2.1 Lagrangian fibration

We begin with the Gross’s construction in [Reference Gross39]. Define the divisors $D_j=\{z_j=0\}$ ( $1\le \ell \le n$ ) and $\mathscr D=\{z_1z_2\cdots z_n=1\}$ in $\mathbb C^n$ . We set $X=\mathbb C^n\setminus \mathscr D$ . Consider the $T^{n-1}$ -action given by

(10) $$ \begin{align} t \cdot (z_1,\dots, z_n) \mapsto (z_1,\dots, z_{k-1}, e^{it} z_k, z_{k+1},\dots, z_{n-1}, e^{-it} z_n) \qquad 1\le k<n \end{align} $$

for $t\in S^1\cong \mathbb R/2\pi \mathbb Z$ . Take the standard symplectic form $\omega =d\lambda $ on $\mathbb C^n$ . Let $ \bar \mu =(h_1,\dots , h_{n-1}): \mathbb C^n\to \mathbb R^{n-1} $ be a moment map for the $T^{n-1}$ -action in (10). One can check the vector fields

$$\begin{align*}\textstyle \mathfrak X_k= i \big(\bar z_k\frac{\partial}{\partial\bar z_k } - z_k\frac{\partial}{\partial z_k}\big) - i \big(\bar z_n\frac{\partial}{\partial\bar z_n } - z_n\frac{\partial}{\partial z_n}\big) \end{align*}$$

satisfy $ \iota (\mathfrak X_k)\omega = d h_k $ for the Hamiltonian functions

$$\begin{align*}h_k(z)=\tfrac{1}{2} \big(|z_k|^2- |z_n|^2 \big) \qquad 1\le k<n. \end{align*}$$

Define $h_n(z)=\log |z_1\cdots z_n-1|$ and $B=\mathbb R^n$ . The Gross Lagrangian fibration (with respect to the holomorphic n-form $\Omega =(z_1\cdots z_n-1)^{-1} dz_1\wedge \cdots \wedge dz_n$ , cf. [Reference Gross39]) refers to the following:

$$\begin{align*}\pi\equiv (\bar\mu, h_n): X \to B, \quad (z_1,\dots, z_n)\mapsto \big( h_1,\dots, h_{n-1} , h_n \big). \end{align*}$$

Note that the $\pi $ maps X onto B. Note also that the $\pi $ -fibers are preserved by the complex conjugation $z_i\mapsto \bar z_i$ . The $\pi $ -fiber over $q\in B$ is denoted by $L_q:=\pi ^{-1}(q)$ , and we write

$$\begin{align*}q=(\bar q, q_n)=(q_1,\dots, q_{n-1},q_n) \end{align*}$$

for the standard Euclidean coordinates in $B \equiv \mathbb R^n$ . Let $ P_{ij}=\{z_i=z_j=0\}=D_i\cap D_j $ and $\Delta _{ij}=\pi (P_{ij})$ . The discriminant locus of $\pi $ is precisely $\Delta =\bigcup \Delta _{ij}$ , and the smooth locus of $\pi $ is $B_0:=B\setminus \Delta $ . We denote the restriction of $\pi $ over $B_0$ by

$$\begin{align*}\pi_0:=\pi|_{B_0} :X_0\to B_0 \end{align*}$$

where $X_0:=\pi _0^{-1}(B_0)\subset X$ . Moreover, we actually have $\Delta =\Pi \times \{0\}$ , where $\Pi $ is the tropical hyperplane that consists of those $\bar q \in \mathbb R^{n-1}$ such that $\min \{0,q_1,\dots , q_{n-1}\}$ is attained twice. The tropical hyperplane $\Pi $ separates the subset $H: =(\mathbb R^{n-1}\setminus \Pi )\times \{0\}$ of the base B into n different connected components

(11) $$ \begin{align} H_k= \{ \bar q \in \mathbb R^{n-1}\setminus \Pi \mid \min (0,q_1,\dots, q_{n-1})=q_k\} \end{align} $$

for $1\le k<n$ and

$$\begin{align*}H_n=\{ \bar q \in\mathbb R^{n-1}\setminus \Pi \mid \min(0,q_1,\dots, q_{n-1}) =0\}. \end{align*}$$

For example, when $n{\kern-1pt}={\kern-1pt}3$ , we have $H_1{\kern-1.2pt}={\kern-1.2pt}\{q_1{\kern-1.2pt}<{\kern-1.2pt}0, q_1{\kern-1.2pt}<{\kern-1.2pt}q_2\}$ , $H_2{\kern-1.2pt}={\kern-1.2pt}\{q_2{\kern-1.2pt}<{\kern-1.2pt}0, q_1{\kern-1.2pt}>{\kern-1.2pt}q_2\}$ and $H_3{\kern-1.2pt}={\kern-1.2pt}\{q_1{\kern-1.2pt}>{\kern-1.2pt}0, q_2{\kern-1.2pt}>{\kern-1.2pt}0\}$ (see the left side of Figure 3). When $n=2$ , we have $H_1=(-\infty , 0)$ and $H_2=(0,+\infty )$ . Notice that $\bar H_i\cap \bar H_j=\Delta _{ij}=\pi (D_i\cap D_j)$ , and $H_j\subset \pi (D_j)$ for $1\le j\le n$ .

We usually call the subset H in $B_0$ the wall because a torus fiber $L_q$ over $q=(\bar q, q_n)\in B_0$ bounds a nontrivial Maslov index zero holomorphic disk if and only if $q_n=0$ (i.e., $q\in H$ ) (see, for example [Reference Auroux6, Example 3.3.1] or [Reference Auroux5, §5]). Next, we define

$$\begin{align*}B_+=\{q=(q_1,\dots, q_n )\mid q_n>0\}=\mathbb R^{n-1}\times (0,+\infty) \end{align*}$$

$$\begin{align*}B_-=\{q=(q_1, \dots, q_n)\mid q_n<0\} =\mathbb R^{n-1}\times (-\infty,0). \end{align*}$$

Then, $B_0=B_+ \sqcup B_-\sqcup \bigsqcup _{j=1}^n H_j$ . We call the $B_+$ (resp. $B_-$ ) the chamber of Clifford tori (resp. Chekanov tori). We say the torus fiber $L_q$ is of Clifford type if $q_n>0$ and it is of Chekanov type if $q_n<0$ , which is introduced in [Reference Chekanov17, Reference Eliashberg and Polterovich27] and many others.

The Arnold-Liouville’s theorem implies the existence of the action-angle coordinates near any Lagrangian torus fiber. Let $(M,\omega )$ be a symplectic manifold of dimension $2n$ . Let $M_U\subset M$ be an open subset. The following two theorems are standard; see [Reference Duistermaat24] and [Reference Evans28].

We call $I=(I_1,\dots , I_n)$ and $\alpha =(\alpha _1,\dots ,\alpha _n)$ the action-angle coordinates. Two sets of them differ by an integral affine transformation. If $\pi :M_U\to U$ is a smooth Lagrangian torus fibration in $M_U\subset M$ over a contractible domain U, then a set of the action coordinates is decided by a choice of the $\mathbb Z$ -basis of $H_1(\pi ^{-1}(q); \mathbb Z)$ for some $q\in U$ . In reality, we have the following.

We will call such a $\chi $ an integral affine chart. If we pick a different $\lambda '$ with $d\lambda '=d\lambda $ , then by Stokes’ theorem, $\int _{\sigma _i}(\lambda '-\lambda )$ is constant, and the action coordinates just differ by constants. The $\psi _i$ ’s in Theorem 2.2 are also called the flux maps. The formula for a set of action coordinates can be complicated in general (cf. [Reference Ngoc60], see also [Reference Evans28, Theorem 6.7]). A key idea used in this paper is that the action coordinates can be often described by linear combinations of symplectic areas of disks in some (possibly larger) ambient space; see §2.3 below. The advantage is that one can intuitively see an integral affine transformation in terms of disk bubbling.

2.2 Topological disks

Now, we go back to our situation. To connect with the Floer theory, we are interested in whether the loops in $\pi _1(L_q)$ can be realized as (linear combinations of) the boundaries of disks in $X=\mathbb C^n\setminus \mathscr D$ or in other reasonable larger ambient symplectic manifold $\overline X$ (e.g., $\mathbb C^n$ or $\mathbb {CP}^n$ ). Beware that at this moment, we just focus on topological disks rather than holomorphic disks. We consider the following local systems over $B_0$ :

(12) $$ \begin{align} \mathscr R_1:=R^1\pi_*(\mathbb Z)\equiv \bigcup_{q\in B_0} \pi_1(L_q), \qquad \mathscr R_2 :=\mathscr R_2(\overline X):=\bigcup_{q\in B_0} \pi_2( \overline X,L_q). \end{align} $$

We must understand the monodromy behavior of the above local systems across multiple components of the walls (cf. Figure 3). In contrast, most of existing literature only studies these disks across a single component. Although the latter is sufficient for mirror space identifications (cf. [Reference Abouzaid, Auroux and Katzarkov1, Reference Chan, Lau and Leung16] and §1.2), we must understand the former monodromy data for mirror fibration realizations. Note also that the monodromy for $\mathscr R_2$ contains more information than the monodromy for $\mathscr R_1$ or the integral affine structure. This justifies why we need to provide many additional details here, although it may be not so difficult.

The local system $\mathscr R_2=\mathscr R_2(\overline X)$ generally depends on the ambient space $\overline X$ . But, we will often make this point implicit. Taking the boundaries gives rise to a morphism $\partial : \mathscr R_2\to \mathscr R_1$ . A class in $\pi _1(L_q)$ or $\pi _2(\overline X,L_q)$ determines a local section of $\mathscr R_1$ or $\mathscr R_2$ over some contractible domain that contains q. Recall that we have a natural Hamiltonian $T^{n-1}$ -action that preserves the Lagrangian torus fibers. Then, for $q\in B_0$ , we use $\sigma _k=\sigma _k(q)$ ( $1\le k\le n-1$ ) to denote the class of the orbit of the $S^1$ -action in (10). They can be regarded as the global sections of $\mathscr R_1$ .

