Advertisement

Semi-simple enlargement of the \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 \) algebra from a \( \mathfrak{so}\left(2,\ 2\right)\oplus \mathfrak{so}\left(2,\ 1\right) \) Chern-Simons theory

  • Patrick ConchaEmail author
  • Nelson Merino
  • Evelyn Rodríguez
  • Patricio Salgado-Rebolledo
  • Omar Valdivia
Open Access
Regular Article - Theoretical Physics
  • 26 Downloads

Abstract

In this work we present a BMS-like ansatz for a Chern-Simons theory based on the semi-simple enlargement of the Poincaré symmetry, also known as AdS-Lorentz algebra. We start by showing that this ansatz is general enough to contain all the relevant stationary solutions of this theory and provides with suitable boundary conditions for the corresponding gauge connection. We find an explicit realization of the asymptotic symmetry at null infinity, which defines a semi-simple enlargement of the \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 \) algebra and turns out to be isomorphic to three copies of the Virasoro algebra. The flat limit of the theory is discussed at the level of the action, field equations, solutions and asymptotic symmetry.

Keywords

Conformal and W Symmetry Space-Time Symmetries Gauge-gravity correspondence Classical Theories of Gravity 

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]
    J.D. Edelstein, M. Hassaine, R. Troncoso and J. Zanelli, Lie-algebra expansions, Chern-Simons theories and the Einstein-Hilbert Lagrangian, Phys. Lett. B 640 (2006) 278 [hep-th/0605174] [INSPIRE].
  2. [2]
    F. Izaurieta, E. Rodriguez and P. Salgado, Eleven-dimensional gauge theory for the M algebra as an Abelian semigroup expansion of osp(32|1), Eur. Phys. J. C 54 (2008) 675 [hep-th/0606225] [INSPIRE].
  3. [3]
    F. Izaurieta, E. Rodriguez, P. Minning, P. Salgado and A. Perez, Standard General Relativity from Chern-Simons Gravity, Phys. Lett. B 678 (2009) 213 [arXiv:0905.2187] [INSPIRE].
  4. [4]
    N. González, P. Salgado, G. Rubio and S. Salgado, Einstein-Hilbert action with cosmological term from Chern-Simons gravity, J. Geom. Phys. 86 (2014) 339 [arXiv:1605.00325] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. Salgado, R.J. Szabo and O. Valdivia, Topological gravity and transgression holography, Phys. Rev. D 89 (2014) 084077 [arXiv:1401.3653] [INSPIRE].
  6. [6]
    P.K. Concha, D.M. Penafiel, E.K. Rodriguez and P. Salgado, Chern-Simons and Born-Infeld gravity theories and Maxwell algebras type, Eur. Phys. J. C 74 (2014) 2741 [arXiv:1402.0023] [INSPIRE].
  7. [7]
    P.K. Concha, D.M. Peñafiel, E.K. Rodríguez and P. Salgado, Generalized Poincaré algebras and Lovelock-Cartan gravity theory, Phys. Lett. B 742 (2015) 310 [arXiv:1405.7078] [INSPIRE].
  8. [8]
    O. Fierro, F. Izaurieta, P. Salgado and O. Valdivia, Minimal AdS-Lorentz supergravity in three-dimensions, Phys. Lett. B 788 (2019) 198 [arXiv:1401.3697] [INSPIRE].
  9. [9]
    P.K. Concha, O. Fierro, E.K. Rodríguez and P. Salgado, Chern-Simons supergravity in D = 3 and Maxwell superalgebra, Phys. Lett. B 750 (2015) 117 [arXiv:1507.02335] [INSPIRE].
  10. [10]
    L. Avilés, E. Frodden, J. Gomis, D. Hidalgo and J. Zanelli, Non-Relativistic Maxwell Chern-Simons Gravity, JHEP 05 (2018) 047 [arXiv:1802.08453] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Schrader, The Maxwell group and the quantum theory of particles in classical homogeneous electromagnetic fields, Fortsch. Phys. 20 (1972) 701 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    H. Bacry, P. Combe and J.L. Richard, Group-theoretical analysis of elementary particles in an external electromagnetic field. 1. the relativistic particle in a constant and uniform field, Nuovo Cim. A 67 (1970) 267 [INSPIRE].
  13. [13]
    J. Gomis and A. Kleinschmidt, On free Lie algebras and particles in electro-magnetic fields, JHEP 07 (2017) 085 [arXiv:1705.05854] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    D.V. Soroka and V.A. Soroka, Gauge semi-simple extension of the Poincaré group, Phys. Lett. B 707 (2012) 160 [arXiv:1101.1591] [INSPIRE].
