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

Heavy-Heavy-Light-Light correlators in Liouville theory