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Sheaf quantization from exact WKB analysis

Tatsuki Kuwagaki

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 Schrödinger 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.

Sheaf quantization from exact WKB analysis

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 Schrödinger 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 -enhanced Riemann--Hilbert correspondence.

Paper Structure

This paper contains 40 sections, 42 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Microlocal category
  3. Positively-microsupported sheaves
  4. Equivariant sheaves
  5. Operations for equivariant sheaves
  6. Positively microsupported equivariant sheaves
  7. Operations for positively microsupported equivariant sheaves
  8. Novikov ring action
  9. Non-conic microsupport
  10. Sheaf quantization
  11. Maslov covering and graded Lagrangian
  12. Relative Pin structure
  13. Microsheaves along Lagrangian
  14. Microlocalization
  15. Brane structure and microlocalization
  16. ...and 25 more sections

Key Result

Theorem \oldthetheorem

Let $C$ be a Riemann surface and $\mathcal{M}$ be a rank 2 flat meromorphic $\hbar$-connection with Assumption ourassumption. For a sufficiently small and generic ${\saturn} \in \mathbb{C}^{\times}_\hbar$, there exists a sheaf quantization $S_{\mathcal{M}}^{{\footnotesize{\saturn}}}$ of the spectral

Figures (3)

  • Figure 8.1: Stokes diagram
  • Figure 8.2: Edge cycle
  • Figure 9.1: An example of $D^\pm$

Theorems & Definitions (116)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem: Lemma \ref{['lem:Novcomputation']}
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 106 more