# What is the relation between the representation the Higgs field transforms under, the types of couplings in the theory and Higgs/Coulomb branches?

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.

-
Page 12 of Samonn's notes (link in original question) tells us that "Moduli space consists of separate branches touching each other at transition point". This implies that we get from one branch to the another via phase transitions and that each branch has it's own phase. However I'm not sure. – Siraj R Khan Jul 23 '13 at 16:17

Actually both questions looks nearly the same to me. Definitely the two given definitions are connected. This is what I understand...

The N=2 SYM has a vector supermultiplet and a hypermultiplet, the former being necessarily in the adjoint of the gauge group. Therefore the theory has three dynamical scalar fields, one in the ajoint.

The classical moduli space regards scalar field configurations that vanish the scalar potential. The configurations with non vanishing vev for the scalar in the adjoint and vanishing vev for the hypermultiplet scalar form the Coulomb branch. As you mentioned, when we have a non vanishing vev for a scalar field the unbroken gauge group contains a U(1) factor. This gives raise to monopole solutions which are characterized by a 1/r potential.

On the other hand, a vanishing vev for the scalar in the adjoint and a non vanishing vev for the scalar(s) in other representation than the adjoint form the Higgs branch of the theory. With a properly chosen representation for these scalars the gauge group can be completely broken and then the gauge mediators are all massive. This leads to short range interactions, consistent with a exponentially decaying potential (Yukawa type) which at long distances goes to zero, apart from an additive constant.

-
Excellent answer, thanks @Diracology. – Siraj R Khan Jan 5 at 22:25