2.1 Definition

Let N be a finitely generated abelian group, and let $N_{\mathbb {R}} =N\otimes _{\mathbb {Z}} \mathbb {R} $ . We have a short exact sequence of (additive) abelian groups:

$$ \begin{align*}0 \to N_ {\mathrm{tor}} \to N \to \bar{N}=N/N_ {\mathrm{tor}} \to 0, \end{align*} $$

where $N_ {\mathrm {tor}} $ is the subgroup of torsion elements in N. Then $N_ {\mathrm {tor}} $ is a finite abelian group and $\bar {N}= \mathbb {Z} ^n$ , where $n=\dim _{\mathbb {R}} N_{\mathbb {R}} $ . The natural projection $N\to \bar {N}$ is denoted $b\mapsto \bar {b}$ . A smooth toric DM stack is an extension of toric varieties [Reference Fulton39, Reference Borisov, Chen and Smith7]. A smooth toric DM stack is given by the following data:

• ○ $b_1,\dots , b_{r'} \in N$ that generate a subgroup of N of finite index, and

• ○ a simplicial fan $\Sigma $ in $N_{\mathbb {R}} $ such that the set of $1$ -cones is

  $$ \begin{align*}\{\rho_1,\dots,\rho_{r'}\}, \end{align*} $$
  where $\rho _i= \mathbb {R} _{\ge 0} \bar b_i$ , $i=1,\dots , r'$ .

The datum $\mathbf \Sigma =(\Sigma , (b_1,\dots , b_{r'}))$ is a stacky fan in the sense of [Reference Borisov, Chen and Smith7]. The vectors $b_1,\dots , b_{r'}$ may or may not generate N; if they do not, we choose additional vectors $b_{r'+1},\dots , b_r$ such that $b_1,\dots , b_r$ generate N. There is a surjective group homomorphism

$$ \begin{align*} \phi : {\widetilde{N}} :=\oplus_{i=1}^r \mathbb{Z} & {\tilde{b}} _i \longrightarrow N,\\ & {\tilde{b}} _i \mapsto b_i. \end{align*} $$

Define $\mathbb {L} :=\mathrm {Ker}(\phi ) \cong \mathbb {Z} ^k$ , where $k:=r-n$ . Then we have the following short exact sequence of finitely generated abelian groups:

(1) $$ \begin{align} 0\to \mathbb{L} \stackrel{\psi }{\longrightarrow} {\widetilde{N}} \stackrel{\phi}{\longrightarrow} N\to 0. \end{align} $$

Applying $ - \otimes _{\mathbb {Z}} \mathbb {C} ^*$ to equation (1), we obtain an exact sequence of abelian groups

(2) $$ \begin{align} 1 \to K \to G \to {\widetilde{ \mathbb{T} }} \to \mathbb{T} \to 1, \end{align} $$

where

$$ \begin{align*} \mathbb{T} &:= N\otimes_{\mathbb{Z}} \mathbb{C} ^* =\bar{N}\otimes_{\mathbb{Z}} \mathbb{C} ^* \cong ( \mathbb{C} ^*)^n,\\ {\widetilde{ \mathbb{T} }} &:= {\widetilde{N}} \otimes_{\mathbb{Z}} \mathbb{C} ^* \cong ( \mathbb{C} ^*)^r,\\ G &:= \mathbb{L}\otimes_{\mathbb{Z}} \mathbb{C} ^* \cong ( \mathbb{C} ^*)^k,\\ K &:= {\mathrm{Tor}} _1^{ \mathbb{Z} }(N, \mathbb{C} ^*)\cong N_ {\mathrm{tor}}. \end{align*} $$

The action of $ {\widetilde { \mathbb {T} }} $ on itself extends to a $ {\widetilde { \mathbb {T} }} $ -action on $ \mathbb {C} ^r = \mathrm {Spec} \mathbb {C} [Z_1,\dots , Z_r]$ . The torus G acts on $ \mathbb {C} ^r$ via the group homomorphism $G\to {\widetilde { \mathbb {T} }} $ in equation (2), so $K\subset G$ acts on $ \mathbb {C} ^r$ trivially. The isomorphism $K\cong N_ {\mathrm {tor}} $ is not canonical.

With the above preparation, we are now ready to define a smooth toric DM stack $\mathcal {X}$ . Let

$$ \begin{align*}\mathcal{A}=\{I\subset \{1,\dots, r\}: \sum_{i\notin I} \mathbb{R} _{\ge 0} \bar b_i\ \text{is a cone of}\ \Sigma\} \end{align*} $$

be the set of anti-cones; note that $\{r'+1,\ldots , r\} \subset I$ for any anti-cone $I\subset \mathcal {A}$ . Given $I\subset \{1,\ldots , r\}$ , let $ \mathbb {C} ^I$ be the subvariety of $ \mathbb {C} ^r$ defined by the ideal in $ \mathbb {C} [Z_1,\ldots , Z_r]$ generated by $\{ Z_i \mid i\notin I\}$ . Define the smooth toric DM stack $\mathcal {X}$ as the quotient stack

$$ \begin{align*}\mathcal{X}:=[U_{\mathcal{A}}/ G], \end{align*} $$

where

$$ \begin{align*}U_{\mathcal{A}}:= \mathbb{C} ^r \backslash \bigcup_{I\notin \mathcal A} \mathbb{C} ^I = \bigcap_{I\notin \mathcal{A}} \left( \mathbb{C} ^r \backslash \mathbb{C} ^I\right). \end{align*} $$

Note that for $i=r'+1,\ldots ,r$ , $ \mathbb {R} _{\geq 0} b_i$ is not a cone in $\Sigma $ , so $\{ i\}' := \{1,\ldots , r\} \backslash \{i\} \notin \mathcal {A}$ . Therefore,

$$ \begin{align*}U_{\mathcal{A}} \subset \bigcap_{i=r'+1}^r \left( \mathbb{C} ^r\backslash \mathbb{C} ^{\{i\}'}\right) = \mathbb{C} ^{r'}\times ( \mathbb{C} ^*)^{r-r'}. \end{align*} $$

The stack $\mathcal {X}$ contains the DM torus $\mathcal {T} := [ {\widetilde { \mathbb {T} }} /G]$ as a dense open subset, and the $ {\widetilde { \mathbb {T} }} $ -action on $U_{\mathcal {A}}$ descends to a $\mathcal {T}$ -action on $\mathcal {X}$ . The smooth toric DM stack $\mathcal {X}$ is a toric orbifold if the G-action on $ {\widetilde { \mathbb {T} }} $ is free.

Let $G^ {\mathrm {rig}} = G/K$ . Then $G^ {\mathrm {rig}} $ acts freely on $ {\widetilde { \mathbb {T} }} $ and $ {\widetilde { \mathbb {T} }} /G^ {\mathrm {rig}} = \mathbb {T} $ . The rigidification of the smooth toric DM stack $\mathcal {X}$ is the toric orbifold

$$ \begin{align*}\mathcal{X}^{ {\mathrm{rig}} } = [U_{\mathcal{A}}/G^ {\mathrm{rig}} ]. \end{align*} $$

The coarse moduli space of the stack $\mathcal {X}$ is the simplical toric variety $X_\Sigma $ defined by the simplicial fan $\Sigma $ in $N_{\mathbb {R}} \cong \mathbb {R} ^n$ . By [Reference Fantechi, Mann and Nironi37, Theorem I], the morphism $\mathcal {X}\to X_\Sigma $ factors canonically via toric morphisms

(3) $$ \begin{align} \mathcal{X} \longrightarrow \mathcal{X}^{ {\mathrm{rig}} } \longrightarrow \mathcal{X}^{ {\mathrm{can}} }\longrightarrow X_{\Sigma}, \end{align} $$

where

• ○ $\mathcal {X}\longrightarrow \mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ is a K-gerbe over $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ ;

• ○ $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} \longrightarrow \mathcal {X}^ {\mathrm {can}} $ is a fibred product of roots of toric divisors;

• ○ $\mathcal {X}^ {\mathrm {can}} \longrightarrow X_\Sigma $ is the minimal orbifold having $X_\Sigma $ as coarse moduli space.

Restricting equation (3) to the open substack $\mathcal {T} \subset \mathcal {X}$ , one obtains $\mathcal {T} \cong \mathbb {T} \times \mathcal {B} K \longrightarrow \mathbb {T} \longrightarrow \mathbb {T} \longrightarrow \mathbb {T} $ , where $ \mathbb {T} \times \mathcal {B} K \longrightarrow \mathbb {T} $ is the projection to the first factor and $ \mathbb {T} \longrightarrow \mathbb {T} $ is the identity map.

Let M, $ {\widetilde {M}} $ , and $\mathbb {L}^\vee $ be the character lattices of the tori $ \mathbb {T} $ , $ {\widetilde { \mathbb {T} }} $ and G, respectively:

$$ \begin{align*} M &= \mathrm{Hom}(N, \mathbb{Z} ) = \mathrm{Hom}( \mathbb{T} , \mathbb{C} ^*), \\ {\widetilde{M}} &= \mathrm{Hom}( {\widetilde{N}} , \mathbb{Z} )= \mathrm{Hom}( {\widetilde{ \mathbb{T} }} , \mathbb{C} ^*),\\ \mathbb{L}^\vee &= \mathrm{Hom}(\mathbb{L}, \mathbb{Z} ) =\mathrm{Hom}(G, \mathbb{C} ^*). \end{align*} $$

Applying $\mathrm {Hom}(-, \mathbb {Z} )$ to equation (1), we obtain the following exact sequence of (additive) abelian groups:

(4) $$ \begin{align} 0 \longrightarrow M \stackrel{\phi^\vee}{\longrightarrow} {\widetilde{M}} \stackrel{\psi^\vee}{\longrightarrow} \mathbb{L}^\vee \longrightarrow \mathrm{Ext}^1(N, \mathbb{Z} ) \longrightarrow 0. \end{align} $$

Therefore, the group homomorphism $\psi ^\vee : {\widetilde {M}} \longrightarrow \mathbb {L}^\vee $ is surjective if and only if $N_ {\mathrm {tor}} =0$ .

We now consider a class of examples of 3-dimensional Calabi-Yau smooth toric DM stacks of the form $[ \mathbb {C} ^3/ \mathbb {Z} _3]$ . Let $\omega =e^{\frac {2\pi \sqrt {-1}}{3}}$ be the generator of $ \mathbb {Z} _3$ . Given $i,j,k\in \{0,1,2\}$ such that $i+j+k\in 3 \mathbb {Z} $ , we define $\mathcal {X}_{i,j,k}$ to be the quotient stack of the following $ \mathbb {Z} _3$ -action on $ \mathbb {C} ^3$ :

$$ \begin{align*}\omega \cdot (Z_1,Z_2, Z_3) = (\omega^i Z_1, \omega^j Z_2, \omega^k Z_3). \end{align*} $$

In the following example, we consider

$$ \begin{align*}\mathcal{X}_{1,1,1},\quad \mathcal{X}_{1,2,0}=[ \mathbb{C} ^2/ \mathbb{Z} _3]\times \mathbb{C} ,\quad \mathcal{X}_{0,0,0}= \mathbb{C} ^3\times \mathcal{B} \mathbb{Z} _3. \end{align*} $$

2.2 Equivariant line bundles and torus-invariant Cartier divisors

A character $\chi \in {\widetilde {M}} $ gives a $ {\widetilde { \mathbb {T} }} $ -action on $ \mathbb {C} ^r \times \mathbb {C} $ by

$$ \begin{align*}(\tilde{t}_1,\ldots, \tilde{t}_r)\cdot (Z_1,\ldots, Z_r, u) =(\tilde{t}_1 Z_1,\ldots, \tilde{t}_r Z_r, \chi(\tilde{t}_1,\ldots, \tilde{t}_r) u), \end{align*} $$

where

$$ \begin{align*}(\tilde{t}_1,\ldots, \tilde{t}_r)\in {\widetilde{ \mathbb{T} }} \cong ( \mathbb{C} ^*)^r, \quad (Z_1,\ldots, Z_r)\in \mathbb{C} ^r,\quad u\in \mathbb{C}. \end{align*} $$

Therefore $ \mathbb {C} ^r\times \mathbb {C} $ can be viewed as the total space of a $ {\widetilde { \mathbb {T} }} $ -equivariant line bundle $ {\widetilde {L}} _\chi $ over $ \mathbb {C} ^r$ . If

$$ \begin{align*}\chi(\tilde{t}_1,\ldots, \tilde{t}_r) =\prod_{i=1}^r \tilde{t}_i^{c_i}, \end{align*} $$

where $c_1,\ldots , c_r\in \mathbb {Z} $ , then

$$ \begin{align*}{\widetilde{L}} _{\chi}= \mathcal{O}_{ \mathbb{C} ^r}(\sum_{i=1}^r c_i {\widetilde{\mathcal{D}}}_i), \end{align*} $$

where $ {\widetilde {\mathcal {D}}}_i$ is the $ {\widetilde { \mathbb {T} }} $ -divisor in $ \mathbb {C} ^r$ defined by $Z_i=0$ . We have

$$ \begin{align*}{\widetilde{M}} \cong {\mathrm{Pic}} _{ {\widetilde{ \mathbb{T} }} }( \mathbb{C} ^r) \cong H^2_{ {\widetilde{ \mathbb{T} }} }( \mathbb{C} ^r; \mathbb{Z} ), \end{align*} $$

where the first isomorphism is given by $\chi \mapsto {\widetilde {L}} _\chi $ and the second isomorphism is given by the $ {\widetilde { \mathbb {T} }} $ -equivariant first Chern class $(c_1)_{ {\widetilde { \mathbb {T} }} }$ . Define

$$ \begin{align*}D_i^{\mathcal{T}}:= (c_1)_{ {\widetilde{ \mathbb{T} }} }(\mathcal{O}_{ \mathbb{C} ^r}( {\widetilde{\mathcal{D}}}_i)) \in H^2_{ {\widetilde{ \mathbb{T} }} }( \mathbb{C} ^r; \mathbb{Z} ) \cong H^2_{\mathcal{T}}([ \mathbb{C} ^r/G]; \mathbb{Z} ). \end{align*} $$

Then $\{ D_1^{\mathcal {T}},\ldots , D_r^{\mathcal {T}} \}$ is a $ \mathbb {Z} $ -basis of $H^2_{ {\widetilde { \mathbb {T} }} }( \mathbb {C} ^r; \mathbb {Z} )\cong {\widetilde {M}} $ dual to the $ \mathbb {Z} $ -basis $\{ {\tilde {b}} _1,\ldots , {\tilde {b}} _r\}$ of $ {\widetilde {N}} $ . We have a commutative diagram

where $\iota _{\mathcal {T}}^*$ is a surjective group homomorphism induced by the inclusion $\iota : U_{\mathcal {A}}\hookrightarrow \mathbb {C} ^r$ , and

$$ \begin{align*}\mathrm{Ker}(\iota_{\mathcal{T}}^*)= \bigoplus_{i=r'+1}^r \mathbb{Z} D_i^{\mathcal{T}}. \end{align*} $$

Therefore,

$$ \begin{align*}{\mathrm{Pic}} _{\mathcal{T}}(\mathcal{X})\cong H^2_{\mathcal{T}}(\mathcal{X}; \mathbb{Z} ) \cong {\widetilde{M}} /\oplus_{i=r'+1}^r \mathbb{Z} D^{\mathcal{T}}_i. \end{align*} $$

Let $\bar {D}_i^{\mathcal {T}} := \iota _{\mathcal {T}}^* D_i^{\mathcal {T}}$ . Then

$$ \begin{align*}\bar{D}_i^{\mathcal{T}} =0,\quad i=r'+1,\ldots, r, \end{align*} $$

and

$$ \begin{align*}H^2_{\mathcal{T}}(\mathcal{X}; \mathbb{Z} ) = \bigoplus_{i=1}^{r'} \mathbb{Z} \bar{D}^{\mathcal{T}}_i \cong \mathbb{Z} ^{r'}. \end{align*} $$

For $i=1,\ldots ,r'$ , $ {\widetilde {\mathcal {D}}}_i\cap U_{\mathcal {A}}$ is a $ {\widetilde { \mathbb {T} }} $ -divisor in $U_{\mathcal {A}}$ , and it descends to a $ \mathbb {T} $ -divisor $\mathcal {D}_i$ in $\mathcal {X}$ . We have

$$ \begin{align*}\bar{D}_i^{\mathcal{T}}=(c_1)_{\mathcal{T}}(\mathcal{O}_{\mathcal{X}}(\mathcal{D}_i)),\quad i=1,\ldots, r'. \end{align*} $$

For $i=r'+1,\ldots , r$ , $ {\widetilde {\mathcal {D}}}_i\cap U_{\mathcal {A}}$ is empty, so it is the zero $ {\widetilde { \mathbb {T} }} $ -divisor.

2.3 Line bundles and Cartier divisors

We have group isomorphisms

$$ \begin{align*}\mathbb{L}^\vee \cong {\mathrm{Pic}} _G( \mathbb{C} ^r) \cong H^2_G( \mathbb{C} ^r; \mathbb{Z} ), \end{align*} $$

where the first isomorphism is given by $\chi \in \mathbb {L}^\vee =\mathrm {Hom}(G, \mathbb {C} ^*)\mapsto {\widetilde {L}} _\chi $ and the second isomorphism is given by the G-equivariant first Chern class $(c_1)_G$ . We have a commutative diagram

where $\iota ^*$ is a surjective group homomorphism induced by the inclusion $\iota : U_{\mathcal {A}}\hookrightarrow \mathbb {C} ^r$ . The surjective map $H^2_G( \mathbb {C} ^r; \mathbb {Z} )\to H^2(\mathcal {X}; \mathbb {Z} )$ is the restriction of the Kirwan map

$$ \begin{align*}\kappa: H^*_G( \mathbb{C} ^r; \mathbb{Z} )\longrightarrow H^*(\mathcal{X}; \mathbb{Z} ). \end{align*} $$

Define

$$ \begin{align*}D_i:= (c_1)_G(\mathcal{O}_{ \mathbb{C} ^r}( {\widetilde{\mathcal{D}}}_i)) \in H^2_G( \mathbb{C} ^r; \mathbb{Z} ) \cong H^2([ \mathbb{C} ^r/G]; \mathbb{Z} ). \end{align*} $$

Then

$$ \begin{align*}\mathrm{Ker}(\iota^*)= \bigoplus_{i=r'+1}^r \mathbb{Z} D_i. \end{align*} $$

Therefore,

$$ \begin{align*}{\mathrm{Pic}} (\mathcal{X})\cong H^2(\mathcal{X}; \mathbb{Z} ) \cong \mathbb{L}^\vee/\oplus_{i=r'+1}^r \mathbb{Z} D_i. \end{align*} $$

Recall that $\psi ^\vee : {\widetilde {M}} \longrightarrow \mathbb {L}^\vee $ is surjective if and only if $N_ {\mathrm {tor}} =0$ . Let

$$ \begin{align*}\bar{D}_i = c_1(\mathcal{O}_{\mathcal{X}}(\mathcal{D}_i))\in H^2(\mathcal{X}; \mathbb{Z} ),\quad i=1,\ldots, r. \end{align*} $$

The map

$$ \begin{align*}\bar{\psi}^\vee: {\mathrm{Pic}} _{\mathcal{T}}(\mathcal{X})\cong H^2_{\mathcal{T}}(\mathcal{X}; \mathbb{Z} ) \to {\mathrm{Pic}} (\mathcal{X})\cong H^2(\mathcal{X}; \mathbb{Z} ), \end{align*} $$

given by

$$ \begin{align*}\bar{D}_i^{\mathcal{T}} \mapsto \bar{D}_i,\quad i=1,\ldots, r', \end{align*} $$

is surjective if and only if $N_ {\mathrm {tor}} =0$ . In general, $\mathrm {Coker}(\psi ^\vee )\cong \mathrm {Coker}(\bar {\psi }^\vee )$ is a finite abelian group.

Pick a $ \mathbb {Z} $ -basis $\{e_1,\ldots , e_k\}$ of $\mathbb {L}\cong \mathbb {Z} ^k$ , and let $\{e_1^\vee ,\ldots , e_k^\vee \}$ be the dual $ \mathbb {Z} $ -basis of $\mathbb {L}^\vee $ . For each $a\in \{1,\ldots ,k\}$ , we define a charge vector

$$ \begin{align*}l^{(a)}= (l^{(a)}_1,\ldots, l^{(a)}_r) \in \mathbb{Z} ^r \end{align*} $$

by

$$ \begin{align*}\psi(e_a)=\sum_{i=1}^r l^{(a)}_i {\tilde{b}} _i, \end{align*} $$

where $\psi : \mathbb {L} \longrightarrow {\widetilde {N}} $ is the inclusion map. Then

$$ \begin{align*}D_i=\psi^\vee(D_i^{\mathcal{T}})=\sum_{a=1}^k l^{(a)}_i e_a^\vee,\quad i=1,\ldots,r, \end{align*} $$

and

$$ \begin{align*}\sum_{i=1}^r l^{(a)}_{i} b_{i} = \phi\circ \psi(e_a)=0,\quad a=1,\ldots, k. \end{align*} $$

2.4 Torus invariant subvarieties and their generic stabilisers

Let $\Sigma (d)$ be the set of d-dimensional cones. For each $ {\sigma } \in \Sigma (d)$ , we define

$$ \begin{align*}I_ {\sigma} :=\{ i\in \{1,\ldots,r\}\mid \rho_i\not\subset {\sigma} \} \in \mathcal{A}, \quad I_ {\sigma} ' := \{1,\ldots, r\}\setminus I_ {\sigma}. \end{align*} $$

Then $|I_ {\sigma } '|=d$ and $|I_ {\sigma } |=r-d$ . Let $\tilde {V}( {\sigma } )\subset U_{\mathcal {A}}$ be the closed subvariety defined by the ideal of $ \mathbb {C} [Z_1,\ldots , Z_r]$ generated by

$$ \begin{align*}\{Z_i=0\mid \rho_i\subset {\sigma} \} = \{ Z_i=0\mid i \in I^{\prime}_ {\sigma} \}. \end{align*} $$

Then $\mathcal {V}( {\sigma } ) := [ {\widetilde {V}} ( {\sigma } )/G]$ is an $(n-d)$ -dimensional $\mathcal {T}$ -invariant closed substack of $\mathcal {X}= [U_{\mathcal {A}}/G]$ .

The group homomorphism $G\cong ( \mathbb {C} ^*)^k \longrightarrow {\widetilde { \mathbb {T} }} \cong ( \mathbb {C} ^*)^r$ is given by

$$ \begin{align*}g\mapsto (\chi_1(g),\ldots, \chi_r(g)), \end{align*} $$

where $\chi _i\in \mathrm {Hom}(G, \mathbb {C} ^*)=\mathbb {L}^\vee $ is given by

$$ \begin{align*}\chi_i(u_1,\ldots, u_k)= \prod_{a=1}^k u_a^{ l_i^{(a)} }. \end{align*} $$

Let

$$ \begin{align*}G_ {\sigma} := \{ g\in G\mid g\cdot z = z \mathrm{\ for\ all\ } z\in {\widetilde{V}} ( {\sigma} )\} =\bigcap_{i\in I_ {\sigma} } \mathrm{Ker}(\chi_i). \end{align*} $$

Then $G_ {\sigma } $ is the generic stabiliser of $\mathcal {V}( {\sigma } )$ . It is a finite subgroup of G. If $\tau \subset {\sigma } $ , then $I_ {\sigma } \subset I_\tau $ , so $G_\tau \subset G_ {\sigma } $ . There are two special cases:

• ○ Let $\{0\}$ be the unique 0-dimensional cone. Then $G_{\{0\}}=K$ is the generic stabiliser of $\mathcal {V}(\{0\})=\mathcal {X}$ .

• ○ If $ {\sigma } \in \Sigma (n)$ , where $n=\dim _{\mathbb {C}} \mathcal {X}$ , then $\mathfrak {p}_ {\sigma } := \mathcal {V}( {\sigma } )$ is a $\mathcal {T}$ -fixed point in $\mathcal {X}$ , and $\mathfrak {p}_ {\sigma } =\mathcal {B} G_ {\sigma } $ .

Define the set of flags in $\Sigma $ to be

$$ \begin{align*}F(\Sigma)=\{(\tau, {\sigma} )\in \Sigma(n-1)\times \Sigma(n): \tau \subset {\sigma} \}. \end{align*} $$

Given $(\tau , {\sigma } )\in F(\Sigma )$ , let $\mathfrak {l}_\tau :=\mathcal {V}(\tau )$ be the 1-dimensional $\mathcal {T}$ -invariant subvariety of $\mathcal {X}$ . Then $\mathfrak {p}_ {\sigma } $ is contained in $\mathfrak {l}_\tau $ . There is a unique $i\in \{1,\ldots , r'\}$ such that $i\in I^{\prime }_ {\sigma } \setminus I^{\prime }_\tau $ . The representation of $G_ {\sigma } $ on the tangent line $T_{\mathfrak {p}_ {\sigma } }\mathfrak {l}_\tau $ to $\mathfrak {l}_\tau $ at the stacky point $\mathfrak {p}_ {\sigma } $ is given by $\chi _{(\tau , {\sigma } )}:= \chi _i|_{G_ {\sigma } }: G_ {\sigma } \to \mathbb {C} ^*$ . The image $\chi _i(G_ {\sigma } ) \subset \mathbb {C} ^*$ is a cyclic subgroup of $ \mathbb {C} ^*$ ; we define the order of this group to be $r(\tau , {\sigma } )$ . Then there is a short exact sequence of finite abelian groups:

$$ \begin{align*}1\to G_\tau\longrightarrow G_ {\sigma} \stackrel{\chi_{(\tau, {\sigma} )} }{\longrightarrow} \boldsymbol{\mu}_{r(\tau, {\sigma} )}\to 1, \end{align*} $$

where $\boldsymbol {\mu }_a=\{ z\in \mathbb {C} ^*\mid z^a=1\}$ is the group of ath roots of unity.

2.5 The extended nef cone and the extended Mori cone

In this paragraph, $ \mathbb {F} = \mathbb {Q} $ , $ \mathbb {R} $ or $ \mathbb {C} $ . Given a finitely generated abelian group $\Lambda $ with $\Lambda /\Lambda _ {\mathrm {tor}} \cong \mathbb {Z} ^m$ , define $\Lambda _{\mathbb {F}} = \Lambda \otimes _{\mathbb {Z}} \mathbb {F} \cong \mathbb {F} ^m$ . We have the following short exact sequences of vector spaces:

$$ \begin{align*} & 0\to \mathbb{L}_{\mathbb{F}} \to {\widetilde{N}} _{\mathbb{F}} \to N_{\mathbb{F}} \to 0,\\ & 0\to M_{\mathbb{F}} \to {\widetilde{M}} _{\mathbb{F}} \to \mathbb{L}^\vee_{\mathbb{F}} \to 0. \end{align*} $$

We also have the following isomorphisms of vector spaces over $ \mathbb {F} $ :

$$ \begin{align*} & H^2(\mathcal{X}; \mathbb{F} )\cong H^2(X; \mathbb{F} ) \cong \mathbb{L}^\vee_{\mathbb{F}} /\oplus_{i=r'+1}^r \mathbb{F} D_i,\\ & H^2_{\mathcal{T}}(\mathcal{X}; \mathbb{F} ) \cong H^2_{\mathbb{T}} (X; \mathbb{F} ) \cong {\widetilde{M}} _{\mathbb{F}} /\oplus_{i=r'+1}^r \mathbb{F} D_i^{\mathcal{T}}, \end{align*} $$

where X is the coarse moduli space of $\mathcal {X}$ .

From now on, we assume all the maximal cones in $\Sigma $ are n-dimensional, where $n=\dim _{\mathbb {C}} \mathcal {X}$ . Given a maximal cone $ {\sigma } \in \Sigma (n)$ , we define

$$ \begin{align*}\mathbb{K}_ {\sigma} ^\vee := \bigoplus_{i\in I_ {\sigma} } \mathbb{Z} D_i. \end{align*} $$

Then $\mathbb {K}_ {\sigma } ^\vee $ is a sublattice of $\mathbb {L}^\vee $ of finite index. We define the extended $ {\sigma } $ -nef cone to be

$$ \begin{align*}\widetilde{ {\mathrm{Nef}} }_ {\sigma} = \sum_{i\in I_ {\sigma} } \mathbb{R} _{\geq 0} D_i, \end{align*} $$

which is a k-dimensional cone in $\mathbb {L}^\vee _{\mathbb {R}} \cong \mathbb {R} ^k$ . The extended nef cone of the extended stacky fan $(\Sigma ,b_1,\ldots , b_r)$ is

$$ \begin{align*}\widetilde{ {\mathrm{Nef}} }_{\mathcal{X}}:=\bigcap_{ {\sigma} \in \Sigma(n)} \widetilde{ {\mathrm{Nef}} }_ {\sigma}. \end{align*} $$

The extended $ {\sigma } $ -Kähler cone $ {\widetilde {C}} _ {\sigma } $ is defined to be the interior of $\widetilde { {\mathrm {Nef}} }_ {\sigma } $ ; the extended Kähler cone of $\mathcal {X}$ , $ {\widetilde {C}} _{\mathcal {X}}$ , is defined to be the interior of the extended nef cone $\widetilde { {\mathrm {Nef}} }_{\mathcal {X}}$ .

Let $\mathbb {K}_ {\sigma } $ be the dual lattice of $\mathbb {K}_ {\sigma } ^\vee $ ; it can be viewed as an additive subgroup of $\mathbb {L}_{\mathbb {Q}} $ :

$$ \begin{align*}\mathbb{K}_ {\sigma} =\{ \beta\in \mathbb{L}_{\mathbb{Q}} \mid \langle D, \beta\rangle \in \mathbb{Z} \ \forall D\in \mathbb{K}_ {\sigma} ^\vee \}, \end{align*} $$

where $\langle -, -\rangle $ is the natural pairing between $\mathbb {L}^\vee _{\mathbb {Q}} $ and $\mathbb {L}_{\mathbb {Q}} $ . Define

$$ \begin{align*}\mathbb{K}:= \bigcup_{ {\sigma} \in \Sigma(n)} \mathbb{K}_ {\sigma}. \end{align*} $$

Then $\mathbb {K}$ is a subset (which is not necessarily a subgroup) of $\mathbb {L}_{\mathbb {Q}} $ , and $\mathbb {L}\subset \mathbb {K}$ .

We define the extended $ {\sigma } $ -Mori cone $\widetilde { {\mathrm {NE}} }_ {\sigma } \subset \mathbb {L}_{\mathbb {R}} $ to be the dual cone of $\widetilde { {\mathrm {Nef}} }_ {\sigma } \subset \mathbb {L}_{\mathbb {R}} ^\vee $ :

$$ \begin{align*}\widetilde{ {\mathrm{NE}} }_ {\sigma} =\{ \beta \in \mathbb{L}_{\mathbb{R}} \mid \langle D,\beta\rangle \geq 0 \ \forall D\in \widetilde{ {\mathrm{Nef}} }_ {\sigma} \}. \end{align*} $$

It is a k-dimensional cone in $\mathbb {L}_{\mathbb {R}} $ . The extended Mori cone of the extended stacky fan $(\Sigma ,b_1,\ldots , b_r)$ is

$$ \begin{align*}\widetilde{ {\mathrm{NE}} }_{\mathcal{X}}:= \bigcup_{ {\sigma} \in \Sigma(n)} \widetilde{ {\mathrm{NE}} }_ {\sigma}. \end{align*} $$

Finally, we define

$$ \begin{align*}\mathbb{K}_{\mathrm{eff}, {\sigma} }:= \mathbb{K}_ {\sigma} \cap \widetilde{ {\mathrm{NE}} }_ {\sigma} ,\quad \mathbb{K}_{\mathrm{eff}}:= \mathbb{K}\cap \widetilde{ {\mathrm{NE}} }(\mathcal{X})= \bigcup_{ {\sigma} \in \Sigma(n)} \mathbb{K}_{\mathrm{eff}, {\sigma} }. \end{align*} $$

2.6 Smooth toric DM stacks as symplectic quotients

Let $G_{\mathbb {R}} \cong U(1)^k$ be the maximal compact subgroup of $G\cong ( \mathbb {C} ^*)^k$ . Then the Lie algebra of $G_{\mathbb {R}} $ is $\mathbb {L}_{\mathbb {R}} $ . Let

$$ \begin{align*}{\widetilde{\mu}} : \mathbb{C} ^r\to \mathbb{L}^\vee_{\mathbb{R}} =\bigoplus_{a=1}^k \mathbb{R} e_a ^\vee \end{align*} $$

be the moment map of the Hamiltonian $G_{\mathbb {R}} $ -action on $ \mathbb {C} ^r$ , equipped with the Kähler form

$$ \begin{align*}\sqrt{-1} \sum_{i=1}^r dZ_i\wedge d\bar{Z}_i. \end{align*} $$

Then

$$ \begin{align*}{\widetilde{\mu}} (Z_1,\ldots, Z_r) = \sum_{i=1}^r \sum_{a=1}^k l_i^{(a)}|Z_i|^2 e^\vee_a. \end{align*} $$

If $\mathbf {r}=\sum _{a=1}^k r_a e_a^\vee $ is in the extended Kähler cone of $\mathcal {X}$ , then

$$ \begin{align*}\mathcal{X} =[ {\widetilde{\mu}} ^{-1}(\mathbf{r})/G_{\mathbb{R}} ]. \end{align*} $$

The generic stabiliser K (which is a finite subgroup of $G\cong ( \mathbb {C} ^*)^k$ ) is contained in the maximal compact subgroup $G_{\mathbb {R}} $ of G. The quotient $G_{\mathbb {R}} ^ {\mathrm {rig}} := G_{\mathbb {R}} /K\cong U(1)^k$ is the maximal compact subgroup of $G^ {\mathrm {rig}} =G/K\cong ( \mathbb {C} ^*)^k$ , and

$$ \begin{align*}\mathcal{X}^ {\mathrm{rig}} =[ {\widetilde{\mu}} ^{-1}(\mathbf{r})/G_{\mathbb{R}} ^ {\mathrm{rig}} ] \end{align*} $$

as a symplectic quotient.

The real numbers $r_1,\ldots , r_k$ are extended Kähler parameters. The symplectic structure $\omega (\mathbf {r})$ depends on $\mathbf {r}$ . The map $\mathbf {r}\mapsto [\omega (\mathbf {r})]$ is given by $\mathbb {L}^\vee _{\mathbb {R}} \to H^2(\mathcal {X}; \mathbb {R} )$ . Let $T_a=-r_a+\sqrt {-1}\theta _a$ be complexified extended Kähler parameters of $\mathcal {X}$ .

2.7 The inertia stack and the Chen-Ruan orbifold cohomology

Given $ {\sigma } \in \Sigma $ , define

$$ \begin{align*}\mathrm{Box}( {\sigma} ):=\Big\{ v\in N: \bar{v}=\sum_{i\in I^{\prime}_ {\sigma} } c_i \bar{b}_i, \quad 0\leq c_i <1\Big\}. \end{align*} $$

Then $N_ {\mathrm {tor}} \subset \mathrm {Box}( {\sigma } )\subset N$ . If $ {\sigma } $ is a d-dimensional cone, then the set $\{ \sum _{i\in I^{\prime }_ {\sigma } } c_i \bar {b}_i: c_i\in \mathbb {R} , 0\leq c_i<1 \}$ is a fundamental domain of the action of $\displaystyle { \bar {N}_ {\sigma } =\oplus _{i\in I^{\prime }_ {\sigma } } \mathbb {Z} \bar {b}_i \cong \mathbb {Z} ^d }$ on $\displaystyle { N_ {\sigma } \otimes _{ \mathbb {Z} } \mathbb {R} = \oplus _{i\in I^{\prime }_{ {\sigma } }} \mathbb {R} \bar {b}_i\cong \mathbb {R} ^d}$ . If $\tau \subset \sigma $ , then $I^{\prime }_\tau \subset I^{\prime }_ {\sigma } $ , so $\mathrm {Box}(\tau )\subset \mathrm {Box}( {\sigma } )$ .

Let $ {\sigma } \in \Sigma (n)$ be a maximal cone in $\Sigma $ . We have a short exact sequence of abelian groups

$$ \begin{align*}0\to \mathbb{K}_ {\sigma} /\mathbb{L}\to \mathbb{L}_{\mathbb{R}} /\mathbb{L}\to \mathbb{L}_{\mathbb{R}} /\mathbb{K}_ {\sigma} \to 0, \end{align*} $$

which can be identified with the following short exact sequence of multiplicative abelian groups

$$ \begin{align*}1\to G_ {\sigma} \to G_{\mathbb{R}} \to (G/G_ {\sigma} )_{\mathbb{R}} \to 1, \end{align*} $$

where $G_{\mathbb {R}} \cong U(1)^k$ is the maximal compact subgroup of $G\cong ( \mathbb {C} ^*)^k$ , and $(G/G_ {\sigma } )_{\mathbb {R}} \cong U(1)^k$ is the maximal compact subgroup of $(G/G_ {\sigma } )\cong ( \mathbb {C} ^*)^k$ .

Given a real number x, we recall some standard notation: $\lfloor x \rfloor $ is the greatest integer less than or equal to x, $\lceil x \rceil $ is the least integer greater or equal to x and $\{ x\} = x-\lfloor x \rfloor $ is the fractional part of x. Define $v: \mathbb {K}_ {\sigma } \to N$ by

$$ \begin{align*}v(\beta)= \sum_{i=1}^r \lceil \langle D_i,\beta\rangle\rceil b_i. \end{align*} $$

Then

$$ \begin{align*}\overline{v(\beta)} = \sum_{i\in I^{\prime}_ {\sigma} } \{ -\langle D_i,\beta\rangle \}\bar{b}_i, \end{align*} $$

so $v(\beta )\in \mathrm {Box}( {\sigma } )$ . Indeed, v induces a bijection $K_ {\sigma } /\mathbb {L}\cong \mathrm {Box}( {\sigma } )$ .

For any $\tau \in \Sigma $ , there exists $ {\sigma } \in \Sigma (n)$ such that $\tau \subset {\sigma } $ . The bijection $G_ {\sigma } \to \mathrm {Box}( {\sigma } )$ restricts to a bijection $G_\tau \to \mathrm {Box}(\tau )$ .

Define

$$ \begin{align*}\mathrm{Box}(\mathbf \Sigma):=\bigcup_{ {\sigma} \in \Sigma}\mathrm{Box}( {\sigma} ) =\bigcup_{ {\sigma} \in\Sigma(n)}\mathrm{Box}( {\sigma} ). \end{align*} $$

Then $N_ {\mathrm {tor}} \subset \mathrm {Box}(\mathbf \Sigma ) \subset N$ . There is a bijection $\mathbb {K}/\mathbb {L}\to \mathrm {Box}(\mathbf \Sigma )$ .

Given $v\in \mathrm {Box}( {\sigma } )$ , where $ {\sigma } \in \Sigma (d)$ , define $c_i(v)\in [0,1)\cap \mathbb {Q} $ by

$$ \begin{align*}\bar{v}= \sum_{i\in I^{\prime}_ {\sigma} } c_i(v) \bar{b}_i. \end{align*} $$

Suppose that $k \in G_ {\sigma } $ corresponds to $v\in \mathrm {Box}( {\sigma } )$ under the bijection $G_ {\sigma } \cong \mathrm {Box}( {\sigma } )$ . Then

$$ \begin{align*}\chi_i(k) = \begin{cases} 1, & i\in I_ {\sigma} ,\\ e^{2\pi\sqrt{-1} c_i(v)},& i \in I^{\prime}_ {\sigma}. \end{cases} \end{align*} $$

Define

$$ \begin{align*}\mathrm{age}(k)=\mathrm{age}(v)= \sum_{i\notin I_ {\sigma} } c_i(v). \end{align*} $$

Let $IU=\{(z,k)\in U_{\mathcal {A}}\times G\mid k\cdot z = z\}$ , and let G act on $IU$ by $h\cdot (z,k)= (h\cdot z,k)$ . The inertia stack $\mathcal {I}\mathcal {X}$ of $\mathcal {X}$ is defined to be the quotient stack

$$ \begin{align*}\mathcal{I}\mathcal{X}:= [IU/G]. \end{align*} $$

Note that $(z=(Z_1,\ldots ,Z_r), k)\in IU$ if and only if

$$ \begin{align*}k\in \bigcup_{ {\sigma} \in \Sigma}G_ {\sigma} \mathrm{\ and\ } Z_i=0 \mathrm{\ whenever\ } \chi_i(k) \neq 1. \end{align*} $$

So

$$ \begin{align*}IU=\bigcup_{v\in \mathrm{Box}(\mathbf \Sigma)} U_v, \end{align*} $$

where

$$ \begin{align*}U_v:= \{(Z_1,\ldots, Z_m)\in U_{\mathcal{A}}: Z_i=0 \mathrm{\ if\ } c_i(v) \neq 0\}. \end{align*} $$

The connected components of $\mathcal {I}\mathcal {X}$ are

$$ \begin{align*}\{ \mathcal{X}_v:= [U_v/G] : v\in \mathrm{Box}(\mathbf \Sigma)\}. \end{align*} $$

The involution $IU\to IU$ , $(z,k)\mapsto (z,k^{-1})$ induces involutions $\mathrm {inv}:\mathcal {I}\mathcal {X}\to \mathcal {I}\mathcal {X}$ and $\mathrm {inv}:\mathrm {Box}(\mathbf \Sigma )\to \mathrm {Box}(\mathbf \Sigma )$ such that $\mathrm {inv}(\mathcal {X}_v)=\mathcal {X}_{\mathrm {inv}(v)}$ .

In the remainder of this subsection, we consider rational cohomology and write $H^*(-)$ instead of $H^*(-; \mathbb {Q} )$ .

As a graded vector space over $ \mathbb {Q} $ (and as the state-space of the relevant quantum theory in physics [Reference Zaslow74]), the Chen-Ruan orbifold cohomology [Reference Chen and Ruan23] is defined to be

$$ \begin{align*}H^*_ {\mathrm{CR}} (\mathcal{X})=\bigoplus_{v\in \mathrm{Box}(\mathbf \Sigma)} H^*(\mathcal{X}_v)[2\mathrm{age}(v)]. \end{align*} $$

Let $\mathbf 1_v$ be the unit in $H^*(\mathcal {X}_v)$ . Then $\mathbf 1_v\in H^{2\mathrm {age}(v)}_ {\mathrm {CR}} (\mathcal {X})$ . In particular,

$$ \begin{align*}H^0_ {\mathrm{CR}} (\mathcal{X}) =\bigoplus_{v\in N_ {\mathrm{tor}} } \mathbb{Q} \mathbf 1_v. \end{align*} $$

Suppose that $\mathcal {X}$ is a proper toric DM stack. Then the orbifold Poincaré pairing on $H^*_ {\mathrm {CR}} (\mathcal {X})$ is defined as

(5) $$ \begin{align} (\alpha,\beta):=\int_{\mathcal{I}\mathcal{X}} \alpha\cup \mathrm{inv}^*(\beta). \end{align} $$

We also have an equivariant pairing on $H^*_{ {\mathrm {CR}} , \mathbb {T} }(\mathcal {X})$ :

(6) $$ \begin{align} (\alpha,\beta)_{ \mathbb{T} } := \int_{\mathcal{I}\mathcal{X}_{ \mathbb{T} }} \alpha\cup \mathrm{inv}^*(\beta), \end{align} $$

where

$$ \begin{align*}\int_{ \mathcal{I}\mathcal{X}_{ \mathbb{T} }}: H_{ {\mathrm{CR}} , \mathbb{T} }^*(\mathcal{X}) \to H_{ \mathbb{T} }^*(\mathrm{point}) = H^*(B \mathbb{T} ) \end{align*} $$

is the equivariant pushforward to a point. When $\mathcal {X}$ is not proper, equation (5) is not defined, but we can still define via equation (6) an equivariant pairing $H^*_{ {\mathrm {CR}} , \mathbb {T} }(\mathcal {X})\otimes H^*_{ {\mathrm {CR}} , \mathbb {T} }(\mathcal {X}) \to \mathcal {Q}_{\mathbb {T}} $ , where $\mathcal {Q}_{\mathbb {T}} $ is the fractional field of the ring $H^*(B \mathbb {T} )$ .

3.1 Rigidification

The rigidification $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ of $\mathcal {X}$ is a toric Calabi-Yau 3-orbifold. The Calabi-Yau condition implies $\mathcal {X}^{ {\mathrm {rig}} }=\mathcal {X}^{ {\mathrm {can}} }$ , where $\mathcal {X}^{ {\mathrm {can}} }$ is determined by the simplicial fan $\Sigma $ and then by choosing each $b_i$ to be the primitive generator of each $1$ -cone (compare to equation (3) in Section 2.1). Let $\mathcal {T}'$ (respectively, $ \mathbb {T} '$ ) be the subtorus of $\mathcal {T}\cong ( \mathbb {C} ^*)^3 \times \mathcal {B} K$ (respectively, $ \mathbb {T} \cong ( \mathbb {C} ^*)^3$ ) preserving the Calabi-Yau 3-form on $\mathcal {X}$ (respectively, $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ ). Then $ \mathbb {T} '\cong ( \mathbb {C} ^*)^2$ and $\mathcal {T}'\cong ( \mathbb {C} ^*)^2 \times \mathcal {B} K$ . There is a primitive $\mathsf {u}_3 \in M=\mathrm {Hom}( \mathbb {T} , \mathbb {C} ^*)\cong \mathbb {Z} ^3$ such that $\mathrm {Ker}(\mathsf {u}_3)= \mathbb {T} '$ . Define $M':=M/\langle \mathsf {u}_3 \rangle \cong \mathbb {Z} ^2$ . Then $\bar {N}':=\mathsf {u}_3^\perp =\{ \mathsf {v} \in \bar {N}: \langle \mathsf {u}_3, \mathsf {v} \rangle =0\}$ is the dual lattice of $M'=\mathrm {Hom}( \mathbb {T} ', \mathbb {C} ^*)$ .

The simplicial fan $\Sigma $ is determined by a convex polytope $\Delta \subset N^{\prime }_{\mathbb {R}} =\bar {N}'\otimes _{\mathbb {Z}} \mathbb {R} $ together with a triangulation of $\Delta $ , where all the vertices are in the lattice $\bar {N}$ . The fan $\Sigma $ is a cone over this triangulation in $N^{\prime }_{1, \mathbb {R} }\subset N_{\mathbb {R}} $ , where $N^{\prime }_{1, \mathbb {R} }=\{ \mathsf {v}\in N_{\mathbb {R}} : \langle \mathsf {u}_3, \mathsf {v}\rangle =1\}$ .

3.2 Toric graph

Let $ \mathbb {T} _{\mathbb {R}} \cong U(1)^3$ (respectively, $ \mathbb {T} ^{\prime }_{\mathbb {R}} \cong U(1)^2$ ) be the maximal compact subgroup of $ \mathbb {T} \cong ( \mathbb {C} ^*)^3$ (respectively, $ \mathbb {T} '\cong ( \mathbb {C} ^*)^2$ ), and we choose an $\mathbf {r}$ in the extended Kähler cone. The $ \mathbb {T} $ -action on $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ restricts to a Hamiltonian $ \mathbb {T} _{\mathbb {R}} $ -action on the Kähler orbifold $(\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} ,\omega (\mathbf {r}))$ . Since $M_{\mathbb {R}} $ (respectively, $M_{\mathbb {R}} '$ ) is canonically identified with the dual of the Lie algebra of $ \mathbb {T} _{\mathbb {R}} $ (respectively, $ \mathbb {T} ^{\prime }_{\mathbb {R}} $ ), the Kähler form $\omega (\mathbf {r})$ determines a moment map $\mu _{ \mathbb {T} _{\mathbb {R}} }: \mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} \longrightarrow M_{\mathbb {R}} $ up to translation by a vector in $M_{\mathbb {R}} $ . The image $\mu _{ \mathbb {T} _{\mathbb {R}} }(\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} )$ is a convex polyhedron. The moment map $\mu _{ \mathbb {T} ^{\prime }_{\mathbb {R}} }:\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} \longrightarrow M^{\prime }_{\mathbb {R}} $ is the composition $\pi \circ \mu _{T_{\mathbb {R}} }$ , where $\pi :M_{\mathbb {R}} \cong \mathbb {R} ^3 \to M^{\prime }_{ \mathbb {R} }\cong \mathbb {R} ^2$ is the projection. The map $\mu _{ \mathbb {T} ^{\prime }_{\mathbb {R}} }$ is surjective. Let $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} _1 \subset \mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ be the union of 0-dimensional and 1-dimensional $ \mathbb {T} $ -orbits in $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ . The toric graph is defined by $\Gamma := \mu _{T_{\mathbb {R}} '}(\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} _1)\subset M^{\prime }_{\mathbb {R}} \cong \mathbb {R} ^2$ . It is determined by the Kähler class $[\omega (\mathbf {r})]\in H^2(\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}}; \mathbb {R} )=H^2(X_\Sigma; \mathbb {R} )$ up to translation by a vector in $M^{\prime }_{\mathbb {R}} $ . The vertices (respectively, edges) of $\Gamma $ are in one-to-one correspondence to $3$ -dimensional (respectively, $2$ -dimensional) cones in $\Sigma $ . Conversely, the Kähler class $[\omega (\mathbf {r})]\in H^2(\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}}; \mathbb {R} )$ is determined by the toric graph.

Pulling back under the map $\mathcal {X}\longrightarrow \mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ defines a one-to-one correspondence between Kähler forms/classes on $\mathcal {X}$ and on its rigidification $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ .

3.3 Aganagic-Vafa A-branes

In [Reference Aganagic and Vafa6], Aganagic-Vafa introduced a class of Lagrangian submanifolds of semi-projective smooth toric Calabi-Yau 3-folds. In this section, we generalise this construction and define Aganagic-Vafa A-branes in a general 3-dimensional Calabi-Yau smooth toric DM stack with semi-projective coarse moduli space.

Let $\mathcal {X}=[ {\widetilde {\mu }} ^{-1}(\mathbf {r})/G_{\mathbb {R}} ]$ be a 3-dimensional Calabi-Yau smooth toric DM stack, where

$$ \begin{align*}\mathbf{r} =\sum_{a=1}^k r_a e_a^\vee \in {\widetilde{C}} (\mathcal{X})\subset \mathbb{L}^\vee_{\mathbb{R}} , \end{align*} $$

and $ {\widetilde {\mu }} ^{-1}(\mathbf {r})\subset \mathbb {C} ^{k+3}$ is defined by the following equations:

$$ \begin{align*}\sum_{i=1}^{k+3} l_i^{(a)} |X_i|^2 =r_a,\quad a=1,\ldots, k. \end{align*} $$

Write $X_i=\rho _i e^{\sqrt {-1}\phi _i}$ , where $\rho _i=|X_i|$ . An Aganagic-Vafa brane is a Lagrangian suborbifold of $\mathcal {X}$ of the form

$$ \begin{align*}\mathcal{L}=[ {\widetilde{L}} /G_{\mathbb{R}} ], \end{align*} $$

where

$$ \begin{align*}{\widetilde{L}} =\left\{ (X_1,\ldots, X_{k+3})\in {\widetilde{\mu}} ^{-1}(\mathbf{r}): \sum_{i=1}^{k+3}\hat{l}^1_i |X_i|^2=c_1, \sum_{i=1}^{k+3} \hat{l}_i^2 |X_i|^2 = c_2, \sum_{i=1}^{k+3}\phi_i =\mathrm{const} \right\} \end{align*} $$

for some $\hat {l}^\alpha _i \in \mathbb {Z} , \sum _{i=1}^{k+3}\hat {l}^\alpha _i =0$ , $\alpha =1,2$ . Note that the action of $G_{\mathbb {R}} $ on $ \mathbb {C} ^{k+3}$ preserves the subsets $ {\widetilde {\mu }} ^{-1}(\mathbf {r})$ and $ {\widetilde {L}} $ . If we view $\mathcal {X} = [ {\widetilde {\mu }} ^{-1}(\mathbf {r})/G_{\mathbb {R}} ]$ as a Lie groupoid (and in particular a category), then $\mathcal {L}=[ {\widetilde {L}} /G_{\mathbb {R}} ]$ is a full subcategory.

An Aganagic-Vafa brane $\mathcal {L}$ intersects a unique 1-dimensional orbit $\mathfrak {o}_\tau \cong \mathbb {C} ^*\times \mathcal {B} G_\tau $ along $\mathcal {S}_\tau := \mathcal {L}\cap \mathfrak {o}_\tau \cong S^1\times \mathcal {B} G_\tau $ . The inclusion $\mathcal {S}_\tau \subset \mathcal {L}$ is a homotopic equivalence, so the fundamental group of $\mathcal {L}$ is

$$ \begin{align*}\pi_1(\mathcal{L}) \cong \pi_1(S^1\times \mathcal{B} G_\tau) \cong \mathbb{Z} \times G_\tau. \end{align*} $$

In particular, it is abelian, so it is isomorphic to its abelianisation $H_1(\mathcal {L}; \mathbb {Z} )$ .

If $(\tau , {\sigma } )\in F(\Sigma )$ , then there is an inclusion $\iota ^{(\tau , {\sigma } )}: \mathcal {S}_\tau \hookrightarrow \mathcal {X}_ {\sigma } =[ \mathbb {C} ^3/G_ {\sigma } ]$ that induces

$$ \begin{align*}\iota^{(\tau, {\sigma} )}_*: \pi_1(\mathcal{S}_\tau) \cong \mathbb{Z} \times G_\tau \to \pi_1(\mathcal{X}_ {\sigma} ) \cong G_ {\sigma}. \end{align*} $$

3.4 Moduli spaces of stable maps to $(\mathcal {X},\mathcal {L})$

In [Reference Katz and Liu49], Katz-Liu introduced stable maps to a symplectic manifold with Lagrangian boundary conditions at all genera; the domain of such a map is a prestable bordered Riemann surface: that is, a smooth or nodal bordered Riemann surface. (See also [Reference Liu55], [Reference Fukaya, Oh, Ohta and Ono38].) In [Reference Cho and Poddar24, Section 2], Cho-Poddar define stable maps to a symplectic orbifold $\mathcal {X}$ with Lagrangian boundary conditions, under the assumption that the Lagrangian suborbifold $\mathcal {L}$ does not contain any stacky points (so that $\mathcal {L}$ is indeed a smooth manifold); the domain of such a map is a prestable bordered orbifold Riemann surface in the sense of [Reference Cho and Poddar24, Section 2]: that is, a smooth or nodal bordered orbifold Riemann surface, where a stacky point is either an interior marked point or an interior node.

In general, $\mathcal {L}$ is a suborbifold that contains stacky points. To obtain compactness of the moduli spaces when $\mathcal {X}$ and $\mathcal {L}$ are compact, one needs to allow orbifold structures at boundary marked points and boundary nodes. In the present paper, $\mathcal {L}$ may contain stacky points, but we do not need to allow orbifold structures on the boundary of the domain for the following two reasons:

1. (i) Our enumerative problem only requires interior insertions, so we do not need to introduce any boundary marked points.

2. (ii) In our case, $\mathcal {X}$ and $\mathcal {L}$ are noncompact, and we will define and compute open GW invariants by torus localisation on moduli space of stable maps $\mathcal {X}$ with boundaries in $\mathcal {L}$ . If a stable map represents a torus fixed point in the moduli space, then any node in the domain must be mapped to a torus fixed (scheme or stacky) point in $\mathcal {X}$ , but $\mathcal {L}$ does not contain any torus fixed point, so the domain does not contain any boundary nodes.

Let $(\Sigma ,x_1,\ldots , x_n)$ be a prestable bordered orbifold Riemann surface with n interior marked point. Then the coarse moduli space $(\bar {\Sigma },\bar {x}_1,\ldots , \bar {x}_n)$ is a prestable bordered Riemann surface with n interior marked points, defined in [Reference Katz and Liu49, Section 3.6] and [Reference Liu55, Section 3.2]. We define the topological type $(g,h)$ of $\Sigma $ to be the topological type of $\bar {\Sigma }$ (see [Reference Liu55, Section 3.2]).

Let $(\Sigma ,\partial \Sigma )$ be a prestable bordered orbifold Riemann surface of type $(g,h)$ , and let $\partial \Sigma = R_1\cup \cdots \cup R_h$ be union of connected components. Each connected component is a circle that contains no orbifold points. A (bordered) prestable map to the pair $(\mathcal {X},\mathcal {L})$ is a map $u:(\Sigma ,\partial \Sigma )\to (\mathcal {X},\mathcal {L})$ , where $\Sigma $ is a prestable bordered orbifold Riemann surface such that $u\circ \nu :\hat {\Sigma }\to \mathcal {X}$ is holomorphic, where $\nu : \hat {\Sigma }\to \Sigma $ is the normalisation (so $\hat {\Sigma }$ is a possibly disconnected smooth bordered orbifold Riemann surface); a prestable map to $(\mathcal {X},\mathcal {L})$ is stable if its automorphism group is finite. The topological type of a stable map u is given by the degree $\beta '= \bar {u}_*[\bar {\Sigma }]\in H_2(X,L; \mathbb {Z} )$ (where X and L are the coarse moduli spaces of $\mathcal {X}$ and $\mathcal {L}$ respectively, and $\bar {u}:\bar {\Sigma }\to X$ is the map between coarse moduli spaces) and $\bar \mu _i=u_*[R_i] = (\mu _i, \lambda _i)\in H_1(\mathcal {L}; \mathbb {Z} )\cong \mathbb {Z} \times G_\tau $ (where $\mu _i\in \mathbb {Z} $ is the winding number and $\lambda _i\in G_\tau $ is the monodromy). Given $\beta '\in H_2(X,L; \mathbb {Z} )$ and

$$ \begin{align*}{\vec{\mu}} = ( (\mu_1,\lambda_1),\dots, (\mu_h,\lambda_h)) \in H_1(\mathcal{L}; \mathbb{Z} )^h.\end{align*} $$

Let $\overline {\mathcal {M}}_{(g,h),n}(\mathcal {X},\mathcal {L}\mid \beta ', {\vec {\mu }} )$ be the moduli space of stable maps of type $(g,h)$ , degree $\beta '$ , winding numbers $\mu _i\in \mathbb {Z} $ and monodromies $\lambda _i\in G_\tau $ , with n interior marked points.

3.5 The tangent-obstruction complex and the virtual dimension

Similar to [Reference Katz and Liu49, Section 4.2], the tangent space $\mathcal {T}_\xi ^1$ and the obstruction space $\mathcal {T}_\xi ^2$ at a moduli point

$$ \begin{align*}\xi=[u:( (\Sigma, x_1,\ldots,x_n), \partial\Sigma)\to (\mathcal{X},\mathcal{L})] \in \overline{\mathcal{M}}_{(g,h),n}(\mathcal{X},\mathcal{L}\mid \beta', {\vec{\mu}} ) \end{align*} $$

fit into the following exact sequence of real vector spaces:

(7) $$ \begin{align} \begin{aligned} 0 \to & \mathrm{Aut}((\Sigma,x_1,\ldots,x_n),\partial\Sigma)\to H^0(\Sigma,\partial\Sigma,u^*T_{\mathcal{X}}, (u|_{\partial\Sigma})^*T_{\mathcal{L}})\to \mathcal{T}^1_\xi \\ \to & \mathrm{Def}((\Sigma,x_1,\ldots, x_n),\partial\Sigma)\to H^1(\Sigma,\partial\Sigma, u^*T_{\mathcal{X}}, (u|_{\partial\Sigma})^* T_{\mathcal{L}})\to \mathcal{T}^2_\xi, \end{aligned} \end{align} $$

where

• ○ $\mathrm {Aut}((\Sigma ,x_1,\ldots , x_n),\partial \Sigma )$ is the space of infinitesimal automorphism of the domain $((\Sigma ,x_1,\ldots , x_n),\partial \Sigma )$ and is equal to $H^0(\Sigma ,\partial \Sigma , T_{\Sigma }(-\sum _{j=1}^nx_j), T_{\partial \Sigma })$ when $\Sigma $ is a smooth bordered orbifold Riemann surface;

• ○ $\mathrm {Def}((\Sigma ,x_1,\ldots , x_n),\partial \Sigma )$ is the space of infinitesimal deformations of the domain, and is equal to $H^1(\Sigma ,\partial \Sigma ,T_{\Sigma }(-\sum _{j=1}^n x_i), T_{\partial \Sigma })$ when $\Sigma $ is a smooth bordered orbifold Riemann surface;

• ○ $H^0(\Sigma ,\partial \Sigma ,u^*T_{\mathcal {X}}, (u|_{\partial \Sigma })^*T_{\mathcal {L}})$ is the space of infinitesimal deformation of the map for a fixed domain;

• ○ $H^1(\Sigma ,\partial \Sigma ,u^*T_{\mathcal {X}},(u|_{\partial \Sigma })^*T_{\mathcal {L}})$ is the space of obstructions to deforming the map for a fixed domain.

Globally on the moduli space $\overline {\mathcal {M}}_{(g,h),n}(\mathcal {X},\mathcal {L},\beta ', {\vec {\mu }} )$ , there is an exact sequence of sheaves

(8) $$ \begin{align} 0 \to B_1 \to B_2 \to \mathcal{T}^1 \to B_4 \to B_5 \to \mathcal{T}^2 \to 0 \end{align} $$

whose fibre at the moduli point $\xi $ is equation (7).

Let $\mathfrak {M}_{(g,h),n}$ be the moduli of prestable bordered orbifold Riemann surfaces of type $(g,h)$ with n interior marked point. Then $\mathfrak {M}_{(g,h),n}$ is a differentiable stack (with corners) of real dimension

(9) $$ \begin{align} 3(2g-2+h) + 2n = \dim_{\mathbb{R}} \mathrm{Def}((\Sigma,x_1,\ldots, x_n),\partial\Sigma) - \dim_{\mathbb{R}} \mathrm{Aut}((\Sigma,x_1,\ldots, x_n),\partial\Sigma). \end{align} $$

There are evaluation maps (at interior marked points)

$$ \begin{align*}\mathrm{ev}_j:\overline{\mathcal{M}}_{(g,h),n}(\mathcal{X},\mathcal{L}\mid \beta', {\vec{\mu}} ) \to \mathcal{I}\mathcal{X},\quad j=1,\ldots,n. \end{align*} $$

Given $\vec {v}=(v_1,\ldots , v_n)$ , where $v_1,\ldots , v_n\in \mathrm {Box}(\mathbf \Sigma )$ , define

$$ \begin{align*}\overline{\mathcal{M}}_{(g,h),\vec{v}}(\mathcal{X},\mathcal{L}\mid\beta', {\vec{\mu}} ) := \bigcap_{j=1}^n \mathrm{ev}_j^{-1}(\mathcal{X}_{v_j}). \end{align*} $$

Suppose that $\xi \in \overline {\mathcal {M}}_{(g,h),\vec {v}}(\mathcal {X},\mathcal {L}\mid \beta ', {\vec {\mu }} )$ . By the Riemann-Roch theorem for prestable bordered orbifold Riemann surface (which can be derived by combining the proof of the Riemann-Roch theorem for prestable bordered Riemann surfaces and prestable orbifold closed Riemann surfaces),

(10) $$ \begin{align} &\dim_{\mathbb{R}} H^0(\Sigma,\partial\Sigma,u^*T_{\mathcal{X}}, (u|_{\partial\Sigma})^*T_{\mathcal{L}}) -\dim_{\mathbb{R}} H^1(\Sigma,\partial\Sigma, u^*T_{\mathcal{X}}, (u|_{\partial\Sigma})^*T_{\mathcal{L}}) \nonumber \\ &\quad = 3(2-2g-h) - 2\sum_{j=1}^n \mathrm{age}(v_j). \end{align} $$

The above equation (10) is the relative virtual dimension of $\overline {\mathcal {M}}_{(g,h),\vec {v}}(\mathcal {X},\mathcal {L}\mid \beta ', {\vec {\mu }} )\rightarrow \mathfrak {M}_{(g,h),n}$ that sends a stable map to its domain. The virtual (real) dimension of $\overline {\mathcal {M}}_{{ (g,h)},\vec {v}}(\mathcal {X},\mathcal {L}\mid \beta ', {\vec {\mu }} )$ is equal to

$$ \begin{align*}\dim_{\mathbb{R}} \mathcal{T}^1_\xi - \dim_{\mathbb{R}} \mathcal{T}^2_\xi = 2 \sum_{j=1}^n(1-\mathrm{age}(v_j)), \end{align*} $$

where $\mathrm {age}(v_j)\in \{0,1,2\}$ .

