Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

Kazuki Hasebe

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

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.

Détail

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

Bibliographie

[1] Madore J., “The fuzzy sphere”, Classical Quantum Gravity, 9 (1992), 69–87 | DOI | MR | Zbl

[2] Grosse H., Klimčík C., Prešnajder P., “On finite 4D quantum field theory in non-commutative geometry”, Comm. Math. Phys., 180 (1996), 429–438, arXiv: hep-th/9602115 | DOI | MR | Zbl

[3] Grosse H., Klimčík C., Prešnajder P., “Field theory on a supersymmetric lattice”, Comm. Math. Phys., 185 (1997), 155–175, arXiv: hep-th/9507074 | DOI | MR | Zbl

[4] Grosse H., Reiter G., “The fuzzy supersphere”, J. Geom. Phys., 28 (1998), 349–383, arXiv: math-ph/9804013 | DOI | MR | Zbl

[5] Seiberg N., Witten E., “String theory and noncommutative geometry”, J. High Energy Phys., 1999:9 (1999), 032, 93 pp., arXiv: hep-th/9908142 | DOI | MR

[6] de Boer J., Grassi P. A., van Nieuwenhuizen P., “Non-commutative superspace from string theory”, Phys. Lett. B, 574 (2003), 98–104, arXiv: hep-th/0302078 | DOI | MR | Zbl

[7] Berkovits N., Seiberg N., “Superstrings in graviphoton background and $\mathcal N=1/2+3/2$ supersymmetry”, J. High Energy Phys., 2003:7 (2003), 010, 10 pp., arXiv: hep-th/0306226 | DOI | MR

[8] Myers R. C., “Dielectric-branes”, J. High Energy Phys., 1999:12 (1999), 022, 41 pp., arXiv: hep-th/9910053 | DOI | MR

[9] Castelino J., Lee S., Taylor W., “Longitudinal 5-branes as 4-spheres in matrix theory”, Nuclear Phys. B, 526 (1998), 334–350, arXiv: hep-th/9712105 | DOI | MR | Zbl

[10] Iso S., Umetsu H., “Gauge theory on noncommutative supersphere from supermatrix model”, Phys. Rev. D, 69 (2004), 1050033, 7 pp., arXiv: hep-th/0311005 | DOI | Zbl

[11] Constable N. R., Myers R. C., Tafjord O., “Noncommutative bion core”, Phys. Rev. D, 61 (2000), 106009, 14 pp., arXiv: ; Constable N. R., Myers R. C., Tafjord O., “Non-Abelian brane intersections”, J. High Energy Phys., 2001:6 (2001), 023, 37 pp., arXiv: hep-th/9911136hep-th/0102080 | DOI | MR | DOI | MR

[12] Cook P. L. H., de Mello Koch R., Murugan J., “Non-Abelian BIonic brane intersections”, Phys. Rev. D, 68 (2003), 126007, 8 pp., arXiv: hep-th/0306250 | DOI | MR

[13] Bhattacharyya R., de Mello Koch R., “Fluctuating fuzzy funnels”, J. High Energy Phys., 2005:10 (2005), 036, 20 pp., arXiv: hep-th/0508131 | DOI | MR

[14] Ho P.-M., Ramgoolam S., “Higher-dimensional geometries from matrix brane constructions”, Nuclear Phys. B, 627 (2002), 266–288, arXiv: hep-th/0111278 | DOI | MR | Zbl

[15] Kimura Y., “Noncommutative gauge theory on fuzzy four-sphere and matrix model”, Nuclear Phys. B, 637 (2002), 177–198, arXiv: ; Kimura Y., “On higher-dimensional fuzzy spherical branes”, Nuclear Phys. B, 664 (2003), 512–530, arXiv: hep-th/0204256hep-th/0301055 | DOI | MR | Zbl | DOI | MR | Zbl

[16] Ramgoolam S., “Higher dimensional geometries related to fuzzy odd-dimensional spheres”, J. High Energy Phys., 2002:10 (2002), 064, 29 pp., arXiv: hep-th/0207111 | DOI | MR

[17] Nair V. P., Randjbar-Daemi S., “Quantum Hall effect on $S^3$, edge states and fuzzy $S^3/\mathbf Z_2$”, Nuclear Phys. B, 679 (2004), 447–463, arXiv: hep-th/0309212 | DOI | MR | Zbl

[18] Taylor W., Lectures on D-branes, gauge theory and M(atrices), arXiv: hep-th/9801182

[19] Balachandran A. P., “Quantum spacetimes in the year 1”, Pramana J. Phys., 59 (2002), 359–368, arXiv: hep-th/0203259 | DOI

