Abstract
We derive the characters of all unitary irreducible representations of the (d+1)-dimensional de Sitter spacetime isometry algebra \( \mathfrak{so}\left(1,\kern0.5em d+1\right) \), and propose a dictionary between those representations and massive or (partially) massless fields on de Sitter spacetime. We propose a way of taking the flat limit of representations in (anti-) de Sitter spaces in terms of these characters, and conjecture the spectrum resulting from taking the flat limit of mixed-symmetry fields in de Sitter spacetime. We identify the equivalent of the scalar singleton for the de Sitter (dS) spacetime.
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J.M.F. Labastida, Massless particles in arbitrary representations of the Lorentz group, Nucl. Phys. B 322 (1989) 185 [INSPIRE].
J.M.F. Labastida, Massless fermionic free fields, Phys. Lett. B 186 (1987) 365 [INSPIRE].
W. Siegel and B. Zwiebach, Gauge string fields from the light cone, Nucl. Phys. B 282 (1987) 125 [INSPIRE].
X. Bekaert and N. Boulanger, Tensor gauge fields in arbitrary representations of GL(D, R): duality and Poincaré lemma, Commun. Math. Phys. 245 (2004) 27 [hep-th/0208058] [INSPIRE].
X. Bekaert and N. Boulanger, Tensor gauge fields in arbitrary representations of GL(D, R) II. Quadratic actions, Commun. Math. Phys. 271 (2007) 723 [hep-th/0606198] [INSPIRE].
A. Campoleoni, D. Francia, J. Mourad and A. Sagnotti, Unconstrained higher spins of mixed symmetry II. Fermi fields, Nucl. Phys. B 828 (2010) 405 [arXiv:0904.4447] [INSPIRE].
E.D. Skvortsov, Mixed-symmetry massless fields in Minkowski space unfolded, JHEP 07 (2008) 004 [arXiv:0801.2268] [INSPIRE].
R.R. Metsaev, Massless mixed symmetry bosonic free fields in d-dimensional anti-de Sitter space-time, Phys. Lett. B 354 (1995) 78 [INSPIRE].
R.R. Metsaev, Arbitrary spin massless bosonic fields in d-dimensional anti-de Sitter space, Lect. Notes Phys. 524 (1999) 331 [hep-th/9810231] [INSPIRE].
R.R. Metsaev, Fermionic fields in the d-dimensional anti-de Sitter space-time, Phys. Lett. B 419 (1998) 49 [hep-th/9802097] [INSPIRE].
C. Fronsdal, Singletons and massless, integral spin fields on de Sitter space (elementary particles in a curved space 7), Phys. Rev. D 20 (1979) 848 [INSPIRE].
N. Boulanger, C. Iazeolla and P. Sundell, Unfolding mixed-symmetry fields in AdS and the BMV conjecture: I. General formalism, JHEP 07 (2009) 013 [arXiv:0812.3615] [INSPIRE].
N. Boulanger, C. Iazeolla and P. Sundell, Unfolding mixed-symmetry fields in AdS and the BMV conjecture: II. Oscillator realization, JHEP 07 (2009) 014 [arXiv:0812.4438] [INSPIRE].
E.D. Skvortsov, Gauge fields in (A)dS d and connections of its symmetry algebra, J. Phys. A 42 (2009) 385401 [arXiv:0904.2919] [INSPIRE].
E.D. Skvortsov, Gauge fields in (A)dS d within the unfolded approach: algebraic aspects, JHEP 01 (2010) 106 [arXiv:0910.3334] [INSPIRE].
V.E. Lopatin and M.A. Vasiliev, Free massless bosonic fields of arbitrary spin in d-dimensional de Sitter space, Mod. Phys. Lett. A 3 (1988) 257 [INSPIRE].
E. Joung, J. Mourad and R. Parentani, Group theoretical approach to quantum fields in de Sitter space I. The principle series, JHEP 08 (2006) 082 [hep-th/0606119] [INSPIRE].
