Berlin 2024 – wissenschaftliches Programm

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 13: Quantum Field Theory III

MP 13.3: Vortrag

Donnerstag, 21. März 2024, 16:20–16:40, HL 001

Obstructions to higher-dimensional second-order superintegrability — •Andreas Vollmer — Universität Hamburg, Deutschland

Superintegrable systems are crucial models in Physics, such as the harmonic oscillator and the Kepler-Coulomb system. The talk will focus on second-order (maximally) superintegrable systems (of a special non-degenerate type) on the cotangent space of a Riemannian manifold. The classification of such systems is an ongoing problem, and to date only achieved in low dimension.

A novel geometric framework will be outlined, which is manageable for arbitrary dimension (encoding a superintegrable system via a (0,3)-tensor field) and is naturally adapted to conformal rescalings (replacing Stäckel transformations / coupling constant metamorphosis).

The main part of the talk will present concise algebraic obstructions for this type of superintegrability. These are relevant in dimensions starting from four, and they are not present in lower dimensions. In dimension four the obstruction condition leads to an algebraic variety isomorphic to a 10-dimensional spinor variety in a pseudo-Euclidean space with split signature.

Time permitting, affine hypersurfaces that are naturally associated to superintegrable systems will also be briefly discussed.

Joint projects with V. Cortés, H.-C. Graf v. Bothmer, J. Kress and K. Schöbel.

Keywords: second-order superintegrable system; conformal geometry