Non-compact Hopf maps and fuzzy ultra-hyperboloids Original Research Article

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, image, are represented by the coset, image, and exhibit two types of generalized dimensional hierarchy; hyperbolic-type (for image) and hybrid-type (for image). 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.