3.6 Torus action and equivariant invariants

Let $ \mathbb {T} ^{\prime }_{\mathbb {R}} \cong U(1)^2$ be the maximal compact subgroup of $ \mathbb {T} '\cong ( \mathbb {C} ^*)^2$ . For any $t \in \mathbb {T} ^{\prime }_{\mathbb {R}} $ , the map $\phi _t: \mathcal {X} \to \mathcal {X}$ given by $x\mapsto t\cdot x$ is an automorphism of the smooth toric DM stack $\mathcal {X}$ , and $\phi _t(\mathcal {L}) =\mathcal {L}$ , so $ \mathbb {T} ^{\prime }_{\mathbb {R}} $ acts on the moduli spaces $\overline {\mathcal {M}}_{(g,h),n}(\mathcal {X},\mathcal {L} \mid \beta ', {\vec {\mu }} )$ ; here we use the notion of group actions on stacks in [Reference Romagny65]. Let $F \subset \overline {\mathcal {M}}_{(g,h),n}(\mathcal {X},\mathcal {L}\mid \beta ', {\vec {\mu }} )$ be the substack of $ \mathbb {T} ^{\prime }_{\mathbb {R}} $ fixed points. The restriction of the exact sequence equation (8) to the substack F is the direct sum of two exact sequences

(11) $$ \begin{align} 0 \to B_1^f \to B_2^f\to \mathcal{T}^{1,f} \to B_4^f\to B_5^f \to \mathcal{T}^{2,f}\to 0, \end{align} $$

(12) $$ \begin{align} 0 \to B_1^m \to B_2^m \to \mathcal{T}^{1,m} \to B_4^m\to B_5^m \to \mathcal{T}^{2,m}\to 0, \end{align} $$

where equation (11) is the subcomplex fixed by the torus action. The virtual tangent bundle $\mathcal {T}^ {\mathrm {vir}} _F$ of F is

$$ \begin{align*}\mathcal{T}^ {\mathrm{vir}} _F =\mathcal{T}^{1,f}-\mathcal{T}^{2,f} \end{align*} $$

whose ranks can be different on different connected components of F. We will see that each connected component of F is a compact orbifold and that $\mathcal {T}^ {\mathrm {vir}} _F$ is equal to the tangent bundle $\mathcal {T}_F$ of F. So

$$ \begin{align*}[F]^ {\mathrm{vir}} = [F]. \end{align*} $$

The virtual normal bundle $N^ {\mathrm {vir}} $ of F in $\overline {\mathcal {M}}_{(g,h),n}(\mathcal {X},\mathcal {L}\mid \beta ', {\vec {\mu }} )$ is

$$ \begin{align*}N^ {\mathrm{vir}} =\mathcal{T}^{1,m}-\mathcal{T}^{2,m}. \end{align*} $$

Given $\gamma _1,\ldots , \gamma _n \in H^*_{ \mathbb {T} ', {\mathrm {CR}} }(\mathcal {X}; \mathbb {Q} )=H^*_{ \mathbb {T} ^{\prime }_{\mathbb {R}} , {\mathrm {CR}} }(\mathcal {X}, \mathbb {Q} )$ , we define

(13) $$ \begin{align} \langle \gamma_1,\ldots, \gamma_n\rangle^{\mathcal{X},\mathcal{L}, \mathbb{T} _{\mathbb{R}} '}_{g,\beta', {\vec{\mu}} }:= \int_{[F]^{ {\mathrm{vir}} }} \frac{\prod_{j=1}^n (\mathrm{ev}_j^*\gamma_i)|_F}{e_{ \mathbb{T} _{\mathbb{R}} '}(N^ {\mathrm{vir}} )} \in \mathcal{Q}_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }, \end{align} $$

where $\mathcal {Q}_{ \mathbb {T} _{\mathbb {R}} '}$ is the fractional field of $H^*_{ \mathbb {T} _{\mathbb {R}} '}(\mathrm {point}; \mathbb {Q} )$ , and

$$ \begin{align*}\frac{1}{e_{ \mathbb{T} _{\mathbb{R}} '}(N^ {\mathrm{vir}} _F)} =\frac{e_{ \mathbb{T} _{\mathbb{R}} '}(\mathcal{T}^{2,m})}{e_{ \mathbb{T} _{\mathbb{R}} '}(\mathcal{T}^{1,m}) }=\frac{ e_{ \mathbb{T} _{\mathbb{R}} '}(B_1^m) e_{ \mathbb{T} _{\mathbb{R}} '}(B_5^m)}{e_{ \mathbb{T} _{\mathbb{R}} '}(B_2^m)e_{ \mathbb{T} _{\mathbb{R}} '}(B_4^m)}. \end{align*} $$

More precisely, the definition in equation (13) also requires an orientation on the virtual tangent bundle $\mathcal {T}^1 -\mathcal {T}^2$ , which we will specify later.

3.7 Tangent weights: the 3-torus, the Calabi-Yau 2-torus, and the framing 1-torus

Let $\mathfrak {o}_\tau \cong \mathbb {C} ^*\times \mathcal {B} G_\tau $ be the unique 1-dimensional $ \mathbb {T} $ -orbit that intersects the Aganagic-Vafa A-brane $\mathcal {L}$ , where $\tau \in \Sigma (2)$ , as before. Let $\mathfrak {l}_\tau $ be the closure of $\mathfrak {o}_\tau $ , and let $\ell _\tau $ be the coarse moduli of $\mathfrak {l}_\tau $ . Then $\ell _\tau $ is either $ \mathbb {P} ^1$ or $ \mathbb {C} $ .

If $\mathcal {L}$ is an outer brane, let $ {\sigma } \in \Sigma (3)$ be the unique 3-cone such that $(\tau , {\sigma } )\in F(\Sigma )$ ; if $\mathcal {L}$ is an inner brane, we choose $ {\sigma } \in \Sigma (3)$ such that $(\tau , {\sigma } )\in F(\Sigma )$ , and let $ {\sigma } _- \in \Sigma (3)$ denote the other choice, so that $\mathfrak {p}_ {\sigma } $ and $\mathfrak {p}_{ \sigma_{-}}$ are the two torus fixed points in $\mathfrak {l}_\tau $ . For inner branes, we also denote $ {\sigma } _+= {\sigma } $ .

By permuting $b_1,\ldots , b_r$ if necessary, we may assume that $I^{\prime }_{ {\sigma } }=\{1,2,3\}$ , and $(\tau _1,\sigma )= (\tau ,\sigma )$ , $(\tau _2,\sigma )$ and $(\tau _3,\sigma )$ are three flags in the toric graph in the counterclockwise direction such that

$$ \begin{align*}I^{\prime}_{\tau_1}= \{2, 3\},\quad I^{\prime}_{\tau_2}=\{3, 1\},\quad I^{\prime}_{\tau_3}=\{1,2\}. \end{align*} $$

Here we fixed an orientation of $ \mathbb {R} ^2$ . If $\mathcal {L}$ is an inner brane, we assume in addition $I^{\prime }_{ {\sigma } _-}=\{ 2,3, 4\}$ .

Recall from Section 2.4 that for any flag $(\tau , {\sigma } )\in F(\Sigma )$ , $\chi _{(\tau , {\sigma } )}\in \mathrm {Hom}(G_ {\sigma } , \mathbb {C} ^*)$ is the character of the 1-dimensional $G_ {\sigma } $ representation $T_{\mathfrak {p}_ {\sigma } }\mathfrak {l}_\tau $ . Let

$$ \begin{align*}\mathfrak{r}:= r(\tau, {\sigma} ) = |G_ {\sigma} /G_\tau|,\quad \mathfrak{m}:= |G_\tau/K|. \end{align*} $$

Then we have the following two short exact sequences of finite abelian groups:

$$ \begin{align*}1\to G_\tau \longrightarrow G_ {\sigma} \stackrel{\chi_{(\tau, {\sigma} )} }{\longrightarrow} \boldsymbol{\mu}_{\mathfrak{r}}\to 1,\quad 1\to K \longrightarrow G_\tau \stackrel{\chi_{(\tau_3, {\sigma} )}}{\longrightarrow} \boldsymbol{\mu}_{\mathfrak{m}}\to 1. \end{align*} $$

Note that for any $\lambda \in G_\tau $ , $\chi _{(\tau , {\sigma } )}(\lambda )=1$ and $\chi _{(\tau _2, {\sigma } )}(\lambda )\chi _{(\tau _3, {\sigma } )}(\lambda )=1$ . Let $\bar {\lambda }$ denote the unique element in $\{0,1,\ldots ,\mathfrak {m}-1\}$ such that

$$ \begin{align*}\chi_3(\lambda)=e^{2\pi\sqrt{-1}\bar{\lambda}/\mathfrak{m}}. \end{align*} $$

Let $\mathsf {u}_3\in M$ be defined as in Section 3.1, so that $\langle \mathsf {u}_3, \bar {b}_i\rangle =1$ . We may choose a $ \mathbb {Z} $ -basis $\{ \mathsf {v}_1,\mathsf {v}_2, \mathsf {v}_3\}$ of $\bar {N}$ such that $ \langle \mathsf {u}_3,\mathsf {v}_i\rangle =\delta _{i,3}$ , and

$$ \begin{align*}\bar{b}_1 =\mathfrak{r} \mathsf{v}_1 - \mathfrak{s} \mathsf{v}_2 + \mathsf{v}_3,\quad \bar{b}_2 =\mathfrak{m} \mathsf{v}_2 +\mathsf{v}_3,\quad \bar{b}_3=\mathsf{v}_3. \end{align*} $$

Moreover, the choice $(\mathsf {v}_1,\mathsf {v}_2,\mathsf {v}_3)$ is unique if we require $\mathfrak {s} \in \{0,1,\ldots ,\mathfrak {r}-1\}$ . Let $\{ \mathsf {u}_1,\mathsf {u}_2,\mathsf {u}_3\}$ be the $ \mathbb {Z} $ -basis of M that is dual to the $ \mathbb {Z} $ -basis $\{\mathsf {v}_1,\mathsf {v}_2,\mathsf {v}_2\}$ of $\bar {N}$ . Let $\{\mathsf {w}_1,\mathsf {w}_2,\mathsf {w}_3\}$ be the $ \mathbb {Q} $ -basis of $M_{\mathbb {Q}} $ that is dual to the $ \mathbb {Q} $ -basis $\{ \bar {b}_1,\bar {b}_2, \bar {b}_3\}$ of $N_{\mathbb {Q}} = N\otimes _{ \mathbb {Z} } \mathbb {Q} $ . Then

$$\begin{align*}\mathsf{w}_1=\frac{1}{\mathfrak{r}}\mathsf{u}_1,\ \mathsf{w}_2=\frac{\mathfrak{s}}{\mathfrak{r} \mathfrak{m}}\mathsf{u}_1+\frac{1}{\mathfrak{m}}\mathsf{u}_2,\ \mathsf{w}_3=-\frac{\mathfrak{s}+\mathfrak{m}}{\mathfrak{r} \mathfrak{m}}\mathsf{u}_1-\frac{1}{\mathfrak{m}}\mathsf{u}_2+\mathsf{u}_3. \end{align*}$$

Moreover, for $i\in \{1,2,3\}$ ,

$$ \begin{align*}\mathsf{w}_i= e_{\mathcal{T}} (T_{\mathfrak{p}_ {\sigma} }\mathfrak{l}_{\tau_i}) = e_{\mathcal{T}}(\mathcal O_{\mathcal{X}}(\mathcal D_i))\Big|_{\mathfrak{p}_ {\sigma} }. \end{align*} $$

The inclusion $ \mathbb {T} '\subset \mathbb {T} $ induces the following surjective ring homomorphism

(14) $$ \begin{align} H^*(\mathcal{B} \mathbb{T} ;R)=R[\mathsf{u}_1,\mathsf{u}_2,\mathsf{u}_3]\longrightarrow H^*(\mathcal{B} \mathbb{T} ';R)=R[\mathsf{u}^{\prime}_1,\mathsf{u}^{\prime}_2],\quad \mathsf{u}_1\mapsto \mathsf{u}_1',\ \mathsf{u}_2 \mapsto \mathsf{u}_2',\ \mathsf{u}_3 \mapsto 0, \end{align} $$

where $R= \mathbb {Z} $ or $ \mathbb {Q} $ .

Given a framing that is an integer $f\in \mathbb {Z} $ , let $ \mathbb {T} _f \subset \mathbb {T} '$ be the kernel of the character $\mathsf {u}_2'-f\mathsf {u}^{\prime }_1\in \mathrm {Hom}( \mathbb {T} '; \mathbb {C} ^*)$ . Then $ \mathbb {T} _f\cong \mathbb {C} ^*$ is a 1-dimensional subtorus of the Calabi-Yau torus $ \mathbb {T} '$ . The inclusion $ \mathbb {T} _f\subset \mathbb {T} '$ induces a surjective ring homomorphism

(15) $$ \begin{align} H^*(\mathcal{B} \mathbb{T} ';R)=R[\mathsf{u}^{\prime}_1,\mathsf{u}^{\prime}_2] \longrightarrow H^*(\mathcal{B} \mathbb{T} _f;R)=R[\mathsf{u}],\quad \mathsf{u}^{\prime}_1\mapsto \mathsf{u},\ \mathsf{u}^{\prime}_2 \mapsto f\mathsf{u}, \end{align} $$

where $R= \mathbb {Z} $ or $ \mathbb {Q} $ . For $i=1,2,3$ , let $\mathsf {w}^{\prime }_i$ denote the image of $\mathsf {w}_i$ under the ring homomorphism equation (14), and let $\mathsf {w}^f_i$ denote the image of $\mathsf {w}^{\prime }_i$ under the ring homomorphism in (15). Then

(16) $$ \begin{align} \mathsf{w}^{\prime}_1=\frac{1}{\mathfrak{r}}\mathsf{u}^{\prime}_1,\ \mathsf{w}^{\prime}_2=\frac{\mathfrak{s}}{\mathfrak{r} \mathfrak{m}}\mathsf{u}^{\prime}_1+\frac{1}{\mathfrak{m}}\mathsf{u}^{\prime}_2,\ \mathsf{w}^{\prime}_3=-\frac{\mathfrak{s}+\mathfrak{m}}{\mathfrak{r} \mathfrak{m}}\mathsf{u}^{\prime}_1-\frac{1}{\mathfrak{m}}\mathsf{u}^{\prime}_2 =-\mathsf{w}^{\prime}_1-\mathsf{w}^{\prime}_2 \in H^2(\mathcal{B} \mathbb{T} ')= \mathbb{Q} \mathsf{u}^{\prime}_1 \oplus \mathbb{Q} \mathsf{u}^{\prime}_2, \end{align} $$

and $\mathsf {w}^f_i = w_i \mathsf {u}$ , where $w_i\in \mathbb {Q} $ are given by

$$ \begin{align*}w_1=\frac{1}{\mathfrak{r}},\quad w_2= \frac{\mathfrak{s} + \mathfrak{r} f}{\mathfrak{r} \mathfrak{m}},\quad w_3 = -w_1-w_2 = \frac{-\mathfrak{m}-\mathfrak{s}-\mathfrak{r} f}{\mathfrak{r}\mathfrak{m}}. \end{align*} $$

3.8 Disk factors as equivariant open GW invariants

A framed Aganagic-Vafa Lagrangian brane is a pair $(\mathcal {L},f)$ , where $\mathcal {L}$ is an Aganagic-Vafa brane together with a choice of a flag $(\tau , {\sigma } )\in F(\Sigma )$ such that $\mathfrak {o}_\tau $ is the unique 1-dimensional orbit intersecting $\mathcal {L}$ and a choice of framing $f\in \mathbb {Z} $ . Given a framed Aganagic-Vafa Lagrangian brane $(\mathcal {L},f)$ , we choose an isomorphism $\pi _1(\mathcal {L}) \cong \mathbb {Z} \times G_\tau $ such that if $h=\iota _*^{(\tau , {\sigma } )}(d_0,\lambda )$ (where $\iota _*^{(\tau , {\sigma } )}$ is defined in Section 3.3); then

$$ \begin{align*}\chi_{(\tau_1, {\sigma} )}(h) = e^{2\pi\sqrt{-1}d_0 w_1},\quad \chi_{(\tau_2, {\sigma} )}(h) = e^{2\pi\sqrt{-1}d_0 (w_2-\frac{\bar{\lambda}}{\mathfrak{m}})},\quad \chi_{(\tau_3, {\sigma} )}(h) = e^{2\pi\sqrt{-1}d_0 (w_3 +\frac{\bar{\lambda}}{\mathfrak{m}})}. \end{align*} $$

Let $\ell _\tau $ be the coarse moduli of $\mathfrak {l}_\tau $ , as before. Let $p_ {\sigma } \in \ell _\tau $ be the coarse moduli of $\mathfrak {p}_ {\sigma } \cong \mathcal {B} G_ {\sigma } $ , and let $S_\tau := L\cap \ell _\tau \cong S^1$ be the coarse moduli of $\mathcal {S}_\tau =\mathcal {L}\cap \mathfrak {l}_\tau \cong S^1\times \mathcal {B} G_\tau $ .

3.8.1 $(\mathcal {L},f)$ is a framed outer brane

In this case $\ell _\tau = \mathbb {C} $ . Let $D\subset \ell _\tau $ be the disk that contains $p_ {\sigma } $ with boundary $S_\tau $ , oriented by the complex structure on $\ell _\tau $ , and let $b=[D]\in H_2(X,L; \mathbb {Z} )$ . Given $(d_0,\lambda )\in H_1(\mathcal {L}; \mathbb {Z} )\cong \mathbb {Z} \times G_\tau $ , where $d_0>0$ , define

$$ \begin{align*}\overline{\mathcal{M}}(d_0,\lambda):= \overline{\mathcal{M}}_{(0,1),1}(\mathcal{X},\mathcal{L}\mid d_0 b, (d_0,\lambda)). \end{align*} $$

The virtual real dimension of $\overline {\mathcal {M}}(d_0,\lambda )$ is $2(1-\mathrm {age}(h(d_0,\lambda )))$ , where $h(d_0,\lambda ):= \iota _*^{(\tau , {\sigma } )}(d_0,\lambda )\in G_\sigma $ . Define the disk factor

$$ \begin{align*}D_{d_0,\lambda}:= \langle \mathbf{1}_{h(d_0,\lambda)}\rangle_{0, d_0 b, (d_0,\lambda)}^{\mathcal{X},\mathcal{L}, \mathbb{T} ^{\prime}_{\mathbb{R}} }, \end{align*} $$

which is a rational function in $\mathsf {w}_1',\mathsf {w}_2'$ , homogeneous of degree $\mathrm {age}(h(d_0,\lambda ))-1$ . The disk factor is computed in [Reference Brini and Cavalieri11] when $G_ {\sigma } $ is cyclic and in [Reference Ross66, Section 3.3] for general $G_ {\sigma } $ . In our notation, the formula in [Reference Ross66, Section 3.3] saysFootnote ²

(17) $$ \begin{align} \begin{aligned} D_{d_0,\lambda}= \left(\frac{\mathfrak{r}\mathsf{w}^{\prime}_1}{d_0}\right)^{\mathrm{age}(h(d_0,\lambda))-1} \frac{1}{d_0 |G_\tau|} \cdot \frac{ \prod_{a=1}^{ \lfloor \frac{d_0}{\mathfrak{r}}\rfloor+ \mathrm{age}(h(d_0,\lambda)) -1}\left( \frac{d_0 \mathsf{w}^{\prime}_2}{\mathfrak{r}\mathsf{w}^{\prime}_1} + a-c_2\left(h(d_0,\lambda)\right) \right) }{\lfloor \frac{d_0}{\mathfrak{r}}\rfloor!} \end{aligned}, \end{align} $$

where $c_i(\cdot )\in \mathbb {Q} \cap [0,1)$ is defined in Section 2.7. More explicitly,

$$ \begin{align*}c_2(h(d_0,\lambda)) = \left\langle d_0 w_2 -\frac{\bar{\lambda}}{\mathfrak{m}}\right\rangle. \end{align*} $$

3.8.2 $(\mathcal {L},f)$ is a framed inner brane

In this case $\ell _\tau \cong \mathbb {P} ^1$ . It contains two torus fixed points $p_+ = p_ {\sigma } $ and $p_-=p_{ {\sigma } _-}$ , where $ {\sigma } _- \in \Sigma (3)$ . The circle $S_\tau $ is the intersection of two disks $D_+$ and $D_-$ that contain $p_+$ and $p_-$ , respectively. Let

$$ \begin{align*}b=[D]\in H_2(X,L; \mathbb{Z} ),\quad \alpha =[\ell_\tau]\in H_2(X; \mathbb{Z} ). \end{align*} $$

Then $[D_-]=\alpha -b\in H_2(X,L; \mathbb {Z} )$ . Given $(d_0,\lambda )\in H_1(\mathcal {L}; \mathbb {Z} )\cong \mathbb {Z} \times G_\tau $ , where $d_0\neq 0$ , we define

$$ \begin{align*}\overline{\mathcal{M}}(d_0,\lambda) := \begin{cases} \overline{\mathcal{M}}_{(0,1),1}(\mathcal{X},\mathcal{L}\mid d_0 b, (d_0,\lambda)), & d_0>0,\\ \overline{\mathcal{M}}_{(0,1),1}(\mathcal{X},\mathcal{L}\mid -d_0(\alpha-b), (d_0, \lambda)), & d_0<0. \end{cases} \end{align*} $$

Then

$$ \begin{align*}\mathrm{virtual\ dimension\ of\ }\overline{\mathcal{M}}(d_0,\lambda)= \begin{cases} 1 -\mathrm{age}(h^+(d_0,\lambda)), & d_0>0,\\ 1-\mathrm{age}(h^-(d_0,\lambda)), & d_0<0, \end{cases} \end{align*} $$

where $h^{\pm }(d_0,\lambda ) = \iota _*^{(\tau , {\sigma } _\pm )}(d_0,\lambda )\in G_{ {\sigma } _\pm }$ . Define

$$ \begin{align*}D_{d_0,\lambda}:= \begin{cases} \langle 1_{h^+(d_0,\lambda)}\rangle_{0,d_0 b, (d_0,\lambda)}^{\mathcal{X},\mathcal{L}, \mathbb{T} ^{\prime}_{\mathbb{R}} }, & d_0>0,\\ & \\ \langle 1_{h^-(d_0,\lambda)}\rangle_{0,-d_0 (\alpha-b), (d_0,\lambda)}^{\mathcal{X}, \mathcal{L}, \mathbb{T} ^{\prime}_{\mathbb{R}} }, & d_0 <0. \end{cases}. \end{align*} $$

Then $D_{d_0,\lambda }$ is a rational function in $\mathsf {u}^{\prime }_1,\mathsf {u}^{\prime }_2$ , homogeneous of degree $\mathrm {age}(h^\pm (d_0,\lambda ))-1$ if $\pm d_0>0$ .

