Classification of polarized deformation quantizations

Joseph Donin

Abstract We give a classification of polarized deformation quantizations on a symplectic

manifold with a (complex) polarization.

Also, we establish a formula which relates the characteristic class of a polarized deformation quantization to its Fedosov class and the Chern class of the polarization.

Mathematics Subject Classifications (2000): 53D55. Key words: deformation quantization, star-product, symplectic manifold, polarization.

Contents

1 Introduction 2

2 Complex distributions 6 2.1 The Kostant-Hochschild-Rosenberg theorem in presence of a distribution . 7 2.2 Differential operators in presence of a distribution . . . . . . . . . . . . . . 9 2.3 Differential forms in presence of a distribution . . . . . . . . . . . . . . . . 10

3 ℂ-symplectic manifolds and their polarizations 11 3.1 ℂ-symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Deformations of a polarized symplectic manifold 12 4.1 Formal everything . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Local structure of deformed polarizations . . . . . . . . . . . . . . . . . . . 13 4.3 Action of formal automorphisms on a symplectic form . . . . . . . . . . . . 14 5 Polarized symplectic connection and characteristic class of a polarized

symplectic manifold 15 5.1 Polarized symplectic connection . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Characteristic class of a polarized symplectic manifold . . . . . . . . . . . 16

6 Deformations of Poisson brackets 17

7 Deformation quantization on a polarized symplectic manifold 19 7.1 Deformation quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.2 Polarized deformation quantization . . . . . . . . . . . . . . . . . . . . . . 21

8 Characteristic classes of PDQ’s and PSP’s 24 8.1 Extension class associated with a PDQ . . . . . . . . . . . . . . . . . . . . 24 8.2 Characteristic 2-form associated with a PSP . . . . . . . . . . . . . . . . . 25 8.3 Characteristic pairs for PSP’s and PDQ’s . . . . . . . . . . . . . . . . . . . 27

9 Existence of polarized deformation quantizations and relation between the extension and Fedosov classes of a PDQ 28 9.1 Some notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 9.2 The Fedosov algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 9.3 Lie subalgebras in W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9.4 Filtrations on W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 9.5 Fedosov’s construction in the Wick case . . . . . . . . . . . . . . . . . . . . 32 9.6 Existence of PSP’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10 The main theorem and corollaries 38

1 Introduction

Let (M, ω) be a symplectic manifold, T_{M} its complexified tangent bundle. It is known that classes of deformation quantizations on (M, ω) are in one-to-one correspondence with their Fedosov classes, the elements of ω + tH^{2}(M, ℂ[[t]]). The set ω + tH^{2}(M, ℂ[[t]]) may be interpreted in the following way. Let X be the set of formal closed 2-forms on M of the form ω + tω_{1} + t^{2}ω_{2} + · · ·. Let Aut(M) be the group of formal automorphisms of M of the form e^{tX}, where X = X_{0} + tX_{1} + t^{2}X_{2} + ⋯, X_{i} ∈ T_{M} is a formal vector field. Since ω is nondegenerate, it is easy to see that the orbit of an element ω_{t} ∈ X under the action of Aut(M) is ω_{t} + t · d(Γ(M, T_{M}^{∗})). It follows from this that ω + tH^{2}(M, ℂ[[t]]) may be identified with the set of orbits in X under the Aut(M)-action. So, the equivalence classes of deformation quantizations on (M, ω) are in one-to-one correspondence with the orbits in X.

In the paper, we extend that picture to polarized deformation quantizations (PDQ). Let (M, ω, P) be a polarized symplectic manifold, i.e. P is a Lagrangian integrable

subbundle of the complexified tangent bundle to M. A PDQ on (M, ω, P) is a pair (𝔸_{t}, 𝕆_{t}), where 𝔸_{t} is a deformation quantization on (M, ω) and 𝕆_{t} is a commutative tadically complete subalgebra of 𝔸_{t} such that 𝕆_{0} = O_{P}, the algebra of functions constant along P.

Let Y denote the set of pairs (ω_{t}, P_{t}), where ω_{t} ∈ X and P_{t} is a polarization of ω_{t} such that P_{0} = P. Our result is that the equivalence classes of PDQ’s on (M, ω, P) are in one-to-one correspondence with the orbits in Y under the Aut(M)-action. Let us describe this correspondence more precisely.

First, we show that any PDQ is equivalent to a polarized star-product (PSP). By a PSP we mean a triple, (C_{M}^{∞}[[t]], μ_{t}, O_{t}), where (C_{M}^{∞}[[t]], μ_{t}) is a star-product, O_{t} = O_{P_{t}}, the algebra of functions from C_{M}^{∞}[[t]] constant along a deformed polarization P_{t}, and the multiplication μ_{t} satisfies the condition: μ_{t}(f, g) = fg (the usual multiplication) for f ∈ O_{t}, g ∈ C_{M}^{∞}[[t]].

Further, we assign to any PSP (C_{M}^{∞}[[t]], μ_{t}, O_{t}) a pair (ω_{t}, P_{t}) ∈ Y in the following way. We put P_{t} = P_{O_{t}}, the sheaf of formal vector fields annihilating O_{t}. The form ω_{t} is equal, locally, to ∑_{i} dy_{i} ∧dx_{i}, where x_{i} ∈ O_{t}, y_{i} ∈ C_{M}^{∞}[[t]], i = 1, ..., \frac{1}{2} dim M, are Darboux coordinates with respect to [,] = [,]_{μ_{t}}, the commutator of μ_{t}; namely [x_{i}, x_{j}] = [y_{i}, y_{j}] = 0, [y_{i}, x_{j}] = δ_{ij}. It turns out that ω_{t} is well defined, i.e. does not depend on the choice of local Darboux coordinates. Denote the constructed map from PSP’s to Y by τ.

The map τ turns out to descend to an isomorphism between the set of classes of PDQ’s and the set [Y] of orbits in Y. So, we obtain the following commutative diagram of maps:

{PSP’s}  ⟶^{τ} Y
↓  ↓
{classes of PDQ’s} ⟶ [Y],

(1.1)

where the left downward arrow is an epimorphism and the bottom arrow is an isomorphism of sets.

We show that the top arrow τ is an epimorphism with the following properties.

1) Two PSP’s are equivalent if and only if their images with respect to τ lie on the same orbit.

2) Two PSP’s (C_{M}^{∞}[[t]], μ_{t}, O_{t}) and (C_{M}^{∞}[[t]], μ̃_{t}, Õ_{t}) have the same image with respect to τ if and only if O_{t} = Õ_{t} and [,]_{μ_{t}} = [,]μ̃_{t}.

3) Let (ω_{t}, P_{t}) = τ(C_{M}^{∞}[[t]], μ_{t}, O_{t}). Then, the 2-form

θ_{t} = ω_{t} + \frac{t}{2}tr(∇^{2}|_{P_{t}})  (1.2)

represents the Fedosov class of the star-product (C_{M}^{∞}[[t]], μ_{t}). Here ∇ is a connection on M preserving ω_{t}, P_{t}, and flat on P_{t} along P_{t}. We prove that such a connection always exists and for it tr(∇^{2}|_{P_{t}}) belongs to Γ(M, dP_{t}^{⊥}).

By definition, ω_{t} belongs to ω +tΓ(M, dP_{t}^{⊥}). So, it follows from (1.2) that the Fedosov class of the star-product (C_{M}^{∞}[[t]], μ_{t}) can be represented by a 2-form belonging to ω + tΓ(M, dP_{t}^{⊥}), as well. In particular, both (M, ω_{t}, P_{t}) and (M, θ_{t}, P_{t}) are formal polarized symplectic manifolds that are deformations of (M, ω, P).

Another consequence of (1.2) is the following one. Let 𝔸_{t} be a deformation quantization on (M, ω). Suppose its Fedosov class cl_{F} (𝔸_{t}) is represented by the 2-form θ_{t} that has a polarization P_{t}. Then 𝔸_{t} can be extended to a PDQ (𝔸_{t}, 𝕆_{t}), i.e. there exists a commutative subalgebra 𝕆_{t} ⊂ 𝔸_{t} isomorphic to O_{P_{t}}.

There is the following interpretation of the image [ω_{t}] of element ω_{t} in the ℂ[[t]]-module Γ(M, dP_{t}^{⊥})/d(Γ(M, P_{t}^{⊥})). Let F (μ_{t}, O_{t}) = {a ∈ C_{M}^{∞}[[t]]; [a, O_{t}]_{μ_{t}} ⊂ O_{t}}. Then, there is the following exact sequence of O_{t}-module and Lie algebra sheaves:

0 ⟶ O_{t} ⟶ F (μ_{t}, O_{t}) ⟶ Der(O_{t}) ⟶ 0.  (1.3)

According to [BB] and [BK], F (μ_{t}, O_{t}) is called an O_{t}-extension of Der(O_{t}). Equivalence classes of such extensions are described by their extension classes that are elements of Γ(M, dP_{t}^{⊥})/d(Γ(M, P_{t}^{⊥})). We show that [ω_{t}] is just the extension class of (1.3).

Analogously, −[tr(∇^{2}|_{P_{t}})] is the extension class of the extension

0 ⟶ O_{t} ⟶ T̃_{det(P_{t})} ⟶ Der(O_{t}) ⟶ 0,  (1.4)

where T̃_{det(P_{t}}) is the sheaf of O_{t}-differential operators of order at most one on the O_{t}-line bundle det(P_{t}).

Note that −tr(∇^{2}|_{P_{t}}) divided by 2π\sqrt{−1} represents the first Chern class of P, [KN]. So, formula (1.2) gives a relation between the Fedosov and extension classes of a PDQ.

Among results related to ours we mention the following.

In [RY], N. Reshetikhin and M. Yakimov considered the case of a real polarization on M defined by a Lagrangian fiber bundle M → B.

In [Ka1], A. Karabegov constructed star-products with separation of variables on Kähler manifolds. This case corresponds to two polarizations on M defined by holomorphic and anti-holomorphic vector fields. In the case of quantization on a Kähler manifold with separation of variables the class of ω_{t} in H^{2}(M, ℂ[[t]]) coincides with the class defined by Karabegov in [Ka1]. A formula relating the Karabegov and Fedosov classes in case of Kähler manifolds is found in [Ka2], see also [KS], [Ne].

Our proof of the existence of a polarized star-product associated with any orbit in Y uses the Fedosov method adapted for the case with polarization. The analogous method was applied by M. Bordemann and S. Waldmann, [BW], for constructing a quantization with separation of variables on a Kähler manifold.

Another approach to proving a formula relating the Fedosov and extension classes, using the Deligne classes, was presented in [BD]. Unfortunately, there is a deficiency in the proof of Lemma 4.3 of that paper relating the extension and Deligne classes, however, the proof becomes correct for PSP’s with the same polarization.

The paper is organized as follows.

In Section 2, we study cohomologies of the differential Hochschild complex on M in presence of a distribution. Also, we prove a version of the Kostant-Hochschild-Rosenberg theorem for functions constant along a distribution. We use these results latter in proving that any PDQ is equivalent to a PSP.

In Section 3, we introduce a notion of ℂ-symplectic manifold, which will be convenient for our consideration. This notion is a generalization of the notion of symplectic manifold. Namely, we suppose that symplectic form ω on a ℂ-symplectic manifold is a complex one and, locally, there exist complex Darboux coordinates with respect to ω. For a usual symplectic manifold, when ω is real, such coordinates exist by the Darboux theorem. In this section, we establish some facts on ℂ-symplectic manifolds with polarization. By a polarization of ω we mean a Lagrangian subbundle, P, of the complexified tangent bundle on M such that, locally on M, there exist Darboux coordinates x_{i}, y_{i}, i = 1, ..., \frac{1}{2} dim M, where x_{i} ∈ O_{P} for all i. So, (pseudo-)Kähler manifolds as well as purely real polarizations are included in our considerations. Note that from an analog of “Dolbeault Lemma” proved in [Ra] one can derive sufficient conditions for P to be a complex polarization of ω.

In Section 4, we study properties of formal (or deformed) polarized symplectic manifolds. In Section 5, we prove the existence of a polarized symplectic connection on a formal polarized symplectic manifold, (M, ω_{t}, P_{t}), and with the help of it introduce the characteristic class c̃_{1}(M, ω_{t}, P_{t}) of a polarized symplectic manifold.

In Section 6, we prove some technical statements related to deformations of Poisson brackets on M. Such deformations appear, in particular, as commutators of star-products.

In Section 7, we study properties of PDQ’s. In particular, we prove the important fact that any PDQ is equivalent to a PSP.

In Section 8, we define the extension class of a PDQ. Besides, we assign to any PSP an element of Y, and to any class of PDQ’s an orbit in Y. We prove that the later assignment is a monomorphism that, actually, is an isomorphism, as we show in the next section.

In Section 9, we prove that each element of Y corresponds to a PSP. To this end, we adapt the Fedosov method for constructing a PSP corresponding to a given pair (ω_{t}, P_{t}) ∈ Y. By this method, a polarized symplectic connection, ∇, extends to a Fedosov connection on the bundle of Weyl algebras on M. This connection has two scalar curvatures: the Weyl curvature, θ_{t}, and the Wick curvature that turns out to be just ω_{t}. We show that these curvatures differ from each other by \frac{t}{2}tr(∇^{2}|_{P_{t}}), which immediately proves (1.2). Note that in order to consider Weyl and Wick curvatures simultaneously on the same bundle of Weyl algebras, we realize latter as a quotient of the tensor algebra over the cotangent bundle to M. This realization was used in [Do] to give an algebraic version of Fedosov’s construction.

In Section 10, we formulate the main theorem collecting the results of the paper and give some corollaries.

Acknowledgments. I thank J.Bernstein, P.Bressler, B.Fedosov, A.Karabegov, and A.Mudrov for helpful discussions.

2 Complex distributions

For a smooth manifold M we will denote by C_{M}^{∞} the sheaf of complex valued smooth functions on M and by T_{M}^{ℂ} = T_{M} ⊗_{ℝ} ℂ the complexified tangent bundle on M.

We say that a set of smooth functions x_{i}, i = 1, ..., dim M, given on an open subset U ⊂ M form a system of (complex) coordinates on U, if dx_{i} are linearly independent at each point of U.

Since any 1-form on U can be uniquely written as ∑_{i} a_{i}dx_{i}, one can define vector fields ∂/∂x_{i} ∈ T_{M}^{ℂ} in the following way. If f is a function on U and df = ∑_{i} a_{i}dx_{i}, then (∂/∂x_{i})f = a_{i}.

