From Klein to Painleve via Fourier, Laplace and Jimbo

Abstract

We will describe a method for constructing explicit algebraic solutions to the sixth Painleve equation, generalising that of Dubrovin-Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL_2(C) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier-Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painleve VI solutions. (In particular this solves a Riemann-Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein's simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin-Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL_2(C). The results of this paper also yield a simple proof of a recent theorem of Inaba-Iwasaki-Saito on the action of Okamoto's affine D4 symmetry group as well as the correct connection formulae for generic Painleve VI equations.