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

TL;DR

This paper constructs $W$-algebras from Gaussian Free Fields, translating algebraic structures into constraints on correlation functions, leading to new integrability results and differential equations for Dotsenko-Fateev integrals.

Contribution

It provides a novel construction of $W$-algebras via Gaussian Free Fields and applies this to derive new integrability results and differential equations for Dotsenko-Fateev integrals.

Findings

New construction of $W$-algebras from Gaussian Free Fields.

Derivation of Ward identities for Dotsenko-Fateev integrals.

A new Fuchsian differential equation related to the Mukhin-Varchenko conjecture.

Abstract

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 $\mathfrak g=B_2$.