Asymptotic Geometry of the Hitchin Metric

TL;DR

This paper analyzes the asymptotic behavior of the $L^2$ metric on the SU(2) Hitchin moduli space, showing it closely approximates the semiflat metric with polynomial convergence rates, advancing understanding of the metric's geometry.

Contribution

It proves the polynomial rate of convergence of the Hitchin metric to the semiflat metric on the regular part of the Hitchin system, confirming conjectures by Gaiotto, Neitzke, and Moore.

Findings

Hitchin metric approximates semiflat metric asymptotically

Convergence rate is polynomial in certain coordinates

Recent work suggests exponential convergence in some directions

Abstract

We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G = \mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from \cite{gmn13}. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.