From Klein to Painlevé via Fourier, Laplace and Jimbo

Abstract

Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé 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 Painlevé 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 InabaIwasaki-Saito on the action of Okamoto's affine D 4 symmetry group as well as the correct connection formulae for generic Painlevé VI equations. IntroductionKlein's quartic curveis of genus three and has the maximum possible number 84(g − 1) = 168 of holomorphic automorphisms. Klein found these automorphisms explicitly (in terms of 3 × 3 matrices). They constitute Klein's simple group K ⊂ PGL 3 (C) which is isomorphic to PSL 2 (7). Lifting to GL 3 (C) there is a two-fold covering group K ⊂ GL 3 (C) of order 336 which is a complex reflection group-there are complex reflections (1) r 1 , r 2 , r 3 ∈ GL 3 (C) which generate K. (Recall a pseudo-reflection is an automorphism of the form "one plus rank one", a complex reflection is a pseudo-reflection of finite order and a complex reflection group is a finite group generated by complex reflections. Here, each generator r i has order two-as for real reflections.) Using the general tools to be described in this paper we will construct, starting from the Klein complex reflection group K, another algebraic curve with affine equation (2) F (t, y) = 0given by a polynomial F with integer coefficients. This curve will be a seven-fold cover of the t-line branched only at 0, 1, ∞ and such that the function y(t), defined implicitly by (2), solves the Painlevé VI differential equation. One upshot of this will be to construct an explicit rank three Fuchsian system of linear differential equations with four singularities (at 0, t, 1, ∞, for some t) on P 1 , and with 1 2 PHILIP BOALCH monodromy group equal to K in its natural representation (so the monodromy around each of the finite singularities 0, t, 1 is a generating reflection).In general the construction of linear differential equations with finite monodromy group is reasonably straightforward provided one works with rigid representations of the monodromy groups. In our situation the representation is minimally non-rigid; it lives in a complex two-dimensional moduli space, and this is the basic reason the (second order) Painlevé VI equati...

An ergodic study of Painlevé VI

“…The logarithm of this rate will give the entropy of the Poincaré return map γ * . Recently several authors [4,5,9,13,14,25,26] have been interested in finding algebraic solutions, which must have only finitely many branches under the analytic continuations along all loops in Z. On the other hand, in this article we will be concerned with those solutions which are finitely many-valued along a fixed single loop.…”

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

“…We note that the parametrization that was adopted in [29] to write system (3.9) explicitly in terms of y where y(t) is a solution of P 6 is not unique. Alternate parameterizations have been identified by Boalch [3]- [4] in his studies of P 6 . Since this system can be mapped to the irregular 3 × 3 Lax pair of [18] and [29] via the generalized Laplace transform, see equation (3.12) above, it follows that these parameterizations are equivalent to system (3.9) up to a gauge transformation.…”

“…The Fuchs-Garnier pairs associated with each Painlevé equation play a very important role in the theory and applications of the Painlevé equations. Nowadays, as a result of the intensive studies of the Painlevé equations, many different Fuchs-Garnier pairs have been derived [3,4,12,18,20,22,24,26,27,30,32,33,41]. The methods and ideas used in these derivations vary widely.…”

Isomonodromy, Painlevé transcendents and scattering off of black holes

“…Here we list the relevant results in the asymptotics of Painlevé VI as studied by [28] and [46] and listed in [52]. The problem was also considered in [53].…”

“…The dependence on φ has been considered in a number of papers [28,46], and is explicit at the Painlevé singular points t = 0, 1, ∞. Let us take t i → 1 as the asymptotic point for definiteness.…”