Existence of Joyce structures on stability spaces of CY3 triangulated categories

Establish the existence of a Joyce structure on the space M of stability conditions of a three-dimensional Calabi–Yau triangulated category, i.e., a geometric structure on M consisting of a closed holomorphic 2-form taking rational values on a lattice subbundle of TM and, when this 2-form is symplectic, inducing a compatible complex hyper-Kähler structure with homothetic symmetry on the total space X = TM, together with the required homogeneity and lattice invariance conditions.

Background

Bridgeland introduced the notion of a Joyce structure to encode Donaldson–Thomas invariants of three-dimensional Calabi–Yau triangulated categories via geometry on their spaces of stability conditions. Concretely, such a structure equips the base manifold M with a closed holomorphic 2-form that takes rational values on a holomorphically varying lattice in TM. Assuming this 2-form is symplectic, the total space X = TM carries a compatible complex hyper-Kähler metric with a homothetic symmetry, constrained by additional homogeneity and lattice invariance conditions.

This paper develops explicit complex hyper-Kähler metrics compatible with such structures in certain settings, notably via isomonodromic flows associated to deformed polynomial oscillators, and studies projectable hyper-Lagrangian foliations. Nonetheless, the general existence of a Joyce structure on the full space of stability conditions of a three-dimensional Calabi–Yau triangulated category remains conjectural and is an outstanding problem.