Kinematic space and the orbit method

  • Robert F. PennaEmail author
  • Claire Zukowski
Open Access
Regular Article - Theoretical Physics


Kinematic space has been defined as the space of codimension-2 spacelike extremal surfaces in anti de Sitter (AdSd+1) spacetime which, by the Ryu-Takayanagi proposal, compute the entanglement entropy of spheres in the boundary CFTd. It has recently found many applications in holography. Coadjoint orbits are symplectic manifolds that are the classical analogues of a Lie group’s unitary irreducible representations. We prove that kinematic space is a particular coadjoint orbit of the d-dimensional conformal group SO(d, 2). In addition, we show that the Crofton form on kinematic space associated to AdS3, that was shown to compute the lengths of bulk curves, is equal to the standard Kirillov-Kostant symplectic form on the coadjoint orbit. Since kinematic space is Kähler in addition to symplectic, it can be quantized. The orbit method extends the kinematic space dictionary, which was originally motivated through connections to integral geometry, by directly translating geometrical properties of holographic auxiliary spaces into statements about the representation theory of the conformal group.


AdS-CFT Correspondence Black Holes Differential and Algebraic Geometry 


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]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: a boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [hep-th/0506118] [INSPIRE].
  2. [2]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
  3. [3]
    D. Kabat, G. Lifschytz and D.A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83 (2011) 106009 [arXiv:1102.2910] [INSPIRE].
  4. [4]
    D. Kabat and G. Lifschytz, Local bulk physics from intersecting modular Hamiltonians, JHEP 06 (2017) 120 [arXiv:1703.06523] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    W. Donnelly and S.B. Giddings, Observables, gravitational dressing and obstructions to locality and subsystems, Phys. Rev. D 94 (2016) 104038 [arXiv:1607.01025] [INSPIRE].
  6. [6]
    V. Balasubramanian et al., Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [INSPIRE].
  7. [7]
    B. Czech, X. Dong and J. Sully, Holographic reconstruction of general bulk surfaces, JHEP 11 (2014) 015 [arXiv:1406.4889] [INSPIRE].
  8. [8]
    M. Headrick, R.C. Myers and J. Wien, Holographic holes and differential entropy, JHEP 10 (2014) 149 [arXiv:1408.4770] [INSPIRE].
  9. [9]
    R.C. Myers, J. Rao and S. Sugishita, Holographic holes in higher dimensions, JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].
  10. [10]
    B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral geometry and holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
  11. [11]
    J. de Boer, M.P. Heller, R.C. Myers and Y. Neiman, Holographic de Sitter geometry from entanglement in conformal field theory, Phys. Rev. Lett. 116 (2016) 061602 [arXiv:1509.00113] [INSPIRE].
  12. [12]
    B. Czech et al., A stereoscopic look into the bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP 08 (2016) 162 [arXiv:1606.03307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    B. Czech et al., Equivalent equations of motion for gravity and entropy, JHEP 02 (2017) 004 [arXiv:1608.06282] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    B. Mosk, Holographic equivalence between the first law of entanglement entropy and the linearized gravitational equations, Phys. Rev. D 94 (2016) 126001 [arXiv:1608.06292] [INSPIRE].
  16. [16]
    B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor networks from kinematic space, JHEP 07 (2016) 100 [arXiv:1512.01548] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    B. Czech et al., Tensor network quotient takes the vacuum to the thermal state, Phys. Rev. B 94 (2016) 085101 [arXiv:1510.07637] [INSPIRE].
  18. [18]
    C.T. Asplund, N. Callebaut and C. Zukowski, Equivalence of emergent de Sitter spaces from conformal field theory, JHEP 09 (2016) 154 [arXiv:1604.02687] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J.-d. Zhang and B. Chen, Kinematic space and wormholes, JHEP 01 (2017) 092 [arXiv:1610.07134] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    J.C. Cresswell and A.W. Peet, Kinematic space for conical defects, JHEP 11 (2017) 155 [arXiv:1708.09838] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.C. Cresswell, I.T. Jardine and A.W. Peet, Holographic relations for OPE blocks in excited states, JHEP 03 (2019) 058 [arXiv:1809.09107] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    B. Czech, P.H. Nguyen and S. Swaminathan, A defect in holographic interpretations of tensor networks, JHEP 03 (2017) 090 [arXiv:1612.05698] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    A. Karch, J. Sully, C.F. Uhlemann and D.G.E. Walker, Boundary kinematic space, JHEP 08 (2017) 039 [arXiv:1703.02990] [INSPIRE].
