2.1 Invariants

Let $(X,\omega )$ be a closed symplectic manifold. For $\beta \in H_2(X;\mathbb {Z})$ and $l \geq 0,$ let

$$\begin{align*}GW_\beta : H^{*}(X;\mathbb{Q})^{\otimes l} \rightarrow \mathbb{Q} \end{align*}$$

denote the standard closed genus zero Gromov-Witten invariant. This invariant counts J-holomorphic maps $S^2 \to X$ representing the class $\beta $ passing through cycles Poincaré dual to l cohomology classes on $X.$

Let $L \subset X$ be a closed Lagrangian submanifold. Genus zero open Gromov-Witten invariants are analogous invariants that count J-holomorphic maps $(D,\partial D) \to (X,L)$ along with correction terms arising from bounding cochains. To simplify the exposition, we focus on the case that L is connected and spin, with $H^{*}(L;\mathbb {R})\simeq H^{*}(S^n; \mathbb {R})$ and $[L] = 0 \in H_n(X;\mathbb {R})$ . These conditions hold for the Chiang Lagrangian. Similar results are known [Reference Solomon and Tukachinsky46] in the case that L is only relatively spin or $[L] \neq 0 \in H_n(X;\mathbb {R}).$

Consider the subcomplex of differential forms on X consisting of those with trivial integral on $L,$

$$\begin{align*}\widehat{A}^{*}(X,L;\mathbb{R}):=\left\{\eta\in A^{*}(X;\mathbb{R})\Big| \int_L \eta|_L=0\right\}.\end{align*}$$

For an $\mathbb {R}$ -algebra $\Upsilon $ , write

$$\begin{align*}\widehat{A}^{*}(X,L;\Upsilon):= \widehat{A}^{*}(X,L;\mathbb{R})\otimes \Upsilon,\quad\widehat{H}^{*}(X,L;\Upsilon):= H^{*}(\widehat{A}^{*}(X,L;\Upsilon), d).\end{align*}$$

Observe that

$$\begin{align*}\widehat{H}^{*}(X,L;\Upsilon) \simeq H^0(L;\Upsilon)\oplus H^{>0}(X,L;\Upsilon).\end{align*}$$

For $\beta \in H_2(X,L;\mathbb {Z})$ and $k,l \geq 0,$ let

$$\begin{align*}{OGW}_{\beta,k} : \widehat{H}^{*}(X,L;\mathbb{R})^{\otimes l}\rightarrow \mathbb{R} \end{align*}$$

denote the open Gromov-Witten invariants of [Reference Solomon and Tukachinsky46]. The definition is recalled in Section 4.2. For $\beta \in H_2(X,L;\mathbb {Z})$ and $k,l \geq 0,$ let

$$\begin{align*}\overline{OGW}\!_{\beta,k} : \widehat{H}^{*}(X,L;\mathbb{R})^{\otimes l}\rightarrow \mathbb{R} \end{align*}$$

denote the enhanced invariants defined in [Reference Solomon and Tukachinsky48]. The definition is recalled in 8. These invariants combine counts of J-holomorphic disks and J-holomorphic spheres. With the exception of the case that $\beta $ belongs to the image of the natural map

$$\begin{align*}\varpi : H_2(X;\mathbb{Z}) \rightarrow H_2(X,L;\mathbb{Z})\end{align*}$$

and $k = 0,$ we have $\overline {OGW}\! _{\beta ,k} = {OGW}_{\beta ,k}.$ In the exceptional case, ${OGW}_{\beta ,k} = 0,$ but $\overline {OGW}\!_{\beta ,k}$ need not vanish. By the assumption $[L] = 0 \in H_n(X;\mathbb {R}),$ there is a short exact sequence

(2.1) $$ \begin{align} 0 \rightarrow H^n(L;\mathbb{R}) \overset{y}{\longrightarrow} {H}^{n+1}(X,L;\mathbb{R}) \overset{\rho}{\longrightarrow} H^{n+1}(X;\mathbb{R}) \rightarrow 0. \end{align} $$

For $k=0$ and $\beta \in \mathrm {Im \varpi }$ , the definition of the enhanced invariants $\overline {OGW}\!_{\beta ,k}$ depends on the choice of a linear map

$$\begin{align*}P_{\mathbb{R}} :{H}^{n+1}(X,L;\mathbb{R}) \rightarrow H^{n}(L;\mathbb{R})\simeq \mathbb{R} \end{align*}$$

that is a left-inverse to the map y in the short exact sequence. A choice of $P_{\mathbb {R}}$ is equivalent to a splitting of (2.1). We extend $P_{\mathbb {R}}$ to a map $P_{\mathbb {R}}: \widehat {H}^{*}(X,L;\mathbb {R}) \to H^n(L;\mathbb {R})$ by setting it to zero outside $\widehat {H}^{n+1}(X,L;\mathbb {R})$ .

2.2 Axioms

The closed Gromov-Witten invariants satisfy the following properties [Reference Behrend4, Reference Fukaya and Ono22, Reference Kontsevich and Manin34, Reference Li and Tian36, Reference McDuff and Salamon38, Reference McDuff and Salamon37, Reference Ruan and Tian41, Reference Ruan and Tian42].

The enhanced open invariants $\overline {OGW}\!$ satisfy the following properties.

2.3 Bases

Let $\Delta _i \in H^{*}(X;\mathbb {R})$ for $i = 0,\ldots ,N,$ be a basis and let $\Gamma _i \in \ker (P_{\mathbb {R}}) \subset \widehat {H}^{*}(X,L;\mathbb {R})$ be the unique classes such that $\rho (\Gamma _i) = \Delta _i$ . Let $\Gamma _{N+1} = y(1).$ We also write $\Gamma _\diamond = y(1).$ For convenience, set $\Delta _{N+1} = 0.$

2.4 Novikov rings

Let $\mu : H_2(X,L;\mathbb {Z}) \to \mathbb {Z}$ denote the Maslov index. Define Novikov coefficients rings

$$ \begin{gather*} \Lambda:=\left\{\sum_{i=0}^\infty a_iT^{\beta_i}|a_i\in \mathbb{R}, \beta_i\in H_2(X,L;\mathbb{Z}), \omega(\beta_i)\ge0, \lim_{i\rightarrow \infty}{\omega(\beta_i)}=\infty \right\},\\ \Lambda_c:=\left\{\sum_{j=0}^\infty a_jT^{\varpi(\beta_j)}|a_j\in \mathbb{R}, \beta_j\in H_2(X;\mathbb{Z}), \omega(\beta_j)\ge0, \lim_{j\rightarrow \infty}{\omega(\beta_j)}=\infty \right\}\leqslant \Lambda. \end{gather*} $$

Gradings on $\Lambda , \Lambda _c$ are defined by declaring $|T^\beta | = \mu (\beta ).$ Let

$$ \begin{gather*} R_W = \Lambda[[t_0,\ldots,t_{N+1},s]], \\ Q_W = \Lambda_c[[t_0,\ldots,t_{N+1}]], \qquad Q_U = \Lambda_c[[t_0,\ldots,t_N]]. \end{gather*} $$

Gradings on $R_W,Q_W,Q_U,$ are defined by declaring $|t_i| = 2-|\Gamma _i|,\, |s| = 1-n.$

2.5 Potentials

The closed Gromov-Witten potential is given by

$$\begin{align*}\Phi(t_0,\ldots,t_N) = \sum_{\substack{\beta \in H_2(X;\mathbb{Z}) \\ r_i \geq 0}} \frac{T^{\varpi(\beta)} t_N^{r_N} \cdots t_0^{r_0}}{r_N! \cdots r_0!}GW_\beta(\Delta_0^{\otimes r_0} \otimes \cdots \otimes \Delta_N^{\otimes r_N}) \in Q_U.\end{align*}$$

The enhanced superpotential, which encodes the enhanced open Gromov-Witten invariants $\overline {OGW}\!_{\beta ,k},$ is given by

(2.2) $$ \begin{align} \overline{\Omega}(s,t_0,\ldots,t_{N+1}) = \sum_{\substack{\beta\in H_2(X,L;\mathbb{Z})\\k\ge 0\\r_i\ge 0}}\frac{T^\beta s^k t_{N+1}^{r_{N+1}} \cdots t_0^{r_0}}{k! r_{N+1}! \cdots r_0!} \overline{OGW}\!_{\beta,k}(\Gamma_0^{\otimes r_0}\otimes \cdots \otimes \Gamma_{N+1}^{\otimes r_{N+1}}) \in R_W. \end{align} $$

2.6 WDVV equations

Let

$$ \begin{align*}g_{ij} = \int_X \Delta_i \smile \Delta_j, \qquad i, j = 0,\ldots,N, \end{align*} $$

and let $g^{ij}$ denote the inverse matrix. The following WDVV equations hold for the closed Gromov-Witten potential [Reference Behrend4, Reference Fukaya and Ono22, Reference Kontsevich and Manin34, Reference Li and Tian36, Reference McDuff and Salamon38, Reference McDuff and Salamon37, Reference Ruan and Tian41, Reference Ruan and Tian42, Reference Witten53].

Theorem 2.1 (Closed WDVV equations).

For all quadruples of integers $i,j,k,l\in \{0,\ldots , n\}$ , the function $\Phi =\Phi ^{\mathbb {C}P^n}$ satisfies

$$\begin{align*}\sum_{\mu, \nu=0}^N \partial_{t_i} \partial_{t_j}\partial_{t_\nu}\Phi\cdot g^{\nu \mu}\cdot \partial_{t_\mu} \partial_{t_k} \partial_{t_l}\Phi= \sum_{\mu, \nu=0}^N \partial_{t_j} \partial_{t_k}\partial_{t_\nu}\Phi\cdot g^{\nu \mu}\cdot \partial_{t_\mu} \partial_{t_i} \partial_{t_l}\Phi. \end{align*}$$

The following is Corollary 1.6 from [Reference Solomon and Tukachinsky48].

Theorem 2.2 (Open WDVV equations).

For $u, v, w = 0,\ldots , N+1,$ we have the open WDVV equations

(2.3) $$ \begin{align} & \sum_{l,m = 0}^N \partial_{t_u} \partial_{t_l} \overline{\Omega} \cdot g^{lm} \cdot \partial_{t_m} \partial_{t_w} \partial_{t_v} \Phi - \partial_{t_u} \partial_s \overline{\Omega} \cdot \partial_{t_w} \partial_{t_v} \overline{\Omega} \notag\\ & \quad =\sum_{l,m = 0}^N \partial_{t_u} \partial_{t_w} \partial_{t_l} \Phi \cdot g^{lm} \cdot \partial_{t_m} \partial_{t_v} \overline{\Omega} - \partial_{t_u} \partial_{t_w} \overline{\Omega} \cdot \partial_{t_v} \partial_s \overline{\Omega}, \end{align} $$

(2.4) $$ \begin{align} \sum_{l,m = 0}^N \partial_s \partial_{t_l} \overline{\Omega} \cdot g^{lm} \cdot \partial_{t_m} \partial_{t_w} \partial_{t_v} \Phi - \partial_s^2 \overline{\Omega} \cdot \partial_{t_w} \partial_{t_v} \overline{\Omega} = -\partial_s \partial_{t_w} \overline{\Omega}\cdot \partial_{t_v} \partial_s \overline{\Omega}. \end{align} $$

2.7 Wall-crossing

Define

$$\begin{align*}\Gamma_{\diamond}=y(1) \in \widehat{H}^{*}(X,L;\mathbb{R}).\end{align*}$$

The following is Theorem 6 from [Reference Solomon and Tukachinsky48].

Theorem 2.3 (Wall crossing).

Suppose $[L]=0$ . Then the invariants $\overline {OGW}\!_{\beta ,k}$ satisfy

$$ \begin{align*}\overline{OGW}\!_{\beta, k+1}(\eta_1, \ldots, \eta_l)=-\overline{OGW}\!_{\beta,k}(\Gamma_\diamond, \eta_1, \ldots, \eta_l).\end{align*} $$

2.8 Relative quantum cohomology

Abbreviate

$$\begin{align*}\Delta= \sum_{i=0}^N t_i\Delta_i,\quad \Gamma= \sum_{i=0}^{N+1} t_i\Gamma_i.\end{align*}$$

The underlying module of the big relative quantum cohomology is

$$\begin{align*}QH^{*}_{\mathrm{big}}(X,L):= \widehat{H}^{*}(X,L;\Lambda[[t_0,\ldots,t_{N+1}]]). \end{align*}$$

The big relative quantum product

is given by

(2.5)

In Theorem 7 of [Reference Solomon and Tukachinsky48], it is shown that the product is associative and graded commutative. Though in [Reference Solomon and Tukachinsky48] the definition of is based on a chain-level construction, the above explicit formula follows from Lemma 5.11 in [Reference Solomon and Tukachinsky48].

