A Non-Linear Deformation of the Hitchin Dynamical System

TL;DR

This paper explores a deformation of Hitchin's integrable system via Mukai's moduli space of sheaves on a K3 surface, revealing geometric properties like the Lagrangian nature of the nilpotent cone and describing its components.

Contribution

It introduces a novel deformation of Hitchin's system using Mukai's space, connecting sheaf moduli on K3 surfaces with Hitchin's nilpotent cone structure.

Findings

Mukai's space is a deformation of Hitchin's system.

The nilpotent cone in Mukai space is Lagrangian.

Components of the nilpotent cone are described as affine bundles over symmetric products.

Abstract

Mukai's space, parametrizing simple sheaves on a K3 surface S whose numerical invariants are those of a line bundle on a curve C in S, is interpreted as a deformation of Hitchin's system on C. This is used to show that the nilpotent cone in Mukai space is Lagrangian. In rank 2, components of this nilpotent cone are described as affine bundles over symmetric products of the curve. The underlying vector bundles give the corresponding components of the Hitchin nilpotent cone.