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

SIGMA 6 (2010), 063, 47 pages      arXiv:1008.2049      https://doi.org/10.3842/SIGMA.2010.063
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions

Athanasios Chatzistavrakidis a, b and George Zoupanos b
a) Institute of Nuclear Physics, NCSR Demokritos, GR-15310 Athens, Greece
b) Physics Department, National Technical University of Athens, GR-15780 Zografou Campus, Athens, Greece

Received May 06, 2010, in final form July 22, 2010; Published online August 12, 2010

Abstract
Theories defined in higher than four dimensions have been used in various frameworks and have a long and interesting history. Here we review certain attempts, developed over the last years, towards the construction of unified particle physics models in the context of higher-dimensional gauge theories with non-commutative extra dimensions. These ideas have been developed in two complementary ways, namely (i) starting with a higher-dimensional gauge theory and dimensionally reducing it to four dimensions over fuzzy internal spaces and (ii) starting with a four-dimensional, renormalizable gauge theory and dynamically generating fuzzy extra dimensions. We describe the above approaches and moreover we discuss the inclusion of fermions and the construction of realistic chiral theories in this context.

Key words: fuzzy extra dimensions; unified gauge theories; symmetry breaking.

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References

  1. Green M.B., Schwarz J.H., Witten E., Superstring theory. Vol. 1. Introduction, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987.
    Green M.B., Schwarz J.H., Witten E., Superstring theory. Vol. 2. Loop amplitudes, anomalies and phenomenology, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987.
    Polchinski J., String theory. Vol. 1. An introduction to the bosonic string, Cambridge University Press, Cambridge, 1998.
    Polchinski J., String theory. Vol. 2. Superstring theory and beyond, Cambridge University Press, Cambridge, 1998.
    Lüst D., Theisen S., Lectures on string theory, Lecture Notes in Physics, Vol. 346, Springer-Verlag, Berlin, 1989.
  2. Gross D.J., Harvey J.A., Martinec E.J., Rohm R., Heterotic string theory. I. The free heterotic string, Nuclear Phys. B 256 (1985), 253-284.
  3. Forgács P., Manton N.S., Space-time symmetries in gauge theories, Comm. Math. Phys. 72 (1980), 15-35.
  4. Kapetanakis D., Zoupanos G., Coset-space-dimensional reduction of gauge theories, Phys. Rep. 219 (1992), 76 pages.
  5. Kubyshin Yu.A., Mourão J.M., Rudolph G., Volobujev I.P., Dimensional reduction of gauge theories, spontaneous compactification and model building, Lecture Notes in Physics, Vol. 349, Springer-Verlag, Berlin, 1989.
  6. Scherk J., Schwarz J.H., How to get masses from extra dimensions, Nuclear Phys. B 153 (1979), 61-88.
  7. 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.
  8. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  9. Madore J., An introduction to noncommutative differential geometry and its physical applications, 2nd ed., London Mathematical Society Lecture Note Series, Vol. 257, Cambridge University Press, Cambridge, 1999.
  10. Connes A., Lott J., Particle models and noncommutative geometry, Nuclear Phys. B Proc. Suppl. 18 (1991), 29-47.
    Chamseddine A.H., Connes A., The spectral action principle, Comm. Math. Phys. 186 (1997), 731-750, hep-th/9606001.
    Chamseddine A.H., Connes A., Conceptual explanation for the algebra in the noncommutative approach to the standard model, Phys. Rev. Lett. 99 (2007), 191601, 4 pages, arXiv:0706.3690.
  11. Martín C.P., Gracia-Bondía M.J., Várilly J.C., The standard model as a noncommutative geometry: the low-energy regime, Phys. Rep. 294 (1998), 363-406, hep-th/9605001.
  12. 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), 323-330.
  13. Madore J., On a quark-lepton duality, Phys. Lett. B 305 (1993), 84-89.
    Madore J., On a noncommutative extension of electrodynamics, in Spinors, Twistors, Clifford Algebras and Quantum Deformations (Sobotka Castle, 1992), Fund. Theories Phys., Vol. 52, Kluwer Acad. Publ., Dordrecht, 1993, 285-298. hep-ph/9209226.
