Mirror symmetry

Summary

Mirror symmetry is a correspondence, first predicted by string theory, between pairs of Calabi–Yau manifolds $X$ and $\check{X}$. The symplectic geometry of $X$, encoded by its Gromov–Witten invariants, matches the complex geometry of $\check{X}$, encoded by the variation of its Hodge structures. For Calabi–Yau threefolds the Hodge numbers are exchanged: $h^{1,1}(X) = h^{2,1}(\check{X})$ and $h^{2,1}(X) = h^{1,1}(\check{X})$. It is not a single theorem but a program with several precise forms, such as homological mirror symmetry.