Microlocal branes are constructible sheaves

Abstract

Let $X$ be a real analytic manifold, and let $T^{\ast }X$ be its cotangent bundle. In a recent paper with E. Zaslow \cite{NZ}, we showed that the dg category $S{h}_c(X)$ of constructible sheaves on $X$ quasi-embeds into the triangulated envelope $F(T^{\ast }X)$ of the Fukaya category of $T^{\ast }X$. We prove here that the quasi-embedding is in fact a quasi-equivalence. When $X$ is complex, one may interpret this as a topological analogue of the identification of Lagrangian branes in $T^{\ast }X$ and holonomic $D_X$-modules developed by Kapustin and Kapustin-Witten from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in $T^{\ast }X$ (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class in $T^{\ast }X$, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in $T^{\ast }X$ has recently been obtained by Fukaya-Seidel-Smith.

49 pages; minor expository changes

References in corpus (2)

• Electric-Magnetic Duality And The Geometric Langlands Program
• A-branes and Noncommutative Geometry

Cited by in corpus (4)

• Gauge Theory and Langlands Duality
• Geometric Endoscopy and Mirror Symmetry
• Unwrapped continuation invariance in Lagrangian Floer theory: energy and $C^0$ estimates
• Constraints on exact Lagrangians in cotangent bundles of manifolds fibred over the circle