Skip to main content

Box graphs and singular fibers

A preprint version of the article is available at arXiv.

Abstract

We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional \( \mathcal{N} \) = 2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as “flopping” of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E 6, E7 and E 8.

References

  1. [1]

    K. Kodaira, On compact complex analytic surfaces. I, Ann. Math. 71 (1960) 111.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    K. Kodaira, On compact complex analytic surfaces. II, Ann. Math. 77 (1963) 563.

    Article  MATH  Google Scholar 

  3. [3]

    A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux (in French), Inst. Hautes Études Sci. Publ. Math. No. 21 (1964) 128.

    MATH  Google Scholar 

  4. [4]

    A.C. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara, Eleven-dimensional supergravity compactified on Calabi-Yau threefolds, Phys. Lett. B 357 (1995) 76 [hep-th/9506144] [INSPIRE].

    ADS  Article  Google Scholar 

  5. [5]

    S. Ferrara, R.R. Khuri and R. Minasian, M theory on a Calabi-Yau manifold, Phys. Lett. B 375 (1996) 81 [hep-th/9602102] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    S. Ferrara, R. Minasian and A. Sagnotti, Low-energy analysis of M and F theories on Calabi-Yau threefolds, Nucl. Phys. B 474 (1996) 323 [hep-th/9604097] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    K. Becker and M. Becker, M theory on eight manifolds, Nucl. Phys. B 477 (1996) 155 [hep-th/9605053] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    S. Sethi, C. Vafa and E. Witten, Constraints on low dimensional string compactifications, Nucl. Phys. B 480 (1996) 213 [hep-th/9606122] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    D.-E. Diaconescu and S. Gukov, Three-dimensional N = 2 gauge theories and degenerations of Calabi-Yau four folds, Nucl. Phys. B 535 (1998) 171 [hep-th/9804059] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    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] [hep-th/9906070] [INSPIRE].

  15. [15]

    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. I, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. II, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    M. Bershadsky, A. Johansen, T. Pantev and V. Sadov, On four-dimensional compactifications of F-theory, Nucl. Phys. B 505 (1997) 165 [hep-th/9701165] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. [22]

    M. Cvetič, T.W. Grimm and D. Klevers, Anomaly cancellation and Abelian gauge symmetries in F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    H. Hayashi, C. Lawrie and S. Schäfer-Nameki, Phases, flops and F-theory: SU(5) gauge theories, JHEP 10 (2013) 046 [arXiv:1304.1678] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    S. Katz and D.R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Alg. Geom. 1 (1992) 449 [alg-geom/9202002].

    MathSciNet  MATH  Google Scholar 

  25. [25]

    J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yaus, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, arXiv:1107.0733 [INSPIRE].

  29. [29]

    C. Lawrie and S. Schäfer-Nameki, The Tate form on steroids: resolution and higher codimension fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  30. [30]

    D.R. Morrison, The birational geometry of surfaces with rational double points, Math. Ann. 271 (1985) 415.

    MathSciNet  Article  MATH  Google Scholar 

  31. [31]

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. [32]

    T.W. Grimm, S. Krause and T. Weigand, F-theory GUT vacua on compact Calabi-Yau fourfolds, JHEP 07 (2010) 037 [arXiv:0912.3524] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. [33]

    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] [INSPIRE].

    ADS  Google Scholar 

  34. [34]

    S. Krause, C. Mayrhofer and T. Weigand, G 4 flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  35. [35]

    R.P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge U.K. (1999).

  36. [36]

    K. Matsuki, Weyl groups and birational transformations among minimal models, Mem. Amer. Math. Soc. 116 (1995) vi+133.

  37. [37]

    M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases, arXiv:1402.6331 [INSPIRE].

  38. [38]

    A. Björner and F. Brenti, Combinatorics of coxeter groups, Graduate Texts in Mathematics 231, Springer, New York U.S.A. (2005).

  39. [39]

    N. Bourbaki, Groupes et algèbres de Lie (in French), chapters VII and VIII, Hermann, Paris France (1975).

    Google Scholar 

  40. [40]

    V. Lakshmibai and K.N. Raghavan, Standard monomial theory, in Encyclopaedia of Mathematical Sciences 137, Springer-Verlag, Berlin Germany (2008).

  41. [41]

    W. Fulton and J. Harris, Representation theory, Springer-Verlag, Germany (1991).

    MATH  Google Scholar 

  42. [42]

    R.P. Stanley, The Fibonacci lattice, Fibonacci Quart. 13 (1975) 215.

    MathSciNet  MATH  Google Scholar 

  43. [43]

    P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds and F-theory, Adv. Theor. Math. Phys. 4 (2000) 95 [hep-th/0002012] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  44. [44]

    K. Intriligator, H. Jockers, P. Mayr, D.R. Morrison and M.R. Plesser, Conifold transitions in M-theory on Calabi-Yau fourfolds with background fluxes, arXiv:1203.6662 [INSPIRE].

  45. [45]

    D.-E. Diaconescu and R. Entin, Calabi-Yau spaces and five-dimensional field theories with exceptional gauge symmetry, Nucl. Phys. B 538 (1999) 451 [hep-th/9807170] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  46. [46]

    S. Kleiman, Toward a numerical theory of ampleness, Ann. Math. 84 (1966) 293.

    MathSciNet  Article  MATH  Google Scholar 

  47. [47]

    S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982) 133.

