Non-relativistic three-dimensional supergravity theories and semigroup expansion method

Abstract

In this work we present an alternative method to construct diverse non-relativistic Chern-Simons supergravity theories in three spacetime dimensions. To this end, we apply the Lie algebra expansion method based on semigroups to a supersymmetric extension of the Nappi-Witten algebra. Two different families of non-relativistic superalgebras are obtained, corresponding to generalizations of the extended Bargmann superalgebra and extended Newton-Hooke superalgebra, respectively. The expansion method considered here allows to obtain known and new non-relativistic supergravity models in a systematic way. In particular, it immediately provides an invariant tensor for the expanded superalgebra, which is essential to construct the corresponding Chern-Simons supergravity action. We show that the extended Bargmann supergravity and its Maxwellian generalization appear as particular subcases of a generalized extended Bargmann supergravity theory. In addition, we demonstrate that the generalized extended Bargmann and generalized extended Newton-Hooke supergravity families are related through a contraction process.

A preprint version of the article is available at ArXiv.

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References

  1. [1]

    R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [arXiv:1305.6737] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  2. [2]

    E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan supergravity with torsion and Schrödinger supergravity, JHEP 11 (2015) 180 [arXiv:1509.04527] [INSPIRE].

    ADS  MATH  Google Scholar 

  3. [3]

    E.A. Bergshoeff and J. Rosseel, Three-dimensional extended Bargmann supergravity, Phys. Rev. Lett. 116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].

    ADS  Google Scholar 

  4. [4]

    N. Ozdemir, M. Ozkan, O. Tunca and U. Zorba, Three-dimensional extended Newtonian (super)gravity, JHEP 05 (2019) 130 [arXiv:1903.09377] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  5. [5]

    J.A. de Azcárraga, D. Gútiez and J.M. Izquierdo, Extended D = 3 Bargmann supergravity from a Lie algebra expansion, Nucl. Phys. B 946 (2019) 114706 [arXiv:1904.12786] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  6. [6]

    N. Ozdemir, M. Ozkan and U. Zorba, Three-dimensional extended Lifshitz, Schrödinger and Newton-Hooke supergravity, JHEP 11 (2019) 052 [arXiv:1909.10745] [INSPIRE].

    ADS  MATH  Google Scholar 

  7. [7]

    P. Concha, L. Ravera and E. Rodríguez, Three-dimensional Maxwellian extended Bargmann supergravity, JHEP 04 (2020) 051 [arXiv:1912.09477] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  8. [8]

    P. Concha, L. Ravera and E. Rodríguez, Three-dimensional non-relativistic extended supergravity with cosmological constant, Eur. Phys. J. C 80 (2020) 1105 [arXiv:2008.08655] [INSPIRE].

    ADS  Google Scholar 

  9. [9]

    R. Grassie, Generalised Bargmann superalgebras, arXiv:2010.01894 [INSPIRE].

  10. [10]

    E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Ann. Ecole Norm. Sup. 40 (1923) 325.

  11. [11]

    E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie) (suite), Ann. Ecole Norm. Sup. 41 (1924) 1.

  12. [12]

    C. Duval and H.P. Kunzle, Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation, Gen. Rel. Grav. 16 (1984) 333 [INSPIRE].

    ADS  Google Scholar 

  13. [13]

    C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  14. [14]

    C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  15. [15]

    R. Andringa, E. Bergshoeff, S. Panda and M. de Roo, Newtonian gravity and the Bargmann algebra, Class. Quant. Grav. 28 (2011) 105011 [arXiv:1011.1145] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  16. [16]

    R. Banerjee, A. Mitra and P. Mukherjee, Localization of the Galilean symmetry and dynamical realization of Newton-Cartan geometry, Class. Quant. Grav. 32 (2015) 045010 [arXiv:1407.3617] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  17. [17]

    R. Banerjee and P. Mukherjee, Torsional Newton-Cartan geometry from Galilean gauge theory, Class. Quant. Grav. 33 (2016) 225013 [arXiv:1604.06893] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  18. [18]

    E. Bergshoeff, A. Chatzistavrakidis, L. Romano and J. Rosseel, Newton-Cartan gravity and torsion, JHEP 10 (2017) 194 [arXiv:1708.05414] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  19. [19]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  20. [20]

    L. Avilés, J. Gomis and D. Hidalgo, Stringy (Galilei) Newton-Hooke Chern-Simons Gravities, JHEP 09 (2019) 015 [arXiv:1905.13091] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  21. [21]

    D. Chernyavsky and D. Sorokin, Three-dimensional (higher-spin) gravities with extended Schrödinger and l-conformal Galilean symmetries, JHEP 07 (2019) 156 [arXiv:1905.13154] [INSPIRE].

