Model Higgs bundles in exceptional components of the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-character variety

TL;DR

This paper develops a gluing method for Higgs bundles on connected sums of Riemann surfaces, enabling the construction of model Higgs bundles in all exceptional components of the maximal $ ext{Sp(4, extbf{R})}$-Higgs moduli space, and compares invariants with Anosov representations.

Contribution

It introduces a new gluing construction for Higgs bundles in the context of $ ext{Sp(4, extbf{R})}$-Hitchin equations, covering all exceptional components of the moduli space.

Findings

Constructed model Higgs bundles in all exceptional components.

Established a comparison between Higgs bundle invariants and topological invariants of Anosov representations.

Provided a framework for understanding the structure of the $ ext{Sp(4, extbf{R})}$-Higgs moduli space.

Abstract

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exceptional components of the maximal $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundle moduli space, which correspond to components solely consisted of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by O. Guichard and A. Wienhard.