Framed Rank r Torsion-free Sheaves on CP^2 and Representations of the Affine Lie Algebra \hat{gl(r)}

TL;DR

This paper constructs geometric models of certain algebraic structures using moduli spaces of sheaves on CP^2, providing new realizations of affine Lie algebra representations via equivariant cohomology.

Contribution

It introduces geometric realizations of affine Lie algebra representations through moduli spaces of sheaves and quiver varieties, connecting algebraic and geometric frameworks.

Findings

Realization of level one irreducible representations of fhat{gl(r)}

Geometric construction of bosonic and fermionic Fock spaces

Representation of level k irreducible representations on quiver varieties

Abstract

We construct geometric realizations of the r-colored bosonic and fermionic Fock space on the equivariant cohomology of the moduli space of framed rank r torsion-free sheaves on CP^2. Using these constructions, we realize geometrically all level one irreducible representations of the affine Lie algebra \hat{gl(r)}. The cyclic group Z_k acts naturally on the moduli space of sheaves, and the fixed-point components of this action are cyclic Nakajima quiver varieties . We realize level k irreducible representations of \hat{gl(r)} on the equivariant cohomology of these quiver varieties.