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**On the connection between harmonic maps and the self-dual Yang-Mills and the sine-Gordon equations.**
*(English)*
Zbl 0747.58025

This paper develops previous work by the same author [J. Differ. Geom. 30, No. 1, 1-50 (1989; Zbl 0677.58020)]. Its aim is to demonstrate some algebraic connections between the Yang-Mills equations, the sine-Gordon equation and harmonic maps from \(E^ 2\) (or \(S^ 2\)) and \(E^{1,1}\) into a Lie group \(G\). More precisely, the harmonic map equations are shown to be equivalent to the reduced self-dual Yang-Mills equations on \(E^{2,2}\) with structure group \(G\).

For such a reduction to be possible, the unusual signature \(++--\) is required in 4-dimensional space (the standard \(+---\) or \(-+++\) not being compatible with the reality conditions required by a compact gauge group).

For such a reduction to be possible, the unusual signature \(++--\) is required in 4-dimensional space (the standard \(+---\) or \(-+++\) not being compatible with the reality conditions required by a compact gauge group).

Reviewer: A.Ratto (Brest)

### MSC:

58E20 | Harmonic maps, etc. |

35Q53 | KdV equations (Korteweg-de Vries equations) |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

43A99 | Abstract harmonic analysis |

### Citations:

Zbl 0677.58020### References:

[1] | Calabi, E., Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geom., 1, 111-125 (1967) · Zbl 0171.20504 |

[2] | Chern, S. S., On the minimal immersions of the two-sphere in a space of constant curvature, (Problems in Analysis (1970), Princeton University Press: Princeton University Press Princeton, NJ), 27-49 · Zbl 0217.47601 |

[3] | Din, A. M.; Zakrzewski, W. J., General classical solutions in the \(CP^{n−}\) model, Nucl. Phys. B, 174, 397-406 (1980) |

[4] | Glaser, V.; Stora, R., Regular solutions of the \(CP^n\) model and further generalizations, preprint, CERN (1980) |

[5] | Eells, J.; Wood, J. C., Harmonic maps from surfaces to complex projective spaces, Adv. Math., 49, 217-263 (1983) · Zbl 0528.58007 |

[6] | Dolan, L., Kac-Moody algebra is hidden symmetry of chiral models, Phys. Rev. Lett., 49, 1371-1374 (1981) |

[7] | Zakharov, V. E.; Mikhailov, A. V., Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Sov. Phys. JETP, 47, 1017-1027 (1978) |

[8] | Zakharov, V. E.; Shabat, A. B., Integration of nonlinear equations of mathematical physics by the method of inverse scattering II, Func. Anal. Appl., 13, 13-22 (1978) |

[9] | Uhlenbeck, K., Harmonic maps into Lie groups, J. Diff. Geom., 30, 1-50 (1989) · Zbl 0677.58020 |

[10] | Atiyah, M.; Hitchin, N., The Geometry and Dynamics of Magnetic Monopoles (1988), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0671.53001 |

[11] | Braam, P., Magnetic monopoles and Kleinian groups, Ph.D. thesis (1988), Oxford |

[12] | Hitchin, N., The self-duality equations on a Riemann surface, (Proc. London Math. Soc., 55 (1987)), 59-126 · Zbl 0634.53045 |

[13] | Piette, B.; Zait, R.; Zakrzewski, W., Solutions of the \(U(N)\) a models with Wess-Zumino-Witten term, Z. Phys. C, 39, 359 (1988) |

[14] | Valli, G., On the energy spectrum of harmonic 2-spheres in unitary groups, Topology, 27, 129-136 (1988) · Zbl 0744.53027 |

[15] | Pohlmeyer, K., Integrable Hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys., 46, 207-221 (1976) · Zbl 0996.37504 |

[16] | Piette, B.; Stokoe, I.; Zakrzewski, W., On stability of solutions of the \(U (N)\) chiral model in two dimensions, Z. Phys. C, 34, 449 (1988) |

[17] | Zakrzewski, W., Classical solutions of two-dimensional Grassmannian models, J. Geom. Phys., 1, 39-63 (1984) · Zbl 0595.58010 |

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