What Chern–Simons theory assigns to a point

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 ΩG 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 ΩG 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 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 A n .Our work builds on the formalism of coordinate free conformal nets, developed jointly with A. Bartels and C. Douglas.

Conformal nets are factorization algebras

“…In this note, we prove that conformal nets of finite index (Definitions 1.1 and 3.1 in [2]) form an instance of the notion of a factorization algebra. Our main result, Theorem 2, is a key ingredient in the proof, announced in [8], that the category of solitons of a finite index conformal net is a bicommutant category.…”

“…In our recent preprint [8], we introduced higher categorical analogs of von Neumann algebras called bicommutant categories. A bicommutant category is a tensor category which is equivalent to its bicommutant inside Bim(R).…”

Drinfeld center of enriched monoidal categories

“…Remark 4.10. In recent works [He1,He2], certain unitary modular tensor categories (completed by separable Hilbert spaces) were shown to be the (usual) Drinfeld center of certain categories of solitons, and the latter were proposed as a candidate for the value of Chern-Simons theory on a point. We expect that a self-enriched modular tensor category B ♯ could be realized as the value of a fully extended Reshetikhin-Turaev TQFT on a point such that the value on a circle is B (see [Z] for more details).…”

“…More importantly, this notion leads to a positive answer to the following question: given a modular tensor category C, is there any mathematical object whose "center" is C? This question is crucial to the study of 2+1D TQFT such as Chern-Simons theory, Reshetikhin-Turaev extended TQFT [He1,He2,Z] and topological orders with gapless edges [KZ]. We show that a modular tensor category (more generally, a nondegenerate braided fusion category) C can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category (see Corollary 4.9).…”

K-theory of AF-algebras from braided C*-tensor categories

“…We refer the reader to [25] for an overview and further references. It should also be mentioned that TLJ pδq can be recovered as the C*-tensor category of M -bimodules arising from certain subfactors pN Ă M q (cf.…”