Heavenly metrics, hyper-Lagrangians and Joyce structures.(English)Zbl 07947523
Summary: 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 \(CY_3\) 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 anaffine 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 aprojectable 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.
© 2024 The Author(s).Journal of the London Mathematical Society is copyright © London Mathematical Society.
© 2024 The Author(s).Journal of the London Mathematical Society is copyright © London Mathematical Society.
MSC:
| 14H70 | Relationships between algebraic curves and integrable systems |
| 53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |
| 53C28 | Twistor methods in differential geometry |
| 34M56 | Isomonodromic deformations for ordinary differential equations in the complex domain |
Cite
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