# Lagrangian Floer theory on compact toric manifolds I

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

(Date: Sep. 3, 2009)

###### Abstract.

The present authors introduced the notion of _weakly unobstructed_ Lagrangian
submanifolds and constructed their _potential function_ $\mathfrak{PO}$ purely
in terms of $A$-model data in [FOOO3]. In this paper, we carry out explicit
calculations involving $\mathfrak{PO}$ on toric manifolds and study the
relationship between this class of Lagrangian submanifolds with the earlier
work of Givental [Gi1] which advocates that quantum cohomology ring is
isomorphic to the Jacobian ring of a certain function, called the Landau-
Ginzburg superpotential. Combining this study with the results from [FOOO3],
we also apply the study to various examples to illustrate its implications to
symplectic topology of Lagrangian fibers of toric manifolds. In particular we
relate it to Hamiltonian displacement property of Lagrangian fibers and to
Entov-Polterovich’s symplectic quasi-states.

###### Key words and phrases:

Floer cohomology, toric manifolds, weakly unobstructed Lagrangian
submanifolds, potential function, Jacobian ring, balanced Lagrangian fibers,
quantum cohomology

KF is supported partially by JSPS Grant-in-Aid for Scientific Research
No.18104001 and Global COE Program G08, YO by US NSF grant # 0503954, HO by
JSPS Grant-in-Aid for Scientific Research No.19340017, and KO by JSPS Grant-
in-Aid for Scientific Research, Nos. 17654009 and 18340014

###### Contents

  1. 1 Introduction
  2. 2 Compact toric manifolds
    1. 2.1 Complex structure
    2. 2.2 Symplectic structure
  3. 3 Deformation theory of filtered $A_{\infty}$-algebras
  4. 4 Potential function
  5. 5 Examples
  6. 6 Quantum cohomology and Jacobian ring
  7. 7 Localization of quantum cohomology ring at moment polytope
  8. 8 Further examples and remarks
  9. 9 Variational analysis of potential function
  10. 10 Elimination of higher order term in nondegenerate cases
  11. 11 Calculation of potential function
  12. 12 Non-unitary flat connection on $L(u)$
  13. 13 Floer cohomology at a critical point of potential function
  14. 14 Appendix 1 : Algebraically closedness of Novikov fields
  15. 15 Appendix 2 : $T^{n}$-equivariant Kuranishi structure
  16. 16 Appendix 3 : Smooth correspondence via the zero set of multisection

##  1\. Introduction

Floer theory of Lagrangian submanifolds plays an important role in symplectic
geometry since Floer’s invention [Fl] of the Floer cohomology and subsequent
generalization to the class of _monotone_ Lagrangian submanifolds [Oh1]. After
the introduction of $A_{\infty}$ structure in Floer theory [Fu1] and
Kontsevich’s homological mirror symmetry proposal [Ko], it has also played an
essential role in a formulation of mirror symmetry in string theory.

In [FOOO1], we have analyzed the anomaly $\partial^{2}\neq 0$ and developed an
obstruction theory for the definition of Floer cohomology and introduced the
class of _unobstructed_ Lagrangian submanifolds for which one can deform
Floer’s original definition of the ‘boundary’ map by a suitable bounding
cochain denoted by $b$. Expanding the discussion in section 7 [FOOO1] and
motivated by the work of Cho-Oh [CO], we also introduced the notion of _weakly
unobstructed_ Lagrangian submanifolds in Chapter 3 [FOOO3] which turns out to
be the right class of Lagrangian submanifolds to look at in relation to the
mirror symmetry of Fano toric $A$-model and Landau-Ginzburg $B$-model proposed
by physicists (see [Ho], [HV]). In the present paper, we study the
relationship between this class of Lagrangian submanifolds with the earlier
work of Givental [Gi1] which advocates that quantum cohomology ring is
isomorphic to the Jacobian ring of a certain function, which is called the
Landau-Ginzburg superpotential. Combining this study with the results from
[FOOO3], we also apply this study to symplectic topology of Lagrangian fibers
of toric manifolds.

While appearance of bounding cochains is natural in the point of view of
deformation theory, explicit computation thereof has not been carried out. One
of the main purposes of the present paper is to perform this calculation in
the case of fibers of toric manifolds and draw its various applications.
Especially we show that each fiber $L(u)$ at $u\in\mathfrak{t}^{*}$ is weakly
unobstructed for _any_ toric manifold $\pi:X\to\mathfrak{t}^{*}$ (see
Proposition 4.3), and then show that the set of the pairs $(L(u),b)$ of a
fiber $L(u)$ and a weak bounding cochain $b$ with nontrivial Floer cohomology
can be calculated from the quantum cohomology of the ambient toric manifold,
at least in the Fano case. Namely the set of such pairs $(L(u),b)$ is
identified with the set of ring homomorphisms from quantum cohomology to the
relevant Novikov ring. We also show by a variational analysis that for any
compact toric manifold there exists at least one pair of $(u,b)$’s for which
the Floer cohomology of $(L(u),b)$ is nontrivial.

We call a Lagrangian fiber (that is, a $T^{n}$-orbit) balanced, roughly
speaking, if its Floer cohomology is nontrivial. (See Definition 4.11 for its
precise definition.) The main result of this paper is summarized as follows.

  1. (1)

When $X$ is a compact Fano toric manifold, we give a method to locate all the
balanced fibers.

  2. (2)

Even when $X$ is not Fano, we can still apply the same method to obtain a
finite set of Lagrangian fibers. We prove this set coincides with the set of
balanced Lagrangian fibers under certain nondegeneracy condition. This
condition can be easily checked, when a toric manifold is given.

Now more precise statement of the main results are in order.

Let $X$ be an $n$ dimensional smooth compact toric manifold. We fix a
$T^{n}$-equivariant Kähler form on $X$ and let
$\pi:X\to\mathfrak{t}^{*}\cong({\mathbb{R}}^{n})^{*}$ be the moment map. The
image $P=\pi(X)\subset({\mathbb{R}}^{n})^{*}$ is called the moment polytope.
For $u\in\text{\rm Int}\,P$, we denote $L(u)=\pi^{-1}(u)$. The fiber $L(u)$ is
a Lagrangian torus which is an orbit of the $T^{n}$ action. (See section 2. We
refer readers to, for example, [Au], [Ful] for the details on toric
manifolds.) We study the Floer cohomology defined in [FOOO3]. According to
[FOOO1, FOOO3], we need an extra data, the bounding cochain, to make the
definition of Floer cohomology more flexible to allow more general class of
Lagrangian submanifolds. In the current context of Lagrangian torus fibers in
toric manifolds, we use _weak bounding cochains_. Denote by
${\mathcal{M}}_{\text{\rm weak}}(L(u);\Lambda_{0})$ the moduli space of (weak)
bounding cochains for a weakly unobstructed Lagrangian submanifold $L(u)$.
(See the end of section 4.)

In this situation we first show that each element in $H^{1}(L(u);\Lambda_{0})$
gives rise to a weak bounding cochain, i.e., there is a natural embedding

$H^{1}(L(u);\Lambda_{0})\hookrightarrow\mathcal{M}_{\text{\rm
weak}}(L(u);\Lambda_{0}).$ (1.1)

(See Proposition 4.3.) Here we use the universal Novikov ring

$\Lambda=\left\\{\sum_{i=1}^{\infty}a_{i}T^{\lambda_{i}}\,\Big{|}\,a_{i}\in{\mathbb{Q}},\lambda_{i}\in{\mathbb{R}},\lim_{i\to\infty}\lambda_{i}=\infty\right\\}$
(1.2)

where $T$ is a formal parameter. (We do not use the grading parameter $e$ used
in [FOOO3] since it will not play much role in this paper.) Then $\Lambda_{0}$
is a subring of $\Lambda$ defined by

$\Lambda_{0}=\left\\{\sum_{i=1}^{\infty}a_{i}T^{\lambda_{i}}\in\Lambda\,\Big{|}\,\lambda_{i}\geq
0\right\\}.$ (1.3)

We also use another subring

$\Lambda_{+}=\left\\{\sum_{i=1}^{\infty}a_{i}T^{\lambda_{i}}\in\Lambda\,\Big{|}\,\lambda_{i}>0\right\\}.$
(1.4)

We note $\Lambda$ is the field of fractions of $\Lambda_{0}$ and $\Lambda_{0}$
is a local ring with maximal ideal $\Lambda_{+}$. Here we take the universal
Novikov ring over ${\mathbb{Q}}$ but we also use universal Novikov ring over
${\mathbb{C}}$ or other ring $R$ which we denote $\Lambda^{{\mathbb{C}}}$,
$\Lambda^{R}$, respectively. (In case $R$ does not contain ${\mathbb{Q}}$,
Floer cohomology over $\Lambda^{R}$ is defined only in Fano case.)

######  Remark 1.1.

If we strictly follow the way taken in [FOOO3], we only get the embedding
$H^{1}(L(u);\Lambda_{+})\hookrightarrow\mathcal{M}_{\text{\rm weak}}(L(u))$,
not (1.1). Here

$\mathcal{M}_{\text{\rm weak}}(L(u))=\mathcal{M}_{\text{\rm
weak}}(L(u);\Lambda_{+})$

is defined in [FOOO3]. (We note that $\mathcal{M}_{\text{\rm
weak}}(L(u);\Lambda_{+})\neq\mathcal{M}_{\text{\rm weak}}(L(u);\Lambda_{0})$
where the right hand side is defined at the end of section 4.)

However we can modify the definition of weak unobstructedness so that (1.1)
follows, using the idea of Cho [Cho]. See section 12.

Hereafter we use the symbol $b$ for an element of $\mathcal{M}_{\text{\rm
weak}}(L(u);\Lambda_{+})$ and $\mathfrak{x}$ for an element of
$\mathcal{M}_{\text{\rm weak}}(L(u);\Lambda_{0})$.

We next consider the quantum cohomology ring $QH(X;\Lambda)$ with the
universal Novikov ring $\Lambda$ as a coefficient ring. (See section 6.) It is
a commutative ring for the toric case, since $QH(X;\Lambda)$ is generated by
even degree cohomology classes.

######  Definition 1.2.

  1. (1)

We define the set $\text{\rm Spec}(QH(X;\Lambda))(\Lambda^{{\mathbb{C}}})$ to
be the set of $\Lambda$ algebra homomorphisms
$\varphi:QH(X;\Lambda)\to\Lambda^{{\mathbb{C}}}$. (In other words it is the
set of all $\Lambda^{{\mathbb{C}}}$ valued points of the scheme $\text{\rm
Spec}(QH(X;\Lambda))$.

  2. (2)

We next denote by $\mathfrak{M}(\mathfrak{Lag}(X))$ the set of all pairs
$(\mathfrak{x},u)$, $u\in\operatorname{Int}P$, $\mathfrak{x}\in
H^{1}(L(u);\Lambda_{0}^{{\mathbb{C}}})/H^{1}(L(u);2\pi\sqrt{-1}{\mathbb{Z}})$
such that

$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda^{{\mathbb{C}}})\neq\\{0\\}.$

######  Theorem 1.3.

If $X$ is a Fano toric manifold then

$\text{\rm
Spec}(QH(X;\Lambda))(\Lambda^{{\mathbb{C}}})\cong\mathfrak{M}(\mathfrak{Lag}(X)).$

If $QH(X;\Lambda)$ is semi-simple in addition, we have

$\sum_{d}\text{\rm
rank}_{{\mathbb{Q}}}H_{d}(X;{\mathbb{Q}})=\\#\left(\mathfrak{M}(\mathfrak{Lag}(X))\right).$
(1.5)

We remark that a commutative ring that is a finite dimensional vector space
over a field (e.g., $\Lambda$ in our case) is semi-simple if and only if it
does not contain any nilpotent element. We also remark that a compact toric
manifold is Fano if and only if every nontrivial holomorphic sphere has
positive Chern number.

We believe that (1.5) still holds in the non-Fano case but are unable to prove
it at the time of writing this paper. We however can prove that there exists a
fiber $L(u)$ whose Floer cohomology is nontrivial, by a method different from
the proof of Theorem 1.3. Due to some technical reason, we can only prove the
following slightly weaker statement.

######  Theorem 1.4.

Assume the Kähler form $\omega$ of $X$ is rational. Then, there exists
$u\in\operatorname{Int}P$ such that for any
${{\mathcal{N}}}\in{\mathbb{R}}_{+}$ there exists $\mathfrak{x}\in
H^{1}(L(u);\Lambda^{{\mathbb{R}}}_{0})$ with

$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda^{{\mathbb{R}}}_{0}/(T^{{\mathcal{N}}}))\cong
H(T^{n};{{\mathbb{R}}})\otimes_{{\mathbb{R}}}\Lambda_{0}^{{\mathbb{R}}}/(T^{{\mathcal{N}}}).$

We suspect that the rationality assumption in Theorem 1.4 can be removed. It
is also likely that we can prove $\mathfrak{M}(\mathfrak{Lag}(X))$ is nonempty
but its proof at the moment is a bit cumbersome to write down. We can however
derive the following theorem from Theorem 1.4, without rationality assumption.

######  Theorem 1.5.

Let $X$ be an $n$ dimensional compact toric manifold. There exists
$u_{0}\in\text{\rm Int}P$ such that the following holds for any Hamiltonian
diffeomorphism $\psi:X\to X$,

$\psi(L(u_{0}))\cap L(u_{0})\neq\emptyset.$ (1.6)

If $\psi(L(u_{0}))$ is transversal to $L(u_{0})$ in addition, then

$\\#(\psi(L(u_{0}))\cap L(u_{0}))\geq 2^{n}.$ (1.7)

Theorem 1.5 will be proved in section 13.

We would like to point out that (1.6) can be derived from a more general
intersection result, Theorem 2.1 [EP1], obtained by Entov-Polterovich with a
different method using a very interesting notion of partial symplectic quasi-
state constructed out of the spectral invariants defined in [Sc], [Oh3]. (See
also [Vi], [Oh2] for similar constructions in the context of exact Lagrangian
submanifolds.)

######  Remark 1.6.

Strictly speaking, Theorem 2.1 [EP1] is stated under the assumption that $X$
is semi-positive and $\omega$ is rational because the theory of spectral
invariant was developed in [Oh3] under these conditions. The rationality
assumption has been removed in [Oh4], [Us] and the semi-positivity assumption
of $\omega$ removed by Usher [Us]. Thus the spectral invariant satisfying all
the properties listed in [EP1] section 5 is now established for an arbitrary
compact symplectic manifold. By the argument of [EP1] section 7, this implies
the existence of a partial symplectic quasi-state. Therefore the proof of
Theorem 2.1 [EP1] goes through without these assumptions (semi-positivity and
rationality) and hence it implies (1.6). (See the introduction of [Us].) But
the result (1.7) is new.

Our proof of Theorem 1.5 gives an explicit way of locating $u_{0}$, as we show
in section 9. (The method of [EP1] is indirect and does not provide a way of
finding such $u_{0}$. See [EP2]. Below, we will make some remarks concerning
Entov-Polterovich’s approach in the perspective of homological mirror
symmetry.) In various explicit examples we can find more than one element
$u_{0}$ that have the properties stated in this theorem. Following terminology
employed in [CO], we call any such torus fiber $L(u_{0})$ as in Theorem 1.4 a
_balanced_ Lagrangian torus fiber. (See Definition 4.11 for its precise
definition.)

A criterion for $L(u_{0})$ to be balanced, for the case $\mathfrak{x}=0$, is
provided by Cho-Oh [CO] and Cho [Cho] under the Fano condition. Our proofs of
Theorems 1.4, 1.5 are much based on this criterion, and on the idea of Cho
[Cho] of twisting _non-unitary_ complex line bundles in the construction of
Floer boundary operator. This criterion in turn specializes to the one
predicted by physicists [HV], [Ho], which relates the location of $u_{0}$ to
the critical points of the Landau-Ginzburg superpotential.

A precise description of balanced Lagrangian fibers including the data of
bounding cochains involves the notion of a _potential function_ : In [FOOO3],
the authors have introduced a function

$\mathfrak{PO}^{L}:{\mathcal{M}}_{\text{\rm weak}}(L)\to\Lambda_{0}$

for an arbitrary weakly unobstructed Lagrangian submanifold
$L\subset(X,\omega)$. By varying the function $\mathfrak{PO}^{L}$ over
$L\in\\{\pi^{-1}(u)\mid u\in\text{Int}\,P\\}$, we obtain the potential
function

$\mathfrak{PO}:\bigcup_{L\in\\{\pi^{-1}(u)\mid
u\in\text{Int}\,P\\}}{\mathcal{M}}_{\text{\rm weak}}(L)\to\Lambda_{0}.$ (1.8)

This function is constructed purely in terms of $A$-model data of the general
symplectic manifold $(X,\omega)$ _without_ using mirror symmetry.

For a toric $(X,\omega)$, the restriction of $\mathfrak{PO}$ to
$H^{1}(L(u);\Lambda_{+})$ (see (1.1)) can be made explicit when combined with
the analysis of holomorphic discs attached to torus fibers of toric manifolds
carried out in [CO], at least in the Fano case. (In the non-Fano case we can
make it explicit modulo ‘higher order terms’.) This function extends to
$H^{1}(L(u);\Lambda_{0})$.

######  Remark 1.7.

In [EP3] some relationships between quantum cohomology, quasi-state, spectral
invariant and displacement of Lagrangian submanifolds are discussed : Consider
an idempotent i of quantum cohomology. The (asymptotic) spectral invariants
associated to i gives rise to a partial symplectic quasi-state via the
procedure concocted in [EP3], which in turn detects non-displaceability of
certain Lagrangian submanifolds. (The assumption of [EP1] is weaker than
ours.)

In the current context of toric manifolds, we could also relate them to Floer
cohomology and mirror symmetry in the following way : Quantum cohomology is
decomposed into indecomposable factors. (See Proposition 7.7.) Let i be the
idempotent corresponding to one of the indecomposable factors. Let
$L=L(u(1,\text{\bf i}))$ be a Lagrangian torus fiber whose non-displaceability
is detected by the partial symplectic quasi-state obtained from i. We
conjecture that Floer cohomology $HF(L(u(1,\text{\bf
i}),\mathfrak{x}),(L(u(1,\text{\bf i}),\mathfrak{x})))$ is nontrivial for some
$\mathfrak{x}$. (See Remark 5.8.) This bounding cochain $\mathfrak{x}$ in turn
is shown to be a critical point of the potential function $\mathfrak{PO}$
defined in [FOOO3].

On the other hand, i also determines a homomorphism $\varphi_{\text{\bf
i}}:QH(X;\Lambda)\to\Lambda$. It corresponds to some Lagrangian fiber
$L(u(2,\text{\bf i}))$ by Theorem 1.3. Then this will imply via Theorem 4.10
that the fiber $L(u(2,\text{\bf i}))$ is non-displaceable.

We conjecture that $u(1,\text{\bf i})=u(2,\text{\bf i})$. We remark that
$u(2,\text{\bf i})$ is explicitly calculable. Hence in view of the way
$u(1,\text{\bf i})$ is found in [EP1], $u(1,\text{\bf i})=u(2,\text{\bf i})$
will give some information on the asymptotic behavior of the spectral
invariant associated with i.

We fix a basis of the Lie algebra $\mathfrak{t}$ of $T^{n}$ which induces a
basis of $\mathfrak{t}^{*}$ and hence a coordinate of the moment polytope
$P\subset\mathfrak{t}^{*}$. This in turn induces a basis of
$H^{1}(L(u);\Lambda_{0})$ for each $u\in\operatorname{Int}P$ and so
identification $H^{1}(L(u);\Lambda_{0})\cong(\Lambda_{0})^{n}$. We then regard
the potential function as a function

$\mathfrak{PO}(x_{1},\cdots,x_{n};u_{1},\cdots,u_{n}):(\Lambda_{0})^{n}\times\operatorname{Int}P\to\Lambda_{0}$

and prove in Theorem 4.10 that Floer cohomology
$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda)$ with
$\mathfrak{x}=(\mathfrak{x}_{1},\cdots,\mathfrak{x}_{n})$,
$u=(u_{1},\cdots,u_{n})$ is nontrivial if and only if $(\mathfrak{x},u)$
satisfies

$\frac{\partial\mathfrak{PO}}{\partial x_{i}}(\mathfrak{x};u)=0,\quad
i=1,\cdots,n.$ (1.9)

To study (1.9), it is useful to change the variables $x_{i}$ to

$y_{i}=e^{x_{i}}.$

In these variables we can write potential function as a sum

$\mathfrak{PO}(x_{1},\cdots,x_{n};u_{1},\cdots,u_{n})=\sum
T^{c_{i}(u)}P_{i}(y_{1},\cdots,y_{n})$

where $P_{i}$ are Laurent polynomials which do not depend on $u$, and
$c_{i}(u)$ are positive real valued functions. When $X$ is Fano, we can
express the right hand side as a finite sum. (See Theorem 4.5.)

