We investigate possible extensions of the $\mathrm{}(2+1)\mathrm{}$-dimensional ${\mathrm{CP}}^{N-1}$ model to 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.

• Received 21 July 2003

DOI:https://doi.org/10.1103/PhysRevD.69.065012