What Chern-Simons theory assigns to a point

<p>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?”.</p><p> 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 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 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 A1.</p><p> 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 Bim(R), the category of bimodules over a hyperfinite 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 An.</p><p> Our work builds on the formalism of coordinate free conformal nets, developed jointly with A. Bartels and C. Douglas.</p>