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

TL;DR

This paper introduces a new class of dg-algebras derived from surface triangulations, connecting them with stability conditions and quadratic differentials, and generalizing classical Brauer graph algebras.

Contribution

It defines relative graded Brauer graph algebras, explores their Calabi--Yau structures, and links their stability spaces to quadratic differentials, extending previous work on derived categories.

Findings

Introduction of a new class of dg-algebras from surface triangulations

Description of stability condition spaces via quadratic differentials

Connection between algebraic structures and geometric surface data

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.