*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?** 



Do all $\mathcal{N}=2$ Gauge Theories "Descend" from String Theory?
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](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://arxiv.org/abs/1606.02002) for type $A$.


On the other hand, as is explained in [1503.03676](https://arxiv.org/abs/1503.03676), its Hilbert series is a generating function of Donaldson-Thomas type invariants. More precisely, no stability condition is imposed in [1503.03676](https://arxiv.org/abs/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](https://sites.google.com/site/derivedstructures/). 


