2. — Twistor interpretation of instantons.

In this section we shall show how to interpret the self-duality equations for a Yang-Mills field on 8l in terms of complex analysis on the twistor space PS(G).

is determined by g. Finally to satisfy (i) we must take B — — (B")*. This uniquely determines B and hence fixes the potential. Computing the field F in the holomorphic gauge we have

As an algebraic preliminary we shall need to understand the significance of the equations *a> = ±a> for a 2-form m on Bl in terms of complex co­ordinates. We recall that once we have introduced complex coordinates, identifying Bl with (72, a 2-form co can be expressed in terms of its type decomposition:

 

 

We ask how this decomposition corresponds to the *-decomposition

 

 

according to the eigenvalues ±1 of *. This is just a question of linear algebra and it can be viewed in terms of group representations. Equation (2.2) corresponds to decomposing a representation of $0(4) into two irreducible pieces (each of dimension 3), while (2.1) corresponds to decomposing this representation under the subgroup 77(2). The (2, 0) and (0, 2) components have dimension one while the 4-dimensional representation in type (1,1) has a further decomposition:

 

 

Here a is a multiple of the (1,1) form /i corresponding to the hermitian metric and coj'1 is the «primitive » part, orthogonal to fi. The 3-dimensional representation of 77(2) on this primitive part is easily seen to be irreducible and so this must coincide with one of the two irreducible pieces of (2.2). But the form n corresponding to the metric is self-dual. Hence we must have

of 2-forms on -B4 which are of type (1,1) for all complex structures u is invariant under $0(4), contains Q~ but is not the whole space: it must there­for coincide with Q~.

Thus we have established the following algebraic lemma.

Lemma (2.5). A 2-form co on Bl is anti-self-dual if and only if it is of type (1,1) for all compatible complex structures.

 

In particular this shows that the space Q~ of co with *m = m is of type (1,1) for all complex structures (compatible with metric and orienta­tion). The converse is also true, because the space

 

We shall now apply this lemma to a 2-form co on $4 which we lift to ob­tain a 2-form û> on PS(G). This form w is purely horizontal, i.e. m«0= 0 if a or /3 is a fibre direction. To compute the horizontal part of w at a point u we note the interpretation, explained in the preceding section, according

to 'Which u parametrizes (infinitesimally) complex structures on $4 at the point below. Prom this and our Lemma we deduce

Proposition (2.6). A 2-form m on $4 is anti-self-dual if and only if its lift (5 to P3(0) is of type (1,1).

Note that this proposition is a purely local one, valid for any open set 77 in 8l and its counterpart U in P3((7).

Finally we consider a complex vector bundle E on 8l with a unitary structure and connection. Let F be its curvature. If we lift E to give a bundle E with connection on P3((7) its curvature is just the lift E of F. Ap­plying (2.6) to the matrix coefficients of F we deduce

Proposition (2.7). A vector bundle E on 84 with unitary structure and connection has anti-self-dual curvature if and only if the lifted bundle E on P3(G) with the lifted connection has curvature of type (1,1).

Using (2.7) and theorem (1.2) we deduce the important result

(2.8)    E anti-self-dual-^-E holomorphic.

We plan now to make this statement more precise by specifying which holo­morphic bundles on P3{0) arise in this way and how one gets back from E to E.

Note first of all that, restricted to any fibre Px of P3(C) -> $4, E is holo­morphically trivial, a basis of Ex giving rise to a holomorphic basis or gauge of E\PX. Conversely this shows that Ex can be uniquely defined as the space of holomorphic sections of E\PX.

We turn next to consider the unitary structures on E and E. The unitary structure on E can be given by an anti-linear isomorphism r: E —> E* such that {u, tv) is a positive hermitian form (E* denotes the dual of E). Passing to E we use r to define a lifting f of the conjugation a on P3(G), namely we define a commutative diagram

 

 