The underlying module of the small relative quantum cohomology is

$$\begin{align*}QH^{*}(X,L) = \widehat{H}^{*}(X,L;\Lambda). \end{align*}$$

The small relative quantum product

is given by specializing the big relative quantum product to $t_0 = \cdots =t_{N+1} = 0.$ So, it too is associated and graded commutative. Explicitly, is given by

(2.6)

3.1 Geometric construction

The following is largely based on [Reference Smith43]. Consider the fundamental representation V of $\text {SL}(2, \mathbb {C}).$ Projectifying, we obtain an action of $\text {SL}(2, \mathbb {C})$ on $\mathbb {C}P^1 = \mathbb {P}(V).$ Taking the d-fold symmetric product, we obtain an action of $\text {SL}(2, \mathbb {C})$ on $\text {Sym}^d \mathbb {C}P^1 = \mathbb {C}P^d.$ Equivalently, we can consider the d-fold symmetric power $S^d V$ and projectify to obtain an action of $\text {SL}(2, \mathbb {C})$ on $\mathbb {P}(S^dV) = \mathbb {C}P^d.$

Fix $d\ge 3$ , and a configuration C of d distinct points in $\mathbb {C}P^1$ . Then, the $\text {SL}(2, \mathbb {C})$ -orbit of C in $\text {Sym}^d \mathbb {C}P^1 \cong \mathbb {P}S^dV$ is a three-dimensional complex submanifold, of which the $\text {SU}(2)$ -orbit is a three-dimensional totally real submanifold. The stabilizer of C in $\text {SL}(2, \mathbb {C})$ is a finite subgroup of $\text {SU}(2)$ which we denote by $\Gamma _C.$ In [Reference Aluffi and Faber1], Aluffi and Faber identify those configurations C for which the $\text {SL}(2, \mathbb {C})$ -orbit has smooth closure $X_C$ in $\mathbb {P}S^dV$ . There are four cases: the vertices of an equilateral triangle on a great circle in $\mathbb {C}P^1$ , which we denote by $\triangle $ , and the vertices of a regular tetrahedron, octahedron and icosahedron in $\mathbb {C}P^1$ , which we denote by $T, O$ and I, respectively.

In each case, the restriction of the $\text {SU}(2)$ -action to $X_C$ with the Fubini- Study Kähler form is Hamiltonian with the moment map $m: \mathbb {P}S^dV\rightarrow \mathfrak {su}(2)^{*}$ which is defined by

$$\begin{align*}\langle m([z]), \xi \rangle=\frac{i}{2}\frac{z^\dagger\varphi(\xi)z}{z^\dagger z}, \qquad \xi\in \mathfrak{su}(2), \end{align*}$$

where $\varphi :\mathfrak {su}(2)\rightarrow \mathrm {Mat}_{(d+1)\times (d+1)}(\mathbb {C})$ is given by the representation $S^dV.$ The $\text {SU}(2)$ -orbits of the configurations $\triangle , T, O$ and I all lie in the zero sets of the respective moment maps, so they are Lagrangian. We denote these orbits by $L_C$ .

The Chiang Lagrangian is $L_{\triangle } \subset X_{\triangle } = \mathbb {P}S^3V \cong \mathbb {C}P^3$ . It is the space of all the equilateral triangles on great circles in $\mathbb {C}P^1.$ Topologically, $L_{\triangle }$ is the quotient of $\text {SU}(2)$ by the binary dihedral subgroup $\Gamma _\triangle $ of order 12.

To work with the Chiang Lagrangian, we will need the following additional notations. Let $W_\triangle $ denote the Zariski open $\text {SL}(2, \mathbb {C})$ -orbit in $\mathbb {C}P^3$ , and let $Y_\triangle $ denote its complement. $Y_\triangle $ consists of those $3$ -point configurations in $\mathbb {C}P^3$ where at least $2$ of the points coincide. Let $N_\triangle \subset Y_\triangle $ be the subvariety consisting of those configurations where all $3$ points coincide. If $[z_0:\ldots :z_3]$ are standard coordinates on $\mathbb {P}S^3 V$ , then the roots of the polynomial

$$\begin{align*}f(T) := \sum z_j (-T)^j \end{align*}$$

correspond (with multiplicity) to the $3$ -tuple of points obtained by viewing $[z]$ as a point of $\text {Sym}^3 \mathbb {C}P^1$ . We count $\infty $ as a root with multiplicity $3 - \deg f$ . Consequently, $Y_\triangle $ is defined by the vanishing of the discriminant $\triangle (f)$ of f.

3.2 Orientation and spin structure

Let $\alpha ,\beta ,\gamma $ be a basis of $\mathfrak {su}(2)$ corresponding to infinitesimal right-handed rotations about a right-handed set of orthonormal axes. The infinitesimal group action gives rise to a frame for $TL_\triangle ,$

$$\begin{align*}\alpha\cdot z,\beta\cdot z,\gamma\cdot z, \end{align*}$$

where $z\in L_\triangle .$ Let $\mathfrak {o}_\triangle $ be the orientation on $L_\triangle $ such that this frame is positively oriented. Let $\mathfrak {s}_\triangle $ be the spin structure on $L_\triangle $ such that this frame can be lifted to the associated double cover of the frame bundle.

3.3 Topological lemmas

Equip $\mathbb {P}S^3V \cong \mathbb {C}P^3$ with the Fubini-Study Kähler form $\omega $ such that $\int _{\mathbb {C}P^1} \omega =1$ . The following lemma is given in Section 4.3 of [Reference Evans and Lekili10].

Proof. By Poincaré duality, it suffices to show $H^1(L;\mathbb {Z})=0$ . By the universal coefficients theorem, we get the following short exact sequence

$$\begin{align*}0\rightarrow \text{Ext}^1(H_{0}(L_\triangle;\mathbb{Z}),\mathbb{Z})\rightarrow H^1(L_\triangle;\mathbb{Z})\rightarrow\text{Hom}(H_1(L_\triangle;\mathbb{Z}),\mathbb{Z})\rightarrow 0. \end{align*}$$

By Lemma 3.1 $H_1(L_\triangle ;\mathbb {Z})=\mathbb {Z}/4\mathbb {Z}$ . In addition, $L_\triangle $ is connected, so $H_0(L_\triangle ;\mathbb {Z})=\mathbb {Z}$ . Hence, we get an exact sequence $ 0\rightarrow H^1(L_\triangle ;\mathbb {Z})\rightarrow 0, $ which yields $H^1(L_\triangle ;\mathbb {Z})=0$ .

The following is Lemma 4.4.1 from [Reference Evans and Lekili10].

For $\zeta \in H_2(\mathbb {C}P^3;\mathbb {Z})$ , we denote by $c_1(\zeta )$ the evaluation of the first Chern class of $\mathbb {C}P^3$ on $\zeta $ . Let $\xi = [\mathbb {C}P^1]\in H_2(\mathbb {C}P^3;\mathbb {Z})$ , so $c_1(\xi )=4$ .

Lemma 3.4. The long exact sequence of the pair $(\mathbb {C}P^3,L_\triangle )$ gives the short exact sequence

$$\begin{align*}0\rightarrow H_2(\mathbb{C}P^3;\mathbb{Z})\overset{\varpi}{\rightarrow}H_2(\mathbb{C}P^3,L_\triangle;\mathbb{Z})\rightarrow H_1(L_\triangle;\mathbb{Z})\rightarrow 0, \end{align*}$$

with $H_2(\mathbb {C}P^3,L_\triangle ;\mathbb {Z}) \simeq \mathbb {Z} \simeq H_2(\mathbb {C}P^3;\mathbb {Z})$ , and the map $\varpi $ is given by multiplication by 4.

Proof. Consider the long exact sequence

$$\begin{align*}\ldots\rightarrow H_2(L_\triangle;\mathbb{Z})\rightarrow H_2(\mathbb{C}P^3;\mathbb{Z}) \overset{\varpi}{\rightarrow} H_2(\mathbb{C}P^3,L_\triangle;\mathbb{Z})\rightarrow H_1(L_\triangle;\mathbb{Z})\rightarrow H_1(\mathbb{C}P^3;\mathbb{Z})\rightarrow\ldots ,\end{align*}$$

where

$$\begin{align*}H_1(\mathbb{C}P^3;\mathbb{Z})=0, \qquad H_2(\mathbb{C}P^3;\mathbb{Z})\simeq \mathbb{Z}, \end{align*}$$

and Lemma 3.1 gives $H_1(L_\triangle ;\mathbb {Z})\simeq \mathbb {Z}/4\mathbb {Z}.$ By Lemma 3.2, we have $H_2(L_\triangle ;\mathbb {Z})=0$ , so we get the short exact sequence

(3.1) $$ \begin{align} 0\rightarrow \mathbb{Z} \overset{\varpi}{\rightarrow} H_2(\mathbb{C}P^3,L_\triangle;\mathbb{Z}) \rightarrow \mathbb{Z}/4\mathbb{Z}\rightarrow0. \end{align} $$

We have two possibilities:

1. 1. The short exact sequence (3.1) is split, so $H_2(\mathbb {C}P^3,L_\triangle ;\mathbb {Z}) \simeq \mathbb {Z}\oplus \mathbb {Z}/4\mathbb {Z}$ . Thus, (3.1) is isomorphic to

   $$\begin{align*}0\rightarrow \mathbb{Z} \overset{\varpi}{\rightarrow} \mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z} \rightarrow \mathbb{Z}/4\mathbb{Z}\rightarrow0, \end{align*}$$
   where
   $$\begin{align*}\varpi(x)= (x,0). \end{align*}$$

2. 2. The short exact sequence (3.1) is not split, so $H_2(\mathbb {C}P^3,L_\triangle ;\mathbb {Z})=\mathbb {Z}$ . Thus, (3.1) is isomorphic to

   $$\begin{align*}0\rightarrow \mathbb{Z} \overset{\varpi}{\rightarrow} \mathbb{Z} \rightarrow \mathbb{Z}/4\mathbb{Z}\rightarrow0,\end{align*}$$
   where
   $$\begin{align*}\varpi(x)= 4x.\end{align*}$$

In case 1, the Maslov index is given by

$$\begin{align*}\mu: \mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}\rightarrow \mathbb{Z},\end{align*}$$

$$\begin{align*}\mu (a,b)= ka, \qquad k\in\mathbb{Z}. \end{align*}$$

Since $\mu (\varpi (\xi ))=2c_1(\xi ) = 8$ , it follows that $k=8$ is in contradiction to Lemma 3.3. Therefore, we must be in case 2.

In light of Lemma 3.4, let $\tilde {\xi }\in H_2(\mathbb {C}P^3,L_\triangle ;\mathbb {Z})$ be the generator such that $\varpi (\xi )=4\tilde {\xi }$ . Recall $\Gamma _1=[\omega ]\in H^2(\mathbb {C}P^3,L_\triangle ;\mathbb {Z})$ .

Lemma 3.5. We have

$$\begin{align*}\int_{\tilde{\xi}} \Gamma_1 = \frac{1}{4}. \end{align*}$$

Proof. We have

$$\begin{align*}4\int_{\tilde{\xi}}\Gamma_1= \int_{\varpi(\xi)}\omega = \int_\xi\omega=1. \end{align*}$$

4.1 Moduli spaces

Denote by $\mathcal {M}_{k+1,l}(\beta )$ the moduli space of J-holomorphic genus zero open stable maps $u:(D, \partial D) \rightarrow (X,L)$ of degree $\beta $ with one boundary component, $k+1$ boundary marked points and l interior marked points. The boundary points are labeled according to their cyclic order. We denote elements in $\mathcal {M}_{k+1,l}(\beta )$ by $[u;z_0,\ldots ,z_k,w_1,\ldots ,w_l]$ , where $z_i\in \partial D$ and $w_i\in \operatorname {\mathrm {int}} D$ . Denote by

$$\begin{align*}evb^\beta_j: \mathcal{M}_{k+1,l}(\beta) \rightarrow L,\quad j=0,\ldots,k, \end{align*}$$

$$\begin{align*}evi^\beta_j: \mathcal{M}_{k+1,l}(\beta) \rightarrow X,\quad j=1,\ldots,l,\end{align*}$$

the evaluation maps associated to boundary marked points and to the interior marked points respectively.

Denote by $\mathcal {M}^S_{k,l}(\beta )$ the moduli space of genus zero J-holomorphic open stable maps $u:(D, \partial D) \rightarrow (X,L)$ of degree $\beta $ with one boundary component, k unordered boundary points, and l interior marked points. It comes with evaluation maps as in the case of $\mathcal {M}_{k,l}(\beta )$ .

We assume that all J-holomorphic genus zero open stable maps with one boundary component are regular and the moduli spaces $\mathcal {M}_{k+1,l}(\beta ),\mathcal {M}^S_{k,l}(\beta )$ are smooth orbifolds with corners. In addition, we assume that the evaluation maps $evb^\beta _0$ are proper submersions. These assumptions should hold for $(X,L) = (\mathbb {C}P^3,L_\triangle )$ and J the standard complex structure on $\mathbb {C}P^3$ by Remark 1.6 of [Reference Solomon and Tukachinsky47] in light of the transitive action of $\text {SL}(4,\mathbb {C})$ on $\mathbb {C} P^3$ and the transitive action of the subgroup $\text {SU}(2) \subset \text {SL}(4,\mathbb {C})$ on $L_\triangle $ . The arguments of the present section extend to arbitrary targets $(X,\omega ,L)$ and arbitrary $\omega $ -tame almost complex structures J given the virtual fundamental class techniques of [Reference Fukaya12, Reference Fukaya13, Reference Fukaya, Oh, Ohta and Ono14, Reference Fukaya, Oh, Ohta and Ono15, Reference Fukaya, Oh, Ohta and Ono16, Reference Fukaya, Oh, Ohta and Ono17, Reference Fukaya, Oh, Ohta and Ono19, Reference Fukaya, Oh, Ohta and Ono18, Reference Fukaya, Oh, Ohta and Ono20, Reference Fukaya, Oh, Ohta and Ono21, Reference Fukaya and Ono22]. Alternatively, it should be possible to use the polyfold theory of [Reference Hofer, Wysocki and Zehnder24, Reference Hofer, Wysocki and Zehnder27, Reference Hofer, Wysocki and Zehnder26, Reference Hofer, Wysocki and Zehnder25, Reference Li and Wehrheim35]. The relative spin structure $\mathfrak {s}$ determines an orientation on $\mathcal {M}_{k+1,l}(\beta ),\mathcal {M}^S_{k,l}(\beta )$ as in Chapter 8 of [Reference Fukaya, Oh, Ohta and Ono14].

4.2 The Fukaya $A_\infty $ algebra

Denote by $A^{*}(X;\mathbb {R})$ the ring of differential forms on X with coefficients in $\mathbb {R}$ . For $m>0$ , denote by $\widetilde {A}^m(X,L;\mathbb {R})$ differential m-forms on X that vanish on L, and denote by $\widetilde {A}^0(X,L;\mathbb {R})$ functions on X that are constant on L. Define

$$ \begin{align*}\widetilde{H}^{*}(X,L;\mathbb{R}):= H^{*}(\widetilde{A}^{*}(X,L;\mathbb{R}), d).\end{align*} $$

By Lemma 5.14 from[Reference Solomon and Tukachinsky46], if $H^{*}(L;\mathbb {R})\simeq H^{*}(S^n;\mathbb {R})$ , then $\widehat {H}^{*}(X,L;\mathbb {R})\simeq \widetilde {H}^{*}(X,L;\mathbb {R})$ . Denote by $\beta _0$ the zero element of $H_2(X,L;\mathbb {Z})$ .

Recall the definition of the Novikov ring $\Lambda $ from Section 2.4. Let $s,t_0,\ldots ,t_M$ , be formal variables with degrees in $\mathbb {Z}$ . Set

$$\begin{align*}R:= \Lambda[[s,t_0,\ldots,t_M]],\quad Q:=\mathbb{R}[[t_0,\ldots,t_M]] \subset R,\end{align*}$$

thought of as differential graded algebras with trivial differential. Define a valuation

$$\begin{align*}\nu: R\rightarrow \mathbb{R}_{\ge0}\end{align*}$$

by

$$\begin{align*}\nu\left(\sum_{j=0}^\infty a_jT^{\beta_j}s^{k_j}\prod_{a=0}^Mt_a^{l_{aj}}\right) = \inf_{\substack{j\\a_j\ne 0}} \left(\omega(\beta_j)+k_j+\sum_{a=0}^M l_{a_j}\right). \end{align*}$$

Whenever a tensor product (resp. direct sum) of modules with valuation is written, we mean the completed tensor product (resp. direct sum). Set

$$\begin{align*}C:= A^{*}(L)\otimes R, \quad D:= \widetilde{A}^{*}(X,L)\otimes Q. \end{align*}$$

In particular, the gradings on C and D take into account the degrees of $T^\beta , s,t_j$ , and the degrees of the differential forms. The valuation $\nu $ induces valuations on $\Lambda ,Q,C,D$ and their tensor products, which we also denote by $\nu $ . Let

$$\begin{align*}\Lambda^+=\{\alpha\in\Lambda | \ \nu(\alpha)>0 \}.\end{align*}$$

Let

$$\begin{align*}R^+:= R\Lambda^+ \triangleleft R,\quad \mathcal{I}_R:= \{ \alpha\in R| \ \nu(\alpha)>0\}\triangleleft R, \quad \mathcal{I}_Q:= \{\alpha\in Q| \ \nu(\alpha)>0 \}\triangleleft Q. \end{align*}$$

Let $\gamma \in \mathcal {I}_QD$ be a closed form with $|\gamma |=2$ . For example, given closed differential forms $\gamma _j\in \widetilde {A}^{*}(X,L;\mathbb {R})$ for $j=0,\ldots ,M,$ take $t_j$ of degree $2-|\gamma _j|$ and $\gamma :=\sum _{j=0}^Mt_j\gamma _j$ . Define maps

$$\begin{align*}\mathfrak{m}_k^{\gamma,\beta}:C^{\otimes k} \rightarrow C \end{align*}$$

by

$$\begin{align*}\mathfrak{m}^{\gamma,\beta_0}_1(\alpha)=d\alpha,\end{align*}$$

and for $k\ge 0$ when $(k,\beta )\ne (1,\beta _0)$ , by

(4.1) $$ \begin{align} \mathfrak{m}^{\gamma,\beta}_k(\alpha_1,\ldots,\alpha_k):=(-1)^{\sum_{j=1}^kj(|\alpha_j|+1)+1} \sum_{\substack{l\ge0}}\frac{1}{l!}{evb_0^\beta}_* (\bigwedge_{j=1}^l(evi_j^\beta)^{*}\gamma\wedge \bigwedge_{j=1}^k (evb_j^\beta)^{*}\alpha_j ). \end{align} $$

The push-forward $evb^\beta _{0_*}$ is given by integration over the fiber, which is well-defined because $evb^\beta _0$ is a proper submersion. Define also

$$\begin{align*}\mathfrak{m}_k^{\gamma}:C^{\otimes k} \rightarrow C\end{align*}$$

by

$$\begin{align*}\mathfrak{m}^{\gamma}_k:= \sum_{\substack{\beta\in H_2(X,L;\mathbb{Z})}}T^\beta\mathfrak{m}_k^{\gamma,\beta}. \end{align*}$$

Furthermore, define

$$ \begin{align*}\mathfrak{m}^\gamma_{-1} := \sum_{\substack{\beta\in H_2(X,L;\mathbb{Z}), \\ l\ge 0}} \frac{1}{l!} T^\beta \int_{\mathcal{M}_{0,l}(\beta)} \bigwedge_{j=1}^l (evi^\beta_j)^{*}\gamma.\end{align*} $$

The following is Proposition 2.6 from [Reference Solomon and Tukachinsky47].

Theorem 4.1. The operations $\{\mathfrak {m}_k^\gamma \}_{k\ge 0}$ define an $A_\infty $ structure on C. That is, for any $\alpha _1,\ldots ,\alpha _{k+1}\in C$ and $k\in \mathbb {Z}_{\ge 0},$

$$\begin{align*}\sum_{\substack{k_1+k_2=k+1\\ 1\le i \le k_1}}(-1)^{\sum_{j=1}^{i-1}(|\alpha_j|+1)}\mathfrak{m}_{k_1}(\alpha_1,\ldots,\alpha_{i-1},\mathfrak{m}_{k_2}(\alpha_i,\ldots,\alpha_{i+k_2-1}),\alpha_{i+k_2},\ldots,\alpha_k)=0. \end{align*}$$

The following is Proposition 3.1 from [Reference Solomon and Tukachinsky47].

Theorem 4.2. The operations $\mathfrak {m}_k$ are R-multilinear. Namely, for any $\alpha _1,\ldots ,\alpha _{k+1}\in C$ and $a\in R$ ,

(4.2) $$ \begin{align} \mathfrak{m}^\gamma_k(\alpha_1,\ldots,\alpha_{i-1},a\cdot\alpha_i,\ldots,\alpha_k)=(-1)^{|a|\cdot(i+\Sigma_{j=1}^{i-1}|\alpha_j|)}a\cdot\mathfrak{m}^\gamma_k(\alpha_1,\ldots,\alpha_k)+\delta_{1,k}\cdot da\cdot \alpha_1. \end{align} $$

Theorem 4.3. For $k\ge 0$ and $\omega (\beta )=0$ ,

$$\begin{align*}\mathfrak{m}^{\gamma,\beta}_k(\alpha_1,\ldots,\alpha_k)= \begin{cases} d\alpha_1, & (\beta,k)=(\beta_0,1),\\ (-1)^{|\alpha_1|}\alpha_1\wedge\alpha_2, & (\beta,k)=(\beta_0,2),\\ -\gamma_1|_L, & (\beta,k)=(\beta_0, 0),\\ 0, & \text{otherwise}. \end{cases} \end{align*}$$

Let $\langle \ , \ \rangle $ denote the Poincaré pairing

(4.3) $$ \begin{align} \langle \xi, \eta \rangle := (-1)^{|\eta|}\int_L\xi \wedge \eta. \end{align} $$

The following is Proposition 3.1 from [Reference Solomon and Tukachinsky47].

Theorem 4.4. Let $a\in R$ and $\alpha _1,\ldots ,\alpha _{k+1}\in C$ . The Poincaré pairing satisfies the following:

(4.4) $$ \begin{align} \langle a\cdot\alpha_1,\alpha_2\rangle=a \langle \alpha_1,\alpha_2\rangle, \qquad \langle \alpha_1, a\cdot\alpha_2\rangle=(-1)^{|a|\cdot(1+|\alpha_1|)}a\cdot\langle \alpha_1,\alpha_2\rangle. \end{align} $$

The following is Proposition 3.3 from [Reference Solomon and Tukachinsky47].

Theorem 4.5. For $\alpha _1,\ldots ,\alpha _{k+1}\in C$ ,

(4.5) $$ \begin{align} \langle\mathfrak{m}_k^\gamma(\alpha_1,\ldots,\alpha_k), \alpha_{k+1}\rangle=(-1)^{(|\alpha_{k+1}|+1)\sum_{j+1}^k(|\alpha_j|+1)}\langle\mathfrak{m}^\gamma_k(\alpha_{k+1},\alpha_1,\ldots,\alpha_{k-1}),\alpha_k\rangle. \end{align} $$

4.3 Bounding pairs, the superpotential and invariants

Definition 4.1. A bounding pair with respect to J is a pair $(\gamma ,b)$ where $\gamma \in \mathcal {I}_Q D$ is closed with $|\gamma |=2$ and $b\in \mathcal {I}_R C$ with $ |b|=1, $ such that

$$\begin{align*}\sum_{k\ge 0}\mathfrak{m}_k^\gamma(b^{\otimes k})=c\cdot 1, \qquad c\in \mathcal{I}_R,\;|c|=2. \end{align*}$$

In this situation, b is called a bounding cochain for $\mathfrak {m}^\gamma $ .

The definition of bounding pair appeared in [Reference Solomon and Tukachinsky46], and the definition of bounding cochain appeared in [Reference Fukaya, Oh, Ohta and Ono14].

