Homological Mirror Symmetry for Toric del Pezzo Surfaces
Kazushi Ueda
We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus $(\bC^\times)^2$. We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.
toricdp.tex
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\title{Homological Mirror Symmetry for Toric del Pezzo Surfaces}
\author{Kazushi Ueda}
\date{}
\pagestyle{plain}

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\begin{document}

\maketitle

\begin{abstract}

We prove the homological mirror conjecture for toric del Pezzo surfaces.
In this case, the mirror object is a regular function
on an algebraic torus $(\bC^\times)^2$.
We show that the derived Fukaya category of this mirror
coincides with the derived category of coherent sheaves
on the original manifold.

\end{abstract}

\section{Introduction}

Mirror symmetry
started as a mysterious relationship
between complex geometry of a Calabi-Yau 3-fold
and symplectic geometry of another Calabi-Yau 3-fold
called the {\em mirror manifold}.
%See e.g. \cite{Cox-Katz} for a general background
%on mirror symmetry.
In 1994, Kontsevich \cite{Kontsevich_HAMS} proposed
a program to understand various mirror phenomena
as a consequence of the following
{\em homological mirror conjecture}:
Calabi-Yau manifolds always come in pairs
in such a way that
the derived category of coherent sheaves
on one manifold is equivalent
as a triangulated category
to the derived Fukaya category of the other.

Although mirror symmetry was first discovered
for Calabi-Yau manifolds,
there are also variants of these phenomena
for other classes of manifolds.
One such example is the Givental's theorem
\cite{Givental_HGMS}
giving integral representations of
$J$-functions
of toric Fano manifolds.
See also \cite{Batyrev_QCRTM}, \cite{Witten_PN2T2D},
\cite{Eguchi-Hori-Xiong} and \cite{Hori-Vafa}.

We can also formulate the homological mirror conjecture
for toric Fano manifolds.
Let $X$ be a toric Fano manifold of dimension $n$.
Then the mirror partner
is
a regular function $W$
on
an algebraic torus $(\bC^\times)^n$
of the same dimension
equipped with a symplectic structure
(along with an additional data
called a {\em grading}).
Here, the function $W$ is a Newton polynomial
for the convex hull of the generators
of 1-dimensional cone of the fan of $X$.
Coefficients of this polynomial do not matter
as long as they are chosen general enough.
By Kouchnirenko \cite{Kouchnirenko},
$W$ has exactly $\dim H^*(X,\bC)$ critical points.
Take a regular value $t$ of $W$ and
a distinguished basis of
vanishing cycles in $W^{-1}(t)$.
We also have to choose
a {\em grading} and
a {\em spin structure}
on each of these vanishing cycles.
The {\em directed Fukaya category}
$\FukW$
of $W$
(along with the choice of gradings
and spin structures)
is an $A_\infty$-category
whose objects are vanishing cycles
and whose morphisms are Floer complexes.
Roughly speaking, the Floer complex
between two vanishing cycles
are the vector space
spanned by intersection points
between them,
and the compositions of morphisms
are given by ``counting polygons.''
By Seidel \cite{Seidel_VC},
the derived category
$D^b \FukW$ of
$\FukW$
is independent of the choice
of a distinguished basis
of vanishing cycles.

The following is the main result of this paper:

\begin{theorem} \label{th:hms}
The derived category of coherent sheaves
on a toric del Pezzo surface $X$
is equivalent as a triangulated category
to the derived category of
the directed Fukaya category
of the mirror $W$ of $X$;
$$
 D^b \coh(X) \cong D^b \FukW.
$$
\end{theorem}

Our proof is based on
an explicit computation.
The above Theorem \ref{th:hms}
extends the work of
Seidel \cite{Seidel_VC2}
and 
Auroux, Katzarkov and Orlov
\cite{Auroux-Katzarkov-Orlov},
where the cases of
$\bP^2$ and $\bP^1 \times \bP^1$,
and
$\bP^2$ blown-up at one point
is treated.
See also the paper
by Hori, Iqbal and Vafa
\cite{Hori-Iqbal-Vafa},
where mirror symmetry
for Fano manifolds
are discussed
from a physics point of view.

{\bf Acknowledgements}:
We thank O. Fujino, K. Fukaya,
H. Iritani, T. Kawai,
K. Saito and A. Takahashi
for valuable discussions and comments.
The author is supported by
JSPS Fellowships for Young Scientists
No.15-5561.

\section{Derived category of coherent sheaves}

We describe the structure
of the bounded derived category $\DbdP$
of coherent sheaves
on $\dP$ in this section,
where $\dP$ is the projective plane
blown-up
at three points $p_1$, $p_2$ and $p_3$
in general position.
Let $\phi : \dP \rightarrow \bP^2$ be this blow-up
and 
$E_1$, $E_2$ and $E_3$
be the exceptional divisors
corresponding to
$p_1$, $p_2$ and $p_3$ respectively.
It is a toric surface and
the generators of one-dimensional cones
of its fan is drawn
in Figure \ref{fg:dp3}.
%Then we describe the relation between
%$\DbdP$ and the derived category of
%coherent sheaves
%on other toric del Pezzo surfaces.

\begin{figure}[H]
\centering
\psfrag{v1}{$v_1$}
\psfrag{v2}{$v_2$}
\psfrag{v3}{$v_3$}
\psfrag{v4}{$v_4$}
\psfrag{v5}{$v_5$}
\psfrag{v6}{$v_6$}
\psfrag{vequalssomething}{$\begin{array}{rcl}
 v_1 & = & (   1   ,   0   ), \\
 v_2 & = & (   0   ,   1   ), \\
 v_3 & = & (   -1  ,  -1   ), \\
 v_4 & = & (   -1  ,   0   ), \\
 v_5 & = & (   0   ,  -1   ), \\
 v_6 & = & (   1   ,   1   ).
	     \end{array}$}
\includegraphics{dp3.eps}
\caption{the toric data of $\dP$}
\label{fg:dp3}
\end{figure}

\begin{definition} \label{def:exceptional_collection}
\begin{enumerate}
\item 
 An object $\scE$ in a triangulated category
 is exceptional if
  $$
   \Ext^i(\scE,\scE)=\left\{\begin{array}{cl}
      \bC & \mbox{if $i=0$,}\\
       0  & \mbox{otherwise.}\\
   \end{array}\right.
  $$
\item
 An ordered set of objects $(\scE_i)_{i=1}^N$
 in a triangulated category
 is an exceptional collection
 if each $\scE_i$ is exceptional and
 $\Ext^k(\scE_i,\scE_j)=0$ for any $i>j$
 and for any $k$.
\end{enumerate}

\end{definition}

By combining theorems of Beilinson \cite{Beilinson} and
Orlov \cite{Orlov},
we have the following generators
of $\DbdP$:

\begin{theorem}
Let
\begin{eqnarray*}
\scC & = &
( \scE_1, \scE_2, \scE_3, \scE_4, \scE_5, \scE_6 )
\end{eqnarray*}
where
\begin{equation} \label{eq:ec}
 \begin{array}{rclrclrcl}
  \scE_1 & = & \scO_{E_1}(-1)[-1], &
  \scE_2 & = & \scO_{E_2}(-1)[-1], &
  \scE_3 & = & \scO_{E_3}(-1)[-1], \\
  \scE_4 & = & \econe, & 
  \scE_5 & = & \ectwo, &
  \scE_6 & = & \ecthree. \\
 \end{array}
\end{equation}
Then $\scC$ is an exceptional collection
generating $\DbdP$.
\end{theorem}
Here, $\scO_{E_i}(-1)$ is the sheaf supported on
$E_i$ and
is isomorphic to the tautological sheaf $\scO_{\bP^1}(-1)$
on $E_i \cong \bP^1$,
$[\bullet]$ is the shift operator in the derived category,
$\Lphi$ is the derived pull-back,
$\scO_{\bP^2}(-1)$ is the tautological sheaf
on $\bP^2$,
and $\Omega_{\bP^2}(1)$ is the cotangent sheaf of $\bP^2$
tensored with the hyperplane sheaf
$\scO_{\bP^2}(1)$.
%Note that $\scO_{\bP^2}(-1)$ is the exterior power
%$\Omega^2_{\bP^2}(2)$ of
%$\Omega_{\bP^2}(1)$.
%$N$ is six in our case.
%Since $\scC$ generates $\DbdP$,
%its categorical structure
%is wholly determined
%by the structure of morphisms
%between objects
%in $\scC$.

\begin{proposition}
All the non-zero $\Ext$-groups
within the exceptional collection $\scC$ are
% \begin{eqnarray*}
%  \Hom(\scO_{E_i}(-1), \econe) 
%   & = & \bC p_i, \\
%  \Hom(\scO_{E_i}(-1), \ectwo)
%   & = & \Ker(V^\vee \rightarrow (\bC p_i)^\vee), \\
%  \Hom(\scO_{E_i}(-1), \ecthree)
%   & = & \bC, \\
%  \Hom(\econe, \ectwo)
%   & = & \Lambda^2 V^\vee, \\
%  \Hom(\econe, \ecthree)
%   & = & V^\vee, \\
%  \Hom(\ectwo, \ecthree)
%   & = & V,
% \end{eqnarray*}
 \begin{eqnarray*}
  \Hom(\scE_i, \scE_4)
   & = & \bC p_i, \\
  \Hom(\scE_i, \scE_5)
   & = & \Ker(V^\vee \rightarrow (\bC p_i)^\vee), \\
  \Hom(\scE_i, \scE_6)
   & = & \bC, \\
  \Hom(\scE_4, \scE_5)
   & = & \Lambda^2 V^\vee, \\
  \Hom(\scE_4, \scE_6)
   & = & V^\vee, \\
  \Hom(\scE_5, \scE_6)
   & = & V,
 \end{eqnarray*}
where $i=1,2,3$,
$V \cong \bC^3$ is the three-dimensional vector space such that
$\bP^2 = \bP(V)$,
the check denotes the dual vector space,
$\bC p_i \subset V$ is the one-dimensional subspace
corresponding to $p_i \in \bP(V)$,
and the map $V^\vee \rightarrow (\bC p_i)^\vee$
is the dual of the inclusion
$\bC p_i \hookrightarrow V$.
Compositions of morphisms
are given by
$$
\begin{array}{ccc}
 \Hom(\scE_i, \eefour) \times \Hom(\eefour, \eefive)
   & \longrightarrow & \Hom(\scE_i, \eefive)\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (v, \omega ) & \longmapsto & -\iota_v \omega
\end{array},
$$
$$
\begin{array}{ccc}
 \Hom(\scE_i, \eefour) \times \Hom(\eefour, \eesix)
   & \longrightarrow & \Hom(\scE_i, \eesix)\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (v, \omega ) & \longmapsto & \omega(v)
\end{array},
$$
$$
\begin{array}{ccc}
 \Hom(\scE_i, \eefive) \times \Hom(\eefive, \eesix)
   & \longrightarrow & \Hom(\scE_i, \eesix)\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (\omega, v ) & \longmapsto & \omega(v)
\end{array},
$$
$$
\begin{array}{ccc}
 \Hom(\eefour, \eefive) \times \Hom(\eefive, \eesix)
   & \longrightarrow & \Hom(\eefour, \eesix)\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (\omega, v ) & \longmapsto & \iota_v \omega
\end{array}
$$
where $\iota_v : \wedge^2 V^\vee \rightarrow V^\vee$
for $v \in V$ is the interior product.
\end{proposition}

