W(E_10) Symmetry, M-Theory and Painleve Equations

Abstract

The Weyl group symmetry W(E_k) is studied from the points of view of the E-strings, Painleve equations and U-duality. We give a simple reformulation of the elliptic Painleve 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.

14 pages, dedicated to the memory of Sung-Kil Yang, references added

References in corpus (4)

• Hyperbolic Kac-Moody Algebras and Chaos in Kaluza-Klein Models
• Colliding String Waves and Duality
• A Mysterious Duality
• The M5-brane on K3 and del Pezzo's and multi-loop string amplitudes

Cited by in corpus (5)

• Cluster integrable systems, q-Painleve equations and their quantization
• Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains
• Quantum spectral problems and isomonodromic deformations
• Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
• On the lattice-geometry and birational group of the six-point multi-ratio equation