Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 071, 42 pages      arXiv:1009.1192      https://doi.org/10.3842/SIGMA.2010.071
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

Kazuki Hasebe
Kagawa National College of Technology, Mitoyo, Kagawa 769-1192, Japan

Received May 05, 2010, in final form August 19, 2010; Published online September 07, 2010; Note and references are added September 22, 2010

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.

Key words: division algebra; Clifford algebra; Grassmann algebra; Hopf map; non-Abelian monopole; Landau model; fuzzy geometry.

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