These Lectures Notes are an expanded version of the Fermi Lectures which I gave at the Scuola Normale in Pisa in June 1978. They also cover material presented in the spring of 1978 in the Loeb Lectures at Harvard and the Whittemore Lectures at Yale. In all cases I was addressing a mixed audience of mathematicians and physicists and the presentation had to be tailored accordingly. In writing up the lectures I have tried as far as pos­sible to keep this dual audience in mind, and the early chapters in parti­cular attempt to bridge the gap between the two points of view. In the later chapters, where the material becomes more technical, there is a danger of falling between two stools. On the one hand the mathematical jargon may be unintelligible to the physicist, while the presentation may, by mathe­matical standards, be lacking in rigour. This is a risk I have deliberately taken. The initiated mathematician should be able to fill in most of the gaps by himself or by referring to other published papers. Physicists who have survived the early chapters may derive some benefit by being exposed to new mathematical techniques, applied to problems they are familiar with. With this aim in mind I have throughout presented the mathematical ma­terial in a somewhat unorthodox order, following a pattern which I felt would relate the new techniques to familiar ground for physicists.

The main new results presented in the lectures, namely the construction of all multi-instanton solutions of Yang-Mills fields, is the culmination of several years of fruitful interaction between many physicists and mathe­maticians. The major breakthrough came with the observation [42] by E. S. Ward that the complex methods developed by E. Penrose in his «twistor programme» were ideally suited to the study of the Yang-Mills equations. The instanton problem was then seen [4] to be equivalent to a problem in complex analysis and finally to one in algebraic geometry. Using the powerful methods of modern algebraic geometry and the specific results of G. Horrocks and W. Barth it was not long before the problem was finally solved [2].

The first two chapters provide an introduction to the basic concepts, a statement of the problem and an explicit description of the solution. The next two chapters are devoted to the Penrose theory and its application to the Yang-Mills equations. In Chapter V I present Horrocks' construction in algebraic geometry which is equivalent via the Penrose theory to the explicit instanton construction of Chapter II. Chapter VI introduces the important mathematical tool of sheaf cohomology and relates it to physically interesting equations. There are a number of digressions which may help to make the material less mysterious and more understandable. Chapter VII is an account of the theorem of Barth [8] which shows that the Horrocks construction of Chapter II yields all relevant bundles and hence that the construction of Chapter II yields all instantons. Finally in Chapter VIII we discuss some other aspects and open problems concerning the Yang-Mills equations.

Although the presentation is somewhat discursive, and includes much background material, it is also reasonably complete from the mathematical point of view. The one point where the proof is only sketched is the iden­tification in Chapter VI of the sheaf cohomology group H1(P3, E(— 2)) with the solution space of an appropriate Laplace operator. A detailed account of this can be found in various forms in [18] [29] [36]. An alternative presentation of the whole instanton theory is contained in the papers of Drinfeld and Manin [16] [17] [18] [19], and mathematicians, particularly if they are proficient in algebraic geometry, may prefer to read these.

My acquaintance with the geometry of Yang-Mills equations arose from lectures given in Oxford in Autumn 1976 by I. M. Singer, and I am very grateful to him for arousing my interest in this aspect of theoretical physics. We have collaborated since on many topics in this area. I have also, over the past few years, greatly benefited from numerous discussions with E. Pen­rose concerning twist or theory and complex analysis. In developing the mathematical theory of instantons I have throughout worked in close col­laboration with N. J. Hitchin, and these lectures embody the results of our joint efforts. I am in addition greatly indebted to my now numerous friends in the physics community who have helped to give me some small understanding of the fascinating mathematical problems facing elementary particle physics.

Finally I should express my thanks to the Accademia Nazionale dei Lincei and to the Scuola Normale for their invitation to deliver the Fermi Lectures and for their hospitality in Pisa.