Download PDF

Discreteness and integrality in Conformal Field Theory

Journal of High Energy Physics volume 2021, Article number: 64 (2021) Cite this article

Abstract

Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists \( t\ge \frac{c}{12} \) is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions \( \Delta >\frac{c-1}{12} \) is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS3 gravity.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  2. [2]

    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  3. [3]

    J.D. Qualls and A.D. Shapere, Bounds on operator dimensions in 2D conformal field theories, JHEP 05 (2014) 091 [arXiv:1312.0038] [INSPIRE].

    ADS  Google Scholar 

  4. [4]

    C.A. Keller and A. Maloney, Poincaré series, 3D gravity and CFT spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE].

    ADS  MATH  Google Scholar 

  5. [5]

    S. Collier, Y.-H. Lin and X. Yin, Modular bootstrap revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  6. [6]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  7. [7]

    S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the spectral function: on the uniqueness of Liouville and the universality of BTZ, JHEP 09 (2018) 150 [arXiv:1702.00423] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  8. [8]

    J.-B. Bae, S. Lee and J. Song, Modular constraints on conformal field theories with currents, JHEP 12 (2017) 045 [arXiv:1708.08815] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  9. [9]

    T. Anous, R. Mahajan and E. Shaghoulian, Parity and the modular bootstrap, SciPost Phys. 5 (2018) 022 [arXiv:1803.04938] [INSPIRE].

    ADS  Google Scholar 

  10. [10]

    J.-B. Bae, S. Lee and J. Song, Modular constraints on superconformal field theories, JHEP 01 (2019) 209 [arXiv:1811.00976] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  11. [11]

    N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast conformal bootstrap and constraints on 3d gravity, JHEP 05 (2019) 087 [arXiv:1903.06272] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  12. [12]

    S. Ganguly and S. Pal, Bounds on the density of states and the spectral gap in CFT2 , Phys. Rev. D 101 (2020) 106022 [arXiv:1905.12636] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  13. [13]

    B. Mukhametzhanov and A. Zhiboedov, Modular invariance, Tauberian theorems and microcanonical entropy, JHEP 10 (2019) 261 [arXiv:1904.06359] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  14. [14]

    N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Light-cone modular bootstrap and pure gravity, Phys. Rev. D 100 (2019) 066029 [arXiv:1906.04184] [INSPIRE].

  15. [15]

    T. Hartman, D. Mazáč and L. Rastelli, Sphere packing and quantum gravity, JHEP 12 (2019) 048 [arXiv:1905.01319] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  16. [16]

    F. Gliozzi, Modular bootstrap, elliptic points, and quantum gravity, Phys. Rev. Res. 2 (2020) 013327 [arXiv:1908.00029] [INSPIRE].

  17. [17]

    E.M. Brehm and D. Das, Aspects of the S transformation Bootstrap, J. Stat. Mech. 2005 (2020) 053103 [arXiv:1911.02309] [INSPIRE].

  18. [18]

    S. Collier, A. Maloney, H. Maxfield and I. Tsiares, Universal dynamics of heavy operators in CFT2, JHEP 07 (2020) 074 [arXiv:1912.00222] [INSPIRE].

    ADS  MATH  Google Scholar 

  19. [19]

    B. Mukhametzhanov and S. Pal, Beurling-Selberg extremization and modular bootstrap at high energies, SciPost Phys. 8 (2020) 088 [arXiv:2003.14316] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  20. [20]

    N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Twist gap and global symmetry in two dimensions, Phys. Rev. D 101 (2020) 106026 [arXiv:2003.02844] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  21. [21]

    N. Benjamin and Y.-H. Lin, Lessons from the Ramond sector, SciPost Phys. 9 (2020) 065 [arXiv:2005.02394] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  22. [22]

    S. Pal and Z. Sun, High energy modular bootstrap, global symmetries and defects, JHEP 08 (2020) 064 [arXiv:2004.12557] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  23. [23]

    N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].

