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**Heisenberg algebra and Hilbert schemes of points on projective surfaces.**
*(English)*
Zbl 0915.14001

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. The other is the infinite dimensional Heisenberg algebra which is closely related to affine Lie algebras.

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 as the author’s earlier construction [H. Nakajima, Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026) and Int. Math. Res. Not. 1994, No. 2, 61-74 (1994; Zbl 0832.58007)] of representations of affine Lie algebras on homology groups of moduli spaces of “instantons” 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 the two papers cited above to general 4-manifolds.

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 C. Vafa and E. Witten [Nucl. Phys., B 431, No. 1-2, 3-77 (1994)]. They conjectured that it is a modular form under certain conditions. If \(X^{[n]}\) is the Hilbert scheme parameterizing \(n\)-points in \(X\), then the generating function of the Poincaré polynomials is given by \[ \sum^\infty_{n= 0} q^nP_t(X^{[n]}) = \prod^\infty_{m=1} {(1+t^{2m-1} q^m)^{b_1 (X)} (1+t^{2m+1} q^m)^{b_3(X)} \over(1-t^{2m-2} q^m)^{b_0(X)} (1-t^{2m}q^m)^{b_2 (X)} (1-t^{2m+2} q^m)^{b_4(X)}},\tag{1} \] where \(b_i(X)\) is the Betti number of \(X\).

The paper is organized as follows. In section 2 we give preliminaries. We recall the definition of the convolution product in §2(i) with some modifications and describe some properties of the Hilbert scheme \(X^{[n]}\) and the infinite Heisenberg algebra and its representations in §§2(ii), 2(iii). The definition of correspondences and the statement of the main result are given in section 3. The proof will be given in section 4.

While the author was preparing this manuscript, he learned that a similar result was announced by T. Grojnowski [Math. Res. Lett. 3, No. 2, 275-291 (1996; Zbl 0879.17011)] who introduced exactly the same correspondence.

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 as the author’s earlier construction [H. Nakajima, Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026) and Int. Math. Res. Not. 1994, No. 2, 61-74 (1994; Zbl 0832.58007)] of representations of affine Lie algebras on homology groups of moduli spaces of “instantons” 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 the two papers cited above to general 4-manifolds.

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 C. Vafa and E. Witten [Nucl. Phys., B 431, No. 1-2, 3-77 (1994)]. They conjectured that it is a modular form under certain conditions. If \(X^{[n]}\) is the Hilbert scheme parameterizing \(n\)-points in \(X\), then the generating function of the Poincaré polynomials is given by \[ \sum^\infty_{n= 0} q^nP_t(X^{[n]}) = \prod^\infty_{m=1} {(1+t^{2m-1} q^m)^{b_1 (X)} (1+t^{2m+1} q^m)^{b_3(X)} \over(1-t^{2m-2} q^m)^{b_0(X)} (1-t^{2m}q^m)^{b_2 (X)} (1-t^{2m+2} q^m)^{b_4(X)}},\tag{1} \] where \(b_i(X)\) is the Betti number of \(X\).

The paper is organized as follows. In section 2 we give preliminaries. We recall the definition of the convolution product in §2(i) with some modifications and describe some properties of the Hilbert scheme \(X^{[n]}\) and the infinite Heisenberg algebra and its representations in §§2(ii), 2(iii). The definition of correspondences and the statement of the main result are given in section 3. The proof will be given in section 4.

While the author was preparing this manuscript, he learned that a similar result was announced by T. Grojnowski [Math. Res. Lett. 3, No. 2, 275-291 (1996; Zbl 0879.17011)] who introduced exactly the same correspondence.