A Riemann Hilbert correspondence for infinity local systems

Details A Riemann Hilbert correspondence for infinity local systems A Riemann Hilbert correspondence for infinity local systems

2009-08-20

arxiv.org/abs/0908.2843v3

Mathematics

Algebraic Topology

36 pages., significant revision, typos, some errors fixed, changed notation, and sign conventions. Version 2 was a mistaken up

Scientific paper

We describe an $A_\infty$-quasi-equivalence of dg-categories between Block's $\Perf$, corresponding to the de Rham dga $\As$ of a compact manifold $M$ and the dg-category of infinity-local systems on $M$. We understand this as a generalization of the Riemann-Hilbert correspondence to $\Z$-graded connections (superconnections in some circles). In one formulation an infinity-local system is an $(\infty,1)$-functor between the $(\infty,1)$-categories ${\pi}_{\infty}M$ 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 $\Z$-graded connections which is a derivative of Chen's main idea of generalized holonomy. In the appendix we describe some alternate prespectives on these ideas and some technical observations.

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