S. Boukraa, S. Hassani, J. -M. Maillard, B. M. McCoy, J. -A. Weil, N. Zenine
备注：
18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 1905
The sigma form of the Painlev{é} 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{é} 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{é}.
4-dimensional Frobenius manifolds and Painleve' VI
一个偶数维的Frobenius流形具有三Hamilton结构，如果它的谱是最大退化的。我们专注于四维的情况下，并表明，在半简单性的假设下，相应的isomonodromic Fuchsian系统所描述的Painlevé VI $μ$方程。这产生了一个明确的程序相关联的任何半单弗罗贝纽斯流形的三维三哈密尔顿弗罗贝纽斯流形的四。我们进行明确的例子的情况下，Frobenius结构的Hurwitz空间。
The sixth Painleve' equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds
