# Ruan’s Conjecture on Singular symplectic flops

Bohui Chen  Department of Mathematics, Sichuan University, Chengdu,610064,
China<EMAIL_ADDRESS>,  An-Min Li  Department of Mathematics, Sichuan
University, Chengdu,610064, China<EMAIL_ADDRESS>and  Guosong Zhao
Department of Mathematics, Sichuan University, Chengdu,610064, China
<EMAIL_ADDRESS>
###### Abstract.

We prove that the orbifold quantum ring is preserved under singular symplectic
flops. Hence we verify Ruan’s conjecture for this case.

B.C. and A.L. are supported by NSFC, G.Z. is supported by a grant of NSFC and
Qiushi Funding.

###### Contents

  1. 1 Introduction
  2. 2 Relative orbifold Gromov-Witten theory and the degeneration formula
    1. 2.1 The Chen-Ruan Orbifold Cohomologies
    2. 2.2 Orbifold Gromov-Witten invariants
    3. 2.3 Ring structures on $H^{\ast}_{CR}(X)$.
    4. 2.4 Moduli spaces of relative stable maps for orbifold pairs
    5. 2.5 Relative orbifold Gromov-Witten invariants.
    6. 2.6 The degeneration formula
  3. 3 Singular symplectic flops
    1. 3.1 Local models and local flops
    2. 3.2 Torus action.
    3. 3.3 Symplectic orbi-conifolds and singular symplectic flops
    4. 3.4 Ruan cohomology rings
  4. 4 Relative Gromov-Witten theory on $M^{s}_{r}$ and $M^{sf}_{r}$
    1. 4.1 Local models $M^{s}_{r}$ and $M^{sf}_{r}$
    2. 4.2 Relative Moduli spaces for the pair $(M^{s}_{r},Z)$
    3. 4.3 Admissible data $(\Gamma,\mathcal{T},\mathbf{a})$
  5. 5 Vanishing results on relative invariants
    1. 5.1 Localization via the torus action
    2. 5.2 Vanishing results on $I^{\Omega}_{F_{T}}(\lambda,u)$, (I)
    3. 5.3 Vanishing results on $I^{\Omega}_{F_{T}}(\lambda,u)$, (II)
  6. 6 Proof of the Main theorem
    1. 6.1 Reducing the comparison to local models
    2. 6.2 Proof of Proposition 6.3

##  1\. Introduction

One of deep discovery in Gromov-Witten theory is its intimate relation with
the birational geometry. A famous conjecture of Ruan asserts that any two
$K$-equivalent manifolds have isomorphic quantum cohomology rings ([R1]) (see
also [Wang]). Ruan’s conjecture was proved by Li-Ruan for smooth algebraic
3-folds ([LR]) almost ten years ago. Only recently, it was generalized to
simple flops and Mukai flops in arbitrary dimensions by Lee-Lin-Wang ([LLW]).
In a slightly different context, there has been a lot of activities regarding
Ruan’s conjecture in the case of McKay correspondence.

On the other hand, it is well known that the appropriate category to study the
birational geometry is not smooth manifolds. Instead, one should consider the
singular manifolds with terminal singularities. In the complex dimension
three, the terminal singularities are the finite quotients of hypersurface
singularities and hence the deformation of them are orbifolds. It therefore
raises the important questions if Ruan’s conjecture still holds for the
orbifolds where there are several very interesting classes of flops. This is
the main topic of the current article.

Li-Ruan’s proof of the case of smooth 3-folds consists of two steps. The first
step is to interpret flops in the symplectic category, then, they use almost
complex deformation to reduce the problem to the simple flop; the second step
is to calculate the change of quantum cohomology under the simple flop. The
description of a smooth simple flop is closely related to the conifold
singularity

$W_{1}=\\{(x,y,z,t)|xy-z^{2}+t^{2}=0\\}.$

In [CLZZ], we initiate a program to understand the flop associated with the
singularities

$W_{r}=\\{(x,y,z,t)|xy-z^{2r}+t^{2}=0\\}/\mu_{r}(a,-a,1,0).$

$W_{r}$ appears in the list of terminal singularities in [K]. The
singularities without quotient are also studied in [La] and [BKL].

The program is along the same framework of that in [LR]. The first step is to
describe the flops with respect to $W_{r}$ symplectically. This is done in the
previous paper([CLZZ]). Our main theorem in this paper is

######  Theorem 1.1.

Suppose that $Y^{s}$ is a symplectic 3-fold with orbifold singularities of
type $W_{r_{1}},\ldots,W_{r_{n}}$ and $Y^{sf}$ is its singular flop, then

$QH_{CR}(Y^{s})=QH_{CR}(Y^{sf}).$

Theorem 1.1 verifies Ruan’s conjecture in this particular case. We should
mention that Ruan also proposed a simplified version of the above conjecture
in terms of Ruan cohomology $RH_{CR}$ which has been established in [CLZZ] as
well. Furthermore, our previous results enters the proof of this general
conjecture in a crucial way.

The technique of the proof is a combination of the degeneration formula of
orbifold Gromov-Witten invariants, the localization techniques and dimension
counting arguments. The theory of relative orbifold Gromov-Witten invariants
and its degeneration formula involves heavy duty analysis on moduli spaces and
will appear elsewhere ([CLS]).

The paper is organized as following. We first describe the relative orbifold
GW-invariants and state the degeneration formula (without proof)(§2). Then, we
summary the result of [CLZZ] on the singular symplectic flops and Ruan
cohomology (§3). The heart of the proof is a detail analysis of relative
orbifold GW-theory on local models (§4 and §5). The main theorem is proved in
§6.

Acknowledge. We would like to thank Yongbin Ruan for suggesting the problem
and for many valuable discussions. We also wish to thank Qi Zhang for many
discussions.

##  2\. Relative orbifold Gromov-Witten theory and the degeneration formula

###  2.1. The Chen-Ruan Orbifold Cohomologies

Let $X$ be an orbifold. For $x\in X$, if its small neighborhood $U_{x}$ is
given by a uniformization system $(\tilde{U},G,\pi)$, we say $G$ is the
isotropy group of $x$ and denoted by $G_{x}$. Let

$\mathcal{T}=\left(\bigcup_{x\in X}G_{x}\right)/\sim.$

Here $\sim$ is certain equivalence relation. For each $(g)\in\mathcal{T}$, it
defines a twisted sector $X_{(g)}$. At the mean while, the twisted sector is
associated with a degree-shifting number $\iota(g)$. The Chen-Ruan orbifold
cohomology is defined to be

$H^{\ast}_{CR}(X)=H^{\ast}(X)\oplus\bigoplus_{(g)\in\mathcal{T}}H^{\ast-2\iota(g)}(X_{(g)}).$

For details, readers are referred to [CR1].

###  2.2. Orbifold Gromov-Witten invariants

Let $\overline{\mathcal{M}}_{g,n,A}(X)$ be the moduli space of representable
orbifold morphism of genus $g,n$-marked points and $A\in H_{2}(X,\mathbb{Z})$
(cf. [CR2],[CR3]). By specifying the monodromy

$\mathbf{h}=((h_{1}),\ldots,(h_{n}))$

at each marked points, we can decompose

$\overline{\mathcal{M}}_{g,n,A}(X)=\bigsqcup_{\mathbf{h}}\overline{\mathcal{M}}_{g,n,A}(X,\mathbf{h}).$

Let

$ev_{i}:\overline{\mathcal{M}}_{g,n,A}(X,\mathbf{h})\to X_{(h_{i})},1\leq
i\leq n$

be the evaluation maps. The primary orbifold Gromov-Witten invariants are
defined as

$\langle\alpha_{1},\ldots,\alpha_{n}\rangle_{g,A}^{X}=\int^{virt}_{\overline{\mathcal{M}}_{g,n,A}(X,\mathbf{h})}\prod_{i=1}^{m}ev_{i}^{\ast}(\alpha_{i}),$

where $\alpha_{i}\in H^{\ast}(X_{(h_{i})})$.

In particular, set

$\displaystyle\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{CR}=\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{0,0},$
$\displaystyle\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle=\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{CR}+\sum_{A\not=0}\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{0,A}.$

###  2.3. Ring structures on $H^{\ast}_{CR}(X)$.

Let $V$ be a vector space over $R$. Let

$h:V\otimes V\to R$

be a non-degenerate pairing and

$A:V\otimes V\otimes V\to R$

be a triple form. Then it is well known that one can define a product $\ast$
on $V$ by

$h(u\ast v,w)=A(u,v,w).$

Different $A$’s give different products.

######  Remark 2.1.

Suppose we have $(V,h,A)$ and $(V^{\prime},h^{\prime},A^{\prime})$. A map
$\phi:V\to V^{\prime}$ induces an isomorphism (with respect to the product) if
$\phi$ is a group isomorphism and

$\phi^{\ast}h^{\prime}=h,\;\;\;\mbox{and}\;\;\;\phi^{\ast}A^{\prime}=A.$

Now let $V=H^{\ast}_{CR}(X)$ and $h$ be the Poincare pairing on $V$. If

$A(\alpha_{1},\alpha_{2},\alpha_{3})=\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{CR},$

it defines the Chen-Ruan product. If

$A(\alpha_{1},\alpha_{2},\alpha_{3})=\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle,$

it defines the Chen-Ruan quantum product. We denote the ring to be
$QH^{\ast}_{CR}(X)$.

###  2.4. Moduli spaces of relative stable maps for orbifold pairs

For the relative stable maps for the smooth case , there are two equivariant
versions. One is on the symplectic manifolds with respect to cylinder ends,
each of which admits a Hamiltonian $S^{1}$ action ([LR]), the other is on the
closed symplectic manifolds with respect to divisors([LR],[Li]). This is also
true for orbifolds. We adapt the second version here.

Let $X$ be a symplectic orbifold with disjoint divisors

$\\{Z_{1},\ldots,Z_{k}\\}.$

For simplicity, we assume $k=1$ and $Z=Z_{1}$.

By a relative stable map in $(X,Z)$, we mean a stable map
$f\in\mathcal{M}_{g,n,A}(X)$ with additional data that record how it
intersects with $Z$. Be precisely, suppose

$f:(\Sigma_{g},\mathbf{z})\to X.$

The additional data are collected in order:

  * •

Set

$\mathbf{x}=f^{-1}(Z)=\\{x_{1},\ldots,x_{k}\\}.$

We call $x_{i}$ the relative marked points. The rest of marked points are
denoted by

$\mathbf{p}=\\{p_{1},\ldots,p_{m}\\},$

i.e, $\mathbf{z}=\mathbf{x}\cup\mathbf{p}$;

  * •

Let

$\mathbf{g}=((g_{1}),\ldots,(g_{k}))$

denote the monodromy of $f$ (with respect to $Z$) at each point in
$\mathbf{x}$. The rest are denoted by

$\mathbf{h}=((h_{1}),\ldots,(h_{m}))$

which are the monodromy of $f$ (with respect to $X$) at each point in
$\mathbf{p}$.

  * •

the multiplicity of the tangency $\ell_{j}$ of $f$ with $Z$ at
$z_{j}=f(x_{j})$ is defined by the following: Locally, the neighborhood of
$z_{j}$ is given by

$(\tilde{V}\times\mathbb{C}\to\tilde{V})/G_{z_{j}},$

where $\tilde{V}/G_{z_{j}}\subset Z$. Suppose the lift of $f$ is

$\displaystyle\tilde{f}:\tilde{\mathbb{D}}\to\tilde{V}\times\mathbb{C}$
$\displaystyle\tilde{f}(t)=(v(t),u(t))$

Suppose the multiplicity of $u$ is $\alpha$ and $g_{j}\in G_{z_{j}}$ acts on
the fiber over $z_{j}$ with multiplicity $c$. Then the multiplicity is set to
be

$\ell_{j}=\frac{\alpha\cdot c}{|g_{j}|}.$

We say $f$ maps $x_{j}$ to $Z_{g_{j}}$ at $\ell_{j}z_{j}$. Set

$\mathbf{l}=(\ell_{1},\ldots,\ell_{k}).$

We may write

$f^{-1}(Z)=\mathbf{l}\cdot\mathbf{x}=\sum_{j=1}^{k}\ell_{j}x_{j}.$

As a relative object, we say $f$ is in the moduli space of relative map

$\mathcal{M}_{g,n,A}(X,Z,\mathbf{h},\mathbf{g},\mathbf{l}).$

We denote the map by

$f:(\Sigma,\mathbf{p},\mathbf{l}\cdot\mathbf{x})\to(X,Z).$

We now describe the compactification of this moduli space. The construction is
similar to the smooth case([LR]).

The target space of a stable relative map is no longer $X$. Instead, it is
extended in the following sense: let $L\to Z$ be the normal bundle of $Z$ in
$X$ and

$PZ=\mathbb{P}(L\oplus\mathbb{C})$

be its projectification, then given an integer $b\geq 0$, we have an extended
target space

$X^{\sharp}_{b}:=X\cup\bigcup_{1\leq\alpha\leq b}PZ^{\alpha}.$

Here $PZ^{\alpha}$ denotes the $\alpha$-th copy of $PZ$. Let $Z^{\alpha}_{0}$
be the 0-section and $Z^{\alpha}_{\infty}$ be the $\infty$-section of
$PZ^{\alpha}$. $X$ is called the root component of $X^{\sharp}_{b}$.
$Z_{0}^{b}$ is called the divisor of $X^{\sharp}_{b}$ and is (again) denoted
by $Z$.

######  Definition 2.1.

A relative map in $X^{\sharp}_{b}$ consists of following data: on each
component, there is a relative map: on the root component, the map is denoted
by

$f^{0}:(\Sigma_{0},\mathbf{p}^{0},\mathbf{l}^{0}\cdot\mathbf{x}^{0})\to(X,Z);$

and on each component $PZ^{\alpha}$, the map is denoted by

$f^{\alpha}:(\Sigma^{\alpha},\mathbf{p}^{\alpha},\mathbf{l}^{\alpha}\cdot\mathbf{x}^{\alpha}\cup\bar{\mathbf{l}}^{\alpha}\cdot\bar{\mathbf{x}}^{\alpha})\to(PZ^{\alpha},Z_{0}^{\alpha}\cup
Z_{\infty}^{\alpha}).$

Here $\mathbf{l}^{\alpha}\cdot\mathbf{x}^{\alpha}=f^{-1}(Z_{0}^{\alpha})$ and
$\bar{\mathbf{l}}^{\alpha}\cdot\bar{\mathbf{x}}^{\alpha}=f^{-1}(Z_{\infty}^{\alpha}).$
Moreover, we require $f^{\alpha}$ at $Z_{0}^{\alpha}$ matches $f^{\alpha+1}$
at $Z_{\infty}^{\alpha+1}$. (see Remark 2.2.)

We denote such a map by

$\mathbf{f}=(f^{0},f^{1},\ldots,f^{b}).$

Set $\mathbf{x}=\mathbf{x}^{b}$ and

$\displaystyle
g_{j}=g_{x^{b}_{j}},\;\;\;\mathbf{g}=(g_{1},\ldots,g_{|\mathbf{x}|})$
$\displaystyle\ell_{j}=\ell_{j}^{b},\;\;\;\mathbf{l}=(\ell_{1},\ldots,\ell_{|\mathbf{x}|}).$

We say that $\mathbf{f}$ maps $x_{j}$ to the divisor $Z$ of $X^{\sharp}_{b}$
at $\ell_{j}z_{j}\in Z_{(g_{j})}$. Similarly, $\mathbf{h}$ records the twisted
sector for

$\mathbf{p}=\mathbf{p}^{0}\cup\bigcup_{\alpha}\mathbf{p}^{\alpha}.$

The homology class $A$ in $X$ represented by $\mathbf{f}$ can be defined
properly. Collect the data

$\Gamma=(g,A,\mathbf{h},\mathbf{g},\mathbf{l}),\;\;\;\mathcal{T}=(\mathbf{g},\mathbf{l}).$

We say that $f$ is a relative orbifold map in $X^{\sharp}_{b}$ of type
$(\Gamma,\mathcal{T})$. Denote the moduli space by
$\tilde{\mathcal{M}}_{\Gamma,\mathcal{T}}(X^{\sharp}_{b},Z)$.

######  Remark 2.2.

Let $f^{\alpha}$ and $f^{\alpha+1}$ be as in the definition. Suppose that

  * •

$f^{\alpha}$ maps $x^{\alpha}_{j}\in\mathbf{x}^{\alpha}$ to
$(Z_{0}^{\alpha})_{(g^{\alpha}_{j})}$ at $\ell^{\alpha}_{k}z^{\alpha}_{j}$;

  * •

$f^{\alpha+1}$ maps $\bar{x}^{\alpha+1}_{i}\in\bar{\mathbf{x}}^{\alpha+1}$ to
$(Z_{\infty}^{\alpha+1})_{(\bar{g}^{\alpha+1}_{i})}$ at
$\bar{\ell}^{\alpha+1}_{i}\bar{z}^{\alpha+1}_{i}$,

then by saying that $f^{\alpha}$ at $Z_{0}^{\alpha}$ matches $f^{\alpha+1}$ at
$Z_{\infty}^{\alpha+1}$ we mean that

$\ell^{\alpha}_{i}=\bar{\ell}^{\alpha+1}_{i},\;\;\;z^{\alpha}_{i}=\bar{z}^{\alpha+1}_{i},\;\;\;\mbox{and}\;\;\;g^{\alpha}_{i}=\bar{g}^{\alpha+1}_{i}.$

Note that there is a $\mathbb{C}^{\ast}$ action on $PZ^{\alpha}$. Let $T$ be
the product of these $b$ copies of $\mathbb{C}^{\ast}$. Then $T$ acts on
$\tilde{\mathcal{M}}_{\Gamma,\mathcal{T}}(X^{\sharp}_{b},Z)$. Define

$\mathcal{M}_{\Gamma,\mathcal{T}}(X^{\sharp}_{b},Z)=\tilde{\mathcal{M}}_{\Gamma,\mathcal{T}}(X^{\sharp}_{b},Z)/T.$

It is standard to show that

######  Proposition 2.3.

There exists a large integer $B$ which depends on topological data
$(\Gamma,\mathcal{T})$ such that
$\mathcal{M}_{\Gamma,\mathcal{T}}(X^{\sharp}_{b},Z)$ is empty when $b\geq B$.

Hence,

######  Definition 2.2.

The compactified moduli space is

$\overline{\mathcal{M}}_{\Gamma,\mathcal{T}}(X,Z)=\bigcup_{b\in\mathbb{Z}^{\geq
0}}\mathcal{M}_{\Gamma,\mathcal{T}}(X^{\sharp}_{b},Z).$

The following technique theorem is proved in [CLS]

######  Theorem 2.4.

$\overline{\mathcal{M}}_{\Gamma,\mathcal{T}}(X,Z)$ is a smooth compact virtual
orbifold without boundary with virtual dimension

$2c_{1}(A)+2(\dim_{\mathbb{C}}X-3)(1-g)+2\sum_{i=1}^{m}(1-\iota(h_{i}))+2\sum_{j=1}^{n}(1-\iota(g_{j})-[\ell_{j}]),$

where $[\ell_{j}]$ is the largest integer that is less or equal to $\ell_{j}$.

###  2.5. Relative orbifold Gromov-Witten invariants.

Let $\overline{\mathcal{M}}_{\Gamma,\mathcal{T}}(X,Z)$ be the moduli space
given above. There are evaluation maps

$ev_{i}^{X}:\overline{\mathcal{M}}_{\Gamma,\mathcal{T}}(X,Z)\to
X_{(h_{i})},\;\;\;ev_{i}^{X}(f)=f(p_{i}),1\leq i\leq m;$

and

$ev_{j}^{Z}:\overline{\mathcal{M}}_{\Gamma,\mathcal{T}}(X,Z)\to
Z_{(g_{j})},\;\;\;ev_{j}^{Z}(f)=f(x_{j}),1\leq j\leq k.$

Then for

$\alpha_{i}\in H^{\ast}(X_{(h_{i})}),1\leq i\leq m,\;\;\;\beta_{j}\in
H^{\ast}(Z_{(g_{j})}),1\leq j\leq k$

the relative invariant is defined as

$\displaystyle\langle\alpha_{1},\ldots,\alpha_{m}|\beta_{1},\ldots,\beta_{k},\mathcal{T}\rangle_{\Gamma}^{(X,Z)}$
$\displaystyle\;\;\;\;=\frac{1}{|Aut(\mathcal{T})|}\int_{\overline{\mathcal{M}}_{\Gamma,\mathcal{T}}(X,Z)}^{vir}\prod_{i=1}^{m}(ev_{i}^{X})^{\ast}\alpha_{i}\prod_{j=1}^{k}(ev_{j}^{Z})^{\ast}\beta_{j}.$

In this paper, we usually set

$\mathbf{a}=(\alpha_{1},\ldots,\alpha_{m}),\;\;\;\mathbf{b}=(\beta_{1},\ldots,\beta_{k}),$

then the invariant is denoted by
$\langle\mathbf{a}|\mathbf{b},\mathcal{T}\rangle_{\Gamma}^{(X,Z)}$.

