Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 071, 42 pages      arXiv:1009.1192      http://dx.doi.org/10.3842/SIGMA.2010.071
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

Kazuki Hasebe
Kagawa National College of Technology, Mitoyo, Kagawa 769-1192, Japan

Received May 05, 2010, in final form August 19, 2010; Published online September 07, 2010; Note and references are added September 22, 2010

Abstract
This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of ''compounds'' of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.

Key words: division algebra; Clifford algebra; Grassmann algebra; Hopf map; non-Abelian monopole; Landau model; fuzzy geometry.

pdf (515 kb)   ps (271 kb)   tex (44 kb)       [previous version:  pdf (512 kb)   ps (270 kb)   tex (44 kb)]

References

  1. Madore J., The fuzzy sphere, Classical Quantum Gravity 9 (1992), 69-87.
  2. Grosse H., Klimcík C., Presnajder P., On finite 4D quantum field theory in non-commutative geometry, Comm. Math. Phys. 180 (1996), 429-438, hep-th/9602115.
  3. Grosse H., Klimcík C., Presnajder P., Field theory on a supersymmetric lattice, Comm. Math. Phys. 185 (1997), 155-175, hep-th/9507074.
  4. Grosse H., Reiter G., The fuzzy supersphere, J. Geom. Phys. 28 (1998), 349-383, math-ph/9804013.
  5. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  6. de Boer J., Grassi P.A., van Nieuwenhuizen P., Non-commutative superspace from string theory, Phys. Lett. B 574 (2003), 98-104, hep-th/0302078.
  7. Berkovits N., Seiberg N., Superstrings in graviphoton background and N=1/2+3/2 supersymmetry, J. High Energy Phys. 2003 (2003), no. 7, 010, 10 pages, hep-th/0306226.
  8. Myers R.C., Dielectric-branes, J. High Energy Phys. 1999 (1999), no. 12, 022, 41 pages, hep-th/9910053.
  9. Castelino J., Lee S., Taylor W., Longitudinal 5-branes as 4-spheres in matrix theory, Nuclear Phys. B 526 (1998), 334-350, hep-th/9712105.
  10. Iso S., Umetsu H., Gauge theory on noncommutative supersphere from supermatrix model, Phys. Rev. D 69 (2004), 1050033, 7 pages, hep-th/0311005.
  11. Constable N.R., Myers R.C., Tafjord O., Noncommutative bion core, Phys. Rev. D 61 (2000), 106009, 14 pages, hep-th/9911136.
    Constable N.R., Myers R.C., Tafjord O., Non-Abelian brane intersections, J. High Energy Phys. 2001 (2001), no. 6, 023, 37 pages, hep-th/0102080.
  12. Cook P.L.H., de Mello Koch R., Murugan J., Non-Abelian BIonic brane intersections, Phys. Rev. D 68 (2003), 126007, 8 pages, hep-th/0306250.
  13. Bhattacharyya R., de Mello Koch R., Fluctuating fuzzy funnels, J. High Energy Phys. 2005 (2005), no. 10, 036, 20 pages, hep-th/0508131.
  14. Ho P.-M., Ramgoolam S., Higher-dimensional geometries from matrix brane constructions, Nuclear Phys. B 627 (2002), 266-288, hep-th/0111278.
  15. Kimura Y., Noncommutative gauge theory on fuzzy four-sphere and matrix model, Nuclear Phys. B 637 (2002), 177-198, hep-th/0204256.
    Kimura Y., On higher-dimensional fuzzy spherical branes, Nuclear Phys. B 664 (2003), 512-530, hep-th/0301055.
  16. Ramgoolam S., Higher dimensional geometries related to fuzzy odd-dimensional spheres, J. High Energy Phys. 2002 (2002), no. 10, 064, 29 pages, hep-th/0207111.
  17. Nair V.P., Randjbar-Daemi S., Quantum Hall effect on S3, edge states and fuzzy S3/Z2, Nuclear Phys. B 679 (2004), 447-463, hep-th/0309212.
