2.1 Notations for toric geometry

In this paper, we use the following notations for an r-dimensional smooth toric variety Z defined by a fan $\Xi $ in $\mathbb {R}^r$ . The algebraic torus of Z is isomorphic to $( {\mathbb {C}} ^{*})^r$ .

• • For $d = 0, \dots , r$ , let $\Xi (d)$ denote the set of d-dimensional cones in $\Xi $ . For a cone $\sigma \in \Xi (d)$ , let $V(\sigma ) \subseteq Z$ denote the $( {\mathbb {C}} ^{*})^r$ -orbit closure corresponding to $\sigma $ , which is a codimension-d closed subvariety of Z.

• • For a maximal cone $\sigma \in \Xi (r)$ , let $p_\sigma := V(\sigma )$ denote the corresponding $( {\mathbb {C}} ^{*})^r$ -fixed point.

• • For a cone $\tau \in \Xi (r-1)$ , let $l_\tau := V(\tau )$ denote the corresponding $( {\mathbb {C}} ^{*})^r$ -invariant line, which is isomorphic to either $ {\mathbb {C}} $ or $\mathbb {P}^1$ . We set $\Xi (r-1)_c := \{\tau \in \Xi (r-1): l_\tau \cong \mathbb {P}^1\}$ .

• • Let $F(\Sigma ) := \{(\tau , \sigma ) \in \Xi (r-1) \times \Xi (r) : \tau \subset \sigma \}$ denote the set of flags in $\Xi $ .

2.2 Open geometry

Let $N \cong \mathbb {Z}^3$ be a lattice and $M := \mathrm {Hom}(N, \mathbb {Z})$ be the dual lattice. Let X be a smooth toric Calabi-Yau 3-fold specified by a finite fan $\Sigma $ in $N_{\mathbb {R}} := N \otimes \mathbb {R} \cong \mathbb {R}^3$ . We assume that $\Sigma (3)$ is non-empty and every cone in $\Sigma $ is a face of some 3-cone.

Let $R := |\Sigma (1)|$ . Let $\Sigma (1) = \{\rho _1, \dots , \rho _R\}$ be a listing of the rays in $\Sigma $ , and for each $i = 1, \dots , R$ let $b_i \in N$ be the primitive generator of $\rho _i$ . The Calabi-Yau condition on X is equivalent to the existence of $u_3 \in M$ such that $\langle u_3, b_i \rangle = 1$ for all i, where $\langle -,- \rangle $ is the natural pairing between M and N. Let $N' := \ker (u_3: N \to \mathbb {Z}) \cong \mathbb {Z}^2$ .

Let P be the cross-section of the support $|\Sigma |$ of $\Sigma $ in the hyperplane

(2) $$ \begin{align} \{v \in N_{\mathbb{R}} : \langle u_3, v \rangle = 1\} \cong N' \otimes \mathbb{R} \cong \mathbb{R}^2, \end{align} $$

which is a 2-dimensional lattice polytope with a triangulation induced by $\Sigma $ . We assume that P is simple. As in the setup of [Reference Liu and Yu33, Section 2.2], we do not assume that P is convex or equivalently X is semi-projective. There is a toric partial compactification $X \subseteq X'$ by a semi-projective smooth toric Calabi-Yau 3-fold $X'$ determined by a fan $\Sigma '$ which contains $\Sigma $ as a subfan and satisfies $\Sigma '(1) = \Sigma (1)$ . The cross-section of $\Sigma '$ with the hyperplane (2) is the convex hull $P'$ of P, and we have $P' \cap N = P \cap N = \{b_1, \dots , b_R\}$ .

Let $T := N \otimes {\mathbb {C}} ^{*} \cong ( {\mathbb {C}} ^{*})^3$ be the algebraic torus of X, whose character lattice is $\mathrm {Hom}(T, {\mathbb {C}} ^{*}) \cong M$ . We consider a 2-subtorus $T' := \ker (u_3: T \to {\mathbb {C}} ^{*}) = N' \otimes {\mathbb {C}} ^{*} \cong ( {\mathbb {C}} ^{*})^2$ . The fixed points and invariant lines of X under the $T'$ -action are the same as those under the T-action.

Let $L \subset X$ be an Aganagic-Vafa brane in X, which is a Lagrangian submanifold diffeomorphic to $S^1 \times {\mathbb {C}} $ . We refer to [Reference Fang and Liu13, Section 2.4], [Reference Liu and Yu33, Section 2.2] for detailed definitions. The brane L is invariant under the action of the maximal compact subtorus $T_{\mathbb {R}}' \cong U(1)^2$ of $T'$ . Moreover, it intersects a unique T-invariant line $l_{\tau _0}$ in X, where $\tau _0 \in \Sigma (2)$ . Given a semi-projective toric partial compactification $X'$ of X as above, L can be viewed as an Aganagic-Vafa brane in $X'$ , intersecting the T-invariant line in $X'$ corresponding to $\tau _0 \in \Sigma '(2)$ . As in [Reference Liu and Yu33, Assumption 2.3], we make the following assumption on L.

Note that this assumption does not depend on the choice of $X'$ . In particular, $\tau _0 \in \Sigma (2) \setminus \Sigma (2)_c$ and L is also an outer brane in X. Let $\sigma _0 \in \Sigma (3)$ be the unique 3-cone containing $\tau _0$ as a face.

For any cone $\sigma $ in $\Sigma $ , we set

$$ \begin{align*}I^{\prime}_\sigma := \{i \in \{1, \dots, R\} : \rho_i \subseteq \sigma\}, \qquad I_\sigma := \{1, \dots, R\} \setminus I^{\prime}_\sigma. \end{align*} $$

We assume without loss of generality that

$$ \begin{align*}I^{\prime}_{\tau_0} = \{2, 3\}, \qquad I^{\prime}_{\sigma_0} = \{1, 2, 3\} \end{align*} $$

with $b_1, b_2, b_3$ appearing in P in a counterclockwise order. Such labeling determines a unique way to complete $u_3$ into a $\mathbb {Z}$ -basis $\{u_1, u_2, u_3\}$ of M such that under the dual $\mathbb {Z}$ -basis $\{v_1, v_2, v_3\}$ of N, we have the coordinates

$$ \begin{align*}b_1 = (1, 0, 1), \qquad b_2 = (0, 1, 1), \qquad b_3 = (0, 0, 1). \end{align*} $$

For $i = 1, \dots , R$ , we write $(m_i, n_i, 1)$ for the coordinate of $b_i \in N$ under the basis $\{v_1, v_2, v_3\}$ . Assumption 2.1 implies that $m_i \ge 0$ for all i.

Finally, let $f \in \mathbb {Z}$ be a framing on the Aganagic-Vafa brane L. This determines a 1-subtorus $T_f := \ker (u_2-fu_1: T' \to {\mathbb {C}} ^{*}) \subset T' \subset T$ . We take the following notations for the equivariant parameters of the tori:

$$ \begin{align*} & R_T := H^{*}_T(\mathrm{pt}) = {\mathbb{C}} [\mathsf{u}_1, \mathsf{u}_2, \mathsf{u}_3], && S_T:= {\mathbb{C}} (\mathsf{u}_1, \mathsf{u}_2, \mathsf{u}_3),\\ & R_{T'} := H^{*}_{T'}(\mathrm{pt}) = {\mathbb{C}} [\mathsf{u}_1, \mathsf{u}_2], && S_{T'}:= {\mathbb{C}} (\mathsf{u}_1, \mathsf{u}_2),\\ & R_{T_f} := H^{*}_{T_f}(\mathrm{pt}) = {\mathbb{C}} [\mathsf{u}_1], && S_{T_f}:= {\mathbb{C}} (\mathsf{u}_1). \end{align*} $$

2.3 Closed geometry

Under the open/closed correspondence [Reference Mayr38, Reference Liu and Yu33, Reference Liu and Yu34], the closed geometry corresponding to the open geometry $(X,L,f)$ is a smooth toric Calabi-Yau 4-fold $\widetilde {X}$ that takes the form

$$ \begin{align*}\widetilde{X} = \mathrm{Tot}(\mathcal{O}_{X \sqcup D}(-D)), \end{align*} $$

where $X \sqcup D$ is a toric partial compactification of X given by adding an additional toric divisor D corresponding to the ray generated by $(-1, -f, 0) \in N$ .Footnote ³ In $X \sqcup D$ , the T-invariant line $l_{\tau _0} \cong {\mathbb {C}} $ that L intersects is compactified by an additional T-fixed point into a $\mathbb {P}^1$ whose normal bundle is isomorphic to $\mathcal {O}_{\mathbb {P}^1}(f) \oplus \mathcal {O}_{\mathbb {P}^1}(-f-1)$ . There is an inclusion

$$ \begin{align*}\iota: X \to X \sqcup D \to \widetilde{X}. \end{align*} $$

Let $\widetilde {N} := N \oplus \mathbb {Z} \cong \mathbb {Z}^4$ and $\widetilde {T} := \widetilde {N} \otimes {\mathbb {C}} ^{*} \cong ( {\mathbb {C}} ^{*})^4$ . We view N as a sublattice of $\widetilde {N}$ and let $v_4$ be a generator of the additional $\mathbb {Z}$ -component. The toric geometry of $\widetilde {X}$ can be described by a fan $\widetilde {\Sigma } \in \widetilde {N}_{\mathbb {R}} := \widetilde {N} \otimes \mathbb {R} \cong \mathbb {R}^4$ as follows. The rays of $\widetilde {\Sigma }$ are given by

$$ \begin{align*}\widetilde{\Sigma}(1) = \{\widetilde{\rho}_1, \dots, \widetilde{\rho}_R, \widetilde{\rho}_{R+1}, \widetilde{\rho}_{R+2} \}, \end{align*} $$

where under the basis $\{v_1, \dots , v_4\}$ of $\widetilde {N}$ , the primitive generators $\widetilde {b}_i \in \widetilde {N}$ of the rays $\widetilde {\rho }_i$ , $i =1, \dots , R+2$ , have the following coordinates:

$$ \begin{align*}\widetilde{b}_i = (b_i, 0) = (m_i, n_i, 1, 0) \qquad \text{for } i = 1, \dots, R, \end{align*} $$

$$ \begin{align*}\widetilde{b}_{R+1} = (-1, -f, 1, 1), \qquad \widetilde{b}_{R+2} = (0, 0, 1, 1). \end{align*} $$

