Stokes Matrix for the Quantum Cohomology of Cubic Surfaces
Kazushi Ueda
We prove the conjectural relation between the Stokes matrix for the quantum cohomology of $X$ and an exceptional collection generating $\DbX$ when $X$ is a smooth cubic surface. The proof is based on a toric degeneration of a cubic surface, the Givental's mirror theorem for toric manifolds, and the Picard-Lefschetz theory.
toricstokes.tex
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%
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%the coefficient of the polynomial W on the mirror side
\newcommand{\wa}{c}

%the first Chern class of the toric line bundle
\newcommand{\toricu}{\ensuremath{w}}

%the running index for the edges of the fan
\newcommand{\edgea}{\ensuremath{k}}

%the running index for the definition of the I-function
\newcommand{\Irindex}{\ensuremath{l}}

\newcommand{\Izero}{\ensuremath{I^{(0)}}}
\newcommand{\Ionezero}{\ensuremath{I^{(10)}}}
\newcommand{\Ioneone}{\ensuremath{I^{(11)}}}
\newcommand{\unknownY}{Y}
\newcommand{\llangle}{\langle \langle}
\newcommand{\rrangle}{\rangle \rangle}
\newcommand{\pd}{\mathrm{P.D.}}
\newcommand{\pt}{pt}
\newcommand{\Etilde}{\ensuremath{\tilde{E}}}
\newcommand{\identitymatrix}{\ensuremath{1}}
\newcommand{\Jmatrix}{L}
\newcommand{\fsconsec}{\ensuremath{\Phi}}
 % the flat section of the first structure connection
\newcommand{\spint}{\ensuremath{\scI}}
 % stationary-phase integral
\newcommand{\docJ}{\ensuremath{D}}
 % domain of convergence of $J$-function
\newcommand{\pair}[2]{\langle #1, #2 \rangle}
\newcommand{\loopG}{G}
\newcommand{\motonot}{t}
\newcommand{\motonoq}{q}
 % coordinate before mirror transformation
\newcommand{\atonot}{\widetilde{t}}
 % coordinate after mirror transformation
\newcommand{\bZtt}{\ensuremath{\bZ / 3 \bZ \times \bZ / 3 \bZ}}
\newcommand{\blsix}{\ensuremath{Y}}
\newcommand{\Dbblsix}{\ensuremath{{D^b \mathrm{coh}(Y)}}}
\newcommand{\sformal}{\mathrm{formal}}
\newcommand{\integrandw}{w}
\newcommand{\degd}{\ensuremath{d}}
\newcommand{\Lphi}{\phi^*}
\newcommand{\Rphi}{\bR \phi_*}
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\newcommand{\XSigma}{{X_\Sigma}}
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\title{Stokes Matrix for the Quantum Cohomology
of Cubic Surfaces}
\author{Kazushi Ueda}
\date{}
\pagestyle{plain}

%
%-------------------- text stars --------------------
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\begin{document}

\maketitle

\begin{abstract}
We prove the conjectural relation
between the Stokes matrix
for the quantum cohomology
of $X$
and an exceptional collection
generating $\DbX$
when $X$ is a smooth cubic surface.
The proof is based on a toric degeneration
of a cubic surface,
the Givental's mirror theorem for toric manifolds,
and the Picard-Lefschetz theory.
\end{abstract}

\section{Introduction}
Let $X$ be a smooth cubic surface in $\bP^3$.
It is a rational surface
obtained by blowing up $\bP^2$
at six points.
Let
$$
 \phi : X \rightarrow \bP^2
$$
denote this blow-up.
For a given
$\degreed \in H_2(X;\bZ)$,
let
$$
 n(\degreed) = \int_\degreed c_1 (TX) - 1
$$
be the expected dimension
of the moduli space of
stable maps
of genus zero and degree $\degreed$
and
$$
 N_\degreed = \langle 
        \underbrace{[\pt], \ldots, [\pt]}_{n(\degreed) \ \text{times}})
       \rangle_{0,\degreed}
$$
be the Gromov-Witten invariant,
where $[\pt]$ denotes
the Poincar\'{e} dual
of the homology class
of a point.
When $n(\degreed) > 0$,
$N_\degreed$ is
the number of
nodal rational curves in $X$
of degree $\degreed$
passing through $n(\degreed)$ points
in general position.
Although it is possible to count
$N_{\degreed}$
by elementary method
when the degree $\degreed$ is small,
the difficulty grows rapidly
as $\degreed$ increases.

The theory of Gromov-Witten invariants
gives
a new perspective
to this classical problem
of counting $N_{\degreed}$
in enumerative geometry.
Instead of considering $N_\degreed$'s individually,
one can treat them as a whole by
the following generating function
%, which is a formal function
%at the origin
on the cohomology ring $H^*(X;\bC)$:
\begin{equation} \label{eq:dppotential}
 \Fpotential(t_0, \ldots, t_8)
  = \frac{1}{2} t_0^2 t_8 + t_0 (t_1^2 - t_2^2 - \cdots - t_7^2)
     + \sum_{\degreed \in H_2(X;\bZ)}
         N_{\degreed} q^\degreed
          \frac{t_8^{n(\degreed)}}{n(\degreed)!}. 
\end{equation}
Here,
$t_0, t_1, t_2, \ldots, t_7, t_8$
are coordinates
on the total cohomology ring $H^*(X;\bC)$
of $X$
corresponding to the basis
$[X], [H], [E_1], \ldots, [E_6], [\pt]$
of $H^*(X;\bC)$,
where
%$[A]$ denotes
%the Poincar\'{e} dual
%of the cycle $A$;
$[X]$ denotes
the dual
of the fundamental class of $X$,
$[H]$
that of the hyperplane class, and
$[E_i]$
that of 
the exceptional divisor
for $i=1, \ldots, 6$,
and
$
 q^\degreed = \exp[\langle \degreed, [H] \rangle t_1]
              \exp[\langle \degreed, [E_1] \rangle t_2]
              \cdots
              \exp[\langle \degreed, [E_6] \rangle t_7].
$
Then,
this $\Fpotential$
endows $H^*(X;\bC)$
with the structure of a
{\em Frobenius manifold}.
%Take the following function
%on the total cohomology ring of $X$;
%$$
% \Fpotential(t_0, \ldots, t_8)
%  = \frac{1}{2} t_0^2 t_8 + t_0 (t_1^2 - t_2^2 - \cdots - t_7^2)
%     + \sum_{d, n} N_{d, n} q^d \frac{t_8^{m(d)}}{m(d)!}
%$$
%where
%$$
% m(d) = \langle d, -K_X \rangle
%$$
%is the dimension of the moduli space
%of stable maps of genus $0$ and 
%degree $d$ without marked points.
%Here, $t_0$ is the coordinate
%corresponding to the fundamental class
%in $H^4(X;\bZ)$
%(an element of the homology
%gives a coordinate on the cohomology),
%$t_1$ corresponds to
%the hyperplane class in $H_2(X;\bZ)$,
%$t_2, \ldots, t_7$ to the exceptional divisors,
%and $t_8$ to the point class in $H_0(X;\bZ)$.
%Therefore, the knowledge of the function
%$\Phi$ is equivalent
%to the knowledge of all the numbers $N_{d,n}$'s.
%Then this function $\Phi$ makes
%the total cohomology ring $H^*(X;\bC)$
%a {\em Frobenius manifold}.

A Frobenius manifold is a complex manifold
whose holomorphic tangent bundle
has a bilinear form and
an associative commutative product structure
satisfying a series of axioms.
%This bilinear form is called
%the {\em metric}.
%Note that this metric
%is bilinear
%instead of Hermitian.
%It must be flat, i.e.,
%there should exist a local coordinate
%on the Frobenius manifold
%called the {\em flat coordinate}
%such that the components of the metric
%with respect to this coordinate
%are constant.
%This flat coordinate is unique
%up to affine transformations.
%Then, with respect to this coordinate,
%%A part of these compatibility conditions
%%implies that 
%the structure constant
%of these products are given by
%third derivatives
%of a function on the Frobenius manifold
%called
%{\em potential}.
%Another ingredient of a Frobenius manifold
%is the {\em Euler vector field},
%which gives degrees
%to the flat coordinates,
%and every structure on the Frobenius manifold
%must be homogeneous
%with respect to this degree.
%This homogeneity structure
%is important
%in the relation
%between Frobenius manifolds
%and
%isomonodromic deformations.
%
%It follows
%from the axioms of the Gromov-Witten invariants
%and the geometry
%of the moduli space of curves
%that
%the generating function
%in (\ref{eq:dppotential})
%is a potential function
%which makes
%$H^*(X; \bC)$
%into a Frobenius manifold.
%The flat coordinate
%in this case
%is just the coordinate
%on $H^*(X; \bC)$
%as a vector space.
%Since the tangent bundle to $H^*(X; \bC)$
%is just $H^*(X; \bC) \times H^*(X; \bC)$,
%the product structure
%on the tangent bundle
%of $H^*(X; \bC)$ as a Frobenius manifold
%is a product structure
%on $H^*(X; \bC)$
%parametrized by
%$H^*(X; \bC)$ itself.
%This product structure on $H^*(X; \bC)$
%is called the quantum cohomology ring.
In the case of a Frobenius manifold
coming from the Gromov-Witten invariants
of a smooth projective variety,
the bilinear form
is given by
the Poincar\'{e} pairing,
and the structure constant
of the product structure
is given by the third derivative
of the generating function
$\Fpotential$.
This product structure
is called the quantum cohomology ring,
whose associativity gives
a highly non-trivial non-linear
differential equation
for $\Fpotential$,
called the {\em WDVV
(Witten-Dijkgraaf-Verlinde-Verlinde)}
equation,
which sometimes determines
all the genus-zero
Gromov-Witten invariants
from a finite numbers of them
\cite{Kontsevich-Manin_reconstruction}.
%See
%\cite{Goettsche-Pandharipande}
%and references therein
%for details.

An important point
in the theory of Frobenius manifolds
is its connection
%with other fields of mathematics
%such as singularity theory and
%integrable systems.
%The connection with singularity theory
%comes from the theory of primitive forms
%by K. Saito.
%See \cite{KSaito}
%for the theory of primitive forms,
%and
%\cite{Hertling_FM},
%\cite{Sabbah_FM}
%for its relationship
%with the theory
%of Frobenius manifolds.
%This connection
%with singularity theory
%is important
%in mirror symmetry,
%which will be discussed
%later in this Introduction
%and plays a key role in this paper.
%
%There are several connections
%between the theory of Frobenius manifolds
%and integrable systems.
%The most notable of them
%is the Witten's conjecture
%\cite{Witten_2DG}
%proved by Kontsevich
%\cite{Kontsevich_ITMSCMAF}
%relating the generating function
%of the Gromov-Witten invariants
%of a point
%and a $\tau$-function
%of the KdV hierarchy.
%To generalize this result
%to more non-trivial target spaces
%is a vastly open problem,
%although there have been
%a lot of progress in many years,
%especially for the case of $\bP^1$.
%See, e.g.,
%\cite{Eguchi-Hori-Xiong},
%\cite{Dubrovin-Zhang_NFH},
%\cite{Givental_SFSHG},
%\cite{Givental_GWIQQH},
%\cite{Getzler_TC},
%\cite{Okounkov-Pandharipande_EGWTP1}.
%
%Another kind of connection
%between the Gromov-Witten theory
%and integrable systems
%is the relation between
%the quantum cohomology of flag manifolds
%and Toda lattices
%\cite{Givental-Kim},
%\cite{Kim_QCFM},
%\cite{Givental-Lee},
%which has recently
%been given a new impetus
%by the Nekrasov's conjecture
%\cite{Braverman},
%\cite{Braverman-Finkelberg}.
%
%Lastly,
%Frobenius manifolds
%are intimately related
with isomonodromic deformations.
Given a Frobenius manifold,
one can construct
an isomonodromic family
of ordinary differential equations
on $\bP^1$
of the following form:
\begin{equation}
 \hbar \frac{\partial \unknownY}{\partial \hbar}
   + \frac{1}{\hbar} \scU(\unknownY)
   - \scV(\unknownY)
  = 0,
   \label{eq:intro_hbar-direction}
\end{equation}
\begin{equation}
  \hbar \frac{\partial \unknownY}{\partial t_\flata}
  = \frac{\partial}{\partial t_\flata} \circ \unknownY, \quad
  \flata=0,\ldots,N-1.
   \label{eq:intro_t-direction}
\end{equation}

Here, $\unknownY$ is an unknown function on $\bP^1$
times the Frobenius manifold
taking value in the tangent bundle of
the Frobenius manifold,
$\hbar$ is the coordinate on $\bP^1$,
$N$ is the dimension of the Frobenius manifold,
$\{t_\flata\}_{\flata=0}^{N-1}$ is
the {\em flat coordinate}
of the Frobenius manifold,
$\circ$ denotes the product on the tangent bundle,
and $\scU$, $\scV$ are certain operators
%which are independent of $\hbar$,
acting on sections of the tangent bundle.
%See Dubrovin \cite{Dubrovin_G2DTFT}
%for details.
%Note that $z$ loc. cit. is
%$1/\hbar$ in this paper.
(\ref{eq:intro_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:intro_t-direction}) gives
its isomonodromic deformation.
If a point
on the Frobenius manifold
is semisimple,
i.e., if there are no nilpotent elements
in the product structure
on the tangent space
at this point,
one can define the {\em monodromy data}
of (\ref{eq:intro_hbar-direction})
at this point,
consisting of the monodromy matrix at infinity,
the Stokes matrix at the origin,
%(i.e., the connection matrix of two solutions
%defined on two overlapping angular domains
%which asymptote to the same formal solution),
and the connection matrix between
infinity and the origin.
These data do not depend
on the choice of a semisimple point
because of the isomonodromicity.
%Dubrovin's reconstruction theorem
%\cite{Dubrovin_PT2DTFT}
%states that
%in the semi-simple case,
%the Frobenius manifold
%(which encodes the information
%of the Gromov-Witten invariants of $X$)
%is uniquely determined by its monodromy data.
%Stokes matrix is the most important piece
%of these monodromy data.

