4. — Asymptotic conditions and topology.

We now restrict ourselves to the Euclidean 4-space so that the Yang- Mills Lagrangian £ is positive. It is natural to consider potentials for which the action £ is finite, so that the integral over R1 converges. To achieve this we assume that the field F decays sufficiently fast as we go to infinity. If we work in a given gauge this simply means that F^v(x) -> 0 sufficiently fast as \x\ -r oo. At first sight this might seem to require the gauge po­tential Au(x) to have similar decay together with its first derivatives. How­ever, because of gauge freedom, all that is necessary is that, for large \x\, we can find a gauge transformation g(x) so that the potential in the new

gauge should decay. This means that where ~ implies asymptotic behaviour including first derivatives. The im­portant point is that the gauge transformation g(x) need only be defined for large \x\. In fact it may be impossible to extend the definition of g(x) con­tinuously to the whole 4-space. To see this consider the restriction of g(x) to a sphere \x\ = R of large radius, and take for example G = SU(2). Then g gives a continuous map

 

 

and both sides are topologically 3-spheres. Such a map has a well-defined integer invariant, its degree Tc, which counts (with appropriate multiplicities and signs) the number of points x e 8% which map to a given general value in 8U{2). The function g can be extended continuously to \x\ <R if and only if k — 0. The identity map (thinking of both spaces as the standard Ss) has k = + 1) and k = — 1 corresponds to an orientation reversal.

An analogous and more easily visualized situation occurs for O = U(l) and R* replaced by R2 in which case g becomes a map of the circle to itself and the degree k is the «winding number». Note however that if we work on R" with O = U(l) there is no topological invariant since every con­tinuous map of S3 to the circle can be deformed to a constant map.

For any simple non-abelian compact Lie group O a corresponding result holds, namely that continuous maps of S3 into O have an integer topological invariant which classifies the map up to deformation. This integer arises from the fact that every such O contains copies of 811(2) (or 80(3)) as subgroups.

Thus in non-abelian gauge theories on R1, potentials which are asymp­totically flat, i.e. have fields asymptotic to zero, fall into distinct families indexed by the corresponding integer k. This is frequently referred to as a topological quantum number even though at this stage we are only dealing with classical fields.

In dealing with asymptotic properties various technical analytical que­stions arise concerning the precise rate of decay. There is one natural and convenient definition of decay to take arising from conformal invariance. We recall that stereographic projection of a sphere onto flat space is always a conformal map, relating the standard curved metric of the sphere to the flat Euclidean metric. In particular the sphere 84 (unit sphere in R'a) gets mapped conformally onto R1.Alternatively we can say that S1 is the conformal compactification of B* obtained by adding a point at oo. A potential on Bl is then said to decay at oo if it extends to a potential on 8l. If the integer k is non-zero this means that we cannot describe our potential using a single gauge, we need one gauge in the finite region \x\ <i? and another gauge near oo i.e. for \x\ >i2, the two gauges being related on \x\ = B by the gauge transformation g(x) of degree k. This means that our fibre bundle P over 8* is no longer the topo­logical product 84 X O. In fact the topological theory of fibre bundles over spaces like 84 which are not contractible tells us that in this case they are precisely classified by the same integer k. Thus the integer k which appeared in the asymptotic description on B4 is now directly coded into the topology of the space P. For example if G = $17(2) and k — 1, P turns out to be topologically the sphere S7 which is quite different from the product Slx83: this will be explained in more detail later. The potential or connection in our fibre bundle now has a well-defined curvature F on the whole of $4 and the particular role of the base point oo e $4 can now be ignored. Note that F 0 at oo on B1 but F need not be zero on $4 at the point oo because when viewed as a differential form one uses different coordinates on B1 and 8*. Clearly the Lagrangian £ = ||.F||2 computed relative to the curved metric of 84 is necessarily finite because $4 is compact (we always assume enough local differentiability so that F is always continuous at least). More­over because of the conformal invariance of the Yang-Mills functional £ takes the same value whether computed on S4 or It1.

Now on closed manifolds like 84 there are well known theorems of global differential geometry which relate topological invariants to integral expres­sions in the curvature. One might expect such results because if the curvature is everywhere zero the connection is flat and (on a simply-connected space) one obtains a global gauge implying k = 0. In dimension 2 the classical theorem of Gauss expressing the Euler characteristic as the integral of the scalar curvature is the prototype of higher-dimensional generalizations. In our case the formula takes the form

 

 

for the group 8U(2). For other groups the same result holds but with a different normalization, the integer fc being replaced by a suitable multiple: for precise details the reader may refer to [3]. We now have a topological constraint on F and this should be fed into the formula (3.4) for the Lagran­gian. To do this it is convenient to decompose F under the action of * into

its self-dual part F+ and its anti-self-dual part F :

 

 

so that *F+= F+ and *F = — F (recall *2 = 1 for the positive definite case). Then (3.4) and (4.2) can be written

 

 

from which we deduce that C>87ia|ft| and equality holds if and only if *F = (sign k)F. Thus the special solutions (3.7) and (3.8) of the Yang- Mills equations correspond to the absolute minimum of the Lagrangian (assuming the value 87i2|ft|) is attained. This argument is due to Belavin et al. [10] who also showed that for i; = ±1 the minimum is attained, and the term instanton has been coined for such solutions of the Yang-Mills equation. Later other solutions were discovered for all k [14] [31] and these were called multi-instantons.

The general problem to which we shall address ourselves is the deter­mination, in as explicit a fashion as possible, of all multi-instantons, not only for 8U(2) but for all compact classical groups. As we shall see, this problem admits of a surprisingly simple and complete answer, but the proofs require a great deal of sophisticated mathematical machinery.

At this point we shall merely note that any gauge transform of a multi- instanton is again a multi-instanton, and such solutions will be regarded as equivalent. In geometric terms this means that two fibre bundles over $4 with connections (satisfying *F = ± F) which are isomorphic will be iden­tified: they cannot be distinguished geometrically.

In the next chapter we shall describe explicitly how to write down the most general multi-instanton for $77(2), and we shall indicate how this has to be generalized to other groups.