Advertisement

Modular fluxes, elliptic genera, and weak gravity conjectures in four dimensions

  • Seung-Joo LeeEmail author
  • Wolfgang Lerche
  • Timo Weigand
Open Access
Regular Article - Theoretical Physics

Abstract

We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory compactifications with N = 1 supersymmetry. Extending our previous work on nearly tensionless heterotic strings in six dimensions, we show that under certain assumptions a tower of asymptotically massless states arises in the limit of vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower contains super-extremal states whose charge-to-mass ratios are larger than those of certain extremal dilatonic Reissner-Nordström black holes, precisely as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower of super-extremal states does not always populate a charge sub-lattice.

The main tool for our analysis is the elliptic genus of the emergent heterotic string in the chiral N = 1 supersymmetric effective theories. This also governs situations where the heterotic string is non-perturbative. We show how it can be computed in terms of BPS invariants on elliptic four-folds, by making use of various dualities and mirror symmetry. Compared to six dimensions, the geometry of the relevant elliptically fibered four-folds is substantially richer than that of the three-folds, and we classify the possibilities for obtaining critical, nearly tensionless heterotic strings. We find that the (quasi-)modular properties of the elliptic genus crucially depend on the choice of flux background. Our general results are illustrated in a detailed example.

Keywords

F-Theory String Duality Superstrings and Heterotic Strings Topological Strings 

Notes

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

References

  1. [1]
    N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and gravity as the weakest force, JHEP06 (2007) 060 [hep-th/0601001] [INSPIRE].MathSciNetGoogle Scholar
  2. [2]
    C. Cheung and G.N. Remmen, Naturalness and the weak gravity conjecture, Phys. Rev. Lett.113 (2014) 051601 [arXiv:1402.2287] [INSPIRE].ADSGoogle Scholar
  3. [3]
    B. Heidenreich, M. Reece and T. Rudelius, Sharpening the weak gravity conjecture with dimensional reduction, JHEP02 (2016) 140 [arXiv:1509.06374] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    B. Heidenreich, M. Reece and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP08 (2017) 025 [arXiv:1606.08437] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  5. [5]
    S. Andriolo, D. Junghans, T. Noumi and G. Shiu, A tower weak gravity conjecture from infrared consistency, Fortsch. Phys.66 (2018) 1800020 [arXiv:1802.04287] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    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].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    D. Klaewer and E. Palti, Super-planckian spatial field variations and quantum gravity, JHEP01 (2017) 088 [arXiv:1610.00010] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  8. [8]
    E. Palti, The weak gravity conjecture and scalar fields, JHEP08 (2017) 034 [arXiv:1705.04328] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  9. [9]
    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].ADSGoogle Scholar
  10. [10]
    B. Heidenreich, M. Reece and T. Rudelius, Emergence of weak coupling at large distance in quantum gravity, Phys. Rev. Lett.121 (2018) 051601 [arXiv:1802.08698] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    T.W. Grimm, E. Palti and I. Valenzuela, Infinite distances in field space and massless towers of states, JHEP08 (2018) 143 [arXiv:1802.08264] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    R. Blumenhagen, D. Kläwer, L. Schlechter and F. Wolf, The refined swampland distance conjecture in Calabi-Yau moduli spaces, JHEP06 (2018) 052 [arXiv:1803.04989] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    S.-J. Lee, W. Lerche and T. Weigand, Tensionless strings and the weak gravity conjecture, JHEP10 (2018) 164 [arXiv:1808.05958] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  14. [14]
    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].ADSMathSciNetzbMATHGoogle Scholar
  15. [15]
    T.W. Grimm, C. Li and E. Palti, Infinite distance networks in field space and charge orbits, JHEP03 (2019) 016 [arXiv:1811.02571] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    P. Corvilain, T.W. Grimm and I. Valenzuela, The swampland distance conjecture for Kähler moduli, arXiv:1812.07548 [INSPIRE].
