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

Paolo Aschieri,Leonardo Castellani

doi:10.1007/jhep11(2014)103

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.

Highlights

Chern-Simons forms and their noncommutative versionsIf we consider homogeneous forms T , T ′ that are Lie algebra valued, the trace of the ∧⋆-product of forms is still graded cyclic up to total Lie derivative terms:
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

Summary

Chern-Simons forms and their noncommutative versions

If we consider homogeneous forms T , T ′ that are Lie algebra valued, the trace of the ∧⋆-product of forms is still graded cyclic up to total Lie derivative terms:. The check that the exterior derivative of the commutative CS form LC(2Sn−1) gives T r(Rn) is algebraic, and relies only on the Leibniz rule property of the exterior derivative and on the graded cyclicity of the trace. Since the exterior derivative satisfies the Leibniz rule in the noncommutative case, and the graded cyclicity of the trace holds up to total Lie derivatives, we can conclude that the noncommutative Chern-Simons form satisfies the relation dLC(2Sn∗−1) = T r(R∧⋆n) + lC Q(2n) C (2.15). Is invariant under the ⋆-gauge variations (2.17)

The Seiberg-Witten map

The SW variation of NC Chern-Simons forms

Extended CS actions from NC CS actions

B Useful identities