Quasi-Jacobi forms, elliptic genera and strings in four dimensions

Abstract

We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which may encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions.

For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present.

Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.

A preprint version of the article is available at ArXiv.

References

  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].

    ADS  MathSciNet  Article  Google Scholar 

  2. [2]

    T.D. Brennan, F. Carta and C. Vafa, The String Landscape, the Swampland, and the Missing Corner, PoS TASI2017 (2017) 015 [arXiv:1711.00864] [INSPIRE].

  3. [3]

    E. Palti, The Swampland: Introduction and Review, Fortsch. Phys. 67 (2019) 1900037 [arXiv:1903.06239] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  4. [4]

    S.-J. Lee, W. Lerche and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP 10 (2018) 164 [arXiv:1808.05958] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    S.-J. Lee, W. Lerche and T. Weigand, Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions, JHEP 08 (2019) 104 [arXiv:1901.08065] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    S.-J. Lee, W. Lerche and T. Weigand, Emergent Strings, Duality and Weak Coupling Limits for Two-Form Fields, arXiv:1904.06344 [INSPIRE].

  8. [8]

    S.-J. Lee, W. Lerche and T. Weigand, Emergent Strings from Infinite Distance Limits, arXiv:1910.01135 [INSPIRE].

  9. [9]

    F. Baume, F. Marchesano and M. Wiesner, Instanton Corrections and Emergent Strings, JHEP 04 (2020) 174 [arXiv:1912.02218] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    P. Corvilain, T.W. Grimm and I. Valenzuela, The Swampland Distance Conjecture for Kähler moduli, JHEP 08 (2019) 075 [arXiv:1812.07548] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  14. [14]

    A. Joshi and A. Klemm, Swampland Distance Conjecture for One-Parameter Calabi-Yau Threefolds, JHEP 08 (2019) 086 [arXiv:1903.00596] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    D. Erkinger and J. Knapp, Refined swampland distance conjecture and exotic hybrid Calabi-Yaus, JHEP 07 (2019) 029 [arXiv:1905.05225] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    N. Gendler and I. Valenzuela, Merging the Weak Gravity and Distance Conjectures Using BPS Extremal Black Holes, arXiv:2004.10768 [INSPIRE].

  17. [17]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    D. Klaewer and E. Palti, Super-Planckian Spatial Field Variations and Quantum Gravity, JHEP 01 (2017) 088 [arXiv:1610.00010] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  19. [19]

    B. Heidenreich, M. Reece and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, JHEP 02 (2016) 140 [arXiv:1509.06374] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  20. [20]

    M. Montero, G. Shiu and P. Soler, The Weak Gravity Conjecture in three dimensions, JHEP 10 (2016) 159 [arXiv:1606.08438] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. [21]

    B. Heidenreich, M. Reece and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP 08 (2017) 025 [arXiv:1606.08437] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. [22]

    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].

    Article  Google Scholar 

  23. [23]

    M. Kaneko and D. Zagier, A generalized Jacobi Theta function and quasimodular forms, in R.H. Dijkgraaf, C.F. Faber and G.B.M. van der Geer eds., The Moduli Space of Curves, Birkhäuser Boston, Boston, MA, U.S.A. (1995) pp. 165–172.

  24. [24]

    A. Libgober, Elliptic genera, real algebraic varieties and quasi-jacobi forms, arXiv:0904.1026.

  25. [25]

    G. Oberdieck and A. Pixton, Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations, Geom. Topol. 23 (2019) 1415 [arXiv:1709.01481] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  26. [26]

    A.N. Schellekens and N.P. Warner, Anomalies and Modular Invariance in String Theory, Phys. Lett. B 177 (1986) 317 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    A.N. Schellekens and N.P. Warner, Anomalies, Characters and Strings, Nucl. Phys. B 287 (1987) 317 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. [28]

    E. Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    O. Alvarez, T.P. Killingback, M.L. Mangano and P. Windey, String Theory and Loop Space Index Theorems, Commun. Math. Phys. 111 (1987) 1 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. [30]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  31. [31]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  32. [32]

    M.B. Green and J.H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  33. [33]

    T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  34. [34]

    V. Gritsenko, Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms, math/9906190 [INSPIRE].

  35. [35]

    V. Gritsenko, Complex vector bundles and Jacobi forms, 1, 1999, math/9906191 [INSPIRE].

  36. [36]

    A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].

