Bargmann-Fock sheaves on Kähler manifolds
aa r X i v : . [ m a t h . DG ] A ug BARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS
KWOKWAI CHAN, NAICHUNG CONAN LEUNG, AND QIN LIA
BSTRACT . Fedosov used flat sections of the Weyl bundle on a symplectic manifold toconstruct a star product ⋆ which gives rise to a deformation quantization. By extendingFedosov’s method, we give an explicit, analytic construction of a sheaf of Bargmann-Fockmodules over the Weyl bundle of a K¨ahler manifold X equipped with a compatible Fe-dosov abelian connection, and show that the sheaf of flat sections forms a module sheafover the sheaf of deformation quantization algebras defined ( C ∞ X [[ ¯ h ]] , ⋆ ) . This sheaf can beviewed as the ¯ h -expansion of L ⊗ k as k → ∞ , where L is a prequantum line bundle on X and ¯ h = k .
1. I
NTRODUCTION
This paper is the last in a series of papers [5, 6], in which we study the relation betweendeformation quantization and geometric quantization on a K¨ahler manifold X . More pre-cisely, we aim at understanding how deformation quantization acts on geometric quanti-zation via the study of Hilbert space representations of deformation quantization algebrasconstructed from the space of holomorphic sections H ( X , L ⊗ k ) of the tensor powers of aprequantum line bundle L on X .For K¨ahler manifolds, deformation quantizations of Wick type is particularly importantbecause they are compatible with the complex structure. The most well-known one is theBerezin-Toeplitz quantization [4, 19], where one considers a compact K¨ahler manifold X equipped with a prequantum line bundle L , as in the setting of geometric quantization.Asymptotic behavior of Toeplitz operators yields the Berezin-Toeplitz star product ⋆ BT on C ∞ ( X )[[ ¯ h ]] . In [5], we applied the technique of oscillatory integrals to construct a family { H x } of formal Hilbert space representations of the Berezin-Toeplitz quantized algebra ( C ∞ ( X )[[ ¯ h ]] , ⋆ BT ) , parametrized by points x ∈ X . It is natural to ask, as x varies, howthe Hilbert spaces H x are related. One aim of this paper is to answer this question.On the other hand, star products on general symplectic manifolds can be obtained byFedosov’s famous construction [10, 11]. There have also been extensive studies on Fe-dosov’s construction on K¨ahler manifolds [2, 8, 14, 22]. In [6], we constructed a specialfamily of Fedosov abelian connections on a K¨ahler manifold X as a natural quantizationof Kapranov’s L ∞ structure [12]. This gives rise to a star product ⋆ α for any formal closed Mathematics Subject Classification.
Key words and phrases.
Deformation quantization, geometric quantization. The inner product on H x is formal because it takes values in C [[ ¯ h ]] . (
1, 1 ) -form α on X . Since ⋆ α satisfies locality, it defines a sheaf ( C ∞ X [[ ¯ h ]] , ⋆ α ) of algebrason X , which should be viewed as the “structure sheaf” of its quantum geometry . The iden-tification of the sheaf C ∞ X [[ ¯ h ]] with the sheaf of flat sections of the Weyl bundle W X , C isdenoted as f ←→ O f .When the Karabegov form ω ¯ h = √− · ω − α of ⋆ α is real analytic, we consider thesubsheaf ( C ω ¯ h X [[ ¯ h ]] , ⋆ α ) of smooth functions satisfying a real analytic condition (Definition3.12). In Section 3, we explicitly construct a sheaf of Bargmann-Fock modules over W X , C which is equipped with a compatible Fedosov abelian connection. We then prove that thesubsheaf F flat X , α of convergent flat sections, which we call the Bargmann-Fock sheaf F flat X , α (seeDefinition 3.14), forms a module over the sheaf of analytic functions: Theorem 1.1 (=Theorem 3.17) . The Bargmann-Fock sheaf F flatX , α is a sheaf of modules over thesheaf (cid:0) C ω ¯ h X [[ ¯ h ]] , ⋆ α (cid:1) . We denote the action of f ∈ ( C ω ¯ h X [[ ¯ h ]] , ⋆ α ) on F flat X , α by O f ⊛ − . Our next theorem showsthat this action is given by formal Toeplitz operators introduced in [5, Definition 2.24]. Theseare defined as compositions of multiplication and orthogonal projection operators on for-mal Hilbert spaces, so they are formal analogues of the usual Toeplitz operators. Theformal Hilbert space relevant to us here is a subspace V x of the stalk ( F flat X , α ) x at a point x ∈ X , which is isomorphic to the space of germs of formal holomorphic functions at x ,i.e., V x ∼ = O X , x [[ ¯ h ]] (see Proposition 4.7). A germ Ψ s ∈ V x can be identified with a germof holomorphic function s ∈ O X , x [[ h ]] via Ψ s = J s · e β / h ⊗ e x ,where J s is the jet of s at x expressed in K-coordinates. More detailed explanation of thenotations can be found in Section 4.3. Theorem 1.2 (=Theorem 4.10) . Given f ∈ C ω ¯ h X [[ ¯ h ]] and Ψ s ∈ V x for any x ∈ X, we haveO f ⊛ Ψ s = Ψ s ′ , where s ′ is obtained by applying the formal Toeplitz operator associated to f on s, namely,T ( J f ) x , Φ ( J s ) = J s ′ .Our construction and proof of Theorem 1.1, which are analytic in nature, follow Fe-dosov’s original approach closely. Note that the module sheaves in Theorem 1.1 existeven when X is not pre-quantizable. On the other hand, closely related studies on suchmodule sheaves have been carried out using deformation-obstruction theory. In the realsymplectic manifolds context, such constructions were established in the work of Nest-Tsygan [21] and Tsygan [24]. In [1], Baranovsky, Ginzburg, Kaledin and Pecharich gave adeformation theoretic construction of quantizations of line bundles as module sheaves inthe algebraic setting. ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 3
Acknowledgement.
We thank Si Li and Siye Wu for useful discussions. The first named author thanks Mar-tin Schlichenmaier and Siye Wu for inviting him to attend the conference GEOQUANT2019 held in September 2019 in Taiwan, in which he had stimulating and very helpfuldiscussions with both of them as well as Jrgen Ellegaard Andersen, Motohico Mulase,Georgiy Sharygin and Steve Zelditch.K. Chan was supported by grants of the Hong Kong Research Grants Council (ProjectNo. CUHK14302617 & CUHK14303019) and direct grants from CUHK. N. C. Leung wassupported by grants of the Hong Kong Research Grants Council (Project No. CUHK14301117& CUHK14303518) and direct grants from CUHK. Q. Li was supported by GuangdongBasic and Applied Basic Research Foundation (Project No. 2020A1515011220).2. P
RELIMINARIES ON F EDOSOV DEFORMATION QUANTIZATION
Throughout this paper, we assume that X is a K¨ahler manifold of complex dimension n , with ω denoting its K¨ahler form. We first recall the definition of Wick type deformationquantization on a K¨ahler manifold: Definition 2.1. A deformation quantization on X is a R [[ ¯ h ]] -bilinear, associative product ⋆ on C ∞ ( X )[[ ¯ h ]] of the form f ⋆ g = f · g + ∑ k ≥ ¯ h k C k ( f , g ) ,where C k ( · , · ) ’s are bi-differential operators on X , such that f ⋆ g − g ⋆ f = ¯ h { f , g } + O ( ¯ h ) .It is said to be of Wick type if the bi-differential operators C k ( · , · ) ’s take holomorphic andanti-holomorphic derivatives of the first and second arguments respectively.In this section, we give a very brief review of Fedosov’s construction of Wick typedeformation quantizations on a K¨ahler manifold X ; we refer to [10] for more details. Definition 2.2.
The
Wick product on the space W C n : = C [[ z , ¯ z , · · · , z n , ¯ z n ]][[ ¯ h ]] is definedby f ⋆ g : = exp − ¯ h n ∑ i = ∂∂ z i ∂∂ ¯ w i ! ( f ( z , ¯ z ) g ( w , ¯ w )) (cid:12)(cid:12)(cid:12) z = w We assign a Z -grading on W C n by letting the monomial ¯ h k z I ¯ z J to have total degree2 k + | I | + | J | . On a K¨ahler manifold X , we consider the following Weyl bundles: W X : = d Sym T ∗ X [[ ¯ h ]] , W X : = d Sym T ∗ X [[ ¯ h ]] , W X , C : = W X ⊗ W X = d Sym T ∗ X C [[ ¯ h ]] . KWOKWAI CHAN, NAICHUNG CONAN LEUNG, AND QIN LI
The fiberwise Hermitian structure on the complexified tangent bundle TX C enables us todefine a fiberwise (non-commutative) Wick product on W X , C . Under a local holomorphiccoordinate system z = ( z , · · · , z n ) , a section of W X , C is written as α = ∑ I , J α I ¯ J y I ¯ y J ,where the sum is over all multi-indices. Writting ω = ω i ¯ j dz i ∧ d ¯ z j , then we have α ⋆ β : = ∑ k ≥ k ! · √− · ¯ h ! k ω i ¯ j · · · ω i k ¯ j k ∂ k α∂ y i · · · ∂ y i k ∂ k β∂ ¯ y j · · · ∂ ¯ y j k .There is the natural symbol map which takes the constant term of a formal power series: σ : Γ ( X , W X , C ) → C ∞ ( X )[[ ¯ h ]] .We also introduce several operators on A • X ( W X , C ) : Definition 2.3.
