G-flux and Spectral Divisors

TL;DR

This work develops a global method to engineer $G$-flux in singular elliptic Calabi–Yau fourfold F-theory compactifications using a spectral divisor that reduces to the Higgs-bundle spectral cover in the local limit. By applying the construction to an $E_6$ singularity, the authors implement a full resolution, identify Cartan and matter surfaces, and derive fluxes that preserve the $E_6$ symmetry, matching local spectral-cover results. They show equivalence between fluxes obtained from holomorphic surfaces in the resolved geometry and those from the spectral-divisor description, including correct flux quantization and D3-tadpole, as well as the local limit to the spectral cover flux. The explicit computations of chirality on the 27 representation and Yukawa couplings in codimension three demonstrate the consistency of the framework and its potential generalization beyond SU(5) models. Overall, the paper provides a coherent bridge between global Calabi–Yau geometry and local Higgs-bundle data via spectral divisors in F-theory.

Abstract

We propose a construction of G-flux in singular elliptic Calabi-Yau fourfold compactifications of F-theory, which in the local limit allow a spectral cover description. The main tool of construction is the so-called spectral divisor in the resolved Calabi-Yau geometry, which in the local limit reduces to the Higgs bundle spectral cover. We exemplify the workings of this in the case of an E_6 singularity by constructing the resolved geometry, the spectral divisor and in the local limit, the spectral cover. The G-flux constructed with the spectral divisor is shown to be equivalent to the direct construction from suitably quantized linear combinations of holomorphic surfaces in the resolved geometry, and in the local limit reduces to the spectral cover flux.