Frobenius morphisms and stability conditions

TL;DR

This paper extends folding techniques to derived categories of Ginzburg algebras, establishing isomorphisms between stability condition spaces and exploring applications to numerical stability and Gepner stability conditions.

Contribution

It generalizes Deng-Du's folding argument to Ginzburg algebras, linking $F$-stable categories with folded species and analyzing stability spaces for Dynkin types.

Findings

Isomorphism between $F$-stable categories and folded species derived categories.

Description of connected components of numerical stability conditions for specific types.

Relation of $F$-stable stability conditions to Gepner type stability conditions.

Abstract

We generalize Deng-Du's folding argument, for the bounded derived category $\mathcal{D}(Q)$ of an acyclic quiver $Q$, to the finite dimensional derived category $\mathcal{D}(\Gamma Q)$ of the Ginzburg algebra $\Gamma Q$ associated to $Q$. We show that the $F$-stable category of $\mathcal{D}(\Gamma Q)$ is equivalent to the finite dimensional derived category $\mathcal{D}(\Gamma\mathbb{S})$ of the Ginzburg algebra $\Gamma\mathbb{S}$ associated to the species $\mathbb{S}$, which is folded from $Q$. If $(Q,\mathbb{S})$ is of Dynkin type, we prove that $\operatorname{Stab}\mathcal{D}(\mathbb{S})$ (resp. the principal component $\operatorname{Stab}^\circ\mathcal{D}(\Gamma\mathbb{S})$) of the space of the stability conditions of $\mathcal{D}(\mathbb{S})$ (resp. $\mathcal{D}(\Gamma\mathbb{S})$) is canonically isomorphic to $\operatorname{FStab}\mathcal{D}(Q)$ (resp. the principal component $\operatorname{FStab}^\circ\mathcal{D}(\Gamma Q)$) of the space of $F$-stable stability conditions of $\mathcal{D}(Q)$ (resp. $\mathcal{D}(\Gamma Q)$). There are two applications. One is for the space $\operatorname{NStab}\mathcal{D}(\Gamma Q)$ of numerical stability conditions in $\operatorname{Stab}^\circ\mathcal{D}(\Gamma Q)$. We show that $\operatorname{NStab}\mathcal{D}(\Gamma Q)$ consists of $\operatorname{Br} Q/\operatorname{Br} \mathbb{S}$ many connected components, each of which is isomorphic to $\operatorname{Stab}^\circ\mathcal{D}(\Gamma\mathbb{S})$, for $(Q,\mathbb{S})$ is of type $(A_3, B_2)$ or $(D_4, G_2)$. The other is that we relate the $F$-stable stability conditions to the Gepner type stability conditions.