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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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The answer to the question in the title is no.

Everything comes to the idea of the swampland in the string theory context. Not every effective QFT can be UV-completed to a string theory, in particular that appears to be the case for $\mathcal{N}=2$ theories in four dimensions.

One of the swampland conjectures explicitly locate $\mathcal{N}=2,d=4$ supergravities without vector multiplets and theta angle different from $0$ or $\pi$ on the swampland. Reference: Theta-problem and the String Swampland.

Some clarifications are needed to answer the case of four dimensional quiver gauge theories. Strictly speaking the answer is no. Take as a counterexample the case of type IIA superstrings on a Taub-NUT space $ \ TN_{n}$ which locally has the same singularity structure than $\mathbb{C}^{2} / \mathbb{Z}_{n}$ in order to realize an interacting $SU(N)$ theory in four dimensions. The problem is there is no compact Calbi-Yau twofold with such kind of singularities if $n$ is large enough. Reference: Sec. 2.3 of the paper The String Landscape, the Swampland, and the Missing Corner

A very different question is whether a four dimensional supersymmetric field theory can be engineered in the topological string setup. That question is very different since string theory predicts gravity and in topological strings the graviton vertex operator is BRST-exact. Although there is no proof, the answer is more or less yes. In principle there is no obstruction to realize a given toric variety (and the geometrical engeeniering procedure) as the target space for a topological sigma model.

Concerning $\mathcal{N}=4,d=4$ SYM. Again the answer is no. $\mathcal{N}=4$ SYM in $d=4$ is inconsistent with string theory if the rank of its group is bigger than 22. Reference: Four Dimensional N=4 SYM and the Swampland

Finally, what is the relationship between the Vafa-Witten partition function and topological strings? The answer is that the Vafa-Witten partition function (in some limits) can be computed by adding $D4$-branes wrapping ample divisors to the Donaldson-Thomas counting of ideal sheaves in a given toric threefold by counting restricted plane partitions. Reference: Crystals and intersecting branes

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