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.

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.

Equations

$$D^b(Coh(X))\cong Tr(Fuk(\check{X}))$$

$$d{z}^{t}Rd\bar{z}\cdot \mathrm{id},$$

$$\iota :\mathrm{Ob}(Fu{k}_{\mathrm{aff}}(\check{X}))\to \mathrm{Ob}(D{G}_X)$$

$${\iota }^{isom}:{\mathrm{Ob}}^{isom}(Fu{k}_{\mathrm{aff}}(\check{X}))\to {\mathrm{Ob}}^{isom}(D{G}_X),$$

$$F:\mathrm{Ob}(D{G}_X)\to \mathrm{Ob}(Fu{k}_{\mathrm{aff}}(\check{X}))$$

$$Tr(D{G}_X)\cong Tr(Fu{k}_{\mathrm{aff}}(\check{X}))$$

$${\mathcal{C}}_X\cong {\mathcal{C}}_{\check{X}}$$

$$T_{J=T}^{2n}:={\mathbb{C}}^n/2\pi ({\mathbb{Z}}^n\oplus T{\mathbb{Z}}^n).$$

$$\displaystyle O^{l_{1}\cdots l_{n}}_{m_{1}\cdots m_{n}}:=\biggl{\{}\left(\begin{array}[]{ccc}x\\ y\end{array}\right)\in T^{2n}_{J=T}\ |\ \frac{2}{3}\pi(l_{j}-1)-\varepsilon<x_{j}<\frac{2}{3}\pi l_{j}+\varepsilon,$$

$$\displaystyle\hskip 116.6563pt\frac{2}{3}\pi(m_{k}-1)-\varepsilon<y_{k}<\frac{2}{3}\pi m_{k}+\varepsilon,\ j,k=1,\cdots,n\biggr{\}}$$

$$x:=(x_1,\cdots ,x_n{)}^t,y:=(y_1,\cdots ,y_n{)}^t,$$

$$(x_1,\cdots ,x_n,y_1,\cdots ,y_n{)}^t\in {\mathbb{R}}^{2n}.$$

$$\check{x}:=(x^1,\cdots ,x^n{)}^t,\check{y}:=(y^1,\cdots ,y^n{)}^t.$$

$$\tilde{\omega }:=d{\check{x}}^t(-T^{-1}{)}^{t}d\check{y},$$

$$\tilde{\omega }=d{\check{x}}^t\mathrm{Re}(-T^{-1}{)}^{t}d\check{y}+\mathbf{i}d{\check{x}}^t\mathrm{Im}(-T^{-1}{)}^{t}d\check{y},$$

$$\omega :=\mathrm{Im}(-T^{-1}{)}^t,B:=\mathrm{Re}(-T^{-1}{)}^t.$$

$${\check{T}}_{J=T}^{2n}:=(T^{2n},\tilde{\omega }=d{\check{x}}^t(-T^{-1}{)}^{t}d\check{y})$$

$$E_{(r,A,r^{\prime },\mathcal{U},p,q)}\to T_{J=T}^{2n},$$

$$D{G}_{T_{J=T}^{2n}}$$

$$\mathcal{A}A\mathcal{B}=\left(\begin{array}[]{cccccc}\tilde{a_{1}}&&&&&\\ &\ddots&&&&\\ &&\tilde{a_{s}}&&&\\ &&&0&&\\ &&&&\ddots&\\ &&&&&0\end{array}\right),$$

$$\frac{\widetilde{a_i}}{r}=\frac{a_i^{\prime }}{r_i^{\prime }},gcd(r_i^{\prime },a_i^{\prime })=1,$$

$$r^{\prime }:=r_1^{\prime }\cdots r_s^{\prime }\in \mathbb{N}.$$

$${\psi }_{m_1\cdots m_n}^{l_1\cdots l_n}:O_{m_1\cdots m_n}^{l_1\cdots l_n}\to O_{m_1\cdots m_n}^{l_1\cdots l_n}\times {\mathbb{C}}^{r^{\prime }},l_j,m_k=1,2,3$$

$$O_{m_1\cdots m_n}^{(l_1=3)\cdots l_n}\cap O_{m_1\cdots m_n}^{(l_1=1)\cdots l_n},O_{m_1\cdots m_n}^{l_1(l_2=3)\cdots l_n}\cap O_{m_1\cdots m_n}^{l_1(l_2=1)\cdots l_n},\cdots ,O_{m_1\cdots m_n}^{l_1\cdots (l_n=3)}\cap O_{m_1\cdots m_n}^{l_1\cdots (l_n=1)},$$

$$O_{(m_1=3)\cdots m_n}^{l_1\cdots l_n}\cap O_{(m_1=1)\cdots m_n}^{l_1\cdots l_n},O_{m_1(m_2=3)\cdots m_n}^{l_1\cdots l_n}\cap O_{m_1(m_2=1)\cdots m_n}^{l_1\cdots l_n},\cdots ,$$

$$O_{m_1\cdots (m_n=3)}^{l_1\cdots l_n}\cap O_{m_1\cdots (m_n=1)}^{l_1\cdots l_n},$$

$${{\psi }_{m_1\cdots m_n}^{l_1\cdots (l_j=3)\cdots l_n}|}_{O_{m_1\cdots m_n}^{l_1\cdots (l_j=3)\cdots l_n}\cap O_{m_1\cdots m_n}^{l_1\cdots (l_j=1)\cdots l_n}}$$

$$=e^{\frac{\mathbf{i}}{r}{a}_{j}y}{V}_j{{\psi }_{m_1\cdots m_n}^{l_1\cdots (l_j=1)\cdots l_n}|}_{O_{m_1\cdots m_n}^{l_1\cdots (l_j=3)\cdots l_n}\cap O_{m_1\cdots m_n}^{l_1\cdots (l_j=1)\cdots l_n}},$$

$${{\psi }_{m_1\cdots (m_k=3)\cdots m_n}^{l_1\cdots l_n}|}_{O_{m_1\cdots (m_k=3)\cdots m_n}^{l_1\cdots l_n}\cap O_{m_1\cdots (m_k=1)\cdots m_n}^{l_1\cdots l_n}}$$

$$=U_k{{\psi }_{m_1\cdots (m_k=1)\cdots m_n}^{l_1\cdots l_n}|}_{O_{m_1\cdots (m_k=3)\cdots m_n}^{l_1\cdots l_n}\cap O_{m_1\cdots (m_k=1)\cdots m_n}^{l_1\cdots l_n}},$$

$${{\psi }_{m_1\cdots (m_k=3)\cdots m_n}^{l_1\cdots (l_j=3)\cdots l_n}|}_{O_{m_1\cdots (m_k=3)\cdots m_n}^{l_1\cdots (l_j=3)\cdots l_n}\cap O_{m_1\cdots (m_k=1)\cdots m_n}^{l_1\cdots (l_j=1)\cdots l_n}}$$

$$=U_k{{\psi }_{m_1\cdots (m_k=1)\cdots m_n}^{l_1\cdots (l_j=3)\cdots l_n}|}_{O_{m_1\cdots (m_k=3)\cdots m_n}^{l_1\cdots (l_j=3)\cdots l_n}\cap O_{m_1\cdots (m_k=1)\cdots m_n}^{l_1\cdots (l_j=1)\cdots l_n}}$$

$$\displaystyle=\Bigl{(}U_{k}\Bigr{)}\left(e^{\frac{\mathbf{i}}{r}a_{j}y}V_{j}\right)\left.\psi^{l_{1}\cdots(l_{j}=1)\cdots l_{n}}_{m_{1}\cdots(m_{k}=1)\cdots m_{n}}\right|_{O^{l_{1}\cdots(l_{j}=3)\cdots l_{n}}_{m_{1}\cdots(m_{k}=3)\cdots m_{n}}\cap O^{l_{1}\cdots(l_{j}=1)\cdots l_{n}}_{m_{1}\cdots(m_{k}=1)\cdots m_{n}}},$$

$$V_j{V}_k=V_k{V}_j,U_j{U}_k=U_k{U}_j,{\zeta }^{-a_{kj}}{U}_k{V}_j=V_j{U}_k,$$

Full text

1. 1 Introduction

2. 2 Preparations

3. 3 Complex geometry side

4. 3.1 The definition of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$

5. 3.2 The isomorphism classes of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$

6. 4 Symplectic geometry side

7. 4.1 The definition of $\mathscr{L}_{(r,A,p,q)}$

8. 4.2 The isomorphism classes of $\mathscr{L}_{(r,A,p,q)}$

9. 5 Main result

1 Introduction

Let $X$ be an $n$-dimensional complex torus, and we denote by $\check{X}$ a mirror partner of the complex torus $X$. For this mirror pair $(X,\check{X})$, the homological mirror symmetry conjecture [12], which is proposed by Kontsevich in 1994, states that there exists an equivalence

[TABLE]

as triangulated categories. Here, $D^{b}(Coh(X))$ is the bounded derived category of coherent sheaves on $X$, and $Tr(Fuk(\check{X}))$ is the derived category of the Fukaya category $Fuk(\check{X})$ on $\check{X}$ [4] obtained by the Bondal-Kapranov-Kontsevich construction [3], [12]. Historically this conjecture has been first studied when $(X,\check{X})$ is a pair of elliptic curves (see [19], [18], [1] etc.), and after that, when $(X,\check{X})$ is a pair of abelian varieties of higher dimension generalizing the case of elliptic curves [5] (see also [13]).

