Advertisement

d-dimensional SYK, AdS loops, and 6j symbols

  • Junyu LiuEmail author
  • Eric Perlmutter
  • Vladimir Rosenhaus
  • David Simmons-Duffin
Open Access
Regular Article - Theoretical Physics
  • 24 Downloads

Abstract

We study the 6j symbol for the conformal group, and its appearance in three seemingly unrelated contexts: the SYK model, conformal representation theory, and perturbative amplitudes in AdS. The contribution of the planar Feynman diagrams to the three-point function of the bilinear singlets in SYK is shown to be a 6j symbol. We generalize the computation of these and other Feynman diagrams to d dimensions. The 6j symbol can be viewed as the crossing kernel for conformal partial waves, which may be computed using the Lorentzian inversion formula. We provide closed-form expressions for 6j symbols in d = 1, 2, 4. In AdS, we show that the 6j symbol is the Lorentzian inversion of a crossing-symmetric tree-level exchange amplitude, thus efficiently packaging the doubletrace OPE data. Finally, we consider one-loop diagrams in AdS with internal scalars and external spinning operators, and show that the triangle diagram is a 6j symbol, while one-loop n-gon diagrams are built out of 6j symbols.

Keywords

AdS-CFT Correspondence Black Holes Conformal Field Theory 

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. Gadde, In search of conformal theories, arXiv:1702.07362 [INSPIRE].
  2. [2]
    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].
  3. [3]
    A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, in KITP strings seminar, http://online.kitp.ucsb.edu/online/joint98/kitaev/, KITP, University of California, Santa Barbara, CA, U.S.A. 12 February 2015.
  4. [4]
    A. Kitaev, A simple model of quantum holography (part 1), in Entanglement 2015 program, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, KITP, University of California, Santa Barbara, CA, U.S.A. 7 April 2015.
  5. [5]
    A. Kitaev, A simple model of quantum holography (part 2), in Entanglement 2015 program, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, KITP, University of California, Santa Barbara, CA, U.S.A. 7 April 2015.
  6. [6]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
  7. [7]
    V. Rosenhaus, An introduction to the SYK model, arXiv:1807.03334 [INSPIRE].
  8. [8]
    D.J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP 12 (2017) 148 [arXiv:1710.08113] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Murugan, D. Stanford and E. Witten, More on supersymmetric and 2d analogs of the SYK model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams and the Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 046004 [arXiv:1611.08915] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    S. Giombi, I.R. Klebanov and G. Tarnopolsky, Bosonic tensor models at large N and small ϵ, Phys. Rev. D 96 (2017) 106014 [arXiv:1707.03866] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
  13. [13]
    A.L. Fitzpatrick and J. Kaplan, Analyticity and the holographic S-matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    A.L. Fitzpatrick and J. Kaplan, Unitarity and the holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from conformal field theory, JHEP 07 (2017) 036 [arXiv:1612.03891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    L.F. Alday and A. Bissi, Loop corrections to supergravity on AdS 5 × S 5, Phys. Rev. Lett. 119 (2017) 171601 [arXiv:1706.02388] [INSPIRE].
  17. [17]
    F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Quantum gravity from conformal field theory, JHEP 01 (2018) 035 [arXiv:1706.02822] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Loop corrections for Kaluza-Klein AdS amplitudes, JHEP 05 (2018) 056 [arXiv:1711.03903] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    L.F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [arXiv:1711.02031] [INSPIRE].
  20. [20]
    C. Cardona, Mellin-(Schwinger) representation of one-loop Witten diagrams in AdS, arXiv:1708.06339 [INSPIRE].
  21. [21]
    S. Giombi, C. Sleight and M. Taronna, Spinning AdS loop diagrams: two point functions, JHEP 06 (2018) 030 [arXiv:1708.08404] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E.Y. Yuan, Loops in the bulk, arXiv:1710.01361 [INSPIRE].
  23. [23]
