Hilbert Schemes of Points on Surfaces and Heisenberg Algebras

In this article, we throw a bridge between two objects which are unrelated at rst sight. One is the in nite dimensional Heisenberg algebra (simply the Heisenberg algebra, later) which plays a fundamental role in the representation theory of the a ne Lie algebras. The other is the Hilbert schemes of points on a complex surface appearing in the algebraic geometry. As we will explain soon, the Heisenberg algebra has a representation, called the Fock space representation, on the polynomial ring of in nitely many variables. The degrees of polynomials, where variables have di erent degrees, give a direct sum decomposition of the representation (weight space decomposition). On the other hand, the Hilbert scheme of points decomposes into in nitely many connected components according to the number of points. Gottsche [7] computed Betti numbers of the Hilbert scheme. His result says that the sum of the Betti number of the Hilbert scheme of n-points is equal to the dimension of the subspace of the Fock space representation of degree n. If we consider the generating function of the Poincar e polynomials moving number of points, we get the character of the Fock space representation of the Heisenberg algebra. Inspired by this coincidence and the S-duality conjecture by physists Vafa and Witten [22], the author gives an `understanding' of Gottsche's formula, constructing a Heisenberg algebra representation on the homology group of the Hilbert scheme [16]. Special features of this construction are 1) the Heisenberg algebra acts on the homology group, not on the Hilbert scheme itself, and 2) probably more importantly, the action is de ned only after we consider all the components of the Hilbert scheme simultaneously. The construction is very di erent from the representation obtained from an action on a space, like the Borel-Weil theory. We will not discuss the relation between the author's work and the S-duality conjecture in detail. We just explain brie y. The interested reader should see [17]. Both the author's work and the S-duality conjecture have relationship to the theory of modular forms. The character of the Fock space representation of the Heisenberg algebra, or more generally integrable highest weight representations of a ne Lie algebras are known to have modular invariance as proved by KacPeterson[11]. And this phenomenon is naturally `explained' through their relation to partition functions of the conformal eld theory on a torus. In this way, the a ne Lie algebra have close relation to the conformal eld theory [20]. So they are usually understood as a real 2-dimensional = complex 1-dimensional objects. But the author's work shows that the Hilbert schemes on a surface, a complex 2-dimensional = real 4-dimensional object also relates to a ne Lie algebras, and hence the modular invariance property. Vafa-Witten derived, as a consequence of