SYZ conjecture

The SYZ conjecture interprets mirror symmetry as a geometric operation on fibrations of Calabi–Yau manifolds. A Calabi–Yau n-fold is expected, in suitable circumstances, to admit a fibration over a real n-dimensional base whose regular fibers are special Lagrangian n-tori. The mirror is then obtained by replacing each torus fiber with its dual torus, whose points represent flat unitary line bundles on the original fiber. This fiberwise dualization is a higher-dimensional analogue of T-duality and is intended to exchange the complex and symplectic structures that mirror symmetry pairs. The proposal is especially influential for Calabi–Yau threefolds, where it gives a concrete picture of why enumerative geometry on one manifold can correspond to periods and complex geometry on another.

The geometric mechanism has three ingredients: special Lagrangian fibers, a discriminant locus, and dual torus geometry. A special Lagrangian submanifold is Lagrangian for the symplectic form and calibrated by the real part of the holomorphic volume form; consequently, it is minimal and carries a controlled phase. Over the regular part of the base, the fibration behaves like a torus bundle, and fiberwise dualization produces the candidate mirror. Singular fibers cannot simply be dualized in isolation, so the base is naturally divided into a smooth affine region and a discriminant locus where the fibration degenerates.¹ Monodromy around that locus records how cycles in the torus change and helps determine the topology and complex structure of the mirror.

The conjecture has guided constructions and proofs in restricted settings rather than becoming a single general theorem. Work on special Lagrangian fibrations made the topology and affine geometry of the base central to the subject, while later approaches related torus fibrations to homological mirror symmetry and algebraic reconstruction.¹² The naive dual-torus space is usually incomplete: holomorphic disks with boundary on fibers produce instanton corrections that alter the gluing of local mirror charts. These corrections are essential for recovering the correct complex manifold and for matching disk counts with mirror coordinates.³ The Gross–Siebert program reformulates much of this picture using integral affine manifolds, polyhedral decompositions, and tropical geometry, providing an algebraic route to mirror construction.⁴

The conjecture is not a claim that every Calabi–Yau manifold possesses a globally smooth special Lagrangian torus fibration. Such fibrations are difficult to construct, and singularities are part of the expected structure rather than technical accidents. The earliest geometric tests often focused on K3 surfaces, where elliptic fibrations and their degenerations offer a lower-dimensional model for the threefold picture. Another subtlety is that the mirror is not always obtained by ordinary pointwise T-duality: affine monodromy, singular fibers, and disk corrections can change the topology and gluing data. The conjecture therefore functions both as a geometric explanation of mirror symmetry and as a program linking symplectic geometry, algebraic geometry, Fukaya categories, and tropical methods, even when a literal fibration remains unavailable.

Calabi–Yau manifold

A complex manifold with a Ricci-flat Kähler metric in the standard compact setting, together with a holomorphic volume form; Calabi–Yau threefolds are central examples in mirror symmetry.

Special Lagrangian

A Lagrangian submanifold calibrated by a suitable phase of the holomorphic volume form, making it volume-minimizing in its homology class.

T-duality

A duality that exchanges a torus with its moduli space of flat line bundles; in the SYZ picture it is applied fiberwise.

Discriminant locus

The subset of the fibration base over which the torus fibers become singular or otherwise fail to be regular.

Instanton correction

A modification of naive mirror gluing produced by holomorphic disks or related curve-counting data.