Abstract
We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an \( \mathfrak{sp} \)(1)-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric theories on curved backgrounds. We define the Killing superalgebra of bosonic backgrounds and show that their existence is implied by the field equations. The maximally supersymmetric backgrounds are characterised and their Killing superalgebras are explicitly described as filtered deformations of the Poincaré superalgebra.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
J.M. Figueroa-O’Farrill, P. Meessen and S. Philip, Supersymmetry and homogeneity of M-theory backgrounds, Class. Quant. Grav. 22 (2005) 207 [hep-th/0409170] [INSPIRE].
J. Figueroa-O’Farrill and N. Hustler, The homogeneity theorem for supergravity backgrounds, JHEP 10 (2012) 014 [arXiv:1208.0553] [INSPIRE].
J. Figueroa-O’Farrill and A. Santi, On the algebraic structure of Killing superalgebras, Adv. Theor. Math. Phys. 21 (2017) 1115 [arXiv:1608.05915] [INSPIRE].
J. Figueroa-O’Farrill and R. Grassie, Kinematical superspaces, JHEP 11 (2019) 008 [arXiv:1908.11278] [INSPIRE].
B. Zumino, Nonlinear Realization of Supersymmetry in de Sitter Space, Nucl. Phys. B 127 (1977) 189 [INSPIRE].
M. Blau, Killing spinors and SYM on curved spaces, JHEP 11 (2000) 023 [hep-th/0005098] [INSPIRE].
S.-J. Cheng and V.G. Kac, Generalized Spencer cohomology and filtered deformations of ℤ-graded Lie superalgebras, Adv. Theor. Math. Phys. 2 (1998) 1141 [math/9805039] [INSPIRE].
S.-J. Cheng and V.G. Kac, Addendum: Generalized Spencer cohomology and filtered deformations of ℤ-graded Lie superalgebras, Adv. Theor. Math. Phys. 8 (2004) 697.
J. Figueroa-O’Farrill and A. Santi, Spencer cohomology and 11-dimensional supergravity, Commun. Math. Phys. 349 (2017) 627 [arXiv:1511.08737] [INSPIRE].
J. Figueroa-O’Farrill and A. Santi, Eleven-dimensional supergravity from filtered subdeformations of the Poincaré superalgebra, J. Phys. A 49 (2016) 295204 [arXiv:1511.09264] [INSPIRE].
P. de Medeiros, J. Figueroa-O’Farrill and A. Santi, Killing superalgebras for Lorentzian four-manifolds, JHEP 06 (2016) 106 [arXiv:1605.00881] [INSPIRE].
P. de Medeiros, J. Figueroa-O’Farrill and A. Santi, Killing superalgebras for Lorentzian six-manifolds, J. Geom. Phys. 132 (2018) 13 [arXiv:1804.00319] [INSPIRE].
J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis and H.S. Reall, All supersymmetric solutions of minimal supergravity in five- dimensions, Class. Quant. Grav. 20 (2003) 4587 [hep-th/0209114] [INSPIRE].
A. Chamseddine, J.M. Figueroa-O’Farrill and W. Sabra, Supergravity vacua and Lorentzian Lie groups, hep-th/0306278 [INSPIRE].
Y. Kosmann, Dérivées de Lie des spineurs, Ann. Mat. Pura Appl. 91 (1972) 317.
J.M. Figueroa-O’Farrill, On the supersymmetries of Anti-de Sitter vacua, Class. Quant. Grav. 16 (1999) 2043 [hep-th/9902066] [INSPIRE].
J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, D-branes and spinning black holes, Phys. Lett. B 391 (1997) 93 [hep-th/9602065] [INSPIRE].
P. Meessen, A small note on P P wave vacua in six-dimensions and five-dimensions, Phys. Rev. D 65 (2002) 087501 [hep-th/0111031] [INSPIRE].
M. Cahen and N. Wallach, Lorentzian symmetric spaces, Bull. Am. Math. Soc. 76 (1970) 585.
M. Cahen and M. Parker, Parallélismes absolus des variétés lorentziennes, Annales Inst. Fourier 27 (1977) 251.
B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Am. Math. Soc. 80 (1955) 528.
R.P. Geroch, Limits of spacetimes, Commun. Math. Phys. 13 (1969) 180 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2105.05775
Rights and permissions
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.
About this article
Cite this article
Beckett, A., Figueroa-O’Farrill, J. Killing superalgebras for lorentzian five-manifolds. J. High Energ. Phys. 2021, 209 (2021). https://doi.org/10.1007/JHEP07(2021)209
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)209