Abstract
We study a generalization of the D-dimensional Vasiliev theory to include a tower of partially massless fields. This theory is obtained by replacing the usual higher-spin algebra of Killing tensors on (A)dS with a generalization that includes “third-order” Killing tensors. Gauging this algebra with the Vasiliev formalism leads to a fully non-linear theory which is expected to be UV complete, includes gravity, and can live on dS as well as AdS. The linearized spectrum includes three massive particles and an infinite tower of partially massless particles, in addition to the usual spectrum of particles present in the Vasiliev theory, in agreement with predictions from a putative dual CFT with the same symmetry algebra. We compute the masses of the particles which are not fixed by the massless or partially massless gauge symmetry, finding precise agreement with the CFT predictions. This involves computing several dozen of the lowest-lying terms in the expansion of the trilinear form of the enlarged higher-spin algebra. We also discuss nuances in the theory that occur in specific dimensions; in particular, the theory dramatically truncates in bulk dimensions D = 3, 5 and has non-diagonalizable mixings which occur in D = 4, 7.
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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].
T. Basile, X. Bekaert and N. Boulanger, Flato-Fronsdal theorem for higher-order singletons, JHEP 11 (2014) 131 [arXiv:1410.7668] [INSPIRE].
X. Bekaert and M. Grigoriev, Higher-order singletons and partially massless fields, Bulg. J. Phys. 41 (2014) 172 [INSPIRE].
K.B. Alkalaev, M. Grigoriev and E.D. Skvortsov, Uniformizing higher-spin equations, J. Phys. A 48 (2015) 015401 [arXiv:1409.6507] [INSPIRE].
E. Joung and K. Mkrtchyan, Partially-massless higher-spin algebras and their finite-dimensional truncations, JHEP 01 (2016) 003 [arXiv:1508.07332] [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, More on equations of motion for interacting massless fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 285 (1992) 225 [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories: star product and AdS space, hep-th/9910096 [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 four-dimensions, three-dimensions and two-dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [INSPIRE].
X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev, Nonlinear higher spin theories in various dimensions, in Higher spin gauge theories: Proceedings, 1st Solvay Workshop, Brussels Belgium, 12-14 May 2004, pg. 132 [hep-th/0503128] [INSPIRE].
C. Iazeolla, On the algebraic structure of higher-spin field equations and new exact solutions, Ph.D. thesis, Rome U. Tor Vergata, Rome Italy, (2008) [arXiv:0807.0406] [INSPIRE].
V.E. Didenko and E.D. Skvortsov, Elements of Vasiliev theory, arXiv:1401.2975 [INSPIRE].
M.A. Vasiliev, Higher-spin theory and space-time metamorphoses, Lect. Notes Phys. 892 (2015) 227 [arXiv:1404.1948] [INSPIRE].
S. Giombi, TASI lectures on the higher spin — CFT duality, arXiv:1607.02967 [INSPIRE].
C. Brust and K. Hinterbichler, Free □k scalar conformal field theory, arXiv:1607.07439 [INSPIRE].
H. Osborn and A. Stergiou, C T for non-unitary CFTs in higher dimensions, JHEP 06 (2016) 079 [arXiv:1603.07307] [INSPIRE].
A. Guerrieri, A.C. Petkou and C. Wen, The free σCFTs, JHEP 09 (2016) 019 [arXiv:1604.07310] [INSPIRE].
Y. Nakayama, Hidden global conformal symmetry without Virasoro extension in theory of elasticity, Annals Phys. 372 (2016) 392 [arXiv:1604.00810] [INSPIRE].
Z. Péli, S. Nagy and K. Sailer, Phase structure of the O(2) ghost model with higher-order gradient term, Phys. Rev. D 94 (2016) 065021 [arXiv:1605.07836] [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].
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].
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 17th International Seminar on High Energy Physics (Quarks 2012), Yaroslavl Russia, 4-10 June 2012 [arXiv:1209.4876] [INSPIRE].
