  Explicit methods are presented for computing the cohomology of stable,
holomorphic vector bundles on elliptically fibered Calabi-Yau threefolds. The
complete particle spectrum of the low-energy, four-dimensional theory is
specified by the dimensions of specific cohomology groups. The spectrum is
shown to depend on the choice of vector bundle moduli, jumping up from a
generic minimal result to attain many higher values on subspaces of
co-dimension one or higher in the moduli space. An explicit example is
presented within the context of a heterotic vacuum corresponding to an SU(5)
GUT in four-dimensions.
