Hyperkahler manifolds from connections and quivers


Philip Boalch

11:00:00 - 11:50:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

In string theory one is apparently supposed to replace a (Feynman) graph by a (Riemann) surface, to pass from a perturbative picture to a nonperturbative one. In the theory of hyperkahler manifolds there is a class of examples attached to graphs (and some data on the graph)—the Nakajima quiver varieties, and a class of examples attached to Riemann surfaces (and some data on the surface, to specify the boundary conditions)—the wild Hitchin spaces.

I will talk about these “nonperturbative” hyperkahler manifolds attached to surfaces, and how in some cases they are related to graphs. This yields a new theory of “multiplicative quiver varieties”, and enables us to extend work of Okamoto and Crawley-Boevey to see the appearance of many non-affine Kac-Moody Weyl groups and root systems in the theory of connections/Higgs bundles on Riemann surfaces (in contrast to the usual, local, understanding of affine Kac-Moody algebras, in terms of loop algebras).