Standard Model bundles of the heterotic string

TL;DR

The paper develops a constructive framework to realize Standard Model-like spectra in heterotic string theory by using non-simply-connected elliptically fibered Calabi–Yau manifolds with a free ${\bf Z_2}$ action. It combines a global ${\bf B}$-model with a second section, a spectral cover description of $SU(n)$ bundles, and a Fourier–Mukai transform to build an $SU(5)$ bundle whose upstairs model has six generations and that remains invariant under the involution, allowing a downstairs three-generation model on the quotient Calabi–Yau. The authors derive explicit formulas for the bundle's Chern classes, establish generation number constraints via $(1/2)c_3(V)=\lambda\eta(\eta-nc_1)$, and present invariant-bundle conditions including $\mu=1/2$, providing concrete six-generation examples over ${\bf F_2}$. This work provides a concrete, moduli-space-sensitive pathway to obtain Standard Model matter content from heterotic compactifications while maintaining control over anomaly cancellation and involution invariance.

Abstract

We show how to construct supersymmetric three-generation models with gauge group and matter content of the Standard Model in the framework of non-simply-connected elliptically fibered Calabi-Yau manifolds Z. The elliptic fibration on a cover Calabi-Yau, where the model has 6 generations of SU(5) and the bundle is given via the spectral cover description, has a second section leading to the needed free involution. The relevant involution on the defining spectral data of the bundle is identified for a general Calabi-Yau of this type and invariant bundles are generally constructible.