Hyperk\"ahler metrics on the moduli space of weakly parabolic Higgs bundles

TL;DR

This paper constructs a hyperk"ahler metric on the moduli space of weakly parabolic Higgs bundles using Darboux coordinates, showing the metric's exponential convergence to a simpler semiflat metric for large parameters.

Contribution

It introduces a method to explicitly construct a hyperk"ahler metric on the moduli space using the theory of Gaiotto, Moore, and Neitzke, with detailed analysis of the asymptotic behavior.

Findings

Darboux coordinates are dominated by a leading term for large R.

The deviation from the limiting configuration is exponentially suppressed in R.

The constructed twistorial hyperk"ahler metric converges exponentially to the semiflat metric.

Abstract

We use the theory of Gaiotto, Moore and Neitzke to construct a set of Darboux coordinates on the moduli space $\mathcal{M}$ of weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles. For generic Higgs bundles ($\mathcal{E},R\Phi)$ with $R\gg 0$ the coordinates are shown to be dominated by a leading term that is given by the coordinates for a corresponding simpler space of limiting configurations and we prove that the deviation from the limiting term is given by a remainder that is exponentially suppressed in $R$. We then use this result to solve an associated Riemann-Hilbert problem and construct a twistorial hyperk\"ahler metric $g_{\text{twist}}$ on $\mathcal{M}$. Comparing this metric to the simpler semiflat metric $g_{\text{sf}}$, we show that their difference is $g_{\text{twist}}-g_{\text{sf}}=O\left(e^{-\mu R}\right)$, where $\mu$ is a minimal period of the determinant of the Higgs field.