# All tree-level correlators in AdS_{5}×S_{5} supergravity: hidden ten-dimensional conformal symmetry

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## Abstract

We study correlators of four protected (half-BPS) operators in strongly coupled supersymmetric Yang-Mills theory. These are dual to tree-level supergravity amplitudes on AdS_{5}×S_{5} for various spherical harmonics on the five-sphere. We use conformal field theory methods, in particular a recently obtained Lorentzian inversion formula, to analytically bootstrap these correlators. The extracted 1*/N*^{2} double-trace anomalous dimensions confirm a simple pattern recently conjectured by Aprile, Drummond, Heslop and Paul. We explain this pattern by an unexpected ten-dimensional conformal symmetry which appears to be enjoyed by tree-level supergravity (or a suitable subsector of it). The symmetry combines all spherical harmonics into a single ten-dimensional object, and yields compact expressions for the leading logarithmic part of any half-BPS correlator at each loop order.

## Keywords

AdS-CFT Correspondence Conformal Field Theory 1/N Expansion Supersymmetric Gauge Theory## Notes

### **Open Access**

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## References

- [1]I. Heemskerk, J. Penedones, J. Polchinski and J. Sully,
*Holography from Conformal Field Theory*,*JHEP***10**(2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [2]E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli,
*Graviton exchange and complete four point functions in the AdS/CFT correspondence*,*Nucl. Phys.***B 562**(1999) 353 [hep-th/9903196] [INSPIRE]. - [3]G. Arutyunov and S. Frolov,
*Four point functions of lowest weight CPOs in N*= 4*SYM(4) in supergravity approximation*,*Phys. Rev.***D 62**(2000) 064016 [hep-th/0002170] [INSPIRE]. - [4]S. Lee, S. Minwalla, M. Rangamani and N. Seiberg,
*Three point functions of chiral operators in D*= 4*, N*= 4*SYM at large N*,*Adv. Theor. Math. Phys.***2**(1998) 697 [hep-th/9806074] [INSPIRE]. - [5]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]. - [6]M. Gary, S.B. Giddings and J. Penedones,
*Local bulk S-matrix elements and CFT singularities*,*Phys. Rev.***D 80**(2009) 085005 [arXiv:0903.4437] [INSPIRE]. - [7]J. Penedones,
*Writing CFT correlation functions as AdS scattering amplitudes*,*JHEP***03**(2011) 025 [arXiv:1011.1485] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [8]J. Maldacena, D. Simmons-Duffin and A. Zhiboedov,
*Looking for a bulk point*,*JHEP***01**(2017) 013 [arXiv:1509.03612] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [9]G. Arutyunov, F.A. Dolan, H. Osborn and E. Sokatchev,
*Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence*,*Nucl. Phys.***B 665**(2003) 273 [hep-th/0212116] [INSPIRE]. - [10]G. Arutyunov and E. Sokatchev,
*On a large N degeneracy in N*= 4*SYM and the AdS/CFT correspondence*,*Nucl. Phys.***B 663**(2003) 163 [hep-th/0301058] [INSPIRE]. - [11]L. Berdichevsky and P. Naaijkens,
*Four-point functions of different-weight operators in the AdS/CFT correspondence*,*JHEP***01**(2008) 071 [arXiv:0709.1365] [INSPIRE]. - [12]L.I. Uruchurtu,
*Four-point correlators with higher weight superconformal primaries in the AdS/CFT Correspondence*,*JHEP***03**(2009) 133 [arXiv:0811.2320] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [13]L.I. Uruchurtu,
*Next-next-to-extremal Four Point Functions of N*= 4 1*/*2*BPS Operators in the AdS/CFT Correspondence*,*JHEP***08**(2011) 133 [arXiv:1106.0630] [INSPIRE]. - [14]M. Nirschl and H. Osborn,
*Superconformal Ward identities and their solution*,*Nucl. Phys.***B 711**(2005) 409 [hep-th/0407060] [INSPIRE]. - [15]F.A. Dolan, M. Nirschl and H. Osborn,
*Conjectures for large N superconformal N*= 4*chiral primary four point functions*,*Nucl. Phys.***B 749**(2006) 109 [hep-th/0601148] [INSPIRE]. - [16]L. Rastelli and X. Zhou,
