Quantum de Sitter and Analytically Continued Chern Simons Theory

We give a Lefschetz thimble definition of the Chern--Simons--Kodama wavefunctional in Lorentzian self-dual gravity with positive cosmological constant. By complexifying the connection and choosing the Lefschetz thimble attached to the self-dual deSitter saddle, Witten's analytic continuation of Chern--Simons theory replaces the real contour, on which the defining integral is at best conditionally convergent, by an absolutely convergent integration cycle and yields a controlled semiclassical expansion. The relevant integrand is the full holomorphic exponential $e^{I}$, with $I=κY_{\rm CS}+(\text{source})$, and the steepest descent flow is governed by the Morse function $h=\operatorname{Re} I$. In the homogeneous isotropic de Sitter reduction, the construction reproduces the Hartle--Hawking Airy wavefunction. In the BianchiIX truncation, we take the Lefschetz thimble attached to the expanding deSitter saddle (the contracting branch defines a conjugate Lefschetz thimble); within this sector, the symmetric configuration is the only critical point for $Λ>0$. The corresponding Lefschetz thimble contains one trace direction with the FRW Airy structure and two anisotropic shear directions that are real-Gaussian damped at leading quadratic order. This gives a finite BianchiIX wavefunction which, to leading order, is the FRW Airy factor multiplied by a real anisotropy Gaussian. Thus, the Airy behavior is not an artifact of the one-dimensional minisuperspace reduction, but the leading form of the wavefunction on the Lefschetz thimble. We also discuss how the Lefschetz thimble prescription interfaces with Lorentzian reality conditions.

Source: arXiv:2608.07467v1 - http://arxiv.org/abs/2608.07467v1 PDF: https://arxiv.org/pdf/2608.07467v1 Original Link: http://arxiv.org/abs/2608.07467v1