Bosonization of the lowest Landau level in arbitrary dimensions: Edge and bulk dynamics

We discuss the bosonization of nonrelativistic fermions interacting with non-Abelian gauge fields in the lowest Landau level in the framework of higher-dimensional quantum Hall effect. The bosonic action is a one-dimensional matrix action, which can also be written as a noncommutative field theory, invariant under WNWN transformations. The requirement that the usual gauge transformation should be realized as a WNWN transformation provides an analog of a Seiberg–Witten map, which allows us to express the action purely in terms of bosonic fields. The semiclassical limit of this, describing the gauge interactions of a higher-dimensional, non-Abelian quantum Hall droplet, produces a bulk Chern–Simons type term whose anomaly is exactly canceled by a boundary term given in terms of a chiral, gauged Wess–Zumino–Witten action.