Quantum Hall Effect in Higher Dimensions, Matrix Models and Fuzzy Geometry

We give a brief review of 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 whose large $N$ limit produces an effective action describing the gauge interactions of a higher dimensional quantum Hall droplet. The bulk action is a Chern-Simons type term whose anomaly is exactly cancelled by the boundary action given in terms of a chiral, gauged Wess-Zumino-Witten theory suitably generalized to higher dimensions. We argue that the gauge fields in the Chern-Simons action can be understood as parametrizing the different ways in which the large $N$ limit of the matrix theory is taken. The possible relevance of these ideas to fuzzy gravity is explained. Other applications are also briefly discussed.