Principal bundles on elliptic fibrations

A central role in recent investigations of the duality of F-theory and heterotic strings is played by the moduli of principal bundles, with various structure groups G, over a Calabi-Yau manifold (on the heterotic side), and specifically over an elliptically-fibered Calabi-Yau, on the F-theory side. In this note we propose a simple algebro-geometric technique for studying the moduli spaces of principal G-bundles on an arbitrary variety X which is elliptically fibered over a base S: The moduli space itself is naturally fibered over a weighted projective base parametrizing spectral covers $\tilde{S}$ of S, and the fibers are identified as translates of distinguished Pryms of these covers. In nice situations, the generic Prym fiber is isogenous to the product of a finite group and an abelian subvariety of $Pic(\tilde{S})$.