A Note on Disk Counting in Toric Orbifolds

TL;DR

This paper extends methods for computing orbi-disk invariants from toric Calabi-Yau orbifolds to Gorenstein semi-Fano toric orbifolds, showing the orbi-disc potential is analytic over complex numbers.

Contribution

It introduces a new approach for calculating orbi-disk invariants in a broader class of toric orbifolds, expanding the applicability of existing techniques.

Findings

Computed orbi-disk invariants for Gorenstein semi-Fano toric orbifolds.

Established the analyticity of the orbi-disc potential over complex numbers.

Extended methods from Calabi-Yau to semi-Fano toric orbifolds.

Abstract

We compute orbi-disk invariants of compact Gorenstein semi-Fano toric orbifolds by extending the method used for toric Calabi-Yau orbifolds. As a consequence the orbi-disc potential is analytic over complex numbers.

Figures

Tables

$n_{(a,b)}$	$a=0$	$a=1$	$a=2$	$a=3$	$a=4$	$a=5$	$a=6$
$b=0$	$0$	$1$	$0$	$0$	$1/648$	$0$	$0$
$b=1$	$0$	$0$	$-1/18$	$0$	$0$	$-1/29160$	$0$
$b=2$	$1/6$	$0$	$0$	$1/972$	$0$	$0$	$1/3149280$
$b=3$	$0$	$-1/162$	$0$	$0$	$-1/104976$	$0$	$0$
$b=4$	$0$	$0$	$1/11664$	$0$	$0$	$1/18895680$	$0$
$b=5$	$-1/9720$	$0$	$0$	$-1/1574640$	$0$	$0$	$-1/5101833600$
$b=6$	$0$	$1/524880$	$0$	$0$	$1/340122240$	$0$	$0$

Equations

$$\displaystyle\big{(}\Sigma,\{\boldsymbol{b}_{i}\}_{i=0}^{m-1}\cup\{\boldsymbol{b}_{j}\}_{j=m}^{m^{\prime}-1}\big{)}.$$

$$\varphi :\tilde{N}:=\underset{i=0}{\overset{m^{\prime }-1}{⨁}}\mathbb{Z}{e}_i\to N,\varphi (e_i):={\mathbf{b}}_i\text{for i=0,…,m′−1}.$$

$$0⟶\mathbb{L}:=\mathrm{Ker}(\varphi )\stackrel{\psi }{⟶}\tilde{N}\stackrel{\varphi }{⟶}N⟶0.$$

$$0⟶G:=\mathbb{L}{\otimes }_{\mathbb{Z}}{\mathbb{C}}^{\times }⟶\tilde{N}{\otimes }_{\mathbb{Z}}{\mathbb{C}}^{\times }\simeq ({\mathbb{C}}^{\times }{)}^{m^{\prime }}\stackrel{{\varphi }_{{\mathbb{C}}^{\times }}}{⟶}\mathbb{T}:=N{\otimes }_{\mathbb{Z}}{\mathbb{C}}^{\times }\to 0,$$

$$\displaystyle\mathcal{A}:=\left\{I\subset\{0,1,\dots,m^{\prime}-1\}\,\Big{|}\,\sum_{i\notin I}\mathbb{R}_{\geq 0}\boldsymbol{b}_{i}\text{ is a cone in }\Sigma\right\}.$$

$$U_{\mathcal{A}}:={\mathbb{C}}^{m^{\prime }}\setminus \bigcup _{I\notin \mathcal{A}}{\mathbb{C}}^I.$$

$${\mathcal{X}}_{\Sigma }:=[U_{\mathcal{A}}/G].$$

$$\displaystyle{\rm Box}_{\boldsymbol{b}_{\sigma}}:=\left\{\nu\in N\,\Big{|}\,\nu=\sum_{k=1}^{d}t_{k}\boldsymbol{b}_{i_{k}},\,t_{k}\in[0,1)\cap\mathbb{Q}\right\}.$$

$${\mathrm{Box}}_{{\mathbf{b}}_{\sigma }}^{\circ }:{\mathrm{Box}}_{{\mathbf{b}}_{\sigma }}\setminus \bigcup _{\tau ⪵\sigma }{\mathrm{Box}}_{{\mathbf{b}}_{\tau }},\mathrm{Box}(\Sigma ):=\bigcup _{\sigma \in {\Sigma }^{(n)}}{\mathrm{Box}}_{{\mathbf{b}}_{\sigma }}=\underset{\sigma \in \Sigma }{⨆}{\mathrm{Box}}_{{\mathbf{b}}_{\sigma }}^{\circ },$$

$${\mathrm{Box}}^{\prime }(\Sigma )=\mathrm{Box}(\Sigma )\setminus \{0\},$$

$$H_{\mathrm{CR}}^d(\mathcal{X};\mathbb{Q})=\underset{\nu \in \mathrm{Box}}{⨁}{H}^{d-2\mathrm{age}(\nu )}({\mathcal{X}}_{\nu };\mathbb{Q}),$$

$$\mathrm{age}(\nu )=\sum _{k=1}^d{t}_k\in {\mathbb{Q}}_{\ge 0}.$$

$$H_{\mathrm{CR},\mathbb{T}}^d(\mathcal{X};\mathbb{Q})=\underset{\nu \in \mathrm{Box}}{⨁}{H}_{\mathbb{T}}^{d-2\mathrm{age}(\nu )}({\mathcal{X}}_{\nu };\mathbb{Q}).$$

$$\displaystyle 0\longrightarrow M:=N^{\vee}=\operatorname{Hom}(N,\mathbb{Z})\overset{\phi^{\vee}}{\longrightarrow}\widetilde{M}:=\widetilde{N}^{\vee}=\operatorname{Hom}\big{(}\widetilde{N},\mathbb{Z}\big{)}\overset{\psi^{\vee}}{\longrightarrow}\mathbb{L}^{\vee}=\operatorname{Hom}(\mathbb{L},\mathbb{Z})\longrightarrow 0,$$

$$\displaystyle\operatorname{Pic}(\mathcal{X})\simeq\operatorname{Hom}(G,\mathbb{C}^{\times})\simeq\mathbb{L}^{\vee},\qquad\operatorname{Pic}_{\mathbb{T}}(\mathcal{X})\simeq\operatorname{Hom}\big{(}(\mathbb{C}^{\times})^{m^{\prime}},\mathbb{C}^{\times}\big{)}\simeq\widetilde{N}^{\vee}=\widetilde{M}.$$

$$\displaystyle\big{\{}D_{i}:=\psi^{\vee}\big{(}e_{i}^{\vee}\big{)}\,|\,0\leq i\leq m-1\big{\}}\subset\mathbb{L}^{\vee}\simeq\operatorname{Pic}(\mathcal{X})$$

$$\tilde{M}\otimes \mathbb{Q}\stackrel{{\psi }^{\vee }\otimes \mathbb{Q}}{\to }{\mathbb{L}}^{\vee }\otimes \mathbb{Q},$$

$$\displaystyle\big{(}\widetilde{M}\otimes\mathbb{Q}\big{)}\Big{/}\left(\sum_{j=m}^{m^{\prime}-1}\mathbb{Q}D^{\mathbb{T}}_{j}\right)\simeq H_{\mathbb{T}}^{2}(\mathcal{X},\mathbb{Q})\to H^{2}(\mathcal{X},\mathbb{Q})\simeq\big{(}\mathbb{L}^{\vee}\otimes\mathbb{Q}\big{)}\Big{/}\left(\sum_{j=m}^{m^{\prime}-1}\mathbb{Q}D_{j}\right).$$

$$⟨D_i,D_j^{\vee }⟩=\{\begin{array}{ll}1& \text{if }i=j,\\ -c_{ji}& \text{if }i\notin I_j,\\ 0& \text{if }i\in I_j\setminus \{j\}.\end{array}$$

$$\displaystyle\mathbb{L}^{\vee}\otimes\mathbb{Q}=\operatorname{Ker}\big{(}\big{(}D_{m}^{\vee},\dots,D_{m^{\prime}-1}^{\vee}\big{)}\colon\mathbb{L}^{\vee}\otimes\mathbb{Q}\to\mathbb{Q}^{m^{\prime}-m}\big{)}\oplus\bigoplus_{j=m}^{m^{\prime}-1}\mathbb{Q}D_{j}.$$

$${\tilde{C}}_{\mathcal{X}}:=\bigcap _{I\in \mathcal{A}}\left(\sum _{i\in I}{\mathbb{R}}_{>0}{D}_i\right)\subset {\mathbb{L}}^{\vee }\otimes \mathbb{R}.$$

$$\{p_1,\dots ,p_r\}\subset {\mathbb{L}}^{\vee },$$

$${\bar{D}}_i=\sum _{a=1}^{r^{\prime }}{Q}_{ia}{\bar{p}}_a,i=0,\dots ,m-1.$$

$${\bar{D}}_i^{\mathbb{T}}=\sum _{a=1}^{r^{\prime }}{Q}_{ia}{\bar{p}}_a^{\mathbb{T}}+{\lambda }_i,\text{where}{\lambda }_i\in H^2(B\mathbb{T};\mathbb{Q}).$$

$$H_{\mathrm{CR},\mathbb{T}}^0(\mathcal{X},K_{\mathbb{T}})=K_{\mathbb{T}}1,H_{\mathrm{CR},\mathbb{T}}^2(\mathcal{X},K_{\mathbb{T}})=\underset{a=1}{\overset{r^{\prime }}{⨁}}{K}_{\mathbb{T}}{\bar{p}}_a^{\mathbb{T}}\oplus \underset{\nu \in \text{Box},\text{age}(\nu )=1}{⨁}{K}_{\mathbb{T}}{1}_{\nu }.$$

$$\{{\gamma }_1,\dots ,{\gamma }_r\}\subset \mathbb{L},{\gamma }_a=\sum _{i=0}^{m^{\prime }-1}{Q}_{ia}{e}_i\in \tilde{N},$$

$$\mathbb{K}:=\{d\in \mathbb{L}\otimes \mathbb{Q}|\{j\in \{0,1,\dots ,m^{\prime }-1\}|⟨D_j,d⟩\in \mathbb{Z}\}\in \mathcal{A}\},$$

$${\mathbb{K}}_{\mathrm{eff}}:=\{d\in \mathbb{L}\otimes \mathbb{Q}|\{j\in \{0,1,\dots ,m^{\prime }-1\}|⟨D_j,d⟩\in {\mathbb{Z}}_{\ge 0}\}\in \mathcal{A}\}.$$

$$\nu (d):=\sum _{i=0}^{m^{\prime }-1}\lceil ⟨D_i,d⟩\rceil {\mathbf{b}}_i\in N,$$

$$\nu (d)=\sum _{i=0}^{m^{\prime }-1}(\{-⟨D_i,d⟩\}+⟨D_i,d⟩){\mathbf{b}}_i=\sum _{i=0}^{m^{\prime }-1}\{-⟨D_i,d⟩\}{\mathbf{b}}_i=\sum _{i\notin I_d}\{-⟨D_i,d⟩\}{\mathbf{b}}_i.$$

$$w:(\mathbb{D},\partial \mathbb{D})\to (\mathcal{X},L)$$

$$h_p:{\mathbb{Z}}_{m_j}\to G_{w(z_j^{+})}$$

Full text

\FirstPageHeading

\ShortArticleName

A Note on Disk Counting in Toric Orbifolds

\ArticleName

A Note on Disk Counting in Toric Orbifolds

\Author

Kwokwai CHAN ${}^{{\dagger}^{1}}$, Cheol-Hyun CHO ${}^{{\dagger}^{2}}$, Siu-Cheong LAU ${}^{{\dagger}^{3}}$, Naichung Conan LEUNG ${}^{{\dagger}^{4}}$

and Hsian-Hua TSENG ${}^{{\dagger}^{5}}$

\AuthorNameForHeading

K. Chan, C.-H. Cho, S.-C. Lau, N.-C. Leung and H.-H. Tseng

\Address

${}^{{\dagger}^{1}}$ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong \EmailDDkwchan@math.cuhk.edu.hk

\Address

${}^{{\dagger}^{2}}$ Department of Mathematical Sciences, Research Institute in Mathematics,

${}^{{\dagger}^{2}}$ Seoul National University, Gwanak-Gu, Seoul, South Korea \EmailDDchocheol@snu.ac.kr

\Address

${}^{{\dagger}^{3}}$ Department of Mathematics and Statistics, Boston University, Boston, MA, USA \EmailDDlau@math.bu.edu

\Address

${}^{{\dagger}^{4}}$ The Institute of Mathematical Sciences and Department of Mathematics,

${}^{{\dagger}^{4}}$ The Chinese University of Hong Kong, Shatin, Hong Kong \EmailDDleung@math.cuhk.edu.hk

\Address

${}^{{\dagger}^{5}}$ Department of Mathematics, Ohio State University, 100 Math Tower,

${}^{{\dagger}^{5}}$ 231 West 18th Ave., Columbus, OH 43210, USA \EmailDDhhtseng@math.ohio-state.edu

\ArticleDates

Received January 24, 2020, in final form June 11, 2020; Published online June 17, 2020

\Abstract

We compute orbi-disk invariants of compact Gorenstein semi-Fano toric orbifolds by extending the method used for toric Calabi–Yau orbifolds. As a consequence the orbi-disc potential is analytic over complex numbers.

\Keywords

orbifold; toric; open Gromov–Witten invariants; mirror symmetry; SYZ

\Classification

53D37; 14J33

1 Introduction

The mirror map plays a central role in the study of mirror symmetry. It provides a canonical local isomorphism between the Kähler moduli and the complex moduli of the mirror near a large complex structure limit. Such an isomorphism is crucial to counting of rational curves using mirror symmetry.

The mirror map is a transformation from the complex coordinates of the Hori–Vafa mirror moduli to the canonical coordinates obtained from period integrals. In [4] and [5], we derived an enumerative meaning of the inverse mirror maps for toric Calabi–Yau orbifolds and compact semi-Fano toric manifolds in terms of genus 0 open (orbifold) Gromov–Witten invariants (or (orbi-)disk invariants). Namely, we showed that coefficients of the inverse mirror map are equal to generating functions of virtual counts of stable (orbi-)disks bounded by a regular Lagrangian moment map fiber. In particular it gives a way to effectively compute all such invariants.

It is interesting to compare this with the mirror family constructed by Gross–Siebert [19], which is written in canonical coordinates [24]. In [19, Conjecture 0.2], it was conjectured that the wall-crossing functions in their construction are generating functions of open Gromov–Witten invariants. Our results verify this conjecture in the toric setting, namely, we showed that the SYZ mirror family [25], constructed using open Gromov–Witten invariants, is written in canonical coordinates.

In this short note we extend our method in [4] to derive an explicit formula for the orbi-disk invariants in the case of compact Gorenstein semi-Fano toric orbifolds; see Theorem 3.6 for the explicit formulas. This proves [3, Conjecture] for such orbifolds, generalizing [5, Theorem 1.2]:

Theorem 1.1** (open mirror theorem).**

For a compact Gorenstein semi-Fano toric orbifold, the orbi-disk potential is equal to the $($extended$)$ Hori–Vafa superpotential via the mirror map.

See (3.6) for the definition of the orbi-disc potential. We remark that the open crepant resolution conjecture [3, Conjecture 1] may be studied using this computation and techniques of analytical continuation in [4, Appendix A].

Corollary 1.2**.**

There exists an open neighborhood around the large volume limit where the orbi-disk potential converges.

This generalizes [5, Theorem 7.6] to the orbifold case.

2 Preparation

2.1 Toric orbifolds

2.1.1 Construction

Following [2], a stacky fan is the combinatorial data $(\Sigma,\boldsymbol{b}_{0},\dots,\boldsymbol{b}_{m-1}),$ where $\Sigma$ is a simplicial fan contained in the $\mathbb{R}$-vector space $N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$, $N$ is a lattice of rank $n$, and $\{\boldsymbol{b}_{i}\,|\,0\leq i\leq m-1\}\subset N$ are generators of $1$-dimensional cones of $\Sigma$. $\boldsymbol{b}_{i}$ are called the stacky vectors.

Choose $\boldsymbol{b}_{m},\dots,\boldsymbol{b}_{m^{\prime}-1}\in N$ so that they are contained in the support of the fan $\Sigma$ and they generate $N$ over $\mathbb{Z}$. An extended stacky fan in the sense of [22] is the data

[TABLE]

The vectors $\{\boldsymbol{b}_{j}\}_{j=m}^{m^{\prime}-1}$ are called extra vectors.

