Abstract
We develop a formalism to study the implications of causality on OPE coefficients in conformal field theories with large central charge and a sparse spectrum of higher spin operators. The formalism has the interpretation of a new conformal collider-type experiment for these class of CFTs and hence it has the advantage of requiring knowledge only about CFT three-point functions. This is accomplished by considering the holographic null energy operator which was introduced in [1] as a generalization of the averaged null energy operator. Analyticity properties of correlators in the Regge limit imply that the holographic null energy operator is a positive operator in a subspace of the total CFT Hilbert space. Utilizing this positivity condition, we derive bounds on three-point functions 〈TO1O2〉 of the stress tensor with various operators for CFTs with large central charge and a sparse spectrum. After imposing these constraints, we also find that the operator product expansions of all primary operators in the Regge limit have certain universal properties. All of these results are consistent with the expectation that CFTs in this class, irrespective of their microscopic details, admit universal gravity-like holographic dual descriptions. Furthermore, this connection enables us to constrain various inflationary observables such as the amplitude of chiral gravity waves, non-gaussanity of gravity waves and tensor-to-scalar ratio.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Shockwaves from the operator product expansion, arXiv:1709.03597 [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality constraints in conformal field theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, A new spin on causality constraints, JHEP 10 (2016) 141 [arXiv:1601.07904] [INSPIRE].
D.M. Hofman et al., A proof of the conformal collider bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
T. Hartman, S. Kundu and A. Tajdini, Averaged null energy condition from causality, JHEP 07 (2017) 066 [arXiv:1610.05308] [INSPIRE].
C. Cordova, J. Maldacena and G.J. Turiaci, Bounds on OPE coefficients from interference effects in the conformal collider, JHEP 11 (2017) 032 [arXiv:1710.03199] [INSPIRE].
D. Meltzer and E. Perlmutter, Beyond a = c: gravitational couplings to matter and the stress tensor OPE, JHEP 07 (2018) 157 [arXiv:1712.04861] [INSPIRE].
S.D. Chowdhury, J.R. David and S. Prakash, Constraints on parity violating conformal field theories in d = 3, JHEP 11 (2017) 171 [arXiv:1707.03007] [INSPIRE].
M. Brigante et al., The viscosity bound and causality violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [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].
X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [INSPIRE].
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality constraints on corrections to the graviton three-point coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
B. Bellazzini, C. Cheung and G.N. Remmen, Quantum gravity constraints from unitarity and analyticity, Phys. Rev. D 93 (2016) 064076 [arXiv:1509.00851] [INSPIRE].
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].
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].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
C.A. Keller, Phase transitions in symmetric orbifold CFTs and universality, JHEP 03 (2011) 114 [arXiv:1101.4937] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [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].
E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal approximation in AdS/CFT: from shock waves to four-point functions, JHEP 08 (2007) 019 [hep-th/0611122] [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 and J. Penedones, Eikonal approximation in AdS/CFT: resumming the gravitational loop expansion, JHEP 09 (2007) 037 [arXiv:0707.0120] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
G. Mack, D-dimensional conformal field theories with anomalous dimensions as dual resonance models, Bulg. J. Phys. 36 (2009) 214 [arXiv:0909.1024] [INSPIRE].
G. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [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].
I. Heemskerk and J. Sully, More holography from conformal field theory, JHEP 09 (2010) 099 [arXiv:1006.0976] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Analyticity and the holographic S-matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].
A.L. Fitzpatrick et al., A natural language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].
S. El-Showk and K. Papadodimas, Emergent spacetime and holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
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].
A.L. Fitzpatrick and J. Kaplan, AdS field theory from conformal field theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].
V. Gonçalves, J. Penedones and E. Trevisani, Factorization of Mellin amplitudes, JHEP 10 (2015) 040 [arXiv:1410.4185] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS geometry of conformal blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
L.F. Alday and A. Bissi, Unitarity and positivity constraints for CFT at large central charge, JHEP 07 (2017) 044 [arXiv:1606.09593] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Lessons from crossing symmetry at large N, JHEP 06 (2015) 074 [arXiv:1410.4717] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP 12 (2017) 049 [arXiv:1610.09378] [INSPIRE].
M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk phase shift, CFT Regge limit and Einstein gravity, JHEP 06 (2018) 121 [arXiv:1705.02934] [INSPIRE].
M.S. Costa, T. Hansen and J. Penedones, Bounds for OPE coefficients on the Regge trajectory, JHEP 10 (2017) 197 [arXiv:1707.07689] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, arXiv:1805.00098 [INSPIRE].
M. Beccaria and A.A. Tseytlin, C T for higher derivative conformal fields and anomalies of (1, 0) superconformal 6d theories, JHEP 06 (2017) 002 [arXiv:1705.00305] [INSPIRE].
H. Ooguri and C. Vafa, Non-supersymmetric AdS and the swampland, Adv. Theor. Math. Phys. 21 (2017) 1787 [arXiv:1610.01533] [INSPIRE].
T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular hamiltonians for deformed half-spaces and the averaged null energy condition, JHEP 09 (2016) 038 [arXiv:1605.08072] [INSPIRE].
