Elementary Deformations and the HyperKähler-Quaternionic Kähler Correspondence

  • Oscar Macia
  • Andrew Swann
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)


The hyperKähler-quaternionic Kähler correspondence constructs quaternionic Kähler metrics from hyperKähler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We outline how this leads to a uniqueness statement for the above correspondence and indicate how basic examples of c-map constructions may be realised in this context.


Curvature Form Circle Action Twist Function Elementary Deformation Flat Connection 
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.



This work is partially supported by the Danish Council for Independent Research, Natural Sciences, and by the Spanish Agency for Scientific and Technical Research (DGICT) and FEDER project MTM2010-15444.


  1. 1.
    Alekseevsky, D.V., Cortés, V., Dyckmanns, M., Mohaupt, T.: Quaternionic Kähler metrics associated with special Kähler manifolds (2013). arXiv:1305.3549[math.DG]Google Scholar
  2. 2.
    Alekseevsky, D.V., Cortés, V., Mohaupt, T.: Conification of Kähler and hyper-Kähler manifolds. Commun. Math. Phys. 324, 637–655 (2013)CrossRefMATHGoogle Scholar
  3. 3.
    Alexandrov, S., Persson, D., Pioline, B.: Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence. J. High Energy Phys. 2011(12), 027, i, 64 pp. (electronic) (2011)Google Scholar
  4. 4.
    Berger, M.: Sur les groupes d’holonomie des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83, 279–330 (1955)MATHMathSciNetGoogle Scholar
  5. 5.
    Besse, A.L.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 10. Springer, Berlin, Heidelberg and New York (1987)Google Scholar
  6. 6.
    Cecotti, S., Ferrara, S., Girardello, L.: Geometry of type II superstrings and the moduli of superconformal field theories. Int. J. Modern Phys. A 4(10), 2475–2529 (1989)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Ferrara, S., Sabharwal, S.: Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces. Nucl. Phys. B 332, 317–332 (1990)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Freed, D.S.: Special Kähler manifolds. Commun. Math. Phys. 203, 31–52 (1999)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Haydys, A.: HyperKähler and quaternionic Kähler manifolds with S 1-symmetries. J. Geom. Phys. 58(3), 293–306 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Hitchin, N.J.: On the hyperkähler/quaternion Kähler correspondence. Commun. Math. Phys. 324(1), 77–106 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Joyce, D.: Compact hypercomplex and quaternionic manifolds. J. Differ. Geom. 35, 743–761 (1992)MATHMathSciNetGoogle Scholar
  13. 13.
    Macia, O., Swann, A.F.: Twist geometry of the c-map (2014). ☺arXiv:1404.0785[math.DG]Google Scholar
  14. 14.
    Salamon, S.M.: Quaternionic Kähler manifolds. Invent. Math. 67, 143–171 (1982)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Swann, A.F.: Aspects symplectiques de la géométrie quaternionique. C. R. Acad. Sci. Paris 308, 225–228 (1989)MATHMathSciNetGoogle Scholar
  16. 16.
    Swann, A.F.: T is for twist. In: Iglesias Ponte, D., Marrero González, J.C., Martín Cabrera, F., Padrón Fernández, E., Martín, S. (eds.) Proceedings of the XV International Workshop on Geometry and Physics, Puerto de la Cruz, September 11–16, 2006, Publicaciones de la Real Sociedad Matemática Española, vol. 11, pp. 83–94. Spanish Royal Mathematical Society, Madrid (2007)Google Scholar
  17. 17.
    Swann, A.F.: Twisting Hermitian and hypercomplex geometries. Duke Math. J. 155(2), 403–431 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Facultad de Ciencias Matematicas, Departamento de Geometria y TopologiaUniversidad de ValenciaValenciaSpain
  2. 2.Department of MathematicsAarhus UniversityAarhus CDenmark

Personalised recommendations