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Probabilistic approach to Toda conformal field theories
Probabilistic approach to Toda conformal field theories
Toda conformal field theories form a family of two-dimensional quantum field theories indexed by simple and complex Lie algebras. One of their remarkable features is that they enjoy, in addition to conformal invariance, an enhanced level of symmetry referred to as higher-spin symmetry and encoded by W-algebras. They provide natural generalizations of Liouville theory in this context.
The mathematical study of these theories relies on a probabilistic definition of these models, allowing to rigorously recover some of the predictions made in the physics literature such as the derivation of a family of structure constants. Conversely this also leads to new probabilistic results.
Three-point correlation functions in the 𝔰𝔩3 Toda theory II: the Fateev-Litvinov formula, to appear in the Journal of the European Mathematical Society
Three-point correlation functions in the 𝔰𝔩3 Toda theory II: the Fateev-Litvinov formula, to appear in the Journal of the European Mathematical Society
Building on previous works on the topic, this article is dedicated to providing a first step towards integrability of Toda conformal field theories. In this perspective we prove the Fateev-Litvinov formula for a family of three-point correlation functions associated to the 𝔰𝔩3 Toda CFT. This result is the analog of the celebrated DOZZ formula in Liouville CFT. 73 pages.
Three-point correlation functions in the 𝔰𝔩3 Toda theory I: Reflection coefficients, Probability Theory and Related Fields
Three-point correlation functions in the 𝔰𝔩3 Toda theory I: Reflection coefficients, Probability Theory and Related Fields
In this article we propose a probabilistic derivation of the reflection coefficients of Toda conformal field theories, 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, is 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. 53 pages.
Ward identities in the 𝔰𝔩3 Toda conformal field theory, with Yichao Huang, Communications in Mathematical Physics
Ward identities in the 𝔰𝔩3 Toda conformal field theory, with Yichao Huang, Communications in Mathematical Physics
Ward identities are key in the understanding of Toda conformal field theories. Indeed they encode the symmetries of the model and translate them into actual constraints on the correlation functions. In this article we show that such identities are valid for the probabilistic model associated to the 𝔰𝔩3 Toda conformal field theory. As a consequence we also derive BPZ-type differential equations for a family of correlation functions of the theory. 51 pages.
Probabilistic construction of Toda conformal field theories, with Rémi Rhodes and Vincent Vargas, Annales Henri Lebesgue
Probabilistic construction of Toda conformal field theories, with Rémi Rhodes and Vincent Vargas, Annales Henri Lebesgue
In this article we initiate the probabilistic study of Toda conformal field theories. To do so we provide a mathematically rigorous construction of the Toda theories on the sphere using probability theory, and more precisely Gaussian Free Fields and Gaussian Multiplicative Chaos. 30 pages.
The manuscript of my PhD thesis "A probabilistic approach to Toda conformal field theories" gathers all these works in a unified document. It is available here.
Probabilistic approach to boundary Toda conformal field theories
Probabilistic approach to boundary Toda conformal field theories
We have considered above the case of Toda theories based on the Riemann sphere. But the study of Toda conformal field theories on a Riemann surface with boundary reveals some additional features related to the representation theory of W-algebras. In addition of being more challenging at both the technical and conceptual level, boundary Toda theories are much less understood in the physics literature than their counterparts on the sphere.
Higher-spin symmetry in the 𝔰𝔩3 boundary Toda conformal field theory II: singular vectors and BPZ equations, with Nathan Huguenin, Preprint
Higher-spin symmetry in the 𝔰𝔩3 boundary Toda conformal field theory II: singular vectors and BPZ equations, with Nathan Huguenin, Preprint
This is the second and last article in a series dedicated to the mathematical understanding of the symmetries enjoyed by the probabilistic construction of the 𝔰𝔩3 boundary Toda Conformal Field Theory. We describe here some singular vectors of the theory and show that they give rise to higher equations of motion as well as under additional assumptions BPZ-type differential equations. Such results are new compared to the existing physics literature. 60 pages.
Higher-spin symmetry in the 𝔰𝔩3 boundary Toda conformal field theory I: Ward identities, with Nathan Huguenin, Preprint
Higher-spin symmetry in the 𝔰𝔩3 boundary Toda conformal field theory I: Ward identities, with Nathan Huguenin, Preprint
This is the first article of a two-part series dedicated to studying the symmetries enjoyed by the probabilistic construction of the 𝔰𝔩3 boundary Toda Conformal Field Theory. Namely we show in this first chapter that this model enjoys higher-spin symmetry in the form of Ward identities, a fact that was previously unkown in the physics literature. 47 pages.
We provide in this document a probabilistic construction of Toda CFTs on a Riemann surface with or without boundary. In contrast with Liouville CFT and the definition of Toda CFT in the closed case, there may exist non-trivial automorphisms of the associated W-algebra, corresponding to different boundary conditions for the field of the theory. Based on this particular feature, we define different classes of models based on Gaussian Free Fields and Gaussian Multiplicative Chaos. 47 pages.
Higher equations of motion for boundary Liouville Conformal Field Theory from the Ward identities, Communications in Mathematical Physics
Higher equations of motion for boundary Liouville Conformal Field Theory from the Ward identities, Communications in Mathematical Physics
In the boundary case, singular vectors are generically not null vectors and give rise to higher equations of motion instead of BPZ-type differential equations like in the closed case. We prove this prediction for Liouville Conformal Field Theory by exploiting the symmetries of the model through the Ward identities it satisfies. As a corollary a new derivation of BPZ-type differential equations and provide a definition of derivatives (and more generally descendant fields) of the correlation functions with respect to a boundary insertion which was lacking in the existing literature. 40 pages.
Between Algebraic and Probabilistic approaches to conformal field theory
Between Algebraic and Probabilistic approaches to conformal field theory
Many different approaches have been developed to study the notion of two-dimensional Conformal Field Theory (CFT). One successful perspective is that of Vertex Operator Algebras as it provides a natural setting where to make sense of some of the key tools used in the physics. Building a bridge between this viewpoint and that based on probability theory described above would allow to translate results from one community to the other.
Based on the intrinsic connection between Gaussian Free Fields and the Heisenberg Vertex Operator Algebra, we study some aspects of the correspondence between probability theory and W -algebras. This leads to new integrability results concerning Dotsenko-Fateev and Selberg integrals that arise from the Mukhin-Varchenko conjecture, as well as statements concerning representation theory of W-algebras. 60 pages.
Other perspectives on Liouville theory
Other perspectives on Liouville theory
We gather here some earlier works on Liouville theory, concerned with other perspectives on this model.
In this article we initiate the mathematical study of Liouville Conformal Field Theory in dimension higher than two. To do so we rely on conformally invariant operators: Graham-Jenne-Mason-Sparling operators and the Q-curvature. We also describe a generalized uniformization problem in this higher-dimensional setting. 32 pages.
Unit boundary length quantum disk: a study of two different perspectives and their equivalence, ESAIM: Probability and Statistics
Unit boundary length quantum disk: a study of two different perspectives and their equivalence, ESAIM: Probability and Statistics
Liouville theory can be studied based on probability theory in many ways. We study here two different perspectives for an object that naturally arises in this setting: the unit boundary length quantum disk. Namely we show that these two approaches, based either on the mating-of-trees approach of Duplantier-Miller-Sheffield or on the probabilistic interpretation of the path integral by David-Kupiainen-Rhodes-Vargas, actually describe the same object. 26 pages.
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