Frobenius Morphisms and Stability Conditions

Abstract

We generalize Deng–Du’s folding argument, for the bounded derived category $D(Q)$ of an acyclic quiver $Q$, to the finite-dimensional derived category $D(\Gamma Q)$ of the Ginzburg algebra $\Gamma Q$ associated to $Q$. We show that the $F$-stable category of $D(\Gamma Q)$ is equivalent to the finite-dimensional derived category $D(\Gamma \mathbb{S})$ 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 the space $\mathrm{Stab}D(\mathbb{S})$ of the stability conditions on $D(\mathbb{S})$ is canonically isomorphic to the space $\mathrm{FStab}D(Q)$ of $F$-stable stability conditions on $D(Q)$. For the case of Ginzburg algebras, we also prove a similar isomorphism between principal components ${\mathrm{Stab}}^{\circ }D(\Gamma \mathbb{S})$ and ${\mathrm{FStab}}^{\circ }D(\Gamma Q)$. There are two applications. One is for the space $\mathrm{NStab}D(\Gamma Q)$ of numerical stability conditions in ${\mathrm{Stab}}^{\circ }D(\Gamma Q)$. We show that $\mathrm{NStab}D(\Gamma Q)$ consists of $\mathrm{Br}Q/\mathrm{Br}\mathbb{S}$ many connected components, each of which is isomorphic to ${\mathrm{Stab}}^{\circ }D(\Gamma \mathbb{S})$, 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 Gepner-type stability conditions.