    ADS  MATH  Google Scholar 

  24. [24]

    N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat and A. Tajdini, High-dimensional sphere packing and the modular bootstrap, JHEP 12 (2020) 066 [arXiv:2006.02560] [INSPIRE].

  25. [25]

    X. Yin, Aspects of two-dimensional conformal field theories, PoS(TASI2017)003 [INSPIRE].

  26. [26]

    N. Benjamin, S. Collier and A. Maloney, Pure gravity and conical defects, JHEP 09 (2020) 034 [arXiv:2004.14428] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  27. [27]

    S.D. Mathur, S. Mukhi and A. Sen, On the classification of rational conformal field theories, Phys. Lett. B 213 (1988) 303.

    ADS  MathSciNet  Google Scholar 

  28. [28]

    S.D. Mathur, S. Mukhi and A. Sen, Reconstruction of conformal field theories from modular geometry on the torus, Nucl. Phys. B 318 (1989) 483 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  29. [29]

    E.B. Kiritsis, Fuchsian differential equations for characters on the torus: a classification, Nucl. Phys. B 324 (1989) 475 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  30. [30]

    S.G. Naculich, Differential equations for rational conformal characters, Nucl. Phys. B 323 (1989) 423 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  31. [31]

    M.R. Gaberdiel and S. Lang, Modular differential equations for torus one-point functions, J. Phys. A 42 (2009) 045405 [arXiv:0810.0106] [INSPIRE].

  32. [32]

    H.R. Hampapura and S. Mukhi, On 2d conformal field theories with two characters, JHEP 01 (2016) 005 [arXiv:1510.04478] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  33. [33]

    M.R. Gaberdiel, H.R. Hampapura and S. Mukhi, Cosets of meromorphic CFTs and modular differential equations, JHEP 04 (2016) 156 [arXiv:1602.01022] [INSPIRE].

    ADS  MATH  Google Scholar 

  34. [34]

    A.R. Chandra and S. Mukhi, Towards a classification of two-character rational conformal field theories, JHEP 04 (2019) 153 [arXiv:1810.09472] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  35. [35]

    A.R. Chandra and S. Mukhi, Curiosities above c = 24, SciPost Phys. 6 (2019) 053 [arXiv:1812.05109] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  36. [36]

    J.A. Harvey and Y. Wu, Hecke relations in rational conformal field theory, JHEP 09 (2018) 032 [arXiv:1804.06860] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  37. [37]

    S. Mukhi, R. Poddar and P. Singh, Rational CFT with three characters: the quasi-character approach, JHEP 05 (2020) 003 [arXiv:2002.01949] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  38. [38]

    M.C.N. Cheng, T. Gannon and G. Lockhart, Modular exercises for four-point blocks — I, arXiv:2002.11125 [INSPIRE].

  39. [39]

    M. Kaneko and M. Koike, On modular forms arising from a differential equation of hypergeometric type, math/0206022.

  40. [40]

    M. Knopp and G. Mason, Vector-valued modular forms and Poincaré series, Illinois J. Math. 48 (2004) 1345.

    MathSciNet  MATH  Google Scholar 

  41. [41]

    P. Bantay and T. Gannon, Conformal characters and the modular representation, JHEP 02 (2006) 005 [hep-th/0512011] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  42. [42]

    P. Bantay and T. Gannon, Vector-valued modular functions for the modular group and the hypergeometric equation, arXiv:0705.2467.

  43. [43]

    C. Marks and G. Mason, Structure of the module of vector-valued modular forms, J. London Math. Soc. 82 (2010) 32.

    MathSciNet  MATH  Google Scholar 

  44. [44]

    C. Marks, Fourier coefficients of three-dimensional vector-valued modular forms, arXiv:1201.5165.

  45. [45]

    C. Marks, Irreducible vector-valued modular forms of dimension less than six, arXiv:1004.3019 [INSPIRE].

  46. [46]

    M. Kaneko, K. Nagatomo and Y. Sakai, Modular forms and second order ordinary differential equations: applications to vertex operator algebras, Lett. Math. Phys. 103 (2013) 439 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  47. [47]

    C. Franc and G. Mason, Fourier coefficients of vector-valued modular forms of dimension 2, Canadian Math. Bull. 57 (2014) 485.

