4D F-theory Research Articles

Chiral Matter Multiplicities and Resolution-Independent Structure in 4D F-Theory Models

Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our approach is the conjecturally resolution-independent intersection pairing of the vertical part of the integer middle cohomology of smooth elliptic Calabi–Yau fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua. We illustrate our approach by analyzing vertical flux backgrounds for F-theory models with simple, simply-laced gauge groups and generic matter content, as well as models with U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {U}}(1)$$\\end{document} gauge factors. We explicitly analyze resolutions of these F-theory models in which the elliptic fiber is realized as a cubic in P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document} over an arbitrary (e.g., not necessarily toric) smooth base, and confirm the independence of the intersection pairing of the vertical part of the middle cohomology for the resolutions we study. In each model, we find that vertical flux backgrounds can produce nonzero multiplicities for a spanning set of anomaly-free chiral matter field combinations, suggesting that F-theory geometry imposes no additional linear constraints on allowed matter representations beyond those implied by 4D anomaly cancellation.

Chiral spectrum of the universal tuned (SU(3) × SU(2) × U(1))/ℤ6 4D F-theory model

We use the recently developed methods of arXiv:2108.07810 to analyze vertical flux backgrounds and associated chiral matter spectra in the 4D universal (SU(3) × SU(2) × U(1))/ℤ6 model introduced in arXiv:1912.10991, which is believed to describe the most generic family of F-theory vacua with tuned (SU(3) × SU(2) × U(1))/ℤ6 gauge symmetry. Our analysis focuses on a resolution of a particular presentation of the (SU(3) × SU(2) × U(1))/ℤ6 model in which the elliptic fiber is realized as a cubic in ℙ2 fibered over an arbitrary smooth threefold base. We show that vertical fluxes can produce nonzero multiplicities for all chiral matter families that satisfy 4D anomaly cancellation, which include as a special case the chiral matter families of the Minimal Supersymmetric Standard Model.

Large U(1) charges from flux breaking in 4D F-theory models

We study the massless charged spectrum of U(1) gauge fields in F-theory that arise from flux breaking of a nonabelian group. The U(1) charges that arise in this way can be very large. In particular, using vertical flux breaking, we construct an explicit 4D F-theory model with a U(1) decoupled from other gauge sectors, in which the massless/light fields have charges as large as 657. This result greatly exceeds prior results in the literature. We argue heuristically that this result may provide an upper bound on charges for light fields under decoupled U(1) factors in the F-theory landscape. We also show that the charges can be even larger when the U(1) is coupled to other gauge groups.

Gauge symmetry breaking with fluxes and natural Standard Model structure from exceptional GUTs in F-theory

We give a general description of gauge symmetry breaking using vertical and remainder fluxes in 4D F-theory models. The fluxes can break a geometric gauge group to a smaller group and induce chiral matter, even when the larger group admits no chiral matter representations. We focus specifically on applications to realizations of the Standard Model gauge group and chiral matter spectrum through breaking of rigid exceptional gauge groups E7, E6, which are ubiquitous in the 4D F-theory landscape. Supplemented by an intermediate SU(5) group, these large classes of models give natural constructions of Standard Model-like theories with small numbers of generations of matter in F-theory.

Natural F-theory constructions of standard model structure from E7 flux breaking

We describe a broad class of 4D F-theory models in which an ${E}_{7}$ gauge group is broken through fluxes to the standard model gauge group. These models are ubiquitous in the 4D F-theory landscape and can arise from flux breaking of most models with ${E}_{7}$ factors. While in many cases the ${E}_{7}$ breaking leads to exotic matter, there are large families of models in which the standard model gauge group and chiral matter representations are obtained through an intermediate SU(5) group. The number of generations of matter appearing in these models can easily be small. We demonstrate the possibility of getting three generations of chiral matter as the preferred matter content.

