SYZ conjecture

The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled "Mirror Symmetry is T-duality".[1]

Along with the homological mirror symmetry conjecture, it is one of the most explored tools applied to understand mirror symmetry in mathematical terms. While the homological mirror symmetry is based on homological algebra, the SYZ conjecture is a geometrical realization of mirror symmetry.

Formulation

In string theory, mirror symmetry relates type IIA and type IIB theories. It predicts that the effective field theory of type IIA and type IIB should be the same if the two theories are compactified on mirror pair manifolds.

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. Therefore, mirror symmetry will map 0-branes of type IIA theories into a subset of 3-branes of type IIB theories.

By considering supersymmetric conditions, it has been shown that these 3-branes should be special Lagrangian submanifolds.[2][3] On the other hand, T-duality does the same transformation in this case, thus "mirror symmetry is T-duality".

Mathematical statement

The initial proposal of the SYZ conjecture by Strominger, Yau, and Zaslow, was not given as a precise mathematical statement.[1] One part of the mathematical resolution of the SYZ conjecture is to, in some sense, correctly formulate the statement of the conjecture itself. 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.

SYZ Conjecture: Every 6-dimensional Calabi–Yau manifold ${\displaystyle X}$ has a mirror 6-dimensional Calabi–Yau manifold ${\displaystyle {\hat {X}}}$ such that there are continuous surjections ${\displaystyle f:X\to B}$, ${\displaystyle {\hat {f}}:{\hat {X}}\to B}$ to a compact topological manifold ${\displaystyle B}$ of dimension 3, such that

1. There exists a dense open subset ${\displaystyle B_{\text{reg}}\subset B}$ on which the maps ${\displaystyle f,{\hat {f}}}$ are fibrations by nonsingular special Lagrangian 3-tori. Furthermore for every point ${\displaystyle b\in B_{\text{reg}}}$, the torus fibres ${\displaystyle f^{-1}(b)}$ and ${\displaystyle {\hat {f}}^{-1}(b)}$ should be dual to each other in some sense, analogous to duality of Abelian varieties.
2. For each ${\displaystyle b\in B\backslash B_{\text{reg}}}$, the fibres ${\displaystyle f^{-1}(b)}$ and ${\displaystyle {\hat {f}}^{-1}(b)}$ should be singular 3-dimensional special Lagrangian submanifolds of ${\displaystyle X}$ and ${\displaystyle {\hat {X}}}$ respectively.

Diagram of a special Lagrangian torus fibration. The fibres of ${\displaystyle f:X\to B}$ over points in ${\displaystyle B_{\text{reg}}}$ are 3-tori, and over the singular set ${\displaystyle B\backslash B_{\text{reg}}}$ the fibre could be a possibly singular special Lagrangian submanifold ${\displaystyle L}$.

The situation in which ${\displaystyle B_{\text{reg}}=B}$ 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. For example the possible behaviour of the singular set ${\displaystyle B\backslash B_{\text{reg}}}$ is not well understood, and this set could be quite large in comparison to ${\displaystyle B}$. 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.[4]

References

1. Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric (1996), "Mirror symmetry is T-duality", Nuclear Physics B, 479 (1–2): 243–259, arXiv:hep-th/9606040, Bibcode:1996NuPhB.479..243S, doi:10.1016/0550-3213(96)00434-8.
2. Becker, Katrin; Becker, Melanie; Strominger, Andrew (1995), "Fivebranes, membranes and non-perturbative string theory", Nuclear Physics B, 456 (1–2): 130–152, arXiv:hep-th/9507158, Bibcode:1995NuPhB.456..130B, doi:10.1016/0550-3213(95)00487-1.
3. Harvey, Reese; Lawson, H. Blaine, Jr. (1982), "Calibrated geometries", Acta Mathematica, 148 (1): 47–157, doi:10.1007/BF02392726.
4. Gross, M., Huybrechts, D. and Joyce, D., 2012. Calabi-Yau manifolds and related geometries: lectures at a summer school in Nordfjordeid, Norway, June 2001. Springer Science & Business Media.
5. Gross, M., 2012. Mirror symmetry and the Strominger-Yau-Zaslow conjecture. Current Developments in Mathematics, 2012(1), pp.133-191.