Communications in Mathematical Physics

, Volume 284, Issue 3, pp 583–647 | Cite as

Classification of Superpotentials

  • A. DancerEmail author
  • M. Wang


We extend our previous classification [DW4] of superpotentials of “scalar curvature type” for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in [DW4], i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano Kähler-Einstein product.


Interior Point Nonzero Entry Orthogonality Condition Null Vector Einstein Metrics 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BB.
    Bérard Bergery L.: Sur des nouvelles variétés riemanniennes d’Einstein. Publications de l’Institut Elie Cartan, Nancy (1982)Google Scholar
  2. BGGG.
    Brandhuber A., Gomis J., Gubser S., Gukov S.: Gauge theory at large N and new G 2 holonomy metrics. Nucl. Phys. B 611, 179–204 (2001)zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. CGLP1.
    Cvetic̆ M., Gibbons G.W., Lü H., Pope C.N.: Hyperkähler Calabi metrics, L2 harmonic forms, Resolved M2-branes, and AdS 4/CFT3 correspondence. Nucl. Phys. B 617, 151–197 (2001)CrossRefADSGoogle Scholar
  4. CGLP2.
    Cvetic̆ M., Gibbons G.W., Lü H., Pope C.N.: Cohomogeneity one manifolds of Spin(7) and G 2 holonomy. Ann. Phys. 300, 139–184 (2002)CrossRefADSGoogle Scholar
  5. CGLP3.
    Cvetic̆ M., Gibbons G.W., Lü H., Pope C.N.: Ricci-flat metrics, harmonic forms and brane resolutions. Commun. Math. Phys. 232, 457–500 (2003)ADSGoogle Scholar
  6. DW1.
    Dancer A., Wang M.: Kähler-Einstein metrics of cohomogeneity one. Math. Ann. 312, 503–526 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. DW2.
    Dancer A., Wang M.: Integrable cases of the Einstein equations. Commun. Math. Phys. 208, 225–244 (1999)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. DW3.
    Dancer A., Wang M.: The cohomogeneity one Einstein equations from the Hamiltonian viewpoint. J. Reine Angew. Math. 524, 97–128 (2000)zbMATHMathSciNetGoogle Scholar
  9. DW4.
    Dancer A., Wang M.: Superpotentials and the cohomogeneity one Einstein equations. Commun. Math. Phys. 260, 75–115 (2005)zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. DW5.
    Dancer, A., Wang, M.: Notes on Face-listings for “Classification of superpotentials”. Posted at, 2007
  11. GGK.
    Ginzburg V., Guillemin V., Karshon Y.: Moment maps, cobordisms, and Hamiltonian group actions, AMS Mathematical Surveys and Monographs, Vol. 98. Amer, Math, Soc., Providence, RI (2002)Google Scholar
  12. EW.
    Eschenburg J., Wang M.: The initial value problem for cohomogeneity one Einstein metrics. J. Geom. Anal. 10, 109–137 (2000)zbMATHMathSciNetGoogle Scholar
  13. WW.
    Wang J., Wang M.: Einstein metrics on S 2-bundles. Math. Ann. 310, 497–526 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  14. WZ1.
    Wang M., Ziller W.: Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84, 177–194 (1986)zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. Zi.
    Ziegler G.M.: Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152. Springer-Verlag, Berlin- Heidelberg-New York (1995)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Jesus CollegeOxford UniversityOxfordUnited Kingdom
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations