A new construction of strict deformation quantization for Lagrangian fiber bundles
aa r X i v : . [ m a t h . S G ] M a r A NEW CONSTRUCTION OF STRICT DEFORMATIONQUANTIZATION FOR LAGRANGIAN FIBER BUNDLES
MAYUKO YAMASHITA
Abstract.
We give a new construction of strict deformation quantiza-tion of symplectic manifolds equipped with a proper Lagrangian fiberbundle structure, whose representation spaces are the quantum Hilbertspaces obtained by geometric quantization. The construction can be re-garded as a ”lattice approximation of the correspondence between differ-ential operators and principal symbols”. We analyze the correspondingformal deformation quantization. We also investigate into relations be-tween our construction and Berezin-Toeplitz deformation quantization.
Contents
1. Introduction 11.1. Deformation quantizations and geometric quantizations 21.2. A motivation for the construction — The quantization of T ∗ M R n × T n . 73.2. The general case 134. Star products 195. The relation with Berezin-Toeplitz quantization 245.1. On R n × T n . 255.2. On Abelian varieties 286. Appendix 296.1. A proof of Proposition 3.17 296.2. A proof of Theorem 3.33 34Acknowledgment 40References 401. Introduction
In this paper, we give a new construction of strict deformation quantiza-tion of symplectic manifolds equipped with a proper Lagrangian fiber bundlestructure, whose representation spaces are the quantum Hilbert spaces ob-tained by geometric quantization.
Research Institute for Mathematical Sciences, Kyoto University, 606-8502,Kyoto, Japan
E-mail address : [email protected] . Deformation quantizations and geometric quantizations.
Firstwe explain some background. Let ( X n , ω ) be a 2 n -dimensional symplecticmanifold. We have the Poisson algebra structure on C ∞ ( X ). In this paperwe are interested in finding strict deformation quantizations for ( X, ω ). Thenotion of strict (or C ∗ -algebraic ) deformation quantization is introduced in[9]. In this paper we use the following definition. Definition 1.1.
Given a symplectic manifold (
X, ω ), a strict deformationquantization consists of the following data. • A sequence of Hilbert spaces {H k } k ∈ N . • A sequence { Q k } k ∈ N of adjoint-preserving linear maps Q k : C ∞ c ( X ) → B ( H k ) so that for all f, g ∈ C ∞ ( X ), we have(1) k Q k ( f ) k → k f k C as k → ∞ , and(2) k [ Q k ( f ) , Q k ( g )] + √− k Q k ( { f, g } ) k = O ( k ) as k → ∞ .Note that there exists many variants in the definition, and the most gen-eral one uses the notion of continuous fields of C ∗ -algebras as in [9].Another formulation of deformation quantization is formal deformationquantizations defined in [3], where they seek for an associative unital prod-uct ∗ , called a star product , on the set of formal power series C ∞ ( X )[[ ~ ]]satisfying f ∗ g = f g + O ( ~ ) and f ∗ g − g ∗ f = { f, g } ~ + O ( ~ ) . Strict and formal deformation quantizations are related as follows. If wehave a strict deformation quantization { Q k } k ∈ N for ( X, ω ), we expect tosolve the equations Q k ( f ) Q k ( g ) = l X j =0 (cid:18) −√− k (cid:19) j Q k ( C j ( f, g )) + O (cid:18) k l +1 (cid:19) (1.2)recursively in l , where C j ( · , · ) is expected to be given by a differential oper-ator, and get a star product ∗ by f ∗ g = ∞ X j =0 C j ( f, g ) ~ j . On the other hand, given a symplectic manifold (
X, ω ), geometric quan-tization is a process to produce quantum Hilbert spaces , which is, physically,expected to be representation spaces for the Poisson algebra C ∞ ( X ). See[11] for details. The process goes as follows. First we fix a prequantizingline bundle ( L, ∇ ) on X , which is a hermitian line bundle with unitary con-nection, satisfying ∇ = −√− ω . To do this we need the integrality of ω/ (2 π ). Next we choose a polarization P ⊂
T X ⊗ C , which is an integrableLagrangian subbundle of T X ⊗ C . Then, roughly speaking we define thequantum Hilbert space H k for k ∈ N as the space of sections of L k which isparallel, with respect to the connection induced from ∇ , along vectors in P .In this paper we are particularly interested in the polarization comingfrom a proper Lagrangian fiber bundle µ : X n → B n with connected fibers.In this case the fibers are n -dimensional affine torus by Arnold-Liouvilletheorem [1]. A point b ∈ B is called a k - Bohr-Sommerfeld point if the space
EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 3 of fiberwise parallel sections of ( L k , ∇ ), denoted by H ( X b ; L k ), is nontrivial.In this paper we define the quantum Hilbert space by the following. H k = ⊕ b ∈ B k H ( X b ; L k ⊗ | Λ | / X b ) , (1.3)where B k denotes the set of k -Bohr-Sommerfeld points, and | Λ | / X b := | Λ | / ( T ∗ X b ) is the half-density bundle equipped with the canonical flatconnection. Thus, we have a one-dimensional Hilbert space on each k -Bohr-Sommefeld point, and the quantum Hilbert space is their direct sum. ByArnold-Liouville theorem, we can take a local action-angle coordinate whichidentifies the prequantum line bundle with the standard one, ( R n × T n , t dx ∧ dθ, L = C , ∇ = d − √− t xdθ ). In this coordinate, we have B k = Z n k . So weregard the set B k as a ”lattice approximation” of the integral affine manifold B , and H k in (1.3) can be regarded as approximation of L ( B ) as k → ∞ .This observation is the key to our construction below.On the other hand, another well-studied type of polarization is K¨ahlerpolarization , coming from an ω -compatible complex structure J on X . Inthis case P = T , J X , and the quantum Hilbert spaces are L H ( X J ; L k ),the space of L -holomorphic sections on L k .So, the following problem arises naturally. Problem 1.4.
Given a prequantized symplectic manifold ( X, ω, L, ∇ ) anda polarization P , construct a strict deformation quantization { Q k } k , whoserepresentation spaces {H k } k ∈ N are those obtained by the geometric quanti-zation. For K¨ahler quantization, there are natural and well-studied answer toProblem 1.4, namely Berezin-Toeplitz deformation quantization. This isgiven by the multiplication operator composed with the orthogonal pro-jection onto the space of holomorphic sections (see Definition 5.1 below).It was shown that these operators have the correct semiclassical behavior,in the case for compact K¨aler manifolds by Bordemann, Meinrenken, andSchlichenmaier [4], and for a certain class of non-compact K¨ahler manifoldsby Ma and Marinescu [7]. Moreover, Schlichenmaier [10] has shown thatthis induces a star product called Berezin-Toeplitz star product.However, Problem 1.4 in the case where P comes from Lagrangian fiberbundles seems to have not been considered. The purpose of this paper isto give an answer to Problem 1.4 in the case where P comes from properLagrangian fiber bundles, and investigate into some basic properties. Theconstruction can be regarded as a ”lattice approximation of the correspon-dence between differential operators and principal symbols”, where the set B k is regarded as the lattice approximation of B .1.2. A motivation for the construction — The quantization of T ∗ M . Let M n be a smooth manifold. As a motivation for our construction, in thissubsection we recall the usual symbol map and its ”inverse” map, which canbe considered as a quantization procedure of the symplectic manifold T ∗ M equipped with the vertical real polarization T ∗ M → M . M. YAMASHITA
The principal symbol map transforms commutators of differential opera-tors into the Poisson bracket with respect to the canonical symplectic struc-ture on T ∗ M , σ pr ([ D , D ]) = −√− { σ pr ( D ) , σ pr ( D ) } . Let σ ∈ C ∞ ( T ∗ R n ) be a symbol of some pseudodifferential operator on R n . Then, the operator P ∈ Ψ ∗ ( R n ) with symbol σ is, up to smoothingoperators, recovered by the following procedure. Identifying P with itsintegral kernel K ∈ C −∞ ( R n × R n ), we define K ( y, x ) = Z e −√− h y − x,ξ i σ ( x, ξ ) dξ. (1.5)In other words, the inverse map for the principal symbol map is given bythe Fourier transform on each fiber of T ∗ R n . For general manifold M , bypatching the above fiberwise Fourier transform for T ∗ M together, from aprincipal symbol, we can recover the operator up to lower order.If we regard this procedure as a quantization map of T ∗ M induced bythe Lagrangian fibration T ∗ M → M , it is natural to generalize this tothe case of proper Lagrangian fiber bundles. In this case, the fiberwiseFourier transform is replaced by the fiberwise Fourier expansion, producingan operator on the lattice approximation of the base manifold.1.3. Outline of the paper.
This paper is organized as follows. In Section3, we give a construction of strict deformation quantization. First in sub-section 3.1, we give the construction for the model case ( R n × T n , t dx ∧ dθ )equipped with the Lagrangian fiber bundle structure µ : R n × T n → R n inDefinition 3.7, and prove that it is indeed a strict deformation quantiza-tion in Theorem 3.19. Next, we deal with the general case in subsection3.2. The construction is given in Definition 3.7, and prove that it is a strictdeformation quantization in Theorem 3.34.In Section 4, we analyze the formal deformation quantization induced byour strict deformation quantization. Since our construction can be writtenexplicitely in action-angle coordinate locally, in principle we can solve theequation (1.2) explicitly. In nice cases, we show in Theorem 4.3 that thestar product obtained from our strict deformation quantization conincideswith the star product given by the Fedosov’s construction [5]. In generalcases, by an explicite computation we check this coincidence up to secondorder term in Theorem 4.7.In Section 5, we explain a relation between our construction and Berezin-Toeplitz deformation quantization. In general, if we have a prequantizedsymplectic manifold ( X, ω ) equipped with an ω -compatible complex struc-ture as well as a Lagrangian fiber bundle structure, there is no canonicalisomorphism between quantum Hilbert spaces obtained by the two polar-izations. Here we restrict our attention to the case of R n × T n (subsection5.1) and abelian varieties (subsection 5.2) with translation invariant complexstructure. In those cases we have a natural isomorphism between quantumHilbert spaces using theta basis for L -holomorphic sections on L k . We showin Theorem 5.7 and Theorem 5.13 that, as k → ∞ , the operator norm ofthe difference between our deformation quantization and Berezin-Toeplitzdeformation quantization converges to zero in both cases. EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 5
Notations. • We let T := R / (2 π Z ). • We denote the trivial hermitian line bundle over a manifold by C . • For a smooth manifold M , we denote by | Λ | / M := | Λ | / ( T ∗ M )its half-density bundle. • For a Hilbert space H , we denote by B ( H ) its bounded operatoralgebra. • Given a fiber bundle µ : X → B and a subset U ⊂ B , we write X U := µ − ( U ). For a point b ∈ B , we write X b := µ − ( b ). • For a subset U ⊂ B of the base of a Lagrangian fiber bundle, wedenote by H k | U the subspace of the quantum Hilbert space H k (1.3)defined by H k | U := ⊕ b ∈ U ∩ B k H ( X b ; L k ⊗ | Λ | / ( X b )) . We denote by P U the orthogonal projection from H k onto this sub-space. When U consists of a point, { x } = U , we also write H k | x = H k | { x } and P x = P { x } .2. Preliminaries
In this section, we recall basic facts on Lagrangian fiber bundles and itsgeometric quantizations.
Definition 2.1.
Let ( X n , ω ) be a symplectic manifold of dimension 2 n .A regular fiber bundle structure µ : X n → B n is called a Lagrangian fiberbundle if all the fibers are Lagrangian.In what follows, we are interested in the case where a Lagrangian fiberbundle is proper with connected fibers.
Example . Let us consider X = R n × T n equipped with the symplecticform ω = t dx ∧ dθ . Then µ : X → R n , ( x, θ ) → x is a Lagrangian fiberbundle.By Arnold-Liouville theorem [1], any proper Lagrangian fiber bundle withconnected fibers are locally isomorphic to the one in Example 2.2. Indeed,we have the following. Fact 2.3 ([1]) . Let ( X n , ω ) be a symplectic manifold and µ : X → B bea proper Lagrangian fiber bundle structure with connected fibers. For anypoint b ∈ B , there exists an open neighborhood U ⊂ B of b and a symplec-tomorphism ( X U , ω ) ≃ ( U ′ × T n , t dx ∧ dθ ) , where U ′ ⊂ R n , which makes thefollowing diagram commutative. ( X U , ω ) ≃ ( U ′ × T n , t dx ∧ dθ ) ↓ ↓ U ≃ U ′ (2.4)On a Lagrangian fiber bundle, we call a local coordinate ( x, θ ) ∈ R n × T n obtained by a local isomorphism (2.4) an action-angle coordinate . As is easyto see, any two action-angle coordinate are related as follows. M. YAMASHITA
Lemma 2.5.
Any symplectomorphism R n × T n → R n × T n which is com-patible with the fiber bundle structure µ : ( R n × T n , t dx ∧ dθ ) → R n is of theform ( x, θ ) ( t A − x + c, Aθ + α ( x )) , (2.6) where A ∈ GL n ( Z ) , c ∈ R n and α ∈ C ∞ ( R n ; T n ) such that A − ∂α∂x issymmetric. Thus, for a Lagrangian fiber bundle as above, the transformation betweentwo action-angle coordinates is given by the formula of the form (2.6). Fromthis, we see that the base B of a Lagrangian fiber bundle admits a canonicalintegral affine manifold structure, and the fibers admit a canonical affinetorus structure.Next we consider prequantum line bundles. Recall that a prequantumline bundle on a symplectic manifold ( X, ω ) is a hermitian line bundle withunitary connection ( L, ∇ ) whose curvature satisfies ∇ = −√− ω . Example . For the case (
X, ω ) = ( R n × T n , t dx ∧ dθ ), we can set ( L, ∇ ) =( C , d − √− t xdθ )).On a Lagrangian fiber bundle, a prequantum line bundle also admits anice local description, as follows. Lemma 2.8.
