2. — Linear field equations

Equation (1.2) asserts that a holomorphic bundle over P^G) with group G (the additive group) is holomorphically trivial: by exponentiation this cor­responds to bundles with group G* (the multiplicative group) for which the topological invariant ft = 0. We can also consider G as the matrix subgroup of translations in SL(2, G), i.e. matrices of the form

 

 

Then (1.2) says that holomorphic bundles for this group over PJO) are trivial. Geometrically a bundle for this matrix group corresponds to a 2-dimensional holomorphic vector bundle E over Pi(O) having a sub line- bundle N such that N and the quotient EjN are both trivial. The solubility of (1.2) then asserts that N always possess a holomorphic complement. Recall that, unlike unitary bundles, holomorphic complements do not ne­cessarily exist. In fact (1.3) corresponds to a slightly more general situa­tion of this type. To see this note that eq. (1.3) is equivalent to the matrix equation

 

 

If we regard the left-hand side as the gauge transformation, near \z\ = 1, defining a 2-dimensional vector bundle E over Pi(O), then E has a trivial sub-bundle N with EjN         and solubility of (1.4) is equivalent to the existence of a holomorphic complement to N. Thus for m>2 such a com­plement does not generally exist. More precisely any such bundle defines an element of the vector space i?1(P1,im) and a complement exists only if this element is zero. More generally H1(P1, Lm) classifies the extensions E of the trivial bundle by L~m, two such extensions being regarded as equivalent

if there is an isomorphism between the bundles which is the identity on the sub-bundle and quotient bundle.

It is called a 1-coboundary if there exist holomorphic functions hx defined in Ua so that

 

After this introduction we shall now give the precise definition of sheaf cohomology groups on any compact complex manifold X, e.g. P3(G). We begin with the simplest case, corresponding to (1.2). Instead of just the 2 open sets \z\ < R and r < \z\ which cover Pi(O) we now have to allow a finite number of open sets {U.«} to cover X: for P3( G) we could take the 4 sets given by 0 (a = 1, 2, 3, 4). A holomorphic 1-cochain is then a col­lection of holomorphic functions gap defined in Ua O Up (conventionally gpa — — gap). Such a 1-cochain is called a 1-cocycle if it satisfies the transi­tivity condition

 

 

Every 1-coboundary is evidently a 1-cocycle but the converse need not be true. We measure the effect by considering the quotient space H1 of 1-cocycles modulo 1-coboundaries. If the open sets Ua are sufficiently small and well-chosen (technically all finite intersections should be domains of holomorphy) H1 is independent of the covering. It is called the first coho­mology group of H with coefficients in the sheaf of holomorphic functions and is denoted by H^X, 0).

We cannot pursue cohomology theory in any detail, for which we refer to standard texts, and we confine ourselves for the present to a few further comments.

For Pi(G) we used just 2 open sets, hence there was only one gap and every 1-cochain was automatically a 1-cocycle.

H1(X, 0) always classifies holomorphic bundles over X with group G or equivalently extensions of the trivial line-bundle by itself.

For every integer g>0 one can define H"(X, 0): H°(X, 0) consists of global holomorphic functions on X (which are constant for compact X). The cohomology groups are linked together by exact sequences.

If X is not compact similar definitions work but one must allow infinite coverings which are locally finite.

5) For X compact algebraic, as for Pm(G) for any m, cohomology groups can be computed by using only rational functions. This is one of the main results of Serre [38].

It is implicit in our notation that other coefficients than the holomorphic functions 0 can be used to form cohomology groups. Thus in the case of Pi(0) the cohomology group we denoted by H1{PX, Lm) uses local holomorphic sections of the holomorphic line-bundle Lm. Clearly the definition above of j? goes through unaltered for any X and any holomorphic line-bundle L or even vector bundle E over X. The cohomology group H1(X, L) can be viewed as classifying extensions of the line-bundle L by the trivial line- bundle. More generally holomorphic vector bundles E of any dimension which have a given sub-bundle E1 and quotient EjEl = E2 are classified by Hx{X,El (x)Ex) where -E* is the dual bundle of E2.

