The Algebra of Non-Local Charges in Non-Linear Sigma Models

TL;DR

This paper derives the exact algebraic structure of conserved non-local charges in bosonic non-linear sigma models, revealing a cubic deformation of the Kac-Moody algebra and extending the results to models with Wess-Zumino terms.

Contribution

It provides the first exact computation of the Dirac algebra of non-local charges in these models, including the effects of Wess-Zumino terms.

Findings

The algebra is a cubic deformation of the Kac-Moody algebra.

Closed-form expressions for non-linear terms are obtained.

The algebra structure persists with calculable corrections when Wess-Zumino terms are included.

Abstract

We obtain the exact Dirac algebra obeyed by the conserved non-local charges in bosonic non-linear sigma models. Part of the computation is specialized for a symmetry group $O(N)$. As it turns out the algebra corresponds to a cubic deformation of the Kac-Moody algebra. The non-linear terms are computed in closed form. In each Dirac bracket we only find highest order terms (as explained in the paper), defining a saturated algebra. We generalize the results for the presence of a Wess-Zumino term. The algebra is very similar to the previous one, containing now a calculable correction of order one unit lower.