Fukaya categories of orbifold surfaces in representation theory

TL;DR

This paper explores the connection between orbifold surface Fukaya categories and derived categories of graded associative algebras, providing geometric methods to understand algebraic structures in representation theory.

Contribution

It introduces a geometric approach to relate orbifold surface Fukaya categories with derived categories of graded algebras, including new perspectives on local models and algebraic equivalences.

Findings

Derived equivalence between Fukaya categories and perfect derived categories of graded algebras.

Equivalence between derived categories of type D_{n+1} and graded type A_{n-1} quivers.

Insight into the relationship between skew-gentle algebras and orbifold surfaces.

Abstract

We give an introduction to partially wrapped Fukaya categories of surfaces with orbifold singularities. Dissecting an orbifold surface $\mathbf S$ into polygons, certain dissections give rise to formal generators, inducing a triangulated equivalence between the derived Fukaya category of $\mathbf S$ and the perfect derived category of a graded associative algebra. This provides a geometric means for obtaining associative algebras -- conjecturally all -- which are derived equivalent to skew-gentle algebras. We include a new perspective on the partially wrapped Fukaya category of an orbifold disk which serves as a local model for the Fukaya categories of general orbifold surfaces. This perspective yields an equivalence between the perfect derived category of a quiver of type $\mathrm D_{n+1}$ and the perfect derived category of a graded quiver of type $\widetilde{\mathrm A}_{n-1}$, the latter being equipped with quadratic zero relations and a nontrivial A$_\infty$ structure. This equivalence elucidates the relationship between skew-gentle algebras and orbifold surfaces, and the role of deformation theory in this relationship.