Sections, multisections, and U(1) fields in F-theory

TL;DR

This paper explores the geometric structures of genus-one fibrations in F-theory, showing how U(1) gauge fields and their breaking to discrete symmetries relate to multisections and Higgsing processes.

Contribution

It demonstrates that genus-one fibrations without sections are part of the Weierstrass moduli space and connects U(1) gauge symmetries to Higgsing of SU(2) in F-theory models.

Findings

Genus-one fibrations without sections fit into Weierstrass models.

U(1) gauge symmetries arise from Higgsing SU(2) with adjoint matter.

U(1) symmetries in 6D and 4D F-theory models are connected to specific Higgsing processes.

Abstract

We show that genus-one fibrations lacking a global section fit naturally into the geometric moduli space of Weierstrass models. Elliptic fibrations with multiple sections (nontrivial Mordell-Weil rank), which give rise in F-theory to abelian U(1) fields, arise as a subspace of the set of genus-one fibrations with multisections. Higgsing of certain matter multiplets charged under abelian gauge fields in the corresponding supergravity theories break the U(1) gauge symmetry to a discrete gauge symmetry group. We further show that in six dimensions every U(1) gauge symmetry arising in an F-theory model can be found by Higgsing an SU(2) gauge symmetry with adjoint matter, and that a similar structure holds for F-theory geometries giving 4D supergravity theories.