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Note on the space group selection rule for closed strings on orbifolds

  • Saúl Ramos-SánchezEmail author
  • Patrick K. S. Vaudrevange
Open Access
Regular Article - Theoretical Physics
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Abstract

It is well-known that the space group selection rule constrains the interactions of closed strings on orbifolds. For some examples, this rule has been described by an effective Abelian symmetry that combines with a permutation symmetry to a non-Abelian flavor symmetry like D4 or Δ(54). However, the general case of the effective Abelian symmetries was not yet fully understood. In this work, we formalize the computation of the Abelian symmetry that results from the space group selection rule by imposing two conditions only: (i) well-defined discrete charges and (ii) their conservation. The resulting symmetry, which we call the space group flavor symmetry DS, is uniquely specified by the Abelianization of the space group. For all Abelian orbifolds with \( \mathcal{N}=1 \) supersymmetry we compute DS and identify new cases, for example, where DS contains a 2 dark matter-parity with charges 0 and 1 for massless and massive strings, respectively.

Keywords

Discrete Symmetries Superstrings and Heterotic Strings 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [INSPIRE].
  3. [3]
    M. Blaszczyk, S. Groot Nibbelink, O. Loukas and S. Ramos-Sánchez, Non-supersymmetric heterotic model building, JHEP 10 (2014) 119 [arXiv:1407.6362] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. Blaszczyk, S. Groot Nibbelink, O. Loukas and F. Ruehle, Calabi-Yau compactifications of non-supersymmetric heterotic string theory, JHEP 10 (2015) 166 [arXiv:1507.06147] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    O. Lebedev et al., A mini-landscape of exact MSSM spectra in heterotic orbifolds, Phys. Lett. B 645 (2007) 88 [hep-th/0611095] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    O. Lebedev, H.P. Nilles, S. Ramos-Sánchez, M. Ratz and P.K.S. Vaudrevange, Heterotic mini-landscape. (II). Completing the search for MSSM vacua in a Z 6 orbifold, Phys. Lett. B 668 (2008) 331 [arXiv:0807.4384] [INSPIRE].
  7. [7]
    D.K. Mayorga Peña, H.P. Nilles and P.-K. Oehlmann, A zip-code for quarks, leptons and Higgs bosons, JHEP 12 (2012) 024 [arXiv:1209.6041] [INSPIRE].CrossRefGoogle Scholar
  8. [8]
    S. Groot Nibbelink and O. Loukas, MSSM-like models on Z 8 toroidal orbifolds, JHEP 12 (2013) 044 [arXiv:1308.5145] [INSPIRE].CrossRefzbMATHGoogle Scholar
  9. [9]
    H.P. Nilles and P.K.S. Vaudrevange, Geography of fields in extra dimensions: string theory lessons for particle physics, Mod. Phys. Lett. A 30 (2015) 1530008 [arXiv:1403.1597] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    B. Carballo-Pérez, E. Peinado and S. Ramos-Sánchez, Δ(54) flavor phenomenology and strings, JHEP 12 (2016) 131 [arXiv:1607.06812] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S. Ramos-Sánchez, On flavor symmetries of phenomenologically viable string compactifications, J. Phys. Conf. Ser. 912 (2017) 012011 [arXiv:1708.01595] [INSPIRE].CrossRefGoogle Scholar
  12. [12]
    Y. Olguín-Trejo, R. Pérez-Martínez and S. Ramos-Sánchez, Charting the flavor landscape of MSSM-like Abelian heterotic orbifolds, Phys. Rev. D 98 (2018) 106020 [arXiv:1808.06622] [INSPIRE].
  13. [13]
    H.P. Nilles, S. Ramos-Sánchez, M. Ratz and P.K.S. Vaudrevange, A note on discrete R symmetries in Z 6 -II orbifolds with Wilson lines, Phys. Lett. B 726 (2013) 876 [arXiv:1308.3435] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    N.G. Cabo Bizet, T. Kobayashi, D.K. Mayorga Peña, S.L. Parameswaran, M. Schmitz and I. Zavala, Discrete R-symmetries and anomaly universality in heterotic orbifolds, JHEP 02 (2014) 098 [arXiv:1308.5669] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    H.P. Nilles, Stringy origin of discrete R-symmetries, PoS(CORFU2016)017 (2017) [arXiv:1705.01798] [INSPIRE].
  16. [16]
    J. Lauer, J. Mas and H.P. Nilles, Duality and the role of nonperturbative effects on the world sheet, Phys. Lett. B 226 (1989) 251 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    J. Lauer, J. Mas and H.P. Nilles, Twisted sector representations of discrete background symmetries for two-dimensional orbifolds, Nucl. Phys. B 351 (1991) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    L.E. Ibáñez and D. Lüst, Duality anomaly cancellation, minimal string unification and the effective low-energy Lagrangian of 4D strings, Nucl. Phys. B 382 (1992) 305 [hep-th/9202046] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    D. Bailin, A. Love, W.A. Sabra and S. Thomas, Modular symmetries in Z N orbifold compactified string theories with Wilson lines, Mod. Phys. Lett. A 9 (1994) 1229 [hep-th/9312122] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    S. Hamidi and C. Vafa, Interactions on orbifolds, Nucl. Phys. B 279 (1987) 465 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    T. Kobayashi, H.P. Nilles, F. Plöger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].
