The higher Riemann–Hilbert correspondence

Abstract

We construct an A∞-quasi-equivalence of dg-categories between PA — the category of prefect A0-modules with flat Z-connection, associated to the de Rham dga A• of a compact manifold M — and the dg-category of infinity-local systems on M — homotopy-coherent representations of the smooth singular simplicial set of M. We understand this as a generalization of the classical Riemann–Hilbert correspondence to Z-connections (Z-graded superconnections in some circles). This theory makes crucial use of Igusaʼs notion of higher holonomy transport for Z-connections which is a derivative of Chenʼs idea of generalized holonomy.