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Fusion-equivariant stability conditions and Morita duality

Classically, finite symmetries are captured by the action of a finite group. Moving to the quantum world, one has to allow for (possibly non-invertible) quantum symmetries — these are instead captured by the action of a more general algebraic structure, known as a fusion category. Such quantum symmetries are actually ubiquitous in mathematics; for example, given a category with an action of a finite group G (e.g. rep(Q), Coh(X) etc.), its G-equivariant category has instead the action of the category of representations rep(G), where rep(G) has the structure of a fusion category.

The aim of this talk is to study the role of fusion categories as “quantum symmetries” in relation to (Bridgeland) stability conditions. Given a triangulated category equipped an action of a fusion category C, we introduce the notion of “C-equivariant stability conditions”, a generalisation of “G-invariant stability conditions”. The first result is that these stability conditions form a closed submanifold of the stability manifold, just as the G-invariant stability conditions do. Moreover, given a triangulated D with a G-action, so that its G-equivariant category D^G has a rep(G)-action, we will see the following (Morita) duality result for stability conditions: the complex manifold of G-invariant stability conditions (associated to D) is homeomorphic to the complex manifold of rep(G)-equivariant stability conditions (associated to D^G).

This is part of joint work with Hannah Dell and Anthony Licata.

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