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Why do a physicist, particularly a string theorist care about Quivers ?

Essentially what I'm interested to know is the origin of quivers in string theory and why studying quivers is a natural thing in string theory.

I've heard that there is some sort of equivalence between category of D-branes and category of quiver representations in some sense, which I don't understand. It would be very helpful if somebody could explain this.

Also there are quiver type gauge theories, what are those and how are they related to representation theory of quivers.

Thanks.

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This is pretty broad, but I'll give it a shot.

The origin (or at least one origin) of quivers in string theory is that, at a singularity, it is often the case that a D-brane becomes marginally stable against decay into a collection of branes that are pinned to the singularity. These are called "fractional branes". To describe the gauge theory that lives on the D-brane at the singularity, we get a gauge group for each fractional brane, and for the massless string states stretching between the D-brane, we get bifundamental matter. Thus, a quiver gauge theory.

The fractional branes and the bifundamental matter are essentially holomorphic information, so you can get at them by looking at the topological B-model. Since the B-model doesn't care about kaehler deformations, you can take a crepant resolution of the singularity which lets you deal with nice smooth things. The connection to the derived category of coherent sheaves comes about because the B-model (modulo some Hodge theoretic stuff) is essentially equivalent to the derived category (even though it doesn't matter so much any more, I can't resist plugging my paper, 0808.0168).

The equivalence of categories, in some ways, can be thought of as a tool for getting a handle on the derived category (representations are easier to deal with than sheaves) and the fractional branes, but I always thought there was some real physics there. Was never quite able to make those ideas work, though.

For the relation between representations and quiver reps, the easiest thing to say is that a representation of the quiver is the same as giving a vev to all bifundamentals.

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I apologize, @Aaron, but wouldn't it be more logical to refer to Douglas+Moore arxiv.org/abs/hep-th/9603167 - the original paper about this topic with 1,000+ citations - rather than your 2008 paper? –  Luboš Motl Nov 11 '11 at 9:33
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I was referring to my paper for the derived category stuff (and I really ought to be referring to Douglas's original paper for that, too). –  Aaron Nov 11 '11 at 15:28
    
@Aaron Can you give some references to start reading about quivers and quiver gauge theory? Something pedagogical like what will begin from "What is Quiver?" I also recently saw this talk - princeton.edu/~masahito/confs/2011/Pestun_PCTS2011.pdf –  user6818 Dec 14 '11 at 21:12
    
It's a broad, broad subject. Which part are you most interested in? –  Aaron Dec 14 '11 at 22:17
    
@Aaron From where to start to be able to understand the literature like the one I liked in my previous comment? I had earlier seen some exposition on what a quiver is - thought of as a graph whose nodes are vector spaces and paths are homomorphisms but that doesn't seem to be how the recent papers by Gaiotto, Pestun et al are thinking about it! What is a "quiver gauge theory"? –  user6818 Dec 15 '11 at 22:43
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