Fukaya categories of orbifold surfaces in representation theory

TL;DR

The paper develops a comprehensive framework connecting Fukaya categories of orbifold surfaces to graded (skew-)gentle algebras. It shows how admissible dissections and orbit-category formalisms yield explicit A∞-algebras whose derived categories model partially wrapped Fukaya categories of orbifolds, with deformations corresponding to partial compactifications and orbifold points. A key outcome is that orbifold disks produce endomorphism algebras of type D quivers (e.g., D_{n+1}) and that formal dissections give rise to graded skew-gentle algebras, while certain new dissection types lead to non-formal A∞ structures and broader derived equivalence classes. The deformation-theoretic viewpoint ties the geometry of orbifold compactifications to the Hochschild cohomology of these categories, clarifying how orbifold singularities arise from deformations and how skein-like operations relate to skew-gentle algebras. Overall, the work provides geometric mechanisms to realize, classify, and deform algebras derived-equivalent to skew-gentle algebras via orbifold Fukaya categories, with implications for representation theory and cluster-theoretic constructions.

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.

Figures (11)

• Figure 1: An orbifold surface $\mathbf S_\lambda$ (right) arising through deformation of a partially wrapped Fukaya category of a smooth surface $\mathbf S_0$ (left).
• Figure 2: A smooth disk with $2n$ stops graded by the horizontal line field (left) and the graded orbifold disk with $n$ stops obtained as its $\mathbb Z_2$ quotient (right).
• Figure 3: Three types of arcs on an orbifold disk $\mathbf D_3^\times$ and their lifts to the double cover $\mathbf D_6$.
• Figure 4: An admissible dissection of $\mathbf D_3^\times$ (left), where $p_1, p_2, p_3, p_4$ are boundary paths and $q_1, q_2$ are orbifold paths. The right dissection is not admissible, since there are two stops in the upper right polygon, one boundary stop and one orbifold stop.
• Figure 5: An orbifold disk sequence of length $2$ contributing to the differential (left) and a smooth disk sequence of length $2$, where the paths $p_1$ and $p_2$ can be boundary paths or orbifold paths (right).

Theorems & Definitions (51)

• Example 5
• Remark 9
• Remark 10: Homological smoothness and higher structures
• Definition 15: barmeierschrollwang
• Remark 16: Smooth structure on orbifold surfaces
• Remark 17: Symplectic structure and order of orbifold points
• Remark 18: Stops
• Remark 19: Double cover
• Definition 20
• Definition 21