Flat surfaces and stability structures

TL;DR

This paper establishes a correspondence between spaces of half-translation surfaces and stability structures on Fukaya categories, introducing new classification methods and tools for infinite-area surfaces.

Contribution

It provides a novel identification between geometric moduli spaces and categorical stability structures, with new techniques for classifying objects and handling infinite-area surfaces.

Findings

Established a correspondence between half-translation surfaces and Fukaya category stability structures.

Developed a complete classification of objects in these categories.

Introduced tools for analyzing infinite-area surfaces with cluster algebra-like structures.

Abstract

We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is achieved by new methods involving the complete classification of objects in these categories, which are defined in an elementary way. We also introduce a number of tools to deal with surfaces of infinite area, where structures similar to those in cluster algebra appear.