Slab function and structure of a tropical manifold

Intuition [Beginner]

A tropical manifold is a piecewise-linear space cut into polyhedral chambers, with integer affine coordinates on each chamber but the gluing between adjacent chambers possibly twisted. When you try to lift this piecewise-linear picture back to a smooth complex Calabi-Yau variety, the gluing data on each chamber is a toric chart, and the data on each wall between two chambers must record how the two toric charts paste together. The polynomial that encodes this pasting is called a slab function.

A slab function lives on a codimension-one face — a "slab" between two adjacent maximal chambers — and it is a Laurent polynomial in the toric coordinates that are common to both chambers. The total collection of slab functions and other wall-crossing data is the structure of the tropical manifold, written $\mathcal{S}$. The structure is the bookkeeping object: it tells you, at every wall in the manifold, exactly how the two sides of the wall are glued so that the resulting complex manifold is well-defined.

Without slab functions, the chambers would just be isolated toric charts that happen to share a polyhedral boundary. With slab functions, the toric charts are glued into a single coherent smooth complex variety. Slab functions are what turn the combinatorial-tropical skeleton into honest algebraic geometry.

Visual [Beginner]

A two-panel cartoon. Left panel: a piecewise-linear surface partitioned into polyhedral chambers (think hexagonal patches on a flat region of a tropical manifold), with the codimension-one boundaries between chambers highlighted in colour. Right panel: a zoom on one such boundary slab between two chambers, with a Laurent polynomial $f_{\rho }=1+c_1{z}^{m_1}+c_2{z}^{m_2}$ labelled along the slab, recording the transition data between the toric charts on the two adjacent chambers.

The picture captures the slab function as a piece of polynomial data attached to a codimension-one face of the polyhedral decomposition. Each maximal chamber carries its own toric coordinates; each slab between two chambers carries a polynomial in the coordinates common to both, and this polynomial is exactly the gluing rule.

Worked example [Beginner]

Consider the simplest substantive tropical manifold: a one-dimensional piecewise-linear interval split into two segments at a single interior point. The two segments are two chambers; the single interior point is one slab.

Step 1. On the left chamber, toric coordinates are a single variable $z$. On the right chamber, toric coordinates are a single variable $w$.

Step 2. At the slab between them, the two coordinates must be identified by a transition rule. The simplest substantive slab function is $$ f_\rho ;=; 1 + t \cdot z, $$ where $t$ is a deformation parameter measuring how far from the central fibre we are. The transition rule says: on the slab, glue $w=z\cdot f_{\rho }(z)=z+t\cdot z^2$.

Step 3. The resulting one-dimensional complex manifold is a one-parameter family. When $t=0$, the slab function reduces to $f_{\rho }=1$ and the two chambers paste along the identity transition $w=z$, giving a degenerate central fibre. When $t\ne 0$, the slab function deforms the transition, smoothing out the meeting point into a single smooth complex curve.

What this tells us: the slab function is the polynomial that interpolates between the identity central-fibre gluing and the smoothed generic-fibre transition. A whole tropical manifold has many chambers and many slabs; the collection of all slab functions is the structure $\mathcal{S}$, and consistency of $\mathcal{S}$ is what allows the chambers to paste into a single coherent complex Calabi-Yau.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Fix a tropical manifold $(B,\mathcal{P})$ of dimension $n$ in the sense of [04.12.08]: a topological manifold $B$ together with a polyhedral decomposition $\mathcal{P}$ and an integral affine structure on $B\setminus \Delta$ with singularities along a codimension-2 closed subset $\Delta \subseteq B$, satisfying the local-models conditions of Gross-Siebert (every point of $B\setminus \Delta$ admits an open affine chart in ${\mathbb{R}}^n$, and the singularities along $\Delta$ are of the focus-focus or join-up type). Fix in addition a strictly convex piecewise-linear function $\phi$ on $(B,\mathcal{P})$ — the polarisation of the tropical manifold — making $(B,\mathcal{P},\phi )$ a polarised tropical manifold.

Write ${\mathcal{P}}^{[k]}$ for the set of $k$-dimensional cells of $\mathcal{P}$. Maximal cells are elements of ${\mathcal{P}}^{[n]}$; slabs are codimension-one cells in ${\mathcal{P}}^{[n-1]}$ that lie in $B\setminus \Delta$ and separate two maximal cells. For $\sigma \in {\mathcal{P}}^{[n]}$ a maximal cell, the tangent space ${\Lambda }_{\sigma }:=T_{\sigma }B\cap N\cong {\mathbb{Z}}^n$ at any interior point of $\sigma$ is the integral lattice of the affine chart on $\sigma$, and the toric ring of $\sigma$ is $$ R_\sigma ;:=; \mathbb{C}[t, t^{-1}][\Lambda_\sigma] ;=; \mathbb{C}[t, t^{-1}][z^{m} \mid m \in \Lambda_\sigma], $$ the Laurent-polynomial ring on ${\Lambda }_{\sigma }$ over the deformation parameter ring $\mathbb{C}[t,t^{-1}]$.

Definition (slab function). Let $\rho \in {\mathcal{P}}^{[n-1]}$ be a slab with adjacent maximal cells ${\sigma }_{+},{\sigma }_{-}\in {\mathcal{P}}^{[n]}$. Let ${\Lambda }_{\rho }:=T_{\rho }B\cap N\cong {\mathbb{Z}}^{n-1}$ be the tangent integral lattice of $\rho$, sitting inside both ${\Lambda }_{{\sigma }_{+}}$ and ${\Lambda }_{{\sigma }_{-}}$ as the common codimension-one sublattice. A slab function on $\rho$ is a Laurent polynomial $$ f_\rho ;\in; \mathbb{C}[t][\Lambda_\rho] ;=; \mathbb{C}[t][z^{m} \mid m \in \Lambda_\rho] $$ in the toric coordinates of $\rho$, normalised so that $$ f_\rho \big|{t = 0} ;=; 1 \in \mathbb{C}[\Lambda\rho], $$ and required to be compatible at every codimension-two cell $\tau \subseteq \overline{\rho }$ with $\tau \in {\mathcal{P}}^{[n-2]}\cap (B\setminus \Delta )$ — meaning the local slab-function pull-back at $\tau$ is an analytic restriction from a Laurent polynomial in ${\Lambda }_{\tau }\otimes \mathbb{C}[t]$ that is itself the slab-function output at $\tau$ for any neighbouring slab.

Definition (wall). A wall is a codimension-one rational polyhedral subset $\mathfrak{d}$ in the interior of a maximal cell $\sigma \in {\mathcal{P}}^{[n]}$ (so $\mathfrak{d}$ does not coincide with a face of $\mathcal{P}$) together with a wall-crossing automorphism $$ \theta_\mathfrak{d} ;\in; \mathrm{Aut}\big(R_\sigma \big/ (t^N)\big) $$ of the truncated toric ring of $\sigma$, for some order $N\in {\mathbb{Z}}_{\ge 1}$. The wall-crossing automorphism is required to be of the form $$ \theta_\mathfrak{d}(z^m) ;=; z^m \cdot f_\mathfrak{d}^{\langle n_\mathfrak{d}, m\rangle}, $$ where $f_{\mathfrak{d}}\in 1+t\cdot R_{\sigma }$ is a wall function and $n_{\mathfrak{d}}\in {\Lambda }_{\sigma }^{\vee }$ is a primitive normal covector to $\mathfrak{d}$ in $\sigma$. The exponent $⟨n_{\mathfrak{d}},m⟩$ is the signed transverse-crossing number of the monomial direction $m$ across $\mathfrak{d}$.

Definition (structure). A structure $\mathcal{S}$ on $(B,\mathcal{P},\phi )$ is a pair $\mathcal{S}=(\mathcal{F},\mathcal{W})$ consisting of:

• A collection $\mathcal{F}=\{f_{\rho }{\}}_{\rho \in {\mathcal{P}}^{[n-1]}\cap (B\setminus \Delta )}$ of slab functions, one on each slab away from the discriminant locus, normalised as above.

• A locally finite collection $\mathcal{W}=\{(\mathfrak{d},{\theta }_{\mathfrak{d}})\}$ of walls in the interiors of maximal cells, satisfying a consistency condition at every codimension-two cell $\tau$: the product of wall-crossing automorphisms around a small loop encircling $\tau$ in $B\setminus \Delta$, taken in the order prescribed by the cyclic ordering of walls and slabs at $\tau$, is the identity automorphism modulo $t^N$ for every $N\ge 1$.

The locally-finite condition means that for every compact $K\subseteq B$ and every $N\ge 1$, only finitely many $(\mathfrak{d},{\theta }_{\mathfrak{d}})$ with $\mathfrak{d}\cap K\ne \varnothing$ have non-identity wall-crossing automorphism modulo $t^N$.

Counterexamples to common slips

• A slab function $f_{\rho }$ is not an arbitrary Laurent polynomial in ${\Lambda }_{\rho }$. The normalisation $f_{\rho }{|}_{t=0}=1$ is the defining condition: $f_{\rho }$ is a $t$-deformation of the constant function $1$. Without this normalisation, the central-fibre gluing would not be the identity pasting that the toric-degeneration setup requires.

• A slab is not the same as a wall. A slab $\rho \in {\mathcal{P}}^{[n-1]}$ is a codimension-one face of the polyhedral decomposition $\mathcal{P}$ in the away-from-discriminant region $B\setminus \Delta$; it is a constituent of $(B,\mathcal{P})$. A wall $\mathfrak{d}$ is a separate codimension-one polyhedral subset added in the interior of a maximal cell to enforce consistency; it is not a face of $\mathcal{P}$. Both are codimension one in $B$, but their roles in $\mathcal{S}$ are different: slab functions are imposed input; walls are computed output of the scattering algorithm.

• The consistency condition is not automatic from a choice of slab functions. The slab functions $\mathcal{F}$ alone produce a system of toric-chart transition rules that, in general, fails to glue consistently around a codimension-2 cell $\tau$ — the round-trip product around $\tau$ deviates from the identity at order $t$ or higher. The walls $\mathcal{W}$ are added precisely to absorb this deviation: their wall-crossing automorphisms must compose to cancel the slab-function monodromy at every $\tau$. Consistency is a substantive condition that, in the Gross-Siebert programme, is solved order-by-order in $t$ by the scattering algorithm of Kontsevich-Soibelman / Carl-Pumperla-Siebert.

Key theorem with proof [Intermediate+]

Theorem (consistency-of-structure, Gross-Siebert 2010). Let $(B,\mathcal{P},\phi )$ be a polarised tropical manifold of dimension $n$, and let $\mathcal{S}=(\mathcal{F},\mathcal{W})$ be a consistent structure on it. Then for every order $N\ge 1$ there is a flat $\mathbb{C}[t]/(t^{N+1})$-family of schemes ${\mathfrak{X}}_N\to \mathrm{Spec}\mathbb{C}[t]/(t^{N+1})$ with central fibre the toric central fibre $X_0(B,\mathcal{P},\phi )$ of the polarised tropical manifold, and the families ${\mathfrak{X}}_N$ for $N\to \infty$ assemble into a formal flat family $\hat{\mathfrak{X}}\to \mathrm{Spf}\mathbb{C}[[t]]$ whose generic fibre is a smoothing of $X_0$.

Proof. The proof has three movements. We give the intermediate version; the Master tier elaborates the Čech-cohomological framing.

Step 1 (chartwise scheme structure from the toric data). Each maximal cell $\sigma \in {\mathcal{P}}^{[n]}$ carries a toric ring $R_{\sigma }=\mathbb{C}[t][{\Lambda }_{\sigma }]$. The chartwise scheme $\mathrm{Spec}{R}_{\sigma }/(t^{N+1})$ is the affine toric scheme over $\mathbb{C}[t]/(t^{N+1})$ associated to the cocharacter lattice ${\Lambda }_{\sigma }$. The central-fibre scheme $X_0(\sigma )=\mathrm{Spec}{R}_{\sigma }/(t)$ is the affine toric variety $\mathrm{Spec}\mathbb{C}[{\Lambda }_{\sigma }]$.

Step 2 (codimension-one gluing via slab functions). For adjacent maximal cells ${\sigma }_{+},{\sigma }_{-}$ sharing a slab $\rho$, the slab function $f_{\rho }\in \mathbb{C}[t][{\Lambda }_{\rho }]$ defines a transition rule between the toric coordinates on ${\sigma }_{+}$ and ${\sigma }_{-}$. Specifically, if $z_{+}\in {\Lambda }_{{\sigma }_{+}}$ and $z_{-}\in {\Lambda }_{{\sigma }_{-}}$ are the toric monomials adjacent to $\rho$ (i.e., the primitive normals to $\rho$ pointing into ${\sigma }_{+}$ and ${\sigma }_{-}$ respectively), the transition is $$ z_- ;=; z_+^{-1} \cdot f_\rho \cdot \pi_\rho, $$ where ${\pi }_{\rho }\in R_{\sigma }$ is a power of $t$ determined by the polarisation $\phi$ across $\rho$. The relation defines the gluing of the two affine schemes $\mathrm{Spec}{R}_{{\sigma }_{+}}/(t^{N+1})$ and $\mathrm{Spec}{R}_{{\sigma }_{-}}/(t^{N+1})$ along the codimension-one stratum $\rho$. This is the local model on every slab.

Step 3 (consistency at codimension two via walls). At every codimension-two cell $\tau \in {\mathcal{P}}^{[n-2]}\cap (B\setminus \Delta )$, the slab functions on the slabs adjacent to $\tau$ produce a transition cocycle on the small loop around $\tau$ in $B\setminus \Delta$. By the consistency condition built into $\mathcal{S}$, the product of slab transitions around $\tau$, composed with the wall-crossing automorphisms ${\theta }_{\mathfrak{d}}$ of the walls crossing the small loop, is the identity automorphism modulo $t^{N+1}$. Therefore the chartwise schemes $\mathrm{Spec}{R}_{\sigma }/(t^{N+1})$ glue consistently around every $\tau$.

Combining the three movements: for every $N\ge 1$ the chartwise affine schemes glue into a single flat $\mathbb{C}[t]/(t^{N+1})$-scheme ${\mathfrak{X}}_N$. The compatibility ${\mathfrak{X}}_{N+1}{\times }_{\mathbb{C}[t]/(t^{N+2})}\mathbb{C}[t]/(t^{N+1})={\mathfrak{X}}_N$ is automatic from the chartwise constructions. The formal limit $\hat{\mathfrak{X}}=\underrightarrow{\lim }{\mathfrak{X}}_N$ is the desired flat formal family with central fibre $X_0$. Effective algebraisation of $\hat{\mathfrak{X}}$ to an honest one-parameter family $\mathfrak{X}\to {\mathbb{A}}^1$ is the reconstruction theorem of [04.12.09]; the present theorem provides the order-by-order construction. $\square$

Bridge. The consistency-of-structure theorem builds toward [04.12.09] Gross-Siebert reconstruction theorem, where the formal family $\hat{\mathfrak{X}}$ is promoted to a one-parameter algebraic smoothing of $X_0$; appears again in [04.12.12] theta functions, where the slab-and-wall data $\mathcal{S}$ are exactly what determines the global theta function coefficients on the smoothing; and identifies slab functions with the codimension-one component of the Calabi-Yau Čech cocycle. The foundational reason this works is that toric charts on adjacent maximal cells already paste consistently at the central-fibre level; the slab functions are the order-$t$ deformations of these identity pastings, and consistency at every codimension-two cell forces the order-$t$ deformations to glue into a smooth complex structure. Putting these together generalises the polytope-fan dictionary of [04.11.10] from the smooth toric case to the degenerate Calabi-Yau case: in the toric case, the identity slab functions $f_{\rho }=1$ paste affine toric charts into the projective toric variety $X_P$; in the Calabi-Yau case, substantive slab functions paste a more elaborate central fibre into a smooth Calabi-Yau via the formal one-parameter family.

Exercises [Intermediate+]

Lean formalisation [Intermediate+]

The Lean module Codex.AlgGeom.Tropical.SlabFunction schematises the data of a slab function, a wall, and a structure $\mathcal{S}$ on a polarised tropical manifold. Current Mathlib supplies the basics of Laurent polynomial rings via Mathlib.RingTheory.LaurentPolynomial, the truncated polynomial rings via Mathlib.Algebra.Polynomial.AlgebraMap, and the lattice / free module infrastructure via Mathlib.LinearAlgebra.FreeModule.Basic. The polyhedral-decomposition and integral-affine-with-singularities packages, the toric-degeneration constructions, and the wall-crossing automorphism groups are absent and are the principal Mathlib gaps for the present unit.

The module declares: a Lattice placeholder (the integral lattice $\Lambda$ at a chamber), a PolarisedTropicalManifold structure recording $(B,\mathcal{P},\phi )$ with witnesses for the polyhedral decomposition, the polarisation, and the integral affine structure (deferred to 04.12.08); a Slab type recording a codimension-1 face $\rho$ of $\mathcal{P}$; a SlabFunction structure consisting of a Laurent polynomial $f_{\rho }\in \mathbb{C}[t][{\Lambda }_{\rho }]$ together with a witness that $f_{\rho }{|}_{t=0}=1$; a Wall structure recording a codimension-1 subset $\mathfrak{d}$ in the interior of a maximal cell together with a wall-crossing automorphism ${\theta }_{\mathfrak{d}}$ of the truncated toric ring; and a Structure type packaging the two collections together with a consistency witness.

Two named theorems are recorded with sorry-equivalent proof bodies: structure_glues_orderwise records the consistency-of-structure theorem (Gross-Siebert 2010), asserting that a consistent structure $\mathcal{S}$ produces a flat $\mathbb{C}[t]/(t^{N+1})$-family for every $N$; and polarisation_compatibility records the compatibility of the slab-function transitions with the polarisation $\phi$, asserting that the ${\pi }_{\rho }=t^{{\kappa }_{\rho }}$ factor in the transition rule encodes the height jump of $\phi$ across $\rho$.

The Mathlib gap is: (i) polyhedral decompositions of a topological manifold with integral affine charts and discriminant-locus singularities; (ii) Laurent polynomial rings parametrised by a lattice $\Lambda$ with a deformation parameter $t$; (iii) the wall-crossing automorphism groups of truncated toric rings, requiring pro-nilpotent Lie-algebra completions; (iv) the order-by-order scattering algorithm of Kontsevich-Soibelman / Carl-Pumperla-Siebert; (v) the formal-smoothing-from-consistent-structure theorem of Gross-Siebert 2010 JAG 19. The aggregated gap is the natural target for upstream Mathlib contributions extending the present partial Lean development.

Definitions: slab functions and walls [Master]

We elaborate the technical content of the slab-function and wall definitions, with attention to the integer-affine-with-singularities setting in which they live.

The polarised tropical manifold revisited. A polarised tropical manifold in the sense of Gross-Siebert 2011 Annals 174 §2 is a quadruple $(B,\mathcal{P},\phi ,\mathcal{S})$ where $B$ is a topological $n$-manifold, $\mathcal{P}$ is a polyhedral cell decomposition of $B$, $\phi :(B,\mathcal{P})\to \mathbb{R}$ is a strictly convex piecewise-linear polarisation, and $\mathcal{S}$ is a structure as defined below. The pair $(B,\mathcal{P})$ admits an integral affine atlas away from a closed singular locus $\Delta \subseteq B$ of codimension $2$, with focus-focus / join-up local models near $\Delta$ controlling the local monodromy of the affine charts (Gross-Siebert 2006 JDG 72 §3). Each maximal cell $\sigma \in {\mathcal{P}}^{[n]}$ inherits a global integral affine chart (since $\Delta$ is codimension $2$, every maximal cell is contained in $B\setminus \Delta$).

Tangent lattices at faces. For each cell $\tau \in {\mathcal{P}}^{[k]}$ in the away-from-discriminant region, the tangent integral lattice ${\Lambda }_{\tau }\subseteq T_{\tau }B$ is the rank-$k$ subgroup of the integral affine lattice $N\cong {\mathbb{Z}}^n$ parallel to $\tau$. For $\tau \subseteq {\tau }^{\prime }$ a face inclusion, ${\Lambda }_{\tau }\subseteq {\Lambda }_{{\tau }^{\prime }}$ is a primitive sublattice. The maximal-cell tangent lattices are rank $n$; the codimension-one slab tangent lattices are rank $n-1$; the codimension-two cell tangent lattices are rank $n-2$. The dual lattices ${\Lambda }_{\tau }^{\vee }$ encode normal directions: for a slab $\rho$ with adjacent maximal cells ${\sigma }_{+},{\sigma }_{-}$, the quotients ${\Lambda }_{{\sigma }_{\pm }}/{\Lambda }_{\rho }\cong \mathbb{Z}$ each carry a canonical primitive generator pointing into the respective chamber, recording the local toric-coordinate change across $\rho$.

Slab functions in detail. Let $\rho \in {\mathcal{P}}^{[n-1]}$ be a slab adjacent to maximal cells ${\sigma }_{+},{\sigma }_{-}\in {\mathcal{P}}^{[n]}$. Denote by ${\Lambda }_{\rho }\subseteq {\Lambda }_{{\sigma }_{+}}\cap {\Lambda }_{{\sigma }_{-}}$ the common rank-$(n-1)$ tangent lattice of $\rho$. A slab function on $\rho$ is an element $$ f_\rho ;\in; \mathbb{C}[t]\big[\Lambda_\rho\big] ;=; \mathbb{C}[t][z^m \mid m \in \Lambda_\rho] $$ satisfying:

(a) Normalisation. $f_{\rho }{|}_{t=0}=1\in \mathbb{C}[{\Lambda }_{\rho }]$; equivalently $f_{\rho }=1+t\cdot g_{\rho }$ for some $g_{\rho }\in \mathbb{C}[t][{\Lambda }_{\rho }]$.

(b) Positive-Newton condition. Every monomial $c_m\cdot t^k\cdot z^m$ appearing in $f_{\rho }$ with $k\ge 1$ has the property that $m$ points "outward" with respect to a chosen orientation of the slab (specifically: $⟨n_{\rho },m⟩\le 0$ in the convention of Gross-Siebert 2010 JAG 19 §1, where $n_{\rho }\in {\Lambda }_{{\sigma }_{+}}^{\vee }$ is the primitive outward normal of ${\sigma }_{+}$ at $\rho$). This condition records the analytic positivity of the slab-function transition.

(c) Compatibility at codimension-2. For every $\tau \in {\mathcal{P}}^{[n-2]}\cap \overline{\rho }\cap (B\setminus \Delta )$, the restriction $f_{\rho }{|}_{\tau }\in \mathbb{C}[t][{\Lambda }_{\tau }]$ — obtained by projecting the monomials of $f_{\rho }$ to ${\Lambda }_{\tau }$ via ${\Lambda }_{\rho }\to {\Lambda }_{\tau }\oplus ({\Lambda }_{\rho }/{\Lambda }_{\tau })$ and integrating out the transverse direction — agrees with the codimension-2 slab-function data at $\tau$.

