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
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