What Chern-Simons theory assigns to a point

TL;DR

This paper explores the mathematical objects assigned by Chern-Simons theory to a point, identifying them as representations of the based loop group and bicommutant categories, with results supported by conjectures valid in specific cases.

Contribution

It characterizes what Chern-Simons theory assigns to a point using loop group representations and bicommutant categories, advancing the understanding of its categorical and algebraic structures.

Findings

Identifies representations of the based loop group as the objects assigned to a point.

Establishes a connection between the Drinfel'd centre of the representation category and positive energy representations.

Proves that the category of locally normal representations forms a bicommutant category under certain conjectures.

Abstract

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 $\Omega G$ that we locally normal representations. We define the fusion product of such representations and we prove that, modulo certain conjectures, the Drinfel'd centre of that representation category of $\Omega G$ is equivalent to the category of positive energy representations of the free loop group $LG$. 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 higher categorical analogs of von Neumann algebras: they are tensor categories that are equivalent to their bicommutant inside $\mathrm{Bim}(R)$, the category of bimodules over a hyperfinite $\mathit{III}_1$ factor. We prove that, modulo certain conjectures, the category of locally normal 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 $A_n$. Our work builds on the formalism of coordinate free conformal nets, developed jointly with A. Bartels and C. Douglas.