The Swampland Distance Conjecture for Kähler moduli

  • Pierre CorvilainEmail author
  • Thomas W. Grimm
  • Irene Valenzuela
Open Access
Regular Article - Theoretical Physics


The Swampland Distance Conjecture suggests that an infinite tower of modes becomes exponentially light when approaching a point that is at infinite proper distance in field space. In this paper we investigate this conjecture in the Kähler moduli spaces of Calabi-Yau threefold compactifications and further elucidate the proposal that the infinite tower of states is generated by the discrete symmetries associated to infinite distance points. In the large volume regime the infinite tower of states is generated by the action of the local monodromy matrices and encoded by an orbit of D-brane charges. We express these monodromy matrices in terms of the triple intersection numbers to classify the infinite distance points and construct the associated infinite charge orbits that become massless. We then turn to a detailed study of charge orbits in elliptically fibered Calabi-Yau threefolds. We argue that for these geometries the modular symmetry in the moduli space can be used to transfer the large volume orbits to the small elliptic fiber regime. The resulting orbits can be used in compactifications of M-theory that are dual to F-theory compactifications including an additional circle. In particular, we show that there are always charge orbits satisfying the distance conjecture that correspond to Kaluza-Klein towers along that circle. Integrating out the KK towers yields an infinite distance in the moduli space thereby supporting the idea of emergence in that context.


