Noncommutative Nonlinear Sigma Models and Integrability

TL;DR

This paper reviews noncommutative nonlinear sigma models, demonstrates their integrability through conserved currents, and explores solutions, extending classical concepts to noncommutative and supersymmetric contexts.

Contribution

It extends the integrability framework of sigma models to noncommutative and supersymmetric settings, constructing conserved currents and analyzing solutions.

Findings

Noncommutative principal chiral model has an infinite number of conserved currents.

A generalized zero curvature representation confirms integrability of the noncommutative CP^1 submodel.

Explicit solutions are discussed for models with and without supersymmetry.

Abstract

We first review the result that the noncommutative principal chiral model has an infinite tower of conserved currents, and discuss the special case of the noncommutative CP^1 model in some detail. Next, we focus our attention to a submodel of the CP^1 model in the noncommutative spacetime A_\theta(R^2+1). By extending a generalized zero curvature representation to A_\theta(R^2+1) we discuss its integrability and construct its infinitely many conserved currents. Supersymmetric principal chiral model with and without the WZW term and a supersymmetric extension of the CP^1 submodel in noncommutative spacetime (i.e in superspaces A_\theta(R^1+1|2), A_\theta(R^2+1|2)) are also examined in detail and their infinitely many conserved currents are given in a systematic manner. Finally, we discuss the solutions of the aforementioned submodels with or without supersymmetry.