SYZ conjecture

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The SYZ conjecture is an attempt to understand mirror symmetry conjecture in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, Zaslow titled "Mirror Symmetry is T-duality",^[1]. Along with the homological mirror symmetry conjecture, it is one of the most explored methods to understand mirror symmetry in mathematical terms. While homological mirror symmetry is based on homological algebra, the SYZ conjecture is a geometrical realization of mirror symmetry.

[edit] Introduction

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".

[edit] References