Magnetic Charge Lattices, Moduli Spaces and Fusion Rules

TL;DR

This paper studies the structure of magnetic charges in monopoles within Yang-Mills-Higgs theory, revealing a geometric cone structure, their classification, and fusion properties, with implications for understanding monopole moduli spaces.

Contribution

It introduces the Murray cone as a geometric framework for magnetic charges and interprets monopoles as basic or composite based on charge decomposability, linking to moduli space dimensions.

Findings

Magnetic charges form a solid cone called the Murray cone.

Magnetic charge sectors correspond to dominant weights of the dual group.

Classical fusion properties support the monopole interpretation.

Abstract

We analyze the set of magnetic charges carried by smooth BPS monopoles in Yang-Mills-Higgs 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.