Abstract
We discuss universal properties of conformal field theories with holographic duals. A central feature of these theories is the existence of a low-lying sector of operators whose correlators factorize. We demonstrate that factorization can only hold in the large central charge limit. Using conformal invariance and factorization we argue that these operators are naturally represented as fields in AdS as this makes the underlying linearity of the system manifest. In this class of CFTs the solution of the conformal bootstrap conditions can be naturally organized in structures which coincide with Witten diagrams in the bulk. The large value of the central charge suggests that the theory must include a large number of new operators not captured by the factorized sector. Consequently we may think of the AdS hologram as an effective representation of a small sector of the CFT, which is embedded inside a much larger Hilbert space corresponding to the black hole microstates.
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References
N. Seiberg, Emergent spacetime, hep-th/0601234 [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
V. Balasubramanian, S.B. Giddings and A.E. Lawrence, What do CFTs tell us about Anti-de Sitter space-times?, JHEP 03 (1999) 001 [hep-th/9902052] [INSPIRE].
M. Berkooz and H.L. Verlinde, Matrix theory, AdS/CFT and Higgs-Coulomb equivalence, JHEP 11 (1999) 037 [hep-th/9907100] [INSPIRE].
E. Witten, Spacetime reconstruction, talk at JHS 60 Conference, November 3-4, Caltech, U.S.A. (2001), http://quark.caltech.edu/jhs60/witten/1.html.
D. Berenstein, Large-N BPS states and emergent quantum gravity, JHEP 01 (2006) 125 [hep-th/0507203] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
I. Heemskerk and J. Sully, More holography from conformal field theory, JHEP 09 (2010) 099 [arXiv:1006.0976] [INSPIRE].
A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective conformal theory and the flat-space limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
E.P. Verlinde, On the origin of gravity and the laws of newton, JHEP 04 (2011) 029 [arXiv:1001.0785] [INSPIRE].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
R. Jost, The general theory of quantized fields, American Mathematical Society, Providence U.S.A. (1965).
K.-H. Rehren, Algebraic holography, Annales Henri Poincaré 1 (2000) 607 [hep-th/9905179] [INSPIRE].
M. Duetsch and K.-H. Rehren, Generalized free fields and the AdS-CFT correspondence, Annales Henri Poincaré 4 (2003) 613 [math-ph/0209035].
R. Gopakumar, From free fields to AdS, Phys. Rev. D 70 (2004) 025009 [hep-th/0308184] [INSPIRE].
R. Gopakumar, From free fields to AdS. 2, Phys. Rev. D 70 (2004) 025010 [hep-th/0402063] [INSPIRE].
R. Gopakumar, Free field theory as a string theory?, Comptes Rendus Physique 5 (2004) 1111 [hep-th/0409233] [INSPIRE].
R. Gopakumar, From free fields to AdS: III, Phys. Rev. D 72 (2005) 066008 [hep-th/0504229] [INSPIRE].
O. Aharony, J.R. David, R. Gopakumar, Z. Komargodski and S.S. Razamat, Comments on worldsheet theories dual to free large-N gauge theories, Phys. Rev. D 75 (2007) 106006 [hep-th/0703141] [INSPIRE].
O. Aharony and Z. Komargodski, The space-time operator product expansion in string theory duals of field theories, JHEP 01 (2008) 064 [arXiv:0711.1174] [INSPIRE].
I. Klebanov and A. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
A.M. Polyakov, Gauge fields and space-time, Int. J. Mod. Phys. A 17S1 (2002) 119 [hep-th/0110196] [INSPIRE].
B. Sundborg, The hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn/deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
S. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for M N /S(N) orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
O. Lunin and S.D. Mathur, Three point functions for M N /S(N) orbifolds with N = 4 supersymmetry, Commun. Math. Phys. 227 (2002) 385 [hep-th/0103169] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for symmetric product orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].
J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679 [hep-th/9604042] [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].
M.A. Vasiliev, Higher spin gauge theories: star product and AdS space, hep-th/9910096 [INSPIRE].
M. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [INSPIRE].
E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [INSPIRE].
A.C. Petkou, Evaluating the AdS dual of the critical O(N) vector model, JHEP 03 (2003) 049 [hep-th/0302063] [INSPIRE].
