Advances in Theoretical and Mathematical Physics

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

Mboyo Esole and Shing-Tung Yau

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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.

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

First available in Project Euclid: 21 August 2014

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Esole, Mboyo; Yau, Shing-Tung. Small resolutions of SU(5)-models in F-theory. Adv. Theor. Math. Phys. 17 (2013), no. 6, 1195--1253.

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