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


SIGMA 4 (2008), 026, 26 pages      arXiv:0802.3454      https://doi.org/10.3842/SIGMA.2008.026
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Unified Gauge Theories and Reduction of Couplings: from Finiteness to Fuzzy Extra Dimensions

Myriam Mondragón a and George Zoupanos b
a) Inst. de Física, Universidad Nacional Autónoma de México, México
b) Physics Department, National Technical University, Athens, Greece

Received November 01, 2007, in final form January 31, 2008; Published online February 23, 2008

Abstract
Finite Unified Theories (FUTs) are N = 1 supersymmetric Grand Unified Theories, which can be made all-loop finite, both in the dimensionless (gauge and Yukawa couplings) and dimensionful (soft supersymmetry breaking terms) sectors. This remarkable property, based on the reduction of couplings at the quantum level, provides a drastic reduction in the number of free parameters, which in turn leads to an accurate prediction of the top quark mass in the dimensionless sector, and predictions for the Higgs boson mass and the supersymmetric spectrum in the dimensionful sector. Here we examine the predictions of two such FUTs. Next we consider gauge theories defined in higher dimensions, where the extra dimensions form a fuzzy space (a finite matrix manifold). We reinterpret these gauge theories as four-dimensional theories with Kaluza-Klein modes. We then perform a generalized à la Forgacs-Manton dimensional reduction. We emphasize some striking features emerging such as (i) the appearance of non-Abelian gauge theories in four dimensions starting from an Abelian gauge theory in higher dimensions, (ii) the fact that the spontaneous symmetry breaking of the theory takes place entirely in the extra dimensions and (iii) the renormalizability of the theory both in higher as well as in four dimensions. Then reversing the above approach we present a renormalizable four dimensional SU(N) gauge theory with a suitable multiplet of scalar fields, which via spontaneous symmetry breaking dynamically develops extra dimensions in the form of a fuzzy sphere SN2. We explicitly find the tower of massive Kaluza-Klein modes consistent with an interpretation as gauge theory on M4 × S2, the scalars being interpreted as gauge fields on S2. Depending on the parameters of the model the low-energy gauge group can be SU(n), or broken further to SU(n1) × SU(n2) × U(1). Therefore the second picture justifies the first one in a renormalizable framework but in addition has the potential to reveal new aspects of the theory.

Key words: unification; gauge theories; finiteness; higher dimensions; fuzzy sphere; non-commutative gauge theories; renormalizability.

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References

  1. Connes A., Non-commutative geometry, Academic Press, San Diego, 1994.
  2. Madore J., An introduction to non-commutative differential geometry and its physical applications, London Mathematical Society Lecture Note Series, Vol. 257, Cambridge University Press, Cambridge, 1999.
  3. Connes A., Lott J., Particle models and noncommutative geometry (expanded version), Nuclear Phys. B Proc. Suppl. 18 (1991), 29-47.
  4. Martin C.P., Gracia-Bondia M.J., Varilly J.C., The standard model as a noncommutative geometry: the low-energy regime, Phys. Rep. 294 (1998) 363-406, hep-th/9605001.
  5. Dubois-Violette M., Madore J., Kerner R., Gauge bosons in a noncommutative geometry, Phys. Lett. B 217 (1989), 485-488.
    Dubois-Violette M., Madore J., Kerner R., Classical bosons in a noncommutative geometry, Classical Quantum Gravity 6 (1989), 1709-1724.
    Dubois-Violette M., Kerner R., Madore J., Noncommutative differential geometry and new models of gauge theory, J. Math. Phys. 31 (1990), 316-323.
  6. Madore J., On a quark-lepton duality, Phys. Lett. B 305 (1993) 84-89.
    Madore J., On a noncommutative extension of electrodynamics, in Proceedings "Spinors, Twistors, Clifford Algebras and Quantum Deformations" (1992, Wroclaw), 1992, 285-298, hep-ph/9209226.
  7. Connes A., Douglas M.R., Schwarz A., Non-commutative geometry and matrix theory: compactification on tori, J. High Energy Phys. 1998 (1998), no. 2, 003, 41 pages, hep-th/9711162.
  8. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 99 pages, hep-th/9908142.
