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%\received{Received January 5, 2005}
%\received{Received August 25, 2005;\enspace Revised October 20, 2005}
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\begin{document}

\title[A topological mirror symmetry on noncommutative tori]
{A topological mirror symmetry on noncommutative complex two-tori}

\author[E. Kim]{Eunsang Kim}
\address{Eunsang Kim \\ Department of Applied Mathematics \\ Hanyang University \\ Ansan 15588, Korea}
\email{eskim@ihanyang.ac.kr}

\author[H. Kim]{Hoil Kim}
\address{Hoil Kim \\ Department of Mathematics \\ Kyungpook  National University \\ Daegu 41566, Korea}
\email{hikim@knu.ac.kr}
\thanks{This work was financially supported by KRF 2003-041-C20009}

\subjclass[2020]{Primary 58B34, 58J42, 81T75}
\keywords{noncommutative complex torus, mirror symmetry, Kronecker foliation}

\begin{abstract}
In this paper, we study a topological mirror symmetry on
noncommutative complex tori. We show that deformation quantization
of an elliptic curve  is mirror symmetric to an irrational
rotation algebra. From this, we conclude that a mirror reflection
of a noncommutative complex torus is an elliptic curve equipped
with a Kronecker foliation.
\end{abstract}

\maketitle

\section{Introduction}

The noncommutative tori is known to be the most accessible
examples of noncommutative geometry developed by A. Connes
\cite{Co}. It also provides the best examples in applications of
noncommutative geometry to the physics of open strings
\cite{SeiWi, CDS}. A noncommutative torus is a universal
$C^*$-algebra generated by two unitary operators subject only to a
suitable commutation relation. It arises naturally in a number of
different situations. Among others, it can be obtained as a
deformation quantization of the algebra of continuous (smooth)
functions on an ordinary torus. This approach has been used widely
in gauge theories on noncommutative tori (cf.~\cite{Ri1})
and in applications to string theory such as in  \cite{HV}. On the
other hand, one can consider a noncommutative torus as a foliation
$C^*$-algebra for a Kronecker foliation on a torus via Morita
equivalence, \cite{Co}. Such an algebra is known to be an
irrational rotation algebra, \cite{Ri}. Here we will choose these
two approach to define noncommutative two-tori and we will discuss
how they are related with the mirror symmetry \cite{SYZ, Tyu}.

Associated to a topological mirror symmetry, we need to define a
complex structure on a  noncommutative torus. The complex geometry
has been developed by A. Schwarz in \cite{Sch01} and some basic
calculations have been made in \cite{DiSch} for the two-dimensional
case which leads to the study of Kontsevich's homological mirror
symmetry (\cite{Ko}) in \cite{PolSch}, \cite{Kaj} and \cite{Pol}
(see also \cite{Kaj1}).


In this paper, we will mainly concern on an aspect of a
topological mirror symmetry on elliptic curves based on \cite{Tyu}
which compares the moduli spaces of stable bundles and
supercycles. The stable bundles of a certain topological type are
deformed to standard holomorphic bundles on a noncommutative
complex torus, along the deformation quantization procedure. We
show that the deformation is equivalent to define a linear
foliation on a mirror reflection of a given elliptic curve.

In Section 2, we will review some basic facts for noncommutative
complex tori. In Section 3, we show how stable bundles on an
elliptic curve are deformed to standard holomorphic bundles on a
noncommutative complex torus. In Section 4, we find a mirror
reflection of a noncommutative complex torus.


\section{Some preliminaries on noncommutative complex tori}

In this section, we review some basic facts for noncommutative
complex tori and bundles on them, following \cite{Sch01} and
\cite{PolSch}.

\subsection{Noncommutative complex two-tori}

A noncommutative two-torus $T^2_\theta$ is defined by two
unitaries $U_1$, $U_2$ obeying the relation
\begin{equation}\label{comrel}
U_1U_2=\exp(2\pi i\theta)U_2U_1,
\end{equation}
where $\theta\in \mathbb{R}/\mathbb{Z}$. The commutation relation
(\ref{comrel}) defines the presentation of the involutive algebra
\[A_\theta=\left\{\sum_{n_1,n_2\in\mathbb{Z}^2}a_{n_1,n_2}U_1^{n_1}U_2^{n_2}
\mid a_{n_1,n_2}\in\mathcal{S}(\mathbb{Z}^2)\right\},\] where
$\mathcal{S}(\mathbb{Z}^2)$ is the Schwartz space of sequences
with rapid decay at infinity. According to \cite{Co1}, the algebra
$A_\theta$ can be understood as the algebra of smooth functions on
$T^2_\theta$. As was given in \cite{Ri0}, the algebra $A_\theta$
can be defined as a deformation quantization of $C^\infty(T^2)$,
the algebra of smooth functions on the ordinary torus $T^2$. The
action of $T^2$ by translation on $C^\infty(T^2)$ gives an action
of $T^2$ on $A_\theta$. The infinitesimal form of the action
defines a Lie algebra homomorphism $\delta:L\rightarrow {\rm {Der
}}({A}_\theta)$, where $L=\mathbb{R}^2$ is an abelian Lie algebra
and ${\rm {Der}}({A}_\theta)$ is the Lie algebra of derivations of
$A_\theta$. Generators $\delta_1$, $\delta_2$ of ${\rm {Der
}}({A}_\theta)$  act in the following way:
\begin{equation}\label{derivation}
\delta_k\left(\sum_{(n_1,n_2)\in\mathbb{Z}^2}a_{n_1,n_2}U^{n_1}U^{n_2}\right)=2\pi
i\sum_{(n_1,n_2)\in\mathbb{Z}^2}n_ka_{n_1,n_2}U^{n_1}U^{n_2}.
\end{equation}