  37. [37]

    A. Schwimmer and N. Seiberg, Comments on the N = 2, N = 3, N = 4 Superconformal Algebras in Two-Dimensions, Phys. Lett. B 184 (1987) 191 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  38. [38]

    W. Lerche and N.P. Warner, Index Theorems in N = 2 Superconformal Theories, Phys. Lett. B 205 (1988) 471 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  39. [39]

    A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, Nucl. Phys. B Proc. Suppl. 58 (1997) 177 [hep-th/9607139] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  40. [40]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  41. [41]

    B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-Strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  42. [42]

    M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  43. [43]

    B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].

    ADS  Article  Google Scholar 

  44. [44]

    S. Hohenegger and A. Iqbal, M-strings, elliptic genera and \( \mathcal{N} \) = 4 string amplitudes, Fortsch. Phys. 62 (2014) 155 [arXiv:1310.1325] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  45. [45]

    B. Haghighat, G. Lockhart and C. Vafa, Fusing E-strings to heterotic strings: E+E→H, Phys. Rev. D 90 (2014) 126012 [arXiv:1406.0850] [INSPIRE].

    ADS  Article  Google Scholar 

  46. [46]

    K. Hosomichi and S. Lee, Self-dual Strings and 2D SYM, JHEP 01 (2015) 076 [arXiv:1406.1802] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. [47]

    W. Cai, M.-x. Huang and K. Sun, On the Elliptic Genus of Three E-strings and Heterotic Strings, JHEP 01 (2015) 079 [arXiv:1411.2801] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  48. [48]

    J. Kim, S. Kim, K. Lee, J. Park and C. Vafa, Elliptic Genus of E-strings, JHEP 09 (2017) 098 [arXiv:1411.2324] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. [49]

    B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of Minimal 6d SCFTs, Fortsch. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  50. [50]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. [51]

    B. Haghighat, From strings in 6d to strings in 5d, JHEP 01 (2016) 062 [arXiv:1502.06645] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  52. [52]

    S. Hohenegger, A. Iqbal and S.-J. Rey, M-strings, monopole strings, and modular forms, Phys. Rev. D 92 (2015) 066005 [arXiv:1503.06983] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  53. [53]

    J. Kim, S. Kim and K. Lee, Little strings and T-duality, JHEP 02 (2016) 170 [arXiv:1503.07277] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  54. [54]

    M. Honda and Y. Yoshida, Supersymmetric index on T 2 × S2 and elliptic genus, arXiv:1504.04355 [INSPIRE].

  55. [55]

    A. Gadde, B. Haghighat, J. Kim, S. Kim, G. Lockhart and C. Vafa, 6d String Chains, JHEP 02 (2018) 143 [arXiv:1504.04614] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  56. [56]

    B. Haghighat, S. Murthy, C. Vafa and S. Vandoren, F-Theory, Spinning Black Holes and Multi-string Branches, JHEP 01 (2016) 009 [arXiv:1509.00455] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  57. [57]

    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d Dualities and Tao Web Diagrams, JHEP 05 (2019) 203 [arXiv:1509.03300] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  58. [58]

    J. Kim, S. Kim and K. Lee, Higgsing towards E-strings, arXiv:1510.03128 [INSPIRE].

  59. [59]

    H.-C. Kim, S. Kim and J. Park, 6d strings from new chiral gauge theories, arXiv:1608.03919 [INSPIRE].

