The geometric Toda equations for noncompact symmetric spaces

Abstract

This paper has two purposes. The first is to classify all those versions of the Toda equations which govern the existence of $\tau$-primitive harmonic maps from a surface into a homogeneous space $G/T$ for which $G$ is a noncomplex noncompact simple real Lie group, $\tau$ is the Coxeter automorphism which Drinfel'd \& Sokolov assigned to each affine Dynkin diagram, and $T$ is the compact torus fixed pointwise by $\tau$. Here $\tau$ may be either an inner or an outer automorphism. We interpret the Toda equations over a compact Riemann surface $\Sigma$ as equations for a metric on a holomorphic principal $T^\mathbb{C}$-bundle $Q^\mathbb{C}$ over $\Sigma$ whose Chern connection, when combined with holomorphic field $\varphi$, produces a $G$-connection which is flat precisely when the Toda equations hold. The second purpose is to establish when stability criteria for the pair $(Q^\mathbb{C},\varphi)$ can be used to prove the existence of solutions. We classify those real forms of the Toda equations for which this pair is a principal pair and we call these \emph{totally noncompact} Toda pairs: stability theory then gives algebraic conditions for the existence of solutions. Every solution to the geometric Toda equations has a corresponding $G$-Higgs bundle. We explain how to construct this $G$-Higgs bundle directly from the Toda pair and show that Baraglia's cyclic Higgs bundles arise from a very special case of totally noncompact cyclic Toda pairs.