Abstract
We initiate the study of non- and ultra-relativistic higher spin theories. For sake of simplicity we focus on the spin-3 case in three dimensions. We classify all kinematical algebras that can be obtained by all possible Inönü-Wigner contraction procedures of the kinematical algebra of spin-3 theory in three dimensional (anti-) de Sitter space-time. We demonstrate how to construct associated actions of Chern-Simons type, directly in the ultra-relativistic case and by suitable algebraic extensions in the non-relativistic case. We show how to give these kinematical algebras an infinite-dimensional lift by imposing suitable boundary conditions in a theory we call “Carroll Gravity”, whose asymptotic symmetry algebra turns out to be an infinite-dimensional extension of the Carroll algebra.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Bacry and J. Levy-Leblond, Possible kinematics, J. Math. Phys. 9 (1968) 1605 [INSPIRE].
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].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [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, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].
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].
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].
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].
P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [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.T. Son and M. Wingate, General coordinate invariance and conformal invariance in nonrelativistic physics: unitary Fermi gas, Annals Phys. 321 (2006) 197 [cond-mat/0509786] [INSPIRE].
C. Hoyos and D.T. Son, Hall viscosity and electromagnetic response, Phys. Rev. Lett. 108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].
D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
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].
A. Gromov and A.G. Abanov, Thermal Hall effect and geometry with torsion, Phys. Rev. Lett. 114 (2015) 016802 [arXiv:1407.2908] [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].
M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime symmetries of the quantum Hall effect, Phys. Rev. D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].
G. Festuccia, D. Hansen, J. Hartong and N.A. Obers, Torsional Newton-Cartan geometry from the Noether procedure, Phys. Rev. D 94 (2016) 105023 [arXiv:1607.01926] [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. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich, A. Gomberoff and H.A. Gonzalez, The flat limit of three dimensional asymptotically anti-de Sitter spacetimes, Phys. Rev. D 86 (2012) 024020 [arXiv:1204.3288] [INSPIRE].
A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].
A. Bagchi, Correspondence between asymptotically flat spacetimes and nonrelativistic conformal field theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].
A. Bagchi, R. Basu, D. Grumiller and M. Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114 (2015) 111602 [arXiv:1410.4089] [INSPIRE].
J. Hartong, Gauging the Carroll algebra and ultra-relativistic gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [INSPIRE].
A. Bagchi, D. Grumiller and W. Merbis, Stress tensor correlators in three-dimensional gravity, Phys. Rev. D 93 (2016) 061502 [arXiv:1507.05620] [INSPIRE].
J. Hartong, Holographic reconstruction of 3D flat space-time, JHEP 10 (2016) 104 [arXiv:1511.01387] [INSPIRE].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat holography: aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].
A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Superrotation charge and supertranslation hair on black holes, arXiv:1611.09175 [INSPIRE].
L. Donnay, G. Giribet, H.A. Gonzalez and M. Pino, Supertranslations and superrotations at the black hole horizon, Phys. Rev. Lett. 116 (2016) 091101 [arXiv:1511.08687] [INSPIRE].
L. Donnay, G. Giribet, H.A. González and M. Pino, Extended symmetries at the black hole horizon, JHEP 09 (2016) 100 [arXiv:1607.05703] [INSPIRE].
H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
D. Grumiller, A. Perez, S. Prohazka, D. Tempo and R. Troncoso, Higher spin black holes with soft hair, JHEP 10 (2016) 119 [arXiv:1607.05360] [INSPIRE].
H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, arXiv:1611.09783 [INSPIRE].
M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].
X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev, Nonlinear higher spin theories in various dimensions, in the proceedings of the 1st Solvay Workshop on Higher spin gauge theories, May 12-14, Brussels, Belgium (2004), hep-th/0503128 [INSPIRE].
M.A. Vasiliev, Holography, unfolding and higher-spin theory, J. Phys. A 46 (2013) 214013 [arXiv:1203.5554] [INSPIRE].
V.E. Didenko and E.D. Skvortsov, Elements of Vasiliev theory, arXiv:1401.2975 [INSPIRE].
S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
S. Giombi and X. Yin, Higher spins in AdS and twistorial holography, JHEP 04 (2011) 086 [arXiv:1004.3736] [INSPIRE].
S. Giombi and X. Yin, The higher spin/vector model duality, J. Phys. A 46 (2013) 214003 [arXiv:1208.4036] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, Partition functions of holographic minimal models, JHEP 08 (2011) 077 [arXiv:1106.1897] [INSPIRE].
C. Candu and M.R. Gaberdiel, Supersymmetric holography on AdS 3, JHEP 09 (2013) 071 [arXiv:1203.1939] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Minimal model holography, J. Phys. A 46 (2013) 214002 [arXiv:1207.6697] [INSPIRE].
C. Candu, M.R. Gaberdiel, M. Kelm and C. Vollenweider, Even spin minimal model holography, JHEP 01 (2013) 185 [arXiv:1211.3113] [INSPIRE].
M. Beccaria, C. Candu, M.R. Gaberdiel and M. Groher, N = 1 extension of minimal model holography, JHEP 07 (2013) 174 [arXiv:1305.1048] [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
M. Gary, D. Grumiller and R. Rashkov, Towards non-AdS holography in 3-dimensional higher spin gravity, JHEP 03 (2012) 022 [arXiv:1201.0013] [INSPIRE].
H. Afshar, M. Gary, D. Grumiller, R. Rashkov and M. Riegler, Non-AdS holography in 3-dimensional higher spin gravity — General recipe and example, JHEP 11 (2012) 099 [arXiv:1209.2860] [INSPIRE].
