Description
It is fair to say we have a rather complete mathematical framework for describing finite symmetries of field theories. I will discuss recent work on the mathematics of continuous symmetries. In particular, I will introduce a (in fact, several) higher category(ies) of `topological' (in the functional analytic sense) n-vector spaces together with, for each compact Lie group, a topological gauge theory (a BF theory) as a once-categorified fully extended theory with this target. Instead of invoking the cobordism hypothesis, I will write down a functor of n-categories explicitly; the construction may interpreted as a fully extended geometric quantization of the shifted cotangent of the group.