Irreducibility of moduli spaces of vector bundles on K3 surfaces

TL;DR

This paper proves that moduli spaces of stable sheaves with primitive Mukai vectors on K3 surfaces are irreducible symplectic manifolds, relating them to Hilbert schemes, computing their periods, and exploring implications in physics.

Contribution

It establishes the irreducibility of these moduli spaces and connects them to Hilbert schemes, also computing their periods and applying results to physics dualities.

Findings

Moduli spaces are irreducible symplectic manifolds.

They are related to Hilbert schemes of points.

Euler characteristics match physical predictions.

Abstract

In this paper, we show the moduli spaces of stable sheaves on K3 surfaces are irreducible symplectic manifolds, if the associated Mukai vectors are primitive. More precisely, we show that they are related to the Hilbert scheme of points. We also compute the period of these spaces. As an application of our result, we discuss Montonen-Olive duality in Physics. In particular our computations of Euler characteristics of moduli spaces are compatible with Physical computations by Minahan et al.