Let Q be a subbundle in a complex vector bundle E over M. We denote by Q^{⊥} the subbundle in E^{∗}, the complex dual to E, orthogonal to Q. If sections e_{i} form a local frame in Q, we set (e_{i})^{⊥} = Q^{⊥}.

A (complex) distribution on a manifold M is a subbundle of T_{M}^{ℂ}.

Definition 2.1. A distribution P is said to be integrable if, locally on M, there exist (complex valued) functions f_{1}, …, f_{k} such that df_{1}, …, df_{k} give a local frame in P^{⊥}, i.e. df_{i} are linearly independent at each point and P = (df_{i})^{⊥}.

An integrable distribution P is obviously involutive, i.e. [P, P] ⊂ P.

Let P be an integrable distribution on M. We will denote by O_{P} the sheaf of functions on M constant along P, i.e. f ∈ O_{P} if and only if Xf = 0 for any vector field X ∈ P.

2.1 The Kostant-Hochschild-Rosenberg theorem in presence of a distribution

Let M be a smooth manifold. Let D^{n} be the sheaf of n-differential operators on M and D^{•} the corresponding Hochschild complex with differential d. Let ∧^{•}T be the complex of sheaves of polyvector fields on M with zero differential, T = T_{M}^{ℂ}.

There is the following “smooth” version of the Kostant-Hochschild-Rosenberg theorem, [Ko], Thm. 4.6.1.1.

Proposition 2.2. The natural embedding

∧^{•}T → D^{•}  (2.1)

is a quasiisomorphism of complexes. Moreover, if φ ∈ D^{n} is a Hochschild cocycle, then its alternation Alt(φ) is a polyvector field of ∧^{•}T cohomological to φ.

Proof. Arguments of this proof will be used also in proving the next proposition. The proposition is local on M, so it is enough to prove it replacing M by an open set U ⊂ M having complex coordinates x_{i}, i = 1, ..., dim M. Any differential operator on U may be uniquely presented as a polynomial in ∂/∂x_{i} with coefficients being smooth functions on U. Hence, D^{•} coincides over U with the complex C^{•}(T ). Here, for any vector bundle E, we denote by C^{•}(E) the complex (⊗^{•}Sym(E), d) with differential of the form

d : ⊗^{n}Sym(E) → ⊗^{n+1}Sym(E),
d(a_{1} ⊗ ... ⊗ a_{n}) = 1 ⊗ a_{1} ⊗ ... ⊗ a_{n}+

+ ∑_{i=1}^{n}(−1)^{i}a_{1} ⊗ ... ⊗ Δa_{i} ⊗ ... ⊗ a_{n} + (−1)^{n+1}a_{1} ⊗ ... ⊗ a_{n},

where Δ is the comultiplication in the symmetric algebra Sym(E) generated by the rule Δ(a) = a ⊗ 1 + 1 ⊗ a for a ∈ E.

One has the following well known statement (see, for example, the proof of Thm. 4.6.1.1. in [Ko]).

Lemma 2.3. Let E be a line bundle over M. Then the conclusion of Proposition 2.2 holds for the map

∧^{•}E → C^{•}(E).  (2.2)

Applying this lemma to E = T we prove the proposition.

Let (M, P) be a smooth manifold with integrable distribution. We call an n-chain ν ∈ D^{n} polarized if ν(a_{1}, ..., a_{n}) = 0 whenever a_{1}, ..., a_{n} ∈ O_{P}. We call ν strongly polarized, if ν(a_{1}, ..., a_{n}) = 0 whenever a_{1}, ..., a_{n−1} ∈ O_{P} and a_{n} ∈ C_{M}^{∞}.

Proposition 2.4. Let ν ∈ D^{2} be a polarized Hochschild 2-cochain such that dν is strongly polarized. Then, there exists a polarized differential operator b such that ν + db is strongly polarized.

Proof. Since subsheaves of polarized and strongly polarized cochains are subsheaves of C_{M}^{∞}-modules, it is enough to prove the proposition locally on M. So, let U be an open set with coordinates x_{i}, i = 1, ..., dim M such that ∂/∂x_{i}, i = 1, ..., k, form a local frame in P. Let Q be the subbundle in T generated by ∂/∂x_{i}, i = k + 1, ..., n. Thus, T = Q ⊕ P over U. There is the natural isomorphism of complex C^{•}(T ) with the tensor product of complexes C^{•}(Q, P) = C^{•}(Q) ⊗ C^{•}(P), so we can identify C^{•}(T ) with C^{•}(Q, P). Similarly, we identify the complex ∧^{•}(T ) with ∧^{•}(Q, P) = ∧^{•}(Q) ⊗ ∧^{•}(P).

Thus, the map (2.2) generates the map of complexes

∧^{•}(Q, P) → C^{•}(Q, P).  (2.3)

Complex C^{•}(Q, P) decomposes obviously into a direct sum of subcomplexes C_{k,l}^{•}(Q, P), where C_{k,l}^{•}(Q, P) consists of elements of total degree k with respect to P and l with respect to Q. The same is true for Λ^{•}(Q, P). Due to Proposition 2.2, the map (2.3) is a quasiisomorphism of bigraded complexes.

It is clear that an element of C^{n}(Q, P) is polarized, if it is a sum of tensor monomials having degree > 0 with respect to P. An element of C^{n}(Q, P) is strongly polarized if it is a sum of tensor monomials of the form a_{1} ⊗ ⋯ ⊗ a_{n} where a_{1} ⊗ ⋯ ⊗ a_{n−1} is polarized.

So, we may suppose that given ν is a polarized Hochschild cochain on U belonging to C^{2}(Q, P). It can be written as ν = ν_{0} + ν_{1}, where ν_{0} = ∑(a_{i} ⊗ c_{i}), the sum of all tensor monomials in ν such that a_{i} ∈ C^{1}(Q), c_{i} ∈ C^{1}(P). Let us denote b = ∑ a_{i}c_{i}. It is clear that ν′ = ν + db does not contain tensor monomials of that type. The proposition will be proved if we show that ν′ is strongly polarized. Let us prove this.

Suppose ν′ = ν′_{0} + ν′_{1}, where ν′_{0} is not strongly polarized and ν′_{1} is strongly polarized. Then ν′_{0} has the form

ν′_{0} = ∑(a_{i} ⊗ b_{i})(1 ⊗ c_{i}),  (2.4)

where (a_{i} ⊗ b_{i}) ∈ ⊗^{2}Sym(Q) and c_{i} ∈ Sym(P) are linearly independent. Besides, b_{i} are of degree > 0.

The element dν = dν′ = dν′_{0}+dν′_{1} is strongly polarized by hypothesis of the proposition, dν′_{1} being the coboundary of a strongly polarized element ν′_{1} is strongly polarized too. All summands of dν′_{0} with first two factors being of degree zero with respect to P are (d(a_{i} ⊗ b_{i}) + a_{i} ⊗ b_{i} ⊗ 1)(1 ⊗ 1 ⊗ c_{i}). These summands are not strongly polarized. Hence, ∑_{i}(d(a_{i}⊗b_{i})+a_{i}⊗b_{i}⊗1)(1⊗1⊗c_{i}) = 0. Since elements 1⊗1⊗c_{i} are linearly independent, it follows that d(a_{i} ⊗ b_{i}) = −a_{i} ⊗ b_{i} ⊗ 1 for all i, which is only possible if a_{i} ⊗ b_{i} = 0 for all i. It follows from (2.4) that ν′_{0} = 0. So, ν′ is equal to ν′_{1} which is strongly polarized.

2.2 Differential operators in presence of a distribution

Let (M, P) be a smooth manifold with integrable distribution. Let z_{i}, y_{j} be complex coordinates on an open set U ⊂ M such that P = (dz_{i})^{⊥}. Vector fields ∂/∂y_{j} form a local frame in P, since, by definition, O_{P} consists of functions a ∈ C_{M}^{∞} such that da has the form ∑_{i} a_{i}dz_{i}. Since da is closed, ∂a_{i}/∂y_{j} = 0 for all i, j, which implies that a_{i} = ∂a/∂z_{i} ∈ O_{P} for all i.

Let Q be a subbundle in T = T_{M}^{ℂ} generated by ∂/∂z_{j}, so that T = Q ⊕ P over U. The vector bundle T /P may be considered as the sheaf of derivations from O_{P} to C_{M}^{∞},

Der(O_{P}, C_{M}^{∞}). Locally, such derivations can be presented in the form ∑ b_{i}∂/∂z_{i}, b_{i} ∈ C_{M}^{∞}, i.e. as sections of Q. Denote by Der(O_{P}) the O_{P}-submodule of Der(O_{P}, C_{M}^{∞}) consisting of operators which take O_{P} to itself. It is clear that Der(O_{P}, C_{M}^{∞}) = C_{M}^{∞} ⊗_{O_{P}} Der(O_{P}). Locally, elements of Der(O_{P}) have the form ∑_{i} a_{i}∂/∂z_{i}, a_{i} ∈ O_{P}.

Let ∧^{•}(T /P) denote the complex of sheaves of polyvector fields on M from O_{P} to C_{M}^{∞}. Let D^{•}(O_{P}, C_{M}^{∞}) denote the restriction of the Hochschild complex D^{•} to O_{P}. So, the

sheaf D^{n}(O_{P}, C_{M}^{∞}) may be considered as the sheaf of n-differential operators from O_{P} to C_{M}^{∞}. Locally, elements of D^{1}(O_{P}, C_{M}^{∞}), the sheaf of differential operators from O_{P} to C_{M}^{∞}, can be presented as polynomials in ∂/∂z_{i} with smooth coefficients. So, locally on M, complex D^{•}(O_{P}, C_{M}^{∞}) is isomorphic to the complex C^{•}(Q) (see previous subsection).

We will need the following version of the Kostant-Hochschild-Rosenberg theorem.

Proposition 2.5. The natural embedding

∧^{•}(T /P) → D^{•}(O_{P}, C_{M}^{∞})  (2.5)

is a quasiisomorphism of complexes. Moreover, if φ ∈ D^{n}(O_{P}, C_{M}^{∞}) is a cocycle, then Alt(φ) is a polyvector field of ∧^{n}(T /P) cohomological to φ.

Proof. As follows from above, embedding (2.5) is locally isomorphic to the embedding

∧^{•}Q → C^{•}(Q).

Now the proposition follows from Lemma 2.3 when E = Q.

Remark 2.6. All conclusions of Propositions 2.2, 2.4, 2.5 remain true for global sections of the corresponding sheaves, since they are sheaves of C_{M}^{∞}-modules.

2.3 Differential forms in presence of a distribution

Let (M, P) be a smooth manifold with integrable distribution.

The sheaf P^{⊥} = (T /P)^{∗} of differential forms on M which being applied to vector fields from P give zero, may be written as C_{M}^{∞}dO_{P}.

Denote Ω_{O_{P}}^{1} = Hom_{O_{P}}(Der(O_{P}), O_{P}), the sheaf of 1-forms on O_{P}. It is clear that Ω_{O_{P}}^{1} = O_{P}dO_{P}.

Denote by Ω_{O_{P}}^{1cl} the subsheaf of closed forms of Ω_{O_{P}}^{1}.

Lemma 2.7. a) The sequence of sheaves

0 ⟶ ℂ ⟶ O_{P} ⟶^{d} Ω_{O_{P}}^{1cl} ⟶ 0  (2.6)

is exact.

b) The sequence of sheaves

0 ⟶ Ω_{O_{P}}^{1cl} ⟶ P^{⊥} ⟶^{d} dP^{⊥} ⟶ 0  (2.7)

is exact.

Proof. Let functions z_{i}, y_{j} form a local basis on M and P = (dz_{i})^{⊥}. Let us prove a). It is sufficient to establish the exactness at the third term of (2.6). Let α = ∑_{i} a_{i}dz_{i} ∈ Ω_{O_{P}}^{1cl}. Since α is closed, there exists, locally, f ∈ C_{M}^{∞} such that df = α. Since α does not contain terms of the form gdy_{j}, one has ∂f/∂y_{j} = 0 for all j. Hence, f ∈ O_{P}.

To prove b), it is sufficient to establish the exactness at P^{⊥}. To this end, suppose β = ∑_{i} b_{i}dz_{i} ∈ P^{⊥} and dβ = 0. Since β is closed, ∂b_{i}/∂y_{j} = 0 for all i, j. This means that all b_{i} ∈ O_{P}. Thus, β ∈ Ω_{O_{P}}^{1} and closed, i.e. β ∈ Ω_{O_{P}}^{1cl}.

Proposition 2.8. Let (M, P) be a smooth manifold with an integrable distribution. Then, there is the natural isomorphism

H^{1}(M, Ω_{O_{P}}^{1cl}) ⋍ Γ(M, dP^{⊥})/d(Γ(M, P^{⊥})).  (2.8)

Proof. This is an immediate consequence of the cohomological exact sequence for (2.7) and of H^{i}(M, P^{⊥}) = 0 for i > 0.

3 ℂ-symplectic manifolds and their polarizations

3.1 ℂ-symplectic manifolds

Definition 3.1. By a ℂ-symplectic manifold we mean a pair (M, ω), where M is a smooth manifold and ω a closed nondegenerate complex 2-form on M satisfying the following condition: each point of M has a neighborhood U and complex coordinates x_{i}, y_{i}, i = 1, ..., \frac{1}{2} dim M, on U such that the form ω on U can be presented as

ω = ∑_{i} dy_{i} ∧ dx_{i},  (3.1)

If the form ω is real, such a presentation is possible by the Darboux theorem. One has

{x_{i}, x_{j}} = {y_{i}, y_{j}} = 0, {y_{i}, x_{j}} = δ_{ij}  (3.2)

for all i, j, where {·, ·} is the Poisson bracket inverse to ω. We call such functions x_{i}, y_{i} Darboux coordinates with respect to ω (or {·, ·}).

In what follows we only deal with ℂ-symplectic manifolds, so we simply call them symplectic ones.

3.2 Polarization

Definition 3.2. Let (M, ω) be a symplectic manifold. We call a (complex) distribution P on M a polarization of ω, if, locally, there exist (complex) Darboux coordinates, x_{i}, y_{i}, with respect to ω such that P = (dx_{i})^{⊥}, i.e. x_{i} ∈ O_{P}.

We call the triple (M, ω, P) a polarized symplectic manifold (PSM).

It follows that a polarization of ω is, in particular, an integrable distribution and a Lagrangian subbundle with respect to ω.

Proposition 3.3. Let (M, ω, P) be a PSM. Then ω ∈ Γ(M, dP^{⊥}).

Proof. Let x_{i}, y_{i} be local Darboux coordinates on M such that P = (dx_{i})^{⊥} and ω = ∑ dy_{i} ∧ dx_{i}. Then, locally, ω = d(∑ y_{i}dx_{i}) and ∑ y_{i}dx_{i} ∈ P^{⊥}.

