The $\sigma$-Model and Non-commutative Geometry

TL;DR

This paper demonstrates how the nonlinear sigma model can be constructed using non-commutative geometry and gauge theory on a discrete group, introducing a new geometric perspective.

Contribution

It introduces a novel approach connecting non-commutative geometry and gauge theory to derive the nonlinear sigma model from the linear one.

Findings

Non-commutative geometry provides a framework for sigma models.

Gauge theory on Z_2 can generate nonlinear sigma models.

A specific constraint links gauge theory to the sigma model structure.

Abstract

In terms of non-commutative geometry, we show that the $\sigma$--model can be built up by the gauge theory on discrete group $Z_2$. We introduce a constraint in the gauge theory, which lead to the constraint imposed on linear $\sigma$ model to get nonlinear $\sigma$ model .