  18. Taylor W., Lectures on D-branes, gauge theory and M(atrices), hep-th/9801182.
  19. Balachandran A.P., Quantum spacetimes in the year 1, Pramana J. Phys. 59 (2002), 359-368, hep-th/0203259.
  20. Karabali D., Nair V.P., Randjbar-Daemi S., Fuzzy spaces, the M(atrix) model and the quantum Hall effect, hep-th/0407007.
  21. Azuma T., Matrix models and the gravitational interaction, PhD thesis, Kyoto University, 2004, hep-th/0401120.
  22. Balachandran A.P., Kurkcuoglu S., Vaidya S., Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114.
  23. Abe Y., Construction of fuzzy spaces and their applications to matrix models, PhD Thesis, The City University of New York, 2005, arXiv:1002.4937.
  24. Karabali D., Nair V.P., Quantum Hall effect in higher dimensions, matrix models and fuzzy geometry, J. Phys. A: Math. Gen. 39 (2006), 12735-12763, hep-th/0606161.
  25. Hatsuda M., Iso S., Umetsu H., Noncommutative superspace, supermatrix and lowest Landau level, Nuclear Phys. B 671, (2003), 217-242, hep-th/0306251.
  26. Hamilton W.R., On a new species of imaginary quantities connected with a theory of quaternions, Proc. R. Ir. Acad. 2 (1844), 424-434.
  27. Baez J.C., The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 145-205, Errata, Bull. Amer. Math. Soc. (N.S.) 42 (2005), 213-213, math.RA/0105155.
  28. Hopf H., Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931), 637-665.
  29. Hopf H., Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension, Fund. Math. 25 (1935), 427-440.
  30. Clifford W.K., Applications of Grassmann's extensive algebra, Amer. J. Math. 1 (1878), 350-358.
  31. Grassmann H., Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik II, 1844.
  32. Landi G., Marmo G., Extensions of Lie superalgebras and supersymmetric Abelian gauge fields, Phys. Lett. B 193 (1987), 61-66.
  33. Dirac P., Quantised singularities in the electromagnetic field, Proc. R. Soc. Lond. Ser. A 133 (1931), 60-72.
  34. Yang C.N., Generalization of Dirac's monopole to SU2 gauge fields, J. Math. Phys. 19 (1978), 320-328.
  35. Grossman B., Kephart T.W., Stasheff J.D., Solutions to Yang-Mills field equations in eight-dimensions and the last Hopf map, Comm. Math. Phys. 96 (1984), 431-437, Erratum, Comm. Math. Phys. 100 (1985), 311-311.
  36. Nakahara M., Geometry, topology and physics, 2nd ed., Graduate Student Series in Physics, Institute of Physics, Bristol, 2003.
  37. Haldane F.D.M., Fractional quantization of the Hall effects: a hierarchy of incompressible quantum fluid states, Phys. Rev. Lett. 51 (1983), 605-608.
  38. Wu T.T., Yang C.N., Dirac monopole without strings: monopole harmonics, Nuclear Physics B 107 (1976), 365-380.
  39. Hasebe K., Kimura Y., Fuzzy supersphere and supermonopole, Nuclear Phys. B 709 (2005), 94-114, hep-th/0409230.
  40. Bartocci C., Bruzzo U., Landi G., Chern-Simons forms on principal superfiber bundles, J. Math. Phys. 31 (1990), 45-54.
  41. Landi G., Projective modules of finite type over the supersphere S2,2, Differential Geom. Appl. 14 (2001), 95-111, math-ph/9907020.
  42. Pais A., Rittenberg V., Semisimple graded Lie algebras, J. Math. Phys. 16 (1975), 2062-2073, Erratum, J. Math. Phys. 17 (1976), 598-598.
  43. Scheunert M., Nahm W., Rittenberg V., Irreducible representations of the osp(2,1) and spl(2,1) graded Lie algebra, J. Math. Phys. 18 (1977), 155-162.
