Carrollian and Galilean conformal higher-spin algebras in any dimensions

[en] We present higher-spin algebras containing a Poincaré subalgebra and with the same set of generators as the Lie algebras that are relevant to Vasiliev's equations in any space-time dimension D ≥ 3. Given these properties, they can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Carrollian conformal higher-spin symmetries in one less dimension. We build these Lie algebras as quotients of the universal enveloping algebra of iso(1,D-1) and we show how to recover them as Inönü-Wigner contractions of the rigid symmetries of higher-spin gauge theories in Anti de Sitter space or, equivalently, of relativistic conformal higher- spin symmetries. We use the same techniques to also define higher-spin algebras with the same set of generators and containing a Galilean conformal subalgebra, to be interpreted as non-relativistic limits of the conformal symmetries of a free scalar field. We begin by showing that the known flat-space higher-spin algebras in three dimensions can be obtained as quotients of the universal enveloping algebra of iso(1,2) and then we extend the analysis along the same lines to a generic number of space-time dimensions. We also discuss the peculiarities that emerge for D = 5.

Research center :

AGIF - Algèbre, Géométrie et Interactions fondamentales

Disciplines :

Physics

Author, co-author :

Campoleoni, Andrea ; Université de Mons > Faculté des Sciences > Service de Physique de l'Univers, Champs et Gravitation

Pekar, Simon ; Université de Mons > Faculté des Sciences > Service de Physique de l'Univers, Champs et Gravitation

Language :

English

Title :

Carrollian and Galilean conformal higher-spin algebras in any dimensions

Publication date :

17 February 2022

Journal title :

Journal of High Energy Physics

ISSN :

1126-6708

Publisher :

Springer, Heidelberg, Germany

Volume :

Pages :

1-87

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://arxiv.org/abs/2110.07794

Research unit :

S827 - Physique de l'Univers, Champs et Gravitation

Research institute :

R150 - Institut de Recherche sur les Systèmes Complexes

Name of the research project :

4907 - MIS-Campoleoni - HighSpinSymm - Fédération Wallonie Bruxelles

Funders :

F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]

Funding number :

FC.36447; F.4503.20; T.0022.19

Available on ORBi UMONS :

since 31 January 2022

Statistics

Number of views

3 (3 by UMONS)

