The Kontsevich-Segal Criterion in the No-Boundary State Constrains Anisotropy

TL;DR

This paper demonstrates that the Kontsevich-Segal-Witten criterion applied to the no-boundary state restricts anisotropic deformations of de Sitter space, influencing boundary geometries and ensuring wave function normalizability.

Contribution

It shows how the KSW criterion constrains anisotropic deformations in specific boundary geometries within the no-boundary proposal, linking geometric conditions to wave function properties.

Findings

Excludes all negative scalar curvature geometries for squashed S^3 boundaries.

Selects low-temperature regimes for S^1 x S^2 boundaries where S^1 is large.

Ensures the semiclassical wave function is normalizable up to one-loop effects.

Abstract

We show that the Kontsevich-Segal-Witten (KSW) criterion applied to the no-boundary state constrains anisotropic deformations of de Sitter space. We consider squashed $S^3$ and $S^1 \times S^2$ boundaries and find that in both models, the KSW criterion excludes a significant range of homogeneous but anisotropic configurations. For squashed $S^3$ boundaries, the excluded range includes all surface geometries with negative scalar curvature, in line with dS/CFT reasoning. For $S^1 \times S^2$ boundaries, we find that KSW selects the low-temperature regime of configuration space where the $S^1$ is sufficiently large compared to the $S^2$. In both models, the KSW criterion renders the semiclassical wave function normalizable, up to one-loop effects.