More precisely, the disk factor $D_{d_0,\lambda }$ is defined up to a sign depending on choice of orientation of $\overline {\mathcal {M}}(d_0,\lambda )$ , which will be clarified in Section 3.11 by relative GW invariants.

3.9 Normal bundle to $\mathfrak {l}_\tau $

Let $\mathcal {L}$ be an inner brane so that $\mathfrak {l}_\tau $ is a proper smooth toric DM curve. Let $\hat {\mathfrak {l}}_\tau $ be the image of $\mathfrak {l}_\tau $ under the morphism $\mathcal {X}\to \mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ . We have

$$ \begin{align*}\mathfrak{l}_\tau \longrightarrow \hat{\mathfrak{l}}_\tau \longrightarrow \mathfrak{l}_\tau^ {\mathrm{rig}} \longrightarrow \ell_\tau \cong \mathbb{P} ^1, \end{align*} $$

where $\mathfrak {l}_\tau \to \hat {\mathfrak {l}}_\tau $ is a K-banded gerbe, $\hat {\mathfrak {l}}_\tau \to \mathfrak {l}_\tau ^ {\mathrm {rig}} $ is a $\boldsymbol {\mu }_\mathfrak{m}$ -banded gerbe and $\mathfrak {l}_\tau \to \mathfrak {l}_\tau ^ {\mathrm {rig}} $ is a $G_\tau $ -banded gerbe. The normal bundle $\mathfrak {l}_\tau $ in $\mathcal {X}$ is a direct sum of two $ \mathbb {T} $ -equivariant line bundles over $\mathfrak {l}_\tau $ :

$$ \begin{align*}N_{\mathfrak{l}_\tau/\mathcal{X}} =L_2 \oplus L_3, \end{align*} $$

where $L_2=\mathcal {O}_{\mathcal {X}}(\mathcal {D}_2)\big |_{\mathfrak {l}_\tau }$ and $L_3 =\mathcal {O}_{\mathcal {X}}(\mathcal {D}_3)\big |_{\mathfrak {l}_\tau }$ . The total space of $N_{\mathfrak {l}_\tau /\mathcal {X}}$ is a smooth toric DM stack that is isomorphic to the open substack $\mathcal {Y}:= \mathcal {X}_{ {\sigma } }\cup \mathcal {X}_{ {\sigma } _-}$ of $\mathcal {X}$ . Let $\hat {\mathcal {D}}_i$ be the image of $\mathcal {D}_i$ under $\mathcal {X}\to \mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ . Then

$$ \begin{align*}N_{\hat{\mathfrak{l}}_\tau/\mathcal{X}^ {\mathrm{rig}} } = \hat{L}_2\oplus \hat{L}_3, \end{align*} $$

where $\hat {L}_2 =\mathcal {O}_{\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} }(\hat {\mathcal {D}}_2)\big |_{\hat {\mathfrak {l}}_\tau }$ and $\hat {L}_3 =\mathcal {O}_{\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} }(\hat {\mathcal {D}}_3)\big |_{\hat {\mathfrak {l}}_\tau }$ . The total space of $N_{\hat {\mathfrak {l}}_\tau /{\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} }}$ is a toric orbifold that is isomorphic to the open substack $\mathcal {Y}^ {\mathrm {rig}} =\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} _ {\sigma } \cup \mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} _{ {\sigma } _-}$ of the toric orbifold $\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ .

Let $\Sigma _0$ be the simplicial fan in $N_{\mathbb {R}} $ consisting of $ {\sigma } $ , $ {\sigma } _-$ and their subcones. The stacky fan of $\mathcal {Y}^ {\mathrm {rig}} $ is given by $(\Sigma _0, (\bar {b}_1, \bar {b}_2, \bar {b}_3, \bar {b}_4))$ , where

$$ \begin{align*}\bar{b}_1 =\mathfrak{r} \mathsf{v}_1 -\mathfrak{s} \mathsf{v}_2 + \mathsf{v}_3,\quad \bar{b}_2 = \mathfrak{m} \mathsf{v}_2 +\mathsf{v}_3,\quad \bar{b}_3 = \mathsf{v}_3, \quad \bar{b}_4 =-\mathfrak{r}_- \mathsf{v}_1 + c\mathsf{v}_2 +\mathsf{v}_3, \end{align*} $$

where c is some integer. For inner branes, we also denote $\mathfrak {r}_+=\mathfrak {r}$ . We have $\mathcal {Y} =[U/G_0]$ , where

$$ \begin{align*} U&=\{(Z_1,Z_2,Z_3,Z_4)\in \mathbb{C} ^4: (Z_1,Z_4)\neq (0,0)\},\\ G_0&=\{ (\tilde{t}_1,\tilde{t}_2,\tilde{t}_3,\tilde{t}_4)\in ( \mathbb{C} ^*)^4: \tilde{t}_1^{\mathfrak{r}} (\tilde{t}_4)^{-\mathfrak{r}_-} = (\tilde{t}_1)^{-\mathfrak{s}}\tilde{t}_2^{\mathfrak{m}} \tilde{t}_4^c =\tilde{t}_1\tilde{t}_2\tilde{t}_3\tilde{t}_4=1\}. \end{align*} $$

We have a short exact sequence of abelian groups:

$$ \begin{align*}1\to \boldsymbol{\mu}_{\mathfrak{m}} \longrightarrow G_0 \stackrel{\chi_1\times \chi_4}{\longrightarrow} G_{\mathfrak{r}, \mathfrak{r}_-} \to 1, \end{align*} $$

where $G_{\mathfrak {r},\mathfrak {r}_-}=\{ (\tilde {t}_1,\tilde {t}_4)\in ( \mathbb {C} ^*)^2: \tilde {t}_1^{\mathfrak {r}} (\tilde {t}_4)^{-\mathfrak {r}_-}=1\}$ and $\chi _i(\tilde {t}_1,\tilde {t}_2, \tilde {t}_3, \tilde {t}_4)=\tilde {t}_i$ . The subgroup $\boldsymbol {\mu }_{\mathfrak {m}}$ of $G_0$ acts trivially on $V =\{ (Z_1,Z_2,Z_3,Z_4)\in U_{\mathcal {A}}: Z_2=Z_3=0\}$ , so the $G_0$ -action on V factors through a $G_{\mathfrak {r},\mathfrak {r}_-}$ -action on V, and

$$ \begin{align*}\hat{\mathfrak{l}}_\tau = [V/G_0],\quad \mathfrak{l}_{\tau}^ {\mathrm{rig}} = [V/G_{\mathfrak{r},\mathfrak{r}_-}] \cong \mathcal{F}_{\mathfrak{r},\mathfrak{r}_-}, \end{align*} $$

where $\mathcal {F}_{\mathfrak {r},\mathfrak {r}_-}$ denotes the football obtained by glueing $[ \mathbb {C} /\boldsymbol {\mu }_{\mathfrak {r}}]$ and $[ \mathbb {C} /\boldsymbol {\mu }_{\mathfrak {r}-}]$ along $[ \mathbb {C} ^*/\boldsymbol {\mu }_{\mathfrak {r}}] \cong [ \mathbb {C} ^*/\boldsymbol {\mu }_{\mathfrak {r}_-}]\cong \mathbb {C} ^*$ . The two torus fixed points in $\mathfrak {l}_{\tau }^ {\mathrm {rig}} $ are

$$ \begin{align*}\mathfrak{p}_x = [ (\{(0,0,0)\} \times \mathbb{C} ^*)/G_{\mathfrak{r},\mathfrak{r}_-}] \cong \mathcal{B} \boldsymbol{\mu}_{\mathfrak{r}},\quad \mathfrak{p}_y = [ ( \mathbb{C} ^* \times \{(0,0,0)\})/G_{\mathfrak{r},\mathfrak{r}_-}] \cong \mathcal{B} \boldsymbol{\mu}_{\mathfrak{r}_-}, \end{align*} $$

and $\mathfrak {l}_\tau ^ {\mathrm {rig}} - \{ \mathfrak {p}_x, \mathfrak {p}_y\} \cong \mathbb {C} ^*$ . We have a surjective group homomorphism $ \mathbb {Z} \oplus \mathbb {Z} \to {\mathrm {Pic}} (\mathfrak {l}_\tau ^ {\mathrm {rig}} )$ sending $(n_x,n_y)$ to $\mathcal {O}_{\mathfrak {l}_\tau ^ {\mathrm {rig}} }(n_x \mathfrak {p}_x + n_y \mathfrak {p}_y)$ ; the kernel is $ \mathbb {Z} (\mathfrak {r},-\mathfrak {r}_-)$ .

Let $\mathcal {O}(-1)$ denote the tautological line bundle over $\mathcal {B} \mathbb {C} ^*$ associated with the fundamental representation $ \mathbb {C} ^* \to GL(1, \mathbb {C} )$ , $t\mapsto t$ . Given a line bundle L over a DM stack $\mathcal {Z}$ and a positive integer $\mathfrak {m}$ , let $\sqrt [\mathfrak {m}]{L/\mathcal {Z}}$ denote the following fibre product (compare to [Reference Cadman14, Definition 2.2.6]):

where the morphism $\phi _L: \mathcal {Z} \longrightarrow \mathcal {B} \mathbb {C} ^*$ is defined by L (so that $\phi _L^*\mathcal {O}(-1)= L$ ), and $\mathcal {B} \mathbb {C} ^*\to \mathcal {B} \mathbb {C} ^*$ is induced by the $\mathfrak {m}$ th power map from $ \mathbb {C} ^*$ to itself. Then $p_1:\sqrt [\mathfrak {m}]{L/\mathcal {Z}} \to \mathcal {Z}$ is a $\boldsymbol {\mu }_{\mathfrak {m}}$ -banded gerbe. Let $\sqrt [\mathfrak {m}]{L}:=p_2^*\mathcal {O}(-1)\in {\mathrm {Pic}} (\sqrt [\mathfrak {m}]{L/\mathcal {Z}})$ . Then $(\sqrt [\mathfrak {m}]{L})^{\otimes \mathfrak {m}}= p_1^*L$ : that is, $\sqrt [\mathfrak {m}]{L}$ is an $\mathfrak{m}$ th root of $p_1^*L$ .

It is straightforward to check that

• ○ $\hat {\mathfrak {l}}_\tau $ is isomorphic to $\sqrt [\mathfrak {m}]{\mathcal {O}_{\mathfrak {l}_\tau ^ {\mathrm {rig}} }(\mathfrak {s}\mathfrak {p}_x - c\mathfrak {p}_y)/\mathfrak {l}^ {\mathrm {rig}} _\tau }$ as a $\boldsymbol {\mu }_{\mathfrak {m}}$ -banded gerbe over $\mathfrak {l}_\tau ^ {\mathrm {rig}} \cong \mathcal {F}_{\mathfrak {r},\mathfrak {r}_-}$ , and

• ○ $\hat {L}_2 \cong \sqrt [\mathfrak {m}]{\mathcal {O}_{\mathfrak {l}_\tau ^ {\mathrm {rig}} }(\mathfrak {s}\mathfrak {p}_x-c\mathfrak {p}_y)}$ , $\hat {L}_3 = \hat {L}_2^{-1} \otimes p_1^*\mathcal {O}_{\mathfrak {l}_\tau ^ {\mathrm {rig}} }(-\mathfrak {p}_x-\mathfrak {p}_y)$ , where $p_1: \hat {\mathfrak {l}}_\tau \to \mathfrak {l}_\tau ^ {\mathrm {rig}} $ and $\mathcal {O}_{\mathfrak {l}_\tau ^ {\mathrm {rig}} }(-\mathfrak {p}_x-\mathfrak {p}_y)$ is the cotangent bundle of $\mathfrak {l}_\tau ^ {\mathrm {rig}} $ .

3.10 Degeneration

Let $\Sigma _1 =\{ \{0\}, \mathbb {R} _{\geq 0} \mathsf {v}_1, \mathbb {R} _{\geq 0}(-\mathsf {v}_1)\}$ be the complete 1-dimensional fan in $ \mathbb {R} \mathsf {v}_1\cong \mathbb {R} $ , and let $\Sigma _2=\{ \{0\}, \mathbb {R} _{\geq 0} \mathsf {v}_4 , \mathbb {R} _{\geq 0}(-\mathsf {v}_4)\}$ be the complete 1-dimensional fan in $ \mathbb {R} \mathsf {v}_4\cong \mathbb {R} $ . Then $X_{\Sigma _1}=X_{\Sigma _2}= \mathbb {P} ^1$ . The stacky fan $(\Sigma _1, (\mathfrak {r}\mathsf {v}_1,-\mathfrak {r}_-\mathsf {v}_1))$ defines the 1-dimensional toric orbifold $\mathcal {F}_{\mathfrak {r},-\mathfrak {r}_-}$ , and the stacky fan

$$ \begin{align*}\mathbf{\Sigma}_{\Box}= (\Sigma_1\times \Sigma_2, (b_1'=\mathfrak{r}\mathsf{v}_1, b_2'=-\mathfrak{r}_-\mathsf{v}_1, b_3'=\mathsf{v}_4, b_4'= -\mathsf{v}_4)) \end{align*} $$

defines the 2-dimensional toric orbifold $\mathcal {F}_{\mathfrak {r},\mathfrak {r}_-}\times \mathbb {P} ^1$ . The 1-dimensional cones in the fan $\Sigma _1\times \Sigma _2$ are $\{ \rho _i = \mathbb {R} _{\geq 0}b_i': 1\leq i\leq 4\}$ . Let $\Sigma '$ be the fan obtained by adding a 1-dimensional cone $\rho _5 = \mathbb {R} _{\geq 0} b_5'$ , where $b_5'=-\mathsf {v}_1-\mathsf {v}_4$ . Let $\mathcal {S}'$ be the 2-dimensional toric orbifold defined by the stacky fan $\mathbf {\Sigma }'=(\Sigma ',(b_1',b_2',b_3',b_4',b_5'))$ , and let $\mathfrak {l}_i'=\mathcal {V}(\rho _i) \subset \mathcal {S}'$ be 1-dimensional closed toric substack associated with the ray $\rho _i$ . The morphism $\mathbf {\Sigma '}\to \mathbf {\Sigma }_{\Box }$ of stacky fans induces a morphism $\nu : \mathcal {S}' \to \mathcal {F}_{\mathfrak {r},\mathfrak {r}_-} \times \mathbb {P} ^1$ of toric orbifolds; $\nu $ contracts the divisor $\mathfrak {l}^{\prime }_5$ to the torus fixed point $[0,1]\times [0,1] \cong \mathcal {B} \boldsymbol {\mu }_{\mathfrak {r}_-}$ in $\mathcal {F}_{\mathfrak {r},\mathfrak {r}_-}\times \mathbb {P} ^1$ . Let $p: \mathcal {F}_{\mathfrak {r},\mathfrak {r}_-}\times \mathbb {P} ^1\to \mathbb {P} ^1$ be the projection to the second factor. The composition $\pi ':= p\circ \nu $ is a flat morphism, and

$$ \begin{align*}(\pi')^{-1}([0,1]) =\mathfrak{l}^{\prime}_3, \quad (\pi')^{-1}([1,0])=\mathfrak{l}^{\prime}_4\cup \mathfrak{l}^{\prime}_5, \end{align*} $$

where $\mathfrak {l}^{\prime }_3\cong \mathcal {F}_{\mathfrak {r},\mathfrak {r}_-}$ , $\mathfrak {l}^{\prime }_4 \cong \mathcal {F}_{\mathfrak {r},1}$ , and $\mathfrak {l}^{\prime }_5\cong \mathcal {F}_{1,\mathfrak {r}_-}$ . The torus fixed points in $\mathcal {S}'$ are

$$ \begin{align*}\mathfrak{p}_x^0 = \mathfrak{l}_1'\cap \mathfrak{l}_3' \cong \mathcal{B}\boldsymbol{\mu}_{\mathfrak{r}},\ \mathfrak{p}_y^0 = \mathfrak{l}_2'\cap \mathfrak{l}_3'\cong \mathcal{B}\boldsymbol{\mu}_{\mathfrak{r}_-},\ \mathfrak{p}_x^\infty = \mathfrak{l}_1'\cap \mathfrak{l}^{\prime}_4 \cong \mathcal{B} \boldsymbol{\mu}_{\mathfrak{r}},\ \mathfrak{p}_y^\infty = \mathfrak{l}_2'\cap \mathfrak{l}^{\prime}_5 \cong \mathcal{B} \boldsymbol{\mu}_{\mathfrak{r}_-},\ p_z =\mathfrak{l}^{\prime}_4\cap \mathfrak{l}^{\prime}_5, \end{align*} $$

where $p_z$ is a scheme point.

Given any $f\in \mathbb {Z} $ , define

$$ \begin{align*}\hat{\mathcal{S}}= \sqrt[\mathfrak{m}]{ \mathcal{O}_{\mathcal{S}'}(\mathfrak{s}\mathfrak{l}_1' -c\mathfrak{l}_2'+f\mathfrak{l}_5')/\mathcal{S}'}, \end{align*} $$

which is a $\boldsymbol {\mu }_{\mathfrak {m}}$ -banded gerbe over $\mathcal {S}'$ , and let $\hat {q}: \hat {\mathcal {S}} \to \mathcal {S}'=\mathcal {S}^ {\mathrm {rig}} $ be the morphism to the rigidification. Define $\pi := \hat {q}\circ \pi ': \mathcal {S}\to \mathbb {P} ^1$ , and let $\hat {\mathfrak {l}}_i\subset \mathcal {S}$ be the divisor that corresponds to $\mathfrak {l}_i'\subset \mathcal {S}'$ under $\hat {q}:\hat {\mathcal {S}}\to \mathcal {S}'$ . Then $\hat {q}_i:= \hat {q}|_{\mathfrak {l}_i}: \hat {\mathfrak {l}}_i\to \mathfrak {l}^{\prime }_i =\hat {\mathfrak {l}}_i^ {\mathrm {rig}} $ is a $\boldsymbol {\mu }_{\mathfrak {m}}$ -banded gerbe. We have

$$ \begin{align*}\pi^{-1}([0,1]) =\hat{\mathfrak{l}}_3\cong \hat{\mathfrak{l}}_\tau \quad \pi^{-1}([1,0])=\hat{\mathfrak{l}}_4\cup \hat{\mathfrak{l}}_5. \end{align*} $$

Define $\tilde {L}_2, \tilde {L}_3 \in {\mathrm {Pic}} (\hat {\mathcal {S}})$ by

$$ \begin{align*}\tilde{L}_2 := \sqrt[\mathfrak{m}]{ \mathcal{O}_{\mathcal{S}'}(\mathfrak{s}\mathfrak{l}_1' -c\mathfrak{l}_2'+f\mathfrak{l}_5')} \quad \tilde{L}_3 := \tilde{L}_2^{-1}\otimes q^* \mathcal{O}_{\mathcal{S}'}(-\mathfrak{l}_1'-\mathfrak{l}_2'). \end{align*} $$

Then

$$ \begin{align*}\tilde{L}_2\big|_{\hat{\mathfrak{l}}_3}= \sqrt[\mathfrak{m}]{\mathcal{O}_{\mathfrak{l}_3'}(\mathfrak{s}\mathfrak{p}^0_x -c\mathfrak{p}^0_y)} \cong \hat{L}_2, \quad \tilde{L}_3\big|_{\hat{\mathfrak{l}}_3}=\sqrt[\mathfrak{m}]{\mathcal{O}_{\mathfrak{l}_3'}(-\mathfrak{s}\mathfrak{p}^0_x + c\mathfrak{p}^0_y)}\otimes \hat{q}_3^*\mathcal{O}_{\mathfrak{l}_3'}(-\mathfrak{p}_x^0-\mathfrak{p}_y^0) \cong \hat{L}_3. \end{align*} $$

For $i\in \{2,3\}$ , define $\hat {L}_i^+ = \tilde {L}_i\big |_{\hat {\mathfrak {l}}_4}$ and $\hat {L}_i^- =\tilde {L}_i\big |_{\hat {\mathfrak {l}}_5}$ . Then

$$ \begin{align*}\begin{array}{ll} \hat{L}_2^+= \sqrt[\mathfrak{m}]{\mathcal{O}_{\mathfrak{l}_4'}(\mathfrak{s}\mathfrak{p}_x^\infty+fp_z )}, & \hat{L}_3^+ = \sqrt[\mathfrak{m}]{\mathcal{O}_{\mathfrak{l}^{\prime}_4}(-\mathfrak{s}\mathfrak{p}^\infty_x-fp_z)}\otimes \hat{q}_4^*\mathcal{O}_{\mathfrak{l}^{\prime}_4}(-\mathfrak{p}_x^\infty),\\[6pt] \hat{L}_2^-= \sqrt[\mathfrak{m}]{\mathcal{O}_{\mathfrak{l}^{\prime}_5}(-c \mathfrak{p}_y^\infty-fp_z)}, & \hat{L}_3^- = \sqrt[\mathfrak{m}]{\mathcal{O}_{\mathfrak{l}^{\prime}_5}(c \mathfrak{p}^\infty_y + fp_z)}\otimes \hat{q}_5^*\mathcal{O}_{\mathfrak{l}^{\prime}_5}(-\mathfrak{p}_y^\infty). \end{array} \end{align*} $$

To summarise:

• ○ $\hat {\mathcal {S}}$ is a degeneration from $\hat {\mathfrak {l}}_\tau $ to a nodal DM curve $\hat {\mathfrak {l}}_4 \cup \hat {\mathfrak {l}}_5$ , and $\mathcal {S}'=\hat {\mathcal {S}}^ {\mathrm {rig}} $ is a degeneration from the football $\mathfrak {l}_\tau ^ {\mathrm {rig}} \cong \mathcal {F}_{\mathfrak {r},\mathfrak {r}_-}$ to the nodal DM curve $\mathfrak {l}_4'\cup \mathfrak {l}_5'$ .

