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On boundary correlators in Liouville theory on AdS$_{2}$

Matteo Beccaria, Arkady A. Tseytlin

TL;DR

The paper provides strong evidence that Liouville boundary correlators in Euclidean AdS$_2$ match the holomorphic stress-tensor correlators of a CFT with the same central charge, extending this correspondence from tree level to one loop. By computing two-, three-, and four-point boundary functions in both ZZ and AdS formulations, it fixes the normalization between the Liouville boundary operator and the boundary stress tensor, including exact perturbative coefficients and the central charge expansion. The results show no anomalous dimensions for the boundary operator and demonstrate that the entire set of boundary correlators aligns with Virasoro/Weyl symmetry constraints, supporting a microscopic AdS$_2$/CFT$_1$-like duality in this setting. The work also outlines a general strategy for proofs and discusses generalizations to Toda theories and super-Liouville, highlighting the potential universality of boundary–stress-tensor duality on AdS$_2$.

Abstract

We consider the Liouville theory in fixed Euclidean AdS$_2$ background. Expanded near the minimum of the potential the elementary field has mass squared 2 and (assuming the standard Dirichlet b.c.) corresponds to a dimension 2 operator at the boundary. We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor (or the Virasoro generator with the same central charge) on a half-plane or a disc restricted to the boundary. This relation was first observed at the leading semiclassical order (tree-level Witten diagrams in AdS$_2$) in arXiv:1902.10536 and here we demonstrate its validity also at the one-loop level. We also discuss arguments that may lead to its general proof.

On boundary correlators in Liouville theory on AdS$_{2}$

TL;DR

The paper provides strong evidence that Liouville boundary correlators in Euclidean AdS match the holomorphic stress-tensor correlators of a CFT with the same central charge, extending this correspondence from tree level to one loop. By computing two-, three-, and four-point boundary functions in both ZZ and AdS formulations, it fixes the normalization between the Liouville boundary operator and the boundary stress tensor, including exact perturbative coefficients and the central charge expansion. The results show no anomalous dimensions for the boundary operator and demonstrate that the entire set of boundary correlators aligns with Virasoro/Weyl symmetry constraints, supporting a microscopic AdS/CFT-like duality in this setting. The work also outlines a general strategy for proofs and discusses generalizations to Toda theories and super-Liouville, highlighting the potential universality of boundary–stress-tensor duality on AdS.

Abstract

We consider the Liouville theory in fixed Euclidean AdS background. Expanded near the minimum of the potential the elementary field has mass squared 2 and (assuming the standard Dirichlet b.c.) corresponds to a dimension 2 operator at the boundary. We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor (or the Virasoro generator with the same central charge) on a half-plane or a disc restricted to the boundary. This relation was first observed at the leading semiclassical order (tree-level Witten diagrams in AdS) in arXiv:1902.10536 and here we demonstrate its validity also at the one-loop level. We also discuss arguments that may lead to its general proof.

Paper Structure

This paper contains 20 sections, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Liouville theory on Euclidean AdS$_{2}$ background
  3. Two alternative formulations
  4. One-point and two-point correlation functions
  5. Three-point boundary correlation function
  6. Diagrams with dressed propagators:
  7. Diagrams with vertex tadpoles:
  8. Diagrams with cubic and quartic vertices:
  9. Diagram with one-loop vertex correction:
  10. Four-point boundary correlation function
  11. Diagrams with dressed external propagators:
  12. Diagrams with dressed internal propagator:
  13. Diagrams with one-loop 1PI cubic vertex correction:
  14. Diagrams with 1PI quartic vertex corrections:
  15. AdS$_2$ integrals in disc parametrization
  16. ...and 5 more sections

Figures (1)

  • Figure 1: Numerical estimate of $C^{(4)}_{3,1}$ extracted by $10^{7}$ integrand evaluations for $\theta_{1}=\frac{2\pi}{3}$, $\theta_{2}=-\frac{2\pi}{3}$ and variable $\theta_{3}$ with $|\theta_{3}|<\frac{\pi}{3}$. The horizontal blue line is the expected value $-\frac{56}{27}$.