The fan map associated to an extended stacky fan (2.1) is defined by

[TABLE]

$\phi$ is surjective and yields an exact sequence of groups called the fan sequence:

[TABLE]

Clearly $\mathbb{L}\simeq\mathbb{Z}^{m^{\prime}-n}$. Tensoring (2.2) with $\mathbb{C}^{\times}$ yields the following sequence:

[TABLE]

which is exact. Note that $G$ is an algebraic torus.

By definition, the set of anti-cones is

[TABLE]

This terminology is justified because for $I\in\mathcal{A}$, the complement of $I$ in $\{0,1,\dots,m^{\prime}-1\}$ indexes generators of a cone in $\Sigma$. For $I\in\mathcal{A}$, the collection $\{Z_{i}\,|\,i\in I\}$ generates an ideal in $\mathbb{C}[Z_{0},\dots,Z_{m^{\prime}-1}]$, which in turn determines a subvariety $\mathbb{C}^{I}\subset\mathbb{C}^{m^{\prime}}$. Set

[TABLE]

The map $G\to(\mathbb{C}^{\times})^{m^{\prime}}$ in (2.3) defines a $G$-action on $\mathbb{C}^{m^{\prime}}$ and hence a $G$-action on $U_{\mathcal{A}}$. This action is effective and has finite stabilizers, because $N$ is torsion-free (see [22, Section 2]). The toric orbifold associated to \big{(}\Sigma,\{\boldsymbol{b}_{i}\}_{i=0}^{m-1}\cup\{\boldsymbol{b}_{j}\}_{j=m}^{m^{\prime}-1}\big{)} is defined to be the following quotient stack:

[TABLE]

The standard $(\mathbb{C}^{\times})^{m^{\prime}}$-action on $U_{\mathcal{A}}$ induces a $\mathbb{T}$-action on $\mathcal{X}_{\Sigma}$ via (2.3).

The coarse moduli space of the toric orbifold $\mathcal{X}_{\Sigma}$ is the toric variety $X_{\Sigma}$ associated to the fan $\Sigma$. In this paper we assume that $X_{\Sigma}$ is semi-projective, i.e., $X_{\Sigma}$ has a $\mathbb{T}$-fixed point and the natural map $X_{\Sigma}\to\operatorname{Spec}H^{0}(X_{\Sigma},\mathcal{O}_{X_{\Sigma}})$ is projective, or equivalently, $X_{\Sigma}$ arises as a GIT quotient of a complex vector space by an abelian group (see [20, Section 2]). This assumption is required for the toric mirror theorem of [11] to hold. More detailed discussions on semi-projective toric varieties can be found in [12, Section 7.2].

2.1.2 Twisted sectors

Consider a $d$-dimensional cone $\sigma\in\Sigma$ generated by $\boldsymbol{b}_{\sigma}=(\boldsymbol{b}_{i_{1}},\dots,\boldsymbol{b}_{i_{d}})$. Define

[TABLE]

$\{\boldsymbol{b}_{i_{1}},\dots,\boldsymbol{b}_{i_{d}}\}$ generates a submodule $N_{\boldsymbol{b}_{\sigma}}\subset N$. One can check that there is a bijection between ${\rm Box}_{\boldsymbol{b}_{\sigma}}$ and the finite group $G_{\boldsymbol{b}_{\sigma}}:=(N\cap\mathrm{Span}_{\mathbb{R}}\boldsymbol{b}_{\sigma})/N_{\boldsymbol{b}_{\sigma}}$. Furthermore, if $\tau$ is a subcone of $\sigma$, then ${\rm Box}_{\boldsymbol{b}_{\tau}}\subset{\rm Box}_{\boldsymbol{b}_{\sigma}}$. Define

[TABLE]

where $\Sigma^{(n)}$ is the set of $n$-dimensional cones in $\Sigma$.

Following the description of the inertia orbifold of $\mathcal{X}_{\Sigma}$ in [2], for $\nu\in{\rm Box}(\Sigma)$, we denote by $\mathcal{X}_{\nu}$ the corresponding component of the inertia orbifold of $\mathcal{X}:=\mathcal{X}_{\Sigma}$. Note that $\mathcal{X}_{0}=\mathcal{X}_{\Sigma}$ as orbifolds. Elements $\nu\in{\rm Box}^{\prime}(\Sigma)$ correspond to twisted sectors of $\mathcal{X}$, namely non-trivial connected components of the inertia orbifold of $\mathcal{X}$.

Following [7], the direct sum of singular cohomology groups of components of the inertia orbifold of $\mathcal{X}$, subject to a degree shift, is called the Chen–Ruan orbifold cohomology $H^{*}_{\mathrm{CR}}(\mathcal{X};\mathbb{Q})$ of $\mathcal{X}$. More precisely,

[TABLE]

where $\operatorname{age}(\nu)$ is called the degree shifting number111Following Miles Reid, it is now more commonly called age. in [7] of the twisted sector $\mathcal{X}_{\nu}$. In case of toric orbifolds, $\operatorname{age}$ has a combinatorial description [2]: if $\nu=\sum\limits_{k=1}^{d}t_{k}\boldsymbol{b}_{i_{k}}\in{\rm Box}(\Sigma)$ where $\{\boldsymbol{b}_{i_{1}},\dots,\boldsymbol{b}_{i_{d}}\}$ generates a cone in $\Sigma$, then

[TABLE]

Using $\mathbb{T}$-actions on twisted sectors induced from that on $\mathcal{X}$, we can define $\mathbb{T}$-equivariant Chen–Ruan orbifold cohomology $H^{*}_{\mathrm{CR},\mathbb{T}}(\mathcal{X};\mathbb{Q})$ by replacing singular cohomology with $\mathbb{T}$-equivariant cohomology $H^{*}_{\mathbb{T}}(-)$. Namely

[TABLE]

By general properties of equivariant cohomology, $H^{*}_{\mathrm{CR},\mathbb{T}}(\mathcal{X};\mathbb{Q})$ is a module over $H^{*}_{\mathbb{T}}(\text{pt},\mathbb{Q})$ and admits a map $H^{*}_{\mathrm{CR},\mathbb{T}}(\mathcal{X};\mathbb{Q})\to H^{*}_{\mathrm{CR}}(\mathcal{X};\mathbb{Q})$ called non-equivariant limit.

2.1.3 Toric divisors, Kähler cones, and Mori cones

We continue using the notations in Sections 2.1.1 and 2.1.2. Applying $\operatorname{Hom}_{\mathbb{Z}}(-,\mathbb{Z})$ to the fan sequence (2.2), we obtain the following exact sequence:222The map $\psi^{\vee}:\widetilde{M}\to\mathbb{L}^{\vee}$ is surjective since $N$ is torsion-free.

[TABLE]

which is called the divisor sequence. Line bundles on $\mathcal{X}=[\mathcal{U}_{\mathcal{A}}/G]$ correspond to $G$-equivariant line bundles on $\mathcal{U}_{\mathcal{A}}$. In view of (2.3), $\mathbb{T}$-equivariant line bundles on $\mathcal{X}$ correspond to $(\mathbb{C}^{\times})^{m^{\prime}}$-equivariant line bundles on $\mathcal{U}_{\mathcal{A}}$. Because the codimension of $\cup_{I\notin\mathcal{A}}\mathbb{C}^{I}\subset\mathbb{C}^{m^{\prime}}$ is at least $2$, the Picard groups satisfy:

[TABLE]

The natural map $\operatorname{Pic}_{\mathbb{T}}(\mathcal{X})\to\operatorname{Pic}(\mathcal{X})$ is identified with the map $\psi^{\vee}\colon\widetilde{M}\to\mathbb{L}^{\vee}$ appearing in the divisor sequence.

The elements \big{\{}e_{i}^{\vee}\,|\,i=0,1,\dots,m^{\prime}-1\big{\}}\subset\widetilde{M}\simeq\operatorname{Pic}_{\mathbb{T}}(\mathcal{X}) dual to $\{e_{i}\,|\,i=0,1,\dots,m^{\prime}-1\}\subset\widetilde{N}$ correspond to $\mathbb{T}$-equivariant line bundle on $\mathcal{X}$ which we denote by $D_{i}^{\mathbb{T}}$, $i=0,1,\dots,m^{\prime}-1$. The collection

[TABLE]

consists of toric prime divisors corresponding to the generators $\{\boldsymbol{b}_{i}\,|\,0\leq i\leq m-1\}$ of $1$-dimensional cones in $\Sigma$. Elements $D^{\mathbb{T}}_{i}$, $0\leq i\leq m-1$ are $\mathbb{T}$-equivariant lifts of these divisors. There are natural maps

[TABLE]

Together with the natural quotient maps, they fit into a commutative diagram.

