Hyper-Kähler manifolds from Riemann-Hilbert problems I: Ooguri-Vafa-like model geometries

We construct model hyper-Kähler geometries that include and generalize the multi-Ooguri-Vafa model using the formalism of Gaitto, Moore, and Neitzke. This is the first paper in a series of papers making rigorous Gaiotto–Moore–Neitzke’s formalism for constructing hyper-Kähler metrics near semi-flat limits. In that context, this paper describes the assumptions we will make on a sequence of lattices $0\to {\mathrm{\Gamma }}_f\to \widehat{\mathrm{\Gamma }}\to \mathrm{\Gamma }\to 0$ over a complex manifold ${\mathcal{B}}^{\mathrm{\prime }}=\mathcal{B}-{\mathcal{B}}^{\mathrm{\prime }\mathrm{\prime }}$ near the singular locus, ${\mathcal{B}}^{\mathrm{\prime }\mathrm{\prime }}$, in order to define a smooth manifold $\mathcal{M}\to \mathcal{B}$ and hyper-Kähler model geometries on neighborhoods of points of the singular locus. In follow-up papers, we will use a modified version of Gaiotto-Moore-Neitzke’s iteration scheme starting at these model geometries to produce true global hyper-Kähler metrics on $\mathcal{M}$.