On the other hand, the SYZ construction [20] by Strominger, Yau, and Zaslow in 1996, proposes a way of constructing mirror pairs geometrically. By this construction, the mirror pair $(X,\check{X})$ is realized as the trivial special Lagrangian torus fibrations $\pi:X\rightarrow B$ and $\check{\pi}:\check{X}\rightarrow B$ on the same base space $B$ which is homeomorphic to an $n$-dimensional real torus. Here, for each point $b\in B$, the special Lagrangian torus fibers $\pi^{-1}(b)$ and $\check{\pi}^{-1}(b)$ are related by the T-duality. In particular, it is expected that the homological mirror symmetry on the mirror pair $(X,\check{X})$ is realized by the SYZ transform (an analogue of the Fourier-Mukai transform) along the special Lagrangian torus fibers of $\pi:X\rightarrow B$ and $\check{\pi}:\check{X}\rightarrow B$.

Concerning the above discussions, we explain the purpose of this paper. For a given mirror pair $(X,\check{X})$, we regard it as the trivial special Lagrangian torus fibrations $\pi:X\rightarrow B$ and $\check{\pi}:\check{X}\rightarrow B$ in the sense of the SYZ construction. First, in the symplectic geometry side, we consider the full subcategory $Fuk_{\mathrm{aff}}(\check{X})$ of the Fukaya category $Fuk(\check{X})$ consisting of affine Lagrangian multi sections of $\check{\pi}:\check{X}\rightarrow B$ with unitary local systems along them (in this paper, we sometimes call $Fuk_{\mathrm{aff}}(\check{X})$ simply the Fukaya category). Then, according to the discussions in [14] and [2], we can obtain a holomorphic vector bundle on $X$ which admits a constant curvature connection from each object of $Fuk_{\mathrm{aff}}(\check{X})$. This is called the SYZ transform. More precisely, the above constant curvature is expressed locally as

[TABLE]

where $z=(z_{1},\cdots,z_{n})^{t}$ is the local complex coordinates of $X$, and $R$ is a constant matrix of order $n$ (actually, $R$ is a Hermitian matrix of order $n$). On the other hand, for a holomorphic vector bundle on $X$ with the Hermitian connection, if its curvature form is expressed locally as the form (1), such a holomorphic vector bundle admits a projectively flat structure (for example, see [11]). Therefore, we see that each object of $Fuk_{\mathrm{aff}}(\check{X})$ is transformed to a projectively flat bundle on $X$, which in particular becomes simple. However, there are ambiguities of the choices of transition functions of it. In this paper, we consider a DG-category $DG_{X}$ consisting of such simple projectively flat bundles with any compatible transition functions. We expect that this $DG_{X}$ generates $D^{b}(Coh(X))$ though we do not discuss it in this paper. At least, it is known that it split-generates $D^{b}(Coh(X))$ when $X$ is an abelian variety (cf. [17], [1]). In this setting, when we fix a choice of transition functions of holomorphic vector bundles in $DG_{X}$, we obtain a map

[TABLE]

by the SYZ transform. Then, for example, it is also shown in [5, Proposition 13.26] that the map $\iota$ induces an injection

[TABLE]

where $\mathrm{Ob}^{isom}(DG_{X})$ and $\mathrm{Ob}^{isom}(Fuk_{\mathrm{aff}}(\check{X}))$ denote the set of the isomorphism classes of holomorphic vector bundles in $DG_{X}$ and the set of the isomorphism classes of objects of $Fuk_{\mathrm{aff}}(\check{X})$, respectively. In the present paper, we prove that the map $\iota^{isom}$ is actually a bijection by constructing a natural map

[TABLE]

whose direction is opposite to the direction of the map $\iota$.

Of course, though we can also regard this result as the first step to prove the homological mirror symmetry conjecture on $(X,\check{X})$, it is a stronger statement in the following sense. In general, for two $A_{\infty}$-categories $\mathscr{C}$ and $\mathscr{C}^{\prime}$, if there exists an $A_{\infty}$-equivalence $\mathscr{C}\cong\mathscr{C}^{\prime}$, then it is known that there exists an equivalence $Tr(\mathscr{C})\cong Tr(\mathscr{C}^{\prime})$ as triangulated categories. Hence, in order to prove the homological mirror symmetry conjecture

[TABLE]

on $(X,\check{X})$, it is enough to prove that there exists an $A_{\infty}$-equivalence

[TABLE]

for some full subcategories $\mathscr{C}_{X}\subset DG_{X}$, $\mathscr{C}_{\check{X}}\subset Fuk_{\mathrm{aff}}(\check{X})$ which generate the respective triangulated categories $Tr(\mathscr{C}_{X})\cong Tr(DG_{X})$, $Tr(\mathscr{C}_{\check{X}})\cong Tr(Fuk_{\mathrm{aff}}(\check{X}))$ (as such categories $\mathscr{C}_{X}$ and $\mathscr{C}_{\check{X}}$, we would like to take the smallest possible category on each side, see also [13]). Thus, from the viewpoint of the proof of the homological mirror symmetry conjecture on $(X,\check{X})$, if we can construct a bijection $\mathrm{Ob}^{isom}(\mathscr{C}_{X})\rightarrow\mathrm{Ob}^{isom}(\mathscr{C}_{\check{X}})$ and prove the existence of an $A_{\infty}$-equivalence $\mathscr{C}_{X}\cong\mathscr{C}_{\check{X}}$, then the map $\iota^{isom}$ itself need not be bijective. Actually, our original motivation of proving the bijectivity of the map $\iota^{isom}$ is that we are interested in the DG-category $DG_{X}$ itself and the corresponding geometric objects in the mirror dual side. In particular, in [9], exact triangles in $Tr(DG_{X})$ consisting of objects of $DG_{X}$ are studied by using the map $F$ (see also [7]). Although [8] is closely related to [9], we are going to revise [8] by using the main result in the present paper. Furthermore, the map $F$ is also employed in discussions in [10].

This paper is organized as follows. In section 2, we take a complex torus $X$, and explain the definition of a mirror partner $\check{X}$ of the complex torus $X$. In section 3, we define a class of a certain kind of simple projectively flat bundles on $X$, and construct the DG-category $DG_{X}$ consisting of those holomorphic vector bundles. We also study the isomorphism classes of them in section 3. In section 4, we consider the Fukaya category $Fuk_{\mathrm{aff}}(\check{X})$ consisting of affine Lagrangian multi sections of $\check{\pi}:\check{X}\rightarrow B$ with unitary local systems along them, and study the isomorphism classes of objects of $Fuk_{\mathrm{aff}}(\check{X})$. In section 5, we explicitly construct the bijection $\mathrm{Ob}^{isom}(DG_{X})\rightarrow\mathrm{Ob}^{isom}(Fuk_{\mathrm{aff}}(\check{X}))$. This result is given in Theorem 5.1.

2 Preparations

In this section, we define a complex torus $T^{2n}_{J=T}$ and its mirror partner $\check{T}^{2n}_{J=T}$.

