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Odd magical triples and maximal Higgs bundles

Enya Hsiao

TL;DR

This work extends the theory of magical $\mathfrak{sl}_2$-triples by introducing extended magical triples and classifying the odd cases in the Hermitian, nontube setting. It proves that the Slodowy slice associated to an odd extended magical triple coincides with the maximal components of $G^\mathbf{R}$-Higgs bundles, and under a genus threshold provides a geometric criterion for when such triples occur. The authors then establish a Cayley-type correspondence for maximal $G^\mathbf{R}$-Higgs bundles in the nontube case by restricting the Slodowy map to odd magical triples, yielding an injective, open, and closed map onto the maximal component locus; in tube-type subcases this reduces to the classical Cayley isomorphism. Collectively, these results extend the generalized Cayley correspondence, illuminate the structure of maximal representations, and deepen connections between Slodowy data and higher Teichmüller components in the non-tube Hermitian setting.

Abstract

We introduce the notion of extended magical $\mathfrak{sl}_2$-triples, a generalization of the magical $\mathfrak{sl}_2$-triples in Bradlow--Collier--García-Prada--Gothen--Oliveira's work and show that, apart from the known even magical triples, there are precisely three odd triples of nontube type Hermitian Lie algebras that are extended magical. We then show that the Slodowy slice of an odd extended magical triple of $G^\textbf{R}$ in the $G^\textbf{R}$-Higgs bundle moduli space are precisely the maximal components. Finally, assuming that the underlying curve has sufficiently large genus, we give a geometric characterization of extended magical triples and prove a Cayley correspondence for the maximal components of $G^\textbf{R}$-Higgs bundles for nontube type Hermitian Lie groups.

Odd magical triples and maximal Higgs bundles

TL;DR

This work extends the theory of magical -triples by introducing extended magical triples and classifying the odd cases in the Hermitian, nontube setting. It proves that the Slodowy slice associated to an odd extended magical triple coincides with the maximal components of -Higgs bundles, and under a genus threshold provides a geometric criterion for when such triples occur. The authors then establish a Cayley-type correspondence for maximal -Higgs bundles in the nontube case by restricting the Slodowy map to odd magical triples, yielding an injective, open, and closed map onto the maximal component locus; in tube-type subcases this reduces to the classical Cayley isomorphism. Collectively, these results extend the generalized Cayley correspondence, illuminate the structure of maximal representations, and deepen connections between Slodowy data and higher Teichmüller components in the non-tube Hermitian setting.

Abstract

We introduce the notion of extended magical -triples, a generalization of the magical -triples in Bradlow--Collier--García-Prada--Gothen--Oliveira's work and show that, apart from the known even magical triples, there are precisely three odd triples of nontube type Hermitian Lie algebras that are extended magical. We then show that the Slodowy slice of an odd extended magical triple of in the -Higgs bundle moduli space are precisely the maximal components. Finally, assuming that the underlying curve has sufficiently large genus, we give a geometric characterization of extended magical triples and prove a Cayley correspondence for the maximal components of -Higgs bundles for nontube type Hermitian Lie groups.
Paper Structure (16 sections, 19 theorems, 79 equations, 2 tables)

This paper contains 16 sections, 19 theorems, 79 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. $\mathfrak{sl}_2$-triples
  5. Complex $\mathfrak{sl}_2$-triples
  6. Real $\mathfrak{sl}_2$-triples
  7. Maximal Higgs bundles
  8. Hermitian Lie groups $G^\textbf{R}$ of nontube type
  9. Maximal $G^\textbf{R}$-Higgs bundles
  10. The Cayley correspondence
  11. The Slodowy slice of an $\mathfrak{sl}_2$-triple
  12. Magical $\mathfrak{sl}_2$-triples and the Cayley correspondence
  13. Extended magical triples and their classification
  14. Odd magical triples
  15. Geometric characterization of extended magical triples
  16. ...and 1 more sections

Key Result

Theorem 1.1

An $\mathfrak{sl}_2$-triple $\hat{\rho}:\mathfrak{sl}_2\to \mathfrak{g}^\textbf{R}$ is an odd magical triple if and only if $\mathfrak{g}^\textbf{R}$ is one of the following Hermitian Lie algebra of nontube type:

Theorems & Definitions (57)

  • Theorem 1.1: Theorem \ref{['thm:classification_theorem']}
  • Theorem 1.2: Theorem \ref{['thm:maximal_components']}
  • Theorem 1.3: Theorem \ref{['thm:characterization']}
  • Theorem 1.4: Theorem \ref{['thm:cayley_correspondence_odd']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 47 more