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.

arXiv comment18 pages, LaTeX. Added clarifications, added reference. Matches published version on JHEP

References in corpus (23)

• String Theory and Noncommutative Geometry
• Construction of non-Abelian gauge theories on noncommutative spaces
• Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces
• Nonabelian noncommutative gauge theory via noncommutative extra dimensions
• Standard General Relativity from Chern-Simons Gravity
• Noncommutative D=4 gravity coupled to fermions
• Chern-Simons action in noncommutative space
• Gauge Supergravities for all Odd Dimensions
• Chern-Simons Forms in Gravitation Theories
• Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008)
• Seiberg--Witten Maps to All Orders
• Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map
• UV/IR mixing in noncommutative QED defined by Seiberg-Witten map
• Noncommutative gravity at second order via Seiberg-Witten map
• Computing the θ-exact Seiberg-Witten map for arbitrary gauge groups
• Second order expansions of action functionals of noncommutative gauge theories
• Noncommutative Chern-Simons terms and the noncommutative vacuum
• Self-energies on deformed spacetimes
• Noncommutative SO(2,3) gauge theory and noncommutative gravity
• AdS-inspired noncommutative gravity on the Moyal plane
• The Chern-Simons Action in Non-Commutative Geometry
• Noncommutative AdS Supergravity in three Dimensions
• On The Complete Seiberg-Witten Map For Theories With Topological Terms

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