In $\widetilde {X} = \mathrm {Tot}(\mathcal {O}_{X \sqcup D}(-D))$ , the toric divisor $V(\widetilde {\rho }_{R+1})$ is the restriction of the line bundle $\mathcal {O}_{X \sqcup D}(-D)$ to D and $V(\widetilde {\rho }_{R+2}) = X \sqcup D$ is the zero section.

We describe cones $\widetilde {\sigma }$ in $\widetilde {\Sigma }$ by the index sets

$$ \begin{align*}I^{\prime}_{\widetilde{\sigma}} := \{i \in \{1, \dots, R+2\} : \widetilde{\rho}_i \subseteq \widetilde{\sigma}\}, \qquad I_{\widetilde{\sigma}} := \{1, \dots, R+2\} \setminus I^{\prime}_{\widetilde{\sigma}}. \end{align*} $$

First, $\widetilde {\Sigma }$ contains $\Sigma $ as a subfan. Any cone $\sigma \in \Sigma (d)$ , $d = 0, \dots , 3$ , can be viewed as a cone in $\widetilde {\Sigma }(d)$ with $I^{\prime }_{\sigma }$ preserved, and there is a cone $\iota (\sigma ) \in \widetilde {\Sigma }(d+1)$ given by

$$ \begin{align*}I^{\prime}_{\iota(\sigma)} = I^{\prime}_{\sigma} \sqcup \{R+2\}. \end{align*} $$

This induces an injective map $\iota : \Sigma (d) \to \widetilde {\Sigma }(d+1)$ .Footnote ⁴ For maximal cones in $\widetilde {\Sigma }$ , we have

$$ \begin{align*}\widetilde{\Sigma}(4) = \iota(\Sigma(3)) \sqcup \{\widetilde{\sigma}_0\}, \end{align*} $$

where the additional cone $\widetilde {\sigma }_0$ is characterized by

$$ \begin{align*}I^{\prime}_{\widetilde{\sigma}_0} = \{2, 3, R+1, R+2\}. \end{align*} $$

Note that $\widetilde {\sigma }_0$ is the only $4$ -cone that contains the ray $\widetilde {\rho }_{R+1}$ . Moreover, the map $\iota : \Sigma (2) \to \widetilde {\Sigma }(3)$ restricts to an injective map $\iota : \Sigma (2)_c \to \widetilde {\Sigma }(3)_c$ , and we have

$$ \begin{align*}\widetilde{\Sigma}(3)_c = \iota(\Sigma(2)_c) \sqcup \{\iota(\tau_0)\}. \end{align*} $$

Indeed, the $\widetilde {T}$ -invariant line $l_{\iota (\tau _0)} \cong \mathbb {P}^1$ is the compactification of $l_{\tau _0} \cong {\mathbb {C}} \subset X$ described at the beginning of this subsection.

Let $\widetilde {M} := \mathrm {Hom}(\widetilde {N}, \mathbb {Z})$ , which is the character lattice of the 4-torus $\widetilde {T}$ , and $\{u_1, \dots , u_4\}$ be the basis of $\widetilde {M}$ dual to the basis $\{v_1, \dots , v_4\}$ of $\widetilde {N}$ . Here, we abuse notations since $u_1, u_2, u_3 \in \widetilde {M}$ are natural lifts of the corresponding elements of M defined before under the projection $\widetilde {M} \to M$ . We consider a 3-subtorus $\widetilde {T}':= \ker (u_3: \widetilde {T} \to {\mathbb {C}} ^{*}) \cong ( {\mathbb {C}} ^{*})^3$ of $\widetilde {T}$ , which contains $T'$ and $T_f$ as subtori. The fixed points and invariant lines of $\widetilde {X}$ under the $\widetilde {T}'$ -action are the same as those under the $\widetilde {T}$ -action. We introduce the following notations:

$$ \begin{align*} & R_{\widetilde{T}} := H^{*}_T(\mathrm{pt}) = {\mathbb{C}} [\mathsf{u}_1, \mathsf{u}_2, \mathsf{u}_3, \mathsf{u}_4], && S_{\widetilde{T}}:= {\mathbb{C}} (\mathsf{u}_1, \mathsf{u}_2, \mathsf{u}_3, \mathsf{u}_4),\\ & R_{\widetilde{T}'} := H^{*}_{T'}(\mathrm{pt}) = {\mathbb{C}} [\mathsf{u}_1, \mathsf{u}_2, \mathsf{u}_4], && S_{\widetilde{T}'}:= {\mathbb{C}} (\mathsf{u}_1, \mathsf{u}_2, \mathsf{u}_4). \end{align*} $$

2.4 Second homology and effective curve classes

The intersection of L with $l_{\tau _0} \cong {\mathbb {C}} $ in X is isomorphic to $S^1$ and bounds a holomorphic disk B in $l_{\tau _0}$ , oriented by the holomorphic structure of X. The disk B represents a class $[B]$ in $H_2(X, L; \mathbb {Z})$ , and its boundary $\partial B = L \cap l_{\tau _0}$ generates $H_1(L; \mathbb {Z}) \cong \mathbb {Z}[\partial B]$ . We have a splitting

$$ \begin{align*}H_2(X, L; \mathbb{Z}) \cong H_2(X; \mathbb{Z}) \oplus \mathbb{Z} [B]. \end{align*} $$

We introduce the following notations for the semigroups of effective classes:

(3) $$ \begin{align} \begin{aligned} & E(X) := {\mathrm{NE}}(X) \cap H_2(X; \mathbb{Z}),\\ & E(X, L) := E(X) \oplus \mathbb{Z}_{\ge 0} [B] \subset H_2(X, L; \mathbb{Z}),\\ & E(\widetilde{X}) := {\mathrm{NE}}(\widetilde{X}) \cap H_2(\widetilde{X}; \mathbb{Z}). \end{aligned} \end{align} $$

The inclusion $\iota : X \to \widetilde {X}$ induces an isomorphism

$$ \begin{align*}\iota_*: H_2(X, L; \mathbb{Z}) \to H_2(\widetilde{X}; \mathbb{Z}), \qquad \beta + d[B] \mapsto \iota_*(\beta) + d[l_{\iota(\tau_0)}] \end{align*} $$

which restricts to a semigroup isomorphism

$$ \begin{align*}\iota_*: E(X,L) \cong E(\widetilde{X}). \end{align*} $$

We will thus use the coordinates $(\beta , d) \in E(X) \oplus \mathbb {Z}_{\ge 0}$ for both semigroups above. The pairing between $\widetilde {\beta } = (\beta , d) \in E(\widetilde {X})$ and the divisor class $[V(\widetilde {\rho }_{R+1})]$ is

$$ \begin{align*}\widetilde{\beta} \cdot [V(\widetilde{\rho}_{R+1})] = d. \end{align*} $$

2.5 Flags and tangent weights at torus-fixed points

For a flag $(\tau , \sigma ) \in F(\Sigma )$ , let

$$ \begin{align*}\mathbf{w}(\tau, \sigma) := c_1^{T'}(T_{p_\sigma}l_{\tau}) \in H^2_{T'}(\mathrm{pt}; \mathbb{Z}) \end{align*} $$

be the weight of the $T'$ -action on tangent space $T_{p_\sigma }l_{\tau }$ of $l_{\tau }$ at $p_{\sigma }$ . Similarly, for a flag $(\widetilde {\tau }, \widetilde {\sigma }) \in F(\widetilde {\Sigma })$ , let

$$ \begin{align*}\widetilde{\mathbf{w}}(\widetilde{\tau}, \widetilde{\sigma}) := c_1^{\widetilde{T}'}(T_{p_{\widetilde{\sigma}}}l_{\widetilde{\tau}}) \in H^2_{\widetilde{T}'}(\mathrm{pt}; \mathbb{Z}). \end{align*} $$

