Abstract
We introduce a unifying framework for the construction of holographic tensor networks, based on the theory of hyperbolic buildings. The underlying dualities relate a bulk space to a boundary which can be homeomorphic to a sphere, but also to more general spaces like a Menger sponge type fractal. In this general setting, we give a precise construction of a large family of bulk regions that satisfy complementary recovery. For these regions, our networks obey a Ryu-Takayanagi formula. The areas of Ryu-Takayanagi surfaces are controlled by the Hausdorff dimension of the boundary, and consistently generalize the behavior of holographic entanglement entropy in integer dimensions to the non-integer case. Our construction recovers HaPPY-like codes in all dimensions, and generalizes the geometry of Bruhat-Tits trees. It also provides examples of infinite-dimensional nets of holographic conditional expectations, and opens a path towards the study of conformal field theory and holography on fractal spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Abramenko and K.S. Brown, Buildings: Theory and Applications, Springer (2008) [DOI].
A. Bhattacharyya, Z.-S. Gao, L.-Y. Hung and S.-N. Liu, Exploring the Tensor Networks/AdS Correspondence, JHEP 08 (2016) 086 [arXiv:1606.00621] [INSPIRE].
M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. GAFA 7 (1997).
M. Bourdon, H. Pajot, Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings, Proc. Am. Math. Soc. 127 (1999) 2315.
L. Boyle, M. Dickens and F. Flicker, Conformal Quasicrystals and Holography, Phys. Rev. X 10 (2020) 011009 [arXiv:1805.02665] [INSPIRE].
M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, (1999) Springer [DOI].
L. Chen, X. Liu and L.-Y. Hung, Emergent Einstein Equation in p-adic Conformal Field Theory Tensor Networks, Phys. Rev. Lett. 127 (2021) 221602 [arXiv:2102.12022] [INSPIRE].
A. Clais, Conformal dimension on boundary of right-angled hyperbolic buildings, arXiv:1602.08611.
N. Combe, Yu. Manin, M. Marcolli, Moufang Patterns and Geometry of Information, arXiv:2107.07486.
M. Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993) 241.
M. Coornaert, T. Delzant, A. Papadopoulos, Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441, Springer-Verlag (1990) [DOI].
J.M. Corson, Complexes of groups, Proc. Lond. Math. Soc. s3-65 (1992) 199.
M.W. Davis, Groups generated by reflections and aspherical manifolds, Annals Math. 117 (1983) 293.
X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations, and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
T. Faulkner, The holographic map as a conditional expectation, arXiv:2008.04810 [INSPIRE].
E. Gesteau and M.J. Kang, The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics, arXiv:2005.05971 [INSPIRE].
E. Gesteau and M.J. Kang, Thermal states are vital: Entanglement Wedge Reconstruction from Operator-Pushing, arXiv:2005.07189 [INSPIRE].
E. Gesteau and M.J. Kang, Nonperturbative gravity corrections to bulk reconstruction, arXiv:2112.12789 [INSPIRE].
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
M. Heydeman, M. Marcolli, S. Parikh and I. Saberi, Nonarchimedean holographic entropy from networks of perfect tensors, Adv. Theor. Math. Phys. 25 (2021) 591 [arXiv:1812.04057] [INSPIRE].
F. Huber and N. Wyderka, Table of AME states, http://www.tp.nt.uni-siegen.de/+fhuber/ame.html.
F. Huber, C. Eltschka, J. Siewert and O. Gühne, Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity, J. Phys. A 51 (2018) 175301.
T. Januszkiewicz and J. Swiatkowski, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv. 78 (2003) 555.
M.J. Kang and D.K. Kolchmeyer, Holographic Relative Entropy in Infinite-dimensional Hilbert Spaces, arXiv:1811.05482 [INSPIRE].
T. Kohler and T. Cubitt, Toy Models of Holographic Duality between local Hamiltonians, JHEP 08 (2019) 017 [arXiv:1810.08992] [INSPIRE].
G. Moussong, Hyperbolic Coxeter groups, Ph.D. Thesis, Ohio State University, Columbus, U.S.A. (1988).
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].
H.O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals, Springer (2014) [DOI].
L. Potyagailo and E. Vinberg, On right-angled reflection groups in hyperbolic spaces, Comment. Math. Helv. 80 (2005) 63.
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
A. Thomas, Lattices in hyperbolic buildings, Geometry, Topology and Dynamics in Negative Curvature 425, Cambridge Univeristy Press (2016), pp. 345–363 [DOI].
E.B. Vinberg, The non-existence of crystallographic reflection groups in Lobachevskii spaces of large dimension, Trudy Moskov. Mat. Obshch. 47 (1984) 68.
F. Zhu, Ergodicity and equidistribution in strictly convex Hilbert geometry, arXiv:2008.00328.
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: 2202.01788
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
Gesteau, E., Marcolli, M. & Parikh, S. Holographic tensor networks from hyperbolic buildings. J. High Energ. Phys. 2022, 169 (2022). https://doi.org/10.1007/JHEP10(2022)169
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2022)169