  44. Marcu M., The representations of spl(2,1) - an example of representations of basic superalgebras, J. Math. Phys. 21 (1980), 1277-1283.
  45. Frappat L., Sciarrino A., Sorba P., Dictionary on Lie algebras and superalgebras, Academic Press, Inc., San Diego, CA, 2000.
  46. Balachandran A.P., Kürkçüoglu S., Rojas E., The star product on the fuzzy supersphere, J. High Energy Phys. 2002 (2002), no. 7, 056, 22 pages, hep-th/0204170.
  47. Zhang S.-C., Hu J.-P., A four-dimensional generalization of the quantum Hall effect, Science 294 (2001), no. 5543, 823-828, cond-mat/0110572.
  48. 't Hooft G., Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D 14 (1976), 3432-3450.
  49. Iachello F., Lie algebras and applications, Lecture Notes in Physics, Vol. 708, Springer-Verlag, Berlin, 2006.
  50. Yang C.N., SU2 monopole harmonics, J. Math. Phys. 19 (1978), 2622-2627.
  51. Bernevig B.A., Chern C.-H., Hu J.-P., Toumbas N., Zhang S.C., Effective field theory description of the higher-dimensional quantum Hall liquid, Ann. Physics 300 (2002), 185-207, cond-mat/0206164.
  52. Ramgoolam S., On spherical harmonics for fuzzy spheres in diverse dimensions, Nuclear Phys. B 610 (2001), 461-488, hep-th/0105006.
  53. Azuma T., Bagnoud M., Curved-space classical solutions of a massive supermatrix model, Nuclear Phys. B 651 (2003), 71-86, hep-th/0209057.
  54. Abe Y., Construction of fuzzy S4, Phys. Rev. D 70 (2004), 126004, 10 pages, hep-th/0406135.
    Abe Y., Emergence of longitudinal 7-branes and fuzzy S4 in compactification scenarios of M(atrix) theory, hep-th/0512174.
  55. Bernevig B.A., Hu J.-P., Toumbas N., Zhang S.-C., Eight-dimensional quantum Hall effect and "octonions", Phys. Rev. Lett. 91 (2003), 236803, 4 pages, cond-mat/0306045.
  56. Hasebe K., Kimura Y., Dimensional hierarchy in quantum Hall effects on fuzzy spheres, Phys. Lett. B 602 (2004), 255-260, hep-th/0310274.
  57. Fabinger M., Higher-dimensional quantum Hall effect in string theory, J. High Energy Phys. 2002 (2002), no. 5, 037, 12 pages, hep-th/0201016.
  58. Meng G., Geometric construction of the quantum Hall effect in all even dimensions, J. Phys. A: Math. Gen. 36 (2003), 9415-9423, cond-mat/0306351.
  59. Horváth Z., Palla L., Spontaneous compactification and "monopoles" in higher dimensions, Nuclear Phys. B 142 (1978), 327-343.
  60. Tchrakian D.H., N-dimensional instantons and monopoles, J. Math. Phys. 21 (1980), 166-169.
  61. Saclioglu C., Scale invariant gauge theories and self-duality in higher dimensions, Nuclear Phys. B 277 (1986), 487-508.
  62. Kimura Y., Nonabelian gauge field and dual description of fuzzy sphere, J. High Energy Phys. 2004 (2004), no. 4, 058, 29 pages hep-th/0402044.
  63. Hasebe K., Split-quaternionic Hopf map, quantum Hall effect, and twistor theory, Phys. Rev. D 81 (2010), 041702, 5 pages, arXiv:0902.2523.
  64. Ivanov E., Mezincescu L., Townsend P.K., A super-flag Landau model, hep-th/0404108.
  65. Hasebe K., Quantum Hall liquid on a noncommutative superplane, Phys. Rev. D 72 (2005), 105017, 9 pages, hep-th/0503162.
  66. Ivanov E., Mezincescu L., Townsend P.K., Planar super-Landau models, J. High Energy Phys. 2006 (2006), no. 1, 143, 23 pages, hep-th/0510019.