If X is compact all Ha of the type considered here are finite-dimensional vector spaces. This ceases to be true for non-compact X. For example H°(X, 0) consists of all holomorphic functions on X. Examples with H1 being infinite-dimensional -will occur naturally in our applications as we shall see in the next section.Before leaving the case of Maxwell fields we should comment on one spe­cial topological feature, related to the bundle interpretation, which does not generalize to the other cases. If we work with some open set U c P4 and the corresponding open set U c P3(G) then holomorphic line-bundles on U which are trivial on the fibres of P3(G) - > Si correspond to anti- self-dual C*-potentials on U. If U is Ri or a contractible open set then the homology of U is generated by the fibres and the line-bundle on U is then topologically trivial. However if U has 2-dimensional homology, e.g. if U = Rl— R1 then the bundle on V need not be topologically trivial and the corresponding bundle (with connection) on U is then also non- trivial. We cannot in such a case take the logarithm of the gauge trans­formations to reduce from 0* to G. Thus the geometrical correspondence between bundles given in Chapter III contains more information than the correspondence described above between Hl(B, O) and solutions of the anti-self-dual Maxwell equations on U.

The cohomology groups H1(U, 0(— m)) fall naturally into two families according as m> 2 or m< 2. The case m = 2 is in a sense the most basic and the situation is essentially symmetric about this case, as we shall see. For example Hx{0, 0(— 4)) will turn out to correspond to solutions of the self-dual Maxwell equations on V.

 

As explained in Section 1

 

We begin therefore with the case m = 2 and we take any element QeH^V, 0(— 2)). If Px is the fibre of P3(C)~>Sl over the point xeUcS4 we can restrict 0 to Px to get an element

so that if we fixed, in some standard way, a basis of this space <px would become a scalar function of x, defined for x e U. If V cP4 then fJcP3{G) - — Pi(C) has a natural map to Px(C), projecting along parallel planes, and this identifies all Px with x e U. Thus in this case <px gives a scalar function <p defined in TJ. Analogy with the Maxwell fields suggests that cp should satisfy a differential equation and the only candidate with the necessary inva- riance properties is the Laplace operator of Rl. In fact the correspondence <P>(p establishes an isomorphism between HX{U, ©(— 2)) and the space of solutions of the Laplace equation in V. When V = Rl both spaces are acted on by the 11-parameter group of conformal motions of R'1 (the Eu­clidean motions together with scalar magnification about an origin) and the isomorphism is compatible with these actions. In particular the subspace

of H^P^C)- P1(C), 0(- 2)) which is homogeneous of degree n, nnder scale change, corresponds to homogeneous polynomials of degree n on P4 which satisfy the Laplace equation. We shall indicate how this statement can be verified.

Once we have fixed an origin in P4, P3(0) — Pi(0) can be viewed as the normal bundle N of the corresponding projective line. The bundle L @N is trivial or equivalently N ^L* (x) 8", where S~^.C2 is a fixed vector space. As explained in [3] (where a different notation is used), 8~ can be naturally identified with one of the half-spin spaces of R4. Holomorphic functions on N which are homogeneous, in each fibre, of degree n form a sheaf over P1 isomorphic to the sections of Ln @ (Tn(»S'~), where ST" denotes the polynomials of degree n (so (fn(>S'~) ^ 0n+1). Since we want to compute H\N,L2) = H1(N, 0(— 2)) we tensor with L2 and deduce a map

 

