Abstract:In recent years, there has been considerable interest in constructing geometric realizations of certain representation theoretical objects such as Lie algebras, Heisenberg algebras, and their representations. Among such approaches are the Hilbert scheme realization of Heisenberg algebras developed by Grojnowski and Nakajima and the quiver variety realization of Lie algebras developed by Lusztig and Nakajima. Such constructions yield natural bases with remarkable properties and are also related to the theory of crystal bases, instantons and other areas of mathematics and physics.
In this talk, we will examine a connection between the two geometric approaches mentioned above. We will see that the fixed points of a natural torus action on the Hilbert schemes of points in C^2 are quiver varieties of (infinite) type A. The equivariant cohomology of the Hilbert schemes and quiver varieties can be given the structure of bosonic and fermionic Fock spaces respectively. Then the localization 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.