  In this work we extend the well-known spectral cover construction first
developed by Friedman, Morgan, and Witten to describe more general vector
bundles on elliptically fibered Calabi-Yau geometries. In particular, we
consider the case in which the Calabi-Yau fibration is not in Weierstrass form,
but can rather contain fibral divisors or multiple sections (i.e. a higher rank
Mordell-Weil group). In these cases, general vector bundles defined over such
Calabi-Yau manifolds cannot be described by ordinary spectral data. To
accomplish this we employ well established tools from the mathematics
literature of Fourier-Mukai functors. We also generalize existing tools for
explicitly computing Fourier-Mukai transforms of stable bundles on elliptic
Calabi-Yau manifolds. As an example of these new tools we produce novel
examples of chirality changing small instanton transitions. The goal of this
work is to provide a geometric formalism that can substantially increase the
understood regimes of heterotic/F-theory duality.
