Fukaya category and Fourier transform

TL;DR

This paper develops a Fourier transform for families of real tori, linking symplectic geometry and complex geometry through a functor that relates Fukaya categories to holomorphic vector bundles.

Contribution

It introduces a new Fourier transform for symplectic torus families, connecting Fukaya categories with holomorphic vector bundles on dual families.

Findings

Constructed a Fourier transform for families of real tori.

Established a functor from symplectic categories to holomorphic vector bundles.

Connected Fukaya categories with complex geometric structures.

Abstract

We construct a version of Fourier transform for families of real tori. This transform defines a functor from certain category associated with a symplectic family of tori to the category of holomorphic vector bundles on the dual family (the dual family has a natural complex structure). In the 1-dimensional case the former category is closely related to the Fukaya category.