Second-order superintegrable Hamiltonian systems via Frobenius structures

Andreas Vollmer (University of Hamburg)

Tuesday 2nd June 16:00-17:00
Maths 311B

Abstract

Second-order (maximally) superintegrable systems are Hamiltonian systems that admit 2n-1 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, and a complete classification exists to date only in dimension two. Partial classification results exist in dimension three.
My talk will present a geometric approach to the classification problem that, unlike other techniques, remains manageable in arbitrarily high dimension. It builds on a framework developed together with J. Kress and K. Schöbel, which encodes superintegrable Hamiltonian systems under mild assumptions in a tensor field similar to the information-geometric Amari-Chentsov tensor. 
As a first application, we consider second-order superintegrable systems on spaces of constant sectional curvature under a natural genericity assumption. These correspond to (possibly non-unital) Frobenius structures (admitting curvature) that are naturally compatible with a Hessian structure, i.e. the underlying (pseudo-)Riemannian metric can locally be written as the Hessian of a function (partially joint work with J. Armstrong and with V. Cortés) span>. These superintegrable systems therefore yield solutions to the Witten- Dijkgraaf-Verlinde-Verlinde equation, and their classification becomes equivalent to that of Frobenius algebras.

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