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

Kuroush Allameh, Edgar Shaghoulian

TL;DR

The paper proposes a holographic dual for AdS$_3$ gravity with conformal boundary conditions as a 2d CFT coupled to timelike Liouville theory deformed by a gravitationally dressed TTbar operator. It validates the proposal by matching sphere and torus partition functions between the bulk theory and the boundary construction in the semiclassical limit, showing that the Liouville field encodes the finite bulk cutoff and yields a consistent flat-space limit to a 2d CFT. The dual theory features a vanishing conformal anomaly and a modular-invariant spectrum controlled by an effective central charge that decreases from the Brown–Henneaux value as the cutoff is varied. The framework hints at a local, finite-cutoff description of AdS$_3$ holography and points to rich extensions to higher dimensions, dS, and fixed-area observables, motivating further exploration of boundary Liouville dynamics and TTbar dressings in holography.

Abstract

We propose that AdS$_3$ gravity with conformal boundary conditions is described by coupling the holographic CFT 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. By scaling parameters of the Liouville theory we can push our bulk description deep inside the interior of black hole geometries. The same scaling of parameters leads to a flat-space limit with a two-dimensional CFT as its holographic dual theory.

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

TL;DR

The paper proposes a holographic dual for AdS gravity with conformal boundary conditions as a 2d CFT coupled to timelike Liouville theory deformed by a gravitationally dressed TTbar operator. It validates the proposal by matching sphere and torus partition functions between the bulk theory and the boundary construction in the semiclassical limit, showing that the Liouville field encodes the finite bulk cutoff and yields a consistent flat-space limit to a 2d CFT. The dual theory features a vanishing conformal anomaly and a modular-invariant spectrum controlled by an effective central charge that decreases from the Brown–Henneaux value as the cutoff is varied. The framework hints at a local, finite-cutoff description of AdS holography and points to rich extensions to higher dimensions, dS, and fixed-area observables, motivating further exploration of boundary Liouville dynamics and TTbar dressings in holography.

Abstract

We propose that AdS gravity with conformal boundary conditions is described by coupling the holographic CFT 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. By scaling parameters of the Liouville theory we can push our bulk description deep inside the interior of black hole geometries. The same scaling of parameters leads to a flat-space limit with a two-dimensional CFT as its holographic dual theory.

Paper Structure

This paper contains 11 sections, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bulk analysis
  3. Sphere partition function
  4. Torus partition function
  5. Transforming between conformal and Dirichlet ensembles
  6. Deformed timelike Liouville theory
  7. Sphere partition function
  8. Torus partition function
  9. Stress tensor correlation functions
  10. Discussion
  11. $T\bar{T}$-deformed CFT$_2$ on a torus

Figures (2)

  • Figure 1: (a) The Penrose diagram for the non-rotating BTZ geometry. Only black-hole type solutions, represented by the red patch, are possible. (b) The Penrose diagram for the rotating BTZ geometry. We can construct both black-hole patches (in red) and cosmic patches (in blue). The wavy curves at $r=0$ indicate the different nature of the singularity.
  • Figure 2: (a) The Penrose diagram for Rindler space, the flat-space solution with horizon at finite temperature and vanishing angular velocity. The relevant patch (shown in red) is obtained by the flat-space limit $K\ell \rightarrow \infty$ of the non-rotating BTZ black hole, which zooms into the outer horizon. (b) The Penrose diagram for the flat-space solution with horizon at finite temperature and nonzero angular velocity. The relevant patch (shown in blue) is a remnant of the region beyond the inner horizon of the rotating BTZ black hole. By analytically continuing this region beyond the horizons after the flat-space limit, one recovers the null asymptotics. The same Penrose diagram governs hyperbolic and planar horizons in higher-dimensional flat space Banihashemi:2025qqi.