Abstract
We construct Chern-Simons gravities in (2 + 1)-dimensional space-time considering the Stringy Galilei algebra both with and without non-central extensions. In the first case, there is an invariant and non-degenerate bilinear form, however the field equations do not allow to express the spin connections in terms of the dreibeins. In the second case there is no invariant non-degenerate bilinear form. Therefore, in both cases we do not have an ordinary gravity theory. Instead, if we consider the stringy Newton-Hooke algebra with extensions as gauge group we have an invariant non-degenerate metric and from the field equations we express the spin connections in terms of the geometric fields.
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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].
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), Compt. Rendus l’Acad. Sci.257 (1963) 617.
G. Dautcourt, Die Newtonske Gravitationstheorie als Strenger Grenzfall der Allgemeinen Relativitätheorie (in German), Acta Phys. Polon.25 (1964) 637.
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].
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].
K.T. Grosvenor, J. Hartong, C. Keeler and N.A. Obers, Homogeneous nonrelativistic geometries as coset spaces, Class. Quant. Grav.35 (2018) 175007 [arXiv:1712.03980] [INSPIRE].
E. Bergshoeff, A. Chatzistavrakidis, L. Romano and J. Rosseel, Newton-Cartan gravity and torsion, JHEP10 (2017) 194 [arXiv:1708.05414] [INSPIRE].
D.H. Friedan, Nonlinear models in 2 + ϵ dimensions, Annals Phys.163 (1985) 318 [INSPIRE].
C.G. Callan Jr., E.J. Martinec, M.J. Perry and D. Friedan, Strings in background fields, Nucl. Phys.B 262 (1985) 593 [INSPIRE].
C. Batlle, J. Gomis and D. Not, Extended Galilean symmetries of non-relativistic strings, JHEP02 (2017) 049 [arXiv:1611.00026] [INSPIRE].
J.-M. Souriau, Structure des systèmes dynamiques (in French), Dunod (1970), Structure of dynamical systems: a symplectic view of physics, translated by C.H. Cushman-de Vries, Birkhäuser, (1997).
J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys.42 (2001) 3127 [hep-th/0009181] [INSPIRE].
J. Brugues, T. Curtright, J. Gomis and L. Mezincescu, Non-relativistic strings and branes as non-linear realizations of Galilei groups, Phys. Lett.B 594 (2004) 227 [hep-th/0404175] [INSPIRE].
F. Passerini, Corde non relativistiche. Soluzioni classiche e quantizzazione (in Italian), Tesi di Laurea, Università di Firenze, Florence, Italy (2004).
A. Barducci, R. Casalbuoni and J. Gomis, Confined dynamical systems with Carroll and Galilei symmetries, Phys. Rev.D 98 (2018) 085018 [arXiv:1804.10495] [INSPIRE].
E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic string theory and T-duality, JHEP11 (2018) 133 [arXiv:1806.06071] [INSPIRE].
T. Harmark, J. Hartong, L. Menculini, N.A. Obers and Z. Yan, Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence, JHEP11 (2018) 190 [arXiv:1810.05560] [INSPIRE].
J. Gomis, J. Oh and Z. Yan, Nonrelativistic string theory in background fields, arXiv:1905.07315 [INSPIRE].
E.A. Bergshoeff, J. Gomis, J. Rosseel, C. Simsek and Z. Yan, String theory and string Newton-Cartan geometry, arXiv:1907.10668 [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.A. Bergshoeff, K.T. Grosvenor, C. Simsek and Z. Yan, An action for extended string Newton-Cartan gravity, JHEP01 (2019) 178 [arXiv:1810.09387] [INSPIRE].
G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2 + 1 dimensions, JHEP11 (2009) 009 [arXiv:0907.2880] [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].
A. Medina and P. Revoy, Algèbres de Lie et producte scalaire invariant (in French), Ann. Sci. Ècole Norm. Sup.18 (1985) 553.
J.M. Figueroa-O’Farrill and S. Stanciu, On the structure of symmetric selfdual Lie algebras, J. Math. Phys.37 (1996) 4121 [hep-th/9506152] [INSPIRE].
J. Matulich, S. Prohazka and J. Salzer, Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension, JHEP07 (2019) 118 [arXiv:1903.09165] [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].
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].
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys.B 311 (1988) 46 [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge, U.K. (2012) [ISBN:9780521194013].
T. Ortín, Gravity and strings, Cambridge Monographs on Mathematical Physics, Cambridge, U.K. (2004) [ISBN:9780521824750].
E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei gravity, JHEP03 (2017) 165 [arXiv:1701.06156] [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].
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.A. Bergshoeff, W. Merbis and P.K. Townsend, On-shell versus off-shell equivalence in 3D gravity, Class. Quant. Grav.36 (2019) 095013 [arXiv:1812.09205] [INSPIRE].
J. Gomis, A. Kleinschmidt and J. Palmkvist, Galilean free Lie algebras, arXiv:1907.00410 [INSPIRE].
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Avilés, L., Gomis, J. & Hidalgo, D. Stringy (Galilei) Newton-Hooke Chern-Simons gravities. J. High Energ. Phys. 2019, 15 (2019). https://doi.org/10.1007/JHEP09(2019)015
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DOI: https://doi.org/10.1007/JHEP09(2019)015