SYZ conjecture
The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled "Mirror Symmetry is T-duality".
The SYZ conjecture uses this fact to realize mirror symmetry.
It starts from considering BPS states of type IIA theories compactified on X, especially 0-branes that have moduli space X.
It is known that all of the BPS states of type IIB theories compactified on Y are 3-branes.
By considering supersymmetric conditions, it has been shown that these 3-branes should be special Lagrangian submanifolds.
The initial proposal of the SYZ conjecture by Strominger, Yau, and Zaslow, was not given as a precise mathematical statement.
There is no agreed upon precise statement of the conjecture within the mathematical literature, but there is a general statement that is expected to be close to the correct formulation of the conjecture, which is presented here.
[4][5] This statement emphasizes the topological picture of mirror symmetry, but does not precisely characterise the relationship between the complex and symplectic structures of the mirror pairs, or make reference to the associated Riemannian metrics involved.
so that there is no singular locus is called the semi-flat limit of the SYZ conjecture, and is often used as a model situation to describe torus fibrations.
The SYZ conjecture can be shown to hold in some simple cases of semi-flat limits, for example given by Abelian varieties and K3 surfaces which are fibred by elliptic curves.
It is expected that the correct formulation of the SYZ conjecture will differ somewhat from the statement above.
Mirror symmetry is also often phrased in terms of degenerating families of Calabi–Yau manifolds instead of for a single Calabi–Yau, and one might expect the SYZ conjecture to reformulated more precisely in this language.
[6] There should be a relationship between these three interpretations of mirror symmetry, but it is not yet known whether they should be equivalent or one proposal is stronger than the other.
[7] Nevertheless, in simple settings there are clear ways of relating the SYZ and HMS conjectures.
The key feature of HMS is that the conjecture relates objects (either submanifolds or sheaves) on mirror geometric spaces, so the required input to try to understand or prove the HMS conjecture includes a mirror pair of geometric spaces.
To relate the SYZ and HMS conjectures, it is convenient to work in the semi-flat limit.
The important geometric feature of a pair of Lagrangian torus fibrations
which encodes mirror symmetry is the dual torus fibres of the fibration.
has the important interpretation as the moduli space of line bundles on
There are two simple examples of this phenomenon: These two examples produce the most extreme kinds of coherent sheaf, locally free sheaves (of rank 1) and torsion sheaves supported on points.
As a simple example, a Lagrangian multisection (a union of k Lagrangian sections) should be mirror dual to a rank k vector bundle on the mirror manifold, but one must take care to account for instanton corrections by counting holomorphic discs which are bounded by the multisection, in the sense of Gromov-Witten theory.
In this way enumerative geometry becomes important for understanding how mirror symmetry interchanges dual objects.
By combining the geometry of mirror fibrations in the SYZ conjecture with a detailed understanding of enumerative invariants and the structure of the singular set of the base
, it is possible to use the geometry of the fibration to build the isomorphism of categories from the Lagrangian submanifolds of
By repeating this same discussion in reverse using the duality of the torus fibrations, one similarly can understand coherent sheaves on
