Non-Higgsable Gauge Groups in F-Theory

• Distribution of Non-Higgsable Gauge Groups are rigid gauge sectors in F-theory, defined by local geometric constraints on elliptic Calabi–Yau bases.
• Statistical and algorithmic studies show these clusters appear ubiquitously, with occurrence frequency growing linearly with the base's h^(1,1) and forming complex quiver structures.
• Their phenomenological implications include natural Standard Model sectors and hidden dark matter candidates, while machine learning aids in predicting and classifying these gauge groups.

The distribution of non-Higgsable gauge groups refers to the statistical and geometric patterns by which gauge sectors appear rigidly (i.e., cannot be removed or "Higgsed" via geometric or complex structure deformation) in string compactifications—most extensively in F-theory and its landscape of flux vacua. Non-Higgsable clusters occur because the base geometry of an elliptic Calabi–Yau dictates, through vanishing orders of Weierstrass coefficients, the unavoidable presence of certain gauge algebras on seven-branes. The spectrum of non-Higgsable gauge factors, their combinatorial linkage, and frequency in the landscape, as well as associated phenomenological features such as Standard Model realizations and hidden sectors, are central areas of current research.

1. Geometric Origin and Definition of Non-Higgsable Gauge Groups

Non-Higgsable gauge groups in F-theory arise on codimension-one divisors in the base of an elliptically fibered Calabi–Yau, where local geometric constraints force the Weierstrass model coefficients $f,g$ to vanish to certain orders. Kodaira's classification then dictates the fiber type and the associated gauge algebra. If these vanishing orders are dictated locally by geometry and cannot be tuned away by moduli deformations, the corresponding gauge group is "non-Higgsable".

Mathematically, for a divisor $D$, $f$ and $g$ admit expansions \begin{align} f &= f_0 + f_1 z + f_2 z^2 + \cdots \ g &= g_0 + g_1 z + g_2 z^2 + \cdots \end{align} Vanishing orders corresponding to fiber types such as III, IV, I$^*_0$, IV$^*$, etc., enforce the presence of gauge algebras $\mathfrak{su}_2$, $\mathfrak{su}_3$, $\mathfrak{so}_7$, $\mathfrak{e}_6$, $\mathfrak{e}_7$, or $\mathfrak{e}_8$, as per the Kodaira table (Halverson et al., 2015). In these settings, no holomorphic deformation of $f, g$ can erase these singularities.

2. Statistical Distribution Across the Landscape

Systematic studies—via analytic, Monte Carlo, and algorithmic enumeration—reveal that non-Higgsable gauge groups are ubiquitous and often dominate the landscape in complex enough base geometries. Explicitly, in the class of $\mathbb{P}^1$-bundle threefold bases over toric surfaces, over 98% of the roughly 100,000 studied bases host non-Higgsable gauge factors (Halverson et al., 2015). Monte Carlo exploration of toric threefold bases finds that the number of non-Higgsable gauge factors grows approximately linearly with $h^{1,1}(B)$: $N_\mathrm{gauge} \simeq c \cdot h^{1,1}(B)$ with $c \sim 0.35$–$0.4$ in the range $h^{1,1}(B)\in[40,100]$ (Taylor et al., 2015).

Algorithmic universality approaches (in ensembles as large as $10^{755}$ distinct geometries) demonstrate that non-Higgsable clusters appear with probability at least $1-1.01\times 10^{-755}$; high-rank gauge groups and multiple $E_8$ factors occur with very high probability (Halverson et al., 2017).

3. Restricted Gauge Group Types and Quiver Structures

The set of non-Higgsable gauge group algebras realized in 4D F-theory compactifications is limited: nine distinct single nonabelian factors and exactly five distinct two-factor combinations with jointly charged matter are possible, notably including $SU(3)\times SU(2)$ matching the Standard Model (Morrison et al., 2014). Beyond isolated clusters, intersections of divisors yield more intricate quiver diagrams featuring branching, loops, and long chains. For example, linear chains with up to eleven SU(3)-type clusters have been constructed in toric bases (Morrison et al., 2014).

