A geometrical upper bound on the inflaton range

  • Michele CicoliEmail author
  • David Ciupke
  • Christoph Mayrhofer
  • Pramod Shukla
Open Access
Regular Article - Theoretical Physics


We argue that in type IIB LVS string models, after including the leading order moduli stabilisation effects, the moduli space for the remaining flat directions is compact due the Calabi-Yau Kähler cone conditions. In cosmological applications, this gives an inflaton field range which is bounded from above, in analogy with recent results from the weak gravity and swampland conjectures. We support our claim by explicitly showing that it holds for all LVS vacua with h1,1 = 3 obtained from 4-dimensional reflexive polytopes. In particular, we first search for all Calabi-Yau threefolds from the Kreuzer-Skarke list with h1,1 = 2, 3 and 4 which allow for LVS vacua, finding several new LVS geometries which were so far unknown. We then focus on the h1,1 = 3 cases and show that the Kähler cones of all toric hypersurface threefolds force the effective 1-dimensional LVS moduli space to be compact. We find that the moduli space size can generically be trans-Planckian only for K3 fibred examples.


Compactification and String Models Cosmology of Theories beyond the SM Superstring Vacua Flux compactifications 


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]
    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
  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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    F. Baume and E. Palti, Backreacted axion field ranges in string theory, JHEP 08 (2016) 043 [arXiv:1602.06517] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    D. Klaewer and E. Palti, Super-Planckian spatial field variations and quantum gravity, JHEP 01 (2017) 088 [arXiv:1610.00010] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    R. Blumenhagen, I. Valenzuela and F. Wolf, The swampland conjecture and F-term axion monodromy inflation, JHEP 07 (2017) 145 [arXiv:1703.05776] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    T. Rudelius, Constraints on axion inflation from the weak gravity conjecture, JCAP 09 (2015) 020 [arXiv:1503.00795] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Montero, A.M. Uranga and I. Valenzuela, Transplanckian axions!?, JHEP 08 (2015) 032 [arXiv:1503.03886] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    B. Heidenreich, M. Reece and T. Rudelius, Weak gravity strongly constrains large-field axion inflation, JHEP 12 (2015) 108 [arXiv:1506.03447] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  9. [9]
    J. Brown, W. Cottrell, G. Shiu and P. Soler, Fencing in the swampland: quantum gravity constraints on large field inflation, JHEP 10 (2015) 023 [arXiv:1503.04783] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    J. Brown, W. Cottrell, G. Shiu and P. Soler, On axionic field ranges, loopholes and the weak gravity conjecture, JHEP 04 (2016) 017 [arXiv:1504.00659] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  11. [11]
    J.P. Conlon and S. Krippendorf, Axion decay constants away from the lamppost, JHEP 04 (2016) 085 [arXiv:1601.00647] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  12. [12]
    E. Palti, The weak gravity conjecture and scalar fields, JHEP 08 (2017) 034 [arXiv:1705.04328] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle 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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    V. Balasubramanian, P. Berglund, J.P. Conlon and F. Quevedo, Systematics of moduli stabilisation in Calabi-Yau flux compactifications, JHEP 03 (2005) 007 [hep-th/0502058] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    M. Cicoli, J.P. Conlon and F. Quevedo, General analysis of LARGE volume scenarios with string loop moduli stabilisation, JHEP 10 (2008) 105 [arXiv:0805.1029] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    M. Cicoli and F. Quevedo, String moduli inflation: an overview, Class. Quant. Grav. 28 (2011) 204001 [arXiv:1108.2659] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    C.P. Burgess, M. Cicoli, F. Quevedo and M. Williams, Inflating with large effective fields, JCAP 11 (2014) 045 [arXiv:1404.6236] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    J.P. Conlon and F. Quevedo, Kähler moduli inflation, JHEP 01 (2006) 146 [hep-th/0509012] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    M. Cicoli, C.P. Burgess and F. Quevedo, Fibre inflation: observable gravity waves from IIB string compactifications, JCAP 03 (2009) 013 [arXiv:0808.0691] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    B.J. Broy, D. Ciupke, F.G. Pedro and A. Westphal, Starobinsky-type inflation from α-corrections, JCAP 01 (2016) 001 [arXiv:1509.00024] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    C.P. Burgess, M. Cicoli, S. de Alwis and F. Quevedo, Robust inflation from fibrous strings, JCAP 05 (2016) 032 [arXiv:1603.06789] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    M. Cicoli, D. Ciupke, S. de Alwis and F. Muia, αinflation: moduli stabilisation and observable tensors from higher derivatives, JHEP 09 (2016) 026 [arXiv:1607.01395] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    M. Cicoli, F.G. Pedro and G. Tasinato, Poly-instanton inflation, JCAP 12 (2011) 022 [arXiv:1110.6182] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Cicoli, F. Muia and F.G. Pedro, Microscopic origin of volume modulus inflation, JCAP 12 (2015) 040 [arXiv:1509.07748] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    M. Cicoli, F. Muia and P. Shukla, Global embedding of fibre inflation models, JHEP 11 (2016) 182 [arXiv:1611.04612] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    M. Cicoli et al., Chiral global embedding of fibre inflation models, JHEP 11 (2017) 207 [arXiv:1709.01518] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    T.W. Grimm and J. Louis, The effective action of N = 1 Calabi-Yau orientifolds, Nucl. Phys. B 699 (2004) 387 [hep-th/0403067] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    K. Becker, M. Becker, M. Haack and J. Louis, Supersymmetry breaking and αcorrections to flux induced potentials, JHEP 06 (2002) 060 [hep-th/0204254] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [INSPIRE].
  31. [31]
    S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    K. Bobkov, Volume stabilization via alpha-prime corrections in type IIB theory with fluxes, JHEP 05 (2005) 010 [hep-th/0412239] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    M. Cicoli, A. Maharana, F. Quevedo and C.P. Burgess, De Sitter string vacua from dilaton-dependent non-perturbative effects, JHEP 06 (2012) 011 [arXiv:1203.1750] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  34. [34]
    M. Cicoli, M. Kreuzer and C. Mayrhofer, Toric K3-fibred Calabi-Yau manifolds with del Pezzo divisors for string compactifications, JHEP 02 (2012) 002 [arXiv:1107.0383] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    M. Dine, N. Seiberg and E. Witten, Fayet-Iliopoulos terms in string theory, Nucl. Phys. B 289 (1987) 589 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    M. Dine, I. Ichinose and N. Seiberg, F terms and d terms in string theory, Nucl. Phys. B 293 (1987) 253 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    M. Berg, M. Haack and B. Körs, String loop corrections to Kähler potentials in orientifolds, JHEP 11 (2005) 030 [hep-th/0508043] [INSPIRE].CrossRefGoogle Scholar
  38. [38]
    M. Berg, M. Haack and E. Pajer, Jumping through loops: on soft terms from LARGE volume compactifications, JHEP 09 (2007) 031 [arXiv:0704.0737] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    M. Cicoli, J.P. Conlon and F. Quevedo, Systematics of string loop corrections in type IIB Calabi-Yau flux compactifications, JHEP 01 (2008) 052 [arXiv:0708.1873] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    M. Berg, M. Haack, J.U. Kang and S. Sjörs, Towards the one-loop Kähler metric of Calabi-Yau orientifolds, JHEP 12 (2014) 077 [arXiv:1407.0027] [INSPIRE].
  41. [41]
    M. Haack and J.U. Kang, One-loop Einstein-Hilbert term in minimally supersymmetric type IIB orientifolds, JHEP 02 (2016) 160 [arXiv:1511.03957] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  42. [42]
    D. Ciupke, J. Louis and A. Westphal, Higher-derivative supergravity and moduli stabilization, JHEP 10 (2015) 094 [arXiv:1505.03092] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    T.W. Grimm, K. Mayer and M. Weissenbacher, Higher derivatives in Type II and M-theory on Calabi-Yau threefolds, JHEP 02 (2018) 127 [arXiv:1702.08404] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    R. Blumenhagen and M. Schmidt-Sommerfeld, Power towers of string instantons for N = 1 vacua, JHEP 07 (2008) 027 [arXiv:0803.1562] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    R. Blumenhagen, X. Gao, T. Rahn and P. Shukla, A note on poly-instanton effects in type IIB orientifolds on Calabi-Yau threefolds, JHEP 06 (2012) 162 [arXiv:1205.2485] [INSPIRE].
  46. [46]
    R. Blumenhagen, X. Gao, T. Rahn and P. Shukla, Moduli stabilization and inflationary cosmology with poly-instantons in type IIB orientifolds, JHEP 11 (2012) 101 [arXiv:1208.1160] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    M. Bianchi, A. Collinucci and L. Martucci, Magnetized E3-brane instantons in F-theory, JHEP 12 (2011) 045 [arXiv:1107.3732] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    M. Bianchi, A. Collinucci and L. Martucci, Freezing E3-brane instantons with fluxes, Fortsch. Phys. 60 (2012) 914 [arXiv:1202.5045] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    J. Louis, M. Rummel, R. Valandro and A. Westphal, Building an explicit de Sitter, JHEP 10 (2012) 163 [arXiv:1208.3208] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    M. Cicoli et al., Global orientifolded quivers with inflation, JHEP 11 (2017) 134 [arXiv:1706.06128] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    D. Lüst and X. Zhang, Four Kähler moduli stabilisation in type IIB orientifolds with K3-fibred Calabi-Yau threefold compactification, JHEP 05 (2013) 051 [arXiv:1301.7280] [INSPIRE].
  52. [52]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
  54. [54]
    C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand, Mordell-Weil torsion and the global structure of gauge groups in F-theory, JHEP 10 (2014) 16 [arXiv:1405.3656] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    D. Klevers et al., F-Theory on all toric hypersurface fibrations and its Higgs branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  56. [56]
    R. Altman et al., A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list, JHEP 02 (2015) 158 [arXiv:1411.1418] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  57. [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] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. [58]
    R. Blumenhagen, B. Jurke and T. Rahn, Computational tools for cohomology of toric varieties, Adv. High Energy Phys. 2011 (2011) 152749 [arXiv:1104.1187] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  59. [59]
    Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE].
  60. [60]
    G. Cabass et al., Updated constraints and forecasts on primordial tensor modes, Phys. Rev. D 93 (2016) 063508 [arXiv:1511.05146] [INSPIRE].ADSGoogle Scholar
  61. [61]
    C.P. Burgess et al., Non-standard primordial fluctuations and nonGaussianity in string inflation, JHEP 08 (2010) 045 [arXiv:1005.4840] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  62. [62]
    M. Cicoli et al., Modulated reheating and large non-gaussianity in string cosmology, JCAP 05 (2012) 039 [arXiv:1202.4580] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    R. Kallosh and A. Linde, Non-minimal inflationary attractors, JCAP 10 (2013) 033 [arXiv:1307.7938] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    R. Kallosh and A. Linde, Escher in the sky, Comptes Rend. Phys. 16 (2015) 914 [arXiv:1503.06785] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    J.J.M. Carrasco, R. Kallosh and A. Linde, α-attractors: Planck, LHC and dark energy, JHEP 10 (2015) 147 [arXiv:1506.01708] [INSPIRE].
  66. [66]
    M. Cicoli, S. de Alwis and A. Westphal, Heterotic moduli stabilisation, JHEP 10 (2013) 199 [arXiv:1304.1809] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    J. Gray et al., Calabi-Yau manifolds with large volume vacua, Phys. Rev. D 86 (2012) 101901 [arXiv:1207.5801] [INSPIRE].ADSGoogle Scholar
  68. [68]
    R. Altman, Y.-H. He, V. Jejjala and B.D. Nelson, New large volume Calabi-Yau threefolds, Phys. Rev. D 97 (2018) 046003 [arXiv:1706.09070] [INSPIRE].ADSMathSciNetGoogle Scholar
  69. [69]
    S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Dipartimento di Fisica e AstronomiaUniversità di BolognaBolognaItaly
  2. 2.INFN — Sezione di BolognaBolognaItaly
  3. 3.Abdus Salam ICTPTriesteItaly
  4. 4.Arnold Sommerfeld Center for Theoretical PhysicsMünchenGermany
  5. 5.Departamento de Física Teórica and Instituto de Física Teórica UAM/CSICUniversidad Autónoma de MadridMadridSpain

Personalised recommendations