04.12.16 · algebraic-geometry / tropical

The A-model, the B-model, and the mirror symmetry conjecture

shipped3 tiersLean: none

Anchor (Master): Candelas-de la Ossa-Green-Parkes 1991 *Nucl. Phys. B* 359, 21-74 (originator: the genus-0 instanton prediction for the quintic threefold); Kontsevich 1995 *Proc. ICM 1994* (homological mirror symmetry); Strominger-Yau-Zaslow 1996 *Nucl. Phys. B* 479, 243-259 (SYZ geometric explanation); Givental 1996 *Internat. Math. Res. Notices* 1996/13 + Lian-Liu-Yau 1997 *Asian J. Math.* 1 (rigorous proof of the genus-0 mirror theorem); Cox-Katz 1999 *Mirror Symmetry and Algebraic Geometry* (AMS Math. Surv. Mono. 68); Hori-Katz-Klemm-Pandharipande-Thomas-Vafa-Vakil-Zaslow 2003 *Mirror Symmetry* (Clay Math. Mono. 1); Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114); Gross-Siebert 2011 *Annals of Mathematics* 174, 1301-1428 (the tropical/algebraic mechanism)

Intuition Beginner

Mirror symmetry is the discovery that Calabi-Yau shapes come in pairs. Each Calabi-Yau is a special six-dimensional space (three complex dimensions) used to roll up the extra dimensions of string theory. The astonishing claim is that two very different-looking Calabi-Yau shapes can secretly encode the same physics — counting curves on one equals integrating periods on its mirror.

A Calabi-Yau space has two independent kinds of data you can measure. The first kind records its size and the ways you can stretch or shrink the loops inside it. The second kind records its complex shape and the ways you can bend that shape. Mirror symmetry says: take a Calabi-Yau $X$ , and there is a partner $\overset{ˇ}{X}$ whose size-data is the same as the shape-data of $X$ , and whose shape-data is the same as the size-data of $X$ . The two kinds of data are swapped.

This swap is useful because one kind of question is hard and its mirror question is easy. Counting how many curves of a given degree sit inside $X$ is a famously hard counting problem. On the partner $\overset{ˇ}{X}$ , the matching question is a calculus problem about integrals that solve a single differential equation. The hard counting on $X$ becomes an easier integral on $\overset{ˇ}{X}$ .

The headline success came in 1991. Candelas, de la Ossa, Green and Parkes used the partner of the quintic — a degree-five Calabi-Yau living in a five-coordinate projective space — to predict the curve counts on the quintic. There are $2875$ lines, $609250$ conics, and the pattern continues. These integers were later confirmed by independent rigorous proofs. The prediction is the founding example of the whole subject.

Visual Beginner

A two-panel mirror diagram. Left panel, labelled "A-model of $X$ ": a Calabi-Yau drawn as a softly rounded six-dimensional blob, with several curves of low degree drawn sitting inside it — a straight line, a small conic, a twisting cubic — each tagged with its count ( $2875$ lines, $609250$ conics). A small box beside it reads "size and loops: $h^{1, 1}$ ". Right panel, labelled "B-model of $\overset{ˇ}{X}$ ": the partner Calabi-Yau drawn as a different-looking blob carrying a single highlighted volume form, with a one-parameter family of these forms sweeping past as a dial turns, and a box reading "shape and bending: $h^{2, 1}$ ".

Across the middle runs a double-headed arrow labelled "mirror map", passing through a small box that shows two numbers being exchanged: the size-number of the left equals the shape-number of the right, and the shape-number of the left equals the size-number of the right. The picture captures the whole conjecture in one image: two different spaces, two kinds of data, one swap.

Worked example Beginner

The cleanest example to hold in mind is the quintic threefold $X$ and its mirror $\overset{ˇ}{X}$ . The quintic is the set of points in five-coordinate projective space where a single degree-five equation vanishes, for instance $x_{0}^{5} + x_{1}^{5} + x_{2}^{5} + x_{3}^{5} + x_{4}^{5} = 0$ . This is a Calabi-Yau space of three complex dimensions.

Step 1. The size-data of the quintic is small. There is essentially one way to measure size on the quintic — one independent size parameter. In the bookkeeping of Hodge numbers this is recorded as $h^{1, 1} (X) = 1$ .

Step 2. The shape-data of the quintic is large. The quintic can be deformed in $101$ independent complex directions, recorded as $h^{2, 1} (X) = 101$ . So the quintic has a one-dimensional size-moduli and a $101$ -dimensional shape-moduli.

Step 3. The mirror swaps these. The partner $\overset{ˇ}{X}$ has size-data of dimension $101$ and shape-data of dimension $1$ : $h^{1, 1} (\overset{ˇ}{X}) = 101$ and $h^{2, 1} (\overset{ˇ}{X}) = 1$ . The shape-moduli of the mirror is one-dimensional, so the integrals that compute its volume-form periods depend on a single parameter and solve one differential equation.

Step 4. The prediction. Because the mirror's shape-moduli is one-dimensional, the period integrals on $\overset{ˇ}{X}$ are computable. Candelas, de la Ossa, Green and Parkes solved them and read off the curve counts on $X$ : $n_{1} = 2875$ degree-one curves (lines), $n_{2} = 609250$ degree-two curves (conics), $n_{3} = 317206375$ degree-three curves, and so on.

What this tells us: the swap is the point. The hard side of $X$ (counting curves) is matched to the easy side of $\overset{ˇ}{X}$ (one-parameter integrals), so a calculation impossible to do directly becomes routine on the mirror.

Check your understanding Beginner

Exercise (easy, multiple choice).

In mirror symmetry, a mirror pair of Calabi-Yau 3-folds $(X, \overset{ˇ}{X})$ satisfies which exchange of Hodge numbers?

A. $h^{1, 1} (X) = h^{1, 1} (\overset{ˇ}{X})$ and $h^{2, 1} (X) = h^{2, 1} (\overset{ˇ}{X})$ . B. $h^{1, 1} (X) = h^{2, 1} (\overset{ˇ}{X})$ and $h^{2, 1} (X) = h^{1, 1} (\overset{ˇ}{X})$ . C. $h^{1, 1} (X) = h^{2, 1} (X)$ on each space separately. D. All four Hodge numbers are equal to one.

Hint

Mirror symmetry swaps the size-data of one space with the shape-data of the other. The size-data is $h^{1, 1}$ and the shape-data is $h^{2, 1}$ .

Answer

B. The defining numerical signature of a mirror pair is the cross-exchange $h^{1, 1} (X) = h^{2, 1} (\overset{ˇ}{X})$ and $h^{2, 1} (X) = h^{1, 1} (\overset{ˇ}{X})$ . The size-data (Kähler / loop directions, $h^{1, 1}$ ) of one space equals the shape-data (complex-structure / bending directions, $h^{2, 1}$ ) of its mirror. Option A would say the spaces have identical data, which is the opposite of a swap. Option C asks for a coincidence on one space that mirror symmetry does not require. Option D is the special situation of a rigid Calabi-Yau, not the general rule.

Exercise (easy, multiple choice).

The A-model of a Calabi-Yau $X$ is concerned with:

A. The complex-structure moduli and period integrals of the holomorphic volume form. B. The symplectic / Kähler side: counts of rational curves (Gromov-Witten invariants) and quantum cohomology. C. The Riemannian curvature tensor of the Calabi-Yau metric. D. The fundamental group of $X$ .

Hint

The A-model is the enumerative, curve-counting side. The B-model is the period-integral side.

Answer

B. The A-model is the symplectic / Kähler side of a Calabi-Yau: it packages the Gromov-Witten invariants (counts of rational curves) into quantum cohomology, a deformation of ordinary cohomology by curve counts. Option A describes the B-model, the complex-structure / period side. Option C names a metric invariant that is not what either model directly computes. Option D names a topological invariant unrelated to the two models.

Formal definition Intermediate+

Fix a Calabi-Yau 3-fold $X$ : a smooth projective complex variety of dimension $3$ with identity canonical bundle, $ω_{X} ≅ O_{X}$ , and $H^{1} (X, O_{X}) = 0$ . Its Hodge diamond is governed by two numbers, the Kähler (size) number $h^{1, 1} (X) = dim_{C} H^{1, 1} (X)$ and the complex-structure (shape) number $h^{2, 1} (X) = dim_{C} H^{2, 1} (X)$ , where $H^{p, q} (X)$ are the Hodge pieces of 04.09.01.

Definition (A-model data). The A-model of $X$ is the package of symplectic / Kähler-moduli data. Its moduli is a neighbourhood of the large-volume limit in the complexified Kähler cone, of dimension $h^{1, 1} (X)$ . Its enumerative content is the family of genus-0 Gromov-Witten invariants $⟨ \dots ⟩_{0, β}$ of 04.10.35, one for each curve class $β \in H_{2} (X, Z)$ , assembled into the small quantum cohomology product: for cohomology classes $α_{1}, α_{2}$ , $$ \alpha_1 *q \alpha_2 = \sum{\beta \in H_2(X,\mathbb{Z})} \Bigl( \sum_{\gamma} \langle \alpha_1, \alpha_2, \gamma^\vee \rangle_{0,\beta}, \gamma \Bigr) q^\beta, $$ a deformation of the cup product by curve counts, with $q^{β}$ the Kähler parameters. The $β = 0$ term recovers the classical cup product.

Definition (B-model data). The B-model of a Calabi-Yau 3-fold $\overset{ˇ}{X}$ is the package of complex-structure-moduli data. Its moduli is the local deformation space of the complex structure, of dimension $h^{2, 1} (\overset{ˇ}{X})$ . Its content is the variation of Hodge structure on $H^{3} (\overset{ˇ}{X}, C)$ : the Hodge filtration $F^{∙}$ moves with the complex structure, governed by the Gauss-Manin connection, and is recorded by the period integrals $Π_{i} (t) = \int_{γ_{i}} Ω_{t}$ of the holomorphic volume form $Ω_{t}$ 04.12.13. The cubic Yukawa coupling $$ Y(\theta, \theta, \theta) = \int_{\check X} \Omega_t \wedge \nabla_\theta \nabla_\theta \nabla_\theta, \Omega_t $$ in the logarithmic derivative $θ = t d / d t$ is the third-order term of the variation and is the B-model counterpart of the quantum product.