C. de Rham, K. Hinterbichler, R.A. Rosen and A.J. Tolley, Evidence for and obstructions to nonlinear partially massless gravity, Phys. Rev. D 88 (2013) 024003 [arXiv:1302.0025] [INSPIRE].
A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory, J. Phys. A 49 (2016) 183001 [arXiv:1512.00021] [INSPIRE].
Yu. M. Zinoviev, On massive spin 2 interactions, Nucl. Phys. B 770 (2007) 83 [hep-th/0609170] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, On partially massless bimetric gravity, Phys. Lett. B 726 (2013) 834 [arXiv:1208.1797] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Bimetric theory and partial masslessness with Lanczos-Lovelock terms in arbitrary dimensions, Class. Quant. Grav. 30 (2013) 184010 [arXiv:1212.4525] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Higher derivative gravity and conformal gravity from bimetric and partially massless bimetric theory, Universe 1 (2015) 92 [arXiv:1303.6940] [INSPIRE].
S. Deser, M. Sandora and A. Waldron, Nonlinear partially massless from massive gravity?, Phys. Rev. D 87 (2013) 101501 [arXiv:1301.5621] [INSPIRE].
Yu. M. Zinoviev, Massive spin-2 in the Fradkin-Vasiliev formalism I. Partially massless case, Nucl. Phys. B 886 (2014) 712 [arXiv:1405.4065] [INSPIRE].
S. Garcia-Saenz and R.A. Rosen, A non-linear extension of the spin-2 partially massless symmetry, JHEP 05 (2015) 042 [arXiv:1410.8734] [INSPIRE].
K. Hinterbichler, Manifest duality invariance for the partially massless graviton, Phys. Rev. D 91 (2015) 026008 [arXiv:1409.3565] [INSPIRE].
E. Joung, W. Li and M. Taronna, No-go theorems for unitary and interacting partially massless spin-two fields, Phys. Rev. Lett. 113 (2014) 091101 [arXiv:1406.2335] [INSPIRE].
S. Alexandrov and C. Deffayet, On partially massless theory in 3 dimensions, JCAP 03 (2015) 043 [arXiv:1410.2897] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Extended Weyl invariance in a bimetric model and partial masslessness, Class. Quant. Grav. 33 (2016) 015011 [arXiv:1507.06540] [INSPIRE].
K. Hinterbichler and R.A. Rosen, Partially massless monopoles and charges, Phys. Rev. D 92 (2015) 105019 [arXiv:1507.00355] [INSPIRE].
D. Cherney, S. Deser, A. Waldron and G. Zahariade, Non-linear duality invariant partially massless models?, Phys. Lett. B 753 (2016) 293 [arXiv:1511.01053] [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)dS 3 gravity, JHEP 05 (2016) 150 [arXiv:1511.05975] [INSPIRE].
S. Garcia-Saenz, K. Hinterbichler, A. Joyce, E. Mitsou and R.A. Rosen, No-go for partially massless spin-2 Yang-Mills, JHEP 02 (2016) 043 [arXiv:1511.03270] [INSPIRE].
K. Hinterbichler and A. Joyce, Manifest duality for partially massless higher spins, JHEP 09 (2016) 141 [arXiv:1608.04385] [INSPIRE].
L. Apolo and S.F. Hassan, Non-linear partially massless symmetry in an SO(1, 5) continuation of conformal gravity, arXiv:1609.09514 [INSPIRE].
L. Apolo, S.F. Hassan and A. Lundkvist, Gauge and global symmetries of the candidate partially massless bimetric gravity, Phys. Rev. D 94 (2016) 124055 [arXiv:1609.09515] [INSPIRE].
J. Maldacena, Einstein gravity from conformal gravity, arXiv:1105.5632 [INSPIRE].
S. Deser, E. Joung and A. Waldron, Partial masslessness and conformal gravity, J. Phys. A 46 (2013) 214019 [arXiv:1208.1307] [INSPIRE].
