Abstract
This paper revisits the question of reconstructing bulk gauge fields as boundary operators in AdS/CFT. In the presence of the wormhole dual to the thermofield double state of two CFTs, the existence of bulk gauge fields is in some tension with the microscopic tensor factorization of the Hilbert space. I explain how this tension can be resolved by splitting the gauge field into charged constituents, and I argue that this leads to a new argument for the “principle of completeness”, which states that the charge lattice of a gauge theory coupled to gravity must be fully populated. I also claim that it leads to a new motivation for (and a clarification of) the “weak gravity conjecture”, which I interpret as a strengthening of this principle. This setup gives a simple example of a situation where describing low-energy bulk physics in CFT language requires knowledge of high-energy bulk physics. This contradicts to some extent the notion of “effective conformal field theory”, but in fact is an expected feature of the resolution of the black hole information problem. An analogous factorization issue exists also for the gravitational field, and I comment on several of its implications for reconstructing black hole interiors and the emergence of spacetime more generally.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [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].
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].
I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].
I.A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography, JHEP 05 (2014) 053 [arXiv:1403.3426] [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].
I. Heemskerk, Construction of Bulk Fields with Gauge Redundancy, JHEP 09 (2012) 106 [arXiv:1201.3666] [INSPIRE].
K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
D. Harlow, Aspects of the Papadodimas-Raju Proposal for the Black Hole Interior, JHEP 11 (2014) 055 [arXiv:1405.1995] [INSPIRE].
D. Kabat and G. Lifschytz, Decoding the hologram: Scalar fields interacting with gravity, Phys. Rev. D 89 (2014) 066010 [arXiv:1311.3020] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
S.B. Giddings, Hilbert space structure in quantum gravity: an algebraic perspective, JHEP 12 (2015) 099 [arXiv:1503.08207] [INSPIRE].
W. Donnelly and S.B. Giddings, Diffeomorphism-invariant observables and their nonlocal algebra, arXiv:1507.07921 [INSPIRE].
W. Donnelly, D. Marolf and E. Mintun, Combing gravitational hair in 2 + 1 dimensions, Class. Quant. Grav. 33 (2016) 025010 [arXiv:1510.00672] [INSPIRE].
D. Engelhardt, B. Freivogel and N. Iqbal, Electric fields and quantum wormholes, Phys. Rev. D 92 (2015) 064050 [arXiv:1504.06336] [INSPIRE].
D. Marolf and S.F. Ross, Boundary Conditions and New Dualities: Vector Fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115 (2015) 211601 [arXiv:1502.06692] [INSPIRE].
J. Polchinski, Monopoles, duality and string theory, Int. J. Mod. Phys. A 19S1 (2004) 145 [hep-th/0304042] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
J.B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D 11 (1975) 395 [INSPIRE].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
D. Radicevic, Notes on Entanglement in Abelian Gauge Theories, arXiv:1404.1391 [INSPIRE].
K.-W. Huang, Central Charge and Entangled Gauge Fields, Phys. Rev. D 92 (2015) 025010 [arXiv:1412.2730] [INSPIRE].
D. Radicevic, Entanglement in Weakly Coupled Lattice Gauge Theories, arXiv:1509.08478 [INSPIRE].
S.-S. Lee, TASI Lectures on Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems, arXiv:1009.5127 [INSPIRE].
A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear σ-models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].
E. Witten, Instantons, the Quark Model and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
A.M. Polyakov, Gauge Fields and Strings, Contemp. Concepts Phys. 3 (1987) 1.
W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, arXiv:1506.05792 [INSPIRE].
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
B. Heidenreich, M. Reece and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, arXiv:1509.06374 [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, 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].
A.L. Fitzpatrick, J. Kaplan, E. Katz and L. Randall, Decoupling of High Dimension Operators from the Low Energy Sector in Holographic Models, arXiv:1304.3458 [INSPIRE].
M. Dodelson and E. Silverstein, String-theoretic breakdown of effective field theory near black hole horizons, arXiv:1504.05536 [INSPIRE].
D. Marolf and A.C. Wall, Eternal Black Holes and Superselection in AdS/CFT, Class. Quant. Grav. 30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
J. Polchinski and J. Sully, Wilson Loop Renormalization Group Flows, JHEP 10 (2011) 059 [arXiv:1104.5077] [INSPIRE].
S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press, (2013) [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].
T. Senthil, Deconfined Quantum Critical Points, Science 303 (2004) 1490 [INSPIRE].
T. Senthil, L. Balents, S. Sachdev, A. Vishwanath and M.P. Fisher, Quantum criticality beyond the landau-ginzburg-wilson paradigm, Phys. Rev. B 70 (2004) 144407 [cond-mat/0312617].
M.A. Metlitski, M. Hermele, T. Senthil and M.P.A. Fisher, Monopoles in CP**(N-1) model via the state-operator correspondence, Phys. Rev. B 78 (2008) 214418 [arXiv:0809.2816] [INSPIRE].
E. Dyer, M. Mezei, S.S. Pufu and S. Sachdev, Scaling dimensions of monopole operators in the \( \mathbb{C}{\mathrm{\mathbb{P}}}^N{{}^{{}_b}}^{-1} \) theory in 2 + 1 dimensions, JHEP 06 (2015) 037 [arXiv:1504.00368] [INSPIRE].
M. Stone, Lattice Formulation of the \( \mathbb{C}{{\mathrm{\mathbb{P}}}^N}^{-1} \) Nonlinear σ Models, Nucl. Phys. B 152 (1979) 97 [INSPIRE].
E. Rabinovici and S. Samuel, The \( \mathbb{C}{{\mathrm{\mathbb{P}}}^N}^{-1} \) Model: A Strong Coupling Lattice Approach, Phys. Lett. B 101 (1981) 323 [INSPIRE].
P. Senjanovic, Path Integral Quantization of Field Theories with Second Class Constraints, Annals Phys. 100 (1976) 227 [Erratum ibid. 209 (1991) 248] [INSPIRE].
N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].
K.G. Wilson, Confinement of Quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].
E.H. Fradkin and S.H. Shenker, Phase Diagrams of Lattice Gauge Theories with Higgs Fields, Phys. Rev. D 19 (1979) 3682 [INSPIRE].
T. Banks and E. Rabinovici, Finite Temperature Behavior of the Lattice Abelian Higgs Model, Nucl. Phys. B 160 (1979) 349 [INSPIRE].
T. Banks, M. Johnson and A. Shomer, A Note on Gauge Theories Coupled to Gravity, JHEP 09 (2006) 049 [hep-th/0606277] [INSPIRE].
Y. Nakayama and Y. Nomura, Weak gravity conjecture in the AdS/CFT correspondence, Phys. Rev. D 92 (2015) 126006 [arXiv:1509.01647] [INSPIRE].
L. Susskind, Trouble for remnants, hep-th/9501106 [INSPIRE].
S. Dimopoulos, S. Kachru, J. McGreevy and J.G. Wacker, N-flation, JCAP 08 (2008) 003 [hep-th/0507205] [INSPIRE].
X. Calmet, S.D.H. Hsu and D. Reeb, Quantum gravity at a TeV and the renormalization of Newton’s constant, Phys. Rev. D 77 (2008) 125015 [arXiv:0803.1836] [INSPIRE].
E. Rabinovici and M. Smolkin, On the dynamical generation of the Maxwell term and scale invariance, JHEP 07 (2011) 040 [arXiv:1102.5035] [INSPIRE].
C. Cheung and G.N. Remmen, Naturalness and the Weak Gravity Conjecture, Phys. Rev. Lett. 113 (2014) 051601 [arXiv:1402.2287] [INSPIRE].
C. Cheung and G.N. Remmen, Infrared Consistency and the Weak Gravity Conjecture, JHEP 12 (2014) 087 [arXiv:1407.7865] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
A.C. Wall, Maximin Surfaces and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians, arXiv:1412.8465 [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
S. Carlip and C. Teitelboim, The off-shell black hole, Class. Quant. Grav. 12 (1995) 1699 [gr-qc/9312002] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
L. Susskind, New Concepts for Old Black Holes, arXiv:1311.3335 [INSPIRE].
D. Marolf and J. Polchinski, Gauge/Gravity Duality and the Black Hole Interior, Phys. Rev. Lett. 111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].
D. Harlow and D. Stanford, Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, arXiv:1409.1231 [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: 1510.07911
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Harlow, D. Wormholes, emergent gauge fields, and the weak gravity conjecture. J. High Energ. Phys. 2016, 122 (2016). https://doi.org/10.1007/JHEP01(2016)122
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)122