Fusion-equivariant stability conditions and Morita duality

Abstract Given a triangulated category $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> with an action of a fusion category $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> , we study the moduli space $${{\,\textrm{Stab}\,}}_{{\mathcal {C}}}({\mathcal {D}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>Stab</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mi>C</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of fusion-equivariant Bridgeland stability conditions on $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> . The main theorem is that the fusion-equivariant stability conditions form a closed, complex submanifold of the moduli space of stability conditions on $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> . 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 $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> and the space of $${\textsf{rep}}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>rep</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -equivariant stability conditions on the equivariant category $${\mathcal {D}}^G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msup> </mml:math> . 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.