Skew-Gentle Algebras in Orbifold and Fukaya Categories

• Skew-gentle algebras are graded associative algebras defined from orbifold surfaces with boundary, incorporating type D quivers to encode local isotropy.
• They are constructed as endomorphism algebras from partially wrapped Fukaya categories, with quiver assembly reflecting geometric dissection and derived equivalence.
• Their classification via formal surface dissections connects algebraic invariants to orbifold topology, guiding new research in symplectic geometry and representation theory.

A skew-gentle algebra is a class of graded associative algebras appearing as the algebraic models underlying partially wrapped Fukaya categories of graded orbifold surfaces with boundary. These algebras generalize the classical notion of gentle algebras by incorporating the local orbifold structure via type D quiver fragments, encoding orbifold points and their ramifications in the algebraic data. The resulting category-theoretic models are simultaneously controlled by the combinatorial topology of surface dissections and by the local representation theory of the type D quivers, with closure properties under derived equivalence paralleling those of gentle algebras.

1. Constructive Origin: Orbifold Surfaces and Fukaya Categories

Skew-gentle algebras are realized as the endomorphism algebras of explicit generators for the partially wrapped Fukaya category $W(S)$ of a graded orbifold surface $S$ with chosen stops. Each admissible dissection $A$ of the surface yields a corresponding "ribbon complex" (a cell structure incorporating both smooth and orbifold points), to which is assigned a cosheaf of $A_\infty$ categories whose local models are controlled according to the nature of the underlying point:

• Smooth corner of valency $n$: the module category of a quiver of type $A_{n-1}$.
• Punctured corner with boundary stop: a cyclic $A_{n-1}$ quiver.
• Orbifold sector of valency $n$: a quiver of type $D_{n+1}$.

The global Fukaya category is constructed as the homotopy colimit of these local models over the ribbon graph, or, for surfaces with only 2-torsion orbifold points, as the orbit category of the smooth cover (Barmeier et al., 2024).

2. Defining Properties and Explicit Algebraic Structure

A graded skew-gentle algebra associated to an orbifold surface is, up to derived equivalence, an algebra $A=kQ/I$ where:

• The quiver $Q$ is assembled by gluing local quivers of type $A_{n-1}$, cyclic $A_{n-1}$, and $D_{n+1}$ at each respective corner or sector of the surface's ribbon decomposition.
• The relations $I$ and the grading on the arrows are determined by the winding data of a fixed line field and the orientation of the surface.

Formally, a formal generator $G$ in $W(S)$ yields

$\mathrm{End}_{W(S)}^*(G) \cong A,$

an honest graded skew-gentle algebra, such that the Yoneda embedding gives an equivalence $W(S) \simeq \mathrm{Perf}(A)$ of triangulated categories.

Key features distinguishing skew-gentle algebras from classical gentle algebras are:

• The presence of type D quiver components at orbifold points (encoding the local isotropy).
• Precise grading data reflecting geometric winding/tracing at the corners.
• Relations which respect the specific combinatorial patterns induced by orbifold sectors and stops.

3. Derived Equivalence and Local-to-Global Classification

Derived equivalence classes of skew-gentle algebras correspond bijectively to formal dissections of the underlying orbifold surface. The closure property echoes the gentle case (Schröer–Zimmermann): if $A$ is a graded skew-gentle algebra arising from a formal dissection, any graded algebra derived equivalent to $A$ is again a skew-gentle algebra arising from a possibly different formal dissection of the same orbifold surface. Thus, derived equivalence classes are in natural bijection with the decorated surface data—a conjectural generalization of the gentle algebra case, confirmed in full generality for gentle algebras and supported for skew-gentle algebras (Barmeier et al., 2024).

The construction is robust under local modifications: changing the dissection, or altering the orbifold point data (e.g., the valency or type at an orbifold sector), changes the corresponding algebra's presentation within the derived equivalence class, but the global category remains governed by the geometry of the orbifold surface.

4. Local Models: The Role of Type D, A, and Cyclic Quivers

The local algebraic models at each "sector" are dictated by the geometry:

• Type $A_n$ sector: represents a disk with $n$ boundary stops; algebraic model is the path algebra of an $A_n$ quiver with nilpotent relations for paths of length two.
• Cyclic $A_n$ sector: arises from a smooth annulus with stops, producing a cyclic quiver.
• Type $D_{n+1}$ sector: unique to orbifolds, corresponds to an orbifold disk ($D / \mathbb Z_2$) with $n$ stops. The type $D_{n+1}$ quiver introduces two short legs and a central node, reflecting the higher isotropy. The only nontrivial higher $A_\infty$ product is

$\mu_n(sY_{n-1},...,sY_1) = sQ,$

mirroring the unique configuration at an orbifold corner.

This local algebraic structure ensures that the global algebra captures both smooth and singular behaviors and yields the correct "flavor" in the categorical construction.

5. Equivalence to the Perfect Derived Category and A∞-Models

Any explicit algebra $A$ constructed from a formal dissection as above, and whose only nonzero higher $A_\infty$ product is $\mu_2$, can be shown (Theorem 7.13, (Barmeier et al., 2024)) to be quasi-isomorphic to its cohomology. Hence,

$W(S) \simeq \mathrm{Perf}(A),$

establishing that partially wrapped Fukaya categories of graded orbifold surfaces are governed by derived categories of skew-gentle algebras. This equivalence underpins the algebraic–topological dictionary: algebraic manipulations correspond precisely to topological modifications of dissections or orbifold data.

6. Applications and Theoretical Significance

Skew-gentle algebras provide comprehensive algebraic invariants for the Fukaya–type categories of orbifold surfaces, classifying the categories up to derived equivalence. This algebraic description supports systematic exploration of categorical and geometric invariants of orbifold surfaces, such as invariants under mapping class group actions or deformation theory. The classification also supports generalizations to more intricate symplectic objects and guides the study of topological invariants and moduli spaces associated to orbifold surfaces (Barmeier et al., 2024).

The mechanism allows the explicit construction of new families of algebras closed under derived equivalence and encompassing both gentle and type D fragments, supporting further exploration in the representation theory of surface and orbifold-related categories. This construction clarifies the precise correspondence between topological surface data, categories of Lagrangian branes, and their algebraic models.

7. Future Directions and Open Conjectures

The central open conjecture states that the set of formal dissections of an orbifold surface is closed under derived equivalence, i.e., every graded algebra derived equivalent to a skew-gentle algebra arises from a formal dissection of the same surface. This is expected to generalize the classical gentle algebra case and remains a critical avenue for further research, with significant implications for both categorical symplectic geometry and the representation theory of surface-related algebras (Barmeier et al., 2024).