Lobachevskii Journal of Mathematics Vol. 4, 1999, 177–206 \begin{matrix}c \\ ◯ \end{matrix}A. Yoshioka

A. Yoshioka

WEYL MANIFOLD AND QUANTIZED CONNECTION

(submitted by B.N.Shapukov)

Dedicated to Professir Hideki Omori for his 60th birthday

Abstract. This paper is two folds. First we give a brief review of Weyl manifold and Deformation quantization. Poincaré-Cartan class is introduced and complete classification of Weyl manifolds is given. Second, Quantized connection or twisted exterior derivative is discussed. Degree operator field is introduced and plays an important role. Classical coordinate is constructed by means of the degree operator field. Quantized connection is then defined on Weyl manifold. In terms of classical coordinates the quantized connection is shown to be the same as Fedosov connection. Finally, we show the Poincaré-Cartan class is equal to the deRham cohomology class of the curvature of Fedosov connection.

1. Introduction

Deformation quantization was initiated by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer [BFFLS1, BFFLS2] as an idea to give quantization on Poisson manifolds. The point of the concept of deformation quantization is to quantize classical system by deforming commutative algebra of classical observables or functions on the manifold to noncommutative associative algebra with the canonical commutation relation, instead of assigning operators on Hilbert spaces to classical observables. The deformed algebra consists of the formal power series with coefficients in functions algebra on manifold. The existence of formal deformation quantization on symplectic manifolds is given by DeWilde-Lecomte [DL], Fedosov [F] and Omori-Maeda-Yoshioka [OMY1] and as to the case of general Poisson manifolds, Kontsevich [K] has given a proof of existence of deformation quantization.

This work is partially supported by Grand-in-Aid for Scientific Research (No.09640132), Ministry of Education, Science and Culture. .

1991 Mathematical Subject Classification. Primary 05C38, 15A15; Secondary 05A15, 15A18.

Key words and phrases. Weyl manifold, Deformation quantization, Quantized connection.

For symplectic manifolds, there are many approches to deformation quantization, for instance, a cohomology argument (e.g, [DL]) and geometrical method (e.g., [F, OMY1]). Bertelson-Cahen-Gutt [BCG], Xu [X] give a complete classification of deformation quantization based on Fedosov method and Deligne [D] showed the relation between DeWilde-Locomte construction and Fedosov construction. Weinstein [W] gave a clear exposition of deformation quantization and Fedosov connection. Fedosov [F] and Omori-Maeda-Yoshioka [OMY1] are in some sense based on the same spirit; considering Weyl algebra bundle and obtraining deformation quantization by embedding formal functions into the space of sections of the budle. The basic idea of Fedosov is a flat connection on the budle while Omori-Maeda-Yoshioka method is essentially using the contact Lie algbra structure over the bundle.

In this paper, we show a relation between Fedosov connection and the contact Lie algebra structure of the Weyl manifolds (cf. [Y]). Starting from the contact structure of Weyl manifolds [OMY1], we contruct Fedosov connection. The idea is the degree operator field given in Omori-Maeda-Yoshioka [OMY2]. The eigenvalue problem of the degree operator field supplies a filtration of Weyl algebra bundle, which plays the central role in this paper. In [OMMY], we introduced the Poincaré-Cartan class of Weyl manifold which is a complete invariant of Weyl manifolds and then classifies deformation quantization on symplectic manifolds. The Poincaré-Cartan class is given by certain Cĕch 2cocycles which are obtained in gluing locally trivial Weyl algebra bundles. We also show the Poincaré-Cartan class coincides with the cohomology class given by the curvature of the Fedosov connection, which proves the conjuecture in [OMMY].

The author thanks to the organizers of Workshop ”Towards 100 years after Sophus Lie” for their hospitality during the stay in Kazan, 1998. The author is also grateful to Professor John Rawnsley for helpful disscussion.

2. Weyl manifold

In this section, we survey on Weyl manifold (cf. [OMY1]) and Poincaré-Cartan invariant of Weyl manifolds (cf. [OMMY]).

2.1. Weyl algebras and Contact algebras. Weyl manifold is an algebra bundle over a symplectic manifold whose fiber is Weyl algebra. In this section, we intorduce Weyl algebra and contact algebra. With the help of contact algebras, one can deal with the center of Weyl algebra. We also investigate certain automorphisms of both Weyl and contact algebra.

2.1.1. Weyl algebra. The Weyl algebra W is the algebra over ℝ formally generated by ν, Z^{1}, ⋯, Z^{2n}

W =< ν, Z^{1}, Z^{2}, ⋯, Z^{2n} >_{ℝ}

with the canonical commutation relation (the CCR for short)

[ν, Z^{i}] = ν ∗ Z^{i} − Z^{i} ∗ ν = 0, [Z^{i}, Z^{j}] = Z^{i} ∗ Z^{j} − Z^{j} ∗ Z^{i} = νΛ^{ij} (2.1) where i, j = 1, 2, ⋯, 2n and

Λ = (Λ^{ij}) = ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )

Here we write the product by using ∗ as a ∗ b, a, b ∈ W. Then, an element of W is expressed as a formal power series

a = ∑_{l,μ} a_{l,μ}ν^{l}Z_{∗}^{μ}, a_{l,μ} ∈ ℝ (2.2)

where μ = (μ_{1}, ⋯, μ_{2n}) is a multi-index and

Z_{∗}^{μ} = (Z^{1})^{μ_{1}} ∗ (Z^{2})^{μ_{2}} ∗ ⋯ ∗ (Z^{2n})^{μ_{2n}}, (Z^{i})^{μ_{i}} = \underbrace{Z^{i} ∗ ⋯ ∗ Z^{i}}_{μ_{i}} .

The summation in (2.2) is infinite sum, which means the term formally generated. The algebra W is equipped with the formal power series topology, i.e., the sequence {a_{n}}_{n} ⊂ W converges to a certain a iff the coefficients of each monomial ν^{l}Z^{α} of a_{n} converges to the coefficient of a, under which W is a complete topological algebra. We define the degree of monomial ν^{l}Z_{∗}^{μ} by

d(ν^{l}Z_{∗}^{μ}) = 2l + |μ| (2.3)

and we set

W_{k} = {a = ∑_{l,μ} a_{l,μ}ν^{l}Z_{∗}^{μ} | d(ν^{l}Z_{∗}^{μ}) ≥ k} (2.4)

We now introduce symmetric product with respect to generators. Consider a system of elements {W^{1}, W^{2}, ⋯, W^{2n}} of W_{1} satisfying the canonical commutation relations;

[ν, W^{i}] = 0, [W^{i}, W^{j}] = νΛ^{ij}, i, j = 1, 2, ⋯, 2n.

Expanding each W^{i} in the form (2.2), we see

W^{i} = ∑_{j=1}^{2n} a_{j}^{i}Z^{j} + ∑_{2l+|μ|≥2} a_{l,μ}^{i}ν^{l}Z_{∗}^{μ}.

Here W^{i} has no constant term since it is an element of W_{1}. The CCR (2.1) gives that the matrix A = (a_{j}^{i}) is a symplectic matrix, A ∈ Sp(2n, ℝ). We have by a simple calculation

Lemma 2.1. Each Z^{i} is written as a formal sum in terms with ν, W^{i};

Z^{i} = ∑_{j=1}^{2n} a_{j}^{′i}W^{j} + ∑_{2l+|μ|≥2} a_{l,μ}^{′i}ν^{l}W_{∗}^{μ}, i = 1, 2, ⋯, 2n.

Hence, W is generated by the system {ν, W^{1}, W^{2}, ⋯, W^{2n}}. We give a definition

Definition 2.1. A system of elements {ν, W^{1}, W^{2}, ⋯, W^{2n}} of W_{1} with the canonical commutation relations is called a canonical generator system of W.

For every canonical generators system of W, one can construct a commutative product ∘ in W as follows. First we set W^{i} ∘ W^{j} = W^{i} ⊙ W^{j} = \frac{1}{2}(W^{i} ∗ W^{j} + W^{j} ∗ W^{i}), the symmetric product. In general, an associator of the symmetric product ⊙ of an associative algebra satisfies

a ⊙ (b ⊙ c) − (a ⊙ b) ⊙ c = \frac{1}{4}[b, [c, a]].

Since [W^{i}, W^{j}] is in the center of W one sees W^{i}⊙(W^{j}⊙W^{k}) = (W^{i}⊙W^{j})⊙W^{k} and hence

W^{i_{1}} ⊙ (W^{i_{2}} ⊙ (⋯ ⊙ (W^{i_{k−1}} ⊙ W^{i_{k}}) ⋯) (2.5)

= W^{i_{σ(1)}} ⊙ (W^{i_{σ(2)}} ⊙ (⋯ ⊙ (W^{i_{σ(k−1)}} ⊙ W^{i_{σ(k)}}) ⋯) (2.6)

for every k-permutation σ ∈ S_{k}. We set

W^{i_{1}} ∘ (W^{i_{2}} ∘ (⋯ ∘ (W^{i_{k−1}} ∘ W^{i_{k}}) ⋯) =

W^{i_{1}} ⊙ (W^{i_{2}} ⊙ (⋯ ⊙ (W^{i_{k−1}} ⊙ W^{i_{k}}) ⋯)

One can check that each ν^{l}W^{i_{1}} ∗ W^{i_{2}} ∗ ⋯ ∗ W^{i_{k−1}} ∗ W^{i_{k}} is uniquely written in terms with combination of ν^{l′}W^{j_{1}} ∘(W^{j_{2}} ∘(⋯∘(W^{j_{k−1}} ∘W^{j_{l}}) ⋯) by induction with respect to the degree. We write ν^{l}W_{∘}^{I} = ν^{l}W^{i_{1}} ∘ (W^{i_{2}} ∘ (⋯ ∘ (W^{i_{k−1}} ∘ W^{i_{k}}) ⋯) for simplicity, where I = {i_{1}, i_{2}, ⋯, i_{k}}.

Definition 2.2. We define the product of elements by

ν^{l}W_{∘}^{I} ∘ ν^{m}W_{∘}^{J} = ν^{l+m}W_{∘}^{I∪J}.

Using the property (2.5), we can write monomials as

ν^{l}Z_{∘}^{μ} = ν^{l}(Z^{1})^{μ_{1}} ∘ (Z^{2})^{μ_{2}} ∘ ⋯ ∘ (Z^{2n})^{μ_{2n}}, (Z^{i})^{μ_{i}} = \underbrace{Z^{i} ∘ ⋯ ∘ Z^{i}}_{μ_{i}} .

Since every element of W is written as a = ∑_{l,μ} a_{l,μ}ν^{l}W_{∘}^{μ}, this relation gives a commutative associative product ∘ in W,

∑_{l,μ} a_{l,μ}ν^{l}W_{∘}^{μ} ∘ ∑_{l′,μ′} b′_{l′,μ′}ν^{l′}W_{∘}^{μ′} = ∑_{l,μ,l′,μ′} a_{l,μ}b′_{l′,μ′}ν^{l+l′}W_{∘}^{μ+μ′}

We remark here that the commutative product ∘ depends on a system of canonical generators. In what follows, the commutative product given by the system of canonical generators {Z^{1}, Z^{2}, ⋯, Z^{2n}} is specified and denoted by ·.

Writing elements as a = ∑_{l,μ} a_{l,μ}ν^{l}W_{∘}^{μ} and b = ∑_{l,μ} b_{l,μ}ν^{l}W_{∘}^{μ}, we have

Theorem 2.1. The product of a and b is expressed by the Moyal product formula with respect to a canonical generator system {ν, W^{1}, ⋯, W^{2n}}

a ∗ b = a exp (\frac{ν}{2} ∑_{ij} Λ^{ij}\overleftarrow{∂_{W^{ι}}} ∘ \overrightarrow{∂_{W^{j}}}) b = a ∑_{n} \frac{1}{n!} (\frac{ν}{2} ∑_{ij} Λ^{ij}\overleftarrow{∂_{W^{ι}}} ∘ \overrightarrow{∂_{W^{j}}})^{n} b,

where a \overleftarrow{∂_{W^{ι}}} ∘ \overrightarrow{∂_{W^{j}}} b = ∂_{W^{ι}}a ∘ ∂_{W^{j}}b. Conversely, we have

a ∘ b = a exp (−\frac{ν}{2} ∑_{ij} Λ^{ij}\overleftarrow{∂_{W^{ι}}} ∗ \overrightarrow{∂_{W^{j}}}) b.

By the argument above, every element is written as a formal power series of ν, Z^{1}, ⋯, Z^{2n} with coefficient ℝ in term with the symmetric product with respect to ν, Z^{1}, ⋯, Z^{2n};

a = ∑_{l,α} a_{l,α}ν^{l}Z^{α}, a_{l,α} ∈ ℝ.

It is clear the center of W is generated by ν.

Definition 2.3. An element a is called without central elements if the formal power series excludes the monomials of ν, i.e., a is of the form

a = ∑_{l,α,|α|>0} a_{l,α}ν^{l}Z^{α}.

We introduce an involution of W by

\overline{ν} = −ν, \overline{Z}^{i} = Z^{i}, \overline{a ∗ b} = \overline{b} ∗ \overline{a}. (2.7)

It is easy to see if \overline{a} = a, then a is of the form

a = ∑_{m,α} a_{2m,α}ν^{2m}Z^{α}, (2.8)

which will be called a hermitian element.

2.1.2. Contact Lie algebra. In order to pick up information of the center of the Weyl algebra W, we introduce a concept of contact Lie algebra. By using the degree, we define a derivation of the Weyl algebra D_{0} : W → W by the relation

D_{0}(ν) = 2ν = d(ν)ν, D_{0}(Z^{i}) = Z^{i} = d(Z^{i})Z^{i}.

Put an element τ such that

[τ, ν] = νD_{0}(ν) = 2ν^{2}, [τ, Z^{i}] = νD_{0}(Z^{i}) = νZ^{i}. (2.9)

Definition 2.4. We introduce a Lie algebra

𝔊 = ℝτ ⊕ W

with the Lie algebra structure as

[k_{1}τ + F_{1}, k_{2}τ + F_{2}] = k_{1}[τ, F_{2}] − k_{2}[τ, F_{1}] + [F_{1}, F_{2}].

We call 𝔊 a contact Lie algebra.

The contact Lie algebra has an anti-involution by extending the one of W as

\overline{τ} = τ. (2.10)

2.2. ν-automorphisms. In this section, we consider automorphisms of Weyl algebras and contact Lie algebras which preserve ν.

2.2.1. ν-Automorphisms of W. We give a definiton of ν-automorphism of W and its normal form in this subsection.

Definition 2.5. An algebra automorphism Φ : W → W is called a ν-auto-morphism if Φ(ν) = ν. A ν-automorphism Φ satisfies the hermitian property if \overline{Φ(a)} = Φ(\overline{a}) for ∀a ∈ W.

Example 2.1. For A = (a_{l}^{i}) ∈ Sp(2n, ℝ), we obtain a ν-automorphism by extending the correspondence ν ↦ ν, Z^{i} ↦ AZ^{i} = ∑_{l} a_{l}^{i}Z^{l} to the algebra W, which we denote by Af, f ∈ W.

Example 2.2. For F ∈ W, by taking the adjoint action we have a derivation of W which vanishes on the center. We then set the action of exp \frac{1}{ν} ad (F) on W by

exp \frac{1}{ν}ad(F)a = a + \frac{1}{ν}ad(F)a + ⋯ + \frac{1}{k!}(\frac{1}{ν}ad(F))^{k}a + ⋯

which gives a ν-automorphism of the algebra W.

Notice in the example above if exp \frac{1}{ν} ad (F) satisfies the hermitian property in Definition 2.5, then the noncentral parts of F is a hermitian element, i.e., of the form (2.8).

Theorem 2.2. For a ν-automorphism Φ : W → W, there exist a symplectic matrix A ∈ Sp(2n, ℝ), an element without central part

F = ∑_{2l+|α|≥3,|α|>0} a_{l,α}ν^{l}Z^{α} ∈ W_{3}

and a central element c = c_{0} + c_{1}ν + ⋯ ∈ ℝ[[ν]] such that

Φ = A ∘ exp \frac{1}{ν} ad (c + F).

The matrix A and F are uniquely determined while the central element c is arbitrary. If Φ satisfies the hermitian property, then F is hermitian, i.e.,of the form (2.8).

Proof. Note W_{1} is a maximal ideal of W and hence Φ(W_{1}) = W_{1}. Thus, for generators Z^{i}, we have

Φ(Z^{i}) = ∑_{j=1}^{2n} a_{j}^{i}Z^{j} + a^{i}ν + ∑_{jk} a_{jk}^{i}Z^{j}Z^{k} + O(3)

where O(3) means the terms in W_{3}. Substituting this expansion into the relation [Φ(Z^{i}), Φ(Z^{j})] = νΛ^{ij}, we see A = (a_{j}^{i}) ∈ Sp(2n, ℝ). Then, we have

A^{−1} ∘ Φ(Z^{i}) = Z^{i} + b^{i}ν + ∑_{jk} b_{jk}^{i}Z^{j}Z^{k} + O(3), i = 1, 2, ⋯, 2n.

We put g_{2}^{i} = ∑_{jk} b_{jk}^{i}Z^{j}Z^{k}. The CCR implies [Z^{i}, g_{2}^{j}] + [g_{2}^{i}, Z^{j}] = 0, i, j = 1, 2, ⋯, 2n, which gives

\frac{∂}{∂Z^{i}} ∑_{k=1}^{2n} ω_{jk}g^{k} = \frac{∂}{∂Z^{j}} ∑_{k=1}^{2n} ω_{ik}g^{k}.

By the Poincaré Lemma, we see there exists G = ∑_{jkl} g_{ijk}Z^{j}Z^{k}Z^{l} ∈ W_{3} without central part such that g_{2}^{i} = \frac{1}{ν} ad (G)(Z^{i}), i = 1, 2, ⋯, 2n. Further, we see b^{i}ν = \frac{1}{ν} ad (∑_{m,l} b^{m}νω_{ml}Z^{l})(Z^{i}), i = 1, 2, ⋯, 2n. Thus, we have the degree 2 terms are given as

b^{i}ν + g_{2}^{i} = \frac{1}{ν} ad (G_{3})(Z^{i}), i = 1, 2, ⋯, 2n,

where G_{3} = ∑_{m,l} b^{m}ω_{ml}νZ^{l} + G ∈ W_{3} which has no central terms. We then have

exp −\frac{1}{ν} ad (G_{3}) ∘ A^{−1} ∘ Φ(Z^{i}) = Z^{i} + b_{3}^{i}(ν) + g_{3}^{i} + O(4), i = 1, 2, ⋯, 2n,

where b_{3}^{i}(ν) is a central part of degree 3 (hence zero in this case) and g_{3}^{i} is the terms of degree 3 without central parts. Using the CCR gives the similar relations [Z^{i}, g_{3}^{j}] + [g_{3}^{i}, Z^{j}] = 0, i, j = 1, 2, ⋯, 2n and then shows that there exists G ∈ W_{4} without central parts such that g_{3}^{i} = \frac{1}{ν} ad (G)(Z^{i}), i = 1, 2, ⋯, 2n. We also see b_{3}^{i}(ν) = \frac{1}{ν} ad (∑_{m,l} b_{3}^{m}(ν)ω_{ml}Z^{l})(Z^{i}), i = 1, 2, ⋯, 2n. Then we have the identity b_{3}^{i}(ν) + g_{3}^{i} = \frac{1}{ν} ad (G_{4})(Z^{i}), i = 1, 2, ⋯, 2n, where G_{4} = ∑_{m,l} b_{3}^{m}(ν)ω_{ml}Z^{l} + G ∈ W_{4} which has no central terms. Thus, we have

exp −\frac{1}{ν} ad (G_{4}) ∘ exp −\frac{1}{ν} ad(G_{3}) ∘ A^{−1} ∘ Φ(Z^{i})

= Z^{i} + b_{4}^{i}(ν) + g_{4}^{i} + O(5), i = 1, 2, ⋯, 2n,

where b_{4}^{i}(ν) is a central part of degree 4 and g_{4}^{i} is the terms of degree 4 without central parts. Repeating this procedure gives

Φ(Z^{i}) = A ∘ ∏_{k=4}^{∞} exp \frac{1}{ν} ad (G_{k})(Z^{i}), i = 1, 2, ⋯, 2n.

Since ∏_{k=4}^{∞} exp \frac{1}{ν} ad (G_{k}) = exp \frac{1}{ν} ad (c + F_{3}) for a certain F_{3} ∈ W_{3} without central part and a certain c ∈ ℝ[[ν]], we have a proof for the first part. The uniqueness of A is clear. For the uniqueness of F_{3}, we expand

F_{3} = F_{(3)} + F_{(4)} + ⋯,

where F_{(k)} is the term of homogeneous degree k without central terms. Calculating as

exp \frac{1}{ν} ad (F_{3})(Z^{i}) = Z^{i} + \frac{1}{ν}[F_{3}, Z^{i}] + \frac{1}{2!ν} [F_{3}, [F_{3}, Z^{i}]] + ⋯

we see the degree 2 term is \frac{1}{ν}[F_{(3)}, Z^{i}] which shows the uniqueness of F_{(3)}. Similarly we can check the uniqueness of F_{(k)} inductively with respect to the degree k. Thus, we have the uniqueness of F_{3}. If Φ satisfies the hermitian property, the remark after the example above indicates F is of the form (2.8), which completes the proof. (QED)

2.2.2. ν-Automorphisms of 𝔊. We first give a definition

Definition 2.6. A Lie algebra automorphism Ψ : 𝔊 → 𝔊 is called a νautomorphism if (i) the restriction Ψ|_{W} is a ν-automorphism of the Weyl algebra W and (ii) the image of τ is given Ψ(τ) = τ + f for certain f ∈ W. A ν-automorphism Ψ satisfies the hermitian property if \overline{Ψ(cτ + a)} = Ψ(\overline{cτ + a}) for ∀cτ + a ∈ 𝔊.

Remark 2.1. In the definition of ν-automorphisms, the element f such that Ψ(τ) = τ + f is taken as f = c_{0} + c_{1}ν + f′ where c_{0}, c_{1} ∈ ℝ and f′ ∈ W_{3}. In fact, applying Ψ to the identity [τ, Z^{i}] = νZ^{i} gives [τ + f, ΦZ^{i}] = νΦZ^{i} where ΦZ^{i} = ∑_{j} a_{j}^{i}Z^{j} + ∑ a_{jk}^{i}Z^{i}Z^{j} + a_{1}^{i}ν + · · ·. Thus, f satisfies [f, ΦZ^{i}] = −ν ∑ a_{jk}^{i}Z^{j}Z^{k}+⋯ which yields [f, Z^{i}] ∈ νW_{2}, hence we have f = c_{0}+c_{1}ν+f′ for certain c_{0}, c_{1} ∈ ℝ and f′ ∈ W_{3}. If Ψ satisfies the hermitian property, then c_{1} = 0 and f′ satisfies \overline{f′} = f′, hence is a hermitian element, i.e., of the form (2.8).

Example 2.3. For A ∈ Sp(2n, ℝ), we extend the ν-automorphism A of W to 𝔊 by setting Aτ = τ.

Example 2.4. We set

[\frac{1}{ν}, τ] = 2, [\frac{1}{ν}a, τ] = 2a − \frac{1}{ν}[τ, a], a ∈ W. (2.11)

Then ad (\frac{1}{ν}a) = [\frac{1}{ν}a, ] induces a derivation of the Lie algebra 𝔊 whose restriction to the Weyl algebra W is a ν-derivation, i.e., a derivation of W vanishing on the center. For F ∈ W_{3}, we set the action of the exponential exp ad(\frac{1}{ν}F) on 𝔊 by

exp ad(\frac{1}{ν}F)(sτ + a)

= (sτ + a) + ad(\frac{1}{ν}F)(sτ + a) + ⋯ + \frac{1}{k!}(ad(\frac{1}{ν}F))^{k}(sτ + a) + ⋯

which gives a ν-automorphism of the algebra 𝔊.

Example 2.5. We set

[Log ν, τ] = −2ν. (2.12)

Then, we have a derivation ad (Log ν) : 𝔊 → 𝔊. For A ∈ Sp(2n, ℝ) and F ∈ W_{3}, we have a ν-automorphism Ψ : 𝔊 → 𝔊 by

(2.13) Ψ = A ∘ exp ad( \frac{1}{ν} (c_{0} + cν + F) − \frac{c_{1}}{2} Log ν)

where c, c_{i} ∈ ℝ.

We have

Theorem 2.3. (i) For a ν-automrophism of the Weyl algebra Φ : W → W, there exists a ν-automorphism Ψ : 𝔊 → 𝔊 such that Ψ|_{W} = Φ. (ii) A νautomorphism Ψ : 𝔊 → 𝔊 is written as

Ψ = A ∘ exp ad(\frac{1}{ν}(c + F_{3}) − \frac{c_{L}}{2} Log ν),

where A ∈ Sp(2n, ℝ) is a symplectic matrix, F_{3} ∈ W_{3} is without central elements, c ∈ ℝ[[ν]] is a central element and c_{L} is a real constant. Here A, F and c_{L} are uniquely given and c = c_{0} + c_{1}ν + c_{2}ν^{2} + ⋯ is also unique except the coefficient c_{1} of the ν term. Moreover, if Ψ satisfies the hermitian property, then F_{3} is of the form (2.8) and c_{2k+1} = 0, k = 1, 2, ⋯ and c_{L} = 0.

Proof. (i) By the previous theorem about the expression of the ν-automorph-isms of the Weyl algebras, the ν-automorphism Φ can be written as

Φ = A ∘ exp \frac{1}{ν} ad (F_{3})

for the unique A ∈ Sp(2n, ℝ) and F_{3} ∈ W_{3} without central terms. Then, we set a ν-automorphism of 𝔊

Ψ′ = A ∘ exp ad (\frac{1}{ν}F_{3})

which is a desired one. (ii) For the ν-automorphism Ψ, we set a ν-automorphism of W as Φ = Ψ|_{W}. In view of (i), we have an expression Φ = A ∘ exp \frac{1}{ν} ad (F_{3}). Similarly as in (i), we consider a ν-automorphism Ψ′ = A ∘ exp ad (\frac{1}{ν}F_{3}) : 𝔊 → 𝔊. Apply Ψ′ and Ψ to τ. Then we have Ψ′(τ) = τ + f′ and Ψ(τ) = τ + f, for certain f′, f ∈ W. Applying the both automorphism to the relation [τ, Z^{i}] = νZ^{i} gives

[τ + f, W^{i}] = νW^{i}, [τ + f′, W^{i}] = νW^{i}, i = 1, 2, ⋯, 2n,

where we set W^{i} = Φ(Z^{i}). Thus, we have [f′ − f, W^{i}] = νW^{i}, i = 1, 2, ⋯, 2n which yields that the difference f′ − f is a central element, i.e.,

f′ = f + b_{0} + b_{1}ν + b_{2}ν^{2} + ⋯, b_{j} ∈ ℝ.

