Study of the "Non-Abelian" Current Algebra of a Non-linear $\sigma$-Model

TL;DR

This paper investigates a non-linear sigma model with a global gauge invariance, revealing its non-abelian current algebra with field-dependent structure functions, and connects it to noncommutative geometry.

Contribution

It analyzes the non-abelian current algebra structure of a specific non-linear sigma model and links it to noncommutative geometry, providing new insights into its algebraic properties.

Findings

Revealed non-abelian current algebra with field-dependent structure functions.

Reduced the field theory to a non-linear harmonic oscillator.

Established a connection with noncommutative geometry.

Abstract

A particular form of non-linear $\sigma$-model, having a global gauge invariance, is studied. The detailed discussion on current algebra structures reveals the non-abelian nature of the invariance, with {\it{field dependent structure functions}}. Reduction of the field theory to a point particle framework yields a non-linear harmonic oscillator, which is a special case of similar models studied before in \cite{car}. The connection with noncommutative geometry is also established.