Fukaya categories as categorical Morse homology

Details Fukaya categories as categorical Morse homology Fukaya categories as categorical Morse homology

2011-09-22

arxiv.org/abs/1109.4848v1

Mathematics

Symplectic Geometry

preliminary draft, some proofs only briefly indicated, comments welcome!

Scientific paper

The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. We show in analogy with Morse homology that the Fukaya category can be obtained by gluing together Fukaya categories of Weinstein cells. Our main technical result is a d\'evissage pattern for Lagrangian branes parallel to that for constructible sheaves. As an application, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplectic analogue of the well-known algebraic and topological theories of (micro)localization.

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