Definition (mirror pair). Two Calabi-Yau 3-folds $X$ and $\overset{ˇ}{X}$ form a mirror pair if there is an isomorphism of the A-model of $X$ with the B-model of $\overset{ˇ}{X}$ (and of the B-model of $X$ with the A-model of $\overset{ˇ}{X}$ ), implemented near the large-volume / large-complex-structure limits by a local biholomorphism of moduli — the mirror map $q = q (t)$ of 04.12.13 — matching the Kähler parameters of $X$ with the complex-structure parameters of $\overset{ˇ}{X}$ and identifying the quantum product of $X$ with the Yukawa coupling of $\overset{ˇ}{X}$ . A numerical consequence is the Hodge-number exchange $$ h^{1,1}(X) = h^{2,1}(\check X), \qquad h^{2,1}(X) = h^{1,1}(\check X). $$

Counterexamples to common slips

"Every Calabi-Yau has a mirror that is again a smooth Calabi-Yau." A rigid Calabi-Yau 3-fold has $h^{2, 1} (X) = 0$ , so its would-be mirror would need $h^{1, 1} (\overset{ˇ}{X}) = 0$ , which a Kähler 3-fold cannot satisfy. Rigid Calabi-Yaus have no classical geometric mirror; their mirror is realised only after enlarging the category of spaces (for example as a generalised complex or non-commutative object). The exchange of Hodge numbers is the constraint that exposes this.
"The mirror map is the identity $q = t$ ." Only the leading term agrees: $q (t) = t (1 + O (t))$ . The higher-order corrections are exactly the curve-counting content. Treating $q = t$ throws away every Gromov-Witten invariant beyond the classical intersection number and reduces the conjecture to a tautology.
"Mirror symmetry equates the two spaces $X$ and $\overset{ˇ}{X}$ ." The spaces are genuinely different — different topology, different Hodge diamond. What is equated is the A-model of one with the B-model of the other. There is no map $X \to \overset{ˇ}{X}$ of varieties; the equivalence lives at the level of the two models, and the SYZ picture 04.12.10 explains it as fibrewise torus duality, not as an isomorphism of total spaces.

Key theorem with proof Intermediate+

The signature result is the rigorous form of the Candelas-de la Ossa-Green-Parkes prediction, proved independently by Givental and by Lian-Liu-Yau.

Theorem (genus-0 mirror theorem for the quintic). Let $X \subset P^{4}$ be a smooth quintic 3-fold and $\overset{ˇ}{X}$ its mirror, with one-dimensional complex-structure moduli coordinatised by $t$ near the large-complex-structure limit. Let $Π_{0} (t), Π_{1} (t)$ be the fundamental and logarithmic periods of the holomorphic volume form on $\overset{ˇ}{X}$ 04.12.13, let $q (t) = exp (Π_{1} (t) / Π_{0} (t))$ be the mirror map, and let $K (q)$ be the normalised B-model Yukawa coupling pulled back along $q (t)$ . Then $K (q)$ equals the A-model genus-0 quantum-cohomology Yukawa coupling of $X$ , $$ K(q) = 5 + \sum_{d \geq 1} n_d, d^3 \frac{q^d}{1 - q^d}, $$ where the $n_{d}$ are the genus-0 Gromov-Witten instanton numbers of $X$ . In particular $n_{1} = 2875$ , $n_{2} = 609250$ , $n_{3} = 317206375$ .

Proof. The full proof is the content of Givental 1996 and Lian-Liu-Yau 1997; we record its architecture in three steps.

Step 1 (the B-model side is a closed computation). The mirror $\overset{ˇ}{X}$ has $h^{2, 1} (\overset{ˇ}{X}) = 1$ , so its period vector has rank $4$ and is annihilated by the order- $4$ Picard-Fuchs operator $L = θ^{4} - 5 t (5 θ + 1) (5 θ + 2) (5 θ + 3) (5 θ + 4)$ with $θ = t d / d t$ 04.12.13. Solving $L Π = 0$ by the Frobenius method at $t = 0$ yields the fundamental period $Π_{0} (t) = \sum_{n \geq 0} \frac{( 5 n )!}{( n ! ) ^{5}} t^{n}$ and the logarithmic period $Π_{1}$ , hence the mirror map $q (t)$ and the B-model Yukawa coupling $K (q)$ as an explicit $q$ -series with rational coefficients.

Step 2 (the A-model side is the genus-0 Gromov-Witten generating series). On $X$ , the genus-0 invariants $⟨ pt, pt, pt ⟩_{0, β}$ of 04.10.35 assemble into the quantum-cohomology Yukawa coupling, which the multiple-cover formula expresses through the instanton numbers $n_{d}$ as the right-hand series $5 + \sum_{d} n_{d} d^{3} q^{d} / (1 - q^{d})$ . The integers $n_{d}$ are the genuine enumerative counts, the $d^{3}$ and the $1/ (1 - q^{d})$ being the universal contribution of degree- $d$ multiple covers of a fixed rational curve.

Step 3 (the identification). Givental's proof compares the $J$ -function of $X$ (a generating series of A-model gravitational descendants) with the $I$ -function built from the $\overset{ˇ}{X}$ periods, using equivariant localisation on the moduli of stable maps to $P^{4}$ and a mirror-transformation lemma to match them after the change of variable $q = q (t)$ ; Lian-Liu-Yau's mirror principle establishes the same identity by an Euler-data / linking-theorem argument on the same moduli. Both reduce the equality $K (q) =$ (A-model coupling) to a finite, verifiable comparison of localisation contributions. The leading coefficient of $q$ then forces $n_{1} = 2875$ , matching the classical count of lines on the quintic, and the recursion delivers $n_{2} = 609250$ , $n_{3} = 317206375$ , and the full table. $□$

