Electromagnetic interactions of higher dimensional quantum Hall droplets

Using a $W_N$-gauge theory to describe electromagnetic interactions of spinless fermions in the lowest Landau level, where the $W_N$ 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 ${\mathbf{C}\mathbf{P}}^k$ with U(1) background magnetic field. The bulk action is a K"ahler-Chern-Simons term whose anomaly is cancelled by a boundary contribution so that gauge invariance is explicitly satisfied.