An algebra of deformation quantization for star-exponentials on complex symplectic manifolds

Giuseppe Dito and Pierre Schapira

July 9, 2006

Abstract

The cotangent bundle T^{∗}X to a complex manifold X is classically endowed with the sheaf of k-algebras W_{T^{∗}X} of deformation quantization, where k := W_{{pt}} is a subfield of ℂ[[ℏ, ℏ^{−1}]. Here, we construct a new sheaf of k-algebras W_{T^{∗}X}^{t} which contains W_{T^{∗}X} as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of W_{T^{∗}X}, we show that exp(tℏ^{−1}P) is well defined in W_{T^{∗}X}^{t}.

Mathematics subject Classification: 53D55, 32C38.

Introduction

A fundamental tool for spectral analysis in deformation quantization is the star-exponential [1]. However, at the formal level, the star-exponential does not make sense as a formal series in ℏ and ℏ^{−1}. The goal of this article is to construct a new sheaf of algebras on the cotangent bundle T^{∗}X to a complex manifold X in which the star-exponential has a meaning and such that quantized symplectic transformations operate on such algebras.

On the cotangent bundle T^{∗}X to a complex manifold X, there is a well-known sheaf of filtered algebras called deformation quantization algebra by many authors (see [1], [6], etc.). This algebra, denoted Ŵ_{T^{∗}X} here, is constructed in [7] as well as its analytic counterpart W_{T^{∗}X}. The sheaf W_{T^{∗}X} is similar to the sheaf E_{T^{∗}X} of microdifferential operators of [8], but with an extra central parameter ℏ, a substitute to the lack of homogeneity^{1}. Here ℏ belongs to the field k := W_{{pt}}, a subfield of ℂ[[ℏ, ℏ^{−1}]. (Note that the notation τ = ℏ^{−1} is used in [7].) When X is affine and one denotes by (x; u) a point of T^{∗}X, a section P of this sheaf on an open subset U ⊂ T^{∗}X is represented by its total symbol σ_{tot}(P) = ∑_{−∞<j≤m} p_{j}(x; u)ℏ^{−j}, with m ∈ ℤ, p_{j} ∈ O_{T^{∗}X}(U), the p_{j}’s satisfying suitable inequalities and the product being given by the Leibniz formula.

1In this paper, we write E_{T^{∗}X} and W_{T^{∗}X} instead of the classical notations E_{X} and W_{X}.

In this paper, we construct a new sheaf of k-algebras W_{T^{∗}X}^{t}, with an extra central holomorphic parameter t defined in a neighborhood of t = 0, with the property that complex symplectic transformations may be locally quantized as isomorphisms of algebras and there are natural morphisms of k-algebras W_{T^{∗}X} ⟶^{ι} W_{T^{∗}X}^{t} ⟶^{res} W_{T^{∗}X} whose composition is the identity on W_{T^{∗}X}. We give the symbol calculus on W_{T^{∗}X}^{t}, which extends naturally that of W_{T^{∗}X} (however, now we get series in ℏ^{j} with −∞ < j < ∞) and finally we show that, if P is a section of W_{T^{∗}X} of order 0, then exp(tℏ^{−1}P) is well defined in W_{T^{∗}X}^{t}. We also briefly discuss the case where T^{∗}X is replaced with a general symplectic manifold.

Our construction is as follows. First, we add a central holomorphic parameter s ∈ ℂ and consider the sheaf W_{ℂ×T^{∗}X}, the subsheaf of W_{T^{∗}(ℂ×X)} consisting of sections not depending on ∂_{s}. Denoting by a: ℂ × T^{∗}X ⟶ T^{∗}X the projection, we first define an algebra W_{T^{∗}X}^{s} := R^{1}a_{!}W_{ℂ×T^{∗}X}. The algebra structure with respect to the s-variable is given by convolution, as in the case of the space H_{c}^{1}(ℂ; O_{ℂ}). In order to replace this convolution product by an usual product, we define the sheaf W_{T^{∗}X}^{t} as the “formal” Laplace transform with respect to the variables sℏ^{−1} of the algebra W_{T^{∗}X}^{s}.

In a deformation quantization context, the existence of exp(tℏ^{−1}P) in W_{T^{∗}X}^{t} gives a precise meaning to the star-exponential [1] of P which is heuristically related to the Feynman Path Integral of P.

Acknowledgments. We would like to thank Masaki Kashiwara for extremely useful conversations and helpful insights. The first named author thanks Yoshiaki Maeda for warm hospitality at Keio university where this work was finalized, and the JSPS for financial support.

1 Symbols

The fields k̂ and k

We set k̂ := ℂ[[ℏ, ℏ^{−1}]. Hence, an element a ∈ k̂ is a series

a = ∑_{−∞<j≤m} a_{j}ℏ^{−j}, a_{j} ∈ ℂ, m ∈ ℤ.

Consider the following condition on a:

{there exist positive constants C, ε such that |a_{j}| ≤ Cε^{−j}(−j)! for all j < 0.

(1.1)

We denote by k the subfield of k̂ consisting of series satisfying (1.1).

Convention We endow k̂, hence k, with the filtration associated to

ord(ℏ) = −1. (1.2)

The fields k̂ and k are ℤ-filtered^{2} and contain the subrings k̂(0) and k(0), respectively. Note that k̂(0) = ℂ[[ℏ]] and k(0) = k ∩ k̂(0).

The sheaves Ô_{X}^{ℏ} and O_{X}^{ℏ}

Let (X, O_{X}) be a complex manifold.

Definition 1.1. (i) We denote by Ô_{X}^{ℏ} the sheaf O_{X}[[ℏ, ℏ^{−1}]. In other words, Ô_{X}^{ℏ} is the filtered k̂-algebra defined as follows: A section f(x, ℏ) of O_{X}^{ℏ} of order ≤ m (m ∈ ℤ) on an open set U of X is a series

f(x, ℏ) = ∑_{−∞<j≤m} f_{j}(x)ℏ^{−j}, (1.3)

with f_{j} ∈ O_{X}(U).

(ii) We denote by O_{X}^{ℏ} the filtered k-subalgebra of Ô_{X}^{ℏ} consisting of sections f(x, ℏ) as above satisfying:

{for any compact subset K of U there exist positive constants C, ε such that sup_{K} |f_{j}| ≤ Cε^{−j}(−j)! for all j < 0.

(1.4)

Note that

(1.5)  Ô_{X}^{ℏ} ≃ Ô_{X}^{ℏ}(0) ⊗_{k̂(0)} k̂, O_{X}^{ℏ} ≃ O_{X}^{ℏ}(0) ⊗_{k(0)} k.

(To be correct, we should have written k_{X}, the constant sheaf with values in k, instead of k in these formulas, and similarly for k(0), k̂(0) and k̂.)

Also note that there exist isomorphisms of sheaves (not of algebras)

Ô_{X}^{ℏ}(0) ≃ O_{X×ℂ}ˆ|_{X×{0}}, (1.6)

O_{X}^{ℏ}(0) ≃ O_{X×ℂ}|_{X×{0}}, (1.7)

where O_{X×ℂ}ˆ|_{X×{0}} is the formal completion of O_{X×ℂ} along the hypersurface X × {0} of X × ℂ and O_{X×ℂ}|_{X×{0}} is the restriction of O_{X×ℂ} to X × {0}.

Denoting by t the coordinate on ℂ, the isomorphism (1.7) is given by the map

O_{X}^{ℏ}(0) ∋ ∑_{j≤0} f_{j}ℏ^{−j} ↦ ∑_{j≥0} f_{−j} \frac{t^{j}}{j!} ∈ O_{X×ℂ}|_{X×\{0\}}.

2In the sequel, we shall say “filtered” instead of “ℤ-filtered”.

The convolution algebra H_{c}^{1}(ℂ; O_{ℂ})

The results of this subsection are well known and elementary. We recall them for the reader’s convenience.

We consider the complex line ℂ endowed with a holomorphic coordinate s. Using this coordinate, we identify the sheaf O_{ℂ} of holomorphic functions on ℂ and the sheaf Ω_{ℂ} of holomorphic forms on ℂ.

The space H_{c}^{1}(ℂ; O_{ℂ}) is endowed with a structure of an algebra by

H_{c}^{1}(ℂ; O_{ℂ}) × H_{c}^{1}(ℂ; O_{ℂ}) ⟶ H_{c}^{2}(ℂ^{2}; O_{ℂ^{2}})
⟶ H_{c}^{1}(ℂ; O_{ℂ}),

where the first arrow is the cup product and the second arrow is the integration along the fibers of the map ℂ^{2} ⟶ ℂ, (s, s′) ↦ s + s′.

