Nonlinear $\sigma$ model in the case of $N\times\alpha N$ rectangular matrices in two-dimensional Euclidean space

The matrix nonlinear $\sigma$ model in the case of $N\times\alpha N$ rectangular matrices is considered. It is shown that in two-dimensional Euclidean space the model is renormalizable with respect to $\alpha$ and $1/N$. The fulfillment of the chirality identity is demonstrated in the operator expansion for the renormalized theory.