Heisenberg Algebra and Hilbert Schemes of Points on Projective Surfaces

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, ES, Gö1, Gö2]). 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 “correspondences” 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 with author’s construction [Na1, Na4] of representations of affine Lie algebras on homology groups of moduli spaces of “instantons” on ALE spaces which are minimal resolution of simple singularities. Certain correspondences, called Hecke correspondences, were used to define operators. These twist vector bundles along curves (irreducible components of the exceptional set), while ours twist 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 [Na1, Na4] to more general 4-manifolds. The same program was also proposed by Ginzburg, Kapranov and Vasserot [GKV]. Another motivation of our study is the conjecture about the generating function of the Euler number of the moduli spaces of instantons, which was recently proposed by