Intermediate Jacobians and ADE Hitchin Systems TeX

Let $Σ$ be a smooth projective complex curve and $\mathfrakg$ a simple Lie algebra of type $ ADE$ with associated adjoint group $G$. For a fixed pair $(Σ, \mathfrakg)$, we construct a family of quasi-projective Calabi-Yau threefolds parameterized by the base of the Hitchin integrable system associated to $(Σ,\mathfrakg)$. 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.