Abstract
We explicitly rewrite the path integral for the free or critical O(N) (or U(N)) bosonic vector models in d space-time dimensions as a path integral over fields (including massless high-spin fields) living on (d + 1)-dimensional anti-de Sitter space. Inspired by de Mello Koch, Jevicki, Suzuki and Yoon and earlier work, we first rewrite the vector models in terms of bi-local fields, then expand these fields in eigenmodes of the conformal group, and finally map these eigenmodes to those of fields on anti-de Sitter space. Our results provide an explicit (non-local) action for a high-spin theory on anti-de Sitter space, which is presumably equivalent in the large N limit to Vasiliev’s classical high-spin gravity theory (with some specific gauge-fixing to a fixed background), but which can be used also for loop computations. Our mapping is explicit within the 1/N expansion, but in principle can be extended also to finite N theories, where extra constraints on products of bulk fields need to be taken into account.
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References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
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].
S.M. Chester, M.B. Green, S.S. Pufu, Y. Wang and C. Wen, Modular invariance in superstring theory from N = 4 super-Yang-Mills, JHEP 11 (2020) 016 [arXiv:1912.13365] [INSPIRE].
D.J. Binder, S.M. Chester and S.S. Pufu, AdS4/CFT3 from weak to strong string coupling, JHEP 01 (2020) 034 [arXiv:1906.07195] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of planar N = 4 supersymmetric Yang-Mills theory: Konishi dimension at any coupling, Phys. Rev. Lett. 104 (2010) 211601 [arXiv:0906.4240] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].
A. Dei, M.R. Gaberdiel, R. Gopakumar and B. Knighton, Free field world-sheet correlators for AdS3, JHEP 02 (2021) 081 [arXiv:2009.11306] [INSPIRE].
L. Eberhardt, AdS3/CFT2 at higher genus, JHEP 05 (2020) 150 [arXiv:2002.11729] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wadia and X. Yin, Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, d = 3 bosonic vector models coupled to Chern-Simons gauge theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].
M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].
M.A. Vasiliev, More on equations of motion for interacting massless fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 285 (1992) 225 [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [INSPIRE].
X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Towards holographic higher-spin interactions: four-point functions and higher-spin exchange, JHEP 03 (2015) 170 [arXiv:1412.0016] [INSPIRE].
X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Bulk quartic vertices from boundary four-point correlators, in International workshop on higher spin gauge theories, World Scientific, Singapore (2016) [arXiv:1602.08570] [INSPIRE].
E. Skvortsov, Light-front bootstrap for Chern-Simons matter theories, JHEP 06 (2019) 058 [arXiv:1811.12333] [INSPIRE].
C. Sleight and M. Taronna, Higher spin interactions from conformal field theory: the complete cubic couplings, Phys. Rev. Lett. 116 (2016) 181602 [arXiv:1603.00022] [INSPIRE].
N. Boulanger, P. Kessel, E.D. Skvortsov and M. Taronna, Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal, J. Phys. A 49 (2016) 095402 [arXiv:1508.04139] [INSPIRE].
C. Sleight and M. Taronna, Higher-spin gauge theories and bulk locality, Phys. Rev. Lett. 121 (2018) 171604 [arXiv:1704.07859] [INSPIRE].
Y. Neiman, Higher-spin gravity as a theory on a fixed (anti) de Sitter background, JHEP 04 (2015) 144 [arXiv:1502.06685] [INSPIRE].
S.R. Das and A. Jevicki, Large N collective fields and holography, Phys. Rev. D 68 (2003) 044011 [hep-th/0304093] [INSPIRE].
R. de Mello Koch, A. Jevicki, K. Jin and J.P. Rodrigues, AdS4/CFT3 construction from collective fields, Phys. Rev. D 83 (2011) 025006 [arXiv:1008.0633] [INSPIRE].
R. de Mello Koch, A. Jevicki, J.P. Rodrigues and J. Yoon, Canonical formulation of O(N) vector/higher spin correspondence, J. Phys. A 48 (2015) 105403 [arXiv:1408.4800] [INSPIRE].
R. de Mello Koch, A. Jevicki, J.P. Rodrigues and J. Yoon, Holography as a gauge phenomenon in higher spin duality, JHEP 01 (2015) 055 [arXiv:1408.1255] [INSPIRE].
R. de Mello Koch, A. Jevicki, K. Jin, J.P. Rodrigues and Q. Ye, S = 1 in O(N)/HS duality, Class. Quant. Grav. 30 (2013) 104005 [arXiv:1205.4117] [INSPIRE].
R. de Mello Koch, A. Jevicki, K. Suzuki and J. Yoon, AdS maps and diagrams of bi-local holography, JHEP 03 (2019) 133 [arXiv:1810.02332] [INSPIRE].
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].
V.K. Dobrev, G. Mack, I.T. Todorov, V.B. Petkova and S.G. Petrova, On the Clebsch-Gordan expansion for the Lorentz group in n-dimensions, Rept. Math. Phys. 9 (1976) 219 [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].
