ar X iv : h ep - t h / 06 07 15 9 v 2 2 4 Ju l 2 00 6 Intermediate Jacobians and ADE Hitchin Systems

Let Σ be a smooth projective complex curve and g a simple Lie algebra of type ADE with associated adjoint group G. For a fixed pair (Σ, g), we construct a family of quasiprojective Calabi-Yau threefolds parameterized by the base of the Hitchin integrable system associated to (Σ, g). Our main result establishes an isomorphism between the Calabi-Yau integrable system, whose fibers are the intermediate Jacobians of this family of Calabi-Yau threefolds, and the Hitchin system for G, whose fibers are Prym varieties of the corresponding spectral covers. This construction provides a geometric framework for Dijkgraaf-Vafa transitions of type ADE. In particular, it predicts an interesting connection between adjoint ADE Hitchin systems and quantization of holomorphic branes on Calabi-Yau manifolds.