Noncommutative Self-dual Gravity

TL;DR

This paper develops a noncommutative version of Einstein's gravity using a self-dual formulation and Seiberg-Witten map, deriving corrections to the classical action up to second order.

Contribution

It introduces a novel noncommutative gravity theory based on self-dual variables and explicitly computes second-order corrections to the Einstein action.

Findings

Noncommutative torsion constraint is equivalent to vanishing commutative torsion.

Second-order noncommutative corrections to the Einstein action are derived.

The theory maintains consistency with classical gravity in the commutative limit.

Abstract

Starting from a self-dual formulation of gravity, we obtain a noncommutative theory of pure Einstein theory in four dimensions. In order to do that, we use Seiberg-Witten map. It is shown that the noncommutative torsion constraint is solved by the vanishing of commutative torsion. Finally, the noncommutative corrections to the action are computed up to second order.