Geometric Engineering in Toric F-Theory and GUTs with U(1) Gauge Factors

TL;DR

Braun, Grimm, and Keitel develop a toric-geometric framework to construct global Calabi–Yau elliptic fourfolds with specified bases and gauge groups, notably SU(5) GUTs with multiple $U(1)$ factors. They classify toric sections and SU(5) tops, compute the toric Mordell–Weil group to bound the number of $U(1)$ factors, and derive systematic $U(1)$ charge patterns for $oldsymbol{5}$- and $oldsymbol{10}$-matter, including a no-go theorem for different antisymmetric charges in hypersurface models. The construction uses an auxiliary polytope to complete to reflexive polytopes and imposes a flatness constraint to avoid tensionless strings, with explicit flat embeddings over bases such as ${f P}^3$ and ${f P}^1 imes{f P}^2$. The work enables potential large-scale scans of global F-theory GUTs with abelian factors and lays groundwork for incorporating $G$-flux for chirality and exploring complete-intersection generalizations.

Abstract

An algorithm to systematically construct all Calabi-Yau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data consists of a top over each toric divisor in the base together with compactification data giving the embedding into a reflexive polytope. The allowed choices of compactification data are integral points in an auxiliary polytope. In order to ensure the existence of a low-energy gauge theory, the elliptic fibration must be flat, which is reformulated into conditions on the top and its embedding. In particular, flatness of SU(5) fourfolds imposes additional linear constraints on the auxiliary polytope of compactifications, and is therefore non-generic. Abelian gauge symmetries arising in toric F-theory compactifications are studied systematically. Associated to each top, the toric Mordell-Weil group determining the minimal number of U(1) factors is computed. Furthermore, all SU(5)-tops and their splitting types are determined and used to infer the pattern of U(1) matter charges.

Figures (9)

• Figure 1: Systematical approach to constructing compact F-theory backgrounds with specified gauge group and base space.
• Figure 2: The $16$ reflexive polygons. $F_i$ and $F_{17-i}$ are dual for $i=0,\dots,6$, and self-dual for $i=7,\dots,10$. The corresponding toric surfaces are also known as $F_1={\mathop{ {\mathbb{P}}}\nolimits}^2$, $F_2={\mathop{ {\mathbb{P}}}\nolimits}^1\times{\mathop{ {\mathbb{P}}}\nolimits}^1$, $F_3=dP_1$, $F_4={\mathop{ {\mathbb{P}}}\nolimits}^2[1,1,2]$, $F_5=dP_2$, $F_7=dP_3$, $F_{10}={\mathop{ {\mathbb{P}}}\nolimits}^2[1,2,3]$, where $dP_n$ are the del Pezzo surfaces obtained by blowing up $\mathbb{P}^2$ at $n$ points. Vertices defining toric sections are colored red. First derived in Figure 1 of 2012arXiv1201.0930G.
• Figure 3: The $SU(5)$ tops based on the $16$ reflexive polygons. Numbers next to boundary points of the facet in the $z=1$ plane indicate which toric sections intersect the associated exceptional divisor.
• Figure 4: (continued) The $SU(5)$ tops $\tau_{i,j}$ based on the $16$ reflexive polygons. For each reflexive polygon (the fiber polygon at $z=0$), the admissible facets at $z=1$ are listed. Below each the values of $z^*$ on the vertices of the dual polygon (in clockwise order, starting at the "y"-axis) are given, which also an equivalent way of specifying the top. See discussion at the beginning of \ref{['sec:tops']}.
• Figure 5: The different splits for the case in which $\sigma_0$ denotes the zero section and $\sigma_1$ is one of possibly more independent Mordell-Weil generators. In the case of a single $U(1)$, the $i$-$(5-i)$-split and the $(5-i)$-$i$-split are equivalent under to the $\mathbb{Z}_2$ outer automorphism of $\mathfrak{su}(5)$.