Skip to main content
Log in

(Anti)self-dual gauge fields in dimensiond≥4

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

The (anti)self-duality equations for gauge fields in dimensiond=4 and the generalization of these equations ford>4 are considered. The results on solutions of the (anti)self-duality equations ind≥4 are reviewed. Some new classes of solutions of Yang-Mills equations ind≥4 for arbitrary gauge fields are described.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. A. Actor,Rev. Mod. Phys.,51, 461 (1979); A. I. Leznov and M. V. Savel'ev,Fiz. Elem. Chastits At. Yadra,11, 40 (1980).

    Google Scholar 

  2. R. Rajaraman,Solitons and Instantons, North-Holland, Amsterdam (1984).

    Google Scholar 

  3. M. K. Prasad, in:Geometrical Ideas in Physics [Russian translation], Mir, Moscow (1983), p. 64.

    Google Scholar 

  4. M. B. Green, J. H. Schwarz, and E. Witen,Superstring Theory, Vols. 1 and 2, C.U.P., Cambridge (1987).

    Google Scholar 

  5. A. Strominger,Nucl. Phys. B,343, 167 (1990); A. Dabholkar, G. Gibbons, J. A. Harvey, and R. F. Ruiz,Nucl. Phys. B,340, 33 (1990); C. G. Callan, J. A. Harvey, and A. Strominger,Nucl. Phys. B,359, 611 (1991); M. J. Duff and J. X. Lu,Nucl. Phys. B,354, 129, 141 (1991);357, 534 (1991).

    Google Scholar 

  6. J. A. Harvey and A. Strominger,Phys. Rev. Lett.,66, 549 (1991).

    Google Scholar 

  7. A. S. Schwarz,Commun. Math. Phys.,56, 79 (1977); D. B. Fairlie,Phys. Lett. B,82, 97 (1979); N. S. Manton,Nucl. Phys. B,158, 141 (1980); P. Forgacs and N. S. Manton,Commun. Math. Phys. 72, 15 (1980); G. Chapline and N. S. Manton,Nucl. Phys. B,184, 391 (1981); A. S. Schwarz and Yu. S. Tyupkin,Nucl. Phys. B,187, 321 (1981).

    Google Scholar 

  8. E. Corrigan, C. Devchand, D. B. Fairlie, and J. Nuyts,Nucl. Phys. B,214, 452 (1983).

    Google Scholar 

  9. R. S. Ward,Ncul. Phys. B,236, 381 (1984).

    Google Scholar 

  10. D. B. Fairlie and J. Nuyts,J. Phys. A.,17, 2867 (1984); J. Nuyts,Lect. Notes Phys.,201, 306 (1984); C. Devchand and D. B. Fairlie,Phys. Lett. B,141, 73 (1984).

    Google Scholar 

  11. S. Fubini and H. Nicolai,Phys. Lett. B,155, 369 (1985).

    Google Scholar 

  12. A. M. Semikhatov, in:Group-Theoretical Methods in Physics.Proc. of the Third International Seminar, Vol. 1 [in Russian], Nauka, Moscow (1986), p. 156.

    Google Scholar 

  13. E. Corrigan, P. Goddard, and A. Kent,Commun. Math. Phys.,100, 1 (1985).

    Google Scholar 

  14. A. D. Popov,Europhys. Lett.,17, 23 (1992); T. A. Ivanova,Usp. Mat. Nauk,47, 191 (1992); T. A. Ivanova and A. D. Popov,Lett. Math. Phys.,24, 85 (1992).

    Google Scholar 

  15. A. D. Popov,Pis'ma Zh. Eksp. Teor. Fiz.,55, 261 (1992); A. D. Popov,Mod. Phys. Lett. A,7, 2077 (1992).

    Google Scholar 

  16. T. A. Ivanova,Usp. Mat. Nauk,46, 149 (1991);Vestn. Mosk. Univ. Mat. Mekh., Ser. 1, No. 3, 10 (1992);Mat. Zametki,52, 43 (1992).

    Google Scholar 

  17. T. A. Ivanova and A. D. Popov,Lett. Math. Phys.,23, 29 (1991).

    Google Scholar 

  18. A. D. Popov,Pis'ma Zh. Eksp. Teor. Fiz.,54, 71 (1991).

    Google Scholar 

  19. A. D. Popov,Teor. Mat. Fiz.,89, 402 (1991).

    Google Scholar 

  20. L. Berard-Bergery and T. Ochiani, in:Global Riemann. Geom. Symp., New York (1984), p. 52.

  21. Yu. i. Manin,Gauge Fields and Complex Geometry [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  22. M. J. Ablowitz, D. J. Costa, and K. Tenenblat,Stud. Appl. Math.,77, 37 (1987).

    Google Scholar 

  23. M. M. Capria and S. M. Salamon,Nonlinearity,1, 517 (1988); T. Nitta,Tôhoku Math. J.,40, 425 (1988); J.F. Clazebrook,Rep. Math. Phys.,25, No. 2, 141; A. Galperin, E. Ivanov, V. Ogievetsky, and E. Sokatchev,Ann. Phys. (N. Y.),185, 1 (1988); K. Galicki and Y. S. Poon,J. Math. Phys.,32, 1263 (1991).

    Google Scholar 

  24. S. Salamon,Invent. Math.,67, 143 (1982);Ann. Sci. Ec. Norm. Sup.,19, 31 (1986); N. R. O'Brien and J. H. Rawnsley,Ann. Glob. Anal. Geom.,3, 29 (1985).

    Google Scholar 

  25. R. J. Baston and M. G. Eastwood,The Penrose Transform: Its Interaction with Representation Theory, Clarendon Press, Oxford (1989).

    Google Scholar 

  26. D. H. Tchrakian,J. Math. Phys.,21, 166 (1980);Phys. Lett. B,150, 360 (1985);155, 255 (1985); O. Lechtenfeld, W. Nahm, and D. H. Tchrakian,Phys. Lett. B,162, 143 (1985); A. Chakrabarti, T. H. Sherry, and D. H. Tchrakian,Phys. Lett. B,162, 340 (1985); G. M. O'Brien and D. H. Tchrakian,Lett. Math. Phys.,11, 133 (1986);J. Math. Phys.,29, 1242 (1988); D. O'Se and D. H. Tchrakian,Lett. Math. Phys.,13, 211 (1987); Y. Yang,Lett. Math. Phys.,19, 257, 285 (1990); D. H. Tchrakian and A. Chakrabarti,J. Math. Phys.,32, 2532 (1991).

    Google Scholar 

  27. B. Grossman, T. W. Kephart, and J. D. Stasheff, Commun. Math. Phys.,96, 431 (1984); Y. Brihaye, C. Devchand, and J. Nuyts,Phys. Rev. D,32, 990 (1985).

    Google Scholar 

  28. C. Sacliogly,Nucl. Phys. B,277, 487 (1986); K. Fujii,Lett. Math. Phys.,12, 363 (1986).

    Google Scholar 

  29. F. A. Bais and P. Batenburg,Nucl. Phys.B,269, 363 (1986); P. Batenburg and R. H. Rietdijk,Nucl. Phys. B,313, 393 (1989).

    Google Scholar 

  30. S. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Vol. 2, Interscience, New York (1963).

    Google Scholar 

  31. B. De Wit and H. Nicolai,Nucl. Phys. B,231, 506 (1984); A. R. Dündarer, F. Gürsey, and C.-H. Tze,J. Math. Phys.,25, 1496 (1984); A. R. Dündarer and F. Gürsey,J. Math. Phys.,32, 1176 (1991).

    Google Scholar 

  32. S. Rouhani,Phys. Lett. A,104, 7 (1984).

    Google Scholar 

  33. R. S. Ward,Phys. Lett. A,112, 3 (1985).

    Google Scholar 

  34. W. Nahm,Lect. Notes Phys.,180, 456 (1984);201, 189 (1984).

    Google Scholar 

  35. N. J. Hitchin,Commun. Math. Phys.,89, 145 (1983); S. K. Donaldson,Commun. Math. Phys.,96, 387 (1984); J. Hartubise,Commun. Math. Phys.,100 191 (1989);120, 613 (1989); J. Hartubise and M. K. Murray,Commun. Math. Phys.,133, 487 (1990).

