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Fast conformal bootstrap and constraints on 3d gravity

  • Nima Afkhami-JeddiEmail author
  • Thomas Hartman
  • Amirhossein Tajdini
Open Access
Regular Article - Theoretical Physics
  • 13 Downloads

Abstract

The crossing equations of a conformal field theory can be systematically truncated to a finite, closed system of polynomial equations. In certain cases, solutions of the truncated equations place strict bounds on the space of all unitary CFTs. We describe the conditions under which this holds, and use the results to develop a fast algorithm for modular bootstrap in 2d CFT. We then apply it to compute spectral gaps to very high precision, find scaling dimensions for over a thousand operators, and extend the numerical bootstrap to the regime of large central charge, relevant to holography. This leads to new bounds on the spectrum of black holes in three-dimensional gravity. We provide numerical evidence that the asymptotic bound on the spectral gap from spinless modular bootstrap, at large central charge c, is Δ1c/9.1.

Keywords

Conformal Field Theory AdS-CFT Correspondence Models of Quantum Gravity 

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]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333.Google Scholar
  2. [2]
    D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques and Applications, Rev. Mod. Phys. 91 (2019) 15002 [arXiv:1805.04405] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
  4. [4]
    D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
  6. [6]
    S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
  7. [7]
    D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
  8. [8]
    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
  9. [9]
    D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) Models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
  11. [11]
    S.M. Chester and S.S. Pufu, Towards bootstrapping QED 3, JHEP 08 (2016) 019 [arXiv:1601.03476] [INSPIRE].
  12. [12]
    L. Iliesiu, M. Koloğlu, R. Mahajan, E. Perlmutter and D. Simmons-Duffin, The Conformal Bootstrap at Finite Temperature, JHEP 10 (2018) 070 [arXiv:1802.10266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L.V. Delacrétaz, T. Hartman, S.A. Hartnoll and A. Lewkowycz, Thermalization, Viscosity and the Averaged Null Energy Condition, JHEP 10 (2018) 028 [arXiv:1805.04194] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, JHEP 03 (2018) 148 [arXiv:1605.08087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
  18. [18]
    S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean Bootstrap, arXiv:1808.03212 [INSPIRE].
  23. [23]
    F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    F. Gliozzi and A. Rago, Critical exponents of the 3d Ising and related models from Conformal Bootstrap, JHEP 10 (2014) 042 [arXiv:1403.6003] [INSPIRE].
  25. [25]
    W. Li, New method for the conformal bootstrap with OPE truncations, arXiv:1711.09075 [INSPIRE].
  26. [26]
    D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
  28. [28]
    A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  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]
    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].
  31. [31]
    J.D. Qualls and A.D. Shapere, Bounds on Operator Dimensions in 2D Conformal Field Theories, JHEP 05 (2014) 091 [arXiv:1312.0038] [INSPIRE].
  32. [32]
    S. Hellerman and C. Schmidt-Colinet, Bounds for State Degeneracies in 2D Conformal Field Theory, JHEP 08 (2011) 127 [arXiv:1007.0756] [INSPIRE].
  33. [33]
    C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    C.-M. Chang and Y.-H. Lin, Bootstrapping 2D CFTs in the Semiclassical Limit, JHEP 08 (2016) 056 [arXiv:1510.02464] [INSPIRE].
  35. [35]
    E. Shaghoulian, Modular forms and a generalized Cardy formula in higher dimensions, Phys. Rev. D 93 (2016) 126005 [arXiv:1508.02728] [INSPIRE].
  36. [36]
    H. Kim, P. Kravchuk and H. Ooguri, Reflections on Conformal Spectra, JHEP 04 (2016) 184 [arXiv:1510.08772] [INSPIRE].ADSGoogle Scholar
  37. [37]
    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].
  38. [38]
    D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP 11 (2017) 183 [arXiv:1706.04612] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    M. Cho, S. Collier and X. Yin, Genus Two Modular Bootstrap, JHEP 04 (2019) 022 [arXiv:1705.05865] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    E. Dyer, A.L. Fitzpatrick and Y. Xin, Constraints on Flavored 2d CFT Partition Functions, JHEP 02 (2018) 148 [arXiv:1709.01533] [INSPIRE].
  41. [41]
    L. Apolo, Bounds on CFTs with W3 algebras and AdS 3 higher spin theories, Phys. Rev. D 96 (2017) 086003 [arXiv:1705.10402] [INSPIRE].
  42. [42]
    N. Afkhami-Jeddi, K. Colville, T. Hartman, A. Maloney and E. Perlmutter, Constraints on higher spin CFT 2, JHEP 05 (2018) 092 [arXiv:1707.07717] [INSPIRE].
  43. [43]
    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
  44. [44]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    J.-B. Bae, S. Lee and J. Song, Modular Constraints on Superconformal Field Theories, JHEP 01 (2019) 209 [arXiv:1811.00976] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    T. Anous, R. Mahajan and E. Shaghoulian, Parity and the modular bootstrap, SciPost Phys. 5 (2018) 022 [arXiv:1803.04938] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    T. Hartman, D. Mazac and L. Rastelli, to appear.Google Scholar
  48. [48]
    A. Castedo Echeverri, B. von Harling and M. Serone, The Effective Bootstrap, JHEP 09 (2016) 097 [arXiv:1606.02771] [INSPIRE].CrossRefGoogle Scholar
  49. [49]
    V. Gorbenko, S. Rychkov and B. Zan, Walking, Weak first-order transitions and Complex CFTs, JHEP 10 (2018) 108 [arXiv:1807.11512] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    N. Arkani-Hamed, Y.-T. Huang and S.-H. Shao, On the Positive Geometry of Conformal Field Theory, arXiv:1812.07739 [INSPIRE].
  51. [51]
    A. Pinkus, Spectral properties of totally positive kernels and matrices, in Total positivity and its applications, Springer, Heidelberg Germany (1996), pg. 477.Google Scholar
  52. [52]
    A. Belin, C.A. Keller and A. Maloney, String Universality for Permutation Orbifolds, Phys. Rev. D 91 (2015) 106005 [arXiv:1412.7159] [INSPIRE].
  53. [53]
    C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsCornell UniversityIthacaU.S.A.

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