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 Poincaré-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 Poincaré 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.

• Received 30 September 2022
• Accepted 24 October 2022

DOI:https://doi.org/10.1103/PhysRevD.106.106007

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society