A GEOMETRIC BOSON-FERMION CORRESPONDENCE
The fixed points of a natural torus action on the Hilbert schemes of points in C 2 are quiver varieties of type A1. The equivariant cohomology of the Hilbert schemes and quiver varieties can be given the structure of bosonic and fermionic Fock spaces respectively. Then the local- ization theorem, which relates the equivariant cohomology of a space with that of its fixed point set, yields a geometric realization of the important boson-fermion correspondence. R´´ Les points fixes d'une action canonique d'un tore sur le schema de Hilbert de C 2 sont des varietes de quiver de type A1. On peut donner la cohomologieequivariante des schemas de Hilbert et des varietes de quiver la structure desespaces de Fock fermionique et bosonique, respectivement. Alors, la theoreme de localisation, qui lie la cohomologie ´equivariante d'uneavec la cohomologiete de son ensemble des point fixes, nous permet de donner une realisation geometrique de la correspondance bosonique-fermionique. Introduction. Recently there has been substantial interest in geometric constructions in representation theory. Such constructions translate between purely algebraic representation theoretic statements and statements involving geometric objects such as flag varieties, affine Grassmannians, quiver varieties and Hilbert schemes. This often provides one with new geometric techniques to examine various topics in representation theory (such as in the proof of the Kazhdan-Lusztig conjecture) as well as representation theoretic tools to organize and study the structure of various geometric objects. In this paper, we will focus on two particular geometric constructions. One of these involves varieties associated to quivers. Quivers, which are simply di- rected graphs, and their representations have a long history (see (21)). Lusztig (14) associated certain varieties to quivers and used these to provide a geometric realization of half of the universal enveloping algebra (or its quantum analogue) of Kac-Moody algebras. Then Nakajima (17, 19) modified these quiver vari- eties and gave a geometric construction of the representations of these algebras. The underlying vector space of the representation is the homology of the quiver varieties and the action of the Kac-Moody algebra is given by certain correspon- dences in products of these varieties. Nakajima's construction was motivated by his work with Kronheimer on solutions to the anti-self-dual Yang-Mills equations