Intermediate Jacobians and ADE Hitchin Systems

TL;DR

The work addresses how ADE Hitchin systems on a smooth curve $\Sigma$ can be realized geometrically through a family of Calabi–Yau threefolds, providing a bridge between Hitchin moduli and Calabi–Yau geometry in the spirit of large $N$ duality. It constructs a Calabi–Yau integrable system given by the relative intermediate Jacobian $J^{3}({\mathcal{X}}/{\bf L})$ parameterized by the Hitchin base ${\bf L}$ and proves an isomorphism with the Prym fibration of the adjoint ADE Hitchin system, ${\rm Prym}_{G}(\widetilde{\boldsymbol{\Sigma}}/\Sigma)$. This establishes a geometric realization of ${\sf ADE}$ Dijkgraaf–Vafa transitions and suggests a link to quantization of holomorphic branes in Calabi–Yau manifolds. The results generalize the ${\sf A}_{1}$ case and provide a concrete framework for exploring connections between Hitchin systems, cameral covers, and Calabi–Yau integrable dynamics.

Abstract

Let $Σ$ 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 $(Σ, \mathfrak{g})$, we construct a family of quasi-projective Calabi-Yau threefolds parameterized by the base of the Hitchin integrable system associated to $(Σ,\mathfrak{g})$. 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.

Theorems & Definitions (6)

• Remark 2.1
• Remark 2.2
• Proposition 2.3
• Lemma 3.1
• Theorem 3.2
• Lemma 3.3