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Journal of High Energy Physics

, 2017:30 | Cite as

Asymptotic structure of \( \mathcal{N}=2 \) supergravity in 3D: extended super-BMS3 and nonlinear energy bounds

  • Oscar FuentealbaEmail author
  • Javier Matulich
  • Ricardo Troncoso
Open Access
Regular Article - Theoretical Physics

Abstract

The asymptotically flat structure of \( \mathcal{N}=\left(2,0\right) \) supergravity in three spacetime dimensions is explored. The asymptotic symmetries are found to be spanned by an extension of the super-BMS3 algebra, endowed with two independent affine û(1) currents of electric and magnetic type. These currents are associated to U(1) fields being even and odd under parity, respectively. Remarkably, although the U(1) fields do not generate a backreaction on the metric, they provide nontrivial Sugawara-like contributions to the BMS3 generators, and hence to the energy and the angular momentum. Consequently, the entropy of flat cosmological spacetimes endowed with U(1) fields acquires a nontrivial dependence on the zero modes of the û(1) charges. If the spin structure is odd, the ground state corresponds to Minkowski spacetime, and although the anticommutator of the canonical supercharges is linear in the energy and in the electric-like û(1) charge, the energy becomes bounded from below by the energy of the ground state shifted by the square of the electric-like û(1) charge. If the spin structure is even, the same bound for the energy generically holds, unless the absolute value of the electric-like charge is less than minus the mass of Minkowski spacetime in vacuum, so that the energy has to be nonnegative. The explicit form of the global and asymptotic Killing spinors is found for a wide class of configurations that fulfills our boundary conditions, and they exist precisely when the corresponding bounds are saturated. It is also shown that the spectra with periodic or antiperiodic boundary conditions for the fermionic fields are related by spectral flow, in a similar way as it occurs for the \( \mathcal{N}=2 \) super-Virasoro algebra. Indeed, our supersymmetric extension of BMS3 can be recovered from the Inönü-Wigner contraction of the superconformal algebra with \( \mathcal{N}=\left(2,2\right) \), once the fermionic generators of the right copy are truncated.

Keywords

Conformal and W Symmetry Space-Time Symmetries Gauge-gravity correspondence Supergravity Models 

Notes

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.