S. Giombi and X. Yin, On higher spin gauge theory and the critical O(N) model, Phys. Rev. D 85 (2012) 086005 [arXiv:1105.4011] [INSPIRE].
S.H. Shenker and X. Yin, Vector models in the singlet sector at finite temperature, arXiv:1109.3519 [INSPIRE].
M.R. Douglas, L. Mazzucato and S.S. Razamat, Holographic dual of free field theory, Phys. Rev. D 83 (2011) 071701 [arXiv:1011.4926] [INSPIRE].
R.G. Leigh, O. Parrikar and A.B. Weiss, Holographic geometry of the renormalization group and higher spin symmetries, Phys. Rev. D 89 (2014) 106012 [arXiv:1402.1430] [INSPIRE].
R.G. Leigh, O. Parrikar and A.B. Weiss, Exact renormalization group and higher-spin holography, Phys. Rev. D 91 (2015) 026002 [arXiv:1407.4574] [INSPIRE].
E. Mintun and J. Polchinski, Higher spin holography, RG, and the light cone, arXiv:1411.3151 [INSPIRE].
O. Aharony, S.M. Chester, T. Solberg and E.Y. Urbach, to appear.
V. Rosenhaus, An introduction to the SYK model, J. Phys. A 52 (2019) 323001 [arXiv:1807.03334] [INSPIRE].
A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].
S. Giombi and I.R. Klebanov, One loop tests of higher spin AdS/CFT, JHEP 12 (2013) 068 [arXiv:1308.2337] [INSPIRE].
S. Giombi, I.R. Klebanov and B.R. Safdi, Higher spin AdSd+1/CFTd at one loop, Phys. Rev. D 89 (2014) 084004 [arXiv:1401.0825] [INSPIRE].
E.D. Skvortsov and T. Tran, AdS/CFT in fractional dimension and higher spin gravity at one loop, Universe 3 (2017) 61 [arXiv:1707.00758] [INSPIRE].
D. Anninos, T. Hartman and A. Strominger, Higher spin realization of the dS/CFT correspondence, Class. Quant. Grav. 34 (2017) 015009 [arXiv:1108.5735] [INSPIRE].
A. Jevicki and B. Sakita, The quantum collective field method and its application to the planar limit, Nucl. Phys. B 165 (1980) 511 [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
D.J. Binder and S. Rychkov, Deligne categories in lattice models and quantum field theory, or making sense of O(N) symmetry with non-integer N, JHEP 04 (2020) 117 [arXiv:1911.07895] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Harmonic analysis and mean field theory, JHEP 10 (2019) 217 [arXiv:1809.05111] [INSPIRE].
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].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
N.S. Craigie, V.K. Dobrev and I.T. Todorov, Conformally covariant composite operators in quantum chromodynamics, Annals Phys. 159 (1985) 411 [INSPIRE].
M.R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W-symmetry in AdS3, JHEP 02 (2011) 004 [arXiv:1009.6087] [INSPIRE].
M.R. Gaberdiel, D. Grumiller and D. Vassilevich, Graviton 1-loop partition function for 3-dimensional massive gravity, JHEP 11 (2010) 094 [arXiv:1007.5189] [INSPIRE].
R.K. Gupta and S. Lal, Partition functions for higher-spin theories in AdS, JHEP 07 (2012) 071 [arXiv:1205.1130] [INSPIRE].
S. Giombi, Higher spin — CFT duality, in Theoretical Advanced Study Institute in Elementary Particle Physics: new frontiers in fields and strings, World Scientific, Singapore (2016) [arXiv:1607.02967] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFTd/AdSd+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
C. Iazeolla, E. Sezgin and P. Sundell, Real forms of complex higher spin field equations and new exact solutions, Nucl. Phys. B 791 (2008) 231 [arXiv:0706.2983] [INSPIRE].
E. Sezgin and P. Sundell, On an exact cosmological solution of higher spin gauge theory, Bulg. J. Phys. 33 (2006) 506 [hep-th/0511296] [INSPIRE].
R. de Mello Koch and J.P. Rodrigues, Systematic 1/N corrections for bosonic and fermionic vector models without auxiliary fields, Phys. Rev. D 54 (1996) 7794 [hep-th/9605079] [INSPIRE].
M. Mulokwe and J.P. Rodrigues, Large N bilocals at the infrared fixed point of the three dimensional O(N) invariant vector theory with a quartic interaction, JHEP 11 (2018) 047 [arXiv:1808.00042] [INSPIRE].
S.S. Gubser and I.R. Klebanov, A universal result on central charges in the presence of double trace deformations, Nucl. Phys. B 656 (2003) 23 [hep-th/0212138] [INSPIRE].
L. Fei, S. Giombi and I.R. Klebanov, Critical O(N) models in 6 − ϵ dimensions, Phys. Rev. D 90 (2014) 025018 [arXiv:1404.1094] [INSPIRE].
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].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
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Aharony, O., Chester, S.M. & Urbach, E.Y. A derivation of AdS/CFT for vector models. J. High Energ. Phys. 2021, 208 (2021). https://doi.org/10.1007/JHEP03(2021)208
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DOI: https://doi.org/10.1007/JHEP03(2021)208