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

We define and analyse certain $L^2$ norm on moduli space of Higgs bundles over ${\mathbb{C}\mathbb{P}}^1$ 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.