When reading about Higgs and Coulomb 'phases' I came across two separate definitions:
The first tells us that the Higgs/Coulomb phases are determined by the representation that the Higgs field transforms under, as explained on Wikipedia. The Higgs field breaks the gauge symmetry; If the Higgs is in the adjoint then we're left with U(1)s and we have a Coulomb phase, if the Higgs transforms in any other representation then (typically) no U(1)s are left and we have a Higgs phase.
However, the second definition tells us that the these phases are determined by the potential between two electric test charges (as read in Sannino's book). A potential proportional to 1/r is the Coulomb phase and a constant potential gives the Higgs phase. Also, there are other phases: Confining, free-electric, free-magnetic and perhaps others.
Question 1) Is there a link between these two definitions? I can see why the Coulomb phase as described in the Wikipedia article might lead to a 1/r potential (due to the surviving U(1) groups). But why would the Wikipedia definition of a Higgs phase lead to a constant potential? Furthermore, if there is a link, what sort of phase (As in the Wikipedia article) would give us confining, free-electric and free-magnetic potentials?
Then there are Higgs and Coulomb 'branches'. These are the moduli (scalar fields) of the N=2 hyermultiplet and vector multiplet respectively.
Question 2) Do these link in with the phases described above or are the concepts of branches and phases very much distinct?
Christian Samann's notes tell us (on page 13) that the Coulomb branch is what we get when the gauge group breaks to U(1)'s. I think this is just an inconsistency in what different authors mean when they talk about branches and phases.