As explained in [21, Section 3.1.2], there is a canonical splitting of the quotient map $\mathbb{L}^{\vee}\otimes\mathbb{Q}\to H^{2}(\mathcal{X};\mathbb{Q})$. For $m\leq j\leq m^{\prime}-1$, let $I_{j}\in\mathcal{A}$ be the anticone of the cone containing $\boldsymbol{b}_{j}$. This allows us to write $\boldsymbol{b}_{j}=\sum\limits_{i\notin I_{j}}c_{ji}\boldsymbol{b}_{i}$ for $c_{ji}\in\mathbb{Q}_{\geq 0}$.

Tensoring the fan sequence (2.2) with $\mathbb{Q}$, we may find a unique $D_{j}^{\vee}\in\mathbb{L}\otimes\mathbb{Q}$ such that values of the natural pairing $\langle-,-\rangle$ between $\mathbb{L}^{\vee}$ and $\mathbb{L}$ satisfy

[TABLE]

Using $D_{j}^{\vee}$ we get a decomposition

[TABLE]

We can view $H^{2}(\mathcal{X};\mathbb{Q})$ as a subspace of $\mathbb{L}^{\vee}\otimes\mathbb{Q}$ because \operatorname{Ker}\big{(}\big{(}D_{m}^{\vee},\dots,D_{m^{\prime}-1}^{\vee}\big{)}\big{)} can be identified with $H^{2}(\mathcal{X};\mathbb{Q})$ via the map $\mathbb{L}^{\vee}\otimes\mathbb{Q}\to H^{2}(\mathcal{X};\mathbb{Q})$.

Define extended Kähler cone of $\mathcal{X}$ to be

[TABLE]

The Kähler cone $C_{\mathcal{X}}$ is the image of $\widetilde{C}_{\mathcal{X}}$ under $\mathbb{L}^{\vee}\otimes\mathbb{R}\to H^{2}(\mathcal{X};\mathbb{R})$. The splitting (2.5) of $\mathbb{L}^{\vee}\otimes\mathbb{Q}$ yields a splitting $\widetilde{C}_{\mathcal{X}}=C_{\mathcal{X}}+\sum\limits_{j=m}^{m^{\prime}-1}\mathbb{R}_{>0}D_{j}.$

By (2.2), $\mathbb{L}^{\vee}$ has rank equal to $r:=m^{\prime}-n$. The rank of $H_{2}(\mathcal{X};\mathbb{Z})$ is $r^{\prime}:=r-(m^{\prime}-m)=m-n.$ We choose an integral basis

[TABLE]

such that $p_{a}$ is in the closure of $\widetilde{C}_{\mathcal{X}}$ for all $a$ and $p_{r^{\prime}+1},\dots,p_{r}\in\sum\limits_{i=m}^{m^{\prime}-1}\mathbb{R}_{\geq 0}D_{i}$. We get a nef basis $\{\bar{p}_{1},\dots,\bar{p}_{r^{\prime}}\}$ for $H^{2}(\mathcal{X};\mathbb{Q})$ as images of $\{p_{1},\dots,p_{r^{\prime}}\}$ under the quotient map $\mathbb{L}^{\vee}\otimes\mathbb{Q}\to H^{2}(\mathcal{X};\mathbb{Q})$. For $r^{\prime}+1\leq a\leq r$, the images satisfies $\bar{p}_{a}=0$.

We choose equivariant lifts of $p_{a}$’s, namely \big{\{}p_{1}^{\mathbb{T}},\dots,p_{r}^{\mathbb{T}}\big{\}}\subset\widetilde{M}\otimes\mathbb{Q} such that \psi^{\vee}\big{(}p_{a}^{\mathbb{T}}\big{)}=p_{a} for all $a$. We also require that for $a=r^{\prime}+1,\dots,r$ the images $\bar{p}_{a}^{\mathbb{T}}$ of $p_{a}^{\mathbb{T}}$ under the natural map $\widetilde{M}\otimes\mathbb{Q}\to H_{\mathbb{T}}^{2}(\mathcal{X},\mathbb{Q})$ satisfies $\bar{p}_{a}^{\mathbb{T}}=0$.

The coefficients $Q_{ia}\in\mathbb{Z}$ in the equations $D_{i}=\sum\limits_{a=1}^{r}Q_{ia}p_{a}$ assemble to a matrix $(Q_{ia})$. The images333$\bar{D}_{i}$ is the class of the toric prime divisor $D_{i}$. $\bar{D}_{i}$ of $D_{i}$ under the map $\mathbb{L}^{\vee}\otimes\mathbb{Q}\to H^{2}(\mathcal{X};\mathbb{Q})$ can be expressed as

[TABLE]

Their equivariant lifts $\bar{D}^{\mathbb{T}}_{i}$ can be expressed as

[TABLE]

For $i=m,\dots,m^{\prime}-1$, we have $\bar{D}_{i}=0$ in $H^{2}(\mathcal{X};\mathbb{R})$ and $\bar{D}^{\mathbb{T}}_{i}=0$.

Localization gives the following description of $H_{\mathrm{CR},\mathbb{T}}^{\leq 2}$:

[TABLE]

Here $K_{\mathbb{T}}$ is the field of fractions of $H_{\mathbb{T}}^{*}(\text{pt},\mathbb{Q})$, ${\bf 1}\in H^{0}(\mathcal{X},\mathbb{Q})$ and ${\bf 1}_{\nu}\in H^{0}(\mathcal{X}_{\nu},\mathbb{Q})$ are fundamental classes.

Let

[TABLE]

be the basis dual to $\{p_{1},\dots,p_{r}\}\subset\mathbb{L}^{\vee}$. $H_{2}^{\rm eff}(\mathcal{X};\mathbb{Q})$ admits a basis $\{\gamma_{1},\dots,\gamma_{r^{\prime}}\}$, and we have $Q_{ia}=0$ when $m\leq i\leq m^{\prime}-1$ and $1\leq a\leq r^{\prime}$.

Set

[TABLE]

Elements of $\mathbb{K}_{\rm eff}$ should be interpreted as effective curve classes. Elements of $\mathbb{K}_{\rm eff}\cap H_{2}(\mathcal{X};\mathbb{R})$ should be viewed as classes of stable maps $\mathbb{P}(1,m)\to\mathcal{X}$ for some $m\in\mathbb{Z}_{\geq 0}$. See, e.g., [21, Section 3.1] for more details.

Definition 2.1**.**

A toric orbifold $\mathcal{X}$ is called semi-Fano if $c_{1}(\mathcal{X})\cdot\alpha>0$ for every effective curve class $\alpha$, in other words, $-K_{\mathcal{X}}$ is nef.

For $d\in\mathbb{K}$, put444For a real number $\lambda\in\mathbb{R}$, let $\lceil\lambda\rceil$, $\lfloor\lambda\rfloor$ and $\{\lambda\}$ denote the ceiling, floor and fractional part of $\lambda$ respectively.

[TABLE]

and let $I_{d}:=\{j\in\{0,1,\dots,m^{\prime}-1\}\,|\,\langle D_{j},d\rangle\in\mathbb{Z}\}\in\mathcal{A}$. Then $\nu(d)\in{\rm Box}$ because

[TABLE]

2.2 Genus 0 open orbifold GW invariants according to [9]

Let $(\mathcal{X},\omega)$ be a toric Kähler orbifold of complex dimension $n$, equipped with the standard toric complex structure $J_{0}$ and a toric Kähler structure $\omega$. Denote by $(\Sigma,\boldsymbol{b})$ the stacky fan that defines $\mathcal{X}$, where $\boldsymbol{b}=(\boldsymbol{b}_{0},\dots,\boldsymbol{b}_{m-1})$ and $\boldsymbol{b}_{i}=c_{i}v_{i}$.

