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Takuma N.C.T.

Non-compact Hopf Maps, Quantum Hall Effect, and Twistor Theory. Supersymmetries and Quantum Symmetries, 29 Jul. ~ 4.Aug. 2009. Kazuki Hasebe. Takuma N.C.T. Takuma N.C.T. arXiv: 0902.2523, 0905.2792. Introduction. 1. Twistor Theory. (Mathematical Physics:

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Takuma N.C.T.

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  1. Non-compact Hopf Maps, Quantum Hall Effect, and Twistor Theory Supersymmetries and Quantum Symmetries, 29 Jul. ~ 4.Aug. 2009 Kazuki Hasebe Takuma N.C.T. Takuma N.C.T. arXiv: 0902.2523, 0905.2792

  2. Introduction 1. Twistor Theory (Mathematical Physics: Relativistic Quantum Mechanics) Quantization of Space-Time Light has special importance. ADHM Construction, Integrable Models. Twistor String etc. 2. Quatum Hall Effect (Condensed matter: Non-relativistic Quantum Mechanics) Novel Quantum State of Matter Monopole plays an important role. Quatum Spin Hall Effect, Quantum Hall Effect in Graphene etc. There are remarkable close relations between these two independently developed fields !

  3. Brief Introduction of Twistors

  4. Space-Time Twistor Space ``moduli space of light’’ Twistor Program Roger Penrose (1967) Quantization of Space-Time What is the fundamental variables ? Light (massless-paticle) will play the role ! Philosophy Quantized space–time will be induced. Quantize not .

  5. Free particle Massless particle : Incidence Relation : Massless Free Particle Gauge symmetry

  6. Twistor Description Massless limit Fundamental variable Helicity: Fuzzy twistor space

  7. Hopf Maps and QHE

  8. To keep finite, is kept finite. LLL projection ``massless limit’’ Landau Quantization Magnetic Field 2D - plane Cyclotron frequency 2nd LL Landau levels 1st LL LLL Lev Landau (1930)

  9. Quantum Hall Effect and Monopole R. Laughlin (1983) Stereographic projection SO(3) symmetry F.D.M. Haldane (1983) Many-body state on a sphere in a monopole b.g.d.

  10. Dirac Monopole and 1st Hopf Map The 1st Hopf map Dirac Monopole P.A.M. Dirac (1931)

  11. Explicit Realization of 1st Hopf Map Connection of fibre Hopf spinor

  12. LLL Fundamental variable One-particle Mechanics Lagrangian Constraint LLL Lagrangian Constraint

  13. Emergence of Fuzzy Geometry Holomorphic wavefunctions No LLL Physics Fuzzy Sphere

  14. Many-body state Laughlin-Haldane wavefunction The groundstate is invariant under SU(2) isometry of , and does not include complex conjugations. : SU(2) singlet combination of Hopf spinors

  15. QHE with Higher Symmetry

  16. Hopf Maps Topological maps from sphere to sphere with different dimensions. ONLY THREE ! Heinz Hopf(1931,1935) 1st (Complex number) 2nd (Quaternion) 3rd (Octonion)

  17. The 2nd Hop Map & SU(2) Monopole SO(5) global symmetry The 2nd Hopf map Yang Monopole C.N. Yang (1978)

  18. In the LLL 4D QHE and Twistor Many-body problem on a four-sphere in a SU(2) monopole b.g.d. S.C. Zhang, J.P. Hu (2001) Point out relations to Twistor theory D. Karabali,V.P. Nair(2002,2003) S.C. Zhang (2002) In particular, Sparling and his coworkers suggested the use of the ultra-hyperboloid G. Sparling (2002) D. Mihai, G. Sparling, P. Tillman(2004)

  19. Short Summary QHE Hopf Map Monopole 1st U(1) 2D 4D SU(2) 2nd LLL Twistor ?? 8D SO(8) 3rd

  20. QHE with SU(2,2) symmetry

  21. Noncompact Version of the Hopf Map Hopf maps Non-compact Hopf maps ! Non-compact groups James Cockle (1848,49) Complex number Split-Complex number Quaternions Split-Quaternions Octonions Split-Octonions

  22. : Ultra-Hyperboloid with signature (p,q) p q+1 Non-compact Hopf Maps 1st (Split-complex number) 2nd (Split-quaternion) 3rd NO OTHER ! (Split-octonion)

  23. Non-compact 2nd Hopf Map SO(3,2) gamma matrices The fibre : (c.f.)

  24. SO(3,2) Hopf spinor SO(3,2) Hopf spinor generators Incidence Relation Stereographic coordinates

  25. One-particle Mechanics on Hyperboloid SU(1,1) monopole One-particle action constraint SO(3,2) symmetry

  26. LLL projection LLL-limit Fundamental variable Symmetry is Enhanced from SO(3,2) to SU(2,2)! constraint SU(2,2) symmetry

  27. Realization of the fuzzy geometry satisfy SU(2,2) algebra. This demonstrates the philosophy of Twistor ! First. the Hopf spinor space becomes fuzzy. Then, the hyperboloid also becomes fuzzy. The space(-time) non-commutativity comes from that of the more fundamental space.

  28. Twistor QHE Quantize and rather than ! Analogies • Massless Condition SU(2,2) Enhanced Symmetry • More Fundamental Quantity than Space-Time • Complex conjugation = Derivative Noncommutative Geometry, Holomorphic functions

  29. Table Non-compact 4D QHE Twistor Theory Fundamental Quantity Hopf spinor Twistor Quantized value Monopole charge Helicity Base manifold Hyperboloid Minkowski space Original symmetry Poincare Special limit LLL zero-mass Enhanced symmetry Emergent manifold Noncommutative Geometry Fuzzy Hyperboloid Fuzzy Twistor space

  30. Physics of the non-compact 4D QHE

  31. One-particle Problem Landau problem on a ultra-hyperboloid Thermodynamic limit : fixed

  32. Many-body Groudstate The groundstate is invariant under SO(3,2) isometry of , and does not include complex conjugations. On the QH groundstate, particles are distributed uniformly on the basemanifold. Higher D. Laughlin-Haldane wavefunction

  33. Topological Excitations • Topological excitations are generated by flux penetrations. • The flux has SU(1,1) internal structures. Membrane-like excitations !

  34. Particular Features

  35. Uniqueness Everything is uniquely determined by the geometry of the Hopf map ! (For instance) n-c. 2nd Hopf map Gauge Symmetry Fundamental space Base manifold Global symmetry :

  36. Extra-Time Physics ? Base manifold 2T Gauge Symmetry • This set-up exactly corresponds to 2T physics developed by I. Bars ! • Sp(2,R) gauge symmetry is required to eliminate the negative norms. • The present model geometrically fulfills this requirement ! • There may be some kind of ``duality’’ ?? Hull, Khuri (98,00), Andrade, Rojas, Toppan (01)

  37. Magic Dimensions of Space-Time ? Compact Hopf maps Non-compact maps 1st 2nd 3rd

  38. Non-compact Hopf Maps Split-algebras Non-commutative Geometry Higher D. quantum liquid Membrane-like excitation Twistor Theory Uniqueness Magic Dimensions Extra-time physics Exotic Math and Physical Concepts

  39. We have seen close relations between QHE and Twistors. • Immediate Questions • Supertwistors and Super Landau models ? (Prof.Mezincescu’s talk) • Noncompact Super Hopf Maps? (Prof.Toppan’s talk) etc. etc. • Deeper reasons for the analogies? • The Entire Picture is still a Mystery. END

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