Recently I am reading a paper about monopoles. In several cases, it seems that writing fields in adjoint representation of the gauge group makes a difference.
Once it leads to different group after symmetry breaking when using other representation. And I also noticed statement like this, "An important open question is whether an analogous Bogomolny monopole's mass bound can be obtained if the Higgs field is not in the adjoint representation."
Can anyone kindly shed light on this. Thanks!
Update: I reckon any field (either EM field in real space or Higgs field in internal isotopic space) be in a certain type of representation space of the symmetry group associated with the Lagrangian or action. This space also dictates some constraints on the fields, e.g., specific tensor or spinor structures (anything more???). And what representation space you use contains physics as well, that is to say, we have to check it by experiments. Perhaps this question addresses on a particular case. Either does the explicit and concrete 2nd answer.
Is this understanding correct?