Journal of High Energy Physics

, 2019:261 | Cite as

Modular invariance, tauberian theorems and microcanonical entropy

  • Baur MukhametzhanovEmail author
  • Alexander Zhiboedav
Open Access
Regular Article - Theoretical Physics


We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.


Conformal Field Theory AdS-CFT Correspondence 


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


  1. [1]
    M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    F.J. Yndurain, Absolute bound on cross-sections at all energies and without unknown constants, Phys. Lett. B 31 (1970) 368 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    E. Katz, S. Sachdev, E.S. Sorensen and W. Witczak-Krempa, Conformal field theories at nonzero temperature: operator product expansions, Monte Carlo and holography, Phys. Rev. B 90 (2014) 245109 [arXiv:1409.3841] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    S. Caron-Huot, Asymptotics of thermal spectral functions, Phys. Rev. D 79 (2009) 125009 [arXiv:0903.3958] [INSPIRE].ADSGoogle Scholar
  5. [5]
    M.A. Shifman, Quark hadron duality, in At the frontier of particle physics. Handbook of QCD, World Scientific, Singapore (2001), pg. 1447 [hep-ph/0009131] [INSPIRE].
  6. [6]
    J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    A. Sen, Black hole entropy function, attractors and precision counting of microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. Sen, Logarithmic corrections to Schwarzschild and other non-extremal black hole entropy in different dimensions, JHEP 04 (2013) 156 [arXiv:1205.0971] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  12. [12]
    J. Korevaar, Tauberian theory: a century of development, Springer, Germany (2004).zbMATHCrossRefGoogle Scholar
  13. [13]
    D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].ADSGoogle Scholar
  14. [14]
    J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119 [arXiv:1709.00008] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    B. Mukhametzhanov and A. Zhiboedov, Analytic euclidean bootstrap, arXiv:1808.03212 [INSPIRE].
  16. [16]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    S. Carlip, Logarithmi c corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
  18. [18]
    I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the monster, Pure Appl. Math. 134, Academic Press, Boston, MA, U.S.A. (1988) [INSPIRE].
  19. [19]
    E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
  20. [20]
    N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast conformal bootstrap and constraints on 3d gravity, JHEP 05 (2019) 087 [arXiv:1903.06272] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    S. Collier, Y.-H. Lin and X. Yin, Modular bootstrap revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP 02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
  23. [23]
    A.E. Ingham, A tauberian theorem for partitions, Ann. Math. 42 (1941) 1075.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP 11 (2017) 183 [arXiv:1706.04612] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    M.A. Subhankulov, Tauberian theorems with remainder (in Russian), Izdat. Nauka, Moscow, Russia (1976).Google Scholar
  26. [26]
    A. Cappelli, C. Itzykson and J.B. Zuber, Modular invariant partition functions in two-dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge trajectories and the Virasoro analytic bootstrap, JHEP 05 (2019) 212 [arXiv:1811.05710] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    S. Ganguly and S. Pal, Bounds on density of states and spectral gap in CFT 2 , arXiv:1905.12636 [INSPIRE].
  29. [29]
    S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    J.D. Brown and J.W. York, Jr., The microcanonical functional integral. 1. The gravitational field, Phys. Rev. D 47 (1993) 1420 [gr-qc/9209014] [INSPIRE].
  31. [31]
    D. Marolf, Microcanonical path integrals and the holography of small black hole interiors, JHEP 09 (2018) 114 [arXiv:1808.00394] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    P. DiFrancesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York, NY, U.S.A. (1997) [INSPIRE].CrossRefGoogle Scholar
  33. [33]
    A. Belin, J. de Boer, J. Kruthoff, B. Michel, E. Shaghoulian and M. Shyani, Universality of sparse d > 2 conformal field theory at large N, JHEP 03 (2017) 067 [arXiv:1610.06186] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    S. Detournay, T. Hartman and D.M. Hofman, Warped conformal field theory, Phys. Rev. D 86 (2012) 124018 [arXiv:1210.0539] [INSPIRE].ADSGoogle Scholar
  35. [35]
    W. Song and J. Xu, Structure constants from modularity in warped CFT, arXiv:1903.01346 [INSPIRE].
  36. [36]
    E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    D. Das, S. Datta and S. Pal, Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks, Phys. Rev. D 98 (2018) 101901 [arXiv:1712.01842] [INSPIRE].ADSGoogle Scholar
  40. [40]
    Y. Kusuki, Large c Virasoro blocks from monodromy method beyond known limits, JHEP 08 (2018) 161 [arXiv:1806.04352] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    Y. Kusuki, Light cone bootstrap in general 2D CFTs and entanglement from light cone singularity, JHEP 01 (2019) 025 [arXiv:1810.01335] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.ADSCrossRefGoogle Scholar
  43. [43]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051].ADSCrossRefGoogle Scholar
  44. [44]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].ADSCrossRefGoogle Scholar
  45. [45]
    A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    N. Benjamin, E. Dyer, A.L. Fitzpatrick and Y. Xin, The most irrational rational theories, JHEP 04 (2019) 025 [arXiv:1812.07579] [INSPIRE].
  47. [47]
    A. Sen, Arithmetic of quantum entropy function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    I. Mandal and A. Sen, Black hole microstate counting and its macroscopic counterpart, Nucl. Phys. Proc. Suppl. 216 (2011) 147 [Class. Quant. Grav. 27 (2010) 214003] [arXiv:1008.3801] [INSPIRE].
  49. [49]
    A. Belin, A. Castro, J. Gomes and C.A. Keller, Siegel modular forms and black hole entropy, JHEP 04 (2017) 057 [arXiv:1611.04588] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    P. Kraus and A. Maloney, A Cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    S. Pal, Bound on asymptotics of magnitude of three point coefficients in 2D CFT, arXiv:1906.11223 [INSPIRE].
  52. [52]
    P. Kraus and A. Sivaramakrishnan, Light-stat e dominance from the conformal bootstrap, JHEP 08 (2019) 013 [arXiv:1812.02226] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  53. [53]
    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  54. [54]
    J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    M. Cho, S. Collier and X. Yin, Genus two modular bootstrap, JHEP 04 (2019) 022 [arXiv:1705.05865] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    D. Mazac, Analytic bounds and emergence of AdS 2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [I NSPIRE].
  57. [57]
    D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP 02 (2019) 162 [arXiv:1803.10233] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsHarvard UniversityCambridgeU.S.A.
  2. 2.CERN, Theoretical Physics DepartmentGeneva 23Switzerland

Personalised recommendations