These topological and combinatorial features are encoded in rules tracing the dependence of vanishing orders of $f$ and $g$ on the canonical class $K^{(a)}$, normal bundle $N^{(a)}$, intersection curves $C_{ab}$ and "propagated" vanishing orders from neighboring divisors: $$F_k^{(a)} = -4K^{(a)} + (4-k)N^{(a)} - \sum_{b\neq a} \phi_b C_{ab}$$

$$G_k^{(a)} = -6K^{(a)} + (6-k)N^{(a)} - \sum_{b\neq a} \gamma_b C_{ab}$$

(Morrison et al., 2014)

4. Factorization in Flux Compactification Statistics

The distribution of non-Higgsable gauge groups in the flux landscape is quantitatively described via an index density over scanning flux subspaces. For models with fixed gauge group and number of generations $N_\mathrm{gen}$: $d\mu_I = \frac{(2\pi L_*)^{K/2}}{(K/2)!} \, \rho_I$ where $K$ is determined by the geometry (and decreased when extra gauge groups are present), $L_*$ is the available D3-brane charge, and $\rho_I$ encodes the moduli space measure (Braun et al., 2014).

The total vacuum number factorizes as

$$N_\mathrm{vac}\sim \exp\left[\frac{K}{6}\right]\,e^{-c N_{\rm gen}^2}$$

where the exponential $K/6$ term is extremely sensitive to the gauge group type; for SU(5) unification, the reduction of $K$ by about 7000 compared to the "no gauge group" case leads to a suppression factor $\exp(-1000)\ll 1$ in vacuum counting. Thus, higher-rank gauge groups are statistically costly.

5. Phenomenological and Model-building Implications

Non-Higgsable clusters naturally realize the non-abelian sector of the Standard Model, specifically through $SU(3)\times SU(2)$ constructions. There are three geometric scenarios for QCD and electroweak non-Higgsability: IV–III, IV–IV$_m$, and IV–I$_2$ intersection types; the IV–III case uniquely reproduces Standard Model matter content with $(3,2)$ bifundamental, $(3,1)$, $(1,2)$, and singlets (Grassi et al., 2014).

Non-Higgsable clusters also generically give rise to hidden sectors (with gauge factors such as $G_2$, $SO(7)$, additional $SU(2)$, etc.), which may be candidates for dark matter. Their presence is a robust feature of generic base geometries, not fine-tuned or exceptional (Morrison et al., 2014, Halverson et al., 2015).

6. Non-Higgsable Abelian Factors and Global Gauge Group Structure

Non-Higgsable abelian gauge factors (U(1)) are subject to additional geometric constraints. A non-Higgsable U(1) arises when the Newton polytopes for $f$ and $g$ degenerate to one-dimensional alignment, so that very few monomials are present: $$\text{Non-Higgsable U(1):} \quad h^0(-6K) - h^0(-3K) - h^0(-2K)\le 0$$ Toric bases cannot admit non-Higgsable U(1) gauge groups, as such alignment would necessitate singularities that cannot be resolved (Wang, 2016).

Globally, the gauge group in F-theory compactifications is determined by the interplay of non-abelian divisors and Mordell–Weil sections, formalized via the Shioda map. The precise global gauge group is

$$G_{\rm glob} = [U(1)\times \widetilde{G}]/\mathbb{Z}_k$$

where the quotient is fixed by fractional U(1) charge assignments to non-abelian matter (with constraints derived from intersection theory and the structure of the singular fibers) (Cvetic et al., 2017).

7. Machine Learning Classification and Predictive Analytics

Recent approaches leverage machine learning (decision trees) to empirically classify non-Higgsable gauge groups from local intersection data. By training decision trees on triple intersection numbers between divisors and their neighbors, prediction accuracies range from 85% to 98% on various toric divisors (Wang et al., 2018). The extracted analytic rules—for instance, inequalities on intersection numbers—provide interpretable checks and enable the construction of explicit geometric configurations, such as infinite SU(3) chains.

This data-driven approach not only confirms the distribution rules but also paves the way for global classification and deeper understanding of the structure of non-Higgsable clusters in string model landscapes.

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The distribution of non-Higgsable gauge groups emerges as a robust, geometrically dictated phenomenon within the landscape of string compactifications. Their ubiquity, restricted algebra types, intricate quiver structures, and profound implications for phenomenology and model building provide compelling evidence that rigid gauge sectors are generic in realistic F-theory vacua. The statistical cost associated with higher-rank clusters and the precise rules governing abelian factor appearance further refine the understanding of their role in vacua counting and model universality. Machine learning-based classification is becoming instrumental for explicit geometric identification and serves to validate and expand the analytic framework describing these gauge sectors.