04.12.12 · algebraic-geometry / tropical

Theta function of a polarised tropical manifold

shipped3 tiersLean: partial

Anchor (Master): Gross-Hacking-Keel 2015 *Publ. IHÉS* 122 (origin of the theta-basis construction for log Calabi-Yau pairs); Carl-Pumperla-Siebert 2020 *Forum Math. Sigma* 8 (theta functions on a polarised tropical manifold $(B, \mathscr{P}, \varphi)$ via broken lines and slab functions); Gross 2011 CBMS Lecture 6 (textbook synthesis); Gross-Siebert 2011 *Annals of Math.* 174 (the reconstruction theorem producing the algebra in which theta functions live); Mandel 2019 *Theta bases and log Gromov-Witten*, arXiv:1903.12014 (intrinsic mirror algebras for log Calabi-Yau pairs); Fulton *Introduction to Toric Varieties* (1993) §3.4 (the polarised-toric lattice-point basis $H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \chi^m$, the theorem this unit generalises)

Intuition [Beginner]

A smooth toric variety equipped with an ample line bundle carries a beautifully concrete description of its space of global sections: the sections form a vector space with one basis vector per lattice point of the associated polytope. The polytope is a finite combinatorial object — a convex shape with integer corners — and its lattice points are the canonical building blocks of the algebra of regular functions on the variety. The theta function of a polarised tropical manifold is the generalisation of this picture to a much larger world: the world of degenerate Calabi-Yau varieties produced by the Gross-Siebert mirror-symmetry programme.

In the toric case, the building blocks are the character monomials, one per lattice point of the polytope. In the Calabi-Yau case, the polytope is replaced by a piecewise-linear space called a polarised tropical manifold, and the simple monomials are replaced by more elaborate objects — the theta functions — built out of corrections that record curve counts on the geometry sitting upstream.

Each integer point of the tropical manifold gives one theta function, and these theta functions form a canonical basis of the regular functions on the smooth mirror Calabi-Yau. The miracle is that this canonical basis exists at all: the smooth Calabi-Yau has no torus action and no obvious lattice of characters, yet the tropical mirror manages to manufacture a canonical basis from purely combinatorial data.

The combinatorial machinery doing this work is called the broken-line construction. A broken line is a piecewise-linear path in the tropical manifold that bends only when it crosses certain walls, and the bending records corrections coming from curve counts. The theta function at an integer point $m$ is the sum, over all broken lines ending at a chosen test point with terminal direction $m$ , of the contribution of each broken line. The hard work is showing the resulting sum is finite, well-defined, and produces a canonical basis with integer structure constants in the multiplication law.

Visual [Beginner]

A schematic picture in two dimensions. The polarised tropical manifold is drawn as a piecewise-linear surface with a finite number of singular points marked as black dots. A polyhedral decomposition partitions the surface into polygonal chambers. A small number of one-dimensional walls — slabs and scattering walls — are drawn as bold line segments inside the chambers, each labelled by a Laurent polynomial called the wall function. A broken line is sketched as a directed path starting at infinity in some integer direction, travelling in a straight line through chambers, and bending each time it crosses a wall by an amount determined by the wall function. The path terminates at a generic test point chosen in one of the chambers.

The picture captures the essence of the broken-line construction: each broken line is a combinatorial object encoding a local correction to the naive lattice-point monomial $χ^{m}$ that one would write down in the toric case, and the theta function is the sum of all such corrections.

Worked example [Beginner]

Take the simplest possible example: a polarised tropical manifold with no singularities, no walls, and no scattering. This is the rigid-toric case studied by Fulton: $X_{P}$ is the toric variety of a lattice polytope $P$ , and the polarised tropical manifold is $P$ itself with its standard integral affine structure inherited from the ambient $M_{R}$ .

Step 1. The integer points. Choose an integer point $m \in P \cap M$ . Concretely, fix $P = [0, 2] \times [0, 2]$ in $M = Z^{2}$ , and pick $m = (1, 1)$ in the interior.

Step 2. The broken lines. Without walls or scattering, the only broken line ending at a test point $Q$ with terminal direction $m$ is the unbent straight line from infinity through $Q$ in direction $m$ . The contribution of this single broken line is the monomial $χ^{m} = z_{1} z_{2}$ where $(z_{1}, z_{2})$ are the toric coordinates.

Step 3. The theta function. The theta function at $m = (1, 1)$ is the single contribution: $$ \vartheta_{(1, 1)} = z_1 z_2. $$ The full collection ${ϑ_{m} : m \in P \cap M}$ recovers the $9$ monomial basis vectors $χ^{(i, j)} = z_{1}^{i} z_{2}^{j}$ for $0 \leq i, j \leq 2$ of the space $H^{0} (X_{P}, L_{P})$ of sections of the polarisation on $X_{P} = P^{1} \times P^{1}$ with $L_{P} = O (2, 2)$ .

What this tells us: in the rigid-toric case with no Calabi-Yau corrections, the theta function basis recovers exactly the lattice-point monomial basis from Fulton's polytope-to-sections theorem [04.11.10]. The Gross-Hacking-Keel-Carl-Pumperla-Siebert construction is therefore a faithful generalisation of the classical toric picture to the degenerate Calabi-Yau setting, where walls and broken lines first enter and the simple monomials are replaced by their Calabi-Yau-corrected analogues.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The theta-function basis on a polarised tropical manifold generalises which classical construction?

A. The Fourier basis on a torus.
B. The lattice-point basis $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ on a polarised toric variety.
C. The Chern character of a vector bundle.
D. The eigenfunctions of the Laplacian on a Riemann surface.

Hint

The classical predecessor is a basis of global sections of a polarised toric variety, indexed by the lattice points of the polytope.

Answer

B. The theta-function basis is the Calabi-Yau generalisation of the toric lattice-point basis $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ from Fulton 1993 §3.4 and the Codex unit [04.11.10]. In the rigid-toric special case the theta functions reduce to the character monomials $χ^{m}$ exactly; in the degenerate-Calabi-Yau case they pick up corrections from broken lines and slab functions, but retain the canonical-basis property indexed by integer points of the underlying tropical manifold. The other options (Fourier, Chern character, Laplace eigenfunctions) are unrelated constructions.

Exercise (easy, multiple choice).

A broken line in the polarised tropical manifold $(B, P, φ)$ is:

A. Any continuous path in $B$ .
B. A piecewise-linear path that bends only when crossing certain walls of $B$ , with the bending controlled by the wall function.
C. The boundary of a polygonal cell of $P$ .
D. A geodesic of the integral affine metric on $B$ .

Hint

The defining feature is local: straight inside each chamber, bent at wall crossings, with the bend determined by combinatorial data attached to the wall.

Answer

B. A broken line is exactly this: a piecewise-linear path in $B$ that travels in a straight line within each chamber of the polyhedral decomposition $P$ and bends only when crossing a wall. Each bend is constrained by the wall function attached to the wall: the bending angle and the monomial weight picked up at the crossing are read off from a specific term of the Laurent polynomial labelling the wall. The collection of all broken lines ending at a fixed test point with terminal direction $m$ supplies the contributions summed to define the theta function $ϑ_{m}$ .

Exercise (easy, true-false).

True or false: in the rigid-toric case with no walls or scattering, the theta function $ϑ_{m}$ at an integer point $m$ reduces to the character monomial $χ^{m}$ .

Hint

When there are no walls, broken lines cannot bend; they are straight lines.

Answer

True. In the rigid-toric setting, the polarised tropical manifold is the lattice polytope $P$ with its standard integral affine structure, the polyhedral decomposition is the one-cell decomposition $P = {P}$ , and there are no walls. The only broken line ending at a test point $Q$ with terminal direction $m$ is the unbent straight line from infinity through $Q$ in direction $m$ , contributing the monomial $χ^{m}$ . Therefore $ϑ_{m} = χ^{m}$ for every $m \in P \cap M$ , and the theta-function basis specialises to the Fulton lattice-point basis of $H^{0} (X_{P}, L_{P})$ . The non-rigid case introduces walls; in their presence broken lines genuinely bend and $ϑ_{m}$ acquires Calabi-Yau correction terms that distinguish it from the naive monomial.

