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

TL;DR

This paper extends the higher Riemann-Hilbert correspondence to include scalar curvature effects, establishing equivalences between various dg-categories related to curved $ abla$-local systems, twisted sheaves, and projectively flat bundles on manifolds.

Contribution

It introduces a framework connecting curved $ abla$-local systems, graded vector bundles with projectively flat connections, and curved representations, generalizing classical correspondences to curved and non-compact settings.

Findings

Establishes $A_ abla$-quasi equivalences between dg-categories of curved $ abla$-local systems and twisted sheaves.

Reduces to classical flat bundle and representation equivalence in the ungraded case.

Proves an equivalence between cohesive modules over curved Dolbeault algebra and twisted sheaves on complex manifolds.

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.