University of Sydney Algebra Seminar

David Jordan (Edinburgh)

Monday 2 June, 12:05-12:55pm, Place: 157

Quantizing character varieties via a 4D TFT

The character variety Ch_G(X), of a surface X with values in G, is a moduli space of representations pi_1(X)-->G. Character varieties carry a natural Poisson bracket first constructed by Goldman and Atiyah-Bott, and the assignment X --> Ch_G(X) can be made into a local (a.k.a fully extended) topological field theory.

In this talk, I'll explain a technique for quantizing the Goldman bracket on character varieties, which exploits the TFT structure, and gives rise to a 4-dimensional topological field theory extending Reshetikhin-Turaev invariants. A crucial point in the construction is that the value of the TFT on surfaces can be computed very explicitly, using two tools: factorization homology of Francis and Lurie, and the representation theory of tensor categories developed in recent years by many authors. I'll explain in some detail how this all looks for punctured and closed surfaces, where one obtains interesting representations of braid groups and mapping class groups of surfaces. This is all joint work with David Ben-Zvi and Adrien Brochier.

Note: so-called ``Australian category theory" has a crucial role to play in this story! In order to extend Reshetikhin-Turaev, the Deligne tensor product must be replaced with the Kelly tensor product - this is needed even to define the theory. Moreover, many of the tools for studying representation theory of tensor categories can be understood via Day and Street's observation that pivotal tensor categories are really categorified Frobenius algebras.