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Let $E$ be an $n$-dimensional vector bundle on a manifold $M$ and $\nabla: \Gamma(E)\to \Omega^1(M,E)$ be a flat connection on $E$. Classical Riemann-Hilbert correspondence tells us that ker$\nabla$ is locally an $n$-dimensional vector space and it gives a local system on $M$.

Now if we drop the condition that $E$ is finite dimensional, do we still get an infinite dimensional local system in the same way?

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    $\begingroup$ It will depend on what you mean by a flat connection and by a vector bundle. Could you be more specific? $\endgroup$
    – Misha
    Commented Feb 9, 2017 at 0:14
  • $\begingroup$ For regular holomorphic $\mathcal D_X$-modules , there is a Riemann-Hilbert correspondence , for example if $X=\mathbb C^n$ , then $\mathcal D_X$ - is infinite dimensional vector bundle. See ams.org/mathscinet-getitem?mr=0579742 $\endgroup$
    – user21574
    Commented May 23, 2017 at 7:07

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