  9. Chaichian M., Presnajder P., Sheikh-Jabbari M.M., Tureanu A., Noncommutative standard model: model building, Eur. Phys. J. C 29 (2003), 413-432, hep-th/0107055.
  10. Calmet X., Jurco B., Schupp P., Wess J., Wohlgenannt M., The standard model on non-commutative space-time, Eur. Phys. J. C 23 (2002), 363-376, hep-ph/011111.
    Aschieri P., Jurco B., Schupp P., Wess J., Non-commutative GUTs, standard model and C, P, T, Nuclear Phys. B 651 (2003), 45-70, hep-th/0205214.
  11. Jurco B., Schraml S., Schupp P., Wess J., Enveloping algebra valued gauge transformations for non-Abelian gauge groups on non-commutative spaces, Eur. Phys. J. C 17 (2000), 521-526, hep-th/0006246.
    Jurco B., Schupp P., Wess J., Non-Abelian noncommutative gauge theory via noncommutative extra dimensions, Nuclear Phys. B 604 (2001), 148-180, hep-th/0102129.
    Jurco B., Moller L., Schraml S., Schupp S., Wess J., Construction of non-Abelian gauge theories on noncommutative spaces, Eur. Phys. J. C 21 (2001), 383-388, hep-th/0104153.
    Barnich G., Brandt F., Grigoriev M., Seiberg-Witten maps and noncommutative Yang-Mills theories for arbitrary gauge groups, J. High Energy Phys. 2002 (2002), no. 8, 023, 12 pages, hep-th/0206003.
  12. Brandt F., Martin C.P., Ruiz F.R., Anomaly freedom in Seiberg-Witten noncommutative gauge theories, J. High Energy Phys. 2003 (2003), no. 07, 068, 32 pages, hep-th/0307292.
  13. Carlson C.E., Carone C.D., Lebed R.F., Bounding noncommutative QCD, Phys. Lett. B 518 (2001), 201-206, hep-ph/0107291.
    Behr W., Deshpande N.G., Duplancic G., Schupp P., Trampetic J., Wess J., The Z ® gg, gg decays in the noncommutative standard model, Eur. Phys. J. C 29 (2003), 441-446, hep-ph/0202121.
    Hinchliffe I., Kersting N., Ma Y.L., Review of the phenomenology of noncommutative geometry, Internat. J. Modern Phys. A 19 (2004), 179-205, hep-ph/0205040.
    Schupp P., Trampetic J., Wess J., Raffelt G., The photon neutrino interaction in non-commutative gauge field theory and astrophysical bounds, Eur. Phys. J. C 36 (2004), 405-410, hep-ph/0212292.
  14. Forgacs P., Manton N.S., Space-time symmetries in gauge theories, Comm. Math. Phys. 72 (1980), 15-35.
  15. Kapetanakis D., Zoupanos G., Coset space dimensional reduction of gauge theories, Phys. Rep. 219 (1992), 1-76.
  16. Kubyshin Y.A., Volobuev I.P., Mourao J.M., Rudolph G., Dimensional reduction of gauge theories, spontaneous compactification and model building, Lecture Notes in Physics, Vol. 349, Springer Verlag, Heidelberg, 1989.
  17. Bais F.A., Barnes K.J., Forgacs P., Zoupanos G., Dimensional reduction of symmetric gauge theories, in, Proceedings "High Energy Physics, HEP1985" (1985, Bari), 1985, 60-62.
    Forgacs P., Lust D., Zoupanos G., Physics from multidimensional gauge theories?, in Proceedings of the 2nd Hellenic School on Elementary Particle Physics (September 1-20, 1985, Corfu, Greece), Editors E.N. Argyres and G. Zoupanos, World Scientific, 1986, 26 pages.
    Zoupanos G., Farakos K., Kapetanakis D., Koutsoumbas G., Coset space dimensional reduction approach to the standard model, in Proceedings "Electroweak Interactions and Unified Theories (1988, Les Arcs), 1988, 559-565.
    Zoupanos G., Farakos K., Kapetanakis D., Koutsoumbas G., Dimemsional reduction of gauge theories, in Proceedings "ICHEP 1988", 1988, 1137-1141.
