tk, j , (2.10) k

k

for all h ∈ Uq (su(3)). For (n 1 , n 2 ) = (0, 1) the elements u ij

ti,n 1j,n 2

are just the generators

(properly reordered). Let us describe them in general.

Proposition 2.1. We have ti,n 1j,n 2 = X nj 1 ,n 2  {(u 11 )∗ }n 1 (u 33 )n 2  (X in 1 ,n 2 )∗ ,

(2.11)

where X in 1 ,n 2 are given in (2.4). Proof. By the Poincaré-Birkhoff-Witt theorem [13, Thm. 6.24 ], the vector space Uq (su(3)) is spanned by elements of the form h = f K 1r K 2s e, where e is a product of n 1 ,n 2 (h) = 0 E i ’s and f is a product of Fi ’s. Since E i |↑ = ↑| Fi = 0, it follows that ρ↑,↑

n 1 ,n 2 (h) = q 2 (r n 1 +sn 2 ) , since unless e and f have degree zero. For e = f = 1 one gets ρ↑,↑ K i |↑ = q n i /2 |↑ . 1 ,n 2 Let us define u n↑,↑ := {(u 11 )∗ }n 1 (u 33 )n 2 . From the explicit formulæ (2.9) it follows   1 ,n 2 1 ,n 2 1 ,n 2 = 0 unless that E i  u n↑,↑ = 0 = u n↑,↑  Fi . Thus, by using (2.7), f K 1r K 2s e, u n↑,↑ 1

1 ,n 2 1 ,n 2 1 ,n 2 e = f = 1. When e = f = 1, from K i  u n↑,↑ = u n↑,↑  K i = q n i /2 u n↑,↑ it derives     1 n 1 ,n 2 n ,n n ,n (r n +sn ) s r 1 2 1 2 1 2 that K 1 K 2 , t↑,↑ = q 2 . Hence, h, u ↑,↑ = ρ↑,↑ (h) for all h ∈ Uq (su(3)). But this implies that  n 1 ,n 2

 

h, ti, j = ρi,n 1j,n 2 (h) = i|h| j = ↑ |(X in 1 ,n 2 )∗ h X nj 1 ,n 2 | ↑

 n 1 ,n 2 1 ,n 2 = ρ↑,↑ ((X in 1 ,n 2 )∗ h X nj 1 ,n 2 ) = (X in 1 ,n 2 )∗ h X nj 1 ,n 2 , u n↑,↑ 

1 ,n 2 = h, X nj 1 ,n 2  u n↑,↑  (X in 1 ,n 2 )∗ , 1 ,n 2 using the identity (2.8). Thus ti,n 1j,n 2 = X nj 1 ,n 2  u n↑,↑  (X in 1 ,n 2 )∗ , which is just (2.11).  

2.3. The quantum 5-sphere and the quantum projective plane. The most natural way to arrive to CPq2 is via the 5-sphere Sq5 . We shall therefore start from the algebra of coordinate functions on the latter, defined as   A(S 5 ) := a ∈ A(SUq (3)) a  h = (h)a, ∀h ∈ Uq (su(2)) q

and, as such, it is the ∗-subalgebra of A(SUq (3)) generated by elements {u i3 , i = 1, . . . , 3} of the last ‘row’. In [20] it is proved to be isomorphic, through the identification z i = u i3 , to the abstract ∗-algebra with generators z i , z i∗ , i = 1, 2, 3, and relations: z i z j = qz j z i ∀i < j,

z i∗ z j = qz j z i∗ ∀i = j,

[z 1∗ , z 1 ] = 0, [z 2∗ , z 2 ] = (1 − q 2 )z 1 z 1∗ , [z 3∗ , z 3 ] = (1 − q 2 )(z 1 z 1∗ + z 2 z 2∗ ), z 1 z 1∗ + z 2 z 2∗ + z 3 z 3∗ = 1. These relations will be useful later on.

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The ∗-algebra A(CPq2 ) of coordinate functions on the quantum projective plane CPq2 is the fixed point subalgebra of A(SUq (3)) for the right action of Uq (u(2)),   A(CPq2 ) := a ∈ A(SUq (3)) a  h = (h)a, ∀h ∈ Uq (u(2)) ∼ = {a ∈ A(Sq5 ) a  K 1 K 22 = a}.

(2.12)

Clearly, both A(Sq5 ) and A(CPq2 ) are left Uq (su(3))-module ∗-algebras. The ∗-algebra A(CPq2 ) is generated by elements pi j := z i∗ z j = (u i3 )∗ u 3j = p ∗ji and from the relations of A(Sq5 ) one gets analogous commutation relations for A(CPq2 ): pi j pkl = q sign(i−k)+sign(l− j) pkl pi j pi j p jk = q

× p jk pi j − (1 − q 2 ) pi j p ji = q ×

if i = l and j = k,

sign(i− j)+sign(k− j)+1

2sign(i− j)

 l
 l< j

if i = k,

pil plk

p ji pi j + (1 − q ) 2

q 2sign(i− j) p jl pl j −

 l< j

 pil pli

if i = j,

with sign(0) := 0. The elements pi j are the matrix entries of a projection P = ( pi j ), that is P 2 = P = P ∗ . This projection has q-trace: Trq (P) := q 4 p11 + q 2 p22 + p33 = 1. This projection gives the ‘tautological line bundle’ on CPq2 ; general line bundles will be discussed in Sect. 3.2 below. Remark 2.2. The two equalities in (2.12) give algebra inclusions A(CPq2 ) → A(SUq (3)) and A(CPq2 ) → A(Sq5 ). These could be seen as ‘noncommutative principal bundles’ with ‘structure Hopf algebra’ Uq (2) and U (1) respectively. Indeed, the action of Uq (u(2)) on A(SUq (3)) dualizes to a coaction of Uq (2) for which A(CPq2 ) is the algebra of coinvariants. Analogously, the action of Uq (u(1)) on A(Sq5 ) dualizes to a coaction of Uq (1)  U (1) for which again A(CPq2 ) is the algebra of coinvariants. These are noncommutative analogues of the classical U(2)-principal bundle SU(3) → CP2 and U (1)-principal bundle S 5 → CP2 .

3. Hermitian Vector Bundles On the manifold CPq2 we shall select suitable ‘monopole’ bundles. These will come as associated bundles to the principal fibrations on CPq2 mentioned in the previous section. We start with the general construction of associated bundles: the starting idea is to define their modules of ‘sections’ as equivariant vector valued functions on SUq (3).

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

849

3.1. Hermitian bundles of any rank. With any n-dimensional ∗-representation σ : Uq (u(2)) → End(Cn ), one associates an A(CPq2 )-bimodule of equivariant elements: M(σ ) = A(SUq (3))σ Cn   := v ∈ A(SUq (3))n σ (S(h (1) ))(v  h (2) ) = (h)v ; ∀h ∈ Uq (u(2)) , where v = (v1 , . . . , vn )t is a column vector and row by column multiplication is implied. It is easy to see that M(σ ) is also a left A(CPq2 )  Uq (su(3))-module. In particular, A(CPq2 ) = M() is the module associated to the trivial representation given by the counit. A natural A(CPq2 )-valued Hermitian structure on M(σ ) is defined by M(σ ) × M(σ ) → A(CPq2 ),

(v, v  ) → v † · v  ,

(3.1)

where v † is the conjugate transpose of v and again one multiplies row by column. Indeed, if v, v  ∈ M(σ ), for h ∈ Uq (u(2)), and denoting t := S(h)∗ , we have that v † · v   h = (v †  h (1) )(v   h (2) ) = (v †  h (1) )σ (h (2) )σ (S(h (3) ))(v   h (4) ) = (v †  h (1) )σ (h (2) )(h (3) )v  = {σ (h ∗(2) )v  S(h (1) )∗ }∗ v   ∗ = σ (S(t(1) ))v  t(2) ) v  = (t)∗ v † · v  = (h) v † · v  ; thus v † · v  ∈ A(CPq2 ) as it should be. We think of each M(σ ) endowed with this Hermitian structure as the module of sections of an equivariant ‘Hermitian vector bundle’ over CPq2 . Remark 3.1. Composing the Hermitian structure with (the restriction to A(CPq2 ) of) the Haar functional 

ϕ : A(SUq (3)) → C, we get a non-degenerate C-valued inner product on M(σ ), v, v  := ϕ(v † · v  ). This will be used later on in Sect. 5.3 when defining a Hodge operator on forms and gauged Laplacian operators on modules of sections. Clearly M(σ1 ) ⊕ M(σ2 )  M(σ1 ⊕ σ2 ) for any pair of representations σ1 , σ2 . We have also an inclusion M(σ1 ) ⊗A(CPq2 ) M(σ2 ) ⊂ M(σ ), where σ := σ1 ⊗ σ2 is the Hopf tensor product of the two representations. Indeed, for v = (v1 , . . . , vn )t ∈ M(σ1 ) and v  = (v1 , . . . , vn  )t ∈ M(σ2 ), the product w = v ⊗A(CPq2 ) v  satisfies σ (S(h (1) )) · (w  h (2) ) = σ1 (S(h (1) )(1) )(v  h (2)(1) )⊗A(CPq2 ) σ2 (S(h (1) )(2) )(v   h (2)(2) ) = σ1 (S(h (2) ))(v  h (3) ) ⊗A(CPq2 ) σ2 (S(h (1) ))(v   h (4) ) = (h (2) )v ⊗A(CPq2 ) σ2 (S(h (1) ))(v   h (3) ) = v ⊗A(CPq2 ) σ2 (S(h (1) ))(v   h (2) ) = (h)v ⊗A(CPq2 ) v  = (h)w, where we used anti-comultiplicativity of the antipode, S(h)(1) ⊗ S(h)(2) = S(h (2) ⊗ h (1) ). Thus, w ∈ M(σ ). The inclusion is an isometry for the natural inner product



 v1 ⊗A(CPq2 ) v2 , v1 ⊗A(CPq2 ) v2 = v1 , v1 v2 , v2 on M(σ1 ) ⊗A(CPq2 ) M(σ2 ).

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F. D’Andrea, G. Landi

For the representations discussed at the end of Sect. 2.1, σ,N : Uq (u(2)) → End(C2+1 ), with  ∈ 21 N and  + N ∈ Z, we denote ,N := M(σ,N ). All these ,N are finitely-generated and projective as one sided (i.e. both as left or right) A(CPq2 )modules. Now, since there always exists an irreducible representation ρ n 1 ,n 2 of Uq (su(3)) containing σ,N as a summand, and since M(σ,N ) is a direct summand (as a bimodule) in the corresponding bimodule M(ρ n 1 ,n 2 |Uq (u(2)) ), to establish the statement it is enough to show that all the M(ρ n 1 ,n 2 |Uq (u(2)) ) are free both as left and right (but not as bimodules!) A(CPq2 )-modules. But this is easy; indeed, as left A(CPq2 )-modules, we have the following isomorphism φ : A(CPq2 )dim ρ

n 1 ,n 2

→ M(ρ n 1 ,n 2 |Uq (u(2)) ), n 1 ,n 2

φ −1 : M(ρ n 1 ,n 2 |Uq (u(2)) ) → A(CPq2 )dim ρ

φ(a)i :=

, φ −1 (v)i :=

 

n 1 ,n 2 j a j ti, j , n 1 ,n 2 ∗ j v j (t j,i ) ,

where ti,n 1j,n 2 ∈ A(SUq (3)) are the elements defined in (2.11). The maps φ and φ −1 are well defined from (2.10); and clearly, they are left A(CPq2 )-module maps. That they are one the inverse of the other follows from the unitarity of the matrix t n 1 ,n 2 . Similarly, as right A(CPq2 )-modules we have the following isomorphism φ  : A(CPq2 )dim ρ

n 1 ,n 2

→ M(ρ n 1 ,n 2 |Uq (u(2)) ),

φ −1 : M(ρ n 1 ,n 2 |Uq (u(2)) ) → A(CPq2 )dim ρ

n 1 ,n 2

φ  (a)i :=

, φ −1 (v)i :=

 

n 1 ,n 2 j ti, j a j , n 1 ,n 2 ∗ j (t j,i ) v j .

3.2. Line bundles. In the construction of the vector bundles over CPq2 we have in particular 0,0 = A(CPq2 ). Moreover 0,N are ‘sections’ of ‘line bundles’; before we discuss them in more detail in this section, we note that they can equivalently be obtained out of the sphere Sq5 : 0,N  {η ∈ A(Sq5 ) η  K 1 K 22 = q N η}.

(3.2)

We are ready to compute projections PN for the modules 0,N , thus realizing them as 0,N  PN A(CPq2 )d N as right modules, and 0,N  A(CPq2 )d N P−N as left modules (notice the ‘−’ sign; and remember that N labels representations), for a suitable integer dN . Lemma 3.2. For N ≥ 0 we have  

j+k+l=N

[ j, k, l]!(z 1 z 2k z l3 )(z 1 z 2k z l3 )∗ = 1,

j+k+l=N

q 2( j−l) [ j, k, l]!(z 1 z 2k z l3 )∗ (z 1 z 2k z l3 ) = q −2N ,

j

j

j

where [ j, k, l]! are the q-trinomial coefficients in (1.1).

j

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

Proof. We seek coefficients c j,k,l (N ) such that

851



j

j+k+l=N

c j,k,l (N )(z 1 z 2k z l3 )

(z 1 z 2k z l3 )∗ = 1. From the algebraic identity  j j 1= c j,k,l (N + 1)(z 1 z 2k z l3 )(z 1 z 2k z l3 )∗ j+k+l=N +1  j j = c j,k,l (N )(z 1 z 2k z l3 )(z 1 z 1∗ + z 2 z 2∗ + z 3 z 3∗ )(z 1 z 2k z l3 )∗ j+k+l=N  j+1 j+1 = q −2(k+l) c j,k,l (N )(z 1 z 2k z l3 )(z 1 z 2k z l3 )∗ j+k+l=N  j j + q −2l c j,k,l (N )(z 1 z 2k+1 z l3 )(z 1 z 2k+1 z l3 )∗ j+k+l=N  j j k l+1 ∗ + c j,k,l (N )(z 1 z 2k z l+1 3 )(z 1 z 2 z 3 ) , j

j+k+l=N

we get the recursive equations c j,k,l (N + 1) = q −2(k+l) c j−1,k,l (N ) + q −2l c j,k−1,l (N ) + c j,k,l−1 (N ),

(3.3)

where N + 1 = j + k + l and with ‘initial datum’ c0,0,0 (0) = 1. Thus, c j,k,l (N ) are the vertices of a q-Tartaglia tetrahedron. Since [ j + k + l] = q −k−l [ j] + q j−l [k] + q j+k [l], one verifies that c j,k,l (N ) = [ j, k, l]! is a solution by plugging it into (3.3). Similarly, since q 4 z 1∗ z 1 + q 2 z 2∗ z 2 + z 3∗ z 3 = 1, we have the algebraic identity  j j 1= d j,k,l (N + 1)(z 1 z 2k z l3 )∗ (z 1 z 2k z l3 ) j+k+l=N +1  j j d j,k,l (N )(z 1 z 2k z l3 )∗ (q 4 z 1∗ z 1 + q 2 z 2∗ z 2 + z 3∗ z 3 )(z 1 z 2k z l3 ) = j+k+l=N  j+1 j+1 q 4 d j,k,l (N )(z 1 z 2k z l3 )∗ (z 1 z 2k z l3 ) = j+k+l=N  j j q −2 j+2 d j,k,l (N )(z 1 z 2k+1 z l3 )∗ (z 1 z 2k+1 z l3 ) + j+k+l=N  j ∗ j k l+1 q −2( j+k) d j,k,l (N )(z 1 z 2k z l+1 + 3 ) (z 1 z 2 z 3 ), j+k+l=N

that gives the recursive equations on the coefficients d j,k,l (N + 1) = q 4 d j−1,k,l (N ) + q −2 j+2 d j,k−1,l (N ) + q −2( j+k) d j,k,l−1 (N ), with ‘initial datum’ d0,0,0 (0) = 1. The solution is d j,k,l (N ) = q 2N q 2( j−l) [ j, k, l]!, as  one can check by using the identity [ j + k + l] = q k+l [ j] + q l− j [k] + q − j−k [l].  Proposition 3.3. Define  j N (ψ j,k,l )∗ := [ j, k, l]! z 1 z 2k z l3 ,  j N (ψ j,k,l )∗ := q −N + j−l [ j, k, l]! (z 1 z 2k z l3 )∗ ,

if N ≥ 0 and with j + k + l = N , if N ≤ 0 and with j + k + l = −N .

N and P be the projection – of size Let  N be the column vector with components ψ j,k,l N 1 d N := 2 (|N | + 1)(|N | + 2) – given by

PN :=  N  N† .

