  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.
