Frobenius morphisms and 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.