Remarks on the homological mirror symmetry for tori

Abstract

Let us consider an $n$-dimensional complex torus $T^{2n}_{J=T}:=\mathbb{C}^n/2\pi(\mathbb{Z}^n \oplus T\mathbb{Z}^n)$. 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 $T^2_{J=T}$ is defined by using $-\frac{1}{T}$ or $T$. However, if we assume $n \geq 2$ and that $T$ is a singular matrix, we can not define a mirror partner of $T^{2n}_{J=T}$ 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.