Abstract
In this work we study a non-relativistic three dimensional Chern-Simons gravity theory based on an enlargement of the Extended Bargmann algebra. A finite nonrelativistic Chern-Simons gravity action is obtained through the non-relativistic contraction of a particular U(1) enlargement of the so-called AdS-Lorentz algebra. We show that the non-relativistic gravity theory introduced here reproduces the Maxwellian Exotic Bargmann gravity theory when a flat limit ℓ → ∞ is applied. We also present an alternative procedure to obtain the non-relativistic versions of the AdS-Lorentz and Maxwell algebras through the semigroup expansion method.
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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].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett.101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev.D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP07 (2009) 037 [arXiv:0902.1385] [INSPIRE].
A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP08 (2010) 004 [arXiv:0912.1090] [INSPIRE].
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].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography, JHEP01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
M. Taylor, Lifshitz holography, Class. Quant. Grav.33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].
D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
C. Hoyos and D.T. Son, Hall viscosity and electromagnetic response, Phys. Rev. Lett.108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].
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].
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].
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].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev.D 31 (1985) 1841 [INSPIRE].
C. Duval, G.W. Gibbons and P. Horvathy, Celestial mechanics, conformal structures and gravitational waves, Phys. Rev.D 43 (1991) 3907 [hep-th/0512188] [INSPIRE].
C. Duval, On Galileian isometries, Class. Quant. Grav.10 (1993) 2217 [arXiv:0903.1641] [INSPIRE].
R. De Pietri, L. Lusanna and M. Pauri, Standard and generalized Newtonian gravities as ‘gauge’ theories of the extended Galilei group. I. The standard theory, Class. Quant. Grav.12 (1995) 219 [gr-qc/9405046] [INSPIRE].
R. De Pietri, L. Lusanna and M. Pauri, Standard and generalized Newtonian gravities as ‘gauge’ theories of the extended Galilei group. II. Dynamical three space theories, Class. Quant. Grav.12 (1995) 255 [gr-qc/9405047] [INSPIRE].
P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev.D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys.A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].
G. Papageorgiou and B.J. Schroers, Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra, JHEP11 (2010) 020 [arXiv:1008.0279] [INSPIRE].
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].
R. Andringa, E. Bergshoeff, J. Gomis and M. de Roo, ‘Stringy’ Newton-Cartan gravity, Class. Quant. Grav.29 (2012) 235020 [arXiv:1206.5176] [INSPIRE].
E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan (super)gravity as a non-relativistic limit, Class. Quant. Grav.32 (2015) 205003 [arXiv:1505.02095] [INSPIRE].
E.A. Bergshoeff and J. Rosseel, Three-dimensional extended Bargmann supergravity, Phys. Rev. Lett.116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].
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].
E. Bergshoeff, A. Chatzistavrakidis, L. Romano and J. Rosseel, Newton-Cartan gravity and torsion, JHEP10 (2017) 194 [arXiv:1708.05414] [INSPIRE].
D. Chernyavsky and D. Sorokin, Three-dimensional (higher-spin) gravities with extended Schrödinger and l-conformal Galilean symmetries, arXiv:1905.13154 [INSPIRE].
H. Bacry and J. Levy-Leblond, Possible kinematics, J. Math. Phys.9 (1968) 1605 [INSPIRE].
H. Bacry and J. Nuyts, Classification of ten-dimensional kinematical groups with space isotropy, J. Math. Phys.27 (1986) 2455 [INSPIRE].
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].
O. Arratia, M.A. Martin and M.A. Olmo, Classical systems and representations of (2 + 1) Newton-Hooke symmetries, math-ph/9903013.
Y.-H. Gao, Symmetries, matrices and de Sitter gravity, Conf. Proc.C 0208124 (2002) 271 [hep-th/0107067] [INSPIRE].
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].
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].
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].
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys.B 311 (1988) 46 [INSPIRE].
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].
J. Zanelli, Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008), in Proceedings, 7thMexican Workshop on Particles and Fields (MWPF 1999), Merida, Mexico, 10-17 November 1999 [hep-th/0502193] [INSPIRE].
J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys.42 (2001) 3127 [hep-th/0009181] [INSPIRE].
A. Barducci, R. Casalbuoni and J. Gomis, Non-relativistic spinning particle in a Newton-Cartan background, JHEP01 (2018) 002 [arXiv:1710.10970] [INSPIRE].
L. Avilés, E. Frodden, J. Gomis, D. Hidalgo and J. Zanelli, Non-relativistic Maxwell Chern-Simons gravity, JHEP05 (2018) 047 [arXiv:1802.08453] [INSPIRE].
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].
R. Schrader, The Maxwell group and the quantum theory of particles in classical homogeneous electromagnetic fields, Fortsch. Phys.20 (1972) 701 [INSPIRE].
J. Gomis and A. Kleinschmidt, On free Lie algebras and particles in electro-magnetic fields, JHEP07 (2017) 085 [arXiv:1705.05854] [INSPIRE].
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].
J. Gomis, K. Kamimura and J. Lukierski, Deformations of Maxwell algebra and their dynamical realizations, JHEP08 (2009) 039 [arXiv:0906.4464] [INSPIRE].
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].
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].
P. Concha and E. Rodríguez, Generalized pure Lovelock gravity, Phys. Lett.B 774 (2017) 616 [arXiv:1708.08827] [INSPIRE].
P.K. Concha, E.K. Rodríguez and P. Salgado, Generalized supersymmetric cosmological term in N = 1 supergravity, JHEP08 (2015) 009 [arXiv:1504.01898] [INSPIRE].
M.C. Ipinza, P.K. Concha, L. Ravera and E.K. Rodríguez, On the supersymmetric extension of Gauss-Bonnet like gravity, JHEP09 (2016) 007 [arXiv:1607.00373] [INSPIRE].
A. Banaudi and L. Ravera, Generalized AdS-Lorentz deformed supergravity on a manifold with boundary, Eur. Phys. J. Plus133 (2018) 514 [arXiv:1803.08738] [INSPIRE].
D.M. Peñafiel and L. Ravera, Generalized cosmological term in D = 4 supergravity from a new AdS-Lorentz superalgebra, Eur. Phys. J.C 78 (2018) 945 [arXiv:1807.07673] [INSPIRE].
P. Concha, N. Merino, E. Rodríguez, P. Salgado-Rebolledó and O. Valdivia, Semi-simple enlargement of the \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 \)algebra from a so(2, 2) ⊕ \( \mathfrak{so} \)(2, 1) Chern-Simons theory, JHEP02 (2019) 002 [arXiv:1810.12256] [INSPIRE].
F. Izaurieta, E. Rodriguez and P. Salgado, Expanding Lie (super)algebras through Abelian semigroups, J. Math. Phys.47 (2006) 123512 [hep-th/0606215] [INSPIRE].
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].
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].
D.M. Peñafiel and P. Salgado-Rebolledó, Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra, arXiv:1906.02161 [INSPIRE].
J. Diaz et al., A generalized action for (2 + 1)-dimensional Chern-Simons gravity, J. Phys.A 45 (2012) 255207 [arXiv:1311.2215] [INSPIRE].
P. Salgado and S. Salgado, \( \mathfrak{so} \)(D − 1, 1) ⊗ \( \mathfrak{so} \)(D − 1, 2) algebras and gravity, Phys. Lett.B 728 (2014) 5 [INSPIRE].
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].
P. Salgado, R.J. Szabo and O. Valdivia, Topological gravity and transgression holography, Phys. Rev.D 89 (2014) 084077 [arXiv:1401.3653] [INSPIRE].
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, JHEP10 (2018) 079 [arXiv:1805.08834] [INSPIRE].
C.R. Nappi and E. Witten, A WZW model based on a nonsemisimple group, Phys. Rev. Lett.71 (1993) 3751 [hep-th/9310112] [INSPIRE].
J.M. Figueroa-O’Farrill and S. Stanciu, More D-branes in the Nappi-Witten background, JHEP01 (2000) 024 [hep-th/9909164] [INSPIRE].
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].
J.A. de Azcarraga, K. Kamimura and J. Lukierski, Generalized cosmological term from Maxwell symmetries, Phys. Rev.D 83 (2011) 124036 [arXiv:1012.4402] [INSPIRE].
R. Durka, J. Kowalski-Glikman and M. Szczachor, Gauged AdS-Maxwell algebra and gravity, Mod. Phys. Lett.A 26 (2011) 2689 [arXiv:1107.4728] [INSPIRE].
J.A. de Azcarraga, K. Kamimura and J. Lukierski, Maxwell symmetries and some applications, Int. J. Mod. Phys. Conf. Ser.23 (2013) 01160 [arXiv:1201.2850] [INSPIRE].
