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The geometric Toda equations for noncompact symmetric spaces

Ian McIntosh

TL;DR

The work classifies all noncompact real forms compatible with Coxeter automorphisms that yield Toda-type equations for $\tau$-primitive harmonic maps into $G/T$, and provides a geometric reinterpretation in terms of holomorphic $T^{\mathbb{C}}$-bundles and holomorphic data $\varphi$ on a compact surface $\Sigma$. It develops a comprehensive framework connecting affine Toda equations, Coxeter theory, and reductive geometry, and identifies a distinguished class—totally noncompact pairs—for which principal-pair stability theory yields existence and uniqueness results. The authors further relate these Toda systems to $G$-Higgs bundles, showing how solutions correspond to equivariant harmonic maps and how the cyclic/non-cyclic nature of the Toda data controls the Higgs-bundle geometry (including Baraglia’s cyclic Higgs bundles). The results unify multiple strands: affine Toda dynamics, real forms and outer-inner automorphisms, geometric PDEs on Riemann surfaces, and Higgs-bundle theory, offering both existence results and a bridge to Hitchin representations and Baraglia’s constructions. The work thus advances the structural understanding of geometric Toda systems on noncompact symmetric spaces and their Higgs-theoretic interpretations, with clear criteria for when solutions exist and how to construct the associated harmonic maps.

Abstract

This paper has two purposes. The first is to classify all those versions of the Toda equations which govern the existence of $τ$-primitive harmonic maps from a surface into a homogeneous space $G/T$ for which $G$ is a noncomplex noncompact simple real Lie group, $τ$ is the Coxeter automorphism which Drinfel'd \& Sokolov assigned to each affine Dynkin diagram, and $T$ is the compact torus fixed pointwise by $τ$. Here $τ$ may be either an inner or an outer automorphism. We interpret the Toda equations over a compact Riemann surface $Σ$ as equations for a metric on a holomorphic principal $T^\mathbb{C}$-bundle $Q^\mathbb{C}$ over $Σ$ 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.

The geometric Toda equations for noncompact symmetric spaces

TL;DR

The work classifies all noncompact real forms compatible with Coxeter automorphisms that yield Toda-type equations for -primitive harmonic maps into , and provides a geometric reinterpretation in terms of holomorphic -bundles and holomorphic data on a compact surface . It develops a comprehensive framework connecting affine Toda equations, Coxeter theory, and reductive geometry, and identifies a distinguished class—totally noncompact pairs—for which principal-pair stability theory yields existence and uniqueness results. The authors further relate these Toda systems to -Higgs bundles, showing how solutions correspond to equivariant harmonic maps and how the cyclic/non-cyclic nature of the Toda data controls the Higgs-bundle geometry (including Baraglia’s cyclic Higgs bundles). The results unify multiple strands: affine Toda dynamics, real forms and outer-inner automorphisms, geometric PDEs on Riemann surfaces, and Higgs-bundle theory, offering both existence results and a bridge to Hitchin representations and Baraglia’s constructions. The work thus advances the structural understanding of geometric Toda systems on noncompact symmetric spaces and their Higgs-theoretic interpretations, with clear criteria for when solutions exist and how to construct the associated harmonic maps.

Abstract

This paper has two purposes. The first is to classify all those versions of the Toda equations which govern the existence of -primitive harmonic maps from a surface into a homogeneous space for which is a noncomplex noncompact simple real Lie group, is the Coxeter automorphism which Drinfel'd \& Sokolov assigned to each affine Dynkin diagram, and is the compact torus fixed pointwise by . Here may be either an inner or an outer automorphism. We interpret the Toda equations over a compact Riemann surface as equations for a metric on a holomorphic principal -bundle over whose Chern connection, when combined with holomorphic field , produces a -connection which is flat precisely when the Toda equations hold. The second purpose is to establish when stability criteria for the pair 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 -Higgs bundle. We explain how to construct this -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.
Paper Structure (16 sections, 25 theorems, 138 equations, 1 figure, 2 tables)

This paper contains 16 sections, 25 theorems, 138 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction.
  2. Primitive harmonic maps.
  3. Some reductive homogeneous geometry.
  4. $\tau$-primitive harmonic maps.
  5. Coxeter automorphisms and noncompact Lie algebras.
  6. The Coxeter automorphism for an affine diagram.
  7. Real forms compatible with the Coxeter automorphism.
  8. Totally noncompact pairs.
  9. Geometric Toda equations.
  10. Non-cyclic Toda pairs which are not simple non-affine.
  11. Inner versus outer Coxeter automorphisms.
  12. Existence of solutions via principal pairs.
  13. Toda pairs and Higgs bundles.
  14. Baraglia's cyclic Higgs bundles.
  15. Simple affine roots when $\nu$ is not the identity.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Gamma^{(n)}$, $n\in\{1,2,3\}$, be an affine Dynkin diagram with vertex set indexed by $I=\{0,\ldots,r\}$ and Coxeter automorphism $\tau$. Every involution $\sigma$ of $\mathfrak{g}^\mathbb{C}$ which is compatible with $\tau$ is determined by a map $\ell:I\to \{-1,1\}$ for which $-1\in\ell(I)$

Figures (1)

  • Figure 1: Affine Dynkin diagrams. There are $r+1$ vertices for all diagrams for which $r$ is a parameter. The numbers in the circles are the coefficients $m_j$ for which $\sum_{j=0}^rm_j\alpha_j=0$, $m_0=1$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3: Bla
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 50 more