(3.4)

852

F. D’Andrea, G. Landi

Then one has 0,N  PN A(CPq2 )d N as right A(CPq2 )-modules and 0,N  A(CPq2 )d N P−N as left A(CPq2 )-modules. Proof. The column vector  N has entries in number d N = 21 (|N | + 1)(|N | + 2) and  N† is a row vector of the same size. By the previous lemma  N†  N = 1, so PN :=  N  N† is a projection. Next, consider the right A(CPq2 )-module map 

v = (v j,k,l ) →  N† · v =

PN A(CPq2 )d N → A(SUq (3)),

N (ψ j,k,l )∗ v j,k,l .

i+ j+k=N 1

Since each z i is Uq (su(2))-invariant and z i K 2 = q 2 z i , it follows that  N† h = (h) N† for all h ∈ Uq (su(2)) and  N†  K 1 K 22 = q N  N† . Hence, the image of v is in 0,N . The inverse A(CPq2 )-module map is 0,N → PN A(CPq2 )d N ,

a →  N a,

thus proving that we have an isomorphism of right A(CPq2 )-modules. The proof for the left module structure is completely analogous, the two module maps being in this case  †  v → i+ j+k=N v j,k,l ψ −N j,k,l and a  → a−N .  0,N −N ,0 It turns out that what we just computed are the elements t↑,i for N ≥ 0 (resp. t↑,i

if N ≤ 0). The next proposition yields the exact relation with the matrix entries of  N† . Proposition 3.4. It holds that ⎧   0,N ∗ ⎨(t0,i ) , with i = 0, j + k, 21 (k − j) and for all N ≥ 0, N = ψ j,k,l ⎩(t −N ,0 )∗ , with i =  j + k, 0, 1 ( j − k) and for all N ≤ 0 . 0,i 2 Proof. For N ≥ 0 and i = (0, 2l, m) the generic label of the representation (0, N ), definition (2.11) gives [N − 2l]! [l + m]! l−m 2l 1 (0,N ) 0,N F t0,i = Xi  z 3N = F2  z 3N . [2l]! [N ]! [l − m]! 1 Induction on N yields F2  z 3N = q −

N −1 2

[N ]z 2 z 3N −1 ,

and induction on l yields F22l  z 3N = q −

N −1 2

q −l(2l−1)

[N ]! z 2l z N −2l . [N − 2l]! 2 3

Similarly, changing the labels 3 → 2 and 2 → 1: F1l−m  z 22l = q −

2l−1 2

1

q − 2 (l−m)(l−m−1)

[2l]! l−m l+m z z2 . [l + m]! 1

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

853

Thus 0,N = t0,i

=



[N ]! [N −2l]![l+m]![l−m]! N ∗ (ψl−m,l+m,N −2l ) ,

q−

N −1 2

q −l(2l−1) q −

2l−1 2

l+m N −2l q − 2 (l−m)(l−m−1) z l−m 1 z2 z3 1

which establishes the case N ≥ 0. The proof for the case N ≤ 0 is similar.

 

It is computationally useful to introduce the left action h → Lh , of Uq (su(3)) on A(SUq (3)), given by Lh a := a  S −1 (h); it satisfies Lx (ab) = (Lx(2) a)(Lx(1) b) due to the presence of the antipode. Also, it is a unitary action for the inner product on A(SUq (3)) coming from the Haar state ϕ. The proof is a simple computation:         ϕ (Lh ∗ a)∗ b = ϕ {a  S −1 (h ∗ )}∗ b = ϕ {a ∗  h}b = ϕ {a ∗  h (1) }(h (2) )b     = ϕ {a ∗  h (1) }{b  S −1 (h (3) )h (2) } = ϕ {a ∗ (b  S −1 (h (2) ))}  h (1)     = (h (1) )ϕ a ∗ {b  S −1 (h (2) )} = ϕ a ∗ {b  S −1 (h)}   = ϕ a ∗ (Lh b) , for all a, b ∈ A(SUq (3)) and h ∈ Uq (su(3)). With this action the bimodule M(σ ) can be viewed as the set of elements of A(SUq (3)) ⊗ Cdim σ that are invariant under the action Lh (1) ⊗ σ (h (2) ) of h ∈ Uq (u(2)).

4. Characteristic Classes Equivalence classes of finitely generated projective (left or right) modules over an algebra A – the algebraic counterpart of vector bundles – are elements of the group K 0 (A). Equivalence classes of even Fredholm modules – the algebraic counterpart of ‘fundamental classes’ – gives a dual group K 0 (A). The natural non-degenerate pairing between K-theory and K-homology, which pairs projective modules with even Fredholm modules, gives index maps K 0 (A(CPq2 )) → Z. For the quantum projective plane, K 0 (A(CPq2 ))  Z3  K 0 (A(CPq2 )). The result for K-theory can be proved viewing the corresponding C ∗ -algebra as the Cuntz–Krieger algebra of a graph [12]. The group K 0 is given as the cokernel of the incidence matrix canonically associated with the graph; the result for K-homology can be proven using the same techniques: the group K 0 is now given as the kernel of the transposed matrix [3]. It is worth mentioning that K 1 (A(CPq2 )) = K 1 (A(CPq2 )) = 0 with the group K 1 (resp. K 1 ) given as the kernel (resp. the cokernel) of the incidence matrix (resp. the transposed matrix). Thus a finitely generated projective (left or right) module over A(CPq2 ) is uniquely identified by three integers and these are obtained by pairing the corresponding idempotent with the three generators of the K-homology. This section is devoted to the explicit construction of three Fredholm modules that, in the next section will be shown to be generators of the K-homology by pairing them with suitable idempotents.

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4.1. Fredholm modules and their characters. We recall that a (k + 1)-summable even Fredholm module for the algebra A is a triple (π, H, F) consisting of a Z2 -graded Hilbert space H = H+ ⊕ H− , a graded representation π = π + ⊕ π − : A → B(H+ ) ⊕ B(H− ) and an odd operator F such that [F, a0 ][F, a1 ] . . . [F, ak ] is of traceclass for any a0 , . . . , ak ∈ A. With a (k + 1)-summable even Fredholm module there are canonically associated even cyclic cocycles chn(π,H,F) , for 2n ≥ k. The map chn(π,H,F) : A2n+1 → C is given by chn(π,H,F) (a0 , . . . , a2n+1 ) := 21 (−1)n Tr H (γ F[F, a0 ][F, a1 ] . . . [F, a2n ]), where γ is the grading and the symbol π for the representation is understood. We recall that a cyclic 2n-cocycle τn is a C-linear map τn : A2n+1 → C which is cyclic, i.e. it satisfies τn (a0 , a1 , . . . , a2n ) = τn (a2n , a0 , . . . , an−1 ), and is a Hochschild coboundary, i.e. it satisfies b τn = 0, with b the boundary operator: b τn (a0 , . . . , a2n+1 ) :=

2n  (−1) j τ (a0 , . . . , a j a j+1 , . . . , a2n+1 ) j=0

−τ (a2n+1 a0 , a1 , . . . , a2n ). When applied to an idempotent one gets a pairing , : K 0 (A) × K 0 (A) → Z, [(π, H, F)], [e] := chn(π,H,F) ([e]) = 21 (−1)n Tr H⊗Cm (γ F[F, e]2n+1 ),

(4.1)

where m is the size of e and matrix multiplication is understood. The pairing is integer + to valued, being the index of the Fredholm operator πm− (e)Fm πm+ (e) from πm+ (e)Hm − − πm (e)Hm , with the natural extensions Hm = H ⊗ Cm ,

Fm = F ⊗ 1,

forming an even Fredholm module over Mm = A ⊗ Mm (C). The result of the pairing depends only on the classes of e and of the Fredholm module (for details see [2]). To construct three independent Fredholm modules we need (at least) four representations. Four is exactly the number of irreducible representations of the algebra A(CPq2 ), that we describe in the next section. It is peculiar of the quantum case that one needs to consider only irreducible representations: at q = 1 irreducible representations are 1-dimensional and give only one of the generators of the K-homology (the trivial Fredholm module). An additional true ‘quantum effect’ is that for CPq2 one gets three independent 1-summable Fredholm modules corresponding to independent traces on A(CPq2 ). Thus all relevant information leaves in degree zero. In contrast, for the classical CP2 one needs (there exist) cohomology classes in degree zero, two and four. Both at the algebraic and at the C ∗ -algebraic levels there is a sequence A(CPq2 ) → A(CPq1 ) → A({pt}) = C → A(∅) = 0,

(4.2)

of ∗-algebra morphisms. In reverse order: the empty space, the space with one point, the quantum projective line and the quantum projective plane.

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The first map is the quotient of A(CPq2 ) by the ideal generated by p1i and pi1 : roughly speaking, by putting p1i = pi1 = 0 the remaining generators satisfy the relations of CPq1 , the quantum projective line – also known as the standard Podle´s sphere – and any ∗-representation of A(CPq2 ) with p1i in the kernel comes from a representation of CPq1 . The second map is the further quotient by the ideal generated by p2i and pi2 : roughly speaking, in A(CPq2 ) we send pi j → χ0 ( pi j ) = δi3 δ j3 and get the algebra C; this is the only non-trivial character χ0 of the algebra A(CPq2 ), the ‘classical point’ of CPq2 . The last map is simply 1 → 0. 4.2. The rank and the 1st Chern number of a projective module. For A(∅) there is only one irreducible representation, the trivial one, not enough to construct a Fredholm module (and indeed, K 0 (0) = 0). For C = A({pt}) there is the representation coming from the morphism C → 0, and one further faithful irreducible representation given by the identity map c → c. These two are all1 the irreducible ∗-representations of C, and are enough to construct an even Fredholm module,     c 0 0 1 H0 = C ⊕ C, π0 (c) = , F0 = , 0 0 1 0 which generates K 0 (C) = Z . The pairing (4.1) with K 0 (C) = Z (projections in Mat ∞ (C) are equivalent iff they have the same rank) is given by the matrix trace: ch0(π0 ,H0 ,F0 ) ([e]) = Tr(e). The pullback of this Fredholm module to CPq2 just substitutes the representation π0 of C with the character χ0 : A(CPq2 ) → C of A(CPq2 ). Using the same symbol for the character, one gets a map ch0(π0 ,H0 ,F0 ) : K 0 (A(CPq2 )) → Z,

ch0(π0 ,H0 ,F0 ) ([e]) = Tr χ0 (e).

(4.3)

The geometrical meaning is the following: the rank of a vector bundle is the dimension of the fiber at any point x of the space, and this coincides with the trace of the corresponding projection evaluated at x. Here we have only one ‘classical point’, and the map in (4.3) computes the rank of the restriction of the vector bundle to this classical point. Notice that if the module is free, then (4.3) is really the rank of the module. We pass to the next level of the sequence (4.2), that is the algebra A(CPq1 ). There are two irreducible ∗-representations coming from the map in the sequence – the trivial one and the character of the algebra – and one further representation that is faithful and irreducible. The latter is the restriction of the representation χ1 : A(Sq5 ) → B(2 (N)) given by χ1 (z 1 ) = 0, χ1 (z 2 ) |n = q n |n ,  χ1 (z 3 ) |n = 1 − q 2(n+1) |n + 1 .

(4.4a) (4.4b) (4.4c)

1 Irreducible ∗-representations of C are 1-dimensional (it is abelian). By linearity, they are in 1 to 1 correspondence with maps 1 → p, with p ∈ C a projection. Hence p = 0 or 1, and the corresponding representations are the trivial map c  → 0, and the identity map c  → c.

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A potential Fredholm module is given by  χ1 (a) H1 = 2 (N) ⊕ 2 (N), π1 (a) = 0

0 χ0 (a) id2 (N)

However, this is not a Fredholm module over A(Sq5 ), since  1 γ F1 [F1 , π(z 3 )] = {χ1 (z 3 ) − χ0 (z 3 )} 0



 ,

0 1

F1 =

0 1

 1 . 0



is not compact. On the other hand, this is a Fredholm module when restricted to A(CPq2 ) (or A(CPq1 )). Indeed, due to the tracial relation q 4 p11 + q 2 p22 + p33 = 1, a complete set of generators for A(CPq2 ) is made of { p11 , p12 , p13 , p22 , p23 } and their adjoints. On these generators the representation χ1 is χ1 ( p1i ) = 0, ∀i = 1, 2, 3, χ1 ( p22 ) |n = q 2n |n ,  χ1 ( p23 ) |n = q n+1 1 − q 2(n+1) |n + 1 ,

(4.5a) (4.5b) (4.5c)

and χ1 ( pi j ) − χ0 ( pi j ) is of trace class for all i, j. In particular, this means that the Fredholm module is 1-summable. The associated character is ch0(π1 ,H1 ,F1 ) : K 0 (A(CPq2 )) → Z,

ch0(π1 ,H1 ,F1 ) ([e]) = Tr 2 (N)⊗Cm (χ1 − χ0 )(e), (4.6)

where m is the size of the matrix e. The value in (4.6) depends only on the restriction of the ‘vector bundle’ to the subspace CPq1 , and will be called for this reason the 1st Chern number (or also the monopole charge). 4.3. The 2nd Chern number of a projective module. Besides the representations (and the Fredholm modules) coming from the sequence (4.2), the algebra A(CPq2 ) has a further irreducible representation and a further Fredholm module, which is independent from the previous two. The representation is faithful and comes from the representation χ2 : A(Sq5 ) → B(2 (N2 )) given by χ2 (z 1 ) |k1 , k2 := q k1 +k2 |k1 , k2 ,  χ2 (z 2 ) |k1 , k2 := q k1 1 − q 2(k2 +1) |k1 , k2 + 1 ,  χ2 (z 3 ) |k1 , k2 := 1 − q 2(k1 +1) |k1 + 1, k2 .

(4.7a) (4.7b) (4.7c)

The construction of the Fredholm module is a bit involved. Let us use the labels  = 21 (k1 + k2 ) and m = 21 (k1 − k2 ). A new basis for the Hilbert space is then given by |, m with  ∈ 21 N and m = −, − + 1, . . . , . In this basis the representation reads χ2 (z 1 ) |, m = q 2 |, m ,  χ2 (z 2 ) |, m = q +m 1 − q 2(−m+1) | + 21 , m − 21 ,  χ2 (z 3 ) |, m = 1 − q 2(+m+1) | + 21 , m + 21 .

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For the third Fredholm module we take as Hilbert space H2 two copies of the linear span of orthonormal vectors |, m , with  ∈ 21 N and  + m ∈ N. The grading γ2 and the operator F2 are the obvious ones. It remains to describe the representation π2 = π+ ⊕ π− . As subrepresentation π+ we choose π+ (a) |, m :=

χ2 (a) |, m if m ≤ , χ0 (a) |, m if m > .

One checks that modulo traceclass operators: π+ ( p11 ) ∼ π+ ( p12 ) ∼ π+ ( p13 ) ∼ 0, q 2(+m) |, m if m ≤ , 0 if m > ,  q +m+1 1 − q 2(+m+1) |, m + 1 if m ≤  − 1, π+ ( p23 ) |, m ∼ 0 if m ≥ .

π+ ( p22 ) |, m ∼

We define the subrepresentation π− by adding multiplicities to χ1 . On the generators: π− ( p11 ) = π− ( p12 ) = π− ( p13 ) = 0,

 π− ( p22 ) |, m = q 2(+m) |, m , π− ( p23 ) |, m = q +m+1 1−q 2(+m+1) |, m + 1 . On each invariant subspace with a fixed , putting n =  + m one recovers the repre sentation χ1 . Since m> q 2(+m) = (1 − q 4 )−2 is finite, on the subspace m >  the operators π− ( p22 ) and π− ( p23 ) are trace class, and so π+ (a) − π− (a) is of trace class as well for all a ∈ A(CPq2 ): the Fredholm module is 1-summable with corresponding character ch0(π2 ,H2 ,F2 ) : K 0 (A(CPq2 )) → Z,

ch0(π2 ,H2 ,F2 ) ([e]) = Tr H2 ⊗Cm (π+ − π− )(e), (4.8)

where m is the size of the matrix e. The above replaces the 2nd Chern class of the module. Working with generators and relations as done in [11] for quantum spheres, it is not difficult to prove that any irreducible ∗-representation of A(CPq2 ) is equivalent to one of the representations described above. For completeness, we give the proof in App. 8. By iterating the construction, for any positive integer n one obtains the n + 2 irreducible representations of the quantum projective spaces CPqn (only one of these is faithful, the others coming from the morphism A(CPqn ) → A(CPqn−1 )), and the corresponding n + 1 Fredholm modules; details will be reported elsewhere [9]. 4.4. Chern numbers of line bundles. We know from Sect. 3.2 that the bimodule 0,N := M(σ0,N ) is isomorphic to A(CPq2 )d N P−N as left module and to PN A(CPq2 )d N as right module, with d N = 21 (|N | + 1)(|N | + 2) and PN :=  N  N† given by Prop. 3.3. We next compute rank, and 1st and 2nd Chern numbers of PN . We focus the discussion on N ≥ 0, the case N ≤ 0 being similar. Since χ0 (PN ) j,k,l| j  ,k  ,l  = δ j,0 δk,0 δl,N δ j  ,0 δk  ,0 δl  ,N , the rank given by (4.3) results in ch0(π0 ,H0 ,F0 ) ([PN ]) = 1,

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thus justifying the name ‘line bundles’ for the virtual bundles underlying the modules 0,N . The same result is valid when N ≤ 0. Next, we compute Tr Cd N (PN ) =

 j+k+l=N

q −( jk+kl+l j)

[N ]! j j (z l )∗ (z 2k )∗ (z 1 )∗ z 1 z 2k z l3 [ j]![k]![l]! 3

(4.9)

and    [N ]! χ1 (z 3∗ )l χ1 (z 2∗ )k χ1 (z 2 )k χ1 (z 3 )l , q −kl χ1 Tr Cd N (PN ) = k+l=N [k]![l]!   χ0 Tr Cd N (PN ) = χ0 (z 3∗ ) N χ0 (z 3 ) N = 1. The use of (4.4b) and (4.4c) leads to

 xn := n χ1 Tr Cd N (PN ) n  [N ]! 2k(n+l) = q q −kl (1 − q 2(n+1) )(1 − q 2(n+2) ) . . . (1 − q 2(n+l) ) k+l=N [k]![l]! 1−q 2n(N +1) N + O(q) = 1 + O(q) if n > 0, 2kn 1−q 2n q + O(q) = = k=0 N + 1 + O(q) if n = 0. As usual we can compute the index for q → 0+ (see [4,8]); we get   (xn − 1) = lim+ (xn − 1) = (x0 − 1)|q=0 = N . n≥0

q→0

n≥0

The very same result holds for N ≤ 0. So, the 1st Chern number is ch0(π1 ,H1 ,F1 ) ([PN ]) = N . It remains to compute the last Chern number. The representations π± in the third Fredholm module are the restriction of (homonymous) representations of A(Sq5 ) given as follows. With χ0 and χ2 the representation described in the previous section, one has that π+ (z i ) |, m = χ2 (z i ) |, m if m ≤ , and π+ (z i ) |, m = χ0 (z i ) |, m if m > . On the other hand, for any m,  it holds that π− (z 1 ) = 0,

π− (z 2 ) |, m = q +m |, m ,  π− (z 3 ) |, m = 1 − q 2(+m+1) |, m + 1 . The use of (4.9) yields: 



 , m π− Tr Cd N (PN ) , m =

r +s=N

=

N

r =0

q −r s

[N ]! 2r (+m+s) !s q (1 − q 2(+m+k) ) k=1 [r ]![s]!

q 2r (+m) {1 + O(q)} = 1 + N δ,−m + O(q),

for the representation π− ; for the representation π+ one gets instead

 , m π+ Tr Cd N (PN ) , m = 1

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859

if m > , and 

 , m π+ Tr Cd N (PN ) , m =

[N ]! q 2i(2+ j+k) q 2 j (+m+k) [i]![ j]![k]! !k (1 − q 2(−m+r ) ) (1 − q 2(+m+s) )

i+ j+k=N

=

× 

!j

r =1

i+ j+k=N

q −(i j+ jk+ki)

s=1

q

4i+2 j (+m)

+ O(q)

= 1 + N δ,−m + 21 N (N + 1)δ,0 + O(q),

if m ≤ . In the computation we have used things together results in 

[N ]! [i]![ j]![k]!



