Calabi-Yau spaces

A Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class, or (equivalently) which is Ricci flat (by Yau's theorem).

Calabi-Yau manifolds of complex dimension three (i.e. dimension six) appear in (supersymmetric) string theory compactifications to four dimensions. That is, ten conjectural dimensions in string theory are supposed to come as four of which we are aware, carrying some kind of fibration with fiber dimension six.