Abstract
We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of \( \mathbb{Z} \)-graded subalgebras with maximum odd dimension of the N = 1 Poincaré superalgebra in four dimensions. Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions. We prove that such Killing spinors always generate a Lie superalgebra, and that this Lie superalgebra is a filtered deformation of a subalgebra of the N = 1 Poincaré superalgebra in four dimensions. Demanding the flatness of the connection defining the Killing spinors, we obtain equations satisfied by the maximally supersymmetric backgrounds. We solve these equations, arriving at the classification of maximally supersymmetric backgrounds whose associated Killing superalgebras are precisely the filtered deformations we classify in this paper.
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References
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-Theorem: N = 2 Field Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
K. Hosomichi, R.-K. Seong and S. Terashima, Supersymmetric Gauge Theories on the Five-Sphere, Nucl. Phys. B 865 (2012) 376 [arXiv:1203.0371] [INSPIRE].
J. Källén, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere, JHEP 08 (2012) 157 [arXiv:1206.6008] [INSPIRE].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
E. Shuster, Killing spinors and supersymmetry on AdS, Nucl. Phys. B 554 (1999) 198 [hep-th/9902129] [INSPIRE].
M. Blau, Killing spinors and SYM on curved spaces, JHEP 11 (2000) 023 [hep-th/0005098] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
B. Jia and E. Sharpe, Rigidly Supersymmetric Gauge Theories on Curved Superspace, JHEP 04 (2012) 139 [arXiv:1109.5421] [INSPIRE].
H. Samtleben and D. Tsimpis, Rigid supersymmetric theories in 4d Riemannian space, JHEP 05 (2012) 132 [arXiv:1203.3420] [INSPIRE].
C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on Curved Spaces and Holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring Curved Superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].
D. Cassani, C. Klare, D. Martelli, A. Tomasiello and A. Zaffaroni, Supersymmetry in Lorentzian Curved Spaces and Holography, Commun. Math. Phys. 327 (2014) 577 [arXiv:1207.2181] [INSPIRE].
J.T. Liu, L.A. Pando Zayas and D. Reichmann, Rigid Supersymmetric Backgrounds of Minimal Off-Shell Supergravity, JHEP 10 (2012) 034 [arXiv:1207.2785] [INSPIRE].
P. de Medeiros, Rigid supersymmetry, conformal coupling and twistor spinors, JHEP 09 (2014) 032 [arXiv:1209.4043] [INSPIRE].
I.L. Buchbinder and S.M. Kuzenko, Ideas and methods of supersymmetry and supergravity: a walk through superspace, Studies in High Energy Physics Cosmology and Gravitation, revised edition, IOP Publishing Ltd., Bristol U.K. (1998).
B.S. Acharya, J.M. Figueroa-O’Farrill, C.M. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [INSPIRE].
J.P. Gauntlett, R.C. Myers and P.K. Townsend, Supersymmetry of rotating branes, Phys. Rev. D 59 (1998) 025001 [hep-th/9809065] [INSPIRE].
J.P. Gauntlett, R.C. Myers and P.K. Townsend, Black holes of D = 5 supergravity, Class. Quant. Grav. 16 (1999) 1 [hep-th/9810204] [INSPIRE].
P.K. Townsend, Killing spinors, supersymmetries and rotating intersecting branes, hep-th/9901102 [INSPIRE].
J.M. Figueroa-O’Farrill, On the supersymmetries of Anti-de Sitter vacua, Class. Quant. Grav. 16 (1999) 2043 [hep-th/9902066] [INSPIRE].
J.M. Figueroa-O’Farrill and G. Papadopoulos, Homogeneous fluxes, branes and a maximally supersymmetric solution of M-theory, JHEP 08 (2001) 036 [hep-th/0105308] [INSPIRE].
M. Blau, J.M. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, A New maximally supersymmetric background of IIB superstring theory, JHEP 01 (2002) 047 [hep-th/0110242] [INSPIRE].
N. Alonso-Alberca, E. Lozano-Tellechea and T. Ortín, Geometric construction of Killing spinors and supersymmetry algebras in homogeneous space-times, Class. Quant. Grav. 19 (2002) 6009 [hep-th/0208158] [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.M. Figueroa-O’Farrill, E. Hackett-Jones and G. Moutsopoulos, The Killing superalgebra of ten-dimensional supergravity backgrounds, Class. Quant. Grav. 24 (2007) 3291 [hep-th/0703192] [INSPIRE].