, m π+ Tr Cd N (PN ) − π− Tr Cd N (PN ) , m =

= q −(i j+ jk+ki) (1 + O(q)). Putting

O(q) if m > . 1 N (N + 1)δ + O(q) if m ≤ . ,0 2

Again, we compute in the limit for q → 0+ to get ch0(π2 ,H2 ,F2 ) ([PN ]) = lim+ q→0

 ∈ 21 N



1 N (N m=− 2

+ 1)δ,0 + O(q)

= 21 N (N + 1). The same formula is valid for N ≤ 0. We summarize the results in a proposition. Proposition 4.1. For any N ∈ Z, the (right) module 0,N has ‘rank’ ch0(π0 ,H0 ,F0 ) ([PN ]) = 1, ‘1st Chern number’ ch0(π1 ,H1 ,F1 ) ([PN ]) = N , and ‘2nd Chern number’ ch0(π2 ,H2 ,F2 ) ([PN ]) = 21 N (N + 1). Corollary 4.2. A complete set of generators {e1 , e2 , e3 } of K 0 (A(CPq2 )) is given by e1 = [1] the class of the rank one free (left or right) A(CPq2 )-module 0,0 , e2 the class of the left module 0,1 (or equivalently of the right module 0,−1 ), e3 the class of the left module 0,−1 (or equivalently of the right module 0,1 ). Thus, in terms of left modules, e2 is the class of the tautological bundle and e3 the class of the dual vector bundle. A complete set of generators of K 0 (A(CPq2 )) is given by the classes of the Fredholm modules (πi , Hi , Fi ), i = 0, 1, 2, given in Sect. 4.

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Proof. We have already mentioned that K 0 (A(CPq2 ))  K 0 (A(CPq2 ))  Z3 . A set of generators of the abelian group Z3 is the same as a basis of Z3 as a Z-module. Suppose we have three elements ei ∈ Z3 and three elements ϕi : Z3 → Z in the dual space of Z-linear maps; call g ∈ Mat(3, Z) the matrix with elements gi j = ϕi (e j ). ij Assume that det g = 0 and that the inverse g −1 = ((g )) of gi j is an element of G L(3, Z). Then for any linear map ψ the difference ψ − i j ψ(ei )g ϕ j vanishes on all the ei ’s and, by the linear independence over Z of the ei ’s, we deduce that any element ψ ∈ (Z3 )∗ is a sum  ψ(ei )g i j ϕ j ψ= ij

with integer coefficients ψ(ei )g i j ∈ Z. Hence, the ϕ j ’s are a basis of (Z3 )∗ . Similarly  for any v ∈ Z3 the difference v − i j ei g i j ϕ j (v) is in the kernel of all ϕi , meaning that the ei ’s are a basis of Z3 over Z. Now, let ei ∈ K 0 (A(CPq2 )) and ϕi = [(πi−1 , Hi−1 , Fi−1 )] ∈ K 0 (A(CPq2 )), i = 1, 2, 3, be the classes in the corollary. By Prop. 4.1 the matrix g is given by ⎛ ⎞ 1 1 1 g = ⎝ 0 −1 1 ⎠ 0 0 1. It is invertible in G L(3, Z) with inverse ⎛ g −1

1 = ⎝0 0

1 −1 0

⎞ −2 1 ⎠ 1.

This proves that we have generators of K-theory and K-homology.

 

5. The Differential Calculus On the line bundles described previously we aim to define and study (anti-)selfdual connections. To this end we need the full machinery of a differential calculus on CPq2 and additional ingredients like a Hodge star operator. We start with forms. 5.1. Holomorphic and antiholomorphic forms. In [7] we studied the antiholomorphic part of the differential calculus on CPq2 . Antiholomorphic forms were defined as 0,0 (CPq2 ) := A(CPq2 ), 0,1 (CPq2 ) :=  1 , 3 and 0,2 (CPq2 ) := 0,3 . 2 2

We complete the calculus presently. Each bimodule of forms i, j (CPq2 ) will be identified with a suitable bimodule M(σ ) of equivariant elements as described in the previous section. That is to say, i, j (CPq2 ) = M(σ i, j ) = A(SUq (3))σ i, j V i, j for a (not necessarily irreducible) representation σ i, j : Uq (u(2)) → Aut(V i, j ) of the Hopf algebra i, j Uq (u(2)) on V i, j  Cdim σ . The representations σ 0, j are known from [7] and the σ j,0 are obtained by conjugation. The dimension of σ i, j does not depend on q, and for q = 1 22 gives the rank r = i j of i, j (CPq2 ). There is a unique sub-representation of the Hopf tensor product σ i,0 ⊗ σ 0, j with the prescribed dimension: this allows one to identify σ i, j . The representations relevant for the calculus are listed in the following figure:

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

(0, 0)

V 0,0....

. ..... ..... .... ........ .......

V 1,1....

. ..... ..... ..... .......... .....

..... ..... ..... ..... .......

( 21 , 23 )

V 1,0....

. .... ..... ..... ......... . . . . . .

..... ..... ..... ..... .......

V 0,2....

..... ..... ........ ......

..... ..... ..... ..... .......

V 0,1....

. .... ..... ..... ......... . . . . . .

..... ..... ..... ..... .......

V 1,2....

..... ..... ..... ..... .......

V 2,0

. ..... ..... ..... .......... .....

..... ..... ...... ........

... ..... ....... ........

(0, 3)

=

..... ..... ..... ...........

..... ..... ..... ...........

( 21 , − 23 )

... ..... ....... ........

(1, 0) ⊕ (0, 0) ..... ..... ........ ........

( 21 , 23 )

V 2,1

. ..... ..... .... ......... . . . . .

..... ..... ..... ..... .......

861

..... ..... ....... ........

..... ..... ..... ...........

..... ..... ...... ........

(0, −3)

..... ..... ........ ........

( 21 , − 23 )

... ..... ......... ........

(0, 0)

V 2,2

In the diamond on the right, in the position (i, j), we give the values of spin and charge (, N ) of the representation σ i, j . For later use, we list the representations σ 1 ,N and σ1,N explicitly: 2 & 1 '     q2 0 1 0 0 0 , σ (E ) = (F ) = σ 1 ,N (K 1 ) = , σ 1 1 1 1 1 0 0, 1 0, 2 2 ,N 2 ,N 0 q− 2 ⎛ ⎛ ⎞ ⎞ q 0 0 0 1 0 1 0 ⎠ σ1,N (E 1 ) = [2] 2 ⎝ 0 0 1 ⎠ σ1,N (K 1 ) = ⎝ 0 1 0 0 0, 0 0 q −1 , ⎛ ⎞ 0 0 0 1 σ1,N (F1 ) = [2] 2 ⎝ 1 0 0 ⎠ 0 1 0, and furthermore, σ 1 ,N (K 1 K 22 ) (resp. σ1,N (K 1 K 22 )) is q N times the identity matrix. 2

( 5.2. The wedge product. We first make •,• (CPq2 ) = i, j i, j (CPq2 ) a bi-graded associative algebra. ( Let V •,• = i, j V i, j , and suppose we have a bi-graded associative left Uq (u(2))covariant product on V •,• , denoted ∧q . For ω = av and ω = a  v  , with a, a  ∈   A(SUq (3)) and v ∈ V i, j , v  ∈ V i , j , define ω ∧q ω := (a a  ) (v ∧q v  ). Using left Uq (u(2))-covariance of the product on V •,• and the fact that A(SUq (3)) is a Uq (u(2))-bimodule algebra so that aa   S −1 (h) = {a  S −1 (h (2) )}{a  S −1 (h (1) )}, we get 



(Lh (1) ⊗ σ i+i , j+ j (h (2) ))(ω ∧q ω ) = (Lh (2) a)(Lh (1) a  )(σ i, j (h (3) )v) ∧q (σ i, j (h (4) )v  ). If ω is invariant (i.e. it belongs to i, j (CPq2 )), this can be simplified and becomes 



(Lh (1) ⊗ σ i+i , j+ j (h (2) ))(ω ∧q ω ) = a(Lh (1) a  )v ∧q (σ i, j (h (2) )v  ). 



If also ω is invariant (i.e. it belongs to i , j (CPq2 )), we get 



(Lh (1) ⊗ σ i+i , j+ j (h (2) ))(ω ∧q ω ) = (h) aa  (v ∧q v  ) = (h) ω ∧q ω .

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F. D’Andrea, G. Landi 







Thus, ∧q defines a bilinear map i, j (CPq2 ) × i , j (CPq2 ) → i+i , j+ j (CPq2 ). Its associativity follows from associativity of both the products in A(SUq (3)) and V •,• . The datum (•,• (CPq2 ), ∧q ) is automatically a left A(CPq2 )Uq (su(3))-module algebra, since left and right canonical actions commute and A(SUq (3)) is a left A(CPq2 )  Uq (su(3))-module algebra. This means that h  (ω ∧q ω ) = (h (1)  ω) ∧q (h (2)  ω ), for all h ∈ Uq (su(3)) and ω, ω ∈ (CPq2 )•,• . All we need is then a graded associative left Uq (u(2))-covariant product on V •,• . For   all i, j, i  , j  , we shall now construct a left Uq (u(2))-module map ∧q : V i, j × V i , j →   V i+i , j+ j which is unique up to some normalization constants. When q = 1, one can fix the normalization, up to some phase factors and angles, by requiring that these maps are partial isometries. Requiring that vectors with real components form a subalgebra (so that real forms are an algebra), the phases must be ±1; the remaining angles and signs are then fixed by the requirement of associativity and graded commutativity of the product. For q = 1, partial isometries do not give an associative product. We determine in App. 8 the most general value of the normalization constants in order to have a left Uq (u(2))-covariant product on V •,• which is i) associative, ii) graded commutative for q = 1, and iii) it sends real vectors into real vectors. Here we just present the result. Proposition 5.1. A left Uq (u(2))-covariant graded associative product ∧q on V •,• , sending real vectors to real vectors and graded commutative for q = 1, is given by V 0,1 × V 0,1 → V 0,2 , v ∧q w := c0 μ0 (v, w)t , V 0,1 × V 1,0 → V 1,1 ,

v ∧q w := (c1 μ1 (v, w), c2 μ0 (v, w))t ,

V 0,1 × V 2,1 → V 2,2 , V 0,1 × V 1,1 V 1,0 × V 1,0 V 1,0 × V 0,1

v ∧q w := c3 μ0 (v, w)t , c0 c0 → V 1,2 , v ∧q w := μ2 (v, w)t − vw4 , [2]c1 [2]c2 → V 2,0 , v ∧q w := c4 μ0 (v, w)t ,  1 t 3 → V 1,1 , v ∧q w := −q 2 s c1 μ1 (v, w), q − 2 s c2 μ0 (v, w) , c3 c4 μ0 (v, w)t , c0 1 3 c4 c4 := −q − 2 s μ2 (v, w)t − q 2 s vw4 , [2]c1 [2]c2 c3 c4 := μ0 (v, w)t , c0 := c3 μ0 (v, w)t , 1 3 c0 c0 := −q − 2 s μ3 (v, w)t − q 2 s v4 w, [2]c1 [2]c2 c4 c4 := μ3 (v, w)t − v4 w, [2]c1 [2]c2 1 3 c3 c4 c3 c4 := −q − 2 s μ4 (v, w) − q 2 s v4 w4 , 2 [2]|c1 | [2]|c2 |2

V 1,0 × V 1,2 → V 2,2 ,

v ∧q w :=

V 1,0 × V 1,1 → V 2,1 ,

v ∧q w

V 1,2 × V 1,0 → V 2,2 ,

v ∧q w

V 2,1 × V 0,1 → V 2,2 ,

v ∧q w

V 1,1 × V 0,1 → V 1,2 ,

v ∧q w

V 1,1 × V 1,0 → V 2,1 ,

v ∧q w

V 1,1 × V 1,1 → V 2,2 ,

v ∧q w

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

863

where the maps μi ’s are 1

1

1

μ3 : R3 × R2 → R2 ,

μ0 (v, w) := [2]− 2 (q 2 v1 w2 − q − 2 v2 w1 ),   1 1 1 μ1 (v, w) := v1 w1 , [2]− 2 (q − 2 v1 w2 + q 2 v2 w1 ), v2 w2 ,   1 1 1 1 μ2 (v, w) := qv1 w2 − q − 2 [2] 2 v2 w1 , q 2 [2] 2 v1 w3 − q −1 v2 w2 ,   1 1 1 1 μ3 (v, w) := q 2 [2] 2 v1 w2 − q −1 v2 w1 , qv2 w2 − q − 2 [2] 2 v3 w1 ,

μ4 : R3 × R3 → R,

μ4 (v, w) := qv1 w3 − v2 w2 + q −1 v3 w1 .

μ0 : R2 × R2 → R, μ1 : R2 × R2 → R3 , μ2 : R2 × R3 → R2 ,

The parameters c0 , . . . , c4 ∈ R× and s = ±1 are not fixed for the time being. To get an involution we use the fact that the spin 1/2 (resp. spin 1) representation of Uq (su(2)) is quaternionic (resp. real). Rephrased in terms of the representations σ 1 ,N 2 and σ1,N of Uq (u(2)) we have the following lemma, which takes into account the fact that real/quaternionic structures change sign to N . Lemma 5.2. Let V,N = C2+1 be the vector space underlying the representation σ,N of Uq (u(2)). An antilinear map J : V,N → V,−N satisfying J 2 = (−1)2 and such that J σ,N (h) = σ,−N (S(h)∗ )J, for any h ∈ Uq (u(2)), is given, for  = 0, 1

1 2 , 1,

(5.1)

by

1

J a = a ∗ , J (v1 , v2 )t = (−q − 2 v2∗ , q 2 v1∗ )t , J (w1 , w2 , w3 )t = (−q −1 w3∗ , w2∗ , −qw1∗ )t , for any a ∈ V0,N , v ∈ V 1 ,N and w ∈ V1,N respectively. Moreover, if c0 = c4 , the map 2



:V

•,•

→ V •,• ,

(vi, j ) → (v  )i, j := (−1)i J (v j,i ),

is a graded involution, i.e. it satisfies (v  ) = v and 

(v ∧q v  ) = (−1)kk v  ∧q v  , for all v ∈ V i,k−i and v  ∈ V

i  ,k  −i 

(5.2)

.

(−1)2

Proof. The property = is easily checked in all the above mentioned cases, and the condition 2 = id is a direct corollary: 2 |V i, j = (−1)i+ j J 2 , and V i, j is a sum of spaces V,N having 2 with the same parity of i + j. Since J always commutes with σ,N (K 1 K 22 ), and σ,N (K 1 K 22 ) = σ,−N (S(K 1 K 22 )∗ ), the claim (5.1) for h = K 1 K 22 is trivially satisfied. We have to check (5.1) for the remaining generators h = K 1 , E 1 , F1 , and in the not-trivial cases  = 21 , 1. A direct computation yields: & '& ' '& 1 ' & 1 − 12 − 12 −2 2 0 −q 0 0 0 q q q J σ 1 ,N (K 1 )J −1 = = 1 1 1 1 2 0 q− 2 0 q2 q2 0 −q 2 0 J2

= σ 1 ,−N (K 1−1 ), 2

& J σ 1 ,N (E 1 )J 2

−1

=

1

−q − 2 0

0 1 q2

'

= −qσ 1 ,−N (F1 ), 2

0 0

1 0

&

0 1 −q 2

1

q− 2 0

'

 =

0 −q

0 0



864

F. D’Andrea, G. Landi ⎛

0 J σ 1 ,N (F1 )J −1 = ⎝ 1 2

q2

⎞ 1  −q − 2 ⎠ 0 1 0

0 0



⎛ ⎝ 01 −q 2

⎞ 1  q− 2 ⎠ 0 = 0 0

−q −1 0



= −q −1 σ 1 ,−N (E 1 ), 2

⎛

0

J σ1,N (K 1 )J −1 = ⎝ 0

−q

0 1 0

⎞⎛ q −q −1 0 ⎠⎝ 0 0 0

0 1 0

⎞⎛ 0 0 0 ⎠⎝ 0 q −1 −q

⎞ ⎛ −1 q −q −1 0 ⎠=⎝ 0 0 0

0 1 0

0 1 0

⎞ 0 0⎠ q

0 0 −q

⎞ 0 0⎠ 0

= σ1,−N (K 1−1 ), ⎛

0

1 J σ1,N (E 1 )J −1 = [2] 2 ⎝ 0

−q

0 1 0

⎞⎛ 0 −q −1 0 ⎠⎝0 0 0

1 0 0

⎞⎛ 0 0 1⎠⎝ 0 0 −q

0 1 0

⎞⎛ 0 −q −1 0 ⎠ ⎝ −q 0 0

= −qσ1,−N (F1 ), ⎛ J σ1,N (F1 )J −1 =

1 [2] 2

0 ⎝ 0 −q

0 1 0

⎞⎛ 0 −q −1 0 ⎠⎝1 0 0

0 0 1

⎞⎛ 0 0 0⎠⎝ 0 0 −q

0 1 0

⎞ ⎛ 0 −q −1 0 ⎠=⎝ 0 0 0

−q −1 0 0

⎞ 0 −q −1⎠ 0

= −q −1 σ1,−N (E 1 ).

