Fukaya Categories as Categorical Morse Homology

TL;DR

This paper proposes a new perspective on the Fukaya category of Weinstein manifolds, viewing it as a form of categorical Morse homology constructed by gluing local categories, and formalizes this via sheaf theory.

Contribution

It introduces a formal framework relating Fukaya categories to Morse homology through sheaf-theoretic gluing and recollement patterns, advancing understanding of symplectic invariants.

Findings

Fukaya categories can be expressed as global sections of a sheaf on the Weinstein manifold.

A recollement pattern for Lagrangian branes parallels constructible sheaves.

The approach offers a symplectic analogue of (micro)localization theories.

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.