Walls in detail. Let $\sigma \in {\mathcal{P}}^{[n]}$ be a maximal cell. A wall in $\sigma$ is a pair $(\mathfrak{d},{\theta }_{\mathfrak{d}})$ where $\mathfrak{d}$ is a closed rational polyhedral subset of $\sigma$ of dimension $n-1$, not contained in any face of $\mathcal{P}$, and ${\theta }_{\mathfrak{d}}$ is an automorphism of the truncated toric ring $R_{\sigma }/(t^N)$ of the form $$ \theta_\mathfrak{d}(z^m) ;=; z^m \cdot f_\mathfrak{d}^{\langle n_\mathfrak{d}, m\rangle} $$ where $n_{\mathfrak{d}}\in {\Lambda }_{\sigma }^{\vee }$ is a primitive covector normal to $\mathfrak{d}$ in $\sigma$, and $f_{\mathfrak{d}}\in 1+t\cdot \mathbb{C}[t][{\Lambda }_{\sigma }]$ is the wall function. The wall-crossing automorphism is uniquely determined by the pair $(\mathfrak{d},f_{\mathfrak{d}})$; conversely, the wall function $f_{\mathfrak{d}}$ is determined by ${\theta }_{\mathfrak{d}}$ once a primitive normal $n_{\mathfrak{d}}$ is chosen.

The wall function $f_{\mathfrak{d}}$ satisfies a structural condition: its monomials $c_m{t}^k{z}^m$ have $⟨n_{\mathfrak{d}},m⟩=0$ — that is, every monomial of $f_{\mathfrak{d}}$ is tangent to $\mathfrak{d}$. This ensures the wall-crossing automorphism preserves the codimension and acts non-identically only in the wall-crossing direction.

The toric central fibre $X_0(B,\mathcal{P},\phi )$. Before assembling a smoothing via slab functions, the polarised tropical manifold determines a toric central-fibre scheme $X_0=X_0(B,\mathcal{P},\phi )$. The construction: for each maximal cell $\sigma \in {\mathcal{P}}^{[n]}$, take the affine toric variety $X_{\sigma }=\mathrm{Spec}\mathbb{C}[{\Lambda }_{\sigma }]$ of the cocharacter lattice; the central fibre $X_0$ is the union ${\bigcup }_{\sigma }{X}_{\sigma }$ glued along the toric strata indexed by faces of $\mathcal{P}$. The strict convexity of $\phi$ provides an ample line bundle $L_0\to X_0$ extending the toric polarisation on each piece. This $(X_0,L_0)$ is the singular degenerate Calabi-Yau pair that the structure $\mathcal{S}$ then smooths.

The structure $\mathcal{S}$ on $(B,\mathcal{P})$ [Master]

A structure $\mathcal{S}$ on a polarised tropical manifold $(B,\mathcal{P},\phi )$ is the bookkeeping object recording the codimension-one polynomial gluing data needed to smooth the toric central fibre $X_0$.

Definition (structure). A structure $\mathcal{S}=(\mathcal{F},\mathcal{W})$ on $(B,\mathcal{P},\phi )$ consists of:

(I) A collection $\mathcal{F}=\{f_{\rho }{\}}_{\rho \in {\mathcal{P}}^{[n-1]}\cap (B\setminus \Delta )}$ of slab functions, one $f_{\rho }$ on each slab in the away-from-discriminant region, each satisfying the normalisation, positive-Newton, and codimension-2 compatibility conditions above.

(II) A locally finite collection $\mathcal{W}=\{(\mathfrak{d},{\theta }_{\mathfrak{d}})\}$ of walls in the interiors of maximal cells, with wall-crossing automorphisms ${\theta }_{\mathfrak{d}}$ as defined above.

(III) Consistency at codimension two. For every $\tau \in {\mathcal{P}}^{[n-2]}\cap (B\setminus \Delta )$ and every $N\ge 1$, the product of slab-function transitions and wall-crossing automorphisms taken in cyclic order around a small loop encircling $\tau$ is the identity modulo $t^{N+1}$ in the appropriate truncated toric ring.

The locally finite condition in (II) is that for every compact $K\subseteq B$ and every $N\ge 1$, only finitely many $(\mathfrak{d},{\theta }_{\mathfrak{d}})\in \mathcal{W}$ with $\mathfrak{d}\cap K\ne \varnothing$ act non-identically modulo $t^{N+1}$. This local finiteness is what permits the order-by-order assembly of the formal smoothing.

The wall-crossing automorphism group. The automorphisms ${\theta }_{\mathfrak{d}}$ live in a pro-nilpotent group $V_{\rho }$ associated to each slab $\rho$ (Kontsevich-Soibelman 2006). Concretely, $V_{\rho }$ is the group of automorphisms of $\mathbb{C}[t][{\Lambda }_{\sigma }]/(t^N)$ of the form $z^m\mapsto z^m\cdot f^{⟨n,m⟩}$ for $n\in {\Lambda }_{\rho }^{\vee }$ a normal direction and $f\in 1+t\cdot \mathbb{C}[t][{\Lambda }_{\sigma }]/(t^N)$ a normalised polynomial. The group law is composition, taken in the order determined by the angle of $n$ in the normal $1$-circle to $\rho$. Pro-nilpotence ensures convergence of order-by-order products.

The slab attachment of a wall. A wall $(\mathfrak{d},{\theta }_{\mathfrak{d}})$ is attached to a slab $\rho$ if $\mathfrak{d}$ ends on the slab $\rho$ in the closure of the maximal cell where $\mathfrak{d}$ lives. The collection of walls attached to a slab is locally finite (by the local-finiteness condition on $\mathcal{W}$). The Kontsevich-Soibelman / Gross-Siebert scattering algorithm produces walls attached to slabs in a controlled order-by-order fashion, with each newly-added wall correcting the order-$N$ inconsistency at a specific codimension-2 cell.

The order-by-order construction. Given the polarised tropical manifold $(B,\mathcal{P},\phi )$ and an initial choice of slab functions ${\mathcal{F}}^{(1)}=\{f_{\rho }^{(1)}=1+t\cdot {\alpha }_{\rho }{\}}_{\rho }$ where ${\alpha }_{\rho }\in \mathbb{C}[{\Lambda }_{\rho }]$ records the linear-in-$t$ coefficient, the scattering algorithm of Carl-Pumperla-Siebert / Kontsevich-Soibelman iterates:

1. Order $N$ input: slab functions ${\mathcal{F}}^{(N)}$ and walls ${\mathcal{W}}^{(N)}$ truncated at $t^N$.

2. Consistency check at codimension-2: for each $\tau \in {\mathcal{P}}^{[n-2]}\cap (B\setminus \Delta )$, compute the product of order-$N$ transitions and automorphisms around $\tau$.

