The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle

TL;DR

The paper develops a comprehensive dictionary linking the arithmetic of elliptic and genus-one fibrations with gauge theories on a circle in F-theory. It identifies the Mordell-Weil group $MW(Y)$ as encoding large circle gauge transformations, with its free part corresponding to Abelian transformations and torsion to fractional non-Abelian ones, and extends this framework to multi-sections via an extended Mordell-Weil group and a generalized Shioda map. It further introduces an arithmetic structure on fibrations with exceptional divisors, showing how non-Abelian large gauge transformations on the circle lift to divisor actions that reproduce the Mordell-Weil behavior in the Abelian limit through Higgs transitions. The results point to a novel geometric symmetry in F-theory that unifies Abelian and non-Abelian circle dynamics with Higgsing, and provide a robust mechanism for ensuring anomaly cancellation across circle reductions.

Abstract

The geometry of elliptic fibrations translates to the physics of gauge theories in F-theory. We systematically develop the dictionary between arithmetic structures on elliptic curves as well as desingularized elliptic fibrations and symmetries of gauge theories on a circle. We show that the Mordell-Weil group law matches integral large gauge transformations around the circle in Abelian gauge theories and explain the significance of Mordell-Weil torsion in this context. We also use Higgs transitions and circle large gauge transformations to introduce a group law for genus-one fibrations with multi-sections. Finally, we introduce a novel arithmetic structure on elliptic fibrations with non-Abelian gauge groups in F-theory. It is defined on the set of exceptional divisors resolving the singularities and divisor classes of sections of the fibration. This group structure can be matched with certain integral non-Abelian large gauge transformations around the circle when studying the theory on the lower-dimensional Coulomb branch. Its existence is required by consistency with Higgs transitions from the non-Abelian theory to its Abelian phases in which it becomes the Mordell-Weil group. This hints towards the existence of a new underlying geometric symmetry.

Figures (3)

• Figure 1: Schematic depiction of the various geometric configurations considered in this work. The geometries are related by geometric transitions describing Higgsing and unHiggsing processes.
• Figure 2: The zero-$n$-section is chosen to contain the zero-section after unHiggsing to a setting with rational sections only.
• Figure 3: Moving from $\hat{Y}$ to the unHiggsed phase $\hat{Y}_{\rm uH}$ there are in general many ways to add the individual sections. We schematically indicate two choices in (1) and (2), which in general differ by their vertical parts when considering the associated multi-section.