Abstract
Given a maximally symmetric d-dimensional background with isometry algebra \( \mathfrak{g} \), a symmetric and traceless rank-s field ϕa(s) satisfying the massive Klein-Gordon equation furnishes a collection of massive \( \mathfrak{g} \)-representations with spins j ∈ {0, 1, · · · , s}. In this paper we construct the spin-(s, j) projectors, which are operators that isolate the part of ϕa(s) that furnishes the representation from this collection carrying spin j. In the case of an (anti-)de Sitter ((A)dSd) background, we find that the poles of the projectors encode information about (partially-)massless representations, in agreement with observations made earlier in d = 3, 4. We then use these projectors to facilitate a systematic derivation of two-derivative actions with a propagating massless spin-s mode. In addition to reproducing the massless spin-s Fronsdal action, this analysis generates new actions possessing higher-depth gauge symmetry. In (A)dSd we also derive the action for a partially-massless spin-s depth-t field with 1 ≤ t ≤ s. The latter utilises the minimum number of auxiliary fields, and corresponds to the action originally proposed by Zinoviev after gauging away all Stückelberg fields. Some higher-derivative actions are also presented, and in d = 3 are used to construct (i) generalised higher-spin Cotton tensors in (A)dS3; and (ii) topologically-massive actions with higher-depth gauge symmetry. Finally, in four-dimensional \( \mathcal{N} \) = 1 Minkowski superspace, we provide closed-form expressions for the analogous superprojectors.
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References
E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals Math. 40 (1939) 149 [INSPIRE].
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].
P.A.M. Dirac, Relativistic wave equations, Proc. Roy. Soc. Lond. A 155 (1936) 447 [INSPIRE].
M. Fierz, Force-free particles with any spin, Helv. Phys. Acta 12 (1939) 3 [INSPIRE].
M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211 [INSPIRE].
C. Fronsdal, On the theory of higher spin fields, Nuovo Cim. 9 (1958) 416.
R.E. Behrends and C. Fronsdal, Fermi decay of higher spin particles, Phys. Rev. 106 (1957) 345 [INSPIRE].
A.Y. Segal, Conformal higher spin theory, Nucl. Phys. B 664 (2003) 59 [hep-th/0207212] [INSPIRE].
A.P. Isaev and M.A. Podoinitsyn, Two-spinor description of massive particles and relativistic spin projection operators, Nucl. Phys. B 929 (2018) 452 [arXiv:1712.00833] [INSPIRE].
L.P.S. Singh, Covariant propagators for massive arbitrary spin fields, Phys. Rev. D 23 (1981) 2236 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
S.-J. Chang, Lagrange formulation for systems with higher spin, Phys. Rev. 161 (1967) 1308 [INSPIRE].
A. Aurilia and H. Umezawa, Projection operators in quantum theory of relativistic free fields, Nuovo Cim. A 51 (1967) 14.
A. Aurilia and H. Umezawa, Theory of high-spin fields, Phys. Rev. 182 (1969) 1682 [INSPIRE].
K.J. Barnes, Electromagnetic form factors, Ph.D. thesis, unpublished (1963).
R.J. Rivers, Lagrangian theory for neutral massive spin-2 fields, Nuovo Cim. 34 (1964) 386.
P. Van Nieuwenhuizen, On ghost-free tensor lagrangians and linearized gravitation, Nucl. Phys. B 60 (1973) 478 [INSPIRE].
E.I. Buchbinder, S.M. Kuzenko, J. La Fontaine and M. Ponds, Spin projection operators and higher-spin Cotton tensors in three dimensions, Phys. Lett. B 790 (2019) 389 [arXiv:1812.05331] [INSPIRE].
S.M. Kuzenko and M. Ponds, Spin projection operators in (A)dS and partial masslessness, Phys. Lett. B 800 (2020) 135128 [arXiv:1910.10440] [INSPIRE].
D. Hutchings, S.M. Kuzenko and M. Ponds, AdS (super)projectors in three dimensions and partial masslessness, JHEP 10 (2021) 090 [arXiv:2107.12201] [INSPIRE].
E.I. Buchbinder, D. Hutchings, S.M. Kuzenko and M. Ponds, AdS superprojectors, JHEP 04 (2021) 074 [arXiv:2101.05524] [INSPIRE].
D. Hutchings, Superspin projection operators and off-shell higher-spin supermultiplets on Minkowski and anti-de Sitter superspace, Ph.D. thesis, Western Australia U., Crawley, WA, Australia (2023) [arXiv:2401.05621] [INSPIRE].
S.J. Gates Jr. and W. Siegel, (3/2,1) superfield of O(2) supergravity, Nucl. Phys. B 164 (1980) 484 [INSPIRE].
S.J. Gates Jr., S.M. Kuzenko and J. Phillips, The off-shell (3/2,2) supermultiplets revisited, Phys. Lett. B 576 (2003) 97 [hep-th/0306288] [INSPIRE].
W. Siegel and S.J. Gates Jr., Superprojectors, Nucl. Phys. B 189 (1981) 295 [INSPIRE].
S.J. Gates, M.T. Grisaru, M. Rocek and W. Siegel, Superspace or one thousand and one lessons in supersymmetry, hep-th/0108200 [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, Phys. Rev. D 20 (1979) 848 [INSPIRE].
Y.M. Zinoviev, On massive high spin particles in AdS, hep-th/0108192 [INSPIRE].
M.A. Vasiliev, Bosonic conformal higher-spin fields of any symmetry, Nucl. Phys. B 829 (2010) 176 [arXiv:0909.5226] [INSPIRE].
S.M. Kuzenko and M. Ponds, Higher-spin Cotton tensors and massive gauge-invariant actions in AdS3, JHEP 05 (2021) 275 [arXiv:2103.11673] [INSPIRE].
E.A. Bergshoeff et al., A spin-4 analog of 3D massive gravity, Class. Quant. Grav. 28 (2011) 245007 [arXiv:1109.0382] [INSPIRE].
S.M. Kuzenko and M. Ponds, Topologically massive higher spin gauge theories, JHEP 10 (2018) 160 [arXiv:1806.06643] [INSPIRE].
D. Dalmazi and A.L.R. Santos, Higher spin analogs of linearized topologically massive gravity and linearized new massive gravity, Phys. Rev. D 104 (2021) 085023 [arXiv:2107.08879] [INSPIRE].
N. Boulanger, D. Ponomarev, E. Sezgin and P. Sundell, New unfolded higher spin systems in AdS3, Class. Quant. Grav. 32 (2015) 155002 [arXiv:1412.8209] [INSPIRE].
E. Sokatchev, Projection operators and supplementary conditions for superfields with an arbitrary spin, Nucl. Phys. B 99 (1975) 96 [INSPIRE].
A. Salam and J.A. Strathdee, On superfields and Fermi-Bose symmetry, Phys. Rev. D 11 (1975) 1521 [INSPIRE].
D.V. Bulgakova, Y.O. Goncharov and T. Helpin, Construction of the traceless projection of tensors via the Brauer algebra, arXiv:2212.14496 [INSPIRE].
S.M. Kuzenko and I.L. Buchbinder, Ideas and methods of supersymmetry and supergravity or a walk through superspace: a walk through superspace, IOP, Bristol, U.K. (1995) [https://doi.org/10.1201/9780367802530] [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].
M.A. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [INSPIRE].
D. Ponomarev, Basic introduction to higher-spin theories, Int. J. Theor. Phys. 62 (2023) 146 [arXiv:2206.15385] [INSPIRE].
T. Basile, X. Bekaert and N. Boulanger, Mixed-symmetry fields in de Sitter space: a group theoretical glance, JHEP 05 (2017) 081 [arXiv:1612.08166] [INSPIRE].
S. Deser and R.I. Nepomechie, Anomalous propagation of gauge fields in conformally flat spaces, Phys. Lett. B 132 (1983) 321 [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 A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [INSPIRE].
R.R. Metsaev, Gauge invariant formulation of massive totally symmetric fermionic fields in (A)dS space, Phys. Lett. B 643 (2006) 205 [hep-th/0609029] [INSPIRE].
L.P.S. Singh and C.R. Hagen, Lagrangian formulation for arbitrary spin. 1. The boson case, Phys. Rev. D 9 (1974) 898 [INSPIRE].
Y.M. Zinovev, Gauge invariant description of massive high spin particles, preprint no. 83-91, Institute for High Energy Physics, Serpukhov, Russia (1983) [INSPIRE].
S.M. Klishevich and Y.M. Zinovev, On electromagnetic interaction of massive spin-2 particle, Phys. Atom. Nucl. 61 (1998) 1527 [hep-th/9708150] [INSPIRE].
