3. — Linear equations in a Yang-Mills background.

As we saw in Chapter IV a vector bundle E on 8i with anti-self-dual connection lifts to give a bundle E on P3(C) which has a natural holomorphic structure. As in the preceding section we can then form the sheaf coho­mology groups S1(P3, E(— m)) or their counterparts over an open set. It is natural to look for an interpretation of these vector spaces in terms of the original bundle E on 8*. When E is a trivial bundle the results of section 2 tell us that we get spaces of solutions of certain standard linear differential equations. These differential equations have obvious counter­parts for a non-trivial bundle in which the derivatives are replaced by ap­propriate covariant derivatives built from the connection. It is then ex­tremely plausible that our sheaf cohomology groups will correspond to these covariant differential equations, and in this section we shall show that this is indeed the case.

As mentioned in Chapter IV the vector bundle E is trivial along each line Px, for xeS4, and furthermore is trivial along the first formal neigh­bourhood P(-x\ We shall first give the proof of this statement. In fact if V is any holomorphic vector bundle on P3, defined in a neighbourhood of a fixed projective line P1? triviality of V restricted to P1 implies triviality of V restricted to To see this we consider first the exact sequence of sheaves where 0 is the sheaf of holomorphic functions and J is the ideal sheaf of P1 (consisting of functions vanishing on PJ. The three sheaves in (3.1) can then be equivalently described as follows:

0/J = 0(P1), holomorphic functions on P1 0/-72 = 0(P^)), holomorphic «functions» on

J*jJ = sheaf of sections of the co-normal bundle N* of P1 in P3.

If /e 0/J2 then it has a value on P1, namely its image in O/J, and extra components corresponding to the first-order terms in the Taylor ex­pansion normal to P1. These first-order terms lie in J2/J.

Since the normal bundle N ^ L~l© we have N*   L and so

in particular

 

 

(see § 1). From the cohomology exact sequence of (3.1) this implies that the global sections of O/J2 map isomorphically onto those of O/J: in other words a global function on P^1' is constant, as we should expect.

where F<°>, F(1) denote the restriction of F to P1 and P[1] respectively. Since F(0) is trivial we again have

Suppose now we introduce the vector bundle F, assumed trivial on Pu and tensor (3.1) with F. We get an exact sequence:

 

 

 

 

 

and (3.2) then implies as before that global sections of F(1)map isomorphically onto global sections of F(0). This means that a basis of (constant) sections of the trivial bundle F(0) has a unique extension to F(1), showing in parti­cular that F(1' is trivial as claimed.

When F = E arises from a bundle E on 8i as in Chapter IV the fibre Ex can be identified with the space of global sections of F^0) (i.e. the restriction of F to Px). Global sections of give vectors in the first jet space of sections of E at x (i.e. including first derivatives). The fact that every global section of F^0) extends uniquely to a global section of F™ means that we have a unique way of extending vectors from Ex to i.e. we have a con­nection on E. This is essentially Ward's definition of the connection as explained in Chapter IV. Note that the property of quadric cones used in Chapter IV, § 3 for Ward's construction is equivalent to the vanishing of H1(P3, 0(— 1)) as one sees by an exact sequence argument; a more ele­mentary argument can however be used based on the fact that a quadric surface factorizes as P1xP1.

We turn now to the question of interpreting the sheaf cohomology groups S1^, E(— m)) where &cP3(C) corresponds to an open set TJ c $4. We begin by considering the case m > 2. Then for any x e TJ, an element

 

(since E is trivial on each Px). In other words <p is a section of E @ Wm where Wm is the vector-bnndle (with fibre 0m_1) which we met in the previous section. For m > 2 there is a first-order differential equation for sections of Wm whose solutions correspond to elements of B1^, 0(— m)). This differential equation may be identified by computing in the first formal neighbourhood       Repeating this computation with E(— m), using the

triviality of E<£\ and the way in which this triviality defines the connection on E, it follows that <p satisfies the obvious differential equation for sections of E @ Wrn. For example if m = 3, Wm is the 2-component spin bundle, the equation is the (mass-less) Dirac equation and the equation for E®W3 is the Dirac equation extended to E using its connection. In more physical terms this is the Dirac equation in a Yang-Mills background field.

For m< 2, the equations are again first-order, and a similar but slightly more complicated argument leads to the same conclusion.

Finally we come to the case m = 2 which is both the most interesting and technically the most complicated since it involves a second-order oper­ator, the Laplacian. General considerations, and computations of the above type, lead to the conclusion that elements 0eH1(U,E(— 2)) correspond to sections of E(g)W2 over TJ (where W2 is the line-bundle of fourth-roots of densities as in Section 2) which satisfy a second-order differential equation. Using the triviality of E on P^ identifies the first-order and second-order parts of this equation but leaves unidentified the zero-order part. A final calculation is therefore necessary to determine this zero-order part. The most elegant and informative way of performing this calculation is explained in detail in [29]. The idea is to exploit the inter-relation between the dif­ferent cohomology groups 31(U, E{— m)) for various m, arising from the fact that multiplication by each of the linear coordinates            maps

E(—m) to E{— m -f- I). Formally we note that      are a basis of

the holomorphic sections over P3{C) of the bundle Irx. Converting this statement into one on 84 provides a link between the differential equations for various integers m. Knowledge of the equation for m # 2 can then be used to help identify the equation for m = 2. In more concrete terms our second-order operator has to annihilate elements of the form za<p where q> satisfies the appropriate Dirac equation (corresponding to m = 3). As shown in [29] this information is enough to identify our equation as

 

 

where V is the covariant derivative of <p, V* its adjoint relative to the standard metric of 8і, and q> is now considered as a pure scalar function (densities being identified with scalars by using the volume form of 8і).

An important consequence of (3.3) is that it has no global solutions on P3 except (p = 0. This is because the operator V*V + P/6 is positive just as in Section 2. We conclude that

 

 

Unlike the case when E is trivial (3.4) is not a general result of algebraic geometry. There are many vector bundles V which do not satisfy (3.4). All that general theory tells us is that

 

 

where m0 is a sufficiently large integer. The proof of (3.4) depends crucially on the fact that E comes from a unitary bundle on 8*.

For m> 2 by composing the first-order operator with its adjoint one gets a second-order operator which can also be shown to be positive. This leads to the further result that:

 

 

However, as we shall see in Chapter VII, (3.5) follows easily from (3.6). The reason is that multiplication by a monomial of degree m — 2 in zlt ..., shifts us from fl"1(P3(0), E(- m)) to fl"1(P3(0), E(- 2)).

So far we have explained that sheaf cohomology groups correspond to solutions of various linear equations in the background field of Yang-Mills instantons. For quantum calculations one needs, not only the solutions of the homogeneous linear equation, but the full Green's function or «propa­gator ». Using the explicit description of instantons given in Chapter II explicit and fairly simple formulae have been derived for some of these Green's functions [13] [15].