Fuchs versus Painlev\'e

TL;DR

This paper explores the elliptic and isomonodromic representations of Painlevé VI and their generalizations to anisotropic Ising models, revealing deep connections between elliptic functions, differential operators, and integrable systems.

Contribution

It demonstrates that correlation functions' polynomiality stems from specific second order differential operators and extends this framework to anisotropic Ising models using higher-order systems.

Findings

Correlation functions expressed via elliptic integrals are linked to second order differential operators.

Generalization of these operators to anisotropic Ising models involves third-order systems and Appell functions.

Connections between Painlevé equations, Fuchsian ODEs, and elliptic curves are extended to Garnier systems.

Abstract

We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, $ K$ and $ E$, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator $ L_E$ which has $ E$ as solution (or, for off-diagonal correlations to the direct sum of $ L_E$ and $ d/dt$). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator $ L_E$ being replaced by an isomonodromic system of two third-order linear partial differential operators associated with $ \Pi_1$, the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of $ L_E$). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlev\'e VI has to be generalized to an ``Appellian'' representation of Garnier systems.