J. Figueroa-O’Farrill, A Geometric construction of the exceptional Lie algebras F 4 and E 8, Commun. Math. Phys. 283 (2008) 663 [arXiv:0706.2829] [INSPIRE].
J. Figueroa-O’Farrill, E. Hackett-Jones, G. Moutsopoulos and J. Simon, On the maximal superalgebras of supersymmetric backgrounds, Class. Quant. Grav. 26 (2009) 035016 [arXiv:0809.5034] [INSPIRE].
J.M. Figueroa-O’Farrill, The Homogeneity conjecture for supergravity backgrounds, J. Phys. Conf. Ser. 175 (2009) 012002 [arXiv:0812.1258] [INSPIRE].
S.M. Kuzenko, Supersymmetric Spacetimes from Curved Superspace, PoS(CORFU2014)140 [arXiv:1504.08114] [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 N. Hustler, The homogeneity theorem for supergravity backgrounds II: the six-dimensional theories, JHEP 04 (2014) 131 [arXiv:1312.7509] [INSPIRE].
S.-J. Cheng and V. G. Kac, Generalized Spencer cohomology and filtered deformations of \( \mathbb{Z} \) -graded Lie superalgebras, Adv. Theor. Math. Phys. 2 (1998) 1141 [math/9805039].
S.-J. Cheng and V. G. Kac, Addendum: Generalized Spencer cohomology and filtered deformations of \( \mathbb{Z} \) -graded Lie superalgebras, Adv. Theor. Math. Phys. 8 (2004) 697.
J. Figueroa-O’Farrill and A. Santi, Spencer cohomology and eleven-dimensional supergravity, 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].
S.M. Kuzenko, Symmetries of curved superspace, JHEP 03 (2013) 024 [arXiv:1212.6179] [INSPIRE].
D. Butter, G. Inverso and I. Lodato, Rigid 4D \( \mathcal{N} \) = 2 supersymmetric backgrounds and actions, JHEP 09 (2015) 088 [arXiv:1505.03500] [INSPIRE].
S.M. Kuzenko and G. Tartaglino-Mazzucchelli, Nilpotent chiral superfield in N = 2 supergravity and partial rigid supersymmetry breaking, JHEP 03 (2016) 092 [arXiv:1512.01964] [INSPIRE].
S. Sternberg, Lectures on differential geometry, , second edition, Chelsea Publishing Co., New York U.S.A. (1983).
A. Santi and A. Spiro, Super-Poincaré algebras, space-times and supergravities (I), Adv. Theor. Math. Phys. 16 (2012) 1411 [arXiv:1011.2722].
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].
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].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
J.M. Figueroa-O’Farrill, Breaking the M waves, Class. Quant. Grav. 17 (2000) 2925 [hep-th/9904124] [INSPIRE].
H. Baum, Twistor and Killing spinors in Lorentzian geometry, in Global analysis and harmonic analysis, Marseille-Luminy France (1999) [Sémin. Congr. 4 (1999) 35].
F. Leitner, Imaginary Killing spinors in Lorentzian geometry, J. Math. Phys. 44 (2003) 4795.
C. Boubel and L. Bérard Bergery, On pseudo-Riemannian manifolds whose Ricci tensor is parallel, Geom. Dedicata 86 (2001) 1.
A. Chamseddine, J.M. Figueroa-O’Farrill and W. Sabra, Supergravity vacua and Lorentzian Lie groups, hep-th/0306278 [INSPIRE].
J.M. Figueroa-O’Farrill, On parallelizable NS-NS backgrounds, Class. Quant. Grav. 20 (2003) 3327 [hep-th/0305079] [INSPIRE].
C.R. Nappi and E. Witten, A WZW model based on a nonsemisimple group, Phys. Rev. Lett. 71 (1993) 3751 [hep-th/9310112] [INSPIRE].
A. Altomani and A. Santi, Classification of maximal transitive prolongations of super-Poincaré algebras, Adv. Math. 265 (2014) 60.
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de Medeiros, P., Figueroa-O’Farrill, J. & Santi, A. Killing superalgebras for Lorentzian four-manifolds. J. High Energ. Phys. 2016, 106 (2016). https://doi.org/10.1007/JHEP06(2016)106
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DOI: https://doi.org/10.1007/JHEP06(2016)106