Abstract
We consider the operator product expansion (OPE) structure of scalar primary operators in a generic Lorentzian CFT and its dual description in a gravitational theory with one extra dimension. The OPE can be decomposed into certain bi-local operators transforming as the irreducible representations under conformal group, called the OPE blocks. We show the OPE block is given by integrating a higher spin field along a geodesic in the Lorentzian AdS space-time when the two operators are space-like separated. When the two operators are time-like separated however, we find the OPE block has a peculiar representation where the dual gravitational theory is not defined on the AdS space-time but on a hyperboloid with an additional time coordinate and Minkowski space-time on its boundary. This differs from the surface Witten diagram proposal for the time-like OPE block, but in two dimensions we reproduce it consistently using a kinematical duality between a pair of time-like separated points and space-like ones.
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R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics (2016) [DOI] [arXiv:1601.05000].
D. Simmons-Duffin, The Conformal Bootstrap, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., 1–26 June 2015, pp. 1–74 (2017) [DOI] [arXiv:1602.07982] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques and Applications, Rev. Mod. Phys.91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
K.G. Wilson, Nonlagrangian models of current algebra, Phys. Rev.179 (1969) 1499 [INSPIRE].
K.G. Wilson and W. Zimmermann, Operator product expansions and composite field operators in the general framework of quantum field theory, Commun. Math. Phys.24 (1972) 87 [INSPIRE].
A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett.12 (1970) 381 [INSPIRE].
E.J. Schreier, Conformal symmetry and three-point functions, Phys. Rev.D 3 (1971) 980 [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].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys.231 (1994) 311 [hep-th/9307010] [INSPIRE].
J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys.B 483 (1997) 431 [hep-th/9605009] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math.37 (1936) 429.
G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys.53 (1969) 174 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions, Nucl. Phys.B 767 (2007) 327 [hep-th/0611123] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP08 (2007) 019 [hep-th/0611122] [INSPIRE].
L. Cornalba, Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange, arXiv:0710.5480 [INSPIRE].
L. Cornalba, M.S. Costa and J. Penedones, Eikonal approximation in AdS/CFT: Resumming the gravitational loop expansion, JHEP09 (2007) 037 [arXiv:0707.0120] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
T. Hartman, S. Kundu and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP07 (2017) 066 [arXiv:1610.05308] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Shockwaves from the Operator Product Expansion, JHEP03 (2019) 201 [arXiv:1709.03597] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional Conformal Blocks, Phys. Rev. Lett.117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
M. Isachenkov and V. Schomerus, Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory, JHEP07 (2018) 180 [arXiv:1711.06609] [INSPIRE].
M. Isachenkov, P. Liendo, Y. Linke and V. Schomerus, Calogero-Sutherland Approach to Defect Blocks, JHEP10 (2018) 204 [arXiv:1806.09703] [INSPIRE]e.
K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Commun. Math. Phys.31 (1973) 83 [INSPIRE].
K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s Functions. 2., Commun. Math. Phys.42 (1975) 281 [INSPIRE].
M. Lüscher and G. Mack, Global Conformal Invariance in Quantum Field Theory, Commun. Math. Phys.41 (1975) 203 [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
V.M. Braun, G.P. Korchemsky and D. Müller, The Uses of conformal symmetry in QCD, Prog. Part. Nucl. Phys.51 (2003) 311 [hep-ph/0306057] [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Manifestly conformal covariant operator-product expansion, Lett. Nuovo Cim.2S2 (1971) 1363 [INSPIRE].
B. Carneiro da Cunha and M. Guica, Exploring the BTZ bulk with boundary conformal blocks, arXiv:1604.07383 [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP07 (2016) 129 [arXiv:1604.03110] [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. Das, Comments on spinning OPE blocks in AdS3/CFT2, Phys. Lett.B 792 (2019) 397 [arXiv:1811.09375] [INSPIRE].
S. Ferrara, P. Gatto and A.F. Grilla, Conformal algebra in space-time and operator product expansion, Springer Tracts Mod. Phys.67 (1973) 1.
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].
