Exponential Decay for the Asymptotic Geometry of the Hitchin Metric

TL;DR

This paper proves that the Hitchin hyperk"ahler metric closely approximates a simpler semiflat metric with exponentially decaying differences along generic rays, confirming a conjecture by Gaiotto-Moore-Neitzke.

Contribution

It establishes exponential decay of the difference between the Hitchin metric and the semiflat metric along generic rays, advancing understanding of the asymptotic geometry.

Findings

Exponential decay of metric difference proved

Confirms Gaiotto-Moore-Neitzke conjecture

Enhances understanding of Hitchin moduli space geometry

Abstract

We consider Hitchin's hyperk\"ahler metric $g_{L^2}$ on the $SU(n)$-Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric $g_{L^2}$ and a simpler "semiflat" hyperk\"ahler metric $g_{\mathrm{sf}}$ is exponentially-decaying along generic rays in the Hitchin moduli space, as conjectured by Gaiotto-Moore-Neitzke.