Asymptotic Geometry of the Moduli Space of Rank Two Irregular Higgs Bundles over the Projective Line

TL;DR

This paper investigates the asymptotic behavior of the Hitchin hyperk"ahler metric on the moduli space of rank two irregular Higgs bundles over the complex projective line, revealing precise asymptotic relations to semiflat and ALG/ALG$^ ext{*}$ metrics.

Contribution

It establishes the asymptotic equivalence of the Hitchin metric to semiflat and ALG/ALG$^\text{*}$ models with explicit rates, advancing understanding of the metric's geometry at infinity.

Findings

Hitchin metric is asymptotic to the semiflat metric at arbitrary polynomial order.

Exponential rate of convergence when no weakly parabolic singularities.

In four-dimensional cases, the semiflat metric approximates an ALG/ALG$^\text{*}$ model metric.

Abstract

We study the asymptotic behavior of Hitchin's hyperk\"ahler metric on the moduli space of rank two irregular Higgs bundles over $\mathbb{C}P^1$. Along a generic curve, we prove that the Hitchin metric is asymptotic to the semiflat metric at an arbitrary polynomial order. When there are no weakly parabolic singularities, the rate is exponential. In the case of four-dimensional moduli spaces, we prove that the semiflat metric is asymptotic to an ALG/ALG$^\ast$ model metric.