I am confused about the difference between the Coulomb and Higgs branches of the moduli space of supersymmetric gauge theories. It's easy to find a definition for $D=4$, $\mathcal{N}=2$ supersymmetric field theories: The moduli space is locally a direct product between the two branches: $\mathcal{M} = \mathcal{M}_C \times \mathcal{M}_H$. The Coulomb branch corresponds to scalar fields in the vector multiplet getting VEV's, and the Higgs branch corresponds to scalar fields in the hypermultiplet getting VEV's.

I'm interested in this question for the case of $D=4$, $\mathcal{N}=4$ SYM, i.e. the worldvolume theory of D3-branes. Again, it's easy to find statements to the effect that the Coulomb branch corresponds to separating the D3 branes in the transverse directions, which gives mass to the some of the gauge fields and thus Higgses them. My main question is: what is the brane interpretation of the Coulomb branch?

Follow up questions are:

1) what is the rationale behind naming the branch of moduli space where the gauge fields are Higgsed as Coulomb, and the one where they are not as Higgs? Is there a universe in which this naming convention makes sense?!

2) I've read that Coulomb branch vacua cannot support finite temperature (for example in http://arxiv.org/abs/hep-th/0002160). I think this has a nice brane interpretation, at least for the case of many stacked branes so that the SUGRA description is valid. Two parallel but transversely separated stacks of branes are mutually BPS and therefore feel no force between each other. If we heat one of them up, then this no-force condition no longer holds, and the two stacks will gravitationally attract. In the SUGRA description, this is essentially the D3 brane version of the fact that extremal black holes can be superimposed to create multi-center solutions, but any finite temperature ruins this since the gravitational and electric forces are no longer balanced.

Can the Higgs branch vacua support finite temperature, and if so, is there a nice way to understand this via above sort of reasoning?


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