Perverse schobers, stability conditions and quadratic differentials II: relative graded Brauer graph algebras

Abstract

We introduce a class of dg-algebras which generalize the classical Brauer graph algebras. They are constructed from mixed-angulations of surfaces and often admit a (relative) Calabi--Yau structure. We discovered these algebras through two very distinct routes, one involving perverse schobers whose stalks are cyclic quotients of the derived categories of relative Ginzburg algebras, and another involving deformations of partially wrapped Fukaya categories of surfaces. Applying the results of our previous work arXiv:2303.18249, we describe the spaces of stability conditions on the derived categories of these algebras in terms of spaces of quadratic differentials.

Figures (11)

• Figure 1.1: Three examples of weighted marked surfaces with a mixed-angulation (blue) and dual S-graph (red). The left one is an ideal triangulation of a torus with two interior marked points. The central and right examples are mixed-angulations of the disk. The central one has singular points with degrees $1,2$, and $4$ which are the centers of $1$-, $2$-, and $4$-gons, respectively. The right one includes boundary singular points and infinitely many boundary marked points arising from $\infty$-gons.
• Figure 2.1: Example of surface with mixed-angulation (blue) and dual S-graph (red) with its numbers $d(h_1,h_2)$.
• Figure 2.2: The forward flip at a usual arc.
• Figure 2.3: The forward flip at a monogon arc.
• Figure 2.4: Schematic view of the cycle $c_0$ in the case of a trivalent vertex of the S-graph with $n/\deg(v)=2$.

Theorems & Definitions (100)

• Theorem 1.1: \ref{['thm_defstab']}
• Theorem 1.2: \ref{['thm:nonformalgen']}
• Theorem 1.3: \ref{['thm:BGAschober']}
• Corollary 1.4
• Theorem 1.5: CHQ23
• Example 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4