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

Uhlenbeck, Karen

J. Geom. Phys. 8, No. 1-4, 283-316 (1992).

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).

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

Keywords:

harmonic maps; self-dual Yang-Mills equations; sine-Gordon equation

Citations:

Zbl 0677.58020

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