Skip to main content

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

A preprint version of the article is available at arXiv.

Abstract

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.

References

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  3. V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  8. E. Witten, New Issues in Manifolds of SU(3) Holonomy, Nucl. Phys. B 268 (1986) 79 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    ADS  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    ADS  MathSciNet  Google Scholar 

  14. M. Cicoli, S. de Alwis and A. Westphal, Heterotic Moduli Stabilisation, JHEP 10 (2013) 199 [arXiv:1304.1809] [INSPIRE].

    Article  ADS  Google Scholar 

  15. A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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.

  17. H. Partouche and B. Pioline, Rolling among G 2 vacua, JHEP 03 (2001) 005 [hep-th/0011130] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  18. M. Dine and N. Seiberg, Nonrenormalization Theorems in Superstring Theory, Phys. Rev. Lett. 57 (1986) 2625 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    ADS  MathSciNet  Google Scholar 

  21. L. Witten and E. Witten, Large Radius Expansion of Superstring Compactifications, Nucl. Phys. B 281 (1987) 109 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  22. M.B. Green, J. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge Monographs on Mathematical Physics (1988).

  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].

    Article  ADS  Google Scholar 

  24. G. Curio, A. Krause and D. Lüst, Moduli stabilization in the heterotic/ IIB discretuum, Fortsch. Phys. 54 (2006) 225 [hep-th/0502168] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  26. O. Lechtenfeld, C. Nolle and A.D. Popov, Heterotic compactifications on nearly Kähler manifolds, JHEP 09 (2010) 074 [arXiv:1007.0236] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  27. A. Chatzistavrakidis, O. Lechtenfeld and A.D. Popov, Nearly Kähler heterotic compactifications with fermion condensates, JHEP 04 (2012) 114 [arXiv:1202.1278] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  28. P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  29. M. Klaput, A. Lukas and E.E. Svanes, Heterotic Calabi-Yau Compactifications with Flux, JHEP 09 (2013) 034 [arXiv:1305.0594] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Dine, R. Rohm, N. Seiberg and E. Witten, Gluino Condensation in Superstring Models, Phys. Lett. B 156 (1985) 55 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  32. R. Rohm and E. Witten, The Antisymmetric Tensor Field in Superstring Theory, Annals Phys. 170 (1986) 454 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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. M. Becker and D. Constantin, A Note on flux induced superpotentials in string theory, JHEP 08 (2003) 015 [hep-th/0210131] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  37. E. Witten, Symmetry Breaking Patterns in Superstring Models, Nucl. Phys. B 258 (1985) 75 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  39. L. Rozansky, A Large k asymptotics of Wittens invariant of Seifert manifolds, Commun. Math. Phys. 171 (1995) 279 [hep-th/9303099] [INSPIRE].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. H. Nishi, Su(n)-chern-simons invariants of seifert fibered 3-manifolds, Int.J.Math. 09 (1998) 295.

    Article  MathSciNet  Google Scholar 

  41. P. Green and T. Hübsch, Calabi-yau Manifolds as Complete Intersections in Products of Complex Projective Spaces, Commun. Math. Phys. 109 (1987) 99.

    Article  ADS  MATH  Google Scholar 

  42. P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].

    Article  ADS  Google Scholar 

  43. T. Hübsch, Calabi-Yau manifolds: A Bestiary for physicists, World Scientific Publishing, Singapore (1992).

    Book  MATH  Google Scholar 

  44. D. Joyce, Lectures on Calabi-Yau and special Lagrangian geometry, math/0108088 [INSPIRE].

  45. N.J. Hitchin, Lectures on special Lagrangian submanifolds, math/9907034 [INSPIRE].

  46. V. Braun, On Free Quotients of Complete Intersection Calabi-Yau Manifolds, JHEP 04 (2011) 005 [arXiv:1003.3235] [INSPIRE].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  48. A. Hatcher, Notes on Basic 3-Manifold Topology, http://www.math.cornell.edu/~hatcher/3M/3M.pdf (2007).

  49. D.R. Auckly, Topological methods to compute chern-simons invariants, Math. Proc. Camb. Phil. Soc. 115 (1994) 229.

    Article  MATH  MathSciNet  Google Scholar 

  50. M. Jankins and W.D. Neumann, Lectures on Seifert Manifolds, Brandeis Lecture Notes 2, Brandais University, Waltham MA, U.S.A. (1983).

  51. M.G. Brin, Sifert fibered spaces, arXiv:0711.1346.

  52. J. Montesinos, Classical tessellations and three-manifolds, Universitext, Springer-Verlag, Germany (1979).

    Google Scholar 

  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.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  57. D. McDuff and S. Dusa, Introduction to Symplectic Topology, Oxford University Press, Oxford U.K. (1999).

    Google Scholar 

  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].

    Article  MATH  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  60. C.G.A. Harnack, Über Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876) 189.

    Article  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  62. L. Anguelova, C. Quigley and S. Sethi, The Leading Quantum Corrections to Stringy Kähler Potentials, JHEP 10 (2010) 065 [arXiv:1007.4793] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  63. L. Anguelova and C. Quigley, Quantum Corrections to Heterotic Moduli Potentials, JHEP 02 (2011) 113 [arXiv:1007.5047] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  64. E. Palti, Model building with intersecting D6-branes on smooth Calabi-Yau manifolds, JHEP 04 (2009) 099 [arXiv:0902.3546] [INSPIRE].

    Article  ADS  Google Scholar 

  65. F. Denef, (Dis)assembling special Lagrangians, hep-th/0107152 [INSPIRE].

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

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fabio Apruzzi.

Additional information

ArXiv ePrint: 1410.2603

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

Apruzzi, F., Gautason, F.F., Parameswaran, S. et al. Wilson lines and Chern-Simons flux in explicit heterotic Calabi-Yau compactifications. J. High Energ. Phys. 2015, 183 (2015). https://doi.org/10.1007/JHEP02(2015)183

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP02(2015)183

Keywords

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