Chiral Four-Dimensional F-Theory Compactifications With SU(5) and Multiple U(1)-Factors

TL;DR

This work develops a comprehensive geometric framework for four-dimensional F-theory compactifications with a rank-two Mordell–Weil group, focusing on Calabi–Yau fourfolds with a dP2 elliptic fiber. It provides explicit techniques to classify resolved fibrations, determine matter representations for U(1)^2 and SU(5)×U(1)^2, and construct general G4-fluxes, including new insights for non-holomorphic zero sections via M-/F-theory 3D CS duality and KK-state corrections. The authors compute the vertical cohomology ring, Euler numbers, and second Chern classes, and present explicit fluxes and chiralities (with anomaly cancellation) for bases such as B= P^3, including toric realizations. They also address non-flat fiber loci and propose flux-based cures to avoid phenomenologically problematic massless towers. Overall, the paper delivers a robust, non-toric-dependent methodology to realize chiral 4D spectra with multiple U(1) factors and SU(5) GUT sectors in global F-theory models, bridging geometric data with 3D CS analyses and anomaly constraints.

Abstract

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP_2. We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)xU(1) and SU(5)xU(1)xU(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B=P^3. We determine the general G_4-flux that belongs to H^{(2,2)}_V of the resolved Calabi-Yau fourfolds. As a by-product, we derive for the first time all conditions on G_4-flux in general F-theory compactifications with a non-holomorphic zero section. These conditions have to be formulated after a circle reduction in terms of Chern-Simons terms on the 3D Coulomb branch and invoke M-theory/F-theory duality. New Chern-Simons terms are generated by Kaluza-Klein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield non-vanishing results precisely for fourfolds with a non-holomorphic zero section. Taking into account the new Chern-Simons terms, all 4D matter chiralities are determined via 3D M-theory/F-theory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.

Figures (10)

• Figure 1: Fan of $dP_2$. The coordinates corresponding to its rays are indicated.
• Figure 2: Each dot corresponds to a $dP_2$-fibration over $\mathbb{P}^3$ with generic Calabi-Yau $\hat{X}$.
• Figure 3: $I_2$-fiber from resolving a codimension two singularity of the fibration of $\hat{X}$.
• Figure 4: The region of allowed values for $(n_7,n_9)$ from figure \ref{['fig:XP3n7n9region']}. On the entire region, there are two conditions on the flux. In the interior of this region, \ref{['eq:G4solP3']} holds. On the red and the blue boundary, there are only four independent $(2,2)$-forms in the expansion \ref{['eq:G4expansionP3']}. On the blue boundary, $\hat{s}_P$ is holomorphic.
• Figure 5: Each dot represents a $dP_2$-fibration over $\mathbb{P}^3$ with generic Calabi-Yau $\hat{X}_{\text{SU}(5)}$. The red region is the set of $dP_2$-fibration with generic $\hat{X}$ without SU(5), cf. figure \ref{['fig:XP3n7n9region']}.