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Advances in High Energy Physics
Volume 2011, Article ID 513436, 18 pages
http://dx.doi.org/10.1155/2011/513436
Review Article

Toric Methods in F-Theory Model Building

1Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8583, Japan
2Institut für Theoretische Physik, TU Vienna, Wiedner Hauptstraβe 8-10, 1040 Vienna, Austria

Received 12 March 2011; Accepted 2 April 2011

Academic Editor: Yang-Hui He

Copyright © 2011 Johanna Knapp and Maximilian Kreuzer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C. Vafa, “Evidence for F-theory,” Nuclear Physics B, vol. 469, pp. 403–418, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. Donagi and M. Wijnholt, “Model building with F-theory,” http://arxiv.org/abs/0802.2969.
  3. C. Beasley, J. J. Heckman, and C. Vafa, “GUTs and exceptional branes in F-theory—I,” Journal of High Energy Physics, vol. 2009, no. 1, article 058, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. C. Beasley, J. J. Heckman, and C. Vafa, “GUTs and exceptional branes in F-theory—II. Experimental predictions,” Journal of High Energy Physics, vol. 2009, no. 1, article 059, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. B. Andreas and G. Curio, “From local to global in F-theory model building,” Journal of Geometry and Physics, vol. 60, no. 9, pp. 1089–1102, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. R. Blumenhagen, T. W. Grimm, B. Jurke, and T. Weigand, “Global F-theory GUTs,” Nuclear Physics B, vol. 829, no. 1-2, pp. 325–369, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. J. Marsano, N. Saulina, and S. Schafer-Nameki, “Compact F-theory GUTs with U(1)PQ,” Journal of High Energy Physics, vol. 2010, no. 4, article 095, 2010. View at Publisher · View at Google Scholar
  8. W. Grimm, S. Krause, and T. Weigand, “F-theory GUT vacua on compact Calabi-Yau fourfolds thomas,” Journal of High Energy Physics, vol. 2010, no. 7, article 037, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. R. Blumenhagen, A. Collinucci, and B. Jurke, “On instanton effects in F-theory,” Journal of High Energy Physics, vol. 2010, no. 8, article 079, 2010. View at Google Scholar · View at Scopus
  10. M. Cvetic, I. Garcia-Etxebarria, and J. Halverson, “Global F-theory models: instantons and gauge dynamics,” Journal of High Energy Physics, vol. 2011, no. 1, article 073, 2011. View at Google Scholar
  11. 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,” Nuclear Physics B, vol. 840, no. 1-2, pp. 304–348, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. C.-M. Chen and Y.-C. Chung, “Flipped SU(5) GUTs from E8 singularities in F-theory,” Journal of High Energy Physics, vol. 2011, no. 3, article 049, 2011. View at Google Scholar
  13. C.-M. Chen, J. Knapp, M. Kreuzer, and C. Mayrhofer, “Global SO(10) F-theory GUTs,” Journal of High Energy Physics, vol. 2010, no. 10, article 057, 2010. View at Google Scholar
  14. T. W. Grimm and T. Weigand, “Abelian gauge symmetries and proton decay in global F-theory GUTs,” Physical Review D, vol. 82, no. 8, Article ID 086009, 17 pages, 2010. View at Publisher · View at Google Scholar
  15. J. Marsano, N. Saulina, and S. Schäfer-Nameki, “A note on G-fluxes for F-theory model building,” Journal of High Energy Physics, vol. 2010, no. 11, article 088, 2010. View at Publisher · View at Google Scholar
  16. Y.-C. Chung, “On global flipped SU(5) GUTs in F-theory,” Journal of High Energy Physics, vol. 2011, no. 3, article 126, 2011. View at Google Scholar
