4. - The 't Hooft Ansatz.

This section is a digression in which we shall explain how solutions of the anti-self-duality equations can be constructed from solutions of linear equations. This point of view was explained in [4] and it leads to the 't Hooft Ansatz and natural generalizations of it.

We shall consider the group 8L(2, G), ignoring for the present reality constraints. Equivalently we work with 2-dimensional vector bundles with a skew form. According to the results of Chapter IV anti-self-dual 8L(2, G) connections on U c 8* correspond to holomorphic vector bundles on UcP3(C) which are trivial along all fibres Px for xeU. One way to construct such vector bundles is to produce extensions of line-bundles, in other words to use only the triangular matrices in 8L(2, G) for gauge trans­formations. As explained in Section 1 such extensions are classified by ele­ments of a sheaf cohomology group. Thus we can construct 8L(2, G) bundles by first picking a line-bundle N on U and then choosing an element 0 of H 1(U, N'2). This gives a vector bundle V with N as sub-bundle and N* as quotient.

In particular we can take N to be the standard line-bundle L or one of its powers Lm. Then to every 0eH1(U, L2m) = Sx(tj, 0(— 2m)) we get a holomorphic vector bundle over U. To descend to 8i we need this vector bundle to be trivial on each Px with x e U. To investigate this question we consider a fixed projective line P1 and ask which elements of S1(P1, 0(— 2m)) give the trivial bundle. Note for example that the zero element gives the direct sum LmQ L which is never trivial (for m¥= 0). As we shall see, a general element always gives the trivial bundle but special elements give rise to direct sums Lr@L~r for r = 1,2, ..., m.

For simplicity consider first the case m = 1, then H1(P1, 0(- 2)) is one-dimensional and every non-zero 0 gives the trivial bundle [1]. Consider next the case m = 2, then 31(P1, &(— 4)) is a 3-dimensional space. In Section 1 we saw that a basis is given by the transition functions (or co- cycles) s, z2, z3. A more invariant description of the space H1{P1, <D(— 4)) is to say that it is the dual of the space of quadratic forms in the two homo­geneous coordinates (zx, z„) of Px. This arises because of the multiplication

 

 

Using our basis it is easy to verify that this multiplication gives the required duality. This is in fact a very special case of the much more general duality theorem of Serre [39]. Thus if Px= PX(F), i.e. V is the copy of C2 whose lines are represented by points of P1? we have a natural isomorphism

 

 

Hence elements 0eH1(P1,6(— 4)) represent quadratic forms on V*. Then 0 defines the trivial bundle over Px if and only if the corresponding quadratic form on V* is non-singular (i.e. not a perfect square) [1]. If 0=/=O but corresponds to a perfect square the bundle is isomorphic to L © L1.where Xi e C and u( is a linear form on V*. The general result proved in [1] is that 0 defines the trivial bundle unless f can be expressed as a combina­tion of less than m—2 perfect powers (or is a limit of such). More generally if f is a combination of m — 2 — r perfect powers (or a limit of such), with r being maximal, then 0 defines the bundle Lr@L~r.

We know therefore the precise conditions to be imposed on an ele­ment 0eH1(U, ©(—2to)) in order to get a trivial bundle on all Px with xeU and so to descend to a bundle on U with anti-self-dual 8L(2, C)- connection. By the results described in Section 2, 0 corresponds to a section of the vector bundle {T2m-2($+) satisfying the appropriate linear differential equation. Here S+ denotes the half-spin bundle corresponding to instantons, whereas 8~ corresponds to anti-instantons. We have also used the metric on 8i or Ri to identify 8+ with its dual ($+)*, although to exhibit full con- formal invariance this should not be done.

To sum up we have given an implicit construction for anti-self-dual 8L(2, C) connections starting from a section <p of ff2m_2($+) satisfying the relevant differential equation. To get a solution this section must be every­where «general» i.e. not a combination of less than (m — 2) perfect powers.

