F-Theory Vacua with Z_3 Gauge Symmetry

TL;DR

This paper explores F-theory compactifications with Z_3 gauge symmetry, analyzing how different M-theory vacua relate through Higgsing of U(1) symmetries in genus-one fibered Calabi-Yau manifolds.

Contribution

It provides a detailed geometric and physical analysis of Z_3 discrete gauge groups arising from genus-one fibrations and their relation to U(1) symmetries via Higgsing.

Findings

Identification of Higgs fields from vanishing cycles in fibers.

Explicit description of resolved phases of the geometry.

Connection between Tate-Shafarevich group elements and gauge symmetries.

Abstract

Discrete gauge groups naturally arise in F-theory compactifications on genus-one fibered Calabi-Yau manifolds. Such geometries appear in families that are parameterized by the Tate-Shafarevich group of the genus-one fibration. While the F-theory compactification on any element of this family gives rise to the same physics, the corresponding M-theory compactifications on these geometries differ and are obtained by a fluxed circle reduction of the former. In this note, we focus on an element of order three in the Tate-Shafarevich group of the general cubic. We discuss how the different M-theory vacua and the associated discrete gauge groups can be obtained by Higgsing of a pair of five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles in $I_2$-fibers that appear at certain codimension two loci in the base. We explicitly identify all three curves that give rise to the corresponding Higgs fields. In this analysis the investigation of different resolved phases of the underlying geometry plays a crucial r\^ole.