Heavy-Heavy-Light-Light correlators in Liouville theory

Four point functions of two heavy operators and two perturbatively heavy operators in holographic theories have been argued to be dominated by the identity conformal block. Upon approximating to this block however crossing symmetry of the four point function has been lost. In order to restore crossing symmetry we compute the heavy-heavy-light-light correlator in Liouville theory, at large central charge and to linear order in the conformal weight of the perturbatively heavy operator. We do so via a saddle point approximation to the path integral with operator insertions, and via a conformal block expansion. We find agreement between the two methods, a crossing symmetric correlator, and recover in the Lorentzian regime a singularity that would be associated in holographic theories to locality below the AdS scale.