• ○ For $i=2,3$ , the line bundle $\tilde {L}_i$ on $\hat {\mathcal {S}}$ defines a degeneration of the line bundle $\hat {L}_i\to \hat {\mathfrak {l}}_\tau $ to a line bundle on $\hat {\mathfrak {l}}_4\cup \hat {\mathfrak {l}}_5$ that restricts to $\hat {L}_i^+$ on $\hat {\mathfrak {l}}_4$ and $\hat {L}_i^-$ on $\hat {\mathfrak {l}}_5$ .

• ○ $\hat {\mathcal {S}}$ , $\tilde {L}_i$ , $\hat {\mathfrak {l}}_4$ , $\hat {\mathfrak {l}}_5$ , and $\hat {L}^\pm _i$ depend on f, while $\mathcal {S}'$ , $\mathfrak {l}_4'$ and $\mathfrak {l}_5'$ do not.

Moreover, the total space of $\tilde {L}_2\oplus \tilde {L}_3 \to {\hat {\mathcal {S}}} $ is a 4-dimensional toric orbifold $\hat {\mathcal {W}}$ defined by a stacky fan

(18) $$ \begin{align} (\Sigma_f, (\bar{b}_1, \bar{b}_2, \bar{b}_3, \bar{b}_4, \mathsf{v}_4, -\mathsf{v}_4, -\mathsf{v}_1 -f\mathsf{v}_2 -\mathsf{v}_4)), \end{align} $$

where $\Sigma _f$ is a simplicial fan in $\bigoplus _{i=1}^4 \mathbb {R} \mathsf {v}_i$ , and $(\bar {b}_1,\ldots ,-\mathsf {v}_1-f\mathsf {v}_2 -\mathsf {v}_4)$ is a 7-tuple of vectors in $\bar {N}\oplus \mathbb {Z} \mathsf {v}_4 \cong \mathbb {Z} ^4$ . Let $\mathcal {W}$ be the 4-dimensional smooth toric DM stack defined by the the stacky fan

(19) $$ \begin{align} (\Sigma_f, (b_1, b_2, b_3, b_4, \mathsf{v}_4, -\mathsf{v}_4, -\tilde{\mathsf{v}}_1 -f\tilde{\mathsf{v}}_2-\mathsf{v}_4)), \end{align} $$

where $\tilde {\mathsf {v}}_1,\tilde {\mathsf {v}}_2 \in N$ are lifts of $\mathsf {v}_1,\mathsf {v}_2 \in \bar {N}$ , so that $(b_1,\ldots , -\tilde {\mathsf {v}}_1- f\tilde {\mathsf {v}}_2- \mathsf {v}_4)$ is a 7-tuple of elements in $N\oplus \mathbb {Z} \mathsf {v}_4$ . Then $\mathcal {W}$ is a K-banded gerbe over $\hat {\mathcal {W}} =\mathcal {W}^ {\mathrm {rig}} $ and is a degeneration from the total space $\mathcal {Y}$ of $N_{\mathfrak {l}_\tau /\mathcal {X}}$ to the total space $\mathcal {Y}_\infty $ of a direct sum $L_2^\infty \oplus L_3^\infty $ of line bundles over a nodal DM curve $\mathfrak {l}_+\cup \mathfrak {l}_-$ ; $\mathfrak {l}_+\cup \mathfrak {l}_-$ is a K-banded gerbe over $\hat {\mathfrak {l}}_4\cup \hat {\mathfrak {l}}_5$ and a $G_\tau $ -banded gerbe over $\mathfrak {l}_4'\cup \mathfrak {l}_5'$ . For $i=2,3$ , let $L_i^\pm = L_i^\infty |_{\mathfrak {l}_\pm }$ . Then $L_i^\pm $ is the pullback of $\hat {L}_i^\pm $ . Let $\mathfrak {p}_0\cong \mathcal {B} G_\tau $ be the node that is the intersection of $\mathfrak {l}_+$ and $\mathfrak {l}_-$ , and let $\mathfrak {p}_\pm $ be the unique torus fixed point in $\mathfrak {l}_\pm -\{ \mathfrak {p}_0\}$ . Then $\mathfrak {p}_+ \cong \mathcal {B} G_ {\sigma } $ and $\mathfrak {p}_-\cong \mathcal {B} G_{ {\sigma } _-}$ . Define $ \mathbb {T} '$ weights

$$ \begin{align*}\mathsf{w}^\pm_1 := (c_1)_{ \mathbb{T} '}(T_{\mathfrak{p}_\pm}\mathfrak{l}^\pm),\quad \mathsf{w}^\pm_2 := (c_1)_{ \mathbb{T} '}(L_2)|_{\mathfrak{p}_\pm},\quad \mathsf{w}^\pm_3 := (c_1)_{ \mathbb{T} '}(L_3)|_{\mathfrak{p}_\pm} \in H^2(\mathcal{B} \mathbb{T} ')= \mathbb{Q} \mathsf{u}_1'\oplus \mathbb{Q} \mathsf{u}_2'. \end{align*} $$

Then $\mathsf {w}^+_i=\mathsf {w}^{\prime }_i$ is given by equation (16), $\mathsf {w}_1^\pm +\mathsf {w}_2^\pm +\mathsf {w}_3^\pm =0$ , and

(20) $$ \begin{align} \mathsf{w}_1^-=-\frac{1}{\mathfrak{r}_-}\mathsf{u}_1',\quad \mathsf{w}_2^-=\frac{c}{\mathfrak{r}_-\mathfrak{m}}\mathsf{u}_1' +\frac{1}{\mathfrak{m}}\mathsf{u}_2',\quad \mathsf{w}_3^-=\frac{-c+\mathfrak{m}}{\mathfrak{r}_-\mathfrak{m}}\mathsf{u}_1'-\frac{1}{\mathfrak{m}}\mathsf{u}_2'. \end{align} $$

We also have

$$ \begin{align*}(c_1)_{ \mathbb{T} '}(T_{\mathfrak{p}_0}\mathfrak{l}_\pm) = \mp \mathsf{u}_1',\quad (c_1)_{ \mathbb{T} '}(L_2^\pm)_{\mathfrak{p}_0}=\frac{\mathsf{u}^{\prime}_2-f\mathsf{u}^{\prime}_1}{\mathfrak{m}}= -(c_1)_{ \mathbb{T} '}(L_3^\pm)_{\mathfrak{p}_0}. \end{align*} $$

The above weights are summarised in Figure 7 below.

Figure 7 Degenerated $N_{\mathfrak {l}_\tau /\mathcal {X}}$ and the $ \mathbb {T} '$ -weights.

3.11 Disk factors as equivariant relative GW invariants

When $d_0>0$ , let $\overline {\mathcal {M}}_{0,1}(\mathfrak {l}_+/\mathfrak {p}_0,(d_0,\lambda ))$ be the moduli space of relative maps to $(\mathfrak {l}_+,\mathfrak {p}_0)$ with the relative condition $(d_0,\lambda )$ , where $\lambda \in G_\tau $ [Reference Abramovich and Fantechi1]. A relative stable map to $(\mathfrak {l}_+,\mathfrak {p}_0)$ is a morphism to $\mathfrak {l}_+[m]$ that is the union of $\mathfrak {l}_+$ and a chain of m copies of $ \mathbb {P} ^1\times \mathcal {B} G_\lambda $ . Let $\mathcal {M}_{0,1}(\mathfrak {l}_+/\mathfrak {p}_0,(d_0,\lambda ))\subset \overline {\mathcal {M}}_{0,1}(\mathfrak {l}_+/\mathfrak {p}_0,(d_0,\lambda ))$ be the open substack where the target is $\mathfrak {l}_+[0]=\mathfrak {l}_+$ . The tangent space $\mathcal {T}^1_\xi $ and the obstruction space $\mathcal {T}^2_\xi $ at a moduli point $\xi =[u:(\mathcal {C},x,y)\to (\mathfrak {l}_+,\mathfrak {p}_0)]$ in $\mathcal {M}_{0,1}(\mathfrak {l}_+/\mathfrak {p}_0,(d_0,\lambda ))$ (where $u^{-1}(\mathfrak {p}_0)= d_0 y$ as Cartier divisors) fit into the following exact sequence of complex vector spaces:

(21) $$ \begin{align} \begin{aligned} 0 \to & \mathrm{Ext}^0(\Omega_{\mathcal{C}}(x+y), \mathcal{O}_{\mathcal{C}})\to H^0(\mathcal{C}, u^*(T_{\mathfrak{l}_+}(-\mathfrak{p}_0)) \to \mathcal{T}^1_\xi \\ \to & \mathrm{Ext}^1(\Omega_{\mathcal{C}}(x+y),\mathcal{O}_{\mathcal{C}}) \to H^1(\mathcal{C}, u^*(T_{\mathfrak{l}_+}(-\mathfrak{p}_0)) \to \mathcal{T}^2_\xi. \end{aligned} \end{align} $$

Globally on $\mathcal {M}_{0,1}(\mathfrak {l}_+/\mathfrak {p}_0,(d_0,\lambda ))$ , there is an exact sequence of sheaves

(22) $$ \begin{align} 0 \to B_1 \to B_2 \to \mathcal{T}^1 \to B_4 \to B_5 \to \mathcal{T}^2 \to 0 \end{align} $$

whose fibre at the moduli point $\xi $ is equation (21).

Let $\pi :\mathcal {U}_+\to \mathcal {M}_{0,1}(\mathfrak {l}_+/\mathfrak {p}_0 ,(d_0,\lambda ))$ be the universal domain curve, and let $F_+:\mathcal {U}_+\to \mathfrak {l}_+$ be the evaluation map. We define

$$ \begin{align*}V^+_{0,1}:=R^\bullet\pi_*F_+^*(L_2^+\oplus L_3^+)\in K_{ \mathbb{T} '}\left(\mathcal{M}_{0,1}(\mathfrak{l}_+/\mathfrak{p}_0,(d_0,\lambda))\right), \end{align*} $$

where $R^\bullet \pi _*$ is the K-theoretic push-forward.

For $d_0>0$ , we define

$$ \begin{align*} D_{d_0,\lambda}&=\langle 1_{h^+(d_0,\lambda)} \rangle^{\mathcal{X},\mathcal{L},T_{\mathbb{R}} '}_{0,d_0 b,(d_0,\lambda )} \\ &=\int_{[\mathcal{M}_{0,1}(\mathfrak{l}_+,\mathfrak{p}_0,(d_0,\lambda))]^ {\mathrm{vir}} } \mathrm{ev}^*(1_{h^+(d_0,\lambda)})e_{ \mathbb{T} '}(V^+_{0,1}). \end{align*} $$

When $d_0<0$ , let $\overline {\mathcal {M}}_{0,1}(\mathfrak {l}_-/\mathfrak {p}_0, (-d_0,\lambda ^{-1}))$ be the moduli space of relative stable maps to $(\mathfrak {l}_-,\mathfrak {p}_0)$ with relative condition $(-d_0,\lambda ^{-1})$ , and let $\mathcal {M}_{0,1}(\mathfrak {l}_-/\mathfrak {p}_0,(-d_0,\lambda ^{-1}))$ be the open substack where the target is $\mathfrak {l}_-$ . Let $\pi :\mathcal {U}_-\to \mathcal {M}_{0,1}(\mathfrak {l}_-/\mathfrak {p}_0 ,(-d_0,\lambda ^{-1}))$ be the universal domain curve, and let $F_-:\mathcal {U}_-\to \mathfrak {l}_-$ be the evaluation map. We define

$$ \begin{align*}V^-_{0,1}=R^\bullet\pi_* F_-^*(L_2^-\oplus L_3^-) \in K_{ \mathbb{T} '}\left(\mathcal{M}_{0,1}(\mathfrak{l}_-/\mathfrak{p}_0,(-d_0,\lambda^{-1}))\right) \end{align*} $$

and define

$$ \begin{align*} D_{d_0,\lambda}&=\langle 1_{h^-(d_0,\lambda)} \rangle^{\mathcal{X},\mathcal{L},T^{\prime}_{\mathbb{R}} }_{0,-d_0 (\alpha-b), (d_0,\lambda))} \\ &=\int_{[\mathcal{M}_{0,1}(\mathfrak{l}_-,\mathfrak{p}_0,(-d_0,\lambda^{-1}))]^ {\mathrm{vir}} } \mathrm{ev}^*(1_{h^-(d_0,\lambda)})e_{ \mathbb{T} '}(V^-_{0,1}). \end{align*} $$

Let $u:(\mathcal {C},x,y)\to \mathfrak {l}_+ $ be a relative stable map that represents a point in $\mathcal {M}$ . Suppose that u is fixed by the torus action. Recall that $c_i:G_ {\sigma } \to [0,1)\cap \mathbb {Q} $ is defined by $\chi _i(k)=\exp (2\pi \sqrt {-1}c_i(k))$ . In the computation below, let $k^\pm =h^\pm (d_0,\lambda )$ . For $j=1, 2,3$ , let $\epsilon _j= c_j(k^+)$ . Then $\epsilon _1=\langle \frac {d_0}{\mathfrak {r}_+}\rangle $ . We have the following weights

$$ \begin{align*} {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^0(\mathcal{C},u^*L_1^+) \bigr) = \sum_{a=0}^{\lfloor d_0w_1\rfloor} e^{a \frac{\mathsf{u}_1'}{d_0}},&\quad {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^1(\mathcal{C},u^*L_1^+) \bigr) = 0, \\[6pt] {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^0(\mathcal{C},u^*L_2^+\otimes \mathcal{O}_y) \bigr) = \delta_{\langle d_0w_2- \epsilon_2\rangle, 0} e^{\frac{\mathsf{u}_2'-f\mathsf{u}_1'}{\mathfrak{m}}},&\quad {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^1(\mathcal{C},u^*L_2^+\otimes \mathcal{O}_y) \bigr) = 0, \end{align*} $$

$$ \begin{align*} {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^0(\mathcal{C}, u^*L^+_2) \bigr) &= \begin{cases} \displaystyle{ \sum_{a= -\lfloor d_0w_2-\epsilon_2\rfloor}^0 e^{\mathsf{w}^{\prime}_2+(a-\epsilon_2) \frac{\mathsf{u}^{\prime}_1}{d_0 } } }, & f \geq 0,\\ 0, & f<0,\\ \end{cases} \\[6pt] {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^1(\mathcal{C}, u^*L^+_2) \bigr) &= \begin{cases} 0, & f \geq 0,\\ \displaystyle{ \sum_{a=1}^{-\lfloor d_0w_2-\epsilon_2\rfloor-1} e^{\mathsf{w}^{\prime}_2+(a-\epsilon_2)\frac{\mathsf{u}^{\prime}_1}{d_0 } } }, & f<0 \end{cases}\\[6pt] {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^0(\mathcal{C}, u^*L^+_3) \bigr) &= \begin{cases} \displaystyle{ \sum_{a=-\lfloor d_0w_3-\epsilon_3\rfloor}^0 e^{\mathsf{w}^{\prime}_3+(a-\epsilon_3) \frac{\mathsf{u}^{\prime}_1}{d_0 } } }, & f<0,\\ 0, & f\geq 0,\\ \end{cases}\\[6pt] {\mathrm{ch}} _{ \mathbb{T} '}\bigl( H^1(\mathcal{C}, u^*L^+_3) \bigr) &= \begin{cases} 0, & f<0 ,\\ \displaystyle{ \sum_{a=1}^{-\lfloor d_0w_3 - \epsilon_3\rfloor-1} e^{\mathsf{w}^{\prime}_3+(a-\epsilon_3)\frac{\mathsf{u}^{\prime}_1}{d_0 } } }, & f\geq 0. \end{cases} \end{align*} $$

and the following identities

$$ \begin{align*} \sum_{a=-\lfloor w_3d_0-\epsilon_3 \rfloor}^0 e^{\mathsf{w}^{\prime}_3+(a-\epsilon_3) \frac{\mathsf{u}^{\prime}_1}{d_0 } } &= \sum_{a=d_0w_1+\epsilon_2+\epsilon_3}^{-\lfloor d_0 w_2-\epsilon_2\rfloor -1 + \delta_{\langle d_0 w_2-\epsilon_2 \rangle , 0}} e^{-\mathsf{w}^{\prime}_2 +(\epsilon_2-a)\frac{\mathsf{u}^{\prime}_1}{d_0 }} \\[6pt] \sum_{a=1}^{-\lfloor w_3 d_0 +1 - \epsilon_3 \rfloor} e^{\mathsf{w}^{\prime}_3+(a-\epsilon_3)\frac{\mathsf{u}^{\prime}_1}{d_0 } } &= \sum_{a=-\lfloor d_0w_2-\epsilon_2\rfloor +\delta_{\langle d_0w_2-\epsilon_2 \rangle}}^{\frac{d_0 }{s^+_1} +\epsilon_2+\epsilon_3-1} e^{-\mathsf{w}^{\prime}_2 +(\epsilon_2-a)\frac{\mathsf{u}_1'}{d_0 }}\\[6pt] d_0w_1 + \epsilon_2 +\epsilon_3 &= \left\lfloor \frac{d_0 }{s^+_1} \right\rfloor + \mathrm{age}(k^+). \end{align*} $$

The $ \mathbb {T} '$ -equivariant Euler classes are

$$ \begin{align*} e_{ \mathbb{T} '}(B_1^m) &= 1,\\[6pt] e_{ \mathbb{T} '}(B_2^m) &= \lfloor d_0 w_1 \rfloor! \left(\frac{\mathsf{u}^{\prime}_1}{d_0} \right)^{\lfloor d_0w_1 \rfloor}, \\[6pt] \frac{e_{ \mathbb{T} '}(B_5^m)}{e_{ \mathbb{T} '}(B_4^m)} &= (-1)^{\lceil d_0 w_2-\epsilon_2 \rceil +\lfloor d_0w_1 \rfloor +\mathrm{age}(k^+)-1} \prod_{a =1}^{\lfloor d_0w_1 \rfloor + \mathrm{age}(k^+) -1} \left(\mathsf{w}^{\prime}_2 + (a-\epsilon_2)\frac{\mathsf{u}^{\prime}_1}{d_0}\right). \end{align*} $$

Since $|\mathrm {Aut}(f)| = d_0|G_\tau |$ , by localisation,

$$ \begin{align*} D_{d_0,\lambda} &=D(d_0,k^+,k^-) = \frac{1}{|\mathrm{Aut}(f)|} \frac{e_{ \mathbb{T} '}(B_1^m) e_{ \mathbb{T} '}(B_5^m)}{e_{ \mathbb{T} '}(B_2^m) e_{ \mathbb{T} '}(B_4^m)} \\ = &\frac{1}{d_0|G_\tau|}\frac{\prod_{a=1}^{\lfloor d_0 w_1\rfloor + \mathrm{age}(k^+) -1}(\mathsf{w}^{\prime}_2 + (a-\epsilon_2)\frac{\mathsf{u}^{\prime}_1}{d_0})} {\lfloor d_0 w_1\rfloor! (\frac{\mathsf{u}^{\prime}_1}{d_0})^{\lfloor d_0w_1\rfloor}} \cdot (-1)^{\lceil d_0w_2-\epsilon_2\rceil +\lfloor d_0w_1\rfloor +\mathrm{age}(k^+)-1 } \\ =&(-1)^{\lceil d_0w_2-\epsilon_2\rceil +\lfloor d_0w_1\rfloor +\mathrm{age}(k^+)-1 } \Big(\frac{\mathsf{u}_1'}{d_0}\Big)^{\mathrm{age}(k^+)-1} \cdot \frac{1}{d_0|G_\tau|}\frac{\prod_{a=1}^{\lfloor d_0w_1\rfloor +\mathrm{age}(k^+)-1} (\frac{d_0 \mathsf{w}^{\prime}_2}{s^+_1\mathsf{w}^{\prime}_1} +a-\epsilon_2)} {\lfloor d_0w_1\rfloor !} \\ = &- (-1)^{\lfloor d_0w_3+\frac{\bar\lambda}{\mathfrak{m}} \rfloor } \Big(\frac{\mathsf{u}_1'}{d_0}\Big)^{\mathrm{age}(k^+)-1} \cdot \frac{1}{d_0|G_\tau|}\frac{\prod_{a=1}^{\lfloor d_0w_1\rfloor +\mathrm{age}(k^+)-1} (\frac{d_0 \mathsf{w}^{\prime}_2}{s^+_1\mathsf{w}^{\prime}_1} +a-\epsilon_2)} {\lfloor d_0w_1\rfloor !} \end{align*} $$

