Fukaya Categories of Surfaces

Updated 15 October 2025

Fukaya categories of surfaces are A∞-categories constructed from Lagrangian submanifolds and pseudo-holomorphic curve counts, capturing key symplectic invariants.
They utilize combinatorial methods via ideal triangulations, quiver representations, and cosheaf techniques to build explicit algebraic models such as gentle and Ginzburg dg algebras.
These structures connect mirror symmetry, cluster algebras, and stability conditions, providing actionable insights into both geometric and algebraic frameworks.

A Fukaya category of a surface is a triangulated or $A_\infty$ -category constructed from Lagrangian (sub)manifolds and pseudo-holomorphic curve counts, encoding essential symplectic topological invariants. For surfaces—possibly with boundary, punctures, or orbifold points—such categories exhibit deep combinatorial and algebraic structure linked to representation theory, cluster algebras, mirror symmetry, and higher Teichmüller theory. This article provides an integrated perspective on Fukaya categories of surfaces, covering foundational constructions, connections with algebraic models (quiver or gentle algebras), stability conditions, explicit computations, and orbifold extensions.

1. Foundations: Construction and Algebraic Models

For an oriented surface $S$ (possibly with marked points $M$ , boundary, or orbifold data), one defines various versions of Fukaya categories depending on geometric choices and boundary/stop conditions. Key settings include:

Topological and (Partially) Wrapped Fukaya Categories: These are constructed using arc systems or curve systems, either with stops at marked points (bounding the "wrapping") or as global $A_\infty$ -categories generated by isotopy classes of Lagrangian arcs/curves, equipped with grading data from a line field $\eta$ (Grossack, 13 Oct 2025).
Combinatorial Models via Quivers or Gentle Algebras:
- For surfaces decomposed via ideal triangulations (WKB triangulations determined by quadratic differentials), the result is a quiver $Q(T)$ with a potential $W(T)$ encoding the counts of holomorphic polygons. The algebraic model is the Ginzburg dg algebra of $(Q(T), W(T))$ , whose derived category embeds into the Fukaya category of associated symplectic 3-folds (Smith, 2013).
- For general marked surfaces (with or without boundary), the partially wrapped Fukaya category is triangle equivalent to the perfect derived category of a homologically smooth and proper graded gentle algebra (Chang et al., 2022).
Sheaf/Cosheaf Local-to-Global Structures: A crucial methodological advance is realizing the Fukaya category as the global sections of a cosheaf of dg or $A_\infty$ -categories on a ribbon or spanning graph (core skeleton) of the surface (Dyckerhoff, 2015, Barmeier et al., 23 Jul 2024). This allows for a local-to-global description, whereby computations reduce to local models (“disks with stops” or local Weinstein sectors) and are glued via colimit procedures.

2. Local-to-Global and Gluing Principles

The computation and structure of Fukaya categories for surfaces leverage their locality. Key aspects:

Pants Decompositions: For closed surfaces of genus $g\ge2$ , decomposing into pairs-of-pants and cylinders allows explicit reconstruction: the Fukaya category is obtained as a homotopy (inverse) limit over the cover by these pieces, with compatibility data along overlaps (Pascaleff et al., 2021). In combinatorial terms, global categories are assembled from local building blocks via perverse schober descent (Christ, 7 Oct 2025).
Cosheaf Formulations for Orbifolds: For orbifold surfaces, the global category is computed as a cosheaf of $A_\infty$ -categories attached to a Lagrangian core, often involving sectors of “type D” (to account for orbifold points) as well as sectors of classical “type A” (Barmeier et al., 23 Jul 2024). Such cosheaf descriptions are Morita invariant and allow explicit computation of the global category for any generator arising from a suitable admissible dissection.
Cutting and Recollement: Cutting along arcs, stops, or admissible subsets implements recollement diagrams in the derived category, reflecting a functorial ‘localization-and-gluing’ structure at the level of categories (Chang et al., 2022).

3. Stability Conditions, Moduli, and Flat Structures

Bridgeland Stability Spaces: The stability conditions on (partially wrapped) Fukaya categories of surfaces are tightly controlled and admit explicit geometric classification. Every Bridgeland stability condition on such a Fukaya category (of a “fully stopped” surface) arises from a flat surface with suitable singularities, encoded by the moduli of meromorphic quadratic differentials (Takeda, 2018, Smith, 2013). Explicitly,

$\mathcal{M}(\Sigma) \xrightarrow{\sim} \mathrm{Stab}(F(\Sigma)),$

where $\mathcal{M}(\Sigma)$ is the moduli space of flat structures.

Cutting/Inductive Procedures: The calculation of stability spaces reduces to three base cases: the disk, the annulus, and the punctured torus. For each, stability conditions are classified by geometric (HKK) stability conditions coming from flat surfaces. Functorial cutting and gluing operations mirror excision in the surface, allowing for explicit assembly of global stability spaces (Takeda, 2018).
Interplay with Cluster Varieties: The identification of stability conditions with moduli spaces of quadratic differentials has repercussions for wall-crossing, Donaldson–Thomas invariants, and the structure of cluster varieties attached to the surface (Smith, 2013, Christ, 2022).

