  It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can
be written as lagrangian of the scalar field propagating in a curved
noncommutative space. In this interpretation, renormalizability of the model is
related to the interaction with the background curvature which introduces
explicit coordinate dependence in the action. In this paper we construct the
$U_1$ gauge field on the same noncommutative space: since covariant derivatives
contain coordinates, the Yang-Mills action is again coordinate dependent. To
obtain a two-dimensional model we reduce to a subspace, which results in
splitting of the degrees of freedom into a gauge and a scalar. We define the
gauge fixing and show the BRST invariance of the quantum action.
