The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle

TL;DR

This paper explores the deep connection between the arithmetic of elliptic fibrations and gauge theory symmetries in F-theory, revealing new structures and symmetries related to gauge transformations and Higgs transitions.

Contribution

It introduces a novel arithmetic structure on elliptic fibrations with non-Abelian gauge groups, linking geometric divisors to gauge transformations in F-theory.

Findings

Mordell-Weil group law corresponds to large gauge transformations in Abelian theories.

Mordell-Weil torsion plays a significant role in gauge symmetry.

A new group law for genus-one fibrations with multi-sections is proposed.

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.