The following conjecture,
originally due to Kontsevich,
developed by Zaslow \cite{Zaslow}, and
formulated into the following form
by Dubrovin \cite{Dubrovin_GATFM},
reveals a striking connection
between the Gromov-Witten invariants
and the derived category of coherent sheaves:

\begin{conjecture} \label{conj:stokes}

The quantum cohomology of a smooth projective variety $X$
is semisimple
if and only if
the bounded derived category $\DbX$
of coherent sheaves on $X$
is generated as a triangulated category
by an exceptional collection
$( \scE_i )_{i=1}^N$.
In such a case, the Stokes matrix $S$
for the quantum cohomology of $X$
is given by
\begin{equation} \label{eq:conjecture}
S_{ij} = \sum_k (-1)^k \dim \Ext^k(\scE_i, \scE_j).
\end{equation}
\end{conjecture}

An exceptional collection appearing above
is the following:

\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 any $k$.

\end{enumerate}

\end{definition}

%Conjecture \ref{conj:stokes} is interesting
%in that
%the left-hand side of the equality
%depends only on the symplectic structure
%of $X$
%and independent of the complex
%structure of $X$
%whereas the right-hand side
%depends only
%on the complex structure
%of $X$.
%Indeed,
%the right-hand side
%is determined by the
%Gromov-Witten invariants,
%which are invariant
%under the deformation of the complex structures
%by the axiom.
%Note that
%the left-hand side of (\ref{eq:conjecture})
%does not depend on the complex structure of $X$
%whereas the right-hand side
%does not depend on the symplectic structure of $X$.
%Therefore,
%Conjecture \ref{conj:stokes}
%can be considered
%as a mysterious relationship
%between symplectic geometry
%and complex geometry
%of the same manifold $X$.

%To our knowledge,
Conjecture \ref{conj:stokes} was previously known to hold
for projective spaces
\cite{Dubrovin_PT2DTFT},
\cite{Guzzetti}
and Grassmannians
\cite{Ueda_SMQCG}.
%The proofs
%of Dubrovin and Guzzetti use
%Barnes-type integral representations
%of the hypergeometric functions
%which are the solutions
%to (\ref{eq:intro_hbar-direction}),
%and are purely analytic.
The main result in this paper is:

\begin{theorem} \label{th:main_theorem}
 Conjecture \ref{conj:stokes} holds
 for smooth cubic surfaces in $\bP^3$.
\end{theorem}

%Although the WDVV equation of associativity
%gives an efficient way to compute the Gromov-Witten invariants
%of the cubic surface recursively
%starting from low degrees,
%the Stokes matrix
%contains somewhat more global data
%of the corresponding isomonodromic deformation
%is a completely different problem
%and requires another method.

To prove
Theorem \ref{th:main_theorem},
we first use the deformation invariance
of the Gromov-Witten invariants
to reduce the problem
to the case of
a toric surface
$\blsix$
defined in Section \ref{sc:vanishing_cycles}
obtained
by bringing
some of the blowing-up centers
to infinitely-near points.
Then
we can use
the Givental's mirror theorem
\cite{Givental_MTTCI},
which gives an integral representation
of the fundamental solution
to (\ref{eq:intro_hbar-direction}),
(\ref{eq:intro_t-direction})
for $\blsix$
%whose anticanonical bundle is nef.
%and stationary-phase integrals
%Although this mirror theorem
%is applicable only to toric manifolds
%and smooth cubic surfaces are not toric,
%they can be deformed
%to a toric manifold.
%Then we can apply mirror theorem
%to this toric manifold
%which is deformation-equivalent
%to cubic surfaces
%since the Gromov-Witten invariants
%are symplectic invariants
%and is independent of the choice of
%complex structures.
%Then Givental's theorem
%gives an integral representation
%of the flat sections
%of the first structure connection
of the following form:

\begin{equation} \label{eq:int_rep}
 I_\Gamma = \int_{\Gamma}
  \phi(x,y;\hbar) \exp[W(x,y)/\hbar] \frac{dx dy}{x y}
\end{equation}

Here,
$\phi(x,y;\hbar)$
is a certain $H^*(\blsix;\bC)$-valued function
of $x$, $y$, $\hbar$ and
the flat coordinate
$\{ t_\flata \}_{\flata=0}^8$,
and $W(x,y)$ is a Laurent polynomial
of $x$ and $y$
whose coefficients depend on the flat coordinate.
$\Gamma$ runs over
certain cycles in $(\bCx)^2$
depending on $\hbar$
and flat coordinate,
which are descending Morse cycles
for $\Re[W(x,y)/\hbar]$
for a suitable choice of a metric on $(\bCx)^2$.
%
%In general,
%Givental's mirror theorem
%requires a change of coordinate
%called the {\em mirror transformation}.
%When the anticanonical bundle
%of the toric variety is ample,
%then this mirror transformation is trivial,
%whereas when the anticanonical bundle
%is not even nef,
%this mirror transformation can be even
%non-convergent.
%In our case,
%the anticanonical bundle of
%the toric manifold $\blsix$
%is not ample but nef,
%so the mirror transformation
%is non-trivial but convergent
%and we can compute the monodromy
%of the flat sections of the first structure connection
%from that of the stationary-phase integrals.

Since the integrand
in (\ref{eq:int_rep})
is single-valued,
the monodromy of the integral
comes solely from the monodromy
of the integration cycle
$\Gamma$,
which is determined by
intersection numbers of vanishing cycles
by the Picard-Lefschetz theory.
This gives the
left-hand side of (\ref{eq:conjecture}).
On the right-hand side,
one can find an exceptional collection
generating the derived category of coherent sheaves
on $\blsix$
by combining the theorems
of Beilinson \cite{Beilinson}
and Kapranov-Vasserot \cite{Kapranov-Vasserot}.
The $\Ext$-groups between them
can be computed explicitly,
and exhibit a complete agreement
with the result
on the left-hand side.

%Let us sum up:
%\begin{itemize}
% \item The Gromov-Witten invariants are invariants of smooth projective
%       variety, and has enumerative significance.
% \item The theory of Frobenius manifolds allows us
%       to study these invariants systematically
%       using the generating function.
%       For example, the WDVV equation of associativity
%       allows us to compute all the Gromov-Witten invariants
%       recursively from a few initial conditions.
% \item The theory of Frobenius manifolds is closely related
%       to isomonodromic deformations.
%\end{itemize}


{\bf Acknowledgements}:
We thank
H. Iritani
for patiently explaining
mirror symmetry
for toric manifolds.
We also thank
A. Ishii,
T. Kawai,
H. Kawanoue,
H. Nasu,
K. Saito,
A. Takahashi,
and H. Uehara
for valuable discussions and comments.
The author is supported by
JSPS Fellowships for Young Scientists
No.15-5561.

\section{Frobenius manifold and Stokes matrix}
 \label{sc:Frobenius_manifolds}

We review the relation between Frobenius manifolds
and isomonodromic deformations
in this section.
See, e.g.,
\cite{Dubrovin_G2DTFT},
\cite{Dubrovin_PT2DTFT},
\cite{Hertling_FM},
\cite{Hitchin_FM},
\cite{Manin_FM}
for more details.

\begin{definition}
A Frobenius manifold is a quintuple
$(M,\circ,e,E,g)$
satisfying the following axioms:
\begin{enumerate}
 \item $M$ is a complex manifold.
 \item $g:\scT_M \otimes \scT_M \rightarrow \scO_M$ is an
       $\scO_M$-bilinear form.
 \item $\circ:\scT_M \otimes \scT_M \rightarrow \scT_M$ defines
       an associative commutative
       $\scO_M$-algebra structure on $\scT_M$.
 \item $g(X \circ Y, Z) = g(X,Y\circ Z)$
       as an $\scO_M$-linear map
       $
        \scT_M \otimes \scT_M \otimes \scT_M
         \rightarrow \scO_M
       $
       (the Frobenius property).
 \item $g$ is flat, i.e., the Levi-Citiva connection
       $\nabla$ associated to $g$ is flat. \label{cond:flatness}
 \item The $(3,1)$-tensor $\nabla \circ$ is totally symmetric
       (the potentiality condition).
 \item $e \in \Gamma(M, \scT_M)$ is an identity element for $\circ$.
 \item $e$ is flat, i.e., $\nabla e=0$.
 \item $E \in \Gamma(M, \scT_M)$ satisfies
       \begin{equation} \label{eq:linearity}
	\nabla \nabla E=0, % \qquad \text{linearity},
       \end{equation}
       \begin{equation} \label{eq:homogeneity_1}
	\Lie_E(\circ)=\circ,
       \end{equation}
       and there exists a complex number $D$ such that
       \begin{equation} \label{eq:homogeneity_2}
	\Lie_E(g)=D g.
       \end{equation}
        \label{cond:euler}
\end{enumerate}
Here, $\scO_M$ is the structure sheaf of $M$,
$\scT_M$ is the tangent sheaf of $M$,
and $\Lie_E$ is the Lie derivative
with respect to the vector field $E$.
We call $g$ the metric, $E$ the Euler vector field,
and $D$ the charge of the Frobenius manifold $M$.
\end{definition}
%$$
% X g(Y,Z)=g(\nabla_X Y, Z)+ g(Y, \nabla_X Z),
% \quad \mbox{(metric connection)}
%$$
%$$
% \nabla_X Y - \nabla_Y X - [X,Y]=0.
% \quad \mbox{(torsion-free)}
%$$
Let $N$ denote
the dimension of $M$.
Since $g$ is flat,
there exists a local coordinate
$\{t_\flata \}_{\flata=0}^{N-1}$
on $M$,
called the {\em flat coordinate},
satisfying
$$
g(\partial_\flata, \partial_\flatb) = \eta_{\flata \flatb}
$$
which is unique up to affine transformations.
Here, $\eta_{\flata \flatb}$ is a constant matrix
and 
$$
\partial_\flata = \frac{\partial}{\partial t_\flata}.
$$
Since the identity vector field
$e$ is flat,
we can choose
the flat coordinate
$\{ t_\flata \}_{\flata=0}^{N-1}$
so that
$$
 e = \partial_0.
$$
Let $\Fstrconst_{\flata \flatb}^\flatc$ be the structure constant
of the product $\circ$
in the flat coordinate:
$$
 \partial_\flata \circ \partial_\flatb
  = \sum_{\flatc = 0}^{N-1}
     \Fpotential_{\flata \flatb}^\flatc \partial_\flatc.
$$
Then the potentiality condition
is that
$$
\sum_{\flate = 0}^{N-1} \eta_{\flata \flate}
 \partial_\flatb \Fstrconst_{\flatc \flatd}^\flate
$$
is totally symmetric
with respect to $\flata, \flatb, \flatc, \flatd$.
From this condition,
it follows that
locally on $M$,
there exists a holomorphic function
$\Fpotential$
such that
\begin{equation}
\sum_{\flatd=0}^{N-1} \eta_{\flata \flatd} \Fstrconst_{\flatb \flatc}^\flatd
 = \partial_\flata \partial_\flatb \partial_\flatc \Fpotential.
     \label{eq:str_const} 
\end{equation}
This function $\Fpotential$
is called the potential of the Frobenius manifold $M$.
Note that
$$
 \eta_{\flata \flatb}
  = \partial_{0} \partial_{\flata} \partial_{\flatb}
     \Fpotential
$$
follows from the Frobenius property.

The associativity
of $\circ$
gives the following
{\em WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation}
\begin{equation} \label{eq:WDVV}
  \sum_{e,f = 0}^{N-1} \frac{\partial^3 \Fpotential}
  {\partial t_\flata  \partial t_\flatb \partial t_e}
  \eta^{e f} \frac{\partial^3 \Fpotential}
  {\partial t_f \partial t_\flatc \partial t_\flatd}
 =  \sum_{e,f = 0}^{N-1} \frac{\partial^3 \Fpotential}
  {\partial t_\flata \partial t_\flatd \partial t_e}
  \eta^{e f} \frac{\partial^3 \Fpotential}
  {\partial t_f \partial t_\flatb \partial t_\flatc},
\end{equation}
which is a non-linear
partial differential equation
for the potential $\Fpotential$.
Here, $\eta^{\flata \flatb}$
is the inverse matrix
of $\eta_{\flata \flatb}$ :
$
\sum_{\flatc=0}^{N-1} \eta^{\flata \flatc} \eta_{\flatc \flatb}
 = \delta^\flata_\flatb.
$

%The condition (\ref{eq:linearity})
%means that
%$E$ is at most linear
%in the flat coordinate;
%$$
% E = \sum_{\flata=0}^{N-1} \left[ (1-q_\flata) t_\flata
%      + r_\flata \right] \partial_{\flata},
%$$
%where
%$q_\flata, r_\flata \in \bC$,
%$\flata = 0, \ldots, N-1$.
%The condition
%(\ref{eq:homogeneity_1}),
%(\ref{eq:homogeneity_2})
%means that
%if one defines the degree operator
%on functions
%by $E$
%and on vector fields
%by $\id + \Lie_E$,
%then $\circ$ preserves this degree
%and $g$ is homogeneous of degree $D-2$;
%if two vector fields
%$X$ and $Y$ satisfies
%$$
% X + \Lie_E X = p X
%$$
%and
%$$
% Y + \Lie_E Y = q Y,
%$$
%then
%$$
% X \circ Y + \Lie_E (X \circ Y) = (p + q) (X \circ Y)
%$$
%and
%$$
% E g(X, Y) = (p+q+D-2) g(X,Y).
%$$

Now let us define
$\scU, \scV \in \Gamma(\scEnd(\scT_M))$
by
\begin{eqnarray}
 \scU(X) &=& E \circ X, \label{eq:U} \\
 \scV(X) &=& \nabla_X E-\frac{D}{2} X \label{eq:V}, %\\
%          &=& \nabla_E X-[E,X]-\frac{D}{2}X \nonumber,
\end{eqnarray}
and let
$\pi : \bCx \times M \rightarrow M$
be the second projection.
Define a connection
$$
\fscon : \pi^* \scT_M \rightarrow \pi^* \scT_M \otimes
 \Omega_{\bCx \times M}^1
$$
on $\pi^* \scT_M$,
where $\Omega_{\bC^\times \times M}^1$
denotes the cotangent sheaf of
$\bCx \times M$,
by
\begin{eqnarray}
 \fscon_X Y &=& \nabla_X Y
  - \frac{1}{\hbar} X \circ Y,
  \label{eq:X-direction} \\
 \fscon_{\partial_\hbar} Y &=& \partial_\hbar Y
   + \frac{1}{\hbar^2} \scU(Y)
   - \frac{1}{\hbar} \scV(Y),
  \label{eq:hbar-direction}
\end{eqnarray}
for $X$ and $Y$
which are pull-backs
of local sections of $\scT_M$
to $\bCx \times M$
and $\hbar$ is the inhomogeneous coordinate
of $\bP^1$,
and extend it by the Leibniz rule.
Note the $\hbar$ here is the notation by Givental
and corresponds to
$1 / z$ in Dubrovin's papers \cite{Dubrovin_G2DTFT},
\cite{Dubrovin_PT2DTFT}.
$\fscon$ is called
the {\em first structure connection}
of the Frobenius manifold $M$.
The following Theorem \ref{th:fsflat}
is a keystone
in the theory of Frobenius manifolds:
\begin{theorem} \label{th:fsflat}
 $\fscon$ is flat.
\end{theorem}
For the proof,
see the references
at the beginning of this section.
One can see from the proof
that the flatness of $\fscon$
encodes many of the axioms
of Frobenius manifolds.

(\ref{eq:hbar-direction})
can be considered as
a family of meromorphic connections
on $\bP^1$
parametrized by $M$,
and the flatness of $\fscon$
means that
this family is isomonodromic.
One can see from (\ref{eq:hbar-direction})
that this meromorphic connection
has a regular singularity
at infinity
and an irregular singularity
of Poincar\'{e} rank 1
at the origin.

\begin{definition}
 A point $p$ on a Frobenius manifold $M$
 is semisimple
 if the product structure $\circ$
 on the tangent space $T_p M$ of $M$
 at $p$
 is semisimple
 (i.e., if there are no nilpotent elements).
\end{definition}

By Dubrovin \cite{Dubrovin_G2DTFT},
%(3.47), (3.57),
there exists a local coordinate
$\{ u_\canonicala \}_{\canonicala=1}^N$
satisfying
\begin{eqnarray}
 \partial_\canonicala \circ \partial_\canonicalb
  = \delta_{\canonicala \canonicalb} \partial_\canonicala,
 \label{eq:canonical_idempotent} \\
 E = \sum_{\canonicala=1}^N u_\canonicala \partial_\canonicala,
 \label{eq:canonical_euler}
\end{eqnarray}
where
$\partial_\canonicala
 = \frac{\partial}{\partial u_\canonicala}$,
in the neighborhood of a semisimple point.
Since
\begin{eqnarray*}
 g(\partial_\canonicala, \partial_\canonicala)
  &=& g(\partial_\canonicala, \partial_\canonicalb
           \circ \partial_\canonicalb) \\
  &=& g(\partial_\canonicala \circ \partial_\canonicalb,
           \partial_\canonicalb),
\end{eqnarray*}
$\partial_\canonicala$ and 
$\partial_\canonicalb$ are orthogonal
for $\canonicala \neq \canonicalb$.
This coordinate
$\{ u_\canonicala \}_{\canonicala=1}^N$
is called the {\em canonical coordinate},
which is unique up to permutations.
By
(\ref{eq:canonical_idempotent}) and
(\ref{eq:canonical_euler}),
$\{ u_\canonicala \}_{i=1}^N$ is characterized
as the set of eigenvalues of $\scU$.

Let
$$
 f_\canonicala = \Delta_\canonicala^{-1/2}
  \partial_\canonicala,
 \quad i=1, \ldots, N,
$$
be the normalized canonical vector field,
where
$$
 \Delta_\canonicala
  = g(\partial_\canonicala, \partial_\canonicala),
$$
and define an $N \times N$ matrix
$\Psi_{\flata \canonicala}$ by
$$
f_\canonicala
  = \sum_{\canonicala = 1}^N
     \Psi_{\flata \canonicala}  \frac{\partial}{\partial t_\flata}.
$$
$\Psi_{\flata \canonicala}$
is the coordinate transformation matrix
from the normalized canonical coordinate
to the flat coordinate.
%Let
%$$
% U = \diag(u_1, \ldots, u_N)
%$$
%be the matrix
%representing the operator $\scU = E \circ$
%in the normalized canonical basis
%$\{ f_\canonicala \}_{\canonicala=1}^N$.
%Now
%fix a semisimple point $p$
%on a Frobenius manifold.
%and consider the equation
%for the flat section
%of the first structure connection
%in the $\hbar$-direction:
%\begin{equation} \label{eq:hbar-flatness}
% \partial_{\hbar}
%  = \frac{1}{\hbar^2} \scU(Y)
%     + \frac{1}{\hbar} \scV(Y).
%\end{equation}
%
%\begin{lemma}[Dubrovin \cite{Dubrovin_PT2DTFT}, Lemma 4.3.]
% \label{lemma:formal_sol}
%$
% R(\hbar) = (1 + R_1 \frac{1}{\hbar} + R_2 \frac{1}{\hbar^2} + \cdots)
%$
%is a formal series
%\end{lemma}
The operator
$\scU$
defined in (\ref{eq:U})
is diagonal
in the normalized canonical coordinate:
$$
 \scU(\partial_\canonicala)
 = \sum_{\canonicalb=1}^N u_\canonicalb \partial_\canonicalb \circ
    \partial_\canonicala
 = u_\canonicala \partial_\canonicala.
$$
Let $U,V$ be the matrices
representing
the operators $\scU, \scV$
in the normalized canonical coordinate.
Then
$$
 U = \diag(u_1, \ldots, u_N)
$$
and (\ref{eq:hbar-direction}) becomes
\begin{equation} \label{eq:nc_hbar-direction}
 \partial_\hbar Y + \frac{1}{\hbar^2} U Y - \frac{1}{\hbar} V Y = 0.
\end{equation}

By Dubrovin
\cite{Dubrovin_PT2DTFT},
Lemma 4.3.,
there exists a unique
$N \times N$-matrix-valued formal series
$$
 R(\hbar) = 1 + R_1 \hbar + R_2 \hbar^2 + \cdots
$$
satisfying
$$
 R^{t}(\hbar) R(-\hbar) = 1
$$
such that
$$
 Y = R(\hbar) \widetilde{Y}
$$
transforms (\ref{eq:nc_hbar-direction})
to
\begin{equation} \label{eq:gt_hbar-direction}
 \hbar \partial_{\hbar} \widetilde{Y} + U \widetilde{Y} = 0.
\end{equation}
Here, $\bullet^t$ denotes the transpose
of a matrix.
Since (\ref{eq:gt_hbar-direction}) has a fundamental solution
of the form
$$
 \widetilde{Y} = \exp[U / \hbar],
$$
(\ref{eq:hbar-direction})
has a formal fundamental solution
of the form
\begin{equation} \label{eq:formal_solution}
 \Phi_\formal = \Psi R(\hbar) \exp[U / \hbar]
\end{equation}
in the flat coordinate.

\begin{definition}
For $0 \leq \phi < \pi$,
a straight line
$l = \{\hbar \in \bCx \suchthat \arg(\hbar) = \phi, \phi - \pi \}$
passing through the origin
is called {\em admissible}
if the line through $u_\canonicala$ and $u_\canonicalb$
is not orthogonal to $l$
for any $\canonicala \neq \canonicalb$.
\end{definition}

Fix such a 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] \label{fg:admissible_line}
 \centering
 \psfrag{Dl}{$D_\lt$}
 \psfrag{Dr}{$D_\rt$}
 \psfrag{D-}{$D_-$}
 \psfrag{phi}{$\phi$}
 \psfrag{l}{$l$}
 \includegraphics{stokes.eps}
 \caption{An admissible line and angular domains}
\end{figure}
Define
angular domains $D_{\rt}$, $D_{\lt}$ and $D_-$ by
\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 \bCx \suchthat
 \phi - \pi - \epsilon < \arg(\hbar) <\phi + \epsilon \},
 \nonumber \\
D_{\lt} & = & \{\hbar \in \bCx \suchthat
 \phi - \epsilon < \arg(\hbar) <\phi +\pi + \epsilon \}, \\
D_- & = & \{\hbar \in \bCx \suchthat
 \phi - \pi - \epsilon < \arg(\hbar) < \phi - \pi + \epsilon \}.
 \nonumber
\end{eqnarray}
See Figure \ref{fg:admissible_line}.
Since the singularity at the origin
is irregular,
the formal solution $\Phi_\formal(\hbar)$
does not converge.
Nevertheless,
by \cite{Dubrovin_PT2DTFT} Theorem 4.2.,
there exist unique solutions
$\Phi_\rt(\hbar)$ and $\Phi_\lt(\hbar)$,
defined on the angular domains $D_\rt$ and $D_\lt$
respectively,
which asymptote
to the same 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 $D_-$,
there exists a matrix $S$
independent of $\hbar$
such that
\begin{equation*}% \label{eq:stokes_def}
 \Phi_\lt(\hbar) = \Phi_\rt(\hbar) S,
 \qquad %\mbox{for}
 \quad \hbar \in D_-.
\end{equation*}
This matrix $S$ is called the Stokes matrix.
%Although locally on the Frobenius manifold
%this Stokes matrix does not depend
%on the choice of a semisimple point
%by \cite{Dubrovin_PT2DTFT},
%Isomonodromicity Theorem (second part),
%it undergoes a discrete change
%as we vary the point
%on the Frobenius manifold
%so that it crosses the point
%where the line $l$
%we have fixed at the beginning
%is not admissible
%any more.
%This change in the Stokes matrix
%is described
%by an action of the braid group
%$B_N$
%(the number of strands
%is the dimension of the Frobenius manifold).

\section{Gromov-Witten invariants}
 \label{sc:Gromov-Witten_invariants}

Let $X$ be a smooth projective variety
over $\bC$.
For simplicity,
let us assume
that $H^k(X;\bC)=0$
for odd $k$.

\begin{definition}
$(f,C,p_1,\ldots,p_n)$
is a stable map
to $X$
of genus $g$
with $n$ marked points
if $C$ is a complete curve
of arithmetic genus $g$
with at worst ordinary double points,
$(p_1,\ldots,p_n)$
is an ordered set of $n$ points
on $C$,
$f:C \rightarrow X$
is a regular map
and the automorphism group
of $f$ is finite.
In this case,
$f_*[C] \in H_2(X,\bZ)$
is called the degree
of $f$.
\end{definition}

Let $\M_{g,n}(X;\degreed)$ be
the moduli space of
stable maps to $X$
of degree $\degreed$ and
genus $g$
with $n$ marked points.
We have the evaluation map
$$
\begin{array}{cccc}
 \ev_i: & \M_{g,n}(X;\degreed)
   & \longrightarrow & X \\
 & \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 & (f,C,p_1,\ldots,p_n) & \longmapsto &
   f(p_i).
\end{array}
$$
Let $\scL_i$
be the line bundle on $\M_{g,n}(X;\degreed)$
whose fiber over
$(f,C,p_1,\ldots,p_n) \in \M_{g,n}(X;\degreed)$
is the cotangent space
$T_{p_i}^* C$
at the $i$-th marked point $p_i$.
%$$
%\begin{array}{cccc}
% \fgt: & \M_{g,n}(X;\degreed)
%   & \longrightarrow & \M_{g,n} \\
% & \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
% & ((C,(z_1,\ldots,z_n)),\psi) & \longmapsto &
%   (C,(z_1,\ldots,z_n))
%\end{array},
%$$
The virtual dimension
of $\M_{g,n}(X;\degreed)$
is
$$
(1-g)(\dim_\bC X - 3) + \int_\degreed c_1(T X) + n,
$$
where
$c_1(T X)$ is the first Chern class
of the tangent bundle of $X$.
Given $n$ cohomology classes
$\gamma_1, \ldots, \gamma_n \in H^*(X;\bC)$
and non-negative integers
$d_1, \ldots, d_n$,
the Gromov-Witten invariant
is defined by
$$
 \langle \tau_{d_1} \gamma_1, \ldots,
  \tau_{d_n} \gamma_n \rangle_{g, \degreed}
  = \int_{[\M_{g,n}(X;\degreed)]^{\mathrm{virt}}}
     \prod_{i=1}^n
      \left(
       c_1(\scL_i)^{d_i} \cup \ev_i^*(\gamma_i)
      \right).
$$
Here,
$[\M_{g,n}(X;\degreed)]^{\mathrm{virt}}$
is the {\em virtual fundamental class}
\cite{Behrend-Fantechi},
\cite{Li-Tian_virtual}.