We define a function $\mathfrak{PO}^{u}$ of $y_{i}$’s by

$\mathfrak{PO}^{u}(y_{1},\cdots,y_{n})=\mathfrak{PO}(x_{1},\cdots,x_{n};u_{1},\cdots,u_{n})$

as a Laurent polynomial of $n$ variables with coefficient in $\Lambda$. We
denote the set of Laurent polynomials by

$\Lambda[y_{1},\cdots,y_{n},y_{1}^{-1},\cdots,y_{n}^{-1}]$

and consider its ideal generated by the partial derivatives of
$\mathfrak{PO}^{u}$. Namely

$\left(\frac{\partial\mathfrak{PO}^{u}}{\partial y_{i}};i=1,\cdots,n\right).$

######  Definition 1.8.

We call the quotient ring

$Jac(\mathfrak{PO}^{u})=\frac{\Lambda[y_{1},\cdots,y_{n},y_{1}^{-1},\cdots,y_{n}^{-1}]}{\left(\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{i}};i=1,\cdots,n\right)}$

the Jacobian ring of $\mathfrak{PO}^{u}$.

We will prove that the Jacobian ring is independent of the choice of $u$ up to
isomorphism (see the end of section 6) and so we will just write
$Jac(\mathfrak{PO})$ for $Jac(\mathfrak{PO^{u}})$ when there is no danger of
confusion..

######  Theorem 1.9.

If $X$ is Fano, there exists a $\Lambda$ algebra isomorphism

$\psi_{u}:QH(X;\Lambda)\to Jac(\mathfrak{PO})$

from quantum cohomology ring to the Jacobian ring such that

$\psi_{u}(c_{1}(X))=\mathfrak{PO}^{u}.$

Theorem 1.9 (or Theorem 1.12 below) enables us to explicitly determine all the
pairs $(\mathfrak{x},u)$ with
$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda)\neq 0$ out of the quantum
cohomology of $X$. More specifically Batyrev’s presentation of quantum
cohomology in terms of the Jacobian ring plays an essential role for this
purpose. We will explain how this is done in sections 7, 8.

######  Remark 1.10.

  1. (1)

The idea that quantum cohomology ring coincides with the Jacobian ring begins
with a celebrated paper by Givental. (See Theorem 5 (1) [Gi1].) There it was
claimed also that the $D$ module defined by an oscillatory integral with the
superpotential as its kernel is isomorphic to $S^{1}$-equivariant Floer
cohomology of periodic Hamiltonian system. When one takes its WKB limit, the
former becomes the ring of functions on its characteristic variety, which is
nothing but the Jacobian ring. The latter becomes the (small) quantum
cohomology ring under the same limit. _Assuming_ the Ansatz that quantum
cohomology can be calculated by fixed point localization, these claims are
proved in a subsequent paper [Gi2] for, at least, toric Fano manifolds. Then
the required fixed point localization is made rigorous later in [GrPa]. See
also Iritani [Iri1].

In physics literature, it has been advocated that Landau-Ginzburg model of
superpotential (that is, the potential function $\mathfrak{PO}$ in our
situation) calculates quantum cohomology of $X$. A precise mathematical
statement thereof is our Theorem 1.9. (See for example p. 473 [MIRROR].)

Our main new idea entering in the proof of Theorem 1.3 other than those
already in [FOOO3] is the way how we combine them to extract information on
Lagrangian submanifolds. In fact Theorem 1.9 itself easily follows if we use
the claim made by Batyrev that quantum cohomology of toric Fano manifold is a
quotient of polynomial ring by the ideal of relations, called quantum Stanley-
Reisner relation and linear relation. (This claim is now well established.) We
include this simple derivation in section 6 for completeness’ sake, since it
is essential to take the Novikov ring $\Lambda$ as the coefficient ring in our
applications the version of which does not seem to be proven in the literature
in the form that can be easily quoted.

  2. (2)

The proof of Theorem 1.9 given in this paper does not contain a serious study
of pseudo-holomorphic spheres. The argument which we outline in Remark 6.15 is
based on open-closed Gromov-Witten theory, and different from other various
methods that have been used to calculate Gromov-Witten invariant in the
literature. In particular this argument does not use the method of fixed point
localization. We will present this conceptual proof of Theorem 1.9 in a sequel
to this paper.

  3. (3)

The isomorphism in Theorem 1.9 may be regarded as a particular case of the
conjectural relation between quantum cohomology and Hochschild cohomology of
Fukaya category. See Remark 6.15.

  4. (4)

In this paper, we only involve small quantum cohomology ring but we can also
include big quantum cohomology ring. Then we expect Theorem 1.9 can be
enhanced to establish a relationship between the Frobenius structure of the
deformation theory of quantum cohomology (see, for example, [Man]) and that of
Landau-Ginzburg model (which is due to K. Saito [Sa]). This statement (and
Theorems 1.3, 1.9) can be regarded as a version of mirror symmetry between the
toric A-model and the Landau-Ginzburg $B$-model. In various literature on
mirror symmetry, such as [Ab], [AKO], [Ue], the B-model is dealt for Fano or
toric manifolds in which the derived category of coherent sheaves is studied
while the A-model is dealt for Landau-Ginzburg $A$-models where the directed
$A_{\infty}$ category of Seidel [Se2] is studied.

  5. (5)

In [Aur], Auroux discussed a mirror symmetry between the $A$-model side of
toric manifolds and the $B$-model side of Landau-Ginzburg models. The
discussion of [Aur] uses Floer cohomology with ${\mathbb{C}}$-coefficients. In
this paper we use Floer cohomology over the Novikov ring which is more
suitable for the applications to symplectic topology.

  6. (6)

Even when $X$ is not necessarily Fano we can still prove a similar isomorphism

$\psi_{u}:QH^{\omega}(X;\Lambda)\cong Jac(\mathfrak{PO}_{0})$ (1.10)

where the left hand side is the Batyrev quantum cohomology ring (see
Definition 6.4) and the right hand side is the Jacobian ring of some function
$\mathfrak{PO}_{0}$ : it coincides with the actual potential function
$\mathfrak{PO}$ ‘up to higher order terms’. (See (4.9).) In the Fano case
$\mathfrak{PO}_{0}=\mathfrak{PO}$. (1.10) is Proposition 6.8.

  7. (7)

During the final stage of writing this article, a paper [CL] by Chan and Leung
was posted in the Archive which studies the above isomorphism via SYZ
transformations. They give a proof of this isomorphism for the case where $X$
is a product of projective spaces and with the coefficient ring
${\mathbb{C}}$, not with Novikov ring. Leung presented their result [CL] in a
conference held in Kyoto University in January 2008 where the first named
author also presented the content of this paper.

From our definition, it follows that the leading order potential function
$\mathfrak{PO}_{0}$ (see (4.9)) can be extended to the whole product
$(\Lambda^{\mathbb{C}}_{0})^{n}\times{\mathbb{R}}^{n}$ in a way that they are
invariant under the translations by elements in
$(2\pi\sqrt{-1}{\mathbb{Z}})^{n}\subseteq(\Lambda^{\mathbb{C}}_{0})^{n}$.
Hence we may regard $\mathfrak{PO}_{0}$ as a function defined on

$\left(\Lambda^{\mathbb{C}}_{0}/(2\pi\sqrt{-1}{\mathbb{Z}})\right)^{n}\times{\mathbb{R}}^{n}\cong\left(\Lambda^{\mathbb{C}}_{0}/(2\pi\sqrt{-1}{\mathbb{Z}})\right)^{n}\times{\mathbb{R}}^{n}.$

In the non-Fano case, the function $\mathfrak{PO}$ is invariant under the
translations by elements in
$(2\pi\sqrt{-1}{\mathbb{Z}})^{n}\subseteq(\Lambda^{\mathbb{C}}_{0})^{n}$ but
may not extend to
$\Lambda^{\mathbb{C}}_{0}/(2\pi\sqrt{-1}{\mathbb{Z}}))^{n}\times{\mathbb{R}}^{n}$.
This is because the infinite sum appearing in the right hand side of (4.7) may
not converge in non-Archimedean topology for $u\notin\mbox{Int}\,P$.

######  Definition 1.11.

We denote by

$\text{\rm Crit}(\mathfrak{PO}_{0}),\quad(\mbox{respectively }\,\text{\rm
Crit}(\mathfrak{PO}))$

the subset of pairs

$(\mathfrak{x},u)\in(\Lambda_{0}^{{\mathbb{C}}}/(2\pi\sqrt{-1}{\mathbb{Z}}))^{n}\times{\mathbb{R}}^{n},\quad(\mbox{respectively
}\,(\mathfrak{x},u)\in(\Lambda_{0}^{{\mathbb{C}}}/(2\pi\sqrt{-1}{\mathbb{Z}}))^{n}\times\mbox{Int}\,P)$

satisfying the equation

$\frac{\partial\mathfrak{PO}_{0}}{\partial
x_{i}}(\mathfrak{x};u)=0,\quad\left(\mbox{respectively
}\,\frac{\partial\mathfrak{PO}}{\partial x_{i}}(\mathfrak{x};u)=0\right)$

$i=1,\cdots,n$.

We define $\mathfrak{M}(\mathfrak{Lag}(X))$ in Definition 1.2. (We use the
same definition in non-Fano case.) In view of Theorem 1.12 (2) below, we also
introduce the subset

$\mathfrak{M}_{0}(\mathfrak{Lag}(X))=\\{(\mathfrak{x},u)\in\text{\rm
Crit}(\mathfrak{PO}_{0})\mid u\in\text{\rm Int}\,P\\}.$

We also remark $\mathfrak{PO}_{0}=\mathfrak{PO}$ in case $X$ is Fano. The
following is a more precise form of Theorem 1.3.

######  Theorem 1.12.

  1. (1)

There exists a bijection :

$\text{\rm Spec}(QH^{\omega}(X;\Lambda))(\Lambda^{{\mathbb{C}}})\cong\text{\rm
Crit}(\mathfrak{PO}_{0}).$

  2. (2)

There exists a bijection :

$\mathfrak{M}(\mathfrak{Lag}(X))\cong\text{\rm Crit}(\mathfrak{PO}).$

  3. (3)

If $X$ is Fano and $\mathfrak{x}$ is a critical point of
$\mathfrak{PO}_{0}^{u}$ then $u\in\mbox{\rm Int}\,P$.

  4. (4)

If $QH^{\omega}(X;\Lambda)$ is semi-simple, then

$\sum_{d}\text{\rm rank}_{\Lambda}QH^{\omega}(X;\Lambda)=\\#\left(\text{\rm
Crit}(\mathfrak{PO}_{0})\right).$

######  Remark 1.13.

(1) Theorem 1.12 (3) does not hold in non-Fano case. We give a counter example
(Example 8.2) in section 8. In fact, in the case of Example 8.2 some of the
critical points of $\mathfrak{PO}_{0}$ correspond to a point
$u\in{\mathbb{R}}^{n}$ which lies outside the moment polytope.

(2) In the Fano case Theorem 1.12 (3) implies

$\mathfrak{M}(\mathfrak{Lag}(X))=\mbox{Crit}(\mathfrak{PO})=\mbox{Crit}(\mathfrak{PO}_{0}).$

We would like to point out that $\mathfrak{PO}_{0}$ is explicitly computable.
But we do not know the explicit form of $\mathfrak{PO}$. However we can show
that elements of $\text{\rm Crit}(\mathfrak{PO}_{0})$ and of $\text{\rm
Crit}(\mathfrak{PO})$ can be naturally related to each other under a mild
nondegeneracy condition. (Theorem 10.4.) So we can use $\mathfrak{PO}_{0}$ in
place of $\mathfrak{PO}$ in most of the cases. For example we can use it to
prove that the following :

######  Theorem 1.14.

For any $k$, there exists a Kähler form on $X(k)$, the $k$ points blow up of
${\mathbb{C}}P^{2}$, that is toric and has exactly $k+1$ balanced fibers.

See Definition 4.11 for the definition of balanced fibers. Balanced fiber
satisfies the conclusions (1.6), (1.7) of Theorem 1.5. We prove Theorem 1.14
in section 10.

######  Remark 1.15.

The cardinality of $\mathfrak{x}\in
H^{1}(L(u);\Lambda^{{\mathbb{C}}}_{0})/H^{1}(L(u);2\pi\sqrt{-1}{\mathbb{Z}})$
with nonvanishing Floer cohomology is an invariant of Lagrangian submanifold
$L(u)$. This is a consequence of [FOOO3] Theorem G (= [FOOO2] Theorem G).

The organization of this paper is now in order. In section 2 we gather some
basic facts on toric manifolds and fix our notations. Section 3 is a brief
review of Lagrangian Floer theory of [FOOO1, FOOO3]. In section 4, we describe
our main results on the potential function $\mathfrak{PO}$ and on its relation
to the Floer cohomology. We illustrate these theorems by several examples and
derive their consequences in sections 5 \- 10. We postpone their proofs until
sections 11 \- 13.

In section 5, we illustrate explicit calculations involving the potential
functions in such examples as ${\mathbb{C}}P^{n}$, $S^{2}\times S^{2}$ and the
two points blow up of ${\mathbb{C}}P^{2}$. We also discuss a relationship
between displacement energy of Lagrangian submanifold (see Definition 5.9) and
Floer cohomology. In sections 6 and 7, we prove the results that mainly apply
to the Fano case. Especially we prove Theorems 1.3, 1.9 and 1.12 in these
sections. Section 7 contains some applications of Theorem 1.9 especially to
the case of monotone torus fibers and to the ${\mathbb{Q}}$-structure of
quantum cohomology ring. In section 8 we first illustrate usage of (the proof
of) Theorem 1.3 to locate balanced fibers by the example of one point blow up
of ${\mathbb{C}}P^{2}$. We then turn to the study of non-Fano cases and
discuss Hirzebruch surfaces. Section 8 also contains some discussion on the
semi-simplicity of quantum cohomology.

In sections 9 and 10, we prove the results that can be used in all toric
cases, whether they are Fano or not. In section 9, using variational analysis,
we prove existence of a critical point of the potential function, which is an
important step towards the proof of Theorems 1.4 and 1.5. Using the arguments
of this section, we can locate a balanced fiber in any compact toric manifold,
explicitly solving simple linear equalities and inequalities finitely many
times. In section 10, we prove that we can find the solution of (1.9) by
studying its reduction to
${\mathbb{C}}=\Lambda_{0}^{{\mathbb{C}}}/\Lambda_{+}^{{\mathbb{C}}}$, which we
call the leading term equation. This result is purely algebraic. It implies
that our method of locating balanced fibers, which is used in the proof of
Theorem 1.3, can be also used in the non-Fano case under certain nondegeneracy
condition. We apply this method to prove Theorem 1.14. We discuss an example
of blow up of ${\mathbb{C}}P^{n}$ along the high dimensional blow up center
${\mathbb{C}}P^{m}$ in that section. We also give several other examples and
demonstrate various interesting phenomena which occur in Lagrangian Floer
theory. For example we provide a sequence $((X,\omega_{i}),L_{i})$ of pairs
that have nonzero Floer cohomology for some choice of bounding cochains, while
its limit $((X,\omega),L)$ has vanishing Floer cohomology for any choice of
bounding cochain (Example 10.17). We also provide an example of Lagrangian
submanifold $L$ such that it has a nonzero Floer cohomology over
$\Lambda^{{\mathbb{C}}}$ for some choice of bounding cochain, but vanishing
Floer cohomology for any choice of bounding cochain over the field
$\Lambda^{F}$ with a field $F$ of characteristic $3$ (Example 10.19).

In section 11, we review the results on the moduli space of holomorphic discs
from [CO] which are used in the calculation of the potential function. We
rewrite them in the form that can be used for the purpose of this paper. We
also discuss the non-Fano case in this section. (Our result is less explicit
in the non-Fano case, but still can be used to explicitly locate balanced
fibers in most of the cases.) In section 12, we use the idea of Cho [Cho] to
deform Floer cohomology by an element from $H^{1}(L(u);\Lambda_{0})$ rather
than from $H^{1}(L(u);\Lambda_{+})$. This enhancement is crucial to obtain an
optimal result about the non-displacement of Lagrangian fibers. In section 13
we use those results to calculate Floer cohomology and complete the proof of
Theorems 1.4 and 1.5 etc.

We attempted to make this paper largely independent of our book [FOOO1, FOOO3]
as much as possible and also to make the relationship of the contents of the
paper with the general story transparent. Here are a few examples :

  1. (1)

Our definition of potential function for the fibers of toric manifolds in this
paper is given in a way independent of that of [FOOO3] except the statement on
the existence of compatible Kuranishi structures and multi-sections on the
moduli space of pseudo-holomorphic discs which provides a rigorous definition
of Floer cohomology of single Lagrangian fiber. Such details are provided in
section 3.1 [FOOO3] (= section 29 [FOOO2]).

  2. (2)

Similarly the definition of $A_{\infty}$ algebra in this paper on the
Lagrangian fiber of toric manifolds is also independent of the book except the
process going from $A_{n,K}$ structure to $A_{\infty}$ structure, which we
refer to section 7.2 [FOOO3] (= section 30 [FOOO2]). However, for all the
applications in this paper, only existence of $A_{n,K}$ structures is needed.

  3. (3)

The property of the Floer cohomology $HF(L,L)$ detecting Lagrangian
intersection of $L$ with its Hamiltonian deformation relies on the fact that
Floer cohomology of the pair is independent under the Hamiltonian isotopy.
This independence is established in [FOOO3]. In the toric case, its
alternative proof based on the de Rham version is given in [FOOO4] in a more
general form than we need in this paper.

The authors would like to thank H. Iritani and D. McDuff for helpful
discussions. They would also like to thank the referee for various helpful
comments.

##  2\. Compact toric manifolds

In this section, we summarize basic facts on the toric manifolds and set-up
our notations to be consistent with those in [CO], which in turn closely
follow those in Batyrev [B1] and M. Audin [Au].

###  2.1. Complex structure

In order to obtain an $n$-dimensional compact toric manifold $X$, we need a
combinatorial object $\Sigma$, a complete fan of regular cones, in an
$n$-dimensional vector space over ${\mathbb{R}}$.

Let $N$ be the lattice ${\mathbb{Z}}^{n}$, and let
$M=Hom_{\mathbb{Z}}(N,{\mathbb{Z}})$ be the dual lattice of rank $n$. Let
$N_{{\mathbb{R}}}=N\otimes{\mathbb{R}}$ and
$M_{{\mathbb{R}}}=M\otimes{\mathbb{R}}$.

######  Definition 2.1.

A convex subset $\sigma\subset N_{{\mathbb{R}}}$ is called a regular
$k$-dimensional cone $(k\geq 1)$ if there exists $k$ linearly independent
elements $v_{1},\cdots,v_{k}\in N$ such that

$\sigma=\\{a_{1}v_{1}+\cdots+a_{k}v_{k}\mid a_{i}\in{\mathbb{R}},a_{i}\geq
0\\},$

and the set $\\{v_{1},\cdots,v_{k}\\}$ is a subset of some
${\mathbb{Z}}$-basis of $N$. In this case, we call $v_{1},\cdots,v_{k}\in N$
the integral generators of $\sigma$.

######  Definition 2.2.

A regular cone $\sigma^{\prime}$ is called a face of a regular cone $\sigma$
(we write $\sigma^{\prime}\prec\sigma$) if the set of integral generators of
$\sigma^{\prime}$ is a subset of the set of integral generators of $\sigma$.

######  Definition 2.3.

A finite system $\Sigma={\sigma_{1},\cdots,\sigma_{s}}$ of regular cones in
$N_{{\mathbb{R}}}$ is called a complete $n$-dimensional fan of regular cones,
if the following conditions are satisfied.

  1. (1)

if $\sigma\in\Sigma$ and $\sigma^{\prime}\prec\sigma$, then
$\sigma^{\prime}\in\Sigma$;

  2. (2)

if $\sigma,\sigma^{\prime}$ are in $\Sigma$, then
$\sigma^{\prime}\cap\sigma\prec\sigma$ and
$\sigma^{\prime}\cap\sigma\prec\sigma^{\prime}$;

  3. (3)

$N_{{\mathbb{R}}}=\sigma_{1}\cup\cdots\cup\sigma_{s}$.

The set of all $k$-dimensional cones in $\Sigma$ will be denoted by
$\Sigma^{(k)}.$

######  Definition 2.4.

Let $\Sigma$ be a complete $n$-dimensional fan of regular cones. Denote by
$G(\Sigma)=\\{v_{1},\cdots,v_{m}\\}$ the set of all generators of
1-dimensional cones in $\Sigma$ ($m=$ Card $\Sigma^{(1)}$). We call a subset
${\mathcal{P}}=\\{v_{i_{1}},\cdots,v_{i_{p}}\\}\subset G(\Sigma)$ a primitive
collection if $\\{v_{i_{1}},\cdots,v_{i_{p}}\\}$ does not generate
$p$-dimensional cone in $\Sigma$, while for all $k\,(0\leq k<p)$ each
$k$-element subset of ${\mathcal{P}}$ generates a $k$-dimensional cone in
$\Sigma$.

