Abstract
We show that our previous work on Galilei and Carroll gravity, apt for particles, can be generalized to Galilei and Carroll gravity theories adapted to p-branes (p = 0, 1, 2, ⋯). Within this wider brane perspective, we make use of a formal map, given in the literature, between the corresponding p-brane Carroll and Galilei algebras where the index describing the directions longitudinal (transverse) to the Galilei brane is interchanged with the index covering the directions transverse (longitudinal) to the Carroll brane with the understanding that the time coordinate is always among the longitudinal directions. This leads among other things in 3D to a map between Galilei particles and Carroll strings and in 4D to a similar map between Galilei strings and Carroll strings. We show that this formal map extends to the corresponding Lie algebra expansion of the Poincaré algebra and, therefore, to several extensions of the Carroll and Galilei algebras including central extensions. We use this formal map to construct several new examples of Carroll gravity actions. Furthermore, we discuss the symmetry between Carroll and Galilei at the level of the p-brane sigma model action and apply this formal symmetry to give several examples of 3D and 4D particles and strings in a curved Carroll background.
References
H. Bacry and J. Levy-Leblond, Possible kinematics, J. Math. Phys. 9 (1968) 1605 [INSPIRE].
M. Henneaux, Geometry of Zero Signature Space-times, Bull. Soc. Math. Belg. 31 (1979) 47 [INSPIRE].
M. Henneaux, M. Pilati and C. Teitelboim, Explicit Solution for the Zero Signature (Strong Coupling) Limit of the Propagation Amplitude in Quantum Gravity, Phys. Lett. B 110 (1982) 123 [INSPIRE].
T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Class. Quant. Grav. 20 (2003) R145 [hep-th/0212256] [INSPIRE].
C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll symmetry of plane gravitational waves, Class. Quant. Grav. 34 (2017) 175003 [arXiv:1702.08284] [INSPIRE].
D.M. Hofman and B. Rollier, Warped Conformal Field Theory as Lower Spin Gravity, Nucl. Phys. B 897 (2015) 1 [arXiv:1411.0672] [INSPIRE].
T. Banks and W. Fischler, Holographic Space-time, Newton’s Law and the Dynamics of Black Holes, arXiv:1606.01267 [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. VII. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21.
E. Bergshoeff, J. Gomis and G. Longhi, Dynamics of Carroll Particles, Class. Quant. Grav. 31 (2014) 205009 [arXiv:1405.2264] [INSPIRE].
E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei Gravity, JHEP 03 (2017) 165 [arXiv:1701.06156] [INSPIRE].
J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [INSPIRE].
A. Bagchi, A. Mehra and P. Nandi, Field Theories with Conformal Carrollian Symmetry, JHEP 05 (2019) 108 [arXiv:1901.10147] [INSPIRE].
D. Roychowdhury, Carroll membranes, JHEP 10 (2019) 258 [arXiv:1908.07280] [INSPIRE].
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].
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].
J. Kluson, Carroll Limit of Non-BPS Dp-brane, JHEP 05 (2017) 108 [arXiv:1702.08685] [INSPIRE].
C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [arXiv:1402.0657] [INSPIRE].
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 L. Parra, The Symmetries of the Carroll Superparticle, J. Phys. A 49 (2016) 185402 [arXiv:1503.06083] [INSPIRE].
B. Cardona, J. Gomis and J.M. Pons, Dynamics of Carroll Strings, JHEP 07 (2016) 050 [arXiv:1605.05483] [INSPIRE].
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].
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].
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].
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].
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].
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].
D. Hansen, J. Hartong and N.A. Obers, Action Principle for Newtonian Gravity, Phys. Rev. Lett. 122 (2019) 061106 [arXiv:1807.04765] [INSPIRE].
E.A. Bergshoeff, K.T. Grosvenor, C. Simsek and Z. Yan, An Action for Extended String Newton-Cartan Gravity, JHEP 01 (2019) 178 [arXiv:1810.09387] [INSPIRE].
C. Batlle, J. Gomis, L. Mezincescu and P.K. Townsend, Tachyons in the Galilean limit, JHEP 04 (2017) 120 [arXiv:1702.04792] [INSPIRE].
C. Batlle, J. Gomis and D. Not, Extended Galilean symmetries of non-relativistic strings, JHEP 02 (2017) 049 [arXiv:1611.00026] [INSPIRE].
J.-M. Souriau, Structure des systèmes dynamiques, Dunod (1970).
J.-M. Souriau, Structure of Dynamical Systems: A Symplectic View of Physics, translated by C.H. Cushman-de Vries, R.H. Cushman and G.M. Tuynman translation eds., Birkhäuser (1997).
J. de Boer, J. Hartong, N.A. Obers, W. Sybesma and S. Vandoren, Perfect Fluids, SciPost Phys. 5 (2018) 003 [arXiv:1710.04708] [INSPIRE].
E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic String Theory and T-duality, JHEP 11 (2018) 133 [arXiv:1806.06071] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2003.03062
Address after 01-01-20: Van Swinderen Institute, Groningen University (Luca Romano).
Rights and permissions
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.
About this article
Cite this article
Bergshoeff, E., Izquierdo, J.M. & Romano, L. Carroll versus Galilei from a brane perspective. J. High Energ. Phys. 2020, 66 (2020). https://doi.org/10.1007/JHEP10(2020)066
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2020)066
Keywords
- Classical Theories of Gravity
- p-branes
- Sigma Models