For m — 1, <p is a scalar field satisfying the Laplace equation and nowhere zero. For m — 2, <p is a self-dual solution of Maxwell's equations which is nowhere the square of a spinor.

The generalization of this to arbitrary m > 0 is now fairly straightfor­ward. We have a natural isomorphism

 

If 0eH1(P1, 0(— 2m)), let / be the corresponding homogeneous polynomial of degree 2m— 2 on V*. Any such polynomial can be written as a linear combination of perfect powers:

 

The case m — 1 is the Ansatz employed by 't Hooft. The explicit for­mulae for m = 2 are given in [4]. It is important to note that the solution of the non-linear Yang-Mills equations obtained in such a way may have a larger domain of regularity than the linear field (p from which we start. For example it is well-known, for the 't Hooft Ansatz, that a 1/r2 singularity for <f disappears when we construct the Yang-Mills field. Geometrically this corresponds to the fact that the reduction from 8L(2, C) to the triangular matrices may only be valid in a smaller open set. The situation is quite clear if we work on the whole of P3{G) as we shall now explain.

Let W be a 2-dimensional holomorphic vector bundle over P3{G). Although W may have no globally defined holomorphic section, other than zero, general theorems tell us that W(x) L~m = W{m) has a non-zero sec­tion if to is sufficiently large. Let s be such a section, then s = 0 on some closed algebraic subariety E of P3{G) and s^O on the complementary open set. On this open set s generates a trivial sub-line-bundle of W(m) and so exhibits W(m) as an extension. Tensoring back by L"' then shows that, over this open set W has Lm as a sub-line-bundle. If W is an 8L(2, G) bundle then the quotient is necessarily L~m and we are in the standard situa­tion described above. Thus the subvariety E given by s = 0 will appear as a singularity of the extension element 0, or the corresponding linear field (p, but is not a singularity of the vector bundle W.

If to is the least integer for which W(m) has global sections then one can show that E necessarily has complex dimension 1, i.e. it is an algebraic curve. Conversely given any curve E one can describe (a) when it arises from a bundle W as above and (b) how to reconstruct W from E when the condi­tions of (a) are satisfied [23]. In particular when E is connected it uniquely determines W. If E has several connected components Et then W depends on non-zero constants c( (up to a common factor).

When to = 1, E has to be a collection of k -f- 1 disjoint lines (where Jc = c2(W) is the anti-instanton number). If we take these lines to be real, i.e. fibres of P3(C) 8i, we recover the 't Hooft Ansatz depending on {k + 1) points of S1 and corresponding weights ..., ck+1. When to = 2, E has to be a collection of disjoint elliptic curves, whereas for to > 2, E involves rather special curves of high genus.

For the bundles given by the Horrocks construction of Chapter V one can show that one need only take to > V3Jc + 1 — 1. Hartshorne [27] has shown that this bound is best possible in the sense that smaller values will not give all — ft-instantons.

Eestricting to Px and taking cohomology we get an exact sequence of vector spaces:

 

We conclude with a few further comments on the Ansatz for general to. In order to give an explicit formula for the potential one must choose a de­finite gauge and it is desirable to pick the gauge as simply as possible. Geometrically this means that we have to pick a basis of H°(PX, Wx) depend­ing smoothly on a? e It*. Eecall that in the appropriate open set W is given as an extension or exact sequence of bundles

 

 

 

 

 

 

 

 

 

 

Now over Bi the bnndle »S'+ has a natural trivialization and so therefore do ffm(»S'+) and (?m ~2(S+). However the kernel of <px clearly varies with x and so the explicit description of the potential is best given by using the natural basis of the larger space {Tm($+). Note that this only applies for m> 2 because for m = 1 the second space ff'"~2 = 0. These remarks ex­plain the apparent singularity in the formula for m = 2 given in [4]: this occurs because a basis for Ker tpx has been taken by projecting a sub-set of basis vectors of 8+, and different subsets are needed for different positions of x in Bi.