天态度规、超拉格朗日量与 Joyce 结构
Heavenly metrics, hyper-Lagrangians and Joyce structures
AI总结：
本文通过奇次变形振子 Schrödinger 方程的等单值流构造显式位似超 Kähler 度规，并利用可投影超拉格朗日叶状结构线性化 Plebański 天态方程，揭示其与 Joyce 结构的联系。
摘要：
在文献{B3}中，Bridgeland 定义了一种名为 Joyce 结构的几何结构，并猜想其存在于 CY3 三角范畴的稳定性条件空间 M 上。在非退化假设下，该结构的一个特征是，在全纯切丛的全空间 X=TM 上存在一个具有位似对称性的复超 Kähler 度规。我们推广了文献{BM}中导出 A2 Joyce 结构的等单值计算，通过构造具有奇数次 2n+1 次变形多项式振子势的 Schrödinger 方程的等单值流，得到了具有位似对称性的超 Kähler 度规的显式表达式。该度规定义在复维数为 4n 的全空间 X 上，并以一个 2n 维流形 M 为纤维底空间，M 可等同于 A_{2n} 奇点的开折。研究表明，该超 Kähler 结构与 M 上的自然辛结构相容，即其容许文献{BS}所定义的仿射辛纤维化。另一方面，利用 Joyce 结构施加的附加条件，我们研究了支配超 Kähler 条件的 Plebański 天态方程的约化。我们引入了可投影超拉格朗日叶状结构的概念，并证明在四维情形下，X 的这种叶状结构会使天态方程线性化。这里构造的超 Kähler 度规被证明容许这样的叶状结构。
英文摘要：
In https://pubs.ams.org/ebooks/pspum/103.2, 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 https://link.springer.com/article/10.1007/s00208-021-02337-w, 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 https://link.springer.com/article/10.1007/s11005-021-01388-z. 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.
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