Regulated $L^2$ Norm on Certain Wild Higgs Bundles Over $\mathbb{CP}^1$

TL;DR

The paper addresses defining and analyzing a regulated $L^2$ norm for wild (irregular) rank-2 Higgs bundles on $\mathbb{CP}^1$ by working on the Hitchin moduli space $\mathcal{M}_{2,3}$, which decomposes into $\mathcal{M}_{2,3}^{\text{small}}$ and $\mathcal{M}_{2,3}^{\text{big}}$ and carries a central-fiber fixed-point structure. It constructs harmonic metrics on both strata, proves uniform estimates and exponential decay for the small stratum, and shows smooth dependence on deformation parameters; it then extends the regulated norm from fixed points to the entire moduli space, providing explicit formulas on the small stratum and an implicit, yet smooth, construction on the big stratum. The main contributions include the extension of the regulated $L^2$ norm to all of $\mathcal{M}_{2,3}$, the uniform control and limiting behavior as parameters approach the central fiber, and a concrete realization of the norm in terms of a moment-map framework aligned with the central-fiber picture of Fredrickson–Neitzke. These results offer a solid analytic foundation for understanding the Hitchin fibration in singular (wild) settings and pave the way for further generalizations and deeper geometric insight into the central fiber data.

Abstract

We define and analyse certain $L^2$ norm on moduli space of Higgs bundles over $\mathbb{CP}^1$ with certain singularity. We prove that certain limit of our metrics here is the regulated $L^2$ norm proposed on the central fiber, first appearing in Fredrickson-Neitzke's work.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Filtered Bundles (FN)
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• Proposition 2.7: Proposition 2.19 in FN
• Proposition 2.8: Proposition 2.14 of FN