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Universal lowest-twist in CFTs from holography

  • A. Liam Fitzpatrick
  • Kuo-Wei HuangEmail author
Open Access
Regular Article - Theoretical Physics
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Abstract

We probe the conformal block structure of a scalar four-point function in d ≥ 2 conformal field theories by including higher-order derivative terms in a bulk gravitational action. We consider a heavy-light four-point function as the boundary correlator at large central charge. Such a four-point function can be computed, on the gravity side, as a two-point function of the light operator in a black hole geometry created by the heavy operator. We consider analytically solving the corresponding scalar field equation in a near-boundary expansion and find that the multi-stress tensor conformal blocks are insensitive to the horizon boundary condition. The main result of this paper is that the lowest-twist operator product expansion (OPE) coefficients of the multi-stress tensor conformal blocks are universal: they are fixed by the dimension of the light operators and the ratio between the dimension of the heavy operator and the central charge CT. Neither supersymmetry nor unitary is assumed. Higher-twist coefficients, on the other hand, generally are not protected. A recursion relation allows us to efficiently compute universal lowest-twist coefficients. The universality result hints at the potential existence of a higher-dimensional Virasoro-like symmetry near the lightcone. While we largely focus on the planar black hole limit in this paper, we include some preliminary analysis of the spherical black hole case in an appendix.

