Physical meanings or definitions of Higgs Branch and Coulomb Branch

Question

Question: What are the physical meanings or definitions of Higgs Branch and Coulomb Branch (in High Energy Physics in particular)?

There are related questions in the past: http://www.physicsoverflow.org/24063/higgs-branch-and-coulomb-branch?show=24063#q24063

Higgs phase: I know the meaning of Higgs phase where gauge field $A$ (such as U(1)) becomes massive with Higgs field $\Phi$ condensation $\langle \Phi \rangle \neq 0$, so there is a mass term for gauge field in the Lagrangian, and physically it gives rise to the Meissner effect. For example, in superconductor, the Meissner effect causes the magnetic field cannot penetrate through the bulk but exponentially decay.

Coulomb phase: I know Coulomb force and it usually implies a massless gauge field $A$ (such as U(1)), and there is long range interaction, perhaps with some screening effect.

But what precisely are the physical meanings or definitions of Higgs Branch and Coulomb Branch?

Comments

Please correct me if I say anything incorrect or imprecise. Thanks.

Answer 1

Background: Let us consider $\mathcal{N}=2$ global SUSY in $d=4$ with no central charges. Right, then as you probably know we have an $SU(2)_R$ symmetry between the supercharges $Q_{\alpha}$ which are Weyl spinors and transform as a doublet. The on-shell spectrum consists of a hypermultiplet (two complex scalars $\phi, \tilde{\phi}$, and two Weyl fermions $\psi, \tilde{\psi}$) and a vector multiplet (one complex scalar $\varphi$, two Weyl spinors $\lambda, \tilde{\lambda}$ and a vector $A_{\mu}$). $\phi$ and $\tilde{\phi}$ transform as an $SU(2)_R$ doublet. $\varphi$ and $A_{\mu}$ transform in the adjoint of the gauge group and both are singlets of the $SU(2)_R$.

Main point: Now, consider this theory at the IR. There are some interesting phenomena now. One important result for $\mathcal{N}=2$ at the IR is that they have an absence of scalar field potential terms which means that as long as $U(1)$ vector multiplets and/or neutral hypers occur in this region of the RG, $\mathcal{N}=2$ theories have a moduli space (parameter space of vacua which we will see it has the form of a -not alway smooth- manifold). Furthermore, it has been found that there are no kinetic terms mixing fields of the hyper(s) and the vector multiplets.  This means that the moduli space $M$ of the theory can be written as $M_{\text{hyper}} \times M_{\text{vector}}$. Now, $M_{\text{hyper}}$ is the moduli sub-space along which the vevs of the hypermultiplet vary with the vector multiplet being fixed and accordingly for the $M_{\text{vector}}$. Now if $M_{\text{vector}}$ is trivial  then $M = M_{\text{hyper}}$ and the moduli space of the theory is called Higgs branch (or we say that the thory is in the Higgs branch). When $M=M_{\text{hyper}}$ then the moduli space is called Coulomb branch, which follows from the fact that there always are massless $U(1)$ vectors from the vector multiplet(s).  

Some comments: Now be careful, this is the picture classically. See [1] as a good introduction to moduli spaces in supersymmetry. Quantum mechanically things are a bit less trivial. It turns out that in  $\mathcal{N}=2$ SUSY gauge theories we have two important coupling functions the Kahler metric $g_{ij}$ and the holomorphic coupling $\tau_{IJ}$ along with a cut-off scale $\Lambda$ and various fimensionless parameters $\lambda_i$. The theories we are mostly looking are asymptotically free and we want to take the limit where $\Lambda \to 0$ where the strong dynamics take place. It is a long discussion but for now just believe that the Higgs branch metric is given by the classical theory. Therefore we conclude that only the Coulomb branch admits quantum corrections. Furthermore, a theory can have a mixed branch. The nice properties of the Higgs branch are retained even for theories with mixed branches when one goes quantum mechanical. Solving a $\mathcal{N}=2$ theory amounts to finding the metric of its Coulomb branch  (along with understanding the theory along the RG flow and its fixed points of course). A nice book on $\mathcal{N}=2$ theories, with slightly more advance topics is [2] (of the PO user Yuji**).

[1] John Terning, Modern Supersymmetry; Dynamics and Duality

[2] Yuji Tachikawa, Supersymmetric Dynamics for Pedestrians 

P.S. My impression is that we also talk about Higgs and Coulomb branch for $\mathcal{N}=1$ theories where the first corresponds to the chiral supermultiplets and the latter to the vector ones.