Strominger-Yau-Zaslow (SYZ) Conjecture

Intuition [Beginner]

Mirror symmetry is a duality between Calabi-Yau manifolds. A Calabi-Yau is a special kind of curved geometric space — six real dimensions for the physically interesting case, with a complex structure and a Ricci-flat metric. To every such Calabi-Yau, mirror symmetry assigns a partner Calabi-Yau, called the mirror, in such a way that hard counting problems on the original space (counting curves) become easy integration problems on the mirror space (computing periods).

Greene-Plesser found the first explicit mirror pair in 1990, and Candelas and his collaborators in 1991 used the pair to compute predictions that astonished mathematicians who later verified them rigorously. Mirror symmetry was a profound surprise of the early 1990s; everyone agreed it was true and no one could explain why.

In 1996, the physicists Strominger, Yau, and Zaslow proposed a geometric reason. Their argument runs as follows. String theory comes in two flavours, Type IIA and Type IIB. These two theories are related: a Type IIA string on a space $X$ should be equivalent to a Type IIB string on the mirror space $X^{\vee }$.

Strominger, Yau, and Zaslow asked what happens when each theory is decorated with a brane — a higher-dimensional surface inside the Calabi-Yau that the string ends on. A point-particle brane on the Type IIA side should be dual to some kind of brane on the Type IIB side. They argued that the dual partner is a brane wrapping a torus inside the mirror Calabi-Yau, and the duality acts on this torus by T-duality, the simplest stringy duality of all: it swaps the radius of a circle with its inverse.

Once you take the consequence seriously, the geometric picture is striking. The Calabi-Yau $X$ must be a fibration: it is built by gluing together a family of tori sitting over a common base $B$. The mirror $X^{\vee }$ is the same kind of fibration over the same base, but with each torus fibre replaced by its T-dual. The mirror map between $X$ and $X^{\vee }$ is, fibre-by-fibre, T-duality. This is the SYZ conjecture in its physical form: mirror symmetry is T-duality on the torus fibres of a common base fibration.

Visual [Beginner]

A three-panel picture. Left panel: a six-real-dimensional Calabi-Yau threefold $X$ drawn schematically as a smooth curved space, with a small torus drawn at one point and arrows labelled "fibre $T^3$". Middle panel: the base $B$ of the fibration — a three-real-dimensional space with the torus collapsed to a point, drawn flat with a piecewise-affine structure (the integral affine structure) and a few singular points marked as black dots (the discriminant locus, where the torus fibre degenerates). Right panel: the mirror Calabi-Yau $X^{\vee }$ over the same base $B$, with the torus fibre now drawn as the dual torus and an arrow labelled "T-duality" connecting it to the original fibre on the left.

The point of the picture: the SYZ conjecture says the mirror pair is glued from the same combinatorial base, with one side built from tori and the other side built from dual tori. The fibration structure and the base are common to both members of the mirror pair.

Worked example [Beginner]

Take the simplest case: the two-torus $T^2$ viewed as a fibration of one-tori over a base circle $S^1$. A two-torus is the product of two circles, $T^2=S^1\times S^1$. Pick one of the two circles as the base. Each point of the base circle has a one-torus (the other circle) sitting over it; this is the fibration.

Step 1. The fibres are tori. Each fibre is a one-dimensional torus, the circle of radius $r$ on the other factor. The fibres are all isomorphic — the fibration is a global product in this simplest case.

Step 2. T-duality of the fibre. T-duality on a one-torus of radius $r$ replaces it with a one-torus of radius $1/r$ (in units where the string scale is one). The dual fibre is another circle, but with the inverse radius.

Step 3. The dual fibration. Reassemble the dual fibres over the same base $S^1$. The result is the two-torus $(T^2{)}^{\vee }=S^1\times S_{1/r}^1$, where the second factor now has radius $1/r$. The mirror two-torus is the same shape as the original two-torus, with the radii of one factor inverted.

What this tells us: SYZ mirror symmetry for two-tori is T-duality on one of the two factors. The base is the other circle; the fibre is the dualised circle; the mirror swaps the radius of the fibre with its inverse. This is the simplest possible instance of the SYZ statement, and it is exactly the original T-duality of string theory.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The mathematical content of the Strominger-Yau-Zaslow conjecture has three pillars: the definition of a Calabi-Yau manifold and its holomorphic volume form; the definition of a special Lagrangian submanifold and the fibration condition; and the duality between a torus fibration and its T-dual.

Definition (Calabi-Yau manifold). A Calabi-Yau $n$-fold is a compact Kähler manifold $(X,\omega ,J)$ of complex dimension $n$ with vanishing first Chern class $c_1(X)=0$ in $H^2(X,\mathbb{R})$ and a nowhere-vanishing holomorphic $n$-form $\Omega \in H^0(X,{\Omega }_X^n)$. By Yau's theorem (Yau 1978 Comm. Pure Appl. Math. 31), the Kähler form $\omega$ may be normalised so that the metric is Ricci-flat: $\mathrm{Ric}(\omega )=0$. The pair $(\omega ,\Omega )$ determines and is determined by the Ricci-flat Kähler structure together with a phase choice for $\Omega$.

Definition (special Lagrangian submanifold). A real $n$-dimensional submanifold $L\subset X$ is special Lagrangian of phase $\theta \in \mathbb{R}/2\pi \mathbb{Z}$ if:

(i) $L$ is Lagrangian: the symplectic form $\omega$ restricts to zero on $L$, that is, $\omega {|}_L=0$;

(ii) $L$ is special: the imaginary part of the phase-rotated holomorphic volume form vanishes on $L$, that is, $\Im (e^{-i\theta }\Omega ){|}_L=0$, equivalently $\Re (e^{-i\theta }\Omega ){|}_L$ is the Riemannian volume form on $L$ (Harvey-Lawson 1982).

The special Lagrangian condition is the calibrated-geometry condition: $\Re (e^{-i\theta }\Omega )$ is a closed form whose restriction to any oriented tangent $n$-plane has absolute value at most one, and the calibration is saturated on $L$. Calibrated submanifolds are volume-minimising in their homology class (Harvey-Lawson 1982 Acta Math. 148).

Definition (special Lagrangian fibration). A special Lagrangian fibration of a Calabi-Yau $n$-fold $X$ is a map $\pi :X\to B$ to a real $n$-manifold $B$ such that:

(i) the generic fibre $F_b={\pi }^{-1}(b)$ for $b\in B$ is a smooth compact special Lagrangian submanifold of $X$ of phase $\theta (b)$;

(ii) the smooth fibres are diffeomorphic to a real $n$-torus $T^n$;

(iii) the fibration has a discriminant locus $\Delta \subset B$ of codimension at least two, on whose complement the fibration is smooth and the fibres are smooth tori, and on which the fibres degenerate (typically by torus pinching).

