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