Abstract
The Weyl group symmetry W( E k ) 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( E 10) 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( E k ) symmetry to the duality of M-theory on a torus is discussed on the level of string equations of motion.
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