Constraints on GUT 7-brane Topology in F-theory

We study the relation between phenomenological requirements and the topology of the surfaces that GUT 7-branes wrap in F-theory compactifications. In addition to the exotic matter free condition in the hypercharge flux scenario of SU(5)_{GUT} breaking, we analyze a new condition that comes from a discrete symmetry aligning the contributions to low-energy Yukawa matrices from a number of codimension-three singularity points. We see that the exotic matter free condition excludes Hirzebruch surfaces (except ${\mathbb F}_0$) as the GUT surface, correcting an existing proof in the literature. We further find that the discrete symmetry for the alignment of the Yukawa matrices excludes del Pezzo surfaces and a rational elliptic surface as the GUT surface. Therefore, some GUT 7-brane surfaces are good for some phenomenological requirements, but sometimes not for others, and this aspect should be kept in mind in geometry search in F-theory compactifications.