    Zoupanos G., Coset space dimensional reduction of gauge theories, in Proceedings of Warsaw Symposium (1991), 1991, 446-460.
  18. Chapline G., Manton N.S., The geometrical significance of certain Higgs potentials: an approach to grand unification, Nuclear Phys. B 184 (1981), 391-405.
    Bais F.A., Barnes K.J., Forgacs P., Zoupanos G., Dimensional reduction of gauge theories yielding unified models spontaneously broken to SU(3) × U(1), Nuclear Phys. B 263 (1986), 557-590.
    Farakos K., Koutsoumbas G., Surridge M., Zoupanos G., Dimensional reduction and the Higgs potential, Nuclear Phys. B 291 (1987), 128-140.
    Farakos K., Koutsoumbas G., Surridge M., Zoupanos G., Geometrical hierarchy and spontaneous symmetry breaking, Phys. Lett. B 191 (1987), 135-140.
    Kubyshin Yu.A., Mourao J.M., Volobuev I.P., Scalar fields in the dimensional reduction scheme for symmetric spaces, Internat. J. Modern Phys. A 4 (1989), 151-171.
  19. Manton N.S., Fermions and parity violation in dimensional reduction schemes, Nuclear Phys. B 193 (1981), 502-516.
    Chapline G., Slansky R., Dimensional reduction and flavor chirality, Nuclear Phys. B 209 (1982), 461-483.
  20. Wetterich C., Dimensional reduction of Weyl, Majorana and Majorana-Weyl spinors, Nuclear Phys. B 222 (1983), 20-44.
    Palla L., On dimensional reduction of gauge theories: symmetric fields and harmonic expansion, Z. Phys. C 24 (1984), 195-204.
    Pilch K., Schellekens A.N., Formulae for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces, J. Math. Phys. 25 (1984), 3455-3459.
    Forgacs P., Horvath Z., Palla L., Spontaneous compactification to nonsymmetric spaces, Z. Phys. C 30 (1986), 261-266.
    Barnes K.J., Forgacs P., Surridge M., Zoupanos G., On fermion masses in a dimensional reduction scheme, Z. Phys. C 33 (1987), 427-431.
  21. Farakos K., Kapetanakis D., Koutsoumbas G., Zoupanos G., The standard model from a gauge theory in ten-dimensions via CSDR Phys. Lett. B 211 (1988), 322-328. Hanlon B.E., Joshi G.C., A three generation unified model from coset space dimensional reduction, Phys. Lett. B 298 (1993), 312-317.
  22. Manousselis P., Zoupanos G., Dimensional reduction over coset spaces and supersymmetry breaking, J. High Energy Phys. 2002 (2002), no. 3, 002, 32 pages, hep-ph/0111125.
    Manousselis P., Zoupanos G., Soft supersymmetry breaking due to dimensional reduction over non-symmetric coset spaces, Phys. Lett. B 518 (2001), 171-180, hep-ph/0106033.
    Manousselis P., Zoupanos G., Supersymmetry breaking by dimensional reduction over coset spaces, Phys. Lett. B 504 (2001), 122-130, hep-ph/0010141.
    Manousselis P., Zoupanos G., Dimensional reduction of ten-dimensional supersymmetric gauge theories in the N = 1, D = 4 superfield formalism, J. High Energy Phys. 2004 (2004), no. 11, 025, 24 pages, hep-ph/0406207.
    Manousselis P., Prezas N., Zoupanos G., Supersymmetric compactifications of heterotic strings with fluxes and condensates, Nuclear Phys. B 739 (2006), 85-105, hep-th/0511122.
    Chatzistavrakidis A., Manousselis P., Prezas N., Zoupanos G., On the consistency of coset space dimensional reduction, arXiv:0708.3222.
  23. Aschieri P., Madore J., Manousselis P., Zoupanos G., Dimensional reduction over fuzzy coset spaces, J. High Energy Phys. 2004 (2004), 034, 21 pages, hep-th/0310072.
    Aschieri P., Madore J., Manousselis P., Zoupanos G., Unified theories from fuzzy extra dimensions, Fortsch. Phys. 52 (2004), 718-723, hep-th/0401200.
