Scattering diagrams in mirror symmetry

TL;DR

This paper explores the role of scattering diagrams in mirror symmetry, particularly their application to the SYZ and HMS conjectures, highlighting their importance in the reconstruction problem.

Contribution

It introduces the main ideas of scattering diagrams in mirror symmetry and explains their relevance to key conjectures in the field.

Findings

Scattering diagrams are crucial tools in understanding mirror symmetry.

They provide insights into the reconstruction problem in mirror symmetry.

The paper clarifies the connection between scattering diagrams, the SYZ conjecture, and the HMS conjecture.

Abstract

Since the pioneering work of Kontsevich and Soibelman [51], scattering diagrams have started playing an important role in mirror symmetry, in particular in the study of the reconstruction problem. This paper aims at introducing the main ideas on the subject describing the role of scattering diagrams in relation to the SYZ conjecture and the HMS conjecture.

Equations

$$H^1(X,T_X)\cong H^1(X,{\Omega }_X^2{)}^{\vee }\cong H^{2,1}(X{)}^{\vee }$$

$${\mathcal{K}}_X:=\{[\omega ]\in H^{1,1}|[\omega ]\text{ is represented by a Ka¨hler form}\}\underset{open}{\subset }{H}^{1,1}(X,\mathbb{R})$$

$${\mathcal{K}}_X^{\mathbb{C}}:=(H^2(X,\mathbb{R})\oplus i{\mathcal{K}}_X)/H^2(X,\mathbb{Z}).$$

$$\dim H^1(X,T_X)=h^{2,1}(X)$$

$$\dim H^1(\check{X},T_{\check{X}})=h^{2,1}(\check{X})$$

$$X:=T^{\ast }{B}_0/{\Lambda }^{\ast }\to B_0$$

$$\omega =\sum _{i}d{x}^i\wedge d{y}^i$$

$$g_{ij}(x)=\frac{{\partial }^{2}h}{\partial x_i\partial x_j}$$

$$J\left(\frac{\partial }{\partial y_i}\right)=-\sum _j{g}^{ij}(x)\frac{\partial }{\partial x_j}=-\sum _j\frac{{\partial }^{2}h}{\partial x^i\partial x^j}\frac{\partial }{\partial x_j}=-\frac{\partial }{\partial {\check{x}}_i}.$$

$$\check{\omega }\left(\frac{\partial }{\partial {\check{x}}_i},\check{J}\left(\frac{\partial }{\partial {\check{x}}_j}\right)\right):=\check{g}\left(\frac{\partial }{\partial {\check{x}}_i},\frac{\partial }{\partial {\check{x}}_j}\right)$$

$$g_1\cdot g_2\cdot ...\cdot g_k\cdot ...\cdot g_2^{-1}\cdot g_1^{-1}={\mathrm{id}}_{\mathbb{G}}.$$

$${\theta }_j:=\theta [0,{\beta }_j](\tau ,z)=\sum _{m\in \mathbb{Z}}\exp \{2\pi i[\tau \frac{m^2}{2}+m(z+{\beta }_j)]\}.$$

$${\mathcal{D}}^{b}Coh({\check{X}}_q)\cong {\mathcal{F}}^0(X_q)$$

$${\mathfrak{g}}_{>n}=\underset{\gamma \in \Gamma \setminus 0,||\gamma ||>n}{⨁}{\mathfrak{g}}_{\gamma }$$

$${\mathfrak{g}}_{\le n}=\mathfrak{g}/{\mathfrak{g}}_{>n}\cong \underset{\gamma \in \Gamma \setminus 0,||\gamma ||\le n}{⨁}{\mathfrak{g}}_{\gamma }$$

$$\hat{G}=\exp (\hat{\mathfrak{g}})$$

$$\exp (g)\cdot \exp (g^{\prime })=\exp (g+g^{\prime }+\frac{1}{2}[g,g^{\prime }]...)$$

$$\text{Sing}(\mathfrak{D}):=\bigcup _{({\mathfrak{d}}_{\gamma },{\theta }_{\gamma })\in \mathfrak{D}}\partial {\mathfrak{d}}_{\gamma }\cup \bigcup _{{\mathfrak{d}}_1,{\mathfrak{d}}_2}{\mathfrak{d}}_1\cap {\mathfrak{d}}_2$$

$${\Theta }_{\gamma ,\mathfrak{D}}:=\underset{k\to \infty }{\lim }{\Theta }_{\ell ,\mathfrak{D}}^k$$

$$\mathfrak{g}:=\big{(}\mathbb{C}[\Lambda]{\otimes}_{\mathbb{C}}(t)\mathbb{C}[\![t]\!]\big{)}\otimes_{\mathbb{Z}}\Lambda^{*}$$

$$[z^m{\partial }_n,z^{m^{\prime }}{\partial }_{n^{\prime }}]:=z^{m+m^{\prime }}{\partial }_{⟨m^{\prime },n⟩n^{\prime }-⟨m,n^{\prime }⟩n}.$$

$$\mathfrak{h}:=\big{(}\bigoplus_{m\in\Lambda\setminus\{0\}}\mathbb{C}z^{m}\cdot(t)\otimes_{\mathbb{Z}}m^{\perp}\big{)}\hat{\otimes}\mathbb{C}[\![t]\!]$$

$${\theta }_{(a,b),f_{(a,b)}}(x)=f^{-b}x{\theta }_{(a,b),f_{(a,b)}}(y)=f^{a}y.$$

$${\theta }_{(a,b),f_{(a,b)}}=\exp \left(\log (f_{(a,b)}){\partial }_{(-b,a)}\right)\in \mathbb{V}.$$

$$z^m\cdot z^{m^{\prime }}=z^{m+m^{\prime }}.$$

$$\{z^m,z^{m^{\prime }}\}=⟨m,m^{\prime }{⟩}_D{z}^{m+m^{\prime }}$$

$${\hat{z}}^m\cdot {\hat{z}}^{m^{\prime }}=q^{\frac{1}{2}⟨m,m^{\prime }{⟩}_D}{\hat{z}}^{m+m^{\prime }}.$$

$$\hat{\mathfrak{h}}:=(t)\left(\underset{m\in \Lambda \setminus \{0\}}{⨁}\mathbb{C}[q^{\pm \frac{1}{2}}]{\hat{z}}^m\right){\hat{\otimes }}_{\mathbb{C}}\mathbb{C}[[t]]$$

$${\hat{\theta }}_m$$

$$\tilde{\mathfrak{h}}:=\left(\underset{m\in \Lambda \setminus \{0\}}{⨁}\mathbb{C}{z}^m(t){\otimes }_{\mathbb{Z}}\left(\mathfrak{gl}(r,\mathbb{C})\oplus m^{\perp }\right)\right){\hat{\otimes }}_{\mathbb{C}}\mathbb{C}[[t]]$$

$$\begin{gathered} [(A,{\partial }_n)z^m,(A^{\prime },{\partial }_{n^{\prime }})z^{m^{\prime }}{]}_{\sim }:=([A,A^{\prime }{]}_{\mathfrak{gl}}{z}^{m+m^{\prime }}+A^{\prime }⟨m^{\prime },n⟩z^{m+m^{\prime }}-A⟨m,n^{\prime }⟩z^{m+m^{\prime }}, \\ [z^m{\partial }_n,z^{m^{\prime }}{\partial }_{n^{\prime }}{]}_{\mathfrak{h}}). \end{gathered}$$

Full text

1. Introduction

2. 1 Mirror symmetry

3. 1.1 Strominger–Yau–Zaslow conjecture

4. 1.2 Kontsevich–Soibelman approach via non–archimedean geometry

5. 1.3 Gross–Siebert approach via logarithmic geometry

6. 1.4 Fukaya approach via multi-valued Morse theory: closed string conjecture

7. 1.5 Homological Mirror Symmetry of Kontsevich: mirror symmetry as an equivalence of categories

8. 1.6 Fukaya approach via multi-valued Morse theory: open/closed string conjecture

9. 2 Scattering diagrams

10. 2.1 Scattering diagrams in the tropical vertex group

11. 2.2 Scattering diagrams in the quantum tropical vertex

12. 2.3 Scattering diagrams in the extended tropical vertex group

Introduction

First conjectured by string theorists, mirror symmetry has been largely studied and it remains an active field of research in geometry and physics.

