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

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 $g^{\mu\nu}$ and the connection $\Gamma^\lambda_{\mu\nu}$ are treated as independent, with the connection playing the role of an auxiliary field. We show that the gauge algebra is closed and irreducible. We further demonstrate that the Dyson-Schwinger equations in the first-order formulation lead to structural identities that constrain the Green's functions of the auxiliary field and encode the classical equations of motion at the quantum level. 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 Senjanovi\'c 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.