Putting

c = −\frac{b_{0}}{2} + c_{1}ν + \frac{b_{2}}{2} ν^{2} + ⋯ + \frac{b_{2k}}{2(2k − 1)}ν^{2k−1} + ⋯

where c_{1} is arbitrary and putting c_{L} = −b_{1}, we have

exp ad (\frac{1}{ν}c − \frac{c_{L}}{2} Log ν) ∘ Ψ′(τ) = τ + f = Ψ(τ)

and

exp ad (\frac{1}{ν}c − \frac{c_{L}}{2} Log ν) ∘ Ψ′(Z^{i}) = Ψ(Z^{i}), i = 1, 2, ⋯, 2n.

Thus, we see

Ψ = A ∘ exp ad(\frac{1}{ν}(c + F_{3}) − \frac{c_{L}}{2} Log ν).

The uniqueness of A and F_{3} is already shown in the previous theorem and hence if

Ψ = A ∘ exp ad(\frac{1}{ν}(c̃+ F_{3}) − \frac{c̃_{L}}{2} Log ν)

then

\{A ∘ exp ad(\frac{1}{ν}(c + F_{3}) − \frac{c_{L}}{2} Log ν)\}^{−1} ∘ A ∘ exp ad(\frac{1}{ν}(c̃+ F_{3}) − \frac{c̃_{L}}{2} Log ν)(τ)

= exp ad (\frac{1}{ν}(c̃− c) − \frac{c̃_{L} − c_{L}}{2} Log ν)(τ) = τ

hence we have c̃ − c = aν and c̃_{L} − c_{L} = 0 where a ∈ ℝ which yields the uniqueness of the central parts. As to the hermitian property, we see Ψ(\overline{τ}) = Ψ(τ) gives F_{3} satisfies (2.8) and \overline{c} = c, which shows the last assertion. (QED)

For later use, we prepare

Lemma 2.2. For an element c_{0} + c_{1}ν + f where c_{0}, c_{1} ∈ ℝ and f ∈ W_{3}, there exists a ν-automorphism Ψ = exp ad(\frac{1}{ν} \frac{c_{0}}{2} + \frac{1}{ν}g − \frac{c_{1}}{2} Log ν) : 𝔊 → 𝔊 , g ∈ W_{3} such that Ψ(τ) = τ + c_{0} + c_{1}ν + f. Moreover, for an element c_{0} + f where c_{0} ∈ ℝ and f ∈ W_{3} is a hermitian element, i.e., of the form (2.8), then we can take a ν-automorphism Ψ = exp ad(\frac{1}{ν} \frac{c_{0}}{2} + \frac{1}{ν}g) : 𝔊 → 𝔊 for a certain hermitian element g ∈ W_{3} satisfying Ψ(τ) = τ + c_{0} + f.

Proof. Write f = f_{(3)} + O(4) where f_{(3)} is a homogeneous degree 3 term of f. Consider an automorphism Ψ_{3} = exp ad (\frac{1}{ν}f_{(3)}). Then, we have

Ψ_{3}(τ + c_{0} + c_{1}ν + f) = τ + c_{0} + c_{1}ν + f_{(3)} + [\frac{1}{ν}f_{(3)}, τ] + O(4)
= τ + c_{0} + c_{1}ν + f_{(4)} + O(5).

Apply Ψ_{4} = exp ad (\frac{1}{2ν} f_{(4)}). Then we see

Ψ_{4}Ψ_{3}(τ + c_{0} + c_{1}ν + f) = τ + c_{0} + c_{1}ν + f_{(5)} + O(6).

Repeating this procedure, we see the automorphism Ψ̃ = lim_{n→∞} Ψ_{n} ⋯ Ψ_{4}Ψ_{3} satisfies Ψ̃(τ + c_{0} + c_{1}ν + f) = τ + c_{0} + c_{1}ν. Then,

Ψ = Ψ̃^{−1} exp ad(\frac{1}{ν} \frac{c_{0}}{2} − \frac{c_{1}}{2} Log ν) = exp ad(\frac{1}{ν} \frac{c_{0}}{2} + \frac{1}{ν}g − \frac{c_{1}}{2} Log ν)

satisfies Ψ(τ) = τ + c_{0} + c_{1}ν + f where g ∈ W_{3}, which shows the lemma. (QED).

2.3. Weyl diffeomorphisms and Contact Weyl diffeomorphisms. In this section, we consider bundle ν-automorphisms which preserve a certain structure, callled Weyl functions.

2.3.1. Weyl functions. Consider the trivial bundle W_{U} = U ×W where U is an open subset of the canonical symplectic manifold (ℝ^{2n}, σ_{0}) with the canonical coordinate

z = (z^{1}, z^{2}, ⋯, z^{2n}), σ_{0} = ∑_{ij} ω_{ij}dz^{i} ∧ dz^{j}, ω = (ω_{ij}) = Λ^{−1}.

We consider the smooth sections ζ^{1}(z), ζ^{2}(z), ⋯, ζ^{2n}(z) ∈ Γ(W_{U}) of the bundle such that

ζ^{i}(z) = z^{i} + Z^{i}, i = 1, 2, ⋯, 2n.

The sections ζ^{1}(z), ζ^{2}(z), ⋯, ζ^{2n}(z) satisfies the CCR;

[ν, ζ^{i}(z)] = 0, [ζ^{i}(z), ζ^{j}(z)] = νΛ^{ij}.

The set of sections Γ(W_{U}) forms an associative algebra by the pointwise multiplication and is a complete topological algebra with the smooth topology. We consider the algebra over ℝ generated by the sections ν, ζ^{1}(z), ζ^{2}(z) ⋯, ζ^{2n}(z) and we denote it by P(W_{U}), called polynomial algebra of ν, ζ^{1}(z), ζ^{2}(z), ⋯, ζ^{2n}(z).

Definition 2.7. We set the subalgebra F(W_{U}) of Γ(W_{U}) as the closure of the polynomial algebra P(W_{U}). An element of F(W_{U}) is called a local Weyl function.

Using the polynomial approximation theorem, we have (cf. [OMY1])

Theorem 2.4. An element of F(W_{U}) is of the form

f^{#}(z) = ∑_{α} \frac{∂^{α}f}{∂z^{α}} (z)Z^{α}, f ∈ C^{∞}(U)[[ν]].

We call f^{#}(z) the Weyl continuation of the function f ∈ C^{∞}(U)[[ν]]. We have (cf. [OMY1])

Proposition 2.1. The Weyl continuation gives an algebra isomorphism

# : (C^{∞}(U)[[ν]], ∗_{0}) → (F(W_{U}), ∗)

where ∗_{0} is the Moyal product on U.

2.3.2. Weyl diffeomorphisms. We introduce a class of bundle isomorphisms of W_{U}. Let U, U′ ⊂ ℝ^{2n} be open subsets and consider trivial bundles W_{U} and W_{U′}.

Definition 2.8. A bundle isomorphism

Φ : W_{U} → W_{U′}

is called a Weyl diffeomorphism iff

Φ(ν) = ν, Φ^{∗}F(W_{U′}) = F(W_{U})

and

\overline{Φ^{∗}f^{#}} = Φ^{∗}\overline{f^{#}}, ∀f^{#} ∈ F(W_{U′}).

We see

Proposition 2.2. For a Weyl diffeomorphism Φ : W_{U} → W_{U′}, the induced diffeomorphism ϕ : U → U′ is a symplectic diffeomorphism.

We show the converse, i.e., for a given symplectic diffeomorphism there exists a Weyl diffeomorphism lift.

According to Theorem 2.2, a Weyl diffeomorphism Φ : W_{U} → W_{U′} can be written as

Φ = A_{ϕ} ∘ exp \frac{1}{ν} ad (c + F_{3})

where ϕ : U → U′ is the induced symplectic diffeomorphism and A_{ϕ} =^{t} dϕ^{−1}, a unique section F_{3} ∈ Γ(U ×W_{3}) without central part and an arbitrary central element c ∈ C^{∞}(U)[[ν]]. Now we have the existence of Weyl diffeomorphisms (cf. [OMY1]).

Theorem 2.5. Every symplectic diffeomorphism ϕ : U → U′ has a Weyl diffeomorphism lift Φ : W_{U} → W_{U′}. Especially, for the identity ι : U → U, a Weyl diffeomorphism lift Φ : W_{U} → W_{U} is given as the form

Φ = exp \frac{1}{ν} ad (c + h^{#})

where h^{#} is a Weyl function of the form

h^{#} = ν^{2}h_{2}^{#} + ν^{2}h_{4}^{#} + ⋯ + ν^{2k}h_{2k}^{#} + ⋯, h_{2k} ∈ C^{∞}(U)

and c is a central function

c = c_{0} + νc_{1} + ⋯ + ν^{j}c_{j} + ⋯, c_{j} ∈ C^{∞}(U).

The Weyl function h^{#} is unique up to constants in ℝ[[ν^{2}]] while c is arbitrary. Here exp \frac{1}{ν} ad h^{#} is a bundle isomorphism of W_{U} given by the fiberwise operation

exp \frac{1}{ν} ad h^{#}(z)(F) = F + \frac{1}{ν}[h^{#}(z), F] + \frac{1}{2} \frac{1}{ν^{2}} [h^{#}(z), [h^{#}(z), F]] + ⋯,

for ∀F ∈ W_{z} ⊂ W_{U}.

Corollary 2.1. For Weyl diffeomorphisms Φ, Φ′ : W_{U} → W_{U′} with the same induced symplectic diffeomorphism, there exists a local Weyl function h^{#} ∈ F(W_{U}), uniquely up to constants in ℝ[[ν^{2}]], such that

Φ′ = Φ ∘ exp \frac{1}{ν} ad h^{#}

where h^{#} = ν^{2}h_{2}^{#} + ν^{2}h_{4}^{#} + ⋯ + ν^{2k}h_{2k}^{#} + ⋯, h_{2k} ∈ C^{∞}(U).

2.4. Contact Weyl form. Similarly as before, for an open subset U ⊂ ℝ^{2n} consider the trivial bundle 𝔊_{U} = U × 𝔊 and denote by Γ(𝔊_{U}) the set of all smooth sections of 𝔊_{U}.

Definition 2.9. A section

τ̃(z) = τ + ∑_{i,j} ω_{ij}z^{i}Z^{j} ∈ Γ(𝔊_{U})

is called a contact Weyl form on 𝔊_{U} where z = (z^{1}, ⋯, z^{2n}) is the canonical coordinate of (ℝ^{2n}, σ_{0}).

Consider the vector field E such that

Ef = ∑_{i=1}^{2n} z^{i} \frac{∂}{∂z^{i}} f.

Then, by a direct calculation we see (cf. [OMY1])

Proposition 2.3. [τ̃, f^{#}] = ν(Ef)^{#} for ∀f ∈ C^{∞}(U)[[ν]].

2.5. Contact Weyl diffeomorphism. Let Φ : W_{U} → W_{U′} be a Weyl diffeomorphism with the induced symplectic diffeomorphism ϕ : U → U′. By Theorem 2.2, the Weyl diffeomorphism Φ is written as

Φ = A_{ϕ} ∘ exp \frac{1}{ν} ad (c + F_{3})

for certain hemitian element F_{3} ∈ Γ(W_{U,3}) without central elements where W_{U,3} = U × W_{3} and c ∈ C^{∞}(U)[[ν]]. According to Theorem 2.3, a Lie algebra bundle isomorphism Ψ : 𝔊_{U} → 𝔊_{U′} such that Ψ|_{W_{U}} = Φ is given by

Ψ = A_{ϕ} ∘ exp ad (\frac{1}{ν}(c + F_{3}) − \frac{c_{L}}{2} Log ν).

By a calculation we see Ψ^{∗}(τ̃) = τ̃ + G for certain G ∈ Γ(W_{U}). Applying Ψ to the both sides of the identity [τ̃, f^{#}] = ν(Ef)^{#} gives [τ̃ + G, Φ^{∗}(f^{#})] = νΦ^{∗}(Ef)^{#}, ∀f ∈ C^{∞}(U)[[ν]]. Then for functions f^{i} ∈ C^{∞}(U)[[ν^{2}]] such that Φ^{∗}(f^{i#}) = z^{i#}, we have

[z^{i#}, G] = νz^{i#} − νΦ^{∗}(Ef^{i})^{#}.

The Jacobi identity of the commutator [ , ] shows

[z^{i#}, g^{j#}] = [z^{j#}, g^{i#}], i, j = 1, 2, ⋯, 2n,

where we set g^{i#} = z^{i#} − Φ^{∗}(Ef^{i})^{#}. Here we can take g ∈ C^{∞}(U)[[ν^{2}]] since Φ is a Weyl diffeomorphism hence satisfies the hermitian property. Then there exists a function f ∈ C^{∞}(U)[[ν^{2}]] such that

g^{i#} = \frac{1}{ν}[z^{i#}, f^{#}] = \frac{1}{ν}[z^{i#}, G], i = 1, 2, ⋯, 2n.

Thus, we see G = f^{#} + c for certain central element c ∈ C^{∞}(U)[[ν]] and then a contact algebra bundle lift Ψ of Φ satisfies

Ψ^{∗}(τ̃) = τ̃ + f^{#} + c, for certain f ∈ C^{∞}(U)[[ν^{2}]], c ∈ C^{∞}(U)[[ν]].

