Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symetry

Abstract

A Lamé connection is a logarithmic $\mathrm{sl}(2,\mathbb{C})$-connection $(E,\nabla )$ over an elliptic curve $X:\{y^2=x(x-1)(x-t)\}$, $t\ne 0,1$, having a single pole at infinity. When this connection is irreducible, we show that it is invariant by the standart involution and can be pushed down as a logarithmic $\mathrm{sl}(2,\mathbb{C})$-connection over ${\mathbb{P}}^1$ with poles at $0$, $1$, $t$ and $\infty$. Therefore, the isomonodromic deformation $(E_t,{\nabla }_t)$ of an irreducible Lamé connection, when the elliptic curve $X_t$ varry in the Legendre family, is parametrized by a solution $q(t)$ of the Painlevé VI differential equation $P_{VI}$. We compute the variation of the underlying vector bundle $E_t$ along the deformation via Tu moduli map: it is given by another solution $\tilde{q}(t)$ of $P_{VI}$ equation related to $q(t)$ by the Okamoto symetry $s_2{s}_1{s}_2$ (Noumi-Yamada notation). Motivated by the Riemann-Hilbert problem for the classical Lamé equation, the question whether Painlevé transcendents do have poles is raised.

Cited by in corpus (1)

• Ramified covers and tame isomonodromic solutions on curves