Remarks on the homological mirror symmetry for tori

Abstract

Let us consider an $n$-dimensional complex torus $T_{J=T}^{2n}:={\mathbb{C}}^n/2\pi ({\mathbb{Z}}^n\oplus T{\mathbb{Z}}^n)$. Here, $T$ is a complex matrix of order $n$ whose imaginary part is positive definite. In particular, when we consider the case $n=1$, the complexified symplectic form of a mirror partner of $T_{J=T}^2$ is defined by using $-\frac{1}{T}$ or $T$. However, if we assume $n\ge 2$ and that $T$ is a singular matrix, we can not define a mirror partner of $T_{J=T}^{2n}$ as a natural generalization of the case $n=1$ to the higher dimensional case. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.

30 pages