We define the following natural operators acting as derivations on A • X ( W X , C ) : δ a : = dz i ∧ ∂ a ∂ y i , δ a : = d ¯ z j ∧ ∂ a ∂ ¯ y j , δ : = δ + δ ,as well as ( δ ) ∗ a : = y i · ι ∂ zi a , ( δ ) ∗ a : = ¯ y j · ι ∂ ¯ zj a , δ ∗ : = ( δ ) ∗ + ( δ ) ∗ We also define the operators ( δ ) − and ( δ ) − by normalizing ( δ ) ∗ and ( δ ) ∗ re-spectively: ( δ ) − : = p + p ( δ ) ∗ on A p , q X ( W X , C ) p , q , ( δ ) − : = q + q ( δ ) ∗ on A p , q X ( W X , C ) p , q , δ − : = p + q δ ∗ on A pX ( W X , C ) q .Following [10], we define the following extension of the Wick algebra: Definition 2.4 (p.224 in [10]) . The extention W + C n of W C n is defined as follows: • Elements U ∈ W + C n are given by power series, possibly with negative powers of ¯ h . • For any element U ∈ W + C n , the total degree 2 k + | I | + | J | of every term is nonneg-ative. • For any element U ∈ W + C n , there are only a finite number of terms with a givennonnegative total degree. Note that the extension of the Weyl algebra considered in [5] is different from the one here.
ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 5
The extension W + C n is closed under the Wick product, and we can define the corre-sponding extended Weyl bundle W + X , C on X .A Fedosov abelian connection on the Weyl bundle W X , C is a connection of the form D F = ∇ − δ + h [ I , − ] ⋆ which is flat, i.e., D F =
0; here ∇ is the Levi-Civita connection on X , [ − , − ] ⋆ denotes thebracket associated to the Wick product, and I ∈ A X ( W X , C ) . Note that ∇ = h [ R ∇ , − ] ⋆ ,where R ∇ = − √− R mi ¯ jk ω m ¯ l dz i ∧ d ¯ z j ⊗ y k ¯ y l . From this we see that the flatness condition D F = Fedosov equation :(2.1) ∇ I − δ I + h I ⋆ I + R ∇ = − α ∈ ¯ h · A X ( X )[[ ¯ h ]] .Let Γ flat ( X , W X , C ) be the space of flat sections of the Weyl bundle under D F . It is shownin [10] that the symbol map σ : W X , C → C ∞ ( X )[[ ¯ h ]] induces the following isomorphism: Γ flat ( X , W X , C ) ∼ → C ∞ ( X )[[ ¯ h ]] .If we denote by O f the flat section of W X , C corresponding to a formal smooth function f ,then the associated star product can be defined by O f ⋆ g : = O f ⋆ O g .2.1. L ∞ structure on K ¨ahler manifolds: classical and quantum. In this subsection, we first recall Kapranov’s L ∞ -algebra structure on a K¨ahler manifoldand its geometric interpretation. Then we describe its classical and quantum extensions.As discovered in [6], the latter gives rise to a special class of Fedosov connections.Explicitly, the L ∞ structure is equivalent to the following flat connection on W X : D K = ∇ − δ + ∑ n ≥ ˜ R ∗ n ,where the subscript “K” stands for “Kapranov”. Here ˜ R ∗ n ’s are defined by extending thefollowing R ∗ n ’s to A • X -linear derivations on W X : R ∗ = R mi ¯ jk d ¯ z j ⊗ ( y i y k ⊗ ∂ y m ) , R ∗ n = ( δ ) − ◦ ∇ ( R ∗ n − ) for n > R mi ¯ jk ’s are the coefficients of the curvature tensor. We write these R ∗ n ’s locally as R ∗ n = R ji ··· i n ,¯ l d ¯ z l ⊗ ( y i · · · y i n ⊗ ∂ y j ) . Remark . These A • X -linear operators ˜ R ∗ n ’s extend naturally to the complexified Weylbundle A • X ( W X , C ) . Notation 2.6.
To simplify notations in later computations, we introduce two operators:˜ ∇ : = ( δ ) − ◦ ∇ , ˜ ∇ : = ( δ ) − ◦ ∇ . KWOKWAI CHAN, NAICHUNG CONAN LEUNG, AND QIN LI
The symbol map of the Weyl bundle gives rise to an isomorphism:
Proposition 2.7.
For every f ∈ O X ( U ) , there exists a unique flat section J f (where “J” standsfor “jets”) under the connection D K such that σ ( J f ) = f . Explicitly:J f = ∑ k ≥ ( ˜ ∇ ) k ( f ) . Thus the space of (local) flat sections of the holomorphic Weyl bundle with respect to the connectionD K is isomorphic to the space of holomorphic functions. It was shown by Bochner that there exists the following special holomorphic coordinatesystem at each point x ∈ X when the K¨ahler form ω is real analytic. Definition 2.8.
A holomorphic coordinate system ( z , · · · , z n ) centered at x ∈ X is calleda K¨ahler normal coordinate if there exists a unique function ρ x around x such that ∂ ¯ ∂ ( ρ x ) = − √− ω and whose Taylor expansion at x is of the form(2.2) ρ x ( z , ¯ z ) ∼ − √− · ω i ¯ j ( x ) z i ¯ z j + ∑ | I | , | J |≥ | I | ! | J | ! ∂ | I | + | J | ρ x ∂ z I ¯ z J ( x ) z I ¯ z J .If − √− ω i ¯ j ( x ) = δ ij , then we call this a K-coordinate centered at x .The geometric meaning of the connection D K is that the germ ( J f ) x is precisely theTaylor expansion of f at x under the K¨ahler normal coordinates.We now introduce two extensions of D K to W X , C : one quantum and the other classical.For the quantum extension, we first use the K¨ahler form to “lift the last subscript” of R ∗ n and define I n : = − √− · R ji ··· i n ,¯ l ω j ¯ k d ¯ z l ⊗ ( y i · · · y i n ¯ y k ) ∈ A X ( W X , C ) .Then we let I : = ∑ n ≥ I n . In [6], we proved the following theorem: Theorem 2.9.
Suppose α is a representative of a formal cohomology class in ¯ hH dR ( X )[[ ¯ h ]] of type (
1, 1 ) . Let ϕ be a (locally defined) function such that ∂ ¯ ∂ϕ = α and set J α : = ∑ k ≥ ( ˜ ∇ ) k ( ¯ ∂ ϕ ) .Then we have (1) I α : = I + J α ∈ A X ( W X , C ) is a solution of the Fedosov equation, i.e., ∇ I α − δ I α + h I α ⋆ I α + R ∇ = − α . We denote the corresponding Fedosov abelian connection by D F , α and the the correspond-ing Fedosov star product by ⋆ α . (2) The Fedosov connection D F , α is an extension of D K , i.e., D F , α | W X = D K . (3) Every star product on X of Wick type can be obtained from such Fedosov connections.
ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 7
Let γ α = I α + √− ω i ¯ j ( dz i ⊗ ¯ y j − d ¯ z j ⊗ y i ) . Then we can also write the Fedosov abelianconnection D F , α as(2.3) D F , α = ∇ + h [ γ α , − ] ⋆ ,If we put ω ¯ h : = √− ω − α , then the Fedosov equation (2.1) is equivalent to ∇ γ α + h γ α ⋆ γ α + R ∇ = ω ¯ h .On the other hand, the complex conjugate of the connection D K is a flat connection D K on W X . Then D C : = D K ⊗ + ⊗ D K is naturally a flat connection on W X , C = W X ⊗ W X such that D C | W X = D K . This gives theclassical extension of D K (where the subscript C stands for “classical”). The motivationbehind this extension is very simple: since the flat sections with respect to D K correspondto (local) holomorphic functions on X , by adding the anti-holomorphic components in W X , C , we shall see all the smooth functions. This is indeed the case. Proposition 2.10.
There is a one-to-one correspondence between C ∞ ( X )[[ ¯ h ]] and the space of flatsections of the Weyl bundle W X , C with respect to the flat connection D C .Proof. Given any smooth function f , we need to show that there exists a unique J f suchthat σ ( J f ) = f and D C ( J f ) =
0, where σ is the symbol map. The proof is very similar tothat of Theorem 3.3 in Fedosov [10], so we will be brief. For the uniqueness of J f , considera nonzero section s of W X , C with σ ( s ) =
0. Let s be the terms in s of the smallest weight.Then δ ( s ) is nonzero and of smaller weight, so s cannot be flat. For the existence of J f ,consider the filtration on A • X ( W X , C ) induced by the polynomial degrees of terms in W X , C .The fact that the fiberwise de Rham differential δ has cohomology concentrated in degree0 implies the existence of J f , which is uniquely determined by the iterative equation(2.4) J f = f + δ − ( D C + δ )( J f ) . (cid:3) We now give explicit formula for some terms in J f . The first observation from theiterative equation (2.4) is that ( J f ) n ,0 = ( ˜ ∇ ) n ( f ) . Proposition 2.11.
Let D C denote the (
0, 1 ) -part of the flat connection D C . Then there is a one-to-one correspondence between ker ( D C ) and smooth sections of the holomorphic Weyl bundle W X .Proof. Consider the projection map π ∗ ,0 : W X , C → W X . KWOKWAI CHAN, NAICHUNG CONAN LEUNG, AND QIN LI and the filtration induced by the degrees of anti-holomorphic components on the Weylbundle. Then the statement of this proposition is simply that there exists a unique sectionof W X , C annihilated by D C , with a prescribed leading term, the proof of which is againvery similar to that of [10, Theorem 3.3]. (cid:3) Proposition 2.11 and the uniqueness of J f implies the following corollary: Corollary 2.12. J f is determined uniquely and iteratively by the following conditions: • The ( k , 0 ) -component of J f is given by ( ˜ ∇ ) k ( f ) . • For n ≥ , we have (2.5) ( J f ) ∗ , n + = (( δ ) − ◦ ( D C + δ ))(( J f ) ∗ , n ) . Lemma 2.13.
Let f be a smooth function on X, then the ( k ) - and ( k , 1 ) -components of J f aregiven respectively by ( J f ) k = ( ˜ ∇ ) k ◦ ˜ ∇ )( f ) and ( J f ) k ,1 = (( ˜ ∇ ) k ◦ ˜ ∇ )( f ) .Proof. We will only prove the formula for ( J f ) k by induction because the other can beproven similarly. It is obvious for k =
0. Suppose that the statement is valid for k ≤ n .Then from equation (2.5), we have ( J f ) k + = (( δ ) − ◦ ( D C + δ ))(( J f ) ∗ , k )= (( δ ) − ◦ ( ∇ + ∑ n ≥ ˜ R ∗ n ))(( J f ) ∗ , k )= (( δ ) − ◦ ∇ )(( J f ) k ) ,where the last equality follows from the fact that ˜ R ∗ n (( J f ) ∗ , k ) has holomorphic degreegreater than or equal to 2. (cid:3) Sections of W X , C associated to closed (
1, 1 ) -forms. We consider here a section of the Weyl bundle associated to a closed (
1, 1 ) -form on X .We use the symplectic form as an example: Let ϕ be a (locally defined) function on X suchthat ∂ ¯ ∂ρ = ω , which is unique up to the sum of a purely holomorphic and a purely anti-holomorphic function. It follows that the components of J ϕ of mixed type only dependon ω . We denote those mixed terms in J ϕ by Φ ω : Φ ω : = ∑ i , j ≥ ( J ϕ ) i , j .It is clear that ( Φ ω ) = ∂ ϕ∂ z i ∂ ¯ z j y i ¯ y j = ω i ¯ j y i ¯ y j . Lemma 2.14.
We have ( Φ ω ) k = ( Φ ω ) k ,1 = for k ≥ .Proof. By Lemma 2.13, we have ( J ϕ ) k ,1 = (( ˜ ∇ ) k ◦ ˜ ∇ )( ϕ ) = ( ˜ ∇ ) k − ( ω i ¯ j y i ¯ y j ) = ∇ . The vanishing for the ( k ) terms is similar. (cid:3) ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 9
For later computations, we give a formula for ( Φ ω ) n ,2 , n ≥
2. Using Corollary 2.12 andthe fact that ( J ϕ ) n ,1 = n ≥
2, we have ( Φ ω ) n ,2 = ( J ϕ ) n ,2 = ( δ ) − ∇ (( J ϕ ) n ,1 ) + n ∑ i = ˜ R ∗ i (( J ϕ ) n − i + ) ! = ( δ ) − (cid:16) ∇ (( J ϕ ) n ,1 ) + ˜ R n (( J ϕ ) ) (cid:17) = ( δ ) − ◦ ˜ R n (( J ϕ ) )= ( δ ) − (cid:16) ˜ R ∗ n ( ω i ¯ j y i ¯ y j ) (cid:17) = ( δ ) − (cid:18) R ki ··· i n ,¯ l d ¯ z l ⊗ ( y i · · · y i n ) ∂∂ y k ( ω i ¯ j y i ¯ y j ) (cid:19) = ( δ ) − (cid:16) R ki ··· i n ,¯ l ω k ¯ j d ¯ z l ⊗ ( y i · · · y i n ¯ y j ) (cid:17) From the above computation, we obtain(2.6) δ (( Φ ω ) n ,2 ) = R ki ··· i n ,¯ l ω k ¯ j d ¯ z l ⊗ ( y i · · · y i n ¯ y j ) = √− I n for n ≥ α be a representative of a formal cohomology class in ¯ hH dR ( X )[[ ¯ h ]] of type (
1, 1 ) , and let ϕ be a (local) potential of α . We set Φ α : = ∑ i , j ≥ ( J ϕ ) i , j . Then we have J α = δ ∑ k ≥ ( Φ α ) k ,1 ! .The following theorem, whose proof will be given in Appendix A, describes the relationbetween the classical and quantum flat sections: Theorem 2.15.
Let Φ : = √− (cid:16) − ω i ¯ j y i ¯ y j + Φ ω (cid:17) − Φ α . Given a smooth function f , let O f bethe flat section under the Fedosov connection D F , α associated to f , i.e., D F , α ( O f ) = . Then wehave (2.7) J f · e Φ /¯ h = e Φ /¯ h ⋆ O f .The germ ( J f ) x is the Taylor expansion of f at x under K¨ahler normal coordinates andtheir complex conjugates, which is a classical object. On the other hand, the flat section O f is a quantum object which we will explain in Section 4. In particular, for holomorphicfunctions f ∈ O ( U ) , since D K = D C | W X = D K , α | W X , we must have J f = O f (which canalso be seen from equation (2.7)). This says that holomorphic functions do not receive anyquantum corrections . 3. B ARGMANN -F OCK SHEAF
Since a deformation quantization defined via Fedosov abelian connections satisfies lo-cality , it defines a sheaf of algebras on X , which can be viewed as the structure sheaf of the “quantum geometry” on X . The goal of this section is to construct a sheaf of modulesover this structure sheaf, which we call a Bargmann-Fock sheaf .3.1.
Extended holomorphic Weyl bundle and formal line bundles.
In this subsection, we define the extended holomorphic Weyl bundle on a K¨ahler man-ifold X . This is the first step in the construction of a Bargmann-Fock sheaf. We first recallthe Bargmann-Fock representation of the Wick algebra: Definition 3.1.
We define an action of a monomial f = z α · · · z α k ¯ z β · · · ¯ z β l ∈ W C n on s ∈ F C n : = C [[ z , · · · , z n ]][[ ¯ h ]] by(3.1) f ⊛ s : = ¯ h l ∂∂ z β ◦ · · · ◦ ∂∂ z β l ◦ m z α ··· z α k ( s ) ,where m z α ··· z α k denotes the multiplication by z α · · · z α k . It is known that f ⊛ ( g ⊛ s ) = ( f ⋆ g ) ⊛ s ,so this defines an action of the Weyl algebra W C n on F C n , known as the Bargmann-Fockrepresentation (or the
Wick normal ordering in physics literature).Via the fiberwise Bargmann-Fock action, the holomorphic Weyl bundle W X can be re-garded as a sheaf of W X , C -modules. We consider the following extension of W X by al-lowing formal exponentials: Definition 3.2.
We define the sheaf W X , e of extended Weyl algebra with exponentials asfollows: for every open set U ⊂ X , we consider the space of finite sum of pairs k ∑ i = ( f i , e g i /¯ h ) ,where f i , g i ’s are smooth sections of W X on U . We define the multiplication by the linearextension of ( f , e g /¯ h ) · ( f , e g /¯ h ) : = ( f f , e ( g + g ) /¯ h ) These are subject to the equivalence relation that ( f , e g /¯ h ) ∼ ( f , e g /¯ h ) if f = f and g − g ∈ C [[ ¯ h ]] . Then the space W X , e ( U ) of sections of W X , e over U is given by the set ofequivalence classes.There is a sub-sheaf O X , e of W X , e defined as follows: for an open set U ⊂ X , the space O X , e ( U ) consists of equivalence classes of finite sums n ∑ i = ( f i , e g i /¯ h ) ,where f i , g i ∈ O X ( U )[[ ¯ h ]] are all formal holomorphic functions on U . We should pointout that O X is naturally a sub-sheaf of O X , e ( W X , e ) by the inclusion f ( f , e h ) . ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 11
Notation 3.3.
