Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symetry (1410.4976v1)

Abstract: A Lam\'e connection is a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection $(E,\nabla)$ over an elliptic curve $X:{y^2=x(x-1)(x-t)}$, $t\not=0,1$, 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 Lam\'e connection, when the elliptic curve $X_t$ varry in the Legendre family, is parametrized by a solution $q(t)$ of the Painlev\'e 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 Lam\'e equation, the question whether Painlev\'e transcendents do have poles is raised.