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 $X\to B$ and $\check{X}\to B$ 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 multisection of $\check{X}\to B$ 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.