# The orbifold DT/PT vertex correspondence

Yijie Lin  School of Mathematics and Statistics
Fujian Normal University
Fuzhou 350117, P.R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We present an orbifold topological vertex formalism for PT invariants of toric
Calabi-Yau 3-orbifolds with transverse $A_{n-1}$ singularities. We give a
proof of the orbifold DT/PT Calabi-Yau topological vertex correspondence. As
an application, we derive an explicit formula for the PT
$\mathbb{Z}_{n}$-vertex in terms of loop Schur functions and prove the multi-
regular orbifold DT/PT correspondence.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Geometry of toric Calabi-Yau 3-orbifolds
    2. 2.2 Orbifold PT stable pairs
    3. 2.3 The DT partition function
  3. 3 The orbifold PT topological vertex
    1. 3.1 Partitions, labelled box configurations and topological vertices
    2. 3.2 $K$-theory decomposition of orbifold PT stable pairs
    3. 3.3 The PT partition function
    4. 3.4 The PT topological vertex formalism
  4. 4 Orbifold DT theory and the graphical condensation recurrence
    1. 4.1 The dimer model for orbifold DT theory
    2. 4.2 The graphical condensation recurrence for orbifold DT theory
    3. 4.3 Weights for orbifold DT theory
      1. 4.3.1 Weights for the graphical condensation recurrence with $\mathbf{G}=\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)$
      2. 4.3.2 Weights for the graphical condensation recurrence with $\mathbf{G}=\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})$
      3. 4.3.3 Weights for the graphical condensation recurrence with $\mathbf{G}=\mathbf{H}(N;\lambda,\mu^{rc},\nu^{rc})$
  5. 5 Orbifold PT theory and the graphical condensation recurrence
    1. 5.1 Labelled box configurations and AB configurations for orbifold PT theory
    2. 5.2 The double-dimer model for orbifold PT theory
    3. 5.3 The graphical condensation recurrence for orbifold PT theory
    4. 5.4 Weights for orbifold PT theory
      1. 5.4.1 Weights for the graphical condensation recurrence with $(\mathbf{H}(N),\mathbf{N}_{\lambda^{rc},\mu^{rc},\nu}(N))$
      2. 5.4.2 Weights for the graphical condensation recurrence with $(\mathbf{G}=\mathbf{H}(N),\mathbf{N}_{\lambda^{rc},\mu,\nu^{rc}}(N))$
      3. 5.4.3 Weights for the graphical condensation recurrence with $(\mathbf{G}=\mathbf{H}(N),\mathbf{N}_{\lambda,\mu^{rc},\nu^{rc}}(N))$
  6. 6 Orbifold DT/PT correspondence
    1. 6.1 A proof of orbifold DT/PT vertex correspondence
    2. 6.2 Remarks on orbifold GW/DT correspondence and DT crepant resolution conjecture

##  1\. Introduction

The topological vertex has been proposed to be an effective tool for computing
enumerative invariants such as Gromov-Witten invariants, Donaldson-Thomas
invariants and Pandharipande-Thomas invariants of toric Calabi-Yau 3-folds
since it made an appearance in physics for calculating the topological string
amplitudes [1]. The GW topological vertex is developed in [1, 20, 21, 17, 26,
23] as a generating function of Hodge integrals while the DT topological
vertex is a generating function of 3D partitions [28, 22]. Applying the
technique of virtual localization [12], the GW/DT correspondence [22] can be
reduced to verify first the GW/DT vertex correspondence and then the
compatibility with the edge terms in their gluing laws. As a geometric
approach to the reduced DT theory, the PT theory [33] equips with its
topological vertex as a generating function of labelled box configurations
[34], where the conjectured DT/PT vertex correspondence is resolved recently
in [14] for the Calabi-Yau case. There are alternative approaches to the DT/PT
correspondence in [6, 43, 42]. With the rationality property of PT theory [29,
30], the GW/PT correspondence is treated in [31, 32, 25].

For toric CY 3-orbifolds, the orbifold DT and GW topological vertices are
developed in [7] and [36] as generating functions of 3D partitions colored by
irreducible representations of a finite Abelian group and of Hurwitz-Hodge
integrals respectively. The orbifold GW/DT vertex correspondence for toric CY
3-orbifolds with transverse $A_{n-1}$ quotient singularities is studied in
[36, 47, 39, 40, 37]. The foundation of the (relative) orbifold PT theory in
the general case is investigated in [18, 19]. When I had obtained PT
invariants of toric CY 3-orbifolds with transverse $A_{n-1}$ singularities
(see Theorem 3.22) through exploring the orbifold PT topological vertex as in
[7], I was informed by Patrick Lei last October during the email communication
on the progress of PT theory of
$[\mathbb{C}^{2}/\mathbb{Z}_{n+1}]\times\mathbb{P}^{1}$ that Zhang [45] had
also studied the orbifold PT topological vertex and obtained the 1-leg
orbifold PT vertex formula in terms of Schur functions. Thanks to the proof of
[34, Conjecture 2] by Jenne, Webb and Young in [14], Zhang and I had
independently obtained the same formula of the PT partition function, see also
[45, Theorem/Conjecture 4.19] where the 3-leg case is conjectural. However,
there are several differences between ours as follows. We have slightly
different arguments on the description of the $T$-fixed orbifold PT stable
pairs, see [45, Section 4.6] and Section 2.2 respectively. Compared with [45,
Proposition 4.25], the more explicit proof of $T$-equivariant K-theory
decomposition of the finite cokernel $\mathtt{Q}$ as a sum of vertex terms is
presented in Proposition 3.15, which is essential for deriving the sign
formula and hence the PT partition function later. Next, the PT partition
function here is defined by virtual localization as in [34] together with
perfect obstruction theories and virtual fundamental classes developed in [18]
especially for the 3-leg case with non-isolated fixed points, while the PT
invariant defined as Behrend’s weighted Euler characteristic in [45,
Definition 4.4] fails to deal with the full 3-leg case by [4, Theorem 3.4],
see Section 3.3 and [45, Section 4] respectively. Explicitly, we have the
following PT version of [7, Theorem 10].

######  Theorem 1.1.

(see Theorem 3.22 and [45, Theorem/Conjecture 4.19]) Let $\mathcal{X}$ be a
toric CY 3-orbifold with transverse $A_{n-1}$ singularities. Then the PT
partition function $PT(\mathcal{X})$ is

$\displaystyle
PT(\mathcal{X})=\underline{PT}(\mathcal{X})\bigg{|}_{q_{e,0}\to-
q_{e,0},\;q\to-q}$

where

$\displaystyle\underline{PT}(\mathcal{X}):=\sum_{\\{\lambda_{e}\\}\in\Lambda_{\mathcal{X}}}\prod_{e\in
E_{\mathcal{X}}}E_{\lambda_{e}}^{e}\prod_{v\in
V_{\mathcal{X}}}(-1)^{\varXi_{\pi_{v}}}W_{\lambda_{1,v}\lambda_{2,v}\lambda_{3,v}}^{n_{e_{3,v}}}((-1)^{\widetilde{s}(\lambda_{3,v})}\mathbf{q}_{v})$

and

$\displaystyle
E_{\lambda_{e}}^{e}=(-1)^{\mathbf{S}_{\lambda_{e}}^{e}}\cdot\left(\prod_{k=0}^{n_{e}-1}v_{e,k}^{|\lambda_{e}|_{k,n_{e}}}\right)\cdot
q_{e}^{C_{m_{e},m_{e}^{\prime}}^{\lambda_{e}}}\cdot\left(\overline{q_{f_{v_{0}}}^{A_{\lambda_{e}}}}\right)^{\delta_{0,e}}\cdot\left(q_{f^{\prime}_{v_{0}}}^{A_{\lambda^{\prime}_{e}}}\right)^{\delta_{0,e}^{\prime}}\cdot\left(q_{g_{v_{\infty}}}^{A_{\lambda_{e}}}\right)^{\delta_{\infty,e}}\cdot\left(\overline{q_{g^{\prime}_{v_{\infty}}}^{A_{\lambda^{\prime}_{e}}}}\right)^{\delta_{\infty,e}^{\prime}}$

with edges
$(e,f_{v_{0}},f^{\prime}_{v_{0}},g_{v_{\infty}},g^{\prime}_{v_{\infty}})$
oriented as in Figure 1 (see other notations in Section 3).

In [45], the author has proved the 1-leg orbifold DT/PT vertex correspondence
by obtaining the explicit formula for the 1-leg orbifold PT topological
vertex. In this paper, we will extend this result to the full 3-leg case,
i.e., prove the 3-leg orbifold DT/PT vertex correspondence by generalizing the
DT/PT Calabi-Yau topological vertex correspondence in [14] to the orbifold
case.

In [14], the DT topological vertex $V_{\lambda\mu\nu}$ (see Definition 3.2) is
described as some dimer model via the well-known correspondence between 3D
partitions and dimer configurations on the honeycomb graph while the PT
topological vertex $W_{\lambda\mu\nu}$ (see Definition 3.7) has certain
double-dimer model description with the established correspondence between
labelled box configurations and tripartite double-dimer configurations on the
honeycomb graph. With these descriptions, the so-called graphical condensation
discovered by Kuo [16, Theorem 5.1] is applied to derive a graphical
condensation recurrence for the DT vertex while the analogue technique
developed in [13, Theorem 1.0.2] is employed to obtain a graphical
condensation recurrence for the PT vertex. Then the DT/PT vertex
correspondence [14, Theorem 1.0.1] (see also Theorem 3.9) is a consequence of
the same recurrence obtained in both sides together with the base 2-leg case
proved in [34]. This strategy can be extended and carried out for toric CY
3-orbifolds with cyclic orbifold structure explicitly as follows. The orbifold
DT vertex $V^{n}_{\lambda\mu\nu}$ (see Definition 3.4) and the orbifold PT
vertex $W^{n}_{\lambda\mu\nu}$ (see Definition 3.10) as generating functions
of colored 3D partitions and colored labelled box configurations can be also
described as some dimer model and double-dimer model respectively but with the
same generalized weighted rule in Definition 4.3, see Section 4.1 and Section
5.2. Next, the graphical condensation recurrence for orbifold DT theory is
obtained in Section 4.2 as follows

$\displaystyle\frac{V^{n}_{\lambda\mu\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}=\frac{V^{n}_{\lambda^{rc}\mu\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\frac{V^{n}_{\lambda\mu^{rc}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\left(\frac{V^{n}_{\lambda^{rc}\mu^{rc}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\right)^{-1}+q^{K_{1}(\lambda,\mu,\nu)}\cdot\frac{V^{n}_{\lambda^{r}\mu^{c}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\frac{V^{n}_{\lambda^{c}\mu^{r}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\left(\frac{V^{n}_{\lambda^{rc}\mu^{rc}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\right)^{-1}$

which coincides with the graphical condensation recurrence for orbifold PT
theory in Section 5.3

$\displaystyle W^{n}_{\lambda\mu\nu}=W^{n}_{\lambda^{rc}\mu\nu}\cdot
W^{n}_{\lambda\mu^{rc}\nu}\cdot\left(W^{n}_{\lambda^{rc}\mu^{rc}\nu}\right)^{-1}+q^{K_{1}(\lambda,\mu,\nu)}\cdot
W^{n}_{\lambda^{r}\mu^{c}\nu}\cdot
W^{n}_{\lambda^{c}\mu^{r}\nu}\cdot\left(W^{n}_{\lambda^{rc}\mu^{rc}\nu}\right)^{-1}$

where

$\displaystyle
q^{K_{1}(\lambda,\mu,\nu)}:=q_{-\widetilde{d}(\lambda^{\prime})}^{d(\lambda^{\prime})}\cdot
q_{\widetilde{d}(\mu)}^{d(\mu)}\cdot\prod\limits_{i=-\widetilde{d}(\lambda^{\prime})}^{\widetilde{d}(\mu)}q_{i}^{-1}\cdot\prod\limits_{i=\widetilde{d}(\lambda^{\prime})+1}^{\ell(\lambda^{\prime})}\left(\frac{q_{-i}}{q_{-i+1}}\right)^{\lambda^{\prime}_{i}}\cdot\prod\limits_{i=\widetilde{d}(\mu)+1}^{\ell(\mu)}\left(\frac{q_{i}}{q_{i-1}}\right)^{\mu_{i}}.$

That is, both
$Y_{1}(\lambda,\mu,\nu):=\frac{V^{n}_{\lambda\mu\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}$
and $Y_{2}(\lambda,\mu,\nu):=W^{n}_{\lambda\mu\nu}$ are solutions $Y$ of the
following equation:

(1) $\displaystyle Y(\lambda,\mu,\nu)=\frac{Y(\lambda^{rc},\mu,\nu)\cdot
Y(\lambda,\mu^{rc},\nu)}{Y(\lambda^{rc},\mu^{rc},\nu)}+q^{K_{1}(\lambda,\mu,\nu)}\cdot\frac{Y(\lambda^{r},\mu^{c},\nu)\cdot
Y(\lambda^{c},\mu^{r},\nu)}{Y(\lambda^{rc},\mu^{rc},\nu)}.$

The above graphical condensation recurrence (1) generalizes [14, Equation (2)]
if we set $q_{l}:=q$ for all $l$ (or equivalently $n=1$). However, it is not
enough to determine the 3-leg orbifold DT/PT vertex correspondence since we
don’t have the 2-leg correspondence
$Y_{1}(\lambda,\emptyset,\nu)=Y_{2}(\lambda,\emptyset,\nu)$ and
$Y_{1}(\emptyset,\mu,\nu)=Y_{2}(\emptyset,\mu,\nu)$ as the base case via
Remark 4.8. Actually, it is observed that the 2-leg correspondence
$Y_{1}(\lambda,\mu,\emptyset)=Y_{2}(\lambda,\mu,\emptyset)$ can be obtained by
Equation (1), [45, Theorem 5.22] and Remark 4.8. This observation leads us to
explore more graphical condensation recurrences from orbifold DT and PT
theories in order to obtain the desired 2-leg correspondence and hence the
full 3-leg case. Specifically, with different choices of graphs for the dimer
model and double-dimer model, both $Y_{1}(\lambda,\mu,\nu)$ and
$Y_{2}(\lambda,\mu,\nu)$ are also the solutions $Y$ of the following two more
recurrences:

(2) $\displaystyle Y(\lambda,\mu,\nu)=\frac{Y(\lambda^{rc},\mu,\nu)\cdot
Y(\lambda,\mu,\nu^{rc})}{Y(\lambda^{rc},\mu,\nu^{rc})}+q^{K_{2}(\lambda,\mu,\nu)}\cdot\frac{Y(\lambda^{r},\mu,\nu^{c})\cdot
Y(\lambda^{c},\mu,\nu^{r})}{Y(\lambda^{rc},\mu,\nu^{rc})}$

and

(3) $\displaystyle Y(\lambda,\mu,\nu)=\frac{Y(\lambda,\mu^{rc},\nu)\cdot
Y(\lambda,\mu,\nu^{rc})}{Y(\lambda,\mu^{rc},\nu^{rc})}+q^{K_{3}(\lambda,\mu,\nu)}\cdot\frac{Y(\lambda,\mu^{r},\nu^{c})\cdot
Y(\lambda,\mu^{c},\nu^{r})}{Y(\lambda,\mu^{rc},\nu^{rc})}$

where

$\displaystyle
q^{K_{2}(\lambda,\mu,\nu)}:=q_{1-d(\lambda^{\prime})}^{-\lambda^{\prime}_{d(\lambda^{\prime})}}\cdot\prod\limits_{i=1}^{d(\lambda^{\prime})-1}q_{-i}\cdot\prod\limits_{i=1}^{d(\lambda^{\prime})-1}\left(\frac{q_{-i}}{q_{-i+1}}\right)^{\lambda^{\prime}_{i}}\cdot\prod\limits_{i=1}^{\ell(\mu)}\left(\frac{q_{i}}{q_{i-1}}\right)^{\mu_{i}}\cdot\prod\limits_{i=1}^{\nu_{d(\nu)}-d(\nu)}q_{i}^{-1},$
$\displaystyle
q^{K_{3}(\lambda,\mu,\nu)}:=q_{d(\mu)-1}^{-\mu_{d(\mu)}}\cdot\prod\limits_{i=1}^{d(\mu)-1}q_{i}\cdot\prod\limits_{i=1}^{d(\mu)-1}\left(\frac{q_{i}}{q_{i-1}}\right)^{\mu_{i}}\cdot\prod\limits_{i=1}^{\ell(\lambda^{\prime})}\left(\frac{q_{-i}}{q_{-i+1}}\right)^{\lambda^{\prime}_{i}}\cdot\prod_{i=1}^{\widetilde{d}(\nu)-d(\nu)}q_{-i}^{-1},$

see Section 4.2 and Section 5.3 for more details.

When $Y$ is chosen to be $Y_{1}$ or $Y_{2}$, there exists certain symmetry
between Equation (2) and Equation (3) due to symmetries of
$Y_{1}(\lambda,\mu,\nu)$ and $Y_{2}(\lambda,\mu,\nu)$ in Remarks 3.5 and 3.11
together with Lemma 4.12, but Equation (2) or (3) can not be derived from
Equation (1) via symmetries unless $n=1$, see also Section 4.2 and 5.3. With
the 1-leg correspondence from [45, Theorem 5.22] as the base case, it follows
from graphical condensation recurrences (1), (2) and (3) that

######  Theorem 1.2.

(see Theorem 6.2) If $\nu$ is multi-regular, then

$\displaystyle\frac{V^{n}_{\lambda\mu\nu}}{V_{\emptyset\emptyset\emptyset}^{n}}=W_{\lambda\mu\nu}^{n}.$

This completes the proof of the full 3-leg orbifold DT/PT vertex
correspondence, which generalizes DT/PT vertex correspondence in [14, Theorem
1.0.1] if $n=1$. Although we don’t impose the multi-regular condition on $\nu$
in recurrences (1), (2) and (3), the 3-leg correspondence in Theorem 1.2
doesn’t hold for any partition $\nu\neq\emptyset$ with $n>1$, see Remark 6.4.
Now we have the explicit formula for $W^{n}_{\lambda\mu\nu}$ in terms of loop
Schur functions.

######  Corollary 1.3.

(see Corollary 6.3 and Remark 6.5) If $\nu$ is multi-regular, then

$\displaystyle
W_{\lambda\mu\nu}^{n}=q^{-A_{\lambda}}\cdot\overline{q^{-A_{\mu^{\prime}}}}\cdot\left(\prod_{(i,j)\in\nu}q_{i-j}^{-j}\right)\cdot\mathfrak{s}_{\nu}\cdot\sum_{\eta}q_{0}^{-|\eta|}\cdot\overline{s_{\lambda^{\prime}/\eta}(\mathbf{q}_{\bullet-\nu^{\prime}})}\cdot
s_{\mu/\eta}(\mathbf{q}_{\bullet-\nu}).$

Notice that if $\lambda,\mu=\emptyset$ and $\nu\neq\emptyset$, the formula in
Corollary 1.3 is true without multi-regularity of $\nu$ due to [45, Theorem
5.22]. The second application is

######  Corollary 1.4.

(see Corollary 6.6) Let $\mathcal{X}$ be a toric CY 3-orbifold with transverse
$A_{n-1}$ singularities. Then we have

$\displaystyle DT^{\prime}_{mr}(\mathcal{X})=PT_{mr}(\mathcal{X}).$

where $DT^{\prime}_{mr}(\mathcal{X})$ and $PT_{mr}(\mathcal{X})$ are defined
in Section 2.3 and Section 3.3 respectively.

This gives a proof of orbifold DT/PT correspondence [45, Conjecture 4.7] for
toric CY 3-orbifolds with transverse $A_{n-1}$ singularities. Restricted on
any given multi-regular edge assignment (see Section 2.1), the equality in
Corollary 1.4 still holds at the level of formal series without using the
equality of rational functions proposed in the alternative proof [2, Theorem
A]. We may investigate orbifold GW/DT/PT correspondence and DT crepant
resolution conjecture for other CY 3-orbifolds satisfying the hard Lefschetz
condition such as the ones with singularities of type $D$ and $E$.

