Abstract

Following Kronheimer’s construction of the ALE spaces from the ADE affine Dynkin graphs, and Kronheimer-Nakajima’s subsequent extension of the ADHM construction, a large class of hyperkahler manifolds attached to graphs emerged, known as “quiver varieties”. Nakajima has shown they play a central role in representation theory. If the underlying graph is of a special type it turns out that the corresponding quiver varieties have natural partial compactifications, which also admit complete hyperkahler metrics. They arise as spaces of solutions to Hitchin’s equations on Riemann surfaces, with wild boundary conditions. (They were constructed in work with Biquard published in 2004). The class of graphs for which this works are known as “supernova” graphs and includes all the complete multipartite graphs. In particular the square and the triangle are supernova graphs, and so some gravitational instantons arise in this way. In this talk I will review this story focussing on specific examples and recent developments such as the algebraic construction of the underlying holomorphic symplectic manifolds (the “wild character varieties”).