Let $L\subset\mathcal{X}$ be a Lagrangian torus fiber of the moment map $\mu_{0}\colon\mathcal{X}\to M_{\mathbb{R}}:=M\otimes_{\mathbb{Z}}\mathbb{R}$, and let $\beta\in\pi_{2}(\mathcal{X},L)=H_{2}(\mathcal{X},L;\mathbb{Z})$ be a relative homotopy class.

2.2.1 Holomorphic orbi-disks and their moduli spaces

A holomorphic orbi-disk in $\mathcal{X}$ with boundary in $L$ is a continuous map

[TABLE]

satisfying the following conditions:

\big{(}\mathbb{D},z_{1}^{+},\dots,z_{l}^{+}\big{)} is an orbi-disk with interior marked points $z_{1}^{+},\dots,z_{l}^{+}$. More precisely $\mathbb{D}$ is analytically the disk $D^{2}\subset\mathbb{C}$ so that for $j=1,\dots,l$, the orbifold structure at $z_{j}^{+}$ is given by a disk neighborhood of $z_{j}^{+}$ uniformized by the branched covering map $\operatorname{br}\colon z\to z^{m_{j}}$ for some $m_{j}\in\mathbb{Z}_{>0}$. (If $m_{j}=1$, $z_{j}^{+}$ is not an orbifold point.) 2. 2.

For any $z_{0}\in\mathbb{D}$, there is a disk neighborhood of $z_{0}$ with a branched covering map $\operatorname{br}\colon z\allowbreak\to z^{m}$, and there is a local chart \big{(}V_{w(z_{0})},G_{w(z_{0})},\pi_{w(z_{0})}\big{)} of $\mathcal{X}$ at $w(z_{0})$ and a local holomorphic lifting $\widetilde{w}_{z_{0}}$ of $w$ satisfying $w\circ\operatorname{br}=\pi_{w(z_{0})}\circ\widetilde{w}_{z_{0}}.$ 3. 3.

The map $w$ is good (in the sense of Chen–Ruan [6]) and representable. In particular, for each $z_{j}^{+}$, the associated group homomorphism

[TABLE]

between local groups which makes $\widetilde{w}_{z_{j}^{+}}$ equivariant, is injective.

The type of a map $w$ as above is defined to be $\boldsymbol{x}:=(\mathcal{X}_{\nu_{1}},\dots,\mathcal{X}_{\nu_{l}})$. Here $\nu_{j}\in{\rm Box}(\Sigma)$ is the image of the generator $1\in\mathbb{Z}_{m_{j}}$ under $h_{j}$.

There are two notions of Maslov index for an orbi-disk. The desingularized Maslov index $\mu^{\rm de}$ is defined by desingularizing the interior singularities of the pull-back bundle $w^{*}T\mathcal{X}$. Namely, the bundle $w^{*}T\mathcal{X}$ over an orbi-disk \big{(}\mathbb{D},z_{1}^{+},\dots,z_{l}^{+}\big{)} cannot be trivialized due to the orbifold structure, but we can obtain another bundle $|w^{*}T\mathcal{X}|$ by modifying the bundle near orbifold points (see Chen–Ruan [6] for more details). This is called a desingularization of $w^{*}T\mathcal{X}$ and it is a smooth bundle over the orbi-disk, hence is a trivial bundle. We can compute the Maslov index of the boundary Lagrangian loop relative to this trivialization, and it is called the desingularized Maslov index. See [9, Section 3] for more details and [9, Section 5] for an explicit formula in the toric case.

The Chern–Weil $($CW$)$ Maslov index $\mu_{\rm CW}$ is defined as the integral of the curvature of a unitary connection on $w^{*}T\mathcal{X}$ which preserves the Lagrangian boundary condition, see [10] (and also [9, Section 3.3] for a relation with $\mu^{\rm de}$). The following lemma, which appeared as [4, Lemma 3.1], computes the CW Maslov indices of disks. This is an orbifold version of the formula in [1, Lemma 3.1].

Lemma 2.2**.**

Let $(\mathcal{X},\omega,J)$ be a Kähler orbifold of complex dimension $n$. Let $\Omega$ be a non-zero meromorphic $n$-form on $\mathcal{X}$ which has at worst simple poles. Let $D\subset\mathcal{X}$ be the pole divisor of $\Omega$. Suppose also that the generic points of $D$ are smooth. Then for a special Lagrangian submanifold $L\subset\mathcal{X}\setminus D$, the CW Maslov index of a class $\beta\in\pi_{2}(\mathcal{X},L)$ is given by

[TABLE]

Here, $\beta\cdot D$ is defined by writing $\beta$ as a fractional linear combination of homotopy classes of smooth disks.

The classification of orbi-disks in a symplectic toric orbifold has been worked out in [9, Theorem 6.2]. It is similar to the classification of holomorphic discs in toric manifolds [8]. In the classification, the basic disks corresponding to the stacky vectors (and twisted sectors) play a basic role.

Theorem 2.3** ([9, Corollaries 6.3 and 6.4]).**

Let $\mathcal{X}$ be a toric Kähler orbifold and let $L$ be a fiber of the toric moment map.

1. $1.$

The smooth holomorphic disks of Maslov index $2$ $($modulo $\mathbb{T}^{n}$-action and reparametrizations of the domain$)$ are in bijective correspondence with the stacky vectors $\{\boldsymbol{b}_{0},\dots,\boldsymbol{b}_{m-1}\}$. Denote the homotopy classes of these disks by $\beta_{0},\dots,\beta_{m-1}$. 2. $2.$

The holomorphic orbi-disks with one interior orbifold marked point and desingularized Maslov index [math] $($modulo $\mathbb{T}^{n}$-action and reparametrizations of the domain$)$ are in bijective correspondence with the twisted sectors $\nu\in{\rm Box}^{\prime}(\Sigma)$ of the toric orbifold $\mathcal{X}$. Denote the homotopy classes of these orbi-disks by $\beta_{\nu}$.

Lemma 2.4** ([9, Lemma 9.1]).**

For $\mathcal{X}$ and $L$ as above, the relative homotopy group $\pi_{2}(\mathcal{X},L)$ is generated by the classes $\beta_{i}$ for $i=0,\dots,m-1$ together with $\beta_{\nu}$ for $\nu\in{\rm Box}^{\prime}(\Sigma)$.

As in [9], these generators of $\pi_{2}(\mathcal{X},L)$ are called basic disk classes. They are the analogue of Maslov index $2$ disk classes in toric manifolds.

Let

[TABLE]

be the moduli space of good representable stable maps from bordered orbifold Riemann surfaces of genus zero with $k+1$ boundary marked points $z_{0},z_{1},\dots,z_{k}$ and $l$ interior (orbifold) marked points $z_{1}^{+},\dots,z_{l}^{+}$ in the homotopy class $\beta$ of type $\boldsymbol{x}=(\mathcal{X}_{\nu_{1}},\dots,\mathcal{X}_{\nu_{l}})$. The superscript “$main$” is meant to indicate the connected component on which the boundary marked points respect the cyclic order of $S^{1}=\partial D^{2}$. According to [9, Lemma 2.5], $\mathcal{M}^{\rm op,main}_{k+1,l}(\mathcal{X},L,\beta,\boldsymbol{x})$ has real virtual dimension

[TABLE]

By [9, Proposition 9.4], if $\mathcal{M}^{\rm op,main}_{1,1}(\mathcal{X},L,\beta)$ is non-empty and if $\partial\beta$ is not in the sublattice generated by $\boldsymbol{b}_{0},\dots,\boldsymbol{b}_{m-1}$, then there exist $\nu\in{\rm Box}^{\prime}(\Sigma)$, $k_{0},\dots,k_{m-1}\in\mathbb{N}$ and $\alpha\in H_{2}^{\rm eff}(\mathcal{X})$ such that $\beta=\beta_{\nu}+\sum\limits_{i=0}^{m-1}k_{i}\beta_{i}+\alpha,$ where $\alpha$ is realized by a union of holomorphic (orbi-)spheres. The CW Maslov index of $\beta$ written in this way is given by $\mu_{\rm CW}(\beta)=2\operatorname{age}(\nu)+2\sum\limits_{i=0}^{m-1}k_{i}+2c_{1}(\mathcal{X})\cdot\alpha$.

