Split-Quaternionic Hopf Map, Quantum Hall Effect, and Twistor Theory

Abstract

Introducing a non-compact version of the Hopf map, we demonstrate remarkable close relations between quantum Hall effect and twistor theory. We first construct quantum Hall effect on a hyperboloid based on the noncompact 2nd Hopf map of split-quaternions. We analyze a hyperbolic one-particle mechanics, and explore many-body problem, where a many-body groundstate wavefunction and membrane-like excitations are derived explicitly. In the lowest Landau level, the symmetry is enhanced from $SO(3,2)$ to the $SU(2,2)$ conformal symmetry. We point out that the quantum Hall effect naturally realizes the philosophy of twistor theory. In particular, the emergence mechanism of fuzzy space-time is discussed somehow in detail.

5 pages, 1 table, no figures, references added, published version

References in corpus (7)

• Quantum Hall Effect in Higher Dimensions, Matrix Models and Fuzzy Geometry
• Supersymmetric Quantum Hall Effect on Fuzzy Supersphere
• Fuzzy Supersphere and Supermonopole
• Quantum Hall Effect on Higher Dimensional Spaces
• Hyperbolic Supersymmetric Quantum Hall Effect
• Twistors and 2T-physics
• Hyperboloid, instanton, oscillator

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• $SO(5)$ Landau Model and 4D Quantum Hall Effect in The $SO(4)$ Monopole Background
• Extensions of real numbers using coset groups
• Spin-entangled Squeezed State on a Bloch Four-hyperboloid
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry