Instanton Moduli Space on ALE Spaces

Question

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?

Answer 1

For every $k$, $M$ and $N$ positive integers, one can consider the moduli space $\mathcal{M}_M(k,N)$ of $U(N)$ instantons of instanton number (=second Chern class) $k$ on the resolution of the $A_{M-1}$ surface singularity. The case $M=1$ corresponds to instantons on $\mathbb{C}^2$. In particular we have at least three parameters: $k$, $M$, $N$ and things can become confusing if one mixes them or tries to obtain non-trivial relations between them.

On how the moduli space $\mathcal{M}_M(k,N)$ looks like: a fairly concrete description is due to Nakajima ( http://math.mit.edu/~ptingley/QuantumGroupsSpring2011/Nakajima94.pdf ) (generalizing the ADHM construction for $M=1$) by hyperkähler quotient construction, more precisely as Nakajima quiver varieties attached to the $\hat{A}_{M-1}$ Dynkin quiver. This quiver construction has a natural physics interpretation: it is the natural description of the moduli space "from the point of view of the instantons" (to make sense of that, one has to be able to separate the instantons from the four dimensional geometry, which is done naturally in string theory in the $D(-1)-D3$ brane system).

For any smooth complex surface $S$, there is a formula due to Göttsche giving the generating series of Euler characteristics of Hilbert schemes of points on $S$: it is the answer for $S=\mathbb{C}^2$ (the generating function of partitions) to the power $\chi(S)$ where $\chi(S)$ is the topological Euler characteristic of $S$.

In physics, the idea of geometric engineering is to consider a 10-dimensional string theory on a space time of the form $X_4 \times X_6$ and to study the effective gauge theory living on the 4-dimensional space $X_4$ in terms of the geometry of the 6-dimensional space $X_6$. In this context, there are at least two ways to introduce ALE spaces: one could take $X_4$ to be an ALE space or/and one could take some $X_6$ whose geometry contains an ALE space. All these possibilities are interesting and one should not confuse them.

For example, one can take $X_4=\mathbb{C}^2$ and $X_6$ a non-trivial fibration in ALE spaces of type $A_{M-1}$ over the complex projective line $\mathbb{P}^1$. The resulting gauge theory on $\mathbb{C}^2$ has gauge group $SU(N)$ and geometric engineering will give a relation between the Nekrasov partition function of this gauge theory, some generating function of equivariant integrals over moduli spaces of $SU(N)$ instantons over $\mathbb{C}^2$, and topological string on $X_6$. But because $X_6$ is a fibration in $A_{M-1}$ ALE space, topological string on $X_6$ is in fact related to the Hilbert schemes of points on the ALE space (see https://arxiv.org/abs/0802.2737 , this is a key step in the proof of the MNOP conjecture relating Gromov-Witten and Donaldson-Thomas theories for 3-folds).