Wild mapping class groups
~~~~~~~~~~~~~~~~~~~~~~~~~
First some examples of nonlinear (Poisson) braid group actions on
spaces of Stokes data studied by Cecotti-Vafa and Dubrovin will be
described. Next I will explain how to relate this to the quantum Weyl
groups of Lusztig, Soibelman and Kirillov-Reshetikhin. Then many
extensions of this setup will be discussed, leading to the nonlinear
algebraic Poisson actions of the wild mapping class groups on the wild
character varieties. This simultaneously generalises the original
examples and the usual actions of mapping class groups on character
varieties of Riemann surfaces.
Global Weyl groups
~~~~~~~~~~~~~~~~~~
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 on Riemann surfaces (in contrast to the
usual, local, understanding of affine Kac-Moody algebras).
For example the affine Weyl groups arise geometrically when one does
meromorphic gauge transformations of a connection on a disk.
The global Weyl groups fit these local pieces together (and involve
non-affine Kac-Moody Weyl groups).