A Quadrillion Standard Models from F-theory

TL;DR

This paper constructs approximately one quadrillion globally consistent F-theory compactifications that realize the Standard Model spectrum, using algebraic geometry to simplify the global consistency conditions.

Contribution

It provides an explicit method to generate an enormous number of Standard Model-like string vacua via a single geometric criterion on elliptically fibered Calabi--Yau fourfolds.

Findings

Over 10^15 distinct compactifications identified

Global consistency reduces to a single geometric criterion

Criterion depends only on the associated polytope for toric bases

Abstract

We present an explicit construction of ${\cal O}(10^{15})$ globally consistent string compactifications that realize the exact chiral spectrum of the Standard Model of particle physics with gauge coupling unification in the context of F-theory. Utilizing the power of algebraic geometry, all global consistency conditions can be reduced to a single criterion on the base of the underlying elliptically fibered Calabi--Yau fourfolds. For toric bases, this criterion only depends on an associated polytope and is satisfied for at least ${\cal O}(10^{15})$ bases, each of which defines a distinct compactification.

Equations

$$7.6\times 10^{13}\lesssim N_{\text{SM}}^{\text{toric}}\lesssim 1.6\times 10^{16}.$$

$$\begin{array}{r}P:=s_1{u}^3+s_2{u}^{2}v+s_{3}u{v}^2+s_5{u}^{2}w+s_{6}uvw+s_{9}v{w}^2.\end{array}$$

$$\displaystyle\begin{array}[]{lll}[s_{1}]=3\,\overline{\cal K}-{\cal S}_{7}-{\cal S}_{9}\,,&[s_{2}]=2\,\overline{\cal K}-{\cal S}_{9}\,,&[s_{6}]=\overline{\cal K}\,,\\[4.30554pt] [s_{3}]=\overline{\cal K}+{\cal S}_{7}-{\cal S}_{9}\,,&[s_{5}]=2\,\overline{\cal K}-{\cal S}_{7}\,,&[s_{9}]={\cal S}_{9}\,,\end{array}$$

$$(3,2{)}_{\frac{1}{6}},(1,2{)}_{-\frac{1}{2}},(\overline{3},1{)}_{-\frac{2}{3}},(\overline{3},1{)}_{\frac{1}{3}},(1,1{)}_1.$$

$$G_4+\frac{1}{2}{c}_2(Y_4)\in H^{2,2}(Y_4,\mathbb{Z}),$$

$$n_{\text{D3}}=\frac{\chi (Y_4)}{24}-\frac{1}{2}{\int }_{Y_4}{G}_4\wedge G_4\stackrel{!}{\in }{\mathbb{Z}}_{\ge 0}.$$

$$\begin{array}{r}\forall \eta \in H^{1,1}(B_3):{\int }_{Y_4}{G}_4\wedge \sigma \wedge {\pi }^{\ast }\eta \stackrel{!}{=}0.\end{array}$$

$$\chi (\mathbf{R})={\int }_{{\gamma }_{\mathbf{R}}}{G}_4\stackrel{!}{=}3,$$

$$n_{\text{D3}}=12+\frac{5}{8}{\overline{\mathcal{K}}}^3-\frac{45}{2{\overline{\mathcal{K}}}^3}\stackrel{!}{\in }{\mathbb{Z}}_{\ge 0},$$

$${\overline{\mathcal{K}}}^3\in \{2,6,10,18,30,90\}.$$

$$N_{\text{SM}}^{\text{toric}}=\sum _{\Delta \in S}{N}_{\text{FRST}}(\Delta ),$$

$$7.6\times 10^{13}\lesssim N_{\text{SM}}^{\text{toric}}\lesssim 1.6\times 10^{16}.$$

$$g_3^2=g_2^2=\frac{5}{3}{g}_Y^2=\frac{2}{\text{vol}(\overline{\mathcal{K}})},$$

$$\begin{array}{rl}P:=& s_1{e}_1^2{e}_2^2{e}_3{e}_4^4{u}^3+s_2{e}_1{e}_2^2{e}_3^2{e}_4^2{u}^{2}v+s_3{e}_2^2{e}_3^{3}u{v}^2\\ +& s_5{e}_1^2{e}_2{e}_4^3{u}^{2}w+s_6{e}_1{e}_2{e}_3{e}_{4}uvw+s_9{e}_{1}v{w}^2=0,\end{array}$$

