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Supersymmetry in Lorentzian Curved Spaces and Holography


We consider superconformal and supersymmetric field theories on four-dimensional Lorentzian curved space-times, and their five-dimensional holographic duals. As in the Euclidean signature case, preserved supersymmetry for a superconformal theory is equivalent to the existence of a charged conformal Killing spinor. Differently from the Euclidean case, we show that the existence of such spinors is equivalent to the existence of a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, this vector field is further restricted to be Killing. We demonstrate how these results agree with the existing classification of supersymmetric solutions of minimal gauged supergravity in five dimensions.

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  1. Pestun, V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71 (2012). arXiv:0712.2824

  2. Kapustin, A., Willett, B., Yaakov, I.: Exact results for Wilson Loops in superconformal Chern–Simons theories with matter. JHEP 03, 089 (2010). arXiv:0909.4559

  3. Jafferis, D.L.: The exact superconformal R-symmetry extremizes Z. JHEP 1205, 159 (2012). arXiv:1012.3210

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

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

  6. Imamura, Y., Yokoyama, D.: \({{\mathcal{N}}=2}\) supersymmetric theories on squashed three-sphere. Phys. Rev. D85, 025015 (2012). arXiv:1109.4734

  7. Martelli, D., Passias, A., Sparks, J.: The gravity dual of supersymmetric gauge theories on a squashed three-sphere. Nucl. Phys. B 864, 840 (2012). arXiv:1110.6400

  8. Martelli, D., Sparks, J.: The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere. Nucl. Phys. B 866, 72 (2013). arXiv:1111.6930

  9. Gauntlett, J.P., Gutowski, J.B.: All supersymmetric solutions of minimal gauged supergravity in five dimensions. Phys. Rev. D68, 105009 (2003). arXiv:hep-th/0304064

  10. Behrndt, K., Klemm, D.: Black holes in Goedel-type universes with a cosmological constant. Class. Quant. Grav. 21, 4107–4122 (2004). arXiv:hep-th/0401239

  11. Gauntlett, J.P., Gutowski, J.B., Suryanarayana, N.V.: A deformation of \({{ {\rm AdS}}_5 \times S^5}\) . Class. Quant. Grav. 21, 5021ΓÇô5034 (2004). arXiv:hep-th/0406188

  12. Klare, C., Tomasiello, A., Zaffaroni, A.: Supersymmetry on curved spaces and holography. JHEP 1208, 061 (2012). arXiv:1205.1062

  13. Dumitrescu, T.T., Festuccia, G., Seiberg, N.: Exploring curved superspace. JHEP 1208, 141 (2012). arXiv:1205.1115

  14. Keck B.: An alternative class of supersymmetries. J. Phys. A A8, 1819–1827 (1975)

    ADS  Article  MathSciNet  Google Scholar 

  15. Zumino B.: Nonlinear realization of supersymmetry in de Sitter space. Nucl. Phys. B127, 189 (1977)

    ADS  Article  Google Scholar 

  16. Ivanov E., Sorin A.S.: Superfield formulation of OSp(1, 4) supersymmetry. J. Phys. A A13, 1159–1188 (1980)

    ADS  Article  MathSciNet  Google Scholar 

  17. Adams, A., Jockers, H., Kumar, V., Lapan, J.M.: N = 1 sigma models in AdS4. JHEP 1112, 042 (2011). arXiv:1104.3155

  18. Jia, B., Sharpe, E.: Rigidly supersymmetric gauge theories on curved superspace. JHEP 1204, 139 (2012). arXiv:1109.5421

  19. Buchbinder I.L., Kuzenko S.M.: Ideas and Methods of Supersymmetry and Supergravity, or a Walk Through Superspace. IOP Publishing Ltd, Bristol (1995)

    Book  MATH  Google Scholar 

  20. Festuccia, G., Seiberg, N.: Rigid supersymmetric theories in curved superspace. JHEP 06, 114 (2011). arXiv:1105.0689

  21. Kaku M., Townsend P., van Nieuwenhuizen P.: Superconformal unified field theory. Phys. Rev. Lett. 39, 1109 (1977)

    ADS  Article  Google Scholar 

  22. Kaku M., Townsend P., van Nieuwenhuizen P.: Gauge theory of the conformal and superconformal group. Phys. Lett. B69, 304–308 (1977)

    ADS  Article  Google Scholar 

  23. Kaku M., Townsend P.K., van Nieuwenhuizen P.: Properties of conformal supergravity. Phys. Rev. D17, 3179 (1978)

    ADS  Google Scholar 

  24. Ferrara S., Zumino B.: Structure of conformal supergravity. Nucl. Phys. B134, 301 (1978)

    ADS  Article  MathSciNet  Google Scholar 

  25. Sohnius M.F., West P.C.: An alternative minimal off-shell version of \({{\mathcal{N}}=1}\) supergravity. Phys. Lett. B105, 353 (1981)

    ADS  Article  Google Scholar 

  26. Van Proeyen, A.: Superconformal tensor calculus in \({{\mathcal{N}}=1}\) and \({{\mathcal{N}}=2}\) supergravity. In: Proceedings of Karpacz Winter School, Karpacz, Poland, Feb 14–26 (1983)

  27. Das A., Kaku M., Townsend P.K.: A unified approach to matter coupling in Weyl and Einstein supergravity. Phys. Rev. Lett. 40, 1215 (1978)

    ADS  Article  Google Scholar 

  28. Penrose R., Rindler W.: Spinors and Space-Time: Spinor and Twistor Methods in Space-Time Geometry, vol. 2. Cambridge University Press, London (1988)

    Google Scholar 

  29. Lewandowski J.: Twistor equation in a curved spacetime. Class. Quantum Gravity 8(1), L11–L17 (1991)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  30. Tomasiello, A.: Generalized structures of ten-dimensional supersymmetric solutions. JHEP 1203, 073 (2012). arXiv:1109.2603

  31. Gutowski, J.B., Martelli, D., Reall, H.S.: All supersymmetric solutions of minimal supergravity in six dimensions. Class. Quantum Gravity 20, 5049–5078 (2003). arXiv:hep-th/0306235

  32. Baum, H.: Conformal Killing spinors and special geometric structures in Lorentzian geometry—a survey. arXiv:math/0202008

  33. Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245(3), 503–527 (2003). arXiv:math/0206117

  34. Walker M., Penrose R.: On quadratic first integrals of the geodesic equations for type {22} spacetimes. Comm. Math. Phys. 18, 265–274 (1970)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  35. Sohnius M., West P.C.: The tensor calculus and matter coupling of the alternative minimal auxiliary field formulation of \({{\mathcal{N}}=1}\) supergravity. Nucl. Phys. B198, 493 (1982)

    ADS  Article  MathSciNet  Google Scholar 

  36. Leitner, F.: About twistor spinors with zero in Lorentzian geometry. SIGMA Symmetry Integrability Geom. Methods Appl. 5, Paper 079, 12 (2009)

    Google Scholar 

  37. Fefferman C.L.: Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. 103(3), 395–416 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  38. Lee J.M.: The Fefferman metric and pseudohermitian invariants. Trans. Am. Math. Soc. 296(1), 411–429 (1986)

    MATH  Google Scholar 

  39. Chamseddine, A.H., Sabra, W.: Magnetic strings in five-dimensional gauged supergravity theories. Phys. Lett. B 477, 329–334 (2000). arXiv:hep-th/9911195

    Google Scholar 

  40. Klemm, D., Sabra, W.: Supersymmetry of black strings in D =  5 gauged supergravities. Phys. Rev. D62, 024003 (2000). arXiv:hep-th/0001131

  41. Samtleben H., Tsimpis D.: Rigid supersymmetric theories in 4d Riemannian space. JHEP 1205, 132 (2012). arXiv:1203.3420

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Correspondence to Davide Cassani.

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Communicated by N. A. Nekrasov

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Cassani, D., Klare, C., Martelli, D. et al. Supersymmetry in Lorentzian Curved Spaces and Holography. Commun. Math. Phys. 327, 577–602 (2014).

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  • Conformal Killing
  • Superconformal Theory
  • Supersymmetric Solution
  • Conformal Supergravity
  • Minimal Supergravity