\proof
All the computations reduce to those on $\bP^2$
in the following way:
Let $\scE$ and $\scF$ be vector bundles
on $\bP^2$. Then
\begin{eqnarray*}
 \RHom(\Lphi \scE, \Lphi \scF)
 & = & \RHom(\scE, \Rphi \Lphi \scF) \\
 & = & \RHom(\scE, \scF \otimes \Rphi \scO_\dP) \\
 & = & \RHom(\scE, \scF),
\end{eqnarray*}
\begin{eqnarray*}
 \RHom(\Lphi \scE, \scO_{E_i}(-1))
 & = & \RHom(\scE, \Rphi \scO_{E_i}(-1)) \\
 & = & 0,
\end{eqnarray*}
and
\begin{eqnarray*}
\RHom(\scO_{E_i}(-1)[-1], \Lphi \scE)
 & = & \RHom( \{ \scO_\dP \rightarrow \scO_\dP(E_i) \},
   \Lphi \scE ) \\
 & = & \RGamma( \{ \scO_\dP(-E_i) \rightarrow \scO_\dP \} \otimes
   \Lphi \scE ) \\
 & = & \RGamma( \{ \Rphi \scO_\dP(-E_i) \rightarrow
   \Rphi \scO_\dP \} \otimes \scE ) \\
 & = & \RGamma( \{ \scI_{p_i} \rightarrow \scO_{\bP^2} \}
   \otimes \scE ) \\
 & = & \RGamma( \scO_{p_i} \otimes \scE ) \\
 & = & \scE|_{p_i},
\end{eqnarray*}
where
%$\iota : p_i \rightarrow \bP^2$ is the inclusion,
$\scI_{p_i}$ is the ideal sheaf of $p_i$
and the last line is the fiber at $p_i$.
Here, we have used the exact sequence
$$
0 \rightarrow \scO_\dP \rightarrow \scO_\dP(E_i)
  \rightarrow \scO_{E_i}(-1) \rightarrow 0.
$$
We can further use the exact sequence
$$
0 \rightarrow \Omega_{\bP^2}(1) \rightarrow V^\vee \otimes \scO_{\bP^2}
 \rightarrow \scO_{\bP^2}(1) \rightarrow 0
$$
to reduce the computations involving $\Omega_{\bP^2}(1)$
to those involving $\scO_{\bP^2}(i)$, $i \in \bZ$:
\begin{eqnarray*}
\RHom(\scO_{E_i}(-1)[-1], \ectwo)
 & = & \RGamma( \scO_{p_i} \otimes
         \{V^\vee \otimes \scO_{\bP^2}
                \rightarrow \scO_{\bP^2}(1) \} ) \\
 & = & \{V^\vee \rightarrow (\bC p_i)^\vee \},
\end{eqnarray*}
\begin{eqnarray*}
\RHom(\econe, \ectwo)
 & = & \RHom( \scO_{\bP^2}(-1),
         \{ V^\vee \otimes \scO_{\bP^2}
                 \rightarrow \scO_{\bP^2}(1) \} ) \\
 & = & \RGamma( \scO_{\bP^2}(1) \otimes
         \{ V^\vee \otimes \scO_{\bP^2}
                 \rightarrow \scO_{\bP^2}(1) \} ) \\
 & = & \RGamma( \{ V^\vee \otimes \scO_{\bP^2}(1)
                 \rightarrow \scO_{\bP^2}(2) \} ) \\
 & = & \{V^\vee \otimes V^\vee \rightarrow \Sym^2 V^\vee \} \\
 & = & \wedge^2 V^\vee,
\end{eqnarray*}
\begin{eqnarray*}
\RHom(\ectwo, \ecthree)
 & = & \RHom( \{ V^\vee \otimes \scO_{\bP^2}
                 \rightarrow \scO_{\bP^2}(1) \},
               \scO_{\bP^2} ) \\
 & = & \RGamma( \{ \scO_{\bP^2}(-1) \rightarrow
                  V \otimes \scO_{\bP^2} \} ) \\
 & = & V.
\end{eqnarray*}
Compositions of morphisms can be
easily read off
from the above computations.
%Compositions involving sheaves
%supported on an exceptional divisor
%is just the restriction of global Hom's
%between vector bundles on $\bP^2$
%to the fiber above the center of the blow-up:
%The composition
%$$
% \Hom(\scO_{E_i}(-1), \Lphi \scE) \times
% \Hom(\Lphi \scE, \Lphi \scF)
%  \rightarrow \Hom(\scO_{E_i}(-1), \Lphi \scF)
%$$
%gives, for any $\varphi \in \Hom(\scE, \scF)
%\cong \Hom(\Lphi \scE, \Lphi \scF)$,
%a homomorphism
%$$
%\Hom(\scO_{E_i}(-1), \Lphi \scE)
%  = \scE |_{p_i}
% \rightarrow \scF |_{p_i}
%  = \Hom(\scO_{E_i}(-1), \Lphi \scF)
%$$
%which is just the restriction of
%$\Hom(\scE, \scF)$
%to the fiber over $p_i$, and
%the composition
%$$
%\Hom(\econe,\ectwo) \times \Hom(\ectwo, \ecthree)
% \rightarrow \Hom(\econe, \ecthree)
%$$
%is given by the interior product
%$\Lambda^2 V^\vee \times V \rightarrow V^\vee.$
\qed

\section{Fukaya category} \label{Fuk}

First we recall the definition of an $A_\infty$-category.
For a $\bZ$-graded vector space $V$
and $i \in \bZ$,
$V[i]$ denotes the shift of $V$ by $i$;
$V[i]^j=V^{i+j}.$

\begin{definition} \label{def:a_infinity}
An $A_\infty$-category $\scA$ consists of
\begin{enumerate}
\item 
the set of objects $\Ob(\scA)$,
\item
for any $c_1,\; c_2 \in \Ob(\scA)$,
a $\bZ$-graded $\bC$-vector space $\hom_{\scA}(c_1, c_2)$
called
the set of morphisms,
\item
for any positive integer $k$
and for any set of objects $\{c_i\}_{i=0}^k$,
the composition
$$\
 \m_k : \hom_{\scA}(c_0,c_1)[1]\otimes \cdots
                \otimes \hom_{\scA}(c_{k-1},c_k)[1]
                   \longrightarrow \hom_{\scA}(c_0,c_k)[1]
$$
which is a linear map
of degree $1$
\end{enumerate}
such that
for any positive integer $k$,
any set of objects $\{c_i\}_{i=0}^k$
and any set of morphisms $\{a_i\}_{i=1}^k$,
$a_i \in \hom_{\scA}(c_{i-1},c_i)$,
the following {\em $A_\infty$-relations} holds:
\begin{eqnarray*} \label{eq:a_infinity}
 \sum_{i=0}^{k-1} \sum_{j=i+1}^k
  (-1)^{\deg a_1 + \cdots + \deg a_i}
  \m_{i-j+k+1}(a_1 \otimes \cdots \otimes a_i \\
   \otimes 
    \m_{j-i}(a_{i+1} \otimes \cdots \otimes a_j)
   \otimes
    a_{j+1} \otimes \cdots \otimes a_k ) = 0.
\end{eqnarray*}
Here degrees are counted after shifts, i.e., if $a \in V^i$,
$\deg a = i-1$ in $V[1]$.
Since $\m_1^2=0$, we define
$$
 \Hom_{\scA}(c_1,c_2) = H^0(\hom_{\scA}(c_1,c_2),\m_1).
$$
\end{definition}

%\begin{definition}
%Let $\scA$ be an $A_\infty$-category
%and $(c_1, \ldots, c_N)$ be a finite ordered subset
%of $\Ob(\scA)$ such that $\hom_{\scA}(c_i,c_j)$ is
%finite-dimensional for $i<j$.
%The {\em directed $A_\infty$-category} $\scA^{\rightarrow}$
%associated to $(c_1, \ldots, c_N)$
%is an $A_\infty$-category such that
%$$
% \Ob(\scA^{\rightarrow}) = (c_1, \ldots, c_N)
%$$
%and whose morphisms satisfy
%$$
% \hom_{\scA^{\rightarrow}}(c_i,c_j) = \left\{
%  \begin{array}{lc}
%   0 & i>j, \\
%   \bC \cdot \id_{c_i} & i=j, \\
%   \hom_{\scA}(c_i,c_j) & i<j
%  \end{array}
%  \right.
%$$
%and compositions satisfy
%$$
%\m^{\scA^\rightarrow}_k(a_1,\ldots,a_k) = \m^\scA_k(a_1,\ldots,a_k)
%$$
%where $a_j \in \hom_\scA(c_{i_j-1},c_{i_j})$ for $i_o<\cdots<i_k$
%and $\id_{c_i}$ is the strict unit,
%i.e., $\m_k$ involving $\id_{c_i}$ for $k \neq 2$ is zero
%and $\id_{c_i}$ is the unit of $\m_2$.
%\end{definition}

The Fukaya category
of Lagrangian submanifolds
in a symplectic manifold
is defined in
\cite{Fukaya-Oh-Ohta-Ono}.
We use the following adaptation
for exact Morse fibrations
by Seidel \cite{Seidel_VC}.
Let $W: Z \rightarrow \bC$
be a regular function
on an affine algebraic manifold $Z$
of complex dimension $n$
with a K\"{a}hler structure.
%We specialize to the case
%of $\dim Z=2$
%to simplify the exposition.
Assume the following conditions:
\begin{itemize}
 \item The K\"{a}hler metric is complete.
 \item The symplectic form $\omega$
  of $Z$ is exact,
  i.e., there exists a one form $\theta$ on $Z$
  such that $\omega = d \theta$.
 \item At any critical points of $W$,
  the Hessian of $W$ is non-degenerate.
 \item All the critical values are distinct.
\end{itemize}
Such $W$ gives rise to an exact Morse fibration
in the terminology of \cite{Seidel_VC}.
Using the K\"{a}hler structure,
we can define the lift
$\widetilde{c}_p : [0,1] \rightarrow Z$
of a path 
$c : [0,1] \rightarrow \bC$
starting from a point $p \in Z$
such that $W(p) = c(0)$
by using the horizontal distribution
defined as the orthogonal complement
of the tangent space
along the fiber of $W$.

Assume that the origin is
a regular value of $W$ and
fix an order $(p_i)_{i=1}^N$
on the set of critical points of $W$.
A distinguished set $(c_i)_{i=1}^N$
of vanishing paths is a set of smooth paths
$c_i : [0,1] \rightarrow \bC$
satisfying
 \begin{enumerate}
  \item $c_i(0) = 0$, $c_i(1) = W(p_i)$,
  \item $c_i$ has no self-intersection,
  \item images of $c_i$ and $c_j$ intersects only at the origin,
  \item $c_i'(0) \neq 0$, and
  \item $\arg c_{i+1}'(0) < \arg c_i'(0)$, $i=1,\ldots N-1$,
         for some choice of the branch of $\arg(\bullet)$.
 \end{enumerate}
Given a distinguished set
$(c_i)_{i=1}^N$ of vanishing paths,
%from zero to the critical values of $W$,
the corresponding vanishing cycles
$(C_i)_{i=1}^N$
are defined by
$$
 C_i = \{p \in W^{-1}(0) \suchthat
           \lim_{t \rightarrow 1} \widetilde{c}_p(t) = p_i\}.
$$
They are Lagrangian submanifolds
of $W^{-1}(0)$.

The {\em directed Fukaya category} $\FukW$
for $W$ is roughly an $A_\infty$-category
whose objects are vanishing cycles
and whose morphisms are Lagrangian intersection
Floer complexes.
To define a $\bZ$-grading
on the Floer complex,
we need the concept of {\em grading}
on Lagrangian submanifolds
introduced by Kontsevich \cite{Kontsevich_HAMS},
which we now recall.
See also Seidel \cite{S}.

Let $(M,\omega)$ be a symplectic manifold
of dimension $2n$.
An almost complex structure on $M$
is a section $J \in \Gamma(M, \End(T M))$
such that $J^2 = -\id$.
$J$ is called $\omega$-compatible
if $g(V_1,V_2) = \omega(V_1, J V_2)$
%for two vector fields $V_1$ and $V_2$
defines a Hermitian metric on the tangent bundle.
Fix an $\omega$-compatible almost complex structure $J$
of $M$ and
let $\scS$ be the principal $U(1)$-bundle
associated to the complex line bundle
$(\Lambda^n (T^*M, J))^{\otimes 2}$.
The fiber of $\scS$ at $p \in M$ is
$(\Lambda^n (T_p^* M, J))^{\otimes 2} / \bR^{> 0}$.
We assume that the first Chern class
of this complex line bundle vanishes,
so that $\scS$ has a section.
A grading of $M$ is
a choice of a section $\Theta : M \rightarrow \scS$.
Fix a grading $\Theta$ on $M$.
Let $\Lag_M \rightarrow M$ be
the Lagrangian Grassmannian bundle on $M$,
whose fiber at $p \in M$
is the Grassmannian of Lagrangian subspaces
in the symplectic vector space $T_p M$.
Define $\det_\Theta^2 : \Lag_M \rightarrow U(1)$ as follows:
For a Lagrangian subspace $L \subset T_p M$,
pick any basis $\{e_i\}_{i=1}^n$ of $L$
and take the square of their exterior product;
$(e_1 \wedge e_2 \wedge \cdots \wedge e_n)^{\otimes 2}
\in (\Lambda^n (T_p M, J))^{\otimes 2}$.
$\det_\Theta^2(L)$ is the image of this element by $\Theta(p)$.
A Lagrangian submanifold $L \subset M$
gives a canonical section
$s_L : L \ni p \mapsto T_p L \in \Lag_M|_p$.
Denote the composition of $s_L$ and $\det_\Theta^2$ by $\phi_L$.
A grading of a Lagrangian submanifold
is a lift $\widetilde{\phi_L} : L \rightarrow \bR$
of $\phi_L$ to the universal cover $\bR$ of
$U(1) \cong \bR/\bZ$.

Now we define the Maslov index.
A smooth path
$\Lambda : [0,1] \rightarrow \Lag(V,\omega)$
in the Lagrangian Grassmannian
of a fixed symplectic vector space $(V, \omega)$
of dimension $2 n$
is called {\em crossingless}
if $\Lambda(0) \cap \Lambda(t) = \Lambda(0) \cap \Lambda(1)$
for all $t \in (0,1]$.
For a crossingless path $\Lambda_t$,
its differential $\Lambda'(0)$ at $t=0$
gives an element of the tangent space
$T_{\Lambda(0)} \Lag(V,\omega) \subset T_{\Lambda(0)} \Gr(n,V)
  \cong \Hom(\Lambda(0),V/\Lambda(0))$
where $\Gr(n,V)$ is the Grassmannian of $n$-dimensional
subspaces in $V$.
The composition of $\Lambda'(0)$ and $\omega$
defines a quadratic form
$\Lambda(0) \ni v \mapsto \omega(v, \Lambda'(0) v)$
on $\Lambda(0)$,
which descends to a quadratic form on 
$\Lambda(0)/\Lambda(0) \cap \Lambda(1)$
since it vanishes on $\Lambda(0) \cap \Lambda(1)$.
The resulting form on $\Lambda(0)/\Lambda(0) \cap \Lambda(1)$
is called the {\em crossing form} \cite{Robbin-Salamon}.
For an intersection $p \in L_1 \cup L_2$
of two graded Lagrangian submanifolds
$(L_1, \widetilde{\phi_{L_1}})$ and $(L_2, \widetilde{\phi_{L_2}})$,
its Maslov index is defined as follows:
Choose a crossingless path
$\Lambda : [0,1] \rightarrow \Lag_M|_p$
from $T_p L_0$ to $T_p L_1$
such that the corresponding crossing form
at $t=0$ is negative definite.
There is a unique lift 
$\widetilde{\alpha} : [0,1] \rightarrow \bR$
of the composition $\alpha$ of
$\Lambda : [0,1] \rightarrow \Lag_M|_p$ and
$\det_\Theta^2(p) : \Lag_M|_p \rightarrow U(1)$
such that $\widetilde{\alpha}(0)=\widetilde{\phi_{L_0}}(p)$,
and the Maslov index $I(p)$ of the intersection
$p$ of two graded Lagrangian submanifolds
is defined by
$$
 I(p) = \widetilde{\phi_{L_1}}(p) - \widetilde{\alpha}(1).
$$

%We also need to choose a {\em spin structure}
%\cite{Fukaya-Oh-Ohta-Ono}
%which, in the case when the Lagrangian submanifold
%is one-dimensional,
%is equivalent to a choice of a two-fold cover
%of the Lagrangian submanifold \cite{Seidel_K3}.
%A choice of spin structures enables us
%to choose an orientation
%on the moduli space of pseudoholomorphic maps
%from a disk to the symplectic manifold
%whose boundaries are mapped into
%Lagrangian submanifolds.
%To define this, we first need
%an element $\st \in H^2(M, \bZ/ 2 \bZ)$ \ such that its
%restriction to $L$ coincides with the second Stiefel-Whitney
%class of $L$.
%When we have several Lagrangian submanifolds,
%$\st$ must be chosen
%simultaneously
%for all the Lagrangian submanifolds.
%Then, we will choose a simplicial decomposition of $M$ which
%is an extention of a simplicial decomposition of $L$, and
%choose a real vector bundle $V$ on the 3-skelton $M^{(3)}$ whose
%second Stiefel-Whitney class coincides with $\st$.
%Then it follows that the vector bundle $V \oplus TL$
%on $M^{(3)} \cap L$ is trivial, and a relative spin
%structure $\$_L$ of $L$ is a trivialization of this bundle.
%A relative spin structure
%(and some additional data which corresponds
%to a choice of orientation of the stable manifold
%in the finite-dimensional Morse theory)
%determines
%an orientaion
%on the moduli space of pseudoholomorphic maps.
%We abbreviate the triple
%$(L, \widetilde{\phi_L}, \$_{L})$
%by $L^\flat$.
%
%Finally, when the symplectic manifold $M$ is non-compact,
%we need a convexity assumpotion.
%For our purpose, the following condition is sufficient
%\cite{Eliashberg-Gromov}:
%There exists a compatible almost complex structre $J$
%and a sequence
%$\{M_i\}_{i=1}^\infty$ of compact submanifolds
%$M_i \in M$
%with pseudo-convex boundaries
%such that $M_i \subset M_{i+1}$
%and $\cup_i M_i = M$.
%In such a case,
%maximum principle prevents
%$J$-holomorphic disks
%from running away to infinity.

Now we come back to our exact Morse fibration $W$.
A relative Maslov map
is a section $\Theta$
of the second tensor power
$(\Omega_{Z / \bC}^{n-1})^{\otimes 2}$
of the top exterior product
of the relative cotangent bundle
away from the critical points.
Since $0$ is a regular value,
the restriction of $\Theta$
to $W^{-1}(0)$ gives a grading
of $W^{-1}(0)$.
Assume that
all the vanishing cycles can be graded,
and fix a grading on each vanishing cycle.
We also need to choose a spin structure
on each vanishing cycles
in order to orient
the moduli spaces of
pseudoholomorphic maps
\cite{Fukaya-Oh-Ohta-Ono}.
Since
each vanishing cycle is homeomorphic
to a sphere,
the choice of a spin structure
is unique
except for $\dim_\bC W^{-1}(0) = 1$,
where one has as many as
$H^1(S^1,\bZ/2 \bZ)=\bZ / 2 \bZ$
choices.
We take the non-trivial spin structure
in such a case.
Let $C_i^\flat$ denote
the vanishing cycle $C_i$
endowed with the above grading and spin structure.
We assume that vanishing cycles intersects each other
transversally.
This condition can always be met
by moving vanishing cycles
within their Hamiltonian isotopy classes
if necessary.

\begin{definition}
Given a function $W$ on an affine K\"{a}hler manifold $Z$
together with a relative Maslov map $\Theta$
and a choice of a distinguished basis
of vanishing cycles
$(C_i^\flat)_{i=1}^N$
with gradings and spin structures,
its directed Fukaya category $\FukW$
is an $A_\infty$-category such that
\begin{itemize}
 \item the set of objects
        is the distinguished basis of vanishing cycles;
        $$ \Ob(\FukW) = (C_1^\flat, \ldots, C_N^\flat),$$
 \item the set of morphisms between
        $C_i^\flat$ and $C_j^\flat$ is
        the $\bZ$-graded vector space
$$
\hFukW(C_i^\flat, C_j^\flat)
 = \left\{
  \begin{array}{lc}
   0 & i>j, \\
   \bC \cdot \id_{C_i^\flat} & i=j, \\
   \bigoplus_{p \in C_i \cap C_j} \vspan_{\bC}\{p\} & i<j
  \end{array}
  \right.
$$
      where $\deg p= I(p)$ (the Maslov index), and
 \item for a positive integer $k$,
        a set of objects
        $(C_{i_0}^\flat,\ldots,C_{i_k}^\flat)$
        and morphisms $p_l \in C_{i_{l-1}} \cap C_{i_l}$
        for $l = 1,\ldots,k$,
        the composition $\m_k$ is given by
$$
 \m_k(p_1,\ldots,p_k) = \sum_{p_0 \in C_{i_0} \cap C_{i_k}}
     \# \M_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k) p_0.
$$
\end{itemize}
\end{definition}
Here, $\M_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
is the stable compactification of \\
$\scM_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
defined as follows:
A disk with $k+1$ marked points
on the boundary
is a pair $(D^2,(z_0,\ldots,z_k))$
of a closed unit disk
$D^2=\{z \in \bC \suchthat |z| \leq 1\}$
and an ordered set
$(z_0,\ldots,z_k)$
of $k+1$ points on the boundary
respecting the cyclic order.
Let $\partial_l D^2 \in \partial D^2$
be the interval
between $z_l$ and $z_{l+1}$,
where we set $z_{k+1} = z_0$.
Fix an almost complex structure $J$ on $W^{-1}(0)$.
A smooth map $\varphi : D^2 \rightarrow M$
is called pseudoholomorphic if
$$
d \varphi \circ J_{D^2} = J \circ d \varphi
$$
where $J_{D^2}$ is the canonical complex structure
on $D^2$.
$\scM_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$ is
the moduli space of pairs
$((D^2,(z_0,\ldots,z_k)),\varphi)$
such that
\begin{enumerate}
 \item $(D^2,(z_0,\ldots,z_k))$ is a disk
   with $k+1$ marked points on the boundary,
 \item $\varphi : D^2 \rightarrow W^{-1}(0)$
   is a pseudoholomorphic map,
 \item $\varphi(\partial_l D^2) \subset C_{i_l}$ for $l=0,\ldots,k$, and
 \item $\varphi(z_l) = p_l$ for $l=0,\ldots,k$.
\end{enumerate}

Although $\M_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
is not an honest manifold in general
due to subtleties in the definition of
the moduli space of pseudoholomorphic maps,
it has a Kuranishi structure with corners
and it makes sense to ``count'' numbers
$\# \M_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
of points.
See \cite{Fukaya-Oh-Ohta-Ono}
for details.
These numbers are counted with signs
determined by the orientation of the moduli space,
which in turn is determined
by the spin structures on vanishing cycles.
$\# \M_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
is zero if \\
$\dim \M_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k) \neq 0$.

These moduli spaces must be compact
so that it makes sense
to count the numbers of points.
To prove compactness of the moduli space,
one needs the boundedness
of the energy.
Usually, this is achieved
by introducing
the Novikov ring
as the coefficient ring
of the Floer complex
in order to control energies
of pseudoholomorphic maps.
This is not necessary in our case
since vanishing cycles are exact Lagrangian submanifolds,
i.e., $[\theta] = 0 \in H^1(C_i,\bR)$
for the primitive $\theta$ of $\omega$;
$\omega = d \theta$.
To see this,
let $C$ be the vanishing cycle
corresponding to a vanishing path
$c : [0,1] \rightarrow \bC$.
Then $C$ bounds the disk
$D = \bigcup_{p \in C} \widetilde{c_p}([0,1])$;
$\partial D = C$.
Since
$d \theta|_D = \omega|_D = 0$,
$[\theta] = 0 \in H^1(\partial D,\bR)$.
Therefore,
there exists a function $K_i$
on each vanishing cycles
such that
$\theta|_{C_i} = d K_i$.
Then for any $\varphi \in
\M_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$,
the energy of $\varphi$ is
\begin{eqnarray*}
 E(\varphi) & = & \int_{D^2} \varphi^*\omega
  =  \int_{\partial D^2} \varphi^* \theta
  =  \sum_{l=0}^k \int_{\partial_i D^2} \varphi^*\theta \\
 &=& \sum_{l=0}^k (K_{i_l}(p_{l+1})- K_{i_l}(p_l))
\end{eqnarray*}
and does not depend on the homotopy class of $\varphi$.
This also implies the vanishing of $\m_0$.
Non-compactness of $W^{-1}(0)$
does not cause any problem either.
Since $W^{-1}(0)$ is Stein,
the maximum principle prevents
pseudoholomorphic  disks
from running away to infinity,
assuring the compactness of the moduli space.


Seidel proved that although
the directed Fukaya category $\FukW$
depends on the choice of the distinguished basis
of vanishing cycles,
different choices are related by mutations,
hence its derived category
is invariant.

Now we explain the mirror construction
of toric Fano manifolds
after Givental \cite{Givental_HGMS}.
Given a fan of an $n$-dimensional toric Fano manifold,
let $\{v_i\}_{i=1}^r$,
$v_i = (v_{i1},\ldots,v_{in}) \in \bZ^n$,
be the set of generators of
its one-dimensional cones.
Then the mirror object
for this
toric Fano manifold is
the regular function
\begin{equation} \label{eq:generalW}
 W(x_1,\ldots,x_n)=\sum_{i=1}^r q_i x_1^{v_{i1}} \cdots x_n^{v_{in}}
\end{equation}
on the algebraic torus $\Spec \bC[x_i^{\pm 1}]_{i=1}^n$.
Here, $q_i$'s are parameters
corresponding to the deformation
of symplectic structures on the toric Fano manifold.

Therefore, the mirror of our $\dP$
is the regular function
\begin{equation} \label{eq:W}
 W(x,y) = q_1 x + q_2 y + \frac{q_3}{x y}
          + \frac{q_4}{x} + \frac{q_5}{y} + q_6 x y 
\end{equation}
on the algebraic torus 
$(\bC^\times)^2 = \Spec \bC[x,x^{-1},y,y^{-1}]$
(See Figure \ref{fg:dp3}).
The Fukaya category does not depend
on a general choice of $q_i$'s.
We have used
$(q_1,q_2,q_3,q_4,q_5,q_6) = (1,1,1,0.215,0.25,0.3)$
to draw the figures appearing below.
Equip $(\bC^\times)^2$
with the symplectic form
$\frac{d|x|}{|x|} \wedge d(\arg x) + \frac{d|y|}{|y|} \wedge d(\arg y)$
and
the relative Maslov map
$$
 \Theta = (\Res \frac{dx \wedge dy}{x y W(x,y)})^{\otimes 2}
        = (\frac{dx}{\partial_y (x y W(x,y))})^{\otimes 2}.
$$
$W^{-1}(0)$ is an affine elliptic curve,
which can be compactified by adding six points.
We depict the critical values of $W$
and our choice of a distinguished set
of vanishing paths
in Figure \ref{fg:vanishing_paths}.
Figure \ref{fg:vanishing_cycles}
shows the corresponding
%distinguished basis of
vanishing cycles.

\begin{figure}[htbp]
\centering
\psfrag{c1}{$c_1$}
\psfrag{c2}{$c_2$}
\psfrag{c3}{$c_3$}
\psfrag{c4}{$c_4$}
\psfrag{c5}{$c_5$}
\psfrag{c6}{$c_6$}
\includegraphics{vanishing_paths.eps}
\caption{Distinguished basis of vanishing paths}
\label{fg:vanishing_paths}
\end{figure}

\begin{figure}[htbp]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{-x1*}{$-x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{1}{\includegraphics{dp3_vanishing_cycles.eps}}
\caption{Vanishing cycles in the fiber at the origin}
\label{fg:vanishing_cycles}
\end{figure}

The opposite sides of the square
in Figure \ref{fg:vanishing_cycles}
are identified
to form a two-dimensional torus.
The open circles denote the points
which are missing
due to the non-compactness
of $(\bC^\times)^2$.
We have drawn Figure \ref{fg:vanishing_cycles}
in the following way:
First regard $W^{-1}(t)$ as a branched double
cover of $\bC^\times$
by the projection
$\pi_t : W^{-1}(t) \ni (x,y) \mapsto y \in \bC^\times$.
For any $i =1,\ldots,6$,
the branch points of $\pi_{c_i(t)}$ moves in $\bC^\times$
as we vary $t$,
until two of them finally collides
at $t=1$.
The vanishing cycle $C_i$ is
the circle in $W^{-1}(0)$
over this trajectory of collision.
This determines the vanishing cycle $C_i$
up to isotopy.
These vanishing cycles
can be straightened within their Hamiltonian isotopy classes.
Here, being straight refers to
the flat metric on the torus
$\overline{W^{-1}(0)}$,
where $\overline{\bullet}$ denotes
the completion.
Note that this flat metric on
$\overline{W^{-1}(0)}$
has nothing to do
with the K\"{a}hler metric
on $W^{-1}(0)$
induced from that of
$(\bC^\times)^2$.
This determines $C_i$ up to translation.
This translational ambiguity can be fixed
by imposing exactness,
which requires the knowledge of the primitive
$\theta$ of the symplectic form $\omega$
on $W^{-1}(0)$.
We do not try to carry this out
since translations of straight $C_i$'s
do not alter the combinatorial structure
of intersections,
which is all we need in our computation of the
Fukaya category below.
The grading on $W^{-1}(0)$
given by $\Theta$
is the grading
coming from the restriction
of the second tensor power
of the holomorphic 1-form
on $\overline{W^{-1}(0)}$.
Then
one can choose a grading $\widetilde{\phi_{C_i}}$
on each vanishing cycle $C_i$
so that
all the Maslov indices
of $C_i \cap C_j$
is zero for $i < j$.
To give a spin structure on $C_i$
is the same as to give a two-fold covering of $C_i$.
We take the non-trivial cover
such that two branches interchange
on the black dots in Figure \ref{fg:vanishing_cycles}.
Different choices for spin structures
lead to different categories,
and it turns out that
the above choice gives a category
derived equivalent to
the category of coherent sheaves
on the del Pezzo surface $\dP$.
Let $C_i^\flat$ denote the vanishing cycle $C_i$
with the above grading and spin structure.
%Different choices for branch points
%are related simply by
%the sign flips
%of the generators
%of the homomorphisms
%corresponding to
%intersection points
%of vanishing cycles.
Since the Maslov index
of all the intersection points are zero,
the sign for the counting of a triangle
is $-1$ if the boundary of the triangle
hits odd numbers of these dots,
and $+1$ otherwise
(see Seidel \cite{Seidel_K3}).
Note that although the choice of spin structures
is important,
the choice of the positions
of the black dots is irrelevant.
The change of the signs
caused by a change of the positions
of the black dots
can be absorbed
by a redefinition of
the signs of the basis for the Floer cohomologies.
$\m_k$ is non-zero only for $k=2$
(polygons with more than three edges
in Figure \ref{fg:vanishing_cycles}
do not contribute
since $\hFukW(C^\flat_i,C^\flat_j) = 0$ for $i > j$).
%By the assignment of objects
%\begin{eqnarray*}
% C_1 & \longleftrightarrow & \scO_{E_1}(-1), \\
% C_2 & \longleftrightarrow & \scO_{E_2}(-1), \\
% C_3 & \longleftrightarrow & \scO_{E_3}(-1), \\
% C_4 & \longleftrightarrow & \econe, \\
% C_5 & \longleftrightarrow & \ectwo, \\
% C_6 & \longleftrightarrow & \ecthree,
%\end{eqnarray*}
%and that of morphisms
%as in Figure \ref{fg:vanishing_cycles},
%one can check that
%the compositions perfectly match
%the table in the Appendix.
%Since both $\DbdP$ and $D^b \Fuk W$
%are generated by exceptional collections
%with identical morphisms and compositions,
%the equivalence in
%Theorem \ref{th:hms} follows.

\begin{theorem} \label{th:commutativity}
There exists an isomorphism
$$ 
 \phi_{ij} : \Hom_\DbdP(\scE_i, \scE_j)
       \rightarrow \HFukW(C_i^\flat,C_j^\flat)
$$
as a $\bC$-vector space
for $i,j=1,\ldots,6$
such that
the diagrams
\begin{equation} \label{eq:diagram}
\begin{CD}
 \Hom_\DbdP(\scE_i, \scE_j) \times \Hom_\DbdP(\scE_j, \scE_k)
   @>>> \Hom_\DbdP(\scE_i, \scE_k) \\
 @VV{\phi_{ij} \times \phi_{jk}}V @VV{\phi_{ik}}V \\
 \HFukW(C_i^\flat,C_j^\flat) \times \HFukW(C_j^\flat,C_k^\flat)
   @>>> \HFukW(C_i^\flat,C_k^\flat)
\end{CD}
\end{equation}
commute.
\end{theorem}
\proof
We omit the lower suffix of $\Hom(\bullet,\bullet)$
since there is no danger of confusion.
We construct the above $\phi_{ij}$'s explicitly.
Since $\m_1=0$,
$\Hom(C_i^\flat,C_j^\flat)$
is isomorphic to $\hom(C_i^\flat,C_j^\flat)$
and spanned by intersection points
of $C_i$ and $C_j$, to which we assign
the basis of $\Hom(\scE_i,\scE_j)$
as in Figure \ref{fg:vanishing_cycles}.
Notations for the basis of
$\Hom(\scE_i, \scE_j)$
is given in Appendix.
This defines the linear isomorphisms $\phi_{ij}$'s.
The commutativity
of (\ref{eq:diagram})
is verified
by counting triangles.
Let us illustrate this
with a few examples.
Take $x_1 \in \Hom(\scE_1, \scE_4)$
and $x_1^\vee \in \Hom(\scE_4, \scE_6)$,
whose composition is
$e_1 \in \Hom(\scE_1, \scE_6)$.
On the Fukaya side,
we count the number of triangles
whose edges are contained in
$C_1 \cup C_4 \cup C_6$
and two of whose vertices are
$x_1$ and $x_1^\vee$.
Such a triangle exists uniquely,
whose remaining vertex is $e_1$
(Figure \ref{fg:cd1}).
%This matches the computation
%on the derived category of coherent sheaves
%on the del Pezzo surface.
\begin{figure}[H]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{0.7}{\includegraphics{cd1.eps}}
\caption{Counting triangle}
\label{fg:cd1}
\end{figure}

Next we try to compose
$x_3 \in \Hom(C_3^\flat,C_4^\flat)$
and $x_1^\vee \in \Hom(C_4^\flat, C_6^\flat)$
in the Fukaya category.
The only possibility
would be the triangle
with $e_3$ as the remaining vertex.
However, this is not allowed
because of
the missing point
(Figure \ref{fg:cd2}).

\begin{figure}[H]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{0.7}{\includegraphics{cd2.eps}}
\caption{No triangles}
\label{fg:cd2}
\end{figure}

This shows that
the composition of $x_3$
and $x_1^\vee$
in the Fukaya category
is zero,
which matches the table
in Appendix
calculated in $\DbdP$.
Finally, as an example of
counting with signs,
we compute
$$
\Hom(C_1^\flat, C_4^\flat) \times
   \Hom(C_4^\flat,C_5^\flat)
 \ni (x_1, x_1^\vee \wedge x_2^\vee)
  \mapsto -x_2^\vee \in \Hom(C_1^\flat, C_5^\flat).
$$
Indeed, the triangle
whose vertices are $x_1, x_1^\vee \wedge x_2^\vee$,
and $x_2^\vee$ contains one black dot 
on its edge
(Figure \ref{fg:cd3}).
\begin{figure}[H]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{0.7}{\includegraphics{cd3.eps}}
\caption{Counting triangles with signs}
\label{fg:cd3}
\end{figure}
We can do similar analyses
for the rest of the triangles.
The result perfectly agrees with the computations
in $\DbdP$.
\qed

Although the derived category of
an $A_\infty$-category
is usually defined
using twisted complexes,
we adopt the following definition
in this paper
since our $\FukW$ satisfies
$\hFukW^k(C_i^\flat,C_j^\flat)=0$
for $k \neq 0$
and $\m_k=0$
for $k \neq 2$.
\begin{definition}
Let $A=\oplus_{i,j}\HFukW(C_i^\flat,C_j^\flat)$
be the total morphism algebra.
The derived Fukaya category
$D^b \FukW$ is
the bounded derived category
$D^b (\modm A)$
of the category of right
finite-dimensional modules
over the algebra $A$.
\end{definition}
Now we can state our main theorem:
\begin{theorem} \label{th:main_theorem}
There exists an equivalence of triangulated categories
$$
\DbdP \cong D^b \FukW.
$$
\end{theorem}
\proof
From Theorem \ref{th:commutativity},
we have
$$
A \cong \oplus_{i,j} \Hom_\DbdP(\scE_i,\scE_j).
$$
Theorem \ref{th:main_theorem} follows
immediately from the theorem of Bondal
\cite{Bondal}
that $\DbdP \cong D^b(\modm (\oplus_{i,j} \Hom(\scE_i,\scE_j)))$.
\qed

We can perform
similar analyses
for all the other toric del Pezzo surfaces
to obtain Theorem \ref{th:hms}.

\section{Appendix}

Here we give the table of compositions of morphisms
in the derived category of coherent sheaves
on the projective plane
$\bP(V)$
blown-up at
$p_1 = [1:0:0]$, $p_2 = [0:1:0]$, $p_3 = [0:0:1]$.
We use the following notations:
$\scE_i$'s are defined in (\ref{eq:ec}),
$\{x_i\}_{i=1}^3$ is the basis of $V$,
$\{x_i^\vee\}_{i=1}^3$ is the dual basis of $V^\vee$, and
$e_i$ is the generator of
$\Hom(\scE_i, \scE_6)=\bC$ for $i=1,2,3$.

$\Hom(\eeone, \eefour)
 \times \Hom(\eefour, \eefive)
 \rightarrow \Hom(\eeone, \eefive)$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee \wedge x_2^\vee
 & x_2^\vee \wedge x_3^\vee 
 & x_3^\vee \wedge x_1^\vee \\
 \hline
 x_1 & -x_2^\vee & 0 & x_3^\vee
\end{array}
$$
$\Hom(\eeone, \eefour) \times \Hom(\eefour, \eesix)
 \rightarrow \Hom(\eeone, \eesix)$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee & x_2^\vee & x_3^\vee \\
 \hline
 x_1 & e_1 & 0 & 0
\end{array}
$$
$\Hom(\eeone, \eefive) \times \Hom(\eefive, \eesix)
 \rightarrow \Hom(\eeone, \eesix)$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_2^\vee & 0 & e_1 & 0 \\
 \hline
 x_3^\vee & 0 & 0 & e_1 \\
\end{array}
$$
$\Hom(\eetwo, \eefour) \times \Hom(\eefour, \eefive)
 \rightarrow \Hom(\eetwo, \eefive)$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee \wedge x_2^\vee
 & x_2^\vee \wedge x_3^\vee 
 & x_3^\vee \wedge x_1^\vee \\
 \hline
 x_2 & x_1^\vee & -x_3^\vee & 0
\end{array}
$$
$\Hom(\eetwo, \eefour) \times \Hom(\eefour, \eesix)
 \rightarrow \Hom(\eetwo, \eesix)$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee & x_2^\vee & x_3^\vee \\
 \hline
 x_2 & 0 & e_2 & 0
\end{array}
$$
$\Hom(\eetwo, \eefive) \times \Hom(\eefive, \eesix)
 \rightarrow \Hom(\eetwo, \eesix)$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_1^\vee & e_2 & 0 & 0 \\
 \hline
 x_3^\vee & 0 & 0 & e_2
\end{array}
$$
$\Hom(\eethree, \eefour) \times \Hom(\eefour, \eefive)
 \rightarrow \Hom(\eethree, \eefive)$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee \wedge x_2^\vee
 & x_2^\vee \wedge x_3^\vee 
 & x_3^\vee \wedge x_1^\vee \\
 \hline
 x_3 & 0 & x_2^\vee & -x_1^\vee
\end{array}
$$
$\Hom(\eethree, \eefour) \times \Hom(\eefour, \eesix)
 \rightarrow \Hom(\eethree, \eesix)$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee & x_2^\vee & x_3^\vee \\
 \hline
 x_3 & 0 & 0 & e_3
\end{array}
$$
$\Hom(\eethree, \eefive) \times \Hom(\eefive, \eesix)
 \rightarrow \Hom(\eethree, \eesix)$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_1^\vee & e_2 & 0 & 0 \\
 \hline
 x_2^\vee & 0 & e_3 & 0
\end{array}
$$
$\Hom(\eefour, \eefive) \times \Hom(\eefive, \eesix)
 \rightarrow \Hom(\eefour, \eesix)$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_1^\vee \wedge x_2^\vee & x_2^\vee & -x_1^\vee & 0 \\
 \hline
 x_2^\vee \wedge x_3^\vee & 0 & x_3^\vee & -x_2^\vee \\
 \hline
 x_3^\vee \wedge x_1^\vee & -x_3^\vee & 0 & x_1^\vee \\
\end{array}
$$

\bibliographystyle{plain}
\bibliography{bibs}

Research Institute of Mathematical Sciences,
Kyoto University,
Oiwake-cho,
Kitashirakawa,
Sakyo-ku,
Kyoto,
606-8502,
Japan.

{\em e-mail address}\ : \  kazushi@kurims.kyoto-u.ac.jp

\end{document}

\section{Stokes matrix} \label{Stokes}
Let $X$ be a symplectic manifold.
Quantum cohomology determines
the following family of differential equations;
%on $\bP^1$
%called the first structure connection, which is
%defined as follows:
\begin{eqnarray}
 \hbar \frac{\partial \Phi}{\partial t_\flata} 
   & =& \frac{\partial}{\partial t_\flata} \circ \Phi, \quad
  k=1,\ldots,N,  \label{eq:t_direction} \\
 \hbar \frac{\partial \Phi}{\partial \hbar}
  & = & (\frac{1}{\hbar} E \circ \Phi - [E, \Phi].
  \label{eq:hbar_direction}
\end{eqnarray}
Here,
$\Phi$ is defined on $H^*(X;\bC) \times \bP^1$
and takes value in $H^*(X,\bC)$,
$\{t_\flata\}_{\flata=1}^N$ is a coordinate
on $H^*(X;\bC)$,
$\hbar$ is a coordinate
on $\bP^1$,
the circle denotes the product
in the quantum cohomology ring,
and $U$ and $V$ are functions of
only $t_\flata$'s
taking values in $\End_\bC(H^*(X,\bC))$,
see Dubrovin \cite{Dubrovin_G2DTFT}.
%Its solution $\Phi$
%is a cohomology-valued function of $\hbar$
%parametrized by $t_k$'s.
%We consider only the small quantum cohomology here,
%which means that we let the parameters $t_i$'s
%run only on the second cohomology ring.
%The difference between the large quantum cohomology
%and the small quantum cohomology
%is irrelevant
%in the computation of the Stokes matrix.
(\ref{eq:hbar_direction})
is an ordinary differential equation
on $\bP^1$
with a regular singularity at infinity
and an irregular singularity at the origin,
and (\ref{eq:t_direction}) gives
its isomonodromic deformation.
When the quantum cohomology ring
is semisimple,
we can define its Stokes matrix
as follows:

Let $\{u_\canonicala \}_{\canonicala=1}^N$ be
the set of eigenvalues
of the matrix $U$.
A straight line
$l = \{\hbar \in \bC \suchthat \arg(\hbar) = \phi, \phi - \pi \}$,
$0 \leq \phi < \pi$
passing through the origin
is called admissible
if the line through $u_\canonicala$ and $u_\canonicalb$
is not orthogonal to $l$
for any $\canonicala$ and $\canonicalb$.
Fix an admissible line and choose
a small enough number $\epsilon > 0$
so that any line passing through the origin
with angle between $\phi-\epsilon$ and $\phi+\epsilon$
is admissible.
\begin{figure}[h]
 \centering
 \psfrag{Dl}{$D_\lt$}
 \psfrag{Dr}{$D_\rt$}
 \psfrag{D+}{$D_+$}
 \psfrag{phi}{$\phi$}
 \psfrag{l}{$l$}
 \includegraphics{stokes.eps}
 \caption{Angular domains}
 \label{fg:stokes}
\end{figure}
Define 
\begin{eqnarray} \label{eq:D}
%l_{++} & = & \{\hbar \in \bC \suchthat
% \arg(\hbar) = \phi+\epsilon \}, \\
%l_{+-} & = & \{\hbar \in \bC \suchthat
% \arg(\hbar) = \phi+\epsilon - \pi \}, \\
%l_{-+} & = & \{\hbar \in \bC \suchthat
% \arg(\hbar) = \phi-\epsilon \}, \\
%l_{--} & = & \{\hbar \in \bC \suchthat
% \arg(\hbar) = \phi-\epsilon - \pi \}, \\
D_{\rt} & = & \{\hbar \in \bC \suchthat
 \phi - \pi - \epsilon < \arg(\hbar) <\phi + \epsilon \},
 \nonumber \\
D_{\lt} & = & \{\hbar \in \bC \suchthat
 \phi - \epsilon < \arg(\hbar) <\phi +\pi + \epsilon \}, \\
D_+ & = & \{\hbar \in \bC \suchthat
 \phi - \epsilon < \arg(\hbar) < \phi + \epsilon \},
 \nonumber
\end{eqnarray}
as in Figure \ref{fg:stokes}.
Fix a formal fundamental solution
$\Phi_{\small \mathrm{formal}}(\hbar)$
of (\ref{eq:hbar_direction})
around $\hbar = 0$.
Since the singularity at the origin
is irregular,
$\Phi_{\small \mathrm{formal}}(\hbar)$
does not converge.
Nevertheless,
there exist unique convergent solutions
$\Phi_\rt(\hbar)$ and $\Phi_\lt(\hbar)$
defined on the angular domain $D_\rt$ and $D_\lt$
respectively,
which asymptote
to the formal solution:
$$
\Phi_{\rt/\lt} \sim \Phi_{\small \mathrm{formal}}
\ \ \mbox{as $\hbar \rightarrow 0$ in $D_{\rt/\lt}$}.
$$
Since these two solutions
satisfy the same linear differential equation
on $D_+$,
they must be related
by a linear transformation:
$$
\Phi_\lt(\hbar) = \Phi_\rt(\hbar) S,
\ \ \ \hbar \in D_+.
$$
This matrix $S$ is called the Stokes matrix.
It is independent of $\hbar$ by definition,
and is also independent of $t_i$'s
because of the isomonodromicity
\cite{Dubrovin_PT2DTFT}.

When $X$ is a toric Fano manifold,
Givental \cite{Givental_HGMS} gave
the following integral representations
of the fundamental solution
to (\ref{eq:hbar_direction}):
$$
\Phi_{\flata,\canonicala}(\hbar)
  = \int_{\Gamma_\canonicala(\hbar)} e^{\frac{1}{\hbar} W(x)}
 \omega_\flata(x)\, dx_1 \cdots dx_n.
$$
Here,
$W$ is defined in (\ref{eq:generalW}),
$t_\flata$'s in (\ref{eq:t_direction})
are certain functions of $q_i$'s
in (\ref{eq:generalW}),
$\omega_\flata$'s are certain functions
on $(\bC^\times)^n$, and
%for any $\alpha=1,\ldots,N$,
%$\{ \Phi_{k,\alpha} \}$, $k=1,\ldots,N$ are
%components of $\Phi$,
$\Gamma_\canonicala(\hbar)$,
$\canonicala=1,\ldots,N$,
is the Lefschetz thimble
corresponding to
the $\canonicala$-th critical point,
which we define here
as the union
of the trajectories
of the gradient flow of $W/\hbar$
with respect to
the K\"{a}hler metric.
$\flata$ is the suffix for $H^*(X,\bC)$
%where $\Phi(\hbar)$
%takes value,
and $\canonicala$ is the suffix
for the fundamental solution
(i.e.,
$(\Phi_{\flata,\canonicala}(\hbar))_{\flata=1}^N
\in H^*(X;\bC)$)
satisfies
(\ref{eq:hbar_direction})
for each $\canonicala$,
$\canonicala = 1,\ldots,N$).

