We associate to the resolved conifold an affine special K\"{ a}hler (ASK) manifold of complex dimension 1, and an instanton corrected hyperk\"{ a}hler (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 conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.
Categories: math.DG hep-th math-ph math.AG math.MP