Proposition 3.4. Let (M, ω, P) be a PSM. Then, O_{P} is a maximal commutative Lie subalgebra in C_{M}^{∞} with respect to the Poisson bracket ω^{−1}.

Proof. It follows from Definition 3.2 that, locally, the bracket ω^{−1} may be written in the form

{·, ·} = ∑_{i} ¯∂_{i} ∧ ∂_{i},  (3.3)

where ∂_{i} = {y_{i}, ·}, ¯∂_{i} = {·, x_{i}}. The module O_{P} consists, locally, of elements g ∈ C_{M}^{∞} such that {g, x_{i}} = ¯∂_{i}g = 0 for all i. Putting two such elements g_{1}, g_{2} in (3.3), we obtain that {g_{1}, g_{2}} = 0. So O_{P} is commutative. The maximality of O_{P} is obvious. Indeed, if a ∈ C_{M}^{∞} commutes with O_{P}, then, in particular, {x_{i}, a} = 0 for all i, hence a ∈ O_{P}.

4 Deformations of a polarized symplectic manifold

4.1 Formal everything

Let t be a formal parameter. We will consider on M formal functions, formal vector fields, formal forms, etc., which are elements of C_{M}^{∞}[[t]], T_{M}^{ℂ}[[t]], Φ^{k}[[t]], etc. In the formal case all sheaves over M and their morphisms will be sheaves and morphisms of ℂ[[t]]-modules.

Let B be a sheaf over M. We call the map σ : B[[t]] → B, b_{0} + tb_{1} + ⋯ → b_{0}, the symbol map. For a subsheaf F_{t} ⊂ B[[t]], we denote F_{0} = σ(F_{t}) ⊂ B.

Let F_{t} be a subsheaf of B[[t]]. We call F_{t} a t-regular subsheaf if it is complete in t-adic topology, and tb ∈ F_{t}, b ∈ B[[t]] implies b ∈ F_{t}.

For a t-regular subsheaf F_{t} ⊂ B[[t]] the natural map F_{t}/tF_{t} → F_{0} is an isomorphism. In this case we call F_{t} a deformation of F_{0}.

Let B be a vector bundle over M. Then, a subsheaf F_{t} ⊂ B[[t]] is called a (formal) subbundle, if it is a t-regular subsheaf of C_{M}^{∞}[[t]]-modules.

All notions and statements above carry over to the formal case.

For example, a deformation of a distribution P on M is a subbundle, P_{t}, of T_{M}^{ℂ}[[t]] such that P_{0} = P. A system of (formal) coordinates on an open set U ⊂ M is a set of formal functions x_{i} = x_{i0} + tx_{1i} + ⋯, i = 1, ..., dim M, ℂ[[t]]-linearly independent at each point of U. A formal polarized symplectic manifold is a triple (M, ω_{t}, P_{t}), where ω_{t} is a formal symplectic form on M (i.e. a closed 2-form of the form ω_{t} = ω_{0} + tω_{1} + ⋯ with nondegenerate ω_{0} ) and P_{t} is a polarization of ω_{t} in sense of Definition 3.2, i.e., locally, there exist formal Darboux coordinates x_{i}, y_{i} with respect to ω_{t} such that P_{t} = (dx_{i})^{⊥}.

We say that (M, ω_{t}, P_{t}) is a deformation of a polarized symplectic manifold (M, ω, P), if (M, ω_{0}, P_{0}) = (M, ω, P).

The following proposition, which follows from Propositions 6.1 and 6.2 below, shows that any deformation of (polarized) symplectic manifold is a formal (polarized) symplectic manifold.

Proposition 4.1. a) Let ω_{t} be a formal closed 2-form. If ω_{0} admits, locally, Darboux coordinates, then there exist their lifts being formal Darboux coordinates for ω_{t}.

b) Let P_{t} be an integrable distribution and Lagrangian with respect to ω_{t}. Let P_{0} be a polarization of ω_{0}. Then P_{t} is a polarization of ω_{t}.

4.2 Local structure of deformed polarizations

It is clear that formal vector fields tX_{1} +t^{2}X_{2} +⋯ ∈ tT_{M}^{ℂ}[[t]] form a sheaf of pro-nilpotent Lie algebras. It follows that elements e^{tX}, X ∈ T_{M}^{ℂ}[[t]], form a sheaf of pro-unipotent Lie groups of formal automorphisms of M.

Let x_{i} be formal coordinates on U and a_{i}, i = 1, ..., dim M, arbitrary formal functions on U. Then there exists a derivation, D, of C_{M}^{∞}[[t]] that takes x_{i} to a_{i}. Such a derivation is D = ∑_{i} a_{i}∂/∂x_{i}. This implies the following

Lemma 4.2. Let x_{i}, x′_{i}, i = 1, ..., dim M, be two systems of formal coordinates on an open set U ⊂ M, and x_{i} = x′_{i} mod t. Then, there exists a formal automorphism on U that takes x_{i} to x′_{i}.

Proposition 4.3. a) Let P_{t} be an integrable distribution on M which is a deformation of a distribution P. Then, locally, there exists a formal vector field, X, such that e^{tX} gives an isomorphism P_{t} with P[[t]].

b) Let (M, ω_{t}, P_{t}) be a deformation of a polarized symplectic manifold (M, ω, P). Then, each point of M has a neighborhood, U, and a formal vector field X on U such that e^{tX} gives an isomorphism of (M, ω_{t}, P_{t})|_{U} with the trivial deformation (U, ω, P[[t]]).

Proof. a) Locally, there exist functions x_{it} = x_{i0} + tx_{i1} + ⋯, i = 1, ..., k, such that P_{t} = (dx_{it})^{⊥} and hence P = (dx_{i0})^{⊥}. Let us add functions x_{(k+1)0}, ..., x_{n0} in such a way that all x_{i0}, i = 1, ..., n, form a coordinate system. According to Lemma 4.2, there exists a formal automorphism which takes coordinates x_{it}, i = 1, ..., k, x_{j0}, j = k + 1, ..., n, to coordinates x_{i0}, i = 1, ..., n. This formal automorphism gives obviously an isomorphism P_{t} onto P[[t]].

b) Let U be an open set in M where Darboux coordinates x_{it} = x_{i0} + tx_{i1} + ⋯, y_{it} = y_{i0} + tx_{i1} + ⋯, i = 1, ..., \frac{1}{2} dim M, exist, and P_{t} = (dx_{it})^{⊥}. Then, x_{i0}, y_{i0} are Darboux coordinates with respect to ω_{0} = ω such that P = (dx_{i0})^{⊥}. By Lemma 4.2 there exists a formal vector field X on U such that the formal automorphism e^{tX} takes coordinates x_{it}, y_{it} to x_{i0}, y_{i0}. Such X satisfies the conclusion of the proposition.

Let P be an integrable distribution. Denote by O_{P} the sheaf of functions constant along P. Let P_{t} be a deformation of P. It follows from the previous proposition that, locally, the pair (C_{M}^{∞}[[t]], O_{P_{t}}) is isomorphic to the pair (C_{M}^{∞}[[t]], O_{P}[[t]]), hence O_{P_{t}} is a t-regular subalgebra of C_{M}^{∞}[[t]]. One holds the following inverse assertion.

Proposition 4.4. Let P be an integrable distribution on M. Let O_{t} be a t-regular subalgebra of C_{M}^{∞}[[t]] such that O_{0} = O_{P}. Then, there exists a deformation, P_{t}, of P such that O_{t} = O_{P_{t}}.

Proof. Let us prove that dO_{t} ⊂ (T_{M}^{ℂ})^{∗}[[t]] is t-regular. Let b ∈ (T_{M}^{ℂ})^{∗}[[t]] and tb ∈ dO_{t}. Then, there exists a = a_{0} + ta′ ∈ O_{t} such that da = tb. It follows that a_{0} is a constant, so ta′ ∈ O_{t}. Since O_{t} is t-regular, a′ ∈ O_{t}, too. Therefore, b = da′ ∈ dO_{t}, so that dO_{t} is a tregular submodule in (T_{M}^{ℂ})^{∗}[[t]]. This implies that C_{M}^{∞}[[t]]dO_{t} is a subbundle in (T_{M}^{ℂ})^{∗}[[t]], so P_{t} = (dO_{t})^{⊥} is a subbundle in T_{M}^{ℂ}[[t]]. Moreover, P_{0} = P. So, P_{t} is a deformation of P. One has O_{t} ⊂ O_{P_{t}}. Since these two subalgebras are t-regular and coincide at t = 0, we have O_{t} = O_{P_{t}}.

The last proposition shows that there is a one-to-one correspondence between deformations of an integrable distribution P and deformations of O_{P}.

4.3 Action of formal automorphisms on a symplectic form

Let (M, ω_{t}) be a formal symplectic manifold, T = T_{M}^{ℂ}.

The well known formula for the Lie derivative

L_{X} = i(X)d + di(X)

implies

L_{X}ω_{t} = dα(X),  (4.1)

where α : T [[t]] → T^{∗}[[t]] is the map defined by X ↦ ω_{t}(X, ·). Since α is an isomorphism, we have the following lemma.

Lemma 4.5. The orbit of ω_{t} by the action of the group of formal automorphisms e^{tX}, X ∈ Γ(M, T [[t]]), is ω_{t} + td(Γ(M, T^{∗}[[t]])).

The lemma shows that the orbit of ω_{t} coincides with the cohomology class of ω_{t} in ω_{0} + tH^{2}(M, ℂ[[t]]).

Let (M, ω_{t}, P_{t}) be a formal PSM and X ∈ Γ(M, P_{t}). Since α(P_{t}) = P_{t}^{⊥}, one has L_{X}ω_{t} = dα(X) ∈ Γ(M, dP_{t}^{⊥}). The argument as above implies

Lemma 4.6. By the action of the group generated by e^{tX}, X ∈ Γ(M, P_{t}), the orbit of ω_{t} is ω_{t} + td(Γ(M, P_{t}^{⊥})).

The lemma shows that the orbit of ω_{t} corresponds to the cohomology class of ω_{t} in Γ(M, dP_{t}^{⊥})/d(Γ(M, P_{t}^{⊥})).

5 Polarized symplectic connection and characteristic class of a polarized symplectic manifold

5.1 Polarized symplectic connection

Let (M, ω, P) be a (formal) PSM. Denote, for shortness, T = T_{M}^{ℂ}[[t]].

Definition 5.1. We call a connection, ∇, on M a P-symplectic connection if a) it preserves ω and is torsion free, i.e. is a symplectic connection;

b) it preserves P, i.e. ∇_{X}(P) ⊂ P for any X ∈ T ;

c) it is flat on P along P, i.e. for any X, Y ∈ P one has (∇_{X}∇_{Y} −∇_{Y} ∇_{X}−∇_{[X,Y]})(P) = 0.

Proposition 5.2. Let (M, ω, P) be a (formal) PSM. Then, there exists a P-symplectic connection on M.

Proof. Let functions a_{1}, ..., a_{2n}, 2n = dim M, form local Darboux coordinates on an open set U ⊂ M and be such that a_{i} ∈ O_{P} for i = 1, ..., n. Let X_{i} = X_{a_{i}} be the corresponding Hamiltonian vector fields. Then, vector fields X_{i}, i = 1, ..., n, form a local frame in P. Also, all X_{i} commute and form a local frame in T . Let ∇ be the standard flat connection on U associated with coordinates a_{i}. This connection is defined on U by the rule ∇_{X_{i}}X_{j} = 0.

It is easy to see that ∇ is a P-symplectic connection on U. Moreover, since X_{f} ∈ P is equivalent to df ∈ P^{⊥}, the connection ∇ satisfies the following property for Hamiltonian vector fields:

∇_{X_{f}} X_{g} = 0 for X_{f}, Y_{g} ∈ P.  (5.1)

Now, let us prove the existence of a global connection.

Let {U_{α}} is an open covering of M such that on each U_{α} there is a P-symplectic connection ∇_{α} as above. Then ∇_{α} − ∇_{β} defined on U_{α} ∩ U_{β} form a Čech cocycle ψ_{α,β} ∈ Hom(T ⊗ T , T ), ψ_{α,β}(X, Y) = ∇_{αX}Y − ∇_{βX}Y.

Elements ψ_{α,β} satisfy the following properties.

As follows from (5.1),

ψ_{α,β}(X, Y) = 0 for X, Y ∈ P.  (5.2)

Since all ∇_{α} are torsion free, ψ_{α,β} are symmetric. Since all ∇_{α} preserve P, one has ψ_{α,β}(X, Y) ∈ P for Y ∈ P.

In addition, ψ_{α,β} considered as elements from Hom(T , Hom(T , T )), X ↦ ψ_{α,β}(X, ·), belong to Hom(T , 𝔰𝔭(T )), where 𝔰𝔭(T ) consists of endomorphisms of T preserving ω.

Since all the properties above are C_{M}^{∞}[[t]]-linear, one can find tensors ψ_{α} ∈ Hom(T ⊗ T , T ) satisfying all of them and such that ψ_{α} −ψ_{β} = ψ_{α,β}. Then ∇ = ∇_{α} −ψ_{α} = ∇_{β} −ψ_{β} is a globally defined connection. Flatness of ∇ on P along P follows from the fact that for all α tensors ψ_{α} satisfy property (5.2), i.e. ψ_{α}(X, Y) = 0 for X, Y ∈ P. Also, ∇ is torsion free because all ψ_{α} are symmetric. So, ∇ satisfies the proposition.

5.2 Characteristic class of a polarized symplectic manifold

Let ∇ be a P_{t}-symplectic connection on a formal polarized symplectic manifold (M, ω_{t}, P_{t}). Let us denote by ∇^{2}|_{P_{t}} the curvature of ∇ restricted to P_{t}. Then, tr(∇^{2}|_{P_{t}}) is a closed 2-form on M that represents, up to a constant factor, the first Chern class of P_{0}.

Lemma 5.3. Let ∇ be a P_{t}-symplectic connection. Then tr(∇^{2}|_{P_{t}}) belongs to Γ(M, dP_{t}^{⊥}). If ∇_{1} is another P_{t}-symplectic connection, then tr(∇_{1}^{2}|_{P_{t}}) differs from tr(∇^{2}|_{P_{t}}) by an element of d(Γ(M, P_{t}^{⊥})).

Proof. Follows from flatness of P_{t} along P_{t} with respect to the connection.

The lemma allows us to consider the element of Γ(M, dP_{t}^{⊥})/d(Γ(M, P_{t}^{⊥})) represented by the form tr(∇^{2}|_{P_{t}}), where ∇ is a P_{t}-symplectic connection, as a characteristic class of the polarized symplectic manifold (M, ω_{t}, P_{t}).

