Universal Hitchin moduli spaces

Abstract

We study metric aspects of the universal moduli space of solutions to Hitchin's equations as the complex structure $J$ varies over the Teichmüller space $\mathcal{T}$ of a closed surface $Σ$. Our approach is gauge theoretical and builds on the theory of Kähler fibrations and the moment map interpretation of constant scalar curvature Kähler 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-Kähler metrics forming a Kähler fibration with a Kähler 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 Kähler fibration structure discovered by Hitchin. Using symplectic reduction, we build universal moduli spaces of solutions to the harmonicity equations depending on a coupling constant $α$, obtaining natural complex and pseudo-Kähler structures and an explicit Kähler 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 $α\to 0$, inspired by the twistor geometry of Hitchin's hyperkähler moduli space.