Some aspects of the problem in the four-dimensional quantum Hall effect

The remarkable Landau level structure can be generalized to 4-sphere with SU(2) magnetic monopole at the center of the 4-sphere. The bulk states reveal the relativistic nature if we maximally enlarge the gauge symmetry to the one of this system, which is SO(5). When the monopole strength S is less than or equal to 2, they describe the helicity states of massless particles of spin. Moreover, the effective field theory of the lowest Landau level is formulated on CP3, which is the level manifold of the lowest Landau level. The single particle Lagrangian is the Berry's phase over CP3. Using the flux attachment transformation, the hydrodynamic description of this model can be written as a 6+1 dimensional U(1) Chem-Simons theory. Locally, the system can be regarded as three independent 2-dimensional quantum Hall planes. Because the choice of the independent planes is arbitrary up to a gauge transformation, the interchanging phase between membranes can be defined on CP3 by moving one membrane around the other on a 2DQHE plane. They are the elementary excitations of the system and may obey fractional statistics, if the system is fractionally filled.