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Journal of High Energy Physics

, 2017:203 | Cite as

Menagerie of AdS2 boundary conditions

  • Daniel GrumillerEmail author
  • Robert McNees
  • Jakob Salzer
  • Carlos Valcárcel
  • Dmitri Vassilevich
Open Access
Regular Article - Theoretical Physics

Abstract

We consider different sets of AdS2 boundary conditions for the Jackiw-Teitelboim model in the linear dilaton sector where the dilaton is allowed to fluctuate to leading order at the boundary of the Poincaré disk. The most general set of boundary conditions is easily motivated in the gauge theoretic formulation as a Poisson sigma model and has an \( \mathfrak{s}\mathfrak{l}(2) \) current algebra as asymptotic symmetries. Consistency of the variational principle requires a novel boundary counterterm in the holographically renormalized action, namely a kinetic term for the dilaton. The on-shell action can be naturally reformulated as a Schwarzian boundary action. While there can be at most three canonical boundary charges on an equal-time slice, we consider all Fourier modes of these charges with respect to the Euclidean boundary time and study their associated algebras. Besides the (centerless) \( \mathfrak{s}\mathfrak{l}(2) \) current algebra we find for stricter boundary conditions a Virasoro algebra, a warped conformal algebra and a \( \mathfrak{u}(1) \) current algebra. In each of these cases we get one half of a corresponding symmetry algebra in three-dimensional Einstein gravity with negative cosmological constant and analogous boundary conditions. However, on-shell some of these algebras reduce to finite-dimensional ones, reminiscent of the on-shell breaking of conformal invariance in SYK. We conclude with a discussion of thermodynamical aspects, in particular the entropy and some Cardyology.