Number of downloads

14 (0 by UMONS)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

Bibliography

E. S. Fradkin and M. A. Vasiliev, Candidate to the Role of Higher Spin Symmetry, Annals Phys. 177 (1987) 63 [INSPIRE].
M. A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].
M. A. Vasiliev, Extended Higher Spin Superalgebras and Their Realizations in Terms of Quantum Operators, Fortsch. Phys. 36 (1988) 33 [INSPIRE].
S. E. Konstein and M. A. Vasiliev, Extended Higher Spin Superalgebras and Their Massless Representations, Nucl. Phys. B 331 (1990) 475 [INSPIRE].
E. Sezgin and P. Sundell, Doubletons and 5D higher spin gauge theory, JHEP 09 (2001) 036 [hep-th/0105001] [INSPIRE].
M. A. Vasiliev, Cubic interactions of bosonic higher spin gauge fields in AdS5, Nucl. Phys. B 616 (2001) 106 [hep-th/0106200] [INSPIRE].
M. A. Vasiliev, Cubic interactions of bosonic higher spin gauge fields in AdS5, Nucl. Phys. B 652 (2003) 407
E. Sezgin and P. Sundell, 7D bosonic higher spin theory: Symmetry algebra and linearized constraints, Nucl. Phys. B 634 (2002) 120 [hep-th/0112100] [INSPIRE].
M. G. Eastwood, Higher symmetries of the Laplacian, Annals Math. 161 (2005) 1645 [hep-th/0206233] [INSPIRE].
M. A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [INSPIRE].
M. A. Vasiliev, Higher spin gauge theories in various dimensions, Fortsch. Phys. 52 (2004) 702 [PoS jhw2003 (2003) 003 [hep-th/0401177] [INSPIRE].
M. A. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [INSPIRE].
N. Boulanger and E. D. Skvortsov, Higher-spin algebras and cubic interactions for simple mixed-symmetry fields in AdS spacetime, JHEP 09 (2011) 063 [arXiv:1107.5028] [INSPIRE].
E. Joung and K. Mkrtchyan, Partially-massless higher-spin algebras and their finite-dimensional truncations, JHEP 01 (2016) 003 [arXiv:1508.07332] [INSPIRE].
K. B. Alkalaev, M. Grigoriev and E. D. Skvortsov, Uniformizing higher-spin equations, J. Phys. A 48 (2015) 015401 [arXiv:1409.6507] [INSPIRE].
C. Brust and K. Hinterbichler, Partially Massless Higher-Spin Theory, JHEP 02 (2017) 086 [arXiv:1610.08510] [INSPIRE].
M. P. Blencowe, A Consistent Interacting Massless Higher Spin Field Theory in D = (2 + 1), Class. Quant. Grav. 6 (1989) 443 [INSPIRE].
E. Bergshoeff, M. P. Blencowe and K. S. Stelle, Area Preserving Diffeomorphisms and Higher Spin Algebra, Commun. Math. Phys. 128 (1990) 213 [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].
M. Grigoriev, K. Mkrtchyan and E. Skvortsov, Matter-free higher spin gravities in 3D: Partially-massless fields and general structure, Phys. Rev. D 102 (2020) 066003 [arXiv:2005.05931] [INSPIRE].
S. F. Prokushkin and M. A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].
M. Henneaux, Asymptotically anti-de Sitter universes in d = 3, 4 and higher dimensions, in proceedings of the 4th Marcel Grossmann Meeting on General Relativity, Rome, Italy, 17–21 June 1985, R. Ruffini ed., Elsevier Science Publishers (1986), pp. 959–966.
A. Campoleoni, M. Henneaux, S. Hörtner and A. Leonard, Higher-spin charges in Hamiltonian form. I. Bose fields, JHEP 10 (2016) 146 [arXiv:1608.04663] [INSPIRE].
A. Campoleoni, M. Henneaux, S. Hörtner and A. Leonard, Higher-spin charges in Hamiltonian form. II. Fermi fields, JHEP 02 (2017) 058 [arXiv:1701.05526] [INSPIRE].
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].
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].