References

  1. [1]
    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].
  2. [2]
    R.K. Sachs, Gravitational waves in general relativity 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  3. [3]
    G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 [Ann. U. Craiova Phys. 21 (2011) S11] [arXiv:1102.4632] [INSPIRE].
  6. [6]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    F. Cachazo and A. Strominger, Evidence for a new soft graviton theorem, arXiv:1404.4091 [INSPIRE].
  9. [9]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M. Campiglia and A. Laddha, New symmetries for the gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    S.W. Hawking, M.J. Perry and A. Strominger, Superrotation charge and supertranslation hair on black holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S.W. Hawking, The information paradox for black holes, arXiv:1509.01147 [INSPIRE].
  14. [14]
    R. Bousso and M. Porrati, Soft hair as a soft wig, arXiv:1706.00436 [INSPIRE].
  15. [15]
    A. Ashtekar, J. Bicak and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev. D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
  16. [16]
    G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].
  17. [17]
    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    A. Bagchi, Correspondence between asymptotically flat spacetimes and nonrelativistic conformal field theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    A. Bagchi, M. Gary and Zodinmawia, Bondi-Metzner-Sachs bootstrap, Phys. Rev. D 96 (2017) 025007 [arXiv:1612.01730] [INSPIRE].
  20. [20]
    A. Bagchi, M. Gary and Zodinmawia, The nuts and bolts of the BMS bootstrap, Class. Quant. Grav. 34 (2017) 174002 [arXiv:1705.05890] [INSPIRE].
  21. [21]
    A. Bagchi, Tensionless strings and galilean conformal algebra, JHEP 05 (2013) 141 [arXiv:1303.0291] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    E. Casali and P. Tourkine, On the null origin of the ambitwistor string, JHEP 11 (2016) 036 [arXiv:1606.05636] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    A. Bagchi, S. Chakrabortty and P. Parekh, Tensionless superstrings: view from the worldsheet, JHEP 10 (2016) 113 [arXiv:1606.09628] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    I. Mandal and A. Rayyan, Super-GCA from N = (2, 2) super-Virasoro, Phys. Lett. B 754 (2016) 195 [Addendum ibid. B 760 (2016) 832] [arXiv:1601.04723] [arXiv:1607.02439] [INSPIRE].
  25. [25]
    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
  26. [26]
    D. Grumiller, A. Perez, S. Prohazka, D. Tempo and R. Troncoso, Higher spin black holes with soft hair, JHEP 10 (2016) 119 [arXiv:1607.05360] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    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
  28. [28]
    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].ADSCrossRefGoogle Scholar
  29. [29]
    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
  30. [30]
    L. Donnay, G. Giribet, H.A. González and M. Pino, Extended symmetries at the black hole horizon, JHEP 09 (2016) 100 [arXiv:1607.05703] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    S. Deser and J.H. Kay, Topologically massive supergravity, Phys. Lett. B 120 (1983) 97 [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    S. Deser, Quantum theory of gravity: essays in honor of the 60th birthday of Bryce S. DeWitt, Adam Hilger Ltd., U.K., (1984) [INSPIRE].
  33. [33]
    N. Marcus and J.H. Schwarz, Three-dimensional supergravity theories, Nucl. Phys. B 228 (1983) 145 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    G. Barnich, L. Donnay, J. Matulich and R. Troncoso, Asymptotic symmetries and dynamics of three-dimensional flat supergravity, JHEP 08 (2014) 071 [arXiv:1407.4275] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    A. Bagchi and I. Mandal, Supersymmetric extension of galilean conformal algebras, Phys. Rev. D 80 (2009) 086011 [arXiv:0905.0580] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    I. Mandal, Supersymmetric extension of GCA in 2d, JHEP 11 (2010) 018 [arXiv:1003.0209] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  37. [37]
    N. Banerjee, D.P. Jatkar, S. Mukhi and T. Neogi, Free-field realisations of the BMS 3 algebra and its extensions, JHEP 06 (2016) 024 [arXiv:1512.06240] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    I. Lodato and W. Merbis, Super-BMS 3 algebras from N = 2 flat supergravities, JHEP 11 (2016) 150 [arXiv:1610.07506] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    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
  40. [40]
    N. Banerjee, D.P. Jatkar, I. Lodato, S. Mukhi and T. Neogi, Extended supersymmetric BMS 3 algebras and their free field realisations, JHEP 11 (2016) 059 [arXiv:1609.09210] [INSPIRE].Google Scholar
  41. [41]
    P.S. Howe, J.M. Izquierdo, G. Papadopoulos and P.K. Townsend, New supergravities with central charges and Killing spinors in (2 + 1)-dimensions, Nucl. Phys. B 467 (1996) 183 [hep-th/9505032] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    A. Schwimmer and N. Seiberg, Comments on the N = 2, N = 3, N = 4 superconformal algebras in two-dimensions, Phys. Lett. B 184 (1987) 191 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    N. Banerjee, I. Lodato and T. Neogi, N = 4 supersymmetric BMS 3 algebras from asymptotic symmetry analysis, arXiv:1706.02922 [INSPIRE].