  14. 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, 35 pages, hep-th/9711162.
  15. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  16. Chaichian M., Presnajder P., Sheikh-Jabbari M.M., Tureanu A., Non-commutative standard model: model building, Eur. Phys. J. C 29 (2003), 413-432, hep-th/0107055.
  17. 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., Nonabelian noncommutative gauge theory via noncommutative extra dimensions, Nuclear Phys. B 604 (2001), 148-180, hep-th/0102129.
    Jurco B., Möller 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, 13 pages, hep-th/0206003.
  18. 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/0111115.
    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.
  19. Aschieri P., Madore J., Manousselis P., Zoupanos G., Dimensional reduction over fuzzy coset spaces, J. High Energy Phys. 2004 (2004), no. 4, 034, 24 pages, hep-th/0310072.
    Aschieri P., Madore J., Manousselis P., Zoupanos G., Unified theories from fuzzy extra dimensions, Fortschr. Phys. 52 (2004), 718-723, hep-th/0401200.
    Aschieri P., Madore J., Manousselis P., Zoupanos G., Renormalizable theories from fuzzy higher dimensions, in Proceedings of the 3rd Summer School in Modern Mathematical Physics, Editors B. Dragovich, Z. Rakic and B. Sazdovic, Institute of Physics, Belgrade, 2005, 135-146, hep-th/0503039.
  20. 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.
    Aschieri P., Steinacker H., Madore J., Manousselis P., Zoupanos G., Fuzzy extra dimensions: dimensional reduction, dynamical generation and renormalizability, in Proceedings of the 4th Summer School in Modern Mathematical Physics, Editors B. Dragovich and Z. Rakic, Institute of Physics, Belgrade, 2007, 25-42, arXiv:0704.2880.
  21. Steinacker H., Zoupanos G., Fermions on spontaneously generated spherical extra dimensions, J. High Energy Phys. 2007 (2007), no. 9, 017, 35 pages, arXiv:0706.0398.
  22. Chatzistavrakidis A., Steinacker H., Zoupanos G., On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes, Fortschr. Phys. 58 (2010), 537-552, arXiv:0909.5559.
  23. Chatzistavrakidis A., Steinacker H., Zoupanos G., Orbifolds, fuzzy spheres and chiral fermions, J. High Energy Phys. 2010 (2010), no. 5, 100, 27 pages, arXiv:1002.2606.
  24. Madore J., The fuzzy sphere, Classical Quantum Gravity 9 (1992), 69-87.
  25. Balachandran A.P., Dolan B.P., Lee J.H., X. Martin X., O'Connor D., Fuzzy complex projective spaces and their star-products, J. Geom. Phys. 43 (2002), 184-204, hep-th/0107099.
    Carow-Watamura U., Steinacker H., Watamura S., Monopole bundles over fuzzy complex projective spaces, J. Geom. Phys. 54 (2005), 373-399, hep-th/0404130.
    Dolan B.P., Huet I., Murray S., O'Connor D., A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces, J. High Energy Phys. 2008 (2008), no. 3, 029, 21 pages, arXiv:0711.1347.
  26. Trivedi S.P., Vaidya S., Fuzzy cosets and their gravity duals, J. High Energy Phys. 2000 (2000), no. 9, 041, 33 pages, hep-th/0007011.
    Dolan B.P., Jahn O., Fuzzy complex Grassmannian spaces and their star products, Internat. J. Modern Phys. A 18 (2003), 1935-1958, hep-th/0111020.
  27. Madore J., Schraml S., Schupp P., Wess J., Gauge theory on noncommutative spaces, Eur. Phys. J. C 16 (2000) 161-167, hep-th/0001203.
  28. Lechtenfeld O., Popov A.D., Szabo R.J., Rank two quiver gauge theory, graded connections and noncommutative vortices, J. High Energy Phys. 2006 (2006), no. 9, 054, 46 pages, hep-th/0603232.
