Classification of subsystems for the Haag-Kastler nets generated by \(c=1\) chiral current algebras
In the algebraic approach to quantum field theory, the objects under investigation are local nets of von Neumann algebras (Haag-Kastler nets). This paper studies the classification of the \(SL(2, { R})\)-covariant subsystems of Haag-Kastler nets constructed from two models of chiral conformal field theory, the \(\widehat{u}(1)\) and the \(\widehat{su}(2)_1\) chiral current algebras. The author finds that there is a one-to-one correspondence between subsystems of the Haag-Kastler net \({\mathcal F}\) corresponding to the \(\widehat{su}(2)_1\) chiral current algebra and nets \({\mathcal F}^H\) consisting of the local algebras generated by local operators invariant under some closed subgroup \(H\) of the gauge automorphisms group \(SO(3)\) of \({\mathcal F}\). In the case of the Haag-Kastler net \({\mathcal F}_1\) generated by the \(\widehat{u}(1)\) chiral current it is shown that there are exactly two distinct proper subsystems, one generated by the energy-momentum tensor and the gauge invariant subsystem \({\mathcal F}_1^{Z_2}\). The paper concludes with a discussion of the possible generalizations of the approach presented in the paper to other models.
classification of the \(SL(2,R)\)-covariant subsystems
scientific article; zbMATH DE number 1334611