Magnetic charge lattices, moduli spaces and fusion rules

Abstract

We analyze the set of magnetic charges carried by smooth BPS monopoles in Yang-MillsHiggs theory with arbitrary gauge group G spontaneously broken to a subgroup H. The charges are restricted by a generalized Dirac quantization condition and by an inequality due to Murray. Geometrically, the set of allowed charges is a solid cone in the coroot lattice of G, which we call the Murray cone. We argue that magnetic charge sectors correspond to points in the Murray cone divided by the Weyl group of H; hence magnetic charge sectors are labelled by dominant integral weights of the dual group H * . We define generators of the Murray cone modulo Weyl group, and interpret the monopoles in the associated magnetic charge sectors as basic; monopoles in sectors with decomposable charges are interpreted as composite configurations. This interpretation is supported by the dimensionality of the moduli spaces associated to the magnetic charges and by classical fusion properties for smooth monopoles in particular cases. Throughout the paper we compare our findings with corresponding results for singular monopoles recently obtained by Kapustin and Witten.

Parameter counting for singular monopoles on ℝ3

“…It has been proven that these are also necessary conditions for finite energy when the gauge group is SU (2) [14], and this is expected to be true in general [41,49]. (See also the discussion in [50].) When 't Hooft defects are present there are certain boundary terms that should be included in the energy functional.…”

Non-Abelian monopoles in the Higgs phase

“…It is important to notice that the electric orbit is non-trivially fibered on the magnetic one. This fact is related to the difficulties to associate to non-Abelian monopoles well-defined algebraic objects, as required by quantum mechanics [21,48].…”

Towards a non-abelian electric-magnetic symmetry: the skeleton group

“…Using the identification of singular monopoles with 't Hooft operators and computing the operator product expansion (OPE) for the latter, they showed that the fusion rules of purely magnetic monopoles are identical to the fusion rules of the dual gauge group. It was shown in [7] that in an ordinary N = 4 Yang-Mills theory, the classical fusion rules of monopoles, obtained from patching together monopole solutions to the classical field equations, are also consistent with the non-abelian Montonen-Olive conjecture.…”

“…The original treatment of that condition in [1,2] makes use of the identification between t * and t via the Killling form and uses a basis in order to describe both magnetic and electric charges as r-component vectors. We have followed that path in the current paper, and also in [7] where we give a review using the same conventions as in the current paper. It is worth emphasising, however, that the Dirac condition only requires the natural duality between t (magnetic charges) and t * (electric charges) for its formulation, as emphasised in [15].…”