Also, by a direct computation one checks that J μ0 (v, v  ) = −μ0 (J v  , J v), J μ1 (v, v  ) = −μ1 (J v  , J v), J μ2 (v, v  ) = μ3 (J v  , J v), J μ3 (v, v  ) = μ2 (J v  , J v), J μ4 (v, v  ) = μ4 (J v  , J v). With these, (5.2) is straightforwardly established.

 

From now on, we take that c4 = c0 for the coefficients of Prop. 5.1. The composition of the involution on A(SUq (3)) with the  in the previous lemma, yields a map ω → ω∗ sending forms into forms and extending the involution of the algebra A(CPq2 ) = 0,0 (CPq2 ). Indeed, with h ∈ Uq (u(2)), and t := S(h)∗ , from (5.1), we get (Lh (1) ⊗ σ i, j (h (2) ))(ω∗ )i, j = (−1)i (Lh (1) ⊗ σ i, j (h (2) ))(∗ ⊗ J )ω j,i = (−1)i (∗ ⊗ J )(L S(h ∗(1) ) ⊗ σ j,i (S(h (2) )∗ ))ω j,i = (−1)i (∗ ⊗ J )(L S 2 (t(2) ) ⊗ σ j,i (t(1) ))ω j,i . Invariance of ω j,i gives Lh ω j,i = (1 ⊗ σ j,i (S −1 (h (3) )))(Lh (1) ⊗ σ j,i (h (2) ))ω j,i = σ j,i (S −1 (h))ω j,i , and in turn, using (S(h)∗ ) = (h), (Lh (1) ⊗ σ i, j (h (2) ))(ω∗ )i, j = (−1)i (∗ ⊗ J )(1 ⊗ σ j,i (t(1) S(t(2) )))ω j,i = (h)(−1)i (∗ ⊗ J )ω j,i = (h)(ω∗ )i, j .

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

865

Thus, the involution maps invariant elements into invariant elements, i.e. forms into forms. As a consequence of (5.2), (•,• (CPq2 ), ∧q ,∗ ) is a graded ∗-algebra: 

(ω ∧q ω )∗ = (−1)dg(ω)dg(ω ) ω∗ ∧q ω∗ ,

∀ω, ω ∈ •,• (CPq2 ).

(5.3)

Lemma 5.3. The algebra (V •,• , ∧q ) is generated in degree 1, that is any form of degree ≥ 1 can be written as a sum of products of 1-forms. Proof. We need to show that the maps V 0,1 × V i, j → V i, j+1 ,

(v, w) → v ∧q w,

V 1,0 × V i, j → V i+1, j ,

(v, w) → v ∧q w,

and

are surjective for all i, j. We give the proof for the first map, the second being analogous. If w is a scalar, the claim is clearly true. If (i, j) = (0, 1) (resp. (2, 1)) the scalar 1

(1, 0)t ∧q (0, 1)t = c0 [2]− 2 ,

1

resp. (1, 0)t ∧q (0, 1)t = c3 [2]− 2 ,

is a basis V 0,2 (resp. V 2,2 ), and the map is clearly surjective. If (i, j) = (1, 0), the map ∧q is invertible; indeed, v ∧q w = diag(c1 , c1 , c1 , c2 )U (v ⊗ w) with U the unitary matrix in (A.1). Finally, if (i, j) = (1, 1) the vectors (1, 0)t ∧q (0, 0, 0, 1)t = −c0 c2−1 [2]−1 (1, 0)t , (0, 1)t ∧q (0, 0, 0, 1)t = −c0 c2−1 [2]−1 (0, 1)t , form a basis of V 1,2 , and the maps surjective. This concludes the proof.

 

5.3. Hodge star and a closed integral. Having an (associative, graded involutive) algebra of forms, the next steps consist in endowing it with i) two derivations ∂ and ∂¯ giving a double complex, with d := ∂ + ∂¯ the total differential; ii) a closed integral; iii) an Hodge star operator. We postpone to the next section the explicit construction of the exterior differentials and start here with the integral. Recall from Remark 3.1 that an inner product on forms is given by composing the natural Hermitian structure on the module ωi, j of forms with the restriction to A(CPq2 ) of the Haar state ϕ : A(SUq (3)) → C; that is    

ω, ω := ϕ ωi,† j · ωi, j , i, j

where the sum is over all the homogeneous components ωi, j , ωi, j ∈ i, j (CPq2 ) of ω and ω . Both the left canonical action  and the left action L of Uq (su(3)) on A(SUq (3)) are unitary for this inner product, as well as the action of A(CPq2 ) by left multiplication. Now 2,2 (CPq2 ) = A(CPq2 ) is a free module of rank one, with basis a central element which we denote 1 (the volume form). Indeed, we call vol the form with all components

866

F. D’Andrea, G. Landi

equal to zero but for the one in degree 4, which is 1. We think of this as the volume form and define an integral by ) ∀ω ∈ •,• (CPq2 ). (5.4) − ω := vol, ω = ϕ(ω2,2 ), * In particular, −vol = 1. If the differentials ∂ and ∂¯ are given via the (right) action of elements of Uq (su(3)) which are in the kernel of the counit , the integral is automatically closed, i.e. ) ) ¯ = − ∂ω = 0, − ∂ω a simple consequence of the invariance of the Haar state: ϕ(a  h) = (h)ϕ(a). Using the Hermitian structure on •,• , given by  (ω, ω ) := ωi,† j · ωi, j , i, j

the Hodge star operator is defined on real forms via the usual requirement that ω ∧q ω = (∗ H ω, ω )vol. This can be extended to complex forms both linearly (as e.g. in [22]) or antilinearly (as e.g. in [19]). In the context of solutions of the Yang-Mills equations, a curvature, being the square of a connection, is always real and it doesn’t matter which extension we choose. We choose the former. Recalling that the Hermitian structure (, ) is linear in the second entry and antilinear in the first, the Hodge star on complex forms is the linear operator ∗ H : i, j (CPq2 ) → 2− j,2−i (CPq2 ) defined by ω∗ ∧q ω = (∗ H ω, ω )vol.

(5.5)

Applying the Haar states to both sides of previous equation we get the usual relation )

 (5.6) − ω∗ ∧q ω = ∗ H ω, ω . With a ∗-calculus, a closed integral and the graded Leibniz rule for the differential, the equality (5.6) implies that ) )

 ∗ H dω, ω = − (dω)∗ ∧q ω = − − d(ω∗ ) ∧q ω ) ) 

= − − d(ω∗ ∧q ω )+(−1)dg(ω) − ω∗ ∧q dω = 0 + (−1)dg(ω) ∗ H ω, dω , which becomes d† ω = ∗ H d ∗ H ω,

(5.7)

if ∗2H ω = (−1)dg(ω) ω. To obtain this last property, which is automatic when q = 1, one needs suitable constraints on the parameters ci ’s in Prop. 5.1. Proposition 5.4. On any form ω one has ∗2H ω = (−1)dg(ω) ω if 1 1 3 1 1 c2 = ±q 4 s [2]− 4 |c0 |, c3 = ±[2] 2 , c1 = ±q − 4 s [2]− 4 |c0 |, with arbitrary signs. For this choice of parameters, on (anti-)holomorphic 2-forms the Hodge star is the identity, and on (1, 1) forms it is the linear map ∗ H : (w, w4 ) → sign(c3 )(−w, w4 ).

(5.8)

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

867

Proof. If ∗ H ω and ω are homogeneous with different degree, both sides of (5.5) are zero. It is then enough to consider the case ω ∈  j,i (CPq2 ), ω ∈ 2−i,2− j (CPq2 ). From the definition of the involution on forms, for the possible values of the labels, one gets (i, j) = (0, 0), (0, 2), (2, 0), (2, 2) : ω∗ ∧q ω = ω† · ω , 1

(i, j) = (0, 1), (1, 0), (1, 2), (2, 1) : ω∗ ∧q ω = c3 μ0 (ω∗ , ω ) = (−1) j [2]− 2 c3 ω† · ω ,  3 c3 c4  − 1 s −q 2 |c1 |−2 w † , q 2 s |c2 |−2 w4† ω . (i, j) = (1, 1) : ω∗ ∧q ω = [2]

Condition (5.6) is satisfied if (∗ H ω)2−i,2− j = ⎧ if (i, j) = (0, 0), (0, 2), (2, 0), (2, 2), ⎪ ⎨ω j,i 1 if (i, j) = (0, 1), (1, 0), (1, 2), (2, 1), (−1) j [2]− 2 c3 ω j,i ⎪ 3 ⎩1 − 12 s s −2 −2 |c1 | w, q 2 |c2 | w4 ) if (i, j) = (1, 1). [2] c3 c0 (−q The square of ∗ H on ω is verified to be (−1)dg(ω) ω if the ci are those given in the statement of the proposition. With these, (5.8) is immediately checked.   With the previous lemma, all parameters are fixed, but for some arbitrary signs and a global rescaling2 encoded in c0 . On the other hand, fixing the sign of c3 corresponds to fixing an orientation, as flipping the orientation results in exchanging selfdual with anti-selfdual forms. From now on we assume that c3 < 0, so that by (5.8) selfdual (1, 1)-forms are of the type (w, 0), and anti-self dual ones are of the type (0, w4 ). Corollary 5.5. The Hodge star operator is an isometry. Proof. Notice that since ϕ(a ∗ ) = ϕ(a), by graded involutivity of the conjugation of forms we have ) ∗ ) 

 ∗ ∗ H ω, ∗ H ω = ∗ H ω , ∗ H ω = − ω∗ ∧q ∗ H ω = − ω∗ ∧q ∗ H ω )   

= (−1)dg(ω) − (∗ H ω)∗ ∧q ω = (−1)dg(ω) ∗2H ω, ω = ω, ω , where we used the fact that ω and ω have the same degree, and dg(ω) and dg(ω)2 have the same parity.   Remark 5.6. There is an equivalent definition of the Hodge star operator. Firstly, one can define the “exterior multiplication” e : •,• (CPq2 ) → End(•,• (CPq2 )) and the dual “contraction” i : •,• (CPq2 ) → End(•,• (CPq2 )) by the formulæ: e(ω)ω := ω ∧ ω ,

i(ω) := e(ω)† ,

∀ω, ω ∈ •,• (CPq2 ).

Then, from (5.6) and (5.4) one has 







∗ H ω, ω = vol, ω∗ ∧ ω = vol, e(ω∗ )ω = i(ω∗ )vol, ω which, from the non degeneracy of the scalar product, yields ∗ H ω = i(ω∗ )vol for any ω ∈ •,• (CPq2 ). 1

1

2 Notations are simplified by choosing c = [2] 2 . In the notations of [7] we would have c = 2[2]− 2 . 0 0

868

F. D’Andrea, G. Landi

5.4. The exterior derivatives. We are left with the definition of the exterior derivative d. As we recall in App. 8, in order to have a real differential calculus for • (CPq2 ) = ( k 2 that is a map d : A(CPq2 ) → 1 (CPq2 ) obeying the k  (CPq ) one needs a derivation,   Leibniz rule and such that A(CPq2 ) dA(CPq2 ) = 1 (CPq2 ) and da = −(da ∗ )∗ . Then the exterior derivative d is extended uniquely to forms of higher degree. If k (CPq2 ) := ( i, j 2 ¯ i+ j=k  (CPq ), this is equivalent to write d = ∂ + ∂ with two derivations ∂ : 2 1,0 2 2 0,1 2 ¯ ¯ = −(∂ω∗ )∗ ; A(CPq ) →  (CPq ) and ∂ : A(CPq ) →  (CPq ), such that ∂ω again both ∂ and ∂¯ are extended uniquely to forms of higher degree. We write X = i X i ⊗ ei ∈ Uq (su(3)) ⊗ V 1,0 , with e1 = (1, 0)t and e2 = (0, 1)t the basis vectors of V 1,0 . Then, we set  ∂( · ) := L X ∧q ( · ) = k L X k ⊗ ek ∧q ( · ) and determine the conditions on the elements X i ’s that yield a ∂ mapping a (i, j)-form, that is any ω ∈ i, j (CPq2 ) = A(SUq (3)) σ i, j V i, j , to a (i + 1, j)-form; namely we impose that ∂ω ∈ i+1, j (CPq2 ) = A(SUq (3))σ i+1, j V i+1, j . Since the assignment h → Lh is a representation, for all h, x ∈ Uq (su(3)) one gets that Lh Lx = L ad −1 Lh (1) , with the right adjoint action given by xS

(h (2) )

ad

x  h = S(h (1) )xh (2) . In turn, left covariance of the wedge product yields   Lh (1) ⊗ σ i+1, j (h (2) ) ∂( · )  = k Lh (1) L X k ⊗ σ i+1, j (h (2) )ek ∧q ( · )  = k (L ad −1 ⊗ σ 1,0 (h (3) )ek ∧q )(Lh (1) ⊗ σ i, j (h (4) ))( · ), Xk  S

(h (2) )

for any h ∈ Uq (u(2)). Then, for X an element which is invariant for the tensor product of the right adjoint action with σ 1,0 , i.e. for X such that 

k Xk

ad

 S −1 (h (1) ) ⊗ σ 1,0 (h (2) )ek = (h)



k Xk

⊗ ek ,

we conclude that     Lh (1) ⊗ σ i+1, j (h (2) ) ◦ ∂ = ∂ ◦ Lh (1) ⊗ σ i, j (h (2) ) .

(5.9)

This means that ∂ maps invariant elements of A(SUq (3))⊗V i, j into invariant elements of A(SUq (3))⊗V i+1, j , that is forms to forms. Since both Lh and σ i, j are ∗-representations, taking the adjoint of Eq. (5.9) yields     N ∂ † Lh ∗(1) ⊗ σk+1 (h ∗(2) ) = Lh ∗(1) ⊗ σkN (h ∗(2) ) ∂ † , which means that ∂ † maps (i, j)-forms to (i − 1, j)-forms. Similarly, by replacing V 1,0 with the representation V 0,1 , an invariant element Y of Uq (su(3)) ⊗ V 0,1 would give an operator ∂¯ = LY ∧q ( · ) mapping (i, j)-forms to (i, j + 1)-forms and the adjoint ∂¯ † mapping (i, j)-forms to (i, j − 1)-forms.