Moreover, if $\Gamma=\coprod_{\gamma}\Gamma^{\gamma}$, the relative invariants
(with disconnected domain curves) is defined to be the product of each
connected component

$\langle\mathbf{a}|\mathbf{b},\mathcal{T}\rangle_{\Gamma}^{\bullet(X,Z)}=\prod_{\gamma}\langle\mathbf{a}|\mathbf{b},\mathcal{T}\rangle_{\Gamma^{\gamma}}^{(X,Z)}.$

###  2.6. The degeneration formula

The symplectic cutting also holds for orbifolds. Let $X$ be a symplectic
orbifold. Suppose that there is a local $S^{1}$ Hamiltonian action on
$U\subset X$. We assume that

$U\cong Y\times(-1,1)$

and the projection onto the second factor

$\pi_{2}:U\to(-1,1)$

gives the Hamiltonian function. $Y\times\\{0\\}$ splits $X$ into two orbifolds
with boundary $Y$, denoted by $X^{\pm}$. Then the routine symplectic cutting
gives the degeneration

$\pi:X\to\bar{X}^{+}\cup_{Z}\bar{X}^{-}.$

Topologically, $\bar{X}^{\pm}$ is obtained by collapsing the $S^{1}$-orbits of
the boundaries of $X^{\pm}$.

There are maps

$\pi_{\ast}:H_{2}(X)\to
H_{2}(X^{+}\cup_{Z}X^{-}),\;\;\;\pi^{\ast}:H^{\ast}(X^{+}\cup_{Z}X^{-})\to
H^{\ast}(X).$

For $A\in H_{2}(M)$ we set $[A]\subset H_{2}(X)$ to be
$\pi_{\ast}^{-1}(\pi_{\ast}(A))$ and denote $\pi_{\ast}(A)$ by
$(A^{+},A^{-})$. On the other hand, for $\alpha^{\pm}\in H^{\ast}(X^{\pm})$
with $\alpha^{+}|_{Z}=\alpha^{-}|_{Z}$, it defines a class on
$H^{\ast}(X^{+}\cup_{Z}X^{-})$ which is denoted by $(\alpha^{+},\alpha^{-})$.
Let $\alpha=\pi^{\ast}(\alpha^{+},\alpha^{-})$.

######  Theorem 2.5.

Suppose $\pi:X\to X^{+}\cup_{Z}X^{-}$ is the degeneration. Then

(2.1)
$\langle\mathbf{a}\rangle_{\Gamma}^{X}=\sum_{I}\sum_{\eta=(\Gamma^{+},\Gamma^{-},I_{\rho})}C_{\eta}\langle\mathbf{a}^{+}|\mathbf{b}^{I},\mathcal{T}\rangle^{\bullet(X^{+},Z)}_{\Gamma^{+}}\langle\mathbf{a}^{-}|\mathbf{b}_{I},\mathcal{T}\rangle^{\bullet(X^{-},Z)}_{\Gamma^{-}}.$

Notations in the formula are explained in order. $\Gamma$ is a data for
Gromov-Witten invariants, it includes $(g,[A])$;
$(\Gamma^{+},\Gamma^{-},I_{\rho})$ is an admissible triple which consists of
(possible disconnected) topological types $\Gamma^{\pm}$ with the same
relative data $\mathcal{T}$ under the identification $I_{\rho}$ and they glue
back to $\Gamma$. (For instance, one may refer to [HLR] who interpret
$\Gamma$’s as graphs and then the gluing has an obvious geometric meaning);
the relative classes $\beta^{i}\in\mathbf{b}^{I}$ runs over a basis of
$Z_{(g_{i})}$ and at the mean while $\beta_{i}$ runs over the dual basis;
finally

$C_{\eta}=|Aut(\mathcal{T})|\prod_{i=1}^{k}\ell_{i}$

for $\mathcal{T}=(\mathbf{g},\mathbf{l})$.

##  3\. Singular symplectic flops

###  3.1. Local models and local flops

Locally, we are concern those resolutions of

$\tilde{W}_{r}=\\{(x,y,z,t)|xy-z^{2r}+t^{2}=0\\}$

and their quotients. $\tilde{W}_{r}-\\{0\\}$ inherits a symplectic form
$\tilde{\omega}_{r}^{\circ}$ from $\mathbb{C}^{4}$.

By blow-ups, we have two small resolutions of $\tilde{W}_{r}$:

$\displaystyle\tilde{W}^{s}_{r}$ $\displaystyle=$
$\displaystyle\\{((x,y,z,t),[p,q])\in\mathbb{C}^{4}\times\mathbb{P}^{1}$
$\displaystyle|xy-z^{2r}+t^{2}=0,\;\;\frac{p}{q}=\frac{x}{z^{r}-t}=\frac{z^{r}+t}{y}\\}$
$\displaystyle\tilde{W}^{sf}_{r}$ $\displaystyle=$
$\displaystyle\\{((x,y,z,t),[p,q])\in\mathbb{C}^{4}\times\mathbb{P}^{1}$
$\displaystyle|xy-z^{2r}+t^{2}=0,\;\;\frac{p}{q}=\frac{x}{z^{r}+t}=\frac{z^{r}-t}{y}\\}.$

Let

$\tilde{\pi}_{r}^{s}:\tilde{W}^{s}_{r}\to\tilde{W}_{r},\;\;\;\tilde{\pi}_{r}^{sf}:\tilde{W}^{sf}_{r}\to\tilde{W}_{r}$

be the projections. The exceptional curves $(\tilde{\pi}_{r}^{s})^{-1}(0)$ and
$(\tilde{\pi}_{r}^{sf})^{-1}(0)$ are denoted by $\tilde{\Gamma}^{s}_{r}$ and
$\tilde{\Gamma}^{sf}_{r}$ respectively. Both of them are isomorphic to
$\mathbb{P}^{1}$.

Let

$\mu_{r}=\langle\xi\rangle,\xi=e^{\frac{2\pi i}{r}}$

be the cyclic group of $r$-th roots of 1. We denote its action on
$\mathbb{C}^{4}$ by $\mu_{r}(a,b,c,d)$ if the action is given by

$\xi\cdot(x,y,z,t)=(\xi^{a}x,\xi^{b}y,\xi^{c}z,\xi^{d}t).$

Then $\mu_{r}(a,-a,1,0)$ acts on $\tilde{W}_{r}$, and naturally extending to
its small resolutions. Set

$W_{r}=\tilde{W}_{r}/\mu_{r},\;\;\;W^{s}_{r}=\tilde{W}^{s}_{r}/\mu_{r},\;\;\;W^{sf}_{r}=\tilde{W}^{sf}_{r}/\mu_{r}.$

Similarly,

$\Gamma^{s}_{r}=\tilde{\Gamma}^{s}_{r}/\mu_{r}\;\;\;\Gamma^{sf}_{r}=\tilde{\Gamma}^{sf}_{r}/\mu_{r}.$

We call that $W^{s}$ and $W^{sf}$ are the small resolutions of $W_{r}$. We say
that $W^{sf}$ is the flop of $W^{s}$ and vice versa. They are both orbifolds
with singular points on $\Gamma^{s}$ and $\Gamma^{sf}$. Note that the
symplectic form $\tilde{\omega}_{r}^{\circ}$ reduces to a symplectic form
$\omega_{r}^{\circ}$ on $W_{r}$.

It is known that

######  Proposition 3.1.

For $r\geq 2$, the normal bundle of $\tilde{\Gamma}^{s}_{r}$
($\tilde{\Gamma}^{sf}_{r}$) in $\tilde{W}^{s}_{r}$ ($\tilde{W}^{sf}_{r}$) is
$\mathcal{O}\oplus\mathcal{O}(-2)$.

Proof.  We take $\tilde{W}_{r}^{s}$ as an example. For the set

$\Lambda_{p}=\\{q\not=0\\},$

set $u=p/q$. Then $(u,z,y)$ gives a coordinate chart for $\Lambda_{p}$.
Similarly, for the set

$\Lambda_{q}=\\{p\not=0\\},$

set $v=q/p$. Then $(v,z,x)$ gives a coordinate chart for $\Lambda_{q}$. The
transition map is given by

(3.1) $\left\\{\begin{array}[]{l}v=u^{-1};\\\ z=z;\\\
x=-u^{2}y+2uz^{r}.\end{array}\right.$

By linearize this equation, it is easy to get the conclusion. q.e.d.

######  Corollary 3.2.

For $r\geq 2$, the normal bundle of $\Gamma^{s}_{r}$ ($\Gamma^{sf}_{r}$) in
$W^{s}_{r}$ ($W^{sf}_{r}$) is $(\mathcal{O}\oplus\mathcal{O}(-2))/\mu_{r}$.

On $\tilde{\Gamma}^{s}_{r}$ ($\tilde{\Gamma}^{sf}_{r}$), there are two special
points. In term of $[p,q]$ coordinates, they are

$0=[0,1];\;\;\infty=[1,0].$

We denote them by $\mathfrak{p}^{s}$ and $\mathfrak{q}^{s}$
($\mathfrak{p}^{sf}$ and $\mathfrak{q}^{sf}$) respectively. After taking
quotients, they become singular points. By the proof of Proposition 3.1, the
uniformization system of $\mathfrak{p}^{s}$ is

$\\{(p,x,y,z,t)|x=t=0\\}$

with $\mu_{r}$ action given by

$\xi(p,y,z)=(\xi^{a}p,\xi^{-a}y,\xi z).$

At $\mathfrak{p}^{s}$, for each given $\xi^{k}=\exp(2\pi ik/r),1\leq k\leq r$,
there is a corresponding twisted sector([CR1]). As a set, it is same as
$\mathfrak{p}^{s}$. We denote this twisted sector by $[\mathfrak{p}^{s}]_{k}$.
For each twisted sector, a degree shifting number is assigned. We conclude
that

######  Lemma 3.3.

For $\xi^{k}=\exp(2\pi ik/r),1\leq k\leq r$, the degree shifting

$\iota([\mathfrak{p}^{s}]_{k})=1+\frac{k}{r}.$

Proof. This follows directly from the definition of degree shifting. q.e.d.

