- 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",Strominger, Yau, Zaslow, "Mirror Symmetry is T-duality" [http://arxiv.org/pdf/hep-th/9606040 hep-th/9606040] ] .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 state s of type IIA theories compactified on "X", especially0-brane s that havemoduli space "X". It is known that all of the BPS states of type IIB theories compactified on "Y" are3-brane s. 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 submanifold sBecker, Becker, Strominger, "Fivebranes, Membranes and Non-Perturbative String Theory" [http://arxiv.org/abs/hep-th/9507158 hep-th/9507158] ] Harvey, Lawson " [http://www.springerlink.com/content/8451j84w08j28432/ Calibrated Geometry] " Acta Mathematica Volume 148, Number 1 / July, 1982] . On the other hand,T-duality does the same transformation in this case, thus "mirror symmetry is T-duality".References
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