Canonical prescription for the integration contour over complex metrics

Identify a canonical prescription for the integration contour in complexified metric space for the gravitational path integral, determining which complex metrics are admissible saddles and how to integrate over them in a way consistent with quantum gravity and holographic expectations.

Background

Including complex metrics appears necessary in Euclidean treatments of rotating solutions and in supersymmetric contexts. However, the Euclidean Einstein–Hilbert action’s lack of boundedness under conformal rescalings (the conformal factor problem) undermines naive convergence arguments, and presently no canonical contour prescription is available.

A principled contour choice would systematize the inclusion of complex saddles, reconcile Euclidean methods with Lorentzian physics, and align gravitational path integrals with boundary CFT partition functions under holography.