Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 058, 15 pages      arXiv:1205.6036
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Hidden Symmetries of Euclideanised Kerr-NUT-(A)dS Metrics in Certain Scaling Limits

Mihai Visinescu a and Eduard Vîlcu b, c
a) National Institute for Physics and Nuclear Engineering, Department of Theoretical Physics, P.O. Box M.G.-6, Magurele, Bucharest, Romania
b) Petroleum-Gas University of Ploieşti, Department of Mathematical Economics, Bulevardul Bucureşti, Nr. 39, Ploieşti 100680, Romania
c) University of Bucharest, Faculty of Mathematics and Computer Science, Research Center in Geometry, Topology and Algebra, Str. Academiei, Nr. 14, Sector 1, Bucharest 70109, Romania

Received May 29, 2012, in final form July 23, 2012; Published online August 27, 2012

The hidden symmetries of higher dimensional Kerr-NUT-(A)dS metrics are investigated. In certain scaling limits these metrics are related to the Einstein-Sasaki ones. The complete set of Killing-Yano tensors of the Einstein-Sasaki spaces are presented. For this purpose the Killing forms of the Calabi-Yau cone over the Einstein-Sasaki manifold are constructed. Two new Killing forms on Einstein-Sasaki manifolds are identified associated with the complex volume form of the cone manifolds. Finally the Killing forms on mixed 3-Sasaki manifolds are briefly described.

Key words: Killing forms; Einstein-Sasaki space; Calabi-Yau spaces.

pdf (389 kb)   tex (26 kb)


  1. Acharya B.S., Figueroa-O'Farrill J.M., Hull C.M., Spence B., Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1998), 1249-1286, hep-th/9808014.
  2. Agricola I., Friedrich T., Killing spinors in supergravity with 4-fluxes, Classical Quantum Gravity 20 (2003), 4707-4717, math.DG/0307360.
  3. Alekseevsky D.V., Cortés V., Galaev A.S., Leistner T., Cones over pseudo-Riemannian manifolds and their holonomy, J. Reine Angew. Math. 635 (2009), 23-69, arXiv:0707.3063.
  4. Ballmann W., Lectures on Kähler manifolds, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2006.
  5. Bär C., Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509-521.
  6. Besse A.L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 10, Springer-Verlag, Berlin, 1987.
  7. Blair D.E., Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin, 1976.
  8. Blažć N., Paraquaternionic projective space and pseudo-Riemannian geometry, Publ. Inst. Math. (Beograd) (N.S.) 60 (1996), 101-107.
  9. Boyer C., Galicki K., 3-Sasakian manifolds, in Surveys in Differential Geometry: Essays on Einstein Manifolds, Surv. Differ. Geom., Vol. VI, Int. Press, Boston, MA, 1999, 123-184, hep-th/9810250.
  10. Boyer C.P., Galicki K., Sasakian geometry, holonomy, and supersymmetry, in Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., Vol. 16, Eur. Math. Soc., Zürich, 2010, 39-83, math.DG/0703231.
  11. Boyer C.P., Galicki K., Mann B.M., The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183-220.
  12. Caldarella A.V., Pastore A.M., Mixed 3-Sasakian structures and curvature, Ann. Polon. Math. 96 (2009), 107-125, arXiv:0803.1953.
  13. Cariglia M., Krtouš P., Kubizňák D., Dirac equation in Kerr-NUT-(A)dS spacetimes: intrinsic characterization of separability in all dimensions, Phys. Rev. D 84 (2011), 024008, 11 pages, arXiv:1104.4123.
  14. Carter B., Killing tensor quantum numbers and conserved currents in curved space, Phys. Rev. D 16 (1977), 3395-3414.
  15. Carter B., McLenaghan R.G., Generalized total angular momentum operator for the Dirac equation in curved space-time, Phys. Rev. D 19 (1979), 1093-1097.
  16. Chen W., Lü H., Pope C.N., General Kerr-NUT-AdS metrics in all dimensions, Classical Quantum Gravity 23 (2006), 5323-5340, hep-th/0604125.
  17. Cortés V., Mayer C., Mohaupt T., Saueressig F., Special geometry of Euclidean supersymmetry. II. Hypermultiplets and the c-map, J. High Energy Phys. 2005 (2005), no. 6, 025, 37 pages, hep-th/0503094.
  18. Cvetič M., Lü H., Page D.N., Pope C.N., New Einstein-Sasaki and Einstein spaces from Kerr-de Sitter, J. High Energy Phys. 2009 (2009), no. 7, 082, 25 pages, hep-th/0505223.
  19. Dunajski M., West S., Anti-self-dual conformal structures in neutral signature, in Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, 113-148, math.DG/0610280.
  20. Emparan R., Reall H.S., Black holes in higher dimensions, Living Rev. Relativ. 11 (2008), 6, 87 pages, arXiv:0801.3471.
  21. García-Río E., Matsushita Y., Vázquez-Lorenzo R., Paraquaternionic Kähler manifolds, Rocky Mountain J. Math. 31 (2001), 237-260.
  22. Gibbons G.W., Hartnoll S.A., Pope C.N., Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons, Phys. Rev. D 67 (2003), 084024, 24 pages, hep-th/0208031.
  23. Gibbons G.W., Rychenkova P., Cones, tri-Sasakian structures and superconformal invariance, Phys. Lett. B 443 (1998), 138-142, hep-th/9809158.
  24. Houri T., Kubizňák D., Warnick C.M., Yasui Y., Generalized hidden symmetries and the Kerr-Sen black hole, J. High Energy Phys. 2010 (2010), no. 7, 055, 33 pages, arXiv:1004.1032.
  25. Houri T., Kubizňák D., Warnick C., Yasui Y., Symmetries of the Dirac operator with skew-symmetric torsion, Classical Quantum Gravity 27 (2010), 185019, 16 pages, arXiv:1002.3616.
  26. Hull C.M., Actions for (2,1) sigma models and strings, Nuclear Phys. B 509 (1998), 252-272, hep-th/9702067.
  27. Hull C.M., Duality and the signature of space-time, J. High Energy Phys. 1998 (1998), no. 11, 017, 36 pages, hep-th/9807127.
  28. Ianuş S., Mazzocco R., Vîlcu G.E., Real lightlike hypersurfaces of paraquaternionic Kähler manifolds, Mediterr. J. Math. 3 (2006), 581-592.
  29. Ianuş S., Vîlcu G.E., Paraquaternionic manifolds and mixed 3-structures, in Differential Geometry, World Sci. Publ., Hackensack, NJ, 2009, 276-285.
  30. Ianuş S., Vîlcu G.E., Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles, Int. J. Geom. Methods Mod. Phys. 5 (2008), 893-903, arXiv:0707.3360.
  31. Ianuş S., Visinescu M., Vîlcu G.E., Conformal Killing-Yano tensors on manifolds with mixed 3-structures, SIGMA 5 (2009), 022, 12 pages, arXiv:0902.3968.
  32. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Interscience Publishers, New York - London, 1963.
  33. Konishi M., On manifolds with Sasakian 3-structure over quaternion Kaehler manifolds, Kōdai Math. Sem. Rep. 26 (1974/75), 194-200.
  34. Krtouš P., Kubizňák D., Page D.N., Vasudevan M., Constants of geodesic motion in higher-dimensional black-hole spacetimes, Phys. Rev. D 76 (2007), 084034, 8 pages, arXiv:0707.0001.
  35. Kubizňák D., On the supersymmetric limit of Kerr-NUT-AdS metrics, Phys. Lett. B 675 (2009), 110-115, arXiv:0902.1999.
  36. Kubizňák D., Frolov V.P., Stationary strings and branes in the higher-dimensional Kerr-NUT-(A)dS spacetimes, J. High Energy Phys. 2008 (2008), no. 2, 007, 14 pages, arXiv:0711.2300.
  37. Kuo Y., On almost contact 3-structure, Tôhoku Math. J. 22 (1970), 325-332.
  38. Martelli D., Sparks J., Toric Sasaki-Einstein metrics on S2×S3, Phys. Lett. B 621 (2005), 208-212, hep-th/0505027.
  39. Ohnita Y., Stability and rigidity of special Lagrangian cones over certain minimal Legendrian orbits, Osaka J. Math. 44 (2007), 305-334.
  40. Ooguri H., Vafa C., Geometry of N=2 strings, Nuclear Phys. B 361 (1991), 469-518.
  41. Oota T., Yasui Y., Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime, Phys. Lett. B 659 (2008), 688-693, arXiv:0711.0078.
  42. Oota T., Yasui Y., Separability of gravitational perturbation in generalized Kerr-NUT-de Sitter space-time, Internat. J. Modern Phys. A 25 (2010), 3055-3094, arXiv:0812.1623.
  43. Page D.N., Kubizňák D., Vasudevan M., Krtouš P., Complete integrability of geodesic motion in general higher-dimensional rotating black-hole spacetimes, Phys. Rev. Lett. 98 (2007), 061102, 4 pages, hep-th/0611083.
  44. Sasaki S., On differentiable manifolds with certain structures which are closely related to almost contact structure. I, Tôhoku Math. J. 12 (1960), 459-476.
  45. Satō I., On a structure similar to the almost contact structure, Tensor (N.S.) 30 (1976), 219-224.
  46. Semmelmann U., Conformal Killing forms on Riemannian manifolds, Math. Z. 245 (2003), 503-527, math.DG/0206117.
  47. Sergyeyev A., Krtouš P., Complete set of commuting symmetry operators for the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetimes, Phys. Rev. D 77 (2008), 044033, 6 pages, arXiv:0711.4623.
  48. Sharfuddin A., Shahid M.H., Hypersurfaces of almost quaternion manifolds, Soochow J. Math. 20 (1994), 297-308.
  49. Sparks J., Sasaki-Einstein manifolds, in Geometry of Special Holonomy and Related Topics, Surv. Differ. Geom., Vol. 16, Int. Press, Somerville, MA, 2011, 265-324, arXiv:1004.2461.
  50. Vîlcu G.E., Voicu R.C., Curvature properties of pseudo-sphere bundles over paraquaternionic manifolds, Int. J. Geom. Methods Mod. Phys. 9 (2012), 1250024, 23 pages, arXiv:1107.5260.
  51. Yano K., Some remarks on tensor fields and curvature, Ann. of Math. (2) 55 (1952), 328-347.

Previous article  Next article   Contents of Volume 8 (2012)