BRST formalism of Weyl conformal gravity

We present the Becchi-Rouet-Stora-Tyupin (BRST) formalism of a Weyl conformal gravity in Weyl geometry. Choosing the extended de Donder gauge-fixing condition (or harmonic gauge condition) for the general coordinate invariance and the new scalar gauge fixing for the Weyl invariance, we find that there is a Poincar\'e-like $IOSp(10|10)$ supersymmetry as in a Weyl invariant scalar-tensor gravity in Riemann geometry. We also point out that there is a gravitational conformal symmetry in quantum gravity although there is a massive Weyl gauge field as a result of spontaneous symmetry breakdown of Weyl gauge symmetry, and we account for how the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry. The corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also show the unitarity of the physical S-matrix on the basis of the BRST quartet mechanism in the case of an absence of anomalies.