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Supersymmetry of the C-metric and the general Plebanski-Demianski solution

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Abstract

We derive the necessary and sufficient conditions under which the general Plebanski-Demianski (PD) solution of Einstein-Maxwell theory with a negative cosmological constant admits Killing spinors. We consider in detail two different scaling limits of the PD metric. The first of these limits removes the acceleration parameter, and leads to the Carter-Plebanski solution. In this case, the integrability conditions for Killing spinors were obtained by Alonso-Alberca, Meessen and Ortín in hep-th/0003071, and we show that these are not only necessary, but also sufficient for the existence of Killing spinors. This fills also a gap in hep-th/9808097, where the integrability conditions for supersymmetry of the Kerr-Newman-AdS black hole were worked out, but the Killing spinor was not constructed explicitely. The second scaling limit eliminates the rotation parameter, and leads to the cosmological C-metric, which describes accelerated black holes in AdS. Also in this case, the supersymmetry conditions are obtained, and it is shown that they follow from the ones of the general PD solution by scaling the parameters appropriately. In all cases, we determine the three-dimensional base space that appears in the classification scheme of hep-th/0307022, and prove that for the 1/2-supersymmetric Reissner-Nordström-AdS spacetime, this base is unique. A Wick-rotation of our results leads to gravitational instantons that generalize the ones constructed recently by Martelli, Passias and Sparks in arXiv:1212.4618 to U(1) × U(1) symmetry. These instantons are shown to admit an integrable almost complex structure. Finally, our work may open the possibility to systematically construct generalizations of the PD metric that include scalar fields with a potential in matter-coupled gauged supergravity.

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References

  1. J.P. Gauntlett, N. Kim, D. Martelli and D. Waldram, Five-branes wrapped on SLAG three cycles and related geometry, JHEP 11 (2001) 018 [hep-th/0110034] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. J.P. Gauntlett, D. Martelli, S. Pakis and D. Waldram, G structures and wrapped NS5-branes, Commun. Math. Phys. 247 (2004) 421 [hep-th/0205050] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. 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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. J.P. Gauntlett and J.B. Gutowski, All supersymmetric solutions of minimal gauged supergravity in five-dimensions, Phys. Rev. D 68 (2003) 105009 [Erratum ibid. D 70 (2004) 089901] [hep-th/0304064] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  5. J.B. Gutowski and H.S. Reall, General supersymmetric AdS 5 black holes, JHEP 04 (2004) 048 [hep-th/0401129] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. J.B. Gutowski and W. Sabra, General supersymmetric solutions of five-dimensional supergravity, JHEP 10 (2005) 039 [hep-th/0505185] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. J. Bellorın, P. Meessen and T. Ortín, All the supersymmetric solutions of N = 1, D = 5 ungauged supergravity, JHEP 01 (2007) 020 [hep-th/0610196] [INSPIRE].

    Article  ADS  Google Scholar 

  8. J. Bellorın and T. Ortín, Characterization of all the supersymmetric solutions of gauged N = 1, D = 5 supergravity, JHEP 08 (2007) 096[arXiv:0705.2567] [INSPIRE].

    Article  ADS  Google Scholar 

  9. J.B. Gutowski, D. Martelli and H.S. Reall, All supersymmetric solutions of minimal supergravity in six-dimensions, Class. Quant. Grav. 20 (2003) 5049 [hep-th/0306235] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. J.P. Gauntlett and S. Pakis, The geometry of D = 11 Killing spinors, JHEP 04 (2003) 039 [hep-th/0212008] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. J.P. Gauntlett, J.B. Gutowski and S. Pakis, The geometry of D = 11 null Killing spinors, JHEP 12 (2003) 049 [hep-th/0311112] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. M.M. Caldarelli and D. Klemm, All supersymmetric solutions of N = 2, D = 4 gauged supergravity, JHEP 09 (2003) 019 [hep-th/0307022] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. J. Bellorın and T. Ortín, All the supersymmetric configurations of N = 4, D = 4 supergravity, Nucl. Phys. B 726 (2005) 171 [hep-th/0506056] [INSPIRE].

    Article  ADS  Google Scholar 

  14. P. Meessen, T. Ortín and S. Vaulà, All the timelike supersymmetric solutions of all ungauged D = 4 supergravities, JHEP 11 (2010) 072[arXiv:1006.0239] [INSPIRE].

    Article  ADS  Google Scholar 

  15. P. Meessen and T. Ortín, Supersymmetric solutions to gauged N = 2 D = 4 SUGRA: the full timelike shebang, Nucl. Phys. B 863 (2012) 65 [arXiv:1204.0493] [INSPIRE].

    Article  ADS  Google Scholar 

  16. U. Gran, J. Gutowski and G. Papadopoulos, The spinorial geometry of supersymmetric IIB backgrounds, Class. Quant. Grav. 22 (2005) 2453 [hep-th/0501177] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  17. U. Gran, J. Gutowski, G. Papadopoulos and D. Roest, Systematics of IIB spinorial geometry, Class. Quant. Grav. 23 (2006) 1617 [hep-th/0507087] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. S.L. Cacciatori, M.M. Caldarelli, D. Klemm, D.S. Mansi and D. Roest, Geometry of four-dimensional Killing spinors, JHEP 07 (2007) 046 [arXiv:0704.0247] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. J.B. Gutowski and W.A. Sabra, Half-supersymmetric solutions in five-dimensional supergravity, JHEP 12 (2007) 025 [Erratum ibid. 04 (2010) 042] [arXiv:0706.3147] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. J. Grover, J.B. Gutowski and W. Sabra, Null half-supersymmetric solutions in five-dimensional supergravity, JHEP 10 (2008) 103 [arXiv:0802.0231] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. S.L. Cacciatori, D. Klemm, D.S. Mansi and E. Zorzan, All timelike supersymmetric solutions of N = 2, D = 4 gauged supergravity coupled to abelian vector multiplets, JHEP 05 (2008) 097 [arXiv:0804.0009] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. D. Klemm and E. Zorzan, All null supersymmetric backgrounds of N = 2, D = 4 gauged supergravity coupled to abelian vector multiplets, Class. Quant. Grav. 26 (2009) 145018 [arXiv:0902.4186] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  23. D. Klemm and E. Zorzan, The timelike half-supersymmetric backgrounds of N = 2, D = 4 supergravity with Fayet-Iliopoulos gauging, Phys. Rev. D 82 (2010) 045012 [arXiv:1003.2974] [INSPIRE].

    ADS  Google Scholar 

  24. K.-I. Maeda and M. Nozawa, Black hole solutions in string theory, Prog. Theor. Phys. Suppl. 189 (2011) 310 [arXiv:1104.1849] [INSPIRE].

    Article  ADS  Google Scholar 

  25. K. Tod, All metrics admitting supercovariantly constant spinors, Phys. Lett. B 121 (1983) 241 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  26. S.L. Cacciatori, M.M. Caldarelli, D. Klemm and D.S. Mansi, More on BPS solutions of N = 2, D = 4 gauged supergravity, JHEP 07 (2004) 061 [hep-th/0406238] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. L. Romans, Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory, Nucl. Phys. B 383 (1992) 395 [hep-th/9203018] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. M.M. Caldarelli and D. Klemm, Supersymmetry of anti-de Sitter black holes, Nucl. Phys. B 545 (1999) 434 [hep-th/9808097] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. N. Alonso-Alberca, P. Meessen and T. Ortín, Supersymmetry of topological Kerr-Newman-Taub-NUT-AdS space-times, Class. Quant. Grav. 17 (2000) 2783 [hep-th/0003071] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  30. P. van Nieuwenhuizen and N. Warner, Integrability conditions for Killing spinors, Commun. Math. Phys. 93 (1984) 277 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  31. J. Plebanski and M. Demianski, Rotating, charged and uniformly accelerating mass in general relativity, Annals Phys. 98 (1976) 98 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].

    MATH  Google Scholar 

  33. J.F. Plebanski, A class of solutions of Einstein-Maxwell equations, Annals Phys. 90 (1975) 196.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. D. Martelli, A. Passias and J. Sparks, The supersymmetric NUTs and bolts of holography, arXiv:1212.4618 [INSPIRE].

  35. G. Gibbons, C. Hull and N. Warner, The stability of gauged supergravity, Nucl. Phys. B 218 (1983) 173 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. V.A. Kostelecký and M.J. Perry, Solitonic black holes in gauged N = 2 supergravity, Phys. Lett. B 371 (1996) 191 [hep-th/9512222] [INSPIRE].

