# Heavy-Heavy-Light-Light correlators in Liouville theory

## TL;DR

This paper computes specific four-point correlators in Liouville theory using path integral and conformal block methods, revealing independence from operator separation and absence of short-distance singularities, with implications for AdS duality.

## Contribution

It introduces a novel calculation of HHLL correlators in Liouville theory, combining path integral and conformal block approaches, and uncovers unique separation-independent features.

## Key findings

- Leading-order HHLL correlator is separation-independent at equal times.
- No short-distance singularity occurs when light operators approach each other.
- Vacuum block exchange is absent, indicating non-singular intermediate states.

## Abstract

We compute four-point functions of two heavy and two "perturbatively heavy" operators in the semiclassical limit of Liouville theory on the sphere. We obtain these "Heavy-Heavy-Light-Light" (HHLL) correlators to leading order in the conformal weights of the light insertions in two ways: (a) via a path integral approach, combining different methods to evaluate correlation functions from complex solutions for the Liouville field, and (b) via the conformal block expansion. This latter approach identifies an integral over the continuum of normalizable states and a sum over an infinite tower of lighter discrete states, whose contribution we extract by analytically continuing standard results to our HHLL setting. The sum over this tower reproduces the sum over those complex saddlepoints of the path integral that contribute to the correlator. Our path integral computations reveal that when the two light operators are inserted at equal time in radial quantization, the leading-order HHLL correlator is independent of their separation, and more generally that at this order there is no short-distance singularity as the two light operators approach each other. The conformal block expansion likewise shows that in the discrete sum short-distance singularities are indeed absent for all intermediate states that contribute. In particular, the Virasoro vacuum block, which would have been singular at short distances, is not exchanged. The separation-independence of equal-time correlators is due to cancelations between the discrete contributions. These features lead to a Lorentzian singularity that, in conformal theories with anti-de Sitter (AdS) duals, would be associated to locality below the AdS scale.