A. Campoleoni and D. Francia, Maxwell-like Lagrangians for higher spins, JHEP 03 (2013) 168 [arXiv:1206.5877] [INSPIRE].
M. Ponds, Models for (super)conformal higher-spin fields on curved backgrounds, Ph.D. thesis, Western Australia U., Crawley, WA, Australia (2021) [arXiv:2201.10163] [INSPIRE].
R. Marnelius, Lagrangian higher spin field theories from the O(N) extended supersymmetric particle, arXiv:0906.2084 [INSPIRE].
C. Pozrikidis, The fractional Laplacian, CRC Press, Boca Raton, FL, U.S.A. (2016).
S.M. Kuzenko and M. Ponds, Conformal geometry and (super)conformal higher-spin gauge theories, JHEP 05 (2019) 113 [arXiv:1902.08010] [INSPIRE].
C.N. Pope and P.K. Townsend, Conformal higher spin in (2+1)-dimensions, Phys. Lett. B 225 (1989) 245 [INSPIRE].
S.M. Kuzenko, Higher spin super-Cotton tensors and generalisations of the linear-chiral duality in three dimensions, Phys. Lett. B 763 (2016) 308 [arXiv:1606.08624] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982) 975 [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [INSPIRE].
D. Dalmazi and E.L. Mendonca, A new spin-2 self-dual model in D=2+1, JHEP 09 (2009) 011 [arXiv:0907.5009] [INSPIRE].
R. Andringa et al., Massive 3D supergravity, Class. Quant. Grav. 27 (2010) 025010 [arXiv:0907.4658] [INSPIRE].
E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive gravity in three dimensions, Phys. Rev. Lett. 102 (2009) 201301 [arXiv:0901.1766] [INSPIRE].
I.L. Buchbinder, A. Pashnev and M. Tsulaia, Lagrangian formulation of the massless higher integer spin fields in the AdS background, Phys. Lett. B 523 (2001) 338 [hep-th/0109067] [INSPIRE].
X. Bekaert, N. Boulanger, Y. Goncharov and M. Grigoriev, Ambient-space variational calculus for gauge fields on constant-curvature spacetimes, J. Math. Phys. 65 (2024) 042301 [arXiv:2305.02892] [INSPIRE].
A.A. Tseytlin, On partition function and Weyl anomaly of conformal higher spin fields, Nucl. Phys. B 877 (2013) 598 [arXiv:1309.0785] [INSPIRE].
R.R. Metsaev, Arbitrary spin conformal fields in (A)dS, Nucl. Phys. B 885 (2014) 734 [arXiv:1404.3712] [INSPIRE].
T. Nutma and M. Taronna, On conformal higher spin wave operators, JHEP 06 (2014) 066 [arXiv:1404.7452] [INSPIRE].
M. Grigoriev and A. Hancharuk, On the structure of the conformal higher-spin wave operators, JHEP 12 (2018) 033 [arXiv:1808.04320] [INSPIRE].
I.L. Buchbinder, S.M. Kuzenko and A.G. Sibiryakov, Quantization of higher spin superfields in the anti-de Sitter superspace, Phys. Lett. B 352 (1995) 29 [hep-th/9502148] [INSPIRE].
S.M. Kuzenko and A.G. Sibiryakov, Free massless higher superspin superfields on the anti-de Sitter superspace, Phys. Atom. Nucl. 57 (1994) 1257 [arXiv:1112.4612] [INSPIRE].
D. Francia and A. Sagnotti, Minimal local Lagrangians for higher-spin geometry, Phys. Lett. B 624 (2005) 93 [hep-th/0507144] [INSPIRE].
I.L. Buchbinder, A.V. Galajinsky and V.A. Krykhtin, Quartet unconstrained formulation for massless higher spin fields, Nucl. Phys. B 779 (2007) 155 [hep-th/0702161] [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].
A. Sharapov and D. Shcherbatov, On auxiliary fields and Lagrangians for relativistic wave equations, J. Phys. A 57 (2024) 015210 [arXiv:2308.02074] [INSPIRE].
J.H. Fegebank and S.M. Kuzenko, Quantisation of the gauge-invariant models for massive higher-spin bosonic fields, arXiv:2310.00951 [INSPIRE].
L.W. Lindwasser, Covariant actions and propagators for all spins, masses, and dimensions, Phys. Rev. D 109 (2024) 085010 [arXiv:2307.11750] [INSPIRE].
K. Koutrolikos, Superspace formulation of massive half-integer superspin, JHEP 03 (2021) 254 [arXiv:2012.12225] [INSPIRE].