Due to Proposition 2.8, one can also consider this class as an element of H^{1}(M, Ω_{O_{P}_{t}}^{1cl}).

6 Deformations of Poisson brackets

In this section we prove three technical statements which we use through the paper.

Let π_{0} = {·, ·} be a nondegenerate Poisson bracket on a smooth manifold M of dimension 2n. We say that a formal sum π_{t} = π_{0} + tπ_{1} + ⋯ is a deformation of π_{0} if all π_{i} are bidifferential operators on M and π_{t} defines a Lie algebra structure on the sheaf C_{M}^{∞}[[t]]. We will also denote π_{t} by [·, ·]. Let us recall the symbol map σ : C_{M}^{∞}[[t]] → C_{M}^{∞}, a = a_{0}+ta_{1}+⋯ ↦ a_{0}. We call a a lift of a_{0}. We say that functions x̂_{i}, ξ̂_{i}, i = 1, ..., n on an open set U ⊂ M form Darboux coordinates with respect to [·, ·], if [x̂_{j}, x̂_{k}] = [ξ̂_{j}, ξ̂_{k}] = 0, [ξ̂_{j}, x̂_{k}] = δ_{jk} for all j, k. It is clear that then functions x_{i} = σ(x̂_{i}), ξ_{i} = σ(ξ̂_{i}) form Darboux coordinates with respect to {·, ·}.

Proposition 6.1. Let [·, ·] be a deformation of a Poisson bracket {·, ·} on M. Let x̂_{1}^{(i)} …, x̂_{n}^{(i)}, ξ̂_{1}^{(i)}, …, ξ̂_{n}^{(i)} ∈ C_{M}^{∞}[[t]], i = 1, 2, be two systems of Darboux coordinates with respect to [·, ·] on a contractible open set U in M satisfying

σ(x̂_{j}^{(1)}) = σ(x̂_{j}^{(2)}), σ(ξ̂_{j}^{(1)}) = σ(ξ̂_{j}^{(2)})

Then, there exists B ∈ C_{M}^{∞}[[t]] on U such that the automorphism Φ = exp(t·ad(B)), where ad(B) = [B, ·], satisfies Φ(x̂_{j}^{(1)}) = x̂_{j}^{(2)} and Φ(ξ̂_{j}^{(1)}) = ξ̂_{j}^{(2)}.

Proof. Let B_{0} = 0. Assume that B_{m} is such that the automorphism Φ_{m} = exp(t·ad(B_{m})) satisfies the conclusion of the proposition modulo t^{m+1}. This assumption is valid for m = 0.

Then, x̂_{j}^{(2)} = Φ_{m}(x̂_{j}^{(1)}) + t^{m+1}y_{j} mod t^{m+2}, ξ̂_{j}^{(2)} = Φ_{m}(ξ̂_{j}^{(1)}) + t^{m+1}η_{j} mod t^{m+2} for suitable y_{j}, η_{j} ∈ C_{M}^{∞}. The Darboux commutation relations for x̂_{1}^{(i)} …, x̂_{n}^{(i)}, ξ̂_{1}^{(i)}, …, ξ̂_{n}^{(i)} imply that the functions y_{1}, …, y_{n}, η_{1} …, η_{n} satisfy {x_{j}, y_{k}} − {x_{k}, y_{j}} = 0, {ξ_{j}, η_{k}} − {ξ_{k}, η_{j}} = 0, {ξ_{j}, y_{k}} − {x_{k}, η_{j}} = 0, where x_{j} = σ(x̂_{j}^{(1)}) = σ(x̂_{j}^{(2)}), ξ_{j} = σ(ξ̂_{j}^{(1)}) = σ(ξ̂_{j}^{(2)}). Equivalently, the differential form α = ∑_{j} y_{j}dx_{j} + η_{j}dξ_{j} is closed. By Poincaré Lemma there exists f ∈ C_{M}^{∞} such that df = α, equivalently y_{j} = {ξ_{j}, f}, and η_{j} = {f, x_{k}}.

There exists B_{m+1} ∈ C_{M}^{∞} such that

exp(t · ad(B_{m+1}) = exp(ad(t^{m+1}f)) ∘ exp(t · ad(B_{m}))

and B_{m+1} = B_{m} mod t^{m+1}. The limit B = lim_{m→∞}B_{m} exists and satisfies the conclusions of the proposition.

Proposition 6.2. Let [·, ·] be a deformation of a Poisson bracket {·, ·} on M and O_{t} a t-adically complete submodule in C_{M}^{∞}[[t]] being a commutative Lie subalgebra with respect to [·, ·]. Let functions x_{i} ∈ O_{0}, ξ_{i} ∈ C_{M}^{∞} form Darboux coordinates with respect to {·, ·} on a contractible open set U ⊂ M. Then, there exist their lifts x̂_{i} ∈ O_{t}, ξ̂_{i} ∈ C_{M}^{∞} on U which are Darboux coordinates with respect to [·, ·].

Proof. Since, by definition, O_{t} → σ(O_{t}) is surjective, we choose arbitrary lifts x̂_{j} ∈ O_{t} of x_{j}. Note that x̂_{j}, ξ_{j} satisfy the both conclusions modulo t.

Let x̂_{j,1} := x̂_{j}, ξ̂_{j,1} := ξ_{j}. Suppose that m ≥ 2 and x̂_{j} ∈ O_{t}, ξ̂_{j,m} ∈ C_{M}^{∞}[[t]] satisfies the conclusion of the proposition modulo t^{m+1}. The assumption on x_{j,m}, ξ_{j,m} implies that

[x̂_{j}, x̂_{k}] = 0,
[ξ̂_{j,m}, x̂_{k}] = δ_{jk} + y_{jk}t^{m+1} mod t^{m+2},
[ξ̂_{j}, ξ̂_{k}] = t^{m+1}z_{jk} mod t^{m+2}

for suitable y_{jk}, z_{jk} ∈ C_{M}^{∞}. The Jacobi identity and the commutation relations for x_{j}, ξ_{j} imply that

{y_{jk}, x_{l}} − {y_{jl}, x_{k}} = 0,
{z_{lj}, x_{k}} + {ξ_{j}, y_{lk}} − {ξ_{l}, y_{jk}} = 0,
{z_{jk}, ξ_{l}} + {z_{kl}, ξ_{j}} + {z_{lj}, ξ_{k}} = 0.

These identities say that the differential form

α = y_{jk}dx_{j} ∧ dξ_{k} + z_{jk}dx_{j} ∧ dx_{k}

is closed.

By the Poincaré Lemma there exists a 1-form β = ∑ a_{i}dξ_{i} + ∑ b_{i}dx_{i} on U such that dβ = α.

Note that {x_{k}, a_{i}} = 0 for all pairs i, k, since α contains no terms of the form fdξ_{i}∧dξ_{k}. Hence, a_{i} ∈ σ(O_{t}). Let â_{i} be a lift of a_{i} in O_{t}.

It is easy to check that x̂_{i,m+1} = x_{i,m}+t^{m+1}â_{i} and ξ̂_{i,m+1}+t^{m+1}b_{i} satisfy the conclusions of the proposition modulo t^{m+2}. Hence, the limits x̂_{i} = lim_{m→∞}x̂_{i,m}, ξ̂_{i} = lim_{m→∞}ξ̂_{i,m} exist and satisfy the conclusions of the proposition.

We will need a more strong assertion.

Proposition 6.3. Let [·, ·] be a deformation of a Poisson bracket {·, ·} on M and O_{t} a t-regular submodule in C_{M}^{∞}[[t]], which is a commutative Lie subalgebra with respect to [·, ·]. Let functions x_{i} ∈ O_{t}, ξ_{i} ∈ C_{M}^{∞}[[t]] form Darboux coordinates modulo t^{k}, k > 0, with respect to [·, ·] on a contractible open set U ⊂ M. Then, there exist functions a_{i} ∈ O_{t}, b_{i} ∈ C_{M}^{∞}[[t]] on U such that the functions x_{i} + t^{k}a^{i}, ξ_{i} + t^{k}b_{i} form Darboux coordinates with respect to [·, ·].

Proof. The same as of Proposition 6.2.

7 Deformation quantization on a polarized symplectic manifold

7.1 Deformation quantization

Let us recall some definitions and facts about the deformation quantization on a smooth manifold M, see [BFFLS], [De], [Fe].

Definition 7.1. a) Let C_{M}^{∞} be the sheaf of smooth complex valued functions on M. A formal deformation of C_{M}^{∞} is a sheaf of ℂ[[t]]-algebras, 𝔸_{t}, with an epimorphism σ : 𝔸_{t} → C_{M}^{∞} of ℂ[[t]]-algebras (called the symbol map) satisfying the condition: There exists an isomorphism of ℂ[[t]]-modules 𝔸_{t} → C_{M}^{∞}[[t]] commuting with symbol maps. (Recall that the symbol map σ : C_{M}^{∞}[[t]] → C_{M}^{∞} takes f_{0} + tf_{1} + ⋯ to f_{0}.)

b) Two formal deformations 𝔸_{t1} and 𝔸_{t2} of C_{M}^{∞} are equivalent if there exists a map of sheaves of ℂ[[t]]-algebras 𝔸_{t1} → 𝔸_{t2}, commuting with symbol maps.

Suppose 𝔸_{t} is a formal deformation of C_{M}^{∞}. The formula

\{a, b\} = σ(\frac{1}{t}[ã, b̃]),  (7.1)

where a and b are locally defined functions on M and ã, b̃ are their lifts with respect to σ, gives a well defined Poisson bracket on C_{M}^{∞}. It is clear that equivalent formal deformations have the same Poisson bracket.

Definition 7.2. A deformation quantization (DQ) on a symplectic manifold (M, ω) is a formal deformation of C_{M}^{∞} whose Poisson bracket is equal to ω^{−1}.

Definition 7.3. a) A star-product (SP) on (M, ω) is the structure of an associative algebra on the sheaf C_{M}^{∞}[[t]] with the multiplication of the form

f ∗ g = ∑_{i≥0} t^{i}μ_{i}(f, g),  f, g ∈ C_{M}^{∞},  (7.2)

where all μ_{i}, i > 0, are bidifferential operators on M vanishing on constants, i.e. μ_{i}(f, g) = 0 if f or g is a constant, μ_{0}(f, g) = fg, and μ_{1}(f, g)−μ_{1}(g, f) = {f, g}, the Poisson bracket inverse to ω.

b) Two star-products (C_{M}^{∞}[[t]], μ′) and (C_{M}^{∞}[[t]], μ′′) on (M, ω) are equivalent if there exists a power series B = 1 + tB_{1} + ⋯, where B_{i} are differential operators vanishing on constants, such that μ′′(f, g) = Bμ′(B^{−1}f, B^{−1}g).

It is clear that any SP (C_{M}^{∞}[[t]], μ) defines a DQ with the natural symbol map C_{M}^{∞}[[t]] → C_{M}^{∞}, f_{0} + tf_{1} + ⋯ ↦ f_{0}.

Proposition 7.4. The above assignment gives a one-to-one correspondence between the equivalence classes of SP’s and DQ’s.

Proof. Let 𝔸_{t} be a DQ. Let us prove that it is equivalent to a star-product. By definition of DQ, there exists an isomorphism of ℂ[[t]]-modules C_{M}^{∞}[[t]] → 𝔸_{t} commuting with symbol maps. Let μ = μ_{0} + tμ_{1} + ⋯ be the multiplication in the sheaf C_{M}^{∞}[[t]] being the pullback of the multiplication in 𝔸_{t}. In order to prove that each μ_{i} is a bidifferential operator it is enough to prove, according to the Peetre theorem, that supp(μ(f, g)) ⊂ supp(f)∩supp(g) for any functions f and g on M. But this is obvious because if, for example, f = 0 on an open set U, then μ(f, g) = 0 on U, since μ is a map of sheaves. The same argument proves that two equivalent DQ’s correspond to equivalent SP’s.

Example 7.5. Moyal-Weyl star-product. Let U be an open set in a symplectic manifold (M, ω), in which there exist Darboux coordinates, x_{i}, y_{i}, i = 1, ..., \frac{1}{2} dim M, so that ω = ∑_{i} dy_{i} ∧ dx_{i}. Then, ω^{−1} = ∑_{i} ∂/∂y_{i} ∧ ∂/∂x_{i} and for f, g ∈ C_{U}^{∞} the multiplication formula

f ⊗ g ↦ m exp (\frac{t}{2} ∑_{i}(∂/∂y_{i} ⊗ ∂/∂x_{i} − ∂/∂x_{i} ⊗ ∂/∂y_{i})) (f ⊗ g),  (7.3)

where m is the usual multiplication of functions, defines a star-product on (U, ω). This star-product is called the Moyal-Weyl star-product.

The following statement is well known and can be proven with the help of Hochschild and de Rham cohomology.

Proposition 7.6. Locally, any star-product on (M, ω) is equivalent to the Moyal-Weyl star-product.

7.2 Polarized deformation quantization

Definition 7.7. a) A polarized deformation quantization (PDQ) on a polarized symplectic manifold (M, ω, P) is a pair (𝔸_{t}, 𝕆_{t}) where 𝔸_{t} is a DQ on (M, ω) and 𝕆_{t} is a t-adically complete commutative subalgebra in 𝔸_{t} such that σ(𝕆_{t}) = O_{P}, functions constant along P.

b) Two PDQ’s (𝔸_{t1}, 𝕆_{t1}) and (𝔸_{t2}, 𝕆_{t2}) of (M, ω, P) are equivalent if there exists an equivalence map of DQ’s 𝔸_{t1} → 𝔸_{t2} which takes 𝕆_{t1} to 𝕆_{t2}.

Since any DQ is equivalent to a SP, any PDQ is equivalent to a triple (C_{M}^{∞}[[t]], μ_{t}, O_{t}), where (C_{M}^{∞}[[t]], μ_{t}) is a SP and O_{t} is a commutative subalgebra in (C_{M}^{∞}[[t]], μ_{t}).

Proposition 7.8. Let (𝔸_{t}, 𝕆_{t}) be a PDQ. Then, a) 𝕆_{t} is a maximal commutative subalgebra in 𝔸_{t}.

b) 𝕆_{t} is a t-regular subalgebra in 𝔸_{t}.

Proof. a) Locally, the bracket [a, b] = \frac{1}{t}(ab−ba), a, b ∈ 𝔸_{t}, is a deformation of the Poisson bracket ω^{−1}. So, the statement easily follows from Proposition 3.4.

b) Follows from a).

The following definition will play an auxiliary role in the paper.

