Abstract
We consider a non-relativistic (NR) limit of (2 + 1)-dimensional Maxwell Chern-Simons (CS) gravity with gauge algebra [Maxwell] ⊕ u(1) ⊕ u(1). We obtain a finite NR CS gravity with a degenerate invariant bilinear form. We find two ways out of this difficulty: to consider i) [Maxwell] ⊕ u(1), which does not contain Extended Bargmann gravity (EBG); or, ii) the NR limit of [Maxwell] ⊕ u(1)⊕u(1)⊕u(1), which is a Maxwellian generalization of the EBG.
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References
S. Sachdev, Quantum phase transitions, Cambridge University Press, Cambridge U.K., (2011) [ISBN:9780521514682].
Y. Liu, K. Schalm, Y.-W. Sun and J. Zaanen, Holographic duality in condensed matter physics, Cambridge University Press, Cambridge U.K., (2015) [ISBN:9781107080089].
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie) (in French), Annales Sci. École Norm. Sup. 40 (1923) 325.
P. Havas, Four-dimensional formulations of Newtonian mechanics and their relation to the special and the general theory of relativity, Rev. Mod. Phys. 36 (1964) 938 [INSPIRE].
A. Trautman, Sur la théorie newtonienne de la gravitation (in French), Comptes Rendus Acad. Sci. 257 (1963) 617.
G. Dautcourt, Die Newtonske Gravitationstheorie als Strenger Grenzfall der Allgemeinen Relativitätheorie (in German), Acta Phys. Pol. 25 (1964) 637.
K. Kuchar, Gravitation, geometry, and nonrelativistic quantum theory, Phys. Rev. D 22 (1980) 1285 [INSPIRE].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291 [hep-th/9412002] [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].
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].
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].
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].
E.A. Bergshoeff, J. Hartong and J. Rosseel, Torsional Newton-Cartan geometry and the Schrödinger algebra, Class. Quant. Grav. 32 (2015) 135017 [arXiv:1409.5555] [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, JHEP 10 (2017) 194 [arXiv:1708.05414] [INSPIRE].
J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys. 42 (2001) 3127 [hep-th/0009181] [INSPIRE].
J. Gomis, K. Kamimura and P.K. Townsend, Non-relativistic superbranes, JHEP 11 (2004) 051 [hep-th/0409219] [INSPIRE].
C. Batlle, J. Gomis and D. Not, Extended Galilean symmetries of non-relativistic strings, JHEP 02 (2017) 049 [arXiv:1611.00026] [INSPIRE].
A. Barducci, R. Casalbuoni and J. Gomis, Non-relativistic spinning particle in a Newton-Cartan background, JHEP 01 (2018) 002 [arXiv:1710.10970] [INSPIRE].
J.-M. Lévy-Leblond, Galilei group and Galilean invariance, in Group theory and applications, volume II, E.M. Loebl ed., Acad. Press, New York U.S.A., (1972), pg. 222.
J. Hartong, Y. Lei, N.A. Obers and G. Oling, Zooming in on AdS 3 /CFT 2 near a BPS bound, arXiv:1712.05794 [INSPIRE].
R. Schrader, The Maxwell group and the quantum theory of particles in classical homogeneous electromagnetic fields, Fortsch. Phys. 20 (1972) 701 [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].
S. Bonanos and J. Gomis, Infinite sequence of Poincaré group extensions: structure and dynamics, J. Phys. A 43 (2010) 015201 [arXiv:0812.4140] [INSPIRE].
J. Gomis and A. Kleinschmidt, On free Lie algebras and particles in electro-magnetic fields, JHEP 07 (2017) 085 [arXiv:1705.05854] [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].
P. Salgado, R.J. Szabo and O. Valdivia, Topological gravity and transgression holography, Phys. Rev. D 89 (2014) 084077 [arXiv:1401.3653] [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].
J. Beckers and V. Hussin, Minimal electromagnetic coupling schemes. II. Relativistic and nonrelativistic Maxwell groups, J. Math. Phys. 24 (1983) 1295 [INSPIRE].
S. Bonanos and J. Gomis, A note on the Chevalley-Eilenberg cohomology for the Galilei and Poincaré algebras, J. Phys. A 42 (2009) 145206 [arXiv:0808.2243] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982) 975 [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [Annals Phys. 281 (2000) 409] [INSPIRE].
D.K. Wise, Symmetric space Cartan connections and gravity in three and four dimensions, SIGMA 5 (2009) 080 [arXiv:0904.1738] [INSPIRE].
E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive gravity in three dimensions, Phys. Rev. Lett. 102 (2009) 201301 [arXiv:0901.1766] [INSPIRE].
S. Bonanos, J. Gomis, K. Kamimura and J. Lukierski, Maxwell superalgebra and superparticle in constant gauge badkgrounds, Phys. Rev. Lett. 104 (2010) 090401 [arXiv:0911.5072] [INSPIRE].
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].
O. Fuentealba, J. Matulich and R. Troncoso, Asymptotic structure of N = 2 supergravity in 3D: extended super-BMS 3 and nonlinear energy bounds, JHEP 09 (2017) 030 [arXiv:1706.07542] [INSPIRE].
R. Basu, S. Detournay and M. Riegler, Spectral flow in 3D flat spacetimes, JHEP 12 (2017) 134 [arXiv:1706.07438] [INSPIRE].
M. Hassaine and J. Zanelli, Chern-Simons (super) gravity, World Scientific Publishing Co., Singapore, (2016).
R. Andringa, Newton-Cartan gravity revisited, doctoral thesis, Rijksuniversiteit Groningen, Groningen The Netherlands, (2016).
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Avilés, L., Frodden, E., Gomis, J. et al. Non-relativistic Maxwell Chern-Simons gravity. J. High Energ. Phys. 2018, 47 (2018). https://doi.org/10.1007/JHEP05(2018)047
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DOI: https://doi.org/10.1007/JHEP05(2018)047