######  Definition 2.5.

Let ${\mathbb{C}}^{m}$ be an $m$-dimensional affine space over ${\mathbb{C}}$
with the set of coordinates $z_{1},\cdots,z_{m}$ which are in the one-to-one
correspondence $z_{i}\leftrightarrow v_{i}$ with elements of $G(\Sigma)$. Let
${\mathcal{P}}=\\{v_{i_{1}},\cdots,v_{i_{p}}\\}$ be a primitive collection in
$G(\Sigma)$. Denote by $\mathbb{A}({\mathcal{P}})$ the $(m-p)$-dimensional
affine subspace in ${\mathbb{C}}^{n}$ defined by the equations

$z_{i_{1}}=\cdots=z_{i_{p}}=0.$

Since every primitive collection ${\mathcal{P}}$ has at least two elements,
the codimension of $\mathbb{A}({\mathcal{P}})$ is at least 2.

######  Definition 2.6.

Define the closed algebraic subset $Z(\Sigma)$ in ${\mathbb{C}}^{m}$ as
follows

$Z(\Sigma)=\bigcup_{{\mathcal{P}}}\mathbb{A}({\mathcal{P}}),$

where ${\mathcal{P}}$ runs over all primitive collections in $G(\Sigma)$. Put

$U(\Sigma)={\mathbb{C}}^{m}\setminus Z(\Sigma).$

######  Definition 2.7.

Let ${\mathbb{K}}$ be the subgroup in ${\mathbb{Z}}^{m}$ consisting of all
lattice vectors $\lambda=(\lambda_{1},\cdots,\lambda_{m})$ such that

$\lambda_{1}v_{1}+\cdots+\lambda_{m}v_{m}=0.$

Obviously ${\mathbb{K}}$ is isomorphic to ${\mathbb{Z}}^{m-n}$ and we have the
exact sequence:

$0\to{\mathbb{K}}\to{\mathbb{Z}}^{m}\stackrel{{\scriptstyle\pi}}{{\to}}{\mathbb{Z}}^{n}\to
0,$ (2.1)

where the map $\pi$ sends the basis vectors $e_{i}$ to $v_{i}$ for
$i=1,\cdots,m$.

######  Definition 2.8.

Let $\Sigma$ be a complete $n$-dimensional fan of regular cones. Define
$D(\Sigma)$ to be the connected commutative subgroup in
$({\mathbb{C}}^{*})^{m}$ generated by all one-parameter subgroups

$a_{\lambda}:{\mathbb{C}}^{*}\to({\mathbb{C}}^{*})^{m},$
$t\mapsto(t^{\lambda_{1}},\cdots,t^{\lambda_{m}})$

where $\lambda=(\lambda_{1},\cdots,\lambda_{m})\in{\mathbb{K}}$.

It is easy to see from the definition that $D(\Sigma)$ acts freely on
$U(\Sigma)$. Now we are ready to give a definition of the compact toric
manifold $X_{\Sigma}$ associated with a complete $n$-dimensional fan of
regular cones $\Sigma$.

######  Definition 2.9.

Let $\Sigma$ be a complete $n$-dimensional fan of regular cones. Then the
quotient

$X_{\Sigma}=U(\Sigma)/D(\Sigma)$

is called the compact toric manifold associated with $\Sigma$.

There exists a simple open coverings of $U(\Sigma)$ by affine algebraic
varieties.

######  Proposition 2.10.

Let $\sigma$ be a $k$-dimensional cone in $\Sigma$ generated by
$\\{v_{i_{1}},\cdots,v_{i_{k}}\\}.$ Define the open subset
$U(\sigma)\subset{\mathbb{C}}^{m}$ as

$U(\sigma)=\\{(z_{1},\cdots,z_{m})\in{\mathbb{C}}^{m}\mid z_{j}\neq
0\;\;\textrm{for all}\;j\notin\\{i_{1},\cdots,i_{k}\\}\\}.$

Then the open sets $U(\sigma)$ have the following properties:

  1. (1)

$U(\Sigma)=\bigcup_{\sigma\in\Sigma}U(\sigma);$

  2. (2)

if $\sigma\prec\sigma^{\prime}$, then $U(\sigma)\subset U(\sigma^{\prime})$;

  3. (3)

for any two cone $\sigma_{1},\sigma_{2}\in\Sigma$, one has $U(\sigma_{1})\cap
U(\sigma_{2})=U(\sigma_{1}\cap\sigma_{2})$; in particular,

$U(\Sigma)=\sum_{\sigma\in\Sigma^{(n)}}U(\sigma).$

######  Proposition 2.11.

Let $\sigma$ be an $n$-dimensional cone in $\Sigma^{(n)}$ generated by
$\\{v_{i_{1}},\cdots,v_{i_{n}}\\}$, which spans the lattice $M$. We denote the
dual ${\mathbb{Z}}$-basis of the lattice $N$ by
$\\{u_{i_{1}},\cdots,u_{i_{n}}\\}$. i.e.

$\langle v_{i_{k}},u_{i_{l}}\rangle=\delta_{k,l}$ (2.2)

where $\langle\cdot,\cdot\rangle$ is the canonical pairing between lattices
$N$ and $M$.

Then the affine open subset $U(\sigma)$ is isomorphic to
${\mathbb{C}}^{n}\times({\mathbb{C}}^{*})^{m-n}$, the action of $D(\Sigma)$ on
$U(\sigma)$ is free, and the space of $D(\Sigma)$-orbits is isomorphic to the
affine space $U_{\sigma}={\mathbb{C}}^{n}$ whose coordinate functions
$y_{1}^{\sigma},\cdots,y_{n}^{\sigma}$ are $n$ Laurent monomials in
$z_{1},\cdots,z_{m}$:

$\begin{cases}y_{1}^{\sigma}=z_{1}^{\langle v_{1},u_{i_{1}}\rangle}\cdots
z_{m}^{\langle v_{m},u_{i_{1}}\rangle}\\\ \qquad\vdots\\\
y_{n}^{\sigma}=z_{1}^{\langle v_{1},u_{i_{n}}\rangle}\cdots z_{m}^{\langle
v_{m},u_{i_{n}}\rangle}\end{cases}$ (2.3)

The last statement yields a general formula for the local affine coordinates
$y_{1}^{\sigma},\cdots,y_{n}^{\sigma}$ of a point $p\in U_{\sigma}$ as
functions of its “homogeneous coordinates” $z_{1},\cdots,z_{m}$.

###  2.2. Symplectic structure

In the last subsection, we associated a compact manifold $X_{\Sigma}$ to a fan
$\Sigma$. In this subsection, we review the construction of symplectic
(Kähler) manifold associated to a convex polytope $P$.

Let $M$ be a dual lattice, we consider a convex polytope $P$ in
$M_{{\mathbb{R}}}$ defined by

$\\{u\in M_{{\mathbb{R}}}\mid\langle
u,v_{j}\rangle\geq\lambda_{j}\;\textrm{for}\;j=1,\cdots,m\\}$ (2.4)

where $\langle\cdot,\cdot\rangle$ is a dot product of
$M_{{\mathbb{R}}}\cong{\mathbb{R}}^{n}$. Namely, $v_{j}$’s are inward normal
vectors to the codimension 1 faces of the polytope $P$. We associate to it a
fan in the lattice $N$ as follows: With any face $\Gamma$ of $P$, fix a point
$u_{0}$ in the (relative) interior of $\Gamma$ and define

$\sigma_{\Gamma}=\bigcup_{r\geq 0}r\cdot(P-u_{0}).$

The associated fan is the family $\Sigma(P)$ of dual convex cones

$\displaystyle\check{\sigma}_{\Gamma}$ $\displaystyle=$ $\displaystyle\\{x\in
N_{\mathbb{R}}\mid\langle y,x\rangle\geq 0\;\;\forall y\in\sigma_{\Gamma}\\}$
(2.5) $\displaystyle=$ $\displaystyle\\{x\in N_{\mathbb{R}}\mid\langle
u,x\rangle\leq\langle p,x\rangle\;\;\forall p\in P,u\in\Gamma\\}$ (2.6)

where $\langle\cdot,\cdot\rangle$ is dual pairing $M_{\mathbb{R}}$ and
$N_{\mathbb{R}}$. Hence we obtain a compact toric manifold $X_{\Sigma(P)}$
associated to a fan $\Sigma(P)$.

Now we define a symplectic (Kähler) form on $X_{\Sigma(P)}$ as follows. Recall
the exact sequence :

$0\to{\mathbb{K}}\stackrel{{\scriptstyle
i}}{{\to}}{\mathbb{Z}}^{m}\stackrel{{\scriptstyle\pi}}{{\to}}{\mathbb{Z}}^{n}\to
0.$

It induces another exact sequence :

$0\to
K\to{\mathbb{R}}^{m}/{\mathbb{Z}}^{m}\to{\mathbb{R}}^{n}/{\mathbb{Z}}^{n}\to
0.$

Denote by $k$ the Lie algebra of the real torus $K$. Then we have the exact
sequence of Lie algebras:

$0\to
k\to{\mathbb{R}}^{m}\stackrel{{\scriptstyle\pi}}{{\to}}{\mathbb{R}}^{n}\to 0.$

And we have the dual of above exact sequence:

$0\to({\mathbb{R}}^{n})^{*}\to({\mathbb{R}}^{m})^{*}\stackrel{{\scriptstyle
i^{*}}}{{\to}}k^{*}\to 0.$

Now, consider ${\mathbb{C}}^{m}$ with symplectic form $\frac{i}{2}\sum
dz_{k}\wedge d\overline{z}_{k}$. The standard action $T^{n}$ on
${\mathbb{C}}^{n}$ is hamiltonian with moment map

$\mu(z_{1},\cdots,z_{m})=\frac{1}{2}(|z_{1}|^{2},\cdots,|z_{m}|^{2}).$ (2.7)

For the moment map $\mu_{K}$ of the $K$ action is then given by

$\mu_{K}=i^{*}\circ\mu:{\mathbb{C}}^{m}\to k^{*}.$

If we choose a ${\mathbb{Z}}$-basis of ${\mathbb{K}}\subset{\mathbb{Z}}^{m}$
as

$Q_{1}=(Q_{11},\cdots,Q_{m1}),\cdots,Q_{k}=(Q_{1k},\cdots,Q_{mk})$

and $\\{q^{1},\cdots,q^{k}\\}$ be its dual basis of ${\mathbb{K}}^{*}$. Then
the map $i^{*}$ is given by the matrix $Q^{t}$ and so we have

$\mu_{K}(z_{1},\cdots,z_{m})=\frac{1}{2}\left(\sum_{j=1}^{m}Q_{j1}|z_{j}|^{2},\cdots,\sum_{j=1}^{m}Q_{jk}|z_{j}|^{2}\right)\in{\mathbb{R}}^{k}\cong
k^{*}$ (2.8)

in the coordinates associated to the basis $\\{q^{1},\cdots,q^{k}\\}$. We
denote again by $\mu_{K}$ the restriction of $\mu_{K}$ on
$U(\Sigma)\subset{\mathbb{C}}^{m}$.

######  Proposition 2.12 (See Audin [Au]).

Then for any $r=(r_{1},\cdots,r_{m-n})\in\mu_{K}(U(\Sigma))\subset k^{*}$, we
have a diffeomorphism

$\mu_{K}^{-1}(r)/K\cong U(\Sigma)/D(\Sigma)=X_{\Sigma}$ (2.9)

And for each (regular) value of $r\in k^{*}$, we can associate a symplectic
form $\omega_{P}$ on the manifold $X_{\Sigma}$ by symplectic reduction [MW].

To obtain the original polytope $P$ that we started with, we need to choose
$r$ as follows: Consider $\lambda_{j}$ for $j=1,\cdots,m$ which we used to
define our polytope $P$ by the set of inequalities $\langle
u,v_{j}\rangle\geq\lambda_{j}$. Then, for each $a=1,\cdots,m-n$, let

$r_{a}=-\sum_{j=1}^{m}Q_{ja}\lambda_{j}.$

Then we have

$\mu_{K}^{-1}(r_{1},\cdots,r_{m-n})/K\cong X_{\Sigma(P)}$

and for the residual $T^{n}\cong T^{m}/K$ action on $X_{\Sigma(P)}$, and for
its moment map $\pi$, we have

$\pi(X_{\Sigma(P)})=P.$

Using Delzant’s theorem [De], one can reconstruct the symplectic form out of
the polytope $P$ (up to $T^{n}$-equivariant symplectic diffeomorphisms). In
fact, Guillemin [Gu] proved the following explicit closed formula for the
$T^{n}$-invariant Kähler form associated to the canonical complex structure on
$X=X_{\Sigma}(P)$

######  Theorem 2.13 (Guillemin).

Let $P$, $X_{\Sigma(P)}$, $\omega_{P}$ be as above and

$\pi:X_{\Sigma(P)}\to({\mathbb{R}}^{m}/k)^{*}\cong({\mathbb{R}}^{n})^{*}$

be the associated moment map. Define the functions on $({\mathbb{R}}^{n})^{*}$

$\displaystyle\ell_{i}(u)$ $\displaystyle=$ $\displaystyle\langle
u,v_{i}\rangle-\lambda_{i}\,\mbox{ for }\,i=1,\cdots,m$ (2.10)
$\displaystyle\ell_{\infty}(u)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\langle u,v_{i}\rangle=\left\langle
u,\sum_{i=1}^{m}v_{i}\right\rangle.$

Then we have

$\omega_{P}=\sqrt{-1}\partial\overline{\partial}\left(\pi^{*}\Big{(}\sum_{i=1}^{m}\lambda_{i}(\log\ell_{i})+\ell_{\infty}\Big{)}\right)$
(2.11)

on $\text{\rm Int}\,P$.

The affine functions $\ell_{i}$ will play an important role in our description
of potential function as in [CO] since they also measure symplectic areas
$\omega(\beta_{i})$ of the canonical generators $\beta_{i}$ of
$H_{2}(X,L(u);{\mathbb{Z}})$. More precisely we have

$\omega(\beta_{i})=2\pi\ell_{i}(u)$ (2.12)

(see Theorem 8.1 [CO]). We also recall

$P=\\{u\in M_{\mathbb{R}}\mid\ell_{i}(u)\geq 0,\,i=1,\cdots,m\\}$ (2.13)

by definition (2.4).

##  3\. Deformation theory of filtered $A_{\infty}$-algebras

In this section, we provide a quick summary of the deformation and obstruction
theory of Lagrangian Floer cohomology developed in [FOOO1, FOOO3] for readers’
convenience. We also refer readers to the third named author’s survey paper
[Ohta] for a more detailed review, and refer to [FOOO3] for complete details
of the proofs of the results described in this section.

We start our discussion with the classical unfiltered $A_{\infty}$ algebra.
Let $C$ be a graded $R$-module where $R$ is the coefficient ring. We denote by
$C[1]$ its suspension defined by $C[1]^{k}=C^{k+1}$. Define the bar complex
$B(C[1])$ by

$B_{k}(C[1])=\underbrace{C[1]\otimes\cdots\otimes C[1]}_{k},\quad
B(C[1])=\bigoplus_{k=0}^{\infty}B_{k}(C[1]).$

Here $B_{0}(C[1])=R$ by definition. $B(C[1])$ has the structure of graded
coalgebra.

######  Definition 3.1.

The structure of $A_{\infty}$ algebra is a sequence of $R$ module
homomorphisms

$\mathfrak{m}_{k}:B_{k}(C[1])\to C[1],\quad k=1,2,\cdots,$

of degree +1 such that the coderivation
$\widehat{d}=\sum_{k=1}^{\infty}\widehat{\mathfrak{m}}_{k}$ satisfies
$\widehat{d}\,\widehat{d}=0$, which is called the _$A_{\infty}$ -relation_.
Here we denote by $\widehat{\mathfrak{m}}_{k}:B(C[1])\to B(C[1])$ the unique
extension of $\mathfrak{m}_{k}$ as a coderivation on $B(C[1])$, that is

$\widehat{\mathfrak{m}}_{k}(x_{1}\otimes\cdots\otimes
x_{n})=\sum_{i=1}^{k-i+1}(-1)^{*}x_{1}\otimes\cdots\otimes\mathfrak{m}_{k}(x_{i},\cdots,x_{i+k-1})\otimes\cdots\otimes
x_{n}$ (3.1)

where $*=\deg x_{1}+\cdots+\deg x_{i-1}+i-1$.

The relation $\widehat{d}\,\widehat{d}=0$ can be written as

$\sum_{k=1}^{n-1}\sum_{i=1}^{k-i+1}(-1)^{*}\mathfrak{m}_{n-k+1}(x_{1}\otimes\cdots\otimes\mathfrak{m}_{k}(x_{i},\cdots,x_{i+k-1})\otimes\cdots\otimes
x_{n})=0,$

where $*$ is the same as above. In particular, we have
$\mathfrak{m}_{1}\mathfrak{m}_{1}=0$ and so it defines a complex
$(C,\mathfrak{m}_{1})$.

A weak (or curved) $A_{\infty}$-algebra is defined in the same way, except
that it also includes the $\mathfrak{m}_{0}$-term $\mathfrak{m}_{0}:R\to
B(C[1]).$ The first two terms of the $A_{\infty}$ relation for a weak
$L_{\infty}$ algebra are given as

$\mathfrak{m}_{1}(\mathfrak{m}_{0}(1))=0,\quad\mathfrak{m}_{1}\mathfrak{m}_{1}(x)+(-1)^{\deg
x+1}\mathfrak{m}_{2}(x,\mathfrak{m}_{0}(1))+\mathfrak{m}_{2}(\mathfrak{m}_{0}(1),x)=0.$

In particular, for the case of weak $A_{\infty}$ algebras, $\mathfrak{m}_{1}$
will not satisfy boundary property, i.e.,
$\mathfrak{m}_{1}\mathfrak{m}_{1}\neq 0$ in general.

We now recall the notion of _unit_ in $A_{\infty}$ algebra.

######  Definition 3.2.

An element ${\bf e}\in C^{0}=C[1]^{-1}$ is called a unit if it satisfies

  1. (1)

$\mathfrak{m}_{k+1}(x_{1},\cdots,{\bf e},\cdots,x_{k})=0$ for $k\geq 2$ or
$k=0$.

  2. (2)

$\mathfrak{m}_{2}({\bf e},x)=(-1)^{\deg x}\mathfrak{m}_{2}(x,{\bf e})=x$ for
all $x$.

Combining this definition of unit and (3.1), we have the following immediate
lemma.

######  Lemma 3.3.

Consider an $A_{\infty}$ algebra $(C[1],\mathfrak{m})$ over a ground ring $R$
for which $\mathfrak{m}_{0}(1)=\lambda{\bf e}$ for some $\lambda\in R$. Then
$\mathfrak{m}_{1}\mathfrak{m}_{1}=0$.

Now we explain the notion of the _filtered $A_{\infty}$ algebra_. We define
the universal Novikov ring $\Lambda_{0,nov}$ by

$\Lambda_{0,nov}=\left.\left\\{\sum_{i=1}^{\infty}a_{i}T^{\lambda_{i}}e^{n_{i}}\,\,\right|\,\,a_{i}\in
R,n_{i}\in{\mathbb{Z}},\lambda_{i}\in{\mathbb{R}}_{\geq
0},\lim_{i\to\infty}\lambda_{i}=\infty\right\\}.$

This is a graded ring by defining $\deg T=0$, $\deg e=2$. Let
$\Lambda_{0,nov}^{+}$ be its maximal ideal which consists of the elements
$\sum_{i=1}^{\infty}a_{i}T^{\lambda_{i}}e^{n_{i}}$ with $\lambda_{i}>0$.

Let $\bigoplus_{m\in{\mathbb{Z}}}C^{m}$ be the free graded $\Lambda_{0,nov}$
module over the basis $\\{\mathbb{v}_{i}\\}$. We define a filtration
$\bigoplus_{m\in{\mathbb{Z}}}F^{\lambda}C^{m}$ on it such that
$\\{T^{\lambda}\mathbb{v}_{i}\\}$ is a free basis of
$\bigoplus_{m\in{\mathbb{Z}}}F^{\lambda}C^{m}$. Here $\lambda\in{\mathbb{R}}$,
$\lambda\geq 0$. We call this filtration the energy filtration. (Our algebra
$\bigoplus_{m\in{\mathbb{Z}}}C^{m}$ may not be finitely generated. So we need
to take completion.) We denote by $C$ the completion of
$\bigoplus_{m\in{\mathbb{Z}}}C^{m}$ with respect to the energy filtration. The
filtration induces a natural non-Archimedean topology on $C$.

A filtered $A_{\infty}$ algebra $(C,\mathfrak{m})$ is a weak $A_{\infty}$
algebra such that $A_{\infty}$ operators $\mathfrak{m}$ have the following
properties :

  1. (1)

$\mathfrak{m}_{k}$ respect the energy filtration,

  2. (2)

$\mathfrak{m}_{0}(1)\in F^{\lambda}C^{1}$ with $\lambda>0$,

  3. (3)

The reduction
$\mathfrak{m}_{k}\mod\Lambda_{0,nov}^{+}:B_{k}\overline{C}[1]\otimes
R[e,e^{-1}]\to\overline{C}\otimes R[e,e^{-1}]$ does not contain $e$. More
precisely speaking, it has the form
$\overline{\mathfrak{m}}_{k}\otimes\,_{R}R[e,e^{-1}]$ where
$\overline{\mathfrak{m}}_{k}:B_{k}\overline{C}[1]\to\overline{C}$ is an $R$
module homomorphism. (Here $\overline{C}$ is the free $R$ module over the
basis $\mathbb{v}_{i}$.)