%$\langle I_{g,n,\degreed} \rangle$
%is defined by
%\begin{eqnarray*}
%& \langle I_{g,n,\degreed} \rangle \; : \; H^*(X;\bQ)^{\otimes n}
%    \longrightarrow  \bQ \; , & \\
%& \langle I_{g,n,\degreed} \rangle (\alpha_1 \otimes \cdots \otimes \alpha_n)
% = \int_{\M_{g,n}(X;\degreed)}
%     \ev^* (\alpha_1 \otimes \cdots \otimes \alpha_n). & 
%\end{eqnarray*}
%
%Here,
%we have considered the element
%$\alpha_1 \otimes \cdots \otimes \alpha_n$
%of $H^*(X;\bQ)^{\otimes n}$
%as an element of $H^*(X^n;\bQ)$
%by the K\"{u}nneth isomorphism.
%In general,
%$\M_{g,n}(X;\degreed)$ is not smooth
%and its dimension is higher than
%the virtual dimension
%(the dimension of the moduli space
%if the deformation of stable maps is unobstructed,
%i.e., the dimension computed by the Riemann-Roch theorem)
%$$
% (1-g)(\dim_\bC X - 3) - \int_\degreed \omega_X + n.
%$$
%Even in such a case,
%Gromov-Witten invariants
%can be defined by using
%{\em virtual fundamental class}.
%See, e.g.,
%\cite{Behrend-Fantechi},
%\cite{Li-Tian_virtual}
%for details.

Let $\{ T_\flata \}_{\flata=0}^{N-1}$
be a basis of $H^*(X;\bC)$
and let $\{ t_\flata \}_{\flata=0}^{N-1}$
be the corresponding coorinate
of $H^*(X;\bC)$.
Set $\deg t_\flata = \deg T_\flata = k$
when $T_\flata \in H^k(X;\bC)$.
Assume that
the basis $\{ T_\flata \}_{\flata=0}^{N-1}$ are taken
so that $T_0$ is the basis of $H^0(X;\bC)$.
Let
$$
 \gamma = \sum_{\flata = 0}^{N-1} t_\flata T_\flata.
$$
The following formal power series
$\Phi(\gamma)$
in $\{t_i\}_{i=0}^{N-1}$
is called
the {\em Gromov-Witten potential}:
$$
 \Phi(\gamma) = \sum_{n=0}^\infty \sum_{\degreed \in H_2(X,\bZ)}
  \frac{1}{n!}
   \langle
    \underbrace{\gamma, \ldots, \gamma}_{n \ \text{times}}
   \rangle_{0, \degreed}. % \q^\degreed
$$
Since
$$
 \frac{1}{n!}
   \langle
    \underbrace{\gamma, \ldots, \gamma}_{n \ \text{times}}
   \rangle_{0, \degreed}
 = \sum_{|\alpha| = n}
    \langle
     \underbrace{T_0, \ldots, T_0}_{\alpha_0 \ \text{times}},
     \ldots,
     \underbrace{T_{N-1}, \ldots, T_{N-1}}_{\alpha_{N-1} \ \text{times}}
    \rangle_{0, \degreed}
   \frac{t^\alpha}{\alpha !}
$$
where
$\alpha = (\alpha_0 , \ldots, \alpha_{N-1})$
is a multi-index,
$|\alpha|=\alpha_0+\cdots+\alpha_{N-1}$,
$\alpha!=\alpha_0! \cdots \alpha_{N-1}!$
and
$t^\alpha = t_0^{\alpha_0} \cdots t_{N-1}^{\alpha_{N-1}}$,
$\Phi$
is the generating function of
the Gromov-Witten invariants.

\begin{theorem}
When the Gromov-Witten potential $\Phi$ converges
in some domain $D$ in $H^*(X;\bC)$,
$D$ has a structure of a Frobenius manifold
with $\Phi$ as a potential function.
\end{theorem}

See the references at the beginning of section
\ref{sc:Frobenius_manifolds}
for details.
The above theorem means that
although the moduli space of stable maps
of genus zero
are divided into connected components
by the degree and the number of marked points,
the Gromov-Witten invariants
as a whole
have a strong structure
so that together
they turn
$H^*(X;\bC)$
into a Frobenius manifold.
A part of the Frobenius structure
is a product structure
on the tangent bundle.
Since
$T H^*(X;\bC) \cong H^*(X;\bC) \times H^*(X;\bC)$,
it is a product structure
on $H^*(X;\bC)$
parametrized by
$H^*(X;\bC)$ itself.
%which is called
%the {\em quantum cohomology ring}.
%and since the cohomology group
%is only a vector space,
%the tangent space
%is the cohomology group itself,
%Gromov-Witten invariants
%defines a product on the cohomology group
%parametrized by the cohomology group itself.
By the {\em Point Mapping Axiom}
in Gromov-Witten theory,
this product structure is
a deformation of the usual cup product
on the cohomology ring,
and
is called the {\em quantum cohomology ring}.
%Sometimes it is convenient
%to restrict ourselves
%to the subspace
%$H^2(X; \bC) \subset H^*(X; \bC)$
%of the parameter space
%of the quantum cohomology.
%Then the restricted family of product structure
%on $H^*(X; \bC)$ is called the
%{\em small quantum cohomology ring}.
The metric $g$
is given by the Poincar\'{e} pairing
on the cohomology group.
The unit vector field is
$$
 e = \frac{\partial}{\partial t_0},
$$
the Euler vector field is
\begin{equation} \label{eq:gw-euler}
 E = \sum_{\flata=0}^{N-1}
      (1-\frac{\deg t_\flata}{2}) t_\flata
        \frac{\partial}{\partial t_\flata}
     + \sum_{\deg t_\flatb = 2}
        r_\flatb \frac{\partial}{\partial t_\flatb}, 
\end{equation}
and the charge is
$$
 D=2-\dim_\bC X.
$$
Here, $r_\flatb \in \bZ$ is defined by
\begin{equation} \label{eq:rflatb}
 c_1(T X) = \sum_{\deg t_\flatb = 2} r_\flatb T_\flatb.
\end{equation}

%The Euler vector field controls
%the degree of the coordinate
%of the flat coordinates:
%(\ref{eq:gw-euler}) means that
%the coordinate $t_\flata$
%has degree
%$1-\deg t_\flata/2$
%and $q_\flatb = \exp(t_\flatb)$
%for $\deg t_\flatb = 2$
%has degree
%$r_\flatb$.
%
%That $\Lie_E \circ = \circ$ means
%that if we define the degree
%of the flat vector field
%$\partial_\flata$, $\flata=0, \ldots, N-1$,
%by
%$$
% \deg(\frac{\partial}{\partial t_\flata})
%  = - \frac{\deg t_a}{2},
%$$
%then $\circ$ preserves this degree.

The WDVV equation
(\ref{eq:WDVV})
is a highly non-trivial differential equation
for the Gromov-Witten potential
$\Fpotential$,
which allows us
to compute
all the genus-zero Gromov-Witten invariants
from a few initial conditions
in special cases.
Fro the Gromov-Witten invariants
of rational surfaces,
see, e.g.,
\cite{Caporaso-Harris},
\cite{Crauder-Miranda},
\cite{Francesco-Itzykson},
\cite{Gathmann_GWIBU},
\cite{Goettsche-Pandharipande},
\cite{Kontsevich-Manin_reconstruction}
and references therein.

%Now let us define
%the Givental's $J$-function by the formula
%$$
%J^X := e^{\frac{1}{\hbar}\sum_i t_i T_i}
% \sum_\degreed e^{\frac{1}{\hbar}\sum_i t_i \int_\degreed T_i}
% J_\degreed(\hbar)
%$$
%where
%$$
%J_\degreed(\hbar) := \ev_*\left[\frac{1}{\hbar(\hbar-\psi)}\right].
%$$
%Here,
%$\psi \in H^2(\frakM_{0,1}(X;\degreed); \bQ)$
%is the Mumford-Morita-Miller class
%(the first Chern class of the orbi-bundle
%on the moduli space
%whose fiber at
%$((C, z_1), \psi) \in \frakM_{0,1}(X;\degreed)$
%is the cotangent space
%$T_{z_1}^*(C)$
%at the marked point.
%
%Then form the column vector
%$$
%\Jmatrix = 
% \begin{pmatrix}
%  \partial_0 J \\ \partial_a J \\ \vdots \\ \partial_{N-1} J
% \end{pmatrix}
%$$
%and think of it as a matrix
%by regarding an element of the cohomology group
%as a row vector.
%Then
%it follows from the {\em topological recursion relation}
%in genus zero
%that $\Jmatrix$ is a fundamental solution
%to the differential equation
%(\ref{eq:intro_t_direction})
%$$
% \frac{\partial}{\partial t_\flata} \Jmatrix
%  = \frac{1}{\hbar} \frac{\partial}{\partial t_\flata} \circ \Jmatrix,
%$$
%which is a part
%of the equation
%for flat sections
%of the first structure connection.

\section{Mirror symmetry for toric manifolds}
\label{sc:toric_mirror}

In this section,
we review the mirror symmetry for toric varieties.
See also \cite{Batyrev_QCRTM},
\cite{Givental_EGWI},
\cite{Givental_MTTCI},
\cite{Iritani_QDGMT}.

Let $\Sigma$ be a complete fan
in $N \cong \bZ^\dimfan$
and $X_\Sigma$ be the corresponding toric variety.
We denote the primitive generators
of one-dimensional cones of $\Sigma$
by $\{v_\edgea\}_{\edgea=1}^\noedges$,
$
 v_\edgea = (v_{\edgea 1},\ldots,v_{\edgea \dimfan})
   \in \bZ^\dimfan.$
Here,
$\picn$
is the Picard number
of $\XSigma$.
If we put $M=\Hom(N, \bZ)$,
we have an exact sequence
\begin{equation*}
 0 \rightarrow M \rightarrow \bZ^\noedges \rightarrow
    H^2(\XSigma;\bZ) \rightarrow 0,
     \label{eq:htwo} 
\end{equation*}
where
the homomorphism
$M \rightarrow \bZ^\noedges$
is defined by
$$
M \ni m \mapsto (m(v_\edgea))_{\edgea=1}^\noedges \in \bZ^\noedges.
$$
Choose a basis
$\{ T_\flata \}_{\flata=1}^\picn$
of $H^2(X;\bZ)$
and let $t = (t_1, \ldots, t_\picn)$
be the corresponding coordinate
of $H^2(X;\bZ)$.
One-dimensional cones in
$\Sigma$
correspond to line bundles
on $\XSigma$,
and let $\toricu_\edgea$ be
the first Chern class
of the line bundle
corresponding to
the one-dimensional cone
generated by
$v_\edgea$.
Define an $(\noedges) \times \picn$-matrix
$m_{\edgea \flata}$ by
\begin{equation} \label{eq:miflata}
 \toricu_\edgea = \sum_{\flata=1}^\picn m_{\edgea \flata} T_\flata.
\end{equation}

Now,
following Givental,
introduce
an $H^*(\XSigma;\bC)$-valued
formal function
$I(\motonot_0, \motonot; \hbar)$
%and $J(t_0, t; \hbar)$
of
$\motonot_0 \in H^0(\XSigma; \bC)$,
$\motonot \in H^2(\XSigma;\bC)$ and
$\hbar \in \bCx$ by
\begin{equation} \label{eq:defI}
 I(\motonot_0, \motonot; \hbar)
 = e^{(\motonot_0 + T \motonot) / \hbar}
%    \Psi(\motonot_0, \motonot; \hbar),
%\end{equation}
%\begin{equation} \label{eq:defPsi}
% \Psi(\motonot_0, \motonot; \hbar) =
 \sum_{\degreed \in H_2(\XSigma; \bZ)_{\mathrm{eff}}}
  q^\degreed
   \prod_{\edgea=1}^\noedges
    \frac{\prod_{\Irindex=-\infty}^0 (\toricu_\edgea + \Irindex \hbar)}
     {\prod_{\Irindex=-\infty}^{\langle \degreed, \toricu_\edgea \rangle}
       (\toricu_\edgea + \Irindex \hbar)}. 
\end{equation}
%and
%$$
%J^X := e^{\frac{1}{\hbar}\sum_i t_i T_i}
% \sum_\degreed e^{\frac{1}{\hbar}\sum_i t_i \int_\degreed T_i}
% J_\degreed(\hbar)
%$$
%where
%$$
%J_\degreed(\hbar) := \ev_*\left[\frac{1}{\hbar(\hbar-\psi)}\right].
%$$
Here,
$T \motonot = T_1 \motonot_1 + \cdots + T_\picn \motonot_\picn$,
$H_2(\XSigma; \bZ)_{\mathrm{eff}} \subset H_2(\XSigma; \bZ)$
is the cone
spanned by the homology class of curves
and
$$
 q^\degreed = q_1^{\pair{\degreed}{T_1}}
               \cdots q_\picn^{\pair{\degreed}{T_\picn}},
$$
where
$q_\flata=\exp(t_\flata)$,
$\flata=1,\ldots,\picn$ and
$\langle \bullet, \bullet \rangle$
is the pairing
between homology and cohomology.

