The Extended Algebra of the SU(2) Wess-Zumino-Witten Models

TL;DR

This paper constructs and analyzes an extended algebra for the SU(2) Wess-Zumino-Witten models, revealing faithful representations and detailed module structures for specific levels, enhancing understanding of the model's algebraic framework.

Contribution

It introduces a new extended algebra based on simple currents in SU(2) WZW models and provides a detailed representation theory, including explicit bases for certain levels.

Findings

Faithful free-field-type representations of the extended algebra

Explicit module bases for levels k=1 and 2

Illustration of subtleties at levels k=1, 2, and 4

Abstract

The Wess-Zumino-Witten model defined on the group SU(2) has a unique (non-trivial) simple current of conformal dimension k/4 for each level k. The extended algebra defined by this simple current is carefully constructed in terms of generalised commutation relations, and the corresponding representation theory is investigated. This extended algebra approach is proven to realise a faithful ("free-field-type") representation of the SU(2) model. Subtleties in the formulation of the extended theory are illustrated throughout by the k=1, 2 and 4 models. For the first two cases, bases for the modules of the extended theory are given and rigorously justified.