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

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 θ. 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 θ dependent terms are present for D ≥ 5 and give a higher derivative theory on commutative space reducing to classical CS theory for θ → 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 θ-expansion for any gauge group. The example of extended D = 5 CS gravity, where the gauge group is SU(2, 2), is treated explicitly.