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The trinification model SU(3)3 from orbifolds for fuzzy spheres

  • Physics of Elementary Particles and Atomic Nuclei. Theory
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Abstract

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)3 model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)3 spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)3.

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References

  1. P. Forgács and S. N. Manton, “Space-time symmetries in gauge theories,” Comm. Math. Phys. 72, 15–35 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  2. D. Kapetanakis and G. Zoupanos, “Coset space dimensional reduction of gauge theories,” Phys. Rep. 219 (1992).

    Google Scholar 

  3. A. Yu. Kubyshin, M. J. Mourao, G. Rudolph, and P. I. Volobujev, Lect. Notes Phys. 349 (1989).

  4. J. Scherk and H. J. Schwarz, “How to get masses from extra dimensions,” Nucl. Phys. B 153, 61–88 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  5. B. M. Green, H. J. Schwarz, and E. Witten, Superstring Theory, Vol. 1: Introduction, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, 1987)

    Google Scholar 

  6. B. M. Green, H. J. Schwarz, and E. Witten, Superstring Theory, Vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, 1987).

    Google Scholar 

  7. J. Polchinski, String Theory, Vol. 1: An Introduction to the Bosonic String (Cambridge Univ. Press, Cambridge, 1998)

    Book  MATH  Google Scholar 

  8. J. Polchinski, String Theory, Vol. 2: Superstring Theory and Beyond (Cambridge Univ. Press, Cambridge, 1998)

    Book  MATH  Google Scholar 

  9. R. Blumenhagen, D. Lüst, and S. Theisen, Basic Concepts of String Theory (Springer, 2013).

    Book  MATH  Google Scholar 

  10. J. D. Gross, A. J. Harvey, J. E. Martinec, and R. Rohm, “I. The free heterotic string,” Nucl. Phys. B 256, 253 (1985), “Heterotic string,” Phys. Rev. Lett. 54, 502 (1985).

  11. P. Candelas, T. G. Horowitz, A. Strominger, and E. Witten, “Vacuum configurations for superstrings,” Nucl. Phys. B 258, 46 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Connes, Noncommutative Geometry (Academic, San Diego, CA, 1994).

    MATH  Google Scholar 

  13. J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, 2nd ed., Vol. 257 of London Mathematical Society Lecture Note Series (Cambridge Univ. Press, Cambridge, 1999).

    Book  MATH  Google Scholar 

  14. M. Buric, T. Grammatikopoulos, J. Madore, and J. Zoupanos, “Gravity and the structure of non commutative algebras,” J. High Energy Phys. 0604, 054 (2006)

    Article  ADS  Google Scholar 

  15. M. Buric, J. Madore, and G. Zoupanos, “WKB approximation in non commutative gravity,” SIGMA 3, 125 (2007). arXiv:0712.4024 [hep-th].

    ADS  MATH  Google Scholar 

  16. T. Filk, “Divergences in a field theory on quantum space,” Phys. Lett. B 376, 53 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. C. J. Várilly and M. J. Gracia-Bondía, “On the ultraviolet behaviour of quantum fields over noncommutative manifolds,” Int. J. Mod. Phys. A 14, 1305 (1999), hep-th/9804001

    Article  ADS  MATH  Google Scholar 

  18. M. Chaichian, A. Demichev, and P. Presnajder, “Quantum field theory on noncommutative space times and the persistence of ultraviolet divergences,” Nucl. Phys. B 567, 360 (2000), hep-th/9812180

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. S. Minwalla, van M. Raamsdonk, and N. Seiberg, “Noncommutative perturbative dynamics,” J. High Energy Phys. 0002, 020 (2000), hep-th/9912072.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. H. Grosse and R. Wulkenhaar, “Renormalisation of f4-theory on noncommutative to all orders,” Lett. Math. Phys. 71, 13 (2005), hep-th/0403232.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. H. Grosse and H. Steinacker, “Exact renormalization of a noncommutative f3-model in 6 dimensions,” Adv. Theor. Math. Phys. 12, 605 (2008), hep-th/0607235; “Finite gauge theory on fuzzy CP2,” Nucl. Phys. B 707, 145 (2005), hep-th/0407089.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Connes and J. Lott, “Particle models and noncommutative geometry,” Nucl. Phys. B Proc. Suppl. 18, 29–47 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. H. A. Chamseddine, and A. Connes, “The spectral action principle,” Comm. Math. Phys. 186, 731–750 (1997), hep-th/9606001; “Conceptual explanation for the algebra in the noncommutative approach to the standard model,” Phys. Rev. Lett. 99, 191601 (2007), arXiv:0706.3690.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. P. C. Martín, J. M. Gracia-Bondía, and C. J. Várilly, “The standard model as a noncom mutative geometry: the low energy regime,” Phys. Rep. 294, 363–406 (1998), hep-th/9605001.