Bridge. The mirror theorem builds toward the homological mirror conjecture by upgrading the numerical match of curve counts to a structural identity of generating series, and it appears again in the Gross-Siebert programme 04.12.09, where the same instanton numbers reappear as tropical-disk counts read off the polarised tropical manifold. The foundational reason the two sides agree is that the period integrals of $\overset{ˇ}{X}$ and the Gromov-Witten series of $X$ are two analytic continuations of one Frobenius-manifold germ; this is exactly the content the mirror map $q (t)$ of 04.12.13 makes precise, putting these together with the Hodge-number exchange to pin the dimensions of the two moduli. The A-model quantum product is dual to the B-model Yukawa coupling under this germ isomorphism, and the central insight — that an impossible enumerative problem becomes a one-parameter differential equation on the mirror — is the bridge from physics heuristics to the proofs of Givental and Lian-Liu-Yau.

Exercises Intermediate+

Exercise 2 (easy, numeric).

In the instanton expansion $K (q) = 5 + \sum_{d \geq 1} n_{d} d^{3} \frac{q ^{d}}{1 - q ^{d}}$ for the quintic, the constant term $5$ is the classical triple self-intersection of the hyperplane class. What is the genus-0 instanton number $n_{1}$ of degree-1 curves?

Hint

This is the famous count of lines on the quintic threefold, the leading correction to the classical $5$ .

Answer

2875. The degree-1 instanton number is $n_{1} = 2875$ , the number of lines on the quintic 3-fold. It is the leading enumerative correction in the quantum Yukawa coupling and the first term Candelas-de la Ossa-Green-Parkes extracted from the mirror periods. The integer was already known to nineteenth-century enumerative geometry as the count of lines on a generic quintic; mirror symmetry's surprise was that the period computation on $\overset{ˇ}{X}$ reproduces it and then continues to $n_{2} = 609250$ and beyond.

Exercise 3 (medium, symbolic).

State the Picard-Fuchs operator $L$ annihilating the periods of the mirror quintic in the variable $t$ and the logarithmic derivative $θ = t d / d t$ , and write the fundamental period $Π_{0} (t)$ as a hypergeometric series.

Hint

The operator is order $4$ with hypergeometric data $1/5, 2/5, 3/5, 4/5$ . The fundamental period has coefficients $(5 n)! / (n!)^{5}$ .

Answer

Picard-Fuchs operator and fundamental period. The mirror quintic periods are annihilated by $$ L = \theta^4 - 5 t (5\theta + 1)(5\theta + 2)(5\theta + 3)(5\theta + 4), \qquad \theta = t \frac{d}{dt}. $$ The unique solution regular at $t = 0$ with value $1$ is the fundamental period $$ \Pi_0(t) = \sum_{n = 0}^{\infty} \frac{(5n)!}{(n!)^5}, t^n = 1 + 120, t + 113400, t^2 + \cdots, $$ a generalised hypergeometric series $_{4} F_{3}$ with upper parameters $\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}$ and lower parameters $1, 1, 1$ . The logarithmic solution $Π_{1} (t) = Π_{0} (t) lo g t + (regular)$ gives the mirror map $q (t) = exp (Π_{1} / Π_{0}) = t (1 + O (t))$ , and the B-model Yukawa coupling pulled back along $q$ produces the instanton series of the key theorem.

Exercise 5 (medium, multiple choice).

Kontsevich's homological mirror symmetry conjecture asserts an equivalence between:

A. The de Rham cohomology of $X$ and the singular cohomology of $\overset{ˇ}{X}$ . B. The bounded derived category of coherent sheaves $D^{b} Coh (\overset{ˇ}{X})$ and the derived Fukaya category $D^{b} F (X)$ . C. The Picard group of $X$ and the Picard group of $\overset{ˇ}{X}$ . D. The fundamental groups of $X$ and $\overset{ˇ}{X}$ .

Hint

Homological mirror symmetry is categorical: it pairs the B-branes (coherent sheaves) of one space with the A-branes (Lagrangians) of the mirror.

Answer

B. Kontsevich 1994 conjectured that the bounded derived category of coherent sheaves on $\overset{ˇ}{X}$ (the category of B-branes) is equivalent, as a triangulated category, to the derived Fukaya category of $X$ (the $A_{\infty}$ -category of Lagrangian submanifolds, the A-branes). This is the categorical lift of the numerical mirror statement: it implies the Hodge-number and Gromov-Witten matches as shadows of a single equivalence of categories. Options A, C, D name invariants that mirror symmetry does relate numerically but are not the content of the homological conjecture.

Exercise 6 (hard, symbolic).

State the SYZ conjecture and explain how it produces the mirror $\overset{ˇ}{X}$ from $X$ geometrically, and how it accounts for the exchange of A-model and B-model data.

Hint

SYZ posits dual special-Lagrangian torus fibrations over a common base. Dualising a torus fibre is the geometric operation; T-duality swaps the symplectic and complex roles of the fibre directions.

Answer

SYZ conjecture (Strominger-Yau-Zaslow 1996). Near the large-complex-structure limit, a Calabi-Yau 3-fold $X$ admits a fibration $f : X \to B$ over a real $3$ -dimensional base $B$ whose generic fibre is a special-Lagrangian $3$ -torus $T^{3}$ (calibrated by the real part of the holomorphic volume form), with the base carrying an integral-affine structure singular along a codimension- $2$ discriminant. The mirror $\overset{ˇ}{X}$ is obtained by dualising each torus fibre: replacing $T^{3}$ by the dual torus $\overset{ˇ}{T}^{3} = Hom (π_{1} (T^{3}), U (1))$ , giving a fibration $\overset{ˇ}{f} : \overset{ˇ}{X} \to B$ over the same base.