M. R. Gaberdiel and T. Hartman, Symmetries of Holographic Minimal Models, JHEP 05 (2011) 031 [arXiv:1101.2910] [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].
A. Campoleoni, D. Francia and C. Heissenberg, On higher-spin supertranslations and superrotations, JHEP 05 (2017) 120 [arXiv:1703.01351] [INSPIRE].
A. Campoleoni, D. Francia and C. Heissenberg, On asymptotic symmetries in higher dimensions for any spin, JHEP 12 (2020) 129 [arXiv:2011.04420] [INSPIRE].
A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].
X. Bekaert, Higher spin algebras as higher symmetries, Ann. U. Craiova Phys. 16 (2006) 58 [arXiv:0704.0898] [INSPIRE].
C. Iazeolla and P. Sundell, A Fiber Approach to Harmonic Analysis of Unfolded Higher-Spin Field Equations, JHEP 10 (2008) 022 [arXiv:0806.1942] [INSPIRE].
X. Bekaert, Comments on higher-spin symmetries, Int. J. Geom. Meth. Mod. Phys. 6 (2009) 285 [arXiv:0807.4223] [INSPIRE].
E. Joung and K. Mkrtchyan, Notes on higher-spin algebras: minimal representations and structure constants, JHEP 05 (2014) 103 [arXiv:1401.7977] [INSPIRE].
B. L. Feigin, The Lie algebras gl λ and cohomologies of Lie algebras of differential operators, Russ. Math. Surv. 43 (1988) 169.
M. Bordemann, J. Hoppe and P. Schaller, Infinite dimensional matrix algebras, Phys. Lett. B 232 (1989) 199 [INSPIRE].
E. S. Fradkin and V. Y. Linetsky, Infinite dimensional generalizations of simple Lie algebras, Mod. Phys. Lett. A 5 (1990) 1967 [INSPIRE].
N. Boulanger, D. Ponomarev, E. D. Skvortsov and M. Taronna, On the uniqueness of higher-spin symmetries in AdS and CFT, Int. J. Mod. Phys. A 28 (2013) 1350162 [arXiv:1305.5180] [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].
Y. S. Stanev, Constraining conformal field theory with higher spin symmetry in four dimensions, Nucl. Phys. B 876 (2013) 651 [arXiv:1307.5209] [INSPIRE].
V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d = 4, arXiv:1307.8092 [INSPIRE].
V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d > 3 dimensions, JHEP 03 (2016) 044 [arXiv:1510.02535] [INSPIRE].
A. K. H. Bengtsson and I. Bengtsson, Massless higher spin fields revisited, Class. Quant. Grav. 3 (1986) 927 [INSPIRE].
X. Bekaert, N. Boulanger, S. Cnockaert and S. Leclercq, On Killing tensors and cubic vertices in higher-spin gauge theories, Fortsch. Phys. 54 (2006) 282 [hep-th/0602092] [INSPIRE].
E. Joung and M. Taronna, Cubic-interaction-induced deformations of higher-spin symmetries, JHEP 03 (2014) 103 [arXiv:1311.0242] [INSPIRE].
C. Sleight and M. Taronna, Higher-Spin Algebras, Holography and Flat Space, JHEP 02 (2017) 095 [arXiv:1609.00991] [INSPIRE].
D. Ponomarev, Chiral Higher Spin Theories and Self-Duality, JHEP 12 (2017) 141 [arXiv:1710.00270] [INSPIRE].
A. Campoleoni, Higher Spins in D = 2 + 1, in Subnuclear Series 49, World Scientific (2013), pp. 385–396 [arXiv:1110.5841] [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].
M. Ammon, D. Grumiller, S. Prohazka, M. Riegler and R. Wutte, Higher-Spin Flat Space Cosmologies with Soft Hair, JHEP 05 (2017) 031 [arXiv:1703.02594] [INSPIRE].
M. Ammon, M. Pannier and M. Riegler, Scalar Fields in 3D Asymptotically Flat Higher-Spin Gravity, J. Phys. A 54 (2021) 105401 [arXiv:2009.14210] [INSPIRE].
O. Fuentealba, J. Matulich and R. Troncoso, Extension of the Poincaré group with half-integer spin generators: hypergravity and beyond, JHEP 09 (2015) 003 [arXiv:1505.06173] [INSPIRE].