When representing the cohomology classes by holomorphic functions, the convolution product is described as follows.

For a compact subset K of ℂ, we identify the vector space H_{K}^{1}(ℂ; O_{ℂ}) with the quotient space Γ(ℂ \ K; O_{ℂ})/Γ(ℂ; O_{ℂ}) and, if f ∈ Γ(ℂ \ K; O_{ℂ}), we still denote by f its image in H_{K}^{1}(ℂ; O_{ℂ}) or in H_{c}^{1}(ℂ; O_{ℂ}). Let K and L be compact subsets of ℂ, let f ∈ Γ(ℂ \ K; O_{ℂ}) and g ∈ Γ(ℂ \ L; O_{ℂ}). The convolution product f ∗ g is given by

f ∗ g(z) = \frac{1}{2iπ} ∫_{γ} f(z − w)g(w)dw (1.8)

where γ is a counter clockwise oriented circle which contains L and |z| is chosen big enough so that z + K is outside of the disc bounded by γ. It is an easy exercise to show that this definition does not depend on the representatives f and g, and that to interchange the role of f and g in the formula (1.8) modifies the result by a function defined all over ℂ, hence gives the same result in H_{c}^{1}(ℂ; O_{ℂ}). Therefore, we obtain a commutative algebra structure on H_{c}^{1}(ℂ; O_{ℂ}).

Example 1.2.

\frac{1}{z^{n+1}} ∗ \frac{1}{z^{m+1}} = \frac{(n + m)!}{n!m!} \frac{1}{z^{n+m+1}} .

The sheaf O_{X}^{s,ℏ}

From now on, we shall concentrate our study on O_{X}^{ℏ}.

Notation 1.3. We shall often denote by ℂ_{s} the complex line ℂ endowed with the coordinate s.

Lemma 1.4. Let Y be a complex manifold and Z a Stein submanifold of Y. Then H^{j}(Z; O_{Y}^{ℏ} (0)|_{Z}) vanishes for j ≠ 0.

Proof. Using the isomorphism (1.7), we may replace the sheaf O_{Y}^{ℏ} (0) with the sheaf O_{Y ×ℂ_{t}}|_{t=0}. By a theorem of Siu [10], Z × {0} admits a fundamental system of open Stein neighborhoods in Y × ℂ_{t} and the result follows. □

Let X be a complex manifold. The manifold ℂ_{s}×X is thus endowed with the k-filtered sheaf O_{ℂ_{s}×X}^{ℏ}. Let a: ℂ_{s} × X ⟶ X denote the projection.

Lemma 1.5. (i) One has the isomorphism

R^{j}a_{!}O_{ℂ_{s}×X}^{ℏ} ≃ R^{j}a_{!}O_{ℂ_{s}×X}^{ℏ}(0) ⊗_{k(0)} k.

(ii) R^{j}a_{!}O_{ℂ_{s}×X}^{ℏ}(0) ≃ 0 for j ≠ 1.

(iii) Let U ⊂⊂ V ⊂⊂ W be three open subsets of X and assume that W is Stein. Then the natural morphism Γ(W; R^{1}a_{!}O_{ℂ_{s}×X}^{ℏ}) ⟶ Γ(U; R^{1}a_{!}O_{ℂ_{s}×X}^{ℏ}) factorizes through

\underrightarrow{lim}_{K⊂ℂ_{s}} Γ((ℂ_{s} \ K) × V; O_{ℂ_{s}×X}^{ℏ})/Γ(ℂ_{s} × V; O_{ℂ_{s}×X}^{ℏ}),

where K ranges over the family of compact subsets of ℂ.

Proof. (i) follows from the projection formula for sheaves (i.e., Ra_{!}(F ⊗a^{−1}G) ≃ Ra_{!}F ⊗G) and (1.5). (ii) For x ∈ X, we have

H^{j}(Ra_{!}O_{ℂ_{s}×X}^{ℏ}(0))_{x} ≃ \underrightarrow{lim}_{K} H_{K}^{j}(ℂ_{s} × \{x\}; O_{ℂ_{s}×X}^{ℏ}(0)|_{ℂ_{s}×\{x\}}).

Applying the distinguished triangle of functors

RΓ_{K}(ℂ_{s} × {x}; ^{•}) ⟶ RΓ(ℂ_{s} × {x}; ^{•}) ⟶ RΓ((ℂ_{s} \ K) × {x}; ^{•}) ⟶^{+1}

to the sheaf O_{ℂ_{s}×X}^{ℏ}(0)|_{ℂ_{s}×{x}} we get the result by Lemma 1.4 for j > 1 and the case j = 0 follows from the principle of analytic continuation.

(iii) Recall first that if W is a Stein manifold and if W_{1} ⊂⊂ W is open, there exists a Stein open subset W_{2} of W with W_{1} ⊂⊂ W_{2} ⊂⊂ W.

For a compact subset L of X, Γ(L; R^{1}a_{!}O_{ℂ_{s}×X}^{ℏ}) ≃ Γ(L; R^{1}a_{!}O_{ℂ_{s}×X}^{ℏ}(0))⊗_{k(0)} k. Hence, it is enough to prove the result for O_{ℂ_{s}×X}^{ℏ}(0).

By Lemma 1.4, H^{j}(D × U; O_{ℂ_{s}×X}^{ℏ}(0)) vanishes for D open in ℂ_{s}, U Stein open in X and j ≠ 0. Therefore, H_{K×U}^{j}(ℂ_{s} × U; O_{ℂ_{s}×X}^{ℏ}(0)) vanishes for j ≠ 1 and we get the exact sequence:

0 ⟶ Γ(ℂ_{s} × U; O_{ℂ_{s}×X}^{ℏ}(0)) ⟶ Γ((ℂ_{s} \ K) × U; O_{ℂ_{s}×X}^{ℏ}(0))
⟶ H_{K×U}^{1}(ℂ_{s} × U; O_{ℂ_{s}×X}^{ℏ}(0)) ⟶ 0.
□

Definition 1.6. We set O_{X}^{s,ℏ} := R^{1}a_{!}O_{ℂ_{s}×X}^{ℏ}.

Clearly, O_{X}^{s,ℏ} is a sheaf of filtered k-modules. By Lemma 1.5, a section f(s, x, ℏ) of order m of the sheaf O_{X}^{s,ℏ} on a Stein open subset W of X may be written on any relatively compact open subset U of W as a series

f(s, x, ℏ) = ∑_{−∞<j≤m} f_{j}(s, x)ℏ^{−j},

where f_{j}(s, x) is a holomorphic function on (ℂ_{s} \ K_{0}) × U for a compact set K_{0} not depending on j and the f_{j}’s satisfy an estimate (1.4) on each compact subset K of (ℂ_{s} \ K_{0}) × U.

We shall extend the product (1.8) to O_{X}^{s,ℏ} as follows. For two sections f(s, x, ℏ) = ∑_{−∞<j≤m} f_{j}(s, x)ℏ^{−j} and g(s, x, ℏ) = ∑_{−∞<j≤m′} g_{j}(s, x)ℏ^{−j} of O_{X}^{s,ℏ}, we set:

\{f(s, x, ℏ) ∗ g(s, x, ℏ) = ∑_{−∞<j≤m+m′} h_{j}(s, x)ℏ^{−j}, ; h_{k}(s, x) = ∑_{i+j=k} \frac{1}{2iπ} ∫_{γ} f_{i}(s − w, x)g_{j}(w, x)dw. (1.9)

Proposition 1.7. The sheaf O_{X}^{s,ℏ} has a structure of a filtered commutative k-algebra.

Proof. It is easily checked that multiplication by ℏ^{−1} induces an isomorphism of sheaves of k-modules O_{X}^{s,ℏ}(m) \underrightarrow{∼} O_{X}^{s,ℏ}(m + 1). Hence we just need to check that the product of two sections of order 0 is a section of order 0. Let f(s, x, ℏ) = ∑_{−∞<i≤0} f_{i}(s, x)ℏ^{−i} and g(s, x, ℏ) = ∑_{−∞<j≤0} g_{j}(s, x)ℏ^{−j} be in O_{X}^{s,ℏ}(0) and K a compact subset of (ℂ_{s} \ K_{0}) × U. Let γ be a counter clockwise oriented circle which contains K_{0} and s > R big enough so that s + K_{0} does not meet γ. Then for w ∈ γ and x ∈ K ∩ (ℂ_{s} \ K_{0}) × U, we have:

| ∑_{i+j=k,i,j≤0} f_{i}(s − w, x)g_{j}(w, x)| ≤ C^{2}(−k)! ∑_{i+j=k,i,j≤0} ε^{−i−j} \frac{(−i)!(−j)!}{(−k)!} ≤ 3C^{2}ε^{−k}(−k)!.

