How can I understand instantons as sheaves?

Question

In specific, instantons are considered or interpeted as torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two moduli spaces (instantons and torsion free coherent sheaves) are identified?

More background: the moduli space of instantons $M_{r,k}$ where $r$ is the rank of the gauge group or of the vector bundle and $k$ is the instanton number also defined as the second Chern class of the vector bundle. Now, these instantons are said to be framed which corresponds to requiring the field strength $F$ to vanish at infinity or, in other words, the instantons to be in pure gauge at infinity. According to Donaldson this space of "framed" instantons is identified with the moduli space of framed rank $r$-vector bundles on $\mathbb{P}^2 = \mathbb{C}^2 \cup l_{\infty}$ where $l_{\infty} = \mathbb{P}^1 = \mathbb{C}\cup \{ \infty \}$. Framing means that there exists a local trivilization at the line at the infinity ( $l_{\infty}$) such that the vector bundle is trivial there.

Now the space $M_{r,k}$ is also identified with the moduli space of torsion free sheaves $E$ on $\mathbb{P}^2$ such that rank($E$)$=r$ and $c_2(E)=k$ satisfying two conditions

• $E$ is torsion free on $\mathbb{P}^2$ and locally free (projective) in a neighborhood of $l_{\infty}$ (that is that at a neighborhood of infinity the sheaf looks like a vector bundle as above) and
• There exists a framing just like above (for the vector bundle).

What I ask for is intuition on the above objects. In what sense I can see the instantons as sheaves. If the instanon moduli space had no singularities and was smooth it is quite straight forward to understand the vector bundle construction. Sheafs are needed in order to take into consideration these singularities, right? And how is the sheaf theoretic construction related to $\text{Hilb}^n(X)$. I know that the problem is when two point like instantons approach each other a singularity appears and thus the vector bundle construction is not well defined but I do not quite understand how the sheaf theoretic construction resolves the problem.

Answer 1

Okay, I cannot give you a full understanding of what is going on, but I can make the objects we are dealing with more precise:

There are two spaces here:

1. The moduli space $M_\text{sh}(r,k)$ of framed torsion-free coherent sheaves of rank $r$ and second Chern class $k$ on the projective scheme $\mathbb{P}^2$ viewed as a complex analytic space with its structure sheaf of analytic functions.

2. The moduli space $M_\text{in}(r,k)$ of framed instanton bundles of a gauge group of rank $r$ and second Chern class $k$ on the 4-sphere $S^4 = \mathbb{R}^4\cup\{\infty\}$, where $\infty$ is added to make the notion of framing precise.

Framed in the first case means that the sheaf is locally free at the line at infinity, framed in the second case means the gauge field configuration is pure gauge at the point at infinity. These two spaces are not the same. In particular, $M_\text{sh}$ is non-singular while $M_\text{in}$ is singular, and this is precisely the motivation to find the following equivalence apparently proved by Donaldson:

The moduli space of framed instantons $M_\text{in}$ is in bijection to the subset $M^\text{reg}_{0,\text{sh}}\subset M_\text{sh}$ of locally free sheaves.

Heuristically, it is not difficult to see that a vector bundle defines a sheaf. Given a bundle $P\to X$, the corresponding sheaf is defined by $U\mapsto \Gamma(U,P)$, i.e. the sheaf which just associates to every open set its local sections. The proof of the above assertion, however, is far more complicated: $S^4$ is just the $\mathbb{R}^4\cong\mathbb{C}^2$ with a point at infinity, but $\mathbb{P}^2$ is the same with a *line at infinity. You have to show that every locally free sheaf on the latter really defines a bundle on the former, and that the bundles on the former really give a proper sheaf on the latter that also plays nice with the analytic structure sheaf. Lastly, a principal bundle isn't a vector bundle, and the proof of the assertion relies on a proof that the instanton bundles on $S^4$ correspond to framed holomorphic vector bundles on $\mathbb{P}^2$.

Now, given the theorem by Donaldson, we see why the passage from instantons to sheaves resolves the issue of singularities - the space of all torsion-free sheaves is non-singular, so slightly generalizing the notion of instanton/locally free sheaf/holomorphic vector bundle to that of torsion-free sheaf gets rid of the issue.

The relation to Hilbert schemes arises because $$ M_\text{sh}(1,k)\cong \mathrm{Hilb}^k(\mathbb{C}^2)$$

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Donaldson's proof is in

Donaldson, "Instantons and geometric invariant theory", Comm. Math. Phys. 93 (1984)

and relies on prior work by Atiyah and Ward.

Atiyah and Ward, "Instantons and algebraic geometry", Comm. Math. Phys. 55 (1977)

The last correspondence can for example be found in

Nakajima, "Lectures on Hilbert Schemes of Points on Surfaces", AMS