Let $\gamma \in \mathcal {I}_Q D $ , $b\in \mathcal {I}_R C$ . The standard superpotential [Reference Fukaya13] is given by

$$\begin{align*}\widehat{\Omega}(\gamma, b):=\widehat{\Omega}_J(\gamma, b):= (-1)^n\left(\sum_{k\ge 0} \frac{1}{k+1} \langle \mathfrak{m}^\gamma_k(b^{\otimes k}), b\rangle + \mathfrak{m}^\gamma_{-1}\right). \end{align*}$$

Intuitively, $\widehat {\Omega }$ counts J-holomorphic disks with constraints $\gamma $ in the interior and b on the boundary. Modification is necessary in order to avoid J-holomorphic disks the boundary of which can degenerate to a point, forming a J-holomorphic sphere. We say that a monomial element of R is of type $\mathcal {D}$ if it has the form $a\, T^{\beta }s^0t_0^{j_0}\cdots t_N^{j_N}$ with $a\in \mathbb {R}$ and $\beta \in \operatorname {\mathrm {Im}}(\varpi )$ . Following [Reference Solomon and Tukachinsky46], in the present paper, the superpotential is defined by

$$\begin{align*}\Omega(\gamma,b):=\Omega_J(\gamma,b) : = \widehat{\Omega}_J(\gamma,b)-\text{ all monomials of type }\mathcal{D}\text{ in }\widehat{\Omega}_J. \end{align*}$$

Definition 3.12 from [Reference Solomon and Tukachinsky46] gives a notion of gauge equivalence between a bounding pair $(\gamma ,b)$ with respect to J and a bounding pair $(\gamma ',b')$ with respect to another almost complex structure $J'.$ Let $\sim $ denote the resulting equivalence relation. For a graded module M, we denote by $(M)_j$ the degree j part of the module.

Define a map

$$\begin{align*}\varrho:\{\text{bounding pairs}\}/\sim \ \longrightarrow (\mathcal{I}_Q\widetilde{H}^{*}(X,L;Q))_2\oplus (\mathcal{I}_R)_{1-n}\end{align*}$$

by

$$\begin{align*}\varrho([\gamma,b]):=\left([\gamma],\int_Lb\right). \end{align*}$$

By Lemma 3.16 in [Reference Solomon and Tukachinsky46], $\varrho $ is well defined. The following is Theorem 1 from [Reference Solomon and Tukachinsky46].

The following is Theorem 2 from [Reference Solomon and Tukachinsky46].

Fix $\Gamma _0,\ldots ,\Gamma _M$ a basis of $\widetilde {H}^{*}(X,L;\mathbb {R})$ , set $|t_j|=2-|\Gamma _j|$ , and take

$$\begin{align*}\Gamma:=\sum_{j=0}^Mt_j\Gamma_j\in (\mathcal{I}_Q\widetilde{H}^{*}(X,L;\mathbb{R}))_2.\end{align*}$$

By Theorem 4.3, choose a bounding pair $(\gamma ,b)$ such that

$$\begin{align*}\varrho([\gamma,b])=(\Gamma,s).\end{align*}$$

By Theorem 4.2, the superpotential $\Omega =\Omega (\gamma ,b)$ is independent of the choice of $(\gamma ,b)$ .

Definition 4.4. The open Gromov-Witten invariants of $(X,L)$ ,

$$\begin{align*}\text{OGW}_{\beta,k}: \widetilde{H}^{*}(X,L;\mathbb{R})^{\otimes l}\rightarrow \mathbb{R},\end{align*}$$

are defined by setting

$$\begin{align*}\text{OGW}_{\beta,k}(\Gamma_{i_1},\ldots,\Gamma_{i_l}):= \text{ the coefficient of }T^{\beta}\text{ in } \partial_{t_{i_1}}\cdots\partial_{t_{i_l}}\partial_s^k\Omega|_{s=0,t_j=0} \end{align*}$$

and extending linearly to general input.

The following is Proposition 5.2 from [Reference Solomon and Tukachinsky46].

4.4 Changes of spin structure and orientation

The following Lemmas describe the dependence of the invariants ${OGW}_{\beta ,k}$ on the spin structure and the orientation of $L.$

Proof. In the following proof, we add a superscript $\mathfrak {s}$ to the $A_\infty $ operations, superpotential and open Gromov-Witten invariants to indicate the spin structure used to construct them. By Lemma 2.10 of [Reference Solomon45], the orientations of $\mathcal {M}_{k+1,l}(\beta )$ induced by $\mathfrak {s}$ and $\alpha \cdot \mathfrak {s}$ differ by a sign of $(-1)^{\alpha (\partial \beta )}$ . So,

$$\begin{align*}\mathfrak{m}_k^{\gamma,\beta,\mathfrak{s}} = (-1)^{\alpha(\partial \beta)} \mathfrak{m}_k^{\gamma,\beta,\alpha\cdot \mathfrak{s}}. \end{align*}$$

Let $\mathfrak {f}_R : R \to R$ be the $\mathbb {R}$ algebra automorphism that sends $T^\beta $ to $(-1)^{\alpha (\partial \beta )}T^\beta $ and acts trivially on the other formal variables. Then, the R module automorphism

$$\begin{align*}\mathfrak{f} = \mathfrak{f}_R \otimes \operatorname{\mathrm{Id}}_{A^{*}(L)}:C\rightarrow C \end{align*}$$

is a strict $A_\infty $ homomorphism between the $A_\infty $ structures $\mathfrak {m}_k^{\mathfrak {s}}$ and $\mathfrak {m}_k^{\alpha \cdot \mathfrak {s}}$ and preserves the pairing $\langle \cdot ,\cdot \rangle $ . So, if b is a bounding cochain for $\mathfrak {m}_k^{\mathfrak {s}}$ with $\int _L b = s$ , then $\mathfrak {f}(b)$ is a bounding cochain for $\mathfrak {m}_k^{\alpha \cdot \mathfrak {s}}$ with $\int _L \mathfrak {f}(b) = s$ and $\mathfrak {f}(\Omega ^{\mathfrak {s}}(\gamma ,b)) = \Omega ^{\alpha \cdot \mathfrak {s}}(\gamma ,\mathfrak {f}(b)).$ By Definition 4.4, we obtain

$$ \begin{align*} {OGW}^{\alpha\cdot\mathfrak{s}}_{\beta,k}(\Gamma_{i_1},\ldots,\Gamma_{i_l}) & = [T^\beta]\partial_{t_{i_1}}\cdots\partial_{t_{i_l}}\partial_s^k \Omega^{\alpha\cdot\mathfrak{s}}(\gamma,\mathfrak{f}(b)))|_{s=0,t_j=0} \\ &=[T^{\beta}]\partial_{t_{i_1}}\cdots\partial_{t_{i_l}}\partial_s^k\mathfrak{f}(\Omega^{\mathfrak{s}}(\gamma,b))|_{s=0,t_j=0} \\ &=(-1)^{\alpha(\partial\beta)} [T^\beta]\partial_{t_{i_1}}\cdots\partial_{t_{i_l}}\partial_s^k(\Omega^{\mathfrak{s}}(\gamma,b))|_{s=0,t_j=0} \\ &=(-1)^{\alpha(\partial\beta)} {OGW}^{\mathfrak{s}}_{\beta,k}(\Gamma_{i_1},\ldots,\Gamma_{i_l}). \end{align*} $$

Proof. In the following proof, we add a superscript $\mathfrak {o}$ to the $A_\infty $ operations, superpotential, Poincaré pairing, $\int _L$ and open Gromov-Witten invariants to indicate the orientation used to construct them. By Lemma 2.9 from [Reference Solomon45], changing the orientation of L reverses the orientation of $\mathcal {M}_{k+1,l}(\beta )$ . Since the relative orientation is not changed, it follows that

$$\begin{align*}\mathfrak{m}^{\gamma,\beta,\mathfrak{o}}_k=\mathfrak{m}^{\gamma,\beta,-\mathfrak{o}}_k.\end{align*}$$

Let $\mathfrak {f}_R : R \to R$ be the $\mathbb {R}$ algebra automorphism that sends s to $-s$ and acts trivially on the other formal variables. Then, the R module automorphism

$$\begin{align*}\mathfrak{f} = \mathfrak{f}_R \otimes \operatorname{\mathrm{Id}}_{A^{*}(L)}:C\rightarrow C \end{align*}$$

is a strict $A_\infty $ homomorphism between the $A_\infty $ structures $\mathfrak {m}_k^{\mathfrak {o}}$ and $\mathfrak {m}_k^{-\mathfrak {o}}$ that satisfies

$$\begin{align*}\langle\cdot,\cdot \rangle^{\mathfrak{o}}=-\langle\cdot,\cdot \rangle^{-\mathfrak{o}}. \end{align*}$$

Similarly, $\int _L^{\mathfrak {o}} = - \int _L^{-\mathfrak {o}}.$ So, if b is a bounding cochain for $\mathfrak {m}_k^{\mathfrak {o}}$ with $\int _L^{\mathfrak {o}} b=s$ , then $\mathfrak {f}(b)$ is a bounding cochain for $\mathfrak {m}_k^{-\mathfrak {o}}$ with $\int _L^{-\mathfrak {o}} \mathfrak {f}(b)=s$ and $\mathfrak {f}(\Omega ^{\mathfrak {o}}(\gamma ,b))=-\Omega ^{-\mathfrak {o}}(\gamma ,\mathfrak {f}(b))$ . By Definition 4.4, we obtain

$$ \begin{align*} {OGW}^{-\mathfrak{o}}_{\beta,k}(\Gamma_{i_1},\ldots,\Gamma_{i_l})&:= [T^\beta]\partial_{t_{i_1}}\cdots\partial_{t_{i_l}}\partial_s^k(\Omega^{-\mathfrak{o}}(\gamma,\mathfrak{f}(b)))|_{s=0,t_j=0}\\ &=- [T^{\beta}] \partial_{t_{i_1}}\cdots\partial_{t_{i_l}}\partial_s^k\mathfrak{f}(\Omega^{\mathfrak{o}}(\gamma,b))|_{s=0,t_j=0}\\ &=(-1)^{k+1}[T^\beta]\partial_{t_{i_1}}\cdots\partial_{t_{i_l}}\partial_s^k(\Omega^{\mathfrak{o}}(\gamma,b))|_{s=0,t_j=0}\\ &=(-1)^{k+1}{OGW}^{\mathfrak{o}}_{\beta,k}(\Gamma_{i_1},\ldots,\Gamma_{i_l}). \end{align*} $$

We apply the preceding lemmas to prove vanishing results for the invariants $\overline {OGW}\!$ of certain Lagrangian submanifolds $L \subset X.$ The case $n = 3$ and k even of the following corollary has been proved previously in [Reference Chen and Zinger6, Proposition 1.3] by a different method, which goes back to an observation of Mikhalkin [Reference Welschinger51, Remark 2.4].

The following was obtained by a different method in [Reference Hugtenburg and Tukachinsky29, Lemma 6.8].

4.5 Obstruction theory

The goal of the present section is the proof of Lemma 4.18, which refines the existence result for bounding cochains given in Proposition 3.4 of [Reference Solomon and Tukachinsky46]. For this purpose, we recall relevant parts of the obstruction theoretic construction of bounding cochains given there. The basic idea goes back to [Reference Fukaya, Oh, Ohta and Ono14].

Let

$$\begin{align*}T(C):= \bigoplus_{k\ge0} C^{\otimes k}\end{align*}$$

denote the completed tensor algebra. For $x\in \mathcal {I}_RC$ , abbreviate

$$\begin{align*}e^x= 1\oplus x \oplus (x\otimes x)\oplus (x\otimes x \otimes x)\otimes \ldots\in T(C) .\end{align*}$$

Moreover, define

$$\begin{align*}\mathfrak{m}^\gamma:T(C)\rightarrow C\end{align*}$$

by

$$\begin{align*}\mathfrak{m}^\gamma(\otimes_{k\ge 0}\eta_k)=\sum_{k\ge0}\mathfrak{m}^\gamma_k(\eta_k).\end{align*}$$

Any element $\alpha \in C$ can be written as

(4.6) $$ \begin{align} \alpha=\sum_{i=1}^\infty\lambda_i\alpha_i, \qquad \alpha_i\in A^{*}(L),\quad \lambda_i=T^{{\beta}_i}s^{k_i}\prod_{a=0}^M t_a^{l_{a_i}},\quad \lim_{i}\nu(\lambda_i)=\infty. \end{align} $$

Define a filtration $F^E$ on R, C by

$$\begin{align*}\lambda\in F^EC\Longleftrightarrow \nu(\lambda)> E.\end{align*}$$

Definition 4.10. A multiplicative submonoid $G\subset R$ is sababa if it can be written as a list

(4.7) $$ \begin{align} G=\{\pm\lambda_0=\pm T^{\beta_0}, \pm\lambda_1,\pm\lambda_2,\ldots\} \end{align} $$

such that $i<j\Rightarrow \nu (\lambda _i)\le \nu (\lambda _j).$

For $j=1,\ldots ,m,$ and elements $\alpha _j=\sum _i\lambda _{ij}\alpha _{ij}\in C$ decomposed as in (4.6), denote by $G(\alpha _1,\ldots ,\alpha _m)$ the multiplicative monoid generated by $\{\pm T^\beta \,|\,\beta \in \Pi _\infty \}$ , $\{t_j\}_{j=0}^N$ , and $\{\lambda _{ij}\}_{i,j}$ . The following is Lemma 3.3 from [Reference Solomon and Tukachinsky46].

