A construction of hyperkähler metrics through Riemann-Hilbert problems
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In 2009 Gaiotto, Moore and Neitzke presented a new construction of hyperkähler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration M over a base space B, except for a divisor D in B, in which the torus fiber degenerates into a nodal torus. The hyperkähler metric g is obtained via solutions X [subscript gamma] of a Riemann-Hilbert problem. We interpret the Kontsevich-Soibelman Wall Crossing Formula as an isomonodromic deformation of a family of RH problems, therefore guaranteeing continuity of X at the walls of marginal stability. The latter functions are obtained through standard Banach contraction principles. By obtaining uniform estimates on arbitrary derivatives of X [subscript gamma], the smoothness property is obtained. To extend this construction to singular fibers, we use the Ooguri-Vafa case as our model and choose a suitable gauge transformation that allow us to define an integral equation defined at the degenerate fiber, whose solutions are the desired Darboux coordinates X [subscript gamma].