Geometric Analysis and Topology Seminar

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Local Topological Field Theory and Fusion Categories

Speaker: Noah Snyder, Columbia University

Location: Warren Weaver Hall 201

Date: Friday, December 2, 2011, 11 a.m.

Synopsis:

A fusion category is a category that looks like the category of representations of a finite group: it has a tensor product, duals, is semisimple, and has finitely many simple objects. A somewhat mysterious fact about fusion categories (generalizing a theorem of Radford's about Hopf algebras) is that the quadruple dual functor is canonically isomorphic to the identity functor. The goal of this talk is to explain this mystery by showing that it follows directly from the Dirac belt trick. The main technique in this proof is the construction of a local topological field theory attached to any fusion category. Topological field theories are invariants of manifolds which can be computed by cutting along codimension 1 boundaries. Local topological field theories allow cutting along lower codimension boundaries. Since manifolds with corners can be glued together in many different ways, this can be formalized using the language of n-categories. Using Lurie's cobordism hypothesis, we describe local field theories with values in the 3-category of tensor categories. This is joint work with Chris Douglas and Chris Schommer-Pries. I will not assume prior familiarity with topological field theory, fusion categories, or n-categories.