Noncommutative gauge fields coupled to noncommutative gravity

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted star product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and twisted gauge transformations. The Seiberg-Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor D_{IJK}.