A construction of hyperk\"ahler metrics through Riemann-Hilbert problems I

In 2009 Gaiotto, Moore and Neitzke presented a new construction of hyperk"{a}hler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration $\mathcal{M}$ over a base space $\mathcal{B}$, except for a divisor $D$ in $\mathcal{B}$, in which the torus fiber degenerates into a nodal torus. The hyperk"{a}hler metric $g$ is obtained via solutions ${\mathcal{X}}_{\gamma }$ 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 ${\mathcal{X}}_{\gamma }$ at the walls of marginal stability. The technical details about solving the different classes of Riemann-Hilbert problems that arise here are left to a second article. 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 ${\mathcal{X}}_{\gamma }$. We show that these functions yield a holomorphic symplectic form $\varpi (\zeta )$, which, by Hitchin’s twistor construction, constructs the desired hyperk"{a}hler metric.