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Effective field theory description of the higher dimensional quantum Hall liquid

We derive an effective topological field theory model of the four dimensional quantum Hall liquid state recently constructed by Zhang and Hu. Using a generalization of the flux attachment transformation, the effective field theory can be formulated as a U(1) Chern-Simons theory over the total configuration space $CP_3$, or as a SU(2) Chern-Simons theory over $S^4$. The new quantum Hall liquid supports various types of topological excitations, including the 0-brane (particles), the 2-brane (membranes) and the 4-brane. There is a topological phase interaction among the membranes which generalizes the concept of fractional statistics.