  24. [24]
    B. Czech, L. Lamprou, S. Mccandlish and J. Sully, Modular Berry connection for entangled subregions in AdS/CFT, Phys. Rev. Lett. 120 (2018) 091601 [arXiv:1712.07123] [INSPIRE].
  25. [25]
    A. Ashtekar and T.A. Schilling, Geometrical formulation of quantum mechanics, gr-qc/9706069 [INSPIRE].
  26. [26]
    E. Witten, Coadjoint orbits of the Virasoro group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Vergne, Representations of Lie groups and the orbit method, in Emmy Noether in Bryn Mawr, B. Srinivasan and J.D. Sally eds., Springer, Germany (1983).Google Scholar
  28. [28]
    A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Oblak, Berry phases on Virasoro orbits, JHEP 10 (2017) 114 [arXiv:1703.06142] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    G. Barnich, H.A. Gonzalez and P. Salgado-ReboLledó, Geometric actions for three-dimensional gravity, Class. Quant. Grav. 35 (2018) 014003 [arXiv:1707.08887] [INSPIRE].
  31. [31]
    J. Cotler and K. Jensen, A theory of reparameterizations for AdS 3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263] [INSPIRE].
  32. [32]
    A. Alekseev and S.L. Shatashvili, Path integral quantization of the coadjoint orbits of the Virasoro group and 2D gravity, Nucl. Phys. B 323 (1989) 719 [INSPIRE].
  33. [33]
    G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, JHEP 11 (2017) 046 [arXiv:1702.04266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
  35. [35]
    P. Caputa and J.M. Magan, Quantum computation as gravity, Phys. Rev. Lett. 122 (2019) 231302 [arXiv:1807.04422] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys. A 47 (2014) 335204 [arXiv:1403.4213] [INSPIRE].
  37. [37]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: I. Induced representations, JHEP 06 (2014) 129 [arXiv:1403.5803] [INSPIRE].
  38. [38]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP 03 (2015) 033 [arXiv:1502.00010] [INSPIRE].
  39. [39]
    B. Oblak, Characters of the BMS group in three dimensions, Commun. Math. Phys. 340 (2015) 413 [arXiv:1502.03108] [INSPIRE].
  40. [40]
    J.M. Souriau, Structure of dynamical systems: a symplectic view of physics, Springer, Germany (2012).Google Scholar
  41. [41]
    A.A. Kirillov, Lectures on the orbit method, American Mathematical Society Providence, U.S.A. (2004).Google Scholar
  42. [42]
    B. Oblak, BMS particles in three dimensions, Ph.D. thesis, Brussels University, Brussels, Belgium (2016), arXiv:1610.08526.
  43. [43]
    J. Marsden and T. Ratiu, Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, Texts in Applied Mathematics, Springer, Germany (2002).Google Scholar
  44. [44]
    A. Pressley and G.B. Segal, Loop groups, Clarendon Press, U.K. (1986).Google Scholar
  45. [45]
    B.-Y. Chen, Pseudo-Riemannian geometry, delta-invariants and applications, World Scientific, Singapore (2011).Google Scholar
  46. [46]
    V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals Math. 48 (1947) 568 [INSPIRE].
  47. [47]
    H. Chandra, Plancherel formula for the 2 × 2 real unimodular group, Proc. Natl. Acad. Sci. U.S.A. 38 (1952) 337.Google Scholar
  48. [48]
    A. Alekseev and S.L. Shatashvili, Coadjoint orbits, cocycles and gravitational Wess-Zumino, arXiv:1801.07963 [INSPIRE].
  49. [49]
    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].
  50. [50]
    M. Henneaux, L. Maoz and A. Schwimmer, Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity, Annals Phys. 282 (2000) 31 [hep-th/9910013] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    N. Callebaut, The gravitational dynamics of kinematic space, JHEP 02 (2019) 153 [arXiv:1808.10431] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    N. Callebaut and H. Verlinde, Entanglement dynamics in 2D CFT with boundary: entropic origin of JT gravity and Schwarzian QM, JHEP 05 (2019) 045 [arXiv:1808.05583] [INSPIRE].
  53. [53]
    G.W. Gibbons, Holography and the future tube, Class. Quant. Grav. 17 (2000) 1071 [hep-th/9911027] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsColumbia UniversityNew YorkU.S.A.
  2. 2.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations