A gauge invariant formulation for the SU(N) non-linear sigma model in 2+1 dimensions

TL;DR

This paper presents a gauge invariant formulation of the SU(N) non-linear sigma model in 2+1 dimensions, describing its dynamics through a pseudo-vector field with correct degrees of freedom and BRST invariance.

Contribution

It introduces a local, gauge invariant action for the model using a pseudo-vector field, ensuring proper degrees of freedom and establishing equivalence with the standard formulation.

Findings

The model has one polarization of the pseudo-vector field, matching Yang-Mills theory in 2+1 dimensions.

The physical content is equivalent to a massless pseudo-scalar field in the Lie algebra.

A BRST invariant gauge-fixed action is constructed.

Abstract

We derive a local, gauge invariant action for the SU(N) non-linear sigma-model in 2+1 dimensions. In this setting, the model is defined in terms of a self-interacting pseudo vector-field \theta_\mu, with values in the Lie algebra of the group SU(N). Thanks to a non-trivially realized gauge invariance, the model has the correct number of degrees of freedom: only one polarization of \theta_\mu, like in the case of the familiar Yang-Mills theory in 2+1 dimensions. Moreover, since \theta_\mu is a pseudo-vector, the physical content corresponds to one massless pseudo-scalar field in the Lie algebra of SU(N), as in the standard representation of the model. We show that the dynamics of the physical polarization corresponds to that of the SU(N) non-linear sigma model in the standard representation, and also construct the corresponding BRST invariant gauge-fixed action.