In the settings of Fact 2.3, we also assume that ( X, ω ) isequipped with a prequantum line bundle ( L, ∇ ) . For any point b ∈ B ,there exists a contractible open neighborhood U ⊂ R n of b a fiber-preservingsymplectomorphism X U ≃ U ′ × T n as in Fact 2.3, and an isomorphism ( L | X U , ∇| X U ) ≃ ( C , d − √− t xdθ )) which covers the symplectomorphism.Proof. First let us choose an action-angle coordinate chart X U ≃ V × T n with U contractible, and denote the coordinate by ( x ′ , θ ′ ). Since U is contractibleand the fibers are Lagrangian, ω | X U is exact. So L | X U is trivial as a hermitianline bundle, and taking a trivialization we have ( L | X U , ∇| X U ) ≃ ( V × T n × C , d − √− β ), where β ∈ Ω ( V × T n ). Since we have ∇ = −√− ω , we seethat β − t xdθ is closed and defines a class ( τ i ) ni =1 in H dR ( V × T n ; R ) ≃ R n ,and we can write β − t xdθ = n X i =1 τ i dθ i + df for some f ∈ C ∞ ( V × T n ). Setting U := V + ( τ i ) i , we define the bundleisomorphism, V × T n × C → U × T n × C , ( x, θ, v ) ( x + ( τ i ) i , θ, e −√− f ( x,θ ) v ) . This gives the desired isomorphism. (cid:3)
Given a Lagrangian fiber bundle with a prequantum line bundle, therestriction of the prequantum line bundle to each fibers is a flat line bundle.Also remark that, the flat connection on the fiberwise tangent bundle ker dµ induces the canonical flat connection on the vertical half-density bundle | Λ | / (ker dµ ) ∗ . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 7
Definition 2.9.
Assume we are given a pre-quantized symplectic mani-fold (
X, ω, L, ∇ ) equipped with a proper Lagrangian fiber bundle structure µ : X → B with connected fibers.(1) A point b ∈ B is called a k - Bohr-Sommerfeld point if the space offiberwise parallel sections of ( L k , ∇ ) is nontrivial.(2) For each k , let B k ⊂ B denote the set of k -Bohr-Sommerfeld points.We define the quantum Hilbert space of level k associated to the realpolarization ker dµ ⊗ C ⊂ T X ⊗ C by H k = ⊕ b ∈ B k H ( X b ; L k ⊗ | Λ | / X b ) , (2.10) where | Λ | / X b = | Λ | / (ker dµ ) ∗ | X b is the vertical half-density bun-dle, equipped with the canonical flat connection. Example . In the example (
X, ω, L, ∇ ) = ( R n × T n , t dx ∧ dθ, C , d −√− t xdθ ), the set of k -Bohr-Sommerfeld point is given by B k = Z n k ⊂ R n .For each b ∈ B k we have H ( X b ; L k ⊗ | Λ | / X b ) = C · { e √− k h b,θ i √ d ′ θ } , (2.12)where d ′ θ := (2 π ) − n/ dθ be the normalized measure on T n . The quantumHilbert space H k is the direct sum, over B k , of the above one-dimensionalHilbert spaces. 3. The construction
The model case — on R n × T n . In this subsection, as a modelcase, as well as a building block of the deformation quantization solving theProblem 1.4, we consider the following settings. Let X = R n × T n equippedwith the standard symplectic structure t dx ∧ dθ and the Lagrangian fibration µ : R n × T n → R n , ( x, θ ) x . Equip X with the canonical prequantizingline bundle ( L = C , ∇ = d − √− t xdθ ).We denote by { ψ kb } b ∈ B k the orthonormal basis of H k given in (2.12),namely ψ kb := e √− k h b,θ i √ d ′ θ ∈ H k . Now we construct a adjoint-preserving linear map φ k : C ∞ c ( X ) → B ( H k ) . Assume we are given a function f ∈ C ∞ c ( X ). Using the canonical basis of H k given in (2.12), the operator φ k ( f ) is identified by a B k × B k -matrix { K f ( b, c ) } b,c ∈ B k . Matrix elements K f ( b, c ) for b, c ∈ B k is given as follows. K f ( b, c ) := Z T n e −√− k h b − c,θ i f (( b + c ) / , θ ) d ′ θ. (3.1)In other words, K f ( b, c ) is given by the k ( b − c )-th coefficient in the Fourierexpansion of f ( c, θ ). This is regarded as a discrete analogue of the formula(1.5). It is easy to see this formula takes a real-valued function to a self-adjoint operator, thus the linear map φ k is adjoint-preserving. Note that weare using the value of a function at the middle point ( b + c ) /
2, in order tomake this map adjoint-preserving.
M. YAMASHITA
The formula (3.1) gives, a priori, a densely defined possibly unboundedoperator φ k ( f ) on H k . We are going to prove that that this operator isbounded in Lemma 3.3.As a preparation, we give an easy estimate of norms of bounded operators.We are going to use this lemma throughout this paper, in order to estimatenorms of operators whose entries are concentrated near the diagonal. Lemma 3.2.
Fix a positive integer n . Let H be a separable Hilbert space,and assume that we are given an complete orthonormal basis { ψ x } x ∈ Z n for H labelled by Z n . Let A be a possibly unbounded densely defined linearoperator on H , defined in terms of matrix coefficients with respect to thebasis { ψ x } x ∈ Z n , denoted by K ( x, y ) for ( x, y ) ∈ Z n . Then we have kAk ≤ X m ∈ Z n (cid:18) sup x ∈ Z n | K ( x + m, x ) | (cid:19) . Proof.
For x ∈ Z n , let P x ∈ B ( H ) be the orthogonal projection onto theone-dimensional subspace C · ψ x ⊂ H . We decompose A = X m ∈ Z n X x ∈ Z n P x + m A P x ! . For each m ∈ Z n , we define the ”shift by m ” operator S m ∈ U ( H ) by theformula S m ( ψ x ) := ψ x + m , for each x ∈ Z n . For each m , we have X x ∈ Z n P x + m A P x = S m · diag( { K ( x + m, x ) } x ∈ Z n ) , where diag( { K ( x + m, x ) } x ∈ Z n ) means the diagonal operator with respectto the orthonormal basis { ψ x } x , whose x -th entry is given by K ( x + m, x ).So the norm is given by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈ Z n P x + m A P x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k diag( { K ( x + m, x ) } x ∈ Z n ) k = sup x ∈ Z n | K ( x + m, x ) | . So we get kAk ≤ X m ∈ Z n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈ Z n P x + m A P x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X m ∈ Z n (cid:18) sup x ∈ Z n | K ( x + m, x ) | (cid:19) . (cid:3) Lemma 3.3.
For f ∈ C ∞ c ( X ) , the linear operator φ k ( f ) defined in termsof the matrix coefficient in (3.1) is a bounded operator on H k . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 9
Proof.
Let f ( x, θ ) = P m ∈ Z n f m ( x ) e √− h m,θ i be the fiberwise Fourier ex-pansion of f . Since f is smooth and compactly supported, there exists aconstant C such that we have k f m k C ≤ C (1 + | m | ) n +1 for all m ∈ Z n . We apply Lemma 3.2 to the operator φ k ( f ) (the index set B k = Z n k of the orthonormal basis for H k is rescaled to Z n in the obviousway). By (3.1), we have (cid:12)(cid:12)(cid:12) K f (cid:16) x + mk , x (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) f m (cid:16) x + m k (cid:17)(cid:12)(cid:12)(cid:12) ≤ C (1 + | m | ) n +1 , for all x ∈ B k . So we get k φ k ( f ) k ≤ C X m ∈ Z n | m | ) n +1 < + ∞ . (cid:3) Now we consider a more coordinate-free way to express (3.1). First wenote the following. Using the product structure X = R n × T n and the trivi-alization of L , for any points b, c ∈ R n we get the explicit affine isomorphism X b ≃ X ( b + c ) / ≃ X c ≃ T n , (3.4)and the explicit ismomorphism of line bundles L | X b ≃ L | X ( b + c ) / ≃ L | X c ≃ T n × C (3.5)that covers (3.4). Using the isomorphisms (3.4) and (3.5), we can regarda function F b ∈ C ∞ ( X b ) or a section ξ b ∈ C ∞ ( X b ; L | X b ) as an element in C ∞ ( T n ). So we can multiply a section ξ c ∈ C ∞ ( X c ; L | X c ) by a functionon a different fiber, F b ∈ C ∞ ( X b ) to get an elemeent F b · ξ c ∈ C ∞ ( T n ) ≃ C ∞ ( X c ; L | X c ). Since the vertical half-density bundle | Λ | / (ker dµ ) ∗ is equippedwith the canonical flat connection, we also get a well-defined pairing of sec-tions of L ⊗ | Λ | / (ker dµ ) ∗ between different fibers, denoted by h· , ·i T n .After these preparations, we proceed to give a more coordinate-free wayto express (3.1). The following formula follows directly from the definition. Lemma 3.6.
We have K f ( b, c ) = h ψ kb , f | X ( b + c ) / · ψ kc i T n , where the left hand side is defined in (3.1) . Definition 3.7.
For k ∈ N , we define the adjoint-preserving linear map φ k : C ∞ c ( X ) → B ( H k )by the formula φ k ( f )( ψ kc ) = X b ∈ B k h ψ kb , f | X ( b + c ) / · ψ kc i T n · ψ kb . (3.8) Example . Assume f ∈ C ∞ c ( X ) is a pullback of a function f ∈ C ∞ c ( R n )on the base R n , i.e., f does not depend on θ . Then φ k ( f ) is just the diagonalmultiplication operator by the value of f at each point on B k , K f ( b, c ) = ( f ( c ) if b = c, Example . Assume f can be expressed as f ( x, θ ) = f m ( x ) e √− h m,θ i forsome m ∈ Z n and a function f m ∈ C ∞ c ( R n ). Then we have K f ( b, c ) = ( f m ( c + m/ (2 k )) if b = c + m/k, e √− h m,θ i plays the role of ” m/k -shift”, and if welet k → ∞ , the matrix elements of this operator concentrate to the diagonal.More generally, if f can be expressed as f = P | m |≤ M f m ( x ) e √− h m,θ i forsome M < ∞ , we have K f ( b, c ) = 0 only when | b − c | ≤ M/k . So also inthis case the matrix elements concentrate to the diagonal as k → ∞ .In fact, the ”concentration to the diagonal” of the matrix elements of theoperator φ k ( f ) as k → ∞ seen in the above examples holds in general, be-cause the Fourier coefficients of smooth function on T n is rapidly decreasing.Basically, this is why we can extend this construction to general Lagrangianfiber bundles in the next subsection.The goal of the rest of this subsection is to prove that the maps { φ k } k ∈ N is a strict deformation quantization of ( X, ω ). Actually, in this model case,it is easy to explicitely compute the asymptotic behavior of φ k ( f ) φ k ( g ) as k → ∞ for f, g ∈ C ∞ c ( X ). We show that this recovers the standard Moyal-Weyl star product on (cid:0) C ∞ ( R n × T n )[[ ~ ]] , t dx ∧ dθ (cid:1) .First we recall the definition of Moyal-Weyl star product on ( R n , ω ) witha translation invariant symplectic form ω = ω ij dx i ∧ dx j . For functions f, g ∈ C ∞ ( R n ), the Moyal-Weyl star product is defined by( f ∗ g )( x ) := exp (cid:18) ~ ω ij ∂ y i ∂ z j (cid:19) f ( y ) g ( z ) | y = z = x . We denote the coefficient of ~ j of the star product by C j ( f, g ) ∈ C ∞ ( R n ),so that f ∗ g = ∞ X j =0 C j ( f, g ) ~ j . Consider case where R n = R n × R n with the coordinate ( x, y ) and thesymplectic form is the standard one ω = t dx ∧ dy . This induces a starproduct on the quotient space, ( R n × T n , t dx ∧ dθ ), denoted by ∗ std . This isgiven by the formula,( f ∗ std g )( x, θ ) := exp ~ X i (cid:16) ∂ x ′ i ∂ θ ′′ i − ∂ x ′′ i ∂ θ ′ i (cid:17)! f ( x ′ , θ ′ ) g ( x ′′ , θ ′′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ ,θ ′ )=( x ′′ ,θ ′′ )=( x,θ ) . (3.11) EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 11
We still call this star product as the standard Moyal-Weyl star product.We denote by C std j ( f, g ) ∈ C ∞ ( R n × T n ) the corresponding coefficient withrespect to the star product ∗ std for f, g ∈ C ∞ ( R n × T n ). Proposition 3.12.