(See Definition 3.2.20 [FOOO3] = Definition 7.20 [FOOO2].)

######  Remark 3.4.

  1. (1)

In [FOOO3] we assume $\mathfrak{m}_{0}=0$ for (unfiltered) $A_{\infty}$
algebra. On the other hand, $\mathfrak{m}_{0}=0$ is not assumed for filtered
$A_{\infty}$ algebra. Filtered $A_{\infty}$ algebra satisfying
$\mathfrak{m}_{0}=0$ is called to be strict.

  2. (2)

In this section, to be consistent with the exposition given in [FOOO3], we use
the Novikov ring $\Lambda_{0,nov}$ which includes the variable $e$. In
[FOOO3], the variable $e$ is used so that the operations $\mathfrak{m}_{k}$
become to have degree one for all $k$ (with respect to the shifted degree.)
But for the applications of this paper, it is enough to use the
${\mathbb{Z}}_{2}$-grading and so encoding the degree with a formal parameter
is not necessary. Therefore we will use the ring $\Lambda_{0}$ in other
sections, which does not contain $e$. An advantage of using the ring
$\Lambda_{0}$ is that it is a local ring while $\Lambda_{0,nov}$ is not. This
makes it easier to use some standard results from commutative algebra in later
sections. We would like to remark that as a ${\mathbb{Z}}_{2}$ graded complex
a ${\mathbb{Z}}$ graded complex over $\Lambda_{0,nov}$ is equivalent to the
complex over $\Lambda_{0}$.

Next we explain how one can deform the given filtered $A_{\infty}$ algebra
$(C,\mathfrak{m})$ by an element $b\in F^{\lambda}C[1]^{0}$ with $\lambda>0$,
by re-defining the $A_{\infty}$ operators as

$\mathfrak{m}_{k}^{b}(x_{1},\cdots,x_{k})=\mathfrak{m}(e^{b},x_{1},e^{b},x_{2},e^{b},x_{3},\cdots,x_{k},e^{b}).$

This defines a new weak $A_{\infty}$-algebra for arbitrary $b$.

Here we simplify notations by writing $e^{b}=1+b+b\otimes
b+\cdots+b\otimes\cdots\otimes b+\cdots.$ Note that each summand in this
infinite sum has degree 0 in $C[1]$. When the ground ring is
$\Lambda_{0,nov}$, the infinite sum will converge in the non-Archimedean
topology since $b\in F^{\lambda}C[1]^{0}$ with $\lambda>0$

######  Proposition 3.5.

For $A_{\infty}$ algebra $(C,\mathfrak{m}_{k}^{b})$,
$\mathfrak{m}_{0}^{b}\equiv 0\mod\Lambda_{0,nov}\\{\bf e\\}$ if and only if
$b$ satisfies

$\sum_{k=0}^{\infty}\mathfrak{m}_{k}(b,\cdots,b)\equiv
0\mod\Lambda_{0,nov}\\{\bf e\\}.$ (3.2)

We call the equation (3.2) the _$A_{\infty}$ Maurer-Cartan equation_.

######  Definition 3.6.

Let $(C,\mathfrak{m})$ be a filtered $A_{\infty}$ algebra in general and
$BC[1]$ be its bar-complex. An element $b\in F^{\lambda}C[1]^{0}$
($\lambda>0$) is called a _weak bounding cochain_ if it satisfies the equation
(3.2). If the $b$ satisfies the strict equation

$\sum_{k=0}^{\infty}\mathfrak{m}_{k}(b,\cdots,b)=0$

we call it a (strict) _bounding cochain_.

From now on, we will also call a weak bounding cochain just a bounding cochain
since we will mainly concern weak bounding cochains. In general a given
$A_{\infty}$ algebra may or may not have a solution to (3.2).

######  Definition 3.7.

A filtered $A_{\infty}$-algebra is called _weakly unobstructed_ if the
equation (3.2) has a solution $b\in F^{\lambda}C[1]^{0}$ with $\lambda>0$.

One can define a notion of gauge equivalence between two bounding cochains as
described in section 4.3 [FOOO3] (=section 16 [FOOO2]).

The way how a filtered $A_{\infty}$ algebra is attached to a Lagrangian
submanifold $L\subset(M,\omega)$ arises as an $A_{\infty}$ deformation of the
classical singular cochain complex including the instanton contributions. In
particular, when there is no instanton contribution as in the case
$\pi_{2}(M,L)=0$, it will reduce to an $A_{\infty}$ deformation of the
singular cohomology in the chain level including all possible higher Massey
products.

We now describe the basic $A_{\infty}$ operators $\mathfrak{m}_{k}$ in the
context of $A_{\infty}$ algebra of Lagrangian submanifolds. For a given
compatible almost complex structure $J$, consider the moduli space
${\mathcal{M}}_{k+1}(\beta;L)$ of stable maps of genus zero. It is a
compactification of

$\\{(w,(z_{0},z_{1},\cdots,z_{k}))\mid\overline{\partial}_{J}w=0,\,z_{i}\in\partial
D^{2},\,[w]=\beta\,\,\,\mbox{in}\,\pi_{2}(M,L)\\}/\sim$

where $\sim$ is the conformal reparameterization of the disc $D^{2}$. The
expected dimension of this space is given by
$n+\mu(\beta)-3+(k+1)=n+\mu(\beta)+k-2$.

Now given $k$ singular chains $[P_{1},f_{1}],\cdots,[P_{k},f_{k}]\in C_{*}(L)$
of $L$, we put the cohomological grading $\mbox{deg}P_{i}=n-\mathop{\kern
0.0pt{\rm dim}}\nolimits P_{i}$ and regard the chain complex $C_{*}(L)$ as the
cochain complex $C^{\mathop{\kern 0.0pt{\rm dim}}\nolimits L-*}(L)$. We
consider the fiber product

$ev_{0}:{\mathcal{M}}_{k+1}(\beta;L)\times_{(ev_{1},\cdots,ev_{k})}(P_{1}\times\cdots\times
P_{k})\to L,$

where $ev_{i}([w,(z_{0},z_{1},\cdots,z_{k})])=w(z_{i})$.

A simple calculation shows that we have the expected degree

$\mbox{deg}\left[{\mathcal{M}}_{k+1}(\beta;L)\times_{(ev_{1},\cdots,ev_{k})}(P_{1}\times\cdots\times
P_{k}),ev_{0}\right]=\sum_{j=1}^{n}(\mbox{deg}P_{j}-1)+2-\mu(\beta).$

For each given $\beta\in\pi_{2}(M,L)$ and $k=0,\cdots$, we define

$\mathfrak{m}_{k,\beta}(P_{1},\cdots,P_{k})=\left[{\mathcal{M}}_{k+1}(\beta;L)\times_{(ev_{1},\cdots,ev_{k})}(P_{1}\times\cdots\times
P_{k}),ev_{0}\right]$ (3.3)

and $\mathfrak{m}_{k}=\sum_{\beta\in\pi_{2}(M,L)}\mathfrak{m}_{k,\beta}\cdot
T^{\omega(\beta)}e^{\mu(\beta)/2}$.

Now we denote by $C[1]$ the completion of a _suitably chosen countably
generated_ (singular) chain complex with $\Lambda_{0,nov}$ as its coefficients
with respect to the non-Archimedean topology. (We regard $C[1]$ as a cochain
complex.) Then by choosing a system of multivalued perturbations of the right
hand side of (3.3) and a triangulation of its zero sets, the map
$\mathfrak{m}_{k}:B_{k}(C[1])\to C[1]$ is defined, has degree 1 and continuous
with respect to non-Archimedean topology. We extend $\mathfrak{m}_{k}$ as a
coderivation $\widehat{\mathfrak{m}}_{k}:BC[1]\to BC[1]$ by (3.1). Finally we
take the sum

$\widehat{d}=\sum_{k=0}^{\infty}\widehat{\mathfrak{m}}_{k}:BC[1]\to BC[1].$
(3.4)

A main theorem proven in [FOOO1, FOOO3] then is the following coboundary
property

######  Theorem 3.8.

(Theorem 3.5.11 [FOOO3] = Theorem 10.11 [FOOO2]) Let $L$ be an arbitrary
compact relatively spin Lagrangian submanifold of an arbitrary tame symplectic
manifold $(M,\omega)$. The coderivation $\widehat{d}$ is a continuous map that
satisfies the $A_{\infty}$ relation $\widehat{d}\widehat{d}=0$.

The $A_{\infty}$ algebra associated to $L$ in this way has the _homotopy unit_
, not a _unit_. In general a filtered $A_{\infty}$ algebra with homotopy unit
canonically induces another filtered _unital_ $A_{\infty}$-algebra called a
_canonical model_ of the given filtered $A_{\infty}$ algebra. In the geometric
context of $A_{\infty}$ algebra associated to a Lagrangian submanifold
$L\subset M$ of a general symplectic manifold $(M,\omega)$, the canonical
model is defined on the cohomology group $H^{*}(L;\Lambda_{0,nov})$. We refer
to [FOOO5] for a quick explanation of construction by summing over trees of
this canonical model.

Once the $A_{\infty}$ algebra is attached to each Lagrangian submanifold $L$,
we then construct a filtered _$A_{\infty}$ bimodule_ $C(L,L^{\prime})$ for the
transversal pair of Lagrangian submanifolds $L$ and $L^{\prime}$. Here
$C(L,L^{\prime})$ is the free $\Lambda_{0,nov}$ module such that the set of
its basis is identified with $L\cap L^{\prime}$. The filtered $A_{\infty}$
bimodule structure is by definition is a family of operators

$\mathfrak{n}_{k_{1},k_{2}}:B_{k_{1}}(C(L)[1])\,\,\widehat{\otimes}_{\Lambda_{0,nov}}\,\,C(L,L^{\prime})\,\,\widehat{\otimes}_{\Lambda_{0,nov}}\,\,B_{k_{1}}(C(L^{\prime})[1])\to
C(L,L^{\prime})$

for $k_{1},k_{2}\geq 0$. (Here $\widehat{\otimes}_{\Lambda_{0,nov}}$ is the
completion of algebraic tensor product.) Let us briefly describe the
definition of $\mathfrak{n}_{k_{1},k_{2}}$. A typical element of the tensor
product
$B_{k_{1}}(C(L)[1])\,\,\widehat{\otimes}_{\Lambda_{0,nov}}\,\,C(L,L^{\prime})\,\,\widehat{\otimes}_{\Lambda_{0,nov}}\,\,B_{k_{2}}(C(L^{\prime})[1])$
has the form

$P_{1,1}\otimes\cdots,\otimes P_{1,k_{1}}\otimes\langle p\rangle\otimes
P_{2,1}\otimes\cdots\otimes P_{2,k_{2}}$

with $p\in L\cap L^{\prime}$. Then the image $\mathfrak{n}_{k_{1},k_{2}}$
thereof is given by

$\sum_{q,B}T^{\omega(B)}e^{\mu(B)/2}\\#\left({\mathcal{M}}(p,q;B;P_{1,1},\cdots,P_{1,k_{1}};P_{2,1},\cdots,P_{2,k_{2}})\right)\langle
q\rangle.$

Here $B$ denotes homotopy class of Floer trajectories connecting $p$ and $q$,
the summation is taken over all $(q,B)$ with

$\text{vir.dim
}{\mathcal{M}}(p,q;B;P_{1,1},\cdots,P_{1,k_{1}};P_{2,1},\cdots,P_{2,k_{2}})=0,$

and
$\\#\left({\mathcal{M}}(p,q;B;P_{1,1},\cdots,P_{1,k_{1}};P_{2,1},\cdots,P_{2,k_{2}})\right)$
is the ‘number’ of elements in the ‘zero’ dimensional moduli space
${\mathcal{M}}(p,q;B;P_{1,1},\cdots,P_{1,k_{1}};P_{2,1},\cdots,P_{2,k_{2}})$.
Here the moduli space
${\mathcal{M}}(p,q;B;P_{1,1},\cdots,P_{1,k_{1}};P_{2,1},\cdots,P_{2,k_{2}})$
is the Floer moduli space ${\mathcal{M}}(p,q;B)$ cut-down by intersecting with
the given chains $P_{1,i}\subset L$ and $P_{2,j}\subset L^{\prime}$.

######  Theorem 3.9.

(Theorem 3.7.21 [FOOO3] = Theorem 12.21 [FOOO2]) Let $(L,L^{\prime})$ be an
arbitrary relatively spin pair of compact Lagrangian submanifolds. Then the
family $\\{\mathfrak{n}_{k_{1},k_{2}}\\}$ define a left $(C(L),\mathfrak{m})$
and right $(C(L^{\prime}),\mathfrak{m}^{\prime})$ filtered
$A_{\infty}$-bimodule structure on $C(L,L^{\prime})$.

What this theorem means is explained below as Proposition 3.10.

Let
$B(C(L)[1])\,\widehat{\otimes}_{\Lambda_{0,nov}}\,C(L,L^{\prime})\,\widehat{\otimes}_{\Lambda_{0,nov}}\,B(C(L^{\prime})[1])$
be the completion of the direct sum of
$B_{k_{1}}(C(L)[1])\,\widehat{\otimes}_{\Lambda_{0,nov}}\,C(L,L^{\prime})\,\widehat{\otimes}_{\Lambda_{0,nov}}\,B_{k_{2}}(C(L^{\prime})[1])$
over $k_{1}\geq 0$, $k_{2}\geq 0$. We define the boundary operator
$\widehat{d}$ on it by using the maps $\mathfrak{n}_{k_{1},k_{2}}$ and
$\mathfrak{m}_{k}$, $\mathfrak{m}^{\prime}_{k}$ as follows :

$\displaystyle\widehat{d}((x_{1}\otimes\cdots\otimes x_{n})\otimes
x\otimes(x^{\prime}_{1}\otimes\cdots\otimes x^{\prime}_{m}))$
$\displaystyle=\widehat{d}(x_{1}\otimes\cdots\otimes x_{n})\otimes
x\otimes(x^{\prime}_{1}\otimes\cdots\otimes x^{\prime}_{m})$
$\displaystyle\quad+(-1)^{\deg x_{1}+\cdots+\deg x_{n}+\deg
x+n+1}(x_{1}\otimes\cdots\otimes x_{n})\otimes
x\otimes\widehat{d}(x^{\prime}_{1}\otimes\cdots\otimes x^{\prime}_{m})$
$\displaystyle\quad+\sum_{k_{1}\leq n}\sum_{k_{2}\leq m}(-1)^{\deg
x_{1}+\cdots+\deg x_{n-k_{1}}+n-k_{1}}(x_{1}\otimes\cdots\otimes x_{n-k_{1}})$
$\displaystyle\hskip
65.44142pt\otimes\mathfrak{n}_{k_{1},k_{2}}((x_{n-k_{1}+1}\otimes\cdots\otimes
x_{n})\otimes x\otimes(x^{\prime}_{1}\otimes\cdots\otimes
x^{\prime}_{k_{2}}))$ $\displaystyle\hskip
65.44142pt\otimes(x^{\prime}_{k_{2}+1}\otimes\cdots\otimes x^{\prime}_{m}).$

Here $\widehat{d}$ in the second and the third lines are induced from
$\mathfrak{m}$ and $\mathfrak{m}^{\prime}$ by Formula (3.4) respectively.

######  Proposition 3.10.

The map $\widehat{d}$ satisfies $\widehat{d}\,\widehat{d}=0$.

The $A_{\infty}$ bimodule structure, which define a boundary operator on the
bar complex, induces an operator $\delta=\mathfrak{n}_{0,0}$ on a much
smaller, ordinary free $\Lambda_{0,nov}$ module $C(L,L^{\prime})$ generated by
the intersections $L\cap L^{\prime}$. However the boundary property of this
Floer’s ‘boundary’ map $\delta$ again meets obstruction coming from the
obstructions cycles of either $L$, $L^{\prime}$ or of both. We need to deform
$\delta$ using suitable bounding cochains of $L,\,L^{\prime}$.

In the case where both $L,\,L^{\prime}$ are weakly unobstructed, we can carry
out this deformation of $\mathfrak{n}$ using weak bounding chains $b$ and
$b^{\prime}$ of fibered $A_{\infty}$ algebras associated to $L$ and
$L^{\prime}$ respectively, in a way similar to $\mathfrak{m}^{b}$. Namely we
define $\delta_{b,b^{\prime}}:C(L,L^{\prime})\to C(L,L^{\prime})$ by

$\delta_{b,b^{\prime}}(x)=\sum_{k_{1},k_{2}}\mathfrak{n}_{k_{1},k_{2}}(b^{\otimes
k_{1}}\otimes x\otimes b^{\prime\otimes
k_{2}})=\mathfrak{\widehat{n}}(e^{b},x,e^{b^{\prime}}).$

We can generalize the story to the case where $L$ has clean intersection with
$L^{\prime}$, especially to the case $L=L^{\prime}$. In the case
$L=L^{\prime}$ we have
$\mathfrak{n}_{k_{1},k_{2}}=\mathfrak{m}_{k_{1}+k_{2}+1}$. So in this case, we
have $\delta_{b,b^{\prime}}(x)=\mathfrak{m}(e^{b},x,e^{b^{\prime}}).$

In general $\delta_{b,b^{\prime}}$ does not satisfy the equation
$\delta_{b,b^{\prime}}\delta_{b,b^{\prime}}=0$. It turns out that there is an
elegant condition for $\delta_{b,b^{\prime}}\delta_{b,b^{\prime}}=0$ to hold
in terms of the _potential function_ introduced in [FOOO3], which we explain
in the next section. In the case
$\delta_{b,b^{\prime}}\delta_{b,b^{\prime}}=0$ we define Floer cohomology by

$HF((L,b),(L^{\prime},b^{\prime});\Lambda_{0,nov})=\mbox{Ker}\,\delta_{b,b^{\prime}}/\mbox{Im}\,\delta_{b,b^{\prime}}.$

Let $\Lambda_{nov}$ be the field of fraction of $\Lambda_{0,nov}$. We define
$HF((L,b),(L^{\prime},b^{\prime});\Lambda_{nov})$ by extending the coefficient
ring $\Lambda_{0,nov}$ to $\Lambda_{nov}$. Then
$HF((L,b),(L^{\prime},b^{\prime});\Lambda_{nov})$ is invariant under the
Hamiltonian isotopies of $L$ and $L^{\prime}$. Therefore we can use it to
obtain the following result about non-displacement of Lagrangian submanifolds.

######  Theorem 3.11.

(Theorem G [FOOO3] = Theorem G [FOOO2]) Assume
$\delta_{b,b^{\prime}}\delta_{b,b^{\prime}}=0$. Let $\psi:X\to X$ be a
Hamiltonian diffeomorphism such that $\psi(L)$ is transversal to $L^{\prime}$.
Then we have

$\\#(\psi(L)\cap L^{\prime})\geq\text{\rm
rank}_{\Lambda_{nov}}HF((L,b),(L^{\prime},b^{\prime});\Lambda_{nov}).$

The Floer cohomology $HF((L,b),(L^{\prime},b^{\prime});\Lambda_{0,nov})$ with
coefficient $\Lambda_{0,nov}$ is not invariant under the Hamiltonian isotopy.
We however can prove the following Theorem 3.12. There exists an integer $a$
and positive numbers $\lambda_{i}$ ($i=1,\cdots,b$) such that

$HF((L,b),(L^{\prime},b^{\prime});\Lambda_{0,nov})=\Lambda_{0,nov}^{\oplus
a}\oplus\bigoplus_{i=1}^{b}(\Lambda_{0,nov}/T^{\lambda_{i}}\Lambda_{0,nov}).$

(See Theorem 6.1.20 [FOOO3] (= Theorem 24.20 [FOOO2]).) Let $\psi:X\to X$ be a
Hamiltonian diffeomorphism and $\|\psi\|$ is its Hofer distance (see [H]) from
the identity map.

######  Theorem 3.12.

(Theorem J [FOOO3] = Theorem J [FOOO2]) If $\psi(L)$ is transversal to
$L^{\prime}$, we have an inequality

$\\#(\psi(L)\cap L^{\prime})\geq a+2\,\\#\\{i\mid\lambda_{i}\geq\|\psi\|\\}.$

In later sections, we will apply Theorems 3.11,3.12 to study non-displacement
of Lagrangian fibers of toric manifolds.

##  4\. Potential function

The $A_{\infty}$ structure defined on a countably generated chain complex
$C(L,\Lambda_{0})$ itself explained in the previous section is not suitable
for explicit calculations as in our study of toric manifolds. For this
computational purpose, we work with the filtered $A_{\infty}$ structure on the
canonical model defined on $H(L;\Lambda_{0})$ which has a _finite_ rank over
$\Lambda_{0}$. Furthermore this has a strict unit ${\bf e}$ given by the dual
of the fundamental class $PD([L])$. Recall $C(L,\Lambda_{0})$ itself has only
a homotopy-unit.

