# SYZ conjecture

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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".

## 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.