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 $\mathrm{Stab}\mathcal{D}(\mathbb{S})$ (resp. the principal component ${\mathrm{Stab}}^{\circ }\mathcal{D}(\Gamma \mathbb{S})$) of the space of the stability conditions of $\mathcal{D}(\mathbb{S})$ (resp. $\mathcal{D}(\Gamma \mathbb{S})$) is canonically isomorphic to $\mathrm{FStab}\mathcal{D}(Q)$ (resp. the principal component ${\mathrm{FStab}}^{\circ }\mathcal{D}(\Gamma Q)$) 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 $\mathrm{NStab}\mathcal{D}(\Gamma Q)$ of numerical stability conditions in ${\mathrm{Stab}}^{\circ }\mathcal{D}(\Gamma Q)$. We show that $\mathrm{NStab}\mathcal{D}(\Gamma Q)$ consists of $\mathrm{Br}Q/\mathrm{Br}\mathbb{S}$ many connected components, each of which is isomorphic to ${\mathrm{Stab}}^{\circ }\mathcal{D}(\Gamma \mathbb{S})$, 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.

Last version, Pulb. R.I.M.S. to appear

References in corpus (3)

• Exchange graphs and Ext quivers
• Stability conditions and quantum dilogarithm identities for Dynkin quivers
• The modular curve as the space of stability conditions of a CY3 algebra