Abstract
In a conformally flat three-dimensional spacetime, the linearised higher-spin Cotton tensor ℭα(n)(h) is the unique conserved conformal current which is a gauge-invariant descendant of the conformal gauge prepotential hα(n). The explicit form of ℭα(n)(h) is well known in Minkowski space. Here we solve the problem of extending the Minkowskian result to the case of anti-de Sitter (AdS) space and derive a closed-form expression for ℭα(n)(h) in terms of the AdS Lorentz covariant derivatives. It is shown that every conformal higher-spin action \( {S}_{\mathrm{CS}}^{(n)}\left[h\right]\propto \int {\mathrm{d}}^3{xeh}^{\alpha (n)}{\mathrm{\mathfrak{C}}}_{\alpha (n)}(h) \) factorises into a product of (n − 1) first-order operators that are associated with the spin-n/2 partially massless AdS values. Our findings greatly facilitate the on-shell analysis of massive higher-spin gauge-invariant actions in AdS3. The main results are extended to the case of \( \mathcal{N} \) = 1 AdS supersymmetry. In particular, we derive simple expressions for the higher-spin super-Cotton tensors in AdS3.
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L.P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton NJ U.S.A. (1926).
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. B 69 (1977) 304 [INSPIRE].
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Properties of Conformal Supergravity, Phys. Rev. D 17 (1978) 3179 [INSPIRE].
D. Butter, N = 1 Conformal Superspace in Four Dimensions, Annals Phys. 325 (2010) 1026 [arXiv:0906.4399] [INSPIRE].
D. Butter, S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, Conformal supergravity in three dimensions: New off-shell formulation, JHEP 09 (2013) 072 [arXiv:1305.3132] [INSPIRE].
D. Butter, S.M. Kuzenko, J. Novak and S. Theisen, Invariants for minimal conformal supergravity in six dimensions, JHEP 12 (2016) 072 [arXiv:1606.02921] [INSPIRE].
S.M. Kuzenko and M. Ponds, Conformal geometry and (super)conformal higher-spin gauge theories, JHEP 05 (2019) 113 [arXiv:1902.08010] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
E.S. Fradkin and V.Y. Linetsky, A Superconformal Theory of Massless Higher Spin Fields in D = (2 + 1), Annals Phys. 198 (1990) 293 [Mod. Phys. Lett. A 4 (1989) 731] [INSPIRE].
C.N. Pope and P.K. Townsend, Conformal Higher Spin in (2 + 1)-dimensions, Phys. Lett. B 225 (1989) 245 [INSPIRE].
E.S. Fradkin and V.Y. Linetsky, Cubic Interaction in Conformal Theory of Integer Higher Spin Fields in Four-dimensional Space-time, Phys. Lett. B 231 (1989) 97 [INSPIRE].
E.S. Fradkin and V.Y. Linetsky, Superconformal Higher Spin Theory in the Cubic Approximation, Nucl. Phys. B 350 (1991) 274 [INSPIRE].
A.A. Tseytlin, On limits of superstring in AdS5 × S5, Theor. Math. Phys. 133 (2002) 1376 [Teor. Mat. Fiz. 133 (2002) 69] [hep-th/0201112] [INSPIRE].
A.Y. Segal, Conformal higher spin theory, Nucl. Phys. B 664 (2003) 59 [hep-th/0207212] [INSPIRE].
R. Marnelius, Lagrangian conformal higher spin theory, arXiv:0805.4686 [INSPIRE].
R.R. Metsaev, Gauge invariant two-point vertices of shadow fields, AdS/CFT, and conformal fields, Phys. Rev. D 81 (2010) 106002 [arXiv:0907.4678] [INSPIRE].
M.A. Vasiliev, Bosonic conformal higher-spin fields of any symmetry, Nucl. Phys. B 829 (2010) 176 [arXiv:0909.5226] [INSPIRE].
X. Bekaert, E. Joung and J. Mourad, Effective action in a higher-spin background, JHEP 02 (2011) 048 [arXiv:1012.2103] [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].
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. Beccaria, X. Bekaert and A.A. Tseytlin, Partition function of free conformal higher spin theory, JHEP 08 (2014) 113 [arXiv:1406.3542] [INSPIRE].
M. Beccaria, S. Nakach and A.A. Tseytlin, On triviality of S-matrix in conformal higher spin theory, JHEP 09 (2016) 034 [arXiv:1607.06379] [INSPIRE].
