Fukaya categories of orbifold surfaces in representation theory

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 ${\tilde{\mathrm{A}}}_{n-1}$, 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.

36 pages, 11 figures, comments are very welcome