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I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper Topological Quiver Matrix Models and Quantum Foam.

Context: The topological string theory partition function can be understood as computing the Donaldson-Thomas invariants of the variety over which the topological string is defined via the Donaldson-Thomas/Gromov-Witten correspondence. That's ok to me, the problem is that I've started to find fascinating, and apparently isolated examples of connections between quiver representations and the derived category of coherent sheaves of some varieties (especially toric cases) that I dont' fully understand.

The prototypical example of the aforementioned connections, is the well known realization of the moduli space of stable, rank $r$ (and $c_{2}=n$) and torsion-free sheaves on $\mathbb{P}^{2}$ as the $\mathcal{M}(n,r)$ quiver variety of the Jordan quiver under the Geiseker stability condition; the first data is precisely what the Donaldson-thomas theory of $\mathbb{P}^{2}$ actually computes. Another example is the Nakamura computation of the $G$-equivariant Hilbert scheme of points over $\mathbb{C}^{3}/G$ where $G$ is a finite $SL(3,\mathbb{C})$ subgroup as described in the page 14 of the paper "The McKay correspondence" using McKay quivers. This latter fact was used in Crystals and Black holes to enummerate tautological sheaves over a crepant resolution of $\mathbb{C}^{3}/G$ to compute the topological string partition function.

My problem: I supspect that the connections are not accidental but I'm uncapable to see what's the precise relation between moduli problems of quiver representations and those ones of sheaves, or where to start to investigate.

My background: I've sudied algebraic geometry from the first four chapters (Varieties,Schemes,Cohomology and Curves) in Hartshorne's textbook, I'm also familiar with the identification between the derived bounded category of coherent sheaves and the D-branes of the topological string B-model.

My weakness: I know very little about representation theory of quivers.

Questions: In Topological Quiver Matrix Models and Quantum Foam is apparently assumed that we can associate to a given toric variety a quiver whose derived category of representations is isomorphic to the derived bounded category of coherent sheaves of the given toric scheme.

1.-Does anyone know a gentile reference to learn about the mathematical details of how this can be explicity achieved?

2.- What could be a good reference to start learning about quivers focused to understand the papers Topological Quiver Matrix Models and Quantum Foam and Crystal Melting and Black holes given my prior knowledge and physics orientation.

Any comment or reading suggestion is very welcome.

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    $\begingroup$ Without knowing too much about the paper (I do not know enough physics to dig into your primary reference) if the toric variety possesses a strong full exceptional collection then its derived category is equivalent to the derived category of quiver representations. For a more complete discussion, see this post. As for quivers, I have found Kirillov's "Quiver Representations and Quiver Varieties" to be helpful and fairly explicit. $\endgroup$ – DKS Jun 23 at 13:06
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    $\begingroup$ One further comment, full exceptional collections of sheaves on toric varieties have been constructed by Kawamata (he has several papers on this). However there is no mention of when they are strong. Hopefully someone more knowledgeable can chime in. $\endgroup$ – DKS Jun 23 at 13:11
  • $\begingroup$ Thanks for your time, kindness and interesting comments DK. I was unaware about Kawamata's work, and indeed, searching for his name in ArXiv shows a lot of wonderful and truly useful references. That's precisely what I was searching for. Thanks again. $\endgroup$ – Ramiro Hum-Sah Jun 23 at 17:00
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    $\begingroup$ My pleasure, good luck with your work. $\endgroup$ – DKS Jun 23 at 18:04
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First, that review is somewhat depressing in that it's been over ten years since people figured out how to write down explicit boundary conditions in the B-model for objects in the derived category, but it's still talking about 'tachyon condensation' and locally-free resolutions, which do not always exist. I'm partial to the discussion in my old paper, but see also Kapustin et al and Herbst et al.

For what it's worth, the main bolded statement in the review is wrong -- the D-branes in the B-model do not need to be stable. Stability depends on the Kahler information of the target and has to do with the physical D-branes, not the topological ones.

Anyways, to answer your actual question, when you have an equivalence of categories between the derived category of coherent sheaves on noncompact CY and the derived category of representations of a quiver algebra, you often get that a component of the moduli space of representations with a specific dimension vector is the original noncompact CY. Physically, you can think of this as the moduli space of D0-branes being the cone itself. These D0-branes naturally correspond to representations of the quiver with a fixed dimension vector, and you can find the cone in the moduli stack pretty easily. With a little more work, you can get the GIT quotient too. You can see this in my two papers with Nick Proudfoot 1 and 2. There have been generalizations of this work, but I don't know if it has been proven for all the toric stuff (I've been away from this for a while). I'd start by looking at the work of Alastair Craw.

With respect to quivers, I was going to recommend Harm Derksen's lecture notes for a good introduction, but it looks like he took them down at some point. Sorry I don't have any good recommendation there.

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  • $\begingroup$ I'm just a beginner trying to create a background in topological string theory, that's why I really appreciate the enormous value of your comments and clarifications regarding the category of B-branes. What a wonderful bonus surprise was to me to learn that any object in the derived category can be coupled to the B-model. $\endgroup$ – Ramiro Hum-Sah Jun 25 at 4:40
  • $\begingroup$ Concerning your answer to my question. Wonderful, that was precisely what I was searching for. Indeed, some of your papers seem to be very relevant to solve some gaps in the papers I want to understand, an example is the conjecture in page 19 of Crystal Melting and Black Holes and Stability Conditions and Branes at Singularities. Thanks for your time and such a wonderful and illuminating answer. $\endgroup$ – Ramiro Hum-Sah Jun 25 at 5:32

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