Skip to main content
SpringerLink
Log in

Properties of Hyperkähler Manifolds and Their Twistor Spaces

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

Abstract

We describe the relation between supersymmetric σ-models on hyperkähler manifolds, projective superspace, and twistor space. We review the essential aspects and present a coherent picture with a number of new results.

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. Alvarez-Gaumé L., Freedman D.Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric sigma model. Commun. Math. Phys. 80, 443 (1981)

    Article  ADS  Google Scholar 

  2. Gates S.J., Hull C.M., Roček M.: Twisted multiplets and new supersymmetric nonlinear sigma models. Nucl. Phys. B 248, 157 (1984)

    Article  ADS  Google Scholar 

  3. Karlhede A., Lindström U., Roček M.: Selfinteracting tensor multiplets in N = 2 superspace. Phys. Lett. B 147, 297 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  4. Grundberg J., Lindström U.: Actions for linear multiplets in six-Dimensions. Class. Quant. Grav. 2, L33 (1985)

    Article  ADS  Google Scholar 

  5. Karlhede A., Lindström U., Roček M.: Hyperkahler manifolds and nonlinear supermultiplets. Commun. Math. Phys. 108, 529 (1987)

    Article  MATH  ADS  Google Scholar 

  6. Lindström U., Roček M.: New hyperkahler metrics and new supermultiplets. Commun. Math. Phys. 115, 21 (1988)

    Article  MATH  ADS  Google Scholar 

  7. Buscher T., Lindström U., Roček M.: New supersymmetric sigma models with wess-zumino terms. Phys. Lett. B 202, 94 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  8. Lindström U., Roček M.: N = 2 super yang-mills theory in projective superspace. Commun. Math. Phys. 128, 191 (1990)

    Article  MATH  ADS  Google Scholar 

  9. Lindström U., Ivanov I.T., Roček M.: New N = 4 superfields and sigma models. Phys. Lett. B 328, 49 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  10. Lindström U., Kim B.B., Roček M.: The Nonlinear multiplet revisited. Phys. Lett. B 342, 99 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  11. Gonzalez-Rey F., Roček M., Wiles S., Lindström U., von Unge R.: Feynman rules in N = 2 projective superspace. I: Massless hypermultiplets. Nucl. Phys. B 516, 426 (1998)

    Article  MATH  ADS  Google Scholar 

  12. Penrose R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Salamon S.: Quaternionic Kähler manifolds. Invent. Math. 67, 143–171 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Hitchin N.J., Karlhede A., Lindström U., Roček M.: Hyperkahler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987)

    Article  MATH  ADS  Google Scholar 

  15. Kuzenko S.M.: On compactified harmonic / projective superspace, 5D superconformal theories, and all that. Nucl. Phys. B 745, 176–207 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Ivanov I.T., Roček M.: Supersymmetric sigma models, twistors, and the Atiyah-Hitchin metric. Commun. Math. Phys. 182, 291 (1996)

    Article  MATH  ADS  Google Scholar 

  17. Chakravarty S., Mason L., Newman E.T.: Canonical structures on antiselfdual four manifolds and the diffeomorphism group. J. Math. Phys. 32, 1458 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Kuzenko S.M.: Projective superspace as a double-punctured harmonic superspace. Int. J. Mod. Phys. A 14, 1737 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Galperin A.S., Ivanov E.A., Ogievetsky V.I., Sokatchev E.S.: Harmonic Superspace. Cambridge Univ. Pr., Cambridge, UK (2001)

    MATH  Google Scholar 

  20. de Wit B., Roček M., Vandoren S.: Hypermultiplets, hyperkaehler cones and quaternion-Kaehler geometry. JHEP 0102, 039 (2001)

    Article  Google Scholar 

  21. Lindström U., Roček M., von Unge R., Zabzine M.: Generalized Kaehler manifolds and off-shell supersymmetry. Commun. Math. Phys. 269, 833–849 (2007)

