Sheaf quantization from exact WKB analysis

TL;DR

This paper develops a sheaf-theoretic approach to quantize spectral curves using exact WKB analysis, linking sheaf quantizations with spectral data and Riemann-Hilbert correspondence in the context of Schrödinger equations.

Contribution

It introduces a method to sheaf-quantize spectral curves over the Novikov ring via exact WKB analysis, connecting sheaf theory with spectral and coordinate data.

Findings

Sheaf quantization of spectral curves is achieved under specific Stokes curve conditions.

The associated local system matches the Voros--Iwaki--Nakanishi coordinate for Schrödinger equations.

Sheaf quantizations potentially realize the $ abla$-enhanced Riemann--Hilbert correspondence.

Abstract

A sheaf quantization is a sheaf associated to a Lagrangian brane. By using the results of exact WKB analysis, we sheaf-quantize spectral curves over the Novikov ring under some assumptions on the behavior of Stokes curves. For Schr\"odinger equations, we prove that the local system associated to the sheaf quantization (microlocalization a.k.a. abelianization) over the spectral curve can be identified with the Voros--Iwaki--Nakanishi coordinate. We expect that these sheaf quantizations are the object-level realizations of the $\hbar$-enhanced Riemann--Hilbert correspondence.