Jackiw-Teitelboim and Kantowski-Sachs quantum cosmology

TL;DR

This work builds a concrete bridge between $2D$ Jackiw–Teitelboim gravity with $Λ>0$ and Kantowski–Sachs cosmology by establishing a precise minisuperspace correspondence and then computing a Hartle–Hawking no-boundary wave function within KS gravity. Using Euclidean path integrals and Picard–Lefschetz contour methods, the authors derive a semiclassical HH wave function dominated by KS saddles, and show how dimensional reduction to JT yields a normalizable dilaton distribution. They argue that a recently proposed exact JT/WDW wave function (IKTV) is not a genuine WDW solution under their boundary conditions and emphasize the importance of closure conditions at small geometry. The KS→JT reduction then provides a well-defined HH wave function for JT gravity and clarifies boundary-condition choices, offering a platform for future direct JT path-integral formulations and for exploring the tunneling boundary condition in these models.

Abstract

We study quantum cosmology of the $2D$ Jackiw-Teitelboim (JT) gravity with $Λ>0$ and calculate the Hartle-Hawking (HH) wave function for this model in the minisuperspace framework. Our approach is guided by the observation that the JT dynamics can be mapped exactly onto that of the Kantowski-Sachs (KS) model describing a homogeneous universe with spatial sections of $S^1\times S^2$ topology. This allows us to establish a JT-KS correspondence between the wave functions of the models. We obtain the semiclassical Hartle-Hawking wave function by evaluating the path integral with appropriate boundary conditions and employing the methods of Picard-Lefschetz theory. The JT-KS connection formulas allow us to translate this result to JT gravity, define the HH wave function and obtain a probability distribution for the dilaton field.

Figures (6)

• Figure 1: Examples of convergent infinite contours in the complex $N$ plane. The singularities $N=\{0,3\}$ are shown in circles.
• Figure 2: The steepest descent contours for $u>1$ and $Hb=1$. The arrowheads point to the direction where $Re(-\tilde{S}_{E})$ decreases. The saddles $\tilde{N}_{i}$ are marked with solid dots and the singularities with circles. Note the branch cut at $\tilde{N}\in(3^{+},+\infty)$. The HH contour corresponds to the solid curve encircling the singularity $N=0$; it is dominated by saddles $N_{1}$, $N_{2}$.
• Figure 3: The steepest descent contours for $u<1$ and $Hb=1$. In this case, all the saddles are real. The HH contour corresponds to the solid curve encircling the singularity $N=0$ and dominated by saddle $N_{4}$.
• Figure 4: The perturbed steepest descent contours for $u>1$ and $Hb>1$. The HH contour does not pass through the saddles $N_{4}$ and $N_{5}$.
• Figure 5: The perturbed steepest descent contours for $u>1$ and $Hb<1$. The HH contour does not pass through the saddles $N_{4}$ and $N_{5}$.