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In mathematics, and applications to string theory, a Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class. Equivalently, by Yau's theorem, it is a compact Kähler manifold which is Ricci flat.
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. Calabi-Yau manifolds of complex dimension three (i.e. dimension six) appear in (supersymmetric) string theory compactifications, supposed (that is) to have them as a fiber, on a base of four dimensions.
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