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 $\widetilde{\mathrm A}{n-1}$, 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.

• The paper introduces a categorical framework linking partially wrapped Fukaya categories of orbifold surfaces to the classification of (skew-)gentle and derived-tame algebras.
• It employs sectorial gluing, ribbon graphs, and A∞ deformations to encode topological data into explicit algebraic invariants.
• It extends known correspondences by incorporating orbifold singularities, providing criteria for derived equivalence and formality in these enriched settings.

Fukaya Categories of Orbifold Surfaces and Skew-Gentle Algebras in Representation Theory

Overview

The paper "Fukaya categories of orbifold surfaces in representation theory" (2602.17370) provides a rigorous synthesis of categorical methods in symplectic topology and algebra, focused on the structure and gluing properties of partially wrapped Fukaya categories of orbifold surfaces and their applications to the classification and construction of (skew-)gentle and derived-tame algebras. The theoretical developments extend foundational results connecting gentle algebras and Fukaya categories of surfaces to a broader class that encompasses orbifold singularities, with a particular emphasis on the interplay between geometric dissection techniques, $A_\infty$ structures, and derived equivalence classification in algebra.

Gentle, Skew-Gentle Algebras, and Surface Models

The link between (skew-)gentle algebras and oriented surfaces with boundary, introduced via the combinatorial data of quivers with quadratic relations, forms the conceptual core of the approach. When dissecting a (possibly orbifold) surface into polygons, these polygons and their intersections encode both the objects and morphisms of the associated Fukaya category, which, in dimension two, admits formalisms rendering it accessible through explicit generators and algebraic models.

The trivial extension of a gentle algebra surfaces as a Brauer graph algebra, and the passage from surface data to algebraic invariants is facilitated by the use of ribbon graphs corresponding to the surface's decomposition. Importantly, maximum path structures in the quiver can be directly associated with topological features of the surface, enabling a geometric encoding of derived invariants such as those of Avella-Alaminos--Geiss and their categorical enhancements.

The paper outlines a bidirectional correspondence:

• Smooth graded surfaces with boundary stops $\leftrightarrow$ graded gentle algebras, via an equivalence of pretriangulated $A_\infty$ categories between partially wrapped Fukaya categories and categories of twisted complexes over graded gentle algebras.

Sectorial Gluing, Local-to-Global Properties, and Diagrammatic Construction

One critical insight is that equivalences between Fukaya categories and derived categories can often be reduced to their behavior on local pieces (sectors): specifically, surfaces (and thus their associated categories) are assembled from elementary building blocks, each corresponding—under varying perspectives (algebraic/quiver, topological/ribbon graph, symplectic/surface sector)—to linearly and cyclically oriented quivers of type $A_n$ and $\widetilde{A}_{n-1}$. The higher category-theoretic machinery employed involves gluing via homotopy colimits in the (Morita-localized) model category of DG/$A_\infty$ categories, and the sectorial descent theory for wrapped Fukaya categories, with cosheaves over ribbon graphs encoding the gluing data.

The formalism naturally extends to cosheaves of pretriangulated $A_\infty$ categories—generalizing Kontsevich's local-to-global conjecture for Fukaya categories—and structures such as sectorial descent become central: the categorical invariants are reconstructed from the local data, matching the local-to-global philosophy present in both symplectic topology and homological algebra.

Extension to Orbifold Surfaces: New Sector Types and Gluing Patterns

The major contribution lies in extending the above framework to compact oriented surfaces with isolated orbifold singularities (necessarily of order 2 in the $\mathbb{Z}$-graded setting, for compatibility with line fields). The authors introduce a regime in which the surface sector building blocks now include, besides standard polygons, local models for orbifold disks—sectors with a unique orbifold point and a new family of associated $A_\infty$ algebraic structures, parametrized by $n$ stops.

The new local models yield $A_\infty$ structures $(\widetilde{A}_{n-1}, \mu_{2n})$ with nontrivial $2n$-ary higher multiplication, reflecting counts of pseudoholomorphic disks in the symplectic model. These algebras are demonstrated to be Morita equivalent to path algebras of quivers of type $D_{n+1}$ (with suitable relations), thus providing a bridge to the representation theory of skew-gentle algebras and their derived categories.