Hence h(s, x, ℏ) = ∑_{−∞<j≤0} h_{k}(s, x)ℏ^{−k} defined by (1.9) is in O_{X}^{s,ℏ}(0). □

The Laplace transform and the algebra O_{X}^{t,ℏ}

In order to replace the convolution product in the s-variable with the ordinary product, we shall apply a kind of Laplace transform to O_{X}^{s,ℏ}.

Definition 1.8. On a complex manifold X, we denote by O_{X}^{t,ℏ} the filtered sheaf of kmodules defined as follows. A section f(t, x, ℏ) of O_{X}^{t,ℏ}(m) (i.e., a section of order m) on an open set U of X is a series

f(t, x, ℏ) = ∑_{−∞<j<∞} f_{j}(t, x)ℏ^{−j}, f_{j} ∈ Γ(U; O_{ℂ×X|_{t=0}}), (1.10)

with the condition that for any compact subset K of U there exists η > 0 such that f_{j}(t, x) is holomorphic in a neighborhood of {|t| ≤ η} × K and satisfies

{there exist positive constants C, ε such that sup_{x∈K,|t|≤η} |f_{j}(t, x)| ≤ C · ε^{−j}(−j)! for all j < 0,

(1.11)

there exist positive constants M and R such that

sup_{x∈K} |f_{j}(t, x)| ≤ M \frac{R^{j−m}}{(j − m)!}|t|^{j−m} for |t| ≤ η and all j ≥ m. (1.12)

Let f(t, x, ℏ) = ∑_{−∞<j<∞} f_{j}(t, x)ℏ^{−j} and g(t, x, ℏ) = ∑_{−∞<j<∞} g_{j}(t, x)ℏ^{−j} be two sections of O_{X}^{t,ℏ} of order m and m′ respectively. Define formally

h(t, x, ℏ) = ∑_{−∞<j<∞} h_{j}(t, x)ℏ^{−j}, h_{k}(t, x) = ∑_{i+j=k} f_{i}(t, x)g_{j}(t, x). (1.13)

Lemma 1.9. (i) Multiplication by ℏ^{−1} induces an isomorphism of sheaves of k(0)- modules O_{X}^{t,ℏ}(m) \underrightarrow{∼} O_{X}^{t,ℏ}(m + 1).

(ii) The product (1.13) of a section f(t, x, ℏ) ∈ O_{X}^{t,ℏ}(m) and a section g(t, x, ℏ) ∈ O_{X}^{t,ℏ}(m′) is well defined and belongs to O_{X}^{t,ℏ}(m + m′).

Proof. (i) (a) Let f(t, x, ℏ) = ∑_{−∞<j<∞} f_{j}(t, x)ℏ^{−j} ∈ O_{X}^{t,ℏ}(m), then ℏ^{−1}f(t, x, ℏ) = ∑_{−∞<j<∞} f̃_{j}(t, x)ℏ^{−j}, with f̃_{j} = f_{j−1}. For any integer j < 0, we have:

sup_{x∈K,|t|≤η} |f̃_{j}(t, x)| = sup_{x∈K,|t|≤η} |f_{j−1}(t, x)| ≤ Cε^{−j+1}(−j + 1)! ≤ (Cε)(εe)^{−j}(−j)!.

Hence Condition (1.11) is satisfied.

For j ≥ m + 1, we have:

sup_{x∈K} |f̃_{j}(t, x)| = sup_{x∈K} |f_{j−1}(t, x)| ≤ M \frac{R^{j−m−1}}{(j − m − 1)!}|t|^{j−m−1},

which is simply Condition (1.12) for m + 1 and ℏ^{−1}f(t, x, ℏ) ∈ O_{X}^{t,ℏ}(m + 1).

(b) Let ℏf(t, x, ℏ) = ∑_{−∞<j<∞} f̃_{j}(t, x)ℏ^{−j}, with f̃_{j} = f_{j+1}. For any integer j < −1, we have:

sup_{x∈K,|t|≤η} |f̃_{j}(t, x)| = sup_{x∈K,|t|≤η} |f_{j+1}(t, x)| ≤ Cε^{−j−1}(−j − 1)! ≤ \frac{C}{ε} ε^{−j}(−j)!.

For j = −1, we have:

sup_{x∈K,|t|≤η} |f̃_{−1}(t, x)| = sup_{x∈K,|t|≤η} |f_{0}(t, x)| = A ≥ 0,

since f_{0}(t, x) is holomorphic in a neighborhood of \{|t| ≤ η\} × K. Set C′ = max\{\frac{A}{ε}, \frac{C}{ε} \}, then for all integer j < 0, we have:

sup_{x∈K,|t|≤η} |f̃_{j}(t, x)| ≤ C′ε^{−j}(−j)!,

and Condition (1.11) is satisfied.

For j ≥ m − 1, we have:

sup_{x∈K} |f̃_{j}(t, x)| = sup_{x∈K} |f_{j+1}(t, x)| ≤ M \frac{R^{j−m+1}}{(j − m + 1)!}|t|^{j−m+1},

which is Condition (1.12) for m−1 and ℏf(t, x, ℏ) ∈ O_{X}^{t,ℏ}(m−1). Therefore, multiplication by ℏ^{−1} induces an isomorphism O_{X}^{t,ℏ}(m) \underrightarrow{∼} O_{X}^{t,ℏ}(m + 1).

(ii) By (i), we may assume m = m′ = 0. Let f = ∑_{−∞<i<∞} f_{i}(t, x)ℏ^{−i} and g = ∑_{−∞<j<∞} g_{j}(t, x)ℏ^{−j} be in O_{X}^{t,ℏ}(0). Let K be a compact set. There exists η > 0 such that f_{i}(t, x) and g_{j}(t, x) are holomorphic in a neighborhood of {|t| ≤ η}×K. Conditions (1.11) and (1.12) guarantee the existence of the positive constants C_{1}, ε_{1}, M_{1} and R_{1} for the f_{i}’s, and C_{2}, ε_{2}, M_{2} and R_{2} for the g_{j}’s. We set C = max{C_{1}, C_{2}}, ε = max{ε_{1}, ε_{2}}, M = max{M_{1}, M_{2}} and R = max{R_{1}, R_{2}}

We shall show that the product (1.13) is well defined. Let h_{k}(t, x) = ∑_{i+j=k} f_{i}(t, x)g_{j}(t, x).

(a) Consider the case k < 0. The sum defining h_{k} can be divided into three parts:

h_{k} = ∑_{k<i<0} f_{i}g_{k−i} + ∑_{i≥0} f_{i}g_{k−i} + ∑_{j≥0} f_{k−j}g_{j}. (1.14)

The first sum is finite and defines a holomorphic function in a neighborhood of {|t| ≤ η} × K.

In the second sum, k − i is strictly negative and for each term in this sum Conditions (1.11) and (1.12) give the following estimates when x ∈ K and |t| ≤ η:

|f_{i}(t, x)g_{k−i}(t, x)| ≤ M \frac{(Rη)^{i}}{i!} Cε^{i−k}(i − k)!
≤ CMε^{−k}(−k)!(Rηε)^{i}\binom{i − k}{i}

Recall that ∑_{i≥0} α^{i}\binom{n+i}{i} = \frac{1}{(1−α)^{n+1}} for |α| < 1. When Rηε < 1, (Rηε)^{i}\binom{i−k}{i} is the general term of an absolutely convergent series. Let η̃ = min\{η, \frac{1}{2Rε}\}. Then the second sum in (1.14) converges uniformly on {|t| ≤ η̃} × K.

The third sum is handled in a similar way and one gets the estimate:

|f_{k−j}(t, x)g_{j}(t, x)| ≤ CMε^{−k}(−k)!(Rηε)^{j}\binom{j − k}{j}.

It follows from the preceding that h_{k} for k < 0 is a holomorphic function in a neighborhood of {|t| ≤ η̃} × K.

Let now show that h_{k} satisfies Condition (1.11). For x ∈ K and |t| ≤ η̃, the first sum in (1.14) is bounded by:

| ∑_{k<i<0} f_{i}(t, x)g_{k−i}(t, x)| ≤ C^{2} ∑_{k<i<0} ε^{−i}ε^{i−k}(−i)!(i − k)! ≤ C^{2}ε^{−k}(−k)!.

