Second-order superintegrable systems via affine hypersurfaces and Frobenius manifolds

Abstract

Second-order (maximally) superintegrable systems are Hamiltonian systems that admit 2n-2 additional (functionally independent) constants of the motion. Famous examples are the harmonic oscillator and the Kepler-Coulomb system. The classification of second-order superintegrable systems is an open problem, with a complete classification existing only in dimension two. Partial results exist in dimension three. In the talk, I will present a geometric framework that allows one to study the classification space by means of conformal and algebraic geometry (joint work with J. Kress and K. Schöbel). Unlike other techniques, our method is manageable in arbitrarily high dimension. Under mild assumptions, we encode these Hamiltonian systems in a (1,2)-tensor field that is similar to the Amari-Chentsov tensor in information geometry.

As a first application, we consider the second-order superintegrable systems for which the integrability conditions are satisfied generically (abundant systems). It turns out that these systems can be realised as affine hypersurfaces, establishing a correspondence between these systems and a certain subclass of affine hypersurface normalisations (work with V. Cortés). The correspondence respects Stäckel transformations, i.e. conformal transformations.

On spaces of constant sectional curvature, this naturally leads to Hessian structures, i.e. the metric can locally be written as the Hessian of a function (with J. Armstrong).

In fact, all abundant systems on spaces of constant sectional curvature in dimension at least three correspond to (curved) Frobenius manifolds that are consistent with the Hessian structure.