Journal of High Energy Physics

, 2019:182 | Cite as

Superselection sectors of gravitational subregions

  • Joan CampsEmail author
Open Access
Regular Article - Theoretical Physics


Motivated by the problem of defining the entanglement entropy of the graviton, we study the division of the phase space of general relativity across subregions. Our key requirement is demanding that the separation into subregions is imaginary — i.e., that entangling surfaces are not physical. This translates into a certain condition on the symplectic form. We find that gravitational subregions that satisfy this condition are bounded by surfaces of extremal area. We characterise the ‘centre variables’ of the phase space of the graviton in such subsystems, which can be taken to be the conformal class of the induced metric in the boundary, subject to a constraint involving the traceless part of the extrinsic curvature. We argue that this condition works to discard local deformations of the boundary surface to infinitesimally nearby extremal surfaces, that are otherwise available for generic codimension-2 extremal surfaces of dimension ≥ 2.


AdS-CFT Correspondence Classical Theories of Gravity Gauge-gravity correspondence 


Open Access

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  1. [1]
    L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
  2. [2]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    R. Emparan, Black hole entropy as entanglement entropy: A Holographic derivation, JHEP 06 (2006) 012 [hep-th/0603081] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292 [INSPIRE].
  8. [8]
    A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [Erratum ibid. 87 (2013) 069904] [arXiv:1105.3445] [INSPIRE].
  9. [9]
    R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].
  10. [10]
    X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
  11. [11]
    A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. Harlow, Wormholes, Emergent Gauge Fields and the Weak Gravity Conjecture, JHEP 01 (2016) 122 [arXiv:1510.07911] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Harlow and D. Jafferis, The Factorization Problem in Jackiw-Teitelboim Gravity, arXiv:1804.01081 [INSPIRE].
  14. [14]
    C. Crnkovic and E. Witten, Covariant Description Of Canonical Formalism In Geometrical Theories, in Three hundred years of gravitation, S.W. Hawking and W. Israel eds., pp. 676-684 (1986) [INSPIRE].
  15. [15]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A.J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, JHEP 02 (2018) 021 [arXiv:1706.05061] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J. Lin, Ryu-Takayanagi Area as an Entanglement Edge Term, arXiv:1704.07763 [INSPIRE].
  20. [20]
    J. Lin, Entanglement entropy in Jackiw-Teitelboim Gravity, arXiv:1807.06575 [INSPIRE].
  21. [21]
    D. Tong, Lectures on the Quantum Hall Effect, 2016, arXiv:1606.06687 [INSPIRE].
  22. [22]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
  24. [24]
    S. Ghosh, R.M. Soni and S.P. Trivedi, On The Entanglement Entropy For Gauge Theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    R.M. Soni and S.P. Trivedi, Aspects of Entanglement Entropy for Gauge Theories, JHEP 01 (2016) 136 [arXiv:1510.07455] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].
  27. [27]
    W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
  28. [28]
    J. Lin and D. Radičević, Comments on Defining Entanglement Entropy, arXiv:1808.05939 [INSPIRE].
  29. [29]
    W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D 94 (2016) 104053 [arXiv:1506.05792] [INSPIRE].
  31. [31]
    S. Hollands and R.M. Wald, Stability of Black Holes and Black Branes, Commun. Math. Phys. 321 (2013) 629 [arXiv:1201.0463] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    J. Camps, work in progress.Google Scholar
  33. [33]
    J. Kirklin, Subregions, Minimal Surfaces and Entropy in Semiclassical Gravity, arXiv:1805.12145 [INSPIRE].
  34. [34]
    N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Sen, Black Hole Entropy Function, Attractors and Precision Counting of Microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].CrossRefzbMATHGoogle Scholar
  38. [38]
    A. Sen, Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions, JHEP 04 (2013) 156 [arXiv:1205.0971] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    K.-W. Huang, Central Charge and Entangled Gauge Fields, Phys. Rev. D 92 (2015) 025010 [arXiv:1412.2730] [INSPIRE].
  40. [40]
    H. Casini and M. Huerta, Entanglement entropy of a Maxwell field on the sphere, Phys. Rev. D 93 (2016) 105031 [arXiv:1512.06182] [INSPIRE].
  41. [41]
    M. Huerta and L.A. Pedraza, Numerical determination of the entanglement entropy for a Maxwell field in the cylinder, arXiv:1808.01864 [INSPIRE].
  42. [42]
    W. Donnelly, B. Michel and A. Wall, Electromagnetic Duality and Entanglement Anomalies, Phys. Rev. D 96 (2017) 045008 [arXiv:1611.05920] [INSPIRE].
  43. [43]
    E. Witten, A Note On Boundary Conditions In Euclidean Gravity, arXiv:1805.11559 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity College LondonLondonUnited Kingdom

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