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

Abstract

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 W N transformations. The requirement that the usual gauge transformation should be realized as a W N 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 cancelled by a boundary term given in terms of a gauged Wess-Zumino-Witten action.

Geometry of the quantum Hall effect: An effective action for all dimensions

“…This action thus yields the topological terms in the response of the system (or correlation functions of the source currents) to changes in the gauge and gravitational fields. The terms which involve only the gauge field had been obtained earlier for the lowest Landau level in a large N simplification, where N denotes the degeneracy of the Landau level [24]- [26]. Terms which involve both the gauge and the gravitational fields provide a generalization of the well known Wen-Zee term in the (2 + 1) dimensional case.…”

“…The effective action for this has also been obtained in the case of integer filling fraction ν = 1; it is a generalization of the WZW action [18,19,20,23]. Once fluctuations in the gauge field are also introduced, the calculated bulk and boundary actions were shown to be consistent with the mutual cancellation of anomalies in the gauge symmetry [24]. 1 The complete effective action captures the response of the system to various gauge field perturbations and edge fluctuations of the droplet.…”

“…And as in the (2 + 1)-dimensional case, one can consider a bulk effective action which captures the response to fluctuations of the gauge field. The topological part of this effective action is a generalization of the Chern-Simons action to higher dimensions [24,25,26]. Also in analogy with the lower dimensional case, one can consider a quantum Hall droplet which would then allow for edge excitations, even when the background gauge field is fixed, i.e., non-fluctuating.…”

Electromagnetic Excitations of Hall Systems on Four-Dimensional Space

“…Indeed, it was shown that Laughlin states at filling factor 1/k can be provided by an appropriate noncommutative finite Chern-Simons matrix model at level k and hence reproduces the basic features of quantum Hall states [24,25]. In connection with quantum Hall systems in higher dimensions [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], the ideas of the noncommutative geometry were useful to show that the effective action for the edge excitations of a quantum hall droplet is generically given by a chiral boson action [33][34][35][36][37][38][39][40][41][42][43][44]. In relation with these issues, the second main task of this paper concerns the electromagnetic excitations of Hall droplets in four-dimensional complex projective space.…”

Noncommutative Wess–Zumino–Witten actions and their Seiberg–Witten invariance

“…The Wess-Zumino-Witten (WZW) model has been of fundamental importance in various physical contexts. Generalizations of this model applying to different systems, such as theories defined on a noncommutative spacetime [1]- [6], higher dimensional quantum Hall theories [7,8] or higher dimensional bosonization [9,10], have also appeared.…”