Path integral for the Hilbert-Palatini and Ashtekar gravity

TL;DR

This work constructs a BRST path integral for real Hilbert–Palatini gravity and demonstrates its rewriting in Ashtekar variables under a restricted class of gauges, with reality conditions fixing the contour of integration. The analysis shows that, for YM-like gauges, ghost terms align with naive expectations and the Faddeev–Popov determinant matches the standard Ashtekar form, while the reality conditions define the contour. By connecting the real HP formulation to the complex Ashtekar framework, the paper bridges perturbative and non-perturbative approaches and clarifies the role of constraints, gauge fixing, and contour selection in quantum gravity path integrals. The results provide a concrete, if gauge-restricted, route to a contour-based Ashtekar gravity quantization and point to challenges in extending to arbitrary gauges or degenerate triads, with potential implications for non-perturbative quantum gravity studies.

Abstract

To write down a path integral for the Ashtekar gravity one must solve three fundamental problems. First, one must understand rules of complex contour functional integration with holomorphic action. Second, one should find which gauges are compatible with reality conditions. Third, one should evaluate the Faddeev-Popov determinant produced by these conditions. In the present paper we derive the BRST path integral for the Hilbert-Palatini gravity. We show, that for certain class of gauge conditions this path integral can be re-written in terms of the Ashtekar variables. Reality conditions define contours of integration. For our class of gauges all ghost terms coincide with what one could write naively just ignoring any Jacobian factors arising from the reality conditions.