Gauge fields on noncommutative geometries with curvature

Understand

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.