Atiyah’s work on holomorphic vector bundles and gauge theories. (English) Zbl 1476.14002
The aim of this paper is to surveys Atiyah’s work on holomorphic vector bundles and gauge theories. This article is divided into two parts. The first part deals with Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves [Proc. Lond. Math. Soc. (3) 5, 407–434 (1955; Zbl 0174.52804); ibid. (3) 7, 414–452 (1957; Zbl 0084.17305); Trans. Am. Math. Soc. 85, 181–207 (1957; Zbl 0078.16002); Proc. R. Soc. Lond., Ser. A 247, 237–244 (1958; Zbl 0135.21301)]. These works concern extensions and the Atiyah class, bundles over curves, double points in dimensions \(2\) and \(3\) as well as some later developments. In the second part the author discusses Atiyah’s work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology. This part deals with the Yang-Mills equations and self-duality, self-dual manifolds and twistor spaces, the ADHM construction and finally topology and geometry of moduli spaces.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |
| 14-03 | History of algebraic geometry |
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 53C05 | Connections (general theory) |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 14H60 | Vector bundles on curves and their moduli |
| 53-03 | History of differential geometry |
| 58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |
| 01A75 | Collected or selected works; reprintings or translations of classics |
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