This paper is organized as follows. In Section 2, we recollect some
definitions and properties of toric CY 3-orbifolds and definitions of DT
partition functions, and characterize $T$-fixed locus of orbifold PT stable
pairs. We devote Section 3 to present the PT version of the theory of orbifold
topological vertex developed in [7] for completeness where we give the
definition of PT partition functions by virtual localization without using
weighted Euler characteristic including the full 3-leg case which removes
conjectural parts in [45, Conjectures 4.23, 4.16, 4.19]. In Section 4, we
generalize the dimer description of the DT topological vertex in [14, Section
3.2] to the orbifold case and derive three graphical condensation recurrences
for orbifold DT theory, one of which generalizes the one in [14, Section 3.3].
In Section 5, we show that AB configurations and double-dimer configurations
as two combinatorial descriptions of the PT topological vertex in [14, Section
4.2, 4.3 and 4.4] can be extended to the ones for the PT
$\mathbb{Z}_{n}$-vertex, and obtain graphical condensation recurrences for
orbifold PT theory containing the one in [14, Section 4.5] as a special case.
In Section 6, we present a proof of orbifold DT/PT vertex correspondence,
which leads to an explicit formula for the PT $\mathbb{Z}_{n}$-vertex in terms
of loop Schur functions appearing in the proof of orbifold GW/DT
correspondence and DT crepant resolution conjecture in [39, 40, 37, 38], and a
proof of multi-regular orbifold DT/PT correspondence for toric CY 3-orbifolds
with transverse $A_{n-1}$ singularities.

Acknowledgements. The author would like to thank Patrick Lei, a PhD student of
Professor Chiu-Chu Melissa Liu, for bringing to my attention the work of Zhang
[45], and thank Xiaowen Hu for his helpful suggestions.

##  2\. Preliminaries

We first recollect some relevant ingredients of the structure of toric CY
3-orbifolds (with transverse $A_{n-1}$ singularities) in [7] for formulating
PT partition functions in Section 3, and then discuss the relation between
$T$-fixed PT stable pairs and $T$-fixed orbifold PT stable pairs, which give
the description of $T$-fixed orbifold PT stable pairs in Section 2.2, see also
[45, Section 4.6] for comparison as mentioned in Introduction. We also recall
definitions of reduced (multi-regular) DT partition functions in [7] for
describing orbifold DT/PT correspondence later.

###  2.1. Geometry of toric Calabi-Yau 3-orbifolds

In this subsection, we briefly recall some relevant geometry of toric Calabi-
Yau (CY) 3-orbifolds in [7]. A CY 3-orbifold is a smooth quasi-projective
Deligne-Mumford stack $\mathcal{X}$ of dimension 3 over $\mathbb{C}$ with
generically trivial stabilizers and its canonical bundle is trivial, i.e.,
$K_{\mathcal{X}}\cong\mathcal{O}_{\mathcal{X}}$. The local model of
$\mathcal{X}$ at any point $p$ is given by $[\mathbb{C}^{3}/H_{p}]$, where
$H_{p}$ is a finite subgroup of $SL(3,\mathbb{C})$ as the automorphism group
of $p$. It is shown in [7, Lemma 40] that a toric CY 3-orbifold $\mathcal{X}$
is uniquely determined by its coarse moduli space $X$, which is a simplicial
toric variety. The Deligne-Mumford torus of $\mathcal{X}$ defined in [11,
Definition 3.1] is isomorphic to $T=(\mathbb{C}^{*})^{3}\subset\mathcal{X}$
with the naturally extended action on $\mathcal{X}$. More generally, one may
refer to [5, 11] for the theory of toric Deligne-Mumford stacks.

A toric CY 3-orbifolds $\mathcal{X}$ is determined by its combinatorial data
called a web diagram $\Gamma_{\mathcal{X}}$, which is a finite trivalent
planar graph with edges and vertices satisfying certain conditions, see [7,
Appendix B] for more details. Denote by $V_{\mathcal{X}}$ all vertices of
$\Gamma_{\mathcal{X}}$ and by $E_{\mathcal{X}}$ all edges of
$\Gamma_{\mathcal{X}}$. Set
$\Gamma_{\mathcal{X}}:=\\{V_{\mathcal{X}},E_{\mathcal{X}}\\}$, where vertices
and edges correspond to torus fixed points and torus invariant curves of
$\mathcal{X}$ respectively, and regions divided by edges in the plane
correspond to torus invariant divisors. Let $p_{v}$ be the torus fixed point
corresponding to $v\in V_{\mathcal{X}}$. Here, $E_{\mathcal{X}}$ may contain
some non-compact edges. If $e\in E_{\mathcal{X}}$ is a compact edge
corresponding to a torus invariant curve $\mathcal{C}\subset\mathcal{X}$, then
$\mathcal{C}$ is an abelian gerbe [11] over a football
$\mathbb{P}^{1}_{l_{0},l_{\infty}}$ constructed from a $\mathbb{P}^{1}$ with
root construction of order $l_{0}$ and $l_{\infty}$ at $0$ and $\infty$.

Next, we assume that $\mathcal{X}$ is a toric CY 3-orbifold with transverse
$A_{n-1}$ singularities, which means that nontrivial orbifold structure is
supported on a union of disjoint smooth curves and the stabilizer along each
curve is $\mathbb{Z}_{n}$ where $n$ is variable for different curves. Then the
local model of $\mathcal{X}$ at a torus fixed point is of the form
$[\mathbb{C}^{3}/\mathbb{Z}_{n}]$ with the generator of $\mathbb{Z}_{n}$
acting on the coordinates with weights $(1,-1,0)$. Denote by $\mathcal{C}_{e}$
the torus invariant curve (line) corresponding to $e$ and by $n_{e}$ the order
of the stabilizer on $\mathcal{C}_{e}$ for each $e\in E_{\mathcal{X}}$.

An orientation on $\Gamma_{\mathcal{X}}$ is defined by taking a choice of
direction for each edge and labeling three edges incident to each vertex $v$
by $(e_{1,v},e_{2,v},e_{3,v})$ in counterclockwise ordering where if an edge
$e$ is incident to a vertex $v$ with $n_{e}>1$, define $e_{3,v}=e$. On
$\Gamma_{\mathcal{X}}$, we define an edge assignment by choosing a partition
$\lambda_{e}$ for each edge $e$ with the convention that
$\lambda_{e}=\emptyset$ for any non-compact edge. Denoted by
$\Lambda_{\mathcal{X}}$ the set of all edge assignments
$\\{\lambda_{e}\;|\;e\in E_{\mathcal{X}}\\}$. With one edge assignment
$\\{\lambda_{e}\\}$, we define a triple of partitions
$(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ for each vertex $v$ by taking
$\lambda_{i,v}=\lambda_{e_{i,v}}$ for $e_{i,v}$ oriented outward from $v$ and
$\lambda_{i,v}=\lambda_{e_{i,v}}^{\prime}$ for $e_{i,v}$ with the opposite
orientation. Here, $\lambda^{\prime}=\\{(j,i)\;|\;(i,j)\in\lambda\\}$ is the
conjugate of $\lambda$. This orientation convention is used to guarantee the
compatibility between edge partitions and triple vertex outgoing partitions in
Section 3. We call an edge assignment $\\{\lambda_{e}\\}$ multi-regular if for
each $e\in E_{\mathcal{X}}$, we have $|\lambda_{e}|_{l}=\frac{1}{n}|\lambda|$
for all $l\in\\{0,1,\cdots,n-1\\}$ where $|\lambda_{e}|_{l}$ is defined in
Section 3.4, and each partition $\lambda_{e}$ is called multi-regular.

According to the orientation of $e$, let $\mathcal{D}_{e}$ and
$\mathcal{D}_{e}^{\prime}$ be torus invariant divisors corresponding to the
right and left regions incident to $e$ respectively, see Figure 1. Denote by
$p_{0,e}$ and $p_{\infty,e}$ the torus fixed points corresponding to the
initial and final vertices $v_{0}$, $v_{\infty}$ adjacent to $e$. Let
$\mathcal{D}_{0,e}$ and $\mathcal{D}_{\infty,e}$ be the torus invariant
divisors intersecting $\mathcal{C}_{e}$ transversely at $p_{0,e}$ and
$p_{\infty,e}$. Let $\mathcal{D}_{1,v}$, $\mathcal{D}_{2,v}$ and
$\mathcal{D}_{3,v}$ be torus invariant divisors corresponding to the regions
opposite to the edges $e_{1,v}$, $e_{2,v}$ and $e_{3,v}$. Let $p_{e}$ be a
generic point on $\mathcal{C}_{e}$. As in [7], we assume that for each edge
$e$ that

$\displaystyle\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{D}_{e})=\mathcal{O}_{\mathcal{C}_{e}}(m_{e}p_{e}-\delta_{0,e}p_{0,e}-\delta_{\infty,e}p_{\infty,e}),$
$\displaystyle\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{D}_{e}^{\prime})=\mathcal{O}_{\mathcal{C}_{e}}(m^{\prime}_{e}p_{e}-\delta_{0,e}^{\prime}p_{0,e}-\delta_{\infty,e}^{\prime}p_{\infty,e})$

where $\delta_{0,e}=1$ if the edge
$f_{v_{0}}:=\mathcal{D}_{e}\cap\mathcal{D}_{0,e}$ is labelled by $e_{3,v_{0}}$
and $\delta_{0,e}=0$ otherwise, and $\delta^{\prime}_{0,e}=1$ if the edge
$f^{\prime}_{v_{0}}:=\mathcal{D}^{\prime}_{e}\cap\mathcal{D}_{0,e}$ is
labelled by $e_{3,v_{0}}$ and $\delta^{\prime}_{0,e}=0$ otherwise. Let
$g_{v_{\infty}}:=\mathcal{D}_{e}\cap\mathcal{D}_{\infty,e}$ and
$g^{\prime}_{v_{\infty}}:=\mathcal{D}_{e}^{\prime}\cap\mathcal{D}_{\infty,e}$.
One can define $\delta_{\infty,e},\delta_{\infty,e}^{\prime}$ corresponding to
$g_{v_{\infty}},g^{\prime}_{v_{\infty}}$ similarly. Then we have

$\displaystyle\deg\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{D}_{e})=m_{e}-\frac{\delta_{0,e}}{n_{f_{v_{0}}}}-\frac{\delta_{\infty,e}}{n_{g_{v_{\infty}}}},$
$\displaystyle\deg\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{D}^{\prime}_{e})=m_{e}^{\prime}-\frac{\delta^{\prime}_{0,e}}{n_{f^{\prime}_{v_{0}}}}-\frac{\delta^{\prime}_{\infty,e}}{n_{g^{\prime}_{v_{\infty}}}}.$

The Calabi-Yau condition implies that

$\displaystyle
m_{e}+m_{e}^{\prime}-\delta_{0,e}-\delta_{0,e}^{\prime}-\delta_{\infty,e}-\delta^{\prime}_{\infty,e}+2=0.$

Let $K(\mathcal{X})$ be the Grothendieck group of compactly supported coherent
sheaves on $\mathcal{X}$ modulo the numerical equivalence $\sim$, where
$\mathcal{G}_{1}\sim\mathcal{G}_{2}$ if
$\chi(\mathcal{G}_{1}\otimes\mathcal{H})=\chi(\mathcal{G}_{2}\otimes\mathcal{H})$
for any locally free sheave $\mathcal{H}$ on $\mathcal{X}$. For $0\leq d\leq
3$, let $F_{d}K(\mathcal{X})$ be the subgroup of those elements in
$FK(\mathcal{X})$ with support of dimension at most $d$. It is shown in [7,
Section 3.3] that classes in $F_{1}K(\mathcal{X})$ are generated by
$[\mathcal{O}_{p}]$ for a generic point $p$ in $\mathcal{X}$,
$[\mathcal{O}_{p_{e}}\otimes\rho_{k}]$ for any (compact or not) edge $e\in
E_{\mathcal{X}}$, and $[\mathcal{O}_{\mathcal{C}_{e}}(-1)\otimes\rho_{k}]$ for
any compact edge $e\in E_{\mathcal{X}}$. Here, $\rho_{k}$, $0\leq k<n_{e}$,
are the irreducible representations of $\mathbb{Z}_{n_{e}}$ with the
convention that
$\mathcal{O}_{p_{e}}(-k\mathcal{D}_{e})\cong\mathcal{O}_{p_{e}}\otimes\rho_{k}$,
and the classes $[\mathcal{O}_{p}],[\mathcal{O}_{p_{e}}\otimes\rho_{k}]$ and
$[\mathcal{O}_{\mathcal{C}_{e}}(-1)\otimes\rho_{k}]$ are associated to
variables $q$, $q_{e,k}$ and $v_{e,k}$ respectively. Among all these
generators for $F_{1}K(\mathcal{X})$, we have the relation

$\displaystyle[\mathcal{O}_{p}]=[\mathcal{O}_{p_{e}}\otimes
R_{reg}]=\sum_{k=0}^{n_{e}-1}[\mathcal{O}_{p_{e}}\otimes\rho_{k}]$

with the associated relation in variables
$q=\prod\limits_{k=0}^{n_{e}-1}q_{e,k}$ for each edge $e\in E_{\mathcal{X}}$,
where $R_{reg}=\sum\limits_{k=0}^{n_{e}-1}\rho_{k}$ is the regular
representation of $\mathbb{Z}_{n_{e}}$.

###  2.2. Orbifold PT stable pairs

In this subsection, we will recall moduli spaces of orbifold PT stable pairs
in [18, 19] and show that every orbifold PT stable pair has the similar
characterization as in [34]. Let $\mathcal{X}$ be a 3-dimensional smooth
projective Delinge-Mumford stack over $\mathbb{C}$. The definition of every
group in the natural filtration $F_{0}K(\mathcal{X})\subset
F_{1}K(\mathcal{X})\subset F_{2}K(\mathcal{X})\subset
F_{3}K(\mathcal{X})=K(\mathcal{X})$ is the same as the one in Section 2.1.
Given $\beta\in F_{1}K(\mathcal{X})$, let $\mathrm{PT}(\mathcal{X},\beta)$ be
the moduli space of orbifold PT stable pairs
$\varphi:\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$ with $[\mathcal{F}]=\beta$,
where $\mathcal{F}$ is pure of dimension one and $\varphi$ has finite
0-dimensional cokernel. Here, $\mathrm{PT}(\mathcal{X},\beta)$ is denoted by
$\mathrm{PT}^{\beta}_{\mathcal{X}/\mathbb{C}}$ in [19, Remark 5.6], which is a
projective scheme over $\mathbb{C}$. And $\mathrm{PT}(\mathcal{X},\beta)$ is a
fine moduli space, which carries a perfect obstruction theory and has a
virtual fundamental class $[\mathrm{PT}(\mathcal{X},\beta)]^{vir}$, see [18,
Theorem 5.20 and 5.21] for more details. Since
$\dim\mathrm{Supp}\mathcal{F}=1$, we have $\beta\notin F_{0}K(\mathcal{X})$
implicitly throughout this paper.

For an orbifold PT stable pair
$\varphi:\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$, as in [33, Section 1.3], we
have the following exact sequence

$\displaystyle
0\to\mathcal{I}_{\mathcal{C}_{\mathcal{F}}}\to\mathcal{O}_{\mathcal{X}}\to\mathcal{F}\to\mathrm{coker}\,\varphi\to
0,$

where $\mathcal{C}_{\mathcal{F}}=\mathrm{Supp}(\mathcal{F})\subset\mathcal{X}$
is a stacky curve, and $\mathcal{I}_{\mathcal{C}_{\mathcal{F}}}$ is the ideal
sheaf of $\mathcal{C}_{\mathcal{F}}$ in $\mathcal{X}$. Here,
$\mathrm{Im}\,\varphi\subset\mathcal{F}$ is equal to the structure sheaf
$\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$ as a quotient of
$\mathcal{O}_{\mathcal{X}}$, and is also a pure sheaf which implies that
locally in the $\acute{e}$tale topology
$\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$ is a Cohen-Macaulay module by [9,
Appendix C] and [41, IV-B]. Then $\mathcal{C}_{\mathcal{F}}$ is Cohen-
Macaulay. Set $\mathtt{Q}=\mathrm{coker}\,\varphi$. Denote by
$\mathrm{Supp}^{red}(\mathtt{Q})$ the finite $0$-dimensional reduced closed
substack support of $\mathtt{Q}$. Then
$\mathrm{Supp}^{red}(\mathtt{Q})\subset\mathcal{C}_{\mathcal{F}}$. Let
$\mathfrak{m}\subset\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$ be the ideal
sheaf of a finite $0$-dimensional closed substack of
$\mathcal{C}_{\mathcal{F}}$. It can be shown as in [33, Proposition 1.8] that
an orbifold PT stable pair $\varphi:\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$
(or denoted by $(\mathcal{F},\varphi)$) with support
$\mathcal{C}_{\mathcal{F}}$ satisfying
$\mathrm{Supp}^{red}(\mathtt{Q})\subset\mathrm{Supp}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}/\mathfrak{m})$
is equivalent to a subsheaf of
$\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})/\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$
for $r\gg 0$. As in the proof of [33, Proposition 1.8], $\mathtt{Q}$ can be
viewed as a coherent subsheaf of

$\displaystyle\lim\limits_{\longrightarrow}\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})/\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$

and hence $\mathcal{F}$ can be viewed as a coherent subsheaf of
$\lim\limits_{\longrightarrow}\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})$
such that we have the following short exact sequence

$\displaystyle
0\to\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}\to\mathcal{F}\to\mathtt{Q}\to 0.$

Let $\mathcal{X}$ be a toric CY 3-orbifold. We can compactify $\mathcal{X}$ to
the projective one $\overline{\mathcal{X}}$ and use the $T$-equivariant
residue to define the PT invariants of $\mathcal{X}$ similarly as in [27, 24,
46] due to the compactness of $T$-fixed locus
$\mathrm{PT}(\mathcal{X},\beta)^{\mathrm{T}}$. One may view
$\mathrm{PT}(\mathcal{X},\beta)$ as a quasi-projective moduli scheme as the
restriction of $\mathrm{PT}(\overline{\mathcal{X}},\beta)$ by the equivalence
of descriptions of (quasi)-projective Deligne-Mumford stacks [15, Theorem 5.3
and Corollary 5.4]. Now we follow [34, Section 2] to give a description of
$T$-fixed orbifold PT stable pair. Let
$[\mathcal{O}_{\mathcal{X}}\to\mathcal{F}]\in\mathrm{PT}(\mathcal{X},\beta)^{\mathrm{T}}$
be a $T$-fixed orbifold PT stable pair. Assume $\mathtt{Q}$ is the cokernel of
$\varphi:\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$. Then the sheaf
$\mathcal{F}$ is supported on torus invariant curves and $\mathtt{Q}$ must be
supported on torus fixed points. Let $\mathcal{X}^{T}=\\{p_{\alpha}\\}$ be all
the torus fixed points in $\mathcal{X}$. By [7, Lemma 46], for each torus
fixed point $p_{\alpha}$, there is a $T$-invariant open neighborhood
$\mathcal{U}_{\alpha}=[\mathbb{C}^{3}/G_{\alpha}]$. Here, the finite subgroup
$G_{\alpha}\subset T$ act on $\mathbb{C}^{3}$ by
$(z_{1},z_{2},z_{3})\mapsto(t_{1}z_{1},t_{2}z_{2},t_{3}z_{3})$ where
$(t_{1},t_{2},t_{3})\in G_{\alpha}$ and
$(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}$. Denote by
$\varphi_{\alpha}:\mathcal{O}_{\mathcal{U}_{\alpha}}\to\mathcal{F}_{\alpha}$
the restriction of $\varphi:\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$ on
$\mathcal{U}_{\alpha}$, then $\varphi_{\alpha}$ is a $T$-invariant section of
$T$-invariant sheaf $\mathcal{F}_{\alpha}$. Let $\mathcal{C}_{\alpha}$ be the
support of $\mathcal{F}_{\alpha}$. Assume that $\mathcal{C}_{\alpha}$ is
nonempty, then $\mathcal{C}_{\alpha}$ is also Cohen-Macaulay and containing
$T$-fixed point $[(0,0,0)/G_{\alpha}]$, and is supported on at least one of
three abelian gerbes [11] over their corresponding footballs restricted in
$\mathcal{U}_{\alpha}$ along three coordinate directions of
$\mathcal{U}_{\alpha}$ intersecting at their common stacky point
$[(0,0,0)/G_{\alpha}]$. Let
$\mathtt{Q}_{\alpha}=\mathrm{coker}\,\varphi_{\alpha}$, we have the following
exact sequence

$\displaystyle
0\to\mathcal{I}_{\mathcal{C}_{\alpha}}\to\mathcal{O}_{\mathcal{U}_{\alpha}}\to\mathcal{F}_{\alpha}\to\mathtt{Q}_{\alpha}\to
0.$

Since $\mathcal{U}_{\alpha}=[\mathbb{C}^{3}/G_{\alpha}]$, by [44, Example
7.21] sheaves $\mathcal{I}_{\mathcal{C}_{\alpha}}$,
$\mathcal{O}_{\mathcal{C}_{\alpha}}$, $\mathcal{O}_{\mathcal{U}_{\alpha}}$,
$\mathcal{F}_{\alpha}$ and $\mathtt{Q}_{\alpha}$ are equivalent to the
corresponding $G_{\alpha}$-equivariant sheaves on $U_{\alpha}=\mathbb{C}^{3}$
denoted by $\mathcal{I}_{C_{\alpha}}$, $\mathcal{O}_{C_{\alpha}}$,
$\mathcal{O}_{U_{\alpha}}$, $F_{\alpha}$ and $\mathcal{Q}_{\alpha}$
respectively, where $C_{\alpha}(\ni(0,0,0))$ is the support of $F_{\alpha}$
and is supported on the coordinate lines of $U_{\alpha}=\mathbb{C}^{3}$
corresponding to those of $\mathcal{C}_{\alpha}\subset\mathcal{U}_{\alpha}$.
And $C_{\alpha}$ is $T$-fixed subscheme of pure dimension one and hence is
Cohen-Macaulay. By the definition of (orbifold) PT stable pairs, an orbifold
PT stable pair $\mathcal{O}_{\mathcal{U}_{\alpha}}\to\mathcal{F}_{\alpha}$ is
equivalent to $G_{\alpha}$-equivariant PT stable pair
$\mathcal{O}_{U_{\alpha}}\to F_{\alpha}$ fitting into the short exact sequence
$0\to\mathcal{I}_{C_{\alpha}}\to\mathcal{O}_{U_{\alpha}}\to
F_{\alpha}\to\mathcal{Q}_{\alpha}\to 0$ on $U_{\alpha}$ corresponding to [34,
(2-3)]. Since $\mathtt{Q}_{\alpha}$ has finite zero-dimensional support at the
origin of $\mathcal{U}_{\alpha}=[\mathbb{C}^{3}/G_{\alpha}]$, then
$\mathtt{Q}_{\alpha}$ can be viewed as a finite dimensional representation
$\mathcal{Q}_{\alpha}$ of $G_{\alpha}$ by [10, Lemma 5.1.23 or Section 5.2.1].

