Abstract
For two-dimensional holographic CFTs, we demonstrate the role of Berry phases for relating the non-factorization of the Hilbert space to the presence of wormholes. The wormholes are characterized by a non-exact symplectic form that gives rise to the Berry phase. For wormholes connecting two spacelike regions in gravitational spacetimes, we find that the non-exactness is linked to a variable appearing in the phase space of the boundary CFT. This variable corresponds to a loop integral in the bulk. Through this loop integral, non-factorization becomes apparent in the dual entangled CFTs. Furthermore, we classify Berry phases in holographic CFTs based on the type of dual bulk diffeomorphism involved. We distinguish between Virasoro, gauge and modular Berry phases, each corresponding to a spacetime wormhole geometry in the bulk. Using kinematic space, we extend a relation between the modular Hamiltonian and the Berry curvature to the finite temperature case. We find that the Berry curvature, given by the Crofton form, characterizes the topological transition of the entanglement entropy in presence of a black hole.
References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
H. Verlinde, Wormholes in Quantum Mechanics, arXiv:2105.02129 [INSPIRE].
M.V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. Lond. A 392 (1984) 45.
F.S. Nogueira, S. Banerjee, M. Dorband, R. Meyer, J.v.d. Brink and J. Erdmenger, Geometric phases distinguish entangled states in wormhole quantum mechanics, Phys. Rev. D 105 (2022) L081903 [arXiv:2109.06190] [INSPIRE].
A. Bhattacharyya, L.K. Joshi and B. Sundar, Quantum information scrambling: from holography to quantum simulators, Eur. Phys. J. C 82 (2022) 458 [arXiv:2111.11945] [INSPIRE].
B. Oblak, Berry Phases on Virasoro Orbits, JHEP 10 (2017) 114 [arXiv:1703.06142] [INSPIRE].
G. Compère, P. Mao, A. Seraj and M.M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons, JHEP 01 (2016) 080 [arXiv:1511.06079] [INSPIRE].
M.M. Sheikh-Jabbari and H. Yavartanoo, On 3d bulk geometry of Virasoro coadjoint orbits: orbit invariant charges and Virasoro hair on locally AdS3 geometries, Eur. Phys. J. C 76 (2016) 493 [arXiv:1603.05272] [INSPIRE].
J. de Boer, R. Espíndola, B. Najian, D. Patramanis, J. van der Heijden and C. Zukowski, Virasoro entanglement Berry phases, JHEP 03 (2022) 179 [arXiv:2111.05345] [INSPIRE].
M. Henneaux, W. Merbis and A. Ranjbar, Asymptotic dynamics of AdS3 gravity with two asymptotic regions, JHEP 03 (2020) 064 [arXiv:1912.09465] [INSPIRE].
D.L. Jafferis and D.K. Kolchmeyer, Entanglement Entropy in Jackiw-Teitelboim Gravity, arXiv:1911.10663 [INSPIRE].
H. Verlinde, ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge, arXiv:2003.13117 [INSPIRE].
B. Czech, L. Lamprou, S. Mccandlish and J. Sully, Modular Berry Connection for Entangled Subregions in AdS/CFT, Phys. Rev. Lett. 120 (2018) 091601 [arXiv:1712.07123] [INSPIRE].
B. Czech, L. Lamprou and L. Susskind, Entanglement Holonomies, arXiv:1807.04276 [INSPIRE].
B. Czech, J. De Boer, D. Ge and L. Lamprou, A modular sewing kit for entanglement wedges, JHEP 11 (2019) 094 [arXiv:1903.04493] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
A. Alekseev and S.L. Shatashvili, Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity, Nucl. Phys. B 323 (1989) 719 [INSPIRE].
A. Alekseev and S.L. Shatashvili, From geometric quantization to conformal field theory, Commun. Math. Phys. 128 (1990) 197 [INSPIRE].
E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].
P. Caputa and J.M. Magan, Quantum Computation as Gravity, Phys. Rev. Lett. 122 (2019) 231302 [arXiv:1807.04422] [INSPIRE].
J. Erdmenger, M. Gerbershagen and A.-L. Weigel, Complexity measures from geometric actions on Virasoro and Kac-Moody orbits, JHEP 11 (2020) 003 [arXiv:2004.03619] [INSPIRE].
C. Fefferman and C.R. Graham, Conformal invariants, in Élie Cartan et les mathématiques d’aujourd’hui - Lyon, 25-29 juin 1984, in Astérisque. S131 (1985).
