What Chern–Simons theory assigns to a point

Abstract

We answer the questions, “What does Chern–Simons theory assign to a point?” and “What kind of mathematical object does Chern–Simons theory assign to a point?” Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group 𝛀G. We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel’d center of that representation category of 𝛀G is equivalent to the category of positive energy representations of the free loop group LG.† The abovementioned conjectures are known to hold when the gauge group is abelian or of type A1. Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside Bim(R), the category of bimodules over a hyperfinite 𝐼𝐼𝐼1 factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type An.