Massive and massless monopoles with nonabelian magnetic charges

TL;DR

This paper shows that monopoles with nonabelian magnetic charges can be analyzed through the moduli space of BPS multimonopole solutions, revealing that massless nonabelian components are not isolated solitons but nonabelian clouds surrounding massive monopoles; the MSB-to-NUS limit yields a smooth transition in the moduli space, with massless degrees of freedom reinterpreted as cloud parameters and global gauge orientations. The explicit SO(5) example demonstrates an eight-parameter family of solutions whose relative metric interpolates between Taub-NUT in the MSB regime and flat $\mathbb{R}^4$ in the NUS regime, with the cloud size encoded by a parameter $a$ and the unbroken symmetry realized as tri-holomorphic isometries. The work extends to general groups, discusses quantization of cloud degrees of freedom, and discusses implications for electric-magnetic duality and threshold bound states in $N=4$ theories, arguing that massless monopoles correspond to dual massless gauge bosons. Overall, the results provide a concrete geometric framework for nonabelian monopoles and offer insights into duality and the role of massless charged states in nonabelian gauge theories.

Abstract

We use the multimonopole moduli space as a tool for studying the properties of BPS monopoles carrying nonabelian magnetic charges. For configurations whose total magnetic charge is purely abelian, the moduli space for nonabelian breaking can be obtained as a smooth limit of that for a purely abelian breaking. As the asymptotic Higgs field is varied toward one of the special values for which the unbroken symmetry is enlarged to a nonabelian group, some of the fundamental monopoles of unit topological charge remain massive but acquire nonabelian magnetic charges. The BPS mass formula indicates that others should become massless in this limit. We find that these do not correspond to distinct solitons but instead manifest themselves as ``nonabelian clouds'' surrounding the massive monopoles. The moduli space coordinates describing the position and $U(1)$ phase of these massless monopoles are transformed into an equal number of nonabelian global gauge orientation and gauge-invariant structure parameters characterizing the nonabelian cloud. We illustrate this explicitly in a class of $Sp(2N)$ examples for which the full family of monopole solutions is known. We show in detail how the unbroken symmetries of the theory are manifested as isometries of the moduli space metric. We discuss the connection of these results to the Montonen-Olive duality conjecture, arguing in particular that the massless monopoles should be understood as the duals to the massless gauge bosons that appear as the mediators of the nonabelian forces in the perturbative sector.