  67. Curtright T., Ivanov E., Mezincescu L., Townsend P.K. Planar super-Landau models revisited, J. High Energy Phys. 2007 (2007), no. 4, 020, 25 pages, hep-th/0612300.
  68. Beylin A., Curtright T., Ivanov E., Mezincescu L., Townsend P.K., Unitary spherical super-Landau models, J. High Energy Phys. 2008 (2008), no. 10, 069, 46 pages, arXiv:0806.4716.
  69. Beylin A., Curtright T., Ivanov E., Mezincescu L., Generalized N=2 super Landau models, arXiv:1003.0218.
  70. Ivanov E., Mezincescu L., Townsend P.K., Fuzzy CP(n|m) as a quantum superspace, hep-th/0311159.
  71. Murray S., Sämann C., Quantization of flag manifolds and their supersymmetric extensions, Adv. Theor. Math. Phys. 12 (2008), 641-710, hep-th/0611328.
  72. Landi G., Pagani C., Reina C., A Hopf bundle over a quantum four-sphere from the symplectic group, Comm. Math. Phys. 263 (2006), 65-88, math.QA/0407342.
  73. Hasebe K., The split-algebras and non-compact Hopf maps, J. Math. Phys. 51 (2010), 053524, 35 pages, arXiv:0905.2792.
  74. Faria Carvalho L., Kuznetsova Z., Toppan F., Supersymmetric extension of Hopf maps: N=4 σ-models and the S3S2 fibration, Nuclear Phys. B 834 (2010), 237-257, arXiv:0912.3279.
  75. Mkrtchyan R., Nersessian A., Yeghikyan V., Hopf maps and Wigner's little groups, arXiv:1008.2589.
  76. Bellucci S., Nersessian A., Yeranyan A., Hamiltonian reduction and supersymmetric mechanics with Dirac monopole, Phys. Rev. D 74 (2006), 065022, 7 pages, hep-th/0606152.
  77. Gonzales M., Kuznetsova Z., Nersessian A., Toppan F., Yeghikyan V., Second Hopf map and supersymmetric mechanics with Yang monopole, Phys. Rev. D 80 (2009), 025022, 13 pages, arXiv:0902.2682.
  78. Bellucci S., Toppan F., Yeghikyan V., Second Hopf map and Yang-Coulomb system on 5D (pseudo)sphere, J. Phys. A: Math. Theor. 43 (2010), 045205, 12 pages, arXiv:0905.3461.
  79. Fedoruk S., Ivanov E., Lechtenfeld O., OSp(4|2) superconformal mechanics, J. High Energy Phys. 2009 (2009), no. 8, 081, 24 pages, arXiv:0905.4951.
  80. Bellucci S., Krivonos S., Sutulin A., Three dimensional N=4 supersymmetric mechanics with Wu-Yang monopole, arXiv:0911.3257.
  81. Ivanov E.A., Konyushikhin M.A., Smilga A.V., SQM with non-Abelian self-dual fields: harmonic superspace description, J. High Energy Phys. 2010 (2010), no. 5, 033, 14 pages, arXiv:0912.3289.
  82. Ivanov E., Konyushikhin M., N=4, 3D supersymmetric quantum mechanics in non-Abelian monopole background, arXiv:1004.4597.
  83. Krivonos S., Lechtenfeld O., Sutulin A., N=4 supersymmetry and the Belavin-Polyakov-Shvarts-Tyupkin instanton, Phys. Rev. D 81 (2010), 085021, 7 pages, arXiv:1001.2659.
  84. Ishii T., Ishiki G., Shimasaki S., Tsuchiya A., T-duality, fiber bundles and matrices, J. High Energy Phys. 2007 (2007), no. 5, 014, 20 pages, hep-th/0703021.
    Ishii T., Ishiki G., Shimasaki S., Tsuchiya A., Fiber bundles and matrix models, Phys. Rev. D 77 (2008), 126015, 25 pages, arXiv:0802.2782.