Superstring Vacua F-Theory Global Symmetries M-Theory 


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


  1. [1]
    C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
  2. [2]
    H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B 766 (2007) 21 [hep-th/0605264] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    T.W. Grimm, E. Palti and I. Valenzuela, Infinite Distances in Field Space and Massless Towers of States, JHEP 08 (2018) 143 [arXiv:1802.08264] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    T.W. Grimm, C. Li and E. Palti, Infinite Distance Networks in Field Space and Charge Orbits, JHEP 03 (2019) 016 [arXiv:1811.02571] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S.-J. Lee, W. Lerche and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP 10 (2018) 164 [arXiv:1808.05958] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S.-J. Lee, W. Lerche and T. Weigand, A Stringy Test of the Scalar Weak Gravity Conjecture, Nucl. Phys. B 938 (2019) 321 [arXiv:1810.05169] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    E. Palti, On Natural Inflation and Moduli Stabilisation in String Theory, JHEP 10 (2015) 188 [arXiv:1508.00009] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F. Baume and E. Palti, Backreacted Axion Field Ranges in String Theory, JHEP 08 (2016) 043 [arXiv:1602.06517] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Valenzuela, Backreaction Issues in Axion Monodromy and Minkowski 4-forms, JHEP 06 (2017) 098 [arXiv:1611.00394] [INSPIRE].
  10. [10]
    S. Bielleman, L.E. Ibáñez, F.G. Pedro, I. Valenzuela and C. Wieck, Higgs-otic Inflation and Moduli Stabilization, JHEP 02 (2017) 073 [arXiv:1611.07084] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Blumenhagen, I. Valenzuela and F. Wolf, The Swampland Conjecture and F-term Axion Monodromy Inflation, JHEP 07 (2017) 145 [arXiv:1703.05776] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    E. Palti, The Weak Gravity Conjecture and Scalar Fields, JHEP 08 (2017) 034 [arXiv:1705.04328] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Hebecker, P. Henkenjohann and L.T. Witkowski, Flat Monodromies and a Moduli Space Size Conjecture, JHEP 12 (2017) 033 [arXiv:1708.06761] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    M. Cicoli, D. Ciupke, C. Mayrhofer and P. Shukla, A Geometrical Upper Bound on the Inflaton Range, JHEP 05 (2018) 001 [arXiv:1801.05434] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    R. Blumenhagen, D. Kläwer, L. Schlechter and F. Wolf, The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces, JHEP 06 (2018) 052 [arXiv:1803.04989] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    G. Buratti, J. Calderón and A.M. Uranga, Transplanckian axion monodromy!?, JHEP 05 (2019) 176 [arXiv:1812.05016] [INSPIRE].
  17. [17]
    E. Cattani, A. Kaplan and W. Schmid, Degeneration of Hodge Structures, Ann. Math. 123 (1986) 457.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    F. Bonetti and T.W. Grimm, Six-dimensional (1,0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    F. Bonetti, T.W. Grimm and S. Hohenegger, A Kaluza-Klein inspired action for chiral p-forms and their anomalies, Phys. Lett. B 720 (2013) 424 [arXiv:1206.1600] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    F. Bonetti, T.W. Grimm and S. Hohenegger, One-loop Chern-Simons terms in five dimensions, JHEP 07 (2013) 043 [arXiv:1302.2918] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    T.W. Grimm, H. het Lam, K. Mayer and S. Vandoren, Four-dimensional black hole entropy from F-theory, JHEP 01 (2019) 037 [arXiv:1808.05228] [INSPIRE].
  22. [22]
    B. Heidenreich, M. Reece and T. Rudelius, The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff, Eur. Phys. J. C 78 (2018) 337 [arXiv:1712.01868] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N. Arkani-Hamed, S. Dimopoulos and S. Kachru, Predictive landscapes and new physics at a TeV, hep-th/0501082 [INSPIRE].
  24. [24]
    J. Distler and U. Varadarajan, Random polynomials and the friendly landscape, hep-th/0507090 [INSPIRE].
  25. [25]
    S. Dimopoulos, S. Kachru, J. McGreevy and J.G. Wacker, N-flation, JCAP 08 (2008) 003 [hep-th/0507205] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    G. Dvali and M. Redi, Black Hole Bound on the Number of Species and Quantum Gravity at LHC, Phys. Rev. D 77 (2008) 045027 [arXiv:0710.4344] [INSPIRE].ADSGoogle Scholar
  27. [27]
    G. Dvali, Black Holes and Large N Species Solution to the Hierarchy Problem, Fortsch. Phys. 58 (2010) 528 [arXiv:0706.2050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    D. Harlow, Wormholes, Emergent Gauge Fields and the Weak Gravity Conjecture, JHEP 01 (2016) 122 [arXiv:1510.07911] [INSPIRE].
  29. [29]
    D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, arXiv:1810.05338 [INSPIRE].
  30. [30]
    D. Harlow and H. Ooguri, Constraints on Symmetries from Holography, Phys. Rev. Lett. 122 (2019) 191601 [arXiv:1810.05337] [INSPIRE].
  31. [31]
    N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    S. Bielleman, L.E. Ibáñez and I. Valenzuela, Minkowski 3-forms, Flux String Vacua, Axion Stability and Naturalness, JHEP 12 (2015) 119 [arXiv:1507.06793] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  34. [34]
    F. Carta, F. Marchesano, W. Staessens and G. Zoccarato, Open string multi-branched and Kähler potentials, JHEP 09 (2016) 062 [arXiv:1606.00508] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    A. Herraez, L.E. Ibáñez, F. Marchesano and G. Zoccarato, The Type IIA Flux Potential, 4-forms and Freed-Witten anomalies, JHEP 09 (2018) 018 [arXiv:1802.05771] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Kerr, G. Pearlstein and C. Robles, Polarized relations on horizontal SL(2)’s, arXiv:1705.03117.
  37. [37]
    C.-L. Wang, On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds, Math. Res. Lett. 4 (1997) 157.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    A. Gerhardus and H. Jockers, Quantum periods of Calabi-Yau fourfolds, Nucl. Phys. B 913 (2016) 425 [arXiv:1604.05325] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  39. [39]
    C.F. Cota, A. Klemm and T. Schimannek, Modular Amplitudes and Flux-Superpotentials on elliptic Calabi-Yau fourfolds, JHEP 01 (2018) 086 [arXiv:1709.02820] [INSPIRE].
  40. [40]
    P. Candelas, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 2, Nucl. Phys. B 429 (1994) 626 [hep-th/9403187] [INSPIRE].
  41. [41]
    A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
  42. [42]
    M. Alim and E. Scheidegger, Topological Strings on Elliptic Fibrations, Commun. Num. Theor. Phys. 08 (2014) 729 [arXiv:1205.1784] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    M.-x. Huang, S. Katz and A. Klemm, Topological String on elliptic CY 3-folds and the ring of Jacobi forms, JHEP 10 (2015) 125 [arXiv:1501.04891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    B. Haghighat, H. Movasati and S.-T. Yau, Calabi-Yau modular forms in limit: Elliptic Fibrations, Commun. Num. Theor. Phys. 11 (2017) 879 [arXiv:1511.01310] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    I. Garc´ıa-Etxebarria, T.W. Grimm and I. Valenzuela, Special Points of Inflation in Flux Compactifications, Nucl. Phys. B 899 (2015) 414 [arXiv:1412.5537] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    B. Andreas, G. Curio, D.H. Ruiperez and S.-T. Yau, Fourier-Mukai Transform and Mirror Symmetry for D-Branes on Elliptic Calabi-Yau, math.AG/0012196 [INSPIRE].
  47. [47]
    B. Andreas and D.H. Ruipérez, Fourier Mukai transforms and applications to string theory, math.AG/0412328 [INSPIRE].
  48. [48]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    K. Hanaki, K. Ohashi and Y. Tachikawa, Supersymmetric Completion of an R 2 term in Five-dimensional Supergravity, Prog. Theor. Phys. 117 (2007) 533 [hep-th/0611329] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    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].ADSCrossRefGoogle Scholar
  51. [51]
    E. Gonzalo, L.E. Ibáñez and Á.M. Uranga, Modular Symmetries and the Swampland Conjectures, JHEP 05 (2019) 105 [arXiv:1812.06520] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  52. [52]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    H. Ooguri, E. Palti, G. Shiu and C. Vafa, Distance and de Sitter Conjectures on the Swampland, Phys. Lett. B 788 (2019) 180 [arXiv:1810.05506] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, de Sitter Space and the Swampland, arXiv:1806.08362 [INSPIRE].
  55. [55]
    A. Gerhardus and H. Jockers, Quantum periods of Calabi-Yau fourfolds, Nucl. Phys. B 913 (2016) 425 [arXiv:1604.05325] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, Adv. Theor. Math. Phys. 21 (2017) 1373 [arXiv:1601.08181] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Pierre Corvilain
    • 1
    Email author
  • Thomas W. Grimm
    • 1
  • Irene Valenzuela
    • 2
  1. 1.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Department of PhysicsCornell UniversityIthacaU.S.A.

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