  17. [17]
    T. Banks and L.J. Dixon, Constraints on string vacua with space-time supersymmetry, Nucl. Phys.B 307 (1988) 93 [INSPIRE].ADSGoogle Scholar
  18. [18]
    T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev.D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].ADSGoogle Scholar
  19. [19]
    D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, arXiv:1810.05338 [INSPIRE].
  20. [20]
    T. Rudelius, On the possibility of large axion moduli spaces, JCAP04 (2015) 049 [arXiv:1409.5793] [INSPIRE].
  21. [21]
    A. de la Fuente, P. Saraswat and R. Sundrum, Natural inflation and quantum gravity, Phys. Rev. Lett.114 (2015) 151303 [arXiv:1412.3457] [INSPIRE].
  22. [22]
    T. Rudelius, Constraints on axion inflation from the weak gravity conjecture, JCAP09 (2015) 020 [arXiv:1503.00795] [INSPIRE].
  23. [23]
    M. Montero, A.M. Uranga and I. Valenzuela, Transplanckian axions!?, JHEP08 (2015) 032 [arXiv:1503.03886] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  24. [24]
    J. Brown, W. Cottrell, G. Shiu and P. Soler, Fencing in the swampland: quantum gravity constraints on large field inflation, JHEP10 (2015) 023 [arXiv:1503.04783] [INSPIRE].
  25. [25]
    T.C. Bachlechner, C. Long and L. McAllister, Planckian axions and the weak gravity conjecture, JHEP01 (2016) 091 [arXiv:1503.07853] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  26. [26]
    A. Hebecker, P. Mangat, F. Rompineve and L.T. Witkowski, Winding out of the swamp: evading the weak gravity conjecture with F-term winding inflation?, Phys. Lett.B 748 (2015) 455 [arXiv:1503.07912] [INSPIRE].ADSzbMATHGoogle Scholar
  27. [27]
    J. Brown, W. Cottrell, G. Shiu and P. Soler, On axionic field ranges, loopholes and the weak gravity conjecture, JHEP04 (2016) 017 [arXiv:1504.00659] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  28. [28]
    B. Heidenreich, M. Reece and T. Rudelius, Weak gravity strongly constrains large-field axion inflation, JHEP12 (2015) 108 [arXiv:1506.03447] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  29. [29]
    K. Kooner, S. Parameswaran and I. Zavala, Warping the weak gravity conjecture, Phys. Lett.B 759 (2016) 402 [arXiv:1509.07049] [INSPIRE].ADSzbMATHGoogle Scholar
  30. [30]
    L.E. Ibáñez, M. Montero, A. Uranga and I. Valenzuela, Relaxion monodromy and the weak gravity conjecture, JHEP04 (2016) 020 [arXiv:1512.00025] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  31. [31]
    A. Hebecker, F. Rompineve and A. Westphal, Axion monodromy and the weak gravity conjecture, JHEP04 (2016) 157 [arXiv:1512.03768] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  32. [32]
    F. Baume and E. Palti, Backreacted axion field ranges in string theory, JHEP08 (2016) 043 [arXiv:1602.06517] [INSPIRE].
  33. [33]
    B. Heidenreich, M. Reece and T. Rudelius, Axion experiments to algebraic geometry: testing quantum gravity via the weak gravity conjecture, Int. J. Mod. Phys.D 25 (2016) 1643005 [arXiv:1605.05311] [INSPIRE].ADSGoogle Scholar
  34. [34]
    R. Blumenhagen, I. Valenzuela and F. Wolf, The swampland conjecture and F-term axion monodromy inflation, JHEP07 (2017) 145 [arXiv:1703.05776] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  35. [35]
    I. Valenzuela, Backreaction in axion monodromy, 4-forms and the swampland, PoS(CORFU2016)112 [arXiv:1708.07456] [INSPIRE].
