Why Calabi-Yau?
Compactification on Calabi–Yau n-folds are important because they leave some of the original supersymmetry unbroken. More precisely, in the absence of fluxes, compactification on a Calabi–Yau 3-fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken if the holonomy is the full SU(3).
How many Calabi-Yau manifolds are there?
What’s more, there are many different 6D Calabi-Yau manifolds that could fit the string theory bill and, disappointingly, no-one was able to work out which was the “right” one. All this somewhat undermined the manifolds’ standing in physics.
What is mirror symmetry of Calabi-Yau manifolds in string theory?
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.