An element $b\in H^{1}(L;\Lambda_{+})$ is called a _weak bounding cochain_ if
it satisfies the $A_{\infty}$ Maurer-Cartan equation

$\sum_{k=0}^{\infty}\mathfrak{m}_{k}(b,\cdots,b)\equiv 0\mod PD([L])$ (4.1)

where $\\{\mathfrak{m}_{k}\\}_{k=0}^{\infty}$ is the $A_{\infty}$ operators
associated to $L$. $[L]\in H_{n}(L)$ is the fundamental class, and $PD([L])\in
H^{0}(L)$ is its Poincaré dual. We denote by
$\widehat{{\mathcal{M}}}_{\text{weak}}(L)$ the set of weak bounding cochains
of $L$. We say $L$ is weakly unobstructed if
$\widehat{{\mathcal{M}}}_{\text{weak}}(L)\neq\emptyset$. The moduli space
${\mathcal{M}}_{\text{weak}}(L)$ is then defined to be the quotient space of
$\widehat{{\mathcal{M}}}_{\text{weak}}(L)$ by suitable gauge equivalence. (See
chapter 3 and 4 [FOOO3] for more explanations.)

######  Lemma 4.1 (Lemma 3.6.32 [FOOO3] = Lemma 11.32 [FOOO2]).

If $b\in\widehat{{\mathcal{M}}}_{\text{\rm weak}}(L)$ then
$\delta_{b,b}\circ\delta_{b,b}=0$, where $\delta_{b,b}$ is the deformed Floer
operator defined by

$\delta_{b,b}(x)=\mathfrak{m}_{1}^{b}(x)=:\sum_{k,\ell\geq
0}\mathfrak{m}_{k+\ell+1}(b^{\otimes k},x,b^{\otimes\ell}).$

For $b\in\widehat{{\mathcal{M}}}_{\text{weak}}(L)$, we define

$HF((L;b),(L;b))=\frac{\operatorname{Ker}(\delta_{b,b}:C\to
C)}{\operatorname{Im}(\delta_{b,b}:C\to C)},$

where $C$ is an appropriate subcomplex of the singular chain complex of $L$.
When $L$ is weakly unobstructed i.e.,
$\widehat{{\mathcal{M}}}_{\text{weak}}(C)\neq\emptyset$, we define a function

$\mathfrak{PO}:\widehat{{\mathcal{M}}}_{\text{weak}}(C)\to\Lambda_{+}$

by the equation

$\mathfrak{m}(e^{b})=\mathfrak{PO}(b)\cdot PD([L]).$

This is the _potential function_ introduced in [FOOO3].

######  Theorem 4.2 (Proposition 3.7.17 [FOOO3] = Proposition 12.17 [FOOO2]).

For each $b\in\widehat{{\mathcal{M}}}_{\text{\rm weak}}(L)$ and
$b^{\prime}\in\widehat{{\mathcal{M}}}_{weak}(L^{\prime})$, the map
$\delta_{b,b^{\prime}}$ defines a continuous map
$\delta_{b,b^{\prime}}:CF(L,L^{\prime})\to CF(L,L^{\prime})$ that satisfies
$\delta_{b,b^{\prime}}\delta_{b,b^{\prime}}=0$, provided

$\mathfrak{PO}(b)=\mathfrak{PO}(b^{\prime}).$ (4.2)

Therefore for each pair $(b,b^{\prime})$ of
$b\in\widehat{{\mathcal{M}}}_{\text{\rm weak}}(L)$ and
$b^{\prime}\in\widehat{{\mathcal{M}}}_{\text{\rm weak}}(L^{\prime})$ that
satisfy (4.2), we define the _$(b,b^{\prime})$ -Floer cohomology_ of the pair
$(L,L^{\prime})$ by

$HF((L,b),(L^{\prime},b^{\prime});\Lambda_{nov})=\frac{\mbox{Ker}\delta_{b,b^{\prime}}}{\mbox{\rm
Im }\delta_{b,b^{\prime}}}.$

Now in the rest of this section we state the main results concerning the
detailed structure of the potential function for the case of Lagrangian fibers
of toric manifolds.

For the later analysis of examples, we recall from [FOOO1, FOOO3] that
$\mathfrak{m}_{k}$ is further decomposed into

$\mathfrak{m}_{k}=\sum_{\beta\in\pi_{2}(M,L)}\mathfrak{m}_{k,\beta}\otimes
T^{\omega(\beta)}e^{\mu(\beta)/2}.$

Here $\mu$ is the Maslov index.

Firstly we remove the grading parameter $e$ from the ground ring. Secondly to
eliminate many appearance of $2\pi$ in front of the affine function $\ell_{i}$
in the exponents of the parameter $T$ later in this paper, we redefine $T$ as
$T^{2\pi}$. Under this arrangement, we get the formal power series expansion

$\mathfrak{m}_{k}=\sum_{\beta\in\pi_{2}(M,L)}\mathfrak{m}_{k,\beta}\otimes
T^{\omega(\beta)/2\pi}$ (4.3)

which we will use throughout the paper.

Now we restrict to the case of toric manifold. Let $X=X_{\Sigma}$ be
associated a complete regular fan $\Sigma$ (in other words $\Sigma$ is the
normal fan of $X$), and $\pi:X\to\mathfrak{t}^{*}$ be the moment map of the
action of the torus $T^{n}\cong T^{m}/K$. We make the identifications

$\mathfrak{t}=Lie(T^{n})\cong
N_{\mathbb{R}}^{n}\cong{\mathbb{R}}^{n},\,\mathfrak{t}^{*}\cong
M_{\mathbb{R}}\cong({\mathbb{R}}^{n})^{*}.$

We will exclusively use $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$ to be consistent
with the standard notations in toric geometry instead of $\mathfrak{t}$ (or
${\mathbb{R}}^{n}$) and $\mathfrak{t}^{*}$ (or $({\mathbb{R}}^{n})^{*}$) as
much as possible.

Denote the image of $\pi:X\to M_{\mathbb{R}}$ by $P\subset M_{\mathbb{R}}$
which is the moment polytope of the $T^{n}$ action on $X$.

We will prove the following in section 11.

######  Proposition 4.3.

For any $u\in\text{\rm Int}P$, the fiber $L(u)$ is weakly unobstructed.
Moreover we have the canonical inclusion

$H^{1}(L(u);\Lambda_{+})\hookrightarrow{\mathcal{M}}_{\text{\rm weak}}(L(u)).$

Choose an integral basis $\text{\bf e}^{*}_{i}$ of $N$ and $\text{\bf e}_{i}$
be its dual basis on $M$. With this choice made, we identify $M_{\mathbb{R}}$
with ${\mathbb{R}}^{n}$ as long as its meaning is obvious from the context.
Identifying $H_{1}(T^{n};{\mathbb{Z}})$ with $N\cong{\mathbb{Z}}^{n}$ via
$T^{n}={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$, we regard $\text{\bf e}_{i}$ as a
basis of $H^{1}(L(u);{\mathbb{Z}})$. The following immediately follows from
definition.

######  Lemma 4.4.

We write $\pi=(\pi_{1},\cdots,\pi_{n}):X\to M_{\mathbb{R}}$ using the
coordinate of $M_{\mathbb{R}}$ associated to the basis $\text{\bf e}_{i}$. Let
$S^{1}_{i}\subset T^{n}$ be the subgroup whose orbit represents $\text{\bf
e}^{*}_{i}\in H_{1}(T^{n};{\mathbb{Z}})$. Then $\pi_{i}$ is proportional to
the moment map of $S^{1}_{i}$ action on $X$.

Let

$b=\sum x_{i}\text{\bf e}_{i}\in
H^{1}(L(u);\Lambda_{+})\subset{{\mathcal{M}}}_{\text{\rm weak}}(L(u)).$

We study the potential function

$\mathfrak{PO}:H^{1}(L(u);\Lambda_{+})\to\Lambda_{+}.$

Once a choice of the family of bases $\\{\text{\bf e}_{i}\\}$ on
$H^{1}(L(u);{\mathbb{Z}})$ for $u\in\text{\rm Int}\,P$ is made as above
starting from a basis on $N$ , then we can regard this function as a function
of $(x_{1},\cdots,x_{n})\in(\Lambda_{+})^{n}$ and $(u_{1},\cdots,u_{n})\in
P\subset M_{\mathbb{R}}$. We denote its value by
$\mathfrak{PO}(x;u)=\mathfrak{PO}(x_{1},\cdots,x_{n};u_{1},\cdots,u_{n})$. We
put

$y_{i}=e^{x_{i}}=\sum_{k=0}^{\infty}\frac{x_{i}^{k}}{k!}\in\Lambda_{0}.$

Let

$\partial P=\bigcup_{i=1}^{m}\partial_{i}P$

be the decomposition of the boundary of the moment polytope into its faces of
codimension one. ($\partial_{i}P$ is a polygon in an $n-1$ dimensional affine
subspace of $M_{\mathbb{R}}$.)

Let $\ell_{i}$ be the affine functions

$\ell_{i}(u)=\langle u,v_{i}\rangle-\lambda_{i}\,\mbox{ for }\,i=1,\cdots,m$

appearing in Theorem 2.13. Then the followings hold from construction :

  1. (1)

$\ell_{i}\equiv 0$ on $\partial_{i}P$.

  2. (2)

$P=\\{u\in M_{\mathbb{R}}\mid\ell_{i}(u)\geq 0,i=1,\cdots,m\\}.$

  3. (3)

The coordinates of the vectors $v_{i}=(v_{i,1},\cdots,v_{i,n})$ satisfy

$v_{i,j}=\frac{\partial\ell_{i}}{\partial u_{j}}$ (4.4)

and are all integers.

######  Theorem 4.5.

Let $L(u)\subset X$ be as in Theorem 1.5 and $\ell_{i}$ be as above. Suppose
$X$ is Fano. Then we can take the canonical model of $A_{\infty}$ structure of
$L(u)$ over $u\in\text{\rm Int}\,P$ so that the potential function restricted
to

$\bigcup_{u\in\text{\rm Int
}P}H^{1}(L(u);\Lambda_{+})\cong(\Lambda_{+})^{n}\times\text{\rm Int}\,P$

has the form

$\displaystyle\mathfrak{PO}(x;u)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}T^{\ell_{i}(u)}$ (4.5) $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}e^{\langle v_{i},x\rangle}T^{\ell_{i}(u)}$ (4.6)

where $(x;u)=(x_{1},\cdots,x_{n};u_{1},\cdots,u_{n})$ and $v_{i,j}$ is as in
$(\ref{eq:vij})$.

Theorem 4.5 is a minor improvement of a result from [CO] (see (15.1) of [CO])
and [Cho] : The case considered in [CO] corresponds to the case where
$y_{i}\in U(1)\subset\\{z\in{\mathbb{C}}\mid|z|=1\\}$ and the case in [Cho]
corresponds to the one where $y_{i}\in{\mathbb{C}}\setminus\\{0\\}$. The
difference of Theorem 4.5 from the ones thereof is that $y_{i}$ is allowed to
contain $T$, the formal parameter of the universal Novikov ring encoding the
energy.

For the non-Fano case, we prove the following slightly weaker statement. The
proof will be given in section 11.

######  Theorem 4.6.

Let $X$ be an arbitrary toric manifold and $L(u)$ be as above. Then there
exist $c_{j}\in{\mathbb{Q}}$, $e_{j}^{i}\in{\mathbb{Z}}_{\geq 0}$ and
$\rho_{j}>0$, such that $\sum_{i=1}^{m}e_{j}^{i}>0$ and

$\displaystyle\mathfrak{PO}(x_{1},\cdots,x_{n};u_{1},\cdots,u_{n})$
$\displaystyle-$ $\displaystyle\sum_{i=1}^{m}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}T^{\ell_{i}(u)}$ (4.7) $\displaystyle=$
$\displaystyle\sum_{j=1}c_{j}y_{1}^{v^{\prime}_{j,1}}\cdots
y_{n}^{v^{\prime}_{j,n}}T^{\ell^{\prime}_{j}(u)+\rho_{j}},$

where

$v^{\prime}_{j,k}=\sum_{i=1}^{m}e_{j}^{i}v_{i,k},\quad\ell^{\prime}_{j}=\sum_{i=1}^{m}e_{j}^{i}\ell_{i}.$

If there are infinitely many non-zero $c_{j}$’s, we have

$\lim_{j\to\infty}\ell^{\prime}_{j}(u)+\rho_{j}=\infty.$

Moreover $\rho_{j}=[\omega]\cap\alpha_{j}$ for some $\alpha_{j}\in\pi_{2}(X)$
with nonpositive first Chern number $c_{1}(X)\cap[\alpha_{j}]$.

We note that although $\mathfrak{PO}$ is defined originally on
$(\Lambda_{+}^{{\mathbb{C}}})^{n}\times P$, Theorems 4.5 and 4.6 imply that
$\mathfrak{PO}$ extends to a function on
$(\Lambda_{0}^{{\mathbb{C}}})^{n}\times P$ in general, and to one on
$(\Lambda_{0}^{\mathbb{C}})^{n}\times M_{\mathbb{R}}$ for the Fano case.
Furthermore these theorems also imply the periodicity of $\mathfrak{PO}$ in
$x_{i}$’s,

$\mathfrak{PO}(x_{1},\cdots,x_{i}+2\pi\sqrt{-1},\cdots,x_{n};u)=\mathfrak{PO}(x_{1},\cdots,x_{n};u).$
(4.8)

We write

$\mathfrak{PO}_{0}=\sum_{i=1}^{m}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}T^{\ell_{i}(u)}$ (4.9)

to distinguish it from $\mathfrak{PO}$. We call $\mathfrak{PO}_{0}$ the
leading order potential function.

We will concern the existence of the bounding cochain $\mathfrak{x}$ for which
the Floer cohomology $HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x}))$ is not
zero, and prove that critical points of the $\mathfrak{PO}^{u}$ (as a function
of $y_{1},\cdots,y_{n}$) have this property. (Theorem 4.10.)

This leads us to study the equation

$\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{k}}(\mathfrak{y}_{1},\cdots,\mathfrak{y}_{n})=0,\qquad k=1,\cdots,n,$
(4.10)

where $\mathfrak{y}_{i}\in\Lambda_{0}\setminus\Lambda_{+}$.

We regard $\mathfrak{PO}^{u}$ as either a function of $x_{i}$ or of $y_{i}$.
Since the variable ($x_{i}$ or $y_{i}$) is clear from situation, we do not
mention it occasionally.

######  Proposition 4.7.

We assume that the coordinates of the vertices of $P$ are rational. Then there
exists $u_{0}\in\text{\rm Int}P\cap{\mathbb{Q}}^{n}$ such that for each
${\mathcal{N}}$ there exists
$\mathfrak{y}_{1},\cdots,\mathfrak{y}_{n}\in\Lambda_{0}\setminus\Lambda_{+}$
satisfying :

$\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
y_{k}}(\mathfrak{y}_{1},\cdots,\mathfrak{y}_{n})\equiv 0,\mod
T^{{\mathcal{N}}}\qquad k=1,\cdots,n.$ (4.11)

Moreover there exists
$\mathfrak{y}^{\prime}_{1},\cdots,\mathfrak{y}^{\prime}_{n}\in\Lambda_{0}\setminus\Lambda_{+}$
such that

$\frac{\partial\mathfrak{PO}_{0}^{u_{0}}}{\partial
y_{k}}(\mathfrak{y}^{\prime}_{1},\cdots,\mathfrak{y}^{\prime}_{n})=0,\qquad
k=1,\cdots,n.$ (4.12)

We will prove Proposition 4.7 in section 9 .

######  Remark 4.8.

  1. (1)

$u_{0}$ is independent of ${\mathcal{N}}$. But $\mathfrak{y}_{i}$ may depend
on ${\mathcal{N}}$. (We believe it does not depend on ${\mathcal{N}}$, but are
unable to prove it at the time of writing this paper.)

  2. (2)

If $[\omega]\in H^{2}(X;{\mathbb{R}})$ is contained in $H^{2}(X;{\mathbb{Q}})$
then we may choose $P$ so that its vertices are rational.

  3. (3)

We believe that rationality of the vertices of $P$ is superfluous. We also
believe there exists not only a solution of (4.11) or of (4.12) but also of
(4.10). However then the proof seems to become more cumbersome. Since we can
reduce the general case to the rational case by approximation in most of the
applications, we will be content to prove the above weaker statement in this
paper.

We put

$\mathfrak{x}_{i}=\log\mathfrak{y}_{i}\in\Lambda_{0}$

and write

$\mathfrak{x}=\sum_{i}\mathfrak{x}_{i}\text{\bf e}_{i}\in
H^{1}(L(u_{0});\Lambda_{0}).$ (4.13)

Since $\mathfrak{y}_{i}\in\Lambda_{0}\setminus\Lambda_{+}$,
$\log\mathfrak{y}_{i}$ is well-defined (by using non-Archimedean topology on
$\Lambda_{0}$) and is contained in $\Lambda_{0}$.

We remark that $\mathfrak{x}_{i}$ is determined from $\mathfrak{y}_{i}$ up to
addtion by an element of $2\pi\sqrt{-1}{\mathbb{Z}}$. It will follow from
(4.8) that changing $\mathfrak{x}_{i}$ by an element of
$2\pi\sqrt{-1}{\mathbb{Z}}$ does not change corresponding Floer cohomology. So
we take for example $\text{\rm Im}\,\mathfrak{x}_{i}\in[0,2\pi)$. (See also
Definition 1.2 (2).)

Let $\mathfrak{y}_{i,0}\in{\mathbb{C}}\setminus\\{0\\}$ be the zero-order term
of $\mathfrak{y}_{i}$ i.e., the complex number such that

$\mathfrak{y}_{i}-\mathfrak{y}_{i,0}\equiv 0\mod\Lambda_{+}^{{\mathbb{C}}}.$

If we put an additional assumption that $\mathfrak{y}_{i,0}=1$ for
$i=1,\cdots,n$, then $\mathfrak{x}$ lies in

$H^{1}(L(u_{0});\Lambda_{+})\subset H^{1}(L(u_{0});\Lambda_{0}).$

Therefore Proposition 4.3 implies the Floer cohomology
$HF((L(u_{0}),\mathfrak{x}),(L(u_{0}),\mathfrak{x});\Lambda_{0})$ is defined.
Then (4.11) combined with the argument from [CO] (see section 13) imply

$HF((L(u_{0}),\mathfrak{x}),(L(u_{0}),\mathfrak{x});\Lambda_{0}^{{\mathbb{C}}}/(T^{{\mathcal{N}}}))\cong
H(T^{n};\Lambda_{0}^{{\mathbb{C}}}/(T^{{\mathcal{N}}})).$ (4.14)

We now consider the case when $\mathfrak{y}_{i,0}\neq 1$ for some $i$. In this
case, we follow the idea of Cho [Cho] of twisting the Floer cohomology of
$L(u)$ by non-unitary flat line bundle and proceed as follows :

We define $\rho:H_{1}(L(u);{\mathbb{Z}})\to{\mathbb{C}}\setminus\\{0\\}$ by

$\rho(\text{\bf e}_{i}^{*})=\mathfrak{y}_{i,0}.$ (4.15)

Let $\mathfrak{L}_{\rho}$ be the flat complex line bundle on $L(u)$ whose
holonomy representation is $\rho$. We use $\rho$ to twist the operator
$\mathfrak{m}_{k}$ in the same way as [Fu3], [Cho] to obtain a filtered
$A_{\infty}$ algebra, which we write
$(H(L(u);\Lambda_{0}),\mathfrak{m}^{{\rho}})$. It is weakly unobstructed and
${\mathcal{M}}_{\text{\rm
weak}}(H(L(u);\Lambda_{0}),\mathfrak{m}^{{\rho}})\supseteq
H^{1}(L(u);\Lambda_{+})$. (See section 12.)

We deform the filtered $A_{\infty}$ structure $\mathfrak{m}^{{\rho}}$ to
${\mathfrak{m}^{{\rho},b}}$ using $b\in H^{1}(L(u);\Lambda_{+})$ for which
$\mathfrak{m}^{\rho,b}_{1}\mathfrak{m}^{\rho,b}_{1}=0$ holds. Denote by
$HF((L(u_{0}),{\rho},b),(L(u_{0}),{\rho},b);\Lambda_{0}^{{\mathbb{C}}})$ the
cohomology of $\mathfrak{m}^{{\rho},b}_{1}$. We denote the potential function
of $(H(L(u);\Lambda_{0}),\mathfrak{m}^{{\rho}})$ by

$\mathfrak{PO}^{u}_{\rho}:H^{1}(L(u);\Lambda_{+})\to\Lambda_{+}$

which is defined in the same way as $\mathfrak{PO}^{u}$ by using
$\mathfrak{m}^{\rho}$ instead of $\mathfrak{m}$.