S.M. Kuzenko, A.G. Sibiryakov and V.V. Postnikov, Massless gauge superfields of higher half integer superspins, JETP Lett. 57 (1993) 534 [INSPIRE].
S.M. Kuzenko and A.G. Sibiryakov, Massless gauge superfields of higher integer superspins, JETP Lett. 57 (1993) 539 [INSPIRE].
S.J. Gates Jr. and K. Koutrolikos, On 4D, N=1 massless gauge superfields of arbitrary superhelicity, JHEP 06 (2014) 098 [arXiv:1310.7385] [INSPIRE].
I.L. Buchbinder, S.J. Gates and K. Koutrolikos, Hierarchy of supersymmetric higher spin connections, Phys. Rev. D 102 (2020) 125018 [arXiv:2010.02061] [INSPIRE].
V.P. Akulov, D.V. Volkov and V.A. Soroka, On the general covariant theory of calibrating poles in superspace, Theor. Math. Phys. 31 (1977) 285 [INSPIRE].
M.F. Sohnius and P.C. West, An alternative minimal off-shell version of N=1 supergravity, Phys. Lett. B 105 (1981) 353 [INSPIRE].
P.S. Howe, K.S. Stelle and P.K. Townsend, The vanishing volume of N=1 superspace, Phys. Lett. B 107 (1981) 420 [INSPIRE].
S.J. Gates Jr., M. Rocek and W. Siegel, Solution to constraints for n = 0 supergravity, Nucl. Phys. B 198 (1982) 113 [INSPIRE].
I.L. Buchbinder, S.J. Gates Jr., I.I.I.W.D. Linch and J. Phillips, New 4D, N=1 superfield theory: model of free massive superspin 3/2 multiplet, Phys. Lett. B 535 (2002) 280 [hep-th/0201096] [INSPIRE].
Y. Nakayama, Imaginary supergravity or virial supergravity?, Nucl. Phys. B 892 (2015) 288 [arXiv:1411.1057] [INSPIRE].
V. Rittenberg and E. Sokatchev, Decomposition of extended superfields into irreducible representations of supersymmetry, Nucl. Phys. B 193 (1981) 477 [INSPIRE].
E.I. Buchbinder, D. Hutchings, J. Hutomo and S.M. Kuzenko, Linearised actions for N-extended (higher-spin) superconformal gravity, JHEP 08 (2019) 077 [arXiv:1905.12476] [INSPIRE].
D. Hutchings, S.M. Kuzenko and E.S.N. Raptakis, The N=2 superconformal gravitino multiplet, Phys. Lett. B 845 (2023) 138132 [arXiv:2305.16029] [INSPIRE].
I. Buchbinder, E. Ivanov and N. Zaigraev, Unconstrained off-shell superfield formulation of 4D, N=2 supersymmetric higher spins, JHEP 12 (2021) 016 [arXiv:2109.07639] [INSPIRE].
I.V. Gorbunov, S.M. Kuzenko and S.L. Lyakhovich, On the minimal model of anyons, Int. J. Mod. Phys. A 12 (1997) 4199 [hep-th/9607114] [INSPIRE].
D.M. Gitman and I.V. Tyutin, Pseudoclassical description of higher spins in (2+1)-dimensions, Int. J. Mod. Phys. A 12 (1997) 535 [hep-th/9602048] [INSPIRE].
Acknowledgments
The authors are grateful to Sergei Kuzenko for collaboration at an early stage of this project and for comments on the manuscript, and to the referee for useful suggestions. DH would like to thank Mirian Tsulaia for useful discussions and Ruben Manveylan for encouraging the formulation of the (A)dSd projectors for d > 4. A preliminary version of this work was presented by DH at the CQUeST-APCTP workshop Gravity beyond Riemannian Paradigm (Jeju island, South Korea). DH is grateful to the organisers of this workshop for the stimulating scientific atmosphere and for the generous support. DH would also like to acknowledge the warm hospitality extended to him by the Quantum Gravity Unit at Okinawa Institute of Science and Technology where final revisions were completed. This work was supported in part by the Australian Research Council, project No. DP200101944 and DP230101629.
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Hutchings, D., Ponds, M. Spin-(s, j) projectors and gauge-invariant spin-s actions in maximally symmetric backgrounds. J. High Energ. Phys. 2024, 292 (2024). https://doi.org/10.1007/JHEP07(2024)292
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DOI: https://doi.org/10.1007/JHEP07(2024)292