*I asked this on MathStackExchange and was instructed it would be better here.*

I've recently been learning about moduli spaces of instantons on $\mathbb{C}^{2}=\mathbb{R}^{4}$. From what I can gather, one can consider the framed moduli space of torsion-free sheaves on $\mathbb{P}^{2}$ of rank $N$ and second Chern class $k$, which we denote $\mathcal{M}(k,N)$. I believe in the rank one case, we can identify this moduli space with the symmetric product of $\mathbb{C}^{2}$, which of course can be crepantly resolved to the Hilbert scheme. In other words,

$$\text{Hilb}^{k}(\mathbb{C}^{2}) \to \text{Sym}^{k}(\mathbb{C}^{2}) = \mathcal{M}(k,1)$$

From what I can gather, this is what's known as the "instanton moduli space on $\mathbb{C}^{2}$." There is then this whole "geometric engineering" story by Vafa, Hollowood, et. al. where they consider either the $\chi_{y}$ genus of these moduli spaces or the elliptic genus $\text{Ell}_{y,q}$ and construct the instanton partition function:

$$ \sum_{k} p^{k} \chi_{y} (\text{Hilb}^{k}(\mathbb{C}^{2})) \,\,\,\,\,\, \text{or} \,\,\,\,\,\, \sum_{k} p^{k} \text{Ell}_{y,q} (\text{Hilb}^{k}(\mathbb{C}^{2}))$$

One can then show that these partition functions are very remarkably equal to partition functions in topological string theory on certain Calabi-Yau varieties.

So really I'm curious about replacing $\mathbb{C}^{2}$ with the ALE spaces, specifically the $A_{N}$ resolutions of the singularities $\mathbb{C}^{2}/\Gamma$ where $\Gamma$ is a finite subgroup of $SU(2)$. The above story with the Hilbert schemes was only for the rank one case, $N=1$ so it's very tempting to hope that maybe the higher rank moduli spaces $\mathcal{M}(k,N)$ might be related to the $A_{N}$ resolutions somehow? I was hoping someone could help me understand what the moduli space of instantons on ALE spaces looks like, and whether there are nice partition functions like the ones above arising from such a space. I know there is physics literature here (like the Vafa-Witten https://arxiv.org/pdf/hep-th/9408074.pdf) but I'm having serious issues understanding the physics! Does considering the Hilbert scheme of points on the $A_{N}$ resolutions provide anything of physical relevance, or do we need something more complicated perhaps?