D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
H.-Y. Chen, E.-J. Kuo and H. Kyono, Anatomy of Geodesic Witten Diagrams, JHEP05 (2017) 070 [arXiv:1702.08818] [INSPIRE].
A. Castro, E. Llabrés and F. Rejon-Barrera, Geodesic Diagrams, Gravitational Interactions & OPE Structures, JHEP06 (2017) 099 [arXiv:1702.06128] [INSPIRE].
E. Dyer, D.Z. Freedman and J. Sully, Spinning Geodesic Witten Diagrams, JHEP11 (2017) 060 [arXiv:1702.06139] [INSPIRE].
C. Sleight and M. Taronna, Spinning Witten Diagrams, JHEP06 (2017) 100 [arXiv:1702.08619] [INSPIRE].
M. Gillioz, X. Lu and M.A. Luty, Scale Anomalies, States and Rates in Conformal Field Theory, JHEP04 (2017) 171 [arXiv:1612.07800] [INSPIRE].
M. Gillioz, Momentum-space conformal blocks on the light cone, JHEP10 (2018) 125 [arXiv:1807.07003] [INSPIRE].
S. Ferrara, A.F. Grillo and G. Parisi, Nonequivalence between conformal covariant Wilson expansion in euclidean and Minkowski space, Lett. Nuovo Cim.5S2 (1972) 147 [INSPIRE].
S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, Covariant expansion of the conformal four-point function, Nucl. Phys.B 49 (1972) 77 [Erratum ibid.B 53 (1973) 643] [INSPIRE].
V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev.D 13 (1976) 887 [INSPIRE].
D. Sarkar and X. Xiao, Holographic Representation of Higher Spin Gauge Fields, Phys. Rev.D 91 (2015) 086004 [arXiv:1411.4657] [INSPIRE].
J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP08 (2016) 162 [arXiv:1606.03307] [INSPIRE].
A. Gadde, Conformal constraints on defects, JHEP01 (2020) 038 [arXiv:1602.06354] [INSPIRE].
M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects, JHEP01 (2018) 013 [arXiv:1710.11165] [INSPIRE].
S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products, Lett. Nuovo Cim.4S2 (1972) 115 [INSPIRE].
G. Mack, Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory, Commun. Math. Phys.53 (1977) 155 [INSPIRE].
B. Schroer, J.A. Swieca and A.H. Volkel, Global Operator Expansions in Conformally Invariant Relativistic Quantum Field Theory, Phys. Rev.D 11 (1975) 1509 [INSPIRE].
J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS Loops and 6j Symbols, JHEP03 (2019) 052 [arXiv:1808.00612] [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].
M. Guica, Bulk fields from the boundary OPE, arXiv:1610.08952 [INSPIRE].
Y. Satoh and J. Troost, On time dependent AdS/CFT, JHEP01 (2003) 027 [hep-th/0212089] [INSPIRE].
R.S. Erramilli, L.V. Iliesiu and P. Kravchuk, Recursion relation for general 3d blocks, JHEP12 (2019) 116 [arXiv:1907.11247] [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].
I.A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography, JHEP05 (2014) 053 [arXiv:1403.3426] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
S.R. Das, A. Ghosh, A. Jevicki and K. Suzuki, Space-Time in the SYK Model, JHEP07 (2018) 184 [arXiv:1712.02725] [INSPIRE].
T.G. Raben and C.-I. Tan, Minkowski conformal blocks and the Regge limit for Sachdev-Ye-Kitaev-like models, Phys. Rev.D 98 (2018) 086009 [arXiv:1801.04208] [INSPIRE].
V.A. Smirnov, Feynman integral calculus, Springer (2006) [INSPIRE].
H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators, JHEP07 (2018) 136 [arXiv:1805.11107] [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial x Space Technique, Nucl. Phys.B 174 (1980) 345 [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, Academic Press (2014).
H. Bateman, Tables of integral transforms, vol. 2, McGraw-Hill Book Company (1954).
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Chen, HY., Chen, LC., Kobayashi, N. et al. The gravity dual of Lorentzian OPE blocks. J. High Energ. Phys. 2020, 139 (2020). https://doi.org/10.1007/JHEP04(2020)139
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DOI: https://doi.org/10.1007/JHEP04(2020)139