E. Joung, J. Mourad and R. Parentani, Group theoretical approach to quantum fields in de Sitter space II. The complementary and discrete series, JHEP 09 (2007) 030 [arXiv:0707.2907] [INSPIRE].
T. Hirai, On irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962) 258.
F. Schwarz, Unitary irreducible representations of the groups SO0(n, 1), J. Math. Phys. 12 (1971) 131.
V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory, Lect. Notes Phys. 63 (1977) 1 [INSPIRE].
I.T. Todorov, M.C. Mintchev and V.B. Petkova, Conformal invariance in quantum field theory, Scuola normale superiore, Classe di scienze, Pisa Italy, (1978) [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].
K.B. Alkalaev and M. Grigoriev, Unified BRST description of AdS gauge fields, Nucl. Phys. B 835 (2010) 197 [arXiv:0910.2690] [INSPIRE].
A. Campoleoni and D. Francia, Maxwell-like Lagrangians for higher spins, JHEP 03 (2013) 168 [arXiv:1206.5877] [INSPIRE].
G. Barnich and C. Troessaert, Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity, JHEP 11 (2013) 003 [arXiv:1309.0794] [INSPIRE].
A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, Rotating higher spin partition functions and extended BMS symmetries, JHEP 04 (2016) 034 [arXiv:1512.03353] [INSPIRE].
C. Sleight and M. Taronna, Higher-spin algebras, holography and flat space, JHEP 02 (2017) 095 [arXiv:1609.00991] [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.A. Thieleker, The unitary representations of the generalized Lorentz groups, Trans. Amer. Math. Soc. 199 (1974) 327.
A.M. Gavrilik and A.U. Klimyk, Analysis of the representations of the Lorentz and Euclidean groups of n-th order, tech. rep. ITP-75-18-E, Inst. Theor. Phys., Kiev Ukraine, (1975).
A. Knapp, Representation theory of semisimple groups: an overview based on examples, Princeton Mathematical Series, Princeton University Press, Princeton U.S.A., (1986).
A. Knapp, Lie groups beyond an introduction, Progress in Mathematics, Birkhäuser, Basel Switzerland, (2002).
V.K. Dobrev, Intertwining operator realization of the AdS/CFT correspondence, Nucl. Phys. B 553 (1999) 559 [hep-th/9812194] [INSPIRE].
A.O. Barut and R. Raczka, Theory of group representations and applications, World Scientific, Singapore, (1986) [INSPIRE].
T. Hirai, The characters of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 41 (1965) 526.
J. Mickelsson and J. Niederle, Contractions of representations of de Sitter groups, Commun. Math. Phys. 27 (1972) 167 [INSPIRE].
A.M. Gavrilik and A.U. Klimyk, The representations of the groups U(n, 1) and SO(o)(n, 1), (1976) [INSPIRE].
O.V. Shaynkman, I. Yu. Tipunin and M.A. Vasiliev, Unfolded form of conformal equations in M dimensions and o(M + 2) modules, Rev. Math. Phys. 18 (2006) 823 [hep-th/0401086] [INSPIRE].
M. Beccaria, X. Bekaert and A.A. Tseytlin, Partition function of free conformal higher spin theory, JHEP 08 (2014) 113 [arXiv:1406.3542] [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 [Erratum ibid. 43 (2002) 6385] [INSPIRE].
S. Deser and R.I. Nepomechie, Gauge invariance versus masslessness in de Sitter space, Annals Phys. 154 (1984) 396 [INSPIRE].
S. Deser and R.I. Nepomechie, Anomalous propagation of gauge fields in conformally flat spaces, Phys. Lett. B 132 (1983) 321 [INSPIRE].
S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [INSPIRE].
E. Joung and K. Mkrtchyan, Partially-massless higher-spin algebras and their finite-dimensional truncations, JHEP 01 (2016) 003 [arXiv:1508.07332] [INSPIRE].
