Decorated marked surfaces III: The derived category of a decorated marked surface

TL;DR

This paper explores the derived categories associated with Ginzburg dg algebras from triangulated decorated marked surfaces, establishing a canonical identification and analyzing the action of the spherical twist group on stability conditions.

Contribution

It introduces a canonical identification of finite-dimensional derived categories for decorated marked surfaces and studies the faithful action of the spherical twist group on stability conditions.

Findings

Canonical identification of derived categories for decorated surfaces

Faithful action of the spherical twist group on stability conditions

Connections between surface triangulations and derived categories

Abstract

We study the Ginzburg dg algebra $\Gamma_\mathbf{T}$ associated to the quiver with potential arising from a triangulation $\mathbf{T}$ of a decorated marked surface $\mathbf{S}_\bigtriangleup$, in the sense of Qiu. We show that there is a canonical way to identify all finite dimensional derived categories $\mathcal{D}_{fd}(\Gamma_\mathbf{T})$, denoted by $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$. As an application, we show that the spherical twist group $\operatorname{ST}(\mathbf{S}_\bigtriangleup)$ associated to $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$ acts faithfully on its space of stability conditions.