  36. [36]
    L.E. Ibáñez and M. Montero, A note on the WGC, effective field theory and clockwork within string theory, JHEP02 (2018) 057 [arXiv:1709.02392] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  37. [37]
    G. Aldazabal and L.E. Ibáñez, A note on 4D heterotic string vacua, FI-terms and the swampland, Phys. Lett.B 782 (2018) 375 [arXiv:1804.07322] [INSPIRE].ADSzbMATHGoogle Scholar
  38. [38]
    R. Blumenhagen, Large field inflation/quintessence and the refined swampland distance conjecture, PoS(CORFU2017) 175 [arXiv:1804.10504] [INSPIRE].
  39. [39]
    M. Reece, Photon masses in the landscape and the swampland, JHEP07 (2019) 181 [arXiv:1808.09966] [INSPIRE].
  40. [40]
    A. Hebecker and P. Soler, The weak gravity conjecture and the axionic black hole paradox, JHEP09 (2017) 036 [arXiv:1702.06130] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  41. [41]
    M. Montero, A.M. Uranga and I. Valenzuela, A Chern-Simons pandemic, JHEP07 (2017) 123 [arXiv:1702.06147] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  42. [42]
    A. Hebecker, T. Mikhail and P. Soler, Euclidean wormholes, baby universes and their impact on particle physics and cosmology, Front. Astron. Space Sci.5 (2018) 35 [arXiv:1807.00824] [INSPIRE].ADSGoogle Scholar
  43. [43]
    A. Hebecker, S. Leonhardt, J. Moritz and A. Westphal, Thraxions: ultralight throat axions, JHEP04 (2019) 158 [arXiv:1812.03999] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    H. Ooguri and C. Vafa, Non-supersymmetric AdS and the swampland, Adv. Theor. Math. Phys.21 (2017) 1787 [arXiv:1610.01533] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  45. [45]
    L.E. Ibáñez, V. Martin-Lozano and I. Valenzuela, Constraining neutrino masses, the cosmological constant and BSM physics from the weak gravity conjecture, JHEP11 (2017) 066 [arXiv:1706.05392] [INSPIRE].
  46. [46]
    Y. Hamada and G. Shiu, Weak gravity conjecture, multiple point principle and the standard model landscape, JHEP11 (2017) 043 [arXiv:1707.06326] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  47. [47]
    L.E. Ibáñez, V. Martin-Lozano and I. Valenzuela, Constraining the EW hierarchy from the weak gravity conjecture, arXiv:1707.05811 [INSPIRE].
  48. [48]
    D. Lüst and E. Palti, Scalar fields, hierarchical UV/IR mixing and the weak gravity conjecture, JHEP02 (2018) 040 [arXiv:1709.01790] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  49. [49]
    E. Gonzalo and L.E. Ibáñez, The fundamental need for a SM Higgs and the weak gravity conjecture, Phys. Lett.B 786 (2018) 272 [arXiv:1806.09647] [INSPIRE].ADSGoogle Scholar
  50. [50]
    D. Klaewer, D. Lüst and E. Palti, A spin-2 conjecture on the swampland, Fortsch. Phys.67 (2019) 1800102 [arXiv:1811.07908] [INSPIRE].MathSciNetGoogle Scholar
  51. [51]
    N. Craig, I. Garcia Garcia and S. Koren, Discrete gauge symmetries and the weak gravity conjecture, JHEP05 (2019) 140 [arXiv:1812.08181] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  52. [52]
    Y. Nakayama and Y. Nomura, Weak gravity conjecture in the AdS/CFT correspondence, Phys. Rev.D 92 (2015) 126006 [arXiv:1509.01647] [INSPIRE].ADSMathSciNetGoogle Scholar
  53. [53]
    D. Harlow, Wormholes, emergent gauge fields and the weak gravity conjecture, JHEP01 (2016) 122 [arXiv:1510.07911] [INSPIRE].
  54. [54]
    G. Shiu, P. Soler and W. Cottrell, Weak gravity conjecture and extremal black holes, Sci. China Phys. Mech. Astron.62 (2019) 110412 [arXiv:1611.06270] [INSPIRE].Google Scholar
  55. [55]
    N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, Universal bounds on charged states in 2d CFT and 3d gravity, JHEP08 (2016) 041 [arXiv:1603.09745] [INSPIRE].
  56. [56]
    M. Montero, G. Shiu and P. Soler, The weak gravity conjecture in three dimensions, JHEP10 (2016) 159 [arXiv:1606.08438] [INSPIRE].
  57. [57]
    Z. Fisher and C.J. Mogni, A semiclassical, entropic proof of a weak gravity conjecture, arXiv:1706.08257 [INSPIRE].
  58. [58]
    C. Cheung, J. Liu and G.N. Remmen, Proof of the weak gravity conjecture from black hole entropy, JHEP10 (2018) 004 [arXiv:1801.08546] [INSPIRE].
  59. [59]
    Y. Hamada, T. Noumi and G. Shiu, Weak gravity conjecture from unitarity and causality, Phys. Rev. Lett.123 (2019) 051601 [arXiv:1810.03637] [INSPIRE].
  60. [60]
    M. Montero, A holographic derivation of the weak gravity conjecture, JHEP03 (2019) 157 [arXiv:1812.03978] [INSPIRE].
  61. [61]
    A.N. Schellekens and N.P. Warner, Anomalies, characters and strings, Nucl. Phys.B 287 (1987)317 [INSPIRE].ADSMathSciNetGoogle Scholar
  62. [62]
    E. Gonzalo, L.E. Ibáñez and A.M. Uranga, Modular symmetries and the swampland conjectures, JHEP05 (2019) 105 [arXiv:1812.06520] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  63. [63]
    G. Honecker, Massive U(1)s and heterotic five-branes on K3, Nucl. Phys.B 748 (2006) 126 [hep-th/0602101] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  64. [64]
    R. Blumenhagen, S. Moster and T. Weigand, Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds, Nucl. Phys.B 751 (2006) 186 [hep-th/0603015] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  65. [65]
    A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, Nucl. Phys. Proc. Suppl.58 (1997) 177 [hep-th/9607139] [INSPIRE].ADSzbMATHGoogle Scholar
  66. [66]
    B. Haghighat et al., M-strings, Commun. Math. Phys.334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
  67. [67]
    B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of minimal 6d SCFTs, Fortsch. Phys.63 (2015) 294 [arXiv:1412.3152] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  68. [68]
    W. Lerche, Elliptic index and superstring effective actions, Nucl. Phys.B 308 (1988) 102 [INSPIRE].ADSMathSciNetGoogle Scholar
  69. [69]
    B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys.173 (1995) 559 [hep-th/9402119] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  70. [70]
    E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys.22 (1997)1 [hep-th/9609122] [INSPIRE].
  71. [71]
    P. Mayr, Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four folds, Nucl. Phys.B 494 (1997) 489 [hep-th/9610162] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  72. [72]
    A. Klemm, B. Lian, S.S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys.B 518 (1998) 515 [hep-th/9701023] [INSPIRE].ADSzbMATHGoogle Scholar
  73. [73]
    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].ADSMathSciNetzbMATHGoogle Scholar
  74. [74]
    A.P. Braun and T. Watari, The vertical, the horizontal and the rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications, JHEP01 (2015) 047 [arXiv:1408.6167] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  75. [75]
    W. Lerche, Fayet-Iliopoulos potentials from four folds, JHEP11 (1997) 004 [hep-th/9709146] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  76. [76]
    R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional string compactifications with D-branes, orientifolds and fluxes, Phys. Rept.445 (2007) 1 [hep-th/0610327] [INSPIRE].ADSMathSciNetGoogle Scholar
  77. [77]
    H. Jockers, P. Mayr and J. Walcher, On N = 1 4d effective couplings for F-theory and heterotic vacua, Adv. Theor. Math. Phys.14 (2010) 1433 [arXiv:0912.3265] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  78. [78]
    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] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  79. [79]
    T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing brane and flux superpotentials in F-theory compactifications, JHEP04 (2010) 015 [arXiv:0909.2025] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  80. [80]
    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] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  81. [81]
    T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys.B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  82. [82]
    A.P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles, Nucl. Phys.B 856 (2012) 129 [arXiv:1107.5337] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  83. [83]
    T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP03 (2012) 027 [arXiv:1111.1232] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  84. [84]
    S. Krause, C. Mayrhofer and T. Weigand, G 4flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys.B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].ADSzbMATHGoogle Scholar
  85. [85]
    K. Intriligator et al., Conifold transitions in M-theory on Calabi-Yau fourfolds with background fluxes, Adv. Theor. Math. Phys.17 (2013) 601 [arXiv:1203.6662] [INSPIRE].
