Toda frames, harmonic maps and extended Dynkin diagrams

Emma Carberry and Katharine Turner

Abstract

We prove that all immersions of a genus one surface into \(G/T\) possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. Here \(G\) is any simple real Lie group (not necessarily compact), \(T\) is a Cartan subgroup and the \(k\)-symmetric space structure on \(G/T\) is induced from the Coxeter automorphism. We provide necessary and sufficient conditions for the existence of a Toda frame for a harmonic map into \(G/T\) and describe those \(G/T\) to which the theory applies in terms of involutions of extended Dynkin diagrams.

Keywords: Harmonic map, Toda frame, Dynkin diagram.