Now using the contact Weyl form, we define

Definition 2.10. A Lie algebra bundle isomorphism Ψ : 𝔊_{U} → 𝔊_{U′} where U, U′ ⊂ ℝ^{2n} is called a contact Weyl diffeomorphism iff (i) the restriction Ψ|_{W_{U}} gives a Weyl diffeomorphism Ψ|_{W_{U}} : W_{U} → W_{U′} and (ii) the action on the contact Weyl form satisfies Ψ^{∗}τ̃ = τ̃+ f^{#} for a certain local Weyl function f^{#} ∈ F(W_{U}) such that f ∈ C^{∞}(U)[[ν^{2}]].

Remark that a contact Weyl diffeomorphism satisfies the hermitian property since \overline{τ̃ + f^{#}} = τ̃+f^{#}. For a given Weyl diffeomorphism, we have the following lift up theorem.

Theorem 2.6. For a Weyl diffeomorphism Φ : W_{U} → W_{U′}, there exists a contact Weyl diffeomorphism lift Ψ : 𝔊_{U} → 𝔊_{U′} such that Ψ|_{W_{U}} = Φ. Such a lift is unique up to constants c ∈ ℝ[[ν^{2}]], i.e., if Ψ and Ψ′ are contact Weyl lifts of Φ, then there exists uniquely

c = c_{0} + c_{2}ν^{2} + ⋯ + c_{2k}ν^{2k} + ⋯ ∈ ℝ[[ν^{2}]]

and one has

Ψ′ = Ψ ∘ exp ad (\frac{1}{ν}c).

Proof. According to the previous argument, we have a Lie algebra bundle lift Ψ′ such that Ψ^{′∗}(τ̃) = τ̃ + f^{#} + c, f ∈ C^{∞}(U)[[ν^{2}]], c ∈ C^{∞}(U)[[ν]]. The central term is written as c = c_{0} + c_{1}ν + ⋯ + c_{m}ν^{m} + · · ·. Then, the central element b = − ∑_{l=0,l≠1}^{∞} \frac{c_{l}}{2(l−1)}ν^{l} satisfies [\frac{1}{ν} b, τ̃] = c − c_{1}ν. Hence, a bundle isomorphism Ψ ∘ exp − ad (\frac{1}{ν}b) gives (Ψ′ ∘ exp − ad (\frac{1}{ν}b))^{∗}(τ̃) = τ̃ + f^{#} + c_{1}ν. We see easily

(Ψ′ ∘ exp − ad (\frac{1}{ν}b + \frac{c_{1}}{2} Log ν))^{∗}(τ̃) = τ̃ + f^{#}

hence, the isomorphism Ψ = Ψ′ ∘ exp − ad (\frac{1}{ν}b + \frac{c_{1}}{2} Log ν) is a contact Weyl diffeomorphism such that Ψ|_{W_{U}} = Φ. For the uniqueness it suffices to show if a contact Weyl diffeomorphism Ψ induces the identity of W_{U}, then it is written as Ψ = exp ad (\frac{1}{ν}c), where

c = c_{0} + c_{2}ν^{2} + ⋯ + c_{2k}ν^{2k} + ⋯ ∈ ℝ[[ν^{2}]]

and c is uniquely determined. Since Ψ^{∗}(τ̃) = τ̃ + f^{#} for certain local Weyl function f^{#}, f ∈ C^{∞}(U)[[ν^{2}]], we see

νz^{i#} = [τ̃ + f^{#}, z^{i#}] = [τ̃, z^{i#}] + [f^{#}, z^{i#}] = νz^{i#} + [f^{#}, z^{i#}],

which yields [f^{#}, z^{i#}] = 0, i = 1, 2, ⋯, 2n. Hence, we have {f, z^{i}} = 0, i = 1, 2, ⋯, 2n which means f is a constant in ℝ[[ν^{2}]] and is written

f^{#} = f = c_{0} + c_{2}ν^{2} + ⋯ + c_{2k}ν^{2k} + ⋯ ∈ ℝ[[ν^{2}]].

The central element b = ∑_{k=0}^{∞} b_{2k}ν^{2k} ∈ ℝ[[ν^{2}]] satisfying [\frac{1}{ν}b, τ] = c is uniquely determined. In fact, it is given as b_{2k} = −c_{2k}/2(2k − 1), k = 0, 1, 2, · · ·. Thus, we have ad (\frac{1}{ν}b)(τ̃) = [\frac{1}{ν}b, τ] = c which yields Ψ = exp ad (\frac{1}{ν}b). (QED).

For a contact lift Ψ of a Weyl diffeomorphism Φ : W_{U} → W_{U′} with symplectic diffeomorphism ϕ : U → U′, the Weyl function f^{#} given by Ψ^{∗}(τ̃) = τ̃ + f^{#} can be related to ϕ in the following manner. Let f_{0} be the lowest order term of the body part of f^{#} with respect to ν, i.e., f^{#} = f_{0}^{#}+ν^{2}f_{2}^{#}+⋯, f_{k} ∈ C^{∞}(U). Apply Ψ^{∗} to the identity [τ̃, z^{i#}] = νz^{i#}. We have [f_{0}^{#}, ϕ^{i#}] = νϕ^{i#} −ν(Eϕ^{i})^{#}, which gives {f_{0}, ϕ^{i}} = ϕ^{i} − Eϕ^{i}, i = 1, 2, ⋯, 2n. Thus, we have

Lemma 2.3. The function f_{0} is a generating function of the canonical transform ϕ, i.e., df_{0} = ϕ^{∗}θ′ − θ, where θ′ and θ are canonical one forms on U′ and U respectively.

In terms of the local coordinates, the relation above is expressed as

\frac{∂f}{∂z^{k}} = ∑_{ij} ω_{ij}ϕ^{i}(z)\frac{∂ϕ^{j}(z)}{∂z^{k}} − ∑_{i} ω_{ik}z^{i}.

Differentiating gives for k, l = 1, 2, ⋯, 2n

\frac{∂^{2}f}{∂z^{k}∂z^{l}} = ∑_{ij} ω_{ij}ϕ^{i}(z)\frac{∂^{2}ϕ^{j}(z)}{∂z^{k}∂z^{l}}. (2.14)

2.6. Existence of Weyl manifolds and Deformation quantization. Suppose we have a symplectic manifold (M, σ). Let {ϕ_{λ} : V_{λ} → U_{λ}}_{λ} be a system of canonical coordinates where {V_{λ}}_{λ} is an open covering of M and U_{λ} is an open subset of ℝ^{2n} diffeomorphic to an open ball. We write the canonical coordinate on U_{λ} as

ϕ_{λ}(p) = z_{λ}(p) = (z_{λ}^{1}(p), z_{λ}^{2}(p), ⋯, z_{λ}^{2n}(p)).

Then we have ϕ_{λ}^{∗}σ_{0} = σ and σ_{0} = ∑_{i,j} ω_{ij}dz_{λ}^{i} ∧ dz_{λ}^{j}.

Let W_{M} be a locally trivial algebra bundle over (M, σ) with the fiber isomorphic to the Weyl algebra W. Then, for each coordinate neighborhood V_{λ} of M, there is a local trivialization Φ_{λ} : W_{V_{λ}} → W_{U_{λ}}. Definition 2.11. A fiber bundle W_{M} is called a Weyl manifold, if for each V_{λ}, V_{μ} such that V_{λ} ∩ V_{μ} ≠ ∅, the patching transformation

Φ_{λμ} = Φ_{μ}Φ_{λ}^{−1} : U_{λμ} × W → U_{μλ} × W

induces a Weyl diffeomorphism, where U_{λμ} = ϕ_{λ}(V_{λ} ∩ V_{μ}) and ϕ_{λ} : (V_{λ}, σ) → (U_{λ}, σ_{0}) is the induced symplectic diffeomorphism. Each Φ_{λ} : W_{V_{λ}} → W_{U_{λ}} is called a local Weyl chart on W_{M}, and W_{U_{λ}} is called the model algebra over V_{λ}.

We have the following existence theorem (cf. [OMY1]). Theorem 2.7. For every symplectic manifold (M, σ), there exist Weyl manifolds W_{M}.

We introduce a concept of (global) Weyl functions on W_{M} as follows.

Definition 2.12. A section f̃ ∈ Γ(W_{M}) is called a Weyl functions iff for every local Weyl chart Φ_{λ} : W_{V_{λ}} → W_{U_{λ}} the pullback satisfies Φ_{λ}^{∗−1}f̃ ∈ F(W_{U_{λ}}).

Then we have (cf. [OMY1]) Proposition 2.4. There is a ℝ[[ν]]-linear isomorphism

𝔍 :C^{∞}(M)[[ν]] → F(W_{M}).

We then have an associative product ∗ among C^{∞}(M)[[ν]] by setting

f ∗ g = 𝔍^{−1}(𝔍f ∗ 𝔍g), f, g ∈ C^{∞}(M)[[ν]]].

We see Proposition 2.5. Expansion of the product f ∗ g shows

f ∗ g = fg + \frac{ν}{2}\{f, g\} + ν^{2}π_{2}(f, g) + ⋯ + ν^{k}π_{k}(f, g) + ⋯ (2.15)

where {f, g} is the Poisson bracket of a symplectic manifold (M, σ) and π_{k}(f, g) is a bidifferential operator on M.

Hence, the product ∗ among C^{∞}(M)[[ν]] gives a deformation quantization of the Poisson algebra (C^{∞}(M), {, }). We then have the existence of deformation quantization. Corollary 2.2. On every symplectic manifold (M, σ), there exists a deformation quantization of the Poisson algebra (C^{∞}(M), {, }).

An associative product ∗ among C^{∞}(M)[[ν]] with the expansion of the form (2.15) is called a star product of a symplectic manifold (M, σ).

2.7. Poincaré-Cartan class. In this section we consider a classification of Weyl manifolds on a symplectic manifold (M, σ) and give a complete invariant of Weyl manifolds (cf. [OMMY]).

For a Weyl manifold W_{M} over a symplectic manifold (M, σ), by definition we have a system of patching Weyl diffeomorphisms Φ_{λμ} = Φ_{μ} ∘ Φ_{λ}^{−1} : W_{U_{λμ}} → W_{U_{μλ}} where U_{λμ} = ϕ_{λ}(V_{λ} ∩ V_{μ}). Since Φ_{νλ}Φ_{μν}Φ_{λμ} = 1_{W_{λμν}} for V_{λ} ∩ V_{μ} ∩ V_{ν} ≠ ∅ where 1_{W_{λμν}} is the identity bundle isomorphism of W_{U_{λμν}}, U_{λμν} = ϕ_{λ}(V_{λ} ∩ V_{μ} ∩ V_{ν}), their contact Weyl lifts satisfy the relation

(2.16) Ψ_{νλ}Ψ_{μν}Ψ_{λμ} = exp ad \frac{1}{ν}c_{λμν}

for certain

c_{λμν} = c_{λμν}^{(0)} + ν^{2}c_{λμν}^{(2)} + ν^{4}c_{λμν}^{(4)} + ⋯ + ν^{2k}c_{λμν}^{(2k)} + ⋯ ∈ ℝ[[ν^{2}]]. (2.17)

Although the lifts {Ψ_{λμ}}_{λμ} are not unique and hence the choice of c_{λμν} has an ambiguity, we can show

Lemma 2.4. Each coefficient {c_{λμν}^{(2k)}}_{λμν} gives an element of Cěch 2-cocycle for every k = 0, 1, 2, · · ·. The cohomology class of each {c_{λμν}^{(2k)}}_{λμν} is independent of the choice of contact Weyl diffeomorphism lifts. Moreover, the class determined by {c_{λμν}^{(0)}}_{λμν} is equal to the 2nd cohomology class corresponding to the symplectic form σ.

Thus, a Weyl manifold W_{M} determines a series of the 2nd cohomology classes c^{(2k)}(W_{M}) = [{c_{λμν}^{(2k)}}_{λμν}], k = 0, 1, 2, ⋯ where c^{(0)}(W_{M}) = [σ].

Definition 2.13. The formal power series of the 2nd cohomology classes

c(W_{M}) = c^{(0)}(W_{M}) + ν^{2}c^{(2)}(W_{M}) + ⋯ + ν^{2k}c^{(2k)}(W_{M}) + ⋯ ∈ H^{2}(M)[[ν^{2}]]

is called the Poincaré-Cartan class of the Weyl manifold W_{M}.

We have (cf. [OMMY])

Theorem 2.8. The set of all equivalence classes of Weyl manifolds W_{M} over a symplectic manifold (M, σ) is bijective to the set of all formal power series of the 2nd cohomology classes

c = c^{(0)} + ν^{2}c^{(2)} + ⋯ + ν^{2k}c^{(2k)} + ⋯ ∈ H^{2}(M)[[ν^{2}]]

such that c^{(0)} = [σ].

3. Quantized connection

On a Weyl manifold W_{M}, we will introduce a fiberwise derivation D_{M}. We consider an eignvalue problem of D_{M}. The spectrum of D_{M} consists of the set of all nonnegative integers ℕ. The eigenspace with eigenvalue one W_{M}(1) forms a vector bundle isomorphic to the cotangent bundle. The local basis of eigenspace W_{M}(1) also generates locally the Weyl manifold. By means of these local generators of the Weyl manifold, we get a new local trivialization of the Weyl manifold which will be called a local classical chart.

Setting the tensor product bundle Λ_{M} ⊗ W_{M} we also introduce a twisted exterior derivative ∂ acting on the sections Γ(Λ_{M} ⊗ W_{M}), where Λ_{M} is the exterior algebra bundle. The square ∂^{2} identically vanishes on the sections Γ(W_{M}). We can extract the ”central element” of the square by locally extending ∂ to the exterior derivative of Γ(Λ_{V_{λ}} ⊗ 𝔊_{V_{λ}}). We give the local expression of the covariant derivative ∂ in terms with the local classical chart and we see ∂ coincides with a Fedosov connection on the Weyl manifold. We show the Poincaré-Cartan class is equal to the 2nd cohomology class determined by the curvature 2-form of the quantized connection ∂.