  86. [86]
    A.P. Braun, A. Collinucci and R. Valandro, Hypercharge flux in F-theory and the stable Sen limit, JHEP07 (2014) 121 [arXiv:1402.4096] [INSPIRE].
  87. [87]
    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].MathSciNetzbMATHGoogle Scholar
  88. [88]
    C.F. Cota, A. Klemm and T. Schimannek, Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds, JHEP01 (2018) 086 [arXiv:1709.02820] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  89. [89]
    A. Klemm and R. Pandharipande, Enumerative geometry of Calabi-Yau 4-folds, Commun. Math. Phys.281 (2008) 621 [math/0702189] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  90. [90]
    R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].
  91. [91]
    R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
  92. [92]
    J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys.B 463 (1996) 315 [hep-th/9510182] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  93. [93]
    J.A. Minahan, D. Nemeschansky, C. Vafa and N.P. Warner, E strings and N = 4 topological Yang-Mills theories, Nucl. Phys.B 527 (1998) 581 [hep-th/9802168] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  94. [94]
    M. Del Zotto and G. Lockhart, On exceptional instanton strings, JHEP09 (2017) 081 [arXiv:1609.00310] [INSPIRE].
  95. [95]
    J. Kim, K. Lee and J. Park, On elliptic genera of 6d string theories, JHEP10 (2018) 100 [arXiv:1801.01631] [INSPIRE].ADSzbMATHGoogle Scholar
  96. [96]
    A.N. Schellekens and N.P. Warner, Anomalies and modular invariance in string theory, Phys. Lett.B 177 (1986) 317 [INSPIRE].ADSMathSciNetGoogle Scholar
  97. [97]
    M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Germany (1995).zbMATHGoogle Scholar
  98. [98]
    A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing and mock modular forms, arXiv:1208.4074 [INSPIRE].
  99. [99]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d \( \mathcal{N} \) = 2 gauge theories, Commun. Math. Phys.333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].ADSzbMATHGoogle Scholar
  100. [100]
    M. Del Zotto et al., Topological strings on singular elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs, JHEP03 (2018) 156 [arXiv:1712.07017] [INSPIRE].
  101. [101]
    M. Del Zotto and G. Lockhart, Universal features of BPS strings in six-dimensional SCFTs, JHEP08 (2018) 173 [arXiv:1804.09694] [INSPIRE].
  102. [102]
    H. Skogman, On the Fourier expansions of Jacobi forms, Int. J. Math. Math. Sci.2004 (2004) 2583.Google Scholar
  103. [103]
    A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
  104. [104]
    M. Alim and E. Scheidegger, Topological strings on elliptic fibrations, Commun. Num. Theor. Phys.08 (2014) 729 [arXiv:1205.1784] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  105. [105]
    M.-x. Huang, S. Katz and A. Klemm, Topological string on elliptic CY 3-folds and the ring of Jacobi forms, JHEP10 (2015) 125 [arXiv:1501.04891] [INSPIRE].
