Poisson bracket algebra for chiral group elements in the WZNW model

TL;DR

This paper derives the Poisson bracket algebra for chiral group elements in the WZNW model on a circle, applicable to any group and boundary conditions, highlighting gauge dependence and invariance of certain observables.

Contribution

It provides a general computation of the Poisson brackets for chiral group elements without fixing a specific parametrization, extending previous results.

Findings

Poisson brackets are derived for arbitrary groups and boundary conditions.

Gauge fixing is necessary for defining chiral group elements.

Monodromy matrix observables commute in the quantum theory.

Abstract

We examine the Wess-Zumino-Novikov-Witten (WZNW) model on a circle and compute the Poisson bracket algebra for left and right moving chiral group elements. Our computations apply for arbitrary groups and boundary conditions, the latter being characterized by the monodromy matrix. Unlike in previous treatments, they do not require specifying a particular parametrization of the group valued fields in terms of angles spanning the group. We do however find it necessary to make a gauge choice, as the chiral group elements are not gauge invariant observables. (On the other hand, the quadratic form of the Poisson brackets may be defined independent of a gauge fixing.) Gauge invariant observables can be formed from the monodromy matrix and these observables are seen to commute in the quantum theory.