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Algebraic Topology and Physics Seminar, Week 4

Noah Synder, Local Topological Field Theory and Fusion Categories

Title: “Local Topological Field Theory and Fusion Categories”
Speaker: Noah Synder, Columbia University
Date: Tuesday, February 14, 2012
Time: 10:30am – 11:30am
Place: Seminar Room 313, Simons Center

 

 

 

 

 

 

Abstract:

Topological field theories give a close relationship between topology and algebra. Traditionally the main application has been from algebra to topology: using algebraic constructions like quantum groups to produce topological invariants. However, you can also run the applications the other way, using topology to arrange and clarify your knowledge about algebra. The goal of this talk is to explain one such application. More specifically, 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. I will 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 version of the Baez-Dolan 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 fusion categories or n-categories.