Classification of deformation quantization algebroids on complex symplectic manifolds

TL;DR

This paper classifies deformation quantization algebroids on complex symplectic manifolds using cohomological methods, specifically showing they are classified by a second cohomology group with coefficients in a subgroup of invertible Laurent series.

Contribution

It provides a cohomological classification of deformation quantization algebroids on complex symplectic manifolds, linking them to H^2 cohomology groups.

Findings

Deformation quantization algebroids are classified by H^2(X;k^*)

Local models are rings of WKB operators with a central parameter au

Classification involves invertible formal Laurent series in au^{-1}

Abstract

Deformation quantization algebroids over a complex symplectic manifold X are locally given by rings of WKB operators, that is, microdifferential operators with an extra central parameter \tau. In this paper, we will show that such algebroids are classified by H^2(X;k^*), where k^* is a subgroup of the group of invertible formal Laurent series in \tau^-1.