S. Gwak, J. Kim and S.-J. Rey, Massless and massive higher spins from anti-de Sitter space waveguide, JHEP 11 (2016) 024 [arXiv:1605.06526] [INSPIRE].
M. Günaydin, E.D. Skvortsov and T. Tran, Exceptional F (4) higher-spin theory in AdS 6 at one-loop and other tests of duality, JHEP 11 (2016) 168 [arXiv:1608.07582] [INSPIRE].
C. Brust and K. Hinterbichler, Partially massless higher-spin theory, JHEP 02 (2017) 086 [arXiv:1610.08510] [INSPIRE].
C. Brust and K. Hinterbichler, Partially massless higher-spin theory II: one-loop effective actions, JHEP 01 (2017) 126 [arXiv:1610.08522] [INSPIRE].
S. Deser and A. Waldron, Arbitrary spin representations in de Sitter from dS/CFT with applications to dS supergravity, Nucl. Phys. B 662 (2003) 379 [hep-th/0301068] [INSPIRE].
P. de Medeiros and C. Hull, Exotic tensor gauge theory and duality, Commun. Math. Phys. 235 (2003) 255 [hep-th/0208155] [INSPIRE].
E. Joung and K. Mkrtchyan, Weyl action of two-column mixed-symmetry field and its factorization around (A)dS space, JHEP 06 (2016) 135 [arXiv:1604.05330] [INSPIRE].
Yu. M. Zinoviev, First order formalism for massive mixed symmetry tensor fields in Minkowski and (A)dS spaces, hep-th/0306292 [INSPIRE].
Yu. M. Zinoviev, Toward frame-like gauge invariant formulation for massive mixed symmetry bosonic fields, Nucl. Phys. B 812 (2009) 46 [arXiv:0809.3287] [INSPIRE].
Yu. M. Zinoviev, Towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields II. General Young tableau with two rows, Nucl. Phys. B 826 (2010) 490 [arXiv:0907.2140] [INSPIRE].
Yu. M. Zinoviev, Gravitational cubic interactions for a massive mixed symmetry gauge field, Class. Quant. Grav. 29 (2012) 015013 [arXiv:1107.3222] [INSPIRE].
M. Flato and C. Fronsdal, One massless particle equals two Dirac singletons: elementary particles in a curved space. 6, Lett. Math. Phys. 2 (1978) 421 [INSPIRE].
E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N ) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
D. Anninos, T. Hartman and A. Strominger, Higher spin realization of the dS/CFT correspondence, Class. Quant. Grav. 34 (2017) 015009 [arXiv:1108.5735] [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].
E. Sezgin and P. Sundell, Supersymmetric higher spin theories, J. Phys. A 46 (2013) 214022 [arXiv:1208.6019] [INSPIRE].
E. Joung and K. Mkrtchyan, Notes on higher-spin algebras: minimal representations and structure constants, JHEP 05 (2014) 103 [arXiv:1401.7977] [INSPIRE].
R. Raczka, N. Limić and J. Niederle, Discrete degenerate representations of noncompact rotation groups I, J. Math. Phys. 7 (1966) 1861.
P. Breitenlohner and D.Z. Freedman, Positive energy in anti-de Sitter backgrounds and gauged extended supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
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].
F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].
J. Lepowsky, A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Alg. 49 (1977) 496.
B. Oblak, BMS particles in three dimensions, arXiv:1610.08526 [INSPIRE].
W. Fulton and J. Harris, Representation theory: a first course, Grad. Texts Math. 129, Springer New York U.S.A., (1991).
T. Hirai, On infinitesimal operators of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962) 83.
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ArXiv ePrint: 1612.08166
Research Associate of the Fund for Scientific Research — FNRS (Belgium) (Nicolas Boulanger).
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Basile, T., Bekaert, X. & Boulanger, N. Mixed-symmetry fields in de Sitter space: a group theoretical glance. J. High Energ. Phys. 2017, 81 (2017). https://doi.org/10.1007/JHEP05(2017)081
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DOI: https://doi.org/10.1007/JHEP05(2017)081