Definition 7.9. A weakly polarized star-product (wPSP) on a polarized symplectic manifold (M, ω, P) is a triple (C_{M}^{∞}[[t]], μ_{t}, O_{t}), where (C_{M}^{∞}[[t]], μ_{t}) is a SP, O_{t} is a t-adically complete ℂ[[t]]-submodule in C_{M}^{∞}[[t]] satisfying the conditions:

a) σ(O_{t}) = O_{P};

b) O_{t} is a commutative subalgebra in C_{M}^{∞}[[t]] with respect to the usual multiplication in C_{M}^{∞}[[t]];

c) μ_{t} being restricted to O_{t} coincides with the usual multiplication.

Definition 7.10. We say that a wPSP (C_{M}^{∞}[[t]], μ_{t}, O_{t}) is a polarized star-product (PSP), if for any a ∈ O_{t}, b ∈ C_{M}^{∞}[[t]] the product μ_{t}(a, b) coincides with the usual product in C_{M}^{∞}[[t]].

So, both wPSP and PSP are particular cases of a PDQ. We are going to prove that, in fact, any PDQ is equivalent to a PSP. But first we prove the following

Lemma 7.11. Any PDQ is equivalent to a wPSP.

Proof. Since any deformation quantization is equivalent to a star-product, it is enough to prove the following. Let O_{t} be a commutative subalgebra in a star-product (C_{M}^{∞}[[t]], μ_{t}) such that the triple (C_{M}^{∞}[[t]], μ, O_{t}) forms a PDQ, in particular, μ_{t}(a, b) = μ_{t}(b, a) for a, b ∈ O_{t}. Then, there exists a differential operator D_{t} = 1 + tD_{1} + ... on M such that the multiplication μ̃_{t}(f, g) = D_{t}^{−1}μ(D_{t}f, D_{t}g) being restricted to D_{t}^{−1}O_{t} is the usual multiplication.

Suppose μ_{t} being restricted to O_{t} coincides with the usual multiplication modulo t^{n}. Then, μ(a, b) = ab + t^{n}ν(a, b) mod t^{n} for a, b ∈ O_{t}. It follows from Proposition 7.8 b) that being considered modulo t, the bilinear form ν defines a Hochschild cocycle ν̄ ∈ Γ(M, D^{2}(O_{P}, C_{M}^{∞})) (see Proposition 2.5).

Since ν̄ is commutative, it follows from Proposition 2.5 that it is a coboundary. So, there exists a differential operator D̄ ∈ Γ(M, D^{1}(O_{P}, C_{M}^{∞})) such that d_{Hoch}D̄ = ν̄.

Let D̃ be a lift of D̄ to a differential operator on M. Let D_{n} = 1 + t^{n}D̃. It is easy to see that D_{n}^{−1}O_{t} is a commutative subalgebra in the star-product (C_{M}^{∞}[[t]], μ̃_{t}) with μ̃_{t}(a, b) = D_{n}^{−1}μ_{t}(D_{n}a, D_{n}b), and μ̃_{t}(a, b) = ab modulo t^{n+1} for a, b ∈ D_{n}^{−1}O_{t}.

By induction, we construct such differential operators D_{n} for all n. Let D = Π_{n=1}^{∞}D_{n}, O′_{t} = D^{−1}O_{t}, and μ′_{t}(a, b) = D^{−1}μ_{t}(Da, Db). Then D gives an isomorphism between PDQ’s (C_{M}^{∞}[[t]], μ_{t}, O_{t}) and (C_{M}^{∞}[[t]], μ′_{t}, O′_{t}), and the second triple is a wPSP, which proves the proposition.

Proposition 7.12. For any wPSP (C_{M}^{∞}[[t]], μ_{t}, O_{t}), there exists a differential operator D = 1 + tD_{1} + ⋯ on M such that Df = f for f ∈ O_{t} and the multiplication μ′_{t}(a, b) = D^{−1}μ_{t}(Da, Db) defines a PSP (C_{M}^{∞}[[t]], μ′_{t}, O_{t}).

Proof. It is obvious that μ = μ_{t} defines a PSP modulo t. Proceeding by induction, we assume that there exists a wPSP multiplication μ′ equivalent to μ and being a PSP modulo t^{n} with respect to P_{t}. The proposition will be proved if we find a differential operator, D_{n}, such that D_{n}(f) = 0 for all f ∈ O_{0} and the multiplication

μ′′(a, b) = D^{−1}μ′(Da, Db),  (7.4)

where D = 1 + t^{n}D_{n}, defines a PSP modulo t^{n+1} with respect to P_{t}.

Let

μ′ = μ_{0} + tμ′_{1} + ⋯ + t^{n−1}μ′_{n−1} + t^{n}ν mod t^{n+1}.

By our assumption, elements μ′_{1}, ..., μ′_{n−1} are strongly polarized and ν is polarized with respect to O_{t} (see the definition before Proposition 2.4). It follows from associativity of μ′ that

d_{Hoch}ν(a, b, c) = ∑_{i+j=n}(μ′_{i}(a, μ′_{j}(b, c)) − μ′_{i}(μ′_{j}(a, b), c)).

It is easy to check that each term in the right hand side is strongly polarized, since all μ′_{i}, i = 1, ..., n − 1, are such. So, ν satisfies the hypothesis of Proposition 2.4. Hence, there exists a polarized differential operator D_{n} such that ν + d_{Hoch}D_{n} is a strongly polarized bidifferential operator. It is obvious that the multiplication μ′′ defined in (7.4) with D_{n} just constructed is as required.

Corollary 7.13. Any PDQ is equivalent to a PSP.

Proof. Follows from Lemma 7.11 and Proposition 7.12.

Proposition 7.14. a) Let (C_{M}^{∞}[[t]], μ_{t}, O_{t}) be a PSP on (M, ω, P). Let P_{t} = (dO_{t})^{⊥}. Then, P_{t} is a deformation of P and O_{t} = O_{P_{t}}.

b) Let (C_{M}^{∞}[[t]], μ̃_{t}, Õ_{t}) be another PSP on (M, ω, P) equivalent to (C_{M}^{∞}[[t]], μ_{t}, O_{t}) as a PDQ. Then, there exists a formal automorphism of M which takes P_{t} to P̃_{t} = (dÕ_{t})^{⊥}.

c) Let (C_{M}^{∞}[[t]], μ_{t}, O_{t}) and (C_{M}^{∞}[[t]], μ̃_{t}, O_{t}) are two equivalent PSP’s with the same O_{t}. Let D = 1 + tD_{1} + ⋯ gives an equivalence. Then, there exists a decomposition, D = D′e^{tX}, where D′ is a differential operator identical on O_{t} and X is a formal vector field on M taking O_{t} to itself.

Proof. a) follows from Proposition 4.4.

b). Let us put X_{0} = 0. Then the automorphism e^{tX_{0}} = Id takes P_{t} to P̃_{t} modulo t. Suppose we have constructed a formal vector field X_{k} such that the formal automorphism e^{tX_{k−1}} of M takes P_{t} to P̃_{t} modulo t^{k}. Then, replacing P_{t} by e^{tX_{k−1}}P_{t} we may assume that P_{t} and P̃_{t} coincide modulo t^{k}. The proposition will be proved, if we show that it is possible to find a vector field Y such that e^{t^{k}Y} takes P_{t} to P̃_{t} modulo t^{k+1}.

Since, by our assumption, the SP’s (C_{M}^{∞}[[t]], μ_{t}) and (C_{M}^{∞}[[t]], μ̃_{t}) are equivalent and coincide modulo t^{k}, there exists a differential operator 1+t^{k}D_{k} realizing that equivalence. Since 1 + t^{k}D_{k} takes O_{t} to Õ_{t} and on the both of these subalgebras the respecting multiplications μ_{t} and μ̃_{t} are trivial, D_{k} being restricted to O_{0} is a derivation from O_{0} to C_{M}^{∞}. Let Y be an extension of that restricted D_{k} to a derivation on C_{M}^{∞}. It is clear that e^{t^{k}Y} takes P_{t} to P̃_{t} modulo t^{k+1}.

c). The operator D being restricted to O_{t} is a formal automorphism of O_{t}. Since formal automorphisms form a pro-unipotent group, there exists X′ ∈ Der(O_{t}) such that D being restricted to O_{t} is equal to e^{tX′}. Let X ∈ Der(C_{M}^{∞}) be a lift of X′. We put D′ = De^{−tX} which is obviously identical on O_{t}.

Example 7.15. Moyal-Wick PSP. Let (M, ω, P) be a polarized symplectic manifold. Let U be an open set in M where there exist Darboux coordinates, x_{i}, y_{i}, (dx_{i})^{⊥} = P, i = 1, ..., \frac{1}{2} dim M, so that ω = ∑_{i} dy_{i} ∧ dx_{i}. Then, ω^{−1} = ∑_{i} ∂/∂y_{i} ∧ ∂/∂x_{i} and for f, g ∈ C_{U}^{∞} the multiplication formula

f ⊗ g ↦ m exp (t ∑_{i} ∂/∂y_{i} ⊗ ∂/∂x_{i}) (f ⊗ g),  (7.5)

where m is the usual multiplication of functions, defines a PSP on (U, ω, P). This PSP is called the Moyal-Wick polarized star-product.

Note that the functions a_{i}, f_{i} satisfying the Darboux relations with respect to the Poisson bracket ω^{−1} also satisfy the Darboux relations with respect to the deformed bracket \frac{1}{t}[·, ·], where [·, ·] is the commutator of the Moyal-Wick PSP.

Let us remark that the Moyal-Weyl SP from Example 7.5 constructed using the same Darboux coordinates gives just a wPSP but not a PSP.

Proposition 7.16. Locally, any PSP on (M, ω, P) is equivalent to the Moyal-Wick PSP.

Proof. Let us prove that any two PSP’s are locally equivalent. Since any SP’s are locally equivalent, we may suppose that there are given two PSP’s, (C_{M}^{∞}[[t]], μ_{t}, O_{t}) and (C_{M}^{∞}[[t]], μ_{t}, Õ_{t}), with the same multiplication and different polarizations, and we have to prove that they are, locally, equivalent. Let x_{i}, y_{i} are Darboux coordinates with respect to ω such that (dx_{i})^{⊥} = P. By Proposition 6.2, there exist their lifts x_{it}, y_{it} and x′_{it}, y′_{it} which satisfy the Darboux relations with respect to the bracket [a, b] = \frac{1}{t}(μ_{t}(a, b)−μ_{t}(b, a)), and x_{it} ∈ O_{t}, x′_{it} ∈ Õ_{t}. By Proposition 6.1, there exists, locally, an inner automorphism of the SP (C_{M}^{∞}[[t]], μ_{t}) that takes x_{it}, y_{it} to x′_{it}, y′_{it}. It follows that this automorphism takes O_{t} to Õ_{t}.

8 Characteristic classes of PDQ’s and PSP’s

8.1 Extension class associated with a PDQ

Let (𝔸_{t}, 𝕆_{t}) be a PDQ on a polarized symplectic manifold (M, ω, P). Since any PDQ is equivalent to a PSP, the sheaf 𝕆_{t} is isomorphic to O_{P_{t}} for some deformed distribution P_{t}. Thus, the sheaves Der(𝕆_{t}) and Ω_{𝕆_{t}}^{1cl} are well defined (see Section 2). Let

F (𝔸_{t}, 𝕆_{t}) = {b ∈ 𝔸_{t}; [b, 𝕆_{t}] ⊂ 𝕆_{t}},  (8.1)

where [·, ·] denotes the commutator in 𝔸_{t}. It is clear that F (𝔸_{t}, 𝕆_{t}) is a sheaf of Lie algebras with the bracket \frac{1}{t}[·, ·] and the center 𝕆_{t}. Moreover, any element b ∈ F (𝔸_{t}, 𝕆_{t}) determines the derivation \frac{1}{t}[b, ·] of 𝕆_{t} and, due to Proposition 6.2, this correspondence defines an epimorphism σ : F (𝔸_{t}, 𝕆_{t}) → Der(𝕆_{t}).

We consider F (𝔸_{t}, 𝕆_{t}) as a left 𝕆_{t}-module with respect to multiplication in 𝔸_{t}. As a Lie algebra sheaf, F (𝔸_{t}, 𝕆_{t}) is an extension of Der(𝕆_{t}).

So, we have the following exact sequence of Lie algebras and O_{t}-modules:

0 ⟶ 𝕆_{t} ⟶^{ι} F (𝔸_{t}, 𝕆_{t}) ⟶^{σ} Der(𝕆_{t}) ⟶ 0.  (8.2)

According to the terminology of [BB], [BK], F (𝔸_{t}, 𝕆_{t}) is called a 𝕆_{t}-extension of Der(𝕆_{t}).

We say that a map of Lie algebras and 𝕆_{t}-modules, s : Der(𝕆_{t}) → F (𝔸_{t}), given over an open set of M is a splitting of (8.2), if sσ = id. Since (𝔸_{t}, 𝕆_{t}) can be realized as a PSP and, locally, there exist Darboux coordinates with respect to \frac{1}{t}[·, ·], the sequence (8.2) locally splits (see the next subsection, where splittings are presented with the help of Darboux coordinates).

Lemma 8.1. Let s and s′ are two splittings of (8.2) over an open set of M. Then, s − s′ ∈ Ω_{𝕆_{t}}^{1cl}.

Proof. Direct calculation.

Let us define the extension class of (8.1) in the following way. Let {U_{α}} be an open covering of M such that over each U_{α} there is a splitting, s_{α}, of F (𝔸_{t}, 𝕆_{t}). By Lemma 8.1, f_{α,β} = s_{β} − s_{α} is a section of Ω_{𝕆_{t}}^{1cl} over U_{α} ∩ U_{β}. We define cl_{Ext}(F (𝔸_{t}, 𝕆_{t})) as the element of H^{1}(M, Ω_{𝕆_{t}}^{1cl}) represented by the collection {f_{α,β}}. One can prove that given 𝕆_{t}, the extension class cl_{Ext}(F (𝔸_{t}, 𝕆_{t})) determines a 𝕆_{t}-extension of Der(𝕆_{t}) up to equivalence.

We will denote the element cl_{Ext}(F (𝔸_{t}, 𝕆_{t})) by cl_{Ext}(𝔸_{t}, 𝕆_{t}) and call it the extension class of the PDQ (𝔸_{t}, 𝕆_{t}).

In the next subsection, the extension class of a PSP, (C_{M}^{∞}[[t]], μ_{t}, O_{t}), will be represented as an element of Γ(M, dP_{t}^{⊥})/d(Γ(M, P_{t}^{⊥})), P_{t} = P_{O_{t}}, with the help of a characteristic 2form associated with that PSP.