  17. J. J. Heckman, Y. Tachikawa, C. Vafa, and B. Wecht, “N = 1 SCFTs from brane monodromy,” Journal of High Energy Physics, vol. 2010, no. 11, article 132, 2010. View at Publisher · View at Google Scholar
  18. M. Cvetič, I. García-Etxebarria, and J. Halverson, “On the computation of non-perturbative effective potentials in the string theory landscape—IIB/F-theory perspective—,” Fortschritte der Physik, vol. 59, no. 3-4, pp. 243–283, 2011. View at Publisher · View at Google Scholar
  19. S. Cecotti, C. Cordova, J. J. Heckman, and C. Vafa, “T-branes and monodromy,” http://arxiv.org/abs/1010.5780.
  20. J. Marsano, “Hypercharge flux, exotics, and anomaly cancellation in F-theory grand unification,” Physical Review Letters, vol. 106, no. 8, Article ID 081601, 2011. View at Publisher · View at Google Scholar
  21. A. Collinucci and R. Savelli, “On flux quantization in F-theory,” http://arxiv.org/abs/1011.6388.
  22. C.-C. Chiou, A. E. Faraggi, R. Tatar, and W. Walters, “T-branes and Yukawa couplings,” Journal of High Energy Physics, vol. 2011, no. 5, article 023, 2011. View at Publisher · View at Google Scholar
  23. C. Lüdeling, H. P. Nilles, and C. C. Stephan, “The potential fate of local model building,” Physical Review D, vol. 83, no. 8, Article ID 086008, 14 pages, 2001. View at Publisher · View at Google Scholar
  24. J. Knapp, M. Kreuzer, C. Mayrhofer, and N.-O. Walliser, “Toric construction of global F-theory GUTs,” Journal of High Energy Physics, vol. 2011, no. 3, article 188, 2011. View at Google Scholar
  25. M. J. Dolan, J. Marsano, N. Saulina, and S. Schafer-Nameki, “F-theory GUTs with U(1) symmetries: generalities and survey,” http://arxiv.org/abs/1102.0290.
  26. A. Collinucci, “New F-theory lifts,” Journal of High Energy Physics, vol. 2009, no. 8, article 076, 2009. View at Publisher · View at Google Scholar
  27. A. Collinucci, “New F-theory lifts II: permutation orientifolds and enhanced singularities,” Journal of High Energy Physics, vol. 2010, no. 4, article 076, 2010. View at Publisher · View at Google Scholar
  28. R. Blumenhagen, T. W. Grimm, B. Jurke, and T. Weigand, “F-theory uplifts and GUTs,” Journal of High Energy Physics, vol. 2009, no. 9, article 053, 2009. View at Publisher · View at Google Scholar
  29. M. Kreuzer and H. Skarke, “PALP: a package for analysing lattice polytopes with applications to toric geometry,” Computer Physics Communications, vol. 157, no. 1, pp. 87–106, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. F. Denef, “Les Houches lectures on constructing string vacua,” http://arxiv.org/abs/0803.1194.
  31. J. J. Heckman, “Particle physics implications of F-theory,” Annual Review of Nuclear and Particle Science, vol. 60, pp. 237–265, 2010. View at Publisher · View at Google Scholar · View at Scopus
  32. T. Weigand, “Lectures on F-theory compactifications and model building,” Classical and Quantum Gravity, vol. 27, no. 21, Article ID 214004, 2010. View at Publisher · View at Google Scholar
  33. W. Fulton, Introduction to Toric Varieties, vol. 131 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1993.
  34. M. Kreuzer, “Toric geometry and Calabi-Yau compactifications,” Ukrainian Journal of Physics, vol. 55, no. 5, pp. 613–625, 2010. View at Google Scholar · View at Scopus
  35. D. A. Cox, J. B. Little, and H. Schenck, Toric Varieties, American Mathematical Society, 2011, http://www.cs.amherst.edu/%7Edac/toric.html.
  36. K. Kodaira, “On compact analytic surfaces II,” The Annals of Mathematics, vol. 77, no. 3, pp. 563–626, 1963. View at Google Scholar
