Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlev\'e Equations

TL;DR

This paper introduces a new class of holomorphic symplectic manifolds associated with quivers with multiplicities, generalizes reflection functors to relate to non-symmetric Kac-Moody Weyl groups, and connects these to moduli spaces of meromorphic connections and Painlevé equations.

Contribution

It extends Nakajima's quiver varieties to include multiplicities, relates these to non-symmetric Kac-Moody Weyl groups, and describes moduli spaces of connections as these new manifolds.

Findings

Constructed holomorphic symplectic manifolds for quivers with multiplicities.

Generalized reflection functors relate to non-symmetric Kac-Moody Weyl groups.

Connected moduli spaces of meromorphic connections to these manifolds and Painlevé equations.

Abstract

To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann sphere are described as our manifolds. In our picture, the list of Dynkin diagrams for Painlev\'e equations is slightly different from (but equivalent to) Okamoto's.