Let us turn to the $B$-model, and understand how tropical geometry may be visible in the variation of complex structures which is necessary for $B$-model computations.

This problem is closely related to the reconstruction problem, as stated in Question 7.12. If given $(B,\mathscr{P},\varphi)$, one can find an explicit description of a toric degeneration $\mathcal{X}\rightarrow D$, with dual intersection complex $(B,\mathscr{P},\varphi)$, then one could use this explicit description to calculate periods and extract $B$-model predictions for the mirror.

Before describing the solution to this problem, let me give a bit of history of the reconstruction problem. The version as stated in Question 5.6 was first studied by Fukaya in [13]. There he considered directly the question of perturbing the complex structure on $X_{\epsilon}(B_{0})$ by looking at the Kodaira-Spencer equation governing deformations of complex structure. Arguing informally in the case that $\dim B=2$, he suggested that the perturbations should be concentrated along trees made of gradient flow lines, with the lines emanating initially from singular points of $B$. This gave the first hint that a nice solution to the reconstruction problem might actually see something related to curves. However, Fukaya’s work contained no definite theorems, and the analysis looked likely to be very difficult.

In 2004, Siebert and I were considering how to solve the reconstruction problem using our program. Given $(B,\mathscr{P})$, we had shown in [30] how to construct log schemes $X_{0}(B,\mathscr{P},s)^{\dagger}$ along with a morphism to $0^{\dagger}$ which had all the properties one would want for a central fibre of a toric degeneration $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$. Our original hope was that a generalization of the Bogomolov-Tian-Todorov unobstructedness theorem would allow us to show such log schemes smoothed. In particular, Kawamata and Namikawa [50] had had success with this point of view in the normal crossings case. While this approach works easily in dimension 2, we couldn’t make it work in higher dimension. Furthermore, this approach fails to give the explicit description of the smoothing which would be needed to describe the $B$-model. As a consequence, we turned towards a more explicit approach, which involved gluing together explicit local models.

While we were working on this approach, Kontsevich and Soibelman in [53] got around the difficult analysis of Fukaya’s approach by replacing complex manifolds with rigid analytic manifolds. They were able to show that given a tropical affine surface $B$ with $24$ singularities of focus-focus type (the simplest type of singularity which occurs in affine surfaces, to be described shortly) one could construct a rigid analytic K3 surface $X^{an}(B)$. This was done by gluing together standard pieces via automorphisms attached to lines on $B$. These lines were given as gradient flow lines, giving a similar, but much more precise, picture to the one given by Fukaya.

Combining our approach of gluing local models with one of the central ideas of Kontsevich and Soibelman’s work [53], we were then able to complete a construction in all dimensions, giving a satisfactory solution to the reconstruction problem within algebraic geometry. This was carried out in [32].

Before surveying this approach, let me make a philosophical remark. Note that when we were discussing the $A$-model, we observed that tropical curves on $B$ should correspond to holomorphic curves on $X(B)$. If we want to see these same tropical curves playing a role on the $B$-model of the mirror, then we should think of the $B$-model not on a complex manifold of the form $X(B)$, but rather on $\check{X}(B)$. This is a slightly confusing reversal of roles. Normally counting curves is done in the symplectic category, here expected to mean on the symplectic manifold $\check{X}(B)$, while anything having to do with complex structures should be done on $X(B)$. This reversal can be explained as follows. If we were to study pseudo-holomorphic curves on $\check{X}(B)$, we would need to put an almost complex structure on $\check{X}(B)$. One way to do this is to choose a metric on $B$; this induces an almost complex structure on $\check{X}(B)$ constant on fibres of the torus fibration, generalizing the construction of a complex structure from a Hessian metric described in §2. Then in a suitable adiabatic limit where the almost complex structure is rescaled, pseudo-holomorphic curves are expected to tend towards trees of gradient flow lines. If the chosen metric was in fact Hessian, these gradient flow lines would in fact be straight lines with respect to the Legendre dual affine structure, so these trees of gradient flow lines can be viewed as a generalization of tropical curves. However, tropical geometry is linear and much easier to control. We take the attitude that we should work on the side in which tropical geometry appears. Indeed, this turns out to be very helpful.

