Kontsevich-Segal criterion in the no-boundary state constrains anisotropy

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>S</a:mi><a:mn>3</a:mn></a:msup></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>S</c:mi><c:mn>1</c:mn></c:msup><c:mo>×</c:mo><c:msup><c:mi>S</c:mi><c:mn>2</c:mn></c:msup></c:math> boundaries and find that, in both models, the KSW criterion excludes a significant range of homogeneous but anisotropic configurations. For squashed <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>S</e:mi><e:mn>3</e:mn></e:msup></e:math> boundaries, the excluded range includes all surface geometries with negative scalar curvature, in line with dS/CFT reasoning. For <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>S</g:mi><g:mn>1</g:mn></g:msup><g:mo>×</g:mo><g:msup><g:mi>S</g:mi><g:mn>2</g:mn></g:msup></g:math> boundaries, we find that KSW selects the low-temperature regime of configuration space where the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mi>S</i:mi><i:mn>1</i:mn></i:msup></i:math> is sufficiently large compared to the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>S</k:mi><k:mn>2</k:mn></k:msup></k:math>. In both models, the KSW criterion renders the semiclassical wave function normalizable, up to one-loop effects.