O. Fuentealba, J. Matulich and R. Troncoso, Hypergravity in five dimensions, Phys. Rev. D 101 (2020) 124002 [arXiv:1910.03179] [INSPIRE].
S. Weinberg, Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].
X. Bekaert, N. Boulanger and S. Leclercq, Strong obstruction of the Berends-Burgers-van Dam spin-3 vertex, J. Phys. A 43 (2010) 185401 [arXiv:1002.0289] [INSPIRE].
X. Bekaert, N. Boulanger and P. Sundell, How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples, Rev. Mod. Phys. 84 (2012) 987 [arXiv:1007.0435] [INSPIRE].
M. Porrati, Old and New No Go Theorems on Interacting Massless Particles in Flat Space, in proceedings of the 17th International Seminar on High Energy Physics, Yaroslavl, Russia, 4–7 June 2012, arXiv:1209.4876 [INSPIRE].
M. Taronna, On the Non-Local Obstruction to Interacting Higher Spins in Flat Space, JHEP 05 (2017) 026 [arXiv:1701.05772] [INSPIRE].
A. Sagnotti, Notes on Strings and Higher Spins, J. Phys. A 46 (2013) 214006 [arXiv:1112.4285] [INSPIRE].
A. K. H. Bengtsson, I. Bengtsson and L. Brink, Cubic Interaction Terms for Arbitrary Spin, Nucl. Phys. B 227 (1983) 31 [INSPIRE].
A. K. H. Bengtsson, I. Bengtsson and N. Linden, Interacting Higher Spin Gauge Fields on the Light Front, Class. Quant. Grav. 4 (1987) 1333 [INSPIRE].
E. S. Fradkin and R. R. Metsaev, A Cubic interaction of totally symmetric massless representations of the Lorentz group in arbitrary dimensions, Class. Quant. Grav. 8 (1991) L89 [INSPIRE].
R. R. Metsaev, Generating function for cubic interaction vertices of higher spin fields in any dimension, Mod. Phys. Lett. A 8 (1993) 2413 [INSPIRE].
R. R. Metsaev, Cubic interaction vertices of massive and massless higher spin fields, Nucl. Phys. B 759 (2006) 147 [hep-th/0512342] [INSPIRE].
R. R. Metsaev, Cubic interaction vertices for fermionic and bosonic arbitrary spin fields, Nucl. Phys. B 859 (2012) 13 [arXiv:0712.3526] [INSPIRE].
R. Manvelyan, K. Mkrtchyan and W. Rühl, General trilinear interaction for arbitrary even higher spin gauge fields, Nucl. Phys. B 836 (2010) 204 [arXiv:1003.2877] [INSPIRE].
A. Sagnotti and M. Taronna, String Lessons for Higher-Spin Interactions, Nucl. Phys. B 842 (2011) 299 [arXiv:1006.5242] [INSPIRE].
D. Francia, G. L. Monaco and K. Mkrtchyan, Cubic interactions of Maxwell-like higher spins, JHEP 04 (2017) 068 [arXiv:1611.00292] [INSPIRE].
D. Ponomarev and E. D. Skvortsov, Light-Front Higher-Spin Theories in Flat Space, J. Phys. A 50 (2017) 095401 [arXiv:1609.04655] [INSPIRE].
E. D. Skvortsov, T. Tran and M. Tsulaia, Quantum Chiral Higher Spin Gravity, Phys. Rev. Lett. 121 (2018) 031601 [arXiv:1805.00048] [INSPIRE].
L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari and T. Stemberga, HS in flat spacetime. YM-like models, arXiv:1812.05030 [INSPIRE].
L. Bonora and S. Giaccari, HS Yang-Mills-like models: a review, Ann. U. Craiova Phys. 30 (2020) 1 [arXiv:2103.10105] [INSPIRE].
K. Krasnov, E. Skvortsov and T. Tran, Actions for self-dual Higher Spin Gravities, JHEP 08 (2021) 076 [arXiv:2105.12782] [INSPIRE].
E. S. Fradkin and A. A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
A. Y. Segal, Conformal higher spin theory, Nucl. Phys. B 664 (2003) 59 [hep-th/0207212] [INSPIRE].
X. Bekaert and B. Oblak, Higher-spin BMS algebras, talk given at the Quarks Online Workshop-2021 “Integrability, Holography, Higher-Spin Gravity and Strings”, online conference, Russian Federation, 31 May–5 June 2021.