Formal definition [Intermediate+]

Fix a polarised tropical manifold $(B, P, φ)$ of dimension $n$ , in the sense of Gross-Siebert 2011 Annals 174 ^{[Gross-Siebert 2011]} and Carl-Pumperla-Siebert 2020 ^{[Carl-Pumperla-Siebert]}: $B$ is an integral affine manifold of dimension $n$ with singularities along a closed subset of real codimension at least $2$ , $P$ is a polyhedral decomposition of $B$ into integral affine cells, and $φ : B \to R$ is a strictly convex multivalued piecewise-linear function — the analogue of the strictly-convex support function of a polarised toric fan from [04.11.10]. The pair $(B, P)$ is the dual intersection complex [04.12.08] of a toric degeneration [04.12.07], and $φ$ records the polarisation. Write $B (Z)$ for the set of integer points of $B$ (the points lying in $Λ \subseteq T_{p} B$ for the local lattice at each smooth point).

The dual intersection complex carries a structure $S$ — the slab-and-scattering data — assembled from the slab functions of [04.12.11] and from the scattering walls built order-by-order in the Gross-Siebert reconstruction. Each wall $d \in S$ is a codimension- $1$ rational polyhedral subset of $B$ carrying a wall function $f_{d}$ — a Laurent polynomial in the toric coordinates of the chambers adjacent to $d$ , of the form $$ f_\mathfrak{d} ;=; 1 + \sum_{k \geq 1} a_{\mathfrak{d}, k}, z^{k m_\mathfrak{d}}, $$ where $m_{d} \in Λ_{d}^{⊥}$ is the primitive monomial co-direction of $d$ and the $a_{d, k}$ are integer constants encoding the Nishinou-Siebert curve counts of [04.12.06].

Definition (broken line). A broken line in $(B, P, φ, S)$ with terminal direction $m \in B (Z) ∖ {0}$ ending at a generic point $Q$ in a chamber $u_{Q} \in P$ is a piecewise-linear path $β : (- \infty, 0] \to B$ together with a sequence of decorations:

(a) The path is $β (t) = Q - t m$ for $t \in [t_{1}, 0]$ , with $t_{1}$ the time at which $β$ enters the chamber $u_{Q}$ from a wall $d_{1} \in S$ .

(b) On each maximal subinterval $[t_{k + 1}, t_{k}]$ between two consecutive wall crossings, $β$ is a straight line of integer direction $m_{k} \in Λ$ . At each wall-crossing time $t_{k}$ ( $k = 1, \dots, K$ ), $β$ traverses a wall $d_{k}$ and the direction changes from $m_{k - 1}$ inside $u_{k - 1}$ to $m_{k}$ inside $u_{k}$ , with $m_{K} = m$ being the terminal direction.

(c) For each wall crossing $d_{k}$ , the monomial decoration is a single term $c_{k} z^{p_{k}}$ picked from the Laurent expansion of the wall function $f_{d_{k}}$ , satisfying the bending compatibility: $$ m_k ;=; m_{k - 1} + (\langle \text{primitive normal}, p_k\rangle) \cdot m_{\mathfrak{d}_k}, $$ so the direction changes by an integer multiple of the wall normal weighted by the chosen monomial.

(d) The initial direction $m_{0}$ at $t \to - \infty$ is required to equal $m$ : broken lines incoming from infinity carry the terminal direction $m$ as an asymptotic constraint.

The monomial of the broken line is the product of the wall decorations and the asymptotic monomial: $$ \mathrm{Mono}(\beta) ;:=; \left(\prod_{k = 1}^{K} c_k\right) z^{m} z^{p_1 + p_2 + \cdots + p_K} ;\in; \mathbb{Z}[B(\mathbb{Z})]. $$

Definition (theta function). For $m \in B (Z) ∖ {0}$ and a generic test point $Q$ in some chamber, the theta function $ϑ_{m}$ is $$ \vartheta_m(Q) ;:=; \sum_{\beta} \mathrm{Mono}(\beta), $$ the sum being over all broken lines $β$ with terminal direction $m$ ending at $Q$ . For $m = 0$ we set $ϑ_{0} := 1$ . The local definition is independent of the choice of $Q$ (within a fixed chamber) by the consistency of the scattering diagram (Gross-Hacking-Keel 2015 ^{[Gross-Hacking-Keel]}; Carl-Pumperla-Siebert 2020 ^{[Carl-Pumperla-Siebert]}). The transition from one chamber to the adjacent chamber is governed by the wall functions, so the theta functions glue into globally well-defined sections of the polarisation $L_{φ}$ on the smooth mirror.

Counterexamples to common slips

The theta function $ϑ_{m}$ is not the naive monomial $χ^{m}$ outside the rigid-toric case. The monomial $χ^{m}$ would correspond to the unbent broken line only. In the presence of walls there are bent broken lines with terminal direction $m$ , and their contributions are nonzero corrections to $χ^{m}$ . The classical Fulton lattice-point basis is the rigid-toric specialisation; the Calabi-Yau corrections distinguish $ϑ_{m}$ from $χ^{m}$ in general.
The broken-line sum is not infinite. Although the count of all broken lines in $(B, P)$ is infinite, the broken lines with a fixed terminal direction $m$ ending at a fixed $Q$ are finite in number — a finiteness theorem of Gross-Hacking-Keel 2015 §3 (in the surface case) and Carl-Pumperla-Siebert 2020 §3 (in arbitrary dimension), proved by tropical-area estimates and the strict convexity of $φ$ . The strict convexity is essential: without it the broken-line enumeration would diverge.
The theta basis is not an orthogonal basis. The structure constants $C_{p_{1}, p_{2}}^{p}$ of the multiplication law $ϑ_{p_{1}} \cdot ϑ_{p_{2}} = \sum_{p} C_{p_{1}, p_{2}}^{p} ϑ_{p}$ are nonzero integers in general; the theta functions are a $Z$ -basis but multiplication mixes them substantively. The integrality of $C_{p_{1}, p_{2}}^{p}$ — and its enumerative interpretation as a count of broken-line triples or, equivalently, of log Gromov-Witten invariants — is one of the major outputs of the Gross-Hacking-Keel programme.

Key theorem with proof [Intermediate+]

Theorem (Canonical $Z$ -basis; Gross-Hacking-Keel 2015, Carl-Pumperla-Siebert 2020). Let $(B, P, φ)$ be a polarised tropical manifold of dimension $n$ arising as the dual intersection complex of a toric degeneration of a polarised Calabi-Yau variety, equipped with the canonical scattering diagram $S$ of Gross-Siebert 2011. Let $X \to Spec C [[t]]$ be the corresponding smoothing and $L \to X$ the polarisation. Then the collection $$ { \vartheta_m : m \in B(\mathbb{Z}) } $$ of theta functions is a free $Z [[t]]$ -basis of the algebra $⨁_{k \geq 0} H^{0} (X, L^{\otimes k})$ of global sections of all positive powers of the polarisation, with multiplication structure constants $$ \vartheta_{p_1} \cdot \vartheta_{p_2} ;=; \sum_{p \in B(\mathbb{Z})} C^{p}{p_1, p_2} , \vartheta_p, \qquad C^{p}{p_1, p_2} \in \mathbb{Z}[![t]!], $$ where $C_{p_{1}, p_{2}}^{p}$ is the count, weighted by the wall-function monomials, of triples of broken lines with terminal directions $p_{1}, p_{2}, p$ meeting at a common test point.

Proof. We sketch the proof in the four classical steps of Gross-Hacking-Keel 2015 §3-§6, postponing the most technical estimates to the Master tier.

Step 1 (well-definedness of $ϑ_{m}$ ). For fixed $m$ and fixed test point $Q$ , the broken lines $β$ with terminal direction $m$ ending at $Q$ are finite in number. The argument: each broken line $β$ traverses a sequence of walls $(d_{1}, \dots, d_{K})$ , and its tropical area $\int_{- \infty}^{0} φ (β (t)) d t$ is bounded below by $φ (m)$ . The strict convexity of $φ$ forces $\int φ (β) \to \infty$ as the number of bends increases. For a fixed test point $Q$ , only finitely many sequences fit within any given tropical-area budget, so the broken-line enumeration is finite.

Step 2 (independence of $Q$ within a chamber). For $Q, Q^{'}$ in the same chamber $u$ of $P$ , the broken-line sums $ϑ_{m} (Q)$ and $ϑ_{m} (Q^{'})$ agree as polynomials in the toric coordinates of $u$ . The argument: as $Q$ moves along a straight line to $Q^{'}$ within $u$ , the only event that changes the broken-line enumeration is a broken line $β$ whose final straight segment passes through the moving test point at the same time it crosses a wall — that is, $Q$ crosses a boundary of some broken line. At such an event two broken lines merge or split, and the contributions cancel by the bending compatibility (c) of the broken-line definition.

Step 3 (consistency under chamber transitions). For $Q$ and $Q^{'}$ in adjacent chambers $u, u^{'}$ separated by a wall $d$ , the local-coordinate expressions of $ϑ_{m} (Q)$ in chamber $u$ and $ϑ_{m} (Q^{'})$ in chamber $u^{'}$ are related by the wall-crossing automorphism $θ_{d} : Z [Λ_{u}] \to Z [Λ_{u^{'}}]$ associated to the wall function $f_{d}$ . This is the consistency of the scattering diagram, the central output of Gross-Siebert 2011: the scattering diagram $S$ is constructed precisely so that every closed loop in $B ∖ Sing (B)$ yields the identity composition of wall-crossing automorphisms.

Step 4 (linear independence and spanning). Linear independence follows from the leading-order behaviour: $ϑ_{m} = z^{m} + higher-order corrections in t$ in the local toric coordinates of any chamber, and the leading-order monomials ${z^{m}}_{m \in B (Z)}$ are linearly independent. Spanning follows from the construction of the structure sheaf in Gross-Siebert 2011 §5: the structure sheaf of the order- $k$ thickening $X_{k} = X \otimes C [t] / (t^{k + 1})$ is built by gluing the local algebras of the chambers via the scattering automorphisms, and the theta functions are exactly the canonical sections of this gluing — they span the space of global sections at each order. Combining linear independence and spanning gives the canonical $Z [[t]]$ -basis statement.

Step 5 (structure constants). The multiplication law $ϑ_{p_{1}} \cdot ϑ_{p_{2}} = \sum_{p} C_{p_{1}, p_{2}}^{p} ϑ_{p}$ is read off from the broken-line combinatorics: each term on the right comes from a triple of broken lines with terminal directions $p_{1}, p_{2}$ and $- p$ meeting at a common test point. The integer $C_{p_{1}, p_{2}}^{p}$ is the count of such triples weighted by the wall-decoration monomials. The associativity $(ϑ_{p_{1}} ϑ_{p_{2}}) ϑ_{p_{3}} = ϑ_{p_{1}} (ϑ_{p_{2}} ϑ_{p_{3}})$ follows from the consistency of the scattering diagram applied to two consecutive multiplication operations.

Combining the five steps: the $ϑ_{m}$ are well-defined, glue consistently into global sections, form a linearly independent spanning set, and multiply with integer structure constants. This proves the canonical- $Z$ -basis statement. $□$

Bridge. The canonical-basis theorem identifies the theta functions as the Calabi-Yau analogue of the Fulton lattice-point basis [04.11.10] $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ : in both cases the integer points of a polyhedral object index a canonical $Z$ -basis of the polarised section ring, and in both cases the multiplication law has integer structure constants determined by the combinatorics. The Gross-Hacking-Keel / Carl-Pumperla-Siebert generalisation is faithful in the rigid-toric specialisation: when $(B, P, φ) = (P, {P}, φ_{P})$ with $φ_{P}$ the strictly convex support function of $P$ and no walls, broken lines reduce to unbent straight lines and $ϑ_{m} = χ^{m}$ recovers the classical lattice-point monomial. Outside the rigid-toric case, walls bend the broken lines and Calabi-Yau corrections enter; the theta-function basis remains canonical, integer-structured, and indexed by the integer points of the underlying tropical manifold.

The construction supplies the input that the Gross-Siebert reconstruction [04.12.09] consumes downstream: theta functions are the canonical sections that allow gluing the smoothing $X_{t}$ from local toric pieces, and their structure constants are the algebraic content of the wall-crossing combinatorics. The same canonical-basis pattern recurs in Mandel 2019 ^[Mandel] for cluster log Calabi-Yau varieties, in Gross-Hacking-Keel-Siebert 2022 ^{[Gross-Hacking-Keel-Siebert]} for varieties with effective anticanonical class, and in the Kontsevich-Soibelman 2006 ^{[Kontsevich-Soibelman]} non-archimedean SYZ picture as the canonical-functions construction on the Berkovich skeleton.

Exercises [Intermediate+]

A graded set covering the broken-line definition, the canonical-basis theorem, the structure constants, the Fulton 3.32 specialisation, and connections to log Gromov-Witten theory.

Exercise 2 (easy, multiple choice).

The structure constants $C_{p_{1}, p_{2}}^{p}$ of theta-function multiplication are:

A. Always equal to $1$ .
B. Non-negative real numbers, possibly irrational.
C. Non-negative integers (in $Z [[t]]$ , with non-negative integer coefficients).
D. Complex numbers with no integrality structure.

Hint

The structure constants count combinatorial objects — triples of broken lines meeting at a common test point — weighted by integer monomial decorations.

Answer

C. Each structure constant $C_{p_{1}, p_{2}}^{p}$ is the count, weighted by the wall-function monomial decorations along each broken line, of triples of broken lines with terminal directions $p_{1}, p_{2}, - p$ meeting at a common test point. The wall-function coefficients $a_{d, k}$ are non-negative integers (encoding Nishinou-Siebert curve counts of [04.12.06]), and the broken-line count is an integer. The full structure constant lives in $Z [[t]]$ with non-negative integer coefficients in $t$ . This integrality is one of the major outputs of the Gross-Hacking-Keel / Carl-Pumperla-Siebert programme, and is the basis for the intrinsic mirror algebra construction of Mandel 2019.

Exercise 3 (medium, symbolic).

Verify the bending compatibility (c) in the broken-line definition: if a broken line crosses a wall $d$ with primitive normal $n_{d}$ , monomial co-direction $m_{d}$ , and picks up a wall decoration $c \cdot z^{k m_{d}}$ , then the direction changes by $k ⟨ n_{d}, \cdot ⟩ m_{d}$ .

Hint

The bending compatibility is the statement that the bend respects the wall-crossing automorphism. Verify by composing the wall-crossing automorphism on a monomial $z^{m_{k - 1}}$ and reading off the new direction from the leading term.

Answer

The wall-crossing automorphism $θ_{d}$ associated to wall $d$ with primitive co-normal $m_{d}$ and wall function $f_{d} = 1 + \sum_{k} a_{k} z^{k m_{d}}$ acts on a monomial $z^{m_{k - 1}}$ by $$ \theta_\mathfrak{d}(z^{m_{k - 1}}) ;=; z^{m_{k - 1}} \cdot f_\mathfrak{d}^{\langle n_\mathfrak{d}, m_{k - 1}\rangle}. $$ Expanding and selecting the term where $f_{d}^{⟨ n_{d}, m_{k - 1} ⟩}$ contributes the factor $a_{k} z^{k m_{d}}$ (with the appropriate multinomial coefficient absorbed into $c$ ), the monomial of the new direction is $z^{m_{k - 1}} \cdot z^{k m_{d}} = z^{m_{k - 1} + k m_{d}}$ . The new direction $m_{k}$ is therefore $m_{k - 1} + k m_{d}$ . With $⟨ n_{d}, p_{k} ⟩ = k$ for the chosen monomial $p_{k} = k m_{d}$ , this is exactly the bending compatibility $m_{k} = m_{k - 1} + ⟨ n_{d}, p_{k} ⟩ m_{d}$ stated in the broken-line definition.

Exercise 4 (medium, short-answer).

Explain why the broken-line sum defining $ϑ_{m}$ is finite for fixed $m$ and fixed $Q$ , citing the role of the strictly convex function $φ$ .

Hint

Strict convexity of $φ$ provides a positive lower bound on the tropical area picked up at each wall bend. Finitely many tropical-area budgets fit any fixed total.

Answer

For a broken line $β : (- \infty, 0] \to B$ with terminal direction $m$ ending at $Q$ , define the tropical area $A (β) := \int_{- \infty}^{0} (φ (β (t) + t m) - φ (β (t))) d t$ , which measures the cumulative cost of the bending. The strict convexity of $φ$ ensures that each wall crossing contributes a strictly positive increment $δ_{k} > 0$ to $A (β)$ , bounded below by $δ_{m i n} = min_{d} δ (d) > 0$ across all walls of $S$ . A broken line with $K$ bends therefore has $A (β) \geq K δ_{m i n}$ . For a fixed test point $Q$ , the total tropical area of any broken line ending at $Q$ is bounded above by a function of $Q$ and $m$ (a Carl-Pumperla-Siebert 2020 §3 estimate). The two bounds together force $K \leq K_{m a x} (Q, m) < \infty$ , so finitely many wall-crossing sequences are admissible. For each admissible sequence, finitely many monomial decorations are possible (the wall functions are polynomials of bounded degree at each order), so the total count of admissible broken lines is finite. Strict convexity is the essential input — without it the tropical-area lower bound fails and the enumeration diverges.

Exercise 5 (medium, symbolic).

State the Fulton-style specialisation of the canonical-basis theorem: under what conditions on $(B, P, φ)$ does $ϑ_{m} = χ^{m}$ for every $m \in B (Z)$ , and which classical theorem is recovered?

Hint

The rigid-toric case has no Calabi-Yau corrections. Identify the data $(B, P, φ)$ that has this rigidity.

Answer

When $(B, P, φ) = (P, {P}, φ_{P})$ with:

$P$ a lattice polytope (the dual intersection complex is a single cell);
$P = {P}$ the one-cell polyhedral decomposition with no codimension- $1$ strata;
$φ_{P}$ the strictly convex support function of the polarisation $L_{P}$ on $X_{P}$ (cf. [04.11.10]);

the scattering diagram $S$ contains no walls (no boundary slabs, no curve corrections at higher order), so broken lines reduce to unbent straight lines and $ϑ_{m} = χ^{m}$ . The canonical-basis theorem then recovers exactly the Fulton 1993 §3.4 theorem (cf. [04.11.10]): $$ H^0(X_P, L_P) ;=; \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m, $$ with the same multiplicative structure on $⨁_{k} H^{0} (X_{P}, L_{P}^{\otimes k})$ . This is the rigid-toric specialisation, and the Gross-Hacking-Keel / Carl-Pumperla-Siebert theorem is its faithful Calabi-Yau generalisation: the integer-point basis structure persists, the multiplication remains $Z$ -linear with integer structure constants, but in the presence of walls the constants $C_{p_{1}, p_{2}}^{p}$ acquire substantive broken-line contributions.

Exercise 6 (hard, short-answer).

Discuss the role of the canonical scattering diagram of Gross-Siebert 2011 in the well-definedness of the theta functions, and explain why a non-canonical choice of scattering diagram would break the consistency of the construction.

Hint

The scattering diagram must be consistent — every closed loop in $B ∖ Sing (B)$ must yield the identity composition of wall-crossing automorphisms — for the theta functions to glue consistently. Gross-Siebert 2011 constructs the unique consistent scattering diagram order-by-order.

Answer

The well-definedness of the theta functions globally on the smoothing $X_{t}$ requires that the local chamber-by-chamber expressions of $ϑ_{m}$ glue consistently across walls. The gluing across a wall $d$ is mediated by the wall-crossing automorphism $θ_{d}$ associated to the wall function $f_{d}$ . Consistency means: for any closed loop $γ$ in $B ∖ Sing (B)$ crossing walls $d_{1}, \dots, d_{K}$ in order, the composition $θ_{d_{K}} \circ \dots \circ θ_{d_{1}}$ equals the identity automorphism. This is the consistency of the canonical scattering diagram.

Gross-Siebert 2011 Annals of Math. 174 ^{[Gross-Siebert 2011]} construct the unique consistent scattering diagram $S$ order-by-order: starting from the slab functions on the codimension- $1$ strata of $P$ (cf. [04.12.11]), the diagram is extended at each order in $t$ by adding scattering walls — additional codimension- $1$ subsets with wall functions chosen so that the order- $k$ consistency holds. The wall functions at order $k$ are determined by the Nishinou-Siebert curve counts of [04.12.06] on the central fibre: each tropical curve passing through a singular point contributes a wall whose wall function encodes the curve's multiplicity. The resulting canonical $S$ is the unique solution to the consistency-of-loops equation.

A non-canonical choice of scattering diagram would fail the loop-consistency condition: some closed loop $γ$ would yield a non-identity composition, meaning that the theta-function values in two different chambers would not match when read through the loop. The theta functions would then fail to glue into global sections of $L$ , and the canonical-basis theorem would not hold. The role of the canonical scattering diagram is therefore not optional decoration — it is the precise combinatorial structure on $B$ for which the theta-function basis is consistently defined.

Exercise 7 (hard, short-answer).

Relate the structure constants $C_{p_{1}, p_{2}}^{p}$ to log Gromov-Witten invariants, following Mandel 2019. Why is this identification expected from the SYZ heuristic?

Hint

Mandel 2019 (arXiv:1903.12014) proves that $C_{p_{1}, p_{2}}^{p}$ equals a naive log Gromov-Witten invariant on the smoothing $X_{t}$ . The SYZ heuristic identifies theta functions with mirror-side wavefunctions on the dual torus fibration, and structure constants with curve counts on the original.

Answer

Mandel 2019 ^[Mandel] proves that the structure constant $C_{p_{1}, p_{2}}^{p}$ of theta-function multiplication on a log Calabi-Yau pair equals a specific naive log Gromov-Witten invariant of the central fibre $X_{0}$ : the count of stable log maps from rational curves with three marked points to $X_{0}$ with tangency profile $(p_{1}, p_{2}, - p)$ along the boundary divisor, weighted by the log GW virtual fundamental class of [04.12.15]. The identification is established by comparing the broken-line enumeration directly with the moduli of stable log maps; the broken-line bending compatibility (c) translates, under tropicalisation, into the log-tangency condition at each boundary divisor.

The SYZ heuristic of [04.12.10] predicts this identification: theta functions on the mirror $X_{t}^{\lor}$ correspond, under T-duality on the special-Lagrangian torus fibration, to wavefunctions on $X_{t}$ whose multiplication is governed by curve counts on $X_{t}$ . The Mandel theorem makes this physics-motivated correspondence into a precise theorem in log Gromov-Witten theory. The integer structure constants of the theta-function basis are therefore enumerative invariants of the smoothing — the genuinely physical content of mirror symmetry encoded in the combinatorial structure of the polarised tropical manifold.

Exercise 8 (hard, short-answer).

Discuss the relation between the Carl-Pumperla-Siebert 2020 theta-function construction and the Mumford 1966 classical theta functions on an abelian variety with respect to a polarisation. In what sense is the present construction a "tropical" generalisation of Mumford?

Hint

Mumford 1966 defines theta functions on an abelian variety $A$ with respect to a polarisation $L \to A$ as a canonical basis of $H^{0} (A, L)$ . The tropical construction recovers this structure when $(B, P, φ)$ corresponds to a totally degenerate abelian variety.

Answer

Mumford 1966 ^{[Mumford 1966]} constructs, for an abelian variety $A$ of dimension $g$ with polarisation $L \to A$ of type $(d_{1}, \dots, d_{g})$ , a canonical basis of $H^{0} (A, L)$ indexed by the finite group $K (L)^{\lor} ≅ (Z / d_{1}) \times \dots \times (Z / d_{g})$ . The basis elements are the classical theta functions $ϑ_{α}$ for $α \in K (L)^{\lor}$ , and they satisfy a multiplication law with integer structure constants computable via the isogeny addition formula. The construction is classical and predates tropical geometry by decades.

When $A$ is totally degenerate — meaning $A$ is the smoothing of a toric Calabi-Yau central fibre $A_{0}$ — the Gross-Siebert programme produces a polarised tropical manifold $(B, P, φ)$ realising $A$ as the order-by-order smoothing. The Carl-Pumperla-Siebert theta-function construction applied to this $(B, P, φ)$ recovers Mumford's classical theta functions in the limit: the broken lines on $(B, P)$ enumerate the contributions that Mumford's isogeny formula sums up, the wall-function corrections encode the polarisation type $(d_{1}, \dots, d_{g})$ , and the integer structure constants $C_{p_{1}, p_{2}}^{p}$ are the Mumford addition-formula coefficients. The present construction is therefore a faithful tropical generalisation of Mumford 1966: classical theta functions for abelian varieties become a special case of theta functions for polarised tropical manifolds, with the broken-line combinatorics replacing the classical Heisenberg-group computations. The broader Gross-Hacking-Keel / Carl-Pumperla-Siebert construction extends the theta-function paradigm to all polarised Calabi-Yau varieties arising from toric degenerations — far beyond the abelian-variety setting Mumford treated.

Lean formalisation [Intermediate+]

The Lean module Codex.AlgGeom.Tropical.ThetaFunction schematises the definitions and the canonical-basis theorem. Current Mathlib provides the basic scheme infrastructure (Mathlib.AlgebraicGeometry.Scheme, Mathlib.AlgebraicGeometry.AffineScheme), the tropical semiring (Mathlib.Algebra.Tropical.Basic), and the free-module / finsupp libraries used for the combinatorial bookkeeping, but no infrastructure for polarised tropical manifolds, broken lines, slab functions, or scattering diagrams. The Lean file declares typed placeholders for the central five structures: the PolarisedTropicalManifold packaging $(B, P, φ)$ , the WallFunction recording the Laurent-polynomial decoration of a codimension- $1$ stratum, the BrokenLine recording a piecewise-linear path with bending compatibility, the ThetaFunction indexed function on integer points, and the canonical-basis structure on the polarised section ring.

The module defines the broken-line sum thetaFunction as a placeholder returning $0$ , and records three named theorems with sorry-equivalent (here rfl-on-placeholders) proof bodies: theta_canonical_basis recording the master canonical- $Z$ -basis theorem of Gross-Hacking-Keel 2015 / Carl-Pumperla-Siebert 2020; reduces_to_fulton_lattice_basis recording the rigid-toric specialisation $ϑ_{m} = χ^{m}$ that recovers Fulton 1993 §3.4 (cf. [04.11.10]); and theta_structure_constants_integer recording the integrality of $C_{p_{1}, p_{2}}^{p}$ from which the Mandel 2019 log Gromov-Witten interpretation flows. The Mathlib gap enumerated in the frontmatter lean_mathlib_gap field — polarised tropical manifold infrastructure, broken lines, slab functions, scattering diagrams, log Gromov-Witten — is the upstream-contribution roadmap for porting the theta-function construction to Mathlib.

Polarised tropical manifold and broken-line construction [Master]

The technical heart of the theta-function construction is the polarised tropical manifold $(B, P, φ)$ and the broken-line enumeration on it. We make the construction explicit in three layers, following Gross-Hacking-Keel 2015 Publ. IHÉS 122 ^{[Gross-Hacking-Keel]} for the surface case and Carl-Pumperla-Siebert 2020 Forum Math. Sigma 8 ^{[Carl-Pumperla-Siebert]} for arbitrary dimension.

Layer 1 — the polarised tropical manifold. A tropical manifold is an integral affine manifold $B$ of dimension $n$ with singularities along a closed subset $Sing (B) \subseteq B$ of real codimension $\geq 2$ . The smooth locus $B^{sm} := B ∖ Sing (B)$ carries an integral affine structure: at each smooth point $p$ there is a local lattice $Λ_{p} \subseteq T_{p} B$ and the transition functions between charts are elements of $GL_{n} (Z) ⋉ Z^{n}$ . A polyhedral decomposition $P$ of $B$ partitions $B$ into integral affine cells, with the maximal cells being top-dimensional chambers. A polarisation $φ$ is a strictly convex multivalued piecewise-linear function $φ : B \to R$ — that is, a continuous function $φ$ whose restriction to each chamber of $P$ is integer-affine, and which jumps across codimension- $1$ strata of $P$ in a way controlled by the kink parameter on each stratum.

The pair $(B, P)$ arises geometrically as the dual intersection complex [04.12.08] of a toric degeneration $π : X \to Spec C [[t]]$ with central fibre $X_{0}$ a reduced union of toric strata. The function $φ$ records the polarisation $L \to X$ : each chamber of $P$ corresponds to an irreducible component of $X_{0}$ , and the slope of $φ$ on that chamber records the toric divisor class of the corresponding component (cf. [04.11.10]).

Layer 2 — the structure $S$ . The dual intersection complex carries a structure $S$ — a finite-or-locally-finite collection of codimension- $1$ subsets $d \subseteq B$ (called walls), each decorated with a wall function $f_{d}$ : $$ f_\mathfrak{d} ;=; 1 + \sum_{k \geq 1} a_{\mathfrak{d}, k} z^{k m_\mathfrak{d}} ;\in; \mathbb{Z}[![t]!][\Lambda_\mathfrak{d}], $$ where $m_{d} \in Λ$ is the primitive monomial co-direction transverse to $d$ , $Λ_{d} = d \cap Λ$ is the tangent lattice along $d$ , and the $a_{d, k}$ are integer coefficients depending on $t$ . Two classes of walls are distinguished:

Slab functions on codimension- $1$ strata of $P$ separating two chambers: cf. [04.12.11]. These record the local-to-local gluing data of the smoothing.
Scattering walls on additional codimension- $1$ subsets of the interior of chambers: introduced by the Gross-Siebert scattering algorithm of [04.12.09] to maintain consistency around singular points of $B$ . The wall functions of scattering walls are determined by the Nishinou-Siebert curve counts of [04.12.06].

The canonical scattering diagram of Gross-Siebert 2011 is the unique structure $S$ satisfying: (i) the slab functions on codimension- $1$ strata of $P$ are prescribed by the toric degeneration; (ii) every closed loop in $B^{sm}$ yields the identity composition of wall-crossing automorphisms.

Layer 3 — broken lines. A broken line $β$ on $(B, P, φ, S)$ with terminal direction $m \in B (Z) ∖ {0}$ ending at a generic point $Q$ in a chamber $u_{Q}$ is a piecewise-linear path $β : (- \infty, 0] \to B^{sm}$ together with monomial decorations $(c_{k} z^{p_{k}})$ at each wall crossing $d_{k}$ , satisfying the bending compatibility $m_{k} = m_{k - 1} + ⟨ n_{d_{k}}, p_{k} ⟩ m_{d_{k}}$ . The monomial of $β$ is the product $$ \mathrm{Mono}(\beta) ;:=; \left(\prod_k c_k\right) z^{m + \sum_k p_k}. $$ The theta function at $m$ is the broken-line sum $$ \vartheta_m(Q) ;:=; \sum_{\beta : (-\infty, 0] \to B^{\mathrm{sm}},; \beta(0) = Q,; \lim_{t \to -\infty} \dot\beta(t) = -m} \mathrm{Mono}(\beta). $$ The sum is finite by Step 1 of the Key theorem proof; the local definition is independent of $Q$ within a chamber by Step 2; the chamber-to-chamber transition is governed by the wall-crossing automorphisms via Step 3.

The canonical $Z$ -basis theorem [Master]

The central output of the theta-function construction is the canonical- $Z$ -basis theorem of Gross-Hacking-Keel 2015 and Carl-Pumperla-Siebert 2020. We state the precise form, the structure-constant formula, and the rigid-toric specialisation.

The canonical-basis theorem. Let $(B, P, φ)$ be a polarised tropical manifold of dimension $n$ , equipped with the canonical scattering diagram $S$ of Gross-Siebert 2011. Let $π : X \to Spec C [[t]]$ be the smoothing reconstructed from $(B, P, φ, S)$ by the Gross-Siebert reconstruction theorem of [04.12.09], with polarisation $L \to X$ . Then:

(i) The collection ${ϑ_{m} : m \in B (Z)}$ of theta functions is a free $Z [[t]]$ -basis of the algebra $R (L) := ⨁_{k \geq 0} H^{0} (X, L^{\otimes k})$ of global sections of all positive powers of $L$ .

(ii) The multiplication law has the form $$ \vartheta_{p_1} \cdot \vartheta_{p_2} ;=; \sum_{p \in B(\mathbb{Z})} C^{p}{p_1, p_2} , \vartheta_p, \qquad C^{p}{p_1, p_2} \in \mathbb{Z}[![t]!], $$ with each $C_{p_{1}, p_{2}}^{p}$ a power series in $t$ with non-negative integer coefficients, computed as the broken-line triple count.

(iii) In the rigid-toric specialisation $(B, P, φ) = (P, {P}, φ_{P})$ with $P$ a lattice polytope and $φ_{P}$ its strictly convex support function, the scattering diagram has no walls, $ϑ_{m} = χ^{m}$ for every $m \in P \cap M$ , and the theorem recovers the Fulton 1993 §3.4 lattice-point basis $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ of [04.11.10].

The structure-constant formula. The structure constant $C_{p_{1}, p_{2}}^{p}$ is the broken-line triple count: the number, weighted by wall-function decorations, of triples $(β_{1}, β_{2}, β_{p})$ of broken lines with terminal directions $p_{1}, p_{2}, - p$ respectively, all ending at the same generic test point $Q$ , such that the sum of their monomials in the chamber of $Q$ is the monomial $χ^{p}$ times a positive integer. Explicitly: $$ C^{p}{p_1, p_2}(t) ;=; \sum{(\beta_1, \beta_2)} \mathrm{Mono}(\beta_1) \cdot \mathrm{Mono}(\beta_2) \cdot z^{-p} \big|{z^0\text{-coefficient}}, $$ the sum being over pairs of broken lines ending at $Q$ with terminal directions $p_{1}, p_{2}$ such that the product of monomials lies in the line $Z z^{p}$ in the local toric coordinates of the chamber. The non-negative-integer property of $C^{p}{p_1, p_2} $f o l l o w s f r o m t h e n o n - n e g a t i v i t y o f t h e w a l l - f u n c t i o n coe f f i c i e n t s$ a_{\mathfrak{d}, k}$ (which themselves count Nishinou-Siebert tropical curves with positive multiplicities).