B. Czech et al., A stereoscopic look into the bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP 08 (2016) 162 [arXiv:1606.03307] [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].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
N. Engelhardt and S. Fischetti, The gravity dual of boundary causality, Class. Quant. Grav. 33 (2016) 175004 [arXiv:1604.03944] [INSPIRE].
S. Gao and R.M. Wald, Theorems on gravitational time delay and related issues, Class. Quant. Grav. 17 (2000) 4999 [gr-qc/0007021] [INSPIRE].
G.T. Horowitz and N. Itzhaki, Black holes, shock waves and causality in the AdS/CFT correspondence, JHEP 02 (1999) 010 [hep-th/9901012] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [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].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
J. Bonifacio, K. Hinterbichler, A. Joyce and R.A. Rosen, Massive and massless spin-2 scattering and asymptotic superluminality, JHEP 06 (2018) 075 [arXiv:1712.10020] [INSPIRE].
K. Hinterbichler, A. Joyce and R.A. Rosen, Eikonal scattering and asymptotic superluminality of massless higher spin fields, Phys. Rev. D 97 (2018) 125019 [arXiv:1712.10021] [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].
A. Zhiboedov, A note on three-point functions of conserved currents, arXiv:1206.6370 [INSPIRE].
A. Zhiboedov, On conformal field theories with extremal a/c values, JHEP 04 (2014) 038 [arXiv:1304.6075] [INSPIRE].
D. Li, D. Meltzer and D. Poland, Conformal collider physics from the lightcone bootstrap, JHEP 02 (2016) 143 [arXiv:1511.08025] [INSPIRE].
S. Giombi, S. Prakash and X. Yin, A note on CFT correlators in three dimensions, JHEP 07 (2013) 105 [arXiv:1104.4317] [INSPIRE].
L. Bonora, P. Pasti and M. Bregola, Weyl COCYCLES, Class. Quant. Grav. 3 (1986) 635 [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
F. Bastianelli, S. Frolov and A.A. Tseytlin, Conformal anomaly of (2, 0) tensor multiplet in six-dimensions and AdS/CFT correspondence, JHEP 02 (2000) 013 [hep-th/0001041] [INSPIRE].
N. Boulanger, Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions, Phys. Rev. Lett. 98 (2007) 261302 [arXiv:0706.0340] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, On holographic entanglement entropy and higher curvature gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Conformal a-anomaly of some non-unitary 6d superconformal theories, JHEP 09 (2015) 017 [arXiv:1506.08727] [INSPIRE].
D. Butter, S.M. Kuzenko, J. Novak and S. Theisen, Invariants for minimal conformal supergravity in six dimensions, JHEP 12 (2016) 072 [arXiv:1606.02921] [INSPIRE].
D. Butter, J. Novak and G. Tartaglino-Mazzucchelli, The component structure of conformal supergravity invariants in six dimensions, JHEP 05 (2017) 133 [arXiv:1701.08163] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
W. Fischler and S. Kundu, Membrane paradigm, gravitational Θ-term and gauge/gravity duality, JHEP 04 (2016) 112 [arXiv:1512.01238] [INSPIRE].
W. Fischler and S. Kundu, Physical effects of the gravitational Θ-parameter, Int. J. Mod. Phys. D 25 (2016) 1644022 [arXiv:1612.06010].
A. Lue, L.-M. Wang and M. Kamionkowski, Cosmological signature of new parity violating interactions, Phys. Rev. Lett. 83 (1999) 1506 [astro-ph/9812088] [INSPIRE].
S. Alexander and J. Martin, Birefringent gravitational waves and the consistency check of inflation, Phys. Rev. D 71 (2005) 063526 [hep-th/0410230] [INSPIRE].
J.E. Lidsey et al., Reconstructing the inflation potential: an overview, Rev. Mod. Phys. 69 (1997) 373 [astro-ph/9508078] [INSPIRE].
D. Baumann, H. Lee and G.L. Pimentel, High-scale inflation and the tensor tilt, JHEP 01 (2016) 101 [arXiv:1507.07250] [INSPIRE].
J.M. Maldacena and G.L. Pimentel, On graviton non-gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].
J. Soda, H. Kodama and M. Nozawa, Parity violation in graviton non-gaussianity, JHEP 08 (2011) 067 [arXiv:1106.3228] [INSPIRE].
M. Shiraishi, D. Nitta and S. Yokoyama, Parity violation of gravitons in the CMB bispectrum, Prog. Theor. Phys. 126 (2011) 937 [arXiv:1108.0175] [INSPIRE].
N. Bartolo and G. Orlando, Parity breaking signatures from a Chern-Simons coupling during inflation: the case of non-Gaussian gravitational waves, JCAP 07 (2017) 034 [arXiv:1706.04627] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1805.07393
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Afkhami-Jeddi, N., Kundu, S. & Tajdini, A. A conformal collider for holographic CFTs. J. High Energ. Phys. 2018, 156 (2018). https://doi.org/10.1007/JHEP10(2018)156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2018)156