Journal of High Energy Physics

, 2015:183 | Cite as

Wilson lines and Chern-Simons flux in explicit heterotic Calabi-Yau compactifications

  • Fabio Apruzzi
  • Fridrik Freyr Gautason
  • Susha Parameswaran
  • Marco Zagermann
Open Access
Regular Article - Theoretical Physics


We study to what extent Wilson lines in heterotic Calabi-Yau compactifications lead to non-trivial H-flux via Chern-Simons terms. Wilson lines are basic ingredients for Standard Model constructions but their induced H-flux may affect the consistency of the leading order background geometry and of the two-dimensional worldsheet theory. Moreover H-flux in heterotic compactifications would play an important role for moduli stabilization and could strongly constrain the supersymmetry breaking scale. We show how to compute H-flux and the corresponding superpotential, given an explicit complete intersection Calabi-Yau compactification and choice of Wilson lines. We do so by identifying large classes of special Lagrangian submanifolds in the Calabi-Yau, understanding how the Wilson lines project onto these submanifolds, and computing their Chern-Simons invariants. We illustrate our procedure with the quintic hypersurface as well as the split-bicubic, which can provide a potentially realistic three generation model.


Flux compactifications Compactification and String Models Superstrings and Heterotic Strings Superstring Vacua 


Open Access

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  1. [1]
    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].CrossRefADSMathSciNetGoogle Scholar
  2. [2]
    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].CrossRefADSMathSciNetGoogle Scholar
  3. [3]
    V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    L.B. Anderson, J. Gray, Y.-H. He and A. Lukas, Exploring Positive Monad Bundles And A New Heterotic Standard Model, JHEP 02 (2010) 054 [arXiv:0911.1569] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    V. Braun, P. Candelas, R. Davies and R. Donagi, The MSSM Spectrum from (0,2)-Deformations of the Heterotic Standard Embedding, JHEP 05 (2012) 127 [arXiv:1112.1097] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic Line Bundle Standard Models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    V. Braun, Y.-H. He and B.A. Ovrut, Supersymmetric Hidden Sectors for Heterotic Standard Models, JHEP 09 (2013) 008 [arXiv:1301.6767] [INSPIRE].CrossRefADSGoogle Scholar
  8. [8]
    E. Witten, New Issues in Manifolds of SU(3) Holonomy, Nucl. Phys. B 268 (1986) 79 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing the Complex Structure in Heterotic Calabi-Yau Vacua, JHEP 02 (2011) 088 [arXiv:1010.0255] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua, Phys. Rev. D 83 (2011) 106011 [arXiv:1102.0011] [INSPIRE].ADSGoogle Scholar
  11. [11]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, The Atiyah Class and Complex Structure Stabilization in Heterotic Calabi-Yau Compactifications, JHEP 10 (2011) 032 [arXiv:1107.5076] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories, JHEP 07 (2013) 017 [arXiv:1304.2704] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    S. Gukov, S. Kachru, X. Liu and L. McAllister, Heterotic moduli stabilization with fractional Chern-Simons invariants, Phys. Rev. D 69 (2004) 086008 [hep-th/0310159] [INSPIRE].ADSMathSciNetGoogle Scholar
  14. [14]
    M. Cicoli, S. de Alwis and A. Westphal, Heterotic Moduli Stabilisation, JHEP 10 (2013) 199 [arXiv:1304.1809] [INSPIRE].CrossRefADSGoogle Scholar
  15. [15]
    A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    E. Witten, Global Anomalies in String Theory, Symposium on Anomalies, Geometry, Topology, edited by W.A. Bardeen and A.R. White, World Scientific, Singapore (1985), pg. 61–99.Google Scholar
  17. [17]
    H. Partouche and B. Pioline, Rolling among G 2 vacua, JHEP 03 (2001) 005 [hep-th/0011130] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    M. Dine and N. Seiberg, Nonrenormalization Theorems in Superstring Theory, Phys. Rev. Lett. 57 (1986) 2625 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, Heterotic string theory on nonKähler manifolds with H flux and gaugino condensate, Fortsch. Phys. 52 (2004) 483 [hep-th/0310021] [INSPIRE].CrossRefADSGoogle Scholar
  20. [20]
    A.R. Frey and M. Lippert, AdS strings with torsion: Non-complex heterotic compactifications, Phys. Rev. D 72 (2005) 126001 [hep-th/0507202] [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    L. Witten and E. Witten, Large Radius Expansion of Superstring Compactifications, Nucl. Phys. B 281 (1987) 109 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  22. [22]
    M.B. Green, J. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge Monographs on Mathematical Physics (1988).Google Scholar
  23. [23]
    E.A. Bergshoeff and M. de Roo, The Quartic Effective Action of the Heterotic String and Supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    G. Curio, A. Krause and D. Lüst, Moduli stabilization in the heterotic/ IIB discretuum, Fortsch. Phys. 54 (2006) 225 [hep-th/0502168] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  25. [25]
    P. Manousselis, N. Prezas and G. Zoupanos, Supersymmetric compactifications of heterotic strings with fluxes and condensates, Nucl. Phys. B 739 (2006) 85 [hep-th/0511122] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    O. Lechtenfeld, C. Nolle and A.D. Popov, Heterotic compactifications on nearly Kähler manifolds, JHEP 09 (2010) 074 [arXiv:1007.0236] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  27. [27]
    A. Chatzistavrakidis, O. Lechtenfeld and A.D. Popov, Nearly Kähler heterotic compactifications with fermion condensates, JHEP 04 (2012) 114 [arXiv:1202.1278] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  28. [28]
    P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    M. Klaput, A. Lukas and E.E. Svanes, Heterotic Calabi-Yau Compactifications with Flux, JHEP 09 (2013) 034 [arXiv:1305.0594] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  30. [30]
    M. Dine, R. Rohm, N. Seiberg and E. Witten, Gluino Condensation in Superstring Models, Phys. Lett. B 156 (1985) 55 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  31. [31]
    J.P. Derendinger, L.E. Ibáñez and H.P. Nilles, On the Low-Energy D = 4, N = 1 Supergravity Theory Extracted from the D = 10, N = 1 Superstring, Phys. Lett. B 155 (1985) 65 [INSPIRE].CrossRefADSGoogle Scholar
  32. [32]
    R. Rohm and E. Witten, The Antisymmetric Tensor Field in Superstring Theory, Annals Phys. 170 (1986) 454 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  33. [33]
    S. Gukov, C. Vafa and E. Witten, CFTs from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477-478] [hep-th/9906070] [INSPIRE].
  34. [34]
    M. Becker and D. Constantin, A Note on flux induced superpotentials in string theory, JHEP 08 (2003) 015 [hep-th/0210131] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  35. [35]
    M.A. Shifman and A.I. Vainshtein, On Gluino Condensation in Supersymmetric Gauge Theories. SU(N) and O(N) Groups, Nucl. Phys. B 296 (1988) 445 [INSPIRE].CrossRefADSGoogle Scholar
  36. [36]
    G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, BPS action and superpotential for heterotic string compactifications with fluxes, JHEP 10 (2003) 004 [hep-th/0306088] [INSPIRE].CrossRefADSGoogle Scholar
  37. [37]
    E. Witten, Symmetry Breaking Patterns in Superstring Models, Nucl. Phys. B 258 (1985) 75 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  38. [38]
    P.A. Kirk and E.P. Klassen, Chern-simons invariants of 3-manifolds and representation spaces of knot groups, Math. Ann. 287 (1990) 343.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    L. Rozansky, A Large k asymptotics of Wittens invariant of Seifert manifolds, Commun. Math. Phys. 171 (1995) 279 [hep-th/9303099] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  40. [40]
    H. Nishi, Su(n)-chern-simons invariants of seifert fibered 3-manifolds, Int.J.Math. 09 (1998) 295.CrossRefMathSciNetGoogle Scholar
  41. [41]
    P. Green and T. Hübsch, Calabi-yau Manifolds as Complete Intersections in Products of Complex Projective Spaces, Commun. Math. Phys. 109 (1987) 99.CrossRefADSzbMATHGoogle Scholar
  42. [42]
    P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].CrossRefADSGoogle Scholar