The maps $\iota : \Sigma (d) \to \widetilde {\Sigma }(d+1)$ defined in Section 2.3 induce an injective map $\iota : F(\Sigma ) \to F(\widetilde {\Sigma })$ , $(\tau , \sigma ) \mapsto (\iota (\tau ), \iota (\sigma ))$ . We have

$$ \begin{align*}\widetilde{\mathbf{w}}(\iota(\tau), \iota(\sigma)) \big|_{\mathsf{u}_4 = 0} = \mathbf{w}(\tau, \sigma). \end{align*} $$

Each 4-cone $\iota (\sigma ) \in \widetilde {\Sigma }(4)$ with $\sigma \in \Sigma (3) \subset \widetilde {\Sigma }(3)$ belongs to an additional flag $(\sigma , \iota (\sigma )) \in F(\widetilde {\Sigma })$ . We have

$$ \begin{align*}\widetilde{\mathbf{w}}(\sigma, \iota(\sigma)) = \mathsf{u}_4. \end{align*} $$

The additional 4-cone $\widetilde {\sigma }_0 \in \widetilde {\Sigma }(4) \setminus \iota (\Sigma (3))$ belongs to the flags

$$ \begin{align*}(\iota(\tau_0), \widetilde{\sigma}_0), (\widetilde{\tau}_2, \widetilde{\sigma}_0), (\widetilde{\tau}_3, \widetilde{\sigma}_0), (\widetilde{\tau}_4, \widetilde{\sigma}_0) \in F(\widetilde{\Sigma}) \end{align*} $$

where the facets $\widetilde {\tau }_2, \widetilde {\tau }_3, \widetilde {\tau }_4$ of $\widetilde {\sigma }_0$ are given by

$$ \begin{align*}I^{\prime}_{\widetilde{\tau}_2} = \{3, R+1, R+2\}, \quad I^{\prime}_{\widetilde{\tau}_3} = \{2, R+1, R+2\}, \quad I^{\prime}_{\widetilde{\tau}_4} = \{2, 3, R+1\}. \end{align*} $$

The tangent weights are given by

$$ \begin{align*}\widetilde{\mathbf{w}}(\iota(\tau_0), \widetilde{\sigma}_0) = -\mathsf{u}_1, \quad \widetilde{\mathbf{w}}(\widetilde{\tau}_2, \widetilde{\sigma}_0) = -f\mathsf{u}_1 + \mathsf{u}_2, \quad \widetilde{\mathbf{w}}(\widetilde{\tau}_3, \widetilde{\sigma}_0) = f\mathsf{u}_1 - \mathsf{u}_2 - \mathsf{u}_4, \quad \widetilde{\mathbf{w}}(\widetilde{\tau}_4, \widetilde{\sigma}_0) = \mathsf{u}_1 + \mathsf{u}_4. \end{align*} $$

2.6 Equivariant cohomology and bases

We fix an ordering of the $T'$ -fixed points of X by

$$ \begin{align*}p_1, \dots, p_m \end{align*} $$

and denote the corresponding $\widetilde {T}'$ -fixed points of $\widetilde {X}$ by

$$ \begin{align*}\widetilde{p}_1, \dots, \widetilde{p}_m. \end{align*} $$

We denote the additional $\widetilde {T}'$ -fixed point $p_{\widetilde {\sigma }_0}$ of $\widetilde {X}$ by $\widetilde {p}_{m+1}$ .

We consider the basis $\{\phi _1,\dots ,\phi _m\}$ of $H^{*}_{T'}(X) \otimes _{R_{T'}} S_{T'}$ defined as

$$ \begin{align*}\phi_i := \frac{[p_i]}{e_{T'}(T_{p_i}X)}=\frac{[p_i]}{\Delta^{i,T'}},\qquad \Delta^{i,T'} := e_{T'}(T_{p_i}X). \end{align*} $$

Then for $i, j = 1, \dots , m$ , we have

$$ \begin{align*}\phi_i \cup \phi_j=\delta_{ij}\phi_i, \qquad (\phi_i,\phi_j)_{X,T'}=\frac{\delta_{ij}}{\Delta^{i,T'}}, \end{align*} $$

where $(-,-)_{X, T'}$ is the $T'$ -equivariant Poincaré pairing on X. It follows that $\{\phi _1,\dots ,\phi _m\}$ is a canonical basis of the semi-simple Frobenius algebra

$$ \begin{align*}(H^{*}_{T'}(X) \otimes_{R_{T'}} S_{T'},\cup,(-,-)_{X,T'}). \end{align*} $$

Similarly, we define the basis $\{\widetilde {\phi }_1,\dots ,\widetilde {\phi }_m,\widetilde {\phi }_{m+1} \}$ of $H^{*}_{\widetilde {T}'}(\widetilde {X}) \otimes _{R_{\widetilde {T}'}} S_{\widetilde {T}'}$ as

$$ \begin{align*}\widetilde{\phi}_i := \frac{[\widetilde{p}_i]}{e_{\widetilde{T}'}(T_{\widetilde{p}_i}\widetilde{X})} = \frac{[\widetilde{p}_i]}{\Delta^{i,\widetilde{T}'}},\qquad \Delta^{i,\widetilde{T}'} := e_{\widetilde{T}'}(T_{\widetilde{p}_i}\widetilde{X}). \end{align*} $$

Note that for any $i, j = 1, \dots , m$ , we have

(4) $$ \begin{align} \widetilde{\phi}_i \big|_{\widetilde{p}_j} = \phi_i \big|_{p_j} = \delta_{ij}, \qquad \widetilde{\phi}_i \big|_{\widetilde{p}_{m+1}} = 0, \end{align} $$

and

$$ \begin{align*}\mathsf{u}_4^{-1} \Delta^{i,\widetilde{T}'} \big|_{\mathsf{u}_4 = 0} = \Delta^{i,T'}. \end{align*} $$

For $i, j = 1, \dots , m+1$ , we have

$$ \begin{align*}\widetilde{\phi}_i \cup \widetilde{\phi}_j = \delta_{ij}\widetilde{\phi}_i,\qquad (\widetilde{\phi}_i,\widetilde{\phi}_j)_{\widetilde{X},\widetilde{T}'} = \frac{\delta_{ij}}{\Delta^{i,\widetilde{T}'}}, \end{align*} $$

where $(-,-)_{\widetilde {X}, \widetilde {T}'}$ is the $\widetilde {T}'$ -equivariant Poincaré pairing on $\widetilde {X}$ . It follows that $\{\widetilde {\phi }_1,\dots ,\widetilde {\phi }_{m+1}\}$ is a canonical basis of the semi-simple Frobenius algebra

(5) $$ \begin{align} (H^{*}_{\widetilde{T}'}(\widetilde{X}) \otimes_{R_{\widetilde{T}'}} S_{\widetilde{T}'},\cup,(-,-)_{\widetilde{X},\widetilde{T}'}). \end{align} $$

Moreover, for $i = 1, \dots , R+2$ , let

$$ \begin{align*}\widetilde{D}^{\widetilde{T}'}_i := c_1^{\widetilde{T}'}(\mathcal{O}_{\widetilde{X}}(V(\widetilde{\rho}_{i}))) \in H^2_{\widetilde{T}'}(\widetilde{X}) \end{align*} $$

denote the $\widetilde {T}'$ -equivariant Poincaré dual of the divisor $V(\widetilde {\rho }_i)$ . Specifically, we denote

$$ \begin{align*}\widetilde{D} := \widetilde{D}^{\widetilde{T}'}_{R+1}. \end{align*} $$

Since the divisor $V(\widetilde {\rho }_{R+1})$ only contains the $\widetilde {T}'$ -fixed point $p_{\widetilde {\sigma }_0} = \widetilde {p}_{m+1}$ , we have that

(6) $$ \begin{align} \widetilde{D} = \widetilde{D}\big|_{\widetilde{p}_{m+1}} \widetilde{\phi}_{m+1} = -\mathsf{u}_1 \widetilde{\phi}_{m+1}. \end{align} $$

3.1 Closed Gromov-Witten invariants of X and $\widetilde {X}$

We refer to [Reference Liu32] for additional details on virtual localization [Reference Graber and Pandharipande22] in the Gromov-Witten theory of toric varieties.

For $n \in \mathbb {Z}_{\ge 0}$ and effective class $\beta \in E(X)$ (see (3)), let $\overline {\mathcal {M}}_{0,n}(X, \beta )$ be the moduli space of genus-zero, n-pointed, degree- $\beta $ stable maps to X. Given $T'$ -equivariant cohomology classes $\gamma _1, \dots , \gamma _n \in H^{*}_{T'}(X) \otimes _{R_{T'}} S_{T'}$ as insertions, we define the closed Gromov-Witten invariant

$$ \begin{align*}\langle \gamma_1, \dots, \gamma_n \rangle^{X, T'}_{0,n,\beta} := \int_{[\overline{\mathcal{M}}_{0,n}(X, \beta)^{T'}]^{ {\mathrm{vir}} }} \frac{\prod_{i=1}^n \mathrm{ev}_i^{*}(\gamma_i)}{e_{T'}(N^{ {\mathrm{vir}} })} \qquad \in S_{T'} \end{align*} $$

by localization with respect to the torus $T'$ , where for $i = 1, \dots , n$ , $\mathrm {ev}_i: \overline {\mathcal {M}}_{0,n}(X, \beta ) \to X$ is the evaluation map at the i-th marked point.

We now define a generating function of such invariants. The Novikov ring of X is the completion of the semigroup ring of $E(X)$ ,

$$ \begin{align*}\Lambda_X := \left\{\sum_{\beta \in E(X)} c_\beta Q^\beta : c_\beta \in {\mathbb{C}} \right\}, \end{align*} $$

in which we use $Q^\beta $ to denote the semigroup ring element corresponding to $\beta \in E(X)$ . We will also use the equivariant versions

$$ \begin{align*}\Lambda_X^{T'} := S_{T'} \otimes_{ {\mathbb{C}} } \Lambda_X, \qquad \Lambda_X^{T_f} := S_{T_f} \otimes_{ {\mathbb{C}} } \Lambda_X. \end{align*} $$

Consider the basis $\{\phi _1,\dots ,\phi _m\}$ of $H^{*}_{T'}(X) \otimes _{R_{T'}} S_{T'}$ defined in Section 2.6. Let

$$ \begin{align*}t := \sum_{i=1}^m t^i \phi_i, \end{align*} $$

where $t^1, \dots , t^m$ are formal variables viewed as coordinates. The genus-zero, $T'$ -equivariant Gromov-Witten potential of X is the following generating function of closed Gromov-Witten invariants:

Now we set up a parallel theory for $\widetilde {X}$ . For $n \in \mathbb {Z}_{\ge 0}$ and effective class $\widetilde {\beta } \in E(\widetilde {X})$ (see (3)), let $\overline {\mathcal {M}}_{0,n}(\widetilde {X}, \widetilde {\beta })$ be the moduli space of genus-zero, n-pointed, degree- $\widetilde {\beta }$ stable maps to $\widetilde {X}$ . Given $\widetilde {T}'$ -equivariant cohomology classes $\widetilde {\gamma }_1, \dots , \widetilde {\gamma }_n \in H^{*}_{\widetilde {T}'}(\widetilde {X}) \otimes _{R_{\widetilde {T}'}} S_{\widetilde {T}'}$ as insertions, we define the closed Gromov-Witten invariant

$$ \begin{align*}\langle \widetilde{\gamma}_1, \dots, \widetilde{\gamma}_n \rangle^{\widetilde{X}, \widetilde{T}'}_{0,n,\widetilde{\beta}} := \int_{[\overline{\mathcal{M}}_{0,n}(\widetilde{X}, \widetilde{\beta})^{\widetilde{T}'}]^{ {\mathrm{vir}} }} \frac{\prod_{i=1}^n \mathrm{ev}_i^{*}(\widetilde{\gamma}_i)}{e_{\widetilde{T}'}(N^{ {\mathrm{vir}} })} \qquad \in S_{\widetilde{T}'} \end{align*} $$

by localization with respect to the torus $\widetilde {T}'$ , where for $i = 1, \dots , n$ , $\mathrm {ev}_i: \overline {\mathcal {M}}_{0,n}(\widetilde {X}, \widetilde {\beta }) \to \widetilde {X}$ is the evaluation map at the i-th marked point.

The Novikov ring of $\widetilde {X}$ is the completion of the semigroup ring of $E(\widetilde {X})$ ,

$$ \begin{align*}\Lambda_{\widetilde{X}} := \left\{\sum_{\widetilde{\beta} \in E(\widetilde{X})} c_{\widetilde{\beta}} \widetilde{Q}^{\widetilde{\beta}} : c_{\widetilde{\beta}} \in {\mathbb{C}} \right\}, \end{align*} $$

in which we use $\widetilde {Q}^{\widetilde {\beta }}$ to denote the semigroup ring element corresponding to $\widetilde {\beta } \in E(\widetilde {X})$ . We will also use the equivariant version

$$ \begin{align*}\Lambda_{\widetilde{X}}^{\widetilde{T}'} := S_{\widetilde{T}'} \otimes_{ {\mathbb{C}} } \Lambda_{\widetilde{X}}. \end{align*} $$

Consider the basis $\{\widetilde {\phi }_1,\dots ,\widetilde {\phi }_m, \widetilde {\phi }_{m+1}\}$ of $H^{*}_{\widetilde {T}'}(\widetilde {X}) \otimes _{R_{\widetilde {T}'}} S_{\widetilde {T}'}$ defined in Section 2.6. Let

$$ \begin{align*}\tilde{t} := \sum_{i=1}^m t^i \widetilde{\phi}_i, \qquad \hat{t} := \tilde{t} + t^{m+1}\widetilde{\phi}_{m+1}, \end{align*} $$

where $t^1, \dots , t^m$ are formal variables as before and $t^{m+1}$ is an additional formal variable. The genus-zero, $\widetilde {T}'$ -equivariant Gromov-Witten potential of $\widetilde {X}$ is the following generating functions of closed Gromov-Witten invariants:

By (6), we have

$$ \begin{align*}\hat{t} = \tilde{t} - \frac{t^{m+1}}{\mathsf{u}_1}\widetilde{D}. \end{align*} $$

Recall from Section 2.4 that each $\widetilde {\beta } \in E(\widetilde {X})$ can be uniquely expressed as $\iota _*(\beta ) + d[l_{\iota (\tau _0)}]$ for some $\beta \in E(X)$ and $d \in \mathbb {Z}_{\ge 0}$ . The divisor equation then implies that

(7) $$ \begin{align} & F_0^{\widetilde{X},\widetilde{T}'}(t^1, \dots, t^m, t^{m+1}) \nonumber \\ &\quad = \frac{(t^{m+1})^3}{6\Delta^{m+1,\widetilde{T}'}} + \sum_{\widetilde{\beta} = (\beta, d) \in E(\widetilde{X})} \sum_{n \in \mathbb{Z}_{\ge 0}} \frac{\langle \tilde{t}, \dots, \tilde{t} \rangle^{\widetilde{X}, \widetilde{T}'}_{0,n,\widetilde{\beta}}}{n!}\widetilde{Q}^{\iota_*(\beta)}\left(e^{-\frac{t^{m+1}}{\mathsf{u}_1}} \widetilde{Q}^{[l_{\iota(\tau_0)}]} \right)^d. \end{align} $$

Here, the term $\frac {(t^{m+1})^3}{6\Delta ^{m+1,\widetilde {T}'}}$ captures the $t^{m+1}$ -dependence of the (3-pointed) degree- $0$ invariants in $F_0^{\widetilde {X},\widetilde {T}'}$ :

$$ \begin{align*}\frac{\langle t^{m+1}\widetilde{\phi}_{m+1}, t^{m+1}\widetilde{\phi}_{m+1}, t^{m+1}\widetilde{\phi}_{m+1} \rangle^{\widetilde{X}, \widetilde{T}'}_{0,3,0}}{3!} = \frac{(t^{m+1})^3}{6} (\widetilde{\phi}_{m+1} \cup \widetilde{\phi}_{m+1},\widetilde{\phi}_{m+1})_{\widetilde{X},\widetilde{T}'} = \frac{(t^{m+1})^3}{6\Delta^{m+1,\widetilde{T}'}}. \end{align*} $$

Note that $\widetilde {\phi }_i \cup \widetilde {\phi }_{m+1} = 0$ for any $i = 1, \dots , m$ .

3.2 Open Gromov-Witten invariants of $(X, L, f)$

Recall from Section 2.2 that the $T^{\prime }_{\mathbb {R}}$ -action on X preserves the Lagrangian L and may thus be used to define open Gromov-Witten invariants, specifically disk invariants which are virtual counts of open stable maps from genus-zero domains with one boundary component. We now recall the definitions and refer to [Reference Fang and Liu13, Reference Fang, Liu and Tseng14] for additional details.

For $n \in \mathbb {Z}_{\ge 0}$ and effective class $\beta ' = (\beta , d) \in E(X, L)$ (see (3)) with $d \in \mathbb {Z}_{>0}$ , let $\overline {\mathcal {M}}_{(0,1),n}(X,L \mid \beta ', d)$ be the moduli space of degree- $\beta '$ stable maps to $(X,L)$ from domains $(C, \partial C)$ with

• • topological type $(0,1)$ , i.e. C is a nodal Riemann surface of arithmetic genus zero with one open disk removed, and

• • n interior marked points disjoint from $\partial C$ .