For $\alpha _1,\ldots ,\alpha _m\in \mathcal {I}_R$ , write the image of $G=G(\alpha _1,\ldots ,\alpha _m)$ under $\nu $ as the sequence $\{E_0^G=0, E^G_1, E^G_2,\ldots \}$ with $E_i^G<E^G_{i+1}$ . Let $\kappa ^G_i\in \mathbb {Z}_{\ge 0}$ be the largest index such that $\nu (\lambda _{\kappa _i^G})=E^G_i$ . In the future, we omit G from the notation and simply write $E_i,\kappa _i$ since G will be fixed in each instance and no confusion should occur.

Given $\alpha \in C$ , a differential form with coefficients in R, we denote by $(\alpha )_j \in C$ the summand of differential form of degree j in $\alpha $ . Let $\Upsilon '$ be an $\mathbb {R}-$ vector space, let $\Upsilon "=R, Q,$ or $\Lambda ,$ and let $\Upsilon =\Upsilon '\otimes \Upsilon "$ . For $x\in \Upsilon $ and a monomial $\lambda \in \Upsilon ",$ denote by $[\lambda ](x)\in \Upsilon '$ the coefficient of $\lambda $ in x.

Fix a sababa multiplicative monoid $G=\{\lambda _j\}_{j=0}^\infty \subset R$ ordered as in (4.7). Let $l\ge 0.$ Suppose we have $b_{(l)}\in C$ with $|b_{(l)}|=1$ , and

$$\begin{align*}\mathfrak{m}^{\gamma}(e^{b_{(l)}})\equiv c_{(l)}\cdot 1\pmod{F^{E_{l}}C},\qquad c_{(l)}\in (\mathcal{I}_R)_2.\end{align*}$$

Define the obstruction cochains $o_j\in A^{*}(L)$ for $j=\kappa _l+1,\ldots ,\kappa _{l+1}$ to be

$$\begin{align*}o_j:= [\lambda_j](\mathfrak{m}^\gamma(e^{b_{(l)}})).\end{align*}$$

Lemmas 4.12–4.17 are Lemmas 3.5–3.10 from [Reference Solomon and Tukachinsky46].

Lemma 4.16. Suppose for all $j\in \{\kappa _l+1,\ldots ,\kappa _{l+1}\}$ such that $|\lambda _j|\ne 2$ , there exist $b_j\in A^{1-|\lambda _j|}(L;\mathbb {R})$ such that $(-1)^{|\lambda _j|}db_j=-o_j.$ Then

$$\begin{align*}b_{(l+1)}:=b_{(l)}+\sum_{\substack{\kappa_l+1\le j\le \kappa_{l+1}\\ |\lambda_j|\ne 2}}\lambda_jb_j \end{align*}$$

satisfies

$$\begin{align*}\mathfrak{m}^{\gamma}(e^{b_{(l+1)}})\equiv c_{(l+1)}\cdot 1\pmod{F^{E_{l+1}}C},\qquad c_{(l+1)}\in (\mathcal{I}_R)_2. \end{align*}$$

Lemma 4.18. Let $b_0\in C$ such that

$$\begin{align*}|b_0|=1,\qquad (db_0)_n=0, \qquad \mathfrak{m}^\gamma({e^{b_0}})\equiv c\cdot 1 \pmod{R^+C}, \quad c\in (\mathcal{I}_R)_2.\end{align*}$$

Then there exists a bounding cochain b, such that

1. 1. $b\equiv b_0 \pmod {R^+}$ ,

2. 2. $\int _L b=\int _L b_0$ ,

3. 3. $(b)_j=(b_0)_j$ where $j\in \{n-1,n\}$ .

Proof. We follow the proof of Proposition 3.4 from [Reference Solomon and Tukachinsky46]. Define $b_{(0)}:=b_0$ . By Lemma 4.17, the cochain $b_{(0)}$ satisfies

$$\begin{align*}\mathfrak{m}^\gamma(e^{b_{(0)}})\equiv 0=c_{(0)}\cdot 1\pmod{F^{E_0}C},\qquad c_{(0)}=0.\end{align*}$$

Moreover, $|b_{(0)}|=1$ , $\int _L b_{(0)}=\int _L b_0$ , $(db_{(0)})_n=0$ , and $(b_{(0)})_j=(b_0)_j$ where $j\in \{n-1,n\}$ . Observe also that $F^{E_0}C \subset R^+C$ . Proceed by induction. Suppose we have $b_{(l)}\in C$ such that $|b_{(l)}|=1$ , $b_{(l)}\equiv b_0 \pmod {R^+C}$ , and

$$ \begin{gather*} (db_{(l)})_n =0,\qquad\int_L b_{(l)}=\int_L b_0,\\ (b_{(l)})_j=(b_0)_j \quad j\in \{n-1,n\},\qquad \mathfrak{m}^\gamma(e^{b_{(l)}})\equiv c_{(l)}\cdot 1\pmod{F^{E_l}C},\quad c_{(l)}\in (\mathcal{I}_R)_2. \end{gather*} $$

By Lemma 4.13, we have $do_j=0$ , and by Lemma 4.12, we have $o_j\in A^{2-|\lambda _j|}(L;\mathbb {R}).$ In order to apply Lemma 4.16, we need to find forms $b_j\in A^{1-|\lambda _j|}(L;\mathbb {R})$ such that $(-1)^{|\lambda _j|}db_j=-o_j$ for all $j\in \{\kappa _l+1,\ldots ,\kappa _{l+1}\}$ such that $|\lambda _j|\ne 2.$ Since $F^{E_l}C \subset R^+C,$ we have

$$\begin{align*}\mathfrak{m}^\gamma(e^{b_{(l)}})\equiv c_{(l)}\cdot 1 \pmod{R^+C}. \end{align*}$$

It follows that $o_j=0$ when $\lambda _j\not \in R^+$ , so we choose $b_j=0$ . Hence, we may assume $\lambda _j\in R^+$ . If $|\lambda _j|= 2-n,$ since $(db_{(l)})_n =0$ , Lemma 4.15 gives $o_j = 0,$ so we choose $b_j = 0.$ If $2-n < |\lambda _j|<~2,$ then $0 < |o_j| < n.$ The assumption $H^{*}(L;\mathbb {R})\simeq H^{*}(S^n;\mathbb {R})$ implies $[o_j] = 0 \in H^{*}(L;\mathbb {R}),$ so we choose $b_j$ such that $(-1)^{|\lambda _j|}db_j = -o_j.$ For other possible values of $|\lambda _j|,$ degree considerations imply $o_j = 0,$ so we choose $b_j =0.$ By Lemma 4.16, $b_{(l+1)}:=b_{(l)}+\sum _{\substack {\kappa _l+1\le j\le \kappa _{l+1}\\ |\lambda _j|\ne 2}}\lambda _jb_j$ satisfies

$$\begin{align*}\mathfrak{m}^\gamma(e^{b_{(l+1)}})\equiv c_{(l+1)}\cdot 1\pmod{F^{E_{l+1}}C},\qquad c_{(l+1)}\in (\mathcal{I}_R)_2. \end{align*}$$

Since $b_j=0$ when $\lambda _j\not \in R^+$ , it follows that $b_{(l+1)}\equiv b_{(l)}\equiv b_0 \pmod {R^+C}$ . Since $b_j = 0$ when $|\lambda _j| = 2-n$ , it follows that $(b_{(l+1)})_{n-1}=(b_{(l)})_{n-1}=(b_{(0)})_{n-1}$ , and that $(db_{(l+1)})_n = (db_{(l)})_n=0$ . In addition, $b_j = 0$ when $|\lambda _j| = 1-n$ . Thus, $(b_{(l+1)})_{n}=(b_{(l)})_{n}=(b_{(0)})_{n},$ and $\int _L b_{(l+1)}=\int _L b_{(l)}=\int _L b_0$ . The inductive process gives rise to a convergent sequence $\{b_{(l)}\}_{l=0}^\infty $ where $b_{(l)}$ is bounding modulo $F^{E_l}C$ . Taking the limit as l goes to infinity, we obtain

$$\begin{align*}b:=\lim_l b_{(l)},\quad |b|=1,\quad b\equiv b_0 \pmod{R^+C}, \quad \int_L b = \int_L b_0, \quad \end{align*}$$

$$\begin{align*}(b)_j=(b_0)_j \quad j\in \{n-1,n\},\qquad \mathfrak{m}^\gamma(e^b)= c\cdot 1,\quad c=\lim_lc_{(l)}\in (\mathcal{I}_R)_2. \end{align*}$$

4.6 Straightforward counts

In the following, we will be using the following notation conventions. For $k\ge 1$ , denote by $[k]$ the set $[k]:= \{1,\ldots ,k \}.$ For $N>0$ , we denote $t:= (t_1,\ldots ,t_N)$ . Similarly, we denote by $\alpha $ an N-tuple where all the components are taken from the set $\mathbb {Z}_{\ge 0}$ ; that is,

$$\begin{align*}\alpha:= (\alpha_1,\ldots,\alpha_N),\quad \alpha_i\in \mathbb{Z}_{\ge0}. \end{align*}$$

We denote $t^\alpha := t_1^{\alpha _1}\cdots t_N^{\alpha _N}$ . Recall the definition of $P_\beta $ from (1.1).

Theorem 4.19. Let $A_{j}\in \widetilde {H}^{*}(X,L;\mathbb {R})$ . Let $\bar {b}\in A^n(L;\mathbb {R})$ such that $\text {PD}([\bar {b}])=pt$ , and let $a_{j}$ be a representative of $A_{j}$ . Let $\sigma _k = (k-1)!$ for $k \in \mathbb {Z}_{>0}$ and let $\sigma _0 = 1.$ Then,

$$ \begin{align*} {OGW}_{\beta,k}(A_{1},\ldots,A_{l}) &= (-1)^n \sigma_k\int_{\mathcal{M}_{k,l}(\beta)} \bigwedge_{j=1}^{l} evi^{\beta*}_j a_{j} \bigwedge_{j=1}^{k}evb^{\beta*}_j\bar{b} \\ &=(-1)^n\int_{\mathcal{M}^S_{k,l}(\beta)} \bigwedge_{j=1}^{l} evi^{\beta*}_j a_{j} \bigwedge_{j=1}^{k}evb^{\beta*}_j\bar{b} \end{align*} $$

if for every

$$\begin{align*}\tilde {\beta}\in P_\beta, \qquad 0\le j\le k,\qquad I\subset[l], \end{align*}$$

one of the following two conditions is satisfied:

1. 1. $1-\mu (\tilde {\beta }) +j(n-1) + \sum _{i\in I} (|A_{i}|-2)<0,$ or

2. 2. $1-\mu (\tilde {\beta }) +j(n-1) + \sum _{i\in I} (|A_{i}|-2)\ge n-1.$

Proof. Let $\Gamma _0,\ldots ,\Gamma _M$ be a basis of $\widetilde {H}^{*}(X,L;\mathbb {R})$ with $\Gamma _0 = 1$ . From multilinearity of ${OGW}$ and Proposition 4.6, it suffices to prove the lemma for $\mathrm {OGW}_{\beta ,k}(\Gamma _0^{\otimes {r_0}}\otimes \cdots \otimes \Gamma _M^{\otimes {r_M}})$ . By the unit axiom 4, we can assume that $r_0=0$ , and by the zero axiom 5, we can assume that $\beta \ne \beta _0$ . We have $\gamma = \sum _{i=1}^M t_i \gamma _i$ , where $\gamma _i$ is a representative of $\Gamma _i\in \widetilde {H}^{*}(X,L;\mathbb {R})$ . Let $b_{0,1,0}\in A^n(L;\mathbb {R})$ be a representative of the Poincaré dual of a point. Since by Proposition 4.3 we have

$$\begin{align*}\mathfrak{m}^\gamma(e^{b_{0,1,0}})\equiv db_{0,1,0}-b_{0,1,0}\wedge b_{0,1,0}+\gamma|_L\equiv 0 \pmod{R^+C},\end{align*}$$

it follows by Lemma 4.18 that there exists a bounding cochain $b\in C$ such that $b\equiv s\cdot b_{0,1,0} \pmod {R^+C}$ , and

