A microlocal category associated to a symplectic manifold

TL;DR

This paper introduces a new microlocal category associated with symplectic manifolds, connecting deformation quantization modules, Lagrangian submanifolds, and existing frameworks like Tamarkin's sheaves and Fukaya categories.

Contribution

It defines a novel class of modules over deformation quantization algebras linked to symplectic topology, and constructs an infinity local system of morphisms, bridging multiple mathematical theories.

Findings

Construction of special modules from Lagrangian submanifolds

Comparison with Morse theory and Tamarkin's microlocal sheaves

Relation to Fukaya categories in symplectic geometry

Abstract

For a symplectic manifold satisfying some topological condition,we define a special class of modules over the deformation quantization algebra. For any two such modules we construct an infinity local system of morphisms. We construct such special module starting from a Lagrangian submanifold satisfying a topological condition. We compare the result to Morse theory, to the microlocal category of sheaves recently defined by Tamarkin, and to the Fukaya category of the two-dimensional torus. Several appendices explain the motivations that come from asymptotic analysis of pseudo-differential operators and distributions.