New Results in Sasaki—Einstein Geometry

  • James Sparks
Part of the Progress in Mathematics book series (PM, volume 271)


This article is a summary of some of the author’s work on Sasaki–Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki–Einstein geometry, in low dimensions, to superconformal field theory; properties of the latter are therefore reflected in the former, and vice versa. Despite this physical motivation, many recent results are of independent geometrical interest and are described here in purely mathematical terms: explicit constructions of infinite families of both quasi-regular and irregular Sasaki–Einstein metrics; toric Sasakian geometry; an extremal problem that determines the Reeb vector field for, and hence also the volume of, a Sasaki–Einstein manifold; and finally, obstructions to the existence of Sasaki–Einstein metrics. Some of these results also provide new insights into Kähler geometry, and in particular new obstructions to the existence of Kähler–Einstein metrics on Fano orbifolds.


Toric Variety Ahler Manifold Einstein Metrics Einstein Manifold Reeb Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    V. Apostolov, D. M. Calderbank, P. Gauduchon, “The geometry of weakly selfdual Kähler surfaces,” Compositio Math. 135 (2003), 279–322.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Bergman, C. P. Herzog, “The volume of some non-spherical horizons and the AdS/CFT correspondence,” JHEP 0201, 030 (2002) [arXiv:hep-th/0108020].Google Scholar
  3. 3.
    R. L. Bishop, R. J. Crittenden, “Geometry of Manifolds,” Academic Press, New York, 1964.MATHGoogle Scholar
  4. 4.
    C. P. Boyer, K. Galicki, “On Sasakian–Einstein geometry,” Int. J. Math. 11 (2000), no. 7, 873–909 [arXiv:math.DG/9811098].MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    C. P. Boyer, K. Galicki, “A note on toric contact geometry,” J. Geom. Phys. 35 No. 4 (2000), 288–298, [arXiv:math.DG/9907043].MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. P. Boyer, K. Galicki, “Sasakian Geometry, Hypersurface Singularities, and Einstein Metrics,” Supplemento ai Rendiconti del Circolo Matematico di Palermo Serie II. Suppl 75 (2005), 57–87 [arXiv:math.DG/0405256].MathSciNetGoogle Scholar
  7. 7.
    C. P. Boyer, K. Galicki, “Sasakian Geometry,” Oxford Mathematical Monographs, Oxford University Press, ISBN-13: 978-0-19-856495-9.Google Scholar
  8. 8.
    C. P. Boyer, K. Galicki, S. R. Simanca, “Canonical Sasakian Metrics,” Commun. Math. Phys. 279 (2008), 705–733.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    E. Calabi, “Extremal Kähler metrics II,” in “Differential Geometry and Complex Analysis” (ed. I. Chavel and H. M. Farkas), Springer-Verlag, Berlin, 1985.Google Scholar
  10. 10.
    J. Cheeger, G. Tian, “On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay,” Invent. Math. 118 (1994), no. 3, 493–571.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    W. Chen, H. Lu, C. N. Pope, J. F. Vazquez-Poritz, “A note on Einstein–Sasaki metrics in D ≥ 7,” Class. Quant. Grav. 22 (2005), 3421–3430 [arXiv:hep-th/0411218].MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D. Conti, “Cohomogeneity one Einstein-Sasaki 5-manifolds,” arXiv:math.DG/0606323.Google Scholar
  13. 13.
    M. Cvetic, H. Lu, D. N. Page, C. N. Pope, “New Einstein-Sasaki spaces in five and higher dimensions,” Phys. Rev. Lett. 95, 071101 (2005) [arXiv:hep-th/0504225].CrossRefMathSciNetGoogle Scholar
  14. 14.
    M. Cvetic, H. Lu, D. N. Page, C. N. Pope, “New Einstein-Sasaki and Einstein spaces from Kerr-de Sitter,” arXiv:hep-th/0505223.Google Scholar
  15. 15.
    J. J. Duistermaat, G. Heckman, “On the variation in the cohomology of the symplectic form of the reduced space,” Inv. Math. 69, 259–268 (1982).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. J. Duistermaat, G. Heckman, Addendum, Inv. Math. 72, 153–158 (1983).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Falcao de Moraes, C. Tomei, “Moment maps on symplectic cones,” Pacific J. Math. 181 (2) (1997), 357–375.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Th. Friedrich, I. Kath, “Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator,” J. Differential Geom. 29 (1989), 263–279.MATHMathSciNetGoogle Scholar
  19. 19.
    A. Futaki, “An obstruction to the existence of Einstein Kähler metrics,” Invent. Math., 73 (1983), 437–443.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    A. Futaki, H. Ono, G. Wang, “Transverse Kähler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds,” arXiv:math.DG/0607586.Google Scholar
  21. 21.
