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Collected works. Volume 5: Gauge theories. (English) Zbl 0691.53003
Oxford Science Publications. Oxford: Clarendon Press. xxiii, 685 p. £45.00 (1988).
This, the fifth volume of Michael Atiyah’s Collected works (so far!), is in many respects the most valuable. This is not a question of the mathematics contained in it being more up-to-date - the index theorem will never be outmoded - but is more concerned with the much wider audience which the author’s work in gauge theory attracted. His earlier work may well have drawn together topologists, algebraic geometers, differential geometers and number theorists at various times but the addition of the community of theoretical physicists to that list has had a profound effect on his more recent papers. This effect relates to speed of publication, presentation of results and location of the relevant material. For years now graduate students and others keen to work on the interrelationship between geometry and physics have been asking “Where do I read about this?” only to be directed to the physics section of the library or to be hold to write to locations in various parts of the world. Now a single book suffices to provide not only some of the standard references and proofs, but also a variety of informal survey articles which set the whole area in context.
The major papers for which this volume is not the natural source are “Self-duality in four dimensional Riemannian geometry”, the Fermi lectures “Geometry of Yang-Mills fields” and “The Yang-Mills equations over Riemann surfaces”. In the first, the author, together with the reviewer and Singer described the self-dual Yang-Mills equations in the natural context of Riemannian geometry in its global form and at the same time the twistor theory of Penrose in this signature. It was written at an intermediate stage in the study of instantons - between the calculation of the 8k-3 dimensions for the moduli space and the later ADHM construction of all instantons using linear algebra. This left the paper at the time in a slightly odd position, providing a general framework for moduli problems involving the self-duality equations but giving way to an explicit construction in the only situation physicists were seriously interested in. Now, in retrospect, it can be seen as providing the framework for Donaldson’s work on general four-manifolds. The Fermi lecture notes provide a very readable account, using in particular the language of quaternions, of the ADHM construction of instantons on the sphere. This work, previously difficult to obtain now appears in the mainstream of mathematical literature. The third large paper, written with Atiyah’s long-time collaborator Bott, studies the geometry and topology of moduli spaces of stable bundles on Riemann surfaces by using Yang-Mills theory. Apart from the explicit results in the paper, it introduces some key ideas into two-dimensional gauge theory, most notably the formalism of symplectic geometry and equivariant cohomology in infinite dimensions.
These long papers make the volume an essential reference but there is of course much more besides including the relationship between Green’s functions for self-dual manifolds and cohomological algebra, convexity questions related to Lie groups, hyperk√§hler metrics on moduli spaces, a beautiful paper on hyperbolic monopoles and the links between anomalies and the index theorem. These are all set in a personal context by the commentary the author includes describing this evolving interest in the mathematical problems arising from physics (or perhaps one should say physicists).
The story, however, only goes up to 1986 and already in the commentary we read about Atiyah’s contacts with the ideas of Witten. We also perceive a recognition of the difference in pace between mathematics and physics. It need hardly be said that the moving finger having writ moves on, and in the last four years the concepts of quantum field theory have come to play a very important role in the study of low-dimensional manifolds, in a way which not even Michael Atiyah himself could have believed when in 1977 he first began to take seriously physicists’ ideas. He has continued to be in the forefront of the evolution of these new methods, but we need to wait a few years for volume 6 of the current work to gain the perspective which this book offers.
Reviewer: N.J.Hitchin

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53-03 History of differential geometry
01A75 Collected or selected works; reprintings or translations of classics
81T08 Constructive quantum field theory
53C80 Applications of global differential geometry to the sciences
53C05 Connections (general theory)