任意维度中共形超可积性的代数条件
Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension
AI总结：
本文将二阶超可积系统的代数几何分类方法推广到共形情形，证明共形等价由权为二的共形密度刻画，并用调和三次形式的二次约束给出高维分类的代数条件，使分类问题进入代数几何与几何不变理论框架。
摘要：
我们证明，在（伪）黎曼流形上二阶超可积系统的定义会引出一个共形不变的超可积性概念。共形等价是著名的 Stäckel 变换的自然推广，而 Stäckel 变换又源于经典的 Maupertuis-Jacobi 原理。我们将最近发展的用于任意高维二阶超可积系统分类的代数几何方法推广到共形超可积系统，这些系统通过在共形几何中定义的二阶超可积系统的共形尺度选择来呈现。对于常曲率空间上的超可积系统，我们发现 Stäckel 等价系统的共形尺度来自拉普拉斯算子的特征函数，并且它们的等价性由一个权为二的共形密度刻画。我们的方法导出了一个代数方程，该方程支配着一大类二阶共形超可积系统在共形等价下的分类。这一类包含迄今为止所有已知的非退化例子，并且由一个关于一般调和三次形式的简单二次代数约束给出。通过这种方式，尚未解决的分类问题被纳入代数几何和几何不变理论的范畴。特别地，在三维中不存在障碍，因此已知的共形超可积系统分类以无约束单变量六次式的形式被重新得到。在更高维度中，该障碍是新的，从未被传统方法揭示过。
英文摘要：
We show that the definition of a second order superintegrable system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant notion of superintegrability. Conformal equivalence is the natural extension of the well-known Stäckel transform, which in turn originates from the classical Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.
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