To avoid the monodromy ambiguity, we will work with a covering of the base manifold $B_0$ by the contractible domains, such as $B_+\cup H_\ell \cup B_-$ for $1\le \ell \le n$ . Let $\mathscr N_\ell $ denote a sufficiently small neighborhood of $H_\ell $ in the smooth locus $B_0$ . The following two contractible domains can cover $B_0$ :

$$\begin{align*}\textstyle B^{\prime}_\pm := B_\pm \cup \bigsqcup_{\ell=1}^n \mathscr N_\ell. \end{align*}$$

Whenever $q\in B_+$ , the Lagrangian torus fiber $L_q$ can deform into a product torus by a Hamiltonian isotopy inside $(\mathbb C^*)^n$ . So, $\pi _2(\mathbb C^n, L_q)\cong \pi _2(\mathbb C^n, (\mathbb C^*)^n )\cong \mathbb Z^n$ has an obvious basis $\beta _1,\dots ,\beta _n$ such that $\beta _i\cdot D_j=\delta _{ij}$ . We can represent $\partial \beta _j$ by a loop in the form $t\mapsto (z_1^0,\dots , z_j^0 e^{it},\dots , z_n^0)$ in the product torus. But abusing the notations, we still write $\partial \beta _j$ (resp. $\beta _j$ ) for the induced local sections of $\mathscr R_1$ (resp. $\mathscr R_2$ ) over $B^{\prime }_+$ . To emphasize the base point, we may write $\partial \beta _j(q)$ and $\beta _j(q)$ . By (10), we have

(13) $$ \begin{align} \sigma_k|_{B^{\prime}_+}= \partial\beta_k -\partial\beta_n \qquad 1\le k<n. \end{align} $$

However, there is a preferred local section of $\mathscr R_2$ over $B_-$ , denoted by $ \hat \beta : B_- \to \mathscr R_2 $ , such that any Maslov index 2 holomorphic disk u bounding $L_{q}$ for $q\in B_-$ must have $[u]=\hat \beta (q)$ ; see, for example, [Reference Eliashberg and Polterovich27, Proposition 4.2. C], [Reference Chan, Lau and Leung16, Lemma 4.31] or [Reference Yuan63]. It can be represented by a section of $z_1\cdots z_n$ over a disk in $\mathbb C$ . Note that $\hat \beta \cdot D_i=0$ for all i. Abusing the notation, we still denote by $\hat \beta $ the extension section of $\mathscr R_2$ over $B^{\prime }_-$ . By studying the intersection numbers with the various divisors, it is direct to show the following purely topological result (see, for example, [Reference Yuan63, Lemma B.7]):

(14) $$ \begin{align} \beta_\ell|_{\mathscr N_\ell}=\hat\beta|_{\mathscr N_\ell}. \end{align} $$

2.3 Action coordinates

Recall that $ q=(q_1,\dots , q_n)=:(\bar q, q_n) $ are the standard coordinates on B, but we aim to find the different action coordinates. On the one hand, a natural trivialization of $\mathscr R_1|_{B^{\prime }_-}$ is given by the sections

$$\begin{align*}\{\sigma_k|_{B^{\prime}_-}: 1\le k<n \}\cup \{\partial\hat\beta\}. \end{align*}$$

By Theorem 2.2, they give rise to a set of the action coordinates in $B^{\prime }_-$ by the flux maps. The first $n-1$ fluxes are given by $ \frac {1}{2\pi } \int _{\sigma _k} \lambda = \frac {1}{2} ( |z_k|^2- |z_n|^2 ) = h_k(z)=q_k $ . The last flux for $\partial \hat \beta =\partial \hat \beta (q)$ is hard to make explicit, but it can have a geometric meaning by the Stokes’ formula:

$$\begin{align*}\textstyle \psi_-(q):=\frac{1}{2\pi} \int_{\partial\hat\beta} \lambda =\frac{1}{2\pi} \int_{\hat\beta} d\lambda= E(\hat\beta), \end{align*}$$

which is regarded as a map $\psi _-: B^{\prime }_-\to \mathbb R$ . To sum up, we get an integral affine coordinate chart

(15) $$ \begin{align} \chi_- : B^{\prime}_- \to \mathbb R^n ,\qquad q\mapsto (q_1,\dots, q_{n-1}, \psi_-(q) ) = (\bar q, \psi_-(q)). \end{align} $$

On the other hand, a natural trivialization of $\mathscr R_1|_{B^{\prime }_+}$ is given by

$$\begin{align*}\{\sigma_k|_{B^{\prime}_+}: 1\le k<n\} \cup\{\partial\beta_n\}. \end{align*}$$

Recall that $\sigma _k|_{B^{\prime }_+}=\partial \beta _k-\partial \beta _n$ for $1\le k<n$ . The first $n-1$ fluxes are as above, and moreover,

(16) $$ \begin{align} \textstyle E(\beta_k) - E(\beta_n) = \frac{1}{2\pi} \int_{\beta_k-\beta_n} d\lambda =\frac{1}{2\pi} \int_{\partial\beta_k-\partial\beta_n} \lambda= \frac{1}{2\pi } \int_{\sigma_k} \lambda = q_k. \end{align} $$

Define $ \psi _+(q)= E(\beta _n): B^{\prime }_+\to \mathbb R $ . By Theorem 2.2, the associated integral affine coordinate chart is

(17) $$ \begin{align} \chi_+ : B^{\prime}_+ \to \mathbb R^n, \qquad q \mapsto \big(q_1, \dots, q_{n-1}, \psi_+(q) \big) = (\bar q, \psi_+(q)). \end{align} $$

We can use (14) and (16) to conclude that $\psi _+(q)=\psi _-(q)$ on $\mathscr N_n$ and $\psi _+(q)+q_\ell =\psi _-(q)$ on $\mathscr N_\ell $ for $\ell \neq n$ , respectively. By (11), this means that for $ \textstyle q\in B^{\prime }_+\cap B^{\prime }_-\equiv \bigsqcup _\ell \mathscr N_\ell $ ,

(18) $$ \begin{align} \psi_-(q) = \psi_+(q)+\min\{0,q_1,\dots, q_{n-1}\}. \end{align} $$

The two charts $\chi _\pm $ give rise to an integral affine structure on $B_0$ which is different from the standard Euclidean one in $B=\mathbb R^n$ . In particular, the former integral affine structure cannot be extended to B. Define $ \unicode{x3c7} _\ell :=\chi _-\circ \chi _+^{-1}: \ \chi _+(\mathscr N_\ell ) \to \chi _-(\mathscr N_\ell ) $ for each $\ell $ , and we can check

(19)

2.4 The embedding j

We aim to embed the integral affine manifold with singularities B into a higher-dimensional Euclidean space $\mathbb R^{n+1}$ . This is actually inspired by the work of Kontsevich-Soibelman (cf. [Reference Kontsevich and Soibelman50, p27 & p45]). To some degree, this helps us to visualize the integral affine structure.

2.4.1 Symplectic area

Thanks to (18), we have a well-defined continuous function on $B_0$ given by

(20) $$ \begin{align} \psi(q)=\psi(\bar q, q_n) = \begin{cases} \psi_+(q)+\min\{0,q_1,\dots, q_{n-1}\}, & q\in B^{\prime}_+ \\ \psi_-(q) , & q\in B^{\prime}_-. \end{cases} \end{align} $$

If $q\in B^{\prime }_+$ , then $\psi (q)=\min \{E(\beta _1),\dots , E(\beta _n)\}$ ; if $q\in B^{\prime }_-$ , then $\psi (q)=E(\hat \beta )$ . The existence of the Lagrangian fibration with singularities over the whole $B=\mathbb R^n$ tells that the $\psi $ can extend continuously from $B_0$ to B, still denoted by $\psi : B \to \mathbb R $ (cf. the yellow disks in Figure 2). The value of $\psi (q)$ can be represented by the symplectic area of a topological disk

$$\begin{align*}u=u(q): (\mathbb D, \partial\mathbb D) \to (\mathbb C^n, L_q) \end{align*}$$

by $\psi (q)= \frac {1}{2\pi } \int u^*\omega $ (cf. Convention 2.3). It is not necessarily a holomorphic disk at this moment, and the symplectic area is purely topological.

Lemma 2.4. The function $q_n\mapsto \psi (\bar q, q_n)$ is an increasing diffeomorphism from $\mathbb R$ to $(0,\infty )$ .

Proof. We first show the monotonicity. We may assume that $\bar q$ is a regular value of $\bar \mu $ (otherwise, since we only compare the symplectic areas, we may take a sequence of regular values approaching it). We study the symplectic reduction space $\Sigma :=\bar \mu ^{-1}(\bar q) / T^{n-1}$ . It has real dimension 2 and is endowed with the reduction form $\omega _{red}$ . If we write and $p: \bar \mu ^{-1}(\bar q) \to \Sigma $ for the inclusion and quotient maps, then $i^*\omega =p^*\omega _{red}$ . The Lefschetz fibration map $w=z_1\cdots z_n$ is invariant under the $T^{n-1}$ action and naturally induces a diffeomorphism $v: \Sigma \to \mathbb C$ with $w\circ i=v\circ p$ . Note that $L_q$ is contained in $\bar \mu ^{-1}(\bar q)$ , and the topological disk u is a section of w. Up to a homotopy in $\mathbb C^n$ (relative to $L_q$ ), we may require u has its image contained in $\bar \mu ^{-1}(\bar q)$ as well. Recall $h_n=\log | w-1|$ , so $|w-1|=e^{q_n}$ on $L_q$ . Now, we have the following maps of topological space-pairs:

$$\begin{align*}(\mathbb D, \partial\mathbb D) \xrightarrow{u} (\bar\mu^{-1}(\bar q), L_q) \xrightarrow{p} (\Sigma, p(L_q) ) \xrightarrow[\cong]{v} (\mathbb C, \partial D(q_n)), \end{align*}$$

where $p(L_q)$ is identified via v with the boundary circle of the disk $D(q_n)=\{\zeta \in \mathbb C\mid |\zeta -1|\le e^{q_n}\}$ . Let $\check \omega $ is a symplectic form on $\mathbb C$ determined by $v^*\check \omega =\omega _{red}$ . Then, $\int u^*\omega = \int u^* p^*\omega _{red} = \int u^* p^* v^* \check \omega $ . In other words, $2\pi \psi (q)=\int u^*\omega $ can be viewed as the symplectic area of the disk $D(q_n)$ in $(\mathbb C, \check \omega )$ . No matter how $\check \omega $ looks like, we have the subset inclusion $D(q^1_n)\subset D(q^2_n)$ if $q^1_n<q^2_n$ , so the $\check \omega $ -area of $D(q_n)$ is increasing in $q_n$ . Namely, this says $\psi (\bar q,q_n)$ is increasing in $q_n$ . Finally, notice that we also have $\lim _{q_n\to \infty } \psi (\bar q, q_n)=\infty $ and $\lim _{q_n\to -\infty }\psi (\bar q, q_n)=0$ .

