Superselection Sectors of $\son$ Wess-Zumino-Witten Models

TL;DR

This paper analyzes the superselection sectors of $ ext{SO}(n)$ WZW models using algebraic quantum field theory, revealing their structure at levels 1 and 2, and connecting them to CAR algebras, DHR sectors, and coset Virasoro algebra.

Contribution

It provides a detailed algebraic construction of superselection sectors for $ ext{SO}(n)$ WZW models at levels 1 and 2, linking them to CAR algebras and DHR framework.

Findings

Observable algebras at level 1 are constructed from even CAR algebras.

Explicit construction of localized endomorphisms via Bogoliubov transformations.

Decomposition of level 2 sectors into tensor products involving the coset Virasoro algebra.

Abstract

The superselection structure of $\son$ WZW models is investigated from the point of view of algebraic quantum field theory. At level $1$ it turns out that the observable algebras of the WZW theory can be constructed in terms of even CAR algebras. This fact allows to give a formulation of these models close to the DHR framework. Localized endomorphisms are constructed explicitly in terms of Bogoliubov transformations, and the WZW fusion rules are proven using the DHR sector product. At level $2$ it is shown that most of the sectors are realized in $\HNSh=\HNS\otimes\HNS$ where $\HNS$ is the Neveu-Schwarz sector of the level $1$ theory. The level $2$ characters are derived and $\HNSh$ is decomposed completely into tensor products of the sectors of the WZW chiral algebra and irreducible representation spaces of the coset Virasoro algebra. Crucial for this analysis is the DHR decomposition of $\HNSh$ into sectors of a gauge invariant fermion algebra since the WZW chiral algebra as well as the coset Virasoro algebra are invariant under the gauge group $\Oz$.