# On Gromov-Witten theory of projective bundles

Honglu Fan<EMAIL_ADDRESS>and  Yuan-Pin Lee<EMAIL_ADDRESS>Department of Mathematics, University of Utah, Salt Lake City, Utah
84112-0090, U.S.A.

###### Contents

  1. 0 Introduction
    1. 0.1 Statement of main result
    2. 0.2 Motivations
    3. 0.3 Discussions
    4. 0.4 Acknowledgements
  2. 1 Genus zero equivariant Gromov-Witten theory
    1. 1.1 Equivariant twisted theory
    2. 1.2 Lagrangian cone formulation
  3. 2 Virtual localization
    1. 2.1 General facts on torus action
    2. 2.2 $T$-invariant stable maps
    3. 2.3 Decorated graphs and moduli of invariant stable maps
  4. 3 A recursion relation
    1. 3.1 Statement
    2. 3.2 Proof of Theorem 3.5
  5. 4 Gromov-Witten invariants of projective bundles

##  0\. Introduction

###  0.1. Statement of main result

Let $Y$ be a smooth projective variety with a $T$-action and $V$ be a
$T$-equivariant, _not necessarily split_ , vector bundle of rank $r$ over $Y$.
The projective bundle $\mathbb{P}(V)$ naturally carries an induced $T$-action.
The equivariant cohomology of $\mathbb{P}(V)$ has the following presentation

$H_{T}^{*}(\mathbb{P}(V))=\frac{H^{*}_{T}(Y)[h]}{(c_{T}(V)(h))}$

where $h=c_{1}(\mathcal{O}(1))$ and $c_{T}(V)(x)=\sum x^{r-i}c_{i}(V)$. Given
two such equivariant vector bundles $V_{1},V_{2}$ over $Y$ with the same
equivariant Chern classes $c_{T}(V_{1})=c_{T}(V_{2})$, the equivariant
cohomology/Chow are canonically isomorphic by the above presentation

$\mathfrak{F}:H_{T}^{*}(\mathbb{P}(V_{1}))\cong H^{*}_{T}(\mathbb{P}(V_{2})).$

$\mathfrak{F}$ induces an isomorphism between $N_{1}(\mathbb{P}(V_{1}))$ and
$N_{1}(\mathbb{P}(V_{2}))$ by the intersection pairing, and we slightly abuse
the notation and denote the induced isomorphism also by $\mathfrak{F}$. This
isomorphism on curve classes is uniquely characterized by the property:
$(\mathfrak{F}D,\mathfrak{F}\beta)=(D,\beta)$ for any $D\in
N^{1}(\mathbb{P}(V_{1})),\beta\in N_{1}(\mathbb{P}(V_{1}))$.

######  Theorem A (=Theorem 4.2).

If $Y$ is a projective smooth variety with a torus action such that there are
finitely many fixed points and one dimensional orbits, then the $\mathfrak{F}$
induces an isomorphism of $T$-equivariant genus 0 Gromov-Witten invariants
between $\mathbb{P}(V_{1})$ and $\mathbb{P}(V_{2})$

$\langle\psi^{a_{1}}\sigma_{1},\dotsc,\psi^{a_{n}}\sigma_{n}\rangle_{0,n,\beta}^{\mathbb{P}(V_{1})}=\langle\psi^{a_{1}}\mathfrak{F}\sigma_{1},\dotsc,\psi^{a_{n}}\mathfrak{F}\sigma_{n}\rangle_{0,n,\mathfrak{F}\beta}^{\mathbb{P}(V_{2})}.$

Such $Y$ are often called a proper algebraic GKM manifold and examples include
toric varieties, Grassmannians, flag varieties, and certain Hilbert schemes of
points.

###  0.2. Motivations

It is natural to study Gromov–Witten theory of projective bundles, which has
appeared in key steps in the relative Gromov–Witten theory (e.g., [16]) and in
flops and $K$-equivalence (crepant transformation) conjecture ([9, 10, 11]),
among other things. Furthermore, in the study of the _functoriality_ of the
Gromov–Witten theory, the projective bundle also plays a role ([13, 12]).
Namely, if one factors a projective morphism $f:X\to Y$ via embedding and
smooth fibration

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{f}$$\textstyle{\mathbb{P}_{Y}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X}$

(where $E$ is a vector bundle over $Y$), one can study the general
functoriality of Gromov–Witten theory by studying embedding (e.g., quantum
Lefschetz hyperplane theorem) and projective bundles.

However, the study of non-split bundles proves to be difficult. For split
bundles, the fiberwise torus action and the virtual localization formula
proves to be fairly effective general strategies to study the Gromov–Witten
theory of the projective bundle, as in [16, 3, 6]. This does not work for the
non-split bundles, however. Instead, a degeneration argument is employed in
[11] and has had some success in a limited form of _quantum splitting
principle_. Since the Chern roots of a nonsplit vector bundle can be
represented as the first Chern classes of line bundles, _Theorem A can
therefore be regarded as a first step towards a more complete form of quantum
splitting principle_. Once the bundle is split, the results in [3, 6] give us
a complete description of the Gromov–Witten theory of $\mathbb{P}(V)$ in all
genera in terms of the genus zero theory of $Y$.

Note that if there exists an isomorphism between $V_{1}$ and $V_{2}$ as
complex vector bundles (but not necessarily as holomorphic ones), D. Joyce
informs us that $\mathbb{P}(V_{1})$ and $\mathbb{P}(V_{2})$ are symplectically
deformation equivalent. This is often not possible however. The simplest
examples are given for $Y=\mathbb{P}^{3}$ in [1].

###  0.3. Discussions

The result of this paper leads to the following questions.

######  Question 1.

Let $Y$ be a smooth variety and $V$ be a vector bundle. Is the Gromov-Witten
invariants of $\mathbb{P}(V)$ uniquely determined by that of the base $Y$ and
the total Chern class $c(V)$?

If the above question seems hopelessly naive, the following might be more
reasonable.

######  Question 2.

Is Question 1 true if one replace the total Chern class $c(V)$ by the class in
the algebraic K-group $[V]\in K^{0}(V)$?

A note on the references: When the reference [3] is used, we specifically mean
this ArXiv version of the paper, as we are not able to understand a key step
in the published version.

###  0.4. Acknowledgements

The first author wish to thank Yuan Wang and Yang Zhou for helpful
discussions, and to Andrei Mustaţă and Sam Payne for answering related
questions. An answer to the first author’s posts in Mathoverflow is also
proven useful. Both authors are partially supported by the NSF.

##  1\. Genus zero equivariant Gromov-Witten theory

###  1.1. Equivariant twisted theory

Let $T:=(\mathbb{C}^{*})^{m}$ be an algebraic torus and $X$ a smooth
projective variety with a $T$ action. Let $E$ be a $T$-equivariant vector
bundle on $X$. All the cohomology classes and characteristic classes are
understood in the equivariant theories. We have
$H^{*}_{T}(\text{point})\cong\mathbb{C}[C(T)]$ where $C(T)$ is the group of
characters of $T$. Once a basis is chosen, $\mathbb{C}[C(T)]$ can be written
as a polynomial ring $\mathbb{C}[\lambda_{1},\dotsc,\lambda_{m}]$. The
following notations are used throughout the paper:

  * •

$R_{T}=H_{T}^{*}(\\{\text{point}\\})=\mathbb{C}[\lambda_{1},\dotsc,\lambda_{m}]$.

  * •

$S_{T}$ is the localization of $R_{T}$ by the set of non-zero homogeneous
elements.

Let $\overline{\mathcal{M}}_{0,n}(X,d)$ be the moduli of stable maps with
curve class $d\in\operatorname{NE}(X)$. The $T$-action on $X$ induces an
$T$-action on $\overline{\mathcal{M}}_{0,n}(X,d)$ and we have an equivariant
virtual fundamental class
$[\overline{\mathcal{M}}_{0,n}(X,d)]^{\operatorname{vir}}$. The universal
family is naturally identified with the forgetful morphism
$\pi_{0,n,d}:\overline{\mathcal{M}}_{0,n+1}(X,d)\rightarrow\overline{\mathcal{M}}_{0,n}(X,d)$
(forgetting the last marked point) and the universal stable map with
$f_{0,n,d}:\overline{\mathcal{M}}_{0,n+1}(X,d)\rightarrow X$ (evaluating at
the last marked point). Let

$E_{0,n,d}=(\pi_{0,n,d})_{*}(f_{0,n,d}^{*}(E))\in
K_{0}(\overline{\mathcal{M}}_{0,n}(X,d)).$

In the following, we _assume_ that $E_{0,n,d}$ is isomorphic in $K$-theory to
a two-term complex of locally free sheaves

$0\rightarrow E_{0,n,d}^{0}\rightarrow E_{0,n,d}^{1}\rightarrow 0,$

such that the equivariant Euler class $e_{T}(E_{0,n,d}^{0})$ is invertible.
Define

$e_{T}(E_{0,n,d})^{-1}=\displaystyle\frac{e_{T}(E_{0,n,d}^{1})}{e_{T}(E_{0,n,d}^{0})},\quad(\text{and
$e_{T}(E_{0,n,d})=\displaystyle\frac{e_{T}(E_{0,n,d}^{0})}{e_{T}(E_{0,n,d}^{1})}$
when $e_{T}(E_{0,n,d}^{1})$ is invertible}).$

######  Definition 1.1.

Let $\psi_{i}$ be the first chern class of the universal cotangent line
bundle. Assume $e_{T}(E_{0,n,d})^{-1}$ is defined. Given
$\alpha_{1},\dotsc,\alpha_{n}\in H^{*}_{T}(X)$, the twisted equivariant
Gromov-Witten invariant is defined as

$\langle\psi^{k_{1}}\alpha_{1},\dotsc,\psi^{k_{n}}\alpha_{n}\rangle_{0,n,d}^{X,E}=\int_{[\overline{\mathcal{M}}_{0,n}(X,d)]^{vir}}\displaystyle\frac{1}{e_{T}(E_{0,n,d})}\prod\limits_{i=1}^{n}\psi_{i}^{k_{i}}ev_{i}^{*}\alpha_{i}.$

By convention the Euler class of zero sheaf is $1$ and the “untwisted theory”
is therefore considered as a special case.

######  Remark 1.2.

$e_{T}(E_{0,n,d})$ is often not invertible. To remedy this situation, an
auxiliary $\mathbb{C}^{*}$ action, scaling the fiber of $E$ while acting
trivially on $X$, is introduced. We have $R_{T\times\mathbb{C}^{*}}=R_{T}[x]$
where $x$ is the equivariant parameter corresponding to the $\mathbb{C}^{*}$
action. Let $R_{T}[x,x^{-1}]\\!]$ be the ring of Laurent series in $x^{-1}$.
Let $(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))[x])_{\text{loc}}$ be the
localization of $H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))[x]$ by inverting
monic polynomials in $x$. It can be embedded in
$H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))\otimes_{R_{T}}R_{T}[x,x^{-1}]\\!]$
by expanding the denominators. In this set-up,

$e_{T\times\mathbb{C}^{*}}(E_{0,n,d})^{-1}=\displaystyle\frac{e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{1})}{e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{0})}$

is defined in
$(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))[x])_{\text{loc}}$. The
corresponding twisted invariants are defined when the Euler class and the
insertions are embedded into
$H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))\otimes_{R_{T}}R_{T}[x,x^{-1}]\\!]$.
The twisted invariants take values in $R_{T}[x,x^{-1}]\\!]$.

$R_{T}[x,x^{-1}]\\!]$ is sometimes too big to be useful, and we need to make
some changes. To start with,
$\displaystyle\frac{e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{1})}{e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{0})}$
is an element in
$(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))[x])_{\text{loc}}$. In Section
4, we embed
$\displaystyle\frac{e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{1})}{e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{0})}$
into
$(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))[x])_{\text{loc}}\otimes_{R_{T}}S_{T}$,
and take $x=0$ limit in
$H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))\otimes_{R_{T}}S_{T}$. The
existence of the limit means that there are elements
$\mathfrak{e}_{0},\mathfrak{e}_{1}\in
H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))\otimes_{R_{T}}S_{T}$ with
$\mathfrak{e}_{0}$ invertible such that

$\mathfrak{e}_{1}e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{0})-\mathfrak{e}_{0}e_{T\times\mathbb{C}^{*}}(E_{0,n,d}^{1})\in
xH^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X,d))[x]\otimes_{R_{T}}S_{T}.$

The twisted invariants are defined by replacing ${e_{T}(E_{0,n,d})}^{-1}$ in
Definition 1.1 with $\displaystyle\frac{\mathfrak{e}_{1}}{\mathfrak{e}_{0}}$.
Note that the invariants take values in $S_{T}$.