  60. [60]

    M. Del Zotto and G. Lockhart, On Exceptional Instanton Strings, JHEP 09 (2017) 081 [arXiv:1609.00310] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  61. [61]

    J. Gu, M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, Refined BPS invariants of 6d SCFTs from anomalies and modularity, JHEP 05 (2017) 130 [arXiv:1701.00764] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  62. [62]

    H. Hayashi and K. Ohmori, 5d/ 6d DE instantons from trivalent gluing of web diagrams, JHEP 06 (2017) 078 [arXiv:1702.07263] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  63. [63]

    S.-S. Kim and F. Yagi, Topological vertex formalism with O5-plane, Phys. Rev. D 97 (2018) 026011 [arXiv:1709.01928] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  64. [64]

    B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings and their partition functions, Phys. Rev. D 97 (2018) 106004 [arXiv:1710.02455] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  65. [65]

    M. Del Zotto, J. Gu, M.-X. Huang, A.-K. Kashani-Poor, A. Klemm and G. Lockhart, Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs, JHEP 03 (2018) 156 [arXiv:1712.07017] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  66. [66]

    R.-D. Zhu, An Elliptic Vertex of Awata-Feigin-Shiraishi type for M-strings, JHEP 08 (2018) 050 [arXiv:1712.10255] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  67. [67]

    J. Kim, K. Lee and J. Park, On elliptic genera of 6d string theories, JHEP 10 (2018) 100 [arXiv:1801.01631] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  68. [68]

    H.-C. Kim, J. Kim, S. Kim, K.-H. Lee and J. Park, 6d strings and exceptional instantons, Phys. Rev. D 103 (2021) 025012 [arXiv:1801.03579] [INSPIRE].

    ADS  Article  Google Scholar 

  69. [69]

    M. Del Zotto and G. Lockhart, Universal Features of BPS Strings in Six-dimensional SCFTs, JHEP 08 (2018) 173 [arXiv:1804.09694] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  70. [70]

    Z. Duan, J. Gu and A.-K. Kashani-Poor, Computing the elliptic genus of higher rank E-strings from genus 0 GW invariants, JHEP 03 (2019) 078 [arXiv:1810.01280] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  71. [71]

    J. Gu, B. Haghighat, K. Sun and X. Wang, Blowup Equations for 6d SCFTs. I, JHEP 03 (2019) 002 [arXiv:1811.02577] [INSPIRE].

  72. [72]

    J. Gu, A. Klemm, K. Sun and X. Wang, Elliptic blowup equations for 6d SCFTs. Part II. Exceptional cases, JHEP 12 (2019) 039 [arXiv:1905.00864] [INSPIRE].

  73. [73]

    J. Gu, B. Haghighat, A. Klemm, K. Sun and X. Wang, Elliptic blowup equations for 6d SCFTs. Part III. E-strings, M-strings and chains, JHEP 07 (2020) 135 [arXiv:1911.11724] [INSPIRE].

  74. [74]

    D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, in Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, U.S.A. (1999).

    Google Scholar 

  75. [75]

    A. Klemm and R. Pandharipande, Enumerative geometry of Calabi-Yau 4-folds, Commun. Math. Phys. 281 (2008) 621 [math/0702189] [INSPIRE].

  76. [76]

    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].

    MathSciNet  MATH  Article  Google Scholar 

  77. [77]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  78. [78]

    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, AMS/IP Stud. Adv. Math. 1 (1996) 655 [hep-th/9302103] [INSPIRE].

    MATH  Article  Google Scholar 

  79. [79]

    J.A. Minahan, D. Nemeschansky and N.P. Warner, Instanton expansions for mass deformed N = 4 superYang-Mills theories, Nucl. Phys. B 528 (1998) 109 [hep-th/9710146] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  80. [80]

    J.A. Minahan, D. Nemeschansky and N.P. Warner, Partition functions for BPS states of the noncritical E8 string, Adv. Theor. Math. Phys. 1 (1998) 167 [hep-th/9707149] [INSPIRE].

    MATH  Article  Google Scholar 

  81. [81]

    S. Hosono, M.H. Saito and A. Takahashi, Holomorphic anomaly equation and BPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 3 (1999) 177 [hep-th/9901151] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  82. [82]

    A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].

