The Ooguri–Vafa Space as a Moduli Space of Framed Wild Harmonic Bundles

The Ooguri–Vafa space is a $4$-dimensional incomplete hyperkähler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri–Vafa hyperkähler metric should be part of the local model of the hyperkähler metric of the Hitchin moduli spaces, near the most generic kind of singular locus of the Hitchin fibration. In order to relate the Ooguri–Vafa space with the Hitchin moduli spaces, we show that the Ooguri–Vafa space can be interpreted as a set of rank $2$, framed wild harmonic bundles over $\mathbb{C}P^1$, with one irregular singularity. Along the way we show that a certain twistor family of holomorphic Darboux coordinates, which describes the hyperkähler geometry of the Ooguri–Vafa space, has an interpretation in terms of Stokes data associated to our framed wild harmonic bundles.