Journal of High Energy Physics

, 2019:140 | Cite as

AdS3× S3 tree-level correlators: hidden six-dimensional conformal symmetry

  • Leonardo Rastelli
  • Konstantinos Roumpedakis
  • Xinan ZhouEmail author
Open Access
Regular Article - Theoretical Physics


We revisit the calculation of holographic correlators in AdS3. We develop new methods to evaluate exchange Witten diagrams, resolving some technical difficulties that prevent a straightforward application of the methods used in higher dimensions. We perform detailed calculations in the AdS3× S3× K 3 background. We find strong evidence that four-point tree-level correlators of KK modes of the tensor multiplets enjoy a hidden 6d conformal symmetry. The correlators can all be packaged into a single generating function, related to the 6d flat space superamplitude. This generalizes an analogous structure found in AdS5× S5 supergravity.


AdS-CFT Correspondence Conformal Field Theory Scattering Amplitudes 


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]
    L. Rastelli and X. Zhou, Mellin amplitudes for AdS 5× S 5 , Phys. Rev. Lett.118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    L. Rastelli and X. Zhou, How to Succeed at Holographic Correlators Without Really Trying, JHEP04 (2018) 014 [arXiv:1710.05923] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    G. Arutyunov, S. Frolov, R. Klabbers and S. Savin, Towards 4-point correlation functions of any 1-BPS operators from supergravity, JHEP04 (2017) 005 [arXiv:1701.00998] [INSPIRE].ADSzbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    G. Arutyunov, R. Klabbers and S. Savin, Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation, JHEP09 (2018) 118 [arXiv:1808.06788] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    G. Arutyunov, R. Klabbers and S. Savin, Four-point functions of all-different-weight chiral primary operators in the supergravity approximation, JHEP09 (2018) 023 [arXiv:1806.09200] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    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].ADSCrossRefGoogle Scholar
  7. [7]
    F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Quantum Gravity from Conformal Field Theory, JHEP01 (2018) 035 [arXiv:1706.02822] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Unmixing Supergravity, JHEP02 (2018) 133 [arXiv:1706.08456] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    F. Aprile, J. Drummond, P. Heslop and H. Paul, Double-trace spectrum of N = 4 supersymmetric Yang-Mills theory at strong coupling, Phys. Rev.D 98 (2018) 126008 [arXiv:1802.06889] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Caron-Huot and A.-K. Trinh, All tree-level correlators in AdS 5× S 5supergravity: hidden ten-dimensional conformal symmetry, JHEP01 (2019) 196 [arXiv:1809.09173] [INSPIRE].ADSzbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from Conformal Field Theory, JHEP07 (2017) 036 [arXiv:1612.03891] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Loop corrections for Kaluza-Klein AdS amplitudes, JHEP05 (2018) 056 [arXiv:1711.03903] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    L.F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP12 (2018) 017 [arXiv:1711.02031] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    V. Gon¸ Calves, Four point function of \( \mathcal{N} \) = 4 stress-tensor multiplet at strong coupling, JHEP04 (2015) 150 [arXiv:1411.1675] [INSPIRE].
  15. [15]
    L.F. Alday, A. Bissi and E. Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP06 (2019) 010 [arXiv:1809.10670] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    L.F. Alday, On Genus-one String Amplitudes on AdS 5× S 5 , arXiv:1812.11783 [INSPIRE].
  17. [17]
    D.J. Binder, S.M. Chester, S.S. Pufu and Y. Wang, \( \mathcal{N} \) = 4 Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization, arXiv:1902.06263 [INSPIRE].
  18. [18]
    L. Rastelli and X. Zhou, Holographic Four-Point Functions in the (2, 0) Theory, JHEP06 (2018) 087 [arXiv:1712.02788] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    X. Zhou, On Superconformal Four-Point Mellin Amplitudes in Dimension d > 2, JHEP08 (2018) 187 [arXiv:1712.02800] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    P. Heslop and A.E. Lipstein, M-theory Beyond The Supergravity Approximation, JHEP02 (2018) 004 [arXiv:1712.08570] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    X. Zhou, On Mellin Amplitudes in SCFTs with Eight Supercharges, JHEP07 (2018) 147 [arXiv:1804.02397] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    S.M. Chester and E. Perlmutter, M-Theory Reconstruction from (2, 0) CFT and the Chiral Algebra Conjecture, JHEP08 (2018) 116 [arXiv:1805.00892] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    T. Abl, P. Heslop and A.E. Lipstein, Recursion relations for anomalous dimensions in the 6d (2, 0) theory, JHEP04 (2019) 038 [arXiv:1902.00463] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    S.M. Chester, AdS4 /CFT3 for unprotected operators, JHEP07 (2018) 030 [arXiv:1803.01379] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S.M. Chester, S.S. Pufu and X. Yin, The M-theory S-matrix From ABJM: Beyond 11D Supergravity, JHEP08 (2018) 115 [arXiv:1804.00949] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    D.J. Binder, S.M. Chester and S.S. Pufu, Absence of D 4R 4in M-theory From ABJM, arXiv:1808.10554 [INSPIRE].
