To simplify the discussion, we will work in this section only with the simplest rings which occur in the previous section, of the form $R^{k}_{\sigma,\sigma,\sigma}$ where $\sigma$ is a maximal (two-dimensional) cell. This ring is isomorphic to $\mathbb{C}[x^{\pm 1},y^{\pm 1},t]/(t^{k+1})$. Let us work formally instead, setting

This is the ring of formal power series in $t$ with coefficients Laurent polynomials in $x$ and $y$. Let $f\in R$ be of the form

Then this defines an automorphism $\theta_{(a,b),f}$ of $R$ as a $\mathbb{C}\mbox{{[}}t\mbox{{]}}$-algebra given by

Note that $\theta_{(a,b),f}^{-1}=\theta_{(a,b),f^{-1}}$. These automorphisms have the further property that they preserve the holomorphic symplectic form ${dx\over x}\wedge{dy\over y}$.

We define the tropical vertex group $\mathbb{V}$ to be the completion with respect to the maximal ideal $(t)\subseteq\mathbb{C}\mbox{{[}}t\mbox{{]}}$ of the subgroup of $\mathbb{C}\mbox{{[}}t\mbox{{]}}$-algebra automorphisms of $R$ generated by all such automorphisms. Note that infinite products are defined in $\mathbb{V}$ only if only finitely many factors are non-trivial modulo $t^{k}$ for every $k>0$. This is a slight modification of a group originally introduced by Kontsevich and Soibelman in [53].

We now describe a local version of the rays described in the previous section. For convenience, set $M=\mathbb{Z}^{2}$, $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$, and identify $\mathbb{C}[x^{\pm 1},y^{\pm 1}]$ with $\mathbb{C}[M]$.

Given a scattering diagram $\mathfrak{D}$, let

If we are given a path $\gamma:[0,1]\rightarrow M_{\mathbb{R}}\setminus\{0\}$ with $\gamma(0),\gamma(1)\not\in\operatorname{Supp}(\mathfrak{D})$ and $\gamma$ being transversal to each ray it crosses, then we can define the path-ordered product $\theta_{\gamma,\mathfrak{D}}\in\mathbb{V}$ which is a composition of automorphisms associated to each ray that $\gamma$ crosses. If at time $t$ the path $\gamma$ crosses a ray $(\mathfrak{d},f_{\mathfrak{d}})$, let $n\in N=\operatorname{Hom}(M,\mathbb{Z})$ be the unique primitive element which vanishes on $\mathfrak{d}$ and is negative on $\gamma^{\prime}(t)$. Then define $\theta_{t}$ to be the automorphism

Note this is of the form $\theta_{(a,b),f_{\mathfrak{d}}}$ for suitable choice of $(a,b)$. We then define

where the time $t$ increases from right to left in the product. Note that if $\gamma$ crosses two rays at the same time, the order doesn’t matter as one checks easily that two automorphisms commute if they are associated with the same underlying $\mathfrak{d}\subseteq M_{\mathbb{R}}$.

We can then express the essential lemma of [53] in this context:

The proof is very simple and algorithmic; I give a quick outline. One constructs a sequence of scattering diagrams $\mathfrak{D}=\mathfrak{D}_{1},\mathfrak{D}_{2},\ldots$ with the property that $\theta_{\gamma,\mathfrak{D}_{k}}\equiv\operatorname{id}\mod t^{k}$. This is clearly true for $\mathfrak{D}_{1}$, so we proceed inductively, assuming we have constructed $\mathfrak{D}_{k}$. Then one shows (by looking at the Lie algebra of $\mathbb{V}$) that

for integers $a_{i},b_{i}$ (with $a_{i},b_{i}$ not both zero) and $c_{i}\in\mathbb{C}$. Then one obtains $\mathfrak{D}_{k+1}$ by adding rays

with the sign chosen so that when $\gamma$ crosses this ray, it produces the automorphism

Since this automorphism will commute with all other automorphisms in $\mathfrak{D}_{k}$ modulo $t^{k+1}$, inserting these rays will precisely cancel out the contributions to $\theta_{\gamma,\mathfrak{D}_{k}}$ to order $k$, and thus $\theta_{\gamma,\mathfrak{D}_{k+1}}\equiv\operatorname{id}\mod t^{k+1}$.

It is very easy to program this algorithm and explore these scattering diagrams. They appear to have a very rich and fascinating structure. The following simple examples show their complexity.

We will now explain the enumerative interpretation for the functions which arise in $\operatorname{{\mathsf{S}}}(\mathfrak{D})$. To motivate this, let us return to the tropical interpretation of §4. Begin, say, with a tropical manifold $B$ which corresponds to a K3 surface, as depicted in Figure 10, along with what we will call a tropical disk. This is almost a tropical curve, but it just ends at the point $P$ without any balancing condition at $P$; meanwhile, it has other legs terminating at the singularities of $B$. This is legal behaviour as explained at the end of §4. Following the description at the end of §4, one can imagine disks over each leg terminating at a singular point. Where these legs meet, one would like to glue these disks together and continue along a cylinder over the segment adjacent to $P$. Terminating at $P$, we roughly obtain a disk in $X(B)$ with boundary contained in the torus fibre over $P$, as depicted. It is natural to ask how many ways the initial disks (possibly taking multiple covers of these disks) can be glued together to give a new disk.

Now compare this picture with what we have seen on the mirror side. Our explicit degeneration really gives, as generic fibre, something like $\check{X}(B)$. However, it is controlled by similar tropical information: rays emanate from the singularities in the monodromy invariant direction, just as in the case of the tropical curves. They collide, and the Kontsevich-Soibelman result in Proposition 11.2 gives new rays. So one may hope that this process precisely reflects holomorphic disks in $X(B)$ with boundary on fibres of $X(B)\rightarrow B$.

It is also worth mentioning work of Auroux [4], which makes more precise the notion that the complex structure on one side should be determined by holomorphic disks on the other. This also provides a posteriori support for the idea that there must be an enumerative interpretation for the process of generating new rays.

It is usually difficult to work with holomorphic disks. It is often easier to translate problems involving holomorphic disks into problems involving genuine Gromov-Witten invariants. We can do so for the problems being discussed here. Here then is the enumerative interpretation, in the simplest situation, as explained in [27].

Suppose we are given distinct non-zero primitive vectors $m_{1},\ldots,m_{p}\in M$ and positive integers $\ell_{1},\ldots,\ell_{p}$. Consider the scattering diagram

Let $(\mathfrak{d},f_{\mathfrak{d}})\in\operatorname{{\mathsf{S}}}(\mathfrak{D})\setminus\mathfrak{D}$. We can always assume that this is the only ray in $\operatorname{{\mathsf{S}}}(\mathfrak{D})\setminus\mathfrak{D}$ with a given underlying ray $\mathfrak{d}$. This is because if there are rays $(\mathfrak{d}_{1},f_{\mathfrak{d}_{1}}),(\mathfrak{d}_{2},f_{\mathfrak{d}_{2}}),\ldots$ in $\operatorname{{\mathsf{S}}}(\mathfrak{D})\setminus\mathfrak{D}$ with $\mathfrak{d}_{1}=\mathfrak{d}_{2}=\cdots$, we can replace this collection of rays with a single ray $(\mathfrak{d}_{1},\prod f_{\mathfrak{d}_{i}})$ without affecting $\theta_{\gamma,\operatorname{{\mathsf{S}}}(\mathfrak{D})}$. With this assumption, $f_{\mathfrak{d}}$ is uniquely determined by $\mathfrak{D}$. We wish to interpret $f_{\mathfrak{d}}$ enumeratively.

To do this, consider a complete fan $\Sigma$ in $M_{\mathbb{R}}$ whose one-dimensional rays are

Assume for the sake of simplicity in this discussion that $\mathfrak{d}$ does not coincide with the other $p$ rays. Let $X$ be the toric variety defined by $\Sigma$, with toric divisors $D_{1},\ldots,D_{p},D_{\mathrm{out}}$ corresponding to the above rays. Next, choose $\ell_{i}$ general points on the divisor $D_{i}$, say labelled $P_{i1},\ldots,P_{i\ell_{i}}$. Let $\nu:\widetilde{X}\rightarrow X$ be the blow-up of these $\sum_{i}\ell_{i}$ points, with exceptional divisor $E_{ij}$ over $P_{ij}$. Let $\widetilde{D}_{i},\widetilde{D}_{\mathrm{out}}$ denote the proper transforms of $D_{i},D_{\mathrm{out}}$.

In what follows, we will use the notation ${\bf P}_{i}=(p_{i1},\cdots,p_{i\ell_{i}})$ for a partition of length $\ell_{i}$ of some non-negative integer $|{\bf P}_{i}|=p_{i1}+\cdots+p_{i\ell_{i}}$, allowing some of the $p_{ij}$’s to be zero. Fix a class $\beta\in H^{2}(X,\mathbb{Z})$ with the property that $a_{i}:=\beta\cdot D_{i}$ are non-negative and $k:=\beta\cdot D_{\mathrm{out}}$ is positive. It is an easy exercise in toric geometry that this implies a relationship

where $m_{\mathrm{out}}$ is a primitive generator of $\mathfrak{d}$. If one chooses a collection of partitions ${\bf P}=({\bf P}_{1},\ldots,{\bf P}_{p})$ where ${\bf P}_{i}$ is a partition of $a_{i}$, let

This can be thought of as the class of a curve on $X$ which passes through the point $P_{ij}$ precisely $p_{ij}$ times.

We would now like to associate a number to this cohomology class. This will be a Gromov-Witten count of one-pointed rational curves in $\widetilde{X}$ which (1) represent the class $\beta_{\bf P}$; (2) are tangent to $\widetilde{D}_{\mathrm{out}}$ at the marked point with order $k$; and (3) are otherwise disjoint from any of the divisors $\widetilde{D}_{i}$. This is a relative Gromov-Witten invariant. However, the classical theory of relative Gromov-Witten invariants works relative to a smooth divisor, and of course the union of the boundary divisors here is singular. One can instead encode the above conditions using log Gromov-Witten theory. At the time [27] was written, log Gromov-Witten theory was not yet available, and as a consequence, we used a technical work-around to reduce to the classical theory. I give this description here since it does not require knowing log Gromov-Witten theory.

One defines $\widetilde{X}^{o}:=\widetilde{X}\setminus\bigcup_{i=1}^{p}\widetilde{D}_{i}$. One then considers the moduli space $\mathfrak{M}(\widetilde{X}^{o}/\widetilde{D}^{o}_{\mathrm{out}},\beta_{\bf P})$ of relative stable maps of genus zero with target space $\widetilde{X}^{o}$, relative to the divisor $\widetilde{D}^{o}_{\mathrm{out}}=\widetilde{D}_{\mathrm{out}}\cap\widetilde{X}^{o}$. These curves have one marked point with order of tangency $k$ with $\widetilde{D}_{\mathrm{out}}^{o}$. The only problem is that the target space is non-proper, but one shows this doesn’t cause any problems because nevertheless the moduli space is proper. One finds it is virtual dimension zero, and since it carries a virtual fundamental class, we can define

We can then state the enumerative result ([27]):

We end this section with brief additional motivation for Theorem 11.4 and a word about the proof.

Suppose we have a piece of an integral affine manifold as depicted in Figure 11. Here we imagine a situation with two singular points in a surface, with local monodromy around the singularities contained in the horizontal and vertical line segments being $\begin{pmatrix}1&\ell_{1}\\ 0&1\end{pmatrix}$ and $\begin{pmatrix}1&\ell_{2}\\ 0&1\end{pmatrix}$ in suitably chosen bases (different for each segment). This is a slightly more general situation than was considered in §10, where we only discussed singularities with monodromy of the form $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$. Nevertheless, the techniques of that section still apply, but the functions attached to the initial rays emanating from the singularities towards the central vertex $v$ can be taken to be of the form $(1+x^{-1})^{\ell_{1}}$ and $(1+y^{-1})^{\ell_{2}}$. This is roughly the shape of the examples discussed above. Applying the scattering procedure would then produce a smoothing of $\check{X}_{0}(B,\mathscr{P},s)$. However, on the mirror side, we interpret $B$ as a dual intersection complex, which means it should arise from a degeneration $\mathcal{X}\rightarrow D$ where the central fibre $\mathcal{X}_{0}$ has an irreducible component $Y_{v}$ isomorphic to $\mathbb{P}^{2}$ (corresponding to the vertex $v$). Furthermore, the total space $\mathcal{X}$ should have $\ell_{1}+\ell_{2}$ ordinary double points lying on the toric boundary of $Y_{v}$. If one blows up the Weil divisor $Y_{v}$ inside of $\mathcal{X}$, one obtains a small resolution $\widetilde{\mathcal{X}}\rightarrow\mathcal{X}$ of these ordinary double points, and in particular, the proper transform $\widetilde{Y}_{v}$ of $Y_{v}$ is the blow-up of $Y_{v}$ at the points $Y_{v}\cap\operatorname{Sing}(\mathcal{X})$. This operation blows up $\ell_{1}$ points on one coordinate axis of $Y_{v}$ and $\ell_{2}$ on the other. This is exactly the same surface considered in Theorem 11.4.

Now consider the kind of curves on $\tilde{Y}_{v}$ counted by Theorem 11.4. These are curves in $\widetilde{Y}_{v}$ which only intersect the third coordinate axis at one point. These can be viewed as curves in $\widetilde{\mathcal{X}}_{0}$, but not ones which deform to holomorphic curves in a general fibre of the family $\widetilde{\mathcal{X}}\rightarrow D$. Rather, roughly, we expect such curves to deform to holomorphic disks, with the point of intersection with the singular locus of $\widetilde{\mathcal{X}}_{0}$ (i.e., the point of intersection with the third axis of $\widetilde{Y}_{v}$) expanding into an $S^{1}$, giving the boundary of the holomorphic disk. Approximately, we expect this boundary to lie in a fibre of an SYZ fibration on a general fibre of the family $\widetilde{\mathcal{X}}\rightarrow D$. The homology class of this boundary inside the fibre is determined by the order of tangency of the curve with the third axis.

This correspondence between the relative curves considered in Theorem 11.4 is only a moral one; there is no proof yet that we are really counting such holomorphic disks. However, this argument served as the primary motivation for Theorem 11.4.

Finally, as far as the proof is concerned, there are several steps. First, we show that scattering diagrams can be deformed to look like a union of tropical curves, and use a variant of Mikhalkin’s fundamental curve-counting results [61] as developed by Nishinou and Siebert [63] to show that scattering diagrams perform certain curve counts on toric surfaces. This is then related to the Gromov-Witten counts of the blown-up surfaces using Jun Li’s gluing formula [56].