Exchange of data. Fibrewise dualisation is T-duality: it interchanges the symplectic data transverse to the fibre with the complex data along it. The Kähler (symplectic) parameters of $X$ , measuring the sizes of the $T^{3}$ fibres, become complex-structure parameters of $\overset{ˇ}{X}$ , measuring the shapes of the dual fibres $\overset{ˇ}{T}^{3}$ , and conversely. This swaps $h^{1, 1}$ with $h^{2, 1}$ and turns the A-model of $X$ into the B-model of $\overset{ˇ}{X}$ . The shared base $B$ — an integral-affine manifold with singularities — is the object the Gross-Siebert programme 04.12.09 makes algebraic: both $X$ and $\overset{ˇ}{X}$ are reconstructed from $B$ together with its two polarisation conventions, so the geometric T-duality of SYZ becomes a combinatorial operation on the tropical manifold $(B, P, φ)$ .

Advanced results Master

Mirror symmetry has matured from a single prediction into a web of theorems and conjectures organised around the A-model / B-model dichotomy. Three structural strands carry the modern subject.

The first is the enumerative strand, where the 1991 prediction is now a theorem and is understood structurally. Givental 1996 and Lian-Liu-Yau 1997 proved the genus-0 mirror theorem; the statement is most cleanly phrased as an equality of two flat sections of the Dubrovin connection on the Kähler-moduli germ — the A-model $J$ -function and the B-model $I$ -function agree after the mirror map. Both sides are flat sections of a single Frobenius manifold (Dubrovin 1996), the geometric object whose A-model incarnation is quantum cohomology and whose B-model incarnation is the variation of Hodge structure with its Yukawa coupling 04.09.01. Higher-genus mirror symmetry (the Bershadsky-Cecotti-Ooguri-Vafa holomorphic-anomaly equation, proved in many cases by Costello-Li and others) extends the match beyond genus 0, where the analytic structure is far more delicate.

The second is the categorical strand, Kontsevich's homological mirror symmetry ^{[Kontsevich 1995]}: an equivalence $D^{b} Coh (\overset{ˇ}{X}) ≃ D^{b} F (X)$ between the derived category of coherent sheaves (B-branes) and the derived Fukaya category (A-branes). This is now a theorem for elliptic curves (Polishchuk-Zaslow), abelian varieties, the quartic and quintic (Seidel, Sheridan), and toric and toric-degeneration cases (Abouzaid, Gross-Siebert-adjacent work). The numerical mirror statement is recovered as the Hochschild-homology shadow of the categorical equivalence.

The third is the geometric strand, the SYZ conjecture ^{[Strominger-Yau-Zaslow 1996]}: dual special-Lagrangian torus fibrations over a common integral-affine base explain why the exchange holds, by fibrewise T-duality 04.12.10. The Gross-Siebert reconstruction programme 04.12.09 makes this algebraic: starting from a single integral-affine manifold with singularities $(B, P, φ)$ , it reconstructs both members of the mirror pair as toric degenerations 04.12.07, with the wall-and-slab scattering data 04.12.11 encoding the very Gromov-Witten counts the A-model demands. The reflexive-polytope construction of Batyrev 04.11.16 is the toric special case where $B$ comes from a lattice polytope and its polar dual.

Synthesis. The central insight is that the A-model and the B-model are two analytic faces of one Frobenius-manifold germ, and mirror symmetry is the assertion that the A-face of $X$ is the B-face of $\overset{ˇ}{X}$ . The foundational reason the enumerative prediction works is that the period integrals of $\overset{ˇ}{X}$ are flat sections of the same connection whose other flat sections are the Gromov-Witten series of $X$ ; this is exactly what the mirror map $q (t)$ of 04.12.13 aligns, and putting these together with the Hodge-number exchange fixes the dimensions of the two moduli. The categorical statement is dual to the enumerative one — Hochschild homology of the equivalence recovers the curve counts — and the SYZ picture is the bridge that makes both geometric, since fibrewise T-duality over a shared affine base is what the Gross-Siebert tropical reconstruction 04.12.09 turns into an algorithm. This pattern recurs throughout the chapter: the tropical manifold $(B, P, φ)$ is the single object from which the A-model, the B-model, and their equivalence all descend, and it generalises the reflexive-polytope duality of 04.11.16 to the full Calabi-Yau setting.

Full proof set Master

Proposition (Hodge-number exchange forces an Euler-characteristic sign flip). If $(X, \overset{ˇ}{X})$ is a mirror pair of Calabi-Yau 3-folds, with $b_{1} = 0$ and the standard Hodge diamond, then $χ (\overset{ˇ}{X}) = - χ (X)$ .

