On $E_8$ and F-Theory GUTs

TL;DR

This work reevaluates the role of $E_8$ in constraining global F-theory GUTs with $SU(5)$ and Abelian factors. By incorporating 15 beyond-$E_8$ singlets that enable gauge-invariant cubic couplings, the authors extend the standard $E_8$ spectrum to a larger class of theories, and show that all 27 literature fibrations with flat, generic bases embed in this extended set. They develop a systematic classification using a Smith-form analysis of the singlet charges, and find that while only a subset of models lie within Higgsed-$E_8$ realizations, a substantial majority are encompassed by the extended framework. The paper also explores phenomenological implications, notably a $oldsymbol{}$ matter parity, and discusses heterotic dual relations, illustrating that beyond-$E_8$ singlets can correspond to heterotic singularities in some cases but not universally. Overall, the results suggest that a carefully extended $E_8$-based approach can capture the landscape of global F-theory GUT spectra and guide phenomenological model building.

Abstract

We study correlations between the massless field spectra of F-theory fibrations supporting an $SU(5)$ gauge symmetry extended by Abelian symmetries and the spectra that arise from the group $E_8$. The adjoint representation of $E_8$ leads to six different classes of matter spectra upon Higgsing $E_8$ to $SU(5)\times U(1)^n$. Of 27 different smooth F-theory elliptic fibrations constructed in the literature, satisfying certain genericness and flatness criteria, the matter spectrum of only one can be embedded in these six theories, thereby apparently ruling out any connection. We define an extension of the spectrum arising from the adjoint of $E_8$ by introducing new $SU(5)$-singlet fields with Abelian charges such that there exists a cubic gauge invariant coupling between any three representations. Higgsing by these new singlets leads to a further 20 classes of spectra, and we find that all the F-theory fibrations can then be embedded in this extended set. These results show that $E_8$, when extended in this specific way, may still have a role to play in controlling the possible matter spectra in F-theory. We give an explicit geometric example of the presence of the extending singlets and their Higgsing. We discuss some phenomenological applications of the new set of theories, in particular due to the existence of a ${\mathbb Z}_2$ matter parity. Finally we make some comments on the Heterotic duals of the F-theory fibrations which extend $E_8$ in this way.