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Toric construction of global F-theory GUTs

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Abstract

We systematically construct a large number of compact Calabi-Yau fourfolds which are suitable for F-theory model building. These elliptically fibered Calabi-Yaus are complete intersections of two hypersurfaces in a six dimensional ambient space. We first construct three-dimensional base manifolds that are hypersurfaces in a toric ambient space. We search for divisors which can support an F-theory GUT. The fourfolds are obtained as elliptic fibrations over these base manifolds. We find that elementary conditions which are motivated by F-theory GUTs lead to strong constraints on the geometry, which significantly reduce the number of suitable models. The complete database of models is available at [1]. We work out several examples in more detail.

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References

  1. http://hep.itp.tuwien.ac.at/f-theory/.

  2. R. Donagi and M. Wijnholt, Model Building with F-theory, arXiv:0802.2969 [SPIRES].

  3. C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory - I, JHEP 01 (2009) 058 [arXiv:0802.3391] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory - II: Experimental Predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. J.J. Heckman, Particle Physics Implications of F-theory, arXiv:1001.0577 [SPIRES].

  6. B. Andreas and G. Curio, From Local to Global in F-theory Model Building, J. Geom. Phys. 60 (2010) 1089 [arXiv:0902.4143] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. J. Marsano, N. Saulina and S. Schäfer-Nameki, Compact F-theory GUTs with U(1) PQ , JHEP 04 (2010) 095 [arXiv:0912.0272] [SPIRES].

    Article  ADS  Google Scholar 

  9. T.W. Grimm, S. Krause and T. Weigand, F-Theory GUT Vacua on Compact Calabi-Yau Fourfolds, JHEP 07 (2010) 037 [arXiv:0912.3524] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  10. R. Blumenhagen, A. Collinucci and B. Jurke, On Instanton Effects in F-theory, JHEP 08 (2010) 079 [arXiv:1002.1894] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  11. M. Cvetič, I. Garcia-Etxebarria and J. Halverson, Global F-theory Models: Instantons and Gauge Dynamics, JHEP 01 (2011) 073 [arXiv:1003.5337] [SPIRES].

    Article  ADS  Google Scholar 

  12. H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, More on Dimension-4 Proton Decay Problem in F-theory – Spectral Surface, Discriminant Locus and Monodromy, Nucl. Phys. B 840 (2010) 304 [arXiv:1004.3870] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  13. T.W. Grimm and T. Weigand, On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [SPIRES].

    ADS  Google Scholar 

  14. J. Marsano, N. Saulina and S. Schäfer-Nameki, A Note on G-fluxes for F-theory Model Building, JHEP 11 (2010) 088 [arXiv:1006.0483] [SPIRES].

    Article  ADS  Google Scholar 

  15. Y.-C. Chung, On Global Flipped SU(5) GUTs in F-theory, arXiv:1008.2506 [SPIRES].

  16. M. Cvetič, I. Garcia-Etxebarria and J. Halverson, On the computation of non-perturbative effective potentials in the string theory landscape – IIB/F-theory perspective, arXiv:1009.5386 [SPIRES].

  17. S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-Branes and Monodromy, arXiv:1010.5780 [SPIRES].

  18. J. Marsano, Hypercharge Flux, Exotics and Anomaly Cancellation in F-theory GUTs, Phys. Rev. Lett. 106 (2011) 081601 [arXiv:1011.2212] [SPIRES].

    Article  ADS  Google Scholar 

  19. A. Collinucci and R. Savelli, On Flux Quantization in F-theory, arXiv:1011.6388 [SPIRES].

  20. C.-C. Chiou, A.E. Faraggi, R. Tatar and W. Walters, T-branes and Yukawa Couplings, arXiv:1101.2455 [SPIRES].

  21. T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  22. C.-M. Chen, J. Knapp, M. Kreuzer and C. Mayrhofer, Global SO(10) F-theory GUTs, JHEP 10 (2010) 057 [arXiv:1005.5735] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  23. M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [SPIRES].

    MathSciNet  Google Scholar 

  24. M. Kreuzer and H. Skarke, PALP: A Package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun. 157 (2004) 87 [math/0204356].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. K. Kodaira, On compace analytic surfaces II, Ann. M. 77 (1963) 563.

    Article  MATH  Google Scholar 

  26. J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, vol. Modular functions of one variable IV of Lecture Notes in Mathematics, Springer, Berlin/Heidelberg (1975).

    Google Scholar 

  27. V.I. Danilov, The geometry of toric varieties, Russian Math. Survey 33(2) (1978).

  28. W. Fulton, Introduction to toric varieties, Princeton Univ. Press, Princeton U.S.A. (1993).

    MATH  Google Scholar 

  29. D. Cox, Minicourse on Toric Varieties, http://www.amherst.edu/∼dacox.

  30. D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, vol. 68 of Mathematical Surveys and Monographs, American Mathematical Society, Berlin/Heidelberg (1999).

