2. - The basic instanton.

Using the quaternionic notation of the preceding section we shall now exhibit the basic instanton with fc = -J- 1 (and the anti-instanton with

k= — 1).

Consider the 8U(2)-potential A defined by

 

 

The explicit components AM(x) can of course be read off from this succinct formula, for example

 

 

Computing the curvature F of this potential as in (1.4) we get

 

Substituting this in (2.2) and simplifying we get the purely imaginary ex­pression :

 

 

so that, as explained in § 1, F is anti-self-dual, i.e. *F = — F. As \x\ oo we see from (2.1) that

 

 

where <p(x) = xj\x\. This shows that A is asymptotically the gauge trans­form of 0 by the gauge transformation g{x) = rp(x), or equivalently that if we apnly the inverse gauge transformation 9?(ж)_1 to A we get 0 asymp­totically. On the unit sphere \x\ = 1 in quaternion space we have cp(x)-1 = x and the map x —> x of Л'to itself has degree — 1. Thus (2.1) describes an anti-instanton.

Clearly if we replace ж by ж throughout we will obtain an instanton with potential and field given by

 

 

These formulae are of course just those of Belavin et al. [10] written in quaternionic notation.

Writing |«|2 = xx the middle term in this expression gives

 

To examine more carefully the behaviour of the anti-instanton (2.1) as \x\ 00 we change gauge by            corresponding to (2.4), and intro­

duce the quaternion coordinate y = x~1 around the point at 00, regarded now as a point of S*. Taking the imaginary part of the identity

 

 

shows that the anti-instanton extends to S4 and has precisely the same form at 00 as it has near 0. Similar calculations hold naturally for the in­stanton (2.5).

If we simply put у = аг1 in (2.1) we get

 

 

and this describes the anti-instanton in the «singular» or « asymptotic » gauge, namely the gauge in which A - > 0 as \y\ -> co, but which is badly behaved at у = 0, where A(y) is singular. As we have seen this singularity can be removed by the appropriate gauge transformation, but the sin­gular form (2.6) is useful in practice.

It is perhaps worth pointing out that all the above formulae hold for complex numbers instead of quaternions, in which case we obtain 77(1)- gauge potentials and fields on Ii* or S2. Self-duality no longer makes sense in dimension two but the 2-form F given by (2.3) is a multiple of the invariant spherical area. In other words F is invariant under $77(2) acting by frac­tional linear transformations x-+(ax + b)(cx -f- d) of the complex variable x.

Since the equations *F = ± F are conformally invariant it follows that any conformal transformation of $4 into itself will convert the basic instanton into some other instanton. We recall now that, just as the proper (i.e. orientation preserving) conformal group of 8і is 8L(2, 0)/{± 1} acting via fractional linear transformations of a complex variable, so the proper conformal group of 8і is SL(2, B.)j{-V_ 1} acting similarly on a quaternionic variable. Thus the transformations x axb with x, a, b all quaternions and аф0, ЬФ0 generate the rotation group $0(4) together with scale change; x-+x-j-c gives translations while xIjx ==-- x/\x\2 gives a proper inver­sion (i.e. inversion together with a compensating reflection x-+x to restore orientation). Because of the non-commutativity of the quaternions every proper conformal transformation of $4 can be written either using left multiplication or using right multiplication, i.e.

 

 

Taking quaternionic conjugates interchanges these two ways of representing the conformal transformations. More precisely let $, T denote the two transformations above with (a, /?, y, 6) = (a, b, c, d) and let G denote con­jugation         then T = C8G. Note that G, being a reflection, is an im­proper conformal transformation.

We return now to the basic anti-instanton (2.1) and we apply conformal transformations to it. Recall that, up to a gauge transformation, (2.1) is

preserved by the inversion x --> ar1. It is evidently unchanged by x --> ax with \a\ = 1 and x -> xa produces only a constant gauge transformation. Thus it is essentially invariant by $0(4). In fact it is invariant up to gauge transformations by the larger group $0(5) which may be viewed here as $P(2)/{±1} where $p(2) c 8L{2, E) is the compact subgroup leaving norms fixed. This verification is best left till later when the proper geometric interpretation of (2.1) will make this invariance evident. To get new anti- instantons therefore we should use elements representing SL(2, II) mo­dulo 8p(2). Such elements are naturally given by the transformations

 

 

where [i is a positive real scalar and b is a quaternion.

These parameters can be regarded as parametrizing 8L(2, E)/8p(2), the space of quaternion norms on II'1 with volume 1, by associating to (fi, b) the positive self-adjoint matrix

 

 

with juv- \b\2 = 1.

As [i, b vary the transformation (2.7) applied to (2.1) generates a 5-par- ameter family of anti-instantons with « centre » b and « scale » fi. From (2.3) we see that the field density is a maximum at the centre and its strength there is [i2. This shows that no two members of our family can be gauge equivalent. The more difficult result, which will emerge much later, is that every anti-instanton (i.e. it = — 1) is gauge equivalent to one of our family.

It will be convenient to apply inversion to (2.7) (and a sign change) to (2.7) to get

 

 

This transformation applied to (2.3) gives us the general anti-instanton in an asymptotic gauge as explained before. We can also apply it to (2.5) to generate the family of all instantons in an asymptotic gauge.

