Einstein gravity as a theory with a SL(2,C) connection double copy

TL;DR

This paper explores a novel perspective on Einstein gravity by expressing it as a theory with a double copy structure involving SL(2,C) connections, linking it to Yang-Mills theory through a holomorphic trail.

Contribution

It introduces a new formulation of Einstein gravity using SL(2,C) connections and proposes a conjecture connecting the BCJ double copy to a holomorphic trail between gauge theories and gravity.

Findings

Decomposition of Einstein gravity into two SL(2,C) connections.

Proposal of a holomorphic trail linking gauge theories and gravity.

Conjecture that BCJ double copy can be understood via injection maps.

Abstract

Results ranging from Ashtekar variables to the perturbative Bern-Carrasco-Johansson (BCJ) double copy suggest a deep relation between Yang-Mills theory and Einstein gravity. I examine this relation by writing down the tetradic Palatini action for Einstein gravity and covariantly decomposing its variables into two $SL(2,\mathbb{C})$ connections and two soldering forms. This leads to a conjecture that the BCJ double copy can be understood through a "holomorphic trail", a series of injection maps between two copies of $SU(2)$ theories and Einstein gravity.