Abstract
Defining quantum information quantities directly in bulk quantum gravity is a difficult problem due to the fluctuations of spacetime. Some progress was made recently in [1], which provided a bulk interpretation of the Bekenstein Hawking formula for two sided BTZ black holes in terms of the entanglement entropy of gravitational edge modes. We generalize those results to give a bulk entanglement entropy interpretation of the quantum extremal surface formula in AdS3 gravity, as applied to a single interval in the boundary theory. Our computation further supports the proposal that AdS3 gravity can be viewed as a topological phase in which the bulk gravity edge modes are anyons that transform under the quantum group \({{\text{SL}}}_{q}^{+}\left(2,{\mathbb{R}}\right)\). These edge modes appear when we cut open the Euclidean path integral along bulk co-dimension 2 slices, and satisfies a shrinkable boundary condition which ensures that the Gibbons-Hawking calculation gives the correct state counting.
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T.G. Mertens, J. Simón and G. Wong, A proposal for 3d quantum gravity and its bulk factorization, JHEP 06 (2023) 134 [arXiv:2210.14196] [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].
N. Engelhardt and A.C. Wall, Extremal Surface Barriers, JHEP 03 (2014) 068 [arXiv:1312.3699] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
A. Almheiri et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
D.L. Jafferis and D.K. Kolchmeyer, Entanglement Entropy in Jackiw-Teitelboim Gravity, arXiv:1911.10663 [INSPIRE].
S.D. Mathur, Fuzzballs and black hole thermodynamics, arXiv:1401.4097 [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. Blommaert, T.G. Mertens and H. Verschelde, Fine Structure of Jackiw-Teitelboim Quantum Gravity, JHEP 09 (2019) 066 [arXiv:1812.00918] [INSPIRE].
L. McGough and H. Verlinde, Bekenstein-Hawking Entropy as Topological Entanglement Entropy, JHEP 11 (2013) 208 [arXiv:1308.2342] [INSPIRE].
J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263] [INSPIRE].
J. Lin, A new look at the entanglement entropy of a single interval in a 2d CFT, arXiv:2107.12634 [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].
E. Dyer and G. Gur-Ari, 2D CFT Partition Functions at Late Times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].
S. Ghosh, R.M. Soni and S.P. Trivedi, On The Entanglement Entropy For Gauge Theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [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].
I.C.-H. Ip, Representation of the Quantum Plane, its Quantum Double and Harmonic Analysis on \({GL}_{q}^{+}\left(2,R\right)\), arXiv:1108.5365.
D. Kazhdan and G. Lusztig, Tensor structures arising from affine lie algebras. iii, J. Am. Math. Soc. 6 (1993) 335.
B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
W.Z. Chua and Y. Jiang, Hartle-Hawking state and its factorization in 3d gravity, arXiv:2309.05126 [INSPIRE].
W.Z. Chua and T. Hartman, Black hole wavefunctions and microcanonical states, arXiv:2309.05041 [INSPIRE].
L.Y. Hung and G. Wong, Entanglement branes and factorization in conformal field theory, Phys. Rev. D 104 (2021) 026012 [arXiv:1912.11201] [INSPIRE].
W. Donnelly and G. Wong, Entanglement branes, modular flow, and extended topological quantum field theory, JHEP 10 (2019) 016 [arXiv:1811.10785] [INSPIRE].
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].
M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].
S. Biswas, J. Kastikainen, S. Shashi and J. Sully, Holographic BCFT spectra from brane mergers, JHEP 11 (2022) 158 [arXiv:2209.11227] [INSPIRE].
A. May and M. Van Raamsdonk, Interpolating between multi-boundary wormholes and single-boundary geometries in holography, JHEP 04 (2021) 185 [arXiv:2011.14258] [INSPIRE].
T. Takayanagi and T. Uetoko, Chern-Simons Gravity Dual of BCFT, JHEP 04 (2021) 193 [arXiv:2011.02513] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, Conformal boundary conditions and 3-D topological field theory, in the proceedings of the NATO Advanced Research Workshop on Statistical Field Theories, Villa Olmo, Italy, June 18–23 (2001) [hep-th/0110158] [INSPIRE].
S. Elitzur, E. Gross, E. Rabinovici and N. Seiberg, Aspects of Bosonization in String Theory, Nucl. Phys. B 283 (1987) 413 [INSPIRE].
D.S. Freed, Extended structures in topological quantum field theory, hep-th/9306045 [INSPIRE].
G. Wong, A note on entanglement edge modes in Chern Simons theory, JHEP 08 (2018) 020 [arXiv:1706.04666] [INSPIRE].
S. Collier, L. Eberhardt and M. Zhang, Solving 3d gravity with Virasoro TQFT, SciPost Phys. 15 (2023) 151 [arXiv:2304.13650] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Entanglement Scrambling in 2d Conformal Field Theory, JHEP 09 (2015) 110 [arXiv:1506.03772] [INSPIRE].
M. Dupuis et al., On the origin of the quantum group symmetry in 3d quantum gravity, arXiv:2006.10105 [INSPIRE].
L.D. Faddeev, Modular double of quantum group, in the proceedings of the Conference Moshe Flato, Dijon, France, September 05–08 (1999) [math/9912078] [INSPIRE].
C. Akers, R. Soni and W. Annie, in preparation.
W. Donnelly, Y. Jiang, M. Kim and G. Wong, Entanglement entropy and edge modes in topological string theory. Part I. Generalized entropy for closed strings, JHEP 10 (2021) 201 [arXiv:2010.15737] [INSPIRE].
Y. Jiang, M. Kim and G. Wong, Entanglement entropy and edge modes in topological string theory. Part II. The dual gauge theory story, JHEP 10 (2021) 202 [arXiv:2012.13397] [INSPIRE].
Acknowledgments
It is a pleasure to thank Hao Geng, Daniel Jafferis, Ahsan Khan, David Kolchmeyer, Samir Mathur, Thomas Mertens, Joan Simon, and Ronak Soni for discussions related to this paper. GW is supported by the Harvard Center of Mathematical Sciences and Applications at Harvard University, and thanks the Aspen Center for Physics for hospitality.
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Wong, G. A note on the bulk interpretation of the quantum extremal surface formula. J. High Energ. Phys. 2024, 24 (2024). https://doi.org/10.1007/JHEP04(2024)024
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DOI: https://doi.org/10.1007/JHEP04(2024)024