On vector bundles and chiral matter in N = 1 heterotic compactifications

In this note we derive the net number of generations of chiral fermions in heterotic string compactifications on Calabi-Yau threefolds with SU(n) vector bundles, for n odd, using the parabolic approach for bundles. This extends the work of Friedman, Morgan and Witten, who treated SU(n) vector bundles, with n even, which lead to a vanishing net amount of chiral matter. We then compare our results with the spectral cover construction for bundles and make a comment on the net number interpretation in F-theory.