S. Deser, E. Joung and A. Waldron, Gravitational- and self-coupling of partially massless spin 2, Phys. Rev. D 86 (2012) 104004 [arXiv:1301.4181] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
C.M. Hull, Timelike T duality, de Sitter space, large-N gauge theories and topological field theory, JHEP 07 (1998) 021 [hep-th/9806146] [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, in Strings 2001: International Conference, Mumbai India, 5-10 January 2001 [hep-th/0106109] [INSPIRE].
A. Strominger, Inflation and the dS/CFT correspondence, JHEP 11 (2001) 049 [hep-th/0110087] [INSPIRE].
V. Balasubramanian, J. de Boer and D. Minic, Notes on de Sitter space and holography, Class. Quant. Grav. 19 (2002) 5655 [Ann. Phys. 303 (2003) 59] [hep-th/0207245] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
D.J. Gross, High-energy symmetries of string theory, Phys. Rev. Lett. 60 (1988) 1229 [INSPIRE].
C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ triality: from higher spin fields to strings, J. Phys. A 46 (2013) 214009 [arXiv:1207.4485] [INSPIRE].
L. Girardello, M. Porrati and A. Zaffaroni, 3D interacting CFTs and generalized Higgs phenomenon in higher spin theories on AdS, Phys. Lett. B 561 (2003) 289 [hep-th/0212181] [INSPIRE].
M. Bianchi, J.F. Morales and H. Samtleben, On stringy AdS 5 × S 5 and higher spin holography, JHEP 07 (2003) 062 [hep-th/0305052] [INSPIRE].
M. Bianchi, P.J. Heslop and F. Riccioni, More on “La Grande Bouffe”, JHEP 08 (2005) 088 [hep-th/0504156] [INSPIRE].
M.A. Vasiliev, Consistent equations for interacting massless fields of all spins in the first order in curvatures, Annals Phys. 190 (1989) 59 [INSPIRE].
N. Boulanger and P. Sundell, An action principle for Vasiliev’s four-dimensional higher-spin gravity, J. Phys. A 44 (2011) 495402 [arXiv:1102.2219] [INSPIRE].
N. Doroud and L. Smolin, An action for higher spin gauge theory in four dimensions, arXiv:1102.3297 [INSPIRE].
N. Boulanger, N. Colombo and P. Sundell, A minimal BV action for Vasiliev’s four-dimensional higher spin gravity, JHEP 10 (2012) 043 [arXiv:1205.3339] [INSPIRE].
N. Boulanger, E. Sezgin and P. Sundell, 4D higher spin gravity with dynamical two-form as a Frobenius-Chern-Simons gauge theory, arXiv:1505.04957 [INSPIRE].
X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Quartic AdS interactions in higher-spin gravity from conformal field theory, JHEP 11 (2015) 149 [arXiv:1508.04292] [INSPIRE].
R. Bonezzi, N. Boulanger, E. Sezgin and P. Sundell, Frobenius-Chern-Simons gauge theory, arXiv:1607.00726 [INSPIRE].
C. Sleight and M. Taronna, Higher spin interactions from conformal field theory: the complete cubic couplings, Phys. Rev. Lett. 116 (2016) 181602 [arXiv:1603.00022] [INSPIRE].
M.G. Eastwood and T. Leistner, Higher symmetries of the square of the Laplacian, IMA Vol. Math. Appl. 144 (2008) 319 [math/0610610].
S. Giombi and I.R. Klebanov, One loop tests of higher spin AdS/CFT, JHEP 12 (2013) 068 [arXiv:1308.2337] [INSPIRE].
S. Giombi, I.R. Klebanov and B.R. Safdi, Higher spin AdS d+1 /CFT d at one loop, Phys. Rev. D 89 (2014) 084004 [arXiv:1401.0825] [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].
C. Brust and K. Hinterbichler, Partially massless higher-spin theory II: one-loop effective actions, arXiv:1610.08522 [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].