Recall that $q=(\bar q, q_n)$ . We write

(21) $$ \begin{align} \psi_0(\bar q)=\psi(\bar q,0), \end{align} $$

and we define a continuous embedding

(22) $$ \begin{align} j: B \to \mathbb R^{n+1} \qquad q\mapsto (\theta_0(q), \ \theta_1(q), \ \bar q ), \end{align} $$

where

(23) $$ \begin{align} \begin{aligned} \theta_0(q_1,\dots, q_n) &:= \min\{ -\psi(q) , -\psi_0( \bar q ) \} + \min \{0, \bar q \} \\ \theta_1(q_1,\dots, q_n) &:= \min\{\ \ \ \psi(q) , \ \ \ \psi_0( \bar q )\} \end{aligned} \end{align} $$

are continuous maps from $\mathbb R^n$ to $\mathbb R$ . Using Lemma 2.4 implies that j is injective. The manifold structure on B induces a one on the image $j(B)$ . Later, we will see the motivation behind j in Section 3.

2.4.2 Description of the image of j.

We think of $B\equiv \mathbb R^n$ as the union of the $\bar q$ -slices for all $\bar q\in \mathbb R^{n-1}$ . Notice that the map j is ‘slice-preserving’ in the sense that the following diagram commutes:

where the left vertical arrow is $(\bar q, q_n)\mapsto \bar q$ and the right one is $(u_0,u_1,\bar q)\mapsto \bar q$ . Thus, we just need to understand the restriction of j on a fixed slice $\bar q\times \mathbb R$ composed with the projection $\mathbb R^{n+1}\cong \mathbb R^2\times \mathbb R^{n-1}_{\bar q} \to \mathbb R^2$ . After taking $\psi (\bar q, \cdot ) :\mathbb R\cong (0,+\infty )$ in Lemma 2.4, this amounts to study the induced map

(24) $$ \begin{align} r_{\bar q}: (0,+\infty) \to \mathbb R^2 \end{align} $$

defined by

$$\begin{align*}(0,+\infty)\ni c\mapsto (\min\{0,\bar q\}+ \min\{-c, -\psi_0(\bar q)\} , \min\{c, \psi_0(\bar q)\}). \end{align*}$$

Here, c represents $\psi (\bar q, q_n)$ . The image of $r_{\bar q}$ is a (half) broken line, denoted by $R_{\bar q}$ , in $\mathbb R^2$ . Define

(25) $$ \begin{align} A=A(\bar q)= (a_0(\bar q), a_1(\bar q) ) := \big( \min\{0,\bar q\}-\psi_0(\bar q) \ , \ \psi_0(\bar q) \big) \end{align} $$

to be the corner point of $R_{\bar q}$ parameterized by $\bar q\in \mathbb R^{n-1}$ . Note that $(\bar q,0)\in \Delta $ if and only if $\bar q\in \Pi $ . Note also that $a_1(\bar q)=\psi _0(\bar q)>0$ , so this broken line $R_{\bar q}$ always contains the corner point $A(\bar q)$ .

Remark 2.5. Intuitively, we may imagine drawing $R_{\bar q}$ in $\mathbb R^2$ with a pen as follows: as c decreases from $+\infty $ to $0+$ , we draw from $(-\infty , a_1(\bar q) )$ horizontally to the corner point $A(\bar q)$ (see Figure 4). Then, we turn the pen and continue drawing vertically downwards until $(a_0(\bar q), 0)$ , as $c\to 0+$ .

Figure 4 The corner point $A=A(\bar q)$ in the broken line $R_{\bar q}$ .

The image $j(B)$ can be identified with the graph in $\mathbb R^2\times \mathbb R^{n-1}$ of a family of broken lines $R_{\bar q}$ in $\mathbb R^2$ parameterized by $\bar q \in \mathbb R^{n-1}$ (see Figure 1 in the introduction). The image $j(\Delta )$ of the singular locus $\Delta \equiv \Pi \times \{0\}$ consists of the corner points in $R_{\bar q}$ for those $\bar q$ in the tropical hyperplane $\Pi $ . To sum up,

(26) $$ \begin{align} j(B)=\bigcup_{\bar q\in\mathbb R^{n-1}} R_{\bar q}\times \{\bar q\}, \, \qquad j(\Delta)=\{ (A(\bar q), \bar q) \mid \bar q\in \Pi \}. \end{align} $$

In particular, when $n=2$ , the $j(\Delta )$ only consists of a single point (i.e., the blue point S in Figure 1).

Remark 2.6. The ‘singular locus’ $j(\Delta )$ in $j(B)$ is curved just like [Reference Kontsevich and Soibelman50, p27]. Something similar is also considered in [Reference Castaño-Bernard and Matessi12, Example 3.9]. Intuitively, we start the cuts at points on a graph of a continuous function (rather than an affine one) and make the gluing in a one-dimension-higher Euclidean space.

3.1 Tropicalization map

In this paper, we exclusively consider the Novikov field $\Lambda =\mathbb C((T^{\mathbb R}))$ , a non-archimedean field that consists of all the infinite series $\sum _{i=0}^\infty a_i T^{\lambda _i}$ , where $a_i\in \mathbb C$ , T is a formal symbol, and $\{\lambda _i\}$ is a divergent strictly-increasing sequence in $\mathbb R$ . It has a non-archimedean valuation

$$\begin{align*}\operatorname{\mathrm{\mathsf{v}}}: \Lambda \to \mathbb R\cup \{\infty\} \end{align*}$$

defined by sending the above series to the smallest $\lambda _i$ with $a_i\neq 0$ and sending the zero series to $\infty $ . It is equivalent to the non-archimedean norm defined by $|x|=\exp (-\operatorname {\mathrm {\mathsf {v}}}(x))$ . The multiplicative group is the subset $U_{\Lambda }=\operatorname {\mathrm {\mathsf {v}}}^{-1}(0)=\{x\in \Lambda \mid |x|=1\}$ that resembles the subgroup $U(1)\equiv S^1$ in $\mathbb C^*$ .

Consider the tropicalization map

(27) $$ \begin{align} \operatorname{\mathrm{\mathfrak{trop}}}: (\Lambda^*)^n\to\mathbb R^n, \qquad (z_i)\mapsto (\operatorname{\mathrm{\mathsf{v}}}(z_i)). \end{align} $$

It is a continuous map with respect to the analytic topology in $(\Lambda ^*)^n$ and the Euclidean topology. Note that a fiber of $\operatorname {\mathrm {\mathfrak {trop}}}$ is simply a copy of $U_\Lambda ^n\equiv \operatorname {\mathrm {\mathfrak {trop}}}^{-1}(0)$ up to a translation $y_i\mapsto T^{c_i}y_i$ ; cf. (1).

3.2 Non-archimedean integrable system

Following Kontsevich and Soibelman [Reference Kontsevich and Soibelman50, §4], we introduce an analog of the notion of an integrable system in the non-archimedean analytic setting.

Let $\mathcal Y$ be an analytic space over $\Lambda $ of dimension n, and let B be an n-dimensional topological manifold or a CW complex. Let $f: \mathcal Y\to B$ a proper continuous map with respect to the analytic topology and Euclidean topology. We call a point $p \in B$ smooth (or f-smooth) if there is a neighborhood U of p in B such that the fibration $f^{-1}(U)\to U$ is isomorphic to $\operatorname {\mathrm {\mathfrak {trop}}}^{-1}(V)\to V$ for some open subset $V\subset \mathbb R^n$ . Here, $f^{-1}(U)\cong \operatorname {\mathrm {\mathfrak {trop}}}^{-1}(V)$ is an isomorphism of $\Lambda $ -analytic spaces while $U\cong V$ is a homeomorphism.

Let’s call it an affinoid tropical chart, which may be also viewed as a tropical chart in the language of Chambert-Loir and Ducros [Reference Chambert-Loir and Ducros14, (3.1.2)].

Let $B_0$ denote the open subset of f-smooth points of B. We call f an affinoid torus fibration if $B_0=B$ (see [Reference Nicaise, Xu and Yu53, §3.3]). In general, we only have $B_0\subsetneq B$ , but the restriction of f over $B_0$ , denoted by $ f_0: f^{-1}(B_0)\to B_0, $ is always an affinoid torus fibration simply by definition.

The following construction is greatly influenced by Kontsevich-Soibelman [Reference Kontsevich and Soibelman50, §8]. But, we have to substantially generalize and modify it for our T-duality purpose in the sense of Definition 1.4. Let

$$\begin{align*}\psi_0:\mathbb R^ {n-1} \to (0,+\infty) \end{align*}$$

be a continuous function. In practice, we choose $\psi _0(\bar q)=\psi (\bar q,0)$ to be the one in (20, 21); in this case, $\psi _0>0$ as the symplectic area of a holomorphic disk.

Recall that the $\Lambda $ -algebraic variety Y is given by the equation $ x_0 x_1=1+y_1+\cdots +y_{n-1} $ in $\Lambda ^2_{(x_0,x_1)}\times (\Lambda ^*)^{n-1}_{(y_1,\dots ,y_{n-1})}$ . As said, we will not always distinguish Y and its analytification. We define

(28) $$ \begin{align} F=(F_0,F_1; G_1,\dots, G_{n-1}): Y\to \mathbb R^{n+1} \end{align} $$

as follows: given $z=(x_0,x_1,y_1,\dots ,y_{n-1})$ , we set

$$\begin{align*}\begin{aligned} F_0(z)&=\min\{ \operatorname{\mathrm{\mathsf{v}}}(x_0), -\psi_0(\operatorname{\mathrm{\mathsf{v}}}(y_1),\dots, \operatorname{\mathrm{\mathsf{v}}}(y_{n-1}) )+\min\{0,\operatorname{\mathrm{\mathsf{v}}}(y_1),\dots, \operatorname{\mathrm{\mathsf{v}}}(y_{n-1})\} \} \\ F_1(z)&= \min \{ \operatorname{\mathrm{\mathsf{v}}}(x_1), \ \ \ \psi_0(\operatorname{\mathrm{\mathsf{v}}}(y_1),\dots, \operatorname{\mathrm{\mathsf{v}}}(y_{n-1})) \} \\ G_k(z)&=\operatorname{\mathrm{\mathsf{v}}}(y_k) \qquad \text{for} \ 1\le k< n. \end{aligned} \end{align*}$$

This is a tropically continuous map in the sense of Chambert-Loir and Ducros [Reference Chambert-Loir and Ducros14, (3.1.6)]. Roughly, this means F locally takes the form $\varphi (\operatorname {\mathrm {\mathsf {v}}}(f_1),\dots , \operatorname {\mathrm {\mathsf {v}}}(f_n))$ , where $f_i$ ’s are local invertible analytic functions and $\varphi :U\to \mathbb R^m$ is a continuous map for the Euclidean topology for some open subset U of $\mathbb R^n$ . By adding other constraints on the $\varphi $ , one may define the notion of tropically piecewise-linear / $C^k$ , etc.

3.2.1 Description of the image of F.

Fix $\bar q=(q_1,\dots , q_{n-1})$ , and define

$$\begin{align*}S_{\bar q}:=\{(u_0,u_1)\in\mathbb R^2\mid (u_0,u_1,\bar q)\in F(Y)\}. \end{align*}$$

In other words, the image of Y in $\mathbb R^{n+1}\equiv \mathbb R^2\times \mathbb R^{n-1}$ under F is given by

(29) $$ \begin{align} \mathfrak B:=F(Y)=\bigcup_{\bar q\in\mathbb R^{n-1}} S_{\bar q}\times \{\bar q\}. \end{align} $$

It suffices to describe each $S_{\bar q}$ . Just like §2.1, let $\Pi \subset \mathbb R^{n-1}$ be the tropical hyperplane associated to $\min \{0,\bar q\}$ , consisting of those points for which $\min \{0,\bar q\}$ is attained twice. Just as (25), we define

$$ \begin{align*} a_0(\bar q)=\min\{0,\bar q\}-\psi_0(\bar q) \qquad \text{and} \qquad a_1(\bar q)=\psi_0(\bar q). \end{align*} $$

Let $z=(x_0,x_1,y_1,\dots , y_{n-1})$ be an arbitrary point in Y with $q_k:=\operatorname {\mathrm {\mathsf {v}}}(y_k)$ for $1\le k<n$ . We write $p=(u_0,u_1,\bar q)=F(z) $ for the image point.

1. (i) Assume $\bar q\notin \Pi $ . Then, $\operatorname {\mathrm {\mathsf {v}}}(x_0)+\operatorname {\mathrm {\mathsf {v}}}(x_1)=\operatorname {\mathrm {\mathsf {v}}}(1+y_1+\cdots +y_{n-1})=\min \{0,\bar q\}$ . Eliminating $\operatorname {\mathrm {\mathsf {v}}}(x_0)$ , we get $ F_0(z)= \min \{0,\bar q\} +\min \{-\operatorname {\mathrm {\mathsf {v}}}(x_1), -\psi _0(\bar q)\} $ and $F_1(z)=\min \{\operatorname {\mathrm {\mathsf {v}}}(x_1), \psi _0(\bar q)\}$ . Hence, the $S_{\bar q}$ is simply the image of the broken line in $\mathbb R^2$ given by

   (30) $$ \begin{align} s_{\bar q}: \mathbb R\to\mathbb R^2 , \quad c\mapsto \big( \min \{0,\bar q\}+ \min\{-c, -\psi_0(\bar q)\}, \quad \min\{c,\psi_0(\bar q)\} \big) \end{align} $$
   with a corner point at $(a_0(\bar q), a_1(\bar q))$ . Here, c represents $\operatorname {\mathrm {\mathsf {v}}}(x_1)$ . We take a small neighborhood $\bar V$ of $\bar q$ in the complement of $\Pi $ in $\mathbb R^{n-1}$ . Given the c with $s_{\bar q}(c)=(u_0,u_1)$ and a sufficiently small $\epsilon>0$ , we can find a neighborhood U of p in $\mathfrak B$ that is homeomorphic to $V:=(c-\epsilon , c+\epsilon )\times \bar V$ in $\mathbb R^n$ via the various $s_{\bar q'}$ for $\bar q'\in \bar V$ . Then, under this identification $U\cong V$ , $F^{-1}(U)$ is isomorphic to $\operatorname {\mathrm {\mathfrak {trop}}}^{-1}(V)$ by forgetting $x_0$ (i.e., $z\mapsto (x_1, y_1,\dots , y_{n-1})$ ). In this way, the fibration F also agrees with $\operatorname {\mathrm {\mathfrak {trop}}}$ . In conclusion, this means p is an F-smooth point in the sense of §3.2.

2. (ii) Assume $\bar q\in \Pi $ . Recall $F_0(z)=\min \{ \operatorname {\mathrm {\mathsf {v}}}(x_0), a_0(\bar q)\}$ and $F_1(z)=\min \{\operatorname {\mathrm {\mathsf {v}}}(x_1),a_1(\bar q)\}$ . As $\operatorname {\mathrm {\mathsf {v}}}(x_0)+\operatorname {\mathrm {\mathsf {v}}}(x_1)\ge \min \{\operatorname {\mathrm {\mathsf {v}}}(1),\operatorname {\mathrm {\mathsf {v}}}(y_1),\dots ,\operatorname {\mathrm {\mathsf {v}}}(y_{n-1})\}=a_0(\bar q)+a_1(\bar q)$ , one of the following cases must hold:

   1. (ii-a) If $\operatorname {\mathrm {\mathsf {v}}}(x_0)<a_0(\bar q)$ , then $\operatorname {\mathrm {\mathsf {v}}}(x_1)>a_1(\bar q)$ . Hence, $F_0(z)=\operatorname {\mathrm {\mathsf {v}}}(x_0) \equiv u_0$ , and $F_1(z)=a_1(\bar q)$ . Find a neighborhood U of p in $\mathfrak B$ in the form $U=\{(u_0', a_1(\bar q'), \bar q') \mid u_0' \in I, \bar q'\in \bar V\}$ , where a neighborhood $\bar V$ of $\bar q$ and an open interval I centered at $u_0$ are both chosen small enough so that $u_0'<a_0(\bar q')$ always holds. Let $V:=I\times \bar V \cong U$ , and $F^{-1}(U)$ is isomorphic to $\operatorname {\mathrm {\mathfrak {trop}}}^{-1}(V)$ by forgetting $x_1$ . Therefore, p is F-smooth.

   2. (ii-b) If $\operatorname {\mathrm {\mathsf {v}}}(x_1) < a_1(\bar q)$ , then $\operatorname {\mathrm {\mathsf {v}}}(x_0)>a_0(\bar q)$ . Hence, $F_0(z)= a_0(\bar q)$ , and $F_1(z)=\operatorname {\mathrm {\mathsf {v}}}(x_1) \equiv u_1$ . In a similar way, we can show p is also F-smooth.

   3. (ii-c) If both $\mathsf v(x_0)\ge a_0(\bar q)$ and $\mathsf v(x_1)\ge a_1(\bar q)$ , then $(u_0,u_1)=(F_0(z),F_1(z))=(a_0(\bar q), a_1(\bar q))$ is exactly the corner point of the broken line $s_{\bar q}$ in (30). One can also check p is not F-smooth.

   Hence, the $S_{\bar q}$ is still given by the broken line $s_{\bar q}$ defined in the same way as (30). Moreover, by (ii-a) and (ii-b), the set of F-smooth points in $S_{\bar q}\cong S_{\bar q}\times \{\bar q\}$ includes the union of $ S_{\bar q}^+:= (-\infty , a_0(\bar q)) \times \{a_1(\bar q)\} $ and $ S_{\bar q}^-:=\{a_0(\bar q)\} \times (-\infty , a_1(\bar q)) $ .

Combining (i) and (ii) above, we have proved the following structural result:

Theorem 3.2. The map F restricts to an affinoid torus fibration over $\mathfrak B_0 \equiv \mathfrak B\setminus \hat \Delta $ , where

$$\begin{align*}\hat \Delta := \bigsqcup_{\bar q\in \Pi} \{ \big(a_0(\bar q), a_1(\bar q)\big)\} \times \{\bar q\}. \end{align*}$$

3.3 Definition of f

Notice that if is a topological embedding, then $\jmath ^{-1} \circ F$ is also an affinoid torus fibration on its domain, by definition (§3.2). For the base $B\equiv \mathbb R^n$ in §2.1 and the $j:B\to \mathbb R^{n+1}$ in §2.4, a comparison between §2.4.2 and §3.2.1 implies the following

Lemma 3.3. $j(B)$ agrees with the open subset

(31) $$ \begin{align} \hat B:=\{ (u_0,u_1, \bar q)\in \mathfrak B \mid u_1>0\} \subset \mathbb R^{n+1}. \end{align} $$

Moreover, we have $j(\Delta )=\hat \Delta $ . In particular, $B=j^{-1}(\hat B)$ and $B_0=j^{-1}(\hat B \setminus \hat \Delta )$ .

Define $ \mathscr Y :=F^{-1}(\hat B) $ , and it is exactly given by $\operatorname {\mathrm {\mathsf {v}}}(x_1)>0$ or, equivalently, $|x_1|<1$ . In particular, $j(B)=F(\mathscr Y)$ . Using the topological embedding j in (22), we define $f:= j^{-1}\circ F|_{\mathscr Y}$ . By Theorem 3.2, $f_0:=f|_{B_0}$ is an affinoid torus fibration. We set $\mathscr Y_0:=f_0^{-1}(B_0)$ . It is not hard to show $\mathscr Y_0$ is Zariski dense in Y (e.g., by dimension reasons; compare also [Reference Payne55]). Notably, our construction of f here is purely non-archimedean and does not use any Floer theory so far.

(32)

4.1 Statement

Let $(X,\omega , J)$ be a Kähler manifold of real dimension $2n$ . Suppose there is a Lagrangian torus fibration $\pi _0:X_0 \to B_0$ on some open domain $X_0\subset X$ ; we require it is semipositive in the sense that there is no holomorphic stable disk of negative Maslov index bounding a Lagrangian fiber. By [Reference Auroux5, Lemma 3.1], every special Lagrangian (or graded Lagrangian) satisfies this condition. Further, we require that all Lagrangian fibers are weakly unobstructed (see, for example, [Reference Auroux6, Page 7]) in the sense that their associated minimal model $A_\infty $ algebras have vanishing obstruction ideal (slightly different from the Maurer-Cartan equations in the literature). Thanks to Solomon [Reference Solomon57], a nice sufficient condition is when each Lagrangian fiber is preserved by an anti-symplectic involution $\varphi $ (see also [Reference Castaño-Bernard, Matessi and Solomon13]). For example, the Gross Lagrangian fibration in §2.1 or in (2) admits the involution given by the complex conjugations $z_i\mapsto \bar z_i$ . In general, such an involution $\varphi $ gives a pairing on $\pi _2(X, L_q)$ via $\beta \leftrightarrow -\varphi _*\beta $ , inducing a pairwise canceling for the obstruction formal power series. Beware that it does not mean the virtual counts of Maslov-0 disks vanish, and they do still contribute to the homological perturbations for the minimal model $A_\infty $ algebras and the wall-crossing $A_\infty $ homotopy equivalence.

