Symmetric Ghost Lagrange Densities for the Coupling of Gravity to Gauge Theories

Abstract

We derive and present symmetric ghost Lagrange densities for the coupling of General Relativity to Yang--Mills theories. The graviton-ghost is constructed with respect to the linearized de Donder gauge fixing condition and the gauge ghost with respect to the covariant Lorenz gauge fixing condition. Both ghost Lagrange densities together with their accompanying gauge fixing Lagrange densities are obtained from the action of the diffeomorphism and gauge super-BRST differential -- which we define as the composition of the BRST differential with its anti-BRST differential -- on suitable gauge fixing bosons. In addition, we introduce a total gauge fixing boson and show that the complete symmetric ghost and gauge fixing Lagrange density can be generated thereof using the total super-BRST differential. In particular, we generalize two earlier approaches for flat-spacetime Yang--Mills theories to General Relativity and covariant Yang--Mills theories: The original approach by Curci and Ferrari (1976), using the Faddeev--Popov method on non-linear gauge fixings, and the modern approach by Baulieu and Thierry-Mieg (1982), using BRST and anti-BRST symmetries with gauge fixing bosons.