Let $\mathfrak{x}$ be as in (4.13) and put

$\mathfrak{x}_{i,0}=\log\mathfrak{y}_{i,0},\qquad
b=\sum(\mathfrak{x}_{i}-\mathfrak{x}_{i,0})\text{\bf e}_{i}\in
H^{1}(L(u);\Lambda_{+}).$ (4.16)

From the way how the definition goes, we can easily prove

######  Lemma 4.9.

$\mathfrak{PO}^{u}_{\rho}(b)=\mathfrak{PO}^{u}(\mathfrak{x}).$

We note from the remark right after Theorem 4.6 that $\mathfrak{PO}^{u}$ has
been extended to a function on $(\Lambda_{0}^{\mathbb{C}})^{n}$ and hence the
right hand side of the identity in this lemma has a well-defined meaning.
Lemma 4.9 will be proved in section 13.

Now we have :

######  Theorem 4.10.

Let $\mathfrak{x}_{i}$, $\mathfrak{y}_{i}=e^{\mathfrak{x}_{i}}$, and $\rho$
satisfy $(\ref{formula:critical})$, $(\ref{frakxdef})$ and $(\ref{repdef})$.
Let $\mathfrak{x}_{i,0}$ and $b$ be as in $(\ref{bdef})$. Then we have

$HF((L(u_{0}),{\rho},b),(L(u_{0}),{\rho},b),\Lambda_{0}^{{\mathbb{C}}})\cong
H(T^{n};\Lambda_{0}^{{\mathbb{C}}}).$ (4.17)

If $(\ref{formula:criticalweak})$, $(\ref{frakxdef})$, $(\ref{repdef})$ and
$(\ref{bdef})$ are satisfied instead then we have

$HF((L(u_{0}),{\rho},b),(L(u_{0}),{\rho},b),\Lambda_{0}^{{\mathbb{C}}}/(T^{{\mathcal{N}}}))\cong
H(T^{n};\Lambda_{0}^{{\mathbb{C}}}/(T^{{\mathcal{N}}})).$ (4.18)

Theorem 4.10 is proved in section 13. Using this we prove Theorem 1.5 in
section 13. More precisely, we will also discuss the following two points in
that section :

  1. (1)

We need to study the case where $\omega$ is not necessarily rational.

  2. (2)

We only have (4.18) instead of (4.17).

Next we give

######  Definition 4.11.

Let $(X,\omega)$ be a smooth compact toric manifold, $P$ be its moment
polytope. We say that a fiber $L(u_{0})$ at $u_{0}\in P$ is balanced if there
exists a sequence $\omega_{i}$, $u_{i}$ such that

  1. (1)

$\omega_{i}$ is a $T^{n}$ invariant Kähler structure on $X$ such that
$\lim_{i\to\infty}\omega_{i}=\omega$.

  2. (2)

$u_{i}$ is in the interior of the moment polytope $P_{i}$ of $(X,\omega_{i})$.
We make an appropriate choice of moment polytope $P_{i}$ so that they converge
to $P$. Then $\lim_{i\to\infty}u_{i}=u_{0}$.

  3. (3)

For each ${{\mathcal{N}}}$, there exist a sufficiently large $i$ and
$\mathfrak{x}_{i,{{\mathcal{N}}}}\in
H^{1}(L(u_{i});\Lambda^{{\mathbb{C}}}_{0})$ such that

$HF((L(u_{i}),\mathfrak{x}_{i,{{\mathcal{N}}}}),(L(u_{i}),\mathfrak{x}_{i,{{\mathcal{N}}}});\Lambda^{{\mathbb{C}}}/(T^{{{\mathcal{N}}}}))\cong
H(T^{n};{\mathbb{C}})\otimes\Lambda^{{\mathbb{C}}}/(T^{{{\mathcal{N}}}}).$

We say that $L(u_{0})$ is strongly balanced if there exists $\mathfrak{x}\in
H^{1}(L;\Lambda^{{\mathbb{C}}}_{0})$ such that
$HF((L(u_{0}),\mathfrak{x}),(L(u_{0}),\mathfrak{x});\Lambda^{{\mathbb{C}}}_{0})\cong
H(T^{n};{\mathbb{Q}})\otimes\Lambda^{{\mathbb{C}}}_{0}$.

Obviously ‘strongly balanced’ implies ‘balanced’. The converse is not true in
general. See Example 10.17. We also refer readers to Remark 13.9 for other
characterizations of being balanced (or strongly balanced).

Theorem 4.10 implies that $L(u_{0})$ in Proposition 4.7 is balanced.
(Proposition 13.2.) We will prove the next following intersection result in
section 13. Theorem 1.5 will then be a consequence of Propositions 4.7 and
4.12.

######  Proposition 4.12.

If $L(u_{0})$ is a balanced Lagrangian fiber then the following holds for any
Hamiltonian diffeomorphism $\psi:X\to X$.

$\psi(L(u_{0}))\cap L(u_{0})\neq\emptyset.$ (4.19)

If $\psi(L(u_{0}))$ is transversal to $L(u_{0})$ in addition, then

$\\#(\psi(L(u_{0}))\cap L(u_{0}))\geq 2^{n}.$ (4.20)

Denoting $\mathfrak{x}=b+\sum\mathfrak{x}_{i,0}\text{\bf e}_{i}$, we sometime
write $HF((L(u_{0}),\mathfrak{x}),(L(u_{0}),\mathfrak{x}),\Lambda_{0})$ for
$HF((L(u_{0}),{\rho},b),(L(u_{0}),{\rho},b),\Lambda_{0})$ from now on. We also
define

$\mathcal{M}_{\text{\rm
weak}}(L(u);\Lambda_{0}):=\\{(\rho,b)\mid\rho:\pi_{1}L(u)\to{\mathbb{C}}\setminus\\{0\\},b\in\mathcal{M}_{\text{\rm
weak}}(H(L(u)),\mathfrak{m}^{\rho})\\}.$

Namely it is the set of pairs $(\rho,b)$ where $\rho$ is a holonomy of a flat
${\mathbb{C}}$ bundle over $L(u)$ and $b\in H(L(u);\Lambda_{+})$ is a weak
bounding cochain of the filtered $A_{\infty}$ algebra associated to $L(u)$ and
twisted by $\rho$. With this definition of $\mathcal{M}_{\text{\rm
weak}}(L(u);\Lambda_{0})$, we have :

$H^{1}(L(u);\Lambda_{0})\subseteq\mathcal{M}_{\text{\rm
weak}}(L(u);\Lambda_{0}).$

##  5\. Examples

In this section, we discuss various examples of toric manifolds which
illustrate the results of section 4.

######  Example 5.1.

Consider $X=S^{2}$ with standard symplectic form with area $2\pi$. The moment
polytope of the standard $S^{1}$-action by rotations along an axis becomes
$P=[0,1]$ after a suitable translation. We have $\ell_{1}(u)=u$,
$\ell_{2}(u)=1-u$ and

$\mathfrak{PO}(x;u)=e^{x}T^{u}+e^{-x}T^{1-u}=yT^{u}+y^{-1}T^{1-u}.$

The zero of

$\frac{\partial\mathfrak{PO}^{u}}{\partial y}=T^{u}-y^{-2}T^{1-u}$

is $\mathfrak{y}=\pm T^{(1-2u)/2}.$ If $u\neq 1/2$ then

$\log\mathfrak{y}=\frac{1-2u}{2}\log(\pm T)$

is not an element of universal Novikov ring. In particular, there is no
critical point in
$\Lambda^{{\mathbb{C}}}_{0}\setminus\Lambda^{{\mathbb{C}}}_{+}$.

If $u=1/2$ then $\mathfrak{y}=\pm 1$. The case $\mathfrak{y}=1$ corresponds to
$\mathfrak{x}=0$. We have

$HF((L(1/2),0),(L(1/2),0);\Lambda_{0})\cong
H(S^{1};\Lambda^{{\mathbb{C}}}_{0}).$

The other case $\mathfrak{y}=-1$, corresponds to a nontrivial flat bundle on
$S^{1}$.

######  Example 5.2.

We consider $X={\mathbb{C}}P^{n}$. Then

$P=\\{(u_{1},\cdots,u_{n})\mid 0\leq u_{i},u_{1}+\cdots+u_{n}\leq 1\\},$

is a simplex. We have

$\mathfrak{PO}(x_{1},\cdots,x_{n};u_{1},\cdots,u_{n})=\sum_{i=1}^{n}e^{x_{i}}T^{u_{i}}+e^{-\sum
x_{i}}T^{1-\sum u_{i}}.$

We put $u=u_{0}=\left(\frac{1}{n+1},\cdots,\frac{1}{n+1}\right).$ Then

$\mathfrak{PO}^{u_{0}}=(y_{1}+\cdots+y_{n}+y_{1}^{-1}y_{2}^{-1}\cdots
y_{n}^{-1})T^{1/(n+1)}.$

Solutions of the equation (4.10) are given by

$\mathfrak{y}_{1}=\cdots=\mathfrak{y}_{n}=e^{2\pi k\sqrt{-1}/{(n+1)}},\quad
k=0,\cdots,n.$

Hence the conclusion of Theorem 1.5 holds for our torus. The case $k=0$
corresponds to $b=0$. The other cases correspond to appropriate flat bundles
on $T^{n}$.

######  Remark 5.3.

The critical values of the potential function is
$(n+1)e^{2\pi\sqrt{-1}k/(n+1)}$, $k=0,\cdots,n$.

We consider the quantum cohomology ring

$QH({\mathbb{C}}P^{n};\Lambda_{0})\cong\Lambda_{0}[z,T]/(z^{n+1}-T).$

The first Chern class $c_{1}$ is $(n+1)z$. The eigenvalues of the operator

$c:QH({\mathbb{C}}P^{n})\to QH({\mathbb{C}}P^{n}),\alpha\mapsto
c_{1}\cup_{Q}\alpha$

are $(n+1)e^{2\pi\sqrt{-1}k/(n+1)}$, $k=0,\cdots,n$. It coincides with the set
of critical values.

Kontsevich announced this statement at the homological mirror symmetry
conference at Vienna 2006. (According to some physicists, this statement is
known to them before.) See [Aur]. In our situation of Lagrangian fiber of
compact toric manifolds, we can prove it by using Theorem 1.9.

In the rest of this subsection, we discuss 2 dimensional examples.

Let $\text{\bf e}_{1}$, $\text{\bf e}_{2}$ be the basis of
$H^{1}(T^{2};{\mathbb{Z}})$ as in Lemma 4.4. We put $\text{\bf
e}_{12}=\text{\bf e}_{1}\cup\text{\bf e}_{2}\in H^{2}(T^{2};{\mathbb{Z}})$.
Let $\text{\bf e}_{\emptyset}$ be the standard basis of
$H^{0}(T^{2};{\mathbb{Z}})\cong{\mathbb{Z}}$. The proof of the following
proposition will be postponed until section 13.

######  Proposition 5.4.

Let $\mathfrak{x}=\sum\mathfrak{x}_{i}\text{\bf e}_{i}\in
H^{1}(L(u);\Lambda_{0})$, $\mathfrak{y}_{i}=e^{\mathfrak{x}_{i}}$. Then the
boundary operator $\mathfrak{m}_{1}^{\mathfrak{x}}$ is given as follows :

$\left\\{\begin{aligned} \mathfrak{m}_{1}^{\mathfrak{x}}(\text{\bf
e}_{i})&=\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{i}}(\mathfrak{y})\text{\bf e}_{\emptyset},\\\
\mathfrak{m}_{1}^{\mathfrak{x}}(\text{\bf
e}_{12})&=\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{1}}(\mathfrak{y})\text{\bf e}_{2}-\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{2}}(\mathfrak{y})\text{\bf e}_{1},\\\
\mathfrak{m}_{1}^{\mathfrak{x}}(\text{\bf
e}_{\emptyset})&=0.\end{aligned}\right.$ (5.1)

We remark that we do not use the grading parameter $e$, which was introduced
in [FOOO3]. So the boundary operator $\mathfrak{m}_{1}^{\mathfrak{x}}$ is of
degree $-1$ rather than $+1$. (Note we are using cohomology notation.) In
other words, our Floer cohomology is only of ${\mathbb{Z}}_{2}$ graded.

With (5.1) in our disposal, we examine various examples.

######  Example 5.5.

We consider $M={\mathbb{C}}P^{2}$ again. We put $u_{1}=\epsilon+1/3$,
$u_{2}=1/3$. ($\epsilon>0$.) Using (5.1) we can easily find the following
isomorphism for the Floer cohomology with $\Lambda_{0}$ coefficients :

$HF^{odd}((L(u),0),(L(u),0))\cong
HF^{even}((L(u),0),(L(u),0))\cong\Lambda_{0}/(T^{1/3-\epsilon}).$

Let us apply Theorem J [FOOO3] (= Theorem 3.12) in this situation. (See also
Theorem 5.11 below.) We consider a Hamiltonian diffeomorphism
$\psi:{\mathbb{C}}P^{2}\to{\mathbb{C}}P^{2}$. We denote by $\|\psi\|$ the
Hofer distance of $\psi$ from identity. Then we have

$\\#(\psi(L(u))\cap L(u))\geq 4$

if $\|\psi\|<2\pi(\frac{1}{3}-\epsilon)$ and $\psi(L(u))$ is transversal to
$L(u)$. We remark $\omega\cap[{\mathbb{C}}P^{1}]=2\pi$ by (2.12).

We remark that this fact was already proved by Chekanov [Che]. (Actually the
basic geometric idea behind our proof is the same as Chekanov’s.)

######  Example 5.6.

Let $M=S^{2}(\frac{a}{2})\times S^{2}(\frac{b}{2})$, where $S^{2}(r)$ denotes
the 2-sphere with radius $r$. We assume $a<b$.

Then $B=[0,a]\times[0,b]$ and we have :

$\mathfrak{PO}(x_{1},x_{2};u_{1},u_{2})=y_{1}T^{u_{1}}+y_{2}T^{u_{2}}+y_{1}^{-1}T^{a-u_{1}}+y_{2}^{-1}T^{b-u_{2}}.$

Let us take $u_{1}=a/2$, $u_{2}=b/2$. Then

$\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{1}}=(1-y_{1}^{-2})T^{a/2},\quad\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{2}}=(1-y_{2}^{-2})T^{b/2}.$

Therefore $y_{1}=\pm 1$ $y_{2}=\pm 1$ are solutions of (4.10). Hence we can
apply Theorem 4.10 to our torus.

We next put $u_{1}=a/2$, $a<2u_{2}<b$. Then

$\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{1}}=(1-y_{1}^{-2})T^{a/2},\quad\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{2}}=T^{u_{2}}-y_{2}^{-2}T^{b-u_{2}}.$

We put $y_{1}=y_{2}=1$. Then $\frac{\partial\mathfrak{PO}^{u}}{\partial
y_{1}}=0$, $\frac{\partial\mathfrak{PO}^{u}}{\partial y_{2}}\neq 0$. We find
that

$HF^{odd}((L(u),0),(L(u),0))\cong
HF^{even}((L(u),0),(L(u),0))\cong\Lambda_{0}/(T^{u_{2}}).$

Let $\psi:{\mathbb{C}}P^{2}\to{\mathbb{C}}P^{2}$ be a Hamiltonian
diffeomorphism. Then, Theorem J [FOOO3] (= Theorem 3.12) implies that

$\\#(\psi(L(u))\cap L(u))\geq 4$

if $\|\psi\|<2\pi u_{2}$ and $\psi(L(u))$ is transversal to $L(u)$. Note there
exists a pseudo-holomorphic disc with symplectic area $\pi a$ ($<2\pi u_{2}$).
Hence our result improves a result from [Che].

######  Example 5.7.

Let $X$ be the two-point blow up of ${\mathbb{C}}P^{2}$. We may take its
Kähler form so that the moment polytope is given by

$P=\\{(u_{1},u_{2})\mid-1\leq u_{1}\leq 1,-1\leq u_{2}\leq 1,u_{1}+u_{2}\leq
1+\alpha\\},$

where $-1<\alpha<1$ depends on the choice of Kähler form. We have

$\displaystyle\mathfrak{PO}(x_{1},x_{2};u_{1},u_{2})=y_{1}T^{1+u_{1}}$
$\displaystyle+y_{2}T^{1+u_{2}}+y_{1}^{-1}T^{1-u_{1}}$ (5.2)
$\displaystyle+y_{2}^{-1}T^{1-u_{2}}+y_{1}^{-1}y_{2}^{-1}T^{1+\alpha-
u_{1}-u_{2}}.$

Note $X$ is Fano in our case.

(Case 1: $\alpha=0$).

In this case $X$ is monotone. We put $u_{0}=(0,0)$. $L(u_{0})$ is a monotone
Lagrangian submanifold. We have

$\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
y_{1}}=(1-y_{1}^{-2}-y_{1}^{-2}y_{2}^{-1})T,\quad\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
y_{2}}=(1-y_{2}^{-2}-y_{1}^{-1}y_{2}^{-2})T.$

The solutions of (4.10) are given by $y_{2}=\frac{1}{y_{1}^{2}-1}$,
$y_{1}^{5}+y_{1}^{4}-2y_{1}^{3}-2y_{1}^{2}+1=0$ in ${\mathbb{C}}$. (There are
5 solutions.)

(Case 2: $\alpha>0$).

We put $u_{0}=(0,0)$. Then

$\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
y_{1}}=(1-y_{1}^{-2})T-y_{1}^{-2}y_{2}^{-1}T^{1+\alpha},\quad\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
y_{2}}=(1-y_{2}^{-2})T-y_{1}^{-1}y_{2}^{-2}T^{1+\alpha}.$

We consider, for example, the case $y_{1}=y_{2}=\tau$. Then (4.10) becomes

$\tau^{3}-\tau-T^{\alpha}=0.$ (5.3)

The solution of (5.3) with $\tau\equiv 1\mod\Lambda_{+}$ is given by

$\tau=1+\frac{1}{2}T^{\alpha}-\frac{3}{8}T^{2\alpha}+\frac{1}{2}T^{3\alpha}+\sum_{k=4}^{\infty}c_{k}T^{k\alpha}.$

Let us put ${\mathfrak{x}}=x_{1}\text{\bf e}_{1}+x_{2}\text{\bf e}_{2}$ with

$x_{1}=x_{2}=\log\left(1+\frac{1}{2}T^{\alpha}-\frac{3}{8}T^{2\alpha}+\frac{1}{2}T^{3\alpha}+\cdots\right)\in\Lambda_{+}.$

Then by Theorem 4.10 we have

$HF((L(u_{0}),{\mathfrak{x}}),(L(u_{0}),{\mathfrak{x}});\Lambda_{0})\cong
H(T^{2};\Lambda_{0}).$

We like to point out that in this example it is essential to deform Floer
cohomology using an element ${\mathfrak{x}}$ of $H^{1}(L(u_{0});\Lambda_{+})$
containing the formal parameter $T$ to obtain nonzero Floer cohomology.

At $u_{0}$, there are actually 4 solutions such that

$(y_{1},y_{2})\equiv(1,1),(1,-1),(-1,1),(-1,-1)\mod\Lambda_{+},$

respectively.

In the current case there is another point $u^{\prime}_{0}=(\alpha,\alpha)\in
P$ at which $L(u^{\prime}_{0})$ is balanced 111Using the method of spectral
invariants and symplectic quasi-states, Entov and Polterovich discovered some
non-displaceable Lagrangian fiber which was not covered by the criterion given
in [CO] (see section 9 [EP1]). Recently this example, among others, was
explained by Cho [Cho] via Lagrangian Floer cohomology twisted by non-unitary
line bundles.. In fact at $u^{\prime}_{0}=(\alpha,\alpha)$ the equation (5.3)
becomes

$0=-(y_{1}^{-2}y_{2}^{-1}+y_{1}^{-2})T^{1-\alpha}+T^{1+\alpha},\quad
0=-(y_{1}^{-1}y_{2}^{-2}+y_{2}^{-2})T^{1-\alpha}+T^{1+\alpha}.$

we put $\tau=y_{1}=y_{2}$ to obtain

$\tau^{3}T^{2\alpha}-\tau-1=0$

This equation has a unique solution with $\tau\equiv-1\mod\Lambda_{+}$. (The
other solution is $T^{2\alpha}\tau^{3}\equiv 1\mod\Lambda_{+}$, for which
Theorem 4.10 is not applicable.)

The total number of the solutions $({\mathfrak{x}},u)$ is $5$.

(Case 3: $\alpha<0$).

We first consider $u_{0}=(0,0)$. Then

$\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
y_{1}}=-y_{1}^{-2}y_{2}^{-1}T^{1+\alpha}+(1-y_{1}^{-2})T,\quad\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
y_{2}}=-y_{1}^{-1}y_{2}^{-2}T^{1+\alpha}+(1-y_{2}^{-2})T.$

We assume $y_{i}$ satisfies (4.10). It is then easy to see that
$y_{1}^{-1}\equiv 0$, or $y_{2}^{-1}\equiv 0\mod\Lambda_{+}$. In other words,
there is no $(y_{1},y_{2})$ to which we can apply Theorem 4.10. Actually it is
easy to find a Hamiltonian diffeomorphism $\psi:X\to X$ such that
$\psi(L(u_{0}))\cap L(u_{0})=\emptyset$.

