Mirror symmetry

In Physics and mathematics, mirror symmetry is a surprising relation that can exist between two Calabi-Yau manifold. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of string theory. More specifically, mirror symmetry relates two manifolds M and W whose Hodge numbers

: h 1,1 and h 1,2

are swapped; string theory compactified on these two manifolds can be proved to be lead to identical physical phenomena.

The discovery of mirror symmetry is connected with names such as Lance Dixon, Wolfgang Lerche, Cumrun Vafa, Nicholas Warner, Brian Greene, Ronen Plesser, Philip Candelas, Monika Lynker, Rolf Schimmrigk and others. . The simultaneous action of T-duality on all three dimensions of this torus is equivalent to mirror symmetry.

Mirror symmetry allowed the physicists to calculate many quantities that seemed virtually incalculable before, by invoking the mirror description of a given physical situation, which can be often much easier. Mirror symmetry has also become a very powerful tool in mathematics, and although mathematicians have proved many rigorous theorems based on the physicists intuition, a full mathematical understanding of the phenomenon of mirror symmetry is lacking. One possible mathematical framework is provided by the homological mirror symmetry conjecture.