Given $T'$ -equivariant (or equivalently $T^{\prime }_{\mathbb {R}}$ -equivariant) cohomology classes $\gamma _1, \dots , \gamma _n \in H^{*}_{T'}(X) \otimes _{R_{T'}} S_{T'}$ as insertions, we define the disk invariant

$$ \begin{align*}\langle \gamma_1, \dots, \gamma_n \rangle^{X, L}_{(0,1),n,\beta', d} := \int_{[\overline{\mathcal{M}}_{(0,1),n}(X,L \mid \beta', d)^{T^{\prime}_{\mathbb{R}}}]^{ {\mathrm{vir}} }} \frac{\prod_{i=1}^n \mathrm{ev}_i^{*}(\gamma_i)}{e_{T^{\prime}_{\mathbb{R}}}(N^{ {\mathrm{vir}} })} \qquad \in S_{T'} \end{align*} $$

by localization with respect to the compact torus $T^{\prime }_{\mathbb {R}}$ , where for $i = 1, \dots , n$ , $\mathrm {ev}_i: \overline {\mathcal {M}}_{(0,1),n}(X,L \mid \beta ', d) \to X$ is the evaluation map at the i-th marked point. Here, we identify the field of fractions of $H^{*}_{T^{\prime }_{\mathbb {R}}}(\mathrm {pt})$ with $S_{T'}$ . Furthermore, using the framing $f \in \mathbb {Z}$ , we take a weight restriction to define

$$ \begin{align*}\langle \gamma_1, \dots, \gamma_n \rangle^{X, (L,f)}_{(0,1),n,\beta', d} := \langle \gamma_1, \dots, \gamma_n \rangle^{X, L}_{(0,1),n,\beta', d} \big|_{\mathsf{u}_2 - f\mathsf{u}_1 = 0} \qquad \in S_{T_f}. \end{align*} $$

In this paper, we will only need to work with insertions for which the above weight restriction of the disk invariant is defined.

The completion of the semigroup ring of $E(X,L)$ is

in which we introduce the new formal variable $\mathsf {X}_0$ for the last component. Note that the isomorphism $\iota _*:E(X,L) \cong E(\widetilde {X})$ induces an isomorphism $\Lambda _{X,L} \cong \Lambda _{\widetilde {X}}$ under the change of variables $\widetilde {Q}^{\iota _*(\beta )} = Q^\beta $ , $\widetilde {Q}^{[l_{\iota (\tau _0)}]} = \mathsf {X}_0$ . We will also use the equivariant version

Consider the basis $\{\phi _1,\dots ,\phi _m\}$ of $H^{*}_{T'}(X) \otimes _{R_{T'}} S_{T'}$ and $t = \sum _{i=1}^m t^i \phi _i$ as in Section 3.1. Let $t^o$ be an additional formal variable for the open sector. The $T_f$ -equivariant disk potential of $(X,L,f)$ is the following generating functions of disk invariants:

Conceptually, we may view $t^o$ as parameterizing a divisor-like insertion arising from the open sector and $\mathsf {X} := e^{t^o}\mathsf {X}_0$ as parameterizing the winding numbers of disk invariants. Note that $F_{0,1}^{X,(L, f)}$ is supported on the ideal of $\Lambda _{X,L}$ generated by $\mathsf {X}_0$ . For later use, we introduce the following modified version:

where $\int $ is interpreted as taking the antiderivative with respect to $t^o$ . We note that the insertions $\phi _1, \dots , \phi _m$ are homogeneous of degree 0 and do not introduce additional poles along $\mathsf {u}_2-f\mathsf {u}_1$ . Thus, the weight restriction to $\mathsf {u}_2 - f\mathsf {u}_1 = 0$ in the definition of the disks invariants in $F_{0,1}^{X,(L, f)}$ is valid. Similarly, it is valid to apply this weight restriction to the closed invariants of X in $F_0^{X,T'}$ .

3.3 Open/closed correspondence

The open/closed correspondence [Reference Liu and Yu33, Reference Liu and Yu34] identifies the genus-zero open Gromov-Witten theory of $(X,L,f)$ and closed Gromov-Witten theory of $\widetilde {X}$ at the numerical level of invariants as well as the level of generating functions. In this paper, we use the following statement of the correspondence. We introduce the notation

(8) $$ \begin{align} \mathsf{v} := \begin{cases} \widetilde{\mathbf{w}}(\widetilde{\tau}_2, \widetilde{\sigma}_0) = \mathsf{u}_2 - f\mathsf{u}_1 & \text{if } f \ge 0,\\ -\widetilde{\mathbf{w}}(\widetilde{\tau}_3, \widetilde{\sigma}_0) = \mathsf{u}_2 - f\mathsf{u}_1 + \mathsf{u}_4 & \text{if } f < 0. \end{cases} \end{align} $$

Theorem 3.1 [Reference Liu and Yu34].

The Gromov-Witten potential $F_0^{\widetilde {X},\widetilde {T}'}$ of $\widetilde {X}$ can be expanded as

(9) $$ \begin{align} \begin{aligned} F_0^{\widetilde{X},\widetilde{T}'}(t^1, \dots, t^m, t^{m+1}) = & \frac{(t^{m+1})^3}{6\Delta^{m+1,\widetilde{T}'}} + \mathsf{u}_4^{-1}\widetilde{A}(t^1, \dots, t^m) + \mathsf{v}^{-1}\widetilde{B}(t^1, \dots, t^m, t^{m+1}) \\ & + \mathsf{u}_4\mathsf{v}^{-1}\widetilde{C}_1 (t^1, \dots, t^m, t^{m+1}) + \widetilde{C}_2(t^1, \dots, t^m, t^{m+1}), \end{aligned} \end{align} $$

where

1. (a) Each of $\widetilde {A}, \widetilde {B}, \widetilde {C}_1, \widetilde {C}_2$ has a well-defined weight restriction to $\mathsf {u}_4 = 0, \mathsf {u}_2 - f\mathsf {u}_1 = 0$ .

2. (b) $\widetilde {A}$ is supported on the Novikov variables $\{\widetilde {Q}^{\iota _*(\beta )}: \beta \in E(X)\}$ and

   $$ \begin{align*}\widetilde{A}(t^1, \dots, t^m) \big|_{\mathsf{u}_4 = 0} = F_0^{X,T'}(t^1, \dots, t^m) \end{align*} $$
   after the change of variables $\widetilde {Q}^{\iota _*(\beta )} = Q^\beta $ .

3. (c) We have

   $$ \begin{align*} \widetilde{B}(t^1, \dots, t^m, t^{m+1}) \big|_{\mathsf{u}_4 = 0, \mathsf{u}_2 - f\mathsf{u}_1 = 0} = \int F_{0,1}^{X,(L, f)}(t^1, \dots, t^m, t^o) \end{align*} $$
   after the change of variables $\widetilde {Q}^{\iota _*(\beta )} = Q^\beta $ , $\widetilde {Q}^{[l_{\iota (\tau _0)}]} = \mathsf {X}_0$ , and $t^{m+1} = -\mathsf {u}_1 t^o$ .

The statement of Theorem 3.1 differs from the results in [Reference Liu and Yu34], particularly Theorems 4.1 and 5.4 there, in that it uses the classes $\phi _1,\dots ,\phi _m$ and their counterparts $\widetilde {\phi }_1,\dots ,\widetilde {\phi }_m$ to parameterize insertions, and that it also involves closed Gromov-Witten invariants of X. Nevertheless, it directly follows from the localization analysis and vanishing arguments in [Reference Liu and Yu34, Section 4]. We defer the derivation details to Appendix A.1.

4.1 Formal Frobenius and F-manifolds

We first recall the definition of formal Frobenius manifolds over a general base ring R which is a commutative algebra over $ {\mathbb {C}} $ , extending Definition 1.2. We refer to [Reference Lee and Pandharipande30, Chapter 2] for additional details.

A formal Frobenius manifold $\hat {M}$ over R may alternatively be viewed as a relative formal complex Frobenius manifold over the affine base $\mathrm {Spec} (R)$ . Elements in R pull back to constants in the structure sheaf $\mathcal {O}_{\hat {M}}$ .

Given a formal Frobenius manifold as above, the origin is the only point in $\hat {M}$ and $\mathcal {T}_{\hat {M}} \cong K \otimes _R \mathcal {O}_{\hat {M}}$ . The product $\star $ defined by the associativity condition specializes to an R-algebra $(K, \star )$ at the origin.

Similarly, we may define flat formal F-manifolds over the general base ring R, extending Definition 1.4.

4.2 Gromov-Witten case

Let $\mathcal {X}$ be a smooth projective variety. Let $\{T_i\}_{i=1}^m$ be a basis of $H^{*}(\mathcal {X})$ and $t^1,\dots ,t^m$ be the corresponding coordinates. Consider the genus-zero Gromov-Witten potential $F_0^{\mathcal {X}}$ of $\mathcal {X}$ . Let

$$ \begin{align*} g_{ij}=( T_i,T_j )_{\mathcal{X}}=\int_{\mathcal{X}}T_i\cup T_j \end{align*} $$

and $(g^{ij})=(g_{ij})^{-1}$ .

Let $\partial _i := \frac {\partial }{\partial t^i}$ . As stated in Theorem 1.1, for any $i,j,k,l\in \{1,\dots ,m\}$ , the following WDVV equation holds:

$$ \begin{align*}\partial_i\partial_j\partial_\nu F_0^{\mathcal{X}}\cdot g^{\nu\mu}\cdot \partial_\mu\partial_k\partial_l F_0^{\mathcal{X}} =\partial_j\partial_k\partial_\nu F_0^{\mathcal{X}}\cdot g^{\nu\mu}\cdot \partial_\mu\partial_i\partial_l F_0^{\mathcal{X}}. \end{align*} $$

For any $i,j\in \{1,\dots ,m\}$ , we define the quantum product $T_i\star _t T_j$ as

$$ \begin{align*}(T_i\star_t T_j,T_k)_{\mathcal{X}}=\frac{\partial^3 F_0^{\mathcal{X}}}{\partial t^i\partial t^j\partial t^k}. \end{align*} $$

The WDVV equation implies that the quantum product $\star _t$ is associative.

