Einstein gravity as a gauge theory for the conformal group

TL;DR

This paper formulates Einstein gravity as a gauge theory of the conformal group using a first order variational principle, avoiding additional constraints and providing a new perspective on the geometric structure of gravity.

Contribution

It introduces a novel first order action for Einstein geometry based on conformal tractor geometry, extending the Cartan-Palatini formulation without requiring torsion-freeness constraints.

Findings

First order field equations derived without supplementary constraints.

A gauge theory formulation of Einstein gravity using conformal tractor geometry.

Connection to the Cartan-Palatini approach extended to conformal groups.

Abstract

General Relativity in dimension $n = p + q$ can be formulated as a gauge theory for the conformal group $SO(p+1,q+1)$, along with an additional field reducing the structure group down to the Poincar\'e group $ISO(p,q)$. In this paper, we propose a new variational principle for Einstein geometry which realizes this fact. Importantly, as opposed to previous treatments in the literature, our action functional gives first order field equations and does not require supplementary constraints on gauge fields, such as torsion-freeness. Our approach is based on the "first order formulation" of conformal tractor geometry. Accordingly, it provides a straightforward variational derivation of the tractor version of the Einstein equation. To achieve this, we review the standard theory of tractor geometry with a gauge theory perspective, defining the tractor bundle a priori in terms of an abstract principal bundle and providing an identification with the standard conformal tractor bundle via a dynamical soldering form. This can also be seen as a generalization of the so called Cartan-Palatini formulation of General Relativity in which the "internal" orthogonal group $SO(p,q)$ is extended to an appropriate parabolic subgroup $P\subset SO(p+1,q+1)$ of the conformal group.