Adrien Brochier: 3D topological field theories from quantum groups

Abstract 

In this talk, I'll review some joint work with D. Ben-Zvi, D. Jordan and N. Snyder aiming at defining and computing a certain "categorified" 3d TFT from the representation category of the quantum group associated with a reductive complex group G. I'll first recall various ways of defining this category, and then explain how to use factorization homology (a close cousin of Walker's skein categories) to define a certain category valued invariant of surfaces. This produces canonical quantizations of character varieties of surfaces, which can then be computed in various different ways, recovering, organazing and generalizing several well-known constructions in quantum algebra. Finally I'll explain how this machine also gives rise to vector space valued invariant of 3-manifold, leading to a TFT construction of skein modules.

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