Massless monopole clouds and electric-magnetic duality

We qualitatively discuss the Montonen-Olive electric-magnetic duality in N = 4 supersymmetric Yang-Mills theory with non-maximally broken gauge symmetry, where BPS configurations of massless monopole clouds have been found. We argue that these low energy non-Abelian clouds can be identified as the dual of the infrared bremsstrahlung radiations of the non-Abelian massless particles. After we break the N = 4 supersymmetry to N = 1 by adding a superpotential, or to N = 0 by further adding soft breaking terms, these non-Abelian clouds will generally condense and screen the non-Abelian charges of the massive monopole probes. The effective mass of these dual non-Abelian states is likely to persist as we lower the energy to QCD scale if all Higgs are massive. This meets the expectation of the dual Meissner effect that the magnetic theory is in the dual Higgs phase. email address: [email protected] The N = 4 supersymmetric Yang-Mills theory is conjectured to have a remarkable electric-magnetic duality [1,2,3,4]. A special form of this conjecture suggests that the electric theory is dual to the magnetic theory with a dual group and an inverse coupling constant. This conjecture is originated in the study of SU(2) theory with gauge group completely broken by the Higgs mechanism [1,2], where there is only one type of fundamental (anti)monopole. The supersymmetric multiplet based on this monopole is dual to the massive gauge multiplet. If the rank of the gauge group is higher than one, the monopole configurations can be treated as superpositions of the fundamental ones associated with the simple roots [5]; while for elementary particles, each root of the dual group corresponds to a massive gauge multiplet. Studies of the supersymmetric sigma model on the monopole modular spaces [6] show these supersymmetric fundamental monopoles indeed form threshold bound states according to the duality predictions. To illustrate this, we use the monopoles in the SO(5) theory. The root diagrams of SO(5) and its dual group Sp(4) are shown in Fig.1. A unique normalizable anti-self-dual harmonic two-form is found [6] on the modular space of the β and γ monopoles (we have chosen and labeled the simple roots as β and γ). This corresponds to a threshold bound state. The supersymmetric multiplet associated with this bound state in SO(5) is dual to the α gauge multiplet in Sp(4) as predicted by the duality. The situation is more subtle when some non-Abelian subgroups of the gauge symmetry remain unbroken. In such cases, BPS configurations of massless monopole clouds have been found [7,8]. In this paper, we will qualitatively study these configurations in the context of the electric-magnetic duality. We will also discuss its possible role in the dual Meissner effect of the confinement by studying its non-BPS properties. We use the same SO(5) example. The gauge symmetry is now partially broken to SU(2)× U(1) by a Higgs expectation value h orthogonal to the root γ or γ. The corresponding monopoles or elementary particles become massless. A spherically symmetric BPS magnetic monopole solution is found in [7]. It describes a massive monopole embedded in the root β surrounded by a non-Abelian cloud. There is a modular parameter a characterizing the size of the cloud. We will be interested only in the non-Abelian fields which do not exponentially decay outside the massive monopole core m W : Ai(γ) = ǫaimr̂mG(r) , φ a γ = r̂aG(r) , (1) where the subscripts γ mean they belong to the triplet SU(2) generators ta(γ) (a = 1, 2, 3) associated with the root γ, and G(r) = 1 er(1+r/a) . If the cloud size a is at infinity, we only have the massive monopole left carrying both Abelian and non-Abelian charges. If a is finite,