1. — Complex projective 3-space.

Our attack on the instanton problem will rest on complex analytic methods which are part of the general twistor theory of E. Penrose [35]. Yery roughly Penrose's programme consists in re-interpreting physical space-time data in terms of corresponding data in a space of 3 complex variables. This space is the (projective) twistor space and the transformation of data can be called the twistor transform. The Penrose theory when con­tinued to Euclidean 4-space can be developed in a slightly different way. In this section we shall describe the basic geometrical picture and comment on it from a variety of viewpoints.

As in previous sections we shall use the quaternions R and we shall identify 84 with Px(H), the projective line over the quaternions. We shall use left scalars so that (qx, q2) and (Xqx, ).q2) represent the same point of Pi(R). Taking conjugates (which reverses the orientation) would convert to right scalars.

We identify the complex numbers 0 with the subfield of R generated by 1 and i, and R then becomes identified with O2 by writing quaternions in the form «i + j with z1; z2e G. Similarly R2 gets identified with C1. Now consider the complex projective 3-space P3(G), the space parametriz­ing complex lines (through 0) in C4. If to each complex line we associate the quaternion line it generates, we get a mapLeft multiplication by j induces a transformation a on P3(G) which is anti-linear (i.e. anti-holomorphic in local complex coordinates) and satisfies a1 — 1. In homogeneous coordinates

 

 

Clearly a preserves the fibration (1.1), acting trivially on Рг(11), and acting as the anti-podal map on each fibre (on $2). We shall consider a as defining a «real structure» for P3(G) which is different from the usual real structure given by just conjugating all coordinates. This terminology is standard in algebraic geometry and means that, for some suitable algebraic embedding of P3(G) in PN(0), a will be given by conjugating, in the usual way the co­ordinates of PN(0). As a lower-dimensional example consider a single P^O)- fibre with its anti-podal a. This can be embedded in P2(0) as a conic with equation tDj -f co, = 0. Note that this has no real points, correspond­ing to the fact that a has no fixed points.

Although a on P3(G) has no fixed points it does have fixed lines; these are precisely the fibres of (1.1). We call these the real lines. Thus 8i appears as the parameter space of all the real lines.

 

which give a skew-symmetric matrix characterizing the line joining (z) and (co). The six homogeneous variables p12, pls, ры, p2a, p3i, Vn satisfy one quadratic identity

 

Now we recall the famous Klein representation of all lines of Ps(0). Given two distinct points (za), (co«) of PZ(G) we introduce the Plucker coordinates

(this expression is the square root of det (pxp)). Thus the parameter space of all lines in P3(0) is the complex 4-dimensional quadric c P&{G) given by equation (1.3). A real structure on P3(G) induces a real structure on Qt. For the standard real structure on P3(G) the real structure of is given by conjugating the pxp and so Qt has (1.3) as its real equation: this is a qua­dratic form of type (3, 3), i.e. having 3 plus signs and 3 minus signs in its diagonalization. For our real structure given by a on P3(G) we get a dif­ferent real form of corresponding to the signature (5,1). To verify this note that (1.2) has the following effect on the pap

 

 

 

are conjugated normally by (1.2). Rewriting equation (1.3) in terms of the coordinates Xx, ...,XS we get

 

 

confirming that the signature is (5,1). Moreover the real points of (1.4), representing >S'4, are indeed given by the affine equation (taking Xx — X2 = 1)

 

 

Reverting to the complex geometry of the Klein representation we recall that there are two families of projective planes lying on Qi. If the equation of is written

 

 

then the planes are given by the equations Y = AZ where A e 0(3, 0). The two families depend on the sign of det A. One family corresponds to points of P3(G) and the other to planes of P3(0) or equivalently to points of the dual P3(G). This means that as a line I in P3(G) varies through a given point A its representative point L e Qt varies in a plane a. on Similarly if I varies in a plane B its representative L varies in a plane f) on Qt, ft being of the opposite family to a.

With our real structure a every a-plane contains a unique real point, namely the intersection a n a(oc). This corresponds to the real line in P3(G) joining A to a{A), and so describes once more the map P3(G) S1.

Hence the six quantities

All this can be conveniently summarized by introducing the correspond­ence space M5cP3xQi consisting of «incident» pairs (A, L), i.e. A lies on the line I (or equivalently L lies in the plane a*). We then have two fibre maps

 

the penkose twistok space

with fibres P2, Pi as indicated. All structures here are complex algebraic.

If we now pick our real structure a then this defines

the real subspace 84 of Q4

a section s: P3^» M5 which picks out the unique point in the P2-fibre which is real (regarded as a subspace of Qt).

Thus the inverse image of 84 in M& gets identified via (ii) with P3(G) and so we recover our basic fibration P3(G) -> $4. Note that whereas the projection M. P3 is complex analytic the section s: Ps Jf5 is not.