Now, the family Floer mirror construction in [Reference Yuan61] can be stated as follows:

Beware that the mere homotopy invariance of Maurer-Cartan sets is quite insufficient to develop the analytic topology on $X_0^\vee $ for which we must seek more structure and information. In our specific SYZ context, the Maurer-Cartan sets are simply $H^1(L_q;\Lambda _0)$ , which can at most give certain set-theoretic or local approximation. This is one key point missing in [Reference Tu59]; see Remark 4.2, 4.3 for more discussions. Indeed, the new ud-homotopy theory in [Reference Yuan61] is necessary to upgrade the conventional Maurer-Cartan picture to a higher level, matching adic-convergent formal power series instead of just bijection of sets. The virtual counts of Maslov-0 disks lead to an analytic space structure on the above fiber-wise union (33) of the space of $U_\Lambda $ -local systems. Moreover, the counts of Maslov-2 disks give rise to the global potential function $W_0^\vee $ on $X_0^\vee $ .

4.2 Local affinoid tropical charts

Let $U\subset B_0$ be a contractible open subset, and choose a point $q_0$ near U ( $q_0\notin U$ is possible). We require U is sufficiently small and $q_0$ is sufficiently close to U so that the reverse isoperimetric inequalities hold uniformly over a neighborhood of $U\cup \{q_0\}$ . Let

$$\begin{align*}\chi: (U,q_0) \xrightarrow{\cong} (V,c)\subset \mathbb R^n \end{align*}$$

be a (pointed) integral affine coordinate chart such that $\chi (q_0)=c$ . Then, we have an identification

(34) $$ \begin{align} \tau: (\pi_0^\vee)^{-1}(U) \xrightarrow{\cong} \operatorname{\mathrm{\mathfrak{trop}}}^{-1}(V-c) \end{align} $$

with $\operatorname {\mathrm {\mathfrak {trop}}}\circ \ \tau =\chi \circ \pi _0^\vee $ . Let’s call $\tau $ a (pointed) affinoid tropical chart as in §3.2. Recall that the left side is the disjoint union $(\pi _0^\vee )^{-1}(U)\equiv \bigcup _{q\in U} H^1(L_q; U_\Lambda )$ set-theoretically. A closed point $ \mathbf y$ in the dual fiber $H^1(L_q; U_\Lambda )$ can be viewed as a group homomorphism $\pi _1(L_q)\to U_\Lambda $ (i.e., a flat $U_\Lambda $ -connection modulo gauge equivalence), and we have the natural pairing

(35) $$ \begin{align} \pi_1(L_q)\times H^1(L_q; U_\Lambda)\to U_\Lambda, \qquad (\alpha, \mathbf y) \mapsto \mathbf y^\alpha. \end{align} $$

Write $\chi =(\chi _1,\dots , \chi _n)$ , and it gives rise to a continuous family $e_i=e_i(q)$ of $\mathbb Z$ -bases of $\pi _1(L_q)$ for all $q\in U$ , cf. (12). Then, the corresponding affinoid tropical chart $\tau $ has a very concrete description:

(36) $$ \begin{align} \tau(\mathbf y)= (T^{\chi_1(q)} \mathbf y^{e_1(q)},\dots, T^{\chi_n(q)} \mathbf y^{e_n(q)} ). \end{align} $$

4.3 Gluing

In the construction of Theorem 4.1, we start with various local affinoid tropical charts as above, and then we can develop transition maps (or call gluing maps) among them in a choice-independent manner. This process encodes the quantum corrections of the pseudo-holomorphic disks bounded by smooth $\pi $ -fibers but possibly meeting the singular $\pi $ -fibers (Red disks in Figure 2).

Let’s take two pointed integral affine charts. Replacing the two domains by their intersection, we may assume the two charts have the same domain $U\subset B_0$ , but the base points may be different and outside of U. Namely, as before in Section 4.2, we take two pointed integral affine charts

$$\begin{align*}\chi_a=(\chi_{a1},\dots, \chi_{an}):(U,q_a) \to (V_a,c_a) \end{align*}$$

for $a=1,2$ . Then, $\unicode{x3c7} :=\chi _2\circ \chi _1^{-1}:V_1\to V_2$ is an integral affine transformation. Due to (34) above, we have two affinoid tropical charts on the same domain:

$$\begin{align*}\tau_a: (\pi_0^\vee)^{-1}(U)\to \operatorname{\mathrm{\mathfrak{trop}}}^{-1}(V_a-c_a) \end{align*}$$

such that $\chi _a\circ \pi _0^\vee =\operatorname {\mathrm {\mathfrak {trop}}}\circ \ \tau _a$ for $a=1,2$ .

The transition map between the two charts $\tau _1$ and $\tau _2$ is an automorphism map $\phi $ that, concerning the Fukaya’s trick, captures the wall-crossing information of a Lagrangian isotopy from $L_{q_1}$ to $L_{q_2}$ . In brief, we can view it as a fiber-preserving map:

(37) $$ \begin{align} \phi: \bigcup_{q\in U} H^1(L_q; U_\Lambda)\to \bigcup_{q\in U} H^1(L_q;U_\Lambda). \end{align} $$

But, be careful. The source and the target of $\phi $ should correspond to the two different affinoid tropical charts $\tau _1$ and $\tau _2$ , respectively, although they are set-theoretically the same. In particular, the base points $q_a$ for $\tau _a$ matter a lot for the analytic structure. Indeed, by the two affinoid tropical charts $\tau _1$ and $\tau _2$ , the gluing map $\phi $ can be regarded an analytic map between open subdomains in $(\Lambda ^*)^n$ as follows:

(38) $$ \begin{align} \Phi:= \tau_2\circ \phi \circ \tau_1^{-1}: \operatorname{\mathrm{\mathfrak{trop}}}^{-1}(V_1-c_1)\to \operatorname{\mathrm{\mathfrak{trop}}}^{-1}(V_2-c_2). \end{align} $$

By definition, if $\mathbf y $ is a point in $ H^1(L_q; U_\Lambda )$ for some $q\in U$ , then $\tilde { \mathbf y} :=\phi (\mathbf y)$ is a point in the same fiber $H^1(L_q;U_\Lambda )$ and is subject to the following condition:

(39) $$ \begin{align} \tilde{\mathbf y}^\alpha= \mathbf y^\alpha \exp \langle \alpha, \pmb {\mathfrak{F}}(\mathbf y) \rangle, \end{align} $$

where we use the pairing in (35) and $\pmb {\mathfrak {F}}$ is a vector-valued formal power seriesFootnote ¹ in $ \Lambda [[\pi _1(L_{q})]]\hat \otimes H^1(L_{q})$ . Roughly, the $\pmb {\mathfrak {F}}$ is decided by the virtual counts of Maslov-0 disksFootnote ² along a Lagrangian isotopy from $L_{q_1}$ to $L_{q_2}$ . The existence and uniqueness of such a $\pmb {\mathfrak {F}}$ is proved in [Reference Yuan61]. By definition, the Novikov coefficients of $\pmb {\mathfrak {F}}$ have positive valuations, and one can prove the $\exp \langle \alpha , \pmb {\mathfrak {F}}\rangle $ has valuation zero for sufficiently small U by the reverse isoperimetric inequality again. In particular, $\phi $ preserves the fibers.

Remark 4.3. Again, we must work with the category of affinoid algebras to be completely rigorous. The gluing map $\phi \equiv \psi ^*$ comes from an affinoid algebra homomorphism:

(40) $$ \begin{align} \psi: \Lambda \langle U, q_2\rangle \to \Lambda\langle U, q_1\rangle, \qquad Y^{\alpha(q_2)} \mapsto T^{\langle \alpha, q_1-q_2\rangle} Y^{\alpha(q_1)} \exp\langle \alpha, \pmb {\mathfrak{F}}(Y)\rangle \end{align} $$

for

(41) $$ \begin{align} \pmb {\mathfrak{F}}=\sum_{\mu(\beta)=0} T^{E(\beta)} Y^{\partial\beta} \mathfrak f_{0,\beta}, \end{align} $$

where $\mathfrak f=\{\mathfrak f_{k,\beta }\} \ (k\ge 0, \beta \in \pi _2(X,L_q))$ is, up to Fukaya’s trick, some $A_\infty $ (ud-)homotopy equivalence between two $A_\infty $ algebras associated to $L_{q_1}$ and $L_{q_2}$ . We can finally check that the simplified description (39) agrees with (40) using the perspective of Remark 4.2.

The idea of finding (40) is to use the coordinate change in [Reference Fukaya33, (1.6)] to the morphism [Reference Fukaya, Oh, Ohta and Ono35, (3.6.37)], and it is first discovered by J. Tu in [Reference Tu59]. But, we cannot just naively use the homotopy invariance of Maurer-Cartan sets to get the analytic topology on $X_0^\vee $ . Indeed, the idea of Maurer-Cartan invariance is overall correct but needs to be carried out in a more precise level, matching adic-convergent formal power series rather than just bijections of sets. For this, we need the stronger ud-homotopy in [Reference Yuan61]; we also need a uniform version of reverse isoperimetric inequality (cf. Remark 4.2) for the non-archimedean convergence issues when we move between adjacent local tropical charts.

Finally, note that the $A_\infty $ morphism $\mathfrak f$ is obtained by a parameterized moduli space of holomorphic disks and is highly choice-sensitive. But surprisingly, the gluing map $\phi $ is actually unchanged for any such $A_\infty $ (ud-)homotopy equivalence. This is missed in [Reference Tu59] but is proved in [Reference Yuan61] by the ud-homotopy relations. In our opinion, the choice-independence of the gluing maps is the cornerstone of everything about the family Floer mirror construction, including the results in this paper as well as those in [Reference Yuan63] [Reference Yuan62].

4.4 Void wall-crossing

Let $B_1\subset B_0$ be a contractible open set. Let $B_2=\{x\in B_0\mid \operatorname {\mathrm {dist}} (x, B_1) < \epsilon \}$ be a slight thickening of $B_1$ in $B_0$ . We assume it is also contractible and $\epsilon>0$ is a sufficiently small number so that the estimate constant in the reverse isoperimetric inequalities for any Lagrangian fiber over $B_1$ exceeds $\epsilon $ uniformly (cf. Remark 4.2, 4.3, and [Reference Yuan61]). Then, we have the following:

4.5 Superpotential

Assume $\beta \in \pi _2(X,L_{q_0})$ has Maslov index two (i.e., $\mu (\beta )=2$ , and it also induces $\beta \equiv \beta (q)\in \pi _1(L_q)$ for any q in a small contractible neighborhood of $q_0$ in $B_0$ ). Denote by $\mathsf n_\beta \equiv \mathsf n_{\beta (q)}$ the corresponding open Gromov-Witten invariant.Footnote ³ It depends on the base point q and the almost complex structure J in use. For our purpose, unless the Fukaya’s trick is applied, we always use the same J in this paper. Then, due to the wall-crossing phenomenon, one may roughly think the numbers $\mathsf n_{\beta (q)}$ will vary dramatically in a discontinuous manner when we move q.

