Abstract
Recently, an infinite class of holographic generalized complexities was proposed. These gravitational observables display the behavior required to be duals of complexity, in particular, linear growth at late times and switchback effect. In this work, we aim to understand generalized complexities in the framework of Lorentzian threads. We reformulate the problem in terms of thread distributions and measures and present a program to calculate the infinite family of codimension-one observables. We also outline a path to understand, using threads, the more subtle case of codimension-zero observables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
J. Harper and M. Headrick, Bit threads and holographic entanglement of purification, JHEP 08 (2019) 101 [arXiv:1906.05970] [INSPIRE].
N. Bao and I.F. Halpern, Holographic Inequalities and Entanglement of Purification, JHEP 03 (2018) 006 [arXiv:1710.07643] [INSPIRE].
V.E. Hubeny, Bulk locality and cooperative flows, JHEP 12 (2018) 068 [arXiv:1808.05313] [INSPIRE].
C.A. Agón, J. De Boer and J.F. Pedraza, Geometric Aspects of Holographic Bit Threads, JHEP 05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
S.X. Cui et al., Bit Threads and Holographic Monogamy, Commun. Math. Phys. 376 (2019) 609 [arXiv:1808.05234] [INSPIRE].
C.A. Agón and J.F. Pedraza, Quantum bit threads and holographic entanglement, JHEP 02 (2022) 180 [arXiv:2105.08063] [INSPIRE].
N. Bao and J. Harper, Bit threads on hypergraphs, arXiv:2012.07872 [INSPIRE].
C.A. Agón, E. Cáceres and J.F. Pedraza, Bit threads, Einstein’s equations and bulk locality, JHEP 01 (2021) 193 [arXiv:2007.07907] [INSPIRE].
U. Gürsoy, J.F. Pedraza and G.P. Planas, Holographic entanglement as nonlocal magnetism, JHEP 09 (2023) 091 [arXiv:2303.05529] [INSPIRE].
J.F. Pedraza, A. Russo, A. Svesko and Z. Weller-Davies, Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity, JHEP 02 (2022) 093 [arXiv:2106.12585] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
A.R. Brown et al., Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown et al., Complexity, action, and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35 (2018) 10 [arXiv:1710.09516] [INSPIRE].
J.F. Pedraza, A. Russo, A. Svesko and Z. Weller-Davies, Lorentzian Threads as Gatelines and Holographic Complexity, Phys. Rev. Lett. 127 (2021) 271602 [arXiv:2105.12735] [INSPIRE].
A. Belin et al., Does Complexity Equal Anything?, Phys. Rev. Lett. 128 (2022) 081602 [arXiv:2111.02429] [INSPIRE].
A. Belin et al., Complexity equals anything II, JHEP 01 (2023) 154 [arXiv:2210.09647] [INSPIRE].
J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume, and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].
K.P.S. Bhaskara Rao and M. Bhaskara Rao, Theory of charges: a study of finitely additive measures, Academic Press (1983).
M. Headrick and V.E. Hubeny, Covariant bit threads, JHEP 07 (2023) 180 [arXiv:2208.10507] [INSPIRE].
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press (2004) [https://doi.org/10.1017/cbo9780511804441].
J.-H. Treude and J.D.E. Grant, Volume Comparison for Hypersurfaces in Lorentzian Manifolds and Singularity Theorems, Ann. Glob. Anal. Geom. 43 (2013) 233 [arXiv:1201.4249] [INSPIRE].
A.R. Chandra et al., Cost of holographic path integrals, SciPost Phys. 14 (2023) 061 [arXiv:2203.08842] [INSPIRE].
J.E. Marsden and F.J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rept. 66 (1980) 109.
J.F. Pedraza, A. Russo, A. Svesko and Z. Weller-Davies, Computing spacetime, Int. J. Mod. Phys. D 31 (2022) 2242010 [arXiv:2205.05705] [INSPIRE].
R. Carrasco, J.F. Pedraza, A. Svesko and Z. Weller-Davies, Gravitation from optimized computation: Einstein and beyond, JHEP 09 (2023) 167 [arXiv:2306.08503] [INSPIRE].
E. Cáceres et al., Einstein’s equations and generalized complexities, work in progress.
S.E. Aguilar-Gutierrez, M.P. Heller and S. Van der Schueren, Complexity = Anything Can Grow Forever in de Sitter, arXiv:2305.11280 [INSPIRE].
E. Jørstad, R.C. Myers and S.-M. Ruan, Complexity = anything: singularity probes, JHEP 07 (2023) 223 [arXiv:2304.05453] [INSPIRE].
J.C. Feng and R.A. Matzner, The Weiss Variation of the Gravitational Action, Gen. Rel. Grav. 50 (2018) 99 [arXiv:1708.04489] [INSPIRE].
Acknowledgments
We are indebted to José Barbón, Jesús Gambín, Juan Pedraza and Andy Svesko for enlightening and fun discussions. Special thanks to Juan Pedraza for comments on the manuscript. EC thanks the Instituto de Física Teórica UAM-CSIC (IFT UAM-CSIC), for hospitality during the last stages of this project. EC and RC thank the organizers and participants of the “Complexity in Field Theory and Gravity” workshop at IFT UAM-CSIC for a stimulating conference. The work of RC is supported through the grants CEX2020-001007-S and PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033, by ERDF A way of making Europe and through the Spanish FPU grant FPU22/01262. EC and VP are supported by the National Science Foundation under grants No. PHY-2112725 and No. PHY–2210562.
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: 2312.10606
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
Cáceres, E., Carrasco, R. & Patil, V. Lorentzian threads and generalized complexity. J. High Energ. Phys. 2024, 10 (2024). https://doi.org/10.1007/JHEP04(2024)010
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2024)010