Universal Hitchin moduli spaces

TL;DR

This work develops a gauge-theoretic framework for universal Hitchin moduli spaces by placing Hitchin's equations and Higgs-bundle theory into a fibration over Teichmüller space. It constructs two principal universal moduli spaces: one for Hitchin's equations in a varying complex structure, yielding a complex structure and a family of (pre-)Kähler metrics forming a Kähler fibration with a Kähler Ehresmann connection; and a second from flat $G$-connections, producing a $J$-dependent Kähler fibration and, via symplectic reduction, universal moduli spaces of coupled harmonic equations with a coupling constant $oldsymbol{ extalpha}$ and an explicit Kähler potential. A parallel universal Higgs-moduli construction is developed, with a natural Kähler fibration for the universal Higgs field and a corresponding Hamiltonian action yielding coupled Hitchin equations; gauge fixing provides complex structures and (pseudo-)Kähler metrics on these moduli. The paper also discusses holomorphic maps between the harmonic and flat moduli and between Hitchin and Higgs moduli, and conjectures a weak-coupling limit relation guided by Hitchin's twistor geometry. These results illuminate the metric geometry of moduli spaces in families over $oldsymbol{ extT}$ and suggest links between distinct universal moduli spaces via adiabatic and twistor-type limits.

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.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Proposition 2.2
• Remark 2.3
• Proposition 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8