Definition (integral affine structure on the base). Over the smooth locus $B^{\circ }=B\setminus \Delta$, the lattice bundle $H^1(F_b,\mathbb{Z})$ of integer cohomology of the fibres varies covariantly. The dual lattice bundle $H_1(F_b,\mathbb{Z})$ defines a flat ${\mathbb{Z}}^n$-bundle on $B^{\circ }$ whose periods give local coordinates on $B^{\circ }$, equipping it with an integral affine structure: an atlas of charts with transition maps in $\mathrm{GL}(n,\mathbb{Z})⋉{\mathbb{R}}^n$. The Hitchin 1997 semi-flat construction (Hitchin 1997 Annali SNS Pisa 25) explicitly builds this structure.

Definition (T-duality of torus fibres). Given a torus fibration $\pi :X\to B^{\circ }$ with fibre $F_b=V_b/{\Lambda }_b$ for $V_b=H_1(F_b,\mathbb{R})$ and ${\Lambda }_b=H_1(F_b,\mathbb{Z})$, the T-dual fibration ${\pi }^{\vee }:X^{\vee }\to B^{\circ }$ has fibre $F_b^{\vee }=V_b^{\ast }/{\Lambda }_b^{\ast }$, the dual torus, where $V_b^{\ast }=\mathrm{Hom}(V_b,\mathbb{R})$ and ${\Lambda }_b^{\ast }=\mathrm{Hom}({\Lambda }_b,\mathbb{Z})$ is the dual lattice. The duality $F_b\leftrightarrow F_b^{\vee }$ is the SYZ formulation of T-duality on the fibre.

Definition (SYZ mirror pair). A pair $(X,X^{\vee })$ of Calabi-Yau $n$-folds is an SYZ mirror pair if:

(i) $X$ admits a special Lagrangian $T^n$-fibration $\pi :X\to B$ over a real $n$-manifold $B$ with integral affine structure away from a codimension-two discriminant locus $\Delta$;

(ii) $X^{\vee }$ admits a special Lagrangian $T^n$-fibration ${\pi }^{\vee }:X^{\vee }\to B$ over the same base $B$;

(iii) on the smooth locus $B^{\circ }=B\setminus \Delta$, the fibres of $\pi$ and ${\pi }^{\vee }$ are dual tori in the sense of the previous definition, with the mirror map $X⇢X^{\vee }$ given fibrewise by T-duality plus a B-field (a flat $U(1)$-connection) twist.

Counterexamples to common slips

• "The SYZ base is itself a Calabi-Yau." The base $B$ is a real $n$-manifold, not a complex manifold. It carries an integral affine structure on the smooth locus, not a complex structure. The Calabi-Yau structure lives on the total space $X$ of the fibration, not on the base.

• "A special Lagrangian submanifold is the same as a complex submanifold." The Lagrangian condition $\omega {|}_L=0$ already prevents $L$ from being complex (a complex submanifold of half-real-dimension would be a divisor satisfying $\omega {|}_L\ne 0$). Special Lagrangian and complex are orthogonal half-dimensional submanifold types of a Calabi-Yau, exchanged by mirror symmetry.

• "T-duality is a continuous deformation of the metric." T-duality is a discrete duality, not a deformation. On a circle of radius $r$, T-duality sends $r\mapsto 1/r$; on a torus, it acts on each fibre by inversion of the metric duality (it dualises the lattice via $\Lambda \to {\Lambda }^{\ast }=\mathrm{Hom}(\Lambda ,\mathbb{Z})$, not by continuous rotation).

Key theorem with proof [Intermediate+]

The Strominger-Yau-Zaslow conjecture is presently a conjecture in full generality. The most precise theorem proved in its direction is Hitchin's 1997 reformulation, plus McLean's 1998 moduli theorem: these supply the mathematical form of the SYZ statement and prove the smoothness of the moduli space of special Lagrangian submanifolds.

Theorem (Hitchin's reformulation of SYZ; Hitchin 1997 Annali SNS Pisa 25). Let $\pi :X\to B^{\circ }$ be a smooth special Lagrangian torus fibration of a Calabi-Yau $n$-fold $X$ over an open subset $B^{\circ }$ of a real $n$-manifold. There exists an integral affine structure on $B^{\circ }$ such that:

(i) the local periods of $\Omega$ along the homology basis of $F_b$ give integral affine coordinates on $B^{\circ }$;

(ii) the metric on $B^{\circ }$ inherited from the Kähler metric on $X$ is the semi-flat metric of Hitchin, of the form $g_B={\mathrm{Hess}}_g(\varphi )$ for a strictly convex function $\varphi :B^{\circ }\to \mathbb{R}$ in the integral affine coordinates;

(iii) the T-dual fibration ${\pi }^{\vee }:X^{\vee }\to B^{\circ }$, obtained by dualising each torus fibre as in the definition above, is the moduli space of pairs $(F_b,\mathcal{L})$ where $F_b$ is a smooth fibre and $\mathcal{L}$ is a flat unitary line bundle on $F_b$. The total space $X^{\vee }$ inherits a natural Kähler structure ${\omega }^{\vee }$ and a holomorphic volume form ${\Omega }^{\vee }$ making $(X^{\vee },{\omega }^{\vee },{\Omega }^{\vee })$ a Calabi-Yau $n$-fold of complex dimension $n$.

Proof. The proof has three steps: construction of the integral affine structure on $B^{\circ }$, identification of the metric as a Hessian metric, and construction of the dual fibration.

Step 1 (integral affine structure). Fix a base point $b_0\in B^{\circ }$ and a $\mathbb{Z}$-basis ${\gamma }_1,\dots ,{\gamma }_n$ of $H_1(F_{b_0},\mathbb{Z})$. By the Ehresmann fibration theorem applied to the smooth locus of $\pi$, this basis extends locally to a smoothly varying $\mathbb{Z}$-basis of $H_1(F_b,\mathbb{Z})$ for $b$ in a neighbourhood of $b_0$. Define coordinates $y_i:B^{\circ }\to \mathbb{R}$ by $$ y_i(b) := \int_{\gamma_i(b)} \Im(\Omega). $$ The closedness of $\Im (\Omega )$ ensures $y_i$ is locally well-defined modulo periods of $\Im (\Omega )$ over closed cycles; the special Lagrangian condition $\Im (\Omega ){|}_{F_b}=0$ ensures these periods vanish on the fibre, hence $y_i$ descends to a single-valued function on $B^{\circ }$.

Locally, $y_1,\dots ,y_n$ form a smooth coordinate system on $B^{\circ }$. Transition maps between local bases of $H_1(F_b,\mathbb{Z})$ lie in $\mathrm{GL}(n,\mathbb{Z})$, so the transition maps on the $y_i$ lie in $\mathrm{GL}(n,\mathbb{Z})⋉{\mathbb{R}}^n$: the integral affine atlas is constructed.

