W(E10) symmetry, M-theory and Painlevé equations

The Weyl group symmetry W(Ek) is studied from the points of view of the E-strings, Painlevé equations and U-duality. We give a simple reformulation of the elliptic Painlevé equation in such a way that the hidden symmetry W(E10) is manifestly realized. This reformulation is based on the birational geometry of the del Pezzo surface and closely related to Seiberg–Witten curves describing the E-strings. The relation of the W(Ek) symmetry to the duality of M-theory on a torus is discussed on the level of string equations of motion.