Toric Methods in F-theory Model Building

TL;DR

This work surveys how to realize global F-theory GUTs using toric geometry by constructing elliptically fibered Calabi–Yau fourfolds as nef-partition complete intersections in toric ambient spaces and employing Tate-model factorization to realize gauge groups such as $SU(5)$ or $SO(10)$. The authors outline a practical pipeline that starts from a toric base $B$ with a del Pezzo divisor $S$, extends to a Calabi–Yau fourfold $X_4$, and computes matter curves, Yukawa points, and D3-tadpoles while incorporating global flux considerations. Through extensive use and extension of PALP, they perform a large-scale search across thousands of geometries, finding thousands of viable models (e.g., $11{,}275$ SU(5) and $10{,}832$ SO(10) cases) after enforcing decoupling and regularity constraints. The paper demonstrates that global, toric F-theory GUT constructions are feasible and highly constrained, and it highlights concrete software enhancements needed to broaden future classifications of Calabi–Yau fourfolds with GUT structures.

Abstract

In this review article we discuss recent constructions of global F-theory GUT models and explain how to make use of toric geometry to do calculations within this framework. After introducing the basic properties of global F-theory GUTs we give a self-contained review of toric geometry and introduce all the tools that are necessary to construct and analyze global F-theory models. We will explain how to systematically obtain a large class of compact Calabi-Yau fourfolds which can support F-theory GUTs by using the software package PALP.