Let $\mathsf {u}:= \iota _f^* \mathsf {u}^{\prime }_1 \in H^2( \mathbb {T} _f; \mathbb {Z} )$ . Then $H^*( \mathbb {T} _f; \mathbb {Q} )= \mathbb {Q} [\mathsf {u}]$ and $\iota _f^*\mathsf {u}_2'=f\mathsf {u}$ . Define $D_{d_0,\lambda ,f}=\iota _{f}^* D_{d_0,\lambda }$ . Hence when $d_0>0$ ,

$$ \begin{align*} & D_{d_0,\lambda,f} \\ &\quad = -(-1)^{\lfloor d_0w_3+\frac{\bar \lambda}{\mathfrak{m}}\rfloor}\left(\frac{\mathsf{u}}{d_0}\right)^{\mathrm{age}(h^+(d_0,\lambda))-1} \cdot \frac{1}{d_0|G_\tau|}\frac{\prod_{a=1}^{\lfloor d_0w_1\rfloor +\mathrm{age}(h^+(d_0,\lambda))-1} (d_0 w_2 +a-c_2(h^+(d_0,\lambda))} {\lfloor d_0w_1\rfloor !}. \end{align*} $$

If $d_0 <0$ , similar computation shows (notice $\overline { \lambda ^{-1}}=(1-\delta _{\bar \lambda ,0})(\mathfrak{m}-\bar \lambda )\in \{0,\dots ,\mathfrak{m}-1\}$ )

$$ \begin{align*} D_{d_0 ,\lambda,f} &= -(-1)^{\lfloor d_0w_2^- + \left(1-\frac{\bar\lambda}{m}-\delta_{\bar \lambda,0}\right)\rfloor} \left(\frac{\mathsf{u}}{d_0 }\right)^{\mathrm{age}(h^-(d_0,\lambda))-1} \\ &\quad \cdot \frac{1}{-d_0 |G_\tau|}\frac{\prod_{a=1}^{\lfloor d_0 w_1^- \rfloor+\mathrm{age}(h^-(d_0,\lambda))-1}(d_0w_3^--c_3(h^-(d_0,\lambda))+a)}{\lfloor d_0w_1^-\rfloor!}. \end{align*} $$

If $\mathcal {L}$ is an outer brane, it is the same as $d_0>0$ . Define

$$ \begin{align*} D_{d_0,\lambda,f} &= -(-1)^{\lfloor d_0w_3+ \frac{\bar \lambda}{\mathfrak{m}}\rfloor}\big(\frac{\mathsf{u}}{d_0} \big)^{\mathrm{age}(h(d_0,\lambda))-1} \cdot \frac{1}{d_0|G_\tau|}\frac{\prod_{a=1}^{\lfloor d_0w_1\rfloor +\mathrm{age}(h(d_0,\lambda))-1} (d_0 w_2 +a-c_2(h(d_0,\lambda))} {\lfloor d_0w_1\rfloor !}. \end{align*} $$

3.12 Open-closed GW invariants and descendant GW invariants

For any torus fixed point $\mathfrak {p}_ {\sigma } $ of $\mathcal {X}$ , where $ {\sigma } \in \Sigma (3)$ , we have

$$ \begin{align*}H^*_ {\mathrm{CR}} (\mathfrak{p}_ {\sigma} ) =\bigoplus_{k \in G_ {\sigma} } \mathbb{Q} \mathbf{1}_k,\quad H^*_{ {\mathrm{CR}} , \mathbb{T} '}(\mathfrak{p}_ {\sigma} ) =\bigoplus_{k \in G_ {\sigma} } \mathbb{Q} [\mathsf{w}^{\prime}_1,\mathsf{w}^{\prime}_2] \mathbf{1}_k. \end{align*} $$

The inclusion $\iota _ {\sigma } : \mathfrak {p}_ {\sigma } \hookrightarrow \mathcal {X}$ induces

$$ \begin{align*}\iota_{ {\sigma} *} : H^*_{ {\mathrm{CR}} , \mathbb{T} '}(\mathfrak{p}_ {\sigma} ) = H^*_{ \mathbb{T} '}(\mathcal{I}\mathfrak{p}_ {\sigma} ) \to H^*_{ {\mathrm{CR}} , \mathbb{T} '}(\mathcal{X}) = H^*_{ \mathbb{T} '}(\mathcal{I}\mathcal{X}). \end{align*} $$

Define

$$\begin{align*}\phi_{ {\sigma} ,k} = \iota_{ {\sigma} *} \mathbf{1}_k\in H^*_{ {\mathrm{CR}} , \mathbb{T} '}(\mathcal{X}),\ \phi_{ {\sigma} ,k}^f=\iota_f^* \phi_{ {\sigma} ,k}\in H^*_{ {\mathrm{CR}} , \mathbb{T} _f}(\mathcal{X}). \end{align*}$$

Proposition 3.2 (framed inner brane)

Suppose that $(\mathcal {L},f)$ is a framed inner brane and

$$ \begin{align*}{\vec{\mu}} =((\mu_1,\lambda_j),\ldots, (\mu_h,\lambda_h)), \end{align*} $$

where $(\mu _j,\lambda _j)\in H_1(\mathcal {L}; \mathbb {Z} )\cong \mathbb {Z} \times \mathcal {B} G_\tau $ . Let $J_\pm =\{ j\in \{1,\ldots ,h\}: \pm \mu _j>0\}$ , and let $k_j^\pm = h^\pm (\mu _j,\lambda _j)$ . Then

$$ \begin{align*} & \langle \gamma_1,\ldots, \gamma_n\rangle_{g,\beta', {\vec{\mu}} }^{\mathcal{X}, (\mathcal{L},f)}=\prod_{j=1}^h D^{\prime}_{\mu_j,\lambda_j,f} \\ &\quad \cdot \int_{[\overline{\mathcal{M}}_{g,n+h}(\mathcal{X},\beta)]^ {\mathrm{vir}} } \frac{ \big(\prod_{i=1}^n\mathrm{ev}_i^*\gamma_i \prod_{j\in J_+}\mathrm{ev}_{n+j}^*\phi^f_{ {\sigma} _+, (h^+(d_0,\lambda_j))^{-1}} \prod_{j\in J_-} \mathrm{ev}_{n+j}^*\phi^f_{ {\sigma} _-, (h^-(d_0,\lambda_j))^{-1}} \big)} {\prod_{j=1}^h\frac{\mathsf{u}}{\mu_j}(\frac{\mathsf{u}}{\mu_j} -\bar{\psi}_{n+j})}, \end{align*} $$

where

$$ \begin{align*}\beta\in H_2(\mathcal{X}),\quad \beta'= \beta + \Big(\sum_{j\in J_+} \mu_j\Big)b -\Big(\sum_{j\in J_-} \mu_j\Big)(\alpha-b) \in H_2(\mathcal{X},\mathcal{L}). \end{align*} $$

Proof. There exists $\beta \in H_2(\mathcal {X})$ such that

$$ \begin{align*}\beta'=\beta +\Big(\sum_{j\in J_+} \mu_j \Big)b + \sum_{j\in J_-}(-\mu_j)(\alpha-b). \end{align*} $$

Let $\langle k_j^+\rangle $ be the cyclic subgroup generated by $k_j^+$ , and let $r_j$ be the cardinality of $\langle k_j^+\rangle $ for $j\in J_+$ and $r_j$ be the cardinality of $\langle k_j^-\rangle $ for $j\in J_-$ .

We have

$$ \begin{align*} \overline{\mathcal{M}}_{(g,h),n}(\mathcal{X},\mathcal{L}\mid \beta', {\vec{\mu}} )^{ \mathbb{T} ^{\prime}_{\mathbb{R}} } &= \bigcup_{ {\Gamma} \in G_{g,n}(\mathcal{X},\mathcal{L}\mid \beta', {\vec{\mu}} )} F_\Gamma \\ \overline{\mathcal{M}}_{g,n+h}(\mathcal{X}, \beta)^{ \mathbb{T} ^{\prime}_{\mathbb{R}} } = \overline{\mathcal{M}}_{g,n+h}(\mathcal{X}, \beta)^{ \mathbb{T} '} &= \bigcup_{ {\hat{ {\Gamma} }} \in G_{g,n+h}(\mathcal{X},\beta) } F_{ {\hat{ {\Gamma} }} }. \end{align*} $$

In the remaining part of this subsection, we use the following abbreviations:

$$ \begin{align*} & \mathcal{M} = \overline{\mathcal{M}}_{g,n}(\mathcal{X},\mathcal{L}\mid \beta', {\vec{\mu}} ),\quad \hat{\mathcal{M}} = \overline{\mathcal{M}}_{g,n+h}(\mathcal{X}, \beta), \\ & \mathcal{M}_j = \begin{cases} \overline{\mathcal{M}}_{(0,1),1}(\mathcal{X},\mathcal{L}\mid \mu_j b, (\mu_j,\lambda)), & j\in J_+\\ \overline{\mathcal{M}}_{(0,1),1}(\mathcal{X}, \mathcal{L}\mid -\mu_j(\alpha-b), (\mu_j, \lambda)), & j\in J_- \end{cases} \\ & \mathcal{G} = G_{g,n}(\mathcal{X},\mathcal{L}\mid \beta', {\vec{\mu}} ),\quad \hat{\mathcal{G}} = G_{g,n+h}(\mathcal{X},\beta)\\ & \mathbf{x}=(x_1,\ldots, x_n),\quad \mathbf{y}=(y_1,\ldots, y_h). \end{align*} $$

Given $u:(\Sigma ,\mathbf {x},\partial \Sigma )\to (\mathcal {X}, \mathcal {L})$ that represents a point $\xi \in \mathcal {M}^{ \mathbb {T} ^{\prime }_{\mathbb {R}} }$ , we have

$$ \begin{align*}\Sigma = \mathcal{C} \cup \bigcup_{j=1}^h D_j, \end{align*} $$

where $\mathcal {C}$ is an orbicurve of genus g, $x_1,\ldots , x_n \in \mathcal {C}$ , $D_j = [\{ z\in \mathbb {C} \mid |z|\leq 1\}/ \mathbb {Z} _{r_j}]$ , $\mathcal {C}$ and $D_j$ intersect at $y_j = B \mathbb {Z} _{r_j}$ . Let $u_j=u|_{D_j}$ and $\hat {u}= u|_{\mathcal {C}}$ . Then

1. 1. For $j=1,\ldots , h$ , $u_j: (D_j,\partial D_j)\to (\mathcal {X}, \mathcal {L})$ represents a point in $\mathcal {M}_j^{ \mathbb {T} ^{\prime }_{\mathbb {R}} }$ .

2. 2. $\hat {u}:(\mathcal {C},\mathbf {x},\mathbf {y})\to \mathcal {X}$ represents a point $\hat {\xi }\in \hat {\mathcal {M}}^{ \mathbb {T} '}$ , and $\hat {u}(\mathbf {y}_j) =[\mathfrak {p}_\pm , (k_j^\pm )^{-1}] \in \mathcal {I} \mathfrak {p}_\pm \subset \mathcal {I}\mathcal {X}$ if $j\in J_\pm $ .

Let $x_{n+j}=y_j$ . Let $F_ {\Gamma } $ be the connected component of $\mathcal {M}^{ \mathbb {T} '}$ associated to the decorated graph $ {\Gamma } \in \mathcal {G}$ , and let $F_{ {\hat { {\Gamma } }} }$ be the connected component of $\hat {\mathcal {M}}^{ \mathbb {T} ^{\prime }_{\mathbb {R}} }$ associated to the decorated graph $\hat {\Gamma } \in \mathcal {G}$ . Then for any $ {\Gamma } \in \mathcal {G}$ there exists $ {\hat { {\Gamma } }} \in \hat {\mathcal {G}}$ such that

$$ \begin{align*}\mathrm{ev}_{n+j}(F_{ {\hat{ {\Gamma} }} }) = (\mathfrak{p}_\pm,(k_j^\pm)^{-1}) \in \mathcal{I} \mathfrak{p}_\pm \subset \mathcal{I} \mathcal{X} \end{align*} $$

if $j\in J_\pm $ , and $F_{ {\Gamma } }$ can be identified with $F_{ {\hat { {\Gamma } }} }$ up to a finite morphism. More precisely,

$$ \begin{align*} [F_ {\Gamma} ]^ {\mathrm{vir}} &= \prod_{j\in J_+} \frac{|G_{ {\sigma} _+}|}{r_j|\mathrm{Aut}(u_j)|} \prod_{j\in J_-} \frac{|G_{ {\sigma} _-}|}{r_j|\mathrm{Aut}(u_j)|} [F_{ {\hat{ {\Gamma} }} }]^ {\mathrm{vir}} \\ &=\prod_{j\in J_+} \frac{s^+_1}{r_j\mu_j} \prod_{j\in J_-} \frac{s^-_1}{-r_j\mu_j} [F_{ {\hat{ {\Gamma} }} }]^ {\mathrm{vir}}. \end{align*} $$

We have

$$ \begin{align*}\frac{1}{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(N_ {\Gamma} ^ {\mathrm{vir}} )} =\frac{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(B_1^m) e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(B_5^m)}{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(B_2^m) e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(B_4^m)},\quad \frac{1}{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(N_ {\hat{ {\Gamma} }} ^ {\mathrm{vir}} )} =\frac{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(\hat{B}_1^m) e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(\hat{B}_5^m) }{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(\hat{B}_2^m) e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(\hat{B}_4^m)}, \end{align*} $$

where

$$ \begin{align*} e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(B_1^m)&= e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(\hat{B}_1^m), \\ e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(B_4^m) &= e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(\hat{B}_4^m) \prod_{j\in J_+}\left(\frac{\mathsf{u}_1'}{r_j \mu_j} -\frac{\bar{\psi}_j}{r_j}\right) \prod_{j\in J_-}\left(\frac{\mathsf{u}_1'}{r_j \mu_j}-\frac{\bar{\psi}_j}{r_j}\right) \end{align*} $$

For $k=0,1$ and $j=1,\ldots , h$ , let

$$ \begin{align*}H^k(D_j) = H^k\bigl(D_j, \partial D_j, u_j^*T\mathcal{X}, (u_j|_{\partial D_j})^*T\mathcal{L}\bigr). \end{align*} $$

Then there is a long exact sequence

$$ \begin{align*} & 0 \to B_2 \to \hat{B}_2 \oplus \bigoplus_{j=1}^h H^0(D_j) \to \bigoplus_{j\in J_+} (T_{\mathfrak{p}_+}\mathcal{X})^{k_j^+}\oplus \bigoplus_{j\in J_-} (T_{\mathfrak{p}_-}\mathcal{X})^{k_j^-} \\ & \to B_5 \to \hat{B}_5 \oplus \bigoplus_{j=1}^h H^1(D_j)\to 0, \end{align*} $$

where $(T_{\mathfrak {p}_\pm }\mathcal {X})^{k_j^\pm }$ denote the $k_j^\pm $ -invariant part of $T_{\mathfrak {p}_\pm }\mathcal {X}$ . Note that

$$ \begin{align*}(T_{\mathfrak{p}_\pm}\mathcal{X})^{k_j^\pm} = T_{(\mathfrak{p}_\pm, k_j^\pm)} \mathcal{I} \mathcal{X} = T_{(\mathfrak{p}_\pm, k_j^{-1})} \mathcal{I} \mathcal{X} \end{align*} $$

$$ \begin{align*} \frac{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(H^1(D_j)^m)}{e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(H^0(D_j)^m)} = |\mu_j ||G_\tau| D_{\mu_j,\lambda_j}. \end{align*} $$

Let

$$ \begin{align*} e_{ \mathbb{T} _f}(N^ {\mathrm{vir}} _\Gamma) &:= \iota_f^* e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(N^ {\mathrm{vir}} _\Gamma) = e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(N^ {\mathrm{vir}} _\Gamma)\Big|_{\mathsf{u}_1'=\mathsf{u}, \mathsf{u}_2'=f\mathsf{u}},\\ e_{ \mathbb{T} _f}(N^ {\mathrm{vir}} _ {\hat{ {\Gamma} }} ) &:= \iota_f^* e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(N^ {\mathrm{vir}} _ {\hat{ {\Gamma} }} ) = e_{ \mathbb{T} ^{\prime}_{\mathbb{R}} }(N^ {\mathrm{vir}} _ {\hat{ {\Gamma} }} )\Big|_{\mathsf{u}_1'=\mathsf{u}, \mathsf{u}_2'=f\mathsf{u}}. \end{align*} $$

Given $\gamma _1,\ldots ,\gamma _n \in H^*_{ {\mathrm {CR}} , \mathbb {T} _f}(\mathcal {X})$ , we define

$$ \begin{align*} &\langle \gamma_1,\dots,\gamma_h\rangle^{\mathcal{X},(\mathcal{L},f)}_{\beta',((\mu_1,\lambda_1),\dots,(\mu_h,\lambda_h))}:=\int_{[F_ {\Gamma} ]^ {\mathrm{vir}} } \frac{ \big(\prod_{i=1}^n\mathrm{ev}_i^*\gamma_i\big)|_{F_\Gamma}}{e_{ \mathbb{T} _f}(N_ {\Gamma} ^ {\mathrm{vir}} )} \\ =& \prod_{j\in J_+ }\frac{\mathfrak{r}_+}{r_j\mu_j} \prod_{j\in J_-}\frac{\mathfrak{r}_-}{- r_j\mu_j} \prod_{j=1}^h (|\mu_j||G_\tau|D_{\mu_j, \lambda_j,f}) \\ &\cdot \int_{[F_{ {\hat{ {\Gamma} }} }]^ {\mathrm{vir}} } \frac{ \big(\prod_{i=1}^n\mathrm{ev}_i^*\gamma_i \prod_{j\in J_+}\mathrm{ev}_{n+j}^*\phi_{ {\sigma} _+, (k_j^+)^{-1}} \prod_{j\in J_-} \mathrm{ev}_{n+j}^*\phi^f_{ {\sigma} _-, (k_j^-)^{-1}}\big)|_{F_\Gamma}} {\prod_{j\in J_+}(\frac{\mathsf{u}_1'}{r_j \mu_j} -\frac{\bar{\psi}_{n+j}}{r_j}) \prod_{j\in J_-}(\frac{\mathsf{u}_1'}{r_j \mu_j} -\frac{\bar{\psi}_{n+j}}{r_j})e_{ \mathbb{T} _f}(N_ {\hat{ {\Gamma} }} ^ {\mathrm{vir}} )} \\ =& \prod_{j=1}^hD^{\prime}_{\mu_j,\lambda_j,f} \cdot \int_{[F_{ {\hat{ {\Gamma} }} }]^ {\mathrm{vir}} } \frac{ \big(\prod_{i=1}^n\mathrm{ev}_i^*\gamma_i \prod_{j\in J_+}\mathrm{ev}_{n+j}^*\phi^f_{ {\sigma} _+, (k_j^+)^{-1}} \prod_{j\in J_-}\mathrm{ev}_{n+j}^*\phi^f_{ {\sigma} _-, (k_j^-)^{-1}}\big)|_{F_\Gamma}}{ \prod_{j=1}^h\frac{\mathsf{u}}{\mu_j}(\frac{\mathsf{u}}{\mu_j} -\bar{\psi}_{n+j})e_{ \mathbb{T} _f}(N_ {\hat{ {\Gamma} }} ^ {\mathrm{vir}} )} \end{align*} $$

where

$$ \begin{align*}& D^{\prime}_{d_0,\lambda,f} \\ &= \begin{cases} \displaystyle{ -(-1)^{\lfloor d_0w^+_3 + \frac{\bar \lambda}{\mathfrak{m}}\rfloor}\Big(\frac{\mathsf{u}}{d_0}\Big)^{\mathrm{age}(k^+)} \cdot \frac{\mathfrak{r}_+}{d_0}\cdot \frac{\prod_{a=1}^{\lfloor d_0w^+_1\rfloor +\mathrm{age}(k^+)-1} (d_0 w^+_2 +a-c_2(k^+))} {\lfloor d_0w^+_1\rfloor !}}, \quad d_0>0, & \\ & \\ \displaystyle{ -(-1)^{\lfloor d_0w_2^- + (1-\frac{\bar\lambda}{\mathfrak{m}}-\delta_{\bar \lambda,0})\rfloor} \Big(\frac{\mathsf{u}}{d_0 }\Big)^{\mathrm{age}(k^-)} \cdot \frac{\mathfrak{r}_-}{-d_0}\cdot \frac{\prod_{a=1}^{\lfloor d_0 w_1^- \rfloor+\mathrm{age}(k^-)-1}(d_0w_3^--c_3(k^-)+a)}{\lfloor d_0w_1^-\rfloor!} } , \quad d_0<0. & \\ \end{cases} \end{align*} $$

