Bootstrapping ADE M-strings

Abstract

We study elliptic genera of ADE-type M-strings in 6d (2,0) SCFTs from their modularity and explore the relation to topological string partition functions. We find a novel kinematical constraint that elliptic genera should follow, which determines elliptic genera at low base degrees and helps us to conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of related geometries. Using this, we can bootstrap the elliptic genera to arbitrary base degree, including D/E-type theories for which explicit formulas are only partially known. We utilize our results to obtain the 6d Cardy formulas and the superconformal indices for (2,0) theories.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, Prog. Math. 319 (2016) 155 [arXiv:1306.4320] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    E. Witten, Some comments on string dynamics, in Strings ′95: future perspectives in string theory, (1995), pg. 501 [hep-th/9507121] [INSPIRE].

  6. [6]

    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 

  7. [7]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    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].

  9. [9]

    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].

  10. [10]

    J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].

  11. [11]

    J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  12. [12]

    S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  16. [16]

    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    M.-X. Huang, A.-K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid N = 2 theories, Annales Henri Poincaré 14 (2013) 425 [arXiv:1109.5728] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  19. [19]

    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 

  20. [20]

    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].

    ADS  Article  Google Scholar 

  21. [21]

    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 

  22. [22]

    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. [23]

    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. [24]

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

  25. [25]

    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 

  26. [26]

    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 

  27. [27]

    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 

  28. [28]

    J. Kim, S. Kim and K. Lee, Higgsing towards E-strings, JHEP 01 (2021) 110 [arXiv:1510.03128] [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

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

  30. [30]

    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].

  31. [31]

    S.-S. Kim, M. Taki and F. Yagi, Tao probing the end of the world, PTEP 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE].

  32. [32]

    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 

  33. [33]

    O. Foda and R.-D. Zhu, An elliptic topological vertex, J. Phys. A 51 (2018) 465401 [arXiv:1805.12073] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  34. [34]

    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 

  35. [35]

    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 

  36. [36]

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

  37. [37]

    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].

  38. [38]

    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].

  39. [39]

    J. Gu, B. Haghighat, A. Klemm, K. Sun and X. Wang, Elliptic blowup equations for 6d SCFTs. Part IV. Matters, arXiv:2006.03030 [INSPIRE].

  40. [40]

    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].

    ADS  MATH  Google Scholar 

  41. [41]

    M. Del Zotto and G. Lockhart, On exceptional instanton strings, JHEP 09 (2017) 081 [arXiv:1609.00310] [INSPIRE].

  42. [42]

    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 

  43. [43]

    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].

  44. [44]

    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 

  45. [45]

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

  46. [46]

    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 

  47. [47]

    B. Haghighat, W. Yan and S.-T. Yau, ADE string chains and mirror symmetry, JHEP 01 (2018) 043 [arXiv:1705.05199] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  48. [48]

    H. Shimizu and Y. Tachikawa, Anomaly of strings of 6d N = (1, 0) theories, JHEP 11 (2016) 165 [arXiv:1608.05894] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. [49]

    N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir energy and the anomaly polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  50. [50]

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

  51. [51]

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

  52. [52]

    R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, AMS/IP Stud. Adv. Math. 23 (2001) 45 [Adv. Theor. Math. Phys. 3 (1999) 1415] [hep-th/9811131] [INSPIRE].

  53. [53]

    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 

  54. [54]

    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  55. [55]

    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  56. [56]

    M. Aganagic, V. Bouchard and A. Klemm, Topological strings and (almost) modular forms, Commun. Math. Phys. 277 (2008) 771 [hep-th/0607100] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  57. [57]

    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 

  58. [58]

    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 

  59. [59]

    H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [arXiv:1110.2175] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  60. [60]

    M. Bullimore, H.-C. Kim and P. Koroteev, Defects and quantum Seiberg-Witten geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  61. [61]

    C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [Addendum ibid. 04 (2016) 094] [arXiv:1406.6793] [INSPIRE].

  62. [62]

    Y. Hwang, J. Kim and S. Kim, M5-branes, orientifolds, and S-duality, JHEP 12 (2016) 148 [arXiv:1607.08557] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  63. [63]

    S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].

  64. [64]

    J. Kim, S. Kim and J. Song, A 4d N = 1 Cardy formula, JHEP 01 (2021) 025 [arXiv:1904.03455] [INSPIRE].

    ADS  Article  Google Scholar 

  65. [65]

    J. Nahmgoong, 6d superconformal Cardy formulas, arXiv:1907.12582 [INSPIRE].

  66. [66]

    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 

  67. [67]

    R. Pandharipande and R.P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009) 407 [arXiv:0707.2348] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  68. [68]

    S.H. Katz, Gromov-Witten invariants via algebraic geometry, Nucl. Phys. B Proc. Suppl. 46 (1996) 108 [hep-th/9510218] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  69. [69]

    J. Choi, S. Katz and A. Klemm, The refined BPS index from stable pair invariants, Commun. Math. Phys. 328 (2014) 903 [arXiv:1210.4403] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  70. [70]

    K. Lee and J. Nahmgoong, Cardy limits of 6d superconformal theories, arXiv:2006.10294 [INSPIRE].

  71. [71]

    S. Choi, J. Kim, S. Kim and J. Nahmgoong, Comments on deconfinement in AdS/CFT, arXiv:1811.08646 [INSPIRE].

  72. [72]

    S. Kim and J. Nahmgoong, Asymptotic M5-brane entropy from S-duality, JHEP 12 (2017) 120 [arXiv:1702.04058] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  73. [73]

    C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].

  74. [74]

    S. Kim and K. Lee, Indices for 6 dimensional superconformal field theories, J. Phys. A 50 (2017) 443017 [arXiv:1608.02969] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  75. [75]

    H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  76. [76]

    J. Kallen, J.A. Minahan, A. Nedelin and M. Zabzine, N3-behavior from 5D Yang-Mills theory, JHEP 10 (2012) 184 [arXiv:1207.3763] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  77. [77]

    H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].

  78. [78]

    G. Lockhart and C. Vafa, Superconformal partition functions and non-perturbative topological strings, JHEP 10 (2018) 051 [arXiv:1210.5909] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  79. [79]

    H.-C. Kim and K. Lee, Supersymmetric M5 brane theories on R × CP2, JHEP 07 (2013) 072 [arXiv:1210.0853] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  80. [80]

    H.-C. Kim, S. Kim, S.-S. Kim and K. Lee, The general M5-brane superconformal index, arXiv:1307.7660 [INSPIRE].

  81. [81]

    J.A. Minahan, A. Nedelin and M. Zabzine, 5D super Yang-Mills theory and the correspondence to AdS7/CFT6, J. Phys. A 46 (2013) 355401 [arXiv:1304.1016] [INSPIRE].

  82. [82]

    C.-M. Chang, M. Fluder, Y.-H. Lin and Y. Wang, Proving the 6d Cardy formula and matching global gravitational anomalies, arXiv:1910.10151 [INSPIRE].

  83. [83]

    N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir energy and the anomaly polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  84. [84]

    M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y.I. Manin, Construction of instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  85. [85]

    J. Kim, S.-S. Kim, K.-H. Lee, K. Lee and J. Song, Instantons from blow-up, JHEP 11 (2019) 092 [Erratum ibid. 06 (2020) 124] [arXiv:1908.11276] [INSPIRE].

  86. [86]

    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 

  87. [87]

    J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, The 1-2-3 of modular forms, Springer, Berlin, Heidelberg, Germany (2008).

  88. [88]

    J.-P. Serre, A course in arithmetic, Springer, New York, NY, U.S.A. (1973).

    Google Scholar 

  89. [89]

    M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, MA, U.S.A. (1985).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zhihao Duan.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2009.03626

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Duan, Z., Nahmgoong, J. Bootstrapping ADE M-strings. J. High Energ. Phys. 2021, 57 (2021). https://doi.org/10.1007/JHEP02(2021)057

Download citation

Keywords

  • Conformal Field Models in String Theory
  • Topological Strings
  • F-Theory