Abstract
We report on a recent progress in constructing off-shell \(4\)D, \(\mathcal{N}=2\) supersymmetric integer higher-superspin theory in terms of unconstrained harmonic analytic gauge superfields and their cubic interaction with matter hypermultiplets. For even superspins, a new equivalent representation of the hypermultiplet couplings in terms of an analytic \(\omega\) superfield is presented. It involves both cubic and quartic vertices.
Similar content being viewed by others
Notes
See [19] for a brief review.
The expression for \(V^{--}\) can be obtained either by solving the flatness condition with the help of harmonic distributions (see book [17]) or by a direct calculation in the component formalism, starting from the WZ form of \(V^{++}\).
The complete form of the WZ gauge can be found in [15].
The spinor indices are raised and lowered in the standard way, with the help of the antisymmetric tensor \(\varepsilon_{\mu\nu}\), \({\varepsilon_{12} = -\varepsilon^{12} = 1}\), e.g., \(l_{(\nu}^{\;\;\;\mu)} = \varepsilon_{\nu\rho}l^{(\rho\mu)}\).
Hereafter, all Lorentz indices of the same nature in the coefficients of differential operators are assumed to be properly symmetrized with those hidden in the multi-index \(M\).
This can be easily observed in the simplest spin \(\mathbf{1}\) case.
This change of variables is given in book [17].
We use the standard definitions \((D^+)^2 := D^{+ \alpha}D^+_{\alpha}\), \((\bar{D}^+)^2 := \bar{D}^{+}_{\dot\alpha}\bar{D}^{+\dot\alpha}\), and \((D^+)^4 := \frac{1}{16}(D^+)^2(\bar{D}^+)^2\).
References
M. A. Vasiliev, “Higher spin gauge theories in various dimensions,” Fortschr. Phys., 52, 702–717 (2004); arXiv: hep-th/0401177.
X. Bekaert, S. Cnockaert, C. Iazeolla, and M. A. Vasiliev, “Nonlinear higher spin theories in various dimensions,” in: Proceedings of the First Solvay Workshop on Higher Spin Gauge Theories (Brussels, Belgium, 12–14 May, 2004, R. Argurio, G. Barnich, G. Bonelli, and M. Grigoriev, eds.), International Solvay Institutes for Physics and Chemistry, Brussels (2006), pp. 132–197; arXiv: hep-th/0503128.
X. Bekaert, N. Boulanger, and P. Sundell, “How higher-spin gravity surpasses the spin-two barrier,” Rev. Mod. Phys., 84, 987–1009 (2012); arXiv: 1007.0435.
A. Sagnotti, “Notes on strings and higher spins,” J. Phys. A: Math. Theor., 46, 214006, 29 pp. (2013); arXiv: 1112.4285.
V. E. Didenko and E. D. Skvortsov, “Elements of Vasiliev theory,” arXiv: 1401.2975.
X. Bekaert, N. Boulanger, A. Campoleoni, M. Chodaroli, D. Francia, M. Grigoriev, E. Sezgin, and E. Skvortsov, “Snowmass white paper: Higher spin gravity and higher spin symmetry,” arXiv: 2205.01567.
C. Fronsdal, “Massless fields with integer spin,” Phys. Rev. D, 18, 3624–3629 (1978).
J. Fang and C. Fronsdal, “Massless fields with half-integral spin,” Phys. Rev. D, 18, 3630–3633 (1978).
T. Curtright, “Massless field supermultiplets with arbitrary spins,” Phys. Lett. B, 85, 219–224 (1979).
M. A. Vasiliev, “ ‘Gauge’ form of description of massless fields with arbitrary spin,” Soviet J. Nucl. Phys., 32, 439-443 (1980).
S. M. Kuzenko, V. V. Postnikov, and A. G. Sibiryakov, “Massless gauge superfields of higher half integer superspins,” JETP Lett., 57, 534–538 (1993); S. M. Kuzenko, A. G. Sibiryakov, “Massless gauge superfields of higher integer superspins,” JETP Lett., 57, 539–542 (1993); S. M. Kuzenko and A. G. Sibiryakov, “Free massless higher superspin superfields in the anti-de Sitter superspace,” Phys. Atom. Nucl., 57, 1257–1267 (1994); arXiv: 1112.4612.
S. J. Gates, Jr., S. M. Kuzenko, and A. G. Sibiryakov, “Towards a unified theory of massless superfields of all superspins,” Phys. Lett. B, 394, 343–353 (1997), arXiv: hep-th/9611193; “\(\mathcal{N}=2\) supersymmetry of higher superspin massless theories,” Phys. Lett. B, 412, 59–68 (1997); arXiv: hep-th/9609141.
S. J. Gates, Jr. and K. Koutrolikos, “On \(4D\), \(N=1\) massless gauge superfields of arbitrary superhelicity,” JHEP, 06, 098, 47 pp. (2014); arXiv: 1310.7385.
K. Koutrolikos, “Superspace formulation of massive half-integer superspin,” JHEP, 03, 254, 23 pp. (2021); arXiv: 2012.12225.
I. Buchbinder, E. Ivanov, and N. Zaigraev, “Unconstrained off-shell superfield formulation of \(4D\), \(\mathcal{N} = 2\) supersymmetric higher spins,” JHEP, 12, 016, 27 pp. (2021); arXiv: 2109.07639.
A. Gal’perin, E. Ivanov, V. Ogievetskiĭ, and É. Sokatchev, “Harmonic superspace: key to \(N=2\) supersymmetry theories,” JETP Lett., 40, 912–916 (1984); “Unconstrained \(N=2\) matter, Yang–Mills and supergravity theories in harmonic superspace,” Class. Quantum Grav., 1, 469–498 (1984).
A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky, and E. S. Sokatchev, Harmonic Superspace, Cambridge Univ. Press, Cambridge (2001).
I. Buchbinder, E. Ivanov, and N. Zaigraev, “Off-shell cubic hypermultiplet couplings to \(\mathcal{N} = 2\) higher spin gauge superfields,” JHEP, 05, 104, 37 pp. (2022); arXiv: 2202.08196.
I. Buchbinder, E. Ivanov, and N. Zaigraev, “Unconstrained \(\mathcal{N} = 2\) higher-spin gauge superfields and their hypermultiplet couplings,” Phys. Part. Nucl. Lett., 20, 300–305 (2023); arXiv: 2211.09501.
I. Buchbinder, E. Ivanov, and N. Zaigraev, “\(\mathcal{N} = 2\) higher spins: superfield equations of motion, the hypermultiplet supercurrents, and the component structure,” JHEP, 03, 036, 87 pp. (2023); arXiv: 2212.14114.
E. S. Fradkin and M. A. Vasiliev, “Minimal set of auxiliary fields and S-matrix for extended supergravity,” Lett. Nuovo Cimento, 25, 79–87 (1979); “Minimal set of auxiliary fields in SO\((2)\)-extended supergravity,” Phys. Lett. B, 85, 47–51 (1979).
B. De Wit and J. W. van Holten, “Multiplets of linearized SO\((2)\) supergravity,” Nucl. Phys. B, 155, 530–542 (1979).
B. De Wit, J. W. van Holten, and A. Van Proeyen, “Transformation rules of \(N=2\) supergravity multiplets,” Nucl. Phys. B, 167, 186–204 (1980).
A. K. H. Bengtsson, I. Bengtsson, and L. Brink, “Cubic interaction terms for arbitrary spin,” Nucl. Phys. B, 227, 31–40 (1983); “Cubic interaction terms for arbitrary extended supermultiplets,” 41–49.
E. S. Fradkin and R. R. Metsaev, “A cubic interaction of totally symmetric massless representations of the Lorentz group in arbitrary dimensions,” Class. Quantum Grav., 8, L89–L94 (1991).
R. R. Metsaev, “Generating function for cubic interaction vertices of higher spin fields in any dimension,” Modern Phys. Lett. A, 8, 2413–2426 (1993).
R. Manvelyan, K. Mkrtchyan, and W. Rühl, “General trilinear interaction for arbitrary even higher spin gauge fields,” Nucl. Phys. B, 836, 204–221 (2010), arXiv: 1003.2877; “A generating function for the cubic interactions of higher spin fields,” Phys. Lett. B, 696, 410–415 (2011); arXiv: 1009.1054.
A. Fotopoulos, N. Irges, A. C. Petkou, and M. Tsulaia, “Higher spin gauge fields interacting with scalars: The Lagrangian cubic vertex,” JHEP, 10, 021, 27 pp. (2007); arXiv: 0708.1399.
X. Bekaert, E. Joung, and J. Mourad, “On higher spin interactions with matter,” JHEP, 05, 126, 31 pp. (2009); arXiv: 0903.3338.
M. V. Khabarov and Yu. M. Zinoviev, “Massless higher spin cubic vertices in flat four dimensional space,” JHEP, 08, 112, 21 pp. (2020); arXiv: 2005.09851.
M. V. Khabarov and Yu. M. Zinoviev, “Cubic interaction vertices for massless higher spin supermultiplets in \(d = 4\),” JHEP, 02, 167, 17 pp. (2021); arXiv: 2012.00482.
I. L. Buchbinder, S. J. Gates, Jr., and K. Koutrolikos, “Integer superspin supercurrents of matter supermultiplets,” JHEP, 05, 031, 18 pp. (2019); arXiv: 1811.12858; S. J. Gates, Jr. and K. Koutrolikos, “Progress on cubic interactions of arbitrary superspin supermultiplets via gauge invariant supercurrents,” Phys. Lett. B, 797, 134868, 6 pp. (2019); arXiv: 1904.13336.
A. Galperin, N. A. Ky, and E. Sokatchev, “\(\mathcal{N}=2\) supergravity in superspace: solution to the constraints,” Class. Quantum Grav., 4, 1235–1253 (1987).
S. M. Kuzenko and S. Theisen, “Correlation functions of conserved currents in \(\mathcal N=2\) superconformal theory,” Class. Quant. Grav., 17, 665–696 (2000); arXiv: hep-th/9907107.
A. Galperin, E. Ivanov, V. Ogievetsky, and E. Sokatchev, “\(N=2\) supergravity in superspace: Different versions and matter couplings,” Class. Quantum Grav., 4, 1255–1265 (1987).
S. M. Kuzenko and E. S. N. Raptakis, “Extended superconformal higher-spin gauge theories in four dimensions,” JHEP, 12, 210, 26 pp. (2021); arXiv: 2104.10416.
E. I. Buchbinder, J. Hutomo, and S. M. Kuzenko, “Higher spin supercurrents in anti-de Sitter space,” JHEP, 09, 27, 51 pp. (2018); arXiv: 1805.08055.
S. M. Kuzenko, M. Ponds, and E. S. N. Raptakis, “Conformal interactions between matter and higher-spin (super)fields,” Fortsch. Phys., 71, 2200157, 31 pp. (2023); arXiv: 2208.07783.
S. M. Kuzenko and E. S. N. Raptakis, “On higher-spin \( \mathcal{N} = 2\) supercurrent multiplets,” JHEP, 05, 056, 20 pp. (2023); arXiv: 2301.09386.
E. Ivanov, “\(\mathcal N=2\) supergravities in harmonic superspace,” arXiv: 2212.07925.
E. I. Buchbinder, B. A. Ovrut, I. L. Buchbinder, E. A. Ivanov, and S. M. Kuzenko, “Low-energy effective action in \(N = 2\) supersymmetric field theories,” Phys. Part. Nucl., 32, 641–674 (2001).
Acknowledgments
I thank the Organizers of the A. A. Slavnov Memorial Conference for inviting me to give this talk and kind hospitality at the Steklov Mathematical Institute in Moscow. I also thank my coauthors Ioseph Buchbinder and Nikita Zaigraev, on the joint papers with whom this talk is essentially based. I also thank the anonymous referee for the useful remarks aimed at making the exposition more complete and coherent.
Funding
This work was supported by the Russian Science Foundation (project No. 21-12-00129).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 515–532 https://doi.org/10.4213/tmf10527.
Rights and permissions
About this article
Cite this article
Ivanov, E.A. Higher spins in harmonic superspace. Theor Math Phys 217, 1855–1869 (2023). https://doi.org/10.1134/S004057792312005X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S004057792312005X