Faking gauge coupling unification in string theory

Gauge coupling unification misleads infrared observers if new gauge bosons do not simultaneously come into the spectrum. Though easy to engineer in gauge theory, the situation in string theory is nuanced due to moduli dependence. We study the possibility of faking gauge coupling unification in the context of 4D F-theory compactifications. Specifically, we formulate a sufficient condition that we call a strong calibration, under which seven-brane gauge couplings on homologically distinct divisors become equal at codimension one in K\"ahler moduli space. We prove that a strong calibration is preserved under appropriate topological transitions and that a pair of nonintersecting divisors each admitting a contraction can always be strongly calibrated. Within the tree ensemble [J. Halverson et al., Phys. Rev. D 96, 126006 (2017)], we find that $\ensuremath{\approx}77.12%$ of pairs of intersecting toric divisors can be strongly calibrated and $\ensuremath{\approx}3.22%$ can never be calibrated. Physically, this means that gauge coupling unification can be faked in most cases that we study, but in others it can not, which is surprising from a gauge theoretic perspective.

Charge completeness and the massless charge lattice in F-theory models of supergravity

We prove that, for every 6D supergravity theory that has an F-theory description, the property of charge completeness for the connected component of the gauge group (meaning that all charges in the corresponding charge lattice are realized by massive or massless states in the theory) is equivalent to a standard assumption made in F-theory for how geometry encodes the global gauge theory by means of the Mordell-Weil group of the elliptic fibration. This result also holds in 4D F-theory constructions for the parts of the gauge group that come from sections and from 7-branes. We find that in many 6D F-theory models the full charge lattice of the theory is generated by massless charged states; this occurs for each gauge factor where the associated anomaly coefficient satisfies a simple positivity condition. We describe many of the cases where this massless charge sufficiency condition holds, as well as exceptions where the positivity condition fails, and analyze the related global structure of the gauge group and associated Mordell-Weil torsion in explicit F-theory models.

Machine learning and algebraic approaches towards complete matter spectra in 4d F-theory

Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To quantify jumps of these cohomologies, we first generate 1.8 million pairs of line bundles and curves embedded in dP3, for which we compute the cohomologies. A white-box machine learning approach trained on this data provides intuition for jumps due to curve splittings, which we use to construct additional vector-like Higgs-pairs in an F-Theory toy model. We also find that, in order to explain quantitatively the full dataset, further tools from algebraic geometry, in particular Brill-Noether theory, are required. Using these ingredients, we introduce a diagrammatic way to express cohomology jumps across the parameter space of each family of matter curves, which reflects a stratification of the F-theory complex structure moduli space in terms of the vector-like spectrum. Furthermore, these insights provide an algorithmically efficient way to estimate the possible cohomology dimensions across the entire parameter space.

On the elliptic Calabi-Yau fourfold with maximal h1,1

In this paper, we explicitly construct the smooth compact base threefold for the elliptic Calabi-Yau fourfold with the largest known h1,1 = 303 148. It is generated by blowing up a smooth toric “seed” base threefold with (E8, E8, E8) collisions. The 4d F-theory compactification model over it has the largest geometric gauge group, {E}_8^{2561}times {F}_4^{7576}times {G}_2^{20168}times mathrm{SU}{(2)}^{30200}, and the largest number of axions, 181 820, in the known 4d mathcal{N} = 1 supergravity landscape. We also prove that there are at least 110015 048≈ 7.5 × 1045 766 different flip and flop phases of this base threefold. Moreover, we find that many other base threefolds with large h1,1 in the 4d F-theory landscape can be constructed in a similar way as well.

Learning non-Higgsable gauge groups in 4D F-theory

We apply machine learning techniques to solve a specific classification problem in 4D F-theory. For a divisor D on a given complex threefold base, we want to read out the non-Higgsable gauge group on it using local geometric information near D. The input features are the triple intersection numbers among divisors near D and the output label is the non-Higgsable gauge group. We use decision tree to solve this problem and achieved 85%-98% out-of-sample accuracies for different classes of divisors, where the data sets are generated from toric threefold bases without (4,6) curves. We have explicitly generated a large number of analytic rules directly from the decision tree and proved a small number of them. As a crosscheck, we applied these decision trees on bases with (4,6) curves as well and achieved high accuracies. Additionally, we have trained a decision tree to distinguish toric (4,6) curves as well. Finally, we present an application of these analytic rules to construct local base configurations with interesting gauge groups such as SU(3).