  27. [27]
    G. Mack, D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
  28. [28]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP03 (2011) 025 [arXiv:1011.1485] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP10 (2011) 074 [arXiv:1107.1504] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  30. [30]
    A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A Natural Language for AdS/CFT Correlators, JHEP11 (2011) 095 [arXiv:1107.1499] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    E. D’Hoker, D.Z. Freedman and L. Rastelli, AdS/CFT four point functions: How to succeed at z integrals without really trying, Nucl. Phys.B 562 (1999) 395 [hep-th/9905049] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    A. Galliani, S. Giusto and R. Russo, Holographic 4-point correlators with heavy states, JHEP10 (2017) 040 [arXiv:1705.09250] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    A. Bombini, A. Galliani, S. Giusto, E. Moscato and R. Russo, Unitary 4-point correlators from classical geometries, Eur. Phys. J.C 78 (2018) 8 [arXiv:1710.06820] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    S. Giusto, R. Russo and C. Wen, Holographic correlators in AdS 3 , JHEP03 (2019) 096 [arXiv:1812.06479] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    A. Bombini and A. Galliani, AdS3 four-point functions from \( \frac{1}{8} \)-BPS states, JHEP06 (2019) 044 [arXiv:1904.02656] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  36. [36]
    M. Heydeman, J.H. Schwarz, C. Wen and S.-Q. Zhang, All Tree Amplitudes of 6D (2, 0) Supergravity: Interacting Tensor Multiplets and the K 3 Moduli Space, Phys. Rev. Lett.122 (2019) 111604 [arXiv:1812.06111] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    S. Giusto, R. Russo, A. Tyukov and C. Wen, Holographic correlators in AdS3 without Witten diagrams, JHEP09 (2019) 030 [arXiv:1905.12314] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  38. [38]
    F.A. Dolan, L. Gallot and E. Sokatchev, On four-point functions of 1/2-BPS operators in general dimensions, JHEP09 (2004) 056 [hep-th/0405180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP07 (2012) 137 [arXiv:1203.1036] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    S. Deger, A. Kaya, E. Sezgin and P. Sundell, Spectrum of D = 6, N = 4b supergravity on AdS 3× S 3 , Nucl. Phys.B 536 (1998) 110 [hep-th/9804166] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  42. [42]
    J. de Boer, Six-dimensional supergravity on S 3× AdS 3and 2 − D conformal field theory, Nucl. Phys.B 548 (1999) 139 [hep-th/9806104] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    J. de Boer, Large N elliptic genus and AdS/CFT correspondence, JHEP05 (1999) 017 [hep-th/9812240] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    G. Arutyunov, A. Pankiewicz and S. Theisen, Cubic couplings in D = 6 N=4b supergravity on AdS3 × S 3 , Phys. Rev.D 63 (2001) 044024 [hep-th/0007061] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    X. Zhou, Recursion Relations in Witten Diagrams and Conformal Partial Waves, JHEP05 (2019) 006 [arXiv:1812.01006] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, Partial nonrenormalization of the stress tensor four point function in N = 4 SYM and AdS/CFT, Nucl. Phys.B 607 (2001) 191 [hep-th/0009106] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  47. [47]
    Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Supersymmetry Constraints and String Theory on K 3, JHEP12 (2015) 142 [arXiv:1508.07305] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    G. Arutyunov and S. Frolov, On the correspondence between gravity fields and CFT operators, JHEP04 (2000) 017 [hep-th/0003038] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    M. Taylor, Matching of correlators in AdS 3/C F T 2 , JHEP06 (2008) 010 [arXiv:0709.1838] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    J. Maldacena, Einstein Gravity from Conformal Gravity, arXiv:1105.5632 [INSPIRE].
  51. [51]
    V. Gon¸calves, R. Pereira and X. Zhou, 201Five-Point Function from AdS 5× S 5Supergravity, arXiv:1906.05305 [INSPIRE].
  52. [52]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton and gauge boson propagators in AdS(d + 1), Nucl. Phys.B 562 (1999) 330 [hep-th/9902042] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    K. Pilch and A.N. Schellekens, Formulae for the Eigenvalues of the Laplacian on Tensor Harmonics on Symmetric Coset Spaces, J. Math. Phys.25 (1984) 3455 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.C.N. Yang Institute for Theoretical Physics (YITP)Stony Brook UniversityStony BrookU.S.A.
  2. 2.Princeton Center for Theoretical SciencePrinceton UniversityPrincetonU.S.A.
  3. 3.Theoretical Physics DepartmentCERNGenevaSwitzerland

Personalised recommendations