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Nonabelian noncommutative gauge theory via noncommutative extra dimensions. (English) Zbl 0983.81054
Summary: The concept of covariant coordinates on noncommutative spaces leads directly to gauge theories with generalized noncommutative gauge fields of the type that arises in string theory with background \(B\)-fields. The theory is naturally expressed in terms of cochains in an appropriate cohomology; we discuss how it fits into the framework of projective modules. The equivalence of star products that arise from the background field with and without fluctuations and Kontsevich’s formality theorem allow an explicit 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 Dirac-Born-Infeld action with \(B\)-field in the commutative setting and its semi-noncommutative cousin in the intermediate picture. Using noncommutative extra dimensions the construction is extended to noncommutative nonabelian gauge theory for arbitrary gauge groups; an explicit map between abelian and nonabelian gauge fields is given. All constructions are also valid for non-constant \(B\)-field, Poisson structure and metric.

MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
81T75 Noncommutative geometry methods in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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