#  Equivariant Gromov-Witten invariants of
algebraic GKM manifolds

Chiu-Chu Melissa Liu  Chiu-Chu Melissa Liu, Department of Mathematics,
Columbia University, 2990 Broadway, New York, NY 10027
<EMAIL_ADDRESS>and  Artan Sheshmani  Artan Sheshmani, Department of
mathematics at the Ohio State University, 600 Math Tower, 231 West 18th
Avenue, Columbus, OH 43210<EMAIL_ADDRESS>
###### Abstract.

An algebraic GKM manifold is a non-singular algebraic variety equipped with an
algebraic action of an algebraic torus, with only finitely many torus fixed
points and finitely many one-dimensional orbits. In this expository article,
we use virtual localization to express equivariant Gromov-Witten invariants of
any algebraic GKM manifold (which is not necessarily compact) in terms of
Hodge integrals over moduli stacks of stable curves and the GKM graph of the
GKM manifold.

###### Contents

  1. 1 Introduction
    1. 1.1 Gromov-Witten invariants of a smooth projective variety
    2. 1.2 Equivariant Gromov-Witten invariants and virtual localization
    3. 1.3 Algebraic GKM manifolds and their Gromov-Witten invariants
    4. 1.4 Outline
  2. 2 Algebraic GKM manifolds
    1. 2.1 Basic notation
    2. 2.2 GKM graph
  3. 3 Gromov-Witten Theory
    1. 3.1 Moduli of stable curves and Hodge integrals
    2. 3.2 Moduli of stable maps
    3. 3.3 Obstruction theory and virtual fundamental classes
    4. 3.4 Gromov-Witten invariants
    5. 3.5 Equivariant Gromov-Witten invariants
  4. 4 Virtual Localization
    1. 4.1 Torus fixed points and graph notation
    2. 4.2 Virtual tangent and normal bundles
      1. 4.2.1 Automorphisms of the domain
      2. 4.2.2 Deformations of the domain
      3. 4.2.3 Unifying stable and unstable vertices
      4. 4.2.4 Deformation of the map
    3. 4.3 Contribution from each graph
      1. 4.3.1 Virtual tangent bundle
      2. 4.3.2 Virtual normal bundle
      3. 4.3.3 Integrand
      4. 4.3.4 Integral
    4. 4.4 Sum over graphs

##  1\. Introduction

In this paper, we work over $\mathbb{C}$.

###  1.1. Gromov-Witten invariants of a smooth projective variety

Let $X$ be a smooth projective variety. Gromov-Witten invariants of $X$ are
virtual counts of parametrized algebraic curves of $X$. More precisely, let
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the Kontsevich’s moduli space of
$n$-pointed, genus $g$, degree $\beta$ stable maps to $X$, where $\beta\in
H_{2}(X,\beta)$ is an effective curve class. It is a proper Deligne-Mumford
stack with a perfect obstruction theory of virtual dimension

(1) $d^{\mathrm{vir}}=\int_{\beta}c_{1}(T_{X})+(\dim X-3)(1-g)+n,$

where $\int$ stands for the pairing between the (rational) homology and
cohomology. There is a virtual fundamental class [31, 4, 2]

$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}\in
A_{d^{\mathrm{vir}}}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})$

which can also be viewed as an element in
$H_{2d^{\mathrm{vir}}}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})$. The
virtual fundamental class defines a $\mathbb{Q}$-linear map

$\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}}:H^{*}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})\longrightarrow\mathbb{Q}.$

For $i=1,\ldots,n$, let
$\mathrm{ev}_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to X$ be the evaluation
map at the $i$-th marked point. Genus $g$, degree $\beta$ descendant Gromov-
Witten invariants of $X$ are defined by

(2)
$\langle\tau_{a_{1}}(\gamma_{1}),\ldots,\tau_{a_{n}}(\gamma_{n})\rangle^{X}_{g,\beta}:=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}}\prod_{i=1}^{n}(\mathrm{ev}_{i}^{*}\gamma_{i}\cup\psi_{i}^{a_{i}})\in\mathbb{Q}$

where $\gamma_{i}\in H^{*}(X;\mathbb{Q})$, $a_{i}\in\mathbb{Z}_{\geq 0}$, and
$\psi_{i}\in H^{2}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})$ are
$\psi$-classes (to be defined in Section 3.4). If $\gamma_{i}\in
H^{d_{i}}(X;\mathbb{Q})$ then the (2) vanishes unless

$\sum_{i=1}^{n}(d_{i}+2a_{i}-2)=2(\int_{\beta}c_{1}(TX)+(\dim X-3)(1-g)).$

If $X$ is a smooth algebraic variety which is not projective, then
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ is usually not proper, so
$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}$ is not defined. If
$X$ is not projective but $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is proper
for some particular $g,n,\beta$, then
$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}$ exists and the
invariants in (2) are defined for such $g,n,\beta$.

###  1.2. Equivariant Gromov-Witten invariants and virtual localization

Suppose that $T=(\mathbb{C}^{*})^{m}$ acts algebraically on $X$. Then $T$ acts
on $\overline{\mathcal{M}}_{g,n}(X,\beta)$. There exists a $T$-equivariant
perfect obstruction theory, and a $T$-equivariant virtual fundamental class

$[\overline{\mathcal{M}}_{g,n}(X,\beta)]_{T}^{\mathrm{vir}}\in
A_{d^{\mathrm{vir}}}^{T}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q}).$

Let

$R_{T}:=H^{*}_{T}(\\{\textup{point}\\};\mathbb{Q})=H^{*}(BT;\mathbb{Q})=\mathbb{Q}[u_{1},\ldots,u_{m}]$

be the $T$-equivariant cohomology of a point, where $u_{i}\in
H^{2}_{T}(BT;\mathbb{Q})$. There is an $R_{T}$-linear map

$\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]_{T}^{\mathrm{vir}}}:H_{T}^{*}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})\longrightarrow
H_{T}^{*}(\\{\textup{point}\\};\mathbb{Q})=R_{T}.$

Genus $g$, degree $\beta$ $T$-equivariant descendant Gromov-Witten invariants
of $X$ are defined by

(3)
$\langle\tau_{a_{1}}(\gamma_{1}^{T}),\ldots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle^{X_{T}}_{g,\beta}:=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]_{T}^{\mathrm{vir}}}\prod_{i=1}^{n}(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi^{T}_{i})^{a_{i}})\in
R_{T},$

where $\gamma_{i}^{T}\in H^{*}_{T}(X;\mathbb{Q})$, and $\psi^{T}_{i}\in
H^{2}_{T}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})$ is a
$T$-equivariant lift of $\psi_{i}\in
H^{2}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})$ (see Section 3.5). If
$\gamma_{i}^{T}\in H^{d_{i}}_{T}(X;\mathbb{Q})$ then

$\langle\tau_{a_{1}}(\gamma_{1}^{T}),\ldots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle^{X_{T}}_{g,\beta}\in
H^{\sum_{i=1}^{n}(d_{i}+2a_{i}-2)-2d^{\mathrm{vir}}}_{T}(\\{\textup{point}\\};\mathbb{Q}).$

In particular, (3) vanishes if
$\sum_{i=1}^{n}(d_{i}+2a_{i}-2)<2d^{\mathrm{vir}}$.

The $T$-equivariant Gromov-Witten invariants (3) are related to the Gromov-
Witten invariants (2) as follows. Let $\gamma_{i}$ be the image of
$\gamma_{i}^{T}$ under the ring homomorphism
$H^{*}_{T}(X;\mathbb{Q})\longrightarrow H^{*}(X;\mathbb{Q})$. Then

$\langle\tau_{a_{1}}(\gamma_{1}),\ldots,\tau_{a_{n}}(\gamma_{n})\rangle^{X}_{g,\beta}=\langle\tau_{a_{1}}(\gamma_{1}^{T}),\ldots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle^{X_{T}}_{g,\beta}\Big{|}_{u_{1}=\cdots=u_{m}=0}.$

The torus fixed part of the restriction of the $T$-equivariant perfect
obstruction theory to the $T$ fixed substack
$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}\subset\overline{\mathcal{M}}_{g,n}(X,\beta)$
defines a perfect obstruction theory on
$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}$ and a virtual class
$[\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}]^{\mathrm{vir}}_{T}\in
A_{*}^{T}(\overline{\mathcal{M}}_{g,n}(X,\beta)^{T};\mathbb{Q})$; the torus
moving part defines the virtual normal bundle $N^{\mathrm{vir}}$ of the
inclusion
$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}\subset\overline{\mathcal{M}}_{g,n}(X,\beta)$.
By localization of virtual fundamental class [18, 3], the RHS of (3) is equal
to

(4)
$\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}]_{T}^{\mathrm{vir}}}\prod_{i=1}^{n}\frac{i^{*}(\mathrm{ev}_{i}^{*}\gamma^{T}_{i}\cup(\psi_{i}^{T})^{a_{i}})}{e_{T}(N^{\mathrm{vir}})}$

where
$i:\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}\hookrightarrow\overline{\mathcal{M}}_{g,n}(X,\beta)$
is the inclusion, and $e_{T}(N^{\mathrm{vir}})$ is the $T$-equivariant Euler
class of $N^{\mathrm{vir}}$. It is know that $e_{T}(N^{\mathrm{vir}})$ is
invertible in
$H^{*}_{T}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})\otimes_{R_{T}}Q_{T}$,
where $Q_{T}=\mathbb{Q}(u_{1},\ldots,u_{m})$ is the fractional field of
$R_{T}=\mathbb{Q}[u_{1},\ldots,u_{m}]$.

Suppose that $X$ is non-compact, and for some $g,n,\beta$,
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ is not proper but
$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}$ is. Then the RHS of (3) is not
defined, but (4) is. In this case, we define $T$-equivariant Gromov-Witten
invariants
$\langle\tau_{a_{1}}(\gamma_{1}^{T}),\ldots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle_{g,\beta}^{X_{T}}$
by (4), which is an element in $\mathbb{Q}(u_{1},\ldots,u_{m})$ instead of
$\mathbb{Q}[u_{1},\ldots,u_{m}]$.

###  1.3. Algebraic GKM manifolds and their Gromov-Witten invariants

In this paper, an algebraic GKM manifold, named after Goresky-Kottwitz-
MacPherson, is a non-singular algebraic variety equipped with an algebraic
action of $T=(\mathbb{C}^{*})^{m}$, such that there are finitely many torus
fixed points and finitely many one-dimensional orbits. Examples of algebraic
GKM manifolds include toric manifolds, Grassmanians, flag manifolds, etc.

If $X$ is an algebraic GKM manifold then each connected component of
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ is, up to some quasi-finite map, a
product of moduli stacks of pointed stable curves, and the RHS of (4) can be
expressed in terms Hodge integrals on moduli stacks of pointed stable curves.
This algorithm was first described by Kontsevich for genus zero Gromov-Witten
invariants of $\mathbb{P}^{r}$ in 1994 [29], before the construction of
virtual fundamental class and the proof of virtual localization. The moduli
spaces $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)$ of genus zero stable
maps to $\mathbb{P}^{r}$ are proper smooth DM stacks, so there exists a
fundamental class $[\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)]\in
H_{*}(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d);\mathbb{Q})$, and one may
apply the classical Atiyah-Bott localization formula [1] in this case. In [18,
Section 4], T. Graber and R. Pandharipande used their virtual localization
formula to derive an explicit formula for all genus Gromov-Witten invariants
of $\mathbb{P}^{r}$. (See also K. Behrend [3, Section 4].) H. Spielberg
derived a formula of genus zero Gromov-Witten invariants of toric manifolds in
his thesis [44]. Localization computations of all genus equivariant Gromov-
Witten invariants of toric manifolds can be found in [33]. The main purpose of
this paper is to provide details of the virtual localization calculations of
all genus equivariant Gromov-Witten invariants for general algebraic GKM
manifolds described on page 20-21 of [17].

###  1.4. Outline

In Section 2, we define algebraic GKM manifolds and their GKM graphs,
following [19, 20]. In Section 3, we give a brief review of Gromov-Witten
theory. In Section 4, we compute all genus equivariant descendant Gromov-
Witten invariants of an arbitrary algebraic GKM manifold by virtual
localization. Most of Section 4 is straightforward generalization of the
$\mathbb{P}^{r}$ case discussed in [29] (genus 0) and [18, Section 4], [3,
Section 4] (higher genus); see also [21, Chapter 27].