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

869

On CPq2 the only elements of Uq (su(3)) which act on the right as derivations of A(CPq2 ) are the operators E 2 , F2 , [E 1 , E 2 ]q and [F2 , F1 ]q . These must be applied in such a way to get an element of 1 (CPq2 ). Proposition 5.7. One defines two exterior derivations ∂ : A(CPq2 ) → 1,0 (CPq2 ) and ∂¯ : A(CPq2 ) → 0,1 (CPq2 ) by ¯ := (L K 1 K 2 [F2 ,F1 ]q a, L K 2 F2 a)t , ∂a := q − 2 (Lq −1 K 2 E 2 a, L−K 1 K 2 [E 1 ,E 2 ]q a)t , ∂a (5.10) 1

for all a ∈ A(CPq2 ). The map d = ∂ + ∂¯ : A(CPq2 ) → 1 (CPq2 ) satisfies the conditions A(CPq2 )dA(CPq2 ) = 1 (CPq2 ) and da ∗ = −(da)∗ . Proof. The elements X ∈ Uq (su(3)) ⊗ V 1,0 and Y ∈ Uq (su(3)) ⊗ V 0,1 given by X := (q −1 K 2 E 2 , −K 1 K 2 [E 1 , E 2 ]q )t , Y := (K 1 K 2 [F2 , F1 ]q , K 2 F2 )t ,

(5.11)

are invariant, hence by the above discussion the maps ω → L X ∧q ω and ω → LY ∧q ω send i, j (CPq2 ) into i+1, j (CPq2 ), resp. i, j+1 (CPq2 ). The wedge product on zero forms is diagonal multiplication and on functions these maps become (proportional to) (5.10). Since S −1 is anticomultiplicative and Lh a = a  S −1 (h), from the covariance of the right canonical action we get that for all a, b ∈ A(SUq (3)), L X 1 (ab) = (L X 1 a)b+(a  K 2−2 )(L X 1 b), L X 2 (ab) = (L X 2 a)b+(a  K 1−2 K 2−2 )(L X 2 b)+q − 2 (q −q −1 )(a  K 1−1 K 2−2 E 1 )(L X 1 b), 1

LY1 (ab) = (LY1 a)b+(a  K 1−2 K 2−2 )(LY1 b) − q 2 (q − q −1 )(a  K 1−1 K 2−2 F1 )(LY2 b), 1

LY2 (ab) = (LY2 a)b+(a  K 2−2 )(LY2 b). If a, b are indeed in A(CPq2 ), right Uq (u(2))-invariance gives L X (ab) = (L X a)b + a(L X b) and LY (ab) = (LY a)b + a(LY b), so ∂ and ∂¯ are derivations on A(CPq2 ). Reality is a simple check. From (a ∗  h)∗ = a  S(h)∗ it follows that   ¯ + (∂a ∗ )∗ = a  S −1 (Y1 ) − q −1 X 2∗ , S −1 (Y2 ) + X 1∗ ∂a   = a  K 1 K 2 [F2 , F1 ]q + q(K 1 K 2 )−1 [F1 , F2 ]q , (K 2 − K 2−1 )F2   = a  (q − q −1 )F1 F2 , 0 = 0. For the generators pi j = z i∗ z j of A(CPq2 ) one computes that ∂ pi j = −q

−1

(u i3 )∗

 2 uj , u 1j

∂¯ pi j = q −1



1

−q − 2 (u i1 )∗ 1

q 2 (u i2 )∗

 u 3j ,

and shows that d pi j are a generating family for 1,0 (CPq2 ) ⊕ 0,1 (CPq2 ) = M(σ 1 ,− 3 ) ⊕ M(σ 1 , 3 ) 2

2

2 2

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F. D’Andrea, G. Landi

as a left A(CPq2 )-module. Indeed, for any ω = (v1 , v2 )t ⊕ (w1 , w2 )t ∈ 1,0 (CPq2 ) ⊕ 0,1 (CPq2 ) the coefficients   ai j (ω) := −q 1−2 j q 2 v1 (u 2j )∗ + v2 (u 1j )∗ u i3 ,  1  1 bi j (ω) := q 5−2 j −q 2 w1 (u 3j )∗ u i1 + q − 2 w2 (u 3j )∗ u i2 , are right Uq (u(2))-invariant, i.e. ai j (ω), bi j (ω) ∈ A(CPq2 ). Since  j

q 2(a− j) (u aj )∗ u bj =

 j

u aj (u bj )∗ = δa,b ,

 j

u 3j (u 3j )∗ =

 j

q 6−2 j (u 3j )∗ u 3j = 1,

the following algebraic identities  

i, j

i, j



ai j (ω) ∂ pi j = (v1 , v2 )t ,

i, j



ai j (ω) ∂¯ pi j = 0,

i, j

bi j (ω) ∂¯ pi j = (w1 , w2 )t , bi j (ω) ∂ pi j = 0

hold. Assembling all together, for any 1-form ω one finally gets ω=

 i, j

  ai j (ω) + bi j (ω) d pi j ,

proving that the vectors d pi j are a generating family for 1 (CPq2 ) as a left A(CPq2 )module. This concludes the proof.   Remark 5.8. Using the properties a  K 1 = a, a  K 2 = a and a  E 1 = a  F1 = 0, for any element a ∈ A(CPq2 ), simple manipulations in (5.10) yield 3

∂a := −q − 2 (a  E 2 , a  E 2 E 1 )t ,

¯ := −(a  F2 F1 , a  F2 )t . ∂a

Also, modulo a proportionality constant the operator ∂¯ coincides with the one of [7]. We close this section with few remarks on invariant (anti-)selfdual 2-forms that we shall use later on in the paper when dealing with monopole connections. Since 0,2 (CPq2 )  0,3 , 2,0 (CPq2 )  0,−3 , and 1,1 (CPq2 )  1,0 ⊕ 0,0 , by the harmonic decomposition 0,0 

 n∈N

1,0 

 n∈N

(n, n),

0,3 

 n∈N

(n, n + 3),

0,−3 

 (n + 1, n + 1) ⊕ (n, n + 3) ⊕ (n + 3, n),

 (n + 3, n), n∈N

n∈N

a 2-form ω is invariant iff ω ∈ 1,1 (CPq2 ) and it has the form ω = (0, w4 ) with w4 ∈ C. By (5.8) such a 2-form is anti-selfdual, for the choice of orientation (c3 < 0) made above.

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6. Monopoles As mentioned in the Introduction, by a monopole we mean a line bundle over CPq2 , that is to say a ‘rank 1’ finitely generated projective module over the coordinate algebra A(CPq2 ), endowed with a connection having anti-selfdual curvature. In this section we present some of these connections. In Sect. 3.2 we have described at length the isomorphism 0,N → A(CPq2 )d N P−N as left A(CPq2 )-modules and the isomorphism 0,N → PN A(CPq2 )d N as right A(CPq2 )modules. Any of the two isomorphisms may be used to transport the Grassmannian connection on the bimodule 0,N . We use the second one because it is notationally simpler. Recall that the isomorphism φ : 0,N → PN A(CPq2 )d N is given φ(a) :=  N a, with inverse φ −1 : PN A(CPq2 )d N → 0,N , φ −1 (v) =  † · v; and  N is the column vector N given in Prop. 3.3, d := 1 (|N |+1)(|N |+2) and P :=   † . with components ψ j,k,l N N N N 2

6.1. The anti-selfdual connections. The Grassmannian connection on the right A(CPq2 )module E N := PN A(CPq2 )d N , with respect to the differential calculus of Sect. 5 is the map ,N : E N ⊗A(CP2 ) n (CPq2 ) → E N ⊗A(CP2 ) n+1 (CPq2 ), ∇ q q

,N ω := PN dω. ∇

For its curvature we get ,N2 ω = PN d (PN d(PN ω)) = PN d PN ∧q d(PN ω) ∇     = PN d PN ∧q PN dω + d PN ∧q ω = PN d PN ∧q d PN ∧q ω, where we used that ω = PN ω for any ω ∈ E N ⊗A(CPq2 ) n (CPq2 ) and the identity ede = (de)(1 − e),

(6.1)

and so e(de)e = 0, both valid for any idempotent e (and any differential calculus). When transported to equivariant maps, the connection ,N φ : 0,N ⊗A(CP2 ) n (CPq2 ) → 0,N ⊗A(CP2 ) n+1 (CPq2 ), ∇N := φ −1 ∇ q q is readily found to be given by ∇N η =  N† d( N η),

(6.2)

,2 φ becomes the operator of left wedge multiplication by and the curvature ∇N2 = φ −1 ∇ N the 2-form, still denoted ∇N2 , given by ∇N2 =  N† (d PN ∧q d PN ) N . Lemma 6.1. The connection ∇ N is left Uq (su(3))-invariant.

(6.3)

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Proof. From Prop. 3.4 we deduce  N† ⊗  N =



t 0,N i ↑,i

0,N ∗ ⊗ (t↑,i ) .

In turn, for any h ∈ Uq (su(3)), (2.10) yields  0,N 0,N ∗ (h (1)  t↑,i ) ⊗ (S(h (2) )∗  t↑,i ) h (1)   N† ⊗ h (2)   N = i  0,N 0,N 0,N 0,N ∗ = ρ j,i (h (1) )ρi,k (S(h (2) ))t↑, j ⊗ (t↑,k ) i, j,k

= (h)



t 0,N j ↑, j

† 0,N ∗ ⊗ (t↑, j ) = (h) N ⊗  N .

Thus, for any h ∈ Uq (su(3)), h  (∇N η) = h  ( N† d N η) = (h (1)   N† )d(h (2)   N )(h (3)  η) =  N† d N (h  η) = ∇ N (h  η). This concludes the proof.

 

The connection being invariant from Lemma 6.1, its curvature ∇ N2 is invariant as well. Then, from the discussion at the end of Sect. 5, the two-form ∇ N2 ∈ 1,1 (CPq2 ) is necessarily of the type ∇ N2 = (0, w N ) for some w N ∈ R. Hence by (5.8) it is antiselfdual. Lemma 6.2. The connection ∇ N is anti-selfdual, that is to say, its curvature is a (constant) anti-selfdual two-form: ∗ H ∇ N2 = −∇ N2 . For its use later on we compute the constant w N for N ≥ 0; a similar computation being possible for N ≤ 0 as well. By construction N

N  K2 = q − 2 N

 N  h = (h) N ∀h ∈ Uq (u(2)).

and

(6.4)

Since z i  F2 = 0, it also holds that  N  E 2 = 0,

 N†  F2 = 0.

(6.5)

Using the condition  N†  N = 1 and covariance of the action one deduces that ( N†  E 2 ) N = 0,

 N† ( N  F2 ) = 0.

(6.6)

These equations allow one to compute − N2

 N† d PN

− 32



 N†  E 2



∈ 1,0 (CPq2 ),  N†  E 2 E 1   N  N  F2 F1 ∈ 0,1 (CPq2 ). q − 2 (d PN ) N = −  N  F2 q

= −q

(6.7)

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By (2.10) we have 3

1

q −N ∇N2 = −c2 q − 2 s−3 [2]− 2 0,N +q −1 (t↑,i 



0,N 0,N q(t↑,i  E 2 )(t↑,i i 0,N E 2 E 1 )(t↑,i  E 2 E 1 )∗

 E 2 )∗

 0,N 0,N qρ↑, j (E 2 )ρk,↑ (F2 ) i, j,k  0,N 0,N 0,N ∗ +q −1 ρ↑, (E E )ρ (F F ) t 0,N 2 1 1 2 j k,↑ j,i (tk,i )  3 1  0,N 0,N qρ↑, = −c2 q − 2 s−3 [2]− 2 j (E 2 )ρk,↑ (F2 ) j,k  0,N 0,N +q −1 ρ↑, j (E 2 E 1 )ρk,↑ (F1 F2 ) δ j,k 3

1

3

1

= −c2 q − 2 s−3 [2]− 2



0,N = −c2 q − 2 s−3 [2]− 2 ρ↑,↑ (q E 2 F2 + q −1 E 2 E 1 F1 F2 )   3 1 = −c2 q − 2 s−3 [2]− 2 0, 0, 0|q E 2 F2 + q −1 E 2 E 1 F1 F2 |0, 0, 0 ,

with |↑ = |0, 0, 0 the highest weight vector of the representation ρ 0,N . Using the vanishing E 1 F2 |0, 0, 0 = F2 E 1 |0, 0, 0 = 0, one gets that 0, 0, 0|E 2 E 1 F1 F2 |0, 0, 0 = 0, 0, 0|E 2 [E 1 , F1 ]F2 |0, 0, 0 = 0, 0, 0|E 2 =

q K 2 −q −1 K −2 0, 0, 0|E 2 F2 1q−q −1 1 |0, 0, 0 3

K 12 −K 1−2 q−q −1

F2 |0, 0, 0

= 0, 0, 0|E 2 F2 |0, 0, 0 ,

1

giving w N = −q − 2 s−3 c2 [2] 2 q N 0, 0, 0|E 2 F2 |0, 0, 0 . √ 3 1 Since F2 |0, 0, 0 = [N ] |0, 1, 21 we finally get w N = −q − 2 s−3 c2 [2] 2 q N [N ]. Hence ∇ N2 = q N −1 [N ] ∇12 ,

(6.8)

where ∇12 = w1 ∈ R is an irrelevant normalization constant. 6.2. Gauged Laplacians. With monopoles connection on the modules 0,N we can study the corresponding gauged Laplacian operator acting on 0,N . Such an operator describes ‘excitations moving on the quantum projective space’ in the field of a magnetic monopole and in the limit q → 1 provides a model of quantum Hall effect on the projective plane. We know that on both the modules 0,N of sections and n (CPq2 ) of forms there are A(CPq2 )-valued Hermitian structures. These can be combined to get a similar Hermitian structure on their tensor products 0,N ⊗A(CPq2 ) n (CPq2 ) which, when composed with the restriction to A(CPq2 ) of the Haar functional ϕ : A(CPq2 ) → C, gives a nondegenerate C-valued inner product. Using the latter inner product, one generalizes Eq. (5.7) for forms to an analogous statement on connections, that is to say ∇ N† η = ∗ H ∇N ∗ H η.

(6.9)

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F. D’Andrea, G. Landi

Having this, as usual we define the gauged Laplacian acting on elements η ∈ 0,N by ∇ η = ∇ N† ∇ N η =  N† d† [ N  N† d( N η)], where ∇N η =  N† d( N η) is the connection in (6.2). Using the covariance of the action – remember that Lx (ab) = (Lx(2) a)(Lx(1) b) due to the presence of the antipode – and Eq. (6.4), straightforward computations yield N

1

N

1

∂( N η) = q 2 − 2 (L X  N )η + q 2 − 2  N (L X η), ¯ N η) = q 2 (LY  N )η + q 2  N (LY η). ∂( N

N

From (6.5) and (6.6): L X  N = 0 and  N† (LY  N ) = 0; hence   1   ∇ η = q N /2  N† d†  N q − 2 L X ⊕ LY η  1   1   = q N /2  N† q − 2 L X † ⊕ LY †  N q − 2 L X ⊕ LY η . Again, (6.5) and (6.6) yield LY †  N = 0 and  N† (L X †  N ) = 0; hence using the coproduct, ∇ η = q N Lq −1 X † X +Y † Y η = η  S −1 (q −1 X † X + Y † Y ) q N . Now, S −1 (q −1 X † X + Y † Y ) = K 2−2 (q −1 E 2 F2 + q −2 F2 E 2 )   +K 1−2 K 2−2 q[E 2 , E 1 ]q [F1 , F2 ]q + q −2 [F1 , F2 ]q [E 2 , E 1 ]q , which, with the commutation rule . . . [F1 , F2 ]q , [E 2 , E 1 ]q = [F1 , [E 2 , E 1 ]q ], F2 q + F1 , [F2 , [E 2 , E 1 ]q ] q . . = [E 2 , [F1 , E 1 ]]q , F2 q + F1 , [[F2 , E 2 ], E 1 ]q q 0 0 / / K 12 −K 1−2 K 22 −K 2−2 = − [E 2 , q−q ] , F − F , [ , E ] q 2 1 1 q −1 q−q −1 q

q

= −[K 12 E 2 , F2 ]q + [F1 , E 1 K 2−2 ]q = K 12 (q −2 F2 E 2 − E 2 F2 ) + (F1 E 1 − q −2 E 1 F1 )K 2−2 , can be rewritten as S

−1

(q

−1

X X + Y Y) = †

†

K 2−2

  K 22 −K 2−2 −2 [2] q−q −1 + (q + q )F2 E 2

+K 1−2 K 2−2 (q + q −2 )[F1 , F2 ]q [E 2 , E 1 ]q − q K 1−2 K 2−4 (F1 E 1 − q −2 E 1 F1 ). By construction the action of Uq (su(2)) is trivial on η, that is to say η  E 1 = η  F1 = 0 and η  K 1 = η, while η  K 2 = q N /2 η. Then, an intermediate result states that    K 22 −K 2−2 −2 −1 . [2] + (q + q ) F E − q F F [E , E ] ∇ η = η  q−q 2 2 2 1 2 1 q −1

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

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Using the commutator [F2 , E 1 ] = 0 and again η  E 1 = 0, a further simplification comes from the computation η  F2 F1 E 1 E 2 = η  F2 [F1 , E 1 ]E 2 = −η  F2

K 12 −K 1−2 q−q −1

E 2 = η  F2 E 2 ,

(6.10)

leading to:   ∇ η = η  [2][N ] + (1 + q −3 ) ([2]F2 E 2 − F2 F1 E 2 E 1 ) .

(6.11)

Next, we relate the gauged Laplacian operator to the Casimir operator in (2.5). Now, when acting from the right on 0,N , a straightforward computation leads to  1 1 η  Cq = η  [ 13 N ]2 + [ 13 N + 1]2 + [ 23 N + 1]2 + (q 3 N +1 + q − 3 N −1 )F2 E 2  1  N +F2 F1 q − 3 N +1 [E 1 , E 2 ]q − q 3 [E 2 , E 1 ]q , which, using again (6.10) reduces to   1 1 η  Cq = η  [ 13 N ]2 +[ 13 N +1]2 +[ 23 N +1]2 +(q 3 N +q − 3 N ) ([2]F2 E 2 − F2 F1 E 2 E 1 ) . (6.12) The generator L = K 1 K 22 of the ‘structure algebra’ Uq (u(1)) acts on 0,N as η  L = Then, a comparison of (6.11) and (6.12) yields the following proposition.

q N /2 η.

Proposition 6.3. The gauged Laplacian is related to the Casimir operator by  L − L −1 ∇ + [2] q 3 3 q − q −1 q 2 + q− 2  1  1 1 1 2 2 = Cq − (q − q −1 )−1 (L 3 − L − 3 )2 + (q L 3 − q −1 L − 3 )2 + (q L 3 − q −1 L − 3 )2 1

3 2



1

L 3 + L− 3

or 3

q2

q

N 3

+ q− 3

N

3 2

− 32

q +q

(∇ − [2][N ]) = Cq − [ 13 N ]2 − [ 13 N + 1]2 − [ 23 N + 1]2 .

The diagonalization of the gauged Laplacian is made simple by the observation that for the Casimir left and right action is the same: Cq  a = a  Cq for all a ∈ A(SUq (3)), and by the fact that with respect to the left action of Uq (su(3)) there are decompositions:   ρ (n,n+N ) if N ≥ 0, 0,N  ρ (n−N ,n) if N ≤ 0. 0,N  n∈N

n∈N

Proposition 6.4. The eigenvalues of the gauged Laplacian ∇ are given by λn,N = (1 + q −3 )[n][n + N + 2] + [2][N ] λn,N = (1 + q with n ∈ N.

