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

TL;DR

This paper explores the W(E_10) symmetry in relation to M-theory, Painleve equations, and E-strings, revealing a reformulation of the elliptic Painleve equation that makes the symmetry explicit through geometric and string-theoretic frameworks.

Contribution

It provides a new reformulation of the elliptic Painleve equation that explicitly exhibits W(E_10) symmetry using birational geometry and Seiberg-Witten curves.

Findings

Explicit realization of W(E_10) symmetry in Painleve equations

Connection between symmetry, M-theory duality, and string equations

Reformulation based on del Pezzo surface geometry

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.