  23. [23]
    H.P. Nilles, M. Ratz and P.K.S. Vaudrevange, Origin of family symmetries, Fortsch. Phys. 61 (2013) 493 [arXiv:1204.2206] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    H.P. Nilles, M. Ratz, A. Trautner and P.K.S. Vaudrevange, CP violation from string theory, Phys. Lett. B 786 (2018) 283 [arXiv:1808.07060] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  25. [25]
    M. Fischer, M. Ratz, J. Torrado and P.K.S. Vaudrevange, Classification of symmetric toroidal orbifolds, JHEP 01 (2013) 084 [arXiv:1209.3906] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Fischer, M. Ratz, J. Torrado and P.K.S. Vaudrevange, Classification of symmetric toroidal orbifolds webpage, http://users.ph.tum.de/ga57raj/Orbifolds/ClassificationOrbifolds/index.html.
  27. [27]
    T. Araki, Anomaly of discrete symmetries and gauge coupling unification, Prog. Theor. Phys. 117 (2007) 1119 [hep-ph/0612306] [INSPIRE].
  28. [28]
    W. Buchmüller, K. Hamaguchi, O. Lebedev and M. Ratz, Supersymmetric Standard Model from the heterotic string (II), Nucl. Phys. B 785 (2007) 149 [hep-th/0606187] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J.G. Ratcliffe and S.T. Tschantz, Abelianization of space groups, Acta Cryst. A 65 (2008) 18.MathSciNetzbMATHGoogle Scholar
  30. [30]
    M. Blaszczyk, Heterotic particle models from various perspectives, Ph.D. thesis, University of Bonn, Bonn, Germany (2012).Google Scholar
  31. [31]
    M. Blaszczyk and P.-K. Oehlmann, Tracing symmetries and their breakdown through phases of heterotic (2, 2) compactifications, JHEP 04 (2016) 068 [arXiv:1512.03055] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  32. [32]
    B. Petersen, M. Ratz and R. Schieren, Patterns of remnant discrete symmetries, JHEP 08 (2009) 111 [arXiv:0907.4049] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    R. Donagi and K. Wendland, On orbifolds and free fermion constructions, J. Geom. Phys. 59 (2009) 942 [arXiv:0809.0330] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S. Förste, T. Kobayashi, H. Ohki and K.-J. Takahashi, Non-factorisable Z 2 × Z 2 heterotic orbifold models and Yukawa couplings, JHEP 03 (2007) 011 [hep-th/0612044] [INSPIRE].CrossRefGoogle Scholar
  35. [35]
    M. Blaszczyk, S. Groot Nibbelink, M. Ratz, F. Ruehle, M. Trapletti and P.K.S. Vaudrevange, A Z 2 × Z 2 Standard Model, Phys. Lett. B 683 (2010) 340 [arXiv:0911.4905] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    H.P. Nilles, S. Ramos-Sánchez, P.K.S. Vaudrevange and A. Wingerter, The orbifolder: a tool to study the low energy effective theory of heterotic orbifolds, Comput. Phys. Commun. 183 (2012) 1363 [arXiv:1110.5229] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    K. Fujikawa, Path integral measure for gauge invariant fermion theories, Phys. Rev. Lett. 42 (1979) 1195 [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    K. Fujikawa, Path integral for gauge theories with fermions, Phys. Rev. D 21 (1980) 2848 [Erratum ibid. D 22 (1980) 1499] [INSPIRE].
  39. [39]
    T. Araki, T. Kobayashi, J. Kubo, S. Ramos-Sánchez, M. Ratz and P.K.S. Vaudrevange, (Non-)Abelian discrete anomalies, Nucl. Phys. B 805 (2008) 124 [arXiv:0805.0207] [INSPIRE].
  40. [40]
    M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    Z. Kakushadze, G. Shiu and S.-H. Henry Tye, Asymmetric non-Abelian orbifolds and model building, Phys. Rev. D 54 (1996) 7545 [hep-th/9607137] [INSPIRE].ADSMathSciNetGoogle Scholar
  42. [42]
    S.J.H. Konopka, Non Abelian orbifold compactifications of the heterotic string, JHEP 07 (2013) 023 [arXiv:1210.5040] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    M. Fischer, S. Ramos-Sánchez and P.K.S. Vaudrevange, Heterotic non-Abelian orbifolds, JHEP 07 (2013) 080 [arXiv:1304.7742] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Saúl Ramos-Sánchez
    • 1
  • Patrick K. S. Vaudrevange
    • 2
  1. 1.Instituto de FísicaUniversidad Nacional Autónoma de MéxicoMexico CityMéxico
  2. 2.Physik Department T75Technische Universität MünchenGarchingGermany

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