8D conformal gravity with Einstein sector, and its relation to the Q-curvature

TL;DR

This paper constructs and analyzes unique conformal gravity actions in six and eight dimensions that admit Einstein metrics as solutions, relating these to Q-curvature and obstruction tensors, revealing a universal conformally-invariant gravity framework.

Contribution

It explicitly constructs the unique eight-dimensional conformal gravity action admitting Einstein metrics, extending previous six-dimensional results, and connects these to Q-curvature and obstruction tensors.

Findings

Explicit eight-dimensional conformal gravity action constructed.

Established the uniqueness of Einstein solutions in these theories.

Linked conformal gravity to Q-curvature and Fefferman-Graham obstruction tensor.

Abstract

We first streamline the construction of the unique six-dimensional conformal gravity action found by L\"u, Pang and Pope, that admits Einstein metrics as solutions to the field equations. We then prove that there exists a unique eight-dimensional conformal gravity action that admits Einstein metrics as solutions to the field equations, and explicitly build the corresponding action. Finally, we relate these results to Branson's Q-curvature and the Fefferman-Graham obstruction tensor, to conclude that on every even-dimensional space there exists a unique -- up to boundary terms -- conformally-invariant gravity theory that is extremised by Einstein metrics.