Abstract
We construct a generalisation of the three-dimensional Poincaré algebra that also includes a colour symmetry factor. This algebra can be used to define coloured Poincaré gravity in three space-time dimensions as well as to study generalisations of massive and massless free particle models. We present various such generalised particle models that differ in which orbits of the coloured Poincaré symmetry are described. Our approach can be seen as a stepping stone towards the description of particles interacting with a non-abelian background field or as a starting point for a worldline formulation of an associated quantum field theory.
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S. R. Coleman and J. Mandula, All Possible Symmetries of the S Matrix, Phys. Rev. 159 (1967) 1251 [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].
M. P. Blencowe, A Consistent Interacting Massless Higher Spin Field Theory in D = (2 + 1), Class. Quant. Grav. 6 (1989) 443 [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].
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].
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].
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].
L. Avilés, E. Frodden, J. Gomis, D. Hidalgo and J. Zanelli, Non-Relativistic Maxwell Chern-Simons Gravity, JHEP 05 (2018) 047 [arXiv:1802.08453] [INSPIRE].
S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, Rainbow Valley of Colored (Anti) de Sitter Gravity in Three Dimensions, JHEP 04 (2016) 055 [arXiv:1511.05220] [INSPIRE].
S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, Rainbow vacua of colored higher-spin (A)dS3 gravity, JHEP 05 (2016) 150 [arXiv:1511.05975] [INSPIRE].
S. E. Konstein and M. A. Vasiliev, Extended Higher Spin Superalgebras and Their Massless Representations, Nucl. Phys. B 331 (1990) 475 [INSPIRE].
R. M. Wald, Spin-2 Fields and General Covariance, Phys. Rev. D 33 (1986) 3613 [INSPIRE].
C. Cutler and R. M. Wald, A New Type of Gauge Invariance for a Collection of Massless Spin-2 Fields. 1. Existence and Uniqueness, Class. Quant. Grav. 4 (1987) 1267 [INSPIRE].
R. M. Wald, A New Type of Gauge Invariance for a Collection of Massless Spin-2 Fields. 2. Geometrical Interpretation, Class. Quant. Grav. 4 (1987) 1279 [INSPIRE].
N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B 597 (2001) 127 [hep-th/0007220] [INSPIRE].
N. Boulanger and L. Gualtieri, An exotic theory of massless spin two fields in three-dimensions, Class. Quant. Grav. 18 (2001) 1485 [hep-th/0012003] [INSPIRE].
E. Joung and W. Li, Nonrelativistic limits of colored gravity in three dimensions, Phys. Rev. D 97 (2018) 105020 [arXiv:1801.10143] [INSPIRE].
S. F. Prokushkin and M. A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3-D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].
S. Prokushkin and M. A. Vasiliev, 3-D higher spin gauge theories with matter, hep-th/9812242 [INSPIRE].
R. Bonezzi, N. Boulanger, E. Sezgin and P. Sundell, An Action for Matter Coupled Higher Spin Gravity in Three Dimensions, JHEP 05 (2016) 003 [arXiv:1512.02209] [INSPIRE].
P. O. Kazinski, S. L. Lyakhovich and A. A. Sharapov, Lagrange structure and quantization, JHEP 07 (2005) 076 [hep-th/0506093] [INSPIRE].
S. Fredenhagen, O. Krüger and K. Mkrtchyan, Vertex-Constraints in 3D Higher Spin Theories, Phys. Rev. Lett. 123 (2019) 131601 [arXiv:1905.00093] [INSPIRE].
D. A. Vogan, Gelfand-Kirillov dimension for Harish-Chandra module, Invent. Math. 48 (1978) 75.
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].
R. Schrader, The maxwell group and the quantum theory of particles in classical homogeneous electromagnetic fields, Fortsch. Phys. 20 (1972) 701 [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].
S. K. Wong, Field and particle equations for the classical Yang-Mills field and particles with isotopic spin, Nuovo Cim. A 65 (1970) 689 [INSPIRE].
M. R. Brown and M. J. Duff, Exact Results for Effective Lagrangians, Phys. Rev. D 11 (1975) 2124 [INSPIRE].
I. A. Batalin, S. G. Matinyan and G. K. Savvidy, Vacuum Polarization by a Source-Free Gauge Field, Sov. J. Nucl. Phys. 26 (1977) 214 [INSPIRE].
J.-M. Souriau, Structure des systèmes dynamiques, Dunod (1970), English translation: Structure of Dynamical Systems: A Symplectic View of Physics, Birkhäuser (1997), [DOI].
V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists, Cambridge University Press (2003).
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].
N. Boulanger, M. Henneaux and P. van Nieuwenhuizen, Conformal (super)gravities with several gravitons, JHEP 01 (2002) 035 [hep-th/0201023] [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].
F. Izaurieta, E. Rodriguez and P. Salgado, Expanding Lie (super)algebras through Abelian semigroups, J. Math. Phys. 47 (2006) 123512 [hep-th/0606215] [INSPIRE].
J. Matulich, S. Prohazka and J. Salzer, Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension, JHEP 07 (2019) 118 [arXiv:1903.09165] [INSPIRE].
A. Barducci, R. Casalbuoni and J. Gomis, Nonrelativistic k-contractions of the coadjoint Poincaré algebra, Int. J. Mod. Phys. A 35 (2020) 2050009 [arXiv:1910.11682] [INSPIRE].
T. Fukuyama and K. Kamimura, Gauge Theory of Two-dimensional Gravity, Phys. Lett. B 160 (1985) 259 [INSPIRE].
K. Isler and C. A. Trugenberger, A Gauge Theory of Two-dimensional Quantum Gravity, Phys. Rev. Lett. 63 (1989) 834 [INSPIRE].
A. H. Chamseddine and D. Wyler, Topological Gravity in (1+1)-dimensions, Nucl. Phys. B 340 (1990) 595 [INSPIRE].
K. B. Alkalaev, On higher spin extension of the Jackiw-Teitelboim gravity model, J. Phys. A 47 (2014) 365401 [arXiv:1311.5119] [INSPIRE].
K. Alkalaev, E. Joung and J. Yoon, in preparation.
E. Bergshoeff, W. Merbis, A. J. Routh and P. K. Townsend, The Third Way to 3D Gravity, Int. J. Mod. Phys. D 24 (2015) 1544015 [arXiv:1506.05949] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically Massive Gauge Theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [INSPIRE].
A. S. Schwarz, The Partition Function of Degenerate Quadratic Functional and Ray-Singer Invariants, Lett. Math. Phys. 2 (1978) 247 [INSPIRE].
G. T. Horowitz, Exactly Soluble Diffeomorphism Invariant Theories, Commun. Math. Phys. 125 (1989) 417 [INSPIRE].
A. S. Cattaneo and C. A. Rossi, Higher dimensional BF theories in the Batalin-Vilkovisky formalism: the BV action and generalized Wilson loops, Commun. Math. Phys. 221 (2001) 591 [math/0010172] [INSPIRE].
S. R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1, Phys. Rev. 177 (1969) 2239 [INSPIRE].
C. G. Callan Jr., S. R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2, Phys. Rev. 177 (1969) 2247 [INSPIRE].
V. I. Ogievetsky, Non-linear realizations of internal and spacetime symmetries, in Proc. 10th Karpacz Winter School of Theoretical physics (1974).
J. Gomis, K. Kamimura and P. C. West, The construction of brane and superbrane actions using non-linear realisations, Class. Quant. Grav. 23 (2006) 7369 [hep-th/0607057] [INSPIRE].
D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, U.S.A. (1993), [DOI].
A. Barducci, R. Casalbuoni and L. Lusanna, Classical Scalar and Spinning Particles Interacting with External Yang-Mills Fields, Nucl. Phys. B 124 (1977) 93 [INSPIRE].
J. Gomis, A. Kleinschmidt, D. Roest and P. Salgado-ReboLledó, A free Lie algebra approach to curvature corrections to flat space-time, JHEP 09 (2020) 068 [arXiv:2006.11102] [INSPIRE].
P. Salgado and S. Salgado, \( \mathfrak{so}\left(D-1,1\right) \) ⊗ \( \mathfrak{so}\left(D-1,2\right) \) algebras and gravity, Phys. Lett. B 728 (2014) 5 [INSPIRE].
J. Gomis, A. Kleinschmidt and J. Palmkvist, Galilean free Lie algebras, JHEP 09 (2019) 109 [arXiv:1907.00410] [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].
M. Kontsevich, Deformation quantization of Poisson manifolds, I, q-alg/9709040.
M. A. Vasiliev, Relativity, causality, locality, quantization and duality in the S(p)(2M) invariant generalized space-time, hep-th/0111119 [INSPIRE].
D. Sorokin and M. Tsulaia, Higher Spin Fields in Hyperspace. A Review, Universe 4 (2018) 7 [arXiv:1710.08244] [INSPIRE].
E. Joung and K. Mkrtchyan, Notes on higher-spin algebras: minimal representations and structure constants, JHEP 05 (2014) 103 [arXiv:1401.7977] [INSPIRE].
T. Basile, E. Joung, K. Mkrtchyan and M. Mojaza, Dual Pair Correspondence in Physics: Oscillator Realizations and Representations, JHEP 09 (2020) 020 [arXiv:2006.07102] [INSPIRE].
E. Joung and K. Mkrtchyan, Partially-massless higher-spin algebras and their finite-dimensional truncations, JHEP 01 (2016) 003 [arXiv:1508.07332] [INSPIRE].
J. F. Schonfeld, A Mass Term for Three-Dimensional Gauge Fields, Nucl. Phys. B 185 (1981) 157.
M. S. Plyushchay, The model of relativistic particle with torsion, Nucl. Phys. B 362 (1991) 54 [INSPIRE].
L. Mezincescu and P. K. Townsend, Semionic Supersymmetric Solitons, J. Phys. A 43 (2010) 465401 [arXiv:1008.2775] [INSPIRE].
C. Batlle, J. Gomis, K. Kamimura and J. Zanelli, Dynamical sectors for a spinning particle in AdS3, Phys. Rev. D 90 (2014) 065017 [arXiv:1407.2355] [INSPIRE].
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Gomis, J., Joung, E., Kleinschmidt, A. et al. Colourful Poincaré symmetry, gravity and particle actions. J. High Energ. Phys. 2021, 47 (2021). https://doi.org/10.1007/JHEP08(2021)047
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DOI: https://doi.org/10.1007/JHEP08(2021)047