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

TL;DR

This paper proposes a duality between AdS$_3$ gravity with finite cutoff and a boundary theory coupling a CFT$_2$ to timelike Liouville, supported by partition function matching and equations of motion analysis.

Contribution

It introduces a novel holographic description involving timelike Liouville theory for finite-cutoff AdS$_3$ and explores its implications for black hole interiors and flat-space limits.

Findings

Partition functions match between bulk and boundary theories.

Liouville field's equation reproduces bulk Hamiltonian constraint.

Strong coupling limit relates deep interior black hole geometries to flat space duality.

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.