Step 2 (semi-flat metric). By Yau's theorem (Yau 1978), the Kähler form $\omega$ on $X$ may be chosen Ricci-flat. The restriction of the Kähler metric to the horizontal directions of the fibration (i.e., directions orthogonal to the fibres) gives a metric $g_B$ on $B^{\circ }$. Hitchin shows that in the integral affine coordinates $y_i$, the metric takes the form $$ g_B = \sum_{i, j} \phi_{ij}(y) , dy_i , dy_j, $$ where $\varphi :B^{\circ }\to \mathbb{R}$ is a strictly convex smooth function and ${\varphi }_{ij}={\partial }^2\varphi /\partial y_i\partial y_j$ is its Hessian. The semi-flat metric on the total space $X$ is determined by $\varphi$ together with the flat connection on the fibre bundle, and the Ricci-flatness condition $\mathrm{Ric}(\omega )=0$ reduces to the real Monge-Ampère equation $$ \det \left( \frac{\partial^2 \phi}{\partial y_i \partial y_j} \right) = \mathrm{const}, $$ the Hessian of $\varphi$ has constant determinant — a real Monge-Ampère equation on $B^{\circ }$.

Step 3 (dual fibration). The moduli space ${\tilde{X}}^{\vee }$ of pairs $(F_b,\mathcal{L})$ where $\mathcal{L}$ is a flat unitary line bundle on $F_b$ is naturally a torus bundle over $B^{\circ }$: the fibre over $b\in B^{\circ }$ is the moduli space of flat $U(1)$-connections on $F_b$ modulo gauge equivalence, isomorphic to $H^1(F_b,\mathbb{R})/H^1(F_b,\mathbb{Z})=V_b^{\ast }/{\Lambda }_b^{\ast }$, the dual torus. The total space ${\tilde{X}}^{\vee }$ carries a natural symplectic form from the Atiyah-Bott formula on the moduli of flat connections, and Hitchin shows it carries a Kähler structure ${\omega }^{\vee }$ and a holomorphic volume form ${\Omega }^{\vee }$ such that the natural projection to $B^{\circ }$ is a special Lagrangian fibration. Identifying $X^{\vee }:={\tilde{X}}^{\vee }$ gives the dual Calabi-Yau structure.

The semi-flat metric on $X^{\vee }$ is constructed from the Legendre dual of $\varphi$: setting $\check{\varphi }(y^{\vee })={\sum }_i{y}_i^{\vee }{y}_i-\varphi (y)$ where $y_i^{\vee }=\partial \varphi /\partial y_i$, the dual metric is $g_B^{\vee }={\mathrm{Hess}}_{\check{\varphi }}$ and the dual Monge-Ampère equation is $\det ({\check{\varphi }}_{ij})=\mathrm{const}$, which is the same equation under Legendre duality. This is the precise mathematical form of T-duality on the smooth locus. $\square$

Bridge. Hitchin's reformulation builds toward [04.12.08] dual intersection complex, where the integral affine base of a toric degeneration is identified with the SYZ base, and appears again in [04.12.09] Gross-Siebert reconstruction theorem, where the same Legendre-dual / Monge-Ampère pairing is the algebraic-geometric core of the mirror construction. The foundational reason that Hitchin's theorem identifies the moduli space of $(F_b,\mathcal{L})$ with the mirror is that the dualisation of a torus is the same operation as the moduli of flat $U(1)$-connections on it — a piece of classical Pontryagin duality made into a fibrewise construction. This is exactly the formulation that identifies the SYZ mirror $X^{\vee }$ with a moduli space of branes on $X$ (a point on $X^{\vee }$ corresponds to a flat-connection brane on the fibre of $X$, exactly as in the Strominger-Yau-Zaslow 1996 D-brane argument). The bridge from the physical SYZ heuristic to the mathematical statement runs through the Hitchin semi-flat construction; the bridge from Hitchin to Gross-Siebert is the replacement of the differential-geometric semi-flat metric by an algebraic-geometric toric degeneration whose dual intersection complex carries the same integral affine structure.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

The companion file lean/Codex/AlgGeom/Tropical/StromingerYauZaslow.lean records the SYZ mirror-pair predicate as a schematic structure together with the conjecture as a sorry-stubbed theorem. The Mathlib gap enumerated in the frontmatter — Calabi-Yau manifolds, Ricci-flat Kähler metrics from Yau's theorem, special Lagrangian submanifolds, integral affine manifolds with singularities — is substantial, and the formalisable kernel of SYZ at present is the combinatorial-affine side rather than the differential-geometric side.

The file declares four schematic types. First, a CalabiYau structure recording the complex dimension and a placeholder for the Ricci-flat Kähler datum. Second, a SpecialLagrangianFibration predicate over a CalabiYau, recording a base manifold and a fibre torus together with the special Lagrangian condition (currently a Prop placeholder). Third, an IntegralAffineStructure structure on a real manifold, with an atlas of charts and transition maps in $\mathrm{GL}(n,\mathbb{Z})⋉{\mathbb{R}}^n$, plus a discriminant locus of codimension at least two. Fourth, a MirrorPair predicate on a pair of Calabi-Yaus, asserting that both admit special Lagrangian torus fibrations over the same integral affine base with dual torus fibres in the sense of fibrewise T-duality.

The conjecture itself is recorded as theorem syz_conjecture with a sorry body: every mirror pair of Calabi-Yau manifolds (in the sense that they have matching Hodge numbers, mirror periods, equal Gromov-Witten invariants under the mirror map, etc.) admits structures making them a MirrorPair in the SYZ sense. The full proof is open in general — Gross-Wilson 2000 proves the K3 case, and partial results exist for elliptic Calabi-Yau threefolds and toric hypersurface Calabi-Yaus via Gross-Siebert toric degenerations [04.12.09], but the general SYZ statement remains conjectural.

The Lean file also records two auxiliary statements that are formalisable independently of the differential geometry: the Legendre-duality preservation of the real Monge-Ampère equation (proved straightforwardly from $\det (M^{-1})=1/\det (M)$ for invertible matrices) and the dimension count for special Lagrangian moduli (McLean's theorem statement as a sorry-stubbed proposition equating the moduli dimension with $b_1(L)$). These auxiliary results are the combinatorial kernel of the SYZ programme that admits Lean formalisation without first formalising Yau's theorem or McLean's analytic moduli theorem.

Advanced results [Master]

Historical: Strominger-Yau-Zaslow 1996 and the physical setting

The 1996 paper of Strominger, Yau, and Zaslow ^[SYZ1996], titled Mirror symmetry is T-duality, is short — seventeen pages — and entirely structured around a physical argument from supersymmetric brane charges. The paper appeared in Nuclear Physics B 479, the leading venue for high-energy theoretical physics, six years after Greene-Plesser 1990 ^{[GreenePlesser1990]} gave the first explicit construction of mirror Calabi-Yau pairs (the quintic and its $(\mathbb{Z}/5{)}^3$-orbifold mirror) and five years after Candelas-de la Ossa-Green-Parkes 1991 ^{[Candelas1991]} computed Gromov-Witten predictions on the quintic via the periods of its mirror. By 1994 Kontsevich had introduced homological mirror symmetry as a derived-category-of-coherent-sheaves equivalence; mirror symmetry was thus pulling on both physical (Type IIA / Type IIB string duality) and categorical (derived categories, Fukaya categories) descriptions, with no shared geometric structure underlying them.

The SYZ paper offers exactly that geometric structure. Strominger, Yau, and Zaslow argue from the BPS state-counting side of string theory. A D-brane on a Calabi-Yau threefold $X$ is, in Type IIA, a complex submanifold of $X$ (an A-brane is Lagrangian; a B-brane is holomorphic — Type IIA has B-branes wrapping holomorphic cycles). On the mirror Type IIB side, the dual brane on $X^{\vee }$ is a Lagrangian submanifold (Type IIB has A-branes wrapping Lagrangians). The simplest D-brane on the Type IIA side is a D0-brane, a point particle. Its moduli space — the set of positions of the point particle inside $X$ — is the entire Calabi-Yau $X$ itself. By mirror symmetry, this moduli space should equal the moduli space of the dual brane on $X^{\vee }$. That dual brane must therefore be a Lagrangian submanifold of $X^{\vee }$ whose moduli space is the Calabi-Yau $X$ itself.

Strominger, Yau, and Zaslow then ask: what Lagrangian submanifold of $X^{\vee }$ has moduli space equal to $X$? Their answer is a special Lagrangian torus $T^n\subset X^{\vee }$. The McLean theorem (still in preparation at the time, McLean 1998 ^[McLean1998]) had just been announced: the moduli space of special Lagrangian submanifolds in a Calabi-Yau $n$-fold is a smooth manifold of dimension $b_1(L)$. For $L=T^n$ the first Betti number is $n$, so the moduli space is $n$-real-dimensional. To reach $X$ — a $2n$-real-dimensional Calabi-Yau — one additionally takes the moduli space of pairs $(L,\mathcal{L})$, where $\mathcal{L}$ is a flat $U(1)$-connection on $L$ (additional $n$ real degrees of freedom). The total moduli space is $2n$-real-dimensional, matching $X$.

The argument concludes: $X$ is the moduli space of pairs $(L,\mathcal{L})$ for $L\subset X^{\vee }$ a special Lagrangian $T^n$ with flat connection. By symmetry, $X^{\vee }$ is the moduli space of pairs $(L^{\prime },{\mathcal{L}}^{\prime })$ for $L^{\prime }\subset X$ a special Lagrangian $T^n$ with flat connection. The two moduli problems are dual, and the duality is fibre-by-fibre T-duality. The mirror map sends a point $p\in X^{\vee }$ (encoding a Lagrangian $L_p\subset X$ with its flat connection) to a point $p^{\prime }\in X$ (encoding the dual flat connection on the dual torus). T-duality on each fibre torus is the precise statement of the mirror map.

Hitchin's mathematical reformulation, 1997

Hitchin 1997 ^{[Hitchin1997]}, appearing one year after SYZ in Annali Scuola Normale Superiore Pisa 25, gave the SYZ proposal its first mathematical form. Hitchin's contribution was the explicit construction of the semi-flat metric on the moduli space of $(L,\mathcal{L})$ and the identification of the integral affine structure on the SYZ base.

Theorem 1 (Hitchin's semi-flat construction; Hitchin 1997). Let $\pi :X\to B^{\circ }$ be a smooth special Lagrangian $T^n$-fibration of a Calabi-Yau $n$-fold over an open subset of a real $n$-manifold. Then $B^{\circ }$ carries a canonical integral affine structure: an atlas with transition maps in $\mathrm{GL}(n,\mathbb{Z})⋉{\mathbb{R}}^n$, built from the lattice bundle $H_1(F_b,\mathbb{Z})$ over $B^{\circ }$. The Kähler metric on $X$ descends to a Hessian metric $g_B={\mathrm{Hess}}_{\varphi }$ on $B^{\circ }$ in the integral affine coordinates, with Ricci-flatness translating to the real Monge-Ampère equation $\det ({\varphi }_{ij})=\mathrm{const}$.

Proof outline. Step 1 (affine coordinates from periods): the special Lagrangian condition $\Im (\Omega ){|}_F=0$ ensures that the local integrals $y_i(b)={\int }_{{\gamma }_i(b)}\Im (\Omega )$ are single-valued functions on $B^{\circ }$ for ${\gamma }_i$ a local $\mathbb{Z}$-basis of $H_1(F_b,\mathbb{Z})$. Transition maps between local bases lie in $\mathrm{GL}(n,\mathbb{Z})$, producing the integral affine atlas. Step 2 (Hessian metric): the Kähler metric, restricted to horizontal directions of the fibration, has a closed form in the affine coordinates given by the second-derivative matrix of a convex potential $\varphi$ — proved by direct computation in the local frame. Step 3 (Monge-Ampère): Ricci-flatness on the total space, reduced to the base via the semi-flat ansatz, is equivalent to $\det ({\varphi }_{ij})=\mathrm{const}$, a real Monge-Ampère equation. The detailed computation is in Hitchin 1997, with the corrected formulation accounting for the U(1)-bundle Chern class in Hitchin 2001 Quart. J. Math. 52. $\square$

Significance. Hitchin's semi-flat construction is the canonical example of a tropical limit: the integral affine structure on $B^{\circ }$ is the tropical-geometric shadow of the Calabi-Yau total space, and the real Monge-Ampère equation is the tropical limit of the complex Monge-Ampère equation that Yau 1978 solved on Calabi-Yau manifolds. This is the foundational reason the SYZ programme connects directly to tropical geometry and to the Gross-Siebert algebraic programme.

Modern: Gross-Wilson K3, the first SYZ proof

Gross-Wilson 2000 J. Diff. Geom. 55 ^{[GrossWilson2000]} proved SYZ in dimension two. The paper studies the Ricci-flat Kähler-Einstein metric on a polarised K3 surface near a large complex structure limit (a one-parameter family of K3s with maximally unipotent monodromy at $t=0$).

Theorem 2 (Gross-Wilson 2000). Let $\mathcal{X}\to D$ be a one-parameter family of polarised K3 surfaces with maximally unipotent monodromy at $t=0$. The Ricci-flat Kähler-Einstein metric on $X_t$ for $t\to 0$ Gromov-Hausdorff converges (after rescaling the Kähler class) to a McLean-Ricci-flat metric on the topological two-sphere $S^2\setminus \{p_1,\dots ,p_{24}\}$, with $24$ punctures encoding the discriminant locus of the SYZ fibration $X_t\to S^2$. The metric is the semi-flat Hitchin metric in integral affine coordinates around each smooth point, with the punctures being the codimension-two singularities (each a "focus-focus" singularity in Joyce's classification) where the torus fibre $T^2$ pinches a one-cycle to a point.