Proof. For a Calabi-Yau 3-fold $Y$ with $h^{1, 0} (Y) = h^{2, 0} (Y) = 0$ and $h^{3, 0} (Y) = 1$ , the nonzero Hodge numbers are $h^{0, 0} = h^{3, 3} = 1$ , $h^{3, 0} = h^{0, 3} = 1$ , $h^{1, 1} = h^{2, 2}$ , and $h^{2, 1} = h^{1, 2}$ . The Euler characteristic is the alternating sum of Betti numbers $b_{k} = \sum_{p + q = k} h^{p, q}$ , giving $$ \chi(Y) = \sum_{p,q} (-1)^{p+q} h^{p,q}(Y) = 2\bigl(h^{1,1}(Y) - h^{2,1}(Y)\bigr), $$ after the $b_{0}, b_{2}, b_{4}, b_{6}$ terms (each carrying $+ 1$ from the $h^{0, 0}, h^{1, 1}, h^{2, 2}, h^{3, 3}$ contributions and the $\pm 1$ from $h^{3, 0}$ in $b_{3}$ ) are collected and the odd-degree pieces $b_{1} = b_{5} = 0$ , $b_{3} = 2 + 2 h^{2, 1}$ are inserted. Apply this to $\overset{ˇ}{X}$ and substitute the mirror exchange $h^{1, 1} (\overset{ˇ}{X}) = h^{2, 1} (X)$ , $h^{2, 1} (\overset{ˇ}{X}) = h^{1, 1} (X)$ : $$ \chi(\check X) = 2\bigl(h^{1,1}(\check X) - h^{2,1}(\check X)\bigr) = 2\bigl(h^{2,1}(X) - h^{1,1}(X)\bigr) = -\chi(X). \qquad \square $$

Proposition (a rigid Calabi-Yau has no smooth Kähler mirror). If $X$ is a Calabi-Yau 3-fold with $h^{2, 1} (X) = 0$ , then no smooth Kähler Calabi-Yau 3-fold $\overset{ˇ}{X}$ satisfies the mirror exchange of Hodge numbers with $X$ .

Proof. The mirror exchange demands $h^{1, 1} (\overset{ˇ}{X}) = h^{2, 1} (X) = 0$ . But for any compact Kähler manifold the cohomology class of the Kähler form is a nonzero element of $H^{1, 1} (\overset{ˇ}{X}, R)$ , so $h^{1, 1} (\overset{ˇ}{X}) \geq 1$ . The two requirements $h^{1, 1} (\overset{ˇ}{X}) = 0$ and $h^{1, 1} (\overset{ˇ}{X}) \geq 1$ are incompatible, so no such smooth Kähler $\overset{ˇ}{X}$ exists. A mirror for a rigid Calabi-Yau therefore lives outside the category of smooth Kähler 3-folds — realised, for instance, as a generalised complex object or via a non-geometric Landau-Ginzburg model. $□$

Proposition (leading term of the mirror map). With $Π_{0} (t) = 1 + O (t)$ the fundamental period and $Π_{1} (t) = Π_{0} (t) lo g t + R (t)$ the logarithmic period, $R$ regular at $0$ with $R (0) = 0$ , the mirror map $q (t) = exp (Π_{1} (t) / Π_{0} (t))$ satisfies $q (t) = t (1 + O (t))$ , hence is a local coordinate at $t = 0$ .

Proof. Divide: $Π_{1} / Π_{0} = lo g t + R (t) / Π_{0} (t)$ . Since $Π_{0} (0) = 1 \neq = 0$ and $R$ is regular with $R (0) = 0$ , the quotient $R / Π_{0}$ is regular at $0$ and vanishes there, so $R (t) / Π_{0} (t) = O (t)$ . Exponentiating, $q (t) = exp (lo g t) \cdot exp (R / Π_{0}) = t \cdot (1 + O (t))$ . The derivative $q^{'} (0)$ is the leading coefficient $1 \neq = 0$ , so by the inverse function theorem $q$ is a biholomorphism of a punctured neighbourhood of $0$ onto a punctured neighbourhood of $0$ , that is, a local coordinate identifying the complex-structure parameter $t$ of $\overset{ˇ}{X}$ with the Kähler parameter $q$ of $X$ . $□$

Connections Master

Moduli of stable maps and Gromov-Witten invariants 04.10.35. The A-model is built directly from the genus-0 Gromov-Witten invariants of that unit: the instanton numbers $n_{d}$ of the quintic are extracted from the three-point genus-0 invariants, and quantum cohomology is the deformation of the cup product they define. Mirror symmetry is what makes these otherwise-intractable counts computable.
Period integral and the mirror map 04.12.13. The B-model is the period side: the holomorphic volume form of $\overset{ˇ}{X}$ , its Picard-Fuchs equation, and the mirror map $q (t)$ that aligns the two moduli are exactly the machinery of that unit. The present unit states the conjecture those periods are computing.
Hodge decomposition 04.09.01. The Hodge numbers $h^{1, 1}$ and $h^{2, 1}$ that mirror symmetry exchanges are the dimensions of Hodge pieces from that unit, and the B-model variation of Hodge structure is the moving Hodge filtration on $H^{3}$ . The numerical signature of a mirror pair is a statement in Hodge theory.
Reflexive polytope and Batyrev mirror duality 04.11.16. Batyrev's polar duality of reflexive polytopes is the combinatorial mechanism that produces mirror pairs of toric Calabi-Yau hypersurfaces and realises the Hodge-number exchange explicitly. It is the toric special case of the conjecture stated here.
Strominger-Yau-Zaslow conjecture 04.12.10. SYZ supplies the geometric explanation: dual special-Lagrangian torus fibrations over a common affine base, with fibrewise T-duality interchanging the A-model and B-model data. It is the bridge from the bare conjecture to the Gross-Siebert reconstruction that makes it algebraic.

