Quantum Gravity from Timelike Liouville theory

TL;DR

This work defines a nonperturbative, bootstrap-based formulation of timelike Liouville gravity in two dimensions by coupling timelike Liouville to supercritical matter and enforcing a BRST structure with a no-ghost theorem. It separates external BRST-invariant states from an internal spectrum used in correlation-function construction, showing that a well-defined, crossing-symmetric four-point function can be obtained by gluing timelike Liouville three-point functions and employing an analytic continuation strategy inspired by string field theory. The authors derive the explicit BRST cohomology for timelike Liouville with a cosmological term, prove that physical states have positive norm, and establish a robust prescription for analytic continuation that yields convergent amplitudes for all physical external energies. The approach provides a priori definitions of timelike quantum gravity in 2D, reveals deep connections with conformal bootstrap data (structure constants $\widehat{C}$ and $C$), and opens pathways to extensions to higher genus, supersymmetry, and holographic interpretations. The results offer insights into the nonperturbative path integral of gravity in a controlled setting and illuminate how to handle ill-defined timelike sectors via bootstrap-consistent analytic structures.

Abstract

A proper definition of the path integral of quantum gravity has been a long-standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cosmological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three-point functions is well defined and crossing symmetric (numerically) for external Liouville energies corresponding to \textit{all} physical states in the BRST cohomology with the choice of the Ribault-Santachiara contour for the internal energy.

Figures (7)

• Figure 1: Two branches for the spacelike Liouville charge $a$, and the corresponding conformal dimensions $\Delta_a$. Because of the reflection the ranges are halved to $a\in [0,Q/2]$ and $p\geq0$.
• Figure 2: Two branches of the timelike Liouville charge $\alpha$ and the corresponding conformal dimensions $\Delta_E$. Because of the reflection, the ranges are halved to $E \geq 0$ and $E \in \mathrm{i} [0, \infty)$.
• Figure 3: Poles of the four-point function integrand and integration contour for the $c_L \le 1$ Liouville theory. The poles in the $E_s$-plane are located on the imaginary axis (shaded area, only few poles are displayed) and depend only on $\beta$, not on the external momenta $\alpha_1, \alpha_2, \alpha_3$ and $\alpha_4$. The $\mathrm{i}\epsilon$ prescription shifts the contour away from the poles.
• Figure 4: Zeros of the function $\Upsilon_{\beta}(x)$ for $\beta \in \mathbb{C}$. If $\beta \in \mathbb{R}$ (resp. $\beta \in \mathrm{i} \mathbb{R}$), the zeros become all real (resp. pure imaginary).
• Figure 5: $s$-channel decomposition of the four-point function.