By [33, Proposition 1.8], a $T$-fixed orbifold PT stable pair
$\varphi_{\alpha}:\mathcal{O}_{\mathcal{U}_{\alpha}}\to\mathcal{F}_{\alpha}$
corresponds to the $G_{\alpha}$-equivariant subsheaf of

$\displaystyle\lim\limits_{\longrightarrow}\mathcal{H}om(\mathbf{m}_{\alpha}^{r},\mathcal{O}_{C_{\alpha}})/\mathcal{O}_{C_{\alpha}}$

on $U_{\alpha}=\mathbb{C}^{3}$, where $\mathbf{m}_{\alpha}$ is the ideal sheaf
of $(0,0,0)$ in $C_{\alpha}\subset\mathbb{C}^{3}$.

To characterize orbifold PT stable pairs, we now breifly recall some notation
for the description of PT stable pairs $\mathcal{O}_{U_{\alpha}}\to
F_{\alpha}$ in [34]. Since $C_{\alpha}$ is $T$-fixed and Cohen-Macaulay, then
$\mathcal{I}_{C_{\alpha}}$ is a monomial ideal in
$\mathbb{C}[x_{1},x_{2},x_{3}]$ which determines a 3D partition $\pi$ given by
the union of the infinite cylinders on the three nonnegative axes with cross-
sections $\lambda,\mu,\nu$. As $\dim C_{\alpha}=1$, at least one of
$\lambda,\mu,\nu$ is nonempty. Let
$\mathbf{M}=\bigoplus_{i=1}^{3}\mathbf{M}_{i}$ with

$\displaystyle\mathbf{M}_{1}=\mathbb{C}[x_{1},x_{1}^{-1}]\otimes\frac{\mathbb{C}[x_{2},x_{3}]}{\lambda[x_{2},x_{3}]},$
$\displaystyle\mathbf{M}_{2}=\mathbb{C}[x_{2},x_{2}^{-1}]\otimes\frac{\mathbb{C}[x_{3},x_{1}]}{\mu[x_{3},x_{1}]},$
$\displaystyle\mathbf{M}_{3}=\mathbb{C}[x_{3},x_{3}^{-1}]\otimes\frac{\mathbb{C}[x_{1},x_{2}]}{\nu[x_{1},x_{2}]},$

where $\lambda[x_{2},x_{3}]$, $\mu[x_{3},x_{1}]$ and $\nu[x_{1},x_{2}]$ are
monomial ideals determined by partitions $\lambda$, $\mu$ and $\nu$
respectively as in [34, Section 2.2] .

By the similar argument in [34, Section 2], the $T$-fixed orbifold PT stable
pair $(\mathcal{F}_{\alpha},\varphi_{\alpha})$ corresponds to a finitely
generated $T$-invariant $\mathbb{C}[x_{1},x_{2},x_{3}]$-submodule

$\displaystyle\mathcal{Q}_{\alpha}\subset\mathbf{M}/\langle(1,1,1)\rangle$

as a $G_{\alpha}$-module (or a representation of $G_{\alpha}$), and every
finitely generated $T$-invariant $\mathbb{C}[x_{1},x_{2},x_{3}]$-submodule
$\mathcal{Q}_{\alpha}\subset\mathbf{M}/\langle(1,1,1)\rangle$ as a
$G_{\alpha}$-module can be viewed as the restriction of a $T$-fixed orbifold
PT stable pair on $\mathcal{X}$ to $[\mathbb{C}^{3}/G_{\alpha}]$. Here the
$i$-th component of $(1,1,1)$ is the canonical $T$-invariant element in
$\mathbf{M}_{i}$, and it should be intepreted as 0 if $\mathbf{M}_{i}$ is 0.
In [34], $\mathbf{M}/\langle(1,1,1)\rangle$ is also denoted by
$\mathbf{M}/\mathcal{O}_{C_{\alpha}}$. It is shown in [34] that a connected
component of the moduli space of $T$-invariant
$\mathbb{C}[x_{1},x_{2},x_{3}]$-submodules of
$\mathbf{M}/\langle(1,1,1)\rangle$ (or $\mathbf{M}/\mathcal{O}_{C_{\alpha}}$)
is a product of $\mathbb{P}^{1}$’s. This implies that a connected component
$\mathsf{Q}$ of $T$-fixed locus of $\mathrm{PT}(\mathcal{X},\beta)$ is also a
product of $\mathbb{P}^{1}$’s, which is projective and nonsingular. As in the
proof of [34, Proposition 3], the component $\mathsf{Q}$ is corresponding to
some connected component of moduli space of the labelling of a collection of
labelled box configurations, see Lemma 3.13 for more details. See also [45,
Lemma 4.21] for comparison.

###  2.3. The DT partition function

We briefly recall several DT partition functions in [7]. Let $\mathcal{X}$ be
a toric CY 3-orbifold with transverse $A_{n-1}$ singularities. Let
$F_{mr}K(\mathcal{X})$ be the multi-regular part of $F_{1}K(\mathcal{X})$,
each of whose elements is represented by some coherent sheaf such that the
associated representation of the stabilizer group at the generic point is a
multiple of the regular representation, see [7, Section 4.1] for more details.
As $F_{0}K(\mathcal{X})\subset F_{mr}K(\mathcal{X})\subset
F_{1}K(\mathcal{X})$, we define

$\displaystyle DT(\mathcal{X})=\sum_{\alpha\in
F_{1}K(\mathcal{X})}DT_{\alpha}(\mathcal{X})q^{\alpha};\;\;DT_{mr}(\mathcal{X})=\sum_{\alpha\in
F_{mr}K(\mathcal{X})}DT_{\alpha}(\mathcal{X})q^{\alpha};\;\;DT_{0}(\mathcal{X})=\sum_{\alpha\in
F_{0}K(\mathcal{X})}DT_{\alpha}(\mathcal{X})q^{\alpha}$

and

$\displaystyle
DT^{\prime}(\mathcal{X})=\frac{DT(\mathcal{X})}{DT_{0}(\mathcal{X})};\;\;\;\;DT_{mr}^{\prime}(\mathcal{X})=\frac{DT_{mr}(\mathcal{X})}{DT_{0}(\mathcal{X})}$

where $DT_{\alpha}(\mathcal{X})$ is defined by the weighted Euler
characteristic of $\mathrm{Hilb}^{\alpha}(\mathcal{X})$ in [7, Definition 1],
and for some choice of a basis $\mathfrak{b}_{1},\cdots,\mathfrak{b}_{l}$ of
$F_{1}K(\mathcal{X})$ and $\alpha=\sum_{i=1}^{l}c_{i}\mathfrak{b}_{i}$, we
have $q^{\alpha}=q_{1}^{c_{1}}\cdots q_{l}^{c_{l}}$ and the similar definition
of $q^{\alpha}$ for $\alpha\in F_{mr}K(\mathcal{X})$ and $\alpha\in
F_{0}K(\mathcal{X})$. The explicit formula for $DT(\mathcal{X})$ is obtained
as a sum over all edge assignments in [7, Theorem 10] while
$DT_{mr}(\mathcal{X})$ is the sum over all multi-regular edge assignments.

##  3\. The orbifold PT topological vertex

The goal of this section is to derive the PT topological vertex formalism (see
Theorem 3.22) for toric CY 3-orbifolds with transverse $A_{n-1}$
singularities, which is also obtained by Zhang [45, Theorem/Conjecture 4.19]
mentioned in Introduction where we point out some differences between ours. In
[14], the authors prove the nonsingularity of $T_{0}$-fixed PT stable pairs,
which is useful for defining the PT partition function (see Formula (6)) in
Section 3.3 via virtual localization. This completes the proof of [45,
Conjecture 4.23] (and [45, Proposition/Conjecture 4.16]) and hence the proof
of [45, Theorem/Conjecture 4.19] for the full 3-leg case. For completeness and
comparison, we present our argument here and suggest the reader also refer to
[45, Section 4] for his argument. We first recall the (orbifold) DT and PT
topological vertices and the (orbifold) DT/PT vertex correspondence in Section
3.1. We explicitly present the proof of $K$-theory class of the finite
cokernel $\mathtt{Q}$ as a sum over vertex terms in Proposition 3.15 which is
important for deriving the sign formula in Theorems 3.18 and 3.21 and hence PT
partition functions in Section 3.4 following the method in [7, Section 6].

###  3.1. Partitions, labelled box configurations and topological vertices

In this subsection, we recall definitions of 3D partitions and the (orbifold)
DT topological vertex in [7, Section 3.1], and then recollect labelled box
configurations and the PT topological vertex defined in [34], and finally give
the definition of orbifold PT topological vertex that will be used later.

######  Definition 3.1.

([7, Definition 4]) Let $\lambda,\mu,\nu$ be three ordinary 2D partitions. Let
$\pi$ be the subset of $(\mathbb{Z}_{\geq 0})^{3}$ satisfy the following
conditions:
$(i)$ for $i,j,k\geq 0$, if any of $(i+1,j,k)$, $(i,j+1,k)$, $(i,j,k+1)$ is in
$\pi$, then $(i,j,k)$ is also in $\pi$;
$(ii)$ three 2D partitions $\lambda,\mu,\nu$ are associated to $\pi$ such that
$(a)$ $(j,k)\in\lambda$ if and only if $(i,j,k)\in\pi$ for all $i\gg 0$, $(b)$
$(k,i)\in\mu$ if and only if $(i,j,k)\in\pi$ for all $j\gg 0$, $(c)$
$(i,j)\in\nu$ if and only if $(i,j,k)\in\pi$ for all $k\gg 0$;
Then we call $\pi$ a 3D partition asymptotic to $(\lambda,\mu,\nu)$. We say
$\\{(i,j,k)|(j,k)\in\lambda\\}$, $\\{(i,j,k)|(k,i)\in\mu\\}$, and
$\\{(i,j,k)|(i,j)\in\nu\\}$ the leg of $\pi$ in the first, second and third
direction respectively.

For the triple of ordinary 2D partitions $(\lambda,\mu,\nu)$, we employ some
notation in [14, Section 2] and [34, Section 2.5]. Let

$\displaystyle\mathrm{Cyl}_{1}=\\{(i,j,k)\in\mathbb{Z}^{3}|(j,k)\in\lambda\\},$
$\displaystyle\mathrm{Cyl}_{2}=\\{(i,j,k)\in\mathbb{Z}^{3}|(k,i)\in\mu\\},$
$\displaystyle\mathrm{Cyl}_{3}=\\{(i,j,k)\in\mathbb{Z}^{3}|(i,j)\in\nu\\}.$

For $1\leq l\leq 3$, let $\mathrm{Cyl}_{l}^{+}=(\mathbb{Z}_{\geq
0})^{3}\cap\mathrm{Cyl}_{l}$ and
$\mathrm{Cyl}_{l}^{-}=\mathrm{Cyl}_{l}\backslash\mathrm{Cyl}_{l}^{+}$. Set

$\displaystyle\mathrm{I}^{-}=\mathrm{Cyl}_{1}^{-}\cup\mathrm{Cyl}_{2}^{-}\cup\mathrm{Cyl}_{3}^{-},$
$\displaystyle\mathrm{I}^{+}=\left(\mathrm{Cyl}_{1}^{+}\cup\mathrm{Cyl}_{2}^{+}\cup\mathrm{Cyl}_{3}^{+}\right)\backslash(\mathrm{II}\cup\mathrm{III}),$
$\displaystyle\mathrm{II}_{\hat{1}}=\left(\mathrm{Cyl}_{2}\cap\mathrm{Cyl}_{3}\right)\backslash\mathrm{Cyl}_{1},$
$\displaystyle\mathrm{II}_{\hat{2}}=\left(\mathrm{Cyl}_{1}\cap\mathrm{Cyl}_{3}\right)\backslash\mathrm{Cyl}_{2},$
$\displaystyle\mathrm{II}_{\hat{3}}=\left(\mathrm{Cyl}_{1}\cap\mathrm{Cyl}_{2}\right)\backslash\mathrm{Cyl}_{3},$
$\displaystyle\mathrm{II}=\mathrm{II}_{\hat{1}}\cup\mathrm{II}_{\hat{2}}\cup\mathrm{II}_{\hat{3}},$
$\displaystyle\mathrm{III}=\mathrm{Cyl}_{1}\cap\mathrm{Cyl}_{2}\cap\mathrm{Cyl}_{3}.$

Notice that $\mathrm{Cyl}_{l}$, $\mathrm{Cyl}_{l}^{\pm}$, $\mathrm{I}^{\pm}$,
$\mathrm{II}_{\hat{l}}$, $\mathrm{II}$, and $\mathrm{III}$ are defined
associated to $(\lambda,\mu,\nu)$, we also write them as
$\mathrm{Cyl}_{l}(\lambda,\mu,\nu)$,
$\mathrm{Cyl}_{l}^{\pm}(\lambda,\mu,\nu)$,
$\mathrm{I}^{\pm}(\lambda,\mu,\nu)$, $\mathrm{II}_{\hat{l}}(\lambda,\mu,\nu)$,
$\mathrm{II}(\lambda,\mu,\nu)$, and $\mathrm{III}(\lambda,\mu,\nu)$
respectively. Let $\pi$ be a 3D partition asymptotic to $(\lambda,\mu,\nu)$.
Then $\mathrm{Cyl}_{l}^{+}$ is the leg of $\pi$ in the $l$-th direction
defined above. Define the renormalized volume of $\pi$ as

$\displaystyle\|\pi\|=\sum_{(i,j,k)\in\pi}\xi_{\pi}(i,j,k)$

where

$\displaystyle\xi_{\pi}(i,j,k)=1-|\\{l\in\\{1,2,3\\}|(i,j,k)\in\mathrm{Cyl}_{l}^{+}\\}|.$

Then
$\|\pi\|=|\pi\backslash(\mathrm{I}^{+}\cup\mathrm{II}\cup\mathrm{III})|-|\mathrm{II}|-2|\mathrm{III}|$
as in [14, Section 3.1]. Let $\mathcal{P}(\lambda,\mu,\nu)$ be the set of all
3D partitions $\pi$ asymptotic to $(\lambda,\mu,\nu)$. Now we have the
topological vertex defined by Okounkov, Reshetikhin and Vafa as follows.

######  Definition 3.2.

([7, Definition 5]) The $\mathrm{DT}$ topological vertex is defined by

$\displaystyle
V_{\lambda\mu\nu}=\sum_{\pi\in\mathcal{P}(\lambda,\mu,\nu)}q^{\|\pi\|}.$

######  Remark 3.3.

To avoid the confusion, we employ the notation $\|\cdot\|$ to denote the
renormalized volume rather than $|\cdot|$ in [7], which has been used to
denote the number of elements in some set.

Assume that a finite abelian group $G$ acts on $\mathbb{C}^{3}$ with
characters $r_{1},r_{2},r_{3}$ for three coordinate lines. Let $\widehat{G}$
be the set of all the characters of irreducible representations of $G$, i.e,
the set of all nonzero complex-valued functions on $G$. The DT $G$-vertex is
defined in [7] by

$\displaystyle
V_{\lambda\mu\nu}^{G}=\sum_{\pi\in\mathcal{P}(\lambda,\mu,\nu)}\prod_{r\in\widehat{G}}q_{r}^{\|\pi\|_{r}}$

where

$\displaystyle\|\pi\|_{r}=\sum_{\begin{subarray}{c}(i,j,k)\in\pi\\\
r_{1}^{i}r_{2}^{j}r_{3}^{k}=r\end{subarray}}\xi_{\pi}(i,j,k),$

is the renormalized number of boxes $(i,j,k)\in\pi$ with color
$r\in\widehat{G}$. Roughly speaking, the orbifold topolocial vertex
$V_{\lambda\mu\nu}^{G}$ counts the 3D partitions colored with elements of
$\widehat{G}$.

If $G=\mathbb{Z}_{n}$, we have

######  Definition 3.4.

([7, Definition 6]) The $\mathrm{DT}$ $\mathbb{Z}_{n}$-vertex is defined by

$\displaystyle
V_{\lambda\mu\nu}^{n}=\sum_{\pi\in\mathcal{P}(\lambda,\mu,\nu)}q_{0}^{\|\pi\|_{0}}\cdots
q_{n-1}^{\|\pi\|_{n-1}}$

where for $0\leq l<n$,

$\displaystyle\|\pi\|_{l}=\sum_{\begin{subarray}{c}(i,j,k)\in\pi\\\
i-j=l\;\mathrm{mod}\;n\end{subarray}}\xi_{\pi}(i,j,k)$

is the renormalized number of boxes $(i,j,k)\in\pi$ with color $i-j=l\mbox{
mod }n$.

######  Remark 3.5.

There are symmetries for the (orbifold) DT toplogical vertex [7, Section 3.1]
as follows

$\displaystyle
V_{\lambda\mu\nu}=V_{\mu^{\prime}\lambda^{\prime}\nu^{\prime}};\;\;V_{\lambda\mu\nu}=V_{\lambda^{\prime}\nu^{\prime}\mu^{\prime}};\;\;V_{\lambda\mu\nu}=V_{\nu^{\prime}\mu^{\prime}\lambda^{\prime}};$
$\displaystyle
V_{\lambda\mu\nu}^{n}(q_{0},q_{1},\cdots,q_{n-1})=V^{n}_{\mu^{\prime}\lambda^{\prime}\nu^{\prime}}(q_{0},q_{n-1},\cdots,q_{1}),\;\;\;\mbox{i.e.,}\;\;\;V_{\lambda\mu\nu}^{n}=\overline{V}^{n}_{\mu^{\prime}\lambda^{\prime}\nu^{\prime}}.$

where the overline denote the exchange of variables $q_{k}\leftrightarrow
q_{-k}$ with the subscripts in $\mathbb{Z}_{n}$.

The labelled box configurations are introduced in [34] in order to describe
the PT topological vertex. It is shown in [34, Section 2.5] that a finitely
generated $T$-invariant $\mathbb{C}[x_{1},x_{2},x_{3}]$-submodule of
$\mathbf{M}/\langle(1,1,1)\rangle$ associated to $(\lambda,\mu,\nu)$ defined
in Section 2.2 corresponds to a labelled box configuration with outgoing
partitions $(\lambda,\mu,\nu)$ defined as follows.

######  Definition 3.6.

([34, Section 2.5]) A labelled box configuration with outgoing partitions
$(\lambda,\mu,\nu)$ is defined as a finite subset of
$\mathrm{I}^{-}\cup\mathrm{II}\cup\mathrm{III}$ whose elements are called
boxes, where a type $\mathrm{III}$ box $w$ may be labelled by an element of

$\displaystyle\mathbb{P}^{1}=\mathbb{P}\left(\frac{\mathbb{C}\cdot\mathbf{1}_{w}\oplus\mathbb{C}\cdot\mathbf{2}_{w}\oplus\mathbb{C}\cdot\mathbf{3}_{w}}{\mathbb{C}\cdot(1,1,1)_{w}}\right).$

These boxes satisfy the following rule:
$(i)$ If $w=(w_{1},w_{2},w_{3})\in\mathrm{I}^{-}$ and any of

$\displaystyle(w_{1}-1,w_{2},w_{3}),\;\;(w_{1},w_{2}-1,w_{3}),\;\;(w_{1},w_{2},w_{3}-1)$

is a box, then $w$ must be a box.
$(ii)$ If $w\in\mathrm{II}_{\hat{i}}$ and any of

$\displaystyle(w_{1}-1,w_{2},w_{3}),\;\;(w_{1},w_{2}-1,w_{3}),\;\;(w_{1},w_{2},w_{3}-1)$

is a box other than a type $\mathrm{III}$ box labelled by the 1-dimensional
subspace $\mathbb{C}\cdot\mathbf{i}$, then $w$ must be a box.
$(iii)$ If $w\in\mathrm{III}$ and the span of the subspaces of

$\displaystyle\frac{\mathbb{C}\cdot\mathbf{1}_{w}\oplus\mathbb{C}\cdot\mathbf{2}_{w}\oplus\mathbb{C}\cdot\mathbf{3}_{w}}{\mathbb{C}\cdot(1,1,1)_{w}}$

induced by boxes

$\displaystyle(w_{1}-1,w_{2},w_{3}),\;\;(w_{1},w_{2}-1,w_{3}),\;\;(w_{1},w_{2},w_{3}-1)$

is not zero, then $w$ must be a box. And if the span has dimension 2, then $w$
must be an ublabelled box. If the span has dimension 1, then $w$ may be either
a box labelled by the span or an unlabelled box.

Let $\widetilde{\pi}$ be a labelled box configuration with outgoing partitions
$(\lambda,\mu,\nu)$. We define the length of a labelled box configuration
$\widetilde{\pi}$ by the sum of contributions from all boxes in
$\widetilde{\pi}$ as follows: $(i)$ each of type $\mathrm{I}^{-}$ boxes, type
$\mathrm{II}$ boxes, and type $\mathrm{III}$ labelled boxes contributes 1;
$(ii)$ each of type $\mathrm{III}$ unlabelled boxes contributes 2. Let
$\mathrm{I}^{-}(\widetilde{\pi})$ be the set of type $\mathrm{I}^{-}$ boxes in
$\widetilde{\pi}$ and $\mathrm{II}(\widetilde{\pi})$ the set of type
$\mathrm{II}$ boxes in $\widetilde{\pi}$. Denote by
$\mathrm{III}_{lb}(\widetilde{\pi})$ the set of type $\mathrm{III}$ labelled
boxes in $\widetilde{\pi}$ and by $\mathrm{III}_{ulb}(\widetilde{\pi})$ the
set of type $\mathrm{III}$ unlabelled boxes in $\widetilde{\pi}$. Set
$\mathrm{III}(\widetilde{\pi})=\mathrm{III}_{lb}(\widetilde{\pi})\cup\mathrm{III}_{ulb}(\widetilde{\pi})$.
Now we define the length of $\widetilde{\pi}$ by

$\displaystyle\ell(\widetilde{\pi})=\sum_{(i,j,k)\in\widetilde{\pi}}\eta_{\widetilde{\pi}}(i,j,k)$

where

$\displaystyle\eta_{\widetilde{\pi}}(i,j,k)=\left\\{\begin{aligned}
&1,\;\;\mbox{if}\;\;(i,j,k)\in\mathrm{I}^{-}(\widetilde{\pi})\cup\mathrm{II}(\widetilde{\pi})\cup\mathrm{III}_{lb}(\widetilde{\pi});\\\
&2,\;\;\mbox{if}\;\;(i,j,k)\in\mathrm{III}_{ulb}(\widetilde{\pi}).\end{aligned}\right.$

Notice that
$|\widetilde{\pi}|=|\mathrm{I}^{-}(\widetilde{\pi})|+|\mathrm{II}(\widetilde{\pi})|+|\mathrm{III}_{lb}(\widetilde{\pi})|+|\mathrm{III}_{ulb}(\widetilde{\pi})|$
and
$\ell(\widetilde{\pi})=|\widetilde{\pi}|+|\mathrm{III}_{ulb}(\widetilde{\pi})|$.
Let $\underline{\widetilde{\pi}}$ be the set of underlying boxes of
$\widetilde{\pi}$ by forgetting the labellings of
$\mathrm{III}_{lb}(\widetilde{\pi})$. And we call $\widetilde{\pi}$ a
labelling of $\underline{\widetilde{\pi}}$. Let
$\widetilde{\mathcal{P}}(\lambda,\mu,\nu)$ be the set of components of the
moduli space of labelled box configurations with outgoing partitions
$(\lambda,\mu,\nu)$. Let
$[\widetilde{\pi}]\in\widetilde{\mathcal{P}}(\lambda,\mu,\nu)$ be the
component of the moduli space of labellings of $\underline{\widetilde{\pi}}$
containing $\widetilde{\pi}$. It is obvious that all elements in one component
$[\widetilde{\pi}]$ have the same length, which we denoted by
$\ell(\widetilde{\pi})$. Let $\pi_{\mathrm{min}}(\lambda,\mu,\nu)$ be the
minimal 3D partition asymptotic to $(\lambda,\mu,\nu)$, i.e., the union of
infinite cylinders on three nonegative axes with cross-sections
$\lambda,\mu,\nu$. It is proved in [34, Proposition 3] that each component
$[\widetilde{\pi}]\in\widetilde{\mathcal{P}}(\lambda,\mu,\nu)$ corresponding
to some component of moduli space of $T$-fixed
$\mathbb{C}[x_{1},x_{2},x_{3}]$-submodule of
$\mathbf{M}/\langle(1,1,1)\rangle$ is a product of $\mathbb{P}^{1}$’s.

######  Definition 3.7.

([34, Section 5.3]) The $\mathrm{PT}$ topological vertex is defined by

$\displaystyle
W_{\lambda\mu\nu}=\sum_{[\widetilde{\pi}]\in\widetilde{\mathcal{P}}(\lambda,\mu,\nu)}\chi_{\mathrm{top}}([\widetilde{\pi}])q^{\ell(\widetilde{\pi})+\|\pi_{\mathrm{min}}(\lambda,\mu,\nu)\|}$

where $\chi_{\mathrm{top}}([\widetilde{\pi}])$ is topological Euler
characteristic of $[\widetilde{\pi}]$.

######  Remark 3.8.

The PT topological vertex defined here is slightly different from that in [34,
(5-3)] by replacing the variable $-q$ by $q$, but coincides with the
definition in [14, Section 4.1] due to the equality
$\ell(\widetilde{\pi})+\|\pi_{\mathrm{min}}(\lambda,\mu,\nu)\|=|\widetilde{\pi}|+|\mathrm{III}_{ulb}(\widetilde{\pi})|-|\mathrm{II}(\pi_{\mathrm{min}}(\lambda,\mu,\nu))|-2|\mathrm{III}(\pi_{\mathrm{min}}(\lambda,\mu,\nu))|$.
The symmetry of $W_{\lambda\mu\nu}$ is the same as that of
$V_{\lambda\mu\nu}$:

$\displaystyle
W_{\lambda\mu\nu}=W_{\mu^{\prime}\lambda^{\prime}\nu^{\prime}};\;\;W_{\lambda\mu\nu}=W_{\lambda^{\prime}\nu^{\prime}\mu^{\prime}};\;\;W_{\lambda\mu\nu}=W_{\nu^{\prime}\mu^{\prime}\lambda^{\prime}}.$

The DT/PT equivariant topological vertex correspondence is conjectured by
Pandharipande and Thomas in [34, Conjecture 4] and proved in the following
Calabi-Yau case.

######  Theorem 3.9.

([14, Theorem 1.0.1])

$\displaystyle V_{\lambda\mu\nu}=M(q)W_{\lambda\mu\nu}$

where $M(q)$ is the MacMahon function defined by

$\displaystyle M(q)=\prod_{n\geq 1}\frac{1}{(1-q^{n})^{n}}.$

As in the DT side, we define the following PT $G$-vertex

$\displaystyle
W^{G}_{\lambda\mu\nu}=\sum_{[\widetilde{\pi}]\in\widetilde{\mathcal{P}}(\lambda,\mu,\nu)}\chi_{\mathrm{top}}([\widetilde{\pi}])\prod_{r\in\widehat{G}}q_{r}^{\ell(\widetilde{\pi})_{r}+\|\pi_{\mathrm{min}}(\lambda,\mu,\nu)\|_{r}}$

where

$\displaystyle\ell(\widetilde{\pi})_{r}=\sum_{\begin{subarray}{c}(i,j,k)\in\widetilde{\pi}\\\
r_{1}^{i}r_{2}^{j}r_{3}^{k}=r\end{subarray}}\eta_{\widetilde{\pi}}(i,j,k)$

is the length of boxes $(i,j,k)\in\widetilde{\pi}$ colored by
$r\in\widehat{G}$. Again, when $G=\mathbb{Z}_{n}$, one expects the following
definition of PT $\mathbb{Z}_{n}$-vertex used later, see also [45, Definition
4.27].

######  Definition 3.10.

The $\mathrm{PT}$ $\mathbb{Z}_{n}$-vertex is defined by

$\displaystyle
W^{n}_{\lambda\mu\nu}=\sum_{[\widetilde{\pi}]\in\widetilde{\mathcal{P}}(\lambda,\mu,\nu)}\chi_{\mathrm{top}}([\widetilde{\pi}])q_{0}^{\ell(\widetilde{\pi})_{0}+\|\pi_{\mathrm{min}}(\lambda,\mu,\nu)\|_{0}}\cdots
q_{n-1}^{\ell(\widetilde{\pi})_{n-1}+\|\pi_{\mathrm{min}}(\lambda,\mu,\nu)\|_{n-1}}$

where for $0\leq l\leq n-1$,

$\displaystyle\ell(\widetilde{\pi})_{l}=\sum_{\begin{subarray}{c}(i,j,k)\in\widetilde{\pi}\\\
i-j=l\;\mathrm{mod}\;n\end{subarray}}\eta_{\widetilde{\pi}}(i,j,k)$

is the length of boxes $(i,j,k)\in\widetilde{\pi}$ colored by $i-j=l\mbox{ mod
}n$.

######  Remark 3.11.

As in Remark 3.5, we have the following symmetries for the $\mathrm{PT}$
$\mathbb{Z}_{n}$-vertex

$\displaystyle
W_{\lambda\mu\nu}^{n}(q_{0},q_{1},\cdots,q_{n-1})=W^{n}_{\mu^{\prime}\lambda^{\prime}\nu^{\prime}}(q_{0},q_{n-1},\cdots,q_{1})\;\;\;\mbox{i.e.,}\;\;\;W_{\lambda\mu\nu}^{n}=\overline{W}^{n}_{\mu^{\prime}\lambda^{\prime}\nu^{\prime}}.$

which has fewer symmetries than that of $W_{\lambda\mu\nu}$ in Remark 3.8.

As in the manifold case, we expect the following conjecture of the DT/PT
$\mathbb{Z}_{n}$-orbifold topological vertex correspondence.

######  Conjecture 3.12.

([45, Conjecture 5.20]) If $\nu$ is multi-regular, then

$\displaystyle
V^{n}_{\lambda\mu\nu}=V_{\emptyset\emptyset\emptyset}^{n}W_{\lambda\mu\nu}^{n}$

where

$\displaystyle V_{\emptyset\emptyset\emptyset}^{n}=M(1,q)^{n}\prod_{0<a\leq
b<n}M(q_{a}\cdots q_{b},q)M(q_{a}^{-1}\cdots q_{b}^{-1},q)$

and

$\displaystyle M(v,q)=\prod_{m=1}^{\infty}\frac{1}{(1-vq^{m})^{m}}.$

This conjecture will be proved in Section 6.

###  3.2. $K$-theory decomposition of orbifold PT stable pairs

As mentioned in Introduction, the more explicit proof of $T$-equivaraint
$K$-theory decomposition of the finite cokernel $\mathtt{Q}$ is present in
Proposition 3.15, which is essential for the in Section 3.3 and 3.4. One can
refer to [45, Proposition 4.25] for comparison. As a generalization of the
description of $T$-fixed PT stable pairs in [34, Section 2.6], we have the
following orbifold PT version of [7, Lemma 13].

######  Lemma 3.13.

The $T$-fixed points in

$\displaystyle\coprod_{\beta\in
F_{1}K(\mathcal{X})}\mathrm{PT}(\mathcal{X},\beta)$

are in bijective correspondence with the set
$\\{\widetilde{\pi}_{v},\lambda_{e}\\}$ where $\\{\widetilde{\pi}_{v}\;|\;v\in
V_{\mathcal{X}}\\}$ is a collection of labelled box configurations
$\widetilde{\pi}_{v}$ with outgoing partitions
$(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ and $\\{\lambda_{e}\;|\;e\in
E_{\mathcal{X}}\\}$ is an edge assignment which is compatible with triple
outgoing partitions at all vertices as in Section 2.1.

###### Proof.

It follows from the description of $T$-fixed orbifold PT stable pairs in
Section 2.2. See also the proof of [45, Lemma 4.21]. ∎

######  Remark 3.14.

In Lemma 3.13, $T$-fixed points are not isolated when the moduli space of the
labelling of $\widetilde{\pi}_{v}$ for some $v\in V_{\mathcal{X}}$ has
positive dimension, i.e., the moduli space is a $m$-th product of
$\mathbb{P}^{1}$’s for some integer $m\geq 1$. And every labelled box
configuration for each vertice has the color scheme uniquely determined by the
coordinate positions of their boxes and representations of the local group.

In order to formulate the $K$-theory classes of $T$-fixed orbifold PT stable
pairs, we make the same convention as in DT side [34, Remark 14] to identify a
box in a labelled box configuration, or a 3D partition, or an edge partition
with its corresponding divisor as follows. For a labelled box configuration
$\widetilde{\pi}_{v}$, the notation $\mathcal{D}\in\widetilde{\pi}_{v}$
corresponding to a box $(i,j,k)\in\widetilde{\pi}_{v}$ means that
$\mathcal{D}=i\mathcal{D}_{1,v}+j\mathcal{D}_{2,v}+k\mathcal{D}_{3,v}$. This
convention is the same for a 3D partition $\pi_{v}$. For an edge partition
$\lambda_{e}$, the notation $\mathcal{D}\in\lambda_{e}$ corresponding to a box
$(i,j)\in\lambda_{e}$ means
$\mathcal{D}=i\mathcal{D}_{e}+j\mathcal{D}_{e}^{\prime}$. These divisors are
compatible due to the orientation in Section 2.1. Now we have

######  Proposition 3.15.

Given $\beta\in F_{1}K(\mathcal{X})$, let
$[\mathcal{O}_{\mathcal{X}}\xrightarrow{\varphi}\mathcal{F}]\in\mathrm{PT}(\mathcal{X},\beta)^{\mathrm{T}}$
be a $T$-fixed orbifold PT stable pair corresponding to the set
$\\{\widetilde{\pi}_{v},\lambda_{e}\\}$ of labelled box configurations
$\widetilde{\pi}_{v}$ with outgoing partitions
$(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ and edge partitions
$\lambda_{e}$. Let $\mathtt{Q}=\mathrm{coker}\,\varphi$. Then we have the
following $T$-equivariant $K$-theory decomposition:

$\displaystyle\mathtt{Q}=\sum_{v\in
V_{\mathcal{X}}}\sum_{\mathcal{D}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{D})\mathcal{O}_{p_{v}}(-\mathcal{D})$

and

$\displaystyle\mathcal{F}=\sum_{v\in
V_{\mathcal{X}}}\left(\sum_{\mathcal{D}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{D})\mathcal{O}_{p_{v}}(-\mathcal{D})+\sum_{\mathcal{D}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{D})\mathcal{O}_{p_{v}}(-\mathcal{D})\right)+\sum_{e\in
E_{\mathcal{X}}}\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D})$

where
$\overline{\pi}_{v}:=\pi_{\mathrm{min}}(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$
and the set $\\{\overline{\pi}_{v},\lambda_{e}\\}$ is corresponding to a
$T$-invariant Cohen-Macaulay substack $\mathcal{C}_{\mathcal{F}}$ of dimension
one, or equivalently a torus fixed point in
$\mathrm{Hilb}^{\alpha}(\mathcal{X})$ for some $\alpha\in F_{1}K(\mathcal{X})$
as in [7, Lemma 13].

###### Proof.

If $Y\subset\mathcal{X}$ is a $T$-invariant substack of dimension zero, then
$\mathcal{O}_{Y}$ must be supported on the torus fixed points of
$\mathcal{X}$. By [7, Lemma 13] it corresponds to the set
$\\{\widehat{\pi}_{v}\;|\;v\in V_{\mathcal{X}}\\}$ which is a collection of 3D
partitions $\widehat{\pi}_{v}$ asymptotic to
$(\emptyset,\emptyset,\emptyset)$. By [7, Proposition 4] we have

$\displaystyle\mathcal{O}_{Y}=\sum_{v\in
V_{\mathcal{X}}}\sum_{\mathcal{D}\in\widehat{\pi}_{v}}\mathcal{O}_{p_{v}}(-\mathcal{D}).$

It is shown in the proof of [7, Lemma 13] that for each $v$ the ideal sheaf
$\mathcal{I}_{\widehat{\pi}_{v}}\subset\mathcal{O}_{\mathcal{X}}$
corresponding to a $T$-invariant substack of dimension zero
$Y\subset\mathcal{X}$ is generated by the image of the maps

$\displaystyle\mathcal{O}_{\mathcal{X}}(-i\mathcal{D}_{1,v}-j\mathcal{D}_{2,v}-k\mathcal{D}_{3,v})\to\mathcal{O}_{\mathcal{X}},\;\;\;(i,j,k)\notin\widehat{\pi}_{v}.$

For a $T$-invariant open neighborhood
$\mathcal{U}_{v}:=[\mathbb{C}^{3}/G_{v}]$ of $p_{v}$, a finite $T$-invariant
$0$-dimensional sheaf supported on $[(0,0,0)/G_{v}]$ is equivalent to a
finitely generated $T$-invariant $\mathbb{C}[x_{1},x_{2},x_{3}]$-module as a
$G_{v}$-module. Therefore the $K$-class decomposition
$\sum_{\mathcal{D}\in\widehat{\pi}_{v}}\mathcal{O}_{p_{v}}(-\mathcal{D})$
corresponds to the finitely generated $T$-invariant
$\mathbb{C}[x_{1},x_{2},x_{3}]$-module (as a $G_{v}$-module) whose
$\mathbb{C}$-basis is given by monomials $x_{1}^{i}x_{2}^{j}x_{3}^{k}$ for all
$(i,j,k)$ belonging to the 3D partition $\widehat{\pi}_{v}$. This
correspondence holds for any 3D partition $\widehat{\pi}_{v}$ asymptotic to
$(\emptyset,\emptyset,\emptyset)$. And it has the correspondence between
tensor product of elements $\mathcal{O}_{p_{v}}(-\mathcal{D})$ for
$\mathcal{D}=i\mathcal{D}_{1,v}+j\mathcal{D}_{2,v}+k\mathcal{D}_{3,v}$ in
$K$-theory and multiplication of elements $x_{1}^{i}x_{2}^{j}x_{3}^{k}$ in the
$T$-invariant $\mathbb{C}[x_{1},x_{2},x_{3}]$-module as a $G_{v}$-module. This
implies that tensoring with $\mathcal{O}_{p_{v}}(\pm D_{i,v})$ corresponds to
multiplication by $x_{i}^{\mp 1}$ for $1\leq i\leq 3$.