    Google Scholar 

  36. M. Atiyah and N. Hitchin,The Geometry and Dynamics of Magnetic Monopoles, Princeton (1988).

  37. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii,The Theory of Solitons. The Inverse Scatting Method [in Russian], Nauka, Moscow (1980); M. J. Ablowitz and H. Segur,Solitons and the Inverse Scattering Transform (SIAM Studies in Applied Maths. Vol.4), Philadelphia (1981).

    Google Scholar 

  38. A. N. Leznov and M. V. Saveliev,Commun. Math. Phys.,74, 111 (1980); N. Ganoulis, P. Goddard, and D. Olive,Nucl. Phys. B,205, 601 (1982); A. Charkrabarti,Nucl. Phys. B.,248, 209 (1984).

    Google Scholar 

  39. A. M. Perelomov,Integrable Systems of Classical Mechanics and Lie Algebras [in Russian], Nauka, Moscow (1990); A. N. Leznov and M. V. Savel'ev,Group Methods of Integration of Nonlinear Dynamical Systems [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  40. S. Chakravarty, M. J. Ablowitz, and P. A. Clarkson,Phys. Rev. Lett.,65, 1085, 2086E (1990).

    Google Scholar 

  41. H. Pedersen and Y. S. Poon,Commun. Math. Phys.,117, 569 (1988); K. Galicki and Y. S. Poon,J. Math. Phys.,32, 1263 (1991).

    Google Scholar 

  42. N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček,Commun. Math. Phys.,108, 535 (1987); A. Karlhede, U. Lindström, and M. Roček,commun. Math. Phys.,108, 529 (1987); U. Lindström and M. Roček,Commun. Math. Phys.,115, 21 (1988).

    Google Scholar 

  43. A. D. Popov,Mod. Phys. Lett. A,5, 2057 (1990); A. D. Popov and A. G. Sergeev, Communication JINR E2-92-261, Dubna (1992).

    Google Scholar 

  44. R. S. Ward,Phys. Lett. B,234, 81 (1990).

    Google Scholar 

  45. R. S. Ward,Philos. Trans. R. Soc. London, Ser. A,315, 451 (1985); N. J. Hitchin,Proc. London Math. Soc.,55, 59 (1987); A. D. Popov,Yad. Fiz.,51, 883 (1990).

    Google Scholar 

  46. A. D. Popov,Pis'ma Zh. Eksp. Teor. Fiz.,54, 128 (1991).

    Google Scholar 

  47. E. Kovacs and S.-Y., Lo,Phys. Rev. D.,19, 3649 (1979); S. Chakravarty and E. T. Newman,J. Math. Phys.,28, 334 (1987); J. Villarroel.J. Math. Phys.,28, 2610 (1987); L. Hajiivanov and D. Stoyanov,Lett. Math. Phys.,13 93 (1987).

    Google Scholar 

  48. H. J. Vega,Commun. Math. Phys.,116, 659 (1988).

    Google Scholar 

  49. T. T. Wu and C. N. Yang,Phys. Rev. D,12, 3843 (1975);13, 3233 (1976).

    Google Scholar 

  50. I. Ya. Aref'eva and I. V. Volovich,Usp. Fiz. Nauk,146, 655 (1985); D. P. Sorokin and V. I. Tkach,Fiz. Elem. Chastits At. Yadra,18, 1035 (1987); Yu. S. Vladimirov and A. D. Popov, in:Classical Field Theory and The Theory of Gravitation, Vol. 1 [in Russian], VINITI, Moscow (1991), p. 5.

    Google Scholar 

  51. A. D. Popov,Europhys. Lett.,19, 465 (1992).

    Google Scholar 

Download references

Authors

Additional information

Moscow State University; Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 316–342, February, 1993.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ivanova, T.A., Popov, A.D. (Anti)self-dual gauge fields in dimensiond≥4. Theor Math Phys 94, 225–242 (1993). https://doi.org/10.1007/BF01019334

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01019334

Keywords

Navigation