  83. [83]

    M. Alim and E. Scheidegger, Topological Strings on Elliptic Fibrations, Commun. Num. Theor. Phys. 08 (2014) 729 [arXiv:1205.1784] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  84. [84]

    H.-C. Kim, S.S. Razamat, C. Vafa and G. Zafrir, E-String Theory on Riemann Surfaces, Fortsch. Phys. 66 (2018) 1700074 [arXiv:1709.02496] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  85. [85]

    R. Blumenhagen, G. Honecker and T. Weigand, Loop-corrected compactifications of the heterotic string with line bundles, JHEP 06 (2005) 020 [hep-th/0504232] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  86. [86]

    R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].

  87. [87]

    R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].

  88. [88]

    S.H. Katz, A. Klemm and C. Vafa, M theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999) 1445 [hep-th/9910181] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  89. [89]

    E.-N. Ionel and T.H. Parker, The Gopakumar-Vafa formula for symplectic manifolds, arXiv:1306.1516.

  90. [90]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  91. [91]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  92. [92]

    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, JHEP 01 (2015) 047 [arXiv:1408.6167] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  93. [93]

    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].

    ADS  MATH  Article  Google Scholar 

  94. [94]

    W. Lerche, Fayet-Iliopoulos potentials from four folds, JHEP 11 (1997) 004 [hep-th/9709146] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  95. [95]

    S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. 608 (2001) 477] [hep-th/9906070] [INSPIRE].

  96. [96]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  97. [97]

    T. Weigand, F-theory, PoS TASI2017 (2018) 016 [arXiv:1806.01854] [INSPIRE].

  98. [98]

    M. Cvetič and L. Lin, TASI Lectures on Abelian and Discrete Symmetries in F-theory, PoS TASI2017 (2018) 020 [arXiv:1809.00012] [INSPIRE].

  99. [99]

    R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  100. [100]

    C. Lawrie, S. Schäfer-Nameki and T. Weigand, Chiral 2d theories from N = 4 SYM with varying coupling, JHEP 04 (2017) 111 [arXiv:1612.05640] [INSPIRE].

  101. [101]

    N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  102. [102]

    O.J. Ganor and A. Hanany, Small E8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  103. [103]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  104. [104]

    B. Haghighat, Mirror Symmetry and Modularity, arXiv:1712.00601 [INSPIRE].

  105. [105]

    T. Schimannek, Modularity from Monodromy, JHEP 05 (2019) 024 [arXiv:1902.08215] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  106. [106]

    C.F. Cota, A. Klemm and T. Schimannek, Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities, JHEP 11 (2019) 170 [arXiv:1910.01988] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  107. [107]

    S.-J. Lee, W. Lerche, G. Lockhart and T. Weigand, Holomorphic Anomalies, Fourfolds and Fluxes, arXiv:2012.00766 [INSPIRE].

  108. [108]

    K. Ooguiso, On algebraic fiber spaces structures on a Calabi-Yau 3-fold, Int. Jour. Math. 4 (1993) 439.

    MathSciNet  Article  Google Scholar 

  109. [109]

    A. Gathmann, Gromov-Witten invariants of hypersurfaces, habilitation thesis, TU Kaiserslautern, Germany (2003).

  110. [110]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  111. [111]

    A. Lukas, B.A. Ovrut and D. Waldram, On the four-dimensional effective action of strongly coupled heterotic string theory, Nucl. Phys. B 532 (1998) 43 [hep-th/9710208] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  112. [112]

    A. Lukas, B.A. Ovrut and D. Waldram, The Ten-dimensional effective action of strongly coupled heterotic string theory, Nucl. Phys. B 540 (1999) 230 [hep-th/9801087] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  113. [113]

    L. Carlevaro and J.-P. Derendinger, Five-brane thresholds and membrane instantons in four-dimensional heterotic M-theory, Nucl. Phys. B 736 (2006) 1 [hep-th/0502225] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  114. [114]

    G. Honecker, Massive U(1)s and heterotic five-branes on K 3, Nucl. Phys. B 748 (2006) 126 [hep-th/0602101] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  115. [115]

    L.B. Anderson and W. Taylor, Geometric constraints in dual F-theory and heterotic string compactifications, JHEP 08 (2014) 025 [arXiv:1405.2074] [INSPIRE].