3.1. Degree operator field. Remark the derivation D of the Weyl algebra is written in terms of τ ∈ 𝔊 as

D = \frac{1}{ν} ad (τ) = \frac{1}{ν}[τ,]. (3.1)

Let {χ_{λ}}_{λ} be the partition of unity subordinate to the covering {V_{λ}}_{λ}.

Definition 3.1. We set the fiberwise derivation D_{M} of the Weyl algebra as

D_{M} = ∑_{λ} χ_{λ}Φ_{λ}^{−1} ∘ D ∘ Φ_{λ} : W_{M} → W_{M}. (3.2)

We call D_{M} the degree operator field of W_{M}.

Consider the local expression with respect to a Weyl chart Φ_{μ} : W_{V_{μ}} → W_{U_{μ}}

D_{μ} = Φ_{μ∗}D_{M} = Φ_{μ} ∘ D_{M} ∘ Φ_{μ}^{−1} : W_{U_{μ}} → W_{U_{μ}}. (3.3)

Then, we have the covariance D_{κ} = Φ_{κμ∗}D_{μ} = Φ_{μκ} ∘ D_{μ} ∘ Φ_{μκ}^{−1} on W_{U_{κλ}}. Substituting the D_{M} in (3.2) into D_{μ} of (3.3) shows that D_{μ} is given by gathering the degree operators D on W_{U_{λ}} by patching Weyl diffeomorphisms Φ_{λμ}

D_{μ} = ∑_{λ} χ_{λ}Φ_{λμ} ∘ D ∘ Φ_{λμ}^{−1}.

Since D = \frac{1}{ν} ad (τ) and the lifts Ψ_{λμ} and Ψ_{μλ}^{−1} differ by certain constant c ∈ ℝ[[ν^{2}]] by Theorem 2.6, the operator D_{μ} can be written as

D_{μ} = \frac{1}{ν} ad (∑_{λ} χ_{λ}Ψ_{μλ}^{∗}τ) (3.4)

where Ψ_{μλ} : 𝔊_{U_{μλ}} → 𝔊_{U_{λμ}} is a contact Weyl lift of the patching Weyl diffeomorphism Φ_{μλ} of W_{M} and we regard τ as a constant section of the trivial bundle 𝔊_{U_{λ}} = U_{λ} × 𝔊. On 𝔊_{U_{μ}}, we set a section

τ_{μ} = ∑_{λ} χ_{λ}Ψ_{μλ}^{∗}τ ∈ Γ(𝔊_{U_{μ}}).

By Remark 2.1, we see τ_{μ} = τ + c_{μ,0} + f, for certain c_{μ,0} ∈ C^{∞}(U_{μ}) and f ∈ Γ(U_{μ} × W_{3}) which is hermitian, i.e., of the form (2.8). Then by virtue of Lemma 2.2, there exists a ν-automorphism

Ψ_{μ} = exp ad (\frac{1}{ν} \frac{c_{μ,0}}{2} + \frac{1}{ν}g) : 𝔊_{U_{μ}} → 𝔊_{U_{μ}}

where g ∈ Γ(U_{μ} × W_{3}) is hermitian and c_{μ,0} ∈ C^{∞}(U_{μ}), such that Ψ_{μ}(τ) = τ_{μ}. Applying Ψ_{μ} to the CCR relation and the relation (2.9), we have for hermitian elements Z_{μ}^{i} = Ψ_{μ}(Z^{i}) ∈ Γ(W_{U_{λ}}), i = 1, 2, ⋯, 2n Lemma 3.1. On each 𝔊_{U_{μ}}, it holds

[τ_{μ}, Z_{μ}^{i}] = νZ_{μ}^{i}, i = 1, 2, ⋯, 2n

and the CCR

[ν, Z_{μ}^{i}] = 0, [Z_{μ}^{i}, Z_{μ}^{j}] = νΛ^{ij}, i, j = 1, 2, ⋯, 2n.

The spectrum of D acting on the Weyl algebra W is nothing but all degrees of the Weyl algebra and is the set of all nonnegative integers ℕ. Then the covariance of the local degree operator field (3.4) shows

Lemma 3.2. The spectrum of D_{λ} consists of ℕ at each point z ∈ U_{λ}. One can take a basis Z_{λ}^{1}, Z_{λ}^{2}, ⋯, Z_{λ}^{2n} such that the expansion of each element starts from the generator Z^{1}, Z^{2}, ⋯, Z^{2n} respectively; i.e.,

Z_{λ}^{i} = Z^{i} + ∑_{j,k} a_{λ,jk}^{i}Z^{j}Z^{k} + O(3), a_{λ}^{i}, a_{λ,jk}^{i}, ⋯ ∈ C^{∞}(U_{λ}). (3.5)

Then, the basis Z_{λ}^{1}, Z_{λ}^{2}, ⋯, Z_{λ}^{2n} ∈ Γ(W_{U_{λ}}) of the eigenspace with eigenvalue one W_{U_{λ}}(1)_{z} at each point z ∈ U_{λ} satisfies the CCR

[ν, Z_{λ}^{i}] = 0, [Z_{λ}^{i}, Z_{λ}^{j}] = νΛ^{ij}.

The covariance of local expressions yields

Φ_{λκ∗}Z_{λ}^{i} = ∑_{j} \frac{∂ϕ_{κλ}^{i}}{∂z_{κ}^{j}} Z_{κ}^{j} (3.6)

on W_{U_{κλ}}.

Inverting the relation (3.5) gives an expansion

Z^{i} = Z_{λ}^{i} − ∑_{j,k} a_{λ,jk}^{i}Z_{λ}^{j}Z_{λ}^{k} + O(3) (3.7)

Applying Φ_{λκ} to the both sides of (3.7), we see

Proposition 3.1. The coefficients

Γ_{λ,jk}^{i} = 2a_{λ,jk}^{i}

defines a symplectic connection on M.

Proof. Apply Φ_{μλ}^{∗} to both sides of (3.6). Then, we see

∑_{j} \frac{∂ϕ_{μλ}^{i}}{∂z_{μ}^{j}} Z_{μ}^{j} = Φ_{μλ}^{∗}Z^{i} + ∑_{jk} a_{λ,jk}^{i}Φ_{μλ}^{∗}Z^{j}Φ_{μλ}^{∗}Z^{k} + O(3).

Substitute Z_{μ}^{j} = Z^{j} + ∑_{lm} a_{μ,lm}^{j}Z^{l}Z^{m} + O(3) and

Φ_{μλ}^{∗}Z^{i} = ∑_{j} \frac{∂ϕ_{μλ}^{i}}{∂z_{μ}^{j}} Z^{j} + \frac{1}{2} ∑_{lm} \frac{∂^{2}ϕ_{μλ}^{i}}{∂z_{μ}^{l}∂z_{μ}^{m}} Z^{l}Z^{m} + O(3)

into the equality above and compare the quadratic terms. Then we see

∑_{j} \frac{∂ϕ_{μλ}^{i}}{∂z_{μ}^{j}} a_{μ,lm}^{j} = \frac{1}{2} \frac{∂^{2}ϕ_{μλ}^{i}}{∂z_{μ}^{l}∂z_{μ}^{m}} + ∑_{jk} a_{λ,jk}^{i} \frac{∂ϕ_{μλ}^{j}}{∂z_{μ}^{l}} \frac{∂ϕ_{μλ}^{k}}{∂z_{μ}^{m}} .

Thus, Γ_{λ,lm}^{i} = 2a_{λ,lm}^{i} defines a connection. We also substitute the expansion of Z_{μ}^{i} into the CCR relation and we get by looking at the degree 3 terms Γ_{μ,ij}^{m}ω_{αk} + Γ_{μ,ik}^{m}ω_{mj} = 0 which indicates the connection is symplectic. (QED)

3.2. Classical charts. Lemma 3.1 yields the system τ_{μ}, ν, Z_{μ}^{1}, Z_{μ}^{2}, ⋯, Z_{μ}^{2n} ∈ Γ(𝔊_{U_{μ}}) forms generators of 𝔊_{U_{μ}} at each fiber of z ∈ U_{μ} and hence every element G of 𝔊_{U_{μ}} is written as

G = kτ_{μ} + ∑_{l,α} a_{μ,l,α}ν^{l}Z_{μ}^{α}, k, a_{μ,l,α} ∈ ℝ.

Especially the system Z_{λ}^{1}, Z_{λ}^{2}, ⋯, Z_{λ}^{2n} generates W_{U_{λ}} and every element F of the each fiber of W_{U_{λ}} can be expressed as the infinite sum of monomials F = ∑_{k,α} a_{k,α}νZ_{λ}^{α}(z), z ∈ U_{λ} with respect to the generators ν, Z_{λ}^{1}, Z_{λ}^{2}, ⋯, Z_{λ}^{2n}, since the basis Z_{λ}^{1}, Z_{λ}^{2}, ⋯, Z_{λ}^{2n} ∈ Γ(W_{U_{λ}}) satisfies the CCR. Hence we have an another local trivialization

C_{λ} : W_{U_{λ}} → W^{∘}_{U_{λ}}

by setting

C_{λ}(∑_{k,α} a_{k,α}νZ_{λ}^{α}) = ∑_{k,α} a_{k,α}νZ^{α}

and a new local trivialization C̃_{μ} : 𝔊_{U_{μ}} → 𝔊^{∘}_{U_{μ}}by

C̃_{μ}(kτ_{μ} + ∑_{l,α} a_{μ,l,α}ν^{l}Z_{μ}^{α}) = kτ + ∑_{l,α} a_{μ,l,α}ν^{l}Z^{α}

which is the contact algebra isomorphism extension of C_{μ}, i.e. C̃_{μ}|_{𝔊_{Uμ}} = C_{μ}. The expansion (3.7) gives

C_{λ}(Z^{i}) = Z^{i} − ∑_{j,k} a_{λ,jk}^{i}Z^{j}Z^{k} + O(3).

Definition 3.2. We call the local trivialization C_{λ} : W_{U_{λ}} → W^{∘}_{U_{λ}} a classical

chart, C̃_{μ} : 𝔊_{U_{μ}} → 𝔊^{∘}_{U_{μ}}a classical contact chart and we also call W^{∘}_{U_{λ}}a classical model algebra.

We remark here that as a set W^{∘}_{U_{λ}} = U_{λ} × W is the same as W_{U_{λ}} but with

the different transformations C_{λκ} = C_{μ} ∘ Φ_{λκ} ∘ C_{λ}^{−1} : W^{∘}_{U_{λκ}} → W^{∘}_{U_{κλ}}satisfying

C_{λκ}(Z^{i}) = ∑_{j} \frac{∂ϕ_{κλ}^{i}}{∂z_{κ}^{j}} Z^{j}, (3.8)

where ν, Z^{1}, Z^{2}, ⋯, Z^{2n} are generators of the Weyl algebra W. The transformation of classical contact charts are given by

C̃_{λμ} = C̃_{μ} ∘ Ψ_{λμ} ∘ C̃_{λ}^{−1} : 𝔊^{∘}_{U_{λμ}} → 𝔊^{∘}_{U_{μλ}}.

For the transformation of τ, we have Lemma 3.3. For each transformation C̃_{λμ} of classical contact chart, we have

C̃_{λμ}^{∗}τ = τ + h_{λμ}(ν)

where h_{λμ}(ν) ∈ C^{∞}(U_{λμ})[[ν^{2}]] is a central element of Γ(W_{U_{λμ}}).

Proof. Consider the pull-back Ψ_{κμ}^{∗}τ_{μ}. The composition Ψ_{μλ}Ψ_{κμ} is a contact Weyl diffeomorphism lift of Φ_{κλ} and hence by Theorem 2.6 the uniqueness induces that there exists a constant c_{κμλ} ∈ ℝ[[ν^{2}]] such that

Ψ_{μλ}Ψ_{κμ} = Ψ_{κλ}e^{ad ( \frac{1}{ν} c_{κμλ})}. (3.9)

Remark here that {c_{κμλ}}_{κμλ} is the Cĕch 2-cocycle of the Poincaré-Cartan class in (2.16). Thus, we see

Ψ_{κμ}^{∗}τ_{μ} = ∑_{λ} Ψ_{κμ}^{∗}χ_{λ}Ψ_{μλ}^{∗}τ = ∑_{λ} χ_{λ}Ψ_{κλ}^{∗}e^{ad ( \frac{1}{ν} c_{κμλ})}τ

= ∑_{λ} χ_{λ}Ψ_{κλ}^{∗}(τ + K_{κμλ}) = ∑_{λ} χ_{λ}Ψ_{κλ}^{∗}τ + ∑_{λ} χ_{λ}K_{κμλ}

= τ_{κ} + ∑_{λ} χ_{λ}K_{κμλ}

where

K_{κμλ} = ad (\frac{1}{ν}c_{κμλ})τ = 2c_{κμλ} + \frac{1}{ν}[c_{κμλ}, τ]
= 2c_{κμλ}^{(0)} − 2ν^{2}c_{κμλ}^{(2)} − ⋯ − 2(2k − 1)ν^{2k}c_{κμλ}^{(2k)} − ⋯ ∈ ℝ[[ν^{2}]].

Setting

h_{κμ}(ν) = ∑_{λ} χ_{λ}K_{κμλ} (3.10)

we see

(3.11)  C̃_{κμ}^{∗}τ = C̃_{k}^{∗−1}Ψ_{κμ}^{∗}C̃_{μ}^{∗}τ = C̃_{k}^{∗−1}Ψ_{κμ}^{∗}τ_{μ} = C̃_{k}^{∗−1}(τ_{κ} + h_{κμ}(ν))
= τ + h_{κμ}(ν)

which shows the lemma. (QED)

By the lemma above, we have the following transformation rules of classical contact algebra bundles.