M. Grigoriev and A.A. Tseytlin, On conformal higher spins in curved background, J. Phys. A 50 (2017) 125401 [arXiv:1609.09381] [INSPIRE].
S.M. Kuzenko, R. Manvelyan and S. Theisen, Off-shell superconformal higher spin multiplets in four dimensions, JHEP 07 (2017) 034 [arXiv:1701.00682] [INSPIRE].
M. Beccaria and A.A. Tseytlin, On induced action for conformal higher spins in curved background, Nucl. Phys. B 919 (2017) 359 [arXiv:1702.00222] [INSPIRE].
R. Bonezzi, Induced Action for Conformal Higher Spins from Worldline Path Integrals, Universe 3 (2017) 64 [arXiv:1709.00850] [INSPIRE].
R. Manvelyan and G. Poghosyan, Geometrical structure of Weyl invariants for spin three gauge field in general gravitational background in d = 4, Nucl. Phys. B 937 (2018) 1 [arXiv:1804.10779] [INSPIRE].
T. Adamo, S. Nakach and A.A. Tseytlin, Scattering of conformal higher spin fields, JHEP 07 (2018) 016 [arXiv:1805.00394] [INSPIRE].
M. Grigoriev, I. Lovrekovic and E. Skvortsov, New Conformal Higher Spin Gravities in 3d, JHEP 01 (2020) 059 [arXiv:1909.13305] [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].
E.A. Bergshoeff, O. Hohm and P.K. Townsend, On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories, Annals Phys. 325 (2010) 1118 [arXiv:0911.3061] [INSPIRE].
E.A. Bergshoeff, M. Kovacevic, J. Rosseel, P.K. Townsend and Y. Yin, A spin-4 analog of 3D massive gravity, Class. Quant. Grav. 28 (2011) 245007 [arXiv:1109.0382] [INSPIRE].
B.E.W. Nilsson, Towards an exact frame formulation of conformal higher spins in three dimensions, JHEP 09 (2015) 078 [arXiv:1312.5883] [INSPIRE].
B.E.W. Nilsson, On the conformal higher spin unfolded equation for a three-dimensional self-interacting scalar field, JHEP 08 (2016) 142 [arXiv:1506.03328] [INSPIRE].
M. Henneaux, S. Hörtner and A. Leonard, Higher Spin Conformal Geometry in Three Dimensions and Prepotentials for Higher Spin Gauge Fields, JHEP 01 (2016) 073 [arXiv:1511.07389] [INSPIRE].
H. Linander and B.E.W. Nilsson, The non-linear coupled spin 2–spin 3 Cotton equation in three dimensions, JHEP 07 (2016) 024 [arXiv:1602.01682] [INSPIRE].
S.M. Kuzenko and D.X. Ogburn, Off-shell higher spin N = 2 supermultiplets in three dimensions, Phys. Rev. D 94 (2016) 106010 [arXiv:1603.04668] [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.M. Kuzenko and M. Tsulaia, Off-shell massive N = 1 supermultiplets in three dimensions, Nucl. Phys. B 914 (2017) 160 [arXiv:1609.06910] [INSPIRE].
T. Basile, R. Bonezzi and N. Boulanger, The Schouten tensor as a connection in the unfolding of 3D conformal higher-spin fields, JHEP 04 (2017) 054 [arXiv:1701.08645] [INSPIRE].
S.M. Kuzenko and M. Ponds, Topologically massive higher spin gauge theories, JHEP 10 (2018) 160 [arXiv:1806.06643] [INSPIRE].
M. Henneaux, V. Lekeu, A. Leonard, J. Matulich and S. Prohazka, Three-dimensional conformal geometry and prepotentials for four-dimensional fermionic higher-spin fields, JHEP 11 (2018) 156 [arXiv:1810.04457] [INSPIRE].
E.I. Buchbinder, D. Hutchings, J. Hutomo and S.M. Kuzenko, Linearised actions for \( \mathcal{N} \)-extended (higher-spin) superconformal gravity, JHEP 08 (2019) 077 [arXiv:1905.12476] [INSPIRE].
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].
T. Damour and S. Deser, ‘Geometry’ of Spin 3 Gauge Theories, Ann. Inst. H. Poincaré Phys. Theor. 47 (1987) 277 [INSPIRE].
