On Vector Bundles and Chiral Matter in N=1 Heterotic Compactifications

TL;DR

The paper addresses computing the net number of chiral generations $N_{gen}$ in $N=1$ heterotic compactifications on a Calabi–Yau threefold with $SU(n)$ bundles, focusing on odd $n$ and employing the parabolic bundle construction. It derives explicit Chern-class relations, yielding $N_{gen}=rac{1}{2n}igl( ext{η}( ext{η}-n c_1({ m L}))igr)$ with $ ext{η}= ext{π}_*(c_2(V))$ divisible by $n$, and demonstrates equivalence with the spectral-cover construction under a specific twist $ ext{λ}=rac{1}{2n}$. The work highlights how discrete data and flux quantization on the heterotic side correspond to four-flux and brane counts in F-theory, establishing a concrete bridge between the two dual descriptions. It also clarifies how twists on the spectral cover affect $n_5$ and $n_3$, and discusses consistency with recent cross-checks of $c_3(V)$ in spectral-cover approaches.

Abstract

In this note we derive the net number of generations of chiral fermions in heterotic string compactifications on Calabi-Yau threefolds with certain SU(n) vector bundles, for n odd, using the parabolic approach for bundles. We compare our results with the spectral cover construction for bundles and make a comment on the net number interpretation in F-theory.