1. Takuma N.C.T. Graduate School of Mathematics, Nagoya University, 23 Jul. 2009 Non-compact Hopf Maps, Quantum Hall Effect, and Twistor Theory Takuma N.C.T. Kazuki Hasebe arXiv: 0902.2523, 0905.2792

2. Introduction There are remarkable close relations between these two independently developed fields ! 2. Quatum Hall Effect Novel Quantum State of Matter (Condensed matter: Non-relativistic Quantum Mechanics) Quatum Spin Hall Effect, Quantum Hall Effect in Graphene etc. 1. Twistor Theory Quantization of Space-Time (Mathematical Physics: Relativistic Quantum Mechanics) ADHM Construction, Integrable Models. Twistor String etc. Light has special importance. Monopole plays an important role. R. Laughlin (1983) R. Penrose (1967)

3. Landau Quantization 2D - plane Magnetic Field Landau levels LLL 1st LL 2nd LL LLL projection ``massless limit’’ Cyclotron frequency Lev Landau (1930)

4. Quantum Hall Effect and Monopole Stereographic projection F.D.M. Haldane (1983) Many-body state on a sphere in a monopole b.g.d. SO(3) global symmetry ``Edge’’ breaks translational sym.

5. Heuristic Observation: Why Light & Monopole Massless particle with helicity To satisfy the SU(2) algebra Massless particle ``sees’’ a charge monopole in p-space ! spin momentum The position of a massless particle with definite helicity is uncertain ! Bacry (1981) Atre, Balachandran, Govindarajan (1986) If

6. Brief Introduction to Twistor

7. Twistor Program Roger Penrose (1967) Quantization of Space-Time What is the fundamental variables ? Light (massless-paticle) will play the role ! Space-TimeTwistor Space ``moduli space of light’’

8. Massless Free Particle Massless particle Free particle : Gauge symmetry

9. Twistor Description Suggests a fuzzy space-time. Massless limit Fundamental variable Helicity: : Incidence Relation

10. Hopf Maps and QHE

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

12. Connection of bundle Explicit Realization of 1 st Hopf Map Hopf spinor

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

14. LLL Physics Emergence of Fuzzy Geometry Holomorphic wavefunctions

15. Many-body Groudstate Laughlin-Haldane wavefunction On the QH groundstate, particles are distributed uniformly on the basemanifold. The groundstate is invariant under SU(2) isometry of, and does not include complex conjugations. : SU(2) singlet combination of Hopf spinors

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

17. Quaternion and 2 nd Hopf Map Willian R. Hamilton (1843) Quaternion2nd Hopf map Unit 1st Hopf map Unit C C

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

19. 4D QHE and Twistor D. Mihai, G. Sparling, P. Tillman (2004) S.C. Zhang, J.P. Hu (2001) Many-body problem on a four- sphere in a SU(2) monopole b.g.d. In the LLL Point out relations to Twistor theory In particular, Sparling and his coworkers suggested the use of the ultra-hyperboloid G. Sparling (2002) D. Karabali, V.P. Nair (2002,2003) S.C. Zhang (2002)

20. Short Summary QHEHopf MapDivision algebra 2D 4D 8D 1st 2nd 3rd complex numbers quaternions octonions LLL Twistor ??

21. QHE with SU(2,2) symmetry

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

23. Split-Algebras Split-Complex number Split-Quaternions

24. Non-compact Hopf Maps 1st 2nd 3rd Ultra-Hyperboloid with signature (p,q) : pq+1

25. Non-compact 2nd Hopf Map

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

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

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

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

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

31. Summary Table Non-compact 4D QHETwistor Theory Fundamental Quantity Quantized value Monopole chargeHelicity Base manifold HyperboloidMinkowski space Original symmetry Hopf spinorTwistor Fuzzy HyperboloidFuzzy Twistor space Noncommutative Geometry Emergent manifold Enhanced symmetry Special limit Poincare LLLzero-mass

32. Physics of the non-compact 4D QHE

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

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

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

36. Perspectives

37. Uniqueness Everything is uniquely determined by the geometry of the Hopf map ! Base manifold Gauge Symmetry Global Symmetry in LLL (For instance) n-c. 2 nd Hopf map

38. Extra-Time Physics ? Sp(2,R) gauge symmetry is required to eliminate the negative norms. Base manifold Gauge Symmetry 2T The present model fulfills this requirement from the very beginning ! There may be some kind of ``duality’’ ?? This set-up exactly corresponds to 2T physics developed by I. Bars ! C.M. Hull (99)

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

40. After ALL Split-algebras Higher D. quantum liquid Membrane-like excitation Non-compact Hopf Maps Non-commutative Geometry Twistor Theory Uniqueness Extra-time physicsMagic Dimensions

41. END Deep mathematical structure exists behind the model ! Entire picture is still Mystery !