Covariant quantization of the Einstein-Hilbert theory in first-order form

Abstract: We present a covariant quantization of the first-order formulation of the Einstein-Hilbert theory using the path integral and BV formalisms. In this approach, the metric $g^{μν}$ and the connection $Γ^λ_{μν}$ are treated as independent, with the connection playing the role of an auxiliary field. We show that the gauge algebra is closed and irreducible. A novel trivial local symmetry arises, leading to structural identities that relate the Green's functions of the auxiliary field to its classical value. We revisit the quantum equivalence between the first- and second-order formulations of the Einstein-Hilbert theory. By employing a suitable trick, a manifestly covariant form of the Senjanović measure is derived. We also show that the two formulations are equivalent at the level of the effective action when the auxiliary field is on-shell.