Comments on Noncommutative Sigma Models

TL;DR

This paper reviews the noncommutative version of the $ ext{CP}^n$ sigma model, highlighting its soliton solutions, topological charge considerations, and the existence of a non-BPS sector, revealing new complexities compared to the commutative case.

Contribution

It provides a detailed derivation of the noncommutative $ ext{CP}^n$ sigma model and explores its soliton solutions and topological properties, uncovering novel features unique to the noncommutative setting.

Findings

Noncommutative $ ext{CP}^n$ model has soliton solutions similar to scalar field theory.

Topological charge definition requires careful treatment due to surface terms.

The model exhibits a non-BPS sector with unexpected features in the noncommutative case.

Abstract

We review the derivation of a noncommutative version of the nonlinear sigma model on $\CPn$ and it's soliton solutions for finite $\theta$ emphasizing the similarities it bears to the GMS scalar field theory. It is also shown that unlike the scalar theory, some care needs to be taken in defining the topological charge of BPS solitons of the theory due to nonvanishing surface terms in the energy functional. Finally it is shown that, like its commutative analogue, the noncommutative $\CPn$-model also exhibits a non-BPS sector. Unlike the commutative case however, there are some surprises in the noncommutative case that merit further study.