Timelike Liouville theory and AdS$_3$ gravity at finite cutoff

Abstract

We propose that AdS$_3$ gravity with conformal boundary conditions is described by coupling the holographic CFT$_2$ to timelike Liouville theory and deforming by an exactly marginal operator. In this description, the Liouville field controls the finite-cutoff radial wall in the bulk. We check this proposal in the semiclassical limit by matching the sphere and torus partition functions between the bulk and boundary theories. We also show that the Liouville field's equation of motion gives the bulk Hamiltonian constraint. The strong coupling limit of our theory pushes the bulk description deep inside the interior of black hole geometries. This is also the flat-space limit, and it leads to a duality between 3d flat space and 2d CFT.