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The most irrational rational theories

  • Nathan BenjaminEmail author
  • Ethan Dyer
  • A. Liam Fitzpatrick
  • Yuan Xin
Open Access
Regular Article - Theoretical Physics
  • 28 Downloads

Abstract

We propose a two-parameter family of modular invariant partition functions of two-dimensional conformal field theories (CFTs) holographically dual to pure three-dimensional gravity in anti de Sitter space. Our two parameters control the central charge, and the representation of SL(2, ℤ). At large central charge, the partition function has a gap to the first nontrivial primary state of \( \frac{c}{24} \). As the SL(2, ℤ) representation dimension gets large, the partition function exhibits some of the qualitative features of an irrational CFT. This, for instance, is captured in the behavior of the spectral form factor. As part of these analyses, we find similar behavior in the minimal model spectral form factor as c approaches 1.

Keywords

AdS-CFT Correspondence Conformal and W Symmetry Conformal Field Theory Field Theories in Lower Dimensions 

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].
  2. [2]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
  5. [5]
    I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Elsevier Science (1989).Google Scholar
  6. [6]
    M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    D. Gaiotto, Monster symmetry and Extremal CFTs, JHEP 11 (2012) 149 [arXiv:0801.0988] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M.R. Gaberdiel and C.A. Keller, Modular differential equations and null vectors, JHEP 09 (2008) 079 [arXiv:0804.0489] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M.R. Gaberdiel, S. Gukov, C.A. Keller, G.W. Moore and H. Ooguri, Extremal N = (2, 2) 2D Conformal Field Theories and Constraints of Modularity, Commun. Num. Theor. Phys. 2 (2008) 743 [arXiv:0805.4216] [INSPIRE].
  10. [10]
    N. Benjamin, E. Dyer, A.L. Fitzpatrick, A. Maloney and E. Perlmutter, Small Black Holes and Near-Extremal CFTs, JHEP 08 (2016) 023 [arXiv:1603.08524] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
  13. [13]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
  15. [15]
    S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
  17. [17]
    S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    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
  19. [19]
    E. Dyer, A.L. Fitzpatrick and Y. Xin, Constraints on Flavored 2d CFT Partition Functions, JHEP 02 (2018) 148 [arXiv:1709.01533] [INSPIRE].
  20. [20]
    P. Bantay and T. Gannon, Conformal characters and the modular representation, JHEP 02 (2006) 005 [hep-th/0512011] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    V. Balasubramanian, B. Craps, B. Czech and G. Sárosi, Echoes of chaos from string theory black holes, JHEP 03 (2017) 154 [arXiv:1612.04334] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Dyer and G. Gur-Ari, 2D CFT Partition Functions at Late Times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].
  23. [23]
    A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
  24. [24]
    J. Cardy, Thermalization and Revivals after a Quantum Quench in Conformal Field Theory, Phys. Rev. Lett. 112 (2014) 220401 [arXiv:1403.3040] [INSPIRE].
  25. [25]
    L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 239 [arXiv:1509.06411] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    P. Bantay and T. Gannon, Vector-valued modular functions for the modular group and the hypergeometric equation, Commun. Num. Theor. Phys. 1 (2007) 651 [arXiv:0705.2467].
  27. [27]
    P. Bantay, The Dimension of Spaces of Vector-Valued Modular Forms of Integer Weight, Lett. Math. Phys. 103 (2013) 1243 [arXiv:1104.1278].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
  29. [29]
    C.A. Keller and A. Maloney, Poincaré Series, 3D Gravity and CFT Spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE].
  30. [30]
    S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
  31. [31]
    A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    H. Chen, C. Hussong, J. Kaplan and D. Li, A Numerical Approach to Virasoro Blocks and the Information Paradox, JHEP 09 (2017) 102 [arXiv:1703.09727] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    I. Esterlis, A.L. Fitzpatrick and D. Ramirez, Closure of the Operator Product Expansion in the Non-Unitary Bootstrap, JHEP 11 (2016) 030 [arXiv:1606.07458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    J.A. Harvey and Y. Wu, Hecke Relations in Rational Conformal Field Theory, JHEP 09 (2018) 032 [arXiv:1804.06860] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, Modular invariance and uniqueness of \( T\overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
  36. [36]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Stanford Institute for Theoretical PhysicsStanfordU.S.A.
  2. 2.Physics DepartmentBoston UniversityBostonU.S.A.
  3. 3.Princeton Center for Theoretical SciencePrinceton UniversityPrincetonU.S.A.
  4. 4.Department of Physics and AstronomyJohns Hopkins UniversityBaltimoreU.S.A.
  5. 5.GoogleMountain ViewU.S.A.

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