Nonabelian noncommutative gauge theory via noncommutative extra dimensions

TL;DR

This paper develops a framework for noncommutative gauge theories using covariant coordinates, projective modules, and noncommutative extra dimensions, establishing explicit maps and extending to nonabelian gauge groups with applications to string theory actions.

Contribution

It introduces a novel approach to noncommutative gauge theories via noncommutative extra dimensions, providing explicit maps and extending to nonabelian cases with variable background fields.

Findings

Established the equivalence of star products with and without fluctuations.

Constructed the Seiberg-Witten map explicitly.

Extended the framework to nonabelian gauge theories with non-constant backgrounds.

Abstract

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 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 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.