Atiyah, Michael F.; Hitchin, Nigel J.; Drinfeld, V. G.; Manin, Yu. I. Construction of instantons. (English) Zbl 0424.14004 Phys. Lett. A 65, No. 3, 185-187 (1978). 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 end of 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 yielded algebraic 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 11 ReviewsCited in 330 Documents MSC: 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 14F06 Sheaves in algebraic geometry 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 81T08 Constructive quantum field theory 81T13 Yang-Mills and other gauge theories in quantum field theory 53C80 Applications of global differential geometry to the sciences Keywords:self-dual euclidean Yang-Mills fields; instantons; Yang-Mills equations; sheaf cohomology groups PDF BibTeX XML Cite \textit{M. F. Atiyah} et al., Phys. Lett., A 65, No. 3, 185--187 (1978; Zbl 0424.14004) Full Text: DOI OpenURL References: [1] Atiyah, M.F.; Hitchin, N.J.; Singer, I.M., () [2] Atiyah, M.F.; Ward, R.S., Commun. math. phys., 55, 117, (1977) [3] W. Barth, to appear. [4] V.G. Drinfeld and Yu.I. Manin, Funkts. Anal. Prilož, to appear. [5] V.G. Drinfeld and Yu.I. Manin, Usp. Math. Nauk, to appear. [6] R. Penrose, Rept. Math. Phys., to appear. [7] Schwarz, A.S., Phys. lett., 67B, 172, (1977) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.