First, we define an $n$-dimensional complex torus $T^{2n}_{J=T}$ as follows. Let $T$ be a complex matrix of order $n$ such that $\mathrm{Im}T$ is positive definite. We consider the lattice $2\pi(\mathbb{Z}^{n}\oplus T\mathbb{Z}^{n})$ in $\mathbb{C}^{n}$ and define

[TABLE]

Sometimes we regard the $n$-dimensional complex torus $T^{2n}_{J=T}$ as a $2n$-dimensional real torus $\mathbb{R}^{2n}/2\pi\mathbb{Z}^{2n}$. In this paper, we further assume that $T$ is a non-singular matrix. Actually, in our setting described bellow, it turns out that the mirror partner of the complex torus $T^{2n}_{J=T}$ does not exist if $\mathrm{det}T=0$. However, we can avoid this problem and discuss the homological mirror symmetry even if $\mathrm{det}T=0$ by modifying the definition of the mirror partner of the complex torus $T^{2n}_{J=T}$ and the class of holomorphic vector bundles which we treat. This case is discussed in [10]. Here, we fix an $\varepsilon>0$ small enough and let

[TABLE]

be subsets of $T^{2n}_{J=T}$, where $l_{j}$, $m_{k}=1,2,3$,

[TABLE]

and we identify $x_{i}\sim x_{i}+2\pi$, $y_{i}\sim y_{i}+2\pi$ for each $i=1,\cdots,n$. Sometimes we denote $O^{l_{1}\cdots(l_{j}=l)\cdots l_{n}}_{m_{1}\cdots(m_{k}=m)\cdots m_{n}}$ instead of $O^{l_{1}\cdots l_{n}}_{m_{1}\cdots m_{n}}$ in order to specify the values $l_{j}=l$, $m_{k}=m$. Then, $\{O^{l_{1}\cdots l_{n}}_{m_{1}\cdots m_{n}}\}_{l_{j},m_{k}=1,2,3}$ is an open cover of $T^{2n}_{J=T}$, and we define the local coordinates of $O^{l_{1}\cdots l_{n}}_{m_{1}\cdots m_{n}}$ by

[TABLE]

Furthermore, we locally express the complex coordinates $z:=(z_{1},\cdots,z_{n})^{t}$ of $T^{2n}_{J=T}$ by $z=x+Ty$.

Next, we define a mirror partner $\check{T}^{2n}_{J=T}$ of $T^{2n}_{J=T}$. From the viewpoint of the SYZ construction, we should describe $\check{T}^{2n}_{J=T}$ as the trivial special Lagrangian torus fibration. However, here, instead of constructing it, we consider the $2n$-dimensional standard real torus $T^{2n}=\mathbb{R}^{2n}/2\pi\mathbb{Z}^{2n}$ with a modified (non standard) symplectic form. For each point $(x^{1},\cdots,x^{n},y^{1},\cdots,y^{n})^{t}\in T^{2n}$, we identify $x^{i}\sim x^{i}+2\pi$, $y^{i}\sim y^{i}+2\pi$, where $i=1,\cdots,n$. We also denote by $(x^{1},\cdots,x^{n},y^{1},\cdots,y^{n})^{t}$ the local coordinates in the neighborhood of an arbitrary point $(x^{1},\cdots,x^{n},y^{1},\cdots,y^{n})^{t}\in T^{2n}$. Furthermore, we use the same notation $(x^{1},\cdots,x^{n},y^{1},\cdots,y^{n})^{t}$ when we denote the coordinates of the covering space $\mathbb{R}^{2n}$ of $T^{2n}$. For simplicity, we set

[TABLE]

We define a complexified symplectic form $\tilde{\omega}$ on $T^{2n}$ by

[TABLE]

where $d\check{x}:=(dx^{1},\cdots,dx^{n})^{t}$ and $d\check{y}:=(dy^{1},\cdots,dy^{n})^{t}$. We decompose $\tilde{\omega}$ into

[TABLE]

and define

[TABLE]

Here, $\mathbf{i}=\sqrt{-1}$. Sometimes we identify the matrices $\omega$ and $B$ with the 2-forms $d\check{x}^{t}\omega d\check{y}$ and $d\check{x}^{t}Bd\check{y}$, respectively. Then, $\omega$ gives a symplectic form on $T^{2n}$. The closed 2-form $B$ is often called the B-field. This complexified symplectic torus $(T^{2n},\tilde{\omega}=d\check{x}^{t}(-T^{-1})^{t}d\check{y})$ is a mirror partner of the complex torus $T^{2n}_{J=T}$. Hereafter, we denote

[TABLE]

for simplicity.

3 Complex geometry side

The purpose of this section is to explain the complex geometry side in the homological mirror symmetry setting on $(T^{2n}_{J=T},\check{T}^{2n}_{J=T})$. In subsection 3.1, we define a class of holomorphic vector bundles

[TABLE]

and construct a DG-category

[TABLE]

consisting of these holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$. In particular, we first construct $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ as a complex vector bundle, and then discuss when it becomes a holomorphic vector bundle in Proposition 3.1. In subsection 3.2, we study the isomorphism classes of holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ by using the classification result of simple projectively flat bundles on complex tori by Matsushima [15] and Mukai [16].

3.1 The definition of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$

We assume $r\in\mathbb{N}$, $A=(a_{ij})\in M(n;\mathbb{Z})$, and $p$, $q\in\mathbb{R}^{n}$. First, we define $r^{\prime}\in\mathbb{N}$ by using a given pair $(r,A)\in\mathbb{N}\times M(n;\mathbb{Z})$ as follows. By the theory of elementary divisors, there exist two matrices $\mathcal{A}$, $\mathcal{B}\in GL(n;\mathbb{Z})$ such that

[TABLE]

where $\tilde{a_{i}}\in\mathbb{N}$ ($i=1,\cdots,s,1\leq s\leq n$) and $\tilde{a_{i}}|\tilde{a_{i+1}}$ ($i=1,\cdots,s-1$). Then, we define $r_{i}^{\prime}\in\mathbb{N}$ and $a_{i}^{\prime}\in\mathbb{Z}$ ($i=1,\cdots,s$) by

[TABLE]

where $gcd(m,n)>0$ denotes the greatest common divisor of $m$, $n\in\mathbb{Z}$. By using these, we set

[TABLE]

This $r^{\prime}\in\mathbb{N}$ is uniquely defined by a given pair $(r,A)\in\mathbb{N}\times M(n;\mathbb{Z})$, and it is actually the rank of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$. Now, we define the transition functions of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ as follows (although the following notations are complicated, roughly speaking, the transition functions of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ in the cases of $x_{j}\mapsto x_{j}+2\pi$, $y_{k}\mapsto y_{k}+2\pi$ are given by $e^{\frac{\mathbf{i}}{r}a_{j}y}V_{j}$, $U_{k}$, respectively, where $j$, $k=1,\cdots,n$). Let

[TABLE]

be a smooth section of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}|_{O^{l_{1}\cdots l_{n}}_{m_{1}\cdots m_{n}}}$. The transition functions of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ are non-trivial on

[TABLE]

and otherwise are trivial. We define the transition function on $O^{l_{1}\cdots(l_{j}=3)\cdots l_{n}}_{m_{1}\cdots m_{n}}\cap O^{l_{1}\cdots(l_{j}=1)\cdots l_{n}}_{m_{1}\cdots m_{n}}$ by

[TABLE]

where $a_{j}:=(a_{1j},\cdots,a_{nj})\in\mathbb{Z}^{n}$ and $V_{j}\in U(r^{\prime})$. Similarly, we define the transition function on $O^{l_{1}\cdots l_{n}}_{m_{1}\cdots(m_{k}=3)\cdots m_{n}}\cap O^{l_{1}\cdots l_{n}}_{m_{1}\cdots(m_{k}=1)\cdots m_{n}}$ by

[TABLE]

where $U_{k}\in U(r^{\prime})$. In the definitions of these transition functions, actually, we only treat $V_{j}$, $U_{k}\in U(r^{\prime})$ which satisfy the cocycle condition, so we explain the cocycle condition below. When we define

[TABLE]

the cocycle condition is expressed as

[TABLE]

where $\zeta$ is the $r$-th root of 1 and $j,k=1,\cdots,n$. We define a set $\mathcal{U}$ of unitary matrices by

[TABLE]

Of course, how to define the set $\mathcal{U}$ relates closely to (in)decomposability of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$. Here, we only treat the set $\mathcal{U}$ such that $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ is simple. Actually, we can take such a set $\mathcal{U}\not=\emptyset$ for any $(r,A,r^{\prime})\in\mathbb{N}\times M(n;\mathbb{Z})\times\mathbb{N}$, and this fact is discussed in Proposition 3.2. Furthermore, we define a connection $\nabla_{(r,A,r^{\prime},\mathcal{U},p,q)}$ on $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ locally as

[TABLE]

where $dy:=(dy_{1},\cdots,dy_{n})^{t}$ and $d$ denotes the exterior derivative. In fact, $\nabla_{(r,A,r^{\prime},\mathcal{U},p,q)}$ is compatible with the transition functions and so defines a global connection. Then, its curvature form $\Omega_{(r,A,r^{\prime},\mathcal{U},p,q)}$ is expressed locally as

[TABLE]

where $dx:=(dx_{1},\cdots,dx_{n})^{t}$. In particular, this local expression (4) implies that holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ are simple projectively flat bundles (for example, the definition of projectively flat bundles is written in [11]). Moreover, the interpretation for these simple projectively flat bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ by using factors of automorphy is given in section 3 of [8]. Now, we consider the condition such that $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ is holomorphic. We see that the following proposition holds.

Proposition 3.1**.**

For a given quadruple $(r,A,p,q)\in\mathbb{N}\times M(n;\mathbb{Z})\times\mathbb{R}^{n}\times\mathbb{R}^{n}$, the complex vector bundle $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\rightarrow T^{2n}_{J=T}$ is holomorphic if and only if $AT=(AT)^{t}$ holds.

Proof.

A complex vector bundle is holomorphic if and only if the (0,2)-part of its curvature form vanishes, so we calculate the (0,2)-part of $\Omega_{(r,A,\mu,\mathcal{U})}$. It turns out to be

[TABLE]

where $d\bar{z}:=(d\bar{z}_{1},\cdots,d\bar{z}_{n})^{t}$. Thus, $\Omega^{(0,2)}_{(r,A,r^{\prime},\mathcal{U},p,q)}=0$ is equivalent to that $\{T(T-\bar{T})^{-1}\}^{t}A^{t}(T-\bar{T})^{-1}$ is a symmetric matrix, i.e., $AT=(AT)^{t}$. ∎

Concerning the above discussions, here, we mention the simplicity of holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$. The following proposition holds.

Proposition 3.2**.**

For each quadruple $(r,A,p,q)\in\mathbb{N}\times M(n;\mathbb{Z})\times\mathbb{R}^{n}\times\mathbb{R}^{n}$, we can take a set $\mathcal{U}\not=\emptyset$ such that $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ is simple.

Proof.

For a given pair $(r,A)\in\mathbb{N}\times M(n;\mathbb{Z})$, we can take two matrices $\mathcal{A}$, $\mathcal{B}\in GL(n;\mathbb{Z})$ which satisfy the relation (2). Then, note that $r^{\prime}:=r_{1}^{\prime}\cdots r_{s}^{\prime}\in\mathbb{N}$ is uniquely defined in the sense of the relation (3). We fix such matrices $\mathcal{A}$, $\mathcal{B}\in GL(n;\mathbb{Z})$, and set

[TABLE]

By using this $T^{\prime}$, we can consider the complex torus $T^{2n}_{J=T^{\prime}}=\mathbb{C}^{n}/2\pi(\mathbb{Z}^{n}\oplus T^{\prime}\mathbb{Z}^{n})$, and we locally express the complex coordinates $Z:=(Z_{1},\cdots,Z_{n})^{t}$ of $T^{2n}_{J=T^{\prime}}$ by $Z=X+T^{\prime}Y$, where $X:=(X_{1},\cdots,X_{n})^{t}$, $Y:=(Y_{1},\cdots,Y_{n})^{t}$. In particular, the complex torus $T^{2n}_{J=T^{\prime}}$ is biholomorphic to the complex torus $T^{2n}_{J=T}$, and the biholomorphic map

[TABLE]

is actually given by

[TABLE]

Furthermore, when we regard the complex manifolds $T^{2n}_{J=T}$ and $T^{2n}_{J=T^{\prime}}$ as the real differentiable manifolds $\mathbb{R}^{2n}/2\pi\mathbb{Z}^{2n}$, the biholomorphic map $\varphi$ is regarded as the diffeomorphism

[TABLE]

Now, we define a set $\mathcal{U}^{\prime}\not=\emptyset$ as follows. First, we set

[TABLE]

where $\zeta_{i}:=e^{\frac{2\pi\mathbf{i}}{r_{i}^{\prime}}}$, $i=1,\cdots,s$. By using these matrices $\tilde{V}_{i}$, $\tilde{U}_{i}$, we define

[TABLE]

and set

[TABLE]

Then, we can construct the holomorphic vector bundle

[TABLE]

and in particular, $V_{j}^{\prime}$ and $U_{k}^{\prime}$ are used in the definition of the transition functions of $E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$ in the $X_{j}$ and $Y_{k}$ directions, respectively ($j$, $k=1,\cdots,n$). For this holomorphic vector bundle $E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$, we can consider the pullback bundle $\varphi^{*}E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$ by the biholomorphic map $\varphi$, and we can regard the pullback bundle $\varphi^{*}E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$ as the holomorphic vector bundle

[TABLE]

by using a suitable set $\mathcal{U}$ which is defined by employing the data $(\mathcal{U}^{\prime},\mathcal{A},\mathcal{B})$. In particular, since we can check $\mathcal{U}\not=\emptyset$ easily, in order to prove the statement of this proposition, it is enough to prove that $E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$ is simple.

Let

[TABLE]

be an element in $\mathrm{End}(E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)})$. Since the rank of $E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$ is $r^{\prime}=r_{1}^{\prime}\cdots r_{s}^{\prime}$, we can treat $\varphi^{r_{1}^{\prime}\cdots r_{s}^{\prime}}$ as a matrix of order $r^{\prime}$. Then, we can divide $\varphi^{r_{1}^{\prime}\cdots r_{s}^{\prime}}$ as follows.

[TABLE]

Here, each $\varphi_{k_{1}l_{1}}^{r_{2}^{\prime}\cdots r_{s}^{\prime}}$ is a matrix of order $r_{2}^{\prime}\cdots r_{s}^{\prime}$. Similarly, we can divide each $\varphi_{k_{1}l_{1}}^{r_{2}^{\prime}\cdots r_{s}^{\prime}}$ as follows.

[TABLE]

Here, each $\left(\varphi_{k_{1}l_{1}}^{r_{2}^{\prime}\cdots r_{s}^{\prime}}\right)_{k_{2}l_{2}}^{r_{3}^{\prime}\cdots r_{s}^{\prime}}$ is matrix of order $r_{3}^{\prime}\cdots r_{s}^{\prime}$. By repeating the above steps, as a result, we can express $\varphi^{r_{1}^{\prime}\cdots r_{s}^{\prime}}$ as

[TABLE]

where $1\leq k_{i},l_{i}\leq r_{i}^{\prime}$ ($i=1,\cdots,s$). Hereafter, we consider the local expression of each component

[TABLE]

of $\varphi^{r_{1}^{\prime}\cdots r_{s}^{\prime}}$. First, for each $k=1,\cdots,n$, by considering the transition functions of $E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$ in the $Y_{k}$ direction, we see that the morphism $\varphi^{r_{1}^{\prime}\cdots r_{s}^{\prime}}$ must satisfy

[TABLE]

Furthermore, the morphism $\varphi^{r_{1}^{\prime}\cdots r_{s}^{\prime}}$ need to satisfy not only the relation (6) but also the Cauchy-Riemann equation

[TABLE]

Therefore, by the relations (6) and (7), we can give a local expression of the component (5) of $\varphi^{r_{1}^{\prime}\cdots,r_{s}^{\prime}}$ as follows.

