An Algebraic Formulation of Level One Wess-Zumino-Witten Models

TL;DR

This paper develops an algebraic framework for level one Wess-Zumino-Witten models using fermionic representations, connecting representation theory with local von Neumann algebras and fusion rules.

Contribution

It introduces a novel algebraic formulation of WZW models at level one via fermionic CAR algebras, linking sector theory with operator algebras.

Findings

Representation theory of CAR algebras reproduces chiral algebra sectors.

Explicit construction of localized endomorphisms using Bogoliubov transformations.

Fusion rules are proven within the CAR framework.

Abstract

The highest weight modules of the chiral algebra of orthogonal WZW models at level one possess a realization in fermionic representation spaces; the Kac-Moody and Virasoro generators are represented as unbounded limits of even CAR algebras. It is shown that the representation theory of the underlying even CAR algebras reproduces precisely the sectors of the chiral algebra. This fact allows to develop a theory of local von Neumann algebras on the punctured circle, fitting nicely in the Doplicher-Haag-Roberts framework. The relevant localized endomorphisms which generate the charged sectors are explicitly constructed by means of Bogoliubov transformations. Using CAR theory, the fusion rules in terms of sector equivalence classes are proven.