In its early days, a great effort was devoted to rigorous formulating mirror symmetry leading to the Strominger–Yau–Zaslow conjecture [57], the Kontsevich’s Homological Mirror Symmetry conjecture [50], the Gross–Siebert program [37, 39, 40], and others even beyond Calabi–Yau varieties.

In addition, one of the first problems addressed by mathematicians was to understand how to reconstruct the mirror of a given family of Calabi–Yau varieties. For elliptic curves, Polishchuk and Zaslow proved that elliptic curves are self-mirror dual meaning the bounded derived category of coherent sheaves of an elliptic curve is equivalent to a suitably modified Fukaya category of the mirror elliptic curve. Already for K3 surfaces, the construction of the mirror is more involved and it was first computed by Kontsevich and Soilbelman for an analytic K3 over a non–archimedean field, using combinatorial structures later called scattering diagrams [51]. As first conjectured by Strominger, Yau, and Zaslow, mirror pairs of Calabi–Yau varieties come in families, and in a suitable limit, they admit dual torus fibrations. Then, according to Kontsevich–Soibelman, scattering diagrams consist of a collection of walls in the base of the fibration decorated with automorphisms. Furthermore, they must encode enumerative geometric data of the mirror (quantum corrections).

The aim of this paper is to give a gentle introduction to scattering diagrams regarded as one the essential tools to study the reconstruction problem in mirror symmetry. What really characterizes scattering diagrams is the choice of automoprhisms that decorate the walls of the diagram. Depending on the problem, different choices of groups are possible: for instance for log Calabi–Yau surfaces either the tropical vertex group $\mathbb{V}$ of Gross, Pandharipande, and Siebert [36] or the quantum tropical vertex $\hat{\mathbb{V}}$ of Bousseau [9] can be chosen. With the first choice, the quantum corrections are expressed in terms of genus-zero Gromov–Witten invariants, while with the second choice higher genus Gromov–Witten invariants contribute. 111In the latter case, the mirror is the deformation quantization of the mirror constructed by Gross–Hacking–Keel using scattering diagrams in $\mathbb{V}$ [34]. The construction developed in [8] can be considered as a generalization of the reconstruction problem, that goes beyond the one of the Strominger–Yau–Zaslow mirror symmetry. Recently, the author has introduced an extension of $\mathbb{V}$, the so-called extended tropical vertex group $\tilde{\mathbb{V}}$. Morally, it is expected that scattering diagrams in $\tilde{\mathbb{V}}$ should prescribe quantum corrections to reconstruct the mirror of a log Calabi–Yau surface together with a holomorphic vector bundle over it.

The paper is divided into two main sections: in the first one (Section 1) we present a general overview of the mirror symmetry conjectures and of the reconstruction problem. In particular, we are going to present the approach of Kontsevich–Soibelman through non–archimedean geometry, the Gross–Siebert approach through logarithmic geometry, and the analytic approach of Fukaya. The second part (Section 2) is then devoted to scattering diagrams, including the examples of $\mathbb{V},\hat{\mathbb{V}}$ and $\tilde{\mathbb{V}}$.

Acknowledgement

The author wishes to thank professors Jacopo Stoppa and Nicoló Sibilla for their courses respectively on GHK construction and on an introduction to mirror symmetry held at SISSA in Spring 2021. We are also thankful to the organizers of the ”First UMI meeting for PhD” held in Padova in May 2022 for the opportunity to present part of this work. Finally, we thank the anonymous referee for his precious comments and corrections which improved and enriched the exposed material.

1 Mirror symmetry

Mirror symmetry was first conjectured by string theorists222See [12, 30]; for more general introductive material see for example the following books [53, 21, 5, 46]. Other references will be mentioned later on., who observed that mirror pairs of Calabi–Yau333Calabi–Yau manifolds are complex Kähler manifolds which admit Ricci flat Kähler metric. We denote by $\omega$ the Kähler form and by $J$ the complex structure. We refer to [35] for a nice introduction on Calabi–Yau geometry with an eye toward mirror symmetry (in particular in Chapter 2). 3–folds $(X,\omega,J)$ and $(\check{X},\check{J},\check{\omega})$ give the same theory. As a first definition of mirror pairs we may say that $X$ and $\check{X}$ have mirror Hodge diamonds as in Figure 1, namely $h^{1,1}(X)=h^{2,1}(\check{X})$ and $h^{2,1}(X)=h^{1,1}(\check{X})$.

This symmetry underlies a duality between the complex moduli and the complexified Kähler moduli between the mirror pairs: indeed on the one hand the deformations of the complex structure of $X$ are parametrized by $H^{1}(X,T_{X})$ and by Serre duality and Dolbeault’s theorem we have the following chain of equivalences

[TABLE]

On the other hand, the Kähler moduli is the cone of Kähler classes $\omega\in H^{1,1}(X,\mathbb{C})\cap H^{2}(X,\mathbb{R})$ associated to a fixed Kähler metric on $X$

[TABLE]

Since the complex moduli is a complex manifold, it is necessary to consider the complexified Kähler moduli defined as

[TABLE]

Recall that if $X$ is a smooth projective variety, the number $h^{p,q}(X)$ are the dimension of complex vector space $H^{p,q}(X)$. Therefore, having mirror Hodge diamonds reflects the following duality between moduli

[TABLE]

The breakthrough, attracting the interest of the geometry community, was the paper by Candelas, De la Ossa, Green, and Parkers [11] who were able to compute the number of rational curves (up to degree 9) on a quintic 3-fold by studying solutions of Picard–Fuchs equations. The number of rational curves (also known as Gromov–Witten invariants) are invariants that only depend on the symplectic form $\omega$, and with the algebraic-geometric techniques available at that time only the number of curves up to degree 3 was known. While the Picard–Fuchs equations are equations for periods of $\check{X}$ with respect to the holomorphic volume form of $(\check{X},\check{J})$. Hence we see again mirror symmetry as a duality between complex $(\check{X},\check{J})$ and the symplectic $(X,\omega)$ (and vice-versa between $(\check{X},\check{\omega})$ and $(X,J)$).

So far we have seen one of the first aspects of mirror symmetry for Calabi–Yau 3-folds, but later on, different people worked on a geometric formulation of the mirror symmetry conjecture. Among them we will recover the Strominger–Yau–Zaslow (SYZ) conjecture [57] and the Homological Mirror Symmetry of Kontsevich [49].

In addition, geometers started looking for examples of mirror pairs (beyond the 3-folds) by studying the so-called reconstruction problem, namely how to build the mirror of a given Calabi–Yau. We will recover some of the main results addressing the reconstruction problem: Kontsevich–Soibelman mirror symmetry via non–archimedean geometry, the Gross–Siebert program via logarithmic geometry, and Fukaya’s mirror symmetry via multi-valued Morse theory. As discussed in the following sections, their common aspect can be traced back to the use of scattering diagrams.

More recently, other progresses have been pursued in the study of the reconstruction problem from different perspectives and using techniques that go beyond scattering diagrams. We will recall some of them in the exposition but without giving many details.

1.1 Strominger–Yau–Zaslow conjecture

The Strominger–Yau–Zaslow conjecture (SYZ) claims that mirror symmetry is T-duality, namely that pairs of Calabi–Yau 3-folds $X$ and $\check{X}$ carry special Lagrangian torus fibrations (with possible singular fibers) $\pi\colon X\to B$ and $\check{p}\colon\check{X}\to B$. Away from singularities, T-duality acts on the smooth fibers as a map from a Lagrangian fiber $X_{b}=\pi^{-1}(b)$ to the moduli of unitary flat connections on a special Lagrangian fiber of the mirror $H^{1}(\check{X}_{b},\mathbb{R}/\mathbb{Z})$. However, due to the presence of singularities, it is necessary to add corrections 444In the 1996 paper [57] quantum corrections are needed to construct the Ricci-flat Kähler metric on the mirror. Although the metric approach seemed originally out of reach in higher dimensions (handling quantum corrections seemed also difficult due to the fact that already in 3 dimensions the critical locus of Lagrangians fibrations is not of codimension 2 [47]), new progress has been made by Collins, Jacobs and Lin [20, 19] but mostly in 2 dimensions. . These should encode enumerative geometric data such as counting holomorphic discs.