  44. [44]
    M. Bañados, R. Troncoso and J. Zanelli, Higher dimensional Chern-Simons supergravity, Phys. Rev. D 54 (1996) 2605 [gr-qc/9601003] [INSPIRE].
  45. [45]
    M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Chemical potentials in three-dimensional higher spin anti-de Sitter gravity, JHEP 12 (2013) 048 [arXiv:1309.4362] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    C. Bunster, M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Generalized black holes in three-dimensional spacetime, JHEP 05 (2014) 031 [arXiv:1404.3305] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    O. Fuentealba, J. Matulich and R. Troncoso, Asymptotically flat structure of hypergravity in three spacetime dimensions, JHEP 10 (2015) 009 [arXiv:1508.04663] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter spaces, Commun. Math. Phys. 98 (1985) 391 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Asymptotic behavior and Hamiltonian analysis of anti-de Sitter gravity coupled to scalar fields, Annals Phys. 322 (2007) 824 [hep-th/0603185] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    M. Henneaux, C. Martinez and R. Troncoso, Asymptotically anti-de Sitter spacetimes in topologically massive gravity, Phys. Rev. D 79 (2009) 081502 [arXiv:0901.2874] [INSPIRE].ADSMathSciNetGoogle Scholar
  51. [51]
    M. Henneaux, C. Martinez and R. Troncoso, More on asymptotically anti-de Sitter spaces in topologically massive gravity, Phys. Rev. D 82 (2010) 064038 [arXiv:1006.0273] [INSPIRE].ADSGoogle Scholar
  52. [52]
    A. Perez, M. Riquelme, D. Tempo and R. Troncoso, Asymptotic structure of the Einstein-Maxwell theory on AdS 3, JHEP 02 (2016) 015 [arXiv:1512.01576] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    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].
  54. [54]
    G.T. Horowitz and A.R. Steif, Singular string solutions with nonsingular initial data, Phys. Lett. B 258 (1991) 91 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Annals Phys. 88 (1974) 286 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    S. Detournay and M. Riegler, Enhanced asymptotic symmetry algebra of 2 + 1 dimensional flat space, Phys. Rev. D 95 (2017) 046008 [arXiv:1612.00278] [INSPIRE].ADSGoogle Scholar
  57. [57]
    C. Troessaert, Enhanced asymptotic symmetry algebra of AdS 3, JHEP 08 (2013) 044 [arXiv:1303.3296] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. [58]
    R. Basu, S. Detournay and M. Riegler, Spectral flow in 3D flat spacetimes, arXiv:1706.07438 [INSPIRE].
  59. [59]
    J.D. Edelstein, C. Núñez and F.A. Schaposnik, Bogomolnyi bounds and Killing spinors in D = 3 supergravity, Phys. Lett. B 375 (1996) 163 [hep-th/9512117] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. [60]
    S. Deser and C. Teitelboim, Supergravity has positive energy, Phys. Rev. Lett. 39 (1977) 249 [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    C. Teitelboim, Surface integrals as symmetry generators in supergravity theory, Phys. Lett. B 69 (1977) 240 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    E. Witten, A simple proof of the positive energy theorem, Commun. Math. Phys. 80 (1981) 381 [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  63. [63]
    L.F. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  64. [64]
    C.M. Hull, The positivity of gravitational energy and global supersymmetry, Commun. Math. Phys. 90 (1983) 545 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    C. Teitelboim, Manifestly positive energy expression in classical gravity: simplified derivation from supergravity, Phys. Rev. D 29 (1984) 2763 [INSPIRE].ADSMathSciNetGoogle Scholar
  66. [66]
    O. Coussaert and M. Henneaux, Supersymmetry of the (2 + 1) black holes, Phys. Rev. Lett. 72 (1994) 183 [hep-th/9310194] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  67. [67]
    M. Henneaux, A. Pérez, D. Tempo and R. Troncoso, Extended anti-de Sitter hypergravity in 2 + 1 dimensions and hypersymmetry bounds,arXiv:1512.08603 [INSPIRE].
  68. [68]
    M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Hypersymmetry bounds and three-dimensional higher-spin black holes, JHEP 08 (2015) 021 [arXiv:1506.01847] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    C. Krishnan, A. Raju and S. Roy, A Grassmann path from AdS 3 to flat space, JHEP 03 (2014) 036 [arXiv:1312.2941] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    M. Gary, D. Grumiller, M. Riegler and J. Rosseel, Flat space (higher spin) gravity with chemical potentials, JHEP 01 (2015) 152 [arXiv:1411.3728] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    M. Riegler, How general is holography?, arXiv:1609.02733 [INSPIRE].
  72. [72]
    J. Matulich, A. Perez, D. Tempo and R. Troncoso, Higher spin extension of cosmological spacetimes in 3D: asymptotically flat behaviour with chemical potentials and thermodynamics, JHEP 05 (2015) 025 [arXiv:1412.1464] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    K. Ezawa, Transition amplitude in (2 + 1)-dimensional Chern-Simons gravity on a torus, Int. J. Mod. Phys. A 9 (1994) 4727 [hep-th/9305170] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  74. [74]
    L. Cornalba and M.S. Costa, A new cosmological scenario in string theory, Phys. Rev. D 66 (2002) 066001 [hep-th/0203031] [INSPIRE].ADSMathSciNetGoogle Scholar
  75. [75]
    L. Cornalba and M.S. Costa, Time dependent orbifolds and string cosmology, Fortsch. Phys. 52 (2004) 145 [hep-th/0310099] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  76. [76]
    G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [arXiv:1208.4371] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  77. [77]
    A. Bagchi, S. Detournay, R. Fareghbal and J. Simón, Holography of 3D flat cosmological horizons, Phys. Rev. Lett. 110 (2013) 141302 [arXiv:1208.4372] [INSPIRE].ADSCrossRefGoogle Scholar
  78. [78]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  79. [79]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
  80. [80]
    A. Perez, D. Tempo and R. Troncoso, Higher spin gravity in 3D: black holes, global charges and thermodynamics, Phys. Lett. B 726 (2013) 444 [arXiv:1207.2844] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  81. [81]
    A. Perez, D. Tempo and R. Troncoso, Higher spin black hole entropy in three dimensions, JHEP 04 (2013) 143 [arXiv:1301.0847] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  82. [82]
    J. de Boer and J.I. Jottar, Thermodynamics of higher spin black holes in AdS 3, JHEP 01 (2014) 023 [arXiv:1302.0816] [INSPIRE].zbMATHCrossRefGoogle Scholar
  83. [83]
    S. Deser, R. Jackiw and G. ’t Hooft, Three-dimensional Einstein gravity: dynamics of flat space, Annals Phys. 152 (1984) 220 [INSPIRE].
  84. [84]
    S. Deser and R. Jackiw, Three-dimensional cosmological gravity: dynamics of constant curvature, Annals Phys. 153 (1984) 405 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  85. [85]
    B. Chen, J. Long and Y.-N. Wang, Conical defects, black holes and higher spin (super-)symmetry, JHEP 06 (2013) 025 [arXiv:1303.0109] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  86. [86]
    A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical defects in higher spin theories, JHEP 02 (2012) 096 [arXiv:1111.3381] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  87. [87]
    S. Datta and J.R. David, Supersymmetry of classical solutions in Chern-Simons higher spin supergravity, JHEP 01 (2013) 146 [arXiv:1208.3921] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  88. [88]
    A. Campoleoni, T. Prochazka and J. Raeymaekers, A note on conical solutions in 3D Vasiliev theory, JHEP 05 (2013) 052 [arXiv:1303.0880] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  89. [89]
    A. Campoleoni and S. Fredenhagen, On the higher-spin charges of conical defects, Phys. Lett. B 726 (2013) 387 [arXiv:1307.3745] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  90. [90]
    W. Li, F.-L. Lin and C.-W. Wang, Modular properties of 3D higher spin theory, JHEP 12 (2013) 094 [arXiv:1308.2959] [INSPIRE].ADSCrossRefGoogle Scholar
  91. [91]
    J. Raeymaekers, Quantization of conical spaces in 3D gravity, JHEP 03 (2015) 060 [arXiv:1412.0278] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  92. [92]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  93. [93]
    A. Achucarro and P.K. Townsend, Extended supergravities in d = (2 + 1) as Chern-Simons theories, Phys. Lett. B 229 (1989) 383 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  94. [94]
    A. Giacomini, R. Troncoso and S. Willison, Three-dimensional supergravity reloaded, Class. Quant. Grav. 24 (2007) 2845 [hep-th/0610077] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  95. [95]
    G. Barnich, L. Donnay, J. Matulich and R. Troncoso, Super-BMS 3 invariant boundary theory from three-dimensional flat supergravity, JHEP 01 (2017) 029 [arXiv:1510.08824] [INSPIRE].ADSCrossRefGoogle Scholar
  96. [96]
    O. Fuentealba, J. Matulich and R. Troncoso, Extension of the Poincaré group with half-integer spin generators: hypergravity and beyond, JHEP 09 (2015) 003 [arXiv:1505.06173] [INSPIRE].CrossRefGoogle Scholar
  97. [97]
    H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 gravity in three-dimensional flat space, Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768] [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    H.A. Gonzalez, J. Matulich, M. Pino and R. Troncoso, Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP 09 (2013) 016 [arXiv:1307.5651] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  99. [99]
    G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: i. Induced representations, JHEP 06 (2014) 129 [arXiv:1403.5803] [INSPIRE].
  100. [100]
    A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, Rotating higher spin partition functions and extended BMS symmetries, JHEP 04 (2016) 034 [arXiv:1512.03353] [INSPIRE].ADSMathSciNetGoogle Scholar
  101. [101]
    A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, BMS modules in three dimensions, Int. J. Mod. Phys. A 31 (2016) 1650068 [arXiv:1603.03812] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  102. [102]
    B. Oblak, BMS particles in three dimensions, arXiv:1610.08526 [INSPIRE].

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© The Author(s) 2017

Authors and Affiliations

  • Oscar Fuentealba
    • 1
    Email author
  • Javier Matulich
    • 1
  • Ricardo Troncoso
    • 1
  1. 1.Centro de Estudios Científicos (CECs)ValdiviaChile

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