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Geometry of Yang-Mills fields. (English) Zbl 0435.58001
Lezioni Fermiane. Pisa: Accademia Nazionale dei Lincei. Scuola Normale Superiore. 98 p. (1979).
This monograph is based on lectures given by Professor Atiyah at the Scuola Normale Superiore in Pisa in 1978, as well as the Loeb Lectures at Harvard, and Whittemore Lectures at Yale of the same year. They were addressed to audiences composed of physicists and mathematicians, and cover an area of common interest to which the author (in various collaborations) has made substantial contributions both in the past and in recent year. The published version has retained some of the lively flavor of Professor Atiyah’s lectures, and although some of the topics are covered only briefly, they form an excellent introduction to some of the most difficult areas of application of modern differential and algebraic geometry to gauge theories. They should be accessible to mathematical physicists willing to learn some of the more abstract mathematical techniques, and to mathematicians who are willing to be convinced that complex algebraic geometry has some “real life” applications. The volume also contains an account of the Atiyah-Drinfel’d-Hitchin-Manin construction of instantons, which at the time of publication was not available elsewhere.
The first chapter deals with the physics background, and contains a formulation of gauge theory in terms of connections and curvatures of principal bundles. Topology is introduced in connection with asymptotic behavior of Yang-Mills fields. Chapter II gives a quaternionic description of instantons (essentially the results of Atiyah-Drinfel’d-Hitchin-Manin) and its geometric interpretation. Chapter III is devoted to the Penrose twistor spare. Complex projective 3-space is introduced in this chapter.
The following chapter deals with holomorphic bundles and the Atiyah-Ward, and Atiyah-Hitchin-Singer correspondence between self-dual connections and holomorphic bundles. Chapter V deals with the construction of algebraic bundles, based an the extension of the techniques of Horrocks and Barth. Chapter VI discusses linear field equations (fields on a Yang-Mills background). It starts with a brief review of sheaf cohomology (a subject with which most physicists will be unfamiliar) and goes on with a discussion of the linear aspects of the Penrose transform. Sheaf cohomology is then exploited to yield results on linear equations on a Yang-Mills background, the ’t Hooft Ansatz, and the chapter ends with a discussion of the relation between the Penrose transform and Radon transformations. Chapter VII is more technical, and devoted to the Horrocks-Barth construction and the Drinfel’d-Manin description of instantons. Chapter VIII gives a brief discussion of Euclidean instantons, and reviews some recent results (Bourguignon-Lawson-Simons, Witten, K. Uhlenbeck).
Although it is not always easy to read, because of the brevity of exposition, the book is an excellent introduction to this difficult area of mathematical physics, and should be perused by any serious researcher in gauge theories. It stresses the importance of modern mathematics for physicists, and shows mathematicians that there is something to be learned from the “heuristic” reasoning of physicists, and that the dialogue between the two disciplines is fruitful for both.
Reviewer: M. E. Mayer

MSC:
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C28 Twistor methods in differential geometry
53C80 Applications of global differential geometry to the sciences
58D15 Manifolds of mappings
58D30 Applications of manifolds of mappings to the sciences
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
81T13 Yang-Mills and other gauge theories in quantum field theory
81T08 Constructive quantum field theory
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L05 Holomorphic bundles and generalizations
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
58Jxx Partial differential equations on manifolds; differential operators
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems