QM Research Seminar: Singular Nakajima quiver varieties associated with extended Dynkin quivers.

Speaker: Gard Olav Helle (University of Southern Denmark)

Abstract:

The McKay correspondence sets up a bijection between the finite subgroups of SU(2) and the Dynkin graphs of type ADE.
In 1987 Kronheimer exploited this to give a construction of the minimal resolution of the quotient singularity \C^2/G using the method of hyper-Kähler reduction. In fact, he produced a family of hyper-Kähler 4-manifolds, the ALE spaces, where each member is diffeomorphic to the minimal resolution. This work can be rephrased in the language of Nakajima quiver varieties; the ALE spaces are the nonsingular quiver varieties associated with the extended Dynkin quivers and the corresponding minimal imaginary root.

In this talk I will introduce hyper-Kähler reduction and Nakajima quiver varieties with the aim of presenting a recent result providing a classification of the singularities in the above-described family of quiver varieties. This topic is intimately linked with instanton gauge theory, and I hope to elaborate on this relation along the way.