The linear map φ k : C ∞ c ( X ) → B ( H k ) defined in Defi-nition 3.7 satisfies, for all f, g ∈ C ∞ c ( X ) and l ∈ N , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ k ( f ) φ k ( g ) − l X j =0 (cid:18) −√− k (cid:19) j φ k (cid:16) C std j ( f, g ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = O (cid:18) k l +1 (cid:19) as k → ∞ .Proof. Fix f, g and l . We denote the Fourier expansion of f , g by f m , g m as usual. The matrix coefficient of φ k ( f ) φ k ( g ), denoted by K φ k ( f ) φ k ( g ) ( · , · ),is given by K φ k ( f ) φ k ( g ) (cid:16) x + pk , x (cid:17) = X m ∈ Z n K f (cid:16) x + pk , x + mk (cid:17) K g (cid:16) x + mk , x (cid:17) = X m ∈ Z n f p − m (cid:18) x + p + m k (cid:19) g m (cid:16) x + m k (cid:17) . (3.13)The proof is given by performing the Taylor expansion of f p − m and g m around the point x + p k in the above formula.For simplicity, we only give the proof in the case n = 1. The proof isessentially the same for general n . Define the operator B k,lerr on H k by B k,lerr := φ k ( f ) φ k ( g ) − l X j =0 (cid:18) −√− k (cid:19) j φ k (cid:16) C std j ( f, g ) (cid:17) . The standard Moyal-Weyl star product is explicitly given by C std j ( f, g ) = 12 j j X i =0 ( − i − j j !( i − j )! (cid:16) ∂ ix ∂ j − iθ f (cid:17) · (cid:0) ∂ j − ix ∂ iθ g (cid:1) So the p -th Fourier coefficient of this function is given by (cid:16) C std j ( f, g ) (cid:17) p = (cid:18) √− (cid:19) j j X i =0 ( − i − j j !( i − j )! X m ∈ Z ( p − m ) j − i f ( i ) p − m · m i g ( j − i ) m . (3.14)On the other hand, the Taylor expansion of f p − m and g m gives, formally, f p − m (cid:18) x + p + m k (cid:19) = ∞ X i =0 i ! (cid:16) m k (cid:17) i f ( i ) p − m (cid:16) x + p k (cid:17) (formally) g m (cid:16) x + m k (cid:17) = ∞ X i =0 i ! (cid:18) m − p k (cid:19) i g ( i ) m (cid:16) x + p k (cid:17) (formally) . Thus the formula (3.13) admits an expansion, at least formally, K φ k ( f ) φ k ( g ) (cid:16) x + pk , x (cid:17) = ∞ X j =0 (cid:18) k (cid:19) j j X i =0 ( − i − j j !( i − j )! X m ∈ Z ( p − m ) j − i f ( i ) p − m · m i g ( j − i ) m . (3.15) By (3.14) and (3.15), we see that, in the above formal expansion, the matrixelements K B k,lerr ( x + pk , x ) of B k,lerr is O ( k − ( l +1) ) with respect to k .Now we give estimates for the error terms and prove the statement. Forsimplicity we only prove in the case n = 1 and l = 1. The proof is essentiallythe same for the general case. Put A kerr := k B k, err . Namely we have A kerr := k (cid:18) φ k ( f ) φ k ( g ) − φ k ( f g ) − −√− k φ k ( { f, g } ) (cid:19) . We need to show sup k kA kerr k < + ∞ . We denote by K kerr ( x, y ) the matrixelement of A kerr for ( x, y ) ∈ B k × B k .Since f and g are smooth and compactly supported, for each N ∈ N thereexists a constant C N such that, for all m ∈ Z we have k f m k , k g m k , k f ′ m k , k g ′ m k , k f ′′ m k , k g ′′ m k < C N ( | m | + 1) − N , (3.16)where k · k denotes the C -norm.By (3.16), we have (cid:12)(cid:12)(cid:12)(cid:12) f p − m (cid:18) x + p + m k (cid:19) − (cid:16) f p − m (cid:16) x + p k (cid:17) + m k f ′ p − m (cid:16) x + p k (cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | m | k · C (1 + | p − m | ) (cid:12)(cid:12)(cid:12)(cid:12) g m (cid:16) x + m k (cid:17) − (cid:18) g m (cid:16) x + p k (cid:17) − p − m k g ′ m (cid:16) x + p k (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | p − m | k · C (1 + | m | ) We also have the estimate k f p − m k + | m | k k f ′ p − m k ≤ (1 + | m | ) C (1 + | p − m | ) , and k g m k + | p − m | k k g ′ m k ≤ (1 + | p − m | ) C (1 + | m | ) Combining the above estimates and (3.13) and (3.14) for j = 0 ,
1, we have (cid:12)(cid:12)(cid:12) K kerr (cid:16) x + pk , x (cid:17)(cid:12)(cid:12)(cid:12) ≤ k X m ∈ Z (cid:18) | m ( p − m ) | k k f ′ p − m k · k g ′ m k + C C k (1 + | m | ) (1 + | p − m | ) + C C k (1 + | m | ) (1 + | p − m | ) + C C k (1 + | m | ) (1 + | p − m | ) (cid:19) ≤ X m ∈ Z C C (cid:18)
14 + 18 + 18 + 164 (cid:19) | m | ) (1 + | p − m | ) ≤ C ′ (1 + | p | ) , where we put C ′ := C C (cid:18)
14 + 18 + 18 + 164 (cid:19) X m ∈ Z | m | ) < + ∞ and used the inequality (1 + | m | )(1 + | p − m | ) ≥ | p | . So we have, byLemma 3.2, kA kerr k ≤ X p ∈ Z sup x ∈ B k (cid:12)(cid:12)(cid:12) K kerr (cid:16) x + pk , x (cid:17)(cid:12)(cid:12)(cid:12)! ≤ X p ∈ Z C ′ (1 + | p | ) . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 13
This gives a bound for kA kerr k which does not depend on k , so we get theresult. (cid:3) Proposition 3.17.
The linear map φ k : C ∞ c ( X ) → B ( H k ) defined in Defi-nition 3.7 satisfies, for all f ∈ C ∞ c ( X ) , k φ k ( f ) k → k f k C . as k → ∞ .Proof. First we observe the following.
Lemma 3.18.
Let F = P m ∈ Z n F m e √− h m,θ i ∈ C ∞ ( T n ) , and consider theoperator Φ k ( F ) ∈ B ( H k ) such that the corresponding matrix coefficient, de-noted by K F ( x, y ) for ( x, y ) ∈ B k , is given by K F (cid:16) x + mk , x (cid:17) := F m , for all x ∈ B k and m ∈ Z n . Then we have k Φ k ( F ) k = k F k C . Notice that if we regard F as a function on X = R n × T n by ˜ F ( x, θ ) = F ( θ ), then the operator Φ k ( F ) should be regarded as ” φ k ( ˜ F )”, even though φ k ( ˜ F ) is not defined because ˜ F is not compactly supported. Proof.
We have a unitary isomorphism H k ≃ L ( T n ) ψ km/k e √− h m,θ i . Under this isomorphism, the operator Ψ k ( F ) transforms to the multiplica-tion operator by F on L ( T n ), and this operator norm is k F k C . (cid:3) Proposition 3.17 is, very roughly, understood as follows. As k → ∞ , theoperator φ k ( f ) reflects the behavior of f only locally in R n -direction (seeExample 3.10), and on a sufficiently small neighbourhood of a point x ∈ R n ,the function f is close to a function which is invariant in R n -direction, andLemma 3.18 applies asymptotically. Since the proof is long (although veryelementary) and not essential for the rest of the paper, we give a detailedproof in the Appendix. (cid:3) Combining Proposition 3.12 and Proposition 3.17, we get the following.
Theorem 3.19.
The family of adjoint-preserving linear maps { φ k } k ∈ N , φ k : C ∞ c ( R n × T n ) → B ( H k ) , defined in Definition 3.7, is a strict defor-mation quantization for ( R n × T n , t dx ∧ dθ ) . The general case.
In this subsection we generalize the construction ofthe strict deformation quantization for the general case. Let ( X n , ω, L, ∇ , h )be a prequantized symplectic manifold equipped with a proper Lagrangianfiber bundle µ : X → B with connected fibers. The following two additionaldatum are needed for the construction of our strict deformation quantiza-tion. (H) A smooth horizontal distribution H ⊂ T X for the fibration µ , T X = H ⊕ ker dµ, (3.20) which is, choosing any local action-angle coordinate ( x, θ ) ∈ U ′ × T n ,invariant in the T n -direction. This condition does not depend on thechoice of the coordinate, and it is clear that such a splitting alwaysexists.(U) An open covering U of B with the following properties.(U1) U is locally finite. Moreover, each element U ∈ U admits anopen affine embedding into R n and its image is convex in R n .(U2) For any two elements U, V ∈ U , the intersection U ∩ V alsoadmits an open affine embedding into R n with convex image(in particular it is connected).(U3) For each U ∈ U , U is a compact subset of B .We are going to construct a deformation quantization from these datum.As we will see, the choice of the horizontal distribution H is essential for ourconstruction, but the choice of U is only technical, and the different choice of U yields essentially the same deformation quantization (Proposition 3.35).We also remark that we can drop the condition (U3) and just require thateach U admits an affine open embedding into R n (see Remark 3.36). Werequire this condition just in order to simplify the estimates.Given a path γ in B from b ∈ B to c ∈ B , by the splitting (3.20), we getthe parallel transform T γ : X b → X c , (3.21)which preserves the affine structure. Also the connection ∇ on L and thecanonical flat connection on | Λ | / (ker dµ ) ∗ gives the parallel transform T γ : L k | X b ⊗ | Λ | / X b → L k | X c ⊗ | Λ | / X c which covers (3.21). We use the same notation for the parallel transform.This allows us to define a pairing between sections ξ kb ∈ C ∞ ( X b ; L k ⊗| Λ | / X b ) and ξ kc ∈ C ∞ ( X c ; L k ⊗ | Λ | / X c ), denoted by h ξ kb , ξ kc i γ .We say that two points b, c ∈ B are close if there exists an element U ∈ U such that b, c ∈ U . For such b, c ∈ B , we can take the unique affine linearpath γ from b to c in U and define, for sections ξ kb ∈ C ∞ ( X b ; L k ⊗ | Λ | / X b )and ξ kc ∈ C ∞ ( X c ; L k ⊗ | Λ | / X c ), h ξ kb , ξ kc i U := h ξ kb , ξ kc i γ . This is well-defined by our assumptions on U . Definition 3.22.
Given a splitting as (3.20) and a covering U of B as above,we define a adjoint-preserving linear map φ kH, U : C ∞ c ( X ) → B ( H k ) by thefollowing formula. For f ∈ C ∞ c ( X ), we define the operator φ kH, U ( f ) by, for c ∈ B k and an element ψ kc ∈ H ( X c ; L k ⊗ | Λ | / X c ) ⊂ H k , φ kH, U ( f )( ψ kc ) := X b ∈ B k ,b is close to c h ψ kb , f | X ( b + c ) / ψ kc i U · ψ kb , where ψ kb ∈ H ( X b ; L k ⊗| Λ | / ( X b )) ⊂ H k is any element with k ψ kb k = 1 (thisdefinition does not depend on this choice). Here, we denote by ( b + c ) / ∈ B EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 15 the middle point between b and c with respect to the affine structure on anopen set U ∈ U which contains both b and c , and we regard f | X ( b + c ) / ∈ C ∞ ( X ( b + c ) / ) as a function on X c using the parallel transform (3.21) alongthe affine linear path between ( b + c ) / c in U .Again it is easy to see that this map is adjoint-preserving, thanks to thefact that we are using the value of function at the fiber over the middlepoint. Example . If a function f ∈ C ∞ c ( X ) is a pullback of a function on B ,then φ kH, U ( f ) is the multiplication operator by its value at B k . In otherwords, if we write f = µ ∗ f for f ∈ C ∞ c ( B ) we have φ kH, U ( µ ∗ f ) ψ kb = f ( b ) ψ kb for any b ∈ B k and ψ kb ∈ H ( X b ; L k ⊗ | Λ | / X b ) ⊂ H k .The goal of this subsection is to prove that the family of maps defined inDefinition 3.22 is a strict deformation quantization. From now on until theend of this subsection, we fix H and U satisfying the conditions in (H) and(U) above, and moreover, • We fix an action-angle coordinate on X U and a trivialization ( L | X U , ∇| X U ) =( C , d − √− t xdθ ) for each U ∈ U (see Lem 2.8).Let us focus on an element U ∈ U . Since H is invariant in the fiberdirection, using the fixed action-angle coordinate we can write H as H = Span (cid:26) ∂∂x j + A ij ∂∂θ i (cid:27) ≤ j ≤ n , (3.24)for some A j = ( A ij ) ∈ C ∞ ( U ; R n ), j = 1 , · · · , n . We regard A = A j dx j ∈ C ∞ ( U ; T ∗ U ⊗ R n ). Using the flat connection on U ⊂ B , we get ∇ A | U ∈ C ∞ ( U ; T ∗ U ⊗ T ∗ U ⊗ R n ) given by ∇ A i = ∂A ij ∂x l dx l ⊗ dx j . We denote by k∇ A k U the C ( U )-norm of ∇ A with respect to the flat metricon U induced by the Euclidean metric of the action coordinate. Remark that A is only defined on U and depends on the action-angle coordinate chosenon U . Lemma 3.25.
