Instantons and affine algebras I: The Hilbert scheme and vertex operators

TL;DR

This paper explores the action of affine Lie algebras on the moduli space of U(n)-instantons, linking geometric structures with representation theory and extending Nakajima's work to more general settings.

Contribution

It introduces a new framework for understanding the cohomology of instanton moduli spaces via affine Lie algebra actions, generalizing previous results to broader contexts.

Findings

Describes the affine Lie algebra action on U(n)-instanton moduli spaces.

Connects the combinatorial complexity of moduli spaces with representation theory.

Shows the case of U(1)-instantons yields the basic representation of affine Lie algebras.

Abstract

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima, who considered the case when $X$ is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i\.e\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of $U(1)$-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of $ch_2$, $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$ becomes precisely the formula for the eigenvalue of the degree operator, i\.e\. the well known quadratic behaviour of affine Lie algebras.