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Instantons in two and four dimensions

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Abstract

It is shown that Yang-Mills instantons in four dimensions can naturally be identified with the instantons of a two-dimensional theory with values in the loop group.

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References

  1. Atiyah, M.F.: Geometry of Yang-Mills fields. Lezioni Fermiane, Scuola Normale Superiore, Pisa, 1979

  2. Atiyah, M.F.: Magnetic monopoles on hyperbolic space. Proc. International Colloquium on vector bundles, Tata Institute, Bombay 1984 (to appear)

    Google Scholar 

  3. Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., Manin, Yu.I.: Construction of instantons. Phys. Lett.65A, 185–187 (1978)

    Google Scholar 

  4. Atiyah, M.F., Jones, J.D.J.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97–118 (1978)

    Google Scholar 

  5. Chakrabarti, A.: Instanton chains with multimonopole limits: Lax pairs for non-axially-symmetric cases. Phys. Rev. D28, 989 (1983)

    Google Scholar 

  6. Donaldson, S.K.: Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. (to appear)

  7. Donaldson, S.K.: Instantons and geometric invariant theory. Commun. Math. Phys. (to appear)

  8. Donaldson, S.K.: Nahm's equations and the classification of monopoles (to appear)

  9. Forgacs, P., Horvath, Z., Palla, L.: An exact fractionally charged self-dual solution, Hungarian Academy of Sciences. Preprint KFKI 60 (1980)

  10. Hitchin, N.J.: Monopoles and geodesics. Commun. Math. Phys.83, 579–602 (1982)

    Google Scholar 

  11. Jaffe, A., Taubes, C.H.: Vortices and monopoles. Boston: Birkhäuser 1980

    Google Scholar 

  12. Kirillov, A.A.: Elements of the theory of representations. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  13. Maruyama, M.: Stable vector bundles on an algebraic surface. Nagoya Math. J.58, 25–68 (1975)

    Google Scholar 

  14. Nahm, W.: All self-dual multimonopoles for arbitrary gauge groups (preprint), TH. 3172-CERN (1981)

  15. Pressley, A.N.: The energy flow on the loop space of a compact Lie group. J. London Math. Soc. (to appear)

  16. Pressley, A.N.: Decompositions of the space of loops on a Lie group. Topology19, 65–79 (1980)

    Google Scholar 

  17. Pressley, A.N., Segal, G.B.: Loop groups. Oxford: Oxford University Press 1984

    Google Scholar 

  18. Segal, G.B.: Unitary representations of some infinite-dimensional groups. Commun. Math. Phys.80, 301–342 (1981)

    Google Scholar 

  19. Segal, G.B.: The topology of spaces of rational functions. Acta Math.143, 39–72 (1979)

    Google Scholar 

  20. Taubes, C.H.: The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations in ℝ3. Commun. Math. Phys.86, 257–298, 299–320 (1982)

    Google Scholar 

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Communicated by A. Jaffe

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Atiyah, M.F. Instantons in two and four dimensions. Commun.Math. Phys. 93, 437–451 (1984). https://doi.org/10.1007/BF01212288

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