An Invariant Action for Noncommutative Gravity in Four-Dimensions

TL;DR

This paper proposes a new invariant action for noncommutative gravity in four dimensions using a constrained gauge group, avoiding the metric and employing star products, with a first-order deformation analysis.

Contribution

It introduces a novel invariant gravitational action based on the U(2,2) gauge group broken to U(1,1)×U(1,1), applicable to noncommutative geometry without relying on a metric.

Findings

Constructed a noncommutative gravity action in 4D using star products.

Derived the first-order deformation of the action in the noncommutative parameter.

Provided a naturally invariant measure without metric dependence.

Abstract

Two main problems face the construction of noncommutative actions for gravity with star products: the complex metric and finding an invariant measure. The only gauge groups that could be used with star products are the unitary groups. I propose an invariant gravitational action in D=4 dimensions based on the constrained gauge group U(2,2) broken to $U(1,1)\times U(1,1).$ No metric is used, thus giving a naturally invariant measure. This action is generalized to the noncommutative case by replacing ordinary products with star products. The four dimensional noncommutative action is studied and the deformed action to first order in deformation parameter is computed.