Quivers and difference Painlevé equations

Groups and Symmetries: From the Neolithic Scots to John McKay,
CRM Proceedings and Lecture Notes, Volume 47, 2009, Montreal

We will describe natural 'Lax pairs' for the difference Painlevé equations with affine Weyl symmetry groups of types E6, E7 and E8. This shows that these nonlinear difference equations do indeed arise as symmetries of certain Fuchsian systems of linear differential equations (and so answers a question of H. Sakai). The main work involves studying certain blow-ups of the projective plane in 6,7,8,and 9 points lying on a cubic curve, and yields explicit algebraic descriptions of some complex two-dimensional moduli spaces of logarithmic connections related to the E series of affine Weyl groups.