### Acknowledgments

The first author wish to thank Tom Graber for his suggestion of generalizing
the computations for toric manifolds in [33] to GKM manifolds. The second
author would like to thank the Columbia University for hospitality during his
visits. We wish to thank Rahul Pandharipande for his comments on an earlier
version of this paper. This work is partially supported by NSF DMS-1159416 and
NSF DMS-1206667.

##  2\. Algebraic GKM manifolds

In this section, we review the geometry of algebraic GKM manifolds, following
[19], and introduce the GKM graph associated to an algebraic GKM manifold,
following [20]. The GKM graph in this paper can be non-compact since we
consider algebraic GKM manifolds which are not necessarily compact. In Section
4, we will see that the GKM graph contains all the information needed for
computing Gromov-Witten invariants and equivariant Gromov-Witten invariants of
the GKM manifold.

###  2.1. Basic notation

Let $X$ be a non-singular algebraic variety of dimension $r$. We say $X$ is an
algebraic GKM manifold if it is equipped with an algebraic action of a complex
algebraic torus $T=(\mathbb{C}^{*})^{m}$ with only finitely many torus fixed
points and finitely many one-dimensional orbits.

Let $N=\mathrm{Hom}(\mathbb{C}^{*},T)\cong\mathbb{Z}^{m}$ be the lattice of
1-parameter subgroups of $T$, and let $M=\mathrm{Hom}(T,\mathbb{C}^{*})$ be
the lattice of irreducible characters of $T$. Then
$M=\mathrm{Hom}(N,\mathbb{Z})$ is the dual lattice of $N$. Let
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}$ and
$M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$, so that they are dual real
vector spaces of dimension $k$. Let $K=U(1)^{m}$ be the maximal compact
subgroup of $T$. Then $N_{\mathbb{R}}$ can be canonically identified with the
Lie algebra of $K$. Let $N_{\mathbb{Q}}=N\otimes_{\mathbb{Z}}\mathbb{Q}$ and
let $M_{\mathbb{Q}}=M\otimes_{\mathbb{Z}}\mathbb{Q}$. Then $M_{\mathbb{Q}}$
can be canonically identified with $H^{2}_{T}(\textup{point};\mathbb{Q})$.

We make the following assumption on $X$.

######  Assumption 5.

  1. (1)

The set $X^{T}$ of $T$ fixed points in $X$ is non-empty.

  2. (2)

The closure of a one-dimensional orbit is either a complex projective line
$\mathbb{P}^{1}$ or a complex affine line $\mathbb{C}$.

Note that (1) and (2) hold when $X$ is proper. Indeed, if $X$ is proper then
the closure of any one-dimensional orbit is $\mathbb{P}^{1}$.

######  Example 6.

If $X$ is a non-singular toric variety defined by a finite fan, then $X$ is an
algebraic GKM manifold.

######  Example 7 (The Grassmannian $Gr(k,m)$).

Let $Gr(k,m)$ be the set of $k$-dimensional linear subspace of
$\mathbb{C}^{m}$. It is a nonsingular projective variety of dimension
$k(m-k)$. Let $T=(\mathbb{C}^{*})^{m}$ act on $\mathbb{C}^{m}$ by

$(t_{1},\ldots,t_{m})\cdot(z_{1},\ldots,z_{m})=(t_{1}z_{1},\ldots,t_{m}z_{m}).$

Given $t\in T$, let $\phi_{t}:\mathbb{C}^{m}\to\mathbb{C}^{m}$ be defined by
$\phi_{t}(z)=t\cdot z$. Let $T$ act on $Gr(k,m)$ by $t\cdot V=\phi_{t}(V)$,
where $V$ is a $k$-dimensional linear subspace of $\mathbb{C}^{m}$. Given
$J\subset\\{1,\ldots,m\\}$, let $J^{c}:=\\{1,\ldots,m\\}\setminus J$, and
define

$\mathbb{C}^{J}:=\\{(z_{1},\ldots,z_{m})\in\mathbb{C}^{m}:z_{i}=0\textup{ if
}i\in J^{c}\\}\cong\mathbb{C}^{|J|}$

Note that $\phi_{t}(\mathbb{C}^{J})=\mathbb{C}^{J}$ for any $t\in T$,
$J\subset\\{1,\ldots,n\\}$.

The torus-fixed points in $Gr(k,m)$ are

$Gr(k,m)^{T}=\\{\mathbb{C}^{J}:J\subset\\{1,\ldots,n\\},|J|=k\\}.$

So there are ${m\choose k}$ torus-fixed points in $G(k,m)$.

Let $\mathbb{C}^{J}$ and $\mathbb{C}^{J^{\prime}}$ be distinct $T$-fixed
points in $Gr(k,m)$. Then
$\mathbb{C}^{J}\cap\mathbb{C}^{J^{\prime}}=\mathbb{C}^{J\cap J^{\prime}}$.
There is a torus-fixed line connecting $\mathbb{C}^{J}$ and
$\mathbb{C}^{J^{\prime}}$ if and only of $|J\cap J^{\prime}|=k-1$. In this
case, $|J\cup J^{\prime}|=k+1$. The $T$-fixed lines in $G(k,m)$ are

$\\{\ell_{I,K}:I\subset K\subset\\{1,\ldots,m\\},|I|=k-1,|K|=k+1\\}$

where

$\ell_{I,K}=\\{V\in G(k,m):\mathbb{C}^{I}\subset
V\subset\mathbb{C}^{K}\\}\cong\mathbb{P}^{1}.$

Suppose that $I\subset K\subset\\{1,\ldots,m\\}$, and $|I|=k-1$, $|K|=k+1$.
Then $K=I\cup\\{j_{1},j_{2}\\}$, where $j_{1},j_{2}\in I^{c}$. So there are
${m\choose k-1}{m-k+1\choose 2}$ torus-fixed lines in $G(k,m)$.

###  2.2. GKM graph

Let $X$ be an algebraic GKM manifold of dimension $r$, so that
$T=(\mathbb{C}^{*})^{m}$ acts algebraically on $X$.

Following [20], we define a graph $\Upsilon$ as follows. Let $V(\Upsilon)$
(resp. $E(\Upsilon)$) denote the set of vertices (resp. edges) in $\Upsilon$.

  1. (1)

(Vertices) We assign a vertex $\sigma$ to each torus fixed point $p_{\sigma}$
in $X$.

  2. (2)

(Edges) We assign an edge $\epsilon$ to each one-dimensional $O_{\epsilon}$ in
$X$. Let $\ell_{e}$ be the closure of $O_{\epsilon}$.

  3. (3)

(Flags) The set of flags in the graph $\Upsilon$ is given by

$F(\Upsilon)=\\{(\epsilon,\sigma)\in E(\Upsilon)\times
F(\Upsilon):\sigma\in\epsilon\\}=\\{(\epsilon,\sigma)\in E(\Upsilon)\times
F(\Upsilon):p_{\sigma}\in\ell_{\epsilon}\\}.$

The Assumption 5 can be rephrased in terms of the graph $\Upsilon$.

######  Assumption 8.

  1. (1)

$V(\Upsilon)$ is non-empty.

  2. (2)

Each edge in $E(\Upsilon)$ contains at least one vertex.

Let $E(\Upsilon)_{c}=\\{\epsilon\in
E(\Upsilon):\ell_{\epsilon}\cong\mathbb{P}^{1}\\}$ be the set of compact edges
in $\Upsilon$. Note that $E(\Upsilon)_{c}=E(\Upsilon)$ if $X$ is proper.

Given a vertex $\sigma\in V(\Upsilon)$, we denote by $E_{\sigma}$ the set of
edges containing $\sigma$, i.e. $E_{\sigma}:=\\{e\in E:(\epsilon,\sigma)\in
F(\Upsilon)\\}$. Then $|E_{\sigma}|=r$ for all $\sigma\in V(\Upsilon)$, so
$\Upsilon$ is an $r$-valent graph.

Given a flag $(\epsilon,\sigma)\in F(\Upsilon)$, let
${\mathbf{w}}(\epsilon,\sigma)\in M=\mathrm{Hom}(T,\mathbb{C}^{*})$ be the
weight of $T$-action on $T_{p_{\sigma}}\ell_{\epsilon}$, the tangent line to
$\ell_{\epsilon}$ at the fixed point $p_{\sigma}$, namely,

${\mathbf{w}}(\epsilon,\sigma):=c_{1}^{T}(T_{p_{\sigma}}\ell_{\epsilon})\in
H^{2}_{T}(p_{\sigma};\mathbb{Z})\cong M.$

This gives rise to a map ${\mathbf{w}}:F(\Upsilon)\to M$ satisfying the
following properties.

  1. (1)

(GKM hypothesis) Given any $\sigma\in V(\Upsilon)$, and any two distinct edges
$\epsilon,\epsilon^{\prime}\in E_{\sigma}$, ${\mathbf{w}}(\epsilon,\sigma)$
and ${\mathbf{w}}(\epsilon^{\prime},\sigma)$ are linearly independent in
$M_{\mathbb{R}}$.

  2. (2)

Any edge $\epsilon\in E_{\sigma}$ connecting the vertices
$\sigma,\sigma^{\prime}\in V(\Upsilon)$ satisfies the property that:

    1. (a)

${\mathbf{w}}(\epsilon,\sigma)+{\mathbf{w}}(\epsilon,\sigma^{\prime})=0$.

    2. (b)

Let $E_{\sigma}=\\{\epsilon_{1},\ldots,\epsilon_{r}\\}$, where
$\epsilon_{r}=\epsilon$. For any $\epsilon_{i}\in E_{\sigma}$ there exists
$\epsilon_{i}^{\prime}\in E_{\sigma^{\prime}}$ and $a_{i}\in\mathbb{Z}$ such
that

${\mathbf{w}}(\epsilon_{i}^{\prime},\sigma^{\prime})={\mathbf{w}}(\epsilon_{i},\sigma)-a_{i}{\mathbf{w}}(\epsilon,\sigma).$

In particular, $\epsilon^{\prime}_{r}=\epsilon_{r}=\epsilon$ and $a_{r}=2$.

Let $\epsilon$ be as in (2). The normal bundle of
$\ell_{\epsilon}\cong\mathbb{P}^{1}$ in $X$ is given by

$N_{\ell_{\epsilon}/X}\cong L_{1}\oplus\cdots\oplus L_{r-1}$

where $L_{i}$ is a degree $a_{i}$ $T$-equivariant line bundle over
$\ell_{\epsilon}$ such that the weights of the $T$-action on the fibers
$(L_{i})_{p_{\sigma}}$ and $(L_{i})_{p_{\sigma^{\prime}}}$ are
${\mathbf{w}}(\epsilon_{i},\sigma)$ and
${\mathbf{w}}(\epsilon_{i}^{\prime},\sigma^{\prime})$, respectively. The map
${\mathbf{w}}:F(\Upsilon)\to M$ is called the axial function.