Fix an element U ∈ U . Using the fixed action-angle coordi-nate and trivialization on X U as above, take an orthonormal basis { ψ kx } x ∈ Z nk ∩ U of H k | U given by ψ kx = e √− k h x,θ i √ d ′ θ ∈ H ( X x ; L k ⊗ | Λ | / ( X x )) . For a function f ∈ C ∞ c ( X ) , denote the matrix coefficient of φ kH, U ( f ) | H kU withrespect to the above orthonormal basis by { K kf ( x, y ) } x,y ∈ Z nk ∩ U . Denote theFourier expansion of f | X U by f ( x, θ ) = P m ∈ Z n f m ( x ) e √− h m,θ i . Then we have, for any points x, x + mk ∈ Z n k ∩ U such that K kf ( x + mk , x ) = 0 , K kf ( x + mk , x ) f m ( x + m k ) ∈ U (1) , (3.26) and the value of (3.26) does not depend on f as long as f m ( x + m k ) = 0 .Moreover, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) arg K kf ( x + mk , x ) f m ( x + m k ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ A k U | m | k . (3.27) Here A ∈ C ∞ ( U ; T ∗ U ⊗ R n ) is defined in (3.24) .Proof. When m = 0 the result is obvious. Fix k , a point x ∈ Z n k ∩ U and m = 0 ∈ Z n with x + mk ∈ Z n k ∩ U . Let u := m | m | be the unit vector in the m -direction. Denote by γ = [ x, x + mk ] the line segment from x to x + mk in U , and denote by ˜ γ : [ x, x + mk ] → U × T n = X U the horizontal lift of[ x, x + mk ] with respect to the splitting (3.20) which passes through the point( x + m k , α : h − | m | k , | m | k i → T n by˜ γ (cid:16) t u + x + m k (cid:17) = (cid:16) t u + x + m k , α ( t ) (cid:17) . We have α (0) = 0. We regard u := m/ | m | ∈ C ∞ ( U ; T U ), and α satisfies α ′ ( t ) = A ( u ) | t u + x + m/ (2 k ) , (3.28) α ′′ ( t ) = ∇ u A ( u ) | t u + x + m/ (2 k ) . So we get, for all t ∈ [ − | m | k , | m | k ] | α ( t ) − α ′ (0) t | ≤ k∇ A k U t , (3.29) | α ′ ( t ) − α ′ (0) | ≤ k∇ A k U t. We are going to compute the pairings between elements of C ∞ ( X x ; L k ⊗| Λ | / X x ) and C ∞ ( X x + mk ; L k ⊗ | Λ | / X x + mk ) by pulling back to the mid-dle fiber X x + m k . So from now on, only in this proof we write the paralleltransport along subsets of the segment γ by T t : X x + m k → X t u + x + m k ,T t : L | X x + m k → L | X t u + x + m k . Then, for t ∈ h − | m | k , | m | k i and a function g ∈ C ∞ ( X t u + x + m k ), we have( T ∗ t g )( θ ) = g ( θ + α ( t )) . (3.30)Let E ∈ C ∞ ( X U ; L ) be the section which gives the trivialization of L as inthe statement. Then, since it satisfies ∇ E = −√− t xdθ ⊗ E , the section f E k ∈ C ∞ ( X γ ; L k ) given by f E k | X t u + x + m k := exp (cid:18) √− k Z t D s u + x + m k , α ′ ( s ) E ds (cid:19) E k , EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 17 satisfies T ∗ t f E k | X t u + x + m k = E k | X x + m k . We have ψ kx = exp √− k h x, θ i − √− k Z −| m | / k D s u + x + m k , α ′ ( s ) E ds ! f E k √ d ′ θ, (3.31) ψ kx + mk = exp √− k D x + mk , θ E − √− k Z | m | / k D s u + x + m k , α ′ ( s ) E ds ! f E k √ d ′ θ. Combining (3.30) and (3.31), for f ( x, θ ) = P m ∈ Z n f m ( x ) e √− h m,θ i we have K kf ( x + mk , x )= (cid:28)(cid:18) T ∗ | m | k ψ kx + mk (cid:19) , (cid:18) f | X x + m k · T ∗− | m | k ψ kx (cid:19)(cid:29) = exp (cid:16) √− k n − (cid:28) x + mk , θ + α (cid:18) | m | k (cid:19)(cid:29) + Z | m | / k D s u + x + m k , α ′ ( s ) E ds + (cid:28) x, θ + α (cid:18) − | m | k (cid:19)(cid:29) − Z −| m | / k D s u + x + m k , α ′ ( s ) E ds o + √− h m, θ i (cid:17) · f m (cid:16) x + m k (cid:17) = exp √− k ( − (cid:28) m k , α (cid:18) | m | k (cid:19) + α (cid:18) − | m | k (cid:19)(cid:29) + Z | m | / k −| m | / k (cid:10) s u , α ′ ( s ) (cid:11) ds )! · f m (cid:16) x + m k (cid:17) . (3.32)This implies (3.26), as well as the independence of the value (3.26) on f .Moreover, by (3.29), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) m k , α (cid:18) | m | k (cid:19) + α (cid:18) − | m | k (cid:19)(cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ A k U (cid:18) | m | k (cid:19) , | Z | m | / k −| m | / k h s u , α ′ ( s ) i ds | ≤ Z | m | / k −| m | / k k∇ A k U s ds = 2 k∇ A k U (cid:18) | m | k (cid:19) . Thus we get, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) arg K kf ( x + mk , x ) f m ( x + m k ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ A k U | m | k , which proves (3.27). (cid:3) Now we state our main theorem.
Theorem 3.33.
The linear map φ kH, U : C ∞ ( X ) → B ( H k ) given in Defini-tion 3.22 satisfies, for all f, g ∈ C ∞ c ( X ) , (cid:13)(cid:13)(cid:13)(cid:13) φ kH, U ( f ) φ kH, U ( g ) − φ kH, U ( f g ) − −√− k φ kH, U ( { f, g } ) (cid:13)(cid:13)(cid:13)(cid:13) = O (cid:18) k (cid:19) as k → ∞ . Proof.
In this proof, we write φ k = φ kH, U . First, informally we compute,for x, x + pk ∈ U for some U ∈ U , using Lemma 3.25, and denoting by K φ k ( f ) φ k ( g ) ( · , · ) the matrix coefficient of φ k ( f ) φ k ( g ), K fg (cid:16) x + pk , x (cid:17) = (cid:18) O (cid:18) | p | k (cid:19)(cid:19) X m ∈ Z n ( f p − m g m ) | x + p k ,K { f,g } (cid:16) x + pk , x (cid:17) = √− (cid:18) O (cid:18) | p | k (cid:19)(cid:19) X m ∈ Z n {h m, ∇ f p − m i · g m − f p − m · h ( p − m ) , ∇ g m i}| x + p k ,K φ k ( f ) φ k ( g ) (cid:16) x + pk , x (cid:17) = X m f p − m (cid:18) x + p + m k (cid:19) g m (cid:16) x + m k (cid:17) (cid:18) O (cid:18) | p − m | + | m | k (cid:19)(cid:19) + (error)= X m (cid:18) O (cid:18) | p − m | + | m | k (cid:19)(cid:19) (cid:26)(cid:18) (cid:16) f p − m + D m k , ∇ f p − m E(cid:17)(cid:12)(cid:12)(cid:12) x + p k + O (cid:18) | m | k (cid:19)(cid:19) · (cid:18) g m − (cid:28) p − m k , ∇ g m (cid:29)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x + p k + O (cid:18) | p − m | k (cid:19)!) + (error) . Here the last equation uses the Taylor expansion of f p − m and g m . The termsin the sum for K φ k ( f ) φ k ( g ) is taken for m with x + m/k ∈ U , and the term(error) comes from the contributions from those points b ∈ B k \ U which areboth close to x and x + p/k , and we see below that this error term is indeednegligible as k → ∞ . So, at least informally, we see that K φ k ( f ) φ k ( g ) (cid:16) x + pk , x (cid:17) = K fg (cid:16) x + pk , x (cid:17) + −√− k K { f,g } (cid:16) x + pk , x (cid:17) + O (cid:18) k (cid:19) . So, what we have to do is to give appropriate estimates of the operatornorms (not only matrix coefficients) of the error terms. Since the proof islong, we give a detailded proof in the Appendix. (cid:3)
From this, we conclude that the maps { φ kH, U } k ∈ N solves Problem 1.4. Theorem 3.34.
Let ( X n , ω, L, ∇ , h ) be a prequantized symplectic manifoldequipped with a proper Lagrangian fiber bundle µ : X → B with connectedfibers. Assume we are given a horizontal distribution H ⊂ T X satisfyingthe condition in (H) and an open covering U satisfying the conditions in(U). Then, the family of adjoint-preserving linear maps { φ kH, U } k ∈ N , definedin Definition 3.22, is a strict deformation quantization for ( X, ω ) .Proof. The condition (1) in Definition 1.1 can be proved in a similar way asin the proof of Proposition 3.17 and Theorem 3.33, so we leave the detailsto the reader. The condition (2) follows from Theorem 3.33. (cid:3)
As we remarked earlier, the choice of open covering U for B is not essentialin our construction, as follows. EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 19
Proposition 3.35.
Let ( X n , ω, L, ∇ , h ) be a prequantized symplectic man-ifold equipped with a proper Lagrangian fiber bundle µ : X → B with con-nected fibers. Assume we are given a horizontal distribution H ⊂ T X sat-isfying the condition in (H) and two choices of open coverings U and V satisfying the conditions in (U). Then, for any function f ∈ C ∞ c ( X ) , wehave k φ kH, U ( f ) − φ kH, V ( f ) k = O ( k − N ) for all N ∈ N .Proof. This follows easily from the fact that the Fourier coefficient of asmooth function on T n are rapidly decreasing, as in (6.25). Indeed, usingthis fact, we can show that matrix elements of φ kH, U ( f ) − φ kH, V ( f ) are of O ( k − N ) for any N ∈ N by a similar estimate as in the proof of Theorem3.33, and the result follows. We leave the details to the reader. (cid:3) Remark . As should be obvious from the proof of Theorem 3.33 in theAppendix, the assumption (U3) is not essential. Indeed, we may drop thiscondition. We can define φ kH, U in the same way, and show that Theorem 3.33extends to this case. We put the condition (U3) only because it simplifies theproof of Theorem 3.33. Since Proposition 3.35 also extends to this generalcase, we lose nothing by requiring the condition (U3).In particular, in our construction of φ k in the model case R n × T n insubsection 3.1, we used such U , namely we set U = { R n } (and set H = T R n ).Later in section 5, we again use this trivial non-relatively compact coveringfor R n and consider deformation quantizations (corresponding to non-trivial H ). 4. Star products
In this section we analyze the star products induced by the deformationquantization defined in Section 3. Our construction is given by explicitformula locally, so in principle we can compute higher terms of star productsin action-angle coordinates. In the case where the horizontal distribution H is Lagrangian and integrable, we show in Theorem 4.3 that the star productobtained from our strict deformation quantization conincides with the starproduct given by the Fedosov’s construction [5]. In general cases, by anexplicite computation we check this coincidence up to second order term inTheorem 4.7.We consider the settings in subsection 3.2. We are given a prequantizedclosed symplectic manifold ( X n , ω, L, ∇ ) and a proper Lagrangian fiberbundle structure µ : X → B with connected fibers. We fix a horizontal dis-tribution H ⊂ T X and a finite open covering U of B satisfying conditions in(H) and (U) in subsection 3.2. Let us consider the deformation quantization { φ kH, U } k defined in Definition 3.22 from these datum.The horizontal distribution H associates a torsion-free symplectic connec-tion ∇ T X,H on T X , as follows. As a first step, we define a connection e ∇ T X,H on T X , which does not necessarily preserve the symplectic form and possiblyhas torsion. By the identification µ ∗ T B ≃ H and the flat connection on T B , we get the pullback connection on H . Moreover, the vertical tangent bun-dle ker dµ admits the canonical flat connection. We define the connection e ∇ T X,H by the direct sum of these two connections, using
T X = H ⊕ ker dµ .This connection is locally described as follows. Let us focus on one openset U ∈ U . We fix an action-angle coordinate on X U , and express H on X U asin (3.24) using a locally defined R n -valued one-form A ∈ C ∞ ( U ; T ∗ U ⊗ R n ).Denote the Christoffel symbol of e ∇ T X,H with respect to the local frame( ∂ x , · · · , ∂ x n , ∂ θ , · · · , ∂ θ n ) by e Γ ··· . Lemma 4.1.
We have e Γ θ i x l x j = − ∂ x l A ij , and e Γ ··· = 0 for all other components.Proof. The horizontal lift of ∂ x j ∈ C ∞ ( U ; T U ) is given by ˜ ∂ x j = ∂ x j + A ij ∂ θ i .Since we have e ∇ T X,H∂ xl ˜ ∂ x j = 0 and e ∇ T X,H∂ xl ∂ θ i = 0, we have e ∇ T X,H∂ xl ∂ x j = − e ∇ T X,H∂ xl A ij ∂ θ i = − ∂A ji ∂x l ∂ θ i . The vanishing for other cases are obvious. (cid:3)
Note that e ∇ T X,H is symplectic if and only if e Γ θ i x l x j = e Γ θ j x l x i , and torsion-free if and only if e Γ θ i x l x j = e Γ θ i x j x l , for all i, j, l . Now we define a torsion-freesymplectic connection ∇ T X,H by symmetrization, as follows.
Definition 4.2.
