Abstract
We propose a simple unfolded description of free massive higher spin particles in anti-de-Sitter spacetime. While our unfolded equation of motion has the standard form of a covariant constancy condition, our formulation differs from the standard one in that our field takes values in a different internal space, which for us is simply a unitary irreducible representation of the symmetry group. Our main result is the explicit construction, for the case of AdS3, of a map from our formulation to the standard wave equations for massive higher spin particles, as well as to the unfolded description prevalent in the literature. It is hoped that our formulation may be used to clarify the group-theoretic content of interactions in higher spin theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
R. Rahman and M. Taronna, From Higher Spins to Strings: A Primer, arXiv:1512.07932 [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, pp. 132-197 (2004) [hep-th/0503128] [INSPIRE].
V.E. Didenko and E.D. Skvortsov, Elements of Vasiliev theory, arXiv:1401.2975 [INSPIRE].
E.S. Fradkin and M.A. Vasiliev, On the Gravitational Interaction of Massless Higher Spin Fields, Phys. Lett. B 189 (1987) 89 [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].
M.A. Vasiliev, Free Massless Fermionic Fields of Arbitrary Spin in d-dimensional de Sitter Space, Nucl. Phys. B 301 (1988) 26 [INSPIRE].
M.A. Vasiliev, Equations of Motion of Interacting Massless Fields of All Spins as a Free Differential Algebra, Phys. Lett. B 209 (1988) 491 [INSPIRE].
M.A. Vasiliev, Unfolded representation for relativistic equations in (2+1) anti-de Sitter space, Class. Quant. Grav. 11 (1994) 649 [INSPIRE].
A.V. Barabanshchikov, S.F. Prokushkin and M.A. Vasiliev, Free equations for massive matter fields in (2+1)-dimensional anti-de Sitter space from deformed oscillator algebra, Teor. Mat. Fiz. 110N3 (1997) 372 [hep-th/9609034] [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].
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].
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].
N. Boulanger, D. Ponomarev, E. Sezgin and P. Sundell, New unfolded higher spin systems in AdS 3, Class. Quant. Grav. 32 (2015) 155002 [arXiv:1412.8209] [INSPIRE].
Yu.M. Zinoviev, Massive higher spins in d = 3 unfolded, J. Phys. A 49 (2016) 095401 [arXiv:1509.00968] [INSPIRE].
I.L. Buchbinder, T.V. Snegirev and Yu.M. Zinoviev, Unfolded equations for massive higher spin supermultiplets in AdS 3, JHEP 08 (2016) 075 [arXiv:1606.02475] [INSPIRE].
M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions, Phys. Lett. B 243 (1990) 378 [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].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher Spins & Strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Stringy Symmetries and the Higher Spin Square, J. Phys. A 48 (2015) 185402 [arXiv:1501.07236] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, String Theory as a Higher Spin Theory, JHEP 09 (2016) 085 [arXiv:1512.07237] [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS 3 at k = 1, arXiv:1803.04420[INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS 3, JHEP 05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
J. Raeymaekers, On matter coupled to the higher spin square, J. Phys. A 49 (2016) 355402 [arXiv:1603.07845] [INSPIRE].
L. Castellani, R. D’Auria and P. Fre, Supergravity and superstrings: A Geometric perspective. Vol. 1: Mathematical foundations, World Scientific, Singapore, Singapore (1991), pp. 1-603 [INSPIRE].
E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [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].
A. Kitaev, Notes on \( \tilde{\mathrm{SL}}\left(2,\mathrm{\mathbb{R}}\right) \) representations, arXiv:1711.08169 [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically Massive Gauge Theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [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 [hep-th/9704132] [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].
S. Deger, A. Kaya, E. Sezgin and P. Sundell, Spectrum of D = 6, N = 4b supergravity on AdS 3 × S 3, Nucl. Phys. B 536 (1998) 110 [hep-th/9804166] [INSPIRE].
J. Repka, Tensor products of unitary representations of SL 2 (R), Bull. Am. Math. Soc. 82 (1976) 930.
T. Ortín, A Note on Lie-Lorentz derivatives, Class. Quant. Grav. 19 (2002) L143 [hep-th/0206159] [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, W (∞) and the Racah-wigner Algebra, Nucl. Phys. B 339 (1990) 191 [INSPIRE].
M. Ammon, P. Kraus and E. Perlmutter, Scalar fields and three-point functions in D = 3 higher spin gravity, JHEP 07 (2012) 113 [arXiv:1111.3926] [INSPIRE].
A. Campoleoni, T. Prochazka and J. Raeymaekers, A note on conical solutions in 3D Vasiliev theory, JHEP 05 (2013) 052 [arXiv:1303.0880] [INSPIRE].
P. Kessel and J. Raeymaekers, work in progress.
M.A. Vasiliev, Current Interactions and Holography from the 0-Form Sector of Nonlinear Higher-Spin Equations, JHEP 10 (2017) 111 [arXiv:1605.02662] [INSPIRE].
M. Abramowitz and I.A. Stegun Handbook of Mathematical Functions, Dover (1972).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1805.07279
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kessel, P., Raeymaekers, J. Simple unfolded equations for massive higher spins in AdS3. J. High Energ. Phys. 2018, 76 (2018). https://doi.org/10.1007/JHEP08(2018)076
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)076