# Instanton Moduli Space on ALE Spaces

+ 7 like - 0 dislike
299 views

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?

This post imported from StackExchange MathOverflow at 2017-03-03 23:28 (UTC), posted by SE-user spietro
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.