−3

if N ≥ 0,

)[n + 2][n − N ] + [2][N ] if N ≤ 0,

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Proof. Using the above harmonic decomposition of 0,N and the spectrum (2.6) of the Casimir operator, one gets 3

λn,N := q − 2

3 3 q 2 +q − 2 N 3

q +q

− N3

.2 .2 .2 .2  n + 13 |N |+1 + n + 23 |N |+1 − 13 N +1 − 23 N +1 +[2][N ],

with n ∈ N and for any N ∈ Z. This expression can be simplified using the identity [a + b]2 − [b]2 = [a][a + 2b]. For N ≥ 0 this becomes λn,N = q

− 32

3

3

N 3

− N3

q 2 + q− 2 q

+q

. . [n] n + 23 N + 2 + n + 43 N + 2 + [2][N ],

that with a further simplification is the claimed expression in the proposition. One proceeds similarly for the case N ≤ 0.   It is worth stressing that the spectrum of ∇ is not symmetric under the exchange N ↔ −N , not even when exchanging in addition the parameter as q ↔ q −1 . A similar phenomenon was already observed in [14] for gauged Laplacians on the standard Podle´s sphere; the latter can be thought of as the quantum projective line CPq1 . 7. Quantum Characteristic Classes Classically, topological invariants are computed by integrating powers of the curvature of a connection, the result being independent of the particular connection. On the other hand, in order to integrate the curvature of a connection on the quantum projective space CPq2 one needs ‘twisted integrals’; the results are not integers any longer but rather q-analogues, as we shall see in Sect. 7.2. We start with some general result on equivariant K-theory and K-homology and corresponding Chern-Connes characters.

7.1. Equivariant K-theory and K-homology. Classically, the equivariant topological K 0 -group of a manifold is the Grothendieck group of the abelian monoid whose elements are equivalence classes of equivariant vector bundles. It has an algebraic version that can be generalized to noncommutative algebras. One has a bialgebra U and a left U-module algebra A. Equivariant vector bundles are replaced by one sided (left, say) A  U-modules that are finitely generated and projective as left A-modules; these will be simply called ‘equivariant projective modules’. Any such module is given by a pair (e, σ ), where e is an N × N idempotent with entries in A, and σ : U → Mat N (C) is a representation and the following compatibility requirement is satisfied (see e.g. [6, Sect. 2]),   h (1)  e σ (h (2) )t = σ (h)t e, for h ∈ U, (7.1) with ‘ t ’ denoting transposition. The corresponding module E = A N e is made of elements v = (v1 , . . . , v N ) ∈ A N in the range of the idempotent, ve = v, with left-module structure given by (a.v)i := avi ,

(h.v)i :=

N j=1

(h (1)  v j )σi j (h (2) ),

for a ∈ A and h ∈ U .

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An equivalence between any two equivariant modules is simply an invertible left A  U-module map between them; V U (A) will denote the abelian monoid whose elements are equivalence classes of equivariant projective left modules with operation the direct sum, as usual. The equivariant K-theory group K 0U (A) is the Grothendieck group of the abelian monoid V U (A). The equivalence of equivariant projective modules can be rephrased in terms of idempotents. What follows is a direct extension of well known results [1]. We give the proof for completeness. 

Lemma 7.1. Two equivariant projective modules E = A N e and E  = A N e are equivalent iff e = uv and e = vu for some u ∈ Mat N ×N  (A) and v ∈ Mat N  ×N (A) satisfying the equivariance conditions (h (1)  u)σ  (h (2) )t = σ (h)t u, (h (1)  v)σ (h (2) )t = σ  (h)t v. (7.2)  N N N  Proof. Let w• = i wi ei • be the generic  element of A e. If π : A e → A e is a left A-module map, then π(w• ) = i wi π(ei • ), so the map is uniquely determined by its value on rows of the idempotent e, and similarly for π −1 . We call u ∈ Mat N ×N  (A) (resp. v ∈ Mat N  ×N (A)) the matrix with entries u i j := π(ei • ) j (resp. vi j := π −1 (ei • ) j ). Since π maps into the range of e (resp. π −1 maps into the range of e), we have the conditions   π(ei • ) j ejk = π(ei • )k , and π −1 (ei • ) j e jk = π −1 (ei • )k , j

j

that in terms of row vectors become  u i j ej • = u i • , j

and

 j

vi j e j • = vi • .

Next, we apply π −1 to the equation on the left, π to the one on the right, and use A-linearity. Since u i j is the j th component of image, via π , of the i th row of e, and π −1 is the inverse map, we have π(u i • ) = ei • and similarly for π −1 . Thus:    u i j v jk = eik , vi j u jk = eik . j

j

From the definition of u, the equivariance properties of e, and the fact that π is an U-module map, we get   h.u i • = h (1)  u i • σ  (h (2) )t = π(h.ei • ) = π(h (1)  ei • σ (h (2) )t ) = π (σ (h)t e)i • = (σ (h)t u)i • , which is the first equivariance condition in (7.2). Similarly one proves the equivariance condition for v. With this, the ‘only if’ part is proved. Next, assume that e = uv and e = vu for some u and v satisfying (7.2) above. Then, (h (1)  e)σ (h (2) )t = (h (1)  u)(h (2)  v)σ (h (3) )t = (h (1)  u)σ  (h (2) )t v = σ (h)t uv, which means e that satisfies (7.1). Similarly for e .   We define π : A N e → A N e and π −1 : A N e → A N e via the formulæ π(w) := wu and π −1 (w) := wv; we need to show that (i) the maps are well defined, (ii) they are one the inverse of the other, (iii) they are left A  U-module maps. Point (iii) is a consequence of the fact that left and right multiplication commute, and of the

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F. D’Andrea, G. Landi

equivariance conditions for u and v. Point (ii) follows from the identity π −1 π(w) = we (resp. π π −1 (w) = we ), and the fact that w is in the range of e (resp. e ). Finally if w ∈ A N e, we have π(w)e = w(uvu) = (we)u = wu = π(w), 

i.e. π(w) ∈ A N e , and similarly for π −1 : the maps are well defined.

 

There is a natural map from equivariant K-theory to equivariant cyclic homology given for instance in [16]. We adapt that construction to our situation. One starts with HomC (U, An+1 ), the collection of C-linear maps from U to An+1 , and defines operations bn,i : HomC (U, An+1 ) → HomC (U, An ), for i = 0, . . . , n, bn,i (a0 ⊗ a1 ⊗ . . . ⊗ an )(h) := (a0 ⊗ . . . ⊗ ai ai+1 ⊗ . . . ⊗ an )(h), if i = n,   bn,n (a0 ⊗ a1 ⊗ . . . ⊗ an )(h) := (h (1)  an )a0 ⊗ a1 ⊗ . . . ⊗ an−1 (h (2) ), (7.3) and an operation λn : HomC (U, An+1 ) → HomC (U, An+1 ),   λn (a0 ⊗ a1 ⊗ . . . ⊗ an )(h) := (−1)n (h (1)  an ) ⊗ a0 ⊗ a1 ⊗ . . . ⊗ an−1 (h (2) ). (7.4) They are the face operators and the cyclic operator of equivariant cyclic homology as we are going to show. The maps bn,i make up a presimplicial module – one checks that bn−1,i bn, j = bn−1, j−1 bn,i for all 0 ≤ i < j ≤ n – so that n (−1)i bn,i bn := i=0

is a boundary operator [15]. The Hopf algebra U acts on An+1 via the rule h  (a0 ⊗ a1 ⊗ . . . ⊗ an ) := h (1)  a0 ⊗ h (2)  a1 ⊗ . . . ⊗ h (n+1)  an , and CnU (A) will denote the collection of elements ω ∈ HomC (U, An+1 ) which are ‘equivariant’, meaning that (h (1)  ω)(xh (2) ) = ω(hx), for all h, x ∈ U. Next, one establishes that the operators bn,i commute with the action of U (a not completely trivial task for the last one bn,n ), and it makes sense to consider the complex of equivariant maps. The cyclic operator λn commutes with the action of U, thus it descends to an operator on CnU (A) as well. Finally, with bn :=

n−1 i=0

(−1)n bn,i ,

it holds that bn (1 − λn ) = (1 − λn−1 )bn , which says that the boundary operator bn maps U (A)/Im(1 − λ CnU (A)/Im(1 − λn ) into Cn−1 n−1 ). The homology of this last complex is called the ‘U-equivariant cyclic homology’ of A with corresponding homology groups usually denoted HCnU (A). Next, for σ : U → Mat N (C) a representation as above, consider the set     Mat σN (A) := T ∈ Mat N (A) h (1)  T σ (h (2) )t = σ (h)t T, ∀h ∈ U .

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

879

This is a subalgebra of Mat N (A); indeed given any two of its elements T1 , T2 one has:     h (1)  (T1 T2 ) σ (h (2) )t = (h (1)  T1 )(h (2)  T2 ) σ (h (3) )t = (h (1)  T1 ) σ (h (2) )t T2 = σ (h)t T1 T2 . Moreover, σ -equivariant N × N idempotents as in (7.1) are elements of Mat σN (A). Due to the definition of Mat σN (A) there exists a map Tr σ : Mat σN (A)n+1 → CnU (A) given by   ˙ T1 ⊗ ˙ ... ⊗ ˙ Tn σ (h)t Tr σ (T0 ⊗ T1 ⊗ . . . ⊗ Tn )(h) := Tr T0 ⊗  = (T0 )i0 i1 ⊗(T1 )i1 i2 ⊗ . . . ⊗ (Tn )in in+1 σ (h)i0 in+1 , i 0 ,i 1 ,...,i n+1

˙ denotes composition of the tensor product over C with matrix multiplication. where ⊗ Also,   ˙ T0 ⊗ ˙ T1 ⊗ ˙ ... ⊗ ˙ Tn−1 (−1)n λn Tr σ (T0 ⊗ T1 ⊗ . . . ⊗ Tn )(h) = Tr h (1)  Tn σ (h)t ⊗   ˙ T0 ⊗ ˙ T1 ⊗ ˙ ... ⊗ ˙ Tn−1 = Tr σ (h)t Tn ⊗ = Tr σ (Tn ⊗ T0 ⊗ . . . ⊗ Tn−1 )(h), which amounts to say that Tr σ transforms the ordinary cyclic operator for the algebra Mat σN (A) into the ‘U-equivariant’ cyclic operator for A. Since bn,n = bn,0 λn , the map Tr σ is a morphism of differential complexes, mapping the complex of the cyclic homology of Mat σN (A) to the complex of the U-equivariant cyclic homology of A. This construction is completely analogous to the ‘non-equivariant’ case, cf. [15, Cor. 1.2.3]. At this point, one can repeat verbatim the proof of Thm 8.3.2 in [15], replacing the ring R := Mat N (A) there, with Mat σN (A) (which is still a matrix ring) and replacing the generalized trace map there, with Tr σ , to prove the following theorem. Theorem 7.2. A map chn : K 0U (A) → HCnU (A) is defined by chn (e, σ ) := Tr σ (e⊗n+1 ). We give in App. 8 an alternative, more explicit proof for the cases n ≤ 4. We stress that what we denote here HCn and call cyclic homology is Connes’ first version of cyclic homology, i.e. the homology of Connes’ complex denoted Hnλ in [15]. Modulo a normalization, the cycle chn (e, σ )(1) is the usual Chern-Connes character in cyclic homology (and in fact, no σ ’s in the formulæ). On the other hand, chn (e, σ )((K 1 K 2 )−4 ) is what we are about to use for CPq2 . In general, one fixes a group-like element K ∈ U calling η the corresponding automorphism of A, η(a) := K  a for all a ∈ A. Then, one pairs HC•U (A) with the Hochschild cohomology of A with coefficients in η A; the latter is A itself with bimodule structure a  (a)a  = η(a  )aa  . Indeed, the pairing , : HomC (An+1 , C)×HomC (U, An+1 ) → C defined by τ, ω := τ (ω(K )) ,

(7.5)

∗ : Hom (An , C) → Hom (An+1 , C) of the face when used to compute the dual bn,i C C operators introduced in (7.3), yields the formulæ: ∗ bn,i τ (a0 , a1 , . . . , an ) = τ (a0 , . . . , ai ai+1 , . . . , an ),

∗ bn,n τ (a0 , a1 , . . . , an )

:= τ (η(an )a0 , a1 , . . . , an−1 ) .

if i = n,

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These are just the face operators of the Hochschild cohomology H • (A, η A) of A with coefficients in η A (cf. [15]). Thus, the pairing in (7.5) descends to a pairing H n (A, η A) × HCnU (A) → C . 7.2. The example of CPq2 . As mentioned, for CPq2 we take K = (K 1 K 2 )−4 the element implementing the square of the antipode (cf. (2.2)). Now, the Haar state of A(SUq (3)), satisfies (cf. Eq. (11.26) and Eq. (11.36) in [13]) ϕ(ab) = ϕ ((K  b  K )a) ,

for a, b ∈ A(SUq (3)),

which for a, b ∈ A(CPq2 ) results in ϕ(ab) = ϕ ((K  b)a) = ϕ (η(b)a) .

(7.6)

This just means that the restriction of the Haar state of A(SUq (3)) to A(CPq2 ) is the representative of a class in H 0 (A(CPq2 ), η A(CPq2 )). An additional zero cocycle is given by the restriction of the counit  of SUq (3), which on CPq2 yields the ‘classical point’, that is the character χ0 in (4.3). On the other hand, with the integral defined in (5.4) by using the Haar state as well, an element [τ4 ] ∈ H 4 (A(CPq2 ), η A(CPq2 )) is constructed as ) τ4 (a0 , . . . , a4 ) := − a0 da1 ∧q . . . ∧q da4 . Let us check that it is a cocycle. Leibniz rule gives ) b5∗ τ4 (a0 , . . . , a5 ) = − a0 a1 da2 ∧q a3 ∧q a4 ∧q da5 ) − − a0 (a1 da2 + da1 a2 ) ∧q a3 ∧q a4 ∧q da5 ) + − a0 da1 ∧q (a2 da3 + da2 a3 ) ∧q a4 ∧q da5 ) − − a0 da1 ∧q a2 ∧q (a3 da4 + da3 a4 ) ∧q da4 ) + − a0 da1 ∧q a2 ∧q a3 ∧q (a4 da5 + da4 a5 ) ) − − η(a5 )a0 ∧q a1 da2 ∧q a3 ∧q a4 ) ) = − a0 (da1 ∧q . . . ∧q da4 )a5 − − η(a5 )a0 (da1 ∧q . . . ∧q da4 ), which is zero by the modular property (7.6) of the Haar state. A 2-cocycle can be defined in a similar way. Recall that elements of 1,1 (CPq2 ) have the form ω = (α, α4 ), with α4 ∈ A(CPq2 ). Let π : 1,1 (CPq2 ) → A(CPq2 ) be the projection onto the second component π(ω) = α4 , and extend it to a projection

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π : 2 (CPq2 ) → A(CPq2 ) by setting π(ω) = 0 if ω ∈ 0,2 or ω ∈ 2,0 . The map π is an A(CPq2 )-bimodule map. Then, the map τ2 (a0 , a1 , a2 ) := ϕ ◦ π(a0 da1 ∧q da2 ) is the representative of a class [τ2 ] ∈ H 2 (A(CPq2 ), η A(CPq2 )). Indeed, by the Leibniz rule,   b3∗ τ2 (a0 , a1 , a2 , a3 ) = ϕ ◦ π a0 (da1 ∧q da2 )a3 − η(a3 )a0 (da1 ∧q da2 ) . π Being a bimodule map we get in turn   b3∗ τ2 (a0 , a1 , a2 , a3 ) = ϕ a0 π(da1 ∧q da2 )a3 − η(a3 )a0 π(da1 ∧q da2 ) , which is zero by the modular property of the Haar state. Both classes [τ4 ] and [τ2 ] will be proven to be not trivial by pairing them with the monopole projections (3.4). First, Lemma 7.3. The monopole projections PN =  N  N† in (3.4) are equivariant with respect to the representation σ N of Uq (su(3)) defined as σ N (h) :=

ρ 0,N (S(h))t if N ≥ 0, σ N (h) := ρ −N ,0 (S(h))t if N ≤ 0.

Proof. By Prop. 3.4, h   N† =

 N† ρ 0,N (h) if N ≥ 0,  N† ρ −N ,0 (h) if N ≤ 0,

for all h ∈ Uq (su(3)). Using (h  a ∗ )∗ = S(h)∗  a, we get h   N = σ N (h)t  N , and the equivariance of PN follows.   From the construction of the previous section, we can pair ch4 (PN , σ N ) with [τ4 ] and ch2 (PN , σ N ) with [τ2 ]. The pairing of [τ2 ] with ch2 (PN , σ N ) gives     τ2 , ch2 (PN , σ N ) = ϕ Tr PN π(d PN ∧q d PN )σ N (K 1−4 K 2−4 )t   = q −2N ϕ  N† π(d PN ∧q d PN ) N . Since  N are ‘functions’ on the total space of the bundle, we cannot move them inside π (which is an A(CPq2 )-bimodule map, not an A(SUq (3))-bimodule map). Nevertheless – with a little abuse of notations – the form ∇ N2 = π(∇ N2 ) is a constant, and PN d PN ∧q d PN =  N ∇ N2  N† = π(∇ N2 ) N  N† = π(∇ N2 )PN . With this, and using  N† PN  N = 1, we come to the final formula   τ2 , ch2 (PN , σ N ) = q −2N ϕ ◦ π(∇ N2 ).

