A general Cayley correspondence and higher Teichm\"uller spaces

TL;DR

This paper introduces a new class of special $ ext{sl}_2$-triples called magical triples, which help parametrize and understand higher Teichmüller spaces and character varieties for various real Lie groups, including previously unknown components.

Contribution

The paper defines magical $ ext{sl}_2$-triples, establishes their classification, and uses them to explicitly parametrize new and known higher Teichmüller components in moduli spaces of Higgs bundles.

Findings

Recovered known higher Teichmüller components for split and Hermitian groups.

Constructed new components for quaternionic real forms of exceptional groups.

Conjecturally identified all higher Teichmüller spaces via $ ext{Θ}$-positive structures.

Abstract

We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$. Using this decomposition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface $X$ for an associated real Lie group, hence also of the corresponding character variety of representations of $\pi_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $\mathrm{SO}_{p,q}$ with $1<p\leq q$, and also constructs previously unknown components for the quaternionic real forms of $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{F}_4$. The classification of magical $\mathfrak{sl}_2$-triples is shown to be in bijection with the set of $\Theta$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichm\"uller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $\Theta$-positive Anosov representations, which indicate that this conjecture holds.