Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

Georgios Kydonakis

doi:10.24297/jam.v14i2.7444

Georgios Kydonakis

Open Access

https://doi.org/10.24297/jam.v14i2.7444

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Abstract

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the Sp(4,R)-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 Sp(4,R)-Higgs bundle moduli space, which correspond to components solely consisted of Zariski dense representations. This alsoallows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by O. Guichard and A. Wienhard.

Highlights

Let Σ be a closed connected and oriented surface of genus g ≥ 2 and G be a connected semisimple Lie group
Fixing a complex structure J on the surface Σ transforms this into a Riemann surface X = (Σ, J) and opens the way for holomorphic techniques using the theory of Higgs bundles
It is important that a gluing construction of parabolic Higgs bundles over the complex connected sum X# of two distinct compact Riemann surfaces X1 and X2 with a divisor of s-many distinct points on each, is formulated so that the gluing of stable parabolic pairs is providing a polystable Higgs bundle over X#

Summary

Introduction

Let Σ be a closed connected and oriented surface of genus g ≥ 2 and G be a connected semisimple Lie group. It is important that a gluing construction of parabolic Higgs bundles over the complex connected sum X# of two distinct compact Riemann surfaces X1 and X2 with a divisor of s-many distinct points on each, is formulated so that the gluing of stable parabolic pairs is providing a polystable Higgs bundle over X#. Let (E1, Φ1) → X1 and (E2, Φ2) → X2 be parabolic stable Sp(4,R)-Higgs bundles with corresponding solutions to the Hitchin equations (A1, Φ1) and (A2, Φ2) Assume that these solutions agree with model solutions A1m,poid , Φ1m,poid and A2m,qojd , Φ2m,qojd near the points pi ∈. In the case when G = Sp(4,R), considering all possible decompositions of a surface Σ along a simple, closed, separating geodesic is sufficient in order to obtain representations in the desired components of Mmax, which are fully distinguished by the calculation of the degree of a line bundle. These invariants live naturally in different cohomology groups

Producing approximate solutions by gluing

Correcting an approximate solution to an exact solution

The linearization operator for an approximate solution

Gluing theorems

Topological invariants