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arXiv · 2412.12657

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

Baptiste Cerclé
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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 $\mathfrak g=B_2$.

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BibTeXRIS

Baptiste Cerclé. 2024-12-17. W-algebras, Gaussian Free Fields and $\mathfrak{g}$-Dotsenko-Fateev integrals. https://arxiv.org/abs/2412.12657

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