The higher Riemann-Hilbert correspondence

We construct an A(infinity)-quasi-equivalence of dg-categories between P-A - the category of prefect A(0)-modules with flat Z-connection, associated to the de Rham dga A(center dot) 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 Igusa's notion of higher holonomy transport for 7Z-connections which is a derivative of Chen's idea of generalized holonomy. (C) 2013 Elsevier Inc. All rights reserved.