Since $\mathcal{C}_{\mathcal{F}}$ is a $T$-invariant Cohen-Macaulay substack
of dimension one, it follows from [7, Proposition 4] that

$\displaystyle\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}=\sum_{v\in
V_{\mathcal{X}}}\sum_{\mathcal{D}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{D})\mathcal{O}_{p_{v}}(-\mathcal{D})+\sum_{e\in
E_{\mathcal{X}}}\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D})$

where the 3D partition $\overline{\pi}_{v}$ is equal to
$\pi_{\mathrm{min}}(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ as
$\mathcal{C}_{\mathcal{F}}$ is Cohen-Macaulay and the set
$\\{\overline{\pi}_{v},\lambda_{e}\\}$ is corresponding to
$\mathcal{C}_{\mathcal{F}}$ by [7, Lemma 13]. Here, $\overline{\pi}_{v}$ is
determined by $\widetilde{\pi}_{v}$ via its outgoing partitions. Let
$\mathfrak{m}$ be the ideal sheaf of torus fixed points in
$\mathcal{C}_{\mathcal{F}}$. Then on the local model at $p_{v}$ for each $v$,
$\mathfrak{m}\subset\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$ is generated by
the image of the maps

$\displaystyle\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}(-i\mathcal{D}_{1,v}-j\mathcal{D}_{2,v}-k\mathcal{D}_{3,v})\to\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$

where $(i,j,k)\in\overline{\pi}_{v}$ and $i+j+k\geq 1$. Hence
$\mathfrak{m}^{r}\subset\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$ is generated
by the image of the maps

$\displaystyle\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}(-i\mathcal{D}_{1,v}-j\mathcal{D}_{2,v}-k\mathcal{D}_{3,v})\to\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$

where $(i,j,k)\in\overline{\pi}_{v}$ and $i+j+k\geq r$ on the local model at
$p_{v}$ for each $v$. We define
$\mathcal{I}_{r}\subset\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$ to be the
ideal such that for each $v$ it is locally generated by the image of the maps

$\displaystyle\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}(-i\mathcal{D}_{1,v}-j\mathcal{D}_{2,v}-k\mathcal{D}_{3,v})\to\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$

where $(i,j,k)\in\overline{\pi}_{v}\setminus\\{(i,j,k)\,|\,0\leq i,j,k<r\\}$.
Now it is easy to see that

$\displaystyle\lim\limits_{\longrightarrow}\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})=\lim\limits_{\longrightarrow}\mathcal{H}om(\mathcal{I}_{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}).$

As in the direct sum decomposition of
$\lim\limits_{\longrightarrow}\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})$
as modules on the local model [34, Section 2.4], we have the following
decomposition in $K$-theory

$\displaystyle\lim\limits_{\longrightarrow}\mathcal{H}om(\mathcal{I}_{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})=\lim\limits_{\longrightarrow}\sum_{e\in
E_{\mathcal{X}}}\mathcal{H}om(\mathcal{I}_{r}|_{\mathcal{C}_{\lambda_{e}}},\mathcal{O}_{\mathcal{C}_{\lambda_{e}}})$

where $\mathcal{C}_{\lambda_{e}}$ is $T$-invariant substack of
$\mathcal{C}_{\mathcal{F}}$ supported only on $\mathcal{C}_{e}$ with
$\mathcal{O}_{\mathcal{C}_{\lambda_{e}}}=\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D})$
by [7, Proposition 4]. Notice that
$\\{\lambda_{1,v},\lambda_{2,v},\lambda_{3,v}\\}$ and $\\{\lambda_{e}\\}$ are
compatible. Since for $r\gg 0$

$\displaystyle\mathcal{H}om(\mathcal{I}_{r}|_{\mathcal{C}_{\lambda_{e}}},\mathcal{O}_{\mathcal{C}_{\lambda_{e}}})\otimes\mathcal{O}_{\mathcal{X}}(-r\mathcal{D}_{0,e}-r\mathcal{D}_{\infty,e})\cong\mathcal{H}om(\mathcal{I}_{r}|_{\mathcal{C}_{\lambda_{e}}}\otimes\mathcal{O}_{\mathcal{X}}(r\mathcal{D}_{0,e}+r\mathcal{D}_{\infty,e}),\mathcal{O}_{\mathcal{C}_{\lambda_{e}}})\cong\mathcal{O}_{\mathcal{C}_{\lambda_{e}}},$

we have
$\mathcal{H}om(\mathcal{I}_{r}|_{\mathcal{C}_{\lambda_{e}}},\mathcal{O}_{\mathcal{C}_{\lambda_{e}}})=\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D}+r\mathcal{D}_{0,e}+r\mathcal{D}_{\infty,e})$.
Notice that for each $\mathcal{D}\in\lambda_{e}$

$\displaystyle\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D}+r\mathcal{D}_{0,e}+r\mathcal{D}_{\infty,e})=\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D})+\sum_{k=1}^{r}\mathcal{O}_{p_{0,e}}(-\mathcal{D}+k\mathcal{D}_{0,e})+\sum_{k=1}^{r}\mathcal{O}_{p_{\infty,e}}(-\mathcal{D}+k\mathcal{D}_{\infty,e}).$

Now we have

$\displaystyle\sum_{e\in
E_{\mathcal{X}}}\mathcal{H}om(\mathcal{I}_{r}|_{\mathcal{C}_{\lambda_{e}}},\mathcal{O}_{\mathcal{C}_{\lambda_{e}}})=\sum_{v\in
V_{\mathcal{X}}}\sum_{i=1}^{3}\sum_{\mathcal{D}\in\lambda_{e_{i,v}}}\sum_{k=1}^{r}\mathcal{O}_{p_{v}}(-\mathcal{D}+k\mathcal{D}_{i,v})+\sum_{e\in
E_{\mathcal{X}}}\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D}).$

Then

$\displaystyle\lim\limits_{\longrightarrow}\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})/\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$
$\displaystyle=$
$\displaystyle\lim\limits_{\longrightarrow}\mathcal{H}om(\mathcal{I}_{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})/\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$
$\displaystyle=$ $\displaystyle\lim\limits_{\longrightarrow}\left(\sum_{v\in
V_{\mathcal{X}}}\left(\sum_{i=1}^{3}\sum_{\mathcal{D}\in\lambda_{e_{i,v}}}\sum_{k=1}^{r}\mathcal{O}_{p_{v}}(-\mathcal{D}+k\mathcal{D}_{i,v})+\sum_{\mathcal{D}\in\overline{\pi}_{v}}\left(-\xi_{\overline{\pi}_{v}}(\mathcal{D})\right)\mathcal{O}_{p_{v}}(-\mathcal{D})\right)\right).$

As before, for each $v$, one representative of the equivalence class
$[x_{1}^{i}x_{2}^{j}x_{3}^{k}]$ in the $\mathbb{C}$-basis described in [34,
Page 1848] of the $T$-module $\mathbf{M}/\langle(1,1,1)\rangle$ (as a
$G_{v}$-module) corresponds to a unique element
$\mathcal{O}_{p_{v}}(-i\mathcal{D}_{1,v}-j\mathcal{D}_{2,v}-k\mathcal{D}_{3,v})$
in the $v$-term part of $K$-class decomposition of
$\lim\limits_{\longrightarrow}\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})/\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$
above. Here, the $\mathbb{C}$-dimension of $[x_{1}^{i}x_{2}^{j}x_{3}^{k}]$ is
equal to the coefficient of
$\mathcal{O}_{p_{v}}(-i\mathcal{D}_{1,v}-j\mathcal{D}_{2,v}-k\mathcal{D}_{3,v})$.

As the $0$-dimensional sheaf $\mathtt{Q}$ is a coherent subsheaf of
$\lim\limits_{\longrightarrow}\mathcal{H}om(\mathfrak{m}^{r},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})/\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}$,
then the $K$-theory decomposition of $\mathtt{Q}$ follows from the description
of $\mathtt{Q}$ locally as a finitely generated $T$-invariant
$\mathbb{C}[x_{1},x_{2},x_{3}]$-submodule of
$\mathbf{M}/\langle(1,1,1)\rangle$ (as a $G_{v}$-module) corresponding to a
labelled box configuration $\widetilde{\pi}_{v}$ with outgoing partitions
$(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ as shown in [34, Section 2.5].

Now the proof is completed by the following short exact sequence

$\displaystyle
0\to\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}\to\mathcal{F}\to\mathtt{Q}\to 0.$

∎

######  Remark 3.16.

By the definition of $\widetilde{\pi}_{v}$ and $\overline{\pi}_{v}$, the
$K$-theory decomposition of $\mathcal{F}$ can be rewritten as

$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle\sum_{v\in
V_{\mathcal{X}}}\sum_{\mathcal{D}\in\breve{\pi}_{v}}\zeta_{\breve{\pi}_{v}}(\mathcal{D})\mathcal{O}_{p_{v}}(-\mathcal{D})+\sum_{e\in
E_{\mathcal{X}}}\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D})$

where

$\displaystyle\zeta_{\breve{\pi}_{v}}(i,j,k)=\left\\{\begin{aligned}
&1,\;\;\;\;\;\;\;\mbox{if}\;\;(i,j,k)\in\mathrm{I}^{-}(\widetilde{\pi}_{v});\\\
&-1,\;\;\;\mbox{if}\;\;(i,j,k)\in(\mathrm{II}(\overline{\pi}_{v})\setminus\mathrm{II}(\widetilde{\pi}_{v}))\cup\mathrm{III}_{lb}(\widetilde{\pi}_{v});\\\
&-2,\;\;\;\mbox{if}\;\;(i,j,k)\in\mathrm{III}(\overline{\pi}_{v})\setminus\mathrm{III}(\widetilde{\pi}_{v})\end{aligned}\right.$

with

$\displaystyle\breve{\pi}_{v}:=\mathrm{I}^{-}(\widetilde{\pi}_{v})\cup(\mathrm{II}(\overline{\pi}_{v})\setminus\mathrm{II}(\widetilde{\pi}_{v}))\cup\mathrm{III}_{lb}(\widetilde{\pi}_{v})\cup(\mathrm{III}(\overline{\pi}_{v})\setminus\mathrm{III}(\widetilde{\pi}_{v}))$
$\displaystyle\mathrm{II}(\overline{\pi}_{v})=\mathrm{II}(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v}),\;\;\mathrm{III}(\overline{\pi}_{v})=\mathrm{III}(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v}).$

###  3.3. The PT partition function

As mentioned in Introduction, based on the perfect obstruction theory and
virtual fundamental classes developed in [18, Section 5] for orbifold PT
theory, we define the PT partition function via the virtual localization
following [34] as follows.

Let $\mathcal{X}$ be a toric CY 3-orbifold. The DT invariant of $\mathcal{X}$
is defined in [7, Section 5.1] by Behrend’s weighted Euler characteristic [3]
of the moduli space $\mathrm{Hilb}^{\alpha}(\mathcal{X})$ for $\alpha\in
F_{1}K(\mathcal{X})$, which is equal to a signed count over the $T$-fixed
points by [4, Theorem 3.4 and Corollary 3.5]. This definition is also employed
for PT invariants of $\mathcal{X}$ in [45, Section 4]. However, it is not
suitable to take the formula in [4, Theorem 3.4 and Corollary 3.5] directly
for the definition of PT invariant of $\mathcal{X}$ since the $T$-fixed points
of the moduli space $\mathrm{PT}(\mathcal{X},\beta)$ for $\beta\in
F_{1}K(\mathcal{X})$ are not necessary isolated as stated in Remark 3.14. More
generally, as the integration over the virtual fundamental class
$[\mathrm{PT}(\mathcal{X},\beta)]^{vir}$, PT invariants of $\mathcal{X}$ can
be defined by $T$-equivariant residue, which resolves [45, Conjecture 4.23,
4.16, 4.19].

The virtual fundamental class of $\mathrm{PT}(\mathcal{X},\beta)$ has been
defined for a $3$-dimensional smooth projective Deligne-Mumford stack
$\mathcal{X}$ in [18, Theorem 5.21]. However, a toric CY 3-orbifold
$\mathcal{X}$ here is quasi-projective, we will employ the strategy as in [8,
27] by restricting to the $T$-fixed locus of $\mathrm{PT}(\mathcal{X},\beta)$
each of whose connected component is a product of $\mathbb{P}^{1}$’s, and
considering the $T$-equivariant perfect obstruction theories on $T$-fixed
locus and the associated virtual fundamental classes in $T$-equivariant Chow
ring, and then applying the virtual localization formula [12] to define
$T$-equivariant integral for PT invariant of $\mathcal{X}$. Actually, we will
follow the argument in [34, Section 4.2 and Section 5.2].

Let $\mathsf{Q}$ be a connected component of $T$-fixed locus of
$\mathrm{PT}(\mathcal{X},\beta)$. By Lemma 3.13, we have
$\mathsf{Q}=\prod_{v\in V_{\mathcal{X}}}\mathsf{Q}_{v}$ where $\mathsf{Q}_{v}$
is a connected component of moduli space of labelled box configurations
containing $\widetilde{\pi}_{v}$ for each $v\in V_{\mathcal{X}}$. Each
$\mathsf{Q}_{v}$ can be viewed as a connected component
$[\widetilde{\pi}_{v}]$ of moduli space of labellings of
$\underline{\widetilde{\pi}_{v}}$ due to [34, Proposition 3]. Since
$\mathsf{Q}_{v}$ is a product of $\mathbb{P}^{1}$’s, then $\mathsf{Q}$ is also
a product of $\mathbb{P}^{1}$’s which is obviously projective and nonsingular.
By the similar argument in [34, Section 4.1 and 4.2], we have the following
formula in localized $T$-equvariant Chow ring of
$\mathrm{PT}(\mathcal{X},\beta)$:

(4)
$\sum_{\mathsf{Q}\subset\mathrm{PT}(\mathcal{X},\beta)^{T}}\varsigma_{*}\left(e(T_{\mathsf{Q}})\cdot\frac{e(E_{\mathsf{Q}}^{1})}{e(E_{\mathsf{Q}}^{0})}\cap[\mathsf{Q}]\right)=[\mathrm{PT}(\mathcal{X},\beta)]^{\mathrm{vir}},$

where $\varsigma:\mathsf{Q}\hookrightarrow\mathrm{PT}(\mathcal{X},\beta)^{T}$
is the inclusion map and $E_{\mathsf{Q}}^{0}\to E_{\mathsf{Q}}^{1}$ is the
dual of $T$-equivariant perfect obstruction theory restricted on $\mathsf{Q}$.
One may take integration over the left hand side of the equation (4) to define
PT invariant of $\mathcal{X}$ as in [8, 27] although it seems difficult to
follow [34, Section 4] to compute $T$-equivariant class
$\frac{e(E_{\mathsf{Q}}^{1})}{e(E_{\mathsf{Q}}^{0})}$. Alternatively, we will
adopt another virtual localization with a different action, which is easy to
deal with $T$-equivariant Euler classes in our CY case.

Let $T_{0}=\\{(t_{1},t_{2},t_{3})\in T\;|\;t_{1}t_{2}t_{3}=1\\}$. Then $T_{0}$
preserves the canonical form of $\mathcal{X}$ since it acts trivially on the
form $dz_{1}\wedge dz_{2}\wedge dz_{3}$ on the local model
$[\mathbb{C}^{3}/G]$ of each $T$-fixed point due to [7, Lemma 46]. Let
$\mathsf{Q}_{0}$ be a connected component of $T_{0}$-fixed locus of
$\mathrm{PT}(\mathcal{X},\beta)$. The moduli space of $T_{0}$-invariant
submodules of $\mathbf{M}/\langle(1,1,1)\rangle$ as in Section 2.2 is
conjectured to be nonsigular [34, Conjecture 2], and fortunately it is proved
recently in [14]. Then $\mathsf{Q}_{0}$ is also nonsingular. And as above we
have

(5)
$\sum_{\mathsf{Q}_{0}\subset\mathrm{PT}(\mathcal{X},\beta)^{T_{0}}}\varsigma_{0*}\left(e(T_{\mathsf{Q}_{0}})\cdot\frac{e(E_{\mathsf{Q}_{0}}^{1})}{e(E_{\mathsf{Q}_{0}}^{0})}\cap[\mathsf{Q}_{0}]\right)=[\mathrm{PT}(\mathcal{X},\beta)]^{\mathrm{vir}}$

in $A^{T_{0}}_{*}(\mathrm{PT}(\mathcal{X},\beta))_{\mathrm{loc}}$, where
$\varsigma_{0}:\mathsf{Q}_{0}\hookrightarrow\mathrm{PT}(\mathcal{X},\beta)^{T_{0}}$
is the inclusion map and $E_{\mathsf{Q}_{0}}^{0}\to E_{\mathsf{Q}_{0}}^{1}$ is
the dual of $T_{0}$-equivariant symmetric perfect obstruction theory
$E^{-1}\to E^{0}$ (see [18, Section 5.3]) restricted on $\mathsf{Q}_{0}$. By
the self-duality of perfect obstruction theory, we define the following PT
invariant of $\mathcal{X}$ by integration against the left hand side classes
of the formula $\eqref{vir-loc-2}$:

$\displaystyle PT_{\beta}(\mathcal{X})$ $\displaystyle=$
$\displaystyle\sum_{\mathsf{Q}_{0}\subset\mathrm{PT}(\mathcal{X},\beta)^{T_{0}}}\int_{\mathsf{Q}_{0}}e(T_{\mathsf{Q}_{0}})(-1)^{\mathrm{rk}\left(\mathrm{Ext}^{1}(\mathbb{I}_{\mathsf{Q}_{0}}^{\bullet},\mathbb{I}_{\mathsf{Q}_{0}}^{\bullet})_{0}\right)}$
$\displaystyle=$
$\displaystyle\sum_{\mathsf{Q}_{0}\subset\mathrm{PT}(\mathcal{X},\beta)^{T_{0}}}\chi_{\mathrm{top}}(\mathsf{Q}_{0})(-1)^{\mathrm{rk}\left(\mathrm{Ext}^{1}(\mathbb{I}_{\mathsf{Q}_{0}}^{\bullet},\mathbb{I}_{\mathsf{Q}_{0}}^{\bullet})_{0}\right)}$

where $\chi_{\mathrm{top}}(\mathsf{Q}_{0})$ is the topological Euler
chracteristic of $\mathsf{Q}_{0}$, and
$\mathbb{I}^{\bullet}_{\mathsf{Q}_{0}}=\\{\mathcal{O}_{\mathcal{X}\times\mathsf{Q}_{0}}\to\mathbb{F}\\}\in\mathrm{D}^{b}(\mathcal{X}\times\mathsf{Q}_{0})$
is the universal complex with $\mathbb{F}$ flat over $\mathsf{Q}_{0}$. Here,
we have identified each orbifold PT stable pair
$\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$ in $\mathsf{Q}_{0}$ with the complex
$\\{\mathcal{O}_{\mathcal{X}}\to\mathcal{F}\\}$ in
$\mathrm{D}^{b}(\mathcal{X})$ as in [34, Proposition 1.21]. As in [34, Section
5.2], we have the $\mathbb{C}^{*}=T/T_{0}$ action on every $T_{0}$-fixed
component $\mathsf{Q}_{0}$, and every element of $\mathsf{Q}$ is the
$\mathbb{C}^{*}$-fixed locus of a unique component
$\mathsf{Q}_{0}\subset\mathrm{PT}(\mathcal{X},\beta)^{T_{0}}$ while any
$\mathbb{C}^{*}$-fixed lous of $\mathsf{Q}_{0}$ is an element of some
$\mathsf{Q}\subset\mathrm{PT}(\mathcal{X},\beta)^{T}$. Then we have

$\displaystyle\chi_{\mathrm{top}}(\mathsf{Q}_{0})=\sum_{\begin{subarray}{c}\mathsf{Q}\subset\mathsf{Q}_{0}\\\
\mathsf{Q}\subset\mathrm{PT}(\mathcal{X},\beta)^{T}\end{subarray}}\chi_{\mathrm{top}}(\mathsf{Q})$

and

$\displaystyle
PT_{\beta}(\mathcal{X})=\sum_{\mathsf{Q}\subset\mathrm{PT}(\mathcal{X},\beta)^{T}}\chi_{\mathrm{top}}(\mathsf{Q})(-1)^{\mathrm{rk}\left(\mathrm{Ext}^{1}(\mathbb{I}_{\mathsf{Q}}^{\bullet},\mathbb{I}_{\mathsf{Q}}^{\bullet})_{0}\right)}$

where
$\mathbb{I}^{\bullet}_{\mathsf{Q}}=\\{\mathcal{O}_{\mathcal{X}\times\mathsf{Q}}\to\mathbb{F}\\}\in\mathrm{D}^{b}(\mathcal{X}\times\mathsf{Q})$.

######  Definition 3.17.

The PT partition function of $\mathcal{X}$ is defined by

$\displaystyle PT(\mathcal{X})=\sum_{\beta\in
F_{1}K(\mathcal{X})}PT_{\beta}(\mathcal{X})q^{\beta}$

and the multi-regular PT partition function is defined by

$\displaystyle PT_{mr}(\mathcal{X})=\sum_{\beta\in
F_{mr}K(\mathcal{X})}PT_{\beta}(\mathcal{X})q^{\beta}$

where $q^{\beta}$ for $\beta\in F_{1}K(\mathcal{X})$ is defined in Section
2.3.

