Representations of Conformal Nets, Universal C*-Algebras and K-Theory

Abstract

We study the representation theory of a conformal net A on the circle from a K-theoretical point of view using its universal C*-algebra C*(A). We prove that if A satisfies the split property then, for every representation \pi of A with finite statistical dimension, \pi(C*(A)) is weakly closed and hence a finite direct sum of type I_\infty factors. We define the more manageable locally normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C*(A) with finite statistical dimension act on K_A, giving rise to an action of the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.Comment: v2: we added some comments in the introduction and new references. v3: new authors' addresses, minor corrections. To appear in Commun. Math. Phys. v4: minor corrections, updated reference

Superconformal nets and noncommutative geometry

“…It was explicitly defined in [CCHW13] and proven to be σ-weakly closed in the universal locally normal representation if A γ is completely rational. As a consequence, in the latter case, A ∆ may alternatively be chosen as a subalgebra of C * ln (A γ ).…”

“…A general and purely topological approach, dispensing with supersymmetry, has been recently achieved in [CCHW13,CCH13] for completely rational local conformal nets. We expect a deeper relation to the present work in the case where the completely rational net is the even part of a superconformal net.…”

Super-KMS Functionals for Graded-Local Conformal Nets

“…Notice that π ln is quasi-equivalent to the direct sum of irreducible GNS representations with only one representative for each equivalence class, i.e., of m mutually inequivalent elements in ∆. We then obtain the following modification of [CCHW13,Th.3…”

Conformal Nets and KK-Theory

“…In a recent paper [CCHW12] together with Mihály Weiner we have started an investigation of K-theoretical aspects of conformal nets. In the present note we take one further step along these lines and provide an interpretation of some of the results obtained there in the framework of Kasparov KK-theory.…”

“…We recall the results in [CCHW12] relevant for the present analysis, referring the reader to that paper for more details, notation, and references. Let A be a completely rational local conformal net on S 1 .…”