    Article  ADS  MathSciNet  Google Scholar 

  25. M. Dubois Violette, J. Madore, and R. Kerner, “Gauge bosons in a noncommutative geometry,” Phys. Lett. B 217, 485–488 (1989), “Classical bosons in a noncommutative geometry,” Class. Quantum Grav. 6, 1709–1724 (1989), “Noncommutative differential geometry and new models of gauge theory,” J. Math. Phys. 31, 323–330 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. J. Madore, “On a quark lepton duality,” Phys. Lett. B 305, 84–89 (1993), “On a noncommutative extension of electrodynamics,” in Spinors, Twistors, Clifford Algebras and Quantum Deformations, Proceedings of the 2nd Max Born Symposium, Sobotka Castle, Wroclaw, Poland, Sept. 24–27, 1992, Vol. 52 of Fundamental Theories of Physics (Kluwer Academic, Dordrecht,1993), pp. 285–298; hep-ph/9209226.

    Article  ADS  Google Scholar 

  27. A. Connes, R. M. Douglas, and A. Schwarz, “Noncommutative geometry and matrix theory: compactification on tori,” J. High Energy Phys., No. 2, 003 (1998), hep-th/9711162.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. N. Seiberg and E. Witten, “String theory and noncommutative geometry,” J. High Energy Phys., No. 9, 032 (1999), hep-th/9908142.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, “A large N reduced model as super string,” Nucl. Phys. B 498, 467 (1997), arXiv:hep-th/9612115.

    Article  ADS  MATH  Google Scholar 

  30. B. Jurco, S. Schraml, P. Schupp, and J. Wess, “Enveloping algebra-valued gauge transformations for non-Abelian gauge groups on non-commutative spaces,” Eur. Phys. J. C 17, 521–526 (2000), hep-th/0006246; “Nonabelian noncommutative gauge theory via noncommutative extra dimensions,” Nucl. Phys. B 604, 148–180 (2001), hep-th/0102129

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. B. Jurco, L. Moller, S. Schraml, S. Schupp, and J. Wess, “Construction of non-Abelian gauge theories on noncommutative spaces,” Eur. Phys. J. C 21, 383–388 (2001), hepth/0104153

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. G. Barnich, F. Brandt, and M. Grigoriev, “Seiberg witten maps and noncommutative yang mills theories for arbitrary gauge groups,” J. High Energy Phys., No. 8, 023 (2002), hep-th/0206003.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. M. Chaichian, P. Prešnajder, Sheikh M. M. Jabbari, and A. Tureanu, “Non-commutative standard model: model building,” Eur. Phys. J. C 29, 413–432 (2003); hep-th/0107055.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. X. Calmet, B. Jurco, P. Schupp, J. Wess, and M. Wohlgenannt, “The standard model on noncommutative spacetime,” Eur. Phys. J. C 23, 363–376 (2002), hep-ph/0111115

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. P. Aschieri, B. Jurco, P. Schupp, and J. Wess, “Non commutative GUTs, standard model and C, P, T,” Nucl. Phys. B 651, 45–70 (2003), hep-th/0205214

    Article  ADS  MATH  Google Scholar 

  36. W. Behr, G. N. Deshpande, G. Duplancic, P. Schupp, J. Trampetic, and J. Wess, “The Z gamma gamma, gg decays in the noncommutative standard model,” Eur. Phys. J. C 29, 441–446 (2003).

    Article  ADS  Google Scholar 

  37. P. Aschieri, J. Madore, P. Manousselis, and G. Zoupanos, “Dimensional reduction over fuzzy coset spaces,” J. High Energy Phys., No. 4, 034 (2004), hep-th/0310072; “Unified theories from fuzzy extra dimensions,” Fortschr. Phys. 52, 718–723 (2004), hep-th/0401200; “Renormalizable theories from fuzzy higher dimensions,” Conference C04-08-20.1 (2005), pp. 135–146; hep-th/0503039.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. P. Aschieri, T. Grammatikopoulos, H. Steinacker, and G. Zoupanos, “Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking,” J. High Energy Phys., No. 9, 026 (2006), hep-th/0606021

    Article  ADS  MathSciNet  Google Scholar 

  39. P. Aschieri, H. Steinacker, J. Madore, P. Manousselis, and G. Zoupanos, “Fuzzy extra dimensions: dimensional reduction, dynamical generation and renormalizability,” arXiv:0704.2880.

  40. H. Steinacker and G. Zoupanos, “Fermions on spontaneously generated spherical extradimensions,” J. High Energy Phys., No. 9, 017 (2007), arXiv:0706.0398.

    Article  ADS  Google Scholar 

  41. A. Chatzistavrakidis, H. Steinacker, and G. Zoupanos, “On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes,” Fortschr. Phys. 58, 537–552 (2010), arXiv:0909.5559 [hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Chatzistavrakidis, H. Steinacker, and G. Zoupanos, “Orbifolds, fuzzy spheres and chiral fermions,” J. High Energy Phys. 1005, 100 (2010), arXiv:hep-th/1002.2606

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. A. Chatzistavrakidis, and G. Zoupanos, “Higher dimensional unified theories with fuzzy extra dimensions,” SIGMA 6, 063 (2010), arXiv:hep-th/1008.2049.

