Odd magical triples and maximal Higgs bundles

TL;DR

This paper introduces extended magical $ ext{sl}_2$-triples, identifies three odd triples of nontube type Hermitian Lie algebras, and relates them to maximal Higgs bundle components, providing new geometric and algebraic insights.

Contribution

It generalizes the concept of magical $ ext{sl}_2$-triples, classifies odd extended magical triples, and establishes a Cayley correspondence for maximal Higgs bundle components.

Findings

Identified three odd extended magical triples of nontube type Hermitian Lie algebras.

Connected the Slodowy slice of these triples to maximal Higgs bundle components.

Provided a geometric characterization and proved a Cayley correspondence for these components.

Abstract

We introduce the notion of extended magical $\mathfrak{sl}_2$-triples, a generalization of the magical $\mathfrak{sl}_2$-triples in Bradlow--Collier--Garc\'ia-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.