Local hyper-Kahler model metrics built near the singular locus of a torus fibration directly inside the Gaiotto-Moore-Neitzke formalism, rather than by matching against the Gibbons-Hawking ansatz. The input is an exact sequence of local systems of lattices over the complement of the singular locus together with a central charge and a BPS spectrum; a Riemann-Hilbert integral relation turns that data into Darboux coordinates on the torus fibers and hence a projective-line family of holomorphic symplectic two-forms, and a concrete variant of the Hitchin-Karlhede-Lindstrom-Rocek twistor theorem upgrades the family to a genuine hyper-Kahler structure with nondegeneracy proved rather than assumed. In four real dimensions the resulting metrics are the Ooguri-Vafa and multi-Ooguri-Vafa geometries attached to a single I_N Kodaira fiber or to its perturbation into several fibers, treated uniformly so that colliding singular fibers cause no case split; in higher rank they include products with flat space, quotient-like generalizations coming from non-unimodular lattices, and hypertoric examples carrying a triholomorphic circle action. The construction is quantitative: for a given fiber length scale it bounds the size of the neighborhood of a singular point on which the model metric is valid, which is what makes it usable as the starting point of an iteration scheme producing global hyper-Kahler metrics and, ultimately, Gromov-Hausdorff collapse statements for hyper-Kahler manifolds degenerating to their semi-flat limits.
Fredrickson and Zimet open a series of papers whose aim is to make the Gaiotto-Moore-Neitzke (GMN) formalism for constructing hyper-Kahler metrics near semi-flat limits into rigorous differential geo…