    E.Y. Yuan, Simplicity in AdS perturbative dynamics, arXiv:1801.07283 [INSPIRE].
  24. [24]
    I. Bertan and I. Sachs, Loops in anti-de Sitter space, Phys. Rev. Lett. 121 (2018) 101601 [arXiv:1804.01880] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
  28. [28]
    K. Krasnov and J. Louko, SO(1, d + 1) Racah coefficients: type I representations, J. Math. Phys. 47 (2006) 033513 [math-ph/0502017] [INSPIRE].
  29. [29]
    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
  30. [30]
    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    L.F. Alday, A. Bissi and E. Perlmutter, Holographic reconstruction of AdS exchanges from crossing symmetry, JHEP 08 (2017) 147 [arXiv:1705.02318] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    S. Giombi, V. Kirilin and E. Perlmutter, Double-trace deformations of conformal correlations, JHEP 02 (2018) 175 [arXiv:1801.01477] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    C. Cardona and K. Sen, Anomalous dimensions at finite conformal spin from OPE inversion, JHEP 11 (2018) 052 [arXiv:1806.10919] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  34. [34]
    C. Sleight and M. Taronna, Spinning Mellin bootstrap: conformal partial waves, crossing kernels and applications, Fortsch. Phys. 66 (2018) 1800038 [arXiv:1804.09334] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  35. [35]
    C. Sleight and M. Taronna, Anomalous dimensions from crossing kernels, JHEP 11 (2018) 089 [arXiv:1807.05941] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    R. Gurau, Colored group field theory, Commun. Math. Phys. 304 (2011) 69 [arXiv:0907.2582] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    V. Bonzom, R. Gurau, A. Riello and V. Rivasseau, Critical behavior of colored tensor models in the large N limit, Nucl. Phys. B 853 (2011) 174 [arXiv:1105.3122] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    S. Carrozza and A. Tanasa, O(N) random tensor models, Lett. Math. Phys. 106 (2016) 1531 [arXiv:1512.06718] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
  40. [40]
    D.J. Gross and V. Rosenhaus, A line of CFTs: from generalized free fields to SYK, JHEP 07 (2017) 086 [arXiv:1706.07015] [INSPIRE].
  41. [41]
    V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic analysis on the N -dimensional Lorentz group and its application to conformal quantum field theory, Lect. Notes Phys. 63 (1977) 1 [INSPIRE].CrossRefzbMATHGoogle Scholar
  42. [42]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    M. D’Eramo, G. Parisi and L. Peliti, Theoretical predictions for critical exponents at the λ point of Bose liquids, Lett. Nuovo Cim. 2 (1971) 878 [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    O. Gürdoğan and V. Kazakov, New integrable 4D quantum field theories from strongly deformed planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 117 (2016) 201602 [Addendum ibid. 117 (2016) 259903] [arXiv:1512.06704] [INSPIRE].
  45. [45]
    J. Caetano, O. Gürdoğan and V. Kazakov, Chiral limit of N = 4 SYM and ABJM and integrable Feynman graphs, JHEP 03 (2018) 077 [arXiv:1612.05895] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, Integrability of conformal fishnet theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly γ-deformed N = 4 supersymmetric Yang-Mills theory as an integrable conformal field theory, Phys. Rev. Lett. 120 (2018) 111601 [arXiv:1711.04786] [INSPIRE].
  48. [48]
    V. Kazakov and E. Olivucci, Biscalar integrable conformal field theories in any dimension, Phys. Rev. Lett. 121 (2018) 131601 [arXiv:1801.09844] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve and structure constants in N = 4 SYM: cusps in the ladder limit, JHEP 10 (2018) 060 [arXiv:1802.04237] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    B. Basso and D.-L. Zhong, Continuum limit of fishnet graphs and AdS σ-model, JHEP 01 (2019) 002 [arXiv:1806.04105] [INSPIRE].
  51. [51]
    J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
  52. [52]
    D.J. Gross and V. Rosenhaus, A generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093 [arXiv:1610.01569] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, Mean field theory and the shadow transform, to appear.Google Scholar
  54. [54]
