The geometric Toda equations for noncompact symmetric spaces

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 $T^\C$-bundle $Q^\C$ 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 $(Q^\C,\varphi)$ 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.