  For supersymmetric GUT models from heterotic string theory, built from a
stable holomorphic SU(n) vector bundle $V$ on a Calabi-Yau threefold $X$, the
net amount of chiral matter can be computed by a Chern class computation.
Corresponding computations for the number $N_H$ of Higgses lead for the
phenomenologically relevant cases of GUT group SU(5) or SO(10) to consideration
of the bundle $\La^2 V$. In a class of bundles where everything can be computed
explicitly (spectral bundles on elliptic $X$) we find that the computation for
$N_H$ gives a result which is in conflict with expectations. We argue that this
discrepancy has its origin in the subtle geometry of the spectral data for
$\La^2 V$ and that taking this subtlety into account properly should resolve
the problem.
