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

A bstract We first streamline the construction of the unique six-dimensional conformal gravity action found by Lü, 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 conjecture that on every even-dimensional space there exists a unique — up to boundary terms — conformally-invariant gravity theory that is extremised by Einstein metrics.