Frobenius Morphisms and Stability Conditions

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}(Γ Q) of the Ginzburg algebra Γ Q associated to Q . We show that the F -stable category of {D}(Γ Q) is equivalent to the finite-dimensional derived category {D}() of the Ginzburg algebra associated to the species {S} , which is folded from Q . If (Q,{S}) is of Dynkin type, we prove that the space Stab{D}({S}) of the stability conditions on {D}({S}) is canonically isomorphic to the space 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 Stab∘{D}() and FStab∘{D}(Γ Q) . There are two applications. One is for the space NStab{D}(Γ Q) of numerical stability conditions in Stab∘{D}(Γ Q) . We show that NStab{D}(Γ Q) consists of BrQ/Br{S} many connected components, each of which is isomorphic to Stab∘{D}() , for (Q,{S}) is of type (A₃, B₂) or (D₄, G₂) . The other is that we relate the F -stable stability conditions to Gepner-type stability conditions.