What Chern-Simons theory assigns to a point

Understand

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 $LG$.