Time-Space Noncommutative Abelian Solitons

TL;DR

This paper constructs and analyzes solitons in a time-space noncommutative integrable sigma model, revealing unique properties and singularities in the U(1) case, and connecting to noncommutative instantons.

Contribution

It introduces explicit soliton solutions for a time-space noncommutative U(n) sigma model and explores their properties and reductions, including a new integrable equation for the reduced model.

Findings

Constructed explicit solitons in a time-space noncommutative sigma model.

Identified singular behavior in the U(1) single-soliton case in the commutative limit.

Reduced the model to a noncommutative instanton-like configuration governed by a new integrable equation.

Abstract

We demonstrate the construction of solitons for a time-space Moyal-deformed integrable U(n) sigma model (the Ward model) in 2+1 dimensions. These solitons cannot travel parallel to the noncommutative spatial direction. For the U(1) case, the rank-one single-soliton configuration is constructed explicitly and is singular in the commutative limit. The projection to 1+1 dimensions reduces it to a noncommutative instanton-like configuration. The latter is governed by a new integrable equation, which describes a Moyal-deformed sigma model with a particular Euclidean metric and a magnetic field.