For the second and third sums we have:

| ∑_{i≥0} f_{i}(t, x)g_{k−i}(t, x)| ≤ CMε^{−k}(−k)!\frac{1}{(1 − Rη̃ε)^{−k+1}} ,

| ∑_{j≥0} f_{k−j}(t, x)g_{j}(t, x)| ≤ CMε^{−k}(−k)!\frac{1}{(1 − Rη̃ε)^{−k+1}} .

Let ε̃ = max\{ε, \frac{ε}{(1−Rη̃ε)}\}. For x ∈ K and |t| ≤ η̃, we find that:

|h_{k}(t, x)| ≤ (C^{2} + \frac{2CM}{(1 − Rη̃ε)})ε̃^{−k}(−k)!.

Hence h_{k} satisfies Condition (1.11).

(b) The case k ≥ 0. We again split the sum defining h_{k} into three parts:

h_{k} = ∑_{0≤i≤k} f_{i}g_{k−i} + ∑_{i<0} f_{i}g_{k−i} + ∑_{j<0} f_{k−j}g_{j}. (1.15)

The first sum is a holomorphic function in a neighborhood of {|t| ≤ η} × K.

For each term in the second sum, we have the following estimates when x ∈ K and |t| ≤ η:

|f_{i}(t, x)g_{k−i}(t, x)| ≤ Cε^{−i}(−i)!M \frac{R^{k−i}}{(k − i)!}|t|^{k−i} ≤ CM(εRη)^{−i} \frac{R^{k}}{k!} |t|^{k}.

(εRη)^{−i} is the general term of the geometric series, hence the second sum in (1.15) defines a holomorphic function in a neighborhood of \{|t| ≤ η̃\} × K where η̃ = min\{η, \frac{1}{2Rε}\}.

Similarly, for the third sum we have:

|f_{k−j}(t, x)g_{j}(t, x)| ≤ CM(εRη)^{−j} \frac{R^{k}}{k!} |t|^{k}.

Therefore h_{k} for k ≥ 0 is a holomorphic function in a neighborhood of {|t| ≤ η̃} × K.

We now show that h_{k} satisfies Condition (1.12) with m = 0. For x ∈ K and |t| ≤ η̃, the first sum in (1.15) is bounded by:

| ∑_{0≤i≤k} f_{i}(t, x)g_{k−i}(t, x)| ≤ M^{2} \frac{R^{k}}{k!} |t|^{k} ∑_{0≤i≤k} \binom{k}{i} ≤ M^{2} \frac{(2R)^{k}}{k!} |t|^{k}.

For the second and third sums we find:

| ∑_{i<0} f_{i}(t, x)g_{k−i}(t, x)| ≤ CM \frac{Rη̃ε}{1 − Rη̃ε} \frac{R^{k}}{k!} |t|^{k}

| ∑_{j<0} f_{k−j}(t, x)g_{j}(t, x)| ≤ CM \frac{Rη̃ε}{1 − Rη̃ε} \frac{R^{k}}{k!} |t|^{k}.

For x ∈ K and |t| ≤ η̃, we have:

|h_{k}(t, x)| ≤ (M^{2} + 2CM \frac{Rη̃ε}{1 − Rη̃ε})\frac{(2R)^{k}}{k!} |t|^{k}.

Hence h_{k} satisfies Condition (1.12) with m = 0.

The product of f ∈ O_{X}^{t,ℏ}(0) and g ∈ O_{X}^{t,ℏ}(0) is well defined and fg ∈ O_{X}^{t,ℏ}(0). □

Therefore:

Proposition 1.10. The sheaf O_{X}^{t,ℏ} is naturally endowed with a structure of a commutative filtered k-algebra.

Let U be an open subset of X and let f(s, x, ℏ) ∈ Γ((ℂ_{s} \K)×U; O_{ℂ_{s}×X}^{ℏ}). One defines formally the Laplace transform L(f) of f by

L(f)(t, x, ℏ) = \frac{1}{2iπ} ∫_{γ} f(s, x, ℏ) exp(stℏ^{−1}) ds,

where γ is a counter clockwise oriented circle centered at 0 with radius R ≫ 0.

Example 1.11.

L(s^{−n−1}) = ℏ^{−n}t^{n}/n!, L(\frac{1}{s − 1}) = exp(tℏ^{−1}).

Lemma 1.12. The Laplace transform induces a k-linear monomorphism

s^{−1} · O_{X}[[s^{−1}]][[ℏ, ℏ^{−1}] ↪ O_{X}[[t]][[ℏ, ℏ^{−1}]]

Proof. One notices that the Laplace transform is given by:

∑_{−∞<j≤m} ∑_{n≥0} a_{n,j}s^{−n−1}ℏ^{−j} ↦ ∑_{j≤m} ∑_{n≥0} \frac{a_{n,j}}{n!} t^{n}ℏ^{−n−j},

and the result follows. □

Theorem 1.13. The Laplace transform induces a k-linear isomorphism of filtered kalgebras

L: O_{X}^{s,ℏ} \underrightarrow{∼} O_{X}^{t,ℏ}. (1.16)

Proof. (i) By Lemma 1.9, it is enough to check that L induces an isomorphism O_{X}^{s,ℏ}(0) \underrightarrow{∼} O_{X}^{t,ℏ}(0).

(ii) Let W be a Stein open subset of X and let U be a relatively compact open subset of W. Let us develop a section f(s, x, ℏ) of Γ(W; R^{1}a_{!}O_{ℂ_{s}×X}^{ℏ}(0)) with respect to s^{−1} for s > R. We get

f̃(s, x, ℏ) = ∑_{−∞<j≤0} f̃_{j}(s, x)ℏ^{−j}
= ∑_{−∞<j≤0} ∑_{n≥0} f_{j,n}(x)s^{−n−1}ℏ^{−j}

with the following Cauchy’s estimates:

{for any compact subset K of U there exist positive constants C, ε, R such that sup_{x∈K} |f_{j,n}(x)| ≤ Cε^{−j}(−j)!R^{n}.

Applying the Laplace transform to f̃(s, x, ℏ) means to replace s^{−n−1} with \frac{t^{n}}{n!}ℏ^{−n}. Hence, we find

L(f̃)(t, x, ℏ) = ∑_{−∞<j<∞} f_{j}(t, x)ℏ^{−j} = ∑_{−∞<j≤0} ∑_{n≥0} f_{j,n}(x)\frac{t^{n}}{n!}ℏ^{−j−n}

where

f_{j}(t, x) = ∑_{j≤n,0≤n} f_{j−n,n}(x)\frac{t^{n}}{n!}

satisfies

|f_{j}(t, x)| ≤ C ∑_{j≤n,0≤n} ε^{n−j} \frac{(n − j)!}{n!}(|t|R)^{n}.

Let η < (εR)^{−1}. It follows that f_{j}(t, x) is holomorphic in a neighborhood of {|t| ≤ η}×K. Assume j < 0, |t| ≤ η and x ∈ K. We get

|f_{j}(t, x)| ≤ Cε^{−j}(−j)! ∑_{0≤n} \frac{(n − j)!}{(−j)!n!}(ηεR)^{n} ≤ \frac{C}{1 − ηεR}(\frac{ε}{1 − ηεR})^{−j}(−j)!,

hence Condition (1.11) is satisfied.

Assume j ≥ 0. We get for |t| ≤ η and x ∈ K

|f_{j}(t, x)| ≤ C \frac{ε^{−j}}{j!} ∑_{j≤n} \frac{j!(n − j)!}{n!}(|t|Rε)^{n} ≤ \frac{C}{1 − ηεR} \frac{R^{j}}{j!} |t|^{j},

hence Condition (1.12) for m = 0 is satisfied and L(f)(t, x, ℏ) is in O_{X}^{t,ℏ}(0).

(iii) Conversely, let f(t, x, ℏ) be a section of O_{X}^{t,ℏ}(0). We develop f as

f(t, x, ℏ) = ∑_{−∞<j<∞} f_{j}(t, x)ℏ^{−j} = ∑_{−∞<j<∞} ∑_{n≥0} n!f_{j,n}(x)\frac{t^{n}}{n!}ℏ^{−n}ℏ^{−j+n}. (1.17)

For any compact set K, there exists η > 0 such that f_{j}(t, x) is holomorphic in a neighborhood of {|t| ≤ η} × K. Conditions (1.11) and (1.12) give the Cauchy’s estimates

|f_{j,n}(x)| ≤ Cε^{−j}(−j)!η^{−n} for j < 0,

|f_{j,n}(x)| ≤ M \frac{R^{j}}{j!} η^{j−n} for j ≥ 0.

