Advertisement

Journal of High Energy Physics

, 2019:209 | Cite as

Modular constraints on superconformal field theories

  • Jin-Beom BaeEmail author
  • Sungjay Lee
  • Jaewon Song
Open Access
Regular Article - Theoretical Physics
  • 17 Downloads

Abstract

We constrain the spectrum of \( \mathcal{N} \) = (1, 1) and \( \mathcal{N} \) = (2, 2) superconformal field theories in two-dimensions by requiring the NS-NS sector partition function to be invariant under the Γθ congruence subgroup of the full modular group SL(2, ℤ). We employ semi-definite programming to find constraints on the allowed spectrum of operators with or without U(1) charges. Especially, the upper bounds on the twist gap for the noncurrent primaries exhibit interesting peaks, kinks, and plateau. We identify a number of candidate rational (S)CFTs realized at the numerical boundaries and find that they are realized as the solutions to modular differential equations associated to Γθ. Some of the candidate theories have been discussed by Höhn in the context of self-dual extremal vertex operator (super)algebra. We also obtain bounds for the charged operators and study their implications to the weak gravity conjecture in AdS3.

Keywords

Conformal and W Symmetry Field Theories in Lower Dimensions Global Symmetries 

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]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J.-B. Bae, S. Lee and J. Song, Modular constraints on conformal field theories with currents, JHEP 12 (2017) 045 [arXiv:1708.08815] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. Collier, Y.-H. Lin and X. Yin, Modular bootstrap revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    C.A. Keller and H. Ooguri, Modular constraints on Calabi-Yau compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J.D. Qualls and A.D. Shapere, Bounds on operator dimensions in 2D conformal field theories, JHEP 05 (2014) 091 [arXiv:1312.0038] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    J.D. Qualls, Universal bounds on operator dimensions in general 2D conformal field theories, arXiv:1508.00548 [INSPIRE].
  11. [11]
    N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, Universal bounds on charged states in 2d CFT and 3d gravity, JHEP 08 (2016) 041 [arXiv:1603.09745] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L. Apolo, Bounds on CFTs with \( \mathcal{W} \) 3 algebras and AdS 3 higher spin theories, Phys. Rev. D 96 (2017) 086003 [arXiv:1705.10402] [INSPIRE].ADSMathSciNetGoogle Scholar
  13. [13]
    N. Afkhami-Jeddi et al., Constraints on higher spin CFT 2, JHEP 05 (2018) 092 [arXiv:1707.07717] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    E. Dyer, A.L. Fitzpatrick and Y. Xin, Constraints on flavored 2d CFT partition functions, JHEP 02 (2018) 148 [arXiv:1709.01533] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    T. Anous, R. Mahajan and E. Shaghoulian, Parity and the modular bootstrap, SciPost Phys. 5 (2018) 022 [arXiv:1803.04938] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M.R. Gaberdiel et al., Extremal N = (2, 2) 2D conformal field theories and constraints of modularity, Commun. Num. Theor. Phys. 2 (2008) 743 [arXiv:0805.4216] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S. Odake, Character formulas of an extended superconformal algebra relevant to string compactification, Int. J. Mod. Phys. A 5 (1990) 897 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    S. Odake, C = 3 − d conformal algebra with extended supersymmetry, Mod. Phys. Lett. A 5 (1990) 561 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. El-Showk and M.F. Paulos, Bootstrapping conformal field theories with the extremal functional method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    S.D. Mathur, S. Mukhi and A. Sen, On the classification of rational conformal field theories, Phys. Lett. B 213 (1988) 303 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    M.R. Gaberdiel and C.A. Keller, Modular differential equations and null vectors, JHEP 09 (2008) 079 [arXiv:0804.0489] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    P. Bantay, Modular differential equations for characters of RCFT, JHEP 06 (2010) 021 [arXiv:1004.2579] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    H.R. Hampapura and S. Mukhi, On 2d conformal field theories with two characters, JHEP 01 (2016) 005 [arXiv:1510.04478] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    H.R. Hampapura and S. Mukhi, Two-dimensional RCFT’s without Kac-Moody symmetry, JHEP 07 (2016) 138 [arXiv:1605.03314] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    S. Mukhi and G. Muralidhara, Universal RCFT correlators from the holomorphic bootstrap, JHEP 02 (2018) 028 [arXiv:1708.06772] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A.R. Chandra and S. Mukhi, Towards a classification of two-character rational conformal field theories, arXiv:1810.09472 [INSPIRE].
  27. [27]
    J.A. Harvey and Y. Wu, Hecke relations in rational conformal field theory, JHEP 09 (2018) 032 [arXiv:1804.06860] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, arXiv:1610.05865 [INSPIRE].
  29. [29]
    C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches and modular differential equations, JHEP 08 (2018) 114 [arXiv:1707.07679] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Buican and Z. Laczko, Nonunitary Lagrangians and unitary non-Lagrangian conformal field theories, Phys. Rev. Lett. 120 (2018) 081601 [arXiv:1711.09949] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    G. Hoehn, Selbstduale vertexoperatorsuperalgebren und das babymonster (self-dual vertex operator super algebras and the baby monster), Bonner Math. Schr. 286 (1996) 1 [arXiv:0706.0236].ADSGoogle Scholar
  32. [32]
    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
  33. [33]
    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
  34. [34]
    M. Montero, G. Shiu and P. Soler, The weak gravity conjecture in three dimensions, JHEP 10 (2016) 159 [arXiv:1606.08438] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    W. Boucher, D. Friedan and A. Kent, Determinant formulae and unitarity for the N = 2 superconformal algebras in two-dimensions or exact results on string compactification, Phys. Lett. B 172 (1986) 316 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    I. Antoniadis, C. Bachas, C. Kounnas and P. Windey, Supersymmetry Among free fermions and superstrings, Phys. Lett. B 171 (1986) 51 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    P. Windey, Super Kac-Moody algebras and supersymmetric two-dimensional free fermions, Commun. Math. Phys. 105 (1986) 511 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
  42. [42]
    S. Mukhi. private communication.Google Scholar
  43. [43]
    C. Vafa, The string landscape and the swampland, hep-th/0509212 [INSPIRE].
  44. [44]
    H. Ooguri and C. Vafa, On the geometry of the string landscape and the swampland, Nucl. Phys. B 766 (2007) 21 [hep-th/0605264] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2019

Authors and Affiliations

  1. 1.Korea Institute for Advanced StudySeoulKorea

Personalised recommendations