M. Gutperle, E. Hijano and J. Samani, Lifshitz black holes in higher spin gravity, JHEP 04 (2014) 020 [arXiv:1310.0837] [INSPIRE].
M. Gary, D. Grumiller, S. Prohazka and S.-J. Rey, Lifshitz holography with isotropic scale invariance, JHEP 08 (2014) 001 [arXiv:1406.1468] [INSPIRE].
V. Breunhölder, M. Gary, D. Grumiller and S. Prohazka, Null warped AdS in higher spin gravity, JHEP 12 (2015) 021 [arXiv:1509.08487] [INSPIRE].
Y. Lei and S.F. Ross, Connection versus metric description for non-AdS solutions in higher-spin theories, Class. Quant. Grav. 32 (2015) 185005 [arXiv:1504.07252] [INSPIRE].
Y. Lei and C. Peng, Higher spin holography with Galilean symmetry in general dimensions, Class. Quant. Grav. 33 (2016) 135008 [arXiv:1507.08293] [INSPIRE].
H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 gravity in three-dimensional flat space, Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768] [INSPIRE].
H.A. Gonzalez, J. Matulich, M. Pino and R. Troncoso, Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP 09 (2013) 016 [arXiv:1307.5651] [INSPIRE].
D. Grumiller, M. Riegler and J. Rosseel, Unitarity in three-dimensional flat space higher spin theories, JHEP 07 (2014) 015 [arXiv:1403.5297] [INSPIRE].
M. Gary, D. Grumiller, M. Riegler and J. Rosseel, Flat space (higher spin) gravity with chemical potentials, JHEP 01 (2015) 152 [arXiv:1411.3728] [INSPIRE].
J. Matulich, A. Perez, D. Tempo and R. Troncoso, Higher spin extension of cosmological spacetimes in 3D: asymptotically flat behaviour with chemical potentials and thermodynamics, JHEP 05 (2015) 025 [arXiv:1412.1464] [INSPIRE].
C. Sleight and M. Taronna, Higher spin interactions from conformal field theory: the complete cubic couplings, Phys. Rev. Lett. 116 (2016) 181602 [arXiv:1603.00022] [INSPIRE].
C. Sleight and M. Taronna, Higher-spin algebras, holography and flat space, arXiv:1609.00991 [INSPIRE].
D. Ponomarev and E.D. Skvortsov, Light-front higher-spin theories in flat space, arXiv:1609.04655 [INSPIRE].
C. Aragone and S. Deser, Hypersymmetry in D = 3 of coupled gravity massless spin 5/2 system, Class. Quant. Grav. 1 (1984) L9 [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].
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].
E.A. Bergshoeff and J. Rosseel, Three-dimensional extended bargmann supergravity, Phys. Rev. Lett. 116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].
E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. Veldhuis, Carroll versus Galilei Gravity, to appear.
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun.Math.Phys. 104 (1986) 207 [INSPIRE].
A. Ashtekar, J. Bicak and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev. D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].
M. Henneaux and S.-J. Rey, Nonlinear W ∞ as asymptotic symmetry of three-dimensional higher spin Anti-de Sitter gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
E. Inonu and E.P. Wigner, On the contraction of groups and their represenations, Proc. Nat. Acad. Sci. 39 (1953) 510 [INSPIRE].
A. Medina and P. Revoy, Algèbres de lie et produit scalaire invariant, Ann. Sci. École Norm. Supér. 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].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
D. Grumiller and M. Riegler, Most general AdS 3 boundary conditions, JHEP 10 (2016) 023 [arXiv:1608.01308] [INSPIRE].
M. Riegler, How general is holography?, Ph.D. thesis, Technische Universität Wien, Vienna, Austria (2016), arXiv:1609.02733 [INSPIRE].
M. Bañados, Three-dimensional quantum geometry and black holes, hep-th/9901148 [INSPIRE].
T. Regge and C. Teitelboim, Role of surface integrals in the hamiltonian formulation of general relativity, Annals Phys. 88 (1974) 286 [INSPIRE].
A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories, JHEP 09 (2011) 113 [arXiv:1107.0290] [INSPIRE].
T. Griffin, P. Hořava and C.M. Melby-Thompson, Conformal Lifshitz gravity from holography, JHEP 05 (2012) 010 [arXiv:1112.5660] [INSPIRE].
T. Griffin, P. Hořava and C.M. Melby-Thompson, Lifshitz gravity for Lifshitz holography, Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].
E. Kiritsis, Lorentz violation, gravity, dissipation and holography, JHEP 01 (2013) 030 [arXiv:1207.2325] [INSPIRE].
S. Janiszewski and A. Karch, String theory embeddings of nonrelativistic field theories and their holographic Hořava gravity duals, Phys. Rev. Lett. 110 (2013) 081601 [arXiv:1211.0010] [INSPIRE].
S. Janiszewski and A. Karch, Non-relativistic holography from Hořava gravity, JHEP 02 (2013) 123 [arXiv:1211.0005] [INSPIRE].
C. Wu and S.-F. Wu, Hořava-Lifshitz gravity and effective theory of the fractional quantum Hall effect, JHEP 01 (2015) 120 [arXiv:1409.1178] [INSPIRE].
S. Golkar, D.X. Nguyen, M.M. Roberts and D.T. Son, Higher-spin theory of the magnetorotons, Phys. Rev. Lett. 117 (2016) 216403 [arXiv:1602.08499] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1612.02277
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bergshoeff, E., Grumiller, D., Prohazka, S. et al. Three-dimensional spin-3 theories based on general kinematical algebras. J. High Energ. Phys. 2017, 114 (2017). https://doi.org/10.1007/JHEP01(2017)114
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2017)114