The effective action for edge states in higher dimensional quantum Hall systems

We show that the effective action for the edge excitations of a quantum Hall droplet of fermions in higher dimensions is generically given by a chiral bosonic action. We explicitly analyze the quantum Hall effect on complex projective spaces ${\bf CP}^k$ with a U(1) background magnetic field. The edge excitations are described by abelian bosonic fields on $S^{2k-1}$ with only one spatial direction along the boundary of the droplet relevant for the dynamics. Our analysis also leads to an action for edge excitations for the case of the Zhang-Hu four dimensional quantum Hall effect defined on $S^4$ with an SU(2) background magnetic field, using the fact that ${\bf CP}^3$ is an $S^2$ bundle over $S^4$.