Classification of Subsystems for the Haag–Kastler Nets Generated by c = 1 Chiral Current Algebras

Abstract

Let \(\mathcal{F}\) be the Haag--Kastler net generated by the \({s\hat u}\) (2) chiral current algebra at level 1. We classify the SL(2, \(\mathbb{R}\))-covariant subsystems \(\mathcal{B}\) ⊂ \(\mathcal{F}\) by showing that they are all fixed points nets \(\mathcal{F}\)H for some subgroup H of the gauge automorphisms group SO(3) of \(\mathcal{F}\). Then, using the fact that the net \(\mathcal{F}\)1 generated by the \({\hat u}\)(1) chiral current can be regarded as a subsystem of \(\mathcal{F}\), we classify the subsystems of \(\mathcal{F}\)1. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem \(\mathcal{F}_{{\text{1}}^{\mathbb{Z}_{\text{2}} } } \).