Allowable complex metrics in minisuperspace quantum cosmology

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 defined these metrics, Witten has recently suggested their proposal also apply gravity. We explore this in context of gravitational path integrals, simple minisuperspace models, specifically considering de Sitter (dS), no-boundary anti--de (AdS) examples. These examples allow us gain some understanding off-shell structure integrals. In all cases, we find saddle points integral lie right at edge allowable domain even when are or Euclidean. Moreover Lefschetz thimbles, particular steepest descent contours for lapse integral, cut off as they intrude into nonallowable metrics. AdS case, implied restriction integration contour is found physical interpretation. dS forced become asymptotically point out K-S provides reason, proposal, why scalar fields would start evolution local extrema potential.