Allowable complex metrics in minisuperspace quantum cosmology

TL;DR

This paper investigates the Kontsevich-Segal criterion for allowable complex metrics in minisuperspace quantum cosmology, analyzing its implications on gravitational path integrals and saddle points in simple models like de Sitter and Anti-de Sitter spaces.

Contribution

It applies the K-S criterion to minisuperspace models, revealing how it constrains the structure of gravitational path integrals and the nature of saddle points.

Findings

Saddle points lie at the edge of the allowable metric domain.

Lefschetz thimbles are cut off by the criterion, affecting the integral contours.

In AdS, the restriction has a clear physical interpretation.

Abstract

Kontsevich and Segal (K-S) have proposed a criterion to determine which complex metrics should be allowed, based on the requirement that quantum field theories may consistently be defined on these metrics, and Witten has recently suggested that their proposal should also apply to gravity. We explore this criterion in the context of gravitational path integrals, in simple minisuperspace models, specifically considering de Sitter (dS), no-boundary and Anti-de Sitter (AdS) examples. These simple examples allow us to gain some understanding of the off-shell structure of gravitational path integrals. In all cases, we find that the saddle points of the integral lie right at the edge of the allowable domain of metrics, even when the saddle points are complex or Euclidean. Moreover the Lefschetz thimbles, in particular the steepest descent contours for the lapse integral, are cut off as they intrude into the domain of non-allowable metrics. In the AdS case, the implied restriction on the integration contour is found to have a simple physical interpretation. In the dS case, the lapse integral is forced to become asymptotically Euclidean. We also point out that the K-S criterion provides a reason, in the context of the no-boundary proposal, for why scalar fields would start their evolution at local extrema of their potential.