######  Example 9 ($Gr(r,m)$).

The GKM graph of $Gr(r,m)$ is a $k(m-k)$-valent graph $\Upsilon$ such that

$\displaystyle V(\Upsilon)$ $\displaystyle=$
$\displaystyle\\{\sigma_{J}:J\subset\\{1,\ldots,n\\},|J|=k\\}$ $\displaystyle
E(\Upsilon)$ $\displaystyle=$ $\displaystyle
E(\Upsilon)_{c}=\\{\epsilon_{I,K}:I\subset
K\subset\\{1,\ldots,m\\}|I|=k-1,|K|=k+1\\},$ $\displaystyle F(\Upsilon)$
$\displaystyle=$ $\displaystyle\\{(\epsilon_{I,K},\sigma_{J})\in
E(\Upsilon)\times V(\Upsilon):I\subset J\subset K$
${\mathbf{w}}(\epsilon_{IK},\sigma_{I\cup\\{j_{1}\\}})=-{\mathbf{w}}(\epsilon_{IK},\sigma_{I\cup\\{j_{2}\\}})=u_{j_{2}}-u_{j_{1}},\quad
j_{1},j_{2}\in I^{c},K=I\cup\\{j_{1},j_{2}\\}.$

We define the 1-skeleton of $X$ to be the union of 1-dimensional orbit
closures:

$X^{1}:=\bigcup_{\epsilon\in E(\Upsilon)}\ell_{\epsilon}.$

The formal completion $\hat{X}$ of $X$ along the 1-skeleton $X^{1}$, together
with the $T$-action, can be reconstructed from the graph $\Upsilon$ and
${\mathbf{w}}:F(\Upsilon)\to M$. We call $(\Upsilon,{\mathbf{w}})$ the GKM
graph of $X$ with the $T$-action. If $\rho:T^{\prime}\to T$ is a homomorphism
between complex algebraic tori, then $T^{\prime}$ acts on $X$ by
$t^{\prime}\cdot x=\rho(t^{\prime})\cdot x$, where $t^{\prime}\in T^{\prime}$,
$\rho(t^{\prime})\in T$, $x\in X$. The GKM graph of $X$ with this
$T^{\prime}$-action is given by $(\Upsilon,\rho^{*}\circ{\mathbf{w}})$, where
$\rho^{*}:M=\mathrm{Hom}(T,\mathbb{C}^{*})\to\mathrm{Hom}(T^{\prime},\mathbb{C}^{*})$.

##  3\. Gromov-Witten Theory

In this section, we give a brief review of Gromov-Witten theory and
equivariant Gromov-Witten theory.

###  3.1. Moduli of stable curves and Hodge integrals

An $n$-pointed, genus $g$ prestable curve is a connected algebraic curve $C$
of arithmetic genus $g$ together with $n$ ordered marked points
$x_{1},\ldots,x_{n}\in C$, where $C$ has at most nodal singularities, and
$x_{1},\ldots,x_{n}$ are distinct smooth points. An $n$-pointed, genus $g$
prestable curve $(C,x_{1},\ldots,x_{n})$ is stable if its automorphism group
is finite, or equivalently,

$\mathrm{Hom}_{\mathcal{O}_{C}}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C})=0.$

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of $n$-pointed, genus
$g$ stable curves, where $n,g$ are non-negative integers. We assume that
$2g-2+n>0$, so that $\overline{\mathcal{M}}_{g,n}$ is nonempty. Then
$\overline{\mathcal{M}}_{g,n}$ is a proper smooth Deligne-Mumford stack of
dimension $3g-3+n$ [8, 27, 25, 26]. The tangent space of
$\overline{\mathcal{M}}_{g,n}$ at a moduli point
$[(C,x_{1},\ldots,x_{n})]\in\overline{\mathcal{M}}_{g,n}$ is given by

$\mathrm{Ext}^{1}_{\mathcal{O}_{C}}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C}).$

Since $\overline{\mathcal{M}}_{g,n}$ is a proper Deligne-Mumford stack, we may
define

$\int_{\overline{\mathcal{M}}_{g,n}}:A^{*}(\overline{\mathcal{M}}_{g,n};\mathbb{Q})\to\mathbb{Q}.$

We now introduce some classes in $A^{*}(\overline{\mathcal{M}}_{g,n})$. There
is a forgetful morphism
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ given by
forgetting the $(n+1)$-th marked point (and contracting the unstable
irreducible component if there is one):

$[(C,x_{1},\ldots,x_{n},x_{n+1})]\mapsto[(C^{st},x_{1},\ldots,x_{n})]$

where $(C^{st},x_{1},\ldots,x_{n})$ is the stabilization of the prestable
curve $(C,x_{1},\ldots,x_{n})$.
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ can be
identified with the universal curve over $\overline{\mathcal{M}}_{g,n}$.

  * •

($\lambda$ classes) Let $\omega_{\pi}$ be the relative dualizing sheaf of
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$. The Hodge
bundle $\mathbb{E}=\pi_{*}\omega_{\pi}$ is a rank $g$ vector bundle over
$\overline{\mathcal{M}}_{g,n}$ whose fiber over the moduli point
$[(C,x_{1},\ldots,x_{n})]\in\overline{\mathcal{M}}_{g,n}$ is
$H^{0}(C,\omega_{C})$, the space of sections of the dualizing sheaf
$\omega_{C}$ of the curve $C$. The $\lambda$ classes are defined by

$\lambda_{j}=c_{j}(\mathbb{E})\in
A^{j}(\overline{\mathcal{M}}_{g,n};\mathbb{Q}).$

  * •

($\psi$ classes) The $i$-th marked point $x_{i}$ gives rise a section
$s_{i}:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_{g,n+1}$ of the
universal curve. Let $\mathbb{L}_{i}=s_{i}^{*}\omega_{\pi}$ be the line bundle
over $\overline{\mathcal{M}}_{g,n}$ whose fiber over the moduli point
$[(C,x_{1},\ldots,x_{n})]\in\overline{\mathcal{M}}_{g,n}$ is the cotangent
line $T_{x_{i}}^{*}C$ of $C$ at $x_{i}$. The $\psi$ classes are defined by

$\psi_{i}=c_{1}(\mathbb{L}_{i})\in
A^{1}(\overline{\mathcal{M}}_{g,n};\mathbb{Q}).$

Hodge integrals are top intersection numbers of $\lambda$ classes and $\psi$
classes:

(10)
$\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\lambda_{1}^{k_{1}}\cdots\lambda_{g}^{k_{g}}\in\mathbb{Q}.$

By definition, (10) is zero unless

$a_{1}+\cdots+a_{n}+k_{1}+2k_{2}+\cdots+gk_{g}=3g-3+n.$

Using Mumford’s Grothendieck-Riemann-Roch calculations in [37], Faber proved,
in [11], that general Hodge integrals can be uniquely reconstructed from the
$\psi$ integrals (also known as descendant integrals):

(11)
$\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}.$

The descendant integrals can be computed recursively by Witten’s conjecture
which asserts that the $\psi$ integrals (11) satisfy a system of differential
equations known as the KdV equations [46]. The KdV equations and the string
equation determine all the $\psi$ integrals (11) from the initial value
$\int_{\overline{\mathcal{M}}_{0,3}}1=1$. For example, from the initial value
$\int_{\overline{\mathcal{M}}_{0,3}}1=1$ and the string equation, one can
derive the following formula of genus $0$ descendant integrals:

(12)
$\int_{\overline{\mathcal{M}}_{0,n}}\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}=\frac{(n-3)!}{a_{1}!\cdots
a_{n}!}$

where $a_{1}+\cdots+a_{n}=n-3$ [29, Section 3.3.2].

The Witten’s conjecture was first proved by Kontsevich in [28]. By now,
Witten’s conjecture has been reproved many times (Okounkov-Pandharipande [38],
Mirzakhani [35], Kim-Liu [24], Kazarian-Lando [23], Chen-Li-Liu [6], Kazarian
[22], Mulase-Zhang [36], etc.).

###  3.2. Moduli of stable maps

Let $X$ be a nonsingular projective or quasi-projective variety, and let
$\beta\in H_{2}(X;\mathbb{Z})$. An $n$-pointed, genus $g$, degree $\beta$
prestable map to $X$ is a morphism $f:(C,x_{1},\ldots,x_{n})\to X$, where
$(C,x_{1},\ldots,x_{n})$ is an $n$-pointed, genus $g$ prestable curve, and
$f_{*}[C]=\beta$. Two prestable maps

$f:(C,x_{1},\ldots,x_{n})\to X,\quad
f^{\prime}:(C^{\prime},x^{\prime}_{1},\ldots,x^{\prime}_{n})\to X$

are isomorphic if there exists an isomorphism
$\phi:(C,x_{1},\ldots,x_{n})\to(C^{\prime},x^{\prime}_{1},\ldots,x^{\prime}_{n})$
of $n$-pointed prestable curves such that $f=f^{\prime}\circ\phi$. A prestable
map $f:(C,x_{1},\ldots,x_{n})\to X$ is stable if its automorphism group is
finite. The notion of stable maps was introduced by Kontsevich [29].

The moduli space $\overline{\mathcal{M}}_{g,n}(X,\beta)$ of $n$-pointed, genus
$g$, degree $\beta$ stable maps to $X$ is a Deligne-Mumford stack which is
proper when $X$ is projective [5].

###  3.3. Obstruction theory and virtual fundamental classes

The tangent space $T^{1}$ and the obstruction space $T^{2}$ at a moduli point
$[f:(C,x_{1},\ldots,x_{n})\to X]\in\overline{\mathcal{M}}_{g,n}(X,\beta)$ fit
in the tangent-obstruction exact sequence:

(13) $\displaystyle 0\to$
$\displaystyle\mathrm{Ext}^{0}_{\mathcal{O}_{C}}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C})\to
H^{0}(C,f^{*}T_{X})\to T^{1}$ $\displaystyle\to$
$\displaystyle\mathrm{Ext}^{1}_{\mathcal{O}_{C}}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C})\to
H^{1}(C,f^{*}T_{X})\to T^{2}\to 0$

where

  * •

$\mathrm{Ext}^{0}_{\mathcal{O}_{C}}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C})$
is the space of infinitesimal automorphisms of the domain
$(C,x_{1},\ldots,x_{n})$,

  * •

$\mathrm{Ext}^{1}_{\mathcal{O}_{C}}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C})$
is the space of infinitesimal deformations of the domain
$(C,x_{1},\ldots,x_{n})$,

  * •

$H^{0}(C,f^{*}T_{X})$ is the space of infinitesimal deformations of the map
$f$, and

  * •

$H^{1}(C,f^{*}T_{X})$ is the space of obstructions to deforming the map $f$.

$T^{1}$ and $T^{2}$ form sheaves $\mathcal{T}^{1}$ and $\mathcal{T}^{2}$ on
the moduli space $\overline{\mathcal{M}}_{g,n}(X,\beta)$.

Let $X$ be a nonsingular projective variety. We say $X$ is convex if
$H^{1}(C,f^{*}TX)=0$ for all genus $0$ stable maps $f$. Projective spaces
$\mathbb{P}^{n}$, or more generally, generalized flag varieties $G/P$, are
examples of convex varieties. When $X$ is convex and $g=0$, the obstruction
sheaf $\mathcal{T}^{2}=0$, and the moduli space
$\overline{\mathcal{M}}_{0,n}(X,\beta)$ is a smooth Deligne-Mumford stack.

In general, $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is a singular Deligne-
Mumford stack equipped with a perfect obstruction theory: there is a two term
complex of locally free sheaves $E\to F$ on
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ such that

$0\to\mathcal{T}^{1}\to F^{\vee}\to E^{\vee}\to\mathcal{T}^{2}\to 0$

is an exact sequence of sheaves. (See [4] for the complete definition of a
perfect obstruction theory.) The virtual dimension $d^{\mathrm{vir}}$ of
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ is the rank of the virtual tangent
bundle $T^{\mathrm{vir}}=F^{\vee}-E^{\vee}$.

(14) $d^{\mathrm{vir}}=\int_{\beta}c_{1}(T_{X})+(\dim X-3)(1-g)+n$

Suppose that $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is proper. (Recall that
if $X$ is projective then $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is proper
for any $g$, $n$, $\beta$.) Then there is a virtual fundamental class

$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}\in
A_{d^{\mathrm{vir}}}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q}).$

The virtual fundamental class has been constructed by Li-Tian [31], Behrend-
Fantechi [4] in algebraic Gromov-Witten theory. The virtual fundamental class
allows us to define

$\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}}:A^{*}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})\longrightarrow\mathbb{Q},\quad\alpha\mapsto\deg(\alpha\cap[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}).$

###  3.4. Gromov-Witten invariants

Let $X$ be a nonsingular projective variety. Gromov-Witten invariants are
rational numbers defined by applying

$\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}}:A^{*}(\overline{\mathcal{M}}_{g,n}(X,\beta))\to\mathbb{Q}$

to certain classes in $A^{*}(\overline{\mathcal{M}}_{g,n}(X,\beta))$.