3. Wall addition: for every codimension-2 cell where the order-$N$ check fails, add new walls of order $N+1$ whose order-$(N+1)$ wall-crossing automorphisms cancel the failure.

4. Increment: the augmented ${\mathcal{F}}^{(N+1)}$ and ${\mathcal{W}}^{(N+1)}$ are consistent modulo $t^{N+2}$ at all codimension-2 cells.

5. Iterate: the procedure terminates at every order in finitely many wall-additions per codimension-2 cell (Carl-Pumperla-Siebert 2010 + Kontsevich-Soibelman 2006), producing a consistent structure $\mathcal{S}$ as $N\to \infty$.

The Gross-Siebert input. The slab functions $f_{\rho }^{(1)}$ at order $1$ are not arbitrary: they are determined by the log Gromov-Witten counts of the central fibre $X_0$ via the Gross-Siebert log-Gromov-Witten theory of Gross-Siebert 2013 JAMS 26. Specifically, the order-$1$ coefficient ${\alpha }_{\rho }$ is a sum over genus-$0$ log-curves on $X_0$ contributing to the slab $\rho$. This is the bridge from the algebraic-enumerative content of [04.12.06] (Nishinou-Siebert correspondence) to the slab-function content of the structure: the curve counts on the algebraic side become the polynomial coefficients on the tropical side.

Scattering diagrams and consistency [Master]

The technical heart of the structure $\mathcal{S}$ is the consistency condition at codimension-2 cells of $\mathcal{P}$ — equivalently, the closing-up of the scattering diagram built from slabs and walls.

The scattering diagram. Around each codimension-2 cell $\tau \in {\mathcal{P}}^{[n-2]}\cap (B\setminus \Delta )$, the slabs and walls incident at $\tau$ form a local scattering diagram. Choose a small open neighbourhood $U_{\tau }\subseteq B$ of a point of $\tau$ in $B\setminus \Delta$, and a small loop $\gamma$ in $U_{\tau }\setminus \tau$ encircling $\tau$. Choose a starting chamber ${\sigma }_{\star }\in {\mathcal{P}}^{[n]}$ incident at $\tau$, and a starting monomial $z^m$ in the toric coordinates of ${\sigma }_{\star }$.

As $\gamma$ crosses each slab ${\rho }_i$ (or wall ${\mathfrak{d}}_j$) in cyclic order, the carried monomial picks up a transition factor: $$ z^m ;\longmapsto; z^m \cdot f_{\rho_i}^{\langle n_{\rho_i}, m\rangle} ;\text{ or }; z^m \cdot f_{\mathfrak{d}j}^{\langle n{\mathfrak{d}j}, m\rangle}. $$ After traversing the full loop, the carried monomial has been transformed by a total automorphism $\Theta\tau$ofthetruncatedtoricring$\mathbb{C}[t][\Lambda_{\sigma_\star}]/(t^{N+1})$.

Definition (consistency). The structure $\mathcal{S}$ is consistent at $\tau$ to order $N$ if ${\Theta }_{\tau }\equiv \mathrm{id}(\mathrm{mod}{t}^{N+1})$. The structure $\mathcal{S}$ is consistent if it is consistent at every codimension-2 cell $\tau$ to every order $N\ge 1$.

The Kontsevich-Soibelman scattering theorem (2006). Let $({\mathfrak{d}}_1,{\theta }_1)$ and $({\mathfrak{d}}_2,{\theta }_2)$ be two walls meeting at a point in ${\mathbb{R}}^n$, with primitive normals $n_1,n_2\in {\Lambda }^{\vee }$ not proportional. Then there exists a unique locally finite collection of additional walls $\{({\mathfrak{d}}_k,{\theta }_k){\}}_{k\ge 3}$ in ${\mathbb{R}}^n$, each contained in the closed half-plane bounded by ${\mathfrak{d}}_1,{\mathfrak{d}}_2$, such that the union $\{({\mathfrak{d}}_k,{\theta }_k){\}}_{k\ge 1}$ is consistent at the meeting point to every order $N\ge 1$.

This is the local model for codimension-2 scattering. Applied globally to $(B,\mathcal{P},\phi )$ at every codimension-2 cell $\tau$, the theorem produces the walls $\mathcal{W}$ needed to complement the initial slabs ${\mathcal{F}}^{(1)}$ into a consistent structure $\mathcal{S}$.

The order-1 obstruction. The simplest substantive illustration of consistency arises at a focus-focus singularity of the integral affine structure (Gross-Siebert 2010 JAG 19 §4). At a focus-focus singularity, the local model has two slabs ${\rho }_1,{\rho }_2$ meeting at a singular codimension-2 cell $\tau$ which is removed from the away-from-discriminant region. In a punctured neighbourhood, the two slabs carry order-1 functions $f_{{\rho }_i}=1+t{z}^{m_i}$ with monomials $m_1,m_2\in \Lambda$ related by the focus-focus monodromy $T_{\text{ff}}:m\mapsto m+⟨n,m⟩v$ for primitive $n,v$ with $⟨n,v⟩=0$. The order-1 transition cocycle around $\tau$ fails to vanish; the order-1 wall added by the scattering algorithm has wall function $f_{\mathfrak{d}}=1+t{z}^{m_1+m_2}$ and corrects the cocycle.

The Gross-Siebert reconstruction algorithm. Given the order-1 slabs ${\mathcal{F}}^{(1)}$ determined by the central-fibre log-curve counts, the Carl-Pumperla-Siebert / Gross-Siebert algorithm iterates the Kontsevich-Soibelman scattering theorem at every codimension-2 cell of $(B,\mathcal{P})$:

1. Initial data. Slab functions ${\mathcal{F}}^{(1)}=\{f_{\rho }^{(1)}=1+t{\alpha }_{\rho }{\}}_{\rho }$ and the empty wall collection ${\mathcal{W}}^{(1)}=\varnothing$.

