Heavenly metrics, hyper-Lagrangians and Joyce structures

In \cite{B3}, Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space $M$ of stability conditions of a $CY_3$ triangulated category. Given a non-degeneracy assumption, 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. \par Generalising the isomonodromy calculation which leads to the $A_2$ Joyce structure in \cite{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 $A_{2n}$-singularity. The hyper-K\"ahler structure is shown to be compatible with the natural symplectic structure on $M$ in the sense of admitting an \textit{affine symplectic fibration} as defined in \cite{BS}. \par 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 \textit{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-K\"ahler metrics constructed here are shown to admit such a foliation.