Tropicalization of the Mirror Curve near the Large Radius Limit

The mirror of a toric orbifold is an affine curve called the mirror curve. The mirror curve was studied in mirror symmetry by Hori and Vafa for toric Calabi-Yau 3-folds. Let $(X,\omega)$ be a symplectic toric Calabi-Yau 3-fold. There exist three types of mirrors for $(X,\omega)$: Landau-Ginzburg (Givental) mirror, Hori-Vafa mirror, and the mirror curve [11][9].

For the toric Calabi-Yau 3-fold $(X,\omega)$, the mirror B-model is a 3-dimensional Landau-Ginzburg model on $(\mathbb{C^{*}})^{3}$ given by the superpotential

Here $H(x,y,q)=0$ is an affine curve with the parameter $q$ related to the Kähler parameter of $X$ given by the mirror map. Another mirror, Hori-Vafa mirror of $(X,\omega)$ is a non-compact Calabi-Yau 3-fold $\tilde{X}$ with a holomorphic 3-form $\Omega$ on $\tilde{X}$, where $\tilde{X}$ is a hypersurface in $\mathbb{C}^{2}\times(\mathbb{C^{*}})^{2}$, given by

and

These two types of mirrors of a toric Calabi-Yau 3-fold $(X,\omega)$ could be reduced to an affine curve

with the Kähler parameter $q$. We define the curve by the mirror curve.

The mirror curve could be used for the prediction of open-closed Gromov-Witten invariants. For example, in [6], the Eynard-Orantin topological recursion provides an algorithm for calculating higher genus invariants for a spectral curve. We can relate the Eynard-Orantin invariants $\omega_{g,n}$ of the mirror curve to all genus open-closed Gromov-Witten invariants of the symplectic toric Calabi-Yau 3-fold $(X,\omega)$ by Bouchard-Klemm-Mariño-Pasquetti (BKMP) Remodeling Conjecture [3][4][11][7][6].

Besides, the statement of a version of homological mirror symmetry for a toric Calabi-Yau 3-fold is formulated via the mirror curve. Specifically, for a toric Calabi-Yau 3-fold, the B-model is the matrix factorization, while the A-model is a Fukaya-type category on its mirror curve [15] [1][13][2].

For a simplicial toric Calabi-Yau 3-fold $X_{\Sigma}$ defined by a 3-dim fan $\Sigma\subset N\cong\mathbb{Z}^{3}$, all the generators of 1-dim cones lie in a hyperplane $N^{\prime}\subset N$, so a generator corresponds to a lattice point in $N^{\prime}$. Thus $\Sigma$ gives a triangulation of $\Delta$, where $\Delta$ is the convex hull of these lattice points given by 1-dim cones. We define such triangulation by $T_{\Sigma}$. The first result is that the tropicalization of the mirror curve near the large radius limit gives a subdivision of $\mathbb{R}^{2}$. This subdivision is exactly dual to the triangulation $T_{\Sigma}$.

Besides, we recall propositions of a special kind of real algebraic curves determined by some Laurent polynomials namely cyclic-M curves in a real toric surface. This kind of curve has the maximal number of connected components in the corresponding real toric surface and intersects the axes of the toric surface in a cyclic order. Here the maximal number of connected components equals one plus the number of lattice points inside the Newton polytope of Laurent polynomial by Harnack’s inequality[10]. We give examples of a non-cyclic-M curve and a cyclic-M curve in Figure 2 and Figure 2.

Mikhalkin has studied cyclic-M curves in manners of tropical geometry. In [14], he has proved the following statements about cyclic-M curves:

The mirror curve is defined by a Laurent polynomial written as

The second result about a mirror curve of a smooth toric Calabi-Yau 3-fold is that under special choices of real parameters, the mirror curve is a cyclic-M curve.

Finally, we apply Mikhalkin’s result to show the mirror curve of smooth toric Calabi-Yau 3-fold is glued from tubes and pairs of pants, which any vertex in the dual graph of $T_{\Sigma}$ corresponds to a pair of pants and any edge corresponds to a tube. This gluing determines the topology of the mirror curve.

We prove the results in several steps. Beginning in Section 2, we recall tropical geometry and give a definition of the mirror curve. For the first result, we mainly finish the proof by applying the method of tropical geometry. Specifically, in Section 2.6, we will see that generally for a Laurent polynomial of $F(z_{1},...,z_{n})$, we could construct an injective map $\psi_{F}$ from the connected components of $\mathbb{R}^{n}-A_{F}$ to $\Delta_{F}\cap\mathbb{Z}^{n}$. Then in Section 3, we prove that near the large radius limit, for the mirror curve $H(x,y,q)=0$ of a smooth toric Calabi-Yau 3-fold, the previous injective map $\psi_{H}$ is also surjective by constructing the preimage of any lattice point in $\Delta_{H}$. Also in Section 3, by introducing the notation of Ronkin function, we show the amoeba of a Laurent polynomial has a canonical deformation retract named tropical spine. Then near the large radius limit, we could calculate the tropical spine of a mirror curve concretely. Further, such a tropical spine gives a dual subdivision of $T_{\Sigma}$. These steps lead to our first Theorem 3.4.

For the second result, we mainly finish the proof with some symmetry propositions of the distribution rules of coefficients for the mirror curve as given in Section 2.5. In Section 4, we recall Mikhalkin’s result about cyclic-M curves that if $\mathbb{R}A$ is a cyclic-M curve and $A$ is the zeroes of the Laurent polynomial $p$ in $(\mathbb{C^{*}})^{2}$, then the amoeba map $\mu_{A}$ is an embedding on the boundary of amoeba and 2-1 inside the amoeba. This fact explains why we hope the mirror curve is a cyclic-M curve. Because we have already known the deformation retract of a mirror curve’s amoeba, if the mirror curve is a cyclic-M curve, we could locally figure out the topology of the mirror curve. However, generally for the real parameters near the large radius limit, the mirror curve is not a cyclic-M curve. In Section 5, we first give the choice of signs for the parameters in Section 5.1 and explain the motivation for such a choice. Then in Section 5.2, we prove under such a choice of coefficients, any interior lattice point in Newton polytope $\Delta_{H}$ corresponds to a unique bounded connected component of mirror curve in $(\mathbb{R}^{*})^{2}$. Thus the mirror curve is an M-curve. Finally, in Section 5.3, we construct arcs connecting any two adjacent points of intersection of the unbounded component and axes of toric surface $\mathbb{R}X_{\Delta_{p}}$ by the intermediate value theorem. Then we get Corollary 5.3.4.

I would sincerely appreciate my advisor for the undergraduate research Prof. Bohan Fang for his patient guidance and useful suggestions both on the project itself and further math research. I would also thank Prof. Shuai Guo for his frequent discussions with me about the project, his helpful advice, and for inviting me to give the report on such a research project. Besides, I would wish to thank Ce Ji, Kaitai He, Zhiyuan Zhang, Jialiang Lin, and Zhengnan Chen for their valuable discussions about the project. This project is partially supported by an undergraduate research grant at School of Mathematical Sciences, Peking University.

In this section, we recall some results in toric variety and tropical geometry, then give the definition of mirror curve. We mainly follow the notation in [7], and we refer to [5][8] for general toric language.

In this chapter, we mainly introduce some combinatorial languages which are used for the description of the mirror curve and its tropicalization. We follow the symbol of [16].

In a subdivision $T$ of $K$, we call $k$-dimensional subsets $k$-cells. If $K$ is a convex polytope, and all cells of $T$ are convex polytopes, we call $T$ a polytopal subdivision. Besides, if $T$ is a subdivision of a 2-dimensional polytope $\Delta$, and all 2-dimensional cells of $T$ are triangles, we call $T$ a triangulation.

If $\tau\subset\sigma$, and $\tau,\sigma$ are both convex sets, we define the cone generated from $\tau$ and $\sigma$ by

Clearly, this set is a convex cone. If C is a convex cone, its dual is defined to be the cone $C^{v}=\{\xi\in\mathbb{R}^{n}|\langle\xi,x\rangle\leq 0,\forall x\in C\}$. Then we will define dual subdivision.