4. Computability and Explicit Models

Fukaya categories of surfaces admit explicit, finite combinatorial models:

Generating Families and Dissections: Choosing a (full) arc system or a suitable dissection, the Fukaya category is encoded as a (dg or $A_\infty$ -)category presented by a quiver whose vertices are arcs, arrows correspond to intersection data (“angles”), and compositions are defined by gluing polygons. For dissections cutting the surface into “polygons with stops,” all higher compositions vanish and the model becomes purely dg (Grossack, 13 Oct 2025).
Closed Curves and Local Systems: Closed curves are realized as iterated complexes of arcs, with local systems modeled as modules over group algebras of fundamental groups. The chain complex for a closed curve includes explicit monodromy factors in the differentials.
Explicit Endomorphism Algebras: For formal generators coming from admissible dissections (possibly on orbifold surfaces), the endomorphism algebra is shown to be a (graded) gentle or skew–gentle algebra, with derived equivalence classes characterized by the combinatorics of the dissection (Barmeier et al., 23 Jul 2024, Amiot et al., 24 May 2024).
Combinatorial Computation: Hom and Ext spaces are directly computable from intersection numbers and combinatorial data of the arcs and their gradings, with degrees recorded by counting signed intersection indices (Grossack, 13 Oct 2025).

Model Type	Generators	Endomorphism Algebra
Smooth, with arcs/dissection	Curves/arcs	Gentle algebra
Orbifold/with involution	Tagged arcs/bands	Skew–gentle or related algebra

5. Orbifold Extensions and Taggings

Skew-group and Tagged Arc Models: For orbifold surfaces (quotients by finite group actions, typically $G=\mathbb{Z}_2$ ), the partially wrapped Fukaya category is modeled as the split-closure of the skew-group $A_\infty$ -category of the G-action. Indecomposable objects are classified via graded tagged arcs; arcs ending at orbifold points receive “tags” (signs) distinguishing summands (Amiot et al., 24 May 2024, Cho et al., 16 Apr 2024).
Global Sections via Covers and Orbit Categories: A key equivalence is established between the cosheaf construction on a core of an orbifold surface and the global category arising as the fixed (equivariant or orbit) category over a smooth cover, allowing for precise transfer of computations from smooth to orbifold settings (Barmeier et al., 23 Jul 2024).
Derived Equivalences and Classification: The class of formal generators in orbifold Fukaya categories (with their associated endomorphism algebras) appears to be closed under derived equivalence, extending known results for gentle algebras (Barmeier et al., 23 Jul 2024).

6. Applications: Mirror Symmetry, Representation Theory, and Group Actions

Homological Mirror Symmetry (HMS): Mirror symmetry results hinge on these explicit models. For example, under HMS, the Fukaya category of a surface (or its wrapped version) is equivalent to the category of matrix factorizations or coherent sheaves on an appropriate Landau–Ginzburg mirror; all indecomposable objects are realized as geometric curves with possibly higher-rank local systems, and canonical forms for their images (e.g., matrix factorizations) are established (Cho et al., 24 Jun 2024).
Cluster Theory and Higher Teichmüller Theory: The cluster category associated to a marked surface is constructed as the global section of a perverse schober associated to an ideal triangulation, and is equivalent (after suitable periodicization or quotienting) to the $1$-periodic topological Fukaya category (Christ, 2022, Christ, 7 Oct 2025). Under higher Teichmüller-theoretic frameworks, relative higher Calabi–Yau completions of surface data yield cosingularity categories that are equivalent to topological Fukaya categories valued in cluster categories of Dynkin type (Christ, 7 Oct 2025).
Mapping Class Group and Autoequivalences: Faithful actions of mapping class groups (and their graded variants) are established on Fukaya categories of surfaces (Azam et al., 2019, Opper, 13 Oct 2025). Moreover, the derived Picard group of the Fukaya category splits into geometric (mapping class) and deformation-theoretic (Hochschild) components, with structural computations available in both characteristic zero and positive characteristic (under formality hypotheses) (Opper, 13 Oct 2025).

7. Future Directions and Open Problems

Infinite-type Surfaces: Fukaya categories for infinite-type surfaces require careful analytic control via sectorial Lagrangians and exhibit structural differences from inverse limits over finite-type pieces (Choi et al., 2023).
Classification under Derived Equivalence: It is conjectured that all endomorphism algebras arising from “formal” admissible dissections on orbifold surfaces are closed under derived equivalence, thus extending the classification of gentle and skew–gentle algebras to a new class associated with orbifold Fukaya categories (Barmeier et al., 23 Jul 2024).
Extensions to Higher Dimensions: Techniques developed in the surface case—local-to-global, cosheaf models, or perverse schober gluing—are expected to extend to higher-dimensional exact symplectic and Weinstein manifolds, with ongoing research on sectorial descent and perverse schober gluing in higher dimensions.
Stability Conditions and Wall-Crossing: Explicit combinatorial descriptions of stability spaces inspire further paper of wall-crossing, categorified Donaldson–Thomas invariants, and their relation to cluster structures and mirror symmetry, especially in orbifold and higher-rank settings (Smith, 2013, Takeda, 2018).

Fukaya categories of surfaces thus provide a paradigm in which deep connections between symplectic geometry, algebraic/categorical structures (gentle/skew–gentle algebras, cluster categories), and geometric representation theory are fully computable, richly structured, and fundamentally grounded in the combinatorics and topology of surfaces.