Small resolutions of SU(5)-models in F-theory

TL;DR

This paper explicitly resolves singularities in elliptically fibered fourfolds used in F-theory SU(5) models, revealing complex fiber geometries and multiple small resolutions connected by flop transitions, impacting the understanding of gauge symmetry and fiber structures.

Contribution

It provides a detailed small resolution of SU(5) F-theory models, describing the fiber geometry, multiple resolutions, and their interconnections, which was not previously explicitly detailed.

Findings

Six distinct small resolutions connected by flop transitions.

Identification of exotic fibers not fitting traditional Kodaira classifications.

Mapping of fiber singularities to conifold points in the base.

Abstract

We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered fourfolds defined by Weierstrass models admitting a split A_4 singularity over a divisor of the discriminant locus. Such varieties are used to geometrically engineer SU(5) Grand Unified Theories in F-theory. The desingularization is given by a small resolution of singularities. The I_5 fiber naturally appears after resolving the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four dimensional affine binomial variety which leads to six different small resolutions of the the elliptically fibered fourfold. These six small resolutions define distinct fourfolds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base is mapped to conifold points of the threefolds that defines the type IIB orientifold limit of the F-theory. The full resolution have interesting properties, specially for fibers in codimension three: the rank of the singular fiber does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagram.