A 4-dimensional topological field theory.

David Jordan (Edinburgh)

Monday 31st October, 2016 16:00-17:00 Maths 204


A turning point for topological field theory occurred in the 90's, when Witten, Reshetikhin, and Turaev defined a new class of 3-dimensional TFT's, using the theory of quantum groups.  More precisely, these constructions hinged on a certain "modular braided tensor category" constructed from the quantum group when the quantum parameter q is a root of unity.  It was soon thereafter understood that the Witten-Reshetikhin-Turaev TFT was really the shadow of a 4-dimensional TFT introduced by Crane and Yetter.  The Crane-Yetter TFT is "invertible", however, which means in particular it does not itself lead to any new invariants of 3- or 4-manifolds.

In this talk, I'll explain that the Crane-Yetter TFT has a natural extension beyond the modular setting, leading to a new four-dimensional TFT which produces interesting invariants of surfaces and 3-manifolds (and some 4-manifolds), and is connected to many structures in representation theory, most notably the theory of character varieties and their quantizations, and the representation theory of so-called double affine Hecke algebras.  The main tools in the construction are topological factorization homology of Ayala-Francis and Lurie, and an application of the cobordism hypothesis.  This is joint work with David Ben-Zvi, Adrien Brochier, and Noah Snyder.

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