Abstract
In this paper we study the AdS/CFT correspondence in the operator formalism without assuming the GKPW relation. We explicitly show that the low energy spectrum of the large N limit of CFT, which is realized by a strong coupling gauge theory, is identical to the spectrum of the free gravitational theory in the global AdS spacetime under some assumptions which are expected to be valid. Thus, two theories are equivalent for the low energy region under the assumptions. Using this equivalence, the bulk local field is constructed and the GKPW relation is derived.
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J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in Anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, AdS field theory from conformal field theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].
M. Miyaji et al., Continuous multiscale entanglement renormalization ansatz as holographic surface-state correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].
Y. Nakayama and H. Ooguri, Bulk locality and boundary creating operators, JHEP 10 (2015) 114 [arXiv:1507.04130] [INSPIRE].
H. Verlinde, Poking holes in AdS/CFT: bulk fields from boundary states, arXiv:1505.05069 [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].
S. El-Showk and K. Papadodimas, Emergent spacetime and holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].
D. Kabat, G. Lifschytz and D.A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83 (2011) 106009 [arXiv:1102.2910] [INSPIRE].
D. Kabat, G. Lifschytz, S. Roy and D. Sarkar, Holographic representation of bulk fields with spin in AdS/CFT, Phys. Rev. D 86 (2012) 026004 [arXiv:1204.0126] [INSPIRE].
D. Kabat and G. Lifschytz, CFT representation of interacting bulk gauge fields in AdS, Phys. Rev. D 87 (2013) 086004 [arXiv:1212.3788] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of long-distance AdS physics from the CFT bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
D. Kabat and G. Lifschytz, Bulk equations of motion from CFT correlators, JHEP 09 (2015) 059 [arXiv:1505.03755] [INSPIRE].
D. Kabat and G. Lifschytz, Locality, bulk equations of motion and the conformal bootstrap, JHEP 10 (2016) 091 [arXiv:1603.06800] [INSPIRE].
K. Goto, M. Miyaji and T. Takayanagi, Causal evolutions of bulk local excitations from CFT, JHEP 09 (2016) 130 [arXiv:1605.02835] [INSPIRE].
J.-W. Kim, Explicit reconstruction of the entanglement wedge, JHEP 01 (2017) 131 [arXiv:1607.03605] [INSPIRE].
K. Goto and T. Takayanagi, CFT descriptions of bulk local states in the AdS black holes, JHEP 10 (2017) 153 [arXiv:1704.00053] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Positive energy in Anti-de Sitter backgrounds and gauged extended supergravity, Phys. Lett. B 115 (1982) 197.
A. Ishibashi and R.M. Wald, Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time, Class. Quant. Grav. 21 (2004) 2981 [hep-th/0402184] [INSPIRE].
J.D. Qualls, Lectures on conformal field theory, arXiv:1511.04074 [INSPIRE].
S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, arXiv:1601.05000.
D. Simmons-Duffin, The conformal bootstrap, arXiv:1602.07982 [INSPIRE].
M. Duetsch and K.-H. Rehren, Generalized free fields and the AdS-CFT correspondence, Annales Henri Poincaré 4 (2003) 613 [math-ph/0209035] [INSPIRE].
N. Arkani-Hamed, A.G. Cohen and H. Georgi, (De)constructing dimensions, Phys. Rev. Lett. 86 (2001) 4757 [hep-th/0104005] [INSPIRE].
C.T. Hill, S. Pokorski and J. Wang, Gauge invariant effective Lagrangian for Kaluza-Klein modes, Phys. Rev. D 64 (2001) 105005 [hep-th/0104035] [INSPIRE].
N. Ishibashi, The boundary and crosscap states in conformal field theories, Mod. Phys. Lett. A 4 (1989) 251 [INSPIRE].
Y. Nakayama and H. Ooguri, Bulk local states and crosscaps in holographic CFT, JHEP 10 (2016) 085 [arXiv:1605.00334] [INSPIRE].
D. Harlow and D. Stanford, Operator dictionaries and wave functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop partition functions of 3D gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
N. Iizuka, A. Tanaka and S. Terashima, Exact path integral for 3D quantum gravity, Phys. Rev. Lett. 115 (2015) 161304 [arXiv:1504.05991] [INSPIRE].
M. Honda, N. Iizuka, A. Tanaka and S. Terashima, Exact path integral for 3D quantum gravity II, Phys. Rev. D 93 (2016) 064014 [arXiv:1510.02142] [INSPIRE].
S. Sugishita and S. Terashima, Exact results in supersymmetric field theories on manifolds with boundaries, JHEP 11 (2013) 021 [arXiv:1308.1973] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W -symmetry in AdS 3, JHEP 02 (2011) 004 [arXiv:1009.6087] [INSPIRE].
M. Honda, N. Iizuka, A. Tanaka and S. Terashima, Exact path integral for 3D higher spin gravity, Phys. Rev. D 95 (2017) 046016 [arXiv:1511.07546] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
G. ’t Hooft, On the quantum structure of a black hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
N. Iizuka and S. Terashima, Brick walls for black holes in AdS/CFT, Nucl. Phys. B 895 (2015) 1 [arXiv:1307.5933] [INSPIRE].
L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].
J.-G. Demers, R. Lafrance and R.C. Myers, Black hole entropy without brick walls, Phys. Rev. D 52 (1995) 2245 [gr-qc/9503003] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
S.D. Mathur, What the information paradox is not, arXiv:1108.0302 [INSPIRE].
J. Avery, Hyperspherical Harmonics; applications in quantum theory, Kluwer Academic Publishers, Dordrecht The Netherlands (1989).
A. Chodos and E. Myers, Gravitational contribution to the Casimir energy in Kaluza-Klein theories, Annals Phys. 156 (1984) 412 [INSPIRE].
M.A. Rubin and C.R. Ordonez, Symmetric tensor eigen spectrum of the laplacian on n spheres, J. Math. Phys. 26 (1985) 65 [INSPIRE].
M.A. Rubin and C.R. Ordonez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, UTTG-10-83 (1983).
A. Higuchi, Symmetric tensor spherical harmonics on the N sphere and their application to the de Sitter group SO(N, 1), J. Math. Phys. 28 (1987) 1553 [Erratum ibid. 43 (2002) 6385] [INSPIRE].
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Terashima, S. AdS/CFT correspondence in operator formalism. J. High Energ. Phys. 2018, 19 (2018). https://doi.org/10.1007/JHEP02(2018)019
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DOI: https://doi.org/10.1007/JHEP02(2018)019