Sectorial descent for wrapped Fukaya categories

TL;DR

This paper introduces new tools and results for computing and understanding wrapped Fukaya categories, including descent properties, generation criteria, and localization techniques, advancing the algebraic and geometric analysis of symplectic manifolds.

Contribution

It establishes a descent property, generation results, stop removal as localization, and Lefschetz thimble generation for wrapped Fukaya categories, with new formulas and criteria.

Findings

Wrapped Fukaya categories satisfy a descent (cosheaf) property.

Partially wrapped Fukaya categories are generated by cocores and linking disks.

Stop removal corresponds to localization in Fukaya categories.

Abstract

We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a `stop removal equals localization' result, and (4) that the Fukaya--Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a K\"unneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful.