Abstract
We study conformal higher spin (CHS) fields on constant curvature backgrounds. By employing parent formulation technique in combination with tractor description of GJMS operators we find a manifestly factorized form of the CHS wave operators for symmetric fields of arbitrary integer spin s and gauge invariance of arbitrary order t ≤ s. In the case of the usual Fradkin-Tseytlin fields t = 1 this gives a systematic derivation of the factorization formulas known in the literature while for t > 1 the explicit formulas were not known. We also relate the gauge invariance of the CHS fields to the partially-fixed gauge invariance of the factors and show that the factors can be identified with (partially gauge-fixed) wave operators for (partially)-massless or special massive fields. As a byproduct, we establish a detailed relationship with the tractor approach and, in particular, derive the tractor form of the CHS equations and gauge symmetries.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E.S. Fradkin and A.A. Tseytlin, Conformal Supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
A.Y. Segal, Conformal higher spin theory, Nucl. Phys. B 664 (2003) 59 [hep-th/0207212] [INSPIRE].
A.A. Tseytlin, On limits of superstring in AdS 5 × S 5, Theor. Math. Phys. 133 (2002) 1376 [hep-th/0201112] [INSPIRE].
X. Bekaert, E. Joung and J. Mourad, Effective action in a higher-spin background, JHEP 02 (2011) 048 [arXiv:1012.2103] [INSPIRE].
R. Bonezzi, Induced Action for Conformal Higher Spins from Worldline Path Integrals, Universe 3 (2017) 64 [arXiv:1709.00850] [INSPIRE].
R.R. Metsaev, Shadows, currents and AdS, Phys. Rev. D 78 (2008) 106010 [arXiv:0805.3472] [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].
X. Bekaert and M. Grigoriev, Notes on the ambient approach to boundary values of AdS gauge fields, J. Phys. A 46 (2013) 214008 [arXiv:1207.3439] [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].
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, Ordinary-derivative formulation of conformal low spin fields, JHEP 01 (2012) 064 [arXiv:0707.4437] [INSPIRE].
R.R. Metsaev, Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields, JHEP 06 (2012) 062 [arXiv:0709.4392] [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].
S. Deser and R.I. Nepomechie, Gauge Invariance Versus Masslessness in de Sitter Space, Annals Phys. 154 (1984) 396 [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].
R.R. Metsaev, Arbitrary spin conformal fields in (A)dS, Nucl. Phys. B 885 (2014) 734 [arXiv:1404.3712] [INSPIRE].
M. Beccaria and A.A. Tseytlin, On higher spin partition functions, J. Phys. A 48 (2015) 275401 [arXiv:1503.08143] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Iterating free-field AdS/CFT: higher spin partition function relations, J. Phys. A 49 (2016) 295401 [arXiv:1602.00948] [INSPIRE].
T. Nutma and M. Taronna, On conformal higher spin wave operators, JHEP 06 (2014) 066 [arXiv:1404.7452] [INSPIRE].
S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, SIGMA 4 (2008) 036 [arXiv:0803.4331].
E.S. Fradkin and A.A. Tseytlin, One Loop β-function in Conformal Supergravities, Nucl. Phys. B 203 (1982) 157 [INSPIRE].
C.R. Graham, R. Jenne, L.J. Mason and G.A.J. Sparling, Conformally invariant powers of the laplacian, I: Existence, J. London Math. Soc. s2–46 (1992) 557.
A.R. Gover, Laplacian operators and Q-curvature on conformally Einstein manifolds, Math. Ann. 336 (2006) 311 [math.DG/0506037] [INSPIRE].
M.G. Eastwood, Notes on conformal differential geometry, Rend. Circ. Mat. Palermo S43 (1996) 57.
T. Bailey, M.G. Eastwood and A. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mt. J. Math. 24 (1994) 1191.
A. Čap and A.R. Gover, Standard tractors and the conformal ambient metric construction, Annals Global Anal. Geom. 24 (2003) 231 [math/0207016] [INSPIRE].
R.R. Metsaev, Long, partial-short and special conformal fields, JHEP 05 (2016) 096 [arXiv:1604.02091] [INSPIRE].
G. Barnich and M. Grigoriev, Parent form for higher spin fields on anti-de Sitter space, JHEP 08 (2006) 013 [hep-th/0602166] [INSPIRE].
X. Bekaert and M. Grigoriev, Manifestly conformal descriptions and higher symmetries of bosonic singletons, SIGMA 6 (2010) 038 [arXiv:0907.3195] [INSPIRE].
G. Barnich and M. Grigoriev, First order parent formulation for generic gauge field theories, JHEP 01 (2011) 122 [arXiv:1009.0190] [INSPIRE].
M. Grigoriev, Parent formulations, frame-like Lagrangians and generalized auxiliary fields, JHEP 12 (2012) 048 [arXiv:1204.1793] [INSPIRE].
G. Barnich, M. Grigoriev, A. Semikhatov and I. Tipunin, Parent field theory and unfolding in BRST first-quantized terms, Commun. Math. Phys. 260 (2005) 147 [hep-th/0406192] [INSPIRE].
M. Grigoriev and A. Waldron, Massive Higher Spins from BRST and Tractors, Nucl. Phys. B 853 (2011) 291 [arXiv:1104.4994] [INSPIRE].
A.R. Gover, A. Shaukat and A. Waldron, Tractors, Mass and Weyl Invariance, Nucl. Phys. B 812 (2009) 424 [arXiv:0810.2867] [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].
A.R. Gover and L.J. Peterson, Conformally invariant powers of the Laplacian, Q-curvature and tractor calculus, Commun. Math. Phys. 235 (2003) 339 [math-ph/0201030] [INSPIRE].
C. Fefferman and C. Graham, Conformal Invariants, Astérisque Hors série (1985) 95.
C. Fefferman and C.R. Graham, The ambient metric, Ann. Math. Stud. 178 (2011) 1 [arXiv:0710.0919] [INSPIRE].
X. Bekaert, M. Grigoriev and E.D. Skvortsov, Higher Spin Extension of Fefferman-Graham Construction, Universe 4 (2018) 17 [arXiv:1710.11463] [INSPIRE].
B.V. Fedosov, A Simple geometrical construction of deformation quantization, J. Diff. Geom. 40 (1994) 213 [INSPIRE].
B.V. Fedosov, Deformation quantization and index theory, Mathematical Topics Series, volume 9, Akademie Verlag, Berlin Germany (1996) [INSPIRE].
M. Grigoriev, Off-shell gauge fields from BRST quantization, hep-th/0605089 [INSPIRE].
M. Grigoriev and A.A. Tseytlin, On conformal higher spins in curved background, J. Phys. A 50 (2017) 125401 [arXiv:1609.09381] [INSPIRE].
C.R. LeBrun, Ambi-twistors and Einstein’s equations, Class. Quant. Grav. 2 (1985) 555.
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].
M.S. Drew and J.D. Gegenberg, Conformally covariant massless spin-2 field equations, Nuovo Cim. A 60 (1980) 41 [INSPIRE].
A.O. Barut and B.-W. Xu, On conformally covariant spin-2 and spin 3/2 equations, J. Phys. A 15 (1982) L207 [INSPIRE].
J. Erdmenger and H. Osborn, Conformally covariant differential operators: Symmetric tensor fields, Class. Quant. Grav. 15 (1998) 273 [gr-qc/9708040] [INSPIRE].
M.A. Vasiliev, Bosonic conformal higher-spin fields of any symmetry, Nucl. Phys. B 829 (2010) 176 [arXiv:0909.5226] [INSPIRE].
A. Chekmenev and M. Grigoriev, Boundary values of mixed-symmetry massless fields in AdS space, Nucl. Phys. B 913 (2016) 769 [arXiv:1512.06443] [INSPIRE].
K. Alkalaev and M. Grigoriev, Unified BRST approach to (partially) massless and massive AdS fields of arbitrary symmetry type, Nucl. Phys. B 853 (2011) 663 [arXiv:1105.6111] [INSPIRE].
K.B. Alkalaev and M. Grigoriev, Unified BRST description of AdS gauge fields, Nucl. Phys. B 835 (2010) 197 [arXiv:0910.2690] [INSPIRE].
M.G. Eastwood and M. Singer, A conformally invariant Maxwell gauge, Phys. Lett. A 107 (1985) 73 [INSPIRE].
L. Dolan, C.R. Nappi and E. Witten, Conformal operators for partially massless states, JHEP 10 (2001) 016 [hep-th/0109096] [INSPIRE].
E.D. Skvortsov and M.A. Vasiliev, Transverse Invariant Higher Spin Fields, Phys. Lett. B 664 (2008) 301 [hep-th/0701278] [INSPIRE].
A. Campoleoni and D. Francia, Maxwell-like Lagrangians for higher spins, JHEP 03 (2013) 168 [arXiv:1206.5877] [INSPIRE].
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: 1808.04320
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
Grigoriev, M., Hancharuk, A. On the structure of the conformal higher-spin wave operators. J. High Energ. Phys. 2018, 33 (2018). https://doi.org/10.1007/JHEP12(2018)033
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2018)033