Moduli stacks of quiver connections and non-Abelian Hodge theory

TL;DR

This paper extends non-Abelian Hodge theory to moduli stacks of quiver connections, constructing algebraic stacks parametrizing diagrams of bundles with λ-connections, and analyzing their properties.

Contribution

It formalizes the de Rham side of a conjectural extension of non-Abelian Hodge correspondence to diagrammatic moduli stacks, including λ-connections and filtrations.

Findings

Constructed algebraic moduli stacks of diagrams of bundles with λ-connections.

Proved these stacks are algebraic, locally of finite presentation, with affine diagonal.

Extended non-Abelian Hodge theory to quiver connection moduli stacks.

Abstract

In arXiv:2407.11958, a moduli stack parametrizing $I$--indexed diagrams of Higgs bundles over a base stack $X$ was constructed for any finite simplicial set $I$, inspiring speculations about extending the non-Abelian Hodge correspondence to these moduli stacks. In the present work, we formalize the de Rham side of this conjectural extension. We construct moduli stacks parametrizing diagrams of bundles with $\lambda$--connections over a base prestack $X$, where $\lambda$ can be a fixed number or a parameter. Taking $\lambda$ to be $1$ gives a moduli stack parametrizing diagrams of bundles with connection, while taking it to be a parameter gives a version of Simpson's non-Abelian Hodge filtration for digrams of bundles with connection. We show that when $X$ is a smooth and projective scheme over an algebraically closed field $k$ of characteristic $0$, these moduli stacks are algebraic and locally of finite presentation, and have affine diagonal.