It follows from (\ref{eq:defI})
that the $I$-function
satisfies the following
set of hypergeometric equations
of the GKZ (Gelfand-Kapranov-Zelevinski) type:

\begin{equation} \label{eq:GKZ}
 D_\degreed I = 0, \quad
  \degreed \in H_2(\XSigma;\bZ),
\end{equation}
where
$$
 D_\degreed = q^\degreed
  \prod_{\pair{\toricu_\edgea}{\degreed} < 0}
   \prod_{\Irindex=0}^{- \pair{\toricu_\edgea}{\degreed} - 1}
    U_{\edgea \Irindex}
  -
  \prod_{\pair{\toricu_\edgea}{\degreed} > 0}
   \prod_{\Irindex=0}^{\pair{\toricu_\edgea}{\degreed} - 1}
    U_{\edgea \Irindex},
$$
$$
 U_{\edgea \Irindex}
  = \sum_{\flata=1}^\picn m_{\edgea \flata}
     \hbar \frac{\partial}{\partial t_\flata} - \Irindex \hbar.
$$
From now on, we assume that
the anticanonical bundle $-K_{\XSigma}$
of the toric variety $\XSigma$ is nef.
Note that
%$I(t_0, t; \hbar)
% \in \bC[\hbar][[\hbar^{-1}]][[t_0, t]]$
%and
%$J(t_0, t; \hbar)
% \in \bC[[\hbar^{-1}t_0, t]]$.
%In particular,
%i.e.,
although $I$-function is an infinite series
in both positive and negative powers of $\hbar$
for general $\XSigma$,
it contains only non-positive powers of $\hbar$
when $-K_{\XSigma}$ is nef.
This follows from the fact that
$
\exp[-(t_0 + T t)/\hbar] I
$
is homogeneous of degree zero
with respect to the degree
assigned as
\begin{equation} \label{eq:degree_assignment}
 \deg \toricu_\canonicala = \deg \hbar = 1, \ 
 \deg( q^\degreed ) = \langle \degreed, c_1(T \XSigma) \rangle.
\end{equation}
Therefore,
the coefficient of a positive power of $\hbar$
must have a negative degree,
which is impossible
when $-K_{\XSigma}$ is nef.

I learned the following Lemma \ref{lemma:nef_convergence}
from H. Iritani:

\begin{lemma} \label{lemma:nef_convergence}
When the anticanonical bundle $-K_{\XSigma}$
of a toric variety $\XSigma$ is nef,
there exists a domain $D \subset H^2(\XSigma; \bC)$
such that
$I(\motonot_0, \motonot; \hbar)$ converges
for any $\motonot_0 \in H^0(\XSigma; \bC)$,
$\motonot \in D$
and $|\hbar|>1$
\end{lemma}
\proof
Fix a norm $\| \bullet \|$
on $H^*(\XSigma;\bC)$
such that
$\| \omega \tau \| \leq \| \omega \| \| \tau \|$.
Let $C_1$ be a positive number
satisfying
$$
 \sup_{|\hbar|=1, \ \Irindex \ge 1, \ \edgea=1, \ldots, \noedges}
   \left(
    \left\| \toricu_\edgea \right\|,
    \left\| 1+\frac{\toricu_\edgea}{\Irindex \hbar} \right\|,
    \left\| \left( 1 + \frac{\toricu_\edgea}{\Irindex\hbar}
            \right)^{-1} \right\|
   \right)
  \le C_1.
$$
Then
for $\hbar \in \bC$ such that $|\hbar| = 1$,
\begin{align*}
 \left\| \prod_{\edgea=1}^\noedges
  \frac{\prod_{\Irindex=-\infty}^0 (\toricu_\edgea + \Irindex \hbar)}
    {\prod_{\Irindex=-\infty}^{\langle \degreed, \toricu_\edgea \rangle}
     (\toricu_\edgea + \Irindex \hbar)}
 \right\|
& \le
\frac{\prod_{\langle \degreed, \toricu_\edgea \rangle <0}
             |\langle \degreed, \toricu_\edgea \rangle|!}
     {\prod_{\langle \degreed, \toricu_\edgea \rangle >0}
             \langle \degreed, \toricu_\edgea \rangle!}
 C_1^{\sum_{\edgea=1}^\noedges |\langle \degreed, \toricu_\edgea \rangle|}  \\
& \le \frac{ C_2^{|\degreed|} }
           { \langle \degreed, c_1(X) \rangle !}
\end{align*}
Here,
$C_2$ is some positive number and
$|\degreed| = |\langle \degreed, T_1 \rangle|
   + \cdots + |\langle \degreed, T_\picn \rangle|$.
The second inequality follows from
\begin{align*}
 n_1!n_2!\cdots n_{k}! & \le (n_1+\cdots+n_k)!, \\
 \frac{1}{n_1! n_2! \cdots n_k!}
  &= \frac{1}{(n_1+\cdots+n_k)!}
      \binom{n_1+\cdots +n_k}{n_1,\cdots,n_k} \\
  & \le \frac{1}{(n_1+\cdots+n_k)!} k^{n_1+\cdots+n_k}, \\
 \frac{n!}{m!}
  & \le \frac{1}{(m-n)!} \text{\quad if } n \le m.
\end{align*}
and
$$
 \sum_{\edgea=1}^{\noedges} \langle \degreed, \toricu_\edgea \rangle
  = \langle \degreed, -K_\XSigma \rangle \geq 0
$$
since $\degreed$ is an effective class
and $-K_{\XSigma}$ is nef.
Therefore,
if we define
$
 D = \{ t \in H^2(\XSigma;\bC) \suchthat 
        q_\flata < 1 / C_2, \ 
         \flata = 1, \ldots, \picn. \},
$
the $I$-function converges
when $|\hbar|=1$
and $t \in D$.
Since the $I$-function contains
only non-positive powers of $\hbar$,
it also converges
when $|\hbar| > 1$
and $t \in D$.
\qed

Now, let us
introduce
another $H^*(\XSigma;\bC)$-valued
formal function
$J(t_0, t; \hbar)$
of $t_0 \in H^0(\XSigma;\bC)$,
$t \in H^2(\XSigma;\bC)$
and $\hbar \in \bCx$
as follows:
First,
define a set of
$H^*(\XSigma;\bC)$-valued formal functions
$\{ s_\flata \}_{\flata=0}^{N-1}$
of $\{t_\flata \}_{\flata=0}^{\picn}$
by
$$
 s_\flata = T_\flata + \sum_{n=0}^\infty \hbar^{-n-1}
  \sum_{\flatb=0}^{N-1}
   \sum_{k=0}^\infty
    \sum_{\degreed \in H_2(\XSigma; \bC)}
     \frac{1}{k!}
      \langle
       \tau_n T_\flata, T_\flatb,
       \underbrace{\gamma, \ldots, \gamma}_{k \ \text{times}}
      \rangle_{0,\degreed}
     T^\flatb,
$$
where
$$
 \gamma = \sum_{\flata = 0}^\picn t_\flata T_\flata.
$$
Here, $T^\flata$ is the Poincar\'{e} dual
of $T_\flata$:
$$
 g(T_\flata, T^\flatb) = \delta_{\flata \flatb}.
$$
Then it follows from the
{\em topological recursion relation}
that
\begin{equation} \label{eq:trr_flat}
 \hbar \frac{\partial s_\flata}{\partial t_\flatb}
  = T_\flatb \circ s_\flata,
 \quad \flatb = 0, 1, \ldots, \picn.
\end{equation}
See, e.g., \cite{Cox-Katz} Proposition 10.2.1.
Note that
(\ref{eq:trr_flat})
is the $t_\flatb$-part of
the differential equation
for flat sections
of the first structure connection
(\ref{eq:X-direction}).
Now define
the {\em Givental's J-function} by
$$
 J = \sum_{\flata=0}^{N-1} g(s_\flata, 1) T^\flata.
$$

Since $H^*(\XSigma;\bC)$
is generated by
$H^2(\XSigma;\bC)$
as a ring,
there exists a set of polynomials
$\{ P_\flata(x_1, \ldots, x_\picn) \}_{\flata=0}^{N-1}$
of $\picn$ variables
such that
$$
 T_\flata = P_\flata(T_1, \ldots, T_\picn),
  \quad \flata = 0, 1, \ldots, N-1.
$$
Since
$$
 T_\flata \circ T_\flatb = T_\flata \cup T_\flatb + O(q)
$$
by the Point Mapping Axiom,
there exists a set
$\{ \widetilde{P}_\flata(x_1, \ldots, x_\picn)
 \}_{\flata = 0}^{N-1}$
of polynomials
whose coefficient is a formal power series in $q$
such that
$$
 \widetilde{P}_\flata(
      \hbar \frac{\partial}{\partial t_1} + T_1 \circ,
      \ldots,
      \hbar \frac{\partial}{\partial t_\picn} + T_\picn \circ) 1
  = T_\flata, \quad \flata = 0, 1, \ldots, N-1.
$$
Then, since
\begin{align*}
 P(\hbar \partial_1, \ldots, \hbar \partial_\picn) J
%  &= \sum_{\flata=0}^{N-1}
%       g \left(P(\hbar \partial_1, \ldots, \hbar \partial_\picn) s_\flata,
%          1 \right) T^\flata \\
%  &= \sum_{\flata=0}^{N-1}
%       g \left(P(\partial_1 \circ, \ldots, \partial_\picn \circ) s_\flata,
%          1 \right) T^\flata \\
  &= \sum_{\flata=0}^{N-1}
       g \left(s_\flata,
         P(\hbar \frac{\partial}{\partial t_1} + T_1 \circ,
           \ldots,
           \hbar \frac{\partial}{\partial t_\picn} + T_\picn \circ
          ) 1
           \right) T^\flata \\
\end{align*}
for a polynomial $P(x_1, \ldots, x_\picn)$,
%$$
% \widetilde{T}_\flata (\partial_1, \ldots, \hbar \partial_\picn) J
%  = \sum_{\flata=0}^{N-1}
%       g\left(s_\flata,
%         \widetilde{T}_\flata (\hbar \partial_1,
%              \ldots, \hbar \partial_\picn) 1
%           \right) T^\flata, \\
%$$
%hence
$$
 s_\flata
  = \sum_{\flatb = 0}^{N-1} g \left(
     \widetilde{P}_\flatb(\hbar \partial_1, \ldots, \hbar \partial_\picn) J,
     T_\flata \right) T^\flatb.
$$
%Here, we have put
%$
% \widetilde{T}_0 (x_1, \ldots, \hbar x_\picn) = 1
%$
%and
%$
% \widetilde{T}_\flata (x_1, \ldots, \hbar x_\picn) = x_\flata
%$
%for $\flata = 1, \ldots, \picn$.

To sum up,
we have introduced
two $H^*(\XSigma;\bC)$-valued
formal functions
$I(t_0, t; \hbar)$ and $J(t_0, t; \hbar)$.
The $I$-function is defined
in terms of the combinatorial data
of the fan $\Sigma$ defining $\XSigma$
and satisfies a set of hypergeometric
differential equations
(\ref{eq:GKZ}) of GKZ type.
The $J$-function
is defined in terms of the Gromov-Witten invariants
of $\XSigma$
and encodes the information
of the solution
to the equation
(\ref{eq:trr_flat}),
hence the structure of the product $\circ$.

The mirror theorem by Givental
(see \cite{Givental_MTTCI} Theorem 0.2,
Proposition 6.4, and the discussion
at the end of section 7)
gives the relation between
the $I$-function
and the $J$-function
as follows:
Since $-K_\XSigma$ is nef,
all the variables $\motonoq_\flata$'s
have non-negative degrees,
see (\ref{eq:degree_assignment}).
It follows from the homogeneity
of
$\exp[- (\motonot_0 + T \motonot) / \hbar]
  I(\motonot_0, \motonot; \hbar)$
that the $I$-function has the form
\begin{equation} \label{eq:expansion_I}
 I(\motonot_0, \motonot;\hbar)
  = e^{(\motonot_0 + T \motonot)/\hbar}
     \left( \Izero(\motonoq)
        + (\Ionezero(\motonoq) + \Ioneone(\motonoq)) \hbar^{-1}
      + O(\hbar^{-2}) \right),
\end{equation}
where
$\Izero(\motonoq)$ and $\Ionezero(q)$
are $H^0(\XSigma; \bC)$-valued
and
$\Ioneone(q)$ is $H^2(\XSigma; \bC)$-valued.
Then, the $J$-function is given by
\begin{align*}
 J(\atonot_0, \atonot; \hbar)
  &= I(\motonot_0, \motonot; \hbar) / \Izero(\motonoq) \\
  &= \exp[(\motonot_0 + T \motonot)/\hbar]
      \left(
       1 + \Izero(\motonoq)^{-1}
        \left(
         \Ionezero(\motonoq)
          + \Ioneone(\motonoq)
        \right) \hbar^{-1}
       + O(\hbar^{-2})
      \right) \\
  &= \exp
      \left[
       \left(
        \left(
         \motonot_0 + \Izero(\motonoq)^{-1} \Ionezero(e^\motonoq)
        \right)
        + \left(
           \motonot + \Izero(\motonoq)^{-1} \Ioneone(\motonoq)
          \right)
       \right) /\hbar
      \right]
      (1 + O(\hbar^{-2})) \\
  &= \exp \left[
           \left( \atonot_0 + T \atonot \right) /\hbar
          \right]
       (1 + O(\hbar^{-2})).
\end{align*}
Here,
we have performed a
homogeneous coordinate transformation
$$
 \atonot_0
  = \motonot_0 + \Izero(\motonoq)^{-1} \Ionezero(\motonoq),
 \quad
 \atonot = \motonot + \Izero(\motonoq)^{-1} \Ioneone(\motonoq),
$$
called the {\em mirror transformation}.
Since the $I$-function is convergent,
the mirror transformation
and the $J$-function are also convergent.

\section{Stationary-phase integral and Stokes matrix}
 \label{sc:spint}
Let $v_0 = (0, \ldots, 0) \in \bZ^\dimfan$,
and $\{ v_\edgea \}_{\edgea=1}^\noedges \subset \bZ^\dimfan$
be the set of primitive generators
of one-dimensional cones of
the fan $\Sigma$ in $\bZ^\dimfan$
defining a toric variety $\XSigma$
as in section \ref{sc:toric_mirror}.
We assume that
the anticanonical bundle
$-K_{\XSigma}$ of $\XSigma$ is nef.
Define a Laurent polynomial
$W$ by
\begin{equation} \label{eq:toric_mirror} 
 W(x_1,\ldots,x_n)
  = \sum_{\edgea=0}^\noedges \wa_\edgea
     x_1^{v_{\edgea 1}} \cdots x_r^{v_{\edgea r}}.
\end{equation}
$W$ defines a regular function
on the algebraic torus
$(\bCx)^\dimfan$.
By Kouchnirenko \cite{Kouchnirenko},
the number $N$
of critical points
of $W$ is
$\dimfan !$ times the volume
of the Newton polygon
of $W$
(i.e., the convex hull of $\{v_\edgea \}_{\edgea =1}^\noedges$)
when $\wa_\edgea \in \bC$, $\edgea=0,\ldots,\noedges$,
are general enough.
This $N$ is equal to
the rank %$\sum_k \dim H^k(\XSigma,\bR)$
of the total cohomology ring
$H^*(\XSigma,\bC)$
of $\XSigma$.
Let $\{p_\canonicala\}_{\canonicala=1}^N$
be the set of critical points of $W$.
They are functions of
$\{ \wa_\edgea \}_{\edgea = 0}^\noedges$.
Fix a complete K\"{a}hler metric
on the algebraic torus $(\bCx)^\dimfan$.
For a general $\hbar$,
let $\Gamma_\canonicala(\hbar)$
be the descending Morse cycle for
$\Re(W/\hbar)$.
The image of $\Gamma_\canonicala(\hbar)$
by $W$
is a half-line
starting from the critical value
$W(p_\canonicala)$,
and the fiber
above a point $p$
on this half-line
is the cycle
in $W^{-1}(p)$
which vanishes
at $p_\canonicala$
by the parallel transport
along this half-line.
Following Givental
\cite{Givental_EGWIGMC},
for $\canonicala=1,\ldots,N$,
consider the integral
\begin{equation} \label{eq:spint}
 \spint_\canonicala
  = (\pi \hbar)^{- \dimfan / 2}
    \int_{\Gamma_\canonicala(\hbar)} e^{\frac{1}{\hbar} W(x)}
     \frac{d x_1 \wedge \cdots \wedge d x_n}{x_1 \cdots x_\dimfan}.
\end{equation}
This integral is invariant
under the natural action
of the torus $(\bCx)^\dimfan$
on itself defined by
$$
 (\bCx)^\dimfan \ni (\alpha_1,\ldots,\alpha_\dimfan) :
  (x_1,\ldots,x_\dimfan) \mapsto
     (\alpha_1 x_1, \ldots, \alpha_r x_\dimfan).
$$
Therefore,
if we define the variables
$\{ t_\flata \}_{\flata=0}^\picn$
by $t_0 = \wa_0$ and
$$
 \exp({t_\flata})
  = \wa_1^{m_{1 \flata}} \cdots \wa_\noedges^{m_{\noedges, \flata}},
 \quad \flata = 1, \ldots, \picn,
$$
where $m_{\edgea \flata}$'s are defined
in (\ref{eq:miflata}),
$\spint_\canonicala$'s
depend on $\{ \wa_\edgea \}_{\edgea=0}^\noedges$
only through $\{ t_\flata \}_{\flata=0}^\picn$.

%\begin{theorem}[Givental]
%For fixed $\hbar$,
%the stationary-phase integrals $\spint_\canonicala$'s
%satisfies the same differential equation
%as the $I$-functions.
%\end{theorem}

It is easy to see
that $\spint_\canonicala$'s
satisfy the same differential equations
(\ref{eq:GKZ})
as the $I$-function.
Since the solution space to
these differential equation
is an $N$-dimensional $\bC$-vector space
and $\{ \spint_\canonicala \}_{\canonicala=1}^N$ are
linearly independent,
%(this can be seen by, for example,
%at the saddle-point approximation
%in (\ref{eq:saddle-point_method}))
the set of components of the $I$-function and
$\{ \spint_\canonicala \}_{\canonicala=1}^N$ differ
only by an $\hbar$-dependent
linear transformation.

The $\hbar$-dependence
of $\spint_\canonicala$ is simple.
%since
%$W/\hbar$ is homogeneous of degree zero
If we assign
degree $1$ to variables $t_0$, $\hbar$
and $\{ x_i \}_{i=1}^\dimfan$,
and degree $r_\flata$
to the variable $\exp(t_\flata)$
for $\flata=1, \ldots, \picn$
where
$r_\flata$'s are defined
in (\ref{eq:rflatb}),
$\spint_\canonicala$ is homogeneous
of degree $-\dimfan/2$:
\begin{equation} \label{eq:homogeneity}
 \left[ \hbar \frac{\partial}{\partial \hbar}
  + t_0 \frac{\partial}{\partial t_0}
  + \sum_{\flata=1}^\picn
     r_\flata \frac{\partial}{\partial t_\flata}
  + \frac{\dimfan}{2} \right] \spint_\canonicala
  = 0. 
\end{equation}

The important point is that
the $\hbar$-part
(\ref{eq:hbar-direction})
of the differential equation
for the $H^{2 \dimfan}(\XSigma;\bC)$-component
of a flat section
of the first structure connection
is precisely the homogeneity condition
(\ref{eq:homogeneity}):
Assume that a section $Y$ of $\pi^* \scT_M$
satisfies
$$
 \fscon_X Y = \nabla_X Y
  - \frac{1}{\hbar} X \circ Y
  = 0
$$
for any section $X$ of $\pi^* T_M$.
Then
\begin{align*}
 \hbar \fscon_{\partial_\hbar} Y
  &= \hbar \partial_\hbar Y
      + \frac{1}{\hbar} \scU(Y)
      - \scV(Y) \\
  &= \hbar \partial_\hbar Y
      + \frac{1}{\hbar} E \circ Y
      - \scV(Y) \\
  &= \hbar \partial_\hbar Y
      + \nabla_E Y
      - \scV(Y).
\end{align*}
Note that
for the flat coordinates
$\{ t_\flata \}_{\flata=0}^{N-1}$,
\begin{align*}
 \scV(\partial_\flata)
  &= \nabla_{\partial_\flata} E - \frac{D}{2} \partial_\flata \\
  &= \nabla_{\partial_\flata} 
      \left(
        \sum_{\flatb=0}^{N-1}
           (1-\frac{\deg t_\flatb}{2}) t_\flatb
            \partial_\flatb
        + \sum_{\deg t_\flatc = 2}
            r_\flatc \partial_\flatc
      \right)
      - \frac{2 - \dimfan }{2} \partial_\flata \\
  &= \left( \frac{\dimfan - \deg t_\flata}{2} \right)
      \partial_\flata.
\end{align*}
Since $\nabla_E$ counts the degree,
this differential equation
requires
the $H^k(\XSigma; \bC)$-component
of the flat section
of the first structure connection
to be homogeneous of degree
$(\dimfan - k) / 2$.
%Since
%$$
% \begin{pmatrix}
%  \spint \\
%  T_1(\hbar q \partial / \partial q) \spint \\
%  \vdots \\
%  T_{N-1} (\hbar q \partial / \partial q) \spint
% \end{pmatrix}
%  = \sqrt{\pi \hbar}^{\dim \XSigma}
%     \Psi R e^{U/\hbar}
%$$
%is homogeneous of degree $0$
%and $\hbar$ has degree $1$,
%$$
% \hbar \fscon_{\frac{\partial}{\partial \hbar}}
%  \sqrt{\pi \hbar}^{\dim \XSigma}
%     \Psi R e^{U/\hbar} = \frac{\dim \XSigma}{2} 
%     \sqrt{\pi \hbar}^{\dim \XSigma},
%$$
%$$
% \fscon_{\frac{\partial}{\partial \hbar}}
%  = 0.
%$$
%$$
% \Psi R e^{U/\hbar}
%$$
%is homogeneous of degree
%$- \dim \XSigma / 2$.

Now, recall that by Givental's mirror theorem,
$\Izero(\motonoq)^{-1} I(\motonot_0, \motonot; \hbar)$
coincides with the $J$-function,
which have the same dependence
on $\{ t_\flata \}_{\flata=0}^\picn$
with the $H^{2 \dimfan}(\XSigma;\bC)$-components
of the flat section
for the first structure connection.
Then, since the stationary-phase integrals
(\ref{eq:spint})
have the same dependence
on $\{ t_\flata \}_{\flata=0}^\picn$
as the $I$-function
and satisfies the desired homogeneity condition
(\ref{eq:homogeneity}),
$\Izero(\motonoq)^{-1} \spint_\canonicala(\motonot_0, \motonot; \hbar)$
gives the $H^{2 \dimfan}(\XSigma;\bC)$-component
of a flat section
for the first structure connection
for $\canonicala = 1, \ldots, N$.

%Next, note that
By the saddle-point approximation,
\begin{equation} \label{eq:saddle-point_method}
 \Izero(\motonoq)^{-1} \spint_\canonicala
  \sim
  \frac{\Izero(\motonoq)^{-1}}
       {p_{\canonicala,1}\cdots p_{\canonicala,n}}
  \left[ \det \left(
   \frac{\partial^2 W}{\partial x_l \partial x_m} (p_\canonicala)
              \right)_{l,m} \right]^{-\frac{1}{2}}
  \left(
   1+O(\hbar)
  \right)
  e^{\frac{1}{\hbar}W(p_\canonicala)}
\end{equation}
as $|\hbar| \rightarrow 0$
with $\arg \hbar$ fixed,
where
$
 p_\canonicala = (p_{\canonicala,1}, \cdots,
   p_{\canonicala,n}) \in (\bCx)^\dimfan
$
is the $\canonicala$-th critical point
of $W$.
This gives the asymptotic expansion
appearing in (\ref{eq:formal_solution}).
%Therefore,
%$$
% (\hbar \pi)^{-\frac{n}{2}} \spint,
%$$
%which has the desired dependence
%on both $t_\flata$'s
%(which is identical to the $t_\flata$ dependence
%of the $J$-function)
%and $\hbar$
%(which is the homogeneity condition
%of degree $- \dim \XSigma / 2$),
%is the $H^\dimfan(\XSigma;\bC)$-component
%of the flat section
%for the first structure connection.
%Since the canonical coordinate
%is given by the critical values;
%$u_\canonicala = W(p_\canonicala)$,
%we obtain the formal solution
%appearing in (\ref{eq:formal_solution})
%if we continue the $1 / \hbar$-expansion
%in (\ref{eq:saddle-point_method}).