    Lechtenfeld O., Popov A.D., Szabo R.J., SU(3)-equivariant quiver gauge theories and nonabelian vortices, J. High Energy Phys. 2008 (2008), no. 8, 093, 63 pages, arXiv:0806.2791.
    Dolan B.P., Szabo R.J., Dimensional reduction, monopoles and dynamical symmetry breaking, J. High Energy Phys. 2009 (2009), no. 3, 059, 27 pages, arXiv:0901.2491.
    Dolan B.P., Szabo R.J., Equivariant dimensional reduction and quiver gauge theories, Gen. Relativity Gravitation, to appear, arXiv:1001.2429.
  29. Harland D., Kurkçuoglu S., Equivariant reduction of Yang-Mills theory over the fuzzy sphere and the emergent vortices, Nuclear Phys. B 821 (2009), 380-398, arXiv:0905.2338.
  30. Barnes K.J., Forgacs P., Surridge M., Zoupanos G., On fermion masses in a dimensional reduction scheme, Z. Phys. C 33 (1987), 427-431.
  31. Steinacker H., Quantized gauge theory on the fuzzy sphere as random matrix model, Nuclear Phys. B 679 (2004), 66-98, hep-th/0307075.
  32. Steinacker H., Gauge theory on the fuzzy sphere and random matrices, in Particle Physics and the Universe, Springer Proceedings in Physics, Vol. 98, Springer, Berlin, 2005, 307-311, hep-th/0409235.
  33. Carow-Watamura U., Watamura S., Noncommutative geometry and gauge theory on fuzzy sphere, Comm. Math. Phys. 212 (2000), 395-413, hep-th/9801195.
  34. Presnajder P., Gauge fields on the fuzzy sphere, Modern Phys. Lett. A 18 (2003), 2431-2438.
  35. Andrews R.P., Dorey N., Spherical deconstruction, Phys. Lett. B 631 (2005), 74-82, hep-th/0505107.
    Andrews R.P., Dorey N., Deconstruction of the Maldacena-Núñez compactification, Nuclear Phys. B 751 (2006), 304-341, hep-th/0601098.
  36. Azuma T., Nagao K., Nishimura J., Perturbative dynamics of fuzzy spheres at large N, J. High Energy Phys. 2005 (2005), no. 6, 081, 21 pages, hep-th/0410263.
  37. 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, 5 pages, hep-th/0504217.
  38. 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, 36 pages, hep-th/0401038.
  39. 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, 10 pages, Erratum, Phys. Rev. D 71 (2005), 069905, 1 pages, hep-th/0412052.
  40. 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, 17 pages, hep-th/0604093.
  41. 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, 22 pages, hep-ph/0511197.
  42. Lim C.S., Maru N., Hasegawa K., Six dimensional Gauge-Higgs unification with an extra space S2 and the hierarchy problem, J. Phys. Soc. Japan 77 (2008), 074101, 15 pages, hep-th/0605180.
  43. 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, 13 pages, hep-ph/0102307.
  44. Antoniadis I., Benakli K., Quiros M., Supersymmetry and electroweak breaking by extra dimensions, Acta Phys. Polon. B 33 (2002), 2477-2488.
  45. 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.
  46. Grosse H., Presnajder P., The Dirac operator on the fuzzy sphere, Lett. Math. Phys. 33 (1995), 171-181.
  47. Behr W., Meyer F., Steinacker H., Gauge theory on fuzzy S2×S2 and regularization on noncommutative R4, J. High Energy Phys. 0507 (2005), no. 7, 040, 38 pages, hep-th/0503041.
  48. Maalampi J., Roos M., Physics of mirror fermions, Phys. Rep. 186 (1990), 53-96.
  49. Brink L., Schwarz J.H., Scherk J., Supersymmetric Yang-Mills theories, Nuclear Phys. B 121 (1977), 77-92.
    Gliozzi F., Scherk J., Olive D.I., Supersymmetry, supergravity theories and the dual spinor model, Nuclear Phys. B 122 (1977), 253-290.
