Mathematical Research Letters

Volume 14 (2007)

Number 5

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

Pages: 745 – 756

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n5.a3

Authors

D. E. Diaconescu (Rutgers University)

R. Donagi (University of Pennsylvania)

T. Pantev (University of Pennsylvania)

Abstract

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.

Full Text (PDF format)