*Mellin amplitudes for AdS*_{5}×*S*^{5},*Phys. Rev. Lett.***118**(2017) 091602 [arXiv:1608.06624] [INSPIRE]. - [17]A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees,
*A Natural Language for AdS/CFT Correlators*,*JHEP***11**(2011) 095 [arXiv:1107.1499] [INSPIRE]. - [18]G. Arutyunov, S. Frolov, R. Klabbers and S. Savin,
*Towards 4-point correlation functions of any*\( \frac{1}{2} \)*-BPS operators from supergravity*,*JHEP***04**(2017) 005 [arXiv:1701.00998] [INSPIRE]. - [19]G. Arutyunov, R. Klabbers and S. Savin,
*Four-point functions of*1*/*2*-BPS operators of any weights in the supergravity approximation*,*JHEP***09**(2018) 118 [arXiv:1808.06788] [INSPIRE]. - [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]. - [21]Z. Komargodski and A. Zhiboedov,
*Convexity and Liberation at Large Spin*,*JHEP***11**(2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar - [22]L.F. Alday and A. Zhiboedov,
*An Algebraic Approach to the Analytic Bootstrap*,*JHEP***04**(2017) 157 [arXiv:1510.08091] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [23]L.F. Alday,
*Large Spin Perturbation Theory for Conformal Field Theories*,*Phys. Rev. Lett.***119**(2017) 111601 [arXiv:1611.01500] [INSPIRE].ADSCrossRefGoogle Scholar - [24]S. Caron-Huot,
*Analyticity in Spin in Conformal Theories*,*JHEP***09**(2017) 078 [arXiv:1703.00278] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [25]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]. - [26]P. Kravchuk and D. Simmons-Duffin,
*Light-ray operators in conformal field theory*,*JHEP***11**(2018) 102 [arXiv:1805.00098] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [27]L.F. Alday, A. Bissi and E. Perlmutter,
*Holographic Reconstruction of AdS Exchanges from Crossing Symmetry*,*JHEP***08**(2017) 147 [arXiv:1705.02318] [INSPIRE]. - [28]J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin,
*d-dimensional SYK, AdS Loops and*6*j Symbols*, arXiv:1808.00612 [INSPIRE]. - [29]L.F. Alday and A. Bissi,
*Unitarity and positivity constraints for CFT at large central charge*,*JHEP***07**(2017) 044 [arXiv:1606.09593] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [30]Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower,
*Fusing gauge theory tree amplitudes into loop amplitudes*,*Nucl. Phys.***B 435**(1995) 59 [hep-ph/9409265] [INSPIRE]. - [31]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]. - [32]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 - [33]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]. - [34]L.F. Alday and S. Caron-Huot,
*Gravitational S-matrix from CFT dispersion relations*,*JHEP***12**(2018) 017 [arXiv:1711.02031] [INSPIRE].ADSCrossRefGoogle Scholar - [35]N. Beisert et al.,
*Review of AdS/CFT Integrability: An Overview*,*Lett. Math. Phys.***99**(2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [36]N. Gromov, V. Kazakov, S. Leurent and D. Volin,
*Quantum spectral curve for arbitrary state/operator in AdS*_{5}*/CFT*_{4},*JHEP***09**(2015) 187 [arXiv:1405.4857] [INSPIRE]. - [37]T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira,
*Handling Handles: Nonplanar Integrability in*\( \mathcal{N}=4 \)*Supersymmetric Yang-Mills Theory*,*Phys. Rev. Lett.***121**(2018) 231602 [arXiv:1711.05326] [INSPIRE]. - [38]R. Ben-Israel, A.G. Tumanov and A. Sever,
*Scattering amplitudes — Wilson loops duality for the first non-planar correction*,*JHEP***08**(2018) 122 [arXiv:1802.09395] [INSPIRE]. - [39]C. Cardona,
*Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS*, arXiv:1708.06339 [INSPIRE]. - [40]
- [41]
- [42]F. Aprile, J. Drummond, P. Heslop and H. Paul,
*The double-trace spectrum of N*= 4*SYM at strong coupling*,*Phys. Rev.***D 98**(2018) 126008 [arXiv:1802.06889] [INSPIRE]. - [43]F.A. Dolan and H. Osborn,
*Conformal partial waves and the operator product expansion*,*Nucl. Phys.***B 678**(2004) 491 [hep-th/0309180] [INSPIRE]. - [44]A. Bissi and T. Lukowski,
*Revisiting*\( \mathcal{N}=4 \)*superconformal blocks*,*JHEP***02**(2016) 115 [arXiv:1508.02391] [INSPIRE]. - [45]F.A. Dolan and H. Osborn,
*On short and semi-short representations for four-dimensional superconformal symmetry*,*Annals Phys.***307**(2003) 41 [hep-th/0209056] [INSPIRE]. - [46]D. Chicherin et al.,
*Correlation functions of the chiral stress-tensor multiplet in*\( \mathcal{N}=4 \)*SYM*,*JHEP***06**(2015) 198 [arXiv:1412.8718] [INSPIRE]. - [47]F.A. Dolan, L. Gallot and E. Sokatchev,
*On four-point functions of*1*/*2*-BPS operators in general dimensions*,*JHEP***09**(2004) 056 [hep-th/0405180] [INSPIRE]. - [48]F.A. Dolan and H. Osborn,
*Conformal partial wave expansions for N*= 4*chiral four point functions*,*Annals Phys.***321**(2006) 581 [hep-th/0412335] [INSPIRE]. - [49]A.L. Fitzpatrick and J. Kaplan,
*Unitarity and the Holographic S-matrix*,*JHEP***10**(2012) 032 [arXiv:1112.4845] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [50]F.A. Dolan and H. Osborn,
*Superconformal symmetry, correlation functions and the operator product expansion*,*Nucl. Phys.***B 629**(2002) 3 [hep-th/0112251] [INSPIRE]. - [51]B. Basso, F. Coronado, S. Komatsu, H.T. Lam, P. Vieira and D.-l. Zhong,
*Asymptotic Four Point Functions*, arXiv:1701.04462 [INSPIRE]. - [52]R. Doobary and P. Heslop,
*Superconformal partial waves in Grassmannian field theories*,*JHEP***12**(2015) 159 [arXiv:1508.03611] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [53]F. Aprile, J.M. Drummond, P. Heslop and H. Paul,
*Unmixing Supergravity*,*JHEP***02**(2018) 133 [arXiv:1706.08456] [INSPIRE]. - [54]J.M. Drummond, L. Gallot and E. Sokatchev,
*Superconformal Invariants or How to Relate Four-point AdS Amplitudes*,*Phys. Lett.***B 645**(2007) 95 [hep-th/0610280] [INSPIRE]. - [55]E. Witten,
*Anti-de Sitter space and holography*,*Adv. Theor. Math. Phys.***2**(1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [56]C. Beem, L. Rastelli and B.C. van Rees,
*More*\( \mathcal{N}=4 \)*superconformal bootstrap*,*Phys. Rev.***D 96**(2017) 046014 [arXiv:1612.02363] [INSPIRE]. - [57]N. Seiberg,
*Naturalness versus supersymmetric nonrenormalization theorems*,*Phys. Lett.***B 318**(1993) 469 [hep-ph/9309335] [INSPIRE]. - [58]S.B. Giddings and M. Srednicki,
*High-energy gravitational scattering and black hole resonances*,*Phys. Rev.***D 77**(2008) 085025 [arXiv:0711.5012] [INSPIRE]. - [59]F.A. Dolan and H. Osborn,
*Conformal Partial Waves: Further Mathematical Results*, arXiv:1108.6194 [INSPIRE]. - [60]R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha,
*A Mellin space approach to the conformal bootstrap*,*JHEP***05**(2017) 027 [arXiv:1611.08407] [INSPIRE]. - [61]R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha,
*Conformal Bootstrap in Mellin Space*,*Phys. Rev. Lett.***118**(2017) 081601 [arXiv:1609.00572] [INSPIRE]. - [62]L. Rastelli and X. Zhou,
*How to Succeed at Holographic Correlators Without Really Trying*,*JHEP***04**(2018) 014 [arXiv:1710.05923] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar - [63]M. Hogervorst and S. Rychkov,
*Radial Coordinates for Conformal Blocks*,*Phys. Rev.***D 87**(2013) 106004 [arXiv:1303.1111] [INSPIRE]. - [64]
- [65]G. Anastasiou and R. Olea,
*From conformal to Einstein Gravity*,*Phys. Rev.***D 94**(2016) 086008 [arXiv:1608.07826] [INSPIRE]. - [66]F. Loebbert, M. Mojaza and J. Plefka,
*Hidden Conformal Symmetry in Tree-Level Graviton Scattering*,*JHEP***05**(2018) 208 [arXiv:1802.05999] [INSPIRE]. - [67]P. Dai, R.-N. Huang and W. Siegel,
*Covariant propagator in AdS*_{5}×*S*^{5}*superspace*,*JHEP***03**(2010) 001 [arXiv:0911.2211] [INSPIRE]. - [68]J.H. Schwarz,
*New formulation of the type IIB superstring action in AdS*_{5}×*S*^{5},*JHEP***09**(2015) 041 [arXiv:1506.07706] [INSPIRE].