Holomorphic vector bundles and non-perturbative vacua in M-theory

We review the spectral cover formalism for constructing both U(n) and SU(n) holomorphic vector bundles on elliptically fibered Calabi-Yau three-folds which admit a section. We discuss the allowed bases of these three-folds an d show that physical constraints eliminate Enriques surfaces from considera tion. Relevant properties of the remaining del Pezzo and Hirzebruch surface s are presented. Restricting the structure group to SU(n), we derive, in de tail, a set of rules for the construction of three-family particle physics theories with phenomenologically relevant gauge groups. We show that anomal y cancellation generically requires the existence of non-perturbative vacua containing five-branes. We illustrate these ideas by constructing four exp licit three-family non-perturbative vacua.