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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 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.