Constructible Sheaves and the Fukaya Category

TL;DR

This paper constructs a Fukaya $A_$-category for cotangent bundles of real analytic manifolds and shows it contains the category of constructible sheaves, linking symplectic geometry with sheaf theory.

Contribution

It develops a new Fukaya $A_$-category for cotangent bundles and proves the category of constructible sheaves embeds into its twisted complexes.

Findings

Constructs a Fukaya $A_$-category for $T^*X$.

Shows $Sh(X)$ embeds into the category of twisted complexes of the Fukaya category.

Establishes a bridge between sheaf theory and symplectic geometry.

Abstract

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $Tw Fuk(T^*X)$ for the $A_\infty$-triangulated envelope of $Fuk(T^*X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $Sh(X)$ quasi-embeds into $Tw Fuk(T^*X)$ as an $A_\infty$-category. Taking cohomology gives an embedding of the corresponding derived categories.