Reality in Noncommutative Gravity

TL;DR

This paper explores the reality conditions and geometric structures in 4D noncommutative gravity, introducing a consistent framework for real covariant derivatives and analyzing the noncommutative corrections to the Riemann tensor.

Contribution

It provides a construction of real covariant derivatives and a minimal noncommutative Riemann tensor within the twist-deformed geometric formalism of 4D gravity.

Findings

Real covariant derivatives can be built using $oldsymbol{ extstar}$-anticommutators.

The noncommutative Riemann tensor has only even $oldsymbol{ heta}$-corrections.

Pure noncommutative tensor and scalar functions of $h_{mn}$ are identified.

Abstract

We study the problem of reality in the geometric formalism of the 4D noncommutative gravity using the known deformation of the diffeomorphism group induced by the twist operator with the constant deformation parameters $\vt^{mn}$. It is shown that real covariant derivatives can be constructed via $\star$-anticommutators of the real connection with the corresponding fields. The minimal noncommutative generalization of the real Riemann tensor contains only $\vt^{mn}$-corrections of the even degrees in comparison with the undeformed tensor. The gauge field $h_{mn}$ describes a gravitational field on the flat background. All geometric objects are constructed as the perturbation series using $\star$-polynomial decomposition in terms of $h_{mn}$. We consider the nonminimal tensor and scalar functions of $h_{mn}$ of the odd degrees in $\vt^{mn}$ and remark that these pure noncommutative objects can be used in the noncommutative gravity.