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

TL;DR

This paper explores a novel approach to constructing hyperk"ahler metrics on complex integrable systems using Riemann-Hilbert problems, linking wall crossing phenomena with isomonodromic deformations and extending to singular fibers.

Contribution

It interprets the Kontsevich-Soibelman Wall Crossing Formula as an isomonodromic deformation, providing a new framework for hyperk"ahler metric construction via Riemann-Hilbert problems.

Findings

Wall crossing formula interpreted as isomonodromic deformation

Construction of hyperk"ahler metrics on singular fibers

Use of integral equations for degenerate fibers

Abstract

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.