Universal Hitchin moduli spaces

TL;DR

This paper explores the metric and complex structures of universal Hitchin moduli spaces over Teichmüller space, introducing new geometric frameworks and conjecturing relationships between different moduli space families.

Contribution

It establishes a natural complex and K"ahler structure on the universal Hitchin moduli space and constructs a novel family of moduli spaces with a coupled scalar curvature and Higgs field system.

Findings

Universal moduli space carries a natural complex structure.

Construction of a family of moduli spaces with explicit K"ahler potential.

Proposal of a conjectural relationship in the weak-coupling limit.

Abstract

We study metric aspects of the universal moduli space of solutions to Hitchin's equations as the complex structure $J$ varies over the Teichm\"uller space $\mathcal{T}$ of a closed surface $\Sigma$. Our approach is gauge theoretical and builds on the theory of K\"ahler fibrations and the moment map interpretation of constant scalar curvature K\"ahler metrics. Our first main result establishes that, over the moduli space of cscK metrics, the universal moduli space of solutions to Hitchin's equations carries a natural complex structure together with a family of pseudo-K\"ahler metrics forming a K\"ahler fibration with a K\"ahler Ehresmann connection. We then investigate a second universal moduli space, constructed from the space of flat $G$-connections over $\mathcal{T}$, which admits a nontrivial $J$-dependent K\"ahler fibration structure discovered by Hitchin. Using symplectic reduction, we build universal moduli spaces of solutions to the harmonicity equations depending on a coupling constant $\alpha$, obtaining natural complex and pseudo-K\"ahler structures and an explicit K\"ahler potential. The main novelty here is that this moduli space is defined by a system coupling the scalar curvature with a cubic term in the Higgs field. Finally, we propose a conjectural relationship between the two resulting families of moduli spaces in the weak-coupling limit $\alpha\to 0$, inspired by the twistor geometry of Hitchin's hyperk\"ahler moduli space.