For any ring $R$, $R[\\![Q]\\!]$ shall always be understood as the ring of
Novikov variables, i.e., the ring of power series of
$Q^{d},d\in\operatorname{NE}(X)$ with coefficients in $R$. (See, e.g., [14,
Chapter 1].) The notations $R[\\![z,z^{-1}]$ and $R[z,z^{-1}]\\!]$ in this
paper stand for the _ring_ of formal Laurent series, with inifinite powers in
$z$ and $z^{-1}$ respectively (and finite in the other).

###  1.2. Lagrangian cone formulation

This section serves the purpose of fixing the notations. The readers are
referred to [7] for details of Givental’s construction and various properties
of the Lagrangian cones.

Let $\mathcal{H}=H^{*}_{T}(X)\otimes_{R_{T}}S_{T}[z,z^{-1}]\\!][\\![Q]\\!]$.
There is a bilinear pairing $(,)_{E}$ induced by the Poincaré pairing

$(\alpha,\beta)_{E}=\int_{[X]}e_{T}(E)^{-1}\alpha\cup\beta$

on $H^{*}_{T}(X)$. Let
$\mathcal{H}_{+}=H^{*}_{T}(X)\otimes_{R_{T}}S_{T}[z][\\![Q]\\!],~{}\mathcal{H}_{-}=z^{-1}H^{*}_{T}(X)\otimes_{R_{T}}S_{T}[\\![z^{-1}]\\!][\\![Q]\\!]$
$\mathcal{H}$ admits a _polarization_
$\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$ and is indeed naturally
identified as the cotangent bundle $T^{*}\mathcal{H}_{+}$. The canonical
symplectic structure of $\mathcal{H}\cong T^{*}\mathcal{H}_{+}$ is the
$S_{T}[\\![Q]\\!]$-bilinear form

$\Omega(f(z),g(z))=\text{Res}_{z=0}(f(-z),g(z))_{E}dz$

for any $f,g\in\mathcal{H}$.

Givental’s (twisted) Lagrangian cone is defined as the section of
$T^{*}\mathcal{H}_{+}$ by the differential of genus-$0$ Gromov-Witten
descendant potential, defined in a formal neighborhood at
$-1z\in\mathcal{H}_{+}$.

More explicitly, let $\\{\phi_{i}\\}$ be a basis of $R_{T}$-module in
$H^{*}_{T}(X)$ that induce a $\mathbb{C}$-basis in
$H^{*}_{T}(X)/H^{*}_{T}(\\{\text{point}\\})$. Assume $\phi_{0}=1$ and let
$\phi^{i}$ be the dual basis in $H^{*}_{T}(X)\otimes_{R_{T}}S_{T}$ with
respect to $(,)_{E}$. We parametrize cohomology classes by taking
$t_{k}=\sum_{i}t_{k}^{i}\phi_{i}$ with $t_{k}^{i}$ formal variables. Let
$t(z)=t_{0}+t_{1}z+\dotsb+t_{k}z^{k}+\dotsb$ be the formal parameters in
$\mathcal{H}_{+}$, and $(\mathcal{H},-1z)$ be the formal neighborhood at
$-1z\in\mathcal{H}$. A general (formal) familiy of points
$F\in(\mathcal{H},-1z)$ in Givental’s (twisted) Lagrangian cone
$\mathcal{L}_{X,E}\subset(\mathcal{H},-1z)$ is of the form

$\displaystyle F(-z,t)$
$\displaystyle=-1z+t(z)+\sum\limits_{d\in\operatorname{NE}(X)_{\mathbb{Z}}}\sum\limits_{n=0}^{\infty}\sum\limits_{j=1}^{N}\displaystyle\frac{Q^{d}}{n!}\langle\frac{\phi_{j}}{-z-\psi},t(\psi),\dotsc,t(\psi)\rangle_{0,n,d}^{X,E}\phi^{j}$
$\displaystyle=-1z+t(z)+\sum\limits_{d\in\operatorname{NE}(X)_{\mathbb{Z}}}\sum\limits_{n=0}^{\infty}\frac{Q^{d}}{n!}(\text{ev}_{1})_{*}\left[\frac{e_{T}^{-1}(E_{0,n+1,d})}{-z-\psi_{1}}\prod\limits_{i=2}^{n+1}\text{ev}_{i}^{*}t(\psi_{i})\right].$

The definition of Lagrangian cone depends on the polarization. Different
polarizations will be employed as necessary. See Convention 3.1.

######  Remark 1.3.

Note that the various functions only exist in various suitably completed
spaces of $\mathcal{H}$, rather than $\mathcal{H}$ itself. The explanation of
“formal neighborhood” and a more rigorous definition of the theory can be made
using formal schemes as in Appendix B of [5] where it is painstakingly spelt
out (and in different guises in [14]).

##  2\. Virtual localization

Let $X$ be a smooth projective variety admitting an action by a torus
$T=(\mathbb{C}^{*})^{m}$. It induces an action of $T$ on
$\overline{\mathcal{M}}_{0,n}(X,d)$. Let $\overline{\mathcal{M}}_{\alpha}$ be
the connected components of the fixed loci in
$\overline{\mathcal{M}}_{0,n}(X,d)^{T}$ labeled by $\alpha$ with the inclusion
$i_{\alpha}:\overline{\mathcal{M}}_{\alpha}\rightarrow\overline{\mathcal{M}}_{0,n}(X,d)$.
The virtual fundamental class $[\overline{\mathcal{M}}_{0,n}(X,d)]^{vir}$ can
be written as [8]

$[\overline{\mathcal{M}}_{0,n}(X,d)]^{vir}=\sum_{\alpha}(i_{\alpha})_{*}\displaystyle\frac{[\overline{\mathcal{M}}_{\alpha}]^{vir}}{e_{T}(N^{vir}_{\alpha})}$

where $[\overline{\mathcal{M}}_{\alpha}]^{vir}$ is constructed from the fixed
part of the restriction of the perfect obstruction theory of
$\overline{\mathcal{M}}_{0,n}(X,d)$, and the virtual normal bundle
$N^{vir}_{\alpha}$ is the moving part of the two term complex in the perfect
obstruction theory of $\overline{\mathcal{M}}_{0,n}(X,d)$. Note that in our
situation, the indices $\alpha$ are the decorated graphs defined in the
Definition 2.15.

###  2.1. General facts on torus action

In this subsection and the next, a few general facts about torus actions are
stated, many without proofs. These facts are straightforward consequences of
results in [2] and [17]. However, they are not expressed there in exactly the
forms needed for our purpose and we have thus elected to collect them here.

Without further specification, by “invariant”, we always mean $T$-invariant
and an irreducible invariant curve on $X$ is always assumed to be a proper
reduced irreducible $1$-dimensional $T$-invariant subscheme of $X$.

$X^{T}\subset X$ denotes the fixed subscheme in $X$ under the torus action by
$T$. Let $Z_{1},\dotsb,Z_{l}$ be connected components of $X^{T}$. Let
$N_{j}:=N_{Z_{j}/X}$ be the normal bundle of $Z_{j}$ in $X$ and
$\iota_{j}:Z_{j}\rightarrow X$ be the inclusion.

Recall that

$C(T)=Hom_{\mathbb{C}^{*}}(T,\mathbb{C}^{*})$

is the group of characters of $T$. We have the decomposition of the normal
bundle $N_{j}$ on $Z_{j}$ into eigensheaves

$N_{j}=\bigoplus_{\chi\in C(T)}N_{j,\chi}.$

Let $\rho:\mathbb{C}^{*}\rightarrow T$ be a $1$-parameter subgroup.

######  Definition 2.1.

Define $(\rho,\chi)$ to be the weight of the composition
$\mathbb{C}^{*}\xrightarrow{\rho}T\xrightarrow{\chi}\mathbb{C}^{*}$.

Since $N_{j}$ is of finite rank, for a generic $1$-parameter subgroup $\rho$,
we have $(\rho,\chi)=0$ if and only if either $\chi=0$ or $N_{j,\chi}=0$.

######  Proposition 2.2.

There is a unique $T$-invariant reduced and irreducible closed subscheme
$Z_{j}^{\rho}$ such that

  1. (1)

$Z_{j}\subset Z_{j}^{\rho}$,

  2. (2)

$Z_{j}^{\rho}$ is regular along $Z_{j}$,

  3. (3)

$N_{Z_{j}/Z_{j}^{\rho}}=\bigoplus\limits_{(\rho,\chi)>0}N_{j,\chi}$.

This can be seen by first restricting the $T$-action to the $1$-parameter
subgroup determined by $\rho$. One can apply [2, Theorem 4.1] and take
$Z_{j}^{\rho}:=(X_{j})^{+}$ (note that $(X_{j})^{+}$ is under the notation of
[2] when their $G$ is our $T$ and their $(X^{G})_{j}$ is our $Z_{j}$). Notice
that a point $x\in Z_{j}^{\rho}$ is characterized by $\lim_{\lambda\rightarrow
0}\rho(\lambda)\cdot x\in Z_{j}$. Since the group $T$ is commutative,
$Z_{j}^{\rho}$ is also $T$-invariant.

Furthermore, by [2, Theorem 2.5], we are able to find a neighborhood $U$ of
$Z_{j}$ in $Z_{j}^{\rho}$ that is locally isomorphic to $(Z_{j}\cap U)\times
V$ with $V$ a $\mathbb{C}^{*}$ representation. We first of all use this
decomposition to describe irreducible invariant curves in $X$. An irreducible
invariant curve on $X$ must be either contained in a $Z_{j}$, or as the
closure of a $1$-dimensional orbit. Any $1$-dimensional orbit $\mathfrak{o}$
is isomorphic to $\mathbb{C}^{*}$ as it is a quotient of a torus. Suppose $C$
intersect $Z_{j}$ at $p$. If we fix the isomorphism
$\mathfrak{o}\cong\mathbb{C}^{*}$ in a way that the limit toward $0$ is $p$,
then $T\rightarrow\mathfrak{o}\cong\mathbb{C}^{*}$ determines a character
$\chi_{p}$. We call $\chi_{p}$ the _(integral) character of an irreducible
invariant curve near $p$_ in this case. If the irreducible invariant curve is
in $X^{T}$, we set $\chi_{p}=0$.

Now given a $\rho$ as in Proposition 2.2,and an irreducible invariant curve
$C$ with $p\in C\cap X^{T}$.

######  Proposition 2.3.

If $(\rho,\chi_{p})>0$, $C$ is contained in $Z_{j}^{\rho}$.

###### Proof.

$X$ can be covered by $T$-invariant affine open sets by [19]. Let $U$ be such
an open neighborhood of $p$ and let $I_{C}\subset\mathbb{C}[U]$ be the ideal
of $C\cap U\subset U$. Since $U$ is $T$-invariant, $\mathbb{C}[U]$ is graded
by the characters lattice $C(T)$, and $I_{C}$ is a homogeneous ideal. Because
$C\setminus\\{p\\}$ is isomorphic to $\mathbb{C}^{*}$ on which $T$ acts by
$\chi_{p}$,
$\mathbb{C}[U]/I_{C}\subset\mathbb{C}[C\setminus\\{p\\}]\cong\mathbb{C}[t,t^{-1}]$
where $t$ is a homogeneous element graded by $\chi_{p}$. Because the limit
$t\rightarrow 0$ exists (which is $p$), $\mathbb{C}[U]/I_{C}$ has nonempty
$a.\chi_{p}$-graded piece only if $a>0$. In other words, the image of
$\mathbb{C}[U]/I_{C}\subset\mathbb{C}[t,t^{-1}]$ in fact lies in
$\mathbb{C}[t]$.

By the construction in [2], $Z_{j}^{\rho}$ is cut out by homogeneous elements
$u\in\mathbb{C}[U]$ graded by $\chi^{\prime}$ such that
$(\rho,\chi^{\prime})<0$. Now the image of $u$ in $\mathbb{C}[U]/I_{C}$ must
be zero, as $(\rho,\chi_{p})>0$ and the fact that $\mathbb{C}[U]/I_{C}$ has
nonempty $a\chi_{p}$-graded piece only if $a>0$. Therefore any such $t$ lies
in $I_{C}$, i.e., $C$ is a closed subscheme of $Z_{j}^{\rho}$. ∎

######  Corollary 2.4.

Let $C$ be the closure of a $1$-dimensional orbit
$\mathfrak{o}\cong\mathbb{C}^{*}$. The limits of $0,\infty$ land in different
connected components $Z_{j}$, $Z_{j^{\prime}}$ of $X$. Furthermore, in an
neighborhood of $p\in Z_{j}\cap C$, the irreducible invariant curve $C$ can be
parameterized by $t$ as $(c_{1}t^{a_{1}},\ldots,c_{n}t^{a_{n}})$.

