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Obstructions to the Existence of Sasaki–Einstein Metrics

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Abstract

We describe two simple obstructions to the existence of Ricci-flat Kähler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kähler–Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R–charge of a gauge invariant chiral primary operator violates the unitarity bound.

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References

  1. Maldacena, J. M.: “The large N limit of superconformal field theories and supergravity.” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]

  2. Kehagias A. (1998). “New type IIB vacua and their F-theory interpretation”. Phys. Lett. B 435: 337

    Article  MathSciNet  ADS  Google Scholar 

  3. Klebanov I.R., Witten E. (1998). “Superconformal field theory on threebranes at a Calabi-Yau singularity”. Nucl. Phys. B 536: 199

    Article  MathSciNet  ADS  Google Scholar 

  4. Acharya B.S., Figueroa-O’Farrill J.M., Hull C.M., Spence B. (1999). “Branes at conical singularities and holography”. Adv. Theor. Math. Phys. 2: 1249

    MathSciNet  Google Scholar 

  5. Morrison D.R., Plesser M.R. (1999). “Non-spherical horizons. I”. Adv. Theor. Math. Phys. 3: 1

    MATH  MathSciNet  Google Scholar 

  6. Boyer, C.P., Galicki, K.: “Sasakian Geometry, Hypersurface Singularities, and Einstein Metrics.” Supplemento ai Rendiconti del Circolo Matematico di Palermo Serie II. Suppl 75, 57–87 (2005)

    Google Scholar 

  7. Matsushima Y. (1957). “Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kaehlérienne”. Nagoya Math. J. 11: 145–150

    MATH  MathSciNet  Google Scholar 

  8. Futaki A. (1983). “An obstruction to the existence of Einstein Kähler metrics”. Invent. Math. 73: 437–443

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Martelli, D., Sparks, J., Yau, S.-T.: “Sasaki–Einstein Manifolds and Volume Minimisation,” http://arxiv.org/list/hep-th/0603021

  10. Martelli D., Sparks J. (2006). “Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals”. Commun. Math. Phys. 262: 51

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Yau S.-T. (1993). “Open problems in Geometry”. Proc. Symp. Pure Math. 54: 1–28

    Google Scholar 

  12. Donaldson S.K. (1999). “Symmetric spaces, Kähler geometry and Hamiltonian dynamics”. Amer. Math. Soc. Transl. 196: 13–33

    MathSciNet  Google Scholar 

  13. Martelli D., Sparks J., Yau S.-T. (2006). “The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds”. Commun. Math. Phys. 268: 39–65

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Intriligator K., Wecht B. (2003). “The exact superconformal R-symmetry maximizes a”. Nucl. Phys. B 667: 183

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Bishop, R.L., Crittenden, R.J.: “Geometry of manifolds.” New York: Academic Press, 1964

  16. Besse, A.L.: “Einstein Manifolds.” Berlin-Heidelberg-New York: Springer–Verlag, 2nd edition, 1987

  17. Lichnerowicz, A.: “Géometrie des groupes de transformations.” Paris: Dunod, 1958

  18. Cachazo F., Fiol B., Intriligator K.A., Katz S., Vafa C. (2002). “A geometric unification of dualities”. Nucl. Phys. B 628: 3

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Conti, D.: “Cohomogeneity one Einstein-Sasaki 5-manifolds.” http://arxiv.org/list/math.DG/0606323, 2006

  20. Gukov, S., Vafa, C., Witten, E.: “CFT’s from Calabi-Yau four-folds.” Nucl. Phys. B 584, 69 (2000) [Erratum-ibid. B 608, 477 (2001)]

    Google Scholar 

  21. Obata M. (1962). “Certain conditions for a Riemannian manifold to be isometric to a sphere”. J. Math. Soc. Japan 14: 333–340

    Article  MATH  MathSciNet  Google Scholar 

  22. Minakshisundaram S., Pleijel A. (1949). “Some Properties of the Eigenfunctions of the Laplace–Operator on Riemannian Manifolds”. Can. J. Math. 1: 242–256

    MATH  MathSciNet  Google Scholar 

  23. Kollár, J.: “Rational Curves on Algebraic Varieties.” Berlin-Heidelberg-New York: Springer–Verlag. Ergebnisse der Math. Vol 32, 1996

  24. Hwang J.-M. (2003). “On the degrees of Fano four-folds of Picard number 1”. J. Reine Angew. Math. 556: 225–235

    MATH  MathSciNet  Google Scholar 

  25. Gubser S., Nekrasov N., Shatashvili S. (1999). “Generalized conifolds and four dimensional N = 1 superconformal theories”. JHEP 9905: 003

    Article  MathSciNet  ADS  Google Scholar 

  26. Gubser S.S., Klebanov I.R., Polyakov A.M. (1998). “Gauge theory correlators from non-critical string theory”. Phys. Lett. B 428: 105

    Article  MathSciNet  ADS  Google Scholar 

  27. Witten E. (1998). “Anti-de Sitter space and holography”. Adv. Theor. Math. Phys. 2: 253

    MATH  MathSciNet  Google Scholar 

  28. Klebanov I.R., Witten E. (1999). “AdS/CFT correspondence and symmetry breaking”. Nucl. Phys. B 556: 89

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Balasubramanian V., Kraus P., Lawrence A.E. (1999). “Bulk vs. boundary dynamics in anti-de Sitter spacetime”. Phys. Rev. D 59: 046003

