Relative Fukaya Categories via Gentle Algebras

Relative Fukaya categories are hard to construct. In this paper, we provide a very explicit construction in the case of punctured surfaces. The starting point is the gentle algebra $\operatorname{Gtl} Q$ associated with a punctured surface $Q$. Our model for the relative Fukaya category is a deformation $\operatorname{Gtl}_q Q$ which has one formal deformation parameter per puncture. We verify that $\operatorname{Gtl}_q Q$ is a model for the relative Fukaya category by computing a large part of the $A_\infty$-structure on the derived category $\operatorname{H}\operatorname{Tw}\operatorname{Gtl}_q Q$. The core method is the deformed Kadeishvili theorem from arXiv:2308.08026. The paper's technical contributions include (1) a construction for removing curvature from band objects, (2) a method for analyzing Kadeishvili trees, (3) a matching between deformed Kadeishvili trees and relative Fukaya disks. This paper is the second in a series of three, whose aim it is to deform mirror symmetry for punctured surfaces.