(4.8) $$ \begin{align} (b)_n=b_{0,1,0},\qquad (b)_{n-1}=0. \end{align} $$

Write $b=s\cdot b_{0,1,0}+\sum T^{\tilde {\beta }}s^j t^\alpha b_{\tilde {\beta },j,\alpha }$ , where

$$\begin{align*}\tilde{\beta}\in H_2(X,L;\mathbb{Z}),\quad \omega(\tilde{\beta})>0, \quad \alpha=(\alpha_1,\ldots,\alpha_M)\in \mathbb{Z}_{\ge0}^M, \quad j\in \mathbb{Z}_{\ge0},\quad b_{\tilde{\beta},j,\alpha}\in A^{*}(L;\mathbb{R}). \end{align*}$$

Since

$$\begin{align*}|b| = 1,\qquad |s|= 1-n, \qquad |T^{\tilde \beta}| = \mu(\tilde\beta), \qquad |t_i|=2-|\gamma_i|, \end{align*}$$

it follows that

$$\begin{align*}|b_{\tilde {\beta},j, \alpha}|= 1-\mu(\tilde {\beta}) -j(1-n) - \sum_{i=1}^M \alpha_i(2-|\gamma_i|). \end{align*}$$

Hence, if conditions 1 and 2 are satisfied, then by (4.8) we must have $b_{\tilde {\beta },j,\alpha }=0$ when

$$\begin{align*}\tilde{\beta}\in P_\beta,\qquad 0 \leq \alpha_i \leq r_i,\qquad 0\le j\le k. \end{align*}$$

Define

$$\begin{align*}J:=R^+ + ( s^{k+1}, t_0, t_1^{r_1+1},\ldots,t_M^{r_M+1} )\triangleleft R, \end{align*}$$

and denote

$$\begin{align*}b' = s\cdot b_{0,1,0}+ \sum_{\substack{\omega(\tilde{\beta})=\omega(\beta)\\ 0\le j\le k\\ 0\le\alpha_i\le r_i}} T^{\tilde{\beta}}s^j t^\alpha b_{\tilde{\beta},j,\alpha}.\end{align*}$$

Since $b'\equiv b \pmod {JC}$ , it follows that

$$\begin{align*}[T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\partial^k_s\Omega(\gamma,b)|_{s=0,t_j=0})=[T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\partial^k_s\Omega(\gamma,b')|_{s=0,t_j=0}).\end{align*}$$

Hence, it suffices to consider $\Omega (\gamma ,b')$ instead of $\Omega (\gamma ,b)$ in the computation of $\mathrm {OGW}_{\beta ,k}(\Gamma _1^{\otimes {r_1}}\otimes \cdots \otimes \Gamma _M^{\otimes {r_M}})$ . Since

$$\begin{align*}db_{0,1,0}=0, \qquad b_{0,1,0}\wedge b_{0,1,0}=0, \qquad \gamma|_L=0, \end{align*}$$

by Proposition 4.3, for every $m\ge 0$ , $0\le j\le k$ and $\tilde {\beta }\in H_2(X,L;\mathbb {Z})$ such that $\omega (\tilde {\beta })=0$ , we get

$$\begin{align*}\langle \mathfrak{m}^{\gamma,\tilde{\beta}}_m(b_{0,1,0}^{\otimes m}), b_{\beta-\tilde{\beta},j,\alpha}\rangle=0.\end{align*}$$

So, by Proposition 4.5 for every $1\le i\le m$ , we get

$$\begin{align*}\langle\mathfrak{m}^{\gamma,\tilde{\beta}}_m(b_{0,1,0}^{\otimes i-1}\otimes b_{\beta-\tilde{\beta},j,\alpha}\otimes b_{0,1,0}^{\otimes {m-i}}),b_{0,1,0}\rangle=0.\end{align*}$$

Hence,

(4.9) $$ \begin{align} [T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\partial^k_s(\langle\sum_{m\ge0} \mathfrak{m}^{\gamma}_m &(b^{\prime\otimes{m}}), b'\rangle)) =\notag \\ &=(\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\partial^k_s( \langle \mathfrak{m}^{\gamma,\beta}_{k-1}(s\cdot b_{0,1,0},\ldots,s\cdot b_{0,1,0}),s\cdot b_{0,1,0}\rangle)). \end{align} $$

Denote $l=\sum _{i=1}^Mr_i.$ For $k=0$ , we get

$$ \begin{align*} {OGW}_{\beta,0}(\Gamma_1^{\otimes{r_1}}\otimes\cdots\otimes\Gamma_M^{\otimes{r_M}})&= [T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\partial^k_s\Omega|_{s=0,t_j=0})\\ &=(-1)^n [T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\mathfrak{m}^\gamma_{-1})\\ &= (-1)^n\int_{\mathcal{M}_{0,l}(\beta)} \bigwedge_{j=1}^{r_1} evi^{\beta*}_j \gamma_1\cdots\bigwedge_{j=1}^{r_M} evi^{\beta*}_j \gamma_M, \end{align*} $$

where $\mathcal {M}_{0,l}(\beta )=\mathcal {M}^S_{0,l}(\beta ).$ For $k\ne 0$ , we get

$$ \begin{align*} {OGW}_{\beta,k}(\Gamma_1^{\otimes{r_1}}\otimes&\cdots\otimes\Gamma_M^{\otimes{r_M}}) = \\ & = [T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\partial^k_s\Omega|_{s=0,t_j=0})\\ &\!\overset{(4.9)}=(-1)^n \frac{1}{k}\cdot [T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\partial_s^k\langle \mathfrak{m}^\gamma_{k-1} ((s\cdot b_{0,1,0})^{\otimes {k-1}}), s\cdot b_{0,1,0} \rangle)\\ &\!\!\!\!\!\overset{(4.2) + (4.4)}=(-1)^{1+(1-n)\frac{k(k-1)}{2}}\cdot(k-1)!\cdot [T^\beta](\partial_{t_1}^{r_1}\ldots \partial_{t_M}^{r_M}\langle \mathfrak{m}^\gamma_{k-1} ( b_{0,1,0}^{\otimes {k-1}}), b_{0,1,0} \rangle)\\ &\overset{(4.1)}= (k-1)!\cdot \left\langle evb^{\beta}_{0*}\bigwedge_{j=1}^{l} evi^{\beta*}_j \gamma_{a_j} \bigwedge_{j=1}^{k-1}evb^{\beta*}_j(b_{0,1,0})), b_{0,1,0} \right\rangle\\ &\overset{(4.3)}= (-1)^n(k-1)!\int_{\mathcal{M}_{k,l}(\beta)} \bigwedge_{j=1}^{l} evi^{\beta*}_j \gamma_{a_j} \bigwedge_{j=1}^{k}evb^{\beta*}_j(b_{0,1,0}). \end{align*} $$

The diffeomorphism of $\mathcal {M}^S_{k,l}(\beta )$ corresponding to relabeling boundary marked points by a permutation $\sigma \in S_k$ preserves or reverses orientation depending on $sgn(\sigma )$ . Let $\sigma \in S_k$ be a permutation. Denote by $\mathcal {M}_{\sigma (k),l}(\beta )$ the moduli space obtained from $\mathcal {M}_{k,l}(\beta )$ by relabeling boundary marked points by $\sigma $ . So,

$$ \begin{align*} \int_{\mathcal{M}_{k,l}(\beta)} \bigwedge_{j=1}^{l} evi^{\beta*}_j\gamma_{a_j} \bigwedge_{j=1}^{k}evb^{\beta*}_j(b_{0,1,0})&= sgn(\sigma)\int_{\mathcal{M}_{\sigma(k),l}(\beta)} \bigwedge_{j=1}^{l} evi^{\beta*}_j \gamma_{a_j} \bigwedge_{j=1}^{k}evb^{\beta*}_{\sigma(j)}(b_{0,1,0})\\ &=sgn(\sigma)^2\int_{\mathcal{M}_{\sigma(k),l}(\beta)} \bigwedge_{j=1}^{l} evi^{\beta*}_j \gamma_{a_j} \bigwedge_{j=1}^{k}evb^{\beta*}_j(b_{0,1,0})\\ &=\int_{\mathcal{M}_{\sigma(k),l}(\beta)}\bigwedge_{j=1}^{l} evi^{\beta*}_j \gamma_{a_j} \bigwedge_{j=1}^{k}evb^{\beta*}_j(b_{0,1,0}). \end{align*} $$

Therefore, since $\mathcal {M}_{k,l}^S(\beta ) = \coprod _{\sigma \in S_k} \mathcal {M}_{\sigma (k),l}(\beta ),$ it follows that

$$\begin{align*}{OGW}_{\beta,k}(\Gamma_1^{\otimes{r_1}}\otimes\cdots\otimes\Gamma_M^{\otimes{r_M}})= (-1)^n \int_{\mathcal{M}^S_{k,l}(\beta)} \bigwedge_{j=1}^{l} evi^{\beta*}_j \gamma_{a_j} \bigwedge_{j=1}^{k}evb^{\beta*}_j(b_{0,1,0}). \end{align*}$$

5.1 Intersections with the compactification divisor

Recall from Section 3.1 the definition of the Chiang Lagrangian $L_\triangle \subset X_\triangle \simeq \mathbb {C} P^3 \simeq \text {Sym}^3\mathbb {C}P^1.$ Recall also the definitions of the discriminant locus $Y_\triangle \subset X_\triangle $ and the subvariety $N_\triangle \subset Y_\triangle .$

The following is Lemma 3.1 from [Reference Auroux2].

The following is Lemma 3.7 from [Reference Smith43].

5.2 Axial disks

The following is Definition 2.1 from [Reference Smith43].

The following is Definition 2.3 from [Reference Smith43].

Thinking of $\mathbb {C}P^1$ as $\mathbb {C} \cup \{\infty \}$ , we identify $\mathbb {C}P^1 \simeq S^2$ by stereographic projection from the north pole. Thus, the complex structure on the $2$ -sphere is given by left-handed rotation around the outward normal by an angle of $\pi /2.$

For concreteness, we choose $\triangle $ to be the triangle with vertices

$$\begin{align*}c_1=(0,0,1),\quad c_2=\left(\frac{\sqrt{3}}{2},0,-\frac{1}{2}\right),\quad c_3=\left(-\frac{\sqrt{3}}{2},0,-\frac{1}{2}\right).\end{align*}$$

By abuse of notation, we also denote by $\triangle $ the point $[c_1,c_2,c_3]\in \text {Sym}^3\mathbb {C}P^1.$

Let $\xi _v\in \mathfrak {su}(2)$ be the infinitesimal right-handed rotation about the z axis and let $\xi _f\in \mathfrak {su}(2)$ be the infinitesimal right-handed rotation about the y axis scaled so that for $\xi = \xi _v,\xi _f,$ we have $\{t \in \mathbb {R}: e^{2\pi \xi t}\cdot \triangle = \triangle \} =\mathbb {Z}$ . Thus, $\xi _v,\xi _f,$ are generators of rotations about the vertex $c_1$ and the center of the face of the triangle, respectively, as shown in Figure 1.

Figure 1

The choice of $\xi _v, \xi _f$ for the configuration $\triangle $ .

Example 5.10 (Axial Maslov 2 disk).

Consider the homomorphism

$$\begin{align*}R:\mathbb{R}\rightarrow \text{SU}(2) \end{align*}$$

given by

$$\begin{align*}R(\theta)=e^{\theta\xi_v}. \end{align*}$$

Let $u: (D,\partial D) \to (\mathbb {C}P^3,L_\triangle )$ be given by $u(z)=e^{-i\xi _v\log z}\cdot \triangle $ . See below for more explicit formulas. Then u satisfies $u(e^{i\theta }z)=R(\theta )u(z),$ so it is an axial disk of type $\xi _v$ .

