Abstract
We give a brief review of the Quantum Hall effect in higher dimensions and its relation to fuzzy spaces. For a quantum Hall system, the lowest Landau level dynamics is given by a one-dimensional matrix action. This can be used to write down a bosonized noncommutative field theory describing the interactions of higher dimensional nonrelativistic fermions with abelian or nonabelian gauge fields in the lowest Landau level. This general approach is applied explicitly to the case of QHE on CPk. It is shown that in the semiclassical limit the effective action contains 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 suitably generalized to higher dimensions.
Highlights
Quantum Hall effect in two dimensions is a very important physical phenomenon [1]
Our detailed formulation of higher dimensional QHE is based on the Landau problem on CPk, we start with a general matrix formulation of the dynamics of noninteracting fermions in the lowest Landau level, which eventually leads to a bosonization approach in terms of a noncommutative field theory
In the limit where N → ∞ and the number of fermions is large, that the action S separates into a boundary term describing the coupling of the quantum Hall droplet to the external gauge field Aμ, and a purely Aμ-dependent bulk term, which is a Chern-Simons like term
Summary
Quantum Hall effect in two dimensions is a very important physical phenomenon [1]. In addition to a variety of fascinating experimental results it has provided a clear theoretical framework for exploring a number of field/string theory ideas such as bosonization, conformal invariance, topological field theories, noncommutative geometry, D-brane physics, etc. In the case where the fermions have nonabelian degrees of freedom and couple to the full U (k) background gauge field, the wavefunctions have the same fixed U (1)R charge as in (13) but under right rotations transform as a particular SU (k) representation J′ of dimension N ′ = dimJ′ [5]. In this case, Ra Ψm;a′ = Ψm;b′ (Ta)b′a′. The dimension N of the SU (k + 1) representation J depends on the particular J′ representation chosen, but for large n
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
