A general Cayley correspondence and higher Teichmüller spaces
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914642680348672 |
|---|---|
| author | Bradlow, Steve Collier, Brian Garcia-Prada, Oscar Gothen, Peter Oliveira, André |
| author_facet | Bradlow, Steve Collier, Brian Garcia-Prada, Oscar Gothen, Peter Oliveira, André |
| contents | 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 $π_1X$ 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 $Θ$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmüller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Θ$-positive Anosov representations, which indicate that this conjecture holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_09377 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A general Cayley correspondence and higher Teichmüller spaces Bradlow, Steve Collier, Brian Garcia-Prada, Oscar Gothen, Peter Oliveira, André Algebraic Geometry Differential Geometry Geometric Topology 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 $π_1X$ 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 $Θ$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmüller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Θ$-positive Anosov representations, which indicate that this conjecture holds. |
| title | A general Cayley correspondence and higher Teichmüller spaces |
| topic | Algebraic Geometry Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2101.09377 |