    J. P. Gauntlett, D. Martelli, J. Sparks, D. Waldram, “Supersymmetric AdS5 solutions of M-theory,” Class. Quant. Grav. 21, 4335 (2004) [arXiv:hep-th/0402153].MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    J. P. Gauntlett, D. Martelli, J. Sparks, D. Waldram, “Sasaki–Einstein metrics on S2 ×S3,” Adv. Theor. Math. Phys. 8, 711 (2004) [arXiv:hep-th/0403002].MATHMathSciNetGoogle Scholar
  23. 23.
    J. P. Gauntlett, D. Martelli, J. F. Sparks, D. Waldram, “A new infinite class of Sasaki–Einstein manifolds,” Adv. Theor. Math. Phys. 8, 987 (2004) [arXiv:hep-th/0403038].MATHMathSciNetGoogle Scholar
  24. 24.
    J. P. Gauntlett, D. Martelli, J. Sparks, D. Waldram, “Supersymmetric AdS Backgrounds in String and M-theory,” IRMA Lectures in Mathematics and Theoretical Physics, vol. 8, published by the European Mathematical Society [arXiv:hep-th/0411194].Google Scholar
  25. 25.
    J. P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau, “Obstructions to the existence of Sasaki– Einstein metrics,” Commun. Math. Phys. 273, 803–827 (2007) [arXiv:hep-th/0607080].MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    A. Haefliger, “Groupoides d’holonomie et classifiants,” Astérisque 116 (1984), 70–97.MathSciNetGoogle Scholar
  27. 27.
    E. Kähler, “Ü ber eine bemerkenswerte Hermitesche Metrik,” Abh. Math. Sem. Hamburg Univ. 9 (1933), 173–186.CrossRefGoogle Scholar
  28. 28.
    J. Koll á r, Y. Miyaoka, S. Mori, “Rational connectedness and boundedness of Fano manifolds,” J. Diff. Geom. 36, 765–769 (1992).Google Scholar
  29. 29.
    E. Lerman, “Contact toric manifolds,” J. Symplectic Geom. 1 (2003), no. 4, 785–828 [arXiv:math.SG/0107201].MATHMathSciNetGoogle Scholar
  30. 30.
    A. Lichnerowicz, “Géometrie des groupes de transformations,” III, Dunod, Paris, 1958.Google Scholar
  31. 31.
    J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200].MATHMathSciNetGoogle Scholar
  32. 32.
    D. Martelli, J. Sparks, “Toric geometry, Sasaki–Einstein manifolds and a new infinite class of AdS/CFT duals,” Commun. Math. Phys. 262, 51 (2006) [arXiv:hep-th/0411238].MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    D. Martelli, J. Sparks, “Toric Sasaki–Einstein metrics on S 2 × S 3,” Phys. Lett. B 621, 208 (2005) [arXiv:hep-th/0505027].CrossRefMathSciNetGoogle Scholar
  34. 34.
    D. Martelli, J. Sparks, S.-T. Yau, “The geometric dual of a-maximisation for toric Sasaki– Einstein manifolds,” Commun. Math. Phys. 268, 39–65 (2006) [arXiv:hep-th/0503183].MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    D. Martelli, J. Sparks, S.-T. Yau, “Sasaki–Einstein Manifolds and Volume Minimisation,” Commun. Math. Phys. 280 (2008), 611–673 arXiv:hep-th/0603021.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    S. Minakshisundaram, A. Pleijel, “Some properties of the eigenfunctions of the Laplace– operator on Riemannian manifolds,” Canadian J. Math. 1 (1949), 242–256.MATHMathSciNetGoogle Scholar
  37. 37.
    S. B. Myers, “Riemannian manifolds with positive mean curvature,” Duke Math. J. 8 (1941), 401–404.CrossRefMathSciNetGoogle Scholar
  38. 38.
    M. Obata, “Certain conditions for a Riemannian manifold to be isometric to a sphere,” J. Math. Soc. Japan 14 (1962), 333–340.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    S. Sasaki, “On differentiable manifolds with certain structures which are closely related to almost contact structure,” Tôhoku Math. J. 2 (1960), 459–476.CrossRefGoogle Scholar
  40. 40.
    N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge theories,” Nucl. Phys. B 435, 129 (1995) [arXiv:hep-th/9411149].MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    S. Smale, “On the structure of 5-manifolds,” Ann. Math. 75 (1962), 38–46.CrossRefMathSciNetGoogle Scholar
  42. 42.
    S. Tanno, “Geodesic flows on C L-manifolds and Einstein metrics on S 3 × S 2,” in “Minimal Submanifolds and Geodesics” (Proc. Japan-United States Sem., Tokyo, 1977), pp. 283–292, North Holland, Amsterdam and New York, 1979.Google Scholar
  43. 43.
    G. Tian, “On Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0,” Invent. Math. 89 (1987) 225–246.MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    G. Tian, S.-T. Yau, “On Kähler–Einstein metrics on complex surfaces with C 1 > 0,” Commun. Math. Phys. 112 (1987) 175–203.MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    S.-T. Yau, “Open problems in geometry,” Proc. Symp. Pure Math. 54 (1993), 1–28.Google Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media LLC 2009

Authors and Affiliations

  • James Sparks
    • 1
    • 2
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA

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