###### Proof.

The first part is obvious. There is an open neighborhood $U$ of $p$, such that
$Z_{j}^{\rho}\bigcap U$ is some open set times a $\mathbb{C}^{*}$
representation $V$. Since $C$ is invariant, $C\bigcap U\subset\\{p\\}\times
V$. As in the previous proof, embed $\mathbb{C}[V]/I_{C}\hookrightarrow k[t]$.
Let $x_{1},\dotsc,x_{n}$ be linear functions in $\mathbb{C}[V]$ that are
homogeneous with respect to $\mathbb{C}^{*}$ action. The image of $x_{i}$ in
$\mathbb{C}[t]$ must be homogeneous. Thus it is $c_{i}t^{a_{i}}$ for some
$c_{i}\in\mathbb{C},a_{i}\in\mathbb{Z}_{>0}$. ∎

In particular, we have

######  Corollary 2.5.

An irreducible invariant curve is homeomorphic to its normalization.

###  2.2. $T$-invariant stable maps

We now proceed to study genus zero invariant stable maps. The contents of this
subsection follows from results in [17] (and [4]). The $g=0$ assumption is
tacitly imposed throughout the paper, although most of the results hold in
general.

Define $C(T)_{\mathbb{Q}}:=C(T)\otimes_{\mathbb{Z}}\mathbb{Q}$. Let
$f:(C,x_{1},\dotsc,x_{n})\rightarrow X$ be an _invariant_ stable map and
$C_{0}$ an irreducible component of $C$. Let $p\in C_{0}$ be a point such that
$f(p)\in X^{T}$.

######  Definition 2.6.

Let $\chi_{f(p)}$ be the integral character of the irreducible invariant curve
$f(C_{0})$ at $f(p)$ and $k$ be the degree of $f|_{C_{0}}:C_{0}\rightarrow
f(C_{0})$. Define the fractional character of $C_{0}$ at $p$ to be
$\chi_{p}=\chi_{f(p)}/k\in C(T)_{\mathbb{Q}}$. In particular when
$f(C_{0})\subset X^{T}$, $\chi$ is $0$.

######  Remark 2.7.

We would like to emphasize again that there is a big difference of definitions
between the _integral character_ at a fixed point on a one-dimensional orbit
closure, and the _fractional character_ at a point on the domain curve of an
invariant stable map.

######  Proposition 2.8 ([17]).

The fractional character $\chi_{p}$ defined above is deformation invariant for
$T$-_invariant_ stable maps. More explicitly, let
$f:(\mathcal{C},x_{0},\dotsc,x_{n})\rightarrow X$ be a family of invariant
stable maps over a connected base $S$ and $x_{i}:S\rightarrow\mathcal{C}$ the
sections to the corresponding markings. Let $(C_{p})_{0}$ be the unique
irreducible component of the fiber $C_{p}$ containing $x_{0}(p)$. Then the
fractional character of $(C_{p})_{0}$ at $x_{0}(p)$ does not depend on the
choice of $p\in S$.

######  Remark 2.9.

In this proposition, stability is crucial. One can construct a counterexample
for semistable curves by making a family of curves specialized into a curve
with $x_{0}$ lying on a contracted component.

######  Proposition 2.10 ([17]).

Let $f:(C,x_{1},\dotsc,x_{n})\rightarrow X$ be an invariant stable map. Let
$C_{1},C_{2}$ be two irreducible components of $C$ such that $p\in C_{1}\cap
C_{2}$ is a node. The fractional characters of $C_{1},C_{2}$ at $p$ are
denoted as $\chi_{1},\chi_{2}$ respectively. Suppose $C_{1},C_{2}$ are not
contracted under $f$, then the node $p$ can be smoothened in a family of
invariant stable maps only if $\chi_{1}+\chi_{2}=0$.

The condition $\chi_{1}+\chi_{2}=0$ is consistent with [17, Definition 2.7(3)]
and is a necessary condition for smoothability within a component of the fixed
substack $\overline{\mathcal{M}}_{0,n}(X,d)^{T}$. In view of this, we make the
following definition.

######  Definition 2.11.

Notations are the same as the above proposition. $x$ is said to be a node
satisfying condition ($\star$) if

  * ($\star$)

$\chi_{1}\neq 0\,\text{and}\,\chi_{1}+\chi_{2}=0.$

We would like to introduce an important type of invariant stable maps whose
moduli spaces are main building blocks in the components of
$\overline{\mathcal{M}}_{0,n}(X,d)^{T}$, cf. [17, § 3] and [4, § 7.4].

######  Definition 2.12.

Given two fixed loci $Z_{j},Z_{j^{\prime}}$, by an unbroken map of a nonzero
fractional character $\chi\in C(T)_{\mathbb{Q}}$ between
$Z_{j},Z_{j^{\prime}}$, we mean an invariant stable map with $2$-markings
$f:(C,x_{+},x_{-})\rightarrow X$ such that

  1. (1)

$C$ is a chain of $\mathbb{P}^{1}$, with irreducible components
$C_{1},\dotsc,C_{l}$ such that $C_{i}\cap C_{i+1}\neq\emptyset$;

  2. (2)

$x_{+}\in C_{1}$, $x_{-}\in C_{l}$. Also $f(x_{+})\in Z_{j}$, and $f(x_{-})\in
Z_{j^{\prime}}$;

  3. (3)

the fractional character of $C$ at $x_{+}$ is $\chi$, and all nodes in $C$
satisfy ($\star$).

Moduli spaces of unbroken maps are used to describe the fixed substack of
$\overline{\mathcal{M}}_{0,n}(X,d)$ under the torus action. We list all
related definitions and notations below for later references.

######  Notations 2.13.

  * •

Given $Z_{j}$, $Z_{j^{\prime}}$ and a curve class $d\in\operatorname{NE}(X)$,
$\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$ denotes the closed substack of
$\overline{\mathcal{M}}_{0,2}(X,d)$ parametrizing unbroken maps of fractional
character $\chi$ between $Z_{j}$ and $Z_{j^{\prime}}$ whose curve class is
$d$.

  * •

Denote
$\pi_{j,j^{\prime},d,\chi}:\mathcal{C}_{j,j^{\prime},d,\chi}\rightarrow\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$
the universal family and
$f_{j,j^{\prime},d,\chi}:\mathcal{C}_{j,j^{\prime},d,\chi}\rightarrow X$ the
corresponding universal stable map.

  * •

Let $[\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}]^{vir}$ be the virtual
class induced by the restriction of the natural perfect obstruction theory
from $\overline{\mathcal{M}}_{0,2}(X,d)$.

  * •

There are two sections
$s_{j,j^{\prime},d,\chi}^{+},s_{j,j^{\prime},d,\chi}^{-}:\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}\rightarrow\mathcal{C}_{j,j^{\prime},d,\chi}$
mapping an unbroken map $f:(C,x_{+},x_{-})\rightarrow X$ between
$Z_{j},Z_{j^{\prime}}$ to the markings $x_{+}$ and $x_{-}$ respectively.

  * •

$\psi_{+}=(s_{j,j^{\prime},d,\chi}^{+})^{*}\omega_{\pi_{j,j^{\prime},d,\chi}}$,
$\psi_{-}=(s_{j,j^{\prime},d,\chi}^{-})^{*}\omega_{\pi_{j,j^{\prime},d,\chi}}$,
where $\omega_{\pi_{j,j^{\prime},d,\chi}}$ is the relative dualizing sheaf.

  * •

$\text{ev}_{+}=f_{j,j^{\prime},d,\chi}\circ
s_{j,j^{\prime},d,\chi}^{+}:\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}\rightarrow
Z_{j}$ and $\text{ev}_{-}=f_{j,j^{\prime},d,\chi}\circ
s_{j,j^{\prime},d,\chi}^{-}$.

  * •

Let $X_{j,j^{\prime},d,\chi}\subset X$ be the image of
$f_{j,j^{\prime},d,\chi}(\mathcal{C}_{j,j^{\prime},d,\chi})$.

######  Example 2.14.

Let $T=\mathbb{C}^{*}$ acts on $\mathbb{P}^{2}$ by sending
$[x_{0};x_{1};x_{2}]$ to $[\lambda^{-2}x_{0};\lambda^{-1}x_{1};x_{2}]$. By a
slight abuse of notation, also let $\lambda$ be the identity character of
$\mathbb{C}^{*}$. Let $Z_{0}=[1;0;0],Z_{1}=[0;1;0],Z_{2}=[0;0;1]$. The moduli
of unbroken maps between $Z_{0}$ and $Z_{2}$ with character $\lambda$ forms a
$1$-dimensional family that sweeps through $\mathbb{P}^{2}$. A generic member
of such unbroken maps has irreducible domain curve, but a special one
($[a;b;0],a,b\in\mathbb{C}$) breaks in $Z_{1}$ with condition ($\star$)
satisfied.

Let $l$ be the class of a line in $\mathbb{P}^{2}$. In this example,
$\overline{\mathcal{M}}_{0,1,l,\lambda}=\\{\text{pt}\\},\overline{\mathcal{M}}_{0,2,l,\lambda}=\mathbb{P}(1,2)$.
For $\overline{\mathcal{M}}_{0,2,l,\lambda/k}$, they parametrize degree $k$
covers of unbroken maps between $Z_{0},Z_{2}$. Although the coarse moduli
spaces are the same, we have
$\overline{\mathcal{M}}_{0,2,l,\lambda/k}=\mathbb{P}(k,2k)$.

###  2.3. Decorated graphs and moduli of invariant stable maps

We associate graphs to combinatorial types of fixed loci
$\overline{\mathcal{M}}_{0,n}(X,d)^{T}\subset\overline{\mathcal{M}}_{0,n}(X,d)$,
following the notations in [15, Definition 52].

######  Definition 2.15.

A decorated graph $\vec{\Gamma}=(\Gamma,\vec{p},\vec{d},\vec{s},\vec{\chi})$
for a genus-$0$, $n$-pointed, degree $d\in\operatorname{NE}(X)$ invariant
stable map consists of the following data.

  * •

$\Gamma$ a finite connected graph, $V(\Gamma)$ the set of vertices and
$E(\Gamma)$ the set of edges;

  * •

$F(\Gamma)=\\{(e,v)\in E(\Gamma)\times V(\Gamma)~{}|~{}v~{}\text{incident
to}~{}e\\}$ the set of flags;

  * •

the label map $\vec{p}:V(\Gamma)\rightarrow\\{1,\dotsc,l\\}$;

  * •

the degree map $\vec{d}:E(\Gamma)\cup
V(\Gamma)\rightarrow\operatorname{NE}(X)\cup\mathop{\bigcup}_{j}\operatorname{NE}(Z_{j})$,
where $\vec{d}$ sends a vertex $v$ to $\operatorname{NE}(Z_{\vec{p}(v)})$ and
an edge $e$ to $\operatorname{NE}(X)$;

  * •

the marking map $\vec{s}:\\{1,2,\dotsc,n\\}\rightarrow V(\Gamma)$ for $n>0$;

  * •

the fractional character map $\vec{\chi}:F(\Gamma)\rightarrow
C(T)_{\mathbb{Q}}$.

They are required to satisfy the following conditions:

  * •

$V(\Gamma),E(\Gamma),F(\Gamma)$ determine a connected graph.

  * •

$\sum\limits_{e\in E(\Gamma)}\vec{d}(e)+\sum\limits_{v\in
V(\Gamma)}\vec{d}(v)=d$.

  * •

For any vertex $v$ of valence $2$, the two edges $e_{1},e_{2}$ incident to $v$
satisfy $\vec{\chi}((e_{1},v))+\vec{\chi}((e_{2},v))\neq 0$.

We associate a decorated graph $\vec{\Gamma}$ to an invariant stable map $f$
in the following way.

Vertices:

  * •

The connected components in $f^{-1}(X^{T})$ are either curves or points.
Assign a vertex $v$ to a connected component $\mathfrak{c}_{v}$ in
$f^{-1}(X^{T})$ that is either a sub-curve of $C$, a smooth point in $C$, or a
node in $C$ that does _not_ satisfy condition ($\star$).

  * •

Define $\vec{p}(v)=j$ where $1\leq j\leq l$ is the label such that
$f(\mathfrak{c}_{v})\in Z_{j}$.

  * •

Define $\vec{d}(v)=[\mathfrak{c}_{v}]\in\operatorname{NE}(Z_{j})$ the
numerical class $f_{*}([\mathfrak{c}_{v}])$ (notice if $\mathfrak{c}_{v}$ is a
point, $[\mathfrak{c}_{v}]=0$).

  * •

For $i=1,\dotsc,n$, define $\vec{s}(i)=v$ if $x_{i}\in\mathfrak{c}_{v}$.

Edges:

  * •

Assign each component of $C\setminus\mathop{\bigcup}\limits_{v\in
V(\Gamma)}\mathfrak{c}_{v}$ an edge $e$. Let $\mathfrak{c}_{e}$ be the closure
of the corresponding component.

  * •

We write $\vec{d}(e)=f_{*}[\mathfrak{c}_{e}]\in\operatorname{NE}(X)$.

