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

Baptiste Cerclé

TL;DR

The paper develops a probabilistic framework for $ rak{sl}_3$ Toda CFTs by deriving reflection coefficients through a generalized Williams path decomposition in Weyl chambers. It builds a bridge between diffusion processes conditioned by generalized minima, Gaussian Free Fields, and Gaussian Multiplicative Chaos to obtain asymptotic expansions for Toda Vertex Operators and class-one Whittaker functions. The main contributions include a probabilistic formulation of Toda reflection coefficients, their expression in terms of GMC tails and Gamma-function factors, and a coherent scheme that parallels Liouville results while accommodating the richer Weyl-group structure of Toda theories. This advancement lays the groundwork for explicit probabilistic computations of three-point Toda correlators and reinforces the connection between stochastic processes in symmetric spaces and conformal field theory, with implications for the conformal bootstrap in higher-rank models.

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.

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

TL;DR

The paper develops a probabilistic framework for Toda CFTs by deriving reflection coefficients through a generalized Williams path decomposition in Weyl chambers. It builds a bridge between diffusion processes conditioned by generalized minima, Gaussian Free Fields, and Gaussian Multiplicative Chaos to obtain asymptotic expansions for Toda Vertex Operators and class-one Whittaker functions. The main contributions include a probabilistic formulation of Toda reflection coefficients, their expression in terms of GMC tails and Gamma-function factors, and a coherent scheme that parallels Liouville results while accommodating the richer Weyl-group structure of Toda theories. This advancement lays the groundwork for explicit probabilistic computations of three-point Toda correlators and reinforces the connection between stochastic processes in symmetric spaces and conformal field theory, with implications for the conformal bootstrap in higher-rank models.

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 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.
Paper Structure (52 sections, 33 theorems, 262 equations, 6 figures)

This paper contains 52 sections, 33 theorems, 262 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Toda Conformal Field Theories
  3. On a reflection principle: Williams path decomposition and class one Whittaker functions
  4. Williams path decomposition
  5. Class one Whittaker functions
  6. Reminders on reflection groups
  7. Presentation of the main results
  8. A path decomposition for diffusions in Euclidean spaces
  9. Vertex Operators: asymptotic expansions and reflection coefficients
  10. Class-one Whittaker functions and Gaussian Multiplicative Chaos
  11. Overview of the article and perspectives
  12. Organization of the paper
  13. Some future directions of work
  14. Path decomposition and a reflection principle
  15. A path decomposition
  16. ...and 37 more sections

Key Result

Theorem 1

For $\gamma\in[1,\sqrt2)$ and $\mathfrak{g}=\mathfrak{sl}_3$, assume that $\alpha_1$ is as prescribed in FaLi1. Then as soon as $\alpha_0,\alpha_1$ and $\alpha_\infty$ satisfy the Seiberg bounds Toda_construction:

Figures (6)

  • Figure 1: The $A_2$ root system
  • Figure 2: Planar Brownian motion with drift $\nu$
  • Figure 3: Decomposition of the path associated with $A_2$
  • Figure 4: Planar Brownian motion with drift $\nu$ and its path decompositions associated to $\mathcal{D}_n$, $n=4,6,8$.
  • Figure 5: The three different behaviours of the process $\bm{\mathcal{B}}^\nu$ when $\bm x\rightarrow+\infty$.
  • ...and 1 more figures

Theorems & Definitions (47)

  • Theorem 1: Adapted from Theorem 1.2 in Toda_correl2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2: Corollary 1.3 in Toda_correl2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Proposition 2.3
  • ...and 37 more