Microlocal Branes Are Constructible Sheaves

Let X be a compact real analytic manifold, and let T X be its cotangent bundle. In a recent paper with E. Zaslow [28], we showed that the dg category Shc(X) of constructible sheaves on X quasi-embeds into the triangulated envelope F (T X) of the Fukaya category of T X. We prove here that the quasi-embedding is in fact a quasi-equivalence. When X is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes in T X and regular holonomic DX -modules developed by Kapustin [15] and Kapustin-Witten [16] from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in T ∗ 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 X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in T X has recently been obtained by Fukaya-Seidel-Smith [9].