    Aschieri P., Madore J., Manousselis P., Zoupanos G., Renormalizable theories from fuzzy higher dimensions, hep-th/0503039.
  24. Kubo J., Mondragón M., Zoupanos G., Reduction of couplings and heavy top quark in the minimal SUSY GUT, Nuclear Phys. B 424 (1994), 291-307.
  25. Zimmermann W., Reduction in the number of coupling parameters, Comm. Math. Phys. 97 (1985), 211-225.
    Oehme R., Zimmermann W., Relation between effective couplings for asymptotically free models, Comm. Math. Phys. 97 (1985), 569-582.
    Ma E., Modified quantum chromodynamics: exact global color symmetry and asymptotic freedom, Phys. Rev. D 17 (1978), 623-628.
    Ma E., Fixing the Higgs boson mass, Phys. Rev. D 31 (1985), 1143-1145.
  26. Lucchesi C., Piguet O., Sibold K., Vanishing b-functions in N = 1 supersymmetric gauge theories, Helv. Phys. Acta 61 (1988), 321-344.
    Piguet O., Sibold K., Nonrenormalization theorems of chiral anomalies and finiteness in supersymmetric Yang-Mills theories, Internat. J. Modern Phys. A 1 (1986), 913-942.
    Piguet O., Sibold K., Non-renormalization theorems of chiral anomalies and finiteness, Phys. Lett. B 177 (1986), 373-376.
    Lucchesi C., Zoupanos G., All order finiteness in N = 1 SYM theories: criteria and applications, Fortsch. Phys. 45 (1997), 129-143, hep-ph/9604216.
  27. Ermushev A., Kazakov D., Tarasov O., Finite N = 1 supersymmetric grand unified theories, Nuclear Phys. B 281 (1987), 72-84.
    Kazakov D., Finite N = 1 SUSY gauge field theories, Modern Phys. Lett. A 2 (1987), 663-674.
  28. Kapetanakis D., Mondragon M., Zoupanos G., Finite unified models, Z. Phys. C 60 (1993), 181-186, hep-ph/9210218.
    Mondragon M., Zoupanos G., Finite unified theories and the top quark mass, Nuclear Phys. C Proc. Suppl. 37 (1995), 98-105.
  29. Kubo J., Mondragón M., Zoupanos G., Perturbative unification of soft supersymmetry breaking terms Phys. Lett. B 389 (1996), 523-532, hep-ph/9609218.
  30. Jack I., Jones D.R.T., Renormalization group invariance and universal soft supersymmetry breaking, Phys. Lett. B 349 (1995), 294-299, hep-ph/9501395.
  31. Rajpoot S., Taylor J.G., On finite quantum field theories, Phys. Lett. B 147 (1984), 91-95.
  32. Rajpoot S., Taylor J.G., Towards finite quantum field theories, Internat. J. Theoret. Phys. 25 (1986), 117-138.
  33. Parkes A., West P., Finiteness in rigid supersymmetric theories, Phys. Lett. B 138 (1984), 99-104.
    Jones D., Mezincescu L., The chiral anomaly and a class of two loop finite supersymmetric gauge theories, Phys. Lett. B 138 (1984), 293-295.
  34. Jones D.R.T., Mezincescu L., Yao Y.-P., Soft breaking of two loop finite N = 1 supersymmetric gauge theories, Phys. Lett. B 148 (1984), 317-322.
  35. Jack I., Jones D.R.T., Soft supersymmetry breaking and finiteness, Phys. Lett. B 333 (1994), 372-379, hep-ph/9405233.
  36. Leigh R., Strassler M., Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nuclear Phys. B 447 (1995), 95-136, hep-th/9503121.
  37. Novikov V., Shifman M., Vainstein A., Zakharov V., Instanton effects in supersymmetric theories, Nuclear Phys. B 229 (1983), 407-420.
    Novikov V., Shifman M., Vainstein A., Zakharov V., Beta function in supersymmetric gauge theories: instantons versus traditional approach, Phys. Lett. B 166 (1986), 329-333.
    Shifman M., Little miracles of supersymmetric evolution of gauge couplings, Internat. J. Modern Phys. A 11 (1996), 5761-5784, hep-ph/9606281.
