G_4 flux, chiral matter and singularity resolution in F-theory compactifications

TL;DR

This work constructs globally defined chirality-inducing $G_4$-fluxes in F-theory compactifications with gauge group $SU(5)\times U(1)_X$ via a $U(1)$-restricted Tate model. It resolves the associated singularities through blow-ups, identifies the resolved fibre structure over matter curves and Yukawa points, and derives the chiral indices for both $SU(5)$-charged matter and $SU(5)$ singlets charged under $U(1)_X$, all while enforcing global consistency such as D3-tadpole cancellation and flux quantisation. The authors define the flux as $G_4=F_X\wedge w_X$ with $F_X\in H^{1,1}(B_3)$ and show that the resulting chiral index is $\\chi(R_q)= q\int_{{\cal C}_{R}} F_X$, a formula that matches local spectral-cover results for charged matter while revealing global corrections for singlets and tadpoles. They provide a concrete three-generation model that satisfies all global constraints and demonstrate how the flux framework generalises to non-restricted SU(5) models, thereby enabling globally consistent F-theory GUT constructions with chiral matter.

Abstract

We construct a set of chirality inducing G_4-fluxes in global F-theory compactifications on Calabi-Yau four-folds. Special emphasis is put on models with gauge group SU(5) x U(1)_X relevant in the context of F-theory GUT model building, which are described in terms of a U(1)-restricted Tate model. In this type of constructions, the G_4-flux arises in a manner completely analogous to the U(1)_X gauge potential. We describe in detail the resolution by blow-up of the various singularities responsible for the U(1)_X factor and the standard SU(5) gauge group and match the result with techniques applied in the context of toric geometry. This provides an explicit identification of the structure of the resolved fibre over the matter curves and over the enhancement points relevant for Yukawa couplings. The U(1)_X flux induces a chiral matter spectrum. We compute the chiral index both of SU(5) charged matter and of SU(5) singlets charged only under U(1)_X localised on curves which are not contained in the SU(5) locus. We furthermore discuss global consistency conditions such as D3-tadpole cancellation, D-term supersymmetry and Freed-Witten quantisation. The U(1)_X gauge flux is a global extension of a class of split spectral cover bundles. It constitutes an essential ingredient in the construction of globally defined F-theory compactifications with chiral matter. We exemplify this in a three-generation SU(5) x U(1)_X model whose flux satisfies all of the above global consistency conditions. We also extend our results to chiral fluxes in models without U(1) restriction.

Figures (2)

• Figure 1: The $\mathbb{P}^1_{\,}$-structure and splitting process for triangulation $T_{11}$. The dashed lines encircling one or several $\mathbb{P}^1_{\,}$s in the 2nd and 3rd row correspond to the ones of the top diagram, each identified by their colour. Those $\mathbb{P}^1_{\,}$s which are marked by a double index with two numbers always have multiplicity $2$, except $\mathbb{P}^1_{\,24}$ in the $\mathbf{10}$$\mathbf{10}$$\mathbf{5}$-diagram, which has multiplicity $3$. All other $\mathbb{P}^1_{\,}$s have multiplicity $1$.
• Figure 2: Schematic drawing of the intersection of the generic and degenerate elliptic fibre with the divisors $S$ and $Z$ in the triangulation $T_{11}$. The green and blue crosses indicate the intersection points of $Z$ and $S$, respectively, with the fibre. On the $SU(2)$-curve $S$ itself becomes a $\mathbb{P}^1_{\,}$. Note that enhancement points (Yukawa interactions) are ignored in this picture.