Moduli stacks of quiver bundles with applications to Higgs bundles

TL;DR

This paper introduces a general method for constructing moduli stacks of quiver bundles, including Higgs bundles, with applications to algebraic geometry and non-abelian Hodge theory.

Contribution

It provides a unified construction of moduli stacks for quiver bundles and Higgs bundles, establishing algebraicity and introducing new perspectives for categorification and applications.

Findings

Recovered Nakajima quiver varieties

Constructed moduli stacks of Higgs bundles with algebraic structure

Identified potential applications in categorification and homotopy theory

Abstract

We provide a general method for constructing moduli stacks whose points are diagrams of vector bundles over a fixed base, indexed by a fixed simplicial set -- that is, quiver bundles of a fixed shape. We discuss some constraints on the base for these moduli stacks to be Artin and observe that a large class of interesting schemes satisfy these constraints. Using this construction, we recover Nakajima quiver varieties and provide an alternate construction for moduli stacks of Higgs bundles along with a proof of algebraicity following readily from the algebraicity of moduli stacks of quiver bundles. One feature of our approach is that, for each of the moduli stacks we discuss, there are moduli stacks that are Artin, parametrizing morphisms of the objects being classified. We discuss some potential applications of this in categorifying non-abelian Hodge theory in a sense we will make precise. We also discuss potential applications of our methods and perspectives to the subjects of quiver varieties, abstract moduli theory, and homotopy theory.