Fusion-equivariant stability conditions and Morita duality

Abstract

Given a triangulated category $D$ with an action of a fusion category $C$, we study the moduli space $Sta{b}_C(D)$ of fusion-equivariant Bridgeland stability conditions on $D$. The main theorem is that the fusion-equivariant stability conditions form a closed, complex submanifold of the moduli space of stability conditions on $D$. As an application of this framework to finite group actions on categories, we generalise a result of Macrì--Mehrotra--Stellari by establishing a biholomorphism between the space of $G$-invariant stability conditions on $D$ and the space of $rep(G)$-equivariant stability conditions on the equivariant category $D^G$. We also describe applications to the study of stability conditions associated to McKay quivers and to geometric stability conditions on free quotients of smooth projective varieties.

v3: Referee comments incorporated. Homotopy equivalent to accepted version. To appear in Math. Z. v2: Updated the proof of lemma 3.17 (3.16 in previous version); new geometrical applications included; added an appendix dealing with the support property. v1: 30 pages. Comments more than welcomed!