Deformations of partially wrapped Fukaya categories of surfaces

TL;DR

This paper characterizes the deformation theory of partially wrapped Fukaya categories of surfaces, showing all deformations are geometric and related to orbifold compactifications, with implications for understanding Fukaya categories in symplectic geometry.

Contribution

It provides a complete description of A∞ deformation theory for these categories, introducing weak duals and unbounded twisted complexes to handle curvature issues, and links deformations to partial compactifications.

Findings

All A∞ deformations are geometric and correspond to orbifold partial compactifications.

Introduces weak duals and unbounded twisted complexes to address curvature problems.

Results extend to surfaces with orbifold points and relate to stop data in Fukaya categories.

Abstract

We give a complete description of the A$_\infty$ deformation theory of partially wrapped Fukaya categories of graded surfaces. We show that any abstract A$_\infty$ deformation is "geometric", namely it is equivalent to the partially wrapped Fukaya category of an orbifold surface obtained as a partial compactification of the original surface. For certain genus 0 surfaces, these deformations are generically Fukaya categories of compact pillowcases. We introduce the notion of a weak dual and use unbounded twisted complexes to overcome the curvature problem that naturally arises when some of the boundary components are fully wrapped. Our results provide a first account of the relationship between A$_\infty$ deformations of Fukaya categories and partial compactifications, as advocated in P. Seidel's ICM 2002 address, in the presence of stop data. All of our results also hold when the original surface has finitely many order 2 orbifold points.