Fusion asymptotics for Liouville correlation functions

TL;DR

This paper rigorously analyzes Liouville Conformal Field Theory using probabilistic methods, deriving fusion estimates for four-point functions, confirming predictions from physics, and extending results to boundary cases.

Contribution

It introduces a probabilistic framework for LCFT, computes fusion estimates consistent with the conformal bootstrap, and extends results to boundary LCFT cases.

Findings

Fusion estimates align with conformal bootstrap predictions.

Exact formulas and asymptotics for negative GMC moments.

Extension of results to boundary LCFT scenarios.

Abstract

David-Kupiainen-Rhodes-Vargas introduced a probabilistic framework based on the Gaussian Free Field and Gaussian Multiplicative Chaos in order to make sense rigorously of the path integral approach to Liouville Conformal Field Theory (LCFT). We use this setting to compute fusion estimates for the four-point correlation function on the Riemann sphere, and find that it is consistent with predictions from the framework of theoretical physics known as the conformal bootstrap. This result fits naturally into the famous KPZ conjecture which relates the four-point function to the expected density of points around the root of a large random planar map weighted by some statistical mechanics model. From a purely probabilistic point of view, we give non-trivial results on negative moments of GMC. We give exact formulae based on the DOZZ formula in the Liouville case and asymptotic behaviours in the other cases, with a probabilistic representation of the limit. Finally, we show how to extend our results to boundary LCFT, treating the cases of the fusion of two boundary or bulk insertions as well as the absorption of a bulk insertion on the boundary.