Quantum group symmetry of the Quantum Hall effect on the non-flat surfaces

TL;DR

This paper reveals that quantum group symmetries, specifically $su_q(2)$, emerge in the Quantum Hall effect on non-flat surfaces, replacing magnetic translation symmetries, with concrete examples on spheres and Poincaré upper half planes.

Contribution

It demonstrates the existence of quantum group symmetries in QHE on curved surfaces and connects these symmetries to the geometry of the underlying surface.

Findings

Quantum group symmetries appear in QHE on non-flat surfaces.

The $su(2)$ symmetry on a sphere contracts to $su_q(2)$ at the equator.

Ground state wave functions form representations of $su_q(2)$.

Abstract

After showing that the magnetic translation operators are not the symmetries of the QHE on non-flat surfaces , we show that there exist another set of operators which leads to the quantum group symmetries for some of these surfaces . As a first example we show that the $su(2)$ symmetry of the QHE on sphere leads to $su_q(2)$ algebra in the equator . We explain this result by a contraction of $su(2)$ . Secondly , with the help of the symmetry operators of QHE on the Pioncare upper half plane , we will show that the ground state wave functions form a representation of the $su_q(2)$ algebra .