Elliptic fibrations for SU(5) x U(1) x U(1) F-theory vacua

TL;DR

This work constructs elliptically fibered Calabi–Yau 4-folds with Mordell–Weil rank two, yielding F-theory vacua with $U(1) \times U(1)$ gauge symmetry alongside non-Abelian sectors. The authors realize the two abelian factors using a $\mathrm{Bl}^2 \mathbb{P}^2[3]$-fibration and provide a birational Weierstrass form $y^2 = x^3 + f x z^4 + g z^6$, together with a complete resolution to obtain two independent sections whose Shioda map generates the $U(1)$ gauge potentials; they compute the charged singlet spectrum and their Yukawa couplings, and show compatibility with toric tops to realize $SU(5) \times U(1) \times U(1)$. An explicit SU(5) model is constructed via tops on this fibration, giving precise matter curves, their $U(1)$ charges (e.g. $\mathbf{10}_{-1,2}$ and various $\mathbf{5}$s), and codimension-three Yukawa couplings, while ensuring flatness conditions and exploring brane recombination to a $\mathrm{Bl}^1 \mathbb{P}^2[3]$-fibration that yields $SU(5) \times U(1)$. The paper also discusses the relation to $SU(5) \times U(1)$ models and outlines an expanded program of tops, including results to be reported in a companion work. Overall, the framework provides a systematic geometric construction of abelian and non-Abelian gauge sectors in F-theory with detailed spectra and interactions, enabling phenomenologically relevant GUT realizations.

Abstract

Elliptic Calabi-Yau fibrations with Mordell-Weil group of rank two are constructed. Such geometries are the basis for F-theory compactifications with two abelian gauge groups in addition to non-abelian gauge symmetry. We present the elliptic fibre both as a Bl^2P^2[3]-fibration and in the birationally equivalent Weierstrass form. The spectrum of charged singlets and their Yukawa interactions are worked out in generality. This framework can be combined with the toric construction of tops to implement additional non-abelian gauge groups. We utilise the classification of tops to construct SU(5) x U(1) x U(1) gauge symmetries systematically and study the resulting geometries, presenting the defining equations, the matter curves and their charges, the Yukawa couplings and explaining the process in detail for an example. Brane recombination relates these geometries to a Bl^1P^2[3]-fibration with a corresponding class of SU(5) x U(1) models. We also present the SU(5) tops based on the elliptic fibre Bl^1P_[1,1,2][4], corresponding to another class of SU(5) x U(1) models.

Figures (4)

• Figure 1: The toric polygon to $\textmd{Bl}^{2}\mathbb{P}^{2}$ and its dual. On the dual one we only indicated the monomials of the vertices and omitted powers of $s_0$ and $s_1$.
• Figure 2: The fibre structure over the singlet curves $C_{\mathbf 1^{(1)}}$ and $C_{\mathbf 1^{(2)}}$. Green corresponds to the zero section $S_0$, blue to $S_1$ and red to $S_2$.
• Figure 3: The fibre structure over the Yukawa points $\mathbf 1_{-5,0}$$\mathbf 1_{5,10} \mathbf 1_{0,-10}$ and $\mathbf 1_{5,0} \mathbf 1_{-5,-5} \mathbf 1_{0,5}$. Green corresponds to the zero section $S_0$, blue to $S_1$ and red to $S_2$.
• Figure 4: The toric polygon to $\textmd{Bl}^{1}\mathbb P_{[1,1,2]}$ and its dual. On the dual one we only indicated the monomials of the vertices and omitted powers of $s_1$.