## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 17, Number 6 (2013), 1195-1253.

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

Mboyo Esole and Shing-Tung Yau

#### Abstract

We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered four-folds defined by Weierstrass models admitting a split $\tilde{A}_4$ 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 $\tilde{A}_4$ 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 elliptically fibered four-fold. These six small resolutions define distinct four-folds 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 three-folds that defines the type IIB orientifold limit of the F-theory. The full resolution has 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 diagrams.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 17, Number 6 (2013), 1195-1253.

**Dates**

First available in Project Euclid: 21 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1408626541

**Mathematical Reviews number (MathSciNet)**

MR3250767

**Zentralblatt MATH identifier**

1272.81166

#### Citation

Esole, Mboyo; Yau, Shing-Tung. Small resolutions of SU(5)-models in F-theory. Adv. Theor. Math. Phys. 17 (2013), no. 6, 1195--1253. https://projecteuclid.org/euclid.atmp/1408626541