For convenience, we will use the notation f · e g /¯ h for the the pair ( f , e g /¯ h ) in W X , e (and also O X , e ).Similar to the definition holomorphic line bundles, we can define the notions of formalholomorphic line bundles, which are similar to local line bundles in [20] and twistingbundles in [24]. Definition 3.4. A formal line bundle on X is an invertible O X , e -module.We can also introduce the notion of connection and curvature on formal line bundles. Lemma 3.5.
For every formal closed (
1, 1 ) -form α ∈ A closed ( X )[[ ¯ h ]] , there is a formal line bundleL α /¯ h with connection ∇ L α /¯ h whose curvature is given by h · α .Proof. Similar to holomorphic line bundles, we choose a fine cover { U i } of X . On each U i ,we choose a local trivialization e i of L α /¯ h on U , and f i ∈ C ∞ ( U i )[[ ¯ h ]] such that ¯ ∂∂ ( f i ) = α | U i . On each non-empty intersection U i ∩ U j , since f i , f j are both potentials of α | U i ∩ U j ,their difference must be a sum of holomorphic and anti-holomorphic functions: f i − f j = f ij ( z ) + g ij ( ¯ z ) ,and the functions f ij and g ij are unique up to constants (i.e., elements in C [[ ¯ h ]] ). So e f ij ( z ) /¯ h ∈ O X , e ( U i ∩ U j ) are well-defined functions satisfying the cocycle condition: e fij ( z )+ fjk ( z )+ fki ( z ) ¯ h = ∈ O X , e ( U i ∩ U j ∩ U k ) .This defines the desired invertible O X , e -module L α /¯ h . It is equipped with a connection ∇ L α /¯ h which acts locally as ∇ L α /¯ h ( f · e g /¯ h ⊗ e i ) = ( d f + h ( f · dg + f · ∂ f i )) · e g /¯ h ⊗ e i .It is easy to see that the connection is well-defined and has curvature ∇ L α /¯ h = h · α . (cid:3) Let us explain the motivations for introducing the notion of formal line bundles. Fromthe point of view of the Berezin-Toeplitz quantization, the above lemma implies the ex-istence of a formal line bundle whose curvature is ω /¯ h . Since 1/¯ h ∼ k , this formal linebundle corresponds to the asymptotics of the tensor powers L ⊗ k of the prequantum linebundle as k → ∞ . This also explain why we use the name “formal” line bundle. On theother hand, from the point of view of Fedosov’s construction, we need to twist W X , e by aformal line bundle so to admit a Fedosov flat connectionWe now explain the Fedosov viewpoint in more details. The idea is very simple: theFedosov connection on the Weyl bundle W X , C is the sum of the Levi-Civita connection ∇ and the bracket h [ γ α , − ] ⋆ . A naive guess is to replace the bracket by an extended fiberwiseBargmann-Fock action h γ α ⊛ on W X , e . We need to be careful here: for a monomial f as in Definition 3.1, we can extend its action to W X , e by the same differential operator as inequation (3.1). In particular,¯ y j ⊛ ( f · e g /¯ h ) = ¯ h ω i ¯ j √− ∂∂ y i ( f · e g /¯ h ) = ¯ h ω i ¯ j √− (cid:18) ∂ f ∂ y i + h f · ∂ g ∂ y i (cid:19) · e g /¯ h However, for general elements of W X , C on W X , e we could run into infinite sums such asthe following example: when X = C , g = y and f = ∑ k ≥ ¯ y k , f ⊛ e g /¯ h = ∑ k ≥ ¯ y k ! ⊛ e y /¯ h = ∑ k ≥ (cid:0) ¯ h ∂ y (cid:1) k ( e y /¯ h ) = e y /¯ h + e y /¯ h + · · · .If we write f = ∑ k , I , J ¯ h k f I , ¯ J , k y I ¯ y J and g = ∑ I , k ¯ h k g k , I y I , then it is not difficult to see fromthe above example that the infinite sums come from two sources:(1) Those terms g i y i in g which are linear in W X and do not include ¯ h ;(2) The infinite sums ∑ J f k , I , ¯ J ¯ h k y I ¯ y J for fixed indices I , k .Thus we have the following lemma: Lemma 3.6.
We say that a section α = ∑ k , I , J ¯ h k · α k , I , ¯ J y I ¯ y J of W X , C is admissible if it satisfiesthe following finiteness condition: for every fixed I and k , ∑ J ¯ h k α k , I , ¯ J y I ¯ y J is a finite sum.Then for any admissible α and for any section s of W X , e , there is a well-defined α ⊛ s. Definition 3.7.
For a representative α of a formal (
1, 1 ) -class [ α ] ∈ ¯ hH dR ( X )[[ ¯ h ]] , let γ α beas in equation (2.3). Since γ α is admissible as in Lemma 3.6, D α : = (cid:18) ∇ + h γ α ⊛ (cid:19) (cid:16) f · e g /¯ h (cid:17) .defines a connection on W X , e [ ¯ h − ] . Here ∇ denotes the naturally extended Levi-Civitaconnection on W X , e : ∇ ( f · e g /¯ h ) = ∇ ( f ) · e g /¯ h ± (cid:18) h f · ∇ ( g ) (cid:19) · e g /¯ h ,and ⊛ denotes the fiberwise Bargmann-Fock action of W X , C on W X , e . Lemma 3.8.
The curvature of D α is given byD α = h ω ¯ h − Ric X , where Ric X = R ki ¯ jk dz i ∧ d ¯ z j is the Ricci form of X. In particular, the connection D α on W X , e isnot flat.Proof. Let f · e g /¯ h be a section of W X , e . Then ∇ ( f · e g /¯ h ) = ∇ (( ∇ ( f ) + f ∇ ( g /¯ h )) · e g /¯ h )= ( ∇ f + f ∇ ( g /¯ h )) · e g /¯ h . ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 13
On the other hand, we have1¯ h R ∇ ⊛ ( f · e g /¯ h ) = h ( − √− ) R mi ¯ jk ω m ¯ l dz i ∧ d ¯ z j ⊗ y k ¯ y l ⊛ ( f · e g /¯ h )= h ( − √− ) R mi ¯ jk ω m ¯ l dz i ∧ d ¯ z j ⊗ ¯ h ω p ¯ l √− ∂∂ y p ( y k f · e g /¯ h )= R mi ¯ jk dz i ∧ d ¯ z j ⊗ ∂∂ y m ( y k f · e g /¯ h )= R mi ¯ jk dz i ∧ d ¯ z j ⊗ y k ∂∂ y m ( f · e g /¯ h ) + R ki ¯ jk dz i ∧ d ¯ z j · ( f · e g /¯ h )= ∇ ( f · e g /¯ h ) + R ki ¯ jk dz i ∧ d ¯ z j · ( f · e g /¯ h ) .So the curvature of the Levi-Civita connection ∇ on W X , e is given by ∇ = (cid:18) h R ∇ − R ki ¯ jk dz i ∧ d ¯ z j (cid:19) ⊛ .Now we compute (cid:18) ∇ + h γ α ⊛ (cid:19) ( s ) = ∇ ( s ) + h ∇ ( γ α ⊛ s ) + h γ α ⊛ ( ∇ s + h γ α ⊛ s )= ∇ ( s ) + h ( ∇ γ α + h γ α ⋆ γ α ) ⊛ s = h ( ∇ γ α + h γ α ⋆ γ α + R ∇ − ¯ hR ki ¯ jk dz i ∧ d ¯ z j ) ⊛ s = h ( ω ¯ h − ¯ h · R ki ¯ jk dz i ∧ d ¯ z j ) · s . (cid:3) Bargmann-Fock sheaves.Definition 3.9.
For a representative α of a formal (
1, 1 ) -class [ α ] ∈ ¯ hH dR ( X )[[ ¯ h ]] , let ω ¯ h : = √− · ω − α and α ′ : = − ω ¯ h + ¯ h · Ric X . Then we define the sheaf of Bargmann-Fock modules as F X , α : = W X , e ⊗ O X , e L α ′ /¯ h .It is equipped with the connection D B , α : = ( ∇ + h γ α ⊛ − ) ⊗ ∇ L α ′ /¯ h .Lemmas 3.8 and 3.5 imply that D B , α is flat, i.e., D B , α =
0. We have seen that there is awell-defined Bargmann-Fock action of admissible sections in W X , C on F X , α . In fact, thisaction is compatible with the connections on these sheaves: Lemma 3.10.
