A Note on 1/4-BPS States

We study classical solutions of N = 4 super Yang-Mills theories that are invariant under 1/4 of the supersymmetry generators. Expressions for the mass and electric charge of the configurations are derived as functions on the monopole moduli space. These functions also provide a method of determining the number of normalisable bosonic zero modes. Introduction The realisation of field theories as the low-energy world-volume dynamics of configurations of branes in type II string theory and M-theory has led to simple geometrical explanations of many field theoretic phenomena. The archetypal example is that of N = 4 super Yang-Mills which is the world-volume theory of D3-branes. On the Coulomb branch, (p, q) strings stretched between each pair of D3-branes correctly reproduces the expected field theory spectrum of electric, magnetic and dyonic 1/2BPS states [1]. For three or more D3-branes however, there exists the possibility of suspending between them networks of (p, q) strings constructed from three-string junctions [2]. If the D3-branes are planar then such networks preserve 1/4 of the space-time supersymmetry [3, 4, 5] and correspond to dyonic 1/4-BPS states of the Yang-Mills theory [6, 7] thus leading to a much richer spectrum of BPS states than previously suspected for gauge groups of rank greater than one. The distinctive signature of such states is that the magnetic and electric charge vectors are not parallel [8, 6]. These states persist at weak coupling in the field theory where they are amenable to standard semi-classical analysis and, in particular, one expects classical solutions with non-parallel magnetic and electric charges to exist. The equations of motion governing such 1/4-BPS configurations consist of the usual monopole Bogomol’nyi equation, together with a second order equation for an adjoint scalar field derived from Gauss’ law. Explicit solutions to these equations were first given in [9, 10] using a spherically symmetric ansatz. For the special case of the (1, 1) dyon of SU(3) gauge group, solutions have also been found using Nahm data [11]. Moreover, as we shall review below, the authors of [11] also demonstrated that the space of gauge invariant 1/4-BPS solutions with given magnetic charge is determined by the moduli space of 1/2-BPS monopole solutions of the same charge. These monopole moduli spaces have been studied extensively in the past and many of their properties are understood [12]. However, although the existence of 1/4-BPS solutions can be shown, little else is known about these configurations. In particular, neither the mass nor electric charge can be determined without an asymptotic solution to Gauss’ law. In this note, we derive an expression for both the mass and electric charge of 1/4-BPS states as a function over the monopole moduli space. The former is shown to be equal to the norm squared of a particular Killing vector on the moduli space. We demonstrate this technique using the (1, 1) dyon of SU(3) as an example and reproduce the results of [11]. Finally, we discuss the moduli space relevant to the low-energy dynamics of these 1/4-BPS states. The existence of non-normalisable zero-modes means this space differs from the moduli space of solutions. We use the expression for the electric charge to determine the number of normalisable bosonic zero modes. As this work was completed, the preprint [13] appeared, where the number of normalisable bosonic zero modes is also calculated, togther with the nor-