2-representations of compact groups and Chern-Simons theory

I will discuss a 'categorification' of the group ring of a compact group and its representations in the context of constructing Chern-Simons theory, as a topological field theory in dimensions (3,2,1,0), by topological methods. At present, the method works for finite groups and for a torus, whereas for general compact Lie groups we run into some new conjectures for Dirac operators and conjugacy classes. This is a joint project with D. Freed, M. Hopkins and J. Lurie.