  106. [106]
    T. Weigand, F-theory, PoS(TASI2017)016 [arXiv:1806.01854] [INSPIRE].
  107. [107]
    M. Cvetič and L. Lin, TASI lectures on abelian and discrete symmetries in F-theory, PoS(TASI2017)020 [arXiv:1809.00012] [INSPIRE].
  108. [108]
    D.S. Park, Anomaly equations and intersection theory, JHEP01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
  109. [109]
    R. Donagi and M. Wijnholt, Higgs bundles and UV completion in F-theory, Commun. Math. Phys.326 (2014) 287 [arXiv:0904.1218] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  110. [110]
    J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yaus, JHEP11 (2011) 098 [arXiv:1108.1794] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  111. [111]
    M. Cvetič, T.W. Grimm and D. Klevers, Anomaly cancellation and abelian gauge symmetries in F-theory, JHEP02 (2013) 101 [arXiv:1210.6034] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  112. [112]
    R. Blumenhagen, M. Cvetič and T. Weigand, Spacetime instanton corrections in 4D string vacua: The Seesaw mechanism for D-brane models, Nucl. Phys.B 771 (2007) 113 [hep-th/0609191] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  113. [113]
    L.E. Ibáñez and A.M. Uranga, Neutrino Majorana masses from string theory instanton effects, JHEP03 (2007) 052 [hep-th/0609213] [INSPIRE].ADSGoogle Scholar
  114. [114]
    M. Berasaluce-Gonzalez, L.E. Ibáñez, P. Soler and A.M. Uranga, Discrete gauge symmetries in D-brane models, JHEP12 (2011) 113 [arXiv:1106.4169] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  115. [115]
    C. Lawrie, S. Schäfer-Nameki and T. Weigand, Chiral 2d theories from N = 4 SYM with varying coupling, JHEP04 (2017) 111 [arXiv:1612.05640] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  116. [116]
    L. Martucci, Topological duality twist and brane instantons in F-theory, JHEP06 (2014) 180 [arXiv:1403.2530] [INSPIRE].
  117. [117]
    T. Weigand and F. Xu, The Green-Schwarz mechanism and geometric anomaly relations in 2d (0, 2) F-theory Vacua, JHEP04 (2018) 107 [arXiv:1712.04456] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  118. [118]
    D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, American Mathematical Society, U.S.A. (2000).zbMATHGoogle Scholar
  119. [119]
    M. Cvetič et al., Origin of abelian gauge symmetries in heterotic/F-theory duality, JHEP04 (2016) 041 [arXiv:1511.08208] [INSPIRE].
  120. [120]
    K.-S. Choi and H. Hayashi, U(n) spectral covers from decomposition, JHEP06 (2012) 009 [arXiv:1203.3812] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  121. [121]
    P. Hořava and E. Witten, Heterotic and type-I string dynamics from eleven-dimensions, Nucl. Phys.B 460 (1996) 506 [hep-th/9510209] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  122. [122]
    A.P. Braun, C.R. Brodie, A. Lukas and F. Ruehle, NS5-branes and line bundles in heterotic/F-theory duality, Phys. Rev.D 98 (2018) 126004 [arXiv:1803.06190] [INSPIRE].
  123. [123]
    B. Haghighat, G. Lockhart and C. Vafa, Fusing E-strings to heterotic strings: E + EH, Phys. Rev.D 90 (2014) 126012 [arXiv:1406.0850] [INSPIRE].ADSGoogle Scholar
  124. [124]
    W. Lerche, B.E.W. Nilsson, A.N. Schellekens and N.P. Warner, Anomaly cancelling terms from the elliptic genus, Nucl. Phys.B 299 (1988) 91 [INSPIRE].ADSMathSciNetGoogle Scholar
  125. [125]
    M. Haack et al., Gaugino condensates and D-terms from D7-branes, JHEP01 (2007) 078 [hep-th/0609211] [INSPIRE].ADSMathSciNetGoogle Scholar
  126. [126]
    R. Blumenhagen, M. Cvetič, S. Kachru and T. Weigand, D-brane instantons in type II orientifolds, Ann. Rev. Nucl. Part. Sci.59 (2009) 269 [arXiv:0902.3251] [INSPIRE].ADSGoogle Scholar
  127. [127]
    T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, On fluxed instantons and moduli stabilisation in IIB orientifolds and F-theory, Phys. Rev.D 84 (2011) 066001 [arXiv:1105.3193] [INSPIRE].ADSGoogle Scholar
  128. [128]
    T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive abelian gauge symmetries and fluxes in F-theory, JHEP12 (2011) 004 [arXiv:1107.3842] [INSPIRE].