Assume that
$[\mathbf{I}^{\bullet}]\in\mathsf{Q}\subset\mathrm{PT}(\mathcal{X},\beta)^{T}$
where the complex
$\mathbf{I}^{\bullet}=\\{\mathcal{O}_{\mathcal{X}}\to\mathcal{F}\\}$ in
$\mathrm{D}^{b}(\mathcal{X})$ concentrated in degree $0$ and 1 corresponds to
the $T$-invariant orbifold PT stable pair
$\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$, which is the restriction of the
$T$-invariant universal complex $\mathbb{I}^{\bullet}_{\mathsf{Q}}$ on
$[\mathbf{I}^{\bullet}]$. Let $\\{\widetilde{\pi}_{v},\lambda_{e}\\}$ be the
set of labelled box configurations and edge partitions corresponding to
$\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$ (or $\mathbf{I}^{\bullet}$) by Lemma
3.13. Since $\mathsf{Q}=\prod_{v\in V_{\mathcal{X}}}\mathsf{Q}_{v}$ and
$\mathsf{Q}_{v}$ corresponds to a component $[\widetilde{\pi}_{v}]$ of moduli
space of labellings of $\underline{\widetilde{\pi}_{v}}$ for each $v\in
V_{\mathcal{X}}$. Then

$\displaystyle\chi(\mathsf{Q})=\prod_{v\in
V_{\mathcal{X}}}\chi_{\mathrm{top}}(\mathsf{Q}_{v})=\prod_{v\in
V_{\mathcal{X}}}\chi_{\mathrm{top}}([\widetilde{\pi}_{v}]).$

Let $\widetilde{\mathcal{P}}(\beta)$ be the set of all collections
$\\{[\widetilde{\pi}_{v}]\\}$ corresponding to all components
$\mathsf{Q}\subset\mathrm{PT}(\mathcal{X},\beta)^{T}$. Then we have

(6) $\displaystyle PT_{\beta}(\mathcal{X})$ $\displaystyle=$
$\displaystyle\sum_{\mathsf{Q}=\prod\limits_{v\in
V_{\mathcal{X}}}\mathsf{Q}_{v}\subset\mathrm{PT}(\mathcal{X},\beta)^{T}}(-1)^{\dim\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}}\prod_{v\in
V_{\mathcal{X}}}\chi_{\mathrm{top}}(\mathsf{Q}_{v})$ $\displaystyle=$
$\displaystyle\sum_{\\{[\widetilde{\pi}_{v}]\\}\in\widetilde{\mathcal{P}}(\beta)}(-1)^{\dim\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}}\prod_{v\in
V_{\mathcal{X}}}\chi_{\mathrm{top}}([\widetilde{\pi}_{v}]).$

Next, we will compute the pairity of
$\dim\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}$ to
obtain the explicit formula of $PT(\mathcal{X})$. We first employ some
notation from the DT side in [7, Section 6] as follows. Define

$\displaystyle s_{1}(W)=\mathrm{vdim}\,W\mbox{ mod }2,\;\;\;\mbox{ for a
virtual $T$-representation W};$ $\displaystyle s_{2}(M)=M\mbox{ mod
}2,\;\;\;\mbox{ for $M\in\mathbb{Z}$}.$

For a $T$-representation $W$, the shifted dual of $W$ is defined by
$W^{*}=W^{\vee}\otimes\mathbb{C}[-\mu]$ where $W^{\vee}$ is the dual
representation of $W$ and $\mu\in\mathrm{Hom}(T,\mathbb{C}^{*})$ is the
primitive weight defined by
$\omega_{\mathcal{X}}\cong\mathcal{O}_{\mathcal{X}}\otimes_{\mathbb{C}}\mathbb{C}[\mu]$
in [7, Lemma 18]. Here $\mathbb{C}[\mu]$ is a one-dimensional
$T$-representation with weight $\mu$. And one can define
$\sigma(W-W^{*})=s_{1}(W)$ for a virtual $T$-representation $W$ by [7,
Proposition 6] where $\sigma$ is defined on anti-self shifted dual virtual
representations. For
$\mathcal{H}_{1},\mathcal{H}_{2}\in\mathrm{D}^{b}(\mathcal{X})$, we define
$\widetilde{\chi}(\mathcal{H}_{1},\mathcal{H}_{2})=\sum_{i}(-1)^{i}\mathrm{Ext}^{i}(\mathcal{H}_{1},\mathcal{H}_{2})$
in $K$-theory, and
$\chi(\mathcal{H}_{1},\mathcal{H}_{2})=\sum_{i}(-1)^{i}\dim\mathrm{Ext}^{i}(\mathcal{H}_{1},\mathcal{H}_{2})$
with the special case
$\chi(\mathcal{H}_{1})=\chi(\mathcal{O}_{\mathcal{X}},\mathcal{H}_{1})$.
Define $\lambda_{-1}W=\sum_{i}(-1)^{i}\Lambda^{i}W$ for a vector bundle $V$.
Now it follows from the argument in [7, Section 6] that
$s_{2}(\dim\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0})=s_{1}(\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0})$
can be computed as follows, see also the proof in [45, Section 4.7].

######  Theorem 3.18.

Let $\varphi:\mathcal{O}_{\mathcal{X}}\to\mathcal{F}$ be a $T$-invariant
orbifold PT stable pair and let $\\{\widetilde{\pi}_{v},\lambda_{e}\\}$ be the
corresponding set of labelled box configurations and edge partitions. Let
$\overline{\pi}_{v}:=\pi_{\mathrm{min}}(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$
be the 3D partitions determined by outgoing partitions
$(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ of the labelled box
configuration $\widetilde{\pi}_{v}$ for each $v\in V_{\mathcal{X}}$. Let
$\mathbf{I}^{\bullet}=\\{\mathcal{O}_{\mathcal{X}}\to\mathcal{F}\\}\in\mathrm{D}^{b}(\mathcal{X})$
be the complex concentrated in degree $0$ and 1. Then we have

$\displaystyle
s_{1}(\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0})=s_{2}(\chi(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}))+\sum_{v\in
V_{\mathcal{X}}}s_{2}(\mathbf{V}(v))+\sum_{e\in
E(\mathcal{X})}s_{2}(\mathbf{E}(e))$

where

$\displaystyle\mathbf{V}(v)$ $\displaystyle=$
$\displaystyle\ell(\widetilde{\pi}_{v})+\sum_{\mathcal{A}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})h^{0}\left(\mathcal{O}_{p_{v}}(-\mathcal{A})\right)$
$\displaystyle+\sum_{\mathcal{A},\mathcal{B}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})\eta_{\widetilde{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)$
$\displaystyle+\sum_{\mathcal{A}\in\widetilde{\pi}_{v}}\sum_{\mathcal{B}\in\overline{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})\xi_{\overline{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)$
$\displaystyle+\sum_{\mathcal{A}\in\overline{\pi}_{v}}\sum_{\mathcal{B}\in\widetilde{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{A})\eta_{\widetilde{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)$
$\displaystyle+\sum_{i=1}^{3}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{B})h^{0}\left(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B})\otimes\lambda_{-1}(N_{\mathcal{C}_{e_{i,v}}/\mathcal{X}})\right)$
$\displaystyle+\|\overline{\pi}_{v}\|+\sum_{\mathcal{A},\mathcal{B}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{A})\xi_{\overline{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)$
$\displaystyle+\sum_{i=1}^{3}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{B})h^{0}\left(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B})\otimes\lambda_{-1}N_{\mathcal{C}_{e_{i,v}}/\mathcal{X}}\right)$
$\displaystyle+\sum_{i\neq
j}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\lambda_{j,v}}h^{0}(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{j,v}))$

and

$\displaystyle\mathbf{E}(e)=\sum_{\mathcal{A},\mathcal{B}\in\lambda_{e}}\chi(\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{A}-\mathcal{B})+\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{e})).$

###### Proof.

By [18, Lemma 5.16], we have the following equality

$\displaystyle\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}-\mathrm{Ext}^{2}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}=\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathcal{O}_{\mathcal{X}})-\widetilde{\chi}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet}).$

As there exist an exact triangle in $\mathrm{D}^{b}(\mathcal{X})$:

$\displaystyle\mathcal{F}[-1]\to\mathbf{I}^{\bullet}\to\mathcal{O}_{\mathcal{X}}\to\mathcal{F},$

and a short exact sequence

$\displaystyle
0\to\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}\to\mathcal{F}\to\mathtt{Q}\to 0,$

then by [7, Proposition 6-(1)], we have

$\displaystyle\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}-\left(\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}\right)^{*}$
$\displaystyle=$
$\displaystyle\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathcal{O}_{\mathcal{X}})-\widetilde{\chi}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})$
$\displaystyle=$
$\displaystyle\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})-\left(\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})\right)^{*}+\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathtt{Q})-\left(\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathtt{Q})\right)^{*}$
$\displaystyle-\left(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathtt{Q})-\left(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathtt{Q})\right)^{*}\right)-\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})-\widetilde{\chi}(\mathtt{Q},\mathtt{Q})$

and

$\displaystyle
s_{1}(\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0})$
$\displaystyle=$ $\displaystyle
s_{1}(\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}))+s_{1}(\widetilde{\chi}(\mathcal{O}_{\mathcal{X}},\mathtt{Q}))+s_{1}(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathtt{Q}))+\sigma(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}))+\sigma(\widetilde{\chi}(\mathtt{Q},\mathtt{Q}))$
$\displaystyle=$ $\displaystyle
s_{2}(\chi(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}))+s_{2}(\chi(\mathtt{Q}))+s_{1}(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathtt{Q}))+\sigma(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}))+\sigma(\widetilde{\chi}(\mathtt{Q},\mathtt{Q})).$

As in the proof of Proposition 3.15, we have

$\displaystyle\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}=\sum_{v\in
V_{\mathcal{X}}}\sum_{\mathcal{D}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{D})\mathcal{O}_{p_{v}}(-\mathcal{D})+\sum_{e\in
E_{\mathcal{X}}}\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D}).$

It is proved in [7, Section 6.2] that

$\displaystyle\sigma(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}))$
$\displaystyle=$ $\displaystyle\sum_{e\in
E_{\mathcal{X}}}s_{2}\bigg{(}\sum_{\mathcal{A},\mathcal{B}\in\lambda_{e}}\chi(\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{A}-\mathcal{B})+\mathcal{O}_{\mathcal{C}_{e}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{e}))\bigg{)}$
$\displaystyle+\sum_{v\in
V_{\mathcal{X}}}s_{2}\Bigg{(}\|\overline{\pi}_{v}\|+\sum_{\mathcal{A},\mathcal{B}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{A})\xi_{\overline{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)$
$\displaystyle+\sum_{i=1}^{3}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{B})h^{0}\left(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B})\otimes\lambda_{-1}N_{\mathcal{C}_{e_{i,v}}/\mathcal{X}}\right)$
$\displaystyle+\sum_{i\neq
j}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\lambda_{j,v}}h^{0}(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{j,v}))\Bigg{)}.$

By Proposition 3.15 and using the similar argument as in [7, Section 6.2], we
have

$\displaystyle\chi(\mathtt{Q})$ $\displaystyle=$ $\displaystyle\sum_{v\in
V_{\mathcal{X}}}\sum_{\mathcal{A}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})h^{0}\left(\mathcal{O}_{p_{v}}(-\mathcal{A})\right),$
$\displaystyle\sigma\left(\widetilde{\chi}(\mathtt{Q},\mathtt{Q})\right)$
$\displaystyle=$ $\displaystyle\sum_{v\in
V_{\mathcal{X}}}s_{2}\left(\ell(\widetilde{\pi}_{v})+\sum_{\mathcal{A},\mathcal{B}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})\eta_{\widetilde{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)\right)$

and

$\displaystyle
s_{1}(\widetilde{\chi}(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}},\mathtt{Q}))$
$\displaystyle=$ $\displaystyle\sum_{v\in
V_{\mathcal{X}}}s_{2}\left(\sum_{\mathcal{A}\in\widetilde{\pi}_{v}}\sum_{\mathcal{B}\in\overline{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})\xi_{\overline{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)\right)$
$\displaystyle+\sum_{v\in
V_{\mathcal{X}}}s_{2}\left(\sum_{\mathcal{A}\in\overline{\pi}_{v}}\sum_{\mathcal{B}\in\widetilde{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{A})\eta_{\widetilde{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)\right)$
$\displaystyle+\sum_{v\in
V_{\mathcal{X}}}s_{2}\left(\sum_{i=1}^{3}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{B})h^{0}\left(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B})\otimes\lambda_{-1}(N_{\mathcal{C}_{e_{i,v}}/\mathcal{X}})\right)\right).$

Now the proof is completed by combining all the above results. ∎

######  Remark 3.19.

The pairity of
$\dim\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}$ in
Theorem 3.18 have the contribution from the edge part $\mathbf{E}(e)$ which is
the same as $\mathbf{SE}_{\lambda(e)}(e)$ in DT case [7, Theorem 21], while
its vertex part $\mathbf{V}(v)$ is more complicated involving labelled box
configurations $\widetilde{\pi}_{v}$ due to the first 6 terms. The sum of the
last 4 terms in $\mathbf{V}(v)$ is exactly the vertex part of DT side
$\mathbf{SV}_{\pi(v)}(v)$ with $\pi(v)=\overline{\pi}_{v}$. The formula for
$\mathbf{V}(v)$ is arranged for comparing with the DT case and noticing the
computation needed only for the additional terms later. Actually, as in Remark
3.16, the formula $\mathbf{V}(v)$ can be rewritten as

$\displaystyle\mathbf{V}(v)$ $\displaystyle=$
$\displaystyle\ell(\widetilde{\pi}_{v})+\|\overline{\pi}_{v}\|+\sum_{\mathcal{A}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})h^{0}\left(\mathcal{O}_{p_{v}}(-\mathcal{A})\right)$
$\displaystyle+\sum_{\mathcal{A},\mathcal{B}\in\breve{\pi}_{v}}\zeta_{\breve{\pi}_{v}}(\mathcal{A})\zeta_{\breve{\pi}_{v}}(\mathcal{B})h^{0}\left(\sum_{i=1}^{3}\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{i,v})\right)$
$\displaystyle+\sum_{i=1}^{3}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\breve{\pi}_{v}}\zeta_{\breve{\pi}_{v}}(\mathcal{B})h^{0}\left(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B})\otimes\lambda_{-1}(N_{\mathcal{C}_{e_{i,v}}/\mathcal{X}})\right)$
$\displaystyle+\sum_{i\neq
j}\sum_{\mathcal{A}\in\lambda_{i,v}}\sum_{\mathcal{B}\in\lambda_{j,v}}h^{0}(\mathcal{O}_{p_{v}}(\mathcal{A}-\mathcal{B}+\mathcal{D}_{j,v})),$

where
$\ell(\widetilde{\pi}_{v})+\|\overline{\pi}_{v}\|=\sum_{\mathcal{A}\in\breve{\pi}_{v}}\zeta_{\breve{\pi}_{v}}(\mathcal{A})$.
See also [45, Page 94].

######  Remark 3.20.

When $\mathcal{X}$ is a scheme $X$, then $\mathcal{C}_{e}\cong\mathbb{P}^{1}$
for all $e\in E_{\mathcal{X}}$. By Remark 3.19 and [7, Example 22], we have

$\displaystyle s_{2}(\mathbf{E}(e))=m_{e}|\lambda_{e}|\;\mathrm{mod}\;2$

and

$\displaystyle s_{2}(\mathbf{V}(v))$ $\displaystyle=$ $\displaystyle
s_{2}\left(\ell(\widetilde{\pi}_{v})+\sum_{\mathcal{D}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{D})+3\sum_{\mathcal{A},\mathcal{B}\in\widetilde{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})\eta_{\widetilde{\pi}_{v}}(\mathcal{B})+6\sum_{\mathcal{A}\in\widetilde{\pi}_{v}}\sum_{\mathcal{B}\in\overline{\pi}_{v}}\eta_{\widetilde{\pi}_{v}}(\mathcal{A})\xi_{\overline{\pi}_{v}}(\mathcal{B})\right)$
$\displaystyle=$ $\displaystyle
s_{2}\left(2\ell(\widetilde{\pi}_{v})+3\ell(\widetilde{\pi}_{v})^{2}\right)$
$\displaystyle=$ $\displaystyle\ell(\widetilde{\pi}_{v})\;\mathrm{mod}\;2.$

Then

$\displaystyle
s_{1}(\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0})=s_{2}\left(\chi(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})+\sum_{v\in
V_{\mathcal{X}}}\ell(\widetilde{\pi}_{v})+\sum_{e\in
E_{\mathcal{X}}}m_{e}|\lambda_{e}|\right)$

exactly coincides with the sign in [34, Theorem/Conjecture 2] proved in [14]
due to

$\displaystyle\chi(\mathcal{F})=\chi(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})+\chi(\mathtt{Q})=\chi(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})+\sum_{v\in
V_{\mathcal{X}}}\ell(\widetilde{\pi}_{v}).$

Assume that the local model of a torus fixed point $p_{v}$ is of the form
$[\mathbb{C}^{3}/G_{v}]$ with a finite abelian group $G_{v}$. Let
$r_{i}\in\widehat{G_{v}}$ be the character obtained from the action of $G_{v}$
on $\mathcal{O}_{p_{v}}(-D_{i})$ for $1\leq i\leq 3$, different from the
convention in [7, Example 23]. Let $0\in\widehat{G_{v}}$ be the trivial
character where we have identified the group $\widehat{G_{v}}$ of characters
of $G_{v}\subset(\mathbb{C}^{*})^{3}$ with the corresponding additive subgroup
of $\mathbb{Z}^{3}$. Then we have

$\displaystyle s_{2}(\mathbf{V}(v))$ $\displaystyle=$ $\displaystyle
s_{2}(\ell(\widetilde{\pi}_{v})_{0}+\ell(\widetilde{\pi}_{v})+\|\overline{\pi}_{v}\|)+\sum_{i\neq
j}\sum_{r\in\widehat{G_{v}}}s_{2}(|\lambda_{i,v}|_{r}\cdot|\lambda_{j,v}|_{r+r_{j}})$
$\displaystyle+\sum_{r\in\widehat{G_{v}}}s_{2}\bigg{(}(\ell(\widetilde{\pi}_{v})_{r}+\|\overline{\pi}_{v}\|_{r})\sum_{i=1}^{3}(\ell(\widetilde{\pi}_{v})_{r+r_{i}}+\|\overline{\pi}_{v}\|_{r+r_{i}})\bigg{)}$
$\displaystyle+\sum_{i=1}^{3}\sum_{r\in\widehat{G_{v}}}s_{2}\bigg{(}|\lambda_{i,v}|_{r}\bigg{(}\ell(\widetilde{\pi}_{v})_{r}+\|\overline{\pi}_{v}\|_{r}+\sum_{j\in\\{1,2,3\\}\setminus\\{i\\}}(\ell(\widetilde{\pi}_{v})_{r+r_{j}}+\|\overline{\pi}_{v}\|_{r+r_{j}})$
$\displaystyle+\ell(\widetilde{\pi}_{v})_{r-r_{i}+\sum_{j=1}^{3}r_{j}}+\|\overline{\pi}_{v}\|_{r-r_{i}+\sum_{j=1}^{3}r_{j}}\bigg{)}\bigg{)}$

where $|\lambda_{i,v}|_{r}$ is the number of elements in 2D partition
$\lambda_{i,v}\subset\mathbb{Z}^{2}$ colored by $r\in\widehat{G_{v}}$, and
$\ell(\widetilde{\pi}_{v})_{r}$, $\|\overline{\pi}_{v}\|_{r}$ are defined as
in Section 3.1.

As in the DT case [7, Remark 24], one can obtain the general formula for the
PT partition function $PT(\mathcal{X})$ in Definition 3.17 by combining Lemma
3.13, Proposition 3.15, and Theorem 3.18 with the formula (6). The explicit
formula is presented in the following transverse $A_{n-1}$ case.