    D.J. Gross and V. Rosenhaus, The bulk dual of SYK: cubic couplings, JHEP 05 (2017) 092 [arXiv:1702.08016] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    G.P. Korchemsky, Conformal bootstrap for the BFKL Pomeron, Nucl. Phys. B 550 (1999) 397 [hep-ph/9711277] [INSPIRE].
  56. [56]
    A. Bialas, H. Navelet and R.B. Peschanski, High mass diffraction in the QCD dipole picture, Phys. Lett. B 427 (1998) 147 [hep-ph/9711236] [INSPIRE].
  57. [57]
    I. Balitsky, V. Kazakov and E. Sobko, Structure constant of twist-2 light-ray operators in the Regge limit, Phys. Rev. D 93 (2016) 061701 [arXiv:1506.02038] [INSPIRE].ADSMathSciNetGoogle Scholar
  58. [58]
    P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    R.S. Ismagilov, Racah operators for principal series of representations of the group SL(2, C), Sbornik: Math. 198 (2007) 369.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    S.E. Derkachov and V.P. Spiridonov, On the 6j-symbols for SL(2, C) group, Theor. Math. Phys. 198 (2019) 29 [arXiv:1711.07073] [INSPIRE].MathSciNetGoogle Scholar
  61. [61]
    H. Osborn, Conformal blocks for arbitrary spins in two dimensions, Phys. Lett. B 718 (2012) 169 [arXiv:1205.1941] [INSPIRE].
  62. [62]
    L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal approximation in AdS/CFT: conformal partial waves and finite N four-point functions, Nucl. Phys. B 767 (2007) 327 [hep-th/0611123] [INSPIRE].
  63. [63]
    M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
  67. [67]
    D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
  68. [68]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS geometry of conformal blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality constraints on corrections to the graviton three-point coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    D. Meltzer and E. Perlmutter, Beyond a = c: gravitational couplings to matter and the stress tensor OPE, JHEP 07 (2018) 157 [arXiv:1712.04861] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    N. Afkhami-Jeddi, S. Kundu and A. Tajdini, A conformal collider for holographic CFTs, JHEP 10 (2018) 156 [arXiv:1805.07393] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    D. Carmi and S. Caron-Huot, private communication.Google Scholar
  74. [74]
    M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    E.Y. Yuan, Simplicity in AdS perturbative dynamics, arXiv:1801.07283 [INSPIRE].
  76. [76]
    T. Hartman and L. Rastelli, Double-trace deformations, mixed boundary conditions and functional determinants in AdS/CFT, JHEP 01 (2008) 019 [hep-th/0602106] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  77. [77]
    D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  78. [78]
    B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U q(SL(2, R)), Commun. Math. Phys. 224 (2001) 613 [math/0007097] [INSPIRE].
  79. [79]
    B. Ponsot, Recent progresses on Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 311 [hep-th/0301193] [INSPIRE].
  80. [80]
    J. Murakami and M. Yano, On the volume of a hyberbolic and spherical tertrahedron, Commun. Anal. Geom. 13 (2005) 379.CrossRefzbMATHGoogle Scholar
  81. [81]
    O. Aharony, L.F. Alday, A. Bissi and R. Yacoby, The analytic bootstrap for large N Chern-Simons vector models, JHEP 08 (2018) 166 [arXiv:1805.04377] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  82. [82]
    G. Passarino and M.J.G. Veltman, One loop corrections for e + e annihilation into μ + μ in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    K. Bulycheva, I.R. Klebanov, A. Milekhin and G. Tarnopolsky, Spectra of operators in large N tensor models, Phys. Rev. D 97 (2018) 026016 [arXiv:1707.09347] [INSPIRE].ADSMathSciNetGoogle Scholar
  84. [84]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. [85]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
  86. [86]
    D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Junyu Liu
    • 1
    • 2
    Email author
  • Eric Perlmutter
    • 1
  • Vladimir Rosenhaus
    • 3
    • 4
  • David Simmons-Duffin
    • 1
  1. 1.Walter Burke Institute for Theoretical PhysicsCaltechPasadenaU.S.A.
  2. 2.Institute for Quantum Information and MatterCaltechPasadenaU.S.A.
  3. 3.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  4. 4.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

Personalised recommendations