We illustrate this duality in the following toy model example, where we avoid singularities.

Toy model I

Let $B_{0}$ be an integral affine smooth manifold (i.e. with transition functions in $GL(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$). Let $\Lambda$ be an integer lattice subbundle of the tangent bundle of $B_{0}$, $\Lambda\subset TB_{0}$, and its dual lattice $\Lambda^{*}=\operatorname{Hom}(\Lambda,\mathbb{Z})\subset T^{*}B_{0}$. Then we define two torus fibrations

[TABLE]

such that $X$ comes with a natural symplectic form $\omega$ and $\check{X}$ with a natural complex structure $J$; in local affine coordinates $(x^{i},y^{i})$

[TABLE]

Then $(X,\omega)$ and $(\check{X},\check{J})$ are mirror pairs that admit dual torus fibrations and such that the symplectic coordinates of $X$ are the complex coordinates of $\check{X}$. This is only the half of mirror symmetry; if $B_{0}$ admits a Riemannian metric $g$ of hessian type, i.e. there exists a function $h$ such that in local coordinates

[TABLE]

then $\check{x}_{i}:=\frac{\partial h}{\partial x_{i}}$ is a complex coordinate on $X$ with respect to the complex structure $J\left(\frac{\partial}{\partial x_{i}}\right)=\sum_{j}g_{ij}(x)\frac{\partial}{\partial y_{j}}$. Indeed,

[TABLE]

The change of coordinates $\check{x}_{i}:=\frac{\partial h}{\partial x_{i}}$ is a Legendre transform. Furthermore, assuming $\det\left(\frac{\partial^{2}h}{\partial x_{i}\partial x_{j}}\right)=1$, $g$ gives a Kähler metric on $X$ compatible with $J$ and $\omega$. Conversely, let $\check{g}=\sum_{j}d\check{x}^{j}d\check{x}^{j}$ be another metric on $B_{0}$: we define

[TABLE]

to be the compatible Kähler form with respect to $\check{g},\check{J}$. Then $\check{\omega}=\sum_{j}g_{ij}(x)d\check{x}^{i}\wedge dy^{j}$ hence $\check{x}^{i},y^{i}$ are symplectic coordinates on $\check{X}$. Schematically, we represent mirror symmetry for $X$ and $\check{X}$ as follows

{(X,\omega)}$${(X,J)}$${(B_{0},g)}$${(B_{0},\check{g})}$${(\check{X},\check{J})}$${(\check{X},\check{\omega})}mirrormirrorLegendre

1.2 Kontsevich–Soibelman approach via non–archimedean geometry

Following the proposal of the SYZ conjecture, in [50] Kontsevich and Soibelman propose to replace the dual SYZ torus fibration with a non-archimedean torus fibration. In this framework, they address the reconstruction problem of K3 surfaces, whose mirror will be constructed as an analytic space over a non–archimedean field [51].555This approach successfully applies in [60] where the author constructs the mirror of a singular Lagrangian torus fibration as a rigid analytic space obtained by gluing local affine pieces with transition functions encoding counting pseudo-holomorphic discs. However, the mirror space is constructed on the smooth locus of the base. Their construction is built on some additional structures on the base of the SYZ fibration, namely the so-called scattering diagrams. 666In fact the name scattering diagrams does not appear in the original work of Kontsevich and Soibelman, but it was later introduced by Gross and Siebert in [40]. In [51] Section $9$, the authors rather talk about lines on the base of the fibration as gradient lines satisfying certain axioms. In the following, we will refer to rays as the image of gradient lines under a local homeomorphism from $U\subset B$ to $\mathbb{R}^{2}$ (see [51, Axiom 6]).

As suggested by SYZ conjecture, mirror pairs of Calabi–Yau admit locally dual torus fibrations; these arise naturally in both rigid analytic complex and symplectic geometry. On the one hand, they consist of a smooth map $\pi\colon X\to B$ from a variety $X$ defined over non–archimedean field to a CW complex $B$. On the other hand, they are integrable systems. In addition, in both cases, the base manifold $B$ carries an integral affine structure with singularities. Conjecturally, integral affine manifolds with singularities arise as the Gromov–Hausdorff limits of families of Calabi–Yau (see [51, Conjecture 1] and [45, Conjecture 6.2], in fact for K3 surfaces the Conjecture was proved by Gross and Wilson [45, Theorem 6.4]). Hence, away from singularities, it should be possible to locally reconstruct the torus fibration from the base manifold, more precisely from its integral affine structure. However, due to the presence of singularities, it is necessary to consider additional structures in order to construct globally the fibration. Here is where the scattering diagrams come into play.

We now discuss the example of K3 surfaces, following [51]. From the symplectic side, elliptically fibered K3 surface $(X,\Omega_{X})$ gives a complex integrable system $(X,\mathbb{C}\mathbb{P}^{1},\omega)$: a holomorphic fibration $\pi\colon X\to\mathbb{C}\mathbb{P}^{1}$ whose fibers are elliptic curves, $\omega=\mathrm{Re}\,\Omega_{X}$ with $\Omega_{X}$ the holomorphic $2$-form on $X$ and $B=\mathbb{C}\mathbb{P}^{1}$ has 24 singular fibers. The integral affine structure on $B$ is induced from the one of $\mathbb{R}^{2}$ with nontrivial monodromy around the singularities. Then Kontsevich and Soibelman construct the mirror $\check{X}$ as an analytic space over $K=\mathbb{C}(\!(t)\!)$. We recall the heuristic of the main steps of their construction and we refer to [51] for details:

• •

on the smooth locus $B_{0}\subset B$ the mirror is locally isomorphic to the dual torus fibration: the group $SL(2,\mathbb{Z})\ltimes(K^{\times})^{2}$ (induced by the integral affine structure of $B_{0}$) acts by automorphisms of the fibration $\pi_{can}\colon(\mathbb{G}_{m}^{an})^{2}\to\mathbb{R}^{2}$.

• •

in a neighborhood of the singularity $U_{s}\subset B$, the local sheaf of functions $\mathcal{O}^{sing}$ is defined by a special choice of gluing automorphisms.

• •

in order to glue the local model of singularities to the canonical one, they introduce a new structure: consider a ray emanating from the singularity together with an automorphism and deform the sheaf of functions across the ray (as in Figure 2).

• •

assuming that two rays meet transversely, the action of the two automorphisms $g_{1},g_{2}$ must be consistent, i.e. the modified sheaves $\mathcal{O}^{mod}_{1},\mathcal{O}^{mod}_{2}$ must glue together in a neighborhood of the intersection point. If $g_{1},g_{2}$ commute, then $\mathcal{O}^{mod}_{1},\mathcal{O}^{mod}_{2}$ glue well. On the contrary, if $g_{1},g_{2}$ do not commute, then the deformation of $\mathcal{O}^{can}$ must receive other (possibly infinitely many) corrections by $g_{1},g_{2},...,g_{k},...$ with the property that

[TABLE]

The latter property assures that the monodromy of the fiber above the intersection point remains trivial. Locally, the set of lines and rays meeting at a smooth point, together with the automorphisms $g_{k}$ is called a scattering diagram (see Figure 3).

• •

finally they construct a nowhere vanishing top degree analytic form $\Omega$, which extends over the modified sheaves without modifying the $K$-affine structure of $B$.

We conclude this section with a simple example: the elliptic curve.

Example: elliptic curve I

Let $\pi\colon X\to B$ be an affine map of tori with $X:=\mathbb{R}^{2}/\Lambda$ and $\Lambda\cong\mathbb{Z}^{2}$, $B=\mathbb{R}/\Lambda^{\prime}$ and $\Lambda^{\prime}\cong\mathbb{Z}$ (in particular we assume $\Lambda^{\prime}=\alpha\mathbb{Z}$ for $\alpha\in\mathbb{R}\setminus\{0\}$). The base $B$ is smooth and it has a natural affine structure induced by $\Lambda^{\prime}$.Then, the dual torus fibration $\check{\pi}\colon\check{X}\to B$ can be easily built from the integral affine structure of $B$.

