Electromagnetic interactions of higher-dimensional quantum Hall droplets

Using a WN-gauge theory to describe electromagnetic interactions of spinless fermions in the lowest Landau level, where the WN transformations are nonlinear realizations of U(1) gauge transformations, we construct the effective action describing electromagnetic interactions of a higher-dimensional quantum Hall droplet. We also discuss how this is related to the Abelian Seiberg-Witten map. Explicit calculations are presented for the quantum Hall effect on CPk with U(1) background magnetic field. The bulk action is a Kähler-Chern-Simons term whose anomaly is canceled by a boundary contribution so that gauge invariance is explicitly satisfied.