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].
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].
S. Bonanos, J. Gomis, K. Kamimura and J. Lukierski, Maxwell superalgebra and superparticle in constant gauge backgrounds, Phys. Rev. Lett.104 (2010) 090401 [arXiv:0911.5072] [INSPIRE].
J.A. de Azcarraga and J.M. Izquierdo, Minimal D = 4 supergravity from the super-Maxwell algebra, Nucl. Phys.B 885 (2014) 34 [arXiv:1403.4128] [INSPIRE].
P.K. Concha and E.K. Rodríguez, N = 1 supergravity and Maxwell superalgebras, JHEP09 (2014) 090 [arXiv:1407.4635] [INSPIRE].
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].
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].
D.M. Peñafiel and L. Ravera, On the hidden Maxwell superalgebra underlying D = 4 supergravity, Fortsch. Phys.65 (2017) 1700005 [arXiv:1701.04234] [INSPIRE].
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].
L. Ravera, Hidden role of Maxwell superalgebras in the free differential algebras of D = 4 and D = 11 supergravity, Eur. Phys. J.C 78 (2018) 211 [arXiv:1801.08860] [INSPIRE].
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].
P. Concha, L. Ravera and E. Rodríguez, On the supersymmetry invariance of flat supergravity with boundary, JHEP01 (2019) 192 [arXiv:1809.07871] [INSPIRE].
J. Gomis, A. Kleinschmidt and J. Palmkvist, Symmetries of M-theory and free Lie superalgebras, JHEP03 (2019) 160 [arXiv:1809.09171] [INSPIRE].
S. Kibaroğlu, M. ¸enay and O. Cebecioğlu, D = 4 topological gravity from gauging the Maxwell-special-affine group, Mod. Phys. Lett.A 34 (2019) 1950016 [arXiv:1810.01635] [INSPIRE].
S. Kibaroğlu and O. Cebecioğlu, D = 4 supergravity from the Maxwell-Weyl superalgebra, arXiv:1812.09861 [INSPIRE].
P. Concha, N -extended Maxwell supergravities as Chern-Simons theories in three spacetime dimensions, Phys. Lett.B 792 (2019) 290 [arXiv:1903.03081] [INSPIRE].
P. Salgado-Rebolledó, The Maxwell group in 2 + 1 dimensions and its infinite-dimensional enhancements, arXiv:1905.09421 [INSPIRE].
J. Hietarinta, Supersymmetry generators of arbitrary spin, Phys. Rev.D 13 (1976) 838 [INSPIRE].
S. Bansal and D. Sorokin, Can Chern-Simons or Rarita-Schwinger be a Volkov-Akulov Goldstone?, JHEP07 (2018) 106 [arXiv:1806.05945] [INSPIRE].
P.K. Concha and E.K. Rodríguez, Maxwell superalgebras and Abelian semigroup expansion, Nucl. Phys.B 886 (2014) 1128 [arXiv:1405.1334] [INSPIRE].
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].
R. Caroca, P. Concha, O. Fierro and E. Rodríguez, Three-dimensional Poincaré supergravity and N -extended supersymmetric BMS 3algebra, Phys. Lett.B 792 (2019) 93 [arXiv:1812.05065] [INSPIRE].
N. González, G. Rubio, P. Salgado and S. Salgado, Generalized Galilean algebras and Newtonian gravity, Phys. Lett.B 755 (2016) 433 [arXiv:1604.06313] [INSPIRE].
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].
E. Bergshoeff, J.M. Izquierdo, T. Ortín and L. Romano, Lie algebra expansions and actions for non-relativistic gravity, arXiv:1904.08304 [INSPIRE].
J.A. de Azcárraga, D. Gútiez and J.M. Izquierdo, Extended D = 3 Bargmann supergravity from a Lie algebra expansion, arXiv:1904.12786 [INSPIRE].
J. Matulich, S. Prohazka and J. Salzer, Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension, arXiv:1903.09165 [INSPIRE].
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Concha, P., Rodríguez, E. Non-relativistic gravity theory based on an enlargement of the extended Bargmann algebra. J. High Energ. Phys. 2019, 85 (2019). https://doi.org/10.1007/JHEP07(2019)085
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DOI: https://doi.org/10.1007/JHEP07(2019)085