Similar results hold for the singular point $\mathfrak{q}^{s}$. Hence we also
have twisted sector $[\mathfrak{q}^{s}]_{k}$ and

$\iota([\mathfrak{q}^{s}]_{k})=1+\frac{k}{r}.$

Similarly, on $W^{sf}$, there are twisted sectors
$[\mathfrak{p}^{sf}]_{k},[\mathfrak{q}^{sf}]_{k}$ and

$\iota([\mathfrak{p}^{sf}]_{k})=\iota([\mathfrak{q}^{sf}]_{k})=1+\frac{k}{r}.$

###  3.2. Torus action.

We introduce a $T^{2}$-action on $\tilde{W}_{r}$:

$(t_{1},t_{2})(x,y,z,t)=(t_{1}t_{2}^{r}x,t_{1}^{-1}t_{2}^{r}y,t_{2}z,t_{2}^{r}t).$

For an action $t_{1}^{a}t_{2}^{b}\cdot$, we write the weight of action by
$a\lambda+bu$. This action naturally extends to the actions on all models
generated from $\tilde{W}_{r}$, such as $W_{r},W^{s}_{r}$ and $W_{r}^{sf}$.

It then induces an action on $\tilde{\Gamma}^{s}_{r}$
($\tilde{\Gamma}^{sf}_{r}$):

$(t_{1},t_{2})[p,q]=[t_{1}p,q].$

Recall that the normal bundle of $\tilde{\Gamma}^{s}_{r}$ in
$\tilde{W}^{s}_{r}$ is $\mathcal{O}\oplus\mathcal{O}(-2)$.

######  Lemma 3.4.

The action weights at $\mathcal{O}_{p}$ and $\mathcal{O}_{q}$ are $u$. The
action weights at $\mathcal{O}_{p}(-2)$ and $\mathcal{O}_{q}(-2)$ are
$-\lambda+ru$ and $\lambda+ru$.

Proof. This follows directly from the model given by §3.1. q.e.d.

it is easy to verify that $\mathfrak{p}^{s},\mathfrak{q}^{s}$
($\mathfrak{p}^{sf},\mathfrak{q}^{sf}$) are fixed points of the action.

On the other hand, there are four special lines connecting to these points
that are invariant with respect to the action. Let us look at
$\tilde{W}^{s}_{r}$. For the point $\mathfrak{p}^{s}$, two lines are in
$\Lambda_{p}$ and are given by

$\displaystyle\tilde{L}^{s}_{p,y}$ $\displaystyle=$
$\displaystyle\\{x=z=t=0,u=0\\},$ $\displaystyle\tilde{L}^{s}_{p,z}$
$\displaystyle=$ $\displaystyle\\{x=y=0,z^{r}+t=0,u=0\\}.$

To the point $\mathfrak{q}^{s}$, two lines are in $\Lambda_{q}$ and are given
by

$\displaystyle\tilde{L}^{s}_{q,x}$ $\displaystyle=$
$\displaystyle\\{y=z=t=0,v=0\\},$ $\displaystyle\tilde{L}^{s}_{q,z}$
$\displaystyle=$ $\displaystyle\\{x=y=0,z^{r}-t=0,v=0\\}.$

Similarly, for $\tilde{W}^{sf}_{r}$ we have

$\displaystyle\tilde{L}^{sf}_{p,y}$ $\displaystyle=$
$\displaystyle\\{x=z=t=0,u=0\\},$ $\displaystyle\tilde{L}^{sf}_{p,z}$
$\displaystyle=$ $\displaystyle\\{x=y=0,z^{r}-t=0,u=0\\},$
$\displaystyle\tilde{L}^{sf}_{q,x}$ $\displaystyle=$
$\displaystyle\\{y=z=t=0,v=0\\},$ $\displaystyle\tilde{L}^{sf}_{q,z}$
$\displaystyle=$ $\displaystyle\\{x=y=0,z^{r}+t=0,v=0\\}.$

Correspondingly, these lines in $W^{s}_{r}$ and $W^{sf}_{r}$ are denoted by
the same notations without tildes.

######  Remark 3.5.

Note that the defining equations for the pairs $L^{s}_{q,x}$ and
$L^{sf}_{q,x}$, $L^{s}_{p,y}$ and $L^{sf}_{p,y}$ are same.

###  3.3. Symplectic orbi-conifolds and singular symplectic flops

An orbi-conifold ([CLZZ]) is a topological space $\mathcal{Z}$ with a set of
(singular) points

$P=\\{p_{1},\ldots,p_{k}\\}$

such that $\mathcal{Z}-P$ is an orbifold and for each $p_{i}\in P$ there
exists a neighborhood $U_{i}$ that is isomorphic to $W_{r_{i}}$ for some
integer $r_{i}\geq 1$. By a symplectic structure on $\mathcal{Z}$ we mean a
symplectic form $\omega$ on $\mathcal{Z}-P$ and it is $\omega^{\circ}_{r_{i}}$
in $U_{i}$. We call $\mathcal{Z}$ a symplectic orbi-conifold. There exists
$2^{k}$ resolutions of $\mathcal{Z}$. Let $Y^{s}$ be such a resolution, its
flop is defined to be the one that is obtained by flops each local model of
$Y^{s}$. We denote it by $Y^{sf}$. In [CLZZ] we prove that

######  Theorem 3.6.

$Y^{s}$ is a symplectic orbifold if and only if $Y^{sf}$ is.

So $Y^{sf}$ is called the (singular) symplectic flop of $Y^{s}$ and vice
versa.

Now for simplicity, we assume that $\mathcal{Z}$ contains only one singular
point $p$ and is smooth away from $p$. Suppose $Y^{s}$ and $Y^{sf}$ are two
resolutions that are flops of each other and, locally, $Y^{s}$ contains
$W^{s}_{r}$ and $Y^{sf}$ contains $W^{sf}_{r}$. Then

$\displaystyle
H^{\ast}_{CR}(Y^{s})=H^{\ast}(Y^{s})\oplus\bigoplus_{k=1}^{r}\mathbb{C}[\mathfrak{p}^{s}]_{k}\oplus\bigoplus_{k=1}^{r}\mathbb{C}[\mathfrak{q}^{s}]_{k};$
$\displaystyle
H^{\ast}_{CR}(Y^{sf})=H^{\ast}(Y^{sf})\oplus\bigoplus_{k=1}^{r}\mathbb{C}[\mathfrak{p}^{sf}]_{k}\oplus\bigoplus_{k=1}^{r}\mathbb{C}[\mathfrak{q}^{sf}]_{k}.$

######  Lemma 3.7.

There are natural isomorphisms

$\psi_{k}:H^{k}(Y^{s})\to H^{k}(Y^{sf}).$

Proof. We know that

$Y^{s}-\Gamma^{s}=Y^{sf}-\Gamma^{sf}.$

We also have the exact sequence

$\cdots\to H^{k}(Y,Y\setminus\Gamma)\to H^{k}(Y)\to H^{k}(Y\setminus\Gamma)\to
H^{k+1}(Y,Y\setminus\Gamma)\to\cdots$

and

$H^{k}(Y,Y\setminus\Gamma)\cong
H^{k}_{c}(\mathcal{O}\oplus\mathcal{O}(-2))\cong H^{k-4}(\mathbb{P}^{1}).$

$Y$ is either $Y^{s}$ or $Y^{sf}$ and $\Gamma$ is the exceptional curve in
$Y$.

Suppose we have $\omega^{s}\in H^{k}(Y^{s})$. Suppose $X^{s}_{\omega^{s}}$ is
its Poincare dual. If $k>2$, we may require that
$X^{s}_{\omega^{s}}\cap\Gamma^{s}=\emptyset$. Hence $X^{s}_{\omega^{s}}$ is in

$Y^{s}\setminus\Gamma^{s}=Y^{sf}\setminus\Gamma^{sf}.$

Using this, we get a class $\omega^{sf}\in H^{k}(Y^{sf})$. Set
$\psi_{k}(\omega^{s})=\omega^{sf}$.

If $k\leq 2$, since

$H^{m}_{comp}(\mathcal{O}\oplus\mathcal{O}(-2))=0,m\leq 3,$

we have

$H^{k}(Y^{s})\cong H^{k}(Y^{s}\setminus\Gamma^{s})\cong
H^{k}(Y^{sf}\setminus\Gamma^{sf})\cong H^{k}(Y^{sf}).$

The isomorphism gives $\psi_{k}$. q.e.d.

On the other hand, we set

$\psi_{o}([\mathfrak{p}^{s}]_{k})=[\mathfrak{p}^{sf}]_{k},\;\;\;\psi_{o}([\mathfrak{q}^{s}]_{k})=[\mathfrak{q}^{sf}]_{k},$

Totally, we combine $\psi_{k}$ and $\psi_{o}$ to get a map

(3.2) $\Psi^{\ast}:H^{\ast}_{CR}(Y^{s})\to H^{\ast}_{CR}(Y^{sf}).$

It can be shown that

######  Proposition 3.8.

$\Psi^{\ast}$ preserves the Poincare pairing.

Without considering the extra classes from twisted sectors, the proof is
standard. When the cohomology classes from twisted sectors are involved, it is
proved in [CLZZ].

On the other hand, there is a natural isomorphism

$\Psi_{\ast}:H_{2}(Y^{s})\to H_{2}(Y^{sf})$

with $\Psi_{\ast}([\Gamma^{s}_{r}])=-[\Gamma^{sf}_{r}]$.

Now suppose that we do the symplectic cutting on $Y^{s}$ and $Y^{sf}$ at
$W^{s}_{r}$ and $W^{sf}_{r}$ respectively. Then

$\displaystyle\pi_{s}:Y^{s}\xrightarrow{degenerate}Y^{-}\cup_{Z}M^{s}_{r};$
$\displaystyle\pi_{sf}:Y^{sf}\xrightarrow{degenerate}Y^{-}\cup_{Z}M^{sf}_{r}.$

It is clear that $M^{s}_{r}$ and $M^{sf}_{r}$ are flops of each other. Then
similarly, we have a map

$\Psi^{\ast}_{r}:H^{\ast}_{orb}(M^{s}_{r})\to H^{\ast}_{orb}(M^{sf}_{r}).$

It is easy to see that the diagram

(3.3)
$\begin{CD}H^{\ast}_{CR}(Y^{-}\cup_{Z}M^{s}_{r})@>{(id,\sigma^{\ast})}>{}>H^{\ast}_{CR}(Y^{-}\cup_{Z}M^{sf}_{r})\\\
@V{{\pi_{s}^{\ast}}}V{}V@V{}V{\pi_{sf}^{\ast}}V\\\
H^{\ast}_{CR}(Y^{s})@>{\Sigma^{\ast}}>{}>H^{\ast}_{CR}(Y^{sf})\end{CD}$

commutes.

###  3.4. Ruan cohomology rings

As explained in §2.3, the cohomology ring structure is defined via a triple
form $A$. In the current situation, we can define a ring structure on $Y^{s}$
(and $Y^{sf}$) that plays a role between Chen-Ruan (classical) ring structure
and Chen-Ruan quantum ring structure. The triple forms on $Y^{s}$ and $Y^{sf}$
are given by

(3.4)
$\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{R}=\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{CR}+\sum_{A=d[\Gamma_{r}^{s}],d>0}\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{0,A}q_{s}^{d},$
(3.5)
$\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{R}=\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{CR}+\sum_{A=d[\Gamma_{r}^{sf}],d>0}\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{0,A}q_{sf}^{d}$

respectively. Here $q_{s}$ and $q_{sf}$ are formal variables that represent
classes $[\Gamma_{r}^{s}]$ and $[\Gamma_{r}^{sf}]$. They define Ruan rings
$RH(Y^{s})$ and $RH(Y^{sf})$. In [CLZZ], we already proved that

######  Theorem 3.9.

$\Psi^{\ast}$ gives the isomorphism $RH(Y^{s})\cong RH(Y^{sf})$.

##  4\. Relative Gromov-Witten theory on $M^{s}_{r}$ and $M^{sf}_{r}$

###  4.1. Local models $M^{s}_{r}$ and $M^{sf}_{r}$

$M^{s}_{r}$ and $M^{sf}_{r}$ are obtained from $W^{s}_{r}$ and $W^{sf}_{r}$ by
cutting at infinity. We explain this precisely.

We introduce an $S^{1}$ action on $\mathbb{C}^{4}$

$\gamma(x,y,z,t)=(\gamma^{r}x,\gamma^{r}y,\gamma z,\gamma^{r}t).$

Using this action, we collapse $\tilde{W}_{r}$ at $\infty$. The infinity
divisor is identified as

$\tilde{Z}=\frac{\tilde{W}^{s}_{r}\cap S^{7}}{S^{1}}.$

By this way, we get an orbifold with singularity at 0, denoted by
$\tilde{M}_{r}$. By blowing-up $\tilde{M}_{r}$ at 0, we have
$\tilde{M}_{r}^{s}$ and $\tilde{M}^{sf}_{r}$. $\mu_{r}$-action can then
naturally extend to $\tilde{M}_{r}$, $\tilde{M}^{s}_{r}$ and
$\tilde{M}^{sf}_{r}$. By taking quotients, we have $M_{r}$, $M^{s}_{r}$ and
$M^{sf}_{r}$. $M^{s}_{r}$ and $M^{sf}_{r}$ are the collapsing of $W^{s}_{r}$
and $W^{sf}_{r}$ at infinity. Note that the $T^{2}$-action given in §3.2 also
acts on these spaces.

Let $\tilde{\mathbb{P}}:=\mathbb{P}(r,r,1,r,1)$ be the weighted projective
space. Then $\tilde{M}_{r}$ can be embedded in $\tilde{\mathbb{P}}$ and is
given by the equation

$xy-z^{2r}+t^{2}=0.$

The original $\tilde{W}_{r}$ is embedded in $\\{w\not=0\\}$ and $\tilde{Z}$ is
in $\\{w=0\\}$. $\mu_{r}$-action extends to $\tilde{\mathbb{P}}$ by

$\xi(x,y,z,t,w)=(\xi^{a}x,\xi^{-a}y,\xi z,t,w).$

Then $M_{r}$ is embedded in $\mathbb{P}:=\tilde{\mathbb{P}}/\mu_{r}$. Set
$Z=\tilde{Z}/\mu_{r}$. To understand the local behavior of $M^{s}_{r}$ and
$M^{sf}_{r}$ at $Z$, it is sufficient to use this model at $\\{w=0\\}$.

We now study the singular points at $Z$. Combining the $S^{1}$ and $\mu_{r}$
actions, we have

$(\gamma,\xi)(x,y,z,t,w)=(\gamma^{r}\xi^{a}x,\gamma^{r}\xi^{-a}y,\gamma\xi
z,\gamma^{r}t,\gamma w),(\gamma,\xi)\in S^{1}\times\mathbb{Z}_{r}.$

######  Lemma 4.1.

There are four singular points

$\displaystyle\mathfrak{x}=[1,0,0,0,0];$
$\displaystyle\mathfrak{y}=[0,1,0,0,0];$
$\displaystyle\mathfrak{z}^{+}=[0,0,1,1,0];$
$\displaystyle\mathfrak{z}^{-}=[0,0,1,-1,0]$

and a singular set

$S=\\{xy+t^{2}=0,z=0\\}$

on $Z$. Their stabilizers are
$\mathbb{Z}_{r^{2}},\mathbb{Z}_{r^{2}},\mathbb{Z}_{r}$, $\mathbb{Z}_{r}$ and
$\mathbb{Z}_{r}$ respectively.

Proof. Take a point $(x,y,z,t,w)$. We find those points with nontrivial
stabilizers. Case 1, assume that $z\not=0$. Then $\gamma\xi=1$. Therefore,

$(\gamma,\xi)(x,y,z,t,w)=(\gamma^{r}\xi^{a}x,\gamma^{r}\xi^{-a}y,\gamma\xi
z,\gamma^{r}t,0)=(\xi^{a}x,\xi^{-a}y,z,t).$

In order to have nontrivial stabilizers, we must have $x=y=0$. Therefore, only
$\mathfrak{z}^{+}$ and $\mathfrak{z}^{-}$ survive. Their stabilizer are both
$\mathbb{Z}_{r}$. Case 2, assume that $z=0$. If $t\not=0$, $\gamma$ should be
a $r$-root. Then

$(\gamma,\xi)(x,y,z,t,w)=(\xi^{a}x,\xi^{-a}y,0,t,0)=(\xi^{a}x,\xi^{-a}y,z,t).$

Hence, when $xy$ are not 0, the set $\\{xy+t^{2}=0\\}$ has the stabilizer
$\mathbb{Z}_{r}$.

Case 3, assume that $z=t=0$. Then $xy=0$ by the equation. Hence we can only
have $\mathfrak{x}$ and $\mathfrak{y}$. Clearly, their stabilizers are both
$\mathbb{Z}_{r^{2}}$. q.e.d.

We now look at the local models at $\mathfrak{z}^{+}$ and $\mathfrak{z}^{-}$.
The coordinate chart at $\mathfrak{z}^{+}$ is given by $(x,y,w)$ and the
action is

(4.1) $\xi(x,y,w)=(\xi^{-a}x,\xi^{a}y,\xi w),\xi\in\mathbb{Z}_{r}.$

The model at $\mathfrak{z}^{-}$ is same.

Now look at the local models at $\mathfrak{x}$ and $\mathfrak{y}$. At
$\mathfrak{x}$, the local coordinate chart is given by $(z,t,w)$. The action
is given by

(4.2) $\xi(z,t,w)=(\xi\eta z,\xi^{r}t,\xi
w),\;\;\;\mbox{where}\;\;\;\eta^{a}\xi^{r}=1,\xi\in\mathbb{Z}_{r^{2}}.$

Suppose

(4.3) $\eta=\exp{2\pi i\mu},0\leq\mu<1.$

Similarly, at $\mathfrak{y}$, the local coordinate chart is given by
$(z,t,w)$. The action is given by

(4.4) $\xi(z,t,w)=(\xi\eta z,\xi^{r}t,\xi
w),\;\;\;\mbox{where}\;\;\;\eta^{-a}\xi^{r}=1,\xi\in\mathbb{Z}_{r^{2}}.$

For points on $S$, the action on the normal direction is given by

(4.5) $\xi(z,w)=(\xi z,\xi w),\xi\in\mathbb{Z}_{r}.$

Recall that we have four lines described in §3.2. They are now being four
lines in $M^{s}_{r}$ ($M^{sf}_{r}$). Take $M^{s}_{r}$ as an example. We have

$\displaystyle L^{s}_{p,y}:\mathfrak{p}^{s}\leftrightarrow\mathfrak{y};$
$\displaystyle L^{s}_{q,x}:\mathfrak{q}^{s}\leftrightarrow\mathfrak{x};$
$\displaystyle L^{s}_{p,z}:\mathfrak{p}^{s}\leftrightarrow\mathfrak{z}^{-};$
$\displaystyle L^{s}_{q,z}:\mathfrak{q}^{s}\leftrightarrow\mathfrak{z}^{+};.$

In the table, for each line we give the name of the curve and the ends it
connects.

For $M^{sf}_{r}$, we have

$\displaystyle L^{sf}_{p,y}:\mathfrak{p}^{sf}\leftrightarrow\mathfrak{y};$
$\displaystyle L^{sf}_{q,x}:\mathfrak{q}^{sf}\leftrightarrow\mathfrak{x};$
$\displaystyle L^{sf}_{p,z}:\mathfrak{p}^{sf}\leftrightarrow\mathfrak{z}^{+};$
$\displaystyle
L^{sf}_{q,z}:\mathfrak{q}^{sf}\leftrightarrow\mathfrak{z}^{-};.$

Regarding the $T^{2}$-action, we have following two lemmas. The proof is
straightforward, we leave it to readers.

######  Lemma 4.2.

The fixed points on $M^{s}_{r}$ ($M^{sf}_{r}$) are $\mathfrak{p}^{s}$,
$\mathfrak{q}^{s}$ ($\mathfrak{p}^{sf},\mathfrak{q}^{sf}$) on $\Gamma^{s}_{r}$
($\Gamma^{sf}_{r}$) and
$\mathfrak{x},\mathfrak{y},\mathfrak{z}^{+},\mathfrak{z}^{-}$ on $Z$.

######  Lemma 4.3.

In $M^{s}_{r}$, the invariant curves with respect to the torus action are
$L^{s}_{p,y},L^{s}_{q,x},L^{s}_{p,z}$ and $L^{s}_{q,z}$.

###  4.2. Relative Moduli spaces for the pair $(M^{s}_{r},Z)$

We explain the relative moduli spaces for the pair $(M^{s}_{r},Z)$. Similar
explanations can be applied to $(M^{sf}_{r},Z)$.

Let

$\Gamma=(g,A,\mathbf{h},\mathbf{g},\mathbf{l}),\;\;\;\mathcal{T}=(\mathbf{g},\mathbf{l})$

be as before. Let $\overline{\mathcal{M}}_{\Gamma,\mathcal{T}}(M^{s}_{r},Z)$
be the moduli space. Recall that the virtual dimension of the moduli space is

$\dim=c_{1}(A)+\sum_{i=1}^{m}(1-\iota(h_{i}))+\sum_{j=1}^{k}(1-[\ell_{j}]-\iota(g_{j})).$

Here we use the complex dimension.

First, we note that

$[c_{1}(M^{s}_{r})\cdot Z]=(r+2)[c_{1}(L_{Z})\cdot
Z]=\sum_{x\in\mathbf{x}}(r+2)\ell_{x}.$

Therefore

$\dim=\sum_{i=1}^{m}(1-\iota(h_{i}))+\sum_{j=1}^{k}((r+2)\ell_{j}+1-[\ell_{j}]-\iota(g_{j}))$

Set

$\displaystyle\mathfrak{u}_{i}=1-\iota(h_{i}),1\leq i\leq m,$
$\displaystyle\mathfrak{v}_{j}=(r+2)\ell_{j}+1-[\ell_{j}]-\iota(g_{j}),1\leq
j\leq k.$

Then

$\dim=\sum_{i=1}^{m}\mathfrak{u}_{i}+\sum_{j=1}^{k}\mathfrak{v}_{j}.$

For $\mathfrak{u}_{i}$ and $\mathfrak{v}_{j}$ we have following facts:

  1. (1)

if $x_{j}$ is mapped to $\mathfrak{x}$ with $g_{j}=\exp(2\pi
i\frac{\alpha}{r^{2}})$. Then

$\ell_{j}=[\ell_{j}]+\frac{\alpha}{r^{2}}.$

So

$\mathfrak{v}_{j}=(r+2)(\frac{\alpha}{r^{2}}+[\ell_{j}])+1-([\ell_{j}]+\frac{\alpha}{r^{2}}+\\{\frac{\alpha}{r^{2}}+\mu\\}+\\{\frac{\alpha}{r}\\}).$

Here $\\{z\\}:=z-[z]$. Note that this can be

$(r+1)[\ell_{j}]+n-\mu,n\geq 1.$

Here $\mu$ is defined in (4.3). The degree shifting numbers are given by
(4.2).

  2. (2)

If $x_{j}$ is mapped to $\mathfrak{y}$ with $g_{j}=\exp(2\pi
i\frac{\alpha}{r^{2}})$, it is same as the previous case.

  3. (3)

If $x_{j}$ is mapped to $S$ with $g_{j}=\exp(2\pi i\frac{\alpha}{r})$, then

$\mathfrak{v}_{j}=(r+1)[\ell_{j}]+\alpha+1.$

The degree shifting numbers are given by (4.5).

  4. (4)

If $x_{j}$ is mapped to $\mathfrak{z}^{+}$ with $g_{j}=\exp(2\pi
i\frac{\alpha}{r})$. Then

$\ell_{j}=[\ell_{j}]+\frac{\alpha}{r}.$

So

$\mathfrak{v}_{j}=(r+1)[l_{x}]+\alpha+\frac{\alpha}{r}.$

Here the degree shifting numbers are given by (4.1).

  5. (5)

If $x_{j}$ is mapped to $\mathfrak{z}^{-}$ with $g_{j}=\exp(2\pi
i\frac{\alpha}{r})$, then it is same as the previous case.

  6. (6)

whenever $g_{j}=1$

$\mathfrak{v}_{j}=(r+1)\ell_{j}+1.$

  7. (7)

when $h_{i}=\exp(2\pi i\frac{\alpha}{r})$,
$\mathfrak{u}_{i}=-\frac{\alpha}{r}.$

  8. (8)

when $h_{i}=1$, $\mathfrak{u}_{i}=1$.

###  4.3. Admissible data $(\Gamma,\mathcal{T},\mathbf{a})$

Suppose we are computing the relative Gromov-Witten invariant

(4.6) $\langle\mathbf{a}|\mathbf{b},\mathcal{T}\rangle_{\Gamma}^{(M^{s},Z)}.$

Let $|\alpha|$ denote the degree of a form $\alpha$. Set

$N=\dim-\sum_{i=1}^{m}|\alpha_{i}|.$

On the other hand, set

$N^{\prime}=\sum_{j=1}^{k}\dim(Z_{g_{j}}).$

######  Definition 4.1.

The data $(\Gamma,\mathcal{T},\mathbf{a})$ is called admissible if $N\leq
N^{\prime}$.

By the definition, we have

######  Lemma 4.4.

If $(\Gamma,\mathcal{T},\mathbf{a})$ is not admissible, the invariant (4.6) is
0.

In this paper, we may assume that:

######  Assumption 4.5.

(i) $|\alpha_{i}|=0$ for all $p_{i}$, (ii) $|\mathbf{a}|\leq 3$.

Since

$N-N^{\prime}=\sum_{i=1}^{m}\mathfrak{u}_{i}+\sum_{j=1}^{k}(\mathfrak{v}_{j}-\dim(Z_{g_{j}})),$

we make the following definition.

######  Definition 4.2.

we say that $\mathfrak{u}_{i}$ is the contribution of marked point $p_{i}$ to
$N-N^{\prime}$ and $\mathfrak{v}_{j}-\dim(Z_{g_{j}})$ is that of $x_{j}$.

######  Proposition 4.6.

Suppose that Assumption 4.5 holds. If $(\Gamma,\mathcal{T},\mathbf{a})$ is
admissible, then one of the following cases holds:

  1. (1)

$\mathbf{x}$ consists of only one smooth point, then $\mathbf{p}$ consists of
three singular points $(p_{1},p_{2},p_{3})$ such that

$\iota(h_{1})+\iota(h_{2})+\iota(h_{3})=5.$

For this case, $N=N^{\prime}$.

  2. (2)

if $\mathbf{x}$ contains a point $x$ maps to $\mathfrak{z}^{-}$ or
$\mathfrak{z}^{+}$, then one of the following should hold:

$\displaystyle|\mathbf{a}|=2,\iota(h_{1})+\iota(h_{2})=3+\frac{1}{r};$
$\displaystyle|\mathbf{a}|=3,\iota(h_{1})+\iota(h_{2})+\iota(h_{3})=4+\frac{1}{r}.$

  3. (3)

$\mathbf{x}$ consists of only singular points, the multiplicities at
$x\in\mathbf{x}$ are all less than 1. Furthermore, $x$ can not be mapped to
either $\mathfrak{z}^{+}$ or $\mathfrak{z}^{-}$.

Moreover, $\ell_{x}\leq 1$.

Proof.  First, we suppose that $x$ is smooth point. It contributes
$(r+1)l_{x}+1$ to $N$ and contributes $2$ to $N^{\prime}$. Hence its
contribution to $N-N^{\prime}$ is

$(r+1)l_{x}-1\geq r.$

If $x$ a singular point, its contribution to $N-N^{\prime}$ is given by the
following list

  * •

$x\to\mathfrak{x}$ but not in $S$, the contribution is
$(r+1)[l_{x}]+n-\mu,n\geq 1$;

  * •

$x\to S$, the contribution is $(r+1)[l_{x}]+\alpha+1,$

  * •

$x\to\mathfrak{z}^{\pm}$, the contribution is $(r+1)[l_{x}]+\alpha+\alpha/r$.

Note that they are all positive. Hence we conclude that, if $\mathbf{x}$
contains a smooth point $x$, then only the following situation survives:
$|\mathbf{x}|=1$, $r=2,|\mathbf{a}|=3$, and

$\iota(h_{1})+\iota(h_{2})+\iota(h_{3})=5.$

Furthermore, $N=N^{\prime}$.

Now suppose that $\mathbf{x}$ contains a point $x$ mapping to
$\mathfrak{z}^{-}$ (or $\mathfrak{z}^{+}$), only the following situation
survives: $\alpha=1$ and one of the following holds:

$\displaystyle|\mathbf{a}|=2,\iota(h_{1})+\iota(h_{2})=3+\frac{1}{r};$
$\displaystyle|\mathbf{a}|=3,\iota(h_{1})+\iota(h_{2})+\iota(h_{3})=4+\frac{1}{r}.$

The rest admissible data belong to the third case. q.e.d.

##  5\. Vanishing results on relative invariants

###  5.1. Localization via the torus action

The torus action $T^{2}$ on $M^{s}_{r}$ induces an action on the moduli space.
We now study the fix loci of the moduli space with respect to the action. We
use the notations in §2.4 for $X=M^{s}_{r}$.

A relative stable map

$\mathbf{f}=(f^{0},f^{1}_{1},\ldots,f^{b_{1}}_{1},\ldots,f^{k}_{1},\ldots,f^{b_{k}}_{k})$

is invariant if and only if each $f$ is invariant. Since we only consider the
invariants for admissable data, the invariant maps in such moduli spaces only
have $f^{0}$. In fact, the fact $\ell_{x}\leq 1$ in Proposition 4.6, implies
this.

$f^{0}$ is a stable map in $X$ whose components are invariant maps (maybe
constant map) and nodal points are mapped to fix points, which are
$\mathfrak{p}^{s},\mathfrak{q}^{s}$ and
$\mathfrak{x},\mathfrak{y},\mathfrak{z}^{+},\mathfrak{z}^{-}$. The constant
map should also map to these points, while the nontrivial invariant curves
should cover one of those four lines or $\Gamma^{s}_{r}$. Set

$\mbox{FT}=\\{\mathfrak{p}^{s},\mathfrak{q}^{s},\mathfrak{x},\mathfrak{y},\mathfrak{z}^{+},\mathfrak{z}^{-}\\},\;\;\;\mbox{IC}=\\{\Gamma^{s}_{r},L^{s}_{p,y},L^{s}_{q,x},L^{s}_{p,z},L^{s}_{q,z}\\}$

As in the Gromov-Witten theory, we introduce graphs to describe the components
of fix loci. We now describe the graph $T$ for $f^{0}$. Let $V_{T}$ and
$E_{T}$ be the set of vertices and edges of $T$.

  * •

each vertex is assigned to a connected component of the pre-image of FT; on
each vertex, the image point is recorded;

  * •

each edge is assigned to the component that is non-constant map; the image
with multiplicity is recorded;

  * •

on each flag, a twisted sector (or the group element of the sector) is
recorded.

Let $F_{T}$ be the fix loci that correspond to the graph $T$. Since $T$ only
describes $f^{0}$, $F_{T}$ may contain several components. Let $\mathfrak{T}$
be the collection of graphs and $\mathcal{F}$ be the collection of $F_{T}$.

