Conformal nets IV: The 3-category

Bartels, Arthur; Douglas, Christopher L.; Henriques, André

doi:10.48550/arxiv.1605.00662

Cited by 1 publication

(4 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proposal (10) for CS (pt) unifies and generalizes the previous ones ( 3) and (8) where H 4 + (BG, Z) denotes the set of positive levels (Definition 1). The arrow labelled 1 is the well known isomorphism H 4 (BG, Z) ∼ = H 3 (G, U (1)).…”

Section: Cs (Pt) For Disconnected Groupsmentioning

confidence: 71%

“…Such a functor would make our current proposal (8) for the value of Chern-Simons theory on a point 'backwards compatible' with respect to our earlier proposal [6].…”

Section: Definition 8 ([44 §3])mentioning

confidence: 72%

“…Remark 5. One possible objection to out proposal (8) is that the category Rep k l.n. (ΩG) is too big (it is neither rigid nor even abelian; in particular, it falls completely outside of the framework of [22]).…”

Section: The Value On a Point: Rep K (ωG)mentioning

confidence: 99%

“…Remark. In our earlier paper [6], we had suggested using the 3-category of conformal nets, constructed in [7,9,8], as a target category for extended 3-dimensional TQFTs, and to have A G,k be the value of Chern-Simons theory on a point. We conjecture that the construction A → T A extends to a fully faithful but maybe not essentially surjective 3-functor…”

Section: Definition 8 ([44 §3])mentioning

confidence: 99%

See 3 more Smart Citations

What Chern–Simons theory assigns to a point

Henriques

2017

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

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.

show abstract

Section: Cs (Pt) For Disconnected Groupsmentioning

confidence: 71%

“…Such a functor would make our current proposal (8) for the value of Chern-Simons theory on a point 'backwards compatible' with respect to our earlier proposal [6].…”

Section: Definition 8 ([44 §3])mentioning

confidence: 72%