Representation Theory of Non-Abelian Magnetic Charges
Adapted from figure 2.2: The weight and root lattice of (\mathfrak{su}(3)), generated respectively by the
fundamental weights (\mu_{1,2}) and the simple roots (\alpha_{1,2}).
For an theory with a global (\mathrm{SU}(3)) structure electric charges lie in the weight lattice, and magnetic charges lie in the co-root lattice (the inverse lattice of the root lattice, which for (\mathfrak{su}(3)) coincides with the root lattice) while for a theory with global group (\mathrm{SU}(3) / \mathrm{Z}_3) the opposite is true and magnetic charges lie in the root lattice and electric charges lie in the co-root lattice.
Abstract
We discuss the representation theory of non-abelian charges in physics, particularly magnetic monopoles, defined analogously to the well known case of electromagnetism. We start by presenting examples of charges and their emergence from physical symmetries. Next, we present concepts of representation theory that allow to generalise the discussion to non-abelian symmetries. We proceed to discuss charge in gauge theories, the distinction between electric and magnetic charges, and their relation through the Dirac quantisation condition. Lastly, using the tools developed in the previous chapters, we discuss how quantisation conditions can be derived for non-abelian charges.