Fuchs versus Painlevé

Details Fuchs versus Painlevé Fuchs versus Painlevé

2007-01-05

arxiv.org/abs/math-ph/0701014v2

Physics

Mathematical Physics

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

Scientific paper

10.1088/1751-8113/40/42/S06

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.

Affiliated with

Also associated with

No associations

Rating

Fuchs versus Painlevé does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Fuchs versus Painlevé, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Fuchs versus Painlevé will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-16493