  129. [129]
    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys.B 433 (1995) 501 [hep-th/9406055] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  130. [130]
    M. Kaneko and D. Zagier, A generalized jacobi theta function and quasimodular forms, in The moduli space of curves, R.H. Dijkgraaf et al. eds., Boston, U.S.A. (1995).Google Scholar
  131. [131]
    Y. Choie and M. Lee, Quasimodular forms and Jacobi-like forms, Math. Z.280 (2015) 643.MathSciNetzbMATHGoogle Scholar
  132. [132]
    J. Polchinski, Monopoles, duality and string theory, Int. J. Mod. Phys.A 19S1 (2004) 145 [hep-th/0304042] [INSPIRE].
  133. [133]
    R.K. Lazarsfeld, Positivity in algebraic geometry I, Springer, Germany (2004).zbMATHGoogle Scholar
  134. [134]
    G.W. Gibbons and K.-i. Maeda, Black holes and membranes in higher dimensional theories with dilaton fields, Nucl. Phys.B 298 (1988) 741 [INSPIRE].ADSMathSciNetGoogle Scholar
  135. [135]
    W. Lerche, B.E.W. Nilsson and A.N. Schellekens, Heterotic string loop calculation of the anomaly cancelling term, Nucl. Phys.B 289 (1987) 609 [INSPIRE].ADSMathSciNetGoogle Scholar
  136. [136]
    M. Buican et al., D-branes at singularities, compactification and hypercharge, JHEP01 (2007)107 [hep-th/0610007] [INSPIRE].ADSMathSciNetGoogle Scholar
  137. [137]
    E. Plauschinn, The generalized green-schwarz mechanism for type IIB orientifolds with D3-and D7-branes, JHEP05 (2009) 062 [arXiv:0811.2804] [INSPIRE].ADSMathSciNetGoogle Scholar
  138. [138]
    J.P. Conlon, A. Maharana and F. Quevedo, Towards realistic string vacua, JHEP05 (2009) 109 [arXiv:0810.5660] [INSPIRE].
  139. [139]
    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].ADSMathSciNetzbMATHGoogle Scholar
  140. [140]
    H. Jockers and J. Louis, The effective action of D7-branes in N = 1 Calabi-Yau orientifolds, Nucl. Phys.B 705 (2005) 167 [hep-th/0409098] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  141. [141]
    A.P. Braun et al., PALPa user manual, in Strings, gauge fields and the geometry behind: the legacy of Maximilian Kreuzer, A. Rebhan et al. eds., World Scientific, Singapore (2012), arXiv:1205.4147 [INSPIRE].
  142. [142]
    W. Stein et al., Sage Mathematics Software (Version 8.4), The Sage Development Team (2018), http://www.sagemath.org.
  143. [143]
    D.R. Morrison and D.S. Park, F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds, JHEP10 (2012) 128 [arXiv:1208.2695] [INSPIRE].
  144. [144]
    S.-J. Lee, D. Regalado and T. Weigand, 6d SCFTs and U(1) flavour symmetries, JHEP11 (2018) 147 [arXiv:1803.07998] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.CERN, Theory DepartmentGeneva 23Switzerland
  2. 2.PRISMA Cluster of Excellence and Mainz Institute for Theoretical PhysicsJohannes Gutenberg-UniversitätMainzGermany

Personalised recommendations