    ADS  Article  Google Scholar 

  116. [116]

    M. Cvetič, A. Grassi, D. Klevers, M. Poretschkin and P. Song, Origin of Abelian Gauge Symmetries in Heterotic/F-theory Duality, JHEP 04 (2016) 041 [arXiv:1511.08208] [INSPIRE].

    ADS  MATH  Google Scholar 

  117. [117]

    M. Cvetič, A. Grassi and M. Poretschkin, Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry, JHEP 06 (2017) 156 [arXiv:1607.03176] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  118. [118]

    L.B. Anderson, H. Feng, X. Gao and M. Karkheiran, Heterotic/Heterotic and Heterotic/F-theory Duality, Phys. Rev. D 100 (2019) 126014 [arXiv:1907.04395] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  119. [119]

    V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  120. [120]

    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  MATH  Article  Google Scholar 

  121. [121]

    D.S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry, JHEP 01 (2012) 141 [arXiv:1110.5916] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  122. [122]

    D.S. Park, Anomaly Equations and Intersection Theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  123. [123]

    W. Taylor, TASI Lectures on Supergravity and String Vacua in Various Dimensions, arXiv:1104.2051 [INSPIRE].

  124. [124]

    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  MATH  Article  Google Scholar 

  125. [125]

    T.W. Grimm and W. Taylor, Structure in 6D and 4D N = 1 supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  126. [126]

    T. Weigand and F. Xu, The Green-Schwarz Mechanism and Geometric Anomaly Relations in 2d (0,2) F-theory Vacua, JHEP 04 (2018) 107 [arXiv:1712.04456] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  127. [127]

    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  Article  Google Scholar 

  128. [128]

    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  MATH  Article  Google Scholar 

  129. [129]

    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  130. [130]

    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, AMS/IP Stud. Adv. Math. 1 (1996) 545 [hep-th/9406055] [INSPIRE].

    MATH  Article  Google Scholar 

  131. [131]

    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].

  132. [132]

    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2., Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].

  133. [133]

    W. Lerche, P. Mayr and N.P. Warner, Noncritical strings, Del Pezzo singularities and Seiberg-Witten curves, Nucl. Phys. B 499 (1997) 125 [hep-th/9612085] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  134. [134]

    J.A. Minahan, D. Nemeschansky and N.P. Warner, Investigating the BPS spectrum of noncritical E(n) strings, Nucl. Phys. B 508 (1997) 64 [hep-th/9705237] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  135. [135]

    D.R. Morrison and C. Vafa, F-theory and \( \mathcal{N} \)N = 1 SCFTs in four dimensions, JHEP 08 (2016) 070 [arXiv:1604.03560] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  136. [136]

    F. Apruzzi, J.J. Heckman, D.R. Morrison and L. Tizzano, 4D Gauge Theories with Conformal Matter, JHEP 09 (2018) 088 [arXiv:1803.00582] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  137. [137]

    D. Gaiotto and S.S. Razamat, \( \mathcal{N} \) = 1 theories of class \( {\mathcal{S}}_k \), JHEP 07 (2015) 073 [arXiv:1503.05159] [INSPIRE].

  138. [138]

    S. Franco, H. Hayashi and A. Uranga, Charting Class \( {\mathcal{S}}_k \) Territory, Phys. Rev. D 92 (2015) 045004 [arXiv:1504.05988] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  139. [139]

    A. Hanany and K. Maruyoshi, Chiral theories of class \( \mathcal{S} \), JHEP 12 (2015) 080 [arXiv:1505.05053] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  140. [140]

    I. Coman, E. Pomoni, M. Taki and F. Yagi, Spectral curves of \( \mathcal{N} \) = 1 theories of class \( {\mathcal{S}}_k \), JHEP 06 (2017) 136 [arXiv:1512.06079] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  141. [141]

    S.S. Razamat, C. Vafa and G. Zafrir, 4d \( \mathcal{N} \) = 1 from 6d (1, 0), JHEP 04 (2017) 064 [arXiv:1610.09178] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  142. [142]

