Electromagnetic Excitations of Hall Systems on Four Dimensional Space

TL;DR

This paper explores electromagnetic excitations in four-dimensional Hall systems using symplectic geometry, showing how coordinate transformations can simplify gauge fluctuations and analyzing their impact on edge velocities in quantum Hall droplets.

Contribution

It introduces a symplectic geometric approach to noncommutative phase spaces and relates coordinate changes to the Seiberg--Witten map in four-dimensional quantum Hall systems.

Findings

Coordinate maps can locally eliminate gauge fluctuations.

Edge velocities depend on noncommutativity parameters.

Derived electromagnetic action for Hall droplets on CP^2.

Abstract

The noncommutativity of a four-dimensional phase space is introduced from a purely symplectic point of view. We show that there is always a coordinate map to locally eliminate the gauge fluctuations inducing the deformation of the symplectic structure. This uses the Moser's lemma; a refined version of the celebrated Darboux theorem. We discuss the relation between the coordinates change arising from Moser's lemma and the Seiberg--Witten map. As illustration, we consider the quantum Hall systems on CP^2. We derive the action describing the electromagnetic interaction of Hall droplets. In particular, we show that the velocities of the edge field, along the droplet boundary, are noncommutativity parameters-dependents.