Taniguchi Lecture on Principal Bundles on Elliptic Fibrations

TL;DR

This work describes the moduli space ${\cal M}^G_X$ of principal $G$-bundles on an elliptic fibration $X\to S$ via cameral covers and the distinguished Prym varieties, tying the problem to Hitchin-type integrable systems. The core construction expresses ${\cal M}^G_E$ as ${\cal M}^T_E / W$ with ${\cal M}^T_E\cong \mathrm{Hom}(\Lambda,E)$, and, for a fibration, identifies the fiber over a cameral base with the Prym variety $Prym_{\Lambda}(\tilde S)$; the approach generalizes to type $E_n$ via del Pezzo fibrations and relative Deligne cohomology. A central result is the equivalence between regularized $G$-bundles on $X$ and spectral data $(\tilde S, v, {\cal L})$, where $v$ encodes a $W$-equivariant value map and ${\cal L}$ provides the Prym datum, thereby unifying the fiberwise Higgs-bundle picture with a global elliptic-fibration framework. The constructions yield a concrete, modular description of the moduli in terms of cameral covers, spectral data, and regular centralizers, with implications for related dualities and geometric representation theory.

Abstract

In this talk we discuss the description of the moduli space of principal G-bundles on an elliptic fibration X-->S in terms of cameral covers and their distinguished Prym varieties. We emphasize the close relationship between this problem and the integrability of Hitchin's system and its generalizations. The discussion roughly parallels that of [D2], but additional examples are included and some important steps of the argument are illustrated. Some of the applications to heterotic/F-theory duality were described in the accompanying ICMP talk (hep-th/9802093).