  A conic fibration has an associated sheaf of even Clifford algebras on the
base. In this paper, we study the relation between the moduli spaces of modules
over the sheaf of even Clifford algebras and the Prym variety associated to the
conic fibration. In particular, we construct a rational map from the moduli
space of modules over the sheaf of even Clifford algebras to the special
subvarieties in the Prym variety, and check that the rational map is birational
in some cases. As an application, we get an explicit correspondence between
instanton bundles of minimal charge on cubic threefolds and twisted Higgs
bundles on curves.