[20] Karabali D., Nair V. P., Randjbar-Daemi S., Fuzzy spaces, the M(atrix) model and the quantum Hall effect, arXiv: hep-th/0407007 | MR

[21] Azuma T., Matrix models and the gravitational interaction, PhD thesis, Kyoto University, 2004, arXiv: hep-th/0401120

[22] Balachandran A. P., Kurkcuoglu S., Vaidya S., Lectures on fuzzy and fuzzy SUSY physics, arXiv: 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, arXiv: hep-th/0606161 | DOI | MR | Zbl

[25] Hatsuda M., Iso S., Umetsu H., “Noncommutative superspace, supermatrix and lowest Landau level”, Nuclear Phys. B, 671 (2003), 217–242, arXiv: hep-th/0306251 | DOI | MR | Zbl

[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, arXiv: ; Errata Bull. Amer. Math. Soc. (N.S.), 42 (2005), 213–213 math.RA/0105155 | DOI | MR | Zbl | DOI | MR

[28] Hopf H., “Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche”, Math. Ann., 104 (1931), 637–665 | DOI | MR

[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 | DOI | MR

[31] Grassmann H., Die Lineale Ausdehnungslehre, Ein neuer Zweig der Mathematik, v. II, 1844

[32] Landi G., Marmo G., “Extensions of Lie superalgebras and supersymmetric Abelian gauge fields”, Phys. Lett. B, 193 (1987), 61–66 | DOI | MR

[33] Dirac P., “Quantised singularities in the electromagnetic field”, Proc. R. Soc. Lond. Ser. A, 133 (1931), 60–72 | DOI

[34] Yang C. N., “Generalization of Dirac's monopole to $\mathrm{SU}_2$ gauge fields”, J. Math. Phys., 19 (1978), 320–328 | DOI | MR

[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 | DOI | MR | Zbl | DOI | MR

[36] Nakahara M., Geometry, topology and physics, Graduate Student Series in Physics, 2nd ed., Institute of Physics, Bristol, 2003 | MR | Zbl

[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 | DOI | MR

[38] Wu T. T., Yang C. N., “Dirac monopole without strings: monopole harmonics”, Nuclear Physics B, 107 (1976), 365–380 | DOI | MR

[39] Hasebe K., Kimura Y., “Fuzzy supersphere and supermonopole”, Nuclear Phys. B, 709 (2005), 94–114, arXiv: hep-th/0409230 | DOI | MR | Zbl

[40] Bartocci C., Bruzzo U., Landi G., “Chern–Simons forms on principal superfiber bundles”, J. Math. Phys., 31 (1990), 45–54 | DOI | MR | Zbl

[41] Landi G., “Projective modules of finite type over the supersphere $S^{2,2}$”, Differential Geom. Appl., 14 (2001), 95–111, arXiv: math-ph/9907020 | DOI | MR | Zbl

[42] Pais A., Rittenberg V., “Semisimple graded Lie algebras”, J. Math. Phys., 17 (1976), 598–598 ; Erratum J. Math. Phys., 16 (1975), 2062–2073 | DOI | MR | Zbl | DOI | MR | Zbl

[43] Scheunert M., Nahm W., Rittenberg V., “Irreducible representations of the ${\rm osp}(2,1)$ and ${\rm spl}(2,1)$ graded Lie algebra”, J. Math. Phys., 18 (1977), 155–162 | DOI | MR | Zbl

[44] Marcu M.,, “The representations of ${\rm spl}(2,1)$ – an example of representations of basic superalgebras”, J. Math. Phys., 21 (1980), 1277–1283 | DOI | MR | Zbl

[45] Frappat L., Sciarrino A., Sorba P., Dictionary on Lie algebras and superalgebras, Academic Press, Inc., San Diego, CA, 2000 | MR

[46] Balachandran A. P., Kürkçüoǧlu S., Rojas E., “The star product on the fuzzy supersphere”, J. High Energy Phys., 2002:7 (2002), 056, 22 pp., arXiv: hep-th/0204170 | DOI | MR

[47] Zhang S.-C., Hu J.-P., “A four-dimensional generalization of the quantum Hall effect”, Science, 294:5543 (2001), 823–828, arXiv: cond-mat/0110572 | DOI

[48] 't Hooft G., “Computation of the quantum effects due to a four-dimensional pseudoparticle”, Phys. Rev. D, 14 (1976), 3432–3450 | DOI

[49] Iachello F., Lie algebras and applications, Lecture Notes in Physics, 708, Springer-Verlag, Berlin, 2006 | MR | Zbl

[50] Yang C. N., “SU$_2$ monopole harmonics”, J. Math. Phys., 19 (1978), 2622–2627 | DOI | MR

[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, arXiv: cond-mat/0206164 | DOI | MR | Zbl

[52] Ramgoolam S., “On spherical harmonics for fuzzy spheres in diverse dimensions”, Nuclear Phys. B, 610 (2001), 461–488, arXiv: hep-th/0105006 | DOI | MR | Zbl

[53] Azuma T., Bagnoud M., “Curved-space classical solutions of a massive supermatrix model”, Nuclear Phys. B, 651 (2003), 71–86, arXiv: hep-th/0209057 | DOI | MR | Zbl

[54] Abe Y.,, “Construction of fuzzy $S^4$”, Phys. Rev. D, 70 (2004), 126004, 10 pp., arXiv: ; Abe Y., Emergence of longitudinal 7-branes and fuzzy $S^4$ in compactification scenarios of M(atrix) theory, arXiv: hep-th/0406135hep-th/0512174 | DOI | MR | Zbl

[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 pp., arXiv: cond-mat/0306045 | DOI

[56] Hasebe K., Kimura Y., “Dimensional hierarchy in quantum Hall effects on fuzzy spheres”, Phys. Lett. B, 602 (2004), 255–260, arXiv: hep-th/0310274 | DOI | MR

[57] Fabinger M., “Higher-dimensional quantum Hall effect in string theory”, J. High Energy Phys., 2002:5 (2002), 037, 12 pp., arXiv: hep-th/0201016 | DOI | MR

[58] Meng G., “Geometric construction of the quantum Hall effect in all even dimensions”, J. Phys. A: Math. Gen., 36 (2003), 9415–9423, arXiv: cond-mat/0306351 | DOI | MR | Zbl

[59] Horváth Z., Palla L., “Spontaneous compactification and “monopoles” in higher dimensions”, Nuclear Phys. B, 142 (1978), 327–343 | DOI | MR

[60] Tchrakian D. H., “$N$-dimensional instantons and monopoles”, J. Math. Phys., 21 (1980), 166–169 | DOI | Zbl

[61] Saclioǧlu C., “Scale invariant gauge theories and self-duality in higher dimensions”, Nuclear Phys. B, 277 (1986), 487–508 | DOI | MR

[62] Kimura Y., “Nonabelian gauge field and dual description of fuzzy sphere”, J. High Energy Phys., 2004:4 (2004), 058, 29 pp., arXiv: hep-th/0402044 | DOI | MR

[63] Hasebe K., “Split-quaternionic Hopf map, quantum Hall effect, and twistor theory”, Phys. Rev. D, 81 (2010), 041702, 5 pp., arXiv: 0902.2523 | DOI | MR

[64] Ivanov E., Mezincescu L., Townsend P. K., A super-flag Landau model, arXiv: hep-th/0404108 | MR

[65] Hasebe K., “Quantum Hall liquid on a noncommutative superplane”, Phys. Rev. D, 72 (2005), 105017, 9 pp., arXiv: hep-th/0503162 | DOI

[66] Ivanov E., Mezincescu L., Townsend P. K., “Planar super-Landau models”, J. High Energy Phys., 2006:1 (2006), 143, 23 pp., arXiv: hep-th/0510019 | DOI | MR

[67] Curtright T., Ivanov E., Mezincescu L., Townsend P. K., “Planar super-Landau models revisited”, J. High Energy Phys., 2007:4 (2007), 020, 25 pp., arXiv: hep-th/0612300 | DOI | MR

[68] Beylin A., Curtright T., Ivanov E., Mezincescu L., Townsend P. K., “Unitary spherical super-Landau models”, J. High Energy Phys., 2008:10 (2008), 069, 46 pp., arXiv: 0806.4716 | DOI | MR

[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, arXiv: hep-th/0311159

[71] Murray S., Sämann C., “Quantization of flag manifolds and their supersymmetric extensions”, Adv. Theor. Math. Phys., 12 (2008), 641–710, arXiv: hep-th/0611328 | MR | Zbl

[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, arXiv: math.QA/0407342 | DOI | MR | Zbl

[73] Hasebe K., “The split-algebras and non-compact Hopf maps”, J. Math. Phys., 51 (2010), 053524, 35 pp., arXiv: 0905.2792 | DOI | MR

[74] Faria Carvalho L., Kuznetsova Z., Toppan F., “Supersymmetric extension of Hopf maps: $\mathcal N=4$ $\sigma$-models and the $S^3\to S^2$ fibration”, Nuclear Phys. B, 834 (2010), 237–257, arXiv: 0912.3279 | DOI | MR

[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 pp., arXiv: hep-th/0606152 | DOI | MR

[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 pp., arXiv: 0902.2682 | DOI | MR

[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 pp., arXiv: 0905.3461 | DOI | MR | Zbl