  37. J. Tate, “Algorithm for determining the type of a singular fiber in an elliptic pencil,” in Modular Functions of One Variable IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), vol. 476 of Lecture Notes in Mathematics, pp. 33–52, Springer, Berlin, Germany, 1975. View at Google Scholar
  38. C. Cordova, “Decoupling gravity in F-theory,” http://arxiv.org/abs/0910.2955.
  39. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, New York, NY, USA, 1994.
  40. V. Braun, “Discrete Wilson lines in F-theory,” http://arxiv.org/abs/1010.2520.
  41. C.-M. Chen and Y.-C. Chung, “On F-theory E6 GUTs,” Journal of High Energy Physics, vol. 2011, no. 3, article 129, 2011. View at Google Scholar
  42. R. Donagi and M. Wijnholt, “Higgs bundles and UV completion in F-theory,” http://arxiv.org/abs/0904.1218.
  43. V. V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,” Journal of Algebraic Geometry, vol. 3, no. 3, pp. 493–535, 1994. View at Google Scholar
  44. V. V. Batyrev and L. A. Borisov, “Mirror duality and string-theoretic Hodge numbers,” Inventiones Mathematicae, vol. 126, no. 1, pp. 183–203, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. P. Berglund, S. Katz, and A. Klemm, “Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties,” Nuclear Physics B, vol. 456, no. 1-2, pp. 153–204, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. T. Oda and H. Park, “Linear Gale transforms and Gel'fand-Kapranov-Zelevinskij decompositions,” The Tohoku Mathematical Journal, vol. 43, no. 3, pp. 375–399, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  47. M. Kreuzer and N.-O. Walliser, work in progress.
  48. J. Rambau, “TOPCOM: triangulations of point configurations and oriented matroids,” in Mathematical Software (Beijing, 2002), A. M. Cohen, X.-S. Gao, and N. Takayama, Eds., pp. 330–340, World Scientific, River Edge, NJ, USA, 2002. View at Google Scholar
  49. 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.
  50. R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of line bundles: a computational algorithm,” Journal of Mathematical Physics, vol. 51, no. 10, Article ID 103525, 2010. View at Publisher · View at Google Scholar
  51. V. Batyrev and M. Kreuzer, “Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions,” http://arxiv.org/abs/0802.3376.
  52. M. Kreuzer, “The making of Calabi-Yau spaces: beyond toric hypersurfaces,” Fortschritte der Physik, vol. 57, no. 5–7, pp. 625–631, 2009. View at Publisher · View at Google Scholar · View at Scopus
  53. V. V. Batyrev and L. A. Borisov, “Dual cones and mirror symmetry for generalized Calabi-Yau manifolds,” in Mirror Symmetry, II, vol. 1 of AMS/IP Stud. Adv. Math., pp. 71–86, American Mathematical Society, Providence, RI, USA, 1997. View at Google Scholar
  54. M. Kreuzer and H. Skarke, “Complete classification of reflexive polyhedra in four dimensions,” Advances in Theoretical and Mathematical Physics, vol. 4, no. 6, pp. 1209–1230, 2000. View at Google Scholar
  55. M. Kreuzer, “On the statistics of lattice polytopes,” in International Conference on Information Theory and Statistical Learning (ITSL '08), pp. 119–124, Las Vegas, Nev, USA, July 2008. View at Scopus
  56. A. Collinucci, M. Kreuzer, C. Mayrhofer, and N. O. Walliser, “Four-modulus “swiss Cheese” chiral models,” Journal of High Energy Physics, vol. 2009, no. 7, article 074, 2009. View at Publisher · View at Google Scholar · View at Scopus
  57. http://hep.itp.tuwien.ac.at/~kreuzer/CY/.
  58. http://hep.itp.tuwien.ac.at/f-theory/.
  59. A. Braun and N.-O. Walliser, work in progress.