Significance. This is the first complete proof of an SYZ statement, and it establishes the prototype for all subsequent analytic approaches to SYZ. The proof is heavily analytic: it uses Yau's solution of the complex Monge-Ampère equation, plus delicate estimates on the Ricci-flat metric in the collapsing limit. The K3 case is self-mirror — the mirror of a K3 is another K3 — and the SYZ fibration realises this self-mirror duality as fibre-by-fibre T-duality on the punctured two-sphere base.

Non-archimedean: Kontsevich-Soibelman, the analytic skeleton

Kontsevich-Soibelman 2001 ^{[KontsevichSoibelman2001]} and 2006 ^{[KontsevichSoibelman2006]} proposed an alternative reformulation: replace the differential-geometric SYZ statement (Ricci-flat metric, special Lagrangian fibration) with a purely algebraic-geometric statement about the Berkovich analytic skeleton of a degeneration.

Theorem 3 (Kontsevich-Soibelman non-archimedean SYZ). Let $X$ be a Calabi-Yau over the non-archimedean field $K=\mathbb{C}\{\{t\}\}$ with a Calabi-Yau snc (simple normal crossings) model $\mathcal{X}/{\mathcal{O}}_K$. The Berkovich analytic space $X^{\mathrm{an}}$ admits a canonical retraction $r:X^{\mathrm{an}}\to \mathrm{Sk}(X)$ onto a finite simplicial complex $\mathrm{Sk}(X)$ — the analytic skeleton of $X$. The skeleton $\mathrm{Sk}(X)$ is a real $n$-manifold (with singularities of codimension at least two) carrying a canonical integral affine structure induced from the toric structure of each codimension-zero stratum of the special fibre ${\mathcal{X}}_0$.

Non-archimedean SYZ. Kontsevich-Soibelman propose: the SYZ mirror $X^{\vee }$ of $X$ is the non-archimedean Calabi-Yau whose analytic skeleton $\mathrm{Sk}(X^{\vee })$ is identified with $\mathrm{Sk}(X)$ as a topological manifold, but with the Legendre-dual integral affine structure. The mirror correspondence is realised by Legendre duality on the affine structure of the skeleton — the same Legendre duality appearing in the Hitchin / semi-flat metric reformulation.

Significance. The non-archimedean approach side-steps Yau's theorem and the analytic difficulties of Ricci-flat metrics, replacing them with algebraic-geometric data on the skeleton. The skeleton is a finite combinatorial-geometric object; the integral affine structure is induced from the lattice structure of the toric strata. Kontsevich-Soibelman 2006 also develop the scattering diagrams on the affine base — combinatorial data encoding wall-crossing of holomorphic curves — that feed into the Gross-Siebert programme [04.12.09].

Algebraic: Gross-Siebert, mirror symmetry via toric degenerations

Gross-Siebert 2006 J. Algebraic Geom. 15 ^{[GrossSiebert2006]} and 2011 Ann. of Math. 174 ^{[GrossSiebert2011]} develop a parallel algebraic-geometric reformulation of SYZ, which constitutes a constructive proof in many cases.

Theorem 4 (Gross-Siebert reconstruction; statement-level). Given an integral affine manifold $B$ with codimension-two singularities, a polyhedral decomposition $\mathcal{P}$, and a structure $\mathcal{S}$ of slab functions on $B$, there exists a one-parameter family $\mathcal{X}\to \mathrm{Spec}\mathbb{C}\{\{t\}\}$ of Calabi-Yau varieties whose special fibre ${\mathcal{X}}_0$ is a degenerate toric Calabi-Yau realising the polyhedral subdivision $\mathcal{P}$, and whose dual intersection complex [04.12.08] is $(B,\mathcal{P})$. The mirror Calabi-Yau is constructed from the Legendre-dual integral affine structure on $B$.

Significance. The Gross-Siebert programme realises the Kontsevich-Soibelman vision in algebraic geometry: the SYZ base $B$ is identified with the dual intersection complex of a toric degeneration, the integral affine structure with the affine structure on the dual complex, and the mirror with the Legendre-dual reconstruction. This is the algebraic SYZ identification: the SYZ base of a differential-geometric Calabi-Yau is the same combinatorial object as the dual intersection complex of an algebraic-geometric toric degeneration. The identification (highlighted in TGMS, the source-anchor monograph) is the cornerstone of the Gross-Siebert programme as a constructive mirror-symmetry theorem.

Singularities of special Lagrangian fibrations: Joyce's programme

Joyce 2003 Comm. Anal. Geom. 11 ^[Joyce2003] gave the most careful analysis of the singular fibres of a special Lagrangian fibration of a Calabi-Yau threefold.

Theorem 5 (Joyce 2003). Generic singular fibres of a special Lagrangian fibration of a Calabi-Yau threefold over a three-real-dimensional base $B$ have codimension-two-stratum singularities classified by a finite list of model singularities (focus-focus, conifold, $T^2$-pinch). The discriminant locus $\Delta \subset B$ is generically a real one-dimensional graph (with codimension-three vertices). The SYZ conjecture in its original Strominger-Yau-Zaslow form needs revision: smooth special Lagrangian $T^3$-fibrations do not exist in the strict sense on a generic Calabi-Yau threefold; one must allow discrepancy at the codimension-two strata, classifying these strata into Joyce's model singularities.

Significance. Joyce's analysis identifies the technical content of the SYZ conjecture beyond the smooth-locus statement: the singularities of the special Lagrangian fibration are not freely chosen but constrained by the differential geometry of special Lagrangians, and a careful classification of these singularities is necessary for any analytic proof of SYZ. Joyce's work is the foundation of all later analytic-SYZ programmes (Auroux 2007 ^[Auroux2007], Tosatti-Zhang 2019, Li 2020).

Beyond Calabi-Yau: Auroux and SYZ with Lagrangian wall-crossing

Auroux 2007 Surveys in Differential Geometry 13 ^[Auroux2007] extends the SYZ proposal beyond Calabi-Yaus to Fano manifolds (positive-curvature Kähler) by allowing Lagrangian wall-crossing.

Theorem 6 (Auroux 2007, schematic). For a Fano variety $X$ equipped with an anticanonical divisor $D$, the complement $X\setminus D$ admits Lagrangian torus fibrations with singularities, and the SYZ-style mirror $X^{\vee }$ is constructed as a Landau-Ginzburg model — a pair $(X^{\vee },W^{\vee })$ of a complex manifold and a superpotential. The mirror map involves wall-crossing transformations of the Lagrangian fibres along walls in the base where holomorphic discs change their boundary classes.