R.G. Leigh and A.C. Petkou, Holography of the N = 1 higher spin theory on AdS 4, JHEP 06 (2003) 011 [hep-th/0304217] [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, AdS 4 /CFT 3 construction from collective fields, Phys. Rev. D 83 (2011) 025006 [arXiv:1008.0633] [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].
S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel and T. Hartman, Symmetries of holographic minimal models, JHEP 05 (2011) 031 [arXiv:1101.2910] [INSPIRE].
R. Minasian, G.W. Moore and D. Tsimpis, Calabi-Yau black holes and (0, 4) σ-models, Commun. Math. Phys. 209 (2000) 325 [hep-th/9904217] [INSPIRE].
J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, A bound on the entropy of supergravity?, JHEP 02 (2010) 062 [arXiv:0906.0011] [INSPIRE].
J. de Boer, F. Denef, S. El-Showk, I. Messamah and D. Van den Bleeken, Black hole bound states in AdS 3 × S 2, JHEP 11 (2008) 050 [arXiv:0802.2257] [INSPIRE].
E. Kiritsis, Product CFTs, gravitational cloning, massive gravitons and the space of gravitational duals, JHEP 11 (2006) 049 [hep-th/0608088] [INSPIRE].
O. Aharony, A.B. Clark and A. Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys. Rev. D 74 (2006) 086006 [hep-th/0608089] [INSPIRE].
E. Kiritsis and V. Niarchos, (Multi)matrix models and interacting clones of Liouville gravity, JHEP 08 (2008) 044 [arXiv:0805.4234] [INSPIRE].
E. Kiritsis and V. Niarchos, Interacting string multi-verses and holographic instabilities of massive gravity, Nucl. Phys. B 812 (2009) 488 [arXiv:0808.3410] [INSPIRE].
V. Niarchos, Multi-string theories, massive gravity and the AdS/CFT correspondence, Fortsch. Phys. 57 (2009) 646 [arXiv:0901.2108] [INSPIRE].
E. Kiritsis and V. Niarchos, Large-N limits of 2D CFTs, quivers and AdS 3 duals, JHEP 04 (2011) 113 [arXiv:1011.5900] [INSPIRE].
I. Bakas and E. Kiritsis, Bosonic realization of a universal W algebra and Z(∞) parafermions, Nucl. Phys. B 343 (1990) 185 [Erratum ibid. B 350 (1991) 512] [INSPIRE].
I. Bakas and E. Kiritsis, Grassmannian coset models and unitary representations of W (∞), Mod. Phys. Lett. A 5 (1990) 2039 [INSPIRE].
I. Bakas and E. Kiritsis, Beyond the large-N limit: nonlinear W (∞) as symmetry of the SL(2, R)/U(1) coset model, Int. J. Mod. Phys. A 7S1A (1992) 55 [hep-th/9109029] [INSPIRE].
I. Bakas and E. Kiritsis, Target space description of W (∞) symmetry in coset models, Phys. Lett. B 301 (1993) 49 [hep-th/9211083] [INSPIRE].
H. Osborn and A. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
F. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
F. Caracciolo and V.S. Rychkov, Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Central charge bounds in 4D conformal field theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].
A.C. Petkou and N.D. Vlachos, Finite size effects and operator product expansions in a CFT for d > 2, Phys. Lett. B 446 (1999) 306 [hep-th/9803149] [INSPIRE].
A.C. Petkou and N.D. Vlachos, Finite size and finite temperature effects in the conformally invariant O(N) vector model for 2 < d < 4, hep-th/9809096 [INSPIRE].
E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].
M.R. Gaberdiel, S. Gukov, C.A. Keller, G.W. Moore and H. Ooguri, Extremal N = (2, 2) 2D conformal field theories and constraints of modularity, Commun. Num. Theor. Phys. 2 (2008) 743 [arXiv:0805.4216] [INSPIRE].
S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
S. Hellerman and C. Schmidt-Colinet, Bounds for state degeneracies in 2D conformal field theory, JHEP 08 (2011) 127 [arXiv:1007.0756] [INSPIRE].
A. Castro, A. Lepage-Jutier and A. Maloney, Higher spin theories in AdS 3 and a gravitational exclusion principle, JHEP 01 (2011) 142 [arXiv:1012.0598] [INSPIRE].
H. Bloete, J.L. Cardy and M. Nightingale, Conformal invariance, the central charge and universal finite size amplitudes at criticality, Phys. Rev. Lett. 56 (1986) 742 [INSPIRE].
