Mathematical Research Letters

Volume 14, Issue 5, September 2007 pp. 745-756.

Intermediate Jacobians and ${\sf {A}{D}{E}}$ Hitchin Systems

Authors: D. E. Diaconescu (1), R. Donagi (2), and T. Pantev (3)
Author institution: Rutgers University (1), University of Pennsylvania (2), University of Pennsylvania (3)

Summary: 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.

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