Tracing symmetries and their breakdown through phases of heterotic (2,2) compactifications

  • Michael Blaszczyk
  • Paul-Konstantin Oehlmann
Open Access
Regular Article - Theoretical Physics


We are considering the class of heterotic \( \mathcal{N}=\left(2,2\right) \) Landau-Ginzburg orbifolds with 9 fields corresponding to A 1 9 Gepner models. We classify all of its Abelian discrete quotients and obtain 152 inequivalent models closed under mirror symmetry with \( \mathcal{N}=1 \), 2 and 4 supersymmetry in 4D. We compute the full massless matter spectrum at the Fermat locus and find a universal relation satisfied by all models. In addition we give prescriptions of how to compute all quantum numbers of the 4D states including their discrete R-symmetries. Using mirror symmetry of rigid geometries we describe orbifold and smooth Calabi-Yau phases as deformations away from the Landau-Ginzburg Fermat locus in two explicit examples. We match the non-Fermat deformations to the 4D Higgs mechanism and study the conservation of R-symmetries. The first example is a \( {\mathrm{\mathbb{Z}}}_3 \) orbifold on an E6 lattice where the R-symmetry is preserved. Due to a permutation symmetry of blow-up and torus Kähler parameters the R-symmetry stays conserved also in the smooth Calabi-Yau phase. In the second example the R-symmetry gets broken once we deform to the geometric \( {\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_{3,\mathrm{free}} \) orbifold regime.


Extended Supersymmetry Superstrings and Heterotic Strings Discrete Symmetries Superstring Vacua 


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.


  1. [1]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S. Groot Nibbelink, Heterotic orbifold resolutions as (2, 0) gauged linear σ-models, Fortsch. Phys. 59 (2011) 454 [arXiv:1012.3350] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    P. Aspinwall and R. Plesser, Elusive worldsheet instantons in heterotic string compactifications, Proc. Symp. Pure Math. 85 (2012) 33.MathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Blaszczyk, S. Groot Nibbelink and F. Ruehle, Gauged Linear σ-models for toroidal orbifold resolutions, JHEP 05 (2012) 053 [arXiv:1111.5852] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    C. Lütken and G. Ross, Taxonomy of heterotic superconformal field theories, Phys. Lett. B 213 (1988) 152.ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    A. Font, L.E. Ibáñez, M. Mondragon, F. Quevedo and G.G. Ross, (0, 2) Heterotic String Compactifications From N = 2 Superconformal Theories, Phys. Lett. B 227 (1989) 34 [INSPIRE].
  7. [7]
    J. Fuchs, A. Klemm, C. Scheich and M.G. Schmidt, Spectra and Symmetries of Gepner Models Compared to Calabi-Yau Compactifications, Annals Phys. 204 (1990) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Lynker and R. Schimmrigk, String compactifications on G/H Landau-Ginzburg theories, Phys. Lett. B 253 (1991) 83 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Fischer, M. Ratz, J. Torrado and P.K.S. Vaudrevange, Classification of symmetric toroidal orbifolds, JHEP 01 (2013) 084 [arXiv:1209.3906] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Altman, J. Gray, Y.-H. He, V. Jejjala and B.D. Nelson, A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List, JHEP 02 (2015) 158 [arXiv:1411.1418] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    R. Blumenhagen and A. Wisskirchen, Exactly solvable (0,2) supersymmetric string vacua with GUT gauge groups, Nucl. Phys. B 454 (1995) 561 [hep-th/9506104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Gato-Rivera and A.N. Schellekens, Heterotic Weight Lifting, Nucl. Phys. B 828 (2010) 375 [arXiv:0910.1526] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    O. Lebedev, H.P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, P.K.S. Vaudrevange et al., A Mini-landscape of exact MSSM spectra in heterotic orbifolds, Phys. Lett. B 645 (2007) 88 [hep-th/0611095] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    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].CrossRefGoogle Scholar
  17. [17]
    D.K. Mayorga Pena and P.-K. Oehlmann, Lessons from an Extended Heterotic Mini-Landscape, PoS(Corfu2012)096 [arXiv:1305.0566] [INSPIRE].
  18. [18]
    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
  19. [19]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, The Exact MSSM spectrum from string theory, JHEP 05 (2006) 043 [hep-th/0512177] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic Line Bundle Standard Models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    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].ADSCrossRefGoogle Scholar
  22. [22]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  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].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    N.G. Cabo Bizet, T. Kobayashi, D.K. Mayorga Pena, S.L. Parameswaran, M. Schmitz and I. Zavala, R-charge Conservation and More in Factorizable and Non-Factorizable Orbifolds, JHEP 05 (2013) 076 [arXiv:1301.2322] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    N.G. Cabo Bizet, T. Kobayashi, D.K. Mayorga Pena, 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
  26. [26]
    H.P. Nilles, S. Ramos-Sánchez, M. Ratz and P.K.S. Vaudrevange, A note on discrete R symmetries in \( {\mathrm{\mathbb{Z}}}_6 \) -II orbifolds with Wilson lines, Phys. Lett. B 726 (2013) 876 [arXiv:1308.3435] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    E.I. Buchbinder, A. Constantin and A. Lukas, A heterotic standard model with BL symmetry and a stable proton, JHEP 06 (2014) 100 [arXiv:1404.2767] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    C. Lüdeling, F. Ruehle and C. Wieck, Non-Universal Anomalies in Heterotic String Constructions, Phys. Rev. D 85 (2012) 106010 [arXiv:1203.5789] [INSPIRE].ADSGoogle Scholar
  29. [29]
    B.R. Greene and M.R. Plesser, Duality in Calabi-Yau Moduli Space, Nucl. Phys. B 338 (1990) 15 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    P. Candelas, X. C. D. L. Ossa, P. S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    E.J. Chun, J. Mas, J. Lauer and H.P. Nilles, Duality and Landau-Ginzburg Models, Phys. Lett. B 233 (1989) 141 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    S. Kachru and E. Witten, Computing the complete massless spectrum of a Landau-Ginzburg orbifold, Nucl. Phys. B 407 (1993) 637 [hep-th/9307038] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    T. Banks, Effective lagrangian description on discrete gauge symmetries, Nucl. Phys. B 323 (1989) 90.ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic Genera of 2d \( \mathcal{N}=2 \) Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    S. Groot Nibbelink and F. Ruehle, Torus partition functions and spectra of gauged linear σ-models, arXiv:1403.2380 [INSPIRE].
  38. [38]
    P.S. Aspinwall, I.V. Melnikov and M.R. Plesser, (0, 2) Elephants, JHEP 01 (2012) 060 [arXiv:1008.2156] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    D. Gepner, Space-Time Supersymmetry in Compactified String Theory and Superconformal Models, Nucl. Phys. B 296 (1988) 757 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    J. Distler and S. Kachru, Quantum symmetries and stringy instantons, Phys. Lett. B 336 (1994) 368 [hep-th/9406091] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    A. Font, L. Ibáñez and F. Quevedo, String compactifications and N = 2 superconformal coset constructions, Phys. Lett. B 224 (1989) 79.ADSMathSciNetGoogle Scholar
  42. [42]
    A. Klemm and R. Schimmrigk, Landau-Ginzburg string vacua, Nucl. Phys. B 411 (1994) 559 [hep-th/9204060] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    A.N. Schellekens and S. Yankielowicz, New Modular Invariants for N = 2 Tensor Products and Four-dimensional Strings, Nucl. Phys. B 330 (1990) 103 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    R. Blumenhagen and A. Wisskirchen, Exploring the moduli space of (0, 2) strings, Nucl. Phys. B 475 (1996) 225 [hep-th/9604140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    J. Distler and S. Kachru, Singlet couplings and (0, 2) models, Nucl. Phys. B 430 (1994) 13 [hep-th/9406090] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    P.S. Aspinwall and M.R. Plesser, Elusive Worldsheet Instantons in Heterotic String Compactifications, Proc. Symp. Pure Math. 85 (2012) 33 [arXiv:1106.2998] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  48. [48]
    C. Vafa and N.P. Warner, Catastrophes and the Classification of Conformal Theories, Phys. Lett. B 218 (1989) 51 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    F. Beye, T. Kobayashi and S. Kuwakino, Gauge Origin of Discrete Flavor Symmetries in Heterotic Orbifolds, Phys. Lett. B 736 (2014) 433 [arXiv:1406.4660] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  50. [50]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stability Walls in Heterotic Theories, JHEP 09 (2009) 026 [arXiv:0905.1748] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    P.K.S. Vaudrevange, Grand Unification in the Heterotic Brane World, Ph.D. Thesis, Bonn University, Bonn Germany (2008) [arXiv:0812.3503].
  52. [52]
    M. Bertolini, I.V. Melnikov and M.R. Plesser, Accidents in (0, 2) Landau-Ginzburg theories, JHEP 12 (2014) 157 [arXiv:1405.4266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    I. Melnikov, S. Sethi and E. Sharpe, Recent Developments in (0, 2) Mirror Symmetry, SIGMA 8 (2012) 068 [arXiv:1209.1134] [INSPIRE].MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Johannes-Gutenberg-UniversitätMainzGermany
  2. 2.Bethe Center for Theoretical Physics, Physikalisches Institut der Universität BonnBonnGermany

Personalised recommendations