8.2 Characteristic 2-form associated with a PSP

Let (C_{M}^{∞}[[t]], μ_{t}, O_{t}) be a PSP that we denote for shortness by (μ_{t}, O_{t}). We denote P_{t} = P_{O_{t}}. Then the extension F (μ_{t}, O_{t}) coincides as a left O_{t}-module with a O_{t}-submodule of C_{M}^{∞}[[t]] with respect to the usual multiplication in C_{M}^{∞}[[t]]. In this case, the local splittings of F (μ_{t}, O_{t}) are differential forms of Hom_{O_{t}}(Der(O_{t}), C_{M}^{∞}[[t]]) = P^{⊥}. These forms can be described explicitly by Darboux coordinates.

Let {U_{α}} be an open covering of M such that each U_{α} has Darboux coordinates x_{αi}, y_{αi}, x_{αi} ∈ O_{t}, i = 1, ..., n, with respect to the bracket [a, b] = \frac{1}{t}(μ_{t}(a, b) − μ_{t}(b, a)) (in particular, [y_{αi}, x_{αj}] = δ_{ij}). By Proposition 6.2 such a covering exists. Then, on U_{α}, the O_{t}-submodule F (μ_{t}, O_{t}) ⊂ C_{M}^{∞}[[t]] is equal to O_{t} ⊕ O_{t}y_{α1} ⊕ ⋯ ⊕ O_{t}y_{αn}.

Splittings s_{α} may be taken by the condition x_{αi} ↦ y_{αi}, and the corresponding forms are

s_{α} = ∑_{i} y_{αi}dx_{αi}.  (8.3)

By Lemma 8.1, ds_{α} = ds_{β} on U_{α} ∩ U_{β}. So, forms s_{α} define the global 2-form

ω_{t} ∈ ω_{0} + tΓ(M, dP_{t}^{⊥}),  ω_{t} = ds_{α} = ∑_{i} dy_{αi} ∧ dx_{αi}.  (8.4)

Lemma 8.2. This form represents the extension class cl_{Ext}(μ_{t}, O_{t}) by the isomorphism (2.8).

Proof. Clear.

Lemma 8.1 shows that if we take other splittings of F (μ_{t}, O_{t}), the procedure above gives the same form ω_{t}. So, in the case of PSP, it is well defined not only the class cl_{Ext}(μ_{t}, O_{t}) but also its representative in ω_{0} + tΓ(M, dP_{t}^{⊥}) that we call the characteristic 2-form of the PSP and denote it by cl_{P SP} (μ_{t}, O_{t}).

Given a SP, (C_{M}^{∞}[[t]], μ_{t}), let us denote [a, b]_{μ_{t}} = \frac{1}{t}(μ_{t}(a, b) − μ_{t}(b, a)), a, b ∈ C_{M}^{∞}[[t]]. The bracket [,]_{μ_{t}} is a deformation of the initial Poisson bracket on M.

Proposition 8.3. Let (μ_{t}, O_{t}), (μ̃_{t}, O_{t}) be PSP’s with the same O_{t}. Then, cl_{P SP} (μ_{t}, O_{t}) = cl_{P SP} (μ̃_{t}, O_{t}) if and only if [,]_{μ_{t}} = [,]μ̃_{t}.

Proof. If cl_{P SP} (μ_{t}, O_{t}) = cl_{P SP} (μ̃_{t}, O_{t}), then forms (8.3) can be taken to be the same. This implies that there exist common Darboux coordinates with respect to [,]_{μ_{t}} and [,]μ̃_{t}.

Proposition 8.4. Let (μ_{t}, O_{t}), (μ̃_{t}, O_{t}) be two PSP’s with the same O_{t}. Let P_{t} = P_{O_{t}} and ΓP_{t} denote the module of global sections of P_{t}. Then, the 2-forms cl_{P SP} (μ_{t}, O_{t}), cl_{P SP} (μ̃_{t}, O_{t}) are lying on the same orbit of e^{tΓP_{t}} if and only if there exists a formal differential operator D = 1+tD_{1}+⋯ identical on O_{t} such that μ̃_{t}(a, b) = D^{−1}μ_{t}(Da, Db).

Proof. Let us denote ω_{t} = cl_{P SP} (μ_{t}, O_{t}), ω̃_{t} = cl_{P SP} (μ̃_{t}, O_{t}). Assume such D exists. Let s_{α} be forms from (8.3) corresponding to (μ_{t}, O_{t}). Let s̃_{α} be forms obtained from s_{α} by applying D. Lemmas 8.1 and 2.7 imply that there exist f_{α,β} ∈ O_{t} over U_{α} ∩ U_{β} such that s_{α} − s_{β} = df_{α,β}. Since D acts on O_{t} trivially, one has s̃_{α} − s̃_{β} = df_{α,β}, too. This means that s̃_{α} − s_{α} does not depend on α and give a global form, b, of P_{t}^{⊥}. Since ω_{t} = ds_{α}, ω̃_{t} = ds̃_{α}, one has ω̃_{t} = ω_{t} + db. By Lemma 4.6, there exists a formal automorphism, e^{tY}, Y ∈ ΓP_{t}, taking ω_{t} to ω̃_{t}.

Conversely, let us suppose that ω_{t} and ω̃_{t} are on the same orbit of e^{tΓP_{t}}. Assume that we have found a differential operator identical on O_{t} which transforms μ̃_{t} to a multiplication μ̃′_{t} that is equal to μ_{t} modulo t^{k}, i.e.

μ̃′_{t} − μ_{t} = t^{k}ν + · · · .  (8.5)

By the previous part of the proof, the corresponding form ω̃′_{t} is also lying on the same orbit as ω_{t}. The proposition will be proved if we find a differential operator 1 + t^{k}D_{k}, D_{k}(O_{t}) = 0, which transforms μ̃′_{t} to μ_{t} modulo t^{k+1}. Let us prove that.

Since μ̃′_{t} = μ_{t} mod t^{k}, we can choose, by Proposition 6.3, systems of Darboux coordinates with respect to [,]_{μ′_{t}} and [,]_{μ_{t}} that coincides modulo t^{k}. It follows that ω̃′_{t} = ω_{t} mod t^{k}. Hence, there is a formal automorphism e^{t^{k}X}, X ∈ P_{t}, which takes ω̃′_{t} to ω_{t}. Applying e^{t^{k}X} to μ̃′_{t}, we obtain a multiplication that is still equal to μ_{t} modulo t^{k} but whose characteristic form is equal to ω_{t}. So, we come to the situation when the multiplications μ̃′_{t} and μ_{t} have the same characteristic form ω_{t}.

By Proposition 8.3, μ̃′_{t} and μ_{t} have the same commutator. This implies that bidifferential operator ν in (8.5) is commutative. Moreover, it is a Hochschild strongly polarized cocycle (see Section 2). So, there exists a polarized differential operator D_{k} such that d_{Hoch}D = ν. It follows that transformation 1 + t^{k}D_{k} applying to μ̃′_{t} gives a multiplication equal to μ_{t} modulo t^{k+1} and identical on O_{t}.

8.3 Characteristic pairs for PSP’s and PDQ’s

Let (M, ω, P) be a polarized symplectic manifold. Let us denote by Aut(M) the group of formal automorphisms of M and by Y = Y(M, ω, P) the set of pairs (ω_{t}, P_{t}), where ω_{t} = ω_{0} + tω_{1} + ⋯ is a formal symplectic form being a deformation of ω = ω_{0} and P_{t} is a polarization of ω_{t} being a deformation of P. The group Aut(M) naturally acts on Y.

It is natural to assign to a PSP (μ_{t}, O_{t}) on (M, ω, P) a pair (ω_{t}, P_{t}) ∈ Y, where ω_{t} = cl_{P SP} (μ_{t}, O_{t}) and P_{t} = P_{O_{t}}. So, we obtain the map

τ : {PSP’s} → Y.

Proposition 8.5. Two PSP’s (μ_{t}, O_{t}) and (μ̃_{t}, Õ_{t}) are equivalent if and only if τ(μ_{t}, O_{t}) and τ(μ̃_{t}, Õ_{t}) are lying on the same orbit in Y under the Aut(M)-action.

Proof. Let (μ_{t}, O_{t}) and (μ̃_{t}, Õ_{t}) be equivalent. Let us prove that the pairs τ(μ_{t}, O_{t}) and τ(μ̃_{t}, Õ_{t}) are lying on the same orbit. By Proposition 7.14 b), c) one can find a formal automorphism of M such that after its applying we come to the situation when (μ̃_{t}, Õ_{t}) turns into a PSP, (μ̃_{t}, O_{t}), with the same O_{t} as in (μ_{t}, O_{t}) and the equivalence morphism from (μ_{t}, O_{t}) to (μ̃_{t}, O_{t}) is given by a differential operator identical on O_{t}. Now the statement follows from Proposition 8.4.

Conversely, suppose that for (μ_{t}, O_{t}) and (μ̃_{t}, Õ_{t}) the corresponding pairs (ω_{t}, P_{t}), (ω̃_{t}, P̃_{t}) lie on the same orbit. Let us prove that those PSP’s are equivalent. Let B be a formal automorphism of M sending P̃_{t} to P_{t}. Applying B to (μ̃_{t}, Õ_{t}), we come to the case when P̃_{t} = P_{t}. So, we may suppose that (ω̃_{t}, P̃_{t}) = (ω̃_{t}, P_{t}). Since the pairs (ω_{t}, P_{t}), (ω̃_{t}, P_{t}) lie on the same orbit, there exists X ∈ ΓP_{t} such that ω̃_{t} = e^{tX}ω_{t}. Now the statement follows from Proposition 8.4.

Let us denote by [Y] the set of orbits in Y = Y(M, ω, P).

Corollary 8.6. The map τ induces the embedding

τ̄ : {classes of PDQ’s} → [Y].

Proof. Let (𝔸_{t}, 𝕆_{t}) be a PDQ on (M, ω, P). Then, by Corollary 7.13, there exists a PSP, (μ_{t}, O_{t}), equivalent to (𝔸_{t}, 𝕆_{t}). We put τ̄(𝔸_{t}, 𝕆_{t}) = [τ(μ_{t}, O_{t})], the orbit passing through the pair τ(μ_{t}, O_{t}). By Propositions 7.14 and 8.5, this map is correctly defined, i.e. does not depends on the choice of an equivalent PSP. Proposition 8.5 also shows that being descended to equivalence classes of PDQ’s τ̄ is embedding.

In the next section we will prove that any element of Y is equal to τ(μ_{t}, O_{t}) for a PSP (μ_{t}, O_{t}), which implies that the map τ̄ is, in fact, an isomorphism.

9 Existence of polarized deformation quantizations and relation between the extension and Fedosov classes of a PDQ

Let (M, ω_{0}) be a symplectic manifold. It is known that all equivalence classes of deformation quantizations (DQ) on M with the Poisson bracket ω_{0}^{−1} can be obtained by the Fedosov method. According to this method, starting with a symplectic connection, one constructs a flat connection, D, (called the Fedosov connection) on the Weyl algebra that we define as a quotient of the tensor algebra over the cotangent bundle to M. The quantized algebra, 𝔸_{t}, is realized as the subalgebra of flat sections in the Weyl algebra. The Weyl curvature of D (see below), being a closed scalar 2-form of the form θ_{t} = ω_{0} + tω_{1} + ⋯, defines the Fedosov class

cl_{F} (𝔸_{t}) = [θ_{t}] ∈ [ω_{0}] + tH^{2}(M, ℂ[[t]]).  (9.1)

It is also known that the correspondence 𝔸_{t} ↦ cl_{F} (𝔸_{t}) is a bijection between the set of equivalence classes of DQ’s on (M, ω_{0}) and the set [ω_{0}] + tH^{2}(M, ℂ[[t]]), [Fe], [NT], [Xu].

Let (M, ω, P) be a polarized symplectic manifold and (ω_{t}, P_{t}) ∈ Y(M, ω, P) a deformation of the pair (ω, P), see Subsection 8.3. We adapt the Fedosov method to construct a PSP, (μ_{t}, O_{t}), such that τ(μ_{t}, O_{t}) = (ω_{t}, P_{t}). We start with a P_{t}-symplectic connection, ∇, and construct the Fedosov connection on the same Weyl algebra. We will see that by realizing the Fedosov scheme in presence of a polarization, the form ω_{t} appears as a so-called Wick curvature of the Fedosov connection. Moreover, ω_{t} differs from the Weyl curvature of that Fedosov connection by the form \frac{t}{2}tr(∇^{2}|_{P_{t}}).

9.1 Some notations

Let E be a formal vector bundle over M, i.e. a free C_{M}^{∞}[[t]]-module of finite rank over M. Denote by T^{k}(E) the k-th tensor power of E over C_{M}^{∞}[[t]] and by T (E) the corresponding tensor algebra completed in the {E, t}-adic topology. Similarly we define the completed symmetric algebra S(E). For a subbundle P of E, we denote by sym_{P} : S(P) → T (E) the natural embedding of C_{M}^{∞}[[t]]-modules defined by symmetrization.

Let Λ(E) be the exterior algebra of E over C_{M}^{∞}[[t]]. We will consider T (E) ⊗ Λ(E) as a graded super-algebra regarding a section x ∈ T (E) ⊗ Λ^{k}(E) of degree k even (odd) if k is even (odd).

Denote by δ_{T (E)} the continuous C_{M}^{∞}[[t]]-linear derivation of T (E) ⊗ Λ(E) defined by the map T^{1}(E) ⊗ 1 → 1 ⊗ Λ^{1}(E), v ⊗ 1 ↦ 1 ⊗ v, v ∈ E is a section. It is clear that δ_{T (E)} is a derivation of degree 1 and δ_{T (E)}^{2} = 0. It is easy to see that for any subbundle P ⊂ E, the map δ_{T (E)} being restricted to S(P) ⊗ Λ(P) via the embedding sym_{P} ⊗ id_{P} gives a derivation of the algebra S(P) ⊗ Λ(P); we denote it by δ_{P}.

On the algebra S(P) ⊗ Λ(P), there is another derivation, δ_{P}^{∗}, of degree −1 generated by the map 1 ⊗ Λ^{1}(P) → S^{1}(P) ⊗ 1, 1 ⊗ v → v ⊗ 1, v ∈ P. It is easy to check that (δ_{P}^{∗})^{2} = 0 and [δ_{P}, δ_{P}^{∗}] = δ_{P}δ_{P}^{∗} + δ_{P}^{∗}δ_{P} = deg, where deg is the derivation assigning to an element x ∈ S^{p}(P) ⊗ Λ^{q}(P) the element (p + q)x.

