3. — Geometrical interpretation.

 Computing naively we find

There is a geometrical way to construct potentials which is very simple but appears not to be familiar to physicists. Boughly speaking the con-straction is analogous to the way in which Biemannian metrics on a manifold can be constructed by embedding the manifold in a Euclidean space and considering the induced metric. Historically this is of course the way Biemannian metrics first arose (e.g. surfaces in 3-space).

We recall that a vector bundle E over a space X is a family of vector spaces Ex parametrized (continuously) by xeX: for example if X is an 11-dimensional manifold its tangent spaces form such a vector bundle with Ex^Rn. We say that E is embedded in the trivial bundle X XRN if each Ex is embedded in RN, the embedding varying continuously (or differentiably) with x. For example if X is embedded as a manifold in RN its tangent spaces (translated to the origin) get identified with subspaces of RN. When E is embedded in X xRN a section of E, namely a function f(x) taking its values in Ex can be viewed as a function with values in RN. We can then form its partial derivatives dpf, but these need no longer take values in the sub- spaces Ex. However if we let Px be any linear transformation on RN varying smoothly with x and projecting onto Ex (i.e. P%x = PJ we can put

 

 

and we get a covariant derivative defined on E. If E is the tangent bundle of a manifold X and P is orthogonal projection V^ is just the usual cova­riant derivative of Biemannian geometry (given by the Levi-Civita connection).

In general (3.1) just corresponds to a GL(n, R)-connection or potential, but if we impose additional structures, preserved by P, we can get potentials for appropriate subgroups such as 0(n), U(m) or Sp(l) (n — 2m or n — 47 respectively). Thus to get Sp( 1) potentials we would consider quaternionic lines in Ek and use orthogonal projection.

Choosing a gauge for the bundle E will give rise to linear maps ux : Rn-> RN whose image is just Ex c RN. If inner products are fixed throughout so that u is an orthogonal gauge then the orthogonal projection Px onto Ex is given by P = uu*, while u*u = 1. To compute the covariant deriva­tive V in the gauge u we put f = ug where g is now a function on X with values in R" and find

 

 

 

showing that the gauge potential A is given by

 

 

Note that u is here an [N x n) matrix of functions so that A„ is an (n x n) matrix. If N = n then (3.2) asserts that A is gauge equivalent to zero, corresponding to the fact that Ex = BN does not really depend on x. How­ever for N > n (3.2) gives interesting potentials. The formula for the gauge field is

 

 

For many purposes it is unnecessary to pick a gauge since we can always work directly inside the larger BN space which has a natural basis. We shall illustrate this by deriving an alternative expression for the field, expressed directly in terms of the projection operator P and the complementary pro­jection Q = 1 — P. Let us take the GL(N, P)-potential B defined by

 

 

Computing the covariant derivative VB = d + B on functions f which lie in E i.e. satisfying Pf = f or Qf = 0, we see that

 

 

where V is the covariant derivative on E defined by (3.1). Thus VB extends V to all .R^-valued functions.

Now differentiating Q2=Q we get QdQ dQQ — dQ and so

 

 

 

Hence the field FB of VB is given simply by

 

 

Eestricting this to E we see that the field F corresponding to the covariant derivative (3.1) is given by

 

 

The components F^v of F are here linear transformations on the image of P. If we choose an orthogonal gauge u and take P = uu* then (3.6) becomes

 

 

and the term in brackets gives

 

 

The last two terms cancel, since u* du = — du*u, and so we find again formula (3.3).

The bundle E when embedded in TiN has a complementary bundle E~l given by the image of Q. The relationship between E and E1- is symmetrical with the roles of P and Q being interchanged. If v : RN~n > RN is an ortho­gonal gauge for E1- so that Q = vv* we can compute the field F of E from (3.6) by

 

 

the other terms dropping out since Pv = 0.

If v is simply a linear, but not orthogonal, gauge these formulae get modified slightly. We take the polar decomposition v = cog where o2 = v*v and Q = coco* = vo~2v*. Substituting in (3.6) for Q we get

 

 

All these formulae hold unchanged if we replace the real numbers by complex numbers or quaternions. In the quaternion case if we want to consider all our matrix operators as left operators then we should regard HN as a right vector space, i.e. a scalar quaternion q acts on a quaternion vector £ by

We now apply this general construction with X = $4= PX{R), the quaternion projective line. The calculations which follow are similar to those in [13] and [15]. We consider a point of P^H) as given by homogeneous coordinates (x, y) with x,y eH and scalar multiplication on the right so that (x, y) and (xq, yq) denote the same point. Now let

 

 

be a (fc + m) x h matrix of quaternions, 0 and D being constant matrices (independent of the scalar quaternion variables x, y). We now assume

(3.10)  v(x, y) has maximal rank for all (x, y) ^ (0, 0).

