(2+1)D Noncommutative CP$^{N-1}$ Model

TL;DR

This paper explores the extension of the (2+1)D $CP^{N-1}$ model to noncommutative space, demonstrating renormalizability in certain representations and identifying potential infrared divergence issues in others.

Contribution

It proves the renormalizability of the noncommutative $CP^{N-1}$ model in the left fundamental representation and analyzes infrared divergences related to different field transformations.

Findings

Model is renormalizable in the left fundamental representation.

Infrared divergences occur in the adjoint representation.

Infrared issues may cause breakdown of the 1/N expansion at higher orders.

Abstract

We investigate possible extensions of the (2+1) dimensional $CP^{N-1}$ model to the noncommutative space. Up to the leading nontrivial order of 1/N, we prove that the model restricted to the left fundamental representation of the gauge group is renormalizable and does not have dangerous infrared divergences. In contrast, if the basic field $\phi$ transforms in accord with the adjoint representation, infrared singularities are present in the two point function of the auxiliary gauge field and also in the leading correction to the self-energy of the $\phi$ field. These infrared divergences may produce nonintegrable singularities leading at higher orders to a breakdown of the 1/N expansion. Gauge invariance of the renormalization procedure is also discussed.