Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension
Jonathan Kress, Konrad Schöbel, Andreas Vollmer
TL;DR
The paper develops a comprehensive algebraic-geometric framework to classify second-order conformally superintegrable systems in arbitrary dimension by introducing a conformally invariant structure tensor framework. It extends the Stäckel transform to conformal geometry, shows abundant systems correspond to a trace-free cubic form $\Psi_{ijk}$ constrained by a simple quadratic master equation, and derives a nonlinear prolongation governing the structure tensors that is invariant under conformal changes. In constant-curvature geometries, the conformal scale is tightly linked to eigenfunctions of the conformal Laplacian, yielding Helmholtz-type equations that tie geometry, scales, and integrals together; in dimension 3 this reduces to a harmonic-cubic (or univariate sextic) classification, while higher dimensions reveal new obstructions and a rich algebraic-geometric structure. The results place the long-standing classification problem of conformally superintegrable systems within the purview of algebraic geometry and invariant theory, and they establish precise links between geometry, scales, and the associated structure functions, enabling a systematic approach to higher-dimensional cases.
Abstract
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.