# Higgs and Coulomb Branches. What are they?

On Wikipedia, there is an article on Moduli(Physics). The link is the following https://en.wikipedia.org/wiki/Moduli_(physics) . What captures me is Higgs and Coulomb Branch. If you click the links on Wikipedia, for these you notice that there is no article for them. A Google search yields research papers non-suited for the novice. What is a SIMPLE model that illustrates these concepts. Preferably written for a Physics high school student or College Freshman.Is there a video, or tutorial I can be directed to which goes over the concepts and operational steps?

1. I am interested in an operational definition (e.g . . A "X" is a "P" with . . . .)

2. A simple worked out example (e.g In a simple theory of "X" coupled to "Y" . . . in d dims the "K" branch is a . . . . and here are steps that get you from "M" to "Z")

3. I like diagrams and animations because I am an idiot ( e.g The vectors from point "l" are the . . .)

I think I can benefit most from simple expository media, with the ideas explicitly stated. I have to confess, I don't want to read a full book.I am human. I can go over a page or two long article, but I love videos. I am aware of prerequisites to gain such understanding.

• The problem is, you are gonna need to read more than two pages to understand it – user126422 Apr 7 '17 at 20:13
• The spontaneous symmetry breaking article has a picture illustrating a vacuum manifold, but for these more complicated examples in this article it is harder to picture. – AHusain Apr 7 '17 at 21:24
• The definition of the Higgs and Coulomb branch is right there in the article. The moduli space locally factors as the product of the moduli of a vector multiplet and a hypermultiplet, and the first factor is called the Coulomb branch and the second the Higgs branch. What exactly do you want to know about that? For the usage outside of $\mathcal{N}=2$ 4d SYM, I've asked a question about the exact meaning here already. – ACuriousMind Apr 8 '17 at 11:41
• Indeed it says "The N=2 supersymmetry algebra contains two representations with scalars, the vector multiplet which contains a complex scalar and the hypermultiplet which contains two complex scalars." The case of N= 2 in 4d is well known. The general meaning eludes me. I am going to be keeping an eye on your question. I am motivated by curiosity. I am not doing research on this. – user151266 Apr 8 '17 at 20:42

Oh hi there . . .

Symmetry breaking happens

• You can break gauge symmetry via the higgs mechanism. Sometimes different types of higgs fields transform differently.

a) Coulomb phase : broken to a product of U(1) factors

b) Higss Phase : the breaking was so hardcore, all U(1) factors died

Consider N =2 super Yang-Mills as @Dirac,@ACuriousMind, Wikipedia ( and perhaps me, depending on if you had coffee) have pointed out "The moduli space locally factors as the product of the moduli of a vector multiplet and a hypermultiplet, and the first factor is called the Coulomb branch and the second the Higgs branch" -- ACuriousMind

@Diracology also points out that this kind of thinking helps characterize the phases of super-symmetric gauge theory.

For the full treatment ncatlab has a link of references. Some of them might even address @AcuriousMind's comment about meaning outside of N = 2.

Notably Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩=4\mathcal{N} = 4 gauge theories, I (arXiv:1503.03676)

References