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I asked the following question (https://mathoverflow.net/questions/263654/instanton-moduli-space-on-ale-spaces/263816#263816) on MathOverflow, with regards to instanton moduli spaces on certain ALE manifolds. In particular, for the time being, I'm only interested in the $A_{N-1}$ resolutions of $\mathbb{C}^{2}/\mathbb{Z}_{N}$. Within the phenomenal answer to my question above, it was pointed out that in physics, there are two entirely different situations: you can consider an ALE space to be part of the internal six dimensional space in string theory, or in some cases you actually want the ALE space to be the four-dimensional spacetime itself.

I've more or less made peace with the first case. The details are above me, but I think you talk about having these blown-up exceptional divisors as part of your ALE space, and considering branes wrapping these 2-cycles correspond to BPS particles in the spacetime $\mathbb{R}^{4}$. Hence, we get a beautiful relationship between instanton counting on $\mathbb{R}^{4} = \mathbb{C}^{2}$ and a topological string theory on some threefold such that four of the six real dimensions are an ALE space!

I was hoping someone could explain to this poor mathematician what motivation the second case has: in other words, why would physicists want to take an ALE space as the four dimensional spacetime? I'm guessing the answer might include the words Taub-NUT or Eguchi-Hanson in the context of General Relativity, but I'm not sure. In addition, what is the corresponding topological string theory, if any? i.e. what string theory engineers a SUSY gauge theory on this ALE space. Or maybe I'm just totally confused about how all this works.

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