# 1. — The linear complex.

In this chapter we shall give an algebraic construction for vector bundles on P3(G) which, by the results of Chapter IV, will correspond to anti- instantons. We begin in this section by considering in some detail the basic anti-instanton (for — 1 and G = 8U(2)). The algebraic geometry in this case is classical and is, in traditional language, the geometry of a «linear complex » of lines in P3(G). It will turn out that this special case is an illuminating preliminary of the general construction and so it repays careful study.

If we start again from our fibration P3(G)-^8* and if we use the standard metrics of 8i and of P3(G) we can obviously construct a vector bundle over P3(0) with fibre 6'2 by taking the horizontal tangent vectors on P3(G), i.e. vectors orthogonal to the fibre direction. Now we recall that taking orthogonal complements is not a holomorphic process so that we would not expect this horizontal vector bundle to be holomorphic. However the vertical bundle, from which we start, is itself not holomorphic since its definition involves the « real structure» of P3(G). In fact it turns out that the horizontal vectors do form a holomorphic bundle (while the complementary verticals do not). The restriction of this bundle to any real line is clearly the normal bundle of the line in P3{G). This is not trivial but becomes so after tensoring with the standard line bundle L on P3(G). We then end up with a holomorphic vector bundle which satisfies the conditions of Theorem (2.9) of Chapter IV and corresponds to the basic anti- instanton on 8*.

The preceding construction used the metric and real structure of P3{G) or equivalently the metric and quaternion structure of <74 = E2. From these structures we can also extract the natural skew-form and this is essentially what is needed to produce the algebraic vector bundle. If LM c Gi is the line corresponding to the point (z) e Pa(G) we consider its annihilating or

is a 2-dimensional vector space depending algebraically on the point (z)ePs(G). In other words E is an algebraic vector bundle with fibre G2 over P3(G). Moreover the skew-form induces a non-degenerate skew-form on E so that its structure group reduces to 8L(2, G).

In projective terms corresponds to a projective plane in P3(G) passing through the point (z). Such a point-plane correspondence is classically called a null-correlation and the lines that lie in such a plane and pass through its corresponding point form the associated «linear complex». These projective lines correspond in Gl to the 2-dimensional isotropic sub- spaces of the skew-form, i.e. those G2 c Gl on which the skew-form is identically zero. If the skew-form in Gi is given by the skew-symmetric 4x4 matrix au the line (%) (y) belongs to the linear complex provided 0 or equivalently where pals= XaiJn — Xpija are the Pliicker coordinates of the line.

We shall now show that the lines of this linear complex are precisely the jumping lines of the bundle E. First assume that the line (x) (y) does not belong to the linear complex. This means that L{x) does not lie in and hence that L°x)C\ L°W = B is a 2-dimensional subspace of <74 which, for all (z) on the line (%) (y), lies in but does not contain L{z>. This shows that E, restricted to the line (x) (y) is a trivial bundle. Conversely we next show that E is not trivial when restricted to a line I of the linear complex. The line I corresponds to an isotropic 2-dimensional subspace W c G1, and so for all (z) 6 I we have the inclusions

Hence E„ contains WjLM, i.e. E\l contains the line-bundle WjL (here W denotes the trivial bundle with fixed fibre W). Topological considerations then show that W/L^L*. Since L* has holomorphic sections with zeros it cannot be a sub-bundle of a trivial bundle and so E\l is not trivial: in fact E\l corresponds to the integers (1,-1) in the general classification.

polar space L0^ with respect to the (non-degenerate) skew-form. The dimension of L\' y is 3 and, since the form is skew-symmetric, c . Hence the quotient space

We come now to consider real structures so that, by the results of Chapter IV, we can construct anti-instantons on

(1.2) implies that L°{z)—where 1 denotes the orthogonal space with respect to the inner product, and so for any (z) e P3 we have an orthogonal decomposition:

where Bx = n L°jz) and x represents the real line lx = {(z)(jz)}.