Significance. Auroux's framework bridges the SYZ programme (originally for Calabi-Yaus only) with the broader mirror-symmetry programme for Fano varieties and Landau-Ginzburg models. The wall-crossing structure that Auroux makes precise on the symplectic side matches the scattering-diagram structure of Kontsevich-Soibelman 2006 and the slab-function structure of Gross-Siebert on the algebraic side. The unification of these three frameworks — Hitchin-Gross-Wilson differential geometric, Kontsevich-Soibelman non-archimedean, and Gross-Siebert algebraic — into a single mirror-symmetry picture is the current state of the art.

Synthesis. The Strominger-Yau-Zaslow conjecture is the foundational reason mirror symmetry has a geometric explanation: the central insight is that mirror Calabi-Yau pairs share a common combinatorial base $B$ — a real $n$-manifold with integral affine structure — and the mirror map is fibrewise T-duality on the special Lagrangian $T^n$-fibres. This identifies the moduli of complex structures on $X$ (the "B-side" data) with the moduli of symplectic structures on $X^{\vee }$ (the "A-side" data) via the moduli of Lagrangian branes on each side, putting these together as Legendre duality on a single affine base.

Putting these together with the McLean dimension theorem and the Hitchin semi-flat construction, the SYZ statement is exactly the assertion that the moduli of $(L,\mathcal{L})$ on $X^{\vee }$ recovers $X$ — and the bridge is the integral affine structure on the smooth locus of the base, which both sides share. The pattern generalises along three substantive axes: to the K3 case (Gross-Wilson 2000, the first complete proof), to the non-archimedean setting (Kontsevich-Soibelman 2001/2006, where the base is the analytic skeleton), and to the algebraic setting (Gross-Siebert 2006/2011, where the base is the dual intersection complex of a toric degeneration [04.12.08]). Each of these reformulations preserves the foundational structure — integral affine base, dual tori, Legendre duality — while replacing the analytically-difficult differential-geometric content with a more tractable algebraic-geometric or non-archimedean substitute.

This is exactly the structural identification that makes the Gross-Siebert programme a constructive mirror symmetry: the algebraic SYZ base is built combinatorially from a polyhedral subdivision of a polytope, the slab-function structure encodes the scattering / wall-crossing data, and the mirror is built order-by-order in the smoothing parameter $t$ via the reconstruction theorem [04.12.09]. The bridge from the original 1996 physical heuristic to the 2010s algebraic-geometric construction runs through the integral affine manifold $B$ that all three reformulations share. The pattern recurs through the entire tropical-and-mirror-symmetry programme: the integral affine base of a toric Calabi-Yau degeneration is the SYZ base; the tropical curves on this base [04.12.05]-[04.12.06] are the enumerative-geometric data; and the smoothing of the toric central fibre is the Gross-Siebert mirror reconstruction.

Full proof set [Master]

Proposition 7 (Legendre duality preserves the real Monge-Ampère equation). Let $\varphi :U\to \mathbb{R}$ be a smooth strictly convex function on an open subset $U\subseteq {\mathbb{R}}^n$ with $\det ({\varphi }_{ij})=C$ for a positive constant $C$. The Legendre dual function $\check{\varphi }$ on the image $U^{\vee }=\nabla \varphi (U)\subseteq {\mathbb{R}}^n$ satisfies $\det ({\check{\varphi }}_{ij})=1/C$.

Proof. The Legendre dual is $\check{\varphi }(y^{\vee })={\sum }_i{y}_i{y}_i^{\vee }-\varphi (y)$, where $y_i^{\vee }=\partial \varphi /\partial y_i$. Compute the partial derivatives of $\check{\varphi }$ using the chain rule on $y\to y^{\vee }$: $$ \frac{\partial \check{\phi}}{\partial y_j^\vee} = y_j + \sum_i y_i^\vee \frac{\partial y_i}{\partial y_j^\vee} - \sum_i \frac{\partial \phi}{\partial y_i} \frac{\partial y_i}{\partial y_j^\vee} = y_j, $$ using $y_i^{\vee }=\partial \varphi /\partial y_i$ to cancel the second and third terms. Hence the Hessian of $\check{\varphi }$ is $$ \frac{\partial^2 \check{\phi}}{\partial y_j^\vee \partial y_k^\vee} = \frac{\partial y_j}{\partial y_k^\vee}, $$ which by the inverse function theorem is the matrix inverse of the Jacobian $\partial y_k^{\vee }/\partial y_j={\partial }^2\varphi /\partial y_j\partial y_k={\varphi }_{jk}$. So $({\check{\varphi }}_{jk})=({\varphi }_{jk}{)}^{-1}$, and $$ \det(\check{\phi}{jk}) = \frac{1}{\det(\phi{jk})} = \frac{1}{C}. \qquad \square $$

Proposition 8 (special Lagrangian condition on a calibrated plane in ${\mathbb{C}}^n$). Let $L\subset {\mathbb{C}}^n$ be a real $n$-plane and $\Omega =d{z}_1\wedge \cdots \wedge d{z}_n$ the standard holomorphic volume form. Then $L$ is special Lagrangian of phase $\theta$ if and only if $L=e^{i\theta /n}U({\mathbb{R}}^n)$ for some $U\in \mathrm{SU}(n)$, i.e., $L$ is the image of ${\mathbb{R}}^n$ under a unitary rotation of determinant $e^{i\theta }$.

Proof. The set of Lagrangian planes through the origin in ${\mathbb{C}}^n$ is the Lagrangian Grassmannian $\mathrm{LG}(n)=U(n)/O(n)$, of real dimension $n(n+1)/2$. A Lagrangian plane $L=U({\mathbb{R}}^n)$ for $U\in U(n)$ has $\Omega {|}_L=\det (U)\cdot {\mathrm{vol}}_{{\mathbb{R}}^n}$, where ${\mathrm{vol}}_{{\mathbb{R}}^n}=d{x}_1\wedge \cdots \wedge d{x}_n$ is the standard real volume form. (This is a direct computation: pulling back $\Omega$ by $U$ scales by $\det (U)$.)

Hence $\Im (e^{-i\theta }\Omega ){|}_L=\Im (e^{-i\theta }\det (U))\cdot {\mathrm{vol}}_{{\mathbb{R}}^n}$, which vanishes iff $e^{-i\theta }\det (U)\in \mathbb{R}$, i.e., $\det (U)=e^{i\theta }$. Decomposing $U=e^{i\theta /n}V$ with $V\in \mathrm{SU}(n)$ gives the claimed parametrisation. The special Lagrangian condition is thus parametrised by the special Lagrangian Grassmannian $\mathrm{SLG}(n,\theta )=SU(n)/SO(n)$ of dimension $\dim \mathrm{LG}(n)-1=n(n+1)/2-1$. $\square$

Proposition 9 (T-dual torus from flat-connection moduli). Let $F=V/\Lambda$ be a torus with $V$ a real vector space of dimension $n$ and $\Lambda \subset V$ a full-rank lattice. The moduli space of flat $U(1)$-connections on $F$ modulo gauge equivalence is canonically isomorphic to the dual torus $F^\vee = V^/\Lambda^$,where$V^ = \mathrm{Hom}(V, \mathbb{R})$and$\Lambda^* = \mathrm{Hom}(\Lambda, \mathbb{Z}) \subset V^$.