Flags:

  * •

$F(\Gamma)=\\{(e,v)\in E(\Gamma)\times
V(\Gamma)~{}|~{}\mathfrak{c}_{e}\cap\mathfrak{c}_{v}\neq\emptyset\\}$

  * •

Given a flag $(e,v)$, $\mathfrak{c}_{e}\cap\mathfrak{c}_{v}$ must consist of a
single point $x$. Let the fractional character of $\mathfrak{c}_{e}$ at $x$ be
$\chi$. Define $\vec{\chi}((e,v))=\chi$.

######  Remark 2.16.

In the case of toric variety, for any invariant stable map, none of the node
in $f^{-1}(X^{T})$ can satisfy condition ($\star$). Thus we reduce back to the
traditional definition of decorated graphs in, for example, [15].

######  Convention.

To further shorten the expressions, we adopt the following convention when the
decorated graph $\vec{\Gamma}$ is given in the context.

  * •

$p_{v}=\vec{p}(v)$,

  * •

$s_{i}=\vec{s}(i)$,

  * •

$d_{v}=\vec{d}(v)$, $d_{\Gamma}=\sum\limits_{e\in
E(\Gamma)}\vec{d}(e)+\sum\limits_{v\in V(\Gamma)}\vec{d}(v)$,

  * •

$\chi_{e,v}=\vec{\chi}((e,v))$.

Since edge curves with nontrivial $T$-action can only degenerate to create
nodes satisfying condition ($\star$), the associated graph is invariant under
deformations.

######  Definition 2.17.

Given a vertex $v\in V(\Gamma)$, define $E_{v}=\\{e\in E(\Gamma)|(e,v)\in
F(\Gamma)\\}$ to be the set of edges containing $v$. Also define
$S_{v}=\vec{s}^{-1}(v)\subset\\{1,\dotsc,n\\}$. Define $\text{val}(v)=|E_{v}|$
to be the valence of $v$, and $n_{v}=|S_{v}|$ to be the number of marked
points on $v$.

An edge $e$ must connect two vertices. We call the two vertices
$v_{1}(e),v_{2}(e)$ whose order is arbitrary. Given a vertex $v$, we write
$\overline{\mathcal{M}}_{0,E_{v}\cup S_{v}}(Z_{j},d)$ instead of
$\overline{\mathcal{M}}_{0,\text{val}(v)+n_{v}}(Z_{j},d)$ if we want to index
the marked points by edges in $E_{v}$ and markings in $S_{v}$. Under such
notations, the corresponding evaluation map at marked point labeled by $e\in
E_{v}$ is denoted by $\text{ev}_{e}$.

Given a decorated graph $\vec{\Gamma}$, one can construct the corresponding
fixed component in $\overline{\mathcal{M}}_{0,n}(X,d)$ out of fiber products
of $\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$ and
$\overline{\mathcal{M}}_{0,n_{i}}(Z_{j},d)$. Let $\vec{\Gamma}$ be a decorated
graph. Define

$\overline{\mathcal{M}}_{\vec{\Gamma}}=\left[\prod\limits_{v\in
V(\Gamma)}\overline{\mathcal{M}}_{0,E_{v}\cup
S_{v}}(Z_{p_{v}},d_{v})\right]\times^{\prime}\left[\prod\limits_{e\in
E(\Gamma)}\overline{\mathcal{M}}_{p_{v_{1}(e)},p_{v_{2}(e)},d_{e},\chi_{e,v_{1}(e)}}\right]$

where we make the convention that
$\overline{\mathcal{M}}_{0,n}(Z_{j},0):=Z_{j}$ for $n=1,2$. Here the
$\times^{\prime}$ is the fiber product defined as follows. For any flag
$(e,v)$, let $v^{\prime}$ be the other vertex on $e$ different from $v$. The
$\times^{\prime}$ identifies the evaluation map $\text{ev}_{e}$ in
$\overline{\mathcal{M}}_{0,E_{v}\cup S_{v}}(Z_{p_{v}},d_{v})$ with $ev_{+}$ in
$\overline{\mathcal{M}}_{p_{v},p_{v^{\prime}},d_{e},\chi_{e,v_{1}(e)}}$.

Let $\text{Aut}(\Gamma)$ be the automorphism of graphs. Note that automorphism
coming from multiple cover (denoted by $k$) is already accounted in
$\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$. We have:

$[\overline{\mathcal{M}}_{\vec{\Gamma}}/\text{Aut}(\Gamma)]\subset\overline{\mathcal{M}}_{0,n}(X,d)^{T}.$

Moreover, $[\overline{\mathcal{M}}_{\vec{\Gamma}}/\text{Aut}(\Gamma)]$ is a
union of connected components of $\overline{\mathcal{M}}_{0,n}(X,d)^{T}$.

##  3\. A recursion relation

###  3.1. Statement

In this subsection, we state a technical result (Theorem 3.5), generalizing A.
Givental’s theorem in [3, Theorem 2]. It is a way of packaging the virtual
localization into a recursive form and is applied in Section 4 to prove our
main result on Gromov–Witten theory of projective (non-split) bundles. Before
the statement of the theorem, a few definitions are in order.

Recall that, given a meromorphic function $f(z)$, the _principal part_ at
$z=a$, denoted $\mathop{\bf{Prin}}\limits_{z=a}f$, is a polynomial in
$1/(z-a)$ without constant term such that $f-\mathop{\bf{Prin}}\limits_{z=a}f$
has no pole at $z=a$.

Let $F$ be a family of points in the Lagrangian cone introduced earlier and
let $F^{j}:=\iota_{j}^{*}F$. Localization formula implies that

(1)
$F=\sum\limits_{j=1}^{l}(\iota_{j})_{!}\displaystyle\frac{F^{j}}{e_{T}(N_{j})}.$

When we restrict via $\iota_{j}$, the following function spaces and
polarizations are used.

######  Convention 3.1.

Let $\mathcal{H}^{j}$ be

$\mathcal{H}^{j}:=H^{*}(Z_{j};S_{T})\left[z,\frac{1}{z}\right]\left[\mathfrak{S}\right][\\![Q]\\!],$

where $\displaystyle\mathfrak{S}=\left\\{\frac{1}{z+\chi}\right\\}_{\chi\in
C(T)_{\mathbb{Q}}}$. The polarization is given by
$\mathcal{H}^{j}=\mathcal{H}^{j}_{+}\oplus\mathcal{H}^{j}_{-}$ such that
$\mathcal{H}^{j}_{+}$ has no $z^{-1}$ (but has $(z+\chi)^{-1}$) and

$\mathcal{H}^{j}_{-}=z^{-1}H^{*}(Z_{j};S_{T})[z^{-1}][\\![Q]\\!].$

The Lagrangian cone in the function space $\mathcal{H}^{j}$ under the above
polarization is denoted by $\mathcal{L}^{j}_{E}$.

######  Remark 3.2.

Even though the above definition of $\mathcal{H}^{j}$ appears to involve
polynomials of infinite indeterminates, in practice this is not needed. Given
a torus action on a fixed $X$, there are only finitely many integral
characters $\chi\in C(T)$ associated to all one-dimensional orbits in $X$. For
a fixed $d$ in $Q^{d}$, the _fractional characters_ involved is of the form
$\chi/k$, where $\chi$ are the integral characters of one-dimensional orbits,
and $k$ is linearly bounded by the numerical class $d$.

A priori we do not have $F^{j}\in(\mathcal{H}^{j},-1z)$, but the localization
computation in the next subsection allows us to write $F^{j}$ as an element in
the formal neighborhood $(\mathcal{H}^{j},-1z)$. $F$ as a Laurent series in
$z^{-1}$ can be recovered using localization formula (1) by expanding
$(z+\chi)^{-1}$ as power series in $z^{-1}$. We note that $F^{j}$ satisfies
the following properties.

  * •

The Novikov variables $Q^{d}$ still have exponents $d\in\operatorname{NE}(X)$.

  * •

The coefficient of $Q^{d}$ for a fixed $d$ is a polynomial instead of a power
series in $z^{-1}$.

######  Remark 3.3.

The polarization
$\mathcal{H}^{j}=\mathcal{H}^{j}_{+}\oplus\mathcal{H}^{j}_{-}$ is a form of
partial fraction decomposition. That is, any element $p(z)\in\mathcal{H}^{j}$
can be uniquely decomposed into

$p(z)=[p(z)]_{+}+[p(z)]_{\bullet}+[p(z)]_{-},$

where

$\displaystyle[p(z)]_{+}$ $\displaystyle\in H^{*}(Z_{j};S_{T})[z][\\![Q]\\!],$
$\displaystyle[p(z)]_{\bullet}$ $\displaystyle\in\bigoplus\limits_{\chi\in
C(T)_{\mathbb{Q}}}(z+\chi)^{-1}H^{*}(Z_{j};S_{T})[(z+\chi)^{-1}][\\![Q]\\!],~{}\text{and}$
$\displaystyle[p(z)]_{-}$ $\displaystyle\in
z^{-1}H^{*}(Z_{j};S_{T})[z^{-1}][\\![Q]\\!].$

This also induces a decomposition of elements in $(\mathcal{H}^{j},-1z)$.

######  Remark 3.4.

We use notation $\mathcal{L}^{j}_{E}$ other than $\mathcal{L}_{Z_{j},E}$ to
emphasize the different underlying function spaces and polarizations. Under
this new polarization, the points in $\mathcal{L}^{j}_{E}$ allow
$(z+\chi)^{-1}$ terms in $t(z)$, i.e.,

$t(z)\in
H^{*}(Z_{j};S_{T})[\\![t_{i}^{1},\dotsb,t_{i}^{N}]\\!][z,(z+\chi)^{-1}][\\![Q]\\!],~{}i\geq
0,~{}\chi\in C(T)_{\mathbb{Q}},~{}N=\text{dim}(H^{*}(Z_{j};\mathbb{C})).$

We now state the recursion relation.

######  Theorem 3.5.

Assume all $e_{T}(E_{0,n,d})^{-1}$ involved are defined. There exist
equivariant cohomology classes $A_{i,k;j,j^{\prime},d,\chi}\in
H_{T}^{*}(\overline{\mathcal{M}}_{j,j^{\prime},d,\chi})\otimes_{R_{T}}S_{T}$
such that (a formal family of point) $F\in(\mathcal{H},-1z)$ lies in
$\mathcal{L}_{X,E}$ if and only if its restrictions $F^{j}$ can be written as
elements in $(\mathcal{H}^{j},-1z)$ and satisfy the following conditions:

  1. (1)

$F^{j}(-z)\in\mathcal{L}^{j}_{N_{j}\oplus E}$ (under the indicated
polarization);

  2. (2)

The principal parts of $\\{F^{j}(-z)\\}$ satisfy recursively defined
expressions:

$\mathop{\bf{Prin}}\limits_{z=-\chi}F^{j}(z)=\sum\limits_{i,k,j^{\prime},d,\chi}(ev_{+})^{vir}_{*}\left[\frac{Q^{d}A_{i,k;j,j^{\prime},d,\chi}\left.\displaystyle\frac{\partial^{k}}{\partial
w^{k}}\left[(ev_{-})^{*}\left(F^{j^{\prime}}(w)-\mathop{\bf{Prin}}\limits_{w=-\chi}F^{j^{\prime}}(w)\right)\right]\right|_{w=-\chi}}{(z+\chi)^{i}}\right]$

where $ev_{-}$ is the map defined on
$\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$.

Furthermore, $A_{i,k;j,j^{\prime},d,\chi}\in
H_{T}^{*}(\overline{\mathcal{M}}_{j,j^{\prime},d,\chi})$ are uniquely
determined by

  * (i)

the classes of equivariant vector bundles $[(TX\oplus
E)|_{X_{j,j^{\prime},d,\chi}}]\in K^{0}_{T}(X_{j,j^{\prime},d,\chi})$,

  * (ii)

the classes of equivariant vector bundles $[N_{j}\oplus E]\in
K^{0}_{T}(Z_{j})$,

  * (iii)

the moduli stacks $\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$, their
virtual classes, and their universal families
$\mathcal{C}_{j,j^{\prime},d,\chi}$.

Condition (2) is sometimes referred to as the “recursion relation”. The
recursion relation comes from virtual localization, cf., Equation (4) in the
proof below.

######  Remark 3.6.

The above sum for $\mathop{\bf{Prin}}\limits_{z=-\chi}F^{j}(z)$ is finite in
$i,k,j^{\prime},\chi$ for a fixed $d$ when the target variety is of finite
type. That is, for a fixed $d$, all but finitely many
$A_{i,k;j,j^{\prime},d,\chi}$ vanish. This follows from the proof given below.

We state a corollary to be used in the last section. Let $X$ be a smooth
variety with a torus actions by $T$. Let $W$ be a smooth invariant subvariety
of $X$. Assume $e_{T}((N_{W/X})_{0,n,d})^{-1}$ are defined (or defined as a
limit according to Remark 1.2). We compare the recursion relations.