 

where the suffix n on the right denotes the part of the cohomology which is homogeneous of degree n. Ey general theorems one can deduce that (2.1) is actually an isomorphism. Now we saw in Section 1 that 31(P1, Ln+2) was of dimension n -f- 1. With a little more care (see section 4) one can naturally identify it with ffn(*S'+), where $+ is the other half-spin space of R*. Now 8+ (x) 8~ ^ Rl (x) C, the complexification of Ri. Hence (2.1) shows that the space of polynomials (p corresponding to elements 0eH1(P3— P1, 0(— 2))„ consists precisely of the image of

 

 

The classical invariant theory of 80(4) then shows that the image of (2.2) is just the space of harmonic polynomials; for example when n = 2 the left side of (2.2) has dimension 9 while the right side has dimension 10, containing

the extra polynomial r2 = 2 •

The preceding piece of algebra can be used as a basis for the general correspondence 0 <-> <f by associating to each of 0, <p a power series expan­sion in homogeneous terms. Appropriate analytic questions of convergence have then to be examined but these can be dealt with by standard methods. Note that such convergence questions are irrelevant when we work on the whole of 84 since all spaces are then finite-dimensional.

If we work on the whole of S4 the different fibres Px cannot all be iden­tified (the bundle P3(0) S4 being topologically non-trivial) and so more care has to be taken in the interpretation of the correspondence 0<r-xp.We must now view <p as a section of the line-bundle W over $4 whose fibre Wx is the one-dimensional space E1(PX, 0(— 2)). One can show that Wi is the volume of density bundle on 84: thus q> is the fourth root of a density. The Laplace operator has a conformally invariant counterpart [34] which acts on such fourth-roots of densities and Ф <-> <p is an isomorphism between B1(P3, 0(— 2)) and the space of solutions of this conformally invariant Laplace equation on 84. Written in terms of the standard metric of 8* this operator is A -f- В/6, where A is the Laplace-Beltrami operator d*d of $4 and В is the scalar curvature [34], which vanishes on P4 but is po­sitive on 8i.

Since A >0 as an operator, A + P/6>0 and so the equation (Л + P/6)<p = 0 has no global solutions on 8i except <p = 0. This checks with the well-known result of algebraic geometry that B1(P3, 0(— 2)) = 0: in fact the same is true for all 0(— in).

In a later section we shall have more to say about the Laplace operator in relation to the Penrose transform. Beturning for the present to the inter­pretation of other sheaf cohomology groups, we get a very similar picture for the groups H1(U, 0{— mj) m> 2. Any element Ф of this cohomology group gives rise to an element q>x e H1(PX, 0(— m)) for all x e U. As shown in section 1 this vector space (in which <px takes its values) has dimension m— 1. Thus, for U cB*, (p is an (m— l)-component function and it sa­tisfies a first-order differential equation. For m = 3 this is the (mass-less) Dirac equation and for m = 4 it is the (self-dual) Maxwell equation. All these can be exhibited in conformally invariant form.

For m < 2 we have already met the case m = 0 in a different context. For other values one has to proceed slightly differently. Eestricting Ф to any Px now gives zero since H1(PX, 0(— m)) = 0 for m<l. However the isomorphism (2.1) still holds with 2 replaced by m. In particular taking n = 2 — m we see that

 

 

Based on this isomorphism we can now assign to any 0eH1(P3— P1, 0(— m)) a ft-component function q> on B*, where Tc = 3 — m. This is done as follows. Given x e B* we take this as origin for scalar magnification and then pick out the component m of 0. By (2.3) we get a vector in the ft-dimensional space $2~m(8~). An alternative description is to consider the r-th formal neighbourhoods P^ of the line Px: this means that we include not only values on Px but also normal derivatives up to order v. Taking v = 2 — m we restrict 0 to give an element

 

 

It is not hard to show that this is equivalent to the previous definition.

Again one shows that the functions (p arising this way are precisely the solutions of the appropriate first-order differential equation. For m = 1 we again get the (mass-less) Dirac equation but this time for the opposite type of spinor from the case m = 3. In general the equations for m and 4 — m are essentially the same, except that they are of opposite type i.e. switch when we reverse orientation.