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I asked this on PhysicsSE, but I think it also fits here as it's related to algebro-geometric connections to string and gauge theory.

I'm thinking about the beautiful story of "geometrical engineering" by Vafa, Hollowood, Iqbal (https://arxiv.org/pdf/hep-th/0310272.pdf) where various types of $\mathcal{N}=2$ SYM gauge theories on $\mathbb{R}^{4} = \mathbb{C}^{2}$ arise from considering string theory on certain local, toric Calabi-Yau threefolds.

More specifically, the topological string partition function (from Gromov-Witten or Donaldson-Thomas theory via the topological vertex) equals the Yang-Mills instanton partition function which is essentially the generating function of the elliptic genera of the instanton moduli space. (In various settings you replace elliptic genus with $\chi_{y}$ genus, $\chi_{0}$ genus, Euler characteristic, or something more fancy.)

From my rough understanding of the Yang-Mills side, we can generalize this in a few ways. Firstly, we can consider more general $\mathcal{N}=2$ quiver gauge theories where I think the field content of the physics is encoded into the vertices and morphisms of a quiver. And there are various chambers where one can define instanton partition functions in slightly different ways, though they are expected to agree in a non-obvious way. For a mathematical account see (https://arxiv.org/pdf/1410.2742.pdf) Of course, the second way to generalize is to consider not $\mathbb{R}^{4}$, but more general four dimensional manifolds like a K3 surface, a four-torus, or the ALE spaces arising from blowing up the singularities of $\mathbb{R}^{4}/\Gamma$.

My questions are the following:

Is it expected that this general class of $\mathcal{N}=2$ quiver gauge theories comes from string theory? In the sense that their partition functions may equal Gromov-Witten or Donaldson-Thomas theory on some Calabi-Yau threefold. If not, are there known examples or counter-examples? Specifically, I'm interested in instantons on the ALE spaces of the resolved $\mathbb{R}^{4}/\Gamma$.

In a related, but slightly different setting, Vafa and Witten (https://arxiv.org/pdf/hep-th/9408074.pdf) showed that the partition function of topologically twisted $\mathcal{N}=4$ SYM theory on these ALE manifolds give rise to a modular form. Now I know Gromov-Witten and Donaldson-Thomas partition functions often have modularity properties related to S-duality. So I'm wondering, does this Vafa-Witten partition function equal one of these string theory partition functions? Or are they related?

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The answer is Yes, at least when the ALE space (more precisely the ALF space instead) is of type $A$. Very roughly, one consider Donaldson-Thomas invariants for the same noncompact Calabi-Yau space, but not of rank $1$, of higher ranks instead. (I do not know whether they correspond to Gromov-Witten invariants.) I do not know physics literature, but I wonder that it was probably known as the answer is simple.

Here is a little more explanation. We consider the Coulomb branch of a framed affine quiver gauge theory as is defined in my paper 1601.03586 with Braverman, Finkelberg. The Coulomb branch in this case is expected to be the moduli spaces of instantons on an ALF space of type $A$, and is proved when the affine quiver is of type $A$ in 1606.02002. Here the type of the quiver determines the gauge group of instantons, and the type of the ALF space is the level of weights given by $V$, $W$. For example, if the quiver is of type $A$, it is the total sum of dimensions of $W$. When the level is $1$, it is the Taub-NUT space, $\mathbb R^4$ with non-flat hyper-Kaehler metric.

The Hilbert series of the Coulomb branch is nothing but the K-theoretic instanton partition function of pure gauge theory on the ALF space. When both of two weights determined by $V$, $W$ are dominant, one can replace the ALF space of the ALE space, as proved in 1606.02002 for type $A$.

On the other hand, as is explained in 1503.03676, its Hilbert series is a generating function of Donaldson-Thomas type invariants. More precisely, no stability condition is imposed in 1503.03676, but the ordinary Donaldson-Thomas invariants can be deduced from the Hilbert series by stuying Harder-Narashimhan stratification. This HN stratification argument is not written, but was explained in my talk at Oxford.

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  • $\begingroup$ Thanks for your answer! I've been in the process of trying to understand your papers on quiver gauge theories and instantons. I was hoping I could clarify what you meant by "higher rank DT theory?" I know what rank is on the gauge theory side, but not the string theory. It sounds like you mean we take generalized DT invariants for the same CY3 or something like that. We can vary the rank $r$ of the gauge group and the $N$ in $A_{N}$ to get instanton moduli spaces. So all these theories have generating functions equal to (generalized?) DT theory? Unfortunately, I couldn't get your Oxford notes $\endgroup$ – Benighted May 3 '17 at 18:37
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    $\begingroup$ Donaldson-Thomas invariants were defined by virtual counting of coherent sheaves on Calabi-Yau 3-folds. Donaldson-Thomas invariants of higher ranks means counting coherent sheaves of rank $\ge 2$. After Maulik-Nekrasov-Okounkov-Pandharipande, people mostly study DT invariants of rank $1$ (counting ideal sheaves), but the original definition is more general. $\endgroup$ – Hiraku Nakajima May 3 '17 at 21:24
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    $\begingroup$ When I consider DT-invariants of rank $r$ on the type $A_n$ Calabi-Yau appearing in geometric engineering, I get K-theoretic partition function for $SU(n+1)$ instantons on the ALF of type $A_{r-1}$, the quotient $\mathrm{Taub-NUT}/(\mathbb Z/r)$. Note that the rank and type (or level, in the context of the affine Lie algebra) are swapped. $\endgroup$ – Hiraku Nakajima May 3 '17 at 21:35
  • $\begingroup$ Thanks! I am too used to thinking about ideal sheaves in DT theory. The Calabi-Yau threefold here should be the fibrations $A_{n} \to \mathbb{P}^{1}$, correct? In other words, along the lines of the last comment, counting $\text{SU}(n+1)$ instantons on $A_{r-1}$ should be equivalent to studying rank $r$ DT-theory on $A_{n} \to \mathbb{P}^{1}$. $\endgroup$ – Benighted May 3 '17 at 22:06
  • $\begingroup$ I have already answered above. $\endgroup$ – Hiraku Nakajima May 4 '17 at 8:28

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