Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map

Abstract

We use the Seiberg-Witten map (SW map) to expand noncommutative gravity coupled to fermions in terms of ordinary commuting fields. The action is invariant under general coordinate transformations and local Lorentz rotations, and has the same degrees of freedom as the commutative gravity action. The expansion is given up to second order in the noncommutativity parameter θ. A geometric reformulation and generalization of the SW map is presented that applies to any abelian twist. Compatibility of the map with hermiticity and charge conjugation conditions is proven. The action is shown to be real and invariant under charge conjugation at all orders in θ. This implies the bosonic part of the action to be even in θ, while the fermionic part is even in θ for Majorana fermions.

27 pages, LaTeX. Revised version with proof of charge conjugation symmetry of the NC action and its parity under theta --> - theta (see new sect. 2.6, sect. 6 and app. B). References added. arXiv admin note: substantial text overlap with arXiv:0902.3817

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• A Gravity Theory on Noncommutative Spaces
• Noncommutative D=4 gravity coupled to fermions
• The Noncommutative Standard Model at O(theta^2)
• Noncommutative supergravity in D=3 and D=4
• Seiberg--Witten Maps to All Orders
• Twisted Covariant Noncommutative Self-dual Gravity
• Dynamical noncommutativity and Noether theorem in twisted phi^*4 theory

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