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Three-point correlation functions in the \(\mathfrak {sl}_3\) Toda theory I: reflection coefficients

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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 Whitakker functions.

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Notes

  1. Reflection groups will be implicitly taken finite in what follows.

  2. Different choices of simple roots are of course possible, but will be related by conjugation under W: for any two simple systems there is a \(s\in W\) such that \(s e_i=e'_i\) (up to reordering the roots). Such a simple system always exists [42, Section 1.3] and \(r=\dim {{\textbf {V}}} \).

  3. To define the process this hypothesis may be relaxed by taking h excessive, by which is meant that for any x in \({{\textbf {V}}} \) and all time t, \(\mathbb {E}_x\left[ h({{\textbf {X}}} _t)\right] \;\leqslant \;h(x)\) with \(\lim \limits _{t\rightarrow 0}\mathbb {E}_x\left[ h({{\textbf {X}}} _t)\right] = h(x)\).

  4. We omit the dependence in \({{\textbf {M}}} \) for such a process in order to keep the notations concise.

  5. The assumption that \(\gamma <\sqrt{2}\) is the optimal one to ensure that all the GMC measures for \(1\;\leqslant \;i\;\leqslant \;r\) are well-defined (at least in the subcritical regime). The fact that we need to assume that \(\gamma <\sqrt{2}\) instead of the usual bound \(\gamma <2\) stems from the fact that the longest roots have squared norm 2. To recover the usual range of values one would need to rescale \(\gamma \) by a multiplicative factor \(\sqrt{2}\) to take into account the fact that some roots are not normalized.

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Acknowledgements

The author acknowledges that this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant Agreement No. 725967.

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  1. Laboratoire de Mathématiques d’Orsay, Faculté des Sciences d’Orsay, Université Paris-Saclay, Bâtiment 307, 91405, Orsay Cedex, France

    Baptiste Cerclé

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Cerclé, B. Three-point correlation functions in the \(\mathfrak {sl}_3\) Toda theory I: reflection coefficients. Probab. Theory Relat. Fields 188, 89–158 (2024). https://doi.org/10.1007/s00440-023-01219-3

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