What Chern-Simons theory assigns to a point

Andre Henriques

Published 2015-03-21Version 1

In this note, 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 that we call positive energy representations. We define a fusion product on those representations and prove that, modulo certain conjectures, the Drinfel'd centre of the category of positive energy representations of the based loop group is equivalent to the category of positive energy representations of the free loop group. The above mentioned conjectures are known to hold when the gauge group is abelian or of type A_1. Our answer to the second question is bicommutant categories. The latter are a sort of categorification of the notion a von Neumann algebras: they are tensor categories that are equivalent to their bicommutant inside a certain fixed tensor category. We prove that, modulo certain conjectures, the category of positive energy representations of the based loop group is a bicommutant category. Here, the relevant conjectures are known to hold when the gauge group is abelian or of type A_n. Our work builds on the formalism of coordinate free conformal nets, developed jointly with A. Bartels and C. Douglas.