Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Liouville Cosmology

Dionysios Anninos, Thomas Hertog, Joel Karlsson

TL;DR

This work constructs and analyzes a tractable 2D quantum cosmology by coupling a unitary CFT to timelike Liouville gravity in conformal gauge, focusing on disk path integrals with matter insertions. It derives the one-loop Liouville disk wavefunctions, scrutinizes saddle points and conformal structure, and proposes an all-loop conjecture for the wavefunctions, including a cosmological pairing that yields a K-independent inner product on Euclidean histories. The paper also investigates multiple ensembles (fixed K, fixed boundary length, fixed area) and demonstrates a consistent Bessel-function structure that matches exact spacelike results under analytic continuation, shedding light on the quantum cosmology Hilbert space and potential static-patch thermodynamics. Overall, it outlines a framework for understanding quantum cosmology beyond minisuperspace in a controlled setting and points toward higher-dimensional generalizations and deeper inner-product structures for the cosmological state space.

Abstract

We provide a detailed analysis of the disk path integral of timelike Liouville theory, conceived as a tractable and precise toy-model quantum cosmology in two dimensions. Disk path integrals with the insertion of matter field operators, taken along a judiciously chosen complex contour, yield states akin to the Hartle-Hawking wavefunction. Working in the fixed $K$-representation, where $K$ is the trace of the extrinsic curvature, we compute the one-loop wavefunctions and put forward a conjecture for the all-loop expressions. A suitable pairing of Liouville disk path integrals yields a $K$-independent quantity that may form the basis for a well-defined inner product on the space of Euclidean histories. We also consider other ensembles, including one with fixed area, and provide a static patch perspective with a timelike feature.

Quantum Liouville Cosmology

TL;DR

This work constructs and analyzes a tractable 2D quantum cosmology by coupling a unitary CFT to timelike Liouville gravity in conformal gauge, focusing on disk path integrals with matter insertions. It derives the one-loop Liouville disk wavefunctions, scrutinizes saddle points and conformal structure, and proposes an all-loop conjecture for the wavefunctions, including a cosmological pairing that yields a K-independent inner product on Euclidean histories. The paper also investigates multiple ensembles (fixed K, fixed boundary length, fixed area) and demonstrates a consistent Bessel-function structure that matches exact spacelike results under analytic continuation, shedding light on the quantum cosmology Hilbert space and potential static-patch thermodynamics. Overall, it outlines a framework for understanding quantum cosmology beyond minisuperspace in a controlled setting and points toward higher-dimensional generalizations and deeper inner-product structures for the cosmological state space.

Abstract

We provide a detailed analysis of the disk path integral of timelike Liouville theory, conceived as a tractable and precise toy-model quantum cosmology in two dimensions. Disk path integrals with the insertion of matter field operators, taken along a judiciously chosen complex contour, yield states akin to the Hartle-Hawking wavefunction. Working in the fixed -representation, where is the trace of the extrinsic curvature, we compute the one-loop wavefunctions and put forward a conjecture for the all-loop expressions. A suitable pairing of Liouville disk path integrals yields a -independent quantity that may form the basis for a well-defined inner product on the space of Euclidean histories. We also consider other ensembles, including one with fixed area, and provide a static patch perspective with a timelike feature.

Paper Structure

This paper contains 21 sections, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Gravity as a Liouville theory
  3. The Weyl gauge and timelike Liouville theory
  4. Wavefunctions of the Liouville universe
  5. Path integral for Liouville one-point functions
  6. One-loop path integral
  7. Saddle points
  8. Conformal structure of the one-point functions
  9. Quantum fluctuations
  10. Boundary fluctuations
  11. Bulk fluctuations
  12. Wavefunctions at one loop and beyond
  13. Fixed trace of the extrinsic curvature
  14. Fixed boundary length
  15. Fixed area and boundary length
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Illustrations of semiclassical saddle geometries \ref{['eq:Liouville_simple_saddle']}, without conical singularities ($k=0$), and the corresponding ranges for the parameter $\gamma$ and $\ell^2/A$. The geometries have been isometrically embedded in $\mathbb{R}^3$ (a, b) and $\mathbb{R}^{2,1}$ (c).
  • Figure 2: Illustration of the complexified contours for the area producing the Bessel functions in \ref{['eq:Bessel_contours']}. The corresponding branch cuts for the integrand run along the real axis and are marked by dashed lines.