R. Andringa, E.A. Bergshoeff, M. de Roo, O. Hohm, E. Sezgin and P.K. Townsend, Massive 3D Supergravity, Class. Quant. Grav. 27 (2010) 025010 [arXiv:0907.4658] [INSPIRE].
S. Deser and J.H. Kay, Topologically massive supergravity, Phys. Lett. B 120 (1983) 97 [INSPIRE].
G.W. Gibbons, C.N. Pope and E. Sezgin, The General Supersymmetric Solution of Topologically Massive Supergravity, Class. Quant. Grav. 25 (2008) 205005 [arXiv:0807.2613] [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.A. Bergshoeff, O. Hohm, J. Rosseel, E. Sezgin and P.K. Townsend, On Critical Massive (Super)Gravity in AdS3, J. Phys. Conf. Ser. 314 (2011) 012009 [arXiv:1011.1153] [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].
I.V. Tyutin and M.A. Vasiliev, Lagrangian formulation of irreducible massive fields of arbitrary spin in (2 + 1)-dimensions, Teor. Mat. Fiz. 113N1 (1997) 45 [Theor. Math. Phys. 113 (1997) 1244] [hep-th/9704132] [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, Forbidden Mass Range for Spin-2 Field Theory in de Sitter Space-time, Nucl. Phys. B 282 (1987) 397 [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].
L. Brink, R.R. Metsaev and M.A. Vasiliev, How massless are massless fields in AdSd , Nucl. Phys. B 586 (2000) 183 [hep-th/0005136] [INSPIRE].
S. Deser and A. Waldron, Gauge invariances and phases of massive higher spins in (A)dS, Phys. Rev. Lett. 87 (2001) 031601 [hep-th/0102166] [INSPIRE].
S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [INSPIRE].
S. Deser and A. Waldron, Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations, Phys. Lett. B 513 (2001) 137 [hep-th/0105181] [INSPIRE].
L. Dolan, C.R. Nappi and E. Witten, Conformal operators for partially massless states, JHEP 10 (2001) 016 [hep-th/0109096] [INSPIRE].
Y.M. Zinoviev, On massive high spin particles in AdS, hep-th/0108192 [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].
E.D. Skvortsov and M.A. Vasiliev, Geometric formulation for partially massless fields, Nucl. Phys. B 756 (2006) 117 [hep-th/0601095] [INSPIRE].
I.L. Buchbinder, T.V. Snegirev and Y.M. Zinoviev, Gauge invariant Lagrangian formulation of massive higher spin fields in (A)dS3 space, Phys. Lett. B 716 (2012) 243 [arXiv:1207.1215] [INSPIRE].
I.L. Buchbinder, T.V. Snegirev and Y.M. Zinoviev, Frame-like gauge invariant Lagrangian formulation of massive fermionic higher spin fields in AdS3 space, Phys. Lett. B 738 (2014) 258 [arXiv:1407.3918] [INSPIRE].
S. Deger, A. Kaya, E. Sezgin and P. Sundell, Spectrum of D = 6, N = 4b supergravity on AdS3 × S3, Nucl. Phys. B 536 (1998) 110 [hep-th/9804166] [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].
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].
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].
J. Fang and C. Fronsdal, Massless, Half Integer Spin Fields in de Sitter Space, Phys. Rev. D 22 (1980) 1361 [INSPIRE].
D. Hutchings, J. Hutomo and S.M. Kuzenko, Higher-spin gauge models with (1, 1) supersymmetry in AdS3: Reduction to (1, 0) superspace, arXiv:2011.14294 [INSPIRE].
S.M. Kuzenko and G. Tartaglino-Mazzucchelli, Conformal supergravities as Chern-Simons theories revisited, JHEP 03 (2013) 113 [arXiv:1212.6852] [INSPIRE].
B.M. Zupnik and D.G. Pak, Superfield Formulation of the Simplest Three-dimensional Gauge Theories and Conformal Supergravities, Theor. Math. Phys. 77 (1988) 1070 [Teor. Mat. Fiz. 77 (1988) 97] [INSPIRE].
S.M. Kuzenko, Prepotentials for N = 2 conformal supergravity in three dimensions, JHEP 12 (2012) 021 [arXiv:1209.3894] [INSPIRE].
S.M. Kuzenko, U. Lindström and G. Tartaglino-Mazzucchelli, Off-shell supergravity-matter couplings in three dimensions, JHEP 03 (2011) 120 [arXiv:1101.4013] [INSPIRE].
