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

The cotangent bundle T X to a complex manifold X is classically endowed with the sheaf of k-algebras WT∗X of deformation quantization, where k := W{pt} is a subfield of C[[~, ~]. Here, we construct a new sheaf of k-algebras W TX which contains WT∗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 WT∗X , we show that exp(t~ P ) is well defined in W TX . 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 ~ and ~−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 ŴT X here, is constructed in [7] as well as its analytic counterpart WT X . The sheaf WT X is similar to the sheaf ET X of microdifferential operators of [8], but with an extra central parameter ~, a substitute to the lack of homogeneity1. Here ~ belongs to the field k := W{pt}, a subfield of C[[~,~ −1]. (Note that the notation τ = ~−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 In this paper, we write ET∗X and WT∗X instead of the classical notations EX and WX .