Assume we are given a symplectic manifold (
X, ω ) witha proper Lagrangian fiber bundle µ : X → B as well as a horizontal dis-tribution H satisfying the condition in (H) in subsection 3.2. In a locallydefined action-angle coordinate chart X U ≃ U × T n as above, we define aconnection on T X U by requiring its Cristoffel symbol Γ ·· , · with respect tothe local frame ( ∂ x , · · · , ∂ x n , ∂ θ , · · · , ∂ θ n ) to beΓ θ i x l x j = − (cid:16) ∂ l A ij + ∂ l A ji + ∂ i A lj + ∂ i A jl + ∂ j A il + ∂ j A li (cid:17) , and Γ ··· = 0 for other components. This construction does not depend on thechoice of action-angle coordinate, so we get a global torsion-free symplecticconnection on T X . We define ∇ T X,H to be this connection.Indeed, it is easily checked that the above symmetrization procedure ofCristoffel symbols is compatible with the change of action-angle coordinates.In general, for a symplectic manifold (
X, ω ), if we fix a torsion-free sym-plectic connection ∇ T X on X , for each closed element a ∈ ~ Ω ( X )[[ ~ ]],Fedosov’s construction [5] associates a star product on C ∞ ( X )[[ ~ ]], denotedby ∗ H,a . Moreover Nest and Tsygan [8] showed that the set of equiva-lence classes of star products is in one-to-one correspondence with the setof equivalence classes of formal deformations of the symplectic structure, ~ H dR ( X )[[ ~ ]]. In particular if we set a = 0, we get a star product ∗ ∇ , , whichis canonically associated to the torsion-free symplectic connection. Applyingthis to our case, we have a canonical choice of star product corresponding tothe connection ∇ T X,H and 0 ∈ ~ Ω ( M )[[ ~ ]], denoted by ∗ H, := ∗ ∇ TX,H , . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 21
We denote by C H, j ( f, g ) the j -th coefficient of ~ in f ∗ H, g for f, g ∈ C ∞ ( X ).i.e., we have f ∗ H, g = ∞ X j =0 C H, j ( f, g ) ~ j . First, we consider the symplest case when the horizontal distribution H is Lagrangian and integrable. In this case e ∇ T X,H is already torsion-free andsymplectic, and we have e ∇ T X,H = ∇ T X,H . Theorem 4.3.
Assume the horizontal distribution H is Lagrangian andintegrable. Then we have, for all f, g ∈ C ∞ c ( X ) and l ∈ N , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ kH, U ( f ) φ kH, U ( g ) − l X j =0 (cid:18) −√− k (cid:19) j φ kH, U (cid:16) C H, j ( f, g ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = O (cid:18) k l +1 (cid:19) as k → ∞ .Proof. We work on an element U ∈ U . Since H is Lagrangian and integrable,we can choose the action-angle coordinate on X U so that H = Span { ∂ x i } i .Using this coordinate and regarding X U ⊂ R n × T n , the star product ∗ H, co-incides with the standard Moyal-Weyl star product ∗ std (see (3.11)). More-over, our deformation quantization coincides with the one constructed inthe ”model case” ( R n × T n , t dx ∧ dθ ) in Subsection 3.1, modulo contributionfrom the terms coming from outside U . So the result essentially follows fromProposition 3.12. We need to show that the error terms coming from outside U is O ( k − N ) for any N ∈ N , and this is done in the similar way as in theproof of Theorem 3.33. (cid:3) Next we turn to the general case, where H is not necessarily symplectic orintegrable. Also in this case, we are able to show that our strict deformationquantization induces a star product, denoted by ⋆ H , and this star productcoincides with the above star product ∗ H, up to order two in ~ .From now on we compute the order-two term of the expected star product.Let us focus on one element U ∈ U and use the local notations as before.Using the derivatives of A ij , we can explicitely compute the complex phaseappearing in (3.26).Fix x, x + m/k ∈ U ∩ Z n k . We use α : h − | m | k , | m | k i → T n defined in theproof of Lemma 3.25, constructed from the horizontal lift of the line segment[ x, x + m/k ] in U with α (0) = 0. We regard u := m/ | m | ∈ C ∞ ( U ; T U ), and α satisfies α ′ ( t ) = A ( u ) | t u + x + m/ (2 k ) = 1 | m | X j m j A j | t u + x + m/ (2 k ) ,α ′′ ( t ) = 1 | m | X j,l m j m l ∂A j ∂x l (cid:12)(cid:12)(cid:12)(cid:12) t u + x + m/ (2 k ) . We compute the phase term in (3.32) as, − (cid:28) m k , α (cid:18) | m | k (cid:19) + α (cid:18) − | m | k (cid:19)(cid:29) + Z | m | / k −| m | / k (cid:10) s u , α ′ ( s ) (cid:11) ds = − (cid:10) u , α ′′ (0) (cid:11) (cid:18) | m | k (cid:19) + Z | m | / k −| m | / k (cid:10) s u , α ′′ (0) s (cid:11) ds + O (cid:18) | m | k (cid:19) = − | m | k (cid:10) u , α ′′ (0) (cid:11) + O (cid:18) | m | k (cid:19) . So we get, for a function f ∈ C ∞ c ( X ), K kf ( x + mk , x ) /f m ( x + m k ) = exp (cid:18) √− (cid:26) − | m | k h u , α ′′ (0) i + O (cid:18) | m | k (cid:19)(cid:27)(cid:19) (4.4) = exp √− − k X i,j,l m i m j m l ∂ l A ij | x + m/ (2 k ) + O (cid:18) | m | k (cid:19) = 1 − √− k X i,j,l m i m j m l ∂ l A ij | x + m/ (2 k ) + O (cid:18) | m | k (cid:19) . Here we denoted ∂ l A ij := ∂A ij ∂x l . We define a symmetric tree tensor Θ ∈ C ∞ ( U ; S ( T ∗ U )) as,Θ := X i,j,l Θ ijl dx i ⊗ dx j ⊗ dx l , (4.5) Θ ijl := 16 (cid:16) ∂ l A ij + ∂ l A ji + ∂ i A lj + ∂ i A jl + ∂ j A il + ∂ j A li (cid:17) . and we also denote, for v ∈ C ∞ ( U ; T U ),Θ( v ) := Θ( v ⊗ v ⊗ v ) . (4.6)Suppose we are given two functions f, g ∈ C ∞ c ( X ). By (4.4) we have K kfg (cid:16) x + pk , x (cid:17) = (cid:18) − √− k Θ( p ) | x + p k (cid:19) X m ∈ Z n ( f p − m g m ) | x + p k + O ( k − ) ,K k { f,g } (cid:16) x + pk , x (cid:17) = (cid:18) − √− k Θ( p ) | x + p k (cid:19) { f, g } p | x + p k + O ( k − ) , Since we have K kf (cid:16) x + pk , x + mk (cid:17) K kg (cid:16) x + mk , x (cid:17) = (cid:18) − √− k (cid:26) Θ( p − m ) (cid:18) x + p + m k (cid:19) + Θ( m ) (cid:16) x + m k (cid:17)(cid:27)(cid:19) f p − m (cid:18) x + p + m k (cid:19) g m (cid:16) x + m k (cid:17) + O ( k − )= (cid:18) − √− k { Θ( p − m ) + Θ( m ) } | x + p k (cid:19) ( f p − m g m ) | x + m k + O ( k − ) EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 23 and f p − m (cid:18) x + p + m k (cid:19) = (cid:26) f p − m + D m k , ∇ f p − m E + 12 t (cid:16) m k (cid:17) H ( f p − m ) (cid:16) m k (cid:17)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) x + p k + O (cid:18) | m | k (cid:19) ,g m (cid:16) x + m k (cid:17) = (cid:26) g m − (cid:28) p − m k , ∇ g m (cid:29) + 12 t (cid:18) p − m k (cid:19) H ( g m ) (cid:18) p − m k (cid:19)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) x + p k + O (cid:18) | p − m | k (cid:19) . If we denote by K kerr ( · , · ) the matrix coefficients of the operator φ kH, U ( f ) φ kH, U ( g ) − φ kH, U ( f g ) + √− k φ kH, U ( { f, g } ), we get K kerr (cid:16) x + p k , x (cid:17) = 1 k X m (cid:26) − √− { Θ( p − m ) + Θ( m ) − Θ( p ) } f p − m g m + 18 {− h m, ∇ f p − m i · h p − m, ∇ g m i + f p − m · t ( p − m ) H ( g m )( p − m ) + t mH ( f p − m ) m · g m } (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) x + p k + O ( k − ) . = − k (X m √− { Θ( p − m ) + Θ( m ) − Θ( p ) } f p − m g m + (cid:16) C std2 ( f, g ) (cid:17) p )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + p k + O ( k − ) , where C std2 ( f, g ) ∈ C ∞ ( U × T n ) is the coefficient of ~ in the standard Moyal-Weyl star product on R n × T n , see (3.11). Since we have { Θ( p − m ) + Θ( m ) − Θ( p ) } f p − m g m = − √− X i,j,l Θ ijl (cid:18) ∂f p − m ∂θ i ∂ g m ∂θ j ∂θ l + ∂ f p − m ∂θ j ∂θ l ∂g m ∂θ i (cid:19) , we see that the second coefficient in the star product induced by the strictdeformation quantization { φ kH, U } k should be given by( f, g ) X i,j,l Θ ijl (cid:18) ∂f∂θ i ∂ g∂θ j ∂θ l + ∂ f∂θ j ∂θ l ∂g∂θ i (cid:19) + C std2 ( f, g ) . It is clear that we can continue this Taylor expansion with respect to k − forhigher orders, and get the higher coefficient of the star product recursively.Summerizing, we get the followings. Theorem 4.7.
Let ( X n , ω, L, ∇ ) be a prequantized symplectic manifoldequipped with a proper Lagrangian fiber bundle µ : X → B with connectedfibers. Assume we are given a horizontal distribution H satisfying the condi-tion (H) in 3.2. Then, there exists a unique star product ⋆ H on C ∞ ( X )[[ ~ ]] satisfying the followings. (1) If we denote by C Hj ( f, g ) the j -th coefficient of ~ in f ⋆ H g for f, g ∈ C ∞ ( X ) . Then we have, for all f, g ∈ C ∞ c ( X ) and l ∈ N , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ kH, U ( f ) φ kH, U ( g ) − l X j =0 (cid:18) −√− k (cid:19) j φ kH, U (cid:0) C Hj ( f, g ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = O (cid:18) k l +1 (cid:19) as k → ∞ . Here U is any choice of open covering of B satisfyingthe conditions in (U) in subsection 3.2. (2) For h , h ∈ C ∞ ( B ) , we have µ ∗ h ⋆ H µ ∗ h = µ ∗ ( h h ) . In other words, the commutative algebra C ∞ ( B ) canonically embedsinto ( C ∞ ( X )[[ ~ ]] , ⋆ H ) . (3) Up to order , this star product ⋆ H coincides with the Fedosov’sstar product ∗ H, , corresponding to the connection ∇ T X,H defined inDefinition 4.2 and ∈ ~ Ω ( X )[[ ~ ]] , i.e., we have C Hj ( f, g ) = C H, j ( f, g ) for j = 0 , , . The second coefficient is explicitely given by, choosing a local action-angle coordinate, C H ( f, g ) = 18 X i,j,l Θ ijl (cid:18) ∂f∂θ i ∂ g∂θ j ∂θ l + ∂ f∂θ j ∂θ l ∂g∂θ i (cid:19) + C std2 ( f, g ) . (4.8) Here C std2 is the second term in the standard Moyal-Weyl star producton R n × T n , and the locally defined symmetric three tensor Θ isdefined in (4.5) .Proof. For (1), in addition to the above argument, we must estimate theoperator norms of error terms. It is done essentially in the same way asthe proof of Theorem 3.33. (2) follows from the fact that for h ∈ C ∞ c ( B ), φ kH, U ( µ ∗ h ) is a multiplication operator by h , so we have φ kH, U ( µ ∗ h ) φ kH, U ( µ ∗ h ) = φ kH, U ( µ ∗ ( h h )).For (3), the formula (4.8) follows from the above computations. To checkthe desired coincidence, we use the formula for second order term in ∗ ∇ , for torsion-free symplectic connection ∇ on T X (see [6, Proposition 2.13]) f ∗ ∇ , g = f g + ~ { f, g } + ~ ω ij ω kl ∇ ik f ∇ jl g + O ( ~ ) , where ∇ XY f := ( XY − ∇ X Y ) f is the second covariant derivative, and( ω ij ) ij is the inverse matrix of the coefficients in the symplectic form ω = ω ij dx i ∧ dx j . Applying this to our case ∇ = ∇ T X,H , by Definition 4.2, theonly nontrivial contribution from the covariant derivative is the terms − ~ n ω x l θ l ω x j θ j ( ∇ x l ∂ x j ) f · ∂ θ l ∂ θ j g + ω θ l x l ω θ j x j ∂ θ l ∂ θ j f · ( ∇ x l ∂ x j ) g o = ~ ijl (cid:8) ∂ θ i f · ∂ θ l ∂ θ j g + ∂ θ l ∂ θ j f · ∂ θ i g (cid:9) , so we see that C H, ( f, g ) is also given by the right hand side of (4.8), thuswe get the result. (cid:3) The relation with Berezin-Toeplitz quantization
In this section, we explain the relation between Berezin-Toeplitz quanti-zation and our quantizations. Here we restrict our attention to the case of R n × T n (subsection 5.1) and abelian varieties (subsection 5.2) with transla-tion invariant complex structures. In those cases we have a natural isomor-phism between quantum Hilbert spaces using theta basis for L -holomorphicsections on L k . We show in Theorem 5.7 and Theorem 5.13 that, as k → ∞ , EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 25 the operator norm of the difference between our deformation quantizationand the Berezin-Toeplitz deformation quantization converges to zero in bothcases.First we recall the definition of Berezin-Toeplitz deformation quantization([4], [7]). Let (
X, ω, J ) be a symplectic manifold equipped with a compatiblecomplex structure, and ( L, ∇ ) be a prequantizing line bundle. Then thequantum Hilbert space by this K¨ahler polarization is given by the spaces of L -holomorphic sections { L H ( X J ; L k ) } k ∈ N . For each k ∈ N , let us denotethe orthogonal projection for the subspace L H ( X J ; L k ) ⊂ L ( X J ; L k ) byΠ k . Definition 5.1.
In the above settings, for each k ∈ N , define a linear oper-ator T k : C ∞ c ( X ) → B (cid:0) L H ( X J ; L k ) (cid:1) by T k ( f ) := Π k M f Π ∗ k : L H ( X J ; L k ) → L H ( X J ; L k ) , for f ∈ C ∞ c ( X ). Here M f denotes the multiplication operator by f .This sequence has the correct semiclassical behavior in the case where X is compact, as shown by Bordemann, Meinrenken and Schlichenmaier [4]. Fact 5.2 ([4]) . If ( X, ω, J ) is a compact K¨ahler manifold, above sequence { T k } k is a strict deformation quantization of C ∞ ( X ) , called Berezin-Toeplitzdeformation quantization. This fact has been generalized in various ways by Ma and Marinescu [7], inparticular to certain classes of non-compact K¨ahler manifolds and orbifolds.Their result includes the case of R n × T n with translation invariant K¨ahlerstructure, which is of our interest in the following subsection 5.1.5.1. On R n × T n . In this subsection, we explain the convergence in the caseof translation invariant K¨ahler quantizations on R n × T n .First we explain our convention on compatible almost complex structureson R n × T n . Let H n := { Ω ∈ M n ( C ) | Ω = t Ω , ImΩ is positive definite } be the Siegel upper half space. Then, if we have an H n -valued functionΩ ∈ C ∞ ( R n × T n ; H n ), we get an almost complex structure on R n × T n by T , x,θ ) ( R n × T n ) = Span C (cid:26) ∂∂x i + Ω ij ( x, θ ) ∂∂θ j (cid:27) ni =1 . This is compatible with ω = t dx ∧ dθ .In this subsection we only consider translation invariant complex struc-tures, i.e., the case where Ω is constant. We denote this complex structureby J Ω .Let us consider the prequantum line bundle ( L, ∇ ) = ( C , d − √− t xdθ ).If we have a section s ∈ C ∞ ( R n × T n ; L k ), we denote its Fourier expansionsby s ( x, θ ) = X m ∈ Z n s m ( x ) e √− h m,θ i . Let k be a positive integer. It is easy to see that an orthonormal basis { Ψ k Ω , lk } l ∈ Z n of L -holomorphic sections on L k is given by(Ψ k Ω , lk ) m = δ l,m (2 π ) − n/ a k, ImΩ exp (cid:0) √− k/ t ( x − l/k ) Ω ( x − l/k ) (cid:1) , (5.3) where a k, ImΩ > a k, ImΩ ) − = Z R n exp (cid:0) − k t x (ImΩ) x (cid:1) dx. We can write explicitely the Berezin-Toeplitz deformation quantizationusing this basis as follows.
Lemma 5.4.
For a function f ∈ C ∞ c ( R n × T n ) , we write the Fourier ex-pansion of f as f = P m ∈ Z n f m ( x ) e √− h m,θ i . Then we have h Ψ k Ω ,b , T k ( f )Ψ k Ω ,c i = ( a k, ImΩ ) Z x ∈ R n f k ( b − c ) ( x ) exp( √− k/ {− t ( x − b )Ω( x − b ) + t ( x − c )Ω( x − c ) } ) dx. The proof is straightforward.Write Ω = P + √− Q , P, Q ∈ M n ( R ). We consider the splitting T ( R n × T n ) = H P ⊕ ker dµ given by H P | ( x,θ ) = Span (cid:26) ∂∂x i + P ij ∂∂θ j (cid:27) ≤ i ≤ n . (5.5)for all ( x, θ ) ∈ R n × T n .Recall that the quantum Hilbert space by the real polarization µ is givenby H k = ⊕ b ∈ Z nk H ( X b ; L k ⊗ | Λ | / X b ) . and we use the orthonormal basis { ψ kb } b ∈ Z nk , ψ kb = e √− k h b,θ i √ d ′ θ , as in(2.12). We consider the strict deformation quantization in Definition 3.22associated to the horizontal distribution H P and the trivial covering U = { R n } (see Remark 3.36), denoted by { φ kH P } k .For each k ∈ N , we get the canonical isomorphism between quantumHilbert spaces, H k ≃ L H ( X J Ω ; L k ) , ψ kb Ψ k Ω ,b , (5.6)for each b ∈ Z n k . Using this isomorphism, we can describe the relation be-tween Berezin-Toeplitz deformation quantization and our deformation quan-tization as follows. Theorem 5.7.
Consider the complex structure on R n × T n associated with Ω = P + √− Q . For all f ∈ C ∞ c ( R n × T n ) , we have lim k →∞ k φ kH P ( f ) − T k ( f ) k = 0 . Here we use the isomorphism of Hilbert spaces (5.6) .Proof.
By the coordinate change R n × T n → R n × T n , ( x, θ ) ( x, − P x + θ ) , the Berezin-Toeplitz deformation quantization and our deformation quanti-zation, as well as the isomorphism (5.6), map to the ones for Ω = √− Q .So we may assume P = 0.First we note the following. EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 27
Lemma 5.8.
There exists a constant C Q > only depending on Q suchthat, for any function g ∈ C ∞ c ( R n ) , we have (cid:12)(cid:12)(cid:12)(cid:12) g (0) − ( a k,Q ) Z R n g ( x ) exp( − k t xQx ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Q k∇ g k C √ k . Proof.
This follows from the estimate | g ( x ) − g (0) | ≤ k∇ g k| x | and ( a k,Q ) − = Z R n exp( − k t xQx ) dx = k − n/ Z exp( − t xQx ) dx, Z R n | x | exp( − k t xQx ) dx = k − ( n +1) / Z | x | exp( − t xQx ) dx. (cid:3) Lemma 5.9.
Assume we are given a function f = P m ∈ Z n f m ( x ) e √− h m,θ i ∈ C ∞ ( R n × T n ) . For each N ∈ N , there exists a constant C N > , which onlydepends on f and Q , such that, for all k ∈ N , p ∈ Z n and x ∈ Z n k , we have |h Ψ k Ω ,x + p/k , T k ( f )Ψ k Ω ,x i − f p ( x + p/ (2 k )) | ≤ C N √ k (1 + | p | ) N . Proof.
We have − t ( x − ( x + p/k ))Ω( x − ( x + p/k )) + t ( x − x )Ω( x − x )= √− (cid:26) t ( x − ( x + p/ (2 k ))) Q ( x − ( x + p/ (2 k ))) + 12 k t pQp (cid:27) . By Lemma 5.4, we have h Ψ k Ω ,x + p/k , T k ( f )Ψ k Ω ,x i = ( a k,Q ) exp( − (4 k ) − t pQp ) · Z x ∈ R n f p ( x ) exp {− k t ( x − ( x + p/ (2 k ))) Q ( x − ( x + p/ (2 k ))) } dx = ( a k,Q ) exp (cid:18) − k t pQp (cid:19) · Z x ∈ R n f p ( x + ( x + p/ (2 k ))) exp( − k t xQx ) dx By Lemma 5.8, we have (cid:12)(cid:12)(cid:12)(cid:12) f p ( x + p/ (2 k )) − ( a k,Q ) Z x ∈ R n f p ( x + ( x + p/ (2 k ))) exp( − k t xQx ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Q k∇ f p k√ k . Since we have | − e − y | ≤ y for any y ≥
0, we see (cid:12)(cid:12)(cid:12) h Ψ k Ω ,x + p/k , T k ( f )Ψ k Ω ,x i − f p ( x + p/ (2 k )) (cid:12)(cid:12)(cid:12) ≤ C Q k∇ f p k√ k + k f p k · t pQp k ≤ C N √ k (1 + | p | ) N , for some constant C N independent of k , x , p , since the Fourier coefficientsare rapidly decreasing. So we get the result. (cid:3) Now we prove Theorem 5.7. Using the isomorphism (5.6), we regard T k ( f ) as an operator on H k . Using Lemma 3.2 and Lemma 5.9, we have k φ kH P ( f ) − T k ( f ) k ≤ X p ∈ Z n sup x ∈ Z nk n(cid:12)(cid:12)(cid:12) h Ψ k Ω ,x + p/k , T k ( f )Ψ k Ω ,x i − f p ( x + p/ (2 k )) (cid:12)(cid:12)(cid:12)o ≤ C n +1 √ k X p ∈ Z n | p | ) n +1 , so we get the result. (cid:3) On Abelian varieties.
In this subsection we show the relation be-tween Berezin-Toeplitz deformation quantization and our deformation quan-tization in the case of Abelian varieties. For works relating geometric quan-tization on Abelian varieties by different polarizations, see for example [2].Let X = ( R / Z ) n × T n be the 2 n -dimensional torus which is obtained bythe Z n -action on R n × T n considered in the subsection 5.1, where m ∈ Z n acts by the symplectomorphism( x, θ ) ( x + m, θ ) , and consider the induced symplectic structure ω = t dx ∧ dθ on X . We getthe induced Lagrangian fibration µ : X → R n / Z n . The Z n -action lifts to anaction on the prequantizing line bundle ( C , ∇ = d − √− t xdθ ) on R n × T n by ( x, θ, v ) ( x + m, θ, e √− h m,θ i v ) , (5.10)preserving ∇ . So we get the induced prequantizing line bundle on X , de-noted by ( L, ∇ ). In this case, the set of k -Bohr-Sommerfeld point is givenby B k = ( k Z / Z ) n ⊂ ( R / Z ) n . A section s ∈ C ∞ ( X ; L k ) is identified with asection ˜ s ∈ C ∞ ( R n × T n ; C ) with the periodicity property˜ s ( x + m, θ ) = e √− h km,θ i ˜ s ( x, θ ) . Fix an element Ω ∈ H n . From a translation invariant complex structure R n × T n given by Ω as in subsection 5.1, we get the induced translationinvariant ω -compatible complex structure on X , also denoted by J Ω . An or-thonormal basis { Θ k Ω ,b } b ∈ B k of H ( X J Ω ; L k ) is given by the following formulafor its lift e Θ k Ω ,b ∈ C ∞ ( R n × T n ; L k ), e Θ k Ω ,b = X l ∈ Z n , [ lk ]= b Ψ k Ω , lk , (5.11)where Ψ k Ω , lk is the basis of L H (( R n × T n ) J Ω ; L k ) given in (5.3). It is easyto see that this basis coincides with the classical Theta basis (see [2, Section2.3]).On the other hand, the orthonormal basis { ψ kb } b ∈ Z nk for the quantumHilbert space by the real polarization on R n × T n as in (2.12) induces theorthonormal basis for the quantum Hilbert space H k by the real polariza-tion µ on X , since { ψ kb } b satisfies the equivariance property with respect tothe Z n -action (5.10). We denote the induced orthonormal basis on H k by EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 29 { ϑ kb } b ∈ B k . Using these basis, we get the canonical isomorphism of quantumHilbert spaces, H k ≃ H ( X J Ω ; L k ) , ϑ kb Θ k Ω s ,b , (5.12)Corresponding to Ω = P + √− Q , we consider the horizontal distribution H P ⊂ T X induced from (5.5).We have the following relation between Berezin-Toeplitz deformation quan-tization and our deformation quantization for this case.
Theorem 5.13.
Consider the complex structure on X = ( R / Z ) n × T n as-sociated with Ω = P + √− Q . Choose any open covering U of ( R / Z ) n satisfying the conditions in (U) in subsection 3.2, and consider the asso-ciated strict deformation quantization { φ kH P , U } k . For all f ∈ C ∞ c ( X ) , wehave lim k →∞ k φ kH P , U ( f ) − T k ( f ) k = 0 . Here we use the isomorphism of Hilbert spaces (5.12) .Proof.
Let us fix a function f ∈ C ∞ c ( X ). We denote the lift of f to R n × T n by ˜ f ∈ C ∞ ( R n × T n ). Let us denote by ˜ T k : C ∞ c ( R n × T n ) → B (cid:0) L H (( R n × T n ) J Ω ; L k ) (cid:1) the Berezin-Toeplitz deformation quantizationon R n × T n . First we easily observe that, we can define a bounded operator˜ T k ( ˜ f ) ∈ B (cid:0) L H (( R n × T n ) J Ω ; L k ) (cid:1) by the same formula as in Definition5.1, by the periodicity of ˜ f . For b, c ∈ B k = ( k Z / Z ) n , choose any lift¯ b, ¯ c ∈ Z n k . Then we have h Θ kb , T k ( f )Θ kc i = X l ∈ Z n h Ψ k ¯ b + l , ˜ T k ( ˜ f )Ψ k ¯ c i . Note that if we denote the Foourier expansion f ( x, θ ) = P m f m ( x ) e √− h m,θ i ,we have (cid:12)(cid:12)(cid:12) h Ψ kx + lk , ˜ T k ( ˜ f )Ψ kx i (cid:12)(cid:12)(cid:12) ≤ k f l k C . Combining this and Lemma 5.9, as well as the fact that the Fourier coeffi-cients of f are rapidly decreasing, we easily get the result. The details areleft to the reader. (cid:3) Appendix
A proof of Proposition 3.17.
In this subsection, we give a detailedproof of Proposition 3.17.
Proof of Proposition 3.17.
For simplicity we only prove in the case where n = 1; for general n the proof is essentially the same. Since the maps φ k areadjoint-preserving, it is enough to show in the case when f ∈ C ∞ c ( X ; R ). Inthis case, φ k ( f ) is a self-adjoint operator for each k .First we show that sup k →∞ k φ k ( f ) k ≤ k f k C . Assume the contrary, andtake ǫ > ǫ < k f k C so thatsup k →∞ k φ k ( f ) k > k f k C + ǫ. (6.1) Let us denote the Fourier expansion of f in the T -direction as P m ∈ Z f m ( x ) e √− h m,θ i .Since f is smooth and compactly supported, there exists a positive integer M ∈ N such that X | m | >M k f m k C < ǫ . Fix such M . Let ˜ f ∈ C ∞ c ( X ) be the function defined by˜ f := X | m |≤ M f m ( x ) e √− h m,θ i . (6.2)It is easy to see that k f − ˜ f k C < ǫ . (6.3)Moreover, applying Lemma 3.2, for all k ∈ N we have k φ k ( f ) − φ k ( ˜ f ) k ≤ X | m | >M k f m k C < ǫ . So by our assumption (6.1), we havesup k →∞ k φ k ( ˜ f ) k > k f k C + ǫ . (6.4)Since f m ∈ C ∞ c ( R ) is smooth and compactly supported for each m , thereexists α > m ∈ Z with | m | ≤ M , we have, | f m ( x ) − f m ( y ) | < ǫ M + 1) for all | x − y | ≤ α. (6.5)Let us choose k ∈ N such that k ≥ M k f k C ǫα , and k φ k ( ˜ f ) k ≥ k f k C + ǫ . (6.6)This is possible by (6.4). Notice that φ k ( ˜ f ) is a self-adjoint operator on H k , and by (6.2), there exists a finite interval [ a, b ] ⊂ R such that φ k ( ˜ f ) = P [ a,b ] φ k ( ˜ f ) P [ a,b ] . Indeed, it is enough to take [ a, b ] so that supp( f ) ⊂ ( a + m k , b − m k ).So the operator φ k ( ˜ f ) can be regarded as a linear operator on the finitedimensional Hilbert space H k | [ a,b ] . In particular, by (6.6), we can take aneigenvalue λ ∈ R of φ k ( ˜ f ) with | λ | ≥ k f k C + ǫ . (6.7)We take a normalized eigenvector v ∈ H k for λ , φ k ( ˜ f ) v = λv, k v k = 1 . Lemma 6.8.
There exists a closed interval I = [ s, t ] ⊂ R such that • | I | = ( t − s =) α , • If we define the interval J := [ s − Mk , t + Mk ] , we have k P I v k ≥ k αk α + 2 M k P J v k . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 31
Proof.
The proof is given by an easy pigeonhole principle argument. Foreach l ∈ Z , we consider the intervals I l := [ l/k , l/k + α ] , J l := [( l − M ) /k , ( l + M ) /k + α ] . Since k v k = 1, we have X l ∈ Z k P I l v k = ⌊ k α + 1 ⌋ , X l ∈ Z k P J l v k = ⌊ k α + 2 M + 1 ⌋ . So there must exist an integer l ∈ Z with k P I l v k ≥ k αk α + 2 M k P J l v k . So we can set I := I l for such l and get the result. (cid:3) Fix an interval
I, J ⊂ R satisfying the conditions in Lemma 6.8. By (6.2)and the definition of φ k , we have P x + mk φ k ( ˜ f ) P x = 0 for all x ∈ B k and m ∈ Z with | m | > M. so we get λP I v = P I φ k ( ˜ f ) v = P I φ k ( ˜ f ) P J v. From this and Lemma 6.8, we have k P I φ k ( ˜ f ) P J k ≥ λ k P I v kk P J v k ≥ λ r k αk α + 2 M ≥ (cid:16) k f k C + ǫ (cid:17) (cid:18) ǫ k f k C (cid:19) − / , where we used (6.7) and (6.6) for the last inequality. The last expression isestimated as (cid:16) k f k C + ǫ (cid:17) (cid:18) ǫ k f k C (cid:19) − / ≥ (cid:16) k f k C + ǫ (cid:17) (cid:18) − ǫ k f k C (cid:19) > k f k C + ǫ , since we have assumed that ǫ < k f k C . So we have k P I φ k ( ˜ f ) P J k > k f k C + ǫ . (6.9)Let us denote by x ∈ I the middle point of the interval I . We define F ∈ C ∞ ( T ) by F ( θ ) := ˜ f ( x , θ ) . We consider the operator Φ k ( F ) ∈ H k defined in Lemma 3.18. Lemma 6.10.
We have (cid:13)(cid:13)(cid:13) P I (cid:16) Φ k ( F ) − φ k ( ˜ f ) (cid:17) P J (cid:13)(cid:13)(cid:13) ≤ ǫ . Proof.
Recall the definition of Φ k ( F ) in Lemma 3.18, the construction of ˜ f in (6.2) and the estimate (6.5). Applying the estimate in Lemma 3.2, k P I (Φ k ( F ) − φ k ( ˜ f )) P J k ≤ X | m |≤ M sup x ∈ J | f m ( x ) − f m ( x ) | ≤ X | m |≤ M ǫ M + 1) ≤ ǫ . Here we used the inequality | x − x | ≤ | J | / α + Mk ) < α for x ∈ J . Thisfollows from (6.6) and our assumption ǫ < k f k C . (cid:3) By Lemma 3.18 and Lemma 6.10, we have k P I φ k ( ˜ f ) P J k ≤ k P I Φ k ( F ) P J k + ǫ ≤ k Φ k ( F ) k + ǫ k F k C + ǫ ≤ k ˜ f k C + ǫ . By (6.3), we get k P I φ k ( ˜ f ) P J k ≤ k f k C + ǫ . This contradicts with the estimate (6.9), so we getsup k →∞ k φ k ( f ) k ≤ k f k C . (6.11)Next we show inf k →∞ k φ k ( f ) k ≥ k f k C . Assume the contrary, and take δ > δ < k f k C such thatinf k →∞ k φ k ( f ) k < k f k C − δ. We take M ′ ∈ N , ˆ f ∈ C ∞ c ( X ) and α ′ > ǫ replaced by δ , as in the first half of this proof. Namely, we take M ′ > X | m | >M ′ k f m k C < δ . We define ˆ f by ˆ f := X | m |≤ M ′ f m ( x ) e √− h m,θ i . (6.12)This satisfies k f − ˆ f k C < δ , (6.13) k φ k ( f ) − φ k ( ˆ f ) k < δ , inf k →∞ k φ k ( ˆ f ) k < k f k C − δ . (6.14)Take α ′ > m ∈ Z with | m | ≤ M ′ , we have, | f m ( x ) − f m ( y ) | < δ M ′ + 1) for all | x − y | ≤ α ′ . (6.15)Let us choose k ′ ∈ N such that k ′ ≥ M ′ k f k C δα ′ , and k φ k ′ ( ˆ f ) k ≤ k f k C − ǫ . (6.16)This is possible by (6.14). EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 33
Since we have assumed that δ < k f k C , by (6.16), we have k ′− < α ′ . Soby (6.15), we can take ( y , θ ) ∈ B k ′ × T such that | ˆ f ( y , θ ) | ≥ k ˆ f k C − δ . (6.17)Let us define the intervals I ′ , J ′ ⊂ R by I ′ := [ y − α ′ / M ′ /k ′ , y + α ′ / − M ′ /k ′ ] , J ′ := [ y − α ′ / , y + α ′ / . Let us define a function ˆ F ∈ C ∞ ( T ) byˆ F ( θ ) := ˆ f ( y , θ ) . We consider the operator Φ k ′ ( ˆ F ) ∈ H k defined in Lemma 3.18. Similarly asLemma 6.10, we have k P I ′ (cid:16) Φ k ′ ( ˆ F ) − φ k ′ ( ˆ f ) (cid:17) P J ′ k ≤ δ . (6.18)Let us define v ′ ∈ P J ′ H k by v ′ := X | m |≤ k ′ α ′ / e −√− h m,θ i ψ ky + m/k ′ √ d ′ θ. We have k v ′ k = J ′ ∩ B k ′ ). For y + pk ′ ∈ I ′ ∩ B k ′ , where | p | ≤ k ′ α ′ / − M ′ ,we have, denoting the matrix element of Ψ k ′ ( ˆ F ) by K ˆ F ( · , · ), h ψ ky + pk ′ , Φ k ′ ( ˆ F ) v ′ i = X | m |≤ k ′ α ′ / K F ′ (cid:16) y + pk ′ , y + mk ′ (cid:17) e −√− h m,θ i = X | m |≤ k ′ α ′ / ˆ f p − m ( y ) e −√− h m,θ i = X m ∈ Z ˆ f p − m ( y ) e −√− h m,θ i = e −√− h p,θ i ˆ f ( y , θ ) . Here the third equality used the fact that, by (6.12), we have ˆ f p − m = 0 for | p | ≤ k ′ α ′ / − M ′ and | m | ≤ k ′ α/
2. So we get k P I ′ Φ k ′ ( ˆ F ) v ′ k = | ˆ f ( y , θ ) | · I ′ ∩ B k ) . Since P J ′ v ′ = v ′ , we see that k P I ′ Φ k ′ ( ˆ F ) P J ′ k ≥ k P I ′ Φ k ′ ( ˆ F ) v ′ k k v ′ k ≥ | ˆ f ( y , θ ) | I ′ ∩ B k ) J ′ ∩ B k ) ≥ (cid:18) k f k C − δ (cid:19) (cid:18) − M ′ k ′ α ′ (cid:19) ≥ (cid:18) k f k C − δ (cid:19) (cid:18) − δ k f k C (cid:19) ≥ (cid:18) k f k C − δ (cid:19) . where the third inequality used (6.17) and (6.13), the fourth inequality used(6.16), and the last inequality used 0 < δ < k f k C . From this and (6.18),we get k φ k ′ ( ˆ f ) k ≥ k P I ′ φ k ′ ( ˆ f ) P J ′ k ≥ k f k C − δ − δ ≥ k f k C − δ . This contradicts with (6.16), so we getinf k →∞ k φ k ( f ) k ≥ k f k C . (6.19)Combining (6.11) and (6.19), we get the desired result. (cid:3) A proof of Theorem 3.33.
In this subsection, we give a proof ofTheorem 3.33. We work in the settings in subsection 3.2, and use notationsthere.
Proof of Theorem 3.33.
As a preperation, we give a sufficient condition for afamily of operators on {H k } k to be uniformly bounded. For a linear operator F on H k and a subset V ⊂ B , we writesupp( F ) ⊂ V if P V F P V = F . Lemma 6.20.
Suppose we are given a family of linear operators {A k } k ∈ N , A k : H k → H k and a compact subset B ′ ⊂ B such that supp( A k ) ⊂ B ′ forall k . Suppose that we are given a finite subset { U i } i ∈ I ⊂ U , a partition B ′ = ⊔ i ∈ I V i and a positive constant M > such that (A) For each i ∈ I , we have V i ⊂ U i . (B) For each i ∈ I , B i ( V i , M ) := { y ∈ R n | x ∈ V i , | y − x | ≤ M } ⊂ U i .Here we regard U i as a subset in R n by the action coordinate, andthe norm is the Euclidean norm with respect to the action coordinateon U i .Furthermore, we assume that, (1) For each i ∈ I , there exists a constant C i > such that, for all k ∈ N , x ∈ V i and p ∈ Z n with | p | ≤ kM , we have k P x + p/k A k P x k ≤ C i (1 + | p | ) n +1 . (2) There exists a constant C ′ > such that, for all k ∈ N and pairs ( b, c ) ∈ B k × B k which cannot be expressed as ( b, c ) = ( x + p/k, x ) with | p | ≤ kM in the action coordinate on U i with c ∈ V i , we have k P b A k P c k ≤ C ′ k n . Then we have sup k →∞ kA k k < ∞ . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 35
Proof.
First we note that, since B ′ is compact, there exists a positive con-stant a < + ∞ such that for all k ∈ N , B k ∩ B ′ ) < ak n . (6.21)Fix k ∈ N . Since supp( A k ) ⊂ ⊔ i ∈ I V i , we can decompose the operator A k as A k = X i ∈ I P U i A k P V i + X i ∈ I (1 − P U i ) A k P V i . (6.22)First we estimate the first term in (6.22). We have P U i A k P V i = X p ∈ Z n , | p |≤ kM X x ∈ V i ∩ B k P x + p/k A k P x + X x ∈ V i ∩ B k P U i \ B i ( x,M ) A k P x . For each p ∈ Z n with | p | ≤ kM , we have (see Lemma 3.2) k X x ∈ V i ∩ B k P x + p/k A k P x k = max x ∈ V i ∩ B k k P x + p/k A k P x k ≤ C i (1 + | p | ) n +1 , by the condition (1) in the statement. So we have k X p ∈ Z n , | p |≤ kM X x ∈ V i ∩ B k P x + p/k A k P x k ≤ C i X p ∈ Z n , | p |≤ kM | p | ) n +1 ≤ C i X p ∈ Z n | p | ) n +1 = C i C ′′ , for some C ′′ < ∞ which does not depend on k nor i ∈ I . Moreover we have,for each x ∈ V i , using the condition (2) in the statement, k P U i \ B i ( x,M ) A k P x k ≤ B k ∩ B ) · C ′ k n . Combining these, for each i , we have k P U i A k P V i k ≤ C i C ′′ + ( B k ∩ B )) · C ′ k n . Next we estimate the second term in (6.22). Since all the pair ( b, c ) with c ∈ V i and b / ∈ U i satisfies the assumption for the condition (2) in thestatement of the Lemma, we get k X i ∈ I (1 − P U i ) A k P V i k ≤ ( B k ∩ B )) · C ′ k n . Combining these, we get kA k k ≤ C ′′ X i ∈ I C i + ( I + 1) · ( B k ) · C ′ k n ≤ C ′′ X i ∈ I C i + ( I + 1) · a C ′ The last inequality used (6.21). Since we have
I < ∞ , we get the result. (cid:3) Let us fix f, g ∈ C ∞ c ( X ) and let A kerr be the operator on H k defined by A kerr := k (cid:18) φ k ( f ) φ k ( g ) − φ k ( f g ) + √− k φ k ( { f, g } ) (cid:19) . (6.23) It is enough to show that the family of operators {A kerr } k satisfies the con-ditions in Lemma 6.20.Since f, g are compactly supported and the covering U is locally finiteand consists of relatively compact subsets, it is easy to see that there existsa compact subset B ′ ⊂ B such that for all k ∈ N , we havesupp( A kerr ) , supp( φ k ( f )) , supp( φ k ( g )) ⊂ B ′ . We fix such B ′ . Then we fix a finite subset { U i } i ∈ I ⊂ U , a partition a par-tition B ′ = ⊔ i ∈ I V i and a positive constant M >
Remark . There exist a positive constant D > i, j ∈ I , any positive number r > b, c ) ∈ B × B with c ∈ U i ∩ U j such that B i ( c, r ) ⊂ U i , we have b / ∈ B i ( c, r ) ⇒ b / ∈ B j ( c, D r ) . Here B i , B j denotes the Euclidean ball with respect to the action coordinateson U i , U j , respectively. This can be seen as follows. Denoting the transitionfunction of the coordinates on U j and U i by x B ij x + c ij , it is enough to set D := max i,j with U i ∩ U j = φ k B ij k ! − . From now on we fix such D .Since f, g are smooth and compactly supported, for each N ∈ N , we havea positive constant C N > i ∈ I , if weexpress the Fourier coefficients of f on U i with respect to the action-anglecoordinates by f m ( x ), m ∈ Z n , and similarly for other functions, we have k f m k U i , k∇ f m k U i , k g m k U i , k∇ g m k U i , k{ f, g } m k U i , k ( f g ) m k U i ≤ C N (1 + | m | ) N , (6.25) k H ( f m ) k U i , k H ( g m ) k U i ≤ C N (1 + | m | ) N . (6.26)Here we denote by H ( · ) the Hessian of a function, and we abuse the notationsto write k · k U i the C ( U i )-norms with respect to the flat metrics induced bythe Euclidean metric of the action coordinate on U i .First we check the condition (2) in Lemma 6.20. Let us take ( b, c ) ∈ B k × B k which satisfies the assumption in the condition (2). By Remark6.24, we have b / ∈ B i ( c, D M ) for all i ∈ I with c ∈ U i . By Lemma 3.25 and(6.25), we have k P b φ k ( f g ) P c k ≤ C N (1 + kD M ) N , (6.27) k P b φ k ( { f, g } ) P c k ≤ C N (1 + kD M ) N . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 37
Also we have P b φ k ( f ) φ k ( g ) P c = X d ∈ B k ( P b φ k ( f ) P d )( P d φ k ( g ) P c ) . The terms in the sum of the last equation is nonzero only when ( b, d ) and( d, c ) are both close. Let us take i ∈ I so that c ∈ V i . Then by theassumption, either(a) d / ∈ B i ( c, M/ d ∈ B i ( c, M/ B i ( d, M/ ⊂ U i and b / ∈ B i ( d, M/ k P d φ k ( g ) P c k ≤ C N (1+ kD M/ N by Lemma 3.25and (6.25), and we also have k P b φ k ( f ) P d k ≤ C . In the case (b), we have k P b φ k ( f ) P d k ≤ C N (1+ kD M/ N and k P d φ k ( g ) P c k ≤ C similarly. Thus in bothcases we get k ( P b φ k ( f ) P d )( P d φ k ( g ) P c ) k ≤ C C N (1 + kD M/ N . So we have k P b φ k ( f ) φ k ( g ) P c k ≤ B k ∩ B ′ ) · C C N (1 + kD M/ N . (6.28)Since B ′ is compact, B k ∩ B ′ ) = O ( k n ) as k → ∞ , the right hand sideabove is O ( k − N + n ).Combining (6.27) and (6.28), we see that k P b A kerr P c k is of O ( k − N ) forany N ∈ N (with coefficients independent of b or c ), so in particular we havea constant ˜ C > k , such that k P b A kerr P c k ≤ ˜ Ck n , (6.29)for all ( b, c ) ∈ B k × B k satisfying the assumption in (2) of Lemma 6.20.Next, we check the condition (1) in Lemma 6.20. Only in this proof, weuse the following notation. Define the set W i for i ∈ I by W i := (cid:26) ( x, p, k ) ∈ (cid:18) V i ∩ Z n k (cid:19) × Z n × N : | p | ≤ kM (cid:27) . The pairs ( b, c ) ∈ B k × B k satisfying the condition (1) in Lemma 6.20 areprecisely those which can be expressed as ( x + pk , x ) for an element ( x, p, k ) ∈ W i for some i ∈ I . Until the end of this proof, we only consider elements( x, p, k ) belonging to W i for some i ∈ I .Let us fix i ∈ I . As in Lemma 3.25, using the action-angle coordinate on X U i and the trivialization of ( L, ∇ ) | X Ui , we take the orthonormal basis of H k | U i , and express operators on H k by the matrix coefficients with respectto that basis. For any ( x, p, k ) ∈ W i , we have K fg (cid:16) x + pk , x (cid:17) = exp (cid:18) √− C ( x, p, k ) | p | k (cid:19) X m ∈ Z n ( f p − m g m ) | x + p k . Here C ( x, p, k ) is a real number such that | C ( x, p, k ) | ≤ k∇ A k U i by Lemma3.25. First we observe that, for any N ∈ N , there exists a constant C ′ N > such that, for all ( x, p, k ) ∈ W i , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K fg (cid:16) x + pk , x (cid:17) − exp (cid:18) √− C ( x, p, k ) | p | k (cid:19) X m ∈ Z n , | m |≤ kM ( f p − m g m ) | x + p k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ′ N k N . i.e., we may replace the sum over m ∈ Z n with those with | m | ≤ kM .Indeed, the above difference is bounded by X m ∈ Z n , | m | >kM k f p − m k U i · k g m k U i ≤ X m ∈ Z n , | m | >kM C C N (1 + | m | ) N . by (6.25), and this is O ( k − N + n ).Put 1 + ˜ C x,p,k | p | k = exp( √− C ( x, p, k ) | p | k ). We have | ˜ C x,p,k | ≤ k∇ A k U i .We have, for all ( x, p, k ) ∈ W i , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K fg (cid:16) x + pk , x (cid:17) − (cid:18) C x,p,k | p | k (cid:19) X m ∈ Z n , | m |≤ kM ( f p − m g m ) | x + p k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ′ N k N . Moreover, we have, using the inequality (1 + | p − m | )(1 + | m | ) ≥ | p | , forany N ∈ N , p ∈ Z n and k ∈ N , we have X m ∈ Z n , | m |≤ kM k f p − m k U i · k g m k U i ≤ X m ∈ Z n , | m |≤ kM C N C N + n +1 (1 + | p − m | ) N (1 + | m | ) N + n +1 (6.30) ≤ C N C N + n +1 (1 + | p | ) N X m ∈ Z n | m | ) n +1 ≤ D ′ N (1 + | p | ) N , where D ′ N > k , p . Thus we get, forany N ∈ N and ( x, p, k ) ∈ W i , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K fg (cid:16) x + pk , x (cid:17) − X m ∈ Z n , | m |≤ kM ( f p − m g m ) | x + p k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k∇ A k U i | p | k · D ′ N +3 (1 + | p | ) N +3 + C ′ N +2 k N +2 (6.31) ≤ k k∇ A k U i D ′ N +3 + ( M + 1) N C ′ N +2 (1 + | p | ) N . where the last inequality uses the fact that | p | + 1 ≤ kM + 1 ≤ k ( M + 1).We apply the same argument for { f, g } and get that, there exists a constant D ′′ N such that for all ( x, p, k ) ∈ W i , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K { f,g } (cid:16) x + pk , x (cid:17) − √− X m ∈ Z n , | m |≤ kM {h m, ∇ f p − m i g m − f p − m h p − m, ∇ g m i}| x + p k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (6.32) ≤ k D ′′ N (1 + | p | ) N . EFORMATION QUANTIZATION FOR LAGRANGIAN FIBRATIONS 39
We have, for ( x, p, k ) ∈ W i , P x + p/k φ k ( f ) φ k ( g ) P x = X d ∈ B k ( P x + p/k φ k ( f ) P d )( P d φ k ( g ) P x ) . When a point d ∈ B k satisfies d / ∈ B i ( x, M ), we have k ( P x + p/k φ k ( f ) P d )( P d φ k ( g ) P x ) k ≤ C C N (1 + kD M ) N , for any N , similarly as before. Since we have supp( φ k ( g )) ⊂ B ′ for thecompact subset B ′ ⊂ B and we have B ′ ∩ B k ) = O ( k n ), we see that foreach N , there exists a constant C ′′ N such that, for any ( x, p, k ) ∈ W i , wehave (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K φ k ( f ) φ k ( g ) (cid:16) x + pk , x (cid:17) − X m ∈ Z n , | m |≤ kM K f (cid:16) x + pk , x + mk (cid:17) K g (cid:16) x + mk , x (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ′′ N k N . Moreover we have, by Lemma 3.25, for any ( x, p, k ) ∈ W i and m ∈ Z n with | m | ≤ kM , (cid:12)(cid:12)(cid:12)(cid:12) K f (cid:16) x + pk , x + mk (cid:17) K g (cid:16) x + mk , x (cid:17) − f p − m (cid:18) x + p + m k (cid:19) g m (cid:16) x + m k (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | p − m | + | m | k · k∇ A k U i · k f p − m k U i · k g m k U i . By the same argument as (6.30), we have X m ∈ Z n , | m |≤ kM ( | p − m | + | m | ) · k f p − m k U i · k g m k U i ≤ C ′′′ N (1 + | p | ) N for some constant C ′′′ N independent of k . Thus we see that, for each N thereexists a constant D ′′′ N such that, for any ( x, p, k ) ∈ W i , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K φ k ( f ) φ k ( g ) (cid:16) x + pk , x (cid:17) − X m ∈ Z n , | m |≤ kM f p − m (cid:18) x + p + m k (cid:19) g m (cid:16) x + m k (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k D ′′′ N (1 + | p | ) N . (6.33)Moreover we have, by (6.26), (cid:13)(cid:13)(cid:13)(cid:13) f p − m (cid:18) x + m + p k (cid:19) − n f p − m + D m k , ∇ f p − m Eo(cid:12)(cid:12)(cid:12) x + p k (cid:13)(cid:13)(cid:13)(cid:13) ≤ C N (1 + | p − m | ) N | m | k , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g m (cid:16) x + m k (cid:17) − (cid:26) g m − (cid:28) p − m k , ∇ g m (cid:29)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) x + p k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C N (1 + | m | ) N | p − m | k . We estimate C N (1 + | p − m | ) N | m | k · k g m k ≤ C N C N + n +3 k (1 + | p − m | ) N (1 + | m | ) N + n +1 ,C N + n +1 (1 + | m | ) N + n +1 | p − m | k · k f p − m k ≤ C N +2 C N + n +1 k (1 + | p − m | ) N (1 + | m | ) N + n +1 , kh m k , ∇ f p − m ik · kh p − m k , ∇ g m ik ≤ C N +1 C N + n +2 k (1 + | p − m | ) N (1 + | m | ) N + n +1 . So by a similar estimate as in (6.30), we see that, for each N there exists aconstant D ′′′′ N such that, for any ( x, p, k ) ∈ W i we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ Z n , | m |≤ kM f p − m (cid:18) x + p + m k (cid:19) g m (cid:16) x + m k (cid:17) (6.34) − X m ∈ Z n , | m |≤ kM (cid:26) f p − m · g m + D m k , ∇ f p − m E g m − f p − m · (cid:28) p − m k , ∇ g m (cid:29)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) x + p k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k · D ′′′′ N (1 + | p | ) N . Combining (6.31), (6.32), (6.33) and (6.34), there exists a constant ˜ D i such that, for any ( x, p, k ) ∈ W i we have k P x + p/k A kerr P x k ≤ ˜ D i (1 + | p | ) n +1 . (6.35)By (6.29) and (6.35), we see that the family of operators {A kerr } k satisfiesthe conditions of Lemma 6.20, so we get the result. (cid:3) Acknowledgment
The author is grateful to Mikio Furuta, Hajime Fujita, Kota Hattori,Kaoru Ono and Takahiko Yoshida for interesting discussions. This work issupported by Grant-in-Aid for JSPS Fellows Grant Number 19J22404.
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