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arXiv · 2602.17370

Fukaya categories of orbifold surfaces in representation theory

Severin Barmeier·Zhengfang Wang
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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.

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Severin Barmeier, Zhengfang Wang. 2026-02-19. Fukaya categories of orbifold surfaces in representation theory. https://arxiv.org/abs/2602.17370

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