Heisenberg Algebra and Hilbert Schemes of Points on Projective Surfaces

Abstract

The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various people (see e.g. [I], [Br], [ES], [G61], [G62]). The other is the infinite dimensional Heisenberg algebra which is closely related to affine Lie algebras (see e.g. [K]). We shall construct a representation of the Heisenberg algebra on the homology group of the Hilbert scheme. In other words, the homology group will become a Fock space. The basic idea is to introduce certain in the product of the Hilbert scheme. Then they define operators on the homology group by a well-known procedure. They give generators of the Heisenberg algebra, and the only thing we must check is that they satisfy the defining relation. Here we remark that the components of the Hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components. Our correspondences live in the product of the different components. Thus it is quite essential to study all components together. Our construction has the same spirit as the author's construction [Nal], [Na4] of representations of affine Lie algebras on homology groups of moduli spaces of instantons1 on ALE spaces which are minimal resolutions of simple singularities. Certain correspondences, called Hecke correspondences, were used to define operators. These twist instantons along curves (irreducible components of the exceptional set), while ours twist ideals around points. In fact, the Hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles, or more precisely torsion free sheaves. Our construction should be considered as a first step to extend [Nal], [Na4] to general 4-manifolds. The same program was also proposed by Ginzburg, Kapranov, and Vasserot [GKV].