$\hbar$-Riemann-Hilbert correspondence

TL;DR

This paper establishes a new correspondence linking $
$-differential equations with sheaf quantizations, bridging deformation and sheaf quantization of holomorphic cotangent bundles, and connecting to Fukaya categories.

Contribution

It formulates and proves a novel Riemann-Hilbert correspondence between $
$-differential equations and sheaf quantizations, integrating asymptotic analysis and geometric quantization concepts.

Findings

Established a correspondence between $
$-differential equations and sheaf quantizations.

Connected sheaf quantizations to Fukaya categories and Lagrangian intersection theory.

Utilized asymptotic/WKB analysis to underpin the constructions.

Abstract

We formulate and prove a Riemann-Hilbert correspondence between $\hbar$-differential equations and sheaf quantizations, which can be considered as a correspondence between two kinds of quantizations (deformation and sheaf quantization) of holomorphic cotangent bundles. The latter category is expected to be equivalent to a version of Fukaya category, which is a "quantization" of Lagrangian intersection theory. The ideas of the constructions are based on asymptotic/WKB analysis, which is related to geometric quantization.