The BRST Double Complex for the Coupling of Gravity to Gauge Theories
David Prinz
TL;DR
This work develops a BRST double complex for (effective) Quantum General Relativity coupled to the Standard Model, introducing two nilpotent differentials $P$ and $Q$ for diffeomorphism and gauge symmetries and forming the total differential $D = P + Q$ with a corresponding total anti-BRST operator. It proves the mutual anticommutation of BRST and anti-BRST operators, derives a complete gauge-fixing construction via the total gauge-fixing fermion $\Upsilon = \sigma^{(1)} + \digamma_{\{1\}}$, and shows that graviton-ghosts decouple from matter when the Yang–Mills gauge fixing density has tensor density weight $w = 1$. The paper further establishes isomorphisms between BRST cocomplexes and anti-BRST complexes through ghost conjugation, and reveals sign-twisted anti-BRST operators as cochain homotopies, laying groundwork for symmetric (Hermitian) ghost densities and a perturbative BRST cocomplex. Collectively, these results clarify the cohomological structure of gravity–gauge theory quantization and inform future directions in transversality, cosmological-constant cases, and renormalization via a BRST-based Hopf algebra.
Abstract
We consider (effective) Quantum General Relativity coupled to the Standard Model (QGR-SM) and clarify whether graviton-ghosts couple to matter particles. To this end, we examine the corresponding BRST and anti-BRST symmetries, which are generated by infinitesimal diffeomorphisms and infinitesimal gauge transformations. In particular, we study their properties and relations: We find that all differentials mutually anticommute, which implies that they form a double complex. In particular, we introduce the total BRST differential as the sum of the diffeomorphism and gauge BRST differentials and similarly the total anti-BRST differential as the sum of the respective anti-BRST differentials. Furthermore, we identify the functionals in particle fields that are (co)cycles up to total derivatives with respect to the diffeomorphism differentials as scalar tensor densities of weight one: This implies that graviton-ghosts decouple from matter particles if and only if the Yang--Mills gauge fixing Lagrange density has said tensor density weight. Moreover, we discuss the relevant gauge fixing fermions: Starting from the de Donder and Lorenz gauge fixing conditions, we introduce a total gauge fixing fermion that generates the complete gauge fixing and ghost Lagrange density of QGR-SM. Finally, we show that the BRST cocomplexes are isomorphic to their corresponding anti-BRST complexes via ghost conjugation. Notably, this relates the BRST cohomologies to their respective anti-BRST homologies.