Oscilatory modules

TL;DR

This paper introduces oscillatory modules on symplectic manifolds, extending deformation quantization, and compares their category on a torus to the Fukaya category, revealing new insights into symplectic geometry.

Contribution

It defines oscillatory modules with additional structure and relates their category to the Fukaya category on a torus, advancing the understanding of quantization and symplectic categories.

Findings

Oscillatory modules form a new sheaf-based structure on symplectic manifolds.

Comparison shows a relationship between oscillatory modules and Fukaya categories.

Provides a framework connecting deformation quantization with symplectic topology.

Abstract

Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce a notion of an oscillatory module on a symplectic manifold which is a sheaf of modules over the sheaf of deformation quantization algebras with an additional structure. We compare the category of oscillatory modules on a torus to the Fukaya category as computed by Polishchuk and Zaslow.