Journal of High Energy Physics

, 2018:29 | Cite as

Lorentz-diffeomorphism edge modes in 3d gravity

  • Marc Geiller
Open Access
Regular Article - Theoretical Physics


The proper definition of subsystems in gauge theory and gravity requires an extension of the local phase space by including edge mode fields. Their role is on the one hand to restore gauge invariance with respect to gauge transformations supported on the boundary, and on the other hand to parametrize the largest set of boundary symmetries which can arise if both the gauge parameters and the dynamical fields are unconstrained at the boundary. In this work we construct the extended phase space for three-dimensional gravity in first order connection and triad variables. There, the edge mode fields consist of a choice of coordinate frame on the boundary and a choice of Lorentz frame on the bundle, which together constitute the Lorentz-diffeomorphism edge modes. After constructing the extended symplectic structure and proving its gauge invariance, we study the boundary symmetries and the integrability of their generators. We find that the infinite-dimensional algebra of boundary symmetries with first order variables is the same as that with metric variables, and explain how this can be traced back to the expressions for the diffeomorphism Noether charge in both formulations. This concludes the study of extended phase spaces and edge modes in three-dimensional gravity, which was done previously by the author in the BF and Chern-Simons formulations.


Classical Theories of Gravity Gauge Symmetry Global Symmetries 


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.


  1. [1]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].ADSGoogle Scholar
  3. [3]
    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    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].ADSGoogle Scholar
  5. [5]
    H. Casini and M. Huerta, Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice, Phys. Rev. D 90 (2014) 105013 [arXiv:1406.2991] [INSPIRE].ADSGoogle Scholar
  6. [6]
    D. Radicevic, Notes on Entanglement in Abelian Gauge Theories, arXiv:1404.1391 [INSPIRE].
  7. [7]
    W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    L.-Y. Hung and Y. Wan, Revisiting Entanglement Entropy of Lattice Gauge Theories, JHEP 04 (2015) 122 [arXiv:1501.04389] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Ghosh, R.M. Soni and S.P. Trivedi, On The Entanglement Entropy For Gauge Theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba and H. Tasaki, On the definition of entanglement entropy in lattice gauge theories, JHEP 06 (2015) 187 [arXiv:1502.04267] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    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].ADSGoogle Scholar
  13. [13]
    C. Delcamp, B. Dittrich and A. Riello, On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity, JHEP 11 (2016) 102 [arXiv:1609.04806] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    J.R. Fliss et al., Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory, JHEP 09 (2017) 056 [arXiv:1705.09611] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    H. Ruegg and M. Ruiz-Altaba, The Stueckelberg field, Int. J. Mod. Phys. A 19 (2004) 3265 [hep-th/0304245] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    M. Geiller, Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity, Nucl. Phys. B 924 (2017) 312 [arXiv:1703.04748] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    A.J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, arXiv:1706.05061 [INSPIRE].
  18. [18]
    E. Witten, (2 + 1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
  19. [19]
    S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    S. Carlip, The Statistical mechanics of the (2+1)-dimensional black hole, Phys. Rev. D 51 (1995) 632 [gr-qc/9409052] [INSPIRE].
  22. [22]
    A.P. Balachandran, L. Chandar and A. Momen, Edge states in canonical gravity, gr-qc/9506006 [INSPIRE].
  23. [23]
    O. Coussaert, M. Henneaux and P. van Driel, The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
  24. [24]
    D. Grumiller and M. Riegler, Most general AdS 3 boundary conditions, JHEP 10 (2016) 023 [arXiv:1608.01308] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    D. Grumiller, W. Merbis and M. Riegler, Most general flat space boundary conditions in three-dimensional Einstein gravity, Class. Quant. Grav. 34 (2017) 184001 [arXiv:1704.07419] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    A.P. Balachandran, L. Chandar and A. Momen, Edge states in gravity and black hole physics, Nucl. Phys. B 461 (1996) 581 [gr-qc/9412019] [INSPIRE].
  28. [28]
    V. Husain and S. Major, Gravity and BF theory defined in bounded regions, Nucl. Phys. B 500 (1997) 381 [gr-qc/9703043] [INSPIRE].
  29. [29]
    L.B. Szabados, On a class of 2-surface observables in general relativity, Class. Quant. Grav. 23 (2006) 2291 [gr-qc/0511059] [INSPIRE].
  30. [30]
    W. Wieland, Quantum gravity in three dimensions, Witten spinors and the quantisation of length, arXiv:1711.01276 [INSPIRE].
  31. [31]
    T. Jacobson and A. Mohd, Black hole entropy and Lorentz-diffeomorphism Noether charge, Phys. Rev. D 92 (2015) 124010 [arXiv:1507.01054] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  33. [33]
    G. Bimonte, K.S. Gupta and A. Stern, Edge currents and vertex operators for Chern-Simons gravity, Int. J. Mod. Phys. A 8 (1993) 653 [hep-th/9205077] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
  35. [35]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  37. [37]
    A. Ashtekar and R.O. Hansen, A unified treatment of null and spatial infinity in general relativity. I — Universal structure, asymptotic symmetries and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].
  38. [38]
    A. Ashtekar, Geometry and Physics of Null Infinity, arXiv:1409.1800 [INSPIRE].
  39. [39]
    B. Oblak, BMS Particles in Three Dimensions, arXiv:1610.08526 [INSPIRE].
  40. [40]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: I. Induced representations, JHEP 06 (2014) 129 [arXiv:1403.5803] [INSPIRE].
  41. [41]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP 03 (2015) 033 [arXiv:1502.00010] [INSPIRE].
  42. [42]
    E. Buffenoir and K. Noui, Unfashionable observations about three-dimensional gravity, gr-qc/0305079 [INSPIRE].
  43. [43]
    F.A. Bais, N.M. Muller and B.J. Schroers, Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity, Nucl. Phys. B 640 (2002) 3 [hep-th/0205021] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    F.A. Bais and N.M. Muller, Topological field theory and the quantum double of SU(2), Nucl. Phys. B 530 (1998) 349 [hep-th/9804130] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    K. Noui, Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double, J. Math. Phys. 47 (2006) 102501 [gr-qc/0612144] [INSPIRE].
  46. [46]
    Y.N. Obukhov, The palatini principle for manifold with boundary, Class. Quant. Grav. 4 (1987) 1085.ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    N. Bodendorfer and Y. Neiman, Imaginary action, spinfoam asymptotics and the ‘transplanckian’ regime of loop quantum gravity, Class. Quant. Grav. 30 (2013) 195018 [arXiv:1303.4752] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    A. Corichi and I. Rubalcava-García, Energy in first order 2+1 gravity, Phys. Rev. D 92 (2015) 044040 [arXiv:1503.03030] [INSPIRE].ADSMathSciNetGoogle Scholar
  49. [49]
    W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    L. Freidel and A. Perez, Quantum gravity at the corner, arXiv:1507.02573 [INSPIRE].
  51. [51]
    L. Freidel, A. Perez and D. Pranzetti, Loop gravity string, Phys. Rev. D 95 (2017) 106002 [arXiv:1611.03668] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations