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Microlocal branes are constructible sheaves

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Abstract

LetX be a compact real analytic manifold, and letT*X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg categoryShc(X) of constructible sheaves onX quasi-embeds into the triangulated envelopeF(T*X) of the Fukaya category ofT*X. We prove here that the quasi-embedding is in fact a quasi-equivalence. WhenX is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes inT*X and regular holonomic\({{\mathcal D}_X}\) -modules developed by Kapustin (A-branes and noncommutative geometry, arXiv:hep-th/0502212) and Kapustin and Witten (Commun Number Theory Phys 1(1):1–236, 2007) from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians inT*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 inT*X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes inT*X has recently been obtained by Fukaya et al. (Invent Math 172(1):1–27, 2008).

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  1. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    David Nadler

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  1. David Nadler

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Correspondence toDavid Nadler.

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