Path integral for the Hilbert-Palatini and Ashtekar gravity

To write down a path integral for the Ashtekar gravity one must solve three fundamental problems. First, one must understand the 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 Hilbert-Palatini gravity. We show that for a certain class of gauge conditions this path integral can be rewritten 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.