Fusion categories as (quantum) symmetries: on McKay correspondence and stability conditions.
Edmund Heng (IHES)
Wednesday 27th September 16:00-17:00 Maths 110
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” also arise naturally in the mathematical setting: 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 introduce fusion categories and discuss their role as “quantum symmetries” in algebra and geometry. More precisely, I will discuss the following results:
1. A generalised (quantum) McKay correspondence for fusion categories, where once again, an ADE classification appears.
2. An identification (homeomorphism) of G-invariant stability conditions with rep(G)-equivariant stability conditions in the G-equivariant category.
These are a combination of joint works with Hannah Dell, Ben Elias and Anthony Licata.