Notice that Condition (1.12) for j > 0 implies that

f_{j}(0, x) = \frac{∂f_{j}}{∂s} (0, x) = ⋯ = \frac{∂^{j−1}f_{j}}{∂s^{j−1}} (0, x) = 0, (1.18)

or f_{j,n}(x) = 0 for 0 ≤ n ≤ j − 1.

The inverse Laplace transform consists formally in remplacing \frac{t^{n}}{n!}ℏ^{−n} by s^{−n−1} in (1.17), we then get

L^{−1}(f)(s, x, ℏ) = f̃(s, x, ℏ) = ∑_{−∞<j<∞} ∑_{n≥0} n!f_{j+n,n}(x)s^{−n−1}ℏ^{−j}.

Writing f̃(s, x, ℏ) = ∑_{−∞<j<∞} f̃_{j}(s, x)ℏ^{−j}, (1.18) implies that f̃_{j}(s, x) = 0 for j ≥ 1.

Let R_{1} > R large enough so that (ηR_{1})^{−1} ≤ 1. We shall check that the sum f̃_{j}(s, x) = ∑_{n≥0} n!f_{j+n,n}(x)s^{−n−1} defines a holomorphic function in a neighborhood of {|s| ≥ R_{1}}×K for any j ≤ 0.

For j ≤ 0, let us split the sum f̃_{j}(s, x) as

f̃_{j}(s, x) = ∑_{n≥−j} n!f_{j+n,n}(x)s^{−n−1} + ∑_{0≤n<−j} n!f_{j+n,n}(x)s^{−n−1}. (1.19)

In the first sum we have n + j ≤ 0 and for |s| ≥ R_{1} and x ∈ K, we get from the Cauchy’s estimates

(1.20) |n!f_{j+n,n}(x)s^{−n−1}| ≤ n!M \frac{R^{n+j}}{(n + j)!}η^{j}|s|^{−n−1} ≤ \frac{M}{R_{1}} (ηR)^{j}(−j)!\binom{n}{−j}(\frac{R}{R_{1}} )^{n}.

The right-hand side is the general term of a convergent series since R_{1} > R and we get the result by noticing that the second sum in (1.19) is finite.

Finally we shall show that f̃_{j}(s, x) satisfies the required estimates. ¿From (1.20), the first sum in (1.19) is bounded by

| ∑_{n≥−j} n!f_{j+n,n}(x)s^{−n−1}| ≤ \frac{M}{R_{1}} (ηR)^{j}(−j)! ∑_{n≥−j} \binom{n}{−j}(\frac{R}{R_{1}} )^{n}

≤ \frac{M}{R_{1} − R}(\frac{1}{η(R_{1} − R)})^{−j}(−j)!.

Similarly, for the second sum we have

| ∑_{0≤n<−j} n!f_{j+n,n}(x)s^{−n−1}| ≤ \frac{C}{R_{1}} ε^{−j}(−j)! ∑_{0≤n<−j} \frac{1}{\binom{−j}{n}}(\frac{1}{ηR_{1}} )^{n} ≤ \frac{2C}{R_{1}} ε^{−j}(−j)!,

where the last inequality follows from (ηR_{1})^{−1} ≤ 1 and ∑_{0≤n<−j} \frac{1}{\binom{−j}{n}} ≤ 2.

Combining these estimates we get for j ≤ 0

|f̃_{j}(s, x)| ≤ C̃ε̃^{−j}(−j)!,

with C̃ = max\{\frac{M}{R_{1}−R}, \frac{2C}{R_{1}} \} and ε̃ = max\{ε, \frac{1}{η(R_{1}−R)}\}.

Therefore f̃(s, x, ℏ) = ∑_{j≤0} f̃_{j}(s, x)ℏ^{−j} is a section of O_{X}^{s,ℏ}(0) and L(f̃)(t, x, ℏ) = f(t, x, ℏ).

(iv) The fact that L is a morphism of algebras follows easily from Example 1.2. □

The ring gr O_{X}^{t,ℏ}

If A is a filtered sheaf of rings, we denote as usual by gr A the associated graded ring.

Let ℂ_{u} be the complex line endowed with the coordinate u and denote by b: X ×ℂ_{u} ⟶ X the projection.

Definition 1.14. (i) One denotes by O_{X}^{exp u} the subsheaf of ℂ-algebras on X of the sheaf b_{∗}O_{X×ℂ_{u}} whose sections on an open set U ⊂ X are the holomorphic functions f(x, u) on U × ℂ_{u} satisfying:

{for any compact subset K of U there exist positive constants C, R such that sup_{x∈K} |f(x, u)| ≤ C exp(R|u|).

(ii) One sets O_{X}^{exp tℏ^{−1}}[ℏ, ℏ^{−1}] = O_{X}^{exp tℏ^{−1}} ⊗_{ℂ} ℂ[ℏ, ℏ^{−1}].

Proposition 1.15. There is a natural isomorphism of graded sheaves of rings

gr O_{X}^{t,ℏ} ≃ O_{X}^{exp tℏ^{−1}}[ℏ, ℏ^{−1}].

Proof. First note the isomorphism

O_{X}^{s,ℏ}(0)/O_{X}^{s,ℏ}(−1) ≃ R^{1}a_{!}O_{ℂ_{s}×X},

from which we deduce the isomorphism

gr O_{X}^{s,ℏ} ≃ R^{1}a_{!}O_{ℂ_{s}×X} ⊗_{ℂ} ℂ[ℏ, ℏ^{−1}].

The classical Paley-Wiener theorem says that the Laplace transform induces an isomorphism between H_{c}^{1}(ℂ; O_{ℂ}) and the space of entire functions of exponential type. An extension of this result with holomorphic parameters provides an isomorphism

L: R^{1}a_{!}O_{ℂ_{s}×X} \underrightarrow{∼} O_{X}^{exp tℏ^{−1}}

and the result follows. □

The formal case

It is possible to replace O_{X}^{ℏ} with Ô_{X}^{ℏ} in the preceding constructions and to set

Ô_{X}^{s,ℏ} := R^{1}a_{!}Ô_{ℂ_{s}×X}^{ℏ}. (1.21)

However the Laplace transform of Ô_{X}^{s,ℏ} does not seem to have an easy description. Indeed, its sections are no longer germs of holomorphic functions with respect to t as shown in the next example.

Example 1.16. Consider a sequence {c_{j}}_{j≤0} of complex numbers and the section f of Ô_{X}^{s,ℏ} given by

f(s, ℏ) = ∑_{j≤0} \frac{c_{j}}{(s − 1)}ℏ^{−j}.

Then, formally, the Laplace transform of f is given by

L(f)(t, ℏ) = ∑_{j≤0} ∑_{n≥0} c_{j} \frac{t^{n}}{n!}ℏ^{−n−j},

and the coefficient of ℏ^{0} is ∑_{n≥0} c_{−n} \frac{t^{n}}{n!} , which does not belong to O_{ℂ_{t}}|_{t=0} in general.

2 The algebra W_{T^{∗}X}

Let (X, O_{X}) be a complex manifold. The cotangent bundle T^{∗}X is a homogeneous symplectic manifold endowed with the ℂ^{×}-conic sheaf of rings E_{T^{∗}X} of finite-order microdifferential operators. This ring is filtered and contains in particular the subring E_{T^{∗}X}(0) of operators of order ≤ 0. This ring is constructed in [8] and we assume that the reader is familiar with this theory, referring to [5] or [9] for an exposition.

On the symplectic manifold T^{∗}X there exists another (no more conic) useful sheaf of rings constructed as follows (see [7]). Let ℂ be the complex line endowed with the coordinate t and (t; τ) the associated coordinates on T^{∗}ℂ. Set T_{{τ≠0}}^{∗}(X × ℂ) = {(x, t; ξ, τ); τ ≠ 0} and consider the map

ρ: T_{{τ≠0}}^{∗}(X × ℂ) ⟶ T^{∗}X, (x, t; ξ, τ) ↦ (x; ξ/τ). (2.1)

Set

E_{T^{∗}(X×ℂ),t̂}= {P ∈ E_{T^{∗}(X×ℂ)}; [P, ∂/∂_{t}] = 0}. (2.2)

The ring W_{T^{∗}X} on T^{∗}X is given by

W_{T^{∗}X} := ρ_{∗}(E_{T^{∗}(X×ℂ),t̂}).

In the sequel we set

ℏ:= τ^{−1}. (2.3)

The ring W_{T^{∗}X} is filtered and we denote by W_{T^{∗}X}(j) the subsheaf of W_{T^{∗}X} consisting of sections of order less or equal to j. The following result was obtained in [7].

