Quantisation of Monopoles with Non-abelian Magnetic Charge

TL;DR

This paper explores the quantisation of non-abelian magnetic monopoles in SU(3) gauge theory, revealing how stratified moduli spaces influence dyonic state representations and their duality properties.

Contribution

It introduces a detailed analysis of monopole stratification, semi-classical quantisation, and the resulting dyonic state representations in non-abelian gauge theories.

Findings

Dyons are classified by magnetic charge and subgroup representations.

States form representations of the semi-direct product U(2) ⋉ R^4.

Electric-magnetic duality and supersymmetric BPS states are discussed.

Abstract

Magnetic monopoles in Yang-Mills-Higgs theory with a non-abelian unbroken gauge group are classified by holomorphic charges in addition to the topological charges familiar from the abelian case. As a result the moduli spaces of monopoles of given topological charge are stratified according to the holomorphic charges. Here the physical consequences of the stratification are explored in the case where the gauge group SU(3) is broken to U(2). The description due to A. Dancer of the moduli space of charge two monopoles is reviewed and interpreted physically in terms of non-abelian magnetic dipole moments. Semi-classical quantisation leads to dyonic states which are labelled by a magnetic charge and a representation of the subgroup of U(2) which leaves the magnetic charge invariant (centraliser subgroup). A key result of this paper is that these states fall into representations of the semi-direct product $U(2) \semidir R^4$. The combination rules (Clebsch-Gordan coefficients) of dyonic states can thus be deduced. Electric-magnetic duality properties of the theory are discussed in the light of our results, and supersymmetric dyonic BPS states which fill the SL(2,Z)-orbit of the basic massive W-bosons are found.