######  Corollary 3.7.

$X,W$ as above. Fix a $\chi\in C(T)_{\mathbb{Q}}$. Fix a $Z_{j}\subset W$.
Suppose for any $1\leq j^{\prime}\leq l$, $d\in\operatorname{NE}(X)$, we have
either $\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}=\emptyset$ or
$X_{j,j^{\prime},d,\chi}\subset W$. Then $A_{i,k;j,j^{\prime},d,\chi}$ in the
Theorem 3.5 for untwisted theory on $X$ equal to those of the twisted theory
by $N_{W/X}$ on $W$.

###### Proof.

Observe that $TW\oplus N_{W/X}$ and $N_{Z_{j}/W}\oplus N_{W/X}|_{Z_{j}}$ can
be deformed to $TX|_{W}$ and $N_{Z_{j}/X}$ respectively. Hence they represent
the same elements in the $K$-groups. All inputs involved in (i-iii) for
$A_{i,k;j,j^{\prime},d,\chi}$ are therefore identified. ∎

The proof of Theorem 3.5 occupies the next subsection.

###  3.2. Proof of Theorem 3.5

The following proof is parallel to the proof of Givental’s theorem in [3,
Theorem 2].

For notational convenience, introduce

$(ev_{+})^{\operatorname{vir}}_{*}(\alpha):=\operatorname{PD}\left[(ev_{+})_{*}(\alpha\cap[\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}]^{vir})\right]$

where $\operatorname{PD}$ is the Poincaré duality map (The case for $ev_{-}$
on $\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$ or $ev_{i}$ on
$\overline{\mathcal{M}}_{0,n}(X,d)$ should be understood accordingly).

Recall $F(-z)\in\mathcal{L}_{E}$, and $F^{j}(-z):=\iota_{j}^{*}F(-z)$ is, by
definition,

(2)
$F^{j}(-z,t)=-1z+\iota_{j}^{*}t+\iota_{j}^{*}\sum\limits_{n,d}\displaystyle\frac{Q^{d}}{n!}(\text{ev}_{1})^{\operatorname{vir}}_{*}\left[\frac{e_{T}^{-1}(E_{0,n+1,d})}{-z-\psi_{1}}\prod\limits_{i=2}^{n+1}\text{ev}_{i}^{*}t(\psi_{i})\right].$

Later the variable $t$ in “$F^{j}(-z,t)$” is suppressed and change of
variables (for example $F^{j}(w)$) occur only in the “$-z$” variable. We
evaluate the above expression by virtual localization.

Let $\vec{\Gamma}$ be a decorated graph. and let
$N_{\vec{\Gamma}}^{\text{vir}}$ be the virtual normal bundle on
$\overline{\mathcal{M}}_{\vec{\Gamma}}$. Then

$F^{j}=-1z+\iota^{*}_{j}t+\sum\limits_{\begin{subarray}{c}\vec{\Gamma}~{}~{}\text{with}\\\
p_{s_{1}}=j\end{subarray}}\displaystyle\frac{1}{\text{Aut}(\Gamma)}\frac{Q^{d}}{n!}\iota_{j}^{*}(\text{ev}_{1})^{\operatorname{vir}}_{*}\left[\frac{e_{T}^{-1}(E_{0,n+1,d}\oplus
N_{\vec{\Gamma}}^{vir})}{-z-\psi_{1}}\prod\limits_{i=2}^{n+1}\text{ev}_{i}^{*}t(\psi_{i})\right],$

where $s_{1}$ is the vertex where the first marking lies (see Definition
2.15). Given a decorated graph $\vec{\Gamma}$ with $\sum\limits_{v\in
V(\Gamma)}d_{v}=d$, write the contribution of $\vec{\Gamma}$ as

$\text{Cont}_{\vec{\Gamma}}(z)=\displaystyle\frac{1}{n!\text{Aut}(\Gamma)}\iota_{j}^{*}(\text{ev}_{1})^{\operatorname{vir}}_{*}\left[\frac{e_{T}^{-1}(E_{0,n+1,d}\oplus
N_{\vec{\Gamma}}^{vir})}{-z-\psi_{1}}\prod\limits_{i=2}^{n+1}\text{ev}_{i}^{*}t(\psi_{i})\right].$

Since $\overline{\mathcal{M}}_{\vec{\Gamma}}$ is a fiber product of a
collection of $\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$ and
$\overline{\mathcal{M}}_{0,E_{v}\cup S_{v}}(Z_{p_{v}},d_{v})$, this graph
contribution can be computed by integrals on each of these spaces with
suitable pull-backs and push-forwards. We will see in a moment that there is a
recursive structure in the summation over graphs $\vec{\Gamma}$ such that
$p_{s_{1}}=j$ (i.e., the first marked point lies on $Z_{j}$).

Consider first the contribution from graphs when the vertex $s_{1}$ is
incident to a single edge with fractional character $\chi\in
C(T)_{\mathbb{Q}}$. In this case, ${s_{1}}$ must be connected to another
vertex $v^{\prime}$ such that $p_{v^{\prime}}=j^{\prime}$ for some
$j^{\prime}\neq j=p_{s_{1}}$. Let $e$ be the edge connecting $s_{1}$ and
$v^{\prime}$, $d_{s_{1}}:=d_{e}\in\operatorname{NE}(X)$ be the degree
associated to the edge $e$, and $\chi=\chi_{e,s_{1}}$ be the fractional
character at $s_{1}$.

Introduce $t^{j,\chi}(z)$ as

$\displaystyle
t^{j,\chi}(z):=\sum\limits_{\begin{subarray}{c}\vec{\Gamma}~{}~{}\text{with}\\\
d_{s_{1}}=0,\,\chi_{e,s_{1}}=\chi\\\
p_{s_{1}}=j,\,n_{s_{1}}=0,\,\text{val}(s_{1})=1\end{subarray}}Q^{d_{\Gamma}}\text{Cont}_{\vec{\Gamma}}(z)$
$\displaystyle=\sum\limits_{j^{\prime}\neq
j,d}\iota_{j}^{*}(\text{ev}_{+})^{vir}_{*}\left[\displaystyle\frac{Q^{d}\text{Cont}_{(e,s_{1})}}{-z-\psi_{+}+\chi}\sum\limits_{\begin{subarray}{c}\vec{\Gamma}~{}~{}\text{with}~{}d_{s_{1}}=0,\\\
\chi_{e,s_{1}}\neq\chi,p_{s_{1}}=j^{\prime}\end{subarray}}Q^{d_{\Gamma}}\left\\{(\text{ev}_{-})^{*}\text{Cont}_{\vec{\Gamma}}(w)\right\\}_{w=\psi_{-}-\chi}\right].$

Expanding the sum over graphs, we get

(3) $t^{j,\chi}(z)=\sum\limits_{j^{\prime}\neq
j,d}\iota_{j}^{*}(\text{ev}_{+})^{vir}_{*}\left[\displaystyle\frac{Q^{d}\text{Cont}_{(e,s_{1})}}{-z-\psi_{+}+\chi}\left\\{(\text{ev}_{-})^{*}\left[F^{j^{\prime}}(w)-t^{j^{\prime},\chi}(w)\right]\right\\}_{w=\psi_{-}-\chi}\right]$

where the $ev_{+},ev_{-}$ are the maps from
$\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$, and the contribution

$\text{Cont}_{(e,s_{1})}=\displaystyle\displaystyle\frac{e_{T}(\left[\pi_{j,j^{\prime},d,\chi})_{!}T_{\pi_{j,j^{\prime},d,\chi}}(-D_{+})\right]^{m})}{e_{T}(\left[(\pi_{j,j^{\prime},d,\chi})_{!}f_{j,j^{\prime},d,\chi}^{*}(TX\oplus
E)\right]^{m})}.$

In the above, $T_{\pi_{j,j^{\prime},d,\chi}}$ is the relative tangent bundle.
$D_{+}$ is the divisor corresponding to the image of
$s_{j,j^{\prime},d,\chi}^{+}$. Superscript $m$ means the moving part, i.e.,
the subsheaf generated by homogeneous elements of nontrivial $T$-character
([8]). In writing this, we also use the fact that $E_{0,n,d}$ does not have
fixed part due to the assumption that $E_{0,n,d}$ has invertible Euler class.

We interrupt the flow with a few remarks.

######  Remark 3.8.

The reason to subtract $t^{j^{\prime},\chi}(w)$ in equation (3) is to exclude
the case when the other end at $Z_{j^{\prime}}$ is directly connected to
another unbroken map of the same fractional character $\chi$ (thus satisfying
condition ($\star$) and violating the assignment of decorated graphs).

######  Remark 3.9.

It might appear that there is a missing factor coming from the gluing at the
node on the $s_{j,j^{\prime},d,\chi}^{-}$ side. However, it is automatically
taken care of because, in the expression
$(\text{ev}_{-})^{*}\left[F^{j^{\prime}}(w)-t^{j^{\prime},\chi}(w)\right]$,
the definition of $F^{j^{\prime}}(w)-t^{j^{\prime},\chi}(w)$ already involves
the pull-back under $\iota_{j}$ and the push-forward under certain evaluation
maps.

######  Remark 3.10.

Since we pull back along $f_{j,j^{\prime},d,\chi}$, the outcome only depends
on the restriction $(TX\oplus E)|_{X_{j,j^{\prime},d,\chi}}$. One can see the
corresponding euler classes only depends on the class of $TX\oplus E$ in the
$K^{0}(X_{j,j^{\prime},d,\chi})$.

Now let’s reorganize $t^{j,\chi}$. Plugging $w=\psi_{-}-\chi$ into
$(ev_{-})^{*}F^{j^{\prime}}(w)$ in the expression (3), the term
$(ev_{-})^{*}F^{j^{\prime}}(\psi_{-}-\chi)$ can be expanded using the Taylor
expansion

$\displaystyle(ev_{-})^{*}F^{j^{\prime}}(\psi_{-}-\chi)=$
$\displaystyle(ev_{-})^{*}F^{j^{\prime}}(-\chi)$ $\displaystyle+$
$\displaystyle\psi_{-}(ev_{-})^{*}\displaystyle\frac{\partial}{\partial
t}\left.F^{j^{\prime}}(t)\right|_{t=-\chi}+\frac{(\psi_{-})^{2}}{2!}(ev_{-})^{*}\frac{\partial^{2}}{\partial
z^{2}}\left.F^{j^{\prime}}(t)\right|_{t=-\chi}+\dotsc.$

As $\psi_{-}$ is nilpotent in
$H^{*}((\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}))$, the above is a finite
sum.

Applying the above Taylor expansion and expanding
$\displaystyle\frac{1}{-z-\psi_{+}+\chi}$ in terms of $(-z+\chi)^{-1}$, we
have

(4)
$t^{j,\chi}(z)=\iota_{j}^{*}(ev_{+})^{vir}_{*}\left[\sum\limits_{i,k,j^{\prime},d,\chi}\frac{Q^{d}A_{i,k;j,j^{\prime},d,\chi}\left.\displaystyle\frac{\partial^{k}}{\partial
w^{k}}\left[(ev_{-})^{*}(F^{j^{\prime}}(w)-t^{j^{\prime},\chi}(w))\right]\right|_{w=-\chi}}{(-z+\chi)^{i}}\right],$

where $A_{i,j^{\prime},k,d,\chi}\in
H^{*}(\overline{\mathcal{M}}_{j,j^{\prime},d,\chi})$ are some classes whose
exact expressions does not concern us in this paper. In a moment, we will show
that all poles of $F^{j}$ at $-\chi$ come from $t^{j,\chi}$, i.e.,
$\mathop{\bf{Prin}}\limits_{w=-\chi}F^{j}(w)=t^{j,\chi}(w)$.

######  Caution.

There might be a confusion of signs. Notice we use $-z$ in $F^{j}(-z)$ but $w$
in $F^{j}(w)$. However, $t^{j,\chi}(z)$ and $t^{j,\chi}(w)$ are used, where
both $z$ and $w$ variables carry positive sign. As a result, the pole of
$F^{j}(w)$ at $-\chi$ means the pole at $w=-\chi$, while $z=\chi$. This choice
of signs is dictated by the localization expression.

