Wavefunctions and the Point of E8 in F-theory

TL;DR

This work develops a semi-local framework to compute operator coefficients in F-theory GUTs by solving for wavefunctions in a patch around a point of maximal symmetry enhancement to $E_8$. It derives local equations of motion for fluctuations in Higgs and flux backgrounds, classifies local versus global modes via the determinant of the local mass matrix, and provides explicit Landau-level and KK wavefunctions with overlap formulas. It then applies these results to a toy $E_8$-point model embedded in $SU(5)$, computing Yukawa couplings and dimension-five proton decay operators, and discusses phenomenological implications and limitations of the local approach. The study highlights the interplay between Higgs flux, local Wilson lines, and the localisation of wavefunctions, and outlines avenues for extending to more realistic models incorporating monodromies and global consistency.

Abstract

In F-theory GUTs interactions between fields are typically localised at points of enhanced symmetry in the internal dimensions implying that the coefficient of the associated operator can be studied using a local wavefunctions overlap calculation. Some F-theory SU(5) GUT theories may exhibit a maximum symmetry enhancement at a point to E8, and in this case all the operators of the theory can be associated to the same point. We take initial steps towards the study of operators in such theories. We calculate wavefunctions and their overlaps around a general point of enhancement and establish constraints on the local form of the fluxes. We then apply the general results to a simple model at a point of E8 enhancement and calculate some example operators such as Yukawa couplings and dimension-five couplings that can lead to proton decay.