Affinizations of Lorentzian Kac-Moody Algebras and Hilbert Schemes of Points on K3 Surfaces

TL;DR

This paper constructs Lie algebra actions on the cohomology of moduli spaces of sheaves on K3 surfaces, linking geometric, algebraic, and modular perspectives, and extends to quiver varieties for affine ADE quivers.

Contribution

It introduces a new Lie algebra action on moduli spaces of sheaves on K3 surfaces via affinization of Kac-Moody algebras, connecting geometric and algebraic frameworks.

Findings

Lie algebra actions generated by correspondences between moduli spaces

Equivalence with actions from Fourier coefficients of vertex operators

Geometric interpretation of quiver varieties for affine ADE quivers

Abstract

For a class of K3 surfaces, the action of a Lie algebra which is a certain affinization of a Kac-Moody algebra is given on the cohomology of the moduli spaces of rank 1 torsion free sheaves on the surface. This action is generated by correspondences between moduli spaces of Bridgeland stable objects on the surface, and is equivalent to an action defined using Fourier coefficients of vertex operators. Two other results are included: a more general result giving geometric finite dimensional Lie algebra actions on moduli spaces of Bridgeland stable objects on K3 surfaces subject to natural conditions and a geometric modular interpretation of some quiver varieties for affine ADE quivers.