C̃_{λμ}^{∗}τ = τ + h_{λμ}(ν)

C̃_{λμ}^{∗}Z^{i} = ∑_{j} \frac{∂ϕ_{λμ}^{i}}{∂z_{λ}^{j}} Z^{j}, i = 1, 2, ⋯, 2n

where we regard τ, ν, Z^{1}, ⋯, Z^{2n} as constant sections of the bundle 𝔊_{U_{μ}}.

3.3. Twisted exterior derivative. Now we introduce a twisted exterior derivative or a quantized connection ∂. Let us consider the tensor bundle Λ_{M} ⊗ W_{M} where Λ_{M} is the exterior algebra bundle of M. The tensor bundle has an algebraic structure such that ξ ⊗ a ∗ η ⊗ b = ξ ∧ η ⊗ a ∗ b, for ξ, η ∈ Λ_{M}, a, b ∈ W_{M}. By the bracket

[ξ ⊗ a, η ⊗ b] = ξ ⊗ a ∗ η ⊗ b − (−1)^{|ξ||η|}η ⊗ b ∗ ξ ⊗ a (3.12)

the algebra becomes a graded Lie algebra, where |ξ| means the degree of exterior differential form. By a direct calculation, we see

Lemma 3.4. For F_{1}, F_{2}, F_{3} ∈ Λ_{M} ⊗ W_{M}, we have

[F_{1}, F_{2}] = −(−1)^{|F_{1}||F_{2}|}[F_{2}, F_{1}]

and

[F_{1}, [F_{2}, F_{3}]] + (−1)^{|F_{3}||F_{2}|+|F_{3}||F_{1}|}[F_{3}, [F_{1}, F_{2}]]

+ (−1)^{|F_{1}||F_{2}|+|F_{3}||F_{1}|}[F_{2}, [F_{3}, F_{1}]] = 0.

In what follows we omit ⊗ of the expression of elements of Λ_{M} ⊗ W_{M}, e.g., ξa = ξ ⊗ a. For each local Weyl chart W_{U_{λ}}, put

Θ_{λ} = ∑_{ij} ω_{ij}dz_{λ}^{i}Z^{j}. (3.13)

We set a derivation δ_{λ} on Λ_{U_{λ}} ⊗ W_{U_{λ}} as

δ_{λ} = \frac{1}{ν} ad (Θ_{λ}) = \frac{1}{ν}[Θ, ],

especially we have δ_{λ}ν = 0 and

δ_{λ}Z^{k} = ∑_{ij} ω_{ij}dz_{λ}^{i} \frac{1}{ν} ad (Z^{j})(Z^{k}) = dz_{λ}^{k}, k = 1, 2, ⋯, 2n.

We now set a twisted exterior derivative ∂_{λ} : Γ(Λ^{l}(U_{λ})⊗W_{U_{λ}}) → Γ(Λ^{l+1}(U_{λ})⊗ W_{U_{λ}}) by

∂_{λ}Z^{k} = −δ_{λ}Z^{k}, k = 1, 2, ⋯, 2n,
∂_{λ}ν = −δ_{λ}ν = 0

and for an exterior differential form ξ ∈ Λ(U_{λ})

∂_{λ}ξ = dξ.

We write this symbolically as

∂_{λ} = d − δ_{λ}.

Using these equalities and the definition of the local Weyl functions, we see

Proposition 3.2.

∂_{λ}^{2} = 0

and

∂_{λ}f^{#} = 0

for every f^{#} ∈ F(W_{U_{λ}}).

Proof. In fact, ∂_{λ}^{2}Z^{i} = ∂_{λ}(−dz_{λ}^{i}) = −d^{2}z_{λ}^{i} = 0, hence we have ∂_{λ}^{2} = 0. Since z_{λ}^{i#} = z_{λ}^{i} + Z^{i}, we see ∂_{λ}z_{λ}^{i#} = dz_{λ}^{i} − dz_{λ}^{i} = 0 and by the fact that the polynomial of z^{i#′}’s are dense in F(W_{U_{λ}}) and ∂_{λ} is continuous we have ∂_{λ}f^{#} = 0 for ∀f^{#} ∈ F(W_{U_{λ}}). (QED).

We will see the transformation rule of ∂_{λ}’s by the patching Weyl diffeomorphisms of W_{M}. We consider Φ_{λμ∗}∂_{λ} = Φ_{λμ}^{∗−1} ∘ ∂_{λ} ∘ Φ_{λμ}^{∗}. It suffices to check for the generators ν, Z^{1}, ⋯, Z^{2n}. We calculate as

(Φ_{λμ∗}∂_{λ})Z^{k} = Φ_{λμ}^{∗−1} ∘ ∂_{λ} ∘ Φ_{λμ}^{∗}(z_{μ}^{k#} − z_{μ}^{k}) = −dz_{μ}^{k}

and (Φ_{λμ∗}∂_{λ})ν = Φ_{λμ}^{∗−1} ∘ ∂_{λ} ∘ Φ_{λμ}^{∗}ν = 0 since Φ_{λμ} is a Weyl diffeomorphism. Thus, we have

Proposition 3.3. Φ_{λμ∗}∂_{λ} = ∂_{μ} for all λ, μ such that V_{λ} ∩ V_{μ} ≠ ∅.

Hence, we have

Definition 3.3. We define a twisted exterior derivative ∂ : Γ(Λ_{M} ⊗ W_{M}) → Γ(Λ_{M} ⊗ W_{M}) by

Φ_{λ∗}∂ = Φ_{λ}^{∗−1} ∘ ∂ ∘ Φ_{λ}^{∗} = ∂_{λ}

for each local Weyl chart W_{U_{λ}}.

Then, we have

Theorem 3.1. (i) ∂^{2} = 0. (ii) A section s ∈ Γ(W_{M}) is a Weyl function iff ∂s = 0.

3.4. Extended derivation to contact algebra. In this section, we consider to extend the twisted local exterior derivation of Weyl manifold ∂_{λ} : Γ(Λ^{l}(U_{λ})⊗ W_{U_{λ}}) → Γ(Λ^{l+1}(U_{λ}) ⊗ W_{U_{λ}}) to the derivation acting on Γ(Λ(U_{λ}) ⊗ 𝔊_{U_{λ}}).

First we set a derivation δ′_{λ} : Λ^{l}(U_{λ}) ⊗ 𝔊_{U_{λ}} → Λ^{l+1}(U_{λ}) ⊗ 𝔊_{U_{λ}} by

δ′_{λ} = ad (\frac{1}{ν}Θ_{λ}), Θ_{λ} = ∑_{ij} ω_{ij}dz_{λ}^{i}Z^{j}

and set

∂′_{λ} = d − δ′_{λ} : Γ(Λ^{l}(U_{λ}) ⊗ 𝔊_{U_{λ}}) → Γ(Λ^{l+1}(U_{λ}) ⊗ 𝔊_{U_{λ}}).

Here we calculate as [\frac{1}{ν}Z^{i}, s] = \frac{1}{ν}[Z^{ι}, s] for a section of Weyl algebra bundle s ∈ Γ(W_{U_{λ}}) and

[\frac{1}{ν}Z^{i}, τ] = [\frac{1}{ν}, τ]Z^{i} + \frac{1}{ν}[Z^{i}, τ] = 2Z^{i} − Z^{i} = Z^{i}

for i = 1, 2, ⋯, 2n. It easy to see the restriction of ∂′_{λ} to the Weyl algebra bundle is ∂_{λ};

∂′_{λ}|Γ(Λ^{l}(U_{λ}) ⊗ W_{U_{λ}}) = ∂_{λ}

since δ′_{λ}|W_{U_{λ}} = δ_{λ} and to see

∂′_{λ}(τ) = −δ′_{λ}(τ) = −[\frac{1}{ν}Θ_{λ}, τ] = −2Θ_{λ} − \frac{1}{ν}[Θ_{λ}, τ] = −Θ_{λ}.

Now we define the extension ˜∂_{λ} of the twisted local exterior derivation ∂_{λ} on Γ(Λ(U_{λ}) ⊗ W_{U_{λ}}) to Γ(Λ(U_{λ}) ⊗ 𝔊_{U_{λ}}) by Definition 3.4. We define a twisted exterior derivative ˜∂_{λ} : Γ(Λ^{l}(U_{λ})⊗𝔊_{U_{λ}}) → Γ(Λ^{l+1}(U_{λ}) ⊗ 𝔊_{U_{λ}}) by

˜∂_{λ} = ∂′_{λ} + \frac{1}{2}θ_{λ} ad (\frac{1}{ν})

where [\frac{1}{ν}, s] = 0 for s ∈ Γ(W_{U_{λ}}) and [\frac{1}{ν}, τ] = 2 and where

θ_{λ} = ∑_{ij} ω_{ij}z_{λ}^{i}dz_{λ}^{j}

is the canonical one form on U_{λ} ,i.e., the one form satisfying dθ_{λ} = σ_{0}.

Remark that for τ̃ = τ + ∑_{ij} ω_{ij}z_{λ}^{i}Z^{j} we have

∂′_{λ}τ̃ = ∂′_{λ}τ + ∂_{λ}(∑_{ij} ω_{ij}z_{λ}^{i}Z^{j})

= −Θ_{λ} + Θ_{λ} − ∑_{ij} ω_{ij}z_{λ}^{i}dz_{λ}^{j} = −θ_{λ}.

Hence, we have

Proposition 3.4. (i) ˜∂_{λ}^{2} = 0. (ii) ˜∂_{λ}τ̃ = 0 and ˜∂_{λ}f^{#} = 0.

In what follows, we consider the transformation of the extension by contact Weyl lifts. Let Ψ_{λμ} : 𝔊_{U_{λμ}} → 𝔊_{U_{μλ}} be a contact Weyl diffeomorphism lift of the patching Weyl diffeomorphism Φ_{λμ} : W_{U_{μλ}} → W_{U_{μλ}}. We investigate Ψ_{λμ∗} ˜∂_{λ} = Ψ_{λμ}^{∗−1} ∘ ˜∂_{λ} ∘ Ψ_{λμ}^{∗}. First we remark

˜∂_{μ}Z^{i} = ∂_{μ}Z^{i} = ∂_{μ}(z_{μ}^{i#} − z_{μ}^{i}) = −dz_{μ}^{i}

and

˜∂_{μ}τ = ˜∂_{μ}(τ̃ − ∑_{ij} ω_{ij}z_{μ}^{i}Z^{j}) = −∂_{μ} ∑_{ij} ω_{ij}z_{μ}^{i}Z^{j}

= −Θ_{μ} + ∑_{ij} ω_{ij}z_{μ}^{i}dz_{μ}^{j} = −Θ_{μ} + θ_{μ}.

On the other hand, we have

Ψ_{λμ∗} ˜∂_{λ}(Z^{i}) = Φ_{λμ∗}∂_{λ}(Z^{i}) = ∂_{μ}(Z^{i}) = −dz_{μ}^{i}.

We also calculate as

Ψ_{λμ∗} ˜∂_{λ}(τ) = Ψ_{λμ}^{∗−1} ∘ ˜∂_{λ} ∘ Ψ_{λμ}^{∗}(τ̃ − ∑_{ij} ω_{ij}z_{μ}^{i}Z^{j}).

Remark that

Ψ_{λμ}^{∗}(τ̃ − ∑_{ij} ω_{ij}z_{μ}^{i}Z^{j}) = τ̃ + f^{#} − ∑_{ij} ω_{ij}ϕ_{λμ}^{∗}z_{μ}^{i}Φ_{λμ}^{∗}Z^{j}

and then

˜∂_{λ} ∘ Ψ_{λμ}^{∗}(τ̃ − ∑_{ij} ω_{ij}z_{μ}^{i}Z^{j}) = ˜∂_{λ}(τ̃ + f^{#} − ∑_{ij} ω_{ij}ϕ_{λμ}^{∗}z_{μ}^{i}Φ_{λμ}^{∗}Z^{j})

= −∂_{λ} ∑_{ij} ω_{ij}ϕ_{λμ}^{∗}z_{μ}^{i}Φ_{λμ}^{∗}Z^{j}

Thus, we have

Ψ_{λμ∗} ˜∂_{λ}(τ) = Ψ_{λμ∗} ˜∂_{λ}(τ̃ − ∑_{ij} ω_{ij}z_{μ}^{i}Z^{j}) = −Φ_{λμ}^{∗−1}(∂_{λ} ∑_{ij} ω_{ij}ϕ_{λμ}^{∗}z_{μ}^{i}Φ_{λμ}^{∗}Z^{j})

= −Φ_{λμ}^{∗−1}(∑_{ij} ω_{ij}dϕ_{λμ}^{∗}z_{μ}^{i}Φ_{λμ}^{∗}Z^{j} + ∑_{ij} ω_{ij}ϕ_{λμ}^{∗}z_{μ}^{i}∂_{λ}Φ_{λμ}^{∗}Z^{j})

= − ∑_{ij} ω_{ij}dz_{μ}^{i}Z^{j} − ∑_{ij} ω_{ij}z_{μ}^{i}Φ_{λμ}^{∗−1}∂_{λ}Φ_{λμ}^{∗}Z^{j} = −Θ_{μ} + θ_{μ}

since Φ_{λμ}^{∗−1}∂_{λ}Φ_{λμ}^{∗} = Φ_{λμ∗}∂_{λ} = ∂_{μ}. Hence, we have the following covariance of the operators.

