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

TL;DR

This paper introduces a regulated $L^2$ norm on the moduli space of singular Higgs bundles over $ ext{CP}^1$, connecting it to existing work on limits of metrics and providing new analytical tools.

Contribution

It defines and analyzes a specific $L^2$ norm on Higgs bundle moduli spaces with singularities, linking it to prior proposals and limits in the field.

Findings

Established the regulated $L^2$ norm as a meaningful metric on the moduli space.

Proved the limit of the constructed metrics converges to the proposed norm.

Connected the norm to the work of Fredrickson and Neitzke on central fibers.

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.