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

## TL;DR

This paper investigates boundary correlators in Liouville theory on Euclidean AdS$_2$, providing evidence that they match the correlators of the boundary stress tensor, extending previous semiclassical results to one-loop order.

## Contribution

It demonstrates the equivalence of boundary correlators in Liouville theory and boundary stress tensor correlators at one-loop level in AdS$_2$, supporting a conjecture beyond leading order.

## Key findings

- Boundary correlators match the boundary stress tensor correlators at one-loop order.
- The relation holds beyond the semiclassical approximation, confirmed at quantum level.
- Arguments are discussed for a potential general proof of the conjecture.

## 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.