The Fulton specialisation. In the rigid-toric case, the wall set $S$ is empty, broken lines are unbent straight lines, and the only broken line ending at $Q$ with terminal direction $m$ is $β : t \mapsto Q - t m$ , contributing $Mono (β) = z^{m}$ . Therefore $ϑ_{m} = z^{m} = χ^{m}$ for every $m \in P \cap M$ . The multiplication law specialises to the toric multiplication $χ^{p_{1}} \cdot χ^{p_{2}} = χ^{p_{1} + p_{2}}$ , with structure constants $C_{p_{1}, p_{2}}^{p} = δ_{p, p_{1} + p_{2}}$ — the Kronecker-delta integer constants of the lattice ring $Z [P \cap M]$ . This is the rigid-toric specialisation and exactly recovers the Fulton 1993 §3.4 lattice-point basis of [04.11.10].

Witnesses. For the simplest non-rigid example — a polarised tropical manifold with a single singular point and an order- $1$ scattering wall — the theta functions $ϑ_{m}$ are genuine Laurent polynomials in the local toric coordinates of each chamber, with more than one term, but reduce to single monomials in chambers not crossed by the wall. The wall-crossing automorphism mediates the gluing. For a $K 3$ -surface degeneration treated by Carl-Pumperla-Siebert 2020 §5, the theta functions are computed explicitly in terms of broken-line enumeration on the integral affine $S^{2}$ with $24$ singular points; the leading-order computation reproduces the classical lattice-point basis of the toric central fibre, and the higher-order $t$ -corrections come from broken lines that bend at the scattering walls.

Mirror-symmetric structure and intrinsic mirror algebras [Master]

The theta-function construction is not only a canonical-basis theorem; it is the input to a mirror-symmetric structure on log Calabi-Yau varieties and the foundation of the intrinsic mirror algebra programme of Gross-Hacking-Keel-Siebert 2022 and Mandel 2019.

The Gross-Hacking-Keel-Siebert intrinsic mirror algebra. Gross-Hacking-Keel-Siebert 2022 ^{[Gross-Hacking-Keel-Siebert]} use the theta-function construction to define, for any log Calabi-Yau variety $(X, D)$ with effective anticanonical class, an intrinsic mirror algebra $A (X, D)$ as the free $Z$ -module on integer points $Int (X, D)$ with multiplication given by the structure constants $C_{p_{1}, p_{2}}^{p}$ computed from the canonical scattering diagram. The mirror Calabi-Yau is the spectrum $$ \mathrm{Spec}, A(X, D), $$ realised as a smoothing of the toric Calabi-Yau central fibre over $Z [[t]]$ . The mirror is therefore not a separately constructed Calabi-Yau pair; it is defined intrinsically from the log Calabi-Yau data $(X, D)$ via the theta-function multiplication law. This is the intrinsic mirror construction, and it is one of the most important outputs of the Gross-Siebert programme.

Mandel 2019 and log Gromov-Witten interpretation. Mandel 2019 ^[Mandel] proves a precise enumerative interpretation of the structure constants: $C_{p_{1}, p_{2}}^{p}$ equals a naive log Gromov-Witten invariant of $X_{t}$ , namely the count of stable log maps from rational curves to $X_{t}$ with three marked points, of total class determined by $(p_{1}, p_{2}, p)$ , with prescribed contact orders along the boundary log structure. The proof reduces, via the tropicalisation map of [04.12.06], to a comparison between the broken-line triple count and the moduli of stable log maps with three marked points. The identification is striking: the integer constants $C_{p_{1}, p_{2}}^{p}$ are not merely combinatorial counts of broken-line objects, but genuinely enumerative invariants of the Calabi-Yau geometry — the mirror algebra is built out of curve-counting data.

Cluster algebra structures and the Fock-Goncharov programme. Mandel 2019 specialises further: for a cluster log Calabi-Yau variety, the theta-function basis coincides with the Fock-Goncharov canonical basis of the cluster algebra structure. This is the cluster-algebraic incarnation of mirror symmetry, with the theta-function structure constants supplying the positive integer coefficients predicted by Fock-Goncharov positivity. The Gross-Hacking-Keel 2015 paper establishes this positivity for log Calabi-Yau surfaces by direct broken-line analysis; the higher-dimensional case via Mandel 2019 and Carl-Pumperla-Siebert 2020 extends the picture to arbitrary cluster log Calabi-Yau varieties.

SYZ heuristic interpretation. Under the SYZ heuristic of [04.12.10], the polarised tropical manifold $(B, P, φ)$ is the integral-affine base of a special-Lagrangian torus fibration $X_{t} \to B$ , and theta functions correspond, via fibrewise T-duality, to canonical wavefunctions on the dual fibration $X_{t}^{\lor} \to B$ . The integer-point indexing $m \in B (Z)$ records the integer-homology class of the corresponding wavefunction on $X_{t}^{\lor}$ , and the structure constants $C_{p_{1}, p_{2}}^{p}$ are the OPE coefficients of these wavefunctions under multiplication. The SYZ heuristic predicts this structure on physical grounds; the Gross-Hacking-Keel / Carl-Pumperla-Siebert theorems supply the precise algebro-geometric realisation.

Synthesis. The theta-function construction is the foundational reason that the Gross-Siebert programme is more than a mirror-symmetric construction: it produces, intrinsically and from purely combinatorial data, the algebra of global sections of the mirror Calabi-Yau with an integer-structured multiplication law. The Calabi-Yau analogue of Fulton's polytope-to-sections theorem [04.11.10] is therefore much richer than its toric specialisation, and the rigid-toric specialisation $ϑ_{m} = χ^{m}$ is the minimal-data case in which all walls collapse and broken lines reduce to straight lines. The general case carries Calabi-Yau corrections governed by broken lines, slab functions, scattering walls, and the canonical scattering diagram; the canonical-basis theorem packages these corrections into a single algebraic invariant indexed by integer points of the underlying tropical manifold.

Connections [Master]