  85. Pedder C., Sonner J., Tong D., The geometric phase and gravitational precession of D-branes, Phys. Rev. D 76 (2007), 126014, 10 pages, arXiv:0709.2136.
    Pedder C., Sonner J., Tong D., The Berry phase of D0-branes, J. High Energy Phys. 2008 (2008), no. 3, 065, 14 pages, arXiv:0801.1813.
  86. Nastase H., Papageorgakis C., Ramgoolam S., The fuzzy S2 structure of M2-M5 systems in ABJM membrane theories, J. High Energy Phys. 2009 (2009), no. 5, 123, 61 pages, arXiv:0903.3966.
  87. Nastase H., Papageorgakis C., Fuzzy Killing spinors and supersymmetric D4 action on the fuzzy 2-sphere from the ABJM model, J. High Energy Phys. 2009 (2009), no. 12, 049, 52 pages, arXiv:0908.3263.
  88. Arovas D.P., Auerbach A., Haldane F.D.M., Extended Heisenberg models of antiferromagnetism: analogies to the fractional quantum Hall effect, Phys. Rev. Lett. 60 (1988), 531-534.
  89. Arovas D.P., Hasebe K., Qi X.-L., Zhang S.-C., Supersymmetric valence bond solid states, Phys. Rev. B 79 (2009), 224404, 20 pages, arXiv:0901.1498.
  90. Hasebe K., Supersymmetric quantum Hall effect on a fuzzy supersphere, Phys. Rev. Lett. 94 (2005), 206802, 4 pages, hep-th/0411137.
  91. Landi G., Spin-Hall effect with quantum group symmetry, Lett. Math. Phys. 75 (2006), 187-200, hep-th/0504092.
  92. Jellal A., Quantum Hall effect on higher-dimensional spaces, Nuclear Phys. B 725 (2005), 554-576, hep-th/0505095.
  93. Hasebe K., Hyperbolic supersymmetric quantum Hall effect, Phys. Rev. D 78 (2008), 125024, 13 pages, arXiv:0809.4885.
  94. Tu H.-H., Zhang G.-M., Xiang T., Liu Z.-X., Ng T.-K., Topologically distinct classes of valence bond solid states with their parent Hamiltonians, Phys. Rev. B 80 (2009), 014401, 11 pages, arXiv:0904.0550.
  95. Asorey M., Esteve J.G., Pacheco A.F., Planar rotor: the θ-vacuum structure, and some approximate methods in quantum mechanics, Phys. Rev. D 27 (1983), 1852-1868.
  96. Karabali D., Nair V.P., Quantum Hall effect in higher dimensions, Nuclear Phys. B 641 (2002), 533-546, hep-th/0203264.
  97. Perelomov A.M., Coherent states for arbitrary Lie group, Comm. Math. Phys. 26 (1972), 222-236, math-ph/0203002.
  98. Alexanian G., Balachandran A.P., Immirzi G., Ydri B., Fuzzy CP2, J. Geom. Phys. 42 (2002), 28-53, hep-th/0103023.
  99. Balachandran A.P., Dolan B.P., Lee J., Martin X., O'Connor D., Fuzzy complex projective spaces and their star-products, J. Geom. Phys. 43 (2002), 184-204, hep-th/0107099.
  100. Carow-Watamura U., Steinacker H., Watamura S., Monopole bundles over fuzzy complex projective spaces, J. Geom. Phys. 54 (2005), 373-399, hep-th/0404130.
  101. Sheikh-Jabbari M.M., Tiny graviton matrix theory: DLCQ of IIB plane-wave string theory, a conjecture, J. High Energy Phys. 2004 (2004), no. 9, 017, 29 pages, hep-th/0406214.
  102. Sheikh-Jabbari M.M., Torabian M., Classification of all 1/2 BPS solutions of the tiny graviton matrix theory, J. High Energy Phys. 2005 (2005), no. 4, 001, 36 pages, hep-th/0501001.

Previous article   Next article   Contents of Volume 6 (2010)