Construction of Gauge Theories on Curved Noncommutative Spacetime

We present a method where derivations of star-product algebras are used to build covariant derivatives for noncommutative gauge theory. We write down a noncommutative action by linking these derivations to a frame field induced by a nonconstant metric. An example is given where the action reduces in the classical limit to scalar electrodynamics on a curved background. We further use the Seiberg-Witten map to extend the formalism to arbitrary gauge groups. A proof of the existence of the Seiberg-Witten-map for an abelian gauge potential is given for the formality star-product. We also give explicit formulas for the Weyl ordered star-product and its Seiberg-Witten-maps up to second order.