Fuchs versus Painlevé

Abstract

We briefly recall the Fuchs-Painlevé elliptic representation of Painlevé 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 Painlevé non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlevé VI has to be generalized to an ``Appellian'' representation of Garnier systems.

Dedicated to the : Special issue on Symmetries and Integrability of Difference Equations, SIDE VII meeting held in Melbourne during July 2006

References in corpus (5)

• The eight-vertex model and Painleve VI
• Elliptic Schlesinger system and Painlev{é} VI
• Holonomy of the Ising model form factors
• Discrete Painlevé equations for a class of \PVI $\tau$-functions given as U(N) averages
• Painleve versus Fuchs

Cited by in corpus (5)

• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• The hypergeometric series for the partition function of the 2-D Ising model
• The anisotropic Ising correlations as elliptic integrals: duality and differential equations
• The lambda extensions of the Ising correlation functions C(M, N)
• On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model