Moreover, we can define a formal Frobenius manifold as follows. Let

$$ \begin{align*}H := \mathrm{Spec}(\Lambda_{\mathcal{X}} [ t^1,\dots,t^m]), \end{align*} $$

where $\Lambda _{\mathcal {X}}$ is the Novikov ring of $\mathcal {X}$ . Let $\hat {H}$ be the formal completion of H along the origin:

Let $\mathcal {O}_{\hat {H}}$ be the structure sheaf on $\hat {H}$ and $\mathcal {T}_{\hat {H}}$ be the tangent sheaf on $\hat {H}$ . Then $\mathcal {T}_{\hat {H}}$ is a sheaf of free $\mathcal {O}_{\hat {H}}$ -modules of rank N. Given an open set U in $\hat {H}$ , we have

$$ \begin{align*}\mathcal{T}_{\hat{H}}(U) \cong \bigoplus_{i=1}^m \mathcal{O}_{\hat{H}}(U) \frac{\partial}{\partial t^{i}}. \end{align*} $$

The quantum product and the Poincaré pairing define the structure of a formal Frobenius manifold on $\hat {H}$ over $\Lambda _{\mathcal {X}}$ :

$$ \begin{align*}\left(\frac{\partial}{\partial t^i}\star_t \frac{\partial}{\partial t^j},\frac{\partial}{\partial t^k}\right)_{\mathcal{X}}=\frac{\partial^3 F_0^{\mathcal{X}}}{\partial t^i\partial t^j\partial t^k}, \qquad \left(\frac{\partial}{\partial t^i}, \frac{\partial}{\partial t^j}\right)_{\mathcal{X}}=g_{ij}. \end{align*} $$

The generalization of the WDVV equation to the equivariant setting is straightforward. Suppose $\mathcal {X}$ admits an action of a torus $\mathbb {T}$ and let $\{T_i\}_{i=1}^m$ be a basis of $H^{*}_{\mathbb {T}}(\mathcal {X})$ . One only needs to replace $F_0^{\mathcal {X}}$ by the genus-zero $\mathbb {T}$ -equivariant Gromov-Witten potential $F_0^{\mathcal {X},\mathbb {T}}$ and replace $( T_i,T_j)_{\mathcal {X}}$ by the $\mathbb {T}$ -equivariant Poincaré pairing $( T_i,T_j)_{\mathcal {X},\mathbb {T}}$ . Then the WDVV equation (Theorem 1.1) still holds. Moreover, in the equivariant setting, $\mathcal {X}$ can be allowed to be non-compact. We only need $\overline {\mathcal {M}}_{g,n}(\mathcal {X},\beta )^{\mathbb {T}}$ to be compact in order to define $\mathbb {T}$ -equivariant Gromov-Witten invariants of $\mathcal {X}$ .

In the equivariant setting, we can still define a formal Frobenius manifold as follows. Let

$$ \begin{align*}H := \mathrm{Spec}(\Lambda_{\mathcal{X}}^{\mathbb{T}} [ t^1,\dots,t^m]), \end{align*} $$

where $\Lambda _{\mathcal {X}}^{\mathbb {T}}$ is the base change of $\Lambda _{\mathcal {X}}$ to adjoin equivariant parameters of $\mathbb {T}$ . Let $\hat {H}$ be the formal completion of H along the origin:

Let $\mathcal {O}_{\hat {H}}$ be the structure sheaf on $\hat {H}$ and $\mathcal {T}_{\hat {H}}$ be the tangent sheaf on $\hat {H}$ . Then $\mathcal {T}_{\hat {H}}$ is a sheaf of free $\mathcal {O}_{\hat {H}}$ -modules of rank m. Given an open set U in $\hat {H}$ , we have

$$ \begin{align*}\mathcal{T}_{\hat{H}}(U) \cong \bigoplus_{i=1}^m \mathcal{O}_{\hat{H}}(U) \frac{\partial}{\partial t^{i}}. \end{align*} $$

The quantum product and the $\mathbb {T}$ -equivariant Poincaré pairing define the structure of a formal Frobenius manifold on $\hat {H}$ over $\Lambda _{\mathcal {X}}^{\mathbb {T}}$ :

$$ \begin{align*}\left(\frac{\partial}{\partial t^i}\star_t \frac{\partial}{\partial t^j},\frac{\partial}{\partial t^k}\right)_{\mathcal{X},\mathbb{T}}=\frac{\partial^3 F_0^{\mathcal{X},\mathbb{T}}}{\partial t^i\partial t^j\partial t^k}, \qquad \left(\frac{\partial}{\partial t^i}, \frac{\partial}{\partial t^j}\right)_{\mathcal{X},\mathbb{T}}=g_{ij}. \end{align*} $$

4.3 Specializing to X and $\widetilde {X}$

Now we specialize to the toric Calabi-Yau 3-fold X and the toric Calabi-Yau 4-fold $\widetilde {X}$ . Recall from Section 2.6 that we defined the bases $\{\phi _1, \dots , \phi _m\}$ , $\{\widetilde {\phi }_1, \dots , \widetilde {\phi }_{m+1}\}$ of $H^{*}_{T'}(X)$ , $H^{*}_{\widetilde {T}'}(\widetilde {X})$ , respectively. Let

be the induced equivariant formal Frobenius manifolds constructed as in Section 4.2. The quantum products are given by the closed Gromov-Witten potentials $F_0^{X,T'}, F_0^{\widetilde {X},\widetilde {T}'}$ , respectively. The equivariant Poincaré parings are diagonal:

$$ \begin{align*} &g_{ij} := (\phi_i,\phi_j)_{X,T'}=\frac{\delta_{ij}}{\Delta^{i,T'}}, && i,j \in \{1, \dots, m\};\\ &\widetilde{g}_{ij} := (\widetilde{\phi}_i,\widetilde{\phi}_j)_{\widetilde{X},\widetilde{T}'}=\frac{\delta_{ij}}{\Delta^{i,\widetilde{T}'}}, && i,j \in \{1, \dots, m+1\}. \end{align*} $$

Note that for $i = 1, \dots , m$ we have

$$ \begin{align*}\widetilde{g}_{ii} = \frac{1}{\mathsf{u}_4}g_{ii}. \end{align*} $$

Let $(g^{ij})=(g_{ij})^{-1}$ and $(\widetilde {g}^{ij})=(\widetilde {g}_{ij})^{-1}$ . For any $i,j,k,l \in \{1, \dots , m\}$ , the WDVV equation for X reads

(10) $$ \begin{align} \partial_i\partial_j\partial_\nu F_0^{X,T'}\cdot g^{\nu\nu}\cdot \partial_\nu\partial_k\partial_l F_0^{X,T'} =\partial_j\partial_k\partial_\nu F_0^{X,T'}\cdot g^{\nu\nu}\cdot \partial_\nu\partial_i\partial_l F_0^{X,T'} \end{align} $$

where the summation index $\nu $ runs through $1, \dots , m$ . For any $i,j,k,l \in \{1, \dots , m+1\}$ , the WDVV equation for $\widetilde {X}$ reads

(11) $$ \begin{align} \partial_i\partial_j\partial_\nu F_0^{\widetilde{X},\widetilde{T}'}\cdot \widetilde{g}^{\nu\nu}\cdot \partial_\nu\partial_k\partial_l F_0^{\widetilde{X},\widetilde{T}'} =\partial_j\partial_k\partial_\nu F_0^{\widetilde{X},\widetilde{T}'}\cdot \widetilde{g}^{\nu\nu}\cdot \partial_\nu\partial_i\partial_l F_0^{\widetilde{X},\widetilde{T}'}, \end{align} $$

where the summation index $\nu $ runs through $1, \dots , m+1$ .

4.4 Recursive relations for open and closed invariants

Now we combine the WDVV equation (11) for $\widetilde {X}$ and the open/closed correspondence (Theorem 3.1) to obtain the following non-linear partial differential equations for the closed Gromov-Witten potential $F_0^{X,T'}$ of X and the disk potential $F_{0,1}^{X,(L, f)}$ of $(X, L, f)$ . For $i,j \in \{1, \dots , m\}$ , we set

(12) $$ \begin{align} h_{ij} := g_{ij} \big|_{\mathsf{u}_2 - f\mathsf{u}_1 = 0} \end{align} $$

which is well-defined by Assumption 2.2. Let $(h^{ij})=(h_{ij})^{-1}$ .

In particular, (Ic) recovers the WDVV equation (10) for X.

Proof. The proposition directly follows from applying the expansion (9) in Theorem 3.1 to both sides of (11) and reading off appropriate coefficients, under the following rules:

• • For (I), apply with $i,j,k,l$ as they are.

• • For (II), apply with $i,j,k$ as they are and $l = m+1$ .

• • For (III), apply with $i,j$ as they are and $k = l = m+1$ .

• • For (Ia), (IIa), (IIIa), read off the coefficients of $\mathsf {v}^{-1}$ on both sides.

• • For (Ib), (IIb), (IIIb), read off the coefficients of $\mathsf {u}_4\mathsf {v}^{-2}$ on both sides.

• • For (Ic), read off the coefficients of $\mathsf {u}_4^{-1}$ on both sides.

Here, we use the following observation: We have

(13) $$ \begin{align} \widetilde{g}^{(m+1)(m+1)} = \Delta^{m+1,\widetilde{T}'} = (\mathsf{u}_2 - f\mathsf{u}_1)(\mathsf{u}_4 + \mathsf{u}_2 - f\mathsf{u}_1) \mathsf{u}_1(\mathsf{u}_1+\mathsf{u}_4) = (\mathsf{v}^2 \pm \mathsf{u}_4\mathsf{v}) \mathsf{u}_1(\mathsf{u}_1+\mathsf{u}_4), \end{align} $$

where the sign ‘ $\pm $ ’ is ‘ $+$ ’ when $f \ge 0$ and ‘ $-$ ’ when $f<0$ (see (8) for the notation $\mathsf {v}$ ). It has second-order zeroes along $\mathsf {v}, \mathsf {u}_4$ , and thus, the $\nu = m+1$ terms in (11) do not contribute to the result except for case (IIIa), where the triple derivative $\partial _{m+1}\partial _{m+1}\partial _{m+1}\frac {(t^{m+1})^3}{6\Delta ^{m+1,\widetilde {T}'}}$ provides a cancelling factor $\frac {1}{\Delta ^{m+1,\widetilde {T}'}}$ . Moreover, we change from $\partial _{m+1}$ to $\partial _o$ using the relation $\partial _{m+1} = -\frac {\partial _o}{\mathsf {u}_1}$ .

5.1 A formal Frobenius structure

In this section, we construct a Frobenius structure on the formal scheme

over the base ring

$$ \begin{align*}\Lambda_{X,L}^{T_f}[\epsilon] := \Lambda_{X,L}^{T_f} \otimes {\mathbb{C}} [\epsilon]/\langle \epsilon^2 \rangle. \end{align*} $$

Let $\mathcal {O}_{\hat {H}_1}$ be the structure sheaf on $\hat {H}_1$ and $\mathcal {T}_{\hat {H}_1}$ be the tangent sheaf on $\hat {H}_1$ . Then $\mathcal {T}_{\hat {H}_1}$ is a sheaf of free $\mathcal {O}_{\hat {H}_1}$ -modules of rank $m+1$ . Given an open set U in $\hat {H}_1$ , we have

$$ \begin{align*}\mathcal{T}_{\hat{H}_1}(U) \cong \bigoplus_{i=1}^m\mathcal{O}_{\hat{H}_1}(U) \frac{\partial}{\partial t^{i}}\bigoplus\mathcal{O}_{\hat{H}_1}(U)\frac{\partial}{\partial t^o}. \end{align*} $$

We will construct a potential function F involving both the open and closed Gromov-Witten invariants of $(X, L)$ , as well as a pairing $(-,-)$ on $\mathcal {T}_{\hat {H}_1}$ . We prove the associativity of the induced product $\star _t$ on $\mathcal {T}_{\hat {H}_1}$ , which packages identities (Ia), (Ic), (IIa) and (IIIa) of Proposition 4.4. We show that the resulting tuple $(\hat {H}_1,\star _t,(-,-))$ is a semi-simple formal Frobenius manifold.

5.1.1 Potential

Introduce the variable $\epsilon $ with $\epsilon ^2=0$ .

Definition 5.1. We define the potential function F as

(15) $$ \begin{align} F(t^1, \dots, t^m, t^o):= -\frac{\mathsf{u}_1}{6}(t^o)^3 + F_0^{X,T'}(t^1, \dots, t^m)\big|_{\mathsf{u}_2-f\mathsf{u}_1=0}+\epsilon\int F_{0,1}^{X,(L,f)}(t^1, \dots, t^m,t^o). \end{align} $$

5.1.2 Pairing

In (12), we defined the restriction $(h_{ij})$ of the $T'$ -equivariant Poincaré pairing $(g_{ij})$ to $T_f$ . We now extend this pairing to the $t^o$ -direction. Recall that we have the change of variables $t^{m+1}=-\mathsf {u}_1 t^o$ from Theorem 3.1, which identifies $\frac {\partial }{\partial t^o}$ with $-\mathsf {u}_1\frac {\partial }{\partial t^{m+1}}$ . Moreover, we have

$$ \begin{align*}\left(\frac{\partial}{\partial t^{m+1}}, \frac{\partial}{\partial t^{m+1}}\right)_{\widetilde{X},\widetilde{T}'} = \widetilde{g}_{(m+1)(m+1)} =\frac{1}{(\mathsf{v}^2 \pm \mathsf{u}_4\mathsf{v}) \mathsf{u}_1(\mathsf{u}_1+\mathsf{u}_4)} \end{align*} $$

(see (13)). Clearing the second-order poles along $\mathsf {v}, \mathsf {u}_4$ , we set

$$ \begin{align*}h_{oo} := 1, \qquad \qquad h_{io} = h_{oi} := 0, \qquad i = 1, \dots, m. \end{align*} $$

Definition 5.2. We define the pairing $(-,-)$ on $\mathcal {T}_{\hat {H}_1}$ by the following: For any $i,j\in \{1,\dots ,m,o\}$ ,

$$ \begin{align*}\left(\frac{\partial}{\partial t^i},\frac{\partial}{\partial t^j}\right) := h_{ij}. \end{align*} $$

As before, let $(h^{ij})=(h_{ij})^{-1}$ .

5.1.3 WDVV equations

Proposition 5.3. For any $i,j,k,l\in \{1,\dots ,m,o\}$ , the following WDVV equation holds:

(16) $$ \begin{align} \partial_i\partial_j\partial_\nu F\cdot h^{\nu\mu}\cdot \partial_\mu\partial_k\partial_l F =\partial_j\partial_k\partial_\nu F\cdot h^{\nu\mu}\cdot \partial_\mu\partial_i\partial_l F, \end{align} $$

where the summation indices $\nu , \mu $ run through $1, \dots , m, o$ .

Proof. Note that $(h^{ij})$ is diagonal and the summation is over $\nu = \mu $ . The proposition directly follows from identities (Ia), (Ic), (IIa) and (IIIa) of Proposition 4.4, under the following rules:

• • When $i,j,k,l \in \{1, \dots , m\}$ , the $\epsilon ^0$ -term of (16) follows from identity (Ic) and the $\epsilon ^1$ -term follows from (Ia).

• • When $i,j,k \in \{1, \dots , m\}$ , $l = o$ , there is no $\epsilon ^0$ -term in (16) and the $\epsilon ^1$ -term follows from (IIa).

• • When $i,j \in \{1, \dots , m\}$ , $k = l = o$ , again there is no $\epsilon ^0$ -term in (16) and the $\epsilon ^1$ -term follows from (IIIa).

Any other case is either trivial or symmetric to a case above. Here, we note that since $\epsilon ^2 = 0$ , the equation (16) does not contain terms involving a product of two copies of $F_{0,1}^{X,(L,f)}$ (or their antiderivatives).

5.1.4 The formal Frobenius manifold

Definition 5.4. For any $i,j\in \{1,\dots ,m, o\}$ , define the product $\frac {\partial }{\partial t^i}\star _t\frac {\partial }{\partial t^j}$ on $\mathcal {T}_{\hat {H}_1}$ by

$$ \begin{align*}\left(\frac{\partial}{\partial t^i}\star_t\frac{\partial}{\partial t^j},\frac{\partial}{\partial t^k}\right)=\frac{\partial^3F}{\partial t^i\partial t^j\partial t^k}, \end{align*} $$

where k ranges through $1,\dots ,m, o$ .

By Proposition 5.3, the product $\star _t$ is indeed associative. Moreover, it is clear by definition that we have the compatibility condition

$$ \begin{align*}\left(\frac{\partial}{\partial t^i}\star_t\frac{\partial}{\partial t^j},\frac{\partial}{\partial t^k}\right)=\left(\frac{\partial}{\partial t^i},\frac{\partial}{\partial t^j}\star_t\frac{\partial}{\partial t^k}\right). \end{align*} $$

In other words, we have the following result.

Theorem 5.5. The tuple $(\hat {H}_1,\star _t,(-,-))$ is a formal Frobenius manifold over $\Lambda _{X,L}^{T_f}[\epsilon ]$ .