  50. Candelas P., Horowitz G.T., Strominger A., Witten E., Vacuum configurations for superstrings, Nuclear Phys. B 258 (1985), 46-74.
  51. Cardoso G.L., Curio G., Dall'Agata G., Lüst D., Manousselis P., Zoupanos G., Non-Kähler string backgrounds and their five torsion classes, Nuclear Phys. B 652 (2003) 5-34, hep-th/0211118.
  52. Gauntlett J.P., Martelli D., Waldram D., Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004), 086002, 27 pages, hep-th/0302158.
  53. Dixon L.J., Harvey J.A., Vafa C., Witten E., Strings on orbifolds, Nuclear Phys. B 261 (1985), 678-686.
    Dixon L.J., Harvey J.A., Vafa C., Witten E., Strings on orbifolds. II, Nuclear Phys. B 274 (1986), 285-314.
  54. Bailin D., Love A., Orbifold compactifications of string theory, Phys. Rep. 315 (1999), 285-408.
  55. Kachru S., Silverstein E., 4D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998), 4855-4858, hep-th/9802183.
  56. Maldacena J.M., The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231-252, hep-th/9711200.
    Maldacena J.M., The large-N limit of superconformal field theories and supergravity, Internat. J. Theoret. Phys. 38 (1999), 1113-1133.
  57. Douglas M.R., Moore G.W., D-branes, quivers, and ALE instantons, hep-th/9603167.
  58. Douglas M.R., Greene B.R., Morrison D.R., Orbifold resolution by D-branes, Nuclear Phys. B 506 (1997), 84-106, hep-th/9704151.
  59. Aldazabal G., Ibáñez L.E., Quevedo F., Uranga A.M., D-branes at singularities: a bottom-up approach to the string embedding of the standard model, J. High Energy Phys. 2000 (2000), no. 8, 002, 70 pages, hep-th/0005067.
  60. Lawrence A.E., Nekrasov N., Vafa C., On conformal field theories in four dimensions, Nuclear Phys. B 533 (1998), 199-209, hep-th/9803015.
  61. Kiritsis E., D-branes in standard model building, gravity and cosmology, Phys. Rep. 421 (2005), 105-190, Erratum, Phys. Rep. 429 (2006), 121-122, hep-th/0310001.
  62. Djouadi A., The anatomy of electro-weak symmetry breaking. II. The Higgs bosons in the minimal supersymmetric model, Phys. Rep. 459 (2008), 1-241, hep-ph/0503173.
  63. Pati J.C., Salam A., Lepton number as the fourth "color", Phys. Rev. D 10 (1974), 275-289, Erratum, Phys. Rev. D 11 (1975), 703-703.
  64. Antoniadis I., Leontaris G.K., A supersymmetric SU(4)×O(4) model, Phys. Lett. B 216 (1989), 333-335.
    Antoniadis I., Leontaris G.K., Rizos J., A three-generation SU(4)×O(4) string model, Phys. Lett. B 245 (1990), 161-168.
  65. Anastasopoulos P., Leontaris G.K., Vlachos N.D., Phenomenological analysis of D-brane Pati-Salam vacua, J. High Energy Phys. 2010 (2010), no. 5, 011, 44 pages, arXiv:1002.2937.
  66. Ibanez L.E., A chiral D=4, N=1 string vacuum with a finite low energy effective field theory, J. High Energy Phys. 1998 (1998), no. 7, 002, 10 pages, hep-th/9802103.
  67. Ma E., Mondragón M., Zoupanos G., Finite SU(N)k unification, J. High Energy Phys. 2004 (2004), no. 12, 026, 17 pages, hep-ph/0407236.
  68. Glashow S.L., Trinification of all elementary particle forces, in Fifth Workshop on Grand Unification, Editors K. Kang, H. Fried and P. Frampton, World Scientific, Singapore, 1984, 88-94.
  69. Rizov V.A., A gauge model of the electroweak and strong interactions based on the group SU(3)L×SU(3)R×SU(3)C, Bulg. J. Phys. 8 (1981), 461-477.
