  Techniques are presented for computing the cohomology of stable, holomorphic
vector bundles over elliptically fibered Calabi-Yau threefolds. These
cohomology groups explicitly determine the spectrum of the low energy,
four-dimensional theory. Generic points in vector bundle moduli space manifest
an identical spectrum. However, it is shown that on subsets of moduli space of
co-dimension one or higher, the spectrum can abruptly jump to many different
values. Both analytic and numerical data illustrating this phenomenon are
presented. This result opens the possibility of tunneling or phase transitions
between different particle spectra in the same heterotic compactification. In
the course of this discussion, a classification of SU(5) GUT theories within a
specific context is presented.