$$G_4(a,\omega )=a{G}_4^a+{\pi }^{\ast }\omega \wedge \sigma ,$$

$$\begin{array}{rl}{G}_4^a=& [e_4]\wedge ([e_4]+{\pi }^{\ast }[s_6])\\ +& \frac{[e_1]\wedge {\pi }^{\ast }[s_9]}{2}+\frac{{\pi }^{\ast }[s_3]\wedge ([e_2]+2[u])}{3},\end{array}$$

$$\forall \eta \in H^{1,1}(B_3):{\int }_{B_3}\overline{\mathcal{K}}\wedge \eta \wedge (5\omega +a\overline{\mathcal{K}})\stackrel{!}{=}0.$$

$$\chi (\mathbf{R})=-\frac{a}{5}{\int }_{B_3}\overline{\mathcal{K}}\wedge \overline{\mathcal{K}}\wedge \overline{\mathcal{K}}=:-\frac{a}{5}{\overline{\mathcal{K}}}^3.$$

$$\begin{array}{rl}& {\int }_{Y_4}\left(G_4+\frac{1}{2}{c}_2(Y_4)\right)\wedge \text{PD}({\gamma }_{\mathbf{R}}),\\ & {\int }_{Y_4}\left(G_4+\frac{1}{2}{c}_2(Y_4)\right)\wedge \text{PD}(D_1\cdot D_2),\end{array}$$

$${\int }_{B_3}\frac{c_2(B)\wedge \overline{\mathcal{K}}}{2},\frac{{\overline{\mathcal{K}}}^3}{2},$$

$${\int }_{B_3}\frac{\alpha \wedge (c_2(B_3)+{\overline{\mathcal{K}}}^2)}{2}\text{ with }\alpha \in H^{1,1}(B_3,\mathbb{Z}),$$

$$n_{\text{D3}}=12+\frac{5}{8}{\overline{\mathcal{K}}}^3-\frac{45}{2{\overline{\mathcal{K}}}^3}.$$

Full text

††preprint: UPR-1297-T

I Introduction and Summary

As a theory of quantum gravity that naturally gives rise to rich gauge sectors at low energies, string theory is a leading candidate for a unified theory. Achieving unification is an ambitious goal that requires accounting for all aspects of our physical world, which includes not only a rich cosmological history, but also the detailed structure of the Standard Model of particle physics.

In this paper we present an explicit construction that guarantees the existence of ${\cal O}(10^{15})$ fully consistent string compactifications which realize the exact chiral particle spectrum of the minimally supersymmetric Standard Model (MSSM). This construction is performed in the framework of F-theory Vafa (1996), a strongly coupled generalization of type IIB superstring theory. It captures the non-perturbative back-reactions of 7-branes onto the compactification space $B_{3}$ in terms of an elliptically fibered Calabi–Yau fourfold $\pi:Y_{4}\rightarrow B_{3}$ over it. Gauge symmetries, charged matter, and Yukawa couplings are then encoded beautifully by $Y_{4}$’s singularity structures in codimensions one, two, and three, respectively.111We refer the interested reader to Weigand (2018); *Cvetic:2018bni and references therein for recent reviews on F-theory.

In the present work, we consider a class of elliptically fibered Calabi–Yau fourfolds giving rise to precisely the three-generation MSSM spectrum provided certain geometric conditions on the base of the fibration are satisfied. We perform a concrete analysis, finding ${\cal O}(10^{15})$ such bases. All these models come equipped with moduli-dependent quark and lepton Yukawa couplings, as well as gauge coupling unification at the compactification scale.

The existence of a very large number of Standard Model realizations in string theory could perhaps be anticipated within the set of an even larger number of string compactifications (see, e.g., Ashok and Douglas (2004); *Taylor:2015xtz; *Halverson:2017ffz) that form the so-called string landscape. Indeed, though Standard Model realizations within the landscape could potentially be scarce Blumenhagen et al. (2005a); *Gmeiner:2005vz, recent works hint towards an astronomical number of them Anderson et al. (2012); *Anderson:2013xka; *Constantin:2018xkj. Our construction explicitly demonstrates this possibility, increasing the number of concretely known, global Standard Model compactifications in string theory by about ten orders of magnitude.

