I'm thinking about the beautiful story of "geometrical engineering" by Vafa, Hollowood, Iqbal (https://arxiv.org/abs/hep-th/0310272) 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 very mathy account see (https://arxiv.org/abs/1410.2742) 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/abs/hep-th/9408074) 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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