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For $\mathcal{N}=2$ 4D Super-Yang-Mills it is easy to find good, precise definitions of "Higgs branches" and "Coulomb branches" in moduli space. The moduli space (of VEVs) of the theory is locally a direct product between the moduli of the vector multiplet and those of the hypermultiplet, and moving along the first is called moving along the Coulomb branch, and along the other moving along the Higgs branch.

However, these terms are also used in situations where we don't have such a neat factorization of moduli space, and even where we don't actually know the moduli space explicitly at all. As an example, consider the following from "The landscape of M-theory compactification on seven-manifolds with $G_2$ holonomy" by Halverson and Morrison:

Suppose there existed a singular limit of $X$ which realizes a [...] gauge group $G\times\mathrm{U}(1)^k$ [...]. If smoothing the manifold back to $X$ Higgses this theory [carries out symmetry breaking by the Higgs mechanism] in a standard way, then an upper bound on the number of $\mathrm{U}(1)$s is set by the dimension of the maximal torus of the gauge theory on the singular space, that is $$ b_2(X) \leq \mathrm{rk}(G) + k \tag{4.1}$$ Certainly, if a gauge enhanced singular limit exists and $b_2(X) = 0$ then the vacuum obtained from M-theory on $X$ is on a Higgs branch; conversely $b_2(X) \neq 0$ is necessary for it to be on a Coulomb branch.

Here $X$ is some 11-dimensional manifold with an Abelian gauge theory on it that goes to a singular manifold in some limit, and there are general arguments to expect this singular limit to carry a non-Abelian gauge theory, meaning that the reverse direction from the singular limit to the smooth $X$ corresponds to symmetry breaking of some sort. $b_2(X)$ denotes the second Betti number. In general, the exact characteristics of the gauge theory (such as the number and type of multiplets) are difficult to determine and rarely explicitly known, which means the naive definition of Coulomb and Higgs branches from the 4D case does not apply since we don't know the full moduli space.

So, what is the general definition of these two branches that can also be applied to higher-dimensional theories with arbitrary $\mathcal{N}$? (In this case it would be $\mathcal{N}=1$, if that's relevant)

I suspect that the "Higgs branches" are just cases where the gauge theory is completely broken and that the "Coulomb branches" are those where the non-Abelian symmetry gets broken to a $\mathrm{U}(1)^n$.


There's a related question here but I can't really tell whether it is asking the same or what the answer is trying to say.

There's also this unanswered question which asks however for the reason for the nomenclature and specific properties of the branches, while I am solely interested in their actual definition for now.

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  • $\begingroup$ "I suspect that the "Higgs branches" are just cases where the gauge theory is completely broken and that the "Coulomb branches" are those where the non-Abelian symmetry gets broken to a $U(1)^n$." I always assumed that this was the definition of Higgs phase/branches and Coulomb phase/branches. $\endgroup$ – Nogueira Jul 31 '18 at 19:49
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This is probably not a satisfactory answer, but I give it since nobody seems to have an opinion on the question.

I would say that a vacuum is on the Higgs branch is there are scalar fields which are not connected by supersymmetry to any gauge field that have a non-zero VEV. It is called "Higgs" because this is what happens in the Standard Model BEH mechanism.

A vacuum is on the Coulomb branch if there are scalar fields that are connected by supersymmetry to a gauge field (or more generally that are adjoint valued) that have a non-zero VEV, and that break the gauge symmetry to an abelian one. This is the origin of the "Coulomb" name.

A vacuum in which both types of scalars would have a non-zero vev would be on both branches, or depending on your taste, on what you can call a mixed branch.

I think that in some situations, one indeed studies these branches as manifolds, compute the structures and dimensions, etc, and in other situations, the use of this terminology is much less strict, and refers roughly to the type of vacuum under consideration.

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  • $\begingroup$ In the $\mathcal{N}=1$ SUSY QCD in 4d, the scalar field $\phi$ is not adjoint valued. It belongs to a $\mathcal{N}=1$ chiral multiplet that takes values in the fundamental representation. So that part of the answer is wrong. $\endgroup$ – Marion Oct 7 '17 at 16:54
  • $\begingroup$ I refer to 4d $\mathcal{N}=2$ multiplets here. For 4d $\mathcal{N}=1$, as far as I know, there is no notion of Coulomb branch, in general there is a discrete set of vacua. $\endgroup$ – Antoine Apr 23 '18 at 17:49

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