Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

Details Hopf Maps, Lowest Landau Level, and Fuzzy Spheres Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

2010-09-07

arxiv.org/abs/1009.1192v3

SIGMA 6 (2010), 071, 42 pages

Physics

High Energy Physics

High Energy Physics - Theory

v2: note and references added; v3: references added

Scientific paper

10.3842/SIGMA.2010.071

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.

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