# Quivers Representations in SUSY gauge theories

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I would like to hear some reasons and ideas on how quivers are useful in SUSY gauge theories. There is a nice answer about the case of D-branes here but it is not clear on their appearance in gauge theory independently of the D-branes. More specifically I have heard that quivers can describe BPS states. Is this correct? And why so?

This post imported from StackExchange Physics at 2015-02-07 11:48 (UTC), posted by SE-user user39726
SUSY quivers are dissected here arxiv.org/abs/hep-th/0201205

This post imported from StackExchange Physics at 2015-02-07 11:48 (UTC), posted by SE-user Autolatry

Indeed, quivers first appeared in the context of D-branes at (conifold) singularities (there are various nice expositions in Klebanov-Witten theory reviews) where the D-branes "conspire" to give a $\mathcal{N}=1$ SYM theory. Additionally, gauge theories are strongly encoded inside the physics of D-branes, so I am not sure in what way you can "separate" these notions. Usually, quivers are used to describe the physics of BPS bound states of $\mathcal{N}=2$ susy and sugra. I will say a few words on this as an example of quivers in gauge theories. So let us consider $\mathcal{N}=2$ theory in four dimensions. As you will probably know this theory has a moduli space with a Coulomb and a Higgs branch. Let us consider a point $u$ in the Coulomb branch $\mathcal{C}$ of the moduli space. There we have a gauged $U(1)^r$ symmetry group together with a lattice $\Gamma$ from which the various BPS states take their charges $(p,q)$. From Seiberg-Witten theory we know how to consider the above on an elliptic curve $\Sigma_u$ that varies along $\mathcal{C}$. It is very well known that the homology classes of 1-cycles along the tori we are considering can be identified with $\Gamma$. This is all standard Seiberg-Witten stuff. Seiberg-Witten is of course solved in the IR. To study the BPS states at some specific point $u \in \mathcal{C}$ we need to introduce the quiver. These theories also have a central charge $Z$. Now, we take half the plain of the plane on which the central charge $Z$ takes values and we name it $H$. On this plane there exist a set of $2r+f$ (where $f$ is the number of flavors of the theory) states which are customary to denote as $\gamma_i$ (that we can naively consider them as particles). It turns out that such a basis, if it exists, it is the only possible one, it is unique. Using this basis, the set $\{ \gamma_i \}$ we can construct a quiver. For every $\gamma_i$ we draw a node and for every pair we draw arrows that connect them. Then we can use quiver quantum mechanics to find the BPS bound states of the BPS "particles" $\gamma_i$. So the moral/summary is the following: In the $\mathcal{N}=2$ theory consider a point $u$ of the Coulomb branch and use its data to form (if possible) a basis $\{ \gamma_i \}$ of the hypers. Then put a node on each one, and arrows between them. Then use quiver quantum mechanics to find the BPS bound states.
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