The connection D B , α is compatible with the Fedosov connection D F , α , i.e., if O ∈W X , C is an admissible section and s is a section of F X , α , then we haveD B , α ( O ⊛ s ) = D F , α ( O ) ⊛ s + ( − ) | O | O ⊛ ( D B , α ( s )) . In particular, if O and s are flat sections, then O ⊛ s is also a flat section.Proof. This actually comes from the compatibility between D α and D F , α , namely, for O ∈W X , C and s ∈ W X , e , we have D α ( O ⊛ s ) = (cid:18) ∇ + h γ α ⊛ (cid:19) ( O ⊛ s )= ∇ ( O ) ⊛ s + ( − ) | O | O ⊛ ∇ ( s ) + h ( γ α ⋆ O ) ⊛ s = ∇ ( O ) ⊛ s + ( − ) | O | O ⊛ ∇ ( s ) + h [ γ α , O ] ⋆ ⊛ s + ( − ) | O | O ⊛ (cid:18) h γ α ⊛ s (cid:19) = D F , α ( O ) ⊛ s + ( − ) | O | O ⊛ D α ( s ) . (cid:3) Consider a smooth function f with Taylor-Fedosov series O f = ∑ I , J a I ¯ J y I ¯ y J . If theaction of O f on F X , α is well-defined, then the above compatibility implies that this actionpreserves flat sections under D B , α . However, we will need certain convergence propertyor analyticity in order to define an action of W X , C on F X , α , since, as we mentioned rightbefore Lemma 3.6, there will be certain infinite sums as in the following example: ∑ J a
0, ¯ J ¯ y J ! ⊛ e β i y i /¯ h = a
0, ¯ J ¯ h | J | ∂ J ∂ y j · · · y j | J | ! ( e β i y i /¯ h )= ∑ J a
0, ¯ J β j · · · β j | J | · e β i y i /¯ h .From now on we will assume that the (
1, 1 ) -form ω ¯ h = √− · ω − α is real analytic, andwe define a function on X which measures its analyticity: Lemma/Definition 3.11.
Suppose ω ¯ h = √− · ω − α is real analytic. We define a functionr : X → ( ∞ ] by letting r ( x ) be the radius of convergence of ω ¯ h under a K-coordinate centeredat x . The function r is lower semi-continuous, and is independent of the choice of K-coordinatesbecause different choices differ only by a U ( n ) transformation. Equivalently, suppose r ( x ) > r ,then there exists a neighborhood x ∈ U, such that r ( x ) > r for all x ∈ U. We define the following sub-class of real analytic functions, which roughly speakingconsists of analytic functions with analyticity at least the same as that of ω ¯ h . Definition 3.12.
For every open set U ⊂ X , let C ω ¯ h X ( U )[[ ¯ h ]] denote the set of real analyticfunctions on X such that at every point x ∈ X , the radius of convergence is greater thanor equal to r ( x ) under a K -coordinate centered at x . ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 15
Lemma 3.13.
The spaces C ω ¯ h X ( U )[[ ¯ h ]] of functions define a sheaf C ω ¯ h X [[ ¯ h ]] of algebras on X underthe Fedosov star product ⋆ α .Proof. Since this convergence property of functions is defined pointwise, it is clear that thespaces define a sub-sheaf of the sheaf of real analytic functions. For every point x ∈ X ,we fix a K -coordinate centered at x . From the Fedosov construction of the Wick typedeformation quantization, we can see that the coefficients of the bi-differential operators C i ( − , − ) are either the Christoffel symbols of the Levi-Civita connection, the coefficientsof ω ¯ h or their derivatives, which all have at least the same convergence property as theformal closed (
1, 1 ) -form ω ¯ h . (cid:3) Definition 3.14.
The
Bargmann-Fock sheaf F flat X , α is defined as the sub-sheaf of F X , α whichconsists of flat sections that are finite sums of the following form: α · e β /¯ h ⊗ e U , where wecan write β = ∑ | I |≥ β I y I locally. We require that the coefficients of the degree 1 terms,i.e., β i , 1 ≤ i ≤ n , satisfy the following boundedness condition:(3.2) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ i = β i y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x < r ( x ) ,where the norm is defined using the Hermitian metric on T ∗ X .Notice that this definition is independent of the local trivializations of the holomorphicWeyl bundle and the formal line bundle. Lemma 3.15.
Consider f ∈ C ω ¯ h ( X ) with Taylor-Fedosov series given locally by O f = ∑ I , J a I ¯ J y I ¯ y J under a K-coordinate centered at x . For every multi-index I , the series (3.3) ∑ J a I ¯ J y I ¯ y J ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = x , ( y , ··· , y n )=( ξ , ··· , ξ n ) ,(3.4) ∇ ∑ J a I ¯ J y I ¯ y J ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = x , ( y , ··· , y n )=( ξ , ··· , ξ n ) converge for ξ = ( ξ , · · · , ξ n ) ∈ C n with || ξ || < r ( x ) .Proof. We can apply the following iterative equation for the Taylor-Fedosov series of f : O f = f + δ − ( ∇ O f + h [ I α , O f ] ⋆ ) .A simple observation is that all terms in δ − ◦ (cid:16) ∇ + h [ I α , − ] ⋆ (cid:17) , except δ − ◦ ∇ , willincrease either the holomorphic degree in W X , C or the degree of ¯ h . Thus all but finitelymany terms in ∑ J a I ¯ J y I ¯ y J can be obtained by applying the operator δ − ◦ ∇ a numberof times to a function which is a product of f , the Christoffel symbols, coefficients of ω ¯ h and the curvature tensor, and also their derivatives. These terms are exactly the purely anti-holomorphic part of the Taylor expansion of these functions under the K -coordinatesat x , and the convergence of the series (3.3) follows.The proof for the convergence of the series (3.4) is similar. From the discussion inthe previous paragraph, we can assume, without loss of generality, that I = ∑ J a I ¯ J y I ¯ y J = ∑ k ≥ ( ˜ ∇ ) k g , where g is a function which is a product of f , the Levi-Civitaconnection, coefficients of ω ¯ h , and their derivatives. We can split the series (3.4) into its (
1, 0 ) - and (
0, 1 ) -parts. It is then clear that the (
0, 1 ) -part has the desired convergenceproperty. Thus we only need to show that ∑ k ≥ ∇ ◦ ( ˜ ∇ ) k ( g ) (cid:12)(cid:12)(cid:12) z = x , ( y , ··· , y n )=( ξ , ··· , ξ n ) converges for || ξ || x ≤ r ( x ) . Notice that for every fixed k ≥
0, we have (cid:16) ∇ ◦ ( ˜ ∇ ) k (cid:17) ( g ) =[ ∇ , ˜ ∇ ] ◦ ( ˜ ∇ ) k − ( g ) ± ˜ ∇ ◦ [ ∇ , ˜ ∇ ] ◦ ( ˜ ∇ ) k − ( g ) + · · ·± ( ˜ ∇ ) k − ◦ [ ∇ , ˜ ∇ ] ± ( ˜ ∇ ) k ◦ ∇ ( g ) ,and the bracket [ ∇ , ˜ ∇ ] contributes the coefficients of the curvature. Thus, when eval-uated at z = x , y = ξ , the absolute values of these terms are bounded by k times thecomponents of the Taylor series of functions with radius of convergence at least r ( x ) .Now the convergence of (3.4) follows. (cid:3) We also need the following lemma from elementary analysis:
Lemma 3.16.
Let f n : U → C , n ∈ N be a sequence of smooth functions on U, and D be a dif-ferential operator on U. Suppose that the two series ∑ ∞ i = f n and ∑ ∞ i = D ( f n ) converge uniformlyto S : U → C and g : U → C respectively. Then g = D ( S ) . In other words, the infinite sumcommutes with the differential operator D. Theorem 3.17.
The Bargmann-Fock sheaf F flatX , α is a sheaf of modules over (cid:0) C ω ¯ h X [[ ¯ h ]] , ⋆ α (cid:1) .Proof. Let O f = ∑ I , J a I , ¯ J y I ¯ y J be the Taylor-Fedosov series of a function f ∈ C ω ¯ h X ( U )[[ ¯ h ]] .We only need to construct a well-defined O f ⊛ e β /¯ h with β = β I y I and show that D B , α ( O f ⊛ e β /¯ h ) =
0. For every x ∈ U , we first prove the convergence of (cid:0) O f ( e β /¯ h ) (cid:1) | x . We choosea K -coordinate centered at x , and assume, without loss of generality, that the fixed holo-morphic index I =
0. Then the inequality (3.2) becomes || ( β ( x ) , · · · , β n ( x )) || ≤ r ( x ) under the standard norm on C n , and we obtain the following infinite sum: (cid:16) a J ¯ y J (cid:17) ( e β i y i /¯ h ) (cid:12)(cid:12) x = a J ¯ h | J | ∂ J ∂ y j · · · y j | J | ! ( e β i y i /¯ h ) (cid:12)(cid:12) x = ∑ J a J ( x ) β j ( x ) · · · β j | J | ( x ) · e β i y i /¯ h (cid:12)(cid:12) x . ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 17
The absolute convergence of this series follows from the inequality (3.2) and Lemma 3.15.Now convergence of the series (3.4) and Lemma 3.16 imply that the connection D B , α com-mutes with the infinite sum, thus we have D B , α ( O f ⊛ e β /¯ h ) = (cid:3)
4. F
ORMAL B EREZIN -T OEPLITZ QUANTIZATIONS
In this section, we consider the case when X is pre-quantizable and equipped with aprequantum line bundle L . We show that the Bargmann-Fock sheaf in this situation givesrise to a micro-local description of the asymptotics of Toeplitz operators on holomorphicsections of L ⊗ k . This can be generalized to all Wick type star products on K¨ahler mani-folds whose Karabegov forms are real analytic.4.1. Formal Toeplitz operators.
