Simply-laced isomonodromy systems

TL;DR

This paper introduces a new class of isomonodromy equations with Kac-Moody Weyl group symmetries, extending classical Painleve equations and suggesting connections to Hitchin systems and hyperbolic Kac-Moody root systems.

Contribution

It generalizes isomonodromy systems to include hyperbolic Kac-Moody root systems and links these to higher order Painleve systems and geometric structures.

Findings

Kac-Moody Weyl group symmetries in new isomonodromy equations

Higher order Painleve systems from hyperbolic Dynkin diagrams

Conjecture relating to Hilbert schemes of points on Hitchin systems

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 Painleve equations, and shows where such Kac-Moody 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 Painleve systems lying over each of the classical Painleve equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.