Let $\Sigma$ be a smooth projective complex curve and $\mathfrak{ g}$ a simple Lie algebra of type ${ \sf ADE}$ 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 ${ \sf ADE}$. In particular, it predicts an interesting connection between adjoint ${ \sf ADE}$ Hitchin systems and quantization of holomorphic branes on Calabi-Yau manifolds.
Categories: hep-th math.AG