Special Joyce structures and hyperk\"ahler metrics

Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a three-dimensional Calabi-Yau category and its associated Donaldson-Thomas invariants. In subsequent work, T. Bridgeland and I. Strachan showed that Joyce structures satisfying a certain non-degeneracy condition encode a complex hyperk\"{a}hler structure on the tangent bundle of the base of the Joyce structure. In this work we give a definition of an analogous structure over an affine special K\"{a}hler (ASK) manifold, which we call a special Joyce structure. Furthermore, we show that it encodes a real hyperk\"{a}hler (HK) structure on the tangent bundle of the ASK manifold, possibly of indefinite signature. Particular examples include the semi-flat HK metric associated to an ASK manifold (also known as the rigid c-map metric) and the HK metrics associated to certain uncoupled variations of BPS structures over the ASK manifold. Finally, we relate the HK metrics coming from special Joyce structures to HK metrics on the total space of algebraic integrable systems.