Since the Stokes matrix acts
on all the components of
the fundamental solution
equally,
it is enough to compute
its action on the $H^{2 \dimfan}(\XSigma;\bC)$-component
computed above.
The integration cycle
$\Gamma_{\canonicala}(\hbar)$
in (\ref{eq:spint})
undergoes a discontinuous change
when $\hbar$ crosses the line
such that
the half-line
starting from the critical value
$u_\canonicala$
in the direction of $- \hbar^{-1}$
passes through another critical value
$u_\canonicalb$.
Therefore,
$\spint_\canonicala$
is not holomorphic
on such a line.
Since $\spint_\canonicala$
has a monodromy,
it is impossible to obtain
a holomorphic function
on the whole $\hbar$-plane
by analytic continuation,
but one can obtain a holomorphic function
on the two angular domains
$D_\lt$ and $D_\rt$
defined in Section \ref{sc:Frobenius_manifolds}
as follows:
Consider the local system
on $\bCx$
whose fiber over $\hbar \in \bCx$ is
the relative homology group
$H_n \left( (\bC^\times)^n, \Re(W/\hbar) \ll 0; \bZ \right)$.
%Here,
%$H_n((\bC^\times)^n, \Re(W/\hbar) \ll 0)$
%can be considered
%as a shorthand notation
%for
%$
% \displaystyle{\lim_{\rihtarrow}}
% H_n((\bC^\times)^n, \Re(W/\hbar) \ll M)
%$
Define
two sets of sections
$\{ \Gamma_{\canonicala, \lt}(\hbar) \}_{\canonicala=1}^N$
and
$\{ \Gamma_{\canonicala, \rt}(\hbar) \}_{\canonicala=1}^N$
of this local system
on $D_\lt$ and $D_\rt$
by the condition that
they coincide with the classes
defined by
the descending Morse cycles
at $\hbar = \exp[ (\phi + \pi / 2) \sqrt{-1}]$
and $\hbar = \exp[ (\phi - \pi / 2) \sqrt{-1}]$
respectively;
\begin{align*}
 \Gamma_{\canonicala, \lt}(\exp[ (\phi + \pi / 2) \sqrt{-1} ])
  &= \Gamma_{\canonicala}(\exp[ (\phi + \pi / 2) \sqrt{-1} ]), \\
  \Gamma_{\canonicala, \rt}(\exp[ (\phi - \pi / 2) \sqrt{-1} ])
  &= \Gamma_{\canonicala}(\exp[ (\phi - \pi / 2) \sqrt{-1} ]).
\end{align*}
Since $D_\lt$ and $D_\rt$ are simply-connected,
these conditions determine
$\Gamma_{\canonicala, \lt}(\hbar)$
and
$\Gamma_{\canonicala, \rt}(\hbar)$
uniquely.
Define $\spint_{\canonicala, \lt}$
(resp. $\spint_{\canonicala, \rt}$)
by
the integral (\ref{eq:spint})
with $\Gamma_{\canonicala, \lt}(\hbar)$
(resp. $\Gamma_{\canonicala, \rt}(\hbar)$)
as the integration cycle.
Then $\spint_{\canonicala, \lt}$
(resp. $\spint_{\canonicala, \rt}$)
are holomorphic
and
have the asymptotic behaviors
(\ref{eq:saddle-point_method})
as $\hbar \rightarrow 0$
in $D_\lt$
(resp. $D_\rt$).

\begin{figure}[htbp]
\begin{minipage}{0.5 \textwidth}
\centering
\psfrag{p1}{$W(p_1)$}
\psfrag{p2}{$W(p_2)$}
\psfrag{p3}{$W(p_3)$}
\psfrag{pN}{$W(p_N)$}
\psfrag{G1}{$\Gamma_1$}
\psfrag{G2}{$\Gamma_2$}
\psfrag{G3}{$\Gamma_3$}
\psfrag{GN}{$\Gamma_N$}
\psfrag{Re}{$\Re$}
\psfrag{Im}{$\Im$}
\includegraphics[width=\textwidth]{stokes1.eps}
\caption{
%\begin{parbox}{35mm}
 Integration cycles
 at $\hbar = 1$
%\end{parbox}
}
\label{fg:stokes1} 
\end{minipage}
\begin{minipage}{0.5 \textwidth}
\centering
\psfrag{p1}{$W(p_1)$}
\psfrag{p2}{$W(p_2)$}
\psfrag{p3}{$W(p_3)$}
\psfrag{pN}{$W(p_N)$}
\psfrag{Gr1}{$\Gamma_{1, \rtt}$}
\psfrag{Gr2}{$\Gamma_{2, \rtt}$}
\psfrag{Gr3}{$\Gamma_{3, \rtt}$}
\psfrag{GrN}{$\Gamma_{N, \rtt}$}
\psfrag{Gl1}{$\Gamma_{1, \ltt}$}
\psfrag{Gl2}{$\Gamma_{2, \ltt}$}
\psfrag{Gl3}{$\Gamma_{3, \ltt}$}
\psfrag{GlN}{$\Gamma_{N, \ltt}$}
\psfrag{Re}{$\Re$}
\psfrag{Im}{$\Im$}
\includegraphics[width=\textwidth]{stokes2.eps}
\caption{
%\begin{minipage}{35mm}
 Integration cycles
 at $\hbar = -\sqrt{-1}$
%\end{minipage}
}
\label{fg:stokes2}
\end{minipage}
\end{figure}

Figure \ref{fg:stokes1}
shows the images $\Gamma_\canonicala$'s
of $\Gamma_{\canonicala, \rt}(\hbar)$'s
by $W$
at $\hbar = 1$.
Here,
we have assumed that
the imaginary axis
$\Re(\hbar) = 0$ is admissible,
and have chosen it
as an admissible line $l$.
Figure \ref{fg:stokes2}
shows
the images
$\Gamma_{\canonicala, \ltt}$'s
and $\Gamma_{\canonicala, \rtt}$'s
of
$\Gamma_{\canonicala, \lt}(\hbar)$'s
and $\Gamma_{\canonicala, \rt}(\hbar)$'s
by $W$
at $\hbar = - \sqrt{-1}$.

Since the integrand is single-valued,
the Stokes matrix is given
by the transformation matrix
between these two sets of cycles.
To describe it,
let us first choose
a regular value
$p \in \bC$
such that $\Re(p)$ is small enough
so that the line segment
$c_\canonicala$
from the $\canonicala$-th critical value
$u_\canonicala$
to $p$
is above $c_\canonicalb$
for any $\canonicalb > \canonicala$
(i.e., if $x \in c_\canonicala$,
$y \in c_\canonicalb$
and $\Re(x) = \Re(y)$,
then $\Im(x) > \Im(y)$).
Then,
the ordered set of cycles
$(C_\canonicala)_{\canonicala = 1}^N$
in $W^{-1}(p)$,
which vanish along the paths
$(c_\canonicala)_{\canonicala = 1}^N$
respectively,
forms a
{\em distinguished basis of vanishing cycles},
see e.g. \cite{Arnold--Gusein-Zade--Varchenko}.
It is obvious that
$$
\Gamma_{1, \lt}(\hbar)
 = \Gamma_{1, \rt}(\hbar). \\
$$
in $H_n((\bC^\times)^n, \Re[W/\hbar] \ll 0; \bZ)$
in the neighborhood of $\hbar = - \sqrt{-1}$.
As for $\Gamma_2(\hbar)$,
it follows from the Picard-Lefschetz formula
that
$$
 \Gamma_{2, \lt}(\hbar)
  = \Gamma_{2,\rt}(\hbar)
     - (C_1,C_2) \Gamma_{1, \rt}(\hbar),
$$
where
$(\bullet, \bullet)$
is the intersection form
in $H_\dimfan(W^{-1}(p); \bZ)$.
In the same way,
we have
\begin{eqnarray}
\Gamma_{\canonicala,\lt}(\hbar) 
 &=& \Gamma_{\canonicala,\rt}(\hbar)
   - (C_{\canonicala-1}, C_{\canonicala}) \Gamma_{\canonicala-1,\rt}(\hbar)
   - \cdots - (C_1, C_\canonicala) \Gamma_{1,\rt}(\hbar),
  \nonumber
%\\
\end{eqnarray}
by successive use of
the Picard-Lefschetz formula,
from which
\begin{eqnarray}
 S_{\canonicala \canonicalb} &=& \left\{\begin{array}{cl}
       1      & \mbox{if $\canonicala=\canonicalb$,} \\
  - (C_\canonicala, C_\canonicalb)
              & \mbox{if $\canonicala<\canonicalb$,} \\
       0      & \mbox{otherwise}
   \end{array}\right. \label{eq:Picard-Lefschetz_Stokes}
\end{eqnarray}
follows.

\section{Intersection numbers of vanishing cycles}
 \label{sc:vanishing_cycles}

Now we consider the toric variety $\blsix$
which is the projective plane
blown-up at 6 points including infinitely-near points.
Figure \ref{fg:blsix} shows the generators
of the one-dimensional cones
of the fan
defining $\blsix$.

\begin{figure}[H]
\centering
\psfrag{v0}{$v_0$}
\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{v7}{$v_7$}
\psfrag{v8}{$v_8$}
\psfrag{v9}{$v_9$}
\psfrag{vequalssomething}{$\begin{array}{rcl}
 v_0 & = & (   0   ,   0   ), \\
 v_1 & = & (   1   ,   0   ), \\
 v_2 & = & (   0   ,   1   ), \\
 v_3 & = & (  -1   ,  -1   ), \\
 v_4 & = & (  -1   ,   0   ), \\
 v_5 & = & (   0   ,  -1   ), \\
 v_6 & = & (  -1   ,   1   ), \\
 v_7 & = & (   1   ,  -1   ), \\
 v_8 & = & (  -1   ,   2   ), \\
 v_9 & = & (   2   ,  -1   ).
	     \end{array}$}
\includegraphics{blsix.eps}
\caption{the toric data of $\blsix$}
\label{fg:blsix}
\end{figure}

$\blsix$ is deformation-equivalent
to cubic surfaces, and
$-K_\blsix$ is not ample
but nef.
The mirror for $\blsix$ is defined
by (\ref{eq:toric_mirror}),
which we specialize to
%$
% a_0 = 0,
% a_1 = 0,
% a_2 = 0,
% a_3 = 1,
% a_4 = 0,
% a_5 = 0,
% a_6 = 0,
% a_7 = 0,
% a_8 = 1,
% a_9 = 1
%$
$\wa_3 = \wa_8 = \wa_9 =1$ and
$\wa_i = 0$ for $i \neq 3,8,9$,
to obtain
\begin{equation} \label{eq:bl6_mirror}
 W(x,y) = \frac{x^2}{y} + \frac{y^2}{x} + \frac{1}{x y},
\end{equation}
which is generic in the sense
that all the critical points
are non-degenerate.
In this case,
$W^{-1}(0)$ is a Fermat curve in $(\bCx)^2$:
$$
 W^{-1}(0) = \{(x,y) \in (\bCx)^2 \suchthat x^3 + y^3 + 1 = 0 \}.
$$

$W$ has nine critical points
$\{p_{ij}\}_{i,j=0}^2$,
$p_{ij}= (\omega^{i+j-1}, \omega^{2 j})$,
where we have fixed
a primitive cubic root
$\omega$ of unity.
Let $C_{ij}$ be the vanishing cycle
in $W^{-1}(0)$
which vanishes at the critical point $p_{ij}$
along the straight line
from $0$
to the critical value $3 \omega^{1-i}$.
Then
$
(
C_{00},
C_{01},
C_{02},
C_{10},
C_{11},
C_{12},
C_{20},
C_{21},
C_{22}
)
$
is a distinguished basis
of vanishing cycles.

\begin{figure}
 \centering
 \psfrag{C00}{$C_{00}$}
 \psfrag{C01}{$C_{01}$}
 \psfrag{C02}{$C_{02}$}
 \psfrag{C10}{$C_{10}$}
 \psfrag{C11}{$C_{11}$}
 \psfrag{C12}{$C_{12}$}
 \psfrag{C20}{$C_{20}$}
 \psfrag{C21}{$C_{21}$}
 \psfrag{C22}{$C_{22}$}
 \includegraphics{vanishing_cycles.eps}
 \caption{Vanishing cycles in $W^{-1}(0)$}
 \label{fg:vanishing_cycles}
\end{figure}

%There exists a $\bZtt$-action
%on $W^{-1}(0)$
%define by
%$$
% \bZtt \ni [i,j] : W^{-1}(0) \in (x, y)
%  \mapsto (\omega^i x, \omega^j y) \in W^{-1}(0)
%$$
%where $\omega = \exp[2 \pi \sqrt{-1} / 3]$.
%The action of the first factor of $\bZtt$
%is the deck transformation
%of the covering $\pi$
%and that of the second factor
%is the rotation of the $y$-plane
%by $2/3 \pi$.
%Since this action
%is transitive
%on the set of vanishing cycles,
%the intersection of vanishing cycles
%can be read off
%from Figure \ref{fg:Cij}:
%$C_{00}$ intersects
%\begin{itemize}
% \item $C_{02}$ at $s$ with intersection number $-1$,
% \item $C_{01}$ at $q$ with intersection number $+1$,
% \item $C_{20}$ at $p$ with intersection number $-1$,
% \item $C_{10}$ at $p$ with intersection number $+1$,
% \item $C_{11}$ at $r$ with intersection number $-1$, and
% \item $C_{22}$ at $t$ with intersection number $+1$.
%\end{itemize}
%\begin{itemize}
% \item $C_{02}$ at $s$ with intersection number $-1$,
% \item $C_{22}$ at $t$ with intersection number $+1$,
% \item $C_{20}$ at $p$ with intersection number $-1$,
% \item $C_{01}$ at $q$ with intersection number $-1$,
% \item $C_{11}$ at $r$ with intersection number $+1$.
%\end{itemize}

Figure \ref{fg:vanishing_cycles}
shows these vanishing cycles
with orientations
chosen by hand.
Opposite sides of the square is identified
to form a torus,
and the circles denote nine points
which are missing in $W^{-1}(0)$
since it is not compact.
Note that
the quotient
of $W^{-1}(0)$
by the action of $\bZ / (3 \bZ)$
generated by
$$
 (x, y) \mapsto (\omega x, \omega^2 y).
$$
is
$$
 \left\{
  (X, Y) \in (\bCx)^2 \left| X + Y + \frac{1}{X Y} = 0 \right.
 \right\}
$$
appearing in
Seidel's work
\cite{Seidel_VC2}.
This action is given by the translation
in the direction of the diagonal line
from the lower-left corner
to the upper-right corner
of the square
in Figure \ref{fg:vanishing_cycles}
by one-third of the length
of the diagonal line,
and the quotient
by this action is exactly
Figure 2 in \cite{Seidel_VC2}.

