Self-dual monopoles and calorons

Abstract

The study of nonlinear partial differential equations remained outside the mainstream of mathematics, because their solution spaces seemed to be rather arbitrary and complicated. But physicists discovered that some of those equations occur naturally, and a closer study by both physicists and mathematicians reveals more and more beautiful structures. Among them, the Yang-Mills equations in four dimensions are today the most outstanding ones. Results on general solutions are still scanty, but quite a lot is known about the more specialized solutions of the self-duality equation for YangMills fields on euclidean four-manifolds. Indeed, this equation already became a valuable tool in the study of differentiable four-manifolds. A basic step in the investigation of this equation was the ADHM construction I) of all instantons, i.e. of all gauge potentials in R 4 with self-dual and square integrable field strengths. The construction uses the cohomology of certain sheaves over the twistdr space, which does not yet belong to the tool kit of many physicists. Thus we shall give an elementary modification of it, which also has the advantage of being easily geoeralizable to self-dual monopoles and calorons. We only consider the gauge group SU(n), but it is easy to specialize to the other classical Lie groups. For the space coordinates and the covariant derivatives we use the standard quaternionic notation