    ADS  MATH  Google Scholar 

  22. [22]

    P. Concha and E. Rodríguez, Non-relativistic gravity theory based on an enlargement of the extended Bargmann algebra, JHEP 07 (2019) 085 [arXiv:1906.00086] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  23. [23]

    T. Harmark, J. Hartong, L. Menculini, N.A. Obers and G. Oling, Relating non-relativistic string theories, JHEP 11 (2019) 071 [arXiv:1907.01663] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  24. [24]

    D. Hansen, J. Hartong and N.A. Obers, Non-relativistic gravity and its coupling to matter, JHEP 06 (2020) 145 [arXiv:2001.10277] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  25. [25]

    M. Ergen, E. Hamamci and D. Van den Bleeken, Oddity in nonrelativistic, strong gravity, Eur. Phys. J. C 80 (2020) 563 [Erratum ibid. 80 (2020) 657] [arXiv:2002.02688] [INSPIRE].

  26. [26]

    O. Kasikci, N. Ozdemir, M. Ozkan and U. Zorba, Three-dimensional higher-order Schrödinger algebras and Lie algebra expansions, JHEP 04 (2020) 067 [arXiv:2002.03558] [INSPIRE].

    ADS  MATH  Google Scholar 

  27. [27]

    P. Concha, M. Ipinza and E. Rodríguez, Generalized Maxwellian exotic Bargmann gravity theory in three spacetime dimensions, Phys. Lett. B 807 (2020) 135593 [arXiv:2004.01203] [INSPIRE].

    MathSciNet  Google Scholar 

  28. [28]

    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].

    ADS  Google Scholar 

  29. [29]

    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  30. [30]

    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  31. [31]

    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  32. [32]

    A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  33. [33]

    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].

    ADS  MATH  Google Scholar 

  34. [34]

    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  35. [35]

    J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  36. [36]

    J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  37. [37]

    J. Hartong, E. Kiritsis and N.A. Obers, Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum, JHEP 08 (2015) 006 [arXiv:1502.00228] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  38. [38]

    M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  39. [39]

    C. Hoyos and D.T. Son, Hall viscosity and electromagnetic response, Phys. Rev. Lett. 108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].

    ADS  Google Scholar 

  40. [40]

    D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].

  41. [41]

    A.G. Abanov and A. Gromov, Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field, Phys. Rev. B 90 (2014) 014435 [arXiv:1401.3703] [INSPIRE].

    ADS  Google Scholar 

  42. [42]

    M. Geracie, K. Prabhu and M.M. Roberts, Curved non-relativistic spacetimes, Newtonian gravitation and massive matter, J. Math. Phys. 56 (2015) 103505 [arXiv:1503.02682] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  43. [43]

    A. Gromov, K. Jensen and A.G. Abanov, Boundary effective action for quantum Hall states, Phys. Rev. Lett. 116 (2016) 126802 [arXiv:1506.07171] [INSPIRE].

    ADS  Google Scholar 

  44. [44]

    D.R. Grigore, The projective unitary irreducible representations of the Galilei group in (1 + 2)-dimensions, J. Math. Phys. 37 (1996) 460 [hep-th/9312048] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  45. [45]

    S.K. Bose, The Galilean group in (2 + 1) space-times and its central extension, Commun. Math. Phys. 169 (1995) 385 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  46. [46]

    C. Duval and P.A. Horvathy, The ‘Peierls substitution’ and the exotic Galilei group, Phys. Lett. B 479 (2000) 284 [hep-th/0002233] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  47. [47]

    R. Jackiw and V.P. Nair, Anyon spin and the exotic central extension of the planar Galilei group, Phys. Lett. B 480 (2000) 237 [hep-th/0003130] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  48. [48]

    G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2 + 1 dimensions, JHEP 11 (2009) 009 [arXiv:0907.2880] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  49. [49]

    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].