Let $K=\mathbb{C}(\!(q)\!)$ be the ring of Laurent polynomial in $q$, and let $\rho\colon\mathbb{Z}\to K^{\times}$ be a group homomorphism such that the image of the composition $\mathit{val}\circ\rho\colon\mathbb{Z}\to\mathbb{R}$ is the lattice $\Lambda^{\prime}\subset\mathbb{R}$. Then $q^{\Lambda^{\prime}}=\{q^{n}\}_{n\in\Lambda^{\prime}}$ acts on $\mathbb{C}^{\times}$ by translations and the quotient is a K-analytic space called the Tate curve $\check{X}:=\mathbb{C}^{\times}/q^{\Lambda^{\prime}}$. The dual fibration is $\check{\pi}\colon\check{X}\to B$.

Since every elliptic curve over $\mathbb{C}$ is isomorphic to $\mathbb{C}/\Lambda_{\tau}$ with $\Lambda_{\tau}=\mathbb{Z}+\tau\mathbb{Z}$ and $\tau$ in the upper half-plane, the map $\mathbf{e}\colon z\to e^{2\pi iz}$ induces an isomorphism $\mathbb{C}/\Lambda_{\tau}\cong\check{X}$ with $q=\mathbf{e}(\frac{\tau}{\alpha})$.

1.3 Gross–Siebert approach via logarithmic geometry

Scattering diagrams appear also in the Gross–Siebert program [37, 39, 40] 777The reader may find useful the following readings [42, 29]. which solves the reconstruction problem through logarithmic geometry and in all dimensions.888Although running in a very different direction, the recent work of Groman and Varulgones [31, 32] addresses the reconstruction problem in the spirit of the Gross–Sibert program. In particular, they replace the set of rays of [51] (which is limited to dimension $2$) by a codimension one polyhedral complex of walls.

In particular, an interesting class of examples is given by log Calabi–Yau pairs. In two dimensions the reconstruction of the mirror of log Calabi–Yau surfaces has been studied in [34, 8] (among which some toric examples are computed in [7]) and of non-compact 3-folds as local $\mathbb{P}^{2}$ (see [38, Examples 5.1-5.2] and [41]). In higher dimension, some examples of computing the mirror of log Calabi–Yau pairs are given in [3]. We review below the construction of [34], which reinterpret the construction of the Gross–Siebert program in terms of curve counting (crucially using [36]), broken lines and generalized theta functions (introduced in [33]).

Let $(Y,D=D_{1}+...+D_{N})$ where $Y$ is a smooth projective surface and $D\in|-K_{Y}|$ is a cycle of rational curves, then $X=Y\setminus D$ is a log Calabi–Yau surface and it admits a holomorphic symplectic form $\Omega_{X}$ with simple poles along the divisor $D$. So $(X,\Omega_{X})$ is non compact Calabi–Yau. The toy model examples of log CY are toric surfaces and their toric boundary such as $\mathbb{P}^{2},\mathbb{P}^{1}\times\mathbb{P}^{1}$ and $dP_{7}$ (which is the blow-up of $\mathbb{P}^{2}$ in two distinct points). More generally, for a given log CY surface $(Y,D)$ there exists a toric blow-up $(\tilde{Y},\tilde{D})$ which has a toric model $(\bar{Y},\bar{D})$ (see the diagram below). Examples of non-toric log CY are the del Pezzo surfaces $dP_{5},dP_{4},dP_{3},dP_{2}$.

Schematically, we summarize the Gross–Siebert program as follows: it takes place at two different levels, namely the mirror map between families of Calabi–Yau $\mathcal{X}\longleftrightarrow\check{\mathcal{X}}$ is already incarnated at the tropical level through a discrete Legendre transform $B\longleftrightarrow\check{B}$.

{\mathcal{X}}$${\check{\mathcal{X}}}$${(B,\varphi)}$${(\check{B},\check{\varphi})}mirror**Legendre

Most important, in order to make sense of the vertical arrows of the diagram it is necessary to use logarithmic geometry. Furthermore, log geometry plays a fundamental role in proving the numerical correspondence between Gromov–Witten invariants and Hodge periods of the mirror.

On the one hand, Gross and Siebert define a discrete Legendre transform as mirror map between $(B,\mathcal{P},\varphi)$ and $(\check{B},\check{\mathcal{P}},\check{\varphi})$ where $B$ is a tropical affine manifold with singularities, $\mathcal{P}$ is a polyhedral decomposition of $B$ and $\varphi$ is a convex multi-valued piecewise linear function on $B$. On the other hand, they show how to build a smooth family $\check{\mathcal{X}}$ from the data of $(\check{B},\check{\mathcal{P}})$ and of a gluing map $\check{s}$ and vice-versa how to recover the data of $(B,\mathcal{P},\varphi)$ and $s$ from the family $\mathcal{X}$. In particular, from $(\check{B},\check{\mathcal{P}},\check{s})$ they construct a log scheme $\check{X}_{0}(\check{B},\check{\mathcal{P}},\check{s})^{\dagger}$ which can be smoothed into a formal family $\check{\mathfrak{X}}$. Furthermore, if the family admits a polarization, then there will be an underlying scheme $\check{\mathcal{X}}$. Conversely, starting with a family $\mathcal{X}$ of Calabi–Yau varieties which admit a toric degeneration i.e. whose central fiber $X_{0}^{\dagger}$ is a union of toric varieties and the map to the base is log smooth away from a singular locus, they recover $({B},{\mathcal{P}})$ and a gluing data $s$ such that $X_{0}=X_{0}(B,\mathcal{P},s)$. If $\mathcal{X}$ is polarized then $B$ comes equipped with a convex multi-valued piecewise linear function $\varphi$. Pictorially, we may sketch the Gross–Siebert construction as follows

{(\mathcal{X},\mathcal{L}_{\blacktriangle})}$${(\check{\mathcal{X}},\check{\mathcal{L}})}$${(X_{0}^{\dagger},s)}$${(\check{X}_{0}^{\dagger},\check{s}_{\bullet})}$${(B,\mathcal{P},\varphi_{\blacktriangle})}$${(\check{B},\check{\mathcal{P}},\check{\varphi})}mirrortoric degen.Legendre$\scriptstyle{\bullet}$

where $\blacktriangle$ denotes a choice of polarization and $\bullet$ a choice of gluing $\check{s}$.

Although, in full generality the Gross–Siebert program is quite technical, in the case when $X=Y\setminus D$ is a toric log CY surface and $D=D_{1}+...+D_{N}$, it simplifies as sketched below: let $\Sigma$ be the fan of the toric surface $Y$ and $\Delta_{(Y,D)}$ be the dual intersection complex

• •

the base $B$ is the topological space underlying $\Delta_{(Y,D)}$, and it admits an integral affine structure that depends on the intersection matrix of $D$.

• •

the base of mirror family is $\operatorname{Spec\,}\mathbb{C}[\mathrm{NE}(Y)]$ where $\mathrm{NE}(Y)$ is the cone of effective curve classes (which is finitely generated by toric divisors)

• •

the construction of the mirror family $\check{\mathcal{X}}\to\operatorname{Spec\,}\mathbb{C}[\mathrm{NE}(Y)]$ mimics toric Mumford’s degenerations [38], and it is obtained as a smoothing of the $N$-vertex $\mathbb{V}_{N}$999The $N$-vertex $\mathbb{V}_{N}:=\mathbb{A}^{2}_{x_{1},x_{2}}\cup...\cup\mathbb{A}^{2}_{x_{N-1},x_{N}}\cup\mathbb{A}^{2}_{x_{N},x_{1}}$ is defined as the union of $N$ coordinates hyperplanes.. In particular, the smooth fibers are algebraic tori $\mathbb{C}^{*}\times\mathbb{C}^{*}$ (and they are indeed mirror of $X$).

