Heavenly metrics, hyper-Lagrangians and Joyce structures

In [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 $M$ of stability conditions of a $C{Y}_3$ 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 $X=TM$ of the holomorphic tangent bundle. Generalising the isomonodromy calculation which leads to the $A_2$ 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 $2n+1$. The metric is defined on a total space $X$ of complex dimension $4n$ and fibres over a $2n$-dimensional manifold $M$ which can be identified with the unfolding of the $A_{2n}$-singularity. The hyper-Kähler structure is shown to be compatible with the natural symplectic structure on $M$ 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 $X$ leads to a linearisation of the heavenly equation. The hyper-Kähler metrics constructed here are shown to admit such a foliation.