2.2.2 Orbi-disk invariants

Pick twisted sectors $\mathcal{X}_{\nu_{1}},\dots,\mathcal{X}_{\nu_{l}}$ of the toric orbifold $\mathcal{X}$. Consider the moduli space

[TABLE]

of good representable stable maps from bordered orbifold Riemann surfaces of genus zero with one boundary marked point and $l$ interior orbifold marked points of type $\boldsymbol{x}=(\mathcal{X}_{\nu_{1}},\dots,\mathcal{X}_{\nu_{l}})$ representing the class $\beta\in\pi_{2}(\mathcal{X},L)$. According to [9], $\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X},L,\beta,\boldsymbol{x})$ can be equipped with a virtual fundamental chain, which has an expected dimension $n$ if the following equality holds:

[TABLE]

Throughout the paper, we make the following assumptions.

Assumption 2.5**.**

We assume that the toric orbifold $\mathcal{X}$ is semi-Fano $($see Definition 2.1)* and Gorenstein.555This means that $K_{\mathcal{X}}$ is Cartier. Moreover, we assume that the type $\boldsymbol{x}$ consists of twisted sectors with $\operatorname{age}\leq 1$.666This assumption does not impose any restriction in the construction of the SYZ mirror over $H^{\leq 2}_{\mathrm{CR}}(\mathcal{X})$. We do not discuss mirror construction in this paper.*

Then the age of every twisted sector of $\mathcal{X}$ is a non-negative integer. Since a basic orbi-disk class $\beta_{\nu}$ has Maslov index $2\operatorname{age}(\nu)$, we see that every non-constant stable disk class has at least Maslov index $2$.

Moreover, the virtual fundamental chain \big{[}\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X},L,\beta,\boldsymbol{x})\big{]}^{\rm vir} has expected dimension $n$ when $\mu_{\rm CW}(\beta)=2$, and in fact we get a virtual fundamental cycle because $\beta$ attains the minimal Maslov index, thus preventing disk bubbling to occur. Therefore the following definition of genus [math] open orbifold GW invariants (also known as orbi-disk invariants) is independent of the choice of perturbations of the Kuranishi structures:777In the general case one may restrict to torus-equivariant perturbations, as did in [14, 15, 17].

Definition 2.6** (orbi-disk invariants).**

Let $\beta\in\pi_{2}(\mathcal{X},L)$ be a relative homotopy class with Maslov index given by (2.6). Suppose that the moduli space $\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X},L,\beta,\boldsymbol{x})$ has a virtual fundamental cycle of dimension $n$. Then we define

[TABLE]

where $\operatorname{ev}_{0}\colon\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X},L,\beta,\boldsymbol{x})\to L$ is the evaluation map at the boundary marked point, $[{\rm pt}]_{L}\in H^{n}(L;\mathbb{Q})$ is the point class of $L$, and $\mathbf{1}_{\nu_{j}}\in H^{0}(\mathcal{X}_{\nu_{j}};\mathbb{Q})\subset H^{2\operatorname{age}(\nu_{j})}_{\mathrm{CR}}(\mathcal{X};\mathbb{Q})$ is the fundamental class of the twisted sector $\mathcal{X}_{\nu_{j}}$.

Remark 2.7**.**

The Kuranishi structures in this paper are the same as those defined in [14, 15], incorporating the works [6, 7] for the interior orbifold marked points. This has been explained in [9, Section 10]. We also refer the readers to [13, Appendix] and [16] for the detailed construction, and to [23] (and its forthcoming sequels) for a different approach.

The moduli spaces considered here are in fact much simpler than those in [14, 15] (and [13]) because we only need to consider stable disks with just one disk component which is minimal, and hence disk bubbling does not occur. In particular, we do not have codimension-one boundary components, and hence the above definition is independent of choices of Kuranishi perturbations.

For a basic (orbi-)disk with at most one interior orbifold marked point, the corresponding moduli space $\mathcal{M}^{\rm op,main}_{1,0}(\mathcal{X},L,\beta_{i})$ (or $\mathcal{M}^{\rm op,main}_{1,1}(\mathcal{X},L,\beta_{\nu},\nu)$ when $\beta_{\nu}$ is a basic orbi-disk class) is regular and can be identified with $L$. Thus the associated invariants are evaluated as follows [9]:

For $\nu\in{\rm Box}^{\prime}$, we have $n_{1,1,\beta_{\nu}}^{\mathcal{X}}([{\rm pt}]_{L};\mathbf{1}_{\nu})=1.$ 2. 2.

For $i\in\{0,\dots,m-1\}$, we have $n_{1,0,\beta_{i}}^{\mathcal{X}}([{\rm pt}]_{L})=1.$

When there are more interior orbifold marked points or when the disk class is not basic, the corresponding moduli space is in general non-regular and virtual theory is involved in the definition, making the invariant much more difficult to compute.

3 Geometric constructions

Let $\beta\in\pi_{2}(\mathcal{X},L)$ be a disk class with $\mu_{\rm CW}(\beta)=2$. By the discussion in Section 2.2, we can write

[TABLE]

with $\alpha\in H_{2}(\mathcal{X},\mathbb{Z})$, $c_{1}(\mathcal{X})\cdot\alpha=0$ and either $\beta_{\mathbf{d}}\in\{\beta_{0},\dots,\beta_{m-1}\}$ or $\beta_{\mathbf{d}}\in\text{Box}^{\prime}(\mathcal{X})$ is of age $1$. Denote by $\boldsymbol{b}_{d}\in N$ the element corresponding to $\beta_{\mathbf{d}}$.

Recall that the fan polytope $\mathcal{P}\subset N_{\mathbb{R}}$ is the convex hull of the vectors $\boldsymbol{b}_{0},\dots,\boldsymbol{b}_{m-1}$. Note that $\boldsymbol{b}_{d}\in\mathcal{P}$. Denote by $F(\boldsymbol{b}_{d})$ the minimal face of the fan polytope $\mathcal{P}$ that contains the vector $\boldsymbol{b}_{d}$. Let $F$ be a facet of $\mathcal{P}$ that contains $F(\boldsymbol{b}_{d})$. Let $\Sigma_{\beta_{\mathbf{d}}}\subset\Sigma$ be the minimal convex subfan containing all $\{\boldsymbol{b}_{0},\dots,\boldsymbol{b}_{m-1}\}\cap F$. The vectors

[TABLE]

determine a fan map $\mathbb{Z}^{p}\to N$ (where $p$ is the number of vectors above). Let

[TABLE]

be the associated toric suborbifold (of the same dimension $n$).

Lemma 3.1**.**

$\mathcal{X}_{\beta_{\mathbf{d}}}$* is a toric Calabi–Yau orbifold.*

Proof.

All the generators in (3.1) lie in the hyperplane containing $F$. Since $\mathcal{X}$ is Gorenstein, this hyperplane has a defining equation $\nu=0$ for some primitive vector $\nu\in M$. Hence $\mathcal{X}_{\beta_{\mathbf{d}}}$ is toric Calabi–Yau. ∎

Example 3.2**.**

Consider $\mathbb{P}^{2}/\mathbb{Z}_{3}$, whose fan is shown in the left of Fig. 1. If $\beta_{\mathbf{d}}$ corresponds to the vector $(1,0)$ (which is marked as ‘113’ in the figure), then $\Sigma_{\beta_{\mathbf{d}}}$ is the cone spanned by $v_{2}=(2,-1)$ and $v_{3}=(-1,2)$. If $\beta_{\mathbf{d}}$ corresponds to the vector $v_{3}$, then $\Sigma_{\beta_{\mathbf{d}}}$ can be taken to be the cone spanned by $v_{2}$, $v_{3}$, or the cone spanned by $v_{1}$, $v_{3}$. In both cases, the corresponding toric Calabi–Yau orbifold is $\mathbb{C}^{2}/\mathbb{Z}_{3}$.

Note that $\mathcal{X}_{\beta_{\mathbf{d}}}$ depends on the choice of the face $F$, not just $\beta_{\mathbf{d}}$. We use $\mathcal{X}_{\beta_{\mathbf{d}}}$ to compute open Gromov–Witten invariants of $\mathcal{X}$ in class $\beta=\beta_{\mathbf{d}}+\alpha$.

In what follows we show that $\mathcal{X}_{\beta_{\mathbf{d}}}\subset\mathcal{X}$ contains all stable orbi-disks of $\mathcal{X}$ of class $\beta$. First, we have the following analogue of [5, Proposition 5.6].