D. Grumiller, A. Pérez, M. M. Sheikh-Jabbari, R. Troncoso and C. Zwikel, Spacetime structure near generic horizons and soft hair, Phys. Rev. Lett. 124 (2020) 041601 [arXiv:1908.09833] [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].
T. Basile, X. Bekaert and E. Joung, Conformal Higher-Spin Gravity: Linearized Spectrum = Symmetry Algebra, JHEP 11 (2018) 167 [arXiv:1808.07728] [INSPIRE].
M. Grigoriev, I. Lovrekovic and E. Skvortsov, New Conformal Higher Spin Gravities in 3d, JHEP 01 (2020) 059 [arXiv:1909.13305] [INSPIRE].
A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].
A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [INSPIRE].
A. Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].
C. Fronsdal, Massless Fields with Integer Spin, Phys. Rev. D 18 (1978) 3624 [INSPIRE].
C. Fronsdal, Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space VII, Phys. Rev. D 20 (1979) 848 [INSPIRE].
X. Bekaert and N. Boulanger, Gauge invariants and Killing tensors in higher-spin gauge theories, Nucl. Phys. B 722 (2005) 225 [hep-th/0505068] [INSPIRE].
G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986) 2693.
G. Barnich, N. Bouatta and M. Grigoriev, Surface charges and dynamical Killing tensors for higher spin gauge fields in constant curvature spaces, JHEP 10 (2005) 010 [hep-th/0507138] [INSPIRE].
R. G. McLenaghan, R. Milson and R. G. Smirnov, Killing tensors as irreducible representations of the general linear group, C. R. Acad. Sci. Paris Ser. I 339 (2004) 621.
M. A. Vasiliev, Free Massless Fields of Arbitrary Spin in the de Sitter Space and Initial Data for a Higher Spin Superalgebra, Fortsch. Phys. 35 (1987) 741 [Yad. Fiz. 45 (1987) 1784] [INSPIRE].
M. Karapetyan, R. Manvelyan and G. Poghosyan, On special quartic interaction of higher spin gauge fields with scalars and gauge symmetry commutator in the linear approximation, Nucl. Phys. B 971 (2021) 115512 [arXiv:2104.09139] [INSPIRE].
A. K. H. Bengtsson, A Unified Action for Higher Spin Gauge Bosons From Covariant String Theory, Phys. Lett. B 182 (1986) 321 [INSPIRE].
M. Henneaux and C. Teitelboim, First and second quantized point particles of any spin, in Quantum Mechanics of Fundamental Systems 2, Series of the Centro de Estudios Científicos de Santiago, C. Teitelboim and J. Zanelli eds., Springer, Boston U.S.A. (1989), pp. 113–152.
D. Francia and A. Sagnotti, On the geometry of higher spin gauge fields, Class. Quant. Grav. 20 (2003) S473 [Comment. Phys. Math. Soc. Sci. Fenn. 166 (2004) 165] [PoS jhw2003 (2003) 005] [hep-th/0212185] [INSPIRE].
A. Sagnotti and M. Tsulaia, On higher spins and the tensionless limit of string theory, Nucl. Phys. B 682 (2004) 83 [hep-th/0311257] [INSPIRE].
A. Campoleoni and D. Francia, Maxwell-like Lagrangians for higher spins, JHEP 03 (2013) 168 [arXiv:1206.5877] [INSPIRE].
M. A. Vasiliev, Higher spin gauge theories: Star product and AdS space, hep-th/9910096 [INSPIRE].
K. B. Alkalaev, Global and local properties of AdS 2 higher spin gravity, JHEP 10 (2014) 122 [arXiv:1404.5330] [INSPIRE].
M. R. Gaberdiel and R. Gopakumar, Minimal Model Holography, J. Phys. A 46 (2013) 214002 [arXiv:1207.6697] [INSPIRE].
S. Fernando and M. Günaydin, Minimal unitary representation of SU(2, 2) and its deformations as massless conformal fields and their supersymmetric extensions, J. Math. Phys. 51 (2010) 082301 [arXiv:0908.3624] [INSPIRE].
