Irregular connections and Kac-Moody root systems

TL;DR

This paper links moduli spaces of irregular connections on the Riemann sphere to Nakajima quiver varieties, establishing connections with Kac-Moody root systems and revealing numerous isomorphisms between different moduli spaces.

Contribution

It introduces a novel association between irregular connection moduli spaces and Kac-Moody root systems, expanding the understanding of their symmetries and isomorphisms.

Findings

Identification of moduli spaces with Nakajima quiver varieties

Association of Kac-Moody root systems to irregular connections

Extension of Crawley-Boevey's results on stable connections

Abstract

Some moduli spaces of irregular connections on the trivial bundle over the Riemann sphere will be identified with Nakajima quiver varieties. In particular this enables us to associate a Kac-Moody root system to such connections (yielding many isomorphisms between such moduli spaces, via the reflection functors for the corresponding Weyl group). The possibility of 'reading' a quiver in different ways also yields numerous isomorphisms between such moduli spaces, often between spaces of connections on different rank bundles and with different polar divisors. Finally some results of Crawley-Boevey on the existence of stable connections will be extended to this more general context.