This shows first that a real line lx for x e >S'4 is never a jumping line for the algebraic bundle E = L°IL on P3(0). It also shows that j induces on E a real structure, i.e. an anti-linear map o:E-±E covering a on P3((7) satisfying a2 = — 1. Finally this real structure is positive, meaning that (u, av) induces a positive inner product on the sections of E restricted to any real line. This last statement follows from the fact that, restricted to the real line {z)(jz), the sections of E can be identified with B and the inner product is just that coming, as in (1.2), from the inner product of Gu.

If we take the standard skew-form ( , ) on <74 = H2 this is related to j and the positive inner product <,> by

and in particular j preserves the skew-form in the sense that

Thus the bundle E on P3(0) together with its real structure a gives precisely the data which, according to Chapter IV, corresponds to an anti- instanton on ,S'4. The bundle B on $4 has as fibre at the point x the vector space Bx in the decomposition (1.4) with its natural inner product. It remains to specify the connection it inherits. According to Chapter IV the connection is uniquely determined by the condition that, on P3(G), it should be compatible with both the unitary and holomorphic structures. Now quite generally if V c W are holomorphic vector bundles with compatible unitary structures then the canonical connection on V is induced by orthogonal projection from that of W. This is clear because a holomorphic section / of V is also holomorphic in W and so satisfies = 0 (V^ being the canonical covariant derivative of W), hence PV^/ = 0 where P is orthogonal projection from W to V. Since this property plus unitarity characterizes the canonical connection we must have PVw=Vv. Dualizing, the same applies to holomorphic quotient bundles. Applying these observations first to the sub-bundle c P3 x O4 and then to the quotient bundle E of L° we see that the canonical connection on E coincides with the connection induced on L° n L1- from the trivial connection of P3 x G*.From (1.4) we see that Bx and so the connection on B is

that induced from 8ixGi by orthogonal projection.

be a linear map depending linearly on 2 — (zlt z2, z3, 24). Thus A(z) =

= 2 -A-iZi where each Af: TF-> F is a constant linear map. We let i-l

In quaternionic notation the decomposition (1.4) can also be written

where cceSl = Pi(H) and Bx is the quaternionic line in H parametrized by x. Thus B and are the standard quaternionic line-bundles over 8discussed in Chapter II and the connection we have given B is the natural one explained in Chapter II. With our present sign conventions B is an anti-instanton while B1- is an instanton. Both are clearly acted on by the compact symplectic group Bp (2) ^ Spin (5).

If we fix the metric on 8* as we have done here then we get a unique anti- instanton bundle B. However by applying a conformai transformation we will get new anti-instantons. The space of moduli is then 8L(2, H)[8p(2) and parametrizes the metrics on 8l in its standard conformai structure. Alternatively the moduli space can be viewed as the interior of the unit ball in jB5 (the hyperbolic 5-space). This unit ball can be thought of as the interior component of the real part of P5(0) — Qs, the complex moduli space. Note that the exterior component corresponds to a gauge field on 8with real singularities on an equatorial 3-sphere (the polar section of the external point).

TJZ = A(z) W c V be the image space and we assume:

(2.2) for all s/0, the space Uz is ^-dimensional and isotropic for the skew-form on V.

With this assumption we have Uz c f/°, the polar space, and hence Ez = ~U°J~UZ is a vector space depending algebraically on (z) e P3(G). Since dim TJZ = k, dim U°z= 2k + 2 — k = k + 2 and so dimEz = 2. Moreover Ez inherits a non-degenerate skew-form and so E is an algebraic vector bundle over P3(G) with group SL(2, C).

Note that, when k = 1, XJZ is one-dimensional and automatically isotropic. For k > 1 however the isotropic condition, which can be written in matrix form as

where A is the transpose and J is the matrix of the skew-form, gives a system of quadratic equations on the coefficients of the 4 matrices A11 A2, A3, J.4. The number of coefficients considerably exceeds the number of equations so that we certainly get solutions.

This implies that

is a 2-dimensional complement to TJZ in XJ° for all points (z) on the line I. This shows that E\l is trivial and so I is not a jumping line.