[79] Fedoruk S., Ivanov E., Lechtenfeld O., “$OSp(4|2)$ superconformal mechanics”, J. High Energy Phys., 2009:8 (2009), 081, 24 pp., arXiv: 0905.4951 | DOI | MR | Zbl

[80] Bellucci S., Krivonos S., Sutulin A., Three dimensional $N=4$ supersymmetric mechanics with Wu–Yang monopole, arXiv: 0911.3257

[81] Ivanov E., Konyushikhin M., $N=4$, 3D supersymmetric quantum mechanics in non-Abelian monopole background, arXiv: 1004.4597

[82] Krivonos S., Lechtenfeld O., Sutulin A., “$N=4$ supersymmetry and the Belavin–Polyakov–Shvarts–Tyupkin instanton”, Phys. Rev. D, 81 (2010), 085021, 7 pp., arXiv: 1001.2659 | DOI

[83] Ishii T., Ishiki G., Shimasaki S., Tsuchiya A., “$T$-duality, fiber bundles and matrices”, J. High Energy Phys., 2007:5 (2007), 014, 20 pp., arXiv: ; Ishii T., Ishiki G., Shimasaki S., Tsuchiya A., “Fiber bundles and matrix models”, Phys. Rev. D, 77 (2008), 126015, 25 pp., arXiv: hep-th/07030210802.2782 | DOI | MR | DOI | MR

[84] Pedder C., Sonner J., Tong D., “The geometric phase and gravitational precession of D-branes”, Phys. Rev. D, 76 (2007), 126014, 10 pp., arXiv: ; Pedder C., Sonner J., Tong D., “The Berry phase of D0-branes”, J. High Energy Phys., 2008:3 (2008), 065, 14 pp., arXiv: 0709.21360801.1813 | DOI | MR | DOI | MR

[85] Nastase H., Papageorgakis C., Ramgoolam S., “The fuzzy $S^2$ structure of M2–M5 systems in ABJM membrane theories”, J. High Energy Phys., 2009:5 (2009), 123, 61 pp., arXiv: 0903.3966 | DOI | MR

[86] Nastase H., Papageorgakis C., “Fuzzy Killing spinors and supersymmetric D4 action on the fuzzy 2-sphere from the ABJM model”, J. High Energy Phys., 2009:12 (2009), 049, 52 pp., arXiv: 0908.3263 | DOI | MR

[87] 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 | DOI | MR

[88] Arovas D. P., Hasebe K., Qi X.-L., Zhang S.-C., “Supersymmetric valence bond solid states”, Phys. Rev. B, 79 (2009), 224404, 20 pp., arXiv: 0901.1498 | DOI

[89] Hasebe K., “Supersymmetric quantum Hall effect on a fuzzy supersphere”, Phys. Rev. Lett., 94 (2005), 206802, 4 pp., arXiv: hep-th/0411137 | DOI | MR | Zbl

[90] Landi G., “Spin-Hall effect with quantum group symmetry”, Lett. Math. Phys., 75 (2006), 187–200, arXiv: hep-th/0504092 | DOI | MR | Zbl

[91] Jellal A., “Quantum Hall effect on higher-dimensional spaces”, Nuclear Phys. B, 725 (2005), 554–576, arXiv: hep-th/0505095 | DOI | MR | Zbl

[92] Hasebe K., “Hyperbolic supersymmetric quantum Hall effect”, Phys. Rev. D, 78 (2008), 125024, 13 pp., arXiv: 0809.4885 | DOI | MR

[93] 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 pp., arXiv: 0904.0550 | DOI

[94] Asorey M., Esteve J. G., Pacheco A. F., “Planar rotor: the $\theta$-vacuum structure, and some approximate methods in quantum mechanics”, Phys. Rev. D, 27 (1983), 1852–1868 | DOI | MR

[95] Karabali D., Nair V. P., “Quantum Hall effect in higher dimensions”, Nuclear Phys. B, 641 (2002), 533–546, arXiv: hep-th/0203264 | DOI | MR | Zbl

[96] Perelomov A. M., “Coherent states for arbitrary Lie group”, Comm. Math. Phys., 26 (1972), 222–236, arXiv: math-ph/0203002 | DOI | MR | Zbl

[97] Alexanian G., Balachandran A. P., Immirzi G., Ydri B., “Fuzzy $\mathbb C\mathrm P^2$”, J. Geom. Phys., 42 (2002), 28–53, arXiv: hep-th/0103023 | DOI | MR | Zbl

[98] 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, arXiv: hep-th/0107099 | DOI | MR | Zbl

[99] Carow-Watamura U., Steinacker H., Watamura S., “Monopole bundles over fuzzy complex projective spaces”, J. Geom. Phys., 54 (2005), 373–399, arXiv: hep-th/0404130 | DOI | MR | Zbl