Painleve versus Fuchs

TL;DR

This paper explores special Fuchsian solutions of the Painlevé VI equation, especially in the context of the Ising model, revealing connections to elliptic integrals and algebraic curves.

Contribution

It demonstrates the existence of finite-order Fuchsian differential equations for certain Painlevé VI solutions related to the Ising model, linking them to elliptic integrals and algebraic geometry.

Findings

Fuchsian equations of order N+1 for C(N,N)

Equivalence to the Nth symmetric power of elliptic integral equations

Connections to rational algebraic curves with Riccati structure

Abstract

The sigma form of the Painlev{\'e} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{\'e} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{\'e}.