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## Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids

### Abstract

Fuzzy hyperboloids naturally emerge in the geometries of D-branes, twistor theory, and higher spin theories. In this work, we perform a systematic study of higher dimensional fuzzy hyperboloids (ultra-hyperboloids) based on non-compact Hopf maps. Two types of non-compact Hopf maps; split-type and hybrid-type, are introduced from the cousins of division algebras. We construct arbitrary even-dimensional fuzzy ultra-hyperboloids by applying the Schwinger operator formalism and indefinite Clifford algebras. It is shown that fuzzy hyperboloids, \$H_F^{2p,2q}\$, are represented by the coset, \$H_F^{2p,2q}\simeq SO(2p,2q+1)/U(p,q)\$, and exhibit two types of generalized dimensional hierarchy; hyperbolic-type (for \$q\neq 0\$) and hybrid-type (for \$q=0\$). Fuzzy hyperboloids can be expressed as fibre-bundle of fuzzy fibre over hyperbolic basemanifold. Such bundle structure of fuzzy hyperboloid gives rise to non-compact monopole gauge field. Physical realization of fuzzy hyperboloids is argued in the context of lowest Landau level physics.Comment: 1+59 pages, 2 tables, explanations added in Sec.2.2, references adde

Topics: High Energy Physics - Theory, Mathematical Physics, Quantum Physics
Year: 2012
DOI identifier: 10.1016/j.nuclphysb.2012.07.017
OAI identifier: oai:arXiv.org:1207.1968