E-string spectrum and typical F-theory geometry

TL;DR

This work investigates strongly coupled sectors in typical 4D F-theory geometries by resolving a non-minimal Weierstrass model with a non-flat fiber, identifying the 6D E-string spectrum from M2-brane wrapping modes on a gdP2 fiber, and analyzing the 4D spectrum obtained by compactifying these modes on a curve with flux. It links landscape features—non-Higgsable groups and dominant gauge factors—to conformal matter, providing explicit branching through $E_8$ weights and a topological-twist framework for 4D reductions. The results illuminate how E-string sectors can be embedded into 4D models, offering a concrete path to include strongly coupled matter in F-theory constructions and highlighting the central role of non-flat fibrations in realizing these degrees of freedom. The study also sets up a methodology combining toric geometry, non-flat fiber resolutions, and M-theory dual pictures to connect geometric data with 6D/4D spectra and their potential phenomenological implications.

Abstract

In recent scans of 4D F-theory geometric models, it was shown that a dominant majority of the base geometries only support SU(2), $G_2$, $F_4$ and $E_8$ gauge groups. Moreover, most of these gauge groups are shown to couple to strongly coupled "conformal matter" sectors. For example, the $E_8$ gauge group can couple to the compactification of 6D E-string theory on a complex curve. In this paper, we initiate the investigation of these strongly coupled sectors by studying the spectrum of 6D E-string theory. We construct a resolved elliptic Calabi-Yau threefold of a non-minimal Weierstrass model, which contains a non-flat fiber with the topology of generalized del Pezzo surface. The spectrum of E-string theory then arises from M2 brane wrapping modes on various 2-cycles on the non-flat fiber. Finally, we discuss the compactification of these fields to 4D.

Figures (8)

• Figure 1: The factor $d_k=\frac{N_{\rm up}(a_k)}{N_{\rm down}(a_{k+1})}$ in terms of $h^{1,1}(B)=k$.
• Figure 2: In the left picture, we have a 4D F-theory geometry with two stack of $E_6$ branes intersecting each other at the complex curve $\Sigma$. In the decoupling limit, the localized 4D theory $\mathcal{T}_\Sigma$ becomes a 6D theory in $\mathbb{R}^{5,1}$, see the right picture. This 6D theory is exactly the 6D $(E_6,E_6)$ SCFT localized at the intersection point $p$ of two curves carrying $E_6$ gauge groups.
• Figure 3: The intersection relation of curves on the vertical divisor $D_1$. Each node denotes a curve which is the intersection of $D_1$ with the corresponding divisor $D$. These curves then form an affine $E_8$ Dynkin diagramon $D_1$. The exceptional divisors are labelled by $E_1,E_2,\dots,E_8$.
• Figure 4: The intersection of curves on the non-flat fiber $S_{nf}$. Each node denotes a rational curve which is the intersection of $S_{nf}$ with the corresponding divisor $D$. The number on a node corresponds to the triple intersection number $S_{nf}\cdot D^2$, and the edges between nodes correspond to an intersection point.
• Figure 5: The intersection of the non-flat fiber with the exceptional divisors of $E_8$.