Monopoles and four-manifolds

Recent developments in the understanding of N = 2 supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining fourmanifold invariants by counting SU (2) instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a "dual" equation in which the gauge group is the dual of the maximal torus of SU (2). The new viewpoint suggests many new results about the Donaldson invariants.