Surface Models of Brauer Graph Algebras

• Surface models of Brauer graph algebras are algebraic frameworks that encode ring structures via decorated surfaces with marked points and mixed-angulations.
• The construction employs differential graded algebras and Calabi–Yau properties, linking geometric data and algebraic invariants through derived equivalences.
• Geometric methods using perverse schobers and partially wrapped Fukaya categories reveal stability conditions and deformations, aiding practical insights in algebraic topology.

A surface model of a Brauer graph algebra encodes the algebraic structure of the algebra via the combinatorics and geometry of a suitably decorated surface—such as a compact oriented surface with marked points and, potentially, boundary. The theory realizes Brauer graph algebras (and their graded, relative analogues) as arising from the topology of surfaces with certain data (ribbon graphs, mixed-angulations, or arc systems) and establishes deep links with Calabi–Yau structures, perverse schobers, Fukaya categories, and stability conditions. This realization provides a dictionary between algebraic constructs and geometric/topological objects.

1. Weighted Marked Surfaces and Mixed-Angulations

Classical Brauer graph algebras are defined via a ribbon graph with multiplicities, where the cyclic order at each vertex specifies the local orientation and the multiplicity prescribes the length of certain cycles in the corresponding quiver. In the modern surface model, one begins with a weighted marked surface $({\bf S}, M, \Delta, w, \nu)$, where:

• ${\bf S}$ is a compact, oriented surface (possibly with boundary).
• $M \subset {\bf S}$ is a finite nonempty set of marked points meeting every boundary component.
• $\Delta \subset {\bf S}$ is a finite set of singular points, each with integer degree $d(x)=w(x)+2 \in \{1,2, \ldots, \infty\}$.
• $\nu$ is a grading (line field) on ${\bf S} \setminus (M \cup \Delta)$ compatible with combinatorial data.

A mixed-angulation $A$ is a collection of non-compact, properly embedded arcs (with endpoints in $M$) cutting ${\bf S}$ into polygons, each ascribed a unique interior singularity $x \in \Delta$ of degree matching the polygon's number of edges. The gradings are compatible such that consecutive polygon edges meet in degree zero.

The S-graph dual to the mixed-angulation has vertices indexed by $\Delta$ and edges by the arcs; at an internal vertex $v$ of degree $m$, the cyclically ordered half-edges record the combinatorics of the local configuration, with integer labels $d(i,i+1)$ defined by the number of boundary edges between consecutive arcs.

2. Construction of Relative Graded Brauer Graph Algebras

The relative graded Brauer graph algebra $A(S, n)$ is a (curved) differential graded algebra whose quiver and relations are determined by the surface data:

• Vertices: edges $e$ of the S-graph.
• Arrows: for each corner $(i,i+1)$, an arrow $a_i: i \to i+1$ of degree $d(i,i+1)$; for boundary half-edges $i$, a loop $\tau_i$ of degree $n-1$.
• Relations:

1. $a_j a_i = 0$ if $i+1 \neq j$ but $i, j$ share the same edge.
2. Around each internal edge $\{i, j\}$, the cycles $c_i$ and $c_j$ satisfy $c_i = (-1)^{n-1} c_j$.
3. $\tau_i^2 = 0$ on boundary edges.
4. $a_i \tau_i = (-1)^{|a_i|} \tau_{i+1} a_i$ at boundary corners.
5. On all-boundary edges, $\tau_i = (-1)^n \tau_j$.

The only nonzero differential occurs for $\tau_j$ at edges with one end internal, given by $d(\tau_j) = (-1)^n c_i$ (where $i$ is the internal half–edge). In the absence of boundary, the algebra reduces to the classical graded Brauer graph algebra with vanishing differential.

3. Calabi–Yau and Relative Calabi–Yau Structures

For orientable surfaces or odd $n$, $A(S, n)$ admits an $n$–Calabi–Yau structure: a homogeneous trace

$$\mathrm{tr} : A(S, n)^n \to k,\quad \mathrm{tr}(c_i)=1$$

such that $\mathrm{tr}(ab)=(-1)^{|a||b|} \mathrm{tr}(ba)$ and the pairing is nondegenerate. In dg settings with boundaries (infinite degree polygons), a relative Calabi–Yau structure emerges in the sense of Brav–Dyckerhoff, realized as a nondegenerate, cyclic class in relative Hochschild homology $\mathrm{HH}_\bullet(A(S, n), \partial)$. The surface construction enables gluing of local Calabi–Yau structures associated to Ginzburg algebras of polygons into a global relative structure on the full algebra (Christ et al., 2024).

