Fuzzy spaces, the M(atrix) model and the quantum Hall effect

TL;DR

This paper reviews recent developments connecting fuzzy spaces, the M(atrix) model, and the quantum Hall effect, highlighting the role of large matrices and edge excitations on complex projective spaces.

Contribution

It provides an introduction to fuzzy spaces, explores their large matrix limits, and discusses their relation to quantum Hall physics and M(atrix) theory.

Findings

Fuzzy spaces can be constructed from large matrices approximating continuous manifolds.

Edge excitations in quantum Hall systems relate to properties of fuzzy spaces.

Connections between fuzzy geometries and M(atrix) theory are elucidated.

Abstract

This is a short review of recent work on fuzzy spaces in their relation to the M(atrix) theory and the quantum Hall effect. We give an introduction to fuzzy spaces and how the limit of large matrices is obtained. The complex projective spaces ${\bf CP}^k$, and to a lesser extent spheres, are considered. Quantum Hall effect and the behavior of edge excitations of a droplet of fermions on these spaces and their relation to fuzzy spaces are also discussed.