We next take $u^{\prime}_{0}=(\alpha/3,\alpha/3)$. Then

$\displaystyle\frac{\partial\mathfrak{PO}^{u^{\prime}_{0}}}{\partial y_{1}}$
$\displaystyle=(1-y_{1}^{-2}y_{2}^{-1})T^{1+\alpha/3}-y_{1}^{-2}T^{1-\alpha/3},$
$\displaystyle\frac{\partial\mathfrak{PO}^{u^{\prime}_{0}}}{\partial y_{2}}$
$\displaystyle=(1-y_{1}^{-1}y_{2}^{-2})T^{1+\alpha/3}-y_{2}^{-2}T^{1-\alpha/3}.$

By putting $y_{1}=y_{2}=\tau$ for example, (4.10) becomes

$\tau^{3}-T^{-2\alpha/3}\tau-1=0.$ (5.4)

Let us put ${\mathfrak{x}}=x_{1}\text{\bf e}_{1}+x_{2}\text{\bf e}_{2}$ with

$x_{1}=x_{2}=\log\tau=\log\left(1+\frac{1}{3}T^{-2\alpha/3}-\frac{1}{81}T^{-6\alpha/3}+\cdots\right)\in\Lambda_{+},$

where $\tau$ solves (5.4). Theorem 4.10 is applicable. (There are actually 3
solutions of (4.10) corresponding to the 3 solutions of (5.4).)

There are two more points $u=(\alpha+1,\alpha),(\alpha,\alpha+1)$ where (4.10)
has a solution in $(\Lambda_{0}\setminus\Lambda_{+})$. Each $u$ has one
solution $b$.

Thus the total number of the pair $({\mathfrak{x}},u)$ is again 5. We remark

$5=\sum\text{\rm rank}\,H^{k}(X;{\mathbb{Q}}).$

This is not just a coincidence but an example of general phenomenon stated as
in Theorem 1.3.

We remark that as $\alpha\to 1$ our $X$ blows down to $S^{2}(1)\times
S^{2}(1)$. On the other hand, as $\alpha\to-1$ our $X$ blows down to
${\mathbb{C}}P^{2}$. The situation of the case $\alpha>0$ can be regarded as a
perturbation of the situation of $S^{2}(1)\times S^{2}(1)$, by the effect of
exceptional curve corresponding to the segment $u_{1}+u_{2}=1+\alpha$. The
situation of the case $\alpha<0$ can be regarded as a perturbation of the
situation of ${\mathbb{C}}P^{2}$ by the effect of the two exceptional curves
corresponding to the segments $u_{1}=1$ and $u_{2}=1$. An interesting phase
change occurs at $\alpha=0$.

We remark that $H^{2}(X;{\mathbb{R}})$ is two dimensional. So there are
actually two parameter family of symplectic structures. We study two points
blow up of ${\mathbb{C}}P^{2}$ more in Example 10.17.

The discussion of this section strongly suggests that Lagrangian Floer theory
(Theorems G, J [FOOO3] = Theorems 3.11, 3.12) gives the optimal result for the
study of non-displacement of Lagrangian fibers in toric manifolds.

######  Remark 5.8.

Let $X$ be a compact toric manifold and $L(u)=\pi^{-1}(u)$, $u\in\text{\rm
Int}P$. We consider the following two conditions :

  1. (1)

There exists no Hamiltonian diffeomorphism $\psi:X\to X$ such that
$\psi(L(u))\cap L(u)=\emptyset.$

  2. (2)

$L(u)$ is balanced.

$(2)\Rightarrow(1)$ follows from Proposition 4.12. In many cases
$(1)\Rightarrow(2)$ can be proved by the method of [Mc]. However there is a
case $(1)\Rightarrow(2)$ does not follow, as we will mention in Remark 10.18
and will prove in a sequel of this series of papers, using the bulk
deformation of Lagrangian Floer cohomology. We conjecture that after including
this wider class of Floer cohomology, we can detect all the non-displacable
Lagrangian fibers in toric manifolds, by Floer cohomology.

Using the argument employed in Example 5.6 we can discuss the relationship
between the Hofer distance and displacement. First we introduce some notations
for this purpose. We denote by $Ham(X,\omega)$ the group of Hamiltonian
diffeomorphisms of $(X,\omega)$. For a time-dependent Hamiltonian
$H:[0,1]\times X\to{\mathbb{R}}$, we denote by $\phi_{H}^{t}$ the time $t$-map
of Hamilton’s equation $\dot{x}=X_{H}(t,x)$. The Hofer norm of $\psi\in
Ham(X,\omega)$ is defined to be

$\|\psi\|=\inf_{H;\phi_{H}^{1}=\psi}\int_{0}^{1}(\max H_{t}-\min H_{t})\,dt.$

(See [H].)

######  Definition 5.9.

Let $Y\subset X$. We define the displacement energy $e(Y)\in[0,\infty]$ by

$e(Y):=\inf\\{\|\psi\|\mid\psi\in
Ham(X,\omega),\,\psi(Y)\cap\overline{Y}=\emptyset\\}.$

We put $e(Y)=\infty$ if there exists no $\psi\in Ham(X,\omega)$ with
$\psi(Y)\cap\overline{Y}=\emptyset$.

Let us consider
$\mathfrak{PO}(y_{1},\cdots,y_{n};u_{1},\cdots,u_{n}):\Lambda_{0}^{n}\times
P\to\Lambda_{+}$ as in Theorem 4.5.

######  Definition 5.10.

We define the number $\mathfrak{E}(u)\in(0,\infty]$ as the supremum of all
$\lambda$ such that there exists
$\mathfrak{y}_{1},\cdots,\mathfrak{y}_{n}\in(\Lambda_{0}\setminus\Lambda_{0}^{+})^{n}$
satisfying

$\frac{\partial\mathfrak{PO}}{\partial
y_{i}}(\mathfrak{y}_{1},\cdots,\mathfrak{y}_{n};u)\equiv 0\mod T^{\lambda}$
(5.5)

for $i=1,\cdots,n$. (Here we consider universal Novikov ring with
${\mathbb{C}}$-coefficients.) We call $\mathfrak{E}(u)$ the _$\mathfrak{PO}$
-threshold_ of the fiber $L(u)$. We put

$\overline{\mathfrak{E}}(u)=\limsup_{\omega_{i}\to\omega,u_{i}\to
u}{\mathfrak{E}}(u_{i}).$

Here limsup is taken over all sequences $\omega_{i}$ and $u_{i}$ such that
$\omega_{i}$ is a sequence of $T^{n}$ invariant symplectic structures on $X$
with $\lim_{i\to\infty}\omega_{i}=\omega$ and $u_{i}$ is a sequence of points
of moment polytopes $P_{i}$ of $(X,\omega_{i})$ such that $P_{i}$ converges to
$P$ and $u_{i}$ converges to $u$.

Clearly $\overline{\mathfrak{E}}(u)\geq{\mathfrak{E}}(u)$. We will give an
example where $\overline{\mathfrak{E}}(u)\neq{\mathfrak{E}}(u)$ in Example
10.17 (10.16).

######  Theorem 5.11.

For any compact toric manifold $X$ and $L(u)=\pi^{-1}(u)$, $u\in\text{\rm
Int}P$, we have

$e(L(u))\geq 2\pi\overline{\mathfrak{E}}(u).$ (5.6)

We will prove Theorem 5.11 in section 13.

######  Remark 5.12.

The equality in $(\ref{displaceineq})$ holds in various examples. However
there are cases that the equality in $(\ref{displaceineq})$ does not hold. The
situation is the same as Remark 5.8.

We like to remark $\mathfrak{E}(u)$, $\overline{\mathfrak{E}}(u)$ can be
calculated in most of the cases once the toric manifold $X$ is given
explicitly. In fact the leading order potential function $\mathfrak{PO}_{0}$
is explicitly calculated by Theorem 4.5. We can then find the maximal value
$\lambda$ for which the polynomial equations

$\frac{\partial{\mathfrak{PO}}_{0}}{\partial
y_{i}}(u;\mathfrak{y}_{1},\cdots,\mathfrak{y}_{n})\equiv 0\mod T^{\lambda}$

has a solution $\mathfrak{y}_{i}\in\Lambda_{0}\setminus\Lambda_{+}$. In a
weakly degenerate case this value of $\lambda$ for $\mathfrak{PO}(u;\cdots)$
is the same as the value for $\mathfrak{PO}_{0}(u;\cdots)$. (See section 10.)

######  Remark 5.13.

Appearance of a new family of pseudo-holomorphic discs with Maslov index 2
after blow up, which we observed in Examples 5.7 can be related to the
operator $\mathfrak{q}$ that we introduced in section 3.8 [FOOO3] (= section
13 [FOOO2]) in the following way.

We denote by $\mathcal{M}^{\text{\rm main}}_{k+1,l}(\beta)$ the moduli space
of stable maps $f:(\Sigma,\partial\Sigma)\to(X,L)$ from bordered Riemann
surface $\Sigma$ of genus zero with $l$ interior and $k+1$ boundary marked
points and in homology class $\beta$. (See section 3 [FOOO1] (= subsection
2.1.2 [FOOO3]). The symbol main means that we require the boundary marked
points to respect the cyclic order of $\partial\Sigma$.) Let us consider the
case when Maslov index of $\beta$ is $2n$. More precisely we take the
following class $\beta$. We use notation introduced at the beginning of
section 11. We put $\beta=\beta_{i_{1}}+\cdots+\beta_{i_{n}}$, where
$\partial_{i_{1}}P\cap\cdots\cap\partial_{i_{n}}P=\overline{p}$ is an vertex
of $P$. We assume $[f]\in\mathcal{M}^{\text{\rm main}}_{0+1,1}(\beta)$ and
$\mathcal{M}^{\text{\rm main}}_{0+1,1}(\beta)$ is Fredholm regular at $f$. The
virtual dimension of $\mathcal{M}^{\text{\rm main}}_{0+1,1}(\beta)$ is $3n$.
Let us take the unique point $p\in X$ such that $\pi(p)=\overline{p}$. $p$ is
the $T^{n}$ fixed point. We assume moreover $f(0)=p$. We blow up $X$ at a
point $p=f(0)\in X$ and obtain $\widehat{X}$. Let $[E]\in
H_{2n-2}(\widehat{X})$ be the homology class of the exceptional divisor
$E=\pi^{-1}(p)$. Now $f$ induces a map
$\widehat{f}:(\Sigma;\partial\Sigma)\to(\widehat{X},L)$. The Maslov index of
the homology class $[\widehat{f}]\in H_{2}(\widehat{X},L)$ becomes $2$. We put
$\widehat{\beta}=[\widehat{f}]$.

Since $p$ is a fixed point of $T^{n}$ action, a $T^{n}$-invariant perturbation
lifts to a perturbation of the moduli space $\mathcal{M}^{\text{\rm
main}}_{0+1,0}(\widehat{\beta})$. Then any $T^{n}$-orbit of the moduli space
$\mathcal{M}^{\text{\rm main}}_{0+1,0}(X;\beta)$ of holomorphic discs passing
through $p$ corresponds to the $T^{n}$-orbit of $\mathcal{M}^{\text{\rm
main}}_{0+1,0}(\widehat{X};\widehat{\beta})$ and vice versa. Namely we have an
isomorphism

$\mathcal{M}^{\text{\rm
main}}_{0+1,1}(\beta)\,\,{}_{ev}\times_{X}\\{p\\}\cong\mathcal{M}^{\text{\rm
main}}_{0+1,0}(\widehat{\beta}).$ (5.7)

Here $ev$ in the left hand side is the evaluation map at the interior marked
point. (Actually we need to work out analytic detail of gluing construction
etc.. It seems very likely that we can do it in the same way as the argument
of Chapter 10 [FOOO2]. See also [LiRu].)

Using (5.7) we may prove :

$\mathfrak{q}_{1,k;\beta}(PD([p]);b,\cdots,b)=\mathfrak{m}_{k,\widehat{\beta}}(b,\cdots,b),$

where

$\mathfrak{q}_{1,k;\beta}(Q;P_{1},\cdots,P_{k})=ev_{0*}\left(\mathcal{M}^{\text{\rm
main}}_{k+1,1}(\beta)\times_{(X\times L^{k})}(Q\times P_{1}\times\cdots\times
P_{k})\right)$

is defined in section 3.8 [FOOO3] (=section 13 [FOOO2]). (Here $Q$ is a chain
in $X$ and $P_{i}$ are chains in $L(u)$, and $ev_{0}:\mathcal{M}^{\text{\rm
main}}_{k+1,1}(\beta)\to X$ is the evaluation map at the $0$-th boundary
marked point. In the right hand side, we take fiber product over $X\times
L^{k}$.) This is an example of a blow-up formula in Lagrangian Floer theory.

##  6\. Quantum cohomology and Jacobian ring

In this section, we prove Theorem 1.9. Let $\mathfrak{PO}_{0}$ be the leading
order potential function. (Recall if $X$ is Fano, we have
$\mathfrak{PO}_{0}=\mathfrak{PO}$.) We define the monomial

$\overline{z}_{i}(u)=y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}T^{\ell_{i}(u)}\in\Lambda_{0}[y_{1},\cdots,y_{n},y^{-1}_{1},\cdots,y^{-1}_{n}].$
(6.1)

Compare this with (2.3). It is also suggestive to write $\overline{z}_{i}$
also as

$\overline{z}_{i}(u)=e^{\langle
x,v_{i}\rangle}T^{\ell_{i}(u)},\,x=(x_{1},\cdots,x_{n}),\,y_{i}=e^{x_{i}}.$
(6.2)

By definition we have

$\displaystyle\mathfrak{PO}_{0}^{u}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\overline{z}_{i}(u)$ (6.3) $\displaystyle
y_{j}\frac{\partial\overline{z}_{i}}{\partial y_{j}}$ $\displaystyle=$
$\displaystyle v_{i,j}\overline{z}_{i}(u).$ (6.4)

The following is a restatement of Theorem 1.9. Let $z_{i}\in
H^{2}(X;{\mathbb{Z}})$ be the Poincaré dual of the divisor
$\pi^{-1}(\partial_{i}P)$.

######  Theorem 6.1.

If $X$ is Fano, there exists an isomorphism

$\psi_{u}:QH(X;\Lambda)\cong Jac(\mathfrak{PO})$

such that $\psi_{u}(z_{i})=\overline{z}_{i}(u)$.

Since $c_{1}(X)=\sum_{i=1}^{m}z_{i}$ (see [Ful]) and
$\mathfrak{PO}_{0}^{u}=\sum_{i=1}^{m}\overline{z}_{i}(u)$ by definition,
Theorem 1.9 follows from Theorem 6.1.

In the rest of this section, we prove Theorem 6.1. We remark that $z_{i}$
($i=1,\cdots,m$) generates the quantum cohomology ring $QH(X;\Lambda)$ as a
$\Lambda$-algebra (see Theorem 6.6 below). Therefore it is enough to prove
that the assignment $\tilde{\psi}_{u}(z_{i})=\overline{z}_{i}(u)$ extends to a
homomorphism $\tilde{\psi}_{u}:\Lambda[z_{1},\cdots,z_{m}]\to
Jac(\mathfrak{PO}_{0}^{u})$ that induces an isomorphism in $QH(X;\Lambda)$. In
other words, it suffices to show that the relations among the generators in
$\Lambda[z_{1},\cdots,z_{m}]$ and in $Jac(\mathfrak{PO}_{0}^{u})$ are mapped
to each other under the assignment
$\tilde{\psi}_{u}(z_{i})=\overline{z}_{i}(u)$. To establish this
correspondence, we now review Batyrev’s description of the relations among
$z_{i}$’s.

We first clarify the definition of quantum cohomology ring over the universal
Novikov rings $\Lambda_{0}$ and $\Lambda$. Let $(X,\omega)$ be a symplectic
manifold and $\alpha\in\pi_{2}(X)$. Let $\mathcal{M}_{3}(\alpha)$ be the
moduli space of stable map with homology class $\alpha$ of genus $0$ with $3$
marked points. Let $ev:\mathcal{M}_{3}(\alpha)\to X^{3}$ be the evaluation
map. We can define the virtual fundamental class
$ev_{*}[\mathcal{M}_{3}(\alpha)]\in H_{d}(X^{3};{\mathbb{Q}})$ where
$d=2(\mathop{\kern 0.0pt{\rm
dim}}\nolimits_{{\mathbb{C}}}X+c_{1}(X)\cap\alpha).$ Let $a_{i}\in
H^{*}(X;{\mathbb{Q}})$. We define $a_{1}\cup_{Q}a_{2}\in H^{*}(X;\Lambda_{0})$
by the following formula.

$\langle
a_{1}\cup_{Q}a_{2},a_{3}\rangle=\sum_{\alpha}T^{\omega\cap\alpha/2\pi}ev_{*}[\mathcal{M}_{3}(\alpha)]\cap(a_{1}\times
a_{2}\times a_{3}).$ (6.5)

Here $\langle\cdot,\cdot\rangle$ is the Poincaré duality. Extending this
linearly we obtain the quantum product

$\cup_{Q}:H(X;\Lambda_{0})\otimes H(X;\Lambda_{0})\to H(X;\Lambda_{0}).$

Extending the coefficient ring further to $\Lambda$, we obtain the (small)
quantum cohomology ring $QH(X;\Lambda)$.

Now we specialize to the case of compact toric manifolds and review Batyrev’s
presentation of quantum cohomology ring. We consider the exact sequence

$0\longrightarrow\pi_{2}(X)\longrightarrow\pi_{2}(X;L(u))\longrightarrow\pi_{1}(L(u))\longrightarrow
0.$ (6.6)

We note $\pi_{2}(X;L(u))\cong{\mathbb{Z}}^{m}$ and choose its basis adapted to
this exact sequence as follows : Consider the divisor
$\pi^{-1}(\partial_{i}P)$ and take a small disc transversal to it. Each such
disc gives rise to an element

$[\beta_{i}]\in H_{2}(X;\pi^{-1}(\text{\rm Int}P))\cong
H_{2}(X;L(u))\cong\pi_{2}(X,L(u)).$ (6.7)

The set of $[\beta_{i}]$ with $i=1,\cdots,m$ forms a basis of
$\pi_{2}(X;L(u))\cong{\mathbb{Z}}^{m}$. The boundary map
$[\beta]\mapsto[\partial\beta]:\pi_{2}(X;L(u))\to\pi_{1}(L(u))$ is identified
with the corresponding map $H_{2}(X;L(u))\to H_{1}(L(u))$. Using the basis
chosen in Lemma 4.4 on $H_{1}(L(u))$ we identify
$H_{1}(L(u))\cong{\mathbb{Z}}^{n}$. Then this homomorphism maps $[\beta_{i}]$
to

$[\partial\beta_{i}]\cong v_{i}=(v_{i,1},\cdots,v_{i,n}),$ (6.8)

where $v_{i,j}$ is as in (4.4). By the exactness of (6.6), we have an
isomorphism

$H_{2}(X)\cong\\{\beta\in H_{2}(X;L(u))\mid[\partial\beta]=0\\}.$ (6.9)

######  Lemma 6.2.

We have

$\omega\cap\left[\sum k_{i}\beta_{i}\right]=2\pi\sum k_{i}\ell_{i}(u).$ (6.10)

If $\left[\sum k_{i}\partial\beta_{i}\right]=0$ then

$\sum k_{i}\frac{d\ell_{i}}{du_{j}}=0.$ (6.11)

In particular, the right hand side of $(\ref{form:symparea})$ is independent
of $u$.

###### Proof.

(6.10) follows from the area formula (2.12),
$\omega(\beta_{i})=2\pi\ell_{i}(u).$ On the other hand if $\left[\sum
k_{i}\partial\beta_{i}\right]=0$, we have

$\sum_{i=1}^{m}k_{i}v_{i}=0.$

By the definition of $\ell_{i}$, $\ell_{i}(u)=\langle
u,v_{i}\rangle-\lambda_{i}$, from Theorem 2.13, this equation is precisely
(6.11) and hence the proof. ∎

Let $\mathcal{P}\subset\\{1,\cdots,m\\}$ be a primitive collection (see
Definition 2.4). There exists a unique subset
$\mathcal{P}^{\prime}\subset\\{1,\cdots,m\\}$ such that
$\sum_{i\in\mathcal{P}}v_{i}$ lies in the interior of the cone spanned by
$\\{v_{i^{\prime}}\mid i^{\prime}\in\mathcal{P}^{\prime}\\}$, which is a
member of the fan $\Sigma$. (Since $X$ is compact, we can choose such
$\mathcal{P}^{\prime}$. See section 2.4 [Ful].) We write

$\sum_{i\in\mathcal{P}}v_{i}=\sum_{i^{\prime}\in\mathcal{P}^{\prime}}k_{i^{\prime}}v_{i^{\prime}}.$
(6.12)

Since $X$ is assumed to be nonsingular $k_{i^{\prime}}$ are all positive
integers. (See p.29 of [Ful].) We put

$\omega(\mathcal{P})=\sum_{i\in\mathcal{P}}\ell_{i}(u)-\sum_{i^{\prime}\in\mathcal{P}^{\prime}}k_{i^{\prime}}\ell_{i^{\prime}}(u).$
(6.13)

It follows from (6.10) that $2\pi\omega(\mathcal{P})$ is the symplectic area
of the homotopy class

$\beta(\mathcal{P})=\sum_{i\in\mathcal{P}}\beta_{i}-\sum_{i^{\prime}\in\mathcal{P}^{\prime}}k_{i^{\prime}}\beta_{i^{\prime}}\in\pi_{2}(X).$
(6.14)

######  Lemma 6.3.

Let $\mathcal{P}$ be any primitive collection. Then $\omega(\mathcal{P})>0$.

