A four-dimensional generalization of the quantum Hall effect.

We construct a generalization of the quantum Hall effect, where particles move in four dimensional space under a SU(2) gauge field. This system has a macroscopic number of degenerate single particle states. At appropriate integer or fractional filling fractions the system forms an incompressible quantum liquid. Gapped elementary excitation in the bulk interior and gapless elementary excitations at the boundary are investigated.