Simply-laced isomonodromy systems

Philip Boalch

doi:10.1007/s10240-012-0044-8

Philip Boalch ¹

¹ École Normale Supérieure et CNRS 45 rue d’Ulm, 75005, Paris France

Publications Mathématiques de l'IHÉS, Volume 116 (2012), pp. 1-68

Abstract

A new class of isomonodromy equations will be introduced and shown to admit Kac–Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painlevé equations, and shows where such Kac–Moody Weyl groups and root systems occur “in nature”. A key point is that one may go beyond the class of affine Kac–Moody root systems. As examples, by considering certain hyperbolic Kac–Moody Dynkin diagrams, we find there is a sequence of higher order Painlevé systems lying over each of the classical Painlevé equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.

Received: 2011-08-01
Accepted: 2012-10-04
Online First: 2012-10-25
Published online: 2012-11-07

MR Zbl

DOI: 10.1007/s10240-012-0044-8

Author's affiliations:

Philip Boalch ¹

¹ École Normale Supérieure et CNRS 45 rue d’Ulm, 75005, Paris France

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Philip Boalch. Simply-laced isomonodromy systems. Publications Mathématiques de l'IHÉS, Volume 116 (2012), pp. 1-68. doi: 10.1007/s10240-012-0044-8

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