%We choose $\Gamma_i$
%to be the Lefschetz thimble
%corresponding to the $i$-th critical point,
%which is the descending Morse cycle
%(unstable manifold)
%for the function
%$\Re W / \hbar$
%with respect to the K\"{a}hler metric.
%Figure \ref{fg:cycles} shows their images
%in the $W$-plane.
%They are straight lines
%in the direction of $-1/\hbar$
%starting from the critical values,
%and the fiber over a point
%on these lines
%is corresponding vanishing cycle.

Now let us explain
how to compute the Stokes matrix.
Suppose, for simplicity,
that we have only two critical points
$p_1$ and $p_2$ such that
$W(p_1)$
and $W(p_2)$
are on the imaginary axis.
Then the imaginary axis is admissible,
and the image of the Lefschetz thimble
for $\hbar = 1$
looks as in Figure \ref{fg:lefschetz1}.
As one rotates $\hbar$ counterclockwise,
the Lefschetz thimbles vary accordingly
(Figure \ref{fg:lefschetz2}),
giving a flat section
of the Gauss-Manin connection
on the relative homology group
$H_n(\bC^\times, \Re(W/\hbar) \ll 0)$,
until $\hbar = \sqrt{-1}$,
where they undergo
a discontinuous change.
To compute the Stokes matrix,
first fix $\epsilon$
such that $0 < \epsilon < \pi/2$
and define $D_{\rt/\lt/+}$
as in (\ref{eq:D}).
Define $\Gamma_{\canonicala,\rt}(\hbar)$
(resp. $\Gamma_{\canonicala,\lt}(\hbar)$)
to be the extension
of $\Gamma_\canonicala(\hbar)$
defined on the right half plane
(resp. the left half plane)
to $D_\rt$
(resp. $D_\lt$)
as a flat section
with respect to the Gauss-Manin connection.
Then
$$
\Phi_{\flata,\canonicala, \rt/\lt}(\hbar)
 = \int_{\Gamma_{\canonicala, \rt/\lt}(\hbar)} e^{\frac{1}{\hbar} W(x)}
 \omega_\flata(x) \, dx_1 \cdots dx_n
$$
give holomorphic solutions on $D_{\rt/\lt}$.
Both of them has the same asymptotics
given by the saddle point method:
$$
\Phi_{\flata, \canonicala, \rt/\lt}(\hbar) \sim
 e^{W(p_\canonicala)/\hbar} \omega_\flata(p_\canonicala)
 \pi^{\frac{n}{2}} \Hess[W(p_\canonicala)]^{-\frac{1}{2}}
 (1+O(\hbar))).
$$
When $\hbar \in D_+$,
the relation between $\Gamma_{\canonicala, \rt}(\hbar)$
and $\Gamma_{\canonicala, \lt}(\hbar)$
(as in Figure \ref{fg:lefschetz3})
is given by the Picard-Lefschetz theory:
\begin{eqnarray*}
\Gamma_{1, \lt}(\hbar)
 & = & \Gamma_{1, \rt}(\hbar), \\
\Gamma_{2, \lt}(\hbar)
 & = & (C_1,C_2) \Gamma_{1, \rt}(\hbar)
     + \Gamma_{2,\rt}(\hbar).
\end{eqnarray*}
Here, $(C_1,C_2)$ is the intersection number
between vanishing cycles
corresponding to $p_1$ and $p_2$.
Since the integrand is single-valued,
this determines the monodromy of $\Phi(\hbar)$:
\begin{eqnarray*}
 \Phi_\lt(\hbar) & = & \Phi_\rt(\hbar) S, \\
 S & = & \left(\begin{array}{cc}
          1 & (C_1, C_2) \\
          0 &      1 \\
	 \end{array}\right).
\end{eqnarray*}

%there are two choice for the Lefschetz thimbles,
%the one continued from $\hbar = 1$
%and the other continued from $\hbar = -1$,
%and the relation between them
%precisely gives the Stokes matrix.
%It is clear that $\Gamma_2 = \Gamma_2'$.
%As for $\Gamma_1$, think of
%deforming the cycle as in
%Figure \ref{fg:lefschetz4}.
%Two curves $\Gamma_1$ and
%$\Gamma_1'$ has
%the same image below
%$p_2$,
%although their fibers are different
%since the paths above $p_1$ are different.
%The relation between their fibers
%$C_1$ and $C_1'$ over the point $p$,
%which are vanishing cycles
%corresponding to each path,
%is
%$$
%C_1' = C_1 + (C_1, C_2) C_2
%$$
%by the Picard-Lefschetz theory.
%Therefore,
%the relation between the Lefschetz thimbles
%are also given by
%$$
%\Gamma_2' = \Gamma_1 + (C_1, C_2) \Gamma_2.
%$$

\begin{figure}[htbp]
\centering
\psfrag{p1}{$W(p_1)$}
\psfrag{p2}{$W(p_2)$}
\psfrag{c1}{$\Gamma_1$}
\psfrag{c2}{$\Gamma_2$}
\includegraphics{lefschetz1.eps}
\caption{Lefschetz thimbles at $\hbar = 1$}
\label{fg:lefschetz1}
\end{figure}
\begin{figure}[htbp]
\centering
\psfrag{p1}{$W(p_1)$}
\psfrag{p2}{$W(p_2)$}
\psfrag{c1}{$\Gamma_1$}
\psfrag{c2}{$\Gamma_2$}
\psfrag{e}{$\epsilon$}
\includegraphics{lefschetz2.eps}
\caption{Lefschetz thimbles at $\hbar = e^{\sqrt{-1} \epsilon }$}
\label{fg:lefschetz2}
\end{figure}
\begin{figure}[htbp]
\centering
\psfrag{p1}{$W(p_1)$}
\psfrag{p2}{$W(p_2)$}
\psfrag{c1}{$\Gamma_{1, \rt}$}
\psfrag{c2}{$\Gamma_{2, \rt}$}
\psfrag{d1}{$\Gamma_{1, \lt}$}
\psfrag{d2}{$\Gamma_{2, \lt}$}
\includegraphics{lefschetz3.eps}
\caption{continued cycles at $\hbar \in D_+$}
\label{fg:lefschetz3}
\end{figure}
%\begin{figure}[htbp]
%\psfrag{p1}{$W(p_1)$}
%\psfrag{p2}{$W(p_2)$}
%\psfrag{p}{$p$}
%\includegraphics{lefschetz4.eps}
%\caption{Comparing two cycles at $\hbar = \sqrt{-1}$}
%\label{fg:lefschetz4}
%\end{figure}

In the case with more critical points,
one obtains in the same way
$$
\Gamma_{i,\lt}(\hbar) 
 = \Gamma_{i,\rt}(\hbar) + (C_i, C_{i+1}) \Gamma_{i+1,\rt}(\hbar)
   + \cdots + (C_i, C_N) \Gamma_{N,\rt}(\hbar),
$$
from which one can read off the entries $S_{ij}$
of the Stokes matrix as
$$
S_{ij} = \left\{\begin{array}{cl}
       1      & \mbox{if $i=j$,} \\
   (C_i, C_j) & \mbox{if $i<j$,} \\
       0      & \mbox{otherwise.}
   \end{array}\right.
$$
Since
\begin{eqnarray*}
 (C_i, C_j) & = & \sum_k (-1)^k \dim \HFukW^k(C_i,C_j) \\
            & = & \sum_k (-1)^k \dim \Ext^k(\scE_i,\scE_j),
\end{eqnarray*}
Conjecture \ref{conj:dubrovin}
on the relation between
the Stokes matrix
and the Euler pairing
follows from Theorem \ref{th:hms}.
%In our case, concretely,
%\begin{eqnarray*}
%S & = & \left(\begin{array}{cccccc}
% 1 & (C_1,C_2) & (C_1,C_3) & (C_1,C_4) & (C_1,C_5) & (C_1, C_6)\\
% 0 &     1     & (C_2,C_3) & (C_2,C_4) & (C_2,C_5) & (C_2, C_6)\\
% 0 &     0     &     1     & (C_3,C_4) & (C_3,C_5) & (C_3, C_6)\\
% 0 &     0     &     0     &     1     & (C_4,C_5) & (C_4, C_6)\\
% 0 &     0     &     0     &     0     &     1     & (C_5, C_6)\\
% 0 &     0     &     0     &     0     &     0     &     0     \\
%    \end{array}\right) \\
% & = & \left(\begin{array}{cccccc}
% 1 &     0     &     0     &     1     &     2     &     1     \\
% 0 &     1     &     0     &     1     &     2     &     1     \\
% 0 &     0     &     1     &     1     &     2     &     1     \\
% 0 &     0     &     0     &     1     &     3     &     3     \\
% 0 &     0     &     0     &     0     &     1     &     3     \\
% 0 &     0     &     0     &     0     &     0     &     0     \\
%    \end{array}\right)
%\end{eqnarray*}


\section{Products and local mirror symmetry}
In this section, we prove that
homological mirror symmetry for Fano manifolds
$X$ and $Y$ implies that for the product $X \times Y$.
Then we discuss the relationship between
homological mirror symmetry for Fano manifolds
and local mirror symmetry,
which is a mirror symmetry
for the total space of the canonical bundle
of a manifold.
Local mirror symmetry for
canonical bundle of del Pezzo surfaces
are important,
due to its relationship with the Seiberg-Witten theory.
We find that the homological version of
the local mirror symmetry
follows directly
from the homological mirror symmetry
for the base manifold.

\begin{proposition}

Let $X$ and $Y$ be Fano manifolds
of dimensions $m$ and $n$,
and $F:\bC^m \rightarrow \bC$ and
$G:\bC^n \rightarrow \bC$ be their mirrors.
We assume:
\begin{enumerate}

\item $X$ and $Y$ have complete exceptional collections
 $\{\scE_i\}_i$ and $\{\scF_j\}_j$,

\item $F$ and $G$ have vanishing cycles
 $\{c_i\}_i$ and $\{c_j\}_j$ corresponding to
 critical values $\{p_i\}_i$ and $\{q_j\}_j$, and

\item Floer cohomologies of $c_i$'s and $d_j$'s
 coincides with $\Ext$-groups of $\scE_i$'s and $\scF_j$'s,
 including the product structure on both sides.

\end{enumerate}

Then $F + G : \bC^m \times \bC^n \rightarrow \bC$
is a homological mirror partner for $X \times Y$.

