# Quantum Gravity from Timelike Liouville theory

## TL;DR

This paper proposes a consistent definition of two-dimensional quantum gravity using timelike Liouville theory, addressing longstanding issues with the path integral and demonstrating well-defined, crossing-symmetric correlation functions.

## Contribution

It introduces a conformal bootstrap approach for timelike Liouville coupled to supercritical matter, proving a no-ghost theorem and constructing well-defined four-point functions.

## Key findings

- Proved a no-ghost theorem for BRST cohomology states.
- Constructed crossing-symmetric four-point functions.
- Demonstrated the well-defined nature of the path integral in this framework.

## 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.