We first give a brief review of formal Hilbert spaces and the associated formal Toeplitzoperators defined in [5]. First of all, the Wick algebra has the following analytic interpre-tation. In the flat case when X = C n (and with trivial prequantum line bundle L ), theHilbert space on which the Toeplitz operators act is the well-known Bargmann-Fock space H L ( C n , µ ¯ h ) , which consists of L integrable entire holomorphic functions with respect tothe density µ ¯ h ( z ) = ( π ¯ h ) − n e −| z | /¯ h ; here ¯ h is regarded as a positive real number.It is easy to see, by direct computations, that the holomorphic polynomials z I p I !¯ h | I | ,where I runs over all multi-indices, form an orthonormal basis of H L ( C n , µ ¯ h ) . Toeplitzoperators associated to polynomials are defined by multiplying by a polynomial f ∈ C [ z , ¯ z ] , which is in general non-holomorphic, and then projecting back to the holomorphicsubspace. For example, when n =
1, we have T z = m z , T ¯ z = ¯ h ddz , T f ( z ) f ( ¯ z ) = f (cid:18) ¯ h ddz (cid:19) ◦ m f ( z ) ,For any f , g ∈ C [ z , ¯ z ] , we have T f ◦ T g = T f ⋆ g .By regarding ¯ h as a formal variable instead, we can interpret the T f ’s as Toeplitz opera-tors on W C n , where the formal inner product is defined using Feynman graph expansions: h f , g i : = h n · Z f ¯ g · e −| y | h ∈ C [[ ¯ h ]] .More generally, we may allow perturbations of the Gaussian measure e −| y | /¯ h by interac-tion terms , and define formal Hilbert spaces: Definition 4.1.
Suppose that all the terms in φ ( y , ¯ y ) ∈ W C n have weight at least 3. Thenfor f , g ∈ W C n (( √ ¯ h )) , we define their formal inner product as the formal integral(4.1) h f , g i : = h n · Z f ¯ g · e −| y | + φ ( y , ¯ y ) ¯ h ∈ C (( √ ¯ h )) , which is in turn defined using Feynman graph expansions. Definition 4.2.
The orthogonal projection π φ : W C n → F C n = C [[ y , · · · , y n ]][[ ¯ h ]] is defined by requiring that h f , y I i = h π φ ( f ) , y I i for all multi-indices I ; here h− , −i is the formal inner product defined in (4.1).The formal Toeplitz operator T f , φ associated to f ∈ W C n is defined as the composition ofmultiplication by f and the orthogonal projection π φ : T f , φ : = π φ ◦ m f . Proposition 4.3 (Theorem 2.2 and its proof in [5]) . For any f ∈ W C n , there exists a uniqueO f in the Wick algebra such that (1) T f , φ ( s ) = O f ⋆ s, for any s ∈ F C n ; (2) Let f be a monomial, then the leading term of O f is exactly f , i.e.,O f = f + · · · , where the dots denote terms of degree greater than deg ( f ) . (3) O f is the unique solution of the following equation: (4.2) f · e φ /¯ h = e φ /¯ h ⋆ O f .4.2. Prequantum Bargmann-Fock sheaves.
In this subsection, we consider the following sheaf of Bargmann-Fock modules: F X , α = W X , e ⊗ O X , e L − √− ω /¯ h ,which we call a prequantum Bargmann-Fock sheaf ; here α = − ¯ h Ric X = − ¯ h · R ki ¯ jk dz i ∧ d ¯ z j .When X is pre-quantizable, we will explain in the next subsection how this prequantumBargmann-Fock sheaf describes the micro-local behavior of the Berezin-Toeplitz operatorson holomorphic sections of L ⊗ k .We choose a holomorphic frame e x of L − √− ω /¯ h satisfying the following condition: ∇ L − √− ω /¯ h ( e x ) = − h ∂ρ x ⊗ e x ,and define a local section of the holomorphic Weyl bundle W X by β = ∑ k ≥ ( ˜ ∇ ) k ( ρ x ) .The following theorem, whose proof will be given in Appendix B, describes some localflat sections of the prequantum Bargmann-Fock sheaf: Theorem 4.4.
Suppose a section of the prequantum Bargmann-Fock sheaf is of the form A · e β /¯ h ⊗ e x around x . Then D B , α (cid:0) A · e β /¯ h ⊗ e x (cid:1) = if and only if D K ( A ) = , or equivalently, A = J s for some holomorphic function s. ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 19
Proposition 4.5.
There exists a neighborhood U of x , such that (4.3) e β /¯ h ⊗ e x ∈ F flatX , α ( U ) . Proof.
By Theorem 4.4, D B , α (cid:0) e β /¯ h ⊗ e x (cid:1) =
0. Thus we only need to prove that the bound-edness condition (3.2) is satisfied. From the Taylor expansion (2.2) of ρ x and the definitionof β , we have β | x =
0. In particular, β i y i | x =
0. Thus the lower semi-continuity of r ( x ) implies that the condition (3.2) holds in a neighborhood U of x . (cid:3) Remark . Proposition 4.5 shows that for a small enough open set U ⊂ X , the space F flat X , α ( U ) is not empty. This can be easily generalized to general Bargmann-Fock sheaves. Proposition 4.7.
There exists a subspace V x of the stalk ( F flatX , α ) x , which is isomorphic to thespace of germs of formal holomorphic functions at x , i.e., V x ∼ = O X , x [[ ¯ h ]] , such that for everyneighborhood U of x , V x is a representation of ( C ω ( U )[[ ¯ h ]] , ⋆ ) .Proof. Let f ∈ C ω ( U ) , then O f ⊛ (cid:0) J s · e β /¯ h ⊗ e x (cid:1) must be of the form A · e β /¯ h ⊗ e x ∈F flat X , α ( U ) . Theorem 4.4 then implies that A = J s ′ for some s ′ ∈ O X ( U )[[ ¯ h ]] . By looking atthe germs of these sections of F flat X , α at x , we obtain the representation. (cid:3) Star products as formal Toeplitz operators.
Suppose that the K¨ahler manifold X is pre-quantizable with the prequantum line bun-dle given by ( L , ∇ L ) . We fix a K-coordinate system ( z , · · · , z n ) at x ∈ X , and also aholomorphic frame e L , x of L around x satisfying − log || e L , x || = ρ x (we call this the K-frame ). The local frame e x of L ω /¯ h is the asymptotics of the local frame e ⊗ kL , x of L ⊗ k as k → ∞ . The prequantum condition implies that ω = √− π ¯ ∂∂ log || e L , x || We first recall the following proposition in Tian’s paper [23] and, in particular, the no-tion of peak sections . We define the function r ( z ) : = p | z | + · · · + | z n | . Proposition 4.8 (Lemma 1.2 in [23]) . For a multi-index p = ( p , · · · , p n ) ∈ Z n + and an integerr > | p | = p + · · · + p n , there exists m > such that, for m > m , there is a holomorphicglobal section S, called a peak section , of the line bundle L ⊗ m , satisfying (4.4) Z X || S || h m dV g = Z X \{ r ( z ) ≤ log m √ m } || S || h m dV g = O (cid:18) m r (cid:19) , and locally at x under the K-coordinates, (4.5) S ( z ) = λ m , p · (cid:16) z p · · · z p n n + O ( | z | r ) (cid:17) e mL , x (cid:18) + O (cid:18) m r (cid:19)(cid:19) , where || · || h m is the norm on L ⊗ m given by h m , and O (cid:16) m r (cid:17) denotes a quantity dominated byC / m r with the constant C depending only on r and the geometry of X, moreover (4.6) λ − m , p = Z r ( z ) ≤ log m / √ m | z p · · · z p n n | · e − m · ρ x ( z ) dV g , where dV g = ω n n ! = ( √− ) n · h ( z , ¯ z ) · dz ∧ d ¯ z ∧ · · · ∧ dz n ∧ d ¯ z n is the volume form. We normalize the peak section S ( z ) in Proposition 4.8 and define S m , p , r : = λ m , p · (cid:18) + O (cid:18) m r (cid:19)(cid:19) · S = (cid:16) z p · · · z p n n + O ( | z | r ) (cid:17) · e mL , x These normalized peak sections S m , p , r are roughly speaking, global holomorphic sectionsof L ⊗ m such that(1) its norm is concentrated around a given point x on the K¨ahler manifold, and(2) modulo higher order terms, its Taylor expansion at x is z p · · · z p n n with respect tothe K-coordinates and K-frame at x .In [5], we constructed a formal Hilbert space H x as a sub-quotient of the vector spacegenerated by these normalized peak sections S m , p , r , and prove that this gives a nice rep-resentation of the Berezin-Toeplitz deformation quantization algebra. Let us briefly re-call the construction here. Let S m , p , r , S m , p , r be normalized peak sections of L ⊗ m with r >> | p | , | p | , we can identify them with holomorphic functions f , f with respect tothe K-frame e L , x . Then the following integral(4.7) m n · Z X h s , s i h m dV g = m n · Z X f ( z ) ¯ f ( z ) · e − m · ρ x ( z , ¯ z ) · √− ! n h ( z , ¯ z ) dz d ¯ z · · · dz n d ¯ z n is also concentrated around x , where it is a Gaussian integral. Thus these integrals haveasymptotics as m → ∞ given by Feynman graph expansions, for which we need to knowthe Taylor expansions of f , f , ρ x and log ( h ( z , ¯ z )) at x . For the first three functions, theirTaylor expansions at x are given by ( J f ) x , ( J f ) x and ( Φ ω ) x respectively. Lemma 4.9.