Keywords

2D Gravity AdS-CFT Correspondence Topological Field Theories 

Notes

Open Access

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References

  1. [1]
    R. Jackiw, Liouville field theory: a two-dimensional model for gravity?, in Quantum theory of gravity, S. Christensen ed., Adam Hilger, Bristol U.K., (1984), pg. 403.Google Scholar
  2. [2]
    C. Teitelboim, The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly, in Quantum theory of gravity, S. Christensen ed., Adam Hilger, Bristol U.K., (1984), pg. 327.Google Scholar
  3. [3]
    G. Mandal, A.M. Sengupta and S.R. Wadia, Classical solutions of two-dimensional string theory, Mod. Phys. Lett. A 6 (1991) 1685 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  4. [4]
    S. Elitzur, A. Forge and E. Rabinovici, Some global aspects of string compactifications, Nucl. Phys. B 359 (1991) 581 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  6. [6]
    R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C.G. Callan Jr., S.B. Giddings, J.A. Harvey and A. Strominger, Evanescent black holes, Phys. Rev. D 45 (1992) R1005 [hep-th/9111056] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    S.D. Odintsov and I.L. Shapiro, One loop renormalization of two-dimensional induced quantum gravity, Phys. Lett. B 263 (1991) 183 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    J.G. Russo and A.A. Tseytlin, Scalar tensor quantum gravity in two-dimensions, Nucl. Phys. B 382 (1992) 259 [hep-th/9201021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    W. Kummer, H. Liebl and D.V. Vassilevich, Exact path integral quantization of generic 2D dilaton gravity, Nucl. Phys. B 493 (1997) 491 [gr-qc/9612012] [INSPIRE].
  11. [11]
    J. Brown, Lower dimensional gravity, World Scientific, Singapore, (1988) [INSPIRE].
  12. [12]
    D. Grumiller, W. Kummer and D.V. Vassilevich, Dilaton gravity in two-dimensions, Phys. Rept. 369 (2002) 327 [hep-th/0204253] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Grumiller and R. Meyer, Ramifications of lineland, Turk. J. Phys. 30 (2006) 349 [hep-th/0604049] [INSPIRE].Google Scholar
  14. [14]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, IRMA Lect. Math. Theor. Phys. 8 (2005) 73 [hep-th/0404176] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  21. [21]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].ADSGoogle Scholar
  22. [22]
    A. Kitaev, A simple model of quantum holography (part 1), in KITP strings seminars, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, Kavli Institute for Theoretical Physics, University of California, Santa Barbara U.S.A., 7 April 2015.
  23. [23]
    A. Kitaev, A simple model of quantum holography (part 2), in KITP strings seminars, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, Kavli Institute for Theoretical Physics, University of California, Santa Barbara U.S.A., 27 May 2015.
  24. [24]
    S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
  25. [25]
    S. Sachdev, Holographic metals and the fractionalized Fermi liquid, Phys. Rev. Lett. 105 (2010) 151602 [arXiv:1006.3794] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    Y.-Z. You, A.W.W. Ludwig and C. Xu, Sachdev-Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry protected topological states, Phys. Rev. B 95 (2017) 115150 [arXiv:1602.06964] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    K. Jensen, Chaos in AdS 2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-Sitter space, Prog. Theor. Exp. Phys. 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
  31. [31]
    J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    L. García- Álvarez, I.L. Egusquiza, L. Lamata, A. del Campo, J. Sonner and E. Solano, Digital quantum simulation of minimal AdS/CFT, Phys. Rev. Lett. 119 (2017) 040501 [arXiv:1607.08560] [INSPIRE].
  34. [34]
    A. Jevicki and K. Suzuki, Bi-local holography in the SYK model: perturbations, JHEP 11 (2016) 046 [arXiv:1608.07567] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  35. [35]
    Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    D.J. Gross and V. Rosenhaus, A generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093 [arXiv:1610.01569] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    M. Berkooz, P. Narayan, M. Rozali and J. Simón, Higher dimensional generalizations of the SYK model, JHEP 01 (2017) 138 [arXiv:1610.02422] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    A.M. García-García and J.J.M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 126010 [arXiv:1610.03816] [INSPIRE].ADSGoogle Scholar
  39. [39]
    S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transition from fast to slow scrambling, Phys. Rev. B 95 (2017) 134302 [arXiv:1610.04619] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [Addendum ibid. D 95 (2017) 069904] [arXiv:1610.08917] [INSPIRE].
  41. [41]
    E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
  42. [42]
    J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams and the Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 046004 [arXiv:1611.08915] [INSPIRE].
  44. [44]
