Mirror symmetry (string theory)

In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds. 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. The classical formulation of mirror symmetry relates two Calabi-Yau threefolds M and W whose Hodge numbers h^1,1 and h^1,2 are swapped; string theory compactified on these two manifolds lead to identical effective field theories.

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. Andrew Strominger, Shing-Tung Yau, and Eric Zaslow have showed that mirror symmetry is a special example of T-duality: the Calabi-Yau manifold may be written as a fiber bundle whose fiber is a three-dimensional torus. 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 still being developed. Most of the physicist's examples could be conceptionalized by the Batyrev-Borisov mirror construction which uses the duality of reflexive polytopes and nef partitions. In their construction the mirror partners appear as anticanonically embedded hypersurfaces or certain complete intersections in Fano toric varieties. The Gross-Siebert mirror construction generalizes this to non-embedded cases by looking at degenerating families of Calabi-Yau manifolds. This point of view also includes the T-duality. Another mathematical framework is provided by the homological mirror symmetry conjecture.

References

• Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric, "Mirror Symmetry is T-duality" hep-th/9606040
• Cox, David A.; Katz, Sheldon, Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, 68. American Mathematical Society, Providence, RI, 1999. xxii+469 pp. ISBN 0-8218-1059-6
• Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Vakil, Ravi; Zaslow, Eric Mirror symmetry. Clay Mathematics Monographs, 1. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003. xx+929 pp. ISBN 0-8218-2955-6
• Victor Batyrev; Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties J. Algebraic Geom. 3 (1994), no. 3, 493--535
• Mark Gross; Toric Degenerations and Batyrev-Borisov Duality: