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періодичних видань НАН України
періодичних видань НАН України
Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2010, том 6
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Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
Інший ID:
2010 Mathematics Subject Classification: 17B70; 58B34; 81V70
DOI:10.3842/SIGMA.2010.071
DOI:10.3842/SIGMA.2010.071
Посилання:
Hopf Maps, Lowest Landau Level, and Fuzzy Spheres / K. Hasebe // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 102 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The ful`l collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html.
I would like to thank Yusuke Kimura for collaborations. Many crucial ingredients in this review are based on the works with him. I am also indebted to Takehiro Azuma for email correspondence about mathematics of fuzzy spheres. Since this article is a review-type, many important works not performed by the author are included. Hereby, I express my gratitude to the researchers whose works are reviewed in the paper.
Дата:
2010
Переглядів:
540
Завантажень:
324
Анотація:
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.
