SL(2,C) gravity on noncommutative space with Poisson structure

TL;DR

This paper develops an SL(2,C) gauge theory formulation of gravity on a noncommutative space with Poisson structure, incorporating the Seiberg-Witten map to relate noncommutative and commutative variables.

Contribution

It introduces a novel noncommutative gravity model based on SL(2,C) gauge theory using covariant coordinates and first-order Seiberg-Witten map corrections.

Findings

Constructed a gauge-invariant action for noncommutative gravity.

Expressed noncommutative degrees of freedom in terms of commutative variables.

Demonstrated the first-order effects of noncommutativity on gravity.

Abstract

The Einstein's gravity theory can be formulated as an SL(2,C) gauge theory in terms of spinor notations. In this paper, we consider a noncommutative space with the Poisson structure and construct an SL(2,C) formulation of gravity on such a space. Using the covariant coordinate technique, we build a gauge invariant action in which, according to the Seiberg-Witten map, the physical degrees of freedom are expressed in terms of their commutative counterparts up to the first order in noncommutative parameters.