arXiv subjects

Nathan Huguenin

Publications and source records attributed to Nathan Huguenin.

3 recordsLinked to original sources

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

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↗