Theorem 2.1. (i) The sheaf W_{T^{∗}X} is naturally endowed with a structure of a filtered k-algebra and gr W_{T^{∗}X} ≃ O_{T^{∗}X}[ℏ, ℏ^{−1}].

(ii) Consider two complex manifolds X and Y, two open subsets U_{X} ⊂ T^{∗}X and U_{Y} ⊂ T^{∗}Y and a symplectic isomorphism ψ : U_{X} \underrightarrow{∼} U_{Y}. Then, locally, ψ may be quantized as an isomorphism of filtered k-algebras Ψ: W_{T^{∗}X} \underrightarrow{∼} W_{T^{∗}Y} such that the isomorphism induced on the graded algebras coincides with the isomorphism O_{T^{∗}X}[ℏ, ℏ^{−1}] \underrightarrow{∼} O_{T^{∗}Y} [ℏ, ℏ^{−1}] induced by ψ.

Total symbols

Assume that X is affine of dimension n, that is, X is open in some ℂ-vector space V of dimension n.

Theorem 2.2. Assume X is affine. There is an isomorphism of filtered sheaves of kmodules (not of algebras), called the “total symbol” morphism:

σ_{tot} : W_{T^{∗}X} \underrightarrow{∼} O_{T^{∗}X}^{ℏ}. (2.4)

The total symbol of a product is given by the Leibniz formula. Denote by (x) a local coordinate system on X and denote by (x, u) the associated local symplectic coordinate system on T^{∗}X. If Q is an operator of total symbol σ_{tot}(Q), then

(2.5) σ_{tot}(P ∘ Q) = ∑_{α∈ℕ^{n}} \frac{ℏ^{|α|}}{α!} ∂_{u}^{α}σ_{tot}(P) · ∂_{x}^{α}σ_{tot}(Q).

The total symbol of a section P ∈ W_{T^{∗}X}(U) is thus written as a formal series:

σ_{tot}(P) = ∑_{−∞≤j≤m} p_{j}(x; u)ℏ^{−j}, m ∈ ℤ, p_{j} ∈ O_{T^{∗}X}(U), (2.6)

with the condition (1.4).

Note that (2.5) does not depend of the choice of a local coordinate system on X but only on the affine structure of V. Indeed, (2.5) may be rewritten as

σ_{tot}(P ∘ Q) = (exp(ℏ⟨d_{u}, d_{y}⟩)σ_{tot}(P)(x, u)σ_{tot}(Q)(y, v))|_{x=y,u=v}.

where ⟨d_{u}, d_{y}⟩ = ∑_{i=1}^{n} ∂_{u_{i}}∂_{y_{i}} does not depend on the affine coordinate system.

Remark 2.3. Let us identify X with the zero section of T^{∗}X. Then the sheaf O_{X}^{ℏ} (see Def. 1.1) is isomorphic to the left coherent W_{T^{∗}X}-module obtained as the quotient of W_{T^{∗}X} by the left ideal generated the vector fields on X.

3 The algebra W_{T^{∗}X}^{s}

Operations on W

Let S be a complex manifold of complex dimension d_{S}. One defines the sheaf W_{S×T^{∗}X} on S × T^{∗}X as the subsheaf of W_{T^{∗}(S×X)} consisting of sections which commute with the holomorphic functions on S. Heuristically, W_{S×T^{∗}X} is the sheaf W_{T^{∗}X} with holomorphic parameters on S. For a morphism of complex manifolds f: S ⟶ Z we shall still denote by f the map S × X ⟶ Z × X, as well as the map S × T^{∗}X ⟶ Z × T^{∗}X. One denotes as usual by Ω_{S} the sheaf of holomorphic forms of maximal degree and one sets for short:

W_{S×T^{∗}X}^{(d_{S})} = W_{S×T^{∗}X} ⊗_{O_{S}} Ω_{S}. (3.1)

Let us recall well-known operations of the theory of microdifferential operators. Although these results do not seem to be explicitly written in the literature, their proofs are straightforward and will not be given here.

Let f: S ⟶ Z be a morphism of complex manifolds. The usual operations of inverse image f^{∗}: f^{−1}O_{Z} ⟶ O_{S} and of direct image ∫_{f}: Rf_{!}Ω_{S}[d_{S}] ⟶ Ω_{Z}[d_{Z}] extend to W_{S×T^{∗}X}. More precisely, there exist morphisms of sheaves of k-modules (the second morphism holds in the derived category D^{b}(k_{Z×T^{∗}X})):

f^{∗}: f^{−1}W_{Z×T^{∗}Z} ⟶ W_{S×T^{∗}X}, (3.2)

∫_{f}: Rf_{!}(W_{S×T^{∗}X}^{(d_{S})} [d_{S}]) ⟶ W_{Z×T^{∗}X}^{(d_{Z})} [d_{Z}], (3.3)

these morphisms having the following properties:

• they are functorial with respect to f, that is, for a morphism of complex manifolds g: Z ⟶ W, one has (g ∘ f)^{∗} ≃ f^{∗} ∘ g^{∗} and ∫_{g∘f} = ∫_{g} ∘ ∫_{f}, and moreover the inverse (resp. direct) image of the identity morphism is the identity,

• when X is affine, f^{∗} and ∫_{f} commute with the total symbol morphism (2.4).

As a convention, we choose the morphism in (3.3) so that the integral of \frac{ds}{s} ∈

H_{c}^{1}(ℂ_{s}; Ω_{ℂ_{s}}) is 1. In other words,

∫_{a} \frac{1}{s} = \frac{1}{2iπ} ∫_{γ} \frac{ds}{s},

where γ is a counter clockwise oriented circle around the origin.

The algebra W_{T^{∗}X}^{s}

Denote by

a: ℂ_{s} × T^{∗}X ⟶ T^{∗}X (3.4)

the projection. Then, after identifying the sheaves O_{ℂ_{s}} and Ω_{ℂ_{s}} by f(s) ↦ f(s)ds, the sheaf R^{1}a_{!}W_{ℂ_{s}×T^{∗}X} is endowed with a structure of a filtered k-algebra by

H_{c}^{1}(ℂ_{s} × T^{∗}X; W_{ℂ_{s}×T^{∗}X}) × H_{c}^{1}(ℂ_{s′} × T^{∗}X; W_{ℂ_{s′}×T^{∗}X})
⟶ H_{c}^{2}(ℂ_{s,s′}^{2} × T^{∗}X; W_{ℂ_{s,s′}^{2}×T^{∗}X})
⟶ H_{c}^{1}(ℂ_{s}; W_{ℂ_{s}×T^{∗}X}),

where the first arrow is the cup product and the second arrow is the integration along the fibers of the map ℂ^{2} ⟶ ℂ, (s, s′) ↦ s + s′.

Definition 3.1. The sheaf W_{T^{∗}X}^{s} of k-modules on T^{∗}X is given by

W_{T^{∗}X}^{s} = R^{1}a_{!}(W_{ℂ_{s}×T^{∗}X}). (3.5)

After identifying the holomorphic function \frac{1}{s} with the cohomology class it defines in

H_{c}^{1}(ℂ_{s}; O_{ℂ_{s}}), we define the morphism of sheaves

ι: W_{T^{∗}X} ⟶ W_{T^{∗}X}^{s}, P ↦ \frac{1}{s}P. (3.6)

Clearly, the morphism (3.6) is a monomorphism of sheaves of k-algebras.

We define the morphism of sheaves

res: W_{T^{∗}X}^{s} ⟶ W_{T^{∗}X} (3.7)

by the integration morphism (3.3) associated to the map (3.4). Clearly, the morphism (3.7) is a morphism of sheaves of k-algebras. Hence:

Theorem 3.2. (i) The sheaf W_{T^{∗}X}^{s} is naturally endowed with a structure of a filtered k-algebra and gr W_{T^{∗}X}^{s} ≃ R^{1}a_{!}O_{ℂ_{s}×T^{∗}X}[ℏ, ℏ^{−1}].

(ii) The monomorphism ι in (3.6) is a morphism of filtered k-algebras, the integration morphism res in (3.7) is a morphism of filtered k-algebras and the composition res ∘ ι: W_{T^{∗}X} ⟶ W_{T^{∗}X}^{s} ⟶ W_{T^{∗}X} is the identity.

