T-duality orbifolds of heterotic Narain compactifications

Abstract

To obtain a unified framework for symmetric and asymmetric heterotic orbifold constructions we provide a systematic study of Narain compactifications orbifolded by finite order T -duality subgroups. We review the generalized vielbein that parametrizes the Narain moduli space (i.e. the metric, the B-field and the Wilson lines) and introduce a convenient basis of generators of the heterotic T -duality group. Using this we generalize the space group description of orbifolds to Narain orbifolds. This yields a unified, crystallographic description of the orbifold twists, shifts as well as Narain moduli. In particular, we derive a character formula that counts the number of unfixed Narain moduli after orbifolding. More-over, we develop new machinery that may ultimately open up the possibility for a full classification of Narain orbifolds. This is done by generalizing the geometrical concepts of and affine classes from the theory of crystallography to the Narain case. Finally, we give a variety of examples illustrating various aspects of Narain orbifolds, including novel T -folds.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, The heterotic string, Phys. Rev. Lett. 54 (1985) 502 [INSPIRE].

  2. [2]

    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic string theory. 1. The free heterotic string, Nucl. Phys. B 256 (1985) 253 [INSPIRE].

  3. [3]

    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic string theory. 2. The interacting heterotic string, Nucl. Phys. B 267 (1986) 75 [INSPIRE].

  4. [4]

    L.E. Ibáñez and A.M. Uranga, String theory and particle physics: an introduction to string phenomenology, Cambridge University Press, Cambridge U.K., (2012) [INSPIRE].

    Google Scholar 

  5. [5]

    L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].

  6. [6]

    L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [INSPIRE].

  7. [7]

    M. Fischer, M. Ratz, J. Torrado and P.K.S. Vaudrevange, Classification of symmetric toroidal orbifolds, JHEP 01 (2013) 084 [arXiv:1209.3906] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    L.E. Ibáñez, H.P. Nilles and F. Quevedo, Orbifolds and Wilson lines, Phys. Lett. B 187 (1987) 25 [INSPIRE].

  9. [9]

    L.E. Ibáñez, J.E. Kim, H.P. Nilles and F. Quevedo, Orbifold compactifications with three families of SU(3) × SU(2) × U(1)n, Phys. Lett. B 191 (1987) 282 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. [10]

    J.A. Casas and C. Muñoz, Three generation SU(3) × SU(2) × U(1) Y models from orbifolds, Phys. Lett. B 214 (1988) 63 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  11. [11]

    J.A. Casas, E.K. Katehou and C. Muñoz, U(1) charges in orbifolds: anomaly cancellation and phenomenological consequences, Nucl. Phys. B 317 (1989) 171 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. [12]

    A. Font, L.E. Ibáñez, F. Quevedo and A. Sierra, The construction ofrealisticfour-dimensional strings through orbifolds, Nucl. Phys. B 331 (1990) 421 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. [13]

    D. Bailin and A. Love, Orbifold compactifications of string theory, Phys. Rept. 315 (1999) 285 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. [14]

    S. Förste, H.P. Nilles, P.K.S. Vaudrevange and A. Wingerter, Heterotic brane world, Phys. Rev. D 70 (2004) 106008 [hep-th/0406208] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  15. [15]

    T. Kobayashi, S. Raby and R.-J. Zhang, Searching for realistic 4d string models with a Pati-Salam symmetry: orbifold grand unified theories from heterotic string compactification on a Z 6 orbifold, Nucl. Phys. B 704 (2005) 3 [hep-ph/0409098] [INSPIRE].

  16. [16]

    W. Buchmüller, K. Hamaguchi, O. Lebedev and M. Ratz, Supersymmetric Standard Model from the heterotic string, Phys. Rev. Lett. 96 (2006) 121602 [hep-ph/0511035] [INSPIRE].

  17. [17]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    J.E. Kim and B. Kyae, Flipped SU(5) from Z 12−I orbifold with Wilson line, Nucl. Phys. B 770 (2007) 47 [hep-th/0608086] [INSPIRE].

  19. [19]

    O. Lebedev et al., A mini-landscape of exact MSSM spectra in heterotic orbifolds, Phys. Lett. B 645 (2007) 88 [hep-th/0611095] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    J.E. Kim, J.-H. Kim and B. Kyae, Superstring Standard Model from Z 12−I orbifold compactification with and without exotics and effective R-parity, JHEP 06 (2007) 034 [hep-ph/0702278] [INSPIRE].