###  3.4. The PT topological vertex formalism

Let $\mathcal{X}$ be a toric CY 3-orbifold with transverse $A_{n-1}$
singularities. We present the explicit formula for PT partition function of
$\mathcal{X}$ for full 3-leg case as follows. We first recall some notations
in [7, Section 3.3] that are used later. For a 2D ordinary partition
$\lambda$, define

$\displaystyle\lambda[k,n]:=\\{(i,j)\in\lambda\;|\;i-j\equiv k\mbox{ mod
}n\\}\;\;\;\mbox{and}\;\;\;|\lambda|_{k,n}:=|\lambda[k,n]|$

where we also write $|\lambda|_{k,n}$ as $|\lambda|_{k}$ for simplicity. We
call a partition $\eta$ multi-regular if $|\eta|_{k}=\frac{|\eta|}{n}$ for all
$k\in\\{0,1,\cdots,n-1\\}$. With some abuse of notation, we use the notation
$|\cdot|_{k}$ not only for edge partitions $\lambda_{e}$ with the above
definition, but also for outgoing partitions
$(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ of labelled box configurations
$\widetilde{\pi}_{v}$ with the slightly different convention: we define the
color of $(j,k)\in\lambda_{1,v}$, $(k,i)\in\lambda_{2,v}$, and
$(i,j)\in\lambda_{3,v}$ by $-j\mbox{ mod }n$, $i\mbox{ mod }n$, and $i-j\mbox{
mod }n$ respectively due to transverse $A_{n-1}$ singularities and hence
$|\lambda_{i,v}|_{l}$ is defined to be the number of boxes in $\lambda_{i,v}$
colored by $l\mbox{ mod }n$ for $1\leq i\leq 3$ where $n=n_{e_{3,v}}$. Define

$\displaystyle\varXi_{\widetilde{\pi}_{v}}=\sum_{k=0}^{n_{e_{3,v}}-1}|\lambda_{3,v}|_{k}(|\lambda_{1,v}|_{k}+|\lambda_{2,v}|_{k}+|\lambda_{1,v}|_{k+1}+|\lambda_{2,v}|_{k-1}).$

Let

$\displaystyle
C_{m,m^{\prime}}^{\lambda}:=\sum_{(i,j)\in\lambda}\left(-mi-m^{\prime}j+1\right),$
$\displaystyle
C_{m,m^{\prime}}^{\lambda}[k,n]:=\sum_{(i,j)\in\lambda[k,n]}\left(-mi-m^{\prime}j+1\right),$
$\displaystyle
A_{\lambda}(k,n):=\sum_{(i,j)\in\lambda}\left\lfloor\frac{i+k}{n}\right\rfloor,$

and

$\displaystyle
q_{e}^{C_{m_{e},m_{e}^{\prime}}^{\lambda_{e}}}=:\prod_{k=0}^{n_{e}-1}q_{e,k}^{C_{m_{e},m_{e}^{\prime}}^{\lambda_{e}}[k,n_{e}]},\;\;\;\;\;q_{e}^{A_{\lambda}}:=\prod_{k=0}^{n_{e}-1}q_{e,k}^{A_{\lambda}(k,n_{e})}.$

We define for each edge $e\in E_{\mathcal{X}}$

$\displaystyle\mathbf{q}_{e}=(q_{e,0},q_{e,1},\cdots,q_{e,n_{e}-1}),\;\;\;\overline{\mathbf{q}_{e}}=(q_{e,0},q_{e,n_{e}-1},\cdots,q_{e,1})$

and

$\displaystyle\mathbf{S}_{\lambda_{e}}^{e}=\sum_{k=0}^{n_{e}-1}\bigg{(}C_{m_{e},m^{\prime}_{e}}^{\lambda_{e}}[k,n_{e}](|\lambda_{e}|_{k-1}-|\lambda_{e}|_{k+1})+|\lambda_{e}|_{k}\big{(}1+(1+m_{e}+\delta_{0,e}+\delta_{\infty,e})|\lambda_{e}|_{k-1}\big{)}\bigg{)}.$

Let

$\displaystyle\mathbf{q}_{v}=\left\\{\begin{aligned}
&\mathbf{q}_{e_{3,v}},\;\;\;\mbox{if $e_{3,v}$ is oriented outward from
$v$};\\\ &\overline{\mathbf{q}_{e_{3,v}}},\;\;\;\mbox{if $e_{3,v}$ is oriented
inward toward $v$}.\end{aligned}\right.$

and $(-1)^{\widetilde{s}_{k}(\lambda_{3,v})}\mathbf{q}_{v}$ be the vector
$\mathbf{q}_{v}$ with its each component $q_{e,k}$ multiplied by the sign
$(-1)^{\widetilde{s}_{k}(\lambda_{3,v})}$ where
$\widetilde{s}_{k}(\lambda_{3,v})=|\lambda_{3,v}|_{k-1}+|\lambda_{3,v}|_{k+1}$
and $e=e_{3,v}$.

With these notation, we have

######  Theorem 3.21.

Let $\mathcal{X}$ be a toric CY 3-orbifold with transverse
$A_{n-1}$-singularities. Then the pairity of
$\dim\mathrm{Ext}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0}$ in Theorem
3.18 is

$\displaystyle
s_{1}(\mathrm{Ext}^{1}(\mathbf{I}^{\bullet},\mathbf{I}^{\bullet})_{0})=s_{2}(\chi(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}))+\sum_{v\in
V_{\mathcal{X}}}s_{2}(\mathbf{V}(v))+\sum_{e\in
E_{\mathcal{X}}}s_{2}(\mathbf{E}(e))$

where

$\displaystyle\mathbf{E}(e)=\left\\{\begin{aligned}
&|\lambda_{e}|(m_{e}+\delta_{0}+\delta_{\infty}),\;\;\;\mbox{if}\;\;n_{e}=1;\\\
&\sum_{k=0}^{n-1}C_{m_{e},m^{\prime}_{e}}^{\lambda_{e}}[k,n](|\lambda_{e}|_{k-1}-|\lambda_{e}|_{k+1})+|\lambda_{e}|_{k}(1+(1+m_{e})|\lambda_{e}|_{k-1}),\;\;\mbox{if}\;\;n_{e}>1.\end{aligned}\right.$

and

$\displaystyle\mathbf{V}(v)$ $\displaystyle=$
$\displaystyle\ell(\widetilde{\pi}_{v})_{0}+\sum_{k=0}^{n_{e_{3,v}}-1}(\ell(\widetilde{\pi}_{v})_{k}+\|\overline{\pi}_{v}\|_{k})(|\lambda_{3,v}|_{k-1}+|\lambda_{3,v}|_{k+1})$
$\displaystyle+\sum_{k=0}^{n_{e_{{3,v}}}-1}|\lambda_{3,v}|_{k}(|\lambda_{1,v}|_{k}+|\lambda_{2,v}|_{k}+|\lambda_{1,v}|_{k+1}+|\lambda_{2,v}|_{k-1}).$

###### Proof.

It follows from Theorem 3.18 and the similar argument in the proof of [7,
Theorem 25]. See also the proof of [45, Proposition 4.29]. ∎

Let $\Gamma_{\mathcal{X}}$ be the web diagram of $\mathcal{X}$. Given an edge
assignment $\\{\lambda_{e}\;|\;e\in
E_{\mathcal{X}}\\}\in\Lambda_{\mathcal{X}}$, there should be many collections
of labelled box configurations where each collection
$\\{\widetilde{\pi}_{v}\;|\;v\in V_{\mathcal{X}}\\}$ of labelled box
configurations $\widetilde{\pi}_{v}$ with outgoing partitions
$(\lambda_{1,v},\lambda_{2,v},\lambda_{3,v})$ determined by
$\\{\lambda_{e}\\}$ via the compatibility in Section 2.1. And each set
$\\{\widetilde{\pi}_{v},\lambda_{e}\\}$ corresponds to one $T$-fixed point in
$\coprod_{\beta\in F_{1}K(\mathcal{X})}\mathrm{PT}(\mathcal{X},\beta)$ by
Lemma 3.13. Therefore, for a given edge assignment
$\\{\lambda_{e}\\}\in\Lambda_{\mathcal{X}}$, there are corresponding several
connected components of $T$-fixed locus of $\coprod\limits_{\beta\in
F_{1}K(\mathcal{X})}\mathrm{PT}(\mathcal{X},\beta)$ where each component is of
the form $\prod\limits_{v\in V_{\mathcal{X}}}[\widetilde{\pi}_{v}]$ satisfying
some compatible conditions.

$e$$v_{\infty}$$v_{0}$$\mathcal{D}_{e}$$\mathcal{D}_{e}^{\prime}$$f^{\prime}_{v_{0}}$$f_{v_{0}}$$\mathcal{D}_{0,e}$$g^{\prime}_{v_{\infty}}$$g_{v_{\infty}}$$\mathcal{D}_{\infty,e}$
Figure 1.  Orientations of the edge $e$ and its adjacent edges.

Now we have the following PT version of [7, Theorem 10], see also [45,
Theorem/Conjecture 4.19].

######  Theorem 3.22.

Let $\mathcal{X}$ be a toric CY 3-orbifold with transverse $A_{n-1}$
singularities. Define

$\displaystyle\underline{PT}(\mathcal{X}):=\sum_{\\{\lambda_{e}\\}\in\Lambda_{\mathcal{X}}}\prod_{e\in
E_{\mathcal{X}}}E_{\lambda_{e}}^{e}\prod_{v\in
V_{\mathcal{X}}}(-1)^{\varXi_{\pi_{v}}}W_{\lambda_{1,v}\lambda_{2,v}\lambda_{3,v}}^{n_{e_{3,v}}}((-1)^{\widetilde{s}(\lambda_{3,v})}\mathbf{q}_{v})$

where

$\displaystyle
E_{\lambda_{e}}^{e}=(-1)^{\mathbf{S}_{\lambda_{e}}^{e}}\cdot\left(\prod_{k=0}^{n_{e}-1}v_{e,k}^{|\lambda_{e}|_{k,n_{e}}}\right)\cdot
q_{e}^{C_{m_{e},m_{e}^{\prime}}^{\lambda_{e}}}\cdot\left(\overline{q_{f_{v_{0}}}^{A_{\lambda_{e}}}}\right)^{\delta_{0,e}}\cdot\left(q_{f^{\prime}_{v_{0}}}^{A_{\lambda^{\prime}_{e}}}\right)^{\delta_{0,e}^{\prime}}\cdot\left(q_{g_{v_{\infty}}}^{A_{\lambda_{e}}}\right)^{\delta_{\infty,e}}\cdot\left(\overline{q_{g^{\prime}_{v_{\infty}}}^{A_{\lambda^{\prime}_{e}}}}\right)^{\delta_{\infty,e}^{\prime}}$

with edges
$(e,f_{v_{0}},f^{\prime}_{v_{0}},g_{v_{\infty}},g^{\prime}_{v_{\infty}})$ are
the oriented edges presented as in Figure 1, and the overline denotes the
exchange of variables $q_{k}\leftrightarrow q_{-k}$. Then we have

$\displaystyle
PT(\mathcal{X})=\underline{PT}(\mathcal{X})\bigg{|}_{q_{e,0}\to-
q_{e,0},\;q\to-q}$

that is, $PT(\mathcal{X})$ is obtained from $\underline{PT}(\mathcal{X})$ by
replacing varibles $q_{e,0}$ by $-q_{e,0}$ for all $e\in E_{\mathcal{X}}$ and
hence replacing $q$ by $-q$.

###### Proof.

Since

$\displaystyle\mathcal{O}_{\mathcal{C}_{\mathcal{F}}}=\sum_{v\in
V_{\mathcal{X}}}\sum_{\mathcal{D}\in\overline{\pi}_{v}}\xi_{\overline{\pi}_{v}}(\mathcal{D})\mathcal{O}_{p_{v}}(-\mathcal{D})+\sum_{e\in
E_{\mathcal{X}}}\sum_{\mathcal{D}\in\lambda_{e}}\mathcal{O}_{\mathcal{C}_{e}}(-\mathcal{D})$

then

$\displaystyle\chi(\mathcal{O}_{\mathcal{C}_{\mathcal{F}}})$ $\displaystyle=$
$\displaystyle\sum_{v\in V_{\mathcal{X}}}\|\overline{\pi}_{v}\|_{0}+\sum_{e\in
E_{\mathcal{X}}}C_{m_{e},m_{e}^{\prime}}^{\lambda_{e}}[0,n_{e}]+\sum_{e\in
E_{\mathcal{X}}}\bigg{(}\delta_{0,e}\cdot A_{\lambda_{e}}(0,n_{f_{v_{0}}})$
$\displaystyle+\delta_{0,e}^{\prime}\cdot
A_{\lambda_{e}^{\prime}}(0,n_{f^{\prime}_{v_{0}}})+\delta_{\infty,e}\cdot
A_{\lambda_{e}}(0,n_{g_{v_{\infty}}})+\delta^{\prime}_{\infty,e}\cdot
A_{\lambda^{\prime}_{e}}(0,n_{g^{\prime}_{v_{\infty}}})\bigg{)}.$

The proof is completed by using Definitions 3.10, 3.17, the formula (6),
Proposition 3.15 for $\beta=\mathcal{F}$, Theorem 3.21, and [7, Lemma 15 and
Proposition 5] via the similar argument in [7, Section 6.3]. ∎

######  Remark 3.23.

The terms involving $f_{v_{0}}$ and $g^{\prime}_{v_{\infty}}$ in the
expression of $E_{\lambda_{e}}^{e}$ are added by the overline since the choice
of orientations of $f_{v_{0}}$ and $g^{\prime}_{v_{\infty}}$ opposite to those
of $f$ and $g^{\prime}$ in [7] exchanges generators
$[\mathcal{O}_{p_{f_{v_{0}}}}\otimes\rho_{k}]\leftrightarrow[\mathcal{O}_{p_{f_{v_{0}}}}\otimes\rho_{-k}]$,
$[\mathcal{O}_{p_{g^{\prime}_{v_{\infty}}}}\otimes\rho_{k}]\leftrightarrow[\mathcal{O}_{p_{g^{\prime}_{v_{\infty}}}}\otimes\rho_{-k}]$
in [7, Lemma 17 and Proposition 5].

##  4\. Orbifold DT theory and the graphical condensation recurrence

In this section, we show the dimer model for DT theory in [14, Section 3.2]
can be extended to our orbifold case in Section 4.1 and generalize the
graphical condensation recurrence for DT theory in [14, Section 3.3] in
Section 4.2, where we also obtain two more recurrences for orbifold DT theory
and exhibit the equivalence between them by the symmetry of DT
$\mathbb{Z}_{n}$-vertex. We compute the weights appearing in these three
recurrences in Section 4.3, which generalize the ones in [14, Section 5.2].

###  4.1. The dimer model for orbifold DT theory

In order to describe the orbifold DT topological vertex as some dimer model,
we recall some relevant notation and definitions in [14, Section 2 and 3] as
follows. Notice that any ordinary 2D partition $\eta$ in this section has its
parts indexed starting from 1 other than 0 in Section 3.

######  Definition 4.1.

([14, Definition 2.0.1 and 2.0.2]) For any partition
$\eta=(\eta_{1},\eta_{2},\cdots,\eta_{l})$, define the Maya diagram of $\eta$
by the set

$\displaystyle S(\eta)=\left\\{\eta_{i}-i+\frac{1}{2}\;\bigg{|}\;i\geq
1\right\\}\subseteq\mathbb{Z}+\frac{1}{2}.$

For a subset $S\subseteq\mathbb{Z}+\frac{1}{2}$, define

$\displaystyle S^{+}=\left\\{t\in
S\;\bigg{|}\;t>0\right\\};\;\;\;\;\;S^{-}=\left\\{t\in\mathbb{Z}+\frac{1}{2}\setminus
S\;\bigg{|}\;t<0\right\\}.$

If $|S^{+}|,|S^{-}|<+\infty$, define the charge of $S$ as

$\displaystyle c(S)=|S^{+}|-|S^{-}|.$

Then the set $\\{t-c(S)\;|\;t\in S\\}$ is the Maya diagram of some partition
$\widehat{\eta}$, i.e.,

$\displaystyle\\{t-c(S)\;|\;t\in S\\}=S(\widehat{\eta})$

and we call $S$ the charge $c(S)$ Maya diagram of $\widehat{\eta}$.

######  Definition 4.2.

([14, Definition 2.0.3]) For any partition $\eta\neq\emptyset$ with the Maya
diagram $S(\eta)$, the partitions $\eta^{r}$, $\eta^{c}$ and $\eta^{rc}$ are
defined by the following equations respectively

$\displaystyle S(\eta^{r})=\\{t+1\;|\;t\in S(\eta)\setminus\\{\min
S(\eta)^{+}\\}\\},$ $\displaystyle S(\eta^{c})=\\{t-1\;|\;t\in
S(\eta)\cup\\{\max S(\eta)^{-}\\}\\},$ $\displaystyle
S(\eta^{rc})=\\{t\;|\;t\in(S(\eta)\setminus\\{\min S(\eta)^{+}\\})\cup\\{\max
S(\eta)^{-}\\}\\}.$

That is, the set $S(\eta)\setminus\\{\min S(\eta)^{+}\\}$ is the charge $-1$
Maya diagram of $\eta^{r}$, the set $S(\eta)\cup\\{\max S(\eta)^{-}\\}$ is the
charge $1$ Maya diagram of $\eta^{c}$, and the set $(S(\eta)\setminus\\{\min
S(\eta)^{+}\\})\cup\\{\max S(\eta)^{-}\\}$ is the (charge $0$) Maya diagram of
$\eta^{rc}$.

As in [14], let $\mathbf{H}(N)$ be the $N\times N\times N$ honeycomb graph
divided into three sectors with some choice of labellings of vertices on the
outer face, which is presented in different ways for the DT and PT case
respectively, see Figure 2 for the graph $\mathbf{H}(4)$. With the convention
in [14], the labelled vertice $v$ in sector $i$ is said to be in sector
$i^{+}$ or $i^{-}$ if the labelled number is positive or negative.

$\frac{1}{2}$$\bullet$$\frac{3}{2}$$\bullet$$\cdots\frac{5}{2}$$\bullet$$\bullet$$-\frac{1}{2}$$\bullet$$-\frac{3}{2}$$\bullet$$-\frac{5}{2}\cdots$$\bullet$$\bullet$3$-\frac{1}{2}$$\bullet$$-\frac{3}{2}$$\bullet$$-\frac{5}{2}$$\bullet$$\bullet$$\vdots$$\frac{1}{2}$$\bullet$$\frac{3}{2}$$\bullet$$\frac{5}{2}$$\bullet$$\bullet$$\cdots$1$\frac{1}{2}$$\bullet$$\frac{3}{2}$$\bullet$$\frac{5}{2}$$\bullet$$\bullet$$\vdots$$-\frac{1}{2}$$\bullet$$-\frac{3}{2}$$\bullet$$-\frac{5}{2}$$\bullet$$\bullet$$\cdots$2$\bullet$$\bullet$$\frac{5}{2}$$\cdots$$\bullet$$\frac{3}{2}$$\bullet$$\frac{1}{2}$$\bullet$$\cdots$$\bullet$$-\frac{5}{2}$$\bullet$$-\frac{3}{2}$$\bullet$$-\frac{1}{2}$3$\bullet$$\vdots$$\bullet$$-\frac{5}{2}$$\bullet$$-\frac{3}{2}$$\bullet$$-\frac{1}{2}$$\bullet$21$\bullet$$\bullet$$\frac{5}{2}$$\vdots$$\bullet$$\frac{3}{2}$$\bullet$$\frac{1}{2}$$\bullet$$\bullet$$\bullet$$\bullet$$-\frac{1}{2}$$\frac{1}{2}$$\frac{3}{2}$$-\frac{3}{2}$$\bullet$$\bullet$$\bullet$$\cdots\frac{5}{2}$$-\frac{5}{2}\cdots$
Figure 2. The left graph $\mathbf{H}(4)$ is divided into 3 sectors with the
labellings of outer vertices for DT case. The right graph is again
$\mathbf{H}(4)$ but divided into another 3 sectors with the different
labellings of outer vertices for PT case.

As is shown in [16, 14], a 3D partition (or a plane partition) visualized as
some collection of boxes can be viewed from a suitable direction as certain
rhombus tiling within a hexagonal region consisting of triangles, denoted by
faces of a finite planar graph. And its dual graph is exactly the beautiful
honeycomb graph, which is the dimer configuration (or the so-called perfect
matching) of this dual graph, see [16, Fig. 17] or [14, Figure 3] for
examples. Labelled box configurations are related to AB configurations which
also have this visualized description in Section 5.2. In order to describe the
orbifold DT or PT topological vertex formula via some weighted configurations
on the modified honeycomb graph as in [14], we assign the weights of edges of
the graph $\mathbf{H}(N)$ as follows.

######  Definition 4.3.

Given $n\in\mathbb{Z}_{>0}$, the honeycomb graph $\mathbf{H}(N)$ is weighted
by the following rule:
$(i)$ all the non-horizontal edges are weighted by 1;
$(ii)$ the $2N$ horizontal edges in the central column are weighted from
bottom to top by

$\displaystyle q_{0}^{0},q_{0}^{1},q_{0}^{2},\cdots,q_{0}^{2N-1};$

$(iii)$ for $1\leq i\leq N-1$, the $(2N-i)$ horizontal edges in the $i$-th
column on the right hand side of the central column (corresponding to $0$-th
column) are weighted from bottom to top by

$\displaystyle q_{-i}^{0},q_{-i}^{1},q_{-i}^{2},\cdots,q_{-i}^{2N-i-1};$

$(iv)$ for $1\leq i\leq N-1$, the $(2N-i)$ horizontal edges in the $i$-th
column on the left hand side of the central column (corresponding to $0$-th
column) are weighted from bottom to top by

$\displaystyle q_{i}^{i},q_{i}^{i+1},q_{i}^{i+2},\cdots,q_{i}^{2N-1},$

where for any $k\in\mathbb{Z}$, the equality $q_{k}=q_{k^{\prime}}$ is used
implicitly through this paper if $k^{\prime}\in\\{0,1,\cdots,n-1\\}$ and
$k^{\prime}\equiv k\mbox{ mod }n$.

