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In this section we shall show how the construction of the preceding section, for 8p(n), can be formulated in terms of quaternions. We then obtain the explicit formulae given in Chapter II.

We recall that we have identified the space (74 with j?2 in such a way that the map a on Cl is given by left multiplication on H2 by the qua­ternion j. Similarly the vector space V of dimension 2k + 2n can be viewed as a left quaternion vector space of dimension h -f n with j given by the anti-linear map a.

The vector space W, of complex dimension Tc, has a with a2 = 1 and so can be viewed as the complexification of the real vector space WR left fixed by a. Then O4 0CW ^H2 WB and a corresponds again to multi­plication by j on the quaternion vector space J?2 WR.

The linear map 1(0): W^-V can now be viewed as a map

The main result of Chapter III was that solutions of the anti-self-dual Yang-Mills equations on 8l converted, via the Penrose transformation, to holomorphic bundles on P3(C). There is a parallel twistor interpretation for solutions of certain important linear differential equations. We shall show in this section that they correspond to elements of appropriate sheaf coho­mology groups.

We begin by considering the case of U(l) Yang-Mills theory. Although there are no global anti-self-dual solutions on S4, solutions certainly exist in B4 corresponding to solutions of the Euclidean Maxwell equations for a 2-form co: dco — 0, *o> = — o>. From Chapter III we know that these cor­respond to holomorphic line-bundles on P3(G) — Pi(0). If we ignore unitarity and work with complex-valued 2-forms the holomorphic line-bundle is unrestricted, except that it must be topologically trivial. As explained in Section 1 such a line-bundle is then described by an element of the coho­mology group H1(P3—Pu 0). Thus this group corresponds under the Penrose transformation to the solutions of the anti- self-dual Maxwell equa­tions on Bl. This is the type of correspondence which we propose to extend. As we shall see all the cohomology groups H1(P3— Plf 0{— m)) correspond in a similar way to solutions of other linear differential equations on JB4.