  38. Avdeev L.V., Kazakov D.I., Kondrashuk I.N., Renormalizations in softly broken SUSY gauge theories, Nuclear Phys. B 510 (1998), 289-312, hep-ph/9709397.
    Kazakov D.I., Exploring softly broken SUSY theories via Grassmannian Taylor expansion, Phys. Lett. B 449 (1999), 201-206, hep-ph/9812513.
  39. Kobayashi T., Kubo J., Zoupanos G., Further all loop results in softly broken supersymmetric gauge theories, Phys. Lett. B 427 (1998), 291-299, hep-ph/9802267.
    Kobayashi T. et al., Finite and gauge-Yukawa unified theories: theory and predictions, AIP Conf. Proc. 490 (1999), 279-309.
  40. Yamada Y., Two loop renormalization group equations for soft SUSY breaking scalar interactions: supergraph method, Phys. Rev. D 50 (1994), 3537-3545, hep-ph/9401241.
  41. Hisano J., Shifman M., Exact results for soft supersymmetry breaking parameters in supersymmetric gauge theories, Phys. Rev. D 56 (1997), 5475-5482, hep-ph/9705417.
  42. Kawamura T., Kobayashi T., Kubo J., Soft scalar mass sum rule in gauge Yukawa unified models and its superstring interpretation, Phys. Lett. B 405 (1997), 64-70, hep-ph/9703320.
  43. Delbourgo R., Superfield perturbation theory and renormalization, Nuovo Cimento A 25 (1975), 646-656.
    Salam A., Strathdee J.A., Feynman rules for superfields, Nuclear Phys. B 86 (1975), 142-152.
    Fujikawa K., Lang W., Perturbation calculations for the scalar multiplet in a superfield formulation, Nuclear Phys. B 88 (1975), 61-76.
    Grisaru M.T., Rocek M., Siegel W., Improved methods for supergraphs, Nuclear Phys. B 159 (1979), 429-450.
  44. Girardello L., Grisaru M., Soft breaking of supersymmetry, Nuclear Phys. B 194 (1982), 65-76.
  45. Jack I., Jones D.R.T., Pickering A., Renormalization invariance and the soft beta functions, Phys. Lett. B 426 (1998), 73-77, hep-ph/9712542.
  46. Kobayashi T., Kubo J., Mondragón M., Zoupanos G., Constraints on finite soft supersymmetry breaking terms, Nuclear Phys. B 511 (1998), 45-68, hep-ph/9707425.
    Kobayashi T., Kubo J., Mondragón M., Zoupanos G., Exact finite and gauge-Yukawa unified theories and their predictions, Acta Phys. Polon. B 30 (1999) 2013-2027.
    Mondragón M., Zoupanos G., Higgs mass prediction in finite unified theories, Acta Phys. Polon. B 34 (2003), 5459-5468.
  47. Jones D.R.T., Raby S., A two loop finite supersymmetric SU(5) theory: towards a theory of fermion masses, Phys. Lett. B 143 (1984), 137-141.
  48. Babu K.S., Enkhbat T., Gogoladze I., Finite grand unified theories and the quark mixing matrix, Phys. Lett. B 555 (2003), 238-247, hep-ph/0204246.
  49. Ma E., Mondragón M., Zoupanos G., Finite SU(N)k unification, J. High Energy Phys. 2004 (2004), no. 12, 026, 14 pages, hep-ph/0407236.
  50. Carena M., Garcia D., Nierste U., Wagner C., Effective Lagrangian for the [`(t)]bH+ interaction in the MSSM and charged Higgs phenomenology, Nuclear Phys. B 577 (2000), 88-120, hep-ph/9912516.
  51. Erler J., Private communication.
  52. Tevatron Electroweak Working Group, hep-ex/0703034.
  53. Kubo J., Mondragón M., Zoupanos G., Testing gauge Yukawa unified models by M(t), Nuclear Phys. B 479 (1996), 25-45, hep-ph/9512435.
  54. Kobayashi T., Kubo J., Mondragón M., Zoupanos G., Finite unification, Surveys High Energ. Phys. 16 (2001), 87-129.
  55. Barate R. et al. [ALEPH Collaboration], A measurement of the inclusive b ® sg branching ratio, Phys. Lett. B 429 (1998), 169-187.
