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

TL;DR

This paper explores the representation theory of conformal nets using K-theory, revealing structural properties of universal C*-algebras and their relation to DHR sectors and fusion algebras.

Contribution

It introduces the locally normal universal C*-algebra and analyzes its structure, linking DHR endomorphisms to K-theory actions and fusion algebra representations.

Findings

C*_ln(A) decomposes into type I_ty factors for rational nets.

K_A has nontrivial K-theory with DHR actions on K_0(K_A).

DHR endomorphisms induce fusion algebra actions on 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.