R. Manvelyan, K. Mkrtchyan, R. Mkrtchyan and S. Theisen, On Higher Spin Symmetries in AdS 5, JHEP 10 (2013) 185 [arXiv:1304.7988] [INSPIRE].
S. Deser and R. I. Nepomechie, Gauge Invariance Versus Masslessness in de Sitter Space, Annals Phys. 154 (1984) 396 [INSPIRE].
A. Higuchi, Symmetric Tensor Spherical Harmonics on the N Sphere and Their Application to the de Sitter Group SO(N, 1), J. Math. Phys. 28 (1987) 1553 [INSPIRE].
[Erratum ibid. 43 (2002) 6385]
S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [INSPIRE].
Y. M. Zinoviev, On massive high spin particles in AdS, hep-th/0108192 [INSPIRE].
E. D. Skvortsov and M. A. Vasiliev, Geometric formulation for partially massless fields, Nucl. Phys. B 756 (2006) 117 [hep-th/0601095] [INSPIRE].
J.-P. Michel, Higher symmetries of the Laplacian via quantization, Ann. Inst. Fourier 64 (2014) 1581.
X. Bekaert and M. Grigoriev, Higher order singletons, partially massless fields and their boundary values in the ambient approach, Nucl. Phys. B 876 (2013) 667 [arXiv:1305.0162] [INSPIRE].
A. R. Gover and J. Silhan, Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds, J. Math. Phys. 53 (2012) 032301 [arXiv:0911.5265].
T. Basile, X. Bekaert and N. Boulanger, Flato-Fronsdal theorem for higher-order singletons, JHEP 11 (2014) 131 [arXiv:1410.7668] [INSPIRE].
C. N. Pope, L. J. Romans and X. Shen, W∞ and the Racah-Wigner Algebra, Nucl. Phys. B 339 (1990) 191 [INSPIRE].
A. Bagchi and R. Fareghbal, BMS/GCA Redux: Towards Flatspace Holography from Non-Relativistic Symmetries, JHEP 10 (2012) 092 [arXiv:1203.5795] [INSPIRE].
A. Campoleoni, H. A. Gonzalez, B. Oblak and M. Riegler, BMS Modules in Three Dimensions, Int. J. Mod. Phys. A 31 (2016) 1650068 [arXiv:1603.03812] [INSPIRE].
M. R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
S. M. Paneitz, All linear representations of the Poincaré group up to dimension 8, Ann. Inst. Henri Poincaré Phys. Theor. 40 (1984) 35.
A. Campoleoni and M. Henneaux, Asymptotic symmetries of three-dimensional higher-spin gravity: the metric approach, JHEP 03 (2015) 143 [arXiv:1412.6774] [INSPIRE].
J. A. Schouten, Uber Differentialkomitanten zweier kontravarianter Grössen, Proc. Kon. Ned. Akad. Wet. Amst. 43 (1940) 449.
A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields, I, Indag. Math. 17 (1955) 390.
A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields, II, Indag. Math. 17 (1955) 398.
M. Dubois-Violette and P. W. Michor, A Common generalization of the Fröhlicher-Nijenhuis bracket and the Schouten bracket for symmetric multivector fields, Indagat. Math. 6 (1995) 51 [alg-geom/9401006] [INSPIRE].
D. Ponomarev, 3d conformal fields with manifest sl(2, ℂ), JHEP 06 (2021) 055 [arXiv:2104.02770] [INSPIRE].
M. Flato and C. Fronsdal, One Massless Particle Equals Two Dirac Singletons. VI: Elementary Particles in a Curved Space, Lett. Math. Phys. 2 (1978) 421 [INSPIRE].
S. M. Kuzenko and A. E. Pindur, Massless particles in five and higher dimensions, Phys. Lett. B 812 (2021) 136020 [arXiv:2010.07124] [INSPIRE].
L. Barannik and W. Fushchich, Casimir operators of the generalised Poincaré and Galilei groups, in Group Theoretical Methods in Physics, proceedings of the Third Yurmala Seminar, M. A. Markov, V. I. Manko and V. V. Dodonov eds., VNU Science Press (1986), pp. 275–282.
