# Yukawa Hierarchies in Global F-theory Models

## TL;DR

This paper demonstrates that global F-theory models naturally produce hierarchical Yukawa couplings and higher rank matrices through geometric features, without relying on instanton effects, as shown in a specific $SU(5) imes U(1)$ example.

## Contribution

It provides a geometric framework for understanding Yukawa hierarchies and matrix ranks in global F-theory compactifications, extending local models to full compact setups.

## Key findings

- Yukawa matrices of rank two from geometric Yukawa points.
- Hierarchies of couplings up to 10^5 observed.
- Explicit tracking of complex structure dependence of couplings.

## Abstract

We argue that global F-theory compactifications to four dimensions generally exhibit higher rank Yukawa matrices from multiple geometric contributions known as Yukawa points. The holomorphic couplings furthermore have large hierarchies for generic complex structure moduli. Unlike local considerations, the compact setup realizes these features all through geometry, and requires no instanton corrections. As an example, we consider a concrete toy model with $SU(5) \times U(1)$ gauge symmetry. From the geometry, we find two Yukawa points for the ${\bf 10}_{-2} \, \bar{\bf 5}_6 \, \bar{\bf 5}_{-4}$ coupling, producing a rank two Yukawa matrix. Our methods allow us to track all complex structure dependencies of the holomorphic couplings and study the ratio numerically. This reveals hierarchies of ${\cal O}(10^5)$ and larger on a full-dimensional subspace of the moduli space.