2-Dimensional Algebra and 3-Dimensional Local Field Theory
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Fri, 11/06/2010 09:00 |
Chris Douglas |
K-Theory Day  |
L3 |
| Witten showed that the Jones polynomial invariants of knots can be computed in terms of partition functions of a (2+1)-dimensional topological field theory, namely the SU(2) Chern-Simons theory. Reshetikhin and Turaev showed that this theory extends to a (1+1+1)-dimensional topological field theory—that is, there is a Chern-Simons-type invariant associated to 3-manifolds, 3-manifolds with boundary, and 3-manifolds with codimension-2 corners. I will explain the notion of a local or (0+1+1+1)-dimensionaltopological field theory, which has, in addition to the structure of a (1+1+1)-dimensional theory, invariants associated to 3-manifolds with codimension-3 corners. I will describe a notion of 2-dimensional algebra that allows us to construct and investigate such local field theories. Along the way I will discuss the geometric classification of local field theories, and explain the dichotomy between categorification and algebraification. | |||