Given this, we can then present a somewhat revised version of the mirror symmetry program:

1. (1)

   We begin with a toric degeneration of Calabi-Yau manifolds $\mathcal{X}\rightarrow D$ with an ample polarization.

2. (2)

   Construct the dual intersection complex $(B,\mathscr{P},\varphi)$ from this data.

3. (3)

   Construct a new toric degeneration $\check{\mathcal{X}}\rightarrow D$ whose intersection complex is $(B,\mathscr{P},\varphi)$. This degeneration should be controlled by tropical data.

4. (4)

   Understand genus $0$ holomorphic curves (or whatever other aspect of the $A$-model one is interested in) on the general fibre of $\mathcal{X}\rightarrow D$ in terms of tropical geometry of $B$.

5. (5)

   Understand the variation of Hodge structures for $\check{\mathcal{X}}\rightarrow D$ in terms of tropical geometry of $B$.

6. (6)

   Use the fact that the $A$- and $B$-models of $\mathcal{X}$ and $\check{\mathcal{X}}$ respectively are controlled by the same tropical geometry on $B$ to prove mirror symmetry.

Here we outline the completion of step (3) as carried out in [32].

The first step is as follows. Given $(B,\mathscr{P},\varphi)$, we wish to construct the central fibre $\mathcal{X}_{0}^{\dagger}$ of the degeneration. This in fact was carried out in §5 of [30], assuming certain genericity assumptions on the singular locus of $B$. As a scheme, it is fairly obvious what $\mathcal{X}_{0}$ should be. For each maximal cell $\sigma$, one has an associated projective toric variety $\mathbb{P}_{\sigma}$ with Newton polytope $\sigma$. Any face $\tau\subseteq\sigma$ specifies a toric strata $\mathbb{P}_{\tau}\subseteq\mathbb{P}_{\sigma}$, and given $\tau=\sigma_{1}\cap\sigma_{2}$ for $\sigma_{1},\sigma_{2}$ maximal, we can glue together the toric strata $\mathbb{P}_{\tau}\subseteq\mathbb{P}_{\sigma_{1}},\mathbb{P}_{\sigma_{2}}$ in a torus equivariant manner. There is of course a whole family of possible gluings, parameterized by what we call closed gluing data in [30]. Given closed gluing data $s$, we obtain a scheme $\check{X}_{0}(B,\mathscr{P},s)$.

Now $\check{X}_{0}(B,\mathscr{P},s)$ cannot be a central fibre of a toric degeneration unless it carries a log structure of the correct sort. There are many reasons this may not happen. If $s$ is poorly chosen, there may be zero-dimensional strata of $\check{X}_{0}(B,\mathscr{P},s)$ which do not have neighbourhoods locally étale isomorphic to the toric boundary of an affine toric variety; this is a minimal prerequisite. As a result, we have to restrict attention to closed gluing data induced by what we call open gluing data. Explicitly, each vertex $v$ of $\mathscr{P}$ defines local models $V(v)\subseteq U(v)$ as follows. The piecewise linear function $\varphi$ is defined locally up to affine linear functions. Choose a representative $\varphi_{v}$ for $\varphi$ in a neighbourhood of $v$ which takes the value $0$ at $v$. By extending the function linearly on each cell, we can view this as a piecewise linear function on the fan $\Sigma_{v}$, viewed as a fan in some $\mathbb{R}^{n}$. We can then set

$$P_{v}:=\{(m,r)\in\mathbb{Z}^{n}\times\mathbb{Z}\,|\,r\geq\varphi_{v}(m)\}.$$

Noting that $(0,1)\in P_{v}$, we set

	$\displaystyle U(v):={}$	$\displaystyle\operatorname{Spec}\mathbb{C}[P_{v}],$
	$\displaystyle V(v):={}$	$\displaystyle\operatorname{Spec}\mathbb{C}[P_{v}]/(z^{(0,1)}).$

