Spectral cover construction for associated bundles

Given an elliptic fibration X → B with section a relatively semistable vector bundle of relative degree zero on X can be constructed using spectral data. These consist of a pair (Cv, Nv) where Cv is a divisor of X which comes equipped with a finite maps to B and Nv is a torsion free sheaf on Cv. Vector bundles on elliptic Calabi-Yau play an important role in compatification of heterotic string theory. In order to answer questions relevant to physicists we have to consider a vector bundle V on a Calabi-Yau and some of the associated bundles constructed from V. All these bundles can be described by spectral data as well. In my thesis I address the problem of reconstructing the spectral data, of any associated bundle in terms of the spectral data of the original bundle. This study is conducted using a relative Fourier-Mukai transform which can be defined on some elliptic fibrations (not necessarely Calabi-Yau). We were able carry out this program for the dual bundle, the tensor product (under some restriction on the elliptic fibration X and the bundle V) and irreducible representations which appear in the decomposition of the tensor product with multeplicity one