The study of “instantons,” i.e. self-dual finite energy solutions of the Yang-Mills equations on Euclidean 4-space has developed rapidly since the initial examples was given by ’t Hooft, Belavin ard Polyakov. Thnis paper marks the endof the first phase, for it shows that every solution can be obtained by a quaternionic matrix construction. Despite the fact that these defining matrices are still difficult to parametrize explicitly, the formalism has since yieldedalgebraic expressions for the scalar Green’s function and the Dirac fields in the Yang-Mīlls background. The proof of completeness of the construction passes from Euclidean space to its conformal compactification the 4-sphere \(S^4\) and thence by Penrose’s twistors to complex proJective 3-space \(\mathbb P^3(\mathbb C)\). Using the recent work of Barth and Horrocks in algebraic geometry, and a crucial vanishing theorem, the result follows. The vanishing theorem itself uses the relationship between sheaf cohomology groups on \(\mathbb P^3(\mathbb C)\) and the conformally invariant Laplacian on \(S^4\), another aspect of twistor theory.

Reviewer: Michael Atiyah

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