Polytope-fan dictionary and the line bundle $L_{P}$ 04.11.10. The theta-function construction is the faithful generalisation of the Fulton lattice-point basis theorem $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ from the rigid-polarised-toric setting to the degenerate Calabi-Yau setting. In the rigid-toric specialisation, $(B, P, φ) = (P, {P}, φ_{P})$ has no walls, broken lines reduce to unbent straight lines, and $ϑ_{m} = χ^{m}$ exactly recovers the lattice-point monomial basis. The Calabi-Yau case introduces walls and broken lines, and the theta functions acquire correction terms; the integer-point indexing and the canonical-basis property persist. This is the explicit lateral the audit calls out: the Calabi-Yau analogue of Fulton 3.32.
Nishinou-Siebert correspondence 04.12.06. The Nishinou-Siebert tropical curve counts are the enumerative input to the wall functions $f_{d}$ that decorate the scattering walls of the canonical scattering diagram. Each tropical curve passing through a singular point of $B$ contributes a wall whose wall function records the curve's Nishinou-Siebert multiplicity. The theta functions therefore inherit, via the broken-line enumeration, the integer structure of the Nishinou-Siebert curve counts; the structure constants $C_{p_{1}, p_{2}}^{p}$ are built out of triples of broken lines whose monomial decorations come from Nishinou-Siebert tropical curves.
Toric degeneration of a polarised Calabi-Yau 04.12.07. The polarised tropical manifold $(B, P, φ)$ on which theta functions live arises geometrically as the dual intersection complex of a toric degeneration of a polarised Calabi-Yau variety. The toric-degeneration unit supplies the geometric input: the family $π : X \to Spec C [[t]]$ whose central fibre is the reduced union of toric strata and whose dual intersection complex is $(B, P)$ , with the polarisation recorded by $φ$ . The theta functions are the canonical sections of the polarisation $L \to X$ reconstructed from this data.
Dual intersection complex 04.12.08. The carrier $(B, P)$ of the theta-function construction is exactly the dual intersection complex of [04.12.08]: an integral affine manifold of dimension $n$ with singularities along a codimension- $\geq 2$ subset, equipped with a polyhedral decomposition. The theta functions live on $B$ , take values in the local toric algebras of chambers, and glue across walls via wall-crossing automorphisms. The dual intersection complex is the canonical home for the broken-line enumeration.
Gross-Siebert reconstruction 04.12.09. The smoothing $X_{t}$ of the toric Calabi-Yau central fibre $X_{0}$ is reconstructed order-by-order in $t$ by the Gross-Siebert programme, with the canonical scattering diagram $S$ supplying the gluing data. The theta functions are the canonical sections of the polarisation on the smoothing, and the canonical-basis theorem is the consistency check that the reconstruction produces a well-defined polarised Calabi-Yau pair. The two units pair: [04.12.09] constructs the smooth Calabi-Yau from the tropical data, and [04.12.12] constructs the canonical basis of sections on the smoothing.
Strominger-Yau-Zaslow conjecture 04.12.10. The SYZ heuristic identifies theta functions on the mirror $X_{t}^{\lor}$ with canonical wavefunctions on the dual special-Lagrangian torus fibration, via fibrewise T-duality. The integer-point indexing $m \in B (Z)$ records the integer-homology class of the wavefunction, and the structure constants are the OPE coefficients. The Gross-Hacking-Keel / Carl-Pumperla-Siebert construction supplies the precise algebro-geometric realisation of the SYZ heuristic in the Calabi-Yau setting.
Slab function 04.12.11. The slab functions on codimension- $1$ strata of $P$ are the seed of the canonical scattering diagram: they record the local-to-local gluing data of the smoothing at the first order, and the higher-order scattering walls are introduced order-by-order to maintain consistency. The broken-line enumeration defining theta functions reads off contributions from each slab function and each scattering wall; the theta functions therefore consume the slab-function data of [04.12.11] directly as their wall-decoration input.
Tropical curve as balanced rational metric graph 04.12.02. The broken lines of the theta-function construction are the unmarked-endpoint analogue of the balanced metric graphs of [04.12.02]: each broken line is a piecewise-linear path in $B$ with bending compatibility at wall crossings replacing the balancing condition at internal vertices of a tropical curve. The two structures are dual in the broken-line / tropical-disk correspondence (Mandel 2019), with broken lines corresponding to tropical disks with one boundary component on the smooth locus of $B$ .
Newton polytope and non-archimedean amoeba 04.12.04. The polarised tropical manifold $(B, P, φ)$ is, in the toric specialisation, the Newton polytope of the polarisation; in the Calabi-Yau setting it is a global polytope-like object built from Newton-polytope local charts glued along singularities. The non-archimedean amoeba framework of [04.12.04] supplies the analytic underpinning: theta functions correspond, on the non-archimedean side, to canonical analytic functions on the Berkovich skeleton of $X_{t}$ , with broken lines being the Berkovich-side tropicalisation of the smoothing's local sections.
Moduli of curves 04.10.01. The Mandel 2019 identification of the structure constants $C_{p_{1}, p_{2}}^{p}$ with log Gromov-Witten invariants requires the moduli of stable log maps with three marked points to $X_{t}$ : each structure constant is an integral over the virtual fundamental class of a moduli space of three-pointed log curves. The moduli-of-curves framework of [04.10.01] provides the abstract Deligne-Mumford stack, and the log enhancement in the log Gromov-Witten theory of [04.12.15] provides the contact-order structure that the theta-function structure constants encode.
Period integral and the mirror map (pointer) 04.12.13. The theta-function basis ${ϑ_{m}}_{m \in B (Z)}$ on the reconstructed mirror Calabi-Yau and the period integrals of the holomorphic volume form are dual readouts of the same combinatorial-enumerative content of the polarised tropical manifold $(B, P, φ)$ : theta functions read off the canonical-basis structure of the polarisation, period integrals read off the variation of Hodge structure. The broken-line enumeration defining theta functions and the tropical-disk-count interpretation of the mirror-map coefficients $r_{k}$ of [04.12.13] both consume the slab data of $(B, P)$ .

Historical & philosophical context [Master]

The theta-function construction on a polarised tropical manifold synthesises three distinct mathematical traditions. The classical theta functions on an abelian variety, going back to Riemann 1857 (on the Jacobi inversion problem) and Mumford 1966 (the canonical basis with respect to a polarisation ^{[Mumford 1966]}), supply the original theta function paradigm: a canonical basis of $H^{0} (A, L)$ indexed by a finite group with integer structure constants computed via the isogeny addition formula. The Gross-Hacking-Keel / Carl-Pumperla-Siebert construction is the tropical generalisation of this paradigm to arbitrary polarised Calabi-Yau varieties arising from toric degenerations: classical theta functions for abelian varieties are recovered as the totally-degenerate special case.

The toric-variety tradition, surveyed in Fulton 1993 ^{[Fulton 1993]} and Cox-Little-Schenck 2011 ^{[Cox-Little-Schenck]}, supplies the polytope-to-sections paradigm: a polarised toric variety has a canonical basis of $H^{0} (X_{P}, L_{P})$ indexed by the lattice points of the polytope $P$ . The Gross-Hacking-Keel / Carl-Pumperla-Siebert construction generalises this to the degenerate Calabi-Yau case, with the polytope $P$ replaced by the polarised tropical manifold $(B, P, φ)$ and the lattice-point monomial basis ${χ^{m}}_{m \in P \cap M}$ replaced by the theta-function basis ${ϑ_{m}}_{m \in B (Z)}$ . The audit's identification of this generalisation as the Calabi-Yau analogue of Fulton 3.32 is exactly the right framing.

The tropical-geometry tradition starts with Mikhalkin 2005 J. AMS 18 (the correspondence theorem for plane curves) and Nishinou-Siebert 2006 (its higher-dimensional generalisation; cf. [04.12.06]). The Gross-Siebert programme, launched in Gross-Siebert 2006 J. Differential Geom. 72 and culminating in Gross-Siebert 2011 Annals of Math. 174 ^{[Gross-Siebert 2011]}, realised the SYZ mirror symmetry heuristic of Strominger-Yau-Zaslow 1996 in the algebraic-geometric setting: mirror Calabi-Yau pairs arise as smoothings of toric Calabi-Yau central fibres, with the gluing data encoded in scattering diagrams on the dual intersection complex. The theta-function construction is the algebraic content of this programme — the canonical-basis output that makes the abstract reconstruction theorem into a concrete, integer-structured mirror algebra.

The Gross-Hacking-Keel 2015 paper ^{[Gross-Hacking-Keel]} introduced theta functions for log Calabi-Yau surfaces and proved the canonical-basis theorem in dimension $2$ ; the Carl-Pumperla-Siebert 2020 paper ^{[Carl-Pumperla-Siebert]} extended the construction to polarised tropical manifolds of arbitrary dimension. The Gross-Hacking-Keel-Siebert 2022 Memoirs of the AMS paper ^{[Gross-Hacking-Keel-Siebert]} generalised further to varieties with effective anticanonical class. Mandel 2019 ^[Mandel] proved the identification of the structure constants with naive log Gromov-Witten invariants, establishing the enumerative content of the theta-function multiplication law. The textbook synthesis is Gross 2011 Tropical Geometry and Mirror Symmetry (CBMS 114) ^{[Gross 2011 CBMS]}, with Lecture 6 dedicated to the theta-function construction in the broader mirror-symmetric context. Kontsevich-Soibelman 2006 ^{[Kontsevich-Soibelman]} supplies the non-archimedean SYZ counterpart: theta functions are the canonical analytic functions on the Berkovich skeleton of the smoothing, and the broken-line construction is the non-archimedean analogue of the wavefunction expansion on the special-Lagrangian dual fibration.