We recall the virtual localization formula. Suppose that $\Omega$ is a form on
$\overline{\mathcal{M}}_{\Delta}(X,Z)$ and $\Omega_{T_{2}}$ is its equivariant
extension if exists. Then

$I(\Omega)=\int_{\overline{\mathcal{M}}_{\Delta}(X,Z)}^{vir}\Omega=\sum_{T\in\mathfrak{T}}\int_{F_{T}}\frac{\Omega_{T^{2}}}{e_{T^{2}}(N^{vir}_{F_{T}})}.$

Here, $N^{vir}_{F_{T}}$ is the virtual normal bundle of $F_{T}$ in the virtual
moduli space, $e_{T^{2}}$ is the $T^{2}$-equivariant Euler class of the
bundle. The right hand side is a function in $(\lambda,u)$, which is rational
in $\lambda$ and polynomial in $u$. We denote each term in the summation as
$I_{F_{T}}^{\Omega}(\lambda,u)$ and the sum by $I^{\Omega}(\lambda,u)$.

######  Lemma 5.1.

Let $(\Gamma,\mathcal{T},\mathbf{a})$ be an admissible data in Proposition
4.6. Then the nontrivial relative invariants

$\langle\mathbf{a}|\mathbf{b}\rangle_{\Gamma,\mathcal{T}}^{(M^{s},Z)}$

can be computed via localization.

Proof. It is sufficient to show that the forms in $\mathbf{a}$ and
$\mathbf{b}$ have equivariant extensions. By our assumption, we always take
$\alpha\in\mathbf{a}$ to be 1. It is already equivariant.

Now suppose that $\beta_{j}$ is assigned to $x_{j}$. If $Z_{(g_{j})}$ is a
single point, $\beta_{j}$ are taken to be 1, which is equivariant. If
$Z_{g_{j}}=S$, $\beta_{j}$ is either a 0 or 2-form, both have equivariant
representatives. The last case is that $x$ is smooth. For this case, since
$N=N^{\prime}$ (cf. case (1) in Proposition 4.6), we must have $\deg(\beta)=4$
to get nontrivial invariants. Since $\beta$ is of top degree, it has an
equivariant representative as well. q.e.d.

###  5.2. Vanishing results on $I^{\Omega}_{F_{T}}(\lambda,u)$, (I)

By localization, we have

$I(\Omega)=I^{\Omega}(\lambda,u)=\sum_{T\in\mathfrak{T}}I^{\Omega}_{F_{T}}(\lambda,u).$

Since the left hand side is independent of $u$, we have

$I(\Omega)=\lim_{u\to
0}I^{\Omega}(\lambda,u)=\sum_{T\in\mathfrak{T}}\lim_{u\to
0}I^{\Omega}_{F_{T}}(\lambda,u).$

######  Theorem 5.2.

Suppose that $|\mathbf{x}|>0$. If $T$ contains an edge $e_{0}$ that records a
map cover $\Gamma^{s}_{r}$, then

$\lim_{u\to 0}I_{F_{T}}(\lambda,u)=0.$

Edge $e_{0}$ records a map:

$f_{0}:S^{2}\to\Gamma^{s}_{r}.$

$f_{0}$ can be either a smooth or an orbifold map. Hence, we restate the
theorem as,

######  Proposition 5.3.

If the map $f_{0}$ for $e_{0}$ is smooth, then

$\lim_{u\to 0}I_{F_{T}}(\lambda,u)=0.$

and

######  Proposition 5.4.

If the map $f_{0}$ for $e_{0}$ is singular, then

$\lim_{u\to 0}I_{F_{T}}(\lambda,u)=0.$

Clearly, Proposition 5.3 and 5.4 imply Theorem 5.2.

Suppose that $T$ is a graph. Let $C$ be a curve in $F_{T}$.

First assume that all components in $C$ are smooth. Then copied from [GP], we
have

$0\to\mathcal{O}_{C}\to\bigoplus_{\mbox{vertices}}\mathcal{O}_{C_{v}}\oplus\bigoplus_{\mbox{edges}}\mathcal{O}_{Ce}\to\bigoplus_{\mbox{flags}}\mathcal{O}_{F}\to
0,$

then (write $E=f^{\ast}TM^{s}_{r}$)

$\displaystyle 0\to
H^{0}(C,E)\to\bigoplus_{\mbox{vertices}}H^{0}(C_{v},E)\oplus\bigoplus_{\mbox{edges}}H^{0}(C_{e},E)$
$\displaystyle\to\bigoplus_{\mbox{flags}}E_{p(F)}\to H^{1}(C,E)$
$\displaystyle\to\bigoplus_{\mbox{edges}}H^{1}(C_{v},E)\oplus\bigoplus_{\mbox{edges}}H^{1}(C_{e},E)\to
0.$

Please refer to [GP] for flags. Here, by $p(F)$ we mean the fixed point
assigned to the flag. Hence the contribution of $H^{1}/H^{0}$ is

(5.1)
$\frac{H^{1}(C,E)}{H^{0}(C,E)}=\frac{\bigoplus_{\mbox{vertices}}E_{p(v)}^{val(v)-1}\oplus\bigoplus_{\mbox{vertices}}H^{1}(C_{e},E)}{\bigoplus_{\mbox{edges}}H^{0}(C_{e},E)}$

We translate each term to the equivariant Euler class, i.e, a polynomial in
$\lambda$ and $u$.

If $C$ is not smooth, each space in the long complex should be replaced by the
invariant subspace with respect to the proper finite group actions. Hence,
each term in the right hand side of (5.1) should be replaced accordingly.

Recall that

(5.2)
$I^{\Omega}_{F_{T}}(\lambda,u)=\int_{F_{T}}\frac{\Omega_{T^{2}}}{e_{T^{2}}(N^{vir}_{F_{T}})}.$

It is known that the equivariant form of $H^{1}/H^{0}$ gives

(5.3) $\frac{1}{e_{T^{2}}(N^{vir}_{F_{T}})}.$

It is easy to show that for each $e$, whose map $f_{e}$ does not
$\Gamma^{s}_{r}$,

$e_{T_{2}}\left(\frac{H^{1}(C_{e},f_{e}^{\ast}E)}{H^{0}(C_{e},f_{e}^{\ast}E)}\right)$

contains no either $u$ or $u^{-1}$. We now focus other terms in $H^{1}/H^{0}$.

Claim 1: if $f_{0}$ is smooth, the equivariant Euler form of the above term
$H^{1}/H^{0}$ contains a positive power of $u$. We count the possible
contributions for $u$. Case 1, there is $v=\mathfrak{p}^{s}$ or
$\mathfrak{q}^{s}$ with $val(v)>1$, we then have $u^{val(v)-1}$; Case 2, there
is a component $C_{e}$ that is a multiple of $\Gamma^{s}_{r}$, we then
actually have $u$ in the denorminator for
$H^{0}(C_{e},f_{0}^{\ast}\mathcal{O})$ and $ru$ in the numerator for
$H^{1}(C_{e},f_{0}^{\ast}\mathcal{O}(-2))$. So they cancel out. Hence we have
the claim. $\Box$

Claim 2: if $f_{0}$ is an orbifold map, the equivariant Euler of $H^{1}/H^{0}$
contains a factor of positive power of $u$.

Suppose $f_{0}$ is given by

$f_{0}:[S^{2}]\to\Gamma^{s}_{r}\subset W^{s}_{r}.$

Such a map can be realized by a map

$\tilde{f}_{0}:S^{2}\to\tilde{\Gamma}^{s}_{r}\subset\tilde{W}^{s}_{r}$

with a quotient by $\mathbb{Z}_{r}$. Here $[S^{2}]=S^{2}/\mathbb{Z}_{r}$.

Unlike the smooth case, $H^{0}([S^{2}],f_{0}^{\ast}E)$ contains no $u$. By the
computations given below, in Corollary 5.7 we conclude that
$H^{1}([S^{2}],f_{0}^{\ast}E)$ contains a factor $u$. $\Box$

We compute $H^{1}([S^{2}],f_{0}^{\ast}E)$ and its weight.

Suppose that $\tilde{f}_{0}$ is a $d$-cover. Then on the sphere
$\tilde{S}^{2}$, the torus action weight at $0$ is $\lambda/d$ and at $\infty$
is $-\lambda/d$, and suppose that $\mathbb{Z}_{r}$ action is $\mu$ at 0 and
$\mu^{-1}$ at $\infty$; for the pull-back bundle $\mathcal{O}(-2d)$ the torus
action weight at fiber over $0$ is $-\lambda+ru$ and at fiber over $\infty$ is
$\lambda+ru$, the $\mathbb{Z}_{r}$ action are $\mu^{-d}$ and $\mu^{d}$ for the
fibers over at $0$ and $\infty$. These data are ready for us to compute the
action on $H^{1}([S^{2}],f_{0}^{\ast}E)$.

By Serre-duality, we have

$H^{1}(S^{2},\mathcal{O}(-2d))=(H^{0}(S^{2},\mathcal{O}(2d-2)))^{\ast}.$

The induced torus action weights on $\mathcal{O}(2d-2)$ at the fibers over $0$
and $\infty$ are

$\frac{d-1}{d}\lambda-ru,\;\;\;\frac{1-d}{d}\lambda-ru$

respectively. The induced $\mathbb{Z}_{r}$ action are $\mu^{d-1}$ and
$\mu^{-d+1}$ for fibers on $\mathcal{O}(2d-2)$ over $0$ and $\infty$.

######  Lemma 5.5.

The sections of $H^{0}(S^{2},\mathcal{O}(2d-2))$ are given by

$\\{x^{a}y^{b}|a+b=2d-2,a,b\geq 0\\}.$

The torus action weight for section $x^{a}y^{b}$ is $\frac{d-1-a}{d}\lambda-
ru$. The action of $\mu$ is $\mu^{d-1-a}$.

Hence

######  Lemma 5.6.

The section $x^{a}y^{b}$ that is $\mathbb{Z}_{r}$-invariant if and only if
$r|d-1-a$, and the torus action weight is $\frac{d-1-a}{d}\lambda-ru$.

######  Corollary 5.7.

$H^{1}([S^{2}],f_{0}^{\ast}\mathcal{O}(-2))$ contains a factor $ru$.

Proof.  By the above lemma, we know that $x^{d-1}y^{d-1}$ is $Z_{r}$-invariant
and it is action weight is $-ru$. By taking the dual, the corresponding factor
is $ru$. q.e.d.

Proof of Proposition 5.3: Claim 1 implies the proposition. Proof of
Proposition 5.4: Claim 2 implies the proposition.

###  5.3. Vanishing results on $I^{\Omega}_{F_{T}}(\lambda,u)$, (II)

Now suppose that $(\Gamma,\mathcal{T},\mathbf{a})$ is an admissible data given
in Proposition 4.6.