P. Kovtun and A. Ritz, Black holes and universality classes of critical points, Phys. Rev. Lett. 100 (2008) 171606 [arXiv:0801.2785] [INSPIRE].
L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [INSPIRE].
J. de Boer, K. Papadodimas and E. Verlinde, Holographic neutron stars, JHEP 10 (2010) 020 [arXiv:0907.2695] [INSPIRE].
X. Arsiwalla, J. de Boer, K. Papadodimas and E. Verlinde, Degenerate stars and gravitational collapse in AdS/CFT, JHEP 01 (2011) 144 [arXiv:1010.5784] [INSPIRE].
I. Bena, On the construction of local fields in the bulk of AdS 5 and other spaces, Phys. Rev. D 62 (2000) 066007 [hep-th/9905186] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: a boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [hep-th/0506118] [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].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: a holographic description of the black hole interior, Phys. Rev. D 75 (2007) 106001 [Erratum ibid. D 75 (2007) 129902] [hep-th/0612053] [INSPIRE].
H. Liu, Scattering in Anti-de Sitter space and operator product expansion, Phys. Rev. D 60 (1999) 106005 [hep-th/9811152] [INSPIRE].
H. Liu and A.A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Comments on 4 point functions in the CFT/AdS correspondence, Phys. Lett. B 452 (1999) 61 [hep-th/9808006] [INSPIRE].
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].
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].
E. D’Hoker, S.D. Mathur, A. Matusis and L. Rastelli, The operator product expansion of N = 4 SYM and the 4 point functions of supergravity, Nucl. Phys. B 589 (2000) 38 [hep-th/9911222] [INSPIRE].
L. Hoffmann, A.C. Petkou and W. Rühl, A note on the analyticity of AdS scalar exchange graphs in the crossed channel, Phys. Lett. B 478 (2000) 320 [hep-th/0002025] [INSPIRE].
L. Hoffmann, A.C. Petkou and W. Rühl, Aspects of the conformal operator product expansion in AdS/CFT correspondence, Adv. Theor. Math. Phys. 4 (2002) 571 [hep-th/0002154] [INSPIRE].
D.E. Diaz and H. Dorn, On the AdS higher spin/O(N) vector model correspondence: degeneracy of the holographic image, JHEP 07 (2006) 022 [hep-th/0603084] [INSPIRE].
J. Polchinski, S matrices from AdS space-time, hep-th/9901076 [INSPIRE].
S.B. Giddings, Flat space scattering and bulk locality in the AdS/CFT correspondence, Phys. Rev. D 61 (2000) 106008 [hep-th/9907129] [INSPIRE].
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].
M. Gary and S.B. Giddings, The flat space S-matrix from the AdS/CFT correspondence?, Phys. Rev. D 80 (2009) 046008 [arXiv:0904.3544] [INSPIRE].
A. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
G. Festuccia and H. Liu, The arrow of time, black holes and quantum mixing of large-N Yang-Mills theories, JHEP 12 (2007) 027 [hep-th/0611098] [INSPIRE].
D.M. Hofman, Higher derivative gravity, causality and positivity of energy in a UV complete QFT, Nucl. Phys. B 823 (2009) 174 [arXiv:0907.1625] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, AdS 7 /CFT 6 , Gauss-Bonnet gravity and viscosity bound, JHEP 03 (2010) 087 [arXiv:0910.5347] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Lovelock gravities and black holes, JHEP 06 (2010) 008 [arXiv:0912.1877] [INSPIRE].
M. Kulaxizi and A. Parnachev, Energy flux positivity and unitarity in CFTs, Phys. Rev. Lett. 106 (2011) 011601 [arXiv:1007.0553] [INSPIRE].
S. Raju, BCFW for Witten diagrams, Phys. Rev. Lett. 106 (2011) 091601 [arXiv:1011.0780] [INSPIRE].
A. Zamolodchikov, Renormalization group and perturbation theory near fixed points in two-dimensional field theory, Sov. J. Nucl. Phys. 46 (1987) 1090 [Yad. Fiz. 46 (1987) 1819] [INSPIRE].
A. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [INSPIRE].
K.A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
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El-Showk, S., Papadodimas, K. Emergent spacetime and holographic CFTs. J. High Energ. Phys. 2012, 106 (2012). https://doi.org/10.1007/JHEP10(2012)106
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DOI: https://doi.org/10.1007/JHEP10(2012)106