    Chen S. et al. [CLEO Collaboration], Branching fraction and photon energy spectrum for b ® sg, Phys. Rev. Lett. 87 (2001), 251807, 11 pages, hep-ex/0108032.
    Koppenburg P. et al. [Belle Collaboration], An inclusive measurement of the photon energy spectrum in b ® sg decays, Phys. Rev. Lett. 93 (2004), 061803, 6 pages, hep-ex/0403004.
  56. Herndon M., Searches for FCNC Decays Bs(d) ® m+m-, Talk given at ICHEP04 (August, 2004, Beijing), available at http://ichep04.ihep.ac.cn/db/paper.php.
  57. LEP Higgs working group, Search for the standard model Higgs boson at LEP, Phys. Lett. B 565 (2003), 61-75, hep-ex/0306033.
    Schael S. et al. [ALEPH Collaboration], Search for neutral MSSM Higgs bosons at LEP, Eur. Phys. J. C 47 (2006), 547-587, hep-ex/0602042.
  58. Heinemeyer S., Hollik W., Weiglein G., FeynHiggs: a program for the calculation of the masses of the neutral CP-even Higgs bosons in the MSSM, Comput. Phys. Comm. 124 (2000), 76-89, hep-ph/9812320, see http://www.feynhiggs.de/.
  59. Heinemeyer S., Hollik W., Weiglein G., The masses of the neutral CP - even Higgs bosons in the MSSM: accurate analysis at the two loop level, Eur. Phys. J. C 9 (1999), 343-366, hep-ph/9812472.
  60. Degrassi G., Heinemeyer S., Hollik W., Slavich P., Weiglein G., Towards high precision predictions for the MSSM Higgs sector, Eur. Phys. J. C 28 (2003), 133-143, hep-ph/0212020.
  61. Frank M., Hahn T., Heinemeyer S., Hollik W., Rzehak H., Weiglein G., The Higgs boson masses and mixings of the complex MSSM in the Feynman-diagrammatic approach, J. High Energy Phys. 2007 (2007), no. 2, 047, 56 pages, hep-ph/0611326.
  62. Goldberg H., Constraint on the photino mass from cosmology, Phys. Rev. Lett. 50 (1983), 1419-1422.
    Ellis J., Hagelin J., Nanopoulos D., Olive K., Srednicki M., Supersymmetric relics from the big bang, Nuclear Phys. B 238 (1984), 453-476.
  63. Bennett C. et al., First year Wilkinson microwave anisotropy probe (WMAP) observations: preliminary maps and basic results, Astrophys. J. Suppl. 148 (2003), 1-42, astro-ph/0302207.
    Spergel D. et al. [WMAP Collaboration], First year Wilkinson microwave anisotropy probe (WMAP) observations: determination of cosmological parameters, Astrophys. J. Suppl. 148 (2003), 175-194, astro-ph/0302209.
  64. Belanger G., Boudjema F., Pukhov A., Semenov A., micrOMEGAs: a program for calculating the relic density in the MSSM, Comput. Phys. Comm. 149 (2002), 103-120, hep-ph/0112278.
    Belanger G., Boudjema F., Pukhov A., Semenov A., micrOMEGAs: version 1.3, Comput. Phys. Comm. 174 (2006), 577-604, hep-ph/0405253.
  65. Heinemeyer S., Mondragón M., Zoupanos G., Confronting finite unified theories with low-energy phenomenology, arXiv:0712.3630.
  66. Davier M., The hadronic contribution to (g - 2)m, Nuclear Phys. Proc. Suppl. 169 (2007), 288-296, hep-ph/0701163.
  67. Djouadi A., Kneur J., Moultaka G., SuSpect: a Fortran code for the supersymmetric and Higgs particle spectrum in the MSSM, Comput. Phys. Comm. 176 (2007) 426-455, hep-ph/0211331.
  68. Aschieri P., Grammatikopoulos T., Steinacker H., Zoupanos G., Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking, J. High Energy Phys. 2006 (2006), no. 9, 026, 26 pages, hep-th/0606021.
    Steinacker H., Zoupanos G., Fermions on spontaneously generated spherical extra dimensions, J. High Energy Phys. 2007 (2007), no. 9, 017, 34 pages, arXiv:0706.0398.
  69. Arkani-Hamed N., Cohen A.G., Georgi H., (De)constructing dimensions, Phys. Rev. Lett. 86 (2001), 4757-4761, hep-th/0104005.
  70. Madore J., The fuzzy sphere, Classical Quantum Gravity 9 (1992), 69-88.
  71. Steinacker H., Quantized gauge theory on the fuzzy sphere as random matrix model, Nuclear Phys. B 679 (2004), 66-98, hep-th/0307075.
  72. Steinacker H., Gauge theory on the fuzzy sphere and random matrices, Springer Proc. Phys. 98 (2005), 307-311, hep-th/0409235.
  73. Carow-Watamura U., Watamura S., Noncommutative geometry and gauge theory on fuzzy sphere, Comm. Math. Phys. 212 (2000), 395-413, hep-th/9801195.
  74. Presnajder P., Gauge fields on the fuzzy sphere, Modern Phys. Lett. A 18 (2003), 2431-2438.
  75. Andrews R.P., Dorey N., Spherical deconstruction, Phys. Lett. B 631 (2005), 74-82, hep-th/0505107.
  76. Andrews R.P., Dorey N., Deconstruction of the Maldacena-Nunez compactification, Nuclear Phys. B 751 (2006), 304-341, hep-th/0601098.
  77. Azuma T., Nagao K., Nishimura J., Perturbative dynamics of fuzzy spheres at large N, J. High Energy Phys. 2005 (2005), no. 6, 081, 16 pagegs, hep-th/0410263.
  78. Azuma T., Bal S., Nishimura J., Dynamical generation of gauge groups in the massive Yang-Mills-Chern-Simons matrix model, Phys. Rev. D 72 (2005), 066005, 4 pages, hep-th/0504217.
  79. Azuma T., Bal S., Nagao K., Nishimura J., Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term, J. High Energy Phys. 2004 (2004), no. 5, 005, 35 pages, hep-th/0401038.
  80. Aoki H., Iso S., Maeda T., Nagao K., Dynamical generation of a nontrivial index on the fuzzy 2-sphere, Phys. Rev. D 71 (2005), 045017, 30 pages, Erratum, Phys. Rev. D 71 (2005), 069905, hep-th/0412052.
  81. Aoki H., Nishimura J., Susaki Y., Suppression of topologically nontrivial sectors in gauge theory on 2d non-commutative geometry, J. High Energy Phys. 2007 (2007), no. 10, 024, 14 pages, hep-th/0604093.
  82. Abel S.A., Jaeckel J., Khoze V.V., Ringwald A., Noncommutativity, extra dimensions, and power law running in the infrared, J. High Energy Phys. 2006 (2006), no. 1, 105, 23 pages, hep-ph/0511197.
  83. Lim C.S., Maru N., Hasegawa K., Six dimensional gauge-Higgs unification with an extra space S2 and the hierarchy problem, hep-th/0605180.
  84. Dvali G.R., Randjbar-Daemi S., Tabbash R., The origin of spontaneous symmetry breaking in theories with large extra dimensions, Phys. Rev. D 65 (2002), 064021, 27 pages, hep-ph/0102307.
  85. Antoniadis I., Benakli K., Quiros M., Supersymmetry and electroweak breaking by extra dimensions, Acta Phys. Polon. B 33 (2002), 2477-2488.
  86. Scrucca C.A., Serone M., Silvestrini L., Electroweak symmetry breaking and fermion masses from extra dimensions, Nuclear Phys. B 669 (2003), 128-158, hep-ph/0304220.
  87. Grosse H., Wulkenhaar R., Renormalisation of f4-theory on noncommutative R4 to all orders, hep-th/0403232.
  88. Steinacker H., Quantized gauge theory on the fuzzy sphere as random matrix model, Nuclear Phys. B 679 (2004), 66-98, hep-th/0307075.
    Grosse H., Steinacker H., Finite gauge theory on fuzzy CP2, Nuclear Phys. B 707 (2005), 145-198, hep-th/0407089.


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