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

Abstract

A Lamé connection is a logarithmic -connection over an elliptic curve , , having a single pole at infinity. When this connection is irreducible, we show that it is invariant under the standard involution and can be pushed down to a logarithmic -connection on with poles at , , and . Therefore the isomonodromic deformation of an irreducible Lamé connection, when the elliptic curve varies in the Legendre family, is parametrized by a solution of the Painlevé VI differential equation . The variation of the underlying vector bundle along the deformation is computed in terms of the Tu moduli map: it is given by another solution of , which is related to by the Okamoto symmetry (Noumi–Yamada notation). Motivated by the Riemann–Hilbert problem for the classical Lamé equation, we raise the question whether the Painlevé transcendents do have poles. Some of these results were announced in [6].