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 [Formula: see text] We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel'd center of that representation category of [Formula: see text] is equivalent to the category of positive energy representations of the free loop group [Formula: see text]^† The abovementioned conjectures are known to hold when the gauge group is abelian or of type [Formula: see text] 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 [Formula: see text], the category of bimodules over a hyperfinite [Formula: see text] 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 [Formula: see text].