Because $cone(\sigma,\sigma)$ is the affine space spanned by $\sigma$, we get $\sigma$ and $\sigma^{*}$ are orthogonal. Moreover, dim $\sigma$+dim $\sigma^{*}=n$. Figure 4 and Figure 4 show a subdivision of $\mathbb{R}^{2}$ and its dual subdivision. The dual subdivision is an example of triangulation.

In this paper, we only consider the simplicial toric Calabi-Yau 3-fold $X_{\Sigma}$ given by a simplicial fan $\Sigma$ in a lattice $N\cong\mathbb{Z}^{3}$. For the convention, we denote $Hom(N,\mathbb{Z})$ by $M$, then $M\cong\mathbb{Z}^{3}$ is also a lattice. We call $M$ the dual lattice of $N$. Besides, by the definition of the toric Calabi-Yau 3-fold, the canonical divisor of $X_{\Sigma}$ is trivial.

It is known that there are two ways to define a general 3-dim toric variety. The first is to use homogeneous coordinates[5]. For any toric 3-fold $X$, we could write

Here $S$ is a fixed subset under the action of open dense complex torus $(\mathbb{C}^{*})^{3}\subset X$, and k $\mathbb{C^{*}}$-actions are given by

The coefficient matrix

is called the toric charge for $X$. Under such definition, the equivalent description of Calabi-Yau condition is that the sum of elements in any fixed line equals zero, i.e.

The second way to define a toric 3-fold is just the original definition given by a 3-dim simplicial fan $\Sigma\subset N\cong\mathbb{Z}^{3}$ [8]. For a toric 3-fold $X_{\Sigma}$ which corresponds to the fan $\Sigma$, we define the d-dimensional cones in $\Sigma$ by $\Sigma(d)$. For the convention, we always write $\Sigma(1)=\{\rho_{1},...,\rho_{k+3}\}$, where $k+3$ is the number of 1-dimensional cones in $\Sigma$. Besides, we denote the generator of $\rho_{i}$ by $b_{i}$.

By definition, $X_{\Sigma}$ contains $\mathbb{T}=N\bigotimes\mathbb{C^{*}}\cong(\mathbb{C^{*}})^{3}$ as an open dense subset. The natural action of $\mathbb{T}$ on itself could be extended to $X_{\Sigma}$. The lattice $N$ could also be canonically identified with $Hom(\mathbb{C}^{*},\mathbb{T})$. $N$ is so called cocharacter lattice of $\mathbb{T}$. $M=Hom(\mathbb{T},\mathbb{C}^{*})$ is called the character lattice.

If we define $\tilde{N}=\bigoplus\limits_{i=1}^{k+3}\mathbb{Z}\tilde{b_{i}}$, then we have a natural group homomorphism $\psi:\tilde{N}\to N$, which sends $\tilde{b_{i}}$ to $b_{i}$. We denote the kernel of $\psi$ by $L$, then we get the exact sequence

Tensoring $\mathbb{C}^{*}$ in the exact sequence, we have the exact sequence

where $\tilde{\mathbb{T}}\cong(\mathbb{C}^{*})^{k+3}$ has an action on itself. We extend the action to $\mathbb{C}^{k+3}=$Spec $\mathbb{C}[Z_{1},$ $...,Z_{3+k}]$. For any $\sigma\in\Sigma$, we consider $Z_{\sigma}=\prod\limits_{\rho_{i}\notin\sigma}Z_{i}$, define $Z(\Sigma)$ to be the closed subvariety defined by the ideal generated by $\{Z_{\sigma}|\sigma\in\Sigma\}$. Then $X_{\Sigma}$ could be identified with the geometric quotient $\mathbb{C}^{3+k}-Z(\Sigma)/G_{\Sigma}$. This is the relation between the two definitions.

Calabi-Yau condition also has a great description under the second definition. The description is that $X$ Calabi-Yau iff all $b_{i}$ lie in a hyperplane of $N$, or equivalently, there exists a vector $e_{3}^{*}\in M$ which makes $\langle e_{3}^{*},b_{i}\rangle=1$ for any $i=1,...,k+3$. Now we choose $e_{1}^{*},e_{2}^{*}$ which make $\{e_{1}^{*},e_{2}^{*},e_{3}^{*}\}$ a basis of $M$. Then under the dual basis $\{e_{1},e_{2},e_{3}\}$, $b_{i}$ has the coordinate $(m_{i},n_{i},1)$ in the lattice $N$. Since then, we always identify the generator of any 1-dimensional cone $b_{i}$ with a lattice point $(m_{i},n_{i})$ in $N^{\prime}$.

For the convention, we define the Calabi-Yau subtorus $\mathbb{T^{\prime}}$ of $\mathbb{T}$ that

Then $N^{\prime}=ker(e_{3}^{*}:N\to Z)$ could be identified with $Hom(\mathbb{C^{*}},\mathbb{T^{\prime}})$. We define $P_{\Sigma}\subset N^{\prime}$ the convex hull of $(m_{i},n_{i})$. Then $P_{\Sigma}$ is a 2-dim convex polytope. The simplicial fan $\Sigma$ also gives a triangulation of $P_{\Sigma}$. For any $\sigma\in\Sigma(3)$, $\sigma\cap N^{\prime}$ is a triangle. Then a subdivision $T_{\Sigma}$ is given by all these triangles and their faces. Since then, we always use a triangulation $T_{\Sigma}$ to represent a simplicial toric Calabi-Yau 3-fold.

For a general triangulation $T_{\Sigma}$, $X_{\Sigma}$ is a toric orbifold. In [12], the author introduced the extended stacky fan to deal with the such orbifold case. Mirror curves are also generally defined for simplicial toric Calabi-Yau 3-folds. However, in this paper, we only consider the smooth toric manifold case. Because $X_{\Sigma}$ is smooth iff any cone of $\Sigma$ forms a $\mathbb{Z}$-basis of $N$, since then we assume any triangle in $T_{\Sigma}$ has area $\frac{1}{2}$. In other words, triangulation $T_{\Sigma}$ is the finest. Figure 5 shows a finest triangulation.

The mirror curve which we will give a detailed definition later is a complex curve in $(\mathbb{C^{*}})^{2}$ as well as a punctured Riemann surface, which is given by a complex Laurent polynomial in two variables with the parameter $q$. Sometimes, we need to compactify it on the toric surface defined by Newton polytope of the polynomial to get a smooth Riemann surface and consider genus or real connected components on the toric surface instead of $(\mathbb{C^{*}})^{2}$ or $(\mathbb{R^{*}})^{2}$. So in this chapter, we review the definition and some basic facts about the toric surface.

Firstly, we follow the definition in Fulton’s book[8]. Let $\Delta$ be a convex polytope in $M_{\mathbb{R}}$ with integer vertices $A_{1},A_{2},...,A_{k}$ in counterclockwise order and $\{\Delta_{i}=A_{i}A_{i+1},i=1,2,...,k\}$ be k sides of $\Delta$, where $A_{1}=A_{k+1}$. For any face of $\Delta$, specifically, all segments, vertices, and the polytope itself (the first two are called proper faces), we consider their dual cones. If $\Delta_{P}$ is a face of $\Delta$, the dual cone $\sigma_{P}=\{v\in N_{\mathbb{R}}|\langle v,u_{1}\rangle\geq\langle v,u_{2}\rangle$, for any $u_{1}\in\Delta_{P},u_{2}\in\Delta\}$. It is easy to see that all dual cones form a fan in $N_{\mathbb{R}}$. Then the fan defines a $2$-dimensional toric variety $X_{\Delta}$, we called this variety the toric surface associated with convex polytope $\Delta$. $X_{\Delta}$ contains $(\mathbb{C^{*}})^{2}$ as an open dense subset.

Next we move on to the real toric surface $\mathbb{R}{X_{\Delta}}$, which is the closure of $(\mathbb{R^{*}})^{2}$ in $X_{\Delta}$. For a non-singular algebraic curve on a toric surface $X_{\Delta}$ whose Newton polytope coincides with the polytope $\Delta$, we could show the genus g of such a curve equals the number of interior lattice points of the Newton polytope by calculating the dimension of global differential forms on the curve[7]. By Harnack’s inequality [10], there are at most g+1 connected components of the curve on the real toric surface.

A more concrete and useful description of the toric surface is given in [7]. Let $\Delta_{1}$, …, $\Delta_{n}$ be the sides of the polytope $\Delta$. We take the dual 1-dimensional cone $\rho_{i}$ of each $\Delta_{i}$. For $\rho_{i}=\mathbb{Z}_{\geq 0}b_{i}$, considering $\tilde{N}=\bigoplus\limits_{i=1}^{n}\mathbb{Z}\tilde{b_{i}}$ and the natural map $\psi:\tilde{N}\to N$, which sends $\tilde{b_{i}}$ to $b_{i}$, we have the exact sequence:

Tensoring $\mathbb{C}^{*}$ on the exact sequence, we have

$G_{\Delta}$ has a natural action on $(\mathbb{C}^{*})^{n}$. We could extend the action to $\mathbb{C}^{n}$. Then we define $X_{\Delta}=\mathbb{C}^{n}/G_{\Delta}$ by the toric surface, with the open dense subset $(\mathbb{C}^{*})^{n}/G_{\Delta}\cong(\mathbb{C}^{*})^{2}$. Moreover, we define $l_{i}=\{(z_{1},...,z_{n})\in\mathbb{R}X_{\Delta}|z_{i}=0\}$ under the homogeneous coordinate $(z_{1},...,z_{n})$ to be the axis of the real toric surface $\mathbb{R}X_{\Delta}$ related to $\Delta_{i}$. For any $i=1,...,n$, the axis $l_{i}$ is a boundary divisor of $\mathbb{R}X_{\Delta}$.

To give the exact definition of the mirror curve, we must firstly introduce the Nef cone with respect to a simplicial fan. One could find the general definition of Nef cone in [5]. But in this paper, still for the convention, we only consider the case of smooth toric Calabi-Yau 3-fold. Recall the exact sequence

and take $Hom(-,Z)$ on the exact sequence of free $Z$-module. We get

We assume $\#\Sigma(1)=p+3$ and then consider the $\mathbb{T}$-equivalent Poincare dual $D_{i}^{\mathbb{T}}$ of the divisor $\{Z_{i}=0\}$ in $\mathbb{C}^{p+3}$. Here $D_{i}^{\mathbb{T}}$ is also the dual basis of $\tilde{b_{i}}$ for $i=1,2,..,p+3$. Next we define $D_{i}=\psi^{v}(D_{i}^{\mathbb{T}})$ in $L^{v}$. For any $\sigma\in\Sigma(3)$, let $I_{\sigma}$ be $\{i|b_{i}\notin\sigma\}$ and $I_{\sigma}^{\prime}$ be $\{i|b_{i}\in\sigma\}$. Then $\{D_{i}|i\in I_{\sigma}\}$ form a basis of $L^{v}$.

In this chapter, we define the mirror curve of a smooth toric Calabi-Yau 3-manifold. For $\Sigma$, we give any lattice point of $P_{\Sigma}$ a monomial with parameter $q=(q_{1},...,q_{p})$. For the convention, we always assume $b_{1}=(1,0)$, $b_{2}=(0,1)$, $b_{3}=(0,0)$. We choose a basis $\{H_{i}|i=1,2,...,p\}$ of $L_{\mathbb{Q}}^{v}$ in the Nef cone of $\Sigma$. Then for any $\sigma\in\Sigma(3)$, because $\{D_{i}|i\in I_{\sigma}\}$ form a basis of $L_{\mathbb{Q}}^{v}$, we could write $H_{i}=\sum\limits_{j\in I_{\sigma}}s_{i,j}^{\sigma}D_{j}$ for $i=1,2,..,p$. Here $s_{i,j}^{\sigma}$ is a non-negative rational number for any $i,j,\sigma$. We now take $H_{1},...,H_{p}$ which make all $s_{i,j}^{\sigma}$ non-negative integers and $S_{\sigma}=[s_{i,j}^{\sigma}]_{p*p}$ non-degenerate. Then for the parameter $q$, we define

A flag $(\tau,\sigma)$ of $\Sigma$ consists of two cones $\sigma\in\Sigma(3)$ and $\tau\in\Sigma(2)$ with $\tau\subset\sigma$. Then for any flag $(\tau,\sigma)$, there exist the unique $i_{1},i_{2},i_{3}$ which make $I_{\sigma}^{\prime}=\{i_{1},i_{2},i_{3}\}$, $I_{\tau}^{\prime}=\{i_{2},i_{3}\}$, $I_{\sigma}=\{1,2,...,p+3\}-I_{\sigma}^{\prime}$, and $I_{\tau}=\{1,2,...,p+3\}-I_{\tau}^{\prime}$, where $b_{i_{1}},b_{i_{2}},b_{i_{3}}$ satisfy the counterclockwise order in $\sigma$. For the convention, we call $b_{i_{3}}$ the origin of flag $(\tau,\sigma)$.

Given a flag $(\tau,\sigma)$, if $\boldsymbol{b_{i_{1}}}-\boldsymbol{b_{i_{3}}}=(m_{i_{1}},n_{i_{1}})$, $\boldsymbol{b_{i_{2}}}-\boldsymbol{b_{i_{3}}}=(m_{i_{2}},n_{i_{2}})$, we define

Then $m_{i_{1}}n_{i_{2}}-m_{i_{2}}n_{i_{1}}=1$. Any lattice point $b_{j}$ of $P_{\Sigma}$ could be written $m_{j}^{\prime}e_{1,(\tau,\sigma)}+n_{j}^{\prime}e_{2,(\tau,\sigma)}+b_{i_{3}}$. We call $(m_{j}^{\prime},n_{j}^{\prime})$ the coordinate of $b_{j}$ under the flag $(\tau,\sigma)$.

Specially, when $(\tau_{1},\sigma_{1})$ satisfies the condition $I_{\sigma_{1}}^{\prime}=\{1,2,3\},I_{\tau_{1}}^{\prime}=\{2,3\}$, for the convention, we write $H_{(\tau_{1},\sigma_{1})}(X_{(\tau_{1},\sigma_{1})},Y_{(\tau_{1},\sigma_{1})},q)$ as $H(X,Y,q)$. From now on, if we talk about a mirror curve without a given flag, then we mean the mirror curve of flag $(\tau_{1},\sigma_{1})$.

There are many good propositions of the mirror curve. The most elementary and important proposition of the mirror curve is the affine equivalence.

By the affine equivalence, when we want to study propositions of one fixed flag or we need a fixed flag, we may do the coordinate change and send the three vertices of the flag to $(0,0),(0,1),(1,0)$.

Two deeper and more useful propositions about mirror curves show the distribution rules of $a_{i}(q)$ at each lattice point.

Figure 6 shows this case.

Figure 7 shows this case.

In this chapter, we recall some basic notations and tools of tropical geometry. First, let us recall the notation of amoeba. Consider the map given below.

We define such a map by the amoeba map.

Amoeba is the core object of tropical geometry. There are many great propositions about the amoeba. Next, we list one and give a brief idea of the proof. If anyone is interested in the details, he could refer to [17].

Since then, for short, if $\psi(C)=\sigma$, we say that the component $C$ has the degree $\sigma$. At the end of this chapter, I will introduce the notation of the tropical Laurent polynomial.

Specifically, the tropical Laurent polynomial with $n$ variables could be written as

where $c_{i_{r}}$ is an integer, and $w_{r}^{c_{i_{r}}}$ is short for $w_{i}\boxplus...\boxplus w_{i}$ which has total $c_{i_{r}}$ $w_{i}$.

Because the tropical Laurent polynomial could be written as $max\{f_{i}\}_{i\in J}$, where $f_{j}=a_{j}+c_{j_{1}}w_{1}+...+c_{j_{n}}w_{n}$, we define the tropical hypersurface of a tropical Laurent polynomial to be the union of all the points in $(\mathbb{R}\cup\{-\infty\})^{n}$ which make at least two $i\in J$, denoted by $i_{1},i_{2}$, satisfy $f_{i_{1}}=f_{i_{2}}=max\{f_{i}\}_{i\in J}$. It is obvious that the tropical hypersurface consists of some segments, rays, and lines. Actually, the tropical hypersurface is exactly the set of critical points of $max\{f_{i}\}_{i\in J}$. Since then, we write the tropical polynomial short for the tropical Laurent polynomial.