  21. [21]

    O. Lebedev et al., The heterotic road to the MSSM with R parity, Phys. Rev. D 77 (2008) 046013 [arXiv:0708.2691] [INSPIRE].

    ADS  Google Scholar 

  22. [22]

    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].

  23. [23]

    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].

    ADS  Article  Google Scholar 

  24. [24]

    D.K. Mayorga Pena, H.P. Nilles and P.-K. Oehlmann, A zip-code for quarks, leptons and Higgs bosons, JHEP 12 (2012) 024 [arXiv:1209.6041] [INSPIRE].

    Article  Google Scholar 

  25. [25]

    S. Groot Nibbelink and O. Loukas, MSSM-like models on Z 8 toroidal orbifolds, JHEP 12 (2013) 044 [arXiv:1308.5145] [INSPIRE].

  26. [26]

    B. Carballo-Pérez, E. Peinado and S. Ramos-Sánchez, Δ(54) flavor phenomenology and strings, JHEP 12 (2016) 131 [arXiv:1607.06812] [INSPIRE].

    ADS  Article  Google Scholar 

  27. [27]

    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A heterotic Standard Model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [INSPIRE].

  28. [28]

    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A Standard Model from the E 8 × E 8 heterotic superstring, JHEP 06 (2005) 039 [hep-th/0502155] [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

    R. Blumenhagen, G. Honecker and T. Weigand, Loop-corrected compactifications of the heterotic string with line bundles, JHEP 06 (2005) 020 [hep-th/0504232] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  30. [30]

    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic line bundle Standard Models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. [31]

    L.B. Anderson, A. Constantin, J. Gray, A. Lukas and E. Palti, A comprehensive scan for heterotic SU(5) GUT models, JHEP 01 (2014) 047 [arXiv:1307.4787] [INSPIRE].

    ADS  Article  Google Scholar 

  32. [32]

    S. Groot Nibbelink, O. Loukas and F. Ruehle, (MS)SM-like models on smooth Calabi-Yau manifolds from all three heterotic string theories, Fortsch. Phys. 63 (2015) 609 [arXiv:1507.07559] [INSPIRE].

  33. [33]

    W. Buchmüller, K. Hamaguchi, O. Lebedev and M. Ratz, Local grand unification, in CP violation and the flavour puzzle: symposium in honour of Gustavo C. Branco. GustavoFest 2005, Lisbon Portugal, July 2005, pg. 143 [hep-ph/0512326] [INSPIRE].

  34. [34]

    T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].

  35. [35]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  36. [36]

    M.T. Mueller and E. Witten, Twisting toroidally compactified heterotic strings with enlarged symmetry groups, Phys. Lett. B 182 (1986) 28 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  37. [37]

    T.H. Buscher, A symmetry of the string background field equations, Phys. Lett. B 194 (1987) 59 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  38. [38]

    A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].

  39. [39]

    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  40. [40]

    K.S. Narain, M.H. Sarmadi and C. Vafa, Asymmetric orbifolds, Nucl. Phys. B 288 (1987) 551 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  41. [41]

    S. Hellerman, J. McGreevy and B. Williams, Geometric constructions of nongeometric string theories, JHEP 01 (2004) 024 [hep-th/0208174] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  42. [42]

    A. Dabholkar and C. Hull, Duality twists, orbifolds and fluxes, JHEP 09 (2003) 054 [hep-th/0210209] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  43. [43]

    J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  44. [44]

    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  45. [45]

    C.M. Hull, Doubled geometry and T-folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  46. [46]

    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  47. [47]

    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  48. [48]

    G. Aldazabal, D. Marqués and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  49. [49]

    D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 12 (2010) 084 [arXiv:1010.1361] [INSPIRE].

    Article  MATH  Google Scholar 

  50. [50]

    C. Condeescu, I. Florakis and D. Lüst, Asymmetric orbifolds, non-geometric fluxes and non-commutativity in closed string theory, JHEP 04 (2012) 121 [arXiv:1202.6366] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  51. [51]

    J.A. Harvey, G.W. Moore and C. Vafa, Quasicrystalline compactification, Nucl. Phys. B 304 (1988) 269 [INSPIRE].

