Abstract
We describe how to introduce dynamics for the holographic states and codes introduced by Pastawski, Yoshida, Harlow and Preskill. This task requires the definition of a continuous limit of the kinematical Hilbert space which we argue may be achieved via the semicontinuous limit of Jones. Dynamics is then introduced by building a unitary representation of a group known as Thompson’s group T, which is closely related to the conformal group conf (ℝ1,1). The bulk Hilbert space is realised as a special subspace of the semicontinuous limit Hilbert space spanned by a class of distinguished states which can be assigned a discrete bulk geometry. The analogue of the group of large bulk diffeomorphisms is given by a unitary representation of the Ptolemy group Pt , on the bulk Hilbert space thus realising a toy model of the AdS/CFT correspondence which we call the Pt /T correspondence.
References
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].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, Lect. Notes Phys.931 (2017) 1 [arXiv:1609.01287] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav.42 (2010) 2323 [Int. J. Mod. Phys.D 19 (2010) 2429] [arXiv:1005.3035] [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev.D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
J.M. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys.61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, Rev. Mod. Phys.88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev.D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
F. Pastawski and J. Preskill, Code properties from holographic geometries, Phys. Rev.X 7 (2017) 021022 [arXiv:1612.00017] [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
N. Bao et al., Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence, Phys. Rev.D 91 (2015) 125036 [arXiv:1504.06632] [INSPIRE].
Z. Yang, P. Hayden and X.-L. Qi, Bidirectional holographic codes and sub-AdS locality, JHEP01 (2016) 175 [arXiv:1510.03784] [INSPIRE].
A. Bhattacharyya, Z.-S. Gao, L.-Y. Hung and S.-N. Liu, Exploring the Tensor Networks/AdS Correspondence, JHEP08 (2016) 086 [arXiv:1606.00621] [INSPIRE].
A. May, Tensor networks for dynamic spacetimes, JHEP06 (2017) 118 [arXiv:1611.06220] [INSPIRE].
D. Goyeneche, D. Alsina, J.I. Latorre, A. Riera and K. Życzkowski, Absolutely maximally entangled states, combinatorial designs and multiunitary matrices, Phys. Rev.A 92 (2015) 032316 [arXiv:1506.08857] [INSPIRE].
M. Enríquez, I. Wintrowicz and K. Życzkowski, Maximally Entangled Multipartite States: A Brief Survey, J. Phys. Conf. Ser.698 (2016) 012003.
Z. Raissi, C. Gogolin, A. Riera and A. Aćın, Optimal quantum error correcting codes from absolutely maximally entangled states, J. Phys.A 51 (2018) 075301 [arXiv:1701.03359] [INSPIRE].
Y. Li, M. Han, M. Grassl and B. Zeng, Invariant Perfect Tensors, New J. Phys.19 (2017) 063029 [arXiv:1612.04504] [INSPIRE].
A. Peach and S.F. Ross, Tensor Network Models of Multiboundary Wormholes, Class. Quant. Grav.34 (2017) 105011 [arXiv:1702.05984] [INSPIRE].
W. Donnelly, B. Michel, D. Marolf and J. Wien, Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers, JHEP04 (2017) 093 [arXiv:1611.05841] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
C. Bény, Causal structure of the entanglement renormalization ansatz, New J. Phys.15 (2013) 023020 [arXiv:1110.4872] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor Networks from Kinematic Space, JHEP07 (2016) 100 [arXiv:1512.01548] [INSPIRE].
X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282 [INSPIRE].
V.F.R. Jones, Some unitary representations of Thompson’s groups F and T , J. Comb. Algebra1 (2017) 1 [arXiv:1412.7740].
V.F.R. Jones, A No-Go Theorem for the Continuum Limit of a Periodic Quantum Spin Chain, Commun. Math. Phys.357 (2018) 295 [arXiv:1607.08769] [INSPIRE].
J.W. Cannon, W.J. Floyd and W.R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math.42 (1996) 215.
J. Belk, Thompson’s Group F, arXiv:0708.3609.
S.S. Gubser, J. Knaute, S. Parikh, A. Samberg and P. Witaszczyk, p-adic AdS/CFT, Commun. Math. Phys.352 (2017) 1019 [arXiv:1605.01061] [INSPIRE].
M. Heydeman, M. Marcolli, I.A. Saberi and B. Stoica, Tensor networks, p-adic fields and algebraic curves: arithmetic and the AdS3/CFT2correspondence, Adv. Theor. Math. Phys.22 (2018) 93 [arXiv:1605.07639] [INSPIRE].
D. Harlow, S.H. Shenker, D. Stanford and L. Susskind, Tree-like structure of eternal inflation: A solvable model, Phys. Rev.D 85 (2012) 063516 [arXiv:1110.0496] [INSPIRE].
L. Schneps and P. Lochak, Geometric Galois Actions. Volume 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge U.K. (1997).
L. Funar and V. Sergiescu, Central extensions of the Ptolemy-Thompson group and quantized Teichmüller theory, J. Topol.3 (2010) 29.
V.F.R. Jones, Scale invariant transfer matrices and Hamiltonians, J. Phys.A 51 (2018) 104001 [arXiv:1706.00515].
V.F.R. Jones, Planar algebras, I, math.QA/9909027.
D. Brill, Black holes and wormholes in (2 + 1)-dimensions, gr-qc/9904083 [INSPIRE].
S. Aminneborg, I. Bengtsson, D. Brill, S. Holst and P. Peldan, Black holes and wormholes in (2 + 1)-dimensions, Class. Quant. Grav.15 (1998) 627 [gr-qc/9707036] [INSPIRE].
S. Carlip, The (2 + 1)-Dimensional black hole, Class. Quant. Grav.12 (1995) 2853 [gr-qc/9506079] [INSPIRE].
S. Carlip, Conformal field theory, (2 + 1)-dimensional gravity and the BTZ black hole, Class. Quant. Grav.22 (2005) R85 [gr-qc/0503022] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett.69 (1992) 1849 [hep-th/9204099] [INSPIRE].
U. Pachner, P.L. Homeomorphic Manifolds are Equivalent by Elementary Shellings, Eur. J. Combinator.12 (1991) 129.
R.C. Penner, Universal Constructions in Teichmüller Theory, Adv. Math.98 (1993) 143.
S. Weinberg, The quantum theory of fields. Volume I, Cambridge University Press, Cambridge U.K. (1996).
M. Schottenloher, A mathematical introduction to conformal field theory, Lect. Notes Phys.759 (2008) 1 [INSPIRE].
R. Bieri and R. Strebel, On Groups of PL-homeomorphisms of the Real Line, Mathematical Surveys and Monographs, volume 215, American Mathematical Society, Providence U.S.A. (2016).
D.E. Stiegemann, Approximating diffeomorphisms by elements of Thompson’s group T , arXiv:1810.11041.
D.E. Stiegemann, Thompson Field Theory, Ph.D. Thesis, Gottfried Wilhelm Leibniz Universität Hannover, Hannover Germany (2019) [arXiv:1907.08442] [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].
H.-J. Matschull, Black hole creation in (2 + 1)-dimensions, Class. Quant. Grav.16 (1999) 1069 [gr-qc/9809087] [INSPIRE].
T.J. Osborne and D.E. Stiegemann, Quantum fields for unitary representations of Thompson’s groups F and T , arXiv:1903.00318 [INSPIRE].
G. Evenbly, Hyperinvariant Tensor Networks and Holography, Phys. Rev. Lett.119 (2017) 141602 [arXiv:1704.04229] [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: 1706.08823
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Osborne, T.J., Stiegemann, D.E. Dynamics for holographic codes. J. High Energ. Phys. 2020, 154 (2020). https://doi.org/10.1007/JHEP04(2020)154
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2020)154
Keywords
- AdS-CFT Correspondence
- Discrete Symmetries
- Conformal and W Symmetry
- Conformal Field Theory