[TABLE]

Here,

[TABLE]

is an arbitrary constant. Finally, for each $j=1,\cdots,n$, we consider the conditions on the transition functions of $E_{(r,\mathcal{A}A\mathcal{B},r^{\prime},\mathcal{U}^{\prime},\mathcal{A}p,\mathcal{B}^{t}q)}$ in the $X_{j}$ direction. By a direct calculation, if $j=1,\cdots,s$, we obtain

[TABLE]

and if $j=s+1,\cdots,n$, we obtain

[TABLE]

In particular, the relation (8) implies

[TABLE]

and by using the relations (9) and (10), we have the condition

[TABLE]

Now, since $\mathrm{Im}T^{\prime-1}$ is positive definite, the condition (11) turns out to be

[TABLE]

Here, recall the relation (8). By the relation (8), it is enough to consider the condition (12) in the case $k_{1}=\cdots=k_{s}=1$. We focus on the first component in the condition (12), i.e.,

[TABLE]

In the case $l_{1}=1$, the condition (13) turns out to be $I_{11}^{1}=0$, so by the relation (8), we see

[TABLE]

We consider the cases $l_{1}=2,\cdots,r_{1}^{\prime}$. Note that $gcd(r_{1}^{\prime},a_{1}^{\prime})=1$ holds by the assumption. Therefore, we have

[TABLE]

by the condition (13). However, this fact contradicts the assumption $l_{1}=2,\cdots,r_{1}^{\prime}$. Thus, for each $l_{1}=2,\cdots,r_{1}^{\prime}$, we obtain

[TABLE]

and by using the relation (8) again, we also obtain

[TABLE]

Similarly, by focusing on the second component in the condition (12), we see

[TABLE]

and the other components $\left(\varphi_{11}^{r_{2}^{\prime}\cdots r_{s}^{\prime}}\right)_{k_{2}l_{2}}^{r_{3}^{\prime}\cdots r_{s}^{\prime}}$ of the matrix $\varphi_{11}^{r_{2}^{\prime}\cdots r_{s}^{\prime}}$ vanish. By repeating the above discussions, as a result, we have

[TABLE]

where

[TABLE]

Thus, we can conclude

[TABLE]

∎

We define a DG-category

[TABLE]

consisting of holomorphic vector bundles $(E_{(r,A,r^{\prime},\mathcal{U},p,q)},\nabla_{(r,A,r^{\prime},\mathcal{U},p,q)})$. This definition is an extension of the case of a pair $(T^{2}_{J=T},\check{T}^{2}_{J=T})$ of elliptic curves in [7] (see section 3) to the higher dimensional case, and it is also written in section 2 of [8]. The objects of $DG_{T^{2n}_{J=T}}$ are holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ with $U(r^{\prime})$-connections $\nabla_{(r,A,r^{\prime},\mathcal{U},p,q)}$. Of course, we assume $AT=(AT)^{t}$. Sometimes we simply denote $(E_{(r,A,\mu,\mathcal{U})},\nabla_{(r,A,\mu,\mathcal{U})})$ by $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$. For any two objects

[TABLE]

the space of morphisms is defined by

[TABLE]

where $\Omega^{0,*}(T^{2n}_{J=T})$ is the space of anti-holomorphic differential forms, and

[TABLE]

is the space of homomorphisms from $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ to $E_{(s,B,s^{\prime},\mathcal{V},u,v)}$. The space of morphisms $\mathrm{Hom}_{DG_{T^{2n}_{J=T}}}(E_{(r,A,r^{\prime},\mathcal{U},p,q)},E_{(s,B,s^{\prime},\mathcal{V},u,v)})$ is a $\mathbb{Z}$-graded vector space, where the grading is defined as the degree of the anti-holomorphic differential forms. In particular, the degree $r$ part is denoted

[TABLE]

We decompose $\nabla_{(r,A,r^{\prime},\mathcal{U},p,q)}$ into its holomorphic part and anti-holomorphic part $\nabla_{(r,A,r^{\prime},\mathcal{U},p,q)}=\nabla^{(1,0)}_{(r,A,r^{\prime},\mathcal{U},p,q)}+\nabla^{(0,1)}_{(r,A,r^{\prime},\mathcal{U},p,q)}$, and define a linear map

[TABLE]

by

[TABLE]

We can check that this linear map is a differential. Furthermore, the product structure is defined by the composition of homomorphisms of vector bundles together with the wedge product for the anti-holomorphic differential forms. Then, these differential and product structure satisfy the Leibniz rule. Thus, $DG_{T^{2n}_{J=T}}$ forms a DG-category.

3.2 The isomorphism classes of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$

In this subsection, we fix $r\in\mathbb{N}$, $A\in M(n;\mathbb{Z})$, and consider the condition such that $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\cong E_{(r,A,r^{\prime},\mathcal{U}^{\prime},p^{\prime},q^{\prime})}$ holds. Here,

[TABLE]

and

[TABLE]

Furthermore, for each $j=1,\cdots,n$, we define $\xi_{j}$, $\theta_{j}$, $\xi^{\prime}_{j}$, $\theta^{\prime}_{j}\in\mathbb{R}$ by

[TABLE]

and set

[TABLE]

Now, in order to consider the condition such that $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\cong E_{(r,A,r^{\prime},\mathcal{U}^{\prime},p^{\prime},q^{\prime})}$ holds, we recall the following classification result of simple projectively flat bundles on complex tori by Matsushima and Mukai (see [15, Theorem 6.1], [16, Proposition 6.17 (1)], and note that the notion of semi-homogeneous vector bundles in [16] is equivalent to the notion of projectively flat bundles).

Theorem 3.3** (Matsushima [15], Mukai [16]).**

For two simple projectively flat bundles $E$, $E^{\prime}$ over a complex torus $\mathbb{C}^{n}/\Gamma$ $(\Gamma\subset\mathbb{C}^{n}$ is a lattice$)$ which satisfy $(\mathrm{rank}E,c_{1}(E))=(\mathrm{rank}E^{\prime},c_{1}(E^{\prime}))$, there exists a line bundle $L\in\mathrm{Pic}^{0}(\mathbb{C}^{n}/\Gamma)$ such that

[TABLE]

By using Theorem 3.3, we obtain the following theorem.

Theorem 3.4**.**

Two holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$, $E_{(r,A,r^{\prime},\mathcal{U}^{\prime},p^{\prime},q^{\prime})}$ are isomorphic to each other,

[TABLE]

if and only if

[TABLE]

holds.

Proof.

In order to prove the statement of this theorem, we again take the biholomorphic map

[TABLE]

in the proof of Proposition 3.2. Of course, we can also regard this biholomorphic map $\varphi$ as the diffeomorphism $\mathbb{R}^{2n}/2\pi\mathbb{Z}^{2n}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{R}^{2n}/2\pi\mathbb{Z}^{2n}$ which is expressed locally as

[TABLE]

Then, by the biholomorphicity of the map $\varphi$,

[TABLE]

holds if and only if

[TABLE]

holds, so we consider the condition such that $(\varphi^{-1})^{*}E_{(r,A,r^{\prime},\mathcal{U},p,q)}\cong(\varphi^{-1})^{*}E_{(r,A,r^{\prime},\mathcal{U}^{\prime},p^{\prime},q^{\prime})}$ holds. Now, by using the suitable sets $\tilde{\mathcal{U}}$ and $\tilde{\mathcal{U}}^{\prime}$ (the definitions of $\tilde{\mathcal{U}}$ and $\tilde{\mathcal{U}}^{\prime}$ depend on the data $(\mathcal{U},\mathcal{A},\mathcal{B})$ and $(\mathcal{U}^{\prime},\mathcal{A},\mathcal{B})$, respectively), we can regard $(\varphi^{-1})^{*}E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ and $(\varphi^{-1})^{*}E_{(r,A,r^{\prime},\mathcal{U}^{\prime},p^{\prime},q^{\prime})}$ as $E_{(r,\tilde{A},r^{\prime},\tilde{\mathcal{U}},\tilde{p},\tilde{q})}$ and $E_{(r,\tilde{A},r^{\prime},\tilde{\mathcal{U}}^{\prime},\tilde{p}^{\prime},\tilde{q}^{\prime})}$, respectively, where

[TABLE]

Then, constant vectors $\xi$, $\theta$, $\xi^{\prime}$, $\theta^{\prime}\in\mathbb{R}^{n}$ are also transformed to constant vectors

