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

TL;DR

This paper develops a second-order expansion of noncommutative gravity coupled to fermions using the Seiberg-Witten map, ensuring invariance and reality conditions, and generalizes the map for abelian twists.

Contribution

It provides a geometric reformulation of the Seiberg-Witten map applicable to abelian twists and proves compatibility with hermiticity and charge conjugation.

Findings

The action remains invariant under coordinate and Lorentz transformations.

The expanded action is real and charge conjugation invariant at all orders.

Bosonic and fermionic parts of the action are even in the noncommutativity parameter { heta}.

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 {\theta}. 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 {\theta}. This implies the bosonic part of the action to be even in {\theta}, while the fermionic part is even in {\theta} for Majorana fermions.