    MathSciNet  Article  MATH  Google Scholar 

  48. [48]

    V. Braun and D.R. Morrison, F-theory on genus-one fibrations, arXiv:1401.7844 [inSPIRE].

  49. [49]

    M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  50. [50]

    A. Degeratu and K. Wendland, Friendly giant meets pointlike instantons? On a new conjecture by John McKay, in Moonshine: the first quarter century and beyond, London Math. Soc. Lecture Note Ser. 372 (2010) 55, Cambridge Univ. Press, Cambridge U.K. (2010).

  51. [51]

    A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  52. [52]

    A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.

    MathSciNet  Article  MATH  Google Scholar 

  53. [53]

    P. Candelas, D.-E. Diaconescu, B. Florea, D.R. Morrison and G. Rajesh, Codimension three bundle singularities in F-theory, JHEP 06 (2002) 014 [hep-th/0009228] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  54. [54]

    H. Pinkham, Factorization of birational maps in dimension 3, in Singularities, P. Orlik ed., Proc. Symp. Pure Math. 40 (1983) 343, American Mathematical Society, Providence U.S.A. (1983).

  55. [55]

    M. Reid, Minimal models of canonical 3-folds, in Algebraic varieties and analytic varieties, S. Iitaka ed., Adv. Stud. Pure Math. 1 (1983) 131, Kinokuniya, Japan (1983).

  56. [56]

    R. Slansky, Group theory for unified model building, Phys. Rept. 79 (1981) 1 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  57. [57]

    P.S. Aspinwall and D.R. Morrison, Nonsimply connected gauge groups and rational points on elliptic curves, JHEP 07 (1998) 012 [hep-th/9805206] [INSPIRE].

    ADS  Article  Google Scholar 

  58. [58]

    A.P. Braun and T. Watari, On singular fibres in F-theory, JHEP 07 (2013) 031 [arXiv:1301.5814] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  59. [59]

    D.R. Morrison and D.S. Park, F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  60. [60]

    C. Mayrhofer, E. Palti and T. Weigand, U(1) symmetries in F-theory GUTs with multiple sections, JHEP 03 (2013) 098 [arXiv:1211.6742] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  61. [61]

    V. Braun, T.W. Grimm and J. Keitel, New global F-theory GUTs with U(1) symmetries, JHEP 09 (2013) 154 [arXiv:1302.1854] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  62. [62]

    J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, Elliptic fibrations for SU(5) × U(1) × U(1) F-theory vacua, Phys. Rev. D 88 (2013) 046005 [arXiv:1303.5054] [INSPIRE].

    ADS  Google Scholar 

  63. [63]

    M. Cvetič, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections, JHEP 06 (2013) 067 [arXiv:1303.6970] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  64. [64]

    T.W. Grimm, A. Kapfer and J. Keitel, Effective action of 6D F-theory with U(1) factors: rational sections make Chern-Simons terms jump, JHEP 07 (2013) 115 [arXiv:1305.1929] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  65. [65]

    V. Braun, T.W. Grimm and J. Keitel, Geometric engineering in toric F-theory and GUTs with U(1) gauge factors, JHEP 12 (2013) 069 [arXiv:1306.0577] [INSPIRE].

    ADS  Article  Google Scholar 

  66. [66]

    M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors, JHEP 04 (2014) 010 [arXiv:1306.3987] [INSPIRE].

    ADS  Article  Google Scholar 

  67. [67]

    J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, SU(5) tops with multiple U(1)s in F-theory, Nucl. Phys. B 882 (2014) 1 [arXiv:1307.2902] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  68. [68]

    M. Cvetič, D. Klevers, H. Piragua and P. Song, Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1) × U(1) × U(1) gauge symmetry, JHEP 03 (2014) 021 [arXiv:1310.0463] [INSPIRE].

    ADS  Article  Google Scholar 

  69. [69]

    M. Kuentzler, C. Lawrie and S. Schäfer-Nameki, Tate trees for elliptic fibrations with rank one Mordell-Weil group, to appear.

  70. [70]

    N. Bourbaki, Groupes et algèbres de Lie (in French), chapters IV, V and VI, Hermann, Paris France (1968).

    Google Scholar 

  71. [71]

    J.F. Adams, Lectures on Lie groups, W.A. Benjamin Inc., New York U.S.A. and Amsterdam The Netherlands (1969).

    MATH  Google Scholar 

  72. [72]

    M. Demazure, Surfaces de Del Pezzo, II, III, IV, V, in Séminaire sur les Singularités des Surfaces, Lect. Notes Math. 777 (1980) 21, Springer-Verlag, Germany (1980).

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

Affiliations

Authors

Corresponding author

Correspondence to Sakura Schafer-Nameki.

Additional information

ArXiv ePrint: 1402.2653

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

Hayashi, H., Lawrie, C., Morrison, D.R. et al. Box graphs and singular fibers. J. High Energ. Phys. 2014, 48 (2014). https://doi.org/10.1007/JHEP05(2014)048

Download citation

Keywords

  • M-Theory
  • F-Theory
  • Differential and Algebraic Geometry
  • Supersymmetric Effective Theories