Heavenly metrics, hyper‐Lagrangians and Joyce structures

Abstract

AbstractIn [Proc. Sympos. Pure Math., American Mathematical Society, Providence, RI, 2021, pp. 1–66], Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space of stability conditions of a triangulated category. Given a non‐degeneracy assumption, a feature of this structure is a complex hyper‐Kähler metric with homothetic symmetry on the total space of the holomorphic tangent bundle. Generalising the isomonodromy calculation which leads to the Joyce structure in [Math. Ann. 385 (2023), 193–258], we obtain an explicit expression for a hyper‐Kähler metric with homothetic symmetry via construction of the isomonodromic flows of a Schrödinger equation with deformed polynomial oscillator potential of odd‐degree . The metric is defined on a total space of complex dimension and fibres over a ‐dimensional manifold which can be identified with the unfolding of the ‐singularity. The hyper‐Kähler structure is shown to be compatible with the natural symplectic structure on in the sense of admitting an affine symplectic fibration as defined in [Lett. Math. Phys. 111 (2021), 54]. Separately, using the additional conditions imposed by a Joyce structure, we consider reductions of Plebański's heavenly equations that govern the hyper‐Kähler condition. We introduce the notion of a projectable hyper‐Lagrangian foliation and show that in dimension four such a foliation of leads to a linearisation of the heavenly equation. The hyper‐Kähler metrics constructed here are shown to admit such a foliation.