Holomorphic Vector Bundles and Non-Perturbative Vacua in M-Theory

TL;DR

This paper reviews the spectral cover formalism for constructing holomorphic vector bundles on elliptically fibered Calabi-Yau three-folds, discusses physical constraints on base surfaces, and constructs explicit three-family non-perturbative vacua with phenomenological relevance.

Contribution

It provides a detailed set of rules for constructing three-family particle physics models with SU(n) bundles and demonstrates the necessity of non-perturbative vacua with five-branes.

Findings

Physical constraints exclude Enriques surfaces as bases.

Explicit construction of four three-family non-perturbative vacua.

Anomaly cancellation requires non-perturbative vacua with five-branes.

Abstract

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 and show that physical constraints eliminate Enriques surfaces from consideration. Relevant properties of the remaining del Pezzo and Hirzebruch surfaces are presented. Restricting the structure group to SU(n), we derive, in detail, a set of rules for the construction of three-family particle physics theories with phenomenologically relevant gauge groups. We show that anomaly cancellation generically requires the existence of non-perturbative vacua containing five-branes. We illustrate these ideas by constructing four explicit three-family non-perturbative vacua.