A complex structure on $T^2_\theta$ is defined in terms of a
complex structure on the Lie algebra $L=\mathbb{R}^2$ which acts
on $A_\theta$. Let us fix a $\tau\in\mathbb{C}$ such that
$\text{Im}(\tau)\ne 0$ and then $\tau$ defines a one-dimensional
subalgebra of ${\rm {Der }}({A}_\theta)$ spanned by the derivation
$\delta_\tau$ given by
\begin{equation}\label{comp}
 \delta_\tau\left(\sum_{(n_1,n_2)\in\mathbb{Z}^2}a_{n_1,n_2}U^{n_1}U^{n_2}\right)
= 2\pi
i\sum_{(n_1,n_2)\in\mathbb{Z}^2}(n_1\tau+n_2)a_{n_1,n_2}U^{n_1}U^{n_2}.
\end{equation}
The noncommutative torus equipped with such a complex structure is
denoted by $T^2_{\theta,\tau}$ and will be called a noncommutative
complex torus.


\subsection{\boldmath Holomorphic bundles on $T^2_{\theta,\tau}$ }

Since the algebra $A_\theta$ is considered as the algebra of
smooth functions on $T^2_\theta$, the vector bundles on
$T^2_\theta$ correspond to finitely generated projective right
$A_\theta$-modules. Such modules can be constructed by the
Heisenberg projective representations and it is known (see
\cite{Ri1}) that for each $g=\left(\smallmatrix
a&b\\c&d\endsmallmatrix \right)\in \text{SL}_2(\mathbb{Z})$ such
that $d+c\theta\ne 0$, a Heisenberg module
$E_g(\theta):=E_{d,c}(\theta)$ over $A_\theta$ is given by the
Schwarz space
$\mathcal{S}(\mathbb{R}\times\mathbb{Z}/c\mathbb{Z})$ equipped
with the right action of $A_\theta$:
\begin{align*}
fU_1(s,k)=f(s-\frac{d+c\theta}{c},k-1),\quad
fU_2(s,k)=\exp(s-\frac{k d}{c})f(s,k),
\end{align*}
where $s\in\mathbb{R}$ and $k\in\mathbb{Z}/c\mathbb{Z}$. For
$g\in\text{SL}_2(\mathbb{Z})$, the modules $E_g(\theta)$  will be
referred as {\it basic modules}.

Connections on a vector bundle on the noncommutative torus
$T^2_\theta$ are defined in terms of derivations. Let $E$ be a
projective right ${A}_\theta$-module, a connection $\nabla$ on $E$
is a linear map from $E$ to $E\otimes L^*$ such that for all $x\in
L$,
\begin{align}
\nabla_x(\xi u)=(\nabla_x\xi)u+\xi\delta_x(u),{\rm { \ \ \
}}\xi\in {E}, u\in {A}_\theta.\notag
\end{align}
The curvature ${F}_\nabla$ of the connection $\nabla$ is a 2-form
on $L$ with values in the algebra of endomorphisms of $E$. That
is, for $x,y\in L$,
$${F}_\nabla(x,y):=[\nabla_x,\nabla_y]-\nabla_{[x,y]}.$$
Since $L$ is abelian, we simply have
${F}_\nabla(x,y)=[\nabla_x,\nabla_y]$.

A {\it holomorphic structure} on a right $A_\theta$-module $E$
compatible with the complex structure on $T^2_{\theta,\tau}$ is a
$\mathbb{C}$-linear map $\overline{\nabla}:E\longrightarrow E$
such that
\[\overline{\nabla}(\xi\cdot u)=\overline{\nabla}(\xi)\cdot
u+\xi\cdot\delta_\tau(u), \ \ \ \xi\in E, u\in A_\theta.\] A
projective right $A_\theta$-module equipped with a holomorphic
structure is called a {\it holomorphic bundle} over the
noncommutative complex torus $T^2_{\theta,\tau}$. In particular,
the basic modules $E_g(\theta)$ equipped with holomorphic
structure are called {\it standard holomorphic bundles} on
$T^2_{\theta,\tau}$.