  52. [52]

    L.E. Ibáñez, J. Mas, H.-P. Nilles and F. Quevedo, Heterotic strings in symmetric and asymmetric orbifold backgrounds, Nucl. Phys. B 301 (1988) 157 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  53. [53]

    K.S. Narain, M.H. Sarmadi and C. Vafa, Asymmetric orbifolds: path integral and operator formulations, Nucl. Phys. B 356 (1991) 163 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  54. [54]

    Y. Imamura, M. Sakamoto, T. Sasada and M. Tabuse, Symmetries between untwisted and twisted strings on asymmetric orbifolds, Nucl. Phys. B 390 (1993) 291 [hep-th/9206042] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  55. [55]

    Y. Imamura, M. Sakamoto, T. Sasada and M. Tabuse, String theories on the asymmetric orbifolds with twist-untwist intertwining currents, Prog. Theor. Phys. Suppl. 110 (1992) 261 [hep-th/9202009] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  56. [56]

    T. Sasada, Fermion currents on asymmetric orbifolds, Phys. Lett. B 343 (1995) 128 [hep-th/9312066] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  57. [57]

    T. Sasada, Space-time supersymmetry in asymmetric orbifold models, hep-th/9403037 [INSPIRE].

  58. [58]

    J. Erler, Asymmetric orbifolds and higher level models, Nucl. Phys. B 475 (1996) 597 [hep-th/9602032] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  59. [59]

    K. Aoki, E. D’Hoker and D.H. Phong, On the construction of asymmetric orbifold models, Nucl. Phys. B 695 (2004) 132 [hep-th/0402134] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  60. [60]

    H.S. Tan, T-duality twists and asymmetric orbifolds, JHEP 11 (2015) 141 [arXiv:1508.04807] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  61. [61]

    Y. Satoh and Y. Sugawara, Lie algebra lattices and strings on T-folds, JHEP 02 (2017) 024 [arXiv:1611.08076] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  62. [62]

    T.R. Taylor, Model building on asymmetric Z 3 orbifolds: nonsupersymmetric models, Nucl. Phys. B 303 (1988) 543 [INSPIRE].

    ADS  Article  Google Scholar 

  63. [63]

    Y. Satoh, Y. Sugawara and T. Wada, Non-supersymmetric asymmetric orbifolds with vanishing cosmological constant, JHEP 02 (2016) 184 [arXiv:1512.05155] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  64. [64]

    Y. Sugawara and T. Wada, More on non-supersymmetric asymmetric orbifolds with vanishing cosmological constant, JHEP 08 (2016) 028 [arXiv:1605.07021] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  65. [65]

    H. Kawai, D.C. Lewellen and S.-H. Henry Tye, Construction of fermionic string models in four-dimensions, Nucl. Phys. B 288 (1987) 1 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  66. [66]

    I. Antoniadis, C.P. Bachas and C. Kounnas, Four-dimensional superstrings, Nucl. Phys. B 289 (1987) 87 [INSPIRE].

  67. [67]

    P. Athanasopoulos, A.E. Faraggi, S. Groot Nibbelink and V.M. Mehta, Heterotic free fermionic and symmetric toroidal orbifold models, JHEP 04 (2016) 038 [arXiv:1602.03082] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  68. [68]

    A.E. Faraggi, D.V. Nanopoulos and K.-J. Yuan, A standard like model in the 4D free fermionic string formulation, Nucl. Phys. B 335 (1990) 347 [INSPIRE].

    ADS  Article  Google Scholar 

  69. [69]

    G.B. Cleaver, A.E. Faraggi and D.V. Nanopoulos, String derived MSSM and M-theory unification, Phys. Lett. B 455 (1999) 135 [hep-ph/9811427] [INSPIRE].

  70. [70]

    A.E. Faraggi, A new standard-like model in the four-dimensional free fermionic string formulation, Phys. Lett. B 278 (1992) 131 [INSPIRE].

    ADS  Article  Google Scholar 

  71. [71]

    A.E. Faraggi, Construction of realistic standard-like models in the free fermionic superstring formulation, Nucl. Phys. B 387 (1992) 239 [hep-th/9208024] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  72. [72]

    F. Beye, T. Kobayashi and S. Kuwakino, Gauge symmetries in heterotic asymmetric orbifolds, Nucl. Phys. B 875 (2013) 599 [arXiv:1304.5621] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  73. [73]

    F. Beye, T. Kobayashi and S. Kuwakino, Three-generation asymmetric orbifold models from heterotic string theory, JHEP 01 (2014) 013 [arXiv:1311.4687] [INSPIRE].