  43. [43]
    T. Hübsch, Calabi-Yau manifolds: A Bestiary for physicists, World Scientific Publishing, Singapore (1992).CrossRefzbMATHGoogle Scholar
  44. [44]
    D. Joyce, Lectures on Calabi-Yau and special Lagrangian geometry, math/0108088 [INSPIRE].
  45. [45]
    N.J. Hitchin, Lectures on special Lagrangian submanifolds, math/9907034 [INSPIRE].
  46. [46]
    V. Braun, On Free Quotients of Complete Intersection Calabi-Yau Manifolds, JHEP 04 (2011) 005 [arXiv:1003.3235] [INSPIRE].CrossRefADSGoogle Scholar
  47. [47]
    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].CrossRefADSGoogle Scholar
  48. [48]
    A. Hatcher, Notes on Basic 3-Manifold Topology, (2007).
  49. [49]
    D.R. Auckly, Topological methods to compute chern-simons invariants, Math. Proc. Camb. Phil. Soc. 115 (1994) 229.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    M. Jankins and W.D. Neumann, Lectures on Seifert Manifolds, Brandeis Lecture Notes 2, Brandais University, Waltham MA, U.S.A. (1983).Google Scholar
  51. [51]
    M.G. Brin, Sifert fibered spaces, arXiv:0711.1346.
  52. [52]
    J. Montesinos, Classical tessellations and three-manifolds, Universitext, Springer-Verlag, Germany (1979).Google Scholar
  53. [53]
    P.A. Kirk and E.P. Klassen, Chern-simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of t2, Comm. Math. Phys. 153 (1993) 521.CrossRefADSzbMATHMathSciNetGoogle Scholar
  54. [54]
    I. Brunner, M.R. Douglas, A.E. Lawrence and C. Romelsberger, D-branes on the quintic, JHEP 08 (2000) 015 [hep-th/9906200] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  55. [55]
    B.A. Ovrut, T. Pantev and R. Reinbacher, Torus fibered Calabi-Yau threefolds with nontrivial fundamental group, JHEP 05 (2003) 040 [hep-th/0212221] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  56. [56]
    V. Braun, B.A. Ovrut, T. Pantev and R. Reinbacher, Elliptic Calabi-Yau threefolds with Z 3 × Z 3 Wilson lines, JHEP 12 (2004) 062 [hep-th/0410055] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  57. [57]
    D. McDuff and S. Dusa, Introduction to Symplectic Topology, Oxford University Press, Oxford U.K. (1999).Google Scholar
  58. [58]
    P. Candelas, X. de la Ossa, Y.-H. He and B. Szendroi, Triadophilia: A Special Corner in the Landscape, Adv. Theor. Math. Phys. 12 (2008) 429 [arXiv:0706.3134] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  59. [59]
    V. Braun, T. Brelidze, M.R. Douglas and B.A. Ovrut, Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP 05 (2008) 080 [arXiv:0712.3563] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  60. [60]
    C.G.A. Harnack, Über Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876) 189.CrossRefMathSciNetGoogle Scholar
  61. [61]
    R. Blumenhagen, S. Moster and T. Weigand, Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds, Nucl. Phys. B 751 (2006) 186 [hep-th/0603015] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  62. [62]
    L. Anguelova, C. Quigley and S. Sethi, The Leading Quantum Corrections to Stringy Kähler Potentials, JHEP 10 (2010) 065 [arXiv:1007.4793] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  63. [63]
    L. Anguelova and C. Quigley, Quantum Corrections to Heterotic Moduli Potentials, JHEP 02 (2011) 113 [arXiv:1007.5047] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  64. [64]
    E. Palti, Model building with intersecting D6-branes on smooth Calabi-Yau manifolds, JHEP 04 (2009) 099 [arXiv:0902.3546] [INSPIRE].CrossRefADSGoogle Scholar
  65. [65]
    F. Denef, (Dis)assembling special Lagrangians, hep-th/0107152 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Fabio Apruzzi
    • 1
    • 3
  • Fridrik Freyr Gautason
    • 1
    • 2
    • 4
  • Susha Parameswaran
    • 1
    • 2
  • Marco Zagermann
    • 1
    • 2
  1. 1.Institut für Theoretische PhysikHannoverGermany
  2. 2.Center for Quantum Engineering and Spacetime ResearchLeibniz Universität HannoverHannoverGermany
  3. 3.Department of Physics, Robeson Hall, Virginia TechBlacksburgU.S.A.
  4. 4.Instituut voor Theoretische FysicaK.U. LeuvenLeuvenBelgium

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