The purely (anti-) holomorphic derivatives of h vanishes at z under the K-coordinates: ∂ | I | h ∂ z I ( x ) = ∂ | J | h ∂ ¯ z I ( x ) = for all mutli-indices with | I | , | J | > , and h ( x ) = .Proof. The equality h ( x ) = x ; the proof for antiholomorphicones is the same. It suffices to show that the statement is valid for functions ω i ¯ j · · · ω i n ¯ j n ,where 2 √− · ω i ¯ j = ∂ ρ x ∂ z i ∂ ¯ z j , ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 21
The Taylor expansion of ρ x in equation (2.2) implies that ∂ | I | + ρ x ∂ z I ∂ ¯ z j ( x ) = | I | ≥ (cid:3) Since h ( z , ¯ z ) is the Hermitian metric on the anti-canonical line bundle K X induced bythe K¨ahler structure under the frame ∂ z ∧ · · · ∧ ∂ z n , there is ∂ ¯ ∂ log ( h ( z , ¯ z )) = R ki ¯ jk dz i ∧ d ¯ z j .Let α : = − ¯ h · ∂ ¯ ∂ ( log h ( z , ¯ z )) = − ¯ h · R ki ¯ jk dz i ∧ d ¯ z j . Then Lemma 4.9 implies that the Taylorexpansion of log ( h ( z , ¯ z )) at x under K -coordinates is exactly given by (cid:18) − h Φ α (cid:19) x .Let Φ : = | y | + √− · ( Φ ω ) x − ( Φ α ) x . Let ¯ h = m , then the asymptotic of the integral(4.7) is given by the following formal integral:1¯ h n · Z ( J f ) x · ( J ¯ f ) x · e ( √− · Φ ω /¯ h + Φ ∂ ¯ ∂ ( log ( h ( z ,¯ z )) ) x = h n · Z ( J f ) x · ( J ¯ f ) x · e −| y | + Φ ¯ h .By taking the asymptotics of S m , p , r ’s as m , r → ∞ , we can ignore the remainder terms andessentially get the monomial z p · · · z p n n . This is roughly how we can define the formalHilbert space: H x ∼ = C [[ z , · · · , z n ]][[ ¯ h ]] ,equipped with formal inner product defined via the above formal integral.The vector space V x in Proposition 4.7 is naturally a subspace of H x consisting of thoseformal power series which are convergent in some neighborhood of 0 ∈ C n . We have thefollowing theorem: Theorem 4.10.
The representation of C ω ¯ h ( X )[[ ¯ h ]] on V x defined in Proposition 4.7 is givenexplicitly as follows: Let f ∈ C ω ¯ h ( X )[[ ¯ h ]] and Ψ s : = J s · e β /¯ h ⊗ e x ∈ V x ⊂ ( F flatX , α ) x wheres ∈ O X , x [[ ¯ h ]] . Then O f ⊛ (cid:16) J s · e β /¯ h ⊗ e x (cid:17) = J s ′ · e β /¯ h ⊗ e x , where s ′ ∈ O X , x [[ ¯ h ]] is determined by its jets J s ′ at x explicitly given byT ( J f ) x , Φ ( J s ) = J s ′ . In other words, s ′ is obtained from the formal Toeplitz operation using f on s.Proof. It is easy to see that the action of W X , C on F X is linear over smooth functions.On the other hand, J s ′ determines s ′ since it is the Taylor expansion of the holomorphicfunction s ′ at x . We have seen that β | x = J s ′ = ( O f ) x ⊛ J s . On the other hand, Theorem 2.15 says that J f · e Φ /¯ h = e Φ /¯ h ⋆ O f .Comparing with equation (4.2), the result follows. (cid:3) It is not difficult to see from the above construction and computation that we do notneed to assume that X admits a prequantum line bundle. For every Wick type starproduct whose Karabegov form is real analytic, there is a subspace in the stalk of theBargmann-Fock sheaf, similar to V x above, such that formal smooth functions act on asformal Berezin-Toeplitz operators.A PPENDIX
A. P
ROOF OF T HEOREM Φ : = √− (cid:16) − ω i ¯ j y i ¯ y j + Φ ω (cid:17) − Φ α . It is clear that e Φ /¯ h : = + Φ /¯ h + ( Φ /¯ h ) + · · · is an invertible section in W + X , C under the Wick product, and we denote by (cid:0) e Φ /¯ h (cid:1) − itsinverse. Lemma A.1.
Let O be any section of the Weyl bundle. Then we have (A.1) ( ∇ − δ ) (cid:16) e Φ /¯ h ⋆ O ⋆ ( e Φ /¯ h ) − (cid:17) = e Φ /¯ h ⋆ D F , α ( O ) ⋆ ( e Φ /¯ h ) − . In other words, the operators D F , α and ∇ − δ differ by the gauge action by e Φ /¯ h .Proof. We will restrict our attention to the case where α =
0; the general case is similar.The operator (cid:0) ∇ − δ (cid:1) is a derivation with respect to both the classical and quantumproduct on W X , C , there is (cid:16) ∇ − δ (cid:17) e Φ /¯ h = h (cid:16) ∇ Φ − δ Φ (cid:17) · e Φ /¯ h = h ∑ k ≥ (cid:16) ∇ Φ ∗ , k − δ Φ ∗ , k (cid:17) · e Φ /¯ h = − h ( δ Φ ∗ ,2 ) · e Φ /¯ h + h ∑ k ≥ (cid:16) ∇ Φ ∗ , k − δ Φ ∗ , k + (cid:17) · e Φ /¯ h = − h e Φ /¯ h ⋆ (cid:16) δ ( Φ ∗ ,2 ) (cid:17) + √− h ω i ¯ j ∂ Φ ∂ y i · ∂∂ ¯ y j (cid:16) δ ( Φ ∗ ,2 ) (cid:17) · e Φ /¯ h + h ∑ k ≥ (cid:16) ∇ Φ ∗ , k − δ Φ ∗ , k + (cid:17) · e Φ /¯ h = − h e Φ /¯ h ⋆ (cid:16) δ ( Φ ∗ ,2 ) (cid:17) ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 23 + h ∑ k ≥ ∇ Φ ∗ , k − δ Φ ∗ , k + + √− ω i ¯ j ∂ Φ ∗ , k ∂ y i · ∂∂ ¯ y j (cid:16) δ ( Φ ∗ ,2 ) (cid:17) · e Φ /¯ h ! · e Φ /¯ h = − h e Φ /¯ h ⋆ (cid:16) δ ( Φ ∗ ,2 ) (cid:17) .In the last line, we have used the following identity: δ ( Φ m , n + ) = ∇ ( Φ m , n ) + √− ω i ¯ j ∂ Φ ∂ y i · ∂∂ ¯ y j (cid:16) δ ( Φ ∗ ,2 ) (cid:17) .Since e Φ /¯ h ⋆ ( e Φ /¯ h ) − =
1, it is easy to show that: ( ∇ − δ )( e Φ /¯ h ) − = h δ ( Φ ∗ ,2 ) ⋆ ( e Φ /¯ h ) − .Using the fact that ∇ − δ is a derivation with respect to ⋆ , we have ( ∇ − δ ) (cid:16) e Φ /¯ h ⋆ O ⋆ ( e Φ /¯ h ) − (cid:17) = e Φ /¯ h ⋆ ( − h δ ( Φ ∗ ,2 )) ⋆ O ⋆ ( e Φ /¯ h ) − + e Φ /¯ h ⋆ ( ∇ O − δ O ) ⋆ ( e Φ /¯ h ) − + e Φ /¯ h ⋆ O ⋆ h ( δ ( Φ ∗ ,2 )) ⋆ ( e Φ /¯ h ) − = e Φ /¯ h ⋆ (cid:18) ∇ O − δ O + h [ − δ ( Φ ∗ ,2 ) , O ] ⋆ (cid:19) ⋆ ( e Φ /¯ h ) − = e Φ /¯ h ⋆ D F , α ( O ) ⋆ ( e Φ /¯ h ) − .In the last line, we have used equation (2.6) to obtain: − δ ( Φ n ,2 ) = − √− · δ ( Φ ω ) n ,2 = − √− √− I n = I n . (cid:3) Proposition A.2.