Proposition 3.5. Ψ_{λμ∗} ˜∂_{λ} = ˜∂_{μ} for each local trivialization Λ(U_{μλ}) ⊗ 𝔊_{U_{μλ}}.

3.5. Classical charts and Fedosov Connection. In this section, we intro-

duce classical charts 𝔊^{∘}_{U_{λ}} of the contact algebra bundle by extending W^{∘}_{U_{λ}} and then we consider to express the local twisted exterior derivative ˜∂_{λ} with respect

to the classical chart 𝔊^{∘}_{U_{λ}}.

3.5.1. Classical coordinate expression of the twisted exterior derivative. In this

section, we consider ˜∂^{∘}_{λ}, the transformation of the twisted exterior derivative to the classical contact chart, given by

˜∂^{∘}_{λ} = C̃_{λ∗} ˜∂_{λ} = C̃_{λ}^{∗−1} ˜∂_{λ}C̃_{λ}^{∗}.

In order to write the shape of ˜∂^{∘}_{λ}, it is enough to investigate the action on the

generators, ˜∂^{∘}_{λ}Z^{i}, i = 1, 2, ⋯, 2n and ˜∂^{∘}_{λ}τ. For this purpose, we calculate as

˜∂^{∘}_{λ}Z^{i} = C̃_{λ}^{∗−1} ˜∂_{λ}C̃_{λ}^{∗}Z^{i} = C̃_{λ}^{∗−1} ˜∂_{λ}Z_{λ}^{i}

and

˜∂^{∘}_{λ}τ = C̃_{λ}^{∗−1} ˜∂_{λ}τ_{λ}.

Recall that on a Weyl chart W_{U_{λ}} the expansion of Z_{λ}^{1}, ⋯, Z_{λ}^{2n} is given in terms of the generators ν, Z^{1}, Z^{2}, ⋯, Z^{2n} as

Z_{λ}^{i} = Z^{i} + ∑_{j,k} a_{λ,jk}^{i}Z^{j}Z^{k} + O(3), a_{λ,jk}^{i}, ⋯ ∈ C^{∞}(U_{λ}).

We now consider the expansion of τ_{λ} in terms of τ and ν, Z^{1}, ⋯, Z^{2n}. From Remark 2.2, we see

τ_{λ} = τ + c_{λ,0} + f_{λ}

where c_{λ,i} ∈ C^{∞}(U_{λ}) and f_{λ} ∈ Γ(W_{U_{λ},3}) is hermitian, W_{U_{λ},3} = U_{λ} × W_{3}. For simplicity, we set Z_{λ}^{i} = Z^{i} + H_{λ}^{i} and τ_{λ} = τ + T_{λ} where we put

H_{λ}^{i} = \frac{1}{2} ∑_{j,k} Γ_{λ,jk}^{i}Z^{j}Z^{k} + O(3) ∈ Γ(W_{U_{λ},2}), a_{λ,jk}^{i}, ⋯ ∈ C^{∞}(U_{λ})

where Γ_{λ,jk}^{i} is the coefficient of the symplectic connection given in Proposition (3.1) and

T_{λ} = c_{λ,0} + f_{λ}.

Apply ˜∂_{λ} to Z_{λ}^{i} and τ_{λ} by using H_{λ}^{i} and T_{λ}. Setting ˜∂_{λ}H_{λ}^{i} = 𝔷_{λ}^{i} + ℨ_{λ}^{i} and ˜∂_{λ}T_{λ} = 𝔱_{λ} + 𝔗_{λ}, we write

˜∂_{λ}Z_{λ}^{i} = ˜∂_{λ}Z^{i} + ˜∂_{λ}H_{λ}^{i} = −dz_{λ}^{i} + 𝔷_{λ}^{i} + ℨ_{λ}^{i},
˜∂_{λ}τ_{λ} = ˜∂_{λ}τ + ˜∂_{λ}T_{λ} = θ_{λ} − Θ_{λ} + 𝔱_{λ} + 𝔗_{λ}

where 𝔷_{λ}^{i} and 𝔱_{λ} are the central part of ˜∂_{λ}H_{λ}^{i} and ˜∂_{λ}T_{λ} respectively and ℨ_{λ}^{i} and 𝔗_{λ} are noncentral part of ˜∂_{λ}H_{λ}^{i} and ˜∂_{λ}T_{λ} respectively. We have 𝔷_{λ}^{i} = da_{λ,(2)}^{i}ν^{2} + O(6) ∈ Γ(ν^{2}Λ_{U_{λ}}^{1}[[ν^{2}]]) for certain a_{λ,(2)}^{i} ∈ C^{∞}(U_{λ}) and ℨ_{λ}^{i} = − ∑_{j,k} Γ_{λ,jk}^{i}dz_{λ}^{j}Z^{k} + O(2) ∈ Γ(Λ_{U_{λ}}^{1} ⊗ W_{U_{λ}}) and

𝔱_{λ} = dc_{λ,0} + O(2) ∈ Γ(Λ_{U_{λ}}^{1}[[ν^{2}]]). (3.14)

We also remark ˜∂_{λ}τ = θ_{λ} − Θ_{λ} where θ_{λ} = ∑_{ij} ω_{ij}z_{λ}^{i}dz_{λ}^{j} is the canonical one form and Θ_{λ} = ∑_{ij} ω_{ij}dz_{λ}^{i}Z^{j}. Consider the relation [Z_{λ}^{i}, Z_{λ}^{j}] = νΛ^{ij}, [τ_{λ}, Z_{λ}^{i}] = νZ_{λ}^{i} and apply ˜∂_{λ} both sides of each equation. We see

[˜∂_{λ}Z_{λ}^{i}, Z_{λ}^{i}] + [Z_{λ}^{i}, ˜∂_{λ}Z_{λ}^{j}] = [ℨ_{λ}^{i}, Z_{λ}^{j}] + [Z_{λ}^{i}, ℨ_{λ}^{j}] = 0.

Then, there exists uniquely the element

γ_{λ}^{(2)} = −\frac{1}{2} ∑_{k,l,m,p} Γ_{λ,kl}^{p}ω_{pm}dz_{λ}^{k}Z_{λ}^{l}Z_{λ}^{m} + γ̃_{λ}^{(3)}, γ̃_{λ}^{(3)} ∈ Γ(Λ_{U_{λ}}^{1} ⊗ W_{U_{λ},3})

where W_{U_{λ},3} = U_{λ} × W_{3} without central elements such that

ℨ_{λ}^{i} = \frac{1}{ν} ad (γ_{λ}^{(2)})(Z_{λ}^{i}), i = 1, 2, ⋯, 2n.

Notice here that the calculation above and in the sequel we use the generators ν, Z_{λ}^{1}, ⋯, Z_{λ}^{2n} instead of ν, Z^{1}, ⋯, Z^{2n} by means of the transformation (3.7). Here we also use that Γ_{λ,kl}^{p}ω_{pm} is completely symmetric in (k, l, m) with respect to canonical coordinate since {Γ_{λ,kl}^{i}} gives a symplectic connection. By a direct calculation we see

\frac{1}{ν} ad (− ∑_{ij} dz_{λ}^{i}ω_{ij}Z_{λ}^{j} + ∑_{ij} 𝔷_{λ}^{i}ω_{ij}Z_{λ}^{j})(Z_{λ}^{k}) = −dz_{λ}^{k} + 𝔷_{λ}^{k}.

Hence, writing

γ_{λ}^{(1)} = − ∑_{ij} dz_{λ}^{i}ω_{ij}Z_{λ}^{j} + ∑_{ij} 𝔷_{λ}^{i}ω_{ij}Z_{λ}^{j} + γ_{λ}^{(2)} ∈ Γ(Λ_{U_{λ}}^{1} ⊗ W_{U_{λ}})

we see

˜∂_{λ}Z_{λ}^{i} = \frac{1}{ν} ad (γ_{λ}^{(1)} + γ_{λ}^{(0)})(Z_{λ}^{i}), i = 1, 2, ⋯, 2n

where γ_{λ}^{(0)} ∈ Γ(Λ_{U_{λ}}[[ν]]) is an arbitrary central element.

A. Yoshioka 202

Next we consider the identity

˜∂_{λ}[τ_{λ}, Z_{λ}^{i}] = [˜∂_{λ}τ_{λ}, Z_{λ}^{i}] + [τ_{λ}, ˜∂_{λ}Z_{λ}^{i}] = ν ˜∂_{λ}Z_{λ}^{i}.

Then the noncentral part −Θ_{λ} + 𝔗_{λ} and the central part θ_{λ} + 𝔱_{λ} of ˜∂_{λ}τ_{λ} are given as follows: Lemma 3.5. We have

ad (\frac{1}{ν}γ_{λ}^{(1)})(τ_{λ}) = [\frac{1}{ν}γ_{λ}^{(1)}, τ_{λ}] = −Θ_{λ} + 𝔗_{λ}.

Moreover, there exists uniquely γ_{λ}^{(0)} ∈ Γ(Λ_{U_{λ}}[[ν^{2}]]) such that

ad (\frac{1}{ν}γ_{λ}^{(0)})(τ_{λ}) = [\frac{1}{ν}γ_{λ}^{(0)}, τ_{λ}] = θ_{λ} + 𝔱_{λ}.

Proof. For the first equation, we see

[\frac{1}{ν}γ_{λ}^{(1)}, τ_{λ}] = 2γ_{λ}^{(1)} + \frac{1}{ν}[γ_{λ}^{(1)}, τ_{λ}]

and we see the terms given by replacing ˜∂_{λ}τ_{λ} with 2γ_{λ}^{(1)}+\frac{1}{ν}[γ_{λ}^{(1)}, τ_{λ}] in [˜∂_{λ}τ_{λ}, Z_{λ}^{i}]+ [τ_{λ}, ˜∂_{λ}Z_{λ}^{i}] are equal to

[2γ_{λ}^{(1)} + \frac{1}{ν}[γ_{λ}^{(1)}, τ_{λ}], Z_{λ}^{i}] + [τ_{λ}, \frac{1}{ν}[γ_{λ}^{(1)}, Z_{λ}^{i}]]

= 2[γ_{λ}^{(1)}, Z_{λ}^{i}] + \frac{1}{ν}[[γ_{λ}^{(1)}, τ_{λ}], Z_{λ}^{i}] − 2[γ_{λ}^{(1)}, Z_{λ}^{i}] + \frac{1}{ν}[τ_{λ}, [γ_{λ}^{(1)}, Z_{λ}^{i}]]

= \frac{1}{ν}[Z_{λ}^{i}, [τ_{λ}, γ_{λ}^{(1)}]] + \frac{1}{ν}[τ_{λ}, [γ_{λ}^{(1)}, Z_{λ}^{i}]] + \frac{1}{ν}[γ_{λ}^{(1)}, [Z_{λ}^{i}, τ_{λ}]] + \frac{1}{ν}[γ_{λ}^{(1)}, [τ_{λ}, Z_{λ}^{i}]]

= \frac{1}{ν}[γ_{λ}^{(1)}, [τ_{λ}, Z_{λ}^{i}]] = [γ_{λ}^{(1)}, Z_{λ}^{i}] = ν ˜∂_{λ}Z_{λ}^{i}

Hence, Θ_{λ} + 𝔗_{λ}, the noncentral part of ˜∂_{λ}τ_{λ} and ad (\frac{1}{ν}γ_{λ}^{(1)})(τ_{λ}) = [\frac{1}{ν}γ_{λ}^{(1)}, τ_{λ}] which is obviously noncentral satisfies the same identity [˜∂_{λ}τ_{λ}, Z_{λ}^{i}]+[τ_{λ}, ˜∂_{λ}Z_{λ}^{i}] = ν ˜∂_{λ}Z_{λ}^{i}, thus we have Θ_{λ} + 𝔗_{λ} = [\frac{1}{ν}γ_{λ}^{(1)}, τ_{λ}].

For the second equality, we calculate

[\frac{1}{ν}γ_{λ}^{(0)}, τ_{λ}] = 2γ_{λ}^{(0)} + \frac{1}{ν}[γ_{λ}^{(0)}, τ_{λ}] = θ_{λ} + 𝔱_{λ}.

Since 𝔱_{λ} ∈ Γ(Λ_{U_{λ}})[[ν^{2}]], there exists uniquely γ_{λ}^{(0)} ∈ Γ(Λ_{U_{λ}})[[ν^{2}]] such that

[\frac{1}{ν}γ_{λ}^{(0)}, τ_{λ}] = θ_{λ} + 𝔱_{λ}.

Thus we get the desired results. (QED).

Notice (3.14) gives θ_{λ} + 𝔱_{λ} = θ_{λ} + dc_{λ,0} + O(ν^{2}). Then we see

γ_{λ}^{(0)} = \frac{1}{2}(θ_{λ} + dc_{λ,0}) + ν^{2}ξ_{λ,2} + ⋯, ξ_{λ,2k} ∈ Λ^{1}(U_{λ}). (3.15)

By the lemma above, we have

˜∂_{λ}τ_{λ} = ad (\frac{1}{ν}(γ_{λ}^{(1)} + γ_{λ}^{(0)}))(τ_{λ}).