Keywords

AdS-CFT Correspondence Conformal Field Theory Field Theories in Higher Dimensions 
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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].MathSciNetzbMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    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].ADSCrossRefGoogle Scholar
  4. [4]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    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].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP08 (2014) 145 [arXiv:1403.6829] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
  9. [9]
    T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
  10. [10]
    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches, JHEP02 (2015) 171 [arXiv:1410.1392] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    B. Chen and J.-q. Wu, Holographic Entanglement Entropy For a Large Class of States in 2D CFT, JHEP09 (2016) 015 [arXiv:1605.06753] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    B. Chen, J.-q. Wu and J.-j. Zhang, Holographic Description of 2D Conformal Block in Semi-classical Limit, JHEP10 (2016) 110 [arXiv:1609.00801] [INSPIRE].
  13. [13]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett.115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
  15. [15]
    T. Anous and J. Sonner, Phases of scrambling in eigenstates, SciPost Phys.7 (2019) 003 [arXiv:1903.03143] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS 3/CFT 2, JHEP05 (2016) 109 [arXiv:1603.08925] [INSPIRE].Google Scholar
  17. [17]
    H. Chen, C. Hussong, J. Kaplan and D. Li, A Numerical Approach to Virasoro Blocks and the Information Paradox, JHEP09 (2017) 102 [arXiv:1703.09727] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP05 (2016) 127 [arXiv:1601.06794] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3gravity, JHEP12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
  20. [20]
    E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP07 (2015) 131 [arXiv:1501.02260] [INSPIRE].
  21. [21]
    T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP09 (2014) 118 [arXiv:1405.5137] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    T. Anous, T. Hartman, A. Rovai and J. Sonner, Black Hole Collapse in the 1/c Expansion, JHEP07 (2016) 123 [arXiv:1603.04856] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    M. Kulaxizi, G.S. Ng and A. Parnachev, Black Holes, Heavy States, Phase Shift and Anomalous Dimensions, SciPost Phys.6 (2019) 065 [arXiv:1812.03120] [INSPIRE].
  24. [24]
    A.L. Fitzpatrick, J. Kaplan, M.T. Walters and J. Wang, Hawking from Catalan, JHEP05 (2016) 069 [arXiv:1510.00014] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    N. Anand, H. Chen, A.L. Fitzpatrick, J. Kaplan and D. Li, An Exact Operator That Knows Its Location, JHEP02 (2018) 012 [arXiv:1708.04246] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    H. Chen, A.L. Fitzpatrick, J. Kaplan and D. Li, The AdS 3propagator and the fate of locality, JHEP04 (2018) 075 [arXiv:1712.02351] [INSPIRE].Google Scholar
  27. [27]
    A.L. Fitzpatrick and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP05 (2016) 075 [arXiv:1512.03052] [INSPIRE].
  28. [28]
    H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys.231 (1994) 311 [hep-th/9307010] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP07 (2011) 023 [arXiv:1007.2412] [INSPIRE].
  31. [31]
    A.L. Fitzpatrick and J. Kaplan, Analyticity and the Holographic S-matrix, JHEP10 (2012) 127 [arXiv:1111.6972] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
  33. [33]
    A.L. Fitzpatrick and J. Kaplan, AdS Field Theory from Conformal Field Theory, JHEP02 (2013) 054 [arXiv:1208.0337] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    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].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP01 (2016) 146 [arXiv:1508.00501] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    E. Dyer, D.Z. Freedman and J. Sully, Spinning Geodesic Witten Diagrams, JHEP11 (2017) 060 [arXiv:1702.06139] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    A. Castro, E. Llabrés and F. Rejon-Barrera, Geodesic Diagrams, Gravitational Interactions & OPE Structures, JHEP06 (2017) 099 [arXiv:1702.06128] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    R.C. Myers and B. Robinson, Black Holes in Quasi-topological Gravity, JHEP08 (2010) 067 [arXiv:1003.5357] [INSPIRE].
  39. [39]
    R.C. Myers, M.F. Paulos and A. Sinha, Holographic studies of quasi-topological gravity, JHEP08 (2010) 035 [arXiv:1004.2055] [INSPIRE].
  40. [40]
    J. Oliva and S. Ray, A new cubic theory of gravity in five dimensions: Black hole, Birkhoff ’s theorem and C-function, Class. Quant. Grav.27 (2010) 225002 [arXiv:1003.4773] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    R.M. Wald, General Relativity, Chicago University Press, Chicago U.S.A. (1984) [INSPIRE].
  42. [42]
    M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter Spaces, Commun. Math. Phys.98 (1985) 391 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    S. Hollands, A. Ishibashi and D. Marolf, Comparison between various notions of conserved charges in asymptotically AdS-spacetimes, Class. Quant. Grav.22 (2005) 2881 [hep-th/0503045] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    E. Keski-Vakkuri, Bulk and boundary dynamics in BTZ black holes, Phys. Rev.D 59 (1999) 104001 [hep-th/9808037] [INSPIRE].ADSMathSciNetGoogle Scholar
  45. [45]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    P. Bueno, P.A. Cano, V.S. Min and M.R. Visser, Aspects of general higher-order gravities, Phys. Rev.D 95 (2017) 044010 [arXiv:1610.08519] [INSPIRE].
  47. [47]
    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].
  48. [48]
    E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
  49. [49]
    L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP02 (2004) 014 [hep-th/0306170] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    J.L. Cardy, Conformal invariance and universality in finite-size scaling, J. Phys.A 17 (1984) L385 [INSPIRE].ADSMathSciNetGoogle Scholar
  51. [51]
    J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys.B 270 (1986) 186 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    H. Chen, A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Degenerate Operators and the 1/c Expansion: Lorentzian Resummations, High Order Computations and Super-Virasoro Blocks, JHEP03 (2017) 167 [arXiv:1606.02659] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    C.P. Herzog and K.-W. Huang, Stress Tensors from Trace Anomalies in Conformal Field Theories, Phys. Rev.D 87 (2013) 081901 [arXiv:1301.5002] [INSPIRE].
  54. [54]
    A. Cappelli and A. Coste, On the Stress Tensor of Conformal Field Theories in Higher Dimensions, Nucl. Phys.B 314 (1989) 707 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    K.-W. Huang, Weyl Anomaly Induced Stress Tensors in General Manifolds, Nucl. Phys.B 879 (2014) 370 [arXiv:1308.2355] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    G.T. Horowitz and N. Itzhaki, Black holes, shock waves and causality in the AdS/CFT correspondence, JHEP02 (1999) 010 [hep-th/9901012] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    C.P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary Central Charge, JHEP10 (2017) 189 [arXiv:1707.06224] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    C.P. Herzog, K.-W. Huang, I. Shamir and J. Virrueta, Superconformal Models for Graphene and Boundary Central Charges, JHEP09 (2018) 161 [arXiv:1807.01700] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  59. [59]
    O. Aharony, O. Bergman, D.L. Jafferis and J.M. Maldacena, \( \mathcal{N}=6 \)superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP10 (2008) 091 [arXiv:0806.1218] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsBoston UniversityBostonU.S.A.

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