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Instantons and affine algebras I The Hilbert scheme and vertex operators.pdf

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Instantons and affine algebras I The Hilbert scheme and vertex operators

a r X i v : a l g - g e o m / 9 5 0 6 0 2 0 v 1 2 5 J u n 1 9 9 5 INSTANTONS AND AFFINE ALGEBRAS I: THE HILBERT SCHEME AND VERTEX OPERATORS I. Grojnowski June 8, 1995 alg-geom/9506020 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 [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 2 c1 · c1 ? c2 becomes precisely the formula for the eigenvalue of the degree operator, i.e. the well known quadratic behaviour of affine Lie algebras. Introduction This is the first in a series of papers devoted to describing the action of an affine Lie algebra on the moduli space of instantons on an algebraic surface X . This paper, which is only an announcement, is concerned with the “boundary” of moduli space; the subsequent papers will describe the action on the interior. We describe the idea briefly. Let X be an algebraic surface, M the moduli space of U(c)-instantons on X (see below for precise definitions). M is not connected; it decomposes into M = ? Mc1,ch2 , where Mc1,ch2 denotes those instantons with fixed first Chern class equal to c1 ∈ H 2(X,Z) and second Chern character equal to ch2 ∈ Q = H 4(X,Q). Let Σ ? X