Vector Bundle Extensions, Sheaf Cohomology, and the Heterotic Standard Model
Volker Braun, Yang-Hui He, Burt A. Ovrut, Tony Pantev
TL;DR
The authors construct stable, holomorphic vector bundles on a torus-fibered, non-simply connected Calabi–Yau threefold X̃ using bundle extensions and equivariant structures, then compute the full sheaf cohomology via Leray spectral sequences to extract the low-energy spectrum. They explicitly prove the existence of necessary equivariant extensions, build a visible SU(4) bundle that cancels anomalies (with either five-branes or an alternative hidden sector), and show the spectrum yields three chiral families plus a Higgs doublet pair with no exotics after Z_3×Z_3 Wilson lines. The work provides a detailed mathematical framework for heterotic standard-model vacua, including stability/simple-ness checks and concrete cohomology computations for V and ∧^2V. The results demonstrate a phenomenologically viable vacuum in the heterotic string with controlled geometric and gauge-data, and outline practical paths for alternative hidden sectors while maintaining anomaly cancellation.
Abstract
Stable, holomorphic vector bundles are constructed on an torus fibered, non-simply connected Calabi-Yau threefold using the method of bundle extensions. Since the manifold is multiply connected, we work with equivariant bundles on the elliptically fibered covering space. The cohomology groups of the vector bundle, which yield the low energy spectrum, are computed using the Leray spectral sequence and fit the requirements of particle phenomenology. The physical properties of these vacua were discussed previously. In this paper, we systematically compute all relevant cohomology groups and explicitly prove the existence of the necessary vector bundle extensions. All mathematical details are explained in a pedagogical way, providing the technical framework for constructing heterotic standard model vacua.