Atiyah M. Geometry of Yang-Mills Fields

Pisa, 1979. — 100 p. English. (OCR-слой).[Accademia Nazionale dei Lince! Scuola Normale Superiore. Lezioni Fermiane].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 R. S. Ward that the complex methods developed by R. Penrose in his "twistor programme" were ideally suited to the iltudy 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 VI 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 R. Penrose concerning twistor 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.Physics background.
The aim of quantum field theory is broadly speaking to put all elementary particles on the same footing as photons. Whereas photons appear as the quanta of classical electromagnetic theory other elementary particles should arise by the quantization of appropriate classical field theories. In recent years gauge theories have appeared the most promising candidates, and the Yang-Mills equation is the generalization of Maxwell's equations (in vacuo). The circle group which embodies the phase factor in Maxwell theory is generalized to a non-abelian compact Lie group G such as SU(2) or SU(3), the choice of group being dictated by the empirically observed symmetries of elementary particles. The non-abelian nature of G leads to non-linearity for the Yang-Mills equations. This non-linearity is of course the source of great mathematical difficulties and the quantization of non-abelian gauge theories is still in its infancy.
One recognized way of attempting to develop the quantum theory is to use the Feynman functional integral approach which involves integrating exp (iS) where Sis the action. If we analytically continue to imaginary time, so that Minkowski space gets replaced by Euclidean 4-space, the Euclidean action is a positive multiple of i and so the integrand exp (iS) becomes a decaying exponential whose maximum value occurs at the minimum of the Euclidean action. It is reasonable therefore to ask for the determination of the classical field configurations in Euclidean space which minimize the action, subject to appropriate asymptotic conditions in 4-space.
These classical solutions are the "instantons" of the Yang-Mills theory, and it will be the primary purpose of these lectures to show how to find all instantons. For further explanations of their physical significance, particularly in relation to tunnelling, I refer to [12] or [30]. From a very general point of view one can also say that a thorough understanding of the classical equations is likely to be a pre-requisite for developing the quantum theory, and one may hope that important structural features will appear at the classical level.
If one were to search ab initio for a non-linear generalization of Maxwell's equation to explain elementary particles, there are various symmetry properties one would require.The Yang-Mills Lagrangian.
Physics background.
Gauge potentials and fields.
The field equations.
Asymptotic conditions and topology.
Description of instantons.
Quaternions.
The basic instanton.
Geometrical interpretation.
The Penrose twistor space.
Complex projective 3-space.
Lie groups.
Complex coordinates in R4
Holomorphic bundles.
Holomorphic and unitary gauges.
Twistor interpretation of instantons.
Bundles over P1(0).
Construction of Algebraic Bundles.
The linear complex.
The Horrocks construction.
Quaternionic formulae.
Linear field equations in a Yang-Mills background.
Bundles and sheaf cohomology.
Linear aspects of the Penrose transform.
Linear equations in a Yang-Mills background.
The't Hooft Ansatz.
Relation with Radon transform.
Theorems on algebraic bundles.
Cohomology of the Horrocks construction.
Theorem of Barth.
Reality constraints.
The Drinfeld-Manin description.
Further Problems (Open problems).
Euclidean approach to instantons.
General solutions of the Yang-Mills equations.
References (45 publ).