Standard Models from Heterotic M-theory

We present a class of N=1 supersymmetric models of particle physics, derived directly from heterotic M-theory, that contain three families of chiral quarks and leptons coupled to the gauge group $SU(3)_C\times SU(2)_{L}\times U(1)_{Y}$. These models are a fundamental form of ``brane-world'' theories, with an observable and hidden sector each confined, after compactification on a Calabi-Yau threefold, to a BPS three-brane separated by a five dimensional bulk space with size of the order of the intermediate scale. The requirement of three families, coupled to the fundamental conditions of anomaly freedom and supersymmetry, constrains these models to contain additional five-branes wrapped around holomorphic curves in the Calabi-Yau threefold. These five-branes ``live'' in the bulk space and represent new, non-perturbative aspects of these particle physics vacua. We discuss, in detail, the relevant mathematical structure of a class of torus-fibered Calabi-Yau threefolds with non-trivial first homotopy groups and construct holomorphic vector bundles over such threefolds, which, by including Wilson lines, break the gauge symmetry to the standard model gauge group. Rules for constructing phenomenological particle physics models in this context are presented and we give a number of explicit examples.