    ADS  MathSciNet  Google Scholar 

  50. [50]

    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].

  51. [51]

    J. Zanelli, Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008), hep-th/0502193 [INSPIRE].

  52. [52]

    J.A. de Azcarraga, J.M. Izquierdo, M. Picón and O. Varela, Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity, Nucl. Phys. B 662 (2003) 185 [hep-th/0212347] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  53. [53]

    F. Izaurieta, E. Rodriguez and P. Salgado, Expanding Lie (super)algebras through Abelian semigroups, J. Math. Phys. 47 (2006) 123512 [hep-th/0606215] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  54. [54]

    M. Hatsuda and M. Sakaguchi, Wess-Zumino term for the AdS superstring and generalized Inonu-Wigner contraction, Prog. Theor. Phys. 109 (2003) 853 [hep-th/0106114] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  55. [55]

    J.A. de Azcarraga, J.M. Izquierdo, M. Picón and O. Varela, Expansions of algebras and superalgebras and some applications, Int. J. Theor. Phys. 46 (2007) 2738 [hep-th/0703017] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  56. [56]

    R. Caroca, I. Kondrashuk, N. Merino and F. Nadal, Bianchi spaces and their three-dimensional isometries as S-expansions of two-dimensional isometries, J. Phys. A 46 (2013) 225201 [arXiv:1104.3541] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  57. [57]

    L. Andrianopoli, N. Merino, F. Nadal and M. Trigiante, General properties of the expansion methods of Lie algebras, J. Phys. A 46 (2013) 365204 [arXiv:1308.4832] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  58. [58]

    M. Artebani, R. Caroca, M.C. Ipinza, D.M. Peñafiel and P. Salgado, Geometrical aspects of the Lie algebra S-expansion procedure, J. Math. Phys. 57 (2016) 023516 [arXiv:1602.04525] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  59. [59]

    M.C. Ipinza, F. Lingua, D.M. Peñafiel and L. Ravera, An analytic method for S-expansion involving resonance and reduction, Fortsch. Phys. 64 (2016) 854 [arXiv:1609.05042] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  60. [60]

    C. Inostroza, I. Kondrashuk, N. Merino and F. Nadal, A Java library to perform S-expansions of Lie algebras, arXiv:1703.04036 [INSPIRE].

  61. [61]

    C. Inostroza, I. Kondrashuk, N. Merino and F. Nadal, On the algorithm to find S-related Lie algebras, J. Phys. Conf. Ser. 1085 (2018) 052011 [arXiv:1802.05765] [INSPIRE].

    Google Scholar 

  62. [62]

    E. Bergshoeff, J.M. Izquierdo, T. Ortín and L. Romano, Lie algebra expansions and actions for non-relativistic gravity, JHEP 08 (2019) 048 [arXiv:1904.08304] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  63. [63]

    L. Romano, Non-relativistic four dimensional p-brane supersymmetric theories and Lie algebra expansion, arXiv:1906.08220 [INSPIRE].

  64. [64]

    A. Fontanella and L. Romano, Lie algebra expansion and integrability in superstring σ-models, JHEP 07 (2020) 083 [arXiv:2005.01736] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  65. [65]

    D.M. Peñafiel and P. Salgado-ReboLledó, Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra, Phys. Lett. B 798 (2019) 135005 [arXiv:1906.02161] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  66. [66]

    J. Gomis, A. Kleinschmidt, J. Palmkvist and P. Salgado-ReboLledó, Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity, JHEP 02 (2020) 009 [arXiv:1912.07564] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  67. [67]

    E. Bergshoeff, J. Gomis and P. Salgado-ReboLledó, Non-relativistic limits and three-dimensional coadjoint Poincaré gravity, Proc. Roy. Soc. Lond. A 476 (2020) 20200106 [arXiv:2001.11790] [INSPIRE].

