2006/12/14 by David Nadler, Nadler, David · 3 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.math/0612399
openalex publication_date 2006/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a real analytic manifold, and let T^*X be its cotangent bundle. In a recent paper with E. Zaslow \citeNZ, 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 complex, one may interpret this as a topological analogue of the identification of Lagrangian branes in T^*X and holonomic DX-modules developed by Kapustin and Kapustin-Witten 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.