    Google Scholar 

  31. M. Kreuzer, Toric Geometry and Calabi-Yau Compactifications, Ukr. J. Phys. 55 (2010) 613 [hep-th/0612307] [SPIRES].

    Google Scholar 

  32. V. Bouchard, Lectures on complex geometry, Calabi-Yau manifolds and toric geometry, hep-th/0702063 [SPIRES].

  33. J. Treutlein, Birationale Eigenschaften generischer Hyperflächen inalgebraischen Tori, PhD thesis, Eberhard-Karls-Universität Tübingen, June (2010), http://tobias-lib.uni-tuebingen.de/.

  34. M. Kreuzer and H. Skarke, PALP: A Package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun. 157 (2004) 87 [math/0204356].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. V.V. Batyrev and L.A. Borisov, Mirror duality and string-theoretic Hodge numbers, alg-geom/9509009 [SPIRES].

  36. L.J. Billera, P. Filliman and B. Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990) 155.

    Article  MathSciNet  MATH  Google Scholar 

  37. I.M. Gelfand, M.M. Kapranov and A.M. Zalevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhauser, Boston (1994).

    Book  MATH  Google Scholar 

  38. A. Klemm, M. Kreuzer, E. Riegler and E. Scheidegger, Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, JHEP 05 (2005) 023 [hep-th/0410018] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  39. T. Oda and H. Park, Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions, Tôhoku Math. J. 43 (1991) 375.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Braun, M. Kreuzer and N.-O. Walliser, An Extension of PALP, work in progress.

  41. W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3-1-2 – A computer algebra system for polynomial computations, http://www.singular.uni-kl.de

  42. C. Cordova, Decoupling Gravity in F-theory, arXiv:0910.2955 [SPIRES].

  43. Calabi Yau data: Tools and data for (toric) Calabi-Yau varieties, Landau-Ginzburg models, and related objects, http://hep.itp.tuwien.ac.at/∼kreuzer/CY/.

  44. V. Braun, Discrete Wilson Lines in F-theory, arXiv:1010.2520 [SPIRES].

  45. H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, Flavor Structure in F-theory Compactifications, JHEP 08 (2010) 036 [arXiv:0910.2762] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  46. M. Cvetič and J. Halverson, TA SI Lectures: Particle Physics from Perturbative and non-perturbative Effects in D-braneworlds, arXiv:1101.2907 [SPIRES].

  47. C.-M. Chen and Y.-C. Chung, Flipped SU(5) GUTs from E 8 Singularities in F-theory, JHEP 03 (2011) 049 [arXiv:1005.5728] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  48. J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory Compactifications for Supersymmetric GUTs, JHEP 08 (2009) 030 [arXiv:0904.3932] [SPIRES].

    Article  ADS  Google Scholar 

  49. C. Mayrhofer, Compactifications of Type IIB String Theory and F-Theory Models by Means of Toric Geometry, PhD thesis, Vienna University of Technology, November (2010), http://aleph.ub.tuwien.ac.at.

  50. R. Donagi and M. Wijnholt, Higgs Bundles and UV Completion in F-theory, arXiv:0904.1218 [SPIRES].

  51. M. Alim et al., Hints for Off-Shell Mirror Symmetry in type-II/F-theory Compactifications, Nucl. Phys. B 841 (2010) 303 [arXiv:0909.1842] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  52. T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing Brane and Flux Superpotentials in F-theory Compactifications, JHEP 04 (2010) 015 [arXiv:0909.2025] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  53. T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Five-Brane Superpotentials and Heterotic/F-theory Duality, Nucl. Phys. B 838 (2010) 458 [arXiv:0912.3250] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. H. Jockers, P. Mayr and J. Walcher, On N = 14d Effective Couplings for F-theory and Heterotic Vacua, arXiv:0912.3265 [SPIRES].

  55. M. Alim et al., Type II/F-theory Superpotentials with Several Deformations and N = 1 Mirror Symmetry, arXiv:1010.0977 [SPIRES].

  56. T.W. Grimm, A. Klemm and D. Klevers, Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds, arXiv:1011.6375 [SPIRES].

  57. R. Blumenhagen, B. Jurke, T. Rahn and H. Roschy, Cohomology of Line Bundles: A Computational Algorithm, J. Math. Phys. 51 (2010) 103525 [arXiv:1003.5217] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  58. H. Skarke, private communication.

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Authors and Affiliations

  1. Institut für Theoretische Physik, TU Vienna, Wiedner Hauptstrasse 8–10, 1040, Vienna, Austria

    Johanna Knapp, Maximilian Kreuzer & Nils-Ole Walliser

  2. Institute for the Physics and the Mathematics of the Universe (IPMU), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277–8583, Japan

    Johanna Knapp

  3. Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, D-69120, Heidelberg, Germany

    Christoph Mayrhofer

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  1. Johanna Knapp
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  2. Maximilian Kreuzer
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  4. Nils-Ole Walliser
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Correspondence to Nils-Ole Walliser.

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ArXiv ePrint: 1101.4908

In memory of Maximilian Kreuzer

Deceased (Maximilian Kreuzer)

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Knapp, J., Kreuzer, M., Mayrhofer, C. et al. Toric construction of global F-theory GUTs. J. High Energ. Phys. 2011, 138 (2011). https://doi.org/10.1007/JHEP03(2011)138

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