Skip to main content
SpringerLink
Log in

Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals

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

Abstract

Recently an infinite family of explicit Sasaki–Einstein metrics Y p,q on S 2×S 3 has been discovered, where p and q are two coprime positive integers, with q<p. These give rise to a corresponding family of Calabi–Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kähler quotients namely the vacua of gauged linear sigma models with charges (p,p,−p+q,−p−q), thereby generalising the conifold, which is p=1,q=0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold for all q<p with fixed p. We hence find that the Y p,q manifolds are AdS/CFT dual to an infinite class of superconformal field theories arising as IR fixed points of toric quiver gauge theories with gauge group SU(N)2 p. As a non–trivial example, we show that Y 2,1 is an explicit irregular Sasaki–Einstein metric on the horizon of the complex cone over the first del Pezzo surface. The dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its IR fixed point using the AdS/CFT correspondence. The value we obtain is a quadratic irrational number and, remarkably, agrees with a recent purely field theoretic calculation using a-maximisation.

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

    MATH  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  12. Herzog, C.P., Walcher, J.: Dibaryons from exceptional collections. JHEP 0309, 060 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  13. Herzog, C.P.: Exceptional collections and del Pezzo gauge theories. JHEP 0404, 069 (2004)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  15. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: A new infinite class of Sasaki-Einstein manifolds. http://arXiv:org/list/hep-th/0403038, 2004

  16. Gauntlett, J.P., Martelli, D., Sparks, J. Waldram, D.: Supersymmetric AdS Backgrounds in String and M-theory. http://arXiv:org/list/hep-th/0411194, 2004

  17. Chen, W., Lu, H., Pope, C.N., Vazquez-Poritz, J.F.: A Note on Einstein–Sasaki Metrics in D ≥7. http://arXiv.org/list/hep-th/0411218, 2004

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  20. Lerman, E.: Contact toric manifolds. J. Symplectic Geom. 1(4), 785–828 (2003)

    MathSciNet  Google Scholar 

  21. Delzant, T.: Hamiltoniens periodiques et images convexes de l'application moment. Bull. Soc. Math. France 116(3), 315–339 (1988)

    MathSciNet  Google Scholar 

  22. Witten, E.: Phases of theories in two dimensions. Nucl. Phys. B 403, 159 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Beasley, C., Greene, B.R., Lazaroiu, C.I., Plesser, M.R.: D3-branes on partial resolutions of abelian quotient singularities of Calabi-Yau threefolds. Nucl. Phys. B 566, 599 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Witten, E.: Baryons and branes in anti de Sitter space. JHEP 9807, 006 (1998)

    ADS  Google Scholar 

  25. Berenstein, D., Herzog, C.P., Klebanov, I.R.: Baryon Spectra and AdS/CFT Correspondence. JHEP 0206, 047 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  26. Smale, S.: On the structure of 5-manifolds. Ann. Math. 75, 38–46 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  27. Friedrich, Th., Kath, I.: Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator. J. Differ. Geom. 29, 263–279 (1989)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  29. Boyer, C.P., Galicki, K.: New Einstein metrics in dimension five. J. Differ. Geom. 57(3), 443–463 (2001)

    MathSciNet  Google Scholar 

  30. Boyer, C.P., Galicki, K., Nakamaye, M.: On the Geometry of Sasakian–Einstein 5-Manifolds. Math. Ann. 325(3), 485–524 (2003)

    Article  MathSciNet  Google Scholar 

  31. Boyer, C.P., Galicki, K., Nakamaye, M.: Sasakian–Einstein structures on 9#(S 2× S 3). Trans. Amer. Math. Soc. 354(8), 2983–2996 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  32. Boyer, C.P., Galicki, K.: New Einstein metrics on 8#(S 2× S 3). Differential Geom. Appl. 19(2), 245–251 (2003)

    Article  MathSciNet  Google Scholar 

  33. Wang, M.Y., Ziller, W.: Einstein metrics on principal torus bundles. J. Diff. Geom. 31, 215 (1990)

    MATH  Google Scholar 

  34. Boyer, C.P., Galicki, K.: 3-Sasakian Manifolds. Surveys Diff. Geom. 7, 123–184 (1999)

    MathSciNet  Google Scholar 

  35. Falcao de Moraes, S., Tomei, C.: Moment maps on symplectic cones. Pacif. J. Math. 181(2), 357–375 (1997)

    Article  Google Scholar 

  36. Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14, 1–15 (1982)

    MATH  MathSciNet  Google Scholar 

  37. Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Lerman, E., Tolman, S.: Hamiltonian torus actions on symplectic orbifolds and toric varieties. http://arXiv.org/list/dg-ga/9511008, 1995

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. Lerman, E.: Homotopy Groups of K-Contact Toric Manifolds. Trans. Amer. Math. Soc. 356(10), 4075–4083 (2004)

    Article  MathSciNet  Google Scholar 

  41. Page, D.N.: A compact rotating gravitational instanton. Phys. Lett. 79B(3), 235–238 (1978)

    Google Scholar 

  42. Feng, B., Franco, S., Hanany, A., He, Y.H.: Symmetries of toric duality. JHEP 0212, 076 (2002)

    Article  ADS  Google Scholar 

  43. Dall'Agata, G.: N = 2 conformal field theories from M2-branes at conifold singularities. Phys. Lett. B 460, 79 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  45. Ceresole, A., Dall'Agata, G., D'Auria, R., Ferrara, S.: M-theory on the Stiefel manifold and 3d conformal field theories. JHEP 0003, 011 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  46. Intriligator, K., Wecht, B.: Baryon charges in 4D superconformal field theories and their AdS duals. Commun. Math. Phys. 245, 407 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  47. Candelas, P., de la Ossa, X.C.: Comments On Conifolds. Nucl. Phys. B 342, 246 (1990)

    Article  ADS  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 and Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02318, U.S.A

    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

Corresponding author

Correspondence to Dario Martelli.

Additional information

Communicated by G.W. Gibbons

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martelli, D., Sparks, J. Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals. Commun. Math. Phys. 262, 51–89 (2006). https://doi.org/10.1007/s00220-005-1425-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1425-3

Keywords

Search

Navigation