Let $\mathrm{ev}_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to X$ be the
evaluation at the $i$-th marked point: $\mathrm{ev}_{i}$ sends
$[f:(C,x_{1},\ldots,x_{n})\to X]\in\overline{\mathcal{M}}_{g,n}(X,\beta)$ to
$f(x_{i})\in X$. Given $\gamma_{1},\ldots,\gamma_{n}\in A^{*}(X)$, define

(15)
$\langle\gamma_{1},\ldots,\gamma_{n}\rangle^{X}_{g,\beta}=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}}\mathrm{ev}_{1}^{*}\gamma_{1}\cup\cdots\cup\mathrm{ev}_{n}^{*}\gamma_{n}\in\mathbb{Q}.$

These are known as the primary Gromov-Witten invariants of $X$. More
generally, we may also view
$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}$ as a class in
$H_{2d}(\overline{\mathcal{M}}_{g,n}(X,\beta))$. Then (15) is defined for
ordinary cohomology classes $\gamma_{1},\ldots,\gamma_{n}\in H^{*}(X)$,
including odd cohomology classes which do not come from
$A^{*}(\overline{\mathcal{M}}_{g,n}(X,\beta))$.

Let
$\pi:\overline{\mathcal{M}}_{g,n+1}(X,\beta)\to\overline{\mathcal{M}}_{g,n}(X,\beta)$
be the universal curve. For $i=1,\ldots,n$, let
$s_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n+1}(X,\beta)$,
be the section which corresponds to the $i$-th marked point. Let
$\omega_{\pi}\to\overline{\mathcal{M}}_{g,n+1}(X,\beta)$ be the relative
dualizing sheaf of $\pi$, and let $\mathbb{L}_{i}=s_{i}^{*}\omega_{\pi}$ be
the line bundle over $\overline{\mathcal{M}}_{g,n}(X,\beta)$ whose fiber at
the moduli point $[f:(C,x_{1},\ldots,x_{n})\to
X]\in\overline{\mathcal{M}}_{g,n}(X,\beta)$ is the cotangent line
$T^{*}_{x_{i}}C$ at the $i$-th marked point $x_{i}$. The $\psi$-classes are
defined to be

$\psi_{i}:=c_{1}(\mathbb{L}_{i})\in
A^{1}(\overline{\mathcal{M}}_{g,n}(X,\beta)),\quad i=1,\ldots,n.$

We use the same notation $\psi_{i}$ to denote the corresponding classes in the
ordinary cohomology group $H^{2}(\overline{\mathcal{M}}_{g,n}(X,\beta))$.

Genus $g$, degree $\beta$ descendant Gromov-Witten invariants of $X$ are
defined by

(16)
$\langle\tau_{a_{1}}(\gamma_{1})\cdots\tau_{a_{n}}(\gamma_{n})\rangle_{g,\beta}^{X}:=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}}\mathrm{ev}_{1}^{*}\gamma_{1}\cup\psi_{1}^{a_{1}}\cup\cdots\cup\mathrm{ev}_{n}^{*}\gamma_{n}\cup\psi_{n}^{a_{n}}\in\mathbb{Q}.$

Suppose that $\gamma_{i}\in H^{d_{i}}(X)$. Then (16) is zero unless

(17) $\sum_{i=1}^{n}(d_{i}+2a_{i}-2)=2\Bigl{(}\int_{\beta}c_{1}(TX)+(\dim
X-3)(1-g)\Bigr{)}.$

###  3.5. Equivariant Gromov-Witten invariants

Let $X$ be a non-singular projective or quasi-projective algebraic variety,
equipped with an algebraic action of $T=(\mathbb{C}^{*})^{m}$. Then $T$ acts
on $\overline{\mathcal{M}}_{g,n}(X,\beta)$ by

$t\cdot[f:(C,x_{1},\ldots,x_{n})\to X]\mapsto[t\cdot
f:(C,x_{1},\ldots,x_{n})\to X]$

where $(t\cdot f)(z)=t\cdot f(z)$, $z\in C$. The evaluation maps
$\mathrm{ev}_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to X$ are
$T$-equivariant and induce $\mathrm{ev}_{i}^{*}:A^{*}_{T}(X;\mathbb{Q})\to
A^{*}_{T}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q})$.

Suppose that $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is proper, so that there
are virtual fundamental classes

$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}\in
A_{d^{\mathrm{vir}}}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q}),\quad[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}_{T}\in
A_{d^{\mathrm{vir}}}^{T}(\overline{\mathcal{M}}_{g,n}(X,\beta);\mathbb{Q}),$

where

$d^{\mathrm{vir}}=\int_{\beta}c_{1}(TX)+(r-3)(1-g)+n.$

Given $\gamma_{i}\in A^{d_{i}}(X;\mathbb{Q})=H^{2d_{i}}(X;\mathbb{Q})$ and
$a_{i}\in\mathbb{Z}_{\geq 0}$, define
$\langle\tau_{a_{i}}(\gamma_{1})\cdots\tau_{a_{n}}(\gamma_{n})\rangle^{X}_{g,\beta}$
as in Section 3.4:

(18)
$\langle\tau_{a_{1}}(\gamma_{1})\cdots\tau_{a_{n}}(\gamma_{n})\rangle^{X}_{g,\beta}=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}\cup\psi_{i}^{a_{i}}\right)\in\mathbb{Q}.$

By definition, (18) is zero unless $\sum_{i=1}^{n}d_{i}=d^{\mathrm{vir}}$. In
this case,

(19)
$\langle\tau_{a_{1}}(\gamma_{1})\cdots\tau_{a_{n}}(\gamma_{n})\rangle^{X}_{g,\beta}=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}_{T}}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi_{i}^{T})^{a_{i}}\right)$

where $\gamma_{i}^{T}\in A^{d_{i}}_{T}(X)$ is any $T$-equivariant lift of
$\gamma_{i}\in A^{d_{i}}(X)$, and $\psi_{i}^{T}\in
A^{1}_{T}(\overline{\mathcal{M}}_{g,n}(X,\beta))$ is any $T$-equivariant lift
of $\psi_{i}\in A^{1}(\overline{\mathcal{M}}_{g,n}(X,\beta))$.

In this paper, we fix a choice of $\psi_{i}^{T}$ as follows. A stable map
$f:(C,x_{1},\ldots,x_{n})\to X$ induces $\mathbb{C}$-linear maps
$T_{x_{i}}C\to T_{f(x_{i})}X$ for $i=1,\ldots,n$. This gives rise to
$\mathbb{L}_{i}^{\vee}\to\mathrm{ev}_{i}^{*}TX$. The $T$-action on $X$ induces
a $T$-action on $TX$, so that $TX$ is a $T$-equivariant vector bundle over
$X$, and $\mathrm{ev}_{i}^{*}TX$ is a $T$-equivariant vector bundle over
$\overline{\mathcal{M}}_{g,n}(X,\beta)$. Let $T$ act on $\mathbb{L}_{i}$ such
that $\mathbb{L}_{i}^{\vee}\to\mathrm{ev}_{i}^{*}TX$ is $T$-equivariant, and
define

$\psi_{i}^{T}=c_{1}^{T}(\mathbb{L}_{i})\in
A^{1}_{T}(\overline{\mathcal{M}}_{g,n}(X,\beta)),\quad i=1,\ldots,n.$

Then $\psi_{i}^{T}$ is a $T$-equivariant lift of
$\psi_{i}=c_{1}(\mathbb{L}_{i})\in
A^{1}(\overline{\mathcal{M}}_{g,n}(X,\beta))$.

Given $\gamma_{i}^{T}\in A_{T}^{d_{i}}(X;\mathbb{Q})$, we define genus $g$,
degree $\beta$ $T$-equivariant descendant Gromov-Witten invariants

(20)
$\displaystyle\langle\tau_{a_{1}}(\gamma_{1}^{T}),\cdots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle_{g,\beta}^{X_{T}}$
$\displaystyle:=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir},T}}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}(\psi_{i}^{T})^{a_{i}}\right)$
$\displaystyle\in\mathbb{Q}[u_{1},\ldots,u_{m}](\sum_{i=1}^{n}d_{i}-d^{\mathrm{vir}}).$

where $\mathbb{Q}[u_{1},\ldots,u_{m}](k)$ is the space of degree $k$
homogeneous polynomials in $u_{1},\ldots,u_{m}$ with rational coefficients. In
particular,

$\langle\tau_{a_{1}}(\gamma_{1}^{T}),\cdots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle_{g,\beta}^{X_{T}}=\begin{cases}0,&\sum_{i=1}^{n}d_{i}<d^{\mathrm{vir}},\\\
\langle\tau_{a_{1}}(\gamma_{1}),\cdots,\tau_{a_{n}}(\gamma_{n})\rangle_{g,\beta}^{X}\in\mathbb{Q},&\sum_{i=1}^{n}d_{i}=d^{\mathrm{vir}}.\end{cases}$

where $\gamma_{i}\in A^{d_{i}}(X;\mathbb{Q})$ is the image of $\gamma_{i}^{T}$
under $A^{d_{i}}_{T}(X;\mathbb{Q})\to A^{d_{i}}(X;\mathbb{Q})$.

Let
$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}\subset\overline{\mathcal{M}}_{g,n}(X,\beta)$
be the substack of $T$-fixed points, and let
$i:\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}\to\overline{\mathcal{M}}_{g,n}(X,\beta)$
be the inclusion. Let $N^{\mathrm{vir}}$ be the virtual normal bundle of
substack $\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}$ in
$\overline{\mathcal{M}}_{g,n}(X,\beta)$; in general, $N^{\mathrm{vir}}$ has
different ranks on different connected components of
$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}$. By virtual localization,

(21)
$\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}_{T}}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi_{i}^{T})^{a_{i}}\right)=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}]^{\mathrm{vir}}_{T}}\frac{i^{*}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi_{i}^{T})^{a_{i}}\right)}{e^{T}(N^{\mathrm{vir}})}.$

If $\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}$ is proper but
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ is not, we define

(22)
$\langle\tau_{a_{1}}(\gamma_{1}^{T}),\ldots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle^{X_{T}}_{g,\beta}=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}]^{\mathrm{vir}}_{T}}\frac{i^{*}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi_{i}^{T})^{a_{i}}\right)}{e^{T}(N^{\mathrm{vir}})}\in
Q_{T}.$

When $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is not proper, the right hand
side of (22) is a rational function (instead of a polynomial) in
$u_{1},\ldots,u_{m}$. It can be nonzero when $\sum d_{i}<d^{\mathrm{vir}}$,
and does not have a nonequivariant limit (obtained by setting $u_{i}=0$) in
general.

##  4\. Virtual Localization

In this section, we compute all genus equivariant descendant Gromov-Witten
invariants of any algebraic GKM manifold by virtual localization. This
generalizes the toric case in [33, Section 5].

Let $X$ be an algebraic GKM manifold of dimension $r$, with an algebraic
action of $T=(\mathbb{C}^{*})^{m}$.

###  4.1. Torus fixed points and graph notation

In this subsection, we describe the $T$-fixed points in
$\overline{\mathcal{M}}_{g,n}(X,\beta)$. Following Kontsevich [29], given a
stable map $f:(C,x_{1},\ldots,x_{n})\to X$ such that

$[f:(C,x_{1},\ldots,x_{n})\to X]\in\overline{\mathcal{M}}_{g,n}(X,\beta)^{T},$

we will associate a decorated graph ${\vec{\Gamma}}$.

We first give a formal definition.

######  Definition 23.

A decorated graph ${\vec{\Gamma}}=(\Gamma,\vec{f},\vec{d},\vec{g},\vec{s})$
for $n$-pointed, genus $g$, degree $\beta$ stable maps to $X$ consists of the
following data.

  1. (1)

$\Gamma$ is a compact, connected 1 dimensional CW complex. We denote the set
of vertices (resp. edges) in $\Gamma$ by $V(\Gamma)$ (resp. $E(\Gamma)$). Let

$F(\Gamma)=\\{(e,v)\in E(\Gamma)\times V(\Gamma)\mid v\in e\\}$

be the set of flags in $\Gamma$.

  2. (2)

The label map $\vec{f}:V(\Gamma)\cup E(\Gamma)\to V(\Upsilon)\cup
E(\Upsilon)_{c}$ sends a vertex $v\in V(\Gamma)$ to a vertex $\sigma_{v}\in
V(\Upsilon)$, and sends an edge $e\in E(\Gamma)$ to an edge $\epsilon_{e}\in
E(\Upsilon)_{c}$. Moreover, $\vec{f}$ defines a map from the graph $\Gamma$ to
the graph $\Upsilon$: if $(e,v)$ is a flag in $\Gamma$ then
$(\epsilon_{e},\sigma_{v})$ is a flag in $\Upsilon$.