Intersection numbers of $C_{ij}$'s
are
$$
 (C_{ij}, C_{ik}) = 0 \quad \text{if $j \neq k$},
$$
and
$$
 (C_{ij}, C_{kl}) = -1 \quad \text{if $i < k$}.
$$
%and $C_{kl}$
%for $i, j, k, l = 0, 1, 2$ and
%$i < k$
%can be read off
%from Figure \ref{fg:vanishing_cycles}:
%$C_{ij}$ intersects
%\begin{itemize}
% \item $C_{i+1,l}$ with intersection number $-1$,
% \item $C_{i+2,l}$ with intersection number $-1$,
%\end{itemize}
%and does not intersect with $C_{il}$
%for any $l$.

\section{Derived category of coherent sheaves}
 \label{sc:derived_category}
Let $\bZ / 3 \bZ$ act on a three-dimensional vector space
$V \cong \bC^3$ by
$$
 \bZ / 3 \bZ \ni [1] : V \ni (x:y:z) \mapsto
  (\omega x : \omega^2 y : z) \in V
$$
where $\omega$ is
a primitive cubic root
of unity.
This action defines an action of $\bZ / 3 \bZ$
on $\bP^2 = \bP(V)$,
and the toric variety
$\blsix$ defined in Section \ref{sc:vanishing_cycles}
is the minimal resolution
of the quotient
$\bP(V)/(\bZ / 3 \bZ)$.
By Beilinson \cite{Beilinson},
$(\scE_0, \scE_1, \scE_2)
  = (\scO_{\bP^2}(-1), \Omega_{\bP^2}(1), \scO_{\bP^2})$
is an exceptional collection generating $D^b \coh \bP^2$,
and by Kapranov-Vasserot \cite{Kapranov-Vasserot},
we have
$$
 D^b \coh \blsix \cong D^b \coh^{\bZ/3\bZ}\bP^2.
$$
Here, $\scO_{\bP^2}(-1)$ is the tautological sheaf,
$\Omega_{\bP^2}(1)$ is the cotangent sheaf
tensored with the dual of $\scO_{\bP^2}(-1)$,
and
$\coh^{\bZ/3\bZ}\bP^2$
is the category of $\bZ / 3 \bZ$-equivariant
coherent sheaves on $\bP^2$.
Now, let
$\scE_{ij}$ be the object
of $D^b \coh \blsix$
corresponding to
the $\bZ / 3 \bZ$-equivariant
coherent sheaf
$\scE_i \otimes \rho_j$ on $\bP^2$
by the above equivalence.
Here, $\rho_j : \bZ / 3 \bZ \ni [1] \mapsto \omega^j \in \bCx$
is a one-dimensional representation of $\bZ / 3 \bZ$.
Then
$(
\scE_{00},
\scE_{01},
\scE_{02},
\scE_{10},
\scE_{11},
\scE_{12},
\scE_{20},
\scE_{21},
\scE_{22}
)$
is an exceptional collection
generating $D^b \coh \blsix$.
The $\Ext$-groups between them
can be calculated as follows:
First,
we can 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 compute the $\Ext$-groups between $\scE_i$'s:
\begin{eqnarray*}
\RHom(\scE_1, \scE_2)
 & = & \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(\scE_1, \scE_3)
 &=& \RHom( \scO_{\bP^2}(-1), \scO_{\bP^2} ) \\
 &=& \RGamma( \scO_{\bP^2}(1)) \\
 &=& V^\vee, \\
%\end{eqnarray*}
%\begin{eqnarray*}
\RHom(\scE_2, \scE_3)
 & = & \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*}
Here, $\RHom$ and $\RGamma$ denote
the right derived functor of $\Hom$ and $\Gamma$
(taking global sections)
respectively.
Then the $\Ext$-groups between $\scE_{ij}$'s are given by
\begin{eqnarray*}
 \Ext^k (\scE_{ij}, \scE_{lm})
  &=& \Ext^k (\scE_{i} \otimes \rho_j, \scE_{l} \otimes \rho_m) \\
  &=& (\Ext^k (\scE_{i}, \scE_{l})
   \otimes \rho_j^\vee \otimes \rho_m)^{\bZ / 3 \bZ},
\end{eqnarray*}
where $\bullet^{\bZ / 3 \bZ}$ denotes
the $\bZ / 3 \bZ$-invariant part.
%Therefore,
%$$
%\begin{cases}
% \dim \Hom(\scE_i \otimes \rho_j, \scE_i \otimes \rho_k)
%  = \begin{cases}
%     1 & \text{if $j \neq k$}, \\
%     0 & \text{if $j = k$},
%    \end{cases} \\
% \dim \Hom(\scE_i \otimes \rho_j, \scE_{i+1} \otimes \rho_k)
%  = 1 & \text{if $i=0,1$},
%\end{cases}
%$$
Therefore,
the dimension of
$\Ext^k(\scE_{ij}, \scE_{lm})$
is non-zero if and only if
$$
 i=l, j=m \ \text{and} \ k=0,
$$
or
$$
 j<l \ \text{and} \ k=0,
$$
and the dimensions in all these cases
are one.
Therefore,
we have
%a perfect agreement
%of the intersection numbers of vanishing cycles
%$\{C_{ij}\}_{i,j=0}^2$
%from Section \ref{sc:vanishing_cycles}
%and Euler form
%on the exceptional collection $(\scE_{ij})_{i,j=0}^2$
%generating $D^b \coh \blsix$:
\begin{eqnarray*}
 -( C_{i j}, C_{l m})
 = \sum_{k=0}^2 (-1)^k \dim \Ext^k (\scE_{i j}, \scE_{l m})
 \quad \text{if $i \leq l$ and $(i,j) \neq (l,m)$},
\end{eqnarray*}
which,
combined with
(\ref{eq:Picard-Lefschetz_Stokes}),
proves Conjecture \ref{conj:stokes}
for $Y$.
\bibliographystyle{plain}
\bibliography{bibs}

Research Institute for 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}

Now,
as a preparation for the discussion of
mirror symmetry
in the next section,
we discuss the solutions
to the differential equation
(\ref{eq:X-direction}).
First, define (formal) functions
$\{ s_\flata \}_{\flata=0}^{N-1}$
of $\{t_\flata \}_{\flata=0}^{N-1}$
with values in $H^*(X; \bC)$
by
$$
 s_\flata = T_\flata + \sum_{n=0}^\infty \hbar^{-n-1}
  \sum_\flatb \llangle \tau_n T_\flata, T_\flatb \rrangle_0 T^\flatb
$$
where
$$
 \llangle \tau_{d_1} \gamma_1, \ldots, \tau_{d_n} \gamma_n \rrangle_g
  = \sum_{k=0}^\infty \sum_{\degreed \in H_2(X; \bZ)} \frac{1}{k!}
     \langle
      \tau_{d_1} \gamma_1, \ldots, \tau_{d_n} \gamma_n,
       \underbrace{\gamma, \ldots, \gamma}_{k \ \text{times}}
     \rangle_{g, \degreed}.
$$

Then, it follows from the
{\em topological recursion relation}
that
$$
 \hbar \frac{\partial s_\flata}{\partial t_\flatb}
  = T_\flatb \circ s_\flata.
$$
See, e.g. \cite{Cox-Katz}, Proposition 10.2.1.
Now define
the {\em Givental's J-function} by
$$
 J = \sum_{\flata=0}^{N-1} \langle s_\flata, 1 \rangle T^\flata.
$$
From this $J$-function,
one can reconstruct the flat sections
$\{ s_\flata \}_{\flata=0}^{N-1}$
as follows:
%When one differentiate $J$ by the flat coordinate $t_\flata$,
%one obtains
Since
\begin{align*}
 \partial_\flata J
  &= \sum_{\flatb=0}^{N-1} \langle \partial_\flata s_\flatb,
       1 \rangle T^\flatb \\
  &= \sum_{\flatb=0}^{N-1} \langle T_\flata \circ s_\flatb,
       1 \rangle T^\flatb \\
  &= \sum_{\flatb=0}^{N-1} \langle s_\flatb,
       T_\flata \rangle T^\flatb, \\
\end{align*}
$$
s_\flatb = \sum_{\flata=0}^{N-1}
 \langle \frac{\partial J}{\partial t_\flata}, T^\flatb \rangle
  T^\flata.
$$
By looking at the differential equation
$$
 \hbar \frac{\partial s_\flata}{\partial t_\flatb}
  = T_\flatb \circ s_\flata,
$$
one can compute the product structure
$\circ$, hence the genus-zero Gromov-Witten invariants,
from the $J$-function.

Now form a row vector
$$
 L = \left(
      T_0 \left(
       \hbar \frac{\partial}{\partial \motonot_1},
       \ldots,
       \hbar \frac{\partial}{\partial \motonot_\picn}
          \right) I,
      \ldots,
      T_{N-1} \left(
       \hbar \frac{\partial}{\partial \motonot_1},
       \ldots,
       \hbar \frac{\partial}{\partial \motonot_\picn}
          \right) I
     \right).
$$
By the definition
(\ref{eq:defI})
of $I$,
there exists a function $S$
defined on $\docJ \times \{ \| \hbar \|=1 \}$
such that
$$
 L = e^{\motonot_0 + p \motonot} S.
$$
By using the basis
$\{ T_i(p_1, \ldots, p_\picn) \}_{i=0}^{N-1}$
of the total cohomology,
$S$ can be considered as an $N \times N$-matrix-valued
function on
$
 %H^0(\XSigma;\bC) \times
 \docJ
 \times \{\hbar \in \bC \suchthat | \hbar | = 1 \}$.
Since
$
 I(\motonot_0, \motonot; \hbar)
  = e^{\motonot_0 + p \motonot}
     \left( 1 + O(q) \right),
$
$$
 S(\motonot; \hbar) = \identitymatrix + O(q)
$$
where $\identitymatrix$ denotes the identity matrix,
which means that
if we fix $\motonot \in H^2(X;\bC)$
such that
$\Re t$ is small enough
so that $q \ll 1$,
$S(\motonot; \hbar)$
gives an element of the loop group of $GL_N(\bC)$
as a function of $\hbar$.
Now take the Birkhoff factorization
$$
 S(t; \hbar) = S_-(t; \hbar) S_+(t; \hbar).
$$
of $S(t; \hbar)$ where
$$
 S_-(t; \hbar) = \identitymatrix + O(\hbar^{-1})
$$
is a loop which can be extended to
$\{ \hbar \in \bC \suchthat | \hbar | > 1 \}$
an
$$
 S_(t; \hbar) = \identitymatrix + O(\hbar)
$$
is another loop which can be extended to
$\{ \hbar \in \bC \suchthat | \hbar | < 1 \}$.
Then by Givental's mirror theorem
(see e.g., \cite{Givental_MTTCI},
\cite{Iritani_QDGMT}),
the $J$-function of $\XSigma$ is
given by the first column
of the matrix
$$
 e^{(\motonot_0
     + p_1 \motonot_1 + \cdots + p_\picn \motonot_\picn
      ) / \hbar} S_-.
$$
This theorem is remarkable
in that
it determines the $J$-function
which encodes the information
of the Gromov-Witten invariants
of $\XSigma$
from the $I$-function
which is explicitly given
by the combinatorial data
of the fan $\Sigma$.
This theorem is also applicable
to a general smooth projective
toric variety
without the assumption
that $-K_\XSigma$ is nef,
where
one can work formally with respect to $q$
although the $I$-function does not converge.

An important point
in the Givental's mirror theorem
is that $\motonot_0$ and $\motonot$ are
not the flat coordinate $\atonot$
of the Frobenius manifold
coming from the Gromov-Witten invariants of $\XSigma$.
The relation between the variables $\motonot_i$'s and
$\atonot_i$'s
is given by the {\em mirror transformation},
defined as follows:

Since $S$ is homogeneous,
\begin{eqnarray*}
 [\Etilde, S] &=& [\Etilde, S_- S_+] \\
  &=& [\Etilde, S_-] S_+ + S_- [\Etilde, S_+] \\
  &=& 0,
\end{eqnarray*}
where $E$ is the Euler vector field.
Therefore,
we have
\begin{eqnarray*}
 - S_-^{-1} [\Etilde, S_-] &=& [\Etilde, S_+] S_+^{-1}. \\
\end{eqnarray*}
Since
$
 S(\motonot_0, \motonot; \hbar) = \identitymatrix + O(q),
$
the left-hand side
and the right-hand side
of the above equality
contains only
negative and positive powers of $\hbar$
respectively,
and
$$
 [\Etilde, S_-] = [\Etilde, S_+] = 0,
$$
follows,
which shows the homogeneity of $S_+$.

toricstokes.bbl
\newcommand{\noop}[1]{}\def\cprime{$'$}
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