Historical & philosophical context Master

Mirror symmetry entered mathematics as a surprise from physics. String theorists studying Calabi-Yau compactifications noticed in the late 1980s that distinct Calabi-Yau spaces gave identical physical theories, with the roles of complex and Kähler moduli interchanged. The decisive moment came in 1991, when Candelas, de la Ossa, Green and Parkes used the mirror of the quintic to predict its genus-0 curve counts and produced the integers $2875$ , $609250$ , $317206375$ ^{[Candelas-de la Ossa-Green-Parkes 1991]}. Enumerative geometers, who had computed the first two of these by classical means, were startled that a one-parameter period integral on a different space reproduced them — and the third was beyond what classical methods had reached. The prediction was a challenge to mathematics: explain why a calculation in Hodge theory on $\overset{ˇ}{X}$ counts curves on $X$ .

The challenge reshaped several fields at once. Kontsevich's 1994 ICM address reframed the conjecture categorically, proposing that the derived category of coherent sheaves on one Calabi-Yau is equivalent to the Fukaya category of its mirror — moving the subject from numbers to categories and connecting it to D-branes in string theory. Strominger, Yau and Zaslow then offered a geometric mechanism in 1996, reading mirror symmetry as T-duality along special-Lagrangian torus fibrations. Givental and Lian-Liu-Yau independently turned the original prediction into a theorem in 1996-1997. The philosophical lesson is that an equivalence invisible at the level of spaces — the two Calabi-Yaus are genuinely different — becomes transparent at the level of the structures they support, whether Frobenius manifolds, triangulated categories, or affine bases. The Gross-Siebert programme, the subject of this chapter, is the most algebraic realisation of that lesson: a single integral-affine manifold with singularities from which both mirrors, and their equivalence, descend.

Bibliography Master

@article{CandelasDeLaOssaGreenParkes1991,
  author  = {Candelas, Philip and de la Ossa, Xenia C. and Green, Paul S. and Parkes, Linda},
  title   = {A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory},
  journal = {Nuclear Physics B},
  volume  = {359},
  number  = {1},
  pages   = {21--74},
  year    = {1991}
}

@incollection{Kontsevich1995,
  author    = {Kontsevich, Maxim},
  title     = {Homological algebra of mirror symmetry},
  booktitle = {Proceedings of the International Congress of Mathematicians (Z\"urich, 1994), Vol. I},
  pages     = {120--139},
  publisher = {Birkh\"auser},
  year      = {1995}
}

@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},
  pages   = {243--259},
  year    = {1996}
}

@article{Givental1996,
  author  = {Givental, Alexander B.},
  title   = {Equivariant {G}romov-{W}itten invariants},
  journal = {International Mathematics Research Notices},
  volume  = {1996},
  number  = {13},
  pages   = {613--663},
  year    = {1996}
}

@article{LianLiuYau1997,
  author  = {Lian, Bong H. and Liu, Kefeng and Yau, Shing-Tung},
  title   = {Mirror principle {I}},
  journal = {Asian Journal of Mathematics},
  volume  = {1},
  number  = {4},
  pages   = {729--763},
  year    = {1997}
}

@book{CoxKatz1999,
  author    = {Cox, David A. and Katz, Sheldon},
  title     = {Mirror Symmetry and Algebraic Geometry},
  publisher = {American Mathematical Society},
  series    = {Mathematical Surveys and Monographs},
  volume    = {68},
  year      = {1999}
}

@book{HoriEtAl2003,
  author    = {Hori, Kentaro and Katz, Sheldon and Klemm, Albrecht and Pandharipande, Rahul and Thomas, Richard and Vafa, Cumrun and Vakil, Ravi and Zaslow, Eric},
  title     = {Mirror Symmetry},
  publisher = {American Mathematical Society / Clay Mathematics Institute},
  series    = {Clay Mathematics Monographs},
  volume    = {1},
  year      = {2003}
}

@book{Gross2011,
  author    = {Gross, Mark},
  title     = {Tropical Geometry and Mirror Symmetry},
  publisher = {American Mathematical Society},
  series    = {CBMS Regional Conference Series in Mathematics},
  volume    = {114},
  year      = {2011}
}

@article{GrossSiebert2011,
  author  = {Gross, Mark and Siebert, Bernd},
  title   = {From real affine geometry to complex geometry},
  journal = {Annals of Mathematics (2)},
  volume  = {174},
  number  = {3},
  pages   = {1301--1428},
  year    = {2011}
}

@book{AspinwallEtAl2009,
  author    = {Aspinwall, Paul S. and Bridgeland, Tom and Craw, Alastair and Douglas, Michael R. and Gross, Mark and Kapustin, Anton and Moore, Gregory W. and Segal, Graeme and Szendr\H{o}i, Bal\'azs and Wilson, P. M. H.},
  title     = {Dirichlet Branes and Mirror Symmetry},
  publisher = {American Mathematical Society / Clay Mathematics Institute},
  series    = {Clay Mathematics Monographs},
  volume    = {4},
  year      = {2009}
}

