Nonabelian noncommutative gauge theory via noncommutative extra dimensions

The concept of covariant coordinates on noncommutative spaces leads directl y to gauge theories with generalized noncommutative gauge fields of the typ e that arises in string theory with background B-fields. The theory is natu rally expressed in terms of cochains in an appropriate cohomology; we discu ss how it fits into the framework of projective modules. The equivalence of star products that arise from the background field with and without fluctu ations and Kontsevich's formality theorem allow an explicitly construction of a map that relates ordinary gauge theory and noncommutative gauge theory (Seiberg-Witten map). As application we show the exact equality of the Dir ac-Born-Infeld action with B-field in the commutative setting and its semi- noncommutative cousin in the intermediate picture. Using noncommutative ext ra dimensions the construction is extended to noncommutative nonabelian gau ge theory for arbitrary gauge groups; an explicit map between abelian and n onabelian gauge fields is given. All constructions are also valid for non-c onstant B-field, Poisson structure and metric. (C) 2001 Published by Elsevi er Science B.V.