A Hyperk\"ahler geometry associated to the BPS structure of the resolved conifold

We associate to the resolved conifold an affine special Kahler (ASK) manifold of complex dimension 1, and an instanton corrected hyperkahler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and relate them to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.