2. — Suppose now

The assumption that dim Uz= k for all z 0 implies that for any two distinct points (oo) and (y) in P3

 

 

To see this we note first that the family of all TJZ gives a ^-dimensional vector bundle over P3 isomorphic to the sum of k copies of L, each copy arising from a basis vector of W. Now any non-zero vector in Vx n Uy would give rise to a holomorphic section (not identically zero) of U restricted to the line I joining (%), (y) in P3. But V\l is isomorphic to k copies of L\l and this bundle has no holomorphic section besides zero (recall that, over any Pm(C), a holomorphic section of L* is a linear form in the coordinates, a section of £-« is a homogeneous polynomial of degree n, and therefore L has only the zero section).

Now let us try to identify the jumping lines of the bundle E on P3. Consider first a line I joining points (oo) (y) such that

that (2.5) is false then the space B given by (2.6) has non-zero intersection with TJy. Let v be a non-zero vector in J? n TJy, then according to (2.4) v$TJx and so v defines an algebraic section of E\I which is zero at y but non­zero at oo. This shows that E\l is not trivial and so Jis a jumping line. Thus the jumping lines are precisely the lines for which (2.5) fails.

 

is positive. Our linear transformation A(z) is now assumed to be compa­tible with a, i.e.

 

where a(z) is, as before, multiplication by the quaternion j. Condition (2.7) implies that

 

and (2.7) implies that

 

To introduce reality conditions we now assume that we have an anti- linear map a acting on W, V with cr2 = -f 1 on W and cr2 = — 1 on V. Moreover we require that a preserve the skew-form on F and be such that the corresponding hermitian form

Hence we get an orthogonal decomposition:

 

 

where Bx — TJ° Pi U°az depends only on the point x e parametrizing the real line (z)(az) of P3. This shows in particular that the real line lx is not a jumping line for E and that E inherits a real structure from a with the properties required, as in Chapter IV, to give an anti-instanton on 8l. Exactly as for the case Tc = 1 we see that the anti-instanton bundle on 84 is B with the connection induced by orthogonal projection from P3 X F.

The instanton number of the bundle B over 8l is — k. This follows from the fact that its orthogonal complement is topologically equivalent to the sum of h copies of the basic 1-instanton bundle.

To generalize from 8p( 1) to 8p(n) all we have to do in the preceding construction is to take V to have dimension 21c -f 2n. For the group 80(n) we have to take F of dimension 2h -)- n and we have to alter the signs. Thus we require F to have a non-degenerate symmetric bilinear form and a2 — + 1 on F and cr2 = — 1 on W. In all cases the instanton number is — ft.

which is quaternion linear. This corresponds to the compatibility of A(z) with a.

We can deal with IJ(n) by regarding it in the standard way as the sub­group of 80(2n) commuting with J where J2 = — 1. We therefore require that V, W should both possess a complex linear automorphism J with J2 = — 1, aJ — — Ja. Moreover J is assumed to preserve the inner product on V or equivalently (Ju, Jv) = — (u, v). Finally the linear transforma­tion A(z)\ TF-> V is required to commute with J. Then the decomposi­tion (2.10) is J-invariant. Hence the bundle B has an action of J and (since J preserves inner products) the canonical connection of B is preserved by J. Hence the connection of B, which is anti-self-dual, reduces from SO(2n) to U(n).

Notice that we could proceed one stage further and regard 8p(n) c SO (in) and so describe 8p(n) solutions as SO (in) solutions with extra structure. However the more direct approach to 8p(n) used earlier is more economical.

Returning now to the symplectic case we see that two triples (W, V, A) (W',V',A') which are isomorphic, in the obvious sense that we have isomorphisms W^W', V^V' commuting with a and the skew-forms on V, V, and taking A to A', will give rise to an isomorphism between the bundles B and B' preserving connections. Thus isomorphic triples in the linear algebra sense yield gauge equivalent solutions of the anti-instanton problem. As we shall prove later the converse is also true.

If we take a real basis of WR and an orthogonal JJ-basis of V, so that V gets identified with Hk+n, then A is described by two matrices G, B of qua­ternions. The row-vectors of G are the image under A of (1, 0)(x) basis vec­tors of WR and D is similarly defined replacing (1, 0) by (0,1) in H'2. We use matrices as right multipliers here since our scalars act on the left, so that G, D are ft X (ft + n) matrices. Regarded as a matrix function of the pair (x, y) of quaternions in H2 we then have

 

 

The non-degeneracy condition on A is just that this matrix have maximal rank for all (x, y) ^ (0, 0). Finally the isotropy condition (2.2) is equivalent, using (2.10), to saying that for any two row-vectors u, v of the matrix (3.1) u and iu are orthogonal to jv and j(iv). This means that

 

 

 

or equivalently that the quaternion uv* is real. Hence (2.2) is equivalent to

 

 

is real.

We thus recover the description given in Chapter II, Section 3 except that we have transposed our matrices since we are now considering left action by scalars. For this reason we now get anti-instantons instead of the instantons of Chapter II.