###### Proof.

Since the cone spanned by
$\\{v_{i^{\prime}}|i^{\prime}\in{\mathcal{P}}^{\prime}\\}$ belongs to the fan
$\Sigma$, we have

$\bigcap_{i^{\prime}\in\mathcal{P}^{\prime}}\pi^{-1}(\partial_{i^{\prime}}P)\neq\emptyset.$

Then for any
$u^{\prime}\in\bigcap_{i^{\prime}\in\mathcal{P}^{\prime}}\partial_{i^{\prime}}P$,
we have $\ell_{i^{\prime}}(u^{\prime})=0$. By the choice of $k_{i^{\prime}}$
in (6.12), we have $\partial\beta(\mathcal{P})=0$. Therefore by Lemma 6.2 and
by the continuity of the right hand side of (6.13) we can evaluate
$\omega(\mathcal{P})$ at a point
$u^{\prime}\in\bigcap_{i^{\prime}\in\mathcal{P}^{\prime}}\pi^{-1}(\partial_{i^{\prime}}P)$.
Then we obtain

$\omega(\mathcal{P})=\sum_{i\in\mathcal{P}}\ell_{i}(u^{\prime}).$

Since $\mathcal{P}$ is a primitive collection, and in particular does not form
a member of the fan, there must be an element $v_{i}\in\mathcal{P}$ such that
$\ell_{i}(u^{\prime})>0$ and so $\omega(\mathcal{P})>0$. This finishes the
proof. ∎

Now we associate formal variables, $z_{1},\cdots,z_{m}$, to
$v_{1},\cdots,v_{m}$, respectively.

######  Definition 6.4 (Batyrev [B1]).

  1. (1)

The quantum Stanley-Reisner ideal $SR_{\omega}(X)$ is the ideal generated by

$z(\mathcal{P})=\prod_{i\in\mathcal{P}}z_{i}-T^{\omega(\mathcal{P})}\prod_{i^{\prime}\in\mathcal{P}^{\prime}}z_{i^{\prime}}^{k_{i^{\prime}}}$
(6.15)

in the polynomial ring $\Lambda\left[z_{1},\cdots,z_{m}\right]$. Here
$\mathcal{P}$ runs over all primitive collections.

  2. (2)

We denote by $P(X)$ the ideal generated by

$\sum_{i=1}^{m}v_{i,j}z_{i}$ (6.16)

for $j=1,\cdots,n$. In this paper we call $P(X)$ the linear relation ideal.

  3. (3)

We call the quotient

$QH^{\omega}(X;\Lambda)=\frac{\Lambda\left[z_{1},\cdots,z_{m}\right]}{(P(X)+SR_{\omega}(X))}$
(6.17)

the Batyrev quantum cohomology ring.

######  Remark 6.5.

We do not take closure of our ideal $P(X)+SR_{\omega}(X)$ here. See
Proposition 8.6.

######  Theorem 6.6 (Batyrev [B1], Givental [Gi2]).

If $X$ is Fano there exists a ring isomorphism from $QH^{\omega}(X;\Lambda)$
to the quantum cohomology ring $QH(X;\Lambda)$ of $X$ such that $z_{i}$ is
sent to the Poincaré dual to $\pi^{-1}(\partial_{i}P)$.

The main geometric part of the proof of Theorem 6.6 is the following.

######  Proposition 6.7.

The Poincaré dual to $\pi^{-1}(\partial_{i}P)$ satisfy the quantum Stanley-
Reisner relation.

We do not prove Proposition 6.7 in this paper. See Remarks 6.15 and 6.16.
However since our choice of the coefficient ring is different from other
literature, we explain here for reader’s convenience how Theorem 6.6 follows
from Proposition 6.7.

Proposition 6.7 implies that we can define a ring homomorphism
$h:QH^{\omega}(X;\Lambda)\to QH(X;\Lambda)$ by sending $z_{i}$ to
$PD(\pi^{-1}(\partial_{i}P))$. Let $F^{k}QH(X;\Lambda)$ be the direct sum of
elements of degree $\leq k$. Let $F^{k}QH^{\omega}(X;\Lambda)$ be the
submodule generated by the polynomial of degree at most $k/2$ on $z_{i}$.
Clearly $h(F^{k}QH^{\omega}(X;\Lambda))\subset F^{k}QH(X;\Lambda)$.

Since $X$ is Fano, it follows that,

$x\cup_{Q}y-x\cup y\in F^{\deg x+\deg y-2}QH(X;\Lambda).$

We also recall the cohomology ring $H(X;{\mathbb{Q}})$ is obtained by putting
$T=0$ in quantum Stanley-Reisner relation. Moreover we find that the second
product of the right hand side of (6.15) has degree strictly smaller than the
first since $X$ is Fano.

Therefore the graded ring

$gr(QH(X;\Lambda))=\bigoplus_{k}F^{k}(QH(X;\Lambda))/F^{k-1}(QH(X;\Lambda)),$

is isomorphic to the (usual) cohomology ring (with $\Lambda$ coefficient) as a
ring. The same holds for $QH^{\omega}(X;\Lambda)$. It follows that $h$ is an
isomorphism. ∎

In the rest of this section, we will prove the following Proposition 6.8.
Theorem 6.1 follows immediately from Proposition 6.8 and Theorem 6.6.

######  Proposition 6.8.

There exists an isomorphism :

$\psi_{u}:QH^{\omega}(X;\Lambda)\cong Jac(\mathfrak{PO}_{0}^{u})$

such that $\psi_{u}(z_{i})=\overline{z}_{i}(u)$.

We remark that we do not assume that $X$ is Fano in Proposition 6.8. We also
remark that for our main purpose to calculate
$\mathfrak{M}_{0}(\mathfrak{Lag}(X))$, Proposition 6.8 suffices. Proposition
6.8 is a rather simple algebraic result whose proof does not require study of
pseudo-holomorphic discs or spheres.

###### Proof of Proposition 6.8.

We start with the following proposition.

######  Proposition 6.9.

The assignment

$\widehat{\psi}_{u}(z_{i})=\overline{z}_{i}(u)$ (6.18)

induces a well-defined ring isomorphism

$\widehat{\psi}_{u}:\frac{\Lambda\left[z_{1},\cdots,z_{m}\right]}{SR_{\omega}(X)}\to\Lambda[y_{1},\cdots,y_{n},y_{1}^{-1},\cdots,y_{n}^{-1}].$
(6.19)

###### Proof.

Let $\mathcal{P}$ be a primitive collection and $\mathcal{P}^{\prime}$,
$k_{i^{\prime}}$ be as in (6.12). We calculate

$\prod_{i\in\mathcal{P}}\overline{z}_{i}(u)=\prod_{i\in\mathcal{P}}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}T^{\ell_{i}(u)}$ (6.20)

by (6.1). On the other hand,

$\displaystyle\prod_{i^{\prime}\in\mathcal{P}^{\prime}}\overline{z}_{i^{\prime}}^{k_{i^{\prime}}}(u)$
$\displaystyle=\prod_{i^{\prime}\in\mathcal{P}^{\prime}}y_{1}^{k_{i^{\prime}}v_{i^{\prime},1}}\cdots
y_{n}^{k_{i^{\prime}}v_{i^{\prime},n}}T^{k_{i^{\prime}}\ell_{i^{\prime}}(u)}$
$\displaystyle=\prod_{i\in\mathcal{P}}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}\prod_{i^{\prime}\in\mathcal{P}^{\prime}}T^{k_{i^{\prime}}\ell_{i^{\prime}}(u)}$

by (6.12). Moreover

$\sum_{i\in\mathcal{P}}\ell_{i}(u)-\sum_{i\in\mathcal{P}^{\prime}}k_{i^{\prime}}\ell_{i^{\prime}}(u)=\omega(\mathcal{P})$

by (6.13). Therefore

$\prod_{i\in\mathcal{P}}\overline{z}_{i}(u)=T^{\omega(\mathcal{P})}\prod_{i^{\prime}\in\mathcal{P}^{\prime}}\overline{z}_{i^{\prime}}^{k_{i^{\prime}}}(u)$

in $\Lambda[y_{1},\cdots,y_{n},y_{1}^{-1},\cdots,y_{n}^{-1}]$. In other words,
(6.18) defines a well-defined ring homomorphism (6.19).

We now prove that $\widehat{\psi}_{u}$ is an isomorphism. Let

$pr:{\mathbb{Z}}^{m}\cong\pi_{2}(X;L(u))\longrightarrow{\mathbb{Z}}^{n}\cong\pi_{1}(L(u))$

be the homomorphism induced by the boundary map $pr([\beta])=[\partial\beta].$
(See (2.1).) We remark $pr(c_{1},\dots,c_{m})=(d_{1},\dots,d_{n})$ with
$d_{j}=\sum_{i}c_{i}v_{i,j}$. Let $A=\sum_{i}c_{i}\beta_{i}$ be an element in
the kernel of $pr$. We write it as

$\sum_{i\in I}a_{i}\beta_{i}-\sum_{j\in J}b_{j}\beta_{j}$

where $a_{i}$ $b_{j}$ are positive and $I\cap J=\emptyset$. We define

$r(A)=\prod_{i\in
I}z_{i}^{a_{i}}-T^{\sum_{i}a_{i}\ell_{i}(u)-\sum_{j}b_{j}\ell_{j}(u)}\prod_{j\in
J}z_{j}^{b_{j}}.$ (6.21)

We remark that a generator of quantum Stanley-Reisner ideal corresponds to
$r(A)$ for which $I$ is a primitive collection $\mathcal{P}$ and
$J=\mathcal{P}^{\prime}$. We also remark that the case $I=\emptyset$ or
$J=\emptyset$ is included.

######  Lemma 6.10.

$r(A)\in SR_{\omega}(X).$

###### Proof.

This lemma is proved in [B1]. We include its proof (which is different from
one in [B1]) here for reader’s convenience. We prove the lemma by an induction
over the values

$E(A)=\sum_{i\in I}a_{i}\ell_{i}(u_{0})+\sum_{j\in J}b_{j}\ell_{j}(u_{0}).$

Here we fix a point $u_{0}\in\text{Int}P$ during the proof of Lemma 6.10.

Since $I\cap J=\emptyset$, at least one of $\\{v_{i}\mid i\in I\\}$,
$\\{v_{j}\mid j\in J\\}$ can not span a cone that is a member of the fan
$\Sigma$. Without loss of generality, we assume that $\\{v_{i}\mid i\in I\\}$
does not span such a cone. Then it contains a subset $\mathcal{P}\subset I$
that is a primitive collection. We take $\mathcal{P}^{\prime}$,
$k_{i^{\prime}}$ as in (6.12) and define

$Z=\prod_{i\in I}z_{i}^{a_{i}}-T^{\omega(\mathcal{P})}\prod_{i\in
I\setminus\mathcal{P}}z_{i}^{a_{i}}\prod_{i\in\mathcal{P}}z_{i}^{a_{i}-1}\prod_{i^{\prime\prime}\in\mathcal{P}^{\prime}}z_{i^{\prime\prime}}^{k_{i^{\prime\prime}}}.$
(6.22)

Then $Z$ lies in $SR_{\omega}(X)$ by construction. We recall from Lemma 6.2
that the values

$\sum_{i\in\mathcal{P}}\ell_{i}(u)-\sum_{i\in\mathcal{P}^{\prime}}k_{i}\ell_{i}(u)=\omega(\mathcal{P})$

are independent of $u$ and positive. By the definitions (6.21), (6.22) of
$r(A)$ and $Z$, we can express

$r(A)-Z=T^{\omega(\mathcal{P})+c}\left(\prod_{h\in K}z_{h}^{n_{h}}\right)r(B)$

for an appropriate $B$ in the kernel of $pr$ and a constant $c$. Moreover we
have

$E(B)+2\sum_{h\in K}n_{h}\ell_{h}(u_{0})+\omega(\mathcal{P})=E(A).$

Since $u_{0}\in\text{Int}\,P$ it follows that $\ell_{h}(u_{0})>0$ which in
turn gives rise to $E(B)<E(A)$. The induction hypothesis then implies $r(B)\in
SR_{\omega}(X)$. The proof of the lemma is now complete. ∎

######  Corollary 6.11.

$z_{i}$ is invertible in

$\frac{\Lambda\left[z_{1},\cdots,z_{m}\right]}{SR_{\omega}(X)}.$

###### Proof.

Since $X$ is compact, the vector $-v_{i}$ is in some cone spanned by $v_{j}$
($j\in I$). Namely

$-v_{i}=\sum_{j\in I}k_{j}v_{j}$

where $k_{j}$ are nonnegative integers. Then

$T^{\ell_{i}(u)+\sum_{j}k_{j}\ell_{j}(u)}=z_{i}\prod_{j\in I}z_{j}^{k_{j}}\mod
SR_{\omega}(X)$

by Lemma 6.10. Since $T^{\ell_{i}(u)+\sum_{j}k_{j}\ell_{j}(u)}$ is invertible
in the field $\Lambda$, it follows that $\prod_{j\in I}z_{j}^{k_{j}}$ defines
the inverse of $z_{i}$ in the quotient ring. ∎

We recall from Lemma 6.2 that $\ell_{i}(u)+\sum_{j}k_{j}\ell_{j}(u)$ is
independent of $u$. We define

$z_{i}^{-1}=T^{-\ell_{i}(u)-\sum_{j}k_{j}\ell_{j}(u)}\prod_{j\in
I}z_{j}^{k_{j}}.$ (6.23)

(Note we have not yet proved that
${\Lambda\left[z_{1},\cdots,z_{m}\right]}/{SR_{\omega}(X)}$ is an integral
domain. This will follow later when we prove Proposition 6.9.)

Since $v_{1},\cdots,v_{m}$ generates the lattice ${\mathbb{Z}}^{n}$, we can
always assume the following by changing the order of $v_{i}$, if necessary.

######  Condition 6.12.

The determinant of the $n\times n$ matrix $(v_{i,j})_{i,j=1,\cdots,n}$ is $\pm
1$.

Let $(v^{i,j})$ be the inverse matrix of $(v_{i,j})$. Namely
$\sum_{j}v^{i,j}v_{j,k}=\delta_{i,k}.$ Condition 6.12 implies that each
$v^{i,j}$ is an integer. Inverting the matrix $(v_{i,j})$, we obtain

$y_{i}=T^{-c_{i}(u)}\prod_{j=1}^{n}\overline{z}_{i}^{v^{i,j}}$ (6.24)

from (6.20) where $c_{i}(u)=\sum v^{i,j}\ell_{j}(u).$ We define using
Corollary 6.11

$\widehat{\phi}_{u}(y_{i}^{\pm 1})=T^{-\pm c_{i}(u)}\prod_{j=1}^{n}z_{j}^{\pm
v^{i,j}}\in\frac{\Lambda\left[z_{1},\cdots,z_{m}\right]}{SR_{\omega}(X)}.$

More precisely, we plug (6.23) here if $\pm v^{i,j}$ is negative.

The identity $\widehat{\psi}_{u}\circ\widehat{\phi}_{u}=id$ is a consequence
of (6.24). We next calculate
$(\widehat{\phi}_{u}\circ\widehat{\psi}_{u})(z_{h})=\widehat{\phi}_{u}(\overline{z}_{h}(u))$
and prove

$(\widehat{\phi}_{u}\circ\widehat{\psi}_{u})(z_{h})=T^{\ell_{h}(u)}\widehat{\phi}_{u}(y_{1}^{v_{h,1}}\cdots
y_{n}^{v_{h,n}})=T^{e(h;u)}\prod_{j=1}^{n}z_{j}^{m_{j}},$

where $m_{j}\geq 0$ and

$v_{h}=\sum m_{j}v_{j},\qquad\ell_{h}(u)=e(h;u)+\sum m_{j}\ell_{j}(u):$ (6.25)

To see (6.25), we consider any monomial $Z$ of
$y_{i},z_{i},\overline{z}_{i},T^{\alpha}$. We define its multiplicative
valuation $\mathfrak{v}_{u}(Z)\in{\mathbb{R}}$ by putting

$\mathfrak{v}_{u}(y_{i})=0,\quad\mathfrak{v}_{u}(z_{i})=\mathfrak{v}_{u}(\overline{z}_{i})=\ell_{i}(u),\quad\mathfrak{v}_{u}(T^{\alpha})=\alpha.$

We also define a (multiplicative) grading $\rho(Z)\in{\mathbb{Z}}^{n}$ by

$\rho(y_{i})=\text{\bf
e}_{i},\quad\rho(z_{i})=\rho(\overline{z}_{i})=v_{i},\quad\rho(T^{\alpha})=0.$

and by $\rho(ZZ^{\prime})=\rho(Z)+\rho(Z^{\prime})$. We remark that
$\mathfrak{v}_{u}$ and $\rho$ are consistent with (6.1). We next observe that
both $\mathfrak{v}_{u}$ and $\rho$ are preserved by $\widehat{\psi}_{u}$,
$\widehat{\phi}_{u}$ and by (6.23). This implies (6.25).

Now we use Lemma 6.10 and (6.25) to conclude

$z_{h}-T^{e(h;u)}\prod_{j=1}^{n}z_{j}^{m_{j}}\in SR_{\omega}(X).$

The proof of Proposition 6.9 is now complete. ∎

Next we prove

######  Lemma 6.13.

Let $P(X)$ be the linear relation ideal defined in Definition 6.4. Then

$\widehat{\psi}_{u}(P(X))=\left(\frac{\partial\mathfrak{PO}_{0}^{u}}{\partial
y_{i}};i=1,\cdots,n\right).$

###### Proof.

Let $\sum_{i=1}^{m}v_{i,j}z_{i}$ be in $P(X)$. Then we have

$\widehat{\psi}_{u}\left(\sum_{i=1}^{m}v_{i,j}z_{i}\right)=\sum_{i=1}^{m}v_{i,j}\overline{z}_{i}=\sum_{i=1}^{m}y_{j}\frac{\partial\overline{z}_{i}}{\partial
y_{j}}=y_{j}\frac{\partial\mathfrak{PO}_{0}^{u}}{\partial y_{j}}$

by (6.1) and (6.4). Since $y_{j}$’s are invertible in
$\Lambda[y_{1},\cdots,y_{n},y_{1}^{-1},\cdots,y^{-1}_{n}]$, this identity
implies the lemma. ∎

The proof of Theorem 6.1 and of Proposition 6.8 is now complete. ∎

######  Remark 6.14.

Proposition 6.8 holds over $\Lambda^{R}$ coefficient for arbitrary commutative
ring $R$ with unit. The proof is the same.

We define

$\psi_{u^{\prime},u}:Jac(\mathfrak{PO}_{0}^{u})\to
Jac(\mathfrak{PO}_{0}^{u^{\prime}})$

by

$\psi_{u^{\prime},u}(\overline{z}_{i}(u))=\overline{z}_{i}(u^{\prime})=T^{\ell_{i}(u^{\prime})-\ell_{i}(u)}\overline{z}_{i}(u).$
(6.26)

It is an isomorphism. We have

$\psi_{u^{\prime},u}\circ{\psi}_{u}={\psi}_{u^{\prime}}.$

The well-definedness $\psi_{u^{\prime},u}$ is proved from this formula or by
checking directly.

In case no confusion can occur, we identify $Jac(\mathfrak{PO}_{0}^{u})$,
$Jac(\mathfrak{PO}_{0}^{u^{\prime}})$ by $\psi_{u^{\prime},u}$ and denote them
by $Jac(\mathfrak{PO}_{0})$. Since
$\psi_{u^{\prime},u}(\overline{z}_{i}(u))=\overline{z}_{i}(u^{\prime})$ we
write them $\overline{z}_{i}$ when we regard it as an element of
$Jac(\mathfrak{PO}_{0})$. Note $\psi_{u^{\prime},u}(y_{i})\neq y_{i}$. In case
we regard $y_{i}\in Jac(\mathfrak{PO}_{0}^{u})$ as an element of
$Jac(\mathfrak{PO}_{0})$ we write it as $y_{i}(u):=\psi_{u}(y_{i})$.

######  Remark 6.15.

The above proof of Theorem 6.1 uses Batyrev’s presentation of quantum
cohomology ring and is not likely generalized beyond the case of compact toric
manifolds. (In fact the proof is purely algebraic and do not contain serious
study of pseudo-holomorphic curve, except Proposition 6.7, which we quote
without proof and Theorem 4.5, which is a minor improvement of an earlier
result of [CO].) There is an alternative way of constructing the ring
homomorphism $\psi_{u}$ which is less computational. (This will give a new
proof of Proposition 6.7.) We will give this conceptual proof in a sequel to
this paper.

We use the operations

$\mathfrak{q}_{1,k;\beta}:H(X;{\mathbb{Q}})[2]\otimes
B_{k}H(L(u);{\mathbb{Q}})[1]\to H(L(u);{\mathbb{Q}})[1]$