A note on non-linear $\sigma$-models in noncommutative geometry

TL;DR

This paper investigates non-linear sigma-models on noncommutative tori, revealing that trivial harmonic unitaries are local but not global minima, and explores symmetries and solutions related to noncommutative geometries.

Contribution

It demonstrates that trivial harmonic unitaries are not global minima in noncommutative sigma-models and introduces a $ abla^2$-action on field maps affecting solutions.

Findings

Trivial harmonic unitaries are local minima but not global minima.

Symmetric unitaries from instanton solutions serve as lower energy configurations.

A $ abla^2$-action on field maps influences solutions of Euler-Lagrange equations.

Abstract

We study non-linear $\sigma$-models defined on noncommutative torus as a two dimensional string world-sheet. We consider a quantum group as a noncommutative space-time as well as two points, a circle, and a noncommutative torus. Using the establised results we show that the trivial harmonic unitaries of the noncommutative chiral model, which are known as local minima, are not global minima by comparing those with the symmetric unitaries coming from instanton solutions of noncommutative Ising model, which corresponds to the two points target space. In addition,we introduce a $\mathbb{Z}^2$-action on field maps to noncommutative torus and show how it acts on solutions of various Euler-Lagrange equations.