"Calabi-Yau manifolds admit Kähler metrics with vanishing Ricci curvatures. They are solutions of the Einstein field equation with no matter. The theory of motions of circles inside of a Calabi-Yau manifold provide a model of a conformal field theory. (It is called a σ-model in physics.) Because of this, Calabi-Yau manifolds are pivotal in superstring theory. ... It has long been argued that, in order to solve certain classic problems of unified gauge theories such as the gauge hierarchy problem, the 4-dimensional effective theory should admit an N = 1 supersymmetry. In a fundamental paper, Candelas-Horowitz-Strominger-Witten ... analyzed what the constraint of that N = 1 supersymmetry would mean for the geometry of the internal space X. They found that, for the most basic product models with N = 1 supersymmetry, the space X must be a Calabi-Yau manifold of complex dimension 3. Shortly afterwards, Strominger ... considered slightly more general models, allowing warped products. For these models, the N = 1 supersymmetry constraint results in a modification of the Ricci-flat equation of the earlier model."