Skip to main content
SpringerLink
Log in

Sasaki–Einstein Manifolds and Volume Minimisation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat–Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kähler–Einstein metric.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Maldacena, J.M. (1999) 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  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Tian G. (1987). On Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0. Invent. Math. 89: 225–246

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Tian G. and Yau S.T. (1987). On Kähler–Einstein metrics on complex surfaces with C 1 > 0. Commun. Math. Phys. 112: 175–203

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  9. Gauntlett J.P., Martelli D., Sparks J. and Waldram D. (2004). Supersymmetric AdS(5) solutions of M-theory. Class. Quant. Grav. 21: 4335

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Gauntlett J.P., Martelli D., Sparks J.F. and Waldram D. (2006). A new infinite class of Sasaki-Einstein manifolds. Adv. Theor. Math. Phys. 8: 987

    MathSciNet  Google Scholar 

  12. Cvetic M., Lu H., Page D.N. and Pope C.N. (2005). New Einstein-Sasaki spaces in five and higher dimensions. Phys. Rev. Lett. 95: 071101

    Article  ADS  MathSciNet  Google Scholar 

  13. Martelli D. and Sparks J. (2005). Toric Sasaki-Einstein metrics on S 2 × S 3 . Phys. Lett. B 621: 208

    Article  ADS  MathSciNet  Google Scholar 

  14. Cvetic, M., Lu, H., Page, D.N., Pope, C.N.: New Einstein-Sasaki and Einstein spaces from Kerr-de Sitter. http://arxiv.org/list/hep-th/0505223, 2005

  15. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Supersymmetric AdS Backgrounds in String and M-theory. Proceedings of the 73rd Meeting between Physicists and Mathematicians “(A)dS/CFT correspondence”, Strasbourg, September 11-13, 2003. Available at http://arxiv.org/list/hep-th/0411194, 2004

  16. Chen W., Lu H., Pope C.N. and Vazquez-Poritz J.F. (2005). A note on Einstein-Sasaki metrics in D ≥ 7. Class. Quant. Grav. 22: 3421

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Lu H., Pope C.N. and Vazquez-Poritz J.F. (2007). A new construction of Einstein-Sasaki metrics in D ≥ 7. Phys. Rev. D75: 026005

    ADS  MathSciNet  Google Scholar 

  18. Cheeger J. and Tian G. (1994). On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118(3): 493–571

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Intriligator K. and Wecht B. (2003). The exact superconformal R-symmetry maximizes a. Nucl. Phys. B 667: 183

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Henningson M. and Skenderis K. (1998). The holographic Weyl anomaly. JHEP 9807: 023

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Benvenuti S., Franco S., Hanany A., Martelli D. and Sparks J. (2006). An infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals. JHEP 0506: 064

    ADS  Google Scholar 

  25. Bertolini M., Bigazzi F. and Cotrone A.L. (2004). New checks and subtleties for AdS/CFT and a-maximization. JHEP 0412: 024

    Article  ADS  MathSciNet  Google Scholar 

  26. Feng B., Hanany A. and He Y.H. (2001). D-brane gauge theories from toric singularities and toric duality. Nucl. Phys. B 595: 165

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Franco S., Hanany A., Kennaway K.D., Vegh D. and Wecht B. (2006). Brane dimers and quiver gauge theories. JHEP 0601: 096

    Article  ADS  MathSciNet  Google Scholar 

  28. Franco S., Hanany A., Martelli D., Sparks J., Vegh D. and Wecht B. (2006). Gauge theories from toric geometry and brane tilings. JHEP 0601: 128

    Article  ADS  MathSciNet  Google Scholar 

  29. Butti A. and Zaffaroni A. (2005). R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization. JHEP 0511: 019

    Article  ADS  MathSciNet  Google Scholar 

  30. Hanany, A., Vegh, D.: Quivers, tilings, branes and rhombi. http://arxiv.org/list/hep-th/0511063, 2005

  31. Franco S. and Vegh D. (2006). Moduli spaces of gauge theories from dimer models: Proof of the correspondence. JHEP 0611: 054

    Article  ADS  MathSciNet  Google Scholar 

  32. Butti A., Forcella D. and Zaffaroni A. (2005). The dual superconformal theory for L p,q,r manifolds. JHEP 0509: 018

    Article  ADS  MathSciNet  Google Scholar 

  33. Benvenuti S. and Kruczenski M. (2006). From Sasaki-Einstein spaces to quivers via BPS geodesics: L(p,q|r). JHEP 0604: 033

    Article  ADS  MathSciNet  Google Scholar 

  34. Futaki, A., Ono, H., Wang, G.: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. http://arxiv.org/list/math.DG/0607586, 2006

  35. Tachikawa Y. (2006). Five-dimensional supergravity dual of a-maximization. Nucl. Phys. B 733: 188

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Barnes E., Gorbatov E., Intriligator K. and Wright J. (2006). Current correlators and AdS/CFT geometry. Nucl. Phys. B 732: 89

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Lee S. and Rey S.J. (2006). Comments on anomalies and charges of toric-quiver duals. JHEP 0603: 068

    Article  ADS  MathSciNet  Google Scholar 

  38. Boyer C.P., Galicki K. and Matzeu P. (2006). On Eta-Einstein Sasakian Geometry. Commun. Math. Phys. 262: 177–208

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Barnes E., Gorbatov E., Intriligator K., Sudano M. and Wright J. (2005). The exact superconformal R-symmetry minimizes τ RR . Nucl. Phys. B 730: 210