Note that $z^{(0,1)}$ vanishes to order one on every toric divisor of $U(v)$, so in fact $V(v)$ is the toric boundary of $U(v)$. It turns out, as we show in [30], that a necessary condition for $\check{X}_{0}(B,\mathscr{P},s)$ to be the central fibre of a toric degeneration is that it is obtained by dividing out $\coprod_{v\in\mathscr{P}}V(v)$ by an equivalence relation. In other words, we are gluing together the $V(v)$’s along Zariski open subsets to obtain a scheme.²² 2 [30] allowed the case that the cells of $\mathscr{P}$ self-intersect. As a consequence, the equivalence relation is merely étale and one obtains an algebraic space. Again, there is some choice of gluing, but now the gluing data are given by equivariant identifications of open subsets of the various $V(v)$’s. We call this open gluing data.

The advantage of using open gluing data is that each $V(v)$ carries a log structure induced by the divisorial log structure $V(v)\subseteq U(v)$. These log structures are not identified under the open gluing maps, but the ghost sheaves of the log structures are isomorphic. So the ghost sheaves $\overline{\mathcal{M}}_{V(v)}$ glue to give a ghost sheaf of monoids $\overline{\mathcal{M}}_{\check{X}_{0}(B,\mathscr{P},s)}$. Thus we see how $\varphi$ influences the log structure.

One then tries to construct a log structure with this ghost sheaf. This is done in [30] by building suitable extensions of the ghost sheaf with $\mathcal{O}_{\check{X}_{0}(B,\mathscr{P},s)}^{\times}$, and this extension depends on some moduli (which may in general be empty). The good situation is that one can find a closed subset $Z\subseteq\check{X}(B,\mathscr{P},s)$ of codimension at least two and a log structure on $\check{X}_{0}(B,\mathscr{P},s)$ along with a morphism $\check{X}_{0}(B,\mathscr{P},s)^{\dagger}\rightarrow 0^{\dagger}$ which is log smooth away from $Z$. Furthermore, the ghost sheaf on $\check{X}_{0}(B,\mathscr{P},s)\setminus Z$ should be the given ghost sheaf of monoids $\overline{\mathcal{M}}_{\check{X}_{0}(B,\mathscr{P},s)}$ restricted to $\check{X}_{0}(B,\mathscr{P},s)\setminus Z$. We call such a log scheme with morphism to $0^{\dagger}$ a log Calabi-Yau space.

The technical heart of [30] is an explicit classification of log Calabi-Yau spaces with given intersection complex $(B,\mathscr{P},\varphi)$, modulo some assumptions on the singularities of $B$ called simplicity. The definition of simplicity is rather involved, so we will not give it here, but it essentially says that not too much topology of $\check{X}(B)$ (or $X(B)$) can be hiding over the singular locus of $B$.

A main result of [30], (Theorem 5.4) is then

So the moduli space is an algebraic torus (or a disjoint union of algebraic tori) of dimension equal to $\dim_{\mathbb{C}}H^{1}(B,i_{*}\check{\Lambda}\otimes\mathbb{C})$. In [31], we in fact show the dimension of this torus is the dimension of $H^{1}(\mathcal{X}_{t},\mathcal{T}_{\mathcal{X}_{t}})\cong H^{n-1,1}(\mathcal{X}_{t})$ for a smooth fibre $\mathcal{X}_{t}$ of a smoothing $\mathcal{X}\rightarrow D$ of $X_{0}(B,\mathscr{P},s)^{\dagger}$. This is the expected dimension, as this latter vector space is the tangent space to the moduli space of $\mathcal{X}_{t}$.