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [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].
M.M. Sheikh-Jabbari and H. Yavartanoo, Excitation entanglement entropy in two dimensional conformal field theories, Phys. Rev. D 94 (2016) 126006 [arXiv:1605.00341] [INSPIRE].
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [INSPIRE].
A. Kundu, Wormholes and holography: an introduction, Eur. Phys. J. C 82 (2022) 447 [arXiv:2110.14958] [INSPIRE].
J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263] [INSPIRE].
J. Cotler and K. Jensen, AdS3 gravity and random CFT, JHEP 04 (2021) 033 [arXiv:2006.08648] [INSPIRE].
D. Harlow and D. Jafferis, The Factorization Problem in Jackiw-Teitelboim Gravity, JHEP 02 (2020) 177 [arXiv:1804.01081] [INSPIRE].
K. Papadodimas and S. Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys. Rev. D 93 (2016) 084049 [arXiv:1503.08825] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians, JHEP 09 (2016) 068 [arXiv:1412.8465] [INSPIRE].
J.J. Bisognano and E.H. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16 (1975) 985 [INSPIRE].
J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
C.T. Asplund, N. Callebaut and C. Zukowski, Equivalence of Emergent de Sitter Spaces from Conformal Field Theory, JHEP 09 (2016) 154 [arXiv:1604.02687] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
T. Faulkner, M. Li and H. Wang, A modular toolkit for bulk reconstruction, JHEP 04 (2019) 119 [arXiv:1806.10560] [INSPIRE].
B. Czech and L. Lamprou, Holographic definition of points and distances, Phys. Rev. D 90 (2014) 106005 [arXiv:1409.4473] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
X. Huang and C.-T. Ma, Berry Curvature and Riemann Curvature in Kinematic Space with Spherical Entangling Surface, Fortsch. Phys. 69 (2021) 2000048 [arXiv:2003.12252] [INSPIRE].
T. Hartman and N. Afkhami-Jeddi, Speed Limits for Entanglement, arXiv:1512.02695 [INSPIRE].
Y.O. Nakagawa, G. Sárosi and T. Ugajin, Chaos and relative entropy, JHEP 07 (2018) 002 [arXiv:1805.01051] [INSPIRE].
D. Blanco, A. Garbarz and G. Pérez-Nadal, Entanglement of a chiral fermion on the torus, JHEP 09 (2019) 076 [arXiv:1906.07057] [INSPIRE].
P. Fries and I.A. Reyes, Entanglement Spectrum of Chiral Fermions on the Torus, Phys. Rev. Lett. 123 (2019) 211603 [arXiv:1905.05768] [INSPIRE].
M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
R. Abt, J. Erdmenger, H. Hinrichsen, C.M. Melby-Thompson, R. Meyer, C. Northe et al., Topological Complexity in AdS3/CFT2, Fortsch. Phys. 66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].
R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP 01 (2019) 012 [arXiv:1805.10298] [INSPIRE].
C. Chowdhury, V. Godet, O. Papadoulaki and S. Raju, Holography from the Wheeler-DeWitt equation, JHEP 03 (2022) 019 [arXiv:2107.14802] [INSPIRE].
S. Raju, Failure of the split property in gravity and the information paradox, Class. Quant. Grav. 39 (2022) 064002 [arXiv:2110.05470] [INSPIRE].
S. Leutheusser and H. Liu, Causal connectability between quantum systems and the black hole interior in holographic duality, arXiv:2110.05497 [INSPIRE].
S. Leutheusser and H. Liu, Emergent times in holographic duality, arXiv:2112.12156 [INSPIRE].
E. Witten, Gravity and the Crossed Product, arXiv:2112.12828 [INSPIRE].
R. Emparan, B. Grado-White, D. Marolf and M. Tomasevic, Multi-mouth Traversable Wormholes, JHEP 05 (2021) 032 [arXiv:2012.07821] [INSPIRE].
R.F. Penna and C. Zukowski, Kinematic space and the orbit method, JHEP 07 (2019) 045 [arXiv:1812.02176] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2202.11717
Rights and permissions
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.
About this article
Cite this article
Banerjee, S., Dorband, M., Erdmenger, J. et al. Berry phases, wormholes and factorization in AdS/CFT. J. High Energ. Phys. 2022, 162 (2022). https://doi.org/10.1007/JHEP08(2022)162
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2022)162
Keywords
- AdS-CFT Correspondence
- Gauge-Gravity Correspondence