Constraints on GUT 7-brane Topology in F-theory

TL;DR

This paper investigates how phenomenological constraints in F-theory GUT models restrict the topology of 7-brane surfaces, excluding certain surfaces like Hirzebruch and del Pezzo, thus guiding geometric searches.

Contribution

It identifies specific topological restrictions on GUT 7-brane surfaces imposed by phenomenological conditions, correcting previous assumptions and highlighting the importance of surface choice.

Findings

Excludes Hirzebruch surfaces (except ${f F}_0$) as GUT surfaces due to exotic matter constraints.

Disallows del Pezzo and rational elliptic surfaces based on discrete symmetry requirements.

Shows some surfaces are suitable for certain phenomenological features but not others.

Abstract

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.