Upcoming events

Algebra seminar: Edmund Heng (IHES Paris) - Fusion categories as quantum symmetries: on Bridgeland stability conditions

Location: Bayes 5.46

Description: Abstract: 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. A-mod, Coh(X)), its G-equivariant category (A#G-mod, Coh(X//G) resp.) has instead the action of the category of G-representations rep(G), which has the structure of a fusion category. There are also other more “exotic” fusion categories, which nonetheless capture “hidden” symmetries on familiar (non-“exotic”) categories.The aim of this talk is to introduce fusion categories and to discuss their role as quantum symmetries. Our main application will be on Bridgeland stability conditions. I will discuss how the fusion-equivariant stability conditions — a generalisation of G-invariant stability conditions (i.e. G-fixed points) — form a closed submanifold of the Bridgeland stability (complex) manifold. More importantly, we will see the following duality result inspired by (1-categorical) Morita duality: let D be a triangulated category with a G-action, so that its G-equivariant category D^G has a rep(G)-action. The manifold of G-invariant stability conditions (associated to D) is homeomorphic to the manifold of rep(G)-equivariant stability conditions (associated to D^G).This is part of joint work with Hannah Dell and Anthony Licata