Now assume given a log Calabi-Yau space $X_{0}:=X_{0}(B,\mathscr{P},s)^{\dagger}\rightarrow 0^{\dagger}$. Our goal is to use the log structure to provide “initial conditions” to produce $k$-th order deformations $X_{k}\rightarrow\operatorname{Spec}\mathbb{C}[t]/(t^{k+1})$, order by order. To do so, we will glue together standard thickenings of “pieces” of $X_{0}$, modifying standard gluings by a complicated system of data we call a structure.

First, the “pieces” of $X_{0}$ we consider are toric open affine subsets of strata of $X_{0}$. Recall that strata of $X_{0}$ are indexed by cells $\tau\in\mathscr{P}$, corresponding to a projective toric variety $\mathbb{P}_{\tau}$. Recall also that if $\omega\subseteq\tau$, the normal cone to $\tau$ along $\omega$ is a cone in the fan defining $\mathbb{P}_{\tau}$ and hence defines an open affine subset of $\mathbb{P}_{\tau}$. We call this open affine subset $V_{\omega,\tau}\subseteq\mathbb{P}_{\tau}$; note

$$V_{\omega,\tau}=\mathbb{P}_{\tau}\setminus\bigcup_{\rho\subseteq\tau\atop\omega\not\subseteq\rho}\mathbb{P}_{\rho}.$$

For example, if $\omega$ is a vertex of $\tau$, then $V_{\omega,\tau}$ is the standard toric open affine subset of $\mathbb{P}_{\tau}$ containing the zero-dimensional stratum of $\mathbb{P}_{\tau}$ corresponding to $\omega$.

Second, what are the thickenings of the sets $V_{\omega,\tau}$? These can be described explicitly as follows. Choose a point $x$ in the interior of $\omega$ not contained in the singular locus $\Gamma$ of $B$. We obtain a fan $\Sigma_{x}$ in the tangent space $\Lambda_{x}\otimes_{\mathbb{Z}}\mathbb{R}$ of not necessarily strictly convex cones consisting of the tangent cones at $x$ of each cell $\sigma$ containing $\omega$. We can choose a representative $\varphi_{x}$ for $\varphi$ in a small neighbourhood of $x$ which is zero along $\omega$, and this can then be extended linearly on each cone of $\Sigma_{x}$ to view $\varphi_{x}$ as a piecewise linear function $\varphi_{x}:\Lambda_{x}\otimes\mathbb{R}\rightarrow\mathbb{R}$. This in turn defines a monoid

$$P_{x}:=\{(m,r)\in\Lambda_{x}\times\mathbb{Z}\,|\,r\geq\varphi_{x}(m)\},$$

completely analogous to the definition of $P_{v}$.

For each maximal cell $\sigma$ containing $\tau$, let $n_{\sigma}\in\check{\Lambda}_{x}$ denote the slope of $\varphi_{x}$ restricted to the tangent cone of $\sigma$. We then define a monomial ideal in the ring $\mathbb{C}[P_{x}]$ given by

$$I_{\omega,\tau}^{>k}=\langle z^{(m,r)}\,|\,\hbox{ $(m,r)\in P_{x}$, $r-\langle n_{\sigma},m\rangle>k$ for some $\sigma\in\mathscr{P}_{\max}$ with $\sigma\supseteq\tau$}\rangle.$$

Then the desired standard thickening of $V_{\omega,\tau}$ is

$$V^{k}_{\omega,\tau}:=\operatorname{Spec}\mathbb{C}[P_{x}]/I_{\omega,\tau}^{>k}.$$

One checks easily that if $k=0$, this recovers $V_{\omega,\tau}$, and if $k>0$, then the reduced space of $V^{k}_{\omega,\tau}$ is $V_{\omega,\tau}$. Thus this is indeed a thickening of $V_{\omega,\tau}$.

There is one point we have to be quite careful about. This definition would appear to depend on the point $x$, and identifications of different tangent spaces $\Lambda_{x}$, $\Lambda_{x^{\prime}}$ via parallel transport depend on the path because of the presence of the singular locus. We deal with this issue not by choosing a specific point $x$, but choosing a specific maximal reference cell $\sigma$ containing $\tau$. We then can identify any $\Lambda_{x}$ with $\Lambda_{\sigma}$, the well-defined tangent space to $\sigma$, via parallel transport from $x$ directly into $\sigma$. We will notate this additional choice of reference cell by writing $V^{k}_{\omega,\tau,\sigma}$. A different choice of reference cell $\sigma^{\prime}$ gives a space $V^{k}_{\omega,\tau,\sigma^{\prime}}$ abstractly, but not canonically, isomorphic to $V^{k}_{\omega,\tau,\sigma}$. This will prove important below. We also use the notation for the coordinate rings

$$R^{k}_{\omega,\tau,\sigma}:=\mathbb{C}[P_{x}]/I_{\omega,\tau}^{>k},$$

again keeping in mind this choice of reference cell.

There are also natural gluings between these various thickened schemes. One notes that given $\tau_{1}\subseteq\tau_{2}\subseteq\tau_{3}$ there are natural surjections

$$R^{k}_{\tau_{1},\tau_{3},\sigma}\rightarrow R^{k}_{\tau_{1},\tau_{2},\sigma}$$

giving a closed embedding $V^{k}_{\tau_{1},\tau_{2},\sigma}\rightarrow V^{k}_{\tau_{1},\tau_{3},\sigma}$, and natural inclusions

$$R^{k}_{\tau_{1},\tau_{3},\sigma}\rightarrow R^{k}_{\tau_{2},\tau_{3},\sigma},$$

giving open embeddings $V^{k}_{\tau_{2},\tau_{3},\sigma}\rightarrow V^{k}_{\tau_{1},\tau_{3},\sigma}$.

If $B$ has no singularities, then the reference cell $\sigma$ is not important, and we drop this from the notation in this case. In particular, it is easy to check that if we take, say, $\tau_{1}$ to be a fixed vertex $v$, and we take the limit of the directed system $\{V^{k}_{v,\tau}\,|\,v\in\tau\}$ of schemes, we obtain a $k$-th order thickening $V^{k}(v)$ of $V(v)$ given by $V^{k}(v)=U(v)\times_{\mathbb{A}^{1}}\operatorname{Spec}\mathbb{C}[t]/(t^{k+1})$, with $U(v)\rightarrow\mathbb{A}^{1}$ the morphism given by $z^{(0,1)}$. This is precisely the kind of vanilla smoothing the log structure leads us to expect. Note we can write this direct limit of schemes as

$$\operatorname{Spec}\lim_{\longleftarrow\atop\tau}R^{k}_{v,\tau}.$$

The basic idea then will be to modify the various maps above by some additional data.

To understand why we need these modifications, let us consider the single most important example, that of an isolated singularity of focus-focus type in a two-dimensional $B$.

We suppose $\mathscr{P}$ contains two maximal cells $\sigma_{1},\sigma_{2}$, with $\sigma_{1}\cap\sigma_{2}=\tau$, as depicted in Figure 5. Note that the intersection of the two coordinate charts is $(\sigma_{1}\cup\sigma_{2})\setminus\tau$, and the transition map is then the identity on $\sigma_{1}\setminus\tau$ and is given by the linear transformation $\begin{pmatrix}1&0\\ 1&1\end{pmatrix}$ on $\sigma_{2}\setminus\tau$. Together, these two charts define an integral affine structure on $(\sigma_{1}\cup\sigma_{2})\setminus\Gamma$, where $\Gamma=\{p\}$ is the common point of the two cuts.

We then take $\varphi$ to be single-valued, identically $0$ on $\sigma_{1}$ and taking the value $1$ at the right-hand vertex.

