Noncommutative nonlinear sigma models and integrability

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 $\mathbb{C}{P}^1$ model in some detail. Next, we focus our attention to a submodel of the $\mathbb{C}{P}^1$ model in the noncommutative spacetime ${\mathcal{A}}_{\theta }({\mathbb{R}}^{2+1})$. By extending a generalized zero-curvature representation to ${\mathcal{A}}_{\theta }({\mathbb{R}}^{2+1})$ we discuss its integrability and construct its infinitely many conserved currents. A supersymmetric principal chiral model with and without the Wess-Zumino-Witten term and a supersymmetric extension of the $\mathbb{C}{P}^1$ submodel in noncommutative spacetime [i.e., in superspaces ${\mathcal{A}}_{\theta }({\mathbb{R}}^{1+1|2})$, ${\mathcal{A}}_{\theta }({\mathbb{R}}^{2+1|2})$] 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.