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

TL;DR

This paper develops a geometric Seiberg-Witten map to expand noncommutative Chern-Simons theories in odd dimensions, revealing higher-derivative corrections and their relation to classical theories, especially in five dimensions.

Contribution

It introduces a geometric Seiberg-Witten map for noncommutative Chern-Simons theories, providing explicit expansions and insights into their structure across dimensions.

Findings

Noncommutative CS actions reduce to classical CS in low dimensions.

Higher derivative terms appear in dimensions five and above.

In slowly varying fields, noncommutative and commutative CS actions coincide.

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 $D\geq 5$ 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.