    I. Bah, A. Hanany, K. Maruyoshi, S.S. Razamat, Y. Tachikawa and G. Zafrir, 4d \( \mathcal{N} \) = 1 from 6d \( \mathcal{N} \) = (1, 0) on a torus with fluxes, JHEP 06 (2017) 022 [arXiv:1702.04740] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  143. [143]

    V. Mitev and E. Pomoni, 2D CFT blocks for the 4D class \( {\mathcal{S}}_k \) theories, JHEP 08 (2017) 009 [arXiv:1703.00736] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  144. [144]

    T. Bourton and E. Pomoni, Instanton counting in class \( {\mathcal{S}}_k \), J. Phys. A 53 (2020) 165401 [arXiv:1712.01288] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  145. [145]

    H.-C. Kim, S.S. Razamat, C. Vafa and G. Zafrir, D-type Conformal Matter and SU/USp Quivers, JHEP 06 (2018) 058 [arXiv:1802.00620] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  146. [146]

    S.S. Razamat and G. Zafrir, Compactification of 6d minimal SCFTs on Riemann surfaces, Phys. Rev. D 98 (2018) 066006 [arXiv:1806.09196] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  147. [147]

    S.S. Razamat and E. Sabag, A freely generated ring for \( \mathcal{N} \) = 1 models in class \( {\mathcal{S}}_k \), JHEP 07 (2018) 150 [arXiv:1804.00680] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  148. [148]

    S.S. Razamat and G. Zafrir, N = 1 conformal dualities, JHEP 09 (2019) 046 [arXiv:1906.05088] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  149. [149]

    J. Chen, B. Haghighat, S. Liu and M. Sperling, 4d N = 1 from 6d D-type N = (1, 0), JHEP 01 (2020) 152 [arXiv:1907.00536] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  150. [150]

    S. Pasquetti, S.S. Razamat, M. Sacchi and G. Zafrir, Rank Q E-string on a torus with flux, SciPost Phys. 8 (2020) 014 [arXiv:1908.03278] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  151. [151]

    S.S. Razamat and E. Sabag, Sequences of 6d SCFTs on generic Riemann surfaces, JHEP 01 (2020) 086 [arXiv:1910.03603] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  152. [152]

    O. Sela and G. Zafrir, Symmetry enhancement in 4d Spin(n) gauge theories and compactification from 6d, JHEP 12 (2019) 052 [arXiv:1910.03629] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  153. [153]

    M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics book series vol. 55, Birkhäuser, Boston, MA (1995).

  154. [154]

    T. Kawai, String duality and enumeration of curves by Jacobi forms, in Taniguchi Symposium on Integrable Systems and Algebraic Geometry, pp. 282–314, 4, 1998, hep-th/9804014 [INSPIRE].

  155. [155]

    V. Gritsenko and H. Wang, Graded rings of integral Jacobi forms, J. Number Theor. 214 (2020) 382 [arXiv:1810.09392] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  156. [156]

    M. Kreuzer and H. Skarke, PALP: A Package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun. 157 (2004) 87 [math/0204356] [INSPIRE].

  157. [157]

    A.P. Braun, J. Knapp, E. Scheidegger, H. Skarke and N.-O. Walliser, PALP - a User Manual, in A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger eds. Strings, gauge fields, and the geometry behind : The legacy of Maximilian Kreuzer, (2012), pp. 461–550, https://doi.org/10.1142/9789814412551_0024 [arXiv:1205.4147] [INSPIRE].

  158. [158]

    W. Stein et al., Sage Mathematics Software (Version 8.4), The Sage Development Team (2018), http://www.sagemath.org.

  159. [159]

    M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

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Lee, SJ., Lerche, W., Lockhart, G. et al. Quasi-Jacobi forms, elliptic genera and strings in four dimensions. J. High Energ. Phys. 2021, 162 (2021). https://doi.org/10.1007/JHEP01(2021)162

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Keywords

  • F-Theory
  • String Duality
  • Superstrings and Heterotic Strings
  • Topological Strings