Let E be presented as a direct sum of C_{M}^{∞}[[t]]-submodules, E = P ⊕ Q. It is obvious that the derivations δ_{P}, δ_{P}^{∗}, δ_{Q}, δ_{Q}^{∗} induce derivations on the algebra S(P, Q) = (S(P) ⊗ S(Q)) ⊗ (Λ(P) ⊗ Λ(Q)) that we will identify in the natural way with the algebra (S(P) ⊗ S(Q)) ⊗ Λ(E). We put δ_{P,Q} = δ_{P} + δ_{Q} and δ_{P,Q}^{∗} = δ_{P}^{∗} + δ_{Q}^{∗}.

Let us define the operator δ_{P,Q}^{−1} on S(P, Q) in the following way. We put δ_{P,Q}^{−1}(x) = 0 for x ∈ C_{M}^{∞}[[t]] and δ_{P,Q}^{−1}(x) = (1/(p + r + q)δ_{P,Q}^{∗}(x) for x ∈ (S^{p}(P) ⊗ S^{r}(Q)) ⊗ Λ^{q}(E), p + r + q > 0. There is the obvious relation

δ_{P,Q}δ_{P,Q}^{−1} + δ_{P,Q}^{−1}δ_{P,Q} = projection on S^{+}(P, Q) along C_{M}^{∞}[[t]],  (9.2)

where S^{+}(P, Q) is the closure of ⊕_{p+r+q>0}(S^{p}(P) ⊗ S^{r}(Q)) ⊗ Λ^{q}(E).

One has the embedding

sym_{P} ⊗ sym_{Q} ⊗ id : (S(P) ⊗ S(Q)) ⊗ Λ(E) → T (E) ⊗ Λ(E).  (9.3)

It is clear that δ_{P,Q} coincides with the restriction of δ_{T (E)} to S(P, Q) via this embedding.

9.2 The Fedosov algebra

Let φ : E ⊗ E → C_{M}^{∞}[[t]] be a C_{M}^{∞}[[t]]-linear skew-symmetric nondegenerate form and I the closed ideal in T (E) generated by relations

x ⊗ y − y ⊗ x = tφ(x, y).  (9.4)

We call W(E) = T (E)/I the Weyl algebra and W = W(E) = W ⊗ Λ(E) the Fedosov algebra over E. The derivation δ_{T (E)} on T (E) ⊗ Λ(E) induces a derivation of W. Indeed, δ_{T (E)} applied to the both sides of (9.4) gives zero. We denote this derivation by δ.

Let E = P ⊕ Q be a decomposition into C_{M}^{∞}[[t]]-modules.

Define the Wick map, w_{P,Q}, as the composition S(P, Q) → T (E) ⊗ Λ(E) → W, where the first map is (9.3) and the second one is the projection. By the PBW theorem w_{P,Q} is an isomorphism of C_{M}^{∞}[[t]]-modules.

Due to the isomorphism w_{P,Q}, the operators δ_{P,Q} and δ_{P,Q}^{−1} are carried over from S(P, Q) to W. We retain for them the same notation. Note that while δ_{P,Q} does not depend on the decomposition of E and coincides with the derivation δ induced from T (E)⊗Λ(E), the operator δ_{P,Q}^{−1} is not a derivation and does depend on the decomposition. In particular, one can suppose that the decomposition is trivial, E = E ⊕ 0. In this case we denote δ_{E}^{−1} = δ_{E,0}^{−1}.

Proposition 9.1. One has

H(W, δ) = C_{M}^{∞}[[t]].

Moreover, if x ∈ W(E)⊗Λ^{k>0}(E) then y = δ_{P,Q}^{−1}x is such that δy = x for any decomposition E = P ⊕ Q.

Proof. Follows from (9.2).

9.3 Lie subalgebras in W

Let E = P ⊕ Q be a decomposition. We say that x ∈ W has w_{P,Q}-degree (p, q) if w_{P,Q}^{−1}(x) ∈ (S^{p}(P ⊗ S^{q}(Q)) ⊗ Λ(E). We say that x ∈ W has w_{P,Q}-degree k if w_{P,Q}^{−1}(x) ∈ ⊕_{p+q=k}(S^{p}(P)⊗S^{q}(Q))⊗Λ(E). We define the w_{E}-degree as the w_{E,0}-degree for the trivial decomposition E = E ⊕ 0.

Let 𝔤 be a sheaf of Lie algebras acting on E. We call a C_{M}^{∞}[[t]]-linear map λ : 𝔤 → W a realization of 𝔤, if it is a Lie algebra morphism (W is considered as a Lie algebra with respect to the commutator \frac{1}{t}[·, ·]) and for any x ∈ 𝔤 and v ∈ E one has x(v) = \frac{1}{t}[λ(x), v]. It is easy to check that any two realizations differ by a Lie algebra morphism of 𝔤 to the center of W, so if 𝔤 is a sheaf of semisimple Lie algebras, there is not more than one realization of 𝔤.

Denote by 𝔰𝔭(E) the sheaf of symplectic Lie algebras with respect to φ. Since 𝔰𝔭(E) is semisimple, there is a unique realization ρ_{E} : 𝔰𝔭(E) → W. The image of this realization consists of elements having w_{E}-degree two.

Let E = P ⊕ Q be a decomposition into Lagrangian subsheaves. Denote by 𝔰𝔭(P, E) the subsheaf of 𝔰𝔭(E) preserving P. It is easy to check that 𝔰𝔭(P, E) can be realized as the subset of elements of W having w_{P,Q}-degree (1, 1) and (2, 0). Denote this realization by ρ_{P,E} : 𝔰𝔭(P, E) → W. On the other hand, 𝔰𝔭(P, E) is realized in W by ρ_{E}.

Lemma 9.2. Let b ∈ 𝔰𝔭(P, E). Then

ρ_{E}(b) = ρ_{P,E}(b) + \frac{t}{2}trace(b̄),

where b̄ is b restricted to P.

Proof. Direct computation using the fact that ρ_{P,E}(b̄) is (1, 1) w_{P,Q}-degree component of ρ_{P,E}(b) in any decomposition E = P ⊕ Q.

9.4 Filtrations on W

We define two decreasing filtrations on W numbered by nonnegative integers.

The T -filtration F_{•}^{T} W is defined as follows. We ascribe to the elements of E degree 1 and to t degree 2. Then F_{n}^{T} W consists of elements of W having the leading term of total degree ≥ n.

The P-filtration, F_{•}^{P}W, is firstly defined on S(P) ⊗ S(Q) by the subsets F_{n}^{P} = ⊕_{i≥n}S^{i}(P) ⊗ S(Q), n = 0, 1, ..., and carried over to W via the Wick isomorphism.

We extend those filtrations to W in the natural way standing, for example, F_{n}^{T} W = F_{n}^{T} W ⊗ Λ(E). We will use the following mnemonic notation. To point out, for example, that a section x ∈ W belongs to F_{n}^{T} W we write F^{T} (x) ≥ n.

In the following we denote S(P) = S(P) ⊗ Λ(E) embedded in S(P, Q) as (S(P) ⊗ 1) ⊗ Λ(E).

Proposition 9.3. Let E = P ⊕ Q be a decomposition of E into Lagrangian subsheaves. Then

a) The Wick map w_{P,Q} : S(P, Q) → W has the following property: for a ∈ S(P) and arbitrary c ∈ S(P, Q) one has ac = w(a)w(c).

The filtrations on W have the properties:

b) for x, y ∈ W, if F^{P}(x) ≥ k, then F^{P}(xy) ≥ k;

c) F^{P}(δ_{P,Q}^{−1}x) ≥ F^{P}(x);

d) F^{T} (δ_{P,Q}^{−1}x) > F^{T} (x).

Proof. Clear.

9.5 Fedosov’s construction in the Wick case

Let (M, ω_{t}, P_{t}) be a formal polarized symplectic manifold. We will write for shortness ω = ω_{t}, P = P_{t}. Let us denote T = T_{M}^{ℂ}[[t]]. It is easy to prove that there exists on M a Lagrangian subbundle Q ⊂ T such that T = P ⊕ Q.

In the following we set E = T^{∗} and consider the Fedosov algebra W = W(E) with respect to φ being the Poisson bracket inverse to ω. The decomposition E = P^{⊥} ⊕ Q^{⊥} is Lagrangian with respect to this φ.

Let ∇ be a P-symplectic connection on M (see Section 5). Then the induced connection ∇ : E → E ⊗ Λ^{1}E on E preserves P^{⊥}, i.e. ∇(P^{⊥}) ⊂ P^{⊥} ⊗ Λ^{1}(E), and is flat on P^{⊥} along P, i.e. for any X, Y ∈ P one has (∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇_{[X,Y]})(P^{⊥}) = 0.

We will identify P with P^{⊥} and Q with Q^{⊥} by the isomorphism x ↦ ω(x, ·) between T and E. So, we will allow us to write E = P ⊕ Q.

The connection ∇ gives a derivation of the Fedosov algebra W, which is an extension of the de Rham differential on functions. Analogously, ∇ gives such derivations of the algebras T (E)⊗Λ(E), S(E)⊗Λ(E), and (S(P)⊗S(Q))⊗Λ(E). These derivations commute with the maps (9.3) and w_{P,Q}.

For convenience, we will mark the elements of the Fedosov algebra lying in E ⊗ 1 by letters with hat over them (x̂), while by dx we will denote the copy of x̂ lying in 1 ⊗ Λ^{1}E.

Let ω = ω_{ij}dx_{i} ∧ dx_{j} in some local coordinates. It is easy to check that for δ̃ = ω_{ij}x̂_{i} ⊗ dx_{j} one has

\begin{aligned}δ & = \frac{1}{t}ad(δ̃), \\ δ̃^{2} & = tω. \end{aligned}  (9.5)

Since the torsion of ∇ is equal to zero,

∇(δ̃) = 0.  (9.6)

Since ∇^{2} is a C_{M}^{∞}[[t]]-linear derivation of degree 0 preserving P, there is an element R ∈ ρ_{P,E}(𝔰𝔭(P, E)) ⊗ Λ^{2}(E) such that ∇^{2} = \frac{1}{t}ad(R). In particular, one has to be

F^{P}(R) ≥ 1.  (9.7)

According to Fedosov, [Fe], we also define R^{F} ∈ ρ_{E}(𝔰𝔭(P, E)) ⊗ Λ^{2}(E) satisfying ∇^{2} = \frac{1}{t}ad(R^{F}). It follows from (9.6)

δ(R) = δ(R^{F}) = 0.  (9.8)

Following to Fedosov, we will consider connections on W of the form

D = ∇ + \frac{1}{t}ad(γ),  γ ∈ W ⊗ Λ^{1}(E).  (9.9)

We define the Wick curvature of D as

Ω_{D} = R + ∇(γ) + \frac{1}{t}γ^{2}.

According to Fedosov, we also define the Weyl (or Fedosov) curvature of D as

Ω_{D}^{F} = R^{F} + ∇(γ) + \frac{1}{t}γ^{2}.  (9.10)

Since by Lemma 9.2 R^{F} = R + \frac{t}{2}tr(∇^{2}|_{P}), we have

Ω_{D}^{F} = Ω_{D} + \frac{t}{2}tr(∇^{2}|_{P}).  (9.11)

One checks

D^{2} = \frac{1}{t}ad(Ω_{D}) = \frac{1}{t}ad(Ω_{D}^{F}).  (9.12)

Let us take γ in the form

γ = δ̃+ r,  r ∈ W ⊗ Λ^{1}(E), F^{T} (r) ≥ 3.  (9.13)

Then the connection D has the form

∇ + δ + \frac{1}{t}ad(r).  (9.14)

Using (9.5) and (9.6), we obtain that its Wick curvature is

Ω_{D} = R + ∇(δ̃+ r) + \frac{1}{t}(δ̃+ r)^{2} = ω + δr + R + ∇r + \frac{1}{t}r^{2}.  (9.15)

Proposition 9.4. There exists an element r ∈ W(E) ⊗ Λ^{1}(E) such that a) F^{T} (r) ≥ 3;

b) F^{P}(r) ≥ 1;

c) the connection D = ∇ + δ + \frac{1}{t}ad(r) is flat, i.e. D^{2} = 0;

d) for its Wick curvature one has

Ω_{D} = ω;

e) its Weyl curvature Ω_{D}^{F} belong to Γ(M, dP^{⊥}) and there is the formula

Ω_{D}^{F} = ω + \frac{t}{2}tr(∇^{2}|_{P}).  (9.16)

Proof. First of all, we apply the Fedosov method, ([Fe], Theorem 5.2.2), to find r satisfying d). According to (9.15), r must obey the equation

δr = −(R + ∇r + \frac{1}{t}r^{2})  (9.17)

Let us look for r being the limit of the sequence, r = lim r_{k}, where r_{k} ∈ W(E) ⊗ Λ^{1}(E), k = 3, 4, ..., and F^{T} (r_{k} − r_{k−1}) ≥ k. As in Lemma 5.2.3 of [Fe], using Proposition 9.3 d) and the fact that F^{T} (R) ≥ 2, such r_{k} can be calculated recursively:

\begin{aligned}r_{3} & = −δ_{P,Q}^{−1}(R) \\ r_{k+3} & = −(r_{3} + δ_{P,Q}^{−1}(∇r_{k+2} + \frac{1}{t}r_{k+2}^{2})). \end{aligned}  (9.18)

So, a) and d) are proven.

Let us prove that F^{P}(r_{k}) ≥ 1 for all k. The inequality F^{P}(r_{3}) ≥ 1 follows from the fact that F^{P}(R) ≥ 1 and from Proposition (9.3) c). Suppose that F^{P}(r_{i}) ≥ 1 for i < k+3, k > 0. Then F^{P}(∇r_{k+2}) ≥ 1 because ∇ preserves P. On the other hand, F^{P}(r_{k+2}^{2}) ≥ 1 because of Proposition (9.3) b), therefore, as follows from (9.18), F^{P}(r_{k+3}) ≥ 1 as well. So, we have that r being the limit of the convergent sequence r_{k} satisfies the conditions a), b), and d) of the proposition.

c) obviously follows from d) and (9.12).

e) follows immediately from d), (9.11), and Lemma 5.3.

Proposition 9.5. Let ˜∇ be another P-symplectic connection on M. Let r̃ ∈ W(E) ⊗ Λ^{1}(E), satisfy the conclusions of Proposition 9.4, in particular, the connection D̃ = ˜∇ + δ + \frac{1}{t}ad(r̃) is flat. Then, there exists an element B ∈ W(E) such that

a) F^{T} (B) ≥ 3;

b) F^{P}(B) ≥ 1

c) e^{\frac{1}{t} adB}D = D̃.