    MathSciNet  MATH  Google Scholar 

  48. [48]

    C. Franc and G. Mason, Hypergeometric series, modular linear differential equations and vector-valued modular forms, Ramanujan J. 41 (2016) 233.

    MathSciNet  MATH  Google Scholar 

  49. [49]

    M. Kaneko, K. Nagatomo and Y. Sakai, The third order modular linear differential equations, J. Algerba 485 (2017) 332.

    MathSciNet  MATH  Google Scholar 

  50. [50]

    C. Franc and G. Mason, Constructions of vector-valued modular forms of rank four and level one, arXiv:1810.09408.

  51. [51]

    G. Mason, K. Nagatomo and Y. Sakai, Vertex operator algebras with two simple modules — The Mathur-Mukhi-Sen theorem revisited, arXiv:1803.11281.

  52. [52]

    C. Franc and G. Mason, Classification of some vertex operator algebras of rank 3, Alg. Numb. Theor. 14 (2020) 1613.

    MathSciNet  MATH  Google Scholar 

  53. [53]

    T. Gannon, The theory of vector-valued modular forms for the modular group, in Conformal field theory, automorphic forms and related topics, W. Kohnen and R. Weissauer eds., Springer, Germany (2014).

  54. [54]

    S. Mukhi, Classification of RCFT from holomorphic modular bootstrap: a status report, arXiv:1910.02973 [INSPIRE].

  55. [55]

    J. . Cardy, Continuously varying exponents and the value of the central charge, J. Phys. A 20 (1987) L891.

  56. [56]

    Z. Komargodski and D. Simmons-Duffin, The random-bond Ising model in 2.01 and 3 dimensions, J. Phys. A 50 (2017) 154001 [arXiv:1603.04444] [INSPIRE].

  57. [57]

    V. Bashmakov, M. Bertolini and H. Raj, On non-supersymmetric conformal manifolds: field theory and holography, JHEP 11 (2017) 167 [arXiv:1709.01749] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  58. [58]

    C. Behan, Conformal manifolds: ODEs from OPEs, JHEP 03 (2018) 127 [arXiv:1709.03967] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  59. [59]

    K. Sen and Y. Tachikawa, First-order conformal perturbation theory by marginal operators, arXiv:1711.05947 [INSPIRE].

  60. [60]

    M.R. Gaberdiel and R. Gopakumar, An AdS3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].

  61. [61]

    A. Castro, M.R. Gaberdiel, T. Hartman, A. Maloney and R. Volpato, The gravity dual of the Ising model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].

  62. [62]

    C.-M. Jian, A.W.W. Ludwig, Z.-X. Luo, H.-Y. Sun and Z. Wang, Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions, JHEP 10 (2020) 129 [arXiv:1907.06656] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  63. [63]

    A. Maloney and E. Witten, Averaging over Narain moduli space, JHEP 10 (2020) 187 [arXiv:2006.04855] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  64. [64]

    A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].

  65. [65]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  66. [66]

    T.M. Apostol, Modular functions and Dirichlet series in number theory, Springer, Germany (2012).

    Google Scholar 

  67. [67]

    K. Conrad, SL2(ℤ), https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf.

  68. [68]

    A.O.L. Atkin and H.P.F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics 19 (1971) 1.

    MathSciNet  MATH  Google Scholar 

  69. [69]

    G. Mason, On the Fourier coefficients of 2-dimensional vector-valued modular forms, arXiv:1009.0781.

  70. [70]

    C. Franc and G. Mason, Three-dimensional imprimitive representations of the modular group and their associated modular forms, J. Numb. Theor. 160 (2016) 186.

