Higher Riemann-Hilbert correspondence for foliations

Abstract: This paper explores foliated differential graded algebras (dga) and their role in extending fundamental theorems of differential geometry to foliations. We establish an $A_{\infty}$ de Rham theorem for foliations, demonstrating that the classical quasi-isomorphism between singular cochains and de Rham forms lifts to an $A_{\infty}$ quasi-isomorphism in the foliated setting. Furthermore, we investigate the Riemann-Hilbert correspondence for foliations, building upon the established higher Riemann-Hilbert correspondence for manifolds. By constructing an integration functor, we prove a higher Riemann-Hilbert correspondence for foliations, revealing an equivalence between $\infty$-representations of $L_{\infty}$-algebroids and $\infty$-representations of Lie $\infty$-groupoids within the context of foliations. This work generalizes the classical Riemann-Hilbert correspondence to foliations, providing a deeper understanding of the relationship between representations of Lie algebroids and Lie groupoids in this framework.