Next we consider the case when the vertex $s_{1}$ is incident with more than
one edge. Write

$t^{j}(z)=\iota_{j}^{*}t(z)+\sum\limits_{\chi}t^{j,\chi}(z).$

Later $t^{j}$ will fit into Equation (2). Let us start by applying virtual
localization to Equation (2):

$\displaystyle F^{j}(-z)$ $\displaystyle=$
$\displaystyle-1z+\iota_{j}^{*}t+\sum\limits_{\vec{\Gamma}~{}\text{with}~{}p_{s_{1}}=j}Q^{d_{\Gamma}}\text{Cont}_{\vec{\Gamma}}(z)$
$\displaystyle=$
$\displaystyle-1z+\iota_{j}^{*}t+\sum\limits_{\begin{subarray}{c}d,n,\\\
j_{1},\dotsc,j_{n+1}\end{subarray}}\iota_{j}^{*}(\text{ev}_{1})^{\operatorname{vir}}_{*}\cdot$
$\displaystyle\qquad\cdot\left[\displaystyle\frac{Q^{d}}{n!}\frac{\text{Cont}_{s_{1}}}{(-z-\psi_{1})}\prod\limits_{i=2}^{n+1}(\text{ev})_{i}^{*}\left\\{\iota_{j}^{*}t(\psi_{+}-\chi)+\sum\limits_{\begin{subarray}{c}\vec{\Gamma}~{}\text{with}~{}d_{s_{1}}=0,\\\
p_{s_{1}}=j_{i},\,n_{s_{1}}=0,\,\text{val}(s_{1})=1\end{subarray}}Q^{d_{\Gamma}}\text{Cont}_{\vec{\Gamma}}(\psi_{+}-\chi)\right\\}\right]$

where $\text{ev}_{i}$ are the evaluation maps in
$\overline{\mathcal{M}}_{0,n+1}(Z_{j},d)$ and

$\text{Cont}_{s_{1}}=e_{T}^{-1}(E_{0,n+1,d}\oplus(N_{j})_{0,n+1,d}).$

This expression follows from localization computation by observing the
following points. Firstly, the vertex $s_{1}$ contributes
$e_{T}^{-1}((N_{j})_{0,n+1,d})$ to the Euler class of virtual normal sheaf.
Secondly, the contribution coming from smoothing the nodes is included in
$\text{Cont}_{\vec{\Gamma}}(\psi_{1}-\chi)$.

Since

$t^{j}=\iota_{j}^{*}t+\sum\limits_{\begin{subarray}{c}\vec{\Gamma}~{}~{}\text{with}~{}d_{s_{1}}=0,\\\
p_{s_{1}}=j,n_{s_{1}}=0,\text{val}(s_{1})=1\end{subarray}}Q^{d_{\Gamma}}\text{Cont}_{\vec{\Gamma}}(z),$

we have

$\displaystyle F^{j}(-z)$
$\displaystyle=-1z+t^{j}(z)+\sum\limits_{n,d}\frac{Q^{d}}{n!}\iota_{j}^{*}(ev_{1})^{vir}_{*}\left[\frac{e_{T}^{-1}(E\oplus(N_{j})_{0,n+1,d})}{-z-\psi_{1}}\prod\limits_{i=2}^{n+1}\text{ev}_{i}^{*}t^{j}(\psi_{1})\right].$

######  Remark 3.11.

Notice in the above expression, $e_{T}^{-1}(E\oplus(N_{j})_{0,n+1,d})$ only
depends on the class $[E\oplus N_{j}]\in K^{0}_{T}(Z_{j})$. Furthermore, since
$\psi_{1}$ is nilpotent for a fixed degree of Novikov variablse $Q^{d}$, the
last term is a polynomial in $z^{-1}$ since there are finitely many graphs of
curve class $\leq d$.

Observe now the only term contributing poles not at $z=0$ or $z=\infty$ is
$t^{j}(z)$. In particular, the pole at $z=-\chi$ is of the following form

(5)
$t^{j,\chi}(z)=\iota_{j}^{*}(ev_{+})^{vir}_{*}\left[\sum\limits_{i,j^{\prime},d,k,\chi}\frac{Q^{d}A_{i,k;j,j^{\prime},d,\chi}\left.\displaystyle\frac{\partial^{k}}{\partial
w^{k}}\left[(ev_{-})^{*}\left(F^{j^{\prime}}(w)-\mathop{\bf{Prin}}\limits_{z=-\chi}F^{j^{\prime}}(w)\right)\right]\right|_{w=-\chi}}{(-z+\chi)^{i}}\right]$

This is exactly $\mathop{\bf{Prin}}\limits_{z=-\chi}F^{j}(z)$ in Theorem 3.5.

Let us summarize what we have obtained so far. If $F\in\mathcal{L}_{E}$, there
exists $F^{j}\in\mathcal{L}_{N_{j}\oplus E}^{j}$, satisfying (1),(2) in
Theorem 3.5, such that when expanded in $1/z$, $\iota_{j}^{*}F=F^{j}$ as
Laurent series in $1/z$. The coefficients $A_{i,j,j^{\prime},k,\chi}$ only
depend on the classes of vector bundle $(TX\oplus
E)|_{X_{j,j^{\prime},d,\chi}}$, $N_{j}\oplus E|_{Z_{j}}$ in their
corresponding K-groups, $\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$,
$[\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}]^{vir}$ and their universal
families $\mathcal{C}_{j,j^{\prime},d,\chi}$

This is the forward implication of the theorem. We now proceed to prove the
converse. Given $\\{F^{j}\\}$ satisfying (1), (2) in Theorem 3.5, we set

$F:=\sum\limits_{j=1}^{l}(\iota_{j})_{!}\displaystyle\frac{F^{j}}{e_{T}(N_{j})}.$

It follows from localization that $F^{j}=\iota_{j}^{*}F$. Our goal is to show
that $F\in\mathcal{L}_{E}$. On the other hand, there is always a point
$F^{\prime}\in\mathcal{L}_{E}$ such that $[F^{\prime}]_{+}=[F]_{+}$. By
previous localization computation, we can find $F^{\prime j}$ such that
$F^{\prime j}=\iota_{j}^{*}F^{\prime}$ as formal functions. It follows that
$[F^{\prime j}]_{+}=[F^{j}]_{+}$. The subsequent argument shows that
_$[F^{j}]_{\bullet}$ and $[F^{j}]_{-}$ are uniquely determined by
$[F^{j}]_{+}$ and the recursion relation_ (condition (2) of Theorem 3.5),
_assuming $F^{j}\in\mathcal{L}^{j}_{N_{j}\oplus E}$_ (condition (1) of Theorem
3.5). Since $F^{j}$, $F^{\prime j}$ both satisfy conditions (1) and (2) in
Theorem 3.5, it follows that $[F^{j}]_{\bullet}=[F^{\prime j}]_{\bullet}$,
$[F^{j}]_{-}=[F^{\prime j}]_{-}$, and therefore, $F=F^{\prime}$. To highlight
the function space we are working on, the following convention is adapted.

We work on the formal neighborhood of the space $\mathcal{H}^{j}$ with
polarization, defined similarly to that of $\mathcal{H}$ in Convention 3.1.
The decomposition
$\mathcal{H}^{j}=\mathcal{H}^{j}_{+}\oplus\mathcal{H}^{j}_{-}$ induces a
decomposition
$(\mathcal{H}^{j},-1z)=(\mathcal{H}^{j},-1z)_{+}\oplus(\mathcal{H}^{j},-1z)_{-}$.

To simplify the expressions, we slightly abuse the notation in the following
way.

$\mathcal{H}^{j}:=(\mathcal{H}^{j},-1z),\quad\mathcal{H}^{j}_{+}:=(\mathcal{H}^{j},-1z)_{+},\quad\mathcal{H}^{j}_{-}:=(\mathcal{H}^{j},-1z)_{-}.$

Since $\\{F^{j}\\}$ lie in the corresponding Lagrangian cones by assumption,
there exists a unique $u^{j}\in H^{j}:=H^{*}(Z_{j})$ such that
$G^{j}(z):=S_{u^{j}}(-z)F^{j}(z)\in z\mathcal{H}^{j}_{+}$ for a given $j$.
(See [3, §2.3].) Notice that $F^{j}$ can be recovered according to
$S_{u^{j}}^{*}(z)G^{j}(z)=F^{j}(z)$. We proceed to show that there is a
recursive relation for $G^{j}$ determined by $[F^{j}]_{+}$.

First of all we have the $S$ matrix $S_{u^{j}}(-z)=1+O(1/z)\in
1+\mathcal{H}^{j}_{-}$, a formal series in $1/z$ but nevertheless a
_polynomial_ in $1/z$ for each $Q^{d}$ term. It follows from formal
manipulation of formal series in $z^{-1}$ that the Taylor polynomial of order
$K$ at $z=-\chi$ for the $S$-matrix can be written as

$\displaystyle
S_{u^{j}}(-z)=\left(S_{u^{j}}(\chi)+\displaystyle\frac{\partial}{\partial
z}S_{u^{j}}(\chi)(-z-\chi)+\dotsc+\displaystyle\frac{1}{K!}\frac{\partial^{K}}{\partial
z^{K}}S_{u^{j}}(\chi)(-z-\chi)^{K}\right)+(-z-\chi)^{K+1}R(z^{-1})$

where $R(z^{-1})\in\mathcal{H}^{j}_{-}$ is the remainder term.

Setting

$\bar{P}_{i,\chi}^{j}=\iota_{j}^{*}(ev_{+})^{vir}_{*}\left[\sum\limits_{j^{\prime},k,d}Q^{d}A_{i,k;j,j^{\prime},d,\chi}\left.\displaystyle\frac{\partial^{k}}{\partial
z^{k}}\left[(ev_{-})^{*}\left[S_{u^{j^{\prime}}}^{*}(z)G^{j^{\prime}}(z)-\mathop{\bf{Prin}}\limits_{z=-\chi}\left(S_{u^{j^{\prime}}}^{*}(z)G^{j^{\prime}}(z)\right)\right]\right]\right|_{z=-\chi}\right],$

we have

$\mathop{\bf{Prin}}\limits_{z=-\chi}F^{j}(z)=\sum\limits_{i=0}^{\infty}\frac{\bar{P}_{i,\chi}^{j}}{(z+\chi)^{i}}.$

Notice that this sum has finite terms for any fixed power of Novikov
variables. That is, the coefficient of $Q^{d}$ has finite order pole .
Therefore it lives in the function space $\mathcal{H}^{j}$. The following is
the recursion relation of $G^{j}$ promised earlier.

######  Lemma 3.12.

We have the following recursion relation of $G^{j}(z)$

(6)
$G^{j}(z)=[G^{j}(z)]_{+}+\sum\limits_{\chi}\left(\sum\limits_{i=0}^{\infty}\displaystyle\frac{\bar{P}_{i,\chi}^{j}}{(z+\chi)^{i}}\sum\limits_{l=0}^{i-1}\displaystyle\frac{\partial^{i}}{\partial
z^{l}}S_{u^{j}}(\chi)(-z-\chi)^{l}\right).$

Notice that this becomes a finite sum for fixed Novikov variables.

###### Proof.

The sum $\sum\limits_{l=0}^{i-1}\displaystyle\frac{\partial^{i}}{\partial
z^{l}}S_{u^{j}}(\chi)(-z-\chi)^{l}$ is the Taylor polynomial for $S(-z)$ of
order $i-1$. Now use the above fact that
$S_{u^{j}}(-z)-\sum\limits_{l=0}^{i-1}\displaystyle\frac{\partial^{i}}{\partial
z^{l}}S_{u^{j}}(\chi)(-z-\chi)^{l}=(-z-\chi)^{i}R(z^{-1})$. Simple calculation
shows that the difference of the LHS and of the RHS of Equation (6) is a
$z^{-1}$ series (and a polynomial in $z^{-1}$ in each coefficient of $Q^{d}$
with fixed $d$). However, since both sides lie in $\mathcal{H}^{j}_{+}$ and so
does the difference. Hence the difference must vanish. ∎

This gives us a recursive relation for $G^{j}(z)$ with initial condition
$[G^{j}(z)]_{+}\in H^{*}(Z_{j};S_{T})[z][\\![Q]\\!]$ and $u^{j}\in H^{j}$. We
note that the $[G^{j}(z)]_{+}$ term is denoted by $q^{\alpha}(z)$ in the proof
of [3, Theorem 2].

The rest of the proof is identical to the corresponding part of _loc. cit._.
Setting $Q=0$, $\iota_{j}^{*}t$ are related to $[G^{j}(z)]_{+}$ in the
following way

$-z+\iota_{j}^{*}t=\left[S^{-1}_{u^{j}}(-z)[G^{j}(z)]_{+}\right]_{+}.$

We are left to show $\\{u^{j}\\}$ are determined by $\\{\iota_{j}^{*}t\\}$ in
view of the fact that $G^{j}(z)\in z\mathcal{H}^{j}_{+}$, cf. [3, §2.3]. Again
modulo Novikov variables, we have $G^{j}(z)=[G^{j}(z)]_{+}$,
$S_{u^{j}}(-z)=e^{-u^{j}/z}$.

$[G^{j}(z)]_{+}=\left[e^{-u^{j}/z}(z+\iota_{j}^{*}t(-z))\right]_{+}\in
z\mathcal{H}^{j}_{+}.$

The constant term being zero provides us with an equation between $u^{j}$ and
$\iota_{j}^{*}t(z)$. Using formal implicit function theorem, one can see that
$u^{j}$ is uniquely determined by successive $Q$-adic approximations.

