The complex Liouville string: the gravitational path integral

TL;DR

We address the problem of formulating quantum gravity with a positive cosmological constant in two dimensions by realizing sine dilaton gravity as the gravitational sector of the complex Liouville string. The authors map the worldsheet theory to a sine dilaton action and compute the gravitational path integral, uncovering a rich spectrum of saddles including constant AdS2 and dS2 vacua and piecewise-constant nodal saddles that realize vacuum transitions in a third-quantized framework. They compare the gravitational path integral to the exact worldsheet answer, finding striking structural similarities but also important nonperturbative effects such as saddle recombination and divergent sphere partition functions, highlighting subtleties in nonperturbative de Sitter quantum gravity. The work further develops sphere and disk observables, the genus expansion, and a third-quantized baby-universe perspective, while clarifying relations to matrix-model duals and potential connections to higher-dimensional de Sitter physics.

Abstract

We give a rigorous definition of sine dilaton gravity in terms of the worldsheet theory of the complex Liouville string arXiv:2409.17246. The latter has a known exact solution that we leverage to explore the gravitational path integral of sine dilaton gravity - a quantum deformation of dS JT gravity that admits both AdS$_2$ and dS$_2$ vacua. We uncover that the gravitational path integral receives contributions from new saddles describing transitions between vacua in a third-quantized picture. We also discuss the sphere and disk partition function in this context and contrast our findings with other recent work on this theory.

Figures (6)

• Figure 1: Indicated is the potential $W(\Phi)= \sin(2\pi\Phi)/\pi$ and its derivative. The latter sets the on-shell value of the curvature of the solutions to the equations of motion. The red dots indicate the alternating value of the scalar curvature $\mathcal{R}_* = \pm 2$ as imposed by the equations of motion (\ref{['eq: AdS saddle']}).
• Figure 2: A selection of (possibly degenerate) saddles contributing to the genus-two path integral. The different colors indicate the possibility of different constant values for the dilaton.
• Figure 3: Possible degenerations of the four-punctured sphere and their interpretation in terms of stable graphs. The labels of the vertices denote the genera of the components of the surface; for more details on the notation see paper2.
• Figure 4: The sphere one-point amplitude $\mathsf{A}_{0,1}^{(b)}(p)$ computes the 2d gravity path integral on the hemisphere, which prepares the Hartle-Hawking state. The norm of the Hartle-Hawking state is computed by the zero-point amplitude $\mathsf{A}_{0,0}^{(b)}$, which is computed by gluing two hemispheres together.
• Figure 5: The sphere three-point amplitude $\mathsf{A}_{0,3}^{(b)}(p,p',p")$ may be interpreted as the leading contribution to the transition amplitude between one initial universe labelled by $p$ and two final universes labelled by $p'$ and $p"$.