The most basic form of the Riemannâ€“Hilbert correspondence states that the category of flat vector bundles on a suitable space is equivalent to the category of local systems.
While Hilbert's 21st problem has a negative solution, there is a generalized sheaf-theoretical formulation which leads to an equivalence of categories discovered by Mebkhout and a bit later also by Kashiwara.
Alain Connes, Matilde Marcolli Noncommutative Geometry, Quantum Fields and Motives
Wikipedia, Riemann-Hilbert correspondence
Z. Mebkhout, Le formalisme des six opĂ©rations de Grothendieck pour les $\mathcal{D}_X$-modules cohĂ©rents, Travaux en Cours 35. Hermann, Paris, 1989. x+254 pp. MR90m:32026
Andrea D'Agnolo?, Masaki Kashiwara, Riemann-Hilbert correspondence for holonomic D-modules, arxiv/1311.2374
The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic D-modules and that of constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkinâ€™s work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.
Generalization to flat infinity-connections via the dg-nerve and iterated integrals is discussed in
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