Moduli stacks of quiver connections and non-Abelian Hodge theory

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 $λ$--connections over a base prestack $X$, where $λ$ can be a fixed number or a parameter. Taking $λ$ 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.