Constructible Sheaves and the Fukaya Category

Abstract

Let $X$ be a compact real analytic manifold, and let $T^{\ast }X$ 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 $Fuk(T^{\ast }X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $TwFuk(T^{\ast }X)$ for the $A_{\infty }$-triangulated envelope of $Fuk(T^{\ast }X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $Sh(X)$ quasi-embeds into $TwFuk(T^{\ast }X)$ as an $A_{\infty }$-category. Taking cohomology gives an embedding of the corresponding derived categories.

56 pages; to appear in JAMS

Cited by in corpus (3)

• Gauge Theory and Langlands Duality
• Geometric Endoscopy and Mirror Symmetry
• Masaki Kashiwara and Algebraic Analysis