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

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 should describe 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 [LMNS]) 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 ch2, ch2, = 1/2c1 · c1 - c2 becomes precisely the formula for the eigenvalue of the degree operator, i.e. the well known quadratic behaviour of affine Lie algebras.