The weighted rule in Definition 4.3 is shown in Figure 3 for $\mathbf{H}(4)$
as an example.

$q^{0}_{-3}$$q^{1}_{-3}$$q^{2}_{-3}$$q^{3}_{-3}$$q^{4}_{-3}$$q^{0}_{-2}$$q^{1}_{-2}$$q^{2}_{-2}$$q^{3}_{-2}$$q^{4}_{-2}$$q^{5}_{-2}$$q^{0}_{-1}$$q^{1}_{-1}$$q^{2}_{-1}$$q^{3}_{-1}$$q^{4}_{-1}$$q^{5}_{-1}$$q^{6}_{-1}$$q^{0}_{0}$$q^{1}_{0}$$q^{2}_{0}$$q^{3}_{0}$$q^{4}_{0}$$q^{5}_{0}$$q^{6}_{0}$$q^{7}_{0}$$q^{1}_{1}$$q^{2}_{1}$$q^{3}_{1}$$q^{4}_{1}$$q^{5}_{1}$$q^{6}_{1}$$q^{7}_{1}$$q^{2}_{2}$$q^{3}_{2}$$q^{4}_{2}$$q^{5}_{2}$$q^{6}_{2}$$q^{7}_{2}$$q^{3}_{3}$$q^{4}_{3}$$q^{5}_{3}$$q^{6}_{3}$$q^{7}_{3}$
Figure 3. The graph $\mathbf{H}(4)$ with weights as assigned in Definition
4.3.

######  Remark 4.4.

The weighted rule is defined such that if one add a box $(i,j,k)$ to a 3D
partition asymptotic to $(\lambda,\mu,\nu)$, the weight of the corresponding
dimer configuration should be multiplied by a factor of $q_{i-j}$, and
removing a box $(i,j,k)$ will multiply the weight by a factor $q_{i-j}^{-1}$.
If all $q_{l}$ are equal to $q$, the weighted rule here generalize the one in
[16, Section 6] (see also [14, Definition 2.0.4]).

Let $(\lambda,\mu,\nu)$ be a triple of partitions with their Maya diagrams

$\displaystyle S_{1}:=S(\lambda),\;\;\;\;S_{2}:=S(\mu),\;\;\;\;S_{3}:=S(\nu).$

The modified honeycomb graph $\mathbf{H}(N;\lambda,\mu,\nu)$ is defined by
removing vertices labelled by elements in $S_{i}^{+}\cup S_{i}^{-}$ from the
$i$-th sector of $\mathbf{H}(N)$. If we view any 3D partition asymptotic to
$(\lambda,\mu,\nu)$ as a collection of boxes, it is shown in [14, Section 3.2]
that there is a bijective correspondence between all these box collections and
all the dimer configurations on (or perfect matchings of)
$\mathbf{H}(N;\lambda,\mu,\nu)$. One may refer to [14, Figure 3] as an example
for this correspondence.

Let $\mathbf{Z}^{D}(\mathbf{G})$ be the sum of the weights of all dimer
configurations on the graph $\mathbf{G}$. There is a unique minimal dimer
configuration on $\mathbf{H}(N;\lambda,\mu,\nu)$ corresponding to the 3D
partition $\pi_{\min}(\lambda,\mu,\nu)$ defined in Section 3. By the weighted
rule, the dimer configuration being minimal here means that if we set all
$q_{l}$ to be equal to $q$, the weight of this dimer configuration with
respect to $q$ is minimal as in [14], or that this dimer configuration has
minimal weight with respect to each $q_{l}$. Denote this minimal dimer
configuration by $\mathbf{D}_{\min}(\lambda,\mu,\nu;N)$ and its weight by
$q^{\omega_{\min}(\lambda,\mu,\nu;N)}$, which is a polynomial of variables
$q_{1-N},q_{2-N},\cdots,q_{0},\cdots,q_{N-2},q_{N-1}$. For any subset
$\mathbf{S}\subseteq\mathbb{Z}^{3}$, we define

$\displaystyle|\mathbf{S}|_{l}=\\{(i,j,k)\in\mathbf{S}\;|\;i-j\equiv l\mbox{
mod }n\\}.$

Then we have

$\displaystyle\lim_{N\to\infty}\left(q^{\omega_{\min}(\lambda,\mu,\nu;N)}\right)^{-1}\cdot\mathbf{Z}^{D}(\mathbf{H}(N;\lambda,\mu,\nu))=V^{n}_{\lambda\mu\nu}\cdot\prod_{l=0}^{n-1}q_{l}^{|\mathrm{II}(\lambda,\mu,\nu)|_{l}}\cdot
q_{l}^{2|\mathrm{III}(\lambda,\mu,\nu)|_{l}}.$

This implies the following orbifold version of [14, Theorem 3.2.1].

######  Theorem 4.5.

We have the following limit in the sense of formal Laurent series

$\displaystyle\lim_{N\to\infty}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu,\nu))=V^{n}_{\lambda\mu\nu}(q_{0},q_{1},\cdots,q_{n-1})$

where

$\displaystyle\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu,\nu)):=\left(q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\right)^{-1}\cdot\mathbf{Z}^{D}(\mathbf{H}(N;\lambda,\mu,\nu))$

and

$\displaystyle
q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}:=q^{\omega_{\min}(\lambda,\mu,\nu;N)}\cdot\prod\limits_{l=0}^{n-1}q_{l}^{|\mathrm{II}(\lambda,\mu,\nu)|_{l}}\cdot
q_{l}^{2|\mathrm{III}(\lambda,\mu,\nu)|_{l}}.$

###  4.2. The graphical condensation recurrence for orbifold DT theory

The graphical condensation is discovered by Kuo in [16], and is successfully
applied in [14] to derive the recurrence for DT topological vertex. In this
section, we will obtain graphical condensation recurrences for DT
$\mathbb{Z}_{n}$-vertex with the weighted rule in Definition 4.3 by the
following technique.

######  Theorem 4.6.

([16, Theorem 5.1]) Let
$\mathbf{G}=(\mathbf{V}_{1},\mathbf{V}_{2},\mathbf{E})$ be a given edge-
weighted plane bipartite graph with $|\mathbf{V}_{1}|=|\mathbf{V}_{2}|$. Let
vertices $a,b,c$ and $d$ appear on a face of $\mathbf{G}$ in a cyclic order.
If $a,c\in\mathbf{V}_{1}$ and $b,d\in\mathbf{V}_{2}$, then

$\displaystyle\mathbf{Z}^{D}(\mathbf{G})\mathbf{Z}^{D}(\mathbf{G}-\\{a,b,c,d\\})=\mathbf{Z}^{D}(\mathbf{G}-\\{a,b\\})\mathbf{Z}^{D}(\mathbf{G}-\\{c,d\\})+\mathbf{Z}^{D}(\mathbf{G}-\\{a,d\\})\mathbf{Z}^{D}(\mathbf{G}-\\{b,c\\}).$

As in [14], we choose $\mathbf{G}$ to be
$\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)$ and let

$\displaystyle a:=\max S_{1}^{-},\;\;\;b:=\min S_{1}^{+},\;\;\;c=\max
S_{2}^{-},\;\;\;d=\min S_{2}^{+}.$

Then we have

$\displaystyle\mathbf{G}-\\{a,b,c,d\\}=\mathbf{H}(N;\lambda,\mu,\nu),$
$\displaystyle\mathbf{G}-\\{a,b\\}=\mathbf{H}(N;\lambda,\mu^{rc},\nu),$
$\displaystyle\mathbf{G}-\\{c,d\\}=\mathbf{H}(N;\lambda^{rc},\mu,\nu).$

It is an important observation [14] that the graph $\mathbf{G}-\\{a,d\\}$ is
derived from the charge $-1$ Maya diagram of $\lambda^{r}$, the charge 1 Maya
diagram of $\mu^{c}$, and (the charge 0) Maya diagram of $\nu$, and that 3D
partitions asymptotic to $(\lambda^{r},\mu^{c},\nu)$ are corresponding to
dimer configurations on $\mathbf{G}-\\{a,d\\}$ but with the orgin in
$\mathbb{Z}^{3}$ shifted to the face directly above the central face of
$\mathbf{H}(N)$. While the graph $\mathbf{G}-\\{b,c\\}$ is obtained from the
charge 1 Maya diagram of $\lambda^{c}$, the charge $-1$ Maya diagram of
$\mu^{r}$, and Maya diagram of $\nu$, together with 3D partitions asymptotic
to $(\lambda^{c},\mu^{r},\nu)$ corresponding to dimer configurations on
$\mathbf{G}-\\{b,c\\}$ but with the orgin in $\mathbb{Z}^{3}$ shifted to the
face directly below the central face of $\mathbf{H}(N)$. One can refer to [14,
Figure 4 and 5] for example. Let
$\mathbf{D}_{\min}^{\uparrow}(\lambda^{r},\mu^{c},\nu;N)$ and
$\mathbf{D}_{\min}^{\downarrow}(\lambda^{c},\mu^{r},\nu;N)$ be the minimal
dimer configurations on $\mathbf{G}-\\{a,d\\}$ and $\mathbf{G}-\\{b,c\\}$ with
their weights denoted by $q^{\omega_{\min}^{U}(\lambda^{r},\mu^{c},\nu;N)}$
and $q^{\omega_{\min}^{D}(\lambda^{c},\mu^{r},\nu;N)}$ respectively. Let

$\displaystyle
q^{\widetilde{\omega}_{\min}^{U}(\lambda^{r},\mu^{c},\nu;N)}=q^{\omega_{\min}^{U}(\lambda^{r},\mu^{c},\nu;N)}\cdot\prod\limits_{l=0}^{n-1}q_{l}^{|\mathrm{II}(\lambda^{r},\mu^{c},\nu)|_{l}}\cdot
q_{l}^{2|\mathrm{III}(\lambda^{r},\mu^{c},\nu)|_{l}},$ $\displaystyle
q^{\widetilde{\omega}_{\min}^{D}}(\lambda^{c},\mu^{r},\nu;N)=q^{\omega_{\min}^{D}(\lambda^{c},\mu^{r},\nu;N)}\cdot\prod\limits_{l=0}^{n-1}q_{l}^{|\mathrm{II}(\lambda^{c},\mu^{r},\nu)|_{l}}\cdot
q_{l}^{2|\mathrm{III}(\lambda^{c},\mu^{r},\nu)|_{l}}.$

Set

$\displaystyle\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{a,d\\}):=\left(q^{\widetilde{\omega}_{\min}^{U}(\lambda^{r},\mu^{c},\nu;N)}\right)^{-1}\cdot\mathbf{Z}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{a,d\\}),$
$\displaystyle\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{b,c\\}):=\left(q^{\widetilde{\omega}_{\min}^{D}(\lambda^{c},\mu^{r},\nu;N)}\right)^{-1}\cdot\mathbf{Z}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{b,c\\}).$

As in Section 4.1, we have

$\displaystyle\lim_{N\to\infty}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{a,d\\})=V^{n}_{\lambda^{r}\mu^{c}\nu},$
$\displaystyle\lim_{N\to\infty}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{b,c\\})=V^{n}_{\lambda^{c}\mu^{r}\nu}.$

Now it follows from Theorem 4.6 that

$\displaystyle q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu^{rc},\nu;N)}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu,\nu))\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu))$
$\displaystyle=$ $\displaystyle
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda,\mu^{rc},\nu;N)}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu))\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu^{rc},\nu))$
$\displaystyle+$ $\displaystyle
q^{\widetilde{\omega}_{\min}^{U}(\lambda^{r},\mu^{c},\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}^{D}(\lambda^{c},\mu^{r},\nu;N)}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{a,d\\})\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{b,c\\}).$

Then

$\displaystyle\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu,\nu))\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu))$
$\displaystyle=$
$\displaystyle\frac{q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda,\mu^{rc},\nu;N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu^{rc},\nu;N)}}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu))\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu^{rc},\nu))$
$\displaystyle+$
$\displaystyle\frac{q^{\widetilde{\omega}_{\min}^{U}(\lambda^{r},\mu^{c},\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}^{D}(\lambda^{c},\mu^{r},\nu;N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu^{rc},\nu;N)}}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{a,d\\})\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)-\\{b,c\\}).$

It is proved in Section 4.3.1 that

$\displaystyle\frac{q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda,\mu^{rc},\nu;N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu^{rc},\nu;N)}}=1\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\frac{q^{\widetilde{\omega}_{\min}^{U}(\lambda^{r},\mu^{c},\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}^{D}(\lambda^{c},\mu^{r},\nu;N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu^{rc},\nu;N)}}=q^{K_{1}(\lambda,\mu,\nu)}$

where $q^{K_{1}(\lambda,\mu,\nu)}$ is defined in Lemma 4.17 and independent of
$N$. Now take $N\to\infty$, we have the following graphical condensation
recurrence

$\displaystyle V^{n}_{\lambda\mu\nu}\cdot
V^{n}_{\lambda^{rc}\mu^{rc}\nu}=V^{n}_{\lambda^{rc}\mu\nu}\cdot
V^{n}_{\lambda\mu^{rc}\nu}+q^{K_{1}(\lambda,\mu,\nu)}\cdot
V^{n}_{\lambda^{r}\mu^{c}\nu}\cdot V^{n}_{\lambda^{c}\mu^{r}\nu}.$

Then we have

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$\displaystyle\frac{V^{n}_{\lambda\mu\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}=\frac{V^{n}_{\lambda^{rc}\mu\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\frac{V^{n}_{\lambda\mu^{rc}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\left(\frac{V^{n}_{\lambda^{rc}\mu^{rc}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\right)^{-1}+q^{K_{1}(\lambda,\mu,\nu)}\cdot\frac{V^{n}_{\lambda^{r}\mu^{c}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\frac{V^{n}_{\lambda^{c}\mu^{r}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\cdot\left(\frac{V^{n}_{\lambda^{rc}\mu^{rc}\nu}}{V^{n}_{\emptyset\emptyset\emptyset}}\right)^{-1}$

which generalizes the one in [14, Section 3.3] by Remark 4.18.

However, this recurrence is still not enough to determine orbifold DT/PT
vertex correspondence for the full 3-leg case unless one has 2-leg
correspondence as the base case via Remark 4.8. As explained in Introduction,
due to the fewer symmetries of $V^{n}_{\lambda\mu\nu}$ (resp.
$W^{n}_{\lambda\mu\nu}$) than $V_{\lambda\mu\nu}$ (resp. $W_{\lambda\mu\nu}$),
one need to explore more graphical condensation recurrences for our orbifold
case.

To derive the next graphical condensation recurrence, we take $\mathbf{G}$ to
be $\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})$ and

$\displaystyle a:=\max S_{1}^{-},\;\;\;b:=\min S_{1}^{+},\;\;\;c:=\max
S_{3}^{-},\;\;\;d:=\min S_{3}^{+}.$

Then as above we have

$\displaystyle\mathbf{G}-\\{a,b,c,d\\}=\mathbf{H}(N;\lambda,\mu,\nu),$
$\displaystyle\mathbf{G}-\\{a,b\\}=\mathbf{H}(N;\lambda,\mu,\nu^{rc}),$
$\displaystyle\mathbf{G}-\\{c,d\\}=\mathbf{H}(N;\lambda^{rc},\mu,\nu).$

Similarly, the graph $\mathbf{G}-\\{a,d\\}$ is obtained from the charge $-1$
Maya diagram of $\lambda^{r}$, the Maya diagram of $\mu$, and the charge 1
Maya diagram of $\nu^{c}$, while the graph $\mathbf{G}-\\{b,c\\}$ is obtained
from the charge 1 Maya diagram of $\lambda^{c}$, the Maya diagram of $\mu$,
and the charge $-1$ Maya diagram of $\nu^{r}$. However, slightly different
from the first case, 3D partitions asymptotic to $(\lambda^{r},\mu,\nu^{c})$
correspond to dimer configurations on $\mathbf{G}-\\{a,d\\}$ but with the
orgin in $\mathbb{Z}^{3}$ shifted directly to the face on the left-up side of
the central face of $\mathbf{H}(N)$, and 3D partitions asymptotic to
$(\lambda^{c},\mu,\nu^{r})$ correspond to dimer configurations on
$\mathbf{G}-\\{b,c\\}$ but with the orgin in $\mathbb{Z}^{3}$ shifted directly
to the face on the right-down side of the central face of $\mathbf{H}(N)$. We
omit the figures and one may take an exmple to draw (or modify) the pictures
as in [14, Figure 5] to witness the required shift.

Let $\mathbf{D}_{\min}^{\nwarrow}(\lambda^{r},\mu,\nu^{c};N)$ and
$\mathbf{D}_{\min}^{\searrow}(\lambda^{c},\mu,\nu^{r};N)$ be the minimal dimer
configurations on $\mathbf{G}-\\{a,d\\}$ and $\mathbf{G}-\\{b,c\\}$ with their
weights denoted by $q^{\omega_{\min}^{LU}(\lambda^{r},\mu,\nu^{c};N)}$ and
$q^{\omega_{\min}^{RD}(\lambda^{c},\mu,\nu^{r};N)}$ respectively. Let

$\displaystyle\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{a,d\\}):=\left(q^{\widetilde{\omega}_{\min}^{LU}(\lambda^{r},\mu,\nu^{c};N)}\right)^{-1}\cdot\mathbf{Z}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{a,d\\}),$
$\displaystyle\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{b,c\\}):=\left(q^{\widetilde{\omega}_{\min}^{RD}(\lambda^{c},\mu,\nu^{r};N)}\right)^{-1}\cdot\mathbf{Z}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{b,c\\}),$

where

$\displaystyle
q^{\widetilde{\omega}_{\min}^{LU}(\lambda^{r},\mu,\nu^{c};N)}:=q^{\omega_{\min}^{LU}(\lambda^{r},\mu,\nu^{c};N)}\cdot\prod\limits_{l=0}^{n-1}q_{l}^{|\mathrm{II}(\lambda^{r},\mu,\nu^{c})|_{l}}\cdot
q_{l}^{2|\mathrm{III}(\lambda^{r},\mu,\nu^{c})|_{l}},$ $\displaystyle
q^{\widetilde{\omega}_{\min}^{RD}}(\lambda^{c},\mu,\nu^{r};N):=q^{\omega_{\min}^{RD}(\lambda^{c},\mu,\nu^{r};N)}\cdot\prod\limits_{l=0}^{n-1}q_{l}^{|\mathrm{II}(\lambda^{c},\mu,\nu^{r})|_{l}}\cdot
q_{l}^{2|\mathrm{III}(\lambda^{c},\mu,\nu^{r})|_{l}}.$

As in the first case, we have

$\displaystyle\lim_{N\to\infty}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{a,d\\})=V^{n}_{\lambda^{r}\mu\nu^{c}},$
$\displaystyle\lim_{N\to\infty}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{b,c\\})=V^{n}_{\lambda^{c}\mu\nu^{r}}.$

and

$\displaystyle\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu,\nu))\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc}))$
$\displaystyle=$
$\displaystyle\frac{q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu^{rc};N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu^{rc};N)}}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu))\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda,\mu,\nu^{rc}))$
$\displaystyle+$
$\displaystyle\frac{q^{\widetilde{\omega}_{\min}^{LU}(\lambda^{r},\mu,\nu^{c};N)}\cdot
q^{\widetilde{\omega}_{\min}^{RD}(\lambda^{c},\mu,\nu^{r};N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu^{rc};N)}}\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{a,d\\})\widetilde{\mathbf{Z}}^{D}(\mathbf{H}(N;\lambda^{rc},\mu,\nu^{rc})-\\{b,c\\})$

It is shown in Section 4.3.2 that

$\displaystyle\frac{q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu^{rc};N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu^{rc};N)}}=1\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\frac{q^{\widetilde{\omega}_{\min}^{LU}(\lambda^{r},\mu,\nu^{c};N)}\cdot
q^{\widetilde{\omega}_{\min}^{RD}(\lambda^{c},\mu,\nu^{r};N)}}{q^{\widetilde{\omega}_{\min}(\lambda,\mu,\nu;N)}\cdot
q^{\widetilde{\omega}_{\min}(\lambda^{rc},\mu,\nu^{rc};N)}}=q^{K_{2}(\lambda,\mu,\nu)}$

where $q^{K_{2}(\lambda,\mu,\nu)}$ defined in Lemma 4.26 is independent of
$N$. Now let $N\to\infty$, we have

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