S.M. Carroll, Spacetime and geometry: an introduction to general relativity, Addison-Wesley, San Francisco U.S.A., (2004) [INSPIRE].
X. Bekaert and N. Boulanger, The unitary representations of the Poincaré group in any spacetime dimension, in 2nd Modave Summer School in Theoretical Physics, Modave Belgium, 6-12 August 2006 [hep-th/0611263] [INSPIRE].
W.K. Tung, Group theory in physics, World Scientific, Singapore, (1985) [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 R.I. Nepomechie, Gauge invariance versus masslessness in de Sitter space, Annals Phys. 154 (1984) 396 [INSPIRE].
A. Higuchi, Forbidden mass range for spin-2 field theory in de Sitter space-time, Nucl. Phys. B 282 (1987) 397 [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].
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, Stability of massive cosmological gravitons, Phys. Lett. B 508 (2001) 347 [hep-th/0103255] [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].
Yu. 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].
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].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].
G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].
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].
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].
A. Higuchi, Massive symmetric tensor field in space-times with a positive cosmological constant, Nucl. Phys. B 325 (1989) 745 [INSPIRE].
L. Dolan, C.R. Nappi and E. Witten, Conformal operators for partially massless states, JHEP 10 (2001) 016 [hep-th/0109096] [INSPIRE].
E. Joung and K. Mkrtchyan, Notes on higher-spin algebras: minimal representations and structure constants, JHEP 05 (2014) 103 [arXiv:1401.7977] [INSPIRE].
K. Hinterbichler and A. Joyce, Goldstones with extended shift symmetries, Int. J. Mod. Phys. D 23 (2014) 1443001 [arXiv:1404.4047] [INSPIRE].
T. Griffin, K.T. Grosvenor, P. Hořava and Z. Yan, Scalar field theories with polynomial shift symmetries, Commun. Math. Phys. 340 (2015) 985 [arXiv:1412.1046] [INSPIRE].
M.A. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [INSPIRE].
N. Boulanger, P. Kessel, E.D. Skvortsov and M. Taronna, Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal, J. Phys. A 49 (2016) 095402 [arXiv:1508.04139] [INSPIRE].
C. Sleight and M. Taronna, Higher-spin algebras, holography and flat space, arXiv:1609.00991 [INSPIRE].
K. Hallowell and A. Waldron, Constant curvature algebras and higher spin action generating functions, Nucl. Phys. B 724 (2005) 453 [hep-th/0505255] [INSPIRE].
A.R. Gover, E. Latini and A. Waldron, Metric projective geometry, BGG detour complexes and partially massless gauge theories, Commun. Math. Phys. 341 (2016) 667 [arXiv:1409.6778] [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 within the unfolded approach: algebraic aspects, JHEP 01 (2010) 106 [arXiv:0910.3334] [INSPIRE].
D.S. Ponomarev and M.A. Vasiliev, Frame-like action and unfolded formulation for massive higher-spin fields, Nucl. Phys. B 839 (2010) 466 [arXiv:1001.0062] [INSPIRE].
V.E. Didenko and E.D. Skvortsov, Towards higher-spin holography in ambient space of any dimension, J. Phys. A 46 (2013) 214010 [arXiv:1207.6786] [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].
R. Manvelyan, K. Mkrtchyan, R. Mkrtchyan and S. Theisen, On higher spin symmetries in AdS 5, JHEP 10 (2013) 185 [arXiv:1304.7988] [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].
K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].
C. de Rham, Massive gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological collider physics, arXiv:1503.08043 [INSPIRE].
H. Lee, D. Baumann and G.L. Pimentel, Non-Gaussianity as a particle detector, JHEP 12 (2016) 040 [arXiv:1607.03735] [INSPIRE].
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Brust, C., Hinterbichler, K. Partially massless higher-spin theory. J. High Energ. Phys. 2017, 86 (2017). https://doi.org/10.1007/JHEP02(2017)086
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DOI: https://doi.org/10.1007/JHEP02(2017)086