Mirror symmetry and deformation quantization

TL;DR

This paper explores the relationship between the Fukaya category and the category of holonomic modules over quantized algebras on symplectic manifolds, proposing a conjectural equivalence after a specific transformation.

Contribution

It introduces a conjecture that these two categories are $A_{ olinebreak{}_{ ext{infty}}}$-equivalent after a certain integral transformation, linking mirror symmetry and deformation quantization.

Findings

Proposes a conjectural $A_{ ext{infty}}$-equivalence between categories

Connects Fukaya category with holonomic modules in quantized setting

Suggests a specific integral transformation as the bridge

Abstract

The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We conjecture that these categories become $A_{\infty}$-equivalent after a twist by a kind of integral transformation.