Three-point correlation functions in the $\mathfrak{sl}_3$ Toda theory I: Reflection coefficients

TL;DR

This paper advances the probabilistic understanding of $rak{sl}_3$ Toda conformal field theory by deriving reflection coefficients and exploring new diffusion process decompositions, contributing to the computation of three-point correlation functions.

Contribution

It provides the first probabilistic derivation of reflection coefficients in $rak{sl}_3$ Toda CFTs and introduces a novel path decomposition for diffusion processes.

Findings

Derived reflection coefficients for $rak{sl}_3$ Toda CFTs.

Introduced a new path decomposition for diffusion processes.

Analyzed joint tail expansions of Gaussian Multiplicative Chaos measures.

Abstract

Toda Conformal Field Theories (CFTs) form a family of 2d CFTs indexed by semisimple and complex Lie algebras. They are natural generalizations of the Liouville CFT in that they enjoy an enhanced level of symmetry encoded by W-algebras. These theories can be rigorously defined using a probabilistic framework that involves the consideration of correlated Gaussian Multiplicative Chaos measures. This document provides a first step towards the computation of a class of three-point correlation functions, that generalize the celebrated DOZZ formula and whose expressions were predicted in the physics literature by Fateev-Litvinov, within the probabilistic framework associated to the $\mathfrak{sl}_3$ Toda CFT. Namely this first article of a two-parts series is dedicated to the probabilistic derivation of the reflection coefficients of general Toda CFTs, which are essential building blocks in the understanding of Toda correlation functions. Along the computations of these reflection coefficients a new path decomposition for diffusion processes in Euclidean spaces, based on a suitable notion of minimum and that generalizes the celebrated one-dimensional result of Williams, will be unveiled. As a byproduct we describe the joint tail expansion of correlated Gaussian Multiplicative Chaos measures together with an asymptotic expansion of class one Whittaker functions.