Proof. Note that ∇− ˜∇ can be presented as \frac{1}{t}adR_{0}, where R_{0} ∈ ρ_{P,E}(𝔰𝔭(P, E)). Therefore, F^{T} (R_{0}) ≥ 2 and F^{P}(R_{0}) ≥ 1. Let us put R_{1} = r −r̃. Then, F^{T} (R_{1}) ≥ 3 and F^{P}(R_{1}) ≥ 1. We have

D̃ = D − \frac{1}{t}ad(R_{0} + R_{1}).

Since Ω_{D} = ΩD̃ = ω, using (9.6) we obtain

δ(R_{0}) = 0.

It follows that the element B_{0} = δ_{P,Q}^{−1}(R_{0}) is such that δ(B_{0}) = R_{0} and F^{T} (B_{0}) ≥ 3, F^{P}(B_{0}) ≥ 1.

Replacing D by e^{\frac{1}{t} adB_{0}}D we obtain

D̃ = D − \frac{1}{t}ad(R′_{0} + R′_{1}),

where F^{T} (R′_{0}) ≥ 3, F^{P}(R′_{0}) ≥ 1 and F^{T} (R′_{1}) ≥ 4, F^{P}(R′_{1}) ≥ 1.

Proceeding by induction on F^{T} -filtration, we obtain a sequence B_{i} ∈ W(E) with increasing F^{T} -filtration and such that Π_{i=0}^{∞}e^{\frac{1}{t} adB_{i}}(D) = D̃.

Since elements e^{\frac{1}{t} adB′}, B′ ∈ W(E), F^{T} (B) ≥ 3, form a pro-unipotent Lie group, there exists an element B ∈ W(E), F^{T} (B) ≥ 3, F^{P}(B) ≥ 1, such that Π_{i=0}^{∞}e^{\frac{1}{t} adB_{i}} = e^{\frac{1}{t} adB}.

Let D be a connection satisfying Proposition 9.4. Denote by W_{D} the subsheaf of W consisting of flat sections a, i.e. such that Da = 0. Since D is a derivation of W, it is clear that W_{D} is a sheaf of subalgebras. Let σ = id − (δδ_{P,Q}^{−1} + δ_{P,Q}^{−1}δ). Then, as follows from (9.2), σ : W → C_{M}^{∞}[[t]] is a projection, where C_{M}^{∞}[[t]] is considered as the center of the algebra W.

Proposition 9.6. a) The map σ : W_{D} → C_{M}^{∞}[[t]] is a bijection.

b) The inverse map η : C_{M}^{∞}[[t]] → W_{D} has the form η(f) = f +f̂, there F^{T} (f̂) > F^{T} (f).

c) If df ∈ P, then F^{P}(f̂) ≥ 1.

d) If df ∈ P, then σ(η(f)η(g)) = fg for any g ∈ C_{M}^{∞}[[t]].

Proof. Again, we apply the Fedosov iteration procedure. According to [Fe], Theorem 5.2.4, we look for η(f) as a limit, η(f) = lim a_{k}, there a_{k} ∈ W can be calculated recursively:

\begin{aligned}a_{0} & = f \\ a_{k+1} & = a_{0} + δ_{P,Q}^{−1}(∇a_{k} + \frac{1}{t}adr(a_{k})). \end{aligned}  (9.19)

Put f̂ = η(f) − a_{0}. As in [Fe], Theorem 5.2.4, one proves that such η(f) and f̂ satisfy a) and b). Now observe that a_{1} − a_{0} = δ_{P,Q}^{−1}(1 ⊗ df) and if df ∈ P, then F^{P}(a_{1} − a_{0}) ≥ 1. By induction, we conclude that F^{P}(a_{k} − a_{0}) ≥ 1 for all k ≥ 1. So F^{P}(a − a_{0}) ≥ 1 as well, which proves c).

Let us prove d). We have η(f)η(g) = fη(g) + f̂η(g). Since by c) F^{P}(f̂) ≥ 1, F^{P}(f̂η(g)) ≥ 1 as well. It follows that σ(f̂η(g)) = 0 and σ(η(f)η(g)) = σ(fη(g)) = fg, because σ is a C_{M}^{∞}[[t]]-linear map and σ(η(g)) = g.

9.6 Existence of PSP’s

Let (M, ω, P) be a polarized symplectic manifold. Recall that in Subsection 8.3 we have assigned to any PSP (μ_{t}, O_{t}) on (M, ω, P) an element τ(μ_{t}, O_{t}) ∈ Y(M, ω, P), which is a pair (ω_{t}, P_{t}) being a deformation of the pair (ω, P). The form ω_{t} represents the extension class cl_{Ext}(μ_{t}, O_{t}).

We show now that any element of Y(M, ω, P) corresponds to a PSP.

Proposition 9.7. a) For any pair (ω_{t}, P_{t}) ∈ Y(M, ω, P), there exists a PSP, (μ_{t}, O_{t}), such that

τ(μ_{t}, O_{t}) = (ω_{t}, P_{t}).  (9.20)

b) The Fedosov class of the corresponding star-product (C_{M}^{∞}[[t]], μ_{t}) is represented by the form of ω + tΓ(M, d(P_{t}^{⊥})) equal to

θ_{t} = ω_{t} + \frac{t}{2}tr(∇^{2}|_{P_{t}}),  (9.21)

where ∇ is a P_{t}-symplectic connection on the formal symplectic manifold (M, ω_{t}, P_{t}).

Proof. Let W be the Fedosov algebra on M corresponding to the symplectic form ω_{t}. Let ∇ be a P_{t}-symplectic connection on M corresponding to ω_{t} and D the flat connection on W constructed in Proposition 9.4 c). Let W_{D} be the sheaf of flat sections of W. Define a star-product (C_{M}^{∞}[[t]], μ_{t}) on M carrying over the multiplication from W_{D} to C_{M}^{∞}[[t]] via the map σ from Proposition 9.6. Point d) of that proposition shows that, in fact, this starproduct present the PSP (C_{M}^{∞}[[t]], μ_{t}, O_{P_{t}}). We are going to prove that this star-product is as required. In the following we identify W_{D} with the corresponding PSP via σ.

Let us prove (9.20). Let (U_{α}) be an open covering of M such that on each U_{α} there exist formal Darboux coordinates x_{αi}, y_{αi}, x_{αi} ∈ O_{P_{t}} with respect to ω_{t}.

Denote by ∇_{α} the standard flat P_{t}-symplectic connections over U_{α} such that the forms dx_{αi}, dy_{αi} are flat sections in P_{t}^{⊥}. Then, the connections D_{α} = ∇_{α} + δ satisfy on U_{α} Proposition 9.4 with r = 0. Let W_{D_{α}} be the star-product on U_{α} constructed in Proposition 9.6 via flat sections of D_{α}. It is easy to see that W_{D_{α}} coincides with the Moyal-Wick PSP with respect to the Darboux coordinates x_{αi}, y_{αi} (see Example 7.15), so x_{αi}, y_{αi} are also Darboux coordinates for the bracket \frac{1}{t}[·, ·] in W_{D_{α}}.

Since D and D_{α} have the same Wick curvature ω_{t}, there exist, by Proposition 9.5, elements B_{α} ∈ W such that e^{\frac{1}{t} adB_{α}}D_{α} = D. It is clear that e^{\frac{1}{t} adB_{α}} acting on W takes W_{D_{α}} to W_{D}, and point b) of that Proposition implies that it is identical on O_{P_{t}}.

Let Ψ_{α,β} = e^{− \frac{1}{t} adB_{α}}e^{\frac{1}{t} adB_{β}}. These may be considered as isomorphisms over U_{α} ∩ U_{β} gluing star-products W_{D_{α}} to a global star-product on M that is obviously isomorphic to W_{D}. We see, that functions x_{αi}, y_{αi} form local Darboux coordinates corresponding to that star-product. So, the characteristic 2-form cl_{P SP} (W_{D}) (see Subsection 8.3) is locally represented as dy_{αi} ∧ dx_{αi}. On the other hand, this form is equal to ω_{t}, since from very beginning the functions x_{αi}, y_{αi} have been chosen as Darboux coordinates for it. Hence, τ(W_{D}) = (ω_{t}, P_{t}).

b) Follows from Proposition 9.4 e) and Lemma 5.3.

10 The main theorem and corollaries

Let (M, ω, P) be a polarized symplectic manifold. Denote by Y the set of pairs (ω_{t}, P_{t}), where ω_{t} = ω + tω_{1} + ⋯ is a deformed symplectic form and P_{t}, P_{0} = P, its polarization. Let Aut(M) be the group of formal automorphisms of M.

Theorem 10.1. a) The equivalence classes of PDQ’s on (M, ω, P) are in one-to-one correspondence with the orbits in Y under the Aut(M)-action.

b) Let the pair (ω_{t}, P_{t}) be a point on the orbit corresponding to a PDQ (𝔸_{t}, 𝕆_{t}). Then, (𝔸_{t}, 𝕆_{t}) is isomorphic to a PSP, (C_{M}^{∞}[[t]], μ_{t}, O_{t}), where O_{t} consists of functions constant along P_{t} and the multiplication μ_{t} satisfies the condition

μ_{t}(f, g) = fg for f ∈ O_{t}, g ∈ C_{M}^{∞}[[t]].

c) The form ω_{t} represents the extension class cl_{Ext}(𝔸_{t}, 𝕆_{t}) ∈ H^{1}(M, Ω_{O_{t}}^{1cl}), associated with (𝔸_{t}, 𝕆_{t}).

d) Under the hypothesis of b), the Fedosov class of the deformation quantization 𝔸_{t} can be represented by a form θ_{t} that is a deformation of ω and polarized by P_{t}. It is defined by the formula

θ_{t} = ω_{t} + \frac{t}{2}tr(∇^{2}|_{P_{t}}),  (10.1)

where ∇ is a P_{t}-symplectic connection on the formal symplectic manifold (M, ω_{t}, P_{t}).

Proof. Parts a) and b) follow from Proposition 8.5, Corollary 8.6, and Proposition 9.7 a). Part c) follows from Lemma 8.2. Part d) is the same as Proposition 9.7 b).

Remark 10.2. We have interpreted the form ω_{t} from (10.1) as a representative of the extension class associated to a PDQ (see Section 8). The form tr(∇^{2}|_{P_{t}}) is the curvature form of the connection induced by ∇|_{P_{t}} on the complex line bundle det(P_{t}). Actually, det(P_{t}) can be presented as a line O_{t}-bundle, i.e. as a locally free sheaf of O_{t}-modules of rank one. Indeed, det(P_{t}) = C_{M}^{∞}[[t]] ⊗_{O_{t}} L, where L = Ω_{O_{t}}^{n}, n = \frac{1}{2} dim M. The form −tr(∇^{2}|_{P_{t}}), as well as minus curvature form of any other connection on L, can be interpreted as a representative of the extension class of an O_{t}-extension of Der(O_{t}) associated with L. Indeed, let T̃_{L} denote the sheaf of O_{t}-differential operators on L of order at most one. Then T̃_{L} equipped with the left O_{t}-module structure and the Lie bracket given by the commutator naturally forms a O_{t}-extension of Der(O_{t}). Splittings of this extension are flat connection on L. Let d_{α} be local flat connections on L in some open covering {U_{α}} of M. Then, d_{α} − d_{β} are closed 1-forms of Ω_{O_{t}}^{1} that form a Čech cocycle.

Hence, there exist smooth 1-forms f_{α} ∈ P_{t}^{⊥} such that d_{α} − d_{β} = f_{α} − f_{β}. Differential operators d_{α} − f_{α} form a global connection on L, ∇_{L}, with the curvature locally equal to −df_{α}. On the other hand, by definition (see Section 8), the extension class of T̃_{L} is represented by the form df_{α} ∈ dP_{t}^{⊥}, i.e. −∇_{L}^{2} . So, projecting the equality (10.1) to Γ(M, dP_{t}^{⊥})/d(Γ(M, P_{t}^{⊥})), we obtain

[θ_{t}] = cl_{Ext}(μ_{t}, O_{t}) − \frac{t}{2}cl_{Ext}(T̃_{det(P_{t})}).  (10.2)

Details are left to the reader.

Note that the form −\frac{1}{2π\sqrt{−1}}tr(∇^{2}|_{P_{t}}) represents the first Chern class of P, [KN].

Corollary 10.3. Let 𝔸_{t} be a deformation quantization on (M, ω). Suppose its Fedosov class cl_{F} (𝔸_{t}) is represented by the form θ_{t} that has a polarization P_{t}. Then 𝔸_{t} can be extended to a PDQ (𝔸_{t}, 𝕆_{t}), where 𝕆_{t} is isomorphic to O_{P_{t}}.

Proof. Let ∇ be a P_{t}-symplectic connection on the formal symplectic manifold (M, θ_{t}, P_{t}). Let (C_{M}^{∞}[[t]], μ_{t}, O_{t}) be a PSP such that

τ(C_{M}^{∞}[[t]], μ_{t}, O_{t}) = (θ_{t} − \frac{t}{2}tr(∇^{2}|_{P_{t}}), P_{t}).

By (10.1), cl_{F} (C_{M}^{∞}[[t]], μ_{t}) = [θ_{t}] = cl_{F} (𝔸_{t}), therefore star-products 𝔸_{t} and (C_{M}^{∞}[[t]], μ_{t}) are equivalent.

Remark 10.4. All constructions of the paper can be extended to the case when M is a formal manifold, M_{λ}, which is M endowed with the function sheaf C_{M}^{∞}[[λ]], λ a formal parameter. A formal polarized symplectic manifold is a triple, (M_{λ}, ω_{λ}, P_{λ}), where ω_{λ} is a formal symplectic form on M_{λ} and P_{λ} its polarization. The above construction of a polarized star-product applied to a formal polarized symplectic manifold (M_{λ}, ω_{λ}, P_{λ}) gives the following

Proposition 10.5. Let (𝔸_{t}, 𝕆_{t}) be a PDQ and (ω_{t}, P_{t}) an element on the orbit corresponding to (𝔸_{t}, 𝕆_{t}). Then, there exists on (M_{λ}, ω_{λ}, P_{λ}) a PSP

(C_{M}^{∞}[[λ]][[t]], μ_{λ,t}, O_{λ,t})

such that (𝔸_{t}, 𝕆_{t}) is equivalent to the diagonal sub-family (C_{M}^{∞}[[t]], μ_{t,t}, O_{t,t}) obtained by the substitution λ = t.

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address: Dept. of Math., Bar-Ilan Univ., 52900 Ramat Gan, Israel e-mail: donin@macs.biu.ac.il