Intermediate Jacobians and ADE Hitchin Systems

Abstract

Let $\Sigma$ be a smooth projective complex curve and $\mathfrak{g}$ a simple Lie algebra of type $\mathsf{ADE}$ with associated adjoint group $G$. For a fixed pair $(\Sigma ,\mathfrak{g})$, we construct a family of quasi-projective Calabi-Yau threefolds parameterized by the base of the Hitchin integrable system associated to $(\Sigma ,\mathfrak{g})$. 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 $\mathsf{ADE}$. In particular, it predicts an interesting connection between adjoint $\mathsf{ADE}$ Hitchin systems and quantization of holomorphic branes on Calabi-Yau manifolds.

14 pages, LaTeX2e, typos fixed, acknowledgments added