(iii) Consider two complex manifolds X and Y, two open subsets U_{X} ⊂ T^{∗}X and U_{Y} ⊂ T^{∗}Y and a symplectic isomorphism ψ : U_{X} \underrightarrow{∼} U_{Y}. Then, locally, ψ may be quantized as an isomorphism of filtered k-algebras Ψ: W_{T^{∗}X}^{s} \underrightarrow{∼} W_{T^{∗}Y}^{s} such that the isomorphism induced on the graded algebras coincides with the isomorphism R^{1}a_{!}O_{ℂ_{s}×T^{∗}X}[ℏ, ℏ^{−1}] \underrightarrow{∼} R^{1}a_{!}O_{ℂ_{s}×T^{∗}Y} [ℏ, ℏ^{−1}] induced by ψ.

(iv) Assume X is affine. There is an isomorphism of filtered sheaves of k-modules (not of algebras), called the “total symbol” morphism:

σ_{tot} : W_{T^{∗}X}^{s} \underrightarrow{∼} O_{T^{∗}X}^{s,ℏ}. (3.8)

The total symbol of a product is given by the Leibniz formula with a convolution product in the s variable (see (3.10)).

Proof. These results follow immediately from Theorem 2.1. □

Assume that X is affine. For each Stein open subset W of T^{∗}X and each relatively compact open subset U ⊂⊂ W, a section P of W_{T^{∗}X}^{s} on W admits a total symbol

(3.9)  σ_{tot}(P)(s, x, u) = ∑_{−∞<j≤m} p_{j}(s, x; u)ℏ^{−j}, m ∈ ℤ

where p_{j} belongs to Γ((ℂ_{s} \ K_{0}) × U; O_{ℂ_{s}×T^{∗}X}), for a compact subset K_{0} of ℂ_{s} which depends only on P and U, and the p_{j}’s satisfy an estimate as in (1.4) on each compact subset K of (ℂ_{s} \ K_{0}) × U.

Consider now two sections P and Q of W_{T^{∗}X}^{s} on a Stein open set W with total symbols as in (3.9) (replacing p_{j} with q_{j} and m with m′ for Q). Then the total symbol of P ∘ Q is given by the Leibniz formula:

(3.10) σ_{tot}(P ∘ Q) = ∑_{α∈ℕ^{n}} \frac{ℏ^{|α|}}{α!} ∂_{u}^{α}σ_{tot}(P) ∗ ∂_{x}^{α}σ_{tot}(Q),

where, setting f(s, x, u) = ∂_{u}^{α}σ_{tot}(P)(s, x; u) and g(s, x, u) = ∂_{x}^{α}σ_{tot}(Q)(s, x; u), the product f ∗ g is given by (1.9).

4 The Laplace transform and the algebra W_{T^{∗}X}^{t}

The filtered k-algebra W_{T^{∗}X}^{t} on T^{∗}X is the algebra W_{T^{∗}X}^{s}, but with a different symbol calculus.

Definition 4.1. We set W_{T^{∗}X}^{t} := W_{T^{∗}X}^{s}. For X affine, the total symbol morphism of k-modules (not of algebras)

σ_{tot} : W_{T^{∗}X}^{t} \underrightarrow{∼} O_{T^{∗}X}^{t,ℏ} (4.1)

is the composition W_{T^{∗}X}^{s} ∼ ; ⟶ ; σ_{tot} O_{T^{∗}X}^{s,ℏ} ∼ ; ⟶ ; L O_{T^{∗}X}^{t,ℏ}.

For P a section of W_{T^{∗}X}^{t} on a Stein open subset V of T^{∗}X and an open subset U ⊂⊂ V, σ_{tot}(P) is written as a series

σ_{tot}(P)(t, x, u, ℏ) = ∑_{−∞<j<∞} p_{j}(t, x, u)ℏ^{−j}, p_{j} ∈ O_{ℂ×T^{∗}X|_{t=0}}(U)

satisfying (1.11) and (1.12).

Applying Theorem 3.2, we get:

Theorem 4.2. (i) W_{T^{∗}X}^{t} is a filtered k-algebra and gr W_{T^{∗}X}^{t} ≃ O_{T^{∗}X}^{exp tℏ^{−1}}[ℏ, ℏ^{−1}] (see Definition 1.14).

(ii) The morphism ι in (3.6) induces a monomorphism of filtered k-algebras ι: W_{T^{∗}X} ↪ W_{T^{∗}X}^{t}, the morphism res in (3.7) induces a morphism of filtered k-algebras res: W_{T^{∗}X}^{t} ⟶ W_{T^{∗}X} and the composition W_{T^{∗}X} ⟶ W_{T^{∗}X}^{t} ⟶ W_{T^{∗}X} is the identity.

(iii) Consider two complex manifolds X and Y, two open subsets U_{X} ⊂ T^{∗}X and U_{Y} ⊂ T^{∗}Y and a symplectic isomorphism ψ : U_{X} \underrightarrow{∼} U_{Y}. Then, locally, ψ may be quantized as an isomorphism of filtered k-algebras Ψ: W_{T^{∗}X}^{t} \underrightarrow{∼} W_{T^{∗}Y}^{t} such that the isomorphism induced on the graded algebras coincides with the isomorphism O_{T^{∗}X}^{exp tℏ^{−1}}[ℏ, ℏ^{−1}] \underrightarrow{∼} O_{T^{∗}Y}^{exp tℏ^{−1}}[ℏ, ℏ^{−1}] induced by ψ.

(iv) Assume X is affine. There is an isomorphism of filtered sheaves of k-modules (not of algebras), called the “total symbol” morphism:

σ_{tot} : W_{T^{∗}X}^{t} \underrightarrow{∼} O_{T^{∗}X}^{t,ℏ}. (4.2)

The total symbol of a product is given by the Leibniz formula.

For P and Q two sections of W_{T^{∗}X}^{t} on an open subset U of T^{∗}X, with X affine, the total symbol of P ∘ Q is thus given by the formula:

(4.3) σ_{tot}(P ∘ Q) = ∑_{α∈ℕ^{n}} \frac{ℏ^{|α|}}{α!} ∂_{u}^{α}σ_{tot}(P) · ∂_{x}^{α}σ_{tot}(Q),

where the product ∂_{u}^{α}σ_{tot}(P) · ∂_{x}^{α}σ_{tot}(Q) is given by the usual commutative algebra structure of O_{T^{∗}X}^{t,ℏ} of Lemma 1.9.

Remark 4.3. In Theorem 4.2, the monomorphism W_{T^{∗}X} ⟶ W_{T^{∗}X}^{t} is given on symbols by σ_{tot}(P) ↦ σ_{tot}(P) and the morphism W_{T^{∗}X}^{t} ⟶ W_{T^{∗}X} is given on symbols by σ_{tot}(P)(t, x; u, ℏ) ↦ σ_{tot}(P)(0, x; u, ℏ).

The above constructions also work when replacing the sheaf W_{T^{∗}X} with its formal counterpart, the sheaf Ŵ_{T^{∗}X}. Let us briefly explain it.

Let X be a complex manifold, as above. Replacing the sheaf of rings E_{T^{∗}X} on T^{∗}X with the sheaf of rings Ê_{T^{∗}X} of formal microdifferential operators and proceeding as for W_{T^{∗}X}, we get the sheaf of rings Ŵ_{T^{∗}X} of finite-order formal WKB-operators on T^{∗}X. It is defined by

Ŵ_{T^{∗}X} := ρ_{∗}(Ê_{T^{∗}(X×ℂ),t̂}).

When X is affine of dimension n, the total symbol morphism induces an isomorphism of k̂-modules

σ_{tot} : Ŵ_{T^{∗}X} \underrightarrow{∼} Ô_{T^{∗}X}^{ℏ}

and the symbol σ_{tot}(P ∘ Q) is given by the Leibniz formula (2.5). Then by a similar construction as for W_{T^{∗}X}^{s} we construct the filtered sheaf of k̂-algebras Ŵ_{T^{∗}X}^{s}. Namely, we set

Ŵ_{T^{∗}X}^{s} := R^{1}a_{!}Ŵ_{ℂ×T^{∗}X}.

If X is affine, the total symbol morphism induces an isomorphism of k̂-modules Ŵ_{T^{∗}X}^{s} \underrightarrow{∼} Ô_{T^{∗}X}^{s,ℏ} and the product is again given by the Leibniz formula (3.10).

However, as already noticed, the Laplace transform does not seem to behave as well for the formal case as for the analytic case, and we shall not construct the Laplace transform of Ô_{T^{∗}X}^{s,ℏ}.