Scanning the skeleton of the 4D F-theory landscape

Using a one-way Monte Carlo algorithm from several different starting points, we get an approximation to the distribution of toric threefold bases that can be used in four-dimensional F-theory compactification. We separate the threefold bases into “resolvable” ones where the Weierstrass polynomials (f, g) can vanish to order (4, 6) or higher on codimension-two loci and the “good” bases where these (4, 6) loci are not allowed. A simple estimate suggests that the number of distinct resolvable base geometries exceeds 103000, with over 10250 “good” bases, though the actual numbers are likely much larger. We find that the good bases are concentrated at specific “end points” with special isolated values of h1,1 that are bigger than 1,000. These end point bases give Calabi-Yau fourfolds with specific Hodge numbers mirror to elliptic fibrations over simple threefolds. The non-Higgsable gauge groups on the end point bases are almost entirely made of products of E8, F4, G2 and SU(2). Nonetheless, we find a large class of good bases with a single non-Higgsable SU(3). Moreover, by randomly contracting the end point bases, we find many resolvable bases with h1,1(B) ∼ 50-200 that cannot be contracted to another smooth threefold base.

ℙ 1 $$ {\mathrm{\mathbb{P}}}^1 $$ -bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models

We explore a large class of F-theory compactifications to four dimensions. We find evidence that gauge groups that cannot be Higgsed without breaking supersymmetry, often accompanied by associated matter fields, are a ubiquitous feature in the landscape of ${\cal N} = 1$ 4D F-theory constructions. In particular, we study 4D F-theory models that arise from compactification on threefold bases that are $\mathbb{P}^1$ bundles over certain toric surfaces. These bases are one natural analogue to the minimal models for base surfaces for 6D F-theory compactifications. Of the roughly 100,000 bases that we study, only 80 are weak Fano bases in which there are no automatic singularities on the associated elliptic Calabi-Yau fourfolds, and 98.3% of the bases have geometrically non-Higgsable gauge factors. The $\mathbb{P}^1$-bundle threefold bases we analyze contain a wide range of distinct surface topologies that support geometrically non-Higgsable clusters. Many of the bases that we consider contain $SU(3)\times SU(2)$ seven-brane clusters for generic values of deformation moduli; we analyze the relative frequency of this combination relative to the other four possible two-factor non-Higgsable product groups, as well as various other features such as geometrically non-Higgsable candidates for dark matter structure and phenomenological ($SU(2)$-charged) Higgs fields.

Non-Higgsable clusters for 4D F-theory models

We analyze non-Higgsable clusters of gauge groups and matter that can arise at the level of geometry in 4D F-theory models. Non-Higgsable clusters seem to be generic features of F-theory compactifications, and give rise naturally to structures that include the nonabelian part of the standard model gauge group and certain specific types of potential dark matter candidates. In particular, there are nine distinct single nonabelian gauge group factors, and only five distinct products of two nonabelian gauge group factors with matter, including $SU(3) \times SU(2)$, that can be realized through 4D non-Higgsable clusters. There are also more complicated configurations involving more than two gauge factors; in particular, the collection of gauge group factors with jointly charged matter can exhibit branchings, loops, and long linear chains.

A note on zeroes of superpotentials in F-theory

We discuss the dependence of superpotential terms in 4D F-theory on moduli parameters. Two cases are studied: the dependence on world-filling 3-brane positions and the dependence on 2-form VEVs. In the first case there is a zero when the 3-brane hits the divisor responsible for the superpotential. In the second case, which has been extensively discussed by Witten in 3D M-theory, there is a zero for special values of 2-form VEVs when the M-theory divisor contains non-trivial 3-cycles. We give an alternative derivation of this fact for the special case of F-theory.

Dualities and phase transitions for Calabi-Yau threefolds and fourfolds

We review several aspects of heterotic, type II, F-theory, and M-theory compactifications on Calabi-Yau three-folds and fourfolds. In the context of dualities we focus on the heterotic gauge structure determined by the various types of fibration relevant in the framework of heterotic/type II duality in D=4 as well as 4D F-theory. We also consider transitions between Calabi-Yau manifolds in both three and four dimensions and review some of the consequences for the behavior of the superpotential.