A Riemann--Hilbert correspondence for infinity local systems

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Block, Jonathan
Smith, Aaron M.
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Abstract
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We describe an $A_\infty$-quasi-equivalence of dg-categories between the first authors' $\mathcal{P}_{\mathcal{A}}$ ---the category of category of prefect $A^0$-modules with flat $\Z$-connection, corresponding to the de Rham dga $\mathcal{A}$ of a compact manifold $M$--- and the dg-category of \emph{infinity-local systems} on $M$ ---homotopy coherent representations of the smooth singular simplicial set of $M$, $\Pinf$. We understand this as a generalization of the Riemann--Hilbert correspondence to $\Z$-connections ($\Z$-graded superconnections in some circles). In one formulation an infinity-local system is simplicial map between the simplicial sets ${\pi}_{\infty}M$ and a repackaging of the dg-category of cochain complexes by virtue of the simplicial nerve and Dold-Kan. This theory makes crucial use of Igusa's notion of higher holonomy transport for $\Z$-connections which is a derivative of Chen's main idea of generalized holonomy.
Comment: 26 pages., the previous version was missing the bibliography
Keywords
Mathematics - Algebraic Topology, Mathematics - Differential Geometry, 53C29, 35Q15, 57R15
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