One now finds

	$\displaystyle R^{k}_{\tau,\sigma_{1},\sigma_{1}}={}$	$\displaystyle\mathbb{C}[x,y,w^{\pm 1}]/(y^{k+1})$
	$\displaystyle R^{k}_{\tau,\sigma_{2},\sigma_{2}}={}$	$\displaystyle\mathbb{C}[x,y,w^{\pm 1}]/(x^{k+1})$
	$\displaystyle R^{k}_{\tau,\tau,\sigma_{i}}={}$	$\displaystyle\mathbb{C}[x,y,w^{\pm 1}]/(x^{k+1},y^{k+1}).$

Here, if we use the chart on the left, i.e., choose a point $s$ below $p$ and work in $P_{s}\subseteq\Lambda_{s}\oplus\mathbb{Z}$, the variables $x,y$ and $w$ are identified with elements of $\mathbb{C}[P_{s}]$ as

$$x=z^{(-1,0,0)},\quad y=z^{(1,0,1)},\quad w=z^{(0,1,0)}.$$

We have the natural surjections $R^{k}_{\tau,\sigma_{i},\sigma_{i}}\rightarrow R^{k}_{\tau,\tau,\sigma_{i}}$, and we identify $R^{k}_{\tau,\tau,\sigma_{1}}$ with $R^{k}_{\tau,\tau,\sigma_{2}}$ by identifying $\Lambda_{\sigma_{1}}$ and $\Lambda_{\sigma_{2}}$ by parallel transport through $s$. Since we have written everything in the left-hand chart, where $\Lambda_{\sigma_{1}}$ and $\Lambda_{\sigma_{2}}$ are identified via parallel transport through $s$, this identification is the trivial one. We can thus glue together the coordinate rings of the thickenings as

$$R^{k}_{\tau,\sigma_{1},\sigma_{1}}\times_{R^{k}_{\tau,\tau,\sigma_{i}}}R^{k}_{\tau,\sigma_{2},\sigma_{2}}.$$

This fibred product of rings is easily seen to be isomorphic to the ring

$$\mathbb{C}[X,Y,W^{\pm 1},t]/(t-XY,t^{k+1}),$$

where $X=(x,x)$, $Y=(y,y)$, $W=(w,w)$, and $t=(xy,xy)$ as elements of the Cartesian product of rings.

On the other hand, suppose we instead identified $R^{k}_{\tau,\tau,\sigma_{1}}$ and $R^{k}_{\tau,\tau,\sigma_{2}}$ by parallel transport through a point $s^{\prime}$ lying above $p$. To do this, we can work in the right-hand chart. Again, $x,y$ and $w$ are defined using the tangent vectors $(-1,0),(1,0)$ and $(0,1)$ in $\sigma_{1}$, and these are transported to the same tangent vectors in $\sigma_{2}$ in the second chart. However, to compare this with our original description of $R^{k}_{\tau,\tau,\sigma_{2}}$, we need to think of these as tangent vectors in $\sigma_{2}$ in the original chart, i.e., the left-hand chart. There, these tangent vectors are $(-1,1)$, $(1,-1)$ and $(0,1)$ respectively. Thus we obtain an isomorphism $R^{k}_{\tau,\tau,\sigma_{1}}\rightarrow R^{k}_{\tau,\tau,\sigma_{2}}$ given by

(10.1)

$$x\mapsto xw,\quad y\mapsto yw^{-1},\quad w\mapsto w.$$

Using this identification, we obtain a composed map $R^{k}_{\tau,\sigma_{1},\sigma_{1}}\rightarrow R^{k}_{\tau,\tau,\sigma_{1}}\rightarrow R^{k}_{\tau,\tau,\sigma_{2}}$, leading to a fibred product

$$R^{k}_{\tau,\sigma_{1},\sigma_{1}}\times_{R^{k}_{\tau,\tau,\sigma_{2}}}R^{k}_{\tau,\sigma_{2},\sigma_{2}}\cong\mathbb{C}[X,Y,W^{\pm 1},t]/(XY-tW,t^{k+1}),$$

where now

$$X=(x,xw),\quad Y=(yw,y),\quad W=(w,w),\quad t=(xy,xy).$$

Note that while this new ring is abstractly isomorphic to the previous ring, there is no isomorphism as $\mathbb{C}[t]/(t^{k+1})$-algebras.

So the gluing is not well-defined, and this is caused by the singularities of $B$. The correct smoothing in this case will depend on the choice of log structure, but in any event we expect it should be a family of the form $\operatorname{Spec}\mathbb{C}[X,Y,W^{\pm 1},t]/(XY-f(W)t)$ for some function $f(W)$ which vanishes along the $W$-axis precisely at the points where the given log structure on $X_{0}(B,\mathscr{P},s)$ is not fine. Clearly $f$ is then determined by the log structure up to invertible functions. Let us take for the sake of this example the function $f(W)=1+W$, noting that $f(W)=1+W^{-1}$ would do just as well. We can now modify the gluings using Figure 6.

In this figure, we have drawn two rays contained in $\tau$ emanating from the singular point, and labelled these two arrows with the functions $1+w^{-1}$ and $1+w$ respectively. These rays tell us that if we try to identify $R^{k}_{\tau,\tau,\sigma_{1}}$ with $R^{k}_{\tau,\tau,\sigma_{2}}$ using parallel transport between the two maximal cells, we need to modify the identification via an automorphism given by the crossing of one of these rays. Here, we will get different automorphisms depending on whether we cross above or below the singularity $p$. If we cross below, the ray tells us to use an automorphism of $R^{k}_{\tau,\tau,\sigma_{1}}$ given by

(10.2)

$$x\mapsto x(1+w),\quad y\mapsto y(1+w)^{-1},\quad w\mapsto w,$$

while if we cross above the singularity, we use the automorphism

(10.3)

$$x\mapsto x(1+w^{-1}),\quad y\mapsto y(1+w^{-1})^{-1},\quad w\mapsto w.$$

Actually, note that $1+w$ or $1+w^{-1}$ is not invertible in $R^{k}_{\tau,\tau,\sigma_{i}}$, so we need to modify this ring by localizing it at $1+w$ (or equivalently $1+w^{-1}$). Let’s see how this affects the fibred products $R^{k}_{\tau,\sigma_{1},\sigma_{1}}\times_{R^{k}_{\tau,\tau,\sigma_{2}}}R^{k}_{\tau,\sigma_{2},\sigma_{2}}$.

If we use parallel transport below the singular point, then the map $R^{k}_{\tau,\sigma_{1},\sigma_{1}}\rightarrow R^{k}_{\tau,\tau,\sigma_{2}}$ is just given by (10.2), while $R^{k}_{\tau,\sigma_{2},\sigma_{2}}\rightarrow R^{k}_{\tau,\tau,\sigma_{2}}$ remains the canonical one. One then finds

$$R^{k}_{\tau,\sigma_{1},\sigma_{1}}\times_{R^{k}_{\tau,\tau,\sigma_{2}}}R^{k}_{\tau,\sigma_{2},\sigma_{2}}\cong\mathbb{C}[X,Y,W^{\pm},t]/(XY-(1+W)t,t^{k+1}),$$

with

$$X=(x,x(1+w)),\quad Y=(y(1+w),y),\quad W=(w,w),\quad t=(xy,xy).$$

On the other hand, if we use parallel transport above the singular point, we need to compose the automorphism (10.3) with the isomorphism (10.1), giving a map $R^{k}_{\tau,\sigma_{1},\sigma_{1}}\rightarrow R^{k}_{\tau,\tau,\sigma_{2}}$ given by

$$x\mapsto xw(1+w^{-1})=x(1+w),\quad y\mapsto yw^{-1}(1+w^{-1})^{-1}=y(1+w)^{-1},\quad w\mapsto w.$$

Thus this map is exactly the same as (10.2), and hence we get the same fibred product. The glued thickenings are independent of choices. The introduction of the extra automorphisms removes the problems caused by monodromy.

This is a very local situation. The next problem which arises is that more globally, we need to propagate the automorphisms attached to the rays. Indeed, imagine now that the picture we are looking at is contained in a more complex situation, as on the left-hand side of Figure 7. Here we have two singularities, and rays emanate in each direction from the singularity. Let us follow the rule that any identification of rings which involves parallel transport through a ray must be modified by the appropriate automorphism as described above. Then looking at the vertex $v_{1}$, say, we need to glue together five irreducible components, but only one of these gluings is modified. These gluings would not be compatible. To correct for this, one can extend the ray indefinitely, and “parallel transport” the automorphism along the ray. There is a precise sense in which this can be done. This is shown on the right-hand picture in Figure 7, with the dotted lines showing the extension of the rays. Now if crossing a ray in one direction produces the inverse of the automorphism given by crossing the ray the other direction, one finds that gluing at the vertices $v_{1}$ and $v_{2}$ have now become compatible.

A new problem arises, however, at the intersection point of the two rays. Again, when we try to identify various rings using parallel transport and automorphisms induced by crossing rays, we don’t want the choice to depend on the particular path we take. Because in general the two automorphisms attached to the rays don’t commute, we again have trouble at the point of intersection.

This is in fact where our thinking stood in early 2004, shortly before the release of Kontsevich and Soibelman’s paper [53]. The solution to this problem, really the key part of Kontsevich and Soibelman’s argument, is to add new rays emanating from the point of intersection of the old rays, as depicted in Figure 8. These rays are added in such a way as to guarantee that the composition of automorphisms given by a loop around the intersection point is in fact the identity, and thus the identifications will be independent of the choice of path.

The description here is somewhat vague, but demonstrates the basic idea. We’ve seen how we obtain our degeneration by gluing together basic pieces. Other than these different basic pieces, in two dimensions, the main distinction between our approach and the one taken by Kontsevich and Soibelman in [53] is that we work in the affine structure dual to the one [53] works with. They propogate automorphisms along gradient flow lines, but we are able to propogate automorphisms along straight lines with respect to the affine structure. This saves a great deal of trouble in higher dimensions, where gradient flow lines will be much more difficult to control. That makes it possible for us to obtain results in all dimensions.

We of course have not made it particularly clear how we really encode automorphisms and how they propagate, but we will make at least the first point clearer in the next section. For the second point, the main thing is that they propagate along straight lines; this in fact is crucial for guaranteeing that the automorphisms don’t start to involve monomials with poles on irreducible components of $X_{0}$. So here we see something which looks tropical already, with the union of rays looking like a tropical tree. Again, we will make this more precise in the next section.

In higher dimensions, the argument becomes much more subtle. Instead of rays carrying automorphisms, codimension one wall carry automorphisms, and one needs to be very careful about how these walls propagate. Furthermore, there are great technical difficulties concerning convergence of the algorithm near the discriminant locus. This was handled in [53] in two dimensions via an argument showing new rays added can be guaranteed to avoid a neighbourhood of each singularity, but this is done by choosing the metric carefully. In higher dimensions, this is not true, and instead we used algebraic methods to prove convergence. All these difficulties were overcome in [32].

In [25] I wrote down a complete version of the proof in two dimensions; this has the advantage of avoiding most of the really technical issues. Hopefully, [25] provides a gentler entry point into the ideas outlined here than the main paper [32].

We now turn to a more precise description of the automorphisms involved, and give evidence that the description of the explicit deformations (which we view as $B$-model information) really encodes $A$-model information on the mirror.