Curved $\infty$-Local Systems And Projectively Flat Riemann-Hilbert Correspondence

Abstract

We generalize the higher Riemann-Hilbert correspondence in the presence of scalar curvature for a (possibly non-compact) smooth manifold $M$. We show that the dg-category of curved $\infty$-local systems, the dg-category of graded vector bundles with projectively flat $\mathbb Z$-graded connections and the dg-category of curved representations of the singular simplicial set of the based loop space of $M$ are all $A_\infty$-quasi equivalent. They provide dg-enhancements of the subcategory of the bounded derived category of twisted sheaves whose cohomology sheaves are locally constant and have finite-dimensional fibers. In the ungraded case, we reduce to an equivalence between projectively flat vector bundles and a subcategory of projective representations of $\pi_1(M; x_0)$. As an application of our general framework, we also prove that the category of cohesive modules over the curved Dolbeault algebra of a complex manifold $X$ is equivalent to a subcategory of the bounded derived category of twisted sheaves of $\mathcal O_X$-modules which generalizes a theorem due to Block to possibly non-compact complex manifolds.