I asked this on MathStackExchange and was instructed it would be better here.
I've recently been learning about moduli spaces of instantons on C2=R4. From what I can gather, one can consider the framed moduli space of torsion-free sheaves on P2 of rank N and second Chern class k, which we denote M(k,N). I believe in the rank one case, we can identify this moduli space with the symmetric product of C2, which of course can be crepantly resolved to the Hilbert scheme. In other words,
Hilbk(C2)→Symk(C2)=M(k,1)
From what I can gather, this is what's known as the "instanton moduli space on C2." There is then this whole "geometric engineering" story by Vafa, Hollowood, et. al. where they consider either the χy genus of these moduli spaces or the elliptic genus Elly,q and construct the instanton partition function:
∑kpkχy(Hilbk(C2))or∑kpkElly,q(Hilbk(C2))
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 C2 with the ALE spaces, specifically the AN resolutions of the singularities C2/Γ where Γ 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 M(k,N) might be related to the AN 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 AN 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