The construction of all self-dual multimonopoles by the ADHM method

From geometry it is well known that a Reimannian manifold can either be described internally, giving e.g. the metric as a function of suitable co-ordinates, or by an embedding into some Euclidean space. Embeddings often allow a better visualization of the manifold. On the other hand, they tend to be rather arbitrary. However, this inconvenience does not exist, if one has a canonical embedding, like that of S/sup n/ into R/sup n+1/. If one walks along a closed path in a Reimannian space S, trying to conserve one's orientation at every step, the orientation at the end of the path nevertheless will be different from the one at the start. This phenomenon is described by the Riemannian connection, which can also be visualized with the help of an embedding of S into some R/sup k/. In fact, the latter also yields an embedding of the fibre bundle of tangent planes into S*R/sup k/. Gauge fields are also described by a fibre bundle and a connection. They only differ in that the fibres are not related to the geometry of the base space. However, the connection can be described by an embedding, too. The author considers gauge fields on a Euclidean base space R/sup 4/, which takes the role of S. The fibre bundle is described by an embedding into R/sup 4/*M, where M is a Hermitian vector space.