SUPERSELECTION STRUCTURE AND LOCALIZED CONNES’ COCYCLES

Abstract

Let ρ be a localized endomorphism of the universal algebra of observables of a chiral conformal quantum field theory on a circle, see [16, 17, 23] or Chapter 1. Then ρ transforms covariant under the Möbius group. As was pointed out by D. Guido and R. Longo, [23], the covariance transformations are implemented by [Formula: see text] where Ad ∆it are modular groups to local algebras w.r.t. the vacuum vector, ut is a Connes-Radon-Nikodym-Cocycle. Using the localization property of ρ, one gets, at least for regular nets, localization properties of the cocycles. In this work we will do some steps into the opposite direction. Given a localized Connes’ cocycle of a local algebra. We will construct a localized endomorphism on the whole net. The features of this approach are twofold. Firstly sectors of finite and infinite statistical dimensions are handled on the same footing. Secondly it is a local theory right from the beginning. Moreover, soliton-like sectors can easily be incorporated. We will sketch on the last part. The program is carried through for a special class of conformal quantum field theories, the strongly additive ones.