On the existence of states saturating the Bogomolʹnyi bound in N = 4 supersymmetry

We give an argument showing that in N = 4 supersymmetric gauge theories there exists at least one bound state saturating the Bogomolʹnyi bound with electric charge p and magnetic charge q, for each p and q relatively prime, and we comment on the uniqueness of such state. This result is a necessary condition for the existence of an exact S-duality in N = 4 supersymmetric theories.