The Topological Structure of the SU(2) Chern-Simons Topological Current in the Four-Dimensional Quantum Hall Effect

In the light of the decomposition of the SU(2) gauge potential for I=1/2, we obtain the SU(2) Chern-Simons current over S^4, i.e. the vortex current in the effective field for the four-dimensional quantum Hall effect. Similar to the vortex excitations in the two-dimensional quantum Hall effect (2D FQH) which are generated from the zero points of the complex scalar field, in the 4D FQH, we show that the SU(2) Chern-Simons vortices are generated from the zero points of the two-component wave functions Ψ, and their topological charges are quantized in terms of the Hopf indices and Brouwer degrees of Φ-mapping under the condition that the zero points of field Ψ are regular points.