Noncommutative D=4 gravity coupled to fermions

TL;DR

This paper develops a noncommutative version of four-dimensional gravity coupled to fermions using twist-deformed space-time, maintaining geometric invariance and reducing to classical gravity in the commutative limit.

Contribution

It introduces a noncommutative extension of Einstein-Hilbert gravity with fermions, employing a general twist-deformed space-time and ensuring gauge invariance and a proper commutative limit.

Findings

Constructed a noncommutative gravity action invariant under diffeomorphisms and Lorentz star-gauge transformations.

Coupled the noncommutative gravity to fermions, including a consistent Majorana condition.

Discussed the noncommutative MacDowell-Mansouri action quadratic in curvatures.

Abstract

We present a noncommutative extension of Einstein-Hilbert gravity in the context of twist-deformed space-time, with a $\star$-product associated to a quite general triangular Drinfeld twist. In particular the $\star$-product can be chosen to be the usual Groenewald-Moyal product. The action is geometric, invariant under diffeomorphisms and centrally extended Lorentz $\star$-gauge transformations. In the commutative limit it reduces to ordinary gravity, with local Lorentz invariance and usual real vielbein. This we achieve by imposing a charge conjugation condition on the noncommutative vielbein. The theory is coupled to fermions, by adding the analog of the Dirac action in curved space. A noncommutative Majorana condition can be imposed, consistent with the $\star$-gauge transformations. Finally, we discuss the noncommutative version of the Mac-Dowell Mansouri action, quadratic in curvatures.