Some Properties of Non-linear $\sigma$-Models in Noncommutative Geometry

TL;DR

This paper develops non-linear sigma-models within noncommutative geometry, focusing on the noncommutative torus, and constructs instantons and analogues of Wess-Zumino-Witten models, revealing their unique features.

Contribution

It introduces non-linear sigma-models in noncommutative geometry, specifically on the noncommutative torus, and constructs instantons and Wess-Zumino-Witten analogues, highlighting their properties.

Findings

Constructed a sigma-model instanton with topological charge 1

Defined a noncommutative Wess-Zumino-Witten model

Illustrated features of models on noncommutative spaces

Abstract

We introduce non-linear $\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some characteristic features of the corresponding $\sigma$-models. In particular we construct a $\sigma$-model instanton with topological charge equal to 1. We also define and investigate some properties of a noncommutative analogue of the Wess-Zumino-Witten model.