The geometric Toda equations for noncompact symmetric spaces (2406.02323v1)

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 and $T$ is a maximal compact torus, i.e., a maximal torus inside a maximal compact subgroup $H < G$. Here $\tau$ is the Coxeter automorphism which Drinfel'd & Sokolov assigned to each affine Dynkin diagram. This allows $\tau$ to be either an inner or an outer automorphism. We show that, up to equivalence, the real forms $G<G^\mathbb{C}$ which are compatible with $\tau$ can be classified using a simple labelling of the corresponding affine diagram. The second purpose is to establish when stability criteria can be used to prove the existence of solutions. 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 $\Sigma$. The corresponding Chern connection, when combined with a holomorphic field $\varphi$, produces a $G$-connection which is flat precisely when the Toda equations hold. We call the pair $(Q^\mathbb{C},\varphi)$ a Toda pair. We classify those real forms of the Toda equations for which the Toda pair is a principal pair (in the sense of Bradlow et al.) and we call these totally noncompact Toda pairs. Using the stability theory for principal pairs we prove that for totally noncompact cyclic Toda pairs $(Q^\mathbb{C},\varphi)$ the corresponding Toda equations always admit 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.