If $X=Y\setminus D$ is a log CY surface (non-toric) and $D=D_{1}+...+D_{N}$, the previous construction has to be modified. First of all, recall that $\mathrm{NE}_{\geq 0}(Y)$ is not always a finitely generated monoid (for instance if the adjacency matrix is not negetive semi-definite, then $\mathrm{NE}_{\geq 0}(Y)$ is finitely generated). So more generally, one has to consider $\sigma_{P}\subset A_{1}(Y,\mathbb{R})$ a strictly convex rational polyhedral cone containing $\mathrm{NE}(Y)_{\mathbb{R}}$ and its associated monoid $P=\sigma_{P}\cap A_{1}(Y,\mathbb{Z})$. Then, by modifying Mumford degeneration, the mirror family is constructed as a smoothing of $\mathbb{V}_{N}^{0}:=\mathbb{V}_{N}\setminus\{0\}$. However, the smooth fibers must receive corrections from consistent scattering diagrams in order to glue them together with the singular fiber at the origin (details in [34]). The outcome of this procedure gives a family $\check{X}_{I}\to\operatorname{Spec\,}R/I$ where $R:=\mathbb{C}[P]$ and $I\subset R$ is a monomial ideal. Most importantly, $\check{X}_{I}$ is defined explicitly as the spectrum of an algebra on a vector space with a canonical basis (the basis of theta functions) and a product defined in terms of the Gromov–Witten theory of $(Y,D)$. Finally, the resulting family $\check{\mathcal{X}}\to\mathrm{Spf}\hat{R}$ is a formal family and $\hat{R}$ is the completion of $R$ with respect to the ideal $\sqrt{I}$.

Remark 1**.**

After the development of punctured Gromov–Witten invariants [2], in [43] the authors address the problem of reconstructing the mirror of a log Calabi–Yau pair by constructing directly the ring of theta functions; this program goes under name of intrinsic mirror symmetry101010An equivalent construction using non–archimedean methods (on the symplectic side) is due to Keel and Yu [48] when $X$ contains an algebraic torus.. It is remarkable, as proved in [44], that intrinsic mirror symmetry for log Calabi–Yau pair fits into the Gross–Siebert program, namely it constructs the same mirror as the one obtained using scattering diagrams in the spirit of the GHK construction we briefly discussed (but in all dimensions).

Remark 2**.**

In the GHK construction, the role of consistent scattering diagrams is essential to obtain the formal mirror family $\check{\mathcal{X}}\to\mathrm{Spf}\hat{R}$ by smoothing the $N$-vertex $\mathbb{V}_{N}$. More generally, degeneration of Calabi–Yau manifolds (beyond log Calabi–Yau surfaces), gives a singular Calabi–Yau variety equipped with a natural log structure. In [24], using a deformation theory approach with differential graded Batalin–Vilkovisky algebras, it has been recently proved that the smoothing is possible for toroidal crossing spaces (both $d$-semistable log smooth Calabi–Yau varieties and maximally degenerate ones are examples of toroidal crossing spaces and their smoothing was also studied in [13]).

1.4 Fukaya approach via multi-valued Morse theory:

closed string conjecture

In [26], the author proposed a new approach to SYZ mirror symmetry based on the asymptotics of the solutions of Maurer–Cartan equation. As already discussed, one of the main difficulties in reconstructing the mirror manifold is due to the presence of singularities on $B$, and Fukaya proposes to study the corrections via multi-valued Morse theory. Also in this case, the new structure consists of some combinatorial data on the integral affine manifold $B$ (or of its Legendre dual $\check{B}$) which plays the role of scattering diagrams. Roughly speaking, in Fukaya’s picture the rays of scattering diagrams are replaced by trivalent graphs embedded in $B$ with roots at the singular points, and the automorphisms which decorate the rays (of scattering diagrams) are replaced by gradient flow vector fields associated with multi-valued functions $f$ on $B$. Furthermore, Fukaya conjectures that this combinatorial structure on the one hand encodes pseudo-holomorphic discs bounding the fibers of $X$ and on the other hand, it governs infinitesimal deformations of the mirror $\check{X}$. This goes under the name of closed string conjecture (see [26, Conjecture 2.2]).

Recently, K. Chan, N. C. Leung and Z. N. Ma proposed a new version of Fukaya’s closed string conjecture by replacing multi-valued Morse theory by scattering diagrams in $B$. In [14], they first study the relationship between scattering diagrams and infinitesimal deformations of $\check{X}:=TB_{0}/\Lambda$ locally on an open affine subset of the smooth locus $B_{0}\subset B$. In [16], the authors proved that solutions of Maurer–Cartan equation which governs deformations of $\check{X}$ encodes tropical discs counting. The global picture is then obtained in [15] by relating scattering diagrams and solutions of Maurer–Cartan equation which governs deformations of log Calabi–Yau variety (in the spirit of the Gross–Siebert program). Therefore all together these results give a new formulation of Fukaya’s closed-string conjecture.

1.5 Homological Mirror Symmetry of Kontsevich: mirror symmetry as an equivalence of categories

In [49], Kontsevich proposed a categorical formulation of mirror symmetry, the Homological Mirror Symmetry (HMS), namely an equivalence between the bounded derived category of coherent sheaves $\mathcal{D}^{b}(\check{X})$ and the Fukaya category of the mirror $\mathcal{F}(X)$. A precise statement of HMS conjecture needs actually some explanations as on the one hand $\mathcal{D}^{b}$ is a triangulated category, and on the other hand, $\mathcal{F}$ is an $A_{\infty}$ category. We are not going to give a precise statement in the general case, and we refer to [5, 50, 6, 56], but we are going to present the example of the elliptic curve where the statement simplifies. Looking simply at the objects of the aforementioned categories, HMS predicts a duality between coherent sheaves on $\check{X}$ and Lagrangians on the mirror $X$ 111111 In string theory, one will refer to this duality as a duality between A-branes (Lagrangians with flat connections) and B-branes (coherent sheaves) [5]. hence, at a first glance we will enhance the SYZ toy model by considering Lagrangians as mirrors to holomorphic vector bundles.

Toy model II

Let $B_{0}$ be an integral affine smooth $1$-dimensional manifold. Let $\Lambda\subset TB_{0}$ be a lattice subbundle of the tangent bundle of $B_{0}$ and $\Lambda^{*}:=\operatorname{Hom}(\Lambda,\mathbb{Z})$ be the dual lattice. On the one hand, we can define a Lagrangian torus fibration $\pi\colon X:=T^{*}B_{0}/\Lambda^{*}\to B_{0}$ and $r$ horizontal Lagrangians $L_{1},...,L_{r}$ in $X$ with unitary local systems. On the other hand, the dual torus fibration is $\check{\pi}\colon\check{X}:=TB_{0}/\Lambda\to B_{0}$ and the mirror of $L_{1}\cup L_{2}\cup...\cup L_{r}$ is a holomorphic trivial rank $r$ vector bundle $\check{E}=\mathcal{O}_{\check{X}}\oplus...\oplus\mathcal{O}_{\check{X}}\to\check{X}$.

If $X=\mathbb{R}^{2}/\mathbb{Z}\oplus\mathbb{Z}$ is a symplectic torus with $\omega=dx\wedge dy$, then we can represent the horizontal Lagrangians $L_{j}$ as in Figure 5. Each Lagrangian $L_{j}$ comes equipped with a flat line bundle with connection $2\pi i\beta_{j}dx$ where $x\sim x+1$ is a local coordinate on $L_{j}$.

Following [55], the mirror of $X$ is the elliptic curve $\check{X}=\mathbb{C}/\mathbb{Z}+i\mathbb{Z}$ and the trivial rank $r$ bundle $\check{E}\to\check{X}$ whose base of global sections is generated by theta functions $\theta_{j}$:

[TABLE]

More generally, let $X_{q}$ be a symplectic torus with $\omega=Adx\wedge dy$, $[b]\in H^{2}(X_{q};\mathbb{R})/H^{2}(X_{q};\mathbb{Z})$ and the complexified Kähler parameter $q=e^{2\pi i(Ai+b)}$. Let $\check{X}_{\tau}$ be an elliptic curve with complex structure parametrized by $\tau$, then Polishchuk and Zaslow proved the existence of equivalence of categories

[TABLE]

with $q=e^{2\pi i\tau}$ and where $\mathcal{F}^{0}$ denotes a modified Fukaya category (we refer to [55] for details). In the simplified set-up where $\check{X}=\mathbb{C}/\mathbb{Z}\oplus i\mathbb{Z}$ and $X=\mathbb{R}^{2}/\mathbb{Z}\oplus\mathbb{Z}$ with $\omega=dx\wedge dy$, the mirror functor $\Phi$ is defined as follows:

• •

it maps line bundles $\check{E}=\mathcal{O}_{\check{X}}(d\mathrm{e})$ of degree $d$ to lines of slope $d$ in $X$ (see Figure 6);

• •

it maps the theta function $\theta[0,0](\tau,z)$ (that generates global sections of $\mathcal{O}_{\check{X}}(\mathrm{e})$, and $\mathrm{e}$ is the identity for the group structure) to $\{\mathrm{e}_{1}\}=L_{2}\cap L_{1}=L_{3}\cap L_{2}$. Then $\theta[0,0](2\tau,2z)$ is mapped to $\mathrm{e}_{1}$ too, and $\theta[1/2,0](2\tau,2z)$ is mapped to $\mathrm{e}_{2}$, where $\{\mathrm{e}_{1},\mathrm{e}_{2}\}=L_{1}\cap L_{3}$.

• •

the compatibility of the composition morphism $m_{2}$ in $\mathcal{F}$ ($m_{2}\colon\operatorname{Hom}(L_{1},L_{2})\otimes\operatorname{Hom}(L_{2},L_{3})\to\operatorname{Hom}(L_{1},L_{3})$) with the one in $\mathcal{D}^{b}$ follows from counting pseudo–holomorphic discs bounding the Lagrangians $L_{j}$ and the addition formula for theta functions.

1.6 Fukaya approach via multi-valued Morse theory:

open/closed string conjecture

In the same paper [26], Fukaya proposes a mirror symmetry conjecture for a pair $(\check{X},\check{E})$ of a Calabi–Yau manifold $\check{X}$ together with a holomorphic vector bundle $\check{E}$. According to HMS conjecture, the mirror of $(\check{X},\check{E})$ is an object in the Fukaya category, so a pair $(X,L)$ where $L$ is a Lagrangian in $X$ with a unitary local system. Also in this case, Fukaya’s main conjecture asserts that mirror symmetry is incarnated at the level of the integral affine manifold $B$ and that multi-valued Morse theory would control both the deformations theory of the pair $(\check{X},\check{E})$ and the holomorphic discs on the mirror $(X,L)$.

Generalizing the construction of [14], in [23] we prove that solutions of the Maurer–Cartan equation that governs deformations of holomorphic pairs $(\check{X}=TB_{0}/\Lambda,\check{E}=\bigoplus_{j=1}^{r}\mathcal{O}_{\check{X}})$ give rise to consistent scattering diagrams in $B$. Trying to establish a connection between scattering diagrams and pseudo-holomorphic discs counting seems quite difficult in general and working with tropical discs seems rather useful. Indeed, Suen and collaborators introduced tropical Lagrangian multi-sections as combinatorial data which allows reconstructing the mirror bundle (see [58, 17, 54, 59]).

A natural question is whether the data of a tropical Lagrangian multi-section can be encoded in scattering diagrams like the one that governs deformations of holomorphic pairs. We leave the discussion to future works.

Remark 3**.**

In absence of singularities, Abouzaid in [1] constructs the mirror of a Lagrangian submanifold $L\subset X$ as a twisted coherent sheaf on $\check{X}$. His construction uses Fukaya’s notion of family of Floer cohomology and not the multivalued Morse theory approach mentioned above. However, as claimed by Fukaya in [26] (Remark 4.4) the two points of view coincide. Along the same line of ideas, recent progress in this direction is due to Yuan [62, 61].

2 Scattering diagrams

Naively, $2$ dimensional scattering diagrams121212There exists a definition of higher dimension scattering diagrams [40]. are collections of lines and rays in $\mathbb{R}^{2}$ decorated with automorphisms. The group structure of scattering diagrams carries the main information, and we are going to discuss it in this section.

The first aspect to take into consideration is that the automorphisms attached to a ray must belong to a pro-nilpotent Lie group $\hat{G}$. Furthermore, different choices of the group describe different phenomena: for example,

• •

if $\hat{G}$ is the tropical vertex group $\mathbb{V}$ (whose name was introduced by Gross, Pandharipande and Siebert [36], but the group itself first appeared in [51]), in [36] the authors show that scattering diagrams in $\mathbb{V}$ compute genus zero Gromov–Witten invariants of $\mathbb{P}^{2}$ or blow-up of $\mathbb{P}^{2}$.131313In higher dimensions, scattering diagrams in the so-called higher dimensional tropical vertex compute punctured log Gromov-Witten invariants of the log Calabi-Yau variety, as proved in [4] Moreover, scattering diagrams in the Gross–Siebert program [40] (as well as in the GHK construction [34]) are a generalization of scattering diagrams in $\mathbb{V}$;

• •

if $\hat{G}$ is the quantum tropical vertex group $\hat{\mathbb{V}}$ (whose name is due to Bousseau [9], but in fact, the group itself was introduced by Kontsevich and Soibelman in [52]), scattering diagrams in $\hat{\mathbb{V}}$ compute higher genus log Gromov–Witten invariants for log Calabi–Yau surfaces [9]141414In fact, before the work of Bousseau [9], scattering diagrams in $\hat{\mathbb{V}}$ appeared in [25] as $q$-deformed diagrams that encode some refined tropical invariants. Bousseau’s contribution in [9] is to relate these tropical invariants to higher genus curve invariants for log Calabi–Yau surfaces.. In addition, in [8] Bousseau generalizes the GHK construction, promoting scattering diagrams in $\mathbb{V}$ to diagrams in $\hat{\mathbb{V}}$. In fact, he obtained the deformation quantization of the GHK family $\check{\mathcal{X}}\to\mathrm{Spf}\hat{R}$;

• •

if $\hat{G}$ is the extended tropical vertex group $\tilde{\mathbb{V}}$ introduced in [23], conjecturally, scattering diagrams in $\tilde{\mathbb{V}}$ should contribute to the reconstruction problem for holomorphic pairs, in the spirit of [40].

Before giving the definitions of the aforementioned groups, we defined scattering diagrams more generally for a pro-nilpotent Lie group $\hat{G}$.

Let $\Gamma$ be a lattice, and $\mathfrak{g}=\bigoplus_{\gamma\in\Gamma}\mathfrak{g}_{\gamma}$ be a $\Gamma$-graded Lie algebra. Assuming the existence of a quadratic form $||-||$ on $\Gamma_{\mathbb{R}}=\Gamma\otimes_{\mathbb{Z}}\mathbb{R}$, we define

[TABLE]

[TABLE]

The completion of $\mathfrak{g}_{\leq n}$ is a pro-nilpotent Lie algebra $\hat{\mathfrak{g}}=\varprojlim_{n}\mathfrak{g}_{\leq n}$ and a pro-nilpotent Lie group is defined via the exponential map

[TABLE]

and it is a group with the Backer–Campbell–Hausdorff product (BCH)

[TABLE]

Then we define scattering diagrams in $\hat{G}$

Definition 1**.**

Let $\Gamma$ be a rank $2$ lattice and let $\hat{G}$ be a pro-nilpotent Lie group defined from a $\Gamma$-graded Lie algebra. Scattering diagram $\mathfrak{D}$ in $\hat{G}$ is a collection of walls $(\mathfrak{d}_{\gamma},\theta_{\gamma})$ such that

• •

$\gamma\in\Gamma$**

• •

$\mathfrak{d}_{\gamma}$* is a line (or a ray) $\mathfrak{d}_{\gamma}=\xi_{0}+\gamma\mathbb{R}\subset\mathbb{R}^{2}$ (or $\mathfrak{d}_{\gamma}=\xi_{0}+\gamma\mathbb{R}_{\geq 0}\subset\mathbb{R}^{2}$)*

• •

$\theta_{\gamma}\in\exp(\varprojlim_{n}\bigoplus_{||k\gamma||\leq n}\mathfrak{g}_{k\gamma})\subset\hat{G}$, in particular $\theta_{\gamma}$ is the automorphism associated to the line (ray) $\mathfrak{d}_{\gamma}$

If $\theta_{\gamma}=\mathsf{id}$ the wall $(\mathfrak{d}_{\gamma},\theta_{\gamma})$ does not belong to the diagram $\mathfrak{D}$.

We remark that $\hat{G}$ being pro-nilpotent is crucial in the previous definition because, on the one hand, the $\Gamma$-grading allows to associate an automorphism to a line of the diagram, and on the other hand, the completion allows to consider products of automorphisms associated to different rays. Notice also that ignoring trivial automorphisms guarantees that products are well defined because modulo $n$ only finitely many rays give meaningful contributions.

Among all scattering diagrams, the important ones in the reconstruction problem are the so-called consistent scattering diagrams $\mathfrak{D}_{\infty}$ (see Figure 3). These can be recursively constructed as proved by Kontsevich and Soibelman in [51].

Denote by $\emph{Sing}(\mathfrak{D})$ the singular set of $\mathfrak{D}$:

[TABLE]

where $\partial\mathfrak{d}=\xi_{0}$ if $\mathfrak{d}_{\gamma}$ is a ray and zero otherwise.

There is a notion of order product for the automorphisms associated with each line of a given scattering diagram, and it is defined as follows:

Definition 2** (Path order product).**

Let $\ell:[0,1]\rightarrow\Gamma_{\mathbb{R}}\setminus\emph{Sing}(\mathfrak{D})$ be a smooth immersion with the starting point that does not lie on a ray of the scattering diagram $\mathfrak{D}$ and such that it intersects transversely the rays of $\mathfrak{D}$. For each power $k>0$, there are times $0<\tau_{1}\leq...\leq\tau_{s}<1$ and rays $\mathfrak{d}_{i}\in\mathfrak{D}$ such that $\ell(\tau_{j})\cap\mathfrak{d}_{j}\neq 0$. Then, define $\Theta_{\ell,\mathfrak{D}}^{k}:=\prod_{j=1}^{s}\theta_{j}$. The path order product is given by:

[TABLE]

Definition 3** (Consistent scattering diagram).**

A scattering diagram $\mathfrak{D}_{\infty}$ is consistent if for any closed path $\ell$ intersecting $\mathfrak{D}_{\infty}$ generically, $\Theta_{\gamma,\mathfrak{D}_{\infty}}=\mathsf{id}_{\hat{G}}$.

The following theorem by Kontsevich and Soibelman is an existence (ad uniqueness) result of consistent scattering diagrams:

Theorem 1** ([51]).**

Let $\mathfrak{D}$ be a scattering diagram with two non-parallel walls. There exists a unique minimal151515The diagram is minimal meaning that we do not consider rays with trivial automorphisms. scattering diagram $\mathfrak{D}_{\infty}\supseteq\mathfrak{D}$ such that $\mathfrak{D}_{\infty}\setminus\mathfrak{D}$ consists only of rays, and it is consistent.

The proof of the Theorem is based on a factorization property for elements of $\hat{G}$, namely working order by order in $n$, computing the product of the $\theta_{\gamma}$ for $||\gamma||\leq n$ and then factorizing the result with respect to different $\gamma$ will give a prescription for new rays.

We say that the consistent scattering diagram $\mathfrak{D}_{\infty}$ of Theorem 1 saturates the initial scattering diagram $\mathfrak{D}$.

2.1 Scattering diagrams in the tropical vertex group

In this section, we recall the definition of the tropical vertex group following the treatment of [36].

The tropical vertex group $\mathbb{V}$ is a subgroup of symplectomorphisms of the formal algebraic torus $\mathbb{T}_{\Lambda}=\operatorname{Spec\,}\mathbb{C}[\Lambda][\![t]\!]$, $\Lambda\simeq\mathbb{Z}^{2}$ being a rank two lattice. Let $e_{1}$ and $e_{2}$ be a basis for $\Lambda\cong\mathbb{Z}^{2}$. The group ring $\mathbb{C}[\Lambda]$ is the ring of Laurent polynomial in the variable $z^{m}$, with the convention that $z^{e_{1}}=:x$ and $z^{e_{2}}=:y$. We denote by $\Lambda^{*}=\operatorname{Hom}(\Lambda,\mathbb{Z})$ be the dual lattice and set $\langle-,-\rangle\colon\Lambda^{*}\times\Lambda\to\mathbb{Z}$ be the symmetric pairing. Every elements $n\in\Lambda^{*}$ is associated with a derivation $\partial_{n}\in\operatorname{Der}(\mathbb{C}[\Lambda])$ whose action is defined as $z^{m}\partial_{n}(z^{m^{\prime}}):=z^{m+m^{\prime}}\langle n,m^{\prime}\rangle$. We are going to introduce the formal Lie algebra of derivations:

[TABLE]

where the natural Lie bracket on $\mathfrak{g}$ is

[TABLE]

In particular $\mathfrak{g}$ has a Lie sub-algebra $\mathfrak{h}\subset\mathfrak{g}$ defined by:

[TABLE]

where $m^{\perp}\in\Lambda^{*}$ is identified with the derivation $\partial_{n}$ for a unique primitive vector $n\in\Lambda^{*}$ such that $\langle n,m\rangle=0$ and it is positively oriented according with the orientation induced by $\Lambda_{\mathbb{R}}:=\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$. By exponentiating the Lie algebra $\mathfrak{h}$ we get a subgroup of the group of formal automorphisms of the algebraic torus, i.e. the tropical vertex group

Definition 4**.**

The tropical vertex group $\mathbb{V}$ is the sub-group of \operatorname{Aut}_{\mathbb{C}[\![t]\!]}\big{(}\mathbb{C}[\Lambda]\hat{{\otimes}}_{\mathbb{C}}\mathbb{C}[\![t]\!]\big{)}, such that $\mathbb{V}:=\exp(\mathfrak{h})$. The product on $\mathbb{V}$ is defined by the BCH formula.

Equivalently, $\mathbb{V}$ can be defined as the group generated by formal one-parameter families of symplectomorphisms of the algebraic torus $\mathbb{T}_{\Lambda}=\operatorname{Spec\,}\mathbb{C}[\Lambda]$ defined as follows: let $f_{(a,b)}=1+tx^{a}y^{b}\cdot g(x^{a}y^{b},t)\in\mathbb{C}[\Lambda]\hat{\otimes}_{\mathbb{C}}\mathbb{C}[\![t]\!]$ for some $(a,b)\in\Lambda$ and for a polynomial $g(x^{a}y^{b},t)\in\mathbb{C}[x^{a}y^{b},t]$, then $\mathbb{V}$ is the group generated by all possible $\theta_{(a,b),f_{(a,b)}}$ defined as

[TABLE]

In particular, $\theta_{(a,b),f_{(a,b)}}^{*}\omega=\omega$ with respect to the holomorphic symplectic form on $\omega=\frac{dx}{x}\wedge\frac{dy}{y}$.

The two definitions are equivalent as one can easily check that

[TABLE]

We can now state the definition of scattering diagrams:

Definition 5** (Scattering diagram).**

A scattering diagram $\mathfrak{D}$ (in $\mathbb{V}$) is a collection of walls $\textsf{w}_{i}=(\mathfrak{d}_{i},\theta_{i})$, where

• •

$\mathfrak{d}_{i}$* can be either a line through $\xi_{0}$, i.e. $\mathfrak{d}_{i}=\xi_{0}-m_{i}\mathbb{R}$ or a ray (half line) $\mathfrak{d}_{i}=\xi_{0}-m_{i}\mathbb{R}_{\geq 0}$,*

• •

$\theta_{i}\in\mathbb{V}$* is such that $\log(\theta_{i})=\sum_{j,k}a_{jk}t^{j}z^{km_{i}}\partial_{n_{i}}\in\mathfrak{h}$.*

Moreover for any $N>0$ there are finitely many $\theta_{i}$ such that $\theta_{i}\not\equiv 1$ mod $t^{N}$.

Remark 4**.**

Notice that $\mathfrak{h}$ is both $\Lambda$-granded and $t$-graded. However, in the definition of $\mathbb{V}$ we choose the completion in the $t$ parameter and not with respect to $\Lambda$ as in the general definition. That explains why we add the latter assumption in the definition of scattering diagrams in $\mathbb{V}$, as otherwise, we would not be able to define products in $\mathbb{V}$.

In [14], the authors prove that the Lie algebra $\mathfrak{h}$ embeds to the degree zero part of the Kodaira–Spencer DGLA which governs infinitesimal deformations of $\check{X}:=TB/\Lambda\to B$ with $B$ being a tropical affine smooth $2$-dimensional manifold (as the toy model example for SYZ).161616In [14], they do not restrict to $2$ dimensional manifolds and they work with $n$-dimensional scattering diagrams. In particular, the group $\mathbb{V}$ acts as the Gauge group on the solutions of the Maurer–Cartan equation. Hence, they show that consistent scattering diagrams in $\mathbb{V}$ can be constructed by studying the asymptotics of formal solutions of the Maurer–Cartan equation. In [16], they also explain how consistent scattering diagrams in $\mathbb{V}$ encode tropical discs counting. The latter should correspond to count pseudo–holomorphic discs on $X$, so summarizing, we have the following picture

Remark 5**.**

The group $\mathbb{V}$ do also appear in [52] (see Example 4 in Section 2.3, when $\Gamma$ is $2$-dimensional) as the pronilpotent Lie group of the torus Lie algebra (which up to a sign, is equivalent to $\mathfrak{h}$). In particular, one can consider automorphisms in $\mathbb{V}$ which encode DT-invariants and define the so-called wall-crossing structure. The latter appears in [27], where they allow to compute the BPS spectrum of certain theories. Finally, consistent scattering diagrams in $\mathbb{V}$ are studied in [10] in relation to quiver representations.

2.2 Scattering diagrams in the quantum tropical vertex

In [52], the authors first considered the quantization of the Poisson algebra of functions of a torus $\mathbb{T}_{\Lambda}=\operatorname{Spec\,}\mathbb{C}[\Lambda]$, $\Lambda\cong\mathbb{Z}^{2}$ in the context of wall–crossing formulas for DT invariants on 3d Calabi–Yau categories. We briefly recall the construction following [9]. Let $\mathcal{O}_{\mathbb{T}}$ be the ring of functions of $\mathbb{T}_{\Lambda}$ with $\Gamma(\mathcal{O}_{\mathbb{T}})=\bigoplus_{m\in\Lambda}\mathbb{C}\,z^{m}$ and

[TABLE]

Then the Poisson bracket is

[TABLE]

with $\langle-,-\rangle_{D}$ being an unimodular antisymmetric bilinear form on $\Lambda$. The ring of functions of the quantum torus $\hat{\mathbb{T}}$ is defined as $\Gamma(\mathcal{O}_{\hat{\mathbb{T}}}):=\bigoplus_{m\in\Lambda}\mathbb{C}[q^{\pm 1/2}]\,\hat{z}^{m}$ and

[TABLE]

Analogously to the tropical vertex group, $\hat{\mathbb{V}}$ is the group of formal automorphisms of $\hat{\mathbb{T}}$: let us first consider the pro-nilpotent, $\Lambda$-graded Lie algebra

[TABLE]

whose Lie bracket simply reads $[\hat{z}^{m},\hat{z}^{m^{\prime}}]=(q^{\frac{1}{2}\langle m,m^{\prime}\rangle_{D}}-q^{-\frac{1}{2}\langle m,m^{\prime}\rangle_{D}})\hat{z}^{m+m^{\prime}}$.

Definition 6**.**

The quantum tropical vertex group $\hat{\mathbb{V}}$ is the subgroup of the formal automorphisms $\operatorname{Aut}_{\mathbb{C}[\![t]\!]}\left(\mathbb{C}[q^{\pm\frac{1}{2}}][\Lambda]\hat{\otimes}_{\mathbb{C}}\mathbb{C}[\![t]\!]\right)$ such that $\hat{\mathbb{V}}=\exp(\hat{\mathfrak{h}})$. The product is defined by the BCH formula.

In particular, elements of $\hat{\mathbb{V}}$ can be defined as

[TABLE]

for $\hat{H}_{m}:=\sum_{k,j=1}^{\infty}a_{k,j}(q)t^{j}\hat{z}^{mk}\in\hat{\mathfrak{h}}$.

2.3 Scattering diagrams in the extended tropical vertex group

In [23], the author studied the relationship between deformations of holomorphic pairs and scattering diagrams. The former consists of a rank $r$ holomorphic vector bundle $\check{E}$ over a semi-flat manifold $\check{X}:=TB/\Lambda\to B$, namely the toy model torus fibration over a $2$ dimensional smooth affine tropical manifold $B$. The construction generalizes the one of Chan, Conan Leung and Ma [14] and we introduce the extended tropical vertex group $\tilde{\mathbb{V}}$ where the scattering diagrams are defined.

The group $\tilde{\mathbb{V}}$ is defined by taking the exponential of the Lie algebra $\tilde{\mathfrak{h}}$ which is a $\mathfrak{gl}$-extension of $\mathfrak{h}$

[TABLE]

with Lie bracket $[-,-]_{\sim}\colon\tilde{\mathfrak{h}}\times\tilde{\mathfrak{h}}\to\tilde{\mathfrak{h}}$ is

[TABLE]

The definition of the Lie algebra $\tilde{\mathfrak{h}}$ comes from the Kodaira Spencer DGLA which governs infinitesimal deformations of holomorphic pairs [18]. We refer to [23] for more details about the construction.

Definition 7**.**

The extended tropical vertex group $\tilde{\mathbb{V}}$ is defined as the pro-nilpotent group $\tilde{\mathbb{V}}=\exp(\tilde{\mathfrak{h}})$. The product on $\tilde{\mathbb{V}}$ is defined by the BCH formula.

Following Fukaya’s approach to mirror symmetry, we expect consistent scattering diagrams in $\tilde{\mathbb{V}}$ to govern the reconstruction of the mirror pair $(L_{1},...,L_{r},X)$ conjecturally given by the mirror manifold $X$ and a collection of $r$ Lagrangians $L_{j}$ in $X$ (as previously discussed in Section 1.5). In addition, we expect that holomorphic discs bounding $L_{1},...,L_{r}$ and the fibers of $X$ are encoded in the commutators of some elements in $\tilde{\mathbb{V}}$. In [22] we consider some ad-hoc scattering diagrams in $\tilde{\mathbb{V}}$ and computing commutators we recover genus zero Gromov–Witten invariants of blow-up of $\mathbb{P}^{2}$ maximally tangent to its boundary divisor as well as genus zero Gromov–Witten invariants relative to the toric boundary divisor. Alternatively, we consider $2d$-$4d$ scattering diagrams in $\tilde{\mathbb{V}}$ which seems more naturally related to the reconstruction problem for the mirror pair.171717Recall that pure $4d$ scattering diagrams are defined in the tropical vertex group $\mathbb{V}$, hence when the $2d$ component is trivial, we end up studying the reconstruction problem of $X$ itself. For instance, let $\mathfrak{D}$ be the scattering diagram consisting of two initial non-parallel walls $\mathsf{w}_{i}=(\mathfrak{d}_{i},\theta_{i})$ $i=1,2$

[TABLE]

where $E_{ij}\in\mathfrak{gl}(r,\mathbb{C})$ is an elementary matrix with non zero entries in position $i<j$, and $m_{1},m_{2}\in\Lambda$. Notice that type $2d$ consists only of a matrix component while type $4d$ consists of a derivative component (namely an element of the Lie algebra $\mathfrak{h}$). Indeed the labels $2d$ and $4d$ refer to the formalism of $2d$-$4d$ wall–crossing formula introduced by Gaiotto–Moore–Neitzke [28]. In [23] (Example 1) we compute the consistent scattering diagram $\mathfrak{D}_{\infty}$ that saturates $\mathfrak{D}$, namely $\mathfrak{D}_{\infty}=\mathfrak{D}\cup\{\mathsf{w}_{3}\}$

[TABLE]

We may interpret this result by saying that in presence of a singular torus fiber $X_{s}$ and of two ramified horizontal Lagrangians $L_{i},L_{j}$, the scattering diagram $\mathfrak{D}_{\infty}$ encodes the number of holomorphic discs bounding $L_{i},L_{j}$ and $X_{s}$.

Furthermore, in [23], we prove that consistent scattering diagrams in $\tilde{\mathbb{V}}$ are equivalent to $2d$-$4d$ wall–crossing formulas (WCF) which govern BPS state counting in $2d$-$4d$ coupled systems.

Summing up, the results of [23, 22] propose a new approach to the original Fukaya’s open–closed string conjecture, with scattering diagrams playing the role of gradient flow trajectories; but we do not know yet an enumerative geometric interpretation.