arXiv subjects

Baptiste Cerclé

Publications and source records attributed to Baptiste Cerclé.

13 recordsLinked to original sources

arXiv2026

On the local conformal structure of Imaginary Liouville theory

Imaginary Liouville theory was recently proposed as a path integral construction of a non-rational conformal field theory (CFT) with central charge less than one (Usciati et al. 2026). In the present work, we establish Ward identities and Belavin-Polyakov-Zamolodchikov differential equations in this framework, assuming a precise conjecture on the decay of the Laplace transform of Imaginary Gaussian multiplicative chaos. These results provide a first step towards a rigorous implementation of the conformal bootstrap program for conformal field theories with central charge lower than one. This work is the first to investigate the intrinsic properties of Imaginary Liouville theory beyond exact computations.

math-ph↗

arXiv2025

Boundary Toda Conformal Field Theory from the path integral

Toda Conformal Field Theories (CFTs hereafter) are generalizations of Liouville CFT where the underlying field is no longer scalar but takes values in a finite-dimensional vector space induced by a complex simple Lie algebra. The goal of this document is to provide a probabilistic construction of such models on all compact hyperbolic Riemann surfaces with or without boundary. To do so, we rely on a probabilistic framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos. In the presence of a boundary, one major difference with Liouville CFT is the existence of non-trivial outer automorphisms of the underlying Lie algebra. The main novelty of our construction is to associate to such an outer automorphism a type of boundary conditions for the field of the theory, leading to the definition of two different classes of models, with either Neumann or Cardy boundary conditions. This in turn has implications on the algebra of symmetry of the models, which are given by $W$-algebras.

math.PR↗

arXiv2025

Around the semi-classical limit of boundary Liouville conformal field theory

Liouville conformal field theory describes a random geometry that fluctuates around a deterministic one: the unique solution of the problem of finding, within a given conformal class, a Riemannian metric with prescribed scalar and geodesic curvatures as well as conical singularities and corners. The level of randomness in Liouville theory is measured by the coupling constant $γ\in(0,2)$, the semi-classical limit corresponding to taking $γ\to0$. Based on the probabilistic definition of Liouville theory, we prove that this semi-classical limit exists and does give rise to this deterministic geometry. At second order this limit is described in terms of a massive Gaussian free field with Robin boundary conditions. This in turn allows to implement CFT-inspired techniques in a deterministic setting: in particular we define the classical stress-energy tensor, show that it can be expressed in terms of accessory parameters (written as regularized derivatives of the Liouville action), and that it gives rise to classical higher equations of motion.

math-ph↗

arXiv2025

W-algebras, Gaussian Free Fields and $\mathfrak{g}$-Dotsenko-Fateev integrals

Based on the intrinsic connection between Gaussian Free Fields and the Heisenberg vertex algebra, we study some aspects of the correspondence between probability theory and $W$-algebras. This is first achieved by providing a construction of the $W$-algebra associated to a complex simple Lie algebra $\mathfrak g$ by means of Gaussian Free Fields. This correspondence in turn allows to translate algebraic statements into actual constraints for free-field correlation functions. This leads to new integrability results for Dotsenko-Fateev integrals associated to $\mathfrak g$, such as Ward identities and the derivation of a new Fuchsian differential equation for deformations of $B_2$-Dotsenko-Fateev integrals arising from the Mukhin-Varchenko conjecture. Along the proof of this statement we also provide new results on representation theory of $W$-algebras such as the description of some singular vectors for the $W$-algebra associated to $\mathfrak g=B_2$.

math-ph↗

arXiv2025

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

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.

math.PR↗

arXiv2025

Higher equations of motion for boundary Liouville Conformal Field Theory from the Ward identities

In this document we prove higher equations of motion at the level 2 for boundary Liouville Conformal Field Theory. As a corollary we present a new derivation of the Belavin-Polyakov-Zamolodchikov differential equations. Our method of proof does not rely on the mating of trees machinery but rather exploits the symmetries of the model through the Ward identities it satisfies. To do so we provide a definition of derivatives of the correlation functions with respect to a boundary insertion which was lacking in the existing literature, and introduce a new notion of descendant fields related to these Ward identities.

math.PR↗

arXiv2025

Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory II: Singular vectors and BPZ equations

This article is the second chapter of a two-part series dedicated to the mathematical study of the higher-spin symmetry enjoyed by the $\mathfrak{sl}_3$ boundary Toda Conformal Field Theory. Namely, based on a probabilistic definition of this model and building on the framework introduced in the first article of this series, we compute some singular vectors of the theory which at the level of correlation functions give rise to higher equations of motion. Under additional assumptions these become BPZ-type differential equations satisfied by the correlation functions, a key feature in the perspective of a rigorous derivation of the structure constants of the theory. Such equations of motion and differential equations were previously unknown in the physics literature.

