The bijectivity of mirror functors on 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.