5.1.5 Semi-simplicity of $\hat {H}_1$

Let $S=\mathcal {O}_{\hat {H}_1}(\hat {H}_1)$ . Consider the global Frobenius algebra $A=(\mathcal {T}_{\hat {H}_1}(\hat {H}_1),\star _t,(,))$ and let $I\subset S$ be the ideal generated by Q and $\mathsf {X}_0$ . Then A is a free S-module of rank $m+1$ . Let

$$ \begin{align*}S_n:=S/I^n,\quad A_n:=A\otimes_{S}S_n. \end{align*} $$

Then $A_n$ is a free $S_n$ -module of rank $m+1$ , and the ring structure $\star _t$ on A induces a ring structure $*_{\underline {n}}$ on $A_n$ . Note that $A_1$ encodes the classical product. From the construction, the semi-simplicity of the (classical) Frobenius algebra (5) associated to $\widetilde {X}$ implies that $A_1$ is semi-simple and $\{\xi _1^{(1)}:=\frac {\partial }{\partial t^1},\dots ,\xi _{m}^{(1)}:=\frac {\partial }{\partial t^{m}},\xi _o^{(1)}:=\frac {\partial }{\partial t^o}\}$ is a system of idempotent basis of $A_1$ . For $n\geq 1$ , let $\{ \xi _{i}^{(n+1)}:i=1,\dots ,m, o\}$ be the unique idempotent basis of $(A_{n+1},\star _{\underline {n+1}})$ which is the lift of the idempotent basis $\{ \xi _{i}^{(n)}:i=1,\dots ,m, o\}$ of $(A_n,\star _{\underline {n}})$ [Reference Lee and Pandharipande30, Lemma 16]. Then

$$ \begin{align*}\{ \xi_{i}(t):=\lim \xi_{i}^{(n)}: i=1,\dots,m, o\} \end{align*} $$

is an idempotent basis of $(A,\star _t)$ . Therefore, we have the following result.

Theorem 5.6. The formal Frobenius manifold $(\hat {H}_1,\star _t,(-,-))$ is semi-simple.

Remark 5.7. As discussed in Remark 1.7, the structural morphism

$$ \begin{align*}\hat{H}_1 \to \mathrm{Spec}(\Lambda_{X,L}^{T_f}[\epsilon]) \end{align*} $$

may be viewed as a submersion of (formal) supermanifolds over $\Lambda _{X,L}^{T_f}$ with $\epsilon $ viewed as an odd variable. Taking $\epsilon = 0$ , we obtain a Frobenius structure on the underlying reduced formal manifold, which we denote by $\hat {H}_{1, \mathrm {red}}$ . The induced global Frobenius algebra of $\hat {H}_{1, \mathrm {red}}$ decomposes as the direct sum of the global Frobenius algebra of $\hat {H}_X^f$ (defined in (14)) and a 1-dimensional Frobenius algebra over $\Lambda _{X,L}^{T_f}$ generated by $\frac {\partial }{\partial t^o}$ , and the decomposition is consistent with the semi-simplicity description above. In particular, $\hat {H}_{1, \mathrm {red}}$ is semi-simple over $\Lambda _{X,L}^{T_f}$ , and $\hat {H}_1$ may be viewed as an infinitesimal deformation of $\hat {H}_{1, \mathrm {red}}$ .

5.2 A flat formal F-manifold structure

In this section, we construct a flat F-manifold structure on the formal scheme

over the base ring $\Lambda _{X,L}^{T_f}$ , where as compared to $\hat {H}_1$ introduced in Section 5.1, we drop the variable $\epsilon $ . Let $\mathcal {O}_{\hat {H}_2}$ be the structure sheaf on $\hat {H}_2$ and $\mathcal {T}_{\hat {H}_2}$ be the tangent sheaf on $\hat {H}_2$ . Then $\mathcal {T}_{\hat {H}_2}$ is a sheaf of free $\mathcal {O}_{\hat {H}_2}$ -modules of rank $m+1$ . Given an open set U in $\hat {H}_2$ , we have

$$ \begin{align*}\mathcal{T}_{\hat{H}_2}(U) \cong \bigoplus_{i=1}^m\mathcal{O}_{\hat{H}_2}(U) \frac{\partial}{\partial t^{i}}\bigoplus\mathcal{O}_{\hat{H}_2}(U)\frac{\partial}{\partial t^o}. \end{align*} $$

We will construct a vector potential $\overline {F} = (F^1, \dots , F^m, F^o)$ whose second derivatives give structural coefficients for a product $\star _t$ on $\mathcal {T}_{\hat {H}_1}$ . We prove the associativity of $\star _t$ , which packages identities (Ia), (Ib), (Ic), (IIa) and (IIb) of Proposition 4.4.

5.2.1 Vector potential

Let $(h^{ij})$ be as defined in (12).

Definition 5.8. We define the vector potential $\overline {F} = (F^1, \dots , F^m, F^o)$ by

$$ \begin{align*}F^i(t^1, \dots, t^m, t^o) := h^{ii} \partial_i \left(F_0^{X,T'}(t^1, \dots, t^m)\big|_{\mathsf{u}_2-f\mathsf{u}_1=0}+\int F_{0,1}^{X,(L,f)}(t^1, \dots, t^m,t^o=0)\right) \end{align*} $$

for $i = 1, \dots , m$ and

$$ \begin{align*}F^o(t^1, \dots, t^m, t^o) := F_{0,1}^{X,(L,f)}(t^1, \dots, t^m,t^o = 0). \end{align*} $$

All components of $\overline {F}$ are functions that are independent of the variable $t^o$ . As discussed in Remark 1.10, as we set $t^o = 0$ in the definitions, conceptually $\overline {F}$ has no insertions from the open sector. The $t^o$ -direction may also be viewed as an auxiliary direction in addition to the original m directions; see Remark 5.12.

5.2.2 Open WDVV equations

Proposition 5.9. For any $i,j,k,l\in \{1,\dots ,m, o\}$ , the following open WDVV equation holds:

(17) $$ \begin{align} \partial_i\partial_\mu F^j \cdot \partial_k\partial_l F^\mu =\partial_k\partial_\mu F^j \cdot \partial_i\partial_l F^\mu, \end{align} $$

where the summation index $\mu $ runs through $1, \dots , m, o$ .

Proof. Recall that the vector potential $\overline {F}$ consists of functions that are independent of $t^o$ . Thus, the two sides of (17) are zero if at least one of $i, k, l$ is o. For the remaining case $i, k, l \in \{1, \dots , m\}$ , first note that the term in (17) corresponding to $\mu = o$ is again zero. Then the case $j \in \{1, \dots , m\}$ follows from identities (Ia), (Ib) and (Ic) of Proposition 4.4, and the case $j = o$ follows from identities (IIa) and (IIb).

5.2.3 The flat formal F-manifold

Let $\nabla $ be the flat connection on $\mathcal {T}_{\hat {H}_2}$ under which $\frac {\partial }{\partial t^1},\dots ,\frac {\partial }{\partial t^m}$ , $\frac {\partial }{\partial t^o}$ are flat. Moreover, we define the following product.

Definition 5.10. For any $i,j\in \{1,\dots ,m, o\}$ , define the product $\frac {\partial }{\partial t^i}\star _t\frac {\partial }{\partial t^j}$ on $\mathcal {T}_{\hat {H}_2}$ by

$$ \begin{align*}\frac{\partial}{\partial t^i}\star_t\frac{\partial}{\partial t^j} = \frac{\partial^2F^k}{\partial t^i\partial t^j} \frac{\partial}{\partial t^k}, \end{align*} $$

where the summation index k runs through $1, \dots , m, o$ .

Since the components of the vector potential $\overline {F}$ are independent of $t^o$ , the above definition implies that

$$ \begin{align*}\frac{\partial}{\partial t^i} \star_t \frac{\partial}{\partial t^o} = 0 \end{align*} $$

for any $i = 1, \dots , m, o$ . Thus, $\frac {\partial }{\partial t^o}$ is nilpotent. Moreover, the product $\star _t$ does not admit an identity field, which means that the induced structure on $\hat {H}_2$ will be an formal F-manifold without unit. This is different from the case studied by [Reference Horev and Solomon26, Reference Solomon and Tukachinsky43] (see Theorem 1.3), and the difference is reflected by that our $F^o = F_{0,1}^{X,(L, f)}$ is supported on the ideal of $\Lambda _{X,L}$ generated by $\mathsf {X}_0$ while the disk potential of [Reference Horev and Solomon26, Reference Solomon and Tukachinsky43] has a constant term. The difference is discussed from the perspective of open WDVV equations in Remark 4.5.

Summarizing the above, we arrive at the following result.

Theorem 5.11. The tuple $(\hat {H}_2, \nabla , \star _t)$ is a flat formal F-manifold without the unit over $\Lambda _{X,L}^{T_f}$ in which the $t^o$ -direction is nilpotent.

Remark 5.12. The flat formal F-manifold $\hat {H}_2$ is a rank-1 extension of the formal Frobenius manifold $\hat {H}_X^f$ (defined in (14)) in the sense of, for example, [Reference Alcolado1, Chapter 3], [Reference Basalaev and Buryak4, Section 4]. In other words, there is a surjective homomorphism from the global algebra of $\hat {H}_2$ to that of $\hat {H}_X^f$ whose kernel is the rank-1 algebra over $\Lambda _{X,L}^{T_f}$ generated by the nilpotent element $\frac {\partial }{\partial t^o}$ .