Enhanced Euclidean supersymmetry, 11D supergravity and SU(∞) Toda equation

Article
First Online:
Received:
Revised:
Accepted:

Abstract

We show how to lift solutions of Euclidean Einstein-Maxwell equations with non-zero cosmological constant to solutions of eleven-dimensional supergravity theory with non-zero fluxes. This yields a class of 11D metrics given in terms of solutions to SU(∞) Toda equation. We give one example of a regular solution and analyse its supersymmetry.

We also analyse the integrability conditions of the Killing spinor equations of N = 2 minimal gauged supergravity in four Euclidean dimensions. We obtain necessary conditions for the existence of additional Killing spinors, corresponding to enhancement of supersymmetry. If the Weyl tensor is anti-self-dual then the supersymmetric metrics satisfying these conditions are given by separable solutions to the SU(∞) Toda equation. Otherwise they are ambi-Kähler and are conformally equivalent to Kähler metrics of Calabi type or to product metrics on two Riemann surfaces.

Keywords

Differential and Algebraic Geometry Supergravity Models 
This is a preview of subscription content, log in to check access.

References

  1. [1]
    V. Apostolov and P. Gauduchon, The Riemannian Goldberg-Sachs theorem, Int. J. Math. 8 (1997) 421.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    V. Apostolov, D.M.J. Calderbank and P. Gauduchon, Ambi-Kähler geometry, ambitoric surfaces and Einstein 4-orbifolds, arXiv:1010.0992.
  3. [3]
    C. Boyer and I. Finley, Killing vectors in selfdual, euclidean Einstein spaces, J. Math. Phys. 23 (1982) 1126 [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  4. [4]
    A. Derdzinski, Self-dual Kahler manifolds and Einstein manifolds of dimension four, Comp. Math. 49 (1983) 405.MathSciNetMATHGoogle Scholar
  5. [5]
    M. Dunajski, Solitons, instantons and twistors, Oxford Graduate Texts in Mathematics 19, Oxford University Press, Oxford U.K. (2009).Google Scholar
  6. [6]
    M. Dunajski and S.A. Hartnoll, Einstein-Maxwell gravitational instantons and five dimensional solitonic strings, Class. Quant. Grav. 24 (2007) 1841 [hep-th/0610261] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  7. [7]
    M. Dunajski, J.B. Gutowski, W.A. Sabra and K.P. Tod, Cosmological Einstein-Maxwell instantons and euclidean supersymmetry: anti-self-dual solutions, Class. Quant. Grav. 28 (2011) 025007 [arXiv:1006.5149] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    M. Dunajski, J.B. Gutowski, W.A. Sabra and K.P. Tod, Cosmological Einstein-Maxwell instantons and euclidean supersymmetry: beyond self-duality, JHEP 03 (2011) 131 [arXiv:1012.1326] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    M. Dunajski and P. Tod, Self-dual conformal gravity, arXiv:1304.7772 [INSPIRE].
  10. [10]
    J.M. Figueroa-O’Farrill and G. Papadopoulos, Maximally supersymmetric solutions of ten-dimensional and eleven-dimensional supergravities, JHEP 03 (2003) 048 [hep-th/0211089] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    J.P. Gauntlett and S. Pakis, The geometry of D = 11 Killing spinors, JHEP 04 (2003) 039 [hep-th/0212008] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    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].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    G. Gibbons and P. Rychenkova, Single sided domain walls in M-theory, J. Geom. Phys. 32 (2000) 311 [hep-th/9811045] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  14. [14]
    U. Gran, J. Gutowski, G. Papadopoulos and D. Roest, N = 31 is not IIB, JHEP 02 (2007) 044 [hep-th/0606049] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    U. Gran, G. Papadopoulos and D. Roest, Systematics of M-theory spinorial geometry, Class. Quant. Grav. 22 (2005) 2701 [hep-th/0503046] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  16. [16]
    J. Gutowski and W. Sabra, Gravitational instantons and euclidean supersymmetry, Phys. Lett. B 693 (2010) 498 [arXiv:1007.2421] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    J. Lukierski and W.J. Zakrzewski, Euclidean supersymmetrization of instantons and selfdual monopoles, Phys. Lett. B 189 (1987) 99 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    D. Martelli and J. Sparks, The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere, Nucl. Phys. B 866 (2013) 72 [arXiv:1111.6930] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    D. Martelli, A. Passias and J. Sparks, The gravity dual of supersymmetric gauge theories on a squashed three-sphere, Nucl. Phys. B 864 (2012) 840 [arXiv:1110.6400] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    D. Martelli, A. Passias and J. Sparks, The supersymmetric NUTs and bolts of holography, arXiv:1212.4618 [INSPIRE].
  21. [21]
    L. Martina, M.B. Sheftel and P. Winternitz, Group foliation and non-invariant solutions of the heavenly equation, J. Phys. A 34 (2001) 9243 [math-ph/0108004].MathSciNetADSGoogle Scholar
  22. [22]
    C. Pope, Eigenfunctions and Spinc structures in CP 2, Phys. Lett. B 97 (1980) 417 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    C.N. Pope, Consistency of truncations in Kaluza-Klein, in The Santa Fe meeting, T. Goldman and M.M. Nieto eds., World Scientific, Singapore (1984).Google Scholar
  24. [24]
    C.N. Pope, A. Sadrzadeh and S.R. Scuro, Timelike Hopf duality and type IIA ∗ string solutions, Class. Quant. Grav. 17 (2000) 623 [hep-th/9905161] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  25. [25]
    K.P. Tod, The SU(∞)-Toda field equation and special four-dimensional metrics, in Geometry and physics, J.E. Andersen et al. eds., New York, Marcel Dekker (1997).Google Scholar
  26. [26]
    K.P. Tod, Cohomogeneity-one metrics with self-dual Weyl tensor, in Twistor theory, S. Huggett ed., Dekker, New York U.S.A. (1995).Google Scholar
  27. [27]
    K.P. Tod, Scalar-flat Kähler and hyper-Kähler metrics from Painlevé III, Class. Quant. Grav 12 (1995) 1535.MathSciNetADSCrossRefMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeU.K
  2. 2.Department of MathematicsUniversity of SurreyGuildfordU.K
  3. 3.Centre for Advanced Mathematical Sciences and Physics DepartmentAmerican University of BeirutBeirutLebanon

Personalised recommendations