Moduli of flat connections on smooth varieties

This paper is a companion to our paper [Poisson geometry of the moduli of local systems on smooth varieties, Publ. RIMS 57 (2021), no. 3-4, 959-991]. We study the moduli functor of flat bundles on the smooth, possibly non-proper, algebraic variety X over a field of characteristic zero. For this we introduce the notion of a formal boundary of X, denoted by (partial derivative) over capX, which is a formal analog of the boundary at 1 of the Betti topological space associated with X. We explain how to construct two derived moduli stacks Vect(del) (X) and Vectr(del)((partial derivative) over capX) of flat bundles on X and on (partial derivative) over capX, respectively, as well as a restriction map R : Vect(del)(X) -> Vect(del)((partial derivative) over capX). This work contains two main results. First we prove that the morphism R comes equipped with a canonical shifted Lagrangian structure. This result can be understood as the de Rham analog of the existence of Poisson structures on moduli of local systems on X. As a second statement, we prove that the geometric fibers of R are representable by quasi-algebraic spaces, a slight weakening of the notion of algebraic spaces.