The map f is anti-holomorphic, i.e. if we give E* its opposite complex structures it becomes a holomorphic isomorphism. This follows from the fact that x preserves the unitary structure and connection and hence (by the uniqueness part of Theorem (1.2)) it preserves the holomorphic structures.Restricting to the fibre Px our map f induces a similar map on holo­morphic sections of E and in this way we recover rx. Lifting back this gives the unitary structure of E which together with the holomorphic structure yields, by (1.1), a unique connection of type (1,1). Our final step is to show that this connection descends to a connection on_E7: by (2.7) this will neces­sarily be anti-self-dual. The condition that the connection on E descend from PS(G) to 81 is that the curvature should be purely horizontal, i.e. that Fap= 0 if a is a vertical direction (i.e. along the fibres). If a and ft are both vertical this is clear because, restricted to a fibre, E is trivial holomorphically and unitarily. The stronger statement, when only a is vertical, follows from the triviality of E restricted to the first formal neighbourhood of a fibre. More concretely this triviality means that we can pick a gauge of E, near a fixed fibre Px, which is holomorphic and unitary on Px and up to first derivatives normal to Px. The verification of this triviality property of E is best postponed until later (see Chapter VI, § 3) since it involves complex analytical machinery. We note in passing that E is not trivial up to second derivatives unless the whole curvature vanishes.

To summarize our results in a convenient form we shall make the fol­lowing definitions. Let F be a holomorphic vector bundle over Pa(G), then an anti-linear isomorphism p: F->- F* covering a on PS(G) such that

 

 

will be called a real form on F. If F is further assumed to be trivial on all real lines of P3(G) then p induces a non-degenerate hermitian form on the space of holomorphic sections of F restricted to any real line. If this her­mitian form is positive we say that our real form is positive. Two vector bundles F, W with real forms are called isomorphic if there is a complex analytic isomorphism from F to W commuting with p.

Our conclusion can now be stated as follows:

Theorem (2.9). There is a natural (1-1) correspondence between

anti-self-dual U(n)-potentials over 81 (up to gauge equivalence) and

holomorphic vector bundles with fibre G" over Pa(G) with a positive real form (up to isomorphism).

Remarks.

1) This theorem is purely local in character. It holds for any open set U of 84 and its counterpart tl on PS(G).In a purely complex form this theorem is originally due to E, S Ward [42]. We shall later discuss Ward's proof. The present version is stated briefly in [4] and elaborated in [3].

It should be emphasized that in (ii) most of the information is already contained in the holomorphic vector bundle. The positive real form when it exists is generally unique.

Both in (i) and (ii) there is an integer topological invariant. In (i) it is the anti-instanton number, while in (ii) it is the second Chern class. These integers are equal (cf. [3]).

Theorem (2.9) can easily be generalized to the orthogonal and symplectic groups. We shall consider the symplectic case in detail, the orthogonal case is quite similar except for sign changes.

Becall that the compact symplectic group >Sp(n) is the group of norm- preserving automorphisms of the quaternion vector space Hn. It can be identified with the subgroup of 77(2n) commuting with the action of the quaternion j on II" = C'2". Alternatively it is the sugbroup of U(2n) com­muting with the skew bilinear form (, ) on 02n defined by (u, jv) = (u, v) where <,) is the hermitian inner product. Hence an $p(w)-potential can be represented geometrically by a vector bundle IJ with fibre O2" having a unitary connection, together with an isomorphism a: E -^E* which is skew i.e. (u, xv) — — (v, au) and preserves connection (E* being endowed with the connection inherited from E by duality).

Suppose now we have an $p(«)-potential on Sl which is anti-self-dual. Then using (2.9) for 77(2«) we see first of all that E on P3(C) is holomorphic. Moreover the isomorphism a: IJ -> E* induces a holomorphic isomorphism a: E -> E* which is also skew. Combining a with the anti-linear isomorphism a:E-^E* given by (2.9) we obtain an anti-linear isomorphism E-^E covering a on P3(G). We shall denote this map on E also by a: it satisfies <r2 = — 1, and is compatible with a or equivalently with the skew form on E defined by a. Thus an anti-self-dual 8p(n)-potential on S1 corresponds to a holomorphic vector bundle E, with fibre 02n, over P3(C) which has two further structures:

a holomorphic non-degenerate skew form on E

an anti-linear map a: E lifting a on PS(C) such that a2 = - -1 and compatible with the skew form i.e. (au, ov) = (u,v).

Moreover the bundle E is holomorphically trivial on all real lines of P3(G) and the hermitian form induced by (u, ov) on sections of E, restricted to a real line, is positive definite.

In the special case of Sp(l) ^ #/7(2) condition (i) reduces to a topo­logical constraint, namely that the first Chern class <:k(E) should vanish. The non-degenerate skew form is then unique up to a constant factor. Moreover the anti-linear map a: E-^-E is unique (unless h — 0) because two such a differ by a holomorphic automorphism of E and (as we shall see later) E has no automorphisms except scalars: the condition a2 = -1 and positivity of the hermitian form then uniquely fix a. We thus recover the version of the $17(2) theorem given in [4].