    Article  MathSciNet  ADS  Google Scholar 

  30. Ceresole A., Dall’Agata G., D’Auria R., Ferrara S. (2000). “Spectrum of type IIB supergravity on AdS(5) x T(11): Predictions on N = 1 SCFT’s”. Phys. Rev. D 61: 066001

    Article  MathSciNet  ADS  Google Scholar 

  31. Fabbri D., Fre P., Gualtieri L., Termonia P. (1999). “M-theory on AdS(4) x M(111): The complete Osp(2|4) x SU(3) x SU(2) spectrum from harmonic analysis”. Nucl. Phys. B 560: 617

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. Fabbri D., Fre’ P., Gualtieri L., Reina C., Tomasiello A., Zaffaroni A., Zampa A. (2000). “3D superconformal theories from Sasakian seven-manifolds: New nontrivial evidences for AdS(4)/CFT(3)”. Nucl. Phys. B 577: 547

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. Kim H.J., Romans L.J., Nieuwenhuizen P. (1985). “The Mass Spectrum Of Chiral N=2 D = 10 Supergravity On S 5”. Phys. Rev. D 32: 389

    Article  MathSciNet  ADS  Google Scholar 

  34. Castellani L., D’Auria R., Fre P., Pilch K., Nieuwenhuizen P. (1984). “The Bosonic Mass Formula For Freund-Rubin Solutions Of D = 11 Supergravity On General Coset Manifolds”. Class. Quant. Grav. 1: 339

    Article  ADS  Google Scholar 

  35. D’Auria R., Fre P. (1985). “Universal Bose-Fermi Mass Relations In: Kaluza-Klein Supergravity And Harmonic Analysis On Coset Manifolds With Killing Spinors”. Annals Phys. 162: 372

    Article  MathSciNet  ADS  Google Scholar 

  36. Henningson M., Skenderis K. (1998). “The holographic Weyl anomaly”. JHEP 9807: 023

    Article  MathSciNet  ADS  Google Scholar 

  37. Gubser S.S. (1999). “Einstein manifolds and conformal field theories”. Phys. Rev. D 59: 025006

    Article  MathSciNet  ADS  Google Scholar 

  38. Gauntlett J.P., Martelli D., Sparks J., Waldram D. (2006). “Supersymmetric AdS(5) solutions of type IIB supergravity”. Class. Quant. Grav. 23: 4693

    Article  MATH  MathSciNet  ADS  Google Scholar 

  39. Randell, R.C.: “The homology of generalized Brieskorn manifolds.” Topology 14, no. 4, 347–355 (1975)

  40. Bergman, A., Herzog, C.P.: “The Volume of some Non-spherical Horizons and the AdS/CFT Correspondence.” JHEP 0201, 030 (2002)

  41. Nekrasov N., Shadchin S. (2004). “ABCD of instantons”. Commun. Math. Phys. 252: 359

    Article  MATH  MathSciNet  ADS  Google Scholar 

  42. Smith I., Thomas R.P. (2003). “Symplectic surgeries from singularities”. Turkish J. Math. 27: 231–250

    MATH  MathSciNet  Google Scholar 

  43. Gauntlett J.P., Martelli D., Sparks J., Waldram D. (2004). “Sasaki-Einstein metrics on S 2 × S 3”. Adv. Theor. Math. Phys. 8: 711

    MATH  MathSciNet  Google Scholar 

  44. Gauntlett J.P., Martelli D., Sparks J., Waldram D. (2004). “Supersymmetric AdS5 solutions of M-theory”. Class. Quant. Grav. 21: 4335

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. Dancer A.S., Strachan I.A.B. (1994). “Kähler–Einstein metrics with SU(2) action”. Math. Proc. Camb. Phil. Soc. 115: 513

    Article  MATH  MathSciNet  Google Scholar 

  46. Kollár J. (1989). “Flops”. Nagoya Math. J. 113: 15–36

    MATH  MathSciNet  Google Scholar 

  47. Laufer, H.B.: “On \({\mathbb{CP}}^1\) as an exceptional set.” In: Recent developments in several complex variables, Tokyo/Princeton, NJ: Princeton University Press and University of Tokyo Press, 1981

  48. Corrado R., Halmagyi N. (2005). “N = 1 field theories and fluxes in IIB string theory”. Phys. Rev. D 71: 046001

    Article  MathSciNet  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Blackett Laboratory, Imperial College, London, SW7 2AZ, U.K.

    Jerome P. Gauntlett

  2. The Institute for Mathematical Sciences, Imperial College, London, SW7 2PG, U.K.

    Jerome P. Gauntlett

  3. Department of Physics, CERN Theory Unit, 1211, Geneva 23, Switzerland

    Dario Martelli

  4. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA, 02138, U.S.A.

    James Sparks & Shing-Tung Yau

  5. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, U.S.A.

    James Sparks

Authors
  1. Jerome P. Gauntlett
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  2. Dario Martelli
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  3. James Sparks
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  4. Shing-Tung Yau
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Correspondence to James Sparks.

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Communicated by G.W. Gibbons

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Gauntlett, J.P., Martelli, D., Sparks, J. et al. Obstructions to the Existence of Sasaki–Einstein Metrics. Commun. Math. Phys. 273, 803–827 (2007). https://doi.org/10.1007/s00220-007-0213-7

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  • DOI: https://doi.org/10.1007/s00220-007-0213-7

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