SO(p,q)-Higgs bundles and higher Teichm\"uller components

TL;DR

This paper explores new exotic components in the moduli spaces of SO(p,q)-Higgs bundles and surface group representations, revealing their geometric significance and relation to positive Anosov representations.

Contribution

It introduces novel examples of exotic components in SO(p,q)-Higgs bundle moduli spaces and analyzes their connection to positive Anosov representations.

Findings

Identification of new exotic components in moduli spaces

Complete count of connected components for most SO(p,q) cases

Relation established between exotic components and positive Anosov representations

Abstract

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic' components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q> 3).