\subsection{The Chern character and the slope of basic modules}


The $K$-theory group $K_0(A_\theta)$ classifies the finitely
generated projective $A_\theta$-modules and there is a natural
map, the Chern character which takes the values in the Grassmann
algebra $\wedge^\bullet L^* $, where $L^*$ is the dual vector
space of the Lie algebra $L$. Since there is a lattice $\Gamma$ in
$L$, there should be elements of $\wedge^\bullet \Gamma^*$ which
are integral, where $\Gamma^*$ is the dual lattice of $\Gamma$.
The Chern character is the map ${\rm { \ Ch
}}:K_0({A}_\theta)\rightarrow\wedge^{\rm{ev}}(L^*)$ defined by
\begin{align}
{\rm{Ch}}(\mathcal{E})=e^{-i(\theta)}\nu(\mathcal{E}),\label{Ch}
\end{align}
where $i(\theta)$ denotes the contraction with the deformation
parameter $\theta$ regarded as an element of $\wedge^2L$ and
$\nu(\mathcal{E}) \in\wedge^{\text{even}}\Gamma^*$. See \cite{Ri1}
for details for the definition of the Chern character. For a given
$g=\left(\smallmatrix a&b\\c&d \endsmallmatrix\right) \in
\text{SL}_2(\mathbb{Z})$ such that $d+c\theta >0$, we have
\begin{align}
{\rm{Ch}}(E_g(\theta))=(d+c\theta)+c \ dx_1\wedge dx_2.
\end{align}
Let us define
\[\text{deg}(g)=\text{deg}(E_g(\theta))=c \text{ \ \ and \ \ }
\text{rk}(g,\theta)=c\theta+d=\text{rank}(E_g(\theta)).\] As in
the classical case, we may define the {\it slope} of the basic
module $E_g(\theta)$ by the numbers
\[\mu(E_g(\theta))=\frac{\text{deg}(g)}{\text{rk}(g,\theta)}=\frac{c}{c\theta+d}.\]

\section{Stable bundles on an elliptic curve and standard
holomorphic bundles}

In this section we construct a standard holomorphic bundle over
$T^2_{\theta,\tau}$ from a stable bundle over an elliptic curve
$X_\tau$. We will show that the moduli space of holomorphic stable
bundles is naturally identified with the moduli space of standard
holomorphic bundles associated to a matrix in
$\text{SL}_2({\mathbb{Z}})$ which determines a topological type on
both bundles.

\subsection{Stable bundles on an elliptic curve}

Let $X_\tau=\mathbb{C}/\mathbb{Z}+\tau \mathbb{Z}$ be an elliptic
curve whose complex structure is specified by $\tau\in\mathbb{C}$,
Im $\tau\ne 0$. For $X_\tau$, the algebraic cohomology ring is
\[A(X_\tau)=H^0(X_\tau,\mathbb{Z})\oplus
H^2(X_\tau,\mathbb{Z})\cong\mathbb{Z}\oplus\mathbb{Z}.\] The Chern
character of a holomorphic vector bundle $E$ on $X_\tau$ takes the
value in $A(X_\tau)$:
\[\text{Ch}(E)=(\text{rank }E, \text{deg }E)\in
H^0(X_\tau,\mathbb{Z})\oplus H^2(X_\tau,\mathbb{Z}),\] where
$\text{deg }E=c_1(E)$. The slope of a vector bundle $E$ is defined
by
\[\mu(E)=\frac{\text{deg
}E}{\text{rank }E}.\] A bundle $E$ is said to be stable if, for
every proper subbundle $E'$ of $E$, $0<\text{rank }E'<\text{rank
}E$, we have
\[\mu(E')<\mu(E).\]
Every stable bundles carries a projectively flat Hermitian
connection $\nabla^E$. In other words, there is a complex 2-form
$\lambda$ on $X_\tau$ such that the curvature of $\nabla^E$ is
\[R_{\nabla^E}=\lambda\cdot \text{Id}_E,\] where  $\text{Id}_E$ is
the identity endomorphism of $E$. Since
\begin{align*}
c_1(E)=\frac{i}{2\pi}\text{Tr }R_{\nabla^E}
=\frac{i}{2\pi}\lambda\cdot\text{ rank }E,
\end{align*}
we have
\[\lambda=\frac{2\pi}{i} \frac{c_1(E)}{\text{rank
}E}=\frac{2\pi}{i}\mu(E).\] Thus
\begin{equation}\label{curvature}
R_{\nabla^E}=-2\pi i \mu(E) \text{Id}_E.
\end{equation}
Note that if $E$ is stable, then the topological type of $E$ is
given by the pair $(\text{rank }E, \text{deg }E)$ which is
relatively prime. Thus we may extend the pair to a matrix in
$\text{SL}_2({\mathbb{Z}})$. For $g=\left(\smallmatrix a&b\\c&d
\endsmallmatrix\right)\in \text{SL}_2(\mathbb{Z})$, let us denote
by $\mathcal{M}_g^s$ the moduli space of holomorphic stable
bundles of rank $d$ and degree $c$ on $X_\tau$. Since every stable
bundle $E$ on an elliptic curve $X_\tau$ is uniquely determined up
to translation by its topological type $(d,c)$, the moduli space
is
\[\mathcal{M}_{g}^s\cong X_\tau.\]

\subsection{Holomorphic deformations of stable bundles}

For a matrix $g=\left(\smallmatrix a&b\\c&d
\endsmallmatrix\right)\in \text{SL}_2(\mathbb{Z})$, the Chern
character of a stable bundle $E$ in $\mathcal{M}_{g}^s$ is of the
form $\text{Ch}(E)=(d,c)\in A(X_\tau)$ and it defines an integral
element $d+cdx^{12}\in\wedge^2 L^*$. Let us consider
\[e^{-i(\theta)}(d+cdx^{12})=(d+c\theta)+c \ dx_1\wedge dx_2\] and let
\[\mu=\frac{c}{c\theta+d}.\] Then we have

\begin{proposition}
For a stable bundle $E$ on $X_\tau$
whose topological type is specified by a matrix
$g=\left(\smallmatrix a&b\\c&d \endsmallmatrix\right)\in {\rm
SL}_2(\mathbb{Z})$, there is a basic $A_\theta$-module
$E_g(\theta)$ equipped with a connection whose curvature is $-2\pi
i\mu$.
\end{proposition}

\begin{proof}
Associated to the curvature condition
(\ref{curvature}) on the stable bundle $E$ on $X_\tau$, we define
a Heisenberg commutation relation by
\begin{equation}\label{2}
F_\nabla=[\nabla_1,\nabla_2]=-2\pi i\mu.
\end{equation}
By the Stone-von Neuman theorem, the above relation has a unique
representation. As discussed in \cite{CoR}, the representation is
just $c$-copies of the Schr\"odinger representation of the
Heisenberg Lie group $\mathbb{R}^3$ on $L^2(\mathbb{R})$, where
the product on $\mathbb{R}^3$ is given by
\[(r,s,t)\cdot(r',s',t')=(r+r',s+s',t+t'+sr').\]
Then the operators $\nabla_1$ and $\nabla_2$ are the infinitesimal
form of the representation and is given by
\begin{align}\label{con1}
(\nabla_1f)(s,k)&=2\pi i\mu sf(s,k),
\end{align}
\begin{align}
(\nabla_2f)(s,k)&=\frac{df}{ds}(s,k)\label{con2}
\end{align}
acting on the Schwartz space
$\mathcal{S}(\mathbb{R}\times\mathbb{Z}/c\mathbb{Z})\cong
\mathcal{S}(\mathbb{R})\otimes \mathbb{C}^c$. Let
$E_g(\theta)=\mathcal{S}(\mathbb{R})\otimes \mathbb{C}^c$. Then
one sees that (\ref{con1}) and (\ref{con2}) are in fact desired
connection on $E_g(\theta)$. To specify the module $E_g(\theta)$,
we need to define a module action which is compatible with the
relation (\ref{comrel}) for $T^2_\theta$. Let us first consider
unitary operators $W_1$, $W_2$ on
$\mathcal{S}(\mathbb{Z}/c\mathbb{Z})=\mathbb{C}^c$ defined by
\begin{align*}
W_1f(k)=f(k-d),\quad W_2f(k)=e^{-2\pi i\frac{k}{c}}f(k).
\end{align*}
Then
\[W_1W_2=e^{2\pi i\frac{d}{c}}W_2W_1.\]
In other words, $W_1$ and $W_2$ provide a representation of the
Heisenberg commutation relations for the finite group
$\mathbb{Z}/c\mathbb{Z}$. Associated to the connections
(\ref{con1}) and (\ref{con2}), we have Heisenberg representations
$V_1$ and $V_2$ on the space $\mathcal{S}(\mathbb{R})$ as
\begin{align*}
V_1f(s)=e^{2\pi i(\frac{d}{c}-\theta)s}f(s),\quad V_2f(s)=f(s+1).
\end{align*}
The operators obey the relation
\[V_1V_2=e^{-2\pi i(\frac{d}{c}-\theta)}V_2V_1.\]
Finally, the operators
\begin{align}\label{3}
U_1=V_1\otimes W_1 \ \ \ \text{ and } \ \ \ U_2=V_2\otimes W_2
\end{align} acting on the
space $\mathcal{S}(\mathbb{R}\times \mathbb{Z}/c\mathbb{Z})$
satisfy the relation
\[U_1U_2=e^{2\pi i\theta}U_2U_1.\]
This completes the construction of basic module $E_g(\theta)$
equipped with a constant curvature connection $\nabla_1$,
$\nabla_2$ such that the curvature is given by
(\ref{2}).
\end{proof}

The basic module $E_g(\theta)$ constructed in Proposition 3.1
admits a constant curvature connection (\ref{con1}, \ref{con2})
and all other constant curvature connections on $E_g(\theta)$
which satisfies the relation (\ref{2}) are given as
\begin{align}
(\nabla_1f)(s)&=2\pi i\mu(E_g(\theta))f(s)+2\pi i\alpha,\label{rcon1}
\end{align}
\begin{align}
(\nabla_2f)(s)&=\frac{\partial f}{\partial s}+2\pi
i\beta,\label{rcon2}
\end{align}
where $\alpha$ and $\beta$ are real numbers. Let us fix a complex
number $\tau$ such that Im $\tau<0$. The parameter $\tau$ defines
a complex structure on $T^2_\theta$ via derivation
$\delta_\tau=\tau\delta_1+\delta_2$ spanning
$\text{Der}(A_\theta)$ as given in subsection 2.1. Then a
holomorphic structure on $E_g(\theta)$ is specified by
$\overline{\nabla}=\tau\nabla_1+\nabla_2$. Along with the
connections in (\ref{rcon1}, \ref{rcon2}), and for
$z\in\mathbb{C}$, let
\[(\overline{\nabla}_z)(f)(s,k)=\frac{\partial f}{\partial
s}(s,k)+2\pi i(\tau\mu(E_g(\theta))s +z)f(s,k).\] Then
$\overline{\nabla}_z$ defines a standard holomorphic structure on
$E_g(\theta)$. Other holomorphic structures are determined by
translations of $\alpha$ and $\beta$ in (\ref{rcon1}) and
(\ref{rcon2}). In other words, all the holomorphic structures are
determined by the complex number $z=\tau \alpha+\beta$. A basic
module $E_g(\theta)$ equipped with a holomorphic structure
$\overline{\nabla}_z$ is called a standard holomorphic bundle and is
denoted by $E_g^z(\theta)$. Let $\mathcal{M}_g^s(\theta)$ be the
moduli space of holomorphic constant curvature connections on
$E_g(\theta)$ which satisfy the relation (\ref{2}). Then our
discussion above shows that the following:

\begin{proposition}
With notations above, we have
\[\mathcal{M}_g^s(\theta)\cong X_\tau\cong\mathcal{M}_{g}^s.\]
\end{proposition}



\section{Mirror symmetry and noncommutative tori}

In this section, we consider a mirror torus $\widehat{X}_\tau$ of
the elliptic curve  $X_\tau$. We first review the construction of
$\widehat{X}_\tau$ following the lines of \cite{Tyu}. Then we will
show that a Kronecker foliation on $\widehat{X}_\tau$ is mirror
symmetric to a noncommutative complex torus.


\subsection{Supercycles}

Let $\tau\in\mathbb{C}$ be an element in the lower half-plane as
in Subsection 3.2. The complex number defines an elliptic curve
$X_\tau=\mathbb{C}^*/q^\mathbb{Z}$, $q=\exp(-2\pi i\tau)$. A
complex orientation of an elliptic curve $X_\tau$ is given by a
holomorphic 1-form $\Omega$, which determines a Calabi-Yau
manifold structure on $X_\tau$. A special Lagrangian cycle of
$X_\tau$ is a 1-dimensional Lagrangian submanifold $\mathcal{L}$
such that the restriction of $\Omega$ satisfies
\[\text{Im }\Omega|_\mathcal{L}=0 \ \ \text{ and } \ \ \text{Re
}\Omega|_\mathcal{L}=\text{Vol}(\mathcal{L}),\] where the volume
form is determined by the Euclidean metric on $X_\tau$. A special
Lagrangian cycle is just a closed geodesic and hence it is
represented by a line with rational slope on the universal
covering space of $X_\tau$. Let us fix a smooth decomposition
\[X_\tau=S^1_+\times S^1_-\] which induces a decomposition of
cohomology group
\[H^1(X_\tau,\mathbb{Z})\cong \mathbb{Z}_+\oplus\mathbb{Z}_-.\]
Let $[B]\in\mathbb{Z}_-$ and $[F]\in\mathbb{Z}_+$ be generators of
the cohomology group such that the cycles representing $[B]$ and
$[F]$ have no self-intersection and two cycles have one
intersection point. Then the cohomology class $[F]\in
H^1(X_\tau,\mathbb{Z})$ is represented by a smooth cycle in
$X_\tau$ and is a special Lagrangian cycle. The family of special
Lagrangian cycles representing the class $[F]$ gives a smooth
fibration
\begin{equation}\label{fibration}
\pi:X_\tau\longrightarrow S_-^1:=B
\end{equation} and the base space $B=S_-^1$ is just the moduli space of
special Lagrangian cycles associated to $[F]\in
H^1(X_\tau,\mathbb{Z})$. The unitary flat connections on the
trivial line bundle $S_+^1\times\mathbb{C}\to S_+^1$ are
parameterized by $\widehat{S^1}:=\text{
Hom}(\pi_1(S^1_+),\text{U}(1))$, up to gauge equivalences. Thus we
have the dual fibration
\begin{equation}\label{dual}
\widehat{\pi}:\widehat{X}_\tau\longrightarrow B=S_-^1
\end{equation} with fibers
\[\widehat{\pi}^{-1}(b)=
\text{Hom}(\pi_1(\pi^{-1}(b)),\text{U}(1))=\widehat{S^1}.\] The
dual fibration admits the section $s_0\in \widehat{X}_\tau$ with
\[s_0\cap\widehat{\pi}^{-1}(b)=1\in\text{Hom}(\pi_1(\pi^{-1}(b)),
\text{U}(1)),\] so that we have a decomposition
\[\widehat{X}_\tau=\widehat{S^1}\times S_-^1.\] Hence, associated to the class
$[F]\in H^1(X_\tau,\mathbb{Z})$, the space $\widehat{X}_\tau$ is
the moduli space of special Lagrangian cycles  endowed with
unitary flat line bundles. Furthermore, $\widehat{X}_\tau$ admits
a Calabi-Yau manifold structure. In other words,
$\widehat{X}_\tau$ is the mirror reflection of $X_\tau$ in the
sense of \cite{SYZ}. Under the K\"ahler-Hodge mirror map (see
\cite{Tyu}), a complexfied K\"ahler parameter $\rho=b+ik$ defines
a complex structure on $\widehat{X}_\tau$, where $k$ is a K\"ahler
form on ${X_\tau}$ and $b$ defines a class in
$H^2(X_\tau,\mathbb{R})/H^2(X_\tau,\mathbb{Z})$. Then
$\widehat{X}_\tau$ is the elliptic curve $\mathbb{C}^*/e^{2\pi
i\rho\mathbb{Z}}$. Similarly, the modular parameter $\tau$ of
$X_\tau=\mathbb{C}^*/q^\mathbb{Z}$, $q=\exp(-2\pi i\tau)$, $\text{
Im } \tau<0$, corresponds to a complexfied K\"ahler parameter
$\widehat{\rho}$ on $\widehat{X}_\tau$.

\subsection{The moduli space of supercycles}

\begin{definition}
A {\it supercycle } or a {\it brane} on
$\widehat{X}_\tau$ is given by a pair $(\mathcal{L},A)$, where
$\mathcal{L}$ is a special Lagrangian submanifold of
$\widehat{X}_\tau$ and $A$ a flat connection on the trivial line
bundle $\mathcal{L}\times \mathbb{C}\to\mathcal{L}$.
\end{definition}

A special Lagrangian cycle in
$\widehat{X}_\tau\cong\mathbb{R}^2/\mathbb{Z}\oplus\mathbb{Z}$ is
represented by a line of rational slope, so can be given by a pair
of relatively prime integers and we extend the pairs to a matrix
in $\text{SL}_2(\mathbb{Z})$. The lines of a fixed rational slope
are parameterized by the points of interception with the line
$y=0$ on the universal covering space $\mathbb{R}^2$ of
$\widehat{X}_\tau$. For $g=\left(\smallmatrix a&b\\c&d
\endsmallmatrix\right)\in \text{SL}_2(\mathbb{Z})$, let
$\mathcal{L}_g$ be a special Lagrangian submanifold of
$\widehat{X}_\tau$ given by
\begin{equation}\label{spec}
\mathcal{L}_g=\{(ds+\alpha,cs)\mid s\in\mathbb{R}/\mathbb{Z}\},
\end{equation}
so that the line has slope $\frac{c}{d}$ and $x$-intercept
$\alpha$. The shift of $\mathcal{L}_g$ is represented by the
translation of $\alpha$. Thus the moduli space of special
Lagrangian cycles are $S^1$. Note that a  unitary flat line bundle
on $\mathcal{L}_g$ is specified by the monodromy around the
circle. On the trivial line bundle $\mathcal{L}_g\times
\mathbb{C}\to\mathcal{L}_g$, we have a  constant real valued
connection one-form  on $\mathbb{R}^2$, restricted to
$\mathcal{L}_g$, given by
\begin{equation}\label{conne}
A=2\pi i\beta dx, \ \ \ x\in \mathbb{R}^2, \ \
\beta\in\mathbb{R}/\mathbb{Z},
\end{equation}
so that the monodromy between points $(x_1,y_1)$ and $(x_2,y_2)$
is given by $\exp[2\pi i $ $\beta(x_2-x_1)]$. Thus, the shift of
connections is represented by monodromies. Combining the result in
Subsection 3.1, we have the following:

\begin{proposition}
Let $\mathcal{SM}_g$ be the moduli
space of supercycles on $\widehat{X}_\tau$, whose special
Lagrangian cycle is specified by the matrix $g\in{\rm
SL}_2(\mathbb{Z})$. Then we have
\[\mathcal{SM}_g\cong X_\tau\cong \mathcal{M}_g^s. \]
\end{proposition}

\begin{remark}
In \cite{Tyu}, the identification
$\mathcal{SM}_g\cong \mathcal{M}_g^s$ was shown in a geometric
way. By Proposition 3.2.2, we also have
\begin{equation}\label{equiv}
\mathcal{SM}_g\cong\mathcal{M}_g^s\cong\mathcal{M}_g^s(\theta).
\end{equation}
\end{remark}

However in the above identification, one may not see how
deformation quantization is related to a mirror symmetry. In
below, we shall reprove the identification (\ref{equiv}) in a
geometric way.

\subsection{\boldmath A mirror reflection of $T^2_{\theta,\tau}$}

There are many ways to define a noncommutative torus such as a
deformation quantization or as an irrational rotation algebra,
etc. In Section 2, we have considered $A_\theta$ as a deformation
quantization of $C^\infty(T^2)$. An irrational rotation algebra
can be obtained from a Kronecker foliation on a torus. We show
that those two algebras are related by a mirror symmetry.

\begin{theorem}
For an irrational number $\theta$, the
noncommutative torus $A_\theta$ obtained from a deformation
quantization of $C^{\infty}(T^2)$ is mirror symmetric to an
irrational rotation algebra which is defined by the Kronecker
foliation on $\widehat{X}_\tau$ whose leaves are represented by
the lines of slope $\theta^{-1}$ on the universal covering space
of $\widehat{X}_\tau$.
\end{theorem}

\begin{proof}
Let us  consider the case when $g=\left(\smallmatrix
1&0\\0&1
\endsmallmatrix\right)$. It corresponds to
trivial line bundles on $X_\tau$. The sections of a trivial line
bundle is identified with $C^\infty(X_\tau)$ and it is deformed to
a free module over $A_\theta$ of rank 1 along the proof of
Proposition 3.2.1. Thus we have $A_\theta$ as a deformation
quantization of $C^\infty(X_\tau)$ with a holomorphic structure
specified by the derivations on it.

On the mirror side, let $\widehat{X}_{\tau,\theta^{-1}}$ be a
foliated torus defined by the differential equation
$dy=\theta^{-1}dx$ with natural coordinate $(x,y)$ on the flat
torus determined by the symplectic form on $\widehat{X}_\tau$.
Such a foliation is called a Kronecker foliation or a linear
foliation ({\it cf.} \cite{CN}). On the covering space of
$\widehat{X}_\tau$ the leaves of the foliation are represented by
straight lines with fixed slope $\theta^{-1}$ and every closed
geodesic of $\widehat{X}_\tau$ yields a compact transversal which
meets every leaves. For $g=\left(\smallmatrix 1&0\\0&1
\endsmallmatrix\right)$,
a trivial line bundle is mapped to a special Lagrangian cycle
represented by the line $y=0$ under the identification
$\mathcal{M}_g^s\cong \mathcal{SM}_g$. The line $y=0$ is a compact
transversal for the $\theta^{-1}$-linear foliation and each leaf
meets the line countably many points. Associated to the
intersection points, each leaf defines the rotation through the
angle $\theta$ on the circle $S^1$, which gives a
$\mathbb{Z}$-action on $S^1$. The action defines the crossed
product of $C^\infty(S^1)$ by $\mathbb{Z}$ which is called the
irrational rotation algebra (see \cite{Ri}, \cite{Bl}). Thus a
trivial line bundle on ${X}_\tau$ naturally defines  the rotation
algebra through $\theta$. This completes the proof.
\end{proof}

The rotation algebra considered in the proof of Theorem 4.3.1 is
Morita equivalent to the foliation $C^*$-algebra for
$\widehat{X}_{\tau,\theta^{-1}}$ (see \cite{Co} for details). Thus
we may conclude the following:

\begin{corollary}
The foliation $C^*$-algebra for the
$\theta^{-1}$-linear foliation is a mirror reflection of the
algebra of functions on $T^2_{\theta,\tau}$. Equivalently, the
foliated, complex torus $\widehat{X}_{\tau,\theta^{-1}}$ is a
mirror reflection of the noncommutative torus
$T^2_{\theta,\tau}$.
\end{corollary}

On the other hand, for any $g\in \text{SL}_2(\mathbb{Z})$, the
leaves of the $\theta^{-1}$-linear foliation rotate the special
Lagrangian cycle $\mathcal{L}_g$ in a different angle $\theta'$
from $\theta$. Thus the leaf action on $\mathcal{L}_g$ defines
another noncommutative torus $A_{\theta'}$ and two irrational
rotation algebras are related by a strongly Morita equivalence
(see \cite{Co}), in other words,
$A_{\theta'}\cong\text{End}_{A_\theta}(E_g(\theta))$. Furthermore,
two deformation parameters $\theta$ and $\theta'$ are in the same
orbit of $\text{SL}_2(\mathbb{Z})$-action;
$\theta'=g\theta=\frac{a\theta+b}{c\theta+d}$. Thus we can extend
Corollary 4.3.2. to a more general foliated manifolds:

\begin{corollary}
The foliated, complex torus
$\widehat{X}_{\tau,\theta'}$ is a mirror reflection of the
noncommutative torus $T^2_{\theta,\tau}$, where $\theta'=g\theta$
for $g\in {\rm SL}_2(\mathbb{Z})$.
\end{corollary}

\begin{remark}
It was suggested in \cite{Fu} that the
relation of $C^*$-algebra of foliation and a $C^*$-algebra
obtained from a deformation quantization of a torus can be
regarded as a mirror symmetry. From a physical point of view, it
was argued in \cite{Kaj} that those two $C^*$-algebras are related
by the T-duality, which is equivalent to the mirror symmetry on
tori. Thus, our results, Theorem 4.3.1. and its Corollaries, may
be considered as a mathematical interpretation of \cite{Kaj}.
\end{remark}

Finally, we show that a standard holomorphic bundle
$E^z_g(\theta)$ on $T^2_{\theta,\tau}$ is obtained geometrically,
using the $\theta^{-1}$-linear foliation structure of
$\widehat{X}_{\tau,\theta^{-1}}$. Our argument here is essentially
based on \cite{Co}. For a matrix $g=\left(\smallmatrix a&b\\c&d
\endsmallmatrix\right)\in
\text{SL}_2(\mathbb{Z})$, the finitely generated projective
$A_\theta$-module $E_g(\theta)=\mathcal{S}(\mathbb{R})\otimes
\mathbb{C}^c$ gives a strong Morita equivalence between $A_\theta$
and $A_{g\theta}$. As we have shown above, the noncommutative tori
$A_\theta$ and $A_{g\theta}$ correspond to special Lagrangian
cycles $\mathcal{L}_I$, $I=\left(\smallmatrix 1&0\\0&1
\endsmallmatrix\right)$,
and $\mathcal{L}_g$, respectively. Since the cycle $\mathcal{L}_I$
is represented by the line $y=0$ on
$\widehat{X}_\tau=\mathbb{R}^2/\mathbb{Z}^2$, the leaves of
$\theta^{-1}$-linear foliation on $\widehat{X}_\tau$ are the lines
given by
\[\{(\theta t+x,t)\mid t\in\mathbb{R}\}.\] On the other hand, the
special Lagrangian cycle $\mathcal{L}_g$ is the line
\[\mathcal{L}_g=\left\{(\frac{d}{c}t+\alpha,t)\mid
\alpha\in\mathbb{R}/\mathbb{Z}, \ t\in\mathbb{R}\right\}.\] Then
the space of leaves starting at $\mathcal{L}_I$ and ending at
$\mathcal{L}g$ is determined by the equation
\begin{equation}\label{deter}
\frac{d}{c}t+\alpha=\theta t+x, \ \text{ or } \
\left(\frac{d}{c}-\theta\right)t=x-\alpha \ \ \text{ mod } 1.
\end{equation}
Let $\mathcal{E}_g=\{((x,0),t)\in \widehat{X}_\tau\times
\mathbb{R}\mid (\frac{d}{c}-\theta)t=x-\alpha \ \ \text{ mod } 1
\}$. Since the line $\mathcal{L}_g$ cuts the line $\mathcal{L}_I$
$c$-times, one finds that the manifold $\mathcal{E}_g$ is the
disjoint union of $c$-copies of $\mathbb{R}$, i.e.,
$\mathcal{E}_g=\mathbb{R}\times (\mathbb{Z}/c\mathbb{Z})$. Then
the algebra of compactly supported smooth functions on
$\mathcal{E}_g$ is
$C_c^\infty(\mathcal{E}_g)=\mathcal{S}(\mathbb{R}\times
\mathbb{Z}/c\mathbb{Z})=\mathcal{S}(\mathbb{R})\otimes\mathbb{C}^c$.
Let $W_1$ and $W_2$ be unitary operators on $\mathbb{C}^c$ such
that $W_1^c=W_2^c=1$ and $W_1W_2=\exp(2\pi i\frac{d}{c})W_2W_1$.
These operators reflect the structure of a stable bundle on
$X_\tau$ ({\it cf.} \cite{hooft}). Associated to the equation
(\ref{deter}), we define operators $V_1$ and $V_2$ on
$\mathcal{S}(\mathbb{R})$ by
\begin{align}
(V_1f)(t)&=\exp(2\pi \alpha)\exp(2\pi
i(\frac{d}{c}-\theta)t)f(t),\label{e1}
\end{align}
\begin{align}
(V_2f)(t)&=f(t+1)\label{e2}.
\end{align}
Then $U_i=V_i\otimes W_i$, ($i=1,2$), gives a left module action
on $C_c^\infty(\mathcal{E}_g)$. The module action (\ref{e1},
\ref{e2}) also determine a constant curvature connection
$\nabla_1$, $\nabla_2$ by the formula given in (\ref{rcon1},
\ref{rcon2}), where the translation of $\nabla_2$ is determined by
the monodromy data or a complex phase $\exp(2\pi i\beta)$,
$\beta\in\mathbb{R}/\mathbb{Z}$, given in (\ref{conne}). Thus we
see that the special Lagrangian cycle $\mathcal{L}_g$ determines a
basic module $E_g(\theta)=C_c^\infty(\mathcal{E}_g)$ equipped with
a constant curvature connection. The monodromy data specified in
(\ref{rcon2}) determines the holomorphic structure on
$E_g(\theta)$ by  $\overline{\nabla}=\tau\nabla_1+\nabla_2$. This
completes the geometric construction of $E_g^z(\theta)$, where
$z=\tau\alpha+\beta$. From this construction we get the following,
which can be interpreted as a noncommutative version of
Proposition 4.2.2.

\begin{proposition}
For a matrix $g\in {\rm
SL}_2(\mathbb{Z})$, let $\mathcal{SM}_g(\theta)$ be the moduli
space of compact transversals for $\theta^{-1}$-linear foliation
on $\widehat{X}_\tau$ together with the monodromy around the
circles $($compact transversals$)$ which is given by a complex
phase $\exp(2\pi i\beta)$, $\beta\in\mathbb{R}/\mathbb{Z}$. Then
we have
\[ \mathcal{SM}_g(\theta)\cong \mathcal{M}_g^s(\theta). \]
\end{proposition}

\begin{remark}
The moduli space $\mathcal{SM}_g(\theta)$
given in Proposition 4.3.5 is essentially the same as the moduli
space of supercycles $\mathcal{SM}_g$ given in Proposition 4.2.2.
Thus Proposition 4.3.5 can be seen as a geometric construction of
the identification of (\ref{equiv}) stated in Remark 4.2.3.
\end{remark}



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\end{document}