######  Theorem 5.8.

If $(\Gamma,\mathcal{T},\mathbf{a})$ is of case (1) and (2) in Proposition
4.6,

$\langle\mathbf{a}|\mathbf{b}\rangle_{\Gamma,\mathcal{T}}^{(M^{s},Z)}=0.$

Proof. Suppose that

$\langle\mathbf{a}|\mathbf{b}\rangle_{\Gamma,\mathcal{T}}^{(M^{s},Z)}=I(\Omega)$

for some $\Omega$ which has equivariant extension $\Omega_{T^{2}}$. (cf. Lemma
5.1).

Let $F_{T}$ be a fix component of the torus action. It contributes 0 to the
invariant unless the fixed curves $C\in F_{T}$ contains no component covering
$\Gamma^{s}$ (cf. Theorem 5.2). It is easy to conclude that $C$ must contains
a ghost map

$f:(S^{2},q_{1},q_{2},\cdots,q_{l})\to M^{s}$

such that the image of $f$ is either $\mathfrak{p}^{s}$ or $\mathfrak{q}^{s}$
and sum of the degree shifting numbers for all twisted sectors defined by
$q_{i}$ is $2+l$. Suppose $\mathfrak{p}^{s}=f(S^{2})$. Again, we claim that
$I^{\Omega}_{F_{T}}(\lambda,u)$ contains factor $u$. In fact,

$e_{T^{2}}(H^{1}(S^{2},f^{\ast}\mathcal{O}_{\mathfrak{p}}^{s}))=u.$

This is proved in [CH]. Hence, it is easy to see the claim follows. q.e.d.

######  Definition 5.1.

If

$\lim_{u\to 0}I^{\Omega}_{F_{T}}(\lambda,u)=0,$

we say the component $F_{T}$ contributes trivial to the invariant $I(\Omega)$.

######  Corollary 5.9.

Let $(\Gamma,\mathcal{T},\mathbf{a})$ be admissible, and

$I(\Omega)=\langle\mathbf{a}|\mathbf{b}\rangle_{\Gamma,\mathcal{T}}^{(M^{s},Z)}.$

If $F_{T}$ contributes nontrivial to the invariant, then for any curve $C\in
F_{T}$, all its connected components in the root component $M^{s}$ must cover
$L^{s}_{p,y}$ or $L^{s}_{q,x}$.

Proof.  By the previous theorem, the admissible data must be of the 3rd case
in Proposition 4.6. Hence, points in $C\cap Z$ must be either $\mathfrak{x}$
or $\mathfrak{y}$. Therefore, the invariant curves that $C$ lives on must be
$L^{s}_{p,y}$ and $L^{s}_{q,x}$. q.e.d.

##  6\. Proof of the Main theorem

Combining §2.3 and Theorem 3.9, we reduce Theorem 1.1 to

######  Theorem 6.1.

$\sum_{A\not\in\mathbb{Z}\Gamma^{s}_{r}}\langle\alpha_{1}^{s},\alpha_{2}^{s},\alpha_{3}^{s}\rangle_{A}=\sum_{A\not\in\mathbb{Z}\Gamma^{sf}_{r}}\langle\Psi^{\ast}\alpha_{1}^{s},\Psi^{\ast}\alpha_{2}^{s},\Psi^{\ast}\alpha_{3}^{s}\rangle_{A}.$

The rest of the section is devoted to the proof of this theorem.

######  Remark 6.2.

We should point out that ”$=$” is rather strong from the point of view of
Ruan’s conjecture. Usually, it is conjectured that

$\langle\alpha_{1}^{s},\alpha_{2}^{s},\alpha_{3}^{s}\rangle\cong\langle\Psi^{\ast}\alpha_{1}^{s},\Psi^{\ast}\alpha_{2}^{s},\Psi^{\ast}\alpha_{3}^{s}\rangle.$

By ”$\cong$”, we mean that both sides equal up to analytic continuations. This
is necessary when classes $[\Gamma^{s}]$ and $[\Gamma^{sf}]$ involved. For
example, in [CLZZ] we proved

$\langle\alpha_{1}^{s},\alpha_{2}^{s},\alpha_{3}^{s}\rangle_{R}\cong\langle\Psi^{\ast}\alpha_{1}^{s},\Psi^{\ast}\alpha_{2}^{s},\Psi^{\ast}\alpha_{3}^{s}\rangle_{R}$

for Theorem 3.9.

But for the invariants that correspond to $A\not=d[\Gamma^{s}]$, it turns out
that we do not need the analytic continuation argument, the reason is because
of Theorem 5.2: as long as $[\Gamma^{s}]$ (so is $[\Gamma^{sf}]$) appears, the
invariant vanishes.

###  6.1. Reducing the comparison to local models

We now apply the degeneration formula to reduce the comparing three point
functions only on local models. We explain this: consider a three point
function

$\langle\alpha_{1}^{s},\alpha_{2}^{s},\alpha_{3}^{s}\rangle_{0,A_{s}}.$

First, we observe that $[A_{s}]=A_{s}$ since $\pi_{\ast}$ has no kernel (cf
[LR]). Denote the topology data by

$\Gamma^{s}=(0,A_{s})$

and forms by $\mathbf{a}^{s}=(\alpha_{1}^{s},\alpha^{s}_{2},\alpha^{s}_{3})$.
Correspondingly, on $Y^{sf}$ we introduce

$\displaystyle\mathbf{a}^{sf}=(\Psi^{\ast})^{-1}\mathbf{a}^{s},$
$\displaystyle A_{sf}=\Psi_{\ast}A_{s},$
$\displaystyle\Gamma^{sf}=(0,A_{sf}).$

We write $\Gamma^{sf}=\Psi(\Gamma^{s})$.

Consider the degenerations

$\displaystyle\pi_{s}:Y^{s}\xrightarrow{degenerate}M^{s}_{r}\cup_{Z}Y^{-};$
$\displaystyle\pi_{sf}:Y^{sf}\xrightarrow{degenerate}M^{sf}_{r}\cup_{Z}Y^{-}.$

Let $\eta^{s}=(\Gamma^{+,s},\Gamma^{-},I_{\rho})$ be a possible splitting of
$\Gamma^{s}$. Correspondingly,

$\Psi(\eta^{s}):=(\Psi_{r}(\Gamma^{+,s}),\Gamma^{-},I_{\rho})$

gives a splitting of $\Psi(\Gamma^{s})$. On the other hand, suppose that

$\mathbf{a}^{s}=\pi_{s}^{\ast}(\mathbf{a}^{+,s},\mathbf{a}^{-}).$

Then by the diagram (3.3),

$\mathbf{a}^{sf}:=\Psi^{\ast}(\mathbf{a}^{s})=\pi_{sf}^{\ast}(\Psi^{\ast}_{r}(\mathbf{a}^{+,s}),\mathbf{a}^{-}).$

######  Proposition 6.3.

Suppose that $\mathbf{a}^{s}$ and $\mathbf{b}$ are given on $M^{s}_{r}$, then

(6.1)
$\langle\mathbf{a}^{+,s}|\mathbf{b},\mathcal{T}\rangle^{\ast(M^{s},Z)}_{\Gamma^{+,s}}=\langle\Psi^{\ast}_{r}\mathbf{a}^{+,s}|\mathbf{b},\mathcal{T}\rangle^{\ast(M^{s},Z)}_{\Psi_{r}(\Gamma^{+,s})}.$

Unlike $\langle\rangle^{\bullet(M^{s},Z)}$, here
$\langle\rangle^{\ast(M^{s},Z)}$ only sums over all admissible data.

######  Proposition 6.4.

Proposition 6.3 $\Rightarrow$ Theorem 6.1.

Proof. Applying the degeneration formula to
$\langle\mathbf{a}^{s}\rangle_{\Gamma^{s}}$, we have

$\langle\mathbf{a}^{s}\rangle_{\Gamma^{s}}^{Y^{s}}=\sum_{I}\sum_{\eta=(\Gamma^{+,s},\Gamma^{-},I_{\rho})}C_{\eta}\langle\mathbf{a}^{+,s}|\mathbf{b}^{I},\mathcal{T}\rangle^{\bullet(M^{s},Z)}_{\Gamma^{+,s}}\langle\mathbf{a}^{-}|\mathbf{b}_{I},\mathcal{T}\rangle^{\bullet(Y^{-},Z)}_{\Gamma^{-}}.$

Similarly,

$\langle\mathbf{a}^{sf}\rangle_{\Gamma^{sf}}^{Y^{sf}}=\sum_{I}\sum_{\eta^{sf}=(\Gamma^{+,sf},\Gamma^{-},I_{\rho})}C_{\eta}\langle\mathbf{a}^{+,sf}|\mathbf{b}^{I},\mathcal{T}\rangle^{\bullet(M^{s},Z)}_{\Gamma^{+,sf}}\langle\mathbf{a}^{-}|\mathbf{b}_{I},\mathcal{T}\rangle^{\bullet(Y^{-},Z)}_{\Gamma^{-}}.$

Here $\ast^{sf}$ is always the correspondence of $\ast^{s}$ via $\Psi$ or
$\Psi_{r}$.

Since only admissible data contributes on the right hand sides of two
equations, (6.1) implies

$\langle\mathbf{a}^{s}\rangle_{\Gamma^{s}}^{Y^{s}}=\langle\mathbf{a}^{sf}\rangle_{\Gamma^{sf}}^{Y^{sf}}.$

which is exactly what Theorem 6.1 asserts.

###  6.2. Proof of Proposition 6.3

We now proceed to proof Proposition 6.3. Since the moduli spaces in local
models admit torus actions. By localizations, we know the contributions only
come from those fix loci. So it is sufficient to compare fix loci and the
invariants they contribute.

For $(M^{s},Z)$, let $T$ be a graph, and $F_{T}^{s}$ be the component of fix
loci. Then by Corollary 5.9, $F_{T}^{s}$ makes a nontrivial contribution only
when each curves in $F_{T}$ consists of only components on $L^{s}_{p,y}$ and
$L^{s}_{q,x}$.

Similarly, for $(M^{sf},Z)$, $F_{T}^{sf}$ makes a nontrivial contribution only
when each curves in $F_{T}^{sf}$ consists of only components on $L^{sf}_{p,y}$
and $L^{sf}_{q,x}$.

Since, the flop identifies

$L^{s}_{p,y}\leftrightarrow L^{sf}_{p,y},L^{s}_{q,x}\leftrightarrow
L^{sf}_{q,x}$

and their normal bundles, therefore the flop identifies $F_{T}^{s}$ and
$F_{T}^{sf}$ and their virtual normal bundles in their moduli spaces. Hence,

$I_{F_{T}^{s}}(\lambda,u)=I_{F_{T}^{sf}}(\lambda,u).$

Proposition 6.3 then follows.

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