Higgs bundles and higher Teichmüller spaces

Abstract

This paper is a survey on the role of Higgs bundle theory in the study of higher Teichmüller spaces. Recall that the Teichmüller space of a compact surface can be identified with a certain connected component of the moduli space of representations of the fundamental group of the surface into $\mathrm{PSL}(2,\mathbb{R})$. Higher Teichmüller spaces correspond to special components of the moduli space of representations when one replaces $\mathrm{PSL}(2,\mathbb{R})$ by a real non-compact semisimple Lie group of higher rank. Examples of these spaces are provided by the Hitchin components for split real groups, and maximal Toledo invariant components for groups of Hermitian type. More recently, the existence of such components has been proved for $\mathrm{SO}(p,q)$, in agreement with the conjecture of Guichard and Wienhard relating the existence of higher Teichmüller spaces to a certain notion of positivity on a Lie group that they have introduced. We review these three different situations, and end up explaining briefly the conjectural general picture from the point of view of Higgs bundle theory.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1511.07751

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• Non-Abelian Hodge Theory and Related Topics
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