##  4\. Gromov-Witten invariants of projective bundles

Let $Y$ be a nonsingular variety equipped with an algebraic action by
$T=(\mathbb{C}^{*})^{m}$.

######  Definition 4.1.

$Y$ is said to be an algebraic GKM manifold if it has finitely many fixed
points and finitely many one-dimensional orbits under the $T$ action.

In this section, we assume $Y$ to be proper algebraic GKM manifold. In this
case the closure of any one-dimensional orbit in $Y$ is $\mathbb{P}^{1}$.
Examples include proper toric varieties, partial flag varieties including
Grassmannians, etc.

Let $V$ be a $T$-equivariant vector bundle of rank $r$ over $Y$ and let
$c_{T}(V)(x):=\sum c_{i}(V)x^{r-i}$ be the total Chern polynomial. The
classical result gives a presentation of the equivariant cohomology of the
projective bundle $\mathbb{P}_{Y}(V)$ as

$H_{T}^{*}(\mathbb{P}(V))=\frac{H^{*}_{T}(Y)[h]}{(c_{T}(V)(h))}$

where $h:=c_{1}(\mathcal{O}(1))$. Let $V_{1},V_{2}$ be two equivariant vector
bundles over $Y$ having the same equivariant Chern classes, i.e.,
$c_{T}(V_{1})=c_{T}(V_{2})$. Their equivariant cohomology rings are
canonically isomorphic using the above presentation

$\mathfrak{F}:H_{T}^{*}(\mathbb{P}(V_{1}))\cong H^{*}_{T}(\mathbb{P}(V_{2})),$

where the isomorphism $\mathfrak{F}$ sends corresponding
$c_{1}(\mathcal{O}(1))$ to $c_{1}(\mathcal{O}(1))$. Furthermore,
$\mathfrak{F}$ induces an isomorphism between $N_{1}(\mathbb{P}(V_{1}))$ and
$N_{1}(\mathbb{P}(V_{2}))$ by Poincaré pairing. By a slight abuse of
notations, we still denote the induced isomorphism by $\mathfrak{F}$. In other
words, given a curve class $\beta\in N_{1}(\mathbb{P}(V_{1}))$,
$\mathfrak{F}\beta$ is the unique curve class in $N_{1}(\mathbb{P}(V_{2}))$
such that $(\mathfrak{F}D,\mathfrak{F}\beta)=(D,\beta)$ for any $D\in
N^{1}(\mathbb{P}(V_{1}))$.

The following is the main result of this paper.

######  Theorem 4.2.

The isomorphism $\mathfrak{F}$ between equivariant cohomology rings and
numerical curve classes induces an isomorphism of the full genus zero
equivariant Gromov-Witten invariants between $\mathbb{P}(V_{1})$ and
$\mathbb{P}(V_{2})$. More precisely,

$\langle\psi^{a_{1}}\sigma_{1},\dotsc,\psi^{a_{n}}\sigma_{n}\rangle_{0,n,\beta}=\langle\psi^{a_{1}}\mathfrak{F}\sigma_{1},\dotsc,\psi^{a_{n}}\mathfrak{F}\sigma_{n}\rangle_{0,n,\mathfrak{F}\beta}.$

The rest of this section is devoted to the proof of this theorem. By
definition of algebraic GKM manifolds, the fixed loci of $\mathbb{P}(V_{i})$
are the fibers over the $T$-fixed points of $Y$. Let $p$ be a fixed point in
$Y$. For a given $1\leq j\leq l$, Since $c_{T}(V_{1})=c_{T}(V_{2})$, we have

$c_{T}(V_{1}|_{p})=c_{T}(V_{2}|_{p})\in R_{T}=H^{*}_{T}(\\{\text{point}\\}),$

where $V_{i}|_{p}$ are the fibers at $p$. Interpreting $R_{T}$ as the
representation ring of $T$, the equality is also saying that $V_{i}|_{p}$ are
isomorphic as $T$-representations. Thus
$\mathbb{P}(V_{1})|_{p},\mathbb{P}(V_{2})|_{p}$ are $T$-equivariantly
isomorphic. Hence, the fixed loci of $\mathbb{P}(V_{1})$ and
$\mathbb{P}(V_{2})$ are naturally identified.

Given $Z\subset\mathbb{P}(V_{1})|_{p},Z^{\prime}\subset\mathbb{P}(V_{2})|_{p}$
two fixed loci that are identified under $\mathfrak{F}$. One sees that there
is a $T$-equivariant short exact sequence

$0\rightarrow N_{Z/(\mathbb{P}(V_{1})|_{p})}\rightarrow
N_{Z/\mathbb{P}(V_{1})}\rightarrow\pi^{*}N_{\\{p\\}/Y}\rightarrow 0.$

Thus $N_{Z/\mathbb{P}(V_{1})}$ can be $T$-equivariantly deformed to
$N_{Z/(\mathbb{P}(V_{1})|_{p})}\oplus\pi^{*}N_{\\{p\\}/Y}$, where
$\pi:\mathbb{P}(V_{1})\rightarrow Y$ is the projection, by sending the
$T$-equivariant extension class to zero. Similar observations conclude that
$N_{Z/\mathbb{P}(V_{1})}$ and $N_{Z^{\prime}/\mathbb{P}(V_{2})}$ represent the
same element in $K^{0}_{T}(Z)\cong K^{0}_{T}(Z^{\prime})$.

Now in order to identify Lagrangian cones $\mathcal{L}_{\mathbb{P}(V_{1})}$
and $\mathcal{L}_{\mathbb{P}(V_{2})}$, it suffices to show that the recursion
relations (2) in Theorem 3.5 are identified under $\mathfrak{F}$. Let’s
analyze the one-dimensional orbit closures in $Y$ first.

######  Lemma 4.3.

Let $p\in Y$ be a fixed point and $l_{1},l_{2}$ be two distinct 1-dimensional
orbit closures passing through $p$. The (integral) characters
$\chi_{1},\chi_{2}$ of $l_{1},l_{2}$ at $p$ do not lie on the same ray in
$C(T)_{\mathbb{Q}}$ .

###### Proof.

Suppose we have these two orbit closures $l_{1},l_{2}$ such that
$\chi_{1},\chi_{2}$ lie on the same ray in $C(T)_{\mathbb{Q}}$. Write
$\chi_{0}$ to be the first $\mathbb{Z}$-point along the ray of $\chi_{1}$ and
$\chi_{2}$ in $C(T)_{\mathbb{Q}}$. Decompose $T_{p}X$ into irreducible
$T$-representations $\bigoplus\limits_{\chi\in C(T)}T_{\chi}$. The assumption
implies that $\bigoplus\limits_{a\in\mathbb{Z}_{\geq 0}}T_{a\chi_{0}}$ is at
least of dimension $2$.

Choosing $\rho$ such that $(\rho,\chi_{0})>0$, there is a subvariety of
$Z^{\rho}$ containing $p$ that is $T$-isomorphic to
$\bigoplus\limits_{(\rho,\chi)>0}T_{\chi}$ by [2, Theorem 2.5] (note that $p$
is a discrete fixed point). $Z^{\rho}$ contains the subrepresentation
$\bigoplus\limits_{a\in\mathbb{Z}_{\geq 0}}T_{a\chi_{0}}$. However, an
elementary verification shows that there are infinitely many $1$-dimensional
orbits in $\bigoplus\limits_{a\in\mathbb{Z}_{\geq 0}}T_{a\chi_{0}}$ since its
dimension is at least $2$. This is a contradiction to the assumption that $Y$
is algebraic GKM. ∎

From now on, we are mostly concerned with general properties of a projective
bundle over an algebraic GKM manifold. To simplify the notation, set
$\pi:X:=\mathbb{P}_{Y}(V)\to Y$. Recall the notations defined in Section 2.
$Z_{1},\dotsc,Z_{n}$ are the fixed loci of $X$ under $T$-action, and
$X_{j,j^{\prime},d,\chi}=f_{j,j^{\prime},d,\chi}(\mathcal{C}_{j,j^{\prime},d,\chi})$
can be seen as the union of the images of all unbroken maps for fixed
$j,j^{\prime},d,\chi$. (See Notations 2.13.)

######  Lemma 4.4.

Fix $\chi\in C(T)_{\mathbb{Q}}$ and $Z_{j}\subset X$ such that
$\pi(Z_{j})=\\{p_{j}\\}\subset Y$. For any $d\in\operatorname{NE}(X)$ and
$Z_{j^{\prime}}$, we have either
$\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}=\emptyset$ or $\pi\circ
f_{j,j^{\prime},d,\chi}(\mathcal{C}_{j,j^{\prime},d,\chi})\subset l$, where
$l\subset Y$ is the closure of a one-dimensional orbit.

###### Proof.

Fix $\chi$ and $Z_{j}$ and pick any $d,Z_{j^{\prime}}$. If
$\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}\neq\emptyset$,
$\pi(X_{j,j^{\prime},d,\chi})$ must be either the fixed point $\\{p_{j}\\}$ or
some one dimensional orbit closure $l\subset Y$. In the latter case, by
definition $\chi$ must be proportional to the fractional character of $l$ at
$p_{j}$. By the previous lemma, there is at most one orbit closure $l$ passing
through $p_{j}$ such that its fractional character is proportional to $\chi$.
Since $\chi$ is fixed at the beginning, such a universal $l$ can be easily
found for all $d$ and $Z_{j^{\prime}}$. ∎

Let $l$ be a one dimensional orbit closure in $Y$ and
$X|_{l}:=\pi^{-1}(l)\subset X$. With the help of the previous lemma, when
there is a fixed $\chi$ and $Z_{j}$, we just found an $l\subset Y$ such that
in the recursion condition for $\mathcal{L}_{X}$, all possible unbroken map
that are used in computing $\mathop{\bf{Prin}}\limits_{z=-\chi}F^{j}(z)$ are
contained in $X|_{l}$. We are left to match the recursion conditions on
$\mathop{\bf{Prin}}\limits_{z=-\chi}F^{j}(z)$ when we change the bundle from
$V_{1}$ to $V_{2}$. For that purpose, we need to make sense of
$\mathcal{L}_{X|_{l},N_{X|_{l}/X}}$. We follow Remark 1.2 by adding an
auxiliary $\mathbb{C}^{*}$ action that acts on $X|_{l}$ trivially, but on
$N_{X|_{l}/X}$ by scaling. Let $R_{T\times\mathbb{C}^{*}}=R_{T}[x]$ where $x$
is the equivariant parameter for the extra $\mathbb{C}^{*}$ action. Let
$R_{T}[x,x^{-1}]\\!]$ be the ring of Laurent series in $x^{-1}$. Let
$\pi_{0,n,d}:\mathcal{C}_{0,n,d}\rightarrow\overline{\mathcal{M}}_{0,n}(X|_{l},d)$
be the universal curve and $f_{0,n,d}:\mathcal{C}_{0,n,d}\rightarrow X$ be the
universal stable map. Let

$0\rightarrow N^{0}\rightarrow N^{1}\rightarrow 0$

be a two term complex of locally free sheaves, whose cohomology are
$R^{i}(\pi_{0,n,d})_{*}f_{0,n,d}^{*}N_{X|_{l}/X}$. Recall we define

$e_{T\times\mathbb{C}^{*}}((N_{X|_{l}/X})_{0,n,d})=\displaystyle\frac{e_{T\times\mathbb{C}^{*}}(N^{0})}{e_{T\times\mathbb{C}^{*}}(N^{1})}\in(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X|_{l},d))[x])_{\text{loc}}$

where $(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X|_{l},d))[x])_{\text{loc}}$ is
the localization of $H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X|_{l},d))[x]$ by
inverting all monic polynomials in $x$. Embedding it into
$H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X|_{l},d)\otimes_{R_{T}}R_{T}[x,x^{-1}]\\!]$
as a subspace renders $\mathcal{L}_{X|_{l},N_{X|_{l}/X}}$ well defined under
the extra $\mathbb{C}^{*}$ action. We will prove, in Proposition 4.6, that the
$x=0$ limit exists in the sense of Remark 1.2.

Since $N_{X|_{l}/X}=\pi^{*}N_{l/Y}$, we first analyze the $T$ action on
$N_{l/Y}$.

######  Lemma 4.5.

There is a splitting $T=T^{\prime}\times\mathbb{C}^{*}$ such that $T^{\prime}$
acts on $l$ trivially. Furthermore, there is no nontrivial $T^{\prime}$-fixed
subsheaf in $N_{l/Y}$.