    Google Scholar 

  68. [68]

    P. Concha, L. Ravera, E. Rodríguez and G. Rubio, Three-dimensional Maxwellian extended Newtonian gravity and flat limit, JHEP 10 (2020) 181 [arXiv:2006.13128] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  69. [69]

    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].

    ADS  MathSciNet  Google Scholar 

  70. [70]

    J. Diaz, O. Fierro, F. Izaurieta, N. Merino, E. Rodriguez, P. Salgado et al., A generalized action for (2 + 1)-dimensional Chern-Simons gravity, J. Phys. A 45 (2012) 255207 [arXiv:1311.2215] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  71. [71]

    P.K. Concha, D.M. Peñafiel, E.K. Rodríguez and P. Salgado, Even-dimensional general relativity from Born-Infeld gravity, Phys. Lett. B 725 (2013) 419 [arXiv:1309.0062] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  72. [72]

    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].

  73. [73]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  74. [74]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  75. [75]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  76. [76]

    P.K. Concha and E.K. Rodríguez, N = 1 supergravity and Maxwell superalgebras, JHEP 09 (2014) 090 [arXiv:1407.4635] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  77. [77]

    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].

    ADS  Google Scholar 

  78. [78]

    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].

    Google Scholar 

  79. [79]

    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].

    ADS  MATH  Google Scholar 

  80. [80]

    R. Caroca, P. Concha, E. Rodríguez and P. Salgado-ReboLledó, Generalizing the bms3 and 2D-conformal algebras by expanding the Virasoro algebra, Eur. Phys. J. C 78 (2018) 262 [arXiv:1707.07209] [INSPIRE].

    ADS  Google Scholar 

  81. [81]

    R. Caroca, P. Concha, O. Fierro and E. Rodríguez, Three-dimensional Poincaré supergravity and N -extended supersymmetric BMS3 algebra, Phys. Lett. B 792 (2019) 93 [arXiv:1812.05065] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  82. [82]

    R. Caroca, P. Concha, O. Fierro and E. Rodríguez, On the supersymmetric extension of asymptotic symmetries in three spacetime dimensions, Eur. Phys. J. C 80 (2020) 29 [arXiv:1908.09150] [INSPIRE].

    ADS  Google Scholar 

  83. [83]

    C.R. Nappi and E. Witten, A WZW model based on a nonsemisimple group, Phys. Rev. Lett. 71 (1993) 3751 [hep-th/9310112] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  84. [84]

    J.M. Figueroa-O’Farrill and S. Stanciu, More D-branes in the Nappi-Witten background, JHEP 01 (2000) 024 [hep-th/9909164] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  85. [85]

    E. Inonu and E.P. Wigner, On the contraction of groups and their represenations, Proc. Nat. Acad. Sci. 39 (1953) 510 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  86. [86]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  87. [87]

    R. Schrader, The Maxwell group and the quantum theory of particles in classical homogeneous electromagnetic fields, Fortsch. Phys. 20 (1972) 701 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  88. [88]

    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].

  89. [89]

    J. Gomis and A. Kleinschmidt, On free Lie algebras and particles in electro-magnetic fields, JHEP 07 (2017) 085 [arXiv:1705.05854] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  90. [90]

    P.K. Concha and E.K. Rodríguez, Maxwell superalgebras and Abelian semigroup expansion, Nucl. Phys. B 886 (2014) 1128 [arXiv:1405.1334] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  91. [91]

    R. Aldrovandi, A.L. Barbosa, L.C.B. Crispino and J.G. Pereira, Non-relativistic spacetimes with cosmological constant, Class. Quant. Grav. 16 (1999) 495 [gr-qc/9801100] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  92. [92]

    G.W. Gibbons and C.E. Patricot, Newton-Hooke space-times, Hpp waves and the cosmological constant, Class. Quant. Grav. 20 (2003) 5225 [hep-th/0308200] [INSPIRE].

    ADS  MATH  Google Scholar 

  93. [93]

    J. Brugues, J. Gomis and K. Kamimura, Newton-Hooke algebras, non-relativistic branes and generalized pp-wave metrics, Phys. Rev. D 73 (2006) 085011 [hep-th/0603023] [INSPIRE].