  3. (3)

The degree map $\vec{d}:E(\Gamma)\to\mathbb{Z}_{>0}$ sends an edge $e\in
E(\Gamma)$ to a positive integer $d_{e}$.

  4. (4)

The genus map $\vec{g}:V(\Gamma)\to\mathbb{Z}_{\geq 0}$ sends a vertex $v\in
V(\Gamma)$ to a non-negative integer $g_{v}$.

  5. (5)

The marking map $\vec{s}:\\{1,2,\ldots,n\\}\to V(\Gamma)$ is defined if $n>0$.

The above maps satisfy the following two constraints:

  1. (i)

(topology of the domain) $\displaystyle{\sum_{v\in
V(\Gamma)}g_{v}+|E(\Gamma)|-|V(\Gamma)|+1=g}$.

  2. (ii)

(topology of the map) $\displaystyle{\sum_{e\in
E(\Gamma)}d_{e}[\ell_{\epsilon_{e}}]=\beta}$.

Let $G_{g,n}(X,\beta)$ be the set of all decorated graphs
${\vec{\Gamma}}=(\Gamma,\vec{f},\vec{d},\vec{g},\vec{s})$ satisfying the above
constraints.

We now describe the geometry and combinatorics of a stable map
$f:(C,x_{1},\ldots,x_{n})\to X$ which represents a $T$-fixed point in
$\overline{\mathcal{M}}_{g,n}(X,\beta)$.

For any $t\in T$, there exists an automorphism
$\phi_{t}:(C,x_{1},\ldots,x_{n})$ such that $t\cdot f(z)=f\circ\phi_{t}(z)$
for any $z\in C$. Let $C^{\prime}$ be an irreducible component of $C$, and let
$f^{\prime}=f|_{C^{\prime}}:C^{\prime}\to X$. There are two possibilities:

  1. Case 1:

$f^{\prime}$ is a constant map, and $f(C^{\prime})=\\{p_{\sigma}\\}$, where
$p_{\sigma}$ is a fixed point in $X$ associated to some $\sigma\in
V(\Upsilon)$

  2. Case 2:

$C^{\prime}\cong\mathbb{P}^{1}$ and $f(C^{\prime})=\ell_{\epsilon}$, where
$\ell_{\epsilon}$ is a $T$-invariant $\mathbb{P}^{1}$ in $X$ associated to
some $\epsilon\in E(\Upsilon)_{c}$.

We define a decorated graph ${\vec{\Gamma}}$ associated to
$f:(C,x_{1},\ldots,x_{n})\to X$ as follows.

  1. (1)

(Vertices) We assign a vertex $v$ to each connected component $C_{v}$ of
$f^{-1}(X^{T})$.

    1. (a)

(label) $f(C_{v})=\\{p_{\sigma}\\}$ for some $\sigma\in V(\Upsilon)$; we
define $\vec{f}(v)=\sigma_{v}=\sigma$.

    2. (b)

(genus) $C_{v}$ is a curve or a point. If $C_{v}$ is a curve then we define
$\vec{g}(v)=g_{v}$ to be the arithmetic genus of $C_{v}$; if $C_{v}$ is a
point then we define $\vec{g}(v)=g_{v}=0$.

    3. (c)

(marking) For $i=1,\ldots,n$, define $\vec{s}(i)=v$ if $x_{i}\in C_{v}$.

  2. (2)

(Edges) For any $\epsilon\in E(\Upsilon)$, let
$O_{\epsilon}\cong\mathbb{C}^{*}$ be the 1-dimensional orbit whose closure is
$\ell_{\epsilon}$. Then

$X^{1}\setminus X^{T}=\bigsqcup_{\epsilon\in E(\Upsilon)}O_{\epsilon}$

where the right hand side is a disjoint union of connected components. We
assign an edge $e$ to each connected component $O_{e}\cong\mathbb{C}^{*}$ of
$f^{-1}(X^{1}\setminus X^{T})$.

    1. (a)

(label) Let $C_{e}\cong\mathbb{P}^{1}$ be the closure of $O_{e}$. Then
$f(C_{e})=\ell_{\epsilon}$ for some $\epsilon\in E(\Upsilon)_{c}$; we define
$\vec{f}(e)=\epsilon_{e}=\epsilon$.

    2. (b)

(degree) We define $\vec{d}(e)=d_{e}$ to be the degree of the map
$f|_{C_{e}}:C_{e}\cong\mathbb{P}^{1}\to\ell_{\epsilon}\cong\mathbb{P}^{1}$.

  3. (3)

(Flags) The set of flags in the graph $\Gamma$ is defined by

$F(\Gamma)=\\{(e,v)\in E(\Gamma)\times V(\Gamma)\mid C_{e}\cap
C_{v}\neq\emptyset\\}.$

The above (1), (2), (3) define a decorated graph
${\vec{\Gamma}}=(\Gamma,\vec{f},\vec{d},\vec{g},\vec{s})$ satisfying the
constraints (i) and (ii) in Definition 23. Therefore ${\vec{\Gamma}}\in
G_{g,n}(X,\beta)$. This gives a map from
$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}$ to the discrete set
$G_{g,n}(X,\beta)$. Let
$\mathcal{F}_{\vec{\Gamma}}\subset\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}$
denote the preimage of ${\vec{\Gamma}}$. Then

$\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}=\bigsqcup_{{\vec{\Gamma}}\in
G_{g,n}(X,\beta)}\mathcal{F}_{\vec{\Gamma}}$

where the right hand side is a disjoint union of connected components. We next
describe the fixed locus $\mathcal{F}_{\vec{\Gamma}}$ associated to each
decorated graph ${\vec{\Gamma}}\in G_{g,n}(X,\beta)$. For later convenience,
we introduce some definitions.

######  Definition 24.

Given a vertex $v\in V(\Gamma)$, we define

$E_{v}=\\{e\in E(\Gamma)\mid(e,v)\in F(\Gamma)\\},$

the set of edges emanating from $v$, and define
$S_{v}=\vec{s}^{-1}(v)\subset\\{1,\ldots,n\\}$. The valency of $v$ is given by
$\mathrm{val}(v)=|E_{v}|$. Let $n_{v}=|S_{v}|$ be the number of marked points
contained in $C_{v}$. We say a vertex is stable if
$2g_{v}-2+\mathrm{val}(v)+n_{v}>0$. Let $V^{S}(\Gamma)$ be the set of stable
vertices in $V(\Gamma)$. There are three types of unstable vertices:

$\displaystyle V^{1}(\Gamma)$ $\displaystyle=$ $\displaystyle\\{v\in
V(\Gamma)\mid g_{v}=0,\mathrm{val}(v)=1,n_{v}=0\\},$ $\displaystyle
V^{1,1}(\Gamma)$ $\displaystyle=$ $\displaystyle\\{v\in V(\Gamma)\mid
g_{v}=0,\mathrm{val}(v)=n_{v}=1\\},$ $\displaystyle V^{2}(\Gamma)$
$\displaystyle=$ $\displaystyle\\{v\in V(\Gamma)\mid
g_{v}=0,\mathrm{val}(v)=2,n_{v}=0\\}.$

Then $V(\Gamma)$ is the disjoint union of $V^{1}(\Gamma)$, $V^{1,1}(\Gamma)$,
$V^{2}(\Gamma)$, and $V^{S}(\Gamma)$.

The set of stable flags is defined to be

$F^{S}(\Gamma)=\\{(e,v)\in F(\Gamma)\mid v\in V^{S}(\Gamma)\\}.$

Given a decorated graph
${\vec{\Gamma}}=(\Gamma,\vec{f},\vec{d},\vec{g},\vec{s})$, the curves $C_{e}$
and the maps $f|_{C_{e}}:C_{e}\to\ell_{\epsilon_{e}}\subset X$ are determined
by ${\vec{\Gamma}}$. If $v\notin V^{S}(\Gamma)$ then $C_{v}$ is a point. If
$v\in V^{S}(\Gamma)$ then $C_{v}$ is a curve, and $y(e,v):=C_{e}\cap C_{v}$ is
a node of $C$ for $e\in E_{v}$.

$\bigl{(}C_{v},\\{y(e,v):e\in E_{v}\\}\cup\\{x_{i}\mid i\in S_{v}\\}\bigr{)}$

is a $(\mathrm{val}(v)+n_{v})$-pointed, genus $g_{v}$ curve, which represents
a point in $\overline{\mathcal{M}}_{g_{v},\mathrm{val}(v)+n_{v}}$. We call
this moduli space $\overline{\mathcal{M}}_{g_{v},E_{v}\cup S_{v}}$ instead of
$\overline{\mathcal{M}}_{g_{v},\mathrm{val}(v)+n_{v}}$ because we would like
to label the marked points on $C_{v}$ by $E_{v}\cup S_{v}$ instead of
$\\{1,2,\ldots,\mathrm{val}(v)+n_{v}\\}$. Then

$\mathcal{M}_{{\vec{\Gamma}}}=\prod_{v\in
V^{S}(\Gamma)}\overline{\mathcal{M}}_{g_{v},E_{v}\cup S_{v}}.$

The automorphism group $A_{{\vec{\Gamma}}}$ for any point
$[f:(C,x_{1},\ldots,x_{n})\to X]\in\mathcal{F}_{{\vec{\Gamma}}}$ fits in the
following short exact sequence of groups:

$1\to\prod_{e\in E(\Gamma)}\mathbb{Z}_{d_{e}}\to
A_{{\vec{\Gamma}}}\to\mathrm{Aut}({\vec{\Gamma}})\to 1$

where $\mathbb{Z}_{d_{e}}$ is the automorphism group of the degree $d_{e}$
morphism

$f|_{C_{e}}:C_{e}\cong\mathbb{P}^{1}\to\ell_{\epsilon_{e}}\cong\mathbb{P}^{1},$

and $\mathrm{Aut}({\vec{\Gamma}})$ is the automorphism group of the decorated
graph ${\vec{\Gamma}}=(\Gamma,\vec{f},\vec{d},\vec{g},\vec{s})$. There is a
morphism
$i_{{\vec{\Gamma}}}:\mathcal{M}_{{\vec{\Gamma}}}\to\overline{\mathcal{M}}_{g,n}(X,\beta)$
whose image is the fixed locus $\mathcal{F}_{{\vec{\Gamma}}}$ associated to
${\vec{\Gamma}}\in G_{g,n}(X,\beta)$. The morphism $i_{{\vec{\Gamma}}}$
induces an isomorphism
$[\mathcal{M}_{{\vec{\Gamma}}}/A_{{\vec{\Gamma}}}]\cong\mathcal{F}_{{\vec{\Gamma}}}$.

###  4.2. Virtual tangent and normal bundles

Given a decorated graph ${\vec{\Gamma}}\in G_{g,n}(X,\beta)$ and a stable map
$f:(C,x_{1},\ldots,x_{n})\to X$ which represents a point in the fixed locus
$\mathcal{F}_{{\vec{\Gamma}}}$ associated to ${\vec{\Gamma}}$, let

$\displaystyle
B_{1}=\mathrm{Hom}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C}),\quad
B_{2}=H^{0}(C,f^{*}TX)$ $\displaystyle
B_{4}=\mathrm{Ext}^{1}(\Omega_{C}(x_{1}+\cdots+x_{n}),\mathcal{O}_{C}),\quad
B_{5}=H^{1}(C,f^{*}TX)$

$T$ acts on $B_{1},B_{2},B_{4},B_{5}$. Let $B_{i}^{m}$ and $B_{i}^{f}$ be the
moving and fixed parts of $B_{i}$. We have the following exact sequences:

(25) $0\to B_{1}^{f}\to B_{2}^{f}\to T^{1,f}\to B_{4}^{f}\to B_{5}^{f}\to
T^{2,f}\to 0$ (26) $0\to B_{1}^{m}\to B_{2}^{m}\to T^{1,m}\to B_{4}^{m}\to
B_{5}^{m}\to T^{2,m}\to 0$

The irreducible components of $C$ are

$\\{C_{v}\mid v\in V^{S}(\Gamma)\\}\cup\\{C_{e}\mid e\in E(\Gamma)\\}.$

The nodes of $C$ are

$\\{y_{v}=C_{v}\mid v\in V^{2}(\Gamma)\\}\cup\\{y(e,v)\mid(e,v)\in
F^{S}(\Gamma)\\}$

####  4.2.1. Automorphisms of the domain

Given any $(e,v)\in F(\Gamma)$, let $y(e,v)=C_{e}\cap C_{v}$, and define

$w_{(e,v)}:=e^{T}(T_{y(e,v)}C_{e})=\frac{{\mathbf{w}}(\epsilon_{e},\sigma_{v})}{d_{e}}\in
H_{T}^{2}(y(e,v);\mathbb{Q})=M\otimes_{\mathbb{Z}}\mathbb{Q}.$

We have

$\displaystyle B_{1}^{f}$ $\displaystyle=$
$\displaystyle\bigoplus_{\tiny\begin{array}[]{c}e\in E(\Gamma)\\\
(e,v),(e,v^{\prime})\in
F(\Gamma)\end{array}}\mathrm{Hom}(\Omega_{C_{e}}(y(e,v)+y(e,v^{\prime})),\mathcal{O}_{C_{e}})$
$\displaystyle=$ $\displaystyle\bigoplus_{\tiny\begin{array}[]{c}e\in
E(\Gamma)\\\ (e,v),(e,v^{\prime})\in
F(\Gamma)\end{array}}H^{0}(C_{e},TC_{e}(-y(e,v)-y(e,v^{\prime}))$
$\displaystyle B_{1}^{m}$ $\displaystyle=$ $\displaystyle\bigoplus_{v\in
V^{1}(\Gamma),(e,v)\in F(\Gamma)}T_{y(e,v)}C_{e}$

####  4.2.2. Deformations of the domain

Given any $v\in V^{S}(\Gamma)$, define a divisor ${\mathbf{x}}_{v}$ of $C_{v}$
by

${\mathbf{x}}_{v}=\sum_{i\in S_{v}}x_{i}+\sum_{e\in E_{v}}y(e,v).$

Then

$\displaystyle B_{4}^{f}$ $\displaystyle=$ $\displaystyle\bigoplus_{v\in
V^{S}(\Gamma)}\mathrm{Ext}^{1}(\Omega_{C_{v}}({\mathbf{x}}_{v}),\mathcal{O}_{C})=\bigoplus_{v\in
V^{S}(\Gamma)}T\overline{\mathcal{M}}_{g_{v},E_{v}\cup S_{v}}$ $\displaystyle
B_{4}^{m}$ $\displaystyle=$ $\displaystyle\bigoplus_{v\in
V^{2}(\Gamma),E_{v}=\\{e,e^{\prime}\\}}T_{y_{v}}C_{e}\otimes
T_{y_{v}}C_{e^{\prime}}\oplus\bigoplus_{(e,v)\in
F^{S}(\Gamma)}T_{y(e,v)}C_{v}\otimes T_{y(e,v)}C_{e}$

where

$\displaystyle e^{T}(T_{y_{v}}C_{e}\otimes T_{y_{v}}C_{e^{\prime}})$
$\displaystyle=$ $\displaystyle w_{(e,v)}+w_{(e^{\prime},v)},\quad v\in
V^{2}(\Gamma)$ $\displaystyle e^{T}(T_{y(e,v)}C_{v}\otimes T_{y(e,v)}C_{e})$
$\displaystyle=$ $\displaystyle w_{(e,v)}-\psi_{(e,v)},\quad v\in
V^{S}(\Gamma)$

####  4.2.3. Unifying stable and unstable vertices

From the discussion in Section 4.2.1 and Section 4.2.2,

(29) $\displaystyle\frac{e^{T}(B_{1}^{m})}{e^{T}(B_{4}^{m})}=$
$\displaystyle\prod_{v\in V^{1}(\Gamma),(e,v)\in
F(\Gamma)}w_{(e,v)}\prod_{v\in
V^{2}(\Gamma),E_{v}=\\{e,e^{\prime}\\}}\frac{1}{w_{(e,v)}+w_{(e^{\prime},v)}}$
$\displaystyle\cdot\prod_{v\in V^{S}(\Gamma)}\frac{1}{\prod_{e\in
E_{v}}(w_{(e,v)}-\psi_{(e,v)})}.$

Recall that

$\mathcal{M}_{\vec{\Gamma}}=\prod_{v\in
V^{S}(\Gamma)}\overline{\mathcal{M}}_{g_{v},E_{v}\cup S_{v}}.$

To unify the stable and unstable vertices, we use the following convention for
the empty sets $\overline{\mathcal{M}}_{0,1}$ and
$\overline{\mathcal{M}}_{0,2}$. Let $w_{1},w_{2}$ be formal variables.

  1. (i)

$\overline{\mathcal{M}}_{0,1}$ is a $-2$ dimensional space, and

(30) $\int_{\overline{\mathcal{M}}_{0,1}}\frac{1}{w_{1}-\psi_{1}}=w_{1}.$

  2. (ii)

$\overline{\mathcal{M}}_{0,2}$ is a $-1$ dimensional space, and

(31)
$\int_{\overline{\mathcal{M}}_{0,2}}\frac{1}{(w_{1}-\psi_{1})(w_{2}-\psi_{2})}=\frac{1}{w_{1}+w_{2}}$
(32) $\int_{\overline{\mathcal{M}}_{0,2}}\frac{1}{w_{1}-\psi_{1}}=1.$

  3. (iii)

$\displaystyle{\mathcal{M}_{{\vec{\Gamma}}}=\prod_{v\in
V(\Gamma)}\overline{\mathcal{M}}_{g_{v},E_{v}\cup S_{v}}}$.

With the above conventions (i), (ii), (iii), we may rewrite (29) as

(33) $\frac{e^{T}(B_{1}^{m})}{e^{T}(B_{4}^{m})}=\prod_{v\in
V(\Gamma)}\frac{1}{\prod_{e\in E_{v}}(w_{(e,v)}-\psi_{(e,v)})}.$

The following lemma shows that the conventions (i) and (ii) are consistent
with the stable case $\overline{\mathcal{M}}_{0,n}$, $n\geq 3$.

######  Lemma 34.

For any positive integer $n$ and formal variables $w_{1},\ldots,w_{n}$, we
have

  1. (a)

$\displaystyle{\int_{\overline{\mathcal{M}}_{0,n}}\frac{1}{\prod_{i=1}^{n}(w_{i}-\psi_{i})}=\frac{1}{w_{1}\cdots
w_{n}}(\frac{1}{w_{1}}+\cdots\frac{1}{w_{n}})^{n-3}}$.

  2. (b)

$\displaystyle{\int_{\overline{\mathcal{M}}_{0,n}}\frac{1}{w_{1}-\psi_{1}}=w_{1}^{2-n}}$.

###### Proof.

(a) The cases $n=1$ and $n=2$ follow from the definitions (30) and (31),
respectively. For $n\geq 3$, we have

$\displaystyle\int_{\overline{\mathcal{M}}_{0,n}}\frac{1}{\prod_{i=1}^{n}(w_{i}-\psi_{i})}=\frac{1}{w_{1}\cdots
w_{n}}\int_{\overline{\mathcal{M}}_{0,n}}\frac{1}{\prod_{i=1}^{n}(1-\frac{\psi_{i}}{w_{i}})}$
$\displaystyle=\frac{1}{w_{1}\cdots
w_{n}}\sum_{a_{1}+\cdots+a_{n}=n-3}w_{1}^{-a_{1}}\cdots
w_{n}^{-a_{n}}\int_{\overline{\mathcal{M}}_{0,n}}\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}$

where

$\int_{\overline{\mathcal{M}}_{0,n}}\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}=\frac{(n-3)!}{a_{1}!\cdots
a_{n}!}.$

So

$\int_{\overline{\mathcal{M}}_{0,n}}\frac{1}{\prod_{i=1}^{n}(w_{i}-\psi_{i})}=\frac{1}{w_{1}\cdots
w_{n}}(\frac{1}{w_{1}}+\cdots\frac{1}{w_{n}})^{n-3}.$

(b) The cases $n=1$ and $n=2$ follow from the definitions (30) and (32),
respectively. For $n\geq 3$, we have

$\int_{\overline{\mathcal{M}}_{0,n}}\frac{1}{w_{1}-\psi_{1}}=\frac{1}{w_{1}}\int_{\overline{\mathcal{M}}_{0,n}}\frac{1}{1-\frac{\psi_{1}}{w_{1}}}=\frac{1}{w_{1}}w_{1}^{3-n}=w_{1}^{2-n}$

∎

####  4.2.4. Deformation of the map

Consider the normalization sequence

(35) $\displaystyle 0$ $\displaystyle\to\mathcal{O}_{C}\to\bigoplus_{v\in
V^{S}(\Gamma)}\mathcal{O}_{C_{v}}\oplus\bigoplus_{e\in
E(\Gamma)}\mathcal{O}_{C_{e}}$ $\displaystyle\to\bigoplus_{v\in
V^{2}(\Gamma)}\mathcal{O}_{y_{v}}\oplus\bigoplus_{(e,v)\in
F^{S}(\Gamma)}\mathcal{O}_{y(e,v)}\to 0.$

We twist the above short exact sequence of sheaves by $f^{*}TX$. The resulting
short exact sequence gives rise a long exact sequence of cohomology groups

$\displaystyle 0$ $\displaystyle\to$ $\displaystyle B_{2}\to\bigoplus_{v\in
V^{S}(\Gamma)}H^{0}(C_{v})\oplus\bigoplus_{e\in E(\Gamma)}H^{0}(C_{e})$
$\displaystyle\to$ $\displaystyle\bigoplus_{v\in
V^{2}(\Gamma)}T_{f(y_{v})}X\oplus\bigoplus_{(e,v)\in
F^{S}(\Gamma)}T_{f(y(e,v))}X$ $\displaystyle\to$ $\displaystyle
B_{5}\to\bigoplus_{v\in V^{S}(\Gamma)}H^{1}(C_{v})\oplus\bigoplus_{e\in
E(\Gamma)}H^{1}(C_{e})\to 0.$

where

$\displaystyle H^{i}(C_{v})$ $\displaystyle=$ $\displaystyle
H^{i}(C_{v},(f|_{C_{v}})^{*}TX)\cong H^{i}(C_{v},\mathcal{O}_{C_{v}})\otimes
T_{p_{\sigma_{v}}}X,$ $\displaystyle H^{i}(C_{e})$ $\displaystyle=$
$\displaystyle H^{i}(C_{e},(f|_{C_{e}})^{*}TX)$

for $i=0,1$. We have

$\displaystyle H^{0}(C_{v})$ $\displaystyle=$ $\displaystyle
T_{p_{\sigma_{v}}}X$ $\displaystyle H^{1}(C_{v})$ $\displaystyle=$
$\displaystyle H^{0}(C_{v},\omega_{C_{v}})^{\vee}\otimes T_{p_{\sigma_{v}}}X.$

######  Lemma 36.

Let $\sigma\in V(\Upsilon)$, so that $p_{\sigma}$ is a $T$-fixed point in $X$.
Define

$\displaystyle{\mathbf{w}}(\sigma)$ $\displaystyle=$ $\displaystyle
e^{T}(T_{p_{\sigma}}X)\in H^{2r}_{T}(\textup{point};\mathbb{Q})$
$\displaystyle{\mathbf{h}}(\sigma,g)$ $\displaystyle=$
$\displaystyle\frac{e^{T}(\mathbb{E}^{\vee}\otimes
T_{p_{\sigma}}X)}{e^{T}(T_{p_{\sigma}}X)}\in
H^{2r(g-1)}_{T}(\overline{\mathcal{M}}_{g,n};\mathbb{Q}).$

Then

(37) ${\mathbf{w}}(\sigma)=\prod_{\epsilon\in
E_{\sigma}}{\mathbf{w}}(\epsilon,\sigma).$ (38)
${\mathbf{h}}(\sigma,g)=\prod_{\epsilon\in
E_{\sigma}}\frac{\Lambda_{g}^{\vee}({\mathbf{w}}(\epsilon,\sigma))}{{\mathbf{w}}(\epsilon,\sigma)}$

where
$\displaystyle{\Lambda^{\vee}_{g}(u)=\sum_{i=0}^{g}(-1)^{i}\lambda_{i}u^{g-i}}$.