Proof. A flat $U(1)$-connection on $F$ is a one-form $A\in {\Omega }^1(F)$ with $dA=0$, modulo gauge transformations $A\to A+d\chi$ for $\chi :F\to U(1)$ a smooth function. Since $F$ is a torus, $H^1(F,\mathbb{R})=V^{\ast }$ — the space of closed one-forms modulo exact ones is exactly the dual vector space.

The moduli of gauge equivalence classes is then $V^{\ast }/\{\text{gauge transformations}\}$, and the gauge transformations $\chi :F\to U(1)$ are classified by their winding numbers around the cycles $\gamma \in H_1(F,\mathbb{Z})=\Lambda$. Each gauge transformation contributes an integer-valued period ${\int }_{\gamma }d\chi \in 2\pi \mathbb{Z}$ to the period of $A$, hence modifies $A$ by an element of ${\Lambda }^{\ast }\subset V^{\ast }$ (the dual lattice). The moduli space is therefore $V^{\ast }/{\Lambda }^{\ast }=F^{\vee }$, the dual torus. $\square$

Connections [Master]

• Lagrangian submanifold 05.05.01. The geometric backbone of the SYZ statement: a special Lagrangian fibre is first a Lagrangian submanifold, with the symplectic-form-vanishing condition $\omega {|}_L=0$ supplemented by the special calibration condition $\Im (e^{-i\theta }\Omega ){|}_L=0$. The Lagrangian-submanifold framework is the entry point to the SYZ programme — a Calabi-Yau threefold's SYZ fibration is a special class of Lagrangian fibration, and the moduli of Lagrangians with flat connection is the symplectic-side mirror data. The bridge from [05.05.01] to SYZ is the calibration condition that selects special Lagrangians from general Lagrangians and the requirement that they fibre over a real base of complementary dimension.

• Toric degeneration of a Calabi-Yau variety 04.12.07. The algebraic-geometric incarnation of SYZ via the Gross-Siebert programme: a toric degeneration $\mathcal{X}\to \mathrm{Spec}\mathbb{C}\{\{t\}\}$ of a Calabi-Yau provides an algebraic substitute for the SYZ fibration, with the dual intersection complex of the central fibre playing the role of the SYZ base. The SYZ identification — that the differential-geometric base of a special Lagrangian fibration equals the dual intersection complex of a toric degeneration — is the cornerstone of the Gross-Siebert programme as a constructive proof of mirror symmetry, and a toric degeneration as in [04.12.07] is the algebraic-geometric carrier of this identification.

• Dual intersection complex (B, P) of a toric degeneration 04.12.08. The combinatorial-affine object identified with the SYZ base in the algebraic setting: the dual intersection complex $(B,\mathcal{P})$ of a Calabi-Yau toric degeneration is a real $n$-manifold with integral affine structure on the complement of a codimension-two discriminant locus. The SYZ base of a smooth special Lagrangian fibration is — by the Gross-Siebert / Kontsevich-Soibelman identification — the same combinatorial object as the dual intersection complex. The reciprocal cross-pointer from [04.12.08] back to the SYZ unit is the bridge from the algebraic geometry to the differential geometry of the Calabi-Yau.

• Gross-Siebert reconstruction theorem (statement) 04.12.09. The constructive mirror-symmetry theorem built directly on the SYZ identification: given an integral affine manifold $B$ with singularities plus a structure $\mathcal{S}$ of slab functions and a polyhedral subdivision $\mathcal{P}$, Gross-Siebert reconstruct a smooth Calabi-Yau and its mirror as deformations of a toric central fibre, order by order in the smoothing parameter $t$. The reconstruction theorem [04.12.09] is the algebraic-geometric realisation of the SYZ conjecture for the broad class of Calabi-Yaus admitting toric degenerations, and the SYZ base $B$ is its input data.

• Mikhalkin's correspondence theorem 04.12.05. The enumerative-geometric prototype of the tropical / SYZ paradigm: Mikhalkin 2005 shows the count of complex algebraic curves on a toric surface equals the count of tropical curves on the corresponding integral affine base, with both counts matching exactly. The SYZ conjecture is the structural ancestor of this correspondence — a Calabi-Yau pair's mirror duality is mediated by the same kind of combinatorial-affine base on which Mikhalkin's tropical curves are enumerated, and the Gromov-Witten invariants on $X$ predicted by SYZ are computed combinatorially via tropical curves on the SYZ base.

• Nishinou-Siebert correspondence 04.12.06. The higher-dimensional tropical-curve correspondence on which the Gross-Siebert mirror-symmetry programme builds: Nishinou-Siebert 2006 extends Mikhalkin's enumeration to toric varieties of arbitrary dimension via toric degenerations, and the tropical curves on the dual intersection complex are the wall-crossing data assembled by Gross-Siebert into the mirror Calabi-Yau. The Nishinou-Siebert correspondence is the enumerative-geometric input that converts the SYZ identification into a constructive mirror-symmetry algorithm.

• Tropical curve as balanced rational metric graph 04.12.02. The combinatorial object enumerated on the SYZ base in the tropical-curve interpretation of mirror symmetry: balanced rational metric graphs on the integral affine base $B$ are precisely the tropical curves that compute Gromov-Witten invariants of the Calabi-Yau via the SYZ identification. The reciprocal cross-pointer is to the SYZ unit, where these tropical curves are interpreted as the perturbative wall-crossing data on the mirror side.

• Period integral and the mirror map (pointer) 04.12.13. Under the SYZ identification, the mirror map is the symplectic-area-to-complex-coordinate change-of-variable on the integral affine base: the symplectic Kähler parameter on the A-side equals (asymptotically) the complex-structure parameter on the B-side, via the period asymptotics at the large-complex-structure limit. The pointer unit [04.12.13] records the quantitative form of this identification — the mirror map as the canonical-coordinate exponential of period ratios — that the SYZ conjecture predicts heuristically.

• Log Gromov-Witten invariants (pointer) 04.12.15. Log Gromov-Witten counts are the algebraic-enumerative shadow of the symplectic disk-counts of the SYZ programme; the Gross-Siebert / Gross-Hacking-Keel / Mandel-Ruddat constructions are the algebraic realisations of the SYZ heuristic, all of which consume log GW data as their enumerative input. The pointer unit [04.12.15] records the log GW infrastructure that the SYZ programme requires for a precise mirror-symmetry statement.

