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

TL;DR

This paper proves that the Hitchin metric on the moduli space of Higgs bundles converges exponentially fast to the semi-flat metric along certain rays, revealing detailed asymptotic behavior of these hyperk"ahler metrics.

Contribution

It establishes the exponential decay rate of the difference between the Hitchin and semi-flat metrics on the moduli space as the parameter tends to infinity.

Findings

Exponential decay of the metric difference along rays

Asymptotic analysis of the Hitchin metric

Insights into the geometry of Higgs bundle moduli space

Abstract

The moduli space of stable Higgs bundles of degree $0$ is equipped with the hyperk\"ahler metric, called the Hitchin metric. On the locus where the spectral curves are smooth, there is the hyperk\"ahler 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 $(E,t\theta)$ as $t\to\infty$.