A remark on the asymptotic form of BPS multi-dyon solutions and their conserved charges

TL;DR

This paper demonstrates that the conserved charges for BPS multi-dyon solutions in Yang-Mills-Higgs theory are topological in nature and can be derived from integral equations, not differential ones.

Contribution

It reveals that many topological charges are actually dynamical and conserved due to integral properties of Yang-Mills equations, providing new insights into their nature.

Findings

Conserved charges correspond to eigenvalues of the Higgs field's asymptotic form.

These charges match the topological charges traditionally considered in the literature.

Conservation of these charges cannot be derived from differential Yang-Mills equations.

Abstract

We evaluate the gauge invariant, dynamically conserved charges, recently obtained from the integral form of the Yang-Mills equations, for the BPS multi-dyon solutions of a Yang-Mills-Higgs theory associated to any compact semi-simple gauge group G. Those charges are shown to correspond to the eigenvalues of the next-to-leading term of the asymptotic form of the Higgs field at spatial infinity, and so coinciding with the usual topological charges of those solutions. Such results show that many of the topological charges considered in the literature are in fact dynamical charges, which conservation follows from the global properties of classical Yang-Mills theories encoded into their integral dynamical equations. The conservation of those charges can not be obtained from the differential form of Yang-Mills equations.