2024/02/22 by Maciej Dunajski, Dunajski, Maciej, Timothy Moy +1 · 3 citations
Arts and Humanities · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Modernist Literature and Criticism #Samuel Beckett and Modernism
paper · pdf · doi:10.48550/arxiv.2402.14352
openalex publication_date 2024/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In \citeB3, Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space M of stability conditions of a CY3 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. \par Generalising the isomonodromy calculation which leads to the A2 Joyce structure in \citeBM, 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 A2n-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 \citeBS. \par 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.