1. — 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 O such as 8U(2) or 8U(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 8 is 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 quan- turn 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. These are

external symmetries under the Lorentz and Poincar6 groups and under the conformal group if one is taking the rest-mass to be zero,

internal symmetries under groups like 8U(2) or 8U(3) to account for the known features of elementary particles,

covariance or the ability to be coupled to gravitation by working on a curved space-time.

Gauge theories satisfy these basic requirements because they are geometric in character. In fact on the mathematical side gauge theory is a well estab­lished branch of differential geometry known as the theory of fibre bundles with connection. It has much in common with Eiemannian geometry which provided Einstein with the basis for his theory of general relativity. As is well known Einstein spent many years on a fruitless search for a unified field theory, a search which most physicists regarded as a chimera. If the current expectations of Yang-Mills theory are eventually fulfilled, it will in some measure justify Einstein's point of view that the basic laws of physics should all be combined in geometrical form.

Gauge theory first appeared in physics in the early attempt by H. Weyl [43] to unify general relativity and electro-magnetism. Weyl had noticed the conformal invariance of Maxwell's equations and sought to exploit this fact by interpreting the Maxwell field as the distortion of relativistic length produced by moving round a closed path. Weyl's inter­pretation was disputed by Einstein and never generally accepted. However after the advent of quantum mechanics with its all-important complex wave-functions it became clear that phase rather than scale was the cor­rect concept for Maxwell's equations, or in modern language that the gauge group was the circle rather than the multiplicative numbers. Unfortunately, while scale changes could be fitted into Einstein's theory by replacing the metric with a conformal structure, there was no room for phase to be incor­porated into general relativity. Eather the gauge theory had to be super­imposed as an additional structure on space-time and the unification sought by Weyl then disappeared.

Non-abelian gauge theories were introduced in 1954 by Yang and Mills [32] and have been increasingly studied by physicists since that time. The relation with the mathematical theory of fibre bundles was either

ignored or considered irrelevant until comparatively recently, when non- perturbative aspects related to instantons have come to the fore. Mathemati­cally this involves global questions of fibre bundle theory incorporating both topology and analysis, as opposed to the purely local theory of classical differential geometry. A great deal of modern geometry of a sophisticated character is involved in dealing with such global problems and the techniques developed by mathematicians are unfamiliar to physicists. One purpose of these lectures is to try to bridge the gap between mathematicians and physicists by explaining the relevant techniques as simply as possible and illustrating how they apply to the determination of instantons. The fact that so many new mathematical tools are naturally involved with this prob­lem may lead to some optimism concerning the ultimate aim of developing the quantized form of gauge theories.