    MathSciNet  MATH  Google Scholar 

  71. [71]

    F. Diamond and J. Shurman, A first course in modular forms, Springer, Germany (2005).

    MATH  Google Scholar 

  72. [72]

    D. Schultz, Notes on modular forms, https://faculty.math.illinois.edu/∼schult25/ModFormNotes.pdf.

  73. [73]

    M. Knopp and G. Mason, Logarithmic vector-valued modular forms, arXiv:0910.3976.

  74. [74]

    P. Bantay, The dimension of spaces of vector-valued modular forms of integer weight, Lett. Math. Phys. 103 (2013) 1243.

    ADS  MathSciNet  MATH  Google Scholar 

  75. [75]

    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer, Germany (1997) [INSPIRE].

  76. [76]

    M. Headrick, A. Maloney, E. Perlmutter and I.G. Zadeh, Rényi entropies, the analytic bootstrap, and 3D quantum gravity at higher genus, JHEP 07 (2015) 059 [arXiv:1503.07111] [INSPIRE].

    ADS  MATH  Google Scholar 

  77. [77]

    J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  78. [78]

    R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  79. [79]

    N. Afkhami-Jeddi, K. Colville, T. Hartman, A. Maloney and E. Perlmutter, Constraints on higher spin CFT2, JHEP 05 (2018) 092 [arXiv:1707.07717] [INSPIRE].

    ADS  MATH  Google Scholar 

  80. [80]

    L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  81. [81]

    A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  82. [82]

    A. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.

    Google Scholar 

  83. [83]

    K.B. Alkalaev, R.V. Geiko and V.A. Rappoport, Various semiclassical limits of torus conformal blocks, JHEP 04 (2017) 070 [arXiv:1612.05891] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  84. [84]

    C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].

  85. [85]

    C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, JHEP 08 (2018) 114 [arXiv:1707.07679] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  86. [86]

    A. Arabi Ardehali, High-temperature asymptotics of supersymmetric partition functions, JHEP 07 (2016) 025 [arXiv:1512.03376] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  87. [87]

    C. Beem, encrypted communication.

  88. [88]

    L. Bianchi and M. Lemos, Superconformal surfaces in four dimensions, JHEP 06 (2020) 056 [arXiv:1911.05082] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  89. [89]

    I. Runkel and G.M.T. Watts, A nonrational CFT with c = 1 as a limit of minimal models, JHEP 09 (2001) 006 [hep-th/0107118] [INSPIRE].

    ADS  Google Scholar 

  90. [90]

    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].

  91. [91]

    N. Benjamin, E. Dyer, A.L. Fitzpatrick and Y. Xin, The most irrational rational theories, JHEP 04 (2019) 025 [arXiv:1812.07579] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  92. [92]

    F. Brown, A class of non-holomorphic modular forms III: real analytic cusp forms for SL(2, ℤ), arXiv:1710.07912.

  93. [93]

    E. D’Hoker and J. Kaidi, Modular graph functions and odd cuspidal functions. Fourier and Poincaré series, JHEP 04 (2019) 136 [arXiv:1902.04180] [INSPIRE].

  94. [94]

    E. D’Hoker, M.B. Green, O. Gürdogan and P. Vanhove, Modular graph functions, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  95. [95]

    E. D’Hoker and M.B. Green, Identities between modular graph forms, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  96. [96]

    E. D’Hoker and J. Kaidi, Hierarchy of modular graph identities, JHEP 11 (2016) 051 [arXiv:1608.04393] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  97. [97]

    E. D’Hoker and W. Duke, Fourier series of modular graph functions, J. Number Theor. 192 (2018) 1 [arXiv:1708.07998] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  98. [98]

    J.E. Gerken and J. Kaidi, Holomorphic subgraph reduction of higher-point modular graph forms, JHEP 01 (2019) 131 [arXiv:1809.05122] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  99. [99]

    J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings, JHEP 01 (2019) 052 [arXiv:1811.02548] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  100. [100]

    J.E. Gerken, A. Kleinschmidt and O. Schlotterer, All-order differential equations for one-loop closed-string integrals and modular graph forms, JHEP 01 (2020) 064 [arXiv:1911.03476] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  101. [101]

    S. Datta, P. Kraus and B. Michel, Typicality and thermality in 2d CFT, JHEP 07 (2019) 143 [arXiv:1904.00668] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  102. [102]

    E. Dyer and G. Gur-Ari, 2D CFT partition functions at late times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  103. [103]

    A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS3/CFT2, JHEP 05 (2016) 109 [arXiv:1603.08925] [INSPIRE].

    ADS  Google Scholar 

  104. [104]

    L.F. Alday and J.-B. Bae, Rademacher expansions and the spectrum of 2d CFT, JHEP 11 (2020) 134 [arXiv:2001.00022] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  105. [105]

    G.W. Moore and N. Seiberg, Naturality in conformal field theory, Nucl. Phys. B 313 (1989) 16 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  106. [106]

    N.J. Iles and G.M.T. Watts, Modular properties of characters of the W3 algebra, JHEP 01 (2016) 089 [arXiv:1411.4039] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  107. [107]

    P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].

  108. [108]

    J. Cotler and K. Jensen, AdS3 gravity and random CFT, arXiv:2006.08648 [INSPIRE].

  109. [109]

    A. Pérez and R. Troncoso, Gravitational dual of averaged free CFT’s over the Narain lattice, JHEP 11 (2020) 015 [arXiv:2006.08216] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  110. [110]

    A. Belin and J. de Boer, Random statistics of OPE coefficients and Euclidean wormholes, arXiv:2006.05499 [INSPIRE].

  111. [111]

    J. Cotler and K. Jensen, AdS3 wormholes from a modular bootstrap, JHEP 11 (2020) 058 [arXiv:2007.15653] [INSPIRE].

    ADS  MATH  Google Scholar 

  112. [112]

    A. Maloney, Caltech hep-th informal discussion.

  113. [113]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  114. [114]

    H. Maxfield and G.J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01 (2021) 118 [arXiv:2006.11317] [INSPIRE].

    ADS  MATH  Google Scholar 

  115. [115]

    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].

    ADS  Google Scholar 

  116. [116]

    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  117. [117]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  118. [118]

    R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A black hole Farey tail, hep-th/0005003 [INSPIRE].

  119. [119]

    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  120. [120]

    J. de Boer, F. Denef, S. El-Showk, I. Messamah and D. Van den Bleeken, Black hole bound states in AdS3 × S2, JHEP 11 (2008) 050 [arXiv:0802.2257] [INSPIRE].

  121. [121]

    I. Bena, B.D. Chowdhury, J. de Boer, S. El-Showk and M. Shigemori, Moulting black holes, JHEP 03 (2012) 094 [arXiv:1108.0411] [INSPIRE].

  122. [122]

    A. Coste and T. Gannon, Congruence subgroups and rational conformal field theory, math/9909080 [INSPIRE].

  123. [123]

    A. Maloney, H. Maxfield and G.S. Ng, A conformal block Farey tail, JHEP 06 (2017) 117 [arXiv:1609.02165] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  124. [124]

    H. Iwaniec, Spectral methods of automorphic forms, Graduate studies in mathematics, American Mathematical Society, U.S.A. (2002).

Download references

Author information

Affiliations

  1. Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY, 11794, USA

    Justin Kaidi

  2. Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, UCLA, Los Angeles, CA, 90095, USA

    Justin Kaidi

  3. Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, CA, 91125, USA

    Eric Perlmutter

Authors
  1. Justin Kaidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric Perlmutter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Perlmutter.

Additional information

Publisher’s Note

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

ArXiv ePrint: 2008.02190

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

Kaidi, J., Perlmutter, E. Discreteness and integrality in Conformal Field Theory. J. High Energ. Phys. 2021, 64 (2021). https://doi.org/10.1007/JHEP02(2021)064

Download citation

Keywords

  • Conformal Field Theory
  • AdS-CFT Correspondence