Various topological excitations in the SOÂ (4) gauge field in higher dimensions

Following the original analysis of Zhang and Hu for the 4-dimensional generalization of Quantum Hall effect, there has been much work from different viewpoints on the higher dimensional condensed matter systems. In this paper, we discuss three kinds of topological excitations in the SO (4) gauge field of condensed matter systems in 4-dimension-the instantons and anti-instantons, the 't Hooft-Polyakov monopoles, and the 2-membranes. Using the ϕ-mapping topological theory, it is revealed that there are 4-, 3-, and 2-dimensional topological currents inhering in the SO (4) gauge field, and the above three kinds of excitations can be directly and explicitly derived from these three kinds of currents, respectively. Moreover, it is shown that the topological charges of these excitations are characterized by the Hopf indices and Brouwer degrees of ϕ-mapping.