5 Remark: The algebra W_{𝔛}^{s} on a symplectic manifold 𝔛

The complex case

Consider a complex symplectic manifold 𝔛. There exists an open covering 𝔛 = ⋃_{i} U_{i} and complex symplectic isomorphisms φ_{i}: U_{i} \underrightarrow{∼} V_{i} where the V_{i}’s are open in some cotangent bundles T^{∗}X_{i} of complex manifolds X_{i}. Set W_{U_{i}} := φ_{i}^{−1}W_{T^{∗}X_{i}|_{V_{i}}}. In general, the W_{U_{i}}’s do not glue in order to give a globally defined sheaf of algebras W_{𝔛} on 𝔛. However the prestack 𝔖 on 𝔛 (roughly speaking, a prestack is a sheaf of categories) U_{i} ↦ Mod(W_{U_{i}}) is a stack and the category Mod(W_{𝔛}) := 𝔖(𝔛) is well defined. Moreover, one can give a precise meaning to W_{𝔛} by replacing the notion of a sheaf of algebras with that of an algebroid. We refer to [4] for the construction of (an analogue of) this stack in the contact complex case and to [6] in the symplectic complex case for Ŵ_{𝔛} and for the definition of an algebroid. See also [7] for a construction of W_{𝔛} (by a different method). By adapting the construction of [7], one easily constructs the algebroid W_{𝔛}^{s} associated with the locally defined sheaves of algebras W_{U_{i}}^{s}. Details are left to the reader.

The real case

Let M be a real analytic manifold, X a complexification of M and denote by ω_{X} the canonical 2-form on T^{∗}X. The conormal bundle T_{M}^{∗}X is Lagrangian for Re ω_{X} and symplectic for Im ω_{X}. In particular, the real manifold T_{M}^{∗}X is symplectic. For an open subset U of T_{M}^{∗}X, we set W_{U} := W_{T^{∗}X}|_{U}.

Now, consider a real analytic symplectic manifold 𝔐. It is well known that it is possible to construct a globally defined sheaf of algebras W_{𝔐} on 𝔐 such that:

• there exists an open covering 𝔐 = ⋃_{i∈I} U_{i} and real symplectic isomorphisms φ_{i}: U_{i} \underrightarrow{∼} V_{i} where the V_{i}’s are open in the conormal bundles T_{M_{i}}^{∗}X_{i} for some real manifolds M_{i} with complexification X_{i},

• W_{𝔐}|_{V_{i}} ≃ φ_{i}^{−1}W_{V_{i}} for all i ∈ I.

Replacing 𝔐 with ℂ_{s} × 𝔐, one easily contructs the sheaf of algebras W_{ℂ_{s}×𝔐} of sections with holomorphic parameter s ∈ ℂ_{s}. Setting

W_{𝔐}^{s} := R^{1}a_{!}W_{ℂ_{s}×𝔐}

we get a filtered k-algebra similar to the algebra W_{T^{∗}X}^{s} of Definition 3.1. Then, if P

belongs to W_{𝔐} and has order 0, the section \frac{1}{s − P} is well defined in W_{𝔐}^{s}.

6 Applications

As an application, let us construct the exponential of sections of order 0 of W_{T^{∗}X}.

Consider a section P of W_{T^{∗}X}(0) on an open subset U of T^{∗}X. For each compact subset K of U, there exists R > 0 such that the section s − P of W_{T^{∗}X}^{s} defined on ℂ_{s} × U is invertible on (ℂ_{s} \ D(0, R)) × K, where D(0, R) denotes the closed disc centered at 0

with radius R. Therefore \frac{1}{s − P} defines an element of H_{c}^{1}(ℂ_{s} × U; W_{ℂ_{s}×T^{∗}X}), hence, an

element of Γ(U; W_{T^{∗}X}^{s}). We still denote this section of W_{T^{∗}X}^{s} on U by \frac{1}{s − P}.

By developing \frac{1}{s − P} as ∑_{n≥0} \frac{P^{n}}{s^{n+1}} and applying the Laplace transform, we get for-

mally: L(\frac{1}{s−P}) = exp(tℏ^{−1}P).

Notation 6.1. We denote by exp(tℏ^{−1}P) the image in W_{T^{∗}X}^{t} of the section \frac{1}{s − P} of W_{T^{∗}X}^{s}.

Proposition 6.2. For P ∈ W_{T^{∗}X}(0), there is a section exp(tℏ^{−1}P) ∈ W_{T^{∗}X}^{t} such that, when X is affine:

σ_{tot}(exp(tℏ^{−1}P)) = ∑_{n≥0} \frac{(tℏ^{−1}σ_{tot}(P))^{⋆n}}{n!} ,

where the star-product f^{⋆n} means the product given by the Leibniz formula (2.5).

Remark 6.3. The Leibniz formula (2.5) is nothing but the standard or normal or Wick star-product and Proposition 6.2 tells us that the star-exponential [1] of P makes sense in W_{T^{∗}X}^{t}.

In a holomorphic deformation quantization context, the star-exponential of P is heuristically related to the Feynman Path Integral FPI(P) of P. Indeed, the Feynman Path Integral of a Hamiltonian H is the symbol of the evolution operator associated to H, the precise relation being given (see [2]) by

exp(−xuℏ^{−1})FPI(P) = σ_{tot}(exp(tℏ^{−1}P)).

Example 6.4. As a simple example, take X = ℂ and P ∈ W_{T^{∗}X}(0) with σ_{tot}(P) = p_{0}(t, x; u) = θxu, θ ∈ ℂ. Up to a change of holomorphic symplectic coordinates, σ_{tot}(P) represents the Hamiltonian of the harmonic oscillator in the holomorphic representation. Clearly P is in W_{ℂ}(0), and the total symbol of exp(tℏ^{−1}P) is easily computed:

\frac{∂}{∂t}σ_{tot}(exp(tℏ^{−1}P)) =σ_{tot}(ℏ^{−1}P ∘ exp(tℏ^{−1}P))
=ℏ^{−1}(σ_{tot}(P)σ_{tot}(exp(tℏ^{−1}P)) + ℏ\frac{∂}{∂u}σ_{tot}(P)\frac{∂}{∂x}σ_{tot}(exp(tℏ^{−1}P)))
=ℏ^{−1}θuxσ_{tot}(exp(tℏ^{−1}P)) + θx\frac{∂}{∂x}σ_{tot}(exp(tℏ^{−1}P)).

Since σ_{tot}(exp(tℏ^{−1}P))|_{t=0} = 1, the solution to the preceding equation is:

σ_{tot}(exp(tℏ^{−1}P)) = exp ((exp(θt) − 1)xuℏ^{−1}).

The Feynman Path Integral for the harmonic oscillator is well known in the Physics literature and is given by exp ( exp(θt)xuℏ^{−1}) [3].

References

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[2] J. Dito, Star product approach to quantum field theory: The free scalar field, Lett. Math. Phys. 20 125–134 (1990).

[3] L. D. Faddeev and A. A. Slanov, Gauge Fields. Introduction to Quantum Theory, Benjamin Cummings Publishing, (1980).

[4] M. Kashiwara, Quantization of contact manifolds, Publ. RIMS, Kyoto Univ. 32 p. 1–5 (1996).

[5] M. Kashiwara, D-modules and Microlocal Calculus, Translations of Mathematical Monographs, 217 American Math. Soc. (2003).

[6] M. Kontsevich, Deformation quantization of algebraic varieties, in: EuroConférence Moshé Flato, Part III (Dijon, 2000) Lett. Math. Phys. 56 (3) (2001), p. 271–294.

[7] P. Polesello and P. Schapira, Stacks of quantization-deformation modules over complex symplectic manifolds, Int. Math. Res. Notices 49 p. 2637–2664 (2004).

[8] M. Sato, T. Kawai, and M. Kashiwara, Microfunctions and pseudo-differential equations, in Komatsu (ed.), Hyperfunctions and pseudo-differential equations, Proceedings Katata 1971, Lecture Notes in Math. Springer-Verlag 287 p. 265–529 (1973).

[9] P. Schapira, Microdifferential Systems in the Complex Domain, Grundlehren der Math. Wiss. 269 Springer-Verlag (1985).

[10] Y. T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 p. 89–100 (1976/77).

Giuseppe Dito Institut de Mathématiques de Bourgogne Université de Bourgogne B.P. 47870, 21078 Dijon Cedex, France mailto: giuseppe.dito@u-bourgogne.fr http://www.u-bourgogne.fr/monge/g.dito/ Pierre Schapira Institut de Mathématiques Université Pierre et Marie Curie 175, rue du Chevaleret, 75013 Paris, France mailto: schapira@math.jussieu.fr http://www.math.jussieu.fr/∼schapira/