Standard-model bundles
Ron Donagi, Burt Ovrut, Tony Pantev, Dan Waldram
TL;DR
The paper addresses constructing non-simply-connected Calabi–Yau threefolds $Z$ with $\pi_{1}(Z)\cong \mathbb{Z}/2$ and a family of Mumford-stable bundles $V$ whose symmetry group is $SU(3)\times SU(2)\times U(1)$ while achieving $c_{3}=6$ and an effective difference $c_{2}(Z)-c_{2}(V)$. The authors realize this by first forming a Schoen-type CY threefold $X$ as a fiber product of two rational elliptic surfaces, then quotientting by a freely acting involution $\tau_{X}$ to obtain $Z=X/\tau_{X}$; bundles $V$ are constructed on $X$ via a relative Fourier–Mukai (spectral) approach, organized as a non-split extension $0\to V_{2}\to V\to V_{3}\to0$, with $V_{i}$ built from reducible spectral covers and Hecke transforms to gain infinite families. A careful analysis of invariance under the involutions, together with a detailed translation of stability and Chern-class conditions into numerical constraints, yields explicit infinite families of solutions, including precise integrality and ampleness requirements; these give rise to moduli spaces of substantial dimension (e.g., $>90$). The resulting quadruples $(X,\tau_{X},H,V)$ descend to $Z$ where the bundle data reproduce a Standard-Model-like gauge symmetry in Heterotic M-theory, providing phenomenologically relevant vacua with controlled anomaly cancellation and chiral index.
Abstract
We describe a family of genus one fibered Calabi-Yau threefolds with fundamental group ${\mathbb Z}/2$. On each Calabi-Yau $Z$ in the family we exhibit a positive dimensional family of Mumford stable bundles whose symmetry group is the Standard Model group $SU(3)\times SU(2)\times U(1)$ and which have $c_{3} = 6$. We also show that for each bundle $V$ in our family, $c_{2}(Z) - c_{2}(V)$ is the class of an effective curve on $Z$. These conditions ensure that $Z$ and $V$ can be used for a phenomenologically relevant compactification of Heterotic M-theory.