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In (6.8), we have already shown that ∇ N2 = q N −1 [N ] ∇12 with ∇12 = (0, w1 ). Thus the corresponding quantum characteristic class is proportional to q −N [N ]:     τ2 , ch2 (PN , σ N ) = ϕ ◦ π(∇12 ) q −N −1 [N ] . At q = 1 the integral of the curvature is (modulo a global normalization constant) the monopole number of the bundle; it is the same as the first Chern number. As for the pairing of ch4 (PN , σ N ) with τ4 ,   )   4 N τ4 , ch (PN , σ ) = − Tr PN (d PN )4 σ N (K 1−4 K 2−4 )t . Using the modular properties of the Haar state, which means ) )   − Tr( N V ) = − Tr V (K 1 K 2 )4   N  (K 1 K 2 )−4 )   −2N =q − Tr V σ N (K 14 K 24 )t  N and is valid for any row vector V with entries in A(SUq (3)), we get: )   4 N −2N τ4 , ch (PN , σ ) = q −  N† (d PN )4  N . We need to compute the top form  N† (d PN )4  N . From the identity (6.1) it follows that PN (d PN )2  N = (d PN )2 PN  N = (d PN )2  N , and  N† (d PN )4  N =  N† (d PN )2 ∧q PN (d PN )2  N =  N† (d PN )2  N ∧q  N† (d PN )2  N = ∇ N2 ∧q ∇ N2 , leading to 



τ4 , ch (PN , σ ) = q 4

N

−2N

) − ∇ N2 ∧q ∇ N2 .

Using ∇ N2 = q N −1 [N ] ∇12 the corresponding quantum characteristic class is found to be proportional to [N ]2 :  )    τ4 , ch4 (PN , σ N ) = q −2 − ∇12 ∧q ∇12 [N ]2 . At q = 1, the integral of the square of the curvature is (modulo a global normalization constant) the instanton number of the bundle. The pairing of a projection with the third Fredholm module as in Sect. 4.3 does not give the ‘classical’ instanton number, that is N 2 , but rather the 2nd Chern number which is a combination of the instanton number and of the monopole number. By pairing ch0 (PN , σ N ) with the Haar state, and using its modular property, one gets     ϕ, ch0 (PN , σ N ) = ϕ Tr PN σ N (K 1−4 K 2−4 ) = q −2N ϕ( N†  N ) = q −2N .

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On the other hand, the pairing with the classical point χ0 yields     χ0 , ch0 (PN , σ N ) = Tr χ0 (PN )σ N (K 1−4 K 2−4 ) = 0, 0, 0|K 14 K 24 |0, 0, 0 = q 2N , with |0, 0, 0 the highest weight of the representation ρ 0,N if N ≥ 0 or ρ −N ,0 if N ≤ 0. As a byproduct, we see that if q is transcendental, the equivariant K 0 -group has U (su(3)) (A(CPq2 )) ⊃ Z∞ , i.e. all [PN ] (at least) a countable number of generators: K 0 q are independent. Indeed, were the classes [PN ] not independent, there  would exist a sequence {k N } of integers – all zero but for finitely many – such that N k N q 2N = 0, and q would be the root of a non-zero polynomial with integer coefficients. The results above are analogous to the ones for the standard Podle´s sphere (cf. Prop. 5.1 and 5.2 in [21]). In fact they are instances of the general fact (cf. [16, Thm. 3.6]) that the equivariant K 0 group is a free abelian group and generators are, for the present case, in bijection with equivalence classes of irreducible corepresentations of Uq (u(2)). 8. Concluding Remarks On a four-dimensional manifold (anti)self-dual connections are stationary points (usually minima) of the Yang-Mills action functional, i.e. they are solutions of the corresponding equations of motion. In dimension greater than four, ‘generalized’ instantons can be defined as solutions of Hermitian Yang-Mills equations. On CPn a basic instanton solution is associated to the canonical (universal) connection on the Stiefel bundle U(n) → U(n + 1)/U(1) → CPn . The extension of this construction to quantum complex projective spaces – using the differential calculus in [5] – will be explored in future works, and it should help, in particular, to understand how to generalize Hermitian Yang-Mills equations to noncommutative spaces. Appendix A. Proof of Proposition 5.1 In this appendix, we determine the most general value of the normalization constants in order to have a left Uq (u(2))-covariant product on V •,• which is i) associative, ii) graded commutative for q = 1, and iii) it sends real vectors into real vectors. Indeed, as a way of illustration, let us start by considering the cases V 0,1 × V 1,0 → 1,1 V and V 0,1 × V 0,1 → V 0,2 . As vector spaces V 0,1  V 1,0  C2 . For v, w ∈ C2 we order the components of v ⊗ w as: v ⊗ w = (v1 w1 , v1 w2 , v2 w1 , v2 w2 )t . A unitary equivalence U between σ 1 ,N ⊗ σ 1 ,N  and σ1,N +N  ⊕ σ0,N +N  is given by 2

2

⎛ 1 ⎜ ⎜0 U =⎜ ⎜0 ⎝

q − 2 [2]− 2

q 2 [2]− 2

0

0

0

q 2 [2]− 2

0 1

1

⎞ 0 ⎟ 0⎟ ⎟. 1⎟ ⎠

0 1

1

1

1

1

1

−q − 2 [2]− 2

(A.1)

0

It is easy to check that U (σ 1 ,N (h (1) )v ⊗ σ 1 ,N  (h (2) )w) = (σ1,N +N  (h) ⊕ σ0,N +N  (h)) 2

2

U (v⊗w) for all h ∈ Uq (u(2)) by doing it explicitly on all generators. For h = K 1 , K 1 K 22

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this is trivial. For h = E 1 , and omitting the representation symbols, we have ⎛ ⎞ 1 q − 2 v2 w2 ⎜ ⎟ 1 ⎜ ⎟ q 2 v2 w2 U (E 1 v ⊗ K 1 w + K 1−1 v ⊗ E 1 w) = U ⎜ ⎟ 0 ⎝ ⎠ 1 1 q − 2 v1 w2 + q 2 v2 w1 ⎞ ⎛ 1 1 q − 2 v1 w2 + q 2 v2 w1 1 ⎟ ⎜ [2] 2 v2 w2 ⎟, =⎜ ⎠ ⎝ 0 0 and ⎛ 0 1 ⎜0 E 1 U (v ⊗ w) = [2] 2 ⎝ 0 0

1 0 0 0

0 1 0 0

⎛ ⎞ 0 0 ⎜ 0⎟ ⎜ U (v ⊗ w) = ⎝0 0⎠ 0 0 0

1

q− 2 0 0 0

1

q2 0 0 0

⎞ 0 1⎟ [2] 2 ⎟ (v ⊗ w). 0 ⎠ 0

Thus U (E 1 ) = E 1 U . Then the statement holds for F1 since F1 = E 1∗ . The map ∧q : V 0,1 × V 1,0 → V 1,1 is then v ∧q w = diag(c1 , c1 , c1 , c2 )U (v ⊗ w), where c1 , c2 ∈ R are arbitrary for the time being. When ci = ±1 we would get partial isometries; by composing U with the orthogonal projection onto the last component, we would get a partial isometry from V 0,1 × V 0,1 → V 0,2 . The general situation is listed in the following proposition. Proposition A.1. The most general left Uq (u(2))-covariant graded product ∧q on V •,• , sending real vectors to real vectors, is given by V 0,1 × V 0,1 → V 0,2 , v ∧q w := c0 μ0 (v, w)t , V 0,1 × V 1,0 → V 1,1 ,

v ∧q w := (c1 μ1 (v, w), c2 μ0 (v, w))t ,

V 0,1 × V 2,1 → V 2,2 ,

v ∧q w := c3 μ0 (v, w)t ,

V 0,1 × V 1,1 → V 1,2 ,

v ∧q w := c01 [3]− 2 μ2 (v, w)t + c02 vw4 ,

1

V 1,0 × V 1,0 → V 2,0 , v ∧q w := c4 μ0 (v, w)t , V 1,0 × V 0,1 → V 1,1 ,

v ∧q w := (−d0 μ1 (v, w), d1 μ0 (v, w))t ,

V 1,0 × V 1,2 → V 2,2 ,

v ∧q w := d2 μ0 (v, w)t ,

V 1,0 × V 1,1 → V 2,1 ,

v ∧q w := c11 [3]− 2 μ2 (v, w)t + c12 vw4 ,

V 1,2 × V 1,0 → V 2,2 ,

v ∧q w := d3 μ0 (v, w)t ,

V 2,1 × V 0,1 → V 2,2 ,

v ∧q w := d4 μ0 (v, w)t ,

V 1,1 × V 0,1 → V 1,2 ,

v ∧q w := −c21 [3]− 2 μ3 (v, w)t + c22 v4 w,

V 1,1 × V 1,0 → V 2,1 ,

v ∧q w := −c31 [3]− 2 μ3 (v, w)t + c32 v4 w,

V 1,1 × V 1,1 → V 2,2 ,

v ∧q w := c41 [3]− 2 μ4 (v, w) + c42 v4 w4 .

1

1 1

1

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j

The coefficients ci , d j and ci ∈ R are arbitrary for the time being; and the maps μi ’s are 1

1

1

μ3 : R3 × R2 → R2 ,

μ0 (v, w) := [2]− 2 (q 2 v1 w2 − q − 2 v2 w1 ),   1 1 1 μ1 (v, w) := v1 w1 , [2]− 2 (q − 2 v1 w2 + q 2 v2 w1 ), v2 w2 ,   1 1 1 1 μ2 (v, w) := qv1 w2 − q − 2 [2] 2 v2 w1 , q 2 [2] 2 v1 w3 − q −1 v2 w2 ,   1 1 1 1 μ3 (v, w) := q 2 [2] 2 v1 w2 − q −1 v2 w1 , qv2 w2 − q − 2 [2] 2 v3 w1 ,

μ4 : R3 × R3 → R,

μ4 (v, w) := qv1 w3 − v2 w2 + q −1 v3 w1 .

μ0 : R2 × R2 → R, μ1 : R2 × R2 → R3 , μ2 : R2 × R3 → R2 ,

On the other hand, there is no need to specify the multiplication rule by elements in V 0,0 = V 0,2 = V 2,0 = V 2,2 = C, being scalars. When all the coefficients ci ’s and di ’s are equal to ±1 and (ci1 )2 + (ci2 )2 = 1, the maps in the proposition are partial isometries. In general, not all the choices give an j associative product. Associativity fixes the value of the parameters di ’s and ci ’s. Lemma A.2. We have μ4 (μ1 (v, v  ), w) = μ0 (μ2 (v, w), v  ) = μ0 (μ3 (w, v), v  ) = μ2 (v, μ1 (v  , v  )) = μ3 (μ1 (v, v  ), v  ) =

μ0 (v, μ2 (v  , w)), μ0 (v, μ3 (w, v  )), μ4 (w, μ1 (v, v  )), [2] μ0 (v, v  )v  + v μ0 (v  , v  ), μ0 (v, v  )v  + [2] v μ0 (v  , v  ),

(A.2a) (A.2b) (A.2c) (A.2d) (A.2e)

for all v, v  , v  ∈ C2 and w ∈ C3 . Proof. By direct computation. Both sides of (A.2a) are equal to 1

1

1

qv1 v1 w3 − [2]− 2 (q − 2 v1 v2 + q 2 v2 v1 )w2 + q −1 v2 v2 w1 , both sides of (A.2b) are equal to 3

1

3

1

q 2 [2]− 2 v1 w2 v2 − v1 w3 v1 − v2 w1 v2 + q − 2 [2]− 2 v2 w2 v1 , both sides of (A.2c) are equal to 1

1

1

qv1 w2 v2 − [2]− 2 v2 (q − 2 w1 v2 + q 2 w2 v1 ) + q −1 v3 w1 v1 , both sides of (A.2d) are equal to & 1 1 3 1 1 1 q 2 [2]− 2 v1 v1 v2 + q 2 [2]− 2 v1 v2 v1 − q − 2 [2] 2 v2 v1 v1 1

1

3

1

1

'

1

q 2 [2] 2 v1 v2 v2 − q − 2 [2]− 2 v2 v1 v2 − q − 2 [2]− 2 v2 v2 v1

,

both sides of (A.2e) are equal to & 1 1 ' 3 1 1 1 q 2 [2] 2 v1 v1 v2 − q − 2 [2]− 2 v1 v2 v1 − q − 2 [2]− 2 v2 v1 v1 . 1 1 3 1 1 1 q 2 [2]− 2 v1 v2 v2 + q 2 [2]− 2 v2 v1 v2 − q − 2 [2] 2 v2 v2 v1  

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Proposition A.3. The map ∧q in Prop. A.1 is a graded associative product on V •,• if 1

d0 = s1 q 2 s2 c1 , √ [3] c0 1 , c1 = [2] c1 1 c0 c12 = − , [2] c2 √ 1 [3] c4 c21 = −s1 q − 2 s2 , [2] c1 3 1 c4 c22 = −s1 q 2 s2 , [2] c2 √ 1 [3] c3 c4 c51 = −s1 q − 2 s2 , [2] |c1 |2 c3 c4 , d2 = c0 d4 = c3 ,

3

d1 = s1 q − 2 s2 c2 , √ 1 − 12 s2 [3] c0 c3 = s1 q , [2] c1 3 1 c0 c32 = −s1 q 2 s2 , [2] c2 √ [3] c4 c41 = − , [2] c1 1 c4 c42 = − , [2] c2 3 1 c3 c4 c52 = −s1 q 2 s2 , [2] |c2 |2 c3 c4 d3 = , c0

(A.3a) (A.3b) (A.3c) (A.3d) (A.3e) (A.3f) (A.3g) (A.3h)

where {s1 , s2 } ∈ {±1} are arbitrary signs. The algebra is graded commutative in the q → 1 limit iff s1 = 1. Proof. We seek solutions with all parameters different from zero. The product is graded by construction. We impose the condition: (v ∧q v  ) ∧q v  = v ∧q (v  ∧q v  ). If one of the three vectors v, v  , v  is a scalar, i.e. an element of V 0,0 , V 0,2 , V 2,0 or V 2,2 , the equality follows by the bilinearity of ∧q . The non-trivial cases are when all three vectors are not scalars. If the total degree is greater than 4, we get 0 = 0. The remaining non-trivial cases are first (v, v  , v  ) ∈ V 0,1 × V 0,1 × V 1,0 and (two) permutations: (v ∧q v  ) ∧q v  = c0 μ0 (v, v  )v  , 1

v ∧q (v  ∧q v  ) = c1 c11 [3]− 2 μ2 (v, μ1 (v  , v  )) + c2 c12 vμ0 (v  , v  ), 1

(v ∧q v  ) ∧q v  = −c1 c31 [3]− 2 μ3 (μ1 (v, v  ), v  ) + c2 c32 μ0 (v, v  )v  , 1

v ∧q (v  ∧q v  ) = −d0 c11 [3]− 2 μ2 (v, μ1 (v  , v  )) + d1 c12 vμ0 (v  , v  ), 1

(v  ∧q v) ∧q v  = d0 c31 [3]− 2 μ3 (μ1 (v  , v), v  ) + d1 c32 μ0 (v  , v)v  , v  ∧q (v ∧q v  ) = c0 v  μ0 (v, v  ). Using (A.2d–A.2e) one sees that associativity is just the conditions (A.3a-A.3c) respectively. Then, (v, v  , v  ) ∈ V 1,0 × V 1,0 × V 0,1 and (two) permutations:

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

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(v ∧q v  ) ∧q v  = c4 μ0 (v, v  )v  , 1

v ∧q (v  ∧q v  ) = −d0 c21 [3]− 2 μ2 (v, μ1 (v  , v  )) + d1 c22 vμ0 (v  , v  ), 1

(v ∧q v  ) ∧q v  = d0 c41 [3]− 2 μ3 (μ1 (v, v  ), v  ) + d1 c42 μ0 (v, v  )v  , 1

v ∧q (v  ∧q v  ) = c1 c21 [3]− 2 μ2 (v, μ1 (v  , v  )) + c2 c22 vμ0 (v  , v  ), 1

(v  ∧q v) ∧q v  = −c1 c41 [3]− 2 μ3 (μ1 (v  , v), v  ) + c2 c42 μ0 (v  , v)v  , v  ∧q (v ∧q v  ) = c4 v  μ0 (v, v  ). Using again (A.2d-A.2e) one sees that associativity is just the condition (A.3d-A.3e) respectively. Finally, (v, v  , v  ) ∈ V 0,1 × V 1,0 × V 1,1 and (five) permutations: 1

{v ∧q v  } ∧q v  = c2 c52 μ0 (v, v  )w4 + c1 c51 [3]− 2 μ4 (μ1 (v, v  ), v  ), 1

v ∧q {v  ∧q v  } = c3 c22 μ0 (v, v  w4 ) + c3 c21 [3]− 2 μ0 (v, μ2 (v  , v  )), 1

{v ∧q v  } ∧q v  = c12 d3 μ0 (vw4 , v  ) + d3 c11 [3]− 2 μ0 (μ2 (v, v  ), v  ), 1

v ∧q {v  ∧q v  } = c3 c42 μ0 (v, w4 v  ) − c3 c41 [3]− 2 μ0 (v, μ3 (v  , v  )), 1

{v  ∧q v} ∧q v  = d1 c52 μ0 (v  , v)w4 − d0 c51 [3]− 2 μ4 (μ1 (v  , v), v  ), 1

v  ∧q {v ∧q v  } = d2 c12 μ0 (v  , vw4 ) + d2 c11 [3]− 2 μ0 (v  , μ2 (v, v  )), 1

{v  ∧q v  } ∧q v = d4 c22 μ0 (v  w4 , v) + d4 c21 [3]− 2 μ0 (μ2 (v  , v  ), v), 1

v  ∧q {v  ∧q v} = d2 c32 μ0 (v  , w4 v) − d2 c31 [3]− 2 μ0 (v  , μ3 (v  , v)), 1

{v  ∧q v} ∧q v  = d3 c32 μ0 (w4 v, v  ) − d3 c31 [3]− 2 μ0 (μ3 (v  , v), v  ), 1

v  ∧q {v ∧q v  } = c2 c52 w4 μ0 (v, v  ) + c1 c51 [3]− 2 μ4 (v  , μ1 (v, v  )), 1

{v  ∧q v  } ∧q v = d4 c42 μ0 (w4 v  , v) − d4 c41 [3]− 2 μ0 (μ3 (v  , v  ), v), 1

v  ∧q {v  ∧q v} = d1 c52 w4 μ0 (v  , v) − d0 c51 [3]− 2 μ4 (v  , μ1 (v  , v)). Using (A.2), associativity for the first couple is shown to be equivalent to (A.3f), for the second couple to the second condition in (A.3g), for the third couple to the first condition in (A.3g), for the fourth couple to (A.3h). For the last two couples the associativity is automatically satisfied. Graded anticommutativity in the q → 1 limit is a simple check, based on the observation that for q = 1: μ0 is antisymmetric, μ1 is symmetric, μ2 (v  , v  ) = −μ3 (v  , v  ), and μ4 is symmetric. This concludes the proof.   Appendix B. Some General Facts on Calculi In this appendix we review the material that leads to the construction of the differential calculus (• (CPq2 ), d) on CPq2 , in particular the fact that it is enough to define things for one-forms and then extend them in a natural and unique (by universality) way.

