2024/12/17 by Baptiste Cerclé, Cerclé, Baptiste
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Mathematical Analysis and Transform Methods #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.12657
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 B2-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=B2.