Heavenly metrics, hyper-Lagrangians and Joyce structures

In {B3}, Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space M of stability conditions of a CY₃ triangulated category. A feature of this structure is a complex hyper-K\"ahler metric with homothetic symmetry on the total space $X = TM$ of the holomorphic tangent bundle. Generalising the isomonodromy calculation which leads to the A₂ Joyce structure in {BM}, we obtain an explicit expression for a hyper-K\"ahler metric with homothetic symmetry via construction of the isomonodromic flows of a Schr\"odinger 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 A₂ₙ-singularity. The hyper-K\"ahler 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 {BS}. Separately, using the additional conditions imposed by a Joyce structure, we consider reductions of Pleba\'nski's heavenly equations that govern the hyper-K\"ahler 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.