[TABLE]

respectively. For these holomorphic vector bundles $E_{(r,\tilde{A},r^{\prime},\tilde{\mathcal{U}},\tilde{p},\tilde{q})}$ and $E_{(r,\tilde{A},r^{\prime},\tilde{\mathcal{U}}^{\prime},\tilde{p}^{\prime},\tilde{q}^{\prime})}$, by Theorem 3.3, we see that there exists a holomorphic line bundle $E_{(1,O,1,\mathcal{V},u,v)}$ such that

[TABLE]

Here,

[TABLE]

and for simplicity, we set

[TABLE]

In particular, for each $j=1,\cdots,n$, we assume that $e^{\mathbf{i}\tau_{j}}$ and $e^{\mathbf{i}\sigma_{j}}$ are the transition functions of $E_{(1,O,1,\mathcal{V},u,v)}$ in the $x_{j}$ direction and the transition function of $E_{(1,O,1,\mathcal{V},u,v)}$ in the $y_{j}$ direction, respectively. Therefore, since

[TABLE]

holds if and only if

[TABLE]

holds, our first goal is to find the relation on the parameters $u$, $v$, $\tau$, $\sigma\in\mathbb{R}^{n}$ such that the relation (16) holds. By using Theorem 3.3 again, we see that there exists a holomorphic line bundle $E_{(1,O,1,\mathcal{V}^{\prime},u^{\prime},v^{\prime})}$ such that

[TABLE]

where

[TABLE]

Here, note that the definitions of $V_{1}^{\prime},\cdots,V_{n}^{\prime},U_{1}^{\prime},\cdots,U_{n}^{\prime}\in U(r^{\prime})$ are given in the proof of Proposition 3.2. Thus, since we can rewrite the relation (16) to

[TABLE]

as a result, it is enough to consider the condition such that

[TABLE]

holds. By a direct calculation, we can actually check that

[TABLE]

holds if and only if

[TABLE]

holds, and in particular, we can regard the relation (17) as

[TABLE]

Hence, we see that the relation (16) holds if and only if the relation (30) holds. Here, note that there exists an isomorphism

[TABLE]

with

[TABLE]

Now, we consider the condition such that the relation (15) holds. One necessary condition for the relation (15) is that

[TABLE]

holds, and by a direct calculation, we can rewrite the relation (31) to the following :

[TABLE]

Therefore, since the definition of $E_{(1,O,1,\mathcal{V},u,v)}$ is given by

[TABLE]

and the relations (16) and (30) are equivalent, by considering the relation (32), we can give the condition such that the relation (15) holds as follows.

[TABLE]

Thus, by considering the pullback bundles $\varphi^{*}E_{(r,\tilde{A},r^{\prime},\tilde{\mathcal{U}},\tilde{p},\tilde{q})}$ and $\varphi^{*}E_{(r,\tilde{A},r^{\prime},\tilde{\mathcal{U}}^{\prime},\tilde{p}^{\prime},\tilde{q}^{\prime})}$, we can conclude that

[TABLE]

holds if and only if

[TABLE]

holds. ∎

Remark 3.5**.**

When we work over a pair $(T_{J=T}^{2},\check{T}_{J=T}^{2})$ of elliptic curves, Theorem 3.4 implies that there exists a one-to-one correspondence between the set of the isomorphism classes of holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ and the set of the isomorphism classes of holomorphic line bundles $\mathrm{det}E_{(r,A,r^{\prime},\mathcal{U},p,q)}$.

4 Symplectic geometry side

In this section, we define the objects of the Fukaya category corresponding to holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\rightarrow T^{2n}_{J=T}$, and study the isomorphism classes of them. The discussions in this section are based on the SYZ construction (SYZ transform) [20] (see also [14], [2]).

4.1 The definition of $\mathscr{L}_{(r,A,p,q)}$

In this subsection, we define a class of pairs of affine Lagrangian submanifolds

[TABLE]

in $\check{T}^{2n}_{J=T}$ and unitary local systems

[TABLE]

First, we recall the definition of objects of the Fukaya categories following [5, Definition 1.1]. Let $(M,\Omega)$ be a symplectic manifold $(M,\omega)$ together with a closed 2-form $B$ on $M$. Here, we put $\Omega=\omega+\sqrt{-1}B$ (note $-B+\sqrt{-1}\omega$ is used in many of the literatures). Then, we consider pairs $(L,\mathcal{L})$ with the following properties :

[TABLE]

In this context, $F_{\nabla^{\mathcal{L}}}$ denotes the curvature form of the connection $\nabla^{\mathcal{L}}$. We define objects of the Fukaya category on $(M,\Omega)$ by pairs $(L,\mathcal{L})$ which satisfy the properties (33), (34).

Let us consider the following $n$-dimensional submanifold $\tilde{L}_{(r,A,p)}$ in $\mathbb{R}^{2n}$ :

[TABLE]

We see that this $n$-dimensional submanifold $\tilde{L}_{(r,A,p)}$ satisfies the property (33), namely, $\tilde{L}_{(r,A,p)}$ becomes a Lagrangian submanifold in $\mathbb{R}^{2n}$ if and only if $\omega A=(\omega A)^{t}$ holds. Then, for the covering map $\pi:\mathbb{R}^{2n}\rightarrow\check{T}^{2n}_{J=T}$,

[TABLE]

defines a Lagrangian submanifold in $\check{T}^{2n}_{J=T}$. On the other hand, we can also regard the complexified symplectic torus $\check{T}^{2n}_{J=T}$ as the trivial special Lagrangian torus fibration $\check{\pi}:\check{T}^{2n}_{J=T}\rightarrow\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$, where $\check{x}$ is the local coordinates of the base space $\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$ and $\check{y}$ is the local coordinates of the fiber of $\check{\pi}:\check{T}^{2n}_{J=T}\rightarrow\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$. Then, we can interpret each affine Lagrangian submanifold $L_{(r,A,p)}$ in $\check{T}^{2n}_{J=T}$ as the affine Lagrangian multi section

[TABLE]

of $\check{\pi}:\check{T}^{2n}_{J=T}\rightarrow\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$.

Remark 4.1**.**

As explained in section 3, while $r^{\prime}:=r_{1}^{\prime}\cdots r_{s}^{\prime}\in\mathbb{N}$ is the rank of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\rightarrow T^{2n}_{J=T}$ $($see the relations $(\ref{matAB})$ and $(\ref{r'}))$, in the symplectic geometry side, this $r^{\prime}\in\mathbb{N}$ is interpreted as follows. For the affine Lagrangian submanifold $L_{(r,A,p)}$ in $\check{T}^{2n}_{J=T}$ which is defined by a given data $(r,A,p)\in\mathbb{N}\times M(n;\mathbb{Z})\times\mathbb{R}^{n}$, we regard it as the affine Lagrangian multi section $s(\check{x})=\frac{1}{r}A\check{x}+\frac{1}{r}p$ of $\check{\pi}:\check{T}^{2n}_{J=T}\rightarrow\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$. Then, for each point $\check{x}\in\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$, we see

[TABLE]

and this indicates that $s(\check{x})$ consists of $r^{\prime}$ points. Thus, we can regard $r^{\prime}\in\mathbb{N}$ as the multiplicity of $s(\check{x})=\frac{1}{r}A\check{x}+\frac{1}{r}p$.

We then consider the trivial complex line bundle

[TABLE]

with the flat connection

[TABLE]

where $q\in\mathbb{R}^{n}$ is the unitary holonomy of $\mathcal{L}_{(r,A,p,q)}$ along $L_{(r,A,p)}\approx T^{n}$. We discuss the property (34) for this pair $(L_{(r,A,p)},\mathcal{L}_{(r,A,p,q)})$ :

[TABLE]

Here, $\Omega_{\mathcal{L}_{(r,A,p,q)}}$ is the curvature form of the flat connection $\nabla_{\mathcal{L}_{(r,A,p,q)}}$, i.e., $\Omega_{\mathcal{L}_{(r,A,p,q)}}=0$. Hence, we see

[TABLE]

so one has $BA=(BA)^{t}$. Note that $\omega A=(\omega A)^{t}$ and $BA=(BA)^{t}$ hold if and only if $AT=(AT)^{t}$ holds. Hereafter, for simplicity, we set

[TABLE]

By summarizing the above discussions, we obtain the following proposition. In particular, the condition $AT=(AT)^{t}$ in the following proposition is also the condition such that a complex vector bundle $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\rightarrow T^{2n}_{J=T}$ becomes a holomorphic vector bundle (see Proposition 3.1).

