The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions to the self-duality equations. In this paper we construct self-duality solutions for strongly parabolic \(\mathfrak {sl}(2,{{\mathbb {C}}})\) Higgs fields on a 4-punctured sphere with parabolic weights \(t \sim 0\) using complex analytic methods. We identify the rescaled limit hyper-Kähler moduli space \({\mathcal {M}}_t\) at \(t=0\) to be the completion of the nilpotent orbit in \(\mathfrak {sl}(2, {{\mathbb {C}}})\) modulo a \({{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2\) action, equipped with the Eguchi–Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson’s \(\lambda \) -connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form \(\varpi _t\) on \({\mathcal {M}}_t\) at \(t=0\) which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of \({\mathcal {M}}_t\) lead to some identities of certain MPLs which we believe deserve further investigations.
