# The bijectivity of mirror functors on tori

## TL;DR

This paper proves a bijection between certain objects in the symplectic and complex geometric categories of mirror pairs of tori, resolving ambiguities in the SYZ transform and establishing a functorial correspondence.

## Contribution

It demonstrates the bijectivity of mirror functors on tori by resolving transition function ambiguities in the SYZ transform.

## Key findings

- Established a bijection between isomorphism classes of objects in mirror categories.
- Resolved ambiguities in the transition functions of the SYZ transform.
- Confirmed the existence of a functor between symplectic and complex categories for tori.

## Abstract

By the SYZ construction, a mirror pair $(X,\check{X})$ of a complex torus $X$ and a mirror partner $\check{X}$ of the complex torus $X$ is described as the special Lagrangian torus fibrations $X \rightarrow B$ and $\check{X} \rightarrow B$ on the same base space $B$. Then, by the SYZ transform, we can construct a simple projectively flat bundle on $X$ from each affine Lagrangian multi section of $\check{X} \rightarrow B$ with a unitary local system along it. However, there are ambiguities of the choices of transition functions of it, and this causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this paper, we prove that there exists a bijection between the set of the isomorphism classes of their objects by solving this problem.