Advances in Theoretical and Mathematical Physics

Volume 17 (2013)

Number 6

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

Pages: 1195 – 1253

DOI: https://dx.doi.org/10.4310/ATMP.2013.v17.n6.a1

Authors

Mboyo Esole (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.; Jefferson Physical Laboratory, Harvard University, Cambridge, Mass., U.S.A.)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Mass., U.S.A. )

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.

Full Text (PDF format)

Published 29 April 2014