Abstract
We present a method where derivations of ⋆-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 ⋆-product. We also give explicit formulas for the Weyl-ordered ⋆-product and its Seiberg–Witten maps up to second order.
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