Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map

Abstract

We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension $D$ and at first order in the noncommutativity parameter $\theta$. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives. A simple explanation of the equality between noncommutative and commutative CS actions in $D=1$ and $D=3$ is obtained. The $\theta$ dependent terms are present for $D\ge 5$ and give a higher derivative theory on commutative space reducing to classical CS theory for $\theta \to 0$. These terms depend on the field strength and not on the bare gauge potential. In particular, as for the Dirac-Born-Infeld action, these terms vanish in the slowly varying field strength approximation: in this case noncommutative and commutative CS actions coincide in any dimension. The Seiberg-Witten map on the $D=5$ noncommutative CS theory is explored in more detail, and we give its second order $\theta$-expansion for any gauge group. The example of extended $D=5$ CS gravity, where the gauge group is $SU(2,2)$, is treated explicitly.

18 pages, LaTeX. Added clarifications, added reference. Matches published version on JHEP

References in corpus (3)

• Noncommutative D=4 gravity coupled to fermions
• Seiberg--Witten Maps to All Orders
• Noncommutative SO(2,3) gauge theory and noncommutative gravity

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