Now, we describe the superpotential $W^\vee :=W_0^\vee $ in Theorem 4.1. Fix a pointed integral affine chart $\chi :(U,q_0)\to (V,c)$ , and pick an affinoid tropical chart $\tau $ that covers $\chi $ as in (34). Recall that the domain U must be sufficiently small. Then, the local expression of $W^\vee $ with respect to $\tau $ is given by

(42) $$ \begin{align} W^\vee|_\tau : \bigcup_{q\in U} H^1(L_q; U_\Lambda)\to \Lambda, \qquad \mathbf y\mapsto \sum_{\beta\in \pi_2(X,L_q), \mu(\beta)=2} T^{E(\beta)} \mathbf y^{\partial\beta} \mathsf n_{\beta(q_0)}, \end{align} $$

where $\mathbf y\in H^1(L_q; U_\Lambda )$ for any $q\in U$ and we use $\mathsf n_{\beta (q_0)}$ for the fixed $q_0$ . Alternatively, by (34), one may think of $W^\vee |_\tau $ as

$$\begin{align*}\mathcal W_\tau \equiv W^\vee \circ \tau^{-1} : \operatorname{\mathrm{\mathfrak{trop}}}^{-1}(V-c)\to (\pi_0^\vee)^{-1}(U)\to \Lambda. \end{align*}$$

The delicate story of the wall-crossing uncertainty can be well narrated by the gluing maps (39) among the atlas of various affinoid tropical charts (34) in view of Theorem 4.1. Specifically, we take two pointed integral affine charts $\chi _a: (U,q_a)\to (V_a, c_a)$ and two corresponding affinoid tropical charts $\tau _a: (\pi _0^\vee )^{-1}(U)\to \operatorname {\mathrm {\mathfrak {trop}}}^{-1}(V_a-c_a)$ for $a=1,2$ as before in §4.3. Let $\phi $ be the gluing map from the chart $\tau _1$ to the $\tau _2$ as in (37, 39). Then, we must have $ W^\vee |_{\tau _2} (\phi (\mathbf y)) = W^\vee |_{\tau _1} (\mathbf y)$ . Equivalently, if we set $\Phi =\tau _2\circ \phi \circ \tau _1^{-1}$ like (38), this means

(43) $$ \begin{align} \mathcal W_{\tau_2}(\Phi(y))=\mathcal W_{\tau_1}(y). \end{align} $$

4.6 Maslov-0 determinism

In our Floer-theoretic mirror construction, the counts of the Maslov-0 disks are overwhelmingly more important than that of Maslov-2 disks. Although the mirror superpotential $W^\vee $ is given by the counts of Maslov-2 disks locally in each chart, the local-to-global gluing among the various local expressions (42) is given by the counts of Maslov-0 disks.

In practice, there is a very useful observation as follows: The Lagrangian fibration $\pi _0$ can be placed in different ambient symplectic manifolds, say $\overline X_1$ and $\overline X_2$ . It often happens that the Maslov-0 disks are the same in both situations, and then the mirrors associated to $(X_i,\pi _0)$ , $i=1,2$ , in Theorem 4.1 can be denoted by $(X_0^\vee , W_i^\vee , \pi _0^\vee )$ , sharing the same mirror analytic space $X^\vee _0$ and the same dual affinoid torus fibration $\pi _0^\vee :X_0^\vee \to B_0$ but having different superpotentials $W_i^\vee $ .

The ‘Maslov-0 open Gromov-Witten invariant’ should be all the counting as a whole rather than any single of them. All the virtual counts of the Maslov-0 disks along an isotopy, only taken together, can form an invariant. Roughly, it forms the ud-homotopy class of a morphism in the category $\mathscr {UD}$ in [Reference Yuan61].

5.1 Affinoid tropical charts for the Clifford and Chekanov chambers

In §2.3, we have introduced the two integral affine charts $\chi _\pm $ on $B^{\prime }_\pm $ . By Proposition 4.4, we have the following two affinoid tropical charts on $(\pi )^{-1}(B_\pm ')$ : (cf. (36))

(44) $$ \begin{align} \tau_+: \bigcup_{q\in B^{\prime}_+} H^1(L_q; U_\Lambda) \to (\Lambda^*)^n, \qquad \mathbf y\mapsto \big( T^{q_1} \mathbf y^{\sigma_1},\dots, T^{q_{n-1}} \mathbf y^{\sigma_{n-1}}, T^{\psi_+(q)} \mathbf y^{\partial\beta_n} \big) \end{align} $$

(45) $$ \begin{align} \tau_- : \bigcup_{q\in B^{\prime}_-} H^1(L_q; U_\Lambda) \to (\Lambda^*)^n, \qquad \mathbf y \mapsto \big(T^{q_1} \mathbf y^{\sigma_1},\dots, T^{q_{n-1}} \mathbf y^{\sigma_{n-1}}, T^{\psi_-(q)} \mathbf y^{\partial\hat\beta} \big). \end{align} $$

The images of $\tau _\pm $ in $(\Lambda ^*)^n$ are just given by the explicit integral affine charts $\chi _\pm $ (15, 17) as follows:

(46) $$ \begin{align} T_\pm:=\operatorname{\mathrm{\mathfrak{trop}}}^{-1}(\chi_\pm(B^{\prime}_\pm)), \end{align} $$

which are the analytic open subdomains in $(\Lambda ^*)^n$ . Clearly, we have $ \operatorname {\mathrm {\mathfrak {trop}}}\circ \ \tau _\pm = \chi _\pm \circ \pi _0^\vee $ ; that is,

5.2 Gluing with a symmetry

By Theorem 4.1, the above two local expressions over $B^{\prime }_\pm $ must be glued by some nontrivial automorphisms over $B^{\prime }_+\cap B^{\prime }_-\equiv \bigsqcup _{1\le \ell \le n}\mathscr N_\ell $ . We denote them by

(47) $$ \begin{align} \phi_\ell : \bigcup_{q\in \mathscr N_\ell} H^1(L_q; U_\Lambda) \to \bigcup_{q\in\mathscr N_\ell} H^1(L_q; U_\Lambda). \end{align} $$

Be cautious that, despite of the same underlying sets, the two sides should refer to the two affinoid tropical charts $\tau _+$ and $\tau _-$ separately; in fact, we adopt a simplified expression as in (37). Given a point $\mathbf y\in H^1(L_q; U_\Lambda )$ , the image point $\tilde {\mathbf y}:=\phi _\ell (\mathbf y)$ is contained in $H^1(L_q; U_\Lambda )$ and satisfies (cf. (39))

(48) $$ \begin{align} \tilde{\mathbf y}^\alpha =\mathbf y^\alpha\exp \langle \alpha, {\pmb{\mathfrak{F}}}_\ell (\mathbf y) \rangle \end{align} $$

for a formal power series ${\pmb {\mathfrak {F}}}_\ell (Y)$ . Basically, the mirror analytic space $X^\vee _0$ is completely determined by these gluing maps $\phi _\ell $ . There is no general algorithm for the gluing maps, and only the existence and uniqueness are proved in [Reference Yuan61]. But, in our case, we first have a natural fiber-preserving $T^{n-1}$ -action (10), and the $T^{n-1}$ -symmetry makes the gluing maps much simpler: (cf. [Reference Yuan63] or [Reference Abouzaid, Auroux and Katzarkov1, Theorem 8.4])

Lemma 5.1. $\langle \sigma _k , {\pmb {\mathfrak {F}}}_\ell \rangle =0$ , for $1\le k\le n-1$ and $1\le \ell \le n$ . In particular, if we set $\tilde {\mathbf y}=\phi _\ell (\mathbf y)$ , then

$$\begin{align*}\tilde{\mathbf y}^{\sigma_k}=\mathbf y^{\sigma_k}. \end{align*}$$

We can express the $\phi _\ell $ ’s explicitly with respect to the two affinoid tropical charts $\tau _\pm $ . Define

(49) $$ \begin{align} \Phi_\ell:= \tau_-\circ \phi_\ell \circ \tau_+^{-1} : T_{+}^\ell \to T_{-}^\ell, \end{align} $$

where $T_\pm ^\ell :=\operatorname {\mathrm {\mathfrak {trop}}}^{-1}(\chi _\pm (\mathscr N_\ell ))$ are analytic open subdomains in $T_\pm \subset (\Lambda ^*)^n$ . Specifically, there exist formal power series $f_\ell (y_1,\dots , y_{n-1})$ for $1\le \ell \le n$ such that

(50) $$ \begin{align} \Phi_\ell(y_1,\dots, y_n)= \begin{cases} (y_1,\dots, y_{n-1}, y_ky_n\exp(f_k(y_1,\dots, y_{n-1})) & \text{if} \ (y_1,\dots, y_n) \in T_+^k, \ 1\le k<n \\ (y_1,\dots, y_{n-1}, y_n \exp(f_n(y_1,\dots, y_{n-1})) & \text{if} \ (y_1,\dots, y_n)\in T_+^n. \end{cases} \end{align} $$

Indeed, the first $n-1$ coordinates are preserved by Lemma 5.1. Since the boundary of any Maslov-0 disk is spanned by $\sigma _k$ ( $1\le k<n$ ), the definition formula (41) implies that each $f_\ell $ does not involve $y_n$ . We also recall that $\partial \hat \beta =\partial \beta _n$ over $\mathscr N_n$ but $\partial \hat \beta =\sigma _k+\partial \beta _n$ over $\mathscr N_n$ .

5.3 Mirror analytic space

As there are no Maslov-2 holomorphic disks in $X=\mathbb C^n\setminus \mathscr D$ bounded by the $\pi $ -fibers, the mirror Landau-Ginzburg superpotential vanishes $W^\vee \equiv 0$ identically. But, as indicated in §4.6, we can choose some larger ambient symplectic manifold $\overline X$ without adding new Maslov-0 disks. No matter what $\overline X$ is, the structure of the mirror affinoid torus fibration $(X_0^\vee , \pi _0^\vee )$ will stay the same. In contrast, there can be new Maslov-2 disks that give rise to a new mirror superpotential $W^\vee $ on $X_0^\vee $ .