    R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: the Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    C. Peng, M. Spradlin and A. Volovich, A supersymmetric SYK-like tensor model, JHEP 05 (2017) 062 [arXiv:1612.03851] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    C. Krishnan, S. Sanyal and P.N. Bala Subramanian, Quantum chaos and holographic tensor models, JHEP 03 (2017) 056 [arXiv:1612.06330] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    G. Turiaci and H. Verlinde, Towards a 2d QFT analog of the SYK model, arXiv:1701.00528 [INSPIRE].
  48. [48]
    F. Ferrari, The large D limit of planar diagrams, arXiv:1701.01171 [INSPIRE].
  49. [49]
    Z. Bi, C.-M. Jian, Y.-Z. You, K.A. Pawlak and C. Xu, Instability of the non-Fermi liquid state of the Sachdev-Ye-Kitaev model, Phys. Rev. B 95 (2017) 205105 [arXiv:1701.07081] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    T. Li, J. Liu, Y. Xin and Y. Zhou, Supersymmetric SYK model and random matrix theory, JHEP 06 (2017) 111 [arXiv:1702.01738] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    R. Gurau, Quenched equals annealed at leading order in the colored SYK model, arXiv:1702.04228 [INSPIRE].
  52. [52]
    G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, arXiv:1702.04266 [INSPIRE].
  53. [53]
    D.J. Gross and V. Rosenhaus, The bulk dual of SYK: cubic couplings, JHEP 05 (2017) 092 [arXiv:1702.08016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    C. Krishnan and K.V.P. Kumar, Towards a finite-N hologram, JHEP 10 (2017) 099 [arXiv:1706.05364] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    H.K. Kunduri, J. Lucietti and H.S. Reall, Near-horizon symmetries of extremal black holes, Class. Quant. Grav. 24 (2007) 4169 [arXiv:0705.4214] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    A. Strominger, AdS 2 quantum gravity and string theory, JHEP 01 (1999) 007 [hep-th/9809027] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    J.M. Maldacena, J. Michelson and A. Strominger, Anti-de Sitter fragmentation, JHEP 02 (1999) 011 [hep-th/9812073] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    M. Brigante, S. Cacciatori, D. Klemm and D. Zanon, The asymptotic dynamics of two-dimensional (anti-)de Sitter gravity, JHEP 03 (2002) 005 [hep-th/0202073] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    M. Astorino, S. Cacciatori, D. Klemm and D. Zanon, AdS 2 supergravity and superconformal quantum mechanics, Annals Phys. 304 (2003) 128 [hep-th/0212096] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  61. [61]
    H.L. Verlinde, Superstrings on AdS 2 and superconformal matrix quantum mechanics, hep-th/0403024 [INSPIRE].
  62. [62]
    R.K. Gupta and A. Sen, AdS 3 /CFT 2 to AdS 2 /CFT 1, JHEP 04 (2009) 034 [arXiv:0806.0053] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    M. Alishahiha and F. Ardalan, Central charge for 2D gravity on AdS 2 and AdS 2 /CFT 1 correspondence, JHEP 08 (2008) 079 [arXiv:0805.1861] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    A. Sen, Quantum entropy function from AdS 2 /CFT 1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [INSPIRE].
  65. [65]
    T. Hartman and A. Strominger, Central charge for AdS 2 quantum gravity, JHEP 04 (2009) 026 [arXiv:0803.3621] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    A. Castro, D. Grumiller, F. Larsen and R. McNees, Holographic description of AdS 2 black holes, JHEP 11 (2008) 052 [arXiv:0809.4264] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    V. Balasubramanian, J. de Boer, M.M. Sheikh-Jabbari and J. Simón, What is a chiral 2d CFT? And what does it have to do with extremal black holes?, JHEP 02 (2010) 017 [arXiv:0906.3272] [INSPIRE].
  68. [68]
    A. Castro and F. Larsen, Near extremal Kerr entropy from AdS 2 quantum gravity, JHEP 12 (2009) 037 [arXiv:0908.1121] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    D. Grumiller, J. Salzer and D. Vassilevich, AdS 2 holography is (non-)trivial for (non-)constant dilaton, JHEP 12 (2015) 015 [arXiv:1509.08486] [INSPIRE].ADSGoogle Scholar
  70. [70]
    D. Grumiller and R. McNees, Thermodynamics of black holes in two (and higher) dimensions, JHEP 04 (2007) 074 [hep-th/0703230] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  71. [71]
    M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
  72. [72]
    D. Grumiller, M. Leston and D. Vassilevich, Anti-de Sitter holography for gravity and higher spin theories in two dimensions, Phys. Rev. D 89 (2014) 044001 [arXiv:1311.7413] [INSPIRE].
  73. [73]
    D. Grumiller, R. McNees and J. Salzer, Cosmological constant as confining U(1) charge in two-dimensional dilaton gravity, Phys. Rev. D 90 (2014) 044032 [arXiv:1406.7007] [INSPIRE].
  74. [74]
    W. Kummer, H. Liebl and D.V. Vassilevich, Integrating geometry in general 2D dilaton gravity with matter, Nucl. Phys. B 544 (1999) 403 [hep-th/9809168] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    W. Kummer and D.V. Vassilevich, Hawking radiation from dilaton gravity in (1 + 1)-dimensions: a pedagogical review, Annalen Phys. 8 (1999) 801 [gr-qc/9907041] [INSPIRE].
  76. [76]
    D. Grumiller, W. Kummer and D.V. Vassilevich, The virtual black hole in 2D quantum gravity, Nucl. Phys. B 580 (2000) 438 [gr-qc/0001038] [INSPIRE].
  77. [77]
    P. Fischer, D. Grumiller, W. Kummer and D.V. Vassilevich, S-matrix for s-wave gravitational scattering, Phys. Lett. B 521 (2001) 357 [Erratum ibid. B 532 (2002) 373] [gr-qc/0105034] [INSPIRE].
