Skip to main content
SpringerLink
Log in

Hermitian Geometry and Complex Space-Time

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

Abstract

We consider a complex Hermitian manifold of complex dimensions four with a Hermitian metric and a Chern connection. It is shown that the action that determines the dynamics of the metric is unique, provided that the linearized Einstein action coupled to an antisymmetric tensor is obtained, in the limit when the imaginary coordinates vanish. The unique action is of the Chern-Simons type when expressed in terms of the Kähler form. The antisymmetric tensor field has gauge transformations coming from diffeomorphism invariance in the complex directions. The equations of motion must be supplemented by boundary conditions imposed on the Hermitian metric to give, in the limit of vanishing imaginary coordinates, the low-energy effective action for a curved metric coupled to an antisymmetric tensor.

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.

Similar content being viewed by others

References

  1. Synge, J.L.: Proc. R. Irish. Acad. 62, 1 (1961)

    MATH  MathSciNet  Google Scholar 

  2. Newman, E.: J. Math. Phys. 2, 324 (1961)

    Article  MATH  Google Scholar 

  3. Penrose, R.: J. Math. Phys. 8, 345 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  4. Penrose, R., Rindler, W.: Spinors and Space-Time, Cambridge: Cambridge University Press, 1986

  5. Plebanski, J.F.: J. Math. Phys. 16, 2396 (1975)

    ADS  Google Scholar 

  6. Flaherty, E.J.: Hermitian and Kählerian Geometry in General Relativity. Lecture Notes in Physics, Volume 46, Heidelberg: Springer, 1976

  7. Flaherty, E.J.: Complex Variables in Relativity. In: General Relativity and Gravitation, One Hundred Years after the Birth of Albert Einstein, A. Held, (ed.), New York: Plenum, 1980

  8. Witten, E.: Phys. Rev. Lett. 61, 670 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  9. Gross, D., Mende, P.: Phys. Lett B197, 129 (1987)

    Google Scholar 

  10. Connes, A.: Noncommutative Geometry, London-New York:Academic Press, 1994

  11. Connes, A., Douglas, M., Schwarz, A.: JHEP 9802, 003 (1998)

  12. Seiberg, N., Witten, E.: JHEP 9909, 032 (1999)

  13. Chamseddine, A.H.: Commun. Math. Phys. 218, 283 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Einstein, A., Strauss, E.: Ann. Math. 47, 731 (1946)

    MATH  MathSciNet  Google Scholar 

  15. Goldberg, S.I.: Ann. Math. 63, 64 (1956)

    MATH  Google Scholar 

  16. Yano, K.: Differential Geometry on Complex and Almost Complex Manifolds. New York: Pergamon Press, 1965

  17. Gauduchon, P.: Math. Ann. 267, 495 (1984)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Advanced Mathematical Sciences (CAMS) and Physics Department, American University of Beirut, Lebanon

    A.H. Chamseddine

Authors
  1. A.H. Chamseddine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A.H. Chamseddine.

Additional information

Communicated by A. Connes

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chamseddine, A. Hermitian Geometry and Complex Space-Time. Commun. Math. Phys. 264, 291–302 (2006). https://doi.org/10.1007/s00220-005-1466-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1466-7

Keywords

Search

Navigation