Recall we denote by $c_1,c_2,c_3$ the vertices of the triangle $\triangle $ . The stereographic projection from the north pole

$$\begin{align*}p: S^2\to \mathbb{C}\cup \{\infty\} \end{align*}$$

is given by

$$\begin{align*}p(x,y,z)= \left\{\begin{array}{ll} \frac{x+iy}{1-z}, & z\ne1\\ \infty, & z=1. \end{array} \right. \end{align*}$$

We have

$$\begin{align*}\overline{c}_1:=p(c_1)=\infty, \quad \overline{c}_2:=p(c_2)=\frac{1}{\sqrt{3}}, \quad \overline{c}_3:=p(c_3)=\frac{-1}{\sqrt{3}}. \end{align*}$$

Since $\xi _v$ is the generator of a rotation about the vertex $c_1$ , the flow of $\xi _v$ is given by

$$\begin{align*}\varphi^t_{\xi_v}(w):=e^{\xi_v t}\cdot w=e^{\chi i t}w, \qquad w\in \mathbb{C}\cup \{\infty\} \end{align*}$$

for some $\chi \in \mathbb {R}.$ Since $\xi _v$ is normalized so that

$$\begin{align*}\mathbb{Z} = \{t\in\mathbb{R}: e^{2\pi\xi_v t}\cdot\triangle=\triangle \} = \{t \in \mathbb{R} : \varphi^{2\pi t}_{\xi_v}(\bar c_2) = \bar c_2 \text{ or } \bar c_3\} = \{t\in \mathbb{R} : 2\pi\chi t \in \pi \mathbb{Z}\}, \end{align*}$$

we have $\chi = \frac {1}{2}$ and

$$\begin{align*}\varphi^t_{\xi_v}(w) = e^{\frac{1}{2} i t}w, \qquad w\in \mathbb{C}\cup \{\infty\}. \end{align*}$$

Hence, for $z\in D$ , we get

$$ \begin{align*} u(z)&= e^{-i\xi_v\log z}\cdot \triangle\\ &=\big[ \ [1:\varphi_{\xi_v}^{-i\log z}(\overline{c}_1)], \ [1:\varphi_{\xi_v}^{-i\log z}(\overline{c}_2)], \ [1: \varphi_{\xi_v}^{-i\log z}(\overline{c}_3)]\ \big]\\ &=\big[ \ [0:1], \ [1:\sqrt{\frac{z}{3}}] , \ [1:-\sqrt{\frac{z}{3}}] \ \big]\in \text{Sym}^3(\mathbb{C}P^1). \end{align*} $$

Hence, we can write $u(z)= [0:1:0: -\frac {z}{3}]\in \mathbb {C}P^3.$

We can describe u geometrically as follows. The boundary of u is obtained by the action of $R(\theta )$ on $\triangle $ . Call an isosceles triangle narrow if the congruent sides are longer than the base. A point in the interior of u is described by a narrow isosceles triangle on a great circle where the north pole is the apex. For example, the triangle $c_1d_2d_3$ in Figure 2 represents a point in the interior of u. Note that as $z\rightarrow 0$ , the vertices $c_2$ and $c_3$ move toward the south pole, so u intersects $Y_{\triangle }$ in a single point. So, $[u]\cdot [Y_{\triangle }]\ge 1$ . The proof of Lemma 3.8 in [Reference Smith43] shows that equality holds. Hence, by Lemma 5.5, u is a Maslov 2 disk.

Figure 2

A Maslov 2 disk passing through $\triangle $ .

Example 5.11 (Axial Maslov 4 disk).

Consider the homomorphism

$$\begin{align*}R:\mathbb{R}\rightarrow \text{SU}(2), \end{align*}$$

given by

$$\begin{align*}R(\theta)=e^{\theta\xi_f}. \end{align*}$$

The disk $u:(D,\partial D) \to (\mathbb {C}P^3,L_\triangle )$ given by $u(z)=e^{-i\xi _f\log z}\cdot \triangle $ satisfies $u(e^{i\theta }z)=R(\theta )u(z),$ so it is axial of type $\xi _f$ .

We can describe u geometrically as follows. The boundary of u is obtained by the action of $R(\theta )$ on $\triangle $ . The interior of u is given by all the equilateral triangles on northern lines of latitude if we take $r:=(0,-1,0)$ for the north pole. For example, the triangle $s_1s_2s_3$ in Figure 3 represents an interior point of $u.$ Since u intersects $N_\triangle $ at the unique point $[r,r,r]$ , it follows by Lemma 5.6 that $[u]\cdot [Y_{\triangle }] \geq 2.$ The proof of Lemma 3.8 in [Reference Smith43] shows that equality holds. Hence, by Lemma 5.5, u is a Maslov 4 disk.

Figure 3

A Maslov 4 disk passing through $\triangle $ .

5.3 Classification of holomorphic disks

The following definition is given in Section 4 from [Reference Smith43].

The following is Definition 4.6 from [Reference Smith43]. Recall from Section 3.1 the definition of $\Gamma _\triangle .$

Definition 5.13. A pole germ is the germ (at 0) of a holomorphic map $u,$ from an open neighborhood of $0$ in $\mathbb {C}$ to $\mathbb {C}P^3,$ such that $u^{-1}(Y_{\triangle })$ contains $0$ as an isolated point. More generally, for a Riemann surface $\Sigma $ and a point $a\in \Sigma ,$ one can speak of a pole germ at $a.$ If we do not specify ‘at a’, then we are implicitly working at $0$ in $\mathbb {C}.$ We define an equivalence relation on pole germs at a by $u_1\sim u_2$ if and only if there exists a germ of holomorphic map $A,$ from a neighborhood of a in $\Sigma $ to $\mathrm {SL}(2,\mathbb {C}),$ such that $u_2=A\cdot u_1.$ The principal part of a pole germ u is its equivalence class $[u]_a$ under this relation.

We say a pole germ u is of type $\xi \in \mathfrak {su}(2)$ and order $k\in \mathbb {Z}_{\ge 1}$ if its principal part is

$$\begin{align*}[z\mapsto e^{-ik\xi\log z}\cdot \triangle]_0,\end{align*}$$

and $\xi $ is scaled so that $\{t\in \mathbb {R}: e^{2\pi \xi t}\in \Gamma _\triangle \}=\mathbb {Z}$ .

The following is Lemma 4.10 from [Reference Smith43].

The following is Corollary 4.13 from [Reference Smith43].

Lemma 5.16. All Maslov index $2$ holomorphic disks $u: (D,\partial D)\to (\mathbb {C}P^3,L_\triangle )$ are, up to reparametrization, of the form

$$\begin{align*}u(z)=A\cdot e^{-i\xi_v \log z}\cdot \triangle \end{align*}$$

for $A\in \text {SU}(2)$ . In particular, they are all axial.

The following is Corollary 4.14 from [Reference Smith43].

Recall the notation for moduli spaces of holomorphic disks from Section 4.1. The following is Corollary 4.21 from [Reference Smith43].

Lemma 5.18. Let $u : (D,\partial D) \to (\mathbb {C}P^3,L_\triangle )$ be an axial disk of type $\xi _f$ . Then for any $w \in \operatorname {\mathrm {int}} D$ , the evaluation map

$$\begin{align*}evi_1: \mathcal{M}_{0,1}(2) \to \mathbb{C}P^3 \end{align*}$$

is a submersion at $[u;w]$ .

5.4 Riemann-Hilbert pairs

Let $(E,F)$ be a Riemann-Hilbert pair. Since D is contractible, we identify E with the trivial bundle $\underline {\mathbb {C}}^n$ , and thus, the pair is determined by the subbundle F. A totally real subspace of $\mathbb {C}^n$ is described by $A\cdot \mathbb {R}^n$ where $A\in GL(n,\mathbb {C})$ . Moreover, $A' \in GL(n,\mathbb {C})$ gives the same totally real subspace if and only if $A^{-1}A'\in GL(n,\mathbb {R})$ . The fibers of a totally real subbundle $F \subset \underline {\mathbb {C}}^n$ can be written as $F_z=A(z)\cdot \mathbb {R}^n$ where $A(z)\in GL(n,\mathbb {C})$ , so such a family of matrices $\{A(z)\}_{z\in \partial D}$ determines a Riemann-Hilbert pair.

Let $F^\kappa \subset \underline {\mathbb {C}}$ denote the totally real subbundle given by $F^\kappa _z= z^{\kappa /2}\mathbb {R}$ and let $F^{\kappa _1,\kappa _2}\subset \underline {\mathbb {C}}^2$ denote the totally real subbundle given by $F^{\kappa _1,\kappa _2}_z= z^{\kappa _1/2}\mathbb {R}\oplus z^{\kappa _2/2}\mathbb {R}.$ The following is Theorem 1 from [Reference Oh39].

Theorem 5.23. Any Riemann-Hilbert pair $(E,F)$ splits as a direct sum of Riemann-Hilbert pairs

$$\begin{align*}(E,F)\simeq \oplus_{i=1}^n(\underline{\mathbb{C}},F^{\kappa_i}),\end{align*}$$

where $\kappa _i\in \mathbb {Z}$ .

If the boundary real subbundle F is orientable, which is equivalent to having even total index, by Theorem 5.23, we can write

$$\begin{align*}(E,F)\simeq \oplus_{i=1}^m(\underline{\mathbb{C}},F^{\kappa_i})\oplus_{i=1}^l(\underline{\mathbb{C}}^2,F^{{\kappa_1}_i,{\kappa_2}_i}),\end{align*}$$

where $\kappa _i$ and ${\kappa _1}_i+{\kappa _2}_i$ are even numbers. Hence, each summand is orientable.

For a Riemann-Hilbert pair $(E,F),$ let

$$\begin{align*}\bar{\partial}_{(E,F)}:\Gamma((D,\partial D),(E,F))\rightarrow \Gamma(D, \Omega^{0,1}(E)) \end{align*}$$

denote the Cauchy-Riemann operator determined by the holomorphic structure on E. Let $\det \bar {\partial }_{(E,F)}$ denote the Fredholm determinant. Abbreviate

$$\begin{align*}\bar{\partial}_{\kappa}:=\bar{\partial}_{(\underline{\mathbb{C}},F^{\kappa})}, \quad \bar{\partial}_{\kappa_1,\kappa_2}:=\bar{\partial}_{(\underline{\mathbb{C}}^2,F^{\kappa_1,\kappa_2})}.\end{align*}$$

The following is a special case of Proposition 2.8 from [Reference Solomon45].

Theorem 5.1. Let $(E,F)$ be a spin Riemann-Hilbert pair. Then $\det \bar {\partial }_{(E,F)}$ admits a canonical orientation. If $\phi :(E,F)\rightarrow (E',F')$ is an isomorphism of spin Riemann-Hilbert pairs, then the induced isomorphism

$$\begin{align*}\phi:\det\bar{\partial}_{(E,F)}\rightarrow \det\bar{\partial}^{\prime}_{(E',F')}\end{align*}$$

preserves the canonical orientation. Furthermore, the canonical orientation varies continuously in a family of Cauchy-Riemann operators.

The following is a consequence of the proof of a special case of Lemma 8.1 from [Reference Solomon45].

Lemma 5.24. Suppose

$$\begin{align*}0\rightarrow V' \rightarrow V\rightarrow V"\rightarrow 0 \end{align*}$$

is a short exact sequence of real vector bundles over a base $B.$ Then an orientation on any two of $V, V' , V"$ naturally induces an orientation on the third. A spin structure on any two of $V, V' , V"$ naturally induces a spin structure on the third.

Suppose now that $V',V"$ are oriented and spin and equip $V = V' \oplus V"$ with the induced orientation and spin structure. If $\xi ^{\prime }_1,\ldots ,\xi ^{\prime }_{k'}$ (resp. $\xi ^{\prime \prime }_1,\ldots ,\xi ^{\prime \prime }_{k"})$ is an oriented frame of $V'$ (resp. $V"$ ) that lifts to the associated spin bundle, then $\xi ^{\prime }_1 \oplus 0,\ldots ,\xi ^{\prime }_{k'} \oplus 0, 0 \oplus \xi ^{\prime \prime }_1,\ldots ,0\oplus \xi ^{\prime \prime }_{k"}$ is an oriented frame of $V' \oplus V"$ that lifts to the associated spin bundle.

In light of Lemma 5.24, the direct sum of spin Riemann-Hilbert pairs is naturally a spin Riemann-Hilbert pair. The following can be deduced from the proof of Proposition 8.4 from [Reference Solomon45].

Lemma 5.25. Let $(E,F)$ be a spin Riemann-Hilbert pair. If $(E,F)$ splits as a direct sum of spin Riemann-Hilbert pairs $(E',F')\oplus (E",F")$ , then the isomorphism

$$\begin{align*}\det\bar{\partial}_{(E,F)}\simeq\det\bar{\partial}_{(E',F')}\otimes \det\bar{\partial}_{(E",F")} \end{align*}$$

is orientation-preserving.

The proof of the following lemma is given as a part of the proof of Theorem C.4.1 in [Reference McDuff and Salamon38].

