Renormalizability of the supersymmetric Yang-Mills theories on the noncommutative torus

We argue that Yang-Mills theory on noncommutative torus, expressed in the Fourier modes, is described by a gauge theory in a usual commutative space, the gauge group being a generalization of the area-preserving diffeomorphisms to the noncommutative case. In this way, performing the loop calculation in this gauge theory in the continuum limit we show that this theory is renormalizable, and discuss the UV and IR limits. The moduli space of the vacua of the noncommutative super Yang-Mills theories in (2+1) dimensions is discussed. (author)