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Recall that a first order differential calculus on a unital ∗-algebra A is a pair (1 A, d), where 1 A is an A-bimodule giving the space of one-forms and d : A → 1 A is the exterior differential – a linear map satisfying the Leibniz rule, d(ab) = a(db) + (da)b

for all a, b ∈ A.

We also assume that 1 A = A(dA) and that we are dealing with a ∗-calculus with the ∗-structure on 1 A given (uniquely) by (da)∗ = −d(a ∗ ) for all a ∈ A. To go beyond one forms a differential graded ∗-algebra (• = ⊕k k A, d) is defined as follows. The set • A is a graded algebra, with 0 A = A and graded product denoted ∧q : k A × l A → k+l A. The universal differential calculus of A is obtained when ∧q is the tensor product, 1 A is the kernel of the multiplication map m(a ⊗ b) = ab and da = a ⊗ 1 − 1 ⊗ a. With a general associative (unital) graded multiplication ∧q , the calculus will always be a quotient of the universal calculus by a differential ideal. Elements in k A (that is k-forms) are (sums of) products of k 1-forms: ω = v1 ∧q . . . ∧q vk . The differential is extended to k A by requiring that its square vanishes, d2 = 0, and that it is a graded derivation of degree 1, that is d : k A → k+1 A and d(ω ∧q ω ) = dω ∧q ω + (−1)dg(ω) ω ∧q dω , for any two forms ω and ω . By universality, these two properties uniquely identify d. It is the map given on 1-forms by d(adb) = da ∧q db = −d(da b) on any product of k 1-forms by d(v1 ∧q . . . ∧q vk ) =

k 

(−1)i−1 v1 ∧q . . . ∧q dvi ∧q . . . ∧q vk ,

i=1

and extended by linearity. Finally, a ∗-structure on a differential calculus is given by a graded involution which anti-commutes with the differential. Given a ∗-structure on A, a ∗-structure on (• A, d) is uniquely defined first on 1-forms by (adb)∗ = −(db∗ )a ∗ , and then by induction on k-forms by (v ∧q w)∗ = (−1)k−1 w ∗ ∧q v ∗ , where v is a 1-form and w is a k − 1 form. With this, one easily checks, again by induction, that dω∗ = −(dω)∗ for any form ω. An equivalent way to give a ∗-differential calculus (• A, d) is via a differential ¯ That is to say, starting with a bigradation k A := double complex (•,• A, ∂, ∂). ( i, j A with a bigraded product, the following statements are equivalent:  i+ j=k i) there is a graded derivation d : • A → •+1 A satisfying d2 = 0

and

dω = −(dω∗ )∗ ;

ii) there are two graded derivations ∂ : •,• A → •+1,• A and ∂¯ : •,• A → •,•+1 A, satisfying ¯ =0 ∂ 2 = ∂¯ 2 = ∂ ∂¯ + ∂∂

and

¯ = −(∂ω∗ )∗ . ∂ω

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Appendix C. An Alternative Proof of Theorem 7.2 for n ≤ 4 We give an alternative proof that the map chn : K 0U (A) → HCnU (A) given by ˙

chn (e, σ )(h) := Tr(e ⊗ n+1 σ (h)t ),

(C.1)

is well-defined. This proof is valid for n ≤ 4 and for it we use the results of Lemma 7.1 stating the equivalence of equivariant projective modules in terms of idempotents. Thus, we aim at proving that for any [(e, σ )] ∈ K 0U (A), the expression in (C.1) is an equivariant cyclic n-cycle, and that if (e, σ ) is in the same class of (e , σ  ), the difference chn (e, σ ) − chn (e , σ  ) is a boundary. In fact, when n is odd (C.1), chn (e, σ ) = 21 (1 − λn )chn (e, σ ), and chn (e, σ ) reduces in CnU (A)/Im(1 − λn ) to the trivial equivariant cyclic cocycle 0. For n even, one finds that bn chn (e, σ ) = chn−1 (e, σ ) which vanishes in CnU (A)/ Im(1 − λn ) since n − 1 is now odd and the class of chn−1 (e, σ ) is zero as before. Thus, chn (e, σ ) is an equivariant cyclic cocycle. For the last part of the proof, ch0 (e, σ ) − ch0 (e , σ  ) is the boundary of the function   ˙ vσ (h)t ; Tr u ⊗ ch2 (e, σ ) − ch2 (e , σ  ) is the boundary (modulo the image of 1 − λ2 ) of       ˙ ˙ e ⊗˙ 2 ⊗ ˙ v⊗ ˙ u⊗ ˙ vσ (h)t ; ˙ u⊗ ˙ vσ (h)t + Tr u ⊗ ˙ vσ (h)t + 21 Tr u ⊗ Tr e ⊗ 2 ⊗ and ch4 (e, σ ) − ch4 (e , σ  ) is the boundary (modulo the image of 1 − λ4 ) of     ˙ ˙ e ⊗˙ 4 ⊗ ˙ u⊗ ˙ vσ (h)t + Tr u ⊗ ˙ vσ (h)t Tr e ⊗ 4 ⊗     ˙ ˙ e ⊗˙ 2 ⊗ ˙ u⊗ ˙ v⊗ ˙ u⊗ ˙ vσ (h)t − 21 Tr u ⊗ ˙ v⊗ ˙ u⊗ ˙ vσ (h)t − 21 Tr e ⊗ 2 ⊗     ˙ ˙ v⊗ ˙ u⊗ ˙ v⊗ ˙ u⊗ ˙ vσ (h)t . ˙ u⊗ ˙ e ⊗˙ 2 ⊗ ˙ vσ (h)t + 16 Tr u ⊗ + 21 Tr e ⊗ 2 ⊗ This concludes the proof. In general, for an arbitrary high n, it is not an easy task to write chn (e, σ ) − chn (e , σ  ) explicitly as the boundary of something else. Appendix D. Irreducible ∗-Representations of CPq2 In this appendix we prove the result mentioned at the end of Sect. 4.3, namely that any non-trivial irreducible ∗-representation (irrep for short) of A(CPq2 ) (by bounded operators on a Hilbert space) is unitarily equivalent to one of the representations χ0 , χ1 or χ2 described in the paper. Lemma D.1. Let π : A(CPq2 ) → B(H) be a ∗-representation such that ker π( p11 ) = {0}. Then there exists Z i ∈ B(H), i = 1, 2, 3, such that Z 1 is positive and ⎞ ⎛ Z 12 q Z2 Z1 q Z3 Z1 ⎟ ⎜ (D.1) π(P) = ⎝q Z 1 Z 2∗ Z 2∗ Z 2 Z 2∗ Z 3 ⎠ . q Z 1 Z 3∗

Z 3∗ Z 2

Z 3∗ Z 3

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With this, a ∗-representation π˜ : A(Sq5 ) → B(H) is defined by π˜ (z i ) = Z i and it satisfies ker π˜ (z 1 ) = {0}. Moreover, Z i∗ Z j = π( pi j ), so that the restriction of π˜ to A(CPq2 ) is exactly π . The ∗-representation π˜ is irreducible if π is irreducible. Proof. Recall that the generators pi j of A(CPq2 ) satisfy the commutation relations in  Sect. 2.3, the quadratic relations j pi j p ji = pii , i = 1, 2, 3, and the real structure is given pi j = p ∗ji . They are related to the generators z i of A(Sq5 ) by pi j = z i∗ z j .   ∗ Since π is a ∗-representation, π( pii ) = j π( pi j )π( p ji ) = j π( p ji ) π( p ji ) is a sum of positive operators and then it is positive as well. The positive operators 1 Ai = π( pii ) 2 are well defined and are mutually commuting since pii are mutually commuting.  We have (q 4 p11 + q 2 p22 + p33 ) pii = pii = j pi j p ji , for all i = 1, 2, 3. Using the commutation rules between pi j and p ji yields q 2 p11 p22 + p11 p33 = p21 p12 + q −2 p31 p13 , q 4 p11 p22 + p22 p33 = q 2 p21 p12 + (q −2 − 1) p31 p13 + q −2 p32 p23 , q 4 p11 p33 + q 2 p22 p33 = p31 p13 + p32 p23 . By solving these equations we get pi1 p1i = q 2 p11 pii ,

i = 2, 3,

(D.2)

and a third relation for p32 p23 that we don’t need. Similarly, using the commutation 2 4 2 rule [ p33 , p23 ] = (1 − q )( p21 p13 + p22 p23 ), the relation (q p11 + q p22 + p33 ) p23 = p23 = i p2i pi3 can be rewritten as p11 p23 = q −2 p21 p13 .

(D.3)

Writing T = V |T | for the polar decomposition of a bounded operator T , for the polar decomposition of π( p1i ), i = 2, 3, from (D.2) we get |π( p1i )|2 = q 2 A21 Ai2 = (q A1 Ai )2 . Therefore π( p1i ) = q Vi A1 Ai . Applying π to both sides of Eq. (D.3) yields A1 π( p23 )A1 = A1 A2 V2∗ V3 A3 A1 ,

(D.4)

where we used the fact that p11 and p23 commute, and so do A1 and π( p23 ). For any bounded operator T , it holds that ker(T )⊥ = range(T ∗ ). Being A1 positive with ker(A1 ) = {0}, we have range(A1 ) = H. Since the range of A1 is dense in H, we can simplify the A1 factor on the right of both sides of (D.4), and since ker(A1 ) = {0}, we can simplify the A1 on the left too. We get π( p23 ) = A2 V2∗ V3 A3 . Setting A1 =: Z 1 , V2 A2 =: Z 2 and V3 A3 =: Z 3 concludes the proof of (D.1). Next, we show that π(z ˜ i ) := Z i defines a ∗-representation of A(Sq5 ), i.e. that the Z i ’s satisfy the commutation rules of the algebra A(Sq5 ). Then the rest of the proof is straightforward: if the relations of A(Sq5 ) are satisfied, some algebraic manipulation immediately gives from (D.1) that Z i∗ Z j = π( pi j ) with ker Z 1 = ker π( p11 ) = {0} by hypothesis. Also, since π˜ (A(Sq5 ))H ⊃ π(A(CPq2 ))H, if the latter space is dense in H, so it must be the former: so if π is irreducible, π˜ is irreducible too. Among the defining relations of CPq2 we have p11 p1i = q 2 p1i p11 for i = 2, 3. Applying π to both sides and using (D.1) it becomes Z 1 (Z 12 Z i − q 2 Z i Z 12 ) = 0. This

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reduces to Z 12 Z i = q 2 Z i Z 12 , since ker Z 1 = {0}, allowing to simplify the factor Z 1 on the left and get. Being Z 1 positive it can be diagonalized, and by the previous relation Z i intertwines the eigenspace of Z 12 with eigenvalue λ ≥ 0 with the eigenspace of Z 12 √ √ √ with eigenvalue q 2 λ, i.e. it maps the eigenvalue λ of Z 1 to q λ (−q λ is excluded by the positivity of Z 1 ). This proves Z1 Zi = q Zi Z1,

i = 2, 3.

(D.5)

Since Z 1 = Z 1∗ , by conjugating we get Z 1∗ Z i = q Z i Z 1∗ . The defining relation p12 p13 = qp13 p12 yields Z 12 (Z 2 Z 3 − q Z 3 Z 2 ) = 0, using (D.1) and (D.5). Again Z 12 can be simplified having kernel {0} and so Z 2 Z 3 = q Z 3 Z 2 . Similarly from p21 p13 = q 3 p13 p21 we deduce Z 2∗ Z 3 = q Z 3 Z 2∗ – and by conjugation Z 3∗ Z 2 = q Z 2 Z 3∗ . From Z 1 = Z 1∗ we get the relation [Z 1∗ , Z 1 ] = 0; from q −2 p21 p12 − 2 we get [Z ∗ , Z ] = (1 − q 2 )Z Z ∗ , from q −2 p p − p p = p12 p21 = (1 − q 2 ) p11 2 1 1 31 13 13 31 2 2 2 (1 − q )( p11 + p12 p21 ) we get [Z 3∗ , Z 3 ] = (1 − q 2 )(Z 1 Z 1∗ + Z 2 Z 2∗ ). Using previous formulas for [Z 2∗ , Z 2 ] and [Z 3∗ , Z 3 ], the tracial relation Trq π(P) = 1 gives the spherical  relation i Z i Z i∗ = 1. With this, all the defining relations of Sq5 are satisfied and the proof is complete.   It is in general not true that a representation on a Hilbert space of a subalgebra of a given algebra can be extended to a representation of the full algebra on the same Hilbert space: one needs at least to extend the Hilbert space. The point here is that we can extend the representation from A(CPq2 ) to A(Sq5 ) without enlarging the Hilbert space. By the previous lemma any irrep π : A(CPq2 ) → B(H) with ker π( p11 ) = {0} is the restriction of an irrep π˜ : A(Sq5 ) → B(H) with ker π(z 1 ) = {0}. On the other hand any irrep π : A(CPq2 ) → B(H) with π( p1i ) = 0 for all i = 1, 2, 3 is the pullback of an irrep of the standard Podle´s sphere, the ∗-algebra morphism A(CPq2 ) → A(CPq1 ) being the map p1 j , p j1 → 0∀ j,

p22 → A,

p23 → B ∗ ,

p32 → B,

p33 → 1 − q 2 A,

where A and B are the original generators of Podle´s [17]. The next lemma shows that only these two cases are possible. Lemma D.2. For any irrep π of A(CPq2 ) it is either ker π( p11 ) = {0} or π( p1i ) = 0 for all i = 1, 2, 3. Proof. Suppose there is a vector v such that π( p11 )v = 0. The element p11 commutes with the generators pi j for all i, j > 1, while p11 p1i = q 2 p1i p11 and p11 pi1 = q −2 pi1 p11 for all i > 1. This means that for any polynomial a in the generators π( p11 )π(a)v ∝ π(a)π( p11 )v = 0 and by linearity π( p11 )π(a)v = 0 for any a ∈ A(CPq2 ). We conclude that the kernel of π( p11 ) carries a subrepresentation of π , but π is irreducible, thus we have only two cases: ker π( p11 ) = {0} or π( p11 ) = 0. In the latter case    ∗ i π( pi1 ) π( pi1 ) = π i p1i pi1 = π( p11 ) = 0. A sum of positive operators π( pi1 )∗ π( pi1 ) is zero iff each one is zero, and for a bounded operator T the condition T ∗ T = 0 implies T = T ∗ = 0. Thus π( p11 ) = 0 implies that π( p12 ) = π( p13 ) = 0 as well.  

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We need to recall the representation theory of quantum spheres. Irreps of quantum orthogonal spheres A(Sqn ) are classified in [11] – and in particular we are interested in A(Sq5 ) – while the case of the standard Podle´s sphere CPq1 is in [17]. The generators z i of A(Sq5 ) are related to the xi ’s used in [11] by xi = z i∗ and by the replacement q → q −1 . We collect some results of [11,17] into the following proposition, adapted to our notations. Proposition D.3. Any non-trivial irrep of A(CPq1 ) is unitarily equivalent to one of the following two representations (cf. [17, Prop. 4.I], with parameter c = 0). The first is one-dimensional, χ0 ( pi j ) = δi3 δ j3 (and χ0 (1) = 1), the second is just the map χ1 in (4.5). Any irrep of A(Sq5 ) with z 1 not in the kernel is unitarily equivalent to one of the following family, parametrized by λ ∈ U (1) (cf. [11], Eq. (3.10), n = 2): ψλ5 (z 1 ) |k1 , k2 := λq k1 +k2 |k1 , k2 ,  ψλ5 (z 2 ) |k1 , k2 := q k1 1 − q 2(k2 +1) |k1 , k2 + 1 ,  ψλ5 (z 3 ) |k1 , k2 := 1 − q 2(k1 +1) |k1 + 1, k2 . 5 Note that ker ψλ5 (z 1 ) = {0}. For λ = 1 we get the representation ψλ=1 = χ2 in (4.7).

Notice that restricted to A(CPq2 ) all the representations ψλ5 are unitarily equivalent. Indeed, let U be the unitary transformation U |k1 , k2 := λk1 +k2 |k1 , k2 ; one easily 5 (a) = χ (a) for all a ∈ A(CP2 ) (indeed it is enough to checks that U ψλ5 (a)U ∗ = ψλ=1 2 q prove it for a = pi j ). By Lemma D.1 and Lemma D.2 any non-trivial irrep of A(CPq2 ) is either the pullback of one of A(CPq1 ) – and then unitarily equivalent to χ0 or χ1 – or it is the restriction of an irrep π˜ of A(Sq5 ) such that ker π˜ (z 1 ) = {0} – and by Prop. D.3 it is unitarily equivalent to (the restriction of) χ2 . Acknowledgement. GL was partially supported by the ‘Italian project Cofin06 - Noncommutative geometry, quantum groups and applications’.

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