Proposition 4.2**.**

For a given quadruple $(r,A,p,q)\in\mathbb{N}\times M(n;\mathbb{Z})\times\mathbb{R}^{n}\times\mathbb{R}^{n}$, $\mathscr{L}_{(r,A,p,q)}$ gives an object of the Fukaya category on $\check{T}^{2n}_{J=T}$ if and only if $AT=(AT)^{t}$ holds.

Definition 4.3**.**

We denote the full subcategory of the Fukaya category on $\check{T}^{2n}_{J=T}$ consisting of objects $\mathscr{L}_{(r,A,p,q)}$ which satisfy the condition $AT=(AT)^{t}$ by $Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T})$.

4.2 The isomorphism classes of $\mathscr{L}_{(r,A,p,q)}$

The discussions in this subsection correspond to the discussions in subsection 3.2, so throughout this subsection, we fix $r\in\mathbb{N}$, $A\in M(n;\mathbb{Z})$, and consider the condition such that $\mathscr{L}_{(r,A,p,q)}\cong\mathscr{L}_{(r,A,p^{\prime},q^{\prime})}$ holds, where

[TABLE]

We explain the definition of the equivalence of objects of the Fukaya categories (cf. [5, Definition 1.4]). Let us consider a pair $(M,\Omega)$ consisting of an even dimensional differentiable manifold $M$ and a complexified symplectic form $\Omega$. We take two objects $\mathscr{L}:=(L,\mathcal{L})$, $\mathscr{L}^{\prime}:=(L^{\prime},\mathcal{L}^{\prime})$ of the Fukaya category $Fuk(M,\Omega)$, where $L$, $L^{\prime}$ are Lagrangian submanifolds in $(M,\Omega)$, and $\mathcal{L}\rightarrow L$, $\mathcal{L}^{\prime}\rightarrow L^{\prime}$ are unitary local systems. Then, if there exists a symplectic automorphism $\Phi:(M,\Omega)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(M,\Omega)$ such that

[TABLE]

we say that $\mathscr{L}$ is isomorphic to $\mathscr{L}^{\prime}$, and write $\mathscr{L}\cong\mathscr{L}^{\prime}$.

We consider the isomorphism classes of $\mathscr{L}_{(r,A,p,q)}$, namely, we consider the condition such that $\mathscr{L}_{(r,A,p,q)}\cong\mathscr{L}_{(r,A,p^{\prime},q^{\prime})}$ holds as an analogue of Theorem 3.4. Actually, the following theorem holds.

Theorem 4.4**.**

Two objects $\mathscr{L}_{(r,A,p,q)}$, $\mathscr{L}_{(r,A,p^{\prime},q^{\prime})}$ are isomorphic to each other,

[TABLE]

if and only if

[TABLE]

hold.

Proof.

First, in order to prove the statement of this theorem, we prepare some notations. Since we considered the complex torus $T^{2n}_{J=T^{\prime}}=\mathbb{C}^{n}/2\pi(\mathbb{Z}^{n}\oplus\mathcal{B}^{-1}T\mathcal{A}^{t}\mathbb{Z}^{n})$ which is biholomorphic to the complex torus $T^{2n}_{J=T}$ in Theorem 3.4 (and Proposition 3.2), we take a mirror partner of the complex torus $T^{2n}_{J=T^{\prime}}$. Here, we consider the complexified symplectic torus $\check{T}_{J=T^{\prime}}^{2n}:=T_{\tilde{\omega}=(-T^{\prime-1})^{t}}^{2n}$ as a mirror partner of the complex torus $T^{2n}_{J=T^{\prime}}$. We denote the local coordinates of $\check{T}^{2n}_{J=T^{\prime}}$ by $(X^{1},\cdots,X^{n},Y^{1},\cdots,Y^{n})^{t}$, and set

[TABLE]

Let us consider a diffeomorphism $\phi:\check{T}^{2n}_{J=T}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\check{T}^{2n}_{J=T^{\prime}}$ which is expressed locally as

[TABLE]

By a direct calculation, we see

[TABLE]

where $d\check{X}:=(dX^{1},\cdots,dX^{n})^{t}$, $d\check{Y}:=(dY^{1},\cdots,dY^{n})^{t}$, so this diffeomorphism $\phi$ is a symplectomorphism.

Now, we define

[TABLE]

and let us consider the condition such that

[TABLE]

holds. Namely, our first goal is to consider when it is possible to construct a symplectic automorphism $\Phi:\check{T}^{2n}_{J=T^{\prime}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\check{T}^{2n}_{J=T^{\prime}}$ such that

[TABLE]

We consider the condition (35). Since $r\in\mathbb{N}$, $A\in M(n;\mathbb{Z})$ are fixed, we can take the map

[TABLE]

as a symplectic automorphism $\Phi:\check{T}^{2n}_{J=T^{\prime}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\check{T}^{2n}_{J=T^{\prime}}$ which satisfies the condition (35), in the case $(\phi^{-1})^{-1}(L_{(r,A,p)})=(\phi^{-1})^{-1}(L_{(r,A,p^{\prime})})$ only. Moreover,

[TABLE]

holds if and only if

[TABLE]

holds. Then, the condition (36) becomes

[TABLE]

so hereafter, we consider when $(\phi^{-1})^{*}\mathcal{L}_{(r,A,p,q)}\cong(\phi^{-1})^{*}\mathcal{L}_{(r,A,p^{\prime},q^{\prime})}$ holds on $(\phi^{-1})^{-1}(L_{(r,A,p)})=(\phi^{-1})^{-1}(L_{(r,A,p^{\prime})})$ by computing an isomorphism

[TABLE]

explicitly. The morphism $\psi$ need to satisfy the differential equation

[TABLE]

In particular, since the differential equation (38) turns out to be

[TABLE]

by solving the differential equation (39), we obtain a solution

[TABLE]

where $\lambda\in\mathbb{C}$ is an arbitrary constant. Furthermore, since $(\phi^{-1})^{*}\mathcal{L}_{(r,A,p,q)}$ and $(\phi^{-1})^{*}\mathcal{L}_{(r,A,p^{\prime},q^{\prime})}$ are trivial, this morphism $\psi$ must satisfy

[TABLE]

By a direct calculation, we see that

[TABLE]

hold, where $\mathcal{B}_{i}:=(\mathcal{B}_{1i},\cdots,\mathcal{B}_{ni})^{t}$, so we obtain

[TABLE]

Clearly, these relations are equivalent to

[TABLE]

where

[TABLE]

and then, by the formula (40), the isomorphism $\psi$ is expressed locally as

[TABLE]

with $\lambda\not=0\in\mathbb{C}$. Hence,

[TABLE]

holds on $(\phi^{-1})^{-1}(L_{(r,A,p)})=(\phi^{-1})^{-1}(L_{(r,A,p^{\prime})})$ if and only if

[TABLE]

holds. Thus, by the relations (37) and (41), we can conclude that

[TABLE]

holds if and only if

[TABLE]

hold. ∎

Hence, by comparing Theorem 3.4 with Theorem 4.4, we can expect that the isomorphism classes of holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\rightarrow T^{2n}_{J=T}$ correspond to the isomorphism classes of objects $\mathscr{L}_{(r,A,p,q)}$ of the Fukaya category $Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T})$. Actually, by a direct calculation, we can check that there exists such a correspondence.

5 Main result

In this section, we prove that there exists a bijection between the set of the isomorphism classes of holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\rightarrow T^{2n}_{J=T}$ and the set of the isomorphism classes of objects $\mathscr{L}_{(r,A,p,q)}$ of the Fukaya category $Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T})$.

First, we prepare two notations. We denote the set of the isomorphism classes of objects of the DG-category $DG_{T^{2n}_{J=T}}$ (i.e., the set of the isomorphism classes of holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$) by

[TABLE]

Similarly, we denote the set of the isomorphism classes of objects $\mathscr{L}_{(r,A,p,q)}$ of the Fukaya category $Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T})$ by

[TABLE]

Now, in order to state the main theorem, we define a map $F:\mathrm{Ob}(DG_{T^{2n}_{J=T}})\rightarrow\mathrm{Ob}(Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T}))$ as follows. Clearly, we need four parameters $r$, $A$, $p$, $q$ when we define objects $\mathscr{L}_{(r,A,p,q)}$ of $Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T})$. On the contrary, we need five parameters $r$, $A$, $p$, $q$, $\mathcal{U}$ when we define objects $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ of $DG_{T^{2n}_{J=T}}$. Hence, when we define a map $\mathrm{Ob}(DG_{T^{2n}_{J=T}})\rightarrow\mathrm{Ob}(Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T}))$, we must transform not only the information about four parameters $r$, $A$, $p$, $q$ but also the information about $\mathcal{U}$. For example, let us consider a map $\mathrm{Ob}(DG_{T^{2n}_{J=T}})\rightarrow\mathrm{Ob}(Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T}))$ which is simply defined by

[TABLE]

Then, this map does not induce a bijection between $\mathrm{Ob}^{isom}(DG_{T^{2n}_{J=T}})$ and $\mathrm{Ob}^{isom}(Fuk_{\mathrm{aff}}(\check{T}^{2n}_{J=T}))$ unfortunately, and we can check it as follows. We set

[TABLE]

It is clear that $AT=(AT)^{t}$ holds and $r^{\prime}=2$. For this quadruple $(r,A,p,q)\in\mathbb{N}\times M(2;\mathbb{Z})\times\mathbb{R}^{2}\times\mathbb{R}^{2}$, we define mutually distinct $\mathcal{U}$ and $\mathcal{U}^{\prime}$ by

[TABLE]

In this situation, we can easily check that $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\not\cong E_{(r,A,r^{\prime},\mathcal{U^{\prime}},p,q)}$ holds by using Theorem 3.4, and actually, the above map $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\mapsto\mathscr{L}_{(r,A,p,q)}$ sends both $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ and $E_{(r,A,r^{\prime},\mathcal{U}^{\prime},p,q)}$ to the same object $\mathscr{L}_{(r,A,p,q)}$. Thus, by concerning these facts, here, we define a map

[TABLE]

by

[TABLE]

where $\xi$, $\theta\in\mathbb{R}^{n}$ denote the vectors associated to $\mathcal{U}$ in the sense of the definition (14). The following is the main theorem of this paper.

Theorem 5.1**.**

The map $F$ induces a bijection between $\mathrm{Ob}^{isom}(DG_{T^{2n}_{J=T}})$ and $\mathrm{Ob}^{isom}(Fuk_{\mathrm{aff}}(\check{T}_{J=T}^{2n}))$.

Proof.

In this proof, for a given object $E_{(r,A,r^{\prime},\mathcal{U},p,q)}\in\mathrm{Ob}(DG_{T^{2n}_{J=T}})$, we denote by $\xi$, $\theta\in\mathbb{R}^{n}$ the vectors associated to $\mathcal{U}$ in the sense of the definition (14). Similarly, for a given object $E_{(s,B,s^{\prime},\mathcal{V},u,v)}\in\mathrm{Ob}(DG_{T^{2n}_{J=T}})$, we denote by $\tau$, $\sigma\in\mathbb{R}^{n}$ the vectors associated to $\mathcal{V}$ in the sense of the definition (14). We denote the induced map from the map $F$ by

[TABLE]

Explicitly, it is defined by

[TABLE]

where, of course, $[E_{(r,A,r^{\prime},\mathcal{U},p,q)}]$ and $[F(E_{(r,A,r^{\prime},\mathcal{U},p,q)})]$ denote the isomorphism class of $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$ and the isomorphism class of $F(E_{(r,A,r^{\prime},\mathcal{U},p,q)})$, respectively.

First, we check the well-definedness of the map $F^{isom}$. We take two arbitrary objects $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$, $E_{(s,B,s^{\prime},\mathcal{V},u,v)}\in\mathrm{Ob}(DG_{T^{2n}_{J=T}})$ and assume

[TABLE]

By considering the $i$-th Chern characters $ch_{i}(E_{(r,A,r^{\prime},\mathcal{U},p,q)})$, $ch_{i}(E_{(s,B,s^{\prime},\mathcal{V},u,v)})$ of the holomorphic vector bundles $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$, $E_{(s,B,s^{\prime},\mathcal{V},u,v)}$ for each $i\in\mathbb{N}$, we see

[TABLE]

We consider the equality (42) in the cases $i=0,1$. Then, we obtain

[TABLE]

so one has

[TABLE]

Hence, we see that there exists a $k\in\mathbb{N}$ such that

[TABLE]

Therefore, since we can regard $E_{(s,B,s^{\prime},\mathcal{V},u,v)}$ as $E_{(kr,kA,r^{\prime},\mathcal{V},u,v)}=E_{(r,A,r^{\prime},\mathcal{V},\frac{1}{k}u,\frac{1}{k}v)}$, by Theorem 3.4, we see

[TABLE]

Thus, by Theorem 4.4 and the relations (43), (44), (51), (58), we can conclude

[TABLE]

namely,

[TABLE]

Next, we prove that $F^{isom}$ is injective. We take two arbitrary objects $E_{(r,A,r^{\prime},\mathcal{U},p,q)}$, $E_{(s,B,s^{\prime},\mathcal{V},u,v)}\in\mathrm{Ob}(DG_{T^{2n}_{J=T}})$ and assume

[TABLE]

namely,

[TABLE]

Then, we see that there exists a $k\in\mathbb{N}$ which satisfies the relations (43) and (44). Here, we take two matrices $\mathcal{A}$, $\mathcal{B}\in GL(n;\mathbb{Z})$ such that

[TABLE]

where $\tilde{a_{i}}\in\mathbb{N}$ ($i=1,\cdots,t$, $1\leq t\leq n$) and $\tilde{a_{i}}|\tilde{a_{i+1}}$ ($i=1,\cdots,t-1$). Therefore, since the relation (44) holds, we obtain

[TABLE]

In particular, the relations (43), (44), (59), (60) imply

[TABLE]

Hence, by Theorem 4.4, the relations (51) and (58) hold. Now, note that we can regard $E_{(s,B,s^{\prime},\mathcal{V},u,v)}$ as $E_{(kr,kA,r^{\prime},\mathcal{V},u,v)}=E_{(r,A,r^{\prime},\mathcal{V},\frac{1}{k}u,\frac{1}{k}v)}$ by the relations (43), (44), (61). Thus, by Theorem 3.4 and the relations (51) and (58), we see that

[TABLE]

holds, and this relation indicates

[TABLE]

Finally, we prove that $F^{isom}$ is surjective. We take an arbitrary quadruple $(r,A,p,q)\in\mathbb{N}\times M(n;\mathbb{Z})\times\mathbb{R}^{n}\times\mathbb{R}^{n}$, and consider the element

[TABLE]

In particular, a representative of $[\mathscr{L}_{(r,A,p,q)}]$ is expressed as

[TABLE]

by using a pair

[TABLE]

(see Theorem 4.4). For the element $[\mathscr{L}_{(r,A,p,q)}]$, we consider the element

[TABLE]

where $\xi$, $\theta\in\mathbb{R}^{n}$ are the vectors associated to $\mathcal{U}$ in the sense of the definition (14). Here, note that how to choose a set $\mathcal{U}$ is not unique even if we fix a quadruple $(r,A,p,q)\in\mathbb{N}\times M(n;\mathbb{Z})\times\mathbb{R}^{n}\times\mathbb{R}^{n}$. Therefore, a representative of $[E_{(r,A,r^{\prime},\mathcal{U},p+\frac{r}{r^{\prime}}\theta,q-\frac{r}{r^{\prime}}\xi)}]$ is expressed as

[TABLE]

by using a set $\mathcal{U}^{\prime}$ with the associated vectors $\xi^{\prime}$, $\theta^{\prime}\in\mathbb{R}^{n}$ and a pair

[TABLE]

(see Theorem 3.4). Then, by a direct calculation, we see

[TABLE]

This completes the proof. ∎

Acknowledgment

I would like to thank Hiroshige Kajiura for various advices in writing this paper. I also would like to thank Masahiro Futaki and Atsushi Takahashi for helpful comments. Finally, I am grateful to the referee for useful suggestions. This work was supported by Grant-in-Aid for JSPS Research Fellow 18J10909.