Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

TL;DR

This review explores the deep connections between monopoles, Landau levels, and fuzzy spheres through Hopf maps and algebraic generalizations, revealing hierarchical structures and supersymmetric extensions.

Contribution

It introduces a graded Hopf map using Grassmann algebra and relates fuzzy spheres to Landau models in higher dimensions, extending previous mathematical frameworks.

Findings

Hierarchical structure of higher-dimensional fuzzy spheres.

Relation between Hopf maps and monopoles.

Introduction of graded Hopf map and fuzzy supersphere.

Abstract

This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of "compounds" of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.