Chern-Simons functional and the no-boundary proposal in Bianchi IX quantum cosmology

TL;DR

This paper explores the role of the Chern-Simons functional as a solution to the Ashtekar-Hamilton-Jacobi equation in Bianchi IX quantum cosmology, revealing families of Euclidean solutions and their implications for the no-boundary wave function.

Contribution

It demonstrates that the Chern-Simons functional generates classical solutions in Bianchi IX cosmology, connecting them to Euclidean Taub-NUT-de Sitter metrics and the no-boundary proposal.

Findings

Identifies a two-parameter family of Euclidean solutions with NUT-type closure.

Shows reduction to Taub-NUT-de Sitter metrics under symmetry conditions.

Provides a semiclassical estimate for the Bianchi IX no-boundary wave function.

Abstract

The Chern-Simons functional $S_{\rm CS}$ is an exact solution to the Ashtekar-Hamilton-Jacobi equation of general relativity with a nonzero cosmological constant. In this paper we consider $S_{\rm CS}$ in Bianchi type IX cosmology with $S^3$ spatial surfaces. We show that among the classical solutions generated by~$S_{\rm CS}$, there is a two-parameter family of Euclidean spacetimes that have a regular NUT-type closing. When two of the three scale factors are equal, these spacetimes reduce to a one-parameter family within the Euclidean Taub-NUT-de~Sitter metrics. For a nonzero cosmological constant, $\exp(iS_{\rm CS})$ therefore provides a semiclassical estimate to the Bianchi~IX no-boundary wave function in Ashtekar's variables.