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

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 ($\mathcal{E},R\mathrm{\Phi })$ 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.