  15. [15]
    J. Gomis, K. Kamimura and J. Lukierski, Deformations of Maxwell algebra and their Dynamical Realizations, JHEP 08 (2009) 039 [arXiv:0906.4464] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Hoseinzadeh and A. Rezaei-Aghdam, (2 + 1)-dimensional gravity from Maxwell and semisimple extension of the Poincaré gauge symmetric models, Phys. Rev. D 90 (2014) 084008 [arXiv:1402.0320] [INSPIRE].
  17. [17]
    P. Concha, N. Merino, O. Mišković, E. Rodríguez, P. Salgado-ReboLledó and O. Valdivia, Asymptotic symmetries of three-dimensional Chern-Simons gravity for the Maxwell algebra, JHEP 10 (2018) 079 [arXiv:1805.08834] [INSPIRE].
  18. [18]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    G. Arcioni and C. Dappiaggi, Exploring the holographic principle in asymptotically flat space-times via the BMS group, Nucl. Phys. B 674 (2003) 553 [hep-th/0306142] [INSPIRE].
  22. [22]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [arXiv:1208.4371] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Ashtekar, Geometry and Physics of Null Infinity, arXiv:1409.1800 [INSPIRE].
  26. [26]
    L. Donnay and G. Giribet, Holographic entropy of Warped-AdS 3 black holes, JHEP 06 (2015) 099 [arXiv:1504.05640] [INSPIRE].
  27. [27]
    B. Oblak, BMS Particles in Three Dimensions, Ph.D. Thesis, Brussels U. (2016) [DOI: https://doi.org/10.1007/978-3-319-61878-4] [arXiv:1610.08526] [INSPIRE].
  28. [28]
    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
  29. [29]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    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].
  31. [31]
    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].
  32. [32]
    G. Barnich and H.A. Gonzalez, Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity, JHEP 05 (2013) 016 [arXiv:1303.1075] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    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].
  34. [34]
    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
  35. [35]
    C. Troessaert, Enhanced asymptotic symmetry algebra of AdS 3, JHEP 08 (2013) 044 [arXiv:1303.3296] [INSPIRE].
  36. [36]
    H.A. Gonzalez and M. Pino, Boundary dynamics of asymptotically flat 3D gravity coupled to higher spin fields, JHEP 05 (2014) 127 [arXiv:1403.4898] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    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
  38. [38]
    O. Fuentealba, J. Matulich and R. Troncoso, Asymptotically flat structure of hypergravity in three spacetime dimensions, JHEP 10 (2015) 009 [arXiv:1508.04663] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    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].
  40. [40]
    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].
  41. [41]
    M.R. Setare and H. Adami, Enhanced asymptotic BM S 3 algebra of the flat spacetime solutions of generalized minimal massive gravity, Nucl. Phys. B 926 (2018) 70 [arXiv:1703.00936] [INSPIRE].
  42. [42]
    O. Fuentealba, J. Matulich and R. Troncoso, Asymptotic structure of \( \mathcal{N}=2 \) supergravity in 3D: extended super-BM S 3 and nonlinear energy bounds, JHEP 09 (2017) 030 [arXiv:1706.07542] [INSPIRE].
  43. [43]
    N. Banerjee, A. Bhattacharjee, I. Lodato and T. Neogi, Maximally \( \mathcal{N} \) -extended super-BM S 3 algebras and generalized 3D gravity solutions, JHEP 01 (2019) 115 [arXiv:1807.06768] [INSPIRE].
  44. [44]
    R. Caroca, P. Concha, E. Rodríguez and P. Salgado-ReboLledó, Generalizing the \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 \) and 2D-conformal algebras by expanding the Virasoro algebra, Eur. Phys. J. C 78 (2018) 262 [arXiv:1707.07209] [INSPIRE].
  45. [45]
    D.V. Soroka and V.A. Soroka, Semi-simple extension of the (super)Poincaré algebra, Adv. High Energy Phys. 2009 (2009) 234147 [hep-th/0605251] [INSPIRE].CrossRefzbMATHGoogle Scholar
  46. [46]
    P. Salgado and S. Salgado, \( \mathfrak{so}\left(D - 1,1\right)\otimes \mathfrak{so}\left(D - 1,2\right) \) algebras and gravity, Phys. Lett. B 728 (2014) 5 [INSPIRE].
  47. [47]
    F. Izaurieta, E. Rodriguez and P. Salgado, Expanding Lie (super)algebras through Abelian semigroups, J. Math. Phys. 47 (2006) 123512 [hep-th/0606215] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    J. Diaz et al., A generalized action for (2 + 1)-dimensional Chern-Simons gravity, J. Phys. A 45 (2012) 255207 [arXiv:1311.2215] [INSPIRE].
  49. [49]
    P.K. Concha, O. Fierro and E.K. Rodríguez, Inönü-Wigner contraction and D = 2 + 1 supergravity, Eur. Phys. J. C 77 (2017) 48 [arXiv:1611.05018] [INSPIRE].
  50. [50]
    P. Concha, D.M. Peñafiel and E. Rodríguez, On the Maxwell supergravity and flat limit in 2+1 dimensions, Phys. Lett. B 785 (2018) 247 [arXiv:1807.00194] [INSPIRE].
  51. [51]
    R. Caroca, P. Concha, O. Fierro, E. Rodríguez and P. Salgado-ReboLledó, Generalized Chern-Simons higher-spin gravity theories in three dimensions, Nucl. Phys. B 934 (2018) 240 [arXiv:1712.09975] [INSPIRE].
  52. [52]
    M.C. Ipinza, P.K. Concha, L. Ravera and E.K. Rodríguez, On the Supersymmetric Extension of Gauss-Bonnet like Gravity, JHEP 09 (2016) 007 [arXiv:1607.00373] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  53. [53]
    A. Banaudi and L. Ravera, Generalized AdS-Lorentz deformed supergravity on a manifold with boundary, Eur. Phys. J. Plus 133 (2018) 514 [arXiv:1803.08738] [INSPIRE].CrossRefGoogle Scholar
  54. [54]
    P. Concha, L. Ravera and E. Rodríguez, On the supersymmetry invariance of flat supergravity with boundary, JHEP 01 (2019) 192 [arXiv:1809.07871] [INSPIRE].
  55. [55]
    P.K. Concha, R. Durka, C. Inostroza, N. Merino and E.K. Rodríguez, Pure Lovelock gravity and Chern-Simons theory, Phys. Rev. D 94 (2016) 024055 [arXiv:1603.09424] [INSPIRE].
  56. [56]
    P.K. Concha, N. Merino and E.K. Rodríguez, Lovelock gravities from Born-Infeld gravity theory, Phys. Lett. B 765 (2017) 395 [arXiv:1606.07083] [INSPIRE].
  57. [57]
    P. Concha and E. Rodríguez, Generalized Pure Lovelock Gravity, Phys. Lett. B 774 (2017) 616 [arXiv:1708.08827] [INSPIRE].
  58. [58]
    P.K. Concha, R. Durka, N. Merino and E.K. Rodríguez, New family of Maxwell like algebras, Phys. Lett. B 759 (2016) 507 [arXiv:1601.06443] [INSPIRE].
  59. [59]
    B. Julia and S. Silva, Currents and superpotentials in classical gauge invariant theories. 1. Local results with applications to perfect fluids and general relativity, Class. Quant. Grav. 15 (1998) 2173 [gr-qc/9804029] [INSPIRE].
  60. [60]
    S.-S. Feng, B. Wang and X.-H. Meng, Conservative currents of boundary charges in AdS(2 + 1) gravity, Commun. Theor. Phys. 36 (2001) 33 [hep-th/9902108] [INSPIRE].
  61. [61]
    G. Barnich, A. Gomberoff and H.A. Gonzalez, The Flat limit of three dimensional asymptotically anti-de Sitter spacetimes, Phys. Rev. D 86 (2012) 024020 [arXiv:1204.3288] [INSPIRE].
  62. [62]
    T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    C. Batlle, V. Campello and J. Gomis, Canonical realization of (2 + 1)-dimensional Bondi-Metzner-Sachs symmetry, Phys. Rev. D 96 (2017) 025004 [arXiv:1703.01833] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Patrick Concha
    • 2
    Email author
  • Nelson Merino
    • 3
  • Evelyn Rodríguez
    • 4
  • Patricio Salgado-Rebolledo
    • 2
  • Omar Valdivia
    • 1
  1. 1.Facultad de Ingeniería y ArquitecturaUniversidad Arturo PratIquiqueChile
  2. 2.Instituto de FísicaPontificia Universidad Católica de ValparaísoValparaisoChile
  3. 3.Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de PhysiqueLyonFrance
  4. 4.Departamento de Ciencias, Facultad de Artes LiberalesUniversidad Adolfo IbáñezViña del MarChile

Personalised recommendations