There are also explicit constructions of the Standard Model in other corners of string theory. Some of the early examples of globally consistent intersecting brane models Cvetič et al. (2001a); *Cvetic:2001tj in type II compactifications (see also Blumenhagen et al. (2005b) and references therein) were strongly constrained by global consistency conditions such as tadpole cancellation. In the heterotic string, the typical difficulties of constructions like Braun et al. (2005); *Braun:2005nv; Bouchard and Donagi (2006); *Bouchard:2006dn; Anderson et al. (2007); *Anderson:2009mh arise from having a stable hidden bundle and the existence of Yukawa couplings. These issues are solved elegantly in F-theory through the geometrization of non-perturbative stringy effects: (almost all222In F-theory, D3-tadpole cancellation requires extra care, and will be a major theme in our constructions.) global conditions analogous to tadpole cancellation or bundle stability are automatically taken care of by having a compact, elliptic Calabi–Yau fourfold $Y_{4}$, and the presence or absence of Yukawa couplings can be easily read off from codimension three singularities of $Y_{4}$.

Despite these advantages, only a handful Cvetič et al. (2015, 2018) of F-theory compactifications that realize the exact chiral spectrum of the MSSM are currently known, due to focusing on a very simple base, $B_{3}={\mathbb{P}}^{3}$. This limitation will be avoided in the current work by instead studying smooth toric varieties, which provide a much larger class Halverson and Tian (2017); *Carifio:2017bov; Altman et al. (2019) of geometries. To take advantage of this large ensemble, we first construct a class of elliptic fibrations (based on the class ${\mathbb{P}}_{F_{11}}$ in Klevers et al. (2015)) that can be consistently fibered over all such toric threefolds.

Every such fibration realizes the precise Standard Model gauge group $[SU(3)\times SU(2)\times U(1)]/{\mathbb{Z}}_{6}$ as well as its matter representations and Yukawa couplings Klevers et al. (2015); Cvetič et al. (2015); Cvetič and Lin (2018b). Moreover, all models exhibit gauge coupling unification at the compactification scale, compatible with the existence of a complex structure deformation to a geometry realizing the Pati–Salam model with unified gauge coupling Klevers et al. (2015); Cvetič et al. (2015).

Furthermore, for each compatible $B_{3}$ there exists a $G_{4}$-flux that induces three families of chiral fermions. These models have a particularly pleasant feature: all global consistency conditions on the flux (including quantization and D3-tadpole cancellation) can be reduced to a single criterion on the intersection number $\overline{\cal K}^{3}$ of the anti-canonical class $\overline{\cal K}$ of the base $B_{3}$. For toric threefolds which have a description in terms of a reflexive polytope $\Delta$, $\overline{\cal K}^{3}$ depends only on the point configuration of $\Delta$ and not its triangulation. On the other hand, for a single polytope there can be multiple different toric threefolds associated with the different fine regular star triangulations (FRSTs) of $\Delta$, the number of which grows exponentially with the number of lattice points in the polytope Halverson and Tian (2017). Putting together these different components, we find that the number $N_{\text{SM}}^{\text{toric}}$ of globally consistent three-family Standard Models in our construction is

[TABLE]

We emphasize that this number is construction dependent; F-theory could realize more Standard Models.

The detailed derivation of this count first requires the construction in section II of a class of elliptic fibrations with a flux inducing three chiral families. All flux consistency conditions reduce to a single criterion on the base $B_{3}$. To count how many $B_{3}$ satisfy this criterion, we discuss the methods to construct FRSTs of 3D polytopes in section III, which ultimately lead us to ${\cal O}(10^{15})$ possibilities. We close in section IV with some geometric and physical comments, as well as future directions.

II Universally Consistent Fibrations with Three Families

We now present our construction of three-family Standard Model vacua; see appendix A for further details.

For that, we first consider an elliptic curve that is a specialized cubic inside $\mathbb{P}^{2}$ with homogeneous coordinates $[u:v:w]$, given by the vanishing of the polynomial

[TABLE]

By promoting the coefficients $s_{i}$ to rational functions over a Kähler threefold $B_{3}$, one obtains a singular, elliptically fibered fourfold $\pi:Y^{(s)}_{4}\rightarrow B_{3}$. For $Y_{4}^{(s)}$ to be Calabi–Yau, the functions $s_{i}$ have to be holomorphic sections of line bundles on $B_{3}$ with first Chern classes $[s_{i}]\in H^{1,1}(B_{3},{\mathbb{Z}})$ given by Klevers et al. (2015); Cvetič et al. (2015):

[TABLE]

where $\overline{\cal K}\equiv c_{1}(B_{3})$ is the anti-canonical class of $B_{3}$. The classes ${\cal S}_{7,9}\in H^{1,1}(B_{3},{\mathbb{Z}})$ parametrize different fibrations over the same base, on which $\{s_{i}=0\}$ define effective divisors.

When all $s_{i}$ are generic, (that is, irreducible and $s_{i}\neq s_{j}$ for $i\neq j$), F-theory compactified on $Y_{4}^{(s)}$ has the gauge symmetry $[SU(3)\times SU(2)\times U(1)]/\mathbb{Z}_{6}$ Klevers et al. (2015); Cvetič and Lin (2018b). The global gauge group structure is reflected in the precise agreement between the geometrically realized matter representations and those of the Standard Model:

[TABLE]

These data can be extracted via the M-/F-theory duality from an explicit resolution $Y_{4}$ of $Y_{4}^{(s)}$, which preserves the Calabi–Yau structure (see appendix A).

A chiral spectrum in F-theory requires a non-zero flux $G_{4}\in H^{2,2}(Y_{4})$, which must also be specified. For the relevant subspace of so-called primary vertical $G_{4}$-fluxes, there is by now a large arsenal of computational methods Lin et al. (2016); Lin and Weigand (2016) (see also Cvetič et al. (2014, 2015); Mayorga Pena and Valandro (2018)) that allows us to determine base-independently the most general flux on $Y_{4}$.

For physical consistency, this $G_{4}$-flux has to satisfy certain conditions. The first condition is a proper quantization Dasgupta et al. (1999); *Witten:1996md; Collinucci and Savelli (2012):

[TABLE]

where $c_{2}(Y_{4})$ is the second Chern class of $Y_{4}$. Heuristically, this condition ensures that the notion of fermions (that requires a flux-dependent spin structure on subspaces of $Y_{4}$) is well-defined. Since explicitly verifying this condition for concrete geometries is difficult, we will content ourselves with the usual necessary consistency checks Intriligator et al. (2013); Cvetič et al. (2014, 2015); Lin and Weigand (2016); Cvetič et al. (2018). The second consistency condition is a D3-tadpole satisfying Sethi et al. (1996),

[TABLE]

While integrality follows as a consequence of the quantization condition (7), positivity aids in ensuring the stability of the compactification.

We must also impose phenomenological constraints on the flux. A massless electroweak hypercharge $U(1)_{Y}$ is guaranteed if the D-terms vanish Grimm (2011); Grimm et al. (2011):

[TABLE]

Here, $\sigma$ is the $(1,1)$-form Poincaré-dual to the so-called Shioda-divisor associated with the $U(1)$ Park (2012); *Morrison:2012ei. A three-family chiral Standard Model requires that Donagi and Wijnholt (2011); *Braun:2011zm; *Marsano:2011hv; *Krause:2011xj; *Grimm:2011fx,

[TABLE]

for all representations ${\bf R}$ in (6). The geometric data $c_{2}(Y_{4})$, $\chi(Y_{4})$, and the matter surfaces $\gamma_{\bf R}$ were computed in Klevers et al. (2015); Cvetič et al. (2015). In the Appendix, we provide the explicit expression of the generic vertical flux in the resolution $Y_{4}$ presented in Klevers et al. (2015) and explain in detail how the above conditions can be checked using well-studied topological methods.

We now present our main result, on how these consistency conditions can be satisfied for a large ensemble of explicit geometries. For that, we first consider the flux configuration (16) on (smooth) fibrations $Y_{4}$ with ${\cal S}_{7,9}=\overline{\cal K}$, which simplifies the expressions for the topological quantities (8)–(10). In fact, one can show that all consistency conditions are reduced to a single criterion on $B_{3}$ from the D3-tadpole:

[TABLE]

where $\overline{\cal K}^{3}$ denotes the triple self-intersection number of the anti-canonical class $\overline{\cal K}$ of the base. This dramatic simplification only requires $\overline{\cal K}^{3}$ of appropriate value and a base that allows irreducible and distinct $s_{i}$, all of which are sections of the anti-canonical class.

In summary, we have constructed a class of elliptically fibered Calabi–Yau fourfolds which gives rise in F-theory to the Standard Model gauge group and matter representations with three chiral generations. The only consistency requirement that guarantees flux quantization and D3-tadpole cancellation is that the base $B_{3}$ of the fibration is a smooth Kähler threefold with non-rigid irreducible anti-canonical divisors that satisfy (11). In fact, the only values $\overline{\cal K}^{3}$ can take such that $n_{\text{D3}}\in\mathbb{Z}_{\geq 0}$ are

[TABLE]

III Counting Standard Model Geometries

Any smooth threefold $B_{3}$ with non-rigid anti-canonical divisors satisfying (12) realizes a globally consistent three-family MSSM in F-theory. A subset of such spaces, which can be enumerated combinatorially, is the set of weak Fano toric threefolds encoded by 3D reflexive polytopes $\Delta$. While there are “only” 4319 such polytopes Kreuzer and Skarke (1998), each $\Delta$ can specify inequivalent manifolds $B_{3}$ through different fine-regular-star triangulations (FRSTs) of the polytope, whose numbers can be very large Halverson and Tian (2017).

What makes this ensemble particularly attractive for our purpose is the fact that the intersection number $\overline{\cal K}^{3}$ is determined solely by the polytope $\Delta$, and is completely triangulation-independent. Therefore any $B_{3}$ associated to an FRST of $\Delta$ gives rise to a consistent chiral three-generation MSSM by our construction, provided that the triangulation-independent constraint on $\overline{\cal K}^{3}$ is satisfied. In fact, there is a set $S$ with 708 polytopes that satisfy (12). By our construction we immediately have

[TABLE]

where $N_{\text{FRST}}(\Delta)$ is the number of FRSTs of $\Delta$.

Hence, the problem of counting the number of consistent F-theory models that admit the chiral MSSM spectrum by our construction reduces to counting FRSTs of reflexive polytopes.

Since $N_{\text{FRST}}(\Delta)$ grows exponentially with the number of lattice points in $\Delta$, the set of consistent threefolds $B_{3}$ is dominated by triangulations of the largest polytope Halverson and Tian (2017), labelled $\Delta_{8}$ in the list of Kreuzer and Skarke (1998). The FRSTs of this polytope (with $\overline{\cal K}^{3}=6$ and 39 lattice points) cannot be all constructed explicitly using the standard computer programs such as SageMath Stein et al. (2018). To enumerate them, we therefore follow the strategy put forward in Halverson and Tian (2017) to provide bounds on $N_{\text{FRST}}(\Delta_{8})$.

The idea is to reduce the complexity by first counting the number of fine-regular triangulations (FRTs) of each facet of a polytope $\Delta$. Since the facets are two dimensional polytopes, it is possible to use brute-force on the combinatorics of FRTs for (almost333For facets with more than 15 lattice points, brute-forcing FRTs also becomes computationally too costly. For these facets, we use different methods outlined in Halverson and Tian (2017) to obtain lower and upper bounds for the number of FRTs. ) all polytopes’ facets. By virtue of the reflexivity of $\Delta$, any combination of FRTs of all its facets yields fine star triangulation of $\Delta$.

The drawback of this approach is that the triangulation of $\Delta_{8}$ obtained this way is not guaranteed to be regular. To tackle this issue, we randomly pick $1.3\times 10^{4}$ samples out of ${\cal O}(10^{9})$ fine-star triangulations constructed by gluing together FRTs of the facets $\Delta_{8}$. Out of these samples, we find roughly $\frac{2}{3}$ to be also regular triangulations. Combining the factor $\frac{2}{3}$ with the bounds of fine-star triangulations (FSTs) for $\Delta_{8}$ Halverson and Tian (2017), we then obtain $2.6\times 10^{13}\leq N_{\text{FRST}}(\Delta_{8})\leq 1.6\times 10^{16}$.

For the other polytopes in $S$ (i.e., those leading to threefolds satisfying (12)) we can either compute all FRSTs, or we can resort to a similar estimation as with $\Delta_{8}$ if the polytope is too large to use brute force all FRSTs. We find that these other polytopes sum up to “only” $\sim 5\times 10^{13}$ FRSTs, which confirms the dominance of $\Delta_{8}$. In total, we therefore expect the number of consistent three-family F-theory Standard Models in our construction over toric threefold bases to be

[TABLE]

IV Discussion and Outlook

We have presented a construction that ensures the existence of ${\cal O}(10^{15})$ explicit, globally consistent string compactifications having the exact chiral spectrum of the Standard Model within the framework of F-theory. To our knowledge, this is the largest such ensemble in the literature, outnumbering existing results by about $10$ orders of magnitude. The models arise by varying the base of one “universal” class of elliptic fibrations introduced in Klevers et al. (2015); Cvetič et al. (2015). We have only focused on the set of toric bases, which already produces around a quadrillion examples. However, we expect that the ensemble of Standard Models arising from our construction is of orders of magnitude larger than this, as might be shown, for instance, by including non-toric bases.

All these models have in common that the Higgs and lepton doublets are localized on the same matter curve. As such, this curve must have non-zero genus to allow for the existence of vector-like pairs Bies et al. (2014); *Bies:2017fam. Given the homology class of the doublet curve Cvetič et al. (2015) and our restriction ${\cal S}_{7,9}=\overline{\cal K}$, the genus in question is indeed $g=1+9/2\overline{\cal K}^{3}>0$, since $\overline{\cal K}^{3}\geq 2$ by (12). It would be very interesting, albeit extremely difficult with current methods, to study the precise complex structure dependence of the number of Higgs doublets and other charged vector-like pairs in this ensemble.

Furthermore, since our models have no additional (possibly massive) abelian gauge symmetries, all Yukawa couplings relevant for the Standard Model are automatically realized perturbatively, as can be shown by an explicit study of codimension three singularities Klevers et al. (2015). However, this in turn also implies that certain proton decay operators compatible with the Standard Model gauge group will in general be present Cvetič et al. (2015). We expect that in some corners of the moduli space, which incidentally could also support high-scale supersymmetry breaking, these operators can be suppressed. Another avenue could be to instead focus on “F-theory Standard Models” that have additional ($U(1)$ Lin and Weigand (2015, 2016) or R-parity Cvetič et al. (2018)) selection rules, and estimate their numbers in the toric base landscape. We leave this for future work.

One interesting aspect of our ensemble is gauge coupling unification without a manifest GUT-origin at the compactification scale. It can be easily read off geometrically from the divisors on $B_{3}$, which the 7-branes supporting the gauge symmetries in the type IIB picture wrap. Due to our restriction ${\cal S}_{7,9}=\overline{\cal K}$, both $SU(3)$ and $SU(2)$ gauge symmetries are realized on anti-canonical divisors $\{s_{9}=0\}$ and $\{s_{3}=0\}$ with class $\overline{\cal K}$. Therefore, the gauge couplings are $g_{3,2}^{2}=2/\text{vol}(\overline{\cal K})$ Grimm (2011); Bonetti and Grimm (2012).444 The factor of 2 arises because in F-theory, the normalization dictated by the geometry is one where the Cartan generators satisfy $\text{tr}_{\text{fund}}(T_{i}\,T_{j})=C_{ij}$ with $C$ the Cartan matrix. On the other hand, the particle physics convention necessary to determine the coupling is $\text{tr}_{\text{fund}}(T_{i}\,T_{j})=\frac{\delta_{ij}}{2}$.

The $U(1)_{Y}$ coupling is the inverse volume of the so-called height-pairing divisor $b\subset B_{3}$ Cvetič et al. (2013), which has been computed in Klevers et al. (2015); Cvetič et al. (2001b) and reduces to $b=5\overline{\cal K}/6$ in our ensemble. Therefore, we have the standard MSSM gauge coupling unification,

[TABLE]

which for our models is achieved at the compactification scale. While this scale as well as the actual values of the couplings will depend on the details of moduli stabilization, the relationship (14) is independent of Kähler moduli. It would be interesting to see if this relationship originates from an honest geometric realization of a GUT-structure. Given the known connection of our ensemble to a Pati–Salam $[SU(4)\times SU(2)^{2}]/{\mathbb{Z}}_{2}$ model Klevers et al. (2015); Cvetič et al. (2015), we expect an underlying $SO(10)$.

Our results may provide phenomenological motivation for the study of new moduli stabilization scenarios. Specifically, though gauge coupling unification is automatic in our ensemble, it is natural to ask whether the correct value $\alpha_{\text{GUT}}\simeq.03$ can be obtained in canonical moduli stabilization schemes. For instance, the KKLT and Large Volume scenarios Kachru et al. (2003); *Balasubramanian:2005zx assume that cycles are at sufficiently large volume to safely ignore string worldsheet instanton corrections to the Kähler potential. This is essential because it is not known how to systematically compute and control all instanton contributions in $\mathcal{N}=1$ backgrounds. A necessary condition for safely ignoring these corrections is to have $\text{vol}(C)>1$ (in string units) for all curves $C\subset B_{3}$. This condition defines a stretched out subset of the Kähler cone Demirtas et al. (2018), where it was also shown that the Kähler cones become increasingly narrow for increasing $h^{1,1}$. In effect, this forces toric divisors to be increasingly large in order to safely ignore worldsheet instantons, leading to smaller gauge couplings, because on toric $B_{3}$ the class $\overline{\cal K}$ is the sum of all toric divisors. Brief calculations suggest that the correct $\alpha_{\text{GUT}}$ cannot be obtained in this controlled regime, in which case realistic models in our scenario are not consistent with the KKLT or LVS scenarios. Firmly concluding this requires a more detailed study, but we emphasize that it would not rule out our models, and instead motivate the study of new moduli stabilization scenarios that allow for the observed value of gauge couplings.

Our compactifications also exhibit D3-branes. These sectors generically give rise to $U(1)$ gauge theories that could be cosmologically relevant as dark photons. Each has its own open string moduli, the position of the D3-brane, which are massless at tree level but may be stabilized by non-perturbative effects due to their appearance in instanton prefactors Ganor (1997); *Baumann:2006th. However, since all but one of the toric divisors are rigid in the geometries we study, it is likely that there are many instanton corrections to the superpotential. Each instanton acts with an attractive force on the D3-brane, pulling it toward the associated divisor, but the existence of many such contributions would provide competing forces that stabilize the D3-brane away from each toric divisor. In particular, due to these competing effects we see a priori no reason that the D3-branes should be stabilized anywhere near the $SU(3)$ or $SU(2)$ 7-branes, in which case jointly charged matter in the form of $3$-$7$ strings decouple from the spectrum. Such a scenario gives rise to numerous dark photon sectors that have cosmological effects only through kinetic mixing with the visible sector and with one another. It would be interesting to study these sectors further, in light of current and future dark photon experiments.

We note that gravity cannot be decoupled in our ensemble since the Standard Model gauge divisors are in the anti-canonical class, yielding a non-trivial interplay between gravity and the visible sector. This interplay arises due to the details of our construction and could lead to other interesting interactions between particle physics and cosmology. At the level of toric geometry, the models of our ensemble differ from one another by how the facets are triangulated. This does not affect the structure of the anti-canonical divisors that realize $SU(3)$ and $SU(2)$, and thus the particle physics is relatively insensitive to details of the triangulation; it is, after all, what gives rise to the large number of Standard Models in our construction. The triangulation is critical, however, for moduli stabilization. For instance, the classical Kähler potential on Kähler moduli is determined by triangulation-dependent topological intersections. This affects numerous aspects of the cosmology of these models, including inflation.

This visible sector universality in the midst of cosmological diversity might lead one to question the extent to which these should be counted as truly different models. Though a natural question, it has a clear answer: since the geometries are different they lead to distinct four-dimensional effective theories below the Kaluza–Klein scale, each of which could give rise to numerous metastable vacua. Instead, our view is that the universal structure in the visible sector provides some evidence for a long-held hope in the string landscape, that, despite large numbers of vacua, there could exist semi-universal features that lead to meaningful statistical predictions.

Acknowledgements.

We thank Craig Lawrie and Cody Long for helpful discussions and Jonathan Carifio for providing the dataset of Altman et al. (2019). MC and LL are supported by DOE Award DE-SC0013528. MC further acknowledges the support by the Fay R. and Eugene L. Langberg Endowed Chair and the Slovenian Research Agency. JH is supported by NSF Grant PHY-1620526.

Appendix A Technical Aspects of the Construction

While the physics is fully determined by the singular elliptic fibration $\pi^{(s)}:Y_{4}^{(s)}\rightarrow B_{3}$, extracting the relevant information is most easily done in a crepant resolution, i.e., a resolution of the singularities which preserve the Calabi–Yau condition Vafa (1996); Weigand (2018); Cvetič and Lin (2018a). One crepant resolution $\pi:Y_{4}\rightarrow B_{3}$ of the singular fibration $Y^{(s)}$ obtained from the specialized cubic (2) can be conveniently described by toric blow-ups on the ambient $\mathbb{P}^{2}$. This introduces exceptional divisors $\{e_{i}=0\}$, $i=1,...,4$, whose coordinates $e_{i}$ are subject to additional scaling relations amongst each other and the $\mathbb{P}^{2}$ coordinates $[u:v:w]$. These relations, together with intersection properties, are encoded in a reflexive 2d polygon (labelled $F_{11}$ in Klevers et al. (2015)), which also determines the polynomial,

[TABLE]

whose vanishing locus defines the resolved elliptic fibration $Y_{4}$ Klevers et al. (2015). The coefficients $s_{i}$ remain unchanged compared to (5) as functions on the base $B_{3}$.

On the resolved fibration $\pi:Y_{4}\rightarrow B_{3}$, we can determine the most generic vertical $G_{4}$-flux base-independently Cvetič et al. (2015); Lin et al. (2016); Lin and Weigand (2016); Cvetič et al. (2014); Mayorga Pena and Valandro (2018). It is parametrized by $a\in\mathbb{Q}$ and $\omega\in H^{1,1}(B_{3})$ and can be written as

[TABLE]

where $\sigma$ is the $(1,1)$-form Poincaré-dual to the Shioda-divisor associated with the $U(1)$ (see Klevers et al. (2015) for its explicit expression). The $(2,2)$-form $G_{4}^{a}$ is

[TABLE]

where $[x]$ denotes the $(1,1)$-form Poincaré-dual to the divisor $\{x=0\}\subset Y_{4}$.

In the following, we show how for the restriction ${\cal S}_{7,9}=\overline{\cal K}$, the flux satisfies the necessary consistency conditions. This is based on topological computation methods developed in Lin et al. (2016); Lin and Weigand (2016); Cvetič et al. (2014); Mayorga Pena and Valandro (2018). These allow to reduce the relevant integral expressions in the fibration $Y_{4}$ to intersection numbers on the base $B_{3}$.

First, we reduce the masslessness condition (9) for $U(1)_{Y}$ with these methods, which becomes

[TABLE]

This can always be satisfied if $\omega=-5\overline{\cal K}/a$, regardless of the base. Inserting this into the condition (10) for three chiral families, we then find

[TABLE]

For exactly three families, we thus need $a=-15/\overline{\cal K}^{3}$. This fixes the flux parameters $(a,\omega)$ completely.

A necessary condition for this flux to satisfy the quantization condition (7) is for the following integrals to be integers:

[TABLE]

where the integrands contain the Poincaré-duals (PD) of all matter surfaces $\gamma_{\bf R}$ as well as intersection products $D_{1}\cdot D_{2}$ of all integral divisors $D_{i}$. For the smooth elliptic fibrations, $D_{i}$ are known to be independent fibral divisors $\{u=0\},\{v=0\},\{w=0\}$, $\{e_{i}=0\}$, and $\pi^{-1}(D_{B})$ for $D_{B}$ any integral divisor of $B_{3}$. The data for $\gamma_{\bf R}$ and $c_{2}(Y_{4})$ can be found in Klevers et al. (2015); Cvetič et al. (2015). After reducing these integrals to intersection numbers on $B_{3}$, the only non-manifestly integer quantities are

[TABLE]

where $c_{2}(B_{3})$ is the second Chern class of $B_{3}$. For smooth threefolds $B_{3}$ that appear as a base of a smooth elliptic Calabi–Yau fourfold, it is known Collinucci and Savelli (2012) that $\int_{B_{3}}c_{2}(B_{3})\wedge\overline{\cal K}=24$ and that $c_{2}(B_{3})+\overline{\cal K}^{2}$ is an even class.

The remaining condition of $\overline{\cal K}^{3}$ being even is covered by the D3-tadpole (8), which with the above simplifications becomes

[TABLE]

Since this is integer if and only if $\overline{\cal K}^{3}$ is even, all necessary consistency conditions on $G_{4}$ are satisfied if $n_{\text{D3}}\in{\mathbb{Z}}_{\geq 0}$.