Probabilistic construction of Toda conformal field theories

TL;DR

This paper rigorously constructs Toda conformal field theories using probability theory, confirming key properties like Weyl anomaly and Seiberg bounds, thus bridging physics intuition with mathematical rigor.

Contribution

It provides the first rigorous probabilistic construction of Toda conformal field theories based on path integral formulation.

Findings

Confirmed Weyl anomaly formula for Toda CFTs

Established existence of Seiberg bounds for correlation functions

Reconciled physical and mathematical perspectives on Toda CFTs

Abstract

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.