    ADS  Article  Google Scholar 

  74. [74]

    F. Beye, T. Kobayashi and S. Kuwakino, Dilaton stabilization in three-generation heterotic string model, Phys. Lett. B 760 (2016) 63 [arXiv:1603.08313] [INSPIRE].

    ADS  Article  Google Scholar 

  75. [75]

    W. Lerche, D. Lüst and A.N. Schellekens, Chiral four-dimensional heterotic strings from selfdual lattices, Nucl. Phys. B 287 (1987) 477 [INSPIRE].

    ADS  Article  Google Scholar 

  76. [76]

    K.S. Narain, New heterotic string theories in uncompactified dimensions < 10, Phys. Lett. B 169 (1986) 41 [INSPIRE].

  77. [77]

    D. Gepner, Space-time supersymmetry in compactified string theory and superconformal models, Nucl. Phys. B 296 (1988) 757 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  78. [78]

    D. Gepner, Exactly solvable string compactifications on manifolds of SU(N) holonomy, Phys. Lett. B 199 (1987) 380 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  79. [79]

    B. Gato-Rivera and A.N. Schellekens, Asymmetric Gepner models: revisited, Nucl. Phys. B 841 (2010) 100 [arXiv:1003.6075] [INSPIRE].

  80. [80]

    B. Gato-Rivera and A.N. Schellekens, Asymmetric Gepner models II: heterotic weight lifting, Nucl. Phys. B 846 (2011) 429 [arXiv:1009.1320] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  81. [81]

    A.N. Schellekens, Big numbers in string theory, arXiv:1601.02462 [INSPIRE].

  82. [82]

    D. Israël and V. Thiéry, Asymmetric Gepner models in type-II, JHEP 02 (2014) 011 [arXiv:1310.4116] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  83. [83]

    D. Israël, Nongeometric Calabi-Yau compactifications and fractional mirror symmetry, Phys. Rev. D 91 (2015) 066005 [Erratum ibid. D 91 (2015) 129902] [arXiv:1503.01552] [INSPIRE].

  84. [84]

    R. Blumenhagen, M. Fuchs and E. Plauschinn, The asymmetric CFT landscape in D = 4, 6, 8 with extended supersymmetry, Fortsch. Phys. 65 (2017) 1700006 [arXiv:1611.04617] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  85. [85]

    R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke and C. Schmid, The intriguing structure of non-geometric frames in string theory, Fortsch. Phys. 61 (2013) 893 [arXiv:1304.2784] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  86. [86]

    R. Blumenhagen and R. Sun, T-duality, non-geometry and Lie algebroids in heterotic double field theory, JHEP 02 (2015) 097 [arXiv:1411.3167] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  87. [87]

    R. Vaidyanathaswamy, Integer-roots of the unit matrix, J. Lond. Math. Soc. 3 (1928) 121.

    MathSciNet  Article  MATH  Google Scholar 

  88. [88]

    J. Polchinski, String theory. Vol. 2: superstring theory and beyond, Cambridge University Press, Cambridge U.K., (2007) [INSPIRE].

  89. [89]

    S. Bittanti, A.J. Laub and J.C. Willems, The Riccati equation, Springer Science & Business Media, Germany, (2012).

  90. [90]

    M. Cvetič, J. Louis and B.A. Ovrut, A string calculation of the Kähler potentials for moduli of Z N orbifolds, Phys. Lett. B 206 (1988) 227 [INSPIRE].

    ADS  Article  Google Scholar 

  91. [91]

    A. Dabholkar and J.A. Harvey, String islands, JHEP 02 (1999) 006 [hep-th/9809122] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Patrick K.S. Vaudrevange.

Additional information

ArXiv ePrint: 1703.05323

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nibbelink, S.G., Vaudrevange, P.K. T-duality orbifolds of heterotic Narain compactifications. J. High Energ. Phys. 2017, 30 (2017). https://doi.org/10.1007/JHEP04(2017)030

Download citation

Keywords

  • String Duality
  • Superstring Vacua
  • Superstrings and Heterotic String