Structure of Coset Models

Abstract

We study inclusions of local, chiral, conformal quantum theories C which are contained in an ambient theory B and commute with another given subtheory A. These subtheories C are called Coset models. Most of our results are model-independent, although our analysis is motivated by the inclusions of current algebras and their Coset models. We prove that to every given A contained in B there is a unique, inner representation U^A which implements conformal symmetry on the subnet. The local observables of B which commute with U^A form the maximal Coset model C_max. Assuming U^A to be generated by integrals of a quantum field affiliated with the subnet A, we show: The inclusion of the subnet and of its Coset models is directly analogous to the inclusion of chiral observables in a local, conformal theory in 1+1 dimensions. The local observables of the maximal Coset model associated with a given region are found to be characterised by their commuting with the local observables of A associated with the very same region. We give applications and discuss possible generalisations of our methods.