Fusion-equivariant stability conditions and Morita duality

TL;DR

This paper explores the structure of fusion-equivariant stability conditions on triangulated categories with fusion category actions, revealing their geometric properties and establishing a Morita duality relating G-invariant and representation-equivariant stability conditions.

Contribution

It introduces the concept of fusion-equivariant stability conditions, proves they form a closed submanifold, and generalizes a known result to establish a Morita duality between G-invariant and representation-equivariant stability conditions.

Findings

Fusion-equivariant stability conditions form a closed, complex submanifold.

Established a biholomorphism between G-invariant and rep(G)-equivariant stability conditions.

Applications to McKay quivers and geometric stability conditions on quotients.

Abstract

Given a triangulated category $D$ with an action of a fusion category $C$, we study the moduli space $Stab_{C}(D)$ 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 Macr\`{i}--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.