Here, we are mainly interested in the case $\overline X=\mathbb C^n$ . But, we will study many others like $\overline X=\mathbb {CP}^n$ in §A. We place the Gross Lagrangian fibration $\pi _0:X_0\to B_0$ (§2.1) in $\mathbb C^n$ . By §4.6, the mirror space and the affinoid torus fibration associated to $(X,\pi _0)$ and $(\mathbb C^n, \pi _0)$ is actually the same, denoted by $(X_0^\vee , \pi _0^\vee )$ . But, the latter is equipped with an extra superpotential $W^\vee :=W^\vee _{\mathbb C^n}$ .

For $q\in B_+$ , the fiber $L_q$ is Hamiltonian isotopic to a product torus in $(\mathbb C^*)^n$ . It follows from [Reference Cho and Oh21] that the open GW invariants (§4.5) are $\mathsf n_{\beta _j}=1$ for the disks $\beta _j$ ’s (§2.2). For $q\in B_-$ , we use the maximal principle to show the only nontrivial open GW invariant is $\mathsf n_{\hat \beta }=1$ (see [Reference Auroux5] [Reference Chan, Lau and Leung16, Lemma 4.31]). Now, by (42), the restrictions $W^\vee _{\pm }$ of $W^\vee $ on the two chambers $(\pi _0^\vee )^{-1}(B_\pm )$ are as follows:

(51) $$ \begin{align} \begin{aligned} W^\vee_{+}(\mathbf y) &= \textstyle \sum_{j=1}^n T^{E(\beta_j)} \mathbf y^{\partial\beta_j} \mathsf n_{\beta_j} = \textstyle T^{\psi_+(q)} \mathbf y^{\partial\beta_n} \big(1+ \sum_{k\neq n} T^{q_k} \mathbf y^{\sigma_k}\big) \\ W^\vee_{-}(\mathbf y) &= T^{E(\hat\beta)} \mathbf y^{\partial\hat\beta} \mathsf n_{\hat\beta} = T^{\psi_-(q)} \mathbf y^{\partial\hat\beta}. \end{aligned} \end{align} $$

Moreover, in view of Proposition 4.4, both of them can be extended slightly to the thickened domains $(\pi _0^\vee )^{-1}(B^{\prime }_\pm )$ . Then, for the affinoid tropical charts $\tau _\pm $ (44), we write $ \mathcal W_{\pm }:= W^\vee _{\pm }\circ \tau _\pm ^{-1} $ and obtain

(52) $$ \begin{align} \begin{aligned} \mathcal W_{+} (y) &= y_n(1+y_1+\cdots+y_{n-1}) && \text{if} \ y=(y_1,\dots, y_n) \in T_+\\ \mathcal W_{-} (y) &= y_n && \text{if} \ y=(y_1,\dots, y_n) \in T_-. \end{aligned} \end{align} $$

According to the wall-crossing property (43), we must have $\mathcal W_-(\Phi _\ell (y))=\mathcal W_+(y)$ for any $y\in T_+^\ell $ . Along with (50), this completely determines all $\Phi _\ell $ for any $1\le \ell \le n$ as follows:

(53) $$ \begin{align} \Phi_\ell: T_{+}^\ell \to T_{-}^\ell \quad \quad (y_1,\dots, y_n) \mapsto \big(y_1,\dots, y_{n-1}, y_n(1+y_1+\cdots +y_{n-1}) \big). \end{align} $$

Remark that although the $\Phi _\ell $ ’s have the same formula, the domains and targets differ and depend on $\ell $ . Note that $\Phi _\ell \circ \tau _+=\tau _-\circ \phi _\ell $ and $\operatorname {\mathrm {\mathfrak {trop}}}\circ \ \Phi _\ell =\unicode{x3c7} _\ell \circ \operatorname {\mathrm {\mathfrak {trop}}}$ .

In conclusion, the mirror analytic space $X_0^\vee $ is isomorphic to the quotient of the disjoint union $T_+\sqcup T_-$ modulo the relation $\sim $ : we say $y\sim y'$ if there exists some $1\le \ell \le n$ such that $y\in T_{+}^\ell $ , $y'\in T_{-}^\ell $ , and $\Phi _\ell (y)=y'$ . That is to say, we have an identification

(54) $$ \begin{align} X_0^\vee \textstyle \cong T_+\sqcup T_-/ \sim, \end{align} $$

which is the adjunction space obtained by gluing $T_+$ and $T_-$ via all these $\Phi _\ell $ ’s. By (19), one can check $ \operatorname {\mathrm {\mathfrak {trop}}} \circ \ \Phi _\ell = \chi _-\circ \chi _+^{-1}\circ \operatorname {\mathrm {\mathfrak {trop}}} $ on the domains, so the dual affinoid torus fibration $\pi _0^\vee :X_0^\vee \to B_0$ can be identified with the gluing of the two maps $\chi _\pm ^{-1}\circ \operatorname {\mathrm {\mathfrak {trop}}}$ via the $\Phi _\ell $ ’s. Under this identification, if we write $\pi _0^\vee =(\pi ^\vee _1,\dots , \pi ^\vee _n)$ , then for $y=(y_1,\dots , y_n)\in T_\pm $ , one can check

(55) $$ \begin{align} \mathsf v(y_k) = \pi_k^\vee (y) \quad (1\le k< n) \qquad \text{and} \quad \operatorname{\mathrm{\mathsf{v}}}(y_n)= \psi_\pm(\pi_0^\vee(y)). \end{align} $$

5.4 The analytic embedding g

From now on, we will always identify the $X_0^\vee $ with $T_+\sqcup T_-/\sim $ and identify the $\pi _0^\vee $ with the one obtained as above (54, 55). See the above diagram. Next, we define

(56) $$ \begin{align} g_+: T_+ \to \Lambda^2 \times (\Lambda^*)^{n-1} \end{align} $$

$$\begin{align*}(y_1,\dots,y_{n-1},y_n) \mapsto \left( \frac{1}{y_n} \ , \ y_n \ h \ , \ y_1,\dots, y_{n-1} \right) \end{align*}$$

and define

(57) $$ \begin{align} g_-: T_-\to \Lambda^2 \times (\Lambda^*)^{n-1} \end{align} $$

$$\begin{align*}(y_1,\dots,y_{n-1},y_n)\mapsto \left( \frac{h}{y_n} \ , \ y_n \ , \ y_1,\dots, y_{n-1} \right). \end{align*}$$

Recall that $T_\pm \subsetneq (\Lambda ^*)^n$ correspond to the Clifford and Chekanov tori, respectively. For all $1\le \ell \le n$ , it is direct to check that $ g_+=g_-\circ \Phi _\ell $ on their various domains. Hence, by (54), we can glue $g_\pm $ to obtain an embedding analytic map

(58) $$ \begin{align} g :X_0^\vee\to \Lambda^2\times (\Lambda^*)^{n-1} \end{align} $$

such that the following diagrams commute:

The image $g(X_0^\vee )$ is clearly contained in the algebraic variety $ Y$ defined by $x_0 x_1 =1+y_1+\cdots +y_{n-1}$ .

5.5 Fibration preserving

Recall $ F=(F_0,F_1, G_1,\dots , G_{n-1}):Y\to \mathbb R^{n+1}$ is defined in §3.2 as follows: for $z=(x_0,x_1,y_1,\dots ,y_{n-1})$ ,

$$\begin{align*}\begin{aligned} F_0(z)&=\min\{ \operatorname{\mathrm{\mathsf{v}}}(x_0), -\psi_0(\operatorname{\mathrm{\mathsf{v}}}(y_1),\dots, \operatorname{\mathrm{\mathsf{v}}}(y_{n-1}) )+\min\{0,\operatorname{\mathrm{\mathsf{v}}}(y_1),\dots, \operatorname{\mathrm{\mathsf{v}}}(y_{n-1})\} \} \\ F_1(z)&= \min \{ \operatorname{\mathrm{\mathsf{v}}}(x_1), \ \ \ \psi_0(\operatorname{\mathrm{\mathsf{v}}}(y_1),\dots, \operatorname{\mathrm{\mathsf{v}}}(y_{n-1})) \} \\ G_k(z)&=\operatorname{\mathrm{\mathsf{v}}}(y_k) \qquad \text{for} \ 1\le k< n. \end{aligned} \end{align*}$$

The next is the key result that puts all the previous constructions together and will prove Theorem 1.5.

Theorem 5.4. $F\circ g=j\circ \pi _0^\vee $ . Namely, we have the following commutative diagram

Proof. Fix $\mathbf y$ in $X_0^\vee \equiv T_+\cup T_-/\sim $ (54), and set $q=\pi _0^\vee (\mathbf y)$ . Then, $ j \circ \pi _0^\vee (\mathbf y)=j(q)=(\theta _0(q),\theta _1(q), \bar q) $ , where (recalling (22))

$$ \begin{align*} \theta_0(q) &:= \min\{ -\psi(q) , -\psi_0( \bar q ) \} + \min \{0, \bar q \} \\ \theta_1(q) &:= \min\{\ \ \ \psi(q) , \ \ \ \psi_0( \bar q )\}, \end{align*} $$

where $\psi (q)$ and $\psi _0(\bar q) \equiv \psi (\bar q,0)$ are given in (20,21). Recall that $\Pi $ is the tropical hypersurface associated to $\min \{0, \bar q\}$ (see §2.1). We aim to check $(\theta _0(q),\theta _1(q),\bar q)$ always agrees with $F\circ g(\mathbf y)$ :

1. 1. If $q\in B^{\prime }_+$ , then $\mathbf y$ is identified with a point $y=(y_1,\dots , y_n)$ in $T_+$ . By (55), $\operatorname {\mathrm {\mathsf {v}}}(y_k)=q_k$ for $1\le k<n$ and $\operatorname {\mathrm {\mathsf {v}}}(y_n)=\psi _+(q)$ . Therefore, as desired, we get $G_k(g_+(y))=q_k$ , and by (20),

   $$ \begin{align*} F_0(g_+(y)) =\min\{ -\psi_+(q), -\psi_0(\bar q)+\min\{0,\bar q\} \} = \min\{-\psi(q),-\psi_0(\bar q)\} + \min\{0,\bar q\}=\theta_0(q). \end{align*} $$

   However,

   $$ \begin{align*} F_1(g_+(y)) = \min\{ \psi_+( q) +\operatorname{\mathrm{\mathsf{v}}}(1+y_1+\cdots+y_{n-1}), \ \ \ \psi_0(\bar q)\}, \end{align*} $$
   and we need to further deal with the ambiguity of $\operatorname {\mathrm {\mathsf {v}}}(1+y_1+\cdots +y_{n-1})$ as follows:

   1. (1a) If $\bar q\in \Pi $ , then since $q\in B^{\prime }_+$ , we must have $q_n>0$ . Using the non-archimedean triangle inequality and Lemma 2.4 infers that $\psi _+(q)+\operatorname {\mathrm {\mathsf {v}}}(1+y_1+\cdots +y_{n-1}) \ge \psi _+(q)+\min \{0,\bar q\} \equiv \psi (q)> \psi _0(\bar q)$ . Hence,

      $$\begin{align*}F_1(g_+(y))=\psi_0(\bar q)=\min\{\psi(q),\psi_0(\bar q)\}=\theta_1(q). \end{align*}$$

   2. (1b) If $\bar q\notin \Pi $ , then the minimum of the values $\operatorname {\mathrm {\mathsf {v}}}(y_k)=q_k$ for $1\le k<n$ cannot be attained twice. Thus,

      $$\begin{align*}\operatorname{\mathrm{\mathsf{v}}}(1+y_1+\cdots+y_{n-1})=\min\{ \operatorname{\mathrm{\mathsf{v}}}(1), \operatorname{\mathrm{\mathsf{v}}}(y_1),\dots, \operatorname{\mathrm{\mathsf{v}}}(y_{n-1})\}=\min\{0,\bar q\} \end{align*}$$
      and $F_1(g_+(y))=\theta _1(q)$ .

2. 2. If $q\in B^{\prime }_-$ , then $\mathbf y$ is identified with a point $y=(y_1,\dots , y_n)$ in $T_-$ . By (55), $\operatorname {\mathrm {\mathsf {v}}}(y_k)=q_k$ for $1\le k<n$ and $\operatorname {\mathrm {\mathsf {v}}}(y_n)=\psi _-(q)$ . Hence, as desired, we also get $G_k(g_-(y))=q_k$ and

   $$\begin{align*}F_1(g_-(y))=\min\{ \psi_-(q), \psi_0(\bar q) \} \equiv \min\{ \psi(q),\psi_0(\bar q) \} =\theta_1(q). \end{align*}$$
   However, in turn,
   $$\begin{align*}F_0(g_-(y))= \min\{ - \psi_-(q) + \operatorname{\mathrm{\mathsf{v}}}(1+y_1+\cdots+y_{n-1}), -\psi_0(\bar q)+\min\{0,\bar q\}\} \end{align*}$$
   has some ambiguity, and we similarly argue by cases:

   1. (2a) If $\bar q\in \Pi $ , then $q_n<0$ . By the non-archimedean triangle inequality and Lemma 2.4, we similarly obtain $-\psi _-(q) + \operatorname {\mathrm {\mathsf {v}}}(1+y_1\cdots +y_{n-1}) \ge -\psi _0(\bar q)+ \min \{0,\bar q\}$ . Hence,

      $$\begin{align*}F_0(g_-(y))=-\psi_0(\bar q)+\min\{0,\bar q\} = \min\{-\psi(q), -\psi_0(\bar q)\} +\min\{0,\bar q\}=\theta_0(q). \end{align*}$$

   2. (2b) If $\bar q\notin \Pi $ , then we similarly get $ \operatorname {\mathrm {\mathsf {v}}}(1+y_1+\cdots +y_{n-1})=\min \{0,\bar q\}$ and $F_0(g_-(y))=\theta _0(q)$ .

(In the sense of Remark 5.3, all the proof here does not rely on any family Floer theory in §4.)

5.6 Dual singular fiber is not a Maurer-Cartan set

It has been long expected that, at least set-theoretically, the mirror dual fiber of a Lagrangian fiber L should be the set $\mathcal {MC}(L)$ of Maurer-Cartan solutions (also known as the bounding cochains) for an $A_\infty $ algebra associated to L. This is mostly correct for the smooth fibers, although we need to be more careful for the analytic topology as indicated in Remark 4.2, 4.3. This is also a basic point of the original family Floer homology program. Thus, it is natural for us to believe the ‘dual singular fibers’ are also the corresponding Maurer-Cartan sets.

Nevertheless, our result responds negatively to this expected Maurer-Cartan picture.

We have obtained the non-archimedean analytic fibration f over B, extending the affinoid torus fibration $f_0\cong \pi _0^\vee $ tropically continuously. Besides, as explained in §1.4.2, its construction is compatible with the above Maurer-Cartan picture over $B_0$ [Reference Yuan61] and is meanwhile backed up by lots of evidence [Reference Abouzaid, Auroux and Katzarkov1, Reference Abouzaid and Sylvan2, Reference Auroux5, Reference Auroux6, Reference Gammage37, Reference Gross, Hacking and Keel41, Reference Kontsevich and Soibelman49, Reference Kontsevich and Soibelman50]. Therefore, the f-fibers over the singular locus $\Delta =B\setminus B_0$ are basically the only reasonable candidates for the ‘dual singular fibers’. But unfortunately, they are indeed larger than the Maurer-Cartan sets as shown in (59) below.

Let’s elaborate this as follows. For clarity, let’s assume $n=2$ ; then, $B=\mathbb R^2$ and the singular locus $\Delta $ consists of a single point $0=(0,0)$ . Let’s also forget about the analytic topology again and just look at the sets of closed points. Notice that $ f^{-1}(0)=F^{-1}(j(0))=F^{-1}(0,-\psi _0, \psi _0) $ , where $\psi _0:=\psi _0(0)>0$ is represented by the symplectic area of a holomorphic disk bounding the immersed Lagrangian $L_0=\pi ^{-1}(0)$ (Figure 2, yellow). By definition, this consists of points $(x_0,x_1,y)$ in the variety $Y=\{x_0x_1=1+y\}$ in $\Lambda ^2\times \Lambda ^*$ so that $\operatorname {\mathrm {\mathsf {v}}}(y)=0$ , $\min \{\operatorname {\mathrm {\mathsf {v}}}(x_0), -\psi _0\}=-\psi _0$ and $\min \{\operatorname {\mathrm {\mathsf {v}}}(x_1), \psi _0\}=\psi _0$ . The last two conditions mean $\operatorname {\mathrm {\mathsf {v}}}(x_0)\ge -\psi _0$ and $\operatorname {\mathrm {\mathsf {v}}}(x_1)\ge \psi _0$ . We can take the coordinate change $z_0:=T^{\psi _0} x_0$ , $z_1:=T^{-\psi _0}x_1$ so that $z_0z_1=x_0x_1$ . Then, the dual singular fiber becomes

$$\begin{align*}\mathbf S:= f^{-1}(0)=\{ (z_0,z_1,y)\in Y \mid \operatorname{\mathrm{\mathsf{v}}}(z_0)\ge 0, \ \operatorname{\mathrm{\mathsf{v}}}(z_1)\ge 0, \ \operatorname{\mathrm{\mathsf{v}}}(y)=0\}. \end{align*}$$

Recall that $\Lambda _0=\{z\in \Lambda \mid \operatorname {\mathrm {\mathsf {v}}}(z)\ge 0\}$ denotes the Novikov ring and $\Lambda _+ =\{z\in \Lambda \mid \operatorname {\mathrm {\mathsf {v}}}(z)>0\}$ is its maximal ideal. Recall also that $U_\Lambda =\{z\in \Lambda \mid \operatorname {\mathrm {\mathsf {v}}}(z)=0\}$ , so the condition $\operatorname {\mathrm {\mathsf {v}}}(y)=0$ means $y\in U_\Lambda \equiv \mathbb C^*\oplus \Lambda _+$ . Next, we decompose $\mathbf S$ by considering the two cases of the variable $y\in U_\Lambda $ :

• • If $y\in -1+\Lambda _+$ , then $\operatorname {\mathrm {\mathsf {v}}}(z_0)+\operatorname {\mathrm {\mathsf {v}}}(z_1)=\operatorname {\mathrm {\mathsf {v}}}(1+y)>0$ , and $(z_0,z_1)\in \Lambda _0\times \Lambda _+ \cup \Lambda _+\times \Lambda _0$ . In turn, such a pair $(z_0,z_1)$ determines $y=-1+z_0z_1$ in $-1+\Lambda _+$ .

• • If $y\notin -1+\Lambda _+$ , then $0\le \operatorname {\mathrm {\mathsf {v}}}(z_0)+\operatorname {\mathrm {\mathsf {v}}}(z_1)=\operatorname {\mathrm {\mathsf {v}}}(1+y)=0$ , so $z_0,z_1\in U_\Lambda $ . In turn, for all these pairs $(z_0,z_1)$ , the number $y=-1+z_0z_1$ can run over all the values in $\mathbb C^*\setminus \{ -1\} \oplus \Lambda _+$ . If we write $\bar z_0,\bar z_1\in \mathbb C^*$ for the reductions of $z_0,z_1$ with respect to $U_\Lambda \twoheadrightarrow \mathbb C^*$ , then this says $\bar z_0\bar z_1\neq 1$ .

Therefore,

$$\begin{align*}\mathbf S=\mathbf S_1 \sqcup \mathbf S_2, \end{align*}$$

where $\mathbf S_1$ and $\mathbf S_2$ correspond to the above two bullets and are thus given by

$$ \begin{align*} \mathbf S_1 &\cong \Lambda_0\times \Lambda_+ \cup \Lambda_+\times \Lambda_0 \\ \mathbf S_2 & \cong \{ (z_0, z_1)\in U_\Lambda\times U_\Lambda \mid \bar z_0\bar z_1\neq 1\} \cong U_\Lambda\times \big(\mathbb C^*\setminus \{-1\} \oplus \Lambda_+\big). \end{align*} $$

Going back to the A side, we can define the Maurer-Cartan set $\mathcal {MC}(L_0)$ by the Akaho-Joyce theory [Reference Akaho and Joyce3]. Moreover, Hong, Kim and Lau [Reference Hong, Kim and Lau45, §3.2] prove that we only have

(59) $$ \begin{align} \mathcal {MC}(L_0)= \mathbf S_1 \ \ \subsetneq f^{-1}(0) \end{align} $$

(compare also many related papers [Reference Cho, Hong and Lau18, Reference Cho, Hong and Lau19, Reference Cho, Hong and Lau20]). In conclusion, as the points in $\mathbf S_2$ are missed, the Maurer-Cartan picture fails over the singular locus. One possibility is that we need additional ‘deformation data’ for the conventional Maurer-Cartan sets. Moreover, as discussed in §1.4.2, the non-archimedean analytic topology may be more relevant for the singular part.

The framework of Definition 1.4 offers a preliminary attempt and should be sufficient when the mirror space is expected to be algebraic rather than transcendental. But admittedly, an ultimate answer would require further researches.