math.PR↗

arXiv2025

Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory I: Ward identities

This article is the first of a two-part series dedicated to studying the symmetries enjoyed by the probabilistic construction of the $\mathfrak{sl}_3$ boundary Toda Conformal Field Theory. Namely in the present document we show that this model enjoys higher-spin symmetry in the form of Ward identities, both local and global. To do so we consider the $\mathfrak{sl}_3$ Toda theory on the upper-half plane and rigorously define the descendant fields associated to the Vertex Operators. We then show that we can express local as well as global Ward identities based on them, for both the stress-energy tensor and the higher-spin current that encodes this enhanced level of symmetry. This answers a question raised in the physics literature as to whether Toda theory still enjoys higher-spin symmetry in the boundary case. The second part of this series will be dedicated to computing the singular vectors of the theory and showing that they give rise to higher equations of motion as well as, under additional assumptions, BPZ-type differential equations for the correlation functions.

math.PR↗

arXiv2024

Three-point correlation functions in the $\mathfrak{sl}_3$ Toda theory II: the Fateev-Litvinov formula

Toda Conformal Field Theories (CFTs) form a family of two-dimensional CFTs indexed by semisimple and complex Lie algebras. One of their remarkable features is that they are natural generalizations of Liouville CFT that enjoy an enhanced level of symmetry, prescribed by $W$-algebras. They likewise admit a probabilistic formulation in terms of Gaussian Multiplicative Chaos. Based on this probabilistic framework, this second article in a two-part series is dedicated to providing a first step towards integrability of these theories. In this perspective we prove the Fateev-Litvinov formula for a family of three-point correlation functions associated to the $\mathfrak{sl}_3$ Toda CFT. This result is the analog of the celebrated DOZZ formula in Liouville CFT. Our method of proof features techniques inspired by the physics literature together with probabilistic ones that naturally arise within the setting of Toda theories.

math.PR↗

arXiv2022

Probabilistic construction of Toda conformal field theories

Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on two-dimensional conformal field theories, Toda conformal field theories were introduced in the physics literature as a family of two-dimensional conformal field theories that enjoy, in addition to conformal symmetry, an extended level of symmetry usually referred to as W-symmetry or higher-spin symmetry. More precisely Toda conformal field theories provide a natural way to associate to a finite-dimensional simple and complex Lie algebra a conformal field theory for which the algebra of symmetry contains the Virasoro algebra. In this document we use the path integral formulation of these models to provide a rigorous mathematical construction of Toda conformal field theories based on probability theory. By doing so we recover expected properties of the theory such as the Weyl anomaly formula with respect to the change of background metric by a conformal factor and the existence of Seiberg bounds for the correlation functions.

math-ph↗

arXiv2022

Liouville Conformal Field Theory on even-dimensional spheres

Initiated by Polyakov in his 1981 seminal work, the study of two-dimensional Liouville Conformal Field Theory has drawn considerable attention over the past decades. Recent progress in the understanding of conformal geometry in dimension higher than two have naturally led to a generalization of Polyakov formalism to higher dimensions, based on conformally invariant operators: Graham-Jenne-Mason-Sparling operators and the $\mathcal{Q}$-curvature. This document is dedicated to providing a rigorous construction of Liouville Conformal Field Theory on even-dimensional spheres. This is done at the classical level in terms of a generalized \textit{Uniformization} problem, and at the quantum level thanks to a probabilistic construction based on log-correlated fields and Gaussian Multiplicative Chaos. The properties of the objects thus defined are in agreement with the ones expected in the physics literature.

math.PR↗

arXiv2022

Ward identities in the $\mathfrak{sl}_3$ Toda conformal field theory

Toda conformal field theories are natural generalizations of Liouville conformal field theory that enjoy an enhanced level of symmetry. In Toda conformal field theories this higher-spin symmetry can be made explicit, thanks to a path integral formulation of the model based on a Lie algebra structure. The purpose of the present document is to explain how this higher level of symmetry can manifest itself within the rigorous probabilistic framework introduced by R. Rhodes, V. Vargas and the first author. One of its features is the existence of holomorphic currents that are introduced via a rigorous derivation of the Miura transformation. More precisely, we prove that the spin-three Ward identities, that encode higher-spin symmetry, hold in the $\mathfrak{sl}_3$ Toda conformal field theory; as an original input we provide explicit expressions for the descendent fields which were left unidentified in the physics literature. This representation of the descendent fields provides a new systematic method to find the degenerate fields of the $\mathfrak{sl}_3$ Toda (and Liouville) conformal field theory, which in turn implies that certain four-point correlation functions are solutions of an hypergeometric differential equation of the third order.

math.PR↗

arXiv2021

Unit boundary length quantum disk: a study of two different perspectives and their equivalence

The theory of the 2-dimensional Liouville Quantum Gravity, first introduced by Polyakov in his 1981 work has become a key notion in the study of random surfaces. In a series of articles, David, Huang, Kupiainen, Rhodes and Vargas, on the one hand, and Duplantier, Miller and Sheffield on the other hand, investigated this topic in the realm of probability theory, and both provided definitions for fundamentals objects of the theory: the unit area quantum sphere and the unit boundary length quantum disk. In a recent article, Aru, Huang and Sun showed that the definitions given in the case of the sphere coincide. We study here the two different perspectives provided for the unit boundary length quantum disk and show that they define the same probabilistic objects by considering two similar limiting procedures giving rise to them.

math.PR↗