Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles

TL;DR

This work analyzes the Hitchin metric $g_H$ on the Higgs bundle moduli space against the semi-flat metric $g_{sf}$ on the smooth spectral-curve locus. By constructing large-scale (asymptotic) solutions $h_t$ to the Hitchin equation and developing a robust horizontal/vertical deformation calculus, the authors obtain harmonic representatives that enable sharp exponential decay estimates for the difference $g_H-g_{sf}$ along the ray $(E,t\theta)$. The main result confirms a weak GMN-type conjecture in this setting and advances the understanding of hyperkähler geometry on $\mathcal{M}_H$ by linking the Hitchin metric to the integrable-system induced semi-flat metric. The techniques combine local fiducial-model analysis, symmetric pairings, and a maximum-principle-based global construction to achieve uniform exponential convergence on compact subsets of the moduli space. This has implications for the asymptotic geometry of the Hitchin system and for related mirror-symmetric structures in higher rank.

Abstract

The moduli space of stable Higgs bundles of degree $0$ is equipped with the hyperkähler metric, called the Hitchin metric. On the locus where the spectral curves are smooth, there is the hyperkähler metric called the semi-flat metric, associated with the algebraic integrable systems with the Hitchin section. We prove the exponentially rapid decay of the difference between the Hitchin metric and the semi-flat metric along the ray $(E,tθ)$ as $t\to\infty$.

Theorems & Definitions (113)

• Theorem 1.1: Theorem \ref{['thm;23.5.11.20']}
• Remark 1.2
• Remark 1.3
• Proposition 1.4: Corollary \ref{['cor;23.5.12.1']}
• Remark 1.5
• Remark 1.6
• Lemma 2.1
• Corollary 2.2
• Lemma 2.3
• Proposition 2.4