Fukaya Categories as Categorical Morse Homology

Abstract

The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, 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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Cited by in corpus (12)

• Legendrian knots and constructible sheaves
• Triangulated surfaces in triangulated categories
• The nonequivariant coherent-constructible correspondence for toric stacks
• On the combinatorics of exact Lagrangian surfaces
• A stable infinity-category of Lagrangian cobordisms
• Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions
• Calabi-Yau structures on topological Fukaya categories
• Morse decomposition for D-module categories on stacks
• Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties
• HMS for punctured tori and categorical mapping class group actions
• Mirror Symmetry in dimension one and Fourier-Mukai equivalences
• Representing the big tilting sheaves as holomorphic Morse branes