W-algebras, Gaussian Free Fields and $\mathfrak{g}$-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 $W$-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 $\mathfrak{g}=B_2$.