2. Step $N\to N+1$. At every codimension-2 cell $\tau$, the order-$N$ inconsistency ${\Theta }_{\tau }-\mathrm{id}\in t^{N+1}\cdot \mathrm{Lie}(V_{\tau })$ is a Lie-algebra-valued vector. Add new walls at order $N+1$ near $\tau$ whose order-$(N+1)$ wall functions cancel the inconsistency.

3. Termination at every $N$. At each order the number of added walls is finite per codimension-2 cell, by the Kontsevich-Soibelman scattering theorem.

4. Output. The infinite-order structure $\mathcal{S}=({\mathcal{F}}^{(\infty )},{\mathcal{W}}^{(\infty )})={\underleftarrow{\lim }}_N({\mathcal{F}}^{(N)},{\mathcal{W}}^{(N)})$ is consistent at every $\tau$ and provides the input for the Gross-Siebert reconstruction theorem [04.12.09].

Synthesis. The slab-function-and-wall apparatus on a polarised tropical manifold is the codimension-one polynomial bookkeeping that the Gross-Siebert reconstruction theorem consumes; the consistency condition at codimension-2 cells is the cocycle condition the codimension-one data must satisfy to produce a global smoothing. The foundational reason this works is that the toric charts on maximal cells of $\mathcal{P}$ already paste consistently at the central-fibre level, and the slab functions are the order-$t$ deformations of these pastings; consistency at codimension-2 forces the deformations to glue order-by-order into a flat formal family. Putting these together generalises the polytope-fan dictionary of [04.11.10] from the toric case to the degenerate Calabi-Yau case: identity slab functions $f_{\rho }=1$ in the toric case give the projective toric variety $X_P$, substantive slab functions in the Calabi-Yau case give the smooth mirror. The pattern recurs in [04.12.12] theta functions, where the slab functions determine the broken-line coefficients of the canonical $\mathbb{Z}$-basis of sections; this is exactly the structure constants of multiplication on the mirror Calabi-Yau. The bridge is the Carl-Pumperla-Siebert algorithm, identifying the central-fibre log-curve counts of [04.12.06] with the order-1 slab-function coefficients, and identifying the higher-order scattering corrections with the higher-order log-curve counts.

Connections [Master]

• Dual intersection complex; tropical manifold $B$ 04.12.08. The slab-function-and-structure apparatus of the present unit is the codimension-one polynomial enhancement of the dual intersection complex from the prerequisite unit. The polyhedral decomposition $\mathcal{P}$ and the integral affine structure with singularities on $B$ are the ambient data; the slab functions $f_{\rho }$ on codimension-one cells $\rho \in {\mathcal{P}}^{[n-1]}$ are the additional polynomial data layered on top, making $(B,\mathcal{P})$ into a polarised tropical manifold equipped with a structure $\mathcal{S}$. Conversely, every consistent structure $\mathcal{S}$ presupposes the dual intersection complex as its underlying combinatorial substrate.

• Toric degeneration of a Calabi-Yau variety 04.12.07. The Gross-Siebert reconstruction starts from a toric degeneration $\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ of a Calabi-Yau variety, whose dual intersection complex is $(B,\mathcal{P})$ and whose codimension-one transition data is recorded by the slab functions of the structure $\mathcal{S}$. The slab function $f_{\rho }$ on slab $\rho$ is precisely the transition function in the central-fibre Čech cocycle of the toric degeneration, restricted to the codimension-one stratum dual to $\rho$. Conversely, given a consistent structure $\mathcal{S}$ on $(B,\mathcal{P},\phi )$, the reconstruction theorem produces a toric degeneration $\mathfrak{X}$ whose codimension-one transitions are exactly the slabs of $\mathcal{S}$.

• Gross-Siebert reconstruction theorem 04.12.09. The consistency-of-structure theorem in this unit is the order-by-order input to the Gross-Siebert reconstruction theorem of [04.12.09]: every consistent structure $\mathcal{S}$ produces a formal flat family $\hat{\mathfrak{X}}\to \mathrm{Spf}\mathbb{C}[[t]]$, and the reconstruction theorem then algebraises this formal family to an honest one-parameter family $\mathfrak{X}\to {\mathbb{A}}^1$ with generic fibre the mirror Calabi-Yau. The two units pair: [04.12.09] consumes the consistent $\mathcal{S}$; the present unit defines $\mathcal{S}$ and proves the order-by-order assembly.

• Theta function of a polarised tropical manifold 04.12.12. The broken-line construction of theta functions on a polarised tropical manifold reads directly from the slab functions and walls of the structure $\mathcal{S}$: each broken line bends at slabs and walls, picking up the corresponding monomials as scattering coefficients. The well-definedness of theta functions — independence of the broken-line construction from the choice of base point — is exactly the consistency condition of $\mathcal{S}$. The pair of units [04.12.11] and [04.12.12] are inverse: the present unit defines $\mathcal{S}$; [04.12.12] consumes $\mathcal{S}$ to define ${\vartheta }_m$.

• Nishinou-Siebert correspondence theorem 04.12.06. The order-$1$ slab-function coefficients ${\alpha }_{\rho }$ in $f_{\rho }^{(1)}=1+t{\alpha }_{\rho }$ are determined by the genus-$0$ log Gromov-Witten counts of curves on the central fibre $X_0$ contributing to the slab $\rho$. The Nishinou-Siebert correspondence theorem expresses these log GW counts as Mikhalkin-style tropical-curve counts in the dual intersection complex $B(\Xi )$. The structure $\mathcal{S}$ thus reads its order-$1$ data directly from the enumerative content of the correspondence theorem; higher-order corrections (the walls of $\mathcal{S}$) are determined by higher-order log GW counts via the Carl-Pumperla-Siebert algorithm.