###### Proof.

$\displaystyle{T_{p_{\sigma}}X=\bigoplus_{\epsilon\in
E_{\sigma}}T_{p_{\sigma}}\ell_{\epsilon}}$, where
$e^{T}(T_{p_{\sigma}}\ell_{\epsilon})={\mathbf{w}}(\epsilon,\sigma)$. So

$\displaystyle e^{T}(T_{p_{\sigma}})$ $\displaystyle=$
$\displaystyle\prod_{\epsilon\in E_{\sigma}}{\mathbf{w}}(\epsilon,\sigma),$
$\displaystyle\frac{e^{T}(\mathbb{E}^{\vee}\otimes
T_{p_{\sigma}}\ell_{\epsilon})}{e^{T}(T_{p_{\sigma}}\ell_{\epsilon})}$
$\displaystyle=$ $\displaystyle\prod_{\epsilon\in
E_{\sigma}}\frac{e^{T}(\mathbb{E}^{\vee}\otimes
T_{p_{\sigma}}\ell_{\epsilon})}{{\mathbf{w}}(\epsilon,\sigma)},$

where

$e^{T}(\mathbb{E}^{\vee}\otimes
T_{p_{\sigma}}\ell_{\epsilon})=\sum_{i=0}^{g}(-1)^{i}c_{i}(\mathbb{E})c_{1}^{T}(T_{p_{\sigma}}\ell_{\epsilon})^{g-i}=\sum_{i=0}^{g}(-1)^{i}\lambda_{i}{\mathbf{w}}(\epsilon,\sigma)^{g-i}.$

∎

The map $B_{1}\to B_{2}$ sends $H^{0}(C_{e},TC_{e}(-y(e,v)-y(e^{\prime},v)))$
isomorphically to $H^{0}(C_{e},(f|_{C_{e}})^{*}T\ell_{\epsilon_{e}})^{f}$, the
fixed part of $H^{0}(C_{e},(f|_{C_{e}})^{*}T\ell_{\epsilon_{e}})$.

######  Lemma 39.

Given $d\in\mathbb{Z}_{>0}$ and $\epsilon\in E(\Upsilon)_{c}$, define
$\sigma,\sigma^{\prime},\epsilon_{i},\epsilon_{i}^{\prime},a_{i}$ as in
Section 2.2, and let
$f_{d}:\mathbb{P}^{1}\to\ell_{\epsilon}\cong\mathbb{P}^{1}$ be the unique
degree $d$ map totally ramified over the two $T$-fixed points $p_{\sigma}$ and
$p_{\sigma^{\prime}}$ in $\ell_{\epsilon}$. Define

${\mathbf{h}}(\epsilon,d)=\frac{e^{T}(H^{1}(\mathbb{P}^{1},f_{d}^{*}TX)^{m})}{e^{T}(H^{0}(\mathbb{P}^{1},f_{d}^{*}TX)^{m})}.$

Then

(40)
${\mathbf{h}}(\epsilon,d)=\frac{(-1)^{d}d^{2d}}{(d!)^{2}{\mathbf{w}}(\epsilon,\sigma)^{2d}}\prod_{i=1}^{r-1}b(\frac{{\mathbf{w}}(\epsilon,\sigma)}{d},{\mathbf{w}}(\epsilon_{i},\sigma),da_{i})$

where

(41) $b(u,w,a)=\begin{cases}\prod_{j=0}^{a}(w-ju)^{-1},&a\in\mathbb{Z},a\geq
0,\\\ \prod_{j=1}^{-a-1}(w+ju),&a\in\mathbb{Z},a<0.\end{cases}$

###### Proof.

We use the notation in Section 2.2. We have

$N_{\ell_{\epsilon}/X}=L_{1}\oplus\cdots\oplus L_{r-1}.$

The weights of $T$-actions on $(L_{i})_{p_{\sigma}}$ and
$(L_{i})_{p_{\sigma}}$ are ${\mathbf{w}}(\epsilon_{i},\sigma)$ and
${\mathbf{w}}(\epsilon_{i},\sigma)-a_{i}{\mathbf{w}}(\epsilon,\sigma)$,
respectively. The weights of $T$-actions on $T_{0}\mathbb{P}^{1}$,
$T_{\infty}\mathbb{P}^{1}$, $(f_{d}^{*}L_{i})_{0}$,
$(f_{d}^{*}L_{i})_{\infty}$ are $u:=\frac{{\mathbf{w}}(\epsilon,\sigma)}{d}$,
$-u$, $w_{i}:={\mathbf{w}}(\epsilon_{i},\sigma)$, $w_{i}-da_{i}u$,
respectively. By Example [33, Example 19],

$\mathrm{ch}_{T}(H^{0}(\mathbb{P}^{1},f_{d}^{*}L_{i})-H^{1}(\mathbb{P}^{1},f_{d}^{*}L_{i}))=\begin{cases}\sum_{j=0}^{da_{i}}e^{w_{i}-ju},&a_{i}\geq
0,\\\ \sum_{j=1}^{-da_{i}-1}e^{w_{i}+ju},&a_{i}<0.\end{cases}$

Note that $w_{i}+ju$ is nonzero for any $j\in\mathbb{Z}$ since $w_{i}$ and $u$
are linearly independent for $i=1,\ldots,n-1$. So

$\frac{e^{T}\left(H^{1}(\mathbb{P}^{1},f_{d}^{*}L_{i})\right)}{e^{T}\left(H^{0}(\mathbb{P}^{1},f_{d}^{*}L_{i})\right)}=\frac{e^{T}\left(H^{1}(\mathbb{P}^{1},f_{d}^{*}L_{i})^{m}\right)}{e^{T}\left(H^{0}(\mathbb{P}^{1},f_{d}^{*}L_{i})^{m}\right)}=b(u,w_{i},da_{i})$

where $b(u,w,a)$ is defined by (41). By Example [33, Example 19],

$\mathrm{ch}_{T}(H^{0}(\mathbb{P}^{1},f_{d}^{*}T\ell_{\epsilon})-H^{1}(\mathbb{P}^{1},f_{d}^{*}T\ell_{\epsilon}))=\sum_{j=0}^{2d}e^{du-
ju}=1+\sum_{j=1}^{d}(e^{j{\mathbf{w}}(\epsilon,\sigma)/d}+e^{-j{\mathbf{w}}(\epsilon,\sigma)/d}).$

So

$\frac{e^{T}(H^{1}(\mathbb{P}^{1},f_{d}^{*}T\ell_{\epsilon})^{m})}{e^{T}(H^{0}(\mathbb{P}^{1},f_{d}^{*}T\ell_{\epsilon})^{m})}=\prod_{j=1}^{d}\frac{-d^{2}}{j^{2}{\mathbf{w}}(\epsilon,\sigma)^{2}}=\frac{(-1)^{d}d^{2d}}{(d!)^{2}{\mathbf{w}}(\epsilon,\sigma)^{2d}}.$

Therefore,

$\displaystyle\frac{e^{T}(H^{1}(\mathbb{P}^{1},f_{d}^{*}TX)^{m})}{e^{T}(H^{0}(\mathbb{P}^{1},f_{d}^{*}TX)^{m})}$
$\displaystyle=$
$\displaystyle\frac{e^{T}(H^{1}(\mathbb{P}^{1},f_{d}^{*}T\ell_{\epsilon})^{m})}{e^{T}(H^{0}(\mathbb{P}^{1},f_{d}^{*}T\ell_{\epsilon})^{m})}\cdot\prod_{i=1}^{r-1}\frac{e^{T}(H^{1}(\mathbb{P}^{1},f_{d}^{*}L_{i})^{m})}{e^{T}(H^{0}(\mathbb{P}^{1},f_{d}^{*}L_{i})^{m})}$
$\displaystyle=$
$\displaystyle\frac{(-1)^{d}d^{2d}}{(d!)^{2}{\mathbf{w}}(\epsilon,\sigma)^{2d}}\prod_{i=1}^{r-1}b(\frac{{\mathbf{w}}(\epsilon,\sigma)}{d},{\mathbf{w}}(\epsilon_{i},\sigma),da_{i}).$

∎

Finally, $f(y_{v})=p_{\sigma_{v}}=f(y(e,v))$, and

$e^{T}(T_{p_{\sigma_{v}}}X)={\mathbf{w}}(\sigma_{v}).$

From the above discussion, we conclude that

$\displaystyle\frac{e^{T}(B_{5}^{m})}{e^{T}(B_{2}^{m})}$ $\displaystyle=$
$\displaystyle\prod_{v\in
V^{2}(\Gamma)}{\mathbf{w}}(\sigma_{v})\cdot\prod_{(e,v)\in
F^{S}(\Gamma)}{\mathbf{w}}(\sigma_{v})\cdot\prod_{v\in
V^{S}(\Gamma)}{\mathbf{h}}(\sigma_{v},g_{v})\cdot\prod_{e\in
E(\Gamma)}{\mathbf{h}}(\epsilon_{e},d_{e})$ $\displaystyle=$
$\displaystyle\prod_{v\in
V(\Gamma)}\bigl{(}{\mathbf{h}}(\sigma_{v},g_{v})\cdot{\mathbf{w}}(\sigma_{v})^{\mathrm{val}(v)}\bigr{)}\cdot\prod_{e\in
E(\Gamma)}{\mathbf{h}}(\epsilon_{e},d_{e})$

where ${\mathbf{w}}(\sigma)$, ${\mathbf{h}}(\sigma,g)$, and
${\mathbf{h}}(\epsilon,d)$ are defined by (37), (38), (40), respectively.

###  4.3. Contribution from each graph

####  4.3.1. Virtual tangent bundle

We have $B_{1}^{f}=B_{2}^{f}$, $B_{5}^{f}=0$. So

$T^{1,f}=B_{4}^{f}=\bigoplus_{v\in
V^{S}(\Gamma)}T\overline{\mathcal{M}}_{g_{v},E_{v}\cup S_{v}},\quad
T^{2,f}=0.$

We conclude that

$[\prod_{v\in V^{S}(\Gamma)}\overline{\mathcal{M}}_{g_{v},E_{v}\cup
S_{v}}]^{\mathrm{vir}}=\prod_{v\in
V^{S}(\Gamma)}[\overline{\mathcal{M}}_{g_{v},E_{v}\cup S_{v}}].$

####  4.3.2. Virtual normal bundle

Let $N^{\mathrm{vir}}_{{\vec{\Gamma}}}$ be the pull back of the virtual normal
bundle of $\mathcal{F}_{{\vec{\Gamma}}}$ in
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ under
$i_{\vec{\Gamma}}:\mathcal{M}_{{\vec{\Gamma}}}\to\mathcal{F}_{{\vec{\Gamma}}}$.
Then

$\frac{1}{e^{T}(N^{\mathrm{vir}}_{\vec{\Gamma}})}=\frac{e^{T}(B_{1}^{m})e^{T}(B_{5}^{m})}{e^{T}(B_{2}^{m})e^{T}(B_{4}^{m})}=\prod_{v\in
V(\Gamma)}\frac{{\mathbf{h}}(\sigma_{v},g_{v})\cdot{\mathbf{w}}(\sigma_{v})^{\mathrm{val}(v)}}{\prod_{e\in
E_{v}}(w_{(e,v)}-\psi_{(e,v)})}\cdot\prod_{e\in
E(\Gamma)}{\mathbf{h}}(\epsilon_{e},d_{e})$

####  4.3.3. Integrand

Given $\sigma\in V(\Upsilon)$, let

$i_{\sigma}^{*}:A^{*}_{T}(X)\to
A^{*}_{T}(p_{\sigma})=\mathbb{Q}[u_{1},\ldots,u_{r}]$

be induced by the inclusion $i_{\sigma}:p_{\sigma}\to X$. Then

(42) $\displaystyle
i_{\vec{\Gamma}}^{*}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi_{i}^{T})^{a_{i}}\right)$
$\displaystyle=$ $\displaystyle\prod_{\tiny\begin{array}[]{c}v\in
V^{1,1}(E)\\\
S_{v}=\\{i\\},E_{v}=\\{e\\}\end{array}}i^{*}_{\sigma_{v}}\gamma_{i}^{T}(-w_{(e,v)})^{a_{i}}\cdot\prod_{v\in
V^{S}(\Gamma)}\Bigl{(}\prod_{i\in
S_{v}}i^{*}_{\sigma_{v}}\gamma_{i}^{T}\prod_{e\in
E_{v}}\psi_{(e,v)}^{a_{i}}\Bigr{)}$

To unify the stable vertices in $V^{S}(\Gamma)$ and the unstable vertices in
$V^{1,1}(\Gamma)$ , we use the following convention: for $a\in\mathbb{Z}_{\geq
0}$,

(43)
$\int_{\overline{\mathcal{M}}_{0,2}}\frac{\psi_{2}^{a}}{w_{1}-\psi_{1}}=(-w_{1})^{a}.$

In particular, (32) is obtained by setting $a=0$. With the convention (43), we
may rewrite (42) as

(44)
$i_{\vec{\Gamma}}^{*}\prod_{i=1}^{n}\left(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi_{i}^{T})^{a_{i}}\right)=\prod_{v\in
V(\Gamma)}\Bigl{(}\prod_{i\in
S_{v}}i^{*}_{\sigma_{v}}\gamma_{i}^{T}\prod_{e\in
E_{v}}\psi_{(e,v)}^{a_{i}}\Bigr{)}.$

The following lemma shows that the convention (43) is consistent with the
stable case $\overline{\mathcal{M}}_{0,n}$, $n\geq 3$.