Lemma 5.27. If $\kappa \ge -1$ , then $\ker \bar {\partial }_{\kappa }$ can be written explicitly as

$$\begin{align*}\ker\bar{\partial}_{\kappa}=\{\sum_{j=0}^{\kappa}a_jz^j| \ a_j\in\mathbb{C} \text{ and } a_j=\bar{a}_{\kappa-j}\} \end{align*}$$

and $\operatorname {\mathrm {Coker}}(\bar {\partial }_{\kappa }) = 0.$ In particular, $\dim \ker \bar {\partial }_{\kappa } = \kappa +1.$

Below, we often use the fact that if $(E,F)$ is a Riemann-Hilbert pair with $\operatorname {\mathrm {Coker}} \bar {\partial }_{(E,F)} = 0,$ then an orientation of $\det \bar {\partial }_{(E,F)}$ is the same as an orientation of $\ker \bar {\partial }_{(E,F)}.$

Lemma 5.28. Evaluation at 1 defines an orientation-preserving isomorphism

$$\begin{align*}f_0:\ker\bar{\partial}_0\rightarrow F^0_1. \end{align*}$$

The following is Lemma A.2.1 from [Reference Smith44].

Lemma 5.29. Let $(E,F)$ be a rank 1 Riemann-Hilbert pair of Maslov index $2$ . Evaluation at $0$ and $1$ defines an isomorphism

$$\begin{align*}f_2:\ker\bar{\partial}_{(E,F)}\rightarrow E_0\oplus F_1. \end{align*}$$

Moreover, for any choice of orientation on $F,$ this isomorphism is orientation-preserving if the codomain is oriented by the complex structure on $E_0$ and the orientation on $F_1$ .

We choose an orientation on $F^2$ such that the frame z is positively oriented. By Remark 5.26, this choice of orientation determines a spin structure on $F^2.$

Lemma 5.30. The orientation on $\ker \bar {\partial }_2$ is determined by the ordered basis

$$\begin{align*}(z^2+1), \quad i(1-z^2), \quad z. \end{align*}$$

Proof. By Lemma 5.27, we have

$$\begin{align*}\ker\bar{\partial}_2=span_{\mathbb{R}}\{(z^2+1), i(1-z^2), z\}.\end{align*}$$

Consider the map

$$\begin{align*}f_2:\ker\bar{\partial}_2\rightarrow \underline{\mathbb{C}}_0\oplus F_1^2\end{align*}$$

from Lemma 5.29. Orient the fiber $\underline {\mathbb {C}}_0$ by its complex structure and the fiber $F^2_1 = \mathbb {R} \subset \underline {\mathbb {C}}_1$ by the basis $1$ . By Lemma 5.29, the map $f_2$ is orientation-preserving. Since

$$\begin{align*}f_2((z^2+1))=(1,2),\quad f_2(i(1-z^2))=(i,0), \quad f_2(z)=(0,1) \end{align*}$$

is an oriented basis, it follows that the basis

$$\begin{align*}(z^2+1), \quad i(1-z^2), \quad z \end{align*}$$

determines the orientation on $\ker \bar {\partial }_2.$

We choose the orientation on $F^{1,1}$ such that the frame

$$\begin{align*}\zeta(z)=\begin{pmatrix} \frac{z+1}{4}\\ \frac{-i(1-z)}{4} \end{pmatrix}, \quad \eta(z)=\begin{pmatrix} \frac{i(1-z)}{4}\\ \frac{z+1}{4} \end{pmatrix}\end{align*}$$

is positively oriented. In addition, we choose the spin structure on $F^{1,1}$ such that this frame can be lifted to the associated double cover of the frame bundle.

Lemma 5.31. Evaluation at zero

$$\begin{align*}f_{1,1}:\ker\bar{\partial}_{1,1}\rightarrow \underline{\mathbb{C}}^2_0\end{align*}$$

defines an isomorphism. If the codomain is oriented by the complex structure on $\underline {\mathbb {C}}^2_0,$ this isomorphism is orientation-preserving.

Proof. By Lemma 5.27, the map $f_{1,1}$ defines an isomorphism. We degenerate $(\underline {\mathbb {C}}^2,F^{1,1})$ to $(\underline {\mathbb {C}}^2,F^{0,2})$ in order to determine the orientation on $\ker \bar {\partial }_{1,1}$ . Consider the family of loops

$$\begin{align*}A_t:\partial D\rightarrow GL(2,\mathbb{C}), \quad t\in[0,1] \end{align*}$$

given by

$$\begin{align*}A_t(z)=\begin{pmatrix} {\frac{1+t^2z}{2}} & \frac{t(z-1)}{2i}\\ \frac{t(1-z)}{2i} & {\frac{t^2+z}{2}} \end{pmatrix}.\end{align*}$$

We claim that the family of boundary conditions $F(t)$ given by $F(t)_z = A_t(z) \cdot \mathbb {R}^2$ is a degeneration from $F^{1,1}$ to $F^{0,2}.$ Indeed,

$$\begin{align*}A_0(z)=\begin{pmatrix} 1 & 0\\ 0 & z \end{pmatrix}\begin{pmatrix} \frac{1}{2} & 0\\ 0 & \frac{1}{2} \end{pmatrix}, \qquad A_1(z)=\begin{pmatrix} z^{\frac{1}{2}} & 0\\ 0 & z^{\frac{1}{2}} \end{pmatrix}\begin{pmatrix} {\frac{z^{-\frac{1}{2}}+z^{\frac{1}{2}}}{2}} & \frac{z^{\frac{1}{2}}-z^{-\frac{1}{2}}}{2i}\\ \frac{z^{-\frac{1}{2}}-z^{\frac{1}{2}}}{2i} & {\frac{z^{-\frac{1}{2}}+z^{\frac{1}{2}}}{2}} \end{pmatrix},\end{align*}$$

where the matrix

$$\begin{align*}\begin{pmatrix} {\frac{z^{-\frac{1}{2}}+z^{\frac{1}{2}}}{2}} & \frac{z^{\frac{1}{2}}-z^{-\frac{1}{2}}}{2i}\\ \frac{z^{-\frac{1}{2}}-z^{\frac{1}{2}}}{2i} & {\frac{z^{-\frac{1}{2}}+z^{\frac{1}{2}}}{2}} \end{pmatrix}\end{align*}$$

is an invertible real matrix for every $z\in \partial D.$ The family of frames given by the columns of $A_t$ gives rise to a continuous family of orientations and spin structures on $F(t)$ . The orientation on $F(1)$ coincides with the orientation on $F^{1,1}.$ The orientation on $F(0)$ coincides with the direct sum orientation $F^{0,2} = F^0 \oplus F^2.$ The spin structure on $F(1)$ coincides with the spin structure on $F^{1,1}$ and the spin structure on $F(0)$ coincides with the direct sum spin structure on $F^{0,2}$ as in Lemma 5.24. Let

$$\begin{align*}R(t):=\begin{pmatrix} 1 & 0\\ 0 & t \end{pmatrix}, \quad B(t,z):=\frac{1}{4}\begin{pmatrix} (3+t^2)+(t^2-1)z & i(1-t^2)+i(t^2-1)z\\ i(\frac{1}{t}-t)+i(t-\frac{1}{t})z & (3t+\frac{1}{t})+(\frac{1}{t}-t)z \end{pmatrix}.\end{align*}$$

Then $B(t,z)$ is an invertible holomorphic matrix on D for every $t\in (0,1]$ , and the matrices satisfy

$$\begin{align*}B(t,z)|_{\partial D}A_1(z)R(t)=A_t(z),\quad t\in(0,1], \ z\in\partial D.\end{align*}$$

Hence, $B(t,z)$ is an isomorphism between $(\underline {\mathbb {C}}^2, F(1))$ and $(\underline {\mathbb {C}}^2, F(t))$ for every $t\in (0,1]$ . Denote by $\bar {\partial }(t)$ the family of the Cauchy-Riemann operators obtained from $(\underline {\mathbb {C}}^2, F(t))$ for $t\in (0,1]$ . Note that $\ker \bar {\partial }(1)=\ker \bar {\partial }_{1,1}$ . By Lemma 5.27, the vectors

$$\begin{align*}\begin{pmatrix} 1+z\\ 0 \end{pmatrix},\quad \begin{pmatrix} i(1-z)\\ 0 \end{pmatrix},\quad \begin{pmatrix} 0\\ 1+z \end{pmatrix},\quad \begin{pmatrix} 0\\ i(1-z) \end{pmatrix} \end{align*}$$

form a basis for $\ker \bar {\partial }_{1,1}.$ So, the following vectors form a basis for $\ker \bar {\partial }(t)$ :

$$\begin{align*}v_{1,t}(z):=B(t,z)\begin{pmatrix} 1+z\\ 0 \end{pmatrix}, \quad v_{2,t}(z):=B(t,z)\begin{pmatrix} i(1-z)\\ 0 \end{pmatrix}, \end{align*}$$

$$\begin{align*}v_{3,t}(z):=B(t,z)\begin{pmatrix} 0\\ 1+z \end{pmatrix}, \quad v_{4,t}(z):=B(t,z)\begin{pmatrix} 0\\ i(1-z) \end{pmatrix}.\end{align*}$$

Let

$$ \begin{align*} w_{1,t}(z)&:=v_{1,t}(z)-v_{4,t}(z)=\begin{pmatrix} zt^2+1\\ izt-it \end{pmatrix}, \\ w_{2,t}(z)&:=v_{1,t}(z)+v_{4,t}(z)=\frac{1}{2}\begin{pmatrix} z^2(t^2-1)+2z+t^2+1\\ iz^2(t-\frac{1}{t})+i(t+\frac{1}{t})-2izt \end{pmatrix},\\ w_{3,t}(z)&:=v_{2,t}(z)-v_{3,t}(z)=\frac{1}{2}\begin{pmatrix} iz^2(1-t^2)+(1+t^2)i-2iz\\ z^2(t-\frac{1}{t})-t-\frac{1}{t}-2zt \end{pmatrix}, \\ w_{4,t}(z)&:=v_{2,t}(z)+v_{3,t}(z)=\begin{pmatrix} -iz+i\\ t+\frac{z}{t} \end{pmatrix}. \end{align*} $$

For $i=2,3,4,$ define $u_{i,t}(z):=tw_{i,t}(z)$ when $t>0,$ and $u_{i,0}(z):=\lim _{t\rightarrow 0}tw_{i,t}(z).$ In addition, define $u_{1,t}(z):=w_{1,t}(z).$ Note that the bases

$$\begin{align*}\{v_{1,t}, v_{2,t}, v_{3,t}, v_{4,t}\}, \qquad \{w_{1,t}, w_{2,t}, w_{3,t}, w_{4,t}\}, \qquad \{u_{1,t}, u_{2,t}, u_{3,t}, u_{4,t}\} \end{align*}$$

determine the same orientation on $\ker \bar {\partial }(t)$ for $t\in (0,1]$ . We have

$$ \begin{align*} u_{1,0}(z)&= \begin{pmatrix} 1\\ 0 \end{pmatrix},\\ u_{2,0}(z)&= \frac{1}{2}\begin{pmatrix} 0\\ i(1-z^2) \end{pmatrix},\\ u_{3,0}(z)&= -\frac{1}{2}\begin{pmatrix} 0\\ z^2+1 \end{pmatrix},\\ u_{4,0}(z)&= \begin{pmatrix} 0\\ z \end{pmatrix}. \end{align*} $$

By Lemma 5.30, the basis $\{-u_{3,0},u_{2,0},u_{4,0}\}$ determines the orientation on $\ker \bar {\partial }_2$ , and by Lemma 5.28, the vector $u_{1,0}(z)$ determines the orientation on $\ker \bar {\partial }_0.$ Hence, by Lemma 5.25, the basis

$$\begin{align*}\{ u_{1,0},u_{2,0},u_{3,0},u_{4,0}\} \end{align*}$$

determines the orientation on $\ker \bar {\partial }(0)$ . Continuity implies that $\{u_{1,t}, u_{2,t}, u_{3,t}, u_{4,t}\}$ is positively oriented for every $t\in [0,1],$ and so is $\{v_{1,t}, v_{2,t}, v_{3,t}, v_{4,t}\}$ for $t\in (0,1]$ . Hence, the basis

$$\begin{align*}v_{1,1}= \begin{pmatrix} 1+z\\ 0 \end{pmatrix}, \quad v_{2,1}= \begin{pmatrix} i(1-z)\\ 0 \end{pmatrix}, \quad v_{3,1}= \begin{pmatrix} 0\\ 1+z \end{pmatrix}, \quad v_{4,1}= \begin{pmatrix} 0\\ i(1-z) \end{pmatrix}\end{align*}$$

is positively oriented. Therefore, $f_{1,1}$ is orientation-preserving.