P.S. Howe, J.M. Izquierdo, G. Papadopoulos and P.K. Townsend, New supergravities with central charges and Killing spinors in (2 + 1)-dimensions, Nucl. Phys. B 467 (1996) 183 [hep-th/9505032] [INSPIRE].
S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, N = 6 superconformal gravity in three dimensions from superspace, JHEP 01 (2014) 121 [arXiv:1308.5552] [INSPIRE].
D. Butter, S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, Conformal supergravity in three dimensions: Off-shell actions, JHEP 10 (2013) 073 [arXiv:1306.1205] [INSPIRE].
P. van Nieuwenhuizen, D = 3 Conformal Supergravity and Chern-Simons Terms, Phys. Rev. D 32 (1985) 872 [INSPIRE].
M. Roček and P. van Nieuwenhuizen, N ≥ 2 supersymmetric Chern-Simons terms as D = 3 extended conformal supergravity, Class. Quant. Grav. 3 (1986) 43 [INSPIRE].
M. Nishimura and Y. Tanii, N = 6 conformal supergravity in three dimensions, JHEP 10 (2013) 123 [arXiv:1308.3960] [INSPIRE].
U. Lindström and M. Roček, Superconformal Gravity in Three-dimensions as a Gauge Theory, Phys. Rev. Lett. 62 (1989) 2905 [INSPIRE].
H. Nishino and S.J. Gates Jr., Chern-Simons theories with supersymmetries in three-dimensions, Int. J. Mod. Phys. A 8 (1993) 3371 [INSPIRE].
S.J. Gates Jr., M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].
B.M. Zupnik and D.G. Pak, Differential and Integral Forms in Supergauge Theories and Supergravity, Class. Quant. Grav. 6 (1989) 723 [INSPIRE].
U. Lindström and M. Roček, A super-Weyl-invariant spinning membrane, Phys. Lett. B 218 (1989) 207 [Conf. Proc. C 8903131 (1989) 341] [INSPIRE].
S.M. Kuzenko, U. Lindström and G. Tartaglino-Mazzucchelli, Three-dimensional (p, q) AdS superspaces and matter couplings, JHEP 08 (2012) 024 [arXiv:1205.4622] [INSPIRE].
S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, Higher derivative couplings and massive supergravity in three dimensions, JHEP 09 (2015) 081 [arXiv:1506.09063] [INSPIRE].
I.L. Buchbinder, T.V. Snegirev and Y.M. Zinoviev, Lagrangian formulation of the massive higher spin supermultiplets in three dimensional space-time, JHEP 10 (2015) 148 [arXiv:1508.02829] [INSPIRE].
I.L. Buchbinder, T.V. Snegirev and Y.M. Zinoviev, Lagrangian description of massive higher spin supermultiplets in AdS3 space, JHEP 08 (2017) 021 [arXiv:1705.06163] [INSPIRE].
I.L. Buchbinder, T.V. Snegirev and Y.M. Zinoviev, Supersymmetric higher spin models in three dimensional spaces, Symmetry 10 (2017) 9 [arXiv:1711.11450] [INSPIRE].
Y.M. Zinoviev, Frame-like gauge invariant formulation for massive high spin particles, Nucl. Phys. B 808 (2009) 185 [arXiv:0808.1778] [INSPIRE].
A.A. Tseytlin, Effective action in de Sitter space and conformal supergravity (in Russian), Yad. Fiz. 39 (1984) 1606 [Sov. J. Nucl. Phys. 39 (1984) 1018] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Instanton zero modes and β-functions in supergravities. 2. Conformal supergravity, Phys. Lett. B 134 (1984) 307 [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.A. Tseytlin, On partition function and Weyl anomaly of conformal higher spin fields, Nucl. Phys. B 877 (2013) 598 [arXiv:1309.0785] [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].
M. Grigoriev and A. Hancharuk, On the structure of the conformal higher-spin wave operators, JHEP 12 (2018) 033 [arXiv:1808.04320] [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].
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].
E.I. Buchbinder, D. Hutchings, S.M. Kuzenko and M. Ponds, AdS superprojectors, JHEP 04 (2021) 074 [arXiv:2101.05524] [INSPIRE].
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Kuzenko, S.M., Ponds, M. Higher-spin Cotton tensors and massive gauge-invariant actions in AdS3. J. High Energ. Phys. 2021, 275 (2021). https://doi.org/10.1007/JHEP05(2021)275
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DOI: https://doi.org/10.1007/JHEP05(2021)275