Orbit Categories and Derived Equivalence

A further key component is the construction and analysis of $A_\infty$ orbit categories. Every orbifold surface $\mathbf{S}$ (with data $(S, \Sigma, \eta)$) admits a smooth double cover $\widetilde{\mathbf{S}}$, and the partially wrapped Fukaya category of $\mathbf{S}$ is realized as an $A_\infty$ orbit category of the category associated to $\widetilde{\mathbf{S}}$ (modulo $\mathbb{Z}_2$-action). This framework elucidates how skew-gentle algebras emerge as explicit endomorphism algebras of formal generators in the resulting orbit categories.

It is shown that every algebra derived equivalent to a skew-gentle algebra arises from a formal generator associated to a dissection of an orbifold surface, motivating a conjecture that the set of such algebras coincides with the full class of derived equivalences of skew-gentle algebras.

$A_\infty$ Deformations: Deformation Theory Meets Geometry

The deformation-theoretic aspect is explored via the construction of semi-universal families of $A_\infty$ deformations for the partially wrapped Fukaya categories of surfaces. Each such deformation corresponds to a partial compactification of boundary components into orbifold points, and the algebraic deformation parameterizes the family by the Hochschild cohomology $\mathrm{HH}^2$ class of the category.

A significant outcome is that orbifold points necessarily appear in the symplectic model whenever a full description of the deformation theory (ala Seidel's vision) is required, with every strict $A_\infty$ deformation geometrically realized as partially compactifying a boundary to an order-2 orbifold singularity, extending the theory profoundly beyond the previously understood smooth/gentle case.

Formal Generators and Classification of Derived Skew-Gentle Algebras

The paper gives a systematic treatment of the notion of formal generators, i.e., formal dissections of surfaces (including orbifold cases) where the resulting endomorphism $A_\infty$ algebra is formal. It is demonstrated that any graded associative algebra derived equivalent to a skew-gentle algebra can be realized as the cohomology of the endomorphism algebra for such a formal dissection. This provides a method of constructing, classifying, and understanding derived equivalence classes of skew-gentle algebras distinctly from the purely algebraic classification via tilting complexes.

Notable is the provision of explicit criteria (in terms of properties of dissections and the underlying polygons) for when an $A_\infty$ category becomes formal, supporting classification results for higher-dimensional cases, and illustrated by examples for orbifold disks with four stops.

Implications and Future Research

Theoretical implications: The results articulate a comprehensive, unified picture of how surfaces with orbifold points encode the full derived equivalence class of skew-gentle and related algebras, interfacing surface topology, symplectic geometry, and representation theory. The fine structure of $A_\infty$ deformations and their geometric realization signal important progress for the classification of derived-tame and cluster-tilted categories, and for the advancement of geometric approaches to representation theory.

Practical impact: The construction of algebra models via sectorial gluing and the explicit description of higher $A_\infty$ operations delivers tools for computational and categorical approaches to surface invariants, spectral invariants in wrapped Fukaya categories, and the homological mirror symmetry program in dimension two. In cluster theory and the theory of Jacobian algebras, the geometric models presented here extend to the punctured case, vital for understanding cluster categories of surfaces.

Prospects for future work:

• Extending classification to higher-order orbifold points and exploring the invariance of $A_\infty$ structures in other gradings or under more general group actions.
• Further investigation into the geometric realization of more general derived-tame algebras via sectorial construction, with implications for categorification problems in low-dimensional topology.
• Exploiting the deformation theory to study moduli spaces of $A_\infty$ categories and connections to mirror symmetry, particularly in non-simply connected and non-orientable settings.

Conclusion

This work advances the synthesis of symplectic and categorical representation theory by systematically extending the Fukaya category framework to orbifold surfaces and unveiling an explicit geometric–algebraic correspondence for skew-gentle and derived-tame algebras. With precise categorical gluing formulas, novel sector types, and a transparent deformation-theoretic interpretation, the results deepen the understanding of homological invariants and set a foundation for further research at the intersection of symplectic topology and algebraic representation theory.