A universal Higgs bundle moduli space

TL;DR

This paper constructs a holomorphic family of Higgs bundle moduli spaces over a curve using differential geometry, linking harmonic map energy to complex structures and revealing hyperk"ahler metric properties.

Contribution

It introduces a novel differential geometric approach to parametrize Higgs bundle moduli spaces over Teichm"uller space via a harmonic map energy function.

Findings

Construction of a holomorphic family over Teichm"uller space

Definition of flat Ehresmann connections parametrized by the circle

Insights into hyperk"ahler metrics of the moduli spaces

Abstract

We give a differential geometric construction of the holomorphic family of Higgs bundle moduli spaces over a curve C as a fibration over Teichm\"uller space. The method uses a function f defined on the character variety, essentially the energy of a harmonic map, which is dependent on the complex structure of C. Using f we define a natural family of flat Ehresmann connections parametrized by the circle which reveal various aspects of these moduli spaces and their hyperk\"ahler metrics.