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Duistermaat J.J. and Heckman G. (1982). On the variation in the cohomology of the symplectic form of the reduced space. Inv. Math. 69: 259–268

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Duistermaat J.J. and Heckman G. (1983). Addendum, Inv. Math. 72: 153–158

    MathSciNet  MATH  Google Scholar 

  42. Herzog C.P. and Karp R.L. (2006). Exceptional collections and D-branes probing toric singularities. JHEP 0602: 061

    Article  ADS  MathSciNet  Google Scholar 

  43. Hanany A., Herzog C.P. and Vegh D. (2006). Brane tilings and exceptional collections. JHEP 0607: 001

    Article  ADS  MathSciNet  Google Scholar 

  44. Vinberg E.B. (1963). The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12: 303–358

    MathSciNet  Google Scholar 

  45. Oda T. (1988). Convex bodies and algebraic geometry. Springer-Verlag, Berlin-Heidelberg-New York

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  47. Romelsberger C. (2006). Counting chiral primaries in N = 1 d = 4 superconformal field theories. Nud. Phys. B747: 329–353

    Article  ADS  MathSciNet  Google Scholar 

  48. Kinney, J., Maldacena, J., Minwalla, S., Raju, S.: An index for 4 dimensional super conformal theories. http://arxiv.org/list/hep-th/0510251, 2005

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Falcão S., Tomei C. and Moraes B. de (1997). Moment maps on symplectic cones. Pacific J. Math. 181(2): 357–375

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. Boyer C.P. and Galicki K. (2000). A Note on Toric Contact Geometry. J. Geom. Phys. 35(4): 288–298

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Besse A.L. (1987). Einstein Manifolds. Berlin-Heidelberg-New York, Spinger-Verlag

    MATH  Google Scholar 

  54. Lawson H.B. and Michelsohn M.-L. (1989). Spin Geometry. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  55. Bär C. (1993). Real Killing spinors and Holomony. Commun. Math. Phys. 154: 509–521

    Article  ADS  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  57. Calabi E. (1985). Extremal Kähler Metrics II. In: Chavel I., Farkas H.M. (eds) Differential Geometry and Complex Analysis. Berlin-Heidelberg-New York, Springer-Verlag

    Google Scholar 

  58. Mabuchi, T.: An Algebraic Character associated with the Poisson Brackets. Advanced Studies in Pure Mathematics 18-I, Recent Topics in Differential and Analytic Geometry, Tokyo and Boston: Kinokuniya and Acad. Press, 1990, pp. 339–358

  59. Atiyah M.F. and Singer I.M. (1968). The index of elliptic operators III. Ann. Math. 87: 546–604

    Article  MathSciNet  Google Scholar 

  60. Bott R. and Tu L. (1982). Differential forms in algebraic topology. Springer-Verlag, Berlin-Heidelberg-New York

    MATH  Google Scholar 

  61. Vergne M. (1996). The equivariant index formula on orbifolds. Duke Math. J. 82: 637–652

    Article  MathSciNet  MATH  Google Scholar 

  62. Cox, D.: Minicourse on Toric Varieties. Available at http://www.amherst.edu/~dacox/

  63. Berenstein D., Herzog C.P., Ouyang P. and Pinansky S. (2005). Supersymmetry breaking from a Calabi-Yau singularity. JHEP 0509: 084

    Article  ADS  MathSciNet  Google Scholar 

  64. Pinansky S. (2006). Quantum deformations from toric geometry. JHEP 0603: 055

    Article  ADS  MathSciNet  Google Scholar 

  65. Lasserre J.B. (1999). Integration and homogeneous functions. Proc. Amer. Math. Soc. 127: 813

    Article  MathSciNet  MATH  Google Scholar 

  66. Nekrasov N. and Shadchin S. (2004). ABCD of instantons. Commun. Math. Phys. 252: 359

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. Grassi P.A. and Morales Morera J.F. (2006). Partition functions of pure spinors. Nucl. Phys. B 751: 53

    Article  ADS  MathSciNet  MATH  Google Scholar 

  68. Grassi, P.A., Policastro, G.: Curved beta-gamma systems and quantum Koszul resolution. http://arxiv.org/list/hep-th/0602153, 2006

  69. Boyer C.P. and Galicki K. (1999). 3-Sasakian Manifolds. Surveys Diff. Geom. 7: 123

    MathSciNet  Google Scholar 

  70. El Kacimi–Alaoui A. (1990). Opérateurs transversalement elliptiques sur un feuilletage Riemannien et applications. Comp. Math. 79: 57–106

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, CERN Theory Division, 1211, Geneva 23, Switzerland

    Dario Martelli

  2. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA, 02138, USA

    James Sparks & Shing-Tung Yau

  3. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

    James Sparks

Authors
  1. Dario Martelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James Sparks
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shing-Tung Yau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James Sparks.

Additional information

Communicated by G.W. Gibbons

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martelli, D., Sparks, J. & Yau, ST. Sasaki–Einstein Manifolds and Volume Minimisation. Commun. Math. Phys. 280, 611–673 (2008). https://doi.org/10.1007/s00220-008-0479-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0479-4

Keywords

Search

Navigation