Historical & philosophical context [Master]

Strominger, Yau, and Zaslow 1996 Nuclear Physics B 479 ^[SYZ1996] proposed the SYZ conjecture in a short physics-style paper, written in response to the puzzle posed by Greene-Plesser 1990 Nuclear Physics B 338 ^{[GreenePlesser1990]} (the first explicit mirror Calabi-Yau pair, the quintic and its orbifold mirror) and Candelas-de la Ossa-Green-Parkes 1991 Nuclear Physics B 359 ^{[Candelas1991]} (the first quantitative mirror-symmetry computation of Gromov-Witten invariants of the quintic via the periods of its mirror). The 1996 paper distilled the physical heuristic into a precise geometric statement: mirror Calabi-Yau pairs are built as dual special Lagrangian torus fibrations over a common base, with the mirror map carried by fibrewise T-duality. The argument relies on the BPS state-counting of D-branes in Type IIA / Type IIB string theory and on McLean's then-unpublished 1998 Communications in Analysis and Geometry 6 ^[McLean1998] theorem on the moduli of special Lagrangians.

Hitchin 1997 Annali Scuola Normale Superiore Pisa 25 ^{[Hitchin1997]} gave the SYZ proposal its first mathematical formulation, constructing the semi-flat metric on the base and identifying the integral affine structure. Gross-Wilson 2000 Journal of Differential Geometry 55 ^{[GrossWilson2000]} proved SYZ for K3 surfaces, the first complete proof in any dimension. Kontsevich-Soibelman 2001 ^{[KontsevichSoibelman2001]} and 2006 ^{[KontsevichSoibelman2006]} reformulated SYZ in non-archimedean analytic geometry via the Berkovich skeleton, opening the algebraic-geometric path that Gross-Siebert 2006 Journal of Algebraic Geometry 15 ^{[GrossSiebert2006]} and 2011 Annals of Mathematics 174 ^{[GrossSiebert2011]} developed into a constructive mirror-symmetry programme via toric degenerations. Joyce 2003 Communications in Analysis and Geometry 11 ^[Joyce2003] analysed the singularities of special Lagrangian fibrations, providing the technical framework for analytic SYZ in dimension three. Auroux 2007 Surveys in Differential Geometry 13 ^[Auroux2007] extended the SYZ picture beyond Calabi-Yaus to Fano targets via Lagrangian wall-crossing and Landau-Ginzburg mirrors. The textbook treatment is Gross 2011 Tropical Geometry and Mirror Symmetry (CBMS 114) ^[Gross2011] Chapter 6.

The SYZ conjecture is the conceptual centre of three decades of mirror-symmetry mathematics. Its physical heuristic — D-brane charges and T-duality on torus fibres — gives a reason mirror symmetry should exist, complementing the categorical reason (Kontsevich's homological mirror symmetry, 1994) and the enumerative reason (Candelas-de la Ossa-Green-Parkes 1991 period computations). The mathematical formulations of SYZ — Hitchin semi-flat, Kontsevich-Soibelman non-archimedean, Gross-Siebert algebraic — give constructions of mirror pairs from combinatorial-affine data. The unifying object across all formulations is the integral affine manifold $B$ with codimension-two singularities, and the SYZ programme is the search for the most natural realisation of $B$ for any given mirror pair. The conjecture remains open in its original Strominger-Yau-Zaslow form for general Calabi-Yau threefolds; it is proved by Gross-Wilson 2000 for K3, by Gross-Siebert in many algebraic-geometric cases, and partially by Joyce, Auroux, Tosatti-Zhang, and others in analytic and symplectic-geometric formulations.

Bibliography [Master]

@article{SYZ1996,
  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},
  note    = {arXiv:hep-th/9606040}
}

@article{Hitchin1997,
  author  = {Hitchin, Nigel},
  title   = {The moduli space of special {L}agrangian submanifolds},
  journal = {Annali Scuola Normale Superiore Pisa},
  volume  = {25},
  year    = {1997},
  pages   = {503--515}
}

@article{McLean1998,
  author  = {McLean, Robert},
  title   = {Deformations of calibrated submanifolds},
  journal = {Communications in Analysis and Geometry},
  volume  = {6},
  year    = {1998},
  pages   = {705--747}
}

@article{GrossWilson2000,
  author  = {Gross, Mark and Wilson, P. M. H.},
  title   = {Large complex structure limits of {K3} surfaces},
  journal = {Journal of Differential Geometry},
  volume  = {55},
  year    = {2000},
  pages   = {475--546}
}

@incollection{KontsevichSoibelman2001,
  author    = {Kontsevich, Maxim and Soibelman, Yan},
  title     = {Homological mirror symmetry and torus fibrations},
  booktitle = {Symplectic Geometry and Mirror Symmetry ({S}eoul, 2000)},
  publisher = {World Scientific},
  year      = {2001},
  pages     = {203--263}
}

@incollection{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{\"a}user},
  year      = {2006},
  pages     = {321--385}
}

@article{Joyce2003,
  author  = {Joyce, Dominic},
  title   = {Singularities of special {L}agrangian fibrations and the {SYZ} conjecture},
  journal = {Communications in Analysis and Geometry},
  volume  = {11},
  year    = {2003},
  pages   = {859--907}
}

@article{GrossSiebert2006,
  author  = {Gross, Mark and Siebert, Bernd},
  title   = {Mirror symmetry via logarithmic degeneration data {I}},
  journal = {Journal of Algebraic Geometry},
  volume  = {15},
  year    = {2006},
  pages   = {451--552}
}

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

@article{Yau1978,
  author  = {Yau, Shing-Tung},
  title   = {On the {R}icci curvature of a compact {K}{\"a}hler manifold and the complex {M}onge-{A}mp{\`e}re equation {I}},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {31},
  year    = {1978},
  pages   = {339--411}
}

@article{Auroux2007,
  author  = {Auroux, Denis},
  title   = {Special {L}agrangian fibrations, wall-crossing, and mirror symmetry},
  journal = {Surveys in Differential Geometry},
  volume  = {13},
  year    = {2009},
  pages   = {1--47}
}

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

@article{GreenePlesser1990,
  author  = {Greene, Brian R. and Plesser, M. Ronen},
  title   = {Duality in {C}alabi-{Y}au moduli space},
  journal = {Nuclear Physics B},
  volume  = {338},
  year    = {1990},
  pages   = {15--37}
}

@article{Candelas1991,
  author  = {Candelas, Philip and de la Ossa, Xenia and Green, Paul S. and Parkes, Linda},
  title   = {A pair of {C}alabi-{Y}au manifolds as an exactly soluble superconformal theory},
  journal = {Nuclear Physics B},
  volume  = {359},
  year    = {1991},
  pages   = {21--74}
}

@article{HarveyLawson1982,
  author  = {Harvey, Reese and Lawson, H. Blaine},
  title   = {Calibrated geometries},
  journal = {Acta Mathematica},
  volume  = {148},
  year    = {1982},
  pages   = {47--157}
}