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.
These talks are based on joint work with Arthur Bartels and Andre Henriques.