X. Bekaert and N. Boulanger, The unitary representations of the Poincaré group in any spacetime dimension, SciPost Phys. Lect. Notes 30 (2021) 1 [hep-th/0611263] [INSPIRE].
F. A. Berends, G. J. H. Burgers and H. van Dam, On the Theoretical Problems in Constructing Interactions Involving Higher Spin Massless Particles, Nucl. Phys. B 260 (1985) 295 [INSPIRE].
X. Bekaert, Notes on Higher-Spin Diffeomorphisms, Universe 7 (2021) 508 [arXiv:2108.09263] [INSPIRE].
J. Gomis, J. Gomis and K. Kamimura, Non-relativistic superstrings: A New soluble sector of AdS 5 × S 5, JHEP 12 (2005) 024 [hep-th/0507036] [INSPIRE].
E. Joung and K. Mkrtchyan, A note on higher-derivative actions for free higher-spin fields, JHEP 11 (2012) 153 [arXiv:1209.4864] [INSPIRE].
D. Francia, Generalised connections and higher-spin equations, Class. Quant. Grav. 29 (2012) 245003 [arXiv:1209.4885] [INSPIRE].
L. Brink, R. R. Metsaev and M. A. Vasiliev, How massless are massless fields in AdS(d), Nucl. Phys. B 586 (2000) 183 [hep-th/0005136] [INSPIRE].
M. S. Drew and J. D. Gegenberg, Conformally covariant massless spin-2 field equations, Nuovo Cim. A 60 (1980) 41 [INSPIRE].
B. de Wit and D. Z. Freedman, Systematics of Higher Spin Gauge Fields, Phys. Rev. D 21 (1980) 358 [INSPIRE].
D. Francia and A. Sagnotti, Free geometric equations for higher spins, Phys. Lett. B 543 (2002) 303 [hep-th/0207002] [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].
J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [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].
N. Boulanger, S. Cnockaert and M. Henneaux, A note on spin s duality, JHEP 06 (2003) 060 [hep-th/0306023] [INSPIRE].
M. Henneaux and P. Salgado-ReboLledó, Carroll contractions of Lorentz-invariant theories, JHEP 11 (2021) 180 [arXiv:2109.06708] [INSPIRE].
A. Bagchi, J. Chakrabortty and A. Mehra, Galilean Field Theories and Conformal Structure, JHEP 04 (2018) 144 [arXiv:1712.05631] [INSPIRE].
A. Bagchi, A. Mehra and P. Nandi, Field Theories with Conformal Carrollian Symmetry, JHEP 05 (2019) 108 [arXiv:1901.10147] [INSPIRE].
L. Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos and K. Siampos, Flat holography and Carrollian fluids, JHEP 07 (2018) 165 [arXiv:1802.06809] [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].
J. Figueroa-O’Farrill and S. Prohazka, Spatially isotropic homogeneous spacetimes, JHEP 01 (2019) 229 [arXiv:1809.01224] [INSPIRE].
E. Bergshoeff, D. Grumiller, S. Prohazka and J. Rosseel, Three-dimensional Spin-3 Theories Based on General Kinematical Algebras, JHEP 01 (2017) 114 [arXiv:1612.02277] [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. Gary, D. Grumiller, S. Prohazka and S.-J. Rey, Lifshitz Holography with Isotropic Scale Invariance, JHEP 08 (2014) 001 [arXiv:1406.1468] [INSPIRE].
S. Prohazka and M. Riegler, Higher Spins Without (Anti-)de Sitter, Universe 4 (2018) 20 [arXiv:1710.11105] [INSPIRE].
D. Chernyavsky and D. Sorokin, Three-dimensional (higher-spin) gravities with extended Schrödinger and l-conformal Galilean symmetries, JHEP 07 (2019) 156 [arXiv:1905.13154] [INSPIRE].
A. V. Korybut, Covariant structure constants for a deformed oscillator algebra, Theor. Math. Phys. 193 (2017) 1409 [arXiv:1409.8634] [INSPIRE].
T. Basile, N. Boulanger and F. Buisseret, Structure constants of shs[λ]: the deformed-oscillator point of view, J. Phys. A 51 (2018) 025201 [arXiv:1604.04510] [INSPIRE].