We now come to the more difficult question of constructing multi- instantons for larger values of Tc. For this purpose we introduce the space Hk consisting of column vectors u with quaternion components ua (a = 1,..., Tc), and we define an $U(2)-potential on the space IIk = Iiik by a formula quite similar to (2.1), namely

 

where «* stands for the transposed conjugate of the column vector u

and u*du stands for the matrix product, so that

 

 

and ]«|2 = u*u = 2 lM«l2 is the Euclidean norm.

X

For our function f(x) we now take the matrix analogue of (2.8), combined with a conjugation (to give instantons rather than anti-instantons), namely

 

 

(II) For every xeH the equations

 

Note that (2.9) restricts to (2.1) on each coordinate axis (all ua= 0 except a = fj) and is unchanged by the group Sp (7c) acting on IIk. Thus it restricts to (2.1) on any one-dimensional -ET-subspace of IIk. It has there­fore a high degree of symmetry and we shall in due course explain its geom­etrical significance. For the present we regard (2.9) as simply an auxiliary formula used to construct potentials on H = R* by using suitable functions u = f{x), i.e. maps/: II- >IIk. Given any such / we substitute in (2.9) and obtain the potential

 

 

Here B is a symmetric kxk matrix of quaternions, A is a row vector (Alt..., Ak) of quaternions and x stands for the scalar quaternion xl where I is the unit kxk matrix. For k = 1 the parameters were arbitrary except that A had to be invertible. In the general case however the parameters A, B will have to satisfy algebraic constraints as follows:

Condition (I) asserts that the coefficients of i, j, k in the kxk quaternion matrix B*B -f A* A all vanish. This gives a system of quadratic relations on the coefficients of B and A.

Condition (II) is a non-degeneracy or open condition which can also be

formulated as saying that the (fc-j-l)xft matrix ^ has maximal

rank fc for all xeH. Condition (I) is the crucial algebraic one which willensure that the potential A^s(x) defined by substituting (2.11) in (2.10) is self-dual. Condition (II) will ensure the solution is non-degenerate and in particular that the points x for which (B — x) is singular give singularities of the potential which can be removed by a gauge transformation.

In principle, given gauge potentials A^s(x) defined as above it is a direct matter of computation to verify that the resulting field F satisfies *F = — F. However the computations can be carried out more elegantly once we have explained the geometrical meaning of some of the formulae. This will be done in the next section. Much more difficult is the proof that our construc­tion gives all self-dual fields. This requires the introduction of quite new ideas and techniques which will be explained in subsequent chapters.

The matrix transformation (2.11) includes, as a specially simple case, the obvious choice of a diagonal matrix B (with diagonal entries bx, ..., bk e II) and real positive scalars Alf ..., A,.. Provided the bt are all distinct, condi­tions (I) and (II) will be satisfied. The resulting ft-instanton therefore looks like a superposition of h instantons with scales A{ and centres b(. These special solutions were discovered by 't Hooft and others [14] [31]. The general solution cannot however be put into this form (even after a con­formal transformation).

If we replace the vector A = (Aj, ..., /.,,) in (2.11) by qA where q is a quaternion of unit norm then the resulting potential A given by (2.9) is unaltered. Similarly if we replace A by AT and B by T~XBT, where T is a (real) orthogonal TcXlc matrix, then the potential gets conjugated by the constant matrix T which simply gives a gauge transformation. Eote that both these alterations of (A, B) preserve the conditions (I) and (II) above. We shall in due course prove that no other transformations of the par­ameters (A, B), except those just described, give gauge equivalent potentials. Thus the main theorem to be proved can be stated as follows [2] [18] [19]

Theorem. Every h-instanton for 8U(2) arises from parameters (A,B) satisfying (I) and (II), the potential being given in an asymptotic gauge by formula (2.9) where u{x) is defined by (2.11). The potentials defined by (A, B) and (A', B') are gauge-equivalent if and only if A' — q/.T, B' — T~XBT with qeSp (1) and TeO(k).

real parameters. The number of real equations involved in (I) is 3fc(fc — l)/2 while the groups 8p( 1) and 0{Tc) have dimensions 3 and ^7c(7c — 1) respect-

It is a simple matter to count the number of effective parameters in­volved in our construction. The initial data of a pair (A, B) involves

 

 

 

 

as the number of effective parameters. This checks with the calculation by infinitesimal variation methods [3] [37]. If we start from a't Hooft solution (B diagonal, 1 real) and consider small perturbations of the solution these parameters can be interpreted as follows. Each bt has 4 parameters and each l{eE has 4 parameters, giving 8Tc in all: we subtract 3 because of the action of Sp(l). However we cannot satisfy condition (I) with B diagonal and the Aj not real. What happens is that condition (I) requires non-diagonal terms to enter in B so as to cancel the imaginary contributions from /. Near a 't Hooft solution one can actually solve by power series expansion for these off-diagonal terms so that (bt... bk, ^ ... Ak) modulo 8p(l) give local parameters for the space of Tc-instantons. However these are not global parameters and the global structure (even topologically) of this « moduli» space is quite complicated (see [5]).

 

We then define the 7c x «.-matrix IJ as a function of x e II by

 

Finally the potential A(x) is defined in terms of TJ(x) by a formula general­izing (2.9):

 

For the group 8p(n) (consisting of n X n quaternion matrices which also preserve the norm of S") there is an entirely analogous solution of the multi-instanton problem. We replace (1, B) by (A, B) where A is an n x 7c- matrix satisfying the analogues of (I) and (II), namely

where a = (1 + 11*11)-* is now a self-adjoint n X w-matrix. Note that, when n = 1, a is a real scalar and (2.13) coincides with the alternative form (2.9).