Skip to main content
SpringerLink
Log in

Four-dimensional sigma-model coupled to the metric tensor field

Четырехмерная сигма-модель, связанная с метрическим тензорным полем

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

We discuss the four-dimensional nonlinear sigma-model with an internalO(n) invariance coupled to the metric tensor field satisfying Einstein equations. We derive a bound on the coupling constant between the sigma-field and the metric tensor using the theory of harmonic maps. A special attention is paid to Einstein spaces and some new explicit solutions of the model are constructed.

Riassunto

Si discute il modello sigma non lineare quadridimensionale con un'invarianza internaO(n) accoppiato al campo tensoriale metrico che soddisfa le equazioni di Einstein. Si deriva un limite sulla costante di accoppiamento tra il campo σ e il tensore metrico, usando la teoria delle mappe armoniche. Si rivolge una speciale attenzione agli spazi di Einstein e si costruiscono alcune nuove soluzioni esplicite del modello.

Резюме

Мы обсуждаем четырехмерную нелинейную сигма-модель с внутреннейO(n) инвариантностью, связанной с метрическим тензорным полем, которое удовлетворяет уравнениям Эйнштейна. Мы выводим границу для постоянной связи между сигма-полем и метрическим тензором, используя теорию гармонических отображений. Особое внимание уделяется пространствам Эйнштейна и конструируются некоторые новые решения предложенной модели.

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. See, for example,D. J. Gross:Nucl. Phys. B,132, 439 (1978);P. Tataru-Mihai:On the two-dimensional O(3) σ-nonlinear model, preprint Starnberg University (1978);V. de Alfaro, S. Fubini andG. Furlan:Nonlinear sigma model and classical solutions, preprint ICTP-Trieste 78/69 (1978);H. Eichenherr:Nucl. Phys. B,146, 215 (1978);H. J. Borchers andW. D. Garber:Analyticity of solutions of the O(N) nonlinear σ-model, preprint University of Göttingen (July 1979);Local theory of solutions for the O(2k+1) σ-model, preprint University of Göttingen (October 1979);D. Maison:Some facts about classical nonlinear σ-models, preprint Max-Planck Institut für Physik und Astrophysik, München, MPI-PAE/PTh 52/79 (November 1979).

    Article  ADS  Google Scholar 

  2. V. De Alfaro, S. Fubini andG. Furlan:Nuovo Cimento A,50, 523 (1979).

    Article  ADS  Google Scholar 

  3. S. M. Christensen andM. J. Duff:Quantizing gravity with a cosmological constant, preprint University of California, Santa Barbara ITP-79-01 (October 1979).

  4. J. A. Wheeler: inRelativity Groups and Topology, edited byB. S. De Witt andC. M. De Witt (New York, N. Y., 1964).

  5. S. W. Hawking:Nucl. Phys. B,144, 349 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  6. G. W. Gibbons andS. W. Hawking:Commun. Math. Phys.,66, 291 (1979).

    Article  MathSciNet  ADS  Google Scholar 

  7. For a recent review seeJ. Eells andL. Lemaire:Bull. London Math. Soc.,10, 1 (1978).

    Article  MathSciNet  Google Scholar 

  8. C. W. Misner:Phys. Rev.,18, 4510 (1978).

    MathSciNet  ADS  Google Scholar 

  9. M. Forger:Instantons in nonlinear σ-models, gauge theories and general relativity, preprint Freie Universität Berlin (July 1978);W. D. Garber, S. N. M. Ruijenaars andE. Seiler:Ann. Phys. (N. Y.),119, 305 (1979);P. Tataru-Mihai:Nuovo Cimento A,51, 169 (1979);F. Gürsey, M. A. Jafarizadeh andH. C. Tze:Phys. Lett. B,88, 282 (1979).

  10. G. Ghika andM. Visinescu:Vacuum degeneracies in terms of harmonic maps, preprint Central Institute of Physics, Bucharest FT-164 (1979);Phys. Rev. D,21, 1538 (1980).

  11. A. Eris:J. Math. Phys. (N. Y.),18, 824 (1977);Y. Nutku andM. Halil:Phys. Rev. Lett.,39, 1379 (1977).

    Article  ADS  MATH  Google Scholar 

  12. S. Bochner:Trans. Am. Math. Soc.,47, 146 (1940).

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Eells andJ. H. Sampson:Am. J. Math.,86, 109 (1964).

    Article  MathSciNet  Google Scholar 

  14. L. P. Eisenhart:Riemannian Geometry (Princeton, N. J., 1949).

  15. K. Yano:Integral Formulas in Riemannian Geometry (New York, N. Y., 1970).

  16. J. Milnor:Morse Theory (Princeton, N. J., 1963).

  17. T. Takahashi:J. Math. Soc. Jpn.,18, 380 (1966).

    Article  MATH  Google Scholar 

  18. N. R. Wallach: inSymmetric Spaces, edited byW. M. Boothby andG. L. Weiss (New York, N. Y., 1972).

  19. W. Y. Hsiang:Proc. Nat. Acad. Sci. USA,56, 5 (1966).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. See, for example,V. L. Golo andA. M. Perelomov:Lett. Math. Phys.,2, 477 (1978);E. Gava, R. Jengo andC. Omero:Phys. Lett. B,81, 187 (1979);E. Gava, R. Jengo, C. Omero andR. Percacci:Nucl. Phys. B,151, 457 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Fundamental Physics, Institute for Physics and Nuclear Engineering, P.O. Box 5206, Bucharest, Romania

    G. Ghika & M. Vişinescu

Authors
  1. G. Ghika
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Vişinescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Traduzione a cura della Redazione.

Переведено редакцией.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ghika, G., Vişinescu, M. Four-dimensional sigma-model coupled to the metric tensor field. Nuov Cim B 59, 59–74 (1980). https://doi.org/10.1007/BF02739046

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02739046

Search

Navigation