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

TL;DR

This paper constructs and explicitly describes a hyperk"ahler geometry linked to the resolved conifold's BPS structure, revealing a smoothing of the semi-flat metric and connections to twistor theory and Riemann-Hilbert problems.

Contribution

It introduces a novel hyperk"ahler geometry associated with the resolved conifold and relates it to ASK geometry, twistor coordinates, and a Riemann-Hilbert problem.

Findings

Explicit description of ASK and HK geometries for the resolved conifold

Realization of an Ooguri-Vafa-like smoothing of the semi-flat HK metric

Conjectured relation between twistor coordinates and a Riemann-Hilbert problem

Abstract

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.