######  Lemma 45.

Let $n,a$ be integers, $n\geq 2$, $a\geq 0$. Then

$\int_{\overline{\mathcal{M}}_{0,n}}\frac{\psi_{2}^{a}}{w_{1}-\psi_{1}}=\begin{cases}\displaystyle{\frac{\prod_{i=0}^{a-1}(n-3-i)}{a!}w_{1}^{a+2-n},}&n=2\textup{
or }0\leq a\leq n-3,\\\ 0,&\textup{otherwise}.\end{cases}$

###### Proof.

The case $n=2$ follows from (43). For $n\geq 3$,

$\displaystyle\int_{\overline{\mathcal{M}}_{0,n}}\frac{\psi_{2}^{a}}{w_{1}-\psi_{1}}=\frac{1}{w_{1}}\int_{\overline{\mathcal{M}}_{0,n}}\frac{\psi_{2}^{a}}{1-\frac{\psi_{1}}{w_{1}}}=w_{1}^{a+2-n}\int_{\overline{\mathcal{M}}_{0,n}}\psi_{1}^{n-3-a}\psi_{2}^{a}$
$\displaystyle=w_{1}^{a+2-n}\frac{(n-3)!}{(n-3-a)!a_{!}}=\frac{\prod_{i=0}^{a-1}(n-3-i)}{a!}w_{1}^{a+2-n}.$

∎

####  4.3.4. Integral

The contribution of

$\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)^{T}]^{\mathrm{vir},T}}\frac{i^{*}\prod_{i=1}^{n}(\mathrm{ev}_{i}^{*}\gamma_{i}^{T}\cup(\psi_{i}^{T})^{a_{i}})}{e^{T}(N^{\mathrm{vir}})}$

from the fixed locus $\mathcal{F}_{\vec{\Gamma}}$ is given by

$\displaystyle\frac{1}{|A_{{\vec{\Gamma}}}|}\prod_{e\in
E(\Gamma)}{\mathbf{h}}(\epsilon_{e},d_{e})\prod_{v\in
V(\Gamma)}\Bigl{(}{\mathbf{w}}(\sigma_{v})^{\mathrm{val}(v)}\prod_{i\in
S_{v}}i_{\sigma_{v}}^{*}\gamma_{i}^{T}\Bigr{)}$
$\displaystyle\quad\quad\cdot\prod_{v\in
V(\Gamma)}\int_{\overline{\mathcal{M}}_{g_{v},E_{v}\cup
S_{v}}}\frac{{\mathbf{h}}(\sigma_{v},g_{v})\cdot\prod_{e\in
E_{v}}\psi^{a_{i}}_{(e,v)}}{\prod_{e\in E_{v}}(w_{(e,v)}-\psi_{(e,v)})}$

where $|A_{{\vec{\Gamma}}}|=|\mathrm{Aut}({\vec{\Gamma}})|\cdot\prod_{e\in
E(\Gamma)}d_{e}$.

###  4.4. Sum over graphs

Summing over the contribution from each graph ${\vec{\Gamma}}$ given in
Section 4.3.4 above, we obtain the following formula.

######  Theorem 46.

(47)
$\displaystyle\langle\tau_{a_{1}}(\gamma_{1}^{T})\cdots\tau_{a_{n}}(\gamma_{n}^{T})\rangle^{X_{T}}_{g,\beta}$
$\displaystyle=$ $\displaystyle\sum_{{\vec{\Gamma}}\in
G_{g,n}(X,\beta)}\frac{1}{|\mathrm{Aut}({\vec{\Gamma}})|}\prod_{e\in
E(\Gamma)}\frac{{\mathbf{h}}(\epsilon_{e},d_{e})}{d_{e}}\prod_{v\in
V(\Gamma)}\Bigl{(}{\mathbf{w}}(\sigma_{v})^{\mathrm{val}(v)}\prod_{i\in
S_{v}}i_{\sigma_{v}}^{*}\gamma_{i}^{T}\Bigr{)}$ $\displaystyle\cdot\prod_{v\in
V(\Gamma)}\int_{\overline{\mathcal{M}}_{g,E_{v}\cup
S_{v}}}\frac{{\mathbf{h}}(\sigma_{v},g_{v})\prod_{i\in
S_{v}}\psi_{i}^{a_{i}}}{\prod_{e\in E_{v}}(w_{(e,v)}-\psi_{(e,v)})}.$

where ${\mathbf{h}}(\epsilon,d)$, ${\mathbf{w}}(\sigma)$,
${\mathbf{h}}(\sigma,g)$ are given by (40), (37), (38), respectively, and we
have the following convention for the $v\notin V^{S}(\Gamma)$:

$\displaystyle\int_{\overline{\mathcal{M}}_{0,1}}\frac{1}{w_{1}-\psi_{2}}=w_{1},\quad\int_{\overline{\mathcal{M}}_{0,2}}\frac{1}{(w_{1}-\psi_{1})(w_{2}-\psi_{2})}=\frac{1}{w_{1}+w_{2}},$
$\displaystyle\int_{\overline{\mathcal{M}}_{0,2}}\frac{\psi_{2}^{a}}{w_{1}-\psi_{1}}=(-w_{1})^{a},\quad
a\in\mathbb{Z}_{\geq 0}.$

Given $g\in\mathbb{Z}_{\geq 0}$, $r$ weights
$\vec{w}=\\{w_{1},\ldots,w_{r}\\}$, $r$ partitions
$\vec{\mu}=\\{\mu^{1},\ldots,\mu^{r}\\}$, and
$a_{1},\ldots,a_{k}\in\mathbb{Z}$, let $\ell(\mu^{i})$ be the length of
$\mu^{i}$, and let $\ell({\vec{\mu}})=\sum_{i=1}^{r}\ell(\mu^{i})$. We define

$\langle\tau_{a_{1}},\ldots,\tau_{a_{k}}\rangle_{g,{\vec{\mu}},\vec{w}}=\int_{\overline{\mathcal{M}}_{g,\ell({\vec{\mu}})+k}}\prod_{i=1}^{r}\Bigl{(}\frac{\Lambda_{g}^{\vee}(w_{i})w_{i}^{\ell({\vec{\mu}})-1}}{\prod_{j=1}^{\ell(\mu^{i})}\frac{w_{i}}{\mu^{i}_{j}}-\psi^{i}_{j})}\Bigr{)}\prod_{b=1}^{k}\psi_{b}^{a_{i}}.$

Given $v\in V(\Gamma)$, define
$\vec{w}(v)=\\{{\mathbf{w}}(\epsilon,\sigma_{v})\mid(\epsilon,\sigma_{v})\in
F(\Upsilon)\\}$. Given $v\in V(\Gamma)$, and $\epsilon\in E_{\sigma_{v}}$, let
$\mu^{v,\epsilon}$ be a (possibly empty) partition defined by $\\{d_{e}\mid
e\in E_{v},\vec{f}(e)=\epsilon\\}$, and define
${\vec{\mu}}(v)=\\{\mu^{v,\epsilon}\mid(\epsilon,\sigma_{v})\in
F(\Upsilon)\\}$. Then (47) can be rewritten as

(48)
$\displaystyle\langle\tau_{a_{1}}(\gamma_{1}^{T})\cdots\tau_{a_{n}}(\gamma_{n}^{T})\rangle^{X_{T}}_{g,\beta}$
$\displaystyle=$ $\displaystyle\sum_{{\vec{\Gamma}}\in
G_{g,n}(X,\beta)}\frac{1}{|\mathrm{Aut}({\vec{\Gamma}})|}\prod_{e\in
E(\Gamma)}\frac{{\mathbf{h}}(\epsilon_{e},d_{e})}{d_{e}}\prod_{v\in
V(\Gamma)}\Bigl{(}\prod_{i\in
S_{v}}i_{\sigma_{v}}^{*}\gamma_{i}\langle\prod_{i\in
S_{v}}\tau_{a_{i}}\rangle_{g_{v},{\vec{\mu}}(v),\vec{w}(v)}\Bigr{)}.$

Recall that

$g=\sum_{v\in V(\Gamma)}g_{v}+|E(\Gamma)|-|V(\Gamma)|+1$

so

$2g-2=\sum_{v\in V(\Gamma)}(2g_{v}-2+\mathrm{val}(v)).$

Given ${\vec{\Gamma}}=(\Gamma,\vec{f},\vec{d},\vec{g},\vec{s})$, let
${\vec{\Gamma}}^{\prime}=(\Gamma,\vec{f},\vec{d},\vec{s})$ be the decorated
graph obtained by forgetting the genus map. Let
$G_{n}(X,\beta)=\\{{\vec{\Gamma}}^{\prime}\mid{\vec{\Gamma}}\in\cup_{g\geq
0}G_{g,n}(X,\beta)\\}$. Define

(49)
$\langle\tau_{a_{1}}(\gamma_{1}^{T}),\cdots,\tau_{a_{n}}(\gamma_{n}^{T})\mid
u\rangle^{X_{T}}_{\beta}=\sum_{g\geq
0}u^{2g-2}\langle\tau_{a_{1}}(\gamma_{1}^{T}),\cdots,\tau_{a_{n}}(\gamma_{n}^{T})\rangle^{X_{T}}_{g,\beta}$
(50) $\langle\tau_{a_{1}},\ldots,\tau_{a_{k}}\mid
u\rangle_{{\vec{\mu}},\vec{w}}=\sum_{g\geq
0}u^{2g-2+\ell({\vec{\mu}})}\langle\tau_{a_{1}},\ldots,\tau_{a_{k}}\rangle_{g,{\vec{\mu}},\vec{w}}.$

Then we have the following formula for the generating function (49).

######  Theorem 51.

(52)
$\displaystyle\langle\tau_{a_{1}}(\gamma_{1}^{T})\cdots\tau_{a_{n}}(\gamma_{n}^{T})\mid
u\rangle^{X_{T}}_{\beta}=$ $\displaystyle\sum_{{\vec{\Gamma}}^{\prime}\in
G_{n}(X,\beta)}\frac{1}{|\mathrm{Aut}({\vec{\Gamma}})|}\prod_{e\in
E(\Gamma)}\frac{{\mathbf{h}}(\epsilon_{e},d_{e})}{d_{e}}$
$\displaystyle\cdot\prod_{v\in V(\Gamma)}\Bigl{(}\prod_{i\in
S_{v}}i_{\sigma_{v}}^{*}\gamma_{i}^{T}\langle\prod_{i\in
S_{v}}\tau_{a_{i}}\mid u\rangle_{{\vec{\mu}}(v),\vec{w}(v)}\Bigr{)}.$

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