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 kinds, 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 LE which has E as solution (or for off-diagonal correlations to the direct sum of LE 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 LE is replaced by an isomonodromic system of two third-order linear partial differential operators associated with Π1, 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 LE). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlevé nonlinear ODEs, Fuchsian linear ODEs and elliptic curves. In particular, the elliptic representation of Painlevé VI has to be generalized to an ‘Appellian’ representation of Garnier systems.