An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
Jonathan Kress, Konrad Schöbel, Andreas Vollmer
TL;DR
The paper tackles the problem of classifying second-order superintegrable systems in arbitrary dimension by replacing PDE-heavy approaches with an algebraic-geometric framework. It introduces a single central object, the structure tensor $T_{ijk}$, and shows that the space of irreducible non-degenerate systems forms a quasi-projective variety inside a Grassmannian of Killing tensors, with explicit polynomial constraints, at least for constant-curvature manifolds. A key achievement is an explicit algebraic equation for the class of non-degenerate systems on constant curvature spaces, linking cubic forms $\Psi_{ijk}x^ix^jx^k$ to a simple curvature-dependent relation, and demonstrating that abundant systems correspond to flat structure connections. The framework also recasts integrability conditions in terms of Codazzi tensors and a nonlinear prolongation of the structure tensor, paving the way for a finite, constructive pathway to classify abundant systems via algebraic geometry and moduli-space-like structures. Overall, the work shifts the classification of second-order superintegrable systems from calculus-based PDE techniques to algebraic geometry, enabling systematic exploration of higher-dimensional cases and connections to quadratic algebras and special functions.
Abstract
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem - based on a proof that the classification space for irreducible non-degenerate second-order superintegrable systems is naturally endowed with the structure of a quasi-projective variety with a linear isometry action. On constant curvature manifolds our approach leads to a single, simple and explicit algebraic equation defining the variety classifying superintegrable Hamiltonians that satisfy all relevant integrability conditions generically. In particular, this includes all non-degenerate superintegrable systems known to date and shows that our approach is manageable in arbitrary dimension. Our work establishes the foundations for a complete classification of second-order superintegrable systems in arbitrary dimension, derived from the geometry of the classification space, with many potential applications to related structures such as quadratic symmetry algebras and special functions.