Thus, we have

(3.16)  ˜∂_{λ} = d − ad (\frac{1}{ν} ∑_{ij} dz_{λ}^{i}ω_{ij}Z_{λ}^{j}) − ad (\frac{1}{ν} \frac{1}{2} ∑_{ijkl} Γ_{λ,jk}^{p}ω_{pi}dz_{λ}^{i}Z_{λ}^{j}Z_{λ}^{k})

+ ad (\frac{1}{ν}(∑_{ij} 𝔷_{λ}^{i}ω_{ij}Z_{λ}^{j} + γ̃_{λ}^{(3)})) + ad (\frac{1}{ν}(γ_{λ}^{(0)}))

for certain γ̃_{λ}^{(3)} ∈ Γ(Λ_{U_{λ}} ⊗ W_{U_{λ},3}) and γ_{λ}^{(0)} ∈ Γ(Λ_{U_{λ}}[[ν^{2}]]), which gives the desired form. We set γ_{λ}^{(3)} = ∑_{ij} 𝔷_{λ}^{i}ω_{ij}Z_{λ}^{j} + γ̃_{λ}^{(3)} ∈ Γ(Λ_{U_{λ}} ⊗ W_{U_{λ},3}). Remark

C̃_{λ}^{∗−1}Z_{λ}^{i} = Z^{i} on 𝔊^{∘}_{U_{λ}}. Then, applying 𝔊^{∘}_{U_{λ}} to (3.16), we have

Theorem 3.2. On the classical contact chart 𝔊^{∘}_{U_{λ}}, the operator ˜∂^{∘}_{λ} = C̃_{λ∗} ˜∂_{λ} is expressed as

˜∂_{λ}^{∘} = d − δ̃_{λ} − ad (\frac{1}{ν} \frac{1}{2}Γ_{λ}^{(2)}) + ad (\frac{1}{ν}γ_{λ})

where δ̃_{λ} =ad (\frac{1}{ν} ∑_{ij} dz_{λ}^{i}ω_{ij}Z^{j}) and

Γ_{λ}^{(2)} = ∑_{ijkl} Γ_{λ,jk}^{p}ω_{pi}dz_{λ}^{i}Z_{λ}^{j}Z_{λ}^{k}, γ_{λ} = γ_{λ}^{(3)} + γ_{λ}^{(0)},

with certain non central element γ_{λ}^{(3)} ∈ Γ(Λ_{U_{λ}} ⊗ W^{∘}_{U_{λ},3}) and central element γ_{λ}^{(0)} = \frac{1}{2}(θ_{λ} + dc_{λ,0}) + ν^{2}ξ_{λ,2} + ⋯ ∈ Γ(Λ_{U_{λ}}[[ν^{2}]]).

3.5.2. Chart transformation. In this section, we consider the chart transfor-

mation of ˜∂^{∘}_{λ}. The identity C̃_{λμ} = C̃_{μ} ∘ Ψ_{λμ} ∘ C̃_{λ}^{−1} : 𝔊^{∘}_{U_{λμ}} → 𝔊^{∘}_{U_{μλ}}gives

C̃_{λμ∗} ˜∂^{∘}_{λ} = C̃_{λμ}^{∗−1} ˜∂^{∘}_{λ}C̃_{λμ}^{∗} = C̃_{μ}^{∗−1}Ψ_{λμ}^{∗−1}C̃_{λ}^{∗} ˜∂^{∘}_{λ}C̃_{λ}^{∗−1}Ψ_{λμ}^{∗}C̃_{μ}^{∗}

= C̃_{μ}^{∗−1}Ψ_{λμ}^{∗−1} ˜∂_{λ}Ψ_{λμ}^{∗}C̃_{μ}^{∗} = C̃_{μ}^{∗−1} ˜∂_{μ}C̃_{μ}^{∗} = ˜∂^{∘}_{μ}.

Substituting the identity C̃_{λμ}^{∗}τ = τ + h_{λμ}(ν) in equation (3.11), we have

˜∂^{∘}_{μ}τ = C̃_{λμ∗} ˜∂^{∘}_{λ}τ = C̃_{λμ}^{∗−1} ˜∂^{∘}_{λ}C̃_{λμ}^{∗}τ

= C̃_{λμ}^{∗−1} ˜∂^{∘}_{λ}(τ + h_{λμ}(ν)) = C̃_{λμ}^{∗−1}(˜∂^{∘}_{λ}τ + dh_{λμ}(ν))

Hence, we see

˜∂^{∘}_{μ}τ = −δ̃_{μ}τ + ad (\frac{1}{ν}γ_{μ})τ

= C̃_{λμ}^{∗−1}(−δ̃_{λ}τ + ad (\frac{1}{ν}γ_{λ})τ + ad (\frac{1}{ν}dh_{λμ}(ν))τ)

= −δ̃_{μ}τ + ad (\frac{1}{ν}C_{μλ}^{∗}γ_{λ} + \frac{1}{ν}ϕ_{μλ}^{∗}dh_{λμ}(ν))τ

Comparing the both side for the central part and noncentral part of the kernel yields

C̃_{μλ}^{∗}(γ_{λ}^{(3)}) = γ_{μ}^{(3)}

and

ϕ_{μλ}^{∗}γ_{λ}^{(0)} + ϕ_{μλ}^{∗}dh_{λμ}(ν) = γ_{μ}^{(0)}. (3.17)

Here we remark from (3.10) each coefficient of h_{κμ}(ν) is given as

h_{κμ}(ν) = ∑_{k} h_{κμ}^{(2k)}ν^{2k}, h_{κμ}^{(2k)} = −2(2k − 1) ∑_{λ} χ_{λ}c_{κμλ}^{(2k)} (3.18)

3.5.3. The derivative ˜∂_{λ} and Fedosov connections. Notice that ∑_{ij} dz_{λ}^{i}ω_{ij}Z_{λ}^{i} ∈ Γ(Λ_{U_{λ}} ⊗ W_{U_{λ}}) defines global section, hence the operator δ = δ̃_{λ}|_{Γ(Λ_{U_{λ}}⊗W_{U_{λ}})} is also global. We set ▽ = d − \frac{1}{ν} ad (\frac{1}{2}Γ_{λ}^{(2)}) , which also defines a symplectic connection extended to the bundle W_{M}. Thus, as is stated above the noncentral parts {γ_{λ}^{(3)}} defines a global section γ^{(3)} ∈ Γ(Λ_{M} ⊗W_{M}) and we have a globally defined operator, which is given in Definition 3.3 called the twisted exterior derivative, expressed as

∂ = ˜∂|_{Γ(W_{U_{λ}})} = ▽ − δ + \frac{1}{ν} ad (γ^{(1)}) : Γ(Λ_{M} ⊗ W_{M}) → Γ(Λ_{M} ⊗ W_{M})

with the property; ∂^{2} = 0, ∂|F(W_{M}) = 0. Thus, we have

Theorem 3.3. The twisted exterior derivative ∂ is a Fedosov connection.

3.6. Poincaré-Cartan class and Fedosov connection. Let us set ˜∂^{∘}_{λ} =

D̃^{∘}_{λ} + Ñ^{∘}_{λ}, where

D̃^{∘}_{λ} = d − δ̃_{λ} − ad (\frac{1}{ν} \frac{1}{2}Γ_{λ}^{(2)}) + ad (\frac{1}{ν}γ_{λ}^{(3)})

is the noncentral part of the operator and

Ñ^{∘}_{λ} = ad (\frac{1}{ν}γ_{λ}^{(0)})

is the central part of Ñ^{∘}_{λ} = ad (\frac{1}{ν}γ_{λ}^{(0)}). Notice D̃^{∘}_{λ}(τ) ∈ Γ(Λ_{U_{λ}}^{1} ⊗ W^{∘}_{υ_{ξs}}), and

Ñ^{∘}_{λ}(τ) ∈ Γ(Λ_{U_{λ}}^{1}[[ν^{2}]]). Since D̃^{∘}_{λ}Ñ^{∘}_{λ}(τ) = dÑ^{∘}_{λ}(τ) and Ñ^{∘}_{λ}|_{Γ(Λ_{U_{λ}}^{1}⊗W^{∘}_{υ_{ξs}})} = 0, we

see the square of ˜∂^{∘}_{λ} satisfies

0 = ˜∂^{∘}_{λ}^{2}(τ) = ˜∂^{∘}_{λ}(D̃^{∘}_{λ}(τ) + Ñ^{∘}_{λ}(τ)) = D̃^{∘}_{λ}^{2}(τ) + dÑ^{∘}_{λ}(τ). (3.19)

We also see 0 = ˜∂^{∘}_{λ}^{2}(Z^{i}) = D̃^{∘}_{λ}^{2}(Z^{i}), i = 1, 2, ⋯, 2n. We calculate as

D̃^{∘}_{λ}^{2}(Z^{i}) = D̃^{∘}_{λ}(\frac{1}{ν}[Ξ_{λ}, Z^{i}]) = \frac{1}{ν}[D̃^{∘}_{λ}Ξ_{λ}, Z^{i}] − \frac{1}{ν}[Ξ_{λ}, D̃^{∘}_{λ}Z^{i}]

= \frac{1}{ν}[dΞ_{λ} + \frac{1}{ν}[Ξ_{λ}, Ξ_{λ}], Z^{i}] − \frac{1}{ν}[Ξ_{λ}, \frac{1}{ν}[Ξ_{λ}, Z^{i}]]

where Ξ_{λ} = −Θ_{λ} − \frac{1}{2}Γ_{λ}^{(2)} + γ_{λ}^{(3)}. Using Lemma 3.4, we see [Ξ_{λ}, [Ξ_{λ}, Z^{i}]] = \frac{1}{2}[[Ξ_{λ}, Ξ_{λ}], Z^{i}] and we have

D̃^{∘}_{λ}^{2}(Z^{i}) = \frac{1}{ν}[dΞ_{λ} + \frac{1}{2} \frac{1}{ν}[Ξ_{λ}, Ξ_{λ}], Z^{i}] = 0, yi = 1, 2, ⋯, 2n.

Hence, the term dΞ_{λ} + \frac{1}{2} \frac{1}{ν}[Ξ_{λ}, Ξ_{λ}] is a central 2-form, i.e., an element of Λ^{2}(U_{λ})[[ν^{2}]]. In fact, dΞ_{λ} = −\frac{1}{2}dΓ_{λ}^{(2)} + dγ_{λ}^{(3)} and

\frac{1}{2} \frac{1}{ν}[Ξ_{λ}, Ξ_{λ}] = −σ_{0} + \frac{1}{8}[Γ_{λ}^{(2)}, Γ_{λ}^{(2)}] − δ̃_{λ}(γ_{λ}^{(3)})

− ad (\frac{1}{2ν} Γ_{λ}^{(2)}) + \frac{1}{2}[γ_{λ}^{(3)}, γ_{λ}^{(3)}].

Thus, we see dΞ_{λ} + \frac{1}{2} \frac{1}{ν}[Ξ_{λ}, Ξ_{λ}] = −\frac{1}{2}σ_{0} + R − δ̃_{λ}γ_{λ}^{(3)} + ▽γ_{λ}^{(3)} + \frac{1}{2ν} [γ_{λ}^{(3)}, γ_{λ}^{(3)}], which is the curvature of the Fedosov connection, where R = \frac{1}{4} ∑ ω_{im}R_{jkl}^{i}dz_{λ}^{k} ∧ dz_{λ}^{l}Z^{i}Z^{j} and R_{jkl}^{i} is the curvature of the symplectic connection. Now we set

dΞ_{λ} + \frac{1}{2} \frac{1}{ν}[Ξ_{λ}, Ξ_{λ}] = Ω = −\frac{1}{2}σ_{0} + ν^{2}σ_{2} + ⋯ + ν^{2k}σ_{2k} + ⋯

where σ_{2k} is a globally defined closed 2-form. The equation (3.19) gives

D̃^{∘}_{λ}^{2}(τ) = [\frac{1}{ν}(−\frac{1}{2}σ_{0} + ν^{2}σ_{2} + ⋯ + ν^{2k}σ_{2k} + ⋯), τ]
= −σ_{0} − 2ν^{2}σ_{2} − ⋯ − 2(2k − 1)ν^{2k}σ_{2k} − ⋯

which is equal to −dÑ^{∘}_{λ}(τ) = −d ad (\frac{1}{ν}γ_{λ}^{(0)})(τ). Using equation (3.15), we see

−d ad (\frac{1}{ν}γ_{λ}^{(0)})(τ) = −d[\frac{1}{2}ν(θ_{λ} + dc_{λ,0}) + νξ_{λ,2} + ⋯ + ν^{2k−1}ξ_{2k} + ⋯, τ]
= −σ_{0} − 2ν^{2}dξ_{λ,2} − ⋯ − 2(2k − 1)ν^{2k}dξ_{λ,2k} − · · · .

Hence,

σ_{2k} = dξ_{λ,2k}, k = 1, 2, · · · . (3.20)

In view of the identity (3.17), we see γ_{μ} − γ_{λ} = dh_{λμ} , i.e.,

ξ_{μ,2k} − ξ_{λ,2k} = dh_{λμ}^{(2k)}, k = 1, 2, ⋯

on V_{μ} ∩ V_{λ}. Here we regard forms and functions as given on open subsets of the manifold. Then, the equations (3.20), (3.10) give the identities of the cohomology classes [σ_{2k}] = −2(2k − 1)[c^{(2k)}], k = 1, 2, · · ·. Thus, we have

A. Yoshioka 206

Theorem 3.4. The Poincaré-Cartan class c(W_{M}) coincides with the 2nd cohomology class [Ω] given by the curvature of Fedosov connection. More precisely, we have

[σ_{2k}] = −2(2k − 1)[c^{(2k)}], k = 1, 2, ⋯

where [Ω] = ∑_{k}[σ_{2k}]ν^{2k} and c(W_{M}) = ∑_{k}[c^{(2k)}]ν^{2k}.

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Department of Math. Faculty of Engineering, Science University of Tokyo, Kagurazaka 1-3, Shinjyuku-ku, 162-8601, Tokyo, Japan, E-mail address: yoshioka@rs.kagu.sut.ac.jp

Received May 5, 1999