G4-Flux and Standard Model Vacua in F-theory

TL;DR

This work develops a base-independent framework to classify vertical $G_4$-fluxes on elliptically fibred Calabi–Yau fourfolds with gauge group $SU(3)\times SU(2)\times U(1)\times U(1)$, using a toric construction with fibre a cubic in ${\rm Bl}_2{\mathbb P}^2$ and tops to realize non-abelian factors. It expresses chiral indices in terms of intersections on the base via matter-surface classes in $H^{(2,2)}_{\rm vert}(Y_4)$, computed with primary decomposition, and shows non-abelian anomalies cancel automatically for any consistent vertical flux while abelian anomalies are handled using Green–Schwarz terms; missing singlet chiralities are inferred by anomaly matching. The authors provide a complete, base-independent flux basis, derive explicit chiralities, and perform a finite scan over simple bases ${\cal B}$ to realize Standard-Model-like spectra, finding a globally consistent flux on ${\cal B}={\rm Bl}_1\mathbb{P}^3$ that yields the SM spectrum plus an extra triplet pair, which can be lifted via recombination. They also present a concrete almost-Standard-Model example with a specific flux and discuss implications for proton stability, Yukawa couplings, and routes to fully realistic models. Overall, the paper delivers a practical, geometry-driven path for constructing and analyzing realistic F-theory vacua with multiple $U(1)$ factors and sheds light on how anomaly cancellation manifests in the vertical flux framework.

Abstract

We study the geometry of gauge fluxes in four-dimensional F-theory vacua with gauge group SU(3)xSU(2)xU(1)xU(1) and its implications for phenomenology. The models are defined by a previously introduced class of elliptic fibrations whose fibre is given as a cubic hypersurface in ${\rm Bl}_2{\mathbb P}^2$, with the non-abelian gauge group factors SU(3)xSU(2) engineered torically via the top construction. To describe gauge fluxes on these fibrations we provide a classification of the primary vertical middle cohomology group in a fashion valid for any choice of base space. Using the ideal theoretic technique of primary decomposition we compute the cohomology classes of the matter surfaces associated with states charged under the non-abelian gauge group. These expressions allow us to interpret the cancellation of the pure and mixed non-abelian anomalies geometrically as a result of the general form of the matter surfaces, without reference to a specific type of gauge flux. Explicit results for the chiral indices of all matter states are obtained in terms of intersection numbers of the base and can be directly applied to any choice of base consistent with the fibration. As a demonstration we scan for globally consistent F-theory vacua on $\mathbb P^3$, ${\rm Bl}_1\mathbb P^3$ and ${\rm Bl}_2 \mathbb{P}^3$, and find a globally consistent flux configuration with the chiral Standard Model spectrum plus an extra triplet pair, which may be lifted by a recombination process.