Quantization of gauge theory on a curved noncommutative space

We study quantization of a gauge analogon of the Grosse-Wulkenhaar scalar field model. We calculate divergent first-order one-loop contributions to the tadpoles and the propagators: we find that there are no UV divergences while all IR divergences are logarithmic. Divergent terms which we obtain can be included in the initial action either by a shift of the vacuum or by adding the Chern-Simons term. Our results, though partial, indicate renormalizability: they need to be complemented by calculation of the second-order divergent contributions to the propagators and vertices.