  70. Lazarides G., Panagiotakopoulos C., MSSM from SUSY trinification, Phys. Lett. B 336 (1994), 190-193, hep-ph/9403317.
  71. Heinemeyer S., Ma E., Mondragon M., Zoupanos G., Finite SU(3)3 model, AIP Conf. Proc. 1200 (2010) 568-571, arXiv:0910.0501.
  72. Babu K.S., He X.G., Pakvasa S., Neutrino masses and proton decay modes in SU(3)×SU(3)×SU(3) trinification, Phys. Rev. D 33 (1986), 763-772.
  73. Leontaris G.K., Rizos J., A D-brane inspired U(3)C×U(3)L×U(3)R model, Phys. Lett. B 632 (2006), 710-716, hep-ph/0510230.
  74. Kim J.E., Z3 orbifold construction of SU(3)3 GUT with sin2(θW0)=3/8, Phys. Lett. B 564 (2003), 35-41, hep-th/0301177.
    Choi K.S., Kim J.E., Three family Z3 orbifold trinification, MSSM and doublet-triplet splitting problem, Phys. Lett. B 567 (2003), 87-92, hep-ph/0305002.
  75. Dienes K.R., Dudas E., Gherghetta T., Extra spacetime dimensions and unification, Phys. Lett. B 436 (1998), 55-65, hep-ph/9803466.
  76. Ghilencea D., Ross G.G., Unification and extra space-time dimensions, Phys. Lett. B 442 (1998), 165-172, hep-ph/9809217.
  77. Kobayashi T., Kubo J., Mondragon M., Zoupanos G., Running of soft parameters in extra space-time dimensions, Nuclear Phys. B 550 (1999) 99-122, hep-ph/9812221.
  78. Kubo J., Terao H., Zoupanos G., Kaluza-Klein thresholds and regularization (in)dependence, Nuclear Phys. B 574 (2000), 495-524, hep-ph/9910277.
  79. Filk T., Divergencies in a field theory on quantum space, Phys. Lett. B 376 (1996), 53-58.
    Várilly J.C., Gracia-Bondía J.M., On the ultraviolet behaviour of quantum fields over noncommutative manifolds, Internat. J. Modern Phys. A 14 (1999), 1305-1323, hep-th/9804001.
    Chaichian M., Demichev A., Presnajder P., Quantum field theory on non-commutative space-times and the persistence of ultraviolet divergences, Nuclear Phys. B 567 (2000), 360-390, hep-th/9812180.
  80. Minwalla S., Van Raamsdonk M., Seiberg N., Noncommutative perturbative dynamics, J. High Energy Phys. 2000 (2000), no. 2, 020, 31 pages, hep-th/9912072.
  81. Grosse H., Wulkenhaar R., Renormalisation of φ4-theory on non-commutative R4 to all orders, Lett. Math. Phys. 71 (2005), 13-26.
  82. Grosse H., Steinacker H., Finite gauge theory on fuzzy CP2, Nuclear Phys. B 707 (2005), 145-198, hep-th/0407089.
  83. Heinemeyer S., Mondragón M., Zoupanos G., Finite unification: theory and predictions, SIGMA 6 (2010), 049, 29 pages, arXiv:1001.0428.
  84. Ishibashi N., Kawai H., Kitazawa Y., Tsuchiya A., A large-N reduced model as superstring, Nuclear Phys. B 498 (1997), 467-491, hep-th/9612115.
  85. Aoki H., Iso S., Suyama T., Orbifold matrix model, Nuclear Phys. B 634 (2002), 71-89, hep-th/0203277.
  86. 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., 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., 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., 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, 23 pages, hep-ph/0406207.
  87. Chatzistavrakidis A., Manousselis P., Zoupanos G., Reducing the Heterotic Supergravity on nearly-Kähler coset spaces, Fortschr. Phys. 57 (2009), 527-534, arXiv:0811.2182.
    Chatzistavrakidis A., Zoupanos G., Dimensional reduction of the heterotic string over nearly-Kähler manifolds, J. High Energy Phys. 2009 (2009), no. 9, 077, 39 pages, arXiv:0905.2398.

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