    MathSciNet  MATH  Google Scholar 

  44. L. Brink, H. J. Schwarz, and J. Scherk, “Supersymmetric Yang-Mills theories,” Nucl. Phys. B 121, 77–92 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  45. F. Gliozzi, J. Scherk, and I. D. Olive, “Supersymmetry, supergravity theories and the dual spinor model,” Nucl. Phys. B 122, 253–290 (1977).

    Article  ADS  Google Scholar 

  46. G. Aldazabal, E. L. Ibáñez, F. Quevedo, and M. A. Uranga, “D-branes at singularities: a bottomup approach to the string embedding of the standard model,” J. High Energy Phys., No. 8, 002 (2000), hep-th/0005067.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. S. Kachru and E. Silverstein, “4d conformal field theories and strings on orbifolds,” Phys. Rev. Lett. 80, 4855–4858 (1998), hep-th/9802183

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. E. A. Lawrence, N. Nekrasov, and C. Vafa, “On conformal field theories in four dimensions,” Nucl. Phys. B 533, 199–209 (1998), hep-th/9803015.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. E. Kiritsis, “D-branes in standard model building, gravity and cosmology,” Phys. Rep. 421, 105–190 (2005), Phys. Rep. 429, 121(E)–122(E) (2006), hep-th/0310001.

    Article  ADS  MathSciNet  Google Scholar 

  50. H. Grosse, F. Lizzi, and H. Steinacker, “Noncommutative gauge theory and symmetry breaking in matrix models,” Phys. Rev. D 81, 085034 (2010), arXiv:1001.2703 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  51. H. Steinacker, “Emergent gravity and noncommutative branes from yang mills matrix models,” Nucl. Phys. B 810, 1 (2009), arXiv:0806.2032 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. L. S. Glashow, “Trinification of all elementary particle forces,” in Proceedings of the 5th Workshop on Grand Unification, Brown University Providence, II, April 12–14, 1984, Providence, no. 0088, pp. 88–94.

    Google Scholar 

  53. A. V. Rizov, “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, 461–477 (1981).

    Google Scholar 

  54. E. Ma, M. Mondragón, and G. Zoupanos, “Finite SU(N)k unification,” J. High Energy Phys., No. 12, 026 (2004), hep-ph/0407236.

    Article  ADS  MathSciNet  Google Scholar 

  55. E. Ma, M. Mondragón, and G. Zoupanos, “Finite SU(N)**k unification,” J. High Energy Phys. 0412, 026 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  56. S. Heinemeyer, E. Ma, M. Mondragon, and G. Zoupanos, “Finite SU(3)**3 model,” AIP Conf. Proc. 1200, 568 (2010), arXiv:0910.0501 [hep-ph].

    Article  ADS  Google Scholar 

  57. G. Lazarides and C. Panagiotakopoulos, “MSSM from SUSY trinification,” Phys. Lett. B 336, 190–193 (1994), hep-ph/9403317.

    Article  ADS  Google Scholar 

  58. S. K. Babu, G. X. He, and S. Pakvasa, “Neutrino masses and proton decay modes in SU(3)× SU(3) × SU(3) trinification,” Phys. Rev. D 33, 763–772 (1986).

    Article  ADS  Google Scholar 

  59. K. G. Leontaris and J. Rizos, “A d-brane inspired U(3)C × U(3)L × U(3)R model,” Phys. Lett. B 632, 710–716 (2006), hep-ph/0510230.

    Article  ADS  Google Scholar 

  60. S. Heinemeyer, M. Mondragón, and G. Zoupanos, “The LHC Higgs boson discovery: implications for finite unified theories,” Int. J. Mod. Phys. A 29, 18 (2014), hep-ph/1430032.

    Article  MATH  Google Scholar 

  61. M. Mondragón, N. Tracas, and G. Zoupanos, “Reduction of couplings in quantum field theories with applications in finite theories and the MSSM,” Springer Proc. Math. Stat. 111, 177–196 (2014), arXiv:1403.7384 [hep-ph].

    Article  MATH  Google Scholar 

  62. S. Heinemeyer, M. Mondragón, and G. Zoupanos, “Finite unification: theory and predictions,” SIGMA 6, 049 (2010), arXiv:1001.0428 [hep-ph].

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Physics Department, National Technical University, Zografou Campus, GR-15780, Athens, Greece

    G. Manolakos & G. Zoupanos

  2. Institut für Theoretische Physik Universität Heidelberg Philosophenweg 16, D-69120, Heidelberg, Germany

    G. Zoupanos

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  1. G. Manolakos
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  2. G. Zoupanos
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Manolakos, G., Zoupanos, G. The trinification model SU(3)3 from orbifolds for fuzzy spheres. Phys. Part. Nuclei Lett. 14, 322–327 (2017). https://doi.org/10.1134/S1547477117020194

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