4. Surface Realization via Perverse Schobers and Fukaya Categories

There are two parallel geometric constructions:

4.1 Perverse Schober Approach

A perverse schober is a constructible sheaf of stable $k$-linear $\infty$-categories on a ribbon graph (here, the S-graph or its augmentation), assigning to each vertex a local dg-category (e.g., the category of modules over a relative Ginzburg algebra $G_n$ for polygons, $\mathbb Z/(n/m)$-quotients $G_{n,m}$ for divisors $m|n$, or one-dimensional spherical functor models for boundary points). The global sections of the resulting schober yield a dg-category Morita equivalent to the Koszul dual $A(S, n)^!$ of the relative graded Brauer graph algebra, producing equivalences

$$\mathrm{Perf}(A(S, n)) \simeq \mathrm{Nil}(G(S, n)) \subset \mathcal D(G(S, n))$$

where $G(S, n)$ is constructed as a colimit over the exit-path category of the S-graph (Christ et al., 2024).

4.2 Partially Wrapped Fukaya Category Approach

Starting from the partially wrapped Fukaya category $\mathcal F(S, M, \nu)$ for the surface, one chooses an arc system cutting $S$ into polygons and builds a strictly unital $A_\infty$-category $\mathcal F_{\mathbb X}$ with morphisms determined by intersection combinatorics (paths along the boundary, $A_\infty$-operations counting immersed polygons). Deforming these categories with curvature to account for marked points $M'$ and winding data, one defines a category of torsion modules $\mathcal C(S, n)$. The subcategory generated by S-graph edges is quasi-equivalent to $\mathrm{Perf} \, A(S, n)$, realizing $A(S, n)$ as an explicit endomorphism algebra in the surface model (Christ et al., 2024, Opper et al., 2021).

5. Derived Equivalence and Surface Moves

Brauer graph algebras realized from different surface decompositions (triangulations, mixed-angulations, or partial triangulations) of the same topological surface are derived equivalent. The fundamental mechanism is the Kauer move (pivoting an edge of the underlying ribbon graph), which corresponds to a flip of an arc in the surface triangulation, or, dually, a Whitehead move on the dual tree. Each such move translates to a tilt at a specific simple module, giving a derived equivalence between the associated Brauer graph algebras [(Marsh et al., 2013); (Schroll, 2016); (Demonet, 2017)].

Invariants classifying derived equivalence classes of Brauer graph algebras are encoded as surface data: the genus and number of boundary components of the underlying surface, the winding numbers of boundary components, and the multiplicities of the cyclic orderings at punctures or singular points. Antipov’s invariants (number of vertices, edges, faces; face-perimeter multiset; vertex multiplicities; bipartiteness) are exactly captured by mapping class group orbits of the surface model with chosen line field (Opper et al., 2021).

6. Embedding in Weighted Surface and Hybrid Algebras

Brauer graph algebras appear as idempotent subalgebras (blocks) of weighted surface algebras and, more generally, as hybrid algebras. The procedure is constructive: any Brauer graph $G$ can be embedded as a ribbon graph in a surface $S$ with a compatible triangulation $\Delta$, assigning weights and parameters accordingly, such that the idempotent block $e A(S, \Delta, m, c) e$ is isomorphic to $B(G)$. Socle-deformations may be necessary to enforce symmetry in low-dimensional cases and are reflected by local monodromy data in the triangulation (Erdmann et al., 2021, Erdmann et al., 2017).

7. Stability Conditions, Quadratic Differentials, and Deformations

Bridgeland stability conditions on the derived category $\mathrm{Perf}\,A(S,n)$ correspond to moduli of framed quadratic differentials on the surface $(S, M, \nu)$. The central charge is given by integrating the square root of a holomorphic quadratic differential along cycles in the surface, providing a geometric realization of stability and wall-crossing phenomena (Christ et al., 2024). Infinitesimal deformations of the algebra—classified by Hochschild cohomology—admit geometric interpretations as surgeries on the surface: filling boundaries, twisting line fields, or introducing orbifold points, each corresponding to a distinct deformation class (Liu et al., 11 Jan 2026).

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Relevant references:

• "Perverse schobers, stability conditions and quadratic differentials II: relative graded Brauer graph algebras" (Christ et al., 2024)
• "Brauer graph algebras" (Schroll, 2016)
• "The geometry of Brauer graph algebras and cluster mutations" (Marsh et al., 2013)
• "Hybrid algebras" (Erdmann et al., 2021)
• "From Brauer graph algebras to biserial weighted surface algebras" (Erdmann et al., 2017)
• "Derived equivalence classification of Brauer graph algebras" (Opper et al., 2021)
• "Introduction to algebras of partial triangulations" (Demonet, 2017)
• "The second Hochschild cohomology and deformations of Brauer graph algebras" (Liu et al., 11 Jan 2026)