###### Proof.

As we have seen in Corollary 2.4, the one dimension orbit is isomorphic to
$\mathbb{C}^{*}$. Picking any point on it, we have an induced group
homomorphism $T\rightarrow\mathbb{C}^{*}$. Let $K$ be its kernel.
$Hom(-,\mathbb{C}^{*})$ is an exact functor among abelian groups because
$\mathbb{C}^{*}$ is injective. $Hom(K,\mathbb{C}^{*})$ decomposes into the
product of a free abelian group and a torsion one. Therefore $K$ decomposes
into $K^{\prime}\times K^{\prime\prime}$ where
$K^{\prime}\cong(\mathbb{C}^{*})^{m-1}$ and $K^{\prime\prime}$ is a finitely
generated torsion abelian group. $K^{\prime}$ is naturally embedded in $K$.
Using $K^{\prime}$ instead of $K$, we have a short exact sequence

$0\rightarrow K^{\prime}\rightarrow T\rightarrow\mathbb{C}^{*}\rightarrow 0.$

Since $K^{\prime}$ fixes points in $l$, there is an induced action of the
$\mathbb{C}^{*}$ on $l$. Now with $K^{\prime}$ being a sub-torus, $T$ splits
into $K^{\prime}\times\mathbb{C}^{*}$ due to injectivity of $K^{\prime}$. We
have shown the first statement with $T^{\prime}=K^{\prime}$.

We now proceed to the second statement. First observe that $N_{l/Y}$ can be
decomposed into $T^{\prime}$-eigensheaves:

(7) $N_{l/Y}=\bigoplus_{\chi\in C(T^{\prime})}N_{\chi}$

such that ach component is locally free. It therefore suffices to think about
$N_{l/Y}|_{\\{p\\}}$ where $p$ is a fixed point on $l$. Similar to Lemma 4.3,
decompose $T_{p}Y$ into irreducible $T$-representations
$\bigoplus\limits_{\chi\in C(T)}T_{\chi}$. The tangent direction along $l$
corresponds to one of the factor $T_{\chi_{0}}$ for some $\chi_{0}\in C(T)$.
This $\chi_{0}$ is proportional to the character corresponding to
$T\rightarrow\mathbb{C}$ from the above short exact sequence. Notice that
$N_{l/Y}|_{\\{p\\}}$ is $T$-isomorphic to $T_{p}Y/T_{\chi_{0}}$. Since $Y$ has
isolated fixed point and $1$-dimensional orbits, $T_{0}=0,T_{\chi_{0}}$ is
$1$-dimensional. Hence there is no fixed subspace in $T_{p}Y/T_{\chi_{0}}$. ∎

We are left to analyze
$e_{T\times\mathbb{C}^{*}}((N_{X|_{l}/X})_{0,n,d})^{-1}$. Let
$R_{T\times\mathbb{C}^{*}}=R_{T}[x]$ and $R_{T}[x,x^{-1}]\\!]$ be the ring of
Laurent series in $x^{-1}$.

By definition,
$e_{T\times\mathbb{C}^{*}}((N_{X|_{l}/X})_{0,n,d})^{-1}\in(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X|_{l},d))[x])_{\text{loc}}$.

######  Proposition 4.6.

$e_{T\times\mathbb{C}^{*}}((N_{X|_{l}/X})_{0,n,d})^{-1}\in(H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X|_{l},d))[x])_{\text{loc}}$

has $x=0$ limit in
$H^{*}_{T}(\overline{\mathcal{M}}_{0,n}(X|_{l},d))\otimes_{R_{T}}S_{T}$ in the
sense of Remark 1.2.

Since we are working on the invariant subvariety $X|_{l}$, in the rest of this
proof, we replace the whole variety $X$ by $X|_{l}$, and still use
$\overline{\mathcal{M}}_{\vec{\Gamma}}$,
$\overline{\mathcal{M}}_{j,j^{\prime},d,\chi}$, etc for the corresponding
notions in $X|_{l}$.

Let
$\pi_{\vec{\Gamma}}:\mathcal{C}_{\vec{\Gamma}}\rightarrow\overline{\mathcal{M}}_{\vec{\Gamma}}$
be the universal family over $\overline{\mathcal{M}}_{\vec{\Gamma}}$ and
$f_{\vec{\Gamma}}:\mathcal{C}_{\vec{\Gamma}}\rightarrow X_{l}$ be the
universal stable map.

######  Lemma 4.7.

$R^{i}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X})$ are locally
free for $i=0,1$.

###### Proof.

This amounts to saying that, given an invariant stable map $f:C\rightarrow
X_{l}$ with $n$ markings assigned with a decorated graph $\vec{\Gamma}$, the
dimensions of $H^{i}(C,f^{*}N_{X|_{l}/X})$ for $i=0,1$ only depend on the
graph $\vec{\Gamma}$ (i.e., when the stable map varies in
$\overline{\mathcal{M}}_{\vec{\Gamma}}$, these dimensions are constant).

Let $v$ be a vertex in $\vec{\Gamma}$. Suppose $v$ corresponds to a sub-curve
$C_{v}$. Under the projection $\pi:X|_{l}\rightarrow l$, $C_{v}$ has to map to
a point. Therefore $N_{X|_{l}/X}|_{C_{v}}$ is a trivial sheaf and the
dimension of $H^{0}(C,f^{*}N_{X|_{l}/X})$ is the rank of $N_{X|_{l}/X}$. There
is no $H^{1}$ since we are working on genus-$0$ curves. If $v$ corresponds to
a point in $C$, the dimension of $H^{0}$ is also the rank of $N_{X|_{l}/X}$.

Let $e$ be an edge in $\vec{\Gamma}$ and $C_{e}$ be the sub-curve
corresponding to $e$. There are two possibilities of the image of $f(C_{e})$
under the projection $\pi:X|_{l}\rightarrow l$. Recall
$d_{e}\in\operatorname{NE}(X|_{l})$ is the curve class corresponding to
$f(C_{e})$.

  1. (1)

If $\pi_{*}d_{e}=0$, $\pi\circ f(C_{e})$ is a point. And the same thing as the
vertex case happens.

  2. (2)

If $\pi_{*}d_{e}\neq 0$, $\pi\circ f(C_{e})$ is $l$. Recall $C_{e}$ is a chain
of $\mathbb{P}^{1}$. Let $C_{1},\dotsc,C_{k}$ be the irreducible components of
$C_{e}$. In this case, there is only one component $C_{i}$ such that $\pi\circ
f(C_{i})=l$ (otherwise the fractional characters on the components violate
$C_{e}$ being an unbroken map ). Suppose $\pi_{*}d_{e}=k[l]$. Then $\pi\circ
f:C\rightarrow l$ is a degree $k$ cover between $\mathbb{P}^{1}$’s. Thus
$H^{i}(C_{i},f^{*}N_{X|_{l}/X})=H^{i}(C_{i},f^{*}\pi^{*}N_{l/Y})$ depends only
on $k$ and the bundle $N_{l/Y}$ over $l$. Other components than $C_{i}$ are
mapped to points under $\pi\circ f$ and won’t contribute extra dimensions to
$H^{i}(C,f^{*}N_{X|_{l}/X})$.

The dimensions of $H^{i}(C,f^{*}N_{X|_{l}/X})$ can then be calculated by
passing to the normalization of $C$. To sum it up,
$\text{dim}(H^{i}(C,f^{*}N_{X|_{l}/X}))$ depends only on the decorated graph
$\vec{\Gamma}$ assigned to the stable map $f:C\rightarrow X_{l}$ and the
normal bundle $N_{l/Y}$ on $l$. This implies the lemma we are proving. ∎

As a result, we have

$\left.e_{T\times\mathbb{C}^{*}}((N_{X|_{l}/X})_{0,n,d})^{-1}\right|_{\overline{\mathcal{M}}_{\vec{\Gamma}}}=\displaystyle\frac{e_{T\times\mathbb{C}^{*}}(R^{1}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X}))}{e_{T\times\mathbb{C}^{*}}(R^{0}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X}))}$

We are left to prove this quotient has $x=0$ limit.

######  Lemma 4.8.

$e_{T}(R^{i}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X}))$ are
invertible elements in
$H^{*}(\overline{\mathcal{M}}_{\vec{\Gamma}})\otimes_{\mathbb{C}}S_{T}$
(notice that we are without the $\mathbb{C}^{*}$ action).

###### Proof.

Notice we have

$H^{*}_{T}(\overline{\mathcal{M}}_{\vec{\Gamma}})\otimes_{R_{T}}\otimes
R_{T}=R_{T}\otimes_{\mathbb{C}}H^{*}(\overline{\mathcal{M}}_{\vec{\Gamma}}).$

One can decompose

$e_{T}(R^{i}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X}))=\sigma_{0}\otimes
1+\sum\limits_{j\geq 1}\sigma_{j}\otimes\tau_{j}\in
R_{T}\otimes_{\mathbb{C}}H^{*}(\overline{\mathcal{M}}_{\vec{\Gamma}})$

where $\sigma_{j}\in R_{T}$ and $\tau_{j}\in
H^{>0}(\overline{\mathcal{M}}_{\vec{\Gamma}})$. If we can prove
$\sigma_{0}\neq 0$, the expression above is invertible after tensoring
$S_{T}$. In order to do this, we show that
$R^{i}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X})$ do not have
non-trivial fixed subsheaves. It suffices to prove it fiberwise, and can be
reduced to the following.

######  Claim 1.

For any proper irreducible invariant curve $C\subset X|_{l}$,
$H^{i}(C,N_{X|_{l}/X})$ does not have any nontrivial fixed subspace for
$i=0,1$.

###### Proof.

It is because of the splitting in equation (7) and the fact that
$N_{X|_{l}/X}=\pi^{*}N_{l/Y}$, and there are two cases.

  1. (1)

If $\pi(C)=l$, since $\pi$ is equivariant, $T^{\prime}$ acts on $C$ trivially.
$\left.N_{X|_{l}/X}\right|_{C}$ splits into eigensheaves where $T^{\prime}$
acts by scaling. Since $N_{0}=0$, the claim follows.

  2. (2)

If $\pi(C)$ is a fixed point, although $T^{\prime}$ might act on $C$ non-
trivially, the underlying vector bundle $\left.N_{X|_{l}/X}\right|_{C}$ is
trivial since it is a pull-back from a point. Only $H^{0}(C,N_{X|_{l}/X})$ can
be nonzero, and it consists of constant sections. $T^{\prime}$ acts on it
without trivial sub-representation again due to $N_{0}=0$.

∎

As a result, $\sigma_{0}\neq 0$.
$e_{T}(R^{i}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X}))$ are
invertible because $H^{*}(\overline{\mathcal{M}}_{\vec{\Gamma}})$ is finite
dimensional. ∎

###### Proof.

(of Proposition 4.6) Now that Lemma 4.8 is proven, the $x=0$ limit for
$e_{T\times\mathbb{C}^{*}}((N_{X|_{l}/X})_{0,n,d})$ can be readily constructed
via localization formula. The limit is invertible because each piece
$e_{T}(R^{i}(\pi_{\vec{\Gamma}})_{*}f^{*}_{\vec{\Gamma}}(N_{X|_{l}/X}))$ on
each connected component of $\overline{\mathcal{M}}_{\vec{\Gamma}}$ is
invertible. ∎

Finally, we use the following fact.

######  Lemma 4.9.

Any two equivariant vector bundles over $\mathbb{P}^{1}$ with the same
equivariant Chern classes can be connected by an equivariant family of vector
bundles over a connected base.

###### Proof.

This lemma amounts to saying that the moduli stack of equivariant vector
bundles with fixed Chern class over $\mathbb{P}^{1}$ is connected. This can be
proven by, for example, adding a framing condition to form the fine moduli of
framed equivariant vector bundles ([18]). This forms a fine moduli space which
is a covering of the moduli stack of equivariant vector bundles with connected
fibers. By the presentation for moduli of framed toric bundles given in [18],
in the case of $\mathbb{P}^{1}$, the moduli space is just a product of flag
varieties which are obviously connected. ∎

To sum it up, Lemma 4.9 shows that $\mathbb{P}(V_{1})|_{l}$ can be
equivariantly deformed to $\mathbb{P}(V_{2})|_{l}$. Thus they have the same
equivariant Gromov-Witten theory. Moreover,
$N_{\mathbb{P}(V_{i})|_{l}/\mathbb{P}(V_{i})}$ are obviously the pull-back of
the same vector bundle $N_{l/Y}$ over $l$. As a result, the Lagrangian cones
$\mathcal{L}_{\mathbb{P}(V_{i})|_{l},N_{\mathbb{P}(V_{i})|_{l}/\mathbb{P}(V_{i})}}$
are identified for $i=1,2$. Using Corollary 3.7 with
$X=\mathbb{P}(V_{i}),W=\mathbb{P}(V_{i})|_{l}$, we are finally able to
identify the recursion relations (2) for any fixed $Z_{j}$ and $\chi$ in
Theorem 3.5 between points in $\mathcal{L}_{\mathbb{P}(V_{1})}$ and
$\mathcal{L}_{\mathbb{P}(V_{2})}$. Theorem 4.2 is now proven.

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