Suppose that $(\mathcal {L},f)$ is a framed outer brane. Define

$$ \begin{align*}D^{\prime}_{d_0,\lambda,f}= -(-1)^{\lfloor d_0w_3+ \frac{\bar \lambda}{\mathfrak{m}}\rfloor}\Big(\frac{\mathsf{u}}{d_0}\Big)^{\mathrm{age}(k)} \cdot \frac{\mathfrak{r}}{d_0}\cdot \frac{\prod_{a=1}^{\lfloor d_0w_1\rfloor +\mathrm{age}(k)-1} (d_0 w_2 +a-c_2(k))} {\lfloor d_0w_1\rfloor !}, \end{align*} $$

where $k=h(d_0,\lambda )$ . By the $d_0>0$ part of the proof of Proposition 3.2, we obtain:

Proposition 3.3 (framed outer brane)

Suppose that $(\mathcal {L},f)$ is a framed inner brane and $ {\vec {\mu }} =((\mu _1,\lambda _1),\ldots , (\mu _h,\lambda _h))$ , where $(\mu _j,\lambda _j)\in H_1(\mathcal {L}; \mathbb {Z} )$ . Then

$$ \begin{align*}\langle \gamma_1,\ldots, \gamma_n\rangle_{g,\beta', {\vec{\mu}} }^{\mathcal{X}, (\mathcal{L},f)} =\prod_{j=1}^h D^{\prime}_{\mu_j,\lambda,f} \cdot \int_{[\overline{\mathcal{M}}_{g,n+h}(\mathcal{X},\beta)]^ {\mathrm{vir}} } \frac{ \big(\prod_{i=1}^n\mathrm{ev}_i^*\gamma_i \prod_{j=1}^h\mathrm{ev}_{n+j}^*\phi^f_{ {\sigma} , (h(d_0,\lambda))^{-1}}\big)} {\prod_{j=1}^h\frac{\mathsf{u}}{\mu_j}(\frac{\mathsf{u}}{\mu_j} -\bar{\psi}_{n+j})}, \end{align*} $$

where

$$ \begin{align*}\beta\in H_2(\mathcal{X}),\quad \beta'= \beta + \Big(\sum_{j=1}^h \mu_j\Big)b. \end{align*} $$

3.13 Generating functions of open-closed GW invariants

From now on, we assume the generic stabiliser K is trivial so that $\mathcal {X}=\mathcal {X}^ {\mathrm {{\kern1.5pt}rig}} $ is a toric Calabi-Yau 3-orbifold. Then

$$ \begin{align*}\chi_3: G_\tau \longrightarrow \boldsymbol{\mu}_{\mathfrak{m}}, \quad \lambda \mapsto e^{2\pi\sqrt{-1}\bar{\lambda}/\mathfrak{m}} \end{align*} $$

is a group isomorphism. We have $\bar {N}=N$ and $\bar {b}_i=\bar {b}_i$ . In particular,

$$ \begin{align*}b_1 = \mathfrak{r} \mathsf{v}_1 -\mathfrak{s} \mathsf{v}_2 + \mathsf{v}_3,\quad b_2 = \mathfrak{m} \mathsf{v}_2 + \mathsf{v}_3,\quad b_3 = \mathsf{v}_3. \end{align*} $$

There exists $m_a,n_a\in \mathbb {Z} $ , such that

$$ \begin{align*}b_{3+a} = m_a \mathsf{v}_1 + n_a \mathsf{v}_2 + \mathsf{v}_3,\quad a=1,\ldots,k. \end{align*} $$

Introduce variables $\{ X_j \mid j=1,\ldots ,h\}$ , and let

$$ \begin{align*}\boldsymbol{\tau}_2 = \sum_{i=1}^m \tau_i u_i, \end{align*} $$

where $u_1,\ldots , u_m$ form a basis of $H^2_ {\mathrm {CR}} (\mathcal {X}; \mathbb {Q} )$ . We choose $ \mathbb {T} '$ -equivariant lifting of $\boldsymbol {\tau }_2$ as follows: for each $u_i\in H^2_ {\mathrm {CR}} (\mathcal {X}; \mathbb {Q} )$ , we choose the unique $ \mathbb {T} '$ -equivariant lifting $u_i^{ \mathbb {T} '} \in H^2_{ {\mathrm {CR}} , \mathbb {T} '}(\mathcal {X}; \mathbb {Q} )$ such that $\iota _ {\sigma } ^* u_i^{ \mathbb {T} '} = 0\in H^2_{ {\mathrm {CR}} , \mathbb {T} '}(\mathfrak {p}_ {\sigma }; \mathbb {Q} )$ , where $\iota _ {\sigma } ^*: H^2_{ {\mathrm {CR}} , \mathbb {T} '}(\mathcal {X}; \mathbb {Q} )\to H^2_{ {\mathrm {CR}} , \mathbb {T} '}(\mathfrak {p}_ {\sigma }; \mathbb {Q} )$ is induced by the inclusion map $\iota _ {\sigma } :\mathfrak {p}_ {\sigma } \to \mathcal {X}$ .

We define $\xi _0:= e^{-\pi \sqrt {-1}/\mathfrak {m}}$ . If $(\mathcal {L},f)$ is a framed outer brane, define

(23) $$ \begin{align} \begin{aligned} & F_{g,h}^{\mathcal{X},(\mathcal{L},f)}(\boldsymbol{\tau}_2, Q^b, X_1,\ldots,X_h)\\ &\quad = \sum_{\beta', n\geq 0}\sum_{(\mu_j, \lambda_j)\in H_1(\mathcal{L}; \mathbb{Z} )} \frac{\langle (\iota_f^*\boldsymbol{\tau}_2) ^n\rangle_{g,\beta,(\mu_1,\lambda_1),\ldots, (\mu_h,\lambda_h)}^{\mathcal{X},(\mathcal{L},f)}}{n!} \prod_{j=1}^h (Q^bX_j)^{\mu_j} \cdot \xi_0^{\bar{\lambda}_1} \mathbf{1}_{\lambda_1^{-1}} \otimes \cdots \otimes \xi_0^{\bar{\lambda}_h} \mathbf{1} _{\lambda_h^{-1}}, \end{aligned} \end{align} $$

which is a function that takes values in $H^*_ {\mathrm {CR}} (\mathcal {B} G_\tau; \mathbb {C} )^{\otimes h}$ , where

$$ \begin{align*}H^*_ {\mathrm{CR}} (\mathcal{B} G_\tau; \mathbb{C} )=\bigoplus_{\lambda \in G_\tau} \mathbb{C} \mathbf{1}_\lambda. \end{align*} $$

When $\lambda =1$ is the identity element of $G_\tau $ , $\mathbf {1}_1=1$ is the unit of $H^*_ {\mathrm {CR}} (\mathcal {B} G_\tau; \mathbb {C} )$ .

If $(\mathcal {L},f)$ is a framed inner brane, define

(24) $$ \begin{align} \begin{aligned} & F_{g,h}^{\mathcal{X},(\mathcal{L},f)}(\boldsymbol{\tau}_2, Q^b, X_1,\ldots, X_h) = \sum_{\beta', n\geq 0}\sum_{(\mu_j,\lambda_j)\in H_1(\mathcal{L}; \mathbb{Z} )} \frac{\langle (\iota_f^*\boldsymbol{\tau}_2)^n\rangle_{g,\beta,(\mu_1,\lambda_1),\ldots, (\mu_h,\lambda_h)}^{\mathcal{X},(\mathcal{L},f)}}{n!} \\ & \quad \quad \cdot \prod_{\substack{ j\in \{1,\ldots, h\}\\ \mu_j>0} } (Q^b X_j)^{\mu_j} \prod_{\substack{ j\in \{1,\ldots, h\}\\ \mu_j<0} } (Q^{b-\alpha}X_j)^{\mu_j} \cdot \xi_0^{\bar{\lambda}_1} \mathbf{1}_{\lambda_1^{-1}} \otimes \cdots \otimes \xi_0^{\bar{\lambda}_h} \mathbf{1}_{\lambda_h^{-1}}, \end{aligned} \end{align} $$

which is a function that takes values in $H^*_ {\mathrm {CR}} (\mathcal {B} G_\tau; \mathbb {C} )^{\otimes h}$ .

3.14 The equivariant J-function and the disk potential

Let $\{u_i\}_{i=1}^N$ be a homogeneous basis of $H^*_{ \mathbb {T} , {\mathrm {CR}} }(\mathcal {X}; \mathbb {Q} )$ and $\{u^i\}_{i=1}^N$ be its dual basis. Define

$$ \begin{align*}\boldsymbol{\tau} =\sum_{i=1}^N \tau_i u_i =\boldsymbol{\tau}_0+ \boldsymbol{\tau}_2 + \boldsymbol{\tau}_{>2}, \end{align*} $$

where

$$ \begin{align*}\boldsymbol{\tau}_0\in H^0_{ \mathbb{T} , {\mathrm{CR}} }(\mathcal{X}; \mathbb{C} ),\quad \boldsymbol{\tau}_2\in H^2_{ \mathbb{T} , {\mathrm{CR}} }(\mathcal{X}; \mathbb{C} ),\quad \boldsymbol{\tau}_{>2} \in H^{>2}_{ \mathbb{T} , {\mathrm{CR}} }(\mathcal{X}; \mathbb{C} ). \end{align*} $$

The J-function [Reference Tseng71, Reference Coates and Givental26, Reference Givental40] is a $H^*_{ \mathbb {T} , {\mathrm {CR}} }(\mathcal {X})$ -valued function:

$$ \begin{align*}J(\boldsymbol{\tau},z):= 1+\sum_{\beta\ge 0, n\ge 0} \frac{1}{n!}\sum_{i=1}^N \left\langle 1,\boldsymbol{\tau}^n, \frac{u_i}{z-\bar \psi}\right\rangle^{\mathcal{X}, \mathbb{T} }_{0,\beta} u^i. \end{align*} $$

Then

$$ \begin{align*}\iota_ {\sigma} ^* J(\boldsymbol{\tau},z)\Big|_{\mathsf{u}_1=\mathsf{u},\, \mathsf{u}_2=f\mathsf{u},\, \mathsf{u}_3=0} =\sum_{k\in G_ {\sigma} } J^f_{ {\sigma} ,k}(\boldsymbol{\tau},z) \mathbf{1}_k, \end{align*} $$

where

$$ \begin{align*}J_{ {\sigma} ,k}^f(\boldsymbol{\tau},z)= \delta_{k,1} +\sum_{\beta\ge 0, n\ge 0} \frac{1}{n!} \sum_{i=1}^N \left\langle 1, (\iota_f^*\boldsymbol{\tau})^n, \frac{|G_ {\sigma} |\phi^f_{ {\sigma} ,k^{-1}}}{z-\bar \psi}\right\rangle^{\mathcal{X}, \mathbb{T} _f}_{0,\beta}. \end{align*} $$

As a special case of Proposition 3.3,

$$ \begin{align*} \langle \gamma_1,\ldots, \gamma_n\rangle_{0,\beta+d_0 b,(d_0 ,k)}^{\mathcal{X},(\mathcal{L},f)} &= D^{\prime}_{d_0 ,\lambda,f} \int_{[\overline{\mathcal{M}}_{0,n+1}(\mathcal{X},\beta)]^ {\mathrm{vir}} } \frac{ \big(\prod_{i=1}^n\mathrm{ev}_i^*\gamma_i\cup \mathrm{ev}_{n+1}^*\phi^f_{(h^+(d_0,\lambda))^{-1}}\big)}{\frac{\mathsf{u}}{d_0}(\frac{\mathsf{u}}{d_0} -\bar{\psi}_{n+1})}\\ &= D^{\prime}_{d_0 ,\lambda,f} \left\langle 1,\gamma_1,\dots,\gamma_n,\frac{\phi^f_{(h^+(d_0,\lambda))^{-1}}}{\frac{\mathsf{u}}{d_0} -\bar{\psi}} \right\rangle^{\mathcal{X}}_{0,\beta}; \end{align*} $$

$$ \begin{align*} F^{\mathcal{X},(\mathcal{L},f)}_{0,1}(\boldsymbol{\tau}_2,Q^b, X_1) &= \sum_{\beta,n\geq 0} \sum_{(d_0 ,\lambda)\in H_1(\mathcal{L}; \mathbb{Z} )} \frac{1}{n!}\langle (\iota_f^*\boldsymbol{\tau}_2)^n\rangle^{\mathcal{X},(\mathcal{L},f)}_{0,\beta+d_0 b,(d_0 ,\lambda)} (Q^bX)^{d_0} \xi_0^{\bar{\lambda}}\mathbf{1}_{\lambda^{-1}}\\ &= \frac{1}{|G_ {\sigma} |}\sum_{(d_0,\lambda)\in H_1(\mathcal{L}; \mathbb{Z} )} (Q^b X_1)^{d_0} D^{\prime}_{d_0 ,\lambda,f} J^f_{ {\sigma} ,h^+(d_0,\lambda)}\Big(\boldsymbol{\tau}_2 ,\frac{\mathsf{u}}{d_0}\Big)\xi_0^{\bar{\lambda}}\mathbf{1}_{\lambda^{-1}}. \end{align*} $$

Proposition 3.4. Let $X=Q^b X_1$ . If $(\mathcal {L},f)$ is a framed outer brane, then

$$ \begin{align*}F^{\mathcal{X},(\mathcal{L},f)}_{0,1}(\boldsymbol{\tau}_2,X) = \frac{1}{|G_ {\sigma} |} \sum_{(d_0 ,\lambda)\in H_1(\mathcal{L}; \mathbb{Z} )} X^{d_0} D^{\prime}_{d_0 ,\lambda,f}J^f_{ {\sigma} ,h^+(d_0,\lambda)}\Big(\boldsymbol{\tau}_2,\frac{\mathsf{u}}{d_0} \Big) \xi_0^{\bar{\lambda}}\mathbf{1}_{\lambda^{-1}}. \end{align*} $$

If $(\mathcal {L},f)$ is a framed inner brane, then

$$ \begin{align*} F^{\mathcal{X},(\mathcal{L},f)}_{0,1}(\boldsymbol{\tau}_2, Q, X) &= \frac{1}{|G_ {\sigma} |}\sum_{(d_0,\lambda)\in H_1(\mathcal{L}; \mathbb{Z} ), d_0>0} X^{d_0} D^{\prime}_{d_0 ,\lambda,f} J^f_{ {\sigma} _+,h^+(d_0,\lambda)}\Big(\boldsymbol{\tau}_2,\frac{\mathsf{u}}{d_0}\Big) \xi_0^{\bar{\lambda}}\mathbf{1}_{\lambda^{-1}} \\ &\quad + \frac{1}{|G_ {\sigma} |} \sum_{(d_0 ,\lambda)\in H_1(\mathcal{L}; \mathbb{Z} ), d_0 <0} X^{d_0} Q^{-d_0 \alpha} D^{\prime}_{d_0,\lambda,f}J^f_{ {\sigma} _-,h^-(d_0,\lambda)}\Big(\boldsymbol{\tau}_2,\frac{\mathsf{u}}{d_0}\Big) \cdot \frac{\mathfrak{r}_+}{\mathfrak{r}_-} \xi_0^{\bar{\lambda}}\mathbf{1}_{\lambda^{-1}}. \end{align*} $$

4.1 The equivariant I-function and the equivariant mirror theorem

We choose $p_1,\ldots , p_k \in \mathbb {L}^\vee \cap \widetilde { {\mathrm {Nef}} }_{\mathcal {X}}$ such that

• ○ $\{p_1,\ldots ,p_k\}$ is a $ \mathbb {Q} $ -basis of $\mathbb {L}^\vee _{\mathbb {Q}} $ .

• ○ $\{\bar {p}_1,\ldots , \bar {p}_{k'}\}$ is a $ \mathbb {Q} $ -basis of $H^2(\mathcal {X}; \mathbb {Q} )$ .

• ○ $p_a=D_{3+a}$ for $a= k'+1,\dots ,k$ .

We define charges $m^{(a)}_i\in \mathbb {Q} $ by $D_i=\sum _{a=1}^k m^{(a)}_i p_a$ .

Let $q^{\prime }_0,q_1,\ldots , q_k$ be $k+1$ formal variables, and define $q^\beta = q_1^{\langle p_1,\beta \rangle } \cdots q_k^{\langle p_k,\beta \rangle }$ for $\beta \in \mathbb {K}$ . We take the equivariant lifting $\bar p^{\mathcal {T}}_a\in H^2_{\mathbb {T}} (\mathcal {X}; \mathbb {Q} )$ of $\bar p_a\in H^2(\mathcal {X}; \mathbb {Q} )$ . The equivariant I-function is an $H^*_{ {\mathrm {CR}} , \mathbb {T} } (\mathcal {X})$ -valued power series defined as follows [Reference Iritani47]:

$$ \begin{align*} I(q_0',q,z) &= e^{\frac{\log q^{\prime}_0+\sum_{a=1}^{k'} \bar p_a^{\mathcal{T}} \log q_a}{z}} \sum_{\beta\in \mathbb{K}_{\mathrm{eff}}} q^\beta \prod_{i=1}^{r'} \frac{\prod_{m=\lceil \langle D_i, \beta \rangle \rceil}^\infty (\bar D^{\mathcal{T}}_i +(\langle D_i,\beta\rangle -m)z)}{\prod_{m=0}^\infty(\bar D^{\mathcal{T}}_i+(\langle D_i,\beta\rangle -m)z)} \\ & \quad \cdot \prod_{i=r'+1}^r \frac{\prod_{m=\lceil \langle D_i, \beta \rangle \rceil}^\infty (\langle D_i,\beta\rangle -m)z} {\prod_{m=0}^\infty(\langle D_i,\beta\rangle -m)z} \mathbf{1}_{v(\beta)}, \end{align*} $$

where $q^\beta =\prod _{a=1}^k q_a^{\langle p_a,\beta \rangle }$ . Note that $\langle p_a,\beta \rangle \geq 0$ for $\beta \in \mathbb {K}_{\mathrm {eff}}$ . The equivariant I-function can be rewritten as

$$ \begin{align*} I(q_0',q,z) &= e^{\frac{\log q^{\prime}_0+\sum_{a=1}^{k'} \bar p_a^{\mathcal{T}} \log q_a}{z}} \sum_{\beta\in \mathbb{K}_{\mathrm{eff}}} \frac{q^\beta}{z^{\langle \hat \rho ,\beta\rangle + \mathrm{age}(v(\beta))} } \prod_{i=1}^{r'} \frac{\prod_{m=\lceil \langle D_i, \beta \rangle \rceil}^\infty ( \frac{\bar D^{\mathcal{T}}_i}{z} +\langle D_i,\beta\rangle -m)} {\prod_{m=0}^\infty(\frac{\bar D^{\mathcal{T}}_i}{z}+ \langle D_i,\beta\rangle -m)} \\ & \quad \cdot \prod_{i=r'+1}^r \frac{\prod_{m=\lceil \langle D_i, \beta \rangle \rceil}^\infty (\langle D_i,\beta\rangle -m)}{\prod_{m=0}^\infty(\langle D_i,\beta\rangle -m)} \mathbf{1}_{v(\beta)}, \end{align*} $$

where $\hat \rho = D_1+\cdots +D_r \in {\widetilde {C}} _{\mathcal {X}}$ .

Since $\mathcal {X}$ is a Calabi-Yau orbifold, $\mathrm {age}(v)$ is an integer for any $v\in \mathrm {Box}(\mathbf \Sigma )$ . Then

$$ \begin{align*}H^{\leq 2}_{ {\mathrm{CR}} , \mathbb{T} }(\mathcal{X}) = H^0_{ {\mathrm{CR}}. \mathbb{T} }(\mathcal{X}) \oplus H^2_{ {\mathrm{CR}} , \mathbb{T} }(\mathcal{X}). \end{align*} $$

Let $\mathcal {Q}= \mathbb {Q} (\mathsf {u}_1,\mathsf {u}_2,\mathsf {u}_3)$ be the fractional field of $H^*_{ \mathbb {T} }(\mathrm {point}; \mathbb {Q} )$ :

$$ \begin{align*} H^0_{ {\mathrm{CR}} , \mathbb{T} }(\mathcal{X};\mathcal{Q}) &= \mathcal{Q} \mathbf{1} , \\ H^2_{ {\mathrm{CR}} , \mathbb{T} }(\mathcal{X};\mathcal{Q}) &= \bigoplus_{a=1}^{k'}\mathcal{Q} {\bar{p}_a^{\mathcal{T}}}\oplus \bigoplus_{\substack{v\in {\mathrm{Box}(\mathbf \Sigma)}\\ \mathrm{age}(v)=1}} \mathcal{Q} \mathbf{1}_v. \end{align*} $$

Recall that the embedding of the stacky fixed point $\mathfrak {p}_\sigma $ is $\iota _\sigma :\mathfrak {p}_\sigma \to \mathcal {X}$ . We choose the lifting $\bar {p}^{\mathcal {T}}_a$ such that $\iota _\sigma ^*\bar {p}^{\mathcal {T}}_a=0$ .

For $i=1,\ldots , r$ , we will define $\Omega _i \subset \mathbb {K}_{\mathrm {eff}}-\{0\}$ and $A_i(q)$ supported on $\Omega _i$ . We observe that if $\beta \in \mathbb {K}_{\mathrm {eff}}$ and $v(\beta )=0$ , then $\langle D_i,\beta \rangle \in \mathbb {Z} $ for $i=1,\ldots , r$ .