    ADS  Google Scholar 

  37. L. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].

    Article  ADS  Google Scholar 

  38. G. Dibitetto and D. Klemm, Magnetic charges in the AdS 4 superalgebra osp(4|2), JHEP 12 (2010) 005 [arXiv:1005.4334] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. K. Hristov, C. Toldo and S. Vandoren, On BPS bounds in D = 4 N = 2 gauged supergravity, JHEP 12 (2011) 014 [arXiv:1110.2688] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. J. Griffiths and J. Podolský, A new look at the Plebanski-Demianski family of solutions, Int. J. Mod. Phys. D 15 (2006) 335 [gr-qc/0511091] [INSPIRE].

    ADS  Google Scholar 

  41. O.J. Dias and J.P. Lemos, Pair of accelerated black holes in anti-de Sitter background: AdS C-metric, Phys. Rev. D 67 (2003) 064001 [hep-th/0210065] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  42. J.B. Gutowski and H.S. Reall, Supersymmetric AdS 5 black holes, JHEP 02 (2004) 006 [hep-th/0401042] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. M. Dunajski, J. Gutowski, W. Sabra and P. Tod, Cosmological Einstein-Maxwell instantons and Euclidean supersymmetry: anti-self-dual solutions, Class. Quant. Grav. 28 (2011) 025007 [arXiv:1006.5149] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. J. Gutowski and W. Sabra, Gravitational instantons and Euclidean supersymmetry, Phys. Lett. B 693 (2010) 498 [arXiv:1007.2421] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  45. M. Dunajski, J. Gutowski, W. Sabra and P. Tod, Cosmological Einstein-Maxwell instantons and Euclidean supersymmetry: beyond self-duality, JHEP 03 (2011) 131 [arXiv:1012.1326] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on curved spaces and holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. 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].

    Article  MathSciNet  ADS  Google Scholar 

  48. D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].

    Article  ADS  Google Scholar 

  49. N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. L. Mason and A. Taghavi-Chabert, Killing-Yano tensors and multi-Hermitian structures, J. Geom. Phys. 60 (2010) 907 [arXiv:0805.3756].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. D. Klemm, Rotating BPS black holes in matter-coupled AdS 4 supergravity, JHEP 07 (2011) 019 [arXiv:1103.4699] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. M. Colleoni and D. Klemm, Nut-charged black holes in matter-coupled N = 2, D = 4 gauged supergravity, Phys. Rev. D 85 (2012) 126003 [arXiv:1203.6179] [INSPIRE].

    ADS  Google Scholar 

  54. F. Dowker, J.P. Gauntlett, D.A. Kastor and J.H. Traschen, Pair creation of dilaton black holes, Phys. Rev. D 49 (1994) 2909 [hep-th/9309075] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  55. C. Charmousis, T. Kolyvaris and E. Papantonopoulos, Charged C-metric with conformally coupled scalar field, Class. Quant. Grav. 26 (2009) 175012 [arXiv:0906.5568] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. A. Anabalón and H. Maeda, New charged black holes with conformal scalar hair, Phys. Rev. D 81 (2010) 041501 [arXiv:0907.0219] [INSPIRE].

    ADS  Google Scholar 

  57. A. Anabalón, Exact black holes and universality in the backreaction of non-linear σ-models with a potential in (A)dS 4, JHEP 06 (2012) 127 [arXiv:1204.2720] [INSPIRE].

    Article  ADS  Google Scholar 

  58. R. Emparan and H.S. Reall, A rotating black ring solution in five-dimensions, Phys. Rev. Lett. 88 (2002) 101101 [hep-th/0110260] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. M.M. Caldarelli, R. Emparan and M.J. Rodriguez, Black rings in (anti)-de Sitter space, JHEP 11 (2008) 011 [arXiv:0806.1954] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Milano, Via Celoria 16, 20133, Milano, Italy

    Dietmar Klemm

  2. INFN, Sezione di Milano, Via Celoria 16, 20133, Milano, Italy

    Dietmar Klemm

  3. Theory Center, IPNS, KEK, 1-1, Oho, Tsukuba, Ibaraki, 305-0801, Japan

    Masato Nozawa

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  1. Dietmar Klemm
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ArXiv ePrint: 1303.3119

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Klemm, D., Nozawa, M. Supersymmetry of the C-metric and the general Plebanski-Demianski solution. J. High Energ. Phys. 2013, 123 (2013). https://doi.org/10.1007/JHEP05(2013)123

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