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The approximation of instantons. (English) Zbl 0778.57010

This expository work gives a simple proof of the following theorem which is in a sense analogous to the classical theorem of Runge that asserts that a meromorphic function defined on a domain in \(\mathbb{C}\) can be approximated over compact subsets by rational functions:
Theorem. There is a sequence of \(\text{SU}(\ell)\) bundles \(P_ n\) over \(S^ 4\), \(n=1,2,\dots\), connections \(A_ n\) on \(P_ n\) which satisfy the instanton equation over \(S^ 4\), and bundle maps \(\rho_ n: P_ U\to P_ n|_ U\) such that the sequence of connections \(\rho^*_ n(A_ n)\) converge in \(C^ \infty\) over a neighbourhood of \(K\) to the connection \(A_ U\).
Here \(U\) is an open set in \(S^ 4\), \(K\) is a compact subset of \(U\) and the connection \(A_ U\) satisfies the anti-self-dual instanton equation \(F(A_ U)+*F(A_ U)\equiv F^ +(A_ U)=0\). The theorem is shown to be a simple by-product of the earlier works of C. H. Taubes [J. Differ. Geom. 19, 337-392 (1984; Zbl 0551.53040); ibid. 29, 163-230 (1989; Zbl 0669.58005)] that were aimed at somewhat, at least superficially, different topological problems.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
58D27 Moduli problems for differential geometric structures
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References:

[1] [A1]M.F. Atiyah, The geometry of Yang-Mills Fields Pisa University, 1979.
[2] [A2]M.F. Atiyah, Instantons in 2 and 4 dimensions, Commun. Math. Phys. 93 (1984), 437–451. · Zbl 0564.58040 · doi:10.1007/BF01212288
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[11] [T2]C.H. Taubes, The stable topology of self-dual moduli spaces, Jour. Differential Geometry 29 (1989), 163–230. · Zbl 0669.58005
[12] [T3]C.H. Taubes, The existence of anti-self-dual conformal structures, Jour. Differential Geometry 36 (1992), 163–254. · Zbl 0822.53006
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