Lemma 3.3**.**

Let $f\colon\mathcal{D}\cup\mathcal{C}\to\mathcal{X}$ be a stable orbi-disk map in the class $\beta=\beta_{\mathbf{d}}+\alpha$, where $\mathcal{D}$ is a $($possibly orbifold$)$ disk and $\mathcal{C}$ is a $($possibly orbifold$)$ rational curve such that $f_{*}[\mathcal{D}]=\beta_{\mathbf{d}}$ and $f_{*}[\mathcal{C}]=\alpha$ with $c_{1}(\alpha)=0$. Then we have

[TABLE]

and $[f(\mathcal{C})]\cdot D_{j}=0$ whenever $\boldsymbol{b}_{j}\notin F(\boldsymbol{b}_{d})$.

Proof.

Since $c_{1}(\alpha)=0$, $f(\mathcal{C})$ should lie in toric divisors of $\mathcal{X}$. Recall that $\beta_{\mathbf{d}}$ achieves the minimal Maslov index $2$, and hence there is no disc bubbling.

Suppose $\beta_{\mathbf{d}}$ is a smooth disk class. Then each sphere component $\mathcal{C}_{0}$ meeting the disk component $\mathcal{D}$ maps into the divisor $D_{\mathbf{d}}$ and it should have non-negative intersection with other toric divisors. By [18, Lemma 4.5] which easily extends to the simplicial setting, we have the desired statement for $f(\mathcal{C}_{0})$.

If $\beta_{\mathbf{d}}$ is an orbi-disk class, then we can write the corresponding $\boldsymbol{b}_{d}\in N$ as $\boldsymbol{b}_{d}=\sum\limits_{\boldsymbol{b}_{i}\in\sigma}c_{i}\boldsymbol{b}_{i}$, with $\sum_{i}c_{i}=1,c_{i}\in[0,1)\cap\mathbb{Q}$. For a sphere component $\mathcal{C}_{0}$ meeting the disk component $\mathcal{D}$, we have $f(\mathcal{C}_{0})\subset\bigcup\limits_{\boldsymbol{b}_{i}\in\sigma}D_{i}$ and each $\boldsymbol{b}_{i}\in\sigma$ satisfies $\boldsymbol{b}_{i}\in F(\boldsymbol{b}_{d})$. Hence $f(\mathcal{C}_{0})\subset\bigcup\limits_{\boldsymbol{b}_{i}\in F(\boldsymbol{b}_{d})}D_{i}$ and $f(\mathcal{C}_{0})\cdot D_{j}=0$ for $\boldsymbol{b}_{j}\notin F(\boldsymbol{b}_{d})$.

Let $\mathcal{C}_{1}\subset\mathcal{C}$ be a sphere component meeting $\mathcal{C}_{0}$, then we have $f(\mathcal{C}_{1})\subset F(\boldsymbol{b}_{j})$ for some $\boldsymbol{b}_{j}\in F(\boldsymbol{b}_{d})$ by the intersection condition. Now, we can follow the proof of [5, Proposition 5.6] shows that $f(\mathcal{C}_{1})\subset\bigcup\limits_{\boldsymbol{b}_{i}\in F(\boldsymbol{b}_{d})}D_{i}$. The result follows by repeating this argument for one sphere component at a time. ∎

Partition $\{\boldsymbol{b}_{0},\dots,\boldsymbol{b}_{m-1}\}\cap F(\boldsymbol{b}_{d})$ into the disjoint union of two subsets,

[TABLE]

where $\boldsymbol{b}_{i}\in F(\boldsymbol{b}_{d})^{c}$ if $D_{i}\subset\mathcal{X}_{\beta_{\mathbf{d}}}$ and $\boldsymbol{b}_{i}\in F(\boldsymbol{b}_{d})^{nc}$ if $D_{i}\not\subset\mathcal{X}_{\beta_{\mathbf{d}}}$.

Lemma 3.4**.**

Let $f\colon\mathcal{D}\cup\mathcal{C}\to\mathcal{X}$ be as in Lemma 3.3. Then we have $f(\mathcal{D}\cup\mathcal{C})\subset\mathcal{X}_{\beta_{\mathbf{d}}}$.

Proof.

Certainly $f(\mathcal{D})\subset\mathcal{X}_{\beta_{\mathbf{d}}}$. We claim that

[TABLE]

from which the lemma follows.

To see (3.2), we write $\mathcal{C}=\mathcal{C}_{c}\cup\mathcal{C}_{nc}$ where $\mathcal{C}_{c}$ consists of components of $\mathcal{C}$ which lie in $\bigcup\limits_{\boldsymbol{b}_{j}\in F(\boldsymbol{b}_{d})^{c}}D_{j}$, and $\mathcal{C}_{nc}$ consists of the remaining components. Set $A:=f_{*}[\mathcal{C}_{c}]$ and $B:=f_{*}[\mathcal{C}_{nc}]$. Then $\alpha=A+B$. Since $-K_{\mathcal{X}}$ is nef and $-K_{\mathcal{X}}\cdot\alpha=0$, we have $-K_{\mathcal{X}}\cdot A=0=-K_{\mathcal{X}}\cdot B$. Write $B=\sum_{k}c_{k}B_{k}$ as an effective linear combination of the classes $B_{k}$ of irreducible $1$-dimensional torus-invariant orbits in $\mathcal{X}$. Again because $-K_{\mathcal{X}}$ is nef, we have $-K_{\mathcal{X}}\cdot B_{k}=0$ for all $k$. Each $B_{k}$ corresponds to an $(n-1)$-dimensional cone $\sigma_{k}\in\Sigma$. In the expression $B=\sum_{k}c_{k}B_{k}$, there is at least one (non-zero) $B_{k}$ which is not contained in $\bigcup\limits_{\boldsymbol{b}_{j}\in F(\boldsymbol{b}_{d})^{c}}D_{j}$. As a consequence, either $\sigma_{k}$ contains a ray $\mathbb{R}_{\geq 0}\boldsymbol{b}_{j}$ with $\boldsymbol{b}_{j}\notin F(\boldsymbol{b}_{d})$, or there exists a $\boldsymbol{b}_{j}\notin F(\boldsymbol{b}_{d})$ such that $\sigma_{k}$ and $\boldsymbol{b}_{j}$ span an $n$-dimensional cone in $\Sigma$.

Since $B_{k}$ is not contained in $\bigcup\limits_{\boldsymbol{b}_{j}\in F(\boldsymbol{b}_{d})^{c}}D_{j}$, we see that if $\boldsymbol{b}_{i}\in F(\boldsymbol{b}_{d})^{c}$ then $\boldsymbol{b}_{i}\notin\sigma_{k}$. Also, $D\cdot B_{k}\geq 0$ for every toric prime divisor $D\subset\mathcal{X}$ not corresponding to a ray in $\sigma_{k}$.

By [18, Lemma 4.5] (which easily extends to the simplical setting), we have $D\cdot B_{k}=0$ for every toric prime divisor $D\subset\mathcal{X}$ corresponding to an element in $\{\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{m}\}\setminus F(\sigma_{k})$, where $F(\sigma_{k})\subset\mathcal{P}$ is the minimal face of $\mathcal{P}$ containing rays in $\sigma_{k}$. Since the divisors $D\subset\mathcal{X}$ corresponding to $\{\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{m}\}\setminus F(\sigma_{k})$ span $H^{2}(\mathcal{X})$, we have $B_{k}=0$, a contradiction. ∎

Let $\boldsymbol{x}=(\mathcal{X}_{\nu_{1}},\dots,\mathcal{X}_{\nu_{l}})$ be an $l$-tuple of twisted sectors of $\mathcal{X}_{\beta_{\mathbf{d}}}$. Then Lemma 3.4 implies that the natural inclusion $\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X}_{\beta_{\mathbf{d}}},L,\beta,\boldsymbol{x})\hookrightarrow\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X},L,\beta,\boldsymbol{x})$ is a bijection. Since $\mathcal{X}_{\beta_{\mathbf{d}}}\subset\mathcal{X}$ is open, the local deformations and obstructions of stable discs in $\mathcal{X}_{\beta_{\mathbf{d}}}$ and their inclusion in $\mathcal{X}$ are isomorphic. It follows that

Proposition 3.5**.**

The moduli spaces $\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X},L,\beta,\boldsymbol{x})$ of disks in $\mathcal{X}$ is isomorphic as Kuranishi spaces to the moduli spaces $\mathcal{M}^{\rm op,main}_{1,l}(\mathcal{X}_{\beta_{\mathbf{d}}},L,\beta,\boldsymbol{x})$ of disks in $\mathcal{X}_{\beta_{\mathbf{d}}}$. Consequently

[TABLE]

Since $\mathcal{X}_{\beta_{\mathbf{d}}}$ is a toric Calabi–Yau orbifold, the open Gromov–Witten invariants $n_{1,l,\beta}^{\mathcal{X}_{\beta_{\mathbf{d}}}}([{\rm pt}]_{L};\allowbreak\mathbf{1}_{\nu_{1}},\dots,\mathbf{1}_{\nu_{l}})$ have been computed in [4]. By Proposition 3.5, this gives open Gromov–Witten invariants of $\mathcal{X}$. Explicitly they are given as follows.

