Loop equations from differential systems

Abstract

To any differential system $d\Psi =\Phi \Psi$ where $\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\Phi$ is a Lie algebra $\mathfrak{g}$ valued 1-form on a Riemann surface $\Sigma$, is associated an infinite sequence of "correlators" $W_n$ that are symmetric $n$-forms on ${\Sigma }^n$. The goal of this article is to prove that these correlators always satisfy "loop equations", the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.

arXiv comment20 pages

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• Weighted Hurwitz numbers and topological recursion: an overview
• Integrable differential systems of topological type and reconstruction by the topological recursion
• The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition
• From the quantum geometry of Fuchsian systems to conformal blocks of W-algebras