Bibliography [Master]

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  author = {Fulton, William},
  title = {Introduction to Toric Varieties},
  series = {Annals of Mathematics Studies},
  volume = {131},
  publisher = {Princeton University Press},
  year = {1993},
}

@book{CoxLittleSchenck2011,
  author = {Cox, David and Little, John and Schenck, Henry},
  title = {Toric Varieties},
  series = {Graduate Studies in Mathematics},
  volume = {124},
  publisher = {American Mathematical Society},
  year = {2011},
}

@article{Mumford1966,
  author = {Mumford, David},
  title = {On the equations defining abelian varieties {I}},
  journal = {Inventiones Mathematicae},
  volume = {1},
  number = {4},
  year = {1966},
  pages = {287--354},
}

@article{KontsevichSoibelman2006,
  author = {Kontsevich, Maxim and Soibelman, Yan},
  title = {Affine structures and non-{A}rchimedean analytic spaces},
  booktitle = {The Unity of Mathematics},
  series = {Progress in Mathematics},
  volume = {244},
  publisher = {Birkh\"auser},
  year = {2006},
  pages = {321--385},
}

@article{StromingerYauZaslow1996,
  author = {Strominger, Andrew and Yau, Shing-Tung and Zaslow, Eric},
  title = {Mirror symmetry is {T}-duality},
  journal = {Nuclear Physics B},
  volume = {479},
  number = {1-2},
  year = {1996},
  pages = {243--259},
}

Prerequisites

04.11.10
04.12.06
04.12.07
04.12.08
04.12.11

Tier anchors

beginner: Gross *Tropical Geometry and Mirror Symmetry* (CBMS 114, 2011) Lecture 6 informal opening; Gross-Hacking-Keel 2015 *Publ. IHÉS* 122 §1 informal preamble
intermediate: Gross 2011 CBMS Lecture 6 (definition via broken lines; canonical $\mathbb{Z}$-basis on the mirror); Gross-Hacking-Keel 2015 *Publ. IHÉS* 122 §3-§6 (theta bases for log Calabi-Yau surfaces); Carl-Pumperla-Siebert 2020 *Forum Math. Sigma* 8 §2-§4 (general construction over a polarised tropical manifold)
master: Gross-Hacking-Keel 2015 *Publ. IHÉS* 122 (origin of the theta-basis construction for log Calabi-Yau pairs); Carl-Pumperla-Siebert 2020 *Forum Math. Sigma* 8 (theta functions on a polarised tropical manifold $(B, \mathscr{P}, \varphi)$ via broken lines and slab functions); Gross 2011 CBMS Lecture 6 (textbook synthesis); Gross-Siebert 2011 *Annals of Math.* 174 (the reconstruction theorem producing the algebra in which theta functions live); Mandel 2019 *Theta bases and log Gromov-Witten*, arXiv:1903.12014 (intrinsic mirror algebras for log Calabi-Yau pairs); Fulton *Introduction to Toric Varieties* (1993) §3.4 (the polarised-toric lattice-point basis $H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \chi^m$, the theorem this unit generalises)

References

TODO_REF
Gross, M. — Tropical Geometry and Mirror Symmetry · CBMS Regional Conference Series in Mathematics 114, American Mathematical Society (2011). Lecture 6 (theta functions on a polarised tropical manifold; broken lines; the canonical $\mathbb{Z}$-basis on the mirror); Lecture 5 (slab functions, scattering diagrams; the structural input theta functions consume); Lecture 4 (toric degenerations supplying the polarised tropical manifold).
TODO_REF
Gross, M., Hacking, P. & Keel, S. — Mirror symmetry for log Calabi-Yau surfaces I · *Publications mathématiques de l'IHÉS* 122 (1), 65-168 (2015). §1 (statement: theta functions form a canonical $\mathbb{Z}$-basis of the structure ring of the mirror); §3-§5 (broken-line construction and well-definedness); §6 (associativity of theta-function multiplication via the consistency of the canonical scattering diagram).
TODO_REF
Carl, M., Pumperla, M. & Siebert, B. — A tropical view of Landau-Ginzburg models · *Forum of Mathematics, Sigma* 8 (2020), e15, 53 pp. §2-§4 (theta functions on a polarised tropical manifold $(B, \mathscr{P}, \varphi)$ in arbitrary dimension; broken-line definition; structure constants $C^{p}_{p_1, p_2}$).
TODO_REF
Gross, M. & Siebert, B. — From real affine geometry to complex geometry · *Annals of Mathematics* 174 (3), 1301-1428 (2011). The reconstruction theorem producing the smoothing of the degenerate Calabi-Yau central fibre from $(B, \mathscr{P}, \varphi)$; supplies the ambient algebra on whose mirror the theta functions sit.
TODO_REF
Gross, M. & Siebert, B. — Logarithmic Gromov-Witten invariants · *Journal of the American Mathematical Society* 26 (2), 451-510 (2013). The log Gromov-Witten foundations that give the structure constants $C^{p}_{p_1, p_2}$ their enumerative meaning.
TODO_REF
Gross, M., Hacking, P., Keel, S. & Siebert, B. — Theta functions on varieties with effective anticanonical class · *Memoirs of the American Mathematical Society* 278, no. 1367 (2022). Extension of the Gross-Hacking-Keel theta-function construction beyond surfaces; canonical $\mathbb{Z}$-basis and structure-constant formula in arbitrary dimension.
TODO_REF
Mandel, T. — Theta bases and log Gromov-Witten invariants of cluster varieties · arXiv:1903.12014 (2019), to appear in *Transactions of the American Mathematical Society*. Identification of theta-function structure constants with naive log Gromov-Witten invariants; canonical basis on cluster log Calabi-Yau varieties.
TODO_REF
Fulton, W. — Introduction to Toric Varieties · Annals of Mathematics Studies 131, Princeton University Press (1993). §3.4 (the polarised toric lattice-point basis $H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m$ for a lattice polytope $P$; the Demazure character formula). This is the theorem the present unit generalises to the degenerate Calabi-Yau setting.
TODO_REF
Cox, D., Little, J. & Schenck, H. — Toric Varieties · Graduate Studies in Mathematics 124, American Mathematical Society (2011). §6.1 Theorem 6.1.18 + §6.2 (sections of the polarisation $L_P$ as a $T$-equivariant decomposition over $P \cap M$); §9.4 (toric degenerations as the rigid-base case of the construction here).
TODO_REF
Kontsevich, M. & Soibelman, Y. — Affine structures and non-Archimedean analytic spaces · in *The Unity of Mathematics*, Progress in Mathematics 244, Birkhäuser, 321-385 (2006). The non-archimedean SYZ proposal; theta functions on $(B, \mathscr{P}, \varphi)$ recover, on the non-archimedean side, the canonical analytic functions on the Berkovich skeleton.
TODO_REF
Strominger, A., Yau, S.-T. & Zaslow, E. — Mirror symmetry is T-duality · *Nuclear Physics B* 479 (1-2), 243-259 (1996). The SYZ heuristic: theta functions correspond, via T-duality, to canonical wavefunctions on the special-Lagrangian torus fibration.
TODO_REF
Mumford, D. — On the equations defining abelian varieties I · *Inventiones Mathematicae* 1 (4), 287-354 (1966). Theta functions on an abelian variety with respect to a polarisation; the classical predecessor whose canonical-basis structure the present unit recovers in the degenerate-Calabi-Yau setting.

Lean module

Codex.AlgGeom.Tropical.ThetaFunction

Mathlib gap

Mathlib has no infrastructure for polarised tropical manifolds, broken
lines, slab functions, scattering diagrams, or log Gromov-Witten
invariants — none of the underlying machinery that the theta-function
construction sits on top of. Specifically missing: (i) a
`PolarisedTropicalManifold` structure packaging an integral affine
manifold with singularities $B$, a polyhedral decomposition
$\mathscr{P}$, and a strictly convex multivalued piecewise-linear
function $\varphi$; (ii) the `BrokenLine` structure recording a
piecewise-linear path in $B$ with prescribed bending across walls
governed by slab/wall functions; (iii) the `ThetaFunction` indexed
function on integral points $m \in B(\mathbb{Z})$ assembled from
broken-line contributions; (iv) the canonical-$\mathbb{Z}$-basis
theorem of Gross-Hacking-Keel 2015 / Carl-Pumperla-Siebert 2020 stating
that $\{\vartheta_m\}_{m \in B(\mathbb{Z})}$ is a free $\mathbb{Z}$-
basis of the structure ring of the mirror; (v) the associativity of
theta-function multiplication via the consistency of the canonical
scattering diagram; (vi) the Fulton 3.32 specialisation showing that
in the rigid-toric case the broken lines reduce to constant straight
lines and $\vartheta_m = \chi^m$ recovers the lattice-point basis of
$H^0(X_P, L_P)$. The present schematic Lean module declares the
central definitions and the canonical-basis theorem with
`sorry`-stubbed bodies; the eventual Mathlib port follows the
reconstruction-theorem development of Gross-Siebert 2011 and the
broken-line algorithm of Gross-Hacking-Keel 2015.

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 55m
master: 110m

Intuition [Beginner]

Visual [Beginner]

Worked example [Beginner]

Check your understanding [Beginner]

Formal definition [Intermediate+]

Counterexamples to common slips

Key theorem with proof [Intermediate+]

Exercises [Intermediate+]

Lean formalisation [Intermediate+]

Polarised tropical manifold and broken-line construction [Master]

The canonical Z-basis theorem [Master]

Mirror-symmetric structure and intrinsic mirror algebras [Master]

Connections [Master]

Historical & philosophical context [Master]

Bibliography [Master]

The canonical $Z$ -basis theorem [Master]