Let O be any section of the Weyl bundle W X , C , and let O q (q for quantization)be the unique solution of the following equation:O · e Φ /¯ h = e Φ /¯ h ⋆ O q . Here Φ is the same as the previous part. Then there is the following identity describing an explicitrelation between the classical and quantum (Fedosov) connections: (A.2) D C ( A ) · e Φ /¯ h = e Φ /¯ h ⋆ D F , α ( A q ) . Proof.
Let A and B be sections of W X and W X respectively, then so are the D C ( A ) and D C ( B ) as the classical connection D C does not change the type in W X , C . For A , there is A q = A by type reason, and there is D C ( A ) · e Φ /¯ h = e Φ /¯ h ⋆ D C ( A q ) = e Φ /¯ h ⋆ D F , α ( A q ) . The last equality follows from the fact that the Fedosov connection D F , α equals D C whenrestricted to W X . For B , there is B ⋆ e Φ /¯ h = B · e Φ /¯ h = e Φ /¯ h ⋆ B q .By Lemma A.1, there is D C ( B ) =( ∇ − δ )( B ) = ( ∇ − δ ) (cid:16) e Φ /¯ h ⋆ B q ⋆ ( e Φ /¯ h ) − (cid:17) = e Φ /¯ h ⋆ D F , α ( B q ) ⋆ ( e Φ /¯ h ) − .In a similar fashion, we can show that D C ( B ) = e Φ /¯ h ⋆ D F , α ( B q ) ⋆ ( e Φ /¯ h ) − . A generalmonomial of W X , C must be a sum of the forms A · B . We first have the following: ( A · B ) · e Φ /¯ h = A · (cid:16) B · e Φ /¯ h (cid:17) = A · (cid:16) e Φ /¯ h ⋆ B q (cid:17) = e Φ /¯ h ⋆ (cid:0) B q ⋆ A (cid:1) ,which implies that ( A · B ) q = B q ⋆ A . And there is D C ( A · B ) · e Φ /¯ h = ( D C ( A ) · B + A · D C ( B )) · e Φ /¯ h = D C ( A ) · B · e Φ /¯ h + A · D C ( B ) · e Φ /¯ h = D C ( A ) · ( e Φ /¯ h ⋆ B q ) + A · ( e Φ /¯ h ⋆ D F , α ( B q ))= ( e Φ /¯ h ⋆ B q ) ⋆ D C ( A ) + A · ( e Φ /¯ h ⋆ D F , α ( B q ))= ( e Φ /¯ h ⋆ B q ) ⋆ D F , α ( A ) + ( e Φ /¯ h ⋆ D F , α ( B q )) ⋆ A = e Φ /¯ h ⋆ ( B q ⋆ D F , α ( A ) + D F , α ( B q ) ⋆ A )= e Φ /¯ h ⋆ D F , α ( B q ⋆ A )= e Φ /¯ h ⋆ D F , α ( A · B ) q . (cid:3) This proposition reduces the proof of Theorem 2.15 to showing that σ ( O f ) = f . Thisfollows the definition of O f and the fact that the section of Φ does not contain any non-trivial purely holomorphic or anti-holomorphic components. Thus we complete the proofof Theorem 2.15. A PPENDIX
B. P
ROOF OF T HEOREM D F , α | W X = D K , we have D B , α ( A · e β /¯ h ⊗ e x ) = D B , α ( A ⊛ e β /¯ h ⊗ e x )= D F , α ( A ) ⊛ ( e β /¯ h ⊗ e x ) + A ⊛ D B , α ( e β /¯ h ⊗ e x )= D K ( A ) · ( e β /¯ h ⊗ e x ) + A ⊛ D B , α ( e β /¯ h ⊗ e x ) .Hence, to prove the theorem, we only need to show that D B , α ( e β /¯ h ⊗ e x ) =
0. We firstrecall that α = − ¯ h · R ki ¯ jk dz i ∧ d ¯ z j . Lemma B.1.
We have ( J α ) n = − ( n + ) ¯ h · R iii ··· i n ,¯ l d ¯ z l ⊗ y i · · · y i n ARGMANN-FOCK SHEAVES ON K ¨AHLER MANIFOLDS 25
Proof.
The proof is by induction on n . For n =
1, we have ( J α ) = ( δ ) − (cid:16) − ¯ h · R ki ¯ jk dz i ∧ d ¯ z j (cid:17) = − h · (cid:18) R ki ¯ jk d ¯ z j ⊗ y i (cid:19) .Then by the induction hypothesis for n −
1, we have ∇ ( J α ) n − = ∇ (cid:16) − n ¯ h · R iii ··· i n − ,¯ l d ¯ z l ⊗ y i · · · y i n − (cid:17) .On the other hand, ∇ (cid:16) n ¯ h · R ji ··· i n ,¯ l d ¯ z l ⊗ y i · · · y i n ⊗ ∂ y j (cid:17) = ( n + ) · n ¯ h · R ji ··· i n + ,¯ l dz i n + ∧ d ¯ z l ⊗ y i · · · y i n ⊗ ∂ y j .Since ∇ is compatible between the contraction between TX and T ∗ X , the above com-putation shows that ( J α ) n = ( δ ) − ( ∇ ( J α ) n − ) = − ( n + ) ¯ h · R i i ··· i n + ,¯ l d ¯ z l ⊗ y i · · · y i n y i n + . (cid:3) Lemma B.2.
The section β satisfies D K ( β ) = √− ω i ¯ j d ¯ z j ⊗ y i − ∂ρ x . Proof.
The function ρ x satisfies the condition that ∂ ¯ ∂ ( ρ x ) = − √− ω . Recall that β = ∑ k ≥ ( ˜ ∇ ) k ( ρ x ) . A straightforward computation shows that D K ( β ) =( − δ + ¯ ∂ )( ˜ ∇ ρ x ) = − ∂ρ x + ¯ ∂ ◦ ( δ ) − ( ∇ ρ x )= − ∂ρ x + ( δ ) − ( ¯ ∂∂ρ x ) = √− ω i ¯ j d ¯ z j ⊗ y i − ∂ρ x . (cid:3) We also have the following:1¯ h I n ⊛ ( e β /¯ h ⊗ e x )= − √− · R ji ··· i n ,¯ l ω j ¯ k d ¯ z l ⊗ ( y i · · · y i n ¯ y k ) ⊛ ( e β /¯ h ⊗ e x )= − √− · R ji ··· i n ,¯ l ω j ¯ k d ¯ z l ⊗ ( ω i ¯ k √− ∂∂ y i )( y i · · · y i n e β /¯ h ⊗ e x )= R ii ··· i n ,¯ l d ¯ z l ⊗ y · · · y n ∂ ( β /¯ h )) ∂ y i · ( e β /¯ h ⊗ e x ) + n · R iii ··· i n − ,¯ l d ¯ z l ⊗ y i · · · y i n − · ( e β /¯ h ⊗ e x )= ˜ R ∗ n ( β /¯ h ) + n · R iii ··· i n − ,¯ l d ¯ z l ⊗ y i · · · y i n − · ( e β /¯ h ⊗ e x ) .Summarizing the above computations, we have D B , α ( e β /¯ h ⊗ e x )= (cid:18) ∇ + h γ α ⊛ (cid:19) ( e β /¯ h ⊗ e x ) + e β /¯ h ⊗ ∇ L ω /¯ h e x = ∇ ( β /¯ h ) + √− h ω i ¯ j ( dz i ⊗ ¯ y i − d ¯ z j ⊗ y i ) ⊛ + h ( I + J α ) ⊛ ! ( e β /¯ h ⊗ e x ) + e β /¯ h ⊗ ∇ L ω /¯ h e x = ∇ ( β /¯ h ) − √− h ω i ¯ j d ¯ z j ⊗ y i + h ∂ρ x ! ( e β /¯ h ⊗ e x )+ h ( √− ω i ¯ j dz i ⊗ ¯ y j + I + J α ) ⊛ ( e β /¯ h ⊗ e x )= ∇ ( β /¯ h ) + ∑ n ≥ ˜ R ∗ n ( β /¯ h ) − √− h ω i ¯ j d ¯ z j ⊗ y i + h ∂ρ x + √− ω i ¯ j dz i ω k ¯ j √− ∂ ( β /¯ h ) ∂ y k ! ( e β /¯ h ⊗ e x )= ∇ ( β /¯ h ) + ∑ n ≥ ˜ R ∗ n ( β /¯ h ) − √− h ω i ¯ j d ¯ z j ⊗ y i + h ∂ρ x − δ ( β /¯ h ) ! ( e β /¯ h ⊗ e x )= h (cid:16) D K ( β ) − √− ω i ¯ j d ¯ z j ⊗ y i + ∂ρ x (cid:17) ( e β /¯ h ⊗ e x )= EFERENCES [1] V. Baranovsky, V. Ginzburg, D. Kaledin, and J. Pecharich,
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