On quantization of complex symplectic manifolds

TL;DR

This paper develops a framework for quantizing complex symplectic manifolds by associating a triangulated category of microdifferential modules, revealing deep connections with contactification and Calabi-Yau structures.

Contribution

It introduces a novel approach to quantization via contactification and constructs a Calabi-Yau category for compact manifolds, linking microdifferential modules to quantization algebroids.

Findings

Unique lift of Lagrangian subvarieties to contactification

Triangulated category of regular holonomic microdifferential modules for X

Realization of modules over a canonical quantization algebroid

Abstract

Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X+1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X.