Prerequisites

04.10.35
04.12.13
04.09.01

Tier anchors

beginner: Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lecture 1 informal opening; Candelas-de la Ossa-Green-Parkes 1991 *Nucl. Phys. B* 359 §1 (the quintic story, informal preamble)
intermediate: Cox-Katz 1999 *Mirror Symmetry and Algebraic Geometry* (AMS Math. Surv. Mono. 68) §1-§2, §8-§11; Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lectures 1-2; Hori et al. 2003 *Mirror Symmetry* (Clay Math. Mono. 1) Ch. 1-2
master: Candelas-de la Ossa-Green-Parkes 1991 *Nucl. Phys. B* 359, 21-74 (originator: the genus-0 instanton prediction for the quintic threefold); Kontsevich 1995 *Proc. ICM 1994* (homological mirror symmetry); Strominger-Yau-Zaslow 1996 *Nucl. Phys. B* 479, 243-259 (SYZ geometric explanation); Givental 1996 *Internat. Math. Res. Notices* 1996/13 + Lian-Liu-Yau 1997 *Asian J. Math.* 1 (rigorous proof of the genus-0 mirror theorem); Cox-Katz 1999 *Mirror Symmetry and Algebraic Geometry* (AMS Math. Surv. Mono. 68); Hori-Katz-Klemm-Pandharipande-Thomas-Vafa-Vakil-Zaslow 2003 *Mirror Symmetry* (Clay Math. Mono. 1); Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114); Gross-Siebert 2011 *Annals of Mathematics* 174, 1301-1428 (the tropical/algebraic mechanism)

References

Candelas, P., de la Ossa, X., Green, P. S. & Parkes, L. — A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory · *Nuclear Physics B* 359 (1), 21-74 (1991). The founding prediction of mirror symmetry: the genus-0 Gromov-Witten invariants (instanton numbers) of the quintic threefold $X \subset \mathbb{P}^4$ are computed from the period integrals of the holomorphic volume form on the mirror $\check X$ via the Picard-Fuchs equation and the mirror map. §1-§2 (the mirror pair and statement); §3-§5 (the instanton-number table, including $n_1 = 2875$ lines and $n_2 = 609250$ conics).
Kontsevich, M. — Homological algebra of mirror symmetry · *Proceedings of the International Congress of Mathematicians (Zürich, 1994)*, Vol. I, Birkhäuser (1995), 120-139. The homological mirror symmetry conjecture: an equivalence of triangulated categories between the bounded derived category of coherent sheaves $D^b\mathrm{Coh}(\check X)$ on one Calabi-Yau and the derived Fukaya category $D^b\mathcal{F}(X)$ of Lagrangian submanifolds on its mirror.
Strominger, A., Yau, S.-T. & Zaslow, E. — Mirror symmetry is T-duality · *Nuclear Physics B* 479 (1-2), 243-259 (1996). The SYZ conjecture: mirror Calabi-Yau pairs $(X, \check X)$ carry dual special-Lagrangian torus fibrations over a common integral-affine base, with the mirror obtained by dualising (T-dualising) the torus fibres. The geometric explanation of the exchange of A-model and B-model data.
Givental, A. B. — Equivariant Gromov-Witten invariants · *International Mathematics Research Notices* 1996 (13), 613-663. One of the two independent proofs that the Candelas-de la Ossa-Green-Parkes period prediction equals the genus-0 Gromov-Witten generating series of the quintic (the mirror theorem), via equivariant localisation and the $J$-function / $I$-function comparison.
Lian, B. H., Liu, K. & Yau, S.-T. — Mirror principle I · *Asian Journal of Mathematics* 1 (4), 729-763 (1997). The second independent proof of the genus-0 mirror theorem for the quintic and a wide class of toric complete intersections, establishing the hypergeometric / Picard-Fuchs prediction as a theorem of Gromov-Witten theory.
Cox, D. A. & Katz, S. — Mirror Symmetry and Algebraic Geometry · Mathematical Surveys and Monographs 68, American Mathematical Society (1999). The canonical algebraic-geometric monograph: §1-§2 (Calabi-Yau and the two models), §6-§8 (Gromov-Witten invariants and quantum cohomology), §11 (toric mirror symmetry and the GKZ/Picard-Fuchs computation for the quintic).
Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R. & Zaslow, E. — Mirror Symmetry · Clay Mathematics Monographs 1, American Mathematical Society / Clay Mathematics Institute (2003). The comprehensive physics-and-mathematics textbook: the A-model and B-model topological field theories, the Yukawa couplings, the quintic computation, and the categorical formulation.
Gross, M. — Tropical Geometry and Mirror Symmetry · CBMS Regional Conference Series in Mathematics 114, American Mathematical Society (2011). Lecture 1 frames the A-model / B-model conjecture; Lectures 2-7 develop the Gross-Siebert programme that makes the exchange algebraic via tropical/affine data. The textbook anchor for the present chapter.
Gross, M. & Siebert, B. — From real affine geometry to complex geometry · *Annals of Mathematics* (2) 174 (3), 1301-1428 (2011). The reconstruction theorem (cf. 04.12.09) that produces mirror Calabi-Yau families from a single integral-affine manifold with singularities, making the A-model/B-model exchange a combinatorial operation on tropical data.
Aspinwall, P. S., Bridgeland, T., Craw, A., Douglas, M. R., Gross, M., Kapustin, A., Moore, G. W., Segal, G., Szendrői, B. & Wilson, P. M. H. — Dirichlet Branes and Mirror Symmetry · Clay Mathematics Monographs 4, American Mathematical Society / Clay Mathematics Institute (2009). The categorical and D-brane synthesis: the derived category as the category of B-branes, the Fukaya category as the category of A-branes, and the homological mirror conjecture as their equivalence.

Estimated time

beginner: 20m
intermediate: 45m
master: 95m