Using the toric data of ${\mathcal{X}_{\beta_{\mathbf{d}}}}$, we define

[TABLE]

Theorem 3.6**.**

If $\beta_{\mathbf{d}}=\beta_{i_{0}}$ is a basic smooth disk class corresponding to the ray generated by $\boldsymbol{b}_{i_{0}}$ for some $i_{0}\in\{0,1,\dots,m-1\}$, then we have

[TABLE]

via the inverse $y=y(q,\tau)$ of the toric mirror map (3.3).

If $\beta_{\mathbf{d}}=\beta_{\nu_{j_{0}}}$ is a basic orbi-disk class corresponding to $\nu_{j_{0}}\in{\rm Box}^{\prime}(\Sigma)^{\operatorname{age}=1}$ for some $j_{0}\in\{m,m+1,\dots,m^{\prime}-1\}$, then we have

[TABLE]

via the inverse $y=y(q,\tau)$ of the toric mirror map (3.3), where $D_{j_{0}}^{\vee}\in\mathbb{K}_{\rm eff}$ is the class defined in (2.4), $I_{j_{0}}\in\mathcal{A}$ is the anticone of the minimal cone containing $\boldsymbol{b}_{j_{0}}=\nu_{j_{0}}$ and $c_{j_{0}i}\in\mathbb{Q}\cap[0,1)$ are rational numbers such that $\boldsymbol{b}_{j_{0}}=\sum\limits_{i\notin I_{j_{0}}}c_{j_{0}i}\boldsymbol{b}_{i}$.

Proof.

By Proposition 3.5, $n_{1,l,\beta}^{\mathcal{X}}([{\rm pt}]_{L};\mathbf{1}_{\nu_{1}},\dots,\mathbf{1}_{\nu_{l}})=n_{1,l,\beta}^{\mathcal{X}_{\beta_{\mathbf{d}}}}([{\rm pt}]_{L};\mathbf{1}_{\nu_{1}},\dots,\mathbf{1}_{\nu_{l}})$, and so the l.h.s. of (3.4) is equal to

[TABLE]

which in turn is equal to \exp\big{(}{-}A^{\mathcal{X}_{\beta_{\mathbf{d}}}}_{i_{0}}(y(q,\tau))\big{)} by [4, Theorem 1.4]. The deduction for (3.5) is similar. ∎

To combine all the invariants into a single expression, one defines the orbi-disc potential

[TABLE]

where $\beta_{\mathbf{d}}$ runs over all the basic smooth or orbi-disc classes, and $Z^{\beta_{\mathbf{d}}}$ are monomials associated to $\beta_{\mathbf{d}}$. See [5, Definition 19] for more detail. The above theorem gives an explicit expression of $W$ via the mirror map.

Example 3.7**.**

$\mathbb{P}^{2}/\mathbb{Z}_{3}$ is a Gorenstein Fano toric orbifold. Its fan and polytope pictures are shown in Fig. 1. It has three toric divisors $D_{1}$, $D_{2}$, $D_{3}$ corresponding to the rays generated by $v_{1}=(-1,-1)$, $v_{2}=(2,-1)$, $v_{3}=(-1,2)$. By pairing with the dual vectors $(1,0)$ and $(0,1)$, the linear equivalence relations are $2D_{2}-D_{3}-D_{1}\sim 0$ and $2D_{3}-D_{2}-D_{1}\sim 0$, and so $D_{1}\sim D_{2}\sim D_{3}$. It has three orbifold points corresponding to the three vertices in the polytope picture. Locally it is $\mathbb{C}^{2}/\mathbb{Z}_{3}$ around each orbifold point.

Fix a Lagrangian torus fiber. $\mathbb{P}^{2}/\mathbb{Z}_{3}$ has nine basic orbi-disk classes corresponding to the nine lattice points on the boundary of the fan polytope. Three of them are smooth disk classes and denote them by $\beta_{1}$, $\beta_{2}$, $\beta_{3}$. The basic orbi-disk classes corresponding to the two lattice points $(2v_{1}+v_{2})/3$ and $(v_{1}+2v_{2})/3$ are denoted by $\beta_{112}$ and $\beta_{122}$, which pass through the twisted sectors $\nu_{112}$ and $\nu_{122}$ respectively. Then $2\beta_{1}+\beta_{2}-3\beta_{112}$ (or $2\beta_{2}+\beta_{1}-3\beta_{122}$) is the class of a constant orbi-sphere passing through the twisted sector $\nu_{112}$ (or $\nu_{122}$ resp.). In particular the area of $\beta_{112}$ equals to $(2\beta_{1}+\beta_{2})/3$. Other basic orbi-disk classes have similar notations.

Theorem 3.6 provides a formula for the open GW invariants n^{\mathcal{X}}_{1,l,\beta_{112}}\Big{(}[{\rm pt}]_{L};\prod\limits_{i=1}^{l}\mathbf{1}_{\nu_{i}}\Big{)} where $\nu_{i}$ is either $\nu_{112}$ or $\nu_{122}$ for each $i$. To write down the invariants more systematically, we consider the open GW potential as follows.

Let $q$ be the Kähler parameter of the smooth sphere class \beta_{1}+\beta_{2}+\beta_{3}\in H_{2}\big{(}\mathbb{P}^{2}/\mathbb{Z}_{3}\big{)}. The basic orbi-disk classes correspond to monomials in the disk potential $q^{\beta}z^{\partial\beta}$, where $q^{\beta_{1}}=q^{\beta_{2}}=q^{\beta_{112}}=q^{\beta_{122}}=1$, $q^{\beta_{3}}=q^{\beta_{1}+\beta_{2}+\beta_{3}}=q$, $q^{\beta_{223}}=q^{(2\beta_{2}+\beta_{3})/3}=q^{\beta_{3}/3}=q^{1/3}$, and similar for other basic orbi-disk classes. The Kähler parameters corresponding to the twisted sectors $\nu_{112}$, $\nu_{122}$ are denoted as $\tau_{112},\tau_{122}$ (and similar for other twisted sectors).

By [4, Example 1, Section 6.5], the open GW potential for $\mathbb{C}^{2}/\mathbb{Z}_{3}$ is given by

[TABLE]

where

[TABLE]

By Proposition 3.5, the disk invariants of $\mathbb{P}^{2}/\mathbb{Z}_{3}$ equal to those of $\mathbb{C}^{2}/\mathbb{Z}_{3}$. Thus the open GW potential of $\mathbb{P}^{2}/\mathbb{Z}_{3}$ is given by

[TABLE]

Then the generating functions of open orbifold GW for $\beta_{112}$ and $\beta_{122}$ are given by the coefficients of $w^{-1}$ and $zw^{-1}$ in $W$ respectively. The first few terms are given by the following table.

In the above table,

[TABLE]

We observe that all invariants satisfy ‘reciprocal integrality’, namely their reciprocals are integers. Moreover, all these integers are divisible by $6$. $n_{(k,k)}=0$. Furthermore, the sign is alternating with respect to $b$.

Acknowledgments

K. Chan was supported by a Hong Kong RGC grant CUHK14314516 and direct grants from CUHK. C.-H. Cho was supported by the NRF grant funded by the Korea government(MSIT) (No. 2017R1A22B4009488). S.-C. Lau was partially supported by the Simons collaboration grant #580648. N.C. Leung was supported by Hong Kong RGC grants CUHK14302215 & CUHK14303516 and direct grants from CUHK. H.-H. Tseng was supported in part by NSF grant DMS-1506551.