    Article  MATH  ADS  Google Scholar 

  22. Ketov, S.V., Lokhvitsky, B.B., Tyutin, I.V.: Hyperkähler sigma models in extended superspace. Theor. Math. Phys. 71, 496 (1987) [Teor. Mat. Fiz. 71, 226 (1987)]

    Google Scholar 

  23. Houghton C.J.: On the generalized Legendre transform and monopole metrics. JHEP 0002, 042 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  24. Bielawski, R.: Twistor quotients of hyperkaehler manifolds. http://arxiv.org/abs/math/0006142v1[math.DG], 2000

  25. Carter B.: Killing tensor quantum numbers and conserved currents in curved space. Phys. Rev. D 16, 3395 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  26. Howe P.S., Karlhede A., Lindström U., Roček M.: The Geometry of duality. Phys. Lett. B 168, 89 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  27. Gates S.J., Grisaru M.T., Roček M., Siegel W.: Superspace, or one thousand and one lessons in supersymmetry. Front. Phys. 58, 1 (1983)

    Google Scholar 

  28. Gates S.J., Kuzenko S.M.: The CNM-hypermultiplet nexus. Nucl. Phys. B 543, 122 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Arai M., Nitta M.: Hyper-Kaehler sigma models on (co)tangent bundles with SO(n) isometry. Nucl. Phys. B 745, 208–235 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Arai M., Kuzenko S.M., Lindström U.: Hyperkaehler sigma models on cotangent bundles of Hermitian symmetric spaces using projective superspace. JHEP 0702, 100 (2007)

    Article  ADS  Google Scholar 

  31. Arai M., Kuzenko S.M., Lindström U.: Polar supermultiplets, Hermitian symmetric spaces and hyperkahler metrics. JHEP 0712, 008 (2007)

    Article  ADS  Google Scholar 

  32. Ionaş, R.A.: Elliptic constructions of hyperkaehler metrics I: The Atiyah-Hitchin manifold. http://arxiv.org/abs/0712.3598v1[math.DG], 2007; Ionaş, R.A.: Elliptic constructions of hyperkaehler metrics II: The quantum mechanics of a Swann bundle. http://arxiv.org/abs/0712.3600v1[math.DG], 2007; Ionaş, R.A.: Elliptic constructions of hyperkaehler metrics III: Gravitons and Poncelet polygons. http://arxiv.org/abs/0712.3601v1[math.DG], 2007

  33. Alexandrov S., Pioline B., Saueressig F., Vandoren S.: Linear perturbations of quaternionic metrics - I. The Hyperkahler case. Lett. Math. Phys. 87, 225–265 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Kuzenko S.M., Lindström U., Roček M., Tartaglino-Mazzucchelli G.: 4D N = 2 supergravity and projective superspace. JHEP 0809, 051 (2008)

    Article  ADS  Google Scholar 

  35. Lindström U., Roček M.: Scalar tensor duality and N = 1, N = 2 nonlinear sigma models. Nucl. Phys. B 222, 285 (1983)

    Article  ADS  Google Scholar 

  36. Hull C.M., Karlhede A., Lindström U., Roček M.: Nonlinear sigma models and their Gauging in and out of superspace. Nucl. Phys. B 266, 1 (1986)

    Article  ADS  Google Scholar 

  37. Ward R.S.: On Selfdual gauge fields. Phys. Lett. A 61, 81 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  38. Roček M., Verlinde E.P.: Duality, quotients, and currents. Nucl. Phys. B 373, 630 (1992)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, Uppsala University, Box 803, SE-751 08, Uppsala, Sweden

    Ulf Lindström

  2. C.N. Yang Institute for Theoretical Physics, SUNY, Stony Brook, NY, 11794-3840, USA

    Martin Roček

Authors
  1. Ulf Lindström
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Roček
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Roček.

Additional information

Communicated by A. Kapustin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lindström, U., Roček, M. Properties of Hyperkähler Manifolds and Their Twistor Spaces. Commun. Math. Phys. 293, 257–278 (2010). https://doi.org/10.1007/s00220-009-0923-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0923-0

Keywords

Search

Navigation