\end{proposition}

\proof

We have to find vanishing cycles for $F + G$.
First note that we can assume
that all the critical values are in the right half plane
by deforming the function $F$ and $G$.
Critical values $\{p_i\}_i$ and $\{q_j\}_j$ are
ordered according to their imaginary parts;
$\Im p_i > \Im p_j$ for $i < j$ and
$\Im q_i > \Im q_j$ for $i < j$.
Critical values of $F + G$ are $p_i + q_j$ for
any $i$ and $j$.
Fiber of $F + G$ at the origin,
$(F + G)^{-1}(0)$,
is a fibration
$\cup_{t \in \bC} F^{-1}(t) \times G^{-1}(-t)$,
and if you choose a path
$\gamma:[0,1] \ni s \mapsto s(p_i+q_j) \in \bC$,
the vanishing cycle along this path
is the cycle fibered over the line segment
between $p_i$ and $-q_j$,
whose fiber over the interior of the interval
is $S^{m-1} \times S^{n-1}$,
the direct product of the vanishing cycles of $F$ and $G$
corresponding to the critical points $p_i$ and $q_j$,
which degenerates to a point times $S^{n-1}$
at $p_i$
and to $S^{m-1}$ times a point at $-q_j$,
which as a whole is topologically an $(m+n-1)$-sphere.
Let us call this cycle $c_{ij}$.
We introduce the lexicographic order
in the set of pairs $(i,j)$.
Then $c_{ij}$ intersects $c_{kl}$
if 1. $i=k$, 2. $j=l$, 3. $i<k$ and $j<l$, or
4. $i>k$ and $j>l$.

From the definition of the Fukaya category,
we consider only the case $(i,j) < (k,l)$.
Then, in the first case, $l$ must be strictly greater
than $k$, and all intersection points are lying
on the fiber over $p_i$, and
there are $\#(d_j \cap d_l)$ of them.
In the second case we must have $i<k$ and
there are $\#(c_i \cap c_k)$ intersection points
over $-q_j$.
In the third case,
two line segments intersects in the interior
and there are $\#(c_i \cap c_k) \times \#(d_j \cap d_l)$
intersection points on the fiber.
Product on the Floer cohomology
is the tensor product
because any holomorphic map with the Lagrangian boundary
condition must map into a fiber
because its image in the $t$-plane
must be a point,
each fiber is a product of fibers of $F$ and $G$,
and map into a product is a product of maps
into each component.

On the mirror side,
the derived category of coherent sheaves
on $X \times Y$ is generated by
the exterior tensor products
$\pi_1^*\scE_i \times_{\scO_{X \times Y}} \pi_2^*\scF_j$.
$\Ext$-groups between them is determined
by the K\"{u}nneth formula
$$
\Ext_r(\pi_1^*\scE_i \times_{\scO_{X \times Y}} \pi_2^*\scF_j,
 \pi_1^*\scE_k \times_{\scO_{X \times Y}} \pi_2^*\scF_l)
 = \sum_{p+q=r} \Ext(\scE_i, \scE_k) \otimes \Ext(\scF_j,\scF_l),
$$
and the Yoneda product is also determined
as a tensor product of Yoneda product
in each component.

Next we discuss a relationship with
local mirror symmetry.
Local mirror symmetry deals with the situation
where a surface is embedded inside a Calabi-Yau 3-fold.

The Frobenius structure is defined by Dubrovin
as a coordinate-free formulation of the
WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation of associativity,
though the same structure has already appeared
in the early 80's
under the name of the ``flat structure''
in singularity theory \cite{Saito}.

\begin{definition}[Frobenius manifold]
A Frobenius structure on a complex manifold $M$ is a tuple
$(g, \circ, e, E)$ of a metric
$$
 g : T^{(1,0)}M \otimes T^{(1,0)}M \rightarrow \bC,
$$
which is a symmetric bilinear form on the holomorphic tangent bundle,
a multiplication
$$
 \circ : T^{(1,0)}M \otimes T^{(1,0)}M \rightarrow T^{(1,0)}M,
$$
the unit vector field
$$
 e \in \Gamma(M, T^{(1,0)}M),
$$
and the Euler vector field
$$
 E \in \Gamma(M, T^{(1,0)}M),
$$
satisfying the following axioms:

\begin{enumerate}
\item
$$
g(X \circ Y, Z) = g(X, Y \circ Z)
\mbox{Frobenius property,}
$$

\item
$g$ is flat,
i.e., there is a local coordinate ${t_i}_{i=1}^N$
($N = \dim_\bC X$)
satisfying
$$
g(\frac{\partial}{\partial t_i}, \frac{\partial}{\partial t_j})
   = \delta_{ij},
$$
where $\delta_{ij}$ is the Kronecker delta.
Such a coordinate is called the flat coordinate,
which is unique up to affine transformations.

We fix a flat coordinate 

\item
For the flat coordinate ${t_i}_{i=1}^N$,
$$
 \frac{\partial}{\partial t_i}
  g(\frac{\partial}{\partial t_j},
    \frac{\partial}{\partial t_k} \circ 
    \frac{\partial}{\partial t_l} ) 
 = \frac{\partial}{\partial t_j}
 g(\frac{\partial}{\partial t_i},
    \frac{\partial}{\partial t_k} \circ 
    \frac{\partial}{\partial t_l})
\mbox{integrability.}
$$
for any $0 \leq i,j,k,l \leq N$.
\end{enumerate}

\begin{remark}
$g$ is often called the metric,
although it is complex linear,
not Hermitian.
Integrability implies the existence
of a function $\scF$
satisfying

\end{remark}

A manifold equipped with a Frobenius structure is called
a Frobenius manifold.

\end{definition}

\section{Frobenius manifolds and
isomonodromic deformations} \label{FM}

Frobenius manifold was defined by Dubrovin
as a coordinate-free formulation
of the WDVV equation
in the two-dimensional topological field theory
\cite{Dubrovin_G2DTFT}.
One important class of Frobenius manifolds
is universal unfoldings of
isolated singularities,
in which case this structure
was already discovered by K. Saito
in the early 80's \cite{KSaito}
before physicists rediscover it
under the name of topological
Landau-Ginzburg theories.
For an excellent exposition, see \cite{Hitchin_FM}.

\begin{definition}
A Frobenius manifold is a quinterple $(M,g,\circ,e,E)$ where
\begin{enumerate}
\item $M$ is a complex manifold of complex dimension $n$,
\item $g$ is a complex bilinear form on
 the tangent bundle $TM$ called the metric,
\item $\circ$ is an associative commutative product on $TM$,
\item $e$ and $E$ are vector fields on $M$
 called the unit vector field and the Euler vector field
\end{enumerate}
satisfying the following axioms:
\begin{enumerate}
\item $g$, $\circ$, $e$, and $E$ are holomorphic on $M$.
\item $g(X \circ Y, Z)=g(X, Y \circ Z)$.
\item the (3,1)-tensor $\nabla \circ$ is totally symmetric,
 where $\nabla$ is the Levi-Civita connection for $g$
 (potentiality).
\item \label{gflat} $g$ is flat.
\item $e \circ X = X$.
\item $e$ is flat.
\item \label{Euler} $\nabla \nabla E=0$, $\mathop{Lie}_E(\circ) = \circ$, and
 $\mathop{Lie}_E(g) = D g$ for some $D \in \bC$.
\end{enumerate}
\end{definition}

\begin{definition}
Define $\scV, \scU \in \Gamma(M, \End(TM))$ by
$$
 \scV(X) := \nabla_X E - \frac{D}{2}X,
$$
$$
 \scU(X) := E \circ X
$$
\end{definition}

\begin{definition}
Frobenius manifold $M$ is called semisimple
at $p \in M$
if the product on the tangent bundle
$T_p M$ at that point
has no nilpotent element.
Semisimplicity is an open condition.
A Frobenius manifold is called semisimple
if it is semisimple on a dense subset.
\end{definition}

\begin{proposition}
A point on a Frobenius manifold is semisimple
if and only if
there is a coordinate $\{ u_{\alpha} \}_\alpha$
called the canonical coordinate
satisfying
$$
\frac{\partial}{\partial u_\alpha} \circ \frac{\partial}{\partial u_\beta}
 = \delta_{\alpha \beta} \frac{\partial}{\partial u_\alpha},
$$
$$
E = \sum_\alpha u_\alpha \frac{\partial}{\partial u_\alpha}.
$$
\end{proposition}

Let $\pi : \bC^{*} \times M \rightarrow M$ be the second projection.
To each Frobenius manifold $M$, we can associate
a flat connection $\hat{\nabla}$
on the lifted tangent bundle
$\pi^*TM$ by
$$
 \hat{\nabla}_{\frac{\partial}{\partial t_i}} X
  = \frac{\partial X}{\partial t_i} + z \frac{\partial}{\partial t_i} \circ X
$$
where $z$ is a coordinate of the $\bC^{*}$.
This connection is called {\em the first structure connection},
and its flatness encodes axioms of the Frobenius structure.
This connection can be considered as an isomonodromic
deformation of ordinary differential operators on $\bP^1$
with a regular singularity at the origin
and an irregular singularity at the infinity,
where the isomonodromicity follows from the flatness.

It is Dubrovin's insight that in semi-simple cases,
one can associate a monodromy data to a Frobenius manifold,
and this monodromy data in turn is enough
to reconstruct the original Frobenius manifold:

\begin{definition}[Stokes matrix]
Fix a radius $R > 0$ and an angle $0 \neq \phi < w \pi$.
Let $l=l_+ + l_-$ divide the complex $\hbar$-plane
into two half planes $\Pi_right$ and $\Pi_left$,
where $l_+ = \{\hbar \suchthat \arg(\hbar) = \phi\}$,
and $l_- = \{\hbar \suchthat \arg(\hbar) = \phi + \pi\}$.

\end{definition}

\begin{theorem}[Dubrovin]
\end{theorem}

\section{Integral representation and Stokes matrix}

We give an integral representation
of flat sections of the first structure connections
for quantum cohomologies of toric del Pezzo surfaces,
due to Givental,
and discuss the relationship between
Stokes matrices and
intersection numbers of vanishing cycles.





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