W‐algebras, Gaussian free fields, and g$\mathfrak {g}$‐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 ‐algebras. This is first achieved by providing a construction of the ‐algebra associated to a complex simple Lie algebra 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 , such as Ward identities and the derivation of a new Fuchsian differential equation satisfied by deformations of ‐Dotsenko–Fateev that arise from the Mukhin–Varchenko conjecture. Along the proof of this statement, we also provide new results on representation theory of ‐algebras such as the description of some singular vectors for the ‐algebra associated to .