  78. [78]
    A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    D. Grumiller and M. Riegler, Most general AdS 3 boundary conditions, JHEP 10 (2016) 023 [arXiv:1608.01308] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    L. Donnay, G. Giribet, H.A. Gonzalez and M. Pino, Supertranslations and superrotations at the black hole horizon, Phys. Rev. Lett. 116 (2016) 091101 [arXiv:1511.08687] [INSPIRE].
  82. [82]
    H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].ADSMathSciNetGoogle Scholar
  83. [83]
    M. Cadoni and S. Mignemi, Asymptotic symmetries of AdS 2 and conformal group in D = 1, Nucl. Phys. B 557 (1999) 165 [hep-th/9902040] [INSPIRE].
  84. [84]
    V. de Alfaro, S. Fubini and G. Furlan, Conformal invariance in quantum mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal structure of 2D quantum gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].
  86. [86]
    F. David, Conformal field theories coupled to 2D gravity in the conformal gauge, Mod. Phys. Lett. A 3 (1988) 1651 [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    J. Distler and H. Kawai, Conformal field theory and 2D quantum gravity, Nucl. Phys. B 321 (1989) 509 [INSPIRE].
  88. [88]
    D. Grumiller, J. Salzer and D. Vassilevich, Aspects of AdS 2 holography with non-constant dilaton, Russ. Phys. J. 59 (2017) 1798 [arXiv:1607.06974] [INSPIRE].
  89. [89]
    A. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  90. [90]
    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
  91. [91]
    P. Schaller and T. Strobl, Poisson structure induced (topological) field theories, Mod. Phys. Lett. A 9 (1994) 3129 [hep-th/9405110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  92. [92]
    N. Ikeda, Two-dimensional gravity and nonlinear gauge theory, Annals Phys. 235 (1994) 435 [hep-th/9312059] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  93. [93]
    D.V. Vassilevich, Holographic duals to Poisson σ-models and noncommutative quantum mechanics, Phys. Rev. D 87 (2013) 104011 [arXiv:1301.7029] [INSPIRE].ADSGoogle Scholar
  94. [94]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  95. [95]
    E. Witten, Coadjoint orbits of the Virasoro group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    B. Oblak, BMS particles in three dimensions, arXiv:1610.08526 [INSPIRE].
  97. [97]
    T.H. Buscher, A symmetry of the string background field equations, Phys. Lett. B 194 (1987) 59 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  98. [98]
    T.H. Buscher, Path integral derivation of quantum duality in nonlinear σ-models, Phys. Lett. B 201 (1988) 466 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  99. [99]
    R.M. Wald and A. Zoupas, A general definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
  100. [100]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
  101. [101]
    T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Annals Phys. 88 (1974) 286 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  102. [102]
    M. Cadoni and S. Mignemi, Symmetry breaking, central charges and the AdS 2 /CFT 1 correspondence, Phys. Lett. B 490 (2000) 131 [hep-th/0002256] [INSPIRE].ADSCrossRefGoogle Scholar
  103. [103]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  104. [104]
    S. Detournay, T. Hartman and D.M. Hofman, Warped conformal field theory, Phys. Rev. D 86 (2012) 124018 [arXiv:1210.0539] [INSPIRE].ADSGoogle Scholar
  105. [105]
    H. Afshar, S. Detournay, D. Grumiller and B. Oblak, Near-horizon geometry and warped conformal symmetry, JHEP 03 (2016) 187 [arXiv:1512.08233] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  106. [106]
    C. Troessaert, Enhanced asymptotic symmetry algebra of AdS 3, JHEP 08 (2013) 044 [arXiv:1303.3296] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  107. [107]
    H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev. D 95 (2017) 106005 [arXiv:1611.09783] [INSPIRE].ADSGoogle Scholar
  108. [108]
    J. Gegenberg, G. Kunstatter and D. Louis-Martinez, Observables for two-dimensional black holes, Phys. Rev. D 51 (1995) 1781 [gr-qc/9408015] [INSPIRE].
  109. [109]
    G. Barnich, H.A. González and P. Salgado-Rebolledo, Geometric actions for three-dimensional gravity, arXiv:1707.08887 [INSPIRE].
  110. [110]
    A. Bagchi, D. Grumiller, J. Salzer, S. Sarkar and F. Schöller, Flat space cosmologies in two dimensions: phase transitions and asymptotic mass-domination, Phys. Rev. D 90 (2014) 084041 [arXiv:1408.5337] [INSPIRE].ADSGoogle Scholar
  111. [111]
    G. Compère, P. Mao, A. Seraj and M.M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS 3 gravity: holographic vs boundary gravitons, JHEP 01 (2016) 080 [arXiv:1511.06079] [INSPIRE].ADSCrossRefGoogle Scholar
  112. [112]
    T. Strobl, Gravity in two space-time dimensions, hep-th/0011240 [INSPIRE].

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Authors and Affiliations

  1. 1.Institute for Theoretical Physics, TU WienViennaAustria
  2. 2.Department of PhysicsLoyola University ChicagoChicagoU.S.A.
  3. 3.CMCC-Universidade Federal do ABCSanto AndréBrazil
  4. 4.Department of PhysicsTomsk State UniversityTomskRussia

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