    ADS  Google Scholar 

  94. [94]

    P.D. Alvarez, J. Gomis, K. Kamimura and M.S. Plyushchay, (2 + 1)D exotic Newton-Hooke symmetry, duality and projective phase, Annals Phys. 322 (2007) 1556 [hep-th/0702014] [INSPIRE].

  95. [95]

    G. Papageorgiou and B.J. Schroers, Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra, JHEP 11 (2010) 020 [arXiv:1008.0279] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  96. [96]

    C. Duval and P. Horvathy, Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes, J. Phys. A 44 (2011) 335203 [arXiv:1104.1502] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  97. [97]

    J. Hartong, Y. Lei and N.A. Obers, Nonrelativistic Chern-Simons theories and three-dimensional Hořava-Lifshitz gravity, Phys. Rev. D 94 (2016) 065027 [arXiv:1604.08054] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  98. [98]

    C. Duval, G. Gibbons and P. Horvathy, Conformal and projective symmetries in Newtonian cosmology, J. Geom. Phys. 112 (2017) 197 [arXiv:1605.00231] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  99. [99]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  100. [100]

    A. Giacomini, R. Troncoso and S. Willison, Three-dimensional supergravity reloaded, Class. Quant. Grav. 24 (2007) 2845 [hep-th/0610077] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  101. [101]

    R. Troncoso and J. Zanelli, Higher dimensional gravity, propagating torsion and AdS gauge invariance, Class. Quant. Grav. 17 (2000) 4451 [hep-th/9907109] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  102. [102]

    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].

    ADS  MATH  Google Scholar 

  103. [103]

    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].

    ADS  MATH  Google Scholar 

  104. [104]

    H.R. Afshar, E.A. Bergshoeff, A. Mehra, P. Parekh and B. Rollier, A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities, JHEP 04 (2016) 145 [arXiv:1512.06277] [INSPIRE].

    ADS  MATH  Google Scholar 

  105. [105]

    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  106. [106]

    G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  107. [107]

    P. Concha, L. Ravera and E. Rodríguez, Three-dimensional exotic Newtonian gravity with cosmological constant, Phys. Lett. B 804 (2020) 135392 [arXiv:1912.02836] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  108. [108]

    D. Hansen, J. Hartong and N.A. Obers, Action principle for Newtonian gravity, Phys. Rev. Lett. 122 (2019) 061106 [arXiv:1807.04765] [INSPIRE].

    ADS  Google Scholar 

  109. [109]

    L. Ravera, AdS Carroll Chern-Simons supergravity in 2 + 1 dimensions and its flat limit, Phys. Lett. B 795 (2019) 331 [arXiv:1905.00766] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  110. [110]

    F. Ali and L. Ravera, \( \mathcal{N} \)-extended Chern-Simons Carrollian supergravities in 2 + 1 spacetime dimensions, JHEP 02 (2020) 128 [arXiv:1912.04172] [INSPIRE].

  111. [111]

    L. Ciambelli, C. Marteau, A.C. Petkou, P.M. Petropoulos and K. Siampos, Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids, Class. Quant. Grav. 35 (2018) 165001 [arXiv:1802.05286] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  112. [112]

    L. Ciambelli, C. Marteau, A.C. Petkou, P.M. Petropoulos and K. Siampos, Flat holography and Carrollian fluids, JHEP 07 (2018) 165 [arXiv:1802.06809] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  113. [113]

    L. Ciambelli and C. Marteau, Carrollian conservation laws and Ricci-flat gravity, Class. Quant. Grav. 36 (2019) 085004 [arXiv:1810.11037] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  114. [114]

    A. Campoleoni, L. Ciambelli, C. Marteau, P.M. Petropoulos and K. Siampos, Two-dimensional fluids and their holographic duals, Nucl. Phys. B 946 (2019) 114692 [arXiv:1812.04019] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

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Concha, P., Ipinza, M., Ravera, L. et al. Non-relativistic three-dimensional supergravity theories and semigroup expansion method. J. High Energ. Phys. 2021, 94 (2021). https://doi.org/10.1007/JHEP02(2021)094

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Keywords

  • Chern-Simons Theories
  • Supergravity Models
  • Gauge Symmetry
  • Classical Theories of Gravity