Superstring theory is consistent only in ten dimensions, so its six surplus spatial dimensions must be rolled up on a compact internal manifold whose geometry dictates the physics observed in four dimensions.
Requiring the four-dimensional effective theory to preserve supersymmetry forces the existence of a covariantly constant spinor on the internal six-manifold, which constrains its holonomy group to and hence identifies it as a Calabi–Yau threefold. This deduction is due to Candelas, Horowitz, Strominger and Witten (1985), and it turned an abstract theorem of algebraic geometry into a criterion for realistic string compactification.
The manifold’s moduli, namely its Kähler and complex-structure deformations counted by the Hodge numbers and , descend to massless scalar fields in the four-dimensional theory, so the choice and shape of the Calabi–Yau fixes the particle content and couplings. A striking feature is mirror symmetry, which exchanges the two Hodge numbers and relates a Calabi–Yau to a mirror partner on which hard computations become tractable; the quintic is the classic worked example, and its cross-section is carved in carving-the-calabi-yau-quintic.