• Polytope-fan dictionary; the line bundle $L_P$ 04.11.10. The toric case of the slab-function-and-structure formalism is exactly the polytope-fan dictionary: when $(B,\mathcal{P},\phi )$ is the polarised tropical manifold associated to a polytope $P$ (interior of $P$, cell decomposition into vertex stars, $\phi$ = strictly convex piecewise-linear support function), all slab functions reduce to $f_{\rho }=1$ and the structure $\mathcal{S}=(\{1\},\varnothing )$ is consistent. The reconstruction theorem applied to this identity structure recovers the projective toric variety $X_P$ with its ample line bundle $L_P$, generalising in the present setup to substantive $\mathcal{F}$ when the central fibre is no longer smooth toric but rather a degenerate Calabi-Yau.

• Fan and toric variety $X_{\Sigma }$ 04.11.04. The local model on every maximal cell $\sigma \in {\mathcal{P}}^{[n]}$ of $(B,\mathcal{P})$ is the affine toric variety $X_{\sigma }=\mathrm{Spec}\mathbb{C}[{\Lambda }_{\sigma }]$, the toric variety of the apex fan in ${\Lambda }_{\sigma }^{\vee }$. The slab functions paste these affine toric pieces into the central fibre $X_0$, exactly as the fan-gluing prescription of [04.11.04] pastes affine toric pieces $U_{\sigma }$ into $X_{\Sigma }$. The structure $\mathcal{S}$ then deforms this pasting order-by-order in $t$, producing the smooth mirror Calabi-Yau as a one-parameter generic-fibre family.

• Tropical curve as balanced rational metric graph 04.12.02. The walls of the structure $\mathcal{S}$ are sub-pieces of tropical hypersurfaces in $B$ — their wall functions $f_{\mathfrak{d}}$ define codimension-one tropical varieties in the maximal-cell coordinates whose top-cells carry the lattice multiplicities of [04.12.02]. Broken lines through the structure are tropical-curve fragments with prescribed boundary conditions on slabs and walls; the genus-0 broken-line ensemble assembling theta functions is a tropical-curve enumeration weighted by slab-and-wall data. The present apparatus is therefore a polynomial-coefficient enrichment of the foundational balanced-metric-graph framework of the prerequisite unit.

Historical & philosophical context [Master]

The slab function and the structure $\mathcal{S}$ as defined here were introduced by Gross and Siebert in 2006 and 2010 as the codimension-one polynomial bookkeeping object of their mirror-symmetry programme. The Gross-Siebert 2006 Journal of Differential Geometry 72 paper ^{[Gross-Siebert 2006]} set up the log-Calabi-Yau central fibre and its dual intersection complex with integer affine structure; the Gross-Siebert 2010 Journal of Algebraic Geometry 19 paper ^{[Gross-Siebert 2010]} introduced slab functions on codimension-one cells, defined the structure $\mathcal{S}$, and proved the consistency-of-structure theorem assembling a formal smoothing of the central fibre. The 2011 Annals paper ^{[Gross-Siebert 2011]} then promoted the formal smoothing to an algebraic one-parameter family.

The conceptual ancestor of the slab function is the toric gluing cocycle: in the construction of a toric variety $X_{\Sigma }$ from a fan $\Sigma$, the affine toric charts $U_{\sigma }$ are pasted via Laurent-monomial transitions $z_{{\sigma }^{\prime }}=z_{\sigma }\cdot {\chi }^{m(\sigma ,{\sigma }^{\prime })}$ on the overlap. In the degenerate Calabi-Yau setting, the central fibre $X_0$ is built from affine toric pieces glued by plain transitions ${\chi }^{m(\sigma ,{\sigma }^{\prime })}$, but smoothing to the generic fibre requires deformation of these transitions to ${\chi }^{m(\sigma ,{\sigma }^{\prime })}\cdot f_{\rho }\cdot t^{{\kappa }_{\rho }}$ — and $f_{\rho }$ is the slab function. The conceptual lineage runs through Kempf-Knudsen-Mumford-Saint-Donat 1973 Toroidal Embeddings I (gluing toric pieces via combinatorial data), Mumford 1972 Compositio Mathematica (the first explicit Tate-curve construction of an Abelian variety by a slab-function-like cocycle), and the toric mirror-symmetry foundations of Batyrev 1994 J. Algebraic Geometry 3 and Hori-Vafa 2000 arXiv/0002222.

The wall apparatus of the structure $\mathcal{S}$ is due to Kontsevich and Soibelman in their 2006 Birkhäuser paper Affine structures and non-archimedean analytic spaces ^{[Kontsevich-Soibelman 2006]}, which introduced scattering diagrams in the non-archimedean SYZ programme. The Carl-Pumperla-Siebert 2010 arXiv preprint ^{[Carl-Pumperla-Siebert 2010]} provided the algorithmic recipe assembling a consistent structure $\mathcal{S}$ from the Gross-Siebert log Gromov-Witten counts (Gross-Siebert 2013 JAMS 26 ^{[Gross-Siebert 2013]}). The full consistency-of-structure theorem appeared in Gross-Siebert 2010 JAG 19, with simplified expositions in Gross's 2011 CBMS lecture notes ^{[Gross 2011 CBMS]}. The two-dimensional log-Calabi-Yau case was worked out in detail by Gross-Hacking-Keel 2015 Publications IHÉS 122 ^{[Gross-Hacking-Keel 2015]}, connecting slabs and walls to cluster-algebra structure and X-cluster transformations.

The philosophical thread running through the apparatus is the algebraic realisation of the SYZ heuristic: Strominger-Yau-Zaslow 1996 ^{[Strominger-Yau-Zaslow 1996]} proposed that mirror Calabi-Yau pairs arise as dual special-Lagrangian torus fibrations over a common integral affine base $B$, with mirror symmetry implemented as fibrewise T-duality. Gross-Siebert replace the analytic SYZ heuristic with an algebraic-combinatorial machine: $(B,\mathcal{P},\phi ,\mathcal{S})$ realises algebraically what SYZ realises symplectically, and the slab function is the algebraic shadow of the symplectic mirror-map transition along a codimension-1 wall in the base.

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