The columns of v(x, y) then span a subspace of Hn+h having dimension Tcand depending only on the ratio coy-1, i.e. on the point of 84. The orthogonal complement is then a subspace T\x<y) of dimension n. We now give this vector bundle E over $4 the covariant derivative induced from Rn+k by orthogonal projection as in (3.1). The field or curvature can then be computed by one of the above formulae. If we restrict to R* c S* where y 0 then we can take affine coordinates (x, 1) and v(x) — v(x, 1) then gives a linear gauge for E\ Substituting for v in (3.8) gives the following expression for the field F:

 

 

is a real matrix for all x e H then the term o~2 in (3.11) commutes with the scalar quaternion dx and shows that F involves only the self-dual expres­sion dxdx. Hence we have verified that (3.9), subject to conditions (3.10) and (3.12), gives rise to a multi-instanton for the group Sp(n). Note that the vector bundle E on which the self-dual potential is defined has the image of v as its orthogonal complement E±.

It is now easy to verify that the instanton number, i.e. the topological invariant, of E is precisely Tc. Since this invariant is additive for direct sums it is equivalent to check that E1- has invariant — Tc. But E-L is, by de­finition, a direct sum of Tc quaternion line-bundles corresponding to the Tc basic vectors of Hk (i.e. the columns to v). Each of these line-bundles can be identified with the standard line-bundle over 8i = Pi(-ff) which asso­ciates to (x, y) the one-dimensional subspace of 7/2 consisting of scalar mul­tiples of (x, y). This has invariant ± 1 depending on conventions. In our case the sign must be — 1 so that the invariant of the self-dual bundle E becomes the positive integer Tc.

 

where q*= v*v = (xC* + D*)(Cx -f B) . If we now assume that

Then A = u*du as in (3.2). Note that conditions (3.13) are preserved if we replace u by ug where g(x) e 8p(n) and this produces a general gauge transformation on A.

If we want to write down explicitly a gauge potential A corresponding to the matrix v(x) we must first pick an orthogonal gauge for the bundle E, i.e. a (Tc n) x n matrix u(x) such that

 

 

The matrices u, v can be put into a sort of normal form which gives more explicit formulae although this will introduce « apparent singularities ». To obtain these normal forms we first decompose v(x, y) into blocks as follows

 

 

where G0, B0 are n x k and C\, D1 are k x k matrices. Since v is assumed by (3.10) to have maximal rank we may, after a change of the (x, y) va­riables (a conformal transformation of $4), assume Gx non-singular. Replac­ing v by RvS where S is a real kxk matrix and R e 8p(n + fc) we can then take G1 — ~I (I the unit JcxJc) matrix and G0 — 0.

Except for singularities at points x where B — xl is singular we can solve for U:

 

Substituting u = ^ a in (3.2) we find the gauge potential given in

Now putting y = 1 we get v(x) in the form

 

 

where A is an n x h matrix and B is a, kxk matrix, both have quaternion entries but are independent of the quaternion variable x. Condition (3.12) is equivalent to requiring both

 

 

But (3.16) is easily seen to be equivalent to requiring (3.17)    B is a symmetric matrix .

We now take u in the form u = ^ ^j a where I is the unit nxn-

matrix, U is an k x w-matrix and a is a self-adjoint n x w-matrix. Equa­tions (3.13) become

 

 

terms of U and a = (1 + 77* 77)-* by

 

 

Equations (3.19) and (3.20) are precisely those given in section 2. We have thus verified that the formulae of section 2 do indeed give ft-instantons. We have also explained why the singularities of section 2 are only «apparent», and due to the particular choice of gauge.

Geometrically our construction of multi-instantons can also be formulated as follows. On the Grassmannian Gkn(H) of w-dimensional subspaces of Hn+k the standard vector bundle with fibre Hn has a standard connection (induced by orthogonal projection) and our instanton connections on 84 are induced by suitable maps /: 8i^Gk>n(H). Equation (3.19) describes / explicitly in terms of appropriate coordinates in the two manifolds. The standard connection on Gk>n(H) is automatically invariant under the group 8p(n + h). In particular for n — Tc = 1 this shows that the basic instanton on 84 is invariant under 8p(2) c^ Spin (5).