Dyons and S-duality in N=4 Supersymmetric Gauge Theory

We analyze the spectrum of dyons in N = 4 supersymmetric Yang-Mills theory with gauge group SU (3) spontaneously broken down to U (1) × U (1). The Higgs fields select a natural basis of simple roots. Acting with S-duality on the W-boson states corresponding to simple roots leads to an orbit of BPS dyon states that are magnetically charged with respect to one of the U (1)'s. The corresponding monopole solutions can be obtained by embedding SU (2) monopoles into SU (3) and the S-duality predictions reduce to the SU (2) case. Acting with S-duality on the W-boson corresponding to a non-simple root leads to an infinite set of new S-duality predictions. The simplest of these corresponds to the existence of a harmonic form on the moduli space of SU (3) monopoles that have magnetic charge (1, 1) with respect to the two U (1)'s. We argue that the moduli space is given by R 3 × (R 1 × M)/Z Z, where M is Euclidean Taub-NUT space, and that the latter admits the appropriate normalizable harmonic two form. We briefly discuss the generalizations to other gauge groups.