Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

Symmetry Integrability and Geometry-Methods and Applications

article

Kazuki Hasebe¹ 

[1] Kagawa National College of Technology

关键词: division algebra;    Clif ford algebra;    Grassmann algebra;    Hopf map;    non-Abelian monopole;    Landau model;    fuzzy geometry;   

DOI  :  10.3842/SIGMA.2010.071

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown