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 $q\ne 0$) 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.

1+59 pages, 2 tables, explanations added in Sec.2.2, references added

References in corpus (11)

• Black Holes, Qubits and Octonions
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras
• Fuzzy Riemann Surfaces
• Hyperbolic Supersymmetric Quantum Hall Effect
• Instantons, Twistors, and Emergent Gravity
• Hyperboloid, instanton, oscillator
• Matrix models and the gravitational interaction
• N=4 chiral supermultiplet interacting with a magnetic field
• Non-Commutative Time, the Quantum Hall Effect and Twistor Theory
• Fuzzy de Sitter space-times via coherent states quantization

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• Cosmological space-times with resolved Big Bang in Yang-Mills matrix models
• Signature change in matrix model solutions
• Quantum (Matrix) Geometry and Quasi-Coherent States
• $Sp(4;\mathbb{R})$ Squeezing for Bloch Four-Hyperboloid